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+{"text":"\\section{Introduction}\n\nThe vertical structure of eddying flow in the oceanic mesoscale is a fundamental\nproblem in geophysical fluid dynamics, one that has has been reinvigorated by the\nneed to interpret altimetric observation of surface velocity fields\n\\citep{ScottWang05,LapeyreJPO09}. Energy cascades and self-organization processes\nare fairly well understood and characterized in two-dimensional turbulence, but\nhow these results generalize to rotating, continuously stratified flows is still\nan open question. We ask in this paper what is the final organization of a freely\nevolving inviscid quasi-geostrophic flow with continuous stratification. In\nparticular, we examine what are the precise conditions for the oft-cited\nbarotropization process to occur. Barotropization refers to the tendency of a\nquasi-geostrophic flow to reach the gravest vertical mode \\citep{Charney71,\nRhines77}. Because the gravest vertical mode is the barotropic one, i.e. a depth\nindependent flow, ``barotropization'' means a tendency toward the formation of\ndepth independent flows. We study in particular the key role played by the beta\neffect (the existence of planetary vorticity gradients) in such barotropization\nprocesses. It has been previously noticed that the presence of these large scale\nplanetary vorticity gradients favors barotropic process, see e.g. \\citet{SmithVallis01}, but this phenomenon remains to our knowledge unexplored and unexplained.\n\n\nContinuously stratified quasi-geostrophic flows take place in three dimensions,\nbut their dynamics are quasi-two-dimensional because the non-divergent advecting\nvelocity field has only horizontal components. The dynamics admits similar\nconservation laws as the two-dimensional Euler equations, including among others\nthe total energy and the enstrophy of each layer. These conservation laws imply\nan inverse energy cascade toward large scales, and weak energy dissipation in\npresence of weak viscosity\n\\citep{Kraichnan_Motgommery_1980_Reports_Progress_Physics}. This situation\ncontrasts with three dimensional turbulence where the energy cascades forward\ntoward small scales, giving rise to finite energy dissipation no matter how weak\nthe viscosity.\n\nJust as for two-dimensional Euler flows, inviscid, freely evolving continuously stratified quasi-geostrophic flows are disordered and an unstable initial condition rapidly evolves into a strongly disordered fine-grained flow with filaments being stretched, folded, and consequently cascading toward smaller and smaller scales. However, the inverse energy cascade leads to the formation of robust, large scale coherent structures filling the domain in which the flow takes place, contrasting strongly with the analogous three dimensional flow with its energy transfer to small scales.\n\nThe aim of this paper is to provide a prediction for the vertical structure of the large scale flow organization resulting from inviscid, freely evolving continuously stratified quasi-geostrophic dynamics using a statistical mechanics approach. The goal of the statistical mechanics approach is to drastically reduce the complexity of the problem of determining the large scale flow organization to the study of a few key parameters given by the dynamical invariants, as for instance the total energy and the fine-grained enstrophy in each layer. Such a theory has been independently proposed by \\cite{Robert:1990_CRAS,SommeriaRobert:1991_JFM_meca_Stat} and \\cite{Miller:1990_PRL_Meca_Stat}, and henceforth will be referred to as MRS theory. It is an equilibrium theory, developed in the case without forcing and dissipation. Our contribution will be to apply this theory to the explicit computation of a class of equilibrium states of the stratified quasi-geostrophic equations.\n\n{A crucial ingredient of the theory comes from the constraints given by dynamical invariants. In the present paper, we will focus on the role played by the energy and the enstrophy constraints. In the framework of the MRS theory, it is justified in a low energy limit, and it is only a first step before more comprehensive studies including the effects of additional invariants, see e.g \\cite{MajdaWangBook,CorvellecBouchet2010}.}\n\n{The phenomenology of two-dimensional and geostrophic turbulence is often explained by considering the energy and enstrophy constraints only, see e.g \\citet{Kraichnan_Motgommery_1980_Reports_Progress_Physics,Salmon_1998_Book}. Actually, earlier statistical mechanics approaches were developed for Galerkin-truncated Euler or quasi-geostrophic dynamics, which allowed for a simpler theoretical treatment than in the continuous dynamics case, see e.g. \\cite{Kraichnan_1967PhFl...10.1417K,SalmonHollowayHendershott:1976_JFM_stat_mech_QG}. In the case of Galerkin-truncated models, only the energy and the enstrophy are conserved quantities, so the statistical mechanics of these systems were called energy-enstrophy theory.}\n\n{A first reason to consider MRS theory rather than earlier energy-enstrophy statistical mechanics approach is that real flows are continuous, not Galerkin-truncated. Because real flows are also forced-dissipated, our first working hypothesis is that the computation of the MRS equilibrium states of the continuous dynamics is a necessary step before taking into account the effect of weak forcing and dissipation. This is for instance the approach followed by \\cite{BouchetSimonnetPRL09}, who found that in the presence of weak forcing and dissipation, the dynamics were close to MRS equilibrium states which were different from energy-enstrophy equilibrium states.}\n\n{One could  argue that numerical models often at best conserve energy and enstrophy, so there would be at first sight no reason to consider higher order invariants to predict the simulated self-organization. However, the truncated dynamics in numerical models may lead to the formation of quasi-stationary states having a life time that tends to infinity as the resolution is refined.  These quasi-stationary states are distinct from the energy-enstrophy equilibria, and thus require more invariants to describe using a statistical theory.  The formation of such quasi-stationary states is a characteristic of long-range interacting systems (e.g.,  \\cite{CampaDauxoisRuffo} and references therein).  Given these results, our second working hypothesis is that the quasi-stationary states found in the truncated dynamics of numerical simulations may be close to a state predicted by MRS theory.}\n\n{There are only a few studies of statistical equilibria of the continuously stratified quasi-geostrophic dynamics. \\cite{Merryfield98JFM} obtained and discussed a class of solutions in the framework of energy-enstrophy statistical theory above topography, for which there is linear relation between potential vorticity $q$ and streamfunction $\\psi$. The solutions obtained were critical points of the theory, but the actual entropy maxima were not selected. Here we consider the particular class of MRS equilibrium states characterized by a linear $q-\\psi$ relation, so we describe the same class of equilibrium states, except that we do not take into account bottom topography. The effect of bottom topography will be addressed in a future work. The novelty of our work is to relate each critical point to the energy and fine-grained enstrophy constraints and to select the actual entropy maxima in different limit cases.}\n\n\\citet{SchecterPRE03} provided a direct comparison between numerical simulations\nand numerical computation of MRS statistical equilibria, focusing on the\nparticular case of initial conditions given by a random field statistically\nhomogeneous in space, as in \\citet{McWilliams94}. {This corresponds to a particular case for which the enstrophy profile is only weakly varying on the vertical.} We will outline in this paper the important consequence of the conservation of enstrophy at each depth. This important role played by the enstrophy constraint  allows us to discuss how the beta effect can favor barotropization. For a given initial streamfunction field, the beta effect modifies the initial distribution of potential vorticity, and therefore modifies the vertical profile of fine-grained enstrophy. More precisely, increasing the beta effect tends to increase the depth independent part of the fine-grained enstrophy profile. We will show that statistical equilibria associated with depth independent fine-grained enstrophy profiles are barotropic, which means that increasing the beta favors barotropization, according to statistical mechanics predictions.\n\nThe paper is organized as follows: the continuously stratified quasi-geostrophic equations are introduced on section \\ref{sec:csqg}. The MRS equilibrium statistical mechanics theory of such systems is presented in section 3. Here it is shown that the class of MRS statistical equilibria characterized by a linear relation between potential vorticity and streamfunction are solutions of a minimum enstrophy problem, which is solved in various limit cases, first without beta effect and then with beta effect. The theory predicts an increase of barotropization with increasing value of beta, which is tested against numerical simulations in section 4. A summary of the main results and a discussion of their application to the ocean is given section 5.\n \n\n\\section{Continuously stratified quasi-geostrophic model} \\label{sec:csqg}\n\n\\subsection{Equations of motion}\n\nWe consider an initial value problem for a freely evolving, unforced,\nundissipated, quasi-geostrophic flow with continuous stratification\n$N(z)$ (see e.g., \\citet{VallisBook}, section 5.4):\n\n\\begin{equation}\n\\partial_{t}q+J(\\psi,q)=0\\ ,\\label{eq:FullDynamics}\\end{equation}\n \\begin{equation}\nq=\\Delta\\psi+\\frac{\\partial}{\\partial z}\\left(\\frac{f_{0}^{2}}{N^{2}}\\frac{\\partial}{\\partial z}\\psi\\right)+\\beta y\\ ,\\label{eq:PV_definition}\\end{equation}\n with $J(\\psi,q)=\\partial_{x}\\psi\\partial_{y}q-\\partial_{y}\\psi\\partial_{x}q$\n, and $f=f_{0}+\\beta y$ is the Coriolis parameter with the $\\beta$-plane\napproximation.\nNeglecting buoyancy and topography variations, the boundary condition at the bottom ($z=-H$, where $H$ is a constant) is given by \n\\begin{equation}\n \\partial_{z}\\psi|_{z=-H}=0.\t\n\\end{equation}\nThe boundary condition at the surface (defined as $z=0$,\nusing the rigid lid approximation), is given by the advection of buoyancy\n\\begin{equation}\n \\partial_{t}b_{s}+J(\\psi,b_{s})=0, \\quad \n    \\frac{f_{0}^{2}}{N^{2}}\\partial_{z}\\psi|_{z=0}=b_{s}.\n\\label{eq:FullDynamicsB}\n\\end{equation}\n\n\n{The reason we focus on quasi-geostrophic dynamics  is that for these flows, there is  no  direct cascade of energy and no dissipative anomaly, see e.g. \\cite{Salmon_1998_Book}, which makes possible a straightforward generalization of statistical mechanics theories developed initially in the framework of two-dimensional Euler equations. As far as the ocean is concerned, quasi-geostrophic dynamics is relevant to describe ``mesoscale turbulence'', but does not account for  ``sub-mesoscale turbulence'', which involve ageostrophic effects, with part of the energy that may cascade forward, see e.g. \\cite{LeithThomas,FerrariWunsch09} and references therein.}\n\n\\subsection{Modal decomposition \\label{sub:Vertical-and-horizontal-eigenmodes}}\n\n\\subsubsection{Laplacian and baroclinic eigenmodes}\n\nWe consider the case of a doubly periodic domain $\\mathcal{D}=[-\\pi\\ \\pi]\\times[-\\pi\\ \\pi]$.\nThe Fourier modes $e_{k,l}(x,y)$ are a complete orthonormal basis\nof the 2D Laplacian $\\Delta e_{k,l}=-K^{2}e_{k,l}$\nwhere $(k,l)$ are the wavenumbers in each directions and $K^2 = k^2 + l^2$.\nIn the vertical, the complete orthonormal basis of barotropic-baroclinic\nmodes is defined by the solutions $F_{m}(z)$ of the Sturm-Liouville\neigenvalue problem \n\\begin{equation}\n\\frac{\\partial}{\\partial z}\\left(\\frac{f_{0}^{2}}{N^{2}}\\frac{\\partial}{\\partial z}F_{m}\\right)=-\\lambda_{m}^{2}F_{m}\\quad\\text{with}\\quad F^{\\prime}(0)=F^{\\prime}(-H)=0,\n\\label{eq:VerticalModes}\n\\end{equation}\n where $m\\ge0$ is an integer, $\\lambda_{m}^{2}$ is the eigenvalue\nassociated with mode $m$, which defines the $m$$^{th}$-baroclinic\nRossby radius of deformation $\\lambda_{m}^{-1}$. The barotropic mode\nis the (depth independent) mode $F_{0}$, associated with $\\lambda_{0}=0$.\n \n\n\n\\subsubsection{Surface quasi-geostrophic (SQG) modes \\label{sub:SQG_modes}}\n\nIn the following, it will be convenient to distinguish the interior\ndynamics from the surface dynamics. For a given potential vorticity\nfield $q(x,y,z,t)$ and surface buoyancy field $b_{s}(x,y,t)$ , the\nstreamfunction can be written $\\psi=\\psi_\\text{int}+\\psi_\\text{surf}$,  where $\\psi_\\text{int}$ is the solution of \n\\begin{equation}\n  \\Delta\\psi+\\frac{\\partial}{\\partial z} \n  \\left(\\frac{f_{0}^{2}}{N^{2}}\\frac{\\partial}{\\partial z}\\psi\\right)=q-\\beta y\\quad\\text{with}\\quad\\partial_{z}\\psi|_{z=0}=\\partial_{z}\\psi|_{z=-H}=0,\n\\label{eq:PV_interior}\n\\end{equation}\n and where $\\psi_\\text{surf}$ is the solution of \n\\begin{equation}\n\\Delta\\psi+\\frac{\\partial}{\\partial z} \\left(\\frac{f_{0}^{2}}{N^{2}}\\frac{\\partial}{\\partial z}\\psi\\right)=0, \\text{~with ~~} \\partial_{z}\\psi|_{z=0}=\\left(\\frac{N(0)}{f_{0}}\\right)^2 b_{s},\n \\quad\\partial_{z}\\psi|_{z=-H}=0 \\ .\\label{eq:PV_surface}\n\\end{equation}\n\nEach projection of $\\psi_\\text{surf}$ on a (horizontal) Laplacian eigenmode\n$(l,k)$ defines a surface quasi-geostrophic eigenmode (SQG mode hereafter). When the stratification $N$ is constant and when $f_{0}\/(NK)\\ll H$, these SQG modes are decreasing exponential functions \n\\begin{equation}\n\\widehat{\\psi}_{\\text{surf},k,l}=A_{k,l}e^{zNK\/f_{0}} + O\\left(\\frac{h_{K}}{H}\\right),\n\\label{eq:SQGmode_ExpDecay}\n\\end{equation}\nwhere the coefficients $A_{k,l}$ are determined using the boundary\ncondition at the surface. When the stratification is not constant,\nbut when $f_{0}\/N(0)K\\ll H,$ the SQG modes are still characterized\nby a decreasing exponential, with an $e$-folding depth $h_{K}=f_{0}\/KN(0)$.\n\nThe distinction between surface and interior streamfunction is sometimes\nuseful to describe the dynamics , see e.g. \\citet{LapeyreKlein06},\nbut a shortcoming of such a decomposition is that $\\psi_\\text{int}$ and\n$\\psi_\\text{surf}$ are not orthogonal.\n\n\\subsection{Delta-sheet approximation and SQG-like modes \\label{sub:SQGlike_modes}}\n\nThe boundary condition (\\ref{eq:FullDynamicsB}) can be formally replaced by the condition of no buoyancy variation ($ \\partial_z \\psi = 0 $ at $z=0$), provided that surface buoyancy anomalies are interpreted as a thin sheet of potential vorticity just below the rigid lid \\citep{Bretherton66}. For this reason, and without loss of generality, we will consider that $b_s=0$ {in the remainder of this paper}.  \n\nLet us now  consider a case with no surface buoyancy, but with a surface intensified potential vorticity field defined as $q=q_\\text{surf}(x,y)\\Theta(z+H_{1})+q_\\text{int}(x,y,z)$, with $H_{1}q_\\text{surf}\\gg Hq_\\text{int}$, $H_{1}\\ll H$ and $\\Theta$ the Heaviside function. Using the linearity of (\\ref{eq:PV_interior}), one can still decompose the streamfunction field into $\\psi=\\psi_\\text{surf}+\\psi_\\text{int}$ where $\\psi_\\text{surf}$ is the flow induced by the PV field $q=q_\\text{surf}\\Theta(z+H_{1})$, obtained by inverting (\\ref{eq:PV_definition}), and $\\psi_\\text{int}$ the flow induced by $q_\\text{int}$. For $z<-H_{1}$, the streamfunction $\\psi_\\text{surf}$ satisfies equation (\\ref{eq:PV_surface}), just as SQG modes. This is why we call it an SQG-like mode. \n\n\n\\section{Equilibrium statistical mechanics of stratified quasi-geostrophic\nflow}\\label{sec:Equilibrium-statistical-mechanics}\n\nIn this section we briefly introduce the MRS equilibrium statistical mechanics theory, and\nintroduce a general method to compute the MRS equilibria. This allows for explicit computation\nof statistical equilibria on the $f-$plane in subsection\n\\ref{sub:Computation-of-the-RSM-f-plane}, and on the $\\beta$-plane in subsection\n\\ref{sub:Including-the-beta}.\n\n\n\\subsection{Theory\\label{sub:Theory}}\n\nLet us consider an initial (fine-grained) PV field $q_{0}(x,y,z)$.\nThe dynamics (\\ref{eq:FullDynamics}) induced by this PV field stirs\nthe field itself, which rapidly leads to its filamentation at smaller\nand smaller scale, with the concomitant effect of stretching and folding.\nThere are two (related) consequences, one in phase space, one in physical\nspace.\n\nIn physical space, the filamentation toward small scales implies that\nafter a sufficiently long time evolution, the PV field can be described\nat a macroscopic level by a probability field\n$\\rho(x,y,z,\\sigma,t)\\mathrm{d}\\sigma$ to measure a given PV level\nbetween $\\sigma$ and $\\sigma+\\mathrm{d}\\sigma$ at a given point\n($x,y,z$). This probability distribution function (pdf hereafter)\n$\\rho$ is normalized locally \\begin{equation}\n\\forall ~ x,y,z\\ \\mathcal{N}[\\rho]=\\int_{\\Sigma}\\mathrm{d}\\sigma\\ \\rho=1,\\label{eq:normalization}\\end{equation}\nwhere the integral is performed over the range $\\Sigma$ of PV levels,\nwhich is prescribed by the initial condition. The field $\\rho$ is\na macrostate because it is associated with a huge number of fine-grained\nPV configurations $q(x,y,z)$ (the microscopic states), in\nthe sense that many realizations of the (microscopic, or fine-grained)\nPV field $q$ leads to the same macroscopic state $\\rho$. \n\nThe phase space is defined by all configurations of the field $q(x,y,z)$.\nThe cornerstone of the theory is to assume that  stirring\nallows the system to explore evenly each possible configuration of\nthe phase space that satisfies the constraints of the dynamics, so\nthat these configurations can be considered equiprobable. When this assumption fails, as in the linear regime discussed in section 4.4, the statistical theory fails. \n\nThe main idea of the theory is then to count the number of microstate\nconfigurations associated with a macrostate (the pdf field $\\rho$),\nand to say that the equilibrium state is the most probable macrostate\nthat satisfies the constraints of the dynamics, i.e. the one that\nis associated with the largest number of microstates satisfying the\nconstraints. One can then show that an overwhelming number of microstates\nare associated with this most probable macrostate \\citep{Michel_Robert_LargeDeviations94C}. An important physical consequence is that an observer who wants to see the actual equilibrium macrostate would neither need to perform ensemble average nor time average, but would simply have to wait a sufficiently long time.\n\n\nNote that the output of the theory is the pdf field $\\rho$, but the\nquantity of interest to describe the flow structure is the coarse-grained PV field defined as \n\\begin{equation}\n\\overline{q}=\\int_{\\Sigma}\\mathrm{d}\\sigma\\ \\sigma\\rho.\n\\label{eq:q_coarse_grained}\n\\end{equation}\nIn practice, counting the number of microstates $q$ associated with\na macrostate $\\rho$ is a difficult task, because $q$ is a continuous\nfield, and it requires the use of large deviation theory{, see e.g. \\cite{BouchetCorvellec10}}. However,\nit can be shown that classical counting arguments \ndo apply here, and that the most probable state $\\rho$ maximizes\nthe usual Boltzmann mixing entropy\n\\begin{equation}\n\\mathcal{S}=-\\int_{\\mathcal{D}}\\mathrm{d}x\\mathrm{d}y\\ \\int_{-H}^{0}\\mathrm{d}z\\ \\int_{\\Sigma}\\mathrm{d}\\sigma\\ \\rho\\ln\\rho\\ ,\n\\label{eq:entropy_mixing}\n\\end{equation}\nsee  \\citet{BouchetCorvellec10} and references therein for further details.\n\n\n\nLet us now collect the constraints that $\\rho$ must satisfy, in addition\nto the local normalization (\\ref{eq:normalization}). \nAn infinite number of constraints are given by conservation\nof the global distribution of PV levels $\\mathcal{P}(\\sigma,z)$, which is prescribed  by the initial global distribution of PV levels $P_{0}(\\sigma,z)$: \n\\begin{equation}\n\\mathcal{P}(\\sigma,z)[\\rho]=\\int_{\\mathcal{D}}\\mathrm{d}x\\mathrm{d}y\\ \\rho=P_{0}(\\sigma,z).\\label{eq:PV_constraint}\\end{equation}\nThese conservation laws include the conservation of the fine-grained enstrophy\n\\begin{equation}\n\\int_{\\mathcal{D}} \\mathrm{d} x  \\mathrm{d} y \\int_{\\Sigma} \\mathrm{d} \\sigma\\ \\sigma^2 \\rho  =Z_0 \\ . \\label{eq:Z0} \n\\end{equation}\nAnother constraint is given by the conservation of the total energy\n  \\begin{equation}\n\\mathcal{E}=\\frac{1}{2}\\int_{-H}^{0}\\mathrm{d}z \\int_{\\mathcal{D}}\\mathrm{d}x\\mathrm{d}y\\ \\left(\\left(\\nabla\\psi\\right)^{2}+\\frac{f_{0}^{2}}{N^{2}}\\left(\\partial_{z}\\psi\\right)^{2}\\right) \\ \n\\label{eq:Energie_definition}\\end{equation}\nBecause an overwhelming number of microstates are close to the equilibrium state $\\rho$, the energy of the fluctuations are negligible with respect to the energy due to the coarse-grained PV field $\\overline{q}$ \\citep{Michel_Robert_LargeDeviations94C}. Integrating \\ref{eq:Energie_definition} by parts and replacing $q$ by $\\overline{q}$ gives the constraint  $E_{0}=\\mathcal{E}[\\rho]$, where $E_{0}$ is the energy of the initial condition, and where \n\\begin{equation}\n\\mathcal{E}=\\frac{1}{2} \\int_{-H}^{0}\\mathrm{d}z \\int_{\\mathcal{D}}\\mathrm{d}x\\mathrm{d}y \\left(\\beta y-\\overline{q}  \\right)\\psi=\\frac{1}{2} \\int_{-H}^{0}\\mathrm{d}z\\int_{\\mathcal{D}}\\mathrm{d}x\\mathrm{d}y \\int_{\\Sigma}\\mathrm{d} \\sigma  \\left( \\beta y-\\sigma \\right)\\psi\\rho\\ .\\label{eq:energy_functional_rho} \n\\end{equation}\nFinally, the MRS theory provides a variational problem\n\\begin{equation}\nS\\left(E_{0},P_{0}(\\sigma,z)\\right)=\\max_{\\rho,\\mathcal{N}[\\rho]=1}\\left\\{ \\mathcal{S}\\left[\\rho\\right]\\ |\\ \\mathcal{E}\\left[\\rho\\right]=E_{0}\\ \\&\\ \\mathcal{P}\\left[\\rho\\right]=P_{0}(\\sigma,z)\\right\\} \\,,\\label{eq:RSM}\\end{equation}\nwhich means that the equilibrium state is the density probability field\n$\\rho_\\text{rsm}$ that satisfies the local normalization constraint (\\ref{eq:normalization}), the energy constraint $\\mathcal{E}\\left[\\rho\\right]=E_{0}$ with $\\mathcal{E}$ given by (\\ref{eq:energy_functional_rho}),  the incompressibility constraint $\\mathcal{P}(\\sigma,z)\\left[\\rho\\right]=P_{0}(\\sigma,z)$ with $\\mathcal{P}$ given by (\\ref{eq:PV_constraint}), and that maximizes the entropy functional $\\mathcal{S}$ given by (\\ref{eq:entropy_mixing}). We see that the parameters of this problem are the values of the constraints, which are prescribed by the initial condition $q_{0}(x,y,z)$.\n\n\n\\subsection{MRS equilibrium states characterized by a linear $q-\\psi$ relation \\label{sub:Energy-enstrophy-equilibrium-states}}\n\n\n\\subsubsection{Simplification of the variational problem \\label{sub:Simplification-of-the}}\n\nIn practice, MRS equilibrium states are difficult to compute, because\nsuch computations involve the resolution of a variational problem\nwith an infinite number of constraints. Computing the critical points\nof (\\ref{eq:RSM}) is straightforward, see e.g. equation (\\ref{eq:rhoChar})\nof appendix A, but showing that second order variations of the entropy\nfunctional are negative is in general difficult. However, recent results\nhave allowed considerable simplifications of the analytical computations\nof these equilibrium states \\citep{Bouchet:2008_Physica_D}.\nThe idea is first to prove equivalence (usually\nfor a restricted range of parameters) between the complicated variational\nproblem (\\ref{eq:RSM}) and other variational problems more simple\nto handle, involving less constraints than the complicated problem,\nbut characterized by the same critical points, and then to explicitly\ncompute solutions of the simpler variational problem.  The equivalence\nbetween the variational problems ensures that solution of the\nsimple problem are also solutions of the more complicated one.\nWe will follow this approach in the following. \n\n\\subsubsection{Minimum enstrophy variational problem \\label{sub:MRS-linear}}\n\nStarting from the MRS variational (\\ref{eq:RSM}), it is shown in appendix A that any MRS equilibrium state characterized by a linear $\\overline{q}-\\psi$ relation is a solution of the variational problem\n\\begin{equation}\nZ_\\text{cg}^\\text{tot}\\left(E_{0},Z_{0}(z)\\right)=\\min_{\\overline{q}}\\left\\{ \\mathcal{Z}_\\text{cg}^\\text{tot}\\left[\\overline{q}\\right]\\ |\\ \\mathcal{E}\\left[\\overline{q}\\right]=E_{0}\\right\\} \\label{eq:MinimumEnstrophyVP}\\end{equation}\nwith\n\\begin{equation}\n\\mathcal{Z}_\\text{cg}^\\text{tot}\\left[\\overline{q}\\right]=\\int_{-H}^{0}\\mathrm{d}z\\ \\frac{\\mathcal{Z}_{cg}}{Z_{0}},\\quad\\mathcal{Z}_{cg}=\\frac{1}{2}\\int_{\\mathcal{D}}\\mathrm{d}x\\mathrm{d}y\\ \\overline{q}^{2},\\label{eq:Enstrophy_coarse-grained}\n\\end{equation}\nwhere $Z_{0}(z)$ is the fine-grained enstrophy profile, $E_{0}$\nthe energy. This variational problem amounts to find the minimizer\n$\\overline{q}_{m}$ (a coarse grained PV field) of the coarse-grained\nenstrophy $\\mathcal{Z}_{cg}^{tot}$ (with $\\mathcal{Z}_{cg}^{tot}\\left[\\overline{q}_{m}\\right]=Z_{cg}^{tot}\\left(E_{0},Z_{0}(z)\\right)$), among all the fields $\\overline{q}$ satisfying the constraint $\\mathcal{E}[\\overline{q}]=E_{0}$ given by (\\ref{eq:Energie_definition}).%\n\nCritical points of the variational problem (\\ref{eq:MinimumEnstrophyVP})\nare computed by introducing a Lagrange multiplier $\\beta_{t}$ associated\nwith the energy constraint, and by solving\\[\n\\delta\\mathcal{Z}_{cg}^{tot}+\\beta_{t}\\delta\\mathcal{E}=0,\\]\n where first variations of the functionals are taken with respect\nto the field $\\overline{q}$. Because $\\beta_{t}$ is the Lagrange\nparameter associated with the energy constraint, it is called inverse\ntemperature%\n\\footnote{The notation $\\beta$ is traditionally used in thermodynamics for\ninverse temperature. We use here the convention $\\beta_{t}$ for this\ninverse temperature, since $\\beta=\\partial_{y}f$ refers to the beta\neffect, i.e. to the variation of the Coriolis parameter with latitude. %\n}. Using $\\delta\\mathcal{Z}_{cg}^{tot}=\\int_{\\mathcal{D}}\\mathrm{d}x\\mathrm{d}y\\ \\int_{-H}^{0}\\mathrm{d}z\\ \\overline{q}\\delta\\overline{q}\/Z_{0}$\nand $\\delta\\mathcal{E}=-\\int\\mathrm{d}x\\mathrm{d}ydz\\ \\psi\\delta\\overline{q}$,\none finds that critical points of this problem are \\begin{equation}\n\\overline{q}=\\beta_{t}Z_{0}\\psi.\\label{eq:q-psi_relation_with_coeff_determined}\\end{equation}\n The flow structure of these critical points can then be computed\nby solving\n\n\\begin{equation}\n\\Delta\\psi+\\frac{\\partial}{\\partial z}\\frac{f_{0}^{2}}{N^{2}}\\frac{\\partial}{\\partial z}\\psi+\\beta y=Z_{0}\\beta_{t}\\psi\\quad\\text{with}\\quad\\partial_{z}\\psi|_{z=0,z=-H}=0.\\label{eq:CriticalPoints}\\end{equation}\nCritical points can then be classified according to the value of their\ninverse temperature $\\beta_{t}$. \nAll the problem is then to find which of these critical point are\nactual minima of the coarse-grained enstrophy $\\mathcal{Z}_{cg}^{tot}[\\overline{q}]$ for a given energy $\\mathcal{E}[\\overline{q}]=E_{0}$. %\n\nMRS equilibrium states characterized by linear $\\overline{q}-\\psi$\nrelations are expected either in a low energy limit or in a strong\nmixing limit, see \\citet{BouchetVenaillePhysRep11} and references\ntherein. \nLet us estimate when the low energy limit is valid. For a given distribution\n$P_{0}(\\sigma,z)$ of PV levels $\\sigma$, one can compute the maximum\nenergy $E_{max}$ among all the energies of the PV fields characterized\nby the distribution $P_{0}(\\sigma,z)$. Low energy states characterized\nby the same distribution $P_{0}(\\sigma,z)$ are therefore those characterized\nby $E\\ll E_{max}$. \nThe strong mixing limit, which can in some case overlap the low energy\nlimit, is obtained by assuming $\\beta_{t}\\sigma\\psi\\ll1$ in the expression\n(\\ref{eq:rhoChar}) of the critical points of the MRS variational\nproblem (\\ref{eq:RSM}), see \\citep{ChavanisSommeria:1996_JFM_Classification}.  {Note that these results would remain valid in the presence of bottom topography. The only change would be the expression of the energy, and the bottom boundary condition.}\n\n\\subsubsection{Link with the phenomenological principle of enstrophy minimization and other equilibrium states}\n\nAt a phenomenological level, the variational problem (\\ref{eq:MinimumEnstrophyVP}) can be interpreted as a generalization of the minimum enstrophy principle\nof \\citet{BrethertonHaidvogel} to the continuous stratified flow:\nthe equilibrium state minimizes the vertical integral of the coarse-grained\nenstrophy normalized by the fine-grained enstrophy in each layer,\nwith a global constraint given by energy conservation. This normalization\nis necessary to give a measure of the degree of mixing comparable\nfrom one depth $z$ to another. \n\nIt has been shown by \\cite{Carnevale_Frederiksen_NLstab_statmech_topog_1987JFM} that solutions of the minimum enstrophy problem of \\cite{BrethertonHaidvogel} for one layer quasi-geostrophic flows, are energy-enstrophy equilibrium states of the Galerkin-truncated dynamics of these models, in the limit when the high-wave number cut-off goes to infinity. These energy-enstrophy equilibrium states were first introduced in the framework of 2D truncated Euler flows by \\cite{Kraichnan_1967PhFl...10.1417K}. \\cite{SalmonHollowayHendershott:1976_JFM_stat_mech_QG} generalized the theory to one or few layers truncated quasi-geostrophic flows. \\cite{FrederiksenSawford} applied the theory for a barotropic flow on a sphere, and \\cite{fredericksen91GAFDa} generalized these results for two-layers quasi-geostrophic flow on a sphere. In this latter case, interesting stability results were obtained by \\cite{FrederiksenB}.\n\nThe case of continuously stratified flow above topography was addressed by \\citet{Merryfield98JFM} for truncated dynamics. \\citet{Merryfield98JFM} computed critical points of the statistical mechanics variational problem, found $\\overline{q}=\\mu_t(z) \\psi$ and discussed the vertical structures of these flows depending on $\\mu_t$. We showed in the previous subsection that the parameter $\\mu_t$ is related to the fine-grained enstrophy profile through $\\mu_t=\\beta_t Z_0(z)$. In addition, the critical states described by  \\citet{Merryfield98JFM} could either be saddle, minima or maxima of the entropy functional. We show in the next section that we can select which of them are actual entropy maxima (or equivalently coarse-grained enstrophy minima), in different limit cases.\n\n\\subsubsection{Transients, mean and sharp equilibrium states}\n\n{In the framework of  energy-enstrophy theory, a distinction is often made between ``transients'' and ``mean'' energy, see e.g. \\cite{SalmonHollowayHendershott:1976_JFM_stat_mech_QG,fredericksen91GAFDa,MajdaWangBook}, among others. Statistical equilibria are called ``sharp'' when the ``transients'' vanish. In earlier works, the ``mean'' quantities were often computed by summing over all the microstates distributed according to the canonical measure, which were implicitly assumed to be equivalent to the microcanonical measure.  In the absence of topography and beta plane, the ``mean'' states defined in this way are always zero in a doubly periodic domain, whatever the resolution of the truncated model, and all the energy is in the ``transients''.}\n\n{By contrast, the equilibrium states computed in the MRS framework in the \\textit{microcanonical ensemble} are always sharp: the energy of the fluctuations necessarily vanish, which can be shown with large deviation theory, see \\cite{BouchetCorvellec10} and references therein. This apparent paradox is solved when one realizes that microcanonical and canonical ensembles are, in general, not equivalent for long-ranged interacting systems, see e.g. \\cite{Ellis00} for general considerations and \\cite{VenailleBouchetJSP11} for application to two-dimensional and geophysical flows.} \n\n{In the case of a doubly periodic domain without bottom topography and beta effect, the MRS equilibrium states (or the energy-enstrophy equilibria) are degenerate, because of the symmetries of the domain, see \\citep{BouchetVenaillePhysRep11} for more details and for a discussion on how to remove the degeneracy, see also \\cite{Corentin} for the case of a spherical domain. In practice, the dynamics breaks the system symmetry by selecting one of the degenerate equilibrium states. Of course, if one considers the average potential vorticity field over all the degenerate equilibria, then one recovers zero, and this is what happens when computing the ``mean'' state in the canonical ensemble. We stress that for either quasi-geostrophic flows or  truncated quasi-geostrophic flows in the limit of infinite resolution, the freely evolving dynamics itself can not jump from one state to another once the symmetry is broken: in the microcanonical ensemble, there are no ``transients''. We also stress that the physically relevant statistical ensemble for an isolated system such as a freely evolving flow is the microcanonical ensemble.}\n\n\\subsection{Enstrophy minima on a $f$-plane \\label{sub:Computation-of-the-RSM-f-plane}}\n\nIn the previous subsection, we have shown that MRS equilibrium states\ncharacterized by a linear $\\overline{q}-\\psi$ relationship are solution\nof a minimum coarse-grained enstrophy variational problem (\\ref{eq:MinimumEnstrophyVP}). Such MRS equilibrium states will be referred to as ``coarse-grained\nenstrophy minima'' in the following. We found in the previous section\nthat the affine relation of these coarse-grained enstrophy minima\nare of the form $\\overline{q}=\\beta_{t}Z_{0}\\psi$, with\na coefficient proportional to the fine-grained enstrophy $Z_{0}$,\nand are solutions of (\\ref{eq:CriticalPoints}). The critical points\nof the variational problem (\\ref{eq:MinimumEnstrophyVP}) are any\nstates $\\overline{q}$ satisfying these conditions. The aim of this\nsubsection is to find which of these critical point $\\overline{q}$\nare the actual coarse-grained enstrophy minima, solutions of variational\nproblem (\\ref{eq:MinimumEnstrophyVP}), when there is no $\\beta-$effect,\nbut for arbitrary stratification $N$.\n\nInjecting (\\ref{eq:q-psi_relation_with_coeff_determined}) in (\\ref{eq:Enstrophy_coarse-grained}),\nand using the expression (\\ref{eq:energy_functional_rho}) with the\nconstraint $E_{0}=\\mathcal{E}[\\rho]$, one finds that the coarse-grained\nenstrophy of each critical point $\\overline{q}$ is given by \\begin{equation}\n\\mathcal{Z}_\\text{cg}^\\text{tot}=-\\beta_{t}E_{0}\\ .\\label{eq:enstrophy_critical_point_simple_case}\n\\end{equation}\n Remarkably, the coarse grained enstrophy of a critical point $\\overline{q}$\ndepends only on the inverse temperature $\\beta_{t}$ . We conclude\nthat coarse-grained enstrophy minimum states are the solutions of\n(\\ref{eq:CriticalPoints}) associated with the largest value $\\beta_{t}$.\n\n \nProjecting (\\ref{eq:CriticalPoints}) on the Laplacian eigenmode\n$e_{l,k}(x,y)$ gives \\begin{equation}\n\\frac{\\partial}{\\partial z}\\left(\\frac{f_{0}^{2}}{N^{2}}\\frac{\\partial}{\\partial z}\\widehat{\\psi}_{k,l}\\right)=\\left(\\beta_{t}Z_{0}+K^{2}\\right)\\widehat{\\psi}_{k,l}\\ ,\n\\label{eq:VertStruc}\n\\end{equation}\nwith\n\\[\n\\partial_{z}\\widehat{\\psi}_{k,l}|_{z=0,-H}=0,\\quad K^{2}=k^{2}+l^{2},\\quad \\psi=\\sum_{k,l}\\widehat{\\psi}_{k,l}e_{kl}.\\]\nWe see that each critical point is characterized by a given wavenumber modulus $K$. Its vertical structure and the corresponding value of $\\beta_t$ must be computed numerically in the case of arbitrary profiles $Z_0(z)$. \nLet us  consider the example shown in Fig.\\ \\ref{fig:e-folding_depth}, for a two-step fine-grained enstrophy profile \n\\begin{equation}\nZ_{0}=Z_\\text{surf}\\Theta\\left(z+H_{1}\\right)+Z_\\text{int}\\Theta\\left(-z-H_{1}\\right),\\quad H_{1}\\ll H,\\label{eq:Z0_profile_PieceWise_TwoLayers}\n\\end{equation}\nwhere $\\Theta$ is the Heaviside function, and for $Z_\\text{int}\/Z_\\text{surf}$\nvarying between $0$ and $1$. \nWe find that the minimum coarse-grained\nenstrophy states are always characterized by the gravest horizontal\nmode on the horizontal ($K=1$). As for the vertical structure, we\nobserve Fig.\\ \\ref{fig:e-folding_depth} a tendency toward more barotropic\nflows when the ratio $Z_\\text{int}\/Z_\\text{surf}$ tends to one.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{z0psi_xfig}\n\\caption{Left panel: three different fine grained enstrophy profiles. Right\npanel: corresponding vertical structure of statistical equilibrium\nstates ($\\psi(z)\/\\psi(0)$ on the left panel), in the case of constant\nstratification ( $f_{0}^{2}\/N^{2}=0.1$). The $e$-folding depth in\ncase A is $h=f_{0}\/NK$ (with here $K=1$ for the statistical equilibrium\nstate). \n\\label{fig:e-folding_depth}}\n\\end{center}\n\\end{figure}\nIt is instructive to consider the limit cases $Z_\\text{surf}=Z_\\text{int}$\nand $Z_\\text{int}=0$, corresponding respectively to case A and C of Fig.\\ \\ref{fig:e-folding_depth},\nfor which minimum enstrophy can be explicitly solved.\nWhen the enstrophy profile is depth independent ($Z_{0}(z)=Z_\\text{int}$,\ncase C of figure \\ref{fig:e-folding_depth}), solutions of (\\ref{eq:VertStruc})\nare given by the modes $F_{m}(z)e_{K}(x,y)$ defined in subsection\n(\\ref{sub:Vertical-and-horizontal-eigenmodes}), and are associated\nwith inverse temperatures $\\beta_{t}=-\\left(\\lambda_{m}^{2}+K^{2}\\right)\/Z_\\text{int}$. We see that the maximum value of $\\beta_{t}$ is reached for the gravest\nhorizontal mode ($K=1$) and the gravest vertical mode $(\\mbox{\\ensuremath{\\lambda}}_{0}=0)$.\nIt means that the coarse-grained enstrophy minimum state is barotropic.\n\nWhen $Z_\\text{int}=0$ (case A of Fig. \\ref{fig:e-folding_depth}), it\nis straightforward to show that critical points, the solutions of\n(\\ref{eq:VertStruc}), are SQG-like modes (see subsection \\ref{sub:SQGlike_modes})\nof $e$-folding depth $h=f_{0}\/N(0)K$, associated with $\\beta_{t}=-N(0)K\/(H_{1}Z_\\text{surf}f_{0})$\nwith corrections of order $H_{1}\/H$ and of order $h\/H$. The coarse-grained\nenstrophy minimum is therefore the SQG-like mode associated with $K=1$\nand with the largest $e-$folding depth $h=Lf\/2\\pi N$, where $L$\nis the domain length scale.\n\nThese examples show the importance of the conservation of fine-grained\nenstrophy to the vertical structure of the equilibrium state. The\nmain result is that statistical mechanics predicts a tendency for\nthe flow to reach the gravest Laplacian mode on the horizontal. The\nvertical structure associated with this state is fully prescribed\nby solving (\\ref{eq:VertStruc}) with $K=1$. Because the barotropic\ncomponent of such flows are larger than solutions of (\\ref{eq:VertStruc})\nwith $K>1$, we can say that the inverse cascade on the horizontal\nis associated with a tendency to reach the gravest vertical mode compatible\nwith the vertical fine-grained enstrophy profile $Z_{0}$. This means\na tendency toward barotropization, although in general, the fact that\nthe profile $Z_{0}$ is non constant prevents complete barotropization.%\n\n{Note that the previous results are presented in the case of a doubly periodic domain, but it would be straightforward to generalize them to channel geometries, or to any other domains with boundaries.  The only change would be the spatial structure of the Laplacian eigenmodes. For these domain geometries, one would also  need to discuss the effect of the circulation values at the boundaries.}\n\n\\subsection{Including the $\\beta$-effect \\label{sub:Including-the-beta}.}\n\nWe now discuss how the results from the previous section generalize in the presence of beta. For a given initial condition $\\psi_{0}(x,y,z)$, increasing $\\beta$ increases the contribution of the (depth independent) available potential enstrophy defined as \n\\begin{equation}\nZ_{p}=\\beta^{2}\\int_{\\mathcal{D}}\\mathrm{d}x\\mathrm{d}y\\ y^{2},\\label{eq:Potential_Enstrophy}\n\\end{equation}\nto the total fine-grained enstrophy profile $Z_{0}(z)=\\int_{\\mathcal{D}}\\mathrm{d}x\\mathrm{d}y\\ q_{0}^{2}$, where $q_{0}$ is the initial PV field that can be computed by injecting $\\psi_{0}$ in (\\ref{eq:PV_definition}). For sufficiently large values of $\\beta$, the PV field is dominated by the beta effect ($q_0 \\approx \\beta y $),  $Z_{0}$ therefore tends to $Z_{p}$ and becomes depth independent. Because statistical equilibria computed in the previous subsection were fully barotropic when the fine-grained enstrophy $Z_{0}$ was depth-independent, we expect a tendency toward barotropization by increasing $\\beta$.\n\nConsider for instance the case where $\\psi_{0}$ is an SQG-like mode\nsuch that $q_{0}=\\beta y$ for $z<-H_{1}$, with $H_{1}\\ll H$ and\nassociated with a surface fine-grained enstrophy $Z_{0}=Z_\\text{surf}$\nfor $z>-H_{1}$. The initial profile of fine-grained enstrophy $Z_{0}$\nis thus given by (\\ref{eq:Z0_profile_PieceWise_TwoLayers}), with\nthe interior enstrophy $Z_\\text{int}=Z_{p}$ prescribed by the available\npotential enstrophy (\\ref{eq:Potential_Enstrophy}). The question\nis then to determine whether the results obtained previously without\nbeta effect still hold in presence of it.\n\n{In oder to apply properly MRS theory in the presence of beta, one must consider southern and northern boundaries. Indeed, MRS theory relies on the conservation of the global fine-grained PV distribution. Strictly speaking, this  distribution is not conserved in the doubly periodic configuration on a beta plane, since any fluid particle that travels $2\\pi$ in the $y$ direction gains a value $-2 \\pi \\beta$, and one can not know a priori how many times a fluid particle will turn over the domain in the $y$ direction before the flow reaches equilibrium. In practice, the theory can  still be used in the doubly periodic case to make qualitative predictions, as explained in the next section.}\n\n{In the channel configuration, the streamfunction is a constant at $y= \\pm \\pi$, and we can compute the statistical equilibria in two limit cases. First, when $\\beta=0$, we recover the configuration with surface intensification of the fine grained enstrophy profile $Z_{0}$, for which we found that statistical equilibria were the SQG-like modes associated with the gravest Laplacian horizontal modes, see subsection \\ref{sub:Computation-of-the-RSM-f-plane}.  Second, with $\\beta$ sufficiently large, such that $Z_{p}\\gg Z_\\text{surf}$, the statistical equilibria can be computed by considering the case of a constant fine-grained enstrophy profile $Z_{0}$. In the general case, the values of the streamfunction at the northern and southern boundaries are different, and are determined using constraints given by mass conservation and  circulation conservation along each boundary, see e.g. \\citep{PedloskyBook}. Here we consider the simple case  $\\psi(x,y=\\pm\\pi,z)=0$, with zero total circulation ($\\int_{\\mathcal{D}}  \\mathrm{d} x \\mathrm{d} y q=0$) at each depth. It is necessary to take into account the conservation of linear momentum (associated with translational invariance of the problem in the $x$ direction), which provides an additional constraint $\\mathcal{L}[\\overline{q}]=L_{0}$, with}\n\\begin{equation}\n\\mathcal{L}[\\overline{q}]= \\int_{-H}^0 \\mathrm{d} z \\ \\int_{\\mathcal{D}} \\mathrm{d}x\\mathrm{d}y\\ \\overline{q}y.\\label{eq:Momentum_y} \\end{equation}\n{Critical points of the variational problem (\\ref{eq:MinimumEnstrophyVP}) with this additional constraint satisfy}\n\\begin{equation}\n\\overline{q}=\\left(\\beta_{t}  \\psi -\\mu y\\right)Z_{0}, \\label{eq:q-psi_with_linear_momentum}\n\\end{equation}\n{where $\\mu$ and $\\beta_t$ are respectively the Lagrange multipliers associated with the linear momentum conservation and the energy conservation. Equilibrium states therefore satisfy} \n\\begin{equation}\nZ_{0}\\left(\\beta_{t} \\psi - \\mu y\\right)=\\Delta\\psi+\\frac{\\partial}{\\partial z}\\left(\\frac{f_{0}^{2}}{N^{2}}\\frac{\\partial}{\\partial z}\\psi \\right) +\\beta y\\ . \\label{eq:critical_points_channel}\n\\end{equation}\n{Taking $\\mu=-\\beta\/Z_{0}$, we recover the $f$-plane case of the previous subsection: solutions of Eq. (\\ref{eq:critical_points_channel}) are barotropic-baroclinic modes on the vertical with eigenvalue $-\\lambda_m^2$, and are Laplacian eigenmodes in the channel (with eigenvalue denoted by $-K^2$), with $\\beta_t Z_0=-K^2+\\lambda_m^2$. These states characterized by $\\beta_{t}=-\\left(K^2+\\lambda_m^2\\right)\/Z_{0}$, $\\mu=-\\beta\/Z_{0}$  are drifting towards the west at speed of  Rossby waves: $U_{drift}=-\\beta\/\\left(\\lambda_m^2+K^2\\right)$. The enstrophy minimizer among these states is the one associated with the smallest value of $\\beta_t$, namely the  barotropic mode associated  with the gravest Laplacian eigenmode consistent with the circulation and momentum constraints. When $\\mu \\ne \\beta\/Z_0$, Eq. (\\ref{eq:critical_points_channel}) can be inverted, and $\\psi$ is barotropic since $(\\beta-\\mu Z_0) y$ is depth-independent. The computation of the minimum enstrophy state is therefore equivalent to computing the minimum enstrophy state of a barotropic flow in a channel. A complete discussion of minimum enstrophy states in a barotropic channel are presented in \\cite{CorvellecPhD}, see also \\cite{MajdaWangBook}, with detailed computations including the effect of bottom topography and of non-zero circulations at the boundaries. The important point here is that in the large beta limit, the statistical equilibrium states are barotropic.}\n\n\\section{Numerical simulations \\label{sec:Numerical-simulations}}\n\n\\subsection{Experimental set-up\\label{sub:Numerical-settings}}\n\nWe consider in this section the final state organization of an initial SQG-like\nmode (defined in subsection \\ref{sub:SQGlike_modes}), varying the values of\n$\\beta$ in a doubly periodic square domain of size $L=2\\pi$. More precisely, we\nconsider a vertical discretization and an horizontal Galerkin truncation of the\ndynamics (\\ref{eq:FullDynamics}), for an initial potential vorticity field\n$q_{0}=q_{0,surf}(x,y)\\Theta\\left(z+H_{1}\\right)+\\beta y$, such that $q_{0}=\\beta\ny$ in the interior ($-H<z<-H_{1}$) and $q_{0}\\approx q_{0surf}$ in a surface layer\n$z>-H_{1}$. The surface PV $q_\\text{surf}(x,y)$ is a random field with random\nphases in spectral space, and a Gaussian power spectrum peaked at wavenumber\n$K_{0}=5$, with variance $\\delta K_{0}=2$, and normalized such that the total\nenergy is equal to one ($E_{0}=1$). This corresponds to the case discussed\nsubsection (\\ref{sub:Including-the-beta}), with a vertical profile of fine-grained\nenstrophy given by (\\ref{eq:Z0_profile_PieceWise_TwoLayers}), where the interior\nenstrophy is given by the available potential enstrophy\n(\\ref{eq:Potential_Enstrophy}): for $z<-H_{1}$, $Z_{0}=4\\pi^{4}\\beta^{2}\/3$.\n\nWe perform simulations of the dynamics by considering a vertical discretization\nwith $10$ layers of equal depth, horizontal discretization of $512^{2}$, $H=1$,\nand $F=(Lf_{0}\/HN)^{2}=1$, using a pseudo-spectral quasi-geostrophic model for the horizontal dynamics\n without small scale dissipation \\citep{SmithVallis01}. We choose $H_{1}=H\/10$ for the initial condition, so that there is non zero enstrophy only in the upper layer in the absence of a beta\neffect. We also perform experiments in the presence of small scale dissipation\n(Laplacian viscosity or hyperviscosity), and found no qualitative differences as\nfar as the large scale horizontal structure and the vertical structure of the flow\nwere concerned. The only difference in presence of small scale dissipation is the\ndecay of the enstrophy in each layer, because small scale dissipation smooths out\nPV filaments. Because of the energy cascade, the energy remains constant at lowest\norder even in presence of small scale dissipation.\n\nTime integration of the freely evolving dynamics proceeds for about $250$ eddy turnover times.\nTypically, i) the unstable initial condition leads to a strong turbulent stirring and the flow\nself-organizes rapidly into a few vortices, which takes a few eddy turnover times ii) same sign\nvortices eventually merge together on a longer time scale (a few dozen of eddy turnover times),\nand iii) the remaining dipole evolves very slowly on a slow time scale (more than hundreds of\neddy turnover times). In absence of $\\beta$-effect, a good indicator of the convergence is given\nby the convergence of the $q-\\psi$ relation to a single functional relation. As we shall see in\nthe following, this phenomenology is complicated by the presence of beta.\n\n{We saw in the previous section that a proper application of the MRS theory in presence of a beta plane would require considering northern or southern boundaries, since the fine-grained PV distribution is not conserved in the doubly-periodic domain case. However, we still can use the theory to interpret qualitatively the simulations in the doubly periodic case.  For small values of $\\beta$, planetary vorticity gradients are negligible with respect to relative vorticity and stretching terms in the PV expression, in which case one can check that the fine grained PV distribution remains nearly the same during the flow evolution, and MRS can be applied. For large values of $\\beta$, the fluid particles can not escape in the meridional direction, they are confined in zonal bands having a width of the order of the Rhines scale\\footnote{Here $E_{0}=1$ is the initial energy, so that $E_{0}^{1\/2}$ is a good metric for typical velocities of the flow (assuming it becomes barotropic).} $L_R = 2 \\pi \\sim E_0^{1\/4}\/\\beta^{1\/2}$, and thus the fine-grained PV distribution remains close to the initial one, and MRS theory can be applied.} \n\n\\subsection{SQG-like dynamics ($\\beta = 0$) \\label{sub:SQG-like-dynamics-(small-beta)}}\n\nThe case with $\\beta=0$ is presented in Fig. \\ref{fig:beta2}.\nThe initial PV and streamfunction fields have horizontal structures\naround wave number $K_{0}=5$, which is associated with an SQG-like\nmode of typical e-folding depth $f_{0}\/(NK_{0})\\approx0.1$. \n\nAn inverse cascade in the horizontal leads to flow structures with an horizontal wavenumber $K$ decreasing with time, associated with a tendency toward barotropization: the $e$-folding depth of the SQG-like mode increases as $f_{0}\/2NK$. According to statistical mechanics arguments of the previous section, the flow should tend to the ``gravest'' horizontal SQG-like mode ($K=1$), which is also the ``gravest'' vertical SQG mode, i.e. the one associated with the largest $e$-folding depth. The concomitant horizontal inverse cascade (most of the kinetic energy is in the gravest horizontal mode $K=1$ at the end of the simulation) and the increase of the $e$-folding depth are observed on Fig. \\ref{fig:beta2}, showing good qualitative agreement between statistical mechanics and numerical simulations.\n\n {There are however quantitative differences between the observed final state organization and the prediction of MRS equilibria characterized by a linear $\\overline{q}-\\psi$ relation. As seen on Fig. \\ref{fig:q-psiBETA0}, the $\\overline{q}-\\psi$ relation is well defined, but it is not linear. Taking into account higher order invariants in the MRS theory would allow to better describe this $\\sinh$-like relation, see e.g. \\citep{BouchetSimonnetPRL09}, but it may not be sufficient to describe the ``unmixed'' cores of the remaining vortex in the final dipole \\citep{SchecterPRE03}.}\n\nThe presence of these ``unmixed''  PV blobs implies the existence of structures smaller than wavenumber $K=1$. Because each projection of the PV field on a different wavenumber is associated with an SQG-like mode associated with a different $e$-folding depth, the vertical structure of the flow is actually a linear combination of SQG-like modes associated with different wavenumbers. A consequence is that the effective e-folding depth of the kinetic energy is $f_{0}\/(2NK^\\text{eff})$. We estimated the coefficient $K_\\text{eff}\\approx3.5$ by considering the linear $q-\\psi$ relation passing through the extremal points of the observed $q-\\psi$ relation of Fig. \\ref{fig:q-psiBETA0}. {Let us conclude on this case by discussing the relaxation towards equilibrium: we observed that the $q-\\psi$ relation did not change at all between $150$ and $250$ eddy turnover times, proving that the system reached a stationary state. In addition, we see in  Fig. \\ref{fig:Emodes}-a that the kinetic energy of each vertical mode (and therefore of each layer) reaches a plateau after a few eddy turnover times. As expected from the MRS theory, the statistical equilibrium is sharp, there is no transient.}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{FigF1BETA0_xfig}\n\\caption{case $\\beta=0$ (SQG-like). Only the fields in upper, middle and lower\nlayer are shown.\n \\label{fig:beta2}}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{scat3beta0}\n\\caption{$q-\\psi$ relation in the upper layer for $t=250$ eddy turnover time.\n\\label{fig:q-psiBETA0}}\n\\end{center}\n\\end{figure}\n\n\\subsection{Effect of beta ($\\beta \\sim 1$)}\n\nWe now consider the free evolution of the same initial condition $\\psi_{0}(x,y,z)$\nas in the previous subsection, but on a beta plane. Because\n$\\beta\\ne0$, the initial condition for potential vorticity is different\nthan in the previous section: the total energy and initial velocity\nfields are the same as before, but the vertical profile of fine grained\nenstrophy is different, since there is now available potential enstrophy\nin the interior. As explained in subsection \\ref{sub:Computation-of-the-RSM-f-plane}, this contribution from interior fine-grained enstrophy shall increase\nthe relative barotropic component of the statistical equilibrium state.\nThis is what is actually observed in the final state organization\nof Fig. \\ref{fig:beta3}. We conclude that in this regime, the beta\neffect is a catalyst\\footnote{By catalyst we do not mean a dynamical effect, as in chemistry, but simply the fact that barotropization in the final state is enhanced in presence of a beta plane.} of barotropization, as predicted by statistical mechanics.\n\n{Additionally, the $q-\\psi$ relation presented in Fig. \\ref{fig:q-psiBETA1} presents a very good agreement with the prediction of our computation of barotropic MRS equilibria carried in subsection \\ref{sub:Including-the-beta}: we observe a linear relation between $(q-\\beta y)$ and $\\psi$ with a slope $\\beta_{t}Z_{0}=-1$. This means that the dipole structure observed in the streamfunction field is the actual gravest horizontal Laplacian eigenmode $K=1$, and that this eigenmode is drifting westward at a speed $\\beta\/K^{2}$. In addition, we see in Fig. \\ref{fig:Emodes} that after a few eddy turnover times, almost all the kinetic energy is in the barotropic mode, and that the energy levels in each vertical mode reach a plateau: there are no transients.}\n\n{We conjecture that the good agreement with MRS theory occurs since the initial distribution of PV levels and the initial energy place the flow in the low energy regime considered in the Appendix. In addition, for  intermediate values of $\\beta$, the Rhines scale $L_R=2\\pi E_O^{1\/4}\/\\beta^{1\/2}$ is of the order of the domain scale. The beta effect limit the meridional displacements of fluid particles at this scale; this is why the computations performed Appendix 2 in a channel geometry are qualitatively relevant to this case.} \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{FigF1BETA2p1_xfig}\n\\caption{case $\\beta=2.1$. See Fig.\\ \\ref{fig:beta2} for legend. Note that\nthe colors scales are changed at each time; this is why the interior `` beta plane`` is not visible in the upper panel of potential\nvorticity. \n\\label{fig:beta3}}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{scat3beta2p1}\n\\caption{{Relation between $q-\\beta y$ and streamfunction in the upper layer, at $t=100$ eddy turnover times, for  $\\beta=2.1$. The fact that this relation is close to be linear shows that the small energy limit considered in this paper is relevant to this case. The coefficient of the slope is approximatively one, meaning that the energy is condensed in the gravest horizontal Laplacian eigenmode, which is propagating westward at the speed of barotropic waves. This is consistent with the computation of minimum enstrophy states on a beta plane, see subsection \\ref{sub:Including-the-beta}}. \\label{fig:q-psiBETA1}}\n\\end{center}\n\\end{figure}\n\n\\subsection{Effect of strong Rossby waves  ($\\beta\\gg1$) \\label{sub:High-values-of-beta}}\n\nFor large values of beta there is still an early stage, lasting a few eddy turnover times, of turbulent stirring of the unstable initial condition. But this initial stirring is limited to scales smaller than the Rhines scale $L_{R}=2\\pi E_{0}^{1\/4}\/\\beta^{1\/2}$, which eventually leads to a zonal PV field perturbed by Rossby waves around this scale, see Fig. \\ref{fig:beta4}. The dynamics is then dominated by Rossby waves, just as in a single layer model \\citet{Rhines77,Rhines75}. \n\n\nBecause the perturbation is confined in the upper layer, irreversible turbulent stirring occurs in the upper layer only, but not at the bottom. We did check that the global distribution of coarse-grained PV levels did not change in the lower layer through time, while this distribution was changed in the upper layer. In this case, the fundamental assumption that the system explores evenly the available phase space through turbulent stirring is not valid. One might reasonably argue that the statistical mechanic theory is incomplete, for it cannot identify the phase space that can be explored. In any case, we  conclude that large values of beta prevent the convergence toward this equilibrium, leaving the question of the convergence itself to future work. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{FigF1BETA10_xfig}\n\n\\caption{case $\\beta=10$, see Fig. \\ref{fig:beta2} for legend. \\label{fig:beta4} }\n\\end{center}\n\\end{figure}\n\n\n\n\\subsection{Turbulent stirring vs wave dynamics \\label{sub:The-key-role-of-beta}}\n\nWe summarize the effect of planetary vorticity gradients on Fig.\\\n\\ref{fig:Ebaro_vs_beta} by considering the ratio $E_\\text{baro}\/E_{0}$\nof barotropic energy to total energy, after $250$ eddy turnover times,\nfor varying values of $\\beta$. The plain red line represents statistical\nmechanics predictions. In order to allow comparison with the numerical\nexperiments, for which the horizontal PV field did present structures\nsmaller than the gravest mode $K=1$ for low values of $\\beta$ (see\nsubsection \\ref{sub:SQG-like-dynamics-(small-beta)}), we computed\nthe vertical structure with equation (\\ref{eq:VertStruc}), using\n$K_\\text{eff}=3.5$, and using the enstrophy profile $Z_{0}(\\beta)$ of\nthe initial condition $\\psi_{0}$. Strictly speaking, this effective\nhorizontal wavenumber $K_\\text{eff}$ should vary with beta between\n$K_\\text{eff}=3.5$ for $\\beta=0$ and $K_\\text{eff}=1$ for  $\\beta>1$, but taking into account this variations would not change much the shape of the red plain curve.\n\nThe critical value of $\\beta$ between the turbulent stirring regime for which statistical mechanics predictions are useful and the wave regime can  be estimated by considering the case in which the Rhines scale $L_{Rh}\\sim2\\pi\\left(E_{0}^{1\/2}\/\\beta\\right)^{1\/2}$ is of the order of the domain scale $L=2\\pi$. Here the total energy is $E_{0}=1$, which renders the critical value $\\beta\\sim1$ in our simulations.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{baro_vs_betaF1_xfig}\n\\caption{{Variation of the Barotropic energy $E_{baro}$ normalized by the total energy $E_0$, varying the ratio of the domain scale with the Rhines scale $L\/L_R=\\beta^{1\/2} E_{0}^{-1\/4}L\/2\\pi$, at the end of each run, for the same initial condition $\\psi_{0}$ (here $L=2\\pi$, $E_{0}=1$, so only $\\beta$ varies). Red continuous line: predictions from statistical mechanics. The vertical structure of the equilibrium state is computed with equation (\\ref{eq:VertStruc}), using $K_\\text{eff}=3.5$, and using the enstrophy profile $Z_{0}(\\beta)$ of the initial condition $\\psi_{0}$. The dashed-dotted line separates the wave regime from the turbulent stirring regime, in which we expect the statistical mechanics predictions to be valid.}  \\label{fig:Ebaro_vs_beta}}\n\\end{center}\n\\end{figure}\n\nBecause the ratio $E_\\text{baro}\/E_{0}$ gives incomplete information on\nthe flow structure, and because dynamical information is useful to\nunderstand where statistical mechanics fails to predict\nthe vertical structure, we represent in Fig. \\ref{fig:Emodes} the\ntemporal evolution of the energies $E_{m}$ of the baroclinic modes%\n\\footnote{Note that with this notation, $E_{m=0}=E_\\text{baro}$ is the energy of\nthe barotropic mode, which is in general different from the total\nenergy of the initial condition $E_{0}$.%\n} defined in (\\ref{eq:VerticalModes}).\n\nFor small values of beta (see the first panel of Fig. \\ref{fig:Emodes}),\nthe dynamics reaches the equilibrium after stirring during\nfew eddy turnover times, and the contribution of each mode $E_{m}$\nsimply reflects the SQG-like structure of the final state. \nFor values of beta of order one (see the second panel of Fig. \\ref{fig:Emodes}),\nthe same stirring mechanism leads to a barotropic flow after\na few eddy turnover times.\nFor values of $\\beta$ larger than one, there is still an initial\n stirring regime, but the duration of this regime tends to\ndiminish with larger values of beta. We observe a slow transfer of\nenergy from first baroclinic to the barotropic mode in the case $\\beta=10$\n(see the third panel of Fig. \\ref{fig:Emodes}), but it is unclear\nif it will eventually lead to a condensation of the energy in the\nbarotropic mode. In the case $\\beta=100$ (see the fourth panel of\nFig. \\ref{fig:Emodes}), there is in average almost no energy transfers\nbetween the vertical modes, after the initial stirring regime. Again,\nit is unclear if interactions between waves would eventually condense\ntheir energy into the barotropic mode in a longer numerical run. The\nstrong temporal fluctuations of the contribution of each vertical\nmode visible on the wave regime of figure Fig. \\ref{fig:Emodes} may\nexplain why they are large variations of $E_\\text{baro}\/E_{tot}$ (in addition\nto the mean tendency of a decay with $\\beta$) in the wave regime\n figure \\ref{fig:Ebaro_vs_beta}. \n\n{To conclude, in the turbulent stirring regime (i.e. $L_R \\sim L$ or $L_R>L$) MRS statistical mechanics predictions are qualitatively correct, and there are no ``transients'' (no temporal fluctuations) once the flow is self-organized. By contrast, in the wave regime (i.e. $L_R \\ll L$) , there are strong temporal fluctuations in the mode amplitudes, and we believe that statistical mechanics (either MRS of energy-enstrophy equilibria for the truncated dynamics) can not account for these observations.}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{modes_xfig}\n\n\\caption{Temporal evolution of the contribution of the energy of each baroclinic\nmode to the total energy, for different values of $\\beta$. The modes\nare indexed by $m$, with $m=0$ the barotropic mode and $m=1$ the\nfirst baroclinic mode. \\label{fig:Emodes}}\n\\end{center}\n\\end{figure}\n\n\n\\section{Discussions and conclusions \\label{sec:Conclusion}}\n\nWe have used  statistical mechanics to predict the\nflow structure resulting from the self-organization of freely evolving\ninviscid stratified quasi-geostrophic flows. The only assumption of\nthe theory is that the flow explores evenly the available phase space.\nIn order to compute explicitly statistical equilibria, and to discuss\nbasic physical properties of these equilibria, we have focused on\nMRS equilibria characterized by linear relation between potential\nvorticity and streamfunction. For such MRS equilibrium states, we need the total energy and the fine-grained enstrophy in each layer, although all the invariants are implicitly taken into account.\n\nWe explained that these states are expected in a low energy limit\nor in a strong mixing limit. By applying a method proposed by \\citep{Bouchet:2008_Physica_D}, we have shown that MRS equilibria characterized by a linear $q-\\psi$ relation are solutions of a minimum coarse-grained enstrophy variational\nproblem, which generalizes the Bretherton-Haidvogel minimum enstrophy\nprinciple to the stratified case.  \n\n\nThe central result of the paper is the elucidation of the physical\nconsequences of the conservation of the fine-grained enstrophy $Z_{0}(z)$\non the structure of the equilibrium states. We showed first that the\naffine coefficients between the coarse-grained PV field and the streamfunction\nare proportional to the fine-grained enstrophy: $\\overline{q}=\\beta_{t}Z_{0}(z)\\psi$.\nThis relation allowed to compute explicitly the statistical equilibria,\nand to discuss their structure depending on the profile $Z_{0}(z)$.\nWhen this profile $Z_{0}$ is depth independent, the equilibrium state\nis fully barotropic, whatever the stratification profile $N$. When\nthis profile $Z_{0}$ is surface intensified, the equilibrium state\nis an SQG-like mode, characterized by an $e$-folding depth $\\sim f_{0}\/NK$,\nwhere $1\/K$ is bounded by the domain scale. Since the $e$-folding\ndepth increases with $K^{-1}$, larger horizontal flow structures\nimply ``more barotropic'' flows. Since statistical equilibria\nare associated with the gravest Laplacian horizontal modes ($K=1$),\nthe equilibrium state is the {}``most barotropic one'' given an\nenstrophy profile $Z_{0}$. \n\nWe conclude that the dynamics leads to the \\emph{gravest\nhorizontal mode on the horizontal (inverse cascade), associated with\nthe gravest vertical structure consistent with the conservation of\nthe fine grained enstrophy. The flow becomes fully barotropic when\nthe fine-grained enstrophy profile is depth independent. }\n\n\nThese results can be used to understand the role of beta in barotropization in the following way: consider an initial SQG-like flow, with $\\beta =0$, prescribed by its streamfunction; its enstrophy profile is surface intensified, and so will be the equilibrium sate. But when one switches on the beta effect, with the same initial streamfunction, the contribution of the depth independent part of the fine-grained enstrophy increases, which means a tendency toward a more barotropic equilibria, according to the previous conclusion. This result reflects the fact that in physical space, the initial SQG-like mode stirs the interior PV field (initially a beta plane), which in turn induces an interior flow, which stirs even more the interior PV field, and so on.\n\nWhen the beta effect becomes large enough, the PV\ndistribution at each depth $z$ becomes prescribed by the initial\nbeta plane, so the most probable state should be also characterized\nby a depth independent PV field, i.e. by a barotropic flow. \n\nTo reach the equilibrium state, the PV field must be stirred\nenough to explore the phase space. Yet large values of beta stabilize the initial condition, the system is trapped in a stable state different from the equilibrium state, and the flow dynamics becomes dominated by the interaction between Rossby waves rather than by turbulent stirring. The reason for this trapping is that the dynamics can not provide by itself sufficient perturbations to escape from the stable state and to explore other regions of the phase space, which would attract the system toward the equilibrium state. We observed in numerical simulations that this wave regime did not lead to a barotropization of the flow after a few hundred turnover times. We do not know if still longer numerical runs would lead to more barotropization. We estimated the critical value of beta between the turbulent stirring regime and the Rossby wave regime as the value such that the Rhines scale is of the order of the domain scale, which gives $\\beta=4\\pi^{2}E_{0}^{1\/2}\/L^{2}$.\n\n{In this paper, we focused on the  effect of $\\beta$ on barotropization process. More generally,  our results show that vertical transfers of energy and momentum are favored by the presence of any lateral potential vorticity gradients, since these gradients provide a  source of available perturbation enstrophy. These lateral PV gradients may for instance be due to large scale mean flows set by external process (large scale wind pattern for the oceans, and large scale temperature gradients for the atmosphere). A similar situation occurs in the framework of interactions between  waves (or eddies) and mean flows: horizontal momentum can be transfered vertically by inviscid form drag if there are horizontal fluxes of potential vorticity, which require themselves the presence of large scale potential vorticity gradients, see e.g. \\cite{VallisBook}.}\n\n{Another source of fine-grained potential vorticity would be provided by the addition of bottom topography. This should in this case play against barotropization, since the topography induces potential fine-grained enstrophy in the lower layer only. In fact, an initially surface intensified flow may evolve towards a bottom trapped current above the topographic anomaly, which can be explained by the statistical mechanics arguments presented in this paper, see \\cite{Venaille12}.}\\\\\n\nWe now discuss how these results may apply to the mesoscale ocean, and in what respect they may provide interesting eddy parametrization. First, we have considered the relaxation of an initial condition by an inviscid flow. Real oceanic flows are forced dissipated. Statistical mechanics predictions should apply if a typical time scale for self-organization is smaller than typical time scale of forcing and dissipation. Issues of forcing and dissipation lead to another difficulty: what is the domain in which the flow self-organizes ? Strictly speaking, it should be the oceanic basin. But forcing and dissipation become dominant at basin scale (with, for instance, the Sverdrup balance setting the gyre structures). In practice, on a phenomenological basis, one could still apply the statistical mechanics predictions by considering an artificial domain scale with prescribed scale of arrest for the inverse cascade, governed say by dissipation processes, but which could not be predicted by statistical mechanics itself.\n\nWe  have also neglected bottom friction, because equilibrium statistical mechanics apply for non-dissipative systems. It is generally believed that bottom friction plays a key role in the vertical structure of oceanic flows \\citet{ArbicFriction,ThompsonYoung2006}. On a phenomenological basis, quasi-statistical equilibria could be computed in the high bottom friction limit by adding a constraint of zero velocity at the bottom, for instance by considering $\\psi(-H)=0$ in equation (\\ref{eq:VertStruc}).\n\nThird, equilibrium statistical mechanics predict the final state organization\nof the freely evolving inviscid dynamics, but neither predicts\nthe route toward this state nor the corresponding typical time scales\nfor the convergence toward this state. Various parametrizations that relax\nthe flow toward the equilibrium state, following a path of maximum\nentropy production have been proposed, see \\citet{KazantsevSommeria98,RobertSommeria92}. These parametrization satisfy basic physical constraints satisfied by the dynamics (PV distribution conservation and energy conservation), but the actual dynamics may in some cases follow a different path than one of maximum entropy production. \n\n{For instance,  \\citet{SmithVallis01,FuFlierl80} reported the existence of two time scales for the flow evolution in a freely evolving quasi-geostrophic turbulent flow with surface intensified stratification. On a fast time scale, the dynamics lead to a inverse cascade of vertical modes toward the first baroclinic modes, which is surface intensified, and observed the tendency toward barotropization at much larger time scale. The existence of these two time scales is a dynamical effect that can not be accounted by statistical mechanics.} \n\nThe results presented in this paper have highlighted the important\nrole of the fine-grained enstrophy profile, for this imposes strong constraints\nfor the eddy structures. On a practical point of view, from the perspective\nof eddy parametrizations, our result suggest that rather than assuming\na vertical structure of eddies given by SQG modes, or given by combination\nof barotropic and first baroclinic modes, one could compute the eddy\nstructure with equation (\\ref{eq:VertStruc}), assuming that $K$\nis the scale of arrest, and $Z_{0}$ an enstrophy profile that could\nbe deduced from the resolved flow. For instance, $Z_{0}$ could be\nset by the structure of the most unstable mode of local baroclinic\ninstabilities.\n\n\nObservations indicate the presence of long-lived surface intensified\neddies in the ocean \\citep{chelton,eddy1,eddy2}, and we here speculate as to how these eddies may be interpreted according to a statistical \nmechanical theory. First, the eddies may be interpreted as local\nstatistical equilibrium states of the continuously stratified dynamics\nassociated with a surface intensified fine-grained enstrophy profile,\nmuch as we considered in the present paper. Second, they could be\ninterpreted as statistical equilibrium states of a 1.5 layer\nquasi-geostrophic model assuming that only the upper layer of the\nocean is active, see e.g. \\cite{VenailleBouchetJPO10}. Third, they\ncould be far from equilibrium states driven by physical mechanisms\npreventing convergence towards statistical equilibrium (dynamical\neffects due to non-uniform stratification, bottom friction, large\nscale forcing). Understanding which hypothesis is relevant to explain\nthe formation of long-lived surface intensified ocean vortices will\nrequire further research. \n\n\\paragraph{Acknowledgments}\n  This work was supported by DoE grant DE-SC0005189 and NOAA grant NA08OAR4320752. The authors sincerely thank Peter Rhines and two other reviewers for their very helpful comments provided during the review process. The authors also warmly thank F. Bouchet and J. Sommeria for fruitful discussions, and K.S. Smith for sharing his QG code. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION} \nLithium is of fundamental astrophysical importance.  This fragile \nlight element is quickly destroyed in stars when \nexposed to $T{\\ge}2.5{\\times}10^6$ K, making it useful in studying \nmatter transport in stars and providing observational feedback on our \nunderstanding of stellar structure and evolution \n\\citep[for a review, see][]{pinsonneaultarap}.  In addition, stellar Li \nabundances have cosmological applications.  Combining the \nA(Li)\\footnote{${\\rm A}(Li){\\equiv}{\\log}N({\\rm Li})$ on the usual scale\nwhere the logarithmic number abundance of hydrogen is given by \n${\\log}{N}({\\rm H}){\\equiv}12$.} \nin \nold stars with a) accurate stellar evolution models to account for stellar \ndestruction, and b) accurate chemical evolution models to account for Galactic \nproduction, should lead to the primordial A(Li), which provides \nconstraints on Big Bang nucleosynthesis \\citep{BS85} and an independent check \non cosmological parameters determined by WMAP \\citep{WMAPp}.  However, \nthe vulnerability of Li to destruction that makes it a good probe of stellar \ninteriors also complicates the derivation of {\\it initial} halo \ndwarf A(Li)--our best connection to the Big Bang A(Li).\n\nThe astrophysical usefulness of Li, therefore, strongly depends on having \ncorrect stellar models that accurately trace the {\\it in situ} history of Li.  \nStandard\\footnote{That is, spherically symmetric models that ignore \nrotation, diffusion, mass loss, magnetic fields, and other physics \nthat could affect real stars \\citep{DDK90,pinsonneaultarap}.} \nstellar models (SSMs) suggest that, for stars of about 1.2$M_\\odot$, \nthe observed surface A(Li) can go down only, \na) during the early pre-main sequence (PMS) when \nsome of the interior Li is actually destroyed, and b) during subgiant \nevolution as the deepening surface convection zone (SCZ) dilutes it.  Standard \ntheory predicts no surface Li depletion during the main sequence (MS).  \nMore specifically, since Li burns at relatively low temperatures, \nanytime the SCZ reaches deep enough into the star's \ninterior, visible Li depletion will occur at the surface.  This occurs during \nthe PMS phase and, for stars of around 1.2$M_\\odot$, the SCZ is not deep \nenough during the MS to carry Li into the interior regions where it can be \nburned appreciably.  Even during late PMS convective burning, such depletion \nis minimal.  For a 1.2$M_\\odot$ star with [Fe\/H]${\\le}-0.1$, the \nmaximum expected depletion on both the MS and the PMS is $<0.1$ dex \n\\citep{pinsonneaultarap}.  Therefore, little Li depletion in stars of this \nmass (or higher) is expected on the PMS, and virtually none on the MS, based \non the predictions from SSMs \\citep{conferencep}. \n\nIn contradistinction, striking Li \\citep[and Be;]{BK02} depletion is observed \nin disk MS stars with $6300{\\leq}T_{\\mathrm{eff}}\\leq6900$ K.  In the Hyades, \\citet{boesgaardp} \ndiscovered that stars with $6500{\\leq}T_{\\mathrm{eff}}\\leq6850$ K were \ndepleted by ${\\geq}0.5-2$ dex as compared to stars merely 200--300 K hotter \nor cooler.  This abrupt F-star Li(Be) depletion (the Li dip) occurs mainly \n{\\it during the MS}, since much younger clusters such as the Pleiades \n(${\\sim}100$ Myr) show a nearly flat Li-$T_{\\mathrm{eff}}$\\ relation \\citep{BBR88,DRS93}.  \nThe Li dip presents a challenge to SSMs, which predict negligible \nLi depletion on the MS, and casts doubt on their ability to \neither predict or backtrack Li abundances to initial values. Instead, \nstellar models are required that can explain the Li dip \naccurately by modeling Li abundances over a star's lifetime.  \nA number of refinements to SSMs have been proposed to explain the Li dip \n\\citep{balachandranp,Deli2000,SR2005}, but here we focus on \nrotationally-induced mixing (RIM) \\citep{RIMp,RIM2p}.  RIM models propose that \na loss of angular momentum, as stars spin down on the MS, could cause slow \nmixing between the photosphere and the hotter denser region beneath the SCZ, \nresulting in the depletion of surface Li.\n\nTo test this theory, \\citet{Litestp} suggest that Li abundances of \nshort-period tidally-locked binaries (SPTLBs) be measured.  RIM and stellar tidal \ntheory \\citep{ZBp} together predict that a SPTLB loses most of its angular momentum \nduring the very early PMS, when interior densities and temperatures are too low to \nburn Li.  Hence, lacking significant angular momentum to lose and drive future\nrequisite mixing, neither does it suffer MS Li depletion responsible for the Li dip. \nThus, one method to test if angular momentum transfer is indeed the causal force \nbehind the Li dip is to find SPTLBs whose ages and temperatures should place them in \nthe Li dip, and measure their Li.  A value of A(Li) higher than otherwise similar Li dip \nstars would support RIM models.  V505 Persei, a Pop I intermediate-age short-period binary \nsystem, is made up of two F stars of nearly equal mass, radius, and $T_{\\mathrm{eff}}$\\ \n\\citep{marschallp, tomasellap}.  Both masses, $T_{\\mathrm{eff}}$, and ${\\sim}4$ day period \nmake them good candidates to test if an early PMS loss of angular momentum can \nlater preserve Li compared to otherwise similar non-SPTLB stars.  We measure \nthe A(Li)s of V505 Per and compare them in the A(Li)\nversus zero-age MS (ZAMS) $T_{\\mathrm{eff}}$\\ plane with stars of intermediate-age open \nclusters, showing in the process that the components of V505 Per have \nA(Li)s higher than single stars of their ZAMS $T_{\\mathrm{eff}}$. \n\n\\section{DATA AND ANALYSIS}\n\\subsection{Observations}\nHigh-resolution ($R=45,000$) spectroscopy of HD 14384 was obtained on 30 \nand 31 August 1997 using Keck\/HIRES.  \nThree nightly exposures totalled 8 and 5.5 \nminutes on the first and second nights, respectively. \nDebiasing, flat-fielding, scattered light removal, order \ntracing\/extraction, and wavelength calibration were carried out with standard \nroutines in IRAF.\n\nGiven the short integration spans relative to the 4.2 d period, \nwe co-added the 3 nightly spectra.  \nThe resulting ${\\lambda}6707$ continuum level Poisson SNR\/pixel \nis 850 and 580 on the first and second \nnights.  Spectra were fit with a low order polynomial to perform \na continuum normalization.  Figure 1 shows our co-added, normalized \nspectra.  The physically similar components of the SB2 are \ndistinguished and identified both by slight \ndifferences in the strengths of the ${\\lambda}6717$ CaI feature and the \nrelative Doppler shifts expected from the orbital ephemeris of \n\\citet{tomasellap}.  The relative Doppler shifts of the components in each \nnight's coadded spectrum, needed for the spectrum synthesis described next, \nwere measured from the centroids of the CaI features.\n\n\\subsection{Syntheses and Comparison}\nTo derive Li and Fe abundances, we conducted spectrum synthesis of the \n${\\lambda}$6707 \\ion{Li}{1} region to account for the line blending between \nthe stellar components.  Such synthesis requires knowledge of $T_{\\mathrm{eff}}$, ${\\log} g$, \nradius ($R$), and microturbulent velocity ($\\xi$) of each star.  The first\nof three of these parameters are taken from \\citet{tomasellap}, who determine radii \nand associated uncertainties from a \\citet{WD71} code-based solution with\nupdated model stellar atmospheres \\citep{MSK92}; these radii agree with \nalternate solutions derived from the radial velocities and $BV$ photometry\nalone to within 0.05 $R_{\\odot}$, which is well within their stated ${\\pm}0.11$ \n$R_{\\odot}$ uncertainty adopted here. The $T_{\\mathrm{eff}}$ values and uncertainties \nof \\citet{tomasellap}, $6512{\\pm}21$ and $6462{\\pm}12$ K for the A and B components, \nare derived from multiparameter ${\\chi}^2$ fitting of 700 {\\AA} of their own high-resolution \n($R{\\sim}$20,000) spectra against synthetic spectra grids.  Encouragingly, the $T_{\\mathrm{eff}}$ {\\ }differences \nof the components they derive as part of the orbital solutions agree to within 18 K of the \nspectroscopically-derived value.  \n\nThe \\citet{tomasellap} masses and radii yield  ${\\log} g=4.33{\\pm}0.02$ for both components.\nWe determined $\\xi$ (1.73 and 1.70 km s$^{-1}$) using the $T_{\\mathrm{eff}}$- and ${\\log}g$-dependent calibration \nof  \\citet{microturbulencep}.  We note, though, the derived Li abundances are insensitive to\nthe value of log $g$ and $\\xi$.  .  While [Fe\/H] was eventually determined from a comparison of \nthe observed and synthetic spectra, we adopted an initial metallicity of [Fe\/H]$=-0.35$ from the \nphotometric determination of \\citet{nordstromp}.  \n\nWe used the $T_{\\mathrm{eff}}$, log g, and [Fe\/H] values to \ninterpolate model atmospheres from the grids of \\citet{kuruczp}.  MOOG\\footnote{http:\/\/as.utexas.edu\/~chris\/moog.html} \nwas used to create a synthetic spectrum for each star using the ${\\lambda}6707$ linelist of \n\\citet{kingp}.  The spectra were smoothed by convolving with a Gaussian with FWHM  measured \nfrom clean, weak lines in the observed spectra.  Each component's synthetic spectrum was Doppler \nshifted and then combined, using the product of the square radii and the Planck function value at \n6707{\\AA} (a pseudo-monochromatic luminosity) as a weighting factor. This correction for flux dilution\nis appropriate for the first night's observations (orbital phase of 0.74) since each star's \nundiminished flux contributes to the spectrum.  The orbital phase (0.97) of second night's observations \nplaces the system on the cusp of primary eclipse; however, the eclipses are very sharp, and the \ntotal system flux diminishment in $B$ at this phase is $\\lesssim$0.01 mag of the ${\\sim}0.5$ mag total at primary eclipse. \n\nFinally, we compared the synthetic spectra to the observed spectra using $\\chi^2$ minimization methods.\nWe used the ${\\lambda}6705$ \\ion{Fe}{1} line to determine [Fe\/H], forcing both \nstars to have an assumed identical Fe abundance.  Once a best-fit value of [Fe\/H] \nhad been determined, we moved on to A(Li), which was allowed to differ in \nthe two components to allow for possible differing Li depletion in the two stars.  \nThese synthetic spectra are compared to the observed data in Figure 1.\n\n\\subsection{Results}\nThe analysis of the first night's data yielded [Fe\/H]$=-0.15{\\pm}0.03$.  The \nquoted error is due to the 1${\\sigma}$ level fitting uncertainties alone; even \nso, our metallicity estimate is in good agreement with \n[M\/H]$=-0.12{\\pm}0.03$ from \\citet{tomasellap}.  The metallicity from \nthe second night was not calculated due to the unfortunate placement of the \n${\\lambda}6705$ FeI feature in the secondary star with a \ndetector\/reduction artifact that can be seen in Figure 1; in carrying out the \nLi syntheses, we assumed the [Fe\/H] value from the first night's data.  \n\nThe Li syntheses (Figure 1) yield average A(Li) of \n$2.67{\\pm}0.1$ and $2.42{\\pm}0.2$ for the primary and secondary components, \nrespectively.  While the best-fit Li abundances differ by only a few hundredths \nof a dex, the larger quoted uncertainties in A(Li) are \ndominated by uncertainties in the continuum location. Contributions from \nfitting uncertainties in the $\\chi^2$ minimization and $T_{\\mathrm{eff}}$\\ uncertainties \namount to only ${\\pm}0.03-0.06$ dex and ${\\pm}0.01-0.02$ dex, respectively; \ncontributions from uncertainties in ${\\xi}$ and log $g$ are similarly small \nor smaller and are ignored.   Abundance uncertainties arising from uncertainties \nin the flux dilution factors of each component, which in turn arise from uncertainties \nin $T_{\\mathrm{eff}}$\\ and the \\citet{tomasellap} stellar radii, are $0.01-0.02$ dex.\n\n\\section{DISCUSSION}\n\n\\subsection{Li versus ZAMS $T_{\\rm eff}$}\nInterpreting the A(Li) of our SPTLB components requires that \nthey be placed in the context of the Li dip morphology defined by other single \n(or non-SPTLB) stars.  In her comparisons of the Li dip in various open \nclusters, \\citet{balachandranp} found that the $T_{\\mathrm{eff}}$\\ at which the Li dip \noccurs varies based on metallicity, but that the ZAMS $T_{\\mathrm{eff}}$\\ at which the Li \ndip is located does not; this provides a means by which the Li dip \nmorphology of different populations of disk stars can be compared.  \nAdditionally, \\citet{balachandranp} found that the morphology of the cool side \nof the Li dip is age-dependent.  Some evidence suggests that the Li dip may \nbegin to form as early as an age of 150 Myr \\citep{SD2004} or even 100 Myr \n\\citep{Marg2007}.  Clearly, comparing our stars with bona fide Li dip stars \nrequires knowledge of our SPTLB components' ZAMS $T_{\\mathrm{eff}}$\\ and age. \n\nFor consistency, we followed the approach of \\citet{balachandranp} to find the \nZAMS $T_{\\mathrm{eff}}$\\ of each star by looking at differences implied by isochrones (and \ntheir assumed color transformations) between our stars in their current \nevolutionary state and on the ZAMS.  This required that we first determine the \nage of our SPTLB stars, which we did by placing the components in the radius \nversus mass plane and comparing these locations with sequences from \n[m\/H]$=-0.14$  Yonsei-Yale isochrones \\citep{Y2isop}.  As shown in Figure 2, \nthis implies an age of $1.15{\\pm}0.15$ Gyr for the system; the majority of the \nage uncertainty comes from uncertainty in the radii estimates.  \n\nWe then used the Yonsei-Yale isochrones with the Green color-temperature relations, \nwhich are also employed by the Revised Yale Isochrones \\citep{isochronep} used \nby \\citet{balachandranp}, to determine the difference \nbetween the $T_{\\mathrm{eff}}$\\ at 1.15 Gyr and on the ZAMS. This $T_{\\mathrm{eff}}$\\ \ndifference was then applied to our current $T_{\\mathrm{eff}}$\\ value from \\citet{tomasellap}, yielding \nZAMS $T_{\\mathrm{eff}}$\\ values of 6483 and 6432K for the primary and secondary \ncomponents.  The $1{\\sigma}$ level uncertainties in our interpolation of the \n$T_{\\mathrm{eff}}$-mass relations of the isochrones are ${\\le}12$ K.\n\nFigure 3 presents the Li-ZAMS $T_{\\mathrm{eff}}$\\ diagram containing: {\\ }the V505 Per \ncomponents; the literature data reanalyzed by \\citet{balachandranp} for the \nopen cluster NGC\\,752, having a $1.45$ Gyr age \\citep{AT2009} and \n[Fe\/H]$=-0.15$ \\citep{NGC752p}; the Li data for the 1.75 Gyr, \n[Fe\/H]$=-0.08$ cluster NGC\\,3680\\ \\citep{AT2009}; and the Li data of \n\\citet{balachandranp} for the 650 Myr [Fe\/H]$=+0.13$ Hyades cluster.  We \ndetermined the NGC\\,3680\\ object masses using a Legendre polynomial relation \nto map the dual $V$ magnitude and mass abscissas of Figure 4 of \\cite{AT2009}.  \nWe then used the Yonsei-Yale isochrones as described above to determine the $T_{\\mathrm{eff}}$\\ \ndifference between the ZAMS and at 1.75 Gyr at a given mass.  This difference \nwas then applied to the {AT2009} $T_{\\mathrm{eff}}$\\ values to yield ZAMS $T_{\\mathrm{eff}}$\\ values.  \n\n\\subsection{v505 Per versus the Hyades}\n\nIf the v505 Per $T_{\\rm eff}$ values and uncertainties of \\citet{tomasellap} are reliable, \nand if the {\\it relative\\\/} uncertainties in the to-ZAMS $T_{\\rm eff}$ corrections for the \nv505 Per components compared to those of the open cluster comparison stars are not \nseveral times the size of the corrections themselves (${\\sim}30$ K for our binary components \nand ${\\sim}40$ K for similar mass stars in the slightly older NGC 752 cluster), then \nFigure 3 indicates that both our SPTLB components \nare positioned inside the Li dip that is well defined by the Hyades data \\citep[or those of the\nor Praesepe data not shown here; see Figure 12 of][]{balachandranp}.  The larger Li abundances \nin the v505 Per components compared to nearly all of the  younger and more metal-rich Hyades \ndata in Figure 3 is especially notable given the metallicity difference and the age difference, \nwhich we discuss in turn, between the two.  \n\nFirst, comparisons of the v505 Per Li abundances with those in cluster stars are most meaningful \nif some account of initial Li abundance differences can be made. An empirical \napproach to parameterize Galactic disk Li production in terms of Fe production is to\nuse the upper envelope of the Li-Fe relation exhibited by large samples of field stars. The field \nstar data over the range $-1{\\leq}\\mathrm{[Fe\/H}{\\leq}0$ in Figure 7 of \n\\citet{LHE} suggest an initial Li-to-Fe (logarithmic by number) relation in \nthe local disk having slope ${\\sim}1$ dex\/dex.  For comparison, the Galactic \nchemical evolution model in Figure 9 of \\citet{Travagliop} produces a slope of \n${\\sim}0.7$ dex\/dex over the same [Fe\/H] range, though this slope may be too \nsmall since it is unable to reproduce the initial solar Li abundance.  Indeed, determinations\nof the slope of the Li-to-Fe relation using Li abundances on the G-dwarf Li peak of various open \nclusters are significantly larger at 1.4 dex\/dex \\citep{Boes91} and 1.0 dex\/dex \\citep{Cumm11}.    \n\nThe Hyades and Praesepe have super-solar metallicities:  [Fe\/H]${\\sim}+0.10$ to $+0.15$ \n\\citep{Boes89,BF90}.  The local disk field and \\citet{Cumm11} open cluster Li-Fe relation \nthus implies the initial Hyades and Praesepe Li abundances were a factor of 2 larger than \nfor the v505 Per components; the \\citet{Boes91} open cluster Li-Fe relation implies initial \nLi abundances a factor of 2.6 larger than for the v505 Per components.  Accounting for initial \nLi differences in this way makes  the observed present-day difference between v505 Per and \nHyades Li abundances even more remarkable.  \n\nSecond, the red side of the Li dip is known to flatten with increasing age, which may be due\nto increasing Li depletion in the dip stars with age \\citep{balachandranp}.  Nevertheless, \ndespite their older age, our SPTLB components exhibit Li abundances a factor of 2 {\\it larger} \nthan nearly all Li detections or upper limits at similar ZAMS {$T_{\\mathrm{eff}}$}  in the younger ($\\sim$650 Myr) \nHyades and Praesepe.  Once reaching the v505 Per age, stars on the red side of the dip \nin these clusters would presumably have even lower Li abundance than at present.  \n\n\\subsection{Comparison with other data}\n  \nFigure 3 also compares V505 Per with stars of more similar [Fe\/H]: those in NGC\\,752\\ \nand NGC 3680.  The A(Li) of our SPTLB components are a factor of 2-5 larger than the upper \nlimits for the NGC\\,752\\ and NGC 3680 Li dip stars (one 3680 star on the steep blue side \nof the dip is within the error bar of our primary component).  Note that \\cite{SRP94} argue \nthat their high-resolution spectroscopy of solar-type dwarfs in NGC\\,752\\ suggests \n[Fe\/H]$=+0.01$ for this cluster, 0.15 dex larger than the canonical value quoted by \\citet{NGC752p}.  \nIf so, the above discussion of the initial Li abudnance versus [Fe\/H] relation suggests an even \nlarger difference between the Li depletion factors of our SPTLB components and the NGC\\,752\\ Li \ndip stars.\n\nFinally, it is important to recall that the preservation of Li by SPTLBs \noccurs if angular momentum loss occurs sufficiently early during the PMS, when \ninterior temperatures and densities are too low to burn Li.  This early \nsynchronization is predicted to occur in systems with periods below some \ncritical period that is a function of stellar mass and metallicity \n\\citep{Zahn94}.  The models of \\citet{Zahn94} for Galactic disk stars of mass \n1.2M$_{\\odot}$ indicate this critical period is 6 days.  The 4.2d period of \nV505 Per falls under this critical period, though the 1.25 and \n1.27M$_{\\odot}$ components slightly exceed the 1.2M$_{\\odot}$ maximum mass \nconsidered by the modeling of \\citet{Zahn94}.  The Hyades SPTLB vB 34 resides \nwithin the blue region of the Hyades Li gap, but does not clearly demonstrate \na Li abundance larger than similar single stars \\citep[see, e.g., Figure 1 of][]{Litestp}.  \nHowever, Yonsei-Yale isochrones ([Fe\/H]$=+0.13$, 650 Myr for the Hyades; \n[Fe\/H]$=-0.14$, 1.15Gyr for v505 Per) indicate that the vB 34 masses are 0.20-0.23M$_{\\odot}$ \nlarger than for v505 Per; thus, the 3.1d period of vB 34 may not be below a necessary \ncritical period for its larger mass components. \n\n\\section{CONCLUSIONS}\nLithium is important for probing stellar interiors and evolution.  It can be used to \nstudy transport of matter in the stars, Galactic chemical evolution, and BBN.  Such uses \ndepend heavily on accurate predictions of Li abundance evolution within stars. \nIt is known that SSMs are unable to explain the Li dip in disk mid-F dwarfs. \nModified stellar models that include the action of rotationally-induced slow \nmixing can explain the Li dip as the result of slow Li mixing in stars \nthat are currently undergoing angular momentum loss and are sufficiently \nmassive that interior temperatures and densities are sufficiently large to \nburn Li as a result of such mixing. \n\nAlthough other types of physical mechanisms have also been proposed as \nexplanations of the Li dip, including diffusion, mass loss, and other types of \nmixing, a variety of observational evidence favors, often quite strongly, the \nRIM-type models over other mechanisms.  This evidence includes (but is not \nlimited to): a) the Li\/Be depletion correlation where {\\it both} elements are \ndepleted, but Li more severely \\citep{Stephensp,Deli98,BAKDSp} {\\,}b) the \nBe\/B depletion correlation \\citep{Boes2005} {\\,}c) subgiants in M67 revealing \nthe size and shape of the (MS) stellar preservation region as they evolve \nout of the cool side of the Li dip \\citep{SD2000} {\\,}d) and the early \nMS formation of the Li dip.\n\nHere, we present an independent observational test of the RIM explanation of \nthe Li dip.  The $T_{\\rm eff}$ values and uncertainties of \\citet{tomasellap} \nindicate that the components of the mildly metal-poor \n([Fe\/H]${\\sim}-0.15$) intermediate-age (${\\sim}1.1$Gyr) short-period \ntidally-locked binary V505 Per both reside in the Pop\\,I Li dip defined by \nopen cluster observations (assuming the differential ${\\sim}10$ K to-ZAMS\n$T_{\\rm eff}$ corrections suggested by isochrones for our \nstars relative to the younger Hyades and older NGC 752 and NGC 3680 clusters \nare not, in fact, an order of magnitude larger).  If angular momentum loss in such a system occurred \nvery early during the pre-main sequence phase, then it would suffer no or \nreduced RIM during the MS compared to non-SPTLB stars occupying the Li dip \nregion; as a result, the V505 Per components would exhibit larger Li \nabundances than otherwise similar stars in the Li gap.  \n\nWe find that \nthe V505 Per components' Li abundances are at least 2-5 times \nlarger than both:  a) the Li upper limits in the ${\\sim}1.5-2$Gyr and \nsimilarly mildly metal-poor ({Fe\/H}${\\sim}-0.15$ and $-0.08$) clusters \nNGC\\,752\\ and 3680, and  b) the upper limits and Li detections in the younger \nmetal-rich Hyades and Praesepe clusters.  If there exists an initial Li-Fe \nrelation of positive slope, as field star data and open cluster observations and Galactic chemical evolution \nmodels each independently suggest, and initial Li abundances indeed scale with [Fe\/H] in the \nrecent disk, then the Li overabundance of V505 Per is even more dramatic in \nthe case of the younger clusters (which presumably started with a higher \ninitial Li abundance) and perhaps in NGC\\,752\\ if one assumes the higher [Fe\/H] \nvalue of \\citet{SRP94}. \n\nOur results suggest, independently, that angular momentum evolution on the \nMS is responsible for the Li dip, confirming the conclusions drawn from the \nvariety of observational evidence listed above, involving both field and \ncluster dwarfs.  SPTLBs with higher-than-normal Li have been found in the \n650-Myr old Hyades \\citep[][see Soderblom et al. 1990 for a related \nidea]{THDP93}, the 4.5Gyr-old M67 \\citep{Deli94}, and in other contexts \n\\citep{Litestp}; and, \nsignificantly, SPTLBs have {\\it normal} Li in clusters, such as the Pleiades, \nwhich are too young for much RIM-related depletion to have occurred \n\\citep{Litestp}.  Our results for V505 Per complement these previous findings \nin that the V505 Per SPTLB stars are the hottest high-Li SPTLBs discovered so \nfar, and are indeed very close to the limiting $T_{\\mathrm{eff}}$\\ beyond which the models \nof \\citet{Zahn94} can no longer synchronize both components sufficiently early \nduring the pre-MS to prevent Li destruction.  The SCZ of hotter \nSPTLBs is too shallow and the their Hayashi paths too short for the components \nto grab onto each other sufficiently effectively so as to cause \ntidal locking during that same evolutionary phase.   \n\nWhile the Li abundances in our V505 Per components likely reflect some Li \ndepletion from a plausible initial abundance in the range \n$A(Li)=3.0-3.3$, which is not unexpected \n\\citep[see section 2.2.1 of][]{Litestp}, they nevertheless suggest that early \ntidal circularization can be efficient in mid-F stars in the Li dip and \nmitigate the effects of RIM on Li depletion.  Identification of additional Li \ndip SPTLBs, analysis of their A(Li), and comparison with that of \nsingle stars of similar Li dip position, metallicity, and age would be a \nprofitable means to extend these conclusions.  \n\n\\acknowledgments\nThis work was supported by NSF awards AST-0239518 and AST-0908342 to JRK, \nand AST-0607567 to CPD. We thank Bruce Twarog for providing us with the \nNGC\\,3680\\ data.  \n\n{\\it Facility:} \\facility{KECK}\n\n\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Background and motivation}\n\\label{sec:background}\n\nIn this section, we review the network by Zhao et al.~ \\cite{Zhao2020JointSR}, since our method relies on it. Then, we discuss the motivation of our method.\n\\subsection{Background}\n\nWe assume the lighting and viewing directions to be identical, as it happens when the flash light is close to the camera lens. \nWe generate four maps to represent the SVBRDF: the diffuse albedo $\\rho_{d} \\in \\mathbb{R}^{3}$, specular albedo $\\rho_{s} \\in \\mathbb{R}$, roughness $\\alpha \\in \\mathbb{R}$, and surface normal $\\mathbf{n} \\in \\mathbb{R}^{3}$. Our goal is to compute  $\\mathbf{u} = {[\\rho_{d},\\rho_{s},\\alpha,\\mathbf{n}]}$. We use the Cook-Torrance reflectance model \\cite{cook1982reflectance} for rendering.\n\n\\textbf{Network architecture.}\nZhao et al.~\\cite{Zhao2020JointSR} proposed a generative adversarial network for SVBRDF recovery and synthesis. The network is unsupervised, thus no dataset is required for training. With a stationary image as an input, the network produces four SVBRDF maps, by training on different cropped tiles from the input image. The network consists of a \\emph{two-stream generator} and a \\emph{patch discriminator}. \n\nThe generator includes an encoder and two decoders, where two groups of maps of the tiles are generated separately: normal and roughness, diffuse and specular. The maps are then used to render an image. Both this rendered image and the tile from the input image are fed to the discriminator to determine the correctness of the generated SVBRDF maps. Regarding the network structures, please see Zhao et al.~\\cite{Zhao2020JointSR} for more details.\n\n\n\\textbf{Loss function.}\nThe loss function in Zhao et al.~\\cite{Zhao2020JointSR} consists of a guessed diffuse map loss and the adversarial loss:\n\\begin{equation}\\label{Lzhao}\n\\mathcal{L}_\\mathrm{\\added{Zhao}} = \\lambda \\mathcal{L}_\\mathrm{GAN}(G,D) + \\mathcal{L}_{d}(G),\n\\end{equation}\n\\begin{equation}\\label{ganlossu}\n\\mathcal{L}_\\mathrm{GAN}(G,D) = \\mathbb{E}[\\log {Dis}(\\mathbf{x})] + \\mathbb{E}[\\log (1-{Dis}(\\mathbf{y})],\n\\end{equation}\n\\begin{equation}\\label{initd}\n\\mathcal{L}_{d}(G) = \\mathbb{E}\\left[\\|\\tilde{\\mathbf{\\rho_{d}}}-\\mathbf{\\rho_{d}}\\|_{1}\\right].\n\\end{equation}\nThe guessed diffuse map $\\tilde{\\mathbf{\\rho_{d}}}$ is obtained via normalizing the input image, and considering the statistical distribution, which is used as the ground truth of the diffuse map, since the ground truth maps are absent. \n\n\\subsection{Analysis}\nWhen highlights exist in the input image, Zhao et al.~\\cite{Zhao2020JointSR} fail to recover satisfactory SVBRDF maps due to the ambiguous highlight spot. The high intensity region is often classified as part of the specular albedo, resulting in wrong results for the roughness and specular maps (see Figure~\\ref{fig:synthetic-brick}). This is an issue for all existing SVBRDF acquisition methods, due to the ambiguity between illumination and material. \n\nIn this paper, we introduce extra information to resolve the ambiguity: the material maps we wish to recover are stationary, that is they repeat themselves after a certain period. As a consequence, the recovered maps should also be stationary. We enforce the stationarity of the recovered maps with a new loss function, based on their Fourier transform. We focus on the acquisition part of Zhao et al.~\\cite{Zhao2020JointSR} and ignore the texture synthesis part, although it can be easily included.\n\nZhao et al.\\cite{Zhao2020JointSR} also has an issue with computation time: as the network is trained from scratch on each individual image, processing can take up to 4 hours for a single image. We solve this issue with a  two-stage training strategy, using a pretrained model for initialization and a fine-tuning stage. The computation time is down to 30~mn for each image. \n\n\n\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\nIn this paper, we improve the unsupervised SVBRDF recovery generative adversarial network, by introducing two new loss functions: a Fourier loss function and a perceptual loss function to enforce the stationarity of SVBRDFs, yielding better quality for the SVBRDF maps, especially for input images with intense highlights. Then, we propose a two-stage training strategy to reduce the training time with 8$\\times$ speedup. In the end, our method is able to generate high-quality SVBRDFs and produce more plausible rendering results, compared with the state-of-the-art methods.\n\n\\section{Introduction}\n\nThe reconstruction of real world material appearance is a long standing problem in computer graphics and vision. The reflectance parameters of opaque materials can be modeled by the 6D spatially-varying bi-directional reflectance distribution function (SVBRDF). It is difficult to recover the SVBRDFs of a real world material because of its high dimensionality and the inherent ambiguity of the unknown parameters: color variations could be caused by changes in any material parameter: albedo, roughness or normal. Several previous methods required complex acquisition equipments to densely sample materials in different light and view directions. These can faithfully capture the appearance parameters of a material, but are also very expensive and time-consuming, limiting the accessibility. \n\nRecent works have shown that it is possible to recover the SVBRDFs from a few photos, or even a single image, of the material~\\cite{deschaintre2018single, li2018materials, gao2019deep, Guo2020MaterialGANRC, Guo2021HighlightawareTN}. These lightweight methods used deep neural networks to capture four SVBRDF maps (diffuse and specular albedo, normal map and roughness parameters) from photographs of a material. They usually rely on convolutional neural networks (CNNs), trained on  synthetic images and corresponding SVBRDF maps, to model the appearance of real world materials. \n\nThese deep learning-based methods are {\\em supervised} and require large training datasets. These datasets are difficult to acquire~\\cite{li2017modeling, deschaintre2018single, aittala2015two}. Existing methods either need professional designers to generate procedural models, or rely on numerous samples of real world materials. A concurrent approach relies on generative adversarial network (GAN)~\\cite{goodfellow2014generative} to avoid heavy work in dataset collection. Zhao et al.~\\cite{Zhao2020JointSR} proposed the first approach to exploit GAN architecture for unsupervised SVBRDF maps recovery. They rely on a two-stream generator to train the SVBRDF maps (diffuse, specular, normal, roughness) and calculate the adversarial loss. Their network is able to predct plausible SVBRDF maps from a single input image, and does not require any dataset. They also provide high quality texture synthesis through a well-designed encoder-decoder structure. \n\nWhen the input image includes an intense specular highlight, it is difficult for acquisition methods to separate between albedo and illumination. As the specular highlight has a large intensity, it gets a strong priority in the learning process, often resulting in bright spots at the center of the albedo maps. To solve this issue, we need to introduce a specific constraint: we make the hypothesis that the material we acquire is stationary (its features repeat themselves after a certain period). We enforce stationarity in the reconstructed SVBRDF maps using a new loss function based on the Fourier transforms of the SVBRDF maps. Also, the rendering loss function based on pixel-wise comparisons do not work well when the exact positions of the camera and the light source are not well known. We introduce a new loss function, based on the perceptual difference between the input image and the reconstructed image~\\cite{johnson2016perceptual}. Combined together, these new loss functions generate high-quality SVBRDF maps from a single input image. In particular, when there are overexposed highlights in the input photograph, our method generates more reasonable SVBRDF maps compare to previous works, leading to more realistic results when re-rendering with different viewing and illumination.\n\nTo speed-up the reconstruction process, we also introduce a two-stage training strategy: we pretrain our network on a single material, which provides the starting parameters for the training on each input image. This pretraining strategy makes the treatment of an image 8 times faster. \n\n\nIn summary, our contributions include:\n\\begin{itemize}\n\\item a Fourier loss function to enforce stationarity in the SVBRDFs, which produces more plausible results when the input image has intense highlights,\n\\item a perceptual loss function that measure the semantic similarity between the input image and re-rendering result, and \n\\item a two-stage training strategy to speedup the train process without quality degradation.\n\n\\end{itemize}\n\nThe rest of the paper is organized as follows. In Sec.~\\ref{sec:related}, we review previous works on SVBRDFs recovery. As our method builds on Zhao et al.\\cite{Zhao2020JointSR}, we present this method in depth in  Sec.~\\ref{sec:background}. We present our method and implementation in Sec.~\\ref{sec:recover}. We show and discuss our results in Sec.~\\ref{sec:results}, and conclude in Sec.~\\ref{sec:conclusion}.\n\n\n\\section{Our method}\n\\label{sec:recover}\n\nIn this section, we propose a novel loss for the SVBRDF GAN~\\cite{Zhao2020JointSR} to enforce the stationarity of SVBRDF maps and relax the pixel-wise connection between the guessed diffuse map and the input image. Then we present a two-stage training strategy to reduce the training time cost. Lastly, we show the implementation details.\n\n\n\\subsection{Stationarity-aware loss function}\n\\label{sec:loss}\n\n\n\nWe propose a joint loss, including a Fourier loss and a perceptual loss, where Fourier loss enforces the stationarity in SVBRDF maps and perceptual loss makes the rerendering result more plausible.\nIn Figure~\\ref{fig:newframework}, we show the difference between our method and Zhao et al.~\\cite{Zhao2020JointSR}.\n\n\n\n\\textbf{Fourier loss.} With a single image, it is difficult to separate between the color changes due to the material and those due to illumination. Without guidance, the network tends to place the highlights as part of the albedo or normal map. We introduce an extra constraint: the material should be stationary. In a stationary texture, variations are high-frequency and illumination effects are low-frequency. We introduce a new loss function based on Fourier analysis to enforce this stationarity: we compute the Fourier transform of the guessed diffuse map $\\tilde{\\mathbf{\\rho_{d}}}$ and of the predicted SVBRDF maps $\\mathbf{u}$(${[\\rho_{d},\\rho_{s},\\alpha,\\mathbf{n}]}$). We compute the Fourier loss function as the  $L_1$ loss in the logarithmic domain:\n\t\\begin{equation}\\label{Lf}\n\t\t\\mathcal{L}_{F}(\\mathbf{u},\\tilde{\\rho_{d}}) = \\log\\mathbb{E}\\left[\\| \\mathrm{FFT}(\\mathbf{u})-\\mathrm{FFT}(\\tilde{\\rho_{d}})\\|_{1}\\right].\n\t\\end{equation}\nWe use the fact that, after normalization, the guessed diffuse map has a stationary distribution of gray scale. \nWith Fourier loss as a guidance on the frequency domain, the predicted SVBRDF maps will be less affected by the illumination and have similar variations as guessed diffuse map.\n\n\n\\textbf{Perceptual loss.} The exact lighting and viewing directions are sometimes unknown, especially for captured photographs. Using loss functions based on pixel-wise difference between input images and rendered images produces poor results, because we cannot guarantee the consistency of our rendering parameters: the rendered image is not rendered with exactly the same viewing and lighting condictions as the input image. To solve this issue, we use a perceptual loss function, to measure the semantic similarity between the input $I$ and the re-rendered result $R$, via a pretrained VGG-19 network:\n\t\\begin{equation}\\label{Lt}\n\t\t\\mathcal{L}_{P}(I,R) = \\mathbb{E}\\left[\\| \\mathrm{VGG}(I)-\\mathrm{VGG}(R)\\|_{1}\\right].\n\t\\end{equation}\nWith this perceptual rendering loss, the re-rendering result of predicted SVBRDF maps is more realistic and reliable.\n\n\\textbf{Summary.} We present a joint loss function combining these three losses:\n\\begin{equation}\\label{Lfinal}\n\\mathcal{L}_\\mathrm{final} = \\mathcal{L}_\\mathrm{Zhao} + \\lambda_{1}\\mathcal{L}_{F}(\\mathbf{u},\\tilde{\\rho_{d}}) +   \\lambda_{2}\\mathcal{L}_{P}(I,R)\n\\end{equation}\nTrained with this joint loss function, the network can achieve better recovery result of SVBRDFs, leading to more reasonable rendering results with novel viewing\/lighting directions, especially when handling images with intense highlights. We will show our recovery results in Sec.~\\ref{sec:recovresult}.\n\n\\textbf{Discussion:} our loss function assumes that all maps being computed for the current SVBRDF have the similarly structured patterns with the same frequency, and that non-stationarity comes from illumination. This is a reasonable expectation for a large class of materials (leather, fabrics, wallpapers), but can be wrong for other materials, with local patterns. We also assume that all maps have the same frequency in their patterns, and that we can use the guessed diffuse map as a guide for learning in the other maps.\n\n\n\n\\begin{figure*}[htbp]\n\t\\centering\n\t\\includegraphics[width=0.9\\linewidth]{synthetic-brick.pdf}\n\t\\caption{\\label{fig:synthetic-brick}%\n\t     SVBRDF maps recovered from synthetic images of $1024 \\times 1024$, compared with Gao et al.~\\cite{gao2019deep}, Guo et al.~\\cite{Guo2020MaterialGANRC}, Guo et al.~\\cite{Guo2021HighlightawareTN} and Zhao et al.~\\cite{Zhao2020JointSR}. Note how the specular highlight at the center of the image is challenging for all acquisition methods, resulting in dark or bright areas in the specular albedo map, and flattened areas in the normal map. We use the network of~\\cite{deschaintre2018single} as initialization for ~\\cite{gao2019deep} and set the number of input images to one. Guo et al.~\\cite{Guo2020MaterialGANRC} can only produce $256 \\times 256$ maps so we scale them to proper size for comparison.\n }\n\\end{figure*}\n\\subsection{Two-stage training strategy}\n\nTraining the network from scratch for each input image is time consuming: all parameters in the network have to be initialized as random value and trained over and over again for each new input. It can take several hours for the network to converge.\n\nTo improve this process, we propose a two-stage training strategy. We first train our network on an image (e.g. red book) for 10,000 steps to get a pretrained model. For a new input (e.g. brick), we use this pretrained model as an initialization of network parameters, and then train on this model for another 3,000 iterations to get the ``plausible'' SVBRDF maps. The key insight is that the generator acts as a prior knowledge about how the four maps look like in general after training for 10,000 steps:\n\\begin{itemize}\n    \\item the RGB value of normal map is close to light blue, due to the planar property of material,\n    \\item  the roughness map looks ``grey'',\n    \\item the specular map is ``dark'', and\n    \\item  the color of diffuse map mostly depends on the color of input image.\n\\end{itemize}\n Besides, as shown in Figure~\\ref{fig:pretrain}, without any extra training, the generator could recover texture information of the new input image in SVBRDF maps (although the color is not ``correct''). Apparently, with the pretrained parameters as a good initialization, it become relatively easier for the network to get a plausible recovery result of SVBRDF maps, comparing to training from scratch.\n\nWe have tried different pretrained models with different input image and find little difference in the final results. We provide some results in Figure~\\ref{fig:diffmodel}.\n\n \\begin{figure*}[htbp]\n\t\\centering\n\t\\includegraphics[width=1\\linewidth]{realphoto-leather.pdf}\n\t\\caption{\\label{fig:photo-leather}%\n     SVBRDF maps recovered from real photos of $1024 \\times 1024$, compared with Gao et al.~\\cite{gao2019deep}, Guo et al.~\\cite{Guo2020MaterialGANRC}, Guo et al.~\\cite{Guo2021HighlightawareTN} and Zhao et al.~\\cite{Zhao2020JointSR}. Note how the specular highlight at the center of the image is challenging for all acquisition methods. Our method produces more stationary SVBRDF maps and more plausible rendering results, compared to previous works.\n }\n\\end{figure*}\n\\subsection{Training and implementation}\n\nWe implemented our framework in TensorFlow. The generator and discriminator were trained using Adam optimizers with a fixed learning rate of 2e-5. We set the hyper-parameter $\\lambda$, $\\lambda_{1}$, $\\lambda_{2}$ to 0.1, 0.1, 0.2 respectively.\nAt the first stage, we train our network on an arbitrary input image for 10000 iterations from scratch to obtain a pretrained model. Then with a new image as input, we fine-tune the pretrained network for another 3000 iterations to get a plausible result. It takes about 2 hours to get the pretrained model, which we then use for any new input images. It then costs about 30 minutes to train the model on a new image, using a RTX 2080Ti GPU. Note that training from scratch for a new input image, rather than using the pretrained model, requires 20,000 iterations with 4 hours, using the same GPU. Hence, the pretrained model represents an approximate 8 times speedup.\n\n\n\n\n\\section{Related work}\n\\label{sec:related}\n\nThe problem of appearance capture has been extensively researched. Please refer to Guarnera et al.~\\cite{guarnera2016brdf} and Gao et al.~\\cite{gao2019deep} for a more comprehensive introduction. In this paper, we focus on light-weight appearance capture, which can be grouped into multi-image methods and single-image methods according to the number of input images.\n\n\n\\subsection{Multi-image appearance modeling} \n\n\\textbf{Non-learning based methods.} With multiple images as input, previous works can capture the SVBRDF based on optimization usually with some domain-specific priors or assumptions, e.g. the known illumination~\\cite{chandraker2014shape,hui2015dictionary,riviere2016mobile}, or sparsity in some domain ~\\cite{hui2017reflectance,dong2014appearance,Xia:2016:Shape}. Aittala et al.~\\cite{aittala2015two} used two photographs (one with flash and one without) to recover the reflectance, assuming the maps are stationary. Xu et al.~\\cite{Xu2016NearField} used two images from a near-field perspective camera, and assume spatial relation for reflectance recovery.\n\n\\textbf{Learning based methods.} Recently, deep learning has been widely used in appearance modeling. Deschaintre et al.~\\cite{Valentin2019Flexible} extracted the feature from each input image via a single-image appearance modeling network (similar to~\\cite{deschaintre2018single}) and then fused the features for SVBRDF recovery, to support arbitrary number of input images. Gao et al.~\\cite{gao2019deep} proposed an auto-encoder to extract the latent space from SVBRDFs as a material 'prior', and then optimized material maps in this latent space to better leverage the inherent connections between maps. However, their method needs an initial value of SVBRDF maps. Guo et al.~\\cite{Guo2020MaterialGANRC} trained a MaterialGAN to produce plausible material maps from a small number (3-7) of images. They used three optimizing strategies for the intermediate vector and noise vector in the latent space to learn the correlations in SVBRDF parameters. In order to tackle the shape\/SVBRDF ambiguity, Boss et al.~\\cite{Boss2020-TwoShotShapeAndBrdf} designed a cascaded network for shape, illumination and SVBRDF estimation, using two images captured by a cellphone with flash both on and off.\n\n\n\\begin{figure*}[htb]\n\t\\centering\n\t\\includegraphics[width=0.993\\linewidth]{newframework.pdf}\n\t\\caption{\\label{fig:newframework}%\n\t\t Our method is similar to Zhao et al.~\\cite{Zhao2020JointSR}, with several key differences: we reduce the number of de-conv layers in the decoder to let the $Generator$ output SVBRDF maps of same resolution as input. The predicted maps are used to calculate the diffuse loss $L_{d}$ and Fourier loss $L_{F}$. The input tile and re-rendered image are feed into discriminator to get the adversial loss $L_{adv}$ and a pretrained VGG-19 network to get the perceptual loss $L_{p}$. Our new loss terms are shown in red in the architecture.\n\t}\n\\end{figure*}\n\n\n\n\\subsection{Single-image appearance modeling} \n\nAnother group of works only use single image as input. Aittala et al.~\\cite{aittala2016reflectance} proposed a convolutional neural network (CNN) to extract a neural Gram-matrix texture descriptor from a single image to estimate the reflectance properties of a stationary textured materials. Under the same assuption, Zhao et al.~\\cite{Zhao2020JointSR} proposed an unsupervised generative adversarial network for joint SVBRDF recovery and synthesis. When the input image has intense highlights, their method confuses material properties and tends to produce maps with a bright spot for the specular albedo. It also takes a long time to process each input image. Our method addresses both issues. \n\nLi et al.~\\cite{li2017modeling} trained a CNN with a novel self-augmentation training strategy, which requires only a small number of labeled SVBRDF training pairs, to learn a large number of unlabeled photos of spatially varying materials. Ye et al.~\\cite{ye2018single} improved on this method and completely eliminated the need for labeled training dataset. Deschaintre et al.~\\cite{deschaintre2018single} proposed a secondary network to extract global features from each stage of an U-net architecture. They also introduce a rendering loss to enhance the estimated reflectance parameters by comparing the appearance rendered of predicted SVBRDF maps with the input image. Li et al.~\\cite{li2018materials} designed an in-network rendering layer to regress SVBRDF maps from single image and a material classifier to constrain the latent representation of a CNN. They also utilized a densely-connected conditional random fields module to further refine the results. A current work by Guo et al.~\\cite{Guo2021HighlightawareTN} proposed a new convolution variant called highlight-aware convolution (HA-convolution). They train the HA-convolution to ``guess'' the saturated pixels (specular highlight area) by the unsaturated area surrounded, making the extracted features more uniform. Their work achieve state-of-the-art performance on single-image SVBRDF acquisition and can well handle images with intense highlights. Compared to all these works, our method avoids the large dataset and learns the maps individually, under the stationary assumption.  \n\nTo remove the limitation of planar materials, Li et al.~\\cite{li2018learning} proposed a cascaded network architecture to recover shape and SVBRDF simultaneously from a single image. This method is further extended to handle complex indoor scenes~\\cite{li2020inverse}. \n \nIn terms of predicting procedural texture parameters, Hu et al.~\\cite{hu2019novel} introduced a novel framework for inverse procedural texture modeling: they trained an unsupervised clustering model to select a most appropriate procedural model and then used a CNN pool to map images to material parameters. \n\n\\begin{figure*}[htbp]\n\t\\centering\n\t\\includegraphics[width=0.993\\linewidth]{pretrainmodel.pdf}\n\t\\caption{\\label{fig:pretrain}%\n\t\tOur two-stage training strategy. At the first stage, we train the network on one image to get the pretrained model. For any new input images, we use this pretrained model as an initialization to start our second stage training. Note that at second stage, texture information already exists in the initialized SVBRDF maps, without extra training. \n\t}\n\\end{figure*}\n\n\\begin{figure*}[htbp]\n\t\\centering\n\t\\includegraphics[width=0.9\\linewidth]{synthetic-book.pdf}\n\t\\caption{\\label{fig:synthetic-book}%\n\t\tSVBRDF maps recovered from synthetic images of $1024 \\times 1024$, compared with Gao et al.~\\cite{gao2019deep}, Guo et al.~\\cite{Guo2020MaterialGANRC}, Guo et al.~\\cite{Guo2021HighlightawareTN} and Zhao et al.~\\cite{Zhao2020JointSR}. \n\t\tNote how the specular highlight at the center of the image is challenging for all acquisition methods, resulting in dark or bright areas in the specular albedo map, and flattened areas in the normal map. We use the network of~\\cite{deschaintre2018single} as initialization for ~\\cite{gao2019deep} and set the number of input images to one. Guo et al.~\\cite{Guo2020MaterialGANRC} can only produce $256 \\times 256$ maps so we scale them to proper size for comparison.\n\t}\n\\end{figure*}\n\n\n\n\n\\section{Results and discussion}\n\\label{sec:results}\n\n\n\\begin{figure*}[htbp]\n\t\\centering\n\t\\includegraphics[width=1\\linewidth]{realphoto-plastic.pdf}\n\t\\caption{\\label{fig:photo-plastic}%\n\tSVBRDF maps recovered from real photos of $1024 \\times 1024$, compared with Gao et al.~\\cite{gao2019deep}, Guo et al.~\\cite{Guo2020MaterialGANRC}, Guo et al.~\\cite{Guo2021HighlightawareTN} and Zhao et al.~\\cite{Zhao2020JointSR}. Note how the specular highlight at the center of the image is challenging for all acquisition methods. Our method produces more stationary SVBRDF maps and more plausible rendering results, comparing to previous works.\n }\n\\end{figure*}\n\n\n\n\\begin{figure*}[htbp]\n\t\\centering\n\t\\includegraphics[width=0.95\\linewidth]{lossabl-leather.pdf}\n\t\\caption{\\label{fig:lossleather}%\n\tAblation study on several maps to valid the the impacts of Fourier loss and perceptual loss in our SVBRDF recovery network. }\n\\end{figure*}\n\n\\begin{figure*}[htbp]\n\t\\centering\n\t\\includegraphics[width=0.9\\linewidth]{differentFFT.pdf}\n\t\\caption{\\label{fig:diffFFT}%\n\tInfluence of Fourier loss on different maps. Without Fourier loss on roughness map (R) or specular map (S), the bright spot still exist.}\n\\end{figure*}\n\n \\begin{figure*}[htbp]\n\t\\centering\n\t\\includegraphics[width=1\\linewidth]{limitation.pdf}\n\t\\caption{\\label{fig:limitation}%\n\t\tFailure case. Images with sharp contrast may not be well treated since our method failed to get an stationary ``guessed'' diffuse map.\n\t}\t\n\\end{figure*}\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\\includegraphics[width=1.0\\linewidth]{differentModel-bookbluegray.pdf}\n\t\\caption{\\label{fig:diffmodel}%\n\tComparison on different pretrained models and different training strategies. There is little difference in the three results. }\n\\end{figure}\n\n\n\n\nWe first compare the results of our network with Gao et al.~\\cite{gao2019deep}, Guo et al.~\\cite{Guo2020MaterialGANRC}, Guo et al.~\\cite{Guo2021HighlightawareTN} and Zhao et al.~\\cite{Zhao2020JointSR} on both synthetic images and captured photos (Sec.~\\ref{sec:recovresult}). Then we show the influence of different loss term by a loss ablation experiment (Sec.~\\ref{sec:ablationStudy}) and the effects of our two-stage training strategy (Sec.~\\ref{sec:twoStageTraining}).\n\n\n\\subsection{Comparison with previous works}\n\\label{sec:recovresult}\n\nWe ran our experiments on images with strong highlights to show the effectiveness of our approach. The input real photos and reference maps are from the two-shot dataset~\\cite{aittala2015two} and a free material website$\\footnotemark$\\footnotetext[1]{https:\/\/texturehaven.com}. For the two-shot dataset, we cropped the $3264 \\times 2448$ SVBRDF maps to $1600 \\times 1600$ and resized them to $1024 \\times 1024$ resolution. We render the input synthetic images using the Cook-Torrance reflectance model~\\cite{cook1982reflectance} with a point light and camera right above the center of the plane, consistent with our re-rendering process in the network.\n\n\n\\textbf{Comparison on synthetic images.} \nIn Figure~\\ref{fig:synthetic-book} and Figure~\\ref{fig:synthetic-brick}, we compare our results on synthetic images with Gao et al.~\\cite{gao2019deep}, Guo et al.~\\cite{Guo2020MaterialGANRC}, Guo et al.~\\cite{Guo2021HighlightawareTN} and Zhao et al.~\\cite{Zhao2020JointSR}. For Gao et al.~\\cite{gao2019deep}, we use the network of Deschaintre et al.~\\cite{deschaintre2018single} as an initialization and set the number of inputs to one for fair comparison. Although \\cite{gao2019deep} achieves plausible rendering results through optimization steps, strong artifacts still exist in the recovered SVBRDF maps, leading to poor performance in novel view rendering. Guo et al.~\\cite{Guo2020MaterialGANRC} also produce unpleasant SVBRDF maps and novel view rendering. Zhao et al.~\\cite{Zhao2020JointSR} preserve some details in SVBRDF maps but still suffer from highlights regions, especially in roughness map and specular map. Our method recovers the stationarity in SVBRDF maps, which are comparable to the reference maps, thus lead to more plausible appearance in novel view rendering. More results are shown in the supplemental materials.\n\n\n\n\n\n\n\\textbf{Comparison on captured photos.} \nIn Figures~\\ref{fig:photo-leather} and \\ref{fig:photo-plastic}, we validate our method on three captured photos, comparing to Gao et al.~\\cite{gao2019deep}, Guo et al.~\\cite{Guo2020MaterialGANRC}, Guo et al.~\\cite{Guo2021HighlightawareTN} and Zhao et al.~\\cite{Zhao2020JointSR}. The photos are from the two-shot dataset~\\cite{aittala2015two}, cropped so that the brightest part of the image is at the center. There are no reference SVBRDFs for the captured photos. As shown in Figure~\\ref{fig:photo-leather}, Gao et al.~\\cite{gao2019deep} and Guo et al.~\\cite{Guo2021HighlightawareTN} produce ``polluted'' SVBRDF maps that are highly affected by highlights regions, while Guo et al.~\\cite{Guo2020MaterialGANRC} produce blurred results that lack detailed structure in SVBRDF maps. Zhao et al.~\\cite{Zhao2020JointSR} produce plausible diffuse maps, but still suffer from highlights regions in the other maps. All the previous works fail to handle the ambiguity in reflectance and illumination, yielding unpleasant novel view rendering results. Our method produces more stationary SVBRDF maps and more plausible rendering results under novel views. As shown in Figure~\\ref{fig:photo-plastic}, our method recovers detailed variations in normal map and suppresses the bright spots in other three maps. Thus, our method better handles the ambiguity in reflectance and illumination, producing more reasonable rendering results under novel view.\n\n\n  \n\\subsection{Ablation study}\n\\label{sec:ablationStudy}\n\nThere are two important components in our SVBRDF recovery network: Fourier loss and perceptual loss. We ran an ablation study to validate the impacts of these components in Figure~\\ref{fig:lossleather}. We compare Zhao et al.~\\cite{Zhao2020JointSR}, our model (with Fourier loss $\\mathcal{L}_{F}$ only), our model (with both Fourier loss $\\mathcal{L}_{F}$ and perceptual loss $\\mathcal{L}_{P}$) and the reference. \n\nZhao et al.~\\cite{Zhao2020JointSR} (first row) suffer from bright spots in the roughness and specular maps, and over-flat normal map, leading to obvious difference from the reference SVBRDF maps, where the material properties are stationary. By introducing the Fourier loss,  the predicted maps (second row) become more uniform and have less artifacts at the center. The novel view rendering results have more pronounced variations in illumination due to the more uniform SVBRDF maps, thus produce more plausible appearance compared to Zhao et al.~\\cite{Zhao2020JointSR}. However, we found that the texture variation is blurred in the highlights region. Further introducing perceptual loss (third row) solves this issue, via measuring the semantic similarity between the input image and re-rendering result. The joint loss function with both Fourier loss and perceptual loss produces the best results. The bright spots in the SVBRDF maps have been removed, making them decoupled from the illumination in the input images, thus leading to more plausible re-rendering results under different viewing\/lighting directions. More results are shown in the supplemental materials.\n\nWe also tried using the Fourier loss on only some of the SVBRDF maps, such as the roughness or specular albedos. The results of this study are shown  in Figure~\\ref{fig:diffFFT}. The maps that were computed without the Fourier loss are highly affected by the bright spot, while the other maps are not. We used the Fourier loss on all 4 maps to  ensure the stationarity in all SVBRDF maps.\n\n\n\\subsection{Validation of the two-stage training strategy}\n\\label{sec:twoStageTraining}\nIn Figure~\\ref{fig:diffmodel}, we compare the SVBRDF maps recovered with two-stage training strategy and one-stage training strategy (without pretraining). For the two-stage training strategy, we show the results recovered from two pretrained models which are trained with different images (book-red and brick). For one-stage training strategy (without pretraining), the training is performed on the input image from scratch, with 20,000 iterations. By comparison, we find that the difference between one-stage and two-stage training strategies is subtle, and the difference with different pretrained models are also not obvious. Thus, we believe that our two-stage training strategy greatly reduces the training time, without any quality degradation and the pretrained model can be trained on arbitrary single image under our assumption.\n\n\n\\subsection{Limitations}\n\n\nWe have identified three main limitations for our\nmethod. First, we assume that all maps have the same frequency in their patterns, and that we can use the guessed diffuse map as a guide for learning in the other maps. If the SVBRDF maps have different frequency, our Fourier loss will provide the wrong guidance. Second, our method does not work well for input images with sharp contrast, as shown in Figure~\\ref{fig:limitation}. The highlights are so strong that the texture information in this region is almost obscured, which prevent us from getting a plausible guessed diffuse map. Third, although the training time has been reduced significantly, compared to the previous work, each image still needs 30 minutes for training. Further reducing the training time will be a valuable work, and we leave it for the future work.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nMagnetic fields generated in laser-matter interactions are of primary interest in high energy density physics \\cite{laserplasma}. For example, magnetic fields generated by the Weibel instability can explain the collisionless shocks that are found in young galaxies and other astrophysical systems \\cite{weibel, astro}. In inertial confinement fusion, magnetic fields are used in one approach to reduce heat losses and thus improve performance of implosions \\cite{magicf}, and in another approach (using cylindrical implosions) as a necessary criterion to reach ignition \\cite{icfcylindrical}. Also, magnetic reconnection is a commonly studied process  which converts some of the magnetic energy of a system into heat, and understanding the heating mechanism well could lead to better hohlraum design for inertial confinement fusion \\cite{magneticreconnection}.\n\nProton radiography is an extensively used technique that characterizes electric and magnetic fields in plasmas over a wide range of field strengths \\cite{roth}. A polyenergetic proton beam, with typical energies on the order of 10 MeV, is usually produced by high intensity laser interaction with solid targets \\cite{hedfrontiers}. This beam then interacts with an object of interest (such as plasmas or shock-compressed matter) and gets deflected as a result of the Lorentz force or collisions with atoms \\cite{radiographyscatter}. The outgoing beam is captured on a radiochromic film (RCF) stack which can resolve both spatial and energy profiles \\cite{rcf}. \n\nVarious methods have been developed in analyzing proton radiographs. Using the principles of differential scattering and stopping, density profiles of dense matter can be retrieved from radiographs \\cite{laserdriven}. Via scaling laws, field strengths of electric and magnetic fields  can be estimated \\cite{measuringeb,quantitative1,quantitative2}. Also, radiographs can be used to qualitatively understand electric and magnetic field structures \\cite{qualitative2,qualitative3,qualitative4}. Furthermore, radiographs can be simulated numerically in order to identify features found in experimental radiographs \\cite{levy,compare}.\n\nIt is only recently that techniques have been developed to reconstruct fields. The relations between the field structures and proton radiographs have been established by Kugland $\\textit{et al}.$ \\cite{Kugland} under certain simplifying assumptions, allowing one to obtain the line-integrated transverse magnetic field from a radiograph by solving a 2-D Poisson equation. Graziani $\\textit{et al.}$ \\cite{morphology} and Kasim $\\textit{et al.}$ \\cite{kasim} provided extensions to this technique, under similar assumptions. As such, radiographs of systems which do not obey any of the assumptions in \\cite{Kugland,morphology,kasim} can only be analyzed qualitatively. \n\nMachine learning, a field of study which enables the performance of a computer (with respect to a certain task) to increase with its experience, has seen many applications in artificial intelligence problems such as image recognition, recommender systems and speech-to-text \\cite{LeCun}. Due to its ability to discover structures in high dimensional data, artificial neural networks (one example of machine learning) has seen many applications in physics, such as analyzing particle accelerator data \\cite{particle}, reconstructing images in optical tomography \\cite{Kamilov:15} and retrieving 3-D potentials in electron scattering \\cite{dynamical}. The flexible nature of artificial neural networks and the prevalence of its usage in image recognition problems prompt us to posit its usage in imaging 3-D magnetic field structures without a need for simplifying assumptions, addressing the gaps found in existing radiograph inversion techniques.\n\nIn this paper, we first review existing work on inverting proton radiographs. We then introduce key ideas of artificial neural networks and review their applications in physics. Next, we outline the new method of using artificial neural networks to reconstruct magnetic fields and retrieving field parameters such as characteristic lengths. Via simulations, we show a proof of concept for the above ideas, and discuss how noise and selection of training data affect our results. Using an example, we highlight the need for proton tomography. Finally, we compare the artificial neural network technique with the existing methods of radiograph inversion and suggest a variety of extensions to our research.\n\n\\section{Theory}\n\n\\subsection{Existing methods of retrieving magnetic fields from radiographs}\n\\label{subsec:existingmethod}\n\nIn this subsection, we will outline the foundational work on proton radiograph inversion by Kugland $\\textit{et al.}$, move on to discuss Graziani $\\textit{et al.}$ and Kasim $\\textit{et al.}$'s extensions, and conclude with the gaps in these methods.\n\nFirst, we go through Kugland $\\textit{et al.}$'s \\cite{Kugland} definitions: The coordinates are defined such that the object is placed at $z=0$ (object plane), and $(x, y)$ refers to the coordinates on the image plane (see Fig. \\ref{fig:setup}). At the object plane, the proton's coordinates are denoted as $(x_0,y_0)$. The distance between the proton source and the object is $l$ while the distance between the object to the image plane (radiochromic film stack) is $L$. $a$, the characteristic length of the object, is assumed to be much smaller than $l$ (paraxial limit) and $L\\gg l$ for high magnification.\n\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{images\/setup.pdf}\n\\caption{Diagram of a typical proton radiography setup. A point source of distance $l$ away from the object emits a beam of protons moving generally in the $z$-direction. $L$ is the distance between the object and image plane.}\n\\label{fig:setup}\n\\end{figure}\n\nIn order to get a tractable result, Kugland $\\textit{et al.}$ have made some simplifying assumptions. We start off with those relating to the proton source: (i) The source can be treated as a point source. Else, the radiograph will be blurred and the resolution of field structures will be affected. (ii) The protons deviate from their straight-line trajectories solely due to the Lorentz force interaction with the object, and we can ignore space-charge effects because the beam is charge-neutral as a result of co-moving electrons \\cite{comovingelectrons}. (iii) The angular width of the beam is much greater than $a\/l$ so that intensity variations in the image plane are due to proton interactions with the object, and not the angular distribution of the proton beam. \n\nConsider the dimensionless parameter\n\n\\begin{eqnarray}\n\\mu\\equiv \\frac{l\\beta}{a},\n\\end{eqnarray}\n\n\\noindent where $a$ is a characteristic length of the electromagnetic field, and $\\beta$ is a characteristic deflection angle. One core assumption in Kugland $\\textit{et al.}$ is that $\\mu\\ll1$ (hence known as the linear regime), where the spatial variation of the intensity on the screen is small. This is in contrast to the non-linear regime ($\\mu$ on the order of 1 or more) where the intensity variations are large, leading to non-linear features. One example of non-linear features is caustics, which occurs when the Jacobian determinant\n\n\\begin{eqnarray}\n \\left|\\frac{\\partial (x,y)}{\\partial (x_0,y_0)}\\right|=0,\n\\end{eqnarray}\n\n\\noindent resulting in features of high intensity (usually multiples of the background intensity).\n\nFurthermore, assuming that the velocity of the proton $\\mathbf{v}$ is approximately constant while the proton is within the object, (trajectories are not perturbed within the plasma so $\\mathrm{dt=d}z_0\/v$), the only relevant component of the magnetic potential is the one in the $z$-direction, $A_z$. Defining the line-integrated potential as\n\n\\begin{eqnarray}\n\\Phi(x_0,y_0)= \\int_{-\\infty}^{\\infty}A_z(x_0,y_0,z_0)\\mathrm{d}z_0,\n\\end{eqnarray}\n\n\\noindent then with all the assumptions listed above, Kugland $et\\:al.$'s formula for radiograph inversion reads:\n\n\\begin{eqnarray}\n \\mathbf{\\nabla_\\perp^2}\\Phi(x_0,y_0) = \\frac{\\sqrt{2m_p K}}{el} \\Big( 1-\\frac{I}{I_0}\\frac{L^2}{l^2}\\Big),\n\\label{eqn:poisson}\n\\end{eqnarray}\n\n\\noindent where $\\nabla_\\perp$ is the gradient with respect to the transverse coordinates $(x_0,y_0)$, $m_p$ is the mass of the proton, $K$ is the (non-relativistic) kinetic energy of the proton, $e$ is the charge of an electron, $I$ is the proton intensity distribution at the image plane and $I_0$ is the proton intensity distribution in the object plane. As such, given the intensity profile at the object plane $I_0(x_0, y_0)$ and radiograph intensity profile $I(x, y)$ (which can be transformed to $I(x_0,y_0)$ via the mapping $x=\\frac{L}{l}x_0,y=\\frac{L}{l}y_0$) in the regime $\\mu\\ll 1$, one can solve a 2-D Poisson equation to get the line-integrated potential $\\Phi(x_0,y_0)$, thereby allowing one to reconstruct the line-integrated transverse magnetic field.\n\nUsing a series of perturbations, and assuming the linear regime $\\mu \\ll 1$, Graziani $\\textit{et al.}$ \\cite{morphology} proposed a correction term in the right hand side of equation (20) in Kugland $\\textit{et al.}$ (equation (\\ref{eqn:poisson}) in this paper) which leads to a second-order non-linear partial differential equation. The authors then conducted a simulation of their proposed equation, and found that their method reconstructed the line-integrated magnetic field accurately at locations near the peak field strength, but was inaccurate at locations where the field strengths are at least 3 orders of magnitude less than the peak field strength. Also, Graziani $\\textit{et al.}$ briefly sketched a method to retrieve the line-integrated magnetic field in the non-linear regime, assuming that the direction of the proton trajectory within the object is nearly constant. \n\nAnother method, based on computational geometry, was implemented by Kasim $\\textit{et al.}$ \\cite{kasim}. This method works well in the beginning of the caustic regime (early part of the non-linear regime where $\\mu\\textgreater 1$), but the relative errors start to become very large in the regime of branching caustics (later part of the non-linear regime). Also,  Kasim $\\textit{et al.}$ demonstrated the large errors that come with solving the Poisson equation in Kugland $\\textit{et al.}$ for a system in the non-linear regime. \n\nSo far, we have seen that existing methods of magnetic field reconstruction require simplifying assumptions in order to get an equation which, when solved, gives the line-integrated transverse magnetic field. This highlights two gaps: (i) In later parts of the non-linear regime (e.g. branching caustics regime), there is no known reconstruction method despite the fact that non-linear features do occur in some experimental radiographs \\cite{nonlinear1,nonlinear2,nonlinear3}. In this regime, experimental radiographs are analyzed by comparison to simulated radiographs of a hypothesized magnetic field structure. (ii) In both regimes, there is no reconstruction method that can give the 3-D magnetic field. As we will demonstrate in the next few subsections, the proposed artificial neural network method can address both gaps.\n\n\\subsection{Artificial neural networks (ANN)}\n\\input{chapters\/ann}\n\n\\section{Methods}\n\\input{chapters\/methods}\n\n\\section{Results and discussion}\n\nWe have simulated special cases of equation (\\ref{eqn:parameters}) as a proof of concept of the idea in section \\ref{subsec:recon}. All results shown here come from applying a trained artificial neural network on the testing set (which is not used in training the artificial neural network), and is indicative of the performance when tested on experimental data. Some of the simulation parameters used in the following subsections can be found in table \\ref{table:params}. \n\n\n\\begin{table*}[]\n\\centering\n\\begin{tabular}{|c|c|c|}\n\\hline\n\\multicolumn{1}{|l|}{} & Value in subsection A-E & Value in subsection F \\\\ \\hline\nEllipsoidal blob parameter $a$\/mm & 0.1 & 0.7 \\\\ \\hline\nEllipsoidal blob parameter $b$\/mm & \\multicolumn{2}{c|}{1} \\\\ \\hline\nFlux rope height\/mm & 2 & 0.3 \\\\ \\hline\nFlux rope parameter $a$\/mm & 0.8 & 0.5 \\\\ \\hline\nDistance between proton source and object $l$\/mm & \\multicolumn{2}{c|}{7} \\\\ \\hline\nDistance between object and screen $L$\/mm & \\multicolumn{2}{c|}{93} \\\\ \\hline\nNumber of neurons in input layer & 2500 & 5000 \\\\ \\hline\nNumber of neurons in output layer & \\multicolumn{2}{c|}{2 for subsections A, B, D, F and 1 for subsections C, E } \\\\ \\hline\nVelocity of protons in the $z$ direction\/ms$^{-1}$ & \\multicolumn{2}{c|}{$10^6$} \\\\ \\hline\nVelocity of protons in the $x,\\:y$ direction\/ms$^{-1}$ & Ranges from -5$\\times 10^4$ to 5$\\times 10^4$ & Ranges from -6.9$\\times 10^5$ to 6.9$\\times 10^5$\\\\ \\hline\n\\end{tabular}\n\\caption{Parameters for simulations in the following subsections. The proton velocities for subsection F refer to protons in the beam fired in the $z$ direction.}\n\\label{table:params}\n\\end{table*}\n\n\\subsection{Reconstructing magnetic fields, a proof of concept}\n\nConsider the following two fields: (a) a magnetic ellipsoidal blob, representative of fields generated by the Weibel instability \\cite{weibel}, that can be written as\n\n\\begin{eqnarray}\nB_\\phi = B_0\\frac{r_0}{a}\\mathrm{exp}(-(\\frac{r_0^2}{a^2}+\\frac{z_0^2}{b^2})),\n\\label{eqn:blob}\n\\end{eqnarray}\n\n\\noindent where $B_0$ is proportional to the peak field strength, $r_0$ is the distance to the center in the $xy$ plane, $z_0$ is the distance to the center along the $z$-axis, and $a,\\:b$ are characteristic lengths of the ellipsoid; (b) a magnetic flux rope of Gaussian cross section, representative of fields due to laser generated plasma flows \\cite{fluxrope}, can be written as\n\n\\begin{eqnarray}\nB_y = B_0\\mathrm{exp}(-\\frac{x_0^2+z_0^2}{a^2}),\n\\end{eqnarray}\n\n\\noindent where $B_0$ is the peak field strength, $x_0$ and $z_0$ are the distances to the center along the $x$- and $z$- axes respectively, and $a$ is a characteristic length of the Gaussian.\n\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{images\/matlabfig\/89b_reg_hist}\n\\caption{Error histograms for the $\\alpha$ coefficients of two basis fields, using an artificial neural network with 1 hidden layer consisting of 10 neurons. The mean errors are 0.34$\\%$ and 2.74$\\%$ while the median errors are 0.20$\\%$ and 1.29$\\%$ for $\\alpha_1$ (ellipsoidal blob) and $\\alpha_2$ (flux rope) respectively. More parameters can be found in table \\ref{table:params}.}\n\\label{fig:twoalphas_reg}\n\\end{figure}\n\nIn terms of equation (\\ref{eqn:parameters}), we assign $\\alpha_1$ to $B_0$ of the magnetic ellipsoidal blob and $\\alpha_2$ to $B_0$ of the magnetic flux rope. $\\alpha_1$ was varied from 5 to 6 T (defocusing) in steps of 0.01 T while $\\alpha_2$ was varied from 2.01 to 3 T in steps of 0.03 T, and all other parameters were kept constant. Radiographs of 50 by 50 pixels were generated for each configuration. As mentioned earlier, 70$\\%$ of these radiographs were randomly chosen to train the artificial neural network, 15$\\%$ were randomly assigned to the validation set to prevent overfitting, and the trained artificial neural network was used to predict the $\\alpha_1$ and $\\alpha_2$ values on the remainder $15\\%$ of the radiographs. The errors, defined as $\\left|\\frac{\\mathrm{predicted\\:value}-\\mathrm{actual\\:value}}{\\mathrm{actual\\:value}}\\right|$, are plotted in Fig. \\ref{fig:twoalphas_reg}. We see that nearly all the errors are less than $5\\%$, suggesting that the full scale implementation outlined in section \\ref{subsec:recon} will work given enough basis fields. Though there are some undesirable outliers in $\\alpha_2$, it is likely to be a result of inadequate data rather than a flaw in the artificial neural network method. This will be discussed in section \\ref{subsec:data_accuracy}.\n\n\\subsection{Obtaining $\\mathbf{B}$ field parameters from a magnetic ellipsoidal blob}\n\\label{subsec:blobparams}\n\n\\begin{figure}\n\\centering\n\\subfloat[ ]{\\includegraphics[width=0.26\\textwidth]{images\/88_radiograph_noncaustic_17aug2016}}\n\\subfloat[ ]{\\includegraphics[width=0.26\\textwidth]{images\/88_radiograph_caustic_16aug2016}}\n  \\caption{(a) Radiograph for a magnetic ellipsoidal blob at B = 0.1 T, $\\sigma$ = 1. This is in the non-caustic regime, where the ring around the center is smeared out. (b) Radiograph for a magnetic ellipsoidal blob at B = 0.3 T, $\\sigma$ = 1. This is in the caustic regime, where most of the protons fall into a very thin ring. The scales are in arbitrary units.}\n\\label{fig:caustic}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{images\/matlabfig\/88_reg_hist}\n\\caption{Error histograms for $\\alpha$ and $\\sigma$ for a magnetic ellipsoidal blob, using an artificial neural network with 1 hidden layer consisting of 50 neurons. The mean errors are 0.26$\\%$ and 0.05$\\%$ while the median errors are 0.20$\\%$ and 0.04$\\%$ for $\\alpha$ and $\\sigma$ respectively.  More parameters can be found in table \\ref{table:params}.}\n\\label{fig:alphasigma_reg}\n\\end{figure}\n\nIn this subsection, we demonstrate that (i) the artificial neural network method works in the non-linear regime, and (ii) the parameter retrieval concept in section \\ref{subsec:retrieval} can be done. Here, we retrieve the field strength coefficient $\\alpha$ and the scaling factor $\\sigma$.\n\nRadiographs for a magnetic ellipsoidal blob were generated with $\\alpha$ (representing $B_0$ in equation (\\ref{eqn:blob})) ranging from 0.1 to 0.3 T (defocusing) in steps of $2\\times 10^{-4}$ T and $\\sigma$ ranging from 0.9 to 1 in steps of 0.02. This spectrum of $\\alpha$ spans both the caustic and non-caustic regime, as can be seen by the radiographs plotted in Fig. \\ref{fig:caustic}. The histogram of errors are plotted in Fig. \\ref{fig:alphasigma_reg}. We can see that the average errors are well below $1\\%$, suggesting that artificial neural networks can be used for parameter retrieval, an alternative to reconstructing entire magnetic fields. We also see that this method works in the non-linear regime, where $\\mu\\approx 2$.\n\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{images\/matlabfig\/92_branchingprofile}\n\\caption{Horizontal profile of a radiograph for a magnetic ellipsoidal blob at 1.5 T, the branching caustics regime. Notice that there are two maxima in the intensity profile, instead of one in the case of the caustic regime.}\n\\label{fig:branchingprofile}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{images\/matlabfig\/92_reg_hist}\n\\caption{Error histogram of $\\alpha$ for a magnetic ellipsoidal blob spanning the linear, caustic and branching caustic regime, using an artificial neural network with 1 hidden layer consisting of 10 neurons. The mean error is 0.06$\\%$ while the median error is 0.04$\\%$.  More parameters can be found in table \\ref{table:params}.}\n\\label{fig:branching}\n\\end{figure}\n\n\\subsection{Branching caustics}\n\\label{subsec:branching_caustics}\n\nThe power diagram method \\cite{kasim} gives relative errors of more than $10\\%$ in the branching caustics regime. Here, we show that the artificial neural network method is flexible enough to accommodate this scenario. We extend the range of field strengths in section \\ref{subsec:blobparams} to range from 0.1 T to 1.5 T in steps of $2\\times 10^{-4}$ T, spanning the linear, caustic and branching caustic regime. As an illustration, the horizontal profile of the radiograph at 1.5 T (branching caustics regime) is plotted in Fig. \\ref{fig:branchingprofile}. The error histogram is plotted in Fig. \\ref{fig:branching} and we can see that all the errors are well below $1\\%$.\n\n\\subsection{Effect of noise on accuracy}\n\n\\begin{figure*}\n\\makebox[\\textwidth][c]{\\includegraphics[width=1.25\\textwidth]{images\/matlabfig\/90_reg_hist}}\n\\caption{Error histograms of $\\alpha$ and $\\sigma$ when 5, 10, 20, and 30 percent noise is introduced into the radiographs for a magnetic ellipsoidal blob. The artificial neural network configurations are: 8 hidden layers with 80 neurons per layer, 5 hidden layers with 50 neurons per layer, 6 hidden layers with 70 neurons per layer and 7 hidden layers with 100 neurons per layer for 5, 10, 20, and 30 percent noise respectively. For $\\alpha$, the mean errors are 0.92$\\%$, 1.14$\\%$, 1.49$\\%$ and 1.91$\\%$ while the median errors are 0.73$\\%$, 0.82$\\%$, 1.19$\\%$ and 1.46$\\%$ for 5, 10, 20, and 30 percent noise respectively. For $\\sigma$, the mean errors are 0.18$\\%$, 0.24$\\%$, 0.31$\\%$ and 0.41$\\%$ while the median errors are 0.14$\\%$, 0.18$\\%$, 0.25$\\%$ and 0.31$\\%$ for 5, 10, 20, and 30 percent noise respectively. Notice that it takes an increase from 5$\\%$ noise to 30$\\%$ noise in order to roughly double the mean and median errors, indicating that the artificial neural network method is robust to noise.  More parameters can be found in table \\ref{table:params}.}\n\\label{fig:alphasigma_noise}\n\\end{figure*}\n\nSo far we have shown that artificial neural networks trained on noise-free radiographs can retrieve quantities from noise-free radiographs with a high accuracy. We proceed to explore the changes in accuracy when noise is introduced into all the radiographs (training, validation and testing sets). Suppose a pixel in the radiograph has a value of $\\chi$ and we want to introduce random noise of $x\\%$. Then each pixel is replaced by a random value from a Gaussian distribution with a mean of $\\chi$ and a standard deviation of $\\chi\\times x\\%$. This was done for $x$ = 5, 10, 20 and 30 percent on the radiographs in section \\ref{subsec:blobparams}, and the entire process of training and prediction was repeated. The error histograms for $\\alpha$ and $\\sigma$ are plotted in Fig. \\ref{fig:alphasigma_noise}. We notice that for both quantities, it takes an increase from 5$\\%$ to 30$\\%$ noise in order to roughly double the mean and median errors. This demonstrates the robustness of artificial neural networks to noise, although noise does occasionally cause very high errors. It is worth noting that the right model to use is Poisson noise, but that model approximates Gaussian noise for a large number of particles per pixel, which is true in our case. The relationship between errors and input noise for various configurations of artificial neural networks is further explored in \\cite{nwnoise1,nwnoise2,nwnoise3,sizenoise}.\n\n\\subsection{Effect of the amount of training data on accuracy}\n\\label{subsec:data_accuracy}\n\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{images\/matlabfig\/93_reg_hist}\n\\caption{Error histogram in $\\alpha$ for a magnetic ellipsoidal blob (more parameters in table \\ref{table:params}) when the step size is increased by a factor of 5, leading to less data. The mean error is 0.20$\\%$ and the median error is 0.16$\\%$, using an artificial neural network with 7 hidden layers with 40 neurons per layer.}\n\\label{fig:stepsize}\n\\end{figure}\n\nWhile the artificial neural network method seems promising so far, it is reliant on the large amounts of training data (specifically, the amount of information in the data, or information entropy) for its accuracy. To elucidate this fact, we generated data for a magnetic ellipsoidal blob and varied only $\\alpha$ between the values 0.1 to 1.5 T, similar to the scenario in section \\ref{subsec:branching_caustics}, except with a larger step size of $10^{-3}$ T. The error histogram is plotted in Fig. \\ref{fig:stepsize}. In comparison to Fig. \\ref{fig:branching}, we see that having a larger step size and thus having less information leads to an increase in errors. This, combined with the universality of the artificial neural network proved in \\cite{hornik}, suggests that extreme outliers in errors can be overcome by generating more data that increases the information entropy of the data set and re-training the neural network. The relationship between errors and size of data set for various configurations of artificial neural networks is further explored in \\cite{nwsize1,nwsize2,nwsize3,sizenoise}.\n\n\\subsection{Limitations of proton radiography and the need for proton tomography}\n\nProton radiographs do not necessarily form one-to-one relationships with field structures: Suppose that at the edge of a plasma that is facing the proton beam, there is a very strong $\\mathbf{B}$ field that deflects the incoming protons before these protons could penetrate further. Then the radiograph formed is independent of the $\\mathbf{B}$ fields in the remainder of the plasma, because no protons interact with it. Due to the lack of information in the radiographs, no method can fully reconstruct the $\\mathbf{B}$ fields. As such, there is a need to modify the experimental set-up to capture more information from the $\\mathbf{B}$ field.\n\nOne possible way to capture more information is to include more probe beams in different directions (tomography). As an example, consider two adjacent field structures, the ellipsoidal blob (field strength parameter assigned to $\\alpha_1$, ranging from 9 to 9.25 T in steps of 0.005 T) and the flux rope (field strength parameter assigned to $\\alpha_2$, ranging from 0.3 to 0.4 T in steps of 0.002 T), with the former obscuring the latter in the $z$ direction by 0.5 mm. When the artificial neural network method was used on radiographs due to a beam fired in the $z$ direction, the errors in the retrieved field strength of the flux rope are very high (top panel, Fig. \\ref{fig:tomography}) due to the lack of protons probing the field structure. When another probe beam was fired in the $x$ direction (with $x$ velocity of $10^6$ ms$^{-1}$ and $y,\\:z$ velocities ranging from -2$\\times 10^5$ ms$^{-1}$ to 2$\\times 10^5$ ms$^{-1}$) and the radiographs were used in addition to the ones from the probe beam in the $z$ direction, errors for both field strengths decrease by at least an order of magnitude (bottom panel, Fig. \\ref{fig:tomography}). This demonstrates two facts: (i) the artificial neural network method (and any other method) cannot fully reconstruct magnetic fields if the radiographs carry insufficient information; (ii) including more information decreases errors, even if the field structure is not obscured, as can be seen by the reduction in errors for $\\alpha_1$ in Fig. \\ref{fig:tomography}. We hope this will inspire future work on theoretical error bounds in artificial neural networks given the lack of information in the data set.\n\n\n\\begin{figure}\n\\makebox[\\columnwidth][c]{\\includegraphics[width=1.15\\columnwidth]{images\/matlabfig\/97_reg_hist}}\n\\caption{Error histograms of $\\alpha_1$ and $\\alpha_2$ for two scenarios. Top panel: Using radiographs generated by proton beams in the $z$ direction as training data, the mean errors are 1.20$\\times 10^{-2}\\%$ and $3.48\\%$ while the median errors are 3.59$\\times 10^{-3}\\%$ and $2.50\\%$ for $\\alpha_1$ and $\\alpha_2$ respectively, using an artificial neural network with 1 hidden layer consisting of 30 neurons. (more parameters in table \\ref{table:params}). The errors for $\\alpha_2$ (field strength of the flux rope) are high because the field structure associated with $\\alpha_1$ (ellipsoidal blob) is deflecting many protons away from the flux rope, causing a lack of information in the resulting radiographs. Bottom panel: Using radiographs generated by proton beams in the $z$ and $x$ directions as training data, the mean errors are $1.81\\times 10^{-4}\\%$ and $0.11\\%$ while the median errors are 1.41$\\times 10^{-4}\\%$ and $8.68\\times 10^{-2}\\%$ for $\\alpha_1$ and $\\alpha_2$ respectively, using an artificial neural network with 1 hidden layer consisting of 110 neurons. We see that upon including data from the proton beam in the $x$ direction, more information for both field structures is added to the data set and errors reduce by at least an order of magnitude.}\n\\label{fig:tomography}\n\\end{figure}\n\n\\subsection{Comparison with the existing radiograph inversion techniques}\n\nThe artificial neural network method addresses the two gaps in existing reconstruction techniques, by being able to work in the non-linear regime (such as the branching caustic regime), and being able to produce 3-D reconstruction of the magnetic field. While existing inversion techniques rely on the paraxial limit for simplicity, the artificial neural network technique does not rely on such a limit, and in fact would benefit more if the paraxial limit was not used--The protons should ideally have non-zero velocities in the $x$ and $y$ directions so that it will be deflected by $\\mathbf{B_\\textit{z}}$, allowing the artificial neural network to capture more information and thus reconstruct the magnetic fields more accurately. \n\nAlso, existing techniques assume that the protons move in a straight line within the plasma, but this assumption does not hold when the $\\mathbf{B}$ field is so strong that deflection occurs within the plasma. As a result, the existing techniques will inevitably fail in the limit of extremely large $\\mathbf{B}$ fields. In comparison, the neural network method will work because it does not require this assumption. \n\nOne major benefit of using artificial neural networks is the long-run computational cost savings. Generating each set of 50 by 50 pixel radiographs (one radiograph for each variation of parameters) takes on the order of hours\/days using 16 cores on one node of the Arcus Phase B supercomputer \\cite{ARC}. Training the artificial neural network takes on the order of minutes\/hours when using a single GPU on the Arcus Phase B, for neural networks with up to 10 hidden layers, with each layer consisting of up to 150 neurons. Reconstruction takes on the order of seconds without using any parallel processing\/GPU. If this project were to go full-scale, we can see that most of the computational cost is in the generation of training data and the training of the artificial neural network, which is a one-off cost. In comparison, existing methods of reconstruction have a recurring cost. As such, over the long run, if the artificial neural network is used to invert sufficiently many radiographs, the artificial neural network method is computationally more efficient.\n\nHowever, the artificial neural network method has some drawbacks. For example, the overall accuracy of the artificial neural network can only be determined empirically, whereas error-propagation can be performed for existing techniques. While the artificial neural network method will allow for computational cost savings over the long run, the minimal start-up computational cost to get it working for a non-trivial field structure is quite high, because the artificial neural network must be trained with many basis fields before it can be used. This is in contrast to existing techniques, where any field, as long as the assumptions are met, can be imaged with the computational cost of solving a differential equation.\n\n\\section{Conclusions and future work}\n\\label{sec:future}\n\nIn conclusion, we have reviewed existing techniques on analyzing $\\mathbf{B}$ fields from proton radiography, and the basics of artificial neural networks. Using the fact that artificial neural networks are highly flexible function approximators, we proposed for the first time the idea of using artificial neural networks to reconstruct arbitrary $\\mathbf{B}$ fields and retrieve important field parameters.\n\nVia simulations, we showed that an artificial neural network can reconstruct $\\mathbf{B}$ fields that can be expressed as linear combinations of two fields, and retrieve useful quantities of $\\mathbf{B}$ fields such as characteristic lengths. We also explored the effects of noise and size of data set on the accuracy of the artificial neural network, and found that artificial neural networks are robust to noise. Artificial neural networks can accommodate a wide variety of scenarios and assumptions which existing techniques cannot, such as the branching caustics part of the non-linear regime. We also highlighted the need for proton tomography as certain field structures cannot be reconstructed fully due to the lack of information from a single radiograph.\n\nAs the usage of artificial neural networks in diagnosing $\\mathbf{B}$ fields in high energy density plasmas is new, there are many avenues where this work can be developed further. There are at least three ways to improve the accuracy of the artificial neural network: (i) experiment with other types of artificial neural network architecture. For example, convolutional neural networks are a type of feedforward artificial neural network where the connections between neurons are inspired by the animal visual cortex \\cite{convnet} and as such, perform very well in image recognition. Since radiograph inversion involves image recognition, convolutional neural networks could offer better performance than the fully connected (dense) feedforward neural network used in this paper. Recurrent neural networks are a class of artificial neural networks where the neuron connections form directed cycles, and such architecture has advantages in analyzing time series data. We could use recurrent neural networks on a time series of proton radiographs to shed light on the dynamics of the $\\mathbf{B}$ field and hence the plasma. (ii) Include energy-resolved radiographs. In our simulations, we only looked at the spatial distribution of the protons, so including extra information on the proton energies could improve accuracy. (iii) Study the effects of the number of pixels on accuracy.  It is interesting to note that promising results were obtained despite the low resolution of the radiographs (50 by 50 pixels). Understanding the effects of discretization noise could help us determine the quality of radiographs to be generated in order to train an artificial neural network to a specific accuracy.\n\nFurthermore, the artificial neural network approach can be extended to similar systems, such as diagnosing electric fields in plasmas or characterizing micromagnetic patterns in magnetic media via electron scattering \\cite{micromag}. Finally, a full scale implementation of an artificial neural network that can reconstruct any $\\mathbf{B}$ field is a possibility we can look forward to.\n\n\\begin{acknowledgements}\n\nThe authors would like to acknowledge the support from the plasma physics HEC Consortium EPSRC grant number EP\/L000237\/1, as well as the Hartree Centre, Daresbury Laboratory, Central Laser Facility,  and the Scientific Computing Department at the Rutherford Appleton Laboratory. The authors would like to acknowledge the use of the University of Oxford Advanced Research Computing (ARC) facility in carrying out this work \\cite{ARC}. N.C. acknowledges financial support from the Singapore government. M.F.K. gratefully thanks the Indonesian Endowment Fund for Education for its support. M.C.L. thanks the Royal Society Newton International Fellowship for support. The authors acknowledge support from OxCHEDS and P.A.N. for his William Penney Fellowship with AWE plc. \n\n\\end{acknowledgements}\n\n\n\n\\subsection{Reconstruction of an arbitrary $\\mathbf{B}$ field}\n\\label{subsec:recon}\n\n\\begin{figure*}\n\\includegraphics[width=1\\textwidth]{images\/process.pdf}\n\\caption{Schematic of the prediction and reconstruction process.}\n\\label{fig:predictreconstructprocess}\n\\end{figure*}\n\nIn this section, we outline the steps to using an artificial neural network to reconstruct any arbitrary $\\mathbf{B}$ field.  First, we expand the magnetic field \\textbf{B(r)} as a linear combination\n\n\\begin{eqnarray}\n\\mathbf{B(r)} =&& \\alpha_1 \\mathbf{B_1}(\\mathbf{\\frac{r-r_1}{\\sigma_1}}) + \\alpha_2 \\mathbf{B_2}(\\mathbf{\\frac{r-r_2}{\\sigma_2}})  + \\ldots  \\\\\n=&&\\sum_{n=1}^{N}  \\alpha_n \\mathbf{B_\\textit{n}}(\\mathbf{\\frac{r-r_\\textit{n}}{\\sigma_\\textit{n}}}),\n\\label{eqn:expansion}\n\\end{eqnarray}\n\n\\noindent where $N$ is the number of terms used in the expansion, $\\alpha_n$ is a scalar coefficient for the $n^\\textrm{th}$ term, $\\mathbf{B_\\textit{n}}$ is a `basis' magnetic field, $\\mathbf{r_\\textit{n}}$ is the position offset of the field and $\\sigma_\\textit{n}$ is a scaling factor. While not necessary, these basis fields should be chosen such that most magnetic fields in plasmas can be represented with as few basis fields as possible, so that we require less training data to train the artificial neural network. One possible way to achieve this is to use principal components analysis (PCA) \\cite{pca} on a large dataset of known $\\mathbf{B}$ fields in plasmas. Principal components analysis looks at a large set of multidimensional vectors and first finds the direction of highest variance in the data (the first principal component), and then finds a set of vectors orthogonal to the first principal component that explains the remainder of the variance. While $\\mathbf{B(r)}$ is a vector field, it can be converted into a vector $\\mathbf{c}$ for principal components analysis by concatenating the magnetic fields at various different points, e.g. for a grid that runs from 0-9 in the $x$, $y$ and $z$ directions, we can write\n\n\\begin{eqnarray}\n\\mathbf{c} = \n\\begin{bmatrix}\n\\mathbf{B(r_{000})} \\\\ \\mathbf{B(r_{001})} \\\\ \\vdots \\\\ \\mathbf{B(r_{999})}\n\\end{bmatrix},\n\\label{eqn:vectorc}\n\\end{eqnarray}\n\n\\noindent where $\\mathbf{r_\\textit{xyz}}$ is the vector ($x, y, z$). If such convenient basis fields cannot be determined, we can use the fact that all magnetic fields can be written in the form of equation (\\ref{eqn:vectorc}), and let each element of the vector correspond to a basis field (i.e. $\\mathbf{B_1}$ corresponds to $\\mathbf{B_x(r_{000})}$, $\\mathbf{B_2}$ corresponds to $\\mathbf{B_y(r_{000})}$ and so on).\n\nNext, generate training data by creating variations of the parameters $\\alpha_n, \\sigma_n \\text{ and } \\mathbf{r_\\textit{n}}$ in the form\n\n\\begin{eqnarray}\n\\mathbf{g} = \n\\begin{bmatrix}\n\\alpha_1 \\\\ \\mathbf{r_1} \\\\ \\sigma_1 \\\\ \\vdots \\\\  \\alpha_N \\\\ \\mathbf{r_\\textit{N}} \\\\ \\sigma_N\n\\end{bmatrix},\n\\label{eqn:parameters}\n\\end{eqnarray}\n\n\\noindent and then conducting numerical simulations (e.g. using software packages mentioned in \\cite{levy,compare}) to obtain the radiograph for each variation of the parameters. The radiograph is expressed as a vector where each element represents the intensity of the protons at a specific pixel.\n\nThen, using the radiograph pixel values as inputs and $\\mathbf{g}$ as the targets, apply backpropagation and optimization algorithms to train the feedforward neural network. After training, the artificial neural network is ready to reconstruct $\\mathbf{B}$ fields: Input the radiograph into the artificial neural network to obtain the predicted parameters (in the form of equation (\\ref{eqn:parameters})), and insert these values into equation (\\ref{eqn:expansion}). See Fig. \\ref{fig:predictreconstructprocess} for a schematic of the training and reconstruction process.\n\n\\subsection{Assumptions, practical considerations and implementation}\n\nIn our simulations, we have made some assumptions for simplicity, but these assumptions are not crucial in the success of our approach. We assumed that the probe beam only interacts with the plasma via the $\\mathbf{B}$ field (no electric fields or collisions with matter). We also assumed that the proton source is a point source, and the probe beam is a planar sheet (velocities in the $z$ direction, before deflection from the plasma, are uniform). As feedforward artificial neural networks are universal function approximators, in principle the technique outlined in the previous section will still work even if the assumptions are violated (e.g. protons interact with the electric field of the plasma, protons collide with the plasma, proton source is of finite size, probe beam follows a specific angular distribution), as long as we include these effects during the production of training data (radiographs). \n\nTo obtain the radiographs we start off with the Lorentz force equation for $\\mathbf{B}$ fields only, given by\n\n\\begin{eqnarray}\n\\frac{\\mathrm{d}\\mathbf{v}}{\\mathrm{d}t}=\\frac{e}{m_p}\\mathbf{v}\\times\\mathbf{B}.\n\\end{eqnarray}\n\n\\noindent This equation, along with $\\frac{\\mathrm{d}\\mathbf{r}}{\\mathrm{d}t}=\\mathbf{v}$ was numerically integrated given the initial conditions of $\\mathbf{r}$ and $\\mathbf{v}$ to get the final positions of the protons on the screen. These final coordinates are then binned in order to produce radiographs.\n\n\nWe used a fully connected (dense) feedforward artificial neural network for simplicity, and we will discuss the possibilities of using other types of artificial neural networks in section \\ref{sec:future}. Scaled conjugate gradient was the optimization algorithm of choice during training because: (i) it is not RAM intensive (this is an important factor because in order to get more accurate results, training with more complicated $\\mathbf{B}$ fields and higher resolution radiographs are required, resulting in an increase in the number of weights in the artificial neural network. If the optimization algorithm does not scale well, an impractical amount of RAM will be required); (ii) it can take advantage of parallel CPU and GPU computing, allowing it to run effectively on supercomputing clusters.\n\nBefore the training process, the entire data set is scaled so that each feature of the input and target (e.g. $\\sigma_1$, $\\alpha_1$, the proton intensity in pixel 1 etc.) falls in the range [-1,1] to prevent features of small magnitude from converging slowly during optimization \\cite{efficientbackprop}, and the scaling is undone afterwards. The objective function was chosen to be the mean squared error (MSE) between the artificial neural network output and the target. \n\nDue to the flexibility of artificial neural networks, overfitting (accidental modeling of noise) is an issue so early stopping and neural network regularization are implemented. In early stopping, training is halted when the errors starts increasing on a data set that was not used in the training process \\cite{earlystopping}. This is done by first splitting the entire simulated data set into training, validation and testing sets at random in the ratio 70\/15\/15. The artificial neural network is applied to the training set, and during each iteration the mean squared error for the validation set is calculated. Initially, after each iteration, the artificial neural network becomes better at modeling the physical phenomenon and the validation mean squared error will decrease. There will come to a point when the artificial neural network starts to model the noise in the training set, and the validation mean squared error will stop decreasing and eventually start increasing (See Fig. \\ref{fig:tvt} for an illustration).  Training is halted after a specified number of iterations fail to decrease the validation mean squared error. In neural network regularization \\cite{weightdecay}, the objective function is modified by adding a term proportional to the mean squared weight, and the constant of proportionality is known as the regularization parameter (chosen via cross-validation). This penalizes the neural network for having large weights or too many neurons, thus encouraging simpler models. \n\nAt this point, the artificial neural network is used to predict quantities on the testing set, and the testing errors are indicative of the artificial neural network's overall performance. For each simulated data set, the training process is run with the number of neurons in the hidden layer ranging from 10 to 100 in steps of 10, and the configuration with the lowest value of the objective function (mean squared error plus regularization term) is initially picked. If the value of the objective function is still decreasing when 100 neurons are used, then the search is extended in steps of 10 till 150 neurons. Once a single layer configuration is picked, another search is performed with multiple hidden layers (in increments of one layer), up to 5 hidden layers. Similarly, if the value of the objective function is still decreasing when 5 hidden layers are used, the search is extended to 10 hidden layers. After this search, the configuration with the lowest value of the objective function is picked and reported in the results section.\n\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{images\/tvt.pdf}\n\\caption{Typical curves of training and validation errors with respect to training iteration. Beyond a certain point, the artificial neural network starts to model noise, causing the validation error to increase. Training should be halted when this happens.}\n\\label{fig:tvt}\n\\end{figure}\n\n\\subsection{Retrieval of specific parameters}\n\\label{subsec:retrieval}\n\nThe idea presented in section \\ref{subsec:recon} requires large amounts of data and processing power, and might be more than necessary if the user only intends to retrieve certain parameters of the $\\mathbf{B}$ field, such as the peak field strength or the full width half max (FWHM) of a Gaussian magnetic flux rope, instead of reconstructing the entire field. This assumes that the user already knows the remainder of the parameters beforehand. For example, if the user only wants to retrieve the peak $\\mathbf{B}$ field strength (proportional to the $\\alpha_i$ coefficient), then the model of the $\\mathbf{B}$ field, the offset $\\mathbf{r_i}$ and the scaling factor $\\sigma_i$ must be known. In this case, the user can repeat the procedure in section \\ref{subsec:recon}, except with the following changes: (i) data is generated by varying only the parameter(s) of interest; (ii) the target vector consists of only the parameter(s) of interest. In fact, this can be applied to parameters other than $\\alpha_i, \\mathbf{r_i}$ and $\\sigma_i$. For example, in an ellipsoidal magnetic blob (which is a spheroid), there are two characteristic lengths, one characterizing the length in the $xy$ plane $a$ and the other characterizing the length along the $z$-axis $b$. If the user knows all other parameters  and wants to retrieve $a$ and $b$, then an artificial neural network trained on simulated radiographs which variation is only due to varying values of $a$ and $b$ will do the job.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn human history, violence is continuously with us, despite our optimistic belief \nthat it is less and less widespread. In our minds, violence of armed against disarmed \npeople is particularly repulsive. However, still it happens in numerous places on earth. \nPerhaps a new element is that all of us are more conscious of the situation, than ever \nbefore. The question is if victims of the violence - treated as a community - can accept \nit. If yes, the situation will remain stable; if not, they will resist, and the violence \nis expected to spread. This duality - to resist or not - is especially inevitable in a ghetto,\nwhere an escape is not possible or at least very difficult. Here we are going to attack \nthis problem by sociophysical methods, i.e. by a construction of an appropriate model.\\\\\n\nAs a model base, we propose two elements. First is the Maslow theory \\cite{maslow}. \nIts basic assumption is \nthat people are going to satisfy their needs one after another, in order from most \nbasic to most sophisticated. Our central question - the decision of the victims about \nresistance - is to be considered in the context of their subsequent decisions how to \nsatisfy their needs. In other words, each vital decision in the situation of violence \nis to be made in relation to this violence. Second element of the model is the mean field\napproach, as applied to a strike by Galam, Gefen and Shapir \\cite{galam}. This \napproach profits from the analogy to ferro-paramagnetic transition in the Ising model. In this\ndescription, the ferromagnetic phase with given orientation of spins is equivalent to\na given decision (as to take part in a strike or not), made by the majority as a consequence\nof social interactions. \\\\\n\nThis author imagines that the goal of this paper is twofold. First is a reconstruction \nof chains of subsequent decisions of people. We apply the decision tree - a concept from \nthe game theory \\cite{straffin}. This concept is modified here in the sense that \nthere is only one player. Still, each decision selects a branch, and getting there \ndefines a new situation. In some cases, estimation of expected payoffs allows to \napply the pruning technique: as the player decides as to get larger payoff, some branches\nwith very low payoff can be {\\it a priori} eliminated. The \nsecond goal is to use the obtained scheme to discuss the problem of resistance in a ghetto.  \\\\\n\nHistorically, \"ghetto\" is the area of iron foundry in Venice, where an enclosed \nneighborhood was created for Jews in 1516 to protect them against persecution from \nRoman Catholic Church. More recently, the term 'ghetto' is explicitly assigned to \nbounded areas in Warsaw, Lodz, Riga, Budapest, Cluj, Terezin and many others under \nNazi rules, where Jewish people were gathered prior to the Holocaust. In social \nsciences, the meaning of the term includes also Jewish diaspora in early modern \nEurope, quarters of black Americans in large cities and some ethnic communities \nin Africa and East Asia \\cite{smelser}. Although this meaning remains under \ndispute \\cite{smelser}, attempts to describe ghettos in current world should at \nleast be remarked \\cite{hannerz, ron}, not pretending to completeness. Social\nprocesses leading to the formation of ghettos was simulated by \\cite{lling} and \nmore recently by \\cite{meyort,sch}. The present work concerns with dynamics of\ndecisions in a ghetto community. For our \npurposes, two traits are to be distinguished: {\\it i)} an attempt of an inhabitant of \nghetto to leave the area makes his situation worse, {\\it ii)} human laws, as understood \nby inhabitants, are broken by an external power. This wide definition applies to \nrefugee camps as well as settlements in countries controlled by army, as in \nPalestine, Tibet, Chechnya and Darfur. Although in most cases ghettos are inhabited \nby ethnic minorities, here we do not need to emphasize this ethnic trait.\\\\\n\nTo refer to the game theory, below we adopt the abbreviations $C$ and $D$ (cooperate or \ndefect), although maybe withdraw or resist would be more appropriate. In two subsequent \nsections we introduce the model and we apply it to the case of ghetto. Last section is \ndevoted to discussion.\n\n\\section{The model}\n\\subsection{Hierarchy of human needs}\nAccording to the theory of Maslow, human needs are arranged hierarchically, from physiological \nneeds, safety, belongingness, to esteem and self-actualization \\cite{maslow}. People \nbecome interested in their safety to the extent of which their physiological \nneeds are satisfied; being safe, they start to struggle for belongingness, and so on. \nThis author and maybe this reader happened to be born in a milieu \nwhere three first needs were satisfied from the very beginning till adulthood. However, \nin numerous cases the situation is less fortuitous. More than often, a human unit has \nto determine a strategy to realize his\/her most basic needs in this or that way. In \nsuch a strategy, one of most important decision is which limitations of human needs \nare to be accepted \\cite{kepinski}. This problem appears to be even more crucial in \nghetto, where the above mentioned limitations are particularly painful.\\\\\n\nTrying to reach its needs in any milieu, a human unit has to consider at each step \nthe context of situation. In particular, in ghetto the problem of any action is if \nit is legal, or - in other words - if this action is allowed by the ruling power. This \nremains true when we ask about actions taken up in order to satisfy human needs \nat all levels. At the physiological level, to cooperate is  equivalent to join common \nlife in frames of the society, using money, sleeping home and eating food bought in \na market. An alternative is to look for a desolate place in a forest or a desert, or \nto form a small community out of or at least at the border of, say, normal civilization. \nTo continue, at the safety level the problem is to accept law or not. Whereas in our \nworld of white collars this alternative is concentrated around payment of taxes, in \nghetto the defection can include uprising, riots, guerilla or terror. At the level of \nbelongingness, \nwe have to select our group of reference. Again, in ghetto the world is sharply divided \ninto two: \"we\" and \"they\". The power is with \"them\", and the quest is to identify with \nwhom? Further, at the level of esteem the problem is, in which group this esteem is \nlooked for? Here we guess that this choice is strongly correlated with the previous \none, and our analysis will be simpler because of this correlation. Finally, reaching \nesteem, our human unit tends to self-actualization. In principle, this again can be \nexpressed as a social action directed against the power or supporting it. However, \nconsequences of these decisions are usually less crucial, human behavior at this \nlevel is much more individualized and it is often affective and expressive rather \nthan aim-oriented \\cite{weber}. That is why the level of self-actualization is \ndifficult for a sociophysical modelling.\n\n\\subsection{Tree of decisions}\nThose who defect at the very root, i.e. at the physiological level, place themselves \nout of frames of the society. It is very hard to defect public access to water, shops \nand houses. The alternative is to live wildly in the forest. Yet this choice happens \nin several places on Earth, where climatic circumstances allow to do it at least \ntemporarily and some strong obstacles prevent to live in accord with law.  This \nkind of defection happens in large social scale only in societies in strongest crisis, \ne.g. during a civil war. Although these situations are of central importance, they \nwill be not discussed here. In a ghetto, there is no possibility to fight in open \nway; main splitting of human behavior happens at higher levels. Then, for the sake \nof our subject all decisions discussed here start from $C$ (cooperate).\\\\\n\nIn the same way we are going to comment further decisions, which form chains as the \none presented in Table 1. This particular chain will be denoted as $CCDDD$ from now on. \nIn this notation, $CD$ means that we discuss the decision $D$ (at the safety level) of \nthose who decided $C$ at the physiological level. \\\\\n\\bigskip\n\n\\begin{tabular}{|l|l|}\n\\hline\nphysiology&C\\\\\nsafety&C\\\\\nbelongingness&D\\\\\nesteem&D\\\\\nself-actualization&D\\\\\n\\hline\n\\end{tabular}\n\n\\bigskip\n\nTable 1. A chain of subsequent decisions of a human unit (man or woman): \ndefect ($D$) or cooperate ($C$) with the power.  \n\n\\bigskip\nIn Fig.1 a part of the resulting tree of decisions is shown. There, the whole branch \nstarting from $D$ at the physiological level is omitted, except its beginning. Also,\nthe decisions $D$ or $C$ at the level of self-actualization are not shown for clarity\nof the figure. \\\\\n\nOmitting the physiological needs, we are going to concentrate on the level of safety.\nA population considered selected $C$ as their first choice, i.e. they decided to live\nwithin the community and to profit its facilities. Now and each time their decision \nis $C$ or $D$, i.e. they wonder if their path is to be $CC$ or $CD$. The probabilities of these\npaths depend on the expected payoffs. Then the choice these people is to decide, if they \nwill be safer when cooperating with the power of when defecting it.\\\\\n\n\n\\begin{figure} \n\\vspace{0.3cm} \n{\\par\\centering \\resizebox*{10cm}{8cm}{\\rotatebox{0}{\\includegraphics{drzewo.eps}}} \\par} \n\\vspace{0.3cm} \n\\caption{Right half of the tree of decisions. Last level (self-actualization) is not shown.\nBeing at the root and selecting $C$, one is placed at node $C$; selecting $D$ as next level one \nis placed at node $CD$ and so on. Then, nodes at the physiology level are indexed with one \nlabel, $D$ and $C$ from left to right, nodes at the safety level - by two labels ($DD$ or \n$CD$ not shown, or $CD$ or $CC$ shown from left to right), and so on. At the esteem level,\nfirst node from the left is indexed as $CDDD$.}  \n\\end{figure}\n\nAs it was indicated by Maslow, people are able to struggle for their safety\nto the extent in which their physiological needs are satisfied. Further, their search for \nbelongingness is limited by their lack of safety, and so on. Maslow gives an example \nwith numbers; an average citizen could have satisfied his successive needs in 85, 70,\n50, 40 and 10 percent, in order as in Table 1. Provided, for example, that the safety needs \nof somebody are not satisfied at all, he will not bother about belongingness, not to speak about \nesteem or self-actualization. It is not clear how the effort for a need of next level \ndepends on the satisfaction of a need in a previous level; the Maslow theory is \nformulated in words, not in numbers. The area is open for speculations, with the only condition \nthat any proposed mathematical formulation reflects the above mentioned rules. On the other hand \nit is obvious that the validity\nof any numbers we can get is limited to statistical considerations. It seems that\nfor this kind of problems, the fundamental equations \\cite{vank} can provide a proper tool. \\\\\n\n\\subsection{Mathematical formulation}\n\nFrom these equations,\nwe expect to obtain the probabilities that people in a given situation (read: at a given\nnode of the tree) select this or that decision. External conditions met by a given community can be\nintroduced as the set of payoffs $\\alpha_X$, describing maximal possible percentage of satisfying\nneeds at node $X$ of the decision tree. As explained in the caption to Fig. 1, the node index $X$ \nis equivalent to a chain of decisions, leading to that node.\nThe root is treated as the chain of zeroth length. Simultaneously, $\\alpha_X$ is \nthe maximal amount of people who are able to struggle for satisfaction of higher needs at nodes\n$XC$ and $XD$. Both these 'maximal' deal with a virtual case when the payoff is limited neither by \nparameters of previous nodes, nor by human decisions at these nodes.\\\\\n\nKeeping the above example of the chain $CCDDD$ as an individual path, \nthe value of satisfaction $s_X$ of a human unit - member of the \ncommunity - at node $C$ (at the physiological level) is then $s_C=\\alpha_C$. At higher level $s_X$\nfulfils an iterative equation\n\n\\begin{equation}\ns_{Y}=s_X \\alpha_{Y},\n\\end{equation}\nfor $Y=XC,XD$. Provided, that the set $s_X$ of satisfaction of our human unit accords with \nthe above exemplary values given by Maslow, we obtain at five successive nodes of the path \n$\\alpha_C=0.85$, $\\alpha_{CC}=0.7\/0.85\\approx 0.82$, $\\alpha_{CCD}=0.5\/0.7\\approx 0.71$, \n$\\alpha_{CCDD}=0.4\/0.5=0.8$ and $\\alpha_{CCDDD}=0.1\/0.4=0.25$. These values of $\\alpha_X$\nallow to reproduce {\\it via} Eq. 1 the above given exemplary chain of individual satisfactions: \n$s_C=0.85$, $s_{CC}=0.7$, $s_{CCD}=0.5$, $s_{CCDD}=0.4$ and $s_{CCDDD}=0.1$. \\\\  \n\n\nThe above vital path consists successive decisions, for our example $CCDDD$, \nas in Table 1. In reality, these decisions are much more detailed than a cooperation\nwith or a defection the ruling power. Actually, the decision can be as specific as to marry\none particular member of a group of revolutionists - or not to marry. However, having defined our \nissue - to withdraw or to resist, we are interested not as in a decision of selecting a detailed person, but - averaging out\nover different possibilities - in a decision to be involved in a revolutionistic group, which\nis equivalent to satisfying some needs by the choice $D$. \\\\\n\nUp to now, we dealt with individual path. Now we can introduce conditional probabilities\n$p(C\\mid X)=p(XC)\/p(X)$ and $p(D\\mid X)=p(XD)\/p(X)$ that, leaving node $X$, a human unit is going \nto $C$ or $D$.\nIn this case the normalization condition should be $p(C\\mid X)+p(D\\mid X)+1-\\alpha_X=1$. This \nshould be not \nconfused with a probability that a human unit will stay at node X with probability $1-\\alpha_X$.\nSuch a formulation would disagree with the original interpretation of Maslow. Instead, the factor\n$1-\\alpha_X$ measures the amount of effort spent inefficiently at node $X$, in the same way as \nit was\nassumed for an individual path. In the latter case, either $p(C\\mid X)=0$ or $p(D\\mid X)=0$, \nand the path\nwas fully determined by subsequent individual decisions. Then, individual satisfaction at \nsubsequent levels depends only on the payoffs $\\alpha_X$, as explained in the second paragraph \nof this section. Instead of using the conditional probabilities, it is simpler to use individual \neffort $w_X$ and averaged effort $W_X$. In the above example of individual path, $CCDDD$, the \nset of individual efforts is: $w_C=1$ at the root, $w_{CC}=1$ at the node $C$, $w_{CCD}=1$ at\nthe node $CC$, $w_{CCDD}=1$ at the node $CCD$ and $w_{CCDDD}=1$ at the node $CCDD$. Other efforts\nare zero, either along the decision (as $w_{CD}$) or because a given node was not reached\nby a given human unit (as $w_{DD})$. \\\\ \n \n\nAveraging over individual paths, we get a set of average amounts of effort $W_X$ \nat all nodes $X$. Then for the physiological level we have $W_C+W_D=1$. The average satisfactions\nat the physiological level, $X=C,D$, are $S_C=W_C\\alpha_C$, and $S_D=W_D\\alpha_D$.  Considering \nthe safety level we take into account that efforts to reach the nodes $CC$ and $CD$ \nare reduced because $\\alpha_C\\le 1$. Then, $W_{CC}+W_{CD}=W_C\\alpha_C$, and similarly \n$W_{DC}+W_{DD}=W_D\\alpha_D$. As a rule,\n  \n\\begin{equation}\nW_{XC}+W_{XD}=W_X\\alpha_X,\n\\end{equation}\nwhere $XC$ and $XD$ are nodes available from node $X$ by decision $C$ or $D$.\nThe whole set $W_X$ is equivalent to a map of social efforts, put into various ways of attempts\nof satisfying the needs. At each node, the average satisfaction $S_X=W_X\\alpha_X\\le W_X$.\nSatisfaction is less or equal than effort, for individual paths as well as in the average.\\\\\n\n\nIn a deterministic picture, people are expected to select always the nodes with larger payoff. \nHowever, it is clear even for a physicist that in reality people have their individual preferences,\nand a common payoff for everybody can be introduced only for a statistical description. This\nintuition on individual character of payoffs is confirmed by the utility theory \\cite{straffin}.\nWorking in statistical physics, we are tempted to use some noise as a measure of, say, lack\nof information of the community members. Then we expect that the ratio $W_{XC}\/W_{XD}$ in stationary \nstate depends on $\\beta(\\alpha_{XC}-\\alpha_{XD})$, where $\\beta=0$ for absolute lack of information\non the payoffs, and $\\beta$ is large when the information is well accessible. From this point, it \nis only one step to mimic the statistical mechanics, writing the stationary probability of \nselecting $C$ from node $X$\n\n\\begin{equation}\np(XC)_{eq}=\\frac{W_{XC}}{\\alpha_X W_X}\\propto \\exp{[\\beta(\\alpha_{XC}-\\alpha_{XD})]},\n\\end{equation}\nand to postulate a dynamic description in the form of fundamental (or Master) equation\n\n\\begin{equation}\n\\frac{dp(XC,t)}{dt}=-r(XC)p(XC,t)+r(XD)[1-p(XC,t)],\n\\end{equation}\nwhere $r(XC)\\propto p(XC)_{eq}$. Here, the constant of proportionality determines the \ntimescale of the dynamics. The dynamics of the probabilities $p(X)$ is equivalent to the dynamics \nof efforts $W_X$.\\\\\n\nIn, say, a standard society the information on the payoffs is well accessible and the \nsuccessive selections are almost deterministic. Then, people who decide to live in a wild forest \nare rare exceptions in the society: almost everybody selects $C$ at the physiological level.\nIt is less clear if the payoffs for those who break law are indeed smaller than for the others.\nIn any case, a great effort is paid to ensure the population that sooner or later this\npayoff will be strongly reduced. Because of this effort, the statistical data on the choice \nof $CD$ are usually less sure. Looking for belongingness and needs of higher order is not directly \nconnected with our issue; anyway, in democratic systems we are partially involved\ninto the ruling power, which cannot then be treated as external and is maybe not entirely against \nour will. Summarizing this section, this author believes that the concept of the decision tree,\nas an adaptation of the Maslow hierarchy, can be useful in many issues.\n\n\\section{The case of ghetto}\n\nAs it is expected to be clear from the definition of ghetto, accepted above, the key point\nof the decision tree is the node $C$, where crucial decision is to be taken: $CC$ or $CD$. The reason \nis as follows. All what we know about ghetto confirms that there, it is almost impossible to \nsatisfy the safety need. The payoff of a useful solution $CC$ is drastically reduced with respect\nto other communities. Examples of this painful truth fill newspapers and TV or, even worse,\nremain unknown if information is prohibited. To bring these examples here, athough justified\nfrom the point of view of the subject, would drive us too far from sociophysics. Instead, let us \nconsider the consequences for the payoff.\\\\\n\nImagine that the safety is strongly reduced in an initially normal society. The reason can be \nwar or revolution, or other abrupt fall of the political system.  It is clear that the accessibility\nof information deteriorates, and in this situation many people do not know what to do. What\nis the payoff if I withdraw? if I resist? who will win? what will be the consequences for me?\nmy family? my assets? and so on. As a rule, a remarkable percentage of people resist, just\nbecause - a physicist would say - large entropy in the system. This thermodynamic \nformulation should not be offending to anyone. Obviously, it does not comprise individual \ndecisions, which are sometimes dramatic and full of unanswered questions. It is a common \nexperience that we\ndecide, not knowing the final results; in most difficult situations, the amount of information\nis too low to allow for a logical reasoning. This experience is encoded in sociology as the law \nof unforeseen consequences \\cite{oxf}. However, here we consider the case when finally \nsome power, external for the ghetto inhabitants, prevails and the information on payoffs becomes \nmore clear. But the above mentioned \ngroup keeps resisting; despite the variety of their motivations, their effort can be translated \ninto numbers and handled by statistical tools.\n They fight against the external power and its supporters - the mechanism\nknown too well, indeed. Relaxing to the stationary state, the system finds that the payoff\nof the choice $CC$ is reduced by an expected repression by the resisting group. The ruling\npower tries to balance this repression by defeating the resistance fighters. Soon, the level of\naggression of both sides becomes equivalent; both find convenient justifications. \\\\\n\nThis author believes that what can be said mathematically, can be said - although longer - in words.\nHere we try the opposite way. Violence bears violence - this sentence is short. In sociophysical\nlanguage, the same content can be expressed as a stability of the solution of Eq. 4, characterized\nby the condition $\\alpha_{CC}<\\alpha_{CD}$. This stability relies on the following premises: \n{\\it i)} the payoff $\\alpha_{CC}$ is drastically lowered by the repressive actions of the resistant\ngroup, {\\it ii)} struggling for their safety, people are not motivated to select $CC$ instead of \n$CD$, if $\\alpha_{CC}<\\alpha_{CD}$, {\\it iii)} selection of $CD$ in a social scale reinforces\nthe resistant group. As we see, this closed circle does not rely on a particular choice of the\nfunctional dependence of the effort $w$ on the payoff $\\alpha$. In fact, the resistant group\ncan be compared to a nucleation center, which initiates the new phase. However, the nucleation\nprocess cannot be described within the simplest version of the mean field theory, used here.\\\\\n\nAs a result, the whole tree becomes degenerated. For those who decided to resist, it is not possible \nto look for belongingness or esteem out of the resistant group. On the other hand, those who \nselect $CC$ remain under fear of, from one side, being accused of treason and, from the other side, \nblind actions of the ruling power. Not being able to get safety, they follow\nthe solidarity with the resistant group, whey they look for belongingness and esteem. As a rule, \nwhen the safety is at risk, no effort can be put to struggle for higher needs. In effect,\nupper branches disappear. \\\\\n\nTrying to illustrate the above processes with some simulations, we need the values of several \nparameters, as the payoffs at the nodes etc. Measurement of these parameters or at least a thorough\ndiscussion of their values far excesses the frames of this work - in social sciences, this is \nalmost an euphemism. Instead, we can present\na qualitative consequences of an abrupt change of ruling power. The event is a special and \nmost simple example of what was discussed before. We limit the calculations to the safety level. \nIn the formalism developed above, the dynamics of this level does not depend on the parameters \non higher levels. In the calculations, the difference of the payoffs consists two factors:\nexternal $\\Delta$ (provided by the ruling power, old or new) and internal, due to interaction \nbetween the community members. The latter is proportional to the actual difference of efforts,\n$W_{CD}-W_{CC}$. This proportionality encodes the above discussed positive feedback between\nthe value of the difference of efforts and its time derivative. In fact, this positive feedback \nis at the core of the mean field theory \\cite{binder}.\\\\\n\nTo simulate the change of the ruling power (for example, from a well-established to an \nexternal), two agents cannot be omitted: a strong decrease of $\\alpha_C$, which is a direct \nconsequence of unavoidable war, and a change of sign of $\\Delta$. Simultaneously, the up-to-now\ncooperators become defectors and the opposite. We keep the node $D$ unoccupied ($W_C=1$ and $W_D=0$); \nthis reflects\nthe assumed fact that nobody can leave the ghetto. For simplicity we keep the parameter $\\beta$\nconstant in time, although it is almost surely not realistic; still we are left with three parameters.\nThe value of $\\alpha_C$ before the political overthrow is assumed to be unity. Its value after \nthe overthrow, kept constant in time, is one of the parameters. The remaining parameters are\n$\\beta$ and $\\Delta$. Initial ratio of the variables $W_{CC}$ and $W_{CD}$ is taken as their ratio \nat equilibrium before the overthrow. As a rule, $W_{CD}>W_{CC}$ at initial time, because most \npeople supported the ancient regime before the overthrow; what was the cooperation, now is treated \nas defection, and the opposite.\n\n\n\\begin{figure} \n\\vspace{0.3cm} \n{\\par\\centering \\resizebox*{10cm}{8cm}{\\rotatebox{270}{\\includegraphics{resistance.eps}}} \\par} \n\\vspace{0.3cm} \n\\caption{The effort $W_{CD}$ put at the resistance at the safety level against the parameter\n$\\alpha_C$.}  \n\\end{figure} \n\nIn Fig. 1 we show the effort $W_{CD}$ put at the resistance, against the satisfaction $\\alpha_C$\nof the physiological needs at node C. These data are for the stationary state. As remarked above, \nwe assume that all the social effort\nat the root is put to satisfy physiological needs within the community. However, these needs can\nbe satisfied only partially. The parameter $\\alpha_C$ measures the level of this satisfaction. \nFurther, it measures also the effort which can be put to struggle for safety, in this ($CC$) or that\n($CD$) way. The parameters for the plot are: $\\beta=3.0$, and $\\Delta=0.2$.  \\\\\n\nAs we see in Fig. 1, there is a jump of the data on $W_{CD}$ near $\\alpha_C=0.7$. Below this value, \nthe effort put to resistance is negligible. Above this value, it is close to its maximal value \n$\\alpha_C$. This means that the initial state of resistance is stable. The results are typical,\nas long as $\\beta$ is not too small, and $\\Delta$ is not too large. Within the magnetic analogy,\nthe results mean that the metastable phase is possible as long as the field ($\\Delta$) and the \ntemperature ($1\/\\beta$) are not too large. Within the sociophysical picture, it means that it is \nadvantageous for the ruling power to keep the whole ghetto community at the limit of starvation\ni.e. with small value of satisfaction $\\alpha_C$ of physiological needs. Then, instead of fighting, \nthey are kept in a queue for water and flour, provided by the army. Then, the best thing is to make \na movie and show it in TV news; those who get water are happy. Please do not blame this author for\nthe invention - it is known for a long time.\n\n\n\\section{Discussion}\n\nOur conclusions are to be divided in three parts. The first is sociophysical. Our mathematical \ndescription is equivalent to the mean-field theory of the ferromagnetic phase, where\ntwo stable solutions coexist \\cite{binder}. This model is well established in applications\nof physics to social sciences \\cite{galam,weidlich}. It is known that the stability of the \nferromagnetic\nphase is overestimated by mean field theory; in fact, it depends on the structure. Here\nwe are faced with the question, what is a realistic structure of a community. Much effort has been\ndone by sociologists to advance our knowledge on the subject; however, even the characteristic \nsize of social networks remains under dispute \\cite{kill,mars}. On the other hand, stability\nof ordered Ising phase at low temperatures has been found in computer simulations for most of \ninvestigated structures \\cite{holyst,tadic,makowiec,malarz}, with directed Albert-Barab\\'asi \nnetworks \\cite{sum,lima} and one-dimensional chains and related \nmodels \\cite{novot} as exceptions. Actually, time dependence of persisting opinion\nof a resistant group was discussed recently by \\cite{aydiner} on the basis of one-dimensional Ising model. \n(We note that the condition of low temperature\nis equivalent to large value of the uncertainty factor $\\beta$ in our considerations.) For social \napplications, the condition of an eternal stability of the ferromagnetic phase can be \nsubstituted by a weaker condition of appearance of long-living ordered domains. We can conclude \nthat sociophysical arguments work for this hypothesis, and not against it.\\\\\n\nThe second conclusion is aimed to be sociological. The results of our analysis indicate, that when\nan external power struggles for control of an isolated community, \nthe problem of safety remains crucial. Obvious aim of the power must be: to guarantee the safety\nfor still neutral part of the community. If this is not possible, war becomes eternal, without\nwinners. Not so rarely, the responsability for safety of isolated communities remains in hands \nof army, without\ncontrol of civil agencies or free press. This is precisely what eliminates the possibility\nof a peaceful solution; army people are trained to fight, not to bring safety.\nIn sociology, the role of safety is known for a long time: first edition of the Maslow's\nbook \\cite{maslow} appeared in 1954. The advantage of this work is to express it in more\nformalized language. One could ask if such a formulation is worthwhile. On the other hand,\nstill some powerful people seem not to recognize the validity of the conclusions of Maslow theory.\nMaybe they will be convinced by mathematics.\\\\\n\nIn my last word I declare to share the opinion that ghettos are shameful for human \ncivilisation. Nevertheless this respectable and rather common opinion, such places exist, \nas we are mercilessly informed by free media. Some people even claim, that some \nof these places are established to protect our laws to free life, where at the last level \nof the tree of decisions we can write our sociophysical papers. If this is done without\ncare about safety of the ghetto inhabitants, the way is destructive and mindless. \\\\\n\n{\\bf Acknowledgements.} The author thanks Ma{\\l}gorzata Krawczyk and Francis Tuerlinckx \nfor their kind help. Thanks are due also to Dietrich Stauffer for helpful criticism and \nreference data.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{INTRODUCTION}\n The distribution of cometary orbital elements in the Oort cloud depends on the dynamical evolution of the solar system's planetesimal disk and the environment in which the solar system formed.  Unfortunately, the vast majority of Oort cloud comets are unobservable, usually being seen only when they are perturbed onto orbits with perihelion $\\lesssim 5$ AU.  Furthermore, the orbits of visible comets have generally been modified by poorly understood non-gravitational forces \\citep{Yeomans04}.  For both reasons it is difficult to infer the properties of the Oort cloud from the statistical distribution of comet orbits.\n\\par \nThis paper describes the expected distribution of orbital elements of nearly isotropic comets (NICs). We define these to be comets that have been perturbed into the planetary region from the Oort cloud.  Theoretical models of the distribution of NICs have been constructed by others e.g., \\citet{Wiegert99}, \\citet{Levison01} and \\citet{Fouchard1, Fouchard2, Fouchard3}; the novel feature of the present study is that we focus on much larger heliocentric distances (up to 45 AU) in anticipation of deep wide-angle sky surveys that are currently under development.  We use a simple physical model that assumes a static spherical Oort cloud with comets uniformly distributed on a surface of constant energy in phase space.  Models of the formation of the Oort cloud \\citep[e.g.,][]{Dones04} suggest that this approximation is reasonable except perhaps at the smallest semi-major axis we examine, $a=5,\\!000$ AU, where the cloud is somewhat flattened.  Furthermore, we assume that these comets evolve solely under the influence of the Galactic tide and perturbations from the giant planets.  We do not consider the stochastic effects of passing stars, as they have effects qualitatively similar to the effects of the Galactic tide \\citep{Heisler87, Collins10}; see \\citet{Fouchard1, Fouchard2, Fouchard3} for a detailed comparison of the influence of these two agents on the Oort Cloud.  We follow the evolution of these comets through N-body simulations for up to 4.5 Gyr and use the results of these orbit integrations to construct a simulated comet catalog. \n\\par\nThis topic is of special interest now as the Large Synoptic Survey Telescope (LSST) will likely see many of these NICs in the outer solar system.  LSST will survey 20,000 square degrees of sky (48\\% of the sphere) about 2,000 times over 10 years, down to an $r$-band magnitude of 24.5 \\citep{LSST09}.   LSST has a flux limit 3.2 magnitudes fainter than, and more than three times the area of, the current leader in finding faint distant solar system objects --- the Palomar Distant Solar System Survey \\citep{Schwamb10}.  It is expected to find tens of thousands of trans-Neptunian objects \\citep{LSST09}; however we are not aware of predictions made specifically for objects originating in the Oort cloud.  \n\\par\n\\citet{Francis05} studied the long-period ($P > 200$ years) comet population using the Lincoln Near-Earth Asteroid Research (LINEAR) survey \\citep{Stokes00}.  Most observed long-period comets likely originated in the Oort cloud.  He found a sample of 51 long-period comets which were either discovered by LINEAR or would have been, had they not previously been discovered by another group.  Fifteen of these had perihelion distances beyond 5 AU, but none beyond 10 AU.  He used this sample to estimate properties of the Oort cloud.  He found a``suggestive\" discrepancy  between the distribution of cometary perihelion distances in the observed sample and in theoretical models \\citep{Tsujii92, Wiegert99}, but cautioned that the difference could be the result of a poor understanding of the rate at which comets brighten as they approach the Sun due to cometary activity.  LSST will address this question by observing many comets at large heliocentric radii where they are inactive (see Section \\ref{sect:disrupt}).\n\\par\n\\citet{Hills81} proposed that the apparent inner edge of the Oort cloud at around 10,000 AU is not due to a lack of comets at smaller semi-major axes, but rather because the perihelion distances of those comets evolve slowly, so they are ejected or evolve to even smaller semi-major axes due to perturbations from the outer planets before they become visible from Earth.  In contrast, comets with semi-major axis $a \\gtrsim$ 10,000 AU have their perihelion distance changed by more than the radius of Saturn's orbit in one orbital period, so they are able to jump across the dynamical barrier of the outer planets, and be seen in the inner solar system \\citep{Hills81}.  This barrier is not 100\\% leak-proof, but as is demonstrated later in the paper, one expects the number density of comets with initial semi-major axes of 10,000 AU to decline by over two orders of magnitude interior to 10 AU.  LSST should detect NICs at distances $>10\\,$-15 AU and so will enable us to estimate the population of this inner Oort cloud directly, because we will be able to see NICs outside the region of phase space from which they are excluded by the giant planets.  The properties of this cloud may contain information about the density and mass distribution in the Sun's birth cluster \\citep{Brasser12}. \n\\par\nObserving NICs far from the Sun also probes in unique ways the parts of the Oort cloud that do send comets near Earth.  Non-gravitational forces due to outgassing when the comet comes near the Sun are the primary source of error in determining the original orbits of these comets \\citep{Yeomans04}.  It is somewhat uncertain at what radius outgassing begins, but a reasonable estimate would be around 5 AU (see discussion in Section \\ref{sect:disrupt}).  Therefore, astrometric observations of comets beyond $\\sim \\!10$ AU should allow much more precise determination of their original orbits (see discussion at the end of Section \\ref{sect:orbel}).\n\\section{SIMULATION DESCRIPTION}\n\\label{desc}\nWe divide phase space into three regions, based on the perihelion distance of the cometary orbit.  We define the ``visibility region\" as containing orbits with perihelion distance $q$ in the range $0 \\; {\\rm AU} < q < 45$ AU.  A ``buffer region\" includes orbits with $45 \\; {\\rm AU} < q < 60\\; {\\rm AU}$.  All other orbits are defined to be in the ``Oort region\". \n \\par\n We simulated orbits with the Rebound package, developed by \\citet{Rein12}.  We used their IAS15 integrator, a 15th order adaptive-timestep integrator that is sufficiently accurate to be symplectic to double precision \\citep{Rein15}.\n \\par\n Our goal is to model the steady-state distribution of NICs with perihelia within 45 AU of the Sun that are produced from an Oort cloud with orbital elements uniformly distributed on an energy surface in phase space (so $dN \\sim \\sin{I} dI de^2$, where $I$ and $e$ are the cometary inclination and eccentricity).  This approximation assumes that perturbations from the Galactic tide, passing stars or molecular clouds over the last four Gyr have been sufficient to isotropize comets both in position space (seen from the Sun) and velocity space at a fixed position.  This has been shown to be roughly true for comets with semi-major axes greater than 2,000 AU \\citep{Duncan87}. \n \\par\n To initialize the simulation, we generated comets at random from this phase-space distribution for four discrete values of initial semi-major axis, $a_i =$ 5,000, 10,000, 20,000, and 50,000 AU, with perihelion distances in the range 60 AU to 60 + $\\Delta$ AU.  Then, using an analytic approximation to the torque from the Galactic tide (see appendix \\ref{Rtorque}), we determined an upper bound on the time $\\tau_{\\rm entry}$ (as a function of $a_i$) such that no comet from outside (60 + $\\Delta$) AU could enter the buffer or visibility regions within the next $\\tau_{\\rm entry}$ years.  We chose $\\Delta = 5$, but it is straightforward to see that the results do not depend on this choice.  We picked $\\tau_{\\rm entry} = 10^7, 2.5 \\cdot 10^6, 6.25 \\cdot 10^5$ and $10^5$ years, for $a_i$ = 5,000, 10,000, 20,000 and 50,000 AU respectively.  These numbers satisfy our upper bound.  \n \\par\nWe then evolved the comets under the influence of the Sun, the Galactic tide, and the four outer planets for $\\tau_{\\rm entry}$.  After $\\tau_{\\rm entry}$ had elapsed, we removed any comet with perihelion greater than 60 AU from the simulation.  The remaining comets were allowed to evolve under the influence of the Galactic tide and gravity from the four giant planets and the Sun.  At fixed intervals $\\tau_{\\rm sample}$ (taken to be 10 years), we recorded the position and velocity of any comet that was within 45 AU of the Sun in a catalog.  This procedure gives us the same expected comet count and distribution of orbital elements as if we had allowed the system to evolve to steady state, and then catalogued the comets visible within 45 AU at an instant in time, and multiplied the flux by $\\tau_{\\rm entry}\/\\tau_{\\rm sample}$.\n\\par\nComets are removed from the simulation if they collide with a planet, come within 0.1 AU of the Sun, move outside 200,000 AU, or are perturbed back into the Oort region ($q >$ 60 AU)\\footnote{Because the boundary between the buffer and Oort region at 60 AU corresponds to a perihelion distance twice the semi-major axis of Neptune's orbit, we expect the planets to have a negligible effect on the orbits of comets in the Oort region.  Therefore, it is reasonable to assume that a comet with an orbit aligned such that the Galactic tide pulls it from the buffer region into the Oort region will not return for a long time. }.  \n\\subsection{Orbital Elements}\n\\label{sect:orbel}\nThe treatment of orbital elements for highly eccentric orbits that pass through the orbits of massive planets is somewhat subtle.  A comet that is having a close encounter with one of the giant planets will undergo large short-term perturbations to its orbital elements that do not reflect changes to its orbit that will last longer than the duration of the encounter.  Short-term perturbations from such encounters are more serious for comets with large semi-major axes because the energy of the comet in the planetary potential well can equal or exceed the total orbital binding energy of the comet.  At distance $r_p$ from a planet with mass $M_p$, the fractional change in energy due to the potential energy of the planet is \n\\begin{equation}\n\\label{perturbation}\n\\frac{\\Delta E}{E} = \\frac{2a_c M_p}{r_p M_\\odot} = 0.19\\; \\frac{a_c}{100 \\; {\\rm AU}} \\frac{1 \\,{\\rm AU}}{r_p} \\frac{M_p}{M_{\\rm Jupiter}},\n\\end{equation}\nwhere $a_c$ is the semi-major axis of the comet prior to the close encounter.  We stress that $a_c$ and $a_i$ are not the same quantity: $a_c$ is the current semi-major axis of the comet, whereas $a_i$ is the semi-major axis of the comet when the simulation was initialized. \n\\par\nTo lessen the short-term planetary perturbations to cometary orbital elements, we adopt the following procedure.  For comets with $a_c < 100$ AU, we simply report heliocentric orbital elements.  These comets have large enough binding energy that they would have to pass close to a planet (within 2.0 AU for Jupiter) to obtain enough extra kinetic energy for $a_c$ to vary by more than 10\\% during the close passage (see Equation \\eqref{perturbation}).  In order to prevent very close planetary approaches from contaminating our results, we discarded any observation in which the comet is currently close enough to a planet that the specific potential energy due to the planet is more than 10\\% of the specific binding energy in an orbit around the Sun with semi-major axis of 100 AU.  This occurs for only $0.004\\%$ of all catalog entries, or $0.3\\%$ of all catalog entries with $R<10$ AU.\n\\par\nComets in the visibility region with $a_c > 100$ AU often have potential energies due to the planets which are comparable to their binding energies.  For this reason, we report the barycentric orbital elements of the comet the last time it was in the range [90 AU, 110 AU].  These elements are well-behaved, since they are calculated far outside the orbits of the giant planets where it is appropriate to represent the solar system as having all its mass located at the barycenter.  \n\\par\nThe ease with which LSST can determine orbital elements for slowly moving nearly unbound objects is also of interest to this study.  To address this question, we searched the JPL small body database\\footnote{\\url{http:\/\/ssd.jpl.nasa.gov\/?horizons}} for objects with semi-major axis greater than 300 AU and perihelion distance greater than 10 AU.  It listed seven objects with a data-arc longer than 5 years.  The estimated errors in $x = 1\/a$ ranged from $1.5\\cdot 10^{-6}$ AU$^{-1}$ to $1.5 \\cdot 10^{-5}$ AU$^{-1}$ for these objects.  It therefore seems reasonable to expect orbits to generally be determined to at least this level of accuracy purely from 10 years of LSST data.\n\\subsection{Disrupted Comets}\n\\label{sect:disrupt}\nThere is a substantial body of evidence suggesting that comets ``fade\" over time (e.g., \\citealt{Fernandez81, Wiegert99}).  A number of processes have been proposed to explain this phenomenon \\citep{Weissman80}: a comet could run out of volatile material, it could have its surface covered with a crust that prevents volatiles from escaping, or it could physically be broken apart by outgassing or tidal stress.  \n\\par\n\\citet{Fernandez05} gives 3 AU as a likely cut-off to comet activity based on the sublimation temperature of water, but cautions that many comets experience some activity outside 3 AU due to the sublimation of more volatile elements.   Comet 67P\/Churyumov-Gerasimenko first showed signs of activity when it was 4.3 AU from the Sun \\citep{Snodgrass13}.  Comet Hale-Bopp showed substantial activity on its approach to perihelion as far out as 7.2 AU \\citep{Weaver97}, and at 27.2 AU after perihelion passage \\citep{Kramer14}.  When we calculate numbers of visible NICs, we restrict ourselves to NICs further than 5 AU from the Sun.  For this reason, we do not consider the effect of comet activity on magnitude, and just calculate the magnitude from the size and albedo of the bare nucleus, see Section \\ref{sect:vis}.  We believe that our assumption that there is negligible activity beyond 5 AU is reasonable, though not certain, given existing observations.  In any event this assumption gives us a conservative estimate of the number of comets that a survey like LSST will discover.\n\\par\nBecause comet activity does not affect brightness in our model, we are only sensitive to physical disruption of comets, not loss of volatiles.  For this reason, throughout this paper, we refer to ``disruption\", rather than ``fading\".\n\nIn the results that follow, we remove a comet after it has made 10 apparitions in the catalog with radius $R < 3$ AU (corresponding to a total exposure to the Sun at $R < 3$ AU of about 100 years, since $\\tau_{\\rm sample} = 10$ years).  Comets are also removed if they ever travel within 0.1 AU of the Sun (even if they do not appear so close in the catalogue) or if they suffer a collision with one of the gas giants.\n\\section{COMET LIFETIMES IN THE SIMULATION REGION}\n\\label{lifetime}\nFigure \\ref{Occurrence} shows the fraction of NICs in our simulations with $a_c > 34.2$ AU (period $>$ 200 years) appearing in the region with $R<45$ AU for more than $t$ years, as a function of $t$.  Different curves correspond to different values of the initial semi major axis $a_i$.  In this plot we terminate each orbit integration after 4.5 Gyr.  The error bars are derived from the re-sampling procedure described in Section \\ref{sect:concentration}.  In this and all subsequent plots, only comets with periods greater than 200 years (corresponding to semi-major axes greater than 34.2 AU) are counted.\n\\par\n\\citet{Yabushita79} argued using a simple random walk model that the number of NICs surviving more than $N_{\\rm peri}$ perihelion passages should scale as $P(>\\! \\!N_{\\rm peri}) \\propto N_{\\rm peri}^{-1\/2}$.  Assuming for the sake of argument that NICs spend a fixed amount of time in the visibility region per perihelion passage, then the number of NICs having a given number of catalog entries is proportional to the number of NICs surviving for more than a given number of perihelion passages.  We should therefore recover the same power-law as \\citet{Yabushita79} if his model is a good approximation to the full physics captured by the simulation.  This plot largely confirms the predictions of \\citet{Yabushita79}, but the exponent of the power-law seems to be slightly steeper than his value of $-1\/2$.  \n  \\par\n Deviations from power-law behavior at short times occur because the visibility region is larger than the region of influence of the planets, so there is some delay before NICs that have entered the visibility region interact with the planets.  Comets with smaller values of $a_i$ experience less torque due to the Galactic tide, so the delay is larger.  They also have more binding energy that must be overcome prior to ejection.  This explains the trend seen in Figure \\ref{Occurrence} that NICs with smaller $a_i$ take longer to be ejected.\n\\par\nThe fact that some of our simulated particles survive for longer than 1 Gyr leads to concern about the physical validity of our assumption that there is a static Oort cloud.  Likely, many of the NICs that will be observed with LSST exited the Oort cloud more than 1 Gyr in the past, when it may have had different physical properties.\n\\par \nEven if the properties of the Oort cloud have not changed over 4.5 Gyr, long-lived comets may bias our simulations, as the following argument shows.  Suppose, as seems reasonable from Figure \\ref{Occurrence}, that the true distribution of time $t$ that a comet spends in the visibility region is given by a power law, i.e.\n\\begin{equation}\n\\label{simpledndt}\ndp\/dt = \\left\\{\n\\begin{array}{ll}\n\\frac{(\\alpha-1) t^{-\\alpha}}{t_{\\rm min}^{1-\\alpha}}  & t > t_{\\rm min} \\\\\n0 & t<t_{\\rm min}\n\\end{array}\n\\right.\n\\end{equation}\nfor some $\\alpha$ in the interval $1<\\alpha < 2$.  Then, if we terminate our integrations at some time $t_{\\rm cutoff}$, the average time that a comet spends in the visibility region during our simulation is\n\\begin{equation}\n\\langle t \\rangle = \\frac{1}{2-\\alpha} \\left[t_{\\rm cutoff} \\left(\\frac{t_{\\rm min}}{t_{\\rm cutoff}}\\right)^{\\alpha-1} + (1-\\alpha) t_{\\rm min}\\right]\n\\end{equation}\nIn the limit that $t_{\\rm cutoff} \\gg t_{\\rm min}$, this becomes \n\\begin{equation}\n\\label{expectedt}\n\\langle t \\rangle = \\frac{t_{\\rm cutoff}}{2-\\alpha}\\left(\\frac{t_{\\rm min}}{t_{\\rm cutoff}}\\right)^{\\alpha - 1} \n\\end{equation}\n\\par\nIn our simulations, (8, 1, 0, 0) particles with $a_i$ = (5,000, 10,000, 20,000, 50,000) AU survive with $q < 45$ AU for the duration of the integration (4.5 Gyr).  For $a_i = 20,\\!000$ AU and $50,\\!000$ AU respectively, the longest lived particles survived for 1.1 and 1.4 Gyr.  Now, consider the case of the simulation with $a_i = 20,\\!000$ AU.  We drew particles with lifetimes from the distribution in Equation \\eqref{simpledndt}.  By chance, we drew no particles with lifetime greater than $t_{\\rm max} = 1.1$ Gyr.  We will assume that we drew randomly from the distribution in Equation \\eqref{simpledndt} subject to the constraint that we draw no particles with lifetimes greater than $t_{\\rm max}$ (this is not quite true, given that we did draw one particle with lifetime exactly $t_{\\rm max}$).  We can renormalize the distribution in Equation \\eqref{simpledndt} subject to the constraint that no comet have lifetime longer than $t_{\\rm max}$, and calculate the mean.  This is given by (assuming $t_{\\rm max} \\gg t_{\\rm min}$)\n\\begin{equation}\n\\label{truncatedt}\n\\langle t \\rangle = \\frac{\\alpha-1}{2-\\alpha} \\left[t_{\\rm max} \\left(\\frac{t_{\\rm min}}{t_{\\rm max}}\\right)^{\\alpha - 1}\\right]\n\\end{equation}\nTaking the ratio of Equations \\eqref{expectedt} to \\eqref{truncatedt}, and assuming that a comet makes a fixed number of appearances in the catalog per unit time spent in the visibility region, we find we have likely underestimated our total flux by something close to a factor of $(4.5\/1.1)^{2-\\alpha}\/(\\alpha-1)$.  This expression is 4.0 if we take $\\alpha = 1.5$, as suggested by \\citet{Yabushita79}, but declines to unity for $\\alpha = 2.0$.  In principle, this bias can be reduced by simulating more comets but the computational resources necessary to reduce the bias substantially are prohibitive.\n\n\n \\begin{figure}\n\\centering\n\\includegraphics[scale = .32]{Occurrence.pdf}\n\\caption{Distribution of the number of NICs having $R<45$ AU for longer than $t$ years.  The curves show four different values of the initial semi-major axis $a_i$, normalized by the number of comets in the simulation which ever enter within $R = 45$ AU.   We terminate the integrations after 4.5 Gyr. The black lines show two power laws with exponents $-0.5$ and $-1$.  Only comets with $a_c > 34.2$ AU (corresponding to long-period comets --- comets with period $>$ 200 years) were counted.}\n\\label{Occurrence}\n\\vspace{-.05cm}\n\\end{figure} \n\\par\nIn the following sections we present simulation data for different values of $t_{\\rm cutoff}$.  $t_{\\rm cutoff}$ is defined relative to the time when the comet first enters within 45 AU {\\it except} when $t_{\\rm cutoff} = 4.5$ Gyr, in which case it is defined relative to the start of the simulation.  We believe that a value of a few Gyr is most observationally relevant, and most of the following discussion is based on such cutoff times, however given the limitations discussed above, we show results for shorter cutoff times as well.\n\\section{CONCENTRATION OF NICS DUE TO THE GIANT PLANETS}\nIn this section we describe how the distribution of NICs is affected by the planets.\n\\subsection{Expected Distribution of $R$ and $q$ with no Planets}\nWe first calculate the expected distribution of orbital radius $R$ and perihelion distance $q$ in the absence of perturbations from the planets, constructing what we call the zero-planet model.  We assume a uniform distribution of cometary orbital elements in phase space at four fixed energies.  These results provide a natural normalization to the plots in the following subsection, which show the distributions of $R$ and $q$ in our simulated catalog.\n\\subsubsection{Distribution in Radius and Perihelion at a Snapshot in Time}\nLet there be $N_0$ comets distributed uniformly on the energy surface corresponding to semi-major axis $a$.  There is no need to distinguish between the initial semi-major axis $a_i$ and the current semi-major axis $a_c$ here, since the semi-major axis is not changed in the absence of planetary perturbations.  Approximating the orbits as parabolic in the visibility region, we find that the radial velocity at radius $R$ of an orbit with perihelion distance $q$ is\n\\begin{equation}\n\\label{vr}\nv_R(q, R) = \\sqrt{\\frac{2GM_\\odot(1-q\/R)}{R}}.\n\\end{equation}\nSince we assume a uniform distribution of comets on the energy surface, the probability density of the squared eccentricity $e^2$ is \n\\begin{equation}\nN(e^2) de^2 = N_0de^2.\n\\end{equation}\nThen, since $q=a(1-e)$, the number of comets per unit perihelion distance $N(q)$ is given by \n\\begin{equation}\n\\label{Nq}\nN(q)dq = 2N_0\\left(1-\\frac{q}{a}\\right )\\frac{dq}{a}\\simeq \\frac{2N_0}{a} dq,\n\\end{equation}\nwhere the last equality holds because we are interested in comets with $q \\ll a$.  A comet on a near-parabolic orbit with perihelion $q$ spends a fraction of its time $f(R,q)dR$ in the radial interval between $R$ and $R+dR$, where \n\\begin{equation}\n\\label{frq}\nf(R, q)dR = \\frac{2 dR}{Pv_R(q, R)},\n\\end{equation}\nand $P$ is the period of the orbit.  Then, using Equations \\eqref{Nq} and \\eqref{frq}, we can solve for the number of comets in a radial interval $N(R)dR$:\n\\begin{equation}\n\\label{Nr}\nN(R) = \\int_{q=0}^R N(q) f(R, q) dq = \\frac{2 \\sqrt{2} N_0 R^{3\/2}}{\\pi a^{5\/2}}.\n\\end{equation}\n\\par\nThe number of comets with perihelion in the range $q$ to $q + dq$ expected to be present out to a maximum value of $R$ is given by \n\\begin{equation}\nN(q|R_{\\rm max})dq = N(q)dq \\int_q^{R_{\\rm max}} f(R', q) dR'.\n\\end{equation}\nThis evaluates to \n\\begin{equation}\n\\label{Nqexpect}\nN(q|R_{\\rm max})dq = \\frac{2 \\sqrt{2}}{3 \\pi a^{5\/2}} \\left(\\frac{R_{\\rm max}}{q}-1\\right)^{1\/2} \\left(2 + \\frac{R_{\\rm max}}{q}\\right)q^{3\/2} dq.\n\\end{equation}\nAs mentioned in Section \\ref{desc}, because of the way we have set up our simulation, and the fact that we sample every $\\tau_{\\rm sample}$ years, we would expect our catalog to contain a number of comets with orbital elements $\\psi$ equal to \n\\begin{equation}\n\\label{sampleToActual}\nN_{\\rm cat}(\\psi|R_{\\rm max})= \\frac{\\tau_{\\rm entry}}{\\tau_{\\rm sample}} N(\\psi|R_{\\rm max}),\n\\end{equation}\nif we had not included any planets in the simulation.\n\\subsubsection{Concentration Factors}\n\\label{sect:concentration}\nIn this subsection we compare our catalog to the one which would be produced had we not included planets in our simulations.  In Figure \\ref{Rconcentration}, we compare our simulated comet catalog with the zero-planet model (Equations \\eqref{Nr} and  \\eqref{sampleToActual}).  We have plotted a ``concentration factor\" --- the ratio of comets appearing in our catalog within a given radial interval to the number calculated from Equations \\eqref{Nr} and \\eqref{sampleToActual}, assuming the same density of comets in the Oort cloud as was used to initialize our simulations.  As stated previously, only comets with periods greater than 200 years are counted.  Each panel corresponds to a different value of the initial cometary semi-major axis $a_i$.  Different colored lines correspond to different values of $\\tau_{\\rm cutoff}$ as shown in the legend. \n\\par\nThese data represent the results from following $N_{\\rm sim}$ comets where $N_{\\rm sim} =$ 10,000, 16,000, 40,000, and 420,000 for $a_i =$ 5,000, 10,000, and 20,000 AU,  and 50,000 AU respectively.   Note that fewer than 10\\% of these ever enter the region $R < 45$ AU (955, 1548, 3949, and 29215 respectively).  Most comets do not evolve to $q < 60$ AU within the entry period and are therefore discarded at the end of the entry period.  Some of those that do come within $q < 60$ AU never reach $q < 45$ AU (if the angular momentum is nearly perpendicular to the torque).\n\\par \nWe would like to know the true distribution of orbital elements in the limit that we simulate a very large number of comets.  To estimate our random error, we employed re-sampling.  For each point on the curve, we drew $N_{\\rm sim}$ comets with replacement from our $N_{\\rm sim}$ simulated comets.  The points are the mean of the re-sampled distribution, and the error bars correspond to the 16$^{\\rm th}$ and 84$^{\\rm th}$ percentiles of the re-sampled distribution (if the distribution were normally distributed, these would be 1-sigma error bars).  The error bars on nearby points are highly correlated in most of our plots because the same comet contributes to several bins in the course of its evolution, hence the low point-to-point scatter relative to the error bars.  \n\\par\nAlthough these data reflect the orbits of thousands of comets, the statistical errors are large in many cases, since often the majority of the contribution to a particular bin comes from only one or two long-lived comets, (see the discussion in \\S \\ref{lifetime}).\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=.5\\textwidth]{Rconcentration.pdf}\n\\caption{Number of comets in the simulated catalog at radius $R$, normalized by the expected number assuming no planets (Equations \\eqref{Nr}, \\eqref{sampleToActual}).  Different curves correspond to different values of the cutoff time $\\tau_{\\rm cutoff}$.  Errors were determined via bootstrapping (see Section \\ref{sect:concentration} for details).  Points and error bars from curves corresponding to different cutoff times have been horizontally displaced slightly for clarity.  }\n\\label{Rconcentration}\n\\vspace{-.05cm}\n\\end{figure} \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=.5\\textwidth]{Qconcentration.pdf}\n\\caption{Number of comets in the catalog at perihelion $q$, normalized by the expected number assuming no planets (Equations \\eqref{Nqexpect}, \\eqref{sampleToActual}).  Different curves correspond to different values of the cutoff time $\\tau_{\\rm cutoff}$.}\n\\label{Qconcentration}\n\\vspace{-.05cm}\n\\end{figure} \nFigure \\ref{Qconcentration} is similar to Figure \\ref{Rconcentration}, except that instead of plotting the number of appearances in our catalog in bins of heliocentric radius $R$, we plot appearances in bins of perihelion $q$.  The distribution of $q$ provides more information about the NIC orbits: a sharp feature in the perihelion distribution is smoothed out when one looks at the distribution of heliocentric radius, since the same orbit can be observed at a range of values of $R$.   \n \\par\nIn Figures \\ref{Rconcentration} and \\ref{Qconcentration}, and in most of the subsequent plots, we have horizontally offset the blue, green, yellow and red curves  curves by $-4.5\\%$, $-1.5\\%$, $1.5\\%$, and $4.5\\%$ respectively in order to make the curves distinguishable in regions where they overlap.\n\\par\nThe following qualitative features of Figures \\ref{Rconcentration} and \\ref{Qconcentration} are straightforward to explain:\n\\begin{itemize}\n\\item The flux of comets with $a_i \\lesssim 20,\\!000$ AU shows a sharp drop-off (in both $N(q)$ and $N(R)$)  interior to 10 AU.  This is because comets originating at small semi-major axes are subjected to weak Galactic tides and change their perihelia slowly.  The majority of kicks given to a comet with $q<10$ AU are large enough to either unbind a comet infalling from outside a few thousand AU, or to reduce $a_c$ to the point that the timescale to change the perihelion distance is much longer than an orbital period.   We therefore expect to see a jump in the number of comets appearing inside 5 AU at values of $a_i$ exceeding that at which a comet can go from perihelion $\\gtrsim 15$ AU (largely unaffected by Jupiter and Saturn) to perihelion $\\lesssim 5$ AU in one orbit.  This occurs at approximately 30,000 AU.  Therefore, in order for a comet starting from inside $\\sim 30,\\!000$ AU to appear in the inner solar system, it needs to have a lucky orientation with respect to Jupiter and Saturn\\footnote{Or a kick from a star that passes unusually close to the Sun, a rare event not included in our model.}.  This lucky orientation can either yield multiple small energy kicks on subsequent perihelion passages, or yield a kick that increases the semi-major axis so that the comet receives a larger torque from the Galaxy \\citep{KaibQuinn09}.  \n \\par   \n Thus we conclude that comets with $a_i \\lesssim 20,\\!000$ AU are mostly ejected by interactions with the outer planets before they reach small heliocentric radii.  \\citet{Hills81} arrives at a similar result considering the effects of passing stars discretely rather than as a smooth Galactic tidal potential.  \\citet{Collins10} provide a discussion of when it is appropriate to treat the influence of the Galaxy as being due to a smooth tidal field, and when it should be modeled as discrete stellar encounters.  \n \\par\n The lower concentration factors for comets with smaller values of $a_i$ do not mean that we will see fewer NICs for a given number of Oort cloud comets at that energy.  This is because the fraction of the total comets that are in the visibility region at a given time in the zero-planet model scales with $a^{-2.5}$ (see Equation \\eqref{Nr}). \n\\item \nThe difference between the green curves and the orange and red curves grows with increasing $q$.  This is because the kicks from the planets are smaller at large $q$, so the comets survive longer.  The exception to this is the curve for $\\tau_{\\rm cutoff} = 100$ Myr and $a_i =$ 5,000 AU.  In this case, comets have mostly had insufficient time to be torqued to small values of $q$.  It should be noted however, that the time to reach a given perihelion distance is not completely determined by the initial semi-major axis because a comet could be scattered to larger $a_i$ by an early encounter with Neptune, and subsequently evolve more rapidly.  \n\\par\nThere is little difference between the results for $t_{\\rm cutoff} = 1$ Gyr, and $t_{\\rm cutoff} = 4.5$ Gyr for comets with $a_i \\geq 20,\\!000$ AU.  As discussed previously, this is likely an artifact of our simulations including insufficiently many comets with these values of $a_i$ to capture the tail of the lifetime distribution (see Section \\ref{lifetime}).\n\\item\nThe concentration factors for large cutoff time in Figure \\ref{Qconcentration} approach unity as $q$ approaches 45 AU, however the concentration factors in Figure \\ref{Rconcentration} are still on the order of 10 at 45 AU.  This is because even a concentration of comets with $q \\ll 45$ AU affects the distribution of comets at $R=45$ AU. \n\\item\nWe do not expect $N(q)$ to drop exactly to unity as soon as $q$ is larger than the extent of the planetary perturbations, because NICs which have interacted with the planets may be systematically carried away from the planetary region by the tide at a different rate than they were carried in (due to a change in orientation or semi-major axis). \n\\end{itemize}\n\\section{DISTRIBUTION OF ORBITAL ELEMENTS FOR VISIBLE COMETS}\nThe above analysis shows the degree to which the giant planets concentrate NICs in the outer solar system and exclude them from inside the orbit of Jupiter.  In this section, we use an estimate of the size distribution of NICs, the relationship between magnitude, size and heliocentric distance, and the concentration effect due to interactions with the planets to calculate the number of NICs expected to be seen in an all-sky survey as a function of the limiting magnitude $m_{\\rm lim}$.   \n\\subsection{Size Distribution}\nComets have so far been observed primarily within a few AU of the Sun, where their brightness is influenced strongly by their activity.  At the larger distances that we focus on here, comets are believed to be generally inactive (see discussion in Section \\ref{sect:disrupt}), so their brightness is determined solely by their size, distance, and albedo.  Let $H$ be the apparent magnitude of a comet 1 AU from the Sun and 1 AU from the observer, seen from zero phase angle.  Based on a sample of long-period comets from about $H = 5$ to $H = 9$, \\citet{Sosa11} derive a relation for active comets between the radius $r$ (in kilometers) and $H$:\n\\begin{equation}\n\\label{rH}\n\\ln{r} = \\alpha+\\beta H,\n\\end{equation}\nwhere $\\alpha$ = 2.072 and $\\beta = -0.2993$.  We caution that our use of this formula requires substantial extrapolation: the largest comet used to determine the formula has a radius of 1.8 km, more than an order of magnitude smaller than the smallest comets detectable at 30 AU in a survey with the limiting magnitude of LSST (see Section \\ref{sect:vis}).  \\citet{Sosa11} note that the relation in Equation \\eqref{rH} predicts a radius (13 km) for comet Hale-Bopp that is somewhat below other estimates (mostly falling in the 20--35 km range).  This discrepancy suggests that Equation \\eqref{rH} may underpredict the radii of large comets, in which case our estimates of the observable comet population will be conservative.  Note that by using Equation \\eqref{rH} we are assuming that long-period comets are mostly the same population as NICs (or at least have the same size distribution).\n\\par\n\\citet{Hughes01} finds that the number $N_{\\rm peri}$ of long-period comets with brightness $H < 6.5$ passing through perihelion in the inner solar system per year per AU of perihelion distance is given by \n\\begin{equation}\n\\frac{dN_{\\rm peri}}{dH} = c_0 e^{\\gamma H},\n\\end{equation}\nwith $c_0 = 2.047 \\cdot 10^{-3}$ and $\\gamma =  0.827$.  We can then transform variables to $r$ using Equation \\eqref{rH}.  We find that\n\\begin{equation}\n\\label{dNdr}\ndN_{\\rm peri}\/dr =  -\\frac{c_0}{\\beta} \\mathlarger{\\mathlarger{e^{-\\gamma \\alpha\/\\beta} r^{\\gamma\/\\beta - 1}}} = 2.09\\cdot r^{-3.76}.\n\\end{equation}\n\\par\nThis distribution holds down to $r(H = 6.5) = 1.1$ km, however, for simplicity we extrapolate it down to $0.9$ km --- the smallest comet visible at 5 AU in our model (see next section).  This size distribution leads to a weak divergence in total mass at the large end of the spectrum.  Nevertheless, we assume that the power-law behavior holds up to several tens of kilometers.  The size distribution in Equation \\eqref{dNdr} is steeper than the relation ($dN_{\\rm peri}\/dr \\sim r^{-2.79}$) estimated in \\citet{Hughes01} because he uses a different relation between $H$ and $r$.  It is also substantially steeper than the relation ($dN_{\\rm peri}\/dr \\sim r^{-2.92}$) found in \\citet{Snodgrass11} for the Jupiter-family comets.  If the size distribution is shallower than we have estimated at large radii, then our estimates of the observable comet population will be conservative. \n\\subsection{Visibility Model}\n\\label{sect:vis}\nIn this section we describe our model for determining how likely a given simulated comet is to be visible.  We assume that comets have an $r$-band geometric albedo $A_g$ of 0.04 as suggested in \\citet{Lamy04}.  We find that the magnitude $m$ of an inactive comet is \n\\begin{align}\n\\label{mag}\nm &= -27.08 - 2.5 \\log{\\left(\\frac{r^2 A_g {\\rm (AU)}^2}{R^4}\\right)} \n\\nonumber\\\\\n&= 24.28 - 2.5 \\log{(A_g\/0.04}) -5 \\log{(r\/{\\rm 1km})} + 10 \\log{(R\/{\\rm 5AU})},\n\\end{align}\nwhere we have used $-27.08$ as the apparent $r$-band magnitude of the Sun.  Equation \\eqref{mag} is only valid for comets far from the Sun, since we have assumed that $R_{\\rm Sun, comet} = R_{\\rm Earth, comet}$, that the phase angle is zero, and most importantly, that we are seeing the bare nucleus of an inactive comet.  \\citet{Lamy04} state that magnitude drops off at about a rate of 0.04 magnitudes\/degree of phase angle, meaning that error due to this nonzero phase angle is limited to at most 0.23 magnitudes for a comet at 10 AU.  Similarly, at 10 AU, the largest possible error arising from the approximation that the Sun-comet distance is equal to the Earth-comet distance also corresponds to a magnitude error of $\\Delta m = 0.23$.\n\\subsection{Weighting of Observations}\nUsing Equation \\eqref{mag} we can solve for $r_{\\rm min}(R)$, the radius in kilometers of the smallest comet visible at distance $R$ in a survey with limiting magnitude $m_{\\rm lim}$, assuming $A_g = 0.04$:\n\\begin{equation}\n\\label{rR}\nr_{\\rm min}(R)  = 0.903 \\cdot 10^{0.2(24.5-m_{\\rm lim})}  \\left(\\frac{R}{\\rm 5\\; AU}\\right)^2.\n\\end{equation}\nThe comet size is not explicitly tracked in our simulations.  We assume that the sizes of comets in our simulation are drawn from the distribution in Equation \\eqref{dNdr}, with a lower cutoff radius of $0.903$ km --- the smallest comet visible at 5 AU in our model.  To account for the fact that not all comets are visible at all orbital radii, we weight a simulated comet appearance at radius $R$ by the fraction of comets that would be visible at the observed value of $R$ given the assumed size distribution in the simulation.  An appearance at high $R$ will receive a low weight, since most comets would not be visible so far away.  We assign weight 1 to observations at $R = 5$ AU, as all comets in our assumed size distribution would be visible at 5 AU.  Then at general $R$, we assign weight \n\\begin{equation}\nW(R) = \\frac{\\int_{r_{\\rm min}(R)}^\\infty r^{-3.76} dr}{\\int_{r_{\\rm min}(5 \\; {\\rm AU})}^\\infty r^{-3.76} dr} = \\left(\\frac{5 \\; {\\rm AU}}{R}\\right)^{5.52}.\n\\end{equation}\n\\par \nWe quote numbers of comets with a given set of orbital elements per $10^{11}$ comets larger than one kilometer in a  spherical distribution at the assumed initial semi-major axis.  To achieve this normalization, we multiply our counts by\n\\begin{equation}\n\\frac{10^{11}f_{60-65}(a_i)}{N_{\\rm init}} \\frac{\\tau_{\\rm sample}}{\\tau_{\\rm entry}} \\frac{\\int_{0.903}^\\infty N(r) dr}{\\int_1^\\infty N(r)dr},\n\\end{equation}\nwhere $f_{60-65}(a_i)$ is the fraction of the phase space of orbits with semi-major axis $a_i$ that consists of orbits with perihelia between 60 and 65 AU, given by \n\\begin{equation}\nf_{60-65}(a_i) = \\frac{(a_i - 60 \\; {\\rm AU})^2 -(a_i - 65 \\; {\\rm AU})^2}{a_i^2},\n\\end{equation}\nand $N_{\\rm init}$ is the number of comets we initialize between 60 and 65 AU.\n\n\n\n\n\n\\subsection{Distribution of Visible Comets}\n\\begin{figure}\n\\centering\n\\includegraphics[width=.5\\textwidth]{Rvisible.pdf}\n\\caption{Number of NICs expected to be seen per logarithmic interval in $R$ at a snapshot of time in an all-sky survey with limiting $r$-band magnitude $m_{\\rm lim} = 24.5$.  This assumes there are $10^{11}$ comets with $r > 1$ km at the value of initial semi-major axis $a_i$ specified in each panel.  Errors were determined via bootstrapping (see Section \\ref{sect:concentration} for details).  Different curves correspond to different values of $t_{\\rm cutoff}$.}\n\\label{Rvisible}\n\\vspace{-.05cm}\n\\end{figure} \n \\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{Qvisible.pdf}\n\\caption{Number of NICs expected to be seen per logarithmic interval in $q$ at a snapshot of time.  This assumes there are $10^{11}$ comets with $r > 1$ km at the value of $a_i$ specified in each panel.  Different curves correspond to different values of $t_{\\rm cutoff}$.}\n\\label{Qvisible}\n\\vspace{-.05cm}\n\\end{figure} \n\\\nIn this section we present results from our simulations showing how many NICs are visible over the whole sky at a given snapshot of time as a function of $R$ and $q$, for observations taken between $R = 5 \\; {\\rm AU}$ and $R = 45 \\; {\\rm AU}$.  In all cases, we assume a limiting $r$-band magnitude $m_{\\rm lim}$ of 24.5, equivalent to the one-exposure limit for LSST \\citep{LSST09}.  The number of distant NICs expected to be discovered by LSST differs from the results presented here for two reasons.  First, LSST is expected to operate for 10 years, so it should see more than just the comets visible in a snapshot, particularly in the case of the closer comets where $R\/v$ is less than 10 years.  Second, LSST will only survey 48\\% of the sky, so will only see about half of the comets that would be visible in an equivalent all-sky survey.  Comets move slowly enough that trailing losses will be insignificant given the 30 second exposure time.  Using Equation (8) from \\citet{Ivezic08}, we estimate a comet at 10 AU will have a limiting magnitude only 0.06 magnitudes brighter due to trailing losses.\n\\par\nFigures \\ref{Rvisible} and \\ref{Qvisible} show the number $N$ of NICs expected to be visible outside $5$ AU per unit of $\\ln{R}$ and $\\ln{q}$ respectively, per $10^{11}$ comets with $r$ greater than 1 km at the labeled initial semi-major axis in the Oort cloud.  The shapes of the curves are substantially different for different values of $a_i$, particularly in the region between 5 and 10 AU, where the statistics are the best.  In the $a_i =$ 5,000 AU case, the expected count {\\it increases} by a factor of 5 from $R=$ 5 AU to $R=$ 10 AU for $t_{\\rm cutoff} \\geq 1$ Gyr.  In the $a_i =$ 50,000 AU case it {\\it decreases} by a factor of $\\approx 5$.  Observations of comets in this regime will therefore allow us to observationally constrain the distribution of $a_i$.  The peak of $RdN\/dR$ moves smoothly from around 15 AU for $a_i = 5,\\!000$ AU to less than 5 AU for $a_i = 50,\\!000$ AU.\n\\par\nAs shown in Appendix \\ref{sect:isovis}, we expect $RdN\/dR$ and $qdN\/dq$ to decline as $R^{-3.02}$ and $q^{-3.02}$ respectively in the zero-planet model.  Deviations from this behavior are due to variation in the concentration factor as shown in Figures \\ref{Rconcentration} and \\ref{Qconcentration}.\n \\begin{figure}\n\\centering\n\\includegraphics[width=.5\\textwidth]{largea.pdf}\n\\caption{Number of NICs with $a > 300$ AU expected to be seen per logarithmic interval in $q$ at a snapshot of time.  This assumes there are $10^{11}$ comets with $r > 1$ km at the value of $a_i$ specified in each panel.}\n\\label{largea}\n\\vspace{-.05cm}\n\\end{figure}  \n\\par\nWe also examined what happened if we broke up the sample into two groups depending on the current semi-major axis of the comet.  Figure \\ref{largea} is identical to Figure \\ref{Qvisible} except that we have only considered those comets that have semi-major axes greater than 300 AU.  The error bars are smaller, because the comets with the most appearances tend to diffuse to smaller values of $a_c$, leaving a population with less spread in number of appearances.  For this reason, this subset of comets, although smaller in number, has more power to discriminate between Oort cloud models.\n \\begin{figure}\n\\centering\n\\includegraphics[width=.5\\textwidth]{smalla.pdf}\n\\caption{Number of NICs with $a < 300$ AU expected to be seen per logarithmic interval in $q$ at a snapshot of time.  This assumes there are $10^{11}$ NICs with $r > 1$ km at the value of $a_i$ specified in each panel.}\n\\label{smalla}\n\\vspace{-.05cm}\n\\end{figure} \nIn Figure \\ref{smalla} we plot $qdN\/dq$ for only the comets in Figure \\ref{Qvisible}, but not Figure \\ref{largea}, i.e., those comets whose orbits have $a_c<300$ AU.  We see that $qdN\/dq$ declines more sharply with $q$ than in the whole sample of long-period comets.  This is because it is difficult for a comet to attain $a_c < 300$ AU at large perihelion, because the kicks are too small.  We also note that the overall numbers are larger by a factor of a few for comets with $a_c < 300$ AU.\n\n\n\\subsection{Distribution in Semi-major Axis}\n\\begin{figure}\n\\centering\n\\includegraphics[width=.5\\textwidth]{avisible.pdf}\n\\caption{Number of NIC appearances per logarithmic interval in semi-major axis for different values of perihelion distance (different color curves) and different values of the initial semi-major axis (different panels).  Each panel assumes that the Oort cloud contains $10^{11}$ comets with $r > 1$ km at the specified value of $a_i$.}\n\\label{avisible}\n\\vspace{-.05cm}\n\\end{figure} \nFigure \\ref{avisible} shows the semi-major axis distribution (number of appearances per unit logarithmic interval in semi-major axis) for all the NICs in a given perihelion bin (denoted by the color of the curve) and initial semi-major axis (panel).  The error bars are generally larger for the points at small semi-major axis, implying that the statistics in these bins are dominated by a few comets. \n\\par\n \\begin{figure}\n\\centering\n\\includegraphics[width=.5\\textwidth]{inversea.pdf}\n\\caption{Number of NIC appearances per linear interval in inverse semi-major axis $x = 1\/a$ for different values of perihelion distance (different colors) and different values of $a_i$ (different panels).  Each panel assumes that the Oort cloud contains $10^{11}$ comets with $r > 1$ km at the specified value of $a_i$. }\n\\label{inversea}\n\\vspace{-.05cm}\n\\end{figure} \n\nWe find it more illuminating to make this plot in the coordinate $x = 1\/a$, which is proportional to the energy.  Doing so allows us to test the predictions of random walk models such as those in \\citet{Yabushita79} and \\citet{Everhart76}.  In their simplest form, one can imagine a comet starting with energy $E = -\\epsilon$.  Every perihelion passage, it takes a step of size $\\epsilon$ towards higher or lower energy.  It is removed if $E = 0$ (it becomes unbound), or if $E < E_{sp}$, where $E_{sp}$ is the critical energy level to be re-classified as a short-period comet.  We ignore the possibility that a short period orbit could be perturbed back to the long-period regime.  Comets are injected near $x = 0$.  In the limit that $-E_{sp}\/\\epsilon \\gg 1$ , one can show that a steady-state distribution of comet energies normalized by the rate of perihelion passage is given by a linear equation of the form\n\\begin{equation}\n\\label{NE}\nN(E) = k (E - E_{sp}),\n\\end{equation}\nwhere $k$ is a constant depending on the comet injection rate and the size of the kick.  \n\\par \nWe see some support for Equation \\eqref{NE} in Figure \\ref{inversea}.  We have only considered comets with $a_c > 1,\\!000$ AU ($x < 0.001$ AU$^{-1}$).  This is a small enough range that we would expect the curves to be nearly flat if Equation \\eqref{NE} were correct (since $E_{\\rm sp}$ is much more negative than the energies shown in Figure \\ref{inversea}).  The curves are generally flat for perihelion distances 5 AU $<q<$ 30 AU.  Inside 5 AU, the error bars are too large to say much.  Outside 30 AU, there is a definite trend favoring energies closer to 0, as would be expected since the diffusion rate at these perihelion distances is so slow that the steady-state energy distribution cannot be achieved. \n\\par\nWe see evidence of a weak ``Oort spike\" when $a_i = 50,\\!000$ AU - an excess of comets with $x < 10^{-4}$ AU$^{-1}$ interpreted as the result of the initial conditions of cometary orbits.  This is seen only for comets with $q < 5$ AU, since for larger values of $q$, the spike is drowned out by the large numbers of comets which remain at more negative energies for many orbits.  As a quantitative measure of the observed Oort spike, \\citet{Fernandez12} find that 30\\% of comets discovered since the year 1900 with $q < 1.3$ AU have $x <$ (10,000 AU)$^{-1}$.  We find that we cannot come close to reproducing this unless we assume some disruption.  We applied a disruption law in which comets were removed after spending $\\tau_{\\rm disrupt}$ years within 5 AU of the sun.  For comets with $a_i = $ 50,000 AU and $q<5$ AU, the fraction seen with $x < 10^{-4}$ AU$^{-1}$ is \\{$0.12 \\pm 0.007$, $0.04 \\pm 0.004$, $0.02 \\pm 0.002$\\}, for $\\tau_{\\rm disrupt}$ = \\{100, 1,000, 10,000 years\\}.  This confirms the well-known result that one cannot reproduce the Oort spike without some model for comet disruption or fading that removes comets after only a few appearances \\citep[e.g.,][]{Wiegert99}.  \n\\par\nWe also notice a broader ``spike\" of comets around the original energy for comets with $a_i = 5,\\!000$ or 10,000 AU, and perihelion distances outside 30 AU.  A typical kick in energy at 30 AU might be on the order of $\\Delta x = 10^{-5}$ AU$^{-1}$ \\citep{Duncan87}, so comets with $a_i$ small enough to not be rapidly carried away by the tide will show a broad peak around their original energy at large perihelion distance.\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=.5\\textwidth]{inclination.pdf}\n\\caption{Distribution of the cosine of the ecliptic inclination angle for different ranges of current semi-major axis (different color curves) and initial semi-major axis (different panels). }\n\\label{inclination}\n\\vspace{-.05cm}\n\\end{figure}\n\n\n\\subsection{Orbital Inclination}\nFigure \\ref{inclination} shows the orbital inclination distribution (in ecliptic coordinates) of the visible NICs.  The longest period NICs are preferentially retrograde.  In a random walk model, the density of comets varies inversely with the step size.  The energy kick is 2-3 times larger for a prograde comet \\citep{Duncan87}, translating into an expected prograde fraction of $0.25$ to $0.33$.    \n\\par\nWe find that our error bars on the prograde fraction are large for the entire comet population, so for the following analysis we split the population into two groups depending on the semi-major axes of the comets.   Once again we find that elements of comets with large values of $a_c$ have less statistical noise.  We obtain prograde fractions among the visible comets of $0.27 \\pm0.03, 0.29 \\pm 0.04, 0.29 \\pm 0.03$, and $0.40 \\pm 0.02$ for $a_i =$ 5,000, 10,000, 20,000, and 50,000 AU respectively, for the comets with $a_c > 1,\\!000$ AU.  For comets with $a_c < 1,\\!000$ AU, we find prograde fractions of $0.42 \\pm 0.15, 0.26 \\pm 0.13, 0.32 \\pm 0.09$, and $0.40 \\pm 0.09$.  These data are consistent with the random walk model.\n\\par\n While the simulation data agree with the random walk model, they contradict the observations.  There is only a slight preference in the observational data for retrograde comets with high perihelion.  64 out of 110 comets (58\\%) with period greater than 200 years and perihelion greater than 5 AU in the database at \\url{http:\/\/ssd.jpl.nasa.gov\/dat\/ELEMENTS.COMET} are retrograde.\n\\subsection{Size Distribution}\nA bigger telescope enables us to see rare large comets because it can search more volume.  It is impossible to say exactly what size distribution of comets to expect in the observed sample, however we can make an estimate based on extrapolation of the size distribution from \\citet{Fernandez12}.  It is instructive to first consider the zero-planet model with a fixed power-law for the size distribution.  \n \\begin{figure}\n\\centering\n\\includegraphics[width=.5\\textwidth]{sizePlot.pdf}\n\\caption{Number of predicted detections of bodies in the range 5 AU $<R<$ 45 AU as a function of $r$ given the size distribution assumed in Equation \\eqref{dNdr11}.  The blue and orange points are unmoved, and the green and red are shifted 3\\% left and right respectively. }\n\\label{sizePlot}\n\\vspace{-.05cm}\n\\end{figure}\nIn Appendix \\ref{sect:isovis}, we calculate the number of NICs greater than a certain size $r$ expected to be seen in the zero-planet model.  We find in Equation \\eqref{eqNL} that the number visible with size greater than $r$ is given by $N = 55.2\/r^{1.50}$ assuming $10^{11}$ comets greater than 1 km at 10,000 AU, but the results depend sensitively on the assumed number density and semi-major axis.   Additionally, due to the concentration at larger orbital radii, we actually expect to see objects much larger than one would infer from the zero-planet model used to derive Equation \\eqref{eqNL}.  \n\\par\nFigure \\ref{sizePlot} shows the expected number of observations of NICs larger than a given size in the distance range 5 AU $<R<$ 45 AU.  This calculation assumes $10^{11}$ comets greater than 1 km at the given initial semi-major axis in the Oort cloud and slope of the size distribution given in Equation \\eqref{dNdr}.  Exclusion of comets with orbital radii less than 5 AU should have a negligible effect on the counts except at the smallest sizes, as the concentration factors are lower there, and it is a negligible fraction of the visible volume for the larger comets.  Exclusion of comets with orbital radii greater than 45 AU has no effect for the size of comets that we have plotted, since none of them would be visible past 45 AU.  We see that assuming the size distribution from Equation \\eqref{dNdr} holds to these sizes, we observe on the order of 10 NICs greater than 80 km in size if the majority of the NICs are coming from $a_i =$ 5,000 AU, and a few if the majority of the NICs are coming from $a_i \\geq $ 10,000 AU.\n\\par\n\\section{CONCLUSIONS}\nWe simulated the evolution of NICs originating in the Oort cloud as they interact with the giant planets.  We used these simulations to create a catalog of simulated comet positions and velocities.  We observe different distributions of orbital elements including perihelion distance, semi-major axis and inclination depending on the semi-major axes at which the NICs originate.  Observations by LSST will therefore let us determine the absolute numbers of comets in the Oort cloud as a function of semi-major axis, and test Oort's standard model for the origin of comets.  The distribution of NICs outside the orbits of Jupiter and Saturn will provide direct evidence for the presence or absence of the hypothesized ``inner Oort cloud\" corresponding to semi-major axes between $5,\\!000$ and $20,\\!000$ AU.\n\\par\nOne surprising result is that we expect at least tens of percent of the comets observed by LSST in the outer solar system to have been interacting with the giant planets for more than 1 Gyr.  This result makes the interpretation of the comet population detected by LSST more difficult and interesting, since the population and spatial distribution of comets in the Oort cloud almost certainly evolves on timescales of a few Gyr.\n\\par\nWe will also get a better measurement of the Oort spike --- the excess of comets in nearly parabolic orbits --- as we will be able to measure high-precision orbits in a regime where comets are likely unaffected by non-gravitational forces.  This will put constraints on models of comet fading, as well as the original semi-major axes of comets.\n\\par\nWe will furthermore be able to constrain the size distribution of Oort cloud comets out conservatively to several tens of kilometers, and perhaps even to larger bodies, depending on the size distribution and number density of comets.  Our results are based on a relatively steep slope for the size distribution of comets, $dN\\propto r^{-3.76}$ (Equation \\ref{dNdr}) and may substantially underestimate the total number of comets that will be discovered at large distances by LSST.\n\\par\nWe thank the referee, Ramon Brasser, for helpful and constructive comments and advice.\n \n \n\n \n\n \n \n\\bibliographystyle{apj}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\\section*{Introduction}\nLet $A$ be a ring (with this term we  will always mean a commutative ring with multiplicative identity) and let ${\\rm Spec}(A)$ be the prime spectrum of $A$, i.e., the set of all prime ideals of $A$. As it is well known, ${\\rm Spec}(A)$ has a natural structure of topological space, whose closed sets are the sets of the form $V(\\mathfrak a):=\\{\\mathfrak p\\in {\\rm Spec}(A):\\mathfrak p\\supseteq \\mathfrak a\\}$, where $\\mathfrak a$ runs in the collection of all ideals of $A$. The topology on ${\\rm Spec}(A)$ obtained in this way is the so called \\emph{Zariski topology}. A topological space is called \\emph{spectral} if it is homeomorphic to ${\\rm Spec}(A)$ (endowed with the Zariski topology), for some ring $A$. In \\cite{ho} the author gives a characterization  of spectral spaces. More precisely, he shows that a topological space is spectral if and only if it is compact (i.e., every open cover has a finite subcover), it has a basis of open and compact subspaces that is closed under finite intersections, and every irreducible closed subset has a unique generic point (i.e., it is the closure of a unique point).  \n\nThus the Zariski topology on the prime spectrum of a ring $A$ is always  T$_0$, but it is almost never Haudorff. More precisely, it satisfies the Hausdorff axiom if and only if the Krull dimension of $A$ is 0. Thus, the following question arises naturally. How can the Zariski topology be refined in order to make ${\\rm Spec}(A)$ an Hausdorff space without losing compactness? The following classical construction (see \\cite[Chapter 3, Exercises 27--30]{AM} and \\cite{ho}) answers completely this question: consider the set ${\\rm Spec}(A)$ and let $\\mathcal P:=\\{D_f:={\\rm Spec}(A)\\setminus V(fA):f\\in A\\}$ denote the collection of the so called \\emph{principal open subsets} of ${\\rm Spec}(A)$ (this is clearly a basis of the Zariski topology). Consider on ${\\rm Spec}(A)$ the coarsest topology for which $\\mathcal P$ is a collection of clopen sets. This new topology is known as \\emph{the constructible} (or  \\emph{patch}) topology on ${\\rm Spec}(A)$, and it makes ${\\rm Spec}(A)$ a compact and Hausdorff space. The reason of the name \\emph{constructible} is mainly historical: indeeed, when ${\\rm Spec}(A)$, with the Zariski topology, is a Noetherian space, then the clopen subsets of the constructible topology are precisely the \\emph{constructible subsets}, in the sense of Chevalley (i.e., are finite unions of locally closed subsets of ${\\rm Spec}(A)$). \n\nIn a very recent paper by Fontana and Loper (see \\cite{folo}) it is shown that the constructible topology is identical to ``another'' topology, defined by using ultrafilters. More precisely, let $Y$ be a subset of ${\\rm Spec}(A)$ and let $\\mathscr U$ be an ultrafilter on $Y$. According to \\cite[Lemma 2.4]{calota}, the subset $\\mathfrak p_{Y,\\mathscr U}:=\\{x\\in A:V(x)\\cap Y\\in \\mathscr U\\}$ of $A$ is a prime ideal of $A$, called \\emph{the ultrafilter limit point of $Y$, with respect to $\\mathscr U$}. Ultrafilter limit points of a subset $Y$ of ${\\rm Spec}(A)$ are not always elements of $Y$: for example, if $A$ is the ring of integers, $Y:={\\rm Max}(A)$ and  $\\mathscr U$ is a nontrivial ultrafilter on $Y$ (i.e., an ultrafilter whose elements are infinite sets), then it is very easy to check that $\\mathfrak p_{Y,\\mathscr U}=(0)$. Thus the following definition is natural: define a subset $Y$ of ${\\rm Spec}(A)$ to be \\emph{closed under ultrafilters} if it contains all its ultrafilter limit points, with respect each ultrafilter on $Y$. Then, in \\cite{folo} it is proved that the ultrafilter closed sets of ${\\rm Spec}(A)$ are the closed sets for a topology on ${\\rm Spec}(A)$, called \\emph{the ultrafilter topology}, and it is shown that the  constructible topology and the ultrafilter topology are the same topology. Recently in \\cite{fifolo1} and \\cite{fifolo2} the authors gave  several applications of the constructible topology on the spectral space ${\\rm Zar}(K|A)$ of all the valuation domains of a field $K$ containing a fixed subring $A$ of $K$. Other important contributions on the algebraic applications of the topological properties of the Riemann--Zariski space ${\\rm Zar}(K|A)$ were given for instance in \\cite{ol} and \\cite{ol1}.\n\nInspired by the idea given in \\cite{folo}, the first goal of this paper is to define the ultrafilter topology in a more general setting. Let $X$ be a set and $\\mathcal F$ be a nonempty collection of subsets of $X$. To any subset $Y$ of $X$ and any ultrafilter $\\mathscr U$ on $Y$, we can associate a  set, namely $Y_{(X,\\mathcal F)}(\\mathscr U)$, whose points are the points $x$ of $X$ such that, for any member $F\\in\\mathcal F$, we have $x\\in F$ if and only if $F\\cap Y\\in \\mathscr U$. As the reader will see in Example \\ref{gioco}(\\ref{giocospec}), in this general setting $X$ plays the role of the space of prime ideals ${\\rm Spec}(A)$ of a ring $A$, $\\mathcal F$ the role of the collection $\\mathcal P$ of the principal open subsets of ${\\rm Spec}(A)$, and $Y_{(X,\\mathcal F)}(\\mathscr U)$ the role of the ultrafilter limit point $\\mathfrak p_{Y,\\mathscr U}$ of $Y$, with respect to $\\mathscr U$. Then we  define the notion of $\\mathcal F-$stable under ultrafilters subset of $X$ (see Definition \\ref{stable}) and we show that the sets that are $\\mathcal F-$stable under ultrafilters are the closed sets for a topology. Thus, given a set $X$ and a collection of subsets $\\mathcal F$ of $X$, we can define a new topology, and we call it the $\\mathcal F-$ultrafilter topology. After several examples, we  discuss the main properties of this kind of topology. For example, we  show that in this topology $\\mathcal F$ is always a collection of clopen sets. Also we describe a closure of a set and characterize when the $\\mathcal F-$ultrafilter topology is compact. From this we deduce the equality of the constructible topology and the ultrafilter topology in the particular case in which $X$ is the prime spectrum of a ring (\\cite[Theorem 8]{folo}). In Section 3 we give an application of the general theory developed before. More precisely, we study the case in which $(X, \\mathcal T)$ is a topological space and $\\mathcal F$ is a basis of this topology. In Proposition \\ref{B-ultra} we describe how the original topology $\\mathcal T$ and the $\\mathcal F-$ultrafilter topology are related. In Corollary \\ref{spectralcriterion} we deduce a new criterion, based on the use of ultrafilters, to decide when a topological space is spectral. In Propositions \\ref{overrings} and \\ref{icoverrings} we apply it to give new examples of spectral spaces. Finally, in Theorem \\ref{scheme-constructible} we show that the constructible topology defined by Grothendiek (see \\cite[pag. 337, (7.2.11)]{EGA}) on the underlying topological space of a scheme can be seen as a particular case of the construction introduced in Section 2. \n\\section{Notation and preliminaries}\nWe begin by giving some notation and preliminary results.  If $X$ is a set, the collection of all the subsets (resp., finite subsets) of $X$ will be denoted by $\\mathcal B(X)$ (resp., $\\mathcal B_{\\tt fin}(X)$). If $\\mathcal F\\subseteq \\mathcal B(X)\\setminus\\{\\emptyset\\}$, we shall denote by $\\bigcap \\mathcal F$ (resp., $\\bigcup \\mathcal F$)  the intersection (resp., the union) of all  members of $\\mathcal F$. If $X$ is a topological space, then we will denote by ${\\rm Clop}(X)$ the collection of all the clopen subsets of $X$. If $Y$ is a subset of $X$, we will denote by ${\\rm Ad}(Y)$ the closure of $Y$.\n\nRecall that given a set $X$ and a nonempty collection $\\mathscr F$ of subsets of $X$, we say\n that $\\mathscr F$ is \\emph{a filter on} $X$ if the following properties hold:\n\\begin{enumerate}[(i)]\n\\item $\\emptyset\\notin \\mathscr F$. \n\\item If $Y,Z\\in \\mathscr F$, then $Y\\cap Z\\in \\mathscr  F$. \n\\item If $Z\\in \\mathscr F$ and $Z\\subseteq Y\\subseteq X$, then $Y\\in \\mathscr F$.\n\\end{enumerate}\nA filter  $\\mathscr F$ is \\emph{an ultrafilter on} $X$ if $\\mathscr F$ is a maximal element (under inclusion) in the set of all filters on $X$. We shall denote by $\\beta X$ the set of all the ultrafilters on $X$. \\\\\\\\\nNow we state without proof some easy and well known properties of ultrafilters on sets (see \\cite{je}). \n\\begin{oss}\\label{beginning}\nLet $X$ be a set.\n\\begin{enumerate}[\\rm (i)]\n\\item If $x\\in X$, then the collection of sets \n$${\\beta_X}^x:=\\{Y\\subseteq X:x\\in Y\\}$$ \nis an ultrafilter on $X$, and it is called {\\it the trivial ultrafilter generated by }$x$.\n\\item\\label{ultracaratt} If  $\\mathscr F$ is a filter on  $X$, the following conditions are equivalent:\n\\begin{enumerate}[\\rm (a)]\n\\item $\\mathscr F$ is an ultrafilter on $X$.\n\\item For each subset $Y$ of $X$, then $Y\\notin \\mathscr F$ implies $X\\setminus Y\\in \\mathscr F$. \n\\item If $Y$ and $Y_0$ are subsets of $X$ such that $Y\\cup Y_0\\in \\mathscr F$, then $Y\\in \\mathscr F$ or $Y_0\\in \\mathscr F$. \n\\end{enumerate}\n\\item\\label{zornultra} If $\\mathscr F$ is a collection of subsets of $X$ with finite intersection property, then there exists an ultrafilter $\\mathscr U$ on $X$ containing $\\mathscr F$ (by a straightforward application of Zorn's Lemma).\n\\item\\label{giu} If $\\mathscr U$ is an ultrafilter on $X$ and $Y\\in \\mathscr U$, then \n$$\n\\mathscr U_{ Y}:=\\{Y\\cap U:U\\in\\mathscr U\\}\n$$\nis an ultrafilter on $Y$ contained in $\\mathscr U$.\n\\item\\label{su}  If $f:X\\longrightarrow Y$ is a function and $\\mathscr U$ is an ultrafilter on $X$, then $\\mathscr U^f:=\\{T\\subseteq Y:f^{-1}(T)\\in\\mathscr U\\}$ is an ultrafilter on $Y$.\\\\\nIn particular, If $Z\\subseteq Y\\subseteq X$ and $\\mathscr U$ is an ultrafilter on $Z$, then \n$$\n\\mathscr U^{ Y}:=\\{T\\subseteq Y:T\\cap Z\\in \\mathscr U\\}\n$$\nis an ultrafilter on $Y$ containing $\\mathscr U$. In fact, if $\\iota:X\\longrightarrow Y$ is the inclusion, then $\\mathscr U^Y=\\mathscr U^\\iota$.\n\\end{enumerate} \n\\end{oss}\n\\section{The construction}\n Let $X$ be a set and $\\mathcal F$ be a nonempty collection of subsets of $X$. For each $Y \\subseteq X$ and each ultrafilter $\\mathscr U$ on  $Y$, we define\n$$ \nY_{(X,\\mathcal F)}(\\mathscr U):=Y_{\\mathcal F}(\\mathscr U):=\\{x\\in X:[\\forall F\\in \\mathcal F, x\\in F\\Longleftrightarrow F\\cap Y\\in \\mathscr U]\\}.\n$$\nSince $\\mathcal F$ will be almost always a fixed collection of subsets of a fixed set $X$, we will denote the set $Y_{(X,\\mathcal F)}(\\mathscr U)$ simply by $Y(\\mathscr U)$, when no confusion can arise.\n\\begin{ex}\\label{gioco}\n Let $X$ be a set, $\\mathcal F$ be a nonempty collection of subsets of $X$ and $Y$ be a subset of $X$. \n\\begin{enumerate}[(1)]\n\\item If  $y\\in Y$ and $\\beta^y_Y$ is, as usual, the trivial ultrafilter on $Y$ generated by $y$, then $y\\in Y_{\\mathcal F}(\\beta^y_Y)$. \n\\item\\label{giocospec} Let $A$ be a ring, $Y$ be a subset of ${\\rm Spec}(A)$ and $\\mathscr U$ be an ultrafilter on $Y$. Set, as before, \n$$\n\\mathfrak p_{\\mathscr U}:=\\{x\\in A:V(x)\\cap Y\\in \\mathscr U\\}.\n$$\nThen, if $\\mathcal P:=\\{D_a:a\\in A\\}$ is the collection of all the principal open subsets of ${\\rm Spec}(A)$, the equality $Y_{\\mathcal P}(\\mathscr U)=\\{\\mathfrak p_{\\mathscr U}\\}$ holds. As a matter of fact, fix $\\mathfrak p\\in {\\rm Spec}(A)$. Then, by definition, $\\mathfrak p\\in Y_{\\mathcal P}(\\mathscr U)$ if and only if the following statement is true:\n$$\n\\mbox{ for any }a\\in A, \\mathfrak p\\in D_a \\Longleftrightarrow D_a\\cap Y\\in \\mathscr U. \n$$\nObviously, the previous statement can be written in the following (equivalent) way:\n$$\n\\mbox{ for any }a\\in A, a\\in \\mathfrak p \\Longleftrightarrow V(a)\\cap Y\\in \\mathscr U. \n$$\nNow it follows immediately that $\\mathfrak p\\in Y_{\\mathcal P}(\\mathscr U)$ if and only if $\\mathfrak p=\\mathfrak p_{\\mathscr U}$. \n\\item\\label{giocozar} Let $K$ be a field and $A$ be a subring of $K$. Denote by ${\\rm Zar}(K|A)$ the set of all the valuation domains of $K$ containing $A$ as a subring. Let $\\mathcal Q:=\\{B_F:={\\rm Zar}(K|A[F]):F\\in \\mathcal B_{\\tt fin}(K)\\}$ be the natural basis of open sets of the Zariski topology of ${\\rm Zar}(K|A)$. If $Y$ is a subset of ${\\rm Zar}(K|A)$ and $\\mathscr U$ is an ultrafilter on $Y$, set\n$$\nA_{\\mathscr U}:=\\{x\\in K:B_{\\{x\\}}\\cap Y\\in \\mathscr U \\}\n$$\nThen, as in (2), we have $Y_{\\mathcal Q}(\\mathscr U)=\\{A_{\\mathscr U}\\}$. \n\\end{enumerate}\n\\end{ex}\n\\begin{oss}\nLet $X$ be a set and $\\mathcal F$ be a nonempty collection of subsets of $X$ that is closed under complements. Then, for any subset $Y$ of $X$ and any ultrafilter $\\mathscr U$ on $Y$, we have\n$$\nY_{\\mathcal F}(\\mathscr U)=\\bigcap \\{F\\in \\mathcal F: F\\cap Y\\in\\mathscr U\\}.\n$$\nThe inclusion $\\subseteq$ is clear, by  definition. Conversely, take an element $x\\in \\bigcap \\{F\\in\\mathcal F:F\\cap Y\\in \\mathscr U\\}$, and fix a set $F\\in \\mathcal F$. If $Y\\cap F\\in\\mathscr U$, then $x\\in F$, by assumption. Conversely, suppose that $x\\in F$. If $Y\\cap F\\notin\\mathscr U$, then $Y{\\setminus} F=(X{\\setminus} F)\\cap Y\\in \\mathscr U$, since $\\mathscr U$ is an ultrafilter on $Y$. Also, by assumption, $X{\\setminus} F\\in \\mathcal F$ and thus $x\\in X{\\setminus} F$, being $x$ an element of the intersection, and this is a contradiction.  \n\\end{oss}\n\\begin{defn}\\label{stable}\nLet $X$ be a set and $\\mathcal F$ be a nonempty collection of subsets of $X$. Then, we say that a subset $Y$ of $X$ is {\\rm $\\mathcal F-$stable under ultrafilters} if $Y(\\mathscr U)\\subseteq Y$, for each ultrafilter $\\mathscr U$ on $Y$. \n\\end{defn}\n\\begin{ex} Let $A$ be a ring (resp. $K$ be a field and $S$ be a subring of $K$). Keeping in mind the notation and the statements  given in Example \\ref{gioco} it follows immediately that a subset $Y$ of ${\\rm Spec}(A)$ (resp. ${\\rm Zar}(K|A)$) is $\\mathcal P-$stable under ultrafilters (resp. $\\mathcal Q-$stable under ultrafilters) if and only if it is closed in the ultrafilter topology of ${\\rm Spec}(A)$ (resp. ${\\rm Zar}(K|A)$). \n\\end{ex}\nThe following easy and technical lemma will allow us to show that the ultrafilter topology is a very particular case of a more general construction. \n\\begin{lem}\\label{technical}\nLet $X$ be a set, $\\mathcal F$ be a given nonempty collection of subsets of $X$ and $ Y\\subseteq Z\\subseteq X$. Let  $\\mathscr U$ be an ultrafilter on $Y$, $T\\in \\mathscr U$ and, as in Remark  \\ref{beginning}(iv,v), set\n$$\n\\mathscr U_T:=\\{U\\cap T:U\\in\\mathscr U\\}\\qquad \\mathscr U^Z:=\\{Z'\\subseteq Z:Z'\\cap Y\\in \\mathscr U\\}\n$$\nThen we have \n$$ \nY(\\mathscr U)=T(\\mathscr U_T)=Z(\\mathscr U^Z).\n$$\n\\end{lem}\n\\begin{proof} We shall prove only the inclusion $Y(\\mathscr U)\\subseteq T(\\mathscr U_T)$. The others are shown with the same straightforward arguments. Let $x\\in Y(\\mathscr U)$ and $F\\in \\mathcal F$. We need to show that $x\\in F$ if and only if $F\\cap T\\in \\mathscr U_T$. Assume $x\\in \\mathcal F$. Then, $F\\cap Y\\in \\mathscr U$ and $F\\cap T=(F\\cap Y)\\cap T\\in \\mathscr U_T$, by definition. Assume $F\\cap T\\in \\mathscr U_T$. Since $\\mathscr U_T\\subseteq \\mathscr U$ and $F\\cap T\\subseteq F\\cap Y$, then $F\\cap Y\\in \\mathscr U$ and thus $x\\in F$. \n\\end{proof}\n\\begin{prop}\\label{F-ultratop}\nLet $X$ be a set and $\\mathcal F$ be a nonempty collection of subsets of $X$. Then, the family of all the subsets of $X$ that are $\\mathcal F-$stable under ultrafilters is the collection of the closed sets for a topology on $X$. We will call it {\\rm the $\\mathcal F-$ultrafilter topology on $X$}, and denote by $X^{^{\\mathcal F-{\\rm ultra}}}$ the set $X$ endowed with the $\\mathcal F-$ultrafilter topology.\n\\end{prop}\n\\begin{proof}\nLet $C,C_0$ be  $\\mathcal F-$stable under ultrafilters subsets of $X$, and $\\mathscr U$ be an ultrafilter on $Y:=C\\cup C_0$. By Remark \\ref{beginning}(ii), we can assume that $C\\in \\mathscr U$. Then, by hypothesis and Lemma \\ref{technical}, we have $Y(\\mathscr U)=C(\\mathscr U_C)\\subseteq C\\subseteq Y$, and thus $Y$ is $\\mathcal F-$stable under ultrafilters. \n\nNow, let $\\mathcal G$ be a collection of  $\\mathcal F-$stable  under ultrafilters subsets of $X$ and let $\\mathscr U$  be an ultrafilter on $Z:=\\bigcap \\mathcal G$. For each $C\\in \\mathcal G$, we have $C(\\mathscr U^C)=Z(\\mathscr U)$ (by Lemma \\ref{technical}), and thus $Z(\\mathscr U)\\subseteq Z$. This completes the proof.\n\\end{proof}\n\\begin{oss}\\label{esempi1}\nLet $X$ be a set.\n\\begin{enumerate}[\\rm (1)]\n\\item The $\\mathcal B(X)-$ultrafilter topology on $X$ is the discrete topology on $X$.\n\\item The $\\{X\\}-$ultrafilter topology on $X$ is the chaotic topology.\n\\item \\label{exspec} Let $A$ be a ring and $\\mathcal P$ be the collection of all the principal open subsets of $X:={\\rm Spec}(A)$. Then, the $\\mathcal P-$ultrafilter topology of $X$ is equal to the ultrafilter topology studied in \\cite{folo}.\n\\item\\label{exzar} Let $K$ be a field, $A$ be a subring of $K$ and $\\mathcal Q$ be the natural basis of open sets for the Zariski topology on ${\\rm Zar}(K|A)$. Then, the $\\mathcal Q-$ultrafilter topology is equal to the ultrafilter topology on ${\\rm Zar}(K|A)$. \n\\item\\label{finer} If $\\mathcal F\\subseteq \\mathcal G$ are collections of subsets of $X$, then the $\\mathcal G-$ultrafilter topology is finer than or equal to the $\\mathcal F-$ultrafilter topology. In fact, for each subset $Y$ of $X$ and each ultrafilter $\\mathscr U$ on $Y$, we have $Y_{\\mathcal G}(\\mathscr U)\\subseteq Y_{\\mathcal F}(\\mathscr U)$. \n\\end{enumerate}\n\\end{oss}\n\\begin{prop}\\label{prebool}\nLet $X$ be a set and $\\mathcal F$ be a nonempty collection of subsets of $X$. Set \n$$\\mathcal F_\\sharp:=\\left\\{\\bigcap\\mathcal  G:\\mathcal G\\in\\mathcal B_{\\tt fin}(\\mathcal F)\\right\\} \\quad \n\\mathcal F^\\sharp:=\\left\\{\\bigcup \\mathcal G:\\mathcal G\\in\\mathcal B_{\\tt fin}(\\mathcal F)\\right\\}\n$$\n$$\n \\mathcal F^-:=\\{X{\\setminus} F:F\\in \\mathcal F\\} .\n$$\nThen, the $\\mathcal F-$ultrafilter topology, the $\\mathcal F_{\\sharp}-$ultrafilter topology and the $\\mathcal F^\\sharp-$ultrafilter topology are the same.\n\\end{prop}\n\\begin{proof}\nBy  Remark \\ref{esempi1}(\\ref{finer}) and the obvious inclusion $\\mathcal F\\subseteq \\mathcal F^\\sharp$, it is enough to show that the $\\mathcal F^\\sharp-$ultrafilter topology is finer or equal than the $\\mathcal F-$ultrafilter topology. Let $Y$ be an $\\mathcal F^\\sharp-$stable under ultrafilter subset of $X$, $\\mathscr U$ be an ultrafilter on $Y$, $x\\in Y_{\\mathcal F}(\\mathscr U)$, $\\mathcal G:=\\{F_1,{\\ldots},F_n\\}\\in\\mathcal B_{\\tt fin}(\\mathcal F)$ and $G:=\\bigcap \\mathcal G$. We want to show that $x\\in G$ if and only if $G\\cap Y\\in \\mathscr U$. If $x\\in G$, then $F_i\\cap Y\\in \\mathscr U$, for $i=1,{\\ldots}, n$, and thus $G\\cap Y\\in \\mathscr U$. Since $G\\cap Y\\subseteq F_i\\cap Y$, for each $i=1,{\\ldots}, n$, if $G\\cap Y\\in \\mathscr U$, then it follows immediately that $F_i\\cap Y\\in\\mathscr U$, for $i=1,{\\ldots},n$, and thus $x\\in G$, by definition. This proves that $Y_{\\mathcal F}(\\mathscr U)\\subseteq Y_{\\mathcal F^\\sharp}(\\mathscr U)$. Thus it is clear that the $\\mathcal F-$ultrafilter topology and the $\\mathcal F_\\sharp-$ultrafilter topology are the same. By a similar argument it can be shown that $Y_{\\mathcal F}(\\mathscr U)=Y_{\\mathcal F^\\sharp}(\\mathscr U)=Y_{\\mathcal F^-}(\\mathscr U) $. Thus the proof is complete.\n\\end{proof}\n\\begin{cor}\\label{bool}\nLet $X$ be a set, $\\mathcal F$ be a nonempty collection of subsets of $X$ and ${\\rm Bool}(\\mathcal F)$ be the boolean subalgebra of $\\mathcal B(X)$ generated by $\\mathcal F$. Then the $\\mathcal F-$ultrafilter topology and the ${\\rm Bool}(\\mathcal F)-$ultrafilter topology are the same. \n\\end{cor}\n\\begin{proof}\nLet $Y$ be a nonempty subset of $X$ and $\\mathscr U$ be an ultrafilter on $Y$. Keeping in mind the proof of Proposition \\ref{prebool}, it follows that $Y_{\\mathcal F}(\\mathscr U)=Y_{\\mathcal F\\cup \\mathcal F^-}(\\mathscr U)$. Thus the $\\mathcal F-$ultrafilter topology and the $(\\mathcal F\\cup \\mathcal F^-)-$ultrafilter topology are the same. Since, obviously, ${\\rm Bool}(\\mathcal F)=\\left(\\left(\\mathcal F\\cup  \\mathcal F^-\\right)_\\sharp\\right)^\\sharp$, the statement follows by using Proposition \\ref{prebool}.\n\\end{proof}\n\\begin{ex}\\label{opencompact}\nPreserve the notation given in Remark \\ref{esempi1}(\\ref{exspec}), and let $\\mathcal K$ be the collection of all the open and compact subspace of $X:={\\rm Spec}(A)$. Since every element of $\\mathcal K$ is a finite union of members of $\\mathcal P$, it follows by Corollary \\ref{bool} that the $\\mathcal P-$ultrafilter topology and the $\\mathcal K-$ultrafilter topology on $X$ are the same. \n\\end{ex}\n\\begin{prop}\\label{Fclopen}\nLet $X$ be a set and $\\mathcal F$ be a nonempty collection of subsets of $X$. Then, the following statements hold.\n\\begin{enumerate}[\\rm (1)]\n\\item\\label{Fclopen1} ${\\rm Bool}(\\mathcal F)\\subseteq {\\rm Clop}(X^{^{\\mathcal F-{\\rm ultra}}})$.\n\\item\\label{FclopenHausdorff} If, for each pair of distinct points $x,y\\in X$ there exists a set $F\\in \\mathcal F$ such that $x\\in F$ and $y\\notin F$, then $X^{^{\\mathcal F-{\\rm ultra}}}$ is an Hausdorff and totally disconnected  space, and $Y(\\mathscr U)$ has at most an element, for each $Y\\subseteq X$ and $\\mathscr U\\in \\beta Y$. \n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\nSince ${\\rm Clop}(X^{^{\\mathcal F-{\\rm ultra}}})$ is a boolean algebra, it is enough to show that $\\mathcal F\\subseteq {\\rm Clop}(X^{^{\\mathcal F-{\\rm ultra}}})$.\nPick a set $E\\in \\mathcal F$. If $\\mathscr U$ is an ultrafilter on $E$ and $x\\in E(\\mathscr U)$, then the statement $x\\in F \\Longleftrightarrow F\\cap E\\in \\mathscr U$ holds for each $F\\in \\mathscr F$, and in particular for $F:=E$. Then, $x\\in E$. Thus $E$ is closed in the $\\mathcal F-$ultrafilter topology.\n\nNow let $\\mathscr V$ be an ultrafilter on $Z:=X{\\setminus} E$ and $x\\in Z(\\mathscr V)$. The statement $x\\in E\\Longleftrightarrow E\\cap Z\\in \\mathscr V$ holds and thus $x\\in Z$. Then, $E$ is clopen. Thus (1) is  proved.\n\n(2) The fact that $X^{^{\\mathcal F-{\\rm ultra}}}$ is an Hausdorff and totally disconnected space  follows immediately by (1) and the extra assumption on $\\mathcal F$.  For the second part of (2), assume, by contradiction, that there exist distinct elements $x,y\\in Y(\\mathscr U)$, and pick, by hypothesis, a set $F\\in\\mathcal F$, such that $x\\in F$ and $y\\notin F$. Thus, $\\emptyset=(F\\cap Y)\\cap (Y{\\setminus} F)\\in \\mathscr U$.\n\\end{proof}\n\\begin{prop}\nLet $X$ be a set, $\\mathcal F$ be a nonempty collection of subsets of $X$, $\\emptyset\\neq Y\\subseteq X$ and $\\mathscr U$ an ultrafilter on $Y$. Then, for each topology on $X$ for which $\\mathcal F$ is a collection of clopen sets,  $Y(\\mathscr U)$ is closed. In particular, $Y(\\mathscr U)$ is closed in the $\\mathcal F-$ultrafilter topology.\n\\end{prop}\n\\begin{proof}\nLet $x_0\\in {\\rm Ad}(Y(\\mathscr U))$ and $E\\in\\mathcal F$. If $x_0\\in E$, then $E$ is an open neighborhood of $x_0$, by assumption, and thus there exists an element $y_1\\in Y(\\mathscr U)\\cap E$. By definition, it follows that, for each $F\\in\\mathcal F$, $y_1\\in F \\Longleftrightarrow F\\cap Y\\in\\mathscr U$, and thus $E\\cap Y\\in \\mathscr U$, in particular. Conversely, assume $x_0\\notin E$. Then, $X{\\setminus} E$ is an open neighborhood of $x_0$, and thus there exists $y_2\\in Y(\\mathscr U){\\setminus} E$. Hence, we have $E\\cap Y\\notin \\mathscr U$. This proves that $Y(\\mathscr U)$ is closed.\n\nThe last part of the statement follows immediately by Proposition \\ref{Fclopen}(\\ref{Fclopen1}).\n\\end{proof}\n\\begin{prop}\\label{closure}\nLet $X$ be a set and $\\mathcal F$ be a nonempty collection of subsets of $X$. Then, for each subspace $Y$ of $X^{^{\\mathcal F-{\\rm ultra}}}$, we have\n$$\n{\\rm Ad}(Y)=\\bigcup\\{Y(\\mathscr U): \\mathscr U\\in \\beta Y\\}\n$$\n\\end{prop}\n\\begin{proof}\nLet $Y\\subseteq X$, $\\mathscr U\\in \\beta Y$,  $x\\in Y(\\mathscr U)$, and $\\Omega$ be an open neighborhood of $x$. If $Y\\cap \\Omega=\\emptyset$, then $Y\\subseteq Z:=X{\\setminus}\\Omega$ and, using Lemma \\ref{technical}, we have $Y(\\mathscr U)=Z(\\mathscr U^Z)\\subseteq Z$, since $Z$ is $\\mathcal F-$stable under ultrafilters. Thus we get a contradiction, since $x\\in \\Omega$. The inclusion $\\supseteq$ follows. Conversely, pick an element $x\\in {\\rm Ad}(Y)$, and set\n$$\n\\mathcal G:=\\{\\Omega\\cap Y:\\Omega \\mbox{ open neighborhood of } x\\}.\n$$\nIt is clear that $\\mathcal G$ is a collection of subsets of $Y$ with the finite intersection property (since $x\\in {\\rm Ad}(Y))$, and thus (by Remark \\ref{beginning}(iii)) there exists an ultrafilter $\\mathscr U^*$ on $Y$ containing $\\mathcal G$. The conclusion will follow if we show that $x\\in Y(\\mathscr U^*)$. Fix $F\\in\\mathcal F$. If $x\\in F$, then $F$ is an open neighborhood of $x$, by Proposition \\ref{Fclopen}(\\ref{Fclopen1}), and thus $F\\cap Y\\in \\mathcal G\\subseteq \\mathscr U^*$. Conversely, assume $F\\cap Y\\in \\mathscr U^*$. If $x\\notin F$, then $X{\\setminus} F$ is an open neighborhood of $x$, again by Proposition \\ref{Fclopen}, and thus $(X{\\setminus} F)\\cap Y\\in \\mathcal G\\subseteq \\mathscr U^*$. It follows that $\\emptyset\\in \\mathscr U^*$, a contradiction. \n\\end{proof}\n\\begin{thm}\\label{compactness}\nLet $X$ be a set and $\\mathcal F$ be a nonempty collection of subsets of $X$. Then, the following conditions are equivalent.\n\\begin{enumerate}[\\rm(i)]\n\\item $X^{^{\\mathcal F-{\\rm ultra}}}$ is a compact topological space.\n\\item $X(\\mathscr U)\\neq \\emptyset$, for each ultrafilter $\\mathscr U$ on $X$.\n\\item If $\\mathcal H$ is a subcollection of $\\mathcal G:=\\mathcal F\\cup\\mathcal F^-$ with the finite intersection property, then $\\bigcap \\mathcal H\\neq \\emptyset$.\n\\end{enumerate}\n\\end{thm}\n\\begin{proof}\n(i)$\\Longrightarrow$(iii). It is enough to use Proposition \\ref{Fclopen}(\\ref{Fclopen1}) and compactness of $X^{\\mathcal F-{\\rm ultra}}$. \n\n(iii)$\\Longrightarrow$(ii). Let $\\mathscr U$ be an ultrafilter on $X$. Assume, by contradiction, that $X(\\mathscr U)=\\emptyset$. This means that, for each $x\\in X$ there exists a set $F_x\\in\\mathcal F$ such that exactly one of the following statements is true\n\\begin{enumerate}[(a)]\n\\item $x\\in F_x$ and $F_x\\notin \\mathscr U$.\n\\item $x\\notin F_x$ and $F_x\\in \\mathscr U$.\n\\end{enumerate}\nNow, for each $x\\in X$, set $C_x:=X{\\setminus} F_x$, if $x\\in F_x$, and $C_x:=F_x$, if $x\\notin F_x$. Then, it is clear that $\\mathcal H:=\\{C_x:x\\in X\\}$ is a subcollection of $\\mathcal G$ and that it has the finite intersection property, since $\\mathcal H\\subseteq \\mathscr U$. Thus, by assumption, there exists $x_0\\in \\bigcap \\mathcal H$. This is a contradiction, since $x\\in X{\\setminus} C_x$, for each $x\\in X$.\n\n(ii)$\\Longrightarrow$(i). Let $\\mathcal C$ be a collection of closed subsets of $X^{\\mathcal F-{\\rm ultra}}$ with the finite intersection property. By Remark \\ref{beginning}(iii), there exists an ultrafilter $\\mathscr U^*$ on $X$ such that $\\mathcal C\\subseteq \\mathscr U^*$. By assumption, we can pick a point $x^*\\in X(\\mathscr U^*)$. Now, let $C\\in \\mathcal C$. Since $C\\in \\mathscr U^*$, we have $x^*\\in X(\\mathscr U^*)=C(\\mathscr {U^*}_C)\\subseteq C$, keeping in mind Lemma \\ref{technical}. Thus $x^*\\in \\bigcap\\mathcal C$. This completes the proof. \n\\end{proof}\n\\begin{ex}\\label{ultra-comp-spec}\nLet $A$ be a ring. By Example \\ref{gioco}(\\ref{giocospec}), Remark \\ref{esempi1}(\\ref{exspec}) and Theorem \\ref{compactness} we get immediately the well known fact that the ultrafilter topology on ${\\rm Spec}(A)$ is compact.\n\\end{ex}\n\\begin{prop}\\label{coarsest}\nLet $X$ be a set and $\\mathcal F$ a nonempty collection of subsets of $X$ such that, for each couple of distinct points $x,y\\in X$, there exists a set $F\\in \\mathcal F$ such that $x\\in F$ and $y\\notin F$. If $X^{^{\\mathcal F-{\\rm ultra}}}$ is a compact topological space, then the $\\mathcal F-$ultrafilter topology is the coarsest topology for which $\\mathcal F$ is a collection of clopen sets.\n\\end{prop}\n\\begin{proof} \nDenote by $X_\\star$ the set $X$ with the coarsest topology for which $\\mathcal F$ is a collection of clopen sets. Then, the identity map $\\mathbf 1:X^{^{\\mathcal F-{\\rm ultra}}}\\longrightarrow X_\\star$ is continuous, by Proposition \\ref{Fclopen}(\\ref{Fclopen1}). Moreover, it is clear that $X_\\star$ is an Hausdorff space, by assumption. Then, $\\mathbf 1$ is an homeomorphism.\n\\end{proof}\n\\begin{cor}{\\rm (\\cite[Theorem 8]{folo})}\\label{P-ultra=cons}\nPreserve the notation of Remark \\ref{esempi1}(\\ref{exspec}). Then, the $\\mathcal P-$ultrafilter topology and the constructible topology of ${\\rm Spec}(A)$ are the same. \n\\end{cor}\n\\begin{proof}\nApply Remark \\ref{esempi1}(\\ref{exspec}) and Proposition \\ref{coarsest}.\n\\end{proof}\n\\begin{prop}\\label{functorial}\nLet $X,Y$ be sets, $\\mathcal F$ (resp. $\\mathcal G$) be a nonempty collection of subsets of $X$ (resp. $Y$). If $f:X\\longrightarrow Y$ is a function such that $\\{f^{-1}(G):G\\in\\mathcal G\\}\\subseteq \\mathcal F$, then $f:X^{^{\\mathcal F-{\\rm ultra}}}\\longrightarrow Y^{^{\\mathcal G-{\\rm ultra}}}$ is a continuous function.\n\\end{prop}\n\\begin{proof}\nLet $C$ be a closed subset of $Y$, set $\\Gamma:=f^{-1}(C)$, and let $\\mathscr U$ be an ultrafilter on $\\Gamma$. Then, it sufficies to show that $\\Gamma(\\mathscr U)\\subseteq \\Gamma$. Let $g:f^{-1}(C)\\longrightarrow C$  be the restriction of $f$ to $f^{-1}(C)$. Now, note that the collection of sets \n$\\mathscr V:=\\{D\\subseteq C:f^{-1}(D)\\in \\mathscr U\\}$ is an ultrafilter on $C$, since $\\mathscr V=\\mathscr U^g$ (see Remark \\ref{beginning}(v)). Now, take an element $x\\in \\Gamma(\\mathscr U)$, and fix a set $G\\in\\mathcal G$. If $f(x)\\in G$, then  $x\\in f^{-1}(G)\\in \\mathcal F$ (by assumption), and thus \n$\nf^{-1}(G\\cap C)=f^{-1}(G)\\cap \\Gamma\\in \\mathscr U,\n$\nsince $x\\in\\Gamma(\\mathscr U)$. This proves that, if $f(x)\\in G$, then $G\\cap C\\in \\mathscr V$. Conversely, if $G\\cap C\\in \\mathscr V$, then $f^{-1}(G)\\cap \\Gamma\\in\\mathscr U$ and, since $f^{-1}(G)\\in \\mathscr U$ and $x\\in \\Gamma(\\mathscr U)$, it follows $f(x)\\in G$. This argument shows that $f(x)\\in C(\\mathscr V)$ and, since $C$ is closed, we have $f(x)\\in C$. Then, the inclusion $\\Gamma(\\mathscr U)\\subseteq \\Gamma$ follows, and  the statement is now clear. \n\\end{proof}\n\\section{Applications}\nAn interesting case is when $\\mathcal F$ is a basis of a topology on $X$. \n\\begin{prop}\\label{B-ultra}\nLet $X$ be a topological space, $\\mathcal T$ the topology and $\\mathcal B$  a basis of open sets of $X$. Then, the following statements hold.\n\\begin{enumerate}[\\rm (1)]\n\\item The $\\mathcal B-$ultrafilter topology on $X$ is finer than or equal to the given topology $\\mathcal T$.\n\\item If $X$ satisfies the $T_0$ axiom, then the $\\mathcal B-$ultrafilter topology is Hausdorff and totally disconnected. In particular, if $X$ satisfies the $T_0$ axiom but it is not Hausdorff, then the  $\\mathcal B-$ultrafilter topology is strictly finer than the given topology $\\mathcal T$.\n\\item Assume that $X$ satisfies the $T_0$-axiom and $X^{^{\\mathcal B-{\\rm ultra}}}$ is compact. \n\\begin{enumerate}[\\rm (a)]\n\\item Then the $\\mathcal B-$ultrafilter topology is the coarsest topology on  $X$  for which $\\mathcal B$ is a collection of clopen sets.\n\\item  Moreover $X$, equipped with the topology $\\mathcal T$, is a spectral space.\n\\end{enumerate}\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\n(1) and (2) are immediate consequences of Proposition \\ref{Fclopen}. Statement (3,a) follows by applying Proposition \\ref{coarsest}. \nBy using  statement (3,a), it follows that $\\mathcal S:=\\mathcal B\\cup \\{X{\\setminus} B:B\\in\\mathcal B\\}$ is a subbasis of open sets for $X^{^{\\mathcal B-\\rm ultra}}$. Moreover, by statement (1), each member of $\\mathcal B$ is compact, with respect to the topology $\\mathcal T$, since it is closed (=compact) in $X^{^{\\mathcal B-\\rm ultra}}$. Now, let $X^{^{\\rm patch}}$ denote the set $X$ endowed with the so-called \\emph{patch topology induced by} $\\mathcal T$, i.e. the topology whose subbasis of open sets is the set $\\mathcal S_0$ of all the open and compact subsets of $X$, with respect to the topology $\\mathcal T$, and their complements in $X$. The following result will be crucial for the last part of the proof. \n\\begin{prop}\\label{patch-referee}\nPreserve the notation and the assumptions of Proposition \\ref{B-ultra}(3). Then the patch topology induced by $\\mathcal T$ is equal to the $\\mathcal B-$ultrafilter topology. \n\\end{prop}\n\\begin{proof}\nIt follows immediately that the patch topology induced by $\\mathcal T$ is finer than or equal to the $\\mathcal B-$ultrafilter topology (in fact $\\mathcal S\\subseteq \\mathcal S_0$). Now, let $S_0\\in\\mathcal S_0$. Since $\\mathcal B$ is a basis of open and compact subspaces of $X$ (with respect to $\\mathcal T$), then there exists a finite subcollection $\\mathcal C\\subseteq \\mathcal B$ such that $S_0=\\bigcup\\mathcal C$, or $S_0=\\bigcap\\mathcal C^-$. Thus $S_0$ is an open set of the $\\mathcal B-$ultrafilter topology. This proves that the patch topology induced by $\\mathcal T$ and the $\\mathcal B-$ultrafilter topology are identical.\n\\end{proof}\nThen, the fact that $X$, endowed with the topology $\\mathcal T$, is a spectral space follows by applying \\cite[Corollary to Proposition 7]{ho}. \n\\end{proof}\n\\begin{cor}\\label{spectralcriterion}\nLet $X$ be a topological space. Then, the following conditions are equivalent.\n\\begin{enumerate}[\\rm (i)]\n\\item $X$ is a spectral space.\n\\item There is a basis $\\mathcal B$ of $X$ such that $X^{^{\\mathcal B-\\rm ultra}}$ is a compact and Hausdorff topological space.\n\\item $X$ satisfies the T$_0$-axiom and there is a basis $\\mathcal B$ of $X$ such that $X_{\\mathcal B}(\\mathscr U)\\neq \\emptyset$, for any ultrafilter $\\mathscr U$ on $X$.\n\\item $X$ satisfies the T$_0$-axiom and there is a subbasis $\\mathcal S$ of $X$ such that $X_{\\mathcal S}(\\mathscr U)\\neq \\emptyset$, for any ultrafilter $\\mathscr U$ on $X$.\n\\end{enumerate}\n\\end{cor}\n\\begin{proof}\n(i)$\\Longrightarrow$(iii) We can assume, without loss of generality, that $X={\\rm Spec}(A)$, for some ring $A$. Let $\\mathscr U$ be an ultrafilter on $X$ and $\\mathcal P$ be the basis of $X$ made up of the principal open subsets. Keeping in mind Remark \\ref{gioco}(\\ref{giocospec}), we have $X_{\\mathcal P}(\\mathscr U)=\\{\\mathfrak p_{\\mathscr U}\\}$. Thus, it sufficies to choose $\\mathcal B:=\\mathcal P$.\n\n(iii)$\\Longrightarrow$(ii). Apply Theorem \\ref{compactness} and Proposition \\ref{B-ultra}(2) to the basis $\\mathcal B$ given in condition (iii).\n\n(ii)$\\Longrightarrow$(i). It is the statement of Proposition \\ref{B-ultra}(3).\n\n(iii)$\\Longleftrightarrow$(iv). It is trivial, by Proposition \\ref{prebool} and Theorem \\ref{compactness}.\n\\end{proof}\n\n\n\n\nThe following example will show that for fixed a topological space $X$ the $\\mathcal B-$ultrafilter topology depends on the choice of the basis $\\mathcal  B$. \n\\begin{ex}\nLet $K$ be a field, $\\{T_n:n\\in\\mathbf N\\}$ be an infinite and countable collection of indeterminates over $K$ and consider the ring $A:=K[\\{T_n:n\\in\\mathbf N\\}]$. Set $X:={\\rm Spec}(A)$ and endow this set with the Zariski topology. As usual, let $\\mathcal P:=\\{D_f:f\\in A\\}$ be the basis of the principal open subsets of ${\\rm Spec}(A)$, and let $\\mathcal T:=\\{D(\\mathfrak a):=X{\\setminus} V(\\mathfrak a): \\mathfrak a \\mbox{ ideal of }A\\}$ (clearly, $\\mathcal T$ is a basis of $X$, being it the topology). We claim that the $\\mathcal P-$ultrafilter topology (i.e. the usual ultrafilter topology on  $X$) and the $\\mathcal T-$ultrafilter topology are different. Let $\\mathfrak m$ be the maximal ideal of $A$ generated by the set $\\{T_n:n\\in\\mathbf N\\}$ and set $\\mathcal F:=\\{V(T_n):n\\in \\mathbf N\\}\\cup\\{X{\\setminus} \\{\\mathfrak m\\}\\}$. It is straightforward that $\\mathcal F$ is a collection of subsets of $X$ with the finite intersection property, and thus there exists an ultrafilter $\\mathscr U$ on $X$ containing $\\mathcal F$, by virtue of Remark \\ref{beginning}(iii). We claim that the set \n$$\nX_{\\mathcal T}(\\mathscr U):=\\{\\mathfrak p\\in X:\\mbox{for each ideal }\\mathfrak a\\mbox{ of }A ,( \\mathfrak p\\in D(\\mathfrak a)\\Longleftrightarrow D(\\mathfrak a)\\in \\mathscr U)\\}\n$$\nis empty. If not, let $\\mathfrak p\\in X_{\\mathcal T}(\\mathscr U)$. Since $\\mathcal F\\subseteq \\mathscr U$, it follows that $V(T_n)\\in \\mathscr U$, for each $n\\in\\mathbf N$, and thus $T_n\\in \\mathfrak p$, for each $n\\in\\mathbf N$ (by the definition of $X_{\\mathcal T}(\\mathscr U)$). This proves that $\\mathfrak p=\\mathfrak m$. On the other hand, if we set $\\mathfrak a:=\\mathfrak m$, we have obviously $\\mathfrak m\\notin D(\\mathfrak a)$, hence $D(\\mathfrak a)\\notin \\mathscr U$, that is, $V(\\mathfrak a)=\\{\\mathfrak m\\}\\in \\mathscr U$. It follows $\\emptyset\\in\\mathscr U$, since $X\\setminus \\{\\mathfrak m\\}\\in \\mathcal F\\subseteq \\mathscr U$, a contradiction. This argument proves that $X_{\\mathcal T}(\\mathscr U)$ is empty, and thus $X^{^{\\mathcal T-{\\rm ultra}}}$ is not compact, by Theorem \\ref{compactness}. It follows that the $\\mathcal T-$ultrafilter topology and the $\\mathcal P-$ultrafilter topology on $X$ are not the same, since the $\\mathcal P-$ultrafilter topology is compact.                                             \n\\end{ex}\\label{basis}\n\\begin{prop}\\label{overrings}\nLet $B|A$ be a ring extension and ${\\rm S}(B|A)$ be the set of all the rings $C$ such that $A\\subseteq C\\subseteq B$. For each $x\\in B$ set \n$$\nU_x:=\\{C\\in {\\rm S}(B|A):x\\in C\\}\n$$\nLet ${\\rm S}(B|A)$ be endowed with the Zariski topology, i.e. the topology for which the collection $\\mathcal R:=\\{U_x:x\\in B\\}$ is a subbasis of open sets.  \nThe following statements hold.\n\\begin{enumerate}[\\rm (1)]\n\\item If $Y$ is a subset of ${\\rm S}(B|A)$ and $\\mathscr U$ is an ultrafilter on $Y$, then $A_{\\mathscr U}:=\\{x\\in B:U_x\\cap Y\\in \\mathscr U\\}$ belongs to ${\\rm S}(B|A)$. \n\\item ${\\rm S}(B|A)$ is a spectral space, endowed with the Zariski topology. \n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\n(1). Let $\\mathscr U$ be an ultrafilter on $Y$. The fact that $A_{\\mathscr U}$ is a ring is proved by using the same argument given in  \\cite{calota}. Moreover, for each $x\\in A$, $U_x\\cap Y=Y$ and thus it belongs to $\\mathscr U$. Now, the inclusion $A\\subseteq A_{\\mathscr U}$ is clear.\n\n(2). Let $\\mathscr U$ be an ultrafilter on $X:={\\rm S}(B|A)$. Keeping in mind (1) and  definitions, it is straightforward to verify that $X_{\\mathcal R}(\\mathscr U)=\\{A_{\\mathscr U}\\}$. Then the conclusion follows immediately by Corollary \\ref{spectralcriterion}.\n\\end{proof}\n\\begin{prop}\\label{icoverrings}\nLet $B|A$ be a ring extension and ${\\rm I}(B|A)$ be the set of all the rings $C\\in {\\rm S}(B|A)$ such that $C$ is integrally closed in $B$. As usual, define on ${\\rm I}(B|A)$ the Zariski topology by taking as subbasis of open sets the collection \n$$\n\\mathcal R':=\\{U_x\\cap {\\rm I}(B|A):x\\in C\\}\n$$\n(i.e. this is the subspace topology, induced by the Zariski topology on ${\\rm S}(B|A)$). Then the following statements hold.\n\\begin{enumerate}[\\rm (1)]\n\\item If $Y$ is a subset of ${\\rm I}(B|A)$ and $\\mathscr U$ is an ultrafilter on $Y$, then $A_{\\mathscr U}:=\\{x\\in C:Y\\cap U_x\\in \\mathscr U\\}$ belongs to ${\\rm I}(B|A)$. \n\\item ${\\rm I}(B|A)$ is a spectral space, endowed with the Zariski topology. \n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\n(1). Proposition \\ref{overrings}(1) implies  $A_{\\mathscr U}\\in {\\rm S}(B|A)$. Now, let $x\\in B$ be an element integral over $A_{\\mathscr U}$. Pick elements  $a_0,{\\ldots},a_{n-1}\\in A_{\\mathscr U}$  such that \n$$\nx^n+a_{n-1}x^{n-1}+{\\ldots}+a_1x+a_0=0.\n$$\nIf $C\\in Y\\cap \\bigcap_{i=0}^{n-1}U_{a_i}$, then $x$ is integral over $C$. Since $Y\\subseteq {\\rm I}(B|A)$, it follows $x\\in C$. This argument shows that $Y\\cap \\bigcap_{i=0}^{n-1}U_{a_i}\\subseteq Y\\cap U_x$. Keeping in mind that $a_0,{\\ldots},a_{n-1}\\in A_{\\mathscr U}$, it follows $Y\\cap \\bigcap_{i=0}^{n-1}U_{a_i}\\in\\mathscr U$, and thus $Y\\cap U_x\\in \\mathscr U$. Then $x\\in A_{\\mathscr U}$, and this proves that $A_{\\mathscr U}$ is integrally  closed in $B$. \n\n(2) Set $X:={\\rm I}(B|A)$. As in Proposition \\ref{overrings}(2), it is immediately verified that $X_{\\mathcal R'}(\\mathscr U)=\\{A_{\\mathscr U}\\}$, for any ultrafilter $\\mathscr U$ on $X$. Then, it suffices to apply Corollary 3.2.\n\\end{proof}\n\n\\begin{prop}\\label{comp-gen-clo}\nLet $X$ be a topological space  and $\\mathcal F$ be a collection of subsets of $X$ containing at least a basis of open sets of $X$. If $Y$ is a compact subspace of $X$, then the generic closure of $Y$ (with respect to the given topology) is closed, with respect to the $\\mathcal F-$ultrafilter topology.  \n\\end{prop}\n\\begin{proof}\nLet us denote by $Y_0$ the generic closure of $Y$ and assume, by contradiction, that $Y_0$ is not a closed subset of $X^{\\mathcal F-{\\rm ultra}}$. By definition, there exist an ultrafilter $\\mathscr U$ on $Y_0$ and a point $x_0\\in Y_0(\\mathscr U){\\setminus} Y_0$. Let $\\mathcal B$ be a basis of $X$ contained in $\\mathcal F$. \nBy the definition of the set $Y_0$, for each  $y\\in Y$ there exists an open neighborhood $\\Omega_y$ of $y$ such that $x_0\\notin\\Omega_y$. Without loss of generality, we can assume that $\\Omega_y\\in \\mathcal B$. By compactness, the open cover $\\{\\Omega_y:y\\in Y\\}$ of $Y$ admits a finite subcover, say $\\{\\Omega_{y_i}:i=1,{\\ldots},n\\}$. Now, by definition, it is immediately verified that $Y_0\\subseteq \\bigcup\\{\\Omega_{y_i}:i=1,{\\ldots},n\\}$, that is $Y_0=\\bigcup\\{\\Omega_{y_i}\\cap Y_0:i=1,{\\ldots},n\\}$, and thus, by Remark \\ref{beginning}(2), it follows that $\\Omega_{y_i}\\cap Y_0\\in \\mathscr U$, for some $i\\in\\{1,{\\ldots},n\\}$. Keeping in mind that, by assumption, $\\Omega_{y_i}\\in \\mathcal F$ and that $x_0\\in Y_0(\\mathscr U)$, it follows $x_0\\in \\Omega_{y_i}$, a contradiction. \n\\end{proof}\nFinally, we will show that the so called constructible topology of a quasi-separated scheme (see \\cite[pag. 337, (7.2.11)]{EGA}) can be presented as a particular case of the construction given in Section 2. \n\\begin{thm}\\label{scheme-constructible}\nLet $X$ be the underlying topological space of a quasi-separated scheme, and let $\\mathcal K$ be the collection of all the open and compact subspaces of $X$. Then, $\\mathcal K$ is a basis of $X$ and the constructible topology on $X$ is equal to the $\\mathcal K-$ultrafilter topology. \n\\end{thm}\n\\begin{proof}\nLet $\\mathcal A$ be any open cover of $X$ that consists only of spectral subspaces of $X$ (such an open cover exists, since $X$ is the underlying topological space of a scheme). For any $U\\in\\mathcal A$, let $\\mathcal B_U$ be the basis of $U$ made up of the open and compact subspaces of $U$, and set $\\mathcal B:=\\bigcup\\{\\mathcal B_U:U\\in\\mathcal A\\}$. Then $\\mathcal B$ is a basis of $X$, and it is straightforward that $\\mathcal B\\subseteq \\mathcal K\\subseteq {\\rm Bool}(\\mathcal B)$ (in particular, also $\\mathcal K$ is a basis of $X$). Thus, in view of Corollary \\ref{bool}, the $\\mathcal K-$ultrafilter topology and the $\\mathcal B-$ultrafilter topology on $X$ are the same. \n Now, let $Y$ be a subset of $X$. Keeping in mind \\cite[Proposition (7.2.3)(iv)]{EGA}, it sufficies to show that the following conditions are equivalent:\n\\begin{enumerate}[(i)]\n\\item $Y$ is $\\mathcal B-$stable under ultrafilters.\n\\item For any $U\\in\\mathcal A$, $Y\\cap U$ is closed, with respect to the constructible topology of $U$.\n\\item[(ii)'] For any $U\\in\\mathcal A$, $Y\\cap U$ is a $\\mathcal B_U-$stable under ultrafilters subset of $U$.\n\\end{enumerate}\nSince the equivalence of (ii) and (ii)' follows immediately by Corollary \\ref{P-ultra=cons},  it is enough to show that (i) is equivalent to (ii)'. \n\n(ii)'$\\Longrightarrow$(i). Let $\\mathscr U$ be an ultrafilter on $Y$ and let $x_0\\in Y_{(X,\\mathcal B)}(\\mathscr U)$. We want to show that $x_0\\in Y$. Pick a spectral subspace $U\\in \\mathcal A$ containing $x_0$, set $T:=Y\\cap U$, and consider the ultrafilter $\\mathscr U_T:=\\{S\\cap T:S\\in\\mathscr U\\}$ on $T$ (see Remark \\ref{beginning}(\\ref{giu})). Keeping in mind that $T$ is a $\\mathcal B_U-$stable under ultrafilter subset of $U$ and applying Lemma \\ref{technical} and Remark \\ref{esempi1}(\\ref{finer}), we have\n$$\nx_0\\in Y_{(X,\\mathcal B)}(\\mathscr U)\\cap U\\subseteq T_{(X,\\mathcal B_U)}(\\mathscr U_T)\\cap U=:T_{(U,\\mathcal B_U)}(\\mathscr U_T)\\subseteq T\n$$\nand thus $x_0\\in Y$. \n\n(i) $\\Longrightarrow$ (ii)'. Suppose that $Y$ is $\\mathcal B-$stable under ultrafilters, fix a spectral subspace $U\\in \\mathcal A$, and let $\\mathscr U$ be an ultrafilter on $Z:=U\\cap Y$. We want to show that $Z_{(U,\\mathcal B_U)}(\\mathscr U)\\subseteq Z$. Take an element $x_0\\in Z_{(U,\\mathcal B_U)}(\\mathscr U)$ and consider the ultrafilter $\\mathscr U^Y:=\\{S\\subseteq Y:S\\cap Z\\in \\mathscr U\\}$ on $Y$ (Remark \\ref{beginning}(\\ref{su})). Since, by assumption, we have \n$$\nY_{(X,\\mathcal B)}(\\mathscr U^Y):=\\{x\\in X:[\\forall B\\in \\mathcal B, x\\in B\\Longleftrightarrow B\\cap Y\\in \\mathscr U^Y]\\}\\subseteq Y\n$$\nit sufficies to show that $x_0\\in Y_{(X,\\mathcal B)}(\\mathscr U^Y)$. Fix an open set $B\\in \\mathcal B$. If $x_0\\in B$, we can pick an open and compact subspace $\\Omega\\in \\mathcal B_U$ of $U$ such that $x_0\\in \\Omega\\subseteq B\\cap U$. Since $x_0\\in Z_{(U,\\mathcal B_U)}(\\mathscr U)$, it follows $(\\Omega\\cap Y)\\cap Z=\\Omega \\cap Z\\in \\mathscr U$, by definition, and thus $\\Omega \\cap Y\\in \\mathscr U^Y$. Moreover $B\\cap Y\\in\\mathscr U^Y$, since $\\Omega\\cap Y\\subseteq B\\cap Y$. Conversely, suppose that $B\\cap Y\\in \\mathscr U^Y$, i.e. $B\\cap Y\\cap U\\in \\mathscr U$. Since $X$ is the underlying topological space of a quasi-separated scheme, $B\\cap U$ is an open and compact subspace of $U$ \\cite[Proposition (6.1.12)]{EGA}, that is $B\\cap U\\in \\mathcal B_U$. Thus we have $x_0\\in B\\cap U$, since $x_0\\in Z_{(U,\\mathcal B_U)}(\\mathscr U)$. The proof is now complete. \n\\end{proof}\n\\begin{center}\n\\textsc{Ackowledgements}\n\\end{center}\nThe author is grateful to the referee for his\/her useful comments which greatly improved the paper. \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \nThe fundamental results of Arthur led to the classification of the automorphic discrete spectrum of  the classical groups. The automorphic representations of a classical group are grouped into global (Arthur) packets. Global  Arthur packets are formed using local Arthur packets. It is very important to have a way  to distinguish representations inside a local  packet, being it Arthur or Langlands packet. The characterization of the representations in a packet by models they have turns out to be very important; let us just mention the landmark work of Gan, Gross, Prasad, Waldspurger and others on restriction problems for classical groups and  existence of Bessel and Fourier-Jacobi models (\\cite{GGP_I},\\cite{GGPW}, etc.). The second use of models for groups over local fields or periods for groups over global fields is their use for the determination of poles of the global L-functions. In that way D. Jiang introduced the generalized Shalika model for the split group $\\operatorname{SO}_{4n}(F),$ where $F$ is a local non-archimedean field of characteristic zero. In more detail, Jiang introduced this model in \\cite{JQ_residues} with the Langlands-Shahidi method to characterize irreducible automorphic cuspidal representations $\\pi$ of $\\operatorname{GL}_{2n}$ whose global L--function $L(s,\\pi,\\Lambda^2)$ has a pole for $s=1.$\nMoreover, Jiang  formulated conjectures about the characterizations of local Arthur packets containing a member having a non-zero generalized Shalika model (cf. the fourth section of \\cite{JNQ}); these conjectures can be viewed as a specific case of on-going research into spherical varieties (cf. \\cite{Sakellaridis_spherical}).\n\nJiang also observed the following: let $\\sigma$ be an irreducible cuspidal  symplectic representation of $\\operatorname{GL}_{2n}(F),$ where $F$ is a non-archimedean fields of characteristic zero. If we induce parabolically from the representation $\\sigma$ twisted by $1\/2$ to a representation of $\\operatorname{SO}_{4n}(F),$ we get a reducible representation whose Langlands quotient has a generalized Shalika model. Similarly, if we induce from $\\sigma$ (twisted buy $1\/2$) parabolically to $\\operatorname{Sp}_{4n}(F),$ the representation is reducible, and its Langlands quotient has a non-zero symplectic linear model. It turns out that these two Langlands quotients are related through theta correspondence. This fact fits nicely into interpretation of  symplecticity of representations of $\\operatorname{GL}_{2n}(F)$ in terms of various functorialities and models existing on members of a dual pair $(\\operatorname{O}_{4n}(F),\\operatorname{Sp}_{4n}(F));$ this is nicely explained in \\cite{JNQ2}, p.~541.\n\nIn this note, we answer a question of Jiang posed in \\cite{JNQ2}, p.~542.  Namely, as we mentioned above, in the specific cases of induction from an irreducible supercuspidal symplectic representation of $\\operatorname{GL}_{2n}(F),$ the corresponding Langlands quotients, which have a non-zero generalized Shalika model, and a non-zero symplectic linear model, respectively, are related through the theta correspondence. We prove that this feature occurs generally; i.e., if an irreducible smooth representation of $\\operatorname{SO}_{4n}(F)$ has a non-zero generalized Shalika model, then its  small theta lift to  $\\operatorname{Sp}_{4n}(F)$ is non-zero and has a non-zero symplectic linear model. This result suggests that the functorialities mentioned in the preceding paragraph (cf. \\cite{JNQ2}, p.~541) can be generalized in an appropriate setting, raising further questions about Gelfand-Graev models and Fourier-Jacobi models of the representations of $\\operatorname{SO}_{4n}(F)$ and  $\\operatorname{SO}_{4n}(F)$ we have studied. \n\n\nOur proof is based on a direct calculation of a twisted Jacquet module of the Weil representation (for a fixed additive character), and not on the more thorough study of the properties of representations having generalized Shalika or symplectic linear model.\n We adopted the latter approach  in a toy  example where we worked out the case of $n=1$ (\\cite{DHL}). Here a slight disambiguation is needed (as we explain in the next subsection), since actually $\\operatorname{O}_{4n}(F)$ and $\\operatorname{Sp}_{4n}(F)$ occur as a dual reductive pair, so we need to extend this irreducible representation of  $\\operatorname{SO}_{4n}(F)$ to an irreducible representation of  $\\operatorname{O}_{4n}(F).$\n\n\n\n\\subsection{Notation and Preliminaries}\n\nLet $F$ be a non-archimedean field of characteristic zero. We use Howe duality conjecture, which is now proved for any residual characteristic (cf.\\cite{Gan_Takeda_Howe_conjecture}), so we do not need any additional assumptions on residual characteristic. We fix a non-trivial additive character $\\psi: F \\rightarrow \\mathbb{C}^*$.\n\nLet \n\\[J_n:= \\begin{pmatrix} & & & 1 \\\\ & & 1 & \\\\ & \\Ddots & & \\\\ 1 & & &  \\\\ \\end{pmatrix} \\in \\operatorname{GL}_{n}(F).\\]\nWe realize the $F$--points of (split) special orthogonal group $\\operatorname{O}_{4n}$ as\n\\[\\operatorname{O}_{4n}(F) = \\{A \\in \\operatorname{GL}_{4n}(F) | A^t J_{4n}A = J_{4n},\\}\\]\nand $\\operatorname{SO}_{4n}(F)$ is realized  a subgroup of $\\operatorname{O}_{4n}(F)$ consisting of matrices of determinant  $1.$ We fix the maximal diagonal torus $T$ and the Borel subgroup $B$ of upper triangular matrices in $\\operatorname{SO}_{4n}(F).$   We let $P=MN$ be a standard maximal parabolic subgroup of $\\operatorname{SO}_{4n}(F)$, whose Levi subgroup $M$ is isomorphic to $\\operatorname{GL}_{2n}(F)$.\\\\\n\n It is embedded via \n\\[\\iota:\\operatorname{GL}_{2n}(F) \\hookrightarrow \\operatorname{SO}_{4n}(F) , \\ g \\mapsto  \\begin{pmatrix} g & 0 \\\\ 0 & J_{2n}{g}^{-t}J_{2n} \\\\\\end{pmatrix}\\]\nand the $F$-points of  its unipotent radical $N$ are given by all matrices\n\\[y(X) = \\begin{pmatrix} I_{2n} & X \\\\ 0 & I_{2n} \\\\\\end{pmatrix}, \\]\nsuch that ${X^t} = - J_{2n} X J_{2n}$. We refer to $P$ as the Siegel subgroup.\nThe subgroup $\\mathcal{H} \\subset P(F)$ generated by all $\\iota(g)$ for $g \\in \\operatorname{Sp}_{2n}(F)$ and all $y \\in N(F)$ is called the \\textit{generalized Shalika subgroup} of $\\operatorname{SO}_{4n}(F)$.\nHere $\\operatorname{Sp}_{2n}(F)$ is the symplectic group realized as\n\\[\\operatorname{Sp}_{2n}(F) = \\left \\{A \\in \\operatorname{GL}_{2n}(F) | A^t\\begin{bmatrix}\n0&J_n\\\\\n-J_n&0\\\\\n\\end{bmatrix}A = \\begin{bmatrix}\n0&J_n\\\\\n-J_n&0\\\\\n\\end{bmatrix}\\right \\}.\\]\nWe consider $\\psi$ to be a character  of $N$ by \n \\[\\psi(y(X)) = \\psi\\left(tr\\left(\\begin{pmatrix} -I_n&0\\\\\n0&I_n\\end{pmatrix}X\\right)\\right)\\] \nand then we  extend it  to a character $\\psi_{\\mathcal{H}}$ of $\\mathcal{H}$ by demanding it is trivial on $\\iota(\\operatorname{Sp}_{2n}(F))$ (this is well defined because it is easily checked that $\\mathcal{H}$ is the stabilizer of a character $\\psi$ in $P$). \n \n\\begin{defn} \nAn irreducible admissible representation $\\pi$ of $\\operatorname{SO}_{4n}(F)$ is said to have a non-zero generalized Shalika model if \n\\[\\mathrm{Hom}_{\\mathcal{H}}(\\pi, \\psi_{\\mathcal{H}}) \\neq 0.\\]\n\\end{defn}\n\nThe group $\\operatorname{Sp}_{2n}(F)\\times \\operatorname{Sp}_{2n}(F)$ injects into $\\operatorname{Sp}_{4n}(F)$ via\n\\begin{equation}\n\\label{eq_SL_Sp4}\n \\left(\\begin{pmatrix}  a & b \\\\  c&d \\\\ \\end{pmatrix}, \\begin{pmatrix}  a_1 & b_1 \\\\  c_1&d_1 \\\\ \\end{pmatrix}\\right)\\mapsto \\begin{pmatrix}  a & & &  b \\\\ & a_1 & b_1& \\\\ & c_1 & d_1 & \\\\ c& & & d \\\\ \\end{pmatrix}.\n \\end{equation}\nHere $a,b,c,d,a_1,b_1,c_1,d_1$ are $n\\times n$ matrices.\n\n\\begin{defn} An irreducible admissible representation $\\pi$ on $\\operatorname{Sp}_{4n}(F)$ has a symplectic linear model if \n\\[ \\mathrm{Hom}_{\\operatorname{Sp}_{2n}(F)\\times \\operatorname{Sp}_{2n}(F)}(\\pi, 1_{\\operatorname{Sp}_{2n}(F) \\times \\operatorname{Sp}_{2n}(F)}) \\neq 0.\\]\n\\end{defn}\n\nSince we need representations of the full orthogonal group to enter the theta correspondence, we recall the following well-known criterion. Let $\\epsilon \\in \\operatorname{O}_{2n}(F)$ be the element \n$$ \\epsilon=  \\begin{pmatrix} I_{n-1} & & &  \\\\ & & 1 &  \\\\ &  1 & & \\\\  & & & I_{n-1}  \\end{pmatrix}. $$\nFor an irreducible admissible representation $\\tau$ of $\\operatorname{SO}_{2n}(F),$ we denote by $\\tau^{\\epsilon}$ representation of $\\operatorname{SO}_{2n}(F)$ on the same space, defined by $\\tau^{\\epsilon}(g)=\\tau(\\epsilon g\\epsilon^{-1}).$ We can pass between irreducible admissible representations of $\\operatorname{O}_{2n}(F)$ and $\\operatorname{SO}_{2n}(F)$ as follows:\n\n\\begin{lem}[cf.\\cite{MVW} 3.II.5, Lemme]\\label{lem:thetacorres} \\\n\\label{lem_SO_O}\n\n\\begin{enumerate}\n\t\\item Let $\\pi$ be an irreducible admissible representation of $\\operatorname{O}_{2n}(F)$. Then \\\\ $\\pi|_{\\operatorname{SO}_{2n}(F)}$ is irreducible if and only if $\\pi \\ncong \\pi \\otimes det$.\n\t\\item Let $\\tau$ be an irreducible admissible representation of $\\operatorname{SO}_{2n}(F)$. Then either \n\t\\begin{itemize}\n\t\\item[(A)] $\\tau \\ncong \\tau^\\epsilon;$ then  $Ind^{\\operatorname{O}_{2n}(F)}_{\\operatorname{SO}_{2n}(F)}(\\tau)=:\\pi$ is irreducible and satisfies $\\pi=\\pi\\otimes det$, or \n\t\\item[(B)] $\\tau \\cong \\tau^\\epsilon;$ then  $Ind^{\\operatorname{O}_{2n}(F)}_{\\operatorname{SO}_{2n}(F)}(\\tau)$ is reducible and the direct sum of two non-equivalent irreducible representations $\\pi$ and $\\pi\\otimes det$. \n\t\\end{itemize}\n\\end{enumerate}\n\\end{lem}\n\nWe use this lemma to adapt the theta correspondence to representations of $\\operatorname{SO}_{2n}(F).$ Let $\\omega_{m,k}$ denote the Weil representation (with respect to an additive character $\\psi'$) of a dual pair consisting of the split orthogonal group $O(V)(F)$  where dimension of $V$ is $2m$ and of the symplectic group $\\operatorname{Sp}(W)$ where the dimension of $W$ is $2k.$ The maximal quotient of $\\omega_{m,k}$ on which  $O(V)(F)=\\operatorname{O}_{2m}(F)$ acts as a multiple of an irreducible representation $\\pi$ decomposes as $\\pi \\otimes \\Theta(\\pi,k),$ where $\\Theta(\\pi,k)$ is a finite-length $\\operatorname{Sp}_{2k}(F)$--module. This module has  the unique irreducible quotient (Howe conjecture) which we denote by $\\theta (\\pi,k).$ We analogously define an irreducible $\\operatorname{Sp}_{2k}(F)$--module $\\theta(\\tau,k)$ for an irreducible representation $\\tau$ of  $\\operatorname{SO}_{2m}(F).$ We have\n\n\\begin{equation}\n\\label{eq_theta1}\n\\theta(\\tau, k) \\cong  \\theta(\\pi,k) \\text { if (A)}\n\\end{equation}\n\\[\\theta(\\tau, k) := \\theta(\\pi,k) \\oplus \\theta(\\pi \\otimes det,k) \\text { if (B)}.\\]\n\\begin{rem}\nIn the remainder of this paper, we are concerned with the theta lifts of irreducible representations of $\\operatorname{SO}_{4n}(F)$ to irreducible representations of $\\operatorname{Sp}_{4n}(F),$ so we are always dealing with the Weil representation $\\omega_{2n,2n}$ so we denote $\\theta (\\tau, 2n)$ by $\\theta (\\tau).$ We retain the notation from Lemma \\ref{lem_SO_O}.\n\nAssume that $\\tau$ is in irreducible representation of $\\operatorname{SO}_{4n}(F)$ such that it satisfies condition $(A)$ from  Lemma \\ref{lem_SO_O} and that it has a non-zero generalized Shalika functional, say $\\lambda.$  Then, $\\pi|_{\\operatorname{SO}_{4n}(F)}=\\tau\\oplus \\tau^{\\epsilon}$ and we can define a Shalika functional on the representation $\\pi$ by prescribing that it is equal to $\\lambda$ on $\\tau$ and zero on $\\tau^{\\epsilon}.$ If $\\tau$ with  a non-zero generalized Shalika model satisfies $(B)$ then the situation is even more straightforward since then $\\pi|_{\\operatorname{SO}_{4n}(F)}=\\tau.$\nSo, we may conclude that we can always extend generalized  Shalika functional from irreducible representation of $\\operatorname{SO}_{4n}(F)$ to irreducible representations of $\\operatorname{O}_{4n}(F)$ in the sense of Lemma \\ref{lem_SO_O}.\n\nWe note that in  (very limited number) of explicitly known representations $\\tau$ with the non-zero generalized  Shalika models (\\cite{JNQ2},\\cite{JQ_residues},\\cite{DHL}), we always had in these examples the situation $(A).$  We know that the following holds (\\cite{Sun_Zhu}):\n\\begin{thm}\n\\label{theta_occurrence}Assume that $\\sigma$ is an irreducible admissible representation of the split $\\operatorname{O}_{2m}(F).$ Then the following holds:\n\\[n(\\sigma)+n(\\sigma\\otimes \\mathrm{det})=2m.\\]\nHere $n(\\sigma)$ denotes the rank of the first non-zero occurrence of the representation $\\sigma$ in theta correspondence.\n\\end{thm}\n\nBecause of that,  if $\\tau$ is irreducible representation of $\\operatorname{SO}_{4n}(F)$ in situation $(A),$ we have that $n(\\pi)=2n$ and $\\theta(\\tau,2n)=\\theta(\\pi,2n)\\neq 0.$ We denote $\\theta'(\\tau)=\\theta(\\tau,2n).$\n\nIf  $\\tau$ is in situation $(B),$ at least one of the representations $\\pi,\\;\\pi \\otimes det$ has a non-zero theta lift to the rank $2n.$ Now, we denote by $\\theta'(\\tau)$ one of the non-zero lifts $\\theta(\\pi,2n)$ or $\\theta(\\pi \\otimes det,2n)$ (and both $\\pi$ and $\\pi \\otimes det$ have a non-zero generalized Shalika model).\n\\end{rem}\n\nWe use $\\mathrm{ind}$ to denote the compact induction, and $\\mathrm{Ind}$ to denote the non-compact induction. By $\\twoheadrightarrow $ we denote a surjective mapping. From now on, we study representations of groups $\\operatorname{SO}_{4n}(F)$ and $\\operatorname{Sp}_{4n}(F)$ for $n\\ge 2,$ since $n=1$ case is resolved in \\cite{DHL}.\n\\section{}\nWe continue to assume that $(\\pi,V)$ is an irreducible representation of $\\operatorname{O}_{4n}(F)$ with a non-zero generalized Shalika model such that $\\theta(\\pi)\\neq 0.$\nWe want to express a property of having non-zero generalized Shalika model in terms of twisted Jacquet modules. We continue to use the notation from the previous section. We form a subspace\n\\[V_{\\psi}(N):=\\operatorname{span}\\{\\pi(n)v-\\psi(n)v:v\\in V,n\\in N\\},\\]\nwhere $N$ is the unipotent radical of the Siegel standard parabolic subgroup of $\\operatorname{SO}_{4n}(F).$\nThen, it is straightforward that  the twisted Jacquet module $R_{\\mathcal{H},\\psi}(\\pi):=V\/V_{\\psi}(N)$ is a $\\operatorname{Sp}_{2n}(F)$--module, since, by definition, $\\operatorname{Sp}_{2n}(F)\\subset \\operatorname{GL}_{2n}\\cong M$ is a stabilizer of a character $\\psi$ of $N.$ Now, the existence of the non-zero generalized Shalika model on $\\pi$ is equivalent to the fact that $R_{\\mathcal{H},\\psi}(\\pi),$ as  a $\\operatorname{Sp}_{2n}(F)$--module, has the trivial quotient, i.e. there exists a non-zero functional $\\lambda$ on $R_{\\mathcal{H},\\psi}(\\pi)$ satisfying\n\\[\\lambda(\\pi(s)v+ V_{\\psi}(N))=\\lambda(v+ V_{\\psi}(N)).\\]\n\n\\subsection{Calculation of $R_{\\mathcal{H},\\psi}(\\omega_{2n,2n})$}\nRecall that we view $\\omega_{2n,2n}$ as a representation of $\\operatorname{O}_{4n}(F)\\times \\operatorname{Sp}_{4n}(F).$ The above discussion motivates us to examine $R_{\\mathcal{H},\\psi}(\\omega_{2n,2n})$ more thoroughly. This is obviously an $\\operatorname{Sp}_{2n}(F)\\times \\operatorname{Sp}_{4n}(F)$-module.   Note that we have a non-trivial additive character $\\psi$ appearing in the definition of the generalized Shalika model; assume that an additive character $\\psi_a(x):=\\psi (ax),$ where $a\\in F^{\\ast},$ enters the definition of theta correspondence (we do not emphasize $\\psi_a$ in the notation of $\\omega_{2n,2n}$). A general description of the twisted Jacquet modules of this kind is given in (\\cite{MVW}, pp.~72, 73). We further elaborate on this description which is given not necessarily for the Weil representation, but in the more general context.\n\nWe study the Schroedinger model of the Weil representation $\\omega_{2n,2n}$ defined in the following way: let $V=V_{2n}'\\oplus V_{2n}''$ be a complete polarization of the  quadratic space $V$ on which $\\operatorname{O}_{4n}(F)$ acts. Let $W$ be $4n$--dimensional skew-symmetric space on which $\\operatorname{Sp}_{4n}(F)$ acts. We denote by  $\\mathbf{W}=V\\otimes W=V_{2n}'\\otimes W\\oplus V_{2n}''\\otimes W.$ Then, the Schroedinger model of  $\\omega_{2n,2n}$ is realized on the Schwartz space $S(V_{2n}'\\otimes W).$ Sometimes we use an isomorphism $V_{2n}'\\otimes W\\cong W^{2n},$ so that given a basis $\\{e_1,\\ldots,e_{2n}\\}$ of an isotropic space $V_{2n}'$ we have \n\\[e_1\\otimes w_1+\\cdots e_{2n}\\otimes w_{2n}\\mapsto (w_1,\\ldots,w_{2n}).\\]\n To be able to directly apply formulas for the Weil representation given in (\\cite{Kudla1}, p. 38) we take a  little bit different matrix realization of $\\operatorname{O}_{4n}(F)$ (isomorphic to ours defined above) where in the definition of $\\operatorname{O}_{4n}(F)$ the symmetric form is defined not by using the matrix $J_{4n}$ but the matrix $\\begin{bmatrix}\n0&I_{2n}\\\\\nI_{2n}&0\\\\\n\\end{bmatrix}.$ Then,\n\\[N=\\left\\{n(S)=\\begin{bmatrix}\nI_{2n}&S\\\\\n0&I_{2n}\\\\\n\\end{bmatrix}:S^{t}=-S\\right\\}.\\] \nNote that then the action of $N$ in $\\omega_{2n,2n}$  is given by the homothety (\\cite{Kudla1}, p. 38)\n\\[\\omega_{2n,2n}(n(S),1)\\phi(w)= \\psi_a( \\frac{1}{2}tr(\\langle w,w\\rangle S))\\phi(w),\\]\nwhere $w=(w_1,\\ldots w_{2n})\\in W^{2n},$ $\\phi \\in S(W^{2n}).$ Here $\\langle x,x\\rangle$ denotes $2n \\times 2n$ skew-symmetric matrix whose $(i,j)$-entry is $\\langle w_i,w_j\\rangle.$\nWe examine (we adopt the notation of \\cite{MVW}, p.~72)\n\\[\\Omega(\\psi)=\\left \\{w\\in W^{2n}:\\psi_a( \\frac{1}{2}tr(\\langle w,w\\rangle S))=\\psi_{\\mathcal{H}}(S)=\\psi(tr(\\begin{bmatrix}\n0&I_{n}\\\\\n-I_{n}&0\\\\\n\\end{bmatrix}S))\\right \\},\\]\nwhere  the action of the Shalika character is adjusted because of the modified definition of $N.$ By changing $a\\mapsto a^{-1},$\nwe get the condition\n\\[\\Omega(\\psi)=\\left \\{w\\in W^{2n}:\\psi (tr(S(\\frac{1}{2}(\\langle w,w\\rangle-a\\begin{bmatrix}\n0&I_{n}\\\\\n-I_{n}&0\\\\\n\\end{bmatrix})))=1,\\forall S^{t}=-S \\in M_n(F)\\right \\}.\\]\n\nBy Lemma on p. 73 of \\cite{MVW}, the restriction on $\\Omega(\\psi)$ gives the isomorphism of $R_{\\mathcal{H},\\psi}(\\omega_{2n,2n})$ with the action of $\\operatorname{Sp}_{2n}(F)\\times \\operatorname{Sp}_{4n}(F)$ on  $S(\\Omega(\\psi)).$ Now we examine this action more throughly.\n\nWe can get rid of $\\psi$ in the above definition of $\\Omega(\\psi).$  We see that in the following calculation. We define an skew-symmetric matrix $A:=\\frac{1}{2}\\langle w,w\\rangle-a\\begin{bmatrix}\n0&I_{n}\\\\\n-I_{n}&0\\\\\n\\end{bmatrix}.$ We put $A=\\begin{bmatrix}x&b\\\\\n-b^t&d\\\\\n\\end{bmatrix},$ where $x^t=-x, d^t=-d$ and $S=\\begin{bmatrix}a_1&b_1\\\\\n-b_1^t&d_1\\\\\n\\end{bmatrix}$ with $a_1^t=-a_1, d_1^t=-d_1.$ The condition becomes\n\\[\\psi(tr(a_1x-b_1b^t-b_1^tb+d_1d))=1,\\]\nfor all $\\begin{bmatrix}a_1&b_1\\\\\n-b_1^t&d_1\\\\\n\\end{bmatrix}.$ If we take  $a_1=d_1=0$ and $b_1=\\lambda e_{i,j},$ where $e_{i,j}$ is a $n\\times n$ matrix whose entries are all zero except  the entry $(i,j)$ which is $1.$\nWe get that  $\\psi(2\\lambda b_{i,j})=1,$ for all $\\lambda \\in F^{\\ast}.$ Since $\\psi$ is non-trivial, we get that $b_{i,j}=0.$ This holds for every $(i,j)$ so that $b=0.$ Since $n\\ge 2,$ we can take $a_1=d_1=\\lambda e_{i,j}-\\lambda e_{j,i},$ for some $i\\neq j.$ Then, the condition becomes\n$\\psi(2\\lambda (x+d)_{j,i})=1,\\forall \\lambda \\in  F^{\\ast}.$ We get that $ (x+d)_{j,i}=0,$ for all $j\\neq i.$ We get that $x+d=0.$ If we take $a_1=-d_1=\\lambda e_{i,j}-\\lambda e_{j,i},$ for some $i\\neq j,$ we analogously get $(x-d)_{j,i}=0$ and then we get $x=d=0.$ Thus,\n\\[A=\\frac{1}{2}\\langle w,w\\rangle-a\\begin{bmatrix}\n0&I_{n}\\\\\n-I_{n}&0\\\\\n\\end{bmatrix}=0.\\] \nThus,\n\\[\\Omega(\\psi)=\\left \\{w\\in W^{2n}:\\frac{1}{2}\\langle w,w\\rangle-a\\begin{bmatrix}\n0&I_{n}\\\\\n-I_{n}&0\\\\\n\\end{bmatrix}=0\\right \\}.\\]\nRecall that the action of $\\operatorname{Sp}_{2n}(F)\\times \\operatorname{Sp}_{4n}(F)$ on $w=e_1\\otimes w_1+\\cdots e_{2n}\\otimes w_{2n}$ is given as follows: for $(g_1,g_2)\\in \\operatorname{Sp}_{2n}(F)\\times \\operatorname{Sp}_{4n}(F)$ we have\n\\[(g_1,g_2)(e_1\\otimes w_1+\\cdots e_{2n}\\otimes w_{2n})=g_1e_1\\otimes g_2 w_2+\\cdots+g_1e_{2n}\\otimes g_2w_{2n}.\\]\nWe put $g_1e_i=\\sum_{l=1}^{2n}a_{l,i}e_l,\\;i=1,\\ldots,2n.$ So we get\n\\begin{equation}\n\\label{eq:action}\n(g_1,g_2)(e_1\\otimes w_1+\\cdots e_{2n}\\otimes w_{2n})=e_1\\otimes (\\sum_{i=1}^{2n}a_{1,i}g_2w_i)+\\cdots+e_{2n}\\otimes (\\sum_{i=1}^{2n}a_{2n,i}g_2w_i).\n\\end{equation} \nWe denote $w_j'=\\sum_{i=1}^{2n}a_{j,i}g_2w_i.$ Now, it is a straightforward that\n\\begin{equation}\n\\label{eq:preserving_product}\n\\langle w_i',w_j' \\rangle=\\langle w_i,w_j \\rangle, \\forall i,j\n\\end{equation}\n (we use that $g_1\\in Sp_{2n}(F),$ where we now realize   $Sp_{2n}(F)$ as\n\\[\\operatorname{Sp}_{2n}(F)=\\left\\{g_1\\in GL_{2n}(F):g_1^t\\begin{bmatrix}\n0&I_{n}\\\\\n-I_{n}&0\\\\\n\\end{bmatrix}g_1=\\begin{bmatrix}\n0&I_{n}\\\\\n-I_{n}&0\\\\\n\\end{bmatrix}\\right\\}).\\]\nThis is, of course, what we knew in advance and it just means that the action of $\\operatorname{Sp}_{2n}(F)\\times \\operatorname{Sp}_{4n}(F)$ preserves $\\Omega(\\psi).$ \n\nWe want to analyze the orbits of this action.\n\n\\begin{lem}\n\\label{lem:transitive}\nThe action of $\\operatorname{Sp}_{2n}(F)\\times \\operatorname{Sp}_{4n}(F)$ on  $\\Omega(\\psi)$ is transitive.\n\\end{lem}\n\\begin{proof} Note that for $w=e_1\\otimes w_1+\\cdots+ e_{2n}\\otimes w_{2n}=(w_1,\\ldots,w_{2n})\\in \\Omega (\\psi)$ the defining relation of $ \\Omega (\\psi)$ guarantees  that the set $\\{w_1,w_2,\\ldots,w_{2n}\\}$ is linearly independent (these vectors form a symplectic basis (up to scalar) of $2n$--dimensional non-degenerate subspace of $W$). An element $g_2\\in \\operatorname{Sp}_{4n}(F)$ turns $\\operatorname{span}\\{w_1,w_2,\\ldots,w_{2n}\\}$ into another non-degenerate $2n$--dimensional subspace of $W$ with a (up to scalar) symplectic basis $\\{g_2 w_1,\\ldots g_2w_{2n}\\},$ and then $g_1$  acts on the $\\{g_2w_1,\\ldots g_2w_{2n}\\}$ by turning it into another basis of the same space.\n\nLet $w=(w_1,\\ldots,w_{2n}), w'=(w_1',\\ldots,w_{2n}')\\in \\Omega(\\psi)$ and denote \n\\[V_1=\\operatorname{span}\\{w_1,\\ldots,w_{2n}\\}\\text{ and } V_2=\\operatorname{span} \\{w_1',\\ldots,w_{2n}'\\}.\\]\nWe define $f:V_1\\to V_2$ with $f(w_i)=w_i',\\;i=1,2,\\ldots,2n.$ It is obvious that $f$ is an isometry. By the  Witt's theorem, there exists an isometry on $W$ (thus an element $g_2 \\in\\operatorname{Sp}_{4n}(F)$) extending $f.$ This means that $(1,g_2)w=w'.$\n\n \\end{proof} \nWe fix $w_0=(w_1,\\ldots,w_{2n})$ in $\\Omega(\\psi)$ and let $G_1\\subset \\operatorname{Sp}_{2n}(F)\\times \\operatorname{Sp}_{4n}(F)$ is a stabilizer of that point.  By the known results (cf.~\\cite{MVW}, p.73), since there is only one orbit for this action on $\\Omega(\\psi),$ we have\n\\begin{equation}\n\\label{eq:the representation_I}\nR_{\\mathcal{H},\\psi}(\\omega_{2n,2n})\\cong \\mathrm{ind}_{G_1}^{\\operatorname{Sp}_{2n}(F)\\times \\operatorname{Sp}_{4n}(F)}\\omega_{w_0}.\t\n\\end{equation}\n\nHere $\\omega_{w_0}$  is a representation of $\\operatorname{Sp}_{2n}(F)\\times \\operatorname{Sp}_{4n}(F)$ satisfying\n\\[(\\omega_{2n,2n})(g_1,g_2)f(w_0)=\\omega_{w_0}(g_1,g_2)f(w_0(g_1,g_2)).\\]\nSince $\\omega_{w_0}$ is a character, it must be equal to 1. Indeed, when we check the formulas from (\\cite{Kudla1}, p.~38) we get\n\\[(\\omega_{2n,2n})(g_1,1)f(w_0)=f(g_1^te_1\\otimes w_1+\\cdots+g_2^te_{2n}\\otimes w_{2n}),\\]\nand\n\\[(\\omega_{2n,2n})(1,g_2)f(w_0)=f(e_1\\otimes g_2^{-1}w_1+\\cdots +e_{2n}\\otimes g_2^{-1}w_{2n}).\\]\n\n\\begin{lem}Let $G_1$ be the stabilizer of $w_0$ with respect to  $\\operatorname{Sp}_{2n}(F)\\times \\operatorname{Sp}_{4n}(F)$ action given by (\\ref{eq:action}). Then, \n\\[G_1\\cong \\operatorname{Sp}_{2n}(F)\\times \\operatorname{Sp}_{2n}(F)\\]\ngiven with\n\\[(g_1,g_2)\\mapsto (g_1^{-t},(g_1,g_2)),\\]\nwhere $(g_1,g_2)$ from the right hand side belongs to $\\operatorname{Sp}_{2n}(F)\\times\\operatorname{Sp}_{2n}(F)\\subset \\operatorname{Sp}_{4n}(F),$ and where $W$ is decomposed as a orthogonal direct sum of non-degenerate symplectic spaces of dimensions $2n$ and each copy of $\\operatorname{Sp}_{2n}(F)$ is the symplectic group of the corresponding subspace.\n\\end{lem}\n\n\\begin{proof} According to the interpretation of this action given in the proof of Lemma \\ref{lem:transitive}, for $(g_1,g_2)\\in \\operatorname{Sp}_{2n}(F)\\times \\operatorname{Sp}_{4n}(F)$ to be in $G_1,$ it is needed that, for $V_1:=\\operatorname{span}\\{w_1,\\ldots,w_{2n}\\},$ we have $g_2(V_1)=V_1.$ Since $V_1$ is non degenerate, we have the orthogonal direct decomposition\n\\[W=V_1\\oplus V_1^{\\perp},\\]\nwhere $V_1^{\\perp}$ denotes the orthogonal complement of $V_1.$ Now, we immediately have $g_2(V_1^{\\perp})=V_1^{\\perp}$ and $g_2\\mapsto (g_2|_{V_1},g_2|_{V_1^{\\perp}})$ is injective. Note that $g_2|_{V_1}$ and $g_2|_{V_1^{\\perp}}$ belong to the symplectic groups of $V_1$ and $V_1^{\\perp},$ respectively. Then, for $g_1$ such that $(g_1,g_2)\\in G_1$ we must have (from (\\ref{eq:action})) that $g_1=(g_2|_{V_1})^{-t}.$\n\\end{proof}\n\nNote that a function $f$  from $\\mathrm{ind}_{G_1}^{\\operatorname{Sp}_{2n}(F)\\times \\operatorname{Sp}_{4n}(F)} 1=\\mathrm{ind}_{\\operatorname{Sp}_{2n}(F)\\times \\operatorname{Sp}_{2n}(F)}^{\\operatorname{Sp}_{2n}(F)\\times \\operatorname{Sp}_{4n}(F)} 1$ satisfies \n\\[f(g_1'^{-t},(g_1',g_2'))(\\alpha,\\beta))=f((\\alpha,\\beta)),\\]\nfor all $(\\alpha,\\beta)\\in \\operatorname{Sp}_{2n}(F)\\times \\operatorname{Sp}_{4n}(F)$ and $(g_1'^{-t},(g_1',g_2'))\\in \\operatorname{Sp}_{2n}(F)\\times \\operatorname{Sp}_{4n}(F)).$ $f$ is also smooth and compactly supported in $\\operatorname{Sp}_{2n}(F)\\times \\operatorname{Sp}_{4n}(F)$ modulo $G_1.$\n Note that this means that $f((\\alpha,\\beta))=f(1,(\\alpha^t,1)\\beta)),$ so that $f$ is completely determined by  its restriction to $\\operatorname{Sp}_{4n}(F).$ We define \n \\[\\phi_f(\\beta)=f(1,\\beta).\\]\n We also note that $\\phi_f:\\operatorname{Sp}_{4n}(F)\\to \\mathbb{C}$ is left $\\operatorname{Sp}_{2n}(F)$-- invariant with respect to the second copy of  $\\operatorname{Sp}_{2n}(F).$\n We get that \n \\[f\\mapsto \\phi_f\\]\n is a bijection from $\\mathrm{ind}_{\\operatorname{Sp}_{2n}(F)\\times \\operatorname{Sp}_{2n}(F)}^{\\operatorname{Sp}_{2n}(F)\\times \\operatorname{Sp}_{4n}(F)} 1$ to $\\mathrm{ind}_{\\operatorname{Sp}_{2n}(F)}^{\\operatorname{Sp}_{4n}(F)} 1$ (we easily get that $\\phi_f$ is smooth and compactly supported modulo the second copy of $\\operatorname{Sp}_{2n}(F)$). The action of $\\operatorname{Sp}_{2n}(F)\\times \\operatorname{Sp}_{4n}(F)$ on $\\mathrm{ind}_{\\operatorname{Sp}_{2n}(F)\\times \\operatorname{Sp}_{2n}(F)}^{\\operatorname{Sp}_{2n}(F)\\times \\operatorname{Sp}_{4n}(F)} 1$ becomes \n \\begin{equation}\n \\label{group_action}\n R(g_1,g_2)\\phi(\\beta)=\\phi((g_1^t,1)\\beta g_2)\n \\end{equation}\n on $\\mathrm{ind}_{\\operatorname{Sp}_{2n}(F)}^{\\operatorname{Sp}_{4n}(F)} 1.$ We have proved\n\n \\begin{prop}$R_{\\mathcal{H},\\psi}(\\omega_{2n,2n})$ is, as a $\\operatorname{Sp}_{2n}(F)\\times \\operatorname{Sp}_{4n}(F)$ module, isomorphic to $\\mathrm{ind}_{\\operatorname{Sp}_{2n}(F)}^{\\operatorname{Sp}_{4n}(F)} 1$ with the action of $\\operatorname{Sp}_{2n}(F)\\times \\operatorname{Sp}_{4n}(F)$ given by (\\ref{group_action}).\n \\end{prop}\n\n Note that the first copy of $\\operatorname{Sp}_{2n}(F)$ acts as the left translation; we denote this action by $\\lambda.$ \n\n Now we want to analyze the biggest quotient of $\\mathrm{ind}_{\\operatorname{Sp}_{2n}(F)}^{\\operatorname{Sp}_{4n}(F)} 1$  on which $\\operatorname{Sp}_{2n}(F)$ (through $\\lambda$) acts trivially. To that end, we define\n  \\[S'=\\operatorname{span}\\{\\lambda (g_2)\\phi-\\phi:g_2\\in \\operatorname{Sp}_{2n}(F),\\phi \\in \\mathrm{ind}_{\\operatorname{Sp}_{2n}(F)}^{\\operatorname{Sp}_{4n}(F)} 1\\}.\\]\n  Obviously, $\\mathrm{ind}_{\\operatorname{Sp}_{2n}(F)}^{\\operatorname{Sp}_{4n}(F)} 1\/S'$ is that quotient; we consider it as a  $\\operatorname{Sp}_{4n}(F)$--module.\n\n \\begin{thm}\n \\label{the_largest_quo_in_Weil}\n There is an isomorphism of  $\\operatorname{Sp}_{4n}(F)$--modules:\n \\[\\mathrm{ind}_{\\operatorname{Sp}_{2n}(F)}^{\\operatorname{Sp}_{4n}(F)} 1\/S'\\cong \\mathrm{ind}_{\\operatorname{Sp}_{2n}(F)\\times \\operatorname{Sp}_{2n}(F)}^{\\operatorname{Sp}_{4n}(F)} 1.\\]\n \\end{thm}\n\\begin{proof}We denote\n\\[T(\\phi)(g)=\\int_{\\operatorname{Sp}_{2n}(F)}\\phi((x,1)g)\\mathrm{d} x.\\]\nFor $\\phi\\in \\mathrm{ind}_{\\operatorname{Sp}_{2n}(F)}^{\\operatorname{Sp}_{4n}(F)} 1$ the integral on the right hand side converges. Indeed, fix $g\\in \\operatorname{Sp}_{4n}(F).$ We know that there exist a compact set $C_1\\subset \\operatorname{Sp}_{4n}(F)$ such that $\\mathrm{supp} \\phi\\subset (\\{1\\}\\times \\operatorname{Sp}_{2n}(F))C_1.$ Assume that $\\phi((x,1)g)\\neq 0,$ which means that $(x,1)\\in (\\{1\\}\\times \\operatorname{Sp}_{2n}(F))C_1g^{-1}.$ We denote $C_1':=C_1g^{-1}.$ Note that $C_1'\\cap \\operatorname{Sp}_{2n}(F)\\times \\operatorname{Sp}_{2n}(F)$ is a compact set in $ \\operatorname{Sp}_{2n}(F)\\times \\operatorname{Sp}_{2n}(F).$ We denote by $p_i,\\;i=1,2$ the projections from $ \\operatorname{Sp}_{2n}(F)\\times \\operatorname{Sp}_{2n}(F)$ to the first and the second copy of $\\operatorname{Sp}_{2n}(F).$\nThis means that \n\\[(x,1)\\in (\\{1\\}\\times \\operatorname{Sp}_{2n}(F))(p_1(C_1')\\times p_2(C_1'))=p_1(C_1')\\times  \\operatorname{Sp}_{2n}(F).\\]\nThis means that $x\\in p_1(C_1'),$ which is a compact set in (the first copy of) $\\operatorname{Sp}_{2n}(F).$ Thus, $x\\mapsto \\phi ((x,1)g)$ is a smooth function with the compact support in $\\operatorname{Sp}_{2n}(F).$ Thus, $T(\\phi)$ is well defined function on $\\operatorname{Sp}_{4n}(F)$. Also, it is smooth. Again, if $C_1$ denotes the compact set  in $\\operatorname{Sp}_{4n}(F)$ related to the support of $\\phi$ as above, then it is easy to see that $\\mathrm{supp} T(\\phi)\\subset (\\operatorname{Sp}_{2n}(F)\\times \\operatorname{Sp}_{2n}(F))C_1.$\nAlso, it is immediate that the following holds\n\\[T(\\phi)((g_1,g_2)g)=T(\\phi)(g),\\forall (g_1,g_2)\\in Sp_{2n}(F)\\times Sp_{2n}(F), g\\in Sp_{4n}(F),\\]\nand \n\\[T(R(g)\\phi)=R(g)T(\\phi).\\]\nThus, $T$ is $Sp_{4n}(F)$--intertwining operator between $\\mathrm{ind}_{\\operatorname{Sp}_{2n}(F)}^{\\operatorname{Sp}_{4n}(F)} 1$ and $\\mathrm{ind}_{\\operatorname{Sp}_{2n}(F)\\times \\operatorname{Sp}_{2n}(F)}^{\\operatorname{Sp}_{4n}(F)} 1.$\nWe immediately see that $T|_{S'}=0.$ \n\nWe now prove the surjectivity of the operator $T.$ We use (\\cite{Bourbaki_Haar}, cf.~\\cite{Casselman_notes}, p.~27) to introduce the mapping\n\\[P_{\\delta_1}:C_c^{\\infty}(\\operatorname{Sp}_{4n}(F))\\to \\mathrm{ind}_{\\operatorname{Sp}_{2n}(F)}^{\\operatorname{Sp}_{4n}(F)} 1\\]\ngiven by\n\\[P_{\\delta_1}(f)(g)=\\int_{\\operatorname{Sp}_{2n}(F)}f((1,x)g)\\mathrm{d} x.\\]\nIt is known that $P_{\\delta_1}$ is surjective (\\cite{Casselman_notes}, p.~27).\nAnalogously we define  a (surjective) mapping\n\\[P_{\\delta_2}:C_c^{\\infty}(\\operatorname{Sp}_{4n}(F))\\to \\mathrm{ind}_{\\operatorname{Sp}_{2n}(F)\\times \\operatorname{Sp}_{2n}(F)}^{\\operatorname{Sp}_{4n}(F)} 1\\]\ngiven by \n\\[P_{\\delta_2}(f)(g)=\\int_{\\operatorname{Sp}_{2n}(F)\\times \\operatorname{Sp}_{2n}(F)}f((x,y)g)\\mathrm{d} x \\mathrm{d} y.\\]\nWe immediately see that \n\\begin{equation}\n\\label{eq_delta1_delta2}\nP_{\\delta_2}(f)(g)=\\int_{\\operatorname{Sp}_{2n}(F)}P_{\\delta_1}(\\lambda(x^t)f)(g)\\mathrm{d} x=\\int_{\\operatorname{Sp}_{2n}(F)}\\lambda(x^t)P_{\\delta_1}(f)(g)\\mathrm{d} x=T(P_{\\delta_1}(f))(g).\n\\end{equation}\nThus, $P_{\\delta_2}(f)=T(P_{\\delta_1}(f))$ and $T$ is surjective.\n\nNow we prove that $\\mathrm{Ker}\\,T=S'.$ Assume that $\\phi\\in \\mathrm{Ker}\\,T.$ Then, there exists $f\\in C_c^{\\infty}(\\operatorname{Sp}_{4n}(F))$ such that $\\phi=P_{\\delta_1}(f).$ Thus, $T(\\phi)=P_{\\delta_2}(f)=0.$ There exist an open compact subgroup $K$ of $\\operatorname{Sp}_{4n}(F),\\;g_1,\\ldots,g_m\\in \\operatorname{Sp}_{4n}(F)$  and $c_1,\\ldots,c_m\\in \\mathbb{C}$ such  that\n\\[f=\\sum_{i=1}^m c_i\\chi_{Kg_i}.\\]\nHere we assume that for $i\\neq j$ $Kg_i\\cap Kg_j=\\emptyset$ and $\\chi_{Kg_i}$ denotes the characteristic function on the right coset $Kg_i.$\nWe examine the first equation in (\\ref{eq_delta1_delta2}). The integrating function,\n \\[x\\mapsto P_{\\delta_1}((x,1)g)=\\sum_{i=1}^mc_i\\mu_{\\{1\\}\\times \\operatorname{Sp}_{2n}(F)}((x^{-1},1)Kg_ig^{-1}\\cap \\{1\\}\\times \\operatorname{Sp}_{2n}(F))\\]\n is locally (uniformly) constant.  Here $\\mu_{\\{1\\}\\times \\operatorname{Sp}_{2n}(F)}$ denotes a Haar measure on $\\{1\\}\\times \\operatorname{Sp}_{2n}(F).$ Indeed, if we denote by $K_0:=K\\cap \\operatorname{Sp}_{2n}(F)\\times \\{1\\},$ which is compact and open in $\\operatorname{Sp}_{2n}(F)\\times \\{1\\},$ we see that the function\n  \\[x\\mapsto\\sum_{i=1}^mc_i\\mu_{\\{1\\}\\times \\operatorname{Sp}_{2n}(F)}((x^{-1},1)Kg_ig^{-1}\\cap \\{1\\}\\times \\operatorname{Sp}_{2n}(F))\\] is a constant on cosets $K_0\\setminus \\operatorname{Sp}_{2n}(F)\\times \\{1\\}.$ Also, we effectively integrate  in  (\\ref{eq_delta1_delta2}) over a compact set. We  integrate over a finite set of different cosets of $K_0\\setminus \\operatorname{Sp}_{2n}(F)\\times \\{1\\}.$ Thus, there exist $x_1,\\ldots,x_l \\in \\operatorname{Sp}_{2n}(F)\\times \\{1\\}$ such that\n\\[0=\\sum_{j=1}^l\\int_{K_0x_j}(\\lambda(x^t)P_{\\delta_1}(f))(g)\\mathrm{d} x=\\mu_{\\{1\\}\\times \\operatorname{Sp}_{2n}(F)}(K_0)\\sum_{j=1}^lP_{\\delta_1}(f)((x_j,1)g),\\]\nfor every $g\\in \\operatorname{Sp}_{4n}(F).$\nThis means\n\\[\\lambda(x_1^t)P_{\\delta_1}(f)=-\\sum_{j=2}^l\\lambda(x_j^t)P_{\\delta_1}(f),\\]\nso that\n\\[P_{\\delta_1}(f)=-\\sum_{j=2}^l\\lambda(x_1^{-t}x_j^t)P_{\\delta_1}(f).\\]\nThis means\n\\[P_{\\delta_1}(f)=\\phi=-\\frac{1}{l}\\sum_{j=2}^l(\\lambda(x_1^{-t}x_j^t)\\phi-\\phi),\\]\nand this means that $\\mathrm{Ker}\\,T=S'.$\n\\end{proof}\n\n\\subsection{Conclusion}\nWe continue to assume that $\\pi$ is an irreducible representation of $\\operatorname{O}_{4n}(F)$ with a non-zero Shalika model such that $\\theta(\\pi)\\neq 0$ is its (irreducible) small theta lift.\nWe thus have\n\\[\\omega_{2n,2n}\\twoheadrightarrow \\pi\\otimes \\theta(\\pi),\\]\nand, since taking a twisted Jacquet module is exact, we have\n\\[R_{\\mathcal{H},\\psi}(\\omega_{2n,2n})\\twoheadrightarrow R_{\\mathcal{H},\\psi}(\\pi)\\otimes \\theta(\\pi).\\]\nSince we assumed that $\\pi$ has a non-zero Shalika model, there is a surjective $\\operatorname{Sp}_{2n}(F)\\times\\operatorname{Sp}_{4n}(F)$ intertwining\n\\[R_{\\mathcal{H},\\psi}(\\omega_{2n,2n})\\twoheadrightarrow 1_{\\operatorname{Sp}_{2n}(F)}\\otimes \\theta(\\pi).\\]\nFrom Theorem \\ref{the_largest_quo_in_Weil} it follows that there is an epimorphism\n\\[\\mathrm{ind}_{\\operatorname{Sp}_{2n}(F)\\times \\operatorname{Sp}_{2n}(F)}^{\\operatorname{Sp}_{4n}(F)} 1\\twoheadrightarrow \\theta(\\pi).\\]\nTaking the smooth adjoint of an epimorphism above, we get that\n\\[\\mathrm{Hom}(\\widetilde{\\theta(\\pi)}, \\mathrm{Ind}_{\\operatorname{Sp}_{2n}(F)\\times \\operatorname{Sp}_{2n}(F)}^{\\operatorname{Sp}_{4n}(F)} 1)\\neq 0,\\]\nsince $\\widetilde{\\mathrm{ind}_{\\operatorname{Sp}_{2n}(F)\\times \\operatorname{Sp}_{2n}(F)}^{\\operatorname{Sp}_{4n}(F)} 1}\\cong  \\mathrm{Ind}_{\\operatorname{Sp}_{2n}(F)\\times \\operatorname{Sp}_{2n}(F)}^{\\operatorname{Sp}_{4n}(F)} 1 .$ This is equivalent to the fact that the representation $\\widetilde{\\theta(\\pi)}$ of $\\operatorname{Sp}_{4n}(F)$ has a non-zero symplectic linear model. But if $\\widetilde{\\theta(\\pi)}$ has this model, the representation $\\theta(\\pi)$ also has it  (cf. the proof of Theorem 17 of \\cite{Ginzburg_Rallis_Soudry_explicit}) and we have proved  the following theorem.\n\\begin{thm}\n\\label{main}\nAssume $\\tau$ is an irreducible smooth representation of $\\operatorname{SO}_{4n}(F)$ having a non-zero generalized Shalika model. Then, the irreducible non-zero representation $\\theta'(\\tau)$ (the small theta lift of $\\tau,$ as explained in Introduction) has a non-zero symplectic linear model.\n\\end{thm}\n\n\n \\smallskip\n\\noindent\\textit{Acknowledgments.}\nWe wish to thank American Institute of Mathematics in Palo Alto (Automorphic forms and harmonic analysis on covering groups 2013.) and CIRM, Luminy (WINE conference 2013.) where the author began to study problems related to theta correspondence and generalized Shalika models.\n\nThis work has been supported in part by Croatian Science Foundation under the\nproject 9364.\n\n\n\\bibliographystyle{siam}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\n\nIt is becoming increasingly impractical to store full copies of large datasets on more than one data center \\cite{Hajjat2012}. As a result, the data for a single job may be located not on multiple machines, but on multiple \\textit{clusters} of machines. \nTo maintain fast response-times and avoid excessive network traffic, it is advantageous to perform computation for such jobs in a completely distributed fashion \\cite{HGY}.\nIn addition, commercial platforms such as AWS Lambda and Microsoft's Azure Service Fabric are demonstrating a trend of centralized cloud computing frameworks in which the user manages neither data flow nor server allocation \\cite{AWSLambda, Azure}. \nIn view of these converging issues, the following scheduling problem arises:\n\n\\textit{If computation is done locally to avoid excessive network traffic, how can individual clusters on the broader grid coordinate schedules for maximum throughput? }\n\nThis was precisely the motivation for Hung, Golubchik, and Yu in their 2015 ACM Symposium on Cloud Computing paper \\cite{HGY}.  \nHung et al. modeled each cluster as having an arbitrary number of identical parallel machines, and choose an objective of average job completion time. \nAs such a problem generalizes the NP-Hard concurrent open shop problem, they proposed a heuristic approach. \nTheir heuristic (called ``SWAG'') runs in $O(n^2m)$ time and performed well on a variety of data sets. Unfortunately, SWAG offers poor worst-case performance, as we show in Section \\ref{sec:TSPT}.\n\nOur contributions to this problem are to extend the model considered by Hung et al. and to introduce the first constant-factor approximation algorithms for this general problem. \nOur extensions of Hung et al.'s model are (1) to allow different machines within the same cluster to operate at different speeds, (2) to incorporate pre-specified ``release times'' (times before which a subjob cannot be processed), and (3) to support \\textit{weighted} average job completion time.\nWe present two algorithms for the resulting problem.\nOur combinatorial algorithm exploits a surprisingly simple mapping to the special case of one machine per cluster, where the problem can be approximated in $O(n^2 + nm)$ time. We also present an LP-rounding approach with strong performance guarantees. E.g., a 2-approximation when machines are of unit speed and subjobs are divided into equally sized (but not necessary \\textit{unit}) tasks.\n\n\\subsection{Formal Problem Statement}\\label{subsec:ps}\n\\begin{definition}[Concurrent Cluster Scheduling]{\\color{white} . } \\hfill \n\\vspace{2pt}\n\\begin{itemize}\n\\item There is a set $M$ of $m$ clusters, and a set $N$ of $n$ jobs. \nFor each job $j \\in N$, there is a set of $m$ ``subjobs'' (one for each cluster).\n\n\\item Cluster $i \\in M$ has $m_i$ parallel machines, and machine $\\ell$ in cluster $i$ has speed $v_{\\ell i}$. \nWithout loss of generality, assume $v_{\\ell i}$ is decreasing in $\\ell$.\n\\footnote{Where we write ``decreasing'', we mean ``non-increasing.'' Where we write ``increasing'', we mean ``non-decreasing''.} \n\n\\item The $i^{\\text{th}}$ subjob for job $j$ is specified by a set of tasks to be performed by machines in cluster $i$, denote this set of tasks $T_{ji}$. \nFor each task $t \\in T_{ji}$, we have an associated processing time $p_{jit}$ (again w.l.o.g., assume $p_{jit}$ is decreasing in $t$). \nWe will frequently refer to ``the subjob of job $j$ at cluster $i$'' as ``subjob $(j,i)$.''\n\n\\item Different subjobs of the same job may be processed concurrently on different clusters. \n\\item Different tasks of the same subjob may be processed concurrently on different machines within the same cluster.\n\\item A subjob is complete when all of its tasks are complete, and a job is complete when all of its subjobs are complete. We denote a job's completion time by ``$C_j$''.\n\\item The objective is to minimize weighted average job completion time (job $j$ has weight $w_j$).\n\\item For the purposes of computing approximation ratios, it is equivalent to minimize $\\sum w_j C_j$. We work with this equivalent objective throughout this paper.\n\\end{itemize}\n\\end{definition}\n\nA machine is said to operate at \\textit{unit speed} it if can complete a task with processing requirement ``$p$'' in $p$ units of time. More generally, a machine with speed ``$v$'' ($v \\geq 1$) processes the same task in $p\/ v$ units of time. Machines are said to be \\textit{identical} if they are all of unit speed, and \\textit{uniform} if they differ only in speed.\n\nIn accordance with Graham et al.'s $\\alpha|\\beta|\\gamma$ taxonomy for scheduling problems \\cite{Graham1979} we take $\\alpha = CC$ to refer to the concurrent cluster environment, and denote our problem by $CC|| \\sum w_jC_j$.\\footnote{ A problem $\\alpha|\\beta|\\gamma$ implies a particular environment $\\alpha$, objective function $\\gamma$, and optional constraints $\\beta$.} Optionally, we may associate a release time $r_{ji}$ to every subjob.  If any subjobs are released after time zero, we write $CC| r | \\sum w_jC_j$.\n\n\\subsubsection{Example Problem Instances}\\label{subsubsec:introToModel}\n\nWe now illustrate our model with several examples (see Figures \\ref{fig:probstm1and2} and \\ref{fig:probstm3and4}). The tables at left have rows labeled to identify jobs, and columns labeled to identify clusters; each entry in these tables specifies the processing requirements for the corresponding subjob. The diagrams to the right of these tables show how the given jobs might be scheduled on clusters with the indicated number of machines. \n\\begin{figure}[ht!]\n\\centering\n  \\includegraphics[width=0.8\\linewidth]{example.png}\n\t\\caption{Two examples of our scheduling model. \\textbf{Left}: Our baseline example. There are 4 jobs and 2 clusters. Cluster 1 has 2 identical machines, and cluster 2 has 3 identical machines. Note that job 4 has no subjob for cluster 1 (this is permitted within our framework). In this case every subjob has at most one task. \\textbf{Right}: Our baseline example with a more general subjob framework : subjob (2,2) and subjob (3,1) both have two tasks. The tasks shown are unit length, but our framework \\textit{does not} require that subjobs be divided into equally sized tasks. }\n  \\label{fig:probstm1and2}\n\\end{figure}\n\\begin{figure}[ht!]\n  \\includegraphics[width=\\linewidth]{figure2.png}\n\t\\caption{Two additional examples of our model. \\textbf{Left}: Our baseline example, with variable machine speeds. Note that the benefit of high machine speeds is only realized for tasks assigned to those machines in the final schedule. \\textbf{Right}: A problem with the peculiar structure that (1) all clusters but one have a single machine, and (2) most clusters have non-zero processing requirements for only a single job. We will use such a device for the total weighted lateness reduction in Section \\ref{sec:relationshipsBetweenProbs}.}\n  \\label{fig:probstm3and4}\n\\end{figure}\n\n\\subsection{Related Work}\n\nConcurrent cluster scheduling subsumes many fundamental machine scheduling problems. For example, if we restrict ourselves to a single cluster (i.e. $m = 1$) we can schedule a set of jobs on a bank of identical parallel machines to minimize makespan ($C_{\\max}$) or total weighted completion time ($\\sum w_j C_j$). With a more clever reduction, we can even minimize \\textit{total weighted lateness} ($\\sum w_j L_j$) on a bank of identical parallel machines (see Section \\ref{sec:relationshipsBetweenProbs}). Alternatively, with $m > 1$ but $\\forall i \\in M, m_i = 1$, our problem reduces to the well-studied ``concurrent open shop'' problem.\n\nUsing Graham et al.'s taxonomy, the concurrent open shop problem is written as $PD||\\sum w_j C_j$. Three groups \\cite{Chen2000, Garg2007, llp} independently discovered an LP-based 2-approximation for $PD||\\sum w_j C_j$ using the work of Queyranne \\cite{Queyranne1993}. The linear program in question has an exponential number of constraints, but can still be solved in polynomial time with a variant of the Ellipsoid method. Our ``strong'' algorithm for concurrent cluster scheduling refines the techniques contained therein, as well as those of Schulz \\cite{Schulz1996, Schulz2012} (see Section \\ref{sec:lpAlg}).\n\nMastrolilli et al. \\cite{mqssu} developed a primal-dual algorithm for $PD || \\sum w_j C_j$ that does not use LP solvers. ``MUSSQ''\\footnote{A permutation of the author's names: Mastrolilli, Queyranne, Schulz, Svensson, and Uhan.} is significant for both its speed and the strength of its performance guarantee : it achieves an approximation ratio of 2 in only $O(n^2 + nm)$ time. Although MUSSQ does not require an LP solver, its proof of correctness is based on the fact that it finds a feasible solution to the dual a particular linear program. Our ``fast'' algorithm for concurrent cluster scheduling uses MUSSQ as a subroutine (see Section \\ref{sec:TSPT}).\n\nHung, Golubchik, and Yu \\cite{HGY} presented a framework designed to improve scheduling across geographically distributed data centers. The scheduling framework had a centralized scheduler (which determined a job ordering) and local dispatchers which carried out a schedule consistent with the controllers job ordering. Hung et al. proposed a particular algorithm for the controller called ``SWAG.'' SWAG performed well in a wide variety of simulations where each data center was assumed to have the same number of identical parallel machines. We adopt a similar framework to Hung et al., but we show in Section \\ref{subsec:swagDegenerate} that SWAG has no constant-factor performance guarantee.\n\n\\subsection{Paper Outline \\& Algorithmic Results}\\label{subsec:outlineAndResults}\n\nAlthough only one of our algorithms requires \\textit{solving} a linear program, both algorithms use the same linear program in their proofs of correctness; we introduce this linear program in Section \\ref{sec:introduceLP} before discussing either algorithm. Section \\ref{sec:listSched} establishes how an ordering of jobs can be processed to completely specify a schedule. This is important because the complex work in both of our algorithms is to generate an ordering of jobs for each cluster.\n\nSection \\ref{sec:lpAlg} introduces our ``strong'' algorithm: CC-LP. CC-LP can be applied to any instance of concurrent cluster scheduling, including those with non-zero release times $r_{ji}$. A key in CC-LP's strong performance guarantees lay in the fact that it allows different permutations of subjobs for different clusters. By providing additional structure to the problem (but while maintaining a generalization of concurrent open shop) CC-LP becomes a 2-approximation. This is significant because it is NP-Hard to approximate concurrent open shop (and by extension, our problem) with ratio $2-\\epsilon$ for any $\\epsilon > 0$ \\cite{nphard2}.\n\nOur combinatorial algorithm (``CC-TSPT'') is presented in Section \\ref{sec:TSPT}. The algorithm is fast, provably accurate, and has the interesting property that it can schedule all clusters using the same permutation of jobs.\\footnote{We call such schedules ``single-$\\sigma$ schedules.'' As we will see later on, CC-TSPT serves as a constructive proof of existence of near-optimal single-$\\sigma$ schedules for all instances of $CC||\\sum w_j C_j$, \\textit{including} those instances for which single-$\\sigma$ schedules are strictly sub-optimal. This is addressed in Section \\ref{sec:discAndConc}.} After considering CC-TSPT in the general case, we show how fine-grained approximation ratios can be obtained in the ``fully parallelizable'' setting of Zhang et al. \\cite{zwl}. We conclude with an extension of CC-TSPT that maintains performance guarantees while offering improved empirical performance.\n\nThe following table summarizes our results for approximation ratios. For compactness, condition $Id$ refers to identical machines (i.e. $v_{\\ell i}$ constant over $\\ell$), condition $A$ refers to $r_{ji} \\equiv 0$, and condition $B$ refers to $p_{jit} \\text{ constant over } t \\in T_{ji}$.\n\\begin{center}\n\\begin{tabular}{l| cccccc}\n\\hline\n \t\t& $(Id,A,B)$ & $(Id, \\neg A, B)$ & $(Id,A,\\neg B)$ & $(Id,\\neg A, \\neg B)$ & $(\\neg Id, A)$ & $(\\neg Id, \\neg A)$ \\\\ \\hline\nCC-LP\t&\t2\t  &     3         &    3         &   4  &  $2+R$     &  $3+R$    \\\\\nCC-TSPT &   3     &     -          &   3         &   -  &  $2+R$      &    -  \\\\ \\hline\n\\end{tabular}\n\\end{center}\nThe term $R$ is the maximum over $i$ of $R_i$, where $R_i$ is the ratio of fastest machine to \\textit{average} machine speed at cluster $i$.\n\nThe most surprising of all of these results is that our scheduling algorithms are remarkably simple. The first algorithm solves an LP,\nand then the scheduling can be done easily on each cluster. The second algorithm is again a rather surprising simple reduction to the\ncase of one machine per cluster (the well understood concurrent open shop problem) and yields a simple combinatorial algorithm. The proof\nof the approximation guarantee is somewhat involved however.\n\nIn addition to algorithmic results, we demonstrate how our problem subsumes that of minimizing total weighted lateness on a bank of identical parallel machines (see Section \\ref{sec:relationshipsBetweenProbs}). Section \\ref{sec:discAndConc} provides additional discussion and highlights our more novel technical contributions.\n\n\n\\section{The Core Linear Program }\\label{sec:introduceLP}\n\n\n\nOur linear program has an unusual form. Rather than introduce it immediately, we conduct a brief review of prior work on similar LP's. All the LP's we discuss in this paper have objective function $\\sum w_j C_j$, where $C_j$ is a decision variable corresponding to the completion time of job $j$, and $w_j$ is a weight associated with job $j$. \n\n\\textit{For the following discussion only, we adopt the notation in which job $j$ has processing time $p_j$. In addition, if multiple machine problems are discussed, we will say that there are $\\mathsf{m}$ such machines (possibly with speeds $s_i, i \\in \\{1,\\ldots, \\mathsf{m}\\}$).} \n\nThe earliest appearance of a similar linear program comes from Queyranne \\cite{Queyranne1993}. In his paper, Queyranne presents an LP relaxation for sequencing $n$ jobs on a single machine where all constraints are of the form $\\sum_{j \\in S} p_j C_j \\geq \\frac{1}{2}\\left[\\left(\\sum_{j \\in S} p_j \\right)^2 + \\sum_{j \\in S} p_j^2\\right]$ where $S$ is an arbitrary subset of jobs. Once a set of optimal $\\{C_j^\\star\\}$ is found, the jobs are scheduled in increasing order of $\\{C_j^\\star\\}$. These results were primarily theoretical, as it was known at his time of writing that sequencing $n$ jobs on a single machine to minimize $\\sum w_j C_j$ can be done optimally in $O(n \\log n)$ time.\n\nQueyranne's constraint set became particularly useful for problems with \\textit{coupling} across distinct machines (as occurs in concurrent open shop). Four separate groups \\cite{Chen2000,Garg2007,llp, mqssu} saw this and used the following LP in a 2-approximation for concurrent open shop scheduling.\n\\begin{equation}\n(\\text{LP0}) ~~ \\min \\sum_{j \\in N} w_j C_j ~~ \\text{s.t.} ~~ \\textstyle\\sum_{j \\in S} p_{ji} C_j \\geq \n\t\t\\frac{1}{2}\n\t\t\\left[ \n\t\t\t\\left(\\textstyle\\sum_{j \\in S} p_{ji}\\right)^2 + \\left(\\textstyle\\sum_{j \\in S} p_{ji}^2\\right) \n\t\t\\right] ~ \\forall ~ \\substack{S \\subseteq N \\\\  i \\in M }\\nonumber\n\\end{equation}\nIn view of its tremendous popularity, we sometimes refer to the linear program above as the \\textit{canonical relaxation} for concurrent open shop.\n\nAndreas Schulz's Ph.D. thesis developed Queyranne's constraint set in greater depth \\cite{Schulz1996}. As part of his thesis, Schulz considered scheduling $n$ jobs on $\\mathsf{m}$ identical parallel machines with constraints of the form $\\sum_{j \\in S} p_j C_j \\geq \\frac{1}{2\\mathsf{m}} \\left(\\sum_{j \\in S} p_j \\right)^2 + \\frac{1}{2}\\sum_{j \\in S} p_j^2$. In addition, Schulz showed that the constraints $\\sum_{j \\in S} p_j C_j \\geq \\left[2 \\sum_{i=1}^{\\mathsf{m}} s_i \\right]^{-1}\\left[\\left(\\sum_{j \\in S} p_j \\right)^2 + \\sum_{j \\in S} p_j^2\\right]$ are satisfied by any schedule of $n$ jobs on $\\mathsf{m}$ uniform machines. In 2012, Schulz refined the analysis for several of these problems \\cite{Schulz2012}. For constructing a schedule from the optimal $\\{C_j^\\star\\}$, Schulz considered scheduling jobs by increasing order of $\\{C_j^\\star\\}$, $\\{C_j^\\star - p_j\/2\\}$, and $\\{C_j^\\star - p_j\/(2\\mathsf{m})\\}$.\n\n\n\\subsection{Statement of LP1}\n\nThe model we consider allows for more fine-grained control of the job structure than is indicated by the LP relaxations above. Inevitably, this comes at some expense of simplicity in LP formulations. In an effort to simplify notation, we define the following constants, and give verbal interpretations for each. \n\\begin{equation}\n \\mu_{i} \\doteq \\textstyle\\sum_{\\ell = 1}^{m_i} v_{\\ell i} \\qquad q_{ji} \\doteq \\min{\\lbrace|T_{ji}|, m_i\\rbrace}  \\qquad \\mu_{ji} \\doteq \\textstyle\\sum_{\\ell = 1}^{q_{ji}} v_{\\ell i} \\qquad p_{ji} \\doteq \\textstyle\\sum_{t \\in T_{ji}} p_{jit}\n\\end{equation}\nFrom these definitions, $\\mu_i$ is the processing power of cluster $i$. For subjob $(j,i)$, $q_{ji}$ is the maximum number of machines that could process the subjob, and $\\mu_{ji}$ is the maximum processing power than can be brought to bear on the same. Lastly, $p_{ji}$ is the total processing requirement of subjob $(j,i)$. In these terms, the core linear program, LP1, is as follows.\n\\begin{align*}\n\t\\text{(LP1) } \\min &\\textstyle\\sum_{j \\in N} w_j C_j \\\\\n\ts.t.\\quad (1A) \\quad & \\textstyle\\sum_{j \\in S} p_{ji} C_j \n\t        \\geq \\frac{1}{2} \n\t        \\left[ \n\t           \\left(\\textstyle\\sum_{j \\in S} p_{ji}\\right)^2\/\\mu_i\n                + \\textstyle\\sum_{j \\in S} p_{ji}^2\/\\mu_{ji} \n            \\right]  \\qquad ~\\forall S \\subseteq N, i \\in M \\\\\n\t(1B) \\quad\t& C_j \\geq p_{jit}\/v_{1i} + r_{ji} \\qquad ~\\forall  i \\in M,~ j \\in N,~ t \\in T_{ji}\\\\\n\t(1C) \\quad\t& C_j \\geq p_{ji}\/\\mu_{ji} + r_{ji} \\qquad ~\\forall j \\in N,~ i \\in M \t \n\\end{align*} \n\nConstraints ($1A$) are more carefully formulated versions of the polyhedral constraints introduced by Queyranne \\cite{Queyranne1993} and developed by Schulz \\cite{Schulz1996}. The use of $\\mu_{ji}$ term is new and allows us to provide stronger performance guarantees for our framework where subjobs are composed of \\textit{sets} of tasks. As we will see, this term is one of the primary factors that allows us to parametrize results under varying machine speeds in terms of maximum to \\textit{average} machine speed, rather than maximum to \\textit{minimum} machine speed. Constraints ($1B$) and ($1C$) are simple lower bounds on job completion time. \n\n\nThe majority of this section is dedicated to proving that LP1 is a valid relaxation of $CC|r|\\sum w_jC_j$. Once this is established, we prove the that LP1 can be solved in polynomial time by providing a separation oracle with use in the Ellipsoid method. Both of these proofs use techniques established in Schulz's Ph.D. thesis \\cite{Schulz1996}. \n\n\\subsection{Proof of LP1's Validity}\n\n\n\n\nThe lemmas below establish the basis for both of our algorithms. Lemma \\ref{lem:sumOfSquaresDiffSpeeds} generalizes an inequality used by Schulz \\cite{Schulz1996}. Lemma \\ref{lem:feasForLP1} relies on Lemma \\ref{lem:sumOfSquaresDiffSpeeds} and cites an inequality mentioned in the preceding section (and proven by Queyranne \\cite{Queyranne1993}). \n\\begin{lemma}\nLet $\\{a_1,\\ldots a_z\\}$ be a set of non-negative real numbers. We assume that $k \\leq z$ of them are positive. Let $b_i$ be a set of decreasing positive real numbers. Then\n\\begin{center}\n$  \\sum_{i = 1}^z a_i^2 \/ b_i  \\geq \\left(\\sum_{i = 1}^z a_i \\right)^2 \/ \\left(\\sum_{i = 1}^k b_i\\right)$.\n\\end{center}\n\\label{lem:sumOfSquaresDiffSpeeds}\n\\end{lemma}\n\\begin{proof}\\footnote{The proceedings version of this paper stated that the proof cites the AM-GM inequality and proceeds by induction from $z=k=2$. We have opted here to demonstrate a different (simpler) proof that we discovered only after the proceedings version was finalized.}\nWe only show the case where $k=z$. Define $\\mathbf{a} = [a_1,\\ldots,a_k] \\in \\mathbb{R}^k_+$, $\\mathbf{b} = [b_1, \\ldots, b_k ] \\in \\mathbb{R}^k_{++}$, and $\\mathbbm{1}$ as the vector of $k$ ones. Now, set $\\mathbf{u} = \\mathbf{a} \/ \\sqrt{\\mathbf{b}}$ and $\\mathbf{w} = \\sqrt{\\mathbf{b}}$ (element-wise), and note that $\\langle \\mathbf{a}, \\mathbbm{1} \\rangle = \\langle \\mathbf{u}, \\mathbf{w} \\rangle$. In these terms, it is clear that $(\\sum_{i=1}^k a_i)^2 = \\langle \\mathbf{u}, \\mathbf{w} \\rangle^2$.\n\nGiven this, one need only cite Cauchy-Schwarz (namely, $\\langle \\mathbf{u}, \\mathbf{w} \\rangle^2 \\leq \\langle \\mathbf{u}, \\mathbf{u} \\rangle \\cdot \\langle \\mathbf{w}, \\mathbf{w} \\rangle$) and plug in the definitions of $\\mathbf{u}$ and $\\mathbf{w}$ to see the desired result.\n\\end{proof}\n\n\n\\begin{lemma}[Validity Lemma]\nEvery feasible schedule for an instance $I$ of $CC|r|\\sum w_jC_j$ has completion times that define a feasible solution to LP1($I$). \\label{lem:feasForLP1}\n\\end{lemma}\n\\begin{proof}\nAs constraints ($1B$) and ($1C$) are clear lower bounds on job completion time, it suffices to show the validity of constraint ($1A$). Thus, let $S$ be a non-empty subset of $N$, and fix an arbitrary but feasible schedule ``$F$'' for $I$. \n\nDefine $C^{F}_{ji}$ as the completion time of subjob $(j,i)$ under schedule $F$. Similarly, define $C^{F}_{ji\\ell}$ as the first time at which tasks of subjob $(j,i)$ scheduled on machine $\\ell$ of cluster $i$ are finished. Lastly, define $p^{\\ell}_{ji}$ as the total processing requirement of job $j$ scheduled on machine $\\ell$ of cluster $i$. Note that by construction, we have $C^{F}_{ji} = \\max_{\\ell \\in \\{1,\\ldots,m_i\\}}{C^{F}_{ji\\ell}}$ and $C^F_j = \\max_{i \\in M}{C^F_{ji}}$. \nSince $p_{ji} = \\sum_{\\ell = 1}^{m_i} p^{\\ell}_{ji}$, we can rather innocuously write\n\\begin{equation}\n\\textstyle\\sum_{j \\in S} p_{ji} C^{F}_{ji} = \\textstyle\\sum_{j \\in S}\\left[ \\textstyle\\sum_{\\ell  = 1}^{m_i}  p^{\\ell}_{ji}  \\right] C^{F}_{ji} . \n\\end{equation} \nBut using $C^{F}_{ji} \\geq C^{F}_{ji\\ell}$, we can lower-bound $\\sum_{j \\in S} p_{ji} C^{F}_{ji}$. Namely,\n\\begin{equation}\n\\textstyle\\sum_{j \\in S} p_{ji} C^{F}_{ji} \\geq \\textstyle\\sum_{j \\in S}\\textstyle\\sum_{\\ell = 1}^{m_i}  p^{\\ell}_{ji} C^{F}_{ji\\ell} =  \\textstyle\\sum_{\\ell = 1}^{m_i} v_{\\ell i}\\textstyle\\sum_{j \\in S} \\left[p^{\\ell}_{ji}\/v_{\\ell i} \\right]C^{F}_{ji\\ell} \\label{eq:specificMachines}\n\\end{equation}\nThe next inequality uses a bound on $\\textstyle\\sum_{j \\in S}\\left[p^{\\ell}_{ji}\/v_{\\ell i} \\right]C^{F}_{ji\\ell}$ proven by Queyranne \\cite{Queyranne1993} for any subset $S$ of $N$ jobs with processing times $\\left[p^{\\ell}_{ji}\/v_{\\ell i} \\right]$ to be scheduled on a single machine.\\footnote{Here, our machine is machine $\\ell$ on cluster $i$.}\n\\begin{equation}\n\\textstyle\\sum_{j \\in S} \\left[p^{\\ell}_{ji}\/v_{\\ell i} \\right]C^{F}_{ji\\ell} \\geq \\frac{1}{2} \\left[\\left(\\textstyle\\sum_{j \\in S} \\left[p^{\\ell}_{ji}\/v_{\\ell i} \\right]\\right)^2 + \\textstyle\\sum_{j \\in S} \\left(\n\\left[p^{\\ell}_{ji}\/v_{\\ell i} \\right]\\right)^2 \\right]\\label{eq:queyranne}\n\\end{equation}\nCombining inequalities \\eqref{eq:specificMachines} and \\eqref{eq:queyranne}, we have the following.\n\\begin{align}\n\\textstyle\\sum_{j \\in S} p_{ji} C^{F}_{ji} &\\geq \\frac{1}{2} \\textstyle\\sum_{\\ell=1}^{m_i} v_{\\ell i} \\left[\\left(\\textstyle\\sum_{j \\in S} \\left[p^{\\ell}_{ji}\/v_{\\ell i} \\right]\\right)^2 + \\textstyle\\sum_{j \\in S} \\left(\n\\left[p^{\\ell}_{ji}\/v_{\\ell i} \\right]\\right)^2 \\right]  \\\\\n& \\geq \\frac{1}{2} \\left[\\textstyle\\sum_{\\ell = 1}^{m_i}\\left(\\sum_{j \\in S} p^\\ell_{j i}\\right)^2 \/ v_{\\ell i} + \\sum_{j \\in S} \\sum_{\\ell = 1}^{m_i} \\left(p^\\ell_{j i}\\right)^2 \/ v_{\\ell i} \\right] \\label{eq:differentBoundForLP1}\n\\end{align}\nNext, we apply Lemma \\ref{lem:sumOfSquaresDiffSpeeds} to the right hand side of inequality \\eqref{eq:differentBoundForLP1} a total of $|S|+1$ times.\n\\begin{align}\n&\\textstyle\\sum_{\\ell = 1}^{m_i} \\left(\\textstyle\\sum_{j \\in S} p^\\ell_{j i}\\right)^2 \/v_{\\ell i} \\geq \\left(\\textstyle\\sum_{\\ell = 1}^{m_i}\\textstyle\\sum_{j \\in S} p^\\ell_{j i}\\right)^2\/ \\textstyle\\sum_{\\ell = 1}^{m_i} v_{\\ell i} = \\left(\\textstyle\\sum_{j \\in S} p_{ji}\\right)^2 \/ \\mu_i \\\\\n&\\textstyle\\sum_{\\ell = 1}^{m_i} \\left(p^\\ell_{j i}\\right)^2 \/ v_{\\ell i} \\geq \\left(\\sum_{\\ell = 1}^{m_i}p^\\ell_{j i}\\right)^2 \/ \\textstyle\\sum_{\\ell = 1}^{q_{j i}} v_{\\ell i} = p_{j i}^2\/\\mu_{j i} ~~\\forall~ j \\in S\n\\end{align}\nCiting $C^{F}_{j} \\geq C^{F}_{ji}$, we arrive at the desired result.\n\\begin{equation}\n\\textstyle\\sum_{j \\in S} p_{ji} C^{F}_{j} \\geq \\frac{1}{2}\\left[\\left(\\textstyle\\sum_{j \\in S} p_{ji} \\right)^2\/\\mu_i + \\textstyle\\sum_{j \\in S}p_{ji}^2\/\\mu_{ji}\\right] \\qquad  \\text{``constraint }(1A)\\text{''}\n\\end{equation}\n\n\\end{proof}\n\n\n\n\\subsection{Theoretical Complexity of LP1}\\label{subsec:thankGodPolyTime}\nAs the first of our two algorithms requires solving LP1 directly, we need to address the fact that LP1 has $m \\cdot (2^n - 1) + n$ constraints. \nLuckily, it is still possible to such solve linear programs in polynomial time with the Ellipsoid method; we introduce the following separation oracle for this purpose.\n\n\\begin{definition}[Oracle LP1]\nDefine the \\textit{violation}\n\\begin{equation}\nV(S,i) = \\frac{1}{2} \\left[ \n\t           \\left(\\textstyle\\sum_{j \\in S} p_{j i}\\right)^2\/\\mu_i \n                + \\textstyle\\sum_{j \\in S} p_{j i}^2\/\\mu_{j i} \n            \\right] - \\textstyle\\sum_{j \\in S} p_{j  i} C_{j}\n\\end{equation}\nLet $\\{C_j\\} \\in \\mathbb{R}^n$ be a \\textit{potentially} feasible solution to LP1. Let $\\sigma_i$ denote the ordering when jobs are sorted in increasing order of $C_j - p_{j i}\/(2\\mu_{ji})$. Find the most violated constraint in $(1A)$ for $i \\in M$ by searching over $V(S_i,i)$ for $S_i$ of the form $\\{\\sigma_i(1),\\ldots,\\sigma_i(j-1),\\sigma_i(j)\\},~ j \\in \\{1,\\ldots,n\\}$. If any of maximal $V(S_i^*,i) > 0$, then return $(S_i^*,i)$ as a violated constraint for ($1A$). Otherwise, check the remaining $n$ constraints $((1B)$ and $(1C))$ directly in linear time.\n\\end{definition} \n\nFor fixed $i$, Oracle-LP1 finds the subset of jobs that maximizes ``violation'' for cluster $i$. That is, Oracle-LP1 finds $S_i^*$ such that $V(S_i^*,i) = \\text{max}_{S \\subset N} V(S,i)$. We prove the correctness of Oracle-LP1 by establishing a necessary and sufficient condition for a job $j$ to be in $S_i^*$.\n\n\\begin{lemma}\nFor $\\mathbb{P}_i(A) \\doteq \\textstyle\\sum_{j \\in A} p_{ji}$, we have $x \\in S_i^* \\Leftrightarrow$ $ C_x - p_{xi}\/(2\\mu_{xi}) \\leq \\mathbb{P}_i(S_i^*)\/\\mu_i$. \n\\label{lem:separation}\n\\end{lemma}\n\\begin{proof}\nFor given $S$ (not necessarily equal to $S_i^*$), it is useful to express $V(S,i)$ in terms of $V(S\\cup x, i)$ or $V(S\\setminus x, i)$ (depending on whether $x \\in S$ or $x \\in N \\setminus S$). Without loss of generality, we restrict our search to $S : x \\in S \\Rightarrow p_{x,i} > 0$.\n\nSuppose $ x \\in S$. By writing $\\mathbb{P}_i(S) = \\mathbb{P}_i(S\\setminus x) + \\mathbb{P}_i(x)$, and similarly decomposing the sum $\\textstyle\\sum_{j \\in S} p_{j i}^2\/(2\\mu_{ji})$, one can show the following.\n\\begin{align}\nV(S, i) = & V(S\\setminus x, i) + p_{xi}\\left(\\frac{1}{2}\\left(\\frac{2\\mathbb{P}_i(S) - p_{xi}}{\\mu_i} + \\frac{p_{xi}}{\\mu_{xi}}\\right) - C_x \\right) \\label{eq:xInS}\n\\end{align}\nNow suppose $ x \\in N\\setminus S$. In the same strategy as above (this time writing $ \\mathbb{P}_i(S) = \\mathbb{P}_i(S\\cup x) - \\mathbb{P}_i(x)$), one can show that\n\\begin{align}\nV(S, i) =&  V(S\\cup x, i) + p_{xi}\\left(C_x - \\frac{1}{2}\\left(\\frac{2\\mathbb{P}_i(S) + p_{xi}}{\\mu_i} + \\frac{p_{xi}}{\\mu_{xi}}\\right) \\right). \\label{eq:xNotInS}\n\\end{align}\nNote that Equations \\eqref{eq:xInS} and \\eqref{eq:xNotInS} hold for all $S$, including $S = S_i^*$. Turning our attention to $S_i^*$, we see that $x \\in S_i^*$ implies that the second term in Equation \\eqref{eq:xInS} is non-negative,  i.e. \n\\begin{equation}\nC_x - p_{xi}\/(2\\mu_{xi}) \\leq \\left(2\\mathbb{P}_i(S_i^*) - p_{xi}\\right)\/(2\\mu_i) < \\mathbb{P}_i(S_i^*)\/\\mu_i.\n\\end{equation}\nSimilarly, $x \\in N \\setminus S_i^*$ implies the second term in Equation \\eqref{eq:xNotInS} is non-negative.\n\\begin{equation}\nC_x - p_{x i}\/(2\\mu_{x i}) \\geq \\left(2\\mathbb{P}_i(S_i^*) + p_{x i}\\right)\/(2\\mu_i) \\geq \\mathbb{P}_i(S_i^*)\/\\mu_i\n\\end{equation}\nIt follows that $x \\in S_i^*$ iff $C_x - p_{x i}\/(2\\mu_{x i}) < \\mathbb{P}_i(S_i^*)\/\\mu_i$.\n\\end{proof}\n\nGiven Lemma \\ref{lem:separation}, It is easy to verify that sorting jobs in increasing order of $C_x - p_{xi}\/(2\\mu_{xi})$ to define a permutation $\\sigma_i$ guarantees that $S_i^*$ is of the form $\\{\\sigma_i(1),\\ldots,\\sigma_i(j-1),\\sigma_i(j)\\}$ for some $j \\in N$. This implies that for fixed $i$, Oracle-LP1 finds $S_i^*$ in $O(n \\log(n))$ time. This procedure is executed once for each cluster, leaving the remaining $n$ constraints in $(1B)$ and $(1C)$ to be verified in linear time. Thus Oracle-LP1 runs in $O(mn\\log(n))$ time.\n\nBy the equivalence of separation and optimization, we have proven the following theorem:\n\\begin{theorem}\nLP1($I$) is a valid relaxation of $I \\in \\Omega_{CC}$, and is solvable in polynomial time. \\label{thm:LP1feasAndSolve}\n\\end{theorem}\n\nAs was explained in the beginning of this section, linear programs such as those in \\cite{Chen2000, Garg2007, llp, Queyranne1993, Schulz1996, Schulz2012} are processed with an appropriate sorting of the optimal decision variables $\\{C^\\star_j\\}$. It is important then to have bounds on job completion times for a particular ordering of jobs. We address this next in Section \\ref{sec:listSched}, and reserve our first algorithm for Section \\ref{sec:lpAlg}.\n\\section{List Scheduling from Permutations}\\label{sec:listSched}\n\n\n\n\n\n\n\n\nThe complex work in both of our proposed algorithms is to generate a \\textit{permutation} of jobs. The procedure below takes such a permutation and uses it to determine start times, end times, and machine assignments for every task of every subjob.\n\\vspace{1em}\n\n\\noindent \\textbf{List-LPT} : Given a single cluster with $m_i$ machines and a permutation of jobs $\\sigma$, introduce $\\text{List}(a,i) \\doteq (p_{ai1}, p_{ai2},\\ldots,p_{ai|T_{ai}|})$ as an ordered set of tasks belonging to subjob $(a,i)$, ordered by longest processing time first. Now define $\\text{List}(\\sigma) \\doteq \\text{List}(\\sigma(1),i) \\oplus \\text{List}(\\sigma(2),i) \\oplus \\cdots \\oplus \\text{List}(\\sigma(n),i)$, where $\\oplus$ is the concatenation operator. \n\nPlace the tasks of $\\text{List}(\\sigma)$ in order- from the largest task of subjob $(\\sigma(1),i)$, to the smallest task of subjob $(\\sigma(n),i)$. When placing a particular task, assign it whichever machine and start time results in the task being completed as early as possible (without moving any tasks which have already been placed). Insert idle time (on all $m_i$ machines) as necessary if this procedure would otherwise start a job before its release time.\n\\vspace{1em}\n\nThe following Lemma is essential to bound the completion time of a set of jobs processed by List-LPT. The proof is adapted from Gonzalez et al. \\cite{Gonzalez1977}.\n\\begin{lemma}\nSuppose $n$ jobs are scheduled on cluster $i$ according to List-LPT($\\sigma$). Then for $ \\bar{v_i} \\doteq \\mu_i\/m_i$, the completion time of subjob $(\\sigma(j),i)$ $($denoted $C_{\\sigma(j)i}$ $)$ satisfies\n\\begin{align}\n&C_{\\sigma(j)i} \\leq \\max_{1\\leq k \\leq j}{r_{\\sigma(k)i}} + p_{\\sigma(j)i1}\/\\bar{v_i} +  \\left(\\textstyle\\sum_{k=1}^{j} p_{\\sigma(k)i} - p_{\\sigma(j)i1}\\right)\/\\mu_i \\label{eq:generalGonzalezLemma}\n\\end{align} \\label{lem:CompTimesOnUniformMachines}\n\\end{lemma}\n\\begin{proof}\nFor now, assume all jobs are released at time zero. Let the task of subjob $(\\sigma(j),i)$ to finish last be denoted $t^*$. If $t^*$ is not the task in $T_{\\sigma(j)i}$ with least processing time, then construct a new set $T'_{\\sigma(j)i} = \\{ t :  p_{\\sigma(j)it^*} \\leq p_{\\sigma(j)it} \\} \\subset T_{\\sigma(j)i}$. Because the tasks of subjob $(\\sigma(j),i)$ were scheduled by List-LPT (i.e. longest-processing-time-first), the sets of potential start times and machines for task $t^*$ (and hence the set of potential completion times for task $t^*$) are the same regardless of whether subjob $(\\sigma(j),i)$ consisted of tasks $T_{\\sigma(j)i}$ or the subset $T'_{\\sigma(j)i}$. Accordingly, reassign $T_{\\sigma(j)i} \\leftarrow T'_{\\sigma(j)i}$ without loss of generality.\n\nLet $D_{\\ell}^j$ denote the total demand for machine $\\ell$ (on cluster $i$) once all tasks of subjobs $(\\sigma(1),i)$ through $(\\sigma(j-1),i)$ and all tasks in the set $T_{\\sigma(j)i}\\setminus \\{t^*\\}$ are scheduled. Using the fact that $C_{\\sigma(j)i}v_{\\ell i} \\leq ({D}_{\\ell}^{j} + p_{\\sigma(j)i t^*}) \\forall \\ell \\in \\{1,\\ldots,m_i\\}$, sum the left and right and sides over $\\ell$. This implies $C_{\\sigma(j)i}\\left( \\textstyle\\sum_{\\ell = 1}^{m_i} v_{\\ell i} \\right) \\leq ~ m_i p_{\\sigma(j) i t^*} + \\textstyle\\sum_{\\ell = 1}^{m_i} {D}_{\\ell}^{j}$. Dividing by the sum of machine speeds and using the definition of $\\mu_i$ yields\n\\begin{equation}\nC_{\\sigma(j)i} \n\t~ \\leq ~  m_i p_{\\sigma(j)i t^*}\/\\mu_i + \\textstyle\\sum_{\\ell = 1}^{m_i} {D}_{\\ell}^j \/\\mu_i ~ \\leq ~ p_{\\sigma(j)i 1}\/\\bar{v_i} + \\left(\\textstyle\\sum_{k = 1}^{j} p_{\\sigma(k)i} - p_{\\sigma(j)i1}\\right)\/\\mu_i \\label{eq:mainGonzalezLemma}\n\\end{equation}\nwhere we estimated $p_{\\sigma(j)i t^*}$ upward by $p_{\\sigma(j)i 1}$. Inequality \\eqref{eq:mainGonzalezLemma} completes our proof in the case when $r_{ji} \\equiv 0$. \n\nNow suppose that some $r_{ji} > 0$. We take our policy to the extreme and suppose that all machines are left idle until every one of jobs $\\sigma(1)$ through $\\sigma(j)$ are released; note that this occurs precisely at time $\\max_{1 \\leq k \\leq j} r_{\\sigma(k)i}$. It is clear that beyond this point in time, we are effectively in the case where all jobs are released at time zero, hence we can bound the remaining time to completion by the right hand side of Inequality \\ref{eq:mainGonzalezLemma}. As Inequality \\ref{eq:generalGonzalezLemma} simply adds these two terms, the result follows.\n\\end{proof}\n\nLemma \\ref{lem:CompTimesOnUniformMachines} is cited directly in the proof of Theorem \\ref{thm:uniformLP1} and Lemma \\ref{lem:TSPTboundInTermsOfPDandOPT}. Lemma \\ref{lem:CompTimesOnUniformMachines} is used implicitly in the proofs of Theorems \\ref{thm:identLP}, \\ref{thm:identLP_2appxWithConstantTasks}, and \\ref{thm:tspt_unit_tasks}.\n\\section{An LP-based Algorithm}\\label{sec:lpAlg}\n\n\n\nIn this section we show how LP1 can be used to construct near optimal schedules for concurrent cluster scheduling both when $r_{ji} \\equiv 0$ and when some $r_{ji} > 0$. Although solving LP1 is somewhat involved, the algorithm itself is quite simple:\n\\vspace{0.5em}\n\n\\noindent \\textbf{Algorithm CC-LP} : Let $I = (T, r, w, v)$ denote an instance of $CC | r | \\sum w_j C_j$. Use the optimal solution $\\{C_j^\\star\\}$ of LP1($I$) to define $m$ permutations $\\{\\sigma_i : i \\in M\\}$ which sort jobs in increasing order of $C^\\star_j - p_{ji}\/(2\\mu_{ji})$. For each cluster $i$, execute List-LPT($\\sigma_i$).\n\\vspace{0.5em}\n\nEach theorem in this section can be characterized by how various assumptions help us cancel an additive term\\footnote{``$+p_{xit^*}$''; see associated proofs.} in an upper bound for the completion time of an arbitrary subjob $(x,i)$. Theorem \\ref{thm:uniformLP1} is the most general, while Theorem \\ref{thm:identLP_2appxWithConstantTasks} is perhaps the most surprising.\n\n\\subsection{CC-LP for Uniform Machines}\\label{sec:unifLP}\n\\begin{theorem}\nLet $\\hat{C}_j$ be the completion time of job $j$ using algorithm CC-LP, and let $R$ be as in Section \\ref{subsec:outlineAndResults}. If $r_{ji} \\equiv 0$, then $ \\textstyle\\sum_{j \\in N} w_j \\hat{C}_j \\leq \\left(2 + R\\right)OPT $. Otherwise, $ \\textstyle\\sum_{j \\in N} w_j \\hat{C}_j \\leq \\left(3 + R\\right)OPT$.\n\\label{thm:uniformLP1}\n\\end{theorem}\n\\begin{proof}\nFor $y \\in \\mathbb{R}$, define $y^+ = \\max\\{y,0\\}$. Now let $x \\in N$ be arbitrary, and let $i \\in M$ be such that $p_{xi} > 0$ (but otherwise arbitrary). Define $t^*$ as the last task of job $x$ to complete on cluster $i$, and let $j_i$ be such that $\\sigma_i(j_i) = x$. Lastly, denote the optimal LP solution $\\{C_j\\}$.\\footnote{We omit the customary $\\star$ to avoid clutter in notation.} Because $\\{C_j\\}$ is a feasible solution to LP1, constraint $(1A)$ implies the following (set $S_i = \\{\\sigma_i(1),\\ldots,\\sigma_i(j_i - 1),x\\}$)\n\\begin{align}\n\\frac{\\left( \\textstyle\\sum_{k = 1}^{j_i} p_{\\sigma_i(k)i} \\right)^2}{2\\mu_i}\n\t&\\leq \\sum_{k = 1}^{j_i} p_{\\sigma_i(k)i}\\left(C_{\\sigma_i(k)} - \\frac{p_{\\sigma_i(k)i}}{2\\mu_{\\sigma_i(k)i}}\\right) \\leq \\left(C_{x} - \\frac{p_{xi}}{2\\mu_{xi}}\\right)\\sum_{k = 1}^{j_i} p_{\\sigma_i(k)i} \\label{eq:compTimeInUnifLP}\n\\end{align}\nwhich in turn implies $\\textstyle\\sum_{k = 1}^{j_i} p_{\\sigma_i(k)i}\/\\mu_i \\leq 2C_x - p_{xi}\/\\mu_{xi}$.\n\nIf all subjobs are released at time zero, then we can combine this with Lemma \\ref{lem:CompTimesOnUniformMachines} and the fact that $p_{xit^*} \\leq p_{xi} = \\textstyle\\sum_{t \\in T_{xi}} p_{xit}$ to see the following (the transition from the first inequality the second inequality uses $C_x \\geq p_{xit^*}\/v_{1i}$ and $R_i = v_{1i}\/\\bar{v}_i$).\n\\begin{align}\n\\hat{C}_{xi} \n\t&\\leq 2C_x - \\frac{p_{xi}}{\\mu_{xi}} + \\frac{p_{xit^*}}{\\bar{v}_i} - \\frac{p_{xit^*}}{\\mu_i} \\leq \n\t\tC_x(2 + \\left[R_i(1 - 2\/m_i)\\right]^+) \\label{eq:generalCompTimeWithOUTReleaseUnif} \n\\end{align}\n\nWhen one or more subjobs are released after time zero, Lemma \\ref{lem:CompTimesOnUniformMachines} implies that it is sufficient to bound  $\\displaystyle\\max_{1 \\leq k \\leq j_i}{\\left\\lbrace r_{\\sigma_i(k)i} \\right\\rbrace}$ by some constant multiple of $C_x$. Since $\\sigma_i$ is defined by increasing $L_{ji} \\doteq C_j - p_{ji}\/(2\\mu_{ji})$, $L_{\\sigma_i(a)i} \\leq L_{\\sigma_i(b)i}$ implies\n\\begin{align}\n&r_{\\sigma_i(a)i} + \\frac{p_{\\sigma_i(a)i}}{2\\mu_{\\sigma_i(a)i}} + \\frac{p_{\\sigma_i(b)i}}{2\\mu_{\\sigma_i(b)i}}  \\leq C_{\\sigma_i(a)} - \\frac{p_{\\sigma_i(a)i}}{2\\mu_{\\sigma_i(a)i}} + \\frac{p_{\\sigma_i(b)i}}{2\\mu_{\\sigma_i(b)i}}  \\leq C_{\\sigma_i(b)} ~\\forall~ a \\leq b\n\\end{align}\nand so $\\max_{1 \\leq k \\leq j_i}{\\left\\lbrace r_{\\sigma_i(k)i} \\right\\rbrace} + p_{xi}\/(2\\mu_{xi}) \\leq C_{x}$. As before, combine this with Lemma \\ref{lem:CompTimesOnUniformMachines} and the fact that $p_{xit^*} \\leq p_{xi} = \\textstyle\\sum_{t \\in T_{xi}} p_{xit}$ to yield the following inequalities\n\\begin{align}\n\\hat{C}_{xi} \n\t&\\leq 3C_x - \\frac{3p_{xi}}{2\\mu_{xi}} + \\frac{p_{xit^*}}{\\bar{v}_i} - \\frac{p_{xit^*}}{\\mu_i} \\leq C_x(3 + \\left[R_i(1 - 5\/(2m_i))\\right]^+)  \\label{eq:generalCompTimeWithReleaseUnif} \n\\end{align}\n-which complete our proof.\n\\end{proof}\n\\subsection{CC-LP for Identical Machines}\\label{sec:identLP}\n\\begin{theorem}\nIf machines are of unit speed, then CC-LP yields an objective that is...\n\\begin{center}\n\\begin{tabular}{l | c c}\n\\hline\n & $r_{ji} \\equiv 0$ & some $r_{ji} > 0$ \\\\ \n \\hline\nsingle-task subjobs & $\\leq$ 2 $OPT$ & $\\leq $ 3 $OPT$ \\\\\nmulti-task subjobs & $\\leq$ 3 $OPT$ & $\\leq$ 4 $OPT$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{thm:identLP}\n\\end{theorem}\n\\begin{proof}\nDefine $[\\cdot]^+$, $x$, $C_x$, $\\hat{C}_x$, $i$, $\\sigma_i$, and $t^*$ as in Theorem \\ref{thm:uniformLP1}. When $r_{ji} \\equiv 0$, one need only give a more careful treatment of the first inequality in \\eqref{eq:generalCompTimeWithOUTReleaseUnif} (using $\\mu_{ji} = q_{ji}$).\n\\begin{align}\n\\hat{C}_{x,i} \n\t&\\leq 2C_x + p_{xit^*} - p_{xit^*}\/m_i - p_{xi}\/q_{xi} \n\t\\leq C_x(2 + \\left[1 - 1\/m_i -1\/q_{xi} \\right]^+) \\label{eq:GeneralIdentCompTimeBound}\n\\end{align}\nSimilarly, when some $r_{ji} > 0$, the first inequality in \\eqref{eq:generalCompTimeWithReleaseUnif} implies the following.\n\\begin{align}\n\\hat{C}_{x,i}\n\t&\\leq 3C_x + p_{xit^*} - p_{xit^*}\/m_i - 3p_{xi}\/(2q_{xi})\n\t\\leq C_x(3 + \\left[1 - 1\/m_i - 3\/(2q_{xi})\\right]^+) \\label{eq:forCstTimeThmWithRelease}\n\\end{align}\n\\end{proof}\nThe key in the refined analysis of Theorem \\ref{thm:identLP} lay in how $-p_{xi}\/q_{xi}$ is used to annihilate $+p_{xit^*}$. While $q_{xi} = 1$ (i.e. single-task subjobs) is sufficient to accomplish this, it is not strictly \\textit{necessary}. The theorem below shows that we can annihilate the $+p_{xit^*}$ term whenever all tasks of a given subjob are of the same length. Note that the tasks need not be \\textit{unit}, as the lengths of tasks across different subjobs can differ.\n\\begin{theorem}\nSuppose $v_{\\ell i} \\equiv 1$. If $p_{jit}$ is constant over $t \\in T_{ji}$ for all $j \\in N$ and $i \\in M$, then algorithm CC-LP is a 2-approximation when $r_{ji} \\equiv 0$, and a 3-approximation otherwise. \\label{thm:identLP_2appxWithConstantTasks}\n\\end{theorem}\n\\begin{proof}\nThe definition of $p_{xi}$ gives $p_{xi}\/q_{xi} = \\textstyle\\sum_{t \\in T_{xi}} p_{xit} \/ q_{xi}$. Using the assumption that $p_{jit}$ is constant over $t \\in T_{ji}$, we see that $p_{xi}\/q_{xi} = (q_{xi} + |T_{xi}| - q_{xi})p_{xit^*} \/ q_{xi} $, where $|T_{xi} |\\geq q_{xi}$. Apply this to Inequality \\eqref{eq:GeneralIdentCompTimeBound} from the proof of Theorem \\ref{thm:identLP}; some algebra yields \n\\begin{align}\n\\hat{C}_{xi} \n\t\\leq& 2C_x  - p_{xit^*}\/m_i - p_{xit^*}\\left(|T_{xi}| - q_{xi}\\right)\/q_{xi} \\leq 2C_x.\n\\end{align}\nThe case with some $r_{ji} > 0$ uses the same identity for $p_{xi}\/q_{xi}$.\n\\end{proof}\nSachdeva and Saket \\cite{nphard2} showed that it is NP-Hard to approximate $CC|m_i \\equiv 1|\\sum w_j C_j$ with a constant factor less than 2. \nTheorem \\ref{thm:identLP_2appxWithConstantTasks} is significant because it shows that CC-LP can attain the same guarantee for \\textit{arbitrary} $m_i$, provided $v_{\\ell i} \\equiv 1$ and $p_{jit}$ is constant over $t$.\n\n\\section{Combinatorial Algorithms}\\label{sec:TSPT}\n\n\n\n\n\n\nIn this section, we introduce an extremely fast combinatorial algorithm with performance guarantees similar to CC-LP for ``unstructured'' inputs (i.e. those for which some $v_{\\ell i} > 1$, or some $T_{ji}$ have $p_{jit}$ non-constant over $t$). \nWe call this algorithm \\textit{CC-TSPT}. \nCC-TSPT uses the MUSSQ algorithm for concurrent open shop (from \\cite{mqssu}) as a subroutine. As SWAG (from \\cite{HGY}) motivated development of CC-TSPT, we first address SWAG's worst-case performance.\n\n\\subsection{A Degenerate Case for SWAG}\\label{subsec:swagDegenerate}\n\\begin{wrapfigure}{r}{0.5\\textwidth}\n\t\\vspace{-0.45cm}\n  \\centering\n    \\begin{minipage}{0.50\\textwidth}\n\t\\begin{algorithm}[H]\n\t\\begin{algorithmic}[1]\n\t\\Procedure{SWAG}{$N,M,p_{ji}$}\n\t\\State $J\\gets \\emptyset$\n\t\\State $q_i\\gets 0,\\forall i\\in M$\n\t\\While{$|J|\\not=|N|$}\n\t\\State mkspn$_j\\gets$ max$_{i\\in M}\\left(\\frac{q_i+p_{ji}}{m_i}\\right)$\n\t\\item[] \\qquad \\qquad $\\forall j\\in N\\setminus J$\n\t\\State nextJob $\\gets$ argmin$_{j \\in N \\setminus J}\\ $mkspn$_j$\n\t\\State $J.$append$($nextJob$)$\n\t\\State $q_i\\gets q_i+ p_{ji}$\n\t\\EndWhile\n\t\\State \\textbf{return} $J$\n\t\\EndProcedure\n\t\\end{algorithmic}\n\t\\end{algorithm}\n  \\end{minipage}\n  \\vspace{-0.45cm}\n\\end{wrapfigure}\nAs a prerequisite for addressing worst-case performance of an existing algorithm, we provide psuedocode and an accompanying verbal description for SWAG.\n\nSWAG computes queue positions for every subjob of every job, supposing that each job was scheduled next. \nA job's potential makespan (``mkspn'') is the largest of the potential finish times of all of its subjobs (considering current queue lengths $q_i$ and each subjob's processing time $p_{ji}$). \nOnce potential makespans have been determined, the job with smallest potential makespan is selected for scheduling. \nAt this point, all queues are updated. \nBecause queues are updated, potential makespans will need to be re-calculated at the next iteration. \nIterations continue until the very last job is scheduled. Note that SWAG runs in $O(n^2m)$ time.\n\n\\begin{theorem}\nFor an instance $I$ of $PD || \\sum C_j$, let $SWAG(I)$ denote the objective function value of SWAG applied to $I$, and let $OPT(I)$ denote the objective function value of an optimal solution to $I$. \nThen for all $L \\geq 1$, there exists an $I \\in \\Omega_{PD || \\sum C_j}$ such that $SWAG(I) \/ OPT(I) > L$.\n\\label{thm:swagBad}\n\\end{theorem}\n\\begin{proof}\nLet $L \\in \\mathbb{N}^+$ be a fixed but arbitrary constant. \nConstruct a problem instance $I_L^m$ as follows: \n\n$N = N_1 \\cup N_2$ where $N_1$ is a set of $m$ jobs, and $N_2$ is a set of $L$ jobs. \nJob $j \\in N_1$ has processing time $p$ on cluster $j$ and zero all other clusters. \nJob $j \\in N_2$ has processing time $p(1-\\epsilon)$ on all $m$ clusters. \n$\\epsilon$ is chosen so that $\\epsilon < 1\/L$\n(see Figure \\ref{fig:swag1}).\n\n\\begin{figure}[ht]\n\\includegraphics[width=\\linewidth]{swag.png}\n\\centering\n\\caption{At left, an input for SWAG example with $m=3$ and $L=2$. At right, SWAG's resulting schedule, and an alternative schedule.}\n\\label{fig:swag1}\n\\end{figure}\n\nIt is easy to verify that SWAG will generate a schedule where all jobs in $N_2$ precede all jobs in $N_1$ (due to the savings of $p \\epsilon$ for jobs in $N_2$). \nWe propose an \\textit{alternative} solution in which all jobs in $N_1$ preceed all jobs in $N_2$. \nDenote the objective value for this alternative solution $ALT(I_L^m)$, noting $ALT(I_L^m) \\geq OPT(I_L^m)$.\n\nBy symmetry, and the fact that all clusters have a single machine, we can see that $SWAG(I_L^m)$ and $ALT(I_L^m)$ are given by the following\n\\begin{align}\nSWAG(I_L^m) &= p(1-\\epsilon)L(L+1)\/2 + p(1-\\epsilon)L m + p m \\\\\nALT(I_L^m) &= p(1-\\epsilon)L(L + 1)\/2 + pL + p m\n\\end{align}\nSince $L$ is fixed, we can take the limit with respect to $m$.\n\\begin{align}\n\\lim_{m \\rightarrow \\infty}{\\frac{SWAG(I_L^m)}{ALT(I_L^m)}} \n\t&= \\lim_{m \\rightarrow \\infty}{\\frac{p(1-\\epsilon)L m + p m}{p m}} = L(1-\\epsilon) + 1 > L\n\\end{align}\nThe above implies the existence of a sufficiently large number of clusters $\\overline{m}$, such that $m \\geq \\overline{m}$ implies $ SWAG(I_L^{m})\/OPT(I_L^{m}) > L $. This completes our proof.\n\\end{proof}\nTheorem \\ref{thm:swagBad} demonstrates that that although SWAG performed well in simulations, it may not be reliable. \nThe rest of this section introduces an algorithm not only with superior runtime to SWAG (generating a permutation of jobs in $O(n^2 + nm)$ time, rather than $O(n^2m)$ time), but also a constant-factor performance guarantee.\n\n\\subsection{CC-TSPT : A Fast 2 + R Approximation}\\label{subsec:fastreduction}\nOur combinatorial algorithm for concurrent cluster scheduling exploits an elegant transformation to concurrent open shop. \nOnce we consider this simpler problem, it can be handled with MUSSQ \\cite{mqssu} and List-LPT. \nOur contributions are twofold: (1) we prove that this intuitive technique yields an approximation algorithm for a decidedly more general problem, and (2) we show that a \\textit{non-intuitive} modification can be made that maintains theoretical bounds while improving empirical performance. \nWe begin by defining our transformation.\n\n\\begin{definition}[The Total Scaled Processing Time (TSPT) Transformation]\nLet $\\Omega_{CC}$ be the set of all instances of $CC || \\sum w_j C_j$, \nand let $\\Omega_{PD}$ be the set of all instances of \n$PD || \\sum w_j C_j$. Note that $\\Omega_{PD} \\subset \\Omega_{CC}$.\nThen the Total Scaled Processing Time Transformation is a mapping\n\\begin{align*}\nTSPT: ~\\Omega_{CC} \\to \\Omega_{PD} \\quad \\text{ with } \\quad (T, v, w) &\\mapsto (X, w) ~:~ x_{ji} = \\textstyle\\sum_{t \\in T_{ji}} p_{jit} \/ \\mu_i\n\\end{align*}\ni.e., $x_{ji}$ is the total processing time required by subjob $(j,i)$, scaled by the sum of machine speeds at cluster $i$. \nThroughout this section, we will use $I = (T, v, w)$ to denote an arbitrary instance of $CC || \\sum w_j C_j$, and $I' = (X, w)$ as the image of $I$ under TSPT. \nFigure \\ref{fig:tspt} shows the result of TSPT applied to our baseline example. \n\\end{definition}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{tspt.png}\n\\caption{An instance $I$ of $CC||\\sum w_j C_j$, and its image $I' = TSPT(I)$. The schedules were constructed with List-LPT using the same permutation for $I$ and $I'$. }\n\\label{fig:tspt}\n\\end{figure}\n\nWe take the time to emphasize the simplicity of our reduction. Indeed, the TSPT transformation is perhaps the first thing one would think of given knowledge of the concurrent open shop problem. What is surprising is how one can attain constant-factor performance guarantees even after such a simple transformation.\n\n\\vspace{1em}\n\\noindent \\textbf{Algorithm CC-TSPT} : Execute MUSSQ on $I'= TSPT(I)$ to generate a permutation of jobs $\\sigma$. List schedule instance $I$ by \n$\\sigma$ on each cluster according to List-LPT.\n\\vspace{1em}\n\nTowards proving the approximation ratio for CC-TSPT, we will establish a critical inequality in Lemma \\ref{lem:TSPTboundInTermsOfPDandOPT}. \nThe intuition behind Lemma \\ref{lem:TSPTboundInTermsOfPDandOPT} requires thinking of every job $j$ in $I$ as having a corresponding representation in $j'$ in $I'$. \nJob $j$ in $I$ will be scheduled in the $CC$ environment, while job $j'$ in $I'$ will be scheduled in the $PD$ environment. \nWe consider what results when the same permutation $\\sigma$ is used for scheduling in both environments. \n\nNow the definitions for the lemma: let $C^{CC}_{\\sigma(j)}$ be the completion time of job $\\sigma(j)$ resulting from List-LPT on an arbitrary permutation $\\sigma$. \nDefine $C^{CC\\star}_{\\sigma(j)}$ as the completion time of job $\\sigma(j)$ in the $CC$ environment in the optimal solution. \nLastly, define $C^{PD,I'}_{\\sigma(j')}$ as the completion time of job $\\sigma(j')$ in $I'$ when scheduling by List-LPT($\\sigma$) in the $PD$ environment.\n\n\\begin{lemma}\nFor $I' = TSPT(I)$, let $j'$ be the job in $I'$ corresponding to job $j$ in $I$. \nFor an arbitrary permutation of jobs $\\sigma$, we have $C^{CC}_{\\sigma(j)} \\leq C^{PD,I'}_{\\sigma(j')} + R\\cdot C^{CC\\star}_{\\sigma(j)}$. \\label{lem:TSPTboundInTermsOfPDandOPT}\n\\end{lemma}\n\\begin{proof}\nAfter list scheduling has been carried out in the $CC$ environment, we may determine $C^{CC}_{\\sigma(j)i}$ - the completion time of subjob $(\\sigma(j),i)$. \nWe can bound $C^{CC}_{\\sigma(j)i}$ using Lemma \\ref{lem:CompTimesOnUniformMachines} (which implies \\eqref{eqInLem:TSPTbound1}), and the serial-processing nature of the $PD$ environment (which implies \\eqref{eqInLem:TSPTbound2}).\n\\begin{align}\n& C^{CC}_{\\sigma(j)i} \\leq p_{\\sigma(j)i1}\\left(1\/\\bar{v} - 1\/\\mu_i\\right) + \\textstyle\\sum_{\\ell = 1}^{j} p_{\\sigma(\\ell)i}\/\\mu_i  \\label{eqInLem:TSPTbound1} \\\\\n& \\textstyle\\sum_{\\ell = 1}^j p_{\\sigma(\\ell)i}\/\\mu_i \\leq C^{PD,I'}_{\\sigma(j')} \\quad \\forall ~i \\in M \\label{eqInLem:TSPTbound2}\n\\end{align}\nIf we relax the bound given in Inequality \\eqref{eqInLem:TSPTbound1} and combine it with Inequality \\eqref{eqInLem:TSPTbound2}, we see that $C^{CC}_{\\sigma(j)i} \\leq C^{PD,I'}_{\\sigma(j')}  +  p_{\\sigma(j)i1}\/\\bar{v}$. \nThe last step is to replace the final term with something more meaningful. Using $p_{\\sigma(j)1}\/\\bar{v} \\leq R \\cdot C^{CC\\star}_{\\sigma(j)}$ (which is immediate from the definition of $R$) the desired result follows.\n\\end{proof}\nWhile Lemma \\ref{lem:TSPTboundInTermsOfPDandOPT} is true for arbitrary $\\sigma$, now we consider $\\sigma = MUSSQ(X, w)$. \nThe proof of MUSSQ's correctness established the first inequality in the chain of inequalities below. \nThe second inequality can be seen by substituting $p_{ji} \/ \\mu_{i}$ for $x_{ji}$  in LP0($I'$) (this shows that the constraints in LP0($I'$) are weaker than those in LP1($I$)). \nThe third inequality follows from the Validity Lemma.\n\\begin{equation}\n\\textstyle\\sum_{j \\in N} w_{\\sigma(j)} C^{PD,I'}_{\\sigma(j)} \n\t\\leq 2 \\textstyle\\sum_{j \\in N} w_j C^{\\text{LP0}(I')}_j\n\t\\leq 2 \\textstyle\\sum_{j \\in N} w_j C^{\\text{LP1}(I)}_j\n\t\\leq 2 OPT(I) \\label{eq:tsptCore}\n\\end{equation}\nCombining Inequality \\eqref{eq:tsptCore} with Lemma \\ref{lem:TSPTboundInTermsOfPDandOPT} allows us to bound the objective in a way that does not make reference to $I'$.\n\\begin{equation}\n\\textstyle\\sum_{j \\in N} w_{\\sigma(j)}C^{CC}_{\\sigma(j)} \n\t\\leq \\textstyle\\sum_{j \\in N} w_{\\sigma(j)}\\left[C^{PD,I'}_{\\sigma(j)} + R\\cdot C^{CC\\star}_{\\sigma(j)}\\right] \\leq ~ 2 \\cdot OPT(I) + R \\cdot OPT(I) \\label{eq:dontReferenceIPrime}\n\\end{equation}\nInequality \\eqref{eq:dontReferenceIPrime} completes our proof of the following theorem.\n\\begin{theorem}\nAlgorithm CC-TSPT is a $2 + R$ approximation for $CC || \\sum w_j C_j$.\n\\label{thm:algCCTspt}\n\\end{theorem}\n\n\\subsection{CC-TSPT with Unit Tasks and Identical Machines}\\label{subsec:tsptOnUnitTasks}\nConsider concurrent cluster scheduling with $v_{\\ell i} = p_{jit} = 1$ (i.e., all processing times are unit, although the size of the collections $T_{ji}$ are unrestricted). In keeping with the work of Zhang, Wu, and Li \\cite{zwl} (who studied this problem in the single-cluster case), we call instances with these parameters ``fully parallelizable,'' and write $\\beta = fps$ for Graham's $\\alpha|\\beta|\\gamma$ taxonomy.\n\nZhang et al. showed that scheduling jobs greedily by  ``Largest Ratio First'' (decreasing $w_j \/ p_{j}$) results in a 2-approximation, where 2 is a tight bound. \nThis comes as something of a surprise since the Largest Ratio First policy is \\textit{optimal} for $1||\\sum w_j C_j~$- which their problem very closely resembles. \nWe now formalize the extent to which $P|fps|\\sum w_j C_j$ resembles $1||\\sum w_j C_j~$: define the \\textit{time resolution} of an instance $I$ of $CC |fps| \\sum w_jC_j$ as $ \\rho_I = \\min_{j \\in N, i \\in M}{\\big\\lceil{p_{ji}\/m_i}\\big\\rceil}$. \nIndeed, one can show that as the time resolution increases, the performance guarantee for LRF on $P | fps | \\sum w_j C_j$ approaches that of LRF on $1||\\sum w_j C_j$. \nWe prove the analogous result for our problem. \n\\begin{theorem}\nCC-TSPT for $CC |fps| \\sum w_jC_j$ is a $(2 + 1\/\\rho_I)-$approximation.\n\\label{thm:tspt_unit_tasks}\n\\end{theorem}\n\\begin{proof}\nApplying techniques from the proof of Lemma \\ref{lem:TSPTboundInTermsOfPDandOPT} under the hypothesis of this theorem, we have $C^{CC}_{\\sigma(j), i} \\leq C^{PD,I'}_{\\sigma(j)} + 1$. \nNext, use the fact that for all $j \\in N$, $C^{CC,OPT}_{\\sigma(j)} \\geq \\rho_I$ by the definition of $\\rho_I$. These facts together imply\n$C^{CC}_{\\sigma(j), i} \\leq C^{PD,I'}_{\\sigma(j)} + C^{CC,OPT} \/ \\rho_I$. Thus\n\\begin{align}\n\\textstyle\\sum_{j \\in N} w_j C^{CC}_{\\sigma(j)} \n\t&\\leq \\textstyle\\sum_{j \\in N} w_j \\left[C^{PD,I'}_{\\sigma(j)} + C^{CC,OPT} \/ \\rho_I\\right] \\leq 2 \\cdot OPT + OPT \/ \\rho_I.\n\\end{align}\n\\end{proof}\n\n\\subsection{CC-ATSPT : Augmenting the LP Relaxation}\nThe proof of Theorem \\ref{thm:algCCTspt} appeals to a trivial lower bound on $C^{CC\\star}_{\\sigma(j)}$, namely $p_{\\sigma(j)1}\/\\bar{v} \\leq R \\cdot C^{CC\\star}_{\\sigma(j)}$. We attain constant-factor performance guarantees in spite of this, but it is natural to wonder how the \\textit{need} for such a bound might come hand-in-hand with empirical weaknesses. Indeed, TSPT can make subjobs consisting of many small tasks look the same as subjobs consisting of a single very long task.\nAdditionally, a cluster hosting a subjob with a single extremely long task might be identified as a bottleneck by MUSSQ, even if that cluster has more machines than it does tasks to process.\n\nWe would like to mitigate these issues by introducing the simple lower bounds on $C_j$ as seen in constraints $(1B)$ and $(1C)$. This is complicated by the fact that MUSSQ's proof of correctness only allows constraints of the form in $(1A)$. For $I \\in \\Omega_{PD}$ this is without loss of generality, since $|S| = 1$ in LP0 implies $C_j \\geq p_{ji}$, but since we apply LP0 to $I' = TSPT(I)$, $C_j \\geq x_{ji}$ is equivalent to $C_j \\geq p_{ji}\/\\mu_i$ (a much weaker bound than we desire). \n\nNevertheless, we can bypass this issue by introducing additional clusters and appropriately defined subjobs. We formalize this with the ``Augmented Total Scaled Processing Time'' (ATSPT) transformation. \nConceptually, ATSPT creates $n$ ``imaginary clusters'', where each imaginary cluster has nonzero processing time for exactly one job.\n\\begin{definition}[The Augmented TSPT Transformation]\nLet $\\Omega_{CC}$ and $\\Omega_{PD}$ be as in the definition for TSPT. Then the Augmented TSPT Transformation is likewise a mapping\n\\begin{align*}\nATSPT: ~\\Omega_{CC} \\to \\Omega_{PD} \\quad \\text{ with } \\quad (T, v, w) &\\mapsto (X, w) ~:~ X = \\big[\\begin{array}{c|c} X_{TSPT(I)} & D \\end{array}\\big].\n\\end{align*}\nWhere $D \\in \\mathbb{R}^{n \\times n}$ is a diagonal matrix with $d_{jj}$ as any valid lower bound on the completion time of job $j$ (such as the right hand sides of constraints ($1B$) and ($1C$) of LP1).\n\\end{definition}\nGiven that $d_{jj}$ is a valid lower bound on the completion time of job $j$, it is easy to verify that for $I' = ATSPT(I)$, LP1($I'$) is a valid relaxation of $I$. \nBecause MUSSQ returns a permutation of jobs for use in list scheduling by List-LPT, these ``imaginary clusters'' needn't be accounted for beyond the computations in MUSSQ.\n\n\\section{A Reduction for Minimizing Total Weighted Lateness on Identical Parallel Machines }\\label{sec:relationshipsBetweenProbs}\nThe problem of minimizing total weighted lateness on a bank of identical parallel machines is typically denoted $P || \\sum w_jL_j$, where the lateness of a job with deadline $d_j$ is $L_j \\doteq \\max{\\{C_j - d_j, 0\\}}$. The reduction we offer below shows that $P || \\sum w_j L_j$ can be stated in terms of $CC || \\sum w_jC_j$ \\textit{at optimality}. Thus while a $\\Delta$ approximation to $CC || \\sum w_jC_j$ does not imply a $\\Delta$ approximation to $P || \\sum w_j L_j$, the reduction below nevertheless provides new insights on the structure of $P || \\sum w_j L_j$.\n\n\\begin{definition}[Total Weighted Lateness Reduction]\nLet $I = (p, d, w, m)$ denote an instance of $P || \\sum w_j L_j$. \n$p$ is the set of processing times, $d$ is the set of deadlines, $w$ is the set of weights, \nand $m$ is the number of identical parallel machines. \nGiven these inputs, we transform $I \\in \\Omega_{P || \\sum w_j L_j}$ \nto $I' \\in \\Omega_{CC}$ in the following way.\n\nCreate a total of $n + 1$ clusters. Cluster 0 has $m$ machines. Job $j$ has processing time $p_j$ on this cluster, and $|T_{j0}| = 1$. Clusters 1 through $n$ each consist of a single machine. Job $j$ has processing time $d_j$ on cluster $j$, and zero on all clusters other than cluster 0 and cluster $j$. Denote this problem $I'$.\n\\end{definition}\nWe refer the reader to Figure \\ref{fig:probstm3and4} for an example output of this reduction.\n\\begin{theorem}\nLet $I$ be an instance of $P || \\textstyle\\sum w_j L_j$. Let $I'$ be an instance of $CC|| \\sum w_j C_j$ resulting from the transformation described above. Any list schedule $\\sigma$ that is optimal for $I'$ is also optimal for $I$.\n\\end{theorem}\n\\begin{proof}\nIf we restrict the solution space of $I'$ to single permutations (which we may do without loss of generality), then any schedule $\\sigma$ for $I$ or $I'$ produces the same value of $\\sum_{j \\in N}  w_j(C_j - d_j)^+$ for $I$ and $I'$.\nThe additional clusters we added for $I'$ ensure that $C_j \\geq d_j$. Given this, the objective for $I$ can be written as $\\sum_{j \\in N} w_j d_j + w_j(C_j - d_j)^+$. Because $w_j d_j$ is a constant, any permutation to solve $I'$ optimally also solves $\\sum_{j \\in N} w_j (C_j - d_j)^+$ optimally. Since $\\sum_{j \\in N} w_j (C_j - d_j)^+ = \\sum_{j \\in N} w_j L_j$, we have the desired result.\n\\end{proof}\n\\section{Closing Remarks}\\label{sec:discAndConc}\nWe now take a moment to address a subtle issue in the concurrent cluster problem: what price do we pay for using the same permutation on all clusters (i.e. single-$\\sigma$ schedules)? For concurrent open shop, it has been shown (\\cite{Sris1993, mqssu}) that single-$\\sigma$ schedules may be assumed without loss of optimality. As is shown in Figure \\ref{fig:singleVsMultiPerm}, this does \\textit{not} hold for concurrent cluster scheduling in the general case. In fact, that is precisely why the strong performance guarantees for algorithm CC-LP rely on clusters having possibly unique permutations.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.7\\textwidth]{singleVsMultiPerm.png}\n\\caption{An instance of $CC||\\sum C_j$ (i.e. $w_j \\equiv 1$) for which there does not exist a single-$\\sigma$ schedule which attains the optimal objective value. In the single-$\\sigma$ case, one of the jobs necessarily becomes delayed by one time unit compared to the multi-$\\sigma$ case. As a result, we see a 20\\% optimality gap even when $v_{\\ell i } \\equiv 1$.}\\label{fig:singleVsMultiPerm}\n\\centering\n\\end{figure}\n\nOur more novel contributions came in our analysis for CC-TSPT and CC-ATSPT. First, we could not rely on the processing time of the last task for a job to be bounded above by the job's completion time variable $C_j$ in LP0($I'$), and so we appealed to a lower bound on $C_j$ that was not stated in the LP itself. The need to incorporate this second bound is critical in realizing the strength of algorithm CC-TSPT, and uncommon in LP rounding schemes. Second, CC-ATSPT is novel in that it introduces constraints that would be redundant for LP0($I$) when $I \\in \\Omega_{PD}$, but become relevant when viewing $LP0(I')$ as a relaxation for $I \\in \\Omega_{CC}$. This approach has potential for more broad applications since it represented effective use of a limited constraint set supported by a known primal-dual algorithm.\n\nWe now take a moment to state some open problems in this area. One topic of ongoing research is developing a factor 2 purely combinatorial algorithm for the special case of concurrent cluster scheduling considered in Theorem \\ref{thm:identLP_2appxWithConstantTasks}. In addition, it would be of broad interest to determine the worst-case loss to optimality incurred by assuming single-permutation schedules for $CC|v\\equiv 1|\\sum w_j C_j$. The simple example above shows that an optimal single-$\\sigma$ schedule can have objective 1.2 times the globally optimal objective. Meanwhile, Theorem \\ref{thm:algCCTspt} shows that there always exists a single-$\\sigma$ schedule with objective no more than 3 times the globally optimal objective. Thus, we know that the worst-case performance ratio is in the interval $[1.2,3]$, but we do not know its precise value. As a matter outside of scheduling theory, it would be valuable to survey primal-dual algorithms with roots in LP relaxations to determine which have constraint sets that are amenable to implicit modification, as in the fashion of CC-ATSPT.\n\n\\subparagraph*{Acknowledgments.}\n\nSpecial thanks to Andreas Schulz for sharing some of his recent work with us \\cite{Schulz2012}. His thorough analysis of a linear program for $P||\\sum w_j C_j$ drives the LP-based results in this paper. Thanks also to Chien-Chung Hung and Leana Golubchik for sharing \\cite{HGY} while it was under review, and to Ioana Bercea and Manish Purohit for their insights on SWAG's performance. Lastly, our sincere thanks to William Gasarch for organizing the REU which led to this work, and to the 2015 CAAR-REU cohort for making the experience an unforgettable one; in the words of Rick Sanchez \\textit{wubalubadubdub!}\n\n\n\\section{Introduction}\\label{sec:intro}\n\nIt is becoming increasingly impractical to store full copies of large datasets on more than one data center \\cite{Hajjat2012}. As a result, the data for a single job may be located not on multiple machines, but on multiple \\textit{clusters} of machines. \nTo maintain fast response-times and avoid excessive network traffic, it is advantageous to perform computation for such jobs in a completely distributed fashion \\cite{HGY}.\nIn addition, commercial platforms such as AWS Lambda and Microsoft's Azure Service Fabric are demonstrating a trend of centralized cloud computing frameworks in which the user manages neither data flow nor server allocation \\cite{AWSLambda, Azure}. \nIn view of these converging issues, the following scheduling problem arises:\n\n\\textit{If computation is done locally to avoid excessive network traffic, how can individual clusters on the broader grid coordinate schedules for maximum throughput? }\n\nThis was precisely the motivation for Hung, Golubchik, and Yu in their 2015 ACM Symposium on Cloud Computing paper \\cite{HGY}.  \nHung et al. modeled each cluster as having an arbitrary number of identical parallel machines, and choose an objective of average job completion time. \nAs such a problem generalizes the NP-Hard concurrent open shop problem, they proposed a heuristic approach. \nTheir heuristic (called ``SWAG'') runs in $O(n^2m)$ time and performed well on a variety of data sets. Unfortunately, SWAG offers poor worst-case performance, as we show in Section \\ref{sec:TSPT}.\n\nOur contributions to this problem are to extend the model considered by Hung et al. and to introduce the first constant-factor approximation algorithms for this general problem. \nOur extensions of Hung et al.'s model are (1) to allow different machines within the same cluster to operate at different speeds, (2) to incorporate pre-specified ``release times'' (times before which a subjob cannot be processed), and (3) to support \\textit{weighted} average job completion time.\nWe present two algorithms for the resulting problem.\nOur combinatorial algorithm exploits a surprisingly simple mapping to the special case of one machine per cluster, where the problem can be approximated in $O(n^2 + nm)$ time. We also present an LP-rounding approach with strong performance guarantees. E.g., a 2-approximation when machines are of unit speed and subjobs are divided into equally sized (but not necessary \\textit{unit}) tasks.\n\n\\subsection{Formal Problem Statement}\\label{subsec:ps}\n\\begin{definition}[Concurrent Cluster Scheduling]{\\color{white} . } \\hfill \n\\vspace{2pt}\n\\begin{itemize}\n\\item There is a set $M$ of $m$ clusters, and a set $N$ of $n$ jobs. \nFor each job $j \\in N$, there is a set of $m$ ``subjobs'' (one for each cluster).\n\n\\item Cluster $i \\in M$ has $m_i$ parallel machines, and machine $\\ell$ in cluster $i$ has speed $v_{\\ell i}$. \nWithout loss of generality, assume $v_{\\ell i}$ is decreasing in $\\ell$.\n\\footnote{Where we write ``decreasing'', we mean ``non-increasing.'' Where we write ``increasing'', we mean ``non-decreasing''.} \n\n\\item The $i^{\\text{th}}$ subjob for job $j$ is specified by a set of tasks to be performed by machines in cluster $i$, denote this set of tasks $T_{ji}$. \nFor each task $t \\in T_{ji}$, we have an associated processing time $p_{jit}$ (again w.l.o.g., assume $p_{jit}$ is decreasing in $t$). \nWe will frequently refer to ``the subjob of job $j$ at cluster $i$'' as ``subjob $(j,i)$.''\n\n\\item Different subjobs of the same job may be processed concurrently on different clusters. \n\\item Different tasks of the same subjob may be processed concurrently on different machines within the same cluster.\n\\item A subjob is complete when all of its tasks are complete, and a job is complete when all of its subjobs are complete. We denote a job's completion time by ``$C_j$''.\n\\item The objective is to minimize weighted average job completion time (job $j$ has weight $w_j$).\n\\item For the purposes of computing approximation ratios, it is equivalent to minimize $\\sum w_j C_j$. We work with this equivalent objective throughout this paper.\n\\end{itemize}\n\\end{definition}\n\nA machine is said to operate at \\textit{unit speed} it if can complete a task with processing requirement ``$p$'' in $p$ units of time. More generally, a machine with speed ``$v$'' ($v \\geq 1$) processes the same task in $p\/ v$ units of time. Machines are said to be \\textit{identical} if they are all of unit speed, and \\textit{uniform} if they differ only in speed.\n\nIn accordance with Graham et al.'s $\\alpha|\\beta|\\gamma$ taxonomy for scheduling problems \\cite{Graham1979} we take $\\alpha = CC$ to refer to the concurrent cluster environment, and denote our problem by $CC|| \\sum w_jC_j$.\\footnote{ A problem $\\alpha|\\beta|\\gamma$ implies a particular environment $\\alpha$, objective function $\\gamma$, and optional constraints $\\beta$.} Optionally, we may associate a release time $r_{ji}$ to every subjob.  If any subjobs are released after time zero, we write $CC| r | \\sum w_jC_j$.\n\n\\subsubsection{Example Problem Instances}\\label{subsubsec:introToModel}\n\nWe now illustrate our model with several examples (see Figures \\ref{fig:probstm1and2} and \\ref{fig:probstm3and4}). The tables at left have rows labeled to identify jobs, and columns labeled to identify clusters; each entry in these tables specifies the processing requirements for the corresponding subjob. The diagrams to the right of these tables show how the given jobs might be scheduled on clusters with the indicated number of machines. \n\\begin{figure}[ht!]\n\\centering\n  \\includegraphics[width=0.8\\linewidth]{example.png}\n\t\\caption{Two examples of our scheduling model. \\textbf{Left}: Our baseline example. There are 4 jobs and 2 clusters. Cluster 1 has 2 identical machines, and cluster 2 has 3 identical machines. Note that job 4 has no subjob for cluster 1 (this is permitted within our framework). In this case every subjob has at most one task. \\textbf{Right}: Our baseline example with a more general subjob framework : subjob (2,2) and subjob (3,1) both have two tasks. The tasks shown are unit length, but our framework \\textit{does not} require that subjobs be divided into equally sized tasks. }\n  \\label{fig:probstm1and2}\n\\end{figure}\n\\begin{figure}[ht!]\n  \\includegraphics[width=\\linewidth]{figure2.png}\n\t\\caption{Two additional examples of our model. \\textbf{Left}: Our baseline example, with variable machine speeds. Note that the benefit of high machine speeds is only realized for tasks assigned to those machines in the final schedule. \\textbf{Right}: A problem with the peculiar structure that (1) all clusters but one have a single machine, and (2) most clusters have non-zero processing requirements for only a single job. We will use such a device for the total weighted lateness reduction in Section \\ref{sec:relationshipsBetweenProbs}.}\n  \\label{fig:probstm3and4}\n\\end{figure}\n\n\\subsection{Related Work}\n\nConcurrent cluster scheduling subsumes many fundamental machine scheduling problems. For example, if we restrict ourselves to a single cluster (i.e. $m = 1$) we can schedule a set of jobs on a bank of identical parallel machines to minimize makespan ($C_{\\max}$) or total weighted completion time ($\\sum w_j C_j$). With a more clever reduction, we can even minimize \\textit{total weighted lateness} ($\\sum w_j L_j$) on a bank of identical parallel machines (see Section \\ref{sec:relationshipsBetweenProbs}). Alternatively, with $m > 1$ but $\\forall i \\in M, m_i = 1$, our problem reduces to the well-studied ``concurrent open shop'' problem.\n\nUsing Graham et al.'s taxonomy, the concurrent open shop problem is written as $PD||\\sum w_j C_j$. Three groups \\cite{Chen2000, Garg2007, llp} independently discovered an LP-based 2-approximation for $PD||\\sum w_j C_j$ using the work of Queyranne \\cite{Queyranne1993}. The linear program in question has an exponential number of constraints, but can still be solved in polynomial time with a variant of the Ellipsoid method. Our ``strong'' algorithm for concurrent cluster scheduling refines the techniques contained therein, as well as those of Schulz \\cite{Schulz1996, Schulz2012} (see Section \\ref{sec:lpAlg}).\n\nMastrolilli et al. \\cite{mqssu} developed a primal-dual algorithm for $PD || \\sum w_j C_j$ that does not use LP solvers. ``MUSSQ''\\footnote{A permutation of the author's names: Mastrolilli, Queyranne, Schulz, Svensson, and Uhan.} is significant for both its speed and the strength of its performance guarantee : it achieves an approximation ratio of 2 in only $O(n^2 + nm)$ time. Although MUSSQ does not require an LP solver, its proof of correctness is based on the fact that it finds a feasible solution to the dual a particular linear program. Our ``fast'' algorithm for concurrent cluster scheduling uses MUSSQ as a subroutine (see Section \\ref{sec:TSPT}).\n\nHung, Golubchik, and Yu \\cite{HGY} presented a framework designed to improve scheduling across geographically distributed data centers. The scheduling framework had a centralized scheduler (which determined a job ordering) and local dispatchers which carried out a schedule consistent with the controllers job ordering. Hung et al. proposed a particular algorithm for the controller called ``SWAG.'' SWAG performed well in a wide variety of simulations where each data center was assumed to have the same number of identical parallel machines. We adopt a similar framework to Hung et al., but we show in Section \\ref{subsec:swagDegenerate} that SWAG has no constant-factor performance guarantee.\n\n\\subsection{Paper Outline \\& Algorithmic Results}\\label{subsec:outlineAndResults}\n\nAlthough only one of our algorithms requires \\textit{solving} a linear program, both algorithms use the same linear program in their proofs of correctness; we introduce this linear program in Section \\ref{sec:introduceLP} before discussing either algorithm. Section \\ref{sec:listSched} establishes how an ordering of jobs can be processed to completely specify a schedule. This is important because the complex work in both of our algorithms is to generate an ordering of jobs for each cluster.\n\nSection \\ref{sec:lpAlg} introduces our ``strong'' algorithm: CC-LP. CC-LP can be applied to any instance of concurrent cluster scheduling, including those with non-zero release times $r_{ji}$. A key in CC-LP's strong performance guarantees lay in the fact that it allows different permutations of subjobs for different clusters. By providing additional structure to the problem (but while maintaining a generalization of concurrent open shop) CC-LP becomes a 2-approximation. This is significant because it is NP-Hard to approximate concurrent open shop (and by extension, our problem) with ratio $2-\\epsilon$ for any $\\epsilon > 0$ \\cite{nphard2}.\n\nOur combinatorial algorithm (``CC-TSPT'') is presented in Section \\ref{sec:TSPT}. The algorithm is fast, provably accurate, and has the interesting property that it can schedule all clusters using the same permutation of jobs.\\footnote{We call such schedules ``single-$\\sigma$ schedules.'' As we will see later on, CC-TSPT serves as a constructive proof of existence of near-optimal single-$\\sigma$ schedules for all instances of $CC||\\sum w_j C_j$, \\textit{including} those instances for which single-$\\sigma$ schedules are strictly sub-optimal. This is addressed in Section \\ref{sec:discAndConc}.} After considering CC-TSPT in the general case, we show how fine-grained approximation ratios can be obtained in the ``fully parallelizable'' setting of Zhang et al. \\cite{zwl}. We conclude with an extension of CC-TSPT that maintains performance guarantees while offering improved empirical performance.\n\nThe following table summarizes our results for approximation ratios. For compactness, condition $Id$ refers to identical machines (i.e. $v_{\\ell i}$ constant over $\\ell$), condition $A$ refers to $r_{ji} \\equiv 0$, and condition $B$ refers to $p_{jit} \\text{ constant over } t \\in T_{ji}$.\n\\begin{center}\n\\begin{tabular}{l| cccccc}\n\\hline\n \t\t& $(Id,A,B)$ & $(Id, \\neg A, B)$ & $(Id,A,\\neg B)$ & $(Id,\\neg A, \\neg B)$ & $(\\neg Id, A)$ & $(\\neg Id, \\neg A)$ \\\\ \\hline\nCC-LP\t&\t2\t  &     3         &    3         &   4  &  $2+R$     &  $3+R$    \\\\\nCC-TSPT &   3     &     -          &   3         &   -  &  $2+R$      &    -  \\\\ \\hline\n\\end{tabular}\n\\end{center}\nThe term $R$ is the maximum over $i$ of $R_i$, where $R_i$ is the ratio of fastest machine to \\textit{average} machine speed at cluster $i$.\n\nThe most surprising of all of these results is that our scheduling algorithms are remarkably simple. The first algorithm solves an LP,\nand then the scheduling can be done easily on each cluster. The second algorithm is again a rather surprising simple reduction to the\ncase of one machine per cluster (the well understood concurrent open shop problem) and yields a simple combinatorial algorithm. The proof\nof the approximation guarantee is somewhat involved however.\n\nIn addition to algorithmic results, we demonstrate how our problem subsumes that of minimizing total weighted lateness on a bank of identical parallel machines (see Section \\ref{sec:relationshipsBetweenProbs}). Section \\ref{sec:discAndConc} provides additional discussion and highlights our more novel technical contributions.\n\n\n\\section{The Core Linear Program }\\label{sec:introduceLP}\n\n\n\nOur linear program has an unusual form. Rather than introduce it immediately, we conduct a brief review of prior work on similar LP's. All the LP's we discuss in this paper have objective function $\\sum w_j C_j$, where $C_j$ is a decision variable corresponding to the completion time of job $j$, and $w_j$ is a weight associated with job $j$. \n\n\\textit{For the following discussion only, we adopt the notation in which job $j$ has processing time $p_j$. In addition, if multiple machine problems are discussed, we will say that there are $\\mathsf{m}$ such machines (possibly with speeds $s_i, i \\in \\{1,\\ldots, \\mathsf{m}\\}$).} \n\nThe earliest appearance of a similar linear program comes from Queyranne \\cite{Queyranne1993}. In his paper, Queyranne presents an LP relaxation for sequencing $n$ jobs on a single machine where all constraints are of the form $\\sum_{j \\in S} p_j C_j \\geq \\frac{1}{2}\\left[\\left(\\sum_{j \\in S} p_j \\right)^2 + \\sum_{j \\in S} p_j^2\\right]$ where $S$ is an arbitrary subset of jobs. Once a set of optimal $\\{C_j^\\star\\}$ is found, the jobs are scheduled in increasing order of $\\{C_j^\\star\\}$. These results were primarily theoretical, as it was known at his time of writing that sequencing $n$ jobs on a single machine to minimize $\\sum w_j C_j$ can be done optimally in $O(n \\log n)$ time.\n\nQueyranne's constraint set became particularly useful for problems with \\textit{coupling} across distinct machines (as occurs in concurrent open shop). Four separate groups \\cite{Chen2000,Garg2007,llp, mqssu} saw this and used the following LP in a 2-approximation for concurrent open shop scheduling.\n\\begin{equation}\n(\\text{LP0}) ~~ \\min \\sum_{j \\in N} w_j C_j ~~ \\text{s.t.} ~~ \\textstyle\\sum_{j \\in S} p_{ji} C_j \\geq \n\t\t\\frac{1}{2}\n\t\t\\left[ \n\t\t\t\\left(\\textstyle\\sum_{j \\in S} p_{ji}\\right)^2 + \\left(\\textstyle\\sum_{j \\in S} p_{ji}^2\\right) \n\t\t\\right] ~ \\forall ~ \\substack{S \\subseteq N \\\\  i \\in M }\\nonumber\n\\end{equation}\nIn view of its tremendous popularity, we sometimes refer to the linear program above as the \\textit{canonical relaxation} for concurrent open shop.\n\nAndreas Schulz's Ph.D. thesis developed Queyranne's constraint set in greater depth \\cite{Schulz1996}. As part of his thesis, Schulz considered scheduling $n$ jobs on $\\mathsf{m}$ identical parallel machines with constraints of the form $\\sum_{j \\in S} p_j C_j \\geq \\frac{1}{2\\mathsf{m}} \\left(\\sum_{j \\in S} p_j \\right)^2 + \\frac{1}{2}\\sum_{j \\in S} p_j^2$. In addition, Schulz showed that the constraints $\\sum_{j \\in S} p_j C_j \\geq \\left[2 \\sum_{i=1}^{\\mathsf{m}} s_i \\right]^{-1}\\left[\\left(\\sum_{j \\in S} p_j \\right)^2 + \\sum_{j \\in S} p_j^2\\right]$ are satisfied by any schedule of $n$ jobs on $\\mathsf{m}$ uniform machines. In 2012, Schulz refined the analysis for several of these problems \\cite{Schulz2012}. For constructing a schedule from the optimal $\\{C_j^\\star\\}$, Schulz considered scheduling jobs by increasing order of $\\{C_j^\\star\\}$, $\\{C_j^\\star - p_j\/2\\}$, and $\\{C_j^\\star - p_j\/(2\\mathsf{m})\\}$.\n\n\n\\subsection{Statement of LP1}\n\nThe model we consider allows for more fine-grained control of the job structure than is indicated by the LP relaxations above. Inevitably, this comes at some expense of simplicity in LP formulations. In an effort to simplify notation, we define the following constants, and give verbal interpretations for each. \n\\begin{equation}\n \\mu_{i} \\doteq \\textstyle\\sum_{\\ell = 1}^{m_i} v_{\\ell i} \\qquad q_{ji} \\doteq \\min{\\lbrace|T_{ji}|, m_i\\rbrace}  \\qquad \\mu_{ji} \\doteq \\textstyle\\sum_{\\ell = 1}^{q_{ji}} v_{\\ell i} \\qquad p_{ji} \\doteq \\textstyle\\sum_{t \\in T_{ji}} p_{jit}\n\\end{equation}\nFrom these definitions, $\\mu_i$ is the processing power of cluster $i$. For subjob $(j,i)$, $q_{ji}$ is the maximum number of machines that could process the subjob, and $\\mu_{ji}$ is the maximum processing power than can be brought to bear on the same. Lastly, $p_{ji}$ is the total processing requirement of subjob $(j,i)$. In these terms, the core linear program, LP1, is as follows.\n\\begin{align*}\n\t\\text{(LP1) } \\min &\\textstyle\\sum_{j \\in N} w_j C_j \\\\\n\ts.t.\\quad (1A) \\quad & \\textstyle\\sum_{j \\in S} p_{ji} C_j \n\t        \\geq \\frac{1}{2} \n\t        \\left[ \n\t           \\left(\\textstyle\\sum_{j \\in S} p_{ji}\\right)^2\/\\mu_i\n                + \\textstyle\\sum_{j \\in S} p_{ji}^2\/\\mu_{ji} \n            \\right]  \\qquad ~\\forall S \\subseteq N, i \\in M \\\\\n\t(1B) \\quad\t& C_j \\geq p_{jit}\/v_{1i} + r_{ji} \\qquad ~\\forall  i \\in M,~ j \\in N,~ t \\in T_{ji}\\\\\n\t(1C) \\quad\t& C_j \\geq p_{ji}\/\\mu_{ji} + r_{ji} \\qquad ~\\forall j \\in N,~ i \\in M \t \n\\end{align*} \n\nConstraints ($1A$) are more carefully formulated versions of the polyhedral constraints introduced by Queyranne \\cite{Queyranne1993} and developed by Schulz \\cite{Schulz1996}. The use of $\\mu_{ji}$ term is new and allows us to provide stronger performance guarantees for our framework where subjobs are composed of \\textit{sets} of tasks. As we will see, this term is one of the primary factors that allows us to parametrize results under varying machine speeds in terms of maximum to \\textit{average} machine speed, rather than maximum to \\textit{minimum} machine speed. Constraints ($1B$) and ($1C$) are simple lower bounds on job completion time. \n\n\nThe majority of this section is dedicated to proving that LP1 is a valid relaxation of $CC|r|\\sum w_jC_j$. Once this is established, we prove the that LP1 can be solved in polynomial time by providing a separation oracle with use in the Ellipsoid method. Both of these proofs use techniques established in Schulz's Ph.D. thesis \\cite{Schulz1996}. \n\n\\subsection{Proof of LP1's Validity}\n\n\n\n\nThe lemmas below establish the basis for both of our algorithms. Lemma \\ref{lem:sumOfSquaresDiffSpeeds} generalizes an inequality used by Schulz \\cite{Schulz1996}. Lemma \\ref{lem:feasForLP1} relies on Lemma \\ref{lem:sumOfSquaresDiffSpeeds} and cites an inequality mentioned in the preceding section (and proven by Queyranne \\cite{Queyranne1993}). \n\\begin{lemma}\nLet $\\{a_1,\\ldots a_z\\}$ be a set of non-negative real numbers. We assume that $k \\leq z$ of them are positive. Let $b_i$ be a set of decreasing positive real numbers. Then\n\\begin{center}\n$  \\sum_{i = 1}^z a_i^2 \/ b_i  \\geq \\left(\\sum_{i = 1}^z a_i \\right)^2 \/ \\left(\\sum_{i = 1}^k b_i\\right)$.\n\\end{center}\n\\label{lem:sumOfSquaresDiffSpeeds}\n\\end{lemma}\n\\begin{proof}\\footnote{The proceedings version of this paper stated that the proof cites the AM-GM inequality and proceeds by induction from $z=k=2$. We have opted here to demonstrate a different (simpler) proof that we discovered only after the proceedings version was finalized.}\nWe only show the case where $k=z$. Define $\\mathbf{a} = [a_1,\\ldots,a_k] \\in \\mathbb{R}^k_+$, $\\mathbf{b} = [b_1, \\ldots, b_k ] \\in \\mathbb{R}^k_{++}$, and $\\mathbbm{1}$ as the vector of $k$ ones. Now, set $\\mathbf{u} = \\mathbf{a} \/ \\sqrt{\\mathbf{b}}$ and $\\mathbf{w} = \\sqrt{\\mathbf{b}}$ (element-wise), and note that $\\langle \\mathbf{a}, \\mathbbm{1} \\rangle = \\langle \\mathbf{u}, \\mathbf{w} \\rangle$. In these terms, it is clear that $(\\sum_{i=1}^k a_i)^2 = \\langle \\mathbf{u}, \\mathbf{w} \\rangle^2$.\n\nGiven this, one need only cite Cauchy-Schwarz (namely, $\\langle \\mathbf{u}, \\mathbf{w} \\rangle^2 \\leq \\langle \\mathbf{u}, \\mathbf{u} \\rangle \\cdot \\langle \\mathbf{w}, \\mathbf{w} \\rangle$) and plug in the definitions of $\\mathbf{u}$ and $\\mathbf{w}$ to see the desired result.\n\\end{proof}\n\n\n\\begin{lemma}[Validity Lemma]\nEvery feasible schedule for an instance $I$ of $CC|r|\\sum w_jC_j$ has completion times that define a feasible solution to LP1($I$). \\label{lem:feasForLP1}\n\\end{lemma}\n\\begin{proof}\nAs constraints ($1B$) and ($1C$) are clear lower bounds on job completion time, it suffices to show the validity of constraint ($1A$). Thus, let $S$ be a non-empty subset of $N$, and fix an arbitrary but feasible schedule ``$F$'' for $I$. \n\nDefine $C^{F}_{ji}$ as the completion time of subjob $(j,i)$ under schedule $F$. Similarly, define $C^{F}_{ji\\ell}$ as the first time at which tasks of subjob $(j,i)$ scheduled on machine $\\ell$ of cluster $i$ are finished. Lastly, define $p^{\\ell}_{ji}$ as the total processing requirement of job $j$ scheduled on machine $\\ell$ of cluster $i$. Note that by construction, we have $C^{F}_{ji} = \\max_{\\ell \\in \\{1,\\ldots,m_i\\}}{C^{F}_{ji\\ell}}$ and $C^F_j = \\max_{i \\in M}{C^F_{ji}}$. \nSince $p_{ji} = \\sum_{\\ell = 1}^{m_i} p^{\\ell}_{ji}$, we can rather innocuously write\n\\begin{equation}\n\\textstyle\\sum_{j \\in S} p_{ji} C^{F}_{ji} = \\textstyle\\sum_{j \\in S}\\left[ \\textstyle\\sum_{\\ell  = 1}^{m_i}  p^{\\ell}_{ji}  \\right] C^{F}_{ji} . \n\\end{equation} \nBut using $C^{F}_{ji} \\geq C^{F}_{ji\\ell}$, we can lower-bound $\\sum_{j \\in S} p_{ji} C^{F}_{ji}$. Namely,\n\\begin{equation}\n\\textstyle\\sum_{j \\in S} p_{ji} C^{F}_{ji} \\geq \\textstyle\\sum_{j \\in S}\\textstyle\\sum_{\\ell = 1}^{m_i}  p^{\\ell}_{ji} C^{F}_{ji\\ell} =  \\textstyle\\sum_{\\ell = 1}^{m_i} v_{\\ell i}\\textstyle\\sum_{j \\in S} \\left[p^{\\ell}_{ji}\/v_{\\ell i} \\right]C^{F}_{ji\\ell} \\label{eq:specificMachines}\n\\end{equation}\nThe next inequality uses a bound on $\\textstyle\\sum_{j \\in S}\\left[p^{\\ell}_{ji}\/v_{\\ell i} \\right]C^{F}_{ji\\ell}$ proven by Queyranne \\cite{Queyranne1993} for any subset $S$ of $N$ jobs with processing times $\\left[p^{\\ell}_{ji}\/v_{\\ell i} \\right]$ to be scheduled on a single machine.\\footnote{Here, our machine is machine $\\ell$ on cluster $i$.}\n\\begin{equation}\n\\textstyle\\sum_{j \\in S} \\left[p^{\\ell}_{ji}\/v_{\\ell i} \\right]C^{F}_{ji\\ell} \\geq \\frac{1}{2} \\left[\\left(\\textstyle\\sum_{j \\in S} \\left[p^{\\ell}_{ji}\/v_{\\ell i} \\right]\\right)^2 + \\textstyle\\sum_{j \\in S} \\left(\n\\left[p^{\\ell}_{ji}\/v_{\\ell i} \\right]\\right)^2 \\right]\\label{eq:queyranne}\n\\end{equation}\nCombining inequalities \\eqref{eq:specificMachines} and \\eqref{eq:queyranne}, we have the following.\n\\begin{align}\n\\textstyle\\sum_{j \\in S} p_{ji} C^{F}_{ji} &\\geq \\frac{1}{2} \\textstyle\\sum_{\\ell=1}^{m_i} v_{\\ell i} \\left[\\left(\\textstyle\\sum_{j \\in S} \\left[p^{\\ell}_{ji}\/v_{\\ell i} \\right]\\right)^2 + \\textstyle\\sum_{j \\in S} \\left(\n\\left[p^{\\ell}_{ji}\/v_{\\ell i} \\right]\\right)^2 \\right]  \\\\\n& \\geq \\frac{1}{2} \\left[\\textstyle\\sum_{\\ell = 1}^{m_i}\\left(\\sum_{j \\in S} p^\\ell_{j i}\\right)^2 \/ v_{\\ell i} + \\sum_{j \\in S} \\sum_{\\ell = 1}^{m_i} \\left(p^\\ell_{j i}\\right)^2 \/ v_{\\ell i} \\right] \\label{eq:differentBoundForLP1}\n\\end{align}\nNext, we apply Lemma \\ref{lem:sumOfSquaresDiffSpeeds} to the right hand side of inequality \\eqref{eq:differentBoundForLP1} a total of $|S|+1$ times.\n\\begin{align}\n&\\textstyle\\sum_{\\ell = 1}^{m_i} \\left(\\textstyle\\sum_{j \\in S} p^\\ell_{j i}\\right)^2 \/v_{\\ell i} \\geq \\left(\\textstyle\\sum_{\\ell = 1}^{m_i}\\textstyle\\sum_{j \\in S} p^\\ell_{j i}\\right)^2\/ \\textstyle\\sum_{\\ell = 1}^{m_i} v_{\\ell i} = \\left(\\textstyle\\sum_{j \\in S} p_{ji}\\right)^2 \/ \\mu_i \\\\\n&\\textstyle\\sum_{\\ell = 1}^{m_i} \\left(p^\\ell_{j i}\\right)^2 \/ v_{\\ell i} \\geq \\left(\\sum_{\\ell = 1}^{m_i}p^\\ell_{j i}\\right)^2 \/ \\textstyle\\sum_{\\ell = 1}^{q_{j i}} v_{\\ell i} = p_{j i}^2\/\\mu_{j i} ~~\\forall~ j \\in S\n\\end{align}\nCiting $C^{F}_{j} \\geq C^{F}_{ji}$, we arrive at the desired result.\n\\begin{equation}\n\\textstyle\\sum_{j \\in S} p_{ji} C^{F}_{j} \\geq \\frac{1}{2}\\left[\\left(\\textstyle\\sum_{j \\in S} p_{ji} \\right)^2\/\\mu_i + \\textstyle\\sum_{j \\in S}p_{ji}^2\/\\mu_{ji}\\right] \\qquad  \\text{``constraint }(1A)\\text{''}\n\\end{equation}\n\n\\end{proof}\n\n\n\n\\subsection{Theoretical Complexity of LP1}\\label{subsec:thankGodPolyTime}\nAs the first of our two algorithms requires solving LP1 directly, we need to address the fact that LP1 has $m \\cdot (2^n - 1) + n$ constraints. \nLuckily, it is still possible to such solve linear programs in polynomial time with the Ellipsoid method; we introduce the following separation oracle for this purpose.\n\n\\begin{definition}[Oracle LP1]\nDefine the \\textit{violation}\n\\begin{equation}\nV(S,i) = \\frac{1}{2} \\left[ \n\t           \\left(\\textstyle\\sum_{j \\in S} p_{j i}\\right)^2\/\\mu_i \n                + \\textstyle\\sum_{j \\in S} p_{j i}^2\/\\mu_{j i} \n            \\right] - \\textstyle\\sum_{j \\in S} p_{j  i} C_{j}\n\\end{equation}\nLet $\\{C_j\\} \\in \\mathbb{R}^n$ be a \\textit{potentially} feasible solution to LP1. Let $\\sigma_i$ denote the ordering when jobs are sorted in increasing order of $C_j - p_{j i}\/(2\\mu_{ji})$. Find the most violated constraint in $(1A)$ for $i \\in M$ by searching over $V(S_i,i)$ for $S_i$ of the form $\\{\\sigma_i(1),\\ldots,\\sigma_i(j-1),\\sigma_i(j)\\},~ j \\in \\{1,\\ldots,n\\}$. If any of maximal $V(S_i^*,i) > 0$, then return $(S_i^*,i)$ as a violated constraint for ($1A$). Otherwise, check the remaining $n$ constraints $((1B)$ and $(1C))$ directly in linear time.\n\\end{definition} \n\nFor fixed $i$, Oracle-LP1 finds the subset of jobs that maximizes ``violation'' for cluster $i$. That is, Oracle-LP1 finds $S_i^*$ such that $V(S_i^*,i) = \\text{max}_{S \\subset N} V(S,i)$. We prove the correctness of Oracle-LP1 by establishing a necessary and sufficient condition for a job $j$ to be in $S_i^*$.\n\n\\begin{lemma}\nFor $\\mathbb{P}_i(A) \\doteq \\textstyle\\sum_{j \\in A} p_{ji}$, we have $x \\in S_i^* \\Leftrightarrow$ $ C_x - p_{xi}\/(2\\mu_{xi}) \\leq \\mathbb{P}_i(S_i^*)\/\\mu_i$. \n\\label{lem:separation}\n\\end{lemma}\n\\begin{proof}\nFor given $S$ (not necessarily equal to $S_i^*$), it is useful to express $V(S,i)$ in terms of $V(S\\cup x, i)$ or $V(S\\setminus x, i)$ (depending on whether $x \\in S$ or $x \\in N \\setminus S$). Without loss of generality, we restrict our search to $S : x \\in S \\Rightarrow p_{x,i} > 0$.\n\nSuppose $ x \\in S$. By writing $\\mathbb{P}_i(S) = \\mathbb{P}_i(S\\setminus x) + \\mathbb{P}_i(x)$, and similarly decomposing the sum $\\textstyle\\sum_{j \\in S} p_{j i}^2\/(2\\mu_{ji})$, one can show the following.\n\\begin{align}\nV(S, i) = & V(S\\setminus x, i) + p_{xi}\\left(\\frac{1}{2}\\left(\\frac{2\\mathbb{P}_i(S) - p_{xi}}{\\mu_i} + \\frac{p_{xi}}{\\mu_{xi}}\\right) - C_x \\right) \\label{eq:xInS}\n\\end{align}\nNow suppose $ x \\in N\\setminus S$. In the same strategy as above (this time writing $ \\mathbb{P}_i(S) = \\mathbb{P}_i(S\\cup x) - \\mathbb{P}_i(x)$), one can show that\n\\begin{align}\nV(S, i) =&  V(S\\cup x, i) + p_{xi}\\left(C_x - \\frac{1}{2}\\left(\\frac{2\\mathbb{P}_i(S) + p_{xi}}{\\mu_i} + \\frac{p_{xi}}{\\mu_{xi}}\\right) \\right). \\label{eq:xNotInS}\n\\end{align}\nNote that Equations \\eqref{eq:xInS} and \\eqref{eq:xNotInS} hold for all $S$, including $S = S_i^*$. Turning our attention to $S_i^*$, we see that $x \\in S_i^*$ implies that the second term in Equation \\eqref{eq:xInS} is non-negative,  i.e. \n\\begin{equation}\nC_x - p_{xi}\/(2\\mu_{xi}) \\leq \\left(2\\mathbb{P}_i(S_i^*) - p_{xi}\\right)\/(2\\mu_i) < \\mathbb{P}_i(S_i^*)\/\\mu_i.\n\\end{equation}\nSimilarly, $x \\in N \\setminus S_i^*$ implies the second term in Equation \\eqref{eq:xNotInS} is non-negative.\n\\begin{equation}\nC_x - p_{x i}\/(2\\mu_{x i}) \\geq \\left(2\\mathbb{P}_i(S_i^*) + p_{x i}\\right)\/(2\\mu_i) \\geq \\mathbb{P}_i(S_i^*)\/\\mu_i\n\\end{equation}\nIt follows that $x \\in S_i^*$ iff $C_x - p_{x i}\/(2\\mu_{x i}) < \\mathbb{P}_i(S_i^*)\/\\mu_i$.\n\\end{proof}\n\nGiven Lemma \\ref{lem:separation}, It is easy to verify that sorting jobs in increasing order of $C_x - p_{xi}\/(2\\mu_{xi})$ to define a permutation $\\sigma_i$ guarantees that $S_i^*$ is of the form $\\{\\sigma_i(1),\\ldots,\\sigma_i(j-1),\\sigma_i(j)\\}$ for some $j \\in N$. This implies that for fixed $i$, Oracle-LP1 finds $S_i^*$ in $O(n \\log(n))$ time. This procedure is executed once for each cluster, leaving the remaining $n$ constraints in $(1B)$ and $(1C)$ to be verified in linear time. Thus Oracle-LP1 runs in $O(mn\\log(n))$ time.\n\nBy the equivalence of separation and optimization, we have proven the following theorem:\n\\begin{theorem}\nLP1($I$) is a valid relaxation of $I \\in \\Omega_{CC}$, and is solvable in polynomial time. \\label{thm:LP1feasAndSolve}\n\\end{theorem}\n\nAs was explained in the beginning of this section, linear programs such as those in \\cite{Chen2000, Garg2007, llp, Queyranne1993, Schulz1996, Schulz2012} are processed with an appropriate sorting of the optimal decision variables $\\{C^\\star_j\\}$. It is important then to have bounds on job completion times for a particular ordering of jobs. We address this next in Section \\ref{sec:listSched}, and reserve our first algorithm for Section \\ref{sec:lpAlg}.\n\\section{List Scheduling from Permutations}\\label{sec:listSched}\n\n\n\n\n\n\n\n\nThe complex work in both of our proposed algorithms is to generate a \\textit{permutation} of jobs. The procedure below takes such a permutation and uses it to determine start times, end times, and machine assignments for every task of every subjob.\n\\vspace{1em}\n\n\\noindent \\textbf{List-LPT} : Given a single cluster with $m_i$ machines and a permutation of jobs $\\sigma$, introduce $\\text{List}(a,i) \\doteq (p_{ai1}, p_{ai2},\\ldots,p_{ai|T_{ai}|})$ as an ordered set of tasks belonging to subjob $(a,i)$, ordered by longest processing time first. Now define $\\text{List}(\\sigma) \\doteq \\text{List}(\\sigma(1),i) \\oplus \\text{List}(\\sigma(2),i) \\oplus \\cdots \\oplus \\text{List}(\\sigma(n),i)$, where $\\oplus$ is the concatenation operator. \n\nPlace the tasks of $\\text{List}(\\sigma)$ in order- from the largest task of subjob $(\\sigma(1),i)$, to the smallest task of subjob $(\\sigma(n),i)$. When placing a particular task, assign it whichever machine and start time results in the task being completed as early as possible (without moving any tasks which have already been placed). Insert idle time (on all $m_i$ machines) as necessary if this procedure would otherwise start a job before its release time.\n\\vspace{1em}\n\nThe following Lemma is essential to bound the completion time of a set of jobs processed by List-LPT. The proof is adapted from Gonzalez et al. \\cite{Gonzalez1977}.\n\\begin{lemma}\nSuppose $n$ jobs are scheduled on cluster $i$ according to List-LPT($\\sigma$). Then for $ \\bar{v_i} \\doteq \\mu_i\/m_i$, the completion time of subjob $(\\sigma(j),i)$ $($denoted $C_{\\sigma(j)i}$ $)$ satisfies\n\\begin{align}\n&C_{\\sigma(j)i} \\leq \\max_{1\\leq k \\leq j}{r_{\\sigma(k)i}} + p_{\\sigma(j)i1}\/\\bar{v_i} +  \\left(\\textstyle\\sum_{k=1}^{j} p_{\\sigma(k)i} - p_{\\sigma(j)i1}\\right)\/\\mu_i \\label{eq:generalGonzalezLemma}\n\\end{align} \\label{lem:CompTimesOnUniformMachines}\n\\end{lemma}\n\\begin{proof}\nFor now, assume all jobs are released at time zero. Let the task of subjob $(\\sigma(j),i)$ to finish last be denoted $t^*$. If $t^*$ is not the task in $T_{\\sigma(j)i}$ with least processing time, then construct a new set $T'_{\\sigma(j)i} = \\{ t :  p_{\\sigma(j)it^*} \\leq p_{\\sigma(j)it} \\} \\subset T_{\\sigma(j)i}$. Because the tasks of subjob $(\\sigma(j),i)$ were scheduled by List-LPT (i.e. longest-processing-time-first), the sets of potential start times and machines for task $t^*$ (and hence the set of potential completion times for task $t^*$) are the same regardless of whether subjob $(\\sigma(j),i)$ consisted of tasks $T_{\\sigma(j)i}$ or the subset $T'_{\\sigma(j)i}$. Accordingly, reassign $T_{\\sigma(j)i} \\leftarrow T'_{\\sigma(j)i}$ without loss of generality.\n\nLet $D_{\\ell}^j$ denote the total demand for machine $\\ell$ (on cluster $i$) once all tasks of subjobs $(\\sigma(1),i)$ through $(\\sigma(j-1),i)$ and all tasks in the set $T_{\\sigma(j)i}\\setminus \\{t^*\\}$ are scheduled. Using the fact that $C_{\\sigma(j)i}v_{\\ell i} \\leq ({D}_{\\ell}^{j} + p_{\\sigma(j)i t^*}) \\forall \\ell \\in \\{1,\\ldots,m_i\\}$, sum the left and right and sides over $\\ell$. This implies $C_{\\sigma(j)i}\\left( \\textstyle\\sum_{\\ell = 1}^{m_i} v_{\\ell i} \\right) \\leq ~ m_i p_{\\sigma(j) i t^*} + \\textstyle\\sum_{\\ell = 1}^{m_i} {D}_{\\ell}^{j}$. Dividing by the sum of machine speeds and using the definition of $\\mu_i$ yields\n\\begin{equation}\nC_{\\sigma(j)i} \n\t~ \\leq ~  m_i p_{\\sigma(j)i t^*}\/\\mu_i + \\textstyle\\sum_{\\ell = 1}^{m_i} {D}_{\\ell}^j \/\\mu_i ~ \\leq ~ p_{\\sigma(j)i 1}\/\\bar{v_i} + \\left(\\textstyle\\sum_{k = 1}^{j} p_{\\sigma(k)i} - p_{\\sigma(j)i1}\\right)\/\\mu_i \\label{eq:mainGonzalezLemma}\n\\end{equation}\nwhere we estimated $p_{\\sigma(j)i t^*}$ upward by $p_{\\sigma(j)i 1}$. Inequality \\eqref{eq:mainGonzalezLemma} completes our proof in the case when $r_{ji} \\equiv 0$. \n\nNow suppose that some $r_{ji} > 0$. We take our policy to the extreme and suppose that all machines are left idle until every one of jobs $\\sigma(1)$ through $\\sigma(j)$ are released; note that this occurs precisely at time $\\max_{1 \\leq k \\leq j} r_{\\sigma(k)i}$. It is clear that beyond this point in time, we are effectively in the case where all jobs are released at time zero, hence we can bound the remaining time to completion by the right hand side of Inequality \\ref{eq:mainGonzalezLemma}. As Inequality \\ref{eq:generalGonzalezLemma} simply adds these two terms, the result follows.\n\\end{proof}\n\nLemma \\ref{lem:CompTimesOnUniformMachines} is cited directly in the proof of Theorem \\ref{thm:uniformLP1} and Lemma \\ref{lem:TSPTboundInTermsOfPDandOPT}. Lemma \\ref{lem:CompTimesOnUniformMachines} is used implicitly in the proofs of Theorems \\ref{thm:identLP}, \\ref{thm:identLP_2appxWithConstantTasks}, and \\ref{thm:tspt_unit_tasks}.\n\\section{An LP-based Algorithm}\\label{sec:lpAlg}\n\n\n\nIn this section we show how LP1 can be used to construct near optimal schedules for concurrent cluster scheduling both when $r_{ji} \\equiv 0$ and when some $r_{ji} > 0$. Although solving LP1 is somewhat involved, the algorithm itself is quite simple:\n\\vspace{0.5em}\n\n\\noindent \\textbf{Algorithm CC-LP} : Let $I = (T, r, w, v)$ denote an instance of $CC | r | \\sum w_j C_j$. Use the optimal solution $\\{C_j^\\star\\}$ of LP1($I$) to define $m$ permutations $\\{\\sigma_i : i \\in M\\}$ which sort jobs in increasing order of $C^\\star_j - p_{ji}\/(2\\mu_{ji})$. For each cluster $i$, execute List-LPT($\\sigma_i$).\n\\vspace{0.5em}\n\nEach theorem in this section can be characterized by how various assumptions help us cancel an additive term\\footnote{``$+p_{xit^*}$''; see associated proofs.} in an upper bound for the completion time of an arbitrary subjob $(x,i)$. Theorem \\ref{thm:uniformLP1} is the most general, while Theorem \\ref{thm:identLP_2appxWithConstantTasks} is perhaps the most surprising.\n\n\\subsection{CC-LP for Uniform Machines}\\label{sec:unifLP}\n\\begin{theorem}\nLet $\\hat{C}_j$ be the completion time of job $j$ using algorithm CC-LP, and let $R$ be as in Section \\ref{subsec:outlineAndResults}. If $r_{ji} \\equiv 0$, then $ \\textstyle\\sum_{j \\in N} w_j \\hat{C}_j \\leq \\left(2 + R\\right)OPT $. Otherwise, $ \\textstyle\\sum_{j \\in N} w_j \\hat{C}_j \\leq \\left(3 + R\\right)OPT$.\n\\label{thm:uniformLP1}\n\\end{theorem}\n\\begin{proof}\nFor $y \\in \\mathbb{R}$, define $y^+ = \\max\\{y,0\\}$. Now let $x \\in N$ be arbitrary, and let $i \\in M$ be such that $p_{xi} > 0$ (but otherwise arbitrary). Define $t^*$ as the last task of job $x$ to complete on cluster $i$, and let $j_i$ be such that $\\sigma_i(j_i) = x$. Lastly, denote the optimal LP solution $\\{C_j\\}$.\\footnote{We omit the customary $\\star$ to avoid clutter in notation.} Because $\\{C_j\\}$ is a feasible solution to LP1, constraint $(1A)$ implies the following (set $S_i = \\{\\sigma_i(1),\\ldots,\\sigma_i(j_i - 1),x\\}$)\n\\begin{align}\n\\frac{\\left( \\textstyle\\sum_{k = 1}^{j_i} p_{\\sigma_i(k)i} \\right)^2}{2\\mu_i}\n\t&\\leq \\sum_{k = 1}^{j_i} p_{\\sigma_i(k)i}\\left(C_{\\sigma_i(k)} - \\frac{p_{\\sigma_i(k)i}}{2\\mu_{\\sigma_i(k)i}}\\right) \\leq \\left(C_{x} - \\frac{p_{xi}}{2\\mu_{xi}}\\right)\\sum_{k = 1}^{j_i} p_{\\sigma_i(k)i} \\label{eq:compTimeInUnifLP}\n\\end{align}\nwhich in turn implies $\\textstyle\\sum_{k = 1}^{j_i} p_{\\sigma_i(k)i}\/\\mu_i \\leq 2C_x - p_{xi}\/\\mu_{xi}$.\n\nIf all subjobs are released at time zero, then we can combine this with Lemma \\ref{lem:CompTimesOnUniformMachines} and the fact that $p_{xit^*} \\leq p_{xi} = \\textstyle\\sum_{t \\in T_{xi}} p_{xit}$ to see the following (the transition from the first inequality the second inequality uses $C_x \\geq p_{xit^*}\/v_{1i}$ and $R_i = v_{1i}\/\\bar{v}_i$).\n\\begin{align}\n\\hat{C}_{xi} \n\t&\\leq 2C_x - \\frac{p_{xi}}{\\mu_{xi}} + \\frac{p_{xit^*}}{\\bar{v}_i} - \\frac{p_{xit^*}}{\\mu_i} \\leq \n\t\tC_x(2 + \\left[R_i(1 - 2\/m_i)\\right]^+) \\label{eq:generalCompTimeWithOUTReleaseUnif} \n\\end{align}\n\nWhen one or more subjobs are released after time zero, Lemma \\ref{lem:CompTimesOnUniformMachines} implies that it is sufficient to bound  $\\displaystyle\\max_{1 \\leq k \\leq j_i}{\\left\\lbrace r_{\\sigma_i(k)i} \\right\\rbrace}$ by some constant multiple of $C_x$. Since $\\sigma_i$ is defined by increasing $L_{ji} \\doteq C_j - p_{ji}\/(2\\mu_{ji})$, $L_{\\sigma_i(a)i} \\leq L_{\\sigma_i(b)i}$ implies\n\\begin{align}\n&r_{\\sigma_i(a)i} + \\frac{p_{\\sigma_i(a)i}}{2\\mu_{\\sigma_i(a)i}} + \\frac{p_{\\sigma_i(b)i}}{2\\mu_{\\sigma_i(b)i}}  \\leq C_{\\sigma_i(a)} - \\frac{p_{\\sigma_i(a)i}}{2\\mu_{\\sigma_i(a)i}} + \\frac{p_{\\sigma_i(b)i}}{2\\mu_{\\sigma_i(b)i}}  \\leq C_{\\sigma_i(b)} ~\\forall~ a \\leq b\n\\end{align}\nand so $\\max_{1 \\leq k \\leq j_i}{\\left\\lbrace r_{\\sigma_i(k)i} \\right\\rbrace} + p_{xi}\/(2\\mu_{xi}) \\leq C_{x}$. As before, combine this with Lemma \\ref{lem:CompTimesOnUniformMachines} and the fact that $p_{xit^*} \\leq p_{xi} = \\textstyle\\sum_{t \\in T_{xi}} p_{xit}$ to yield the following inequalities\n\\begin{align}\n\\hat{C}_{xi} \n\t&\\leq 3C_x - \\frac{3p_{xi}}{2\\mu_{xi}} + \\frac{p_{xit^*}}{\\bar{v}_i} - \\frac{p_{xit^*}}{\\mu_i} \\leq C_x(3 + \\left[R_i(1 - 5\/(2m_i))\\right]^+)  \\label{eq:generalCompTimeWithReleaseUnif} \n\\end{align}\n-which complete our proof.\n\\end{proof}\n\\subsection{CC-LP for Identical Machines}\\label{sec:identLP}\n\\begin{theorem}\nIf machines are of unit speed, then CC-LP yields an objective that is...\n\\begin{center}\n\\begin{tabular}{l | c c}\n\\hline\n & $r_{ji} \\equiv 0$ & some $r_{ji} > 0$ \\\\ \n \\hline\nsingle-task subjobs & $\\leq$ 2 $OPT$ & $\\leq $ 3 $OPT$ \\\\\nmulti-task subjobs & $\\leq$ 3 $OPT$ & $\\leq$ 4 $OPT$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{thm:identLP}\n\\end{theorem}\n\\begin{proof}\nDefine $[\\cdot]^+$, $x$, $C_x$, $\\hat{C}_x$, $i$, $\\sigma_i$, and $t^*$ as in Theorem \\ref{thm:uniformLP1}. When $r_{ji} \\equiv 0$, one need only give a more careful treatment of the first inequality in \\eqref{eq:generalCompTimeWithOUTReleaseUnif} (using $\\mu_{ji} = q_{ji}$).\n\\begin{align}\n\\hat{C}_{x,i} \n\t&\\leq 2C_x + p_{xit^*} - p_{xit^*}\/m_i - p_{xi}\/q_{xi} \n\t\\leq C_x(2 + \\left[1 - 1\/m_i -1\/q_{xi} \\right]^+) \\label{eq:GeneralIdentCompTimeBound}\n\\end{align}\nSimilarly, when some $r_{ji} > 0$, the first inequality in \\eqref{eq:generalCompTimeWithReleaseUnif} implies the following.\n\\begin{align}\n\\hat{C}_{x,i}\n\t&\\leq 3C_x + p_{xit^*} - p_{xit^*}\/m_i - 3p_{xi}\/(2q_{xi})\n\t\\leq C_x(3 + \\left[1 - 1\/m_i - 3\/(2q_{xi})\\right]^+) \\label{eq:forCstTimeThmWithRelease}\n\\end{align}\n\\end{proof}\nThe key in the refined analysis of Theorem \\ref{thm:identLP} lay in how $-p_{xi}\/q_{xi}$ is used to annihilate $+p_{xit^*}$. While $q_{xi} = 1$ (i.e. single-task subjobs) is sufficient to accomplish this, it is not strictly \\textit{necessary}. The theorem below shows that we can annihilate the $+p_{xit^*}$ term whenever all tasks of a given subjob are of the same length. Note that the tasks need not be \\textit{unit}, as the lengths of tasks across different subjobs can differ.\n\\begin{theorem}\nSuppose $v_{\\ell i} \\equiv 1$. If $p_{jit}$ is constant over $t \\in T_{ji}$ for all $j \\in N$ and $i \\in M$, then algorithm CC-LP is a 2-approximation when $r_{ji} \\equiv 0$, and a 3-approximation otherwise. \\label{thm:identLP_2appxWithConstantTasks}\n\\end{theorem}\n\\begin{proof}\nThe definition of $p_{xi}$ gives $p_{xi}\/q_{xi} = \\textstyle\\sum_{t \\in T_{xi}} p_{xit} \/ q_{xi}$. Using the assumption that $p_{jit}$ is constant over $t \\in T_{ji}$, we see that $p_{xi}\/q_{xi} = (q_{xi} + |T_{xi}| - q_{xi})p_{xit^*} \/ q_{xi} $, where $|T_{xi} |\\geq q_{xi}$. Apply this to Inequality \\eqref{eq:GeneralIdentCompTimeBound} from the proof of Theorem \\ref{thm:identLP}; some algebra yields \n\\begin{align}\n\\hat{C}_{xi} \n\t\\leq& 2C_x  - p_{xit^*}\/m_i - p_{xit^*}\\left(|T_{xi}| - q_{xi}\\right)\/q_{xi} \\leq 2C_x.\n\\end{align}\nThe case with some $r_{ji} > 0$ uses the same identity for $p_{xi}\/q_{xi}$.\n\\end{proof}\nSachdeva and Saket \\cite{nphard2} showed that it is NP-Hard to approximate $CC|m_i \\equiv 1|\\sum w_j C_j$ with a constant factor less than 2. \nTheorem \\ref{thm:identLP_2appxWithConstantTasks} is significant because it shows that CC-LP can attain the same guarantee for \\textit{arbitrary} $m_i$, provided $v_{\\ell i} \\equiv 1$ and $p_{jit}$ is constant over $t$.\n\n\\section{Combinatorial Algorithms}\\label{sec:TSPT}\n\n\n\n\n\n\nIn this section, we introduce an extremely fast combinatorial algorithm with performance guarantees similar to CC-LP for ``unstructured'' inputs (i.e. those for which some $v_{\\ell i} > 1$, or some $T_{ji}$ have $p_{jit}$ non-constant over $t$). \nWe call this algorithm \\textit{CC-TSPT}. \nCC-TSPT uses the MUSSQ algorithm for concurrent open shop (from \\cite{mqssu}) as a subroutine. As SWAG (from \\cite{HGY}) motivated development of CC-TSPT, we first address SWAG's worst-case performance.\n\n\\subsection{A Degenerate Case for SWAG}\\label{subsec:swagDegenerate}\n\\begin{wrapfigure}{r}{0.5\\textwidth}\n\t\\vspace{-0.45cm}\n  \\centering\n    \\begin{minipage}{0.50\\textwidth}\n\t\\begin{algorithm}[H]\n\t\\begin{algorithmic}[1]\n\t\\Procedure{SWAG}{$N,M,p_{ji}$}\n\t\\State $J\\gets \\emptyset$\n\t\\State $q_i\\gets 0,\\forall i\\in M$\n\t\\While{$|J|\\not=|N|$}\n\t\\State mkspn$_j\\gets$ max$_{i\\in M}\\left(\\frac{q_i+p_{ji}}{m_i}\\right)$\n\t\\item[] \\qquad \\qquad $\\forall j\\in N\\setminus J$\n\t\\State nextJob $\\gets$ argmin$_{j \\in N \\setminus J}\\ $mkspn$_j$\n\t\\State $J.$append$($nextJob$)$\n\t\\State $q_i\\gets q_i+ p_{ji}$\n\t\\EndWhile\n\t\\State \\textbf{return} $J$\n\t\\EndProcedure\n\t\\end{algorithmic}\n\t\\end{algorithm}\n  \\end{minipage}\n  \\vspace{-0.45cm}\n\\end{wrapfigure}\nAs a prerequisite for addressing worst-case performance of an existing algorithm, we provide psuedocode and an accompanying verbal description for SWAG.\n\nSWAG computes queue positions for every subjob of every job, supposing that each job was scheduled next. \nA job's potential makespan (``mkspn'') is the largest of the potential finish times of all of its subjobs (considering current queue lengths $q_i$ and each subjob's processing time $p_{ji}$). \nOnce potential makespans have been determined, the job with smallest potential makespan is selected for scheduling. \nAt this point, all queues are updated. \nBecause queues are updated, potential makespans will need to be re-calculated at the next iteration. \nIterations continue until the very last job is scheduled. Note that SWAG runs in $O(n^2m)$ time.\n\n\\begin{theorem}\nFor an instance $I$ of $PD || \\sum C_j$, let $SWAG(I)$ denote the objective function value of SWAG applied to $I$, and let $OPT(I)$ denote the objective function value of an optimal solution to $I$. \nThen for all $L \\geq 1$, there exists an $I \\in \\Omega_{PD || \\sum C_j}$ such that $SWAG(I) \/ OPT(I) > L$.\n\\label{thm:swagBad}\n\\end{theorem}\n\\begin{proof}\nLet $L \\in \\mathbb{N}^+$ be a fixed but arbitrary constant. \nConstruct a problem instance $I_L^m$ as follows: \n\n$N = N_1 \\cup N_2$ where $N_1$ is a set of $m$ jobs, and $N_2$ is a set of $L$ jobs. \nJob $j \\in N_1$ has processing time $p$ on cluster $j$ and zero all other clusters. \nJob $j \\in N_2$ has processing time $p(1-\\epsilon)$ on all $m$ clusters. \n$\\epsilon$ is chosen so that $\\epsilon < 1\/L$\n(see Figure \\ref{fig:swag1}).\n\n\\begin{figure}[ht]\n\\includegraphics[width=\\linewidth]{swag.png}\n\\centering\n\\caption{At left, an input for SWAG example with $m=3$ and $L=2$. At right, SWAG's resulting schedule, and an alternative schedule.}\n\\label{fig:swag1}\n\\end{figure}\n\nIt is easy to verify that SWAG will generate a schedule where all jobs in $N_2$ precede all jobs in $N_1$ (due to the savings of $p \\epsilon$ for jobs in $N_2$). \nWe propose an \\textit{alternative} solution in which all jobs in $N_1$ preceed all jobs in $N_2$. \nDenote the objective value for this alternative solution $ALT(I_L^m)$, noting $ALT(I_L^m) \\geq OPT(I_L^m)$.\n\nBy symmetry, and the fact that all clusters have a single machine, we can see that $SWAG(I_L^m)$ and $ALT(I_L^m)$ are given by the following\n\\begin{align}\nSWAG(I_L^m) &= p(1-\\epsilon)L(L+1)\/2 + p(1-\\epsilon)L m + p m \\\\\nALT(I_L^m) &= p(1-\\epsilon)L(L + 1)\/2 + pL + p m\n\\end{align}\nSince $L$ is fixed, we can take the limit with respect to $m$.\n\\begin{align}\n\\lim_{m \\rightarrow \\infty}{\\frac{SWAG(I_L^m)}{ALT(I_L^m)}} \n\t&= \\lim_{m \\rightarrow \\infty}{\\frac{p(1-\\epsilon)L m + p m}{p m}} = L(1-\\epsilon) + 1 > L\n\\end{align}\nThe above implies the existence of a sufficiently large number of clusters $\\overline{m}$, such that $m \\geq \\overline{m}$ implies $ SWAG(I_L^{m})\/OPT(I_L^{m}) > L $. This completes our proof.\n\\end{proof}\nTheorem \\ref{thm:swagBad} demonstrates that that although SWAG performed well in simulations, it may not be reliable. \nThe rest of this section introduces an algorithm not only with superior runtime to SWAG (generating a permutation of jobs in $O(n^2 + nm)$ time, rather than $O(n^2m)$ time), but also a constant-factor performance guarantee.\n\n\\subsection{CC-TSPT : A Fast 2 + R Approximation}\\label{subsec:fastreduction}\nOur combinatorial algorithm for concurrent cluster scheduling exploits an elegant transformation to concurrent open shop. \nOnce we consider this simpler problem, it can be handled with MUSSQ \\cite{mqssu} and List-LPT. \nOur contributions are twofold: (1) we prove that this intuitive technique yields an approximation algorithm for a decidedly more general problem, and (2) we show that a \\textit{non-intuitive} modification can be made that maintains theoretical bounds while improving empirical performance. \nWe begin by defining our transformation.\n\n\\begin{definition}[The Total Scaled Processing Time (TSPT) Transformation]\nLet $\\Omega_{CC}$ be the set of all instances of $CC || \\sum w_j C_j$, \nand let $\\Omega_{PD}$ be the set of all instances of \n$PD || \\sum w_j C_j$. Note that $\\Omega_{PD} \\subset \\Omega_{CC}$.\nThen the Total Scaled Processing Time Transformation is a mapping\n\\begin{align*}\nTSPT: ~\\Omega_{CC} \\to \\Omega_{PD} \\quad \\text{ with } \\quad (T, v, w) &\\mapsto (X, w) ~:~ x_{ji} = \\textstyle\\sum_{t \\in T_{ji}} p_{jit} \/ \\mu_i\n\\end{align*}\ni.e., $x_{ji}$ is the total processing time required by subjob $(j,i)$, scaled by the sum of machine speeds at cluster $i$. \nThroughout this section, we will use $I = (T, v, w)$ to denote an arbitrary instance of $CC || \\sum w_j C_j$, and $I' = (X, w)$ as the image of $I$ under TSPT. \nFigure \\ref{fig:tspt} shows the result of TSPT applied to our baseline example. \n\\end{definition}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{tspt.png}\n\\caption{An instance $I$ of $CC||\\sum w_j C_j$, and its image $I' = TSPT(I)$. The schedules were constructed with List-LPT using the same permutation for $I$ and $I'$. }\n\\label{fig:tspt}\n\\end{figure}\n\nWe take the time to emphasize the simplicity of our reduction. Indeed, the TSPT transformation is perhaps the first thing one would think of given knowledge of the concurrent open shop problem. What is surprising is how one can attain constant-factor performance guarantees even after such a simple transformation.\n\n\\vspace{1em}\n\\noindent \\textbf{Algorithm CC-TSPT} : Execute MUSSQ on $I'= TSPT(I)$ to generate a permutation of jobs $\\sigma$. List schedule instance $I$ by \n$\\sigma$ on each cluster according to List-LPT.\n\\vspace{1em}\n\nTowards proving the approximation ratio for CC-TSPT, we will establish a critical inequality in Lemma \\ref{lem:TSPTboundInTermsOfPDandOPT}. \nThe intuition behind Lemma \\ref{lem:TSPTboundInTermsOfPDandOPT} requires thinking of every job $j$ in $I$ as having a corresponding representation in $j'$ in $I'$. \nJob $j$ in $I$ will be scheduled in the $CC$ environment, while job $j'$ in $I'$ will be scheduled in the $PD$ environment. \nWe consider what results when the same permutation $\\sigma$ is used for scheduling in both environments. \n\nNow the definitions for the lemma: let $C^{CC}_{\\sigma(j)}$ be the completion time of job $\\sigma(j)$ resulting from List-LPT on an arbitrary permutation $\\sigma$. \nDefine $C^{CC\\star}_{\\sigma(j)}$ as the completion time of job $\\sigma(j)$ in the $CC$ environment in the optimal solution. \nLastly, define $C^{PD,I'}_{\\sigma(j')}$ as the completion time of job $\\sigma(j')$ in $I'$ when scheduling by List-LPT($\\sigma$) in the $PD$ environment.\n\n\\begin{lemma}\nFor $I' = TSPT(I)$, let $j'$ be the job in $I'$ corresponding to job $j$ in $I$. \nFor an arbitrary permutation of jobs $\\sigma$, we have $C^{CC}_{\\sigma(j)} \\leq C^{PD,I'}_{\\sigma(j')} + R\\cdot C^{CC\\star}_{\\sigma(j)}$. \\label{lem:TSPTboundInTermsOfPDandOPT}\n\\end{lemma}\n\\begin{proof}\nAfter list scheduling has been carried out in the $CC$ environment, we may determine $C^{CC}_{\\sigma(j)i}$ - the completion time of subjob $(\\sigma(j),i)$. \nWe can bound $C^{CC}_{\\sigma(j)i}$ using Lemma \\ref{lem:CompTimesOnUniformMachines} (which implies \\eqref{eqInLem:TSPTbound1}), and the serial-processing nature of the $PD$ environment (which implies \\eqref{eqInLem:TSPTbound2}).\n\\begin{align}\n& C^{CC}_{\\sigma(j)i} \\leq p_{\\sigma(j)i1}\\left(1\/\\bar{v} - 1\/\\mu_i\\right) + \\textstyle\\sum_{\\ell = 1}^{j} p_{\\sigma(\\ell)i}\/\\mu_i  \\label{eqInLem:TSPTbound1} \\\\\n& \\textstyle\\sum_{\\ell = 1}^j p_{\\sigma(\\ell)i}\/\\mu_i \\leq C^{PD,I'}_{\\sigma(j')} \\quad \\forall ~i \\in M \\label{eqInLem:TSPTbound2}\n\\end{align}\nIf we relax the bound given in Inequality \\eqref{eqInLem:TSPTbound1} and combine it with Inequality \\eqref{eqInLem:TSPTbound2}, we see that $C^{CC}_{\\sigma(j)i} \\leq C^{PD,I'}_{\\sigma(j')}  +  p_{\\sigma(j)i1}\/\\bar{v}$. \nThe last step is to replace the final term with something more meaningful. Using $p_{\\sigma(j)1}\/\\bar{v} \\leq R \\cdot C^{CC\\star}_{\\sigma(j)}$ (which is immediate from the definition of $R$) the desired result follows.\n\\end{proof}\nWhile Lemma \\ref{lem:TSPTboundInTermsOfPDandOPT} is true for arbitrary $\\sigma$, now we consider $\\sigma = MUSSQ(X, w)$. \nThe proof of MUSSQ's correctness established the first inequality in the chain of inequalities below. \nThe second inequality can be seen by substituting $p_{ji} \/ \\mu_{i}$ for $x_{ji}$  in LP0($I'$) (this shows that the constraints in LP0($I'$) are weaker than those in LP1($I$)). \nThe third inequality follows from the Validity Lemma.\n\\begin{equation}\n\\textstyle\\sum_{j \\in N} w_{\\sigma(j)} C^{PD,I'}_{\\sigma(j)} \n\t\\leq 2 \\textstyle\\sum_{j \\in N} w_j C^{\\text{LP0}(I')}_j\n\t\\leq 2 \\textstyle\\sum_{j \\in N} w_j C^{\\text{LP1}(I)}_j\n\t\\leq 2 OPT(I) \\label{eq:tsptCore}\n\\end{equation}\nCombining Inequality \\eqref{eq:tsptCore} with Lemma \\ref{lem:TSPTboundInTermsOfPDandOPT} allows us to bound the objective in a way that does not make reference to $I'$.\n\\begin{equation}\n\\textstyle\\sum_{j \\in N} w_{\\sigma(j)}C^{CC}_{\\sigma(j)} \n\t\\leq \\textstyle\\sum_{j \\in N} w_{\\sigma(j)}\\left[C^{PD,I'}_{\\sigma(j)} + R\\cdot C^{CC\\star}_{\\sigma(j)}\\right] \\leq ~ 2 \\cdot OPT(I) + R \\cdot OPT(I) \\label{eq:dontReferenceIPrime}\n\\end{equation}\nInequality \\eqref{eq:dontReferenceIPrime} completes our proof of the following theorem.\n\\begin{theorem}\nAlgorithm CC-TSPT is a $2 + R$ approximation for $CC || \\sum w_j C_j$.\n\\label{thm:algCCTspt}\n\\end{theorem}\n\n\\subsection{CC-TSPT with Unit Tasks and Identical Machines}\\label{subsec:tsptOnUnitTasks}\nConsider concurrent cluster scheduling with $v_{\\ell i} = p_{jit} = 1$ (i.e., all processing times are unit, although the size of the collections $T_{ji}$ are unrestricted). In keeping with the work of Zhang, Wu, and Li \\cite{zwl} (who studied this problem in the single-cluster case), we call instances with these parameters ``fully parallelizable,'' and write $\\beta = fps$ for Graham's $\\alpha|\\beta|\\gamma$ taxonomy.\n\nZhang et al. showed that scheduling jobs greedily by  ``Largest Ratio First'' (decreasing $w_j \/ p_{j}$) results in a 2-approximation, where 2 is a tight bound. \nThis comes as something of a surprise since the Largest Ratio First policy is \\textit{optimal} for $1||\\sum w_j C_j~$- which their problem very closely resembles. \nWe now formalize the extent to which $P|fps|\\sum w_j C_j$ resembles $1||\\sum w_j C_j~$: define the \\textit{time resolution} of an instance $I$ of $CC |fps| \\sum w_jC_j$ as $ \\rho_I = \\min_{j \\in N, i \\in M}{\\big\\lceil{p_{ji}\/m_i}\\big\\rceil}$. \nIndeed, one can show that as the time resolution increases, the performance guarantee for LRF on $P | fps | \\sum w_j C_j$ approaches that of LRF on $1||\\sum w_j C_j$. \nWe prove the analogous result for our problem. \n\\begin{theorem}\nCC-TSPT for $CC |fps| \\sum w_jC_j$ is a $(2 + 1\/\\rho_I)-$approximation.\n\\label{thm:tspt_unit_tasks}\n\\end{theorem}\n\\begin{proof}\nApplying techniques from the proof of Lemma \\ref{lem:TSPTboundInTermsOfPDandOPT} under the hypothesis of this theorem, we have $C^{CC}_{\\sigma(j), i} \\leq C^{PD,I'}_{\\sigma(j)} + 1$. \nNext, use the fact that for all $j \\in N$, $C^{CC,OPT}_{\\sigma(j)} \\geq \\rho_I$ by the definition of $\\rho_I$. These facts together imply\n$C^{CC}_{\\sigma(j), i} \\leq C^{PD,I'}_{\\sigma(j)} + C^{CC,OPT} \/ \\rho_I$. Thus\n\\begin{align}\n\\textstyle\\sum_{j \\in N} w_j C^{CC}_{\\sigma(j)} \n\t&\\leq \\textstyle\\sum_{j \\in N} w_j \\left[C^{PD,I'}_{\\sigma(j)} + C^{CC,OPT} \/ \\rho_I\\right] \\leq 2 \\cdot OPT + OPT \/ \\rho_I.\n\\end{align}\n\\end{proof}\n\n\\subsection{CC-ATSPT : Augmenting the LP Relaxation}\nThe proof of Theorem \\ref{thm:algCCTspt} appeals to a trivial lower bound on $C^{CC\\star}_{\\sigma(j)}$, namely $p_{\\sigma(j)1}\/\\bar{v} \\leq R \\cdot C^{CC\\star}_{\\sigma(j)}$. We attain constant-factor performance guarantees in spite of this, but it is natural to wonder how the \\textit{need} for such a bound might come hand-in-hand with empirical weaknesses. Indeed, TSPT can make subjobs consisting of many small tasks look the same as subjobs consisting of a single very long task.\nAdditionally, a cluster hosting a subjob with a single extremely long task might be identified as a bottleneck by MUSSQ, even if that cluster has more machines than it does tasks to process.\n\nWe would like to mitigate these issues by introducing the simple lower bounds on $C_j$ as seen in constraints $(1B)$ and $(1C)$. This is complicated by the fact that MUSSQ's proof of correctness only allows constraints of the form in $(1A)$. For $I \\in \\Omega_{PD}$ this is without loss of generality, since $|S| = 1$ in LP0 implies $C_j \\geq p_{ji}$, but since we apply LP0 to $I' = TSPT(I)$, $C_j \\geq x_{ji}$ is equivalent to $C_j \\geq p_{ji}\/\\mu_i$ (a much weaker bound than we desire). \n\nNevertheless, we can bypass this issue by introducing additional clusters and appropriately defined subjobs. We formalize this with the ``Augmented Total Scaled Processing Time'' (ATSPT) transformation. \nConceptually, ATSPT creates $n$ ``imaginary clusters'', where each imaginary cluster has nonzero processing time for exactly one job.\n\\begin{definition}[The Augmented TSPT Transformation]\nLet $\\Omega_{CC}$ and $\\Omega_{PD}$ be as in the definition for TSPT. Then the Augmented TSPT Transformation is likewise a mapping\n\\begin{align*}\nATSPT: ~\\Omega_{CC} \\to \\Omega_{PD} \\quad \\text{ with } \\quad (T, v, w) &\\mapsto (X, w) ~:~ X = \\big[\\begin{array}{c|c} X_{TSPT(I)} & D \\end{array}\\big].\n\\end{align*}\nWhere $D \\in \\mathbb{R}^{n \\times n}$ is a diagonal matrix with $d_{jj}$ as any valid lower bound on the completion time of job $j$ (such as the right hand sides of constraints ($1B$) and ($1C$) of LP1).\n\\end{definition}\nGiven that $d_{jj}$ is a valid lower bound on the completion time of job $j$, it is easy to verify that for $I' = ATSPT(I)$, LP1($I'$) is a valid relaxation of $I$. \nBecause MUSSQ returns a permutation of jobs for use in list scheduling by List-LPT, these ``imaginary clusters'' needn't be accounted for beyond the computations in MUSSQ.\n\n\\section{A Reduction for Minimizing Total Weighted Lateness on Identical Parallel Machines }\\label{sec:relationshipsBetweenProbs}\nThe problem of minimizing total weighted lateness on a bank of identical parallel machines is typically denoted $P || \\sum w_jL_j$, where the lateness of a job with deadline $d_j$ is $L_j \\doteq \\max{\\{C_j - d_j, 0\\}}$. The reduction we offer below shows that $P || \\sum w_j L_j$ can be stated in terms of $CC || \\sum w_jC_j$ \\textit{at optimality}. Thus while a $\\Delta$ approximation to $CC || \\sum w_jC_j$ does not imply a $\\Delta$ approximation to $P || \\sum w_j L_j$, the reduction below nevertheless provides new insights on the structure of $P || \\sum w_j L_j$.\n\n\\begin{definition}[Total Weighted Lateness Reduction]\nLet $I = (p, d, w, m)$ denote an instance of $P || \\sum w_j L_j$. \n$p$ is the set of processing times, $d$ is the set of deadlines, $w$ is the set of weights, \nand $m$ is the number of identical parallel machines. \nGiven these inputs, we transform $I \\in \\Omega_{P || \\sum w_j L_j}$ \nto $I' \\in \\Omega_{CC}$ in the following way.\n\nCreate a total of $n + 1$ clusters. Cluster 0 has $m$ machines. Job $j$ has processing time $p_j$ on this cluster, and $|T_{j0}| = 1$. Clusters 1 through $n$ each consist of a single machine. Job $j$ has processing time $d_j$ on cluster $j$, and zero on all clusters other than cluster 0 and cluster $j$. Denote this problem $I'$.\n\\end{definition}\nWe refer the reader to Figure \\ref{fig:probstm3and4} for an example output of this reduction.\n\\begin{theorem}\nLet $I$ be an instance of $P || \\textstyle\\sum w_j L_j$. Let $I'$ be an instance of $CC|| \\sum w_j C_j$ resulting from the transformation described above. Any list schedule $\\sigma$ that is optimal for $I'$ is also optimal for $I$.\n\\end{theorem}\n\\begin{proof}\nIf we restrict the solution space of $I'$ to single permutations (which we may do without loss of generality), then any schedule $\\sigma$ for $I$ or $I'$ produces the same value of $\\sum_{j \\in N}  w_j(C_j - d_j)^+$ for $I$ and $I'$.\nThe additional clusters we added for $I'$ ensure that $C_j \\geq d_j$. Given this, the objective for $I$ can be written as $\\sum_{j \\in N} w_j d_j + w_j(C_j - d_j)^+$. Because $w_j d_j$ is a constant, any permutation to solve $I'$ optimally also solves $\\sum_{j \\in N} w_j (C_j - d_j)^+$ optimally. Since $\\sum_{j \\in N} w_j (C_j - d_j)^+ = \\sum_{j \\in N} w_j L_j$, we have the desired result.\n\\end{proof}\n\\section{Closing Remarks}\\label{sec:discAndConc}\nWe now take a moment to address a subtle issue in the concurrent cluster problem: what price do we pay for using the same permutation on all clusters (i.e. single-$\\sigma$ schedules)? For concurrent open shop, it has been shown (\\cite{Sris1993, mqssu}) that single-$\\sigma$ schedules may be assumed without loss of optimality. As is shown in Figure \\ref{fig:singleVsMultiPerm}, this does \\textit{not} hold for concurrent cluster scheduling in the general case. In fact, that is precisely why the strong performance guarantees for algorithm CC-LP rely on clusters having possibly unique permutations.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.7\\textwidth]{singleVsMultiPerm.png}\n\\caption{An instance of $CC||\\sum C_j$ (i.e. $w_j \\equiv 1$) for which there does not exist a single-$\\sigma$ schedule which attains the optimal objective value. In the single-$\\sigma$ case, one of the jobs necessarily becomes delayed by one time unit compared to the multi-$\\sigma$ case. As a result, we see a 20\\% optimality gap even when $v_{\\ell i } \\equiv 1$.}\\label{fig:singleVsMultiPerm}\n\\centering\n\\end{figure}\n\nOur more novel contributions came in our analysis for CC-TSPT and CC-ATSPT. First, we could not rely on the processing time of the last task for a job to be bounded above by the job's completion time variable $C_j$ in LP0($I'$), and so we appealed to a lower bound on $C_j$ that was not stated in the LP itself. The need to incorporate this second bound is critical in realizing the strength of algorithm CC-TSPT, and uncommon in LP rounding schemes. Second, CC-ATSPT is novel in that it introduces constraints that would be redundant for LP0($I$) when $I \\in \\Omega_{PD}$, but become relevant when viewing $LP0(I')$ as a relaxation for $I \\in \\Omega_{CC}$. This approach has potential for more broad applications since it represented effective use of a limited constraint set supported by a known primal-dual algorithm.\n\nWe now take a moment to state some open problems in this area. One topic of ongoing research is developing a factor 2 purely combinatorial algorithm for the special case of concurrent cluster scheduling considered in Theorem \\ref{thm:identLP_2appxWithConstantTasks}. In addition, it would be of broad interest to determine the worst-case loss to optimality incurred by assuming single-permutation schedules for $CC|v\\equiv 1|\\sum w_j C_j$. The simple example above shows that an optimal single-$\\sigma$ schedule can have objective 1.2 times the globally optimal objective. Meanwhile, Theorem \\ref{thm:algCCTspt} shows that there always exists a single-$\\sigma$ schedule with objective no more than 3 times the globally optimal objective. Thus, we know that the worst-case performance ratio is in the interval $[1.2,3]$, but we do not know its precise value. As a matter outside of scheduling theory, it would be valuable to survey primal-dual algorithms with roots in LP relaxations to determine which have constraint sets that are amenable to implicit modification, as in the fashion of CC-ATSPT.\n\n\\subparagraph*{Acknowledgments.}\n\nSpecial thanks to Andreas Schulz for sharing some of his recent work with us \\cite{Schulz2012}. His thorough analysis of a linear program for $P||\\sum w_j C_j$ drives the LP-based results in this paper. Thanks also to Chien-Chung Hung and Leana Golubchik for sharing \\cite{HGY} while it was under review, and to Ioana Bercea and Manish Purohit for their insights on SWAG's performance. Lastly, our sincere thanks to William Gasarch for organizing the REU which led to this work, and to the 2015 CAAR-REU cohort for making the experience an unforgettable one; in the words of Rick Sanchez \\textit{wubalubadubdub!}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe Standard Model is flawed by the large number of free parameters, for which there is at present no explanation.\nThere is no prediction of the family replication pattern, nor of the number of families. All the families are really treated on the same footing.\nMost of the Standard Model free parameters reside in ``flavour space'' - with six quark masses, six lepton masses, four quark mixing angles and ditto for the leptonic sector, as well as the strong CP-violating parameter $\\bar{\\Theta}$. \nThe structure of flavour space is determined by the fermion mass matrices, i.e. by the form that the mass matrices take in the ``weak interaction basis'' where mixed fermion states interact weakly, in contrast to the ``mass bases'', where the mass matrices are diagonal.\n\nOne may wonder how one may ascribe such importance to the different bases in flavour space, considering that\nthe information content of a matrix is contained in its matrix invariants, which in the case of a $N\\times N$ matrix $M$ are the $N$ sums and products of the eigenvalues $\\lambda_j$, such as $trace M$, $detM$,\n\\begin{equation}\n  \\def1.1{1.1}\n  \\begin{array}{r@{\\;}l} \nI_1 = &\\sum_j\\lambda_j                     = \\lambda_1+\\lambda_2+\\lambda_3...\\\\\n\nI_2 = &\\sum_{jk}\\lambda_j\\lambda_k          = \\lambda_1\\lambda_2+\\lambda_1\\lambda_3+\\lambda_1\\lambda_4+... \\\\\n\nI_3 = &\\sum_{jkl}\\lambda_j\\lambda_k\\lambda_l = \\lambda_1\\lambda_2\\lambda_3+\\lambda_1\\lambda_2\\lambda_4+...\\\\\n      & \\vdots\\\\\nI_N = &\\lambda_1\\lambda_2 \\cdots \\lambda_N \n\\end{array}\n\\end{equation} \nThese expressions are invariant under permutations of the eigenvalues, which in the context of mass matrices means that they are flavour symmetric, and obviously independent of any choice of flavour space basis.\n\nEven if the information content of a matrix is contained in its invariants,\nthe form of a matrix may also carry information, albeit of another type. The idea - the hope - is that the form that the mass matrices have in the weak interaction basis can give some hint about the origin of the unruly masses. There is a certain circularity to this reasoning; to make a mass matrix ansatz is in fact to define what we take as the weak interaction basis in flavour space.\nWe denote the quark mass matrices of the up- and down-sectors in the weak interaction basis by $M$ and $M'$, respectively. \nWe go from the weak interaction basis to the mass bases by rotating the matrices by the unitary matrices $U$ and $U'$,\n\\begin{equation}\\label{mss}\nM \\rightarrow UMU^{\\dagger} = D = diag(m_u,m_c,m_t)\n\\end{equation}\n\\[\nM' \\rightarrow U'M'U'^{\\dagger} = D' = diag(m_d,m_s,m_b)\\\\\n\\]\n  \\begin{figure}[htb]\n    \\begin{center}\n    \\includegraphics[scale=0.87]{b.png}\n\\end{center}\n    \\end{figure}\n\nThe lodestar in the hunt for the right mass matrices is the family hierarchy, with two lighter particles in the first and second family, and a much heavier particle in the third family. This hierarchy is present in all the charged sectors, with fermions in different families exhibiting very different mass values, ranging from the electron mass to the about $10^{5}$ times larger top mass. It is still an open question whether the neutrino masses also follow this pattern \\cite{neutrino hier}.\n\n\n\n\n\\section{Democratic mass matrices}\nIn the democratic approach  \\cite{demo}, \\cite{koide}, \\cite{Fritzsch} the family hierarchy is taken very seriously. It is assumed that in the weak basis the fermion mass matrices have a form close to the $S(3)_L  \\times  S(3)_R $ symmetric ``democratic'' matrix \n\\begin{equation}\\label{nambu}\n{\\bf{N}}= k\\begin{pmatrix}\n  1 & 1 & 1\\\\\n  1 & 1 & 1\\\\\n  1 & 1 & 1\\\\\n\\end{pmatrix}\n\\end{equation}\nwith the eigenvalues $(0,0,3k)$, reflecting the family hierarchy.\n\nThe underlying philosophy is that in the Standard Model, where the fermions get their masses from the Yukawa couplings by the Higgs mechanism, there is no reason why there should be a different Yukawa coupling for each fermion. \nThe couplings to the gauge bosons of the strong, weak and\nelectromagnetic interactions are identical for all the fermions in a given charge sector, it thus seems like a natural assumption that they should also have identical Yukawa couplings.\nThe difference is that the weak interactions take place in a specific flavour space basis, while the other interactions are flavour independent. \n\nThe democratic assumption is thus that the fermion fields of the same charge initially have the same Yukawa couplings. \nWith three families, the quark mass matrices in the weak interaction basis then have the (zeroth order) form\n\\begin{equation}\\label{nambu2}\nM^{(0)}= k_u\\begin{pmatrix}\n  1 & 1 & 1\\\\\n  1 & 1 & 1\\\\\n  1 & 1 & 1\\\\\n \\end{pmatrix},\\hspace{2cm} \nM'^{(0)}= k_d\\begin{pmatrix}\n  1 & 1 & 1\\\\\n  1 & 1 & 1\\\\\n  1 & 1 & 1\\\\\n \\end{pmatrix}\n\\end{equation}\nwhere $k_u$ and $k_d$ have dimension mass. \nThe corresponding mass spectra $(m_1,m_2,m_3) \\sim (0,0,3k_j)$ reflect\nthe family hierarchy with two light families and a third much heavier family, a mass hierarchy that can be interpreted as the representation ${\\bf{1}}\\oplus {\\bf{2}}$ of $S(3)$.\nIn order to obtain realistic mass spectra with non-zero masses, the $S(3)_L  \\times  S(3)_R $ symmetry must obviously be broken, \nand the different democratic matrix ans\\\"{a}tze correspond to different schemes for breaking the democratic symmetry.\n\n\\subsection{The lepton sector}\nWe can apply the democratic approach to the lepton sector as well, postulating democratic (zeroth order) mass matrices for the charged leptons and the neutrinos, whether they are Fermi-Dirac or Majorana states, \n\\begin{equation}\\label{leptons}\nM^{(0)}_l= k_l\\begin{pmatrix}\n  1 & 1 & 1\\\\\n  1 & 1 & 1\\\\\n  1 & 1 & 1\\\\\n \\end{pmatrix},\\hspace{2cm} \nM^{(0)}_{\\nu}= k_{\\nu}\\begin{pmatrix}\n  1 & 1 & 1\\\\\n  1 & 1 & 1\\\\\n  1 & 1 & 1\\\\\n \\end{pmatrix}\n\\end{equation}\nRelative to the quark ratio \n$k_u\/k_d \\sim m_t\/m_b \\sim  40 - 60$, the leptonic ratio $k_{\\nu}\/k_l < 10^{-8}$ is so extremely small that it seems unnatural. One way out is to simply assume that $k_{\\nu}$ vanishes, meaning that the neutrinos get no mass contribution in the democratic limit \\cite{Frit}. According to the democratic philosophy, then there would be no reason for a hierarchical pattern \\`{a} la the one observed in the charged sectors; the neutrino masses could even be of the same order of magnitude.\n\nData are indeed compatible with a much weaker hierarchical structure for the neutrino masses than the hierarchy displayed by the charged fermion masses.  \n\n\nUnlike the situation for the quark mixing angles, in lepton flavour mixing there are two quite large mixing angles and a third much smaller mixing angle, these large mixing angles can be interpreted as indicating weak hierachy of the neutrino mass spectrum. The neutrino mass spectrum hierarchy could even be inverted; \nif the solar\nneutrino doublet $(\\nu_1,\\nu_2)$ has a mean mass larger than the remaining\natmospheric neutrino $\\nu_3$, the hierarchy is called\n\"inverted\", otherwise it is called \"normal\". \n\nSupposing that the neutrino masses do not emerge from a democratic scheme, a (relatively) flat neutrino mass spectrum could be taken as a support for the idea that the masses in the charged sectors emerge from a democratic scheme.\n\n\n\n\\section{The democratic basis}\nIn the case that both the up- and down-sector mass matrices have a purely democratic texture, the quark mixing matrix is $V = UU'^{\\dagger} = U_{dem}U_{dem}^{\\dagger} = {\\bf{1}}$, where  \n\\begin{equation}\\label{dem}\nU_{dem} =\\frac{1}{\\sqrt{6}}\n \\begin{pmatrix}\n  \\sqrt{3} & -\\sqrt{3} & \\hspace{2mm} 0 \\\\\n  1        &         1 & -2 \\\\\n  \\sqrt{2} & \\sqrt{2}  & \\sqrt{2}\n \\end{pmatrix}\n\\end{equation}\nis the unitary matrix that diagonalizes the democratic matrix (\\ref{nambu}).\n \nWe use this to define the democratic basis, meaning the flavour space basis where the mass matrices are diagonalized by (\\ref{dem}) and the mass Lagrangian is symmetric under permutations of the fermion fields $(\\varphi_1,\\varphi_2,\\varphi_3)$ of a given charge sector.\n\nIn the democratic basis the mass Lagrangian\n\\[\n{\\mathcal{L}}_m =\\bar{\\varphi} M_{(dem)} \\varphi =k\\sum_{jk=1}^3 \\bar{\\varphi}_j \\varphi_k\n\\]\nis symmetric under permutations of the fermion fields $(\\varphi_1,\\varphi_2,\\varphi_3)$, while \nin the mass basis\nwith \n\\[\nM_{(mass)} \n=\\begin{pmatrix}\n \\lambda_1\\\\\n& \\lambda_2\\\\\n&& \\lambda_3\\\\\n\\end{pmatrix}\n\\]\n the mass Lagrangian has the form\n \\begin{equation} \n{\\mathcal{L}}_m =\\lambda_1 \\bar{\\psi}_1 \\psi_1+\\lambda_2 \\bar{\\psi}_2 \\psi_2+\\lambda_3 \\bar{\\psi}_3 \\psi_3\n\\end{equation} \nwhich is clearly not invariant under permutations of ($\\psi_1$,$\\psi_3$,$\\psi_3$). \n\nWe can perform a shift of the democratic matrix, by just adding a unit matrix $diag(a,a,a)$, \n$M_0 \\rightarrow M_1$, \n\\begin{equation}\\label{pert}\nM_1 =\nk \\begin{pmatrix}\n  1 & 1 & 1\\\\\n  1 & 1 & 1\\\\\n  1 & 1 & 1\\\\\n \\end{pmatrix}+\n\\begin{pmatrix}\n a\\\\\n&a\\\\\n&&a\\\\\n\\end{pmatrix}=\n\\begin{pmatrix}\n  k+a & k   & k\\\\\n  k   & k+a & k\\\\\n  k   & k   & k+a\\\\\n \\end{pmatrix}\n\\end{equation}\ncorresponding to the mass spectrum $(a,a,3a+3k)$. The matrix\n$M_1$ has a democratic texture, both because it is diagonalized by $U_{dem}$, and because the mass Lagrangian is invariant under permutations of the quark fields,\n\\begin{equation}{\\mathcal{L}}_{M_1} = (k+a)\\sum \\bar{\\varphi}_j \\varphi_j+ k \\sum_{j\\neq k} \\bar{\\varphi}_j \\varphi_j\n\\end{equation}\n\nIf $M_1$ and $M'_1$ both have a texture like (\\ref{pert}), there is no CP-violation. This is independent of how many families there are, because of the degeneracy of the mass values. \nCP-violation only occurs once there are three or more non-degenerate families, because only then the phases can no longer be defined away.\n\nWe can repeat the democratic scheme with a number $n$ of families, where the fermion mass matrices again are proportional to the $S(n)_L  \\times  S(n)_R $ symmetric democratic matrix \nwhich is diagonalized by a unitary matrix analogous to $U_{dem}$ in (\\ref{dem}). \nTo the $n\\times n$-dimensional democratic matrix term, we can again add a $n\\times n$-dimensional diagonal matrix $diag(a,a,...,a)$, and get a $n\\times n$-dimensional mass spectrum \nwith $n$ massive states, and $n-1$ degenerate masses.\nThe mass matrix still has a democratic texture, and there is still no CP-violation.\n\n\n\\section{Breaking the democratic symmetry}\nIn order to obtain non-degenerate, non-vanishing masses for the physical flavours $(\\psi_1,\\psi_2,\\psi_3)$, the permutation symmetry of the fermion fields $(\\varphi_1,\\varphi_2,\\varphi_3)$ in the democratic basis must be broken. The proposal here is to derive mass matrices with a nearly democratic texture, not by explicitly perturbing the assumed initial democratic form (\\ref{nambu}), but instead by perturbing the matrix $U_{dem}$ which diagonalises the democratic mass matrix. This is done by deriving the unitary rotation matrices $U$, $U'$ for the up- and down- sectors, from a specific parameterisation of the weak mixing matrix $V = UU^{'\\dagger}$.\n\nThe idea is to embed the assumption of democratic symmetry into the Standard Model mixing matrix, by expressing the mixing matrix as a product\n\\begin{equation}\\label{UU}  \nV=UU'^{\\dagger}=  ({\\tilde{U}}U_{dem})(U_{dem}^{\\dagger}{\\tilde{U'}}^{\\dagger})\n\\end{equation}  \nSince both the mixing matrix and its factors, according to the standard parameterisation \\cite{Stand}, are so close to the unit matrix, the rotation matrices $U$, $U'$ are effectively perturbations of the democratic diagonalising matrix (\\ref{dem}). In this way, the weak interaction basis remains close to the democratic basis.\n\n\n\n\\subsection{Factorizing the mixing matrix}   \nThe Cabbibo-Kobayashi-Maskawa (CKM) mixing matrix \\cite{CKM} can of course be parametrized - and factorized - in many different ways, and different factorizations correspond to different\nrotation matrices $U$ and $U'$.\nThe most obvious and ``symmetric'' factorization of the CKM mixing matrix is, following the standard parametrization \\cite{Stand} with three Euler angles $\\alpha$, $\\beta$, $2\\theta$,\n\\begin{equation}\nV =\\begin{pmatrix}\n  c_{\\beta} c_{2\\theta}                                     &s_{\\beta} c_{2\\theta}                                     &s_{2\\theta} e^{-i\\delta}\\\\\n-c_{\\beta} s_{\\alpha} s_{2\\theta} e^{i\\delta}-s_{\\beta}c_{\\alpha}&-s_{\\beta}s_{\\alpha}s_{2\\theta} e^{i\\delta}+c_{\\beta}c_{\\alpha}&s_{\\alpha}c_{2\\theta}\\\\\n-c_{\\beta} c_{\\alpha} s_{2\\theta} e^{i\\delta}+s_{\\beta}s_{\\alpha}&-s_{\\beta}c_{\\alpha}s_{2\\theta} e^{i\\delta}-c_{\\beta}s_{\\alpha}&c_{\\alpha}c_{2\\theta}\\\\\n\\end{pmatrix}= UU^{'\\dagger}\n\\end{equation} \nwith the diagonalizing rotation matrices for the up- and down-sectors \n\\begin{equation}\\label{diagu}\nU =\n \\begin{pmatrix}\n  1 & 0          & 0 \\\\\n  0 &\\cos\\alpha  &\\sin\\alpha \\\\\n  0 &-\\sin\\alpha &\\cos\\alpha \\\\\n \\end{pmatrix}\n\\begin{pmatrix}\n  e^{-i\\gamma} \\\\\n  & 1\\\\\n  && e^{i\\gamma} \\\\\n \\end{pmatrix}\n\\begin{pmatrix}\n  \\cos\\theta     & 0          & \\sin\\theta \\\\\n  0              & 1          & 0\\\\\n  -\\sin\\theta    & 0          & \\cos\\theta \\\\\n \\end{pmatrix}\n \\begin{pmatrix}\n \\frac{1}{\\sqrt{2}}  & -\\frac{1}{\\sqrt{2}} & \\hspace{2mm} 0 \\\\\n  \\frac{1}{\\sqrt{6}} &\\frac{1}{\\sqrt{6}}   & -\\frac{2}{\\sqrt{6}} \\\\\n  \\frac{1}{\\sqrt{3}} &\\frac{1}{\\sqrt{3}}   & \\frac{1}{\\sqrt{3}}\n \\end{pmatrix}\n\\end{equation}\nand\n\\[\nU' =\n \\begin{pmatrix}\n  \\cos\\beta  &-\\sin\\beta &0\\\\\n  \\sin\\beta  &\\cos\\beta  &0\\\\\n   0         & 0         &1\n \\end{pmatrix}\n\\begin{pmatrix}\n  e^{-i\\gamma} \\\\\n  & 1\\\\\n  && e^{i\\gamma} \\\\\n \\end{pmatrix}\n\\begin{pmatrix}\n  \\cos\\theta     & 0          & -\\sin\\theta \\\\\n  0              & 1          & 0\\\\\n   \\sin\\theta    & 0          & \\cos\\theta \\\\\n \\end{pmatrix}\n \\begin{pmatrix}\n \\frac{1}{\\sqrt{2}}  & -\\frac{1}{\\sqrt{2}} & \\hspace{2mm} 0 \\\\\n  \\frac{1}{\\sqrt{6}} &\\frac{1}{\\sqrt{6}}   & -\\frac{2}{\\sqrt{6}} \\\\\n  \\frac{1}{\\sqrt{3}} &\\frac{1}{\\sqrt{3}}   & \\frac{1}{\\sqrt{3}}\n \\end{pmatrix},\n\\]\nrespectively,\nwhere $\\alpha$, $\\beta$, $\\theta$ and $\\gamma$ correspond to the parameters in the standard parametrization in such a way that\n$\\gamma = \\delta\/2$, $\\delta = 1.2 \\pm 0.08$ rad, and $2\\theta = 0.201 \\pm 0.011 ^{\\circ}$, while  \n$\\alpha = 2.38 \\pm 0.06 ^{\\circ}$ \nand $\\beta = 13.04\\pm 0.05 ^{\\circ}$.\n\nFrom the rotation matrices $U$ and $U'$ we then obtain the mass matrices \n$M={U^{\\dagger}}diag(m_u,m_c,m_t)U$ and \n$M'={U'^{\\dagger}}diag(m_d,m_s,m_b)U'$, such that  \n\\begin{equation}\\label{1}\nM=\\frac{1}{6}\\begin{pmatrix}\nX+H                        &\\hat{M}_{12}                    &Z+W\\\\\n\\hat{M}_{12}^*\\hspace{0.5mm}      &X-H                      &Z-W\\\\\n\\hspace{0.5mm}Z^*+W^*      &\\hspace{2mm}Z^*-W^*      &6T-2X\\\\\n  \\end{pmatrix} \n\\end{equation}\nwhere $T$ is the trace $T= m_u+m_c+m_t$, and with $D= \\sqrt{3}s_{\\theta}  -\\sqrt{2}c_{\\theta}$, \n$ C= \\sqrt{3}s_{\\theta}  +\\sqrt{2}c_{\\theta}$, \n$ F= c_{\\alpha}s_{\\alpha}(m_t-m_c)$,\n\\begin{description}\n\\item $ X=\\frac{1}{2}(m_cs_{\\alpha}^2+m_tc_{\\alpha}^2-m_u)(D^2+C^2-2)+F(D-C)\\cos\\gamma+T+3m_u$\n\\item $ H=\\frac{1}{2}(m_cs_{\\alpha}^2+m_tc_{\\alpha}^2-m_u)(D^2-C^2)+F\\cos\\gamma(D+C)$\n\\end{description}\n\\begin{description}\n\\item $W = \\frac{1}{4}(m_cs_{\\alpha}^2+m_tc_{\\alpha}^2-m_u)\\hspace{1mm}(D^2-C^2)-F\\hspace{1mm}(D+C)\\hspace{1mm}e^{-i\\gamma}$\n\\item $Z = (m_cs_{\\alpha}^2+m_tc_{\\alpha}^2-m_u)\\left[2+\\frac{1}{4}(D-C)^2\\right]\\hspace{1mm}+\\frac{F}{2}\\hspace{1mm}(D-C)\\hspace{1mm}(e^{i\\gamma}-2\\hspace{1mm}e^{-i\\gamma})-2T+6\\hspace{1mm}m_u$\n\\item $\\hat{M}_{12}= -(m_cs_{\\alpha}^2+m_tc_{\\alpha}^2-m_u)\\hspace{1mm}(D\\hspace{1mm}C+1)-F\\hspace{1mm}(C\\hspace{1mm}e^{i\\gamma}-D\\hspace{1mm}e^{-i\\gamma})+T-3\\hspace{1mm}m_u$ \n\\end{description}\nSimilarly for the down-sector,\n\\begin{equation}\\label{11}\nM'=\\frac{1}{6}\\begin{pmatrix}\nX'+H'                        &\\hat{M}'_{12}                    &Z'+W'\\\\\n\\hat{M}_{12}^{'*}\\hspace{0.5mm}      &X'-H'                      &Z'-W'\\\\\n\\hspace{0.5mm}Z^{'*}+W^{'*}      &\\hspace{2mm}Z^{'*}-W^{'*}      &6T'-2X'\\\\\n  \\end{pmatrix} \n\\end{equation}\nwith the parameters $T'= m_d+m_s+m_b$, $ G= \\sqrt{2}s_{\\theta}  -\\sqrt{3}c_{\\theta}$, $ J= \\sqrt{2}s_{\\theta}  +\\sqrt{3}c_{\\theta} $,\n\\begin{description}\n\\item$ F'= c_{\\beta}s_{\\beta}(m_b-m_s)$, and \n\\item $X'= \\frac{1}{2}(m_ss_{\\beta}^2+m_bc_{\\beta}^2-m_d)(G^2+J^2-2)-F'(J+G)\\cos\\gamma+T'+3m_b$\n\\item $H'= \\frac{1}{2}(m_ss_{\\beta}^2+m_bc_{\\beta}^2-m_d)(G^2-J^2)+F'(J-G)\\cos\\gamma$\n\\item $W'= \\frac{1}{4}(m_ss_{\\beta}^2+m_bc_{\\beta}^2-m_d)(G^2-J^2)+F'(G-J)e^{i\\gamma}$\n\\item $Z'=(m_ss_{\\beta}^2+m_bc_{\\beta}^2-m_d)\\left[2+\\frac{1}{4}(J+G)^2\\right]+\\frac{F'}{2}(J+G)(2e^{i\\gamma}-e^{-i\\gamma})-2T'+6m_b$\n\\item $\\hat{M}'_{12}= (m_ss_{\\beta}^2+m_bc_{\\beta}^2-m_d)\\hspace{1mm}(G\\hspace{1mm}J-1)-F'\\hspace{1mm}(J\\hspace{1mm}e^{i\\gamma}-G\\hspace{1mm}e^{-i\\gamma})+T'-3\\hspace{1mm}m_b$ \n\\end{description}\n\nIn order to evaluate to what degree these rather opaque matrices are democratic, we calculate numerical matrix elements by inserting numerical mass values.\nFor the up-sector\nwe get the (nearly democratic) matrix texture\n\\begin{equation}\\label{CU}\nM=C_u\\left[\\begin{pmatrix}\n  1\\\\ \n  & k\\hspace{1mm}e^{-i(\\mu+\\rho)} \\\\\n  &&kp\\hspace{1mm} e^{-i\\mu} \\\\\n \\end{pmatrix}\n\\begin{pmatrix}\n1&1&1\\\\\n1&1&1\\\\\n1&1&1\\\\\n\\end{pmatrix}\n\\begin{pmatrix}\n  1\\\\\n  & k\\hspace{1mm}e^{i(\\mu+\\rho)} \\\\\n  &&kp\\hspace{1mm} e^{i\\mu} \\\\\n\\end{pmatrix}+\\Lambda \\right]\n\\end{equation}\nwhere the ``small'' matrix\n\\[\n\\Lambda=\n\\begin{pmatrix}\n0   &                   0&   0\\\\\n0   &\\varepsilon         &\\varepsilon'e^{-i\\rho}\\\\\n0   &\\varepsilon'e^{i\\rho}&\\eta\\\\\n\\end{pmatrix},\n\\]\nwith \n$\\varepsilon \\sim \\varepsilon' \\ll \\eta < k, p$, is what breaks the democratic symmetry, supplying the two lighter families with non-zero masses. With mass values calculated at $\\mu = M_Z$ (Jamin 2014) \\cite{Jamin}, \n\\begin{equation}\\label{jam1}\n(m_u(M_Z),m_c(M_Z),m_t(M_Z)) =(1.24, 624 , 171550 ) MeV,\n\\end{equation}\nwe get \n\\begin{description}\n\\item$\\mu \\sim 2.7895^o$,\\hspace{1mm}  $\\rho \\sim 2.7852^o$,\\hspace{1mm}$C_u=54240.36$ MeV $\\approx m_t\/3 $,\nand\n\\item$k\\approx 1.00438$,\\hspace{1mm} $p\\approx 1.06646$,\\hspace{1mm}$\\varepsilon' \\approx 5.05\\hspace{1mm} 10^{-5}$,\n\\item$\\varepsilon \\approx 4.6\\hspace{1mm} 10^{-5} \\approx 2\\frac{m_u}{C_u}$,\\hspace{1mm}$\\eta = 1.815\\hspace{1mm} 10^{-2}\\approx \\frac{1}{2}\\frac{m_t}{C_u}\\frac{m_c}{C_u}$.\n\\end{description}\nFor the down-sector, with \n\\begin{equation}\\label{jam2}\n(m_d(M_Z), m_s(M_Z), m_b(M_Z))=(2.69, 53.8, 2850) MeV\n\\end{equation}\nwe get another democratic texture,\n\\begin{equation}\nM'=C_d\n\\begin{pmatrix}\nX+A  &  Ye^{-i\\tau}   &\\hspace{2mm} e^{-i\\nu}\\\\    \nYe^{i\\tau}   &  X-A  & (1+2A)e^{i\\kappa}\\\\    \ne^{i\\nu}   & (1+2A)e^{-i\\kappa}  &\\hspace{2mm} X+Y-A-1\\\\    \n\\end{pmatrix}\n\\end{equation}\nwhere \n\\begin{description}\n\\item$C_d=966.5 MeV$, $A=5.6\\hspace{1mm}10^{-3}$, $X=1.0362$, $Y=1.0305$, and\n$\\tau \\leq \\kappa \\sim 0.22^o$ $<$ $\\nu \\sim 0.226^o$.\n\\end{description}\nJust like in the up-sector mass matrix, the matrix elements in $M'$ display a nearly democratic texture. For both the up-sector and the down-sector the mass matrices are thus approximately democratic.\n\n\\section{Calculability}\nIn the mass matrix literature there is an emphasis on ``calculability''. The ideal is to obtain mass matrices that have a manageable form, but there is nothing that forces nature to serve us such user-friendly formalism. It is however tempting to speculate that there are relations between the elements that could make the democratic matrices more calculable, and in the search for matrices that are reasonably transparent and calculable, we look at a more radical factorization of the mixing matrix, viz. \n\\begin{equation}\\label{ddiagun}\nU =\n \\begin{pmatrix}\n  1 & 0          & 0 \\\\\n  0 &\\cos\\alpha  &\\sin\\alpha \\\\\n  0 &-\\sin\\alpha &\\cos\\alpha \\\\\n \\end{pmatrix}\n\\begin{pmatrix}\n  \\cos\\omega                & 0          & \\sin\\omega \\hspace{1mm}e^{-i\\delta} \\\\\n  0                         & 1          & 0\\\\\n  -\\sin\\omega \\hspace{1mm}e^{ i\\delta}    & 0          & \\cos\\omega \\\\\n \\end{pmatrix}\n \\begin{pmatrix}\n \\frac{1}{\\sqrt{2}}  & -\\frac{1}{\\sqrt{2}} & \\hspace{2mm} 0 \\\\\n  \\frac{1}{\\sqrt{6}} &\\frac{1}{\\sqrt{6}}   & -\\frac{2}{\\sqrt{6}} \\\\\n  \\frac{1}{\\sqrt{3}} &\\frac{1}{\\sqrt{3}}   & \\frac{1}{\\sqrt{3}}\n \\end{pmatrix}\n\\end{equation}\nand\n\\[\nU' =\n \\begin{pmatrix}\n  \\cos\\beta  &-\\sin\\beta &0\\\\\n  \\sin\\beta  &\\cos\\beta  &0\\\\\n   0         & 0         &1\n \\end{pmatrix}\n \\begin{pmatrix}\n \\frac{1}{\\sqrt{2}}  & -\\frac{1}{\\sqrt{2}} & \\hspace{2mm} 0 \\\\\n  \\frac{1}{\\sqrt{6}} &\\frac{1}{\\sqrt{6}}   & -\\frac{2}{\\sqrt{6}} \\\\\n  \\frac{1}{\\sqrt{3}} &\\frac{1}{\\sqrt{3}}   & \\frac{1}{\\sqrt{3}}\n \\end{pmatrix}\n\\]\nwhere, as before, $\\delta = 1.2 \\pm 0.08$ rad, and $\\omega=2\\theta = 0.201 \\pm 0.011 ^{\\circ}$, while  \n$\\alpha = 2.38 \\pm 0.06 ^{\\circ}$, \nand $\\beta = 13.04\\pm 0.05 ^{\\circ}$.\nThese rotation matrices are still ``perturbations'' of the democratic diagonalizing matrix (\\ref{dem}), and\nthe up-sector mass matrix has a texture similar to (\\ref{1}),\n\\begin{equation}\\label{4}\nM=\n\\frac{1}{6}\\begin{pmatrix}\nR+Q+S\\hspace{1mm}\\cos\\delta             &R-Q-iS\\hspace{1mm}\\sin\\delta& A-Be^{-i\\delta}\\\\\nR-Q+iS\\hspace{1mm}\\sin\\delta&R+Q-S\\hspace{1mm}\\cos\\delta             & A+Be^{-i\\delta}\\\\\nA-Be^{i\\delta}                &A+Be^{i\\delta}               &T-2(R+Q)\\\\\n  \\end{pmatrix} \n\\end{equation}\nwhere $T$ is the trace, $T= m_u+m_c+m_t$, and\n\\begin{description}\n\\item $R= N \\hspace{1mm}(2\\hspace{1mm}c_{\\omega}^2-1)+T-2\\hspace{1mm}\\sqrt{2}\\hspace{1mm}c_{\\omega}\\hspace{1mm}F$, \\hspace{1mm} $Q= 3\\hspace{1mm}s_{\\omega}^2\\hspace{1mm}N+3\\hspace{1mm}m_u$, \n\\item$S= -2\\sqrt{6}\\hspace{1mm}c_{\\omega}\\hspace{1mm}s_{\\omega}\\hspace{1mm}N+2\\hspace{1mm}\\sqrt{3}s_{\\omega}\\hspace{1mm}F$ \n\\item $A= N\\hspace{1mm}(2\\hspace{1mm}c_{\\omega}^2+2)-2\\hspace{1mm}T+\\sqrt{2}\\hspace{1mm}c_{\\omega}\\hspace{1mm}F+6\\hspace{1mm}m_u$, \\hspace{1mm} $B= \\sqrt{6}\\hspace{1mm}c_{\\omega}\\hspace{1mm}s_{\\omega}\\hspace{1mm}N+2\\hspace{1mm}\\sqrt{3}\\hspace{1mm}F\\hspace{1mm}s_{\\omega}$ \n\\end{description}\nwith\n$N= m_c\\hspace{1mm}s_{\\alpha}^2+m_t\\hspace{1mm}c_{\\alpha}^2-m_u$,\\hspace{1mm} $F= c_{\\alpha}\\hspace{1mm}s_{\\alpha}\\hspace{1mm}(m_t-m_c)$.\nThis matrix can be reformulated in a form similar to (\\ref{CU}), \n\n\\[ \nM_u=\nC_u \\left[\\begin{pmatrix}\n  1\\\\\n  & k\\hspace{1mm}e^{-i\\mu} \\\\\n  &&kp\\hspace{1mm}e^{-i(\\mu-\\rho)}  \\\\\n \\end{pmatrix}\n\\begin{pmatrix}\n1&1&1\\\\\n1&1&1\\\\\n1&1&1\\\\\n\\end{pmatrix}\n\\begin{pmatrix}\n  1\\\\\n  & k\\hspace{1mm}e^{i\\mu} \\\\\n  &&kp\\hspace{1mm}e^{i(\\mu-\\rho)} \\\\ \n\\end{pmatrix}+\\Lambda\\right]\n\\]\nwhere $C_u=R+Q+S\\cos\\delta$, $\\mu=\\arctan\\left[S\\sin\\delta\/(Q-R)\\right]$, $\\rho=\\arctan\\left(B\\sin\\delta\/(A+B\\cos\\delta)\\right)$, and\n\\[\n\\Lambda=\n\\begin{pmatrix}\n0   &                   0&   0\\\\\n0   &\\varepsilon         &\\varepsilon'e^{-i\\rho}\\\\\n0   &\\varepsilon'e^{i\\rho}&\\eta\\\\\n\\end{pmatrix}\n\\]\nwith \n\\begin{description}\n\\item $k=|M_{12}|\/M_{11}=\\frac{|R-Q-iS\\sin\\delta|}{R+Q+S\\cos\\delta}$,\\hspace{4mm}$p=|M_{13}|\/|M_{12}|=\\frac{|A-Be^{-i\\delta}|}{|R-Q-iS\\sin\\delta|}$,\n\\item $\\varepsilon=(|M_{22}||M_{11}|-|M_{12}|^2)\/|M_{11}|^2=\n\\frac{4RQ-S^2}{|R+Q+S\\cos\\delta|^2}$,\n\\item $\\varepsilon'=(|M_{23}||M_{11}|-|M_{13}||M_{12}|)\/|M_{11}|^2$,\n\\item$\\eta=(|M_{33}||M_{11}|-|M_{13}|^2)\/|M_{11}|^2$\n\\end{description}\nInserting the mass values (\\ref{jam1}) gives\n\\begin{description}\n\\item$C_u=53723.5 MeV$,\\hspace{1mm}$k=1.00318$,\\hspace{1mm} $p=1.0828$, and\n\\item $\\varepsilon\\approx4.65\\hspace{1mm}10^{-5} \\approx 2\\frac{m_u}{C_u}$,\\hspace{1mm} $\\varepsilon'\\approx4.44\\hspace{1mm}10^{-5}$,\\hspace{1mm} $\\eta\\approx1.85\\hspace{1mm} 10^{-2}\\approx \\frac{1}{2}\\frac{m_t}{C_u}\\frac{m_c}{C_u}$\n\\end{description}\nFor the down-sector, with \n\\[  \nU' =\n \\begin{pmatrix}\n  \\cos\\beta  &-\\sin\\beta &0\\\\\n  \\sin\\beta  &\\cos\\beta  &0\\\\\n   0         & 0         &1\n \\end{pmatrix}\n \\begin{pmatrix}\n \\frac{1}{\\sqrt{2}}  & -\\frac{1}{\\sqrt{2}} & \\hspace{1mm} 0 \\\\\n  \\frac{1}{\\sqrt{6}} &\\frac{1}{\\sqrt{6}}   & -\\frac{2}{\\sqrt{6}} \\\\\n  \\frac{1}{\\sqrt{3}} &\\frac{1}{\\sqrt{3}}   & \\frac{1}{\\sqrt{3}}\n \\end{pmatrix},\n\\]\nthe mass matrix $U'^{\\dagger} diag(m_d,m_s,m_b)U'$ reads \n\\[\nM'=\nC_d\\begin{pmatrix}\nX+A  &  Y   &\\hspace{1mm} 1\\\\    \nY   &  X-A  &\\hspace{1mm} 1+2A\\\\    \n1   & 1+2A  &\\hspace{1mm} X+Y-A-1\\\\    \n\\end{pmatrix}\n\\]\nwhere\n\\begin{description} \n\\item$C_d= 2(m_d c^2_{\\beta}+m_s s^2_{\\beta})-2\\sqrt{3}c_{\\beta}s_{\\beta}(m_s-m_d))+2(m_b-m_s-m_d)$\n\\item$X= (2m_b+m_s+m_d+2(m_d c^2_{\\beta}+m_s s^2_{\\beta})+2\\sqrt{3}c_{\\beta}s_{\\beta}(m_s-m_d))\/C_d$\n\\item$Y=(2m_b+m_s+m_d-4(m_d c^2_{\\beta}+m_s s^2_{\\beta}))\/C_d$,\n\\item$A=2\\sqrt{3}c_{\\beta}s_{\\beta}(m_s-m_d))\/C_d$. \n\\end{description}\nInserting the mass values (\\ref{jam2}) \nwe moreover get the numerical values \n\\begin{description}\n\\item $C_d= 926.448 MeV \\approx m_b\/3$,\\hspace{1mm}$X = 1.0375$,\\hspace{1mm}$A = 7\\hspace{1mm} 10^{-3}$,\\hspace{1mm}$Y = 1.0318$.\n\\end{description}\n\n\\section{Conclusion}\nBy including the democratic rotation matrix in the parametrization of the weak mixing matrix, we obtain mass matrices with specific democratic textures. In this way we make contact between the democratic hypothesis and the experimentally derived parameters of the CKM mixing matrix, avoiding the introduction of additional concepts. \n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n \nOdor-guided navigation is common across the animal kingdom \\citep{Baker2018-ys}. Olfactory cues inform an animal of its location in a natural environment \\citep{Boie2018-eb}, and allow it to adjust its locomotion to navigate an odor landscape in a goal directed manner \\citep{bargmann1991chemosensory,berg1972chemotaxis,aceves1979learning}. Odor guided navigation is an ethologically relevant task that is important for the animal's survival, and it has been a useful framework with which to study genes and circuits underlying sensory-motor transformations \\citep{calhoun2017quantifying, clark2013mapping}.\nSmall model organisms navigate continuous gradients established by the spread of odorants from their sources due to diffusion and drift. \nHow animals interpret these gradients and use them to inform their actions remains an active and productive area of research especially in genetic model systems like \\textit{C. elegans} and \\textit{Drosophila melanogaster} \\citep{bargmann1991chemosensory,aceves1979learning,Levy2020-oh,mattingly2021escherichia, Gomez-Marin2011-ok,Gepner2015-wm}\n\n\n\n\nA major challenge is to quantitatively relate the animal's behavior to the precise olfactory cue that the animal experiences moment-by-moment.\nTherefore it is critical to precisely control the odor environment and record the sensory cues experienced by these animals. The need to \\emph{control} and \\emph{measure} odorants still pose a formidable challenge. While many techniques exist to either present or measure odors in a lab environment, no technique currently exists for precise control and continuous monitoring of an odor landscape. \n\nAll approaches to generate odor landscapes in a lab environment must contend with the odor's diffusivity and interaction with other substrates. Early approaches to \\textit{control} odor concentration relied on passive diffusion to construct a quasi-stationary spatial odor gradient, for example by adding a droplet of odorant in a petri dish in a ``droplet assay'' \\citep{Louis2008-ju,Iino2009-al,Pierce-Shimomura1999-nt, monte1989characterization}. Diffusion places severe limits on the space of possible landscapes that can be created and on the timescales over which they are stable, and the created odor profile is sensitive to adsorption of odor to surfaces, absorption into the substrate, temperature gradients, and air currents, all parameters that are difficult to measure, model, or control. Microfulidics allow water-soluble odors to be continuously delivered to a chamber in order to provide spatiotemporal control \\citep{chronis2007microfluidics, Albrecht2011-fj, lockery2008artificial}. Microfluidics devices, however, are limited in extent, require water-soluble odors and must be tailored-designed to the specific attributes of the animal's size and locomotion. While a post array has been shown to support \\textit{C. elegans} locomotion, no microfluidic device has been demonstrated to support olfactory navigation of \\textit{Drosophila} larvae, for example. \n\nWe previously reported a macroscopic gas-based active flow cell that uses parallel flow paths to construct temporally stable odor profiles \\citep{Gershow2012-nt}. That approach allows for finer spatiotemporal control of the odor gradient, is compatible with \\textit{Drosophila} larvae, and works with volatile airborne-based odor cues. This device used an array of solenoid valves to generate programmable odor profiles, but perhaps because of its complexity has not been widely adopted. \n\n\n\n\n\n\nMost methods to create an odor landscape do not provide a means for knowing or specifying the spatiotemporal odor concentration. In other words while an experimenter may know that some regions of an area have higher odor concentrations, they cannot quantify the animal's behavior given a precise concentration of the odor. This limits the ability to quantitatively characterize sensorimotor processing. To address this shortcoming, various methods have been proposed to \\textit{measure} odor concentration across space. For example, gas samples at specific locations could be taken and measured offline \\citep{Yamazoe-Umemoto2018-nx}. In one of the most comprehensive measurements to date, Louis and colleagues\n\\citep{Louis2008-ju, tadres2022depolarization} used infra-red spectroscopy to measure the spatial profile of a droplet based odor gradient. \n\nIn all of these cases, measurements were performed offline, not during animal behavior, and the odor concentration was assumed to be the same across repeats of the same experiment, and when animals are present. But even a nominally stable odor landscape is subject to subtle but significant disruptions over time from small changes in airflow, from temperature variation, and from the odor's interaction with the substrate, which can include absorption, adhesion, and reemission \\citep{Gorur-Shandilya2019-me, Yamazoe-Umemoto2018-nx,tanimoto2017calcium,Yamazoe-Umemoto2015-ru}.\nThis is challenging to account for and control within a single behavior experiment, and is even more difficult to account for across multiple instances of such experiments. Additional variability also arises across experiments as a result of the introduction of animals, changes to agar substrates, and alteration in humidity or other environmental conditions. To recover the odor concentration that an animal experienced, there is a need to measure odor concentration and animals' behaviors concurrently.\n\n\nOur previously reported flow cell used a photo-ionization detector (PID) sensor moved across the lid before behavioral experiments to measure the odor concentration across space at a single point in time \\citep{Gershow2012-nt}. During experiments, the total concentration of odor in the chamber was monitored concurrently with measurements of behavior. While this provided some assurances that the overall odor concentration was relatively stable, it did not provide any spatial information concurrently with behavior measurements.\n\n\n\n\n\n\n\n\n\n\n\n\n\n    \n\n\n\n\n\n\n\nHere we present a new flow chamber and a new multi-sensor odor array that addresses these prior limitations and can be used for measurement of the odor gradient with high spatial and temporal resolution. The array of sensors can be used two ways: the full array can be used to measure the generated gradient throughout the extent of the chamber, or parts of the array can be used on the borders to monitor, \\textit{during behavioral experiments}, the odor profile in the chamber. By varying flow rates and the sites of odor introduction, we show a variety of odor profiles can be generated and stabilized.\n\n\nTo demonstrate the utility of the apparatus, we applied this instrument to quantitatively characterize the sensorimotor transformation underlying navigational strategies used by \\textit{C. elegans} and \\textit{D. melanogaster} larva to climb up a butanone odor gradient. Butanone is a water-soluble odorant found naturally in food sources \\citep{worthy2018identification} that is often used in odor-guided navigation studies \\citep{bargmann1993odorant, Levy2020-oh, Cho2016-is, Torayama2007-qi}. We show that the agar gel used during behavioral experiments greatly disrupts an applied butanone gradient, and we demonstrate a pre-equilibration protocol allowing generation of stable gradients taking into consideration the effects of agar. Moreover we monitor these gradients during ongoing behavior measurements via continuous measurements of the odor profile along the boundaries of the arena. %\n\nUsing these stable and continuously measured butanone gradients, we measure odor-guided navigation in animals by tracking their posture and locomotion as they navigate the odor landscape. We record chemotaxis behavior and identify navigation strategies in response to the changing  odor concentration they experience. \nIn \\textit{C. elegans}, we observe the presence of navigational strategies that were reported in other sensory-guided navigation conditions, such as salt chemotaxis \\citep{Iino2009-al, Dahlberg2020-ip, Luo2014-pc}. These two strategies are: a biased random turn, known as a pirouette \\citep{Pierce-Shimomura1999-nt}, and a gradual veering, known as weathervaning \\citep{Iino2009-al, Izquierdo2015-la}. %\nIn \\textit{Drosophila melanogaster} larvae, we identify runs followed by directed turns \\citep{Gershow2012-nt, Louis2008-ju, Gomez-Marin2011-ok}. %\nBy using concurrent measurements of behavior and odor gradient we characterize olfactory navigation in these small animals on agar with known butanone odor concentrations, which for \\textit{C. elegans} has not been reported before.\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{Results}\n\n\nWe developed new methods both for generating and measuring odor gradients which we describe here. The systems are modular, scalable, and flexible. The components, which can be used independently of each other, can be fabricated directly from provided files using online machining services, or the provided plans can be modified for other geometries. \n\n\\subsection{Flow chamber for generating spatiotemporal patterns of airborne odors}\n\nWe first sought to develop a method of creating odor gradients that satisfied the following criteria:\n\\begin{enumerate}\n    \\item The spatial odor profile should be \\textit{controllable}. Varying control parameters (e.g. flow rates, tubing connections) should result in predictable changes to the resulting odor landscape. \n    \\item The odor profile should be \\textit{stable} and \\textit{verifiable}. The same spatial profile should be maintained over the course of an experiment lasting up to an hour, and this should be verifiable via concurrent measurements during behavior experiments.\n    \\item The apparatus should be \\textit{straightforward} to construct and to use, and \\textit{flexible} to adapt to various experimental configurations, including using with either \\textit{C. elegans} or \\textit{Drosophila} larva, and with agar arenas of various sizes.\n\\end{enumerate}\n\n\\begin{figure}\n\\begin{fullwidth}\n\\includegraphics[width=0.85\\linewidth]{Odor_flow_Fig1_v2.png}\n\\caption{\\textbf{Odor flow chamber with controlled and measured odor concentration.} \\textbf{(a)} Schematic of airflow paths. Airflow paths for odor solution and water are controlled separately by mass flow controllers (MFCs) and spatially arranged into the odor chamber. The outflux from the chamber connects to a flow meter and photo-ionization detector (PID). \\textbf{(b)} Flow chamber design. \\textbf{(c)} Odor sensory array (OSA). Seven odor sensor bars are connected to a sensor hub. Each bar has 16 odor sensors (OS) and 8 temperature\/humidity sensors (THS). Measured odor concentrations from the OSA of a spatially patterned butanone odor concentration shown in \\textbf{(d)} for each sensor, \\textbf{(e)} interpolated across the arena with square dashed line indicating the area where agar and animals are placed. \\textbf{(f)} A two-parameter analytic flow model fit to measurement. \\textbf{(g)} Cross-sections from (f) at at 4 different x-axis positions (show as colored arrows). Sensor readouts are overlaid points on the smooth curves from a 1D diffusion model. \\label{fig:fig1}\n}\n\\figsupp[Long-duration calibration confirms stable control and measurements.]{Long-duration calibration confirms stable control and measurements. (a, top) Odor flow rate driven by the MFC, (a, middle) raw reading of an odor sensor (OS) located in the chamber, (a, bottom) odor concentration readout from the photo ionization detector (PID) located at the outlet of the chamber. Note that the measurement is stable across the 90 minute recording. (b) We correct for the time lag between sensors and plot the mapping between PID measurements and OS readings.}{\\includegraphics[width=10cm]{Odor_flow_SI1_2.png}}\n\\label{figsupp:figSI1-2}\n\\end{fullwidth}\n\\end{figure}\n\n\nWe constructed a flow chamber to control odor air flow across an arena (\\FIG{fig1}a). Odor and humidified air are sourced from two bubblers, one containing pure water and the other an aqueous solution of odorant and water. Flow rates are controlled by separate mass flow controllers (MFCs) upstream of the bubblers. Downstream of the bubblers, the odor and air streams are divided into parallel sections of equal lengthed tubing. Each tube is connected to one input port of the flow chamber. The pattern of connections and the flow rates set by the two MFCs determines the shape of the produced odor profile. For instance, if odor is provided at a single central inlet, the resulting profile is a `cone' (\\FIG{fig1}d-g) whose peak concentration and divergence are controlled with the MFCs (e.g. speeding up the odor flow while slowing down the air flow broadens the cone). Temporal gradients can be achieved by varying the odor flow in time, subject to constraints imposed by the odor's absorption into the agar gel.\n\nThe outflux from the flow chamber is connected to a flow meter and photo-ionization detector to monitor the overall flow rate and odor concentration respectively. The geometry of this flow chamber is shown in (\\FIG{fig1}b), where parallel tubings are connected from the side and the chamber is vacuum sealed with a piece of acrylic on top during experiments. The chamber is designed for use with $\\sim$100 mm square agar plates. The extra width (2.5 cm on either side) diminishes the influence of the chamber boundary on the odor profile over the arena. Interchangeable inserts allow for different agar substrates (e.g. circular plates) or for full calibration by odor sensor arrays (\\FIG{fig1}c), discussed in the next section. Metal components are designed for low-cost fabrication by automated mechanisms (either laser cut-able or 3-axis CNC machinable). The fabrication plans for the flow chamber, design for the agar plate inserts, and required components to construct the flow path are publicly available in the \\nameref{ssec:num1} section.\n\n\n\\subsection{Measuring the spatiotemporal odor distributions} A central difficulty in measuring animals' responses to olfactory cues is quantifying airborne odor concentrations that vary in space and time. This difficulty is exacerbated in turbulent environments where odor plumes carry abrupt spatial and temporal jumps in concentration far from the source with fundamentally unpredictable dynamics. But even in laminar flows, boundary conditions, slight changes in temperature, and the presence of absorbing substrates like agar make this challenging. \n\nThere is therefore a need, even for quasi-stationary gradients, to characterize the odor profiles in situ and to monitor these profiles during experiments. Various optical techniques, like laser induced fluorescence or optical absorption \\citep{Louis2008-ju,tadres2022depolarization,demir2020walking}, %\nexist to monitor concentration across planar arenas, but in general, these are incompatible with behavior experiments, expensive to construct, require specially designed arenas, or some combination of these disadvantages. Electronic chemical sensors can reveal the time-varying concentration at a particular point in space. A tiled array of these sensors acts as a `camera' forming a 2D spatiotemporal reading of the concentration. The gold-standard for measurement of odor concentration is the photo-ionization detector (PID), but even the smallest versions of these sensors are both too large ($\\sim$ 2 cm in all dimensions) and too expensive ($\\sim$ \\$500 each) to make an array. Metal-oxide odor sensors, designed to be used in commercial air quality sensors, are available in inexpensive and compact integrated circuit packages. However, in general, commercial metal-oxide sensors are not designed for precision work - they tend to drift due to variations in heater temperature, humidity, adsorption of chemicals and ageing effects. Most such sensors are designed to detect the presence of gas above a particular concentration but not to precisely measure the absolute concentration. We became aware of a newer metal-oxide sensor, the Sensirion SGP30 that was designed for long-term stability and concentration measurement; we wondered if such a sensor could be calibrated for use in an odor sensor array.\n\nTo calibrate the sensor, we created a controllable concentration source by bubbling air through butanone. The odor reservoir contains butanone dissolved in water and is kept below the saturation concentration (11 mM or 110 mM odor sources).\nWe then mixed this odorized air flow into a carrier stream of pure air. We kept the carrier air flow rate constant ($\\sim 400$ mL\/min) and varied the flow rate through the odor source ($0-50$ mL\/min); the odor flow rate was slow enough that the vapor remained saturated, so the concentration of butanone in the mixed stream was proportional to the flow rate through the butanone bubbler, as directly measured with a PID (\\FIGSUPP[fig1]{figSI1-2}). We typically calibrated concentration with continuously ramped flow rate in triangle wave with 500 s period for 2-3 cycles.\nWe found a one-to-one correspondence between the odor sensor reading and the PID reading that persisted over time and showed no hysteresis. We reasoned that after applying this calibration procedure to an array of sensors, we could use the array to measure spatiotemporal odor concentration distributions with accuracy derived from the PID. Continuous calibration for 90 minutes showed that the odor sensors reliably reported concentration across durations (\\FIGSUPP[fig1]{figSI1-2}) much longer than the typical behavioral experiment.\n\n\nWe constructed the sensor array from `odor sensor bars' (OSBs), printed circuit boards each containing 16 sensors in two staggered rows of 8. Each OSB also contained 8 temperature and humidity sensors to allow compensation of the odor sensor readings. The OSBs are mounted orthogonal to the direction of air flow; 7 OSBs fit inside our flow chamber (112 sensors total) allowing a full measurement of the odor profile. Taken together these 112 sensors formed an odor sensor array (OSA), capable of measuring odor concentrations with %\n$\\sim$ 1 cm spatial and 1 second temporal resolution. Prior to all experiments we calibrated the OSA in situ by varying the butanone concentration across the entire anticipated range of measurement while simultaneously recording the odor sensor and PID readings. \n\nTo verify the ability of the OSA to measure concentration gradients we created an artificially simple steady-state odor landscape by flowing odorized air ($\\sim 30$ mL\/min) through a central tube and clean air through the others ($\\sim 400$ mL\/min distributed into 14 surrounding tubings) in an environment without agar and without animals. This results in an air flow velocity $\\sim 5$ mm\/s in the flow chamber.  As the flow rates and concentrations are all known and the flow is non-turbulent, and there is no agar or animals present, the concentration across the chamber should match a convection-diffusion model. After establishing the gradient, we recorded from the discrete odor sensors on the array (\\FIG{fig1}d) and estimated the values in between sensors using spline interpolation with length scale equal to the inter-sensor distances (\\FIG{fig1}e). We compared this stationary profile with a two-parameter convection-diffusion flow model (\\FIG{fig1}f,g) fit to the data and described in the Methods section. \nThe measured concentrations in this artificially simplistic odor gradient show good agreement with the fit convection-diffusion model, especially in the central region where experiments are to be conducted, leading us to conclude the OSA can accurately report the odor concentration. We proceed to consider more complex odor environments. %\n\n\n\n\n\n\n\n\n\n\n\nWe demonstrate several examples of flexible control over the odor profile.  In the configuration shown in \\FIG{fig1} the center-most input provides odorized air and all surrounding inputs provide moisturized clean air to form a cone-shape stationary odor pattern. A narrower cone can be created by increasing the air flow (600 mL\/min) of the surrounding inputs relative to the middle odorized flow (\\FIG{fig2}a). %\nThe cone can be inverted by placing odorized air in the \ntwo most distal inputs, and clean air in all middle inputs (\\FIG{fig2}b). This inverse cone has lower concentration in the middle and higher on the sides. In later animal experiments, we restrict odorized air to one side to form a biased-cone odor landscapes, resulting in a cone with an offset from the middle line of the arena. Many more configurations are possible, demonstrating that the odor flow chamber enables the flexible control of airborne odor landscapes that are much more complex than a single odor point source. To show that the flow control and measurement methods are not restricted to any single odor molecule, we created and measured a cone profile using ethanol (\\FIG{fig2}c).\n\n\\begin{figure}\n\\begin{fullwidth}\n\\includegraphics[width=1.0\\linewidth]{Odor_flow_SI2.png}\n\\caption{\n\\textbf{Flexible control of a steady-state odor landscape.} \\textbf{(a)} Configuration for a narrow cone with butanone. Colorbar shows interpolated odor concentration in ppm as measured by the odor sensor array. \\textbf{(b)} An inverse cone landscape that has higher concentration of butanone on both sides and lower in the middle. \\textbf{(c)}Another stationary odor landscape of a different shape, this time with ethanol.\n\\label{fig:fig2}\n}\n\\end{fullwidth}\n\\end{figure}\n\n\n\\subsection{Odor-agar interactions dominate classical droplet assays}\n\n\\begin{figure}\n\\begin{fullwidth}\n\\includegraphics[width=1.0\\linewidth]{Odor_flow_droplet_2.png}\n\\caption{\\textbf{The presence of agar in the droplet assay alters the time-evolution of butanone odor landscape.}\n\\textbf{(a)} Concentration measured by odor sensor array is reported immediately after butanone droplet is introduced into the arena without agar. Red dot indicates the position of butanone droplet on the lid over the sensor array ($2\\mu L$ of $10\\%$ v\/v butanone in water). \\textbf{(b)} Same but three minutes later.\n\\textbf{(c)} Same measurement as in (a) but now droplet is added onto agar (gray). Two odor sensor bars have been removed to make space for the agar. \\textbf{(d)} Same as (c) but three minutes later. Side view of the configurations with OSB sensors, butanone droplets, and agar gel are shown on the right.}\n\\label{fig:fig3}\n\\figsupp[Concentration measurements with an odor droplet and experimental perturbation.]{Concentration measurements with an odor droplet and experimental perturbation. (a) The initial concentration readout from a droplet of butanone on the lid near the middle of the arena. (b) Same condition as (a), but 20 minutes after the recording (left) and after shortly opening and closing the lid back (right) to mimic perturbation during worm experiments. (c) When there is agar in the chamber, the odor concentration is better maintained in the chamber. Location of the odor droplet is shown with a red dot. (d) After 20 minutes of recording (left) and after shortly opening and closing the lid back (right) to mimic experimental perturbation.}{\\includegraphics[width=10cm]{Odor_flow_SI2-2.png}}\\label{figsupp:figSI3}\n\\videosupp{Time-evolution of odor landscape from a butanone droplet with (right) and without (left) agar. The experimental conditions are the same as \\FIG{fig3}. Blank sensor positions indicate sensors replaced by agar. The video updates every 2 seconds and measures butanone concentration from a droplet in the first 3 minutes. \nVideo available online at  \\href{https:\/\/figshare.com\/articles\/dataset\/Continuous_odor_profile_monitoring_to_study_olfactory_navigation_in_small_animals\/21737303}{10.6084\/m9.figshare.21737303}\n}\\label{videosupp:sv1}\n\\end{fullwidth}\n\\end{figure}\n\nClassic chemotaxis experiments in small animals commonly construct odor environments with odor droplets in a petri dish, usually with a substrate like agar. Our odor delivery instrument is designed to be compatible with a similar environment. To first better understand classical chemotaxis experiments, we sought to characterize the spatiotemporal odor profile from an odor droplet point source using our odor sensor array. We first considered the case without agar. In that case the odor concentration should be governed entirely by gas-phase diffusion. %\nWe placed a $2 \\mu$L droplet of $10\\%$ butanone in water %\non the lid of our instrument centered in the arena above the full OSA and without any airflow (\\FIG{fig3}a,b). Butanone was observed to diffuse across the arena in the first three minutes (supplementary video (\\VIDEOSUPP[fig3]{sv1})\nand the equilibrium concentration is close to uniform across the odor sensors. We note that the final concentration of roughly 100 ppm, and the equilibration timescale both match what we would expect from first principals for $\\sim 10^{-6}$ mol of butanone in a $\\sim 225 \\text{mL}$ arena that have a diffusion rate of $\\sim 0.08 \\text{cm}^2\/$s in air.  \nA uniform odor landscape is not helpful for studying odor guided navigation, but most behavioral experiments are not conducted in a bare flow chamber but contain a biologically compatible substrate, such \nas an gar gel, as is typically used in droplet assays. We therefore sought to investigate the role that agar plays in sculpting the odor landscape.\n\n\n\nWe introduced agar into the droplet assay by removing two sensor bars and replacing them with agar. We placed a butanone droplet directly on the agar, as done classically, and measured the odor landscape over time (\\FIG{fig3}c,d). The odor concentration measured with agar is dramatically different from that measured without agar. Instead of quickly equilibrating to a uniform concentration, in the presence of agar there was instead a local maximum of butanone surrounding the droplet that persists even after 3 minutes. This difference in airborne odor concentration with and without agar persists after experimental perturbation such as removing and replacing the lid over the chamber (\\FIGSUPP[fig3]{figSI3}). More broadly, the odor landscape we observed in the presence of agar would have been hard to predict ahead of time. An important consequence of this finding is that, to create a specific odor landscape (as in \\FIG{fig1} or \\FIG{fig2}) with agar, one will need to account for the effect of agar. We therefore sought to study odor-agar interactions more systematically and in the context of air flow. %\n\n\n\n\n\\subsection{Measuring and compensating odor-agar interactions with flow}\n\nWe first sought to measure whether the presence of agar changed the odor profiles generated due to flow in a bare chamber (\\FIG{fig1}, \\FIG{fig2}). As in \\FIG{fig3}c,d, we replaced two odor sensor bars with a rectangular strip of agar gel or a metal plate as a control, and then measured airborne odor concentration upstream and downstream of the agar under odorized airflow that would normally produce a cone profile (\\FIG{fig4}a-b).\nWhile the agar had little effect on the odor landscape upstream of the agar, it drastically altered the downstream odor landscape (\\FIG{fig4}b), suggesting that the agar absorbs\nthe airborne butanone molecules. This finding is consistent with the odor droplet experiments (\\FIG{fig3}) and to be expected since butanone is highly soluble in water (275 g\/L). Pulse-chase style experiments confirm that agar does indeed absorb and reemit butanone (\\FIGSUPP[fig4]{figSI4-2}b). We also observed disruptions to the odor landscape when we used a full-sized 96 mm square agar plate intended for use with animals, \\FIG{fig4}c. To accommodate the full sized agar plate we measured only the one dimensional odor profiles upstream and downstream of the agar \\FIG{fig4}d. Taken together, these experiments suggest that agar-butanone interaction presents a challenge for setting up and maintaining stable odor landscapes. \n\nWe next sought a method to generate desired odor landscapes even in the presence of agar. We generate the odor profile by constant flow, which continuously replenishes the airborne odor. In principle, the disruption caused by agar should be overcome by constant flow of a sufficiently long duration, after which the agar and airborne odor would be in quasi-equilibrium at all spatial locations, with the concentration of odor dissolved in the gel proportional to the airborne concentration above it. \n\nWe measured odor concentration downstream of the agar and found that the airborne concentration failed to approach equilibrium on the timescales of single experiments \\FIG{fig4}e,f. This suggests that it is not practical to simply wait for the agar and odor to reach equilibrium. Instead we developed a pre-equilibration protocol to more efficiently bring the agar and airborne odor into equilibrium before our experiments.\n    \nTo more rapidly establish a desired airborne odor landscape, we briefly first exposed the agar to an airflow pattern corresponding to higher-than-desired odor concentration, created by replacing the odor reservoir with one containing a higher concentration of butanone.\nWe monitored the odor profile downstream of the agar until it reached the desired concentration and then switched to the original bubbler to maintain that concentration. Using this pre-equilibration protocol, we reached quasi-equilibrium quickly, typically after the order of ten minutes, \\FIGSUPP[fig4]{figSI4-2},c. Note the spatial parameters of the two airflow patterns were the same, only the concentration of the odor source changes. Pre-equilibration allows the generation of airborne odor gradients in the presence of agar that match those in the absence of agar (\\FIG{fig4}b,d,f right vs left column). We modeled the pre-equilibration protocol using a reaction-convection-diffusion model considering first order interactions between odor and agar. Under reasonable assumptions about the absorption rate, reemission rate, and capacity of the agar, simulations of this simple model provided qualitative agreement to our observations (\\FIG{PE_model}a-c).\n\n\\subsection*{Monitoring the boundary determines the odor landscape at quasi-equilibrium}\n\n\\begin{figure}\n\\begin{fullwidth}\n\\includegraphics[width=0.95\\linewidth]{Odor_flow_Fig2_v3.png}\n\\caption{\\textbf{Under flow, agar interacts with odor to disrupt the downstream spatial odor profile, but a  pre-equilibration protocol can coax the system into quasi-equilibrium and restore the odor profile.} %\n\\textbf{(a)} Two odor sensor bars are replaced with agar to observe the effect of introducing agar on the downstream spatial odor profile. %\n\\textbf{(b)} Measured odor profile is shown upstream and downstream of the removed odor sensor bars in the absence (left) and presence of agar (middle).\nTransiently delivering a specific higher odor concentration ahead of time via a pre-equilibration (PE) protocol restores the downstream odor profile even in the presence of agar. \\textbf{(c)} Additional odor sensor bars are removed and replaced with a larger agar, as is typical for animal experiments. \\textbf{(d)} Measurements from the downstream sensor bar under the same three conditions in (b). The dots are sensor measurements and the smooth curve is a Gaussian fit. \\textbf{(e)} The same experimental setup in (a), here focusing on time traces of only three downstream odor-sensors (colored circles for selected OS). \\textbf{(f)} Concentration time series of three sensors color-coded in (e). Traces for three conditions are shown: time aligned to initial flow without agar (left), time aligned to initial flow with agar (middle), and traces after PE (right, with transparent line showing measurements another 20 min after the protocol). The dash-lines indicate the target steady-state concentration for each sensor.\n\\label{fig:fig4}\n}\n\\figsupp[Time series of concentration change that capture effects of agar gel and the PE protocol]{Time series of concentration change that capture effects of agar gel and the PE protocol. (a) Concentration readout from the downstream PID (top) in response to the impulse of air flow rate though the odor bottle controlled by MFC (bottom), with no agar in the flow chamber. The background clean air flow is constant $\\sim$ 400 mL\/min throughout the recording. (b) Same as (a) but with agar plate in the flow chamber. Note that the response time scales to the same impulse are significantly different. (c) The time trace recorded from the PE protocol with agar plates. Odor reservoir with high butanone concentration (110 mM) is applied in the beginning, swapped back to the target concentration (11 mM) at $\\sim$ 200 seconds, the odor concentration readout relaxes and stabilizes after $\\sim$ 900 seconds, which enters steady-state for a duration longer than animal experiments.\n}{\\includegraphics[width=10cm]{Odor_flow_SI_PE2.png}}\n\\label{figsupp:figSI4-2}\n\\end{fullwidth}\n\\end{figure}\n\n\n\\begin{figure}\n\\begin{fullwidth}\n\\includegraphics[width=0.9\\linewidth]{Odor_flow_SI_PE_model.png}\n\\caption{\\textbf{Simulations from a reaction-convection-diffusion model of  odor-agar interaction show that at quasi-equilibrium the airborne odor concentration is the same with or without agar.} \\textbf{(a)} Simulation results of steady-state odor concentration in air without agar and with  flow configured as in \\FIG{fig1}. \\textbf{(b)} The same simulation condition in (a) but now shortly after agar is introduced and before quasi-equilibrium is reached. A schematic of odor-agar interaction model is shown below. When agar is introduced, it absorbs the odor in air and decreases concentration measured downstream, producing a non-equilibrium (NE) concentration profile. \\textbf{(c)} Odor concentration profile in air, with agar present, but after the pre-equilibration (PE) protocol brings this system to quasi-equilibrum. The PE protocol is shown in the schematic below, followed with steady-state (SS) with the stable odor concentration profile shown above. \\textbf{(d)} The absolute difference of concentration profile without agar (a) and with agar after PE (c) is shown. \\textbf{(e)} Upstream and \\textbf{(f)} downstream odor concentrations along the agar boundary are shown for all three conditions.\n    }\n    \\label{fig:PE_model}\n    \\end{fullwidth}\n\\end{figure}\n\n\nIt is critical to monitor the odor landscape during animal experiments because the landscape is sensitive to environmental and experimental conditions which may fluctuate within and between experiments. But it is inconvenient to measure airborne concentration directly over the agar (e.g. because sensors impede optical access and also require heat management). \nFortunately, measuring the odor profile upstream and downstream of the agar places strong constraints on the airborne odor concentration over the agar such that in practice the spatial concentration can be confidently inferred.\n\nIf the airborne odor concentration upstream and downstream of the agar matches the profile in the absence of agar, one can infer that the airborne odor concentration landscape above the agar is also the same. The argument is straightforward: in the absence of sources or sinks, the fact that two concentration distributions obey the same differential equations and share the same conditions on all boundaries means that the the distributions are identical throughout the interior. Given identical measurements of the with-agar and without agar profiles at the inlet and outlets and reflecting boundary conditions on both walls, the only way for the with-agar distribution to differ from that without-agar is for sources and sinks of odor in the agar to be precisely arranged so that the all excess odor emitted from one point is exactly reabsorbed somewhere else before reaching the boundary. Not only is such an arrangement unlikely, it is inherently temporally unstable.  A mathematical version of this argument is presented in the \\hyperref[ssec:appendix]{Appendix}. \n\n\n\n\n\n\n\nThis quasi-equilibrium argument is supported by empirical concentration measurements shown in \\FIG{fig4} and numerical results with the reaction-convection-diffusion model demonstrated in \\FIG{PE_model}. The simulation results show that there is negligible difference between conditions with and without agar at quasi-equilibrium (\\FIG{PE_model}d) when the odor concentration along the boundary  is the same (\\FIG{PE_model}e,f). Together, our numerical estimations, along with empirically observations allow us to safely infer that when measurements along the boundary indicate that they system is in quasi-equilibrium, the odor concentration experienced by animals on agar are the same as the concentrations measured in the absence of agar.\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{Butanone chemotaxis in \\emph{C. elegans}} %\nWe sought to directly quantify \\textit{C. elegans}' navigation strategies for airborne butanone using our odor delivery system.\n\\textit{C. elegans} are known to climb gradients towards butanone \\citep{bargmann1993odorant, Cho2016-is, Levy2020-oh}. Microfluidic environments suggest that they use a biased random walk strategy to navigate in a liquid butanone environment \\citep{Levy2020-oh, Albrecht2011-fj}. Worms are also known to use weathervaning to navigate airborne odor gradients \\citep{Iino2009-al, kunitomo2013concentration} although to our knowledge this has not been specifically investigated for butanone.  \n\n\nWorms were imaged crawling on agar in the flow chamber under an airborne butanone odor landscape illuminated by infrared light. Here 6 recording assays were presented, with approximately $50-100$ animals per assay, and two different odor landscapes were used.\n\\textit{C. elegans} navigated up the odor gradient towards higher concentrations of butanone, as expected (\\FIG{fig5}a,b). Importantly, the odor concentration experienced by the animal at every point in time was inferred from concurrent measurements of the odor profile along the boundary of the agar, \\FIG{fig5}c. On average, animals were more likely to travel in a direction up the local gradient than away from the local gradient, as expected for chemo-attraction \\FIG{fig5}d. We use the term ``bearing to local gradient'' to describe the animal's direction of travel with respect to the local odor gradient that it experiences.\n\nWe find quantitative evidence that the worm exhibits both biased random walk and weathervaning strategies. \nTo investigate biased random walks, we measured the animal's probability of turning (pirouette) depending on its bearing with respect to the local airborne butanone gradient \\FIG{fig5}e.\nWe find that the animal is least likely to turn when it navigates up the local gradient and most likely to turn when it navigates down the gradient, a key signature of the biased random walk strategy \\citep{berg2018random, mattingly2021escherichia}.  \n\nTo test for weathervaning, we measured how the curvature of the animal's trajectory depended on its bearing with respect to the local airborne butanone gradient, \\FIG{fig5}f. \nWhen the animal navigated up the butanone gradient (\\FIG{fig5}f, blue) the distribution of the curvature of its trajectory was roughly symmetric and centered around 0 (straight line trajectory). By contrast, when the animal navigated perpendicular to the gradient (\\FIG{fig5}f, yellow and red) the distribution of the curvature of its trajectories was skewed. The skew was such that it enriched for cases where the animal curved its trajectories towards the local gradient, a key signature of weathervaning.\nBoth the biased random walk and weathervaning behavior was absent in control experiments with flow but not odor, and we observed no evidence of anemotaxis at the $\\sim 5$mm\/s air velocities encountered by the animals   (\\FIGSUPP[fig5]{no_odor_control}).\nWe conclude that \\textit{C. elegans} utilize both biased random walk and weathervaning strategies to navigate butanone airborne odor landscapes. We note that the quantitative analysis needed to make this conclusion relied on knowledge of the local airborne odor gradient experienced by the animal, which was provided by our odor profile measurements.\n\n\n\n\n\n\n\n\n\n\\begin{figure}\n\\begin{fullwidth}\n\\includegraphics[width=1.0\\linewidth]{Odor_flow_Fig3_v3.png}\n\\caption{\\textbf{\\textit{C. elegans} use both biased random walk and weathervaning to navigate in a butanone odor landscape.} Animals on agar were exposed to butanone in the flow chamber.  \\textbf{(a,b)} Measured animal trajectories are shown overlaid on airborne butanone concentration for different odor landscapes. Green dots are each animal's initial positions and red dots are the endpoints. %\n\\textbf{(c)} An animal's trajectory is shown colored by the butanone concentration it experiences at each position (top). Its turning behavior is quantified and plotted over time. Turning bouts are highlighted in gray. \\textbf{(d)} Distribution of the animal's bearing with respect to the local airborne odor gradient is shown. Peak around zero is consistent with chemotaxis. \\textbf{(e)} \nProbability of observing a sharp turn per time is shown as a function of the absolute value of the bearing relative to the local gradient. Modulation of turning is a signature of biased random walk. Error bars show error for counting statistics. Data analyzed from over 9,000 tracks produced from $\\sim$300 worms, resulting in 108 hours of observations. \\textbf{(f)} Probability density of the curvature of the animal's trajectory is shown conditioned on bearing with respect to the local gradient. Weathervaning strategy is evident by a skew in the distribution of trajectory curvature when the animal travels perpendicular to the gradient (yellow and red). Three distributions are significantly different from each other according to two-sample Kolmogorov-Smirnov test ($p < 0.001$). Means are shown as vertical dashed lines.\n\\label{fig:fig5}}\n\\figsupp[Control measurements with air flow and without odor gradient.]{Control measurements with air flow but no odor. (a) Behavioral trajectories overlaid on the odor landscape that would have been expected had odor been present (mock odor landscape). No odor is presented, only  moisturized airflow. (b) Distribution of bearing to the mock gradients under clean air flow. (c) Turn probability at different  bearing conditions to the mock gradient. (d) Curvature conditioned on different  bearing measurements.\n}{\\includegraphics[width=10cm]{Odor_flow_SI_noodor.png}}\n\\label{figsupp:no_odor_control}\n\\end{fullwidth}\n\\end{figure}\n\n\n\n\n\nTo quantify the overall navigational response with respect to local gradients, we further compute the animal's drift velocity as a function of local gradients (\\FIG{fig6}). This captures the animal's overall gradient climbing performance as a result of all the navigational strategies it uses, including the biased random walk and weathervaning. This calculation is only possible with a knowledge of the odor concentration experienced by the animal. \n\n\n\n\n\n\n\n\\begin{figure}\n\\begin{fullwidth}\n\\includegraphics[width=0.7\\linewidth]{Odor_flow_Fig6_v3.png}\n\\caption{\n\\textbf{Tuning curve relating animal drift velocity to experienced odor concentration gradient.} \\textbf{(a)} Schematic of a worm tracked in the odor landscape. The crawling velocity vector $V$, local concentration gradient $\\nabla C$, bearing angle $\\theta$, and drift velocity $V\\cos(\\theta)$ are shown. \\textbf{(b)} Tuning curve shows the drift velocity $V\\cos(\\theta)$\nas a function of the odor concentration gradient. Gray dash line indicates an unbiased performance with zero drift velocity, gray dots are the discrete measurements, and the black line shows the average value within bins. Error bar shows lower and upper quartiles of the measurements. %\n}\n\\label{fig:fig6}\n\\end{fullwidth}\n\\end{figure}\n\n\n\\subsection{Butanone chemotaxis in \\emph{Drosophila} larvae}\nTo further evaluate the utility of the flow chamber and gradient calibration for the study of small animal navigation, we investigated how larval \\textit{Drosophila} navigate butanone. Although butanone is not as commonly used as a stimulus with \\textit{Drosophila} as with \\textit{C. elegans}, butanone is known to be attractive to larval flies \\citep{dubin1995scutoid, dubin1998involvement} and has been variously reported to be attractive \\citep{park2002inactivation} and aversive \\citep{israel2022olfactory, lerner2020differential} to adult flies. \nTo investigate the larva's navigational strategy in a butanone gradient, we created a \"cone\" shaped butanone gradient over the agar substrate using the pre-equilibration protocol, as before, and we confirmed the presence and stability of the gradient by continuously measuring the spatial distribution of butanone upstream and downstream of the agar arena. We monitored the orientation and movement of 59 larvae over 6 separate 10 minute experiments ($\\sim$ 10 larvae per experiment) with an average observation time of 7 min per larva (\\FIG{fig7}). \n\nLarvae moved towards higher concentration of butanone (\\FIG{fig7}a). To analyze the strategy by which they achieved this, we first constructed a coordinate system in which 0 degrees was in the direction of the odor gradient (towards higher concentration) and 180 degrees was directly down-gradient; angles increased counterclockwise when viewed from above.\nWe found that larvae initiated turns at a higher rate when headed down-gradient ($\\pm 180^\\circ$ bearing with respect to the local gradient) than up gradient ($0^\\circ$) (\\FIG{fig7}b). When larvae turned, their reorientations tended to orient up gradient (negative angle changes from $+90^\\circ$ bearing with respect to the local gradient, and positive angles changes from $-90^\\circ$) (\\FIG{fig7}c). Thus \\textit{Drosophila} larvae use similar navigational strategies to \\textit{C. elegans} to move towards butanone \n\n\n\\begin{figure}\n\\begin{fullwidth}\n\\includegraphics[width=0.9\\linewidth]{Odor_flow_Fig7.png}\n\\caption{\n\\textbf{\\emph{D. melanogaster} larvae chemotaxis in the odor flow chamber.} \\textbf{(a)} Trajectories overlaid on the measured butanone odor concentration landscape. Example tracks are highlighted and the initial points are indicated with white dots. \\textbf{(b)} Top: turn rate versus the bearing, which is the instantaneous heading relative to the gradient defined by quadrants shown on top. Error bar show counting statistics. Bottom: average heading change versus the bearing prior to all turns (re-orientation with at least one head cast). Error bars show standard error of the mean. Data analyzed from 6 experiments, 59 animals, with 620 turns over 6.8 hours of observation.\n\\label{fig:fig7}\n} %\n\\end{fullwidth}\n\\end{figure}\n\n\n\n\n\n\\section{Discussion}\n\n\n\n\n\n\n\n\n\n\nWe present a custom-designed flow chamber and odor sensor array that enables us to measure navigation strategies of worms and fly larvae within the context of a controlled and measured odor environment.\nThe key features of this odor delivery system are that (1) the odor concentration profile through space is controlled in the flow chamber, (2) the odor sensor array provides a spatial readout to calibrate and measure the profile, and (3) the odor concentration profile is monitored during animal experiments. This last feature, the ability to monitor the spatial profile of odor concentration on the boundary during experiments, sets this method apart from previous approaches. \n\nThe ability to monitor spatial profile during experiments, along with a quantitative understanding of odor-agar interactions, provides confident knowledge of the odor experienced by the animal over time. This in turn allows us to extract tuning curves that describe the animal's behavioral response to the odor it experiences. In the future, such tuning curves may form the basis of investigations into neural mechanisms driving the sensorimotor transformations underlying navigation. \n\nIn contrast to liquid delivery of odor gradients via microfluidic chips \\citep{Albrecht2011-fj,Larsch2015-xy}, our method allows worms to crawl freely on an agar surface. This allows our behavior measurements to be directly compared against classical chemotaxis assays \\citep{bargmann1993odorant, Louis2008-ju, Pierce-Shimomura1999-nt}. Additionally, the macroscopic odor airflow chamber makes it straightforward to flexibly adjust the the spatial pattern between experiments without the need to redesign the chamber. \n\n\n\nOur setup uses low flow rates corresponding to low wind speed velocities (5 mm\/s) to avoid anemotaxis.  Larger organisms, including adult flies, navigate towards odor sources by combining odor and wind flow measurements \\citep{vergassola_infotaxis_2007, matheson_neural_2022}. Fly larvae exhibit negative anemotaxis at wind speeds 200 to 1000 times higher than those used here \\citep{jovanic_neural_2019}, but previous work showed that they do not  exhibit anemotaxis at lower  windspeeds like the ones used here   \\citep{Gershow2012-nt}. For example, they do not exhibit anemotaxis  at 12 mm\/s which is still higher than the velocities they experience here  \\citep{Gershow2012-nt}. Therefore we do not expect \\textit{Drosophila} larvae to exhibit anemotaxis under our flow conditions.  \\textit{C. elegans} are  not thought  to respond to  airflow. In experiments in aqueous microfluidic chips under flow, \\textit{C. elegans}  move towards higher concentrations of attractant and do not respond to the flow of the liquid \\citep{Albrecht2011-fj}. In agreement, we do not observe any evidence of \\textit{C. elegans} anemotaxis in our chamber in response to  control experiments without odor and with  wind speeds of 5 mm\/s \\FIGSUPP[fig5]{no_odor_control}.\n\nWe focused on the  odor butanone because it is important for a prominent associative learning assay \\citep{Torayama2007-qi, kauffman_c._2011}. Butanone is soluble in water, and therefore it interacts strongly with agar. In this work we showed that this   odor-agar interaction makes it challenging to \\textit{a priori} infer an odor landscape experienced by the animal when agar is present, but that continuously monitoring the odor profile on the boundary  overcomes this challenge. Other  odors may instead have interactions with other substrates, such as glass, aluminum or plastic, which would  also necessitate the use of our continuous monitoring approach. We show that our system is also compatible  with less water-soluble odors, such as ethanol.  \n\n\n\nHere we have addressed the problem of creating airborne odor landscapes. The biophysical processes governing odor sensing in small animals such as \\textit{C. elegans} are not fully understood. The worm carries a thin layer of moisture around its body as it moves on the agar substrate \\citep{Bargmann2006-dy} and it is unclear to what extent the worm pays attention to the concentration of an odorant in the agar below it vs the air above it.\nOur reaction-convection-diffusion model suggests that at the quasi-equilibrium conditions used in our experiments the odor concentration in agar is related to the airborne odor concentration directly above it up to a scalar that we predict to be constant across the agar. Although we have not measured this empirically, this suggests that even in the extreme case that the the animal only senses odor molecules in the agar, the odor concentration experienced by the animal in our experiments should differ by no more than a scaling factor compared to our estimates based on the airborne odor concentration. \n\n\n\n\n\n\n\n\n\n\nKnowing the concentration experienced by the animal is not only useful for measuring navigational strategies more precisely than in classical assays, like the droplet chemotaxis assays. It will also be crucial for studying \\textit{changes} in navigational strategy, such as those in the context of associative learning \\citep{Cho2016-is, Torayama2007-qi}, sensory adaptation \\citep{Levy2020-oh, itskovits2018concerted}, and long time scale behavioral states \\citep{Calhoun2014-aa, Gomez-Marin2011-ok, klein2017exploratory}. In all those cases, it will be critical to disambiguate slight changes to the odor landscape from gradual changes in the navigational strategies. Continuously monitoring the odor landscape during behavior will remove this ambiguity. \n\n\n\n\\section{Methods and Materials}\n\n\\subsection{Odor flow chamber}\n\n\\subsubsection{Flow chamber setup}\n\nThe odor chamber (\\FIG{fig1}b) was machined from \naluminum (CAD file in supplementary \\nameref{ssec:num1} section). %\nThe chamber is vacuum sealed with an acrylic lid. The inner arena contains an aluminum insert that can hold the odor sensor array or a square petri dish lid (96x96 mm). The heading in which air flow  can travel above the insert in the arena is 1 cm tall.\nThe whole setup is mounted on an optical breadboard and enclosed in a black box during imaging.\n\nThe airflow system is connected to a pressurized air source, passing through a particulate filter (Wilkerson F08) and a coalescing filter (Wilkerson M03), then regulated by mass flow controllers (MFCs, Aalborg GFC). \nMFCs are controlled via a Labjack D\/A board from a computer using custom Labview code. \nWe modulate the flow rate bubbling through liquid in enclosed bottles (Duran GL 45). The moisturized or odorized air is then passed into the flow chamber through inlet tubings. The outlets are connected to a copper manifold, then passed to a flow meter to assure that the inlet and outlet flow rate match. An optical flow sensor is fixed on the flow meter to make time stamps for opening and closing of the lid of the flow chamber during animal experiments. A photo-ionization detector (PID, piD-TECH 10.6 eV lamp) is connected to the outlet of airflow, providing calibration for the odor sensor array and detection of air leaks or odor residuals in the system.\nOutput readings from the PID, MFC,   odor sensors described in the next section, and imaging camera, are all captured on the same computer sharing the same clock. Analog signals from the PID readout and MFC readback are digitized via a Labjack and recorded with the Labview program.\n\n\n\\subsubsection{Odor flow control}\nTo construct different odor landscapes tubes from the liquid-odor and water reservoirs are connected to the flow chamber in different configurations. \nFor a centered \"cone-shape\" odor landscape the tubing carrying odorized airflow is connected to the middle inlet. For the \"biased-cone\" landscape, the tubing for odorized air is connected to the inlet 4 cm off-center. For uniform patterns, all are connected to the same source through a manifold. \nFor all experiments the background airflow that carries moisturized clean air is set to $\\sim$ 400 mL\/min, except for \\FIG{fig2} where this value was varied, \nThe odor reservoir contains either a 11mM or 110mM butanone solution in water with $\\sim$ 30 mL\/min airflow bubbling through the liquid.\n\nOverall flow rates across the chamber in experiments were always around or less than $\\sim$ 400 mL\/min to avoid turbulence. We confirmed that this regime had no turbulence by visualizing flow in a prototype chamber using dry ice and dark field illumination. Our empirical observations matched theory: Given that the chamber is 15 cm wide and 1 cm deep, a flow rate up to 1 L\/min corresponds to $\\sim$ 1.1 cm\/s. With kinematic viscosity of air $\\sim 0.15$ cm$^2$\/s, the Reynolds number is 7.3 times the flow rate in L\/min, which is below the turbulence onset (Re=2000). \n\n\\subsection{Odor sensor array}\nA spatial array of metal-oxide based gas sensors (Sensorion, SGP30) along with a relative humidity and temperature sensors (ams, ENS210) was used to measure the odor concentration field in the flow chamber. \nSensors are arranged together into groups of 16 odor sensors and 8 humidity sensors on a custom circuit board (MicroFab, Plano, TX) called an odor sensor bar (OSB). OSB's are in turn  plugged into a second circuit board (OSH Park, Portland, OR) called the odor sensor hub (OSH). OSBs can be added or removed in different arrangements depending on the experiment, for example to make room for agar. Depending on the experiment, up to 112 odor sensors are arranged in a triangular grid such that no sensor directly blocks the flow from its downstream neighbor, accompanied by 56 humidity sensors in a rectangular grid. \n\nSensors are read out via the I2C protocol. Each SGP30 sensor has the same I2C address, as does each ENS210 sensor (different from the SGP30); to address multiple sensors of the same type we use an I2C bus multiplexer (NXP, \nPCA9547\n). Each OSB contains 2 multiplexers for its 16 sensors. The multiplexers are also addressed over I2C and can have one of 8 addresses (3 address bits). On each board, the two multiplexers share two bits (set by DIP switches); the remaining bit is hardwired to be opposite on the two multiplexers. Thus each OSB can have one of 4 addresses set by DIP switches, and 4 OSBs can be shared on one I2C bus. \n\nTo communicate with the sensors, we used a Teensy 4.0 microcontroller (PJRC, Sherwood, OR) running custom Arduino software. While the Teensy has two I2C busses, we found it more straightforward to use two micro-controllers instead. Both micro-controllers communicated via USB serial to a desktop computer running custom LabView software. Measurements from all sensors are saved to computer disk in real time. Readouts from the humidity sensors are also sent to their neighboring odor sensors in real time for an on-chip humidity compensation algorithm.\n\n\n\n\n\n\n\n\n\\subsubsection*{Heat management}\nTo avoid generating thermal gradients, the system has been designed to dissipate heat to the optics table. Each metal oxide odor sensor contains a micro hotplate which consumes 86 mW power during readings. To dissipate this heat the aluminum insert inside the flow chamber serves as a heat sink. Odor sensor bars are connected to the insert using heat conductive tape and thermal paste.  The insert and chamber are in turn in direct thermal contact with the optics table. Temperature and humidity is constantly monitored at 8 locations per OSB via the on-board temperature and humidity sensors during experiments to confirm that there is no thermal or moisture gradient created in the environment.\n\n\\subsubsection{Measurements and calibration}\n\nWe measure from the odor sensors at 1 Hz for both calibration and behavior experiment modes. \nWe sample from the PID at up to 13 Hz. We synchronize and time align the measurements from the odor sensor array, MFC read-back, and PID recording with the same computer clock. %\n\nTo calibrate the odor sensors to the PID as in \\FIGSUPP[fig1]{figSI1-2}, a spatial uniform flow was delivered in a triangle wave or a step pattern.\nTime series from each odor sensor and the downstream PID were aligned by time shifting according to the peak location found via cross-correlation. The time shift was confirmed to be reasonable based on first principle estimates form the flow rate. \n\nAfter measuring odor sensors' baseline response under clean moisturized air for 5 minutes, an odorized air was delivered. \nTo fit calibration curves, the raw sensor readout was fit to the PID measurements with an exponential of form:\n\\begin{equation}\n    \\text{PID}(t) = A \\exp(B*\\text{OS}(t-\\tau))\n\\end{equation}\nwhere $\\text{PID}(t)$ voltage is on the left hand side, the scale factor $A$ and sensitivity $B$ are fitted to match the raw sensor reading $\\text{OS}(t-\\tau)$ that is time shifted by time window $\\tau$. This fitted curve maps from raw readings to odor concentration for each sensor. We validate the fitted curve across different recordings. The distribution of the coefficients $A$ and $B$ are relatively uniform across sensors in the middle of the arena. The sensor mapping are also reliable, so using $\\pm$std of the fitted curve changes less than $10\\%$ of the overall concentration scale of the landscape.\n\n\n\n\n\\subsection{Models for odor flow and odor-agar interaction}\n\nWe use two models in our work: (1) a convection-diffusion model that captures quasi-steady state odor concentration profile measured without agar used for the fits in \\FIG{fig1}f,g and (2) a reaction-convection-diffusion model for odor-agar interaction shown in \\FIG{PE_model}. A version of this second model is also used to justify the pre-equilibration protocol, as discussed in the \\hyperref[ssec:appendix]{Appendix}.\n\n\n\n\n\\subsubsection{Convection-diffusion model for odor flow without agar}\n\nTo model odor flow without agar, for example for the fits in \\FIG{fig1}f,g, we use a two-dimensional convection-diffusing model:\n\n\n\\begin{equation}\n    \\frac{\\partial C(x,y,t)}{\\partial t} = -v\\nabla C + D\\nabla^2 C \\label{eq:convection-diffusion}\n\\end{equation}\nwhere the  concentration across space and time is $C(x,y,t)$, flow velocity is $v$, and the diffusion coefficient of our odor is $D$. In our chamber, at steady state $(\\frac{\\partial C}{\\partial t} =0)$ we have:\n\\begin{equation}\n    v\\frac{\\partial C}{\\partial x} = D\\frac{\\partial^2 C}{\\partial y^2}\n    \\label{eq:steady-state}\n\\end{equation}\nbecause with our configuration flow along the $x$ axis is dominated by convection while flow along the $y$ axis is dominated by diffusion, and therefore $\\frac{\\partial^2 C}{\\partial x^2} \\ll \\frac{\\partial^2 C}{\\partial y^2}$.\n\n\n\n\nThe fit in \\FIG{fig1}f is the solution to equation \\ref{eq:steady-state}:\n\\begin{equation}\n\\label{eq:2dmodelfit}\n    C(x,y) = \\frac{C_o}{2}(1-\\erf(\\frac{x}{2\\sqrt{D\\frac{x}{v}}})) \\exp(-\\frac{y^2}{4D\\frac{x}{v}}),\n\\end{equation}\nwhere $\\erf$ is the error function and $C_o$ is the odor source concentration measured in air.\n\nIn \\FIG{fig1}g we show a fit for a one dimensional slice along $y$ at various positions along $x_c$, for the situation in which there is an\nodor-source at  $(y=0,x=0)$:\n\n\\begin{equation}\n    C(y) = \\frac{C(x_c,y=0)}{\\sqrt{4\\pi D \\frac{x_c}{v}}} \\exp(-\\frac{y^2}{4D\\frac{x_c}{v}})\n\\end{equation}\nwhere $\\frac{x_c}{v}$ is an analogy of time in non-stationary diffusion process at the cross-section at $x_c$. %\n\n\nFor the fits in \\FIG{fig1} the air flow velocity is set to be $v\\sim0.5$ cm\/s based on the flow rate and geometry of the chamber (15 parallel tubes provide around 450 mL\/min of flow into a $\\sim$255 mL chamber with $\\sim$15 cm$^2$ cross section). The diffusion coefficient $D$ is left as a free parameter and the value that minimizes the mean-squared error between the model and the empirical measurement is used. We chose to leave the diffusion coefficient as a free parameter instead of using butanone's nominal diffusion constant of $D\\sim0.08$ cm$^2$\/s, because we expect butanone's effective diffusion coefficient to be different in a confined chamber with background flow.   \nWe note that the fitted profile shown in \\FIG{fig1}g,f and the fitted value agrees with what is expected in a stable convection-diffusion process (Peclet number $\\sim 80$).\n\n\n\n\n\n\n\\subsubsection{Reaction-convection-diffusion model for odor-agar interaction}\n\n\nTo justify the pre-equilibration protocol of \\FIG{fig4} and to show that measurements of odor concentration along the agar's boundary allows us to infer the concentration on the agar, we propose a reaction-convection-diffusion model. This phenomenological model forms the basis of \\FIG{PE_model}. Compared to the convection-diffusion model, we include the \"reaction\" term to account for odor-agar interactions. \n\nThe model used is a 2D generalization of this non-spatial model:\n\\begin{equation} \\label{eqn_flow+agar}\n    \\frac{dC}{dt} = -\\frac{1}{\\tau}(C - C_o) - w \\frac{dA}{dt}\n\\end{equation}\n\n\\begin{equation} \\label{eqn_agar}\n    \\frac{dA}{dt} = k_a C (1-\\frac{A}{M}) - k_d A \n\\end{equation}\nwhere $C$ is a downstream concentration readout after the airflow has surface interaction with the agar gel. The influx odor concentration is $C_o$ and the odor concentration in agar is $A$. Without agar interaction, the flow chamber has its own timescale $\\tau$ and the molecular flux into the agar is weighted by a scalar $w$ (so $w=0$ when there's no agar in the chamber). The association and dissociation constants are $k_a$ and $k_d$ and the maximum capacity of odor concentration that can be absorbed is $M$. This model is similar to the description of odorant pulse kinetics shown in \\citep{Gorur-Shandilya2019-me}.\n\n\n\nIn \\FIG{PE_model} we use the 2D generalization:\n\\begin{equation}\n    \\frac{\\partial}{\\partial t} C(x,y) = \\mathcal{L} C(x,y) - w \\frac{\\partial}{\\partial t} A(x,y)\n\\end{equation}\n\\begin{equation}\n    \\frac{\\partial}{\\partial t} A(x,y) = k_a C(x,y)(1-\\frac{A(x,y)}{M(x,y)}) - k_d A(x,y)\n\\end{equation}\nwhere $\\mathcal{L} =  -v\\nabla  + D\\nabla^2 $ (\\autoref{eq:convection-diffusion}) is a linear operator for the convection-diffusion process and the odor influx is at the boundary $C(x=0,y=0)=C_o$. We perform numerical analysis on the set of 2D equations and permit $A$ to be non-zero only in the region where agar is present. We use a target concentration $C_o$ that is lower than $M$ and $k_a \\gg k_d$ to capture odor absorption into agar. In the simulated pre-equilibration protocol we temporarily increase $C_o$ above $M$ then switch back to the target concentration to efficiently reach a steady-state. \n\nA slightly simplified version of this model forms the basis of the arguments in the \\hyperref[ssec:appendix]{Appendix}.\n\n\n\\subsection{Animal handling}\n\n\\subsubsection{\\emph{C. elegans}}\n\nWild type \\emph{C. elegans} (N2) worms were maintained at 20 C on NGM agar plates with OP50 food patches. Before each chemotaxis experiments, we synchronized batches of worms and conducted measurements on young adults. Worms were rinsed with M9 solution and kept in S. Basal solution for around 30 min, while applying the pre-equilibration protocol to the flow chamber. Experiments were performed on $1.6\\%$ agar pads with chemotaxis solution (5 mM phosphate buffer with pH 6.0, 1 mM CaCl$_2$, 1 mM MgSO$_4$) \\citep{Bargmann1993-is, Bargmann2006-dy} formed in the lid of a 96x96 mm square dish. 50-100 worms were deposited onto the plate by pippetting down droplets of worms and removing excess solution with kimwipes. The plate was then placed in the odor flow chamber to begin recordings.\n\n\\subsubsection{\\emph{D. melanogaster}}\n\nWild type \\emph{D. melanogaster} (NM91) were maintained at 25 C incubator with 12 hr light cycle. Around 20 pairs of male and female flies were introduced into a 60 mm embryo-collection cage. A petri dish with apple juice and yeast paste was fixed at the bottom of the cage and replace every 3 hrs for two rounds during the day time. The collected eggs were kept in the petri dish in the same 25 C environment for another 48-60 hours to grow to second instars. We washed down and sorted out the second instar larva from the plate via $30\\%$ sucrose in water around 10 min before each behavioral experiments. We used a 96x96 mm lid with $2.5\\%$ agar containing $0.75\\%$ activated charcoal for larval experiments \\citep{Gepner2015-wm, Gershow2012-nt}. Around 10-20 larva were rinsed with water in a mesh and placed onto the agar plate with a paint brush. The same imaging setup and flow chamber configuration as the worm experiments were used for \\textit{Drosophila} larva.\n\n\n\n\n\n\n\n\\subsection{Imaging and behavioral analysis}\n\n\\subsubsection{Image acquisition}\n\nAnimals are imaged via a CMOS camera (Baslar, acA4112-30, with Kowa LM16FC lens) suspended above the flow chamber and illuminated by a rectangular arrangement of 850 nm LED lights. The camera acquires $2,500 \\times 3,000$ pixel images at 14 fps. A single pixel corresponded to 32 $\\mu$m on the agar plate. Labview scripts acquired images during experiments.\n\n\n\\subsubsection{\\emph{C. elegans} behavioral analysis}\n\nTo increase contrast for worm imaging, a blackout fabric sheet is placed underneath the agar plate. Custom Matlab scripts based on \\citep{Liu2018-mv} were used to process acquired images after experiments, as linked in the \\nameref{ssec:num2} section. Briefly, the centroid position of worms were found in acquired images via thresholding and binarization. The animal's centerline was found, and its body pose was estimated follwing \\citep{Liu2018-mv}, but in this work only the position and velocity was used. The tracking parameters are adjusted for this imaging setup and we extract the centroid position and velocity of worm.\n\nThe analysis pipeline focuses on the trajectory of animal navigation in the arena. The trajectories are smoothed in space with a third order polynomial in a 0.5 s time window to remove tracking noise. We only consider tracks that appear in the recording for more than 1 minutes and produce displacement larger than 3 mm across the recordings. Trajectories starting at a location with odor concentration higher than $70 \\%$ of the maximum odor concentration in space is removed, since these are likely tracks from animals that have performed chemotaxis already. We calculate the displacement of the center of the worm body in the camera space. The location in pixel space is aligned with the odor landscape constructed with the odor sensor array to compute concentration gradient given a position. To avoid double counting turns when the animal turns slowly, and to mitigate effects of small displacements from tracking noise, we measure the angle change between displacement vectors over 1 s time window and define turns as angle changes larger than 60 degrees. To quantify the curvature of navigation trajectories, we measure the angle between displacement vectors over 1 mm displacement in space. \n\n\\subsubsection{\\emph{D. melanogaster} behavioral analysis}\nAnalysis of fly larvae is performed as previously \\citep{Gepner2015-wm, Gershow2012-nt}.\n\n\n\n\n\n\n\n\n\\subsection{Data sharing}\n\\label{ssec:num1}\nRecordings for odor flow control, concentration measurements, and behavioral tracking data are publicly available: \\href{https:\/\/figshare.com\/articles\/dataset\/Continuous_odor_profile_monitoring_to_study_olfactory_navigation_in_small_animals\/21737303}{10.6084\/m9.figshare.21737303}\n\n\\subsection{Software sharing}\n\\label{ssec:num2}\n\\begin{itemize}\n    \\item Odor sensor array: \\href{https:\/\/github.com\/GershowLab\/OdorSensorArray}{https:\/\/github.com\/GershowLab\/OdorSensorArray}\n    \\item Worm imaging and analysis: \\href{https:\/\/github.com\/Kevin-Sean-Chen\/leifer-Behavior-Triggered-Averaging-Tracker-new}{https:\/\/github.com\/Kevin-Sean-Chen\/leifer-Behavior-Triggered-Averaging-Tracker-new}\n    \\item Larvae imaging: \\href{https:\/\/github.com\/GershowLab\/Image-Capture-Software}{https:\/\/github.com\/GershowLab\/Image-Capture-Software}\n\\end{itemize}\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{Acknowledgments}\nResearch reported in this work was supported by the National Institutes of Health  National Institute of Neurological Disorders and Stroke under New Innovator award number DP2-NS116768 to AML and DP2-EB022359 to MHG; the Simons Foundation under award SCGB \\#543003 to A.M.L.; by the National Science Foundation, through NSF 1455015 to MHG, an NSF CAREER Award to AML (IOS-1845137),  under Grant No. NSF PHY-1748958 and through the Center for the Physics of Biological Function (PHY-1734030). This work was also supported in part by the Gordon and Betty Moore Foundation Grant No. 2919.02. We thank the Kavli Institute for Theoretical Physics at University of California Santa Barbara for hosting us during the completion of this work.\nStrains from this work are being distributed by the CGC, which is funded by the NIH Office of Research Infrastructure Programs (P40 OD010440). We thank the Murthy Lab and Gregor Labs for flies.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nIn this work, we introduce and study the notion of a minimal Cohen-Macaulay complex. Fix a field $k$. Let $\\Delta$ be a simplicial complex. We say $\\Delta$ is minimal Cohen-Macaulay (over $k$) if it is Cohen-Macaulay and removing any facet from the facet list of $\\Delta$ results in a complex which is not Cohen-Macaulay. See Section \\ref{sec2} for precise definitions.\n\nFor the rest of the paper we shall write CM for Cohen-Macaulay. We first observe a crucial fact. \n\n\\begin{theorem}\\label{t1}\nAny CM complex is shelled over a minimal CM complex. $($Theorem \\ref{thm1}$)$ \n\\end{theorem}\n\nThus, in a strong sense, understanding CM complexes amounts to understanding the minimal ones. We support this claim by demonstrating that many interesting examples of CM complexes in combinatorics are minimal. Theorem \\ref{t1} also puts shellable complexes in a broader context: they are precisely complexes shelled over the empty one. Its proof relies on a simple but somewhat surprising statement (Lemma \\ref{isshelled}), which might be of independent interest.\n\nBelow is a collection of our main technical results which establish various necessary and sufficient conditions for a complex to be minimal CM.\n\n\\begin{theorem} The following statements hold.\n\\begin{enumerate}\n\\item[$(1)$] A minimal CM complex is acyclic. $($Corollary \\ref{cor2}$)$\n\\item[$(2)$] Let $\\Delta$ be CM and $i$-fold acyclic. If no facet of $\\Delta$ contains more than $i-1$ boundary ridges, then $\\Delta$ is minimal CM. $($Theorem \\ref{thm3}$)$\n\\item[$(3)$] If $\\Delta$ is a ball, then $\\Delta$ is minimal CM if and only if it is strongly non-shellable in the sense of \\cite{Zi98}. $($Proposition \\ref{strongnonshell}$)$\n\\item[$(4)$] If $\\Delta$ is minimal CM and $\\Gamma$ is CM, then $\\Delta \\star \\Gamma$ is minimal CM. $($Theorem \\ref{thm4}$)$\n\n\\end{enumerate}\n\\end{theorem}\n\n\nIn Section \\ref{sec2}, we give the formal definitions, provide background, and set notation. Section \\ref{main} contains the proofs of Theorem \\ref{thm1}, Theorem \\ref{thm3}, Corollary \\ref{cor2} and Proposition \\ref{strongnonshell}. In Section \\ref{newmCM}, we provide many ways to build new minimal Cohen Macaulay from old ones, such as gluing (Corollary \\ref{glue1}, Proposition \\ref{glue2}) and taking joins (Theorem \\ref{thm4}).  In the last section, we use our results to examine many classical and recent examples of Cohen-Macaulay complexes from the literature and show that they are minimal.  \n\n\n\n\n\n\n\n\n\n\\section{Background and Notation}\\label{sec2}\n\n\nOnce and for all, fix the base field $k$.  We let $\\tilde{H}_i$ denote $i$th simplicial or singular homology, as appropriate, always with coefficients in $k$. We use $\\tilde{\\chi}$ for reduced Euler characteristic. Throughout this paper, we let $\\Delta$ be a simplicial complex of dimension $d-1$ with facet list $\\{F_1,\\dots,F_e\\}$, and we denote by $\\Delta_{F_i}$ the subcomplex of $\\Delta$ with facet list $\\{F_1,\\dots,F_{i-1},F_{i+1},\\dots,F_e\\}$. We write $f_i(\\Delta)$ for the number of $i$-dimensional faces of $\\Delta$, and $h_i(\\Delta)$ for the $i$th entry of the $h$-vector of $\\Delta$; so $h_i(\\Delta)=\\sum^i_{k=0} \\binom{d-k}{i-k}(-1)^{i-k}f_{k-1}(\\Delta)$.  In particular, we note that $h_d(\\Delta)=\\sum^d_{k=0}(-1)^{d-k}f_{k-1}(\\Delta)=(-1)^{d-1}\\tilde{\\chi}(\\Delta)$.  The $\\operatorname{depth}$ of $\\Delta$ is, by definition, the depth of the Stanley-Reisner ring $k[\\Delta]$ of $\\Delta$.  We say $\\Delta$ is CM if $\\operatorname{depth} \\Delta=d$.  The following consequence of Hochster's formula (\\cite[Theorem 5.3.8]{BH98}) is an extension of Reisner's famous criterion for Cohen-Macaulayness (\\cite[Theorem]{Re76}) and gives a combinatorial characterization of $\\operatorname{depth}$. \n\n\\begin{prop}\\label{depth}\n\n$\\operatorname{depth} \\Delta \\ge \\ell$ if and only if $\\tilde{H}_{i-1}(\\operatorname{lk}_{\\Delta}(T))=0$ whenever $i+|T|<\\ell$.\n\n\n\\end{prop}\n\nWe use $\\Delta^{(i)}:=\\{ \\sigma \\in \\Delta \\colon |\\sigma| \\le i+1 \\}$ to denote the $i$-skeleton of $\\Delta$, and we note that $\\operatorname{depth} \\Delta=\\max\\{i \\mid \\Delta^{(i-1)} \\mbox{ is CM}\\}$.\n\n\n\n\n\n\nThe following definition gives the main focus of this paper.\n\n\\begin{defn}\nWe say $\\Delta$ is \\textbf{minimal CM} if $\\Delta$ is CM but $\\Delta_{F_i}$ is not CM for any $i$.\n\\end{defn}\n\n\n\nThe following related concept provides an extension of the notion of shellability.\n\n\n\\begin{defn}\n\nWe say $\\Delta_F$ to $\\Delta$ is a \\textbf{shelling move} if $\\langle F \\rangle \\cap \\Delta_F$ is pure of dimension $|F|-2$.  If $\\Gamma$ is a subcomplex of $\\Delta$ generated by facets of $\\Delta$, we say $\\Delta$ is \\textbf{shelled over $\\Gamma$} if there exists a sequence of shelling moves taking $\\Gamma$ to $\\Delta$.\n\n\n\\end{defn}\n\n\nWe note that shellable complexes are exactly those which are shelled over $\\varnothing$.\n\n\nWe will use the following definitions in the later sections. \n\n\n\\begin{defn}\nWe say that $\\Delta$ is $l$-fold acyclic if $\\operatorname{lk}_{\\Delta}(\\sigma)$ is acyclic whenever $|\\sigma|<l$.\n\\end{defn}\n\n\\begin{defn}\nA \\textit{pseudomanifold} is a simplicial complex that is pure, has a connected dual graph, and every ridge is in exactly two facets.\n\\end{defn}\n\n\n\\begin{defn}[{See \\cite{MT09}}]\n\nWe say $\\Delta$ satisfies Serre's condition $(S_{\\ell})$ if $\\ell=1$ or if $\\ell \\ge 2$ and $\\tilde{H}_{i-1}(\\operatorname{lk}_{\\Delta}(T))=0$ whenever $i+|T|<d$ and $0 \\le i<\\ell$.\n\n\n\n\\end{defn}\n\n\n\n\n\nFinally, we recall Alexander duality and some of its features.\n\n\\begin{defn}\n\nThe \\textbf{Alexander dual} of $\\Delta$ is the simplicial complex $\\Delta^{\\vee}:=\\{F \\subseteq [n] \\mid [n]-F \\notin \\Delta\\}$.\n\\end{defn}\n\n\nWe now list the connection between several combinatorial properties and the corresponding properties of the Alexander dual ideal. \n\n\\begin{theorem}\\label{dualstuff}\n\\\n\\begin{enumerate}\n\n\\item[$(1)$] $\\Delta$ is CM if and only if $I_{\\Delta^{\\vee}}$ has a linear resolution $($\\cite[Theorem 3]{ER98}$)$. \n\n\\item[$(2)$] $\\Delta$ is shellable and pure if and only if $I_{\\Delta^{\\vee}}$ has linear quotients $($\\cite[Theorem 1.4 (c)]{HH04}$)$. \n\n\\item[$(3)$] $\\Delta_F$ to $\\Delta$ is a shelling move if and only if $(I_{\\Delta_F^{\\vee}}:f)$ is generated in degree $1$, where $f=\\prod_{i \\in [n]-F} x_i$ is the monomial corresponding to $F$ in $I_{\\Delta^{\\vee}}$.\n\n\n\\end{enumerate}\n\n\\end{theorem}\n\n\\begin{remark}\nWe remark that this correspondence leads to a dual version of minimal CM: an ideal with linear resolution is minimal if removing any minimal generator results in one which does not have a linear resolution. While we won't pursue this point of view here, it was quite helpful to us; for instance, it suggested the original (algebraic) proof of Lemma \\ref{isshelled}. \n\\end{remark}\n\n\n\n\\section{Main Results}\\label{main}\n\nIn this section we prove most of our main technical results regarding minimal CM complexes. We start with the following lemma.\n\n\n\\begin{lemma}\\label{isshelled}\n\n\nSuppose $\\Delta$ satisfies Serre's condition $(S_2)$.  Then, for any facet $F \\in \\Delta$, $\\Delta$ is shelled over $\\Delta_F$.\n\n\n\n\\end{lemma}\n\nFor the convenience of the reader, we provide two proofs of this lemma, one combinatorial and one algebraic. We begin with the combinatorial proof.\n\n\n\\begin{proof}\n\nThe claim is clear if $\\Delta$ is a simplex, so we suppose this is not the case.\nNoting that $\\Delta$ is pure (since it satisfies $(S_2)$), we want to show that \\(\\langle F \\rangle \\cap \\Delta_{F}\\) is pure of dimension \\(d-2\\).\n\nLet \\(\\sigma\\) be a facet of \\(\\langle F \\rangle \\cap \\Delta_{F}\\). Then \\(F \\setminus \\sigma \\) is a facet of \\(\\operatorname{lk}_\\Delta(\\sigma)\\), but there is also a facet \\(G \\in \\Delta_{F}\\) such that \\(G \\setminus \\sigma\\) is a facet of \\(\\operatorname{lk}_\\Delta(\\sigma)\\). If $v \\in F \\setminus \\sigma \\cap G \\setminus \\sigma$, then $\\sigma \\cup \\{v\\} \\in \\langle F \\rangle \\cap \\Delta_{F}$, contradicting that $\\sigma$ is a facet.  Thus $F \\setminus \\sigma \\cap G \\setminus \\sigma=\\varnothing$, which means \\(\\operatorname{lk}_\\Delta(\\sigma)\\) is disconnected. Since \\(\\Delta\\) satisfies \\((S_2)\\), this must mean that \\(\\sigma\\) is \\(d-2\\)-dimensional. Ergo, \\(\\langle F \\rangle \\cap \\Delta_{F}\\) is pure of dimension $d-2$, and so \\(\\Delta\\) is shelled over \\(\\Delta_{F}\\).  \n\n\n\n\n\n\n\n\\end{proof}\n\n\n\nWe now provide the algebaric proof.\n\n\\begin{proof}\n\n\n\nSet $I=I_{\\Delta_F}$ and let $f$ be the monomial (of degree $c:=n-d$) in $I_{\\Delta^{\\vee}}$ corresponding to the facet $F$.  Then $I_{\\Delta^{\\vee}}=(I,f)$ and we have a short exact sequence of graded $S$-modules\n\n\\[0 \\to S\/(I:f)(-c) \\to S\/I \\to S\/(I,f) \\to 0.\\]\nThis exact sequence induces the following long exact sequence in $\\operatorname{Tor}$: \n\n\\[\\begin{tikzpicture}[descr\/.style={fill=white,inner sep=1.5pt}]\n        \\matrix (m) [\n            matrix of math nodes,\n            row sep=1em,\n            column sep=1.8em,\n            text height=1.5ex, text depth=0.25ex\n        ]\n        { S\/(I:f)(-c) \\otimes_S k & S\/I \\otimes_S k & S\/(I,f) \\otimes_S k & 0 \\\\\n           \\operatorname{Tor}_1^{S}(S\/(I:f)(-c),k) & \\operatorname{Tor}_1^{S}(S\/I,k) & \\operatorname{Tor}_1^{S}(S\/(I,f),k) & \\mbox{} \\\\\n          \\operatorname{Tor}_2^{S}(S\/(I:f)(-c),k)  & \\operatorname{Tor}_2^{S}(S\/I,k) & \\operatorname{Tor}_2^{S}(S\/(I,f),k) & \\mbox{} \\\\\n          \\mbox{}  & \\mbox{} & \\hspace{2cm} & \\mbox{} \\\\\n        };\n\n        \\path[overlay,->, font=\\scriptsize,>=latex]\n        (m-1-1) edge (m-1-2)\n        (m-1-2) edge (m-1-3)\n        (m-1-3) edge (m-1-4);\n        \\path[overlay,->, font=\\scriptsize,>=latex]\n        (m-2-3) edge[out=365,in=185] (m-1-1)\n        (m-2-2) edge (m-2-3)\n        (m-2-1) edge (m-2-2)\n        (m-3-3) edge[out=365,in=185] (m-2-1);\n        \\path[overlay,->, font=\\scriptsize,>=latex]\n        (m-3-2) edge (m-3-3)\n        (m-3-1) edge (m-3-2);\n        \\path[overlay,->, font=\\scriptsize,>=latex]\n        (m-4-3) edge[out=365,in=185,dashed] (m-3-1);   \n\\end{tikzpicture}\\]\n\n\nAs $S\/I \\otimes_S k \\cong S\/(I,f) \\otimes_S k \\cong k$, the map $S\/I \\otimes_S k \\to S\/(I,f) \\otimes_S k$ is an isomorphism. So $\\operatorname{Tor}^S_1(S\/(I,f),k) \\cong k^{\\mu(I)+1}$ and $\\operatorname{Tor}^S_1(S\/(I,f),k) \\cong k^{\\mu(I)}$. By additivity of dimensions, it follows that the map $\\operatorname{Tor}^S_1(S\/I,k) \\to \\operatorname{Tor}^S_1(S\/(I,f),k)$ is injective.  Hence the map $\\operatorname{Tor}_2(S\/(I,f),k) \\to \\operatorname{Tor}_1^S(S\/(I:f)(-c),k)$ is surjective.  But since $\\Delta$ satisfies $(S_2)$, $S\/(I,f)$ has linear first syzygy (see \\cite[Corollary 3.7]{Ya00}), so $\\operatorname{Tor}_2^S(S\/(I,f),k) \\cong k^{\\beta^S_1(I,f)}(-c-1)$. Hence it must be that $\\operatorname{Tor}_1^S(S\/(I:f)(-c),k)$ is generated in degree $-c-1$.  Thus $(I:f)$ is generated in degree $1$, and the claim follows from Theorem \\ref{dualstuff} (3).   \n\n\n\n\\end{proof}\n\n\n\\begin{theorem}\\label{thm1}\nIf \\(\\Delta\\) is a CM complex, then there is a minimal CM complex \\(\\Gamma\\) so that \\(\\Delta\\) is shelled over \\(\\Gamma\\).\n\\end{theorem}\n\n\\begin{proof}\nSince \\(\\Delta\\) is CM, it also satisfies \\((S_2)\\). We can then apply Lemma \\ref{isshelled} to conclude that, for every facet \\(F\\) of \\(\\Delta\\), \\(\\Delta\\) is shelled over \\(\\Delta_{F}\\). If none of these is  CM, then $\\Delta$ is minimal CM by definition. If not, we may continue this process to eventually reach a minimal one. \n\\end{proof}\n\n\\begin{remark}\nIt is not hard to see that a given CM complex can be shelled over two different minimal ones. For instance, let $\\Delta= K_{6,2}$ be the complete two-skeleton of the simplex on $6$ vertices, and let $\\Gamma$ be a triangulation of the projective plane on $6$ vertices.  Then $\\Delta$ is shellable and is also shelled over $\\Gamma$. That $\\Gamma$ is minimal CM follows from Corollary \\ref{twofacet}.\n\\end{remark}\n\nNext, we aim to prove that minimal CM complexes are acyclic. This is accomplished by showing a more general result. \n\n\\begin{theorem}\\label{facetdeath}\nSuppose $\\tilde{H}_{d-1}(\\Delta) \\ne 0$. Then there is a maximal facet $F_i$ of $\\Delta$ so that the following hold: \n\n\\begin{align}\n\\tag{1}\n\\dim \\tilde{H}_{i-1}(\\Delta_{F_i})&=\\begin{cases} \\dim \\tilde{H}_{i-1}(\\Delta) & \\mbox{if } 0 \\le i<d \\\\ \\dim \\tilde{H}_{i-1}(\\Delta)-1 & \\mbox{if } i=d \\end{cases} \\\\ \\tag{2}\nf_{k-1}(\\Delta_{F_i})&=\\begin{cases} f_{k-1}(\\Delta) & \\mbox{if } 0 \\le i<d \\\\ f_{k-1}(\\Delta)-1 & \\mbox{if } i=d \\end{cases}  \\\\ \\tag{3}\n\\operatorname{depth} \\Delta&=\\operatorname{depth} \\Delta_{F_i}. \n\\end{align}\n\n\\end{theorem}\n\n\\begin{proof}\n\nLet $C_{\\bullet}:0 \\rightarrow C_{d-1} \\xrightarrow{\\partial_{d-1}}  \\cdots \\xrightarrow{\\partial_2} C_1 \\xrightarrow{\\partial_1} C_0 \\rightarrow 0$ be the associated chain complex of $\\Delta$.  Choose a nonzero $\\mathfrak{C} \\in \\tilde{H}_{d-1}(\\Delta)=\\ker \\partial_{d-1}$.  Write $\\mathfrak{C}= \\displaystyle \\sum_{i=1}^e r_iF_i$ where, without loss of generality, $r_1 \\ne 0$.  Then $\\partial_{d-1}(\\mathfrak{C})=\\displaystyle \\sum_{i=1}^e r_i\\partial_{d-1}(F_i)=0$ so $\\partial(F_1)=\\displaystyle \\sum^e_{i=2}(-\\dfrac{r_i}{r_1})\\partial_{d-1}(F_i)$. Thus, if we remove $F_1$, its boundary remains, and so it follows that\n\n\\begin{equation}\\label{skels}\n\\Delta^{(d-2)}=\\Delta_{F_1}^{(d-2)}.\n\\end{equation}\n\nSo (1) and (2) are now immediate, and it only remains to show (3).  Suppose $\\sigma \\in \\Delta_{F_1}$.  \nFollowing from (\\ref{skels}), we have $(\\operatorname{lk}_{\\Delta}(\\sigma))^{(\\dim \\operatorname{lk}_{\\Delta}(\\sigma)-1)}=(\\operatorname{lk}_{\\Delta_{F_i}}(\\sigma))^{(\\dim \\operatorname{lk}_{\\Delta_{F_i}}(\\sigma)-1)}$ and so $\\operatorname{lk}_{\\Delta_{F_1}}(\\sigma)$ and $\\operatorname{lk}_{\\Delta}(\\sigma)$ have the same homologies except potentially in top degree.  It follows that $\\operatorname{depth} \\Delta=\\operatorname{depth} \\Delta_{F_1}$. \n\n\n\n\n\\end{proof}\n\n\n\n\n\\begin{cor}\\label{cor2}\n\nMinimal CM complexes are acyclic.\n\n\\end{cor}\n\n\n\\begin{proof}\n\nThis is an immediate consequence of Lemmas \\ref{isshelled} and Theorem \\ref{facetdeath}.\n\\end{proof}\n\n\n\n\n\nWhile highly acyclic CM complexes need not be minimal CM (a stack of simplices is $d-2$-acyclic but are shellable, so can never be minimal CM), one can provide some additional conditions under which they are.\n\n\n\\begin{theorem}\\label{thm3}\nIf $\\Delta$ is an \\(\\ell\\)-fold acyclic CM simplicial complex and \\(F\\) is a facet of $\\Delta$ that contains no more than \\(\\ell-1\\) boundary ridges, then $\\Delta_F$ is not CM. In particular, if every facet of \\(\\Delta\\) contains no more than \\(\\ell-1\\) boundary ridges, then \\(\\Delta\\) is minimal CM.\n\\end{theorem}\n\n\n\n\\begin{proof}\nLet \\(F\\) be a facet of \\(\\Delta\\), and let \\(\\{R_1,\\ldots,R_j\\}\\) be the set of boundary ridges contained in \\(F\\). We define \\(v_i = F \\setminus R_i\\), and \\(\\sigma = v_1\\ldots v_j\\). Since \\(\\Delta\\) is \\(\\ell\\)-fold acyclic with \\(|\\sigma|=j \\leq \\ell-1 < \\ell\\), \\(\\operatorname{lk}_\\Delta(\\sigma)\\) is acyclic. Furthermore, \\(F\\setminus \\sigma\\) has no boundary ridges in \\(\\operatorname{lk}_\\Delta(\\sigma)\\).\n\nWe first note that\n\\[h_{d-j}(\\operatorname{lk}_\\Delta(\\sigma))=\\sum^{d-j}_{i=0} (-1)^{d-j-i}f_{i-1}(\\operatorname{lk}_\\Delta(\\sigma))=(-1)^{d-j-1}\\tilde{\\chi}(\\operatorname{lk}_\\Delta(\\sigma))=0.\\]\n\nSince every ridge in \\(F\\setminus \\sigma\\) in \\(\\operatorname{lk}_\\Delta(\\sigma)\\) is contained in some other facet of \\(\\operatorname{lk}_\\Delta(\\sigma)\\),\n\n\\[f_{i-1}(\\operatorname{lk}_{\\Delta_F}(\\sigma))=\\begin{cases} f_{i-1}(\\operatorname{lk}_\\Delta(\\sigma)) & \\mbox{if } 0 \\le i<d-j \\\\ f_{i-1}(\\operatorname{lk}_\\Delta(\\sigma))-1 & \\mbox{if } i=d-j. \\end{cases}\\]\n\nThis implies \n\\begin{align*}\nh_{d-j}(\\operatorname{lk}_{\\Delta_F}(\\sigma))&=\\sum^{d-j}_{i=0} (-1)^{d-j-i}f_{i-1}(\\operatorname{lk}_{\\Delta_F}(\\sigma))\\\\\n&=\\sum^{d-j-1}_{i=0} (-1)^{d-j-i}f_{i-1}(\\operatorname{lk}_{\\Delta_F}(\\sigma))+f_{d-j}(\\operatorname{lk}_{\\Delta_F}(\\sigma))\\\\\n&=\\sum^{d-j-1}_{i=0} (-1)^{d-j-i}f_{i-1}(\\operatorname{lk}_\\Delta(\\sigma))+f_{d-j}(\\operatorname{lk}_{\\Delta_F}(\\sigma))\\\\\n&=\\sum^{d-j-1}_{i=0} (-1)^{d-j-i}f_{i-1}(\\operatorname{lk}_\\Delta(\\sigma))+f_{d-j}(\\operatorname{lk}_\\Delta(\\sigma))-1\\\\\n&=h_{d-j}(\\operatorname{lk}_\\Delta(\\sigma))-1\\\\\n&=-1.\\\\\n\\end{align*}\n\nThus $\\operatorname{lk}_{\\Delta_F}(\\sigma)$ has a negative entry in its $h$-vector, and so cannot be CM.  In particular, $\\Delta_F$ is not CM, so $\\Delta$ is minimal CM.\n\n\n\\end{proof}\n\nSetting \\(\\ell =1\\) immediately gives the following special case.\n\n\\begin{cor}\\label{twofacet}\nIf \\(\\Delta\\) is an acyclic CM pseudomanifold, then \\(\\Delta\\) is minimal CM.\n\\end{cor}\n\n\nWe now consider the relationship between the minimal CM property and the strongly non-shellable property for balls. Strongly non-shellablility has been used quite frequently in the study of non-shellable balls (see e.g. \\cite{DK78,Ha00,Lu04-2,Lu04,Zi98}). It is defined as follows: \n\n\n\\begin{defn}\nWe say a ball \\(B\\) is strongly non-shellable if \\(B_F\\) is a non-ball for any facet $F \\in B$.\n\n\n\\end{defn}\n\n\\begin{remark}\n\nA strongly non-shellable ball is often defined (as in \\cite{Zi98}) as a ball $B$ that does not contain a free facet, i.e., a facet $F$ such that $\\langle F \\rangle \\cap \\partial B$ is a ball of dimension $d-2$.  It's easy to see this definition is equivalent to the one we provide; it follows immediately from \\cite[Proposition 2.4 (iii)]{Zi98} that any ball with a free facet cannot be strongly non-shellable in our notion.  On the other hand, if \\(B_F\\) is a ball, then, as in the proof of Proposition \\ref{strongnonshell}, $\\langle F \\rangle \\cap \\partial B$ is generated by the ridges not containing $\\sigma$, so it must also be a ball.\n\n\n\n\\end{remark}\n\n\\begin{prop}\\label{strongnonshell}\nFor a ball \\(B\\), the following are equivalent:\n\\begin{enumerate}\n\\item[$(1)$] \\(B\\) is minimal CM.\n\\item[$(2)$] \\(B\\) is strongly non-shellable.\n\\end{enumerate}\n\\end{prop}\n\n\\begin{proof}\nWe first show $(1)$ implies $(2)$. Suppose \\(B\\) is minimal CM. Then removing any facet of \\(B\\) gives a complex that is not CM. This certainly can't be a ball, so $B$ is strongly non-shellable.\n\nWe now show $(2)$ implies $(1)$.  Suppose \\(B\\) is strongly non-shellable. Let \\(F\\) be a facet of \\(B\\). By Lemma \\ref{isshelled}, since \\(B\\) is CM, \\(\\langle F \\rangle \\cap B_F\\) is pure of dimension \\(d-2\\). Then the facets of \\(\\langle F \\rangle \\cap B_F\\) are exactly the ridges of $\\langle F \\rangle$ which are non-boundary ridges in \\(B\\); label these $R_1,\\dots,R_m$ and let \\(\\sigma=\\bigcap_{i=1}^m R_i\\). \n\nEach ridge of $\\langle F \\rangle$ that is not in $B_F$ must then be in $\\langle F \\rangle \\cap \\partial B$, which cannot be a ball by \\cite[Proposition 2.4 (iii)]{Zi98}.  Thus \\(\\langle F \\rangle \\cap \\partial B\\) must have a facet $\\tau$ that is not a ridge of $\\langle F \\rangle$.  It follows that $\\tau \\in \\langle F \\rangle \\cap B_F$.  In particular, we have $\\sigma \\in \\langle F \\rangle \\cap \\partial B$. \n\nNow, note that \n\\[\\operatorname{lk}_{\\langle F \\rangle \\cap B_F}(\\sigma)=\\langle R_1 \\setminus \\sigma,\\dots,R_m \\setminus \\sigma \\rangle=\\partial \\langle F \\setminus \\sigma \\rangle= \\partial \\operatorname{lk}_{\\langle F \\rangle} (\\sigma).\\]\nThis shows that $F \\setminus \\sigma$ is a facet of $\\operatorname{lk}_B(\\sigma)$ which contains no boundary ridges of $\\operatorname{lk}_B(\\sigma)$.  Since $\\sigma \\in \\partial B$, $\\operatorname{lk}_{B}(\\sigma)$ is acyclic, so we may apply Theorem \\ref{thm3} to conclude that $\\operatorname{lk}_B(\\sigma)_{F \\setminus \\sigma}=\\operatorname{lk}_{B_F}(\\sigma)$ is not CM.  Thus \\(B_F\\) is not CM, and we are done. \n\n\n\n\\end{proof}\n\n\n\n\n\n\n\n\\section{Building New Minimal CM Complexes From Old Ones}\\label{newmCM}\n\nIn this section we provide several results which show operations such as gluing or taking joins can be used to construct new examples of minimal CM complexes. \n\nWe begin with results on gluing, starting with the following corollary of Theorem \\ref{thm3}.\n\n\\begin{cor}\\label{glue1}\n\nSuppose $\\Delta_1$ and $\\Delta_2$ are CM acyclic complexes.  Set $\\Gamma=\\Delta_1 \\cap \\Delta_2$.  Suppose $\\partial D_1,\\partial D_2 \\subseteq \\Gamma$, that $\\dim(\\partial \\Delta_1)=\\Gamma=\\dim(\\partial \\Delta_2)$, and that $\\Gamma$ is acyclic and CM.  Then $\\Delta_1 \\cup \\Delta_2$ is minimal CM.\n\n\n\n\\end{cor}\n\n\\begin{proof}\nThe assumptions and construction ensures that any ridge of $\\Delta = \\Delta_1\\cup \\Delta_2$ is contained in at least two facets. Thus we can appeal to \\ref{twofacet}.\n\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nFor our next gluing result, we need the following dual notion of minimal CM. \n\n\n\\begin{defn}\n\nWe say $\\Delta$ is \\textbf{strongly CM} if $\\Delta$ is CM and $\\Delta_{F_i}$ is CM for any $i$.\n\n\n\\end{defn}\n\n\n\n\\begin{prop}\\label{glue2}\n\nIf $\\Delta$ and $\\Gamma$ are minimal CM of dimension $d-1$ and $\\Delta \\cap \\Gamma$ is strongly CM of dimension $d-1$, then $\\Delta \\cup \\Gamma$ is minimal CM. \n\\end{prop}\n\n\\begin{proof}\n\nWe have an exact sequence $0 \\to k[\\Delta \\cup \\Gamma] \\to k[\\Delta] \\oplus k[\\Gamma] \\to k[\\Delta \\cap \\Gamma] \\to 0$ and then $k[\\Delta \\cup \\Gamma]$ is CM by the depth lemma. Now let $F$ be a facet of $\\Delta \\cap \\Gamma$.  Without loss of generality, assume $F \\in \\Delta$.  If $F \\notin \\Delta \\cap \\Gamma$, then we have the exact sequence \n\n\\[0 \\to k[\\Delta_F \\cup \\Gamma] \\to k[\\Delta_F] \\oplus k[\\Gamma] \\to k[\\Delta \\cap \\Gamma] \\to 0.\\] \nSince $k[\\Delta_F]$ is not CM, neither is $k[\\Delta_F \\cup \\Gamma]$. Otherwise, $F \\in \\Delta \\cap \\Gamma$, and we have the exact sequence \n\\[0 \\to k[(\\Delta \\cup \\Gamma)_F] \\to k[\\Delta_F] \\oplus k[\\Gamma_F] \\to k[(\\Delta \\cap \\Gamma)_F] \\to 0.\\]\n\nSince $\\Delta \\cap \\Gamma$ is CM and $k[\\Delta_F] \\oplus k[\\Gamma_F]$ is not, $k[(\\Delta \\cup \\Gamma)_F]$ is not CM, completing the proof.  \n\n\n\\end{proof}\n\n\\begin{remark}\n\nThe proof of Proposition \\ref{glue2} does not actually require $\\Delta \\cap \\Gamma$ to be strongly CM, only that $\\Delta \\cap \\Gamma$ have dimension and depth $d-1$, and that $(\\Delta \\cap \\Gamma)_F$ be CM for every facet $F \\in \\Delta \\cap \\Gamma$.\n\n\n\n\\end{remark}\n\n\n\nWe end this section by showing that the join of a minimal CM complex and another (not necessarily minimal) CM complex is minimal CM. \n\n\\begin{theorem}\\label{thm4}\nIf \\(\\Delta\\) is minimal CM and \\(\\Gamma\\) is CM, then \\(\\Delta \\star \\Gamma\\) is minimal CM.\n\\end{theorem}\n\n\\begin{proof}\nFirst we note that \\(\\Delta \\star \\Gamma\\) is CM. Let \\(F\\) be a facet of \\(\\Delta \\star \\Gamma\\). We may write \\(F = F' \\star G\\) for some facets \\(F'\\) of \\(\\Delta\\) and \\(G\\) of \\(\\Gamma\\). Since \\(\\Delta_{F'}\\) is not CM, there exists \\(\\sigma \\in \\Delta_{F'}\\) such that \\(\\tilde{H}_i(\\operatorname{lk}_{\\Delta_{F'}}(\\sigma)) \\ne 0\\) for some \\(i<\\dim(\\Delta)-|\\sigma|\\).\n\nWe now show that \\(L=\\operatorname{lk}{\\Delta_{F'}}(\\sigma)\\) and \\(L'=\\operatorname{lk}_{(\\Delta\\star\\Gamma)_{F}}(\\sigma \\star G))\\) are isomorphic posets. If \\(\\tau \\in L\\), it is immediate that \\(\\tau \\star \\emptyset \\in L'\\). Furthermore, if \\(\\tau \\star \\emptyset \\in L'\\), then \\(\\tau \\in L\\). Suppose \\(\\tau \\star \\gamma \\in L'\\), then \\(\\gamma \\cup G \\in \\Gamma\\) and \\(\\gamma \\cap G = \\emptyset\\). This implies that \\(\\gamma = \\emptyset\\). So \\(L\\) and \\(L'\\) are isomorphic posets.\n\nWith this isomorphism, we see that \\(\\tilde{H}_i(\\operatorname{lk}_{(\\Delta\\star\\Gamma)_{F}}(\\sigma \\star G)) \\ne 0\\) for some \\(i<\\dim(\\Delta \\star \\Gamma)-|\\sigma \\star G|\\). Thus \\((\\Delta \\star \\Gamma)_F\\) is not CM by Reisner's Criterion (\\cite[Theorem 1]{Re76}), and therefore, \\(\\Delta \\star \\Gamma\\) is minimal CM.\n\\end{proof}\n\n\n\\section{Examples}\\label{examples}\n\nIn this section,  we consider some notable examples of CM complexes from the literature, and show that they are minimal CM. We expect that this is far from a complete list of minimal CM examples currently published. \n\nA large class of minimal CM complexes are those that satisfy the conditions of Corollary \\ref{twofacet}. The following complexes fall in this class:\n\n\\begin{itemize}\n\\item Triangulations of \\(\\mathbb{RP}^{2n}\\) (over \\(k\\) of characteristic not 2)\n\\item The dunce hat\n\\item Bing's House with 2 rooms \\cite{Ha99}\n\\item The pastry \\cite{Do18}\n\\end{itemize}\n\n\nThe following are all strongly non-shellable balls, which are minimal CM by Proposition \\ref{strongnonshell}.\n\n\\begin{itemize}\n\\item Rudin's Ball \\cite{Ru58}\n\\item \\(B^3_{16,48}\\), \\(B^3_{12,37,a}\\), and  \\(B^3_{12,37,b}\\) \\cite{Lu04}\n\\item \\(B_{3,9,18}\\) \\cite{Lu04-2}\n\\item Ziegler's Ball \\cite{Zi98}\n\\end{itemize}\n\nThis next class of minimal CM complexes are constructible complexes which are not themselves balls. Each of these were verified to be minimal CM by applying Theorem \\ref{thm3}. \n\n\\begin{itemize}\n\\item The complex \\(C_3\\) in \\cite{DG16}, a non-partitionable CM complex\n\\item The complex \\(C_3\\) in \\cite{JV17}, a balanced non-partitionable CM complex\n\\item The complex \\(\\Omega_3\\) in \\cite{DG18}, a 2-fold acyclic complex with no decomposition into rank 2 boolean intervals\n\\end{itemize}\n\nEach of these complexes is a counterexample to an associated conjecture in the literature; see the references for more details. They are each the result of gluing many copies of a CM complex along a CM subcomplex, a similar process to  Proposition \\ref{glue2}.\n\n\n\n\n\\section*{Acknowledgments}\nWe thank Bruno Benedetti for insightful conversations regarding constructions of interesting balls and nonshellable complexes. The first author also acknowledges partial support from Simons Collaboration Grant FND0077558.  We thank the Mathematics Department at the University of Kansas for the excellent working and collaborating conditions.\n\n\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} Let the base field $K$ be the field of\ncomplex numbers. Consider a cyclic quiver with $h$ vertices and\ndimension vector ${\\underline d}=(d_1,\\cdots, d_h)$:\n$$\\xymatrix{ 1 \\ar[r] & 2 \\ar[r] & \\cdots & h-2 \\ar[r] &\nh-1\\ar[dll]\\\\ &&h \\ar[ull] & & }$$ Denote $V_i=K^{d_i}$. Let\n$$Z=Hom(V_1,V_{2})\\times \\cdots\\times Hom(V_h,V_{1}),\\\nGL_{{\\underline d}}=\\prod_{1\\le i\\le h}\\,GL{(V_i)}$$ We have a\nnatural action of $GL_{{\\underline d}}$ on $Z$: for\n$g=(g_1,\\cdots,g_h)\\in GL_{{\\underline d}},f=(f_1,\\cdots,f_h)\\in\nZ$,\n$$g\\cdot f=(g_2f_1g_1^{-1},g_3f_2g_2^{-1},\\cdots,g_1f_hg_h^{-1})$$\nLet $$\\mathcal{N}=\\{(f_1,\\cdots,f_h)\\in Z\\,|\\,f_h\\circ\nf_{h-1}\\circ\\cdots\\circ f_1:V_1\\rightarrow V_1{\\mathrm{\\ is\\\nnilpotent}}\\}$$\nNote that $f_h\\circ f_{h-1}\\circ\\cdots\\circ f_1:V_1\\rightarrow\nV_1$ being nilpotent is equivalent to\n\n\\noindent $f_{i-1}\\circ f_{i-2}\\circ\\cdots\\circ f_1\\circ f_h\n\\circ\\cdots\\circ f_{i+1} f_i:V_i\\rightarrow V_i$ being nilpotent.\nClearly $\\mathcal{N}$ is $GL_{{\\underline d}}$-stable. Lusztig\n(cf.\\cite{lus}) has shown that an orbit closure in $\\mathcal{N}$\nis canonically isomorphic to an open subset of a Schubert variety\nin ${\\widehat{SL}}_n\/Q$, where $n=\\sum_{1\\le i\\le h}\\, d_i$, and\n$Q$ is the parabolic subgroup of ${\\widehat{SL}}_n$ corresponding\nto omitting $\\alpha_0,\\alpha_{d_{1}},\n\\alpha_{d_{1}+d_{2}},\\cdots,\\alpha_{d_{1}+\\cdots +d_{h-1}}$\n($\\alpha_i, 0\\le i\\le n-1$ being the set of simple roots for\n${\\widehat{SL}}_n$). Corresponding to $h=1$, we have that\n$\\mathcal{N}$ is in fact the variety of nilpotent elements in\n$M_{d_{1},d_{1}}(K)$, and thus the above isomorphism identifies\n$\\mathcal{N}$ with an open subset of a Schubert variety\n$X_{\\mathcal{N}}$ in ${\\widehat{SL}}_n\/G_0$, $G_0$\nbeing the maximal parabolic subgroup of ${\\widehat{SL}}_n$\ncorresponding to ``omitting\" $\\alpha_0$.\n\nLet now $h=2$\n$$Z_0=\\{(f_1,f_2)\\in Z\\,|\\,f_2\\circ f_1=0,f_1\\circ f_2=0\\}$$\nStrickland (cf. \\cite{st}) has shown that each irreducible\ncomponent of $Z_0$ is the conormal variety to a determinantal\nvariety in $M_{d_1,d_2}(K)$. A determinantal variety in $M_{d_1,d_2}(K)$\nbeing canonically isomorphic to an open subset in a certain\nSchubert variety in $G_{d_2,d_1+d_2}$ (the Grassmannian variety of\n$d_2$-dimensional subspaces of $K^{d_1+d_2}$) (cf.\\cite{g\/p-2}),\nthe above two results of Lusztig and Strickland suggest a\nconnection between conormal varieties to Schubert varieties in the\n(finite-dimensional) flag variety and the affine Schubert\nvarieties. This is the motivation for this article. \nLet $G=SL_n$.\n\nInspired by Lusztig's  embedding of $\\mathcal N$ in $\\widehat {SL_n}\/Q$,\nwe define a family of maps $\\psi_p:T^*G\/B\\hookrightarrow\\widehat{SL}_n\/\\mathcal{B}$, parametrized by polynomials in one variable  with coefficients in $\\mathbb C((t))$, and with $1$ as the constant term.\nFor a particular map $\\phi$ (analogous to Lusztig's map) in this family, we find a $\\kappa_0\\in\\widehat W$ such that the affine Schubert variety $X(\\kappa_0)$ is $G_0$-stable ($G_0$ being as above, the\nmaximal parabolic subgroup of ${\\widehat{SL}}_n$ corresponding to\n``omitting\" $\\alpha_0$) and show that $\\phi$ gives an embedding $T^*G\/B\\hookrightarrow X(\\kappa_0)\\subset{\\widehat{SL}}_n\/\\mathcal{B}$.\nWe thus obtain a ${{SL_n}}$-stable closed subvariety of $X(\\kappa_0)$ as a natural\ncompactification of $T^*G\/B$ (cf. Theorem \\ref{ctgt}). Let\n$\\pi:{\\widehat{SL}}_n\/\\mathcal{B}\\rightarrow\n{\\widehat{SL}}_n\/G_0$  be the canonical projection. Then we show\nthat $\\pi(T^*G\/B)=\\mathcal{N}$, the variety of nilpotent\nmatrices, and that $\\pi\\,|_{T^*G\/B}:T^*G\/B\\rightarrow \\mathcal{N}$\nis in fact the Springer resolution.\n\nFollowing the above ideas, Lakshmibai (cf. \\cite{ls}) has obtained a stronger result for $T^*G_{d,n}$, the cotangent bundle to the Grassmannian variety $G_{d,n}$. She shows that there is an embedding $\\chi$ (analogous to $\\phi$) of $T^*G_{d,n}$ inside a Schubert variety $X(\\iota)\\subset\\widehat {SL_n}\/\\mathcal Q_d$ (where $\\mathcal Q_d$ is the two step parabolic subgroup of $\\widehat {SL_n}$ corresponding to omitting $\\alpha_0,\\alpha_d$) such that $X(\\iota)$ \\textbf{is in fact a compactification of}  $T^*G_{d,n}$.\nThe result of \\cite{ls} has been generalized to $T^*G\/P$ in \\cite{comin}, $G\/P$ being a cominuscule Grasssmannian variety.\n\nIt would be interesting to know if the result of \\cite{ls} could be achieved replacing $P$ with $B$, for a suitable generalization of $\\chi$.\nWe show in \\S\\ref{counter} that this is not possible for any $\\psi_p$ in the above family, even when $n=3$. \nWe think that our result about the embedding $\\phi:T^*G\/B\\hookrightarrow X(\\kappa_0)$ identifying a certain ${{SL_n}}$-stable closed subvariety of $X(\\kappa_0)$ as a natural\ncompactification of $T^*G\/B$ is the best possible in relating $T^*G\/B$ and affine Schubert varieties in $\\widehat{SL_n}\/\\mathcal B$.\n\nThe results of this paper open up other related problems like,  the study of \nline bundles on $T^*G\/B, G=SL_n$ (using the embedding of $T^*G\/B$ into $X(\\kappa_0)$, and realizing line bundles on $T^*G\/B$ as restrictions of suitable line bundles on $X(\\kappa_0)$), establishing similar embeddings of the cotangent bundles to partial flag varieties $G\/Q$ ($G$ semi-simple and $Q$ a parabolic subgroup) etc. Further, the facts that conormal varieties to Schubert varieties in $G\/B$ are closed subvarieties of  $T^*G\/B$, and that the affine Schubert variety $X(\\kappa_0)$ contains a $G$-stable closed subvariety which is a natural\ncompactification of $T^*G\/B$, suggest similar compactifications\nfor conormal varieties to Schubert varieties in $G\/B$ (by\nsuitable affine Schubert varieties in\n${\\widehat{SL}}_n\/\\mathcal{B})$; such a realization could lead to important consequences such as a knowledge of  the equations of the conormal varieties (to Schubert varieties) as subvarieties of  the cotangent bundle. These problems will be dealt with in a subsequent paper.\n\nRegarding results on similar compactifications, we mention Mirkovic-Vybornov's work (cf. \\cite{m-v}), where the authors construct compactifications of Nakajima's quiver varieties of type $\\mathbf A$ inside affine Grassmannians of type $\\mathbf A$. Also, Manivel and Michalek (\\cite{m-m}) have recently studied the local geometry of tangential varieties (which are compactifications of the tangent bundle) to cominuscule Grassmannians.\n\nThe sections are organized as follows. In \\S \\ref{prelim}, we fix\nnotation and recall \\emph{affine Schubert varieties}. In \\S\n\\ref{elt}, we introduce the elements $\\kappa$ and $\\kappa_0$ (in ${\\widehat{W}}$,\nthe affine Weyl group), and prove some properties of $\\kappa$. In\n\\S \\ref{main}, we prove a crucial result on $\\kappa$ needed for\nrealizing the embeddings of $\\mathcal{N}$ and $T^*SL_n\/B$ inside\n${\\widehat{SL}}_n\/G_0$ and ${\\widehat{SL}}_n\/\\mathcal{B}$\nrespectively. In \\S \\ref{lumap}, we spell out Lusztig's\nisomorphism which identifies $\\mathcal{N}$ with an open subset of\n$X_{G_0}(\\kappa)$ (inside ${\\widehat{SL}}_n\/G_0$).\nIn \\S \\ref{bundle}, using the map $\\phi:T^*G\/B\\rightarrow \\widehat{SL}_n\/\\mathcal B$ as above and the natural projection $\\widehat{SL}_n\/\\mathcal B\\rightarrow \\widehat{SL}_n\/G_0$, we recover the Springer resolution of $\\mathcal N$;\nwe also prove the main result that $\\phi$ identifies an $SL_n$-stable closed subvariety of $X(\\kappa_0)$ as a compactification of $T^*G\/B$.\nIn \\S \\ref{counter}, we show that it is not possible, for any choice in the family $\\psi_p$, to realize an affine Schubert variety (in $\\widehat{SL}_3\/\\mathcal B$) as a compactification of the cotangent bundle $T^*SL_3\/B$.\n\n\\vskip.2cm\\noindent\\textbf{Acknowledgement:} The first author wishes to\nthank Chennai Mathematical Institute for the hospitality extended\nto her during her visit in Winter 2014 when this work was started. \nThe authors thank the Macaulay2 team for providing a \nweb client to the same at http:\/\/habanero.math.cornell.edu:3690\/\n\\section{Affine Schubert varieties}\\label{prelim} Let $K=\\mathbb{C}, F=K((t))$,\nthe field of Laurent series, $A=K[[t]]$. Let $G$ be a\nsemi-simple algebraic group over $K$, $T$ a maximal torus in $G$,\n$B$ a Borel subgroup, $B\\supset T$, and let $B^{-}$ be the Borel subgroup opposite to $B$. \nLet $\\mathcal{G}=G(F)$. The natural inclusion\n$K\\hookrightarrow A\\hookrightarrow F$ induces an inclusion\n$$G\\hookrightarrow G(A)\\hookrightarrow\\mathcal{G}$$ The\nnatural projection\n$A\\rightarrow K$ given by $t\\mapsto 0$\ninduces a homomorphism\n$$\\pi:G(A)\\rightarrow G$$ \nThe group $\\mathcal{B}:=\\pi^{-1}(B)$ is a Borel subgroup of $\\mathcal G$. \n\n\n\\subsection{Bruhat decomposition:}  Let ${\\widehat W}=N(K[t,t^{-1}])\/T$, the \\emph{affine Weyl group} of\n$G$ (here, $N$ is the normalizer of $T$ in $G$). \nThe group $\\widehat W$ is a Coxeter group (cf. \\cite{kac}).\nWe have that $$G(F)=\\dot\\cup_{w\\in{\\widehat W}}\\mathcal{B}w\\mathcal{B},\nG(F)\/\\mathcal{B}=\\dot\\cup_{w\\in{\\widehat\nW}}\\mathcal{B}w\\mathcal{B}(mod\\,\\mathcal{B})$$ For $w\\in{\\widehat\nW}$, let $X(w)$ be the \\emph{affine Schubert variety} in\n$G(F)\/\\mathcal{B}$:\n$$X(w)=\\dot\\cup_{\\tau\\le w}\\mathcal{B}\\tau\\mathcal{B}(mod\\,\n\\mathcal{B})$$ It is a projective variety of dimension $\\ell(w)$.\n\n\\subsection{Affine Flag variety,  Affine Grassmannian:}\n\nLet $G=SL(n)$, $\\mathcal{G}=G(F)$, $G_0=G(A)$. \nWe say $g\\in\\mathcal G$ is integral if and only if $g\\in G_0$, i.e viewed as $G$-valued meromorphic function on $\\mathbb C$, it has no poles at $t=0$.\nThe homogeneous space $\\mathcal{G}\/\\mathcal{B}$ is the \\emph{affine Flag variety}, and $\\mathcal{G}\/G_0$ is the \\emph{affine Grassmannian}. \nFurther, \n$$\\mathcal{G}\/G_0=\\dot\\cup_{w\\in{\\widehat W}^{G_0}}\\mathcal{B}w\\,G_0 (mod\\,G_0)$$ \nwhere ${\\widehat W}^{G_0}$ is the set of minimal representatives in ${\\widehat W}$ of ${\\widehat W}\/W_{G_0}$.\n\n\\noindent  Let $${\\widehat\n{Gr(n)}}=\\{A\\texttt{-lattices\\ in\\ } F^n \\}$$ Here, by an\n$A$-lattice in $F^n$, we mean a free $A$-submodule of $F^n$ of\nrank $n$. Let $E$ be the standard lattice, namely, the $A$-span of\nthe standard $F$-basis $\\{e_1,\\cdots,e_n\\}$ for $F^n$. For $V\\in\n{\\widehat {Gr(n)}}$, define\n$$\\mathrm{vdim}(V):=dim_K(V\/V\\cap E)-dim_K(E\/V\\cap E)$$\n One refers to $\\mathrm{vdim}(V)$ as the\n \\emph{virtual dimension of $V$}. For $j\\in \\mathbb{Z}$ denote\n$${\\widehat {Gr_j(n)}}=\\{V\\in{\\widehat {Gr(n)}}\\,|\\,\n\\mathrm{vdim}(V)=j\\}$$ Then ${\\widehat {Gr_j(n)}},j\\in\\mathbb{Z}$\ngive the connected components of ${\\widehat {Gr(n)}}$. We have a\ntransitive action of $GL_n(F)$ on ${\\widehat {Gr(n)}}$ with\n$GL_n(A)$ as the stabilizer at the standard lattice $E$. Further,\nlet $\\mathcal{G}_0$ be the subgroup of $GL_n(F)$, defined as,\n$$\\mathcal{G}_0=\\{g\\in GL_n(F)\\,|\\,\\mathrm{ord(det\\, }g)=0\\}$$\n(here, for a $f\\in F$, say $f=\\sum\\,a_it^i$, \\emph{order f} is the\nsmallest $r$ such that $a_r\\ne 0$). Then $\\mathcal{G}_0$ acts\ntransitively on ${\\widehat {Gr_0(n)}}$ with $GL_n(A)$ as the\nstabilizer at the standard lattice $E$. Also, we have a transitive\naction of $SL_n(F)$ on ${\\widehat {Gr_0(n)}}$ with $SL_n(A)$ as\nthe stabilizer at the standard lattice $E$. Thus we obtain the\nidentifications:\n$$\\begin{gathered}GL_n(F)\/GL_n(A)\\simeq{\\widehat {Gr(n)}}\\\\\n\\mathcal{G}_0\/GL_n(A)\\simeq{\\widehat {Gr_0(n)}},\nSL_n(F)\/SL_n(A)\\simeq{\\widehat {Gr_0(n)}}\n\\end{gathered}\\leqno{(*)}$$ In particular, we obtain\n$$\\mathcal{G}_0\/GL_n(A)\\simeq SL_n(F)\/SL_n(A)\\leqno{(**)}$$\n\n\\subsection{Generators for ${\\widehat W}$:}\\label{gen} \nRecall that the Weyl group $\\widehat W=N(K[t,t^{-1}])\/T$.\nLet $R$ (resp. $R^+$) be the set of roots (resp. positive roots) of $G$ relative to $B$, and let $\\delta$ be the basic imaginary root of the affine Kac-Moody algebra of type $\\widehat{\\mathbf A}_{n-1}$ given by (cf. \\cite{kac})\n$$\\delta=\\alpha_0+\\theta=\\alpha_0+\\cdots+\\alpha_{n-1}$$\nThe set of real roots of $\\mathcal{G}$ is given by $\\left\\{r\\delta+\\beta\\mid r\\in\\mathbb Z, \\beta\\in R\\right\\}$, and the set of positive roots of $\\mathcal{G}$ is given by $\\left\\{r\\delta+\\beta\\mid r>0, \\beta\\in R\\right\\}\\dot\\cup R^+$ (cf. \\cite{kac}).\nFollowing the notation in \\cite{kac}, we shall work with the set of generators for\n${\\widehat W}$ given by $\\{s_0, s_1,\\cdots, s_{n-1}\\}$, where\n$s_i, 0\\le i\\le n-1$ are the reflections with respect to\n$\\alpha_i, 0\\le i\\le n-1$. Note that $\\{\\alpha_i, 1\\le i\\le n-1\\}$\nis simply the set of simple roots of $SL_n$ (with respect to the\nBorel subgroup $B$). \nIn particular, the Weyl group $W$ of $SL_n(\\mathbb C)$ is simply the subgroup of $\\widehat W$ generated by $\\{s_1,\\cdots,s_{n-1}\\}$. \n\n\\def\\ensuremath Q{\\ensuremath Q}\n\\subsection{The Affine Presentation:}\n\\label{affine}\nThe generators $s_i,\\, 1\\leq i \\leq n-1$ have the following canonical lifts to $N(K[t,t^{-1}])$:\n$s_i$ is the permutation\nmatrix $(a_{rs})$, with $a_{jj}=1,j\\ne i,i+1,\\  a_{i\\,i+1}=1,\na_{i+1\\,i}=-1$, and all other entries are $0$. \nA canonical lift for $s_0$ is given by\n$$\\begin{pmatrix}\n0&0&\\cdots & t^{-1}\\\\\n0&1&\\cdots &0\\\\\n\\vdots & \\vdots & \\vdots & \\vdots\\\\\n0&\\cdots &1 &0\\\\\n-t &0&0&0\n\\end{pmatrix}$$\nLet $s_\\theta\\in W$ be the reflection with respect to the longest root $\\theta$ in $\\mathbf A_{n-1}$ given by $\\theta=\\alpha_1+\\cdots+\\alpha_{n-1}$.\nLet $L$ (resp. \\ensuremath Q) be the root (resp. coroot) lattice of $\\mathfrak{sl}_n(=\\operatorname{Lie}(SL_n))$, and let $\\langle\\,,\\,\\rangle$ be the canonical pairing on $L\\times\\ensuremath Q$. \nConsider $\\theta^\\vee\\in\\ensuremath Q$ given by $\\theta^\\vee=\\alpha_1^\\vee+\\cdots+\\alpha_{n-1}^\\vee$.\nThere exists (cf. \\cite{kumar}, \\S 13.1.6) a group isomorphism $\\widehat{W}\\rightarrow W\\ltimes\\ensuremath Q$ given by \\begin{align*}\n    s_i&\\mapsto s_i                                 &\\text{ for }1\\leq i\\leq n-1\\\\\n    s_0&\\mapsto s_\\theta\\lambda_{-\\theta^{\\vee}}    &\n\\end{align*}\nwhere we write $\\lambda_q$ for $(\\operatorname{id},q)\\in W\\ltimes\\ensuremath Q$.\nIn particular, we get $s_0s_\\theta\\mapsto\\lambda_{\\theta^\\vee}$, which we use to compute a lift of $\\lambda_{\\theta^\\vee}$ to $N(K[t,t^{-1}])$: \\begin{align*} \n\\begin{pmatrix}\n0&0&\\cdots & t^{-1}\\\\\n0&1&\\cdots &0\\\\\n\\vdots & \\vdots & \\vdots & \\vdots\\\\\n0&\\cdots &1 &0\\\\\n-t &0&0&0\n\\end{pmatrix}\n\\begin{pmatrix}\n0&0&\\cdots &-1\\\\\n0&1&\\cdots &0\\\\\n\\vdots & \\vdots & \\vdots & \\vdots\\\\\n0&\\cdots &1 &0\\\\\n1 &0&0&0\n\\end{pmatrix\n=\\begin{pmatrix}\nt^{-1}&0&\\cdots &0\\\\\n0&1&\\cdots &0\\\\\n\\vdots & \\vdots & \\vdots & \\vdots\\\\\n0&\\cdots &1 &0\\\\\n0 &0&0&t\n\\end{pmatrix}\n\\end{align*}\nConsider the element $w\\in W$ corresponding to $(1,i)(i+1,n)\\in S_n$, and \nobserve that $w(\\theta^\\vee)=\\alpha_i^\\vee$, the $i^{th}$ simple coroot.  \nIt follows that a lift of $\\lambda_{\\alpha_i^\\vee}=w\\lambda_{\\theta^\\vee}w^{-1}$ is given by\\begin{align*}\nw\\begin{pmatrix}\nt^{-1}&0&\\cdots &0\\\\\n0&1&\\cdots &0\\\\\n\\vdots & \\vdots & \\vdots & \\vdots\\\\\n0&\\cdots &1 &0\\\\\n0 &0&0&t\n\\end{pmatrix}w^{-1\n=\\begin{pmatrix}\n \\ddots& &&\\\\\n & t^{-1} & &\\\\\n  & & t&\\\\\n  & &  &\\ddots\\\\\n\\end{pmatrix}\n\\end{align*}\nwhere in the matrix on the right hand side, the dots are $1$, and the off-diagonal entries are $0$, i.e., the matrix on the right hand side is the diagonal matrix with $i,\\,(i+1)$-th entries being $t^{-1},t$ respectively, and all other diagonal entries being $1$.\n\\\\\n\\\\\nThe (Coxeter) length of $\\lambda_q$ is given by the following formula (cf. \\cite{kumar}, \\S13.1.E(3)):$$\n    l(\\lambda_q)=\\sum\\limits_{\\alpha\\in R^+}\\lvert\\alpha(q)\\rvert,\\qquad q\\in\\ensuremath Q$$ \nwhere $\\alpha(q):=\\langle\\alpha,q\\rangle$. \nThe action of $\\lambda_q$ on the root system of $\\mathcal G$ is determined by the following formulae (cf. \\cite{kumar}, \\S13.1.6):\\begin{align*}\n    \\lambda_q(\\alpha)&=\\alpha-\\alpha(q)\\delta,\\qquad\\text{ for }\\alpha\\in R,q\\in\\ensuremath Q\\\\\n    \\lambda_q(\\delta)&=\\delta\n\\end{align*} \nIn particular, for $\\alpha\\in R^+$, $\\lambda_q(\\alpha)>0$ if and only if $\\alpha(q)\\leq 0$.\n\\begin{cor}\n\\label{count}\nFor $\\alpha\\in R^+,\\,q\\in\\ensuremath Q$, $l(\\lambda_qs_\\alpha)>l(\\lambda_q)$ if and only if $\\alpha(q)\\leq 0$.\n\\end{cor}\n\\begin{proof}\nFollows from the equivalence $ws_\\alpha>w$ if and only if $w(\\alpha)>0$, applied to $w=\\lambda_q$.\n\\end{proof}\n\n\n\\section{The element $\\kappa_0$}\\label{elt} \nOur goal is to give a compactification of the cotangent bundle $T^*G\/B$ as a (left) $SL_n$ stable subvariety of the affine Schubert variety $X(\\kappa_0)$, where $\\kappa_0$ is as defined below:\\begin{align*}\n    \\tau&:=s_{n-1}\\cdots s_2s_1s_0 \\\\\n    \\kappa&:=\\tau^{n-1}\\\\\n    \\kappa_0&:=w'\\tau^{n-1}\n\\end{align*}\nwhere $w'$ is the longest element in the Weyl group generated by $s_1,\\cdots s_{n-2}$.\nWe first prove some properties of $\\kappa$ and $\\tau$ which are consequences of the braid relations\n$$\\begin{gathered}s_is_{i+1}s_{i}=s_{i+1}s_{i}s_{i+1},\n0\\le i\\le n-2,\\\\\ns_0s_{n-1}s_{0}=s_{n-1}s_{0}s_{n-1}\\end{gathered}$$ \nand the commutation relations:\n$$s_is_j=s_js_i, 1\\le i,j\\le n-1, |i-j|>1,\\ \\  s_0s_i=s_is_0,\n2\\le i\\le n-2$$\n\n\\subsection{Some Facts:}\\label{facts}\n\n\\noindent\\textbf{Fact 1:} $\\tau(\\delta)=\\delta$\n\n\\noindent\\textbf{Fact 2:}\n$\\tau(\\alpha_1+\\cdots+\\alpha_{n-1})=2\\delta+\\alpha_{n-1}$\n\n\\noindent\\textbf{Fact 3:} $\\tau(r\\delta+\\alpha_i+\\cdots+\\alpha_{n-1})=\n(r+1)\\delta+\\alpha_{i-1}+\\alpha_i+\\cdots+\\alpha_{n-1},2\\le i\\le\nn-1,r\\in\\mathbb{Z}_+$\n\n\\noindent\\textbf{Fact 4:} $s_{n-1}\\cdots\ns_{j+1}(\\alpha_j)=\\alpha_{j}+\\alpha_{j+1}+\\cdots+\\alpha_{n-1},j\\ne\n0, n-1$\n\n\\noindent\\textbf{Fact 5:} $s_{n-1}\\cdots\ns_{1}(\\alpha_0)=\\delta+\\alpha_{n-1}$\n\n\\noindent\\textbf{Fact 6:}\n$\\tau(\\alpha_{n-1})=\\delta+\\alpha_{n-2}+\\alpha_{n-1}$ (a special\ncase of Fact 3 with $r=0,i=n-1$)\n\n\\noindent\\textbf{Fact 7:} $\\tau(\\alpha_1)=\\alpha_0+\\alpha_{n-1}$\n\n\\noindent\\textbf{Fact 8:} $\\tau(\\alpha_i)=\\alpha_{i-1},i\\ne 1,n-1$\n\n\\noindent\\textbf{Fact 9:} $\\tau(\\alpha_0+\\alpha_{n-1})=\\alpha_{n-2}$\n\n\\begin{remark}\\label{cyclic}\nFacts 7, 8, 9 imply that $(\\alpha_{n-1}+\\alpha_0,\\alpha_{n-2},\\alpha_{n-3},\\ldots,\\alpha_1)$ is a cycle of order $n-1$ for $\\tau$.\nIn particular, each of these roots is fixed by $\\kappa$.\n\\end{remark}\n\n\\subsection{A reduced expression for $\\kappa$}\\label{kap} Let $\\kappa$ be the element in\n${\\widehat{W}}$ defined as above. We may write $\\kappa=\\tau_1\\cdots\\tau_{n-1}$, where\n$\\tau_i$'s are equal, and equal to $\\tau(=s_{n-1}\\cdots\ns_2s_1s_0)$ (we have a specific purpose behind writing $\\kappa$ as\nabove).\n\\begin{lem}\\label{reduced}\n  The expression $\\tau_1\\cdots\\tau_{n-1}$\nfor $\\kappa$ is reduced.\n\\end{lem}\n\\begin{proof}\n\n\\noindent\\textbf{Claim:} $\\tau_1\\cdots\\tau_{i}s_{n-1}\\cdots\ns_{j+1}(\\alpha_j),1\\le i\\le n-2, 0\\le j\\le n-2,\n\\tau_1\\cdots\\tau_{i}(\\alpha_{n-1})$, \n\n\\noindent $1\\le i\\le n-2$ are positive\nreal roots.\n\nNote that the Claim implies the required result. We divide the\nproof of the Claim into the following three cases.\n\n\\noindent\\textbf{Case 1:} \\emph{To show:}\n$\\tau_1\\cdots\\tau_{i}(\\alpha_{n-1}),1\\le i\\le n-2$ is a positive\nreal root.\n\n\\noindent We have\n\n\\noindent $\\tau_1\\cdots\\tau_{i}(\\alpha_{n-1})$\n\n\\noindent $=\\tau_1\\cdots\\tau_{i-1}(\\delta+\\alpha_{n-2}+\\alpha_{n-1})$\n(cf. \\S \\ref{facts}, Fact 6)\n\n\\noindent $=\\tau_1\\cdots\\tau_{i-2}(2\\delta+\\alpha_{n-3}+\\alpha_{n-2}+\n\\alpha_{n-1})$ (cf. \\S \\ref{facts}, Fact 3)\n\n\\noindent $=\\tau_1\\cdots\\tau_{i-k}(k\\delta+\\alpha_{n-k-1}+\\cdots+\n\\alpha_{n-1}), 0\\le k\\le i-1$ (cf. \\S \\ref{facts}, Fact 3)\n\n\\vskip.2cm\\noindent Note that $k\\le i-1$ implies that $n-k-1\\ge n-i\\ge 2$,\nand hence we can apply \\S \\ref{facts}, Fact 3. Corresponding to\n$k=i-1$, we obtain\n$\\tau_1\\cdots\\tau_{i}(\\alpha_{n-1})=\\tau_1((i-1)\\delta+\n\\alpha_{n-i}+\\cdots+\\alpha_{n-1}$ ). Hence once again using \\S\n\\ref{facts}, Fact 3, we obtain\n\n $$\\tau_1\\cdots\\tau_{i}(\\alpha_{n-1})=\ni\\delta+\\alpha_{n-i-1}+\\cdots+\\alpha_{n-1},\\ 1\\le i\\le n-2$$ (note\nthat for $1\\le i\\le n-2$, $n-i-1\\ge 1$).\n\n\\noindent\\textbf{Case 2:} \\emph{To show:}\n$\\tau_1\\cdots\\tau_{i}s_{n-1}\\cdots s_{1}(\\alpha_{0}), 1\\le i\\le\nn-2$ is a positive real root.\n\n\\noindent We have\n\n\\noindent $\\tau_1\\cdots\\tau_{i}s_{n-1}\\cdots s_{1}(\\alpha_{0})$\n\n\\noindent $=\\tau_1\\cdots\\tau_{i}(\\delta+\\alpha_{n-1})$ (cf. \\S\n\\ref{facts}, Fact 5)\n\n\\noindent $=\\tau_1\\cdots\\tau_{i-1}(2\\delta+\\alpha_{n-2}+\\alpha_{n-1})$\n(cf. \\S \\ref{facts}, Fact 6)\n\n\\noindent $=\\tau_1\\cdots\\tau_{i-k}((k+1)\\delta+\\alpha_{n-k-1}+\n\\cdots+\\alpha_{n-1}),0\\le k\\le i-1$ (cf. \\S \\ref{facts}, Fact 3)\n\n\\vskip.2cm\\noindent Note that as in Case 1, for $k\\le i-1$, we have,\n$n-k-1\\ge 2$, and therefore \\S \\ref{facts}, Fact 3 holds.\nCorresponding to $k=i-1$, we have,\n\n\\noindent $\\tau_1\\cdots\\tau_{i}s_{n-1}\\cdots\ns_{1}(\\alpha_{0})=\\tau_1(i\\delta+\\alpha_{n-i}+\\cdots+\\alpha_{n-1})$.\n Hence once again using \\S\n\\ref{facts} Fact 3, we obtain $$\\tau_1\\cdots\\tau_{i}s_{n-1}\\cdots\ns_{1}(\\alpha_{0})=(i+1)\\delta+\\alpha_{n-i-1}+\\cdots+\\alpha_{n-1},\n1\\le i\\le n-2$$ (note that for $1\\le i\\le n-2$, $n-i-1\\ge 1$).\n\n\\noindent\\textbf{Case 3:} \\emph{To show:}\n$\\tau_1\\cdots\\tau_{i}s_{n-1}\\cdots s_{j+1}(\\alpha_j), 1\\le i\\le\nn-2, j\\ne 0,n-1$ is a positive real root.\n\n\\noindent We have $\\tau_1\\cdots\\tau_{i}s_{n-1}\\cdots s_{j+1}(\\alpha_j)=\\tau^i(\\alpha_{j}+\\alpha_{j+1}+\\cdots+\\alpha_{n-1})$\n(cf. \\S \\ref{facts}, Fact 4)\n\\noindent $=\\tau^i(\\alpha_j)+\\ldots+\\tau^i(\\alpha_{n-2})+\\tau^i(\\alpha_{n-1})$\nwhich is positive because each term is positive (cf. Case 1 and Remark \\ref{cyclic}).\n\\end{proof}\n\\begin{cor}\\label{length}\n$\\ell(\\kappa)=n(n-1)$.\n\\end{cor}\n\\subsection{Minimal representative-property for $\\kappa$}\\label{rep}\n\n\\begin{lem}\\label{minimal} $\\kappa(\\alpha_i)$ is a real positive root\nfor all $i\\ne 0$.\n\\end{lem}\n\\begin{proof} \nFor $1\\leq i\\leq n-2$, $\\kappa(\\alpha_i)=\\alpha_i$ is positive from Remark \\ref{cyclic}.\nFurther, $\\tau_1\\cdots\\tau_{n-1}(\\alpha_{n-1})$\n\n\\noindent $=\\tau_1\\cdots\\tau_{n-2}(\\delta+\\alpha_{n-2}+\\alpha_{n-1})$\n(cf. \\S \\ref{facts}, Fact 6))\n\n\\noindent $=\\tau_1\\cdots\\tau_{n-k}((k-1)\\delta+\\alpha_{n-k}+\\cdots+\n\\alpha_{n-1}), 1\\le k\\le n-1$ (cf. \\S \\ref{facts}, Fact 3))\n\n\\noindent Note that for $1\\le k\\le n-2, n-k\\ge 2$ and hence \\S\n\\ref{facts}, Fact 3 holds. Corresponding to $k=n-1$, we get,\n\n\\noindent $\\tau_1\\cdots\\tau_{n-1}(\\alpha_{n-1})$\n\n\\noindent $=\\tau_1((n-2)\\delta+\\alpha_{1}+\\cdots+ \\alpha_{n-1})$\n\n\\noindent $=n\\delta+\\alpha_{n-1}$ (cf.\\S \\ref{facts}, Facts 1,2 )\n\n\\end{proof}\n\n\\begin{cor}\\label{minimal'}\n$\\kappa$ is a minimal representative in ${\\widehat W}\/\\widehat W_{G_0}$.\n\\end{cor}\n\\noindent For $w\\in {\\widehat{W}}$, we shall denote the Schubert variety in\n$\\mathcal{G}\/G_0$ by $X_{G_0}(w)$.\n\\begin{lem}\\label{stable}\n$X_{G_0}(\\kappa)$ is stable for multiplication on the left by\n$G_0$.\n\\end{lem}\n\\begin{proof}\nIt suffices to show that\n$$s_i\\kappa\\le\\kappa(\\,mod\\,\\widehat{W}_{G_0}), 1\\le i\\le n-1\\leqno{(*)}$$ \nThe assertion (*) is clear if $i=n-1$. \nObserve that $ws_\\alpha=s_{w(\\alpha)}w$.\nIn particular, since $\\kappa$ fixes $\\alpha_i$, $1\\leq i\\leq n-2$, it follows $s_i\\kappa=\\kappa s_i=\\kappa(\\,mod\\,\\widehat W_{G_0})$, for $1\\leq i\\leq n-2$.\n\\end{proof}\n\n\\begin{lemma}\n\\label{sn2}\nLet \\ensuremath{\\mathcal P}\\ be the parabolic subgroup of $\\mathcal G$ corresponding to the choice of simple roots $\\left\\{\\alpha_1,\\cdots\\alpha_{n-2}\\right\\}$.\nThe element $\\kappa$ is a minimal representative in $\\widehat W_\\ensuremath{\\mathcal P}\\backslash\\widehat W$.\n\\end{lemma}\n\\begin{proof}\nIt is enough to show that $s_i\\kappa>\\kappa$, or equivalently, $\\kappa^{-1}(\\alpha_i)>0$ for $1\\leq i\\leq n-2$.\nThis follows from Remark \\ref{cyclic}.\n\\end{proof}\n\n\\begin{rem}\nFor the discussion in \\S \\ref{kap}, \\S \\ref{rep}, concerning reduced expressions, minimal-representative property and $G_0$-stability, we have used the expression for elements of $\\widehat{W}$,  $\\widehat{W}$ being considered as a Coxeter group. One may as well carry out the discussion using the permutation presentations  for elements of $\\widehat{W}$.\n\\end{rem}\n\n\\begin{theorem}\n[A reduced expression for $\\kappa_0$]\nThe element $\\kappa_0(=w'\\tau^{n-1})$ is the  maximal representative of $\\kappa$ in $\\widehat W_{G_0}\\backslash\\widehat W$, i.e. the unique element in $\\widehat W$ such that \n$$X(\\kappa_0)={\\overline{G_0\\kappa\\mathcal{B}}}(mod\\,\\mathcal{B})$$\nIn particular, $X(\\kappa_0)$ is (left) $G_0$-stable.\nLet $\\underline w'$ be a reduced expression for the longest element $w'$ in $\\widehat W_\\ensuremath{\\mathcal P}$ and $\\underline\\tau$ the reduced expression $s_{n-1}\\cdots s_1s_0$.\nThen $\\underline w'\\underline\\tau^{n-1}$ is a reduced expression for $\\kappa_0$.\n\\end{theorem}\n\\begin{proof}\nObserve that $\\underline w=\\underline w's_{n-1}\\cdots s_1$ is a reduced expression for the longest element $w$ in $\\widehat W_{G_0}$, and so $w'\\kappa=ws_0\\tau^{n-2}$.\nLemma \\ref{sn2} implies that $\\underline w'\\underline\\tau^{n-1}$ is a reduced expression. \nIn particular, $$l(\\kappa_0)=l(w'\\kappa)=l(w's_{n-1}\\cdots s_1)+l(s_0\\tau^{n-2})=l(w)+l(s_0\\tau^{n-2})$$\nIt remains to show that $w'\\kappa$ is a maximal representative in $\\widehat W_{G_0}\\backslash\\widehat W$, i.e $s_iw'\\kappa<w'\\kappa$, or equivalently $l(s_iw'\\kappa)<l(w'\\kappa)$ for $1\\leq i\\leq n-1$.\nFirst note that $$l(s_iw'\\kappa)=l(s_iws_0\\tau^{n-2})\\leq l(s_iw)+l(s_0\\tau^{n-2})$$\nNow, since $w$ is the longest element in $\\widehat W_{G_0}$, it follows $l(s_iw)<l(w)$ and further \n$$l(s_iw'\\kappa)<l(w)+l(s_0\\tau^{n-2})=l(w'\\kappa)$$\n\\end{proof}\n\n\\section{The main lemma}\\label{main} In this section, we prove one\ncrucial result involving $\\kappa$, which we then use to prove \nthe main result.\n\\begin{lemma}\\label{crucial}\nLet $Y=\\sum_{1\\le i<j\\le n}\\, a_{ij}E_{ij}$, where $E_{ij} $ is the\nelementary $n\\times n$ matrix with $1$ at the $(i,j)$-th place and\n0's elsewhere. Let ${\\underline{Y}}=Id_{n\\times n}+\\sum_{1\\le i\\le\nn-1}\\,t^{-i}Y^i$ (note that $Y^n=0$). Assume that $a_{i i+1}\\ne 0,\n1\\le i\\le n-1$. There exist $g\\in G_0, h\\in\\mathcal{B}$ such that\n$g\\kappa = {\\underline{Y}}h $\n\\end{lemma}\n\\begin{proof}\nChoose $g$ to be the matrix $$g=\\begin{pmatrix} 0 & 0 & 0 &\\cdots\n&\n1\\\\\n -1 & 0 &0  &\\cdots & g_{2n}\\\\\n 0 &-1& 0 & \\cdots & g_{3n}\\\\\n\\vdots & \\vdots & \\vdots & \\vdots\\\\\n0 &0& \\cdots & -1 & g_{nn}\n\\end{pmatrix}$$\nNote that the lower left corner submatrix (i.e.,the $n-1\\times\nn-1$ submatrix with rows $2$-nd through the $(n-1)$-th of $g$, and\nthe first $n-1$ columns of $g$) is $-Id_{n-1\\times n-1}$, and that\ndeterminant of $g$ equals $1$. Hence, we may take $g_{in},2\\le\ni\\le n$ as elements in $K[[t]]$ so that $g\\in G_0$. We shall now\nshow that there exist $g_{in}, 2\\le i\\le n$, and $h_{ij}, 1\\le\ni,j\\le n $ such that $h(\\in\\mathcal{B})$, and $g\\kappa =\n{\\underline{Y}}h $. We have, ${\\underline{Y}}^{-1}= Id_{n\\times\nn}-t^{-1}Y$.  Set\n$$h=(Id_{n\\times n}-t^{-1}{\\underline{Y}})g\\kappa$$ We have\n(by definition of $\\kappa$ (\\S\\ref{elt}), and the choice of lifts\nfor $s_i$ (cf.\\S\\ref{gen}))\n$$\\kappa=diag(t,\\cdots, t, t^{-(n-1)})$$ Note that since we want $h$ to belong to $\\mathcal{B}$,\neach diagonal entry in $h$ (as an element of $K[[t]]$) should have\norder $0$, $h_{ij},i>j$ should have order $>0$, and $h_{ij},i<j$\nshould have order $\\ge 0$ (since $h(0)$ should belong to $B$). Now\nthe diagonal entries in $h$ are\n given by $$h_{ii}=a_{ii+1}, 1\\le i\\le n-1, h_{nn}=\n t^{-(n-1)}g_{nn}$$\n  Hence choosing $g_{nn}$ such that order$\\,g_{nn}=n-1$ (note that\n  since $g\\in G_0$, order$\\,g_{ij}\\ge 0, 1\\le i,j\\le n$,\n  so this choice for $g$ is allowed), we obtain\n  that each diagonal entry in $h$ is in $K[[t]]$, with order equal to\n   $0$. Also, we have\n   $$\\begin{gathered}h_{i+1 i}=-t, 1\\le i\\le n-1,\\\\ h_{ik}=0,k\\le i-2, 3\\le i\\le n-1,\n   h_{ik}=a_{ik+1}, 1\\le i<k\\le n-1\n   \\end{gathered}$$ Thus the entries\n   $h_{ik},k\\le n-1$ satisfy the order conditions mentioned above.\n   Let us then consider $h_{jn}, 1\\le j\\le n$. We have\n$$h_{jn}=t^{-(n-1)}g_{jn}-\\sum_{j+1\\le k\\le n}\\,\nt^{-n}a_{jk}g_{kn}, 1\\le j\\le n \\leqno{(*)}$$ We shall choose\n$g_{in}$ (in $K[[t]]$) so that order of $g_{in}$ equals $i-1$ (\nnote that this agrees with the above choice of $g_{nn}$ - in the\ndiscussion of the diagonal entries in $h$). Let us write\n$$g_{in}=\\sum g_{in}^{(k)}t^k$$ We shall show that with the above\n choice of $g_{in}$, the integrality condition on the $h_{in}$'s\n imposes conditions on $g_{in}^{(k)},\ni-1\\le k\\le n, 1\\le i\\le n$, leading to a linear system in these\n$g_{in}^{(k)}$'s\n (note that, the integrality condition on the $h_{in}$'s,\n $1\\le i\\le n$,  implies \n that $h_{in}$'s should belong to $K[[t]]$, with the additional\ncondition that $h_{nn}$ should have order $0$ - the latter\ncondition having already been accommodated, since $g_{nn}$ has been\nchosen to have order $n-1$). Treating $g_{in}^{(k)}$'s as the\nunknowns, we show that the resulting linear system has a unique\nsolution, thus proving the choice of $g,h$ with the said\nproperties. We shall now describe this linear system. The linear\nsystem will involve $n\\choose 2$ equations in $n\\choose 2$\nunknowns, namely, $g_{in}^{(k)}, i-1\\le k\\le n, 2\\le i\\le n$. The\nlinear system is obtained as follows. The lowest power of $t$\nappearing on the right hand side of (*) above is $-(n-j)$ (note\nthat order of $g_{kn}$ equals $k-1$). Hence equating the\ncoefficients of $t^{-(n-i)}, j\\le i\\le n-1$ on the right hand side\nof (*) to $0$, we obtain\n$$g_{jn}^{(i-1)}-\\sum_{j+1\\le k\\le n}\\,a_{jk}g_{kn}^{(i)}=0,  j\\le\ni\\le n-1, 1\\le j\\le n-1\\leqno{(**)}$$ Note that, corresponding to\n$h_{nn}$, we do not have any conditions, since by our choice of\n$g_{nn}$ (order of $g_{nn}$ is $n-1$), we have that $h_{nn}$ (=\n$t^{-(n-1)}g_{nn}$) is integral. Also, corresponding to $g_{1n}$\n(which is equal to $1$, by our choice of $g$), we have\n$g_{1n}^{(i)}=0, i\\ge 1$, and occurs just in one equation, namely,\nthe equation corresponding to the coefficient of $t^{-(n-1)}$ in\n$h_{1n}$;\n$$g_{1n}-a_{12}g_{2n}^{(1)}=0$$  Rewriting this equation as\n$$-a_{12}g_{2n}^{(1)}=-1$$\n(there is a purpose behind retaining the negative sign in\n$-a_{12}g_{2n}^{(1)}$), we arrive at the linear system\n$$A_nX=B$$ where $A_n$ is a square matrix of size $n\\choose 2$, $X$ is\nthe $n\\choose 2$ column matrix $(g_{jn}^k, j-1\\le k\\le n, 2\\le\nj\\le n)$, and $B$ is the $n\\choose 2$ column matrix with the first\nentry equal to $-1$, and all other entries equal to $0$.\n\n\\noindent \\textbf{Claim:} $A_n$ is invertible, and $|A_n|=(-1)^{n\\choose\n2}\\prod_{1\\le i\\le n-1}a_{i\\,i+1}^{n-i}$.\n\nNote that Claim implies that $(g_{jn}^{(k)}$'s, $j-1\\le k\\le n,\n2\\le j\\le n)$ are uniquely determined, and therefore we may choose\n$g_{jn}$ as elements in $K[[t]]$ with $(g_{jn}^{(k)}$'s, $j-1\\le\nk\\le n$ as the solutions of the above linear system, with\n$(g_{jn}^{(k)}, k>n$ being arbitrary.\n\nWe prove the Claim by induction on $n$. We shall first show that\n$A_{n-1}$ can be identified in a natural way as a submatrix of\n$A_n$. We want to think of the rows of $A_n$ forming $(n-1)$\nblocks (referred to as \\emph{row-blocks} in the sequel) of size\n$n-1,n-2,\\cdots,n-j,\\cdots,1$, namely, the $j$-th block consists\nof $n-j$ rows given by the coefficients occurring on the left hand\nside of (**) for $j\\ge 2$, and for $j=1$, the first block consists\nof $n-1$ rows given by the coefficients occurring on the left hand\nside of the following $n-1$ equations:\n$$-a_{12}g_{2n}^{(1)}=-1,\\ -g_{2n}^{(i)}-\\sum_{3\\le k\\le\nn}\\,a_{2k}g_{kn}^{(i)}=0,2\\le i\\le n-1$$ Similarly, we want to\nthink of the columns of $A_n$ forming $(n-1)$ blocks (referred to\nas \\emph{column-blocks} in the sequel) of size\n$n-1,n-2,\\cdots,n-j,\\cdots,1$, namely, the $j$-th block consisting\nof $n-j$ columns indexed by $g_{jn}^{(i)}, j-1\\le i\\le n$. Then\nindexing the $n-j$ rows in the $j$-th row-block as $j, j+1,\\cdots,\nn-1$, the entries in the rows of the $j$-th row-block have the\nfollowing description:\n\nThe non-zero entries in the $i$-th row in the $j$th row-block\n($j\\ge 2$) are\n\n\\noindent $1, -a_{23},-a_{24}, \\cdots, -a_{2\\,i+1} $ respectively,\noccurring at the columns indexed by\n\n\\noindent $g_{2n}^{(i-1)},g_{3n}^{(i)},\\cdots,g_{i+1\\,n}^{(i)}$.\n\nThe non-zero entries in the $i$-th row in the first row-block\n($j\\ge 2$) are\n\n\\noindent $-a_{12}, -a_{13}, \\cdots, -a_{2\\,i+1} $ respectively,\noccurring at the columns indexed by\n\n\\noindent $g_{2n}^{(i)},g_{3n}^{(i)},\\cdots,g_{i+1\\,n}^{(i)}$.\n\nFrom this it follows that $A_{n-1}$ is obtained from $A_n$ by\ndeleting the first row in each row-block and the first column in\neach column-block. For instance, we describe below $A_5$ and\n$A_4$; for convenience of notation, we denote $b_{ij}=-a_{ij}$. We\nhave,\n$$A_5=\\left(\\begin{array}{>{\\columncolor{lightgray}}cccc>{\\columncolor{lightgray}}ccc>{\\columncolor{lightgray}}cc>{\\columncolor{lightgray}}c}\n\\rowcolor{lightgray}\n{b}_{12}&0&0&0 &0&0&0&0&0&0\\\\\n0&b_{12}&0&0&b_{13}&0&0&0&0&0\\\\\n0&0&b_{12}&0&0 &b_{13}&0&b_{14}&0&0\\\\\n0&0&0&b_{12} &0&0&b_{13}&0&b_{14}&b_{15}\\\\\n\\rowcolor{lightgray}\n1&0&0&0 &b_{23}&0&0&0&0&0\\\\\n0&1&0&0&0&b_{23}&0&b_{24}&0&0\\\\\n0&0&1&0 &0&0&b_{23}&0&b_{24}&b_{25}\\\\\n\\rowcolor{lightgray}\n0&0&0& &1&0&0&b_{34}&0&0\\\\\n0&0&0&0 &0&1&0&0&b_{34}&b_{35}\\\\\n\\rowcolor{lightgray} 0&0&0&0 &0&0&0&1&0&b_{45}\n\\end{array}\\right)$$\n\n$$A_4=\\begin{pmatrix}\n\nb_{12}&0&0&0&0&0\\\\\n0&b_{12}&0 &b_{13}&0&0\\\\\n0&0&b_{12} &0&b_{13}&b_{14}\\\\\n1&0&0&b_{23}&0&0\\\\\n0&1&0 &0&b_{23}&b_{24}\\\\\n0&0&0 &1&0&b_{34}\n\\end{pmatrix}$$\nAs rows (respectively columns) of $A_5$, the positions of the\nfirst row (respectively, the first column) in each of the four\nrow-blocks (respectively columns-blocks) in $A_5$ are given by\n$1,5,8,10$; deleting these rows and columns in $A_5$, we get\n$A_4$. These rows and columns are highlighted in $A_5$.\n\nAs above, let $b_{ij}=-a_{ij}$. Now expanding $A_n$ along the\nfirst row, we have that $|A_n|$ equals $b_{12}|M_{1}|, M_{1}$\nbeing the submatrix of $A_n$ obtained by deleting the first row\nand first column in $A_n$ (i.e., deleting the first row\n(respectively, the first column) in the first row-block\n(respectively, the first column-block)). Now in $M_{1}$,  in the\nfirst row in the second row-block the only non-zero entry is\n$b_{23}$, and it is a diagonal entry in $M_{1}$. Hence expanding\n$M_{1}$ through this row, we get that $|A_n|$ equals\n$b_{12}b_{23}|M_{2}|, M_{2}$ being the submatrix of $A_n$ obtained\nby deleting the first rows (respectively, the first columns)\nin the first two row-blocks (respectively, the first two\ncolumn-blocks) in $A_n$. Now in $M_{2}$, in the first row in the\nthird row-block, the only non-zero entry is $b_{34}$, and it is a\ndiagonal entry in $M_{2}$. Hence expanding $M_{2}$ along this row,\nwe get that $|A_n|$ equals $b_{12}b_{23}b_{34}|M_{3}|, M_{3}$\nbeing the submatrix of $A_n$ obtained by deleting the first\nrows (respectively, the first columns) in the first three\nrow-blocks (respectively, the first three column-blocks) in $A_n$.\nThus proceeding, at the $(n-1)$-th step, we get that $|A_n|$\nequals $b_{12}b_{23}\\cdots b_{n-1\\,n}|A_{n-1}|$. By induction, we\nhave $|A_{n-1}|=(-1)^{n-1\\choose 2} \\prod_{1\\le i\\le\nn-2}a_{i\\,i+1}^{n-1-i}$. Substituting back for $b_{ij}$'s, we\nobtain $|A_n|=(-1)^{n\\choose 2}\\prod_{1\\le i\\le\nn-1}a_{i\\,i+1}^{n-i}$. It remains to verify the statement of the\nclaim when $n=2$ (starting point of induction). In this case, we\nhave $$\\begin{gathered}g=\\begin{pmatrix}0&1\\\\\n-1&g_{22}\\end{pmatrix}, \\kappa=\\begin{pmatrix}t&0\\\\\n0&t^{-1}\\end{pmatrix},\\\\\n{\\underline{Y}^{-1}}=\n\\begin{pmatrix}1&-t^{-1}a_{12}\\\\\n0&1\\end{pmatrix},h=\\begin{pmatrix}a_{12}&t^{-1}-t^{-2}a_{12}g_{22}\\\\\n-t&-t^{-1}g_{22}\\end{pmatrix} \\end{gathered}$$ Hence the linear\nsystem consists of the single equation $$-a_{12}g_{22}^{(1)}=-1$$\nHence $A_2$ is the $1\\times 1$ matrix $(-a_{12})$, and\n$|A_2|=-a_{12}$, as required.\n\\end{proof}\n\n\\section{Lusztig's map}\\label{lumap} Consider $\\mathcal{N}$,\nthe variety of nilpotent elements in $\\frak{g}$ (the Lie algebra\nof $G$). In this section, we spell out (Lusztig's) isomorphism\nwhich identifies $X_{G_0}(\\kappa)$ as a compactification of $\\mathcal{N}$.\n\\subsection{The map $\\psi$:}\\label{luss} Consider the map $$\\psi:\\mathcal{N}\\rightarrow\n \\mathcal{G}\/G_0,\n\\psi(N)=(Id+t^{-1}N+t^{-2}N^2+\\cdots)(mod\\,G_0),N\\in\\mathcal{N}$$\nNote that the sum on the right hand side is finite, since $N$ is\nnilpotent. We now list some properties of $\\psi$.\n\n\\noindent\\textbf{(i) $\\psi$ is injective:} Let $\\psi(N_1)=\\psi(N_2)$.\nDenoting $\\lambda_i:=\\psi(N_i), i=1,2$, we get that\n$\\lambda_2^{-1}\\lambda_1$ belongs to $G_0$. On the other hand, \n$$\\lambda_2^{-1}\\lambda_1=(Id-t^{-1}N_2)(Id+t^{-1}N+t^{-2}N^2+\\cdots)$$\nNow $\\lambda_2^{-1}\\lambda_1$ is integral.\nIt follows that both sides of the above equation equal $Id$. This\nimplies $\\lambda_1=\\lambda_2$ which in turn implies that\n$N_1=N_2$. Hence we obtain the injectivity of $\\psi$.\n\n\\noindent\\textbf{(ii) $\\psi$ is $G$-equivariant:} We have\n\n\\noindent $\\psi(g\\cdot N)=\\psi(gNg^{-1})$\n\n\\noindent $=(Id+t^{-1}gNg^{-1}+t^{-2}gN^2g^{-1}+\\cdots)(mod\\,G_0)$\n\n\\noindent $=g(Id+t^{-1}N+t^{-2}N^2+\\cdots)g^{-1}(mod\\,G_0)$\n\n\\noindent $=g(Id+t^{-1}N+t^{-2}N^2+\\cdots)(mod\\,G_0)$ (since $g^{-1}\\in\nG_0$)\n\n\\noindent $=g\\psi(N)$ \n\n\n\\begin{prop}\\label{lu}\nFor $N\\in \\mathcal{N}, \\psi(N)$ belongs to $X_{G_0}(\\kappa)$.\n\\end{prop}\n\n\\begin{proof}\nWe divide the proof into two cases.\n\n\\noindent \\textbf{Case 1:} Let $N$ be upper triangular, say,\n$$N=\\left(n_{ij}\\right)_{1\\le i,j\\le n}$$ where $n_{ij}=0$,\nfor $i\\ge j$;  note that $N\\in {\\underline{b}_u},\n{\\underline{b}_u}$ being the Lie algebra of $B_u$, the unipotent\nradical of $B$. We may work in the open subset $x_{ii+1}\\ne 0,1\\le i\\le\nn-1$ in ${\\underline{b}_u}$, $\\sum_{1\\le i<j\\le n}\\, x_{ij}E_{ij}$\nbeing a generic element in ${\\underline{b}_u}$. Hence we may\nsuppose that $n_{i\\,i+1}\\ne 0, 1\\le i\\le n-1$. In this case, in\nview of Lemma \\ref{crucial}, we have that there exist $g\\in G_0, h\\in\n\\mathcal{B}$ such that $g\\kappa =\\psi(N)h$. This implies, in view\nof the $G_0$-stability for $X_{G_0}(\\kappa)$ (cf. Lemma\n\\ref{stable}), $\\psi(N)$ belongs to $X_{G_0}(\\kappa)$.\n\n\\noindent \\textbf{Case 2:} Let $M$ be an arbitrary nilpotent matrix.\nThen there exists an upper triangular matrix $N$ in the $G$-orbit\nthrough $N$. Hence there exists a $g\\in G$ such that\n$M=gNg^{-1}(=g\\cdot N)$ with $N$ upper triangular. Now by\n$G$-equivariance of $\\psi$ (cf. (ii) above), we have\n$\\psi(M)=g\\cdot \\psi(N)$. By case 1, $\\psi(N)\\in X_{G_0}(\\kappa)$;\nthis together with the $G_0$-stability for $X_{G_0}(\\kappa)$\nimplies that $\\psi(M)$ belongs to $X_{G_0}(\\kappa)$.\n\\end{proof}\n\n\\begin{theorem}\\label{l-isom}\n$X_{G_0}(\\kappa)$ is a compactification of $\\mathcal N$.\n\\end{theorem}\n\\begin{proof}\nLet $\\overline{\\mathcal N}$ be the closure of $\\mathcal N$ in $\\mathcal G\/G_0$.\nCombining the above Proposition with \\S \\ref{luss}, (i) and\nthe facts that $\\dim \\mathcal{N}=n(n-1)=\\dim X_{G_0}(\\kappa)$ (cf. Corollaries \\ref{length},\\ref{minimal'}), we obtain $\\overline{\\mathcal{N}}=X_{G_0}(\\kappa)$.\n\\end{proof}\n\n\\section{Cotangent bundle}\nIn this section, we first recall the Springer resolution.\nWe then construct a family $\\psi_p$, parametrized by polynomials $p$ in one variable with coefficients in $\\mathbb C((t))$ and constant term $1$, of maps $\\psi_p:T^*G\/B\\rightarrow\\mathcal G\/\\mathcal B$.\nWe show that for a particular choice $\\phi$ in the family, we get an embedding of $T^*G\/B$ inside $\\mathcal G\/\\mathcal B$.\nUsing the natural projection $\\mathcal G\/\\mathcal B\\rightarrow \\mathcal G\/G_0$ and the results of $\\S 5$, we recover the Springer resolution.\nWe then show that $\\phi$ identifies an $SL_n$-stable closed subvariety of $X(\\kappa_0)$ as a compactification of $T^*G\/B$.\n\n\\label{bundle} The cotangent bundle $T^*G\/B$ is a\nvector bundle over $G\/B$, with the fiber at any point $x\\in G\/B$ being\nthe cotangent space to $G\/B$ at $x$; the dimension of $T^*G\/B$\nequals $2\\dim G\/B$. Also, $T^*G\/B$ is the fiber bundle over\n$G\/B$ associated to the principal $B$-bundle $G\\rightarrow G\/B$,\nfor the Adjoint action of $B$ on ${\\underline{b}_u}$ (the Lie\nalgebra of the unipotent radical $B_u$ of $B$). Thus\n$$T^*G\/B=G\\times^B {\\underline{b}_u}=G\\times {\\underline{b}_u}\/\n\\sim$$ where the equivalence relation $\\sim$ is given by $(g,\nY)\\sim(gb, b^{-1}Yb),g\\in G,Y\\in{\\underline{b}_u}, b\\in B$ .\n\\subsection{Springer resolution} Let\n$\\mathcal{N}$ be the variety of nilpotent elements in ${\\frak g}$,\nthe Lie algebra of $G$. Consider the map\n$$\\theta:G\\times^B {\\underline{b}_u}\\rightarrow\nG\/B\\times\\mathcal{N}, \\theta((g, Y))=(gB,gYg^{-1}), g\\in G, Y\\in\n{\\underline{b}_u}$$ We observe the following on the map $\\theta$:\n\n\\noindent \\textbf{(i) $\\theta$ is well defined:} Let $b\\in B$. Consider\n$(gb, b^{-1}Yb) (\\sim (g,b))$. We have, $$\\theta((gb,\nb^{-1}Yb))=(gB,gb(b^{-1}Yb)b^{-1}g^{-1})=(gB,gYg^{-1})=\\theta((g,\nY))$$\n\n\\noindent \\textbf{(ii) $\\theta$ is injective:} Suppose $\\theta((g_1,\nY_1))=\\theta((g_2, Y_2))$. Then\n$(g_1B,g_1Y_1g_1^{-1})=(g_2B,g_2Y_2g_2^{-1})$. This implies\n$$g_1B=g_2B,g_1Y_1g_1^{-1}=g_2Y_2g_2^{-1}$$ Hence we obtain $$\n\\begin{gathered} g_1^{-1}g_2=:b\\in B, Y_2=\ng_2^{-1}g_1Y_1g_1^{-1}g_2\\\\\n\\therefore g_2=g_1b, Y_2=b^{-1}Y_1b\\\\\n\\therefore (g_1, Y_1)=(g_1b, b^{-1}Y_1b)=(g_2, Y_2)\n\\end{gathered}$$ Thus we get  an embedding $$\\theta:T^*G\/B\\hookrightarrow G\/B\\times\\mathcal{N}$$\nThe second projection $$T^*G\/B\\rightarrow \\mathcal{N},(g,Y)\\mapsto\ngYg^{-1}\n$$ is proper and birational and is the celebrated\n\\emph{Springer resolution}\n\n\\subsection{The Maps $\\psi_p$}\\label{maps} \nLet $p(Y)$ be a polynomial in $Y$ with coefficients in $F$, and constant term $1$.\nWe write \\begin{align*}\n    p(Y)=1+\\sum\\limits_{i\\geq1}p_i(t)Y^i\n\\end{align*}\nIt is clear that $p(Y)\\in\\mathcal G$.\nDefine the map $\\psi_p:G\\times^B{\\underline{b}_u}\\rightarrow \\mathcal{G}\/\\mathcal{B}$ by \n$$\\psi_p(g,Y)=gp(Y)(mod\\,\\mathcal{B}),\ng\\in G, Y\\in{\\underline{b}_u}$$ \nThe following calculation shows that $\\psi_p$ is well defined: Let $g\\in G,\\ b\\in B,\\ Y\\in{\\underline{b}_u}$. Then \\begin{align*}\n\\psi_p((gb, b^{-1}Yb)) &=gb\\left(Id+p_1(t)b^{-1}Yb+p_2(t)b^{-1}Y^2b+\\cdots\\right)(mod\\,\\mathcal{B})\\\\\n&=g\\left(Id+p_1(t)Yb+p_2(t)Y^2b+\\cdots\\right)(mod\\,\\mathcal{B})\\\\\n&=g\\left(Id+p_1(t)Y+p_2(t)Y^2+\\cdots\\right)(mod\\,\\mathcal{B})\\\\\n&=\\psi_p(g,Y) \n\\end{align*}\nAlso, it is clear that $\\psi_p$ is $G$-equivariant. \n\n\\subsection{Embedding of $T^*G\/B$ into $\\mathcal{G}\/\\mathcal{B}$:} \n\\label{phi}\nWe consider one particular member $\\phi$ of the family $\\psi_p$: namely $\\phi=\\psi_p$ where $p(Y)$ is the polynomial $(1-t^{-1}Y)^{-1}$;\nobserve that for nilpotent $Y$, the function \\begin{align*}\n    p(Y)&=(1-t^{-1}Y)^{-1}\\\\\n        &=1+t^{-1}Y+t^{-2}Y^2+\\cdots\n\\end{align*}\nis a polynomial, since the sum on the right hand side is finite.\nIn particular, $\\phi:G\\times^B\n{\\underline{b}_u}\\rightarrow \\mathcal{G}\/\\mathcal{B}$ is given by \n$$\\phi(g,Y)=g(Id+t^{-1}Y+t^{-2}Y^2+\\cdots)(mod\\,\\mathcal{B})$$\nIn the sequel, we shall denote $${\\underline{Y}}:=Id+t^{-1}Y+t^{-2}Y^2+\\cdots$$ We\nnow list some facts on the map $\\phi$:\n\n\\noindent\\textbf{(i) $\\phi$ is well-defined} \n\n\\noindent\\textbf{(ii) $\\phi$ is injective:} Let\n$\\phi((g_1,Y_1))=\\phi((g_2,Y_2))$. This implies that\n$g_1{\\underline{Y_1}}\\equiv\ng_2{\\underline{Y_2}}(mod\\,\\mathcal{B})$, where recall that for\n$Y\\in{\\underline{b}_u},{\\underline{Y}}=Id+t^{-1}Y+t^{-2}Y^2+\\cdots$.\nHence, $g_1{\\underline{Y_1}}= g_2{\\underline{Y_2}}x$, for some\n$x\\in\\mathcal{B}$. Denoting $h=:g_2^{-1}g_1$, we have,\n$h{\\underline{Y_1}}={\\underline{Y_2}}x$, and\ntherefore,$$x={\\underline{Y_2}}^{-1}h{\\underline{Y_1}}=\n{\\underline{Y_2}}^{-1}(h{\\underline{Y_1}}\nh^{-1})h={\\underline{Y_2}}^{-1}{\\underline{Y'_1}}h$$ where\n${\\underline{Y'_1}}=h{\\underline{Y_1}} h^{-1}$. Hence\n$$xh^{-1}={\\underline{Y_2}}^{-1}{\\underline{Y'_1}}=(Id-t^{-1}Y_2)\n(Id+t^{-1}hY_1h^{-1}+t^{-2}hY^2_1h^{-1}+\\cdots)$$ \nNow, since $x\\in\\mathcal{B},h(=g_2^{-1}g_1)\\in G$, the left hand\nside is integral, i.e. it does not involve negative powers of $t$.\nHence both sides equal $Id$. This implies\n$${\\underline{Y_2}}={\\underline{Y'_1}},x=h$$ The fact that $x=h$\ntogether with the facts that $x\\in\\mathcal{B},h\\in G$ implies that\n$$h\\in \\mathcal{B}\\cap G(=B)\\leqno{(*)}$$ Further, the fact that\n${\\underline{Y_2}}={\\underline{Y'_1}}$ implies that\n${\\underline{Y_1}}=h^{-1}{\\underline{Y_2}}h$. Hence\n$$Id+t^{-1}Y_1+t^{-2}Y_1^2+\\cdots=Id+t^{-1}h^{-1}Y_2h+t^{-2}h^{-1}Y^2_2h\n+\\cdots$$ From this it follows that $$Y_1=h^{-1}Y_2h\\leqno{(**)}$$\nNow (*), (**) together with the fact that $h=g_2^{-1}g_1$ imply\nthat $$(g_1,Y_1)=(g_2h,h^{-1}Y_2h)\\sim (g_2,Y_2)$$ From this\ninjectivity of $\\phi$ follows.\n\n\\noindent\\textbf{(iii) $G$-equivariance:} It is clear that $\\phi$ is $G$-equivariant.\n\n\\noindent\\textbf{(iv) Springer resolution:} Consider the projection\n$\\pi:\\mathcal{G}\/\\mathcal{B}\\rightarrow \\mathcal{G}\/G_0$. Let\n$x\\in T^*G\/B$, say, $x=(g,Y), g\\in G, Y\\in{\\underline{b}_u}$. We\nhave\n\n\\noindent $\\pi((g,Y))=\\phi((g,Y))(mod\\,G_0)$\n\n\\noindent $=g(Id+t^{-1}Y+t^{-2}Y^2+\\cdots)\n(mod\\,G_0)$\n\n\\noindent $=g(Id+t^{-1}Y+t^{-2}Y^2+\\cdots)g^{-1}(mod\\,G_0)$\n\n\\noindent $=(Id+t^{-1}N+t^{-2}N^2+\\cdots) (mod\\,G_0)$\n\n\\noindent where $N=gYg^{-1}$ is nilpotent. Hence, in view of Lusztig's\nisomorphism (cf. Proposition \\ref{l-isom}), we recover the\nSpringer resolution as\n$$\\pi\\,|_{T^*G\/B}:T^*G\/B\\rightarrow \\mathcal{N}\\hookrightarrow\n\\mathcal{G}\/G_0$$ \n\n\\begin{thm}[Compactification of $T^*G\/B$]\\label{ctgt} \nLet $G=SL_n(\\mathbb C)$ and $\\phi:G\/B\\rightarrow\\mathcal G\/\\mathcal B$ be as in section \\ref{phi}.\nThen $\\phi$ identifies\n${\\overline{ T^*G\/B}}$ (the closure being in\n$\\mathcal{G}\/\\mathcal{B}$) with a $G$-stable closed subvariety of\nthe affine Schubert variety $X(\\kappa_0)$.\n\\end{thm}\n\\begin{proof} Let $(g_0,Y), g_0\\in G, Y\\in{\\underline{b}_u}$. Then\n$\\phi(g_0,Y)=g_0(Id+t^{-1}Y+t^{-2}Y^2+\\cdots)(mod\\,\\mathcal{B})=\ng_0{\\underline{Y}}(mod\\,\\mathcal{B})$, where\n${\\underline{Y}}=Id+t^{-1}Y+t^{-2}Y^2+\\cdots$. Writing\n$Y=\\sum_{1\\le i<j\\le n}\\, a_{ij}E_{ij}$ with $E_{ij} $ as in Lemma\n\\ref{crucial}, we may work in the open subset $x_{ii+1}\\ne 0,1\\le\ni\\le n-1$ in ${\\underline{b}_u}$, $\\sum_{1\\le i<j\\le n}\\,\nx_{ij}E_{ij}$ being a generic element in ${\\underline{b}_u}$. Then\nLemma \\ref{crucial} implies that there exist $g\\in G_0,\nh\\in\\mathcal{B}$ such that $g\\kappa = {\\underline{Y}}h $. Hence\n${\\underline{Y}}$ belongs to\n$X(\\kappa_0)(={\\overline{G_0\\kappa\\mathcal{B}}}(mod\\,\\mathcal{B}))$;\nhence $g_0{\\underline{Y}}$ is also in $X(\\kappa_0)$ (since $g_0$\nis clearly in $G_0$).\n\\end{proof}\n\n\n\\section{Consequences of $\\psi_p$ for $T^*G\/B$}\n\\label{counter}\nIn this section, we show that for any polynomial $p$, the map $\\psi_p$ as defined in \\S\\ref{maps} cannot realize an affine Schubert variety (in $\\widehat{SL}_3\/\\mathcal B$) as a compactification of the cotangent bundle $T^*SL_3(K)\/B$.\n\\begin{prop}\nLet $G$ be the group $SL_3(K)$ and the $B$ the Borel subgroup of upper triangular matrices in $G$. \nLet $p$ be a polynomial as in \\S\\ref{maps}.\nSuppose that the associated map $\\psi_p:T^*G\/B\\rightarrow \\mathcal G\/\\mathcal B$ is injective.\nThen there exist $g\\in G$, $w\\in\\widehat W$ and $Y\\in\\underline b _u$ such that $\\psi_p(g,Y)\\in\\mathcal{B} w\\mathcal{B}$\nand $l(w)>6$. \n\\end{prop}\n\\begin{proof}\nFrom \\S\\ref{maps}, we may assume $p(Y)=1+\\sum\\limits_{i\\geq1}p_i(t)Y^i$. \nWe first claim that $p_1(t)\\notin A$.\nAssume the contrary.\nFor $$Z=\\begin{pmatrix}0&1&0\\\\0&0&0\\\\0&0&0\\end{pmatrix}$$\nwe see that $Z^2=0$, and so $p(Z)=1+p_1(t)Z\\in\\mathcal B$. \nIn particular, $\\psi_p(Z)=\\psi_p(0)$, contradicting the injectivity of $\\psi_p$.\n\\\\\n\\\\\nWe now write\n   $ p(Y)=1-t^{-a}qY-t^{-b}rY^2$\nwhere \\begin{itemize}\n    \\item $q,r\\in A$. \n    \\item $q(0)\\neq 0$. \n    \\item $a\\geq 1$.\n    \\item Either $r=0$ or $r(0)\\neq 0$.\n\\end{itemize}\nWe now fix $Y=         \\begin{pmatrix}\n                    0 & 1 & 0   \\\\\n                    0 & 0 & 1   \\\\\n                    0 & 0 & 0 \n                \\end{pmatrix}$ and \n$g=             \\begin{pmatrix}\n                    0 & 0 &-1   \\\\\n                    0 &-1 & 0   \\\\\n                   -1 & 0 & 0 \n                \\end{pmatrix}$, so that \n$$gp(Y)=\\begin{pmatrix}\n                    0 & 0 & -1\\\\\n                    0 &-1 &t^{-a}q\\\\\n                   -1 &t^{-a}q&t^{-b}r\n                \\end{pmatrix}$$\nOur strategy is to find elements $C,D\\in\\mathcal B$ such that $Cgp(Y)D\\in N(K[t,t^{-1}])$.\nWe can then identify the Bruhat cell containing $gp(Y)$, and so identify the minimal Schubert variety containing $\\psi_p(g,Y)$.\nThe choice of $C,D$ depends on the values of certain inequalities, which we divide into $4$ cases.\nWe draw here a decision tree showing the relationship between the inequalities and the choice $C,D$.\n\n\\tikzstyle{level 1}=[level distance=2.5cm, sibling distance=2cm]\n\\tikzstyle{level 2}=[level distance=3.5cm, sibling distance=1.3cm]\n\n\\tikzstyle{bag} = [text width=4em, text centered]\n\\tikzstyle{end} = [circle, minimum width=6pt,fill, inner sep=0pt]\n\n\\begin{tikzpicture}[grow=right, sloped]\n\\node[end] {}\n    child {\n        node[bag] {Case $1$}        \n            edge from parent \n            node[above] {$r=0$}\n    }\n    child {\n        node[end] {}        \n        child {\n                node[bag] {Case $1$}\n                edge from parent\n                node[below] {$\\qquad b\\leq a$}\n            }\n            child {\n                node[bag] {Case $2$} \n                edge from parent\n                node[above]  {$\\qquad a<b<2a$}\n            }\n            child {\n                node[end] {}\n        child {\n                node[bag] {Case $3$}\n                edge from parent\n                node[below] {$\\qquad q^2+r=0$}\n            }\n        child {\n                node[bag] {Case $4$}\n                edge from parent\n                node[above] {$q^2+r\\neq 0$}\n            }\n                edge from parent\n                node[above] {$\\qquad 2a=b$}\n            }\n            child {\n                node[bag] {Case $4$}\n                edge from parent\n                node[above] {$\\qquad 2a<b$}\n            }\n        edge from parent         \n            node[above] {$r\\neq 0$}\n    };\n\\end{tikzpicture}\n\\\\\n\\\\\nA rational function in $t$ is implicitly equated with its Laurent power series at $0$. \nIn particular, a rational function $f$ belongs to $A$ if and only if $f$ has no poles at $0$, i.e. its denominator is not divisible by $t$.\n\\begin{enumerate}\n\\item\nIf $r=0$ or $b\\leq a$, let \\begin{align*}\n    C&= \\begin{pmatrix}\n        rt^{2a-b}+q^2   & qt^a                      & t^{2a}                        \\\\\n        0               & -\\dfrac{q}{rt^{2a-b}+q^2}  & -\\dfrac{t^a}{rt^{2a-b}+q^2}    \\\\ \n        0               & 0                         & -\\dfrac{1}{q}           \n        \\end{pmatrix}\\\\\n    D&= \\begin{pmatrix}\n        1                           & 0             & 0                     \\\\\n        \\dfrac{qt^a}{rt^{2a-b}+q^2}  & 1             & -\\dfrac{rt^{a-b}}{q}   \\\\  \n        \\dfrac{t^{2a}}{rt^{2a-b}+q^2}& 0             & 1\n        \\end{pmatrix} \n\\end{align*}    \nWe compute \\begin{align*}\n    Cgp(Y)D=    \\begin{pmatrix}\n                    -t^{2a} & 0         & 0          \\\\\n                    0       & 0         & t^{-a}    \\\\\n                    0       & t^{-a}    & 0 \n                \\end{pmatrix}\n\\end{align*}\nIt follows $gp(Y)\\in\\mathcal B\\lambda_qs_2\\mathcal B$, where $q=-2a\\alpha_1^\\vee-a\\alpha_2^\\vee$.\nWe calculate \\begin{align*}\n    l(\\lambda_q)&=\\lvert\\alpha_1(q)\\rvert+\\lvert\\alpha_2(q)\\rvert+\\lvert\\alpha_1(q)+\\alpha_2(q)\\rvert\\\\\n                &=3a+0+3a\\\\\n                &=6a\n\\end{align*}\nIt follows from lemma \\ref{count} that $l(\\lambda_qs_2)>l(\\lambda_q)=6a\\geq 6$.\n\\item\nSuppose $a<b<2a$. \nIn particular, $a\\geq2,b\\geq3$. \nLet \\begin{align*}\n    C&= \\begin{pmatrix}\n            -rt^{2a-b}+q^2  & qt^a                      & t^{2a}                            \\\\\n            0               & -\\dfrac{r}{rt^{2a-b}+q^2} & \\dfrac{-qt^{b-a}}{rt^{2a-b}+q^2}  \\\\\n            0               & 0                         & \\dfrac{1}{r} \n        \\end{pmatrix} \\\\\n    D&= \\begin{pmatrix}\n            1                               & 0                     & 0 \\\\\n            \\dfrac{t^aq}{q^2+t^{2a-b}r}     & 1                     & 0 \\\\\n            \\dfrac{t^{2a}}{q^2+t^{2a-b}r}   & -\\dfrac{qt^{b-a}}{r}  & 1 \n        \\end{pmatrix} \\end{align*}    \nWe compute\\begin{align*}\n    Cgp(Y)R=\\begin{pmatrix}\n                t^{2a}  & 0         & 0         \\\\\n                0       & t^{b-2a}  & 0         \\\\\n                0       & 0         & t^{-b} \n                \\end{pmatrix}\\end{align*}\nIt follows $gp(Y)\\in\\mathcal B\\lambda_q\\mathcal B$, where $q=-2a\\alpha_1^\\vee-b\\alpha_2^\\vee$.\nWe calculate\\begin{align*}\n    l(\\lambda_q)&=\\lvert\\alpha_1(q)\\rvert+\\lvert\\alpha_2(q)\\rvert+\\lvert\\alpha_1(q)+\\alpha_2(q)\\rvert\\\\\n                &=(4a-b)+(2b-2a)+(2a+b)\\\\\n                &=4a+2b\\geq 14\n\\end{align*}\n\\item\nIf $b=2a$ and $q^2+r=0$, let \\begin{align*}\n    C= \\begin{pmatrix}\n            q   & t^a   & 0             \\\\\n            0   & q     & t^a           \\\\\n            0   & 0     & \\dfrac{1}{q^2} \n        \\end{pmatrix},\\qquad\n    D= \\begin{pmatrix}\n            1                   & 0             & 0 \\\\\n            0                   & 1             & 0 \\\\\n            -\\dfrac{t^{2a}}{q^2} & \\dfrac{t^a}{q} & 1 \n        \\end{pmatrix} \\end{align*}    \nWe compute\\begin{align*}\n    Cgp(Y)D=    \\begin{pmatrix}\n                    0   &-t^a   & 0         \\\\\n                    -t^a& 0     & 0         \\\\\n                    0   & 0     &-t^{-2a} \n                \\end{pmatrix}\n\\end{align*}\nIt follows $gp(Y)\\in\\mathcal B\\lambda_qs_1\\mathcal B$, where $q=-a\\alpha_1^\\vee-2a\\alpha_2^\\vee$.\nSimilar to the first case, we see that $l(\\lambda_qs_1)>6$.\n\\item\nSuppose either $b>2a$, or $b=2a$ and $r+q^2\\neq0$. \nIn particular, $r+q^2t^{b-2a}\\neq0$ and $b\\geq2$.\nLet \\begin{align*}\n    C&= \\begin{pmatrix}\n            -r-q^2t^{b-2a}  &-qt^{b-a}                  &-t^b                               \\\\\n            0               &-\\dfrac{r}{r+q^2t^{b-2a}}  & \\dfrac{qt^{b-a}}{r+q^2t^{b-2a}}   \\\\\n            0               & 0                         & \\dfrac{1}{r} \n        \\end{pmatrix}\\\\\n    D&= \\begin{pmatrix}\n            1                               & 0                     & 0 \\\\\n            \\dfrac{qt^{b-a}}{q^2t^{b-2a}+r} & 1                     & 0 \\\\\n            \\dfrac{t^b}{q^2t^{b-2a}+r}      & -\\dfrac{qt^{b-a}}{r}  & 1 \n        \\end{pmatrix} \n\\end{align*}    \nWe compute\\begin{align*}\n    Cgp(Y)D= \\begin{pmatrix}\n                    t^b & 0 & 0     \\\\\n                    0   & 1 & 0     \\\\\n                    0   & 0 & t^{-b} \n                \\end{pmatrix}\n\\end{align*}\nIt follows $gp(Y)\\in\\mathcal B\\lambda_q\\mathcal B$, where $q=-b\\alpha_1^\\vee-b\\alpha_2^\\vee$.\nWe calculate\\begin{align*}\n    l(\\lambda_q)&=\\lvert\\alpha_1(q)\\rvert+\\lvert\\alpha_2(q)\\rvert+\\lvert\\alpha_1(q)+\\alpha_2(q)\\rvert\\\\\n                &=b+b+2b\\\\\n                &=4b\\geq 8\n\\end{align*}\n\\end{enumerate}\n\\end{proof}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\label{sec:intro}\n\n\\subsection{Motivation} Convolutional networks (CNNs) have shown near human performance in image classification~\\cite{Kolesnikov2020BigT} over non-structured dense networks.\nHowever, CNNs are vulnerable to specifically designed adversarial attacks~\\cite{adversarialexamples2015}.  Several papers in adversarial machine learning literature reveal the brittleness of convolutional networks to adversarial examples. For example, gradient based methods \\cite{Goodfellow2018existence,Kurakin2017} design a perturbation by taking steps proportional to the gradient of the loss of the input image $x$ in a given $\\ell_p$ neighborhood.\nThis has led to refined robust training approaches, or defenses, which train the network to see adversarial examples during the training stage and produce the unaltered label corresponding to it \\cite{madry2018towards,trades}.\n\nVision transformers (ViT) were recently introduced \\cite{Dosovitskiy2021AnII}, as a new network architecture inspired by transformers \\cite{Vaswani2017AttentionIA} which have been successfully used for modeling language data. ViTs rely on self attention \\cite{Vaswani2017AttentionIA}, a mechanism that allows the network to find correlations between spatially separated parts of the input data. In the context of vision, these are small non-overlapping \\textit{patches} which serve as \\textit{tokens} to the transformer. ViTs and more recently distillation based Data Efficient Image Transformers (DeIT) \\cite{Touvron2021TrainingDI} have shown to have competitive performance on classification tasks and rely on pre-training on very large datasets. It is of imminent interest to therefore study the robustness of self-attention based networks.\n\nThere has been some preliminary work on adversarial robustness of vision transformers. \\cite{Bhojanapalli2021UnderstandingRO} show that under certain regimes, vision transformers are at least as robust to $\\ell_2$ and $\\ell_\\infty$ PGD attacks as ResNets. While $\\ell_2$ and $\\ell_\\infty$ threat models are useful in understanding fundamental properties of deep networks, they are not realizable in the real world and do not capture actual threats. Transformer based networks also introduce the need for tokenizing the image, leading to an encoded bias in the input. It is therefore important to understand the sensitivity of the architecture to token level changes rather than to the full image.\n\nSpecifically, we attempt to answer:\n\\emph{Are transformers robust to perturbations to a subset of the input tokens?} We present a systemic approach to answer this query by constructing token level attacks by leveraging block sparsity constraints.\n\n\\subsection{Our contributions} \n\nIn this paper, we propose a patch based block sparse attack where the attack budget is defined by the number of tokens the attacker is allowed to perturb. We identify top salient pixels using the magnitude of their loss gradients and perturb them to create attacks. We extend a similar idea to block sparsity by constraining salient pixels to lie in non-overlapping patches. We probe three families of neural architectures using our token attack; self-attention (ViT~\\cite{Dosovitskiy2021AnII}, DeIT~\\cite{Touvron2021TrainingDI}), convolutional (Resnets~\\cite{He2016DeepRL} and WideResNet~\\cite{Zagoruyko2016WideRN}) and MLP based (MLP Mixer~\\cite{Tolstikhin2021MLPMixerAA}).\n\nWe make the following contributions and observations:\n\\begin{enumerate}[nolistsep, left=0pt]\n\\item We propose a new attack which imposes block sparsity constraints, allowing for \\textit{token attacks} for Transformers. \n\n\\item We show classification performance of all architectures on token attacks of varying patch sizes and number of patches.\n\n\\item We demonstrate that for token attacks matching the architecture token size, vision transformers are less resilient to token attacks as compared to MLP Mixers and ResNets.\n\n\\item For token attacks smaller than architecture token size, vision transformers are comparably robust to ResNets.\n\n\\item We also specifically note the shortcomings of previous studies on robustness of transformers~\\cite{Bhojanapalli2021UnderstandingRO}, where ViTs are shown to be more robust than ResNets.  \n\n\\item With our token attacks we can break Vision transformers using only $1\\%$ of pixels as opposed to $\\ell_2$ or $\\ell_\\infty$ attacks which rely on perturbing all image pixels.\n\n\\end{enumerate}\nWe therefore motivate designing attacks adaptively modeled after neural architectures.\n\n\n\\subsection{Related work} \n\n\n\\noindent\\textbf{\\textit{Threat models:}}  Deep networks are vulnerable to imperceptible changes to input images as defined by the $\\ell_\\infty$ distance \\cite{Szegedy2014intriguing}. There exist several test-time attack algorithms with various threat models: $\\ell_p$ constrained~\\cite{adversarialexamples2015, Kurakin2017, Carlini2017cwl2}, black-box~\\cite{Ilyas2018blackbox, Ilyas2018alimitedqueries}, geometric attacks~\\cite{engstrom2019a, Xiao2018SpatiallyTA}, semantic and meaningful attacks~\\cite{joshi2019semantic, zhang2019camou, song2018constructing} and data poisoning based~\\cite{Shafahi2018poisonfrogs}.\n\n \n\\noindent\\textbf{\\textit{Defenses:}} Due to the vast variety of attacks, adversarial defense is a non-trivial problem. Empirical defenses as proposed by \\cite{madry2018towards}, \\cite{trades}, and \\cite{jagatap2020adversarially} rely on adversarial data augmentation and modified loss functions to improve robustness. Several works~\\cite{samangouei2018defensegan, yin2020defense} propose preprocessing operations as defenses. However, such defenses often fail to counter adaptive attacks~\\cite{Athalye2018obfuscated}. \\cite{wong2018provable}, \\cite{cohen2019certified} and \\cite{salman2019provably} provide methods that guarantee robustness in terms of a volume around an input. Such methods often fail or provide trivial certificates for larger networks, and large high resolution images. Apart from algorithmic approaches, newer papers discuss optimal hyper-parameter tuning as well as combination of regularizers from aformentioned techniques, choice of activation functions, choice of architecture and data augmentation to extract best possible robust accuracies using pre-existing algorithms \\cite{Gowal2020UncoveringTL, Pang2021BagOT}.\n\n\\noindent\\textbf{\\textit{Patch attacks:}} Patch attacks~\\cite{Brown2017advpatch} are practically realizable threat model. \\cite{zolfi2021translucent, thys2019fooling, wu2020making} have successfully attacked detectors and classifiers with physically printed patches. In addition, \\cite{croce2019sparse, croce2019sparse} also show that spatially limited sparse perturbations suffice to consistently reduce the accuracy of classification model. This motivates our analysis of the robustness of recently invented architectures towards sparse and patch attacks.\n\n\n\n\\noindent\\textbf{\\textit{Vision transformers}}\nWhile convolutional networks have successfully achieved near human accuracy on massive datasets~\\cite{Kolesnikov2020BigT, Xie2020SelfTrainingWN}, there has been a surge of interest in leveraging self-attention as an alternative approach. Transformers~\\cite{Vaswani2017AttentionIA} have been shown to be extremely successful at language tasks~\\cite{Devlin2019BERTPO, Sanh2019DistilBERTAD, Brown2020LanguageMA}. \\cite{parmar2018image} extend this for image data, where in they use pixels as tokens. While they some success in generative tasks, the models had a large number of parameters and did not scale well. \\cite{Dosovitskiy2021AnII} improve upon this by instead using non-overlapping patches as tokens and show state of the art classification performance on the ImageNet dataset. \\cite{Touvron2021TrainingDI} further leverage knowledge distillation to improve efficiency and performance. Further improvements have been suggested by \\cite{Dai2021CoAtNetMC}, \\cite{Wu2021CvTIC} and \\cite{Touvron2021GoingDW} to improve performance using architectural modifications, deeper networks and better training methods. In parallel, \\cite{Tolstikhin2021MLPMixerAA} instead propose a pure MLP based architecture that achieves nearly equivalent results with faster training time. However, studies on generalization and robust performance of such networks is still limited. We discuss a few recent works below. \n\n\n\\noindent\\textbf{\\textit{Attacks on vision transformers:}\n\\cite{Bhojanapalli2021UnderstandingRO,hendrycks2020pretrained}  analyse the performance of vision transformers in comparison to massive ResNets under various threat models and concur that vision transformers (ViT) are at least as robust as Resnets when pretrained with massive training datasets. \\cite{mahmood2021robustness} show that adversarial examples do not transfer well between CNNs and transformers, and build an ensemble based approach towards adversarial defense. \\cite{paul2021vision} claims that Transformers are robust to a large variety of corruptions due to attention mechanism.\n\n\n\\begin{comment}\n\\section{Problem formulation}\n\nSpecifically, if forward map between the inputs $x \\in \\mathbb{R}^d$ and output logits $y\\in \\mathbb{R}^m$ corresponding to classes $\\{1,\\dots m\\}$, is modelled via a neural network as $y = f(w;x)$; $w$ being the set of trainable weights, then neural prediction $\\hat{y}(x) = f(\\hat{w};x)$, can be very sensitive to changes in $x$. For a bounded perturbation to a test image input, $$\\hat{y}_i = f(\\hat{w};x_i+\\delta_i)$$ where $\\delta_i$ represents the perturbation, the predicted label $\\hat{y_i}$ can be made \\emph{arbitrarily} different from the true label $\\max_j y_i$, $j\\in\\{1,\\dots m\\}$.\n\n\n\n\n\n\n\\subsection{Transformer-based models}\n\n\\textcolor{red}{directly present vision transformer formulation, we dont need to explain how vision transformer itself was derived}\n\nThe Transformer block was introduced by \\cite{Vaswani2017AttentionIA}, for text input. The basic idea of the Transformer model is to leverage an efficient form of ``self-attention''. A standard attention block is formally defined as,\n\\begin{equation}\n    {\\mathbf{x}}_{out} = \\text{Softmax}\\left(\\frac{{\\mathbf{x}}{\\mathbf{W}}_Q {\\mathbf{W}}_k {\\mathbf{x}}^T}{\\sqrt{d}}\\right) {\\mathbf{x}} {\\mathbf{W}}_V,\n    \\label{eq:selfatt}\n\\end{equation}\nwhere ${\\mathbf{x}} \\in {\\mathbb{R}}^{d\\times n}$ is an input string, ${\\mathbf{x}}_{out} \\in {\\mathbb{R}}^{d\\times n}$ is the output of the self-attention block, ${\\mathbf{W}}_Q,~{\\mathbf{W}}_K~\\text{and}{\\mathbf{W}}_V$ are the learnable \\emph{query}, \\emph{key} and the \\emph{value} matrices. Note that ${\\mathbf{x}}$ is actually a concatenation of $n$ ``tokens'' of size $d$, which each represent some part of the input. \n\\emph{Multi-headed self attention} stacks multiple such blocks in a single layer. The Transformer model has multiple such layers followed by a final output attention layer with a \\emph{classification token}. \\cite{Dosovitskiy2021AnII} proposed using using non-overlapping patches of $16\\times16$ as tokens to ensure near state of the art accuracies. \\cite{Touvron2021TrainingDI} propose a data-efficient distillation based method to train Transformers and improves upon both the sample complexity and the performance over Vision Transformers.  \n\\end{comment}\n\n\n\n\n\n\n\n\\section{Token Attacks on Vision transformers}\n\\label{sec:blocksparse}\n\n\n\\noindent\\textbf{Threat Model:} We define the specific threat model that we consider in our analysis. Let ${\\mathbf{x}} \\in {\\mathbb{R}}^d$ be a $d$-dimensional image, and $f:{\\mathbb{R}}^d \\to [m]$ be a classifier that takes ${\\mathbf{x}}$ as input and outputs one of $m$ class labels. For our attacks, we focus on sparsity as the constraining factor. Specifically, we restrict the number of pixels or blocks of pixels that an attacker is allowed to change. We consider ${\\mathbf{x}}$ as a concatenation of $B$ blocks $[{\\bm{x}}_1, \\dots {\\bm{x}}_b, \\dots, {\\bm{x}}_B]$, where each block is of size $p$. In order to construct an attack, the attacker is allowed to perturb up to $K\\leq B$ such blocks for a $K$-token attack. We also assume a white-box threat model, that is, the attacker has access to all knowledge about the model including gradients and preprocessing. We consider two varying attack budgets. In both cases we consider a block sparse token budget, where we restrict the attacker to modifying $K$ patches or ``tokens\" (1) with an unconstrained perturbation allowed per patch (2) a ``mixed norm'' block sparse budget, where the pixelwise perturbation for each token is restricted to an $\\ell_\\infty$ ball with radius $\\epsilon$ defined as $K, \\epsilon$-attack.\n\n\n\n\\noindent\\textbf{Sparse attack:}\nTo begin, consider the simpler case of a sparse ($\\ell_0$) attack. This is a special case of the block sparse attack with block size is \\emph{one}. Numerous such attacks have been proposed in the past (refer to appendix). The general idea behind most such attacks is to analyse which pixels in the input image tend to affect the output the most \n    $S(x_{i}) := \\left |\\frac{\\partial L(f({\\mathbf{x}}, {\\mathbf{y}}))}{\\partial x_{i}}\\right|$,\n   \nwhere $L(\\cdot)$ is the adversarial loss, and $c$ is the true class predicted by the network.\nThe next step is to perturb the top $s$ most salient pixels for a $s$-sparse attack by using gradient descent to create the least amount of change in the $s$ pixels to adversarially flip the label\n\n\\noindent\\textbf{Patchwise token attacks:}Instead of inspecting saliency of single pixel we check the norm of gradients of pixels belonging to non-overlapping patches using patch saliency $S(\\mathbf{x}_b) := \\sqrt{\\sum_{x_i\\in {\\bm{x}}_b} \\left |\\frac{\\partial L(f({\\mathbf{x}}, {\\mathbf{y}}))}{\\partial x_{i}}\\right|^2 }$, for all $b\\in \\{1,\\dots B\\}$. We pick top $K$ blocks according to patch saliency. The effective sparsity is thus $s=K\\cdot p$.\nThese sequence of operations are summarized in \\Algref{alg:attack}. \n\n\\begin{algorithm}[tp]\n\\small\n\\caption{Adversarial Token Attack}\n\\label{alg:attack}\n\\begin{algorithmic}[1]\n\\Require  ${\\mathbf{x}}_0$:Input image, $f(.)$: Classifier, ${\\mathbf{y}}:$ Original label, $K$: Number of patches to be perturbed, $p$: Patch size.\n $i \\gets 0$\n\\State $[b_1\\dots b_K]$= Top-K of $S(\\mathbf{x}_b) = \\sqrt{\\sum_{x_i\\in {\\bm{x}}_b} \\left |\\frac{\\partial L(f({\\mathbf{x}}, {\\mathbf{y}}))}{\\partial x_{i}}\\right|^2 }$,~$\\forall b$.\n\\While $f({\\mathbf{x}}) \\neq y \\text{ OR MaxIter}$\n    \\State ${\\mathbf{x}}_{b_k} = {\\mathbf{x}}_{b_k} + \\nabla_{{\\mathbf{x}}_{b_k}} L;~~\\forall~~b_k~\\in~\\{b_1,\\dots, b_K\\}$\n    \\State $ {\\mathbf{x}}_{b_k} = Project_{\\epsilon_\\infty} ({\\mathbf{x}}_{b_k})$ (optional)\n\\EndWhile\n\\end{algorithmic}\n\\end{algorithm}\n\nWe use non-overlapping patches to understand the effect of manipulating salient tokens instead of arbitrarily choosing patches. In order to further test the robustness of transformers, we also propose to look at the minimum number of patches that would required to be perturbed by an attacker. For this setup, we modify \\Algref{alg:attack} by linearly searching over the range of $1$ to $K$ patches.\n\n\\begin{figure*}\n\\centering\n\\begin{tabular}{c}\n \\hspace{15pt} Original \\hspace{38pt} Adversarial (patch) \\hspace{8pt} Pertubation (patch) \\hspace{10pt} Adversarial (sparse) \\hspace{8pt} Pertubation (sparse)\\\\\n \\includegraphics[height=0.15\\linewidth]{images\/155_1.png} \\includegraphics[height=0.15\\linewidth]{images\/s155_1_deit224_distill_crop.png}\\\\\n\\end{tabular}\n\n\\caption{\\sl\\textbf{Patch and sparse attacks on transformers}: The attack images are generated with a fixed budget of $20$ patches of size $16\\times 16$, or $5120$ pixels for sparse attack on vision transformer (ViT). Note that the perturbations are imperceptible. The third and fifth columns shows the perturbations brightened $10$ times.}\n\\label{fig:vis_blocksparse}\n\\end{figure*}\n\n\\noindent\\textbf{{Mixed-norm attacks:}}\nMost approaches~\\cite{croce2019sparse, croce2020sparse} additionally rely on a mixed $\\ell_2$-norm based sparse attack in order to generate imperceptible perturbations. Motivated by this setting, we propose a mixed-norm version of our modified attack as well. In order to ensure that our block sparse attacks are imperceptible, we enforce an additional $\\ell_\\infty$ projection step post the gradient ascent step. This is enforced via Step 4 in Alg. \\ref{alg:attack}\n\n\n\n\n\n\n\n\n\\section{Experiments and Results}\n\\label{sec:exps}\n\n\n\n\\noindent\\textbf{Setup:}  To ensure a fair comparison, we choose the best models for the Imagenet dataset~\\cite{ILSVRC15} reported in \\cite{Dosovitskiy2021AnII}, \\cite{Touvron2021TrainingDI} and \\cite{Zagoruyko2016WideRN}. The models achieve near state-of-the-art results in terms of classification accuracy. They also are all trained using the best possible hyperparameters for each case. We use these weights and the shared models from the \\texttt{Pytorch Image models}~\\cite{rw2019timm} repository.\nWe restrict our analysis to a fixed subset of $300$ randomly chosen images from the Imagenet validation dataset.\n\n\\begin{comment}\n\\begin{table}[htp]\n    \\centering\n    \\caption{\\sl \\textbf{Clean Accuracies for the sub-sampled ImageNet dataset.}}\n    \\begin{tabular}{c c}\n    \\toprule \\\\\n    \\textbf{Model} & \\textbf{Clean Accuracy} \\\\\n    \\midrule\n     ViT 224     &  88.70\\\\\n     ViT 384   &  90.03\\\\\n     DeIT &  85.71\\\\\n     DeIT (Distilled) & 87.70 \\\\\n     Wide Resnet & 87.04 \\\\\n    \\bottomrule \n    \\end{tabular}\n    \n    \\label{t:clean}\n\\end{table}\n\\end{comment}\n\n\\noindent\\textbf{Models:} In order to compare the robustness of transformer models to standard CNNs, we consider three different families of architectures:(1) Vision Transformer (ViT)~\\cite{Dosovitskiy2021AnII}, Distilled Vision Transformers (DeIT)~\\cite{Touvron2021TrainingDI}, (2) Resnets~\\cite{He2016DeepRL, Zagoruyko2016WideRN} and (3) MLP Mixer \\cite{Tolstikhin2021MLPMixerAA}.\nFor transformers, \\cite{Dosovitskiy2021AnII} show that best performing Imagenet models have a fixed input token size of $16\\times16$.\nIn order to ensure that the attacks are equivalent, we ensure that any norm or patch budgets are appropriately scaled as per the pre-processing used \\footnote{In case of varying image sizes due to pre-processing, we calculate the scaling factor in terms of the number of pixels and appropriately increase or decrease the maximum number of patches.}. We also scale the $\\epsilon$-norm budget for mixed norm attacks to eight gray levels of the input image post normalization. Additionally, we do a hyper parameter search to find the best attacks for each model analysed. Specific details can be found in the Appendix\\footnote{\\url{https:\/\/github.com\/NYU-DICE-Lab\/TokenAttacks_Supplementary.git}}.\n\n\\begin{figure*}[htp]\n    \\centering\n    \\begin{tabular}{c c }\n         \\includegraphics[width=0.39\\linewidth]{plots\/varypatches_16.tex} &\n         \\includegraphics[width=0.5\\linewidth]{plots\/varyingtokens.tex} \n    \\end{tabular}\n    \\caption{(a) \\small \\sl \\textbf{Robustness to Token Attacks with varying budgets ($p=16$).} Vision transformers are less robust than MLP Mixer and ResNets against patch attacks with patch size matching token size of transformer architecture, (b) \\textbf{Token attacks with varying patch sizes.$K=5$} When the attack patch size is smaller than token size of architecture, vision transformers are comparably robust against patch attacks, to MLP and ResNets. Detailed results can be found in the Appendix }\n    \\label{fig:results}\n\\end{figure*}\n\n\n\n\n\\noindent\\textbf{Patch attacks:} We allow the attacker a fixed budget of tokens as per Algorithm \\ref{alg:attack}.\nWe use the robust accuracy as the metric of robustness, where a higher value is better.\nWe start with an attack budget of $1$ token for an image size of $224\\times224$ for the attacker where each token is a patch of the size $16\\times 16$. In order to compensate for the differences in the size of the input, we scale the attack budget for ViT-384 by allowing for more patches ($3$ to be precise) to be perturbed.  However, we do not enforce any imperceptibility constraints. We run the attack on the fixed subset of ImageNet for the network architectures defined above.  \\Figref{fig:results}(a) shows the result of our analysis. Notice that Transformer architectures are more vulnerable to token attacks as compared to ResNets and MLP-Mixer. Further, ViT-384 proves to be the most vulnerable, and ResNet-101 is the most robust model. DeiT which uses a teacher-student network is more robust than ViTs. We therefore conclude that distillation improves robustness to single token attacks. \n\n\\noindent\\textbf{\\textit{Varying the Token budget:}} For this experiment, we start with a block-budget of $1$ patch, and iterate upto $40$ patches to find the minimum number of tokens required to break an image. We then measure the robust accuracy for each constraint and for each model. For this case, we only study attacks for a fixed patch (token) size of $16\\times16$ and represent our findings in \\Figref{fig:results}(a). We clearly observe a difference in the behavior of ViT versus ResNets here. In general, for a given token budget, ResNets outperform all other token based models. In addition, the robust accuracies for Transformers fall to zero for as few as \\emph{two} patches. The advantage offered by distillation for single token attacks is also lost once the token budget increases. \n\n\\noindent\\textbf{\\textit{Varying patch sizes:}}In order to further analyse if these results hold across stronger and weaker block sparse constraints, we further run attacks for varying patch sizes. Smaller patch sizes are equivalent to partial token manipulation. We fix the token budget to be $5$ or $15$ tokens as dictated by the input size. Here, this corresponds to allowing the attacker to perturb $5$ $p\\times p$ patches. As one would expect, a smaller partial token attack is weaker than a full token attack. Surprisingly, the Transformer networks are comparable or better than ResNets for attacks smaller than a single token. This leads us to conclude that Transformers can compensate for adversarial perturbations within a tokens. However, as the patch size approaches the token size, Resnets achieve better robustness. We also see that MLP-Mixers, while also using the token based input scheme, perform better than Transformers as the patch attack size increases.\n\nHowever, this approach allows for unrestricted changes to the tokens. Another approach would be to study the effect of ``mixed norm'' attacks which further constrain the patches to be \\emph{imperceptibly} perturbed.\n\n\n\n\n\n\n\\noindent\\textit{\\textbf{Mixed Norm Attacks:}} For the mixed norm attacks, we analyse the robustness of all networks for a fixed $\\epsilon$ $\\ell_\\infty$ budget of one gray level. We vary the token budgets from $1$ to $5$. Here, almost all the networks show similar robustness for a small token budget ($K$=1,2); refer \\Tableref{t:mixednorm}. However, as the token budget increases, Transformer and MLP Mixer networks are far more vulnerable. Note that this behavior contradicts~\\cite{Bhojanapalli2021UnderstandingRO}, where ViTs outperform ResNets. Since our threat model leverages the token based architecture of the Transformers, our attacks are far more successful at breaking ViTs over Resnets.\n\\begin{small}\n\\begin{table}[h]\n\\centering\n\\begin{minipage}{0.48\\linewidth}\n\\caption{\\small \\textbf{\\sl  Robust Accuracy for Mixed Norm Attacks:} The models are attacked with a $K,(1\/255)$ Patch Attack. Note that for smaller token budgets, the models perform nearly the same. However, as the token budget increases, Resnets are more robust than Transformers.}\n\\label{t:mixednorm}\n\\begin{tabular}{c c c c c} \n  \\toprule\n  \\textbf{Model} & \\textbf{Clean} & \\multicolumn{3}{c}{\\textbf{Token Budget}}  \\\\\n  \\midrule \n  {} & & 1 & 2 & 5 \\\\\n  \\midrule\n   ViT-224 & 88.70 & 68.77 & 50.83 & 15.28 \\\\\n   ViT-384 & \\textbf{90.03} & \\textit{53.48} & \\textit{28.57} & \\textit{4.98} \\\\ \n   \\midrule\n   DeIT & 85.71 & \\textbf{72.42}  & 46.84 & 6.31 \\\\\n   DeIT-Distilled & 87.70 &  68.77 & 54.15 & 16.61 \\\\\n   \\midrule\n   Resnet-101 & 85.71 & 69.10 & 55.14 & \\textbf{32.89} \\\\\n   Resnet-50 & 85.38 &  67.44 & \\textbf{55.81} & 31.22 \\\\\n   Wide Resnet & 87.04 &  54.81 & 32.89 & 11.62 \\\\\n   \\midrule\n   MLP-Mixer & \\textit{83.78} &  63.78 & 37.87 & 5.98 \\\\\n  \\bottomrule    \n\\end{tabular}\n\\end{minipage}\n\\begin{minipage}{0.48\\linewidth}\n \\centering\n    \\small\n    \\caption{\\sl \\textbf{Robust accuracies, $s=256$ sparse and $K=1$, $16\\times16$ patch attack .} }\n    \\begin{tabular}{c c c c}\n        \\toprule\n        \\textbf{Model} & & \\multicolumn{2}{c}{\\textbf{Norm constraint}} \\\\\n        \\midrule \n        {} & Clean & Sparse  & Patch\\\\\n        \\midrule \n        ViT 224 & 88.70 & 5.98 & 13.62 \\\\\n        ViT 384 & \\textbf{90.03} & 3.32 & \\textit{1.33}\\\\\n        \\midrule\n        DeIT & 85.71 & 4.65  & 17.27  \\\\\n        DeIT (Distilled) & 87.70& 14.95  &  17.94\\\\\n        \\midrule\n        MLP Mixer & \\textit{83.72} & 5.98 & 26.91 \\\\\n        \\midrule\n        ResNet 50 & 85.38 & 13.95 & 19.90  \\\\\n        ResNet 101 & 85.71 & \\textbf{23.59} & \\textbf{49.50} \\\\\n        Wide Resnet & 87.04 & \\textit{1.33} & 26.57  \\\\\n        \\bottomrule \\\\\n    \\end{tabular}   \n    \\label{tab:sparse}\n\\end{minipage}\n\\end{table}\n\\end{small}\n\n\\noindent\\textbf{Sparse Attacks:} The sparse variant of our algorithm restricts the patch size to $1\\times 1$. We allow for a sparsity budget of $0.5\\%$ of original number of pixels. In case of the standard $224\\times 224$ ImageNet image, the attacker is allowed to perturb $256$ pixels. We compare the attack success rate of both sparse attack and patch-based token attack at same sparsity budget; to compare we chose $1, 16\\times 16$ patch attack (refer \\Tableref{tab:sparse}).\nWe see that as is the case with token attacks, even for sparse attacks, vision transformers are less robust as compared to ResNets. With the same sparsity budget, sparse attacks are stronger than token attacks; however we stress that sparse threat model is less practical to implement as the sparse coefficients may be scattered anywhere in the image. \n\n\n\\section{Discussion and Conclusion}\nAnalysing the above results, we infer certain interesting properties of transformers. \n\\begin{enumerate}[nolistsep, parsep=0pt]\n \\item We find that Transformers are generally susceptible to token attacks, even for very low token budgets.\n \\item However, Transformers appear to compensate for perturbations to patch attacks smaller than the token size. \n \\item  Further, ResNets and MLP-Mixer outperform Transformers for token attacks consistently.\n\\end{enumerate}\n\nAn interesting direction of follow-up work is to develop strong certifiable defenses for token attacks. Further directions of research also include analysis of the effect of distillation and semi-supervised pre-training.\n \n\\section*{Acknowledgements}\nThe authors were supported in part by the National Science Foundation under grants CCF-2005804 and CCF-1815101, USDA\/NIFA under grant USDA-NIFA:2021-67021-35329, and ARPA-E under grant DE:AR0001215.\n\n\n\n\n\n\n\n\\bibliographystyle{IEEEbib}\n\\begin{small}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn this paper, we investigate the Liouville property of nonnegative solutions to the following Dirichlet problems for elliptic equations in exterior domains\n\\begin{equation}\\label{GPDE}\\\\\\begin{cases}\n(-\\Delta)^{\\frac{\\alpha}{2}}u(x)=f(x,u(x)), \\,\\,\\,\\,\\,\\,\\, u(x)\\geq0, \\,\\,\\,\\,\\,\\,\\,\\, x\\in\\Omega_{r}, \\\\\nu(x)\\equiv0,\\,\\,\\,\\,\\,\\,\\,\\, x\\in\\mathbb{R}^{n}\\setminus\\Omega_{r},\n\\end{cases}\\end{equation}\nwhere the exterior domains $\\Omega_{r}:=\\{x\\in\\mathbb{R}^{n}\\,|\\,|x|>r\\}$ with arbitrary $r>0$, $n\\geq2$, $0<\\alpha\\leq2$ and the nonlinear terms $f:\\, \\Omega_{r}\\times\\overline{\\mathbb{R}_{+}}\\rightarrow \\overline{\\mathbb{R}_{+}}$. When $0<\\alpha<2$, the nonlocal fractional Laplacians $(-\\Delta)^{\\frac{\\alpha}{2}}$ is defined by\n\\begin{equation}\\label{nonlocal defn}\n  (-\\Delta)^{\\frac{\\alpha}{2}}u(x)=C_{\\alpha,n} \\, P.V.\\int_{\\mathbb{R}^n}\\frac{u(x)-u(y)}{|x-y|^{n+\\alpha}}dy:=C_{\\alpha,n}\\lim_{\\epsilon\\rightarrow0}\\int_{|y-x|\\geq\\epsilon}\\frac{u(x)-u(y)}{|x-y|^{n+\\alpha}}dy\n\\end{equation}\nfor functions $u\\in C^{1,1}_{loc}\\cap\\mathcal{L}_{\\alpha}(\\mathbb{R}^{n})$, where the constant $C_{\\alpha,n}=\\big(\\int_{\\mathbb{R}^{n}}\\frac{1-\\cos(2\\pi\\zeta_{1})}{|\\zeta|^{n+\\alpha}}d\\zeta\\big)^{-1}$ and the function spaces\n\\begin{equation}\\label{0-1}\n  \\mathcal{L}_{\\alpha}(\\mathbb{R}^{n}):=\\Big\\{u: \\mathbb{R}^{n}\\rightarrow\\mathbb{R}\\,\\Big|\\,\\int_{\\mathbb{R}^{n}}\\frac{|u(x)|}{1+|x|^{n+\\alpha}}dx<\\infty\\Big\\}.\n\\end{equation}\nFor $0<\\alpha<2$, we assume the solution $u\\in C_{loc}^{1,1}(\\Omega_{r})\\cap C(\\overline{\\Omega_{r}})\\cap\\mathcal{L}_{\\alpha}(\\mathbb{R}^{n})$. For $\\alpha=2$, we assume the solution $u\\in C^{2}(\\Omega_{r})\\cap C(\\overline{\\Omega_{r}})$.\n\nWe say equations \\eqref{GPDE} have critical order if $\\alpha=n$ and non-critical order if $0<\\alpha<n$. The following definitions and assumptions on the nonlinear terms $f(x,u)$ will be necessary.\n\\begin{defn}\\label{defn1}\nIn the non-critical order cases $0<\\alpha<n$, we say that the nonlinear term $f$ has subcritical growth provided that\n\\begin{equation}\\label{e1}\n  \\mu^{\\frac{n+\\alpha}{n-\\alpha}}f(\\mu^{\\frac{2}{n-\\alpha}}x,\\mu^{-1}u)\n\\end{equation}\nis strictly increasing with respect to $\\mu\\geq1$ or $\\mu\\leq1$ for all $(x,u)\\in\\Omega_{r}\\times\\mathbb{R}_{+}$. In the critical order cases $\\alpha=n$, we say that the nonlinear term $f$ has subcritical growth, provided that\n\\begin{equation}\\label{e1}\n  \\mu^{n}f(\\mu x,u)\n\\end{equation}\nis strictly increasing with respect to $\\mu\\geq1$ or $\\mu\\leq1$ for all $(x,u)\\in\\Omega_{r}\\times\\mathbb{R}_{+}$.\n\\end{defn}\n\\begin{defn}\\label{defn2}\nA function $g(x,u)$ is called locally Lipschitz on $u$ in $\\Omega_{r}\\times\\overline{\\mathbb{R}_{+}}$, provided that for any $u_{0}\\in\\overline{\\mathbb{R}_{+}}$ and $\\omega\\subseteq\\Omega_{r}$ bounded, there exists a (relatively) open neighborhood $U(u_{0})\\subset\\overline{\\mathbb{R}_{+}}$ such that $g$ is Lipschitz continuous on $u$ in $\\omega\\times U(u_{0})$.\n\\end{defn}\n\nWe need the following three assumptions on the nonlinear terms $f(x,u)$. \\\\\n$(\\mathbf{f_{1}})$ The nonlinear term $f(x,u)$ is non-decreasing about $u$ in $\\Omega_{r}\\times\\overline{\\mathbb{R}_{+}}$, namely,\n\\begin{equation}\\label{e2}\n  (x,u), \\, (x,v)\\in\\Omega_{r}\\times\\overline{\\mathbb{R}_{+}} \\,\\,\\, \\text{with} \\,\\,\\, u\\leq v \\,\\,\\, \\text{implies} \\,\\,\\, f(x,u)\\leq f(x,v).\n\\end{equation}\n$(\\mathbf{f_{2}})$ There exists a $\\theta<\\frac{\\alpha}{n}$ such that, $(|x|-r)^{\\theta}f(x,u)$ is locally Lipschitz on $u$ in $\\Omega_{r}\\times\\overline{\\mathbb{R}_{+}}$. \\\\\n$(\\mathbf{f_{3}})$ There exist a cone $\\mathcal{C}$ with vertex at $0$, constants $\\overline{C}>0$, $\\sigma>1$, $-\\alpha<\\tau<+\\infty$ and $0<p<\\frac{n+\\alpha+2\\tau}{n-\\alpha}$ if $0<\\alpha<n$, or $-\\alpha\\leq\\tau<+\\infty$ and $0<p<+\\infty$ if $\\alpha=n$ such that, the nonlinear term\n\\begin{equation}\\label{e3}\n  f(x,u)\\geq \\overline{C}|x|^{\\tau}u^{p}\n\\end{equation}\nin $(\\mathcal{C}\\cap\\Omega_{2\\sigma r})\\times\\overline{\\mathbb{R}_{+}}$.\n\n\\begin{rem}\\label{rem0}\nIn particular, assume $0\\leq b<+\\infty$, $-b-\\alpha<a<+\\infty$ and $1\\leq p<\\frac{n+\\alpha+2(a+b)}{n-\\alpha}$ if $0<\\alpha<n$, or $-b-\\alpha\\leq a<+\\infty$ and $1\\leq p<+\\infty$ if $\\alpha=n$, then\n\\begin{equation}\\label{0-2}\n  f(x,u)=|x|^{a}(|x|-r)^{b}u^{p}\n\\end{equation}\nis subcritical and satisfies all the assumptions $(\\mathbf{f_{1}})$, $(\\mathbf{f_{2}})$ and $(\\mathbf{f_{3}})$. Moreover, under the same assumptions, for any $i=1,\\cdots,n$, nonlinearities\n\\begin{equation}\\label{0-3}\n  f(x,u)=|x_{i}|^{a}(|x|-r)^{b}u^{p}, \\quad |x|^{a}(|x_{i}|-r)^{b}u^{p} \\quad \\text{or} \\quad |x_{i}|^{a}(|x_{i}|-r)^{b}u^{p}\n\\end{equation}\nare also subcritical and satisfy all the assumptions $(\\mathbf{f_{1}})$, $(\\mathbf{f_{2}})$ and $(\\mathbf{f_{3}})$.\n\\end{rem}\n\nFor $0<\\alpha\\leq n$, PDEs of the form\n\\begin{equation}\\label{PDE}\n  (-\\Delta)^{\\frac{\\alpha}{2}}u(x)=|x|^{a}u^{p}(x)\n\\end{equation}\nare called the fractional order or higher order H\\'{e}non, Lane-Emden, Hardy equations for $a>0$, $a=0$, $a<0$, respectively. These equations have numerous important applications in conformal geometry and Sobolev inequalities. In particular, in the case $a=0$, \\eqref{GPDE} becomes the well-known Lane-Emden equation, which models many phenomena in mathematical physics and in astrophysics.\n\nThe nonlinear terms in \\eqref{PDE} is called critical if $p=p_{s}(a):=\\frac{n+\\alpha+2a}{n-\\alpha}$ ($:=+\\infty$ if $n=\\alpha$), subcritical if $0<p<p_{s}(a)$ and supercritical if $p_{s}(a)<p<+\\infty$. Liouville type theorems for equations \\eqref{PDE} (i.e., nonexistence of nontrivial nonnegative solutions) in the whole space $\\mathbb{R}^n$, the half space $\\mathbb{R}^n_+$ and bounded domains $\\Omega$ have been extensively studied (see \\cite{BG,BP,CD,CDQ,CFY,CL,CLL,CLZ,DPQ,DQ1,DQ0,DQ,DQ2,DQZ,DZ,GS,Lei,Lin,MP,PS,RW,RZ,WX} and the references therein). For other related properties on PDEs \\eqref{PDE} and Liouville type theorems on systems of PDEs of type \\eqref{PDE} with respect to various types of solutions (e.g., stable, radial, singular, nonnegative, sign-changing, $\\cdots$), please refer to \\cite{BG,CD,CGS,CLO,DFQ,DQ1,DQ,FG,LB,Lei,Lin,M,PQS,WX} and the references therein. These Liouville theorems, in conjunction with the blowing up and re-scaling arguments, are crucial in establishing a priori estimates and hence existence of positive solutions to non-variational boundary value problems for a class of elliptic equations on bounded domains or on Riemannian manifolds with boundaries (see \\cite{CDQ,DPQ,DQ0,DQ2,GS1,MP,PQS,RW}).\n\nIn this paper, by applying the method of scaling spheres developed in \\cite{DQ0}, we will establish Liouville theorems for nonnegative solutions of the generalized equations \\eqref{GPDE} with Dirichlet boundary conditions in unbounded exterior domains.\n\nFirst, by using similar arguments as in \\cite{CFY,ZCCY} (see also \\cite{DFQ,DQ0}), we can deduce the equivalence between PDEs \\eqref{GPDE} and the following integral equations\n\\begin{equation}\\label{IEe}\n  u(x)=\\int_{\\Omega_{r}}G_{\\alpha}(x,y)f(y,u(y))dy,\n\\end{equation}\nwhere\n\\begin{equation}\\label{GREENe-0}\n  G_{\\alpha}(x,y):=\\frac{C_{n,\\alpha}}{|x-y|^{n-\\alpha}}\\int_{0}^{\\frac{(|x|^{2}-r^{2})(|y|^{2}-r^{2})}{r^{2}|x-y|^{2}}}\\frac{b^{\\frac{\\alpha}{2}-1}}{(1+b)^{\\frac{n}{2}}}db \\quad\\quad \\text{if} \\,\\,\\, x,y\\in \\Omega_{r},\n\\end{equation}\nand $G_{\\alpha}(x,y):=0$ if $x$ or $y\\in\\mathbb{R}^{n}\\setminus \\Omega_{r}$ is the Green's function in exterior domain $\\Omega_{r}$ for $(-\\Delta)^{\\frac{\\alpha}{2}}$ with Dirichlet boundary conditions when $0<\\alpha\\leq2$ and $n\\geq2$. That is, we have the following theorem.\n\\begin{thm}\\label{equivalence}\nIf u is a nonnegative solution of \\eqref{GPDE}, then $u$ is also a nonnegative solution of integral equation \\eqref{IEe}, and vice versa.\n\\end{thm}\n\n\\begin{rem}\\label{rem2}\nTheorem \\ref{equivalence} can be proved through entirely similar arguments as in \\cite{CFY,ZCCY}, so we omit the details here.\n\\end{rem}\n\nNext, we consider the integral equations \\eqref{IEe} instead of PDEs \\eqref{GPDE}. We will study the integral equations \\eqref{IEe} via the method of scaling spheres developed by Dai and Qin in \\cite{DQ0}. The method of scaling spheres is essentially a frozen variant of the method of moving spheres, that is, we only dilate or shrink the spheres with respect to one fixed center. The method of moving spheres was initially used by Chen and Li \\cite{CL1}, Li and Zhu \\cite{LZ} and Padilla \\cite{Pa}, which means moving spheres centered at every points in $\\mathbb{R}^{n}$ or $\\partial\\mathbb{R}^{n}_{+}$ in conjunction with calculus and ODE analysis. Later, it was further developed by Li \\cite{Li}, Chen and Li \\cite{CL0}, Jin, Li and Xu \\cite{JLX}. Recently, Chen, Li and Zhang developed a direct method of moving spheres on fractional order equations in \\cite{CLZ}. One should note that, being different from the method of moving spheres, the method of scaling spheres take full advantage of the integral representation formulae of solutions and can be applied to various problems with singularities or without translation invariance on general domains. The method of scaling spheres, in conjunction with the integral representation formulae of solutions and a ``Bootstrap\" iteration process, will provide useful lower bound estimates on the asymptotic behaviour of solutions, which will lead to a contradiction with the integrability of solutions unless the solution $u\\equiv0$.\n\nOur Liouville type result for IEs \\eqref{IEe} is the following theorem.\n\\begin{thm}\\label{Thm0}\nAssume $n\\geq2$, $0<\\alpha\\leq2$, $f$ is subcritical and satisfies the assumptions $(\\mathbf{f_{1}})$, $(\\mathbf{f_{2}})$ and $(\\mathbf{f_{3}})$. If $u\\in C(\\overline{\\Omega_{r}})$ is a nonnegative solution to IEs \\eqref{IEe}, then $u\\equiv 0$ in $\\overline{\\Omega_{r}}$.\n\\end{thm}\n\nAs a consequence of Theorem \\ref{equivalence} and \\ref{Thm0}, we obtain immediately the following Liouville type theorem on PDEs \\eqref{GPDE}.\n\\begin{thm}\\label{Thm1}\nAssume $n\\geq2$, $0<\\alpha\\leq2$, $f$ is subcritical and satisfies the assumptions $(\\mathbf{f_{1}})$, $(\\mathbf{f_{2}})$ and $(\\mathbf{f_{3}})$. Suppose $u$ is a nonnegative solution of PDEs \\eqref{GPDE}, then $u\\equiv0$ in $\\overline{\\Omega_{r}}$.\n\\end{thm}\n\n\\begin{rem}\\label{rem3}\nFor $\\alpha=2$ and $n\\geq3$, Reichel and Zou \\cite{RZ} have obtained some Liouville type theorems for equations \\eqref{GPDE} under some assumptions. Theorem \\ref{Thm1} improved the results in \\cite{RZ} at least in three aspects. First, Reichel and Zou \\cite{RZ} required that there exist some $\\tau>-2$ and $1<p<\\frac{n+\\alpha+2\\tau}{n-\\alpha}$ such that $f(x,u)$ satisfies the lower bound \\eqref{e3} in assumption $(\\mathbf{f_{3}})$ on the whole $\\Omega_{r}\\times\\overline{\\mathbb{R}_{+}}$. But we only need to assume in assumption $(\\mathbf{f_{3}})$ that there exist $-\\alpha<\\tau<+\\infty$, $0<p<\\frac{n+\\alpha+2\\tau}{n-\\alpha}$ if $0<\\alpha<n$ ($-\\alpha\\leq\\tau<+\\infty$, $0<p<+\\infty$ if $\\alpha=n$), a cone $\\mathcal{C}$ with vertex at $0$ and $\\sigma>1$ such that $f(x,u)$ satisfies the lower bound \\eqref{e3} in $(\\mathcal{C}\\cap\\Omega_{2\\sigma r})\\times\\overline{\\mathbb{R}_{+}}$. This allows us to have much more admissible choices of the nonlinearities $f(x,u)$ (see Remark \\ref{rem0} and \\ref{rem1}). Second, Theorem \\ref{Thm1} can also be applied to general fractional order cases $0<\\alpha<2$ with $n\\geq2$ and the critical order cases $\\alpha=n=2$. Third, in assumption $(\\mathbf{f_{2}})$, we only assume $(|x|-r)^{\\theta}f(x,u)$ (not $f(x,u)$ itself) is locally Lipschitz on $u$, this allows $f(x,u)$ to have some singularities near the sphere $S_{r}:=\\{x\\in\\mathbb{R}^{n}\\,|\\,|x|=r\\}$.\n\\end{rem}\n\nIn particular, we consider the following Dirichlet problems for the H\\'{e}non-Hardy type equations in exterior domains\n\\begin{equation}\\label{HPDE}\\\\\\begin{cases}\n(-\\Delta)^{\\frac{\\alpha}{2}}u(x)=|x|^{a}(|x|-r)^{b}u^{p}, \\,\\,\\,\\,\\,\\,\\, u(x)\\geq0, \\,\\,\\,\\,\\,\\,\\,\\, x\\in\\Omega_{r}, \\\\\nu(x)\\equiv0,\\,\\,\\,\\,\\,\\,\\,\\, x\\in\\mathbb{R}^{n}\\setminus\\Omega_{r},\n\\end{cases}\\end{equation}\nwhere $n\\geq2$, $0<\\alpha\\leq2$.\n\nAs a consequence of Theorem \\ref{Thm1} and Remark \\ref{rem0}, we deduce the following corollary.\n\\begin{cor}\\label{Cor1}\nAssume $0\\leq b<+\\infty$, $-b-\\alpha<a<+\\infty$ and $1\\leq p<\\frac{n+\\alpha+2(a+b)}{n-\\alpha}$ if $0<\\alpha<n$, $-b-\\alpha\\leq a<+\\infty$ and $1\\leq p<+\\infty$ if $\\alpha=n=2$. Suppose $u$ is a nonnegative solution of \\eqref{HPDE}, then $u\\equiv0$ in $\\overline{\\Omega_{r}}$.\n\\end{cor}\n\n\\begin{rem}\\label{rem1}\nBy Remark \\ref{rem0}, if we suppose the nonlinearities in \\eqref{GPDE} take the form:\n\\begin{equation}\\label{0-3}\n  f(x,u)=|x_{i}|^{a}(|x|-r)^{b}u^{p}, \\quad |x|^{a}(|x_{i}|-r)^{b}u^{p} \\quad \\text{or} \\quad |x_{i}|^{a}(|x_{i}|-r)^{b}u^{p}\n\\end{equation}\nfor any $i=1,\\cdots,n$, then under the same assumptions as in Corollary \\ref{Cor1}, the Liouville type results in Theorem \\ref{Thm1} are also valid for equations \\eqref{GPDE}.\n\\end{rem}\n\nWe also consider the following higher order integral equations\n\\begin{equation}\\label{HIEe}\n  u(x)=\\int_{\\Omega_{r}}G_{\\alpha}(x,y)f(y,u(y))dy,\n\\end{equation}\nwhere for $\\alpha=2m$ with $1\\leq m<\\frac{n}{2}$ and $n\\geq3$,\n\\begin{equation}\\label{GREENe-H1}\n  G_{\\alpha}(x,y):=C_{n,\\alpha}\\Bigg(\\frac{1}{|x-y|^{n-\\alpha}}-\\frac{1}{\\left|\\frac{rx}{|x|}-\\frac{|x|y}{r}\\right|^{n-\\alpha}}\\Bigg) \\qquad \\text{if} \\,\\,\\, x,y\\in \\Omega_{r},\n\\end{equation}\nwhile for $\\alpha=n$ with $n\\geq2$ even,\n\\begin{equation}\\label{GREENe-H2}\n  G_{\\alpha}(x,y):=C_{n,\\alpha}\\Bigg(\\ln\\frac{1}{|x-y|}-\\ln\\frac{1}{\\left|\\frac{rx}{|x|}-\\frac{|x|y}{r}\\right|}\\Bigg) \\qquad \\text{if} \\,\\,\\, x,y\\in \\Omega_{r},\n\\end{equation}\nand $G_{\\alpha}(x,y):=0$ if $x$ or $y\\in\\mathbb{R}^{n}\\setminus \\Omega_{r}$ is the Green's function in exterior domain $\\Omega_{r}$ for $(-\\Delta)^{\\frac{\\alpha}{2}}$ with Navier boundary conditions when $\\alpha=2m$ with $1\\leq m\\leq\\frac{n}{2}$ and $n\\geq2$. The integral equations \\eqref{HIEe} is closely related to higher order Navier problems in $\\Omega_{r}$.\n\nBy entirely similar arguments as in the proof of Theorem \\ref{Thm0}, we can prove the following Liouville theorem for integral equations \\eqref{HIEe}.\n\\begin{thm}\\label{Thm0-H}\nAssume $n\\geq2$, $\\alpha=2m$ with $1\\leq m\\leq\\frac{n}{2}$, $f$ is subcritical and satisfies the assumptions $(\\mathbf{f_{1}})$, $(\\mathbf{f_{2}})$ and $(\\mathbf{f_{3}})$. If $u\\in C(\\overline{\\Omega_{r}})$ is a nonnegative solution to IEs \\eqref{HIEe}, then $u\\equiv 0$ in $\\overline{\\Omega_{r}}$.\n\\end{thm}\n\n\\begin{rem}\\label{rem5}\nTheorem \\ref{Thm0-H} can be proved through entirely similar arguments as Theorem \\ref{Thm0}, so we omit the details here.\n\\end{rem}\n\n\\begin{rem}\\label{rem4}\nConsider the following Navier problems for higher order elliptic equations in $\\Omega_{r}$:\n\\begin{equation}\\label{NPDE-H}\\\\\\begin{cases}\n(-\\Delta)^{\\frac{\\alpha}{2}} u(x)=f(x,u(x)), \\,\\,\\,\\,\\,\\,\\,\\, u(x)\\geq0, \\,\\,\\,\\,\\, \\,\\,\\, x\\in\\Omega_{r}, \\\\\nu=(-\\Delta)u=\\cdots=(-\\Delta)^{\\frac{\\alpha}{2}-1}u=0 \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\text{on} \\,\\,\\,  S_{r},\n\\end{cases}\\end{equation}\nwhere $u\\in C^{\\alpha}(\\Omega_{r})\\cap C^{\\alpha-2}(\\overline{\\Omega_{r}})$, $n\\geq2$, $\\alpha=2m$ with $1\\leq m\\leq\\frac{n}{2}$, $f$ is subcritical and satisfies the assumptions $(\\mathbf{f_{1}})$, $(\\mathbf{f_{2}})$ and $(\\mathbf{f_{3}})$. Once the equivalence between the Navier problems \\eqref{NPDE-H} and the integral equations \\eqref{HIEe} has been established, we can also derive from Theorem \\ref{Thm0-H} that Liouville theorem for nonnegative solutions to the Navier problems \\eqref{NPDE-H} holds.\n\\end{rem}\n\nIn the following, we will use $C$ to denote a general positive constant that may depend on $n$, $\\alpha$, $\\tau$, $\\sigma$, $\\theta$, $p$, $\\overline{C}$, $u$ and the cone $\\mathcal{C}$, and whose value may differ from line to line.\n\n\n\n\\section{Proof of Theorem \\ref{Thm0}}\n\nIn this section, we will prove Theorem \\ref{Thm0} via contradiction arguments and the method of scaling spheres. Without loss of generality, we may assume the radius $r=1$. We may also assume that the nonlinear term $f(x,u)$ satisfies subcritical conditions in Definition \\ref{defn1} for $\\mu\\leq1$. If $f(x,u)$ satisfies subcritical conditions in Definition \\ref{defn1} for $\\mu\\geq1$, we only need to carry out calculations and estimates inside the ball $B_{\\lambda}(0)$ during the scaling spheres procedure.\n\nNow suppose on the contrary that $u\\geq0$ satisfies the equivalent integral equations \\eqref{IEe} but $u$ is not identically zero, then one can infer from the integral equations \\eqref{IEe} that $u$ is actually a positive solution, i.e., $u>0$ in $\\Omega_{1}$. Next, we will carry out our proof by discussing the non-critical order cases and the critical order case separately.\n\n\\subsection{The non-critical order cases $0<\\alpha<n$}\n\nWe will apply the method of scaling spheres to show the following lower bound estimates for asymptotic behaviour of positive solution $u$ as $|x|\\rightarrow+\\infty$, which will contradict with the integral equations \\eqref{IEe}.\n\\begin{thm}\\label{lower-e}\nAssume $n\\geq2$, $n>\\alpha$, $0<\\alpha\\leq2$, $f(x,u)$ is subcritical and satisfies assumptions $(\\mathbf{f_{1}})$, $(\\mathbf{f_{2}})$ and $(\\mathbf{f_{3}})$. Suppose $u$ is a positive solution to integral equations \\eqref{IEe}, then it satisfies the following lower bound estimates: for all $|x|\\geq2\\sigma$,\n\\begin{equation}\\label{lb1-e}\n  u(x)\\geq C_{\\kappa}|x|^{\\kappa} \\quad\\quad \\forall \\, \\kappa<\\frac{\\alpha+\\tau}{1-p}, \\quad\\quad \\text{if} \\,\\,\\,\\, 0<p<1;\n\\end{equation}\n\\begin{equation}\\label{lb2-e}\n  u(x)\\geq C_{\\kappa}|x|^{\\kappa} \\quad\\quad \\forall \\, \\kappa<+\\infty, \\quad\\quad \\text{if} \\,\\,\\,\\, 1\\leq p<\\frac{n+\\alpha+2\\tau}{n-\\alpha}.\n\\end{equation}\n\\end{thm}\n\\begin{proof}\nGiven any $\\lambda>1$, we define the Kelvin transform of $u$ centered at $0$ by\n\\begin{equation}\\label{Kelvin-e}\n  u_{\\lambda}(x):=\\left(\\frac{\\lambda}{|x|}\\right)^{n-\\alpha}u\\left(\\frac{\\lambda^{2}x}{|x|^{2}}\\right)\n\\end{equation}\nfor arbitrary $x\\in\\{x\\in\\overline{\\Omega_{1}}\\,|\\,1\\leq|x|\\leq\\lambda^{2}\\}$, and define the reflection of $x$ about the sphere $S_{\\lambda}:=\\{x\\in\\mathbb{R}^{n}\\,|\\,|x|=\\lambda\\}$ by $x^{\\lambda}:=\\frac{\\lambda^{2}x}{|x|^{2}}$.\n\nNow, we will carry out the process of scaling spheres in $\\Omega_{1}$ with respect to the origin $0\\in\\mathbb{R}^{n}$.\n\nLet $\\lambda>1$ be an arbitrary real number and let $\\omega^{\\lambda}(x):=u_{\\lambda}(x)-u(x)$ for any $x\\in B_{\\lambda^{2}}(0)\\setminus\\overline{B_{\\lambda}(0)}$. We will first show that, for $\\lambda>1$ sufficiently close to $1$,\n\\begin{equation}\\label{2-7-e}\n  \\omega^{\\lambda}(x)\\leq0, \\,\\,\\,\\,\\,\\, \\forall \\,\\, x\\in B_{\\lambda^{2}}(0)\\setminus\\overline{B_{\\lambda}(0)}.\n\\end{equation}\nThen, we start dilating the sphere $S_{\\lambda}$ from near the unit sphere $S_{1}$ outward as long as \\eqref{2-7-e} holds, until its limiting position $\\lambda=+\\infty$ and derive lower bound estimates on asymptotic behaviour of $u$ as $|x|\\rightarrow+\\infty$. Therefore, the scaling sphere process can be divided into two steps.\n\n\\emph{Step 1. Start dilating the sphere $S_{\\lambda}$ from near $\\lambda=1$.} Define\n\\begin{equation}\\label{2-8-e}\n  (B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}})^{+}:=\\{x\\in B_{\\lambda^{2}}(0)\\setminus\\overline{B_{\\lambda}(0)} \\, | \\, \\omega^{\\lambda}(x)>0\\}.\n\\end{equation}\nWe will show that, for $\\lambda>1$ sufficiently close to $1$,\n\\begin{equation}\\label{2-9-e}\n  (B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}})^{+}=\\emptyset.\n\\end{equation}\n\nSince $u$ is a positive solution to the integral equations \\eqref{IEe}, through direct calculations, we get, for any $\\lambda>1$,\n\\begin{equation}\\label{2-34-e}\n  u(x)=\\int_{|y|>\\lambda}G_{\\alpha}(x,y)f(y,u(y))dy+\\int_{B_{\\lambda^{2}}(0)\\setminus\\overline{B_{\\lambda}(0)}}G_{\\alpha}(x,y^{\\lambda})\n  \\left(\\frac{\\lambda}{|y|}\\right)^{2n}f(y^{\\lambda},u(y^{\\lambda}))dy\n\\end{equation}\nfor any $x\\in\\overline{\\Omega_{1}}$. By direct calculations, one can also verify that $u_{\\lambda}$ satisfies the following integral equation\n\\begin{equation}\\label{2-35-e}\n  u_{\\lambda}(x)=\\int_{|y|>1}G_{\\alpha}(x^{\\lambda},y)\\left(\\frac{\\lambda}{|x|}\\right)^{n-\\alpha}f(y,u(y))dy\n\\end{equation}\nfor any $x\\in\\{x\\in\\overline{\\Omega_{1}}\\,|\\,1\\leq|x|\\leq\\lambda^{2}\\}$, and hence, it follows immediately that\n\\begin{eqnarray}\\label{2-36-e}\n  u_{\\lambda}(x)&=&\\int_{|y|>\\lambda}G_{\\alpha}(x^{\\lambda},y)\\left(\\frac{\\lambda}{|x|}\\right)^{n-\\alpha}f(y,u(y))dy \\\\\n \\nonumber \\quad\\quad &&+\\int_{B_{\\lambda^{2}}(0)\\setminus\\overline{B_{\\lambda}(0)}}G_{\\alpha}(x^{\\lambda},y^{\\lambda})\\left(\\frac{\\lambda}{|x|}\\right)^{n-\\alpha}\n \\left(\\frac{\\lambda}{|y|}\\right)^{2n}f(y^{\\lambda},u(y^{\\lambda}))dy.\n\\end{eqnarray}\nTherefore, we have, for any $x\\in B_{\\lambda^{2}}(0)\\setminus\\overline{B_{1}(0)}$,\n\\begin{eqnarray}\\label{omega-e}\n  && \\omega_{\\lambda}(x)=u_{\\lambda}(x)-u(x) \\\\\n  \\nonumber &=& \\int_{B_{\\lambda^{2}}(0)\\setminus\\overline{B_{\\lambda}(0)}}\\Bigg\\{\\left[\\left(\\frac{\\lambda}{|x|}\\right)^{n-\\alpha}G_{\\alpha}(x^{\\lambda},y^{\\lambda})\n  -G_{\\alpha}(x,y^{\\lambda})\\right]\\left(\\frac{\\lambda}{|y|}\\right)^{2n}f(y^{\\lambda},u(y^{\\lambda})) \\\\\n \\nonumber && -\\left[G_{\\alpha}(x,y)-\\left(\\frac{\\lambda}{|x|}\\right)^{n-\\alpha}G_{\\alpha}(x^{\\lambda},y)\\right]f(y,u(y))\\Bigg\\}dy \\\\\n \\nonumber && +\\int_{|y|>\\lambda^{2}}\\left[\\left(\\frac{\\lambda}{|x|}\\right)^{n-\\alpha}G_{\\alpha}(x^{\\lambda},y)-G_{\\alpha}(x,y)\\right]f(y,u(y))dy.\n\\end{eqnarray}\n\nNow we need the following Lemma on properties of the Green's function $G_{\\alpha}(x,y)$.\n\\begin{lem}\\label{G-e}\nThe Green's function $G_{\\alpha}(x,y)$ satisfies the following point-wise estimates:\n\\begin{flalign}\n\\nonumber &\\text{$(i)\\,\\, 0\\leq G_{\\alpha}(x,y)\\leq\\frac{C'}{|x-y|^{n-\\alpha}}, \\quad\\quad \\forall \\,\\, x,y\\in\\mathbb{R}^{n};$}& \\\\\n\\nonumber &\\text{$(ii) \\,\\, G_{\\alpha}(x,y)\\geq\\frac{C''}{|x-y|^{n-\\alpha}}, \\quad\\quad \\forall \\,\\, |x|,|y|\\geq2;$}& \\\\\n\\nonumber &\\text{$(iii) \\,\\, \\left(\\frac{\\lambda}{|x|}\\right)^{n-\\alpha}G_{\\alpha}(x^{\\lambda},y)-G_{\\alpha}(x,y)\\leq0, \\quad\\quad \\forall \\,\\, \\lambda<|x|<\\lambda^{2}, \\,\\, \\lambda<|y|<+\\infty;$}& \\\\\n\\nonumber &\\text{$(iv) \\,\\, \\left(\\frac{\\lambda^{2}}{|x|\\cdot|y|}\\right)^{n-\\alpha}G_{\\alpha}(x^{\\lambda},y^{\\lambda})-\\left(\\frac{\\lambda}{|y|}\\right)^{n-\\alpha}\n  G_{\\alpha}(x,y^{\\lambda})\\leq G_{\\alpha}(x,y)-\\left(\\frac{\\lambda}{|x|}\\right)^{n-\\alpha}G_{\\alpha}(x^{\\lambda},y),$}& \\\\\n\\nonumber &\\text{$\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\quad \\forall \\,\\, \\lambda<|x|,|y|<\\lambda^{2}.$}&\n\\end{flalign}\n\\end{lem}\n\nLemma \\ref{G-e} can be proved by direct calculations, so we omit the details here.\n\nFrom Lemma \\ref{G-e} and the integral equations \\eqref{omega-e}, one can derive that, for any $x\\in B_{\\lambda^{2}}(0)\\setminus\\overline{B_{\\lambda}(0)}$,\n\\begin{eqnarray}\\label{2-37-e}\n  &&\\omega^{\\lambda}(x)=u_{\\lambda}(x)-u(x) \\\\\n \\nonumber &\\leq&\\int_{\\lambda<|y|<\\lambda^{2}}\\Bigg[G_{\\alpha}(x,y)-\\left(\\frac{\\lambda}{|x|}\\right)^{n-\\alpha}G_{\\alpha}(x^{\\lambda},y)\\Bigg] \\left[\\left(\\frac{\\lambda}{|y|}\\right)^{n+\\alpha}f(y^{\\lambda},u(y^{\\lambda}))-f(y,u(y))\\right]dy\\\\\n\\nonumber &<&\\int_{\\lambda<|y|<\\lambda^{2}}\\Bigg(G_{\\alpha}(x,y)-\\left(\\frac{\\lambda}{|x|}\\right)^{n-\\alpha}G_{\\alpha}(x^{\\lambda},y)\\Bigg) \\left[f(y,u_{\\lambda}(y))-f(y,u(y))\\right]dy\\\\\n\\nonumber &\\leq&C\\int_{\\left(B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}}\\right)^{+}}\\frac{1}{|x-y|^{n-\\alpha}}\\left[f(y,u_{\\lambda}(y))-f(y,u(y))\\right]dy\\\\\n\\nonumber &=&C\\int_{\\left(B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}}\\right)^{+}}\\frac{1}{|x-y|^{n-\\alpha}}\\cdot\\frac{f(y,u_{\\lambda}(y))-f(y,u(y))}{u_{\\lambda}(y)-u(y)}\n\\omega^{\\lambda}(y)dy,\n\\end{eqnarray}\nwhere we have used the subcritical condition on $f(x,u)$ for $\\mu=\\left(\\frac{\\lambda}{|y|}\\right)^{n-\\alpha}<1$ to derive the second inequality and the assumption $(\\mathbf{f_{1}})$ on $f(x,u)$ to derive the third inequality.\n\nBy Hardy-Littlewood-Sobolev inequality and \\eqref{2-37-e}, we have, for any $\\frac{n}{n-\\alpha}<q<\\infty$,\n\\begin{eqnarray}\\label{3-14-e}\n && \\|\\omega^{\\lambda}\\|_{L^{q}((B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}})^{+})}\\leq C\\left\\|\\frac{f(y,u_{\\lambda}(y))-f(y,u(y))}{u_{\\lambda}(y)-u(y)}\n\\omega^{\\lambda}(y)\\right\\|_{L^{\\frac{nq}{n+\\alpha q}}((B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}})^{+})}\\\\\n  \\nonumber &\\leq& C\\left\\|\\frac{f(y,u_{\\lambda}(y))-f(y,u(y))}{u_{\\lambda}(y)-u(y)}\\right\\|_{L^{\\frac{n}{\\alpha}}((B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}})^{+})}\n  \\cdot\\|\\omega^{\\lambda}\\|_{L^{q}((B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}})^{+})}.\n\\end{eqnarray}\nSince $u\\in C(\\overline{\\Omega_{1}})$ and $f(x,u)$ satisfies the assumption $(\\mathbf{f_{2}})$, there exists a $\\epsilon_{0}>0$ small enough, such that\n\\begin{equation}\\label{3-15-e}\n  C\\left\\|\\frac{f(y,u_{\\lambda}(y))-f(y,u(y))}{u_{\\lambda}(y)-u(y)}\\right\\|_{L^{\\frac{n}{\\alpha}}((B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}})^{+})}\\leq\\frac{1}{2}\n\\end{equation}\nfor all $1<\\lambda\\leq1+\\epsilon_{0}$, and hence \\eqref{3-14-e} implies\n\\begin{equation}\\label{3-16-e}\n  \\|\\omega^{\\lambda}\\|_{L^{q}((B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}})^{+})}=0,\n\\end{equation}\nwhich means $(B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}})^{+}=\\emptyset$. Therefore, we have proved for all $1<\\lambda\\leq1+\\epsilon_{0}$, $(B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}})^{+}=\\emptyset$, that is,\n\\begin{equation}\\label{3-17-e}\n  \\omega^{\\lambda}(x)\\leq0, \\,\\,\\,\\,\\,\\,\\, \\forall \\, x\\in B_{\\lambda^{2}}(0)\\setminus\\overline{B_{\\lambda}(0)}.\n\\end{equation}\nThis completes Step 1.\n\n\\emph{Step 2. Dilate the sphere $S_{\\lambda}$ outward until $\\lambda=+\\infty$ to derive lower bound estimates on asymptotic behaviour of $u$ as $|x|\\rightarrow+\\infty$.} Step 1 provides us a start point to dilate the sphere $S_{\\lambda}$ from near $\\lambda=1$. Now we dilate the sphere $S_{\\lambda}$ outward as long as \\eqref{2-7-e} holds. Let\n\\begin{equation}\\label{2-29-e}\n  \\lambda_{0}:=\\sup\\{1<\\lambda<+\\infty\\,|\\, \\omega^{\\mu}\\leq0 \\,\\, in \\,\\, B_{\\mu^{2}}(0)\\setminus\\overline{B_{\\mu}(0)}, \\,\\, \\forall \\, 1<\\mu\\leq\\lambda\\}\\in(1,+\\infty],\n\\end{equation}\nand hence, one has\n\\begin{equation}\\label{2-30-e}\n  \\omega^{\\lambda_{0}}(x)\\leq0, \\quad\\quad \\forall \\,\\, x\\in B_{\\lambda_{0}^{2}}(0)\\setminus\\overline{B_{\\lambda_{0}}(0)}.\n\\end{equation}\nIn what follows, we will prove $\\lambda_{0}=+\\infty$ by contradiction arguments.\n\nSuppose on contrary that $1<\\lambda_{0}<+\\infty$. In order to get a contradiction, we will first prove\n\\begin{equation}\\label{2-31-e}\n  \\omega^{\\lambda_{0}}(x)\\equiv0, \\,\\,\\,\\,\\,\\,\\forall \\, x\\in B_{\\lambda_{0}^{2}}(0)\\setminus\\overline{B_{\\lambda_{0}}(0)}\n\\end{equation}\nby using contradiction arguments.\n\nSuppose on contrary that \\eqref{2-31-e} does not hold, that is, $\\omega^{\\lambda_{0}}\\leq0$ but $\\omega^{\\lambda_{0}}$ is not identically zero in $B_{\\lambda_{0}^{2}}(0)\\setminus\\overline{B_{\\lambda_{0}}(0)}$, then there exists a $x^{0}\\in B_{\\lambda_{0}^{2}}(0)\\setminus\\overline{B_{\\lambda_{0}}(0)}$ such that $\\omega^{\\lambda_{0}}(x^{0})<0$. We will obtain a contradiction with \\eqref{2-29-e} via showing that the sphere $S_{\\lambda}$ can be dilated outward a little bit further, more precisely, there exists a $\\varepsilon>0$ small enough such that $\\omega^{\\lambda}\\leq0$ in $B_{\\lambda^{2}}(0)\\setminus\\overline{B_{\\lambda}(0)}$ for all $\\lambda\\in[\\lambda_{0},\\lambda_{0}+\\varepsilon]$.\n\nFor that purpose, we will first show that\n\\begin{equation}\\label{2-32-e}\n  \\omega^{\\lambda_{0}}(x)<0, \\,\\,\\,\\,\\,\\, \\forall \\, x\\in B_{\\lambda_{0}^{2}}(0)\\setminus\\overline{B_{\\lambda_{0}}(0)}.\n\\end{equation}\nIndeed, since we have assumed there exists a point $x^{0}\\in B_{\\lambda_{0}^{2}}(0)\\setminus\\overline{B_{\\lambda_{0}}(0)}$ such that $\\omega^{\\lambda_{0}}(x^{0})<0$, by continuity, there exists a small $\\delta>0$ and a constant $c_{0}>0$ such that\n\\begin{equation}\\label{2-33-e}\nB_{\\delta}(x^{0})\\subset B_{\\lambda_{0}^{2}}(0)\\setminus\\overline{B_{\\lambda_{0}}(0)} \\,\\,\\,\\,\\,\\, \\text{and} \\,\\,\\,\\,\\,\\,\n\\omega^{\\lambda_{0}}(x)\\leq -c_{0}<0, \\,\\,\\,\\,\\,\\,\\,\\, \\forall \\, x\\in B_{\\delta}(x^{0}).\n\\end{equation}\nSince $f(x,u)$ is subcritical and satisfies the assumption $(\\mathbf{f_{1}})$, one can derive from \\eqref{2-33-e}, Lemma \\ref{G-e} and \\eqref{2-37-e} that, for any $x\\in B_{\\lambda_{0}^{2}}(0)\\setminus\\overline{B_{\\lambda_{0}}(0)}$,\n\\begin{eqnarray}\\label{9-37-e}\n  &&\\omega^{\\lambda_{0}}(x)=u_{\\lambda_{0}}(x)-u(x) \\\\\n  \\nonumber &\\leq&\\int_{\\lambda_{0}<|y|<\\lambda_{0}^{2}}\\Bigg[G_{\\alpha}(x,y)-\\left(\\frac{\\lambda_{0}}{|x|}\\right)^{n-\\alpha}G_{\\alpha}(x^{\\lambda_{0}},y)\\Bigg] \\left[\\left(\\frac{\\lambda_{0}}{|y|}\\right)^{n+\\alpha}f(y^{\\lambda_{0}},u(y^{\\lambda_{0}}))-f(y,u(y))\\right]dy\\\\\n\\nonumber &<&\\int_{B_{\\delta}(x_{0})}\\Bigg(G_{\\alpha}(x,y)-\\left(\\frac{\\lambda_{0}}{|x|}\\right)^{n-\\alpha}G_{\\alpha}(x^{\\lambda_{0}},y)\\Bigg) \\left[f(y,u_{\\lambda_{0}}(y))-f(y,u(y))\\right]dy\\leq0,\n\\end{eqnarray}\nthus we arrive at \\eqref{2-32-e}.\n\nNow, we choose a $0<r_{0}<\\frac{1}{4}\\min\\{\\lambda_{0}^{2}-\\lambda_{0},1\\}$ small enough, such that\n\\begin{equation}\\label{9-0e}\n  C\\left\\|\\frac{f(y,u_{\\lambda}(y))-f(y,u(y))}{u_{\\lambda}(y)-u(y)}\\right\\|_{L^{\\frac{n}{\\alpha}}\\big(A_{\\lambda_{0}+r_{0},r_{0}}\\cup A_{\\lambda_{0}^{2}+r_{0},2r_{0}}\\big)}\\leq\\frac{1}{2}\n\\end{equation}\nfor any $\\lambda\\in[\\lambda_{0},\\lambda_{0}+\\frac{r_{0}}{4}]$, where the constant $C$ is the same as in \\eqref{3-15-e} and the narrow region\n\\begin{equation}\\label{9-1e}\n  A_{r,l}:=\\left\\{x\\in B_{r}(0)\\,\\big|\\,|x|>r-l\\right\\}\n\\end{equation}\nfor $r>0$ and $0<l<r$. By \\eqref{2-37-e}, one can easily verify that inequality as \\eqref{3-14-e} (with the same constant $C$) also holds for any $\\lambda\\in[\\lambda_{0},\\lambda_{0}+\\frac{r_{0}}{4}]$, that is, for any $\\frac{n}{n-\\alpha}<q<\\infty$,\n\\begin{equation}\\label{9-2e}\n  \\|\\omega^{\\lambda}\\|_{L^{q}\\left((B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}})^{+}\\right)}\\leq C\\left\\|\\frac{f(y,u_{\\lambda}(y))-f(y,u(y))}{u_{\\lambda}(y)-u(y)}\\right\\|_{L^{\\frac{n}{\\alpha}}\\left((B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}})^{+}\\right)}\n  \\cdot\\|\\omega^{\\lambda}\\|_{L^{q}\\left((B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}})^{+}\\right)}.\n\\end{equation}\n\nBy \\eqref{2-32-e}, we can define\n\\begin{equation}\\label{2-40-e}\n  M_{0}:=\\sup_{x\\in \\overline{B_{\\lambda_{0}^{2}-r_{0}}(0)}\\setminus B_{\\lambda_{0}+r_{0}}(0)}\\omega^{\\lambda_{0}}(x)<0.\n\\end{equation}\nSince $u$ is uniformly continuous on arbitrary compact set $K\\subset\\overline{\\Omega_{1}}$ (say, $K=\\{x\\in\\overline{\\Omega_{1}}\\,|\\,\\lambda_{0}+r_{0}\\leq|x|\\leq2(\\lambda_{0}^{2}-r_{0})\\}$), we can deduce from \\eqref{2-40-e} that, there exists a $0<\\varepsilon_{0}<\\frac{r_{0}}{4}$ sufficiently small, such that, for any $\\lambda\\in[\\lambda_{0},\\lambda_{0}+\\varepsilon_{0}]$,\n\\begin{equation}\\label{2-41-e}\n  \\omega^{\\lambda}(x)\\leq\\frac{M_{0}}{2}<0, \\,\\,\\,\\,\\,\\, \\forall \\, x\\in \\overline{B_{\\lambda_{0}^{2}-r_{0}}(0)}\\setminus B_{\\lambda_{0}+r_{0}}(0).\n\\end{equation}\n\nFor any $\\lambda\\in[\\lambda_{0},\\lambda_{0}+\\varepsilon_{0}]$, it follows from \\eqref{2-41-e} that\n\\begin{equation}\\label{9-4e}\n  (B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}})^{+}\\subset A_{\\lambda_{0}+r_{0},r_{0}}\\cup A_{\\lambda_{0}^{2}+r_{0},2r_{0}}.\n\\end{equation}\nAs a consequence of \\eqref{9-0e}, \\eqref{9-2e} and \\eqref{9-4e}, we get\n\\begin{equation}\\label{9-5e}\n  \\|\\omega^{\\lambda}\\|_{L^{q}\\left((B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}})^{+}\\right)}=0,\n\\end{equation}\nand hence $(B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}})^{+}=\\emptyset$ for all $\\lambda\\in[\\lambda_{0},\\lambda_{0}+\\varepsilon_{0}]$, that is,\n\\begin{equation}\\label{2-45-e}\n  \\omega^{\\lambda}(x)\\leq0, \\,\\,\\,\\,\\,\\,\\, \\forall \\,\\, x\\in B_{\\lambda^{2}}(0)\\setminus\\overline{B_{\\lambda}(0)},\n\\end{equation}\nwhich contradicts with the definition \\eqref{2-29-e} of $\\lambda_{0}$. As a consequence, in the case $1<\\lambda_{0}<+\\infty$, \\eqref{2-31-e} must hold true, that is,\n\\begin{equation}\\label{2-46-e}\n  \\omega^{\\lambda_{0}}\\equiv0 \\,\\,\\,\\,\\,\\, \\text{in} \\,\\,\\, B_{\\lambda_{0}^{2}}(0)\\setminus\\overline{B_{\\lambda_{0}}(0)}.\n\\end{equation}\n\nHowever, by subcritical condition on $f(x,u)$, the first inequality in \\eqref{9-37-e} and \\eqref{2-46-e}, we arrive at\n\\begin{eqnarray}\\label{2-47-e}\n && 0=\\omega^{\\lambda_{0}}(x)=u_{\\lambda_{0}}(x)-u(x)\\\\\n \\nonumber &\\leq&\\int_{\\lambda_{0}<|y|<\\lambda_{0}^{2}}\\Bigg[G_{\\alpha}(x,y)-\\left(\\frac{\\lambda_{0}}{|x|}\\right)^{n-\\alpha}G_{\\alpha}(x^{\\lambda_{0}},y)\\Bigg] \\left[\\left(\\frac{\\lambda_{0}}{|y|}\\right)^{n+\\alpha}f(y^{\\lambda_{0}},u(y^{\\lambda_{0}}))-f(y,u(y))\\right]dy\\\\\n\\nonumber &<&\\int_{\\lambda_{0}<|y|<\\lambda_{0}^{2}}\\Bigg(G_{\\alpha}(x,y)-\\left(\\frac{\\lambda_{0}}{|x|}\\right)^{n-\\alpha}G_{\\alpha}(x^{\\lambda_{0}},y)\\Bigg) \\left[f(y,u_{\\lambda_{0}}(y))-f(y,u(y))\\right]dy=0\n\\end{eqnarray}\nfor any $x\\in B_{\\lambda_{0}^{2}}(0)\\setminus\\overline{B_{\\lambda_{0}}(0)}$, which is absurd. Thus we must have $\\lambda_{0}=+\\infty$, that is,\n\\begin{equation}\\label{2-48-e}\n  u(x)\\geq\\left(\\frac{\\lambda}{|x|}\\right)^{n-\\alpha}u\\left(\\frac{\\lambda^{2}x}{|x|^{2}}\\right), \\quad\\quad \\forall \\,\\, \\lambda\\leq|x|\\leq\\lambda^{2}, \\quad \\forall \\,\\, 1<\\lambda<+\\infty.\n\\end{equation}\nFor arbitrary $\\sigma\\leq|x|<+\\infty$, let $1<\\sqrt{|x|}<\\lambda:=\\sqrt{\\sigma|x|}\\leq|x|<+\\infty$, then \\eqref{2-48-e} yields that\n\\begin{equation}\\label{2-49-e}\n  u(x)\\geq\\left(\\frac{\\sigma}{|x|}\\right)^{\\frac{n-\\alpha}{2}}u\\left(\\frac{\\sigma x}{|x|}\\right),\n\\end{equation}\nand hence, we arrive at the following lower bound estimate on asymptotic behaviour of $u$ as $|x|\\rightarrow+\\infty$:\n\\begin{equation}\\label{2-50-e}\n  u(x)\\geq\\left(\\min_{x\\in S_{\\sigma}}u(x)\\right)\\left(\\frac{\\sigma}{|x|}\\right)^{\\frac{n-\\alpha}{2}}:=\\frac{C_{0}}{|x|^{\\frac{n-\\alpha}{2}}}, \\quad\\quad \\forall \\,\\, \\sigma\\leq|x|<\\infty.\n\\end{equation}\n\nThe lower bound estimate \\eqref{2-50-e} can be improved remarkably by using the ``Bootstrap\" iteration technique and the integral equations \\eqref{IEe}.\n\nIn fact, let $\\mu_{0}:=\\frac{n-\\alpha}{2}$, we infer from the assumption $(\\mathbf{f_{3}})$ on $f(x,u)$, the integral equations \\eqref{IEe}, Lemma \\ref{G-e} and \\eqref{2-50-e} that, for any $2\\sigma\\leq|x|<+\\infty$,\n\\begin{eqnarray}\\label{2-51-e}\n  u(x)&\\geq&\\overline{C}\\int_{\\mathcal{C}\\cap\\{2|x|\\leq|y|\\leq4|x|\\}}G_{\\alpha}(x,y)|y|^{\\tau}\\frac{1}{|y|^{p\\mu_{0}}}dy \\\\\n  \\nonumber &\\geq&C\\int_{\\mathcal{C}\\cap\\{2|x|\\leq|y|\\leq4|x|\\}}\\frac{1}{|x-y|^{n-\\alpha}}\\cdot\\frac{1}{|y|^{p\\mu_{0}-\\tau}}dy \\\\\n  \\nonumber &\\geq&\\frac{C}{|x|^{n-\\alpha}}\\int^{4|x|}_{2|x|}r^{n-1-p\\mu_{0}+\\tau}dr \\\\\n  \\nonumber &\\geq&\\frac{C_{1}}{|x|^{p\\mu_{0}-(\\alpha+\\tau)}}.\n\\end{eqnarray}\nNow, let $\\mu_{1}:=p\\mu_{0}-(\\alpha+\\tau)$. Due to $0<p<p_{s}(\\tau):=\\frac{n+\\alpha+2\\tau}{n-\\alpha}$, our important observation is\n\\begin{equation}\\label{2-52-e}\n  \\mu_{1}:=p\\mu_{0}-(\\alpha+\\tau)<\\mu_{0}.\n\\end{equation}\nThus we have obtained a better lower bound estimate than \\eqref{2-50-e} after one iteration, that is,\n\\begin{equation}\\label{2-53-e}\n  u(x)\\geq\\frac{C_{1}}{|x|^{\\mu_{1}}}, \\quad\\quad \\forall \\,\\, 2\\sigma\\leq|x|<+\\infty.\n\\end{equation}\n\nFor $k=0,1,2,\\cdots$, define\n\\begin{equation}\\label{2-54-e}\n  \\mu_{k+1}:=p\\mu_{k}-(\\alpha+\\tau).\n\\end{equation}\nSince $0<p<p_{s}(\\tau):=\\frac{n+\\alpha+2\\tau}{n-\\alpha}$, it is easy to see that the sequence $\\{\\mu_{k}\\}$ is monotone decreasing with respect to $k$. Continuing the above iteration process involving the integral equation \\eqref{IEe}, we have the following lower bound estimates for every $k=0,1,2,\\cdots$,\n\\begin{equation}\\label{2-55-e}\n  u(x)\\geq\\frac{C_{k}}{|x|^{\\mu_{k}}}, \\quad\\quad \\forall \\,\\, 2\\sigma\\leq|x|<+\\infty.\n\\end{equation}\nNow Theorem \\ref{lower-e} follows easily from the obvious properties that as $k\\rightarrow+\\infty$,\n\\begin{equation}\\label{2-56-e}\n   \\mu_{k}\\rightarrow-\\frac{\\tau+\\alpha}{1-p} \\quad \\text{if} \\,\\, 0<p<1;\n  \\quad\\quad \\mu_{k}\\rightarrow-\\infty \\quad \\text{if} \\,\\, 1\\leq p<\\frac{n+\\alpha+2\\tau}{n-\\alpha}.\n\\end{equation}\nThis finishes our proof of Theorem \\ref{lower-e}.\n\\end{proof}\n\nOne can easily observe that the lower bound estimates in Theorem \\ref{lower-e} contradicts with the following integrability\n\\begin{equation}\\label{2-57-e}\n  C\\int_{\\mathcal{C}\\cap\\Omega_{2\\sigma}}\\frac{|x|^{a}u^{p}(x)}{|2\\sigma e_{n}-x|^{n-\\alpha}}dx\\leq u(2\\sigma e_{n})<+\\infty\n\\end{equation}\nindicated by the integral equations \\eqref{IEe}, where the unit vector $e_{n}:=(0,\\cdots,0,1)$. Therefore, we must have $u\\equiv0$ in $\\overline{\\Omega_{1}}$, that is, the unique nonnegative solution to IEs \\eqref{IEe} is $u\\equiv0$ in $\\overline{\\Omega_{1}}$. This finishes our proof of Theorem \\ref{Thm0} in the noncritical cases $0<\\alpha<n$.\n\n\n\\subsection{The critical order case $\\alpha=n=2$}\n\nIn the critical order case, the Green's function in the equivalent integral equations \\eqref{IEe} takes the form:\n\\begin{eqnarray}\\label{GREENe-0c}\n  G_{2}(x,y)&:=&C\\int_{0}^{\\frac{(|x|^{2}-1)(|y|^{2}-1)}{|x-y|^{2}}}\\frac{1}{1+b}db=C\\ln\\left[1+\\frac{(|x|^{2}-1)(|y|^{2}-1)}{|x-y|^{2}}\\right] \\\\\n \\nonumber &=&2C\\left(\\ln\\frac{1}{|x-y|}-\\ln\\frac{1}{\\left|\\frac{x}{|x|}-|x|y\\right|}\\right)\\quad\\quad \\text{if} \\,\\,\\, x,y\\in \\Omega_{1},\n\\end{eqnarray}\nand $G_{2}(x,y):=0$ if $x$ or $y\\in\\mathbb{R}^{2}\\setminus \\Omega_{1}$.\n\nNext, we will apply the method of scaling spheres to show the following lower bound estimate of positive solution $u$ for $|x|$ large, which will contradict with the integral equations \\eqref{IEe} through a ``Bootstrap\" iteration procedure.\n\\begin{thm}\\label{lower-ec}\nAssume $\\alpha=n=2$, $f(x,u)$ is subcritical and satisfies assumptions $(\\mathbf{f_{1}})$, $(\\mathbf{f_{2}})$ and $(\\mathbf{f_{3}})$. Suppose $u$ is a positive solution to integral equations \\eqref{IEe}, then it satisfies the following lower bound estimate:\n\\begin{equation}\\label{lb1-ec}\n  \\inf_{|x|\\geq\\sigma}u(x)\\geq C_{0}>0.\n\\end{equation}\n\\end{thm}\n\\begin{proof}\nGiven any $\\lambda>1$, we define the Kelvin transform of $u$ centered at $0$ by\n\\begin{equation}\\label{Kelvin-ec}\n  u_{\\lambda}(x):=u\\left(\\frac{\\lambda^{2}x}{|x|^{2}}\\right)\n\\end{equation}\nfor arbitrary $x\\in\\{x\\in\\overline{\\Omega_{1}}\\,|\\,1\\leq|x|\\leq\\lambda^{2}\\}$.\n\nNow, we will carry out the process of scaling spheres in $\\Omega_{1}$ with respect to the origin $0\\in\\mathbb{R}^{n}$.\n\nLet $\\lambda>1$ be an arbitrary real number and let $\\omega^{\\lambda}(x):=u_{\\lambda}(x)-u(x)$ for any $x\\in B_{\\lambda^{2}}(0)\\setminus\\overline{B_{\\lambda}(0)}$. We will first show that, for $\\lambda>1$ sufficiently close to $1$,\n\\begin{equation}\\label{2-7-ec}\n  \\omega^{\\lambda}(x)\\leq0, \\,\\,\\,\\,\\,\\, \\forall \\,\\, x\\in B_{\\lambda^{2}}(0)\\setminus\\overline{B_{\\lambda}(0)}.\n\\end{equation}\nThen, we start dilating the circle $S_{\\lambda}:=\\{x\\in\\mathbb{R}^{2}\\,|\\,|x|=\\lambda\\}$ from near the unit circle $S_{1}$ outward as long as \\eqref{2-7-ec} holds, until its limiting position $\\lambda=+\\infty$ and derive lower bound estimates of $u$ for $|x|$ large. Therefore, the scaling sphere process can be divided into two steps.\n\n\\emph{Step 1. Start dilating the circle $S_{\\lambda}$ from near $\\lambda=1$.} Define\n\\begin{equation}\\label{2-8-ec}\n  (B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}})^{+}:=\\{x\\in B_{\\lambda^{2}}(0)\\setminus\\overline{B_{\\lambda}(0)} \\, | \\, \\omega^{\\lambda}(x)>0\\}.\n\\end{equation}\nWe will show that, for $\\lambda>1$ sufficiently close to $1$,\n\\begin{equation}\\label{2-9-ec}\n  (B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}})^{+}=\\emptyset.\n\\end{equation}\n\nSince $u$ is a positive solution to integral equations \\eqref{IEe}, through direct calculations, we get, for any $\\lambda>1$,\n\\begin{equation}\\label{2-34-ec}\n  u(x)=\\int_{|y|>\\lambda}G_{2}(x,y)f(y,u(y))dy+\\int_{B_{\\lambda^{2}}(0)\\setminus\\overline{B_{\\lambda}(0)}}G_{2}(x,y^{\\lambda})\n  \\left(\\frac{\\lambda}{|y|}\\right)^{4}f(y^{\\lambda},u_{\\lambda}(y))dy\n\\end{equation}\nfor any $x\\in\\overline{\\Omega_{1}}$. By direct calculations, one can also verify that $u_{\\lambda}$ satisfies the following integral equation\n\\begin{equation}\\label{2-35-ec}\n  u_{\\lambda}(x)=\\int_{|y|>1}G_{2}(x^{\\lambda},y)f(y,u(y))dy\n\\end{equation}\nfor any $x\\in\\{x\\in\\overline{\\Omega_{1}}\\,|\\,1\\leq|x|\\leq\\lambda^{2}\\}$, and hence, it follows immediately that\n\\begin{eqnarray}\\label{2-36-ec}\n  u_{\\lambda}(x)&=&\\int_{|y|>\\lambda}G_{2}(x^{\\lambda},y)f(y,u(y))dy \\\\\n \\nonumber \\quad\\quad &&+\\int_{B_{\\lambda^{2}}(0)\\setminus\\overline{B_{\\lambda}(0)}}G_{2}(x^{\\lambda},y^{\\lambda})\n \\left(\\frac{\\lambda}{|y|}\\right)^{4}f(y^{\\lambda},u_{\\lambda}(y))dy.\n\\end{eqnarray}\nTherefore, we have, for any $x\\in B_{\\lambda^{2}}(0)\\setminus\\overline{B_{1}(0)}$,\n\\begin{eqnarray}\\label{omega-ec}\n  && \\omega^{\\lambda}(x)=u_{\\lambda}(x)-u(x)\\\\\n  \\nonumber &=&\\int_{B_{\\lambda^{2}}(0)\\setminus\\overline{B_{\\lambda}(0)}}\\bigg\\{\\left[G_{2}(x^{\\lambda},y^{\\lambda})\n  -G_{2}(x,y^{\\lambda})\\right]\\left(\\frac{\\lambda}{|y|}\\right)^{4}f(y^{\\lambda},u_{\\lambda}(y)) \\\\\n \\nonumber && -\\left[G_{2}(x,y)-G_{2}(x^{\\lambda},y)\\right]f(y,u(y))\\bigg\\}dy+\\int_{|y|>\\lambda^{2}}\\left[G_{2}(x^{\\lambda},y)-G_{2}(x,y)\\right]f(y,u(y))dy.\n\\end{eqnarray}\n\nNow we need some basic properties about the Green's function $G_{2}(x,y)$. From \\eqref{GREENe-0c}, one can obtain that for any $x,\\,y \\in B_{\\lambda^{2}}(0)\\setminus\\overline{B_{1}(0)}$, $x\\neq y$,\n\\begin{equation}\\label{GP-ec}\nG_{2}(x,y)=C\\ln\\left[1+\\frac{(|x|^{2}-1)(|y|^{2}-1)}{|x-y|^{2}}\\right]\\leq C \\ln{\\left(1+\\frac{\\lambda^4}{|x-y|^2}\\right)}.\n\\end{equation}\nIt is well known that\n\\begin{equation}\\label{wk-ec}\n\\ln{(1+t)}=o(t^\\varepsilon), \\,\\,\\quad \\text{as}\\,\\, t\\rightarrow +\\infty,\n\\end{equation}\nwhere $\\varepsilon$ is an arbitrary positive real number. This implies, for any given $\\varepsilon>0$, there exists a $\\delta(\\varepsilon)>0$ such that\n\\begin{equation}\\label{ln-ec}\n\\ln{(1+t)}\\leq t^\\varepsilon, \\,\\,\\qquad \\forall \\, t>\\frac{1}{{\\delta(\\varepsilon)}^2}.\n\\end{equation}\n\nTherefore, by \\eqref{GP-ec}, \\eqref{ln-ec} and straightforward calculations, we have the following Lemma on properties of the Green's function $G_{2}(x,y)$.\n\\begin{lem}\\label{G-ec}\nThe Green's function $G_{2}(x,y)$ satisfies the following point-wise estimates:\n\\begin{flalign}\n\\nonumber &\\text{$(i)\\,\\, G_{2}(x,y)\\leq C\\lambda^{4\\varepsilon} \\frac{1}{|x-y|^{2\\varepsilon}}, \\,\\,\\qquad \\forall\\,\\,\\,1<|x|,\\,|y|<\\lambda^{2}, \\, |x-y|<\\lambda^{2}\\delta(\\varepsilon);$}& \\\\\n\\nonumber &\\text{$(ii) \\,\\, G_{2}(x,y)\\leq C \\ln\\left(1+\\frac{1}{{\\delta(\\varepsilon)}^2}\\right), \\,\\,\\qquad \\forall\\,\\,\\,1<|x|,\\,|y|<\\lambda^{2}, \\,\n|x-y|\\geq\\lambda^{2}\\delta(\\varepsilon);$}& \\\\\n\\nonumber &\\text{$(iii) \\,\\, G_{2}(x,y)\\geq C>0, \\,\\,\\qquad \\forall\\,\\,\\,|x|,\\,|y|\\geq2;$}& \\\\\n\\nonumber &\\text{$(iv) \\,\\, G_{2}(x^{\\lambda},y)-G_{2}(x,y)\\leq0, \\quad\\quad \\forall \\,\\, \\lambda<|x|<\\lambda^{2}, \\,\\, \\lambda<|y|<+\\infty;$}& \\\\\n\\nonumber &\\text{$(v) \\,\\, G_{2}(x^{\\lambda},y^{\\lambda})-G_{2}(x,y^{\\lambda})\\leq G_{2}(x,y)-G_{2}(x^{\\lambda},y), \\qquad \\forall \\,\\, \\lambda<|x|,|y|<\\lambda^{2}.$}&\n\\end{flalign}\n\\end{lem}\n\nLemma \\ref{G-ec} can be proved by direct calculations, so we omit the details here.\n\nFrom Lemma \\ref{G-ec} and \\eqref{omega-ec}, one can derive that, for any $x\\in B_{\\lambda^{2}}(0)\\setminus\\overline{B_{\\lambda}(0)}$,\n\\begin{eqnarray}\\label{2-37-ec}\n  &&\\omega^{\\lambda}(x)=u_{\\lambda}(x)-u(x) \\\\\n \\nonumber &\\leq&\\int_{\\lambda<|y|<\\lambda^{2}}\\left[G_{2}(x,y)-G_{2}(x^{\\lambda},y)\\right]\n \\left[\\left(\\frac{\\lambda}{|y|}\\right)^{4}f(y^{\\lambda},u_{\\lambda}(y))-f(y,u(y))\\right]dy\\\\\n\\nonumber &<&\\int_{\\lambda<|y|<\\lambda^{2}}\\left(G_{2}(x,y)-G_{2}(x^{\\lambda},y)\\right)\\left(f(y,u_{\\lambda}(y))-f(y,u(y))\\right)dy\\\\\n\\nonumber &\\leq&C\\int_{\\left(B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}}\\right)^{+}}G_{2}(x,y)\\frac{f(y,u_{\\lambda}(y))-f(y,u(y))}{u_{\\lambda}(y)-u(y)}\n\\omega^{\\lambda}(y)dy\\\\\n\\nonumber &\\leq&C\\lambda^{4\\varepsilon}\\int_{\\left(B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}}\\right)^{+}\\cap B_{\\lambda^{2}\\delta(\\varepsilon)}(x)}\\frac{1}{|x-y|^{2\\varepsilon}}\\cdot\\frac{f(y,u_{\\lambda}(y))-f(y,u(y))}{u_{\\lambda}(y)-u(y)}\n\\omega^{\\lambda}(y)dy\\\\\n\\nonumber &&+C(\\delta(\\varepsilon))\\int_{\\left(B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}}\\right)^{+}\\setminus B_{\\lambda^{2}\\delta(\\varepsilon)}(x)}\\frac{f(y,u_{\\lambda}(y))-f(y,u(y))}{u_{\\lambda}(y)-u(y)}\n\\omega^{\\lambda}(y)dy,\n\\end{eqnarray}\nwhere we have used the subcritical condition on $f(x,u)$ for $\\mu=\\left(\\frac{\\lambda}{|y|}\\right)^{2}<1$ to derive the second inequality and the assumption $(\\mathbf{f_{1}})$ on $f(x,u)$ to derive the third inequality.\n\nBy Hardy-Littlewood-Sobolev inequality, H\\\"{o}lder inequality and \\eqref{2-37-ec}, we have, for any $\\frac{1}{\\varepsilon}<q<+\\infty$,\n\\begin{eqnarray}\\label{3-14-ec}\n && \\|\\omega^{\\lambda}\\|_{L^{q}((B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}})^{+})}\\\\\n \\nonumber &\\leq& C\\lambda^{4\\varepsilon}\\left\\|\\frac{f(y,u_{\\lambda}(y))-f(y,u(y))}{u_{\\lambda}(y)-u(y)}\n\\omega^{\\lambda}(y)\\right\\|_{L^{\\frac{nq}{n+(n-2\\varepsilon)q}}((B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}})^{+})}\\\\\n\\nonumber &&+C(\\delta(\\varepsilon))\\,\\left|\\left(B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}}\\right)^{+}\\right|^{\\frac{1}{q}}\n\\int_{\\left(B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}}\\right)^{+}}\\left|\\frac{f(y,u_{\\lambda}(y))-f(y,u(y))}{u_{\\lambda}(y)-u(y)}\\right|\\omega^{\\lambda}(y)dy\\\\\n  \\nonumber &\\leq& C\\lambda^{4\\varepsilon}\n  \\left\\|\\frac{f(y,u_{\\lambda}(y))-f(y,u(y))}{u_{\\lambda}(y)-u(y)}\\right\\|_{L^{\\frac{n}{n-2\\varepsilon}}((B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}})^{+})}\n  \\|\\omega^{\\lambda}\\|_{L^{q}((B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}})^{+})}\\\\\n  \\nonumber &&+C(\\delta(\\varepsilon))\\,\\left|\\left(B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}}\\right)^{+}\\right|^{\\frac{1}{q}}\n  \\left\\|\\frac{f(y,u_{\\lambda}(y))-f(y,u(y))}{u_{\\lambda}(y)-u(y)}\\right\\|_{L^{\\frac{q}{q-1}}((B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}})^{+})}\n  \\|\\omega^{\\lambda}\\|_{L^{q}((B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}})^{+})}.\n\\end{eqnarray}\nSince $\\theta<1$ in the assumption $(\\mathbf{f_{2}})$, we first choose $\\varepsilon>0$ sufficiently small such that $-\\frac{n\\theta}{n-2\\varepsilon}>-1$, then choose $q>\\frac{1}{\\varepsilon}$ sufficiently large such that $-\\frac{q\\theta}{q-1}>-1$. Then, since $u\\in C(\\overline{\\Omega_{1}})$ and $f(x,u)$ satisfies the assumption $(\\mathbf{f_{2}})$, there exists a $\\delta_{0}>0$ small enough, such that\n\\begin{eqnarray}\\label{3-15-ec}\n && C\\lambda^{4\\varepsilon}\n  \\left\\|\\frac{f(y,u_{\\lambda}(y))-f(y,u(y))}{u_{\\lambda}(y)-u(y)}\\right\\|_{L^{\\frac{n}{n-2\\varepsilon}}((B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}})^{+})}\\\\\n \\nonumber &&\\qquad+C(\\delta(\\varepsilon))\\,\\left|\\left(B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}}\\right)^{+}\\right|^{\\frac{1}{q}}\n  \\left\\|\\frac{f(y,u_{\\lambda}(y))-f(y,u(y))}{u_{\\lambda}(y)-u(y)}\\right\\|_{L^{\\frac{q}{q-1}}((B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}})^{+})}\\leq\\frac{1}{2}\n\\end{eqnarray}\nfor all $1<\\lambda\\leq1+\\delta_{0}$, and hence \\eqref{3-14-ec} implies\n\\begin{equation}\\label{3-16-ec}\n  \\|\\omega^{\\lambda}\\|_{L^{q}((B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}})^{+})}=0,\n\\end{equation}\nwhich means $(B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}})^{+}=\\emptyset$. Therefore, we have proved for all $1<\\lambda\\leq1+\\delta_{0}$, $(B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}})^{+}=\\emptyset$, that is,\n\\begin{equation}\\label{3-17-ec}\n  \\omega^{\\lambda}(x)\\leq0, \\,\\,\\,\\,\\,\\,\\, \\forall \\, x\\in B_{\\lambda^{2}}(0)\\setminus\\overline{B_{\\lambda}(0)}.\n\\end{equation}\nThis completes Step 1.\n\n\\emph{Step 2. Dilate the circle $S_{\\lambda}$ outward until $\\lambda=+\\infty$ to derive lower bound estimates of $u$ for $|x|$ large.} Step 1 provides us a start point to dilate the circle $S_{\\lambda}$ from near $\\lambda=1$. Now we dilate the circle $S_{\\lambda}$ outward as long as \\eqref{2-7-ec} holds. Let\n\\begin{equation}\\label{2-29-ec}\n  \\lambda_{0}:=\\sup\\{1<\\lambda<+\\infty\\,|\\, \\omega^{\\mu}\\leq0 \\,\\, in \\,\\, B_{\\mu^{2}}(0)\\setminus\\overline{B_{\\mu}(0)}, \\,\\, \\forall \\, 1<\\mu\\leq\\lambda\\}\\in(1,+\\infty],\n\\end{equation}\nand hence, one has\n\\begin{equation}\\label{2-30-ec}\n  \\omega^{\\lambda_{0}}(x)\\leq0, \\quad\\quad \\forall \\,\\, x\\in B_{\\lambda_{0}^{2}}(0)\\setminus\\overline{B_{\\lambda_{0}}(0)}.\n\\end{equation}\nIn what follows, we will prove $\\lambda_{0}=+\\infty$ by contradiction arguments.\n\nSuppose on contrary that $1<\\lambda_{0}<+\\infty$. In order to get a contradiction, we will first prove\n\\begin{equation}\\label{2-31-ec}\n  \\omega^{\\lambda_{0}}(x)\\equiv0, \\,\\,\\,\\,\\,\\,\\forall \\, x\\in B_{\\lambda_{0}^{2}}(0)\\setminus\\overline{B_{\\lambda_{0}}(0)}\n\\end{equation}\nby using contradiction arguments.\n\nSuppose on contrary that \\eqref{2-31-ec} does not hold, that is, $\\omega^{\\lambda_{0}}\\leq0$ but $\\omega^{\\lambda_{0}}$ is not identically zero in $B_{\\lambda_{0}^{2}}(0)\\setminus\\overline{B_{\\lambda_{0}}(0)}$, then there exists a $x^{0}\\in B_{\\lambda_{0}^{2}}(0)\\setminus\\overline{B_{\\lambda_{0}}(0)}$ such that $\\omega^{\\lambda_{0}}(x^{0})<0$. We will obtain a contradiction with \\eqref{2-29-ec} via showing that the circle $S_{\\lambda}$ can be dilated outward a little bit further, more precisely, there exists a $\\epsilon>0$ small enough such that $\\omega^{\\lambda}\\leq0$ in $B_{\\lambda^{2}}(0)\\setminus\\overline{B_{\\lambda}(0)}$ for all $\\lambda\\in[\\lambda_{0},\\lambda_{0}+\\epsilon]$.\n\nFor that purpose, we will first show that\n\\begin{equation}\\label{2-32-ec}\n  \\omega^{\\lambda_{0}}(x)<0, \\,\\,\\,\\,\\,\\, \\forall \\, x\\in B_{\\lambda_{0}^{2}}(0)\\setminus\\overline{B_{\\lambda_{0}}(0)}.\n\\end{equation}\nIndeed, since we have assumed there exists a point $x^{0}\\in B_{\\lambda_{0}^{2}}(0)\\setminus\\overline{B_{\\lambda_{0}}(0)}$ such that $\\omega^{\\lambda_{0}}(x^{0})<0$, by continuity, there exists a small $\\delta>0$ and a constant $c_{0}>0$ such that\n\\begin{equation}\\label{2-33-ec}\nB_{\\delta}(x^{0})\\subset B_{\\lambda_{0}^{2}}(0)\\setminus\\overline{B_{\\lambda_{0}}(0)} \\,\\,\\,\\,\\,\\, \\text{and} \\,\\,\\,\\,\\,\\,\n\\omega^{\\lambda_{0}}(x)\\leq -c_{0}<0, \\,\\,\\,\\,\\,\\,\\,\\, \\forall \\, x\\in B_{\\delta}(x^{0}).\n\\end{equation}\nSince $f(x,u)$ is subcritical and satisfies the assumption $(\\mathbf{f_{1}})$, one can derive from \\eqref{2-33-ec}, Lemma \\ref{G-ec} and \\eqref{2-37-ec} that, for any $x\\in B_{\\lambda_{0}^{2}}(0)\\setminus\\overline{B_{\\lambda_{0}}(0)}$,\n\\begin{eqnarray}\\label{9-37-ec}\n  &&\\omega^{\\lambda_{0}}(x)=u_{\\lambda_{0}}(x)-u(x) \\\\\n  \\nonumber &\\leq&\\int_{\\lambda_{0}<|y|<\\lambda_{0}^{2}}\\left[G_{2}(x,y)-G_{2}(x^{\\lambda_{0}},y)\\right] \\left[\\left(\\frac{\\lambda_{0}}{|y|}\\right)^{4}f(y^{\\lambda_{0}},u_{\\lambda_{0}}(y))-f(y,u(y))\\right]dy\\\\\n\\nonumber &<&\\int_{B_{\\delta}(x_{0})}\\left(G_{2}(x,y)-G_{2}(x^{\\lambda_{0}},y)\\right)\\left(f(y,u_{\\lambda_{0}}(y))-f(y,u(y))\\right)dy\\leq0,\n\\end{eqnarray}\nthus we arrive at \\eqref{2-32-ec}.\n\nNow, we choose a $0<r_{0}<\\frac{1}{4}\\min\\{\\lambda_{0}^{2}-\\lambda_{0},1\\}$ small enough, such that\n\\begin{eqnarray}\\label{9-0ec}\n && C\\lambda^{4\\varepsilon}\n  \\left\\|\\frac{f(y,u_{\\lambda}(y))-f(y,u(y))}{u_{\\lambda}(y)-u(y)}\\right\\|_{L^{\\frac{n}{n-2\\varepsilon}}\\big(A_{\\lambda_{0}+r_{0},r_{0}}\\cup A_{\\lambda_{0}^{2}+r_{0},2r_{0}}\\big)}\\\\\n \\nonumber &&\\quad+C(\\delta(\\varepsilon))\\,\\left|\\left(B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}}\\right)^{+}\\right|^{\\frac{1}{q}}\n  \\left\\|\\frac{f(y,u_{\\lambda}(y))-f(y,u(y))}{u_{\\lambda}(y)-u(y)}\\right\\|_{L^{\\frac{q}{q-1}}\\big(A_{\\lambda_{0}+r_{0},r_{0}}\\cup A_{\\lambda_{0}^{2}+r_{0},2r_{0}}\\big)}\\leq\\frac{1}{2}\n\\end{eqnarray}\nfor any $\\lambda\\in[\\lambda_{0},\\lambda_{0}+\\frac{r_{0}}{4}]$, where the choices of $\\varepsilon$, $q$ and the constants $C$, $C(\\delta(\\varepsilon))$ are the same as in \\eqref{3-15-ec}. By \\eqref{2-37-ec}, one can easily verify that inequality as \\eqref{3-14-ec} (with the same constants $C$ and $C(\\delta(\\varepsilon))$) also holds for any $\\lambda\\in[\\lambda_{0},\\lambda_{0}+\\frac{r_{0}}{4}]$, that is, for any $\\frac{1}{\\varepsilon}<q<+\\infty$,\n\\begin{eqnarray}\\label{9-2ec}\n && \\|\\omega^{\\lambda}\\|_{L^{q}((B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}})^{+})}\\\\\n \\nonumber &\\leq& C\\lambda^{4\\varepsilon}\n  \\left\\|\\frac{f(y,u_{\\lambda}(y))-f(y,u(y))}{u_{\\lambda}(y)-u(y)}\\right\\|_{L^{\\frac{n}{n-2\\varepsilon}}((B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}})^{+})}\n  \\|\\omega^{\\lambda}\\|_{L^{q}((B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}})^{+})}\\\\\n  \\nonumber &&+C(\\delta(\\varepsilon))\\,\\left|\\left(B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}}\\right)^{+}\\right|^{\\frac{1}{q}}\n  \\left\\|\\frac{f(y,u_{\\lambda}(y))-f(y,u(y))}{u_{\\lambda}(y)-u(y)}\\right\\|_{L^{\\frac{q}{q-1}}((B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}})^{+})}\n  \\|\\omega^{\\lambda}\\|_{L^{q}((B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}})^{+})}.\n\\end{eqnarray}\n\nBy \\eqref{2-32-ec}, we can define\n\\begin{equation}\\label{2-40-ec}\n  M_{1}:=\\sup_{x\\in \\overline{B_{\\lambda_{0}^{2}-r_{0}}(0)}\\setminus B_{\\lambda_{0}+r_{0}}(0)}\\omega^{\\lambda_{0}}(x)<0.\n\\end{equation}\nSince $u$ is uniformly continuous on arbitrary compact set $K\\subset\\overline{\\Omega_{1}}$, we can deduce from \\eqref{2-40-ec} that, there exists a $0<\\epsilon_{1}<\\frac{r_{0}}{4}$ sufficiently small, such that, for any $\\lambda\\in[\\lambda_{0},\\lambda_{0}+\\epsilon_{1}]$,\n\\begin{equation}\\label{2-41-ec}\n  \\omega^{\\lambda}(x)\\leq\\frac{M_{1}}{2}<0, \\,\\,\\,\\,\\,\\, \\forall \\, x\\in \\overline{B_{\\lambda_{0}^{2}-r_{0}}(0)}\\setminus B_{\\lambda_{0}+r_{0}}(0).\n\\end{equation}\n\nFor any $\\lambda\\in[\\lambda_{0},\\lambda_{0}+\\epsilon_{1}]$, it follows from \\eqref{2-41-ec} that\n\\begin{equation}\\label{9-4ec}\n  (B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}})^{+}\\subset A_{\\lambda_{0}+r_{0},r_{0}}\\cup A_{\\lambda_{0}^{2}+r_{0},2r_{0}}.\n\\end{equation}\nAs a consequence of \\eqref{9-0ec}, \\eqref{9-2ec} and \\eqref{9-4ec}, we get\n\\begin{equation}\\label{9-5ec}\n  \\|\\omega^{\\lambda}\\|_{L^{q}\\left((B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}})^{+}\\right)}=0,\n\\end{equation}\nand hence $(B_{\\lambda^{2}}\\setminus\\overline{B_{\\lambda}})^{+}=\\emptyset$ for all $\\lambda\\in[\\lambda_{0},\\lambda_{0}+\\epsilon_{1}]$, that is,\n\\begin{equation}\\label{2-45-ec}\n  \\omega^{\\lambda}(x)\\leq0, \\,\\,\\,\\,\\,\\,\\, \\forall \\,\\, x\\in B_{\\lambda^{2}}(0)\\setminus\\overline{B_{\\lambda}(0)},\n\\end{equation}\nwhich contradicts with the definition \\eqref{2-29-ec} of $\\lambda_{0}$. As a consequence, in the case $1<\\lambda_{0}<+\\infty$, \\eqref{2-31-ec} must hold true, that is,\n\\begin{equation}\\label{2-46-ec}\n  \\omega^{\\lambda_{0}}\\equiv0 \\,\\,\\,\\,\\,\\, \\text{in} \\,\\,\\, B_{\\lambda_{0}^{2}}(0)\\setminus\\overline{B_{\\lambda_{0}}(0)}.\n\\end{equation}\n\nHowever, by subcritical condition on $f(x,u)$, the first inequality in \\eqref{9-37-ec} and \\eqref{2-46-ec}, we arrive at\n\\begin{eqnarray}\\label{2-47-ec}\n && 0=\\omega^{\\lambda_{0}}(x)=u_{\\lambda_{0}}(x)-u(x)\\\\\n \\nonumber &\\leq&\\int_{\\lambda_{0}<|y|<\\lambda_{0}^{2}}\\left[G_{2}(x,y)-G_{2}(x^{\\lambda_{0}},y)\\right] \\left[\\left(\\frac{\\lambda_{0}}{|y|}\\right)^{4}f(y^{\\lambda_{0}},u_{\\lambda_{0}}(y))-f(y,u(y))\\right]dy\\\\\n\\nonumber &<&\\int_{\\lambda_{0}<|y|<\\lambda_{0}^{2}}\\left(G_{2}(x,y)-G_{2}(x^{\\lambda_{0}},y)\\right)\\left(f(y,u_{\\lambda_{0}}(y))-f(y,u(y))\\right)dy=0\n\\end{eqnarray}\nfor any $x\\in B_{\\lambda_{0}^{2}}(0)\\setminus\\overline{B_{\\lambda_{0}}(0)}$, which is absurd. Thus we must have $\\lambda_{0}=+\\infty$, that is,\n\\begin{equation}\\label{2-48-ec}\n  u(x)\\geq u\\left(\\frac{\\lambda^{2}x}{|x|^{2}}\\right), \\quad\\quad \\forall \\,\\, \\lambda\\leq|x|\\leq\\lambda^{2}, \\quad \\forall \\,\\, 1<\\lambda<+\\infty.\n\\end{equation}\nFor arbitrary $\\sigma\\leq|x|<+\\infty$, let $1<\\sqrt{|x|}<\\lambda:=\\sqrt{\\sigma|x|}\\leq|x|<+\\infty$, then \\eqref{2-48-ec} yields that\n\\begin{equation}\\label{2-49-ec}\n  u(x)\\geq u\\left(\\frac{\\sigma x}{|x|}\\right),\n\\end{equation}\nand hence, we arrive at the following lower bound estimate of $u$:\n\\begin{equation}\\label{2-50-ec}\n  u(x)\\geq\\min_{x\\in S_{\\sigma}}u(x):=C_{0}>0, \\quad\\quad \\forall \\,\\, \\sigma\\leq|x|<\\infty.\n\\end{equation}\n\nThis finishes our proof of Theorem \\ref{lower-ec}.\n\\end{proof}\n\nSince $-2\\leq\\tau<+\\infty$ in assumption $(\\mathbf{f_{3}})$, we can deduce from the assumption $(\\mathbf{f_{3}})$ on $f(x,u)$, the integral equations \\eqref{IEe}, Lemma \\ref{G-ec} and Theorem \\ref{lower-ec} that, for any $2\\sigma\\leq|x|<+\\infty$,\n\\begin{eqnarray}\\label{2-51-ec'}\n  u(x)&\\geq&\\overline{C}\\int_{\\mathcal{C}\\cap\\{|y|\\geq2|x|\\}}G_{2}(x,y)|y|^{\\tau}C^{p}_{0}dy \\\\\n  \\nonumber &\\geq&C\\int_{\\mathcal{C}\\cap\\{|y|\\geq2|x|\\}}|y|^{\\tau}dy=+\\infty,\n\\end{eqnarray}\nwhich is a contradiction! Therefore, we must have $u\\equiv0$ in $\\overline{\\Omega_{1}}$, that is, the unique nonnegative solution to IEs \\eqref{IEe} with $\\alpha=n=2$ is $u\\equiv0$ in $\\overline{\\Omega_{1}}$.\n\nThis concludes our proof of Theorem \\ref{Thm0}.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\subsection{NGC 2419}\nNGC 2419 is a stellar aggregate with a number of puzzling characteristics and its nature and origin are yet unclear. \nWith a half-light radius $r_h$ of 21.4 pc it is the fifth-most extended object listed in the 2010-version of the Harris (1996) catalogue, while \nit is also one of the most luminous Globular Clusters (GCs) in the Milky Way (MW; Fig.~1). At a Galactocentric distance of 90 kpc it \nresides in the outermost halo. All these traits have fueled discussions of whether it contains any dark matter or could be affected by non-Newtonian\ndynamics (Baumgardt et al. 2005; Conroy et al. 2011; Ibata et al. 2012).  For instance,  Ibata et al. (2012) argue that its kinematics is incompatible with a dark matter content in excess of some 6\\% of its total mass. \nOverall, these morphological and dynamical considerations beg the question to what extent NGC~2419 has evolved in isolation and \nwhether it could be associated with a once-accreted, larger system like a dwarf (spheroidal) galaxy.\n\\begin{figure}[t!]\n\\resizebox{\\hsize}{!}{\n\\includegraphics[clip=true]{Koch_f1.eps}\n}\n\\caption{\\footnotesize\nMagnitude-half light radius plot for GCs (black dots), luminous dSphs (blue squares) and ultrafaint MW satellites (red circles). NGC~2419 is labeled -- what is this object?}\n\\vspace{-0.5cm}\n\\end{figure}\n\n\nAlso chemically, NGC 2419 has much to offer: Cohen \\& Kirby (2012) and Mucciarelli et al. (2012) identified a population of stars (ca. 30\\% by number) with remarkably low Mg- and high K-abundances, \nwhich could be the result of ``extreme nucleosynthesis'' (Ventura et al. 2012). The question of an abundance-spread has been addressed by several authors using high-resolution spectroscopy and \nlow-resolution measurements of the calcium triplet. However,   the large abundance variation of \nthe electron-donor Mg will upset the commonly used stellar model atmospheres so that any claimed spread in iron-,  Ca-, and thus overall  metallicity needs to be considered with caution. \nHowever, settling exactly this aspect is of prime importance, since any significant spread in heavy elements is a trademark signature of an object with a likely extragalactic origin \n(e.g., Fig.~1 in Koch et al. 2012). \n\nThe color-magnitude diagrams (CMDs) of  Di Criscienzo et al. (2011) show a hint of a color-spread towards the subgiant branch and the presence of a \nhot, faint Horizontal Branch (HB), \nconsistent with a second generation of stars with a strongly increased He-content. Thus, also NGC~2419 does appear to show signs of multiple stellar populations,\nin line with the majority of the MW GC system. \n\\subsection{Str\\\"omgren photometry}\nWhile broad-band filter combinations have succeeded in unveiling multiple stellar populations in sufficiently deep data sets and more massive systems (e.g., Piotto et al. 2007), additional \nobservations in intermediate-band \nStr\\\"omgren filters\nare desirable for a number of reasons:\n\\begin{enumerate}\n\\item[\\em i)] The $c_1 = (u-v)- (v-b)$ index in combination with a color such as $v-y$ is a powerful {\\em dwarf\/giant separator} and can efficiently remove any foreground contamination (e.g., Faria et al. 2007). \nAt $b=25\\degr$ this can be expected to be  less of a problem in NGC~2419, but see, e.g., Ad\\'en et al. (2009) for an impressive demonstration of such a CMD cleaning. \nOur first assessment of the $c_1$-$(b-y)$ plane indicates that the foreground contamination is indeed minimal on the upper RGB (see also Fig.~2). \n\\item[\\em ii)] The index $m_1 = (v-b)- (b-y)$ is a good proxy for stellar {\\em metallicity} and calibrations have been devised by several authors (e.g., Hilker 2000; Calamida et al. 2007; \nAd\\'en et al. 2009). \n\\item[\\em iii)] {\\em Multiple populations} in terms of split red giant branches (RGBs), multiple subgiant branches, and main sequence turnoffs are well separated in CMDs that use combinations \nof Str\\\"omgren filters, e.g., $\\delta_4 = c_1 + m_1$ (Carretta et al. 2011), where optical CMDs based on broad-band filters still show unimodal, ``simple stellar populations''. \n\\item[\\em iv)] This is immediately interlinked with the {\\em chemical abundance variations} in the light chemical elements (e.g., Anthony-Twarog et al. 1995) that accompany the multiple populations, \nmost prominently driven by N-variations. Accordingly, Yong et al. (2008) confirmed linear correlations of  $c_y = c_1 - (b-y)$ with the [N\/Fe] ratio.\n\\end{enumerate}\n\\section{Data and analysis}\nWe obtained imaging in all relevant Str\\\"omgren filters ($u$,$b$,$v$,$y$) using the Wide Field Camera (WFC) at the 2.5-m Isaac Newton Telescope (INT) at La Palma, Spain. \nIts large field of view ($33\\arcmin\\times33\\arcmin$) allows us to trace the large extent of NGC~2419 out to several times its tidal radius ($r_t\\sim7.5\\arcmin$). \n\nInstrumental magnitudes were obtained via PSF-fitting using the \\textsc{Daophot\/Allframe} software packages \n(Stetson 1987).\nThe instrumental magnitudes were transformed to the standard Str\\\"omgren system using ample observations of standard stars \n(Schuster \\& Nissen 1988).\nWe set up transformation equations similar to those given by Grundahl, Stetson \\& Andersen (2002).\n\\section{Preliminary results: CMDs and [M\/H]}\nFig.~2 shows two CMDs of NGC~2419, where we restrict our analysis to the bona-fide region between 1 and 3 half-light radii to avoid potentially crowded, inner regions, yet \nminimizing the field star contamination of the outer parts. \nFor the present analysis, we adopted a constant reddening of E($B-V$)$= 0.061$\\,mag\\footnote{Obtained from \\url{http:\/\/irsa.ipac.caltech.edu\/applications\/DUST}}, \nand its respective transformations to the Str\\\"omgren system (Calamida et al. 2009).\n\\begin{figure*}[t!]\n\\resizebox{\\hsize}{!}{\n\\includegraphics[width=0.02\\hsize]{Koch_f2.eps}\n}\n\\caption{\\footnotesize\nCMDs of NGC 2419 for two possible Str\\\"omgren-filter combinations. Shown are stars between $r_h < r <  3\\,r_h$; no other selection criteria have been applied. \nStars falling within the lines shown right  were used to construct the metallicity distribution in Fig.~3.}\n\\end{figure*}\n\nWhile we do not resolve the main sequence turnoff, our photometry reaches about 1 mag below the HB at $y_{\\rm HB}$$\\equiv$$V_{\\rm HB}$$\\sim$$20.5$ mag. \nAll regions of the CMD are well reproduced, showing \na clear RGB, hints of an AGB and bright AGB (which stand out more clearly in other color indices; Frank et al. in prep.), and a prominent HB. Hess diagrams also highlight the presence of a clear \nRGB bump at $y_0\\sim20.3$ mag. \nMoreover, the extreme, hotter HB stands out in the bluer u-band (left panel), confirming the presence of this  He-rich, secondary population (di Crisicienzo et al. 2011). \n\nTo obtain a first impression of the metallicity distribution function (MDF)  of NGC~2419 (Fig.~3, right panel), we convert our Str\\\"omgren photometry to metallicities, [M\/H], through the \ncalibration by Ad\\'en et al. (2009). \nThis was carried out for stars on the RGB (see ridge lines in Fig.~2, right panel, and Fig.~3, left). As a result, we find \na mean [M\/H] of $-2$ dex. This is  in good agreement with the values listed in the Harris catalogue and the high-resolution data of Mucciarelli et al. (2012) and Cohen \\& Kirby (2012) of [Fe\/H] = $-2.15$ dex. \n\\begin{figure*}[t!]\n\\resizebox{\\hsize}{!}{\n\\includegraphics[clip=true,width=0.55\\hsize]{Koch_f3a.eps}\n\\includegraphics[clip=true,width=0.53\\hsize]{Koch_f3b.eps}\n}\n\\caption{\\footnotesize\nPreliminary metallicity calibration in the $m_1$-vs-$(b-y)$ plane (left panel). Dashed lines indicate iso-metallicity curves based on the calibration of Ad\\'en et al.  (2009) for [M\/H] = $-2.5$ up to +0.5 dex in steps of 0.5 (bottom to top). \nBlack dots are those within the RGB-ridge lines of Fig.~2, used to infer the MDF (light gray) in the right panel. The dark gray MDF in this plot uses a stricter RGB criterion of $(v-y)_0 > 1.4$.}\n\\label{eta}\n\\end{figure*}\n\n\nThe MDF also indicates the presence of a broad metallicity spread, where we find a nominal 1$\\sigma$-spread of 0.5 dex, but this is probably still dominated by \nremaining foreground contaminants and photometric errors. While we cannot exclude the presence of an abundance spread in NGC 2419 from the \npresent data, it is very likely much smaller than the one suggested by Fig.~3.\n\\section{Discussion} \nAlthough our CMD does not allow us to clearly isolate any multiple stellar populations at this stage of our analysis,  we nevertheless   \nfind strong reason to believe in their presence in NGC~2419, bolstered by recent optical images (di Criscienzo et al. 2011). \nThese authors detected a color spread at the base of the RGB and an extreme, hot HB, indicative of an increased He-abundance of a populous second stellar generation.  \nThis HB population is also visible in our intermediate-band CMDs.  \n\nAlthough our first analysis suggests a broad metallicity spread in NGC~2419, this is probably not significant and further CMD filtering is necessary. \nHowever, our derived mean metallicity is in line with the results from high-resolution spectroscopy, which indicates that  our Str\\\"omgren photometry \nis well calibrated. \n\\begin{acknowledgements}\nAK, MF, and NK gratefully acknowledge the Deutsche Forschungsgemeinschaft for funding from  Emmy-Noether grant  Ko 4161\/1. \nThis research has made use of the NASA\/ IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of \nTechnology, under contract with the National Aeronautics and Space Administration.\n\\end{acknowledgements}\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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Since many years \nefforts of theorists and experimentalists were focused on the investigation \nof time-space evolution of the quark-gluon plasma (QGP) and production of\ndifferent species of particles, primarily hadrons \n(pions, kaons, nucleons, hyperons, etc.) emitted in the collision. \nAt high energies, the velocities of such beam nuclei are close to light \nvelocity thus they are often called ultra-relativistic \nvelocities (URV). \n\nCentral collisions are the most interesting in the context of\nthe QGP studies. Plasma is, of course, also produced in more peripheral \ncollisions.\nIn peripheral collisions, the so-called spectators are relatively large \nand have large moving charge. It was realized relatively late that \nthis charge generates strong quickly changing electromagnetic fields \nthat can influence the trajectories and some observables \nfor charged particles.\n\nSuch effects were investigated in previous studies of one of the present\nauthors \\cite{Rybicki:2006qm,Rybicki:2013qla}.\nOn one hand side the EM effects strongly modify the Feynman $x_F$ spectra\nof low-$p_T$ pions, creating a dip for $\\pi^+$ and\nan enhancement for $\\pi^-$ at $x_F \\approx \\frac{m_{\\pi}}{m_N}$.\nIn \\cite{Rybicki:2006qm} a formalism of charged meson evolution\nin the EM field (electric and magnetic) of fast moving nuclei was\ndeveloped. Later on the spectacular effects were confronted with \nthe SPS data \\cite{Rybicki:2009zz} confirming the theoretical predictions.\nThe investigation was done for $^{208}$Pb+$^{208}$Pb at 158~GeV\/nucleon \nenergy ($\\sqrt{{s}_{NN}} =$ 17.3~GeV) at \nCERN Super Proton Synchrotron (SPS) \\cite{Schlagheck:1999aq}.\nIn \\cite{Rybicki:2013qla} the influence of the EM fields\non azimuthal flow parameters ($v_n$) was studied and confronted in\n\\cite{Rybicki:2014rna} with the RHIC data. It was found that the EM field\nleads to a split of the directed flow for opposite e.g.\ncharges. In the initial calculation, a simple model of single initial\ncreation point of pions was assumed for simplicity.\nMore recently, such a calculation was further developed by taking \ninto account also the time-space evolution of the fireball, treated \nas a set of firestreaks \\cite{Ozvenchuk:2019cve}. \nThe distortions of the $\\pi^+$ and $\\pi^-$ distributions allow to\ndiscuss the electromagnetic effects of the spectator and charged pions \nin URV collisions of heavy ions and nicely explain the experimental data. \nThis study was done for non-central collisions where the remaining \nafter collision object, called 'spectator', loses only a small part \nof nucleons of the original beam\/target nucleus.\n \nThe ultra-peripheral heavy-ion collisions (e.g.~$^{208}$Pb+$^{208}$Pb) at \nultra-relativistic energies ($\\sqrt{s_{NN}}\\ge 5~$GeV) \\cite{\nKlusek-Gawenda:2016suk}\nallow to produce particles in a broad region of impact parameter\nspace, even far from ``colliding'' nuclei.\nThe nuclei passing near one to each other with ultrarelativistic energies \nare a source of virtual photons that can collide producing\ne.g. a pair of leptons.\nIn real current experiments (RHIC, LHC), the luminosity is big enough to observe\ne.g. $AA \\rightarrow AA\\rho_0$ and $AA \\rightarrow AAe^+e^-$, \n$AA \\rightarrow AA\\mu^+\\mu^-$ processes. \nOne of the most interesting phenomena is multiple interaction \n\\cite{Klusek-Gawenda:2016suk,vanHameren:2017krz}\nwhich may lead to the production of more than one lepton pair.\n\nThe studies on the creation of positron-electron pairs started in early \n1930'ties with the discovery of positron by Dirac \\cite{Dirac:1934} and works\nof G. Breit and J.A. Wheeler \\cite{Breit:1934zz}, where they calculated \nthe cross section for the production of such pairs in the electric field. \nIt was E.J. Williams who realized \\cite{Williams:1935} that the \nproduction of $e^+ e^-$ pairs is enhanced in the vicinity of the atomic\nnucleus. An overview of the theoretical investigation of \nthe $e^+ e^-$ pairs creation in historical context was presented \nby e.g. J.H. Hubbell \\cite{Hubbel:2006} \nand the detailed discussion about this process in physics and\nastrophysics was written by R. Ruffini et al. \\cite{Ruffini:2009hg}.\n\nThe early analyses were done in the momentum space and therefore\ndid not include all details in the impact parameter space.\nAn example of the calculation where such details are taken into\naccount can be found e.g. in \n\\cite{Klusek-Gawenda:2010vqb,Klusek-Gawenda:2016suk}.\nIn ultra-peripheral collision (UPC), the nuclei do not collide\nwithout loosing, in principle, any nucleon.\nHowever, the electromagnetic interaction induced by fast moving\nnuclei may cause excitation of the nuclei and subsequent emission \nof different particles, in particular neutrons \\cite{Klusek-Gawenda:2013ema}\nthat can be measured both at RHIC and at the LHC.\nMoreover the UPC are responsible not only \nfor Coulomb excitation of the spectator but also for \nthe multiple scattering and production of more than one dielectron\npair \\cite{vanHameren:2017krz}. \nWith large transverse momentum cut typical at RHIC and the LHC the effect\nis not dramatic.\n\nCan the strong EM fields generated at high energies modify \nthe electron\/positron distributions?\nNo visible effect was observed for electrons with $p_T >$ 1~GeV\nas discussed in \\cite{vanHameren:2017krz}, where the ALICE distributions\nwere confronted with the $b$-space equivalent photon approximation (EPA)\nmodel.\nBut the electromagnetic effect is expected rather at \nlow transverse momenta.\nAccording to our knowledge this topic was not discussed \nin the literature.\nAs the spectators, which in ultra-peripheral collisions are almost \nidentical to colliding nuclei, are charged,\nthey can interact electromagnetically with electrons and positrons as \nit was in the case of pions. \nSimilar effects as observed for pions may be expected also for charged \nleptons. The motion of particles in EM field depends not only on their \ncharge but also on their mass. Thus the distortions of $e^+\/e^-$\ndistributions should be different than those for $\\pi^+\/\\pi^-$ distributions. \nAlso the mechanism of production is completely different. \nIn contrast to pion production, where the emission site is well localized,\nthe electron-positron pairs produced by photon-photon fusion can \nbe produced in a broad configuration space around the ``collision''\npoint - point of the closest approach of nuclei. \nA pedagogical illustration of the impact parameter dependence can \nbe found e.g. in \\cite{Klusek-Gawenda:2010vqb}.\n\nFrom one side, the previous works \\cite{Ozvenchuk:2019cve} and reference \ntherein, were connected with the electromagnetic effects caused by \nthe emission of the pions from the fireball region. \nFrom the other side, the model considered \\cite{Klusek-Gawenda:2010vqb,Klusek-Gawenda:2018zfz,Klusek-Gawenda:2020eja} can correctly estimate \nthe localization in the impact parameter space.\nThe present study will be focused on electromagnetic \ninteraction between electrons\/positrons with highly positively charged nuclei.\n\nOur approach consists of two steps. First, the $e^+ e^-$ distributions \nare calculated within EPA in terms of initial distributions in a given space \npoint and at a given initial rapidity and transverse momentum.\nSecondly, the space-time evolution of leptons in the electromagnetic fields\nof fast moving nuclei with URV is performed by solving relativistic \nequation of motion \\cite{Rybicki:2011zz}.\n\nIn Section \\ref{Sect.II} the details of the calculation of \nthe differential cross section of $e^+ e^-$ production will be\npresented.\nWe will not discuss in detail equation of motion which was presented\ne.g. in \\cite{Rybicki:2011zz}. \nThe results of the evolution of electrons\/positrons in\nthe EM field of ``colliding'' nuclei are presented in Section \\ref{Sect.III}. \n\\section{Lepton pair production, equivalent photon approximation}\n\\label{Sect.II}\n\nThe particles originated from photon-photon collisions can be created\nin full space around excited nucleus thus first of all the geometry of \nthe reaction should be defined.  \n\nIn the present study, the ultra-peripheral collisions (UPC) are\ninvestigated in the reaction plane ($b_x,b_y$) which are perpendicular \nto the beam axis taken as $z$-direction.\n\nThe collision point ($b_x=0,b_y=0$) is time-independent center of mass (CM)\nof the reaction as shown in Fig.~\\ref{fig01}. The impact parameter\nis fixed as double radius of each (identical) Pb nucleus b=(13.95~fm, 14.05~fm). \nFor comparison we will show results also for b=(49.95~fm, 50.05~fm).\n\nFour characteristic points ($\\pm$15~fm, 0), (0, $\\pm$15~fm), which are discussed later are also \nmarked in the figure. In the present paper \nwe shall show results for these initial emission points for \nillustrating the effect of evolution of electrons\/positrons in \nthe EM field of nuclei.\n\n\\begin{figure}[!hbt]\n    \\begin{center}\n     \\includegraphics[scale=0.30]{geometra01.pdf}\n     \\caption{ The impact parameter space and the\n       five selected points ($b_x,b_y$):\n     (0, 0), ($\\pm$15~fm,0), (0,$\\pm$15~fm), for which \n      the distribution in rapidity and transverse momentum will be\n      compared latter on in the text, shown in the CM rest frame.\n     \n      }\n     \\label{fig01}\n  \\end{center}\n\n\\end{figure}\n\nUsually the exclusive dilepton production was estimated by using \nthe monopole charge form factor which allows to reproduce correctly \nthe total cross section. \nThe differential cross sections are more sensitive to\ndetails, thus realistic charge form factor (Fourier transform of the\ncharge distribution) has to be employed.\n\\footnote{Double scattering production of positron-electron pairs using \nthe realistic charge form factor has been discussed \nin \\cite{Klusek-Gawenda:2016suk} and \\cite{vanHameren:2017krz}.} \nThe total cross section for the considered process\n($A A \\to A A e^+ e^-$) can be written as:\n\n\\begin{eqnarray}\n&& \\sigma_{A_1A_2\\rightarrow A_1A_2e^+e^-} (\\sqrt{s_{A_1A_2}})=\t\\nonumber \\\\\n &=&\\int \\frac{d\\sigma_{\\gamma\\gamma \\rightarrow e^+e^-}(W_{\\gamma\\gamma}) }{d\\cos{\\theta}} \nN(\\omega_1,b_1) N(\\omega_2,b_2) S^2_{abs}(b) \\nonumber\\\\\n&\\times&  2\\pi bdbd\\overline{b_x} d \\overline{b_y}\n\\frac{W_{\\gamma\\gamma}}{2} d W_{\\gamma\\gamma}d Y_{e^+e^-} \\ d\\cos{\\theta},\n\\label{EPA}\n\\end{eqnarray}\nwhere $N(\\omega_i,b_i)$ are photon fluxes, $W_{\\gamma\\gamma}=M_{e^+e^-}$\n is invariant mass and $Y_{e^+e^-}= (y_{e^+} + y_{e^-})\/2$ is rapidity of \nthe outgoing system and $\\theta$ is the scattering angle in the $\\gamma\\gamma\\rightarrow e^+e^-$ \ncenter-of mass system. The gap survival factor $S^2_{abs}$ assures that \nonly ultra-peripheral reactions are considered.\n\\begin{figure}[!hbt]\n    \\begin{center}\n     \\includegraphics[scale=0.5]{paw_ypt_b14.png}\n     \\caption{The map of differential cross section in rapidity of\n       electron or positron and lepton transverse momentum \n       for ($b_x,b_y$) = (0, 0) which will be called CM point for brevity.}\n\n     \\label{fig02a}\n  \\end{center}\n\\end{figure}\n\nThe\n$\\overline{b_x}=(b_{1x}+b_{2x})\/2$ and\n$\\overline{b_y}=(b_{1y}+b_{2y})\/2$\nquantities are particularly useful for our purposes.\nWe define $\\vec{{\\bar b}} = (\\vec{b}_1 + \\vec{b}_2)\/2$ which is\n(initial) position\nof electron\/positron in the impact parameter space.\nThis will be useful when considering motion of electron\/positron\nin the EM field of nuclei.\nThe energies of photons are included by the relation: \n$\\omega_{1,2}=W_{\\gamma\\gamma}\/2 \\exp(\\pm Y_{e^+e^-})$.\nIn the following for brevity we shall use $b_x, b_y$ instead of \n$\\overline{b_x}, \\overline{b_y}$.\nThen $(b_x, b_y)$ is the position in the impact parameter plane,\nwhere the electron and positron are created.{\\footnote{Expression (\\ref{EPA}) allows to estimate not only \nthe lepton pair production but also a production of any other \nparticle pair \\cite{Klusek-Gawenda:2010vqb}.}}\nThe differential (in rapidity and transverse momentum) cross section \ncould be obtained in each emission point \nin the impact parameter space ($b_x,b_y$).\n\n\\begin{figure}[!hbt]\n    \\begin{center}\n     \\includegraphics[scale=0.60]{rap_e_xy_weight_17GeV.pdf}\n     \\includegraphics[scale=0.60]{pt_e_xy_weight_17GeV.pdf}\n     \\caption{(Color on-line) The differential cross section for various emission points\n       of electrons\/positrons produced in the $^{208}$Pb+$^{208}$Pb\n       reaction at 158~GeV\/nucleon energy ($\\sqrt{{s}_{NN}} =$ 17.3~GeV)\n       at impact parameter 14$\\pm$0.05~fm. The cross section \n       for selected points ($b_x,b_y$): (0, 0), ($\\pm$15~fm, 0), (0, $\\pm$15~fm) and (40~fm, 0)\n       are integrated over $p_T$ (a) and rapidity (b), respectively.}\n     \\label{fig02b}\n  \\end{center}\n\\end{figure}\n\n\nThe calculations will be done assuming the collision of\n$^{208}$Pb+$^{208}$Pb \nat 158~GeV\/nucleon energy ($\\sqrt{{s}_{NN}} =$ 17.3~GeV) corresponding to\nthe CERN SPS and $\\sqrt{{s}_{NN}} =$ 200~GeV of \nthe STAR RHIC at impact parameter 14$\\pm$0.05~fm which is \napproximately twice the radius of the lead nucleus. \nThis is minimal configuration assuring ultra-peripheral collisions.\n\nFigure \\ref{fig02a} illustrates the differential cross section on \nthe plane of rapidity ($y$) vs. transverse momentum ($p_T$). \nRather broad range of rapidity (-5, 5) is chosen, but the distribution in $p_T$ will be limited\nto (0, 0.1~GeV) as the cross section drops at $p_T$ = 0.1~GeV \nalready a few orders of magnitude. The electromagnetic effects may \nbe substantial only in the region of the small transverse momenta.\n\nThus for our exploratory study here we have limited the range for rapidity \nto (-5, 5) and for transverse momentum to $p_T$=(0, 0.1~GeV).\nThe integrated distribution can be seen in Fig.~\\ref{fig02b}(a) and\n(b). There we compare the distributions obtained for \ndifferent emission points ($b_x,b_y$): (0, 0), ($\\pm$15~fm, 0), \n(0, $\\pm$15~fm) as shown in Fig.~\\ref{fig01}.\nThe behavior of the differential cross section is very similar in each \n($b_x,b_y$) point but it differs in normalization as it is shown \nin Fig.~\\ref{fig02b}.\n\n\\begin{figure}[!hbt]\n\\includegraphics[scale=0.4]{dsig_dbm.eps}\n\\caption{\nDistribution of the cross section in the impact parameter $b$\nfor different energies:\n$\\sqrt{s_{NN}}$ = 17.3, 50 and 200~GeV (from bottom to top).\n}\n\\label{fig:dsig_dbm}\n\\end{figure}\n\nIn Fig.~\\ref{fig:dsig_dbm} we show a distribution of the cross section\nin impact parameter $b$ for different collision energies\n$\\sqrt{s_{NN}}$ = 17.3, 50, 200~GeV.\nIn this calculation we have taken $p_T >$ 0~GeV (the cross section\nstrongly depends on the lowest value of lepton transverse momentum $p_T$).\nIn general, the larger collision energy the broader the range of impact\nparameter. However, the cross section for $b \\approx R_{A_1} + R_{A_2}$\nis almost the same. Only taking into account limitation, e.g. on the momentum transfer, makes the difference in the cross section significant even at $b=14$~fm.\n\n\\begin{figure}[!hbt]\n    \\begin{center}\n     \\includegraphics[scale=0.450]{paw_baxbay_b14_check.eps}\\\\\n     \\vspace{-8cm}\\hspace{-5.0cm}{\\bf(a)}\\\\\n     \\vspace{7cm}\n     \\includegraphics[scale=0.450]{paw_baxbay_b50_check.eps}\\\\\n     \\vspace{-8cm}\\hspace{-5.0cm}{\\bf(b)}\\\\\n     \\vspace{7cm}\n     \\caption{ Two-dimensional cross section\n      as a function of $b_x$ and $b_y$ for\n      two values of impact parameter: \n      (a) b=14$\\pm$0.05~fm and (b) b=50$\\pm$0.05~fm.}\n     \\label{fig02d}\n  \\end{center}\n\\end{figure}\n\n\nThe emission point of the electrons\/positrons does not change the\nbehavior (shape) of the cross section on the ($y,p_T$) plane but \nchanges the absolute value of the cross section.\nAs it is visible in Fig.~\\ref{fig02b} (a) and (b) the biggest\ncross section is obtained for the CM emission point. The production of \n$e^+,e^-$ at ($b_x,b_y$) = (40~fm, 0) i.e. far from the CM point, \nis hindered by three orders of magnitude. \nMoreover, the production at $b_x$=$\\pm$15~fm and $b_y$=0 is more\npreferable than the production at $b_x$=0 and $b_y$=$\\pm$15~fm \nwhat is fully understandable taking into account the geometry of \nthe system (see Fig.~\\ref{fig01}). \nAs the system taken here into consideration is fully symmetric\n($A_1 = A_2, Z_1 = Z_2$), \nthus corresponding results are symmetric under the following\nreplacements: $b_x \\to -b_x$ or $b_y \\to -b_y$.\n\n\nFigure~\\ref{fig02d} compares the integrated cross section on reaction \nplane ($b_x,b_y$) for two impact parameters: (a) b=14$\\pm$0.05~fm \n(when nuclei are close to each other) and (b) b=50$\\pm$0.05~fm (when nuclei are well separated). \nThe landscape reflects the position of the nuclei in the moment of\nthe closest approach.\nSimilar plots have also been done for higher $\\sqrt{s_{NN}}$ but \nthe shape is almost unchanged, only the cross section value is\ndifferent. \nThis figure illustrates the influence of the geometry of the reaction. \nRegardless of the impact parameter (b), distance between colliding nuclei, the cross-section has\na maximum at $b_x=0$. The change of $b$ is correlated with the shift \nin a peak at $b_x$.\n\n\\begin{figure}[!hbt]\n    \\begin{center}\n     \\includegraphics[scale=0.60]{rap_e_xy_weight.pdf}\n     \\includegraphics[scale=0.60]{pt_e_xy_weight.pdf}\n     \\caption{(Color on-line) The differential cross section for various emission points\n       of electrons in the $^{208}$Pb+$^{208}$Pb reaction \n       at $\\sqrt{{s}_{NN}} =$ 17.3, 50, 200~GeV at impact parameter 14$\\pm$0.05~fm.}\n     \\label{fig02c}\n  \\end{center}\n\\end{figure}\n\n\nThe calculations confirm that the shape of the electron\/positron \ndistribution, shown in Fig.~\\ref{fig02c} does not depend on \nthe energy of the colliding nuclei. There are visible small differences \nin the magnitude of cross sections \nfor $\\sqrt{{s}_{NN}} =$ 17.3 and 200~GeV\nat least in the selected limited $p_T$=(0, 0.1)~GeV range. \nDependence on the rapidity is even weaker as the differences are visible\nonly for $|y|>$3.\n\nThese cross sections are used as weights in calculation of electromagnetic\neffects between electrons\/positrons and the fast moving nuclei. The\ncorresponding matrix has following dimensions: $b_{x,y}$=(-50~fm, 50~fm) - \n99$\\times$99 points in the reaction plane and 100$\\times$15 in the ($y,p_T$) space.\n\n\\begin{figure}[!hbt]\n\t\\begin{center}\n\t\t\\includegraphics[scale=0.4]{dsig_dxipt.eps}\n\t\t\\caption{\n\t\t\tDistribution of the cross section in $\\log_{10}p_T$\n\t\t\tfor different energies:\n\t\t\t$\\sqrt{s_{NN}}$ = 17.3, 50 and 200~GeV (from bottom to top).\n\t\t}\n\t\t\\label{fig:dsig_dxipt}\n\t\\end{center}\n\\end{figure}\n\nRather small transverse momenta enter such calculation.\nTo illustrate this in Fig.~\\ref{fig:dsig_dxipt} we show a distribution\nin $log_{10}(p_T)$. As seen from the figure the cross section is\nintegrable and we have no problem with this with our Monte Carlo\nroutine \\cite{Lepage:1977sw}.\n\n\n\\section{Electromagnetic interaction effects}  \\label{Sect.III}\n\nThe spectator system are modeled as two uniform spheres in \ntheir respective rest frames that change into disks in the \noverall center-of-mass collision frame.\nThe total charge of nuclei is 82 consistent with UPC. \nThe lepton emission region is reduced to a single point and the time of emission is a free parameter. \n\\begin{figure*}[!hbt]\n\\includegraphics[width=5.8cm,height=5.4cm]{ym_pt_000_17GeV_b14fm.png}\n\\includegraphics[width=5.8cm,height=5.4cm]{ym_pt_0150_17GeV_b14fm.png}\n\\includegraphics[width=5.8cm,height=5.4cm]{ym_pt_1500_17GeV_b14fm.png}\\\\\n\\vspace{-5.3cm}\\hspace{0.cm} {\\bf (a) ($b_x$=0, $b_y$=0) \\hspace{2.cm} (b) ($b_x$=0, $b_y$=15~fm) \\hspace{2.5cm} (c) ($b_x$=15~fm, $b_y$=0)} \\\\\n\\vspace{4.55cm}\n\\includegraphics[width=5.8cm,height=5.4cm]{ym_pt_full_17GeV_b14fm.png}\n\\includegraphics[width=5.8cm,height=5.4cm]{ym_pt_0m150_17GeV_b14fm.png}\n\\includegraphics[width=5.8cm,height=5.4cm]{ym_pt_m1500_17GeV_b14fm.png}\\\\\n\\vspace{-5.3cm}\\hspace{0.cm} {\\bf (d) full ($b_x$, $b_y$) space \\hspace{2.cm} (e) ($b_x$=0, $b_y$=-15~fm) \\hspace{2.cm} (f) ($b_x$=-15~fm, $b_y$=0)} \\\\\n\\vspace{5.cm}\n\\caption{(Color on-line) Rapidity vs $p_T$ distributions for final (subjected to EM\n  effects) electrons for different emission points: \n(a) (0, 0) and (d) full xy plane; (b)\n(0, 15~fm); (c) (15~fm, 0); (e) (0, -15~fm) and (f) (-15~fm, 0).\nThese results are for $\\sqrt{s_{NN}}$ = 17.3~GeV.}\\label{fig03}\n\\includegraphics[width=5.8cm,height=5.4cm]{yp_pt_000_17GeV_b14fm.png}\n\\includegraphics[width=5.8cm,height=5.4cm]{yp_pt_0150_17GeV_b14fm.png}\n\\includegraphics[width=5.8cm,height=5.4cm]{yp_pt_1500_17GeV_b14fm.png}\\\\\n\\vspace{-5.3cm}\\hspace{0.cm} {\\bf (a) ($b_x$=0, $b_y$=0) \\hspace{3.cm} (b) ($b_x$=0,$b_y$=15~fm) \\hspace{2.5cm} (c) ($b_x$=15~fm, $b_y$=0)} \\\\\n\\vspace{4.55cm}\n\\includegraphics[width=5.8cm,height=5.4cm]{yp_pt_full_17GeV_b14fm.png}\n\\includegraphics[width=5.8cm,height=5.4cm]{yp_pt_0m150_17GeV_b14fm.png}\n\\includegraphics[width=5.8cm,height=5.4cm]{yp_pt_m1500_17GeV_b14fm.png}\\\\\n\\vspace{-5.3cm}\\hspace{0.6cm} {\\bf (d) full ($b_x$, $b_y$) space \\hspace{3.0cm} (e) ($b_x$=0, $b_y$=-15~fm) \\hspace{2.6cm} (f) ($b_x$=-15~fm, $b_y$=0)} \\hspace{0cm}\\\\\n\\vspace{5.cm}\n\\caption{(Color on-line) Rapidity vs $p_T$ distributions for final (subjected to EM \neffects) positrons for different emission points: \n(a) (0, 0) and (d) full xy plane; (b) (0, 15~fm); (c) (15~fm, 0);\n(e) (0, -15~fm) and (f) (-15~fm, 0).\nThese results are for $\\sqrt{s_{NN}}$ = 17.3~GeV.}\\label{fig04}\n\\end{figure*}\n\n\n\\begin{figure*}[!hbt]\n\\includegraphics[width=8.6cm,height=6.cm]{yf_m_p_0_0_0_17GeV.png}\n\\hspace{0.cm}\n\\includegraphics[width=8.6cm,height=6.cm]{yf_m_p_0_0_0_17GeV_b50fm.png}\\\\\n\\vspace{-1.5cm}\\hspace{0.6cm} {\\bf (a) $\\sqrt{s_{NN}}$= 17.3~GeV \\hspace{5.2cm} (b) $\\sqrt{s_{NN}}$= 17.3~GeV\\\\\n\\vspace{-1.5cm}\\hspace{0.cm} b=14~fm ($b_x$=0,$b_y$=0) \\hspace{6.3cm} b=50~fm}\\\\\n\\vspace{1.7cm}\n\\includegraphics[width=8.6cm,height=6.cm]{yf_m_p_0_0_0_50GeV.png}\n\\includegraphics[width=8.6cm,height=6.cm]{yf_m_p_0_0_0_200GeV.png}\\\\\n\\vspace{-1.5cm}\\hspace{0.2cm}{\\bf(c)\\hspace{0.5cm} $\\sqrt{s_{NN}}$= 50~GeV \\hspace{4.2cm} (d)\\hspace{1.cm} $\\sqrt{s_{NN}}$= 200~GeV\\\\\n\\vspace{-1.5cm}\\hspace{0.2cm} b=14~fm \\hspace{7.3cm} b=14~fm}\\\\\n\\vspace{2.3cm}\n\\caption{(Color on-line) Reduced rapidity distributions for final electrons (blue) and\n  positrons (red) for fixed $b$ and ($b_x$=0,$b_y$=0) plane of\nemission points at three collision energies: $\\sqrt{s_{NN}}$=17.3, 50 and 200~GeV.\n}\n     \\label{fig05a}\n\\end{figure*}\n\n\nIn this work we assume there is no delay time between collisions of\nnuclei and the start of the EM interactions. \nThe $z$-dependence of the first occurrence of the $e^+ e^-$ pair is\nbeyond the EPA and is currently not known.\nIn our opinion production of $e^+ e^-$ happens when the moving\ncones, fronts of the EM fields, cross each other. This happens\nfor $z \\approx$ 0. \nIn the following we assume $z$ = 0 for simplicity.\n\\footnote{Any other distribution could be taken.}\n\n\\begin{figure*}[!hbt]\n\\includegraphics[width=8.6cm,height=6.cm]{yf_m_p_full_17GeV.png}\n\\includegraphics[width=8.6cm,height=6.cm]{yf_m_p_full_17GeV_b50fm.png}\\\\\n\\vspace{-5.5cm}\\hspace{0cm}{\\bf (a)\\hspace{1.cm} $\\sqrt{s_{NN}}$= 17.3~GeV \\hspace{4.cm} (b)\\hspace{1.cm} $\\sqrt{s_{NN}}$= 17.3~GeV\\\\\n\\vspace{2.5cm}\\hspace{-8cm} full ($b_x$,$b_y$) space \\hspace{12.5cm}  \\\\\n\\vspace{0.5cm}\\hspace{0cm} b=14~fm \\hspace{7.5cm} b=50~fm}\\\\\n\\vspace{0.9cm}\n\\includegraphics[width=8.6cm,height=6.cm]{yf_m_p_full_50GeV.png}\n\\includegraphics[width=8.6cm,height=6.cm]{yf_m_p_full_200GeV.png}\\\\\n\\vspace{-5.5cm}{\\bf (c)\\hspace{0.8cm} $\\sqrt{s_{NN}}$= 50~GeV \\hspace{4.0cm} (d)\\hspace{1.cm} $\\sqrt{s_{NN}}$= 200~GeV\\\\\n\\vspace{3.5cm}\\hspace{0cm} b=14~fm \\hspace{7.5cm} b=14~fm}\\\\\n\\vspace{1.3cm}\n\\caption{(Color on-line) Reduced rapidity distributions for final electrons (blue) \nand positrons (red) for fixed $b$ and full ($b_x$,$b_y$) plane of\nemission points at three collision energies: $\\sqrt{s_{NN}}$=17.3, 50 and 200~GeV.}\n     \\label{fig05b}\n\\end{figure*}\n\n\nThe trajectories of $e^{\\pm}$ in the field of moving nuclei are obtained\nby solving the equation of motion numerically for electrons\/positrons:\n\\begin{equation}\n\\frac{d \\vec{p}_{e^{\\pm}}}{d t} =\n\\vec{F}_{1,e^{\\pm}}(\\vec{r}_1,t) + \\vec{F}_{2,e^{\\pm}}(\\vec{r}_2,t) \\; .\n\\label{equation_of_motion}\n\\end{equation}\n\n\nThe total interaction is a superposition of interactions with both\nnuclei which positions depend on time.\nWe solve the motion of electron\/positron in the overall center \nof mass system, i.e. both position and time are given in this frame.\nIn this frame we have to deal with both electric and magnetic force\n\\cite{Rybicki:2006qm}.\nBecause nuclei are very heavy compared to electrons\/positron\ntheir motion is completely independent and is practically not distorted\nby the EM interaction.\nWe take:\n\\begin{eqnarray}\n\\vec{r}_1(t) =  + {\\hat z} c t + \\vec{b}\/2 \\; , \\nonumber \\\\\n\\vec{r}_2(t) =  - {\\hat z} c t - \\vec{b}\/2 \\; ,\n\\end{eqnarray} \ni.e. assume that the nuclei move along straight trajectories\nindependent of the motion of electron\/positron.\nThe step of integration depends on energy and must be carefully adjusted.\n\nThe rapidity vs $p_T$ distributions of initial leptons are \nobtained by randomly choosing the position on the two-dimensional space. \nThe path of particle in electromagnetic field generated by nuclei\nare traced up to 10 000~fm away from the original interaction point. \nThe Monte Carlo method is used to randomize the \ninitial rapidity and $p_T$ from uniform distribution. \nThe initial rapidity and $p_T$ of electrons\/positrons are randomly\nchosen in the range: $y$=(-5,5) and $p_T$=(0, 0.1)~GeV as fixed\nin the previous section. The ($y,p_T$) distributions for final\n(subjected to the EM evolution) leptons\nare presented in Fig.~\\ref{fig03} for electrons and in Fig.~\\ref{fig04} \nfor positrons. These distributions were obtained by analyzing \nthe EM evolution event-by-event.\nThe number of events taken here is $n_{event}$ = 10$^7$ \nfor each two-dimensional plot.\n\nFor electron production an enhancement and for positron a loss with\nrespect to the neighborhood or\/and flat initial distribution\nis observed for $y \\approx \\pm$ 3. This corresponds to the beam rapidity\nat $\\sqrt{s_{NN}}$= 17.3~GeV  energy.\n\nThese two sets of two-dimensional plots illustrate \neffect of the EM interaction between $e^+$ or $e^-$ and the moving nuclei.  \nThe motion of particles in the EM field of nuclei changes \nthe initial conditions and the final ($y,p_T$) are slightly different.\nFig.~\\ref{fig03} shows the behavior of electrons and \nFig.~\\ref{fig04} of positrons at CM (a) and \ndifferent impact parameter points: (panels c,f) ($\\pm$15~fm,0) and \n(panels b,e) (0,$\\pm$15~fm) marked in Fig.~\\ref{fig01}. \nWe observe that the maximal number of electrons is located \nwhere the cross section for positrons has minimum. The Coulomb effects are well visible as \na missing areas for positrons for $p_T<0.02$~GeV for particles\nemitted from the CM point. The emission of leptons from $b_y$=$\\pm$15~fm gives\nlower effect. The asymmetry in emission is well visible for\n$b_x$=$\\pm$15~fm, \nwhere a larger empty space is for positive rapidity when $b_x$=15~fm and \nfor negative rapidity when $b_x$=-15~fm. \n\nAlthough the EM effects are noticeable for \nelectrons\/positrons in different impact parameter points, \nthe integration over full reaction plane washes out almost totally the effect.\n\nFor comparison the results of integration over full space \n$b_x$=(-50~fm, 50~fm) and $b_y$=(-50~fm, 50~fm) are shown \nin panels (d) of Figs.~\\ref{fig03} and \\ref{fig04}. \nThese results are independent of the source of leptons, thus it could \nbe treated as a general trend and an indication for which \nrapidity-transverse momentum ranges one could observe effects of the EM\ninteraction between leptons and nuclei.\n\\begin{figure*}\n\\includegraphics[width=8.6cm,height=6.cm]{pTf_m_p_full_17GeVa.png}\\hspace{-0.cm}\n\\includegraphics[width=8.6cm,height=6.cm]{pTf_m_p_full_17GeV_b50fma.png}\\\\\n\\vspace{-5.0cm}\\hspace{-2cm}{\\bf (a) $\\sqrt{s_{NN}}$= 17.3~GeV \\hspace{4.9cm} (b) $\\sqrt{s_{NN}}$= 17.3GeV\\\\\n\\vspace{1.5cm}\\hspace{-1.0cm} b=14~fm \\hspace{6.5cm} b=50~fm}\\\\\n\\vspace{2.4cm}\n\\includegraphics[width=8.6cm,height=6.cm]{pTf_m_p_full_50GeVa.png}\\hspace{-0.cm}\n\\includegraphics[width=8.6cm,height=6.cm]{pTf_m_p_full_200GeVa.png}\\\\\n\\vspace{-5.4cm}\\hspace{-2cm}{\\bf(c) $\\sqrt{s_{NN}}$= 50~GeV \\hspace{4.9cm} (d) $\\sqrt{s_{NN}}$= 200~GeV\\\\\n\\vspace{1.5cm}\\hspace{-1.0cm} b=14~fm \\hspace{6.5cm} b=14~fm}\\\\\n\\vspace{3.5cm}\n\\caption{(Color on-line) Transverse momentum distributions for final electrons (blue) \nand positrons (red) for fixed $b$ and full ($b_x$, $b_y$) plane of\nemission points at three collision energies: $\\sqrt{s_{NN}}$=17.3, 50 and 200~GeV.}\n     \\label{fig05c}\n\\end{figure*}\n\nThe distribution in reduced rapidity of final leptons are shown \nin Fig.~\\ref{fig05a} and \\ref{fig05b}. The reduced rapidity\n(dimensionless quantity) is the rapidity $y$ normalized to the beam rapidity $y_{beam}$\n(different for various collision energies)\n\\begin{equation}\ny_{red} = y \/ y_{beam} \\; ,\n\\end{equation}\nwhere\n\\begin{equation}\ny_{beam} = \\pm ln \\left( \\frac{\\sqrt{s_{NN}}}{m_p} \\right)\n\\end{equation}\nand $m_p$ is the proton mass.\nThe results shown above were obtained, somewhat arbitrarily, with\nuniform distribution in $(y,p_T)$.\nThis leads to the observation of peaks or dips at beam rapidities.\nNo such peaks appear for $\\sqrt{s}$ = 200~GeV as here the chosen range\nof rapidity (-5,5) is not sufficient.\nWhether such effects survive when weighting with the b-space EPA\ncross section will be discussed below.\n\nFig.~\\ref{fig05a} is focused on emission from the center of mass point \nand Fig.~\\ref{fig05b} is obtained when integrating over full ($b_x,b_y$) plane. \nThe main differences between electron (blue lines) and \npositron (red lines) distributions are not only at midrapidities \nbut also around the beam rapidity. The effect is more visible \nfor the CM emission point but it is slightly smoothed out when \nthe full $(b_x, b_y)$ plane is taken into consideration.   \nMoreover increasing the impact parameter (panels (a) and (b)) \ndiminishes the difference between rapidity distributions of final \nelectrons and positrons. The beam energy is another crucial parameter. \nThe collision with $\\sqrt{s_{NN}} > 100$~GeV (panel (d)) does not \nallow for sizeable effects of electromagnetic interaction between \nleptons and nuclei, at least at midrapidities.\n\nThe discussion of the EM interaction between $e^+e^-$ and nuclei has \nto be completed by combining with the cross section of lepton production\nas obtained within EPA.\nTaking into account the leptons coming from photon-photon fusion \nthe distributions from Fig.~\\ref{fig03} and \\ref{fig04} are multiplied \nby differential cross section obtained with Eq.(~\\ref{EPA}).\n \nThe details of the method are presented in Ref.~\\cite{Rybicki:2006qm} \nand adapted here from pion emission to electron\/positron emission.\n\\begin{figure*}[!hbt]\n    \\begin{center}\n     \\includegraphics[scale=0.60]{y_p_m_000_EM_pure_weighta.pdf}\n     \\includegraphics[scale=0.60]{y_p_m_full_EM_pure_weighta.pdf}\n     \\caption{(Color on-line)\n     The electron and positron emission cross section normalized to 100\\% \n     in the $^{208}$Pb+$^{208}$Pb reaction at 158~GeV\/nucleon energy \n($\\sqrt{{s}_{NN}} =$ 17.3~GeV) at impact parameter 14$\\pm$0.05~fm\nassuming the $p_T$=(0, 0.1)~GeV produced in the center of mass (0, 0)\npoint (a) and when integrating over full reaction space (b).\nShown are original EPA distributions (dotted line) and results when\nincluding evolution in the EM field of nuclei for positrons (solid line)\nand for electrons (dashed line).\n}\n     \\label{fig07}\n  \\end{center}\n\\end{figure*}\nIn Fig.~\\ref{fig05c} we show the influence of EM interaction on $p_T$ distributions.\nHere we integrate over rapidity and ($b_x,b_y$). One can observe that the EM effects lead to a diffusion of transverse momenta \n(see the diffused edge at $p_T=0.1$~GeV, marked by green vertical line). No spectacular effect is observed when changing the \nimpact parameter or beam energy.\n\nFig.~\\ref{fig07} shows a comparison of rapidity distribution of \nfinal electrons and positrons, assuming the particles are emitted \nfrom (a) the center of mass (0, 0) point and (b) when integrating over \n$b_x,b_y$. \nThe comparison is done between EPA distribution relevant for initial stage (red, dotted line) with the final stage, resulting from the EM interaction of charged leptons with positively charged nuclei. \nIf leptons are produced in the CM point, the electron\ndistributions are almost unchanged but positron distributions \nare squeezed to $|y| < 2$. \nIf the cross section is integrated over full ($b_x,b_y$) parameter space, \nthe positron distribution is still steeper than that for electrons \nbut mainly for $|y| <$ 2. \n\n\n\nEven when the leptons produced in the full ($b_x,b_y$) plane are considered, the\n$e^+$ and $e^-$ distributions are different from the initial ones. The electrons under the EM interactions are focused at midrapidities.\n\n\n\\begin{figure*}[!hbt]\n\\includegraphics[width=7.8cm,height=6.cm]{bx_ym_17GeV_14fm.png}\n\\includegraphics[width=7.8cm,height=6.cm]{bx_yp_17GeV_14fm.png}\\\\\n\\vspace{-6.0cm}{\\bf\\hspace{-5cm} (a) \\hspace{7.0cm} (b)}\\\\\n\\vspace{5.5cm}\n\\includegraphics[width=7.8cm,height=6.cm]{by_ym_17GeV_14fm.png}\n\\includegraphics[width=7.8cm,height=6.cm]{by_yp_17GeV_14fm.png}\\\\\n\\vspace{-6.0cm}{\\bf\\hspace{-5cm} (c) \\hspace{7.0cm} (d)}\\\\\n\\vspace{5.5cm}\n\\caption{ Distribution of electrons ((a), (c)) and positrons ((b), (d)) for $\\sqrt{s_{NN}}$= 17.3\n ~GeV at b=14~fm\n  integrated over ($b_x,b_y$)=(-50~fm, 50~fm),\n  $p_T^{ini}$=(0, 0.1~GeV)}\n     \\label{fig08}\n\\end{figure*}\n\\begin{figure*}\n\\includegraphics[width=7.8cm,height=6.cm]{bx_ym_17GeV_50fm.png}\n\\includegraphics[width=7.8cm,height=6.cm]{bx_yp_17GeV_50fm.png}\\\\\n\\vspace{-6.0cm}{\\bf\\hspace{-5cm} (a) \\hspace{7.0cm} (b)}\\\\\n\\vspace{5.5cm}\n\\includegraphics[width=7.8cm,height=6.cm]{by_ym_17GeV_50fm.png}\n\\includegraphics[width=7.8cm,height=6.cm]{by_yp_17GeV_50fm.png}\\\\\n\\vspace{-6.0cm}{\\bf\\hspace{-5cm} (c) \\hspace{7.0cm} (d)}\\\\\n\\vspace{5.5cm}\n\\caption{ Distribution of electrons ((a), (c)) and positrons ((b), (d)) for $\\sqrt{s_{NN}}$= 17.3\n ~GeV at b=50~fm \n     integrated over ($b_x,b_y$)=(-100~fm, 100~fm), \n   $p_T^{ini}$=(0, 0.1~GeV).}\n     \\label{fig09}\n\\end{figure*}\n\\begin{figure*}[!hbt]\n\\includegraphics[width=7.8cm,height=6.cm]{bx_ym_200GeV_14fm.png}\n\\includegraphics[width=7.8cm,height=6.cm]{bx_yp_200GeV_14fm.png}\\\\\n\\vspace{-6.0cm}{\\bf\\hspace{-5cm} (a) \\hspace{7.0cm} (b)}\\\\\n\\vspace{5.5cm}\n\\includegraphics[width=7.8cm,height=6.cm]{by_ym_200GeV_14fm.png}\n\\includegraphics[width=7.8cm,height=6.cm]{by_yp_200GeV_14fm.png}\\\\\n\\vspace{-6.0cm}{\\bf\\hspace{-5cm} (c) \\hspace{7.0cm} (d)}\\\\\n\\vspace{5.5cm}\n\\caption{ Distribution of electrons ((a), (c)) and positrons ((b), (d)) for $\\sqrt{s_{NN}}$= 200\n ~GeV at b=14~fm \nintegrated over ($b_x,b_y$)=(-50~fm, 50~fm) for $p_T^{ini}$=(0, 0.1~GeV)}\n     \\label{fig11}\n\\end{figure*}\n\nThe dependence on the position of emission and final rapidity allows to \nunderstand how the geometry influences the electromagnetic interaction between \nleptons and nuclei. Figures \\ref{fig08},\\ref{fig09} \nand \\ref{fig11} present the cross section distribution in  \n$b_x$ (top rows) and rapidity for electrons (a) and positrons (b) and (bottom rows)\n$b_y$ and rapidity for electrons (c) and positrons (d).\nFigs.~\\ref{fig08} and \\ref{fig09} are for $\\sqrt{s_{NN}}$= 17.3~GeV but with \nthe impact parameter 14$\\pm$0.05 and 50$\\pm$0.05~fm.\nFig.~\\ref{fig11} is for $\\sqrt{s_{NN}}$= 200~GeV and\nb=14~fm. \nThese plots allow to investigate the anisotropy caused by the\ninteraction between leptons and nuclei. It is more visible for larger \nimpact parameter when the spectators are well separated (Fig.~\\ref{fig09}). \n\\begin{figure*}\n\\includegraphics[width=7.8cm,height=6.cm]{y_m_full_x_weight_17_50_200.pdf}\n\\includegraphics[width=7.8cm,height=6.cm]{y_p_full_x_weight_17_50_200.pdf}\n\\caption{Rapidity distribution of electrons for $\\sqrt{s_{NN}}$= 17.3~GeV (b=14~fm,\n  50~fm) and 50~GeV and 200~GeV with b=14$\\pm$0.05~fm (only) integrated over \n($b_x,b_y$)=(-50~fm, 50~fm), $p_T^{ini}$=(0, 0.1~GeV).}\n     \\label{fig13}\n\\end{figure*}\n\n\\begin{figure}\n\\includegraphics[width=7.8cm,height=6.cm]{y_pm_full_x_weight_17_50_200.pdf}\n\\caption{The ratio of rapidity distributions of positrons and electrons for \n  $\\sqrt{s_{NN}}$= 17.3~GeV and fixed b=14~fm and 50~fm \n  and $\\sqrt{s_{NN}}$ = 50~GeV and 200~GeV with fixed b=14~fm \n  integrated over ($b_x,b_y$)=(-50~fm, 50~fm) and transverse momenta in the interval \n   $p_T^{ini}$=(0, 0.1~GeV).}\n     \\label{fig14}\n\\end{figure}\n\nDistributions for fixed impact parameter and different beam energies \nreflect the behavior seen in Fig.~\\ref{fig05b}. For collision with \n$\\sqrt{s_{NN}}$= 200~GeV (Fig.~\\ref{fig11})\nthe electrons and positrons almost do not feel the presence of the EM\nfields of the nuclei.  \n\nIntegrating over full ($b_x,b_y$) plane one obtains \nthe rapidity distribution of final leptons\nshown in Fig.~\\ref{fig13} (a) separately for electrons and (b) positrons, \nfor two impact parameters: 14~fm (full lines) and 50~fm (dashed lines). \nElectrons have a somewhat wider distribution than positrons and \nthis is independent of the impact parameter.\nWhile positron rapidity distributions only weakly depend on collision energy it is not the case for electrons, where sizeable differences can be observed.\n\nThe ratio of distributions for positrons and electrons (Fig.~\\ref{fig14}) reflects the behavior seen in Figs.~\\ref{fig08} and \\ref{fig09}. \nThis plot shows a combined effect of EPA cross section and \nthe EM interactions of leptons with nuclei. Thus despite the production \ncross section is lower for larger impact parameter, the discussed\nphenomenon should be visible for larger rapidities. \nThe ratio quickly changes with energy and tends to 1 for larger energies (see dotted line for $\\sqrt{{s}_{NN}}$ = 200~GeV).\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusions}\n\nIn the last 15 years the electromagnetic effects due to a large (moving)\ncharge of the spectators on charged pion momentum distributions were \nobserved both in theoretical calculations and experimentally \nin peripheral heavy ion collisions at SPS and RHIC energies.\nInteresting and sometimes spectacular effects were identified.\n\nIn the present paper we have discussed whether such effects could also\nbe observed for the distributions of electrons\/positrons produced\nvia photon-photon fusion in heavy ion UPC.\nThe corresponding cross section can be rather reliably calculated\nand turned out to be large, especially for low transverse\nmomentum electrons\/positrons. The impact parameter equivalent photon\napproximation is well suitable for investigating the electromagnetic\neffects.\nOn the experimental side only rather large transverse momentum\nelectrons\/positrons could be measured so far at RHIC and the LHC, \ntypically larger than 0.5~GeV.\n\nWe have organized calculations that include the EM effects using as an input EPA distributions.\nFirst multidifferential (in momenta and impact parameter) distributions \nfor the diphoton production of the $e^+ e^-$ pairs are prepared in \nthe impact parameter equivalent photon approximation. \nSuch distributions are used next to calculate the propagation of \nthe electrons\/positrons in strong EM fields generated by \nthe quickly moving nuclei.\nThe propagation has been done by solving numerically relativistic \nequation of motion. Strong EM effects have been observed only at \nvery small transverse momenta of electron\/positron. Therefore \nto accelerate calculations we have limited to really small initial \ntransverse momenta $p_T <$ 0.1~GeV.\n\n\nThe shape of the differential cross section in rapidity and transverse\nmomentum does not depend on the energy of the process but rather \non the emission point in the impact parameter plane ($b_x,b_y$). \nWe have investigated effects for different initial conditions, \ni.e. different emission positions in the impact parameter space.\n\nThe leptons interact electromagnetically with charged nuclei which \nchanges their trajectories. The biggest effect has been identified for \nthe CM emission point. \nHowever, the integration over full ($b_x,b_y$) plane washes out this effect to large extent. \nThe range of $p_T$=(0, 0.02~GeV) has turned out the most preferable \nto investigate the influence of the EM effects on leptons originating from various \n($b_x,b_y$) plane points.\n\nMoreover, the impact parameter influences not only the value of \nthe cross section but also the shapes of distributions of final leptons. \n\nThe $AA \\rightarrow AAe^+e^-$ process creates leptons in a broad\nrange of rapidities. We have found that only at small transverse momenta of\nelectrons\/positrons one can observe sizeable EM effects.\n \nThe performed calculations allow to conclude that the maximal \nbeam energy for the Pb+Pb collision, where the EM effects between leptons\nand nuclei are evident at midrapidities, is probably \n$\\sqrt{{s}_{NN}}$=100~GeV.\nObservation of the effect at higher energies may be therefore \nrather difficult, if not impossible.\nThe effect survives even at high energies but close to beam rapidity. \nHowever, this region of the phase space is usually not instrumented\nand does not allow electron\/positron measurement.\n\nSo far the effect of EM interaction was studied for fixed values of \nthe impact parameter (mainly for $b$ = 14~fm). However, the impact parameter\ncannot be measured.\nThe integration over the impact parameter is rather difficult and\ngoes beyond the scope of the present paper.\nSuch an integration will be studied elsewhere.\n\nOn the experimental side a good measurement of electrons and positrons\nat low transverse momenta ($p_T <$ 0.1~GeV is necessary to see \nan effect.\nIn principle, the measurement of the $e^+ \/ e^-$ ratio as a function\nof lepton rapidity and transverse momentum would be useful in \nthis context.\nAccording to our knowledge there are definite plans only for\nhigh energies (ALICE-3 project) where, however, the EM effect should\nbe very small (high collision energy). At CERN SPS energies the effect\nis rather large but at very small transverse momenta.\nRHIC would probably be a good place to observe\nthe effect of the discussed here EM interactions but this would\nrequire a modification of the present apparatus.\n\n{\\bf Acknowledgement}\n\nA.S. is indebted to Andrzej Rybicki for past collaboration on\nelectromagnetic effects in heavy ion collisions.\nThis work is partially supported by\nthe Polish National Science Centre under Grant\nNo. 2018\/31\/B\/ST2\/03537\nand by the Center for Innovation and Transfer of Natural Sciences \nand Engineering Knowledge in Rzesz\\'ow (Poland).\n\n\n\n\\bibliographystyle{spphys}      \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Appendix}\n\\label{sec:appendix}\n\\begin{proof}[Proof of Lemma \\ref{lem:wtforg}]\nTo construct the weight function $w$, we associate a sequence $P(u,v)$ of edges of $G'$ with each edge $(u,v)$ of $G$. Assume that $T'$ is a rooted tree, and root is the highest node in the tree. The heights of all the other nodes are one less than that of their parent. As we mentioned that $T'$ is a tree decomposition of $G$. For an edge $(u,v)$, we know that there are unique highest bags $B_1$ and $B_2$ that contain vertices $u$ and $v$, respectively.\n\\begin{itemize}\n\\item If $B_1 = B_2$ then $P(u,v) = (u_{B_1} , v_{B_2})$.\n\n\\item If $B_1$ is an ancestor of $B_2$ then \\\\$P(u,v) = (u_{B_1}, u_{parent(\\cdots (parent(B_2)))}), \\ldots ,(u_{parent(B_2)}, u_{B_2}),(u_{B_2},v_{B_2})$.\n\n\\item If $B_1$ is a descendant of $B_2$ then \\\\ $P(u,v) = (u_{B_1}, v_{B_1}), (v_{B_1}, v_{parent(B_1)}), \\ldots , (v_{parent(\\cdots (parent(B_1)))},v_{B_2})$.\n\\end{itemize}\n\nThe weight function $w$ for the graph $G$ is defined as follows:\n\\begin{eqnarray*}\nw(u,v) = \\sum_{e\\in P(u,v)} w'(e)\n\\end{eqnarray*}\n\n\nFor a simple cycle $C = e_1,e_2, \\ldots ,e_j$ in $G$,  we define $P(C) = P(e_1),P(e_2), \\ldots ,P(e_j)$. Note that $P(C)$ is a closed walk in $G'$. Let $E'_d(C)$ be the subset of edges of $G'$ such for all edges $e \\in E'_d(C)$ both $e$ and $e^{r}$ appear in $P(C)$, where $e^r$ denotes the edge obtained by reversing the direction of $e$. We prove that if we remove the edges of $E'_d(C)$ from $P(C)$ then the remaining edges $P(C) - E'_d(C)$  form a simple cycle in $G'$. \n\n\\begin{claim}\n\\label{clm:simp}\nEdges in the set $P(C)- E'_d(C)$ form a simple cycle in $G'$. \n\\end{claim}\n\n\\begin{proof}\nNote that the lemma follows trivially if $P(C)$ is a simple cycle. Therefore, assume that $P(C)$ is not a simple cycle. We start traversing the walk $P(C)$ starting from the edges of the sequence $P(e_1)$. Let $P(e_k)$ be the first place where a vertex in the walk $ P(e_1) P(e_2).....P(e_k)$ repeats, i.e., edges in the sequence  $P(e_1) P(e_2).....P(e_{k-1})$ form a simple path, but after adding the edges of  $P(e_k)$ some vertices are visited twice in the walk $P(e_1) P(e_2).....P(e_{k-1})P(e_k)$ for some $k \\leq j$. This implies that some vertices are visited twice in the sequence $P(e_{k-1})P(e_k)$. Let $e_{k-1}=(u,v)$ and $e_k=(v,x)$. This implies that some copies of the vertex $v$ appear twice in the sequence $P(u,v)P(v,x)$. Let $B_1$ and $B_2$ be the highest bags such that $B_1$ contains the copies of vertices $u$ and $v$, and $B_2$ contains the copies of $v$ and $x$. Let bag $B$ be the lowest common ancestor of $B_1$ and $B_2$. We know that $B$ must contain a copy of the vertex $v$, i.e., $v_B$. Let $B'$ be the highest bag containing  a copy of vertex $v$, i.e., $v_{B'}$. First, consider the case when neither $B_1$ is an ancestor of $B_2$ and vice-versa, other cases can be handled similarly. In that case sequence  $P(u,v) = u_{B_1}v_{B_1}  v_{parent(B_1)} \\ldots v_{B} \\ldots v_{B'}$ and $P(v,w) = v_{B'} \\ldots v_{B} \\ldots v_{parent(B_2)} v_{B_2}w_{B_2}$. Note that in $P(u,v)$ a path goes from $v_B$ to $v_{B'}$ and the same path appear in reverse order from $v_{B'}$ to $v_{B}$ in the sequence $P(v,x)$. Therefore if we remove these two paths from $P(u,v)$ and $P(v,w)$ the remaining subsequence of the sequence $P(u,v)P(v,w)$ will be a simple path, i.e., no vertex will appear twice since $B$ is the lowest common ancestor of $B_1$ and $B_2$. Now repeat this procedure for $P_{k+1},P_{k_2} \\ldots$ and so on till $P_{k}$. In the end, we will obtain a simple cycle.\n\\end{proof}\n\n\nSince we assumed that the weight function $w'$ is skew-symmetric, we know that $w'(e) = -w (e^{r})$, for all $e \\in G'$. This implies that $w'(E'_d(C)) = 0 $. Therefore $w(C) = w'(P(C)) = w'(P(C) - E'_d(C))$. From Claim \\ref{clm:simp} we know that edges in the set $P(C) - E'_d(C)$ form a simple cycle and we assumed that $w'$ gives nonzero circulation to every simple cycle therefore, $w'(P(C) - E'_d(C)) \\neq 0$. This implies that $w(C) \\neq 0$. This finishes the proof of Lemma \\ref{lem:wtforg}.\n\\end{proof}\n\n\\begin{proof}[Proof of Claim \\ref{clm:connect}]\nNote that if we treat each vertex in the bags of $T'$ distinctly, then there is a one-to-one correspondence between vertices of $G'$ and vertices in the bags of $T'$. Therefore, in $T'$ a vertex of $G'$ is identified by its corresponding vertex. Note that all the bags which contain vertices of the cycle $C$ form a connected component in $T'$. We will now prove that if a bag $B$ contains some vertices of $C$, then either $B$ has some edges of $C$ associate with it or no bag in the subtree rooted at $B$ has any edge of $C$ associated with it. From this, we can conclude that the bags which have some edges of $C$ associated with them form a connected component in $T'$.\n\nAssume that $B$ is a bag which contains a vertex of $C$ but no edge of $C$ is associated with it. This implies that $C$  never enters in any of the children of $B$. Because, let us assume it enters to some child $B'$ of $B$ through some vertex $v_{B'}$ of $B'$. In that case, there will be an edge $(v_{B},v_{B'})$ of $C$ associated with the bag $B$, which is a contradiction. Therefore subtree rooted at $B$ will not have any edge of the cycle $C$ associated with it. This finishes the proof.\n\\end{proof}\n\\section{Weight function}\n\\label{sec:wtfn}\nIn order to construct the desired weight function for a given graph $G_0 \\in \\langle \\mathcal{G}_{P, W}\\rangle_3$, we modify the component tree $T_0$ of $G_0$ such that it has the following properties.\n\\begin{itemize}\n\\item No two separating sets share a common vertex. \\item A separating set is shared by at most two components.\n\\item any virtual triangle, i.e., the triangle consists of virtual edge, in a planar component is always a face.\n\\end{itemize}\n\nLet $T$ be this modified component tree, and $G$ be the graph represented by $T$. We show that if we have a weight function that gives nonzero circulation to every cycle in $G$, then we can obtain a weight function that will give nonzero circulation to all the cycles in $G_0$.\nArora et al. \\cite{AGGT16} showed how a component tree satisfying these properties can be obtained for $K_{3,3}$-free and $K_5$-free graphs. We give a similar construction below and show that we can modify the components of $T_0$ such that $T$ satisfies the above properties (see Section \\ref{sec:modt}). Note that if the graphs inside two nodes of $T_0$ share a separating set $\\tau$ and they both are constant tree-width graphs, then we can take the clique-sum of these two graphs on the vertices of $\\tau$, and the resulting graph will also be a constant tree-width graph. Therefore, we can assume that if two components share a separating set, then either both of them are planar, or one of them is planar and the other is of constant tree-width.\n\n\\subsection{Modifying the Component Tree}\n\\label{sec:modt}\nIn this section, we show that how we obtain the component tree $T$ from $T_0$ so that it satisfies the above three properties.\n\\subparagraph{(i) No two separating sets share a common vertex:}\nFor a node $D$ in $T_0$, let $G_D$ be the graph inside node $D$. Assume that $G_D$ contains a vertex $v$ which is shared by separating sets $\\tau_1,\\tau_2, \\ldots ,\\tau_k$, where $k>1$, present in $G_D$. We replace the vertex $v$ with a gadget $\\gamma$ defined as follows: $\\gamma $ is a star graph such that $v$ is the center node and $v_1,,v_2, \\ldots ,v_k$ are the leaf nodes of $\\gamma$. The edges which were incident on $v$ and had their other endpoints in $\\tau_i$, will now incident on $v_i$ for all $i \\in[k]$. All the other edges which were incident on $v$ will continue to be incident on $v$. We do this for each vertex which is shared by more than one separating set in $G_{D}$. Let $G_{D'}$ be the graph obtained after replacing each such vertex with gadget $\\gamma$. It is easy to see that if $G_D$ was a planar component, then $G_{D'}$ will also be a planar component. We show that the same holds for constant tree-width components as well.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=1]{figures\/singlecmg}\n\\caption{(Left)A separating set $\\{x_1,x_2\\}$ is shared by components $D_1,D_2$ and $D_3$. (Right) Replace them by adding the gadget $\\beta$ and connect $D_1,D_2$ and $D_3$ to $\\beta$.}\n\\label{fig:sepset}\n\\end{center}\n\\end{figure}\n\n\\begin{claim}\nIf $G_D$ is a constant treewidth graph, then $G_{D'}$ will also be of constant treewidth.\n\\end{claim}\n\\begin{proof}\nLet $T_D$ be a tree decomposition of $G_D$ such that each bag of $T_D$ is of constant size, i.e., contains some constant number of vertices. Let $v$ be a vertex shared by $k$ separating sets $\\{x_i,y_i,v\\}$, for all $i \\in [k]$ in $G_D$. Let $B_1,B_2, \\ldots B_k$ be the bags in $T_D$ that contain separating sets $\\{x_1,y_1,v\\}, \\{x_2,y_2,v\\} , \\ldots ,\\{x_k,y_k,v\\}$ respectively (note that one bag may contain many separating sets). Now we obtain a tree decomposition $T_{D'}$ of the graph $G_{D'}$ using $T_D$ as follows: add the vertices $v_i$ in the bag $B_i$, for all $i \\in [k]$. Repeat this for each vertex $v$ in $G_D$, which is shared by more than one separating set to obtain $T_{D'}$. Note that in each bag of $T_D$ we add at most one new vertex with respect to each separating set contained in the bag in order to obtain $T_{D'}$. Since each bag in $T_D$ can contain vertices of only constant many separating sets, size of each bag remain constant in $T_{D'}$. Also, $T_{D'}$ is a tree decomposition of $G_{D'}$.\n\\end{proof}\n\n\n\\subparagraph{(ii) A separating set is shared by at most two components:} Assume that a separating set of size $t$, $\\tau =\\{x_i\\}_{i \\leq t}$ is shared by $k$ components $D_1,D_2, \\ldots D_k$, for $k>2$, in $T_0$. Let $\\beta$ be a gadget defined as follows: the gadget consists of $t$ star graphs $\\{\\gamma_i\\}_{i \\leq t}$ such that $x_i$ is the center node of $\\gamma_i$ and each $\\gamma_i$ has $k$ leaf nodes $\\{x_i^1,x_i^2, \\ldots x_i^k\\}$. There are virtual cliques present among the vertices $\\{x_i^j\\}_{i \\leq t}$ for all $j \\in [k]$ and among $\\{x_i\\}_{i \\leq t}$ (see Figure \\ref{fig:sepset}). If there is an edge present between any pair of vertices in the set $\\{x_i\\}_{i \\leq t}$ in the original graph, then we add a real edge between respective vertices in $\\beta$. $\\beta$ shares the separating set $\\{x_i^j\\}_{i \\leq t}$ with the component $D_j$ for all $j \\in [k]$.\n\nNote that in this construction, we create new components ($\\beta$) while all the other components in the component tree remain unchanged. Notice that the tree-width of $\\beta$ is constant (at most $5$ to be precise). We can define a tree decomposition of $\\beta$ of tree-width $5$ as follows: $B_0,B_1',B_2', \\ldots ,B_k'$ be the bags in the tree decomposition such that $B_0 = \\{x_1,x_2,x_3\\}$, $B_i'= \\{x_1,x_2,x_3,x^i_1,x^i_2,x^i_3\\}$ and there is an edge from $B_0$ to $B_i'$ for all $i \\in [k]$.\n\n\\subparagraph{(iii) Any virtual triangle, i.e., the triangle consists of virtual edges, in a planar component is always a face:} $3$-cliques in a $3$-clique sum of a planar and a bounded tree-width component is always a face in the planar component. This is because suppose there is a planar component $G_i$ in which the $3$-clique on $u,v,w$ occurs but does not form a face. Then the triangle $u,v,w$ is a separating set in $G_i$, which separates the vertices in its interior $V_1$ from the vertices in its exterior $V_2$. Notice that neither of $V_1,V_2$ is empty by assumption since $u,v,w$ is not a face. However, then we can decompose $G_i$ further.\n\n\\subsection{Preserving nonzero circulation:}\n\\label{subsec:presnzc}\nWe can show that if we replace a vertex with the gadget $\\gamma$, then the nonzero-circulation in the graph remains preserved: let $G_1(V_1,E_1)$ be a graph such that a vertex $v$ in $G_1$ is replaced with the gadget $\\gamma$ (star graph). Let this new graph be $G_2(V_2,E_2)$. We show that if we have a skew-symmetric weight function $w_2$ that gives nonzero circulation to every cycle in $G_2(V_2,\\dvec{E}_2)$, then we can obtain a skew-symmetric weight function $w_1$ that gives nonzero circulation to every cycle in $G_1(V_1,\\dvec{E}_2)$ as follow. Let $u_1,u_2, \\ldots, u_k$ be the neighbors of $v$ in $G_1$. For the sake of simplicity, assume that $v$ is replaced with $\\gamma$ such that $\\gamma$ has only two leaves $v_1$ and $v_2$ and $v$ is the center of $\\gamma$. Now assume that $u_1,u_2, \\ldots ,u_j$ become neighbors of $v_1$ and, $u_{j+1},u_{j+2}, \\ldots , u_k$ become neighbors of $v_2$ in $G_2$, for some $j<k$. We define a function $P$ that maps each edge of $G_1$ to at most two edges of $G_2$ as follows. For edge $(u_i,v)$ in $G_1$,\n\\[\nP(u_i,v)=\n\\begin{cases}\n\\{(u_i,v_1),(v_1,v) \\},& \\text{ if $u_i$ is a neighbor of $v_1$ in $G_2$, }\\\\\n\\{ (u_i,v_2),(v_2,v) \\},& \\text{ if $u_i$ is a neighbor of $v_2$ in $G_2$. }\n\\end{cases}\n\\]\nFor all the other edges $e$ of $G_1$, $P(e) = \\{e \\in G_2\\}$. Now given weight function $w_2$ for $G_2$, we define weight function $w_1$ for $G_1$ as follows:\n\\begin{align*}\nw_1(e) = \\sum_{e' \\in P(e)} w_2(e')\n\\end{align*}\nFor any ${F} \\subseteq \\dvec{E_1}$, we define $P({F}) = ( P(e) \\mid e \\in {F})$. Let $C$ be a simple cycle in $G_1$. Notice that the set of edges in $P(C)$ form a walk in $G_2$. Also, note that for some edges $e$ of $G_2$ both $e$ and $e^r$ may appear in $P(C)$. Let $\\hat{E}_2(C)$ be the set of edges in $P(C)$ such that for each $e \\in \\hat{E}_2(C)$, both $e$ and $e^r$ appear in $P(C)$. Since our weight function is skew-symmetric, we know that $\\sum_{e \\in \\hat{E}_2(C)}w_2(e) = 0$. Also, notice that set of edges in the set $P(C) - \\hat{E}_2(C)$ form a simple cycle in $G_2$ (proof of this is same as the proof of Claim \\ref{clm:simp}, that we prove later in this paper). Let $C'$ be the simple cycle in $G_2$ formed by the edges in the set $P(C) - \\hat{E}_2(C)$. We know that,  $\\sum_{e \\in C}w_1(e) =  \\sum_{e \\in P(C)}w_2(e) = \\sum_{e \\in C'}w_2(e) + \\sum_{e \\in \\hat{E}_2(C)}w_2(e) $ and since $\\sum_{e \\in \\hat{E}_2(C)}w_2(e) = 0$ we have,  $\\sum_{e \\in C}w_1(e) = \\sum_{e \\in C'}w_2(e).$\n\nTherefore, we can say that if $w_2$ gives nonzero circulation to $C'$, then $w_1$ gives nonzero circulation to $C$. To satisfy property $1$ on the component tree, we replace vertices of the graph with $\\gamma$. Furthermore, to satisfy property $2$, we replace vertices with the gadget $\\beta$, which contains nothing but multiple copies of $\\gamma$. Thus from above, we can conclude that these constructions preserve the nonzero circulation. Now we will work with the graph $G$ and the component tree $T$.\n\\subsection{Tree decomposition}\nNote that the component tree $T$ of $G$ is also a tree decomposition of $G$ in the sense that we can consider the nodes of the component tree as bags of vertices. We know that $T$ contains two types of nodes: (i) nodes that contain planar graphs, (ii) nodes that contain constant tree-width graphs. We call them \\textit{p-type } and \\textit{c-type } nodes, respectively.\n\nNow we will construct another tree decomposition $T'$ of $G$ using the component tree $T$. $T'$ have two types of bags: (i) A bag with respect to each \\textit{p-type } node of $T$, which contains the same set of vertices as the \\textit{p-type } node, and (ii) bags obtained from tree decomposition of the component inside each \\textit{c-type } node. For a node $N$ of $T$, let $G_N$ denote the graph inside node $N$. Let $V(G_N)$ denote the vertices in the graph $G_N$ and $T_N$ be a tree decomposition of $G_N$ obtained using Lemma \\ref{lem:boundtw}.\n\n\\begin{itemize}\n\n\\item Bags of $T'$ are defined as follows: $\\{V({G_N}) \\mid N \\in T, \\text{ where $N$ is a \\textit{p-type } node in $T$} \\}\\; \\bigcup \\; \\{B \\mid B \\in T_{N}, \\text{ where $N$ is a \\textit{c-type } node in $T$}\\}$.\nWe know that in a tree decomposition of a graph $H$, for each clique in $H$, there exists a bag in the tree decomposition of $H$ that contains all the vertices of the clique. Let $\\tau$ be a separating set present in the graph contained in a \\textit{c-type } node $N$ of $T$. While constructing a tree decomposition $T_N$, we consider the virtual clique present among the vertices of $\\tau$ as a part of $G_{N}$. This ensures that there exists a bag in $T_N$ that contains all the vertices of $\\tau$.\n\n\\item Edges in $T'$ are defined as follows: (i) for a $\\textit{c-type }$ node $N$, let $B$ and $B'$ be two bags in $T_N$. If an edge in $T_N$ connects $B$ and $B'$, then add an edge between them in $T'$ as well,(ii) let $N'$ and $N''$ be two adjacent nodes in $T$ such that they share a separating set $\\tau$. As mentioned above, we know that either both $N'$ and $N''$ are \\textit{p-type } or one of them is \\textit{p-type } and the other one is a \\textit{c-type } node. If both of them are \\textit{p-type }: we add an edge between the bags in $T'$ that contain the vertices $V(G_{N'})$ and $V(G_{N''})$. If $N'$ is a \\textit{p-type } and $N''$ is a \\textit{c-type } node: remember that we replaced node $N''$ by its tree decomposition. let $B$ be any bag in $T_{N''}$ which contains all the vertices of $\\tau$. We add an edge between the bag containing vertices $V(G_N)$ and $B$.\n\n\\end{itemize}\n\nIt is easy to see that $T'$ is also a tree decomposition of $G$. From Lemma \\ref{lem:boundtw}, we can say that the overall construction of $T'$ remains in logspace. For simplicity, we rename the bags of $T'$ to $B_1,B_2, \\ldots ,B_k$. Let $B_i$ be a bag in $T'$ corresponding to a node $N$ in $T$. If $N$ contains a separating set $\\tau$, then the set of vertices of $\\tau$ is also called a separating set in $B_i$. We will need this notion later on while constructing the weight function.\n\nNote that Arora et al. \\cite{AGGT16} obtained a component tree $\\hat{T}$ of an input graph $\\hat{G}$ such that each node of $\\hat{T}$ contains either a planar graph or a constant size graph. Then they show that for a cycle $C$ in $\\hat{G}$, the nodes which contain edges of $C$ form a connected component of $\\hat{T}$. They use this property to construct the desired weight function. Here we can say that $T'$ is also a component tree of $G$ such that each node of $T'$ contains a planar graph or a constant size graph, in a sense that we assign each edge $e$ of $G$ to one of the bags of $T'$, which contains both the endpoints of $e$. However, we cannot claim that for a cycle $C$, the bags containing edges of $C$ form a connected component in $T'$ (for example, see Figure \\ref{fig:btw}). Thus in this paper, we use the tree decomposition $T'$ to construct another graph $G'$ and associate edges of $G'$ to the bags to $T'$ such that for each cycle $C'$ in $G'$, the bags which have edges of $C'$ associated with them, form a connected component in $T'$. We show that if we can construct a skew-symmetric weight function for $G'$ such that it gives nonzero circulation to every cycle in $G'$, then we can obtain a skew-symmetric weight function for $G$, which gives nonzero circulations to all the cycles in $G$.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=2]{figures\/Btwtd}\n\\caption{Assume $C$ (left) is a constant treewidth component of $G$. We replace $C$ with its tree decomposition (right) in the component tree $T'$ and associated each edge of $C$ with one of the bags in the tree decomposition, as shown in the figure. Notice that for cycle \\textit{abfghedca} the bags ($B_1,B_2,B_5$ and $B_6$), which have edges of the cycle associated with them, do not form a connected component.}\n\\label{fig:btw}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{Construction of G'}\nWe construct $G'(V',E')$ from the tree decomposition $T'$ of $G(V,E)$ as follows. We borrow notation from Datta et al. \\cite{DKMTVZ20}. Without loss of generality, assume that $T'$ is a rooted tree and the parent-child relationship is well defined.\n\\begin{itemize}\n\\item Vertex set $V' = \\{v_{B_i} \\mid B_i \\in T' , v \\in B_i\\}$, i.e., for each vertex $v$ of the $G$, we have copies of $v$ in $G'$ for each bag $B_i$ of $T'$ in which $v$ appears. Copies of vertices of a separating set are also called a separating set in $G'$.\n\\item Edge set $E' = \\{(u_{B_i},v_{B_i}) \\mid (u,v) \\in E, \\; u \\notin parent(B_i) \\; or \\; v \\notin parent(B_i)\\} \\; \\bigcup \\; \\{(v_{B_i},v_{B_j}) \\mid \\text{$B_i$ and $B_j$ are adjacent in $T'$}\\}$. In other words, we add an edge between the two vertices $u$ and $v$ of same bag $B_i$ if there is an edge between those vertices in $G$ and no ancestor of $B_i$ in $T'$ contains both the vertices $u$ and $v$. We add an edge between two copies of a vertex if the bags they belong to, are adjacent in $T'$.\n\\end{itemize}\n\n\\begin{lemma}\n\\label{lem:wtforg}\nGiven a polynomially bounded, skew-symmetric weight function $w'$ that gives a nonzero circulation to every cycle in $G'$, we can find a polynomially bounded, skew-symmetric weight function $w$ for $G$ that gives a nonzero circulation to every cycle in $G$.\n\\end{lemma}\nProof of Lemma \\ref{lem:wtforg} is given in the Appendix. Now the only thing remaining is the logspace construction of the polynomially bounded skew-symmetric weight function $w'$.\n\n\\subsection*{Constructing Weight Function for G'}\\label{subsec:wtandaux} To construct the weight function $w'$ for $G'$, we associate each edge of $G'$ with some bag of $T'$. Let $(u_{B_i},v_{B_j})$ be an edge in $G'$: (i) if $i=j$, i.e., if $B_i$ and $B_j$ are the same bags. In this case associated $(u_{B_i},v_{B_j})$ with that bag, (ii) if $i \\neq j$: by our construction of $G'$ we know that either $B_i$ is the parent of $B_j$ or $B_j$ is the parent of $B_i$ (i.e. $u_{B_i}$ and $v_{B_j}$ are the copies of a same vertex of $G$). In both the cases, associate $(u_{B_i},v_{B_j})$ with the parent bag. We will use the following claim later in the paper.\n\n\\begin{claim}\n\\label{clm:connect}\nFor any cycle, $C$ in $G'$, the bags of $T'$ which have some edge of $C$ associated with them, form a connected component in $T'$.\n\\end{claim}\n\nThe weight function $w'$ is similar to the one constructed for $K_{3,3}$-free and $K_5$ free graphs \\cite{AGGT16}. We assign weights to the edges of the graph $G'$ depending upon the height of the bag they are associated with. The weights assigned to them are exponential in the height of the bags. Therefore, we need the height of a bag to be $O(\\log n)$ to obtain a polynomially bounded weight function. Hence similar to \\cite{AGGT16}, we define an auxiliary tree $A(T')$ of the tree $T'$. In some sense, $A(T')$ is a balanced representation of $T'$, therefore the height of $A(T')$ is $O(\\log n)$. Nodes in $A(T')$ are the same as $T'$, i.e. the bags of $T'$, but the edges between the bags are inserted differently. The weight of an edge of $G'$ associated with a bag $B$ of $T'$ depends upon the height of the bag $B$ in $A(T')$.\n\n\\subparagraph*{Auxiliary Tree:} In order to construct the auxiliary tree from $T'$, first, we find a node called \\textit{center} node $c(T')$. Make this node the root of the $A(T')$. Let $T_1,T_2, \\ldots ,T_l$ be the subtrees obtained by deleting $c(T')$ from $T'$. Recursively apply the same procedure on these subtrees and make $c(T_1),c(T_2), \\ldots, c(T_l)$ children of $c(T)$. If $c(T)$ shares a separating set $\\tau$ with $T_i$ then $c(T_i)$ is said to be attached at $\\tau$ with $c(T)$. Center nodes are chosen in such a way that the resultant tree $A(T')$ has height $O(\\log n)$. Readers are referred to Section 3.3 of Arora et al. \\cite{AGGT16} to see the logspace construction of $A(T')$. We use the same construction here. The height of the root of $A(T')$ is defined as the number of nodes in the longest path from the root $c(T)$ to a leaf node. The heights of other nodes are one less than that of their parent. From now on, we work with $A(T')$. We use the following two properties of the auxiliary tree.\n\\begin{enumerate}[(i)]\n\\item Height of a node in $A(T')$ is $O(\\log n)$.\n\\item If ${\\tilde{T}}$ is a subtree of $T'$, then there exists a bag $B$ in $\\tilde{T}$ such that all the other bags of $\\tilde{T}$ are descendants of $B$ in $A(T')$.\n\\end{enumerate}\nNow we define the weight function $w'$ for the graph $G'$. Note that the graph induced by the set of edges associated with a bag $B_i$ is either planar or constant size; we call these the components of $G'$. $w'$ is a linear combination of two weight functions $w_1$ and $w_2$. $w_1$ gives nonzero circulation to those cycles which are completely contained within a component, and $w_2$ gives nonzero circulation to those cycles which span over at least two components. We define $w_1$ and $w_2$ separately for planar and constant size components. Let $G'_{B_i}$ be the graph induced by the set of edges associated with the bag $B_i$. Let $K$ be a constant such that $K > max(2^{m+2},7)$, where $m$ is the maximum number of edges associated with any constant size component.\n\n\\subparagraph{ If $G'_{B_i}$ is a planar component:}\n$w_1$ for such components is same as the weight function defined in \\cite{BTV09} for planar graphs. We know that given a planar graph $G$, its planar embedding can be computed in logspace \\cite{AllenderMahajan04}.\n\n\\begin{theorem}[\\cite{BTV09}]\nGiven a planar embedding of a graph $H$, there exists a logspace computable function $w$ such that for every cycle $C$ of $H$, circulation of the cycle $w(C) \\neq 0$.\n\\end{theorem}\nThe above weight function gives nonzero circulation to every cycle that is completely contained in a planar component.\n\nThe weight function $w_2$ for planar components is defined as follows. $w_2$ assigns weights to only those faces of the component, which are adjacent to some separating set. For a subtree of $T_s$ of $A(T')$, let $l(T_s)$ and $r(T_s)$ denote the number of leaf nodes in $T_s$ and root node of $T_s$, respectively. For a bag $B_i$, $h(B_i)$ denotes the height of the bag in $A(T')$. If $B_i$ is the only bag in the subtree rooted at $B_i$, then each face in $G'_{B_i}$ is assigned weight zero. Otherwise, let $\\tau$ be a separating set where some subtree $T_i$ is attached to $B_i$. The faces adjacent to $\\tau$ in $G'_{B_i}$ are assigned weight $2\\times K^{h(r(T_i))} \\times l(T_i)$. If a face is adjacent to more than one separating set, then the weight assigned to the face is the sum of the weights due to each separating set. The weight of a face is defined as the sum of the weights of the edges of the face in clockwise order. If we have a skew-symmetric weight function, then the weight of the clockwise cycle will be the sum of the weights of the faces inside the cycle \\cite{BTV09}. Therefore assigning positive weights to every face inside a cycle will ensure that the circulation of the cycle is nonzero. Given weights on the faces of a graph, we can obtain weights for the edges so that the sum of the weights of the edges of a face remains the same as the weight of the face assigned earlier \\cite{Kor09}.\n\n\n\\subparagraph{If $G'_{B_i}$ is a constant size component:} For this type of component, we need only one weight function. Thus we set $w_2$ to be zero for all the edges in $G'_{B_i}$ and $w_1$ is defined as follows. Let $e_1,e_2, \\ldots ,e_k$ be the edges in the component $Q_i$, for some $k \\leq m$. Edge $e_j$ is assigned weight $2^i \\times K^{h(r(T_i))-1} \\times l(T_{i})$ (for some arbitrarily fixed orientation), Where $T_{i}$ is the subtree of $A(T')$ rooted at $B_i$. Note that for any subset of edges of $G'_{B_i}$, the sum of the weight of the edges in that subset is nonzero with respect to $w_1$.\n\n\nThe final weight function is $w' = \\langle w_1 + w_2 \\rangle.$ Since the maximum height of a bag in $A(T')$ is $O(\\log n)$, the weight of an edge is at most $O( n^c)$, for some constant $c>0$.\n\n\\begin{lemma}\n\\label{lem:maxval}\nFor a cycle $C$ in $G'$ sum of the weights of the edges of $C$ associated with the bags in a subtree $T_i$ of $A(T')$ is $< K^{h(r(T_i))} \\times l(T_i) $.\n\\end{lemma}\n\n\\begin{proof}\nLet $w(C_{T_i})$ denotes the sum of the weight of the edges of a cycle $C$ associated with the bags in $T_i$. We prove the Lemma by induction on the height of the root of the subtrees of $A(T')$. Note that the Lemma holds trivially for the base case when the height of the root of a subtree is $1$.\n\n\\textit{Induction hypothesis}: Assume that it holds for all the subtrees such that the height of their root is $<h(r(T_i))$.\n\nNow we will prove it for $T_i$. Let $T_i^1,T_i^2, \\ldots , T_i^k$ be the subtrees attached to $r(T_i)$.\n\\begin{itemize}\n\\item First, consider the case when $G'_{r(T_i)}$ is a constant size graph: In this case, we know that the sum of the weights of the edges of $C$ associated with $r(T_i)$ is $ \\leq \\sum_{j=1}^m 2^j \\times K^{h(r(T_i))-1} \\times l(T_i) $ and by the induction hypothesis, we know that $w(C_{T_i^j}) < K^{h(r(T_i^j))} \\times l(T_i^j)$, for all $j\\in [k]$. Therefore,\n\\begin{eqnarray*}\nw(C_{T_i}) &\\leq& \\sum_{j=1}^m 2^j \\times K^{h(r(T_i))-1} \\times l(T_i) + \\sum_{j=1}^k K^{h(r(T_i^j))} \\times l(T_i^j) \\\\\nw(C_{T_i}) &\\leq& (2^{m+1}-1) \\times K^{h(r(T_i))-1} \\times l(T_i) + K^{h(r(T_i))-1} \\times l(T_i) \\\\\nw(C_{T_i}) &\\leq & (2^{m+1}) \\times (K^{h(r(T_i))-1} \\times l(T_i)) \\hspace{4cm} [ K > 2^{m+2}] \\\\\nw(C_{T_i}) &<& K^{h(r(T_i))} \\times l(T_i)\n\\end{eqnarray*}\n\n\\item When $G'_{r(T_i)}$ is a planar graph: let $\\tau_1,\\tau_2, \\ldots ,\\tau_k$ be the separating sets present in $G'_{r(T_i)}$ such that the subtree $T_i^j$ is attached to $r(T_i)$ at $\\tau_j$, for all $j \\in [k]$. A separating set can be present in at most 3 faces. Thus it can contribute $2 \\times 3 \\times K^{h(r(T_i^j))} \\times l(T_i^j)$ to the circulation of the cycle $C$. Therefore,\n\\begin{eqnarray*}\nw(C_{T_i}) &\\leq & \\sum_{j=1}^k 6 \\times K^{h(r(T_i^j))} \\times l(T_i^j) + \\sum_{j=1}^k K^{h(r(T_i^j))} \\times l(T_i^j) \\\\\nw(C_{T_i}) &\\leq& 7 \\times K^{h(r(T_i))-1} \\times l(T_i)\\hspace{5cm} [K > 7] \\\\\nw(C_{T_i}) &<& K^{h(r(T_i))} \\times l(T_i)\n\\end{eqnarray*}\n\\end{itemize}\n\\end{proof}\n\n\n\\begin{lemma}\n\\label{lem:domwt}\nFor a cycle, $C$ in $G$ let $B_i$ be the unique highest bag in $A(T')$ that have some edges of $C$ associated with it. Then the sum of the weights of the edges of $C$ associated with $B_i$ will be more than that of the rest of the edges of $C$ associated with the other bags.\n\\end{lemma}\n\n\\begin{proof}\nLet $T_i$ be the subtree of $A(T')$ rooted at $B_i$. We know that sum of the weights of the edges of $C$ associated $B_i$ is $\\geq 2 \\times K^{h(r(T_i))-1} \\times l(T_i)$. Let $T_i^1, T_i^2 , \\ldots ,T_i^k$ be the subtree of $T_i$ rooted at children of $B_i$. By Lemma \\ref{lem:maxval}, we know that the sum of the weight of the edges of $C$ associated with the bags in these subtrees is $< \\sum_{j=1}^k K^{h(r(T_i^j))} \\times l(T_i^j) = K^{h(r(T_i))-1} \\times l(T_i)$. Therefore, the lemma follows.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lem:nonzerowt}\nCirculation of a simple cycle $C$ in the graph $G'$ is nonzero with respect to $w'$.\n\\end{lemma}\n\n\\begin{proof}\nIf $C$ is contained within a component, i.e., its edges are associated with a single bag $B_i$, then we know that $w_1$ assigns nonzero circulation to $C$. Suppose the edges of $C$ are associated with more than one bag in $T'$. By Claim \\ref{clm:connect}, we know that these bags form a connected component. By the (ii) property of $A(T')$, we know that there is a unique highest bag $B_i$ in $A(T')$ which have edges of $C$ associated with it. Therefore from Lemma \\ref{lem:domwt} we know that the circulation of $C$ will be nonzero.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:main}]\nProof of Theorem \\ref{thm:main} follows from Lemma \\ref{lem:wtforg} and \\ref{lem:nonzerowt}.\n\\end{proof}\n\n\\section{Conclusion}\n\\label{sec:concl}\nWe have given a construction of a nonzero circulation weight function for the class of graphs that can be expressed as 3-clique-sums of planar and constant treewidth graphs. However, it seems that our technique can be extended to the class of graphs that can be expressed as 3-clique-sums of constant genus and constant treewidth graphs. Further extending our results to larger graph classes would require fundamentally new techniques. This is so because the most significant bottleneck in parallelizing matching algorithms for larger graph classes such as apex minor free graphs or $H$-minor free graphs for a finite $H$ is the absence of a parallel algorithm for the structural decomposition of such families. Thus we would need to\nrevisit the Robertson-Seymour graph minor theory to parallelize it. This paper thus serves the dual purpose of delineating the boundaries of the known regions of parallel (bipartite) matching and reachability and as an invitation to the vast unknown of parallelizing the Robertson-Seymour structure theorems.\n\\section{Introduction}\nDirected graph reachability and perfect matching are two fundamental problems in computer science. The history of the two problems has been inextricably linked together from the inception of computer science (and before!) \\cite{FordF56}. The problems and their variants, such as shortest path \\cite{Dijkstra59} and maximum matching \\cite{Edmonds65} have classically been studied in the sequential model of computation. Since the 1980s, considerable efforts have been spent trying to find parallel algorithms for matching problems spurred on by the connection to reachability which is, of course, parallelizable.\nThe effort succeeded only in part with the discovery of randomized parallel algorithms \\cite{KUW85,MulmuleyVV87}. While we know that the reachability problem is complete for the complexity class $\\NL$, precise characterization has proved to be elusive for matching problems. The 1990s saw attempts in this\ndirection when surprisingly ``small'' upper bounds were proved \\cite{ARZ99}\nfor the perfect matching problem, although in the non-uniform setting.\nAt roughly the same time, parallel algorithms for various\nversions of the matching problem for restricted graph classes like planar\n\\cite{MN95} and bounded genus \\cite{MV00} graphs were\ndiscovered. The last two decades have seen efforts towards pinning down the\nexact\nparallel complexity of reachability and matching related problems in restricted\ngraph classes \\cite{BTV09,KV10,DKR10,DKTV11,AGGT16,KT16,GST19,GST20}.\nMost of these papers are based on the method of constructing \\emph{nonzero circulations}.\n\nThe circulation of a simple cycle is the sum of its edge-weights in a\nfixed orientation (see Section~\\ref{sec:prelims} for the definition)\nand\nwe wish to assign polynomially bounded weights to the edges of a graph, such\nthat every simple cycle has a nonzero circulation.\nAssigning such weights \\emph{isolates} a reachability witness or a matching witness in the graph \\cite{TV12}. Constructing\npolynomially bounded isolating weight function in parallel for general graphs has been\nelusive so far.\nThe last five years\nhave seen rapid progress in the realm of matching problems, starting with\n\\cite{FGT} which showed that the method of nonzero circulations could\nbe extended from topologically restricted (bipartite) graphs to general\n(bipartite) graphs. A subsequent result extended this to all graphs\n\\cite{ST17}. More recently, the endeavour to parallelize planar\nperfect matching has borne fruit \\cite{Sankowski18,AV20} and has been followed up by further exciting work \\cite{AV2}. \n\nWe know that polynomially bounded weight functions that give nonzero circulation to every cycle can be constructed in logspace for planar graphs, bounded genus graphs and bounded treewidth graphs \\cite{BTV09,DKTV11,DKMTVZ20} . Planar graphs are both $K_{3,3}$-free and $K_5$-free graphs. Such a weight function is also known to be constructable in logspace for $K_{3,3}$-free graphs and $K_5$-free graphs, individually \\cite{AGGT16}. A natural question arises if we can construct such a weight function for $H$-minor-free graphs for any arbitrary graph $H$. A major hurdle in this direction is the absence of a space-efficient (Logspace) or parallel algorithm (\\NC) for finding a structural decomposition of $H$-minor free graphs. However, such a decomposition is known when $H$ is a single crossing graph. This induces us to solve the problem for single crossing minor-free (SCM-free) graphs. An SCM-free graph can be decomposed into planar and bounded treewidth graphs. Moreover, $K_{3,3}$ and $K_5$ are single crossing graphs. Hence our result can also be seen as a generalization of the previous results on these classes. There have also been important follow-up works on parallel algorithms for SCM-free graphs \\cite{EV21}. SCM-free graphs have been studied in several algorithmic works (for example \\cite{STW16,CE13,DHNRT04}).\n\n\n\\subsection{Our Result}\nIn this paper, we show that results for previously studied graph classes\n(planar, constant tree-width and $H$-minor free for $H \\in \\{K_{3,3},K_5\\}$)\ncan be extended and unified to yield similar results for SCM-free graphs.\n\\begin{theorem}\n\\label{thm:main}\nThere is a logspace algorithm for computing polynomially-bounded, skew-symmetric nonzero circulation weight function in SCM-free graphs.\n\\end{theorem}\n\nAn efficient solution to the circulation problem for a class of graphs yields better complexity bounds for determining reachability in the directed version of that class and constructing minimum weight maximum-matching in the bipartite version of that class. Theorem~\\ref{thm:main}\nwith the results of \\cite{DKKM18,RA00}, yields the following:\n\\begin{corollary}\\label{cor:stat}\nFor SCM-free graphs, reachability is in $\\UL\\cap\\coUL$ and minimum weight bipartite maximum matching is in $\\SPL$.\n\\end{corollary}\n\nAlso using the result of \\cite{TW10}, we obtain that the \\textit{Shortest path} problem in SCM-free graphs can be solved in $\\UL\\cap \\coUL$.  \n\\subparagraph*{Overview of Our Techniques and Comparison With Previous Results:} We know that for planar graphs and constant treewidth graphs nonzero circulation weights can be constructed in logspace \\cite{BTV09,DKMTVZ20}. We combine these weight functions using the techniques from Arora et al. \\cite{AGGT16}, Datta et al. \\cite{DKKM18} and, Datta et al. \\cite{DKMTVZ20} together with some modifications to obtain the desired weight function. In \\cite{AGGT16}, the authors decompose the given input graph $G$ ($K_{3,3}$-free or $K_5$-free) and obtain a component tree that contains planar and constant size components. They modify the components of the component tree so that they satisfy few properties which they use for constructing nonzero circulation weights (these properties are mentioned at the beginning of Section \\ref{sec:wtfn}). The new graph represented by these modified components preserves the perfect matchings of $G$. Then, they construct a \\emph{working-tree} of height $O(\\log n)$ corresponding to this component tree and use it to assign nonzero circulation weights to the edges of this new graph. The value of the weights assigned to the edges of the new graph is exponential in the height of the working tree.\n\nWhile $K_{3,3}$-free and $K_5$-free graphs can be decomposed into planar and constant size components, an SCM-free graph can be decomposed into planar and constant treewidth components. Thus the component tree of the SCM-free graph would have several non-planar constant treewidth components. While we can construct a working tree of height $O(\\log n)$, this tree would contain constant-treewidth components and hence make it difficult to find nonzero circulation weights. A na\\\"ive idea would be to replace each constant treewidth component with its tree decomposition in the working tree. However, the resultant tree would have the height $O(\\log^2 n)$. Thus the weight function obtained in this way is of $O(\\log^2 n)$-bit. We circumvent this problem as follows: we obtain a component tree $T$ of the given SCM-free graph $G$ and modify its components to satisfy the same property as \\cite{AGGT16} (however, we use different gadgets for modification). Now we replace each bounded treewidth component with its tree decomposition in $T$. Using this new component tree, say $T'$, we define another graph $G'$. We use the technique from \\cite{DKKM18} to show that if we can construct the nonzero circulation for $G'$, then we can \\textit{pull back} nonzero circulation for $G$. Few points to note here: (i) pull back technique works because of the new gadget that we use to modify the components in $T$, (ii) since ultimately we can obtain nonzero circulation for $G$, it allows us to compute maximum matching in $G$ in $\\SPL$, which is not the case in \\cite{AGGT16}.\n \n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Organization of the Paper}\nAfter introducing the definitions and preliminaries in Section~\\ref{sec:prelims}, in Section~\\ref{sec:wtfn} we discuss the weight function that achieves non-zero circulation in single-crossing minor free graphs and its application to maximum matching in Section~\\ref{sec:staticmatch}. Finally, we conclude with Section~\\ref{sec:concl}.\n\n\n\n\\section{Dynamic isolation from static non-zero circulation}\\label{sec:iso}\nIn  previous sections, we have shown that we can compute a non-zero\ncirculation (for Matchings) in a bipartite single crossing minor free graph \nin $\\L$. This allows us to compute if the graph has a perfect matching \nin $\\SPL \\subseteq \\NC$.\n\nIn the dynamic setting, we allow the graph to evolve by insertion and \ndeletion of a small number of edges at every time step under the promise \nthat the graph stays $H$-minor free for a fixed single crossing graph $H$. \nWe would like to maintain if the graph has a perfect matching in $\\DynFO$.\nOne plausible approach would be to maintain the non-zero circulation as\ndescribed in Section~\\ref{sec:wtfn}.\nHowever, this seems non-trivial as we would need to maintain the decomposition\nof the given graph into planar and constant tree-width parts in $\\DynFO$. Even\nif we could somehow do that, the weight of many edges may change due to \neven a single edge change. Thus many more than polylog many entries of the\nadjacency and Tutte matrices may change. This would preclude the use of ``small\nchange'' techniques like the Sherman-Morrison-Woodbury formula.\nThis induces us to side-step the update of the non-zero\ncirculation by the method of Section~\\ref{sec:wtfn}.\n\nInstead we use a result from \\cite{FGT} to convert the given static circulation\nto  dynamic isolating weights. Notice that \\cite{FGT} yields a black box recipe\nto produce isolating weights of quasipolynomial magnitude in the following way.\nGiven a bipartite graph $G$, they first consider a non-zero circulation\nof exponential magnitude viz. $w_0(e) : e \\mapsto 2^e$. Next, they consider a \nlist of $\\ell = O(\\log{n})$ primes $\\vec{p} = \\left<p_,\\ldots,p_{\\ell}\\right>$ \nwhich yield a weight function $w_{\\vec{p}}(e)$. This is defined\nby considering the $\\ell$ weight functions \n$w_0\\bmod{p_i}$ for $i \\in \\{1,\\ldots,\\ell\\}$\nand concatenating them after introducing a \\emph{shift} in the bit\npositions i.e.:\n$w_{\\vec{p}}(e) = \\left<w_0(e) \\bmod{p_1},\\ldots,w_0(e) \\bmod{p_\\ell}\\right>$.\nThis is so that there is no\noverflow from the $i$-th field to the $(i-1)$-st for any \n$i \\in \\{2,\\ldots,\\ell\\}$. \n\nSuppose we start with a graph with static weights ensuring non-zero \ncirculation. In a step, some edges are \ninserted or deleted.\nThe graph after deletion is a subgraph of the original graph; hence the\nnon-zero circulation remains non-zero after a deletion\\footnote{If we merely had isolating\nweights this would not necessarily preserve isolation.}, but we have to do more\nin the case of insertions. We aim to give the newly inserted edges FGT-weights\n (from \\cite{FGT})\nin the higher order bits while giving weight $0$ to all the original edges in \n$G$ again in the higher order bits. Thus the weight of all perfect matchings\nthat survive the deletions in a step remains unchanged. Moreover, if none such\nsurvive but new perfect matchings are introduced (due to insertion of edges)\nthe lightest of them is determined solely by the weights of the newly\nintroduced edges. In this case, our modification of the existential proof \nfrom \\cite{FGT} ensures that the minimum weight perfect matching is unique.\n\nNotice that the FGT-recipe applied to a graph with polylog ($N = \\log^{O(1)}{n}$)\nedges yields\nquasipolylogarithmically ($N^{\\log^{O(1)}{N}}$) \nlarge weights which are therefore still subpolynomial ($2^{(\\log{\\log{n}})^{O(1)}} = 2^{o(\\log{n})} = n^{o(1)}$).\nThus the weights remain polynomial when shifted to accommodate for the old \nweights. Further the number of primes is polyloglog ($\\log^{O(1)}{N} = (\\log{\\log{n}})^{O(1)}$) and so sublogarithmic ($= \\log^{o(1)}{n}$)\nthus the number of possible different weights is subpolynomial hence our \nalgorithm can be derandomized.\n\nBefore getting into technical details, we point out that in \\cite{DKMTVZ20} a\nsimilar scheme is used for reachability and bears the same relation to\n\\cite{KT16} as this section does to \\cite{FGT}. We have the following lemma\n, which we prove in Appendix~\\ref{subsec:isoApp}:\n\\begin{lemma} \\label{lem:combFGT}\nLet $G$ be a bipartite graph with a non-zero circulation\n$w$. Suppose $N = \\log^{O(1)}{n}$ edges are inserted into $G$ to yield $G^{new}$\nthen we can compute polynomially many weight functions in $\\FOar$ that\nhave $O(\\log{n})$ bit weights, and at least one of them,\n$w^{new}$ is isolating. Further the weights of the original edges remain\nunchanged under $w^{new}$.\n\\end{lemma}\n\n\n\\section{The details from Section~\\ref{sec:iso}}\\label{subsec:isoApp}\nWe use the same general strategy as in \\cite{DKMTVZ20} and divide the edges into\n\\emph{real} and \\emph{fictitious} where the former represent the newly inserted\nedges and the latter original undeleted edges\\footnote{We use the terms\nold $\\leftrightarrow$ fictitious and new $\\leftrightarrow$ real interchangeably\nin this section.}.\n\nLet $\\mathcal{C}$ be a set of cycles containing both real and fictitious edges that \noccur in any PM.\nLet $w$ be a  weight \nfunction on the edges that gives non-zero weight only to the real edges.\nDefine $c_w(\\mathcal{C})$ to be the set of all circulations in the cycles of \n$\\mathcal{C}$. Here the circulation is the absolute value of the alternating sum\nof weights $c_w(C) = |w(f_1) - w(f_2) + w(f_3) - \\ldots|$ where \n$C = f_1,f_2,f_3,\\ldots$ is the sequence of edges in the cycle. \n\nWe say that a weight function,\nthat gives non-zero weights to the real edges, \\emph{real isolates} $\\mathcal{M}$ \nfor a set system $\\mathcal{M}$ if the minimum weight set in $\\mathcal{M}$ is unique. In our context $\\mathcal{M}$ will refer to the set of perfect\/maximum\n matchings.\n\nNext we follow the proof idea of \\cite{FGT} but focus on assigning weights\nto real edges which are, say, $N$ in number.\n We do this in $\\log{N}$ stages starting with a graph $G_0 = G$\nand ending with the acyclic graph $G_\\ell$ where $\\ell = \\log{N}$. The inductive\nassumption is that:\n\\begin{invariant}\\label{inv:analogFGT}\n For $i \\geq 1$, $G_i$ contains no cycles with at most $2^{i+1}$ real edges. \n\\end{invariant}\n\n\nNotice that induction starts at $i > 0$.\n\nWe first show how to construct $G_{i+1}$ from $G_i$ such that if $G_i$ satisfies\nthe inductive invariant~\\ref{inv:analogFGT} then so does $G_{i+1}$.\nLet $i > 1$, then in the $i$-th stage, let $\\mathcal{C}_{i+1}$ be the set of cycles that contain \nat most $2^{i+2}$ real edges. For each such cycle \n$C = f_0,f_1,\\ldots$ containing $k \\leq 2^{i+2}$ real edges (with $f_0$ being the\nleast numbered real edge in the cycle) edge-partition it into $4$ consecutive \npaths $P_j(C)$ for $j \\in \\{0,1,2,3\\}$ such that the first $3$ paths contain exactly \n$\\lfloor\\frac{k}{4}\\rfloor$ real edges and the last path contains the rest. In\naddition ensure that the first edge in each path is a real edge. Let the first\nedge of the $4$-paths be respectively $f_0 = f'_0, f'_1, f'_2, f'_3$. We \nhave the following which shows that the associated \n$4$-tuples $\\left<f'_0,f'_1,f'_2,f'_3\\right>$\nuniquely characterise cycles in $\\mathcal{C}_{i+1}$.\n\\begin{claim}\nThere is at most one cycle in $\\mathcal{C}_{i+1}$ that has a given $4$-tuple \n$\\left<f'_0,f'_1,f'_2,f'_3\\right>$ associated with it.\n\\end{claim}\n\\begin{proof}\nSuppose two distinct cycles $C,C' \\in \\mathcal{C}_{i+1}$ have\na tuple $\\left<f'_0,f'_1,f'_2,f'_3\\right>$ \nassociated with them. Then for least one $j \\in \\{0,1,2,3\\}$\n$P_j(C) \\neq P_j(C')$. Then $P_j(C) \\cup P_j(C')$\nis a closed walk in $G_i$ containing at most \n$2\\times \\lceil\\frac{2^{i+2}}{4}\\rceil = 2^{i+1}$ many real edges,\ncontradicting the assumption on $G_i$.\n\\end{proof}\n\nThis lemma shows that there are at most $N^4$ elements in $\\mathcal{C}_i$.\nNext consider the following lemma from \\cite{FKS}:\n\\begin{lemma}[\\cite{FKS}]\nFor every constant $c>0$ there is a constant $c_0>0$ such that for\nevery set $S$ of $m$ bit integers with $|S| \\leq m^c$,\nthe following holds: There is a $ c_0 \\log{m}$  bit prime\nnumber $p$ such that for any $x,y \\in S$ it holds that if $x \\neq y$ then \n$x \\not\\equiv y \\bmod{p}$.\n\\end{lemma}\nWe apply it to the set $c_{w_0}(\\mathcal{C}_i) = \\{c_{w_0}(C) : C \\in \\mathcal{C}_i\\}$. \nHere, the weight\nfunction $w_0$ assigns weights $w_0(e_j) = 2^j$ to the real edges which are\n$e_1,e_2,\\ldots,e_N$ in an arbitrary but fixed order.\nNotice that from the above claim, the size of this set \n$|w_0(\\mathcal{C}_i)| \\leq N^4$.\nAnd $w_0(e_j)$ is $j$-bits long hence $c_{w_0}(C)$ for any cycle\n$C \\in \\mathcal{C}_i$ that has less than $2^{i+2}$ real edges \nis at most $i+j+2 < 4N$-bits long.  Thus, we obtain a prime $p_{i+1}$ \nof length at most $c_0\\log{4N}$ by picking $c = 4$. We define \n$w_{i+1}(e_j) = w_0(e_j) \\bmod{p_{i+1}}$.\n\nNow consider the following crucial lemma from \\cite{FGT}:\n\\begin{lemma}[\\cite{FGT}]\\label{lem:crucialFGT}\nLet $G = (V,E)$ be a bipartite graph with weight function $w$. \nLet $C$ be a cycle in $G$ such that $c_w(C) \\neq 0$. \nLet $E_1$ be the union of all minimum weight perfect matchings in $G$.\nThen the graph $G_1(V,E_1)$ does not contain the cycle $C$.\nMoreover all the perfect matchings in $G_1$ have the same weight.\n\\end{lemma}\nLet $B$ be a large enough constant (though bounded by a polynomial in $N$) \nto be specified later.\nWe shift the original accumulated weight function\n$W_i$ and add the new weight function $w_{i+1}$ to obtain:\n$W_{i+1}(e) = W_{i}(e)B + w_{i+1}(e)$. \nApply $W_{i+1}$ on the graph $G_i$ to obtain the graph $G_{i+1}$.\n Inductively suppose we have\n the invariant~\\ref{inv:analogFGT}\n that  the graph $G_i$ did not have any cycles containing at least \n$2^{i+1}$ real edges. This property is preserved when we take all the\nperfect matchings in $G_i$ and apply $W_{i+1}$ yielding $G_{i+1}$. Moreover \nfrom Lemma~\\ref{lem:crucialFGT} and the construction of $w_{i+1}$ the cycles of\n$\\mathcal{C}_i$ disappear from $G_{i+1}$ restoring the invariant. \n\nNotice that it suffices to take $B$ greater than the number of real edges\ntimes the maximum of $w_i(e)$ over $i,e$.\nShowing that $G_1$ contains no cycle of length at most $4$ mimics the above\nmore general proof and we skip it here.\nWe can now complete the proof of Lemma~\\ref{lem:combFGT}:\n\\begin{lemma*} (Lemma~\\ref{lem:combFGT} restated)\nLet $G$ be a bipartite graph with a non-zero circulation\n$w^{old}$. Suppose $N = \\log^{O(1)}{n}$ edges are inserted into $G$ to yield $G^{new}$\nthen we can compute polynomially many weight functions in $\\FOar$ that\nhave $O(\\log{n})$ bit weights, and at least one of them,\n$w^{new}$ is isolating. Further the weights of the original edges remains\nunchanged under $w^{new}$.\n\\end{lemma*}\n\n\\begin{proof}(of Lemma~\\ref{lem:combFGT})\nFrom the invariant above $G_\\ell$ does not contain\nany cycles. From the construction of $G_\\ell$ if $G$ has a perfect matching then\nso does $G_\\ell$ and hence it is a perfect matching. Notice that $W_\\ell$\nis obtained from $p_1,\\ldots,p_\\ell$ that include $O((\\log{\\log{n}})^2) = o(\\log{n})$ \nmany bits. Thus there are (sub)polynomially many such weighting functions $W_\\ell$, depending on the primes $\\vec{p}$.\nLet $w = B\\cdot W_\\ell + w^{old}$ where we recall that $W_\\ell(e)$ is non-zero only\nfor the new (real) edges and $w^{old}$ is non-zero only for the old (fictitious)\nedges. Thus, any perfect matching that consists of only old edges is lighter\nthan any perfect matching containing at least one new edge. Moreover if \nthe real edges in two matchings differ then from the construction of \n$W_\\ell$ (for some choice of $\\vec{p}$) both matchings cannot be lightest \nas $W_\\ell$ real isolates a matching. Thus the only remaining case is\nthat we have two distinct lightest perfect matchings which differ only in the\nold edges. But the symmetric difference of any two such perfect matchings\nis a collection of cycles consisting of old edges. But each cycle has a\nnon-zero circulation in the old graph and so we can obtain a matching\nof even lesser weight by replacing the edges of one of the matchings in one\ncycle by the edges of the other one. This contradicts that both matchings were\nof least weight. This completes the proof.\n\\end{proof}\n\n\n\\subsection{Language Definitions}\\label{subsec:langDefs}\n\\begin{definition}\n\\begin{itemize}\n\\item $\\PM$ is a set of independent edges covering all the vertices of a graph. \n\\item $\\BPM$ is a $\\PM$ in a bipartite graph.\n\\item $\\PMD$ Given an undirected graph $G(V,E)$ determine if there exists a \nperfect matching in $G$.\n\\item $\\PMS$ Construct a perfect matching for a given graph $G$ if one exists.\n\\item $\\BMWPMS$ In an edge  weighted bipartite graph construct a $\\PM$ of least weight.\n\\item $\\MCM$ is a set of largest size consisting of independent edges.\n\\item $\\BMCM$ is an $\\MCM$ in a bipartite graph.\n\\item $\\MWMCM$ is an $\\MCM$ of least weight in an edge weighted graph.\n\\item $\\BMWMCM$ is an $\\MWMCM$ in a weighted bipartite graph.\n\\item $\\MCMSz$ determine the size of an $\\MCM$.\n\\item $\\BMCMSz$ the $\\MCMSz$ problem in bipartite graphs.\n\\item $\\BMWMCMS$ Construct a $\\BMWMCM$. \n\\item $\\Reach$ Given a directed graph $G(V,E)$ and $s,t \\in V$ does there exist a directed path from $s$ to $t$ in $G$.\n\\item $\\Dist$ Given a directed graph $G(V,E)$, polynomially bounded edge weights and $s,t \\in V$ find the weight of a least weight path from $s$ to $t$.\n\\item $\\Rank$ Given an $m \\times n$  matrix $A$ with integer entries find \nthe rank of $A$ over $\\mathbb{Q}$.\n\\end{itemize}\n\\end{definition}\n\n\n\\section{Preliminaries and Notations}\n\\label{sec:prelims}\n\\subparagraph*{Tree decomposition:} Tree decomposition is a well-studied concept in graph theory. Tree decomposition of a graph, in some sense, reveals the information of how much tree-like the graph is. We use the following definition of tree decomposition.\n\n\\begin{definition}\n\\label{def:treedec}\nLet $G(V,E)$ be a graph and $\\tilde{T}$ be a tree, where nodes of the $\\tilde{T}$ are $\\{B_1, \\ldots ,B_k \\mid B_i \\subseteq V\\}$ (called bags). $T$ is called a tree decomposition of $G$ if the following three properties are satisfied:\n\\begin{itemize}\n\n\\item $B_1 \\cup \\ldots \\cup B_k = V$,\n\\item for every edge $(u,v) \\in E$, there exists a bag $B_i$ which contains both the vertices $u$ and $v$,\n\\item for a vertex $v \\in V$, the bags which contain the vertex $v$ form a connected component in $\\tilde{T}$.\n\\end{itemize}\n\n\\end{definition}\n\nThe width of a tree decomposition is defined as one less than the size of the largest bag. The treewidth of a graph $G$ is the minimum width among all possible tree decompositions of $G$. Given a constant treewidth graph $G$, we can find its tree decomposition $\\tilde{T}$ in logspace such that $\\tilde{T}$ has a constant width \\cite{EJT10}.\n\n\\begin{lemma}[\\cite{EJT10}]\n\\label{lem:boundtw}\nFor every constant $c$, there is a logspace algorithm that takes a graph as input and outputs its tree decomposition of treewidth at most $c$, if such a decomposition exists.\n\\end{lemma}\n\n\\begin{definition}\n\\label{def:cls}\nLet $G_1$ and $G_2$ be two graphs containing cliques of equal size. Then the \\emph{clique-sum} of $G_1$ and $G_2$ is formed from their disjoint union by identifying pairs of vertices in these two cliques to form a single shared clique, and then possibly deleting some of the clique edges.\n\\end{definition}\nFor a constant $k$, a $k$-clique-sum is a clique-sum in which both cliques have at most $k$ vertices. One may also form clique-sums of more than two graphs by repeated application of the two-graph clique-sum operation. For a constant $W$, we use the notation $ \\langle \\mathcal{G}_{P, W}\\rangle_k$ to denote the class of graphs that can be obtained by taking repetitive $k$-clique-sum of planar graphs and graphs of treewidth at most $W$. In this paper, we construct a polynomially bounded skew-symmetric weight function that gives nonzero circulation to all the cycles in a graph $G \\in \\langle \\mathcal{G}_{P, W}\\rangle_3$. Note that if a weight function gives nonzero circulations to all the cycles in the biconnected components of $G$, it will give nonzero circulation to all the cycles in $G$ because no simple cycle can be a part of two different biconnected components of $G$. We can find all the biconnected components of $G$ in logspace by finding all the articulation points. Therefore, without loss of generality, assume that $G$ is biconnected.\n\n\nThe \\emph{crossing number} of a graph $G$ is the lowest number of edge crossings of a plane drawing of $G$. A \\emph{single-crossing} graph is a graph whose crossing number is at most 1. SCM-free graphs are graphs that do not contain $H$ as a minor, where $H$ is a fixed single crossing graph. Robertson and Seymour have given the following characterization of SCM-free graphs.\n\n\\begin{theorem}[\\cite{RS91}]\n\\label{thm:rs}\nFor any single-crossing graph $H$, there is an integer\n$c_H \\geq 4$ (depending only on $H$) such that every graph with no minor isomorphic to $H$ can\nbe obtained as $3$-clique-sum of planar graphs and graphs of treewidth at most $c_H$.\n\\end{theorem}\n\n\\subparagraph*{Component Tree:} In order to construct the desired weight function for a graph $G \\in \\langle \\mathcal{G}_{P, W}\\rangle_3$, we decompose $G$ into smaller graphs and obtain a component tree of $G$ defined as follows: we first find $3$-connected and $4$-connected components of $G$ such that each of these components is either planar or of constant treewidth. We know that these components can be obtained in logspace \\cite{TW09}. Since $G$ can be formed by taking repetitive $3$-clique-sum of these components, the set of vertices involved in a clique-sum is called a separating set. Using these components and separating sets, we define a component tree of $G$. A component tree $T$ of $G$ is a tree such that each node of $T$ contains a $3$-connected or $4$-connected component of $G$, i.e., each node contains either a planar or constant treewidth subgraph of $G$. There is an edge between two nodes of $T$ if the corresponding components are involved in a clique-sum operation. If two nodes are involved in a clique-sum operation, then copies of all the vertices of the clique are present in both components. It is easy to see that $T$ will always be a tree. Within a component, there are two types of edges present, \\textit{real} and \\textit{virtual edges}. Real edges are those edges that are present in $G$. Let $\\{a,b,c\\}$(or $\\{a,b\\}$) be a separating triplet(or pair) shared by two nodes of $T$, then there is a clique $\\{a,b,c\\}$ (or $\\{a,b\\}$) of virtual edges present in both the components. Suppose there is an edge present in $G$ between any pair of vertices of a separating set. In that case, there is a real edge present between that pair of vertices parallel to the virtual edge, in exactly one of the components which share that separating set.\n\n\\subparagraph*{Weight function and circulation:} Let $G(V, E)$ be an undirected graph with vertex set $V$ and edge set $E$. By $\\dvec{E}$, we denote the set of bidirected edges corresponding to $E$. Similarly, by $G(V, \\dvec{E})$, we denote the graph corresponding to $G(V, E)$ where each of its edges is replaced by a corresponding bidirected edge. A weight function $w : \\dvec{E} \\rightarrow \\mathbb{Z}$ is called skew-symmetric if for all $e\\in \\dvec{E}$, $w(e) = -w(e^r)$ (where $e^r$ represent the edge with its direction reversed). We know that if $w$ gives nonzero circulation to every cycle that consists of edges of $\\dvec{E}$ then it isolates a directed path between each pair of vertices in $G(V,\\dvec{E})$. Also, if $G$ is a bipartite graph, then the weight function $w$ can be used to construct a weight function $w^{\\textrm{{\\tiny{und}}}} : E \\rightarrow \\mathbb{Z}$ that isolates a perfect matching in $G$ \\cite{TV12}.\n\n\nA convention is to represent by $\\left<w_1,\\ldots,w_k\\right>$ the weight function that on edge $e$ takes the weight $\\sum_{i=1}^k{w_i(e)B^{k-i}}$ where $w_1,\\ldots,w_k$ are weight functions such that $\\max_{i=1}^k{(nw_i(e))} \\leq B$.\n\n\\subparagraph*{Complexity Classes:} The complexity classes \\Log\\ and \\NL\\ are the classes of languages accepted by deterministic and non-deterministic logspace Turing machines, respectively. $\\UL$ is a class of languages that can be accepted by an $\\NL$ machine that has at most one accepting path on each input, and hence $\\UL \\subseteq \\NL$. $\\SPL$ is the class of languages whose characteristic function can be written as a logspace computable integer determinant. \n\n\n\\section{Maximum Matching}\\label{sec:staticmatch}\nIn this section, we consider the complexity of the maximum matching problem in single crossing minor free graphs. Recently Datta et al.~\\cite{DKKM18} have shown that the bipartite maximum matching can be solved in \\SPL\\ in the planar, bounded genus, $K_{3,3}$-free and $K_{5}$-free graphs.\n\nTheir techniques can be extended to any graph class where nonzero circulation weights can be assigned in logspace. For constructing a maximum matching in $K_{3,3}$-free and $K_{5}$-free bipartite graphs, they use the logspace algorithm of~\\cite{AGGT16} as a black box. Since from Theorem~\\ref{thm:main} nonzero circulation weights can be computed for the more general class of any single crossing minor free graphs, we get the bipartite maximum matching result of Corollary~\\ref{cor:stat}.\n\nIn a related work recently, Eppstein and Vazirani~\\cite{EV21} have shown an $\\NC$ algorithm for the case when the graph is not necessarily bipartite. However, the result holds only for constructing perfect matchings. In non-bipartite graphs, there is no known parallel (e.g., \\NC) or space-efficient algorithm for deterministically constructing a maximum matching even in the case of planar graphs~\\cite{Sankowski18, DKKM18}. Datta et al.~\\cite{DKKM18} givelo an approach to design a \\emph{pseudo-deterministic} \\NC\\ algorithm for this problem. Pseudo-deterministic algorithms are probabilistic algorithms for search problems that produce a unique output for each given input with high probability. That is, they return the same output for all but a few of the possible random choices. We call an algorithm pseudo-deterministic $\\NC$ if it runs in $\\RNC$ and is pseudo-deterministic.\n\nUsing the Gallai-Edmonds decomposition theorem, \\cite{DKKM18} shows that the search version of the maximum matching problem reduces to determining the size of the maximum matching in the presence of algorithms to (a) find a perfect matching and to (b) solve the bipartite version of the maximum matching, all in the same class of graphs. This reduction implies a pseudo-deterministic $\\NC$ algorithm as we only need to use randomization for determining the size of the matching, which always returns the same result. For single crossing minor free graphs, using the $\\NC$ algorithm of \\cite{EV21} for finding a perfect matching and our $\\SPL$ algorithm for finding a maximum matching in bipartite graphs, we have the following result:\n\\begin{theorem}\nMaximum matching in single-crossing minor free graphs (not necessarily bipartite) is in pseudo-deterministic \\NC.\n\\end{theorem}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\nSolar coronal jets are collimated plasma ejections that occur in the solar corona and they offer ways for plasma and particles to enter interplanetary space. They are transient (tens of minutes) but ubiquitous, with a typical height of $\\sim5\\times10^4$ km and a typical width of $\\sim8\\times10^3$ km \\citep[e.g.][]{2007PASJ...59S.771S}. X-ray emissions from coronal jets were first observed by the Soft X-ray Telescope (SXT) onboard \\textit{Yohkoh} in the early 1990s \\citep{1992PASJ...44L.173S}. Since then, coronal jets have been studied in various aspects including morphology, dynamics, driving mechanisms, and more \\citep[see, e.g.,][]{2016SSRv..201....1R, 2021RSPSA.47700217S}. Jets or jet-like events have also been observed in other wavelengths such as extreme ultraviolet (EUV), ultraviolet (UV), and H alpha (those studied in H alpha are historically known as ``surges'') \\citep[e.g.][]{1997Natur.386..811I, 1999ApJ...513L..75C,2009SoPh..259...87N}. These wavebands cover a wide range of plasma temperatures (from chromospheric to coronal), and one important feature of jets is the presence of both hot and cool components in many events \\citep{2010ApJ...720..757M, 2013ApJ...769..134M}.\n\nCurrent models generally suggest that jets are formed by magnetic reconnection between open and closed magnetic field lines; however, the detailed triggering process for such magnetic reconnection is still not fully clear. In the emerging flux model, jets are generated through interchange reconnection when field lines of the newly emerging magnetic flux reach those of the pre-existing open field \\citep{1992PASJ...44L.173S}. As shown in the 2D simulation by \\citet{1995Natur.375...42Y, 1996PASJ...48..353Y}, this model could successfully produce a hot jet and an adjacent cool jet (or surge) simultaneously. The embedded-bipole model developed by \\citet[][etc.]{2015A&A...573A.130P, 2016A&A...596A..36P} considers a 3D fan-spine topology where magnetic reconnection occurs around the 3D null point. In their simulation, straight jets are generated through slow reconnection at the current sheet and driven by magnetic tension, while helical jets are generated through explosive magnetic reconnection triggered by a kink-like instability and driven by a rapid untwisting process of magnetic field lines. Recently, a few studies have reported small-scale filament structures (known as ``minifilaments'') at the base of some coronal jets, leading to the minifilament eruption model \\citep{2015Natur.523..437S, 2016ApJ...821..100S, 2016ApJ...832L...7P, 2017ApJ...844..131P, 2018ApJ...853..189P, 2017Natur.544..452W, 2018ApJ...852...98W, 2018ApJ...859....3M, 2019ApJ...882...16M}. This model suggests that jets are generated through miniature filament eruptions similar to those that drive larger eruptive events such as coronal mass ejections (CMEs). In addition to the external\/interchange magnetic reconnection, this process also involves internal magnetic reconnection inside the filament-carrying field, and the jet bright point (JBP, which corresponds to the solar flare arcade in the larger-scale case) appears underneath the erupting minifilament. Many recent observations have shown that the triggers for these minifilaments eruptions are usually magnetic flux cancellation \\citep{2011ApJ...738L..20H, 2012A&A...548A..62H, 2014ApJ...783...11A,  2016ApJ...832L...7P, 2017ApJ...844..131P, 2018ApJ...853..189P, 2019ApJ...882...16M, 2021ApJ...909..133M}.\n\nHXR observations can also provide helpful insights into jet formation mechanisms by constraining energetic electron populations within coronal jets. \\citet{2011ApJ...742...82K} investigated HXR emissions for 16 flare-related energetic electron events and found that 7 of them showed three distinct HXR footpoints, which was consistent with the interchange reconnection geometry. (In the remaining events, the fact that they showed less then three sources was likely due to instrument limitations.) Also in that study, EUV jets were found in all 6 events that had EUV data coverage. HXR bremsstrahlung emissions could also directly come from coronal jets if there are energetic electrons, but those extended sources are usually much fainter than the footpoint sources and only a few studies \\citep{2009A&A...508.1443B, 2012ApJ...754....9G} have reported such observations. More recently, \\citet{2018ApJ...867...84G} combined HXR observations with microwave emission, EUV emission, and magnetogram data, performing 3D modeling of electron distributions for a flare-related jet. They obtained direct constraints on energetic electron populations within that event. \\citet{2020ApJ...889..183M} carried out a statistical study of 33 flare-related coronal jets using HXR and EUV data, and they observed non-thermal emissions from energetic electrons in 8 of these events. They also studied the relation between jets and the associated flares but found no clear correlations between jet and flare properties. \n\nIn most of the previous studies of coronal jets, hot plasma and HXR emissions were found near the base of the jet (the location of the primary reconnection site) \\citep[e.g.][]{2011ApJ...742...82K, 2016A&A...589A..79M, 2020ApJ...889..183M}. However, for two coronal jets on November 13, 2014, HXR thermal emissions were observed near the far end of the jet spire (hereafter the ``top''). In fact, in the second event which had full HXR coverage, HXR emissions were observed at three different locations: the base of the jet, the top of the jet, and a location to the north of the jet. Here we present a multi-wavelength analysis of these two jets using data from the Atmospheric Imaging Assembly (AIA) onboard the \\textit{Solar Dynamic Observatory} (\\textit{SDO}), the \\textit{Reuven Ramaty High Energy Solar Spectroscopic Imager} (\\textit{RHESSI}), the X-ray Telescope (XRT) onboard \\textit{Hinode}, and the \\textit{Interface Region Imaging Spectrograph} (\\textit{IRIS}). We found that all these different HXR sources showed evidence of mildly accelerated electrons, and particle acceleration also happened near the jet top in addition to the site at the jet base. To our knowledge, this is the most thorough HXR study to date of particle acceleration in coronal jets.\n\nThe paper is structured as follows: In section \\ref{sec:data}, we describe the observations from each instrument. In Section \\ref{sec:analysis}, we show results from differential emission measure (DEM) analysis, imaging spectroscopy, and velocity estimation. In Section \\ref{sec:discussion}, we calculate the energy budget for one of the jets, discuss the interpretations of observational results, and compare them with jet models. Finally, in Section \\ref{sec:summ}, we summarize the key findings of this work. \n\n\n\n\\section{Observations} \\label{sec:data}\nOn November 13, 2014, more than ten recurrent jets were ejected from NOAA Active Region 12209 near the eastern solar limb at different times throughout the day. While most (if not all) of the jets can be identified in one or more AIA channels, only two events, SOL2014-11-13T17:20 and SOL2014-11-13T20:47, were simultaneously observed by AIA and {\\textit{RHESSI}}. We select these two flare-related jets for this study, and we add supporting observations from XRT and {\\textit{IRIS}}. The associated flares are GOES class C1.5-1.7 without background subtraction (see top row of Figure \\ref{fig:t_profile}), or B2.4-3.7 with background subtraction.\n\n\n\\begin{figure}\n\t\\includegraphics[width=0.5\\textwidth]{fig_t_profile1.pdf}\n\t\\includegraphics[width=0.5\\textwidth]{fig_t_profile2.pdf}\n\t\\caption{Time profiles of the $\\sim$17:20 jet (left) and the $\\sim$20:50 jet (right) on November 13, 2014. Top row: GOES light curves in the 1-8 {\\AA} channel. Second row: GOES light curves in the 0.5-4 {\\AA} channel. Third row: {\\textit{RHESSI}} emission in 3-6 keV (black) and 6-12 keV (red), using detectors 3, 6, 8, and 9. The first event only has partial coverage from {\\textit{RHESSI}} due to spacecraft night. Fourth row: Examples of AIA EUV emissions from the jet base\/top. Blue lines show light curves of a 3\"$\\times$3\" box at the base of each jet in the 304 {\\AA} channel, and red lines show light curves of a 3\"$\\times$3\" box at the top of each jet in the 131 {\\AA} channel (boxes are not shown). Bottom row: XRT measurements of selected regions (3\"$\\times$3\", not shown) at the jet base (blue) and the jet top (red) in the thin-Be filter. Both events only have partial coverage from XRT.}\n\t\\label{fig:t_profile}\t\n\\end{figure}\n\n\n\\subsection{AIA data}\n\nThe AIA instrument provides full-disk solar images in ten EUV\/UV\/visible-light channels with a spatial resolution of 1.5 arcsec \\citep{2012SoPh..275...17L}. In this work we use data from the seven EUV channels of AIA: 94 {\\AA}, 131 {\\AA}, 171 {\\AA}, 193 {\\AA}, 211 {\\AA}, 304 {\\AA}, and 335 {\\AA}, which have a cadence of 12 seconds and cover plasma temperatures from $\\sim$0.05 MK up to $\\sim$20 MK.\n\nFigure \\ref{fig:aia_im} shows AIA images of the two jets in the 131 {\\AA} and 304 {\\AA} channels at selected times. At the beginning of each event, a minifilament (pointed by yellow arrows) was identified in multiple AIA channels at the base of the jet. After the minifilamnet eruption, a JBP (pointed by white arrows) appeared underneath the prior minifilament location.\n\nInterestingly, both events showed slightly different jet evolution in cool and hot AIA channels. In cooler channels including 171 {\\AA}, 193 {\\AA}, 211 {\\AA}, 304 {\\AA}, and 335 {\\AA}, the first jet started at $\\sim$17:15 UT, reached its maximum extent at $\\sim$17:23 UT, and lasted about 20 minutes. However, in hot channels that are sensitive to $\\gtrsim$10 MK plasma (94 {\\AA} and 131 {\\AA}, particularly), the jet reached its maximum height within five minutes after the same starting time; then it slightly expanded transversely and gradually faded away in a much longer time. (The 193 {\\AA} channel in principle could also measure hot plasma \\citep{2012SoPh..275...17L}, but its response was dominated by temperatures below 10 MK and thus looked like a cool channel.) Similar behavior was observed in the later jet, which started at $\\sim$20:40 UT and reached its maximum extent at $\\sim$20:47 UT in the hotter 94 {\\AA} and 131 {\\AA} channels or at $\\sim$20:50 UT in the rest of the channels. The jet had already disappeared in those cooler channels before 20:58 UT, but it was visible in 94 {\\AA} and 131 {\\AA} for more than an hour. \n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=0.98\\textwidth]{fig_aia.pdf}\n\t\\caption{AIA 131 {\\AA} and 304 {\\AA} images of the two jets at selected times. The top two rows show the evolution of the earlier jet and the bottom two rows show the evolution of the later jet. The yellow arrows point to the minifilament while the blue arrows point to the JBP in each event. Both jets reached their maximum extents at earlier times and lasted longer in the hotter 131 {\\AA} channel (sensitive to both $\\sim$0.4 MK and $\\sim$10 MK temperatures) compared to the cool 304 {\\AA} channel (sensitive to chromospheric temperatures around 0.05 MK).}\n\t\\label{fig:aia_im}\t\n\\end{figure*}\n\n\n\n\\subsection{RHESSI data} \\label{sec:rhessi}\n\n{\\textit{RHESSI}} was a solar-dedicated HXR observatory launched in 2002 and decommissioned in 2018. It consisted of nine rotating modulation collimators, each placed in front of a cooled germanium detector, and used indirect Fourier imaging techniques. {\\textit{RHESSI}} measured both images and spectra over the full sun in the energy range of 3 keV - 17 MeV and had good spatial and energy resolutions especially for lower energies (2.3 arcsec and $\\sim$1 keV, respectively) \\citep{2002SoPh..210....3L}. \n\n{\\textit{RHESSI}} was in eclipse during 16:46-17:23 UT, so it didn't capture the entire first jet; but {\\textit{RHESSI}} did have full coverage for the later jet. Figure \\ref{fig:rhessi_im} shows {\\textit{RHESSI}} images in 3-12 keV using detectors 3, 6, 8, and 9. All images were produced using the CLEAN algorithm in the HESSI IDL software package. In both events, HXR emissions were observed near the top of the jet. Furthermore, time slices of the later jet show that there were actually three HXR sources in that event. The first HXR source appeared at the base of the jet a few minutes after the jet's starting time and peaked at around 20:46 UT. The location of this source is consistent with the erupting minifilament site where magnetic reconnection took place. Meanwhile, starting from $\\sim$20:46 UT, the second HXR source appeared near the top of the jet and it became dominant during 20:48-20:51 UT. Finally, after the source at the jet top had faded away, another HXR source was observed to the north of the jet which reached its maximum intensity at $\\sim$20:53 UT.\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=0.99\\textwidth]{fig_rhessi.pdf}\n\t\\caption{{\\textit{RHESSI}} contours in 3-12 keV overlaid on the AIA 94 {\\AA} images. Panel (a) shows an image of the earlier event and panels (b)-(j) show time slices of the later event. HXR emissions were observed near the top of the jet in both events. The later event showed three different HXR sources: one at the base of the jet at $\\sim$20:46 UT (panel c), one near the top of the jet at $\\sim$20:50 UT (panel g), and one to the north of the jet at $\\sim$20:53 UT (panel j).}\n\t\\label{fig:rhessi_im}\t\n\\end{figure*}\n\n\n\\subsection{XRT data} \\label{sec:xrt}\nXRT provides additional coverage of high-temperature plasma beyond AIA hot channels and {\\textit{RHESSI}}, though only data in the thin-Be filter for part of each jet are available. This filter is sensitive to plasma temperatures around 10 MK, and shows very similar jet behavior as the AIA 94 {\\AA} and 131 {\\AA} filters. Here we include these data as supplementary observations.\n\nThin-Be filter data were available for the first 15 minutes of the earlier jet (before 17:30 UT) with a cadence of a half minute. The jet started with a very fast flow at $\\sim$17:15 UT, and reached its maximum extent in just a few minutes. Then after $\\sim$17:20 UT, it grew slightly wider and remained visible towards the end of the observation time (Figure \\ref{fig:xrt_iris}). As for the later jet, XRT missed most of its erupting process since no data were available between 20:45 UT and 21:09 UT. But after 21:09 UT, the jet was still visible in the thin-Be filter until it finally faded away at $\\sim$22:00 UT (not shown).\n\n\n\\subsection{IRIS data} \n{\\textit{IRIS}} has full temporal coverage and partial spatial coverage of the earlier jet in its 1330 {\\AA} slit-jaw images. This channel is sensitive to temperatures around 0.02 MK with a spatial resolution of 0.33 arcsec and a cadence of 10 seconds, thus it helps to investigate the dynamics of plasma at chromospheric temperatures. For the earlier event, the jet was at the corner of the field of view and most part of the jet body (but no jet base or top) was captured in these images (Figure \\ref{fig:xrt_iris}). Jet evolution in this channel was similar to that in the AIA 304 {\\AA} channel, and we used these data (in addition to the AIA data) to estimate jet velocities (see Section \\ref{velocities}). However, this channel had much less coverage of the later jet and it was not considered for that event. \n\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=0.8\\textwidth]{fig_xrt_iris.pdf}\n\t\\caption{XRT and {\\textit{IRIS}} images of the two jets at selected times. Panels (a)-(d): XRT Be-thin images of the earlier event. Jet evolution is similar to that in AIA hot filters (94 {\\AA} and 131 {\\AA}). Panels (e)-(g): XRT Be-thin images of the later event. No data was available between 20:44 and 21:09 UT. Panel (h): An {\\textit{IRIS}} SJI 1330{\\AA} image of the earlier event. The jet was located at the corner of the {\\textit{IRIS}} field of view. }\n\t\\label{fig:xrt_iris}\t\n\\end{figure*}\n\n\n\\section{Data analysis} \\label{sec:analysis}\n\n\\subsection{Differential emission measure (DEM) analysis} \\label{sec:DEM}\nWe carried out a differential emission measure (DEM) analysis to investigate the temperature profile for these two events. A DEM describes a plasma distribution with respect to temperature along the line of sight, and it is directly related to the observed flux $F$ for a particular instrument via\n\\begin{equation}\n\tF=\\int{R(T) \\cdot DEM(T) \\, dT},\n\\end{equation} \nwhere $R$ is the temperature response of that instrument. In this analysis, we used the regularization method developed by \\citet{2012A&A...539A.146H, 2013A&A...553A..10H} for DEM inversion. We considered two different data selections: (a) AIA bandpass filter data only, where we used data from six AIA bandpass filters that are sensitive to coronal temperatures: 94 {\\AA}, 131 {\\AA}, 171 {\\AA}, 193 {\\AA}, 211 {\\AA}, and 335 {\\AA}; and (b) a combination of multi-instrument data, where we used the same set of AIA data, together with HXR measurements in 4-5, 5-6, and 6-7 keV bands from {\\textit{RHESSI}}, and thin-Be filter data from XRT if available. The {\\textit{RHESSI}} 4-5 keV and 5-6 keV energy bands were selected because they measure plasma temperature via the bremsstrahlung continuum, and the 6-7 keV energy band was particularly important as it includes the 6.7 keV Fe line complex. The uncertainties for AIA data were estimated via the SSWIDL procedure ``\\texttt{aia\\textunderscore bp\\textunderscore estimate\\textunderscore error.pro}'', added in quadrature with a systematic error of 10\\%. The uncertainties for {\\textit{RHESSI}} and XRT data were both estimated as 20\\%.\n\nThe temperature responses for AIA and XRT filters were generated through SSWIDL routines ``\\texttt{aia\\textunderscore get\\textunderscore response.pro}'' and ``\\texttt{make\\textunderscore xrt\\textunderscore temp\\textunderscore resp.pro}'', respectively. To obtain the temperature responses for {\\textit{RHESSI}} in different energy bands, we first calculated the isothermal HXR spectra as a function of energy for multiple temperatures ranging from 3 MK to 30 MK, using the SSWIDL routine ``\\texttt{f\\textunderscore vth.pro}''. Thus for each energy band, we obtained a series of photon fluxes at different temperatures, which would correspond to the temperature response (in photon space) for that energy band after applying proper normalization. (The {\\textit{RHESSI}} instrument response was already taken into account when producing HXR images, thus it did not need to be included in the temperature response.) In above calculation, coronal abundances were adopted.\n\nWe calculated the DEMs for four regions where HXR emissions were observed, around the times when each HXR source reached its maximum intensity: the top of the earlier jet at 17:24 UT, the base of the later jet at 20:46 UT, the top of the later jet at 20:50 UT, and the loop to the north of the later jet at 20:53 UT. Each region was selected based on contours in AIA 131 {\\AA} images and the observed intensities were averaged over the whole region. We first obtained the DEM results using AIA data only (black lines in Figure \\ref{fig:dem_ave}), as well as the corresponding residuals in data space (asterisks in Figure \\ref{fig:dem_ave}). All these AIA-alone DEMs indicate the existence of multi-thermal plasma, each with a high-temperature component peaking around 10 MK. However, although these AIA-alone DEMs had good enough predictions in the AIA channels that were used for the DEM inversion, they failed to predict the HXR measurements well. As shown by the blue asterisks in the residual plots, the photon fluxes predicted by the AIA-alone DEMs in the {\\textit{RHESSI}} 4-5 keV and 5-6 keV energy bins were always lower than the actual measurements, and the line emissions in the 6-7 keV energy bin were very prominent compared to the bremsstrahlung continuum.\n\nTo have a better understanding of the level of agreement between AIA and {\\textit{RHESSI}}, we carried out a more complete quantitative comparison of HXR fluxes between these two instruments. In this exercise, we predicted the HXR spectrum in the 3-15 keV energy range for the top region of the earlier jet\u00a0using the AIA-alone DEM and compared\u00a0it with the spectrum directly measured by {\\textit{RHESSI}}. The predicted HXR spectrum was calculated according to Eq.(1), with R being the {\\textit{RHESSI}} temperature response. The results are shown in the left panel of Figure \\ref{fig:comp}, where the AIA-predicted HXR fluxes were consistently lower than the {\\textit{RHESSI}} fluxes, indicating a possible cross-calibration factor between AIA and {\\textit{RHESSI}}. And again, the AIA-alone DEM predicted much stronger line emissions in the 6-7 keV energy bin over the continuum as compared to the actual {\\textit{RHESSI}} observation. Reasons for these disagreements could be some instrumental effects that are not well understood, such as the change in {\\textit{RHESSI}} blanketing with respect to time, or could be the possible ``non-standard'' elemental abundances in these events that we are unable to characterize.\n\nBecause of the discrepancies mentioned above, incorporating {\\textit{RHESSI}} data into this DEM analysis is challenging. To obtain a DEM solution that could successfully predict both the HXR continuum and the line feature at the same time, we found that a cross-calibration factor between AIA and {\\textit{RHESSI}} was required and the initial DEM guess must be very carefully chosen. We had the best chance of success when using a ``modified'' AIA-alone DEM as the initial guess, where we substituted the high-temperature component of each AIA-alone DEM with a Gaussian distribution that peaked around the temperature given by {\\textit{RHESSI}} spectroscopy (more details of spectroscopy will be discussed in Section \\ref{sec:imag_spec}). The height, width, and exact peak location of that Gaussian distribution were tested with a series of values, and we selected the most robust ones. In this piece of analysis, we scaled down the photon fluxes from {\\textit{RHESSI}} by a factor of 3.5, but this cross-calibration factor could be in the range of 3-5 and was not well constrained. Incidentally, this factor that we found here is similar to the AIA-{\\textit{RHESSI}} discrepancy found by \\citet{2013ApJ...779..107B}. In addition, in some literature a factor of 2-3 was suggested for cross-calibration between AIA and XRT \\citep[e.g.][]{2015ApJ...806..232S, 2017ApJ...844..132W}. We found that multiplying a factor of 2 to the XRT response would result in a better agreement between the predicted and measured Be-thin filter data; thus that factor was also included here.\n\nThe joint DEMs (red lines) and the data-space residuals (red triangles) are also plotted in Figure \\ref{fig:dem_ave}, along with the AIA-alone DEM results. These joint DEMs are the only set of solutions we found that fit both the line emission and the bremsstrahlung continuum well. For all the selected regions, the joint DEMs have very similar cool components as the AIA-alone DEMs, but the hot components of the joint DEMs tend to be more isothermal and slightly cooler. Particularly, HXR constraints significantly reduced the amount of plasma above $\\sim$15 MK (otherwise the predicted line emission was always too prominent). However, previous studies have seen larger discrepancies between bandpass filter DEMs and the ones that included HXR constraints. For example, in the DEM analysis for a quiescent active region presented by \\citet{2009ApJ...704..863S}, a high-temperature component that peaked around $10^{7.4}$K was found when only using data from XRT filters, but the DEM for that component was reduced by more than one order of magnitude when combining observations from both XRT and {\\textit{RHESSI}}. Compared to that study, the AIA-alone DEMs here are not too far from the joint DEMs that incorporated HXR data.\n\nWe further compared the HXR fluxes predicted by the joint DEM with {\\textit{RHESSI}} measurements (with a cross-calibration factor applied) for the top region of the earlier jet, as shown in the right panel of Figure \\ref{fig:comp}. As expected, the two HXR spectra had a much better agreement at lower energies than the spectra predicted from AIA-only DEMs did, including both the overall continuum and the line feature. Besides, for higher energies around 10 keV, {\\textit{RHESSI}} measurements had systematically higher emissions, suggesting a possible non-thermal component for this source. This is consistent with our later findings through spectral analysis (Section \\ref{sec:imag_spec}). As a side note, later spectral analysis suggests that for some of the HXR sources non-thermal emissions might dominate in the 6-7 keV (and maybe 5-6 keV) energy bin(s). In this scenario, the fluxes from those thermal sources would be even lower and the joint-DEMs shown here provide upper limits for the possible amount of hot plasma. \n\nAs the last part of the DEM analysis, we examined the temporal evolution of the DEM maps in 11-14 MK (i.e. the hot component) for each event (Figures \\ref{fig:em1} and \\ref{fig:em2}). Because of the missing {\\textit{RHESSI}} and XRT data, and the fact that the AIA-alone DEMs in this temperature range qualitatively agree with the joint DEMs, these DEM maps were all generated using AIA data only. In both events, hot plasma first appeared at the base of the jet (which was the same location from where the minifilament erupted and magnetic reconnection occurred); however, as the hot plasma at the jet base gradually cooled down, more and more hot plasma was observed near the top of the jet and that location was mostly stationary. These DEM maps show consistent results with the location and temporal evolution of the {\\textit{RHESSI}} HXR sources.\n\n\\begin{figure}\n\t\\plotone{fig_dem_comp.pdf}\n\t\\caption{DEMs and data-space residuals (defined as (model-data)\/error) for four different selected regions: the top of the earlier jet at 17:24 UT (top left), the base of the later jet at 20:47 UT (top right), the top of the later jet at 20:49 UT (bottom left), and a loop to the north of the later jet at 20:53 UT (bottom right). The black lines show results for the DEM inversion that used AIA data only, and the red lines show results for the DEM inversion that used multi-instrument data from AIA, {\\textit{RHESSI}}, and XRT if available. The AIA-alone DEMs and the joint DEMs agree qualitatively, but the joint DEMs require a more isothermal and slightly cooler high-temperature component for each source. In the residual plots, asterisks stand for results from the AIA-alone DEMs (among which black asterisks show the residuals in AIA channels that were used for the DEM inversion while blue asterisks show the residuals in {\\textit{RHESSI}} energy bands and possibly XRT Be-thin filter predicted by this DEM), and red triangles stand for results from the joint DEMs. The HXR fluxes predicted by the joint DEMs have a much better agreement with the actual data.}\n\t\\label{fig:dem_ave}\t\n\\end{figure}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.8\\textwidth]{fig_hxr_comp.pdf}\n\t\\caption{\\textit{Left}: HXR spectrum for the source at the top of the earlier jet deduced from the AIA-alone DEM (black), compared to {\\textit{RHESSI}} measurements (red). \\textit{Right}:\u00a0HXR spectrum for the same source deduced from the joint DEM (black), compared to {\\textit{RHESSI}} measurements with a cross-calibration factor applied (red).\u00a0With a cross-calibration factor of 3.5, the HXR spectrum from the joint DEM could successfully predict the bremsstrahlung continuum and the line feature at 6.7 keV simultaneously.}\n\t\\label{fig:comp}\t\n\\end{figure}\n\n\\begin{figure}\n\t\\plotone{fig_emmaps1.pdf}\n\t\\caption{Temporal evolution of the DEM maps in 11-14 MK for the earlier jet. Hot plasma appeared at the base of the jet during the first few minutes, but starting from $\\sim$17:20 more and more hot plasma was observed near the top of the jet and the top source became dominant at $\\sim$17:22. Color scale is in units of cm$^{-5}$ K$^{-1}$.}\n\t\\label{fig:em1}\t\n\\end{figure}\n\n\\begin{figure}\n\t\\plotone{fig_emmaps2.pdf}\n\t\\caption{Temporal evolution of the DEM maps in 11-14 MK for the later jet. Similar to the earlier event, hot plasma first appeared at the base of the jet, but was also observed near the top of the jet starting from $\\sim$20:45 and the top source became dominant at $\\sim$20:50. Color scale is in units of cm$^{-5}$ K$^{-1}$.}\n\t\\label{fig:em2}\t\n\\end{figure}\n\n\n\\subsection{Jet velocities} \\label{velocities}\n\nIdentifying different velocities associated with the jet will be helpful to differentiate possible mechanisms behind those jets. A common method for velocity estimation is making time-distance plots \\citep[e.g.][]{2016A&A...589A..79M, 2020ApJ...889..183M}. Such plots are usually produced by putting together time slices of the intensity profile along the direction of the jet, in which case the jet velocities (in the plane of the sky) are the slopes. To take into account everything within the width of the jets, here we selected a rectangular region around each jet and we summed the intensities across the width of this region. \n\nFigure \\ref{fig:td1} shows the time-distance plots for the earlier jet using seven EUV filters of AIA and the slit-jaw 1330{\\AA} filter of {\\textit{IRIS}}. Interestingly, the chromospheric filters (AIA 304 {\\AA} and {\\textit{IRIS}} 1330 {\\AA}) are the ones where the velocities are most clearly identified and show very consistent results. The 304 {\\AA} filter shows multiple upward velocities ranging from 104 km\/s to 226 km\/s, while the 1330 {\\AA} filter shows upward velocities ranging from 83 km\/s to 404 km\/s (the uncertainties for those velocities are on the order of 10-20$\\%$ considering the pixel size and the temporal cadence of the images). Also, both filters clearly indicate that some plasma returned to the solar surface (possibly) along the same trajectory as the original jet, with downflow velocities of $\\sim$110 km\/s. For the rest of the AIA filters, similar upward and downward velocities as mentioned above can partly be seen in the 171 {\\AA} filter (sensitive to $\\sim$0.6 MK plasma), but could barely be seen in other ones. However, there were also some really fast outflows at the beginning of this jet in the 131{\\AA} filter (sensitive to both $\\sim$0.4 MK and $\\sim$10 MK plasma), which has a velocity of $\\sim$700 km\/s.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.75\\textwidth]{fig_td_jet1_paper.pdf}\n\t\\caption{Time-distance plots of the earlier jet in seven AIA EUV filters and the {\\textit{IRIS}} 1330 {\\AA} filter. Slopes for velocity calculation are shown as dashed lines. The chromospheric filters (AIA 304 {\\AA} and {\\textit{IRIS}} 1330 {\\AA}) show various upward velocities around 200 km\/s and downward velocities around 110 km\/s. The AIA 131 {\\AA} filter (sensitive to both $\\sim$0.4 MK and $\\sim$10 MK) shows a faster upward velocity of 694 km\/s at the beginning of the jet. In the rest of the AIA filters, velocities are less apparent. (The black line at $\\sim$17:21 in the  171 {\\AA} plot is due to some instrument issue.)}\n\t\\label{fig:td1}\t\n\\end{figure}\n\nThe time-distance plots for the later jet describe a slightly different picture (Figure \\ref{fig:td2}). The 304 {\\AA} filter again shows multiple upward velocities ranging from 192 km\/s to 251 km\/s, and downward velocities around 130 km\/s (the uncertainty is on the order of 10-20\\%). However, these main upward velocities\u00a0can be clearly identified in all seven AIA filters, including both cool and hot ones. Besides, in the 131 {\\AA} filter, a faster outflow at the beginning of the jet is still identifiable but harder to see compared to the earlier event, and the velocity for this outflow is 377 km\/s. These velocities will be compared to other studies and to models in Section \\ref{sec:v_discuss}.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.75\\textwidth]{fig_td_jet2_paper.pdf}\n\t\\caption{Time-distance plots of the later jet in seven AIA EUV filters. Slopes for velocity calculation are shown as dashed lines. Unlike the earlier event, the main upward velocities can be clearly identified in all filters. The AIA 131 {\\AA} filter still shows a faster upward velocity of 377 km\/s at the beginning of the jet, but this velocity is less apparent.}\n\t\\label{fig:td2}\t\n\\end{figure}\n\n\n\\subsection{Imaging spectroscopy} \\label{sec:imag_spec}\n\nTo study the accelerated electron populations in these events, we performed imaging spectroscopy for the four HXR sources observed by {\\textit{RHESSI}}. For each source, a one-minute time interval during which the source reached its maximum HXR intensity was first selected by eye based on the {\\textit{RHESSI}} images. These images were produced using the CLEAN algorithm and detectors 3, 6, 8, and 9. Then we chose a circular region which contained that source and obtained the spectrum for the selected region. Finally, we carried out spectral fitting using the OSPEX software package in the energy range of 3-15 keV.  \n\nAs mentioned in Section \\ref{sec:DEM}, the comparison of the joint DEM and {\\textit{RHESSI}} measurements suggests a non-thermal component in the HXR spectrum. To further confirm this, we first fitted the spectra with an isothermal model (not shown), but the models always overpredicted the fluxes at the 6.7 keV line complex and had systematically low emissions at energies above 10 keV for all the sources. This indicated that there should be another component in the spectra, either due to a second thermal distribution or a non-thermal distribution. However, the results of fitting for a double thermal model (not shown) had unphysical fit parameters for one of the thermal components, and it again overpredicted the line emission, making this scenario unlikely. Therefore, we confirmed that there should be non-thermal emissions in these events. We then added a thick-target non-thermal component to the fitting (the justification for the thick-target regime will be discussed in Section 4.2), and could obtain good fits across the entire observed energy range. (We used the temperatures from those fits as a reference when generating the initial DEM guess for the joint DEM inversion.) However, due to the limited number of energy bins and the number of free parameters, some of the fit parameters were not well constrained and had uncertainties over 100$\\%$. To further reduce the uncertainties, we performed another fitting with a fixed temperature, which was chosen to be the average temperature of the hot component derived from the joint DEMs. The resulting spectra are shown in Figure \\ref{fig:spec} and the parameters are reported in Table \\ref{tab:spec_fit}. Interestingly, the non-thermal electron power laws in all sources have similar spectral indices around 10 and low energy cutoffs around 9 keV. While these non-thermal power laws are steeper than in most flares, the parameters are consistent with the range found for microflares by \\citet{2008ApJ...677..704H}.\n\n\\begin{figure}\n\t\\plotone{fig_spec_paper.pdf}\n\t\\caption{{\\textit{RHESSI}} spectra for the four HXR sources observed in these two events, each during a one-minute interval when the source approximately reached its maximum HXR intensity. All spectra can be fitted well with an isothermal (blue) plus thick-target (red) model. Note that in these fits, the isothermal temperatures were fixed to be the average temperatures of the hot components derived from the joint DEMs.}\n\t\\label{fig:spec}\t\n\\end{figure}\n\n\\begin{deluxetable*}{lcccccc}\n\t\\tablenum{1}\n\t\\tablecaption{Fit parameters for the four {\\textit{RHESSI}} HXR sources assuming an isothermal plus thick-target model \\label{tab:spec_fit}}\n\t\\tablehead{\n\t\t\\colhead{} & \\colhead{Time} & \n\t\t\\colhead{Emission measure} &\n\t\t\\colhead{Temperature} & \\colhead{spectral index} &\n\t\t\\colhead{low energy cutoff} \\\\\n\t\t\\colhead{} & \\colhead{} &\n\t\t\\colhead{($10^{46} \\mathrm{cm}^{-3}$)} &\n\t\t\\colhead{(MK, fixed)} & \\colhead{} & \\colhead{(keV)} \n\t}\n\t\\startdata\n\tJet 1 top source & 17:23:30-17:24:30 & 1.8 $\\pm$ 0.5  & 11.1 & 11.4 $\\pm$ 3.6 & 9.3 $\\pm$ 1.6 \\\\\n\tJet 2 base source & 20:46:00-20:47:00 & 4.7 $\\pm$ 1.0 & 9.9 & 9.1 $\\pm$ 0.9 & 8.5 $\\pm$ 0.9 \\\\\n\tJet 2 top source & 20:50:00-20:51:00 & 4.1 $\\pm$ 0.5 & 10.5 & 10.3 $\\pm$ 1.4 & 9.4 $\\pm$ 1.2 \\\\\n\tJet 2 northern source & 20:53:00-20:54:00 & 3.9 $\\pm$ 1.3 & 9.6 & 8.5 $\\pm$ 0.8 & 8.7 $\\pm$ 1.0 \n\t\\enddata\n\\end{deluxetable*}\n\n\n\n\\section{Discussion} \\label{sec:discussion}\n\\subsection{Jet velocities and driving mechanisms} \\label{sec:v_discuss}\nIn these two events, we observed two types of upward velocities in jets. One type falls within the range of 80-400 km\/s, most clearly seen in the AIA 304 {\\AA} and {\\textit{IRIS}} 1330 {\\AA} filters (sensitive to chromospheric temperatures) and possibly visible in other filters. This type of velocity is consistent with many previous studies of coronal jets \\citep[e.g.][]{1996PASJ...48..123S, 2007PASJ...59S.771S, 2016A&A...589A..79M, 2016ApJ...832L...7P, 2020ApJ...889..183M} where the jet velocities usually range from a few tens of km\/s to $\\sim$500 km\/s, with an average around 200 km\/s. The other velocities, $\\sim$700 km\/s and $\\sim$400 km\/s respectively for the two jets, could only be identified in the 131 {\\AA} filter (sensitive to $\\sim$0.4 MK and hot temperatures $\\sim$10 MK) at the beginning of each event (though harder for the later jet). The velocity of $\\sim$400 km\/s is still within the common range for coronal jets, but it is faster than the other velocities observed in that jet. The velocity of $\\sim$700 km\/s seems to be faster than most of the observed jets. However, such velocities are not rare and have been reported in a few coronal jet observations by XRT \\citep{2007PASJ...59S.771S, 2007Sci...318.1580C}.\n\nOne possible acceleration mechanism for coronal jets is chromospheric evaporation, which is also the responsible mechanism for some plasma flows in solar flares. In this process, the energy released from magnetic reconnection is deposited in the chromosphere, compresses and heats the plasma there, and produces a pressure-driven evaporation outflow on the order of sound speed. In fact, \\citet{1984ApJ...281L..79F} derived a theoretical upper limit for the velocity of this evaporation outflow to be 2.35 times the local sound speed, where the sound speed $c_s$ can be calculated as: $c_{s}=147\\sqrt{\\frac{T}{\\mathrm{1MK}}}$ km\/s assuming an isothermal model \\citep[e.g.][]{2004psci.book.....A}. In the 304 {\\AA} filter (characteristic temperature $10^{4.7}$ K), this upper limit corresponds to a very low speed of 77 km\/s, indicating that the cool plasma is unlikely driven by chromospheric evaporation. Also, common velocities reported by observations of chromospheric evaporation usually fall within the range of tens of km\/s up to 400 km\/s \\citep[e.g.][]{2013ApJ...767...55D, 2015ApJ...811..139T, 2015ApJ...805..167S}, thus chromospheric evaporation seems not able to explain the very fast flow of the earlier jet observed in the hot 131{\\AA} filter. Furthermore, it is expected that the velocity\u00a0would increase\u00a0with the temperature if a jet is generated\u00a0by chromospheric evaporation \\citep{2012ApJ...759...15M}, but here we have seen very consistent velocities in all seven AIA filters that are sensitive to different temperatures in the later event. For all these reasons, if both jets are driven by the same mechanism, that mechanism is likely \\textit{not} chromospheric evaporation but magnetic tension instead. However, it is not clear why the earlier jet shows more complicated and various velocities (even in a single channel) if both jets are driven similarly.\n\n\n\\subsection{Particle acceleration locations}  \nIn Section \\ref{sec:imag_spec}, we fitted the {\\textit{RHESSI}} spectra of the four HXR sources with an isothermal plus thick-target model. Here we first justify that the thick-target regime is a reasonable approximation.\n\nThe column depth (defined as $N_s=\\int ndz$ where n is the plasma density) to fully stop an electron of energy E (in units of keV) can be calculated as: $N_s=1.5\\times10^{17}\\mathrm{cm^{-2}} E^2$ \\citep[e.g.][]{krucker2008hard}. Based on this formula, Figure \\ref{fig:stopping_d} plots the relation between the stopping distance and the plasma density for a given electron energy. Under the thick-target regime, according to Table \\ref{tab:spec_fit}, the average electron energy for the source at the top of the earlier jet is $\\sim$10 keV and the density there is $6\\times10^{9}$ $\\mathrm{cm}^{-3}$ (derived from the joint DEM), which corresponds to a distance of 36 arcsec in average that the electrons can travel before fully stopped by the ambient plasma. Similar average electron energies around 10 keV are found for the other three HXR sources in the later event, and the densities of those sources are $(0.8-1.9)\\times10^{10}$ $\\mathrm{cm}^{-3}$, resulting in stopping distances of 10-28 arcsec. Moreover, in Section \\ref{sec:DEM}, we report a possible cross-calibration factor around 3.5 between AIA and {\\textit{RHESSI}}. That factor is not included in the above calculation; however, if the cross-calibration factor is included, all the densities above would be multiplied by $\\sqrt{3.5}$, corresponding to even shorter stopping distances of 5-20 arcsec. In general, these stopping distances are comparable to the size of the HXR sources, meaning that accelerated electrons deposit a considerable portion of their energies into each source. Furthermore, if this is a thin-target regime, the spectral indices would be slightly smaller but the average electron energies would still be $\\sim$10 keV. This would result in very similar stopping distances that are comparable to source sizes, which is not consistent with the thin-target assumption. Therefore, we conclude that the HXR sources observed in these events can be approximated as thick targets, and mildly accelerated electrons are found at all these locations.\n\nSince the three HXR sources observed in the later event share similar electron distributions, here comes the next question: were the HXR emissions in the later event produced by the same population of accelerated electrons that traveled to different locations, or were they produced by different groups of accelerated electrons individually? In the standard jet models, reconnection happens near the base of the jet, which would require electrons to be accelerated near the base and travel upwards along the magnetic field lines to produce the HXR source at the jet top. However, from the DEM analysis, we find that the densities in the body of the jet (where there were not many HXR emissions) are $\\sim1\\times10^{10}$ $\\mathrm{cm}^{-3}$, so the stopping distance along the jet body is still 20-30 arcsec (for both jets). This is a few times smaller than the distance from the jet base to the jet top; therefore the HXR source at the top of each jet was produced by electrons that were accelerated very close to this source, rather than electrons that traveled far from the primary reconnection site at the jet base. This finding is in line with a similar one made for the powerful X8.3 class flare on September 10, 2017, obtained with an entirely different methodology that employs microwave imaging spectroscopy \\citep{fleishman2022solar}. \n\nAnother possible explanation for the sources at the jet top could be that the jets were actually ejected along large closed loops perpendicular to the plane of the sky rather than the so-called ``open'' field lines. Then the top of the jet is in fact the apex of the loop, which would have higher emissions purely because of the line-of-sight effect. However, even in this scenario the stopping distance along the jet body would remain the same, thus the conclusion of an additional particle acceleration site near the jet top (or loop apex) still holds regardless of jet geometry.\n\nThe HXR source to the north of the later jet appeared last among the three HXR sources, but still during a time when the jet was visible in EUV filters. It is also likely related to the jet because the formation of the jet would change the magnetic configuration of the active region, but neither the electron path nor the density along the path is clear if the energetic electrons traveled from the jet base to the northern location. The typical coronal density for an active region is about $10^{9}$ $\\mathrm{cm^{-3}}$ \\citep[e.g.][]{1961ApJ...133..983N}, corresponding to a stopping distance of 200 arcsec for electrons of 10 keV. Thus, in general situations energetic electrons could travel a decent distance in the corona, but it is also possible that the density in this active region is larger than that typical value. However, due to lack of data, we couldn't determine which was the case for this source.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.75\\textwidth]{fig_stopping_d.pdf}\n\t\\caption{The relation between collisional stopping distances and ambient plasma densities for electrons of certain energies. Red plus signs mark the values for the densities (without a cross-calibration factor) and average energies for the four observed HXR sources, while brown triangles mark the values with a cross-calibration factor applied. The stopping distances for these sources are less than a few tens of arcsecs, but for lower densities and\/or higher electron energies, accelerated electrons could travel an appreciable distance in the corona. }\n\t\\label{fig:stopping_d}\t\n\\end{figure}\n\n\n\\subsection{Energy budget}\nInvestigating the partition of different energy components can help to understand the energy release process in these events. Such calculations have been done in the past for a number of flares and CMEs \\citep[e.g.][]{2012ApJ...759...71E, 2015ApJ...802...53A, 2016A&A...588A.116W}, but only for a few jets so far \\citep{2013ApJ...776...16P}. Here we present our estimates of various energy components for the later event, including kinetic energy, gravitational energy, thermal energy, and the energy in non-thermal electrons. We calculated the maximum amount of energy that could be converted into each of the forms above.\n\nThe jet's major eruption started from 20:46 UT, which was visible in all AIA filters and had a speed of $\\sim$260 km\/s (Figure \\ref{fig:td2}). The density of this plasma was derived from its DEM, which is $1.3\\times10^{10}$ $\\mathrm{cm}^{-3}$. Assuming the jet body that contained this group of plasma to be a cylinder, the peak kinetic energy of the jet is $5\\times10^{26}$ erg.\n\nThe maximum height of this jet is $\\sim$80 arcsec; however, as the height increases the amount of plasma that traveled there decreases, and it's not clear what fraction of plasma finally reached the maximum height. Therefore, instead of calculating the maximum gravitational energy of the jet, we set an upper limit of $2\\times10^{26}$ erg, which is the gravitational energy if all the plasma of the major eruption reached a height of 80 arcsec. This upper limit is smaller than the kinetic energy of the jet, meaning the eruption is not ballistic. \n\nThe thermal energy, $E_{th} = 3k_BTnV$, is dominated by contributions from HXR sources. Using the joint DEMs, the peak thermal energy for each HXR source is about $5\\times10^{27}$ erg. This value is consistent with the flare thermal energies found for other jets in \\citet{2020ApJ...889..183M}.\n\nThe energy in non-thermal electrons can be simply estimated as $E_{nonth} = N_eE_{e,ave}$ where $N_e$ is the total number of accelerated electrons and $E_{e,ave}$ is the average electron energy. Adopting the thick-target approximation and using the parameters from Table \\ref{tab:spec_fit}, the non-thermal energy for each HXR source is about $(6-11)\\times10^{29}$ erg. However, this value here is calculated based on {\\textit{RHESSI}} measurements. If we apply the cross-calibration factor between AIA and {\\textit{RHESSI}} to match the calculations of other energy forms, the non-thermal energy for each HXR source becomes $(3-6)\\times10^{29}$ erg.\n\nThe energies of HXR sources in this event can be compared to those in previous studies of flare energetics. \\citet{2012ApJ...759...71E} studied 38 eruptive events (all except one were M or X class flares and most flares were accompanied by a CME), and they found that the flare thermal energies were always smaller than the energies in accelerated particles. Similar results were found in a later study by \\citet{2016A&A...588A.116W}, where the median ratio of thermal energies to non-thermal energies in electrons for 24 (C-to-X class) flares was 0.3. In a subclass of ``cold'' flares \\citep{2018ApJ...856..111L},  the thermal energy is equal (within the uncertainties) to the non-thermal energy deposition \\citep{2016ApJ...822...71F, 2020ApJ...890...75M, 2021ApJ...913...97F}. (Theoretically, the thermal energy cannot be less than the non-thermal energy as the latter one decays into the thermal one.) For our jet event that contains low-C class flares, the thermal energies are more than one order of magnitude smaller than the non-thermal energies. Therefore, the conclusion that the non-thermal energy is always larger than or at least equal to the thermal energy is likely consistent across a wide range of flare classes and regardless of whether the flare is associated with a jet\/CME or not. \n \nHowever, the energy partition between the jet and the associated flares is different from the energy partition between a CME and a flare. For this event, the kinetic\/gravitational energy of the jet is more than one order of magnitude smaller than the energy of the flares (thermal\/non-thermal), while in \\citet{2012ApJ...759...71E} the total energy of the CME is usually significantly larger. (The kinetic energy in confined flares is much smaller \\citep{2021ApJ...913...97F}, though.) This variety could be explained in the minifilament eruption scenario that jets and CMEs are still both parts of the same eruptive events but the energy partition changes with scale, or this could also indicate that there are fundamental differences between jets and CMEs. To further answer this question, future studies with more samples of flare-related jets are needed.\n\nLastly, it should also be noted that there are still other forms of energy that were not considered in the calculations above, such as magnetic energy, wave energy, etc. These energies could also be important components of the event energy budget, but are hard to evaluate here due to limited data.\n\n\n\\subsection{Comparison to the current jet models} \\label{comp_models}\nConsidering the locations of hot plasma as well as the HXR sources, these two jets are interesting examples to be compared with current jet models. On the one hand, the source at the base of the jet is consistent with what is expected from jet models. During the minifilament eruption at the jet base, magnetic reconnection happens close to the bottom of the corona, heating the plasma there directly and generating accelerated electrons near the reconnection site. The downward-traveling energetic electrons radiate bremsstrahlung emissions as they collide with the dense chromosphere, producing a HXR source and\/or further heating the ambient plasma at the base of the jet. On the other hand, processes after a jet's eruption are generally not considered by those models; thus the hot plasma and the HXR source at the top of the jet are not expected. Our observations have shown that additional particle acceleration could happen at other locations besides the jet base. In other words, there could be multiple reconnection and energy release sites in a single jet event. Also, despite the significantly different particle acceleration sites (and even two separate events), the non-thermal electrons share very similar energy distributions. The spectral indices around 10 and the low energy cutoffs around 9 keV suggest that jet reconnection typically produces only mild particle acceleration. These low energy cutoffs are similar to those of the cold flares \\citep[e.g.,][]{2020ApJ...890...75M}, while the spectra are much softer in the case of the jets.\n\nAnother interesting point about these events is the relation between hot and cool material. For both jets, the cool ejections observed in the 304 {\\AA} filter were adjacent to the hot ejections observed in the 94 {\\AA} and 131 {\\AA} filters. While past simulations have successfully produced a hot jet and a cool jet (or surge) in a single event, it is generally expected that hot and cool jets are driven through different mechanisms. For example, in the simulation by \\citet{1996PASJ...48..353Y}, the hot jet was accelerated by the pressure gradient while the cool surge was accelerated by magnetic tension. Similarly in an observational study by \\citet{2012ApJ...759...15M}, the hot component (at coronal temperatures) was generated by chromospheric evaporation while the cool component (at chromospheric temperatures) was accelerated by magnetic force. However, though the observation of the earlier jet doesn't conflict with this picture, the later jet had consistent velocities in hot and cool filters, indicating that some of the hot components might be driven by a very similar process as the cool components in that event. Therefore, at least in some cases the hot and cool components must be more closely related, and a jet model should be able to explain this kind of observation as well as those similar to \\citet{2012ApJ...759...15M}.\n\n\n\n\\section{Summary} \\label{sec:summ}\nIn this paper, we present a multi-wavelength analysis of two active region jets that were associated with low C-class flares on November 13, 2014. Key aspects of this study include:\n\n\\begin{enumerate}\n\t\\item In both events, hot ($\\gtrsim$10MK) plasma not only appeared near the base of the jet (which is the location of the primary reconnection site) at the beginning, but also appeared near the top of the jet after a few minutes. \n\t\\item Four {\\textit{RHESSI}} HXR sources were observed: one (at the jet top) in the first event and three (at the jet base, jet top, and a location to the north of the jet) in the later event. All those sources showed evidence of mildly accelerated electrons which had spectral indices around 10 and extended to low energies around 9 keV. \n\t\\item Various jet velocities were identified through time-distance plots, including major upward velocities of $\\sim$250 km\/s and downward velocities of $\\sim$100 km\/s. Fast outflows of $\\sim$700 km\/s or $\\sim$400 km\/s were observed only in the hot AIA 131 {\\AA} filter at the beginning of each jet.\u00a0These velocities indicate that the jets were likely driven by magnetic force.\n\t\\item The HXR source and hot plasma at the base of the jet were expected from current models. However, the HXR sources at the top of the jet were produced by energetic electrons that were accelerated very close to the top location, rather than electrons that were accelerated near the jet base but traveled to the top. This means that there was more than one reconnection and particle acceleration site in each event.\t\n\\end{enumerate}\t\n\nCoronal jets are an important form of solar activity that involves particle acceleration, and they share similarities with larger eruptive events such as CMEs. HXRs can provide important constraints on hot plasma within a coronal jet, as well as unique diagnostics of energetic electron populations. To obtain the best constraints for jet models, observations should take advantage of state-of-the-art instruments in different wavebands, but only a few studies have included HXR observations to date. In future work, we would like to extend the method described in this paper to other coronal jets. Those jets could come from the jet database that will be generated by the citizen science project Solar Jet Hunter \\footnote{https:\/\/www.zooniverse.org\/projects\/sophiemu\/solar-jet-hunter} (which was launched through the Zooniverse platform in December 2021). We expect studies with more jet samples to further advance our understanding of particle acceleration in jets.\n\nFurthermore, as shown in this study, HXR sources that are associated with jets could be found in the corona, and they could be faint in some events, thus not identified by current instruments. One solution is to develop direct focusing instruments, such as that demonstrated by the Focusing Optics X-ray Solar Imager (\\textit{FOXSI}) sounding rocket experiment, which will provide better sensitivity and dynamic range for future HXR observations. \n\n\\acknowledgments\nThis work is supported by NASA Heliophysics Guest Investigator grant 80NSSC20K0718. Y.Z. is also supported by the NASA FINESST program 80NSSC21K1387. N.K.P. acknowledges support from NASA's {\\textit{SDO}}\/AIA and HGI grant. We thank Samaiyah Farid for helpful discussions. We are also grateful to the {\\textit{SDO}}\/AIA, {\\textit{RHESSI}}, {\\textit{Hinode}}\/XRT, and {\\textit{IRIS}} teams for their open data policy. {\\textit{Hinode}} is a Japanese mission developed and launched by ISAS\/JAXA, with NAOJ as domestic partner and NASA and STFC (UK) as international partners. It is operated by these agencies in co-operation with ESA and the NSC (Norway). {\\textit{IRIS}} is a NASA small explorer mission developed and operated by LMSAL with mission operations executed at NASA Ames Research Center and major contributions to downlink communications funded by ESA and the Norwegian Space Centre.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe discrete logarithm problem (DLP) was first proposed as a hard\nproblem in cryptography in the seminal article of Diffie and\nHellman~\\cite{DiHe76}. Since then, together with factorization, it has\nbecome one of the two major pillars of public key cryptography. As a\nconsequence, the problem of computing discrete logarithms has\nattracted a lot of attention. From an exponential algorithm in $1976$,\nthe fastest DLP algorithms have been greatly improved during the past\n$35$ years. A first major progress was the realization that the DLP in\nfinite fields can be solved in subexponential time, i.e. $L(1\/2)$\nwhere $L_N(\\alpha)=\\exp\\left(O((\\log N)^\\alpha(\\log\\log\n  N)^{1-\\alpha})\\right)$. The next step further reduced this to a\nheuristic $L(1\/3)$ running time in the full range of finite fields,\nfrom fixed characteristic finite fields to prime\nfields~\\cite{Adl79,Cop84,Gor93,Adl94,JoLe06,JLVS07}.\n\nRecently, practical and theoretical advances have been\nmade~\\cite{Jo13faster,GGMZ13,Joux13} with an\nemphasis on small to medium characteristic finite fields and composite\ndegree extensions. The most general and efficient\nalgorithm~\\cite{Joux13} gives a complexity of $L(1\/4+o(1))$ when the\ncharacteristic is smaller than the square root of the extension\ndegree. Among the ingredients of this approach, we find the use of a\nvery particular representation of the finite field; the use of the\nso-called {\\em systematic equation}\\footnote{While the terminology is\nsimilar, no parallel is to be made with the systematic equations as\ndefined in early works related to the computation discrete logarithms in\n${\\mathbb F}_{2^n}$, as~\\cite{BlFuMuVa84}.}; and the use of algebraic\nresolution of bilinear polynomial systems in the individual logarithm\nphase.\n\nIn this work, we present a new discrete logarithm algorithm, in\nthe same vein as in~\\cite{Joux13} that uses an asymptotically more\nefficient descent approach. The main result gives a {\\it\n  quasi-polynomial} heuristic complexity for the DLP in finite fields\nof small characteristic. By quasi-polynomial, we mean a complexity of\ntype $n^{O(\\log n)}$ where $n$ is the bit-size of the cardinality of\nthe finite field. Such a complexity is smaller than any\n$L(\\epsilon)$ for $\\epsilon>0$. It remains super-polynomial\nin the size of the input, but offers a major asymptotic improvement\ncompared to $L(1\/4+o(1))$.\n\nThe key features of our algorithm are the following.\n\\begin{itemize}\n\\item We keep the field representation and the systematic equations of~\\cite{Joux13}.\n\\item The algorithmic building blocks are elementary. In particular,\n  we avoid the use of Gr\u00f6bner basis algorithms.\n\\item The complexity result relies on three key heuristics:\nthe existence of a polynomial representation of the appropriate\nform; the fact that the smoothness probabilities of some non-uniformly\ndistributed\npolynomials are similar to the probabilities for uniformly random\npolynomials of the same degree; and the linear independence of some\nfinite field elements related to the action of $\\PGL_2({\\mathbb F}_{q})$.\n\\end{itemize}\n\nThe heuristics are very close to the ones used in~\\cite{Joux13}. In\naddition to the arguments in favor of these heuristics already given\nin~\\cite{Joux13}, we performed some experiments to validate them on\npractical instances.  \\medskip\n\nAlthough we insist on the case of finite fields of small\ncharacteristic, where quasi-polynomial complexity is obtained, our new\nalgorithm improves the complexity of discrete logarithm computations in a\nmuch larger range of finite fields.\n\nMore precisely, in finite fields of the form ${\\mathbb F}_{q^k}$, where $q$\ngrows as $L_{q^k}(\\alpha)$, the complexity becomes\n$L_{q^k}(\\alpha+o(1))$. As a consequence, our algorithm is\nasymptotically faster than the Function Field Sieve algorithm in\nalmost all the range previously covered by this algorithm. Whenever \n$\\alpha<1\/3$, our new algorithm offers the smallest complexity. For\nthe limiting case $L(1\/3,c)$, the Function Field Sieve remains more\nefficient for small values of $c$, and the Number Field Sieve is better\nfor large values of $c$ (see~\\cite{JLVS07}).\n\\bigskip\n\nThis article is organized as follows. In Section~\\ref{sec:main}, we state\nthe main result, and discuss how it can be used to design a complete\ndiscrete logarithm algorithm. In Section~\\ref{sec:csq}, we analyze how\nthis result can be interpreted for various types of finite fields,\nincluding the important case of fields of small characteristic.\nSection~\\ref{sec:descent-one-step} is devoted to the description of our\nnew algorithm. It relies on heuristics that are discussed in\nSection~\\ref{sec:heur}, from a theoretical and a practical point of view.\nBefore getting to the conclusion, in Section~\\ref{sec:improvement}, we\npropose a few variants of the algorithm.\n\n\\section{Main result}\n\\label{sec:main}\n\nWe start by describing the setting in which our algorithm applies. It is\nbasically the same as in~\\cite{Joux13}: we need a large enough subfield,\nand we assume that a sparse representation can be found.  This is\nformalized in the following definition.\n\n\\begin{definition}\n    A finite field $K$ admits a {\\em sparse medium subfield representation} if\n    \\begin{itemize}\n        \\item it has a subfield of $q^2$ elements for a prime power $q$,\n\t\ti.e. $K$ is isomorphic to ${\\mathbb F}_{q^{2k}}$ with $k\\geq1$;\n        \\item there exist two polynomials $h_0$ and $h_1$ over\n            ${\\mathbb F}_{q^2}$ of small degree, such that $h_1X^q-h_0$ has a\n            degree $k$ irreducible factor.\n    \\end{itemize}\n\\end{definition}\n\nIn what follows, we will assume that all the fields under consideration\nadmit a sparse medium subfield representation. Furthermore, we assume that\nthe degrees of the polynomials $h_0$ and $h_1$ are uniformly bounded by a\nconstant $\\delta$.  Later, we will provide heuristic arguments for the\nfact that any finite field of the form ${\\mathbb F}_{q^{2k}}$ with $k \\le q+2$\nadmits a sparse medium subfield representation with polynomials $h_0$ and\n$h_1$ of degree at most 2.  But in fact, for our result to hold, allowing\nthe degrees of $h_0$ and $h_1$ to be bounded by any constant $\\delta$\nindependent of $q$ and $k$ or even allowing $\\delta$ to grow\nslower than $O(\\log q)$ would be sufficient.\n\nIn a field in sparse medium subfield representation, elements will\nalways be represented as polynomials of degree less than $k$ with\ncoefficients in ${\\mathbb F}_{q^2}$. When we talk about the discrete logarithm of\nsuch an element, we implicitly assume that a basis for this discrete\nlogarithm has been chosen, and that we work in a subgroup whose order has\nno small irreducible factor (we refer to the Pohlig-Hellman\nalgorithm~\\cite{PoHe78} to limit ourselves to this case).\n\n\\begin{prop}\\label{prop:onestep}\n    Let $K={\\mathbb F}_{q^{2k}}$ be a finite field that admits a sparse medium subfield\n    representation.\n    Under the heuristics explained below, there exists an algorithm whose\n    complexity is polynomial in $q$ and $k$ and which can be used for the\n    following two tasks. \n\n    \\begin{enumerate}\n    \\item \n        Given an element of $K$ represented by a polynomial\n        $P\\in{\\mathbb F}_{q^2}[X]$ with $2\\leq \\deg P\\leq k-1$,\n        the algorithm returns an expression of\n        $\\log P(X)$ as a linear combination of at most $O(kq^2)$\n        logarithms $\\log P_i(X)$ with $\\deg P_i \\leq \\lceil\n        \\frac12 \\deg P\\rceil$ and of $\\log h_1(X)$.\n\n    \\item\n        The algorithm returns the logarithm of $h_1(X)$ and \n        the logarithms of all the elements of $K$\n        of the form $X+a$, for $a$ in ${\\mathbb F}_{q^2}$.  \n     \\end{enumerate}\n\\end{prop}\n\nBefore the presentation of the algorithm, which is made in Section~\\ref{sec:descent-one-step}, we explain how to use it as a building block for a complete discrete logarithm algorithm.\n\nLet $P(X)$ be an element of $K$ for which we want to compute the discrete\nlogarithm. Here $P$ is a polynomial of degree at most $k-1$ and with\ncoefficients in ${\\mathbb F}_{q^2}$. We start by applying the algorithm of Proposition~\\ref{prop:onestep} to $P$. We obtain a relation of the form\n$$ \\log P = e_0 \\log h_1 + \\sum e_i \\log P_i,$$\nwhere the sum has at most $\\kappa q^2 k$ terms for a constant\n$\\kappa$ and the $P_i$'s have degree at most $\\lceil \\frac12 \\deg P\\rceil$.\nThen, we apply\nrecursively the algorithm to the $P_i$'s, thus creating a descent\nprocedure where at each step, a given element $P$  is expressed as a\nproduct of elements, whose degree is at most half the degree of $P$\n(rounded up)  and the arity of the descent tree is in $O(q^2 k)$.\n\nAt the end of the process, the logarithm of $P$ is expressed as a linear\ncombination of the logarithms of $h_1$ and of the linear polynomials,\nfor which the logarithms are computed with the algorithm in\nProposition~\\ref{prop:onestep} in its second form.\n\nWe are left with the complexity analysis of the descent\nprocess. Each internal node of the descent tree corresponds to one application of\nthe algorithm of Proposition~\\ref{prop:onestep}, therefore each internal\nnode has a cost which is bounded by a polynomial in $q$ and~$k$. The total cost\nof the descent is therefore bounded by the number of nodes in the descent\ntree times a polynomial in $q$ and $k$.  The depth of the descent tree is in\n$O(\\log k)$. The number of nodes of the tree is then less than or equal to\nits arity raised to the power of its depth, which is $(q^2\nk)^{O(\\log k)}$.  Since any polynomial in $q$ and\n$k$ is absorbed in the $O()$ notation in the exponent, we obtain the\nfollowing result.\n\n\\begin{theo}\\label{thm}\n    Let $K={\\mathbb F}_{q^{2k}}$ be a finite field that admits a sparse medium\n    subfield representation. Assuming the same heuristics as in\n    Proposition~\\ref{prop:onestep}, any discrete logarithm in $K$ can be\n    computed in a time bounded by \n    $$ \\max(q,k)^{O(\\log k)}.$$\n\\end{theo}\n\n\n\\section{Consequences for various ranges of parameters}\n\\label{sec:csq}\n\nWe now discuss the implications of Theorem~\\ref{thm} depending on the\nproperties of the finite field ${\\mathbb F}_Q$ where we want to compute discrete\nlogarithms in the first place. The complexities will be expressed in\nterms of $\\log Q$, which is the size of the input.\n\nThree cases are considered. In the first one, the finite field admits a\nsparse medium subfield representation, where $q$ and $k$ are almost\nequal. This is the optimal case. Then we consider the case where the\nfinite field has small (maybe constant) characteristic. And finally, we\nconsider the case where the characteristic is getting larger so that the\nonly available subfield is a bit too large for the algorithm to have an\noptimal complexity.\n\nIn the following, we always assume that for any field of the form\n${\\mathbb F}_{q^{2k}}$, we can find a sparse medium subfield representation.\n\n\\subsection{Case where the field is ${\\mathbb F}_{q^{2k}}$, with $q\\approx k$}\n\nThe finite fields ${\\mathbb F}_Q = {\\mathbb F}_{q^{2k}}$ for which $q$ and $k$ are almost\nequal are tailored for our algorithm. In that case, the complexity of\nTheorem~\\ref{thm} becomes $q^{O(\\log q)}$. Since $Q \\approx q^{2q}$, we\nhave $q=(\\log Q)^{O(1)}$. This gives an expression of the form\n$2^{O\\left((\\log \\log Q)^2\\right)}$, which is sometimes called\nquasi-polynomial in complexity theory.\n\n\\begin{cor}\\label{cor1}\n    For finite fields of cardinality $Q = q^{2k}$ with $q+O(1)\\geq k$\n    and $q=(\\log Q)^{O(1)}$,\n    there exists a heuristic algorithm for computing discrete logarithms\n    in quasi-polynomial time\n    $$ 2^{O\\left((\\log \\log Q)^2\\right)}.$$\n\\end{cor}\n\nWe mention a few cases which are almost directly covered by\nCorollary~\\ref{cor1}. First, we consider the case where $Q=p^n$ with\n$p$ a prime bounded by $(\\log\nQ)^{O(1)}$, and yet large enough so that $n \\le (p+\\delta)$. In this\ncase ${\\mathbb F}_Q$, or possibly ${\\mathbb F}_{Q^2}$ if $n$ is odd, can be represented in\nsuch a way that Corollary~\\ref{cor1} applies.\n\nMuch the same can be said in the case where $n$ is composite and factors\nnicely, so that ${\\mathbb F}_Q$ admits a large enough subfield ${\\mathbb F}_q$ with\n$q=p^m$. This can be used to solve certain discrete logarithms in, say,\n${\\mathbb F}_{2^n}$ for adequately chosen $n$ (much similar to records tackled\nby~\\cite{record1778,record1971,record4080,record6120,record6168}).\n\n\\subsection{Case where the characteristic is polynomial in the input size}\n\nLet now ${\\mathbb F}_Q$ be a finite field whose characteristic $p$ is bounded by\n$(\\log Q)^{O(1)}$, and let $n=\\log Q \/ \\log p$, so that $Q = p^n$. While\nwe have seen that Corollary~\\ref{cor1} can be used to treat some cases,\nits applicability might be hindered by the absence of an appropriately\nsized subfield: $p$ might be as small as $2$,\nand $n$ might not factor adequately. In those cases, we use the same strategy as\nin~\\cite{Joux13} and embed the discrete logarithm problem in ${\\mathbb F}_Q$ into\na discrete logarithm problem in a larger field.\n\nLet $k$ be $n$ if $n$ is odd and $n\/2$ if $n$ is even. Then, we set\n$q = p^{\\lceil \\log_p k \\rceil}$, and we work in the field ${\\mathbb F}_{q^{2k}}$.\nBy construction this field contains ${\\mathbb F}_Q$ (because $p|q$ and $n|2k$) and\nit is in the range of applicability of Theorem~\\ref{thm}. Therefore,\none can solve a discrete logarithm problem in ${\\mathbb F}_Q$ in time\n$\\max(q, k)^{O(\\log k)}$. Rewriting this complexity in terms of $Q$, we get\n$\\log_p(Q)^{O(\\log\\log Q)}$. And finally, we get a similar complexity\nresult as in the previous case. Of course, since we had to embed in a\nlarger field, the constant hidden in the $O()$ is larger than for\nCorollary~\\ref{cor1}.\n\n\\begin{cor}\\label{cor2}\n    For finite fields of cardinality $Q$ and characteristic bounded by\n    $\\log(Q)^{O(1)}$, there exists a heuristic algorithm for \n    computing discrete logarithms in quasi-polynomial time\n    $$ 2^{O\\left((\\log \\log Q)^2\\right)}.$$\n\\end{cor}\n\nWe emphasize that the case ${\\mathbb F}_{2^n}$ for a prime $n$ corresponds to\nthis case. A direct consequence of Corollary~\\ref{cor2} is that discrete logarithms in ${\\mathbb F}_{2^n}$ can be computed in\nquasi-polynomial time $2^{O((\\log n)^2)}$.\n\n\\subsection{Case where $q = L_{q^{2k}}(\\alpha)$}\nIf the characteristic of the base field is not so small compared to\nthe extension degree, the complexity of our algorithm does not keep\nits nice quasi-polynomial form. However, in almost the whole range of\napplicability of the Function Field Sieve algorithm, our algorithm is\nasymptotically better than FFS.\n\nWe consider here finite fields that can be put into the form ${\\mathbb F}_Q =\n{\\mathbb F}_{q^{2k}}$, where $q$ grows not faster than an expression of the form\n$L_Q(\\alpha)$. In the following, we assume that there is equality, which\nis of course the worst case. The condition can then be rewritten as\n$\\log q = O((\\log Q)^\\alpha(\\log\\log\nQ)^{1-\\alpha})$ and therefore $k = \\log Q \/ \\log q = O((\\log Q \/ \\log\\log\nQ)^{1-\\alpha})$. In particular we have $k\\leq q+\\delta$, so that\nTheorem~\\ref{thm} can be applied and gives a complexity of $q^{O(\\log\nk)}$. This yields the following result.\n\n\\begin{cor}\\label{cor3}\n    For finite fields of the form ${\\mathbb F}_Q = {\\mathbb F}_{q^{2k}}$ where $q$ is\n    bounded by $L_Q(\\alpha)$, there exists a heuristic algorithm for computing\n    discrete logarithms in subexponential time\n    $$ L_Q(\\alpha)^{O(\\log \\log Q)}.$$\n\\end{cor}\n\nThis complexity is smaller than $L_Q(\\alpha')$ for any $\\alpha' >\n\\alpha$. Hence, for any $\\alpha<1\/3$, our algorithm is faster than the\nbest previously known algorithm, namely FFS and its variants.\n\n\n\\section{Main algorithm: proof of Proposition~\\ref{prop:onestep}}\n\\label{sec:descent-one-step}\n\nThe algorithm is essentially the same for proving the two points of\nProposition~\\ref{prop:onestep}. The strategy is to find relations between\nthe given polynomial $P(X)$ and its translates by a constant in\n${\\mathbb F}_{q^2}$. Let $D$ be the degree of $P(X)$, that we assume to be at\nleast 1 and at most $k-1$.\n\nThe key to find relations is the {\\em systematic equation}:\n\\begin{equation}\\label{eq:frobenius}\n    X^q-X=\\prod_{a\\in {\\mathbb F}_q}(X-a)\\text.\n\\end{equation}\n\nWe like to view Equation~\\eqref{eq:frobenius} as involving\nthe projective line $\\ensuremath{\\mathbb{P}}^1({\\mathbb F}_q)$. Let $\\ensuremath{\\mathcal S}=\\{(\\alpha,\\beta)\\}$ be a set\nof representatives of the $q+1$ points $(\\alpha:\\beta)\\in\\ensuremath{\\mathbb{P}}^1({\\mathbb F}_q)$,\nchosen adequately so that the following equality holds.\n\\begin{equation}\n    \\label{eq:frobenius-proj}\n    X^qY-XY^q=\\prod_{(\\alpha,\\beta)\\in\\ensuremath{\\mathcal S}}(\\beta X-\\alpha Y)\\text.\n\\end{equation}\n\nTo make translates of $P(X)$ appear, we consider the action of {\\em\nhomographies}. \nAny matrix $m = \\begin{pmatrix}a & b\\\\ c& d\\end{pmatrix}$ acts on $P(X)$\nwith the following formula:\n$$m\\cdot P = \\frac{aP+b}{cP+d}.$$\nIn the following, this action will become trivial if the matrix $m$ has\nentries that are defined over ${\\mathbb F}_q$. This is also the case if $m$\nis non-invertible. Finally, it is clear that multiplying all the\nentries of $m$ by a non-zero constant does not change its action on\n$P(X)$. Therefore the matrices of the homographies that we consider are\ngoing to be taken in the following set of cosets:\n$$ \\ensuremath{\\mathcal{P}}_q = \\PGL({\\mathbb F}_{q^2}) \/ \\PGL({\\mathbb F}_q).$$\n(Note that in general $\\PGL_2({\\mathbb F}_q)$ is not a\nnormal subgroup of $\\PGL_2({\\mathbb F}_{q^2})$, so that $\\ensuremath{\\mathcal{P}}_q$ is not a quotient\ngroup.) \n\nTo each element $m = \\begin{pmatrix}a & b\\\\ c& d\\end{pmatrix}\\in\n\\ensuremath{\\mathcal{P}}_q$, we associate the equation~\\eqref{eq:Em} obtained by substituting $aP+b$\nand $cP+d$ in place of $X$ and $Y$ in \nEquation~\\eqref{eq:frobenius-proj}.\n\\def\\mathop{\\raise-.0125ex\\hbox{x}}{\\mathop{\\raise-.0125ex\\hbox{x}}}\n\\begin{align*}\n    \\tag{$E_m$}\\label{eq:Em}\n(aP+b)^q(cP+d) - (aP+b)(cP+d)^q & =\n    \\prod_{(\\alpha,\\beta)\\in\\ensuremath{\\mathcal S}} \\beta(aP+b) - \\alpha(cP+d) \\\\\n & =\\prod_{(\\alpha,\\beta)\\in\\ensuremath{\\mathcal S}}\n       (-c\\alpha + a\\beta)  P - (d\\alpha - b\\beta) \\\\\n & =\\lambda\\prod_{(\\alpha,\\beta)\\in\\ensuremath{\\mathcal S}}\n       P - \\mathop{\\raise-.0125ex\\hbox{x}}(m^{-1} \\cdot (\\alpha:\\beta))\\text.\n\\end{align*}\nThis sequence of formulae calls for a short comment because of an abuse\nof notation in the last expression. First, $\\lambda$ is the constant in\n${\\mathbb F}_{q^2}$ which makes the leading terms of the two sides match. Then,\nthe term $P-\\mathop{\\raise-.0125ex\\hbox{x}}(m^{-1} \\cdot\n(\\alpha:\\beta))$ denotes $P-u$ when $m^{-1} \\cdot (\\alpha:\\beta)=(u:1)$\n(whence we have $u=\\frac{d\\alpha - b\\beta}{-c\\alpha + a\\beta}$), or $1$\nif $m^{-1} \\cdot (\\alpha:\\beta)=\\infty$. The latter may occur since when\n$a\/c$ is in ${\\mathbb F}_q$, the expression $-c\\alpha + a\\beta$ vanishes for a\npoint $(\\alpha:\\beta)\\in\\ensuremath{\\mathbb{P}}^1({\\mathbb F}_{q})$ so that one of the factors of the\nproduct contains no term in $P(X)$.\n \nHence the right-hand side of Equation~\\eqref{eq:Em} is, up to a\nmultiplicative constant, a product of $q+1$ or $q$ translates of the\ntarget $P(X)$ by elements of\n${\\mathbb F}_{q^2}$. The equation obtained is actually related to the set of\npoints $m^{-1}\\cdot\\ensuremath{\\mathbb{P}}^1({\\mathbb F}_q)\\subset \\ensuremath{\\mathbb{P}}^1({\\mathbb F}_{q^2})$.\n\\medskip\n\n\nThe polynomial on the left-hand side of~\\eqref{eq:Em} can be rewritten as\na smaller degree equivalent. For this, we use the\nspecial form of the defining polynomial: in $K$ we have $X^q \\equiv\n\\frac{h_0(X)}{h_1(X)}$.  Let us denote by $\\tilde{a}$ the element $a^q$ when $a$\nis any element of ${\\mathbb F}_{q^2}$. Furthermore, we write\n$\\tilde{P}(X)$ the polynomial $P(X)$ with all its coefficients\nraised to the power $q$. The left-hand side of~\\eqref{eq:Em} is\n$$(\\tilde{a}\\tilde{P}(X^q)+\\tilde{b})(cP(X)+d)\n- (aP(X) + b)(\\tilde{c}\\tilde{P}(X^q)+\\tilde{d}),$$\nand using the defining equation for the field $K$, it is congruent to\n$$\n\\ensuremath{\\mathcal{L}}_m \\mathrel{:=} \\left(\\tilde{a}\\tilde{P}\\left(\\frac{h_0(X)}{h_1(X)}\\right)+\\tilde{b}\\right)(cP(X)+d)\n- (aP(X) +\nb)\\left(\\tilde{c}\\tilde{P}\\left(\\frac{h_0(X)}{h_1(X)}\\right)+\\tilde{d}\\right).\n$$\nThe denominator of $\\ensuremath{\\mathcal{L}}_m$ is a\npower of~$h_1$ and its numerator has degree at most $(1+\\delta) D$ where\n$\\delta=\\max(\\deg h_0,\\deg h_1)$. We say that $m\\in\\ensuremath{\\mathcal{P}}_q$ yields a\nrelation if this numerator of $\\ensuremath{\\mathcal{L}}_m$ is $\\lceil D\/2 \\rceil$-smooth. \n\nTo any $m\\in\\ensuremath{\\mathcal{P}}_q$, we associate a row vector $v(m)$ of dimension $q^2+1$ in the following\nway. Coordinates are indexed by $\\mu\\in\\ensuremath{\\mathbb{P}}^1({\\mathbb F}_{q^2})$, and the value\nassociated to $\\mu\\in{\\mathbb F}_{q^2}$ is $1$ or $0$ depending on whether\n$P-\\mathop{\\raise-.0125ex\\hbox{x}}(\\mu)$ appears in the right-hand side of Equation~\\eqref{eq:Em}. Note that\nexactly $q+1$ coordinates are $1$ for each $m$. Equivalently, we may write\n\\begin{equation}\\label{eq:v(m)}\nv(m)_{\\mu\\in\\ensuremath{\\mathbb{P}}^1({\\mathbb F}_{q^2})}=\\left\\{\n    \\begin{array}{l}\n        1\\text{ if }\\mu=m^{-1}\\cdot(\\alpha:\\beta) \\text{ with }\n\t    (\\alpha:\\beta)\\in\\ensuremath{\\mathbb{P}}^1({\\mathbb F}_q),\\\\\n        0\\text{ otherwise}.\n    \\end{array}\n\\right.\n\\end{equation}\n\nWe associate to the polynomial $P$ a matrix $H(P)$ whose rows are\nthe vectors $v(m)$ for which $m$ yields a relation, taking at most one\nmatrix $m$ in each coset of $\\ensuremath{\\mathcal{P}}_q$. The validity of Proposition~\\ref{prop:onestep} crucially relies on the\nfollowing heuristic.\n\n\\begin{heuristic}\\label{heu:fullrank}\n    For any $P(X)$, the set of rows $v(m)$ for cosets\n$m\\in\\ensuremath{\\mathcal{P}}_q$ that yield a relation form a matrix which has full rank $q^2+1$.\n\\end{heuristic}\n\nAs we will note in Section~\\ref{sec:heur}, the matrix $H(P)$ is\nheuristically expected to have $\\Theta(q^3)$ rows, where the implicit\nconstant depends on $\\delta$. This means that for our decomposition\nprocedure to work, we rely on the fact that $q$ is large enough\n(otherwise $H(P)$ may have less than $q^2+1$ rows, which precludes the\npossibility that it have rank $q^2+1$).\n\\medskip\n\nThe first point of Proposition~\\ref{prop:onestep}, where we descend a\npolynomial $P(X)$ of degree $D$ at least 2, follows by linear algebra\non this matrix.\nSince we assume that the matrix has full rank, \nthen the vector $(\\ldots,0,1,0,\\ldots)$ with $1$\ncorresponding to $P(X)$ can be written as a linear combination of the rows.\nWhen doing this linear combination on the equations~\\eqref{eq:Em} corresponding to\n$P$ we write $\\log P(X)$ as a linear combination of $\\log P_i$ where\n$P_i(x)$ are the elements occurring in the\nleft-hand sides of the equations. Since there are $O(q^2)$ columns, the elimination process\ninvolves at most $O(q^2)$ rows, and since each row corresponds to\nan equation~\\eqref{eq:Em}, it involves at most $\\deg \\ensuremath{\\mathcal{L}}_m\\leq (1+\\delta)D$ polynomials in the\nleft-hand-side\\footnote{This estimate of the number of irreducible\n  factors is a pessimistic upper bound. In practice, one expects to\n  have only $O(\\log D)$ factors on average. Since the crude estimate\n  does not change the overall complexity, we keep it that way to avoid\n  adding another heuristic.}.  In total, the polynomial $D$ is\nexpressed by a linear combination of at most $O(q^2D)$ polynomials of\ndegree less than $\\lceil D\/2\\rceil$. The logarithm of $h_1(X)$ is also\ninvolved, as a denominator of $\\ensuremath{\\mathcal{L}}_m$. We have not made precise the\nconstant in ${\\mathbb F}_{q^2}^*$ which occurs to take care of the leading\ncoefficients. Since discrete logarithms in ${\\mathbb F}_{q^2}^*$ can certainly be\ncomputed in polynomial time in $q$, this is not a problem.\n\nSince the order of $\\PGL_2({\\mathbb F}_{q^i})$ is $q^{3i}-q^i$, the set of cosets\n$\\ensuremath{\\mathcal{P}}_q$ has $q^3+q$ elements. For each $m \\in\\ensuremath{\\mathcal{P}}_q$, testing\nwhether~\\eqref{eq:Em}\nyields a relation amounts to some polynomial manipulations and a \nsmoothness test. All of them can be done in polynomial time in $q$ and\nthe degree of $P(X)$ which is bounded by $k$. Finally, the linear algebra\nstep can be done in $O(q^{2\\omega})$ using asymptotically fast matrix\nmultiplication algorithms, or alternatively $O(q^5)$ operations using\nsparse matrix techniques.\nIndeed, we have $q+1$\nnon-zero entries per row and a size of $q^2+1$.\nTherefore, the overall cost is polynomial in $q$ and $k$ as claimed.\n\\medskip\n\nFor the second part of Proposition~\\ref{prop:onestep} we replace $P$ by $X$ during the\nconstruction of the matrix. In that case, both sides of the\nequations~\\eqref{eq:Em} involve only linear polynomials.\nHence we obtain a linear system whose unknowns\nare $\\log (X+a)$ with $a\\in{\\mathbb F}_{q^2}$. Since Heuristic~\\ref{heu:fullrank}\nwould give us only the full rank of the system corresponding to the\nright-hand sides of the equations~\\eqref{eq:Em}, we have to rely on a\nspecific heuristic for this step:\n\\begin{heuristic}\\label{heu:linfullrank}\n    The linear system constructed from all the equations~\\eqref{eq:Em}\n    for $P(X)=X$ has full rank.\n\\end{heuristic}\nAssuming that this heuristic holds, we can\nsolve the linear system and obtain the discrete logarithms of the linear\npolynomials and of $h_1(X)$. \n\n\\section{Supporting the heuristic argument in the proof}\n\\label{sec:heur}\n\nFor Heuristic~\\ref{heu:fullrank}, we propose two approaches to support this\nheuristic. Both allow to gain some confidence in the validity of\nthe heuristic, but of course none affect the heuristic nature of this\nstatement.\n\nFor the first line of justification, we denote by $\\ensuremath{\\mathcal{H}}$ the matrix of all\nthe $\\#\\ensuremath{\\mathcal{P}}_q=q^3+q$ vectors $v(m)$ defined as in\nEquation~\\eqref{eq:v(m)}. Associated to a polynomial~$P$,\nSection~\\ref{sec:descent-one-step} defines the matrix\n$H(P)$ formed of the\nrows $v(m)$ such that the numerator of $\\ensuremath{\\mathcal{L}}_m$ is smooth. We will give\nheuristics that $H(P)$ has $\\Theta(q^3)$ rows and then prove that $\\ensuremath{\\mathcal{H}}$ has\nrank $q^2+1$, which of course does not prove that its submatrix $H(P)$ has full rank. \n\nIn order to estimate the number of rows of $H(P)$ we assume that the\nnumerator of $\\ensuremath{\\mathcal{L}}_m$ has\nthe same probability to be $\\lceil \\frac{D}{2}\\rceil$-smooth as a random\npolynomial of same degree. In this paragraph, we assume that the\ndegrees of $h_0$ and $h_1$ are bounded by $2$, merely to avoid awkward\nnotations; the result holds for any constant bound $\\delta$.\nThe degree of the numerator of $\\ensuremath{\\mathcal{L}}_m$ is then bounded by $3D$, so we have\nto estimate\nthe probability that a polynomial in ${\\mathbb F}_{q^2}[X]$ of degree $3D$ is\n$\\lceil \\frac{D}{2}\\rceil$-smooth. For any prime power $q$ and integers\n$1\\leq m\\leq n$, we denote by $N_q(m,n)$ the number of $m$-smooth monic\npolynomials of degree $n$. Using analytic methods, Panario et\nal. gave a precise estimate of this quantity (Theorem~$1$\nof~\\cite{FGP98}):\n\\begin{equation}\\label{eq:Flajolet}\n\tN_{q}(n,m)=q^n \\rho\\left(\\frac{n}{m}\\right)\\left(1+O\\left(\\frac{\\log\nn}{m}\\right)\\right),\n\\end{equation}\nwhere $\\rho$ is Dickman's function defined as the unique continuous\nfunction such that $\\rho(u)=1$ on $[0,1]$ and $u\\rho'(u)=\\rho(u-1)$ for\n$u>1$. We stress that the constant $\\kappa$ hidden in the $O()$ notation\nis independent of $q$.\nIn our case, we are interested in the value of $N_{q^2}(3D, \\lceil\n\\frac{D}{2}\\rceil)$. Let us call $D_0$ the least\ninteger such that $1+\\kappa\\left(\\frac{\\log (3D)}{\\lceil D\/2\\rceil}\\right)$\nis at least $1\/2$. For $D>D_0$, we will use the\nformula~\\eqref{eq:Flajolet}; and for $D\\le D_0$, we will use the crude\nestimate $N_q(n,m) \\ge N_q(n,1) = q^n\/n!$. Hence the smoothness\nprobability of $\\ensuremath{\\mathcal{L}}_m$ is at least\n$\\min\\left(\\frac{1}{2}\\rho(6),1\/(3D_0)!\\right)$.\n\nMore generally, if $\\deg h_0$ and $\\deg h_1$ are bounded by a constant\n$\\delta$ then we have a smoothness probability of $\\rho(2\\delta+2)$ times\nan absolute constant. Since we have $q^3+q$ candidates and a constant\nprobability of success, $H(P)$ has $\\Theta(q^3)$ rows.\n\n\n\nNow, unless some theoretical obstruction\noccurs, we expect a matrix over ${\\mathbb F}_\\ell$ to have full rank with\nprobability at least $1-\\frac{1}{\\ell}$. The matrix $\\ensuremath{\\mathcal{H}}$ is however peculiar, and does enjoy\nregularity properties which are worth noticing.\nFor instance, we have the following proposition.\n\\begin{prop}\n    \\label{prop:bigmat-fullrank}\nLet $\\ell$ be a prime not dividing $q^3-q$. Then the matrix\n$\\ensuremath{\\mathcal{H}}$ over ${\\mathbb F}_\\ell$ has full rank $q^2+1$.\n\\end{prop}\n\\begin{proof}\n    We may obtain this result in two ways. First, \n\\ensuremath{\\mathcal{H}}\\ is\nthe incidence matrix of a $3-(q^2+1,q+1,1)$ combinatorial design called\n\\emph{inversive plane} (see e.g.~\\cite[Theorem 9.27]{Stinson03}). As such\nwe obtain the identity $$\\ensuremath{\\mathcal{H}}^T\\ensuremath{\\mathcal{H}}=(q+1)(J_{q^2+1}-(1-q)I_{q^2+1})$$\n(see~\\cite[Theorem 1.13 and Corollary 9.6]{Stinson03}), where\n$J_n$ is the $n\\times n$ matrix with all entries equal to one, and $I_n$\nis the $n\\times n$ identity matrix. This readily gives the result exactly\nas announced.\n\n    We also provide an elementary proof of the Proposition.\n    We have a\n    bijection between rows of $\\ensuremath{\\mathcal{H}}$ and the different possible image\n    sets of the projective line $\\ensuremath{\\mathbb{P}}^1({\\mathbb F}_q)$ within $\\ensuremath{\\mathbb{P}}^1({\\mathbb F}_{q^2})$,\n    under injections of the form $(\\alpha:\\beta)\\mapsto\n    m^{-1}\\cdot(\\alpha:\\beta)$. All these $q^3+q$ image sets have size\n    $q+1$, and by symmetry all points of $\\ensuremath{\\mathbb{P}}^1({\\mathbb F}_{q^2})$ are reached\n    equally often. Therefore, the sum of all rows of $\\ensuremath{\\mathcal{H}}$ is the\n    vector whose coordinates are all equal to $\\frac1{1+q^2}(q^3+q)(q+1)=q^2+q$.\n\n    Let us now consider the sum of the rows in $\\ensuremath{\\mathcal{H}}$ whose first\n    coordinate is $1$ (as we have just shown, we have $q^2+q$ such rows).\n    Those correspond to image sets of $\\ensuremath{\\mathbb{P}}^1({\\mathbb F}_q)$ which contain one\n    particular point, say $(0:1)$. The value of the sum for any other\n    coordinate indexed by e.g.\\ $Q\\in\\ensuremath{\\mathbb{P}}^1({\\mathbb F}_{q^2})$ is the number of\n    image sets $m^{-1}\\cdot\\ensuremath{\\mathbb{P}}^1({\\mathbb F}_q)$ which contain both $(0:1)$ and\n    $Q$, which we prove is equal to $q+1$ as follows. Without loss of generality, we may assume $Q=\\infty=(1:0)$.\n    We need to count the relevant\n    homographies $m^{-1}\\in\\PGL_2({\\mathbb F}_{q^2})$, modulo\n    $\\PGL_2({\\mathbb F}_q)$-equivalence $m\\equiv hm$. By\n    $\\PGL_2({\\mathbb F}_q)$-equivalence, we may without loss of generality assume\n    that $m^{-1}$ fixes $(0:1)$ and $(1:0)$.\n    Letting $m^{-1}=\n\\begin{pmatrix}a&b\\\\c&d\\end{pmatrix}$, we obtain $(b:d)=(0:1)$ and\n    $(a:c)=(1:0)$, whence $b=c=0$, and both $a,d\\not=0$. We may\n    normalize to $d=1$, and notice that multiplication of $a$ by a scalar\n    in ${\\mathbb F}_q^*$ is absorbed in $\\PGL_2({\\mathbb F}_q)$-equivalence. Therefore\n    the number of suitable $m$ is $\\#{{\\mathbb F}_{q^2}^*}\/{{\\mathbb F}_q^*}=q+1$.\n\n\n    These two facts show that the row span of $\\ensuremath{\\mathcal{H}}$\n    contains the vectors $(q^2+q, \\ldots, q^2+q)$ and $(q^2+q, q+1,\n    \\ldots, q+1)$. The vector $(q^3-q,0,\\ldots,0)$ is obtained as a linear\n    combination of these two vectors, which suffices to prove that\n    $\\ensuremath{\\mathcal{H}}$ has full rank, since the same reasoning holds\n    for any coordinate.\n\n\\end{proof}\n\n\nProposition~\\ref{prop:bigmat-fullrank}, while encouraging, is clearly not\nsufficient. We are, at the moment, unable to provide a proof of a\nmore useful statement. On the experimental side, it is reasonably easy to\nsample arbitrary subsets of the rows of $\\ensuremath{\\mathcal{H}}$ and check for their rank.\nTo this end, we propose the following experiment. We have considered\nsmall values of $q$ in the range $[16,\\ldots,64]$, and made~50 random picks of\nsubsets $S_i\\subset\\ensuremath{\\mathcal{P}}_q$, all of size exactly $q^2+1$. For each we\nconsidered the matrix of the corresponding linear system, which is made of\nselected rows of the matrix \\ensuremath{\\mathcal{H}}, and computed its determinant $\\delta_i$.\nFor all values of $q$ considered,\nwe have observed the following facts.\n\\begin{itemize}\n    \\item First, all square matrices considered had full rank over \\ensuremath{\\mathbb{Z}}.\n        Furthermore, their determinants had no common factor apart\n        possibly from those appearing in the factorization of $q^3-q$ as\n        predicted by Proposition~\\ref{prop:bigmat-fullrank}. In fact,\n        experimentally it seems that only the factors of $q+1$ are\n        causing problems.\n    \\item We also explored the possibility that modulo some primes, the\n        determinant could vanish with non-negligible probability. We thus\n        computed the pairwise GCD of all~50 determinants computed, for\n        each $q$. Again, the only prime factors appearing in the GCDs\n        were either originating from the factorization of $q^3-q$, or\n        sporadically from the birthday paradox.\n\\end{itemize}\n        \\begin{table}\n\\begin{center}\n    \\begin{minipage}[t]{0.5\\textwidth}\n\\begin{tabular}{c|c|l|l}\n    $q$ & \\#trials &  in $\\gcd(\\{\\delta_i\\})$ &\n        in $\\gcd(\\delta_i, \\delta_j)$\\\\\n    \\hline\n16 & 50 & 17 & 691\\\\\n17 & 50 & 2, 3 & 431, 691\\\\\n19 & 50 & 2, 5 & none above $q^2$\\\\\n23 & 50 & 2, 3 & none above $q^2$\\\\\n25 & 50 & 2, 13 & none above $q^2$\\\\\n27 & 50 & 2, 7 & 1327\\\\\n29 & 50 & 2, 3, 5 & none above $q^2$\\\\\n31 & 50 & 2 & 1303, 3209\\\\\n32 & 50 & 3, 11 & none above $q^2$\\\\\n\n    \n\\end{tabular}\n    \\end{minipage}%\n    \\begin{minipage}[t]{0.5\\textwidth}\n\\begin{tabular}{c|c|l|l}\n    $q$ & \\#trials &  in $\\gcd(\\{\\delta_i\\})$ &\n        in $\\gcd(\\delta_i, \\delta_j)$\\\\\n    \\hline\n37 & 50 & 2, 19 & 2879\\\\\n41 & 50 & 2, 3, 7 & none above $q^2$\\\\\n43 & 50 & 2, 11 & none above $q^2$\\\\\n47 & 50 & 2, 3 & none above $q^2$\\\\\n49 & 50 & 2, 5 & none above $q^2$\\\\\n53 & 50 & 2, 3 & none above $q^2$\\\\\n59 & 50 & 2, 3, 5 & none above $q^2$\\\\\n61 & 50 & 2, 31 & none above $q^2$\\\\\n64 & 50 & 5, 13 & none above $q^2$\\\\\n\\end{tabular}\n    \\end{minipage}%\n\n\\caption{\\label{tab:experiment1}Prime factors appearing in determinant of\nrandom square submatrices of \\ensuremath{\\mathcal{H}}\\ (for one given set of random trials)}\n\\end{center}\n        \\end{table}\n        These results are \nsummarized in\ntable~\\ref{tab:experiment1}, where the last column omits small prime\nfactors below $q^2$.\nOf course, we remark that considering square submatrices is a more demanding check than\nwhat Heuristic~\\ref{heu:fullrank} suggests, since our algorithm only\nneeds a slightly larger matrix of size $\\Theta(q^3)\\times(q^2+1)$ to have\nfull rank.\n\\medskip\n\nA second line of justification is more direct and natural, as it is\npossible to implement the algorithm outlined in\nSection~\\ref{sec:descent-one-step}, and verify that it does provide the\ndesired result. A \\textsc{Magma} implementation validates this claim, and\nhas been used to implement descent steps for an example field of\ndegree~$53$ over ${\\mathbb F}_{53^2}$. An example step in this context is given\nfor applying our algorithm to a polynomial of degree~10, attempting to\nreduce it to polynomials of degree~6 or less.\nAmong the 148,930 elements\nof $\\ensuremath{\\mathcal{P}}_q$, it sufficed to consider only 71,944 matrices $m$, of which about 3.9\\%\nled to relations, for a minimum sufficient number of relations equal to\n$q^2+1=2810$ (as more than half of the elements of $\\ensuremath{\\mathcal{P}}_q$ had not even\nbeen examined at this point, it is clear that getting more relations was\neasy---we did not have to).\nAs the defining polynomial\nfor the finite field considered was constructed with $\\delta=\\deg\nh_{0,1}=1$, all left-hand sides involved\nhad degree 20.\nThe polynomials appearing in their\nfactorizations had the\nfollowing degrees (the number in brackets give the number of distinct\npolynomials found for each degree): 1(2098), 2(2652), 3(2552), 4(2463), 5(2546), 6(2683).\nOf course this tiny example size uses no\noptimization, and is only intended to check the validity of\nProposition~\\ref{prop:onestep}.\n\n\\bigskip\n\nAs for Heuristic~\\ref{heu:linfullrank}, it is already present\nin~\\cite{Joux13} and~\\cite{GGMZ13}, so this is not a new heuristic.\nJust like for Heuristic~\\ref{heu:fullrank}, it is based on the fact that\nthe probability that a left-hand side is $1$-smooth and yields a relation\nis constant. Therefore, we have a system with $\\Theta(q^3)$ relations\nbetween $O(q^2)$ indeterminates, and it seems reasonable to expect that\nit has full rank. On the other hand, there is not as much algebraic\nstructure in the linear system as in Heuristic~\\ref{heu:fullrank}, so that\nwe see no way to support this heuristic apart from testing it on several\ninputs. This was already done (including for record computations)\nin~\\cite{Joux13} and~\\cite{GGMZ13}, so we do not elaborate on our own\nexperiments that confirm again that Heuristic~\\ref{heu:linfullrank} seems\nto be valid except for tiny values of $q$.\n\n\\paragraph{An obstruction to the heuristics.}\n\nAs noted by Cheng, Wan and Zhuang~\\cite{traps13}, the irreducible factors\nof $h_1X^q-h_0$ other than the degree $k$ factor that is used to define\n${\\mathbb F}_{q^{2k}}$ are problematic. Let $P$ be such a problematic polynomial.\nThe fact that it divides the defining equation implies that it also\ndivides the $\\ensuremath{\\mathcal{L}}_m$ quantity that is involved when trying to build a\nrelation that relates $P$ to other polynomials. Therefore the first part\nof Proposition~\\ref{prop:onestep} can not hold for this $P$. Similarly, if\n$P$ is linear, its presence will prevent the second part of\nProposition~\\ref{prop:onestep} to hold since the logarithm of $P$ can not\nbe found with the technique of Section~\\ref{sec:descent-one-step}.\nWe present here a technique to deal with the problematic polynomials.\n(The authors of~\\cite{traps13} proposed another solution to keep the\nquasi-polynomial nature of algorithm.)\n\n\n\\begin{prop}\\label{solvetrap}\nFor each problematic polynomial $P$ of degree $D$, we can find a linear relation\nbetween $\\log P$, $\\log h_1$ and $O(D)$ logarithms of polynomials of degree at\nmost $(\\delta-1)D$ which are not problematic.\n\\end{prop}\n\n\\begin{proof}\nLet $P$ be an irreducible factor of $h_1X^q-h_0$ of degree $D$.  Let us\nconsider $P^q$; by reducing modulo $h_1X^q-h_0$ and clearing\ndenominators, there exists a polynomial $A(X)$ such that\n\\begin{equation}\\label{eq:freerel}\nh_1^D P^q = h_1^D\\tilde{P}\\left(\\frac{h_0}{h_1}\\right)\n+(h_1X^q-h_0)A(X).\n\\end{equation}\nSince $P$ divides two of the terms of this equality, it must also\ndivide the third one, namely the polynomial $\\ensuremath{\\mathcal R} =\nh_1^D\\tilde{P}\\left(h_0\/h_1\\right)$. Let $v_P\\ge 1$ be the valuation of\n$P$ in $\\ensuremath{\\mathcal R}$. In the finite field ${\\mathbb F}_{q^{2k}}$ we obtain the following\nequalities between logarithms:\n\\begin{equation*}\n\t(q-v_P)\\log P = -D\\log h_1 +\\sum_i e_i \\log Q_i,\n\\end{equation*}\nwhere $Q_i$ are the irreducible factors of $\\ensuremath{\\mathcal R}$ other than $P$ and $e_i$\ntheir valuation in~$\\ensuremath{\\mathcal R}$. A polynomial $Q_i$ can not be problematic.\nOtherwise, it would divide the right-hand side of\nEquation~\\eqref{eq:freerel}, and therefore, also the left-hand side, which\nis impossible.\nSince $v_P\\leq \\frac{\\deg\n\\ensuremath{\\mathcal R}}{\\deg P}\\leq \\delta <q$, the quantity $q-v_P$ is invertible modulo\n$\\ell$ (we assume, as usual that $\\ell$ is larger than $q$) and we obtain\na relation between $\\log P$, $\\log h_1$ and the\nlogarithms of the non-problematic polynomials $Q_i$.\nThe degree of $\\ensuremath{\\mathcal R}\/P^{v_P}$ is at most $(\\delta-1)D$, which gives the\nclaimed bound on the degrees of the $Q_i$.\n\\end{proof}\n\nIf $\\delta\\le 2$, this proposition solves the issues raised\nby~\\cite{traps13} about\nproblematic polynomials. Indeed, for each problematic polynomial of\ndegree $D>1$, it will be possible to rewrite its logarithm in terms of\nlogarithms of non-problematic polynomials of at most the same degree that\ncan be descended in the usual way. Similarly, each problematic\npolynomial of degree 1 can have its logarithm rewritten in terms of the\nlogarithms of other non-problematic linear polynomials. Adding these\nrelations to the ones obtained in Section~\\ref{sec:descent-one-step}, we\nexpect to have a full-rank linear system.\n\nIf $\\delta>2$, we need to rely on the additional heuristic. Indeed, when\ndescending the $Q_i$ that have a degree potentially larger than the\ndegree of $D$, we could hit again the problematic polynomial we started\nwith, and it could be that the coefficients in front of $\\log P$ in the\nsystem vanishes. More generally, taking into account all the problematic\npolynomials, if when we apply Proposition~\\ref{solvetrap} to them we get\npolynomials $Q_i$ of higher degrees, it could be that descending those we\ncreates loops so that the logarithms of some of the problematic\npolynomials could not be computed. We expect this event to be very\nunlikely. Since in all our experiments it was always possible to obtain\n$\\delta=2$, we did not investigate further.\n\n\\paragraph{Finding appropriate $h_0$ and $h_1$.}\n\nOne key fact about the algorithm is the existence of two polynomials\n$h_0$ and $h_1$ in ${\\mathbb F}_{q^2}[X]$ such that $h_1(X)X^q-h_0(X)$ has an\nirreducible factor of degree $k$. A partial solution is due to\nJoux~\\cite{Joux13} who showed how to construct such polynomials when\n$k\\in\\{q-1,q,q+1\\}$. \nNo such deterministic construction is known in the general\ncase, but experiments show that one can apparently choose $h_0$ and $h_1$ of\ndegree at most $2$. We performed an experiment for every odd prime\npower $q$ in $[3,\\ldots,1000]$ and every $k\\leq q$ and found that we could\nselect $a\\in{\\mathbb F}_{q^2}$ such that $X^q+X^2+a$ has an irreducible factor\nof degree $k$. Finally, note that the result is similar to a commonly made\nheuristic in discrete logarithm algorithms: for fixed\n$f\\in{\\mathbb F}_{q^2}[X,Y]$ and random $g\\in{\\mathbb F}_{q^2}[X,Y]$, the polynomial\n$\\text{Res}_Y(f,g)$ behaves as a random polynomial of same degree with\nrespect to the degrees of its irreducible factors.\n\n\n\n\n\\section{Some directions of improvement}\n\\label{sec:improvement}\nThe algorithm can be modified in several ways. On the one hand one can\nobtain a better complexity if one proves\na stronger result on the smoothness probability. On the other\nhand, without changing the complexity, one can obtain a version which\nshould behave better in practice. \n\n\\subsection{Complexity improvement}\nHeuristic~\\ref{heu:fullrank} tells that a rectangular matrix with $\\Theta(q)$\ntimes more rows than columns has full rank. It seems reasonable to expect\nthat only a constant times more rows than columns would be enough to get\nthe full rank properties (as is suggested by the experiments proposed in\nSection~\\ref{sec:heur}). Then, it means that we expect to have a lot of\nchoices to select the best relations, in the sense that their left-hand\nsides split into irreducible factors of degrees as small as possible.\n\nOn average, we expect to be able to try $\\Theta(q)$ relations for each row\nof the matrix. So, assuming that the numerators of $\\ensuremath{\\mathcal{L}}_m$ behave like\nrandom polynomials of similar degrees, we have to evaluate the\nexpected smoothness that we can hope for after trying $\\Theta(q)$\npolynomials of degree $(1+\\delta)D$ over ${\\mathbb F}_{q^2}$. Set $u=\\log q \/\n\\log\\log q$, so that $u^u\\approx q$. According to~\\cite{FGP98} it is then possible to replace $\\lceil\nD\/2\\rceil$ in\nProposition~\\ref{prop:onestep} by the value $O(D\\log\\log q\/\\log q)$.\n\nThen, the discussion leading to Theorem~\\ref{thm} can be changed to\ntake this faster descent into account. We keep the same estimate for the\narity of each node in the tree, but the depth is now only in $\\log k \/\n\\log\\log q$. Since this depth ends up in the exponent, the resulting\ncomplexity in Theorem~\\ref{thm} is then\n$$ \\max(q, k)^{O(\\log k \/ \\log\\log q)}.$$\n\n\\subsection{Practical improvements}\nBecause of the arity of the descent tree, the breadth eventually\nexceeds the number of polynomials below some degree bound. It\nmakes no sense, therefore, to use the descent procedure beyond\nthis point, as the recovery of discrete logarithms of all these\npolynomials is better achieved as a pre-computation. Note that this\ncorresponds to the computations of the $L(1\/4+\\epsilon)$ algorithm which starts by\npre-computing the logarithms of polynomials up to degree $2$.\nIn our case, we could in principle go up to degree $O(\\log q)$ without\nchanging the complexity.\n\\medskip\n\nWe propose another practical improvement in the case where we would like\nto spend more time descending a given polynomial $P$ in order to improve\nthe quality of the descent tree rooted at $P$. \nThe set of polynomials appearing in the right-hand side of\nEquation~\\eqref{eq:Em} in Section~\\ref{sec:descent-one-step} is\n$\\{P-\\lambda\\}$, because in the factorization of $X^q-X$, we\nsubstitute $X$ with $m\\cdot P$ for homographies~$m$. In fact, we\nmay apply $m$ to $(P:P_1)$ for any polynomial $P_1$ whose degree\ndoes not exceed that of $P$. In the right-hand sides, we will have only\nfactors of form $P - \\lambda P_1$ for $\\lambda$ in ${\\mathbb F}_{q^2}$.\nOn the left-hand sides, we have polynomials of the same degree as before,\nso that the smoothness probability is expected to be the same.\nNevertheless, it is possible to test several $P_1$ polynomials, and to\nselect the one that leads to the best tree.\n\nThis strategy can also be useful in the following context (which will not\noccur for large enough $q$):\nit can\nhappen that for some triples $(q,D,D')$ one has $N_{q^2}(3D,D')\/q^n\\approx 1\/q$. In\nthis case we have no certainty that we can descend a degree-$D$\npolynomial to degree $D'$, but we can hope that at least one of the\n$P_1$ allows to descend.\n\nFinally, if one decides to use several auxiliary $P_1$ polynomials to descend\na polynomial $P$, it might be interesting to take a set of polynomials\n$P_1$ with an arithmetic structure, so that the smoothness tests on the\nleft-hand sides can benefit from a sieving technique.\n\n\\section{Conclusion}\nThe algorithm presented in this article achieves a significant improvement\nof the asymptotic complexity of discrete logarithm in finite fields, in\nalmost the whole range of parameters where the Function Field Sieve was\npresently the most competitive algorithm. Compared to existing\napproaches, and in particular to the line of recent\nworks~\\cite{Jo13faster,GGMZ13}, the practical relevance of our algorithm\nis not clear, and will be explored by further work. \n\nWe note that the analysis of the algorithm presented here is heuristic, as discussed\nin Section~\\ref{sec:heur}. Some of the heuristics we stated,\nrelated to the properties of matrices $H(P)$ extracted from the\nmatrix $\\ensuremath{\\mathcal{H}}$, seem accessible to more solid justification. It seems\nplausible to have the validity of algorithm rely on the sole heuristic of\nthe validity of the smoothness estimates.\n\nThe crossing point between the $L(1\/4)$ algorithm and our quasi-polynomial\none is not determined yet. One of the key factors which hinders the practical\nefficiency of this algorithm is the $O(q^2D)$ arity of the descent tree,\ncompared to the $O(q)$ arity achieved by techniques based on Gr\u00f6bner\nbases~\\cite{Jo13faster} at the expense of a $L(1\/4+\\epsilon)$ complexity.\nAdj et al.~\\cite{AMOR13} proposed to mix the two algorithms\nand deduced that the new descent technique must be used for cryptographic\nsizes. Indeed, by estimating the time required to\ncompute discrete logarithms in ${\\mathbb F}_{3^{6\\cdot 509}}$, they showed the weakness\nof some pairing-based cryptosystems. \n\n\\ifanon\n\\else\n\\section*{Acknowledgements}\nThe authors would like to thank Daniel J. Bernstein for his comments on\nan earlier version of this work, and for pointing out to us the possible\nuse of asymptotically fast linear algebra for solving the linear systems\nencountered.\n\n\\fi\n\n\\bibliographystyle{splncs03}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section*{Abstract}\nIn the current work, a problem-splitting approach and a scheme motivated by transfer learning is applied to a structural health monitoring problem. The specific\nproblem in this case is that of localising damage on an aircraft wing. The original experiment is described, together with the initial approach, in which a neural \nnetwork was trained to localise damage. The results were not ideal, partly because of a scarcity of training data, and partly because of the difficulty in \nresolving two of the damage cases. In the current paper, the problem is split into two sub-problems and an increase in classification accuracy is obtained. The \nsub-problems are obtained by separating out the most difficult-to-classify damage cases. A second approach to the problem is considered by adopting ideas from \ntransfer learning (usually applied in much deeper) networks to see if a network trained on the simpler damage cases can help with feature extraction in the more\ndifficult cases. The transfer of a fixed trained batch of layers between the networks is found to improve classification by making the classes more separable in the feature\nspace and to speed up convergence.\n\n\\textbf{Key words: Structural health monitoring (SHM), machine learning, classification, problem splitting, transfer learning.}\n\n\t\n\\section{Introduction}\n\\label{sec:intro}\n\nStructural health monitoring (SHM) refers to the process of implementing a damage detection strategy for aerospace, civil or mechanical engineering infrastructure \\cite{farrar2012structural}. Here, damage is defined as changes introduced into a system\/structure, either intentionally or unintentionally, that affect current or \nfuture performance of the system. Detecting damage is becoming more and more important in modern societies, where everyday activities depend increasingly on \nengineering systems and structures. One the one hand, safety has to be assured, both for users and for equipment or machinery existing within these structures. On \nthe other hand, infrastructure is often designed for a predefined lifetime and damage occurrence may reduce the expected lifetime and have a huge economic impact \nas a result of necessary repairs or even rebuilding or decommison. Damage can be visible on or in structures, but more often it is not, and has to be inferred from \nsignals measured by sensors placed on them.\n\nAn increasingly useful tool in SHM is {\\em machine learning} (ML) \\cite{farrar2012structural}. In many current applications large sets of data are gathered by sensors \nor generated by models and these can be exploited to gain insight into structural dynamics and materials engineering. Machine learning is employed because of its \nefficiency in classification, function interpolation and prediction using data. Data-driven models are built and used to serve SHM purposes. These models can also be \nused to further understand how structures react to different conditions and explain their physics. However, one of the main drawbacks of such methods is the need for \nlarge datasets. ML models may have many parameters which are established during {\\em training} on data which may need to span all the health conditions of interest for the\ngiven structure or system. Larger datasets assist in better tuning of the models as far as accuracy and generalisation are concerned. However, even if large datasets are available, sometimes there are very few observations on damaged states, which are important in SHM. In the current paper, increased \naccuracy of a data-driven SHM classifier will be discussed in terms of two strategies: splitting the problem into two sub-problems and attempting transfer of information\nbetween the two sub-problems in a manner motivated by transfer learning \\cite{Pan2010}.\n\nTransfer learning is the procedure of taking knowledge from a source domain and task and applying it to a different domain and task to help improve performance on the \nsecond task \\cite{Pan2010}. Transfer learning is useful, as we know that a model trained on a dataset can not naturally be applied on another due to difference in data distribution, but can be further tuned to also apply on the second dataset. An accurate representation of the difference between traditional and transfer learning schemes can be seen in Figure \\ref{fig:ml_schemes}. \nThe SHM problem herein will be addressed using neural networks \\cite{Bishop:1995:NNP:525960}, for which transfer learning has been proven quite efficient (although \nusually in deeper learning architectures \\cite{GabrielPuiCheongFung2006,AlMubaid2006}). Due to the layered structure of the networks, after having created a model for a task, transferring a part of it (e.g.\\ some \nsubset of the layers) is easy. The method is used in many disciplines, such as computer vision \\cite{oquab2014learning,shin2016deep}. The most commonly-used learners\nare Convolutional Neural Networks (CNNs), which can be very slow to train and may need a lot of data, which in many cases can be hard to obtain (e.g.\\ labelled \nimages). These problems can be dealt with by using the fixed initial layers of pre-trained models to extract features of images, and then train only the last layers to \nclassify in the new context. In this way, both the number of trainable parameters and the need for huge datasets and computation time are reduced. Another topic that \ntransfer learning has been used in is natural language processing (NLP) \\cite{DBLP:journals\/corr\/BingelS17}, where the same issues of lack of labelled data and large \namounts of training time are dealt with by transferring of pre-trained models into new tasks. Further examples of the benefits of transfer learning can be found in web \ndocument classification \\cite{GabrielPuiCheongFung2006,AlMubaid2006}; in these cases, in newly-created web sites, lack of labelled data occurs. To address this problem, \neven though the new web sites belong to a different domain than the training domain of the existing sites, the same models can be used to help classify documents in the \nnew websites.\n\nIn the context of the current work, transfer learning is considered in transferring knowledge from one sub-problem to the other by introducing pre-trained layers into\nnew classifiers. The classification problem that will be presented is related to damage class\/location. A model trained to predict a subset of the damage classes (source \ntask) with data corresponding of that subset (source domain), will be used to boost performance of a second classifier trained to identify a different subset of damage\nstates. \n\n\\begin{figure}[ht!]\n\t\\centering\n\t\\begin{subfigure}{.5\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=.95\\linewidth]{Figure_1a}\n\t\t\\caption{}\n\t\t\\label{fig:trad_ml}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{.5\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=.95\\linewidth]{Figure_1b}\n\t\t\\caption{}\n\t\t\\label{fig:transfer_ml}\n\t\\end{subfigure}\\\\\n\t\\caption{Traditional (a) and transfer (b) learning schemes (following \\cite{Pan2010}).}\n\t\\label{fig:ml_schemes}\n\\end{figure}\n\\section{Problem description}\n\\label{sec:problem_desc}\n\nSimilar to the aforementioned applications, in SHM machine learning is also used for classification and regression. In data driven SHM one tries to identify features that will reveal whether a structure is damaged or what type of damage is present and so, labelled data are necessity. Therefore, in SHM applications lack of labelled data about damage location or severity is a drawback. SHM problems can be categorised in many ways but are often broken down according to the hierarchical structure proposed by Rytter \\cite{rytter1993vibrational}:  \n\n\\begin{enumerate}\n\t\\item Is there damage in the system ({\\em existence})?\n\t\\item Where is the damage in the system ({\\em location})?\n\t\\item What kind of damage is present ({\\em type\/classification})?\n\t\\item How severe is the damage ({\\em extent\/severity})?\n\t\\item How much useful (safe) life remains ({\\em prognosis})?\n\\end{enumerate}\n\nA common approach to the first level is to observe the structure in its normal condition and try to find changes in features extracted from measured signals that \nare sensitive to damage. This approach is called {\\em novelty detection} \\cite{Worden1997,WORDEN2000}, and it has some advantages and disadvantages. The main advantage \nis that it is usually an {\\em unsupervised} method, that is only trained on data that are considered to be from the undamaged condition of the structure, without \na specific target class label. These methods are thus trained to detect any {\\em changes} in the behaviour of the elements under consideration, which can be a \ndisadvantage, since structures can change their behaviour for benign reasons, like changes in their environmental or operational conditions; such benign changes or\n{\\em confounding influences} can raise false alarms.\n\nIn this work a problem of damage localisation is considered (at Level 2 in Rytter's hierarchy \\cite{rytter1993vibrational}); the structure of interest being a wing of \na Gnat trainer aircraft. The problem is one of supervised-learning, as the data for all damage cases were collected and a classification model was trained accordingly. Subsequently, the classifier was used to predict the damage class of newly-presented data. The features used as inputs to the classifier were novelty indices calculated \nbetween frequency intervals of the transmissibilities of the normal condition of the structure (undamaged state) and the testing states. The transmissibility between \ntwo points of a structure is given by equation (\\ref{eq:transmissibility}), and this represents the ratio of two response spectra. This feature is useful because it \ndescribes the response of the structure in the frequency domain, without requiring any knowledge of the frequency content of the excitation. The transmissibility is\ndefined as,\n\n\\begin{equation} \n\\label{eq:transmissibility}\n     T_{ij} = \\frac{FRF_i}{FRF_j} =  \n     \\frac{\\frac{\\mathcal{F}_{i}}{\\mathcal{F}_{excitation}}}{\\frac{\\mathcal{F}_{j}}{\\mathcal{F}_{excitation}}} = \\frac{\\mathcal{F}_{i}}{\\mathcal{F}_{j}}\n\\end{equation}\nwhere, $\\mathcal{F}_{i}$ is the Fourier Transform of the signal given by the $i^{th}$ sensor and $FRF_i$ is the {\\em Frequency Response Function} (FRF) at the $i$th \npoint. \n\nThe experiment was set up as described in \\cite{Worden2007}. The wing of the aircraft was excited with a Gaussian white noise using an electrodynamic shaker attached \non the bottom surface of the wing. The configuration of the sensors placed on the wing can be seen in Figure \\ref{fig:sensors}. Responses were measured with \naccelerometers on the upper surface of the wing, and the transmissibilities between each sensor and the corresponding reference sensor were calculated. The \ntransmissibilities were recorded in the 1-2 kHz range, as this interval was found to be sensitive to the damage that was going to be introduced to the structure. Each transmissibility contained 2048 spectral lines.\n\n\\begin{figure}[ht!]\n\t\\centering\n\t\\includegraphics[scale=0.60]{Figure_2.png}\n\t\\caption{Configuration of sensors on the Gnat aircraft wing \\cite{manson2003experimental}.}\n\t\\label{fig:sensors}\n\\end{figure}\n\nInitially, the structure was excited in its normal condition, i.e.\\ with no introduced damage. The transmissibilities of this state were recorded and subsequently, \nto simulate damage, several panels were removed from the wing, one at a time. In each panel removal, the wing was excited again with white Gaussian noise and the transmissibilities were recorded. The panels that were removed are shown in Figure \\ref{fig:panels}. Each panel has a different size, varying from 0.008 to 0.08m$^{2}$ \nand so the localisation of smaller panels becomes more difficult, since their removal affects the transmissibilities less than the bigger panels. The measurements were \nrepeated 200 times for each damage case, ultimately leading to 1800 data points belonging to nine different damage cases\/classes. The data were separated into training, \nvalidation and testing sub sets, each having 66 points per damage case.\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[scale=0.70]{Figure_3.png}\n\t\\caption{Schematic showing wing panels removed to simulate the nine damage cases \\cite{manson2003experimental}.}\n\t\\label{fig:panels}\n\\end{figure}\n\nFor the purposes of damage localisation, features had to be selected which would be sensitive to the panel removals; this was initially done manually \\cite{manson2003experimental}, selecting by visual `engineering judgement' the intervals of the transmissibilities that appeared to be more sensitive to damage and \ncalculating the novelty indices of each state by comparison with the transmissibilities of the undamaged state. The novelty indices were computed using the \nMahalanobis squared-distance (MSD) $D^2_{\\zeta}$ of the feature vectors $\\mathbf{x_{\\zeta}}$, which in this case contained the magnitudes of transmissibility spectral \nlines. The MSD is defined by,\n\n\\begin{equation} \n\\label{eq:mahal_dist}\n     D_{\\zeta}^{2} = (\\mathbf{x_{\\zeta}} - \\mathbf{\\overline{x}})^{T} S^{-1}(\\mathbf{x_{\\zeta}} - \\mathbf{\\overline{x}})\n\\end{equation}\nwere $\\mathbf{\\overline{x}}$ is the sample mean on the normal condition feature data, and $S$ is the sample covariance matrix.\n\nAfter selecting `by eye' the most important features for damage detection \\cite{manson2003experimental}, a genetic algorithm was used \\cite{Worden2008} to choose the \nmost sensitive features, in order to localise\/classify the damage. Finally, nine features were chosen as the most sensitive and an MLP neural network \\cite{Bishop:1995:NNP:525960} with nine nodes in the input layer, ten nodes in the hidden layer and nine nodes in the decision layer was trained. The confusion matrix \nof the resulting classifier is shown in the Table \\ref{Tab:Initial_network_conf_mat}. It can be seen that the misclassification rate is very low and that the damage \ncases that are most confused are the ones where the missing panel is Panel 3 or Panel 6, which were the smallest ones.\n\n\n\\section{Problem splitting}\n\\label{sec:splitting}\n\nAs mentioned in \\cite{tarassenko1998guide}, the rule-of-thumb for a network that generalises well is that it should be trained with at least ten samples per weight \nof the network. The aforementioned network had 180 trainable weights (and another 19 bias terms) so the 596 training samples are not ideal for the neural network. As a solution, \na splitting of the original problem into two sub-problems is considered here to try and reduce the misclassification rate on the testing data even further. The dataset \nis split into two parts, one containing all the damage cases except Panels 3 \\& 6 and the second containing the rest of the data. Subsequently, two neural network \nclassifiers were trained separately on the new datasets. This was thought to be a good practice, since the panels are the smallest, and their removal affects the \nnovelty indices less than the rest of the panel removals. The impact is that the points appear closer to each other in the feature space, and are swamped by points \nbelonging to other classes, so the initial classifier cannot separate them efficiently. By assigning the tasks to different classifiers, an increase in performance \nis expected, especially in the case of separating the two smallest panel classes.\n\n\\begin{table}[h!]\n\t\\centering\n\t\\begin{tabular}{ |p{3cm}||p{1.0cm}|p{1.0cm}|p{1.0cm}|p{1.0cm}|p{1.0cm}|p{1.0cm}|p{1.0cm}|p{1.0cm}|p{1.0cm}| }\n\t\t\\hline\n\t\tPredicted panel & 1 & 2 & 3& 4& 5 & 6 & 7 & 8 & 9\\\\\n\t\t\\hline\n\t\tMissing panel 1& 65 & 0   & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\n\t\tMissing panel 2& 0 & 65  & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n\t\tMissing panel 3& 1 &  0    & 62 & 0 & 0 & 1 & 0 & 1 & 1 \\\\\n\t\tMissing panel 4 & 0 &  0  & 0 & 66 & 0 & 0 & 0 & 0 & 0 \\\\\n\t\tMissing panel 5& 0 &  0  & 0 & 0 & 66 & 0 & 0 & 0 & 0 \\\\\n\t\tMissing panel 6& 0 &  3  & 0 & 0 & 0 & 62 & 0 & 1 & 0 \\\\\n\t\tMissing panel 7& 0 &  0  & 0 & 0 & 0 & 0 & 66 & 0 & 0 \\\\\n\t\tMissing panel 8& 1 &  0  & 0 & 0 & 0 & 0 & 0 & 65 & 0 \\\\\n\t\tMissing panel 9& 0 &  0  & 0 & 0 & 0 & 0 & 0 & 0 & 66 \\\\\n\t\t\\hline\n\t\\end{tabular}\n\t\\caption{Confusion Matrix of neural network classifier, test set, total accuracy: 98.14\\% \\cite{Worden2008}}\n\t\\label{Tab:Initial_network_conf_mat}\n\\end{table}\n\nTo illustrate the data feature space, a visualisation is attempted here. Since the data belong to a nine-dimensional feature space, principal component analysis (PCA) \nwas performed on the data and three of the principal components, explaining 71\\% of total variance, are plotted in scatter plots shown in Figure \\ref{fig:all_data_pcs}. \nPoints referring to data corresponding to the missing panels 3 and 6 (grey and magenta points respectively) are entangled with other class points causing most of the misclassification rate shown above.\n\n\\begin{figure}[ht!]\n\t\\centering\n\t\\begin{subfigure}{.5\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=.98\\linewidth]{Figure_4a}\n\t\t\\caption{}\n\t\t\\label{fig:all_data_pcs}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{.5\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=.98\\linewidth]{Figure_4b}\n\t\t\\caption{}\n\t\t\\label{fig:7_classes_pcs}\n\t\\end{subfigure}\\\\\n\t\\begin{subfigure}{.5\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=.98\\linewidth]{Figure_4c}\n\t\t\\caption{}\n\t\t\\label{fig:2_classes_pcs}\n\t\\end{subfigure}\n\t\\caption{Principal components of all samples (a), samples excepting panels 3 and 6 (b) and samples of panels 3 and 6(c).}\n\t\\label{fig:first_pcs}\t\n\\end{figure}\n\nRandom initialisation was followed for the neural networks. Initial values of the weights and biases of the networks were sampled from a normal zero-mean distribution. The two networks were initialised several times and trained for different sizes of the hidden layer to find the ones with optimal structure for the \nnewly-defined problems. After randomly initialising and training multiple neural networks for both cases and keeping the ones with the minimum loss function value \nthe best architectures were found to be networks with nine nodes in the hidden layer for both cases and seven output nodes for the first dataset and two for the \nsecond. The loss function used in training was the categorical cross-entropy function given by,\n\n\\begin{equation} \n\\label{eq:categorical_crossentropy}\n     L(y, \\hat{y}) = -\\frac{1}{N}\\sum_{i=1}^{N}\\sum_{j=1}^{n_{cl}} [y_{i, j}log \\hat{y}_{i, j} + (1 - y_{i, j}) log (1 - \\hat{y}_{i, j})]\n\\end{equation}\n\nIn Equation (\\ref{eq:categorical_crossentropy}), $N$ is the number of samples during training, $n_{cl}$ is the number of possible classes, $\\hat{y}_{i,j}$ the estimated probability that the $i$th point belongs to the $j$th class and $y_{i, j}$ is 1 if the $i$th sample belongs to the $j$th class, otherwise it is 0.\n\nConfusion matrices on the test sets for the classifiers are shown in Tables \\ref{Tab:First_set_conf_mat} and \\ref{Tab:Second_set_conf_mat}. By splitting the dataset \ninto two subsets the total accuracy is slightly increased from 98.14\\% to 98.82\\%. This is best considered in terms of classification error, which has been reduced from\n1.86\\% to 1.18\\%, and this is an important reduction in SHM terms. Reduction of the number of trainable parameters has certainly contributed to this improvement, since \nthe amount of training data is small. Performance on the task of separating only the two smallest panel classes was also increased because it is an easier task for the \nclassifier than trying to discriminate them among the panel removals with greater impact on the novelty indices. This fact is also clear in Figure \\ref{fig:2_classes_pcs}, \nwhere the principal components of samples belonging to the classes of missing Panels 3 and 6 are clearly separable.\n\n\\begin{table}[ht!]\n\t\\centering\n\t\\begin{tabular}{ |p{3cm}||p{1.0cm}|p{1.0cm}|p{1.0cm}|p{1.0cm}|p{1.0cm}|p{1.0cm}|p{1.0cm}| }\n\t\t\\hline\n\t\tPredicted panel & 1 & 2 & 4& 5 & 7 & 8 & 9\\\\\n\t\t\\hline\n\t\tMissing panel 1& 65 & 1   & 0 & 0 & 0 & 0 & 0 \\\\\n\t\tMissing panel 2& 0 & 63  & 1 & 0 & 0 & 0 &  2\\\\\n\t\tMissing panel 4 & 1 &  0  & 65 & 0 & 0 & 0 & 0 \\\\\n\t\tMissing panel 5& 0 &  0  & 0 & 66 & 0 & 0 & 0 \\\\\n\t\tMissing panel 7& 0 &  0  & 0 & 0 & 66 & 0 & 0 \\\\\n\t\tMissing panel 8& 1 &  0  & 0 & 0 & 0 & 65 & 0 \\\\\n\t\tMissing panel 9& 1 &  0  & 0 & 0 & 0 & 0 & 65 \\\\\n\t\t\\hline\n\t\\end{tabular}\n\t\\caption{Confusion Matrix of neural network classifier trained on the first dataset, test set, total accuracy: 98.48\\%}\n\t\\label{Tab:First_set_conf_mat}\n\\end{table}\n\n\\begin{table}[ht!]\n\t\\centering\n\t\\begin{tabular}{ |p{3cm}||p{1.0cm}|p{1.0cm}|}\n\t\t\\hline\n\t\tPredicted panel & 3 & 6 \\\\\n\t\t\\hline\n\t\tMissing panel 3& 66 & 0   \\\\\n\t\tMissing panel 6& 0 & 66 \\\\\n\t\t\\hline\n\t\\end{tabular}\n\t\\caption{Confusion Matrix of neural network classifier trained on the first dataset, test set, total accuracy: 100\\%}\n\t\\label{Tab:Second_set_conf_mat}\n\\end{table}\n\n\\section{Knowledge transfer between the two problems}\n\\label{sec:transfer}\n\nHaving split the problem into two sub-problems, a scheme motivated by transfer learning in deeper learners was examined. The idea being to establish if the features\nextracted at the hidden layer in one problem, could be used for the other. In transfer learning termology, the seven-class problem specifies the source domain and \ntask, while the two-class problem gives the target domain and task. The transfer is carried out by using the fixed input and hidden layers from the classifier in\nthe source task, as the input and hidden layers of the target task; this means that only the weights between the hidden and output layers remain to be trained for\nthe target task. This strategy reduces the number of parameters considerably. The functional form of the network for the source task is given by,\n\n\\begin{equation} \n\\label{eq:network_output}\n     \\mathbf{y} = f_{0}W_2(f_{1}(W_{1}\\mathbf{x} + b_{1}) + b_{2}\n\\end{equation}\nwhere $f_0$ and $f_1$ are the non-linear activation functions of the output layer and the hidden layer respectively, $W_{1,2}$ are the weight matrices of the \ntransformations between the layers, $b_{1, 2}$ are the bias vectors of the layers, $\\mathbf{x}$ is the input vector and $\\mathbf{y}$ the output vector. The {\\em softmax} \nfunction is chosen to be the activation function of the decision layer, as this is appropriate to a classification problem. The prediction of the network, concerning \nwhich damage class the sample belongs to, is the index that maximises the output vector $\\mathbf{y}$; the outputs are interpreted as the {\\em a posteriori} probabilities\nof class membership, so this leads to a Bayesian decision rule. Loosely speaking, one can think of the transformation between the hidden and ouput layers as the actual\nclassifier, and the transformation between the input layer into the hidden layer as a map to latent states in which the classes are more easily separable. In the context\nof deep networks, the hope is that the earlier layers carry out an automated feature extraction which facilitates an eventual classifier. In the deep context, transfer\nbetween problems is carried out by simply copying the `feature extraction' layers directly into the new network, and only training the later classificatiion layers. The\nsimple idea explored here, is whether that strategy helps in the much more shallow learner considered in this study. The transfer is accomplished by copying the weights $W_1$\nand biases $b_1$ from sub-problem one directly into the network for sub-problem two, and only training the weights $W_2$ and biases $b_2$.\n\nAs before, multiple neural networks were trained on the first dataset. In a transfer learning scheme, it is even more important that models should not be overtrained, \nsince that will make the model too case-specific and it would be unlikely for it to carry knowledge to other problems. To achieve this for the current problem, an early stopping strategy \nwas followed. Models were trained until a point were the value of the loss function decreases less than a percentage of the current value. An example of this can be \nseen in Figure \\ref{fig:early_stoping} where instead of training the neural network for 1000 epochs, training stops at the point indicated with the red arrow.\n\n\\begin{figure}[ht!]\n \t\\centering\n \t\\includegraphics[scale=0.70]{Figure_5}\n \t\\caption{Training and validation loss histories and the point of early stopping (red arrow).}\n \t\\label{fig:early_stoping}\n\\end{figure}\n\nAfter multiple networks were trained following the early stopping scheme above, the network with the lowest value on validation loss was determined and the transfer \nlearning scheme was applied to the second problem. The nonlinear transformation given by the transition from the input layer to the hidden layer was applied on the data \nof the second dataset. Consequently, another neural network was trained on the transformed data, having only one input layer and one output\/decision layer. To comment \non the effect of the transformation, another two-layer network was trained on the original second dataset and the results were compared. \n\n\\begin{table}[ht!]\n\t\\centering\n\t\\begin{tabular}{ |p{3cm}||p{1.0cm}|p{1.0cm}|}\n\t\t\\hline\n\t\tPredicted panel & 3 & 6 \\\\\n\t\t\\hline\n\t\tMissing panel 3& 65 & 1   \\\\\n\t\tMissing panel 6& 2 & 64 \\\\\n\t\t\\hline\n\t\\end{tabular}\n\t\\caption{Confusion Matrix of neural network classifier trained on the original data of the second dataset, test set, total accuracy: 97.72\\%}\n\t\\label{Tab:conf_original_data_2_layer}\n\\end{table}\n\n\\begin{table}[ht!]\n\t\\centering\n\t\\begin{tabular}{ |p{3cm}||p{1.0cm}|p{1.0cm}|}\n\t\t\\hline\n\t\tPredicted panel & 3 & 6 \\\\\n\t\t\\hline\n\t\tMissing panel 3& 65 & 1   \\\\\n\t\tMissing panel 6& 3 & 63 \\\\\n\t\t\\hline\n\t\\end{tabular}\n\t\\caption{Confusion Matrix of neural network classifier trained on the transformed data of the second dataset, test set, total accuracy: 96.96\\%}\n\t\\label{Tab:conf_trans_data_2_layer}\n\\end{table}\n\n\\begin{figure}[ht!]\n\t\\centering\n\t\\begin{subfigure}{.5\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=.8\\linewidth]{Figure_6a}\n\t\t\\caption{}\n\t\t\\label{fig:first_dataset_features_pcs}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{.5\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=.8\\linewidth]{Figure_6b}\n\t\t\\caption{}\n\t\t\\label{fig:first_dataset_features_pcs_transed}\n\t\\end{subfigure}\n\t\\caption{Principal components of original features of the first dataset (a) and transformed features (b).}\n\t\\label{fig:first_pcs_1}\t\n\\end{figure}\n\n\\begin{figure}[ht!]\n\t\\centering\n\t\\begin{subfigure}{.5\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=.8\\linewidth]{Figure_7a}\n\t\t\\caption{}\n\t\t\\label{fig:second_dataset_features_pcs}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{.5\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=.8\\linewidth]{Figure_7b}\n\t\t\\caption{}\n\t\t\\label{fig:second_dataset_trans_features_1_pcs}\n\t\\end{subfigure}\n\t\\caption{Principal components of original features of the second dataset (a) and transformed features (b).}\n\t\\label{fig:second_pcs}\t\n\\end{figure}\n\nThe confusion matrices of the two neural networks on the testing data are given in Tables \\ref{Tab:conf_original_data_2_layer} and \\ref{Tab:conf_trans_data_2_layer}; \nthe misclassification rates are very similar. However, it is interesting to also look at the effect of the transfer on the convergence rate of the network trained \non the transferred data and also to illustrate the feature transformation on the first and the second datasets.\n\n\\begin{figure}[ht!]\n\t\\centering\n\t\\includegraphics[scale=0.5]{Figure_8.png}\n\t\\caption{Loss histories of transferred model: train(blue), validation(cyan) and model trained on initial data: train(red), validation(magenta).}\n\t\\label{fig:loss_histories}\n\\end{figure}\n\nThe training histories of the two models can be seen in Figure \\ref{fig:loss_histories}. It is clear that the loss history of the model with transformed data (blue \nand cyan lines) converges faster, especially in the initial part of the training, and it also reaches a lower minimum value for the loss function in the same number \nof training epochs. This can be explained by looking at the effect of the learnt transformation on the data. In Figures \\ref{fig:second_pcs} and \\ref{fig:first_pcs_1} \nthis effect is illustrated. (Note that the points are different from those in Figure \\ref{fig:first_pcs}, because principal component analysis was performed this time \non the normalised data in the interval [-1, 1] for the neural network training). The transformation spreads out the points of the original problem (first dataset) in \norder to make their separation by the decision layer easier; however, it is clear that it also accomplishes the same result on the second dataset. The points in Figure \\ref{fig:second_dataset_trans_features_1_pcs} are spread out compared to the initial points and thus, their separation by the single layer neural network is easier. \nFurthermore, the points lay further away from the required decision boundary and this explains both the faster training convergence and the lower minimum achieved. \nIn contrast to the transformation of the first dataset, in the second dataset, the transformation does not concentrate points of the same class in specific areas of the feature space. \nIn Figure \\ref{fig:first_dataset_features_pcs_transed} the points are both spread and concentrated closer according to the class they belong. This probably means that \nonly a part of the physics of the problem is transferred in the second problem through this specific transformation.\n\n\n\\section{Discussion and Conclusions}\n\\label{sec:conclusions}\n\nFor the SHM classification (location) problem considered here, splitting the dataset into two subsets contributed to increasing the classification accuracy by a \nsmall percentage. This result was explained by the lesser effect that the small panel removals had on the novelty index features. This issue arose because the \npoints representing these classes were close to each other and also points from other classes -- those corresponding to large panel removal\/damage. By considering \nthe two damage cases as different problem, perfect accuracy was achieved in the task of classifying damage to the small panels, and there was also a small increase \nin the performance of the classifier tasked to identify the more severe damage states.\n\nAn attempt at a crude form of transfer learning was also investigated. Having trained the neural network classifier on the first dataset of the seven damage cases, \ntransfer of knowledge to the second sub-problem was considered. This was accomplished by copying the first two layers of the first classifier -- the `feature \nextraction' layers -- directly into the second classifier and only training the connections from the hidden layer to the output. The result is not particularly \nprofound; the transfer does allow a good classifier, even with the smaller set of trainable parameters, but is not as good as training the network from scratch.\nThe result is interesting, because it is clear that the source network is carrying out a feature clustering and cluster separation on the source data, that is still\nuseful when the target data are presented. This suggests that the main issue with the small-panel damage classification is that the data are masked by the close \npresence of the large-panel data. Separating out the small-panel is the obvious answer. The results are interesting because they illustrate in a `toy' example, how\nthe early layers in deeper networks are manipulating features automatically in order to improve the ultimate classification step. The other benefit of the \nseparation into sub-problems, was the faster convergence of the network training.\n\n\\section{Acknowledgement}\n\\label{sec:ack}\n\nThe authors would like to acknowledge Graeme Manson for providing the data used. Moreover, the authors would like to acknowledge the support of the Engineering and Physical Science Research Council (EPSRC) and the European Union (EU). G.T is supported by funding from the EU's Horizon 2020 research and innovation programme under the Marie Sk\u0142odowska-Curie grant agreement DyVirt (764547). The other authors are supported by EPSRC grants EP\/R006768\/1, EP\/R003645\/1, EP\/R004900\/1 and EP\/N010884\/1.\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\vspace{-0.25cm}\nAn intelligent system that interacts with its environment faces the problem of \\emph{continual learning}, in which new experience is constantly being acquired, while old experience may still be relevant \\cite{ring1997child}. An effective continual learning system must optimize for two potentially conflicting goals. First, when a previously encountered scenario is encountered, performance should immediately be good, ideally as good as it was historically. Second, maintenance of old skills should not inhibit rapid acquisition of a new skill. These simultaneous constraints -- maintaining the old while still adapting to the new -- represent the challenge known as the \\emph{stability-plasticity} dilemma \\cite{grossberg1982does}.\n\nThe quintessential failure mode of continual learning is \\emph{catastrophic forgetting}, in which new knowledge supplants old knowledge, resulting in high plasticity but little stability. Within biological systems, hippocampal replay has been proposed as a systems-level mechanism to reduce catastrophic forgetting and improve generalization, as in the theory of complementary learning systems \\cite{mcclelland1998complementary}. This process complements local consolidation of past experience at the level of individual neurons and synapses, which is also believed to be present in biological neural networks \\cite{benna2016computational, leimer2019synaptic}.\n\nWithin artificial continual learning systems, deep neural networks have become ubiquitous tools and increasingly are applied in situations where catastrophic forgetting becomes relevant. In reinforcement learning (RL) settings, this problem is often circumvented by using massive computational resources to ensure that all tasks are learned simultaneously, instead of sequentially. Namely, in simulation or in self-play RL, fresh data can be generated on demand and trained on within the same minibatch, resulting in a stable data distribution.\n\nHowever, as RL is increasingly applied to continual learning problems in industry or robotics, situations arise where gathering new experience is expensive or difficult, and thus simultaneous training is infeasible. Instead, an agent must be able to learn from one task at a time, and the sequence in which tasks occur is not under the agent's control. In fact, boundaries between tasks will often be unknown -- or tasks will deform continuously and not have definite boundaries at all \\cite{kaplanis2019policy}. Such a paradigm for training eliminates the possibility of simultaneously acting upon and learning from several tasks, and leads to the possibility of catastrophic forgetting.\n\nThere has recently been a surge of interest in methods for preventing catastrophic forgetting in RL, inspired in part by neuroscience \\cite{kaplanis2018continual,kirkpatrick2017overcoming,rusu2016progressive, schwarz2018progress}. Remarkably, however, such work has focused on synaptic consolidation approaches, while possibilities of experience replay for reducing forgetting have largely been ignored, and it has been unclear how replay could be applied in this context within a deep RL framework.\n\nWe here demonstrate that replay can be a powerful tool in continual learning. We propose a simple technique, Continual Learning with Experience And Replay (CLEAR), mixing on-policy learning from novel experiences (for plasticity) and off-policy learning from replay experiences (for stability). For additional stability, we introduce behavioral cloning between the current policy and its past self. While it can be memory-intensive to store all past experience, we find that CLEAR is just as effective even when memory is severely constrained. Our approach has the following advantages:\n\\begin{itemize}\n    \\item \\textbf{Stability and plasticity.} CLEAR performs better than state-of-the-art (Elastic Weight Consolidation, Progress \\& Compress), almost eliminating catastrophic forgetting.\n    \\item \\textbf{Simplicity.} CLEAR is much simpler than prior methods for reducing forgetting and can also be combined easily with other approaches.\n    \\item \\textbf{No task information.} CLEAR does not rely upon the identity of tasks or the boundaries between them, by contrast with prior art. This means that CLEAR can be applied to a much broader range of situations, where tasks are unknown or are not discrete.\n\\end{itemize}\n\n\\vspace{-0.25cm}\n\\section{Related work}\n\\vspace{-0.25cm}\nThe problem of catastrophic forgetting in neural networks has long been recognized \\cite{grossberg1982does}, and it is known that rehearsing past data can be a satisfactory antidote for some purposes \\cite{french1999catastrophic, mcclelland1998complementary}. Consequently, in the supervised setting that is the most common paradigm in machine learning, forgetting has been accorded less attention than in cognitive science or neuroscience, since a fixed dataset can be reordered and replayed as necessary to ensure high performance on all samples.\n\nIn recent years, however, there has been renewed interest in overcoming catastrophic forgetting in RL and in supervised learning from streaming data, where it is not possible simply to reorder the data (see e.g.~\\cite{ hayes2018memory, lopez2017gradient, parisi2018continual, rios2018closed}). Current strategies for mitigating catastrophic forgetting have primarily focused on schemes for protecting the parameters inferred in one task while training on another. For example, in Elastic Weight Consolidation (EWC) \\cite{kirkpatrick2017overcoming}, weights important for past tasks are constrained to change more slowly while learning new tasks. Progressive Networks \\cite{rusu2016progressive} freezes subnetworks trained on individual tasks, and Progress \\& Compress \\cite{schwarz2018progress} uses EWC to consolidate the network after each task has been learned. Kaplanis et al.~\\cite{kaplanis2018continual} treat individual synaptic weights as dynamical systems with latent dimensions \/ states that protect information. Outside of RL, Zenke et al.~\\cite{zenke2017continual} develop a method similar to EWC that maintains estimates of the importance of weights for past tasks, Li and Hoiem \\cite{li2017learning} leverage a mixture of task-specific and shared parameters, and Milan et al.~\\cite{milan2016forget} develop a rigorous Bayesian approach for estimating unknown task boundaries. Notably all these methods assume that task identities or boundaries are known, with the exception of \\cite{milan2016forget}, for which the approach is likely not scalable to highly complex tasks.\n\nRehearsing old data via experience replay buffers is a common technique in RL, but such methods have largely been driven by the goal of data-efficient learning on single tasks \\cite{gu2017interpolated,lin1992self, mnih2015human}. Research in this vein has included prioritized replay for maximizing the impact of rare experiences \\cite{schaul2015prioritized}, learning from human demonstration data seeded into a buffer \\cite{hester2017deep}, and methods for approximating replay buffers with generative models \\cite{shin2017continual}. A noteworthy use of experience replay buffers to protect against catastrophic forgetting was demonstrated in Isele and Cosgun \\cite{isele2018selective} on toy tasks, with a focus on how buffers can be made smaller. Previous works \\cite{gu2017interpolated,o2016combining, wang2016sample} have explored mixing on- and off-policy updates in RL, though these were focused on speed and stability in individual tasks and did not examine continual learning.\n\nWhile it may not be surprising that replay can reduce catastrophic forgetting to some extent, it is remarkable that, as we show, it is powerful enough to outperform state-of-the-art methods. There is a marked difference between reshuffling data in supervised learning and replaying past data in RL. Notably, in RL, past data are typically leveraged best by \\emph{off-policy} algorithms since historical actions may come from an out-of-date policy distribution. Reducing this deviation is our motivation for the behavioral cloning component of CLEAR, inspired by work showing the power of policy consolidation \\cite{kaplanis2019policy}, self-imitation \\cite{oh2018self}, and knowledge distillation \\cite{furlanello2018born}.\n\n\\vspace{-0.25cm}\n\\section{The CLEAR Method}\n\\vspace{-0.25cm}\nCLEAR uses actor-critic training on a mixture of new and replayed experiences. We employ distributed training based on the Importance Weighted Actor-Learner Architecture presented in \\cite{espeholt2018impala}. Namely, a single learning network is fed experiences (both novel and replay) by a number of acting networks, for which the weights are asynchronously updated to match those of the learner. Training proceeds as in \\cite{espeholt2018impala} by the \\emph{V-Trace} off-policy learning algorithm, which uses truncated importance weights to correct for off-policy distribution shifts. While V-Trace was designed to correct for the lag between the parameters of the acting networks and those of the learning network, we find it also successfully corrects for the distribution shift corresponding to replay experience. Our network architecture and training hyperparameters are chosen to match those in~\\cite{espeholt2018impala} and are not further optimized.\n\nFormally, let $\\theta$ denote the network parameters, $\\pi_\\theta$ the (current) policy of the network over actions $a$, $\\mu$ the policy generating the observed experience, and $h_s$ the hidden state of the network at time $s$. Then, the V-Trace target $v_s$ is given by:\n$$v_s:=V(h_s) + \\sum_{t=s}^{s+n-1} \\gamma^{t-s}\\left(\\prod_{i=s}^{t-1} c_i\\right)\\delta_t V,$$\nwhere $\\delta_t V:= \\rho_t\\left(r_t + \\gamma V(h_{t+1}) - V(h_t)\\right)$ for truncated importance sampling weights $c_i := \\min(\\bar{c}, \\frac{\\pi_\\theta(a_i| h_i)}{\\mu(a_i|h_i)})$, and $\\rho_t =  \\min(\\bar{\\rho}, \\frac{\\pi_\\theta(a_t| h_t)}{\\mu(a_t|h_t)})$ (with $\\bar{c}$ and $\\bar{\\rho}$ constants). The policy gradient loss is:\n$$L_\\text{policy-gradient} := -\\rho_s\\log\\pi_\\theta(a_s| h_s)\\left(r_s+\\gamma v_{s+1} - V_\\theta(h_s)\\right).$$\nThe value function update is given by the L2 loss, and we regularize policies using an entropy loss:\n$$L_\\text{value} := \\left(V_\\theta(h_s) - v_s\\right)^2,\\quad L_\\text{entropy} := \\sum_a \\pi_\\theta(a|h_s) \\log \\pi_\\theta(a|h_s).$$\nThe loss functions $L_\\text{policy-gradient}$, $L_\\text{value}$, and $L_\\text{entropy}$ are applied both for new and replay experiences. In addition, we add $L_\\text{policy-cloning}$ and $L_\\text{value-cloning}$ for replay experiences only. In general, our experiments use a 50-50 mixture of novel and replay experiences, though performance does not appear to be very sensitive to this ratio. Further implementation details are given in Appendix \\ref{sec:methods}.\n\nIn the case of replay experiences, two additional loss terms are added to induce behavioral cloning between the network and its past self, with the goal of preventing network output on replayed tasks from drifting while learning new tasks. We penalize (1) the KL divergence between the historical policy distribution and the present policy distribution, (2) the L2 norm of the difference between the historical and present value functions. Formally, this corresponds to adding the loss functions:\n$$L_\\text{policy-cloning} := \\sum_a \\mu(a|h_s) \\log\\frac{\\mu(a|h_s)}{\\pi_\\theta(a|h_s)},\\quad L_\\text{value-cloning} := ||V_\\theta(h_s) - V_\\text{replay}(h_s)||_2^2.$$\nNote that computing $\\text{KL}[\\mu||\\pi_\\theta]$ instead of $\\text{KL}[\\pi_\\theta||\\mu]$ ensures that $\\pi_\\theta(a|h_s)$ is nonzero wherever the historical policy is as well.\n\n\\vspace{-0.25cm}\n\\section{Results}\n\\vspace{-0.25cm}\n\\begin{figure*}[htb]\n    \\centering\n    \\includegraphics[scale=0.33]{figures\/separate.png}\n    \\includegraphics[scale=0.33]{figures\/simultaneous.png}\n    \\includegraphics[scale=0.33]{figures\/sequential.png}\n    \\caption{Separate, simultaneous, and sequential training: the $x$-axis denotes environment steps summed across all tasks and the $y$-axis episode score. In ``Sequential'', thick line segments are used to denote the task currently being trained, while thin segments are plotted by evaluating performance without learning. In simultaneous training, performance on \\texttt{explore\\_object\\_locations\\_small} is higher than in separate training, an example of modest constructive interference. In sequential training, tasks that are not currently being learned exhibit very dramatic catastrophic forgetting. See Appendix \\ref{sec:cumsum} for plots of the same data, showing cumulative performance.}\n    \\label{fig:sequential}\n    \\vspace{-0.5em}\n\\end{figure*}\n\\begin{figure*}[htb]\n    \\centering\n    \\includegraphics[scale=0.5]{figures\/clear.png}\n    \\includegraphics[scale=0.5]{figures\/clear_without_behavioral_cloning.png}\n    \\caption{Demonstration of CLEAR on three DMLab tasks, which are trained cyclically in sequence. CLEAR reduces catastrophic forgetting so significantly that sequential tasks train almost as well as simultaneous tasks (compare to Figure \\ref{fig:sequential}). When the behavioral cloning loss terms are ablated, there is still reduced forgetting from off-policy replay alone. As above, thicker line segments are used to denote the task that is currently being trained. See Appendix \\ref{sec:cumsum} for plots of the same data, showing cumulative performance.}\n    \\vspace{-1.5em}\n    \\label{fig:clear}\n\\end{figure*}\n\n\\subsection{Catastrophic forgetting vs.~interference}\n\\vspace{-0.25cm}\n\\label{subsec:interference}\nOur first experiment (Figure \\ref{fig:sequential}) was designed to distinguish between two distinct concepts that are sometimes conflated, \\emph{interference} and \\emph{catastrophic forgetting}, and to emphasize the outsized role of the latter as compared to the former. Interference occurs when two or more tasks are incompatible (\\emph{destructive interference}) or mutually helpful (\\emph{constructive interference}) within the same model. Catastrophic forgetting occurs when a task's performance goes down not because of incompatibility with another task but because the second task overwrites it within the model. As we aim to illustrate, the two are independent phenomena, and while interference may happen, forgetting is ubiquitous.\\footnote{As an example of (destructive) interference, learning how to drive on the right side of the road may be difficult while also learning how to drive on the left side of the road, because of the nature of the tasks involved. Forgetting, by contrast, results from sequentially training on one and then another task - e.g. learning how to drive and then not doing it for a long time.}\n\nWe considered a set of three distinct tasks within the DMLab set of environments \\cite{beattie2016deepmind}, and compared three training paradigms on which a network may be trained to perform these three tasks: (1) Training networks on the individual tasks \\emph{separately}, (2) training a single network examples from all tasks \\emph{simultaneously} (which permits interference among tasks), and (3) training a single network \\emph{sequentially} on examples from one task, then the next task, and so on cyclically. Across all training protocols, the total amount of experience for each task was held constant. Thus, for separate networks training on separate tasks, the $x$-axis in our plots shows the total number of environment frames summed across all tasks. For example, at three million frames with one million each on tasks one, two, and three.  This allows a direct comparison to simultaneous training, in which the same network was trained on all three tasks.\n\nWe observe that in DMLab, there is very little difference between separate and simultaneous training. This indicates minimal interference between tasks. If anything, there is slight constructive interference, with simultaneous training performing marginally better than separate training. We assume this is a result of (i) commonalities in image processing required across different tasks, and (ii) certain basic exploratory behaviors, e.g.,~moving around, that are advantageous across tasks. (By contrast, destructive interference might result from incompatible behaviors or from insufficient model capacity.) \n\nBy contrast, there is a large difference between either of the above modes of training and sequential training, where performance on a task decays immediately when training switches to another task -- that is, catastrophic forgetting. Note that the performance of the sequential training appears at some points to be greater than that of separate training. This is purely because in sequential training, training proceeds exclusively on a single task, then exclusively on another task. For example, the first task quickly increases in performance since the network is effectively seeing three times as much data on that task as the networks training on separate or simultaneous tasks.\n\\begin{figure*}[htb]\n\\centering\n    \\begin{tabular}{l|c|c|c}\n    &\\small{\\texttt{explore...}}&\\small{\\texttt{rooms\\_collect...}}&\\small{\\texttt{rooms\\_keys...}}\\\\ \\hline\n    Separate&29.24&8.79&19.91\\\\ \\hline\n    Simultaneous&32.35&8.81&20.56\\\\ \\hline \\hline\n    Sequential (no CLEAR)&17.99&5.01&10.87\\\\ \\hline\n    CLEAR (50-50 new-replay) &\\textbf{31.40}&\\textbf{8.00}&18.13\\\\ \\hline\n    CLEAR w\/o behavioral cloning&28.66&7.79&16.63\\\\ \\hline\n    CLEAR, 75-25 new-replay&30.28&7.83&17.86\\\\ \\hline\n    CLEAR, 100\\% replay&31.09&7.48&13.39\\\\ \\hline\n    CLEAR, buffer 5M&30.33&\\textbf{8.00}&18.07\\\\ \\hline\n    CLEAR, buffer 50M&30.82&7.99&\\textbf{18.21}\\\\\n    \\end{tabular}\n    \\caption{Quantitative comparison of the final cumulative performance between standard training (``Sequential (no CLEAR)'') and various versions of CLEAR on a cyclically repeating sequence of DMLab tasks. We also include the results of training on each individual task with a separate network (``Separate'') and on all tasks simultaneously (``Simultaneous'') instead of sequentially (see Figure \\ref{fig:sequential}). As described in Section \\ref{subsec:interference}, these are no-forgetting scenarios and thus present upper bounds on the performance expected in a continual learning setting, where tasks are presented sequentially. Remarkably, CLEAR achieves performance comparable to ``Separate'' and ``Simultaneous'', demonstrating that forgetting is virtually eliminated.  See Appendix \\ref{sec:cumsum} for further details and plots.}\n    \\vspace{-1.5em}\n    \\label{tab:dmlab}\n\\end{figure*}\n\\vspace{-0.25cm}\n\\subsection{Stability}\n\\vspace{-0.25cm}\nWe here demonstrate the efficacy of CLEAR for diminishing catastrophic forgetting (Figure \\ref{fig:clear}). We apply CLEAR to the cyclically repeating sequence of DMLab tasks used in the preceding experiment. Our method effectively eliminates forgetting on all three tasks, while preserving overall training performance (see ``Sequential'' training in Figure \\ref{fig:sequential} for reference). When the task switches, there is little, if any, dropoff in performance when using CLEAR, and the network picks up immediately where it left off once a task returns later in training. Without behavioral cloning, the mixture of new experience and replay still reduces catastrophic forgetting, though the effect is reduced.\n\nIn Figure \\ref{tab:dmlab}, we perform a quantitative comparison of the performance for CLEAR against the performance of standard training on sequential tasks, as well as training on tasks separately and simultaneously. In order to perform a comparison that effectively captures the overall performance during continual learning (including the effect of catastrophic forgetting), the reward shown for time $t$ is the average $(1\/t)\\sum_{s<t} r_s$, thus effectively measuring the area under the curve for plots such as Figure \\ref{fig:clear}. (We replot the results of our main experiments according to the cumulative performance in Appendix \\ref{sec:cumsum}.) We find that CLEAR attains similar cumulative performance to networks trained on tasks separately and simultaneously -- effectively eliminating catastrophic forgetting.\n\n\\vspace{-0.25cm}\n\\subsection{Plasticity}\n\\vspace{-0.25cm}\n\\label{subsec:probe}\nIt is a reasonable worry that relying on a replay buffer could cause new tasks to be learned more slowly as the new task data will make up a smaller and smaller portion of the replay buffer as the buffer gets larger. In this experiment (Figure \\ref{fig:probe}), we find that this is not a problem for CLEAR, relying as it does on a mixture of off- and on-policy learning. Specifically, we find the performance attained on a task is largely independent of the amount of data stored in the buffer and on the identities of the preceding tasks. We consider a cyclically repeating sequence of three DMLab tasks. At different points in the sequence, we insert a fourth DMLab task (\\texttt{natlab\\_varying\\_map\\_randomized}) as a ``probe''. We find that the performance attained on the probe task is independent of the point at which it is introduced within the training sequence. This is true both for normal training and for CLEAR. Notably, CLEAR succeeds in greatly reducing catastrophic forgetting for all tasks, and the effect on the probe task does not diminish as the probe task is introduced later on in the training sequence. We also test the performance of CLEAR using 100\\% (off-policy) replay experience. We observe that, unlike with standard CLEAR, the performance obtained on the probe task \\texttt{natlab\\_varying\\_map\\_randomized} deteriorates markedly as it appears later in the sequence of tasks. For later positions in the sequence, the probe task comprises a smaller percentage of replay experience, thereby impeding purely off-policy learning. This result underlines why CLEAR uses new experience, as well as replay, to allow rapid learning of new tasks.\n\n\\begin{figure*}[ht]\n    \\centering\n    \\includegraphics[scale=0.32]{figures\/probe_task_small.png}\n    \\caption{The DMLab task \\texttt{natlab\\_varying\\_map\\_randomize} (brown line) is presented as a ``probe task'' at different positions within a cyclically repeating sequence of three other DMLab tasks. As in previous experiments, CLEAR greatly reduces catastrophic forgetting. The key point here is that the performance attained on the probe task does not depend on the identities of the preceding tasks and does not degrade as more experiences and tasks are introduced into the buffer (i.e., as the probe task appears later in the task sequence). In the case that all learning is driven by the replay buffer, however, performance on the probe task \\emph{does} degrade as the buffer fills. This justifies our use of a mixture of novel and replay experience in training CLEAR, to allow for fast learning of new tasks, in addition to preventing catastrophic forgetting.\n    }\n    \\vspace{-1em}\n    \\label{fig:probe}\n\\end{figure*}\n\n\\vspace{-0.25cm}\n\\subsection{Balance of on- and off-policy learning}\n\\vspace{-0.25cm}\nIn this experiment (Figure \\ref{fig:buffer_util}), we consider the ratio of new examples to replay examples during training. Using 100\\% new examples is simply standard training, which as we have seen is subject to dramatic catastrophic forgetting. At 75-25 new-replay, there is already significant resistance to forgetting. At the opposite extreme, 100\\% replay examples is extremely resistant to catastrophic forgetting, but at the expense of a (slight) decrease in performance attained. We believe that 50-50 new-replay represents a good tradeoff, combining significantly reduced catastrophic forgetting with no appreciable decrease in performance attained. Unless otherwise stated, our experiments on CLEAR will use a 50-50 split of new and replay data in training.\n\\begin{figure*}[htb]\n    \\centering\n    \\includegraphics[scale=0.33]{figures\/75-25_new-replay.png}\n    \\includegraphics[scale=0.33]{figures\/50-50_new-replay.png}\n    \\includegraphics[scale=0.33]{figures\/all_from_replay.png}\n    \\caption{Comparison between different proportions of new and replay examples while cycling training among three tasks. We observe that with a 75-25 new-replay split, CLEAR eliminates most, but not all, catastrophic forgetting. At the opposite extreme, 100\\% replay prevents forgetting at the expense of reduced overall performance (with an especially noticeable reduction in early performance on the task \\texttt{rooms\\_keys\\_doors\\_puzzle}). A 50-50 split represents a good tradeoff between these extremes. See Appendix \\ref{sec:cumsum} for plots of the same data, showing cumulative performance.}\n    \\label{fig:buffer_util}\n\\end{figure*}\n   \nIt is notable that it is possible to train purely on replay examples, since the network has essentially no on-policy learning. In fact, the figure shows that with 100\\% replay, performance on each task increases throughout, even when on-policy learning is being applied to a different task. Just as Figure \\ref{fig:clear} shows the importance of behavioral cloning for maintaining past performance on a task, so this experiment shows that off-policy learning can actually increase performance from replay alone. Both ingredients are necessary for the success of CLEAR.\n\n\\vspace{-0.25cm}\n\\subsection{Limited-size buffers}\n\\vspace{-0.25cm}\nIn some cases, it may be impractical to store all past experiences in the replay buffer. We therefore test the efficacy of buffers that have capacity for only a relatively small number of experiences (Figure \\ref{fig:buffer_size}). Once the buffer is full, we use reservoir sampling to decide when to replace elements of the buffer with new experiences \\cite{isele2018selective} (see details in Appendix \\ref{sec:methods}). Thus, at each point in time, the buffer contains a (fixed size) sample uniformly at random of all past experiences.\n\\begin{figure*}[ht]\n    \\centering\n    \\includegraphics[scale=0.33]{figures\/buffer_5M.png}\n    \\includegraphics[scale=0.33]{figures\/buffer_50M.png}\n    \\includegraphics[scale=0.33]{figures\/buffer_450M.png}\n    \\caption{Comparison of the performance of training with different buffer sizes, using reservoir sampling to ensure that each buffer stores a uniform sample from all past experience. We observe only minimal difference in performance, with the smallest buffer (storing 1 in 200 experiences) demonstrating some catastrophic forgetting. See Appendix \\ref{sec:cumsum} for plots of the same data, showing cumulative performance.}\n    \\vspace{-1em}\n    \\label{fig:buffer_size}\n\\end{figure*}\n\nWe consider a sequence of tasks with 900 million environmental frames, comparing a large buffer of capacity 450 million to two small buffers of capacity 5 and 50 million. We find that all buffers perform well and conclude that it is possible to learn and reduce catastrophic forgetting even with a replay buffer that is significantly smaller than the total number of experiences. Decreasing the buffer size to 5 million results in a slight decrease in robustness to catastrophic forgetting. This may be due to over-fitting to the limited examples present in the buffer, on which the learner trains disproportionately often.\n\n\\vspace{-0.25cm}\n\\subsection{Comparison to P\\&C and EWC}\n\\vspace{-0.25cm}\nFinally, we compare our method to Progress \\& Compress (P\\&C) \\cite{schwarz2018progress} and Elastic Weight Consolidation (EWC) \\cite{kirkpatrick2017overcoming}, state-of-the-art methods for reducing catastrophic forgetting that, unlike replay, assume that the boundaries between different tasks are known (Figure \\ref{fig:atari}). We use exactly the same sequence of Atari tasks as the authors of P\\&C \\cite{schwarz2018progress}, with the same time spent on each task.  Likewise, the network and hyperparameters we use are designed to match exactly those used in \\cite{schwarz2018progress}.  This is simplified by the authors of P\\&C also using a training paradigm based on that in \\cite{espeholt2018impala}. In this case, we use CLEAR with a 75-25 balance of new-replay experience.\n\\begin{figure*}[htb]\n\\centering\n    \\includegraphics[scale=0.45]{figures\/atari_space_invaders.png}\n    \\includegraphics[scale=0.45]{figures\/atari_krull.png}\n    \\includegraphics[scale=0.45]{figures\/atari_beam_rider.png}\n    \\includegraphics[scale=0.45]{figures\/atari_hero.png}\n    \\includegraphics[scale=0.45]{figures\/atari_star_gunner.png}\n    \\includegraphics[scale=0.45]{figures\/atari_ms_pacman.png}\n    \\vskip .25in\n    \\begin{tabular}{l|c|c|c|c|c|c}\n    &\\small{\\texttt{space\\_invaders}}&\\small{\\texttt{krull}}&\\small{\\texttt{beam\\_rider}}&\\small{\\texttt{hero}}&\\small{\\texttt{star\\_gunner}}&\\small{\\texttt{ms\\_pacman}}\\\\ \\hline \n    Baseline&346.47&5512.36&\\textbf{952.72}&2737.32&1065.20&753.83\\\\ \\hline\n    CLEAR&426.72&\\textbf{8845.12}&585.05&\\textbf{8106.22}&991.74&\\textbf{1222.82}\\\\ \\hline\n    EWC&\\textbf{549.22}&5352.92&432.78&4499.35&704.20&639.75\\\\ \\hline \n    P\\&C&455.34&7025.40&743.38&3433.10&\\textbf{1261.86}&924.52\\\\\n    \\end{tabular}\n    \\caption{(a) Comparison of CLEAR to Progress \\& Compress (P\\&C) and Elastic Weight Consolidation (EWC). We find that CLEAR demonstrates comparable or greater performance than these methods, despite being significantly simpler and not requiring any knowledge of boundaries between tasks. See Appendix \\ref{sec:cumsum} for plots of the same data, showing cumulative performance. (b) Quantitative comparison of the final cumulative performance between these methods. Overall, CLEAR performs comparably to or better than P\\&C, and significantly better than EWC and baseline.\n    }\n    \\label{fig:atari}\n\\end{figure*}\n\nWe find that we obtain slightly better performance than P\\&C and significantly better performance than EWC, despite CLEAR being much simpler and agnostic to the boundaries between tasks. It is worth noting that though the on-policy model (baseline) experiences significant catastrophic forgetting, it also rapidly re-acquires its previous performance after re-exposure to the task; this allows baseline to be cumulatively better than EWC on some tasks (as is noted in the original paper \\cite{kirkpatrick2017overcoming}). An alternative plot of this experiment, showing cumulative performance on each task, is presented in Appendix \\ref{sec:cumsum}.\n\n\\vspace{-0.25cm}\n\\section{Discussion}\n\\vspace{-0.25cm}\nThe ingredients in the CLEAR method are intuitively complementary. The behavioral cloning objective keeps the policy distribution close to historical data, which in turn ensures that the importance-weighted gradient estimates have lower variance than they would with purely off-policy estimation. The behavioral cloning loss acts as regularization while not preventing ultimate convergence; as training proceeds, the behavioral cloning loss increasingly acts to regularize the network towards improved policies, which constitute a greater proportion of the replayed experience. Further study is warranted to study these intuitive claims more formally.\n\nAlgorithms for continual learning may live on a \\emph{Pareto frontier}: different methods may have different regimes of applicability. In cases where storing a memory buffer is truly prohibitive, methods that protect inferred parameters, such as Progress \\& Compress, may be more suitable. When task identities are available or boundaries between tasks are clear, leveraging this information may reduce memory or computational demands or be useful to alert the agent to learn rapidly. Further, there exist training scenarios that are adversarial either to our method or to any method that prevents forgetting. For example, if the action space of a task changed during training, fitting to the old policy's action distribution, through behavioral cloning, off-policy learning, weight protection, or any of a number of other strategies for preventing forgetting, could have a deleterious effect on performance. For such cases, we may need to develop algorithms that selectively \\emph{forget} skills as well as protecting them. \n\nWe anticipate many algorithmic innovations that build on the ideas set forward here. For example, weight-consolidation techniques such as Progress \\& Compress are quite orthogonal to our approach and could be married with it for further performance gains. Moreover, while the V-Trace algorithm we use is effective at off-policy correction for small shifts between the present and past policy distributions, it is possible that off-policy approaches leveraging Q-functions, such as Retrace \\cite{munos2016safe}, may prove more powerful still. We believe the experimental protocols we introduce -- such as comparing separate, simultaneous, and sequential training, and ``probing'' with a novel task at different points in a training sequence -- may prove useful in guiding future work in this area.\n\nWe have described a simple but powerful approach for preventing catastrophic forgetting in continual learning settings. CLEAR uses on-policy learning on fresh experiences to adapt rapidly to new tasks, while using off-policy learning with behavioral cloning on replay experience to maintain and modestly enhance performance on past tasks. Behavioral cloning on replay data further enhances the agent's stability. Our method is simple, scalable, and practical; it takes advantage of the general abundance of memory and storage in modern computers and computing facilities. We believe that the broad applicability and simplicity of the approach make CLEAR a candidate ``first line of defense'' against catastrophic forgetting in many RL contexts. \n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\nThe Firefighter problem was introduced by\nHartnell~\\cite{Hartnell1995} as a natural model for optimal\ninhibition of harmful spreading phenomena on a graph.\nDespite considerable interest in the problem\nand progress on several fronts, our understanding\nof how well this and related problems can be approximated\nis still very limited.\nInterestingly, this is even true when the underlying\ngraph is a spanning tree,\nwhich is one of the most-studied graph\nstructures in this context and also the focus of\nthis paper.\n\n\nThe Firefighter problem on trees is defined as follows.\nWe are given a graph $G=(V,E)$ which is a spanning tree\nand a vertex $r\\in V$, called \\emph{root}.\nThe problem is defined over discretized time steps.\nAt time $0$, a fire starts at $r$ and spreads step\nby step to neighboring vertices.\nDuring each time step $1,2,\\dots$ an arbitrary non-burning\nvertex $u$ can be \\emph{protected}, preventing $u$\nfrom burning in any future time step.\nIn its original form, the goal is to find a protection\nstrategy minimizing the number of vertices that will\ncatch fire.\nA closely related problem, called \\emph{Resource\nMinimization for Fire Containment (RMFC)} on trees,\nwas introduced by\nChalermsook and Chuzhoy~\\cite{ChalermsookChuzhoy2010}.\nHere the task is to determine the\nsmallest number $B\\in \\mathbb{Z}_{>0}$\nsuch that if one can protect $B$ vertices at each time\nstep (instead of just $1$), then there is a protection\nstrategy where none of the leaves of the tree\ncatches fire.\nIn this context, $B$ is referred to as\nthe \\emph{number of firefighters}.\n\n\nBoth the Firefighter problem and RMFC---both restricted\nto trees as defined above---are known to be computationally\nhard problems.\nMore precisely, Finbow, King, MacGillivray and\nRizzi~\\cite{FinbowKingMacGillivrayRizzi2007} showed \nNP-hardness for the Firefighter problem on trees with \nmaximum degree three.\nFor RMFC on trees, it is NP-hard to decide\nwhether one firefighter,\ni.e., $B=1$, is sufficient~\\cite{king_2010_firefighter};\nthus, unless $\\textsc{P}=\\textsc{NP}$, there is\nno (efficient) approximation algorithm with an approximation\nfactor strictly better than $2$.\n\nOn the positive side, several approximation algorithms\nhave been suggested for the Firefighter problem and RMFC.\nHartnell and Li~\\cite{HartnellLi2000} showed that a \nnatural greedy algorithm is a $\\frac{1}{2}$-approximation for the\nFirefighter problem. This approximation guarantee\nwas later improved by Cai, Verbin\nand Yang~\\cite{CaiVerbinYang2008} to $1-\\frac{1}{e}$,\nusing a natural linear programming (LP) relaxation\nand dependent randomized rounding.\nIt was later observed by Anshelevich, Chakrabarty, Hate \nand Swamy~\\cite{AnshelevichChakrabartyHateSwamy2009}\nthat the Firefighter problem on\ntrees can be interpreted as a monotone submodular\nfunction maximization (SFM) problem subject to a\npartition matroid constraint.\nThis leads to alternative ways to obtain a\n$(1-\\frac{1}{e})$-approximation by using a recent\n$(1-\\frac{1}{e})$-approximation for monotone SFM subject\nto a matroid\nconstraint~\\cite{vondrak_2008_optimal,calinescu_2011_maximizing}.\nThe factor $1-\\frac{1}{e}$ was later only improved for\nvarious restricted tree\ntopologies (see~\\cite{IwaikawaKamiyamaMatsui2011})\nand hence, for arbitrary trees,\nthis is the best known approximation factor to date.\n\nFor RMFC on trees, Chalermsook and\nChuzhoy~\\cite{ChalermsookChuzhoy2010} presented an\n$O(\\log^* n)$-approximation, where $n=|V|$ is the\nnumber of vertices.\\footnote{\n$\\log^* n$ denotes the minimum number $k$ of\nlogs of base two that have to be nested such\nthat $\\underbrace{\\log\\log\\dots\\log}_{k \\text{ logs}} n \\leq 1$.}\nTheir algorithm is based on a natural\nlinear program which is a straightforward\nadaptation of the one used in~\\cite{CaiVerbinYang2008}\nto get a $(1-\\frac{1}{e})$-approximation for the\nFirefighter problem on trees.\n\n\n\nWhereas there are still considerable gaps between\ncurrent hardness results and approximation algorithms\nfor both the Firefighter problem and RMFC on trees, \nthe currently best approximations essentially match\nthe integrality gaps of the underlying LPs.\nMore precisely,\nChalermsook and Vaz~\\cite{chalermsook_2016_new}\nshowed that for any $\\epsilon >0$, the canonical LP used\nfor the Firefighter problem on trees has an integrality\ngap of $1-\\frac{1}{e}+\\epsilon$. This generalized\na previous result by Cai, Verbin and Yang~\\cite{CaiVerbinYang2008},\nwho showed the same gap if the integral solution is required\nto lie in the support of an optimal LP solution.\nFor RMFC on trees, the integrality gap\nof the underlying LP\nis~$\\Theta(\\log^* n)$~\\cite{ChalermsookChuzhoy2010}.\n\n\nIt remained open to what extent these integrality\ngaps may reflect the approximation hardnesses of the\nproblems.\nThis question is motivated by two related problems\nwhose hardnesses of approximation indeed matches the\nabove-mentioned integrality gaps for the Firefighter\nproblem and RMFC.\nIn particular, many versions of monotone SFM subject\nto a matroid constraint---which we recall was shown\nin~\\cite{AnshelevichChakrabartyHateSwamy2009}\nto capture the Firefigther problem on\ntrees as a special\ncase---are\nhard to approximate up to a factor of\n$1-1\/e+\\epsilon$ for any constant $\\epsilon >0$.\nThis includes the problem of maximizing an explicitly\ngiven coverage function subject to a single cardinality\nconstraint, as shown by Feige~\\cite{feige_1998_threshold}.\nMoreover, as highlighted in~\\cite{ChalermsookChuzhoy2010},\nthe Asymmetric $k$-center problem is similar in nature\nto RMFC, and has an approximation hardness of\n$\\Theta(\\log^* n)$.\n\n\nThe goal of this paper is to fill the gap between\ncurrent approximation ratios and hardness results\nfor the Firefighter problem and RMFC on trees.\nIn particular, we present approximation ratios\nthat nearly match the hardness results, thus showing\nthat both problems can be approximated to factors\nthat are substantially better than the integrality\ngaps of the natural LPs.\nOur results are based on several new techniques,\nwhich may be of independent interest.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\t\n\\subsection{Our results}\n\n\nOur main results show\nthat both the Firefighter\nproblem and RMFC admit strong approximations\nthat essentially match known hardness bounds,\nshowing that approximation factors can be achieved that\nare substantially stronger than the integrality\ngaps of the natural LPs.\nIn particular, we obtain the following result\nfor RMFC.\n\\begin{theorem}\\label{thm:O1RMFC}\nThere is a $12$-approximation for RMFC.\n\\end{theorem}\nRecalling that RMFC is hard to approximate\nwithin any factor better than $2$, the above\nresult is optimal up to a constant factor,\nand improves on the previously best\n$O(\\log^* n)$-approximation of\nChalermsook and Chuzhoy~\\cite{ChalermsookChuzhoy2010}.\n\n\nMoreover, our main result for the Firefighter problem\nis the following, which, in view of NP-hardness \nof the problem, is essentially best possible in\nterms of approximation guarantee.\n\\begin{theorem}\\label{thm:PtasFF}\nThere is a PTAS for the Firefighter problem\non trees.\\footnote{A polynomial\ntime approximation scheme (PTAS) is\nan algorithm that, for any constant\n$\\epsilon > 0$, returns in polynomial time\na $(1-\\epsilon)$-approximate solution.}\n\\end{theorem}\n\nNotice that the Firefighter problem does not admit\nan FPTAS\\footnote{A fully polynomial time approximation\nscheme (FPTAS) is a PTAS with running\ntime polynomial in the input size and $\\frac{1}{\\epsilon}$.}\nunless $\\textsc{P}=\\textsc{NP}$, since the optimal\nvalue of any Firefighter problem on a tree of\n$n$ vertices is bounded by $O(n)$.\\footnote{\nThe nonexistence of FPTASs unless $\\textsc{P}=\\textsc{NP}$\ncan often be derived easily from strong\nNP-hardness. Notice that the Firefighter problem\nis indeed strongly NP-hard because\nits input size is $O(n)$, in which case NP-hardness is\nequivalent to strong NP-hardness.\n}\nWe introduce several new techniques that allow us\nto obtain approximation factors well beyond \nthe integrality gaps of the natural LPs,\nwhich have been a barrier for previous approaches.\nWe start by providing an overview of these techniques.\n\n\n\\smallskip\n\nDespite the fact that we obtain\napproximation\nfactors beating the integrality gaps, the natural LPs\nplay a central role in our approaches.\nWe start by introducing general transformations\nthat allow for transforming the Firefighter problem\nand RMFC into a more compact and better structured\nform, only losing small factors in terms of\napproximability.\nThese transformations by themselves do not decrease\nthe integrality gaps.\nHowever, they allow us to identify small\nsubstructures, over which we can\noptimize efficiently, and having an optimal solution\nto these subproblems we can define\na residual LP with small integrality gap.\n\nSimilar high-level approaches,\nlike guessing a constant-size\nbut important subset of an optimal solution are well-known\nin various contexts to decrease \nintegrality gaps of natural LPs. The best-known example\nmay be classic PTASs for the knapsack problem,  where the\nintegrality gap of the natural LP can be decreased to\nan arbitrarily small constant by first guessing a constant\nnumber of heaviest elements of an optimal solution.\nHowever, our approach differs substantially \nfrom this standard enumeration idea.\nApart from the above-mentioned transformations which,\nas we will show later, already lead to new results\nfor both RMFC and the Firefighter problem, we\nwill introduce new combinatorial approaches to gain information\nabout a \\emph{super-constant} subset of an optimal solution.\nIn particular, for the RMFC problem we define\na recursive enumeration\nalgorithm which, despite being very slow for enumerating all\nsolutions, can be shown to reach a good subsolution \nwithin a small recursion depth that can be reached in\npolynomial time.\nThis enumeration procedure explores the space step by\nstep, and at each step we first solve an LP that determines\nhow to continue the enumeration in the next step.\nWe think that this LP-guided enumeration\ntechnique may be of independent interest.\nFor the Firefighter problem, we use a well-chosen\nenumeration procedure to identify a polynomial\nnumber of additional constraints to be added to the\nLP, that improves its integrality gap to\n$1-\\epsilon$.\n\n\n\n\n\n\n\n\n\n\\subsection{Further related results}\nIwaikawa, Kamiyama and Matsui~\\cite{IwaikawaKamiyamaMatsui2011}\nshowed that the approximation guarantee of $1-\\frac{1}{e}$ can be improved \nfor some restricted families of trees, in particular of low maximum degree.\nAnshelevich, Chakrabarty, Hate and \nSwamy~\\cite{AnshelevichChakrabartyHateSwamy2009} studied the approximability\nof the Firefighter problem in general graphs, which they prove admits no \n$n^{1-\\epsilon}$-approximation for any $\\epsilon > 0$, unless\n$\\textsc{P}=\\textsc{NP}$. In a \ndifferent model, where the protection also spreads through the graph \n(the \\emph{Spreading Model}), the authors show that the problem \nadmits a polynomial $(1-\\frac{1}{e})$-approximation on general graphs. \nMoreover, for RMFC, an $O(\\sqrt n)$-approximation for general \ngraphs and an $O(\\log n)$-approximation for directed layered\ngraphs is presented.\nThe\nlatter result was obtained independently by Chalermsook and Chuzhoy~\\cite{ChalermsookChuzhoy2010}.\nKlein, Levcopoulos and Lingas~\\cite{KleinLevcopoulosLingas2014} introduced a \ngeometric variant of the Firefighter problem, proved its NP-hardness and provided \na constant-factor approximation algorithm. \nThe Firefighter problem and RMFC are natural special cases\nof the Maximum Coverage Problem with \nGroup Constraints (MCGC)~\\cite{ChekuriKumar2004} and the \nMultiple Set Cover problem (MSC)~\\cite{ElkinKortsarz2006}, respectively. \nThe input in MCGC is a set system consisting of a finite set $X$\nof elements with nonnegative weights, a\ncollection of subsets $\\mathcal{S} = \\{S_1, \\cdots, S_k\\}$ of $X$ and \nan integer $k$. The sets in $\\mathcal{S}$ are partitioned into \ngroups $G_1, \\cdots, G_l\\subseteq \\mathcal{S}$.\nThe goal is to pick a\nsubset $H\\subseteq \\mathcal{S}$ of $k$ \nsets from $\\mathcal{S}$ whose union covers elements of total\nweight as large as possible with the \nadditional constraint that $|H \\cap G_j| \\leq 1$\nfor all $j\\in [l]\\coloneqq \\{1,\\dots, l\\}$. In MSC,\ninstead of the fixed bounds for groups and the parameter $k$, the goal is\nto choose a subset $H\\subseteq \\mathcal{S}$\nthat covers $X$ completely, while \nminimizing $\\max_{j\\in [l]}|H \\cap G_j|$. The Firefighter\nproblem and RMFC can naturally be interpreted as special cases of the latter\nproblems with a laminar set system $\\mathcal{S}$.\n\n\nThe Firefighter problem admits polynomial time algorithms in some restricted classes \nof graphs. Finbow, King, MacGillivray and Rizzi~\\cite{FinbowKingMacGillivrayRizzi2007}\nshowed that, while the problem is NP-hard on trees with maximum degree three, \nwhen the fire starts at a vertex with degree two in a subcubic tree, the problem is solvable \nin polynomial time. Fomin, Heggernes and van Leeuwen~\\cite{FominHeggernes_vanLeeuwen2012}\npresented polynomial algorithms for interval graphs, split graphs,\npermutation graphs and $P_k$-free graphs.\n\nSeveral sub-exponential exact algorithms were developed for the Firefighter\nproblem on trees. Cai, Verbin and Yang~\\cite{CaiVerbinYang2008} presented\na $2^{O(\\sqrt{n}\\log n)}$-time algorithm. \nFloderus, Lingas and Persson~\\cite{FloderusLingasPersson2013} presented a\nsimpler algorithm with a slightly better running time, as well as a\nsub-exponential algorithm for general graphs in the\nspreading model and an $O(1)$-approximation in planar graphs\nunder some further conditions.\n\nAdditional directions of research on the Firefighter problem include parameterized\ncomplexity (Cai, Verbin and Yang~\\cite{CaiVerbinYang2008}, Bazgan, Chopin and \nFellows~\\cite{BazganChopinFellows2011}, Cygan, Fomin and \nvan Leeuwen~\\cite{CyganFomin_vanLeeuwen2012} and \nBazgan, Chopin, Cygan, Fellows, Fomin and \nvan Leeuwen~\\cite{BazganChopinCyganFellowsFomin_van_Leeuwen2014}), generalizations\nto the case of many initial fires and many firefighters (Bazgan, Chopin and \nRies~\\cite{BazganChopinRies2013} and Costa, Dantas, Dourado, Penso and \nRautenbach~\\cite{CostaDantasDouradoPensoRautenbach2013}),\nand the study of potential strengthenings of the canonical LP for\nthe Firefighter problem on trees (Hartke~\\cite{Hartke2006} and Chalermsook and Vaz~\\cite{chalermsook_2016_new}).\n\nComputing the \\emph{Survivability} of a graph is \na further problem closely related to Firefighting\nthat has attracted considerable attention\n(see~\\cite{CaiWang2009,CaiChengVerbinZhou2010,Pralat2013,Esperet_Van_Den_HeuvelMaffraySipma2013,Gordinowicz2013,KongZhangWang2014}).\nFor a graph $G$ and a parameter\n$k\\in \\mathbb{Z}_{\\geq 0}$, the $k$-survivability of $G$ \nis the average fraction of nodes that one can save with $k$ firefighters\nin $G$, when the fire starts at a random node.\n\n\n\nFor further references we refer the reader to the survey of Finbow and \nMacGillivray~\\cite{FinbowMacGillivray2009}.\n\n\n\\subsection{Organization of the paper}\n\n\n\nWe start by introducing the classic linear programming\nrelaxations for the Firefighter problem and RMFC\nin Section~\\ref{sec:preliminaries}.\nSection~\\ref{sec:overview} outlines our main\ntechniques and algorithms. Some \nproofs and additional discussion\nare deferred to later sections, namely\nSection~\\ref{sec:proofsCompression}, providing\ndetails on a compression technique that is\ncrucial for both our algorithms, Section~\\ref{sec:proofsFF},\ncontaining proofs for results related to the\nFirefighter problem, and Section~\\ref{sec:proofsRMFC},\ncontaining proofs for results related to RMFC.\nFinally, Appendix~\\ref{apx:trans} contains some basic\nreductions showing how to reduce different variations\nof the Firefighter problem to each other.\n\n\n\n\n\n\n\n\n\\section{Classic LP relaxations and preliminaries}\n\\label{sec:preliminaries}\n\nInterestingly, despite the fact that we\nobtain approximation factors considerably\nstronger than the known integrality gaps of the\nnatural LPs,\nthese LPs still play a central role in our approaches.\nWe thus start by introducing the natural LPs together with\nsome basic notation and terminology.\n\nLet $L\\in \\mathbb{Z}_{\\geq 0}$ be the \\emph{depth} of\nthe tree, i.e., the largest\ndistance---in terms of number of edges---between $r$\nand any other vertex in $G$.\nHence, after at most $L$ time steps, the fire\nspreading process will halt.\nFor $\\ell\\in [L]:=\\{1,\\dots, L\\}$, let\n$V_\\ell\\subseteq V$ be the set of all vertices of\ndistance $\\ell$ from $r$, which we call the\n\\emph{$\\ell$-th level} of the instance.\nFor brevity, we use $V_{\\leq \\ell} = \\cup_{k=1}^\\ell V_k$,\nand we define in the same spirit $V_{\\geq \\ell}$, \n$V_{< \\ell}$, and $V_{> \\ell}$.\nMoreover, we denote by\n$\\Gamma \\subseteq V$ the set of all leaves\nof the tree, and for any $u\\in V$, the set\n$P_u\\subseteq V\\setminus \\{r\\}$ denotes\nthe set of all vertices on the unique $u$-$r$ path\nexcept for the root $r$.\n\nThe relaxation for RMFC used in~\\cite{ChalermsookChuzhoy2010}\nis the following:\n\\begin{equation}\\label{eq:lpRMFC}\n\\begin{array}{*2{>{\\displaystyle}r}c*2{>{\\displaystyle}l}}\n\\min & B & & \\\\\n     & x(P_u) &\\geq &1   &\\forall u\\in \\Gamma \\\\\n     & x(V_{\\leq \\ell}) &\\leq &B\\cdot \\ell\\hspace*{2em}\n               &\\forall \\ell\\in [L]\\\\\n     & x &\\in &\\mathbb{R}_{\\geq 0}^{V\\setminus \\{r\\}},\n\\end{array}\\tag{$\\mathrm{LP_{RMFC}}$}\n\\end{equation}\nwhere $x(U):=\\sum_{u\\in U} x(u)$ for any $U\\subseteq V\\setminus \\{r\\}$.\nIndeed, if one enforces $x\\in \\{0,1\\}^{V\\setminus \\{r\\}}$\nand $B\\in \\mathbb{Z}$\nin the above relaxation, an exact description of RMFC is\nobtained where $x$ is the characteristic vector of the\nvertices to be protected and $B$ is the number of\nfirefighters:\nThe constraints $x(P_u)\\geq 1$ for $u\\in \\Gamma$ enforce \nthat for each leaf $u$, a vertex between $u$ and $r$\nwill be protected, which makes sure that $u$ will not\nbe reached by the fire;\nmoreover, the\nconstraints $x(V_{\\leq \\ell})\\leq B\\cdot \\ell$\nfor $\\ell\\in [L]$ describe the vertex sets that can be\nprotected given $B$ firefighters per time step\n(see~\\cite{ChalermsookChuzhoy2010} for more details).\nAlso, as already highlighted in~\\cite{ChalermsookChuzhoy2010},\nthere is an optimal solution to RMFC (and also to the Firefighter\nproblem), that protects with the firefighters available at\ntime step $\\ell$ only vertices in $V_\\ell$.\nHence, the above relaxation\ncan be transformed into one with same optimal objective value\nby replacing the constraints\n$x(V_{\\leq \\ell})\\leq B\\cdot \\ell$ \\;$\\forall\\ell\\in [L]$\nby the constraints\n$x(V_\\ell) \\leq B$ \\;$\\forall \\ell\\in [L]$.\n\n\nThe natural LP relaxation for the Firefighter\nproblem, which leads to the previously best\n$(1-1\/e)$-approximation presented in~\\cite{CaiVerbinYang2008},\nis obtained analogously.\nDue to higher generality, and even more importantly\nto obtain more flexibility\nin reductions to be defined later, we work on a slight\ngeneralization of the Firefighter problem on trees,\nextending it in two ways:\n\\begin{enumerate}[nosep,label=(\\roman*)]\n\\item Weighted version: vertices $u\\in V\\setminus \\{r\\}$ have\nweights $w(u)\\in \\mathbb{Z}_{\\geq 0}$, and the goal\nis to maximize the total weight of vertices not catching\nfire.\nIn the classical Firefighter problem all weights are one.\n\n\\item General budgets\/firefighters:\nWe allow for having a different number of\nfirefighters at each time step, say $B_\\ell \\in \\mathbb{Z}_{>0}$\nfirefighters for time step $\\ell\\in [L]$.\\footnote{Without\nloss of generality we exclude $B_\\ell=0$, since a level\nwith zero budget can be eliminated through a simple\ncontraction operation. For more details we refer to\nthe proof of Theorem~\\ref{thm:compressionFF} which,\nas a sub-step, eliminates zero-budget levels.\n}\n\\end{enumerate}\nIndeed, the above generalizations are mostly for convenience\nof presentation, since general budgets can be reduced to\nunit budgets (see Appendix~\\ref{apx:trans} for a proof):\n\\begin{lemma}\\label{lem:genBudgetsToUnit}\nAny weighted Firefighter problem on trees with $n$ vertices\nand general budgets can be transformed efficiently \ninto an equivalent weighted Firefighter problem with\nunit-budgets and $O(n^2)$ vertices.\n\\end{lemma}\nWe also show in Appendix~\\ref{apx:trans} that\nup to an arbitrarily small error in terms\nof objective, any weighted Firefighter instance can be\nreduced to a unit-weighted one.\nIn what follows, we always assume to deal with\na weighted Firefighter instance if not specified\notherwise. Regarding the budgets, we will be\nexplicit about whether we work with unit or\ngeneral budgets, since some techniques are easier\nto explain in the unit-budget case, even though\nit is equivalent to general budgets by\nLemma~\\ref{lem:genBudgetsToUnit}.\n\n\\smallskip\n\nAn immediate extension of the LP relaxation\nfor the unit-weighted unit-budget Firefighter\nproblem used in~\\cite{CaiVerbinYang2008}---which\nis based on an IP formulation\npresented in~\\cite{macgillivray_2003_firefighter}---leads\nto the\nfollowing LP relaxation \nfor the weighted Firefighter problem\nwith general budgets. \nFor $u\\in V$, we denote by $T_u\\subseteq V$\nthe set of all vertices in the subtree starting\nat $u$ and including $u$, i.e., all vertices $v$\nsuch that the unique $r$-$v$ path in $G$ contains $u$.\n\n\\begin{equation}\\label{eq:lpFF}\n\\begin{array}{*2{>{\\displaystyle}r}c*2{>{\\displaystyle}l}}\n\\max & \\sum_{u\\in V\\setminus \\{r\\}} x_u w(T_u) & & \\\\\n     & x(P_u) &\\leq &1   &\\forall u\\in \\Gamma \\\\\n     & x(V_{\\leq \\ell}) &\\leq &\\sum_{i=1}^\\ell B_i\\hspace*{2em}\n               &\\forall \\ell\\in [L]\\\\[1.5em]\n     & x &\\in &\\mathbb{R}_{\\geq 0}^{V\\setminus \\{r\\}}.\n\\end{array}\\tag{$\\mathrm{LP_{FF}}$}\n\\end{equation}\nThe constraints\n$x(P_u)\\leq 1$ exclude redundancies, i.e., a vertex\n$u$ is forbidden of being protected if another vertex above\nit, on the $r$-$u$ path, is already protected. This\nelimination of redundancies allows for writing the objective\nfunction as shown above.\n\nWe recall that the integrality gap of~\\ref{eq:lpRMFC}\nwas shown to be $\\Theta(\\log^* n)$~\\cite{ChalermsookChuzhoy2010},\nand the integrality gap of~\\ref{eq:lpFF} is\nasymptotically  $1-1\/e$\n(when $n\\to \\infty$)~\\cite{chalermsook_2016_new}.\n\n\\smallskip\n\nThroughout the paper, all logarithms are of base $2$ if\nnot indicated otherwise.\nWhen using big-$O$ and related notations\n(like $\\Omega, \\Theta, \\ldots$), we will always\nbe explicit about the dependence on \nsmall error terms $\\epsilon$---as used when talking\nabout $(1-\\epsilon)$-approximations---and not consider\nit to be part of the hidden constant.\nTo make statements where $\\epsilon$ is part of the\nhidden constant, we will use the notation\n$O_{\\epsilon}$ and likewise\n$\\Omega_{\\epsilon}, \\Theta_{\\epsilon},\\ldots$.\n\n\n\n\n\n\n\n\\section{Overview of techniques and algorithms}\n\\label{sec:overview}\n\nIn this section, we present our main technical\ncontributions and outline our algorithms.\nWe start by introducing a compression technique in\nSection~\\ref{subsec:compression} that works\nfor both RMFC and the Firefighter problem and allows\nfor transforming any instance to one on a tree with\nonly logarithmic depth.\nOne key property we achieve with compression,\nis that we can later use (partial)\nenumeration techniques with exponential running\ntime in the depth of the tree. \nHowever, compression on its own already leads to\ninteresting results. In particular, it allows\nus to obtain a QPTAS for\nthe Firefighter problem, and a quasipolynomial time\n$2$-approximation for RMFC.\\footnote{The running time\nof an algorithm is \\emph{quasipolynomial} if it is\nof the form $2^{\\polylog(\\langle \\mathrm{input} \\rangle)}$,\nwhere $\\langle \\mathrm{input} \\rangle$ is the input\nsize of the problem. A QPTAS is an algorithm that,\nfor any constant $\\epsilon >0$, returns\na $(1-\\epsilon)$-approximation in quasipolynomial\ntime.}\nHowever, it seems highly\nnon-trivial to transform these quasipolynomial time\nprocedures to efficient ones. \n\n\n\nTo obtain the claimed results, we develop two \n(partial) enumeration methods to reduce the integrality\ngap of the LP.\nIn Section~\\ref{subsec:overviewFirefighter}, we provide\nan overview of our PTAS for the Firefighter problem,\nand Section~\\ref{subsec:overviewRMFC} presents \nour $O(1)$-approximation for RMFC.\n\n\n\n\n\n\n\n\n\n\n\\subsection{Compression}\\label{subsec:compression}\n\nCompression is a technique that is applicable to both\nthe Firefighter problem and RMFC. It allows for reducing\nthe depth of the input tree at a very small loss in\nthe objective.\nWe start by discussing compression in the context of\nthe Firefighter problem. \n\nTo reduce the depth of the tree, we will\nfirst do a sequence of what we call \\emph{down-pushes}.\nEach down-push acts on two levels $\\ell_1,\\ell_2\\in [L]$\nwith $\\ell_1 < \\ell_2$ of the tree, and moves the budget $B_{\\ell_1}$\nof level $\\ell_1$ down to $\\ell_2$, i.e., the new\nbudget of level $\\ell_2$ will be $B_{\\ell_1}+B_{\\ell_2}$,\nand the new budget of level $\\ell_1$ will be $0$.\nClearly, down-pushes only restrict our options for\nprotecting vertices. However, we can show that\none can do a sequence of down-pushes such that\nfirst, the optimal objective value of the new instance\nis very close to the one of the original instance,\nand second,\nonly $O(\\log L)$ levels have non-zero budgets.\nFinally, levels with $0$-budget can easily be removed\nthrough a simple contraction operation, thus leading\nto a new instance with only $O(\\log L)$ depth.\n\nTheorem~\\ref{thm:compressionFF} below\nformalizes our main compression\nresult for the Firefighter problem, which we state for\nunit-budget Firefighter instances for simplicity.\nSince Lemma~\\ref{lem:genBudgetsToUnit}\nimplies that every general-budget\nFirefighter instance with $n$ vertices can\nbe transformed into a unit-budget Firefighter instance\nwith $O(n^2)$ vertices---and thus $O(n^2)$\nlevels---Theorem~\\ref{thm:compressionFF} can also be used to reduce\nany Firefighter instance on $n$ vertices to one\nwith $O(\\frac{\\log n}{\\delta})$\nlevels, by losing a factor of at most $1-\\delta$\nin terms of objective.\n\n\n\\begin{theorem}\\label{thm:compressionFF}\nLet $\\mathcal{I}$ be a unit-budget Firefighter instance \non a tree with depth $L$, and let $\\delta\\in (0,1)$.\nThen one can efficiently construct a general budget\nFirefighter instance $\\overline{\\mathcal{I}}$ with depth\n$L'=O(\\frac{\\log L}{\\delta})$, and such that\nthe following holds, where $\\val(\\mathsf{OPT}(\\overline{\\mathcal{I}}))$\nand $\\val(\\mathsf{OPT}(\\mathcal{I}))$ are the optimal values of\n$\\overline{\\mathcal{I}}$ and $\\mathcal{I}$, respectively.\n\n\\smallskip\n\n\\begin{enumerate}[nosep,label=(\\roman*)]\n\\item\n$\\val(\\mathsf{OPT}(\\overline{\\mathcal{I}}))\n  \\geq (1-\\delta) \\val(\\mathsf{OPT}(\\mathcal{I}))$, and\n\n\\item any solution to $\\overline{\\mathcal{I}}$ can be\ntransformed efficiently into a solution of $\\mathcal{I}$\nwith same objective value.\n\\end{enumerate}\n\\end{theorem}\n\n\n\n\nFor RMFC we can use a very similar compression technique\nleading to the following.\n\n\\begin{theorem}\\label{thm:compressionRMFC}\nLet $G=(V,E)$ be a rooted tree of depth $L$.\nThen one can construct efficiently a rooted\ntree $G'=(V',E')$ with $|V'|\\leq |V|$ and\ndepth $L'=O(\\log L)$, such that:\n\n\\smallskip\n\n\\begin{enumerate}[nosep,label=(\\roman*)]\n\\item If the RMFC problem on $G$ has a \nsolution with budget $B\\in \\mathbb{Z}_{> 0}$ at\neach level, then the RMFC problem on $G'$\nwith non-uniform budgets, where level $\\ell \\geq 1$\nhas a budget of $B_\\ell=2^{\\ell} \\cdot B$, has a solution.\n\n\\item Any solution to the RMFC problem on $G'$,\nwhere level $\\ell$ has budget $B_\\ell=2^{\\ell} \\cdot B$,\ncan be transformed efficiently into an RMFC solution\nfor $G$ with budget $2B$.\n\\end{enumerate}\n\n\\end{theorem}\n\nInterestingly, the above compression results already\nallow us to obtain strong quasipolynomial approximation\nalgorithms for the Firefighter problem and RMFC,\nusing dynamic programming.\nConsider for example the RMFC problem. We can first guess\nthe optimal budget $B$, which can be done efficiently\nsince $B\\in \\{1,\\dots, n\\}$. Consider now the instance\n$G'$ claimed by Theorem~\\ref{thm:compressionRMFC}\nwith budgets $B_\\ell = 2^{\\ell} B$.\nBy Theorem~\\ref{thm:compressionRMFC},\nthis RMFC instance is feasible\nand any solution to it can be converted to one of\nthe original RMFC problem with budget $2B$.\nIt is not hard to see that, for the fixed budgets $B_\\ell$,\none can solve the RMFC problem on $G'$ in quasipolynomial\ntime using a bottom-up dynamic programming approach.\nMore precisely, starting with the leaves and moving\nup to the root, we compute for each vertex $u\\in V$\nthe following table. Consider a subset of the available\nbudgets, which can be represented as a vector\n$q\\in [B_1]\\times \\dots \\times [B_{L'}]$. For each such\nvector $q$ we want to know whether or not using the sub-budget\ndescribed by $q$ allows for disconnecting $u$ from all\nleaves below it.\nSince $L'=O(\\log L)$ and the size of each budget $B_\\ell$\nis at most the number of vertices, the table size is\nquasipolynomial.\nMoreover, one can check that these tables can be\nconstructed bottom-up in quasipolynomial time.\nHence, this approach leads to a quasipolynomial time\n$2$-approximation for RMFC. We recall that there is no\nefficient approximation algorithm with an approximation\nratio strictly below $2$,\nunless $\\textsc{P}=\\textsc{NP}$.\nA similar dynamic programming approach for the Firefighter\nproblem on a compressed instance leads to a\nQPTAS.\n\nHowever, our focus is on efficient algorithms, and it\nseems non-trivial to transform the\nabove quasipolynomial time dynamic programming approaches\ninto efficient procedures. To obtain our results,\nwe therefore combine\nthe above compression techniques with\nfurther approaches to be discussed next.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Overview of PTAS for Firefighter problem}\n\\label{subsec:overviewFirefighter}\n\nDespite the fact that~\\ref{eq:lpFF} has a large integrality\ngap\n---which can be shown to be the case even after\ncompression\\footnote{This follows from the fact\nthat through compression with some\nparameter $\\delta\\in (0,1)$, both the optimal\nvalue and optimal LP value change at most\nby a $\\delta$-fraction.}---%\nit is a crucial tool in our PTAS.\nConsider a general-budget\nFirefighter instance,\nand let $x$ be a vertex solution to~\\ref{eq:lpFF}.\nWe say that a vertex $u\\in V\\setminus \\{r\\}$ is\n\\emph{$x$-loose}, or simply \\emph{loose}, if\n$u\\in \\operatorname{supp}(x):=\\{v\\in V\\setminus \\{r\\} \\mid x(v) > 0\\}$\nand $x(P_u) < 1$.\nAnalogously, we call a vertex $u\\in V\\setminus \\{r\\}$\n\\emph{$x$-tight}, or simply \\emph{tight}, if\n$u\\in \\operatorname{supp}(x)$ and $x(P_u)=1$.\nHence, $\\operatorname{supp}(x)$ can be partitioned into\n$\\operatorname{supp}(x)=V^{\\mathcal{L}} \\cup V^{\\mathcal{T}}$,\nwhere $V^{\\mathcal{L}}$ and $V^{\\mathcal{T}}$\nare the set of all loose and tight vertices, respectively.\nUsing a sparsity argument for vertex solutions\nof~\\ref{eq:lpFF} we can bound the number of\n$x$-loose vertices.\n\n\n\\begin{lemma}\\label{lem:sparsityFF}\nLet $x$ be a vertex solution to~\\ref{eq:lpFF}\nfor a Firefighter problem with general budgets.\nThen the number of $x$-loose vertices is at\nmost $L$, the depth of the tree.\n\\end{lemma}\n\nHaving a vertex solution $x$ to~\\ref{eq:lpFF},\nwe can consider a simplified LP obtained from~\\ref{eq:lpFF}\nby only allowing to protect vertices that are $x$-tight.\nA simple yet useful property of $x$-tight vertices is that\nfor any $u,v\\in V^{\\mathcal{T}}$ with $u\\neq v$\nwe have $u\\not\\in P_v$. Indeed, if $u\\in P_v$, then\n$x(P_u) \\leq x(P_v) - x(v) < x(P_v)=1$ because $x(v)>0$.\nHence, no two tight vertices lie on the same leaf-root path.\nThus, when restricting~\\ref{eq:lpFF} to $V^{\\mathcal{T}}$,\nthe path constraints $x(P_u) \\leq 1$ for $u\\in \\Gamma$ transform\ninto trivial constraints requiring $x(v)\\leq 1$ for\n$v\\in V^{\\mathcal{T}}$, and one can easily observe that the\nresulting constraint system is totally unimodular because\nit describes a laminar matroid constraint given by the\nbudget constraints (see~\\cite[Volume B]{Schrijver2003} for more details on\nmatroid optimization).\nRe-optimizing over this LP we get an integral solution\nof objective value at least\n$\\sum_{u\\in V\\setminus \\{r\\}} x_u w(T_u)\n  - \\sum_{u\\in V^{\\mathcal{L}}} x_u w(T_u)$,\nbecause the restriction of $x$ to $V^{\\mathcal{T}}$\nis still feasible for the new LP.\n\n\nIn particular, if $\\sum_{u\\in V^{\\mathcal{L}}} x_u w(T_u)$\nwas at most $\\epsilon\\cdot\\val(\\mathsf{OPT})$, where\n$\\val(\\mathsf{OPT})$ is the optimal value of the instance,\nthen this would lead to a PTAS.\nClearly, this is not true in general, since it would contradict\nthe $(1-\\frac{1}{e})$-integrality gap of~\\ref{eq:lpFF}.\nIn the following, we will present techniques to limit\nthe loss in terms of LP-value when re-optimizing\nonly over variables corresponding to tight vertices $V^{\\mathcal{T}}$.\n\n\nNotice that when we work with a compressed instance, by\nfirst invoking Theorem~\\ref{thm:compressionFF} with $\\delta=\\epsilon$,\nwe have $|V^{\\mathcal{L}}|=O(\\frac{\\log N}{\\epsilon})$, where $N$ is the number of\nvertices in the original instance. Hence, a PTAS would\nbe achieved if for all $u\\in V^{\\mathcal{L}}$, we had\n$w(T_u) = \\Theta(\\frac{\\epsilon^2}{\\log N})\\cdot \\val(\\mathsf{OPT})$.\nOne way to achieve this in quasipolynomial time is\nto first guess a subset of $\\Theta(\\frac{\\log N}{\\epsilon^2})$ many\nvertices of an optimal solution with highest impact,\ni.e., among all vertices $u\\in \\mathsf{OPT}$ we\nguess those with largest $w(T_u)$.\nThis techniques has been used in various other settings\n(see for example~\\cite{ravi_1996_constrained,grandoni_2014_new}\nfor further details)\n and leads to another QPTAS for the Firefighter problem.\nAgain, it is unclear how \nthis QPTAS could be turned into an efficient procedure.\n\n\nThe above discussion motivates to investigate vertices\n$u\\in V\\setminus \\{r\\}$ with \n$w(T_u) \\geq \\eta$ for some\n$\\eta = \\Theta(\\frac{\\epsilon^2}{\\log N}) \\val(\\mathsf{OPT})$.\nWe call such vertices \\emph{heavy};\nlater, we will provide an explicit definition of $\\eta$\nthat does not depend on the unknown $\\val(\\mathsf{OPT})$ and\nis explicit about the hidden constant.\nLet $H=\\{u\\in V\\setminus \\{r\\} \\mid w(u) \\geq \\eta\\}$\nbe the set of all heavy vertices.\nObserve that $G[H\\cup \\{r\\}]$---i.e., the induced subgraph\nof $G$ over the vertices $H\\cup\\{r\\}$---is a subtree of\n$G$, which we call the \\emph{heavy tree}.\n\n\nRecall that by the above discussion, if we work on a\ncompressed instance with $L=O(\\frac{\\log N}{\\epsilon})$ levels,\nand if an optimal vertex solution to~\\ref{eq:lpFF}\nhas no loose vertices that are heavy, then an integral\nsolution can be obtained of value at \nleast $1-\\epsilon$ times the LP value.\nHence, if we were able to guess the\nheavy vertices contained in an optimal solution,\nthe integrality gap of the reduced problem\nwould be small\nsince no heavy vertices are left in the LP,\nand can thus not be loose anymore.\n\nWhereas there are too many options\nto enumerate over all possible subsets\nof heavy vertices that an optimal solution\nmay contain, we will do a coarser\nenumeration.\nMore precisely, we will partition\nthe heavy vertices into $O_{\\epsilon}(\\log N)$\nsubpaths and guess for each subpath whether\nit contains a vertex of $\\mathsf{OPT}$.\nFor this to work out we need\nthat the heavy tree has a very\nsimple topology;\nin particular, it should only have\n$O_{\\epsilon}(\\log N)$ leaves.\nWhereas this does not hold in general,\nwe can enforce it by a further transformation\nmaking sure that $\\mathsf{OPT}$ saves a\nconstant-fraction of $w(V)$ which---as we\nwill observe next---indeed limits the\nnumber of leaves of the heavy tree to $O_{\\epsilon}(\\log N)$.\nFurthermore, this transformation is useful to\ncomplete our definition of heavy vertices\nby explicitly defining the threshold $\\eta$.\n\n\n\\begin{lemma}\\label{lem:pruning}\nLet $\\mathcal{I}$ be a general-budget Firefighter instance\non a tree $G=(V,E)$ with weights $w$.\nThen for any $\\lambda\\in \\mathbb{Z}_{\\geq 1}$,\none can efficiently construct\na new Firefighter instance\n$\\overline{\\mathcal{I}}$ on a subtree $G'=(V',E')$ of $G$\nwith same budgets,\nby starting from $\\mathcal{I}$ and applying\nnode deletions and weight reductions, such that\n\\smallskip\n\\begin{enumerate}[nosep, label=(\\roman*)]\n \\item\\label{item:pruningSmallLoss}\n    $\\val(\\mathsf{OPT}(\\overline{\\mathcal{I}})) \\geq\n    \\left(1 - \\frac{1}{\\lambda}\\right)\n     \\val(\\mathsf{OPT}(\\mathcal{I}))$, and\n\n \\item\\label{item:pruningLargeOpt}\n   $\\val(\\mathsf{OPT}(\\overline{\\mathcal{I}})) \\geq\n         \\frac{1}{\\lambda} w'(V')$,\nwhere $w'\\leq w$ are the vertex weights\nin instance $\\overline{\\mathcal{I}}$.\n\\end{enumerate}\n\\smallskip\nThe deletion of $u\\in V$ corresponds to removing the whole\nsubtree below $u$ from $G$, i.e., all vertices in $T_u$.\n\\end{lemma}\n\nSince Lemma~\\ref{lem:pruning} constructs a new instance\nusing only node deletions and weight reductions, any\nsolution to the new instance is also a solution to the\noriginal instance of at least the same objective value.\n\nOur PTAS for the Firefighter problem first applies the\ncompression Theorem~\\ref{thm:compressionFF} with $\\delta=\\epsilon\/3$\nand then Lemma~\\ref{lem:pruning} with\n$\\lambda = \\lceil\\frac{3}{\\epsilon} \\rceil$ to obtain\na general budget Firefighter instance on a tree $G=(V,E)$.\nWe summarize the properties of this new instance $G=(V,E)$\nbelow. As before, to avoid confusion, we denote by $N$\nthe number of vertices of the \noriginal instance.\n\n\n\\begin{property}\\leavevmode\n\\label{prop:preprocessedFF}\n\n\\begin{enumerate}[nosep, label=(\\roman*)]\n\\item The depth $L$ of $G$ satisfies $L=O(\\frac{\\log N}{\\epsilon})$.\n\n\\item $\\val(\\mathsf{OPT}) \\geq \\lceil\\frac{3}{\\epsilon} \\rceil^{-1} w(V)\n  \\geq \\frac{1}{4}\\epsilon w(V)$.\n\n\\item The optimal value $\\val(\\mathsf{OPT})$ of the new instance is\nat least a $(1-\\frac{2}{3}\\epsilon)$-fraction of the\noptimal value of the original instance.\n\n\\item Any solution to the new instance can be transformed\nefficiently into a solution of the original instance\nof at least the same value.\n\\end{enumerate}\n\\end{property}\n\nHence, to obtain a PTAS for the original instance, it\nsuffices to obtain, for any $\\epsilon >0$, a \n$(1-\\frac{\\epsilon}{3})$-approximation for an instance\nsatisfying Property~\\ref{prop:preprocessedFF}.\nIn what follows, we assume to work with an instance\nsatisfying Property~\\ref{prop:preprocessedFF} and show\nthat this is possible.\n\n\nDue to the lower bound on $\\val(\\mathsf{OPT})$ provided\nby Property~\\ref{prop:preprocessedFF}, we now define\nthe threshold\n$\\eta = \\Theta(\\frac{\\epsilon}{\\log N}) \\val(\\mathsf{OPT})$\nin terms of $w(V)$ by\n\\begin{equation*}\n\\eta = \\frac{1}{12} \\frac{\\epsilon^2}{L} w(V),\n\\end{equation*}\nwhich implies that we can afford losing $L$ times a weight\nof $\\eta$, which will sum up to a total loss of at most\n$\\frac{1}{12}\\epsilon^2 w(V) \\leq \\frac{1}{3} \\epsilon \\val(\\mathsf{OPT})$,\nwhere the inequality is due to\nProperty~\\ref{prop:preprocessedFF}.\n\nConsider again the heavy tree $G[H\\cup \\{r\\}]$. Due to\nProperty~\\ref{prop:preprocessedFF} its topology is quite\nsimple. More precisely, the heavy tree has\nonly $O(\\frac{\\log N}{\\epsilon^3})$ leaves.\nIndeed, each leaf $u\\in H$ of the heavy tree fulfills\n$w(T_u) \\geq \\eta$, and two different leaves $u_1,u_2\\in H$\nsatisfy $T_{u_1} \\cap T_{u_2} = \\emptyset$; since the\ntotal weight of the tree is $w(V)$, the heavy tree\nhas at most\n$w(V)\/\\eta = 12 L \/ \\epsilon^2 = O(\\frac{\\log N}{\\epsilon^3})$\nmany leaves.\n\nIn the next step, we define a well-chosen\nsmall subset $Q$ of heavy vertices\nwhose removal (together with $r$) from $G$ will\nbreak $G$ into components of weight at most $\\eta$.\nSimultaneously, we choose $Q$ such that removing\nit together with $r$ from the heavy tree breaks\nit into paths, over which we will do an enumeration\nlater.\n\n\n\\begin{lemma}\\label{lem:setQ}\nOne can efficiently determine a set $Q\\subseteq H$\nsatisfying the following.\n\n\\begin{enumerate}[nosep,label=(\\roman*)]\n\\item $|Q|=O(\\frac{\\log N}{\\epsilon^3})$.\n\\item $Q$ contains all leaves and all vertices\nof degree at least $3$ of the heavy tree,\nexcept for the root $r$.\n\\item Removing $Q\\cup\\{r\\}$ from $G$ leads to a\ngraph $G[V\\setminus (Q\\cup \\{r\\})]$ where each\nconnected component has vertices whose weight\nsums up to at most $\\eta$.\n\\end{enumerate}\n\n\\end{lemma}\n\n\nFor each vertex $q\\in Q$, let $H_q\\subseteq H$ be\nall vertices that are visited when traversing the\npath $P_q$ from $q$ to $r$\nuntil (but not including) the next\nvertex in $Q\\cup \\{r\\}$.\nHence, $H_q$ is a subpath of the heavy tree such\nthat $H_q\\cap Q = \\{q\\}$, which we call for brevity\na \\emph{$Q$-path}.\nMoreover the set of all $Q$-paths partitions $H$.\n\n\nWe use an enumeration procedure to determine on which\n$Q$-paths to protect a vertex. Since $Q$-paths are subpaths\nof leaf-root paths, we can assume that at most one vertex\nis protected in each $Q$-path.\nOur algorithm enumerates over all $2^{|Q|}$ possible\nsubsets $Z \\subseteq Q$, where $Z$\nrepresents the $Q$-paths on which we will protect a\nvertex. Incorporating this guess into~\\ref{eq:lpFF},\nwe get the following linear program~\\ref{eq:lpFFZ}:\n\n\\begin{equation}\\label{eq:lpFFZ}\n\\begin{array}{*2{>{\\displaystyle}r}c*2{>{\\displaystyle}l}}\n\\max & \\sum_{u\\in V\\setminus \\{r\\}} x_u w(T_u) & & \\\\\n     & x(P_u) &\\leq &1   &\\forall u\\in \\Gamma \\\\\n     & x(V_{\\leq \\ell}) &\\leq &\\sum_{i=1}^\\ell B_i\\hspace*{2em}\n               &\\forall \\ell\\in [L]\\\\\n     & x(H_q) &= &1 &\\forall q\\in Z\\\\\n     & x(H_q) &= &0 &\\forall q\\in Q\\setminus Z\\\\\n     & x &\\in &\\mathbb{R}_{\\geq 0}^{V\\setminus \\{r\\}}.\n\\end{array}\\tag{$\\mathrm{LP_{FF}}(Z)$}\\labeltarget{eq:lpFFZtarget}\n\\end{equation}\n\n\nWe start with a simple observation regarding~\\ref{eq:lpFFZ}.\n\\begin{lemma}\\label{lem:isFaceFF}\nThe polytope over which~\\ref{eq:lpFFZ} optimizes is a face\nof the polytope describing the feasible\nregion of~\\ref{eq:lpFF}.\nConsequently, any vertex solution of~\\ref{eq:lpFFZ} is a\nvertex solution of~\\ref{eq:lpFF}.\n\\end{lemma}\n\\begin{proof}\nThe statement immediately follows by observing that\nfor any $q\\in Q$, the inequalities $x(H_q)\\leq 1$\nand $x(H_q)\\geq 0$ are valid inequalities\nfor~\\ref{eq:lpFF}.\nNotice that $x(H_q)\\leq 1$ is a valid inequality\nfor~\\ref{eq:lpFF} because $H_q$ is a subpath of\na leaf-root path, and the load on any leaf-root\npath is limited to $1$ in~\\ref{eq:lpFF}.\n\\end{proof}\n\nAnalogously to~\\ref{eq:lpFF} we define loose and tight\nvertices for a solution to~\\ref{eq:lpFFZ}.\nA crucial implication of Lemma~\\ref{lem:isFaceFF} is that\nLemma~\\ref{lem:sparsityFF} also applies to any vertex\nsolution of~\\ref{eq:lpFFZ}.\n\nWe will show in the following that for any choice of\n$Z\\subseteq Q$, the integrality gap of~\\ref{eq:lpFFZ}\nis small and we can efficiently obtain an integral solution of\nnearly the same value as the optimal value of~\\ref{eq:lpFFZ}.\nOur PTAS then follows by enumerating all $Z\\subseteq Q$\nand considering the set $Z\\subseteq Q$ of \nall $Q$-paths on which $\\mathsf{OPT}$ protects a vertex.\nThe low integrality gap of~\\ref{eq:lpFFZ} will follow from the fact\nthat we can now limit the impact of loose vertices.\nMore precisely, any loose vertex outside of the heavy\ntree has LP contribution at most $\\eta$ by definition of\nthe heavy tree. Furthermore, for each loose vertex $u$\non the heavy tree, which lies on some $Q$-path $H_q$,\nits load $x(u)$ can be moved to the single tight vertex\non $H_q$. As we will show, such a load redistribution\nwill decrease the LP-value by at most $\\eta$, due to our choice of $Q$.\n\n\nWe are now ready to state our $(1-\\frac{\\epsilon}{3})$-approximation\nfor an instance satisfying Property~\\ref{prop:preprocessedFF},\nwhich, as discussed, implies a PTAS for the Firefighter problem.\nAlgorithm~\\ref{alg:FF} describes our\n$(1-\\frac{\\epsilon}{3})$-approximation.\n\n\n\n\\begin{algorithm}\n\n\\begin{enumerate}[rightmargin=1em]\n\\item Determine heavy vertices $H=\\{u\\in V \\mid w(T_u) \\geq \\eta\\}$,\nwhere $\\eta=\\frac{1}{12} \\frac{\\epsilon^2}{L} w(V)$.\n\n\\item Compute $Q\\subseteq H$ using Lemma~\\ref{lem:setQ}.\n\n\\item For each $Z\\subseteq Q$, obtain an optimal vertex solution\nto~\\ref{eq:lpFFZ}. Let $Z^*\\subseteq Q$ be a set for which the\noptimal value\nof~\\hyperlink{eq:lpFFZtarget}{$\\mathrm{LP_{FF}(Z^*)}$}\nis largest among\nall subsets of $Q$, and let $x$ be an optimal vertex\nsolution to~\\hyperlink{eq:lpFFZtarget}{$\\mathrm{LP_{FF}(Z^*)}$}.\n\n\\item\\label{algitem:reoptFF}\nLet $V^{\\mathcal{T}}$ be the $x$-tight vertices.\nObtain an optimal vertex solution\nto~\\ref{eq:lpFF} restricted to variables\ncorresponding to vertices in $V^{\\mathcal{T}}$.\nThe solution will be a $\\{0,1\\}$-vector, being the\ncharacteristic vector of a\nset $U\\subseteq V^{\\mathcal{T}}$ which we return.\n\\end{enumerate}\n\n\n\\caption{A $(1-\\frac{\\epsilon}{3})$-approximation for\na general-budget Firefighter instance satisfying\nProperty~\\ref{prop:preprocessedFF}.}\n\\label{alg:FF}\n\\end{algorithm}\n\n\nThe following statement completes the proof of\nTheorem~\\ref{thm:PtasFF}.\n\\begin{theorem}\\label{thm:PtasFFProp}\nFor any general-budget Firefighter instance satisfying\nProperty~\\ref{prop:preprocessedFF},\nAlgorithm~\\ref{alg:FF} computes efficiently a feasible\nset of vertices $U\\subseteq V\\setminus \\{r\\}$ to protect\nthat is a $(1-\\frac{\\epsilon}{3})$-approximation. \n\\end{theorem}\n\\begin{proof}\nFirst observe that the linear program solved in\nstep~\\ref{algitem:reoptFF} will indeed lead to\na characteristic vector with only $\\{0,1\\}$-components.\nThis is the case since no two $x$-tight vertices\ncan lie on the same leaf-root path. Hence, as discussed\npreviously, the linear program~\\ref{eq:lpFF} restricted\nto variables corresponding to $V^{\\mathcal{T}}$ is totally\nunimodular; indeed, the leaf-root path constraints $x(P_u)\\leq 1$\nfor $u\\in \\Gamma$\nreduce to $x(v)\\leq 1$ for $v\\in V^{\\mathcal{T}}$, and\nthe remaining LP corresponds to a linear program over a laminar\nmatroid, reflecting the budget constraints.\nMoreover, the set $U$ is clearly budget-feasible since \nthe budget constraints are enforced by~\\ref{eq:lpFF}.\nAlso, Algorithm~\\ref{alg:FF} runs in polynomial time\nbecause $|Q|=O(\\frac{\\log N}{\\epsilon^3})$\nby Lemma~\\ref{lem:setQ} and hence,\nthe number of subsets of $Q$ is bounded by\n$N^{O(\\frac{1}{\\epsilon^3})}$.\n\n\nIt remains to show that $U$ is a\n$(1-\\frac{\\epsilon}{3})$-approximation.\nLet $\\mathsf{OPT}$ be an optimal solution to the considered\nFirefighter instance with value $\\val(\\mathsf{OPT})$.\nObserve first that the value $\\nu^*$\nof~\\hyperlink{eq:lpFFZtarget}{$\\mathrm{LP_{FF}(Z^*)}$}\nsatisfies $\\nu^* \\geq \\val(\\mathsf{OPT})$, because\none of the sets $Z\\subseteq Q$ corresponds to\n$\\mathsf{OPT}$, namely $Z=\\{q\\in Q \\mid H_q\\cap \\mathsf{OPT} \\neq \\emptyset\\}$,\nand for this $Z$ the characteristic vector\n$\\chi^{\\mathsf{OPT}}\\in \\{0,1\\}^{V\\setminus \\{r\\}}$\nof $\\mathsf{OPT}$ is feasible\nfor~\\ref{eq:lpFFZ}.\nWe complete the proof of Theorem~\\ref{thm:PtasFFProp}\nby showing that the value $\\val(U)$ of $U$ satisfies\n$\\val(U) \\geq (1-\\frac{\\epsilon}{3}) \\nu^*$.\nFor this we show how to transform an optimal solution\n$x$ of~\\hyperlink{eq:lpFFZtarget}{$\\mathrm{LP_{FF}(Z^*)}$}\ninto a solution $y$\nto~\\hyperlink{eq:lpFFZtarget}{$\\mathrm{LP_{FF}(Z^*)}$}\nwith $\\operatorname{supp}(y) \\subseteq V^{\\mathcal{T}}$\nand such that the objective value $\\val(y)$ of $y$ satisfies\n$\\val(y)\\geq (1-\\frac{\\epsilon}{3}) \\nu^*$.\n\nLet $V^{\\mathcal{L}} \\subseteq \\operatorname{supp}(x)$ be the set of\n$x$-loose vertices, and let $H$ be all heavy vertices,\nas usual. To obtain $y$, we start with $y=x$\nand first set $y(u)=0$ for each $u\\in V^{\\mathcal{L}}\\setminus H$.\nMoreover, for each $u\\in V^{\\mathcal{L}}\\cap H$ \nwe do the following. Being part of the heavy vertices and\nfulfilling $x(u)>0$, the vertex $u$\nlies on some $Q$-path $H_{q_u}$ for some $q_u\\in Z^*$.\nBecause $x(H_{q_u})=1$, there is a tight vertex\n$v\\in H_{q_u}$. We move the $y$-value from vertex\n$u$ to vertex $v$, i.e., $y(v) = y(v)+y(u)$ and\n$y(u)=0$. This finishes the construction of $y$.\nNotice that $y$ is feasible\nfor~\\hyperlink{eq:lpFFZtarget}{$\\mathrm{LP_{FF}(Z^*)}$},\nbecause it was obtained from $x$ by reducing values\nand moving values to lower levels. \n\nTo upper bound the reduction of the LP-value when\ntransforming $x$ into $y$, we show that the modification\ndone for each loose vertex $u\\in V^{\\mathcal{L}}$ decreased\nthe LP-value by at most $\\eta$.\nClearly, for each $u\\in V^{\\mathcal{L}}\\setminus H$,\nsince $u$ is not heavy we have $w(T_u)\\leq \\eta$; thus\nsetting $y(u)=0$ will have an impact of at most $\\eta$\non the LP value.\nSimilarly, for $u\\in V^{\\mathcal{L}}\\cap H$, moving the\n$y$-value of $u$ to $q_u$ decreases the LP objective value\nby\n\\begin{equation*}\ny(u) \\cdot \\left(w(T_u) - w(T_{v})\\right)\n\\leq\nw(T_u) - w(T_v)\n= w(T_u \\setminus T_{v})\n\\leq \\eta,\n\\end{equation*}\nwhere the last inequality follows by observing\nthat $T_u \\setminus T_{v}\\subseteq T_u\\setminus T_{q_u}$\nare vertices in the\nsame connected component of $G[V\\setminus (Q\\cup \\{r\\})]$,\nand thus have a total weight of at most $\\eta$\nby Lemma~\\ref{lem:setQ}.\n\nHence,\n$\\val(x) - \\val(y) \\leq |V^{\\mathcal{L}}|\n  \\cdot \\eta \\leq L\\cdot \\eta$,\nwhere the second inequality follows by\nProperty~\\ref{prop:preprocessedFF}.\nThis completes the proof by \nobserving \nthat $|V^{\\mathcal{L}}| \\leq L$\nby Lemma~\\ref{lem:sparsityFF}, and thus\n\\begin{align*}\n\\val(y) &= \\val(x) + \\left(\\val(y) - \\val(x)\\right)\n\\geq \\val(\\mathsf{OPT}) + \\val(y) - \\val(x)\n\\geq \\val(\\mathsf{OPT}) - L\\cdot \\eta\\\\\n&= \\val(\\mathsf{OPT}) - \\frac{1}{12}\\epsilon^2 w(V)\n\\geq \\left(1-\\frac{1}{3}\\epsilon\\right) \\val(\\mathsf{OPT}),\n\\end{align*}\nwhere the last inequality\nis due to Property~\\ref{prop:preprocessedFF}.\n\n\\end{proof}\n\n\n\n\n\n\\subsection{Overview of $O(1)$-approximation for RMFC}\n\\label{subsec:overviewRMFC}\n\nAlso our $O(1)$-approximation for RMFC uses the natural LP,\ni.e,~\\ref{eq:lpRMFC}, as a crucial tool to guide the algorithm.\nThroughout this section we will work on a compressed instance\n$G=(V,E)$ of RMFC, obtained through Theorem~\\ref{thm:compressionRMFC}.\nHence, the number of levels is $L=O(\\log N)$, where $N$ is the\nnumber of vertices of the original instance. Furthermore, the\nbudget on level $\\ell\\in [L]$ is given by $B_\\ell = 2^{\\ell} B$.\nThe advantage of working with a compressed instance for\nRMFC is twofold.\nFirst, we will again apply sparsity reasonings to limit in certain\nsettings the number of loose (badly structured) vertices by the\nnumber of levels of the instance.\nSecond, the fact that low levels---i.e., levels far away from\nthe root---have high budget, will allow\nus to protect a large number of loose vertices by only\nincreasing $B$ by a constant.\n\n\nFor simplicity, we work with a slight variation \nof~\\ref{eq:lpRMFC}, where we replace, for $\\ell\\in [L]$,\nthe budget constraints\n$x(V_{\\leq \\ell}) \\leq \\sum_{i=1}^{\\ell} B_i$\nby $x(V_\\ell) \\leq B_\\ell$.\nFor brevity, we define\n\\begin{equation*}\nP_B = \\left\\{x\\in \\mathbb{R}_{\\geq 0}^{V\\setminus \\{r\\}}\n    \\;\\middle\\vert\\;\n  x(V_\\ell) \\leq B\\cdot 2^\\ell \\;\\;\\forall \\ell\\in [L]\n   \\right\\}.\n\\end{equation*}\nAs previously mentioned (and shown\nin~\\cite{ChalermsookChuzhoy2010}), the resulting LP\nis equivalent to~\\ref{eq:lpRMFC}.\nFurthermore, since the budget $B$ for a feasible RMFC solution\nhas to be chosen integral, we require $B\\geq 1$.\nHence, the resulting linear relaxation asks to find\nthe minimum $B\\geq 1$ such that \nthe following polytope is non-empty:\n\\begin{equation*}\n\\bar{P}_B = P_B \\cap\n  \\left\\{x\\in \\mathbb{R}^{V\\setminus \\{r\\}}_{\\geq 0}\n\\;\\middle\\vert\\;\nx(P_u)\\geq 1 \\;\\;\\forall u\\in \\Gamma\\right\\}.\n\\end{equation*}\n\n\nWe start by discussing approaches to partially round a\nfractional point $x\\in \\bar{P}_B$, for some fixed budget $B\\geq 1$.\nAny leaf $u\\in \\Gamma$ is fractionally cut off from\nthe root through the $x$-values on $P_u$. A crucial property\nwe derive and exploit is that leaves that are \n(fractionally) cut off from $r$ largely on low levels,\ni.e., there is high $x$-value on $P_u$ on vertices\nfar away from the root, can be cut off from the root\nvia a set of vertices to be protected that are budget-feasible\nwhen increasing $B$ only by a constant.\nTo exemplify the above statement, consider the level\n$h=\\lfloor \\log L \\rfloor$ as a threshold to define\ntop levels $V_\\ell$ as those with indices $\\ell\\leq h$\nand bottom levels when $\\ell > h$. For any leaf\n$u \\in \\Gamma$,\nwe partition the path $P_u$ into its top\npart $P_u \\cap V_{\\leq h}$ and its bottom part\n$P_u \\cap V_{> h}$. Consider all leaves that are cut\noff in bottom levels by at least $0.5$ units:\n$W=\\{u\\in \\Gamma \\mid x(P_u\\cap V_{> h}) \\geq 0.5\\}$.\nWe will show that there is a subset of\nvertices $R\\subseteq V_{>h}$ on bottom levels\nto be protected that\nis feasible for budget $\\bar{B}=2B+1 \\leq 3B$ and cuts off\nall leaves in $W$ from the root.\nWe provide a brief sketch why this result holds,\nand present a formal proof later.\nIf we set all entries of $x$ on top levels $V_{\\leq h}$\nto zero, we get a vector $y$ with $\\operatorname{supp}(y) \\subseteq V_{>h}$\nsuch that $y(P_u) \\geq 0.5$ for $u\\in W$. Hence, $2y$ fractionally\ncuts off all vertices in $W$ from the root and is feasible\nfor budget $2B$. To increase sparsity, we can replace $2y$ by\na vertex $\\bar{z}$ of the polytope\n\\begin{equation*}\nQ=\\left\\{z\\in \\mathbb{R}_{\\geq 0}^{V\\setminus \\{r\\}}\n \\;\\middle\\vert\\;\n z(V_\\ell) \\leq 2B\\cdot 2^\\ell \\;\\;\\forall \\ell\\in [L],\n z(V_{\\leq h}) = 0, z(P_u)\\geq 1 \\;\\;\\forall u\\in W\\right\\},\n\\end{equation*}\nwhich describes possible ways to cut off $W$ from $r$\nonly using levels $V_{> h}$, and $Q$ is non-empty\nsince $2y\\in Q$.\nExhibiting a sparsity reasoning analogous to the\none used for the Firefighter problem, we can show that\n$z$ has no more than $L$ many $z$-loose vertices. \nThus, we can first include all $z$-loose vertices\nin the set $R$ of vertices to be protected by increasing\nthe budget of each level $\\ell > h$ by at most\n$L\\leq 2^{h+1} \\leq 2^\\ell$.\nThe remaining vertices in $\\operatorname{supp}(z)$ are well structures\n(no two of them lie on the same leaf-root path), and an \nintegral solution can be obtained easily.\nThe new budget value is $\\bar{B}=2B+1$, where the ``$+1$''\nterm pays for the loose vertices.\n\nThe following theorem formalizes the above reasoning\nand generalizes it in two ways. First, for a leaf $u\\in \\Gamma$\nto be part of $W$, we required it to have a total $x$-value\nof at least $0.5$ within the bottom levels; we will allow\nfor replacing $0.5$ by an arbitrary threshold $\\mu\\in (0,1]$.\nSecond, the level $h$ defining what is top and bottom\ncan be chosen to be of the form $h=\\lfloor \\log^{(q)} L\\rfloor$\nfor $q\\in \\mathbb{Z}_{\\geq 0}$, where\n$\\log^{(q)} L \\coloneqq\n\\log\\log\\dots\\log L$ is the value obtained by\ntaking $q$ many logs of $L$, and\nby convention we set $\\log^{(0)}L \\coloneqq L$.\nThe generalization in terms of $h$ can be thought of as\niterating the above procedure on the RMFC instance\nrestricted to $V_{\\leq h}$.\n\n\n\\begin{theorem}\\label{thm:bottomCover}\nLet $B\\in \\mathbb{R}_{\\geq 1}$, $\\mu \\in (0,1]$, \n$q\\in \\mathbb{Z}_{\\geq 1}$, and\n$h = \\lfloor \\log^{(q)} L\\rfloor$.\nLet $x\\in P_B$ with $\\operatorname{supp}(x)\\subseteq V_{> h}$,\nand we define $W=\\{u\\in \\Gamma \\mid x(P_u) \\geq \\mu\\}$.\nThen one can efficiently compute\na set $R\\subseteq V_{>h}$ such that\n\\smallskip\n\\begin{enumerate}[nosep,label=(\\roman*)]\n\\item $R\\cap P_u \\neq \\emptyset \\quad \\forall u\\in W$, and\n\\item $\\chi^R \\in P_{B'}$, where $B'= \\frac{q}{\\mu}B + 1$\nand $\\chi^R\\in \\{0,1\\}^{V\\setminus \\{r\\}}$ is the\ncharacteristic vector of $R$.\n\\end{enumerate}\n\n\\end{theorem}\n\n\nTheorem~\\ref{thm:bottomCover} has several interesting\nconsequences.\nIt immediately implies an\nLP-based $O(\\log^* N)$-approximation for RMFC, thus\nmatching the currently best approximation result\nby Chalermsook and Chuzhoy~\\cite{ChalermsookChuzhoy2010}:\nIt suffices to start with an optimal LP solution $B\\geq 1$\nand $x\\in \\bar{P}_B$ and invoke the above theorem with\n$\\mu=1$, $q=1+\\log^* L$.\nNotice that by definition of $\\log^*$ we have\n$\\log^* L = \\min \\{\\alpha \\in \\mathbb{Z}_{\\geq 0} \\mid\n\\log^{(\\alpha)} L \\leq 1\\}$; hence\n$h=\\lfloor \\log^{(1+\\log^* L)} L\\rfloor = 0$, implying that\nall levels are bottom levels.\nSince the integrality gap of the LP\nis~$\\Omega(\\log^* N)=\\Omega(\\log^* L)$,\nTheorem~\\ref{thm:bottomCover} captures the limits of what\ncan be achieved by techniques based on the standard LP.\n\nInterestingly, Theorem~\\ref{thm:bottomCover} also implies\nthat the $\\Omega (\\log^* L)$ integrality gap is only\ndue to the top levels of the instance. More precisely, if,\nfor any $q=O(1)$ and $h=\\lfloor \\log^{(q)} L \\rfloor$,\none would know what vertices an optimal solution $R^*$ protects\nwithin the levels $V_{\\leq h}$, then a constant-factor\napproximation for RMFC follows easily \nby solving an LP on the\nbottom levels $V_{> h}$ and using Theorem~\\ref{thm:bottomCover}\nwith $\\mu=1$\nto round the obtained solution.\n\n\nAlso, using Theorem~\\ref{thm:bottomCover} it is not hard\nto find constant-factor approximation algorithms for RMFC\nif the optimal budget $B_\\mathsf{OPT}$ is large enough, say\n$B \\geq \\log L$.\\footnote{Actually, the argument we\npresent in the following works for any\n$B = \\log^{(O(1))}L$. However, we later only\nneed it for $B\\geq \\log L$ and thus focus \non this case.}\nThe main idea is to solve the LP and define\n$h=\\lfloor \\log L \\rfloor$. Leaves that are largely\ncut off by $x$ on bottom levels can be handled using\nTheorem~\\ref{thm:bottomCover}. For the remaining leaves,\nwhich are cut off mostly on top levels, we can resolve an\nLP only on the top levels $V_{\\leq h}$ to cut them off.\nThis LP solution is sparse and contains at most $h\\leq B$\nloose nodes. Hence, all loose vertices can be selected\nby increasing the budget by at most $h\\leq B$, leading\nto a well-structured residual problem for which one can\neasily find an integral solution.\nThe following theorem summarizes this discussion.\nA formal proof for Theorem~\\ref{thm:bigBIsGood}\ncan be found in Section~\\ref{sec:proofsRMFC}. \n\n\\begin{theorem}\\label{thm:bigBIsGood}\nThere is an efficient algorithm that computes a\nfeasible solution to a (compressed) instance of\nRMFC with budget $B\\leq 3 \\cdot \\max\\{\\log L, B_{\\mathsf{OPT}}\\}$.\n\\end{theorem}\n\n\n\n\n\\medskip\n\nIn what follows, we therefore assume $B_\\mathsf{OPT} < \\log L$\nand present an efficient way to partially\nenumerate vertices to be protected on top levels, \nleading to the claimed $O(1)$-approximation.\n\n\n\\subsubsection*{Partial enumeration algorithm}\n\nThroughout our algorithm, we set \n\\begin{equation*}\nh=\\lfloor\\log^{(2)} L\\rfloor\n\\end{equation*}\nto be the threshold level defining top vertices $V_{\\leq h}$\nand bottom vertices $V_{> h}$.\nWithin our enumeration procedure we will solve LPs\nwhere we explicitly include some vertex set\n$A\\subseteq V_{\\leq h}$ to be part of the protected\nvertices, and also exclude some set $D\\subseteq V_{\\leq h}$\nfrom being protected. Our enumeration works by growing\nthe sets $A$ and $D$ throughout the algorithm.\nWe thus define the following LP for two disjoint\nsets $A,D \\subseteq V_{\\leq h}$:\n\\begin{equation}\\label{eq:lpRMFCAD}\n\\begin{array}{*2{>{\\displaystyle}r}c*2{>{\\displaystyle}l}}\n\\min & B    &     &     & \\\\\n     & x    &\\in  & \\bar{P}_B & \\\\\n     & B    &\\geq &   1 & \\\\\n     & x(u) &=    &   1 & \\quad \\forall u\\in A\\\\\n     & x(u) &=    &   0 & \\quad \\forall u\\in D\\enspace .\\\\\n\\end{array}\\tag{$\\mathrm{LP(A,D)}$}\\labeltarget{eq:lpRMFCADtarget}\n\\end{equation}\nNotice that~\\ref{eq:lpRMFCAD} is indeed an LP even though the\ndefinition of $\\bar{P}_B$ depends on $B$ (but it does so linearly).\n\nThroughout our enumeration procedure, the disjoint\nsets $A, D \\subseteq V_{\\leq h}$ that we consider are\nalways such that for any $u\\in A\\cup D$, we have\n$P_u\\setminus\\{u\\} \\subseteq D$. In other words, the vertices\n$A\\cup D \\cup \\{r\\}$ form the vertex set of a subtree\nof $G$ such that no root-leaf path contains two vertices\nin $A$. We call a disjoint pair  of\nsets $A,D\\subseteq V_{\\leq h}$ with this property\na \\emph{clean pair}.\n\n\n\nBefore formally stating our enumeration procedure,\nwe briefly discuss the main idea behind it.\nLet $\\mathsf{OPT}\\subseteq V\\setminus \\{r\\}$ be an optimal solution\nto our (compressed) RMFC instance corresponding to some\nbudget $B_{\\mathsf{OPT}} \\in \\mathbb{Z}_{\\geq 1}$. We assume without loss\nof generality that $\\mathsf{OPT}$ does not contain redundancies, i.e.,\nthere is precisely one vertex of $\\mathsf{OPT}$ on each leaf-root\npath.\nAssume that we already guessed some clean pair\n$A,D \\subseteq V_{\\leq h}$ of vertex sets to be\nprotected and not to be protected, respectively,\nand that this guess is compatible with $\\mathsf{OPT}$, i.e.,\n$A\\subseteq \\mathsf{OPT}$ and $D\\cap \\mathsf{OPT}=\\emptyset$.\nLet $(x,B)$ be an optimal solution to~\\ref{eq:lpRMFCAD}.\nBecause we assume that the sets $A$ and $D$ are compatible with\n$\\mathsf{OPT}$, we have $B\\leq B_{\\mathsf{OPT}}$ because\n$(B_\\mathsf{OPT}, \\chi^\\mathsf{OPT})$ is feasible for \\ref{eq:lpRMFCAD}. We define\n\\begin{equation*}\nW_x = \\left\\{u\\in \\Gamma \\;\\middle\\vert\\;\n  x(P_u \\cap V_{> h}) \\geq \\frac{2}{3}\\right\\}\n\\end{equation*}\nto be the set of leaves cut off from the root\nby an $x$-load of at least $\\mu=\\frac{2}{3}$\nwithin bottom levels.\nFor each $u\\in \\Gamma\\setminus W_x$,\nlet $f_u\\in V_{\\leq h}$ be the vertex closest\nto the root among all vertices in\n$(P_u \\cap V_{\\leq h}) \\setminus D$, and we define\n\\begin{equation}\\label{eq:defFx}\nF_x = \\{f_u \\mid u\\in \\Gamma\\setminus W_x\\} \\setminus A.\n\\end{equation}\nNotice that by definition, no two vertices of $F_x$ lie on\nthe same leaf-root path.\nFurthermore, every leaf $u\\in \\Gamma\\setminus W_x$\nis part of the subtree\n$T_f$ for precisely one $f\\in F_x$.\nThe main motivation for considering $F_x$ is that to guess\nvertices in top levels, we can show that it suffices\nto focus on vertices\nlying below some vertex in $F_x$, i.e., vertices\nin the set $Q_x = V_{\\leq h} \\cap (\\cup_{f\\in F_x} T_{f})$.\nTo exemplify this, we first consider the special case\n$\\mathsf{OPT}\\cap Q_x = \\emptyset$, which will also play\na central role later in the analysis of our algorithm.\nWe show that for this case we can get an\n$O(1)$-approximation to RMFC, even though we may only\nhave guessed a proper subset $A\\subsetneq \\mathsf{OPT}\\cap V_{\\leq h}$\nof the $\\mathsf{OPT}$-vertices within the top levels.\n\n\n\\begin{lemma}\\label{lem:goodEnum}\nLet $(A, D)$ be a clean pair of\nvertices that is compatible with $\\mathsf{OPT}$, i.e.,\n$A\\subseteq \\mathsf{OPT}, D\\cap \\mathsf{OPT} = \\emptyset$,\nand let $x$ be an optimal solution\nto~\\ref{eq:lpRMFCAD}.\nMoreover, let $(y,\\bar{B})$ be an optimal solution to\n\\hyperlink{eq:lpRMFCADtarget}{$\\mathrm{LP(A,V_{\\leq h} \\setminus A)}$}.\nThen, if $\\mathsf{OPT}\\cap Q_x=\\emptyset$, we have\n$\\bar{B}\\leq \\frac{5}{2} B_{\\mathsf{OPT}}$.\n\nFurthermore, if $\\mathsf{OPT}\\cap Q_x = \\emptyset$,\nby applying Theorem~\\ref{thm:bottomCover}\nto $y\\wedge \\chi^{V_{> h}}$ with $\\mu=1$ and $q=2$, a set \n$R\\subseteq V_{> h}$ is obtained such that\n$R\\cup A$ is a feasible solution to RMFC with respect\nto the budget $6 \\cdot B_{\\mathsf{OPT}}$.\\footnote{For two vectors\n$a,b\\in \\mathbb{R}^n$ we denote by $a\\wedge b\\in \\mathbb{R}^n$\nthe component-wise minimum of $a$ and $b$.}\n\\end{lemma}\n\\begin{proof}\nNotice that $\\mathsf{OPT}\\cap Q_x=\\emptyset$\nimplies that for each $u\\in \\Gamma \\setminus W_x$,\nwe either have $A\\cap P_u \\neq \\emptyset$ and thus\na vertex of $A$ cuts $u$ off from the root, or\nthe set $\\mathsf{OPT}$ contains a vertex on $P_u \\cap V_{>h}$.\nIndeed, consider a leaf $u\\in \\Gamma \\setminus W_x$\nsuch that $A\\cap P_u = \\emptyset$.\nThen\n$\\mathsf{OPT}\\cap Q_x = \\emptyset$ implies that no vertex\nof $T_{f_u}\\cap V_{\\leq h}$ is part of $\\mathsf{OPT}$.\nFurthermore, $P_{f_u}\\setminus T_{f_u} \\subseteq D$\nbecause $(A,D)$ is a clean pair and $f_u$ is the\ntopmost vertex on $P_u$ that is not in $D$.\nTherefore, $\\mathsf{OPT} \\cap P_u \\cap V_{\\leq h} = \\emptyset$,\nand since $\\mathsf{OPT}$ must contain a vertex in $P_u$, we must\nhave $\\mathsf{OPT}\\cap P_u \\cap V_{>h}\\neq \\emptyset$.\n\nHowever, this observation implies\nthat $z=\\frac{3}{2}(x\\wedge \\chi^{V_{>h}})\n+(\\chi^{\\mathsf{OPT}} \\wedge \\chi^{V_{>h}})+\\chi^A$\nsatisfies\n$z(P_u) \\geq 1$ for all $u\\in \\Gamma$.\nMoreover we have $z\\in P_{\\frac{3}{2}B+B_{\\mathsf{OPT}}}$\ndue to the following.\nFirst, $x\\wedge \\chi^{V_{>h}} \\in P_B$ and\n$\\chi^{\\mathsf{OPT}} \\in P_{B_{\\mathsf{OPT}}}$, which implies\n$z-\\chi^A\\in P_{\\frac{3}{2}B+B_{\\mathsf{OPT}}}$.\nFurthermore, $\\chi^A\\in P_B$, and the vertices in\n$A$ are all on levels $V_{\\leq h}$ which are disjoint\nfrom the levels on which vertices in \n$\\operatorname{supp}(z-\\chi^A)\\subseteq V_{>h}$ lie,\nand thus do not compete\nfor the same budget.\nHence, $(z,\\frac{3}{2}B+B_{\\mathsf{OPT}})$ is feasible for\n\\hyperlink{eq:lpRMFCADtarget}{$\\mathrm{LP(A,V_{\\leq h} \\setminus A)}$},\nand thus\n$\\bar{B} \\leq \\frac{3}{2}B + B_{\\mathsf{OPT}} \\leq \\frac{5}{2} B_{\\mathsf{OPT}}$,\nas claimed.\n\nThe second part of the lemma follows in a straightforward\nway from Theorem~\\ref{thm:bottomCover}.\nObserve first that each leaf $u\\in \\Gamma$ is either\nfully cut off from the root by $y$ on only top levels\nor only bottom levels because $y$ is a $\\{0,1\\}$-solution\non the top levels $V_{\\leq h}$, since on top levels it was\nfixed to $\\chi^A$ because it is a solution to\n\\hyperlink{eq:lpRMFCADtarget}{$\\mathrm{LP(A,V_{\\leq h} \\setminus A)}$}.\nReusing the notation in Theorem~\\ref{thm:bottomCover},\nlet $W=\\{u\\in \\Gamma \\mid (y\\wedge \\chi^{V_{> h}})(P_u) \\geq 1\\}$\nbe all leaves cut off from the root by $y\\wedge \\chi^{V_{>h}}$.\nBy the above discussion, every leaf is thus either part of $W$\nor it is cut off from the root by vertices in $A$. \nTheorem~\\ref{thm:bottomCover} guarantees that $R\\subseteq V_{>h}$\ncuts off all leaves in $W$ from the root, and hence, $R\\cup A$\nindeed cuts off all leaves from the root.\nMoreover, by Theorem~\\ref{thm:bottomCover}, the set\n$R\\subseteq V_{> h}$ is feasible with respect to the\nbudget $5B_{\\mathsf{OPT}} +1 \\leq 6 B_{\\mathsf{OPT}}$.\nFurthermore, $A$ is feasible for budget $B_{\\mathsf{OPT}}$ because\nit is a subset of $\\mathsf{OPT}$. Since $A\\subseteq V_{\\leq h}$\nand $R\\subseteq V_{> h}$ are on disjoint levels, the\nset $R\\cup A$ is feasible for the budget $6 B_{\\mathsf{OPT}}$.\n\\end{proof}\n\n\nOur final algorithm is based on a recursive enumeration\nprocedure that computes a polynomial\ncollection of clean pairs $(A,D)$\nsuch that there is one pair $(A,D)$ in the collection\nwith a corresponding LP solution $x$ of \n\\hyperlink{eq:lpRMFCADtarget}{$\\mathrm{LP(A,D)}$}\nsatisfying that the triple $(A,D,x)$ fulfills the conditions of\nLemma~\\ref{lem:goodEnum}, and thus leading to a\nconstant-factor approximation.\nOur enumeration algorithm\n\\hyperlink{alg:enumRMFCtarget}{$\\mathrm{Enum}(A,D,\\gamma)$}\nis described below.\nIt contains a parameter $\\gamma\\in \\mathbb{Z}_{\\geq 0}$\nthat bounds the recursion depth of the enumerations.\n\n\\smallskip\n\n{\n\\renewcommand{\\thealgocf}{}\n\\begin{algorithm}[H]\n\\SetAlgorithmName{$\\bm{\\mathrm{Enum}(A,D,\\gamma)}$\n\\labeltarget{alg:enumRMFCtarget}\n}{}\n\n\\begin{enumerate}[rightmargin=1em]\n\\item Compute optimal solution $(x,B)$ to \n\\hyperlink{eq:lpRMFCADtarget}{$\\mathrm{LP(A,D)}$}.\n\n\\item\\label{item:stopWhenBLarge}\n\\textbf{If} $B > \\log L$\\textbf{:} \\textbf{stop}.\nOtherwise, continue with step~\\ref{item:addTriple}.\n\n\\item\\label{item:addTriple}\nAdd $(A,D,x)$ to the family of triples to be considered.\n\n\\item\\label{item:enumRecCall} \\tm{if}{\\textbf{I}}%\n\\textbf{f} $\\gamma\\neq 0$ \\textbf{:}\n\\hfill \\texttt{\/\/recursion depth not yet reached \\quad}\n\n\\quad \\tm{for}{\\textbf{F}}\\textbf{or $u\\in F_x$:}\n\\hfill \\texttt{\/\/$F_x$ is defined as in~\\eqref{eq:defFx} \\quad}\n\n\\quad\\quad Recursive call to $\\mathrm{Enum}(A\\cup\\{u\\},D,\\gamma-1)$.\\\\\n\\quad\\quad \\tm[overlay]{end}{}Recursive call\nto $\\mathrm{Enum}(A,D\\cup \\{u\\},\\gamma-1)$.\n\n\\begin{tikzpicture}[overlay, remember picture]\n\\draw (if) ++ (0,-0.5em) |- ($(if |- end) + (0.2,-0.2)$);\n\\draw (for) ++ (0,-0.5em) |- ($(for |- end) + (0.2,-0.1)$);\n\\end{tikzpicture}\n\n\\vspace{-1.5em}\n\n\\end{enumerate}\n\n\n\\caption{Enumerating triples $(A,D,x)$ to find one \nsatisfying the conditions of Lemma~\\ref{lem:goodEnum}.\n}\n\\label{alg:enumRMFC}\n\n\\end{algorithm}\n\\addtocounter{algocf}{-1}\n}%\n\n\\smallskip\n\n\n\nNotice that for any clean pair $(A,D)$ and $u\\in F_x$,\nthe two pairs $(A\\cup \\{u\\}, D)$ and $(A, D\\cup \\{u\\})$\nare clean, too. Hence, if we start \n\\hyperlink{alg:enumRMFCtarget}%\n{$\\mathrm{Enum}(A,D,\\gamma)$}\nwith a clean pair $(A,D)$, we will encounter\nonly clean pairs during all recursive calls.\n\nThe key property of the above enumeration procedure\nis that only a small recursion\ndepth $\\gamma$ is needed for the enumeration algorithm \nto explore a good triple $(A,D,x)$, which satisfies\nthe conditions of Lemma~\\ref{lem:goodEnum}, if we\nstart with the trivial clean pair $(\\emptyset, \\emptyset)$.\nFurthermore, due to step~\\ref{item:stopWhenBLarge},\nwe always have $B\\leq \\log L$ whenever the\nalgorithmm is in step~\\ref{item:enumRecCall}. As we will see\nlater, this allows us to prove that $|F_x|$ is small, which\nwill limit the width of our recursive calls, and leads to\nan efficient procedure as highlighted in the following Lemma.\n\n\n\\begin{lemma}\\label{lem:enumWorks}\nLet $\\bar{\\gamma}= 2(\\log L)^2 \\log^{(2)} L$.\nThe enumeration procedure \\hyperlink{alg:enumRMFCtarget}%\n{$\\mathrm{Enum}(\\emptyset,\\emptyset,\\bar{\\gamma})$}\nruns in polynomial time.\nFurthermore, if $B_\\mathsf{OPT} \\leq \\log L$, then\n\\hyperlink{alg:enumRMFCtarget}%\n{$\\mathrm{Enum}(\\emptyset,\\emptyset,\\bar{\\gamma})$} will\nencounter a triple $(A,D,x)$ satisfying\nthe conditions of Lemma~\\ref{lem:goodEnum}, i.e.,\n\\begin{enumerate}[nosep, label=(\\roman*)]\n\\item $(A,D)$ is a clean pair,\n\\item $A\\subseteq \\mathsf{OPT}$,\n\\item $D\\cap \\mathsf{OPT} = \\emptyset$, and\n\\item $\\mathsf{OPT}\\cap Q_x = \\emptyset$.\n\\end{enumerate}\n\\end{lemma}\n\n\nHence, combining Lemma~\\ref{lem:enumWorks} and\nLemma~\\ref{lem:goodEnum} completes our enumeration procedure\nand implies the following result.\n\n\\begin{corollary}\\label{cor:summaryEnum}\nLet $\\mathcal{I}$ be an RMFC instance on $L$ levels\non a graph $G=(V,E)$ with budgets $B_\\ell = 2^\\ell \\cdot B$.\nThen there is a procedure with running time polynomial\nin $2^L$, returning\na solution $(Q,B)$ for $\\mathcal{I}$, where\n$Q\\subseteq V\\setminus \\{r\\}$ is a set of vertices\nto protect that is feasible for budget $B$,\nsatisfying the following:\nIf the optimal budget $B_{\\mathsf{OPT}}$ for $\\mathcal{I}$ satisfies\n$B_{\\mathsf{OPT}} \\leq \\log L$, then $B\\leq 6 B_\\mathsf{OPT}$.\n\\end{corollary}\n\\begin{proof}\nIt suffices to run \n\\hyperlink{alg:enumRMFCtarget}%\n{$\\mathrm{Enum}(\\emptyset,\\emptyset,\\bar{\\gamma})$} to\nfirst efficiently obtain a family of triples\n$(A_i,D_i,x_i)_i$, where $(A_i, D_i)$ is a clean pair,\nand $x_i$ is an optimal solution to\n\\hyperlink{eq:lpRMFCADtarget}{$\\mathrm{LP(A_i,D_i)}$}.\nBy Lemma~\\ref{lem:enumWorks}, one of these triples\nsatisfies the conditions of Lemma~\\ref{lem:goodEnum}.\n(Notice that these conditions cannot be checked since\nit would require knowledge of $\\mathsf{OPT}$.)\nFor each triple $(A_i,D_i,x_i)$ we obtain a corresponding\nsolution for $\\mathcal{I}$ following the construction\ndescribed in Lemma~\\ref{lem:goodEnum}. More precisely,\nwe first compute an optimal solution $(y_i,\\bar{B}_i)$ to \n\\hyperlink{eq:lpRMFCADtarget}{$\\mathrm{LP(A_i,V_{\\leq h} \\setminus A_i)}$}.\nThen, by applying Theorem~\\ref{thm:bottomCover} to\n$y_i\\wedge \\chi^{V_{> h}}$ with $\\mu=1$ and $q=2$,\na set of vertices\n$R_i\\subseteq V_{> h}$ is obtained such that\n$R_i\\cup A_i$ is feasible for $\\mathcal{I}$ for some\nbudget $B_i$.\nAmong all such sets $R_i\\cup A_i$, we return the one\nwith minimum $B_i$.\nBecause Lemma~\\ref{lem:enumWorks} guarantees that\none of the triples $(A_i, D_i, x_i)$ satisfies the\nconditions of Lemma~\\ref{lem:goodEnum}, we have by\nLemma~\\ref{lem:goodEnum} that the best protection\nset $Q=R_j\\cup A_j$ among all $R_i\\cup A_i$ has a\nbudget $B_j$ satisfying $B_j \\leq 6 B_{\\mathsf{OPT}}$.\n\\end{proof}\n\n\n\n\n\n\n\n\\subsection*{Summary of our $O(1)$-approximation for RMFC}\n\nStarting with an RMFC instance $\\mathcal{I}^{\\mathrm{orig}}$\non a tree with $N$ vertices, we\nfirst apply our compression result, Theorem~\\ref{thm:compressionRMFC},\nto obtain an RMFC instance $\\mathcal{I}$ on a graph $G=(V,E)$ with depth\n$L=O(\\log N)$, and non-uniform budgets $B_\\ell = 2^\\ell B$\nfor $\\ell\\in [L]$.\nLet $B_{\\mathsf{OPT}}\\in \\mathbb{Z}_{\\geq 1}$ be the optimal\nbudget value for $B$ for instance $\\mathcal{I}$%\n---recall that $B=B_{\\mathsf{OPT}}$ in instance $\\mathcal{I}$\nimplies that level $\\ell\\in [L]$ has budget $2^{\\ell} \\cdot B_{\\mathsf{OPT}}$---%\nand let $B_{\\mathsf{OPT}}^{\\mathrm{orig}}$\nbe the optimal budget for $\\mathcal{I}^{\\mathrm{orig}}$.\nBy Theorem~\\ref{thm:compressionRMFC}, we have\n$B_{\\mathsf{OPT}} \\leq B_{\\mathsf{OPT}}^{\\mathrm{orig}}$, and any solution\nto $\\mathcal{I}$ using budget $B$ can efficiently be transformed\ninto one of $\\mathcal{I}^{\\mathrm{orig}}$ of budget\n$2B$.\n\nWe now invoke\nTheorem~\\ref{thm:bigBIsGood} and Corollary~\\ref{cor:summaryEnum}.\nBoth guarantee that a solution to $\\mathcal{I}$ with certain properties\ncan be computed efficiently.\nAmong the two solutions derived from Theorem~\\ref{thm:bigBIsGood}\nand Corollary~\\ref{cor:summaryEnum}, we consider the one\n$(Q,B)$ with lower budget $B$, where $Q\\subseteq V\\setminus \\{r\\}$\nis a set of vertices to protect, feasible for budget\n$B$.\nIf $B\\geq \\log L$, then Theorem~\\ref{thm:bigBIsGood} implies\n$B\\leq 3 B_{\\mathsf{OPT}}$, otherwise Corollary~\\ref{cor:summaryEnum}\nimplies $B\\leq 6 B_{\\mathsf{OPT}}$. Hence, in any case we have\na $6$-approximation for $\\mathcal{I}$. As mentioned before,\nTheorem~\\ref{thm:compressionRMFC} implies that the solution\n$Q$ can efficiently be transformed into a solution for the\noriginal instance $\\mathcal{I}^{\\mathrm{orig}}$ that is\nfeasible with respect to the budget\n$2 B \\leq 12 B_{\\mathsf{OPT}} \\leq 12 B^{\\mathrm{orig}}_{\\mathsf{OPT}}$,\nthus implying Theorem~\\ref{thm:O1RMFC}.\n\n\n\n\\section{Details on compression results}\\label{sec:proofsCompression}\n\nIn this section, we present the proofs for our compression results,\nTheorem~\\ref{thm:compressionFF} and Theorem~\\ref{thm:compressionRMFC}.\nWe start by proving Theorem~\\ref{thm:compressionFF}. The same ideas are used\nwith a slight adaptation in the proof of Theorem~\\ref{thm:compressionRMFC}. \n\nWe call an instance $\\overline{\\mathcal{I}}$ obtained from \nan instance $\\mathcal{I}$ by a sequence of down-push operations\na \\emph{push-down of} $\\mathcal{I}$.\nWe prove Theorem~\\ref{thm:compressionFF} by proving\nthe following result, of which Theorem~\\ref{thm:compressionFF}\nis an immediate consequence, as we will soon show.\n\n\\begin{theorem}\\label{thm:compressionDownPush}\nLet $\\mathcal{I}$ be a unit-budget Firefighter instance\nwith depth $L$, and let $\\delta\\in (0,1)$.\nThen one can efficiently construct a push-down\n$\\overline{\\mathcal{I}}$\nof $\\mathcal{I}$ such that\n\\smallskip\n\\begin{enumerate}[nosep,label=(\\roman*)]\n\\item\\label{item:closeToOPT}\n   $\\val(\\mathsf{OPT}(\\overline{\\mathcal{I}}))\n  \\geq (1-\\delta)\\val(\\mathsf{OPT}(\\mathcal{I}))$, and\n\n\\item\n$\\overline{\\mathcal{I}}$ has nonzero budget\non only $O(\\frac{\\log L}{\\delta})$ levels.\n\\end{enumerate}\n\\end{theorem}\n\nBefore we prove Theorem~\\ref{thm:compressionDownPush}, we show\nhow it implies \nTheorem~\\ref{thm:compressionFF}.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:compressionFF}]\nWe start by showing \nhow levels of zero budget can be removed \nthrough the following \\emph{contraction operation}. \nLet $\\ell \\in \\{2,\\dots, L\\}$ be a level whose budget\nis zero. For each vertex\n$u \\in V_{\\ell-1}$ we contract all edges from $u$\nto its children and increase the\nweight $w(u)$ of $u$ by the sum of the weights\nof all of its children.\nFormally, if $u$ has children $v_1, \\dots, v_k\\in V_\\ell$,\nthe vertices $u,v_1, \\dots, u_k$ are replaced by a single\nvertex $z$ with weight $w(z) = w(u) + \\sum_{i=1}^k w(v_i)$,\nand $z$ is adjacent to the parent of $u$ and to all children\nof $v_1,\\dots, v_k$.\nOne can easily observe that this is an ``exact''\ntransformation in the sense that any solution before\nthe contraction remains one after contraction\nand vice versa (when identifying the vertex $z$\nin the contracted version with $v$);\nmoreover, solutions before and\nafter contraction have the same value.\n\nNow, by first applying Theorem~\\ref{thm:compressionDownPush}\nand then applying the latter contraction operations level by\nlevel to all levels\n$\\ell\\in \\{2,\\dots, L\\}$\nwith zero budget (in an arbitrary order),\nwe obtain an equivalent instance with the desired \ndepth, thus satisfying the conditions of\nTheorem~\\ref{thm:compressionFF}.\n\\end{proof}\n\n\nIt remains to prove Theorem~\\ref{thm:compressionDownPush}.\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:compressionDownPush}]\nConsider a unit-budget Firefighter instance on a tree\n$G=(V,E)$ with depth $L$.\nThe push-down $\\overline{\\mathcal{I}}$ that we construct\nwill have nonzero budgets precisely on the following\nlevels $\\mathcal{L} \\subseteq [L]$:\n\\begin{equation*}\n\\mathcal{L} = \\left\\{\\left\\lceil(1+\\delta)^j\\right\\rceil\n    \\;\\middle\\vert\\; j\\in \\left\\{0,\\dots,\n        \\left\\lfloor\\frac{\\log L}{\\log(1+\\delta)}\n           \\right\\rfloor\\right\\}\\right\\}\n   \\cup \\{L\\}.\n\\end{equation*}\nFor simplicity, let $\\mathcal{L}= \\{\\ell_1,\\dots, \\ell_k\\}$\nwith $1=\\ell_1 < \\ell_2 < \\dots < \\ell_k=L$.\nHence,\n$k=O(\\frac{\\log L}{\\log(1+\\delta)})\n  = O(\\frac{\\log L}{\\delta})$. The push-down\n$\\overline{\\mathcal{I}}$ is obtained by pushing\nany budget on a level not in $\\mathcal{L}$ down\nto the next level in $\\mathcal{L}$. Formally,\nfor $i\\in [k]$, the budget $B_{\\ell_i}$\nat level $\\ell_i$ is given by\n$B_{\\ell_i} = \\ell_i - \\ell_{i-1}$, where\nwe set $\\ell_{0}=0$.\nMoreover, $B_\\ell=0$ for\n$\\ell\\in [L]\\setminus \\mathcal{L}$.\nClearly, the instance $\\overline{\\mathcal{I}}$ can be\nconstructed efficiently. Furthermore, the number\nof levels with nonzero budget is equal to\n$k=O(\\frac{\\log L}{\\delta})$ as desired. It remains\nto show point~\\ref{item:closeToOPT}\nof Theorem~\\ref{thm:compressionDownPush}.\n\nTo show~\\ref{item:closeToOPT}, consider an optimal\nredundancy-free solution $S^*\\subseteq V$ of $\\mathcal{I}$; hence,\n$\\val(\\mathsf{OPT}(\\mathcal{I})) = \\sum_{u\\in S^*} w(T_u)$ and\nno two vertices of $S^*$ lie on the same leaf-root path.\nWe will show that there is a feasible solution\n$\\overline{S}$ to $\\overline{\\mathcal{I}}$ such that\n$\\overline{S}\\subseteq S^*$ and the value of\n$\\overline{S}$ is at least $(1-\\delta)\\val(\\mathsf{OPT}(\\mathcal{I}))$.\nNotice that since $S^*$ is redundancy-free, any subset\nof $S^*$ is also redundancy-free. Hence, the value of\nthe set $\\overline{S}$ to construct will be equal\nto $\\sum_{u\\in \\overline{S}} w(T_u)$.\nThe set $S^*$ being (budget-)feasible for $\\mathcal{I}$\nimplies \n\\begin{equation}\\label{eq:SStarFeasible}\n|S^*\\cap V_{\\leq \\ell}| \\leq \\ell\n   \\quad \\forall \\ell\\in [L].\n\\end{equation}\nAnalogously, a set $S\\subseteq V$ is feasible for\n$\\overline{\\mathcal{I}}$ if and only if\n\\begin{equation}\\label{eq:SFeasibleFull}\n|S\\cap V_{\\leq \\ell}| \\leq \\sum_{i=1}^\\ell B_i\n   \\quad \\forall \\ell\\in [L].\n\\end{equation}\nHence, we want to show that there is a set $\\overline{S}$\nsatisfying the above system and such that\n$\\sum_{u\\in \\overline{S}}w(T_u)\n  \\geq (1-\\delta)\\val(\\mathsf{OPT}(\\mathcal{I}))$.\nNotice that in~\\eqref{eq:SFeasibleFull}, the constraint\nfor any $\\ell\\in [L-1]$ such that $B_{l+1}=0$ is\nredundant due to the constraint for level $\\ell+1$\nwhich has the same right-hand side but a larger\nleft-hand side.\nThus, system~\\eqref{eq:SFeasibleFull} is equivalent\nto the following system\n\\begin{equation}\\label{eq:SFeasibleShort}\n\\begin{aligned}\n|S\\cap V_{\\leq \\ell_{i+1}-1}| &\\leq \\ell_{i} \n   \\quad \\forall i\\in [k-1],\\\\\n|S\\cap V| &\\leq L.\n\\end{aligned}\n\\end{equation}\nTo show that there is a good subset\n$\\overline{S}\\subseteq S^*$ that\nsatisfies~\\eqref{eq:SFeasibleShort} we use a\npolyhedral approach.\nObserve that~\\eqref{eq:SFeasibleFull} is the\nconstraint system of a laminar matroid\n(see~\\cite[Volume B]{Schrijver2003} for more information on matroids).\nHence,\nthe convex hull of all characteristic vectors\n$\\chi^S\\in \\{0,1\\}^V$ of sets $S\\subseteq S^*$\nsatisfying~\\eqref{eq:SFeasibleShort} is given\nby the following polytope\n\\begin{equation*}\nP = \\left\\{\nx\\in [0,1]^V \\;\\middle\\vert\\;\n\\begin{minipage}[c]{0.4\\linewidth}\n\\vspace{-1em}\n\\begin{align*}\nx(V_{\\leq \\ell_{i+1}-1}) &\\leq \\ell_{i} \\;\\;\\forall i\\in [k-1],\\\\\nx(V) &\\leq L,\\\\\nx(V\\setminus S^*) &= 0\n\\end{align*}\n\\end{minipage}\n\\right\\}.\n\\end{equation*}\nAlternatively, to see that $P$ indeed\ndescribes the correct polytope,\nwithout relying on matroids, one can observe that its\nconstraint matrix is totally unimodular because it\nhas the consecutive-ones property with respect to the\ncolumns.\n\n\nThus there exists a set $\\overline{S}\\subseteq S^*$ with\n$\\sum_{u\\in \\overline{S}} w(T_u) \\geq (1-\\delta)\\val(\\mathsf{OPT}(\\mathcal{I}))$\nif and only if\n\\begin{equation}\\label{eq:polSb}\n\\max\\left\\{\\sum_{u\\in S^*} x(u)\\cdot\n w(T_u) \\;\\middle\\vert\\;\n    x\\in P\\right\\}\\geq (1-\\delta)\\val(\\mathsf{OPT}(\\mathcal{I})).\n\\end{equation}\nTo show~\\eqref{eq:polSb}, and thus complete the proof,\nwe show that $y=\\frac{1}{1+\\delta} \\chi^{S^*}\\in P$.\nThis will indeed imply~\\eqref{eq:polSb} since the\nobjective value of $y$ satisfies\n\\begin{equation*}\n\\sum_{u\\in S^*} y(u) \\cdot w(T_u) =\n \\frac{1}{1+\\delta}\\val(\\mathsf{OPT}(\\mathcal{I}))\n    \\geq (1-\\delta)\\val(\\mathsf{OPT}(\\mathcal{I})).\n\\end{equation*}\n\nTo see that $y\\in P$, notice that\n$y(V\\setminus S^*)=0$ and\n$y(V) = \\frac{1}{1+\\delta} |S^*|\n\\leq \\frac{1}{1+\\delta} L \\leq L$, where the\nfirst inequality follows by $S^*$\nsatisfying~\\eqref{eq:SStarFeasible} for $\\ell=L$.\nFinally, for $i\\in [k-1]$, we have\n\\begin{align*}\ny(V_{\\leq \\ell_{i+1}-1}) &=\n  \\frac{1}{1+\\delta}\n    |S^* \\cap V_{\\leq \\ell_{i+1}-1}|\n\\leq \\frac{1}{1+\\delta}(\\ell_{i+1}-1),\n\\end{align*}\nwhere the inequality follows from $S^*$\nsatisfying~\\eqref{eq:SStarFeasible}\nfor $\\ell=\\ell_{i+1}-1$.\nIt remains to show $\\ell_{i+1} -1 \\leq (1+\\delta)\\ell_i$\nto prove $y\\in P$.\nLet $\\alpha \\in \\mathbb{Z}_{\\geq 0}$ be the smallest\ninteger for which we have\n$\\ell_{i+1} = \\lceil (1+\\delta)^{\\alpha}\\rceil$. In\nparticular, this implies\n$\\ell_{i}=\\lceil (1+\\delta)^{\\alpha-1}\\rceil$. We\nthus obtain\n\\begin{equation*}\n\\ell_{i+1} - 1 \\leq (1+\\delta)^{\\alpha}\n   = (1+\\delta) (1+\\delta)^{\\alpha-1}\n   \\leq (1+\\delta) \\ell_i,\n\\end{equation*}\nas desired.\n\n\\end{proof}\n\n\n\n\nWe conclude with the proof of Theorem~\\ref{thm:compressionRMFC}.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:compressionRMFC}.]\n\n We start by describing the construction of $G' = (V',E')$. As is the case\nin the proof of Theorem~\\ref{thm:compressionFF}, we first change the \nbudget assignment of the instance and then contract all levels with zero budgets.\nNotice that, for a given budget $B$ per layer,\nwe can consider an RMFC instance as a Firefighter instance,\nwhere each leaf $u\\in \\Gamma$ has weight $w(u)=1$, and all other\nweights are zero. Since our goal is to save all leaves, we want\nto save vertices of total weight $|\\Gamma|$.\n\n\nFor simplicity of presentation we assume that $L$ is a power of $2$. This assumption does\nnot compromise generality, as one can always augment the original tree with one path starting from the root and going down to level\n$2^{\\lceil\\log L\\rceil}$.\n\nThe set of levels in which the transformed instance will have\nnonzero budget is \n\\begin{equation*}\n\\mathcal{L} = \\left\\{2^j-1 \\,\\middle\\vert\\, j\\in \\{1,\\ldots, \\log L \\} \\right\\}.\n\\end{equation*}\nHowever, instead of down-pushes we will do \\emph{up-pushes} were\nbudget is moved upwards. More precisely, \nthe budget of any level $\\ell\\in [L]\\setminus \\mathcal{L}$\nwill be assigned to the first level in $\\mathcal{L}$ that\nis above $\\ell$, i.e., has a smaller index than $\\ell$.\nAs for the Firefighter case, we now remove all $0$-budget\nlevels using contraction, which will lead to a new\nweight function $w'$ on the vertices. Since our goal\nis to save the weight of the whole tree,\nwe can remove for each vertex $u$ with $w'(u) > 0$, the\nsubtree below $u$. This does not change the problem since\nwe have to save $u$, and thus will anyway also save its subtree.\nThis finishes our construction of $G'=(V',E')$, and the task\nis again to remove all leaves of $G'$.\nNotice that $G'$ has $L' \\leq |\\mathcal{L}| = \\log L $\nmany levels, and level $\\ell\\in [L']$ has a budget of\n$B 2^{\\ell}$ as desired.\nAnalogous to the\ndiscussion for compression in the context of the Firefighter \nproblem we have that if the original problem is feasible,\nthen so is the RMFC problem on $G'$ with\nbudgets $B 2^{\\ell}$.\nIndeed, before performing the contraction operations (which\ndo not change the problem), the original RMFC problem was\na push-down of the one we constructed.\n\n\nSimilarly, one can observe that before contraction,\nthe instance we obtained is itself a push-down of\nthe original instance with budgets $2B$ on each level.\nHence, analogously to the compression result for\nthe Firefighter case, any solution to the RMFC problem\non $G'$ can \nefficiently be transformed into a solution to the original\nRMFC problem on $G$ with budgets $2B$ on each level.\n\n\\end{proof}\n\n\n\n\n\n\\section{Missing details for Firefighter PTAS}\\label{sec:proofsFF}\n\nIn this section we present the missing proofs for our PTAS for the\nFirefighter problem.\n\n\nWe start by proving Lemma~\\ref{lem:sparsityFF}, showing that\nany vertex solution $x$ to \\ref{eq:lpFF} has\nfew $x$-loose vertices.\nMore precisely, the proof below shows that the number\nof $x$-loose vertices is upper bounded by the number\nof tight budget constraints.\nThe precise same reasoning used in the proof of\nLemma~\\ref{lem:sparsityFF} can also be applied\nin further contexts, in particular for the RMFC problem.\n\n\n\\subsubsection*{Proof of Lemma~\\ref{lem:sparsityFF}}\n\nLet $x$ be a vertex of the polytope defining the feasible set\nof~\\ref{eq:lpFF}.\nHence, $x$ is uniquely defined by\n$|V\\setminus\\{r\\}|$ many linearly independent and tight\nconstraints of this polytope.\nNotice that the tight constraints can be partitioned into\nthree groups:\n\\begin{enumerate}[label=(\\roman*),nosep]\n\\item Tight nonnegativity constraints, one for\neach vertex in $\\mathcal{F}_1=\\{u\\in V\\setminus \\{r\\} \\mid x(u) = 0\\}$.\n\n\\item Tight budget constraints, one for each level in\n$\\mathcal{F}_2 = \\{\\ell\\in [L] \\mid x(V_{\\leq \\ell})=\\sum_{i=1}^\\ell B_i\\}$.\n\n\\item Tight leaf constraints, one for each vertex in\n$\\mathcal{F}_3 = \\{u\\in \\Gamma \\mid x(P_u) = 1\\}$.\n\\end{enumerate}\nDue to potential degeneracies of the polytope describing\nthe feasible set of~\\ref{eq:lpFF} there may be several\noptions to describe $x$ as the unique solution to\na full-rank linear subsystem of the constraints described\nby $\\mathcal{F}_1 \\cup \\mathcal{F}_2 \\cup \\mathcal{F}_3$.\nWe consider a system that contains all tight\nnonnegativity constraints, i.e.,\nconstraints corresponding to $\\mathcal{F}_1$, and\ncomplement these constraints with arbitrary subsets\n$\\mathcal{F}'_2\\subseteq \\mathcal{F}_2$ and \n$\\mathcal{F}'_3\\subseteq \\mathcal{F}_3$ of\nbudget and leaf constraints that lead to a full rank\nlinear system corresponding to the constraints\n$\\mathcal{F}_1 \\cup \\mathcal{F}'_2 \\cup \\mathcal{F}'_3$.\nHence\n\\begin{equation}\\label{eq:fullRankSys}\n|\\mathcal{F}_1| + |\\mathcal{F}'_2| + |\\mathcal{F}'_3| = |V| - 1.\n\\end{equation}\n\n\nLet $V^{\\mathcal{L}}\\subseteq \\operatorname{supp}(x)$\nand $V^{\\mathcal{T}}\\subseteq \\operatorname{supp}(x)$\nbe the $x$-loose and $x$-tight vertices, respectively.\nWe first show $|\\mathcal{F}'_3|\\leq |V^{\\mathcal{T}}|$.\nFor each leaf $u\\in \\mathcal{F}'_3$, let $f_u\\in V^\\mathcal{T}$ be \nthe first vertex on the unique $u$-root path that is part of\n$\\operatorname{supp}(x)$. In particular, if $u\\in \\operatorname{supp}(x)$ then $f_u=u$.\nClearly, $f_u$ must be an $x$-tight vertex because\nthe path constraint with respect to $u$ is tight.\nNotice that for any distinct vertices $u_1,u_2\\in \\mathcal{F}'_3$,\nwe must have $f_{u_1}\\neq f_{u_2}$. Assume by sake of\ncontradiction that $f_{u_1}= f_{u_2}$. However, this implies\n$\\chi^{P_{u_1}} - \\chi^{P_{u_2}}\\in \\spn(\\{\\chi^{v} \\mid v\\in \\mathcal{F}_1\\})$, since \n$P_{u_1} \\Delta P_{u_2} := (P_{u_1} \\setminus P_{u_2})\\cup (P_{u_2} \\setminus P_{u_1}) \\subseteq \\mathcal{F}_1$, and leads to a contradiction\nbecause we exhibited a linear dependence among the constraints\ncorresponding to $\\mathcal{F}'_3$ and $\\mathcal{F}_1$.\nHence, $f_{u_1}\\neq f_{u_2}$ which implies that the\nmap $u \\mapsto f_u$ from $\\mathcal{F}'_3$ to $V^{\\mathcal{T}}$\nis injective and thus\n\\begin{equation}\\label{eq:boundLeafConstr}\n|\\mathcal{F}'_3| \\leq |V^{\\mathcal{T}}|.\n\\end{equation}\nWe thus obtain\n\\begin{align*}\n|\\operatorname{supp}(x)| &= |V|-1-|\\mathcal{F}_1|\n    && \\text{($\\operatorname{supp}(x)$ consists of all $u\\in V\\setminus \\{r\\}$ with\n               $x(u)\\neq 0$, i.e., $u\\not\\in \\mathcal{F}_1$)}\\\\\n  &= |\\mathcal{F}'_2| + |\\mathcal{F}'_3|\n    && \\text{(by~\\eqref{eq:fullRankSys})}\\\\\n  &\\leq |\\mathcal{F}'_2| + |V^{\\mathcal{T}}|\n    && \\text{(by~\\eqref{eq:boundLeafConstr})},\n\\end{align*}\nwhich leads to the desired result since\n\\begin{equation*}\n|V^{\\mathcal{L}}| = |\\operatorname{supp}(x)| - |V^{\\mathcal{T}}|\n  \\leq |\\mathcal{F}'_2| \\leq L.\n\\end{equation*}\n\n\n\n\n\n\\qed\n\n\n\\subsubsection*{Proof of Lemma~\\ref{lem:pruning}}\nWithin this proof we focus on protection sets where the budget available\nfor any level is spent on the same level (and not a later one).\nAs discussed, there is always an optimal protection set\nwith this property.\n\nLet $B_\\ell \\in \\mathbb{Z}_{\\geq 0}$ be the budget available at level $\\ell\\in [L]$ and let \n$\\lambda_\\ell = \\lambda B_\\ell$.\n We construct the tree $G'$ using the following greedy procedure. Process\nthe levels of $G$ from the first one to the last one. At every level $\\ell\\in [L]$,\npick $\\lambda_\\ell$ vertices $u^\\ell_1, \\cdots, u^\\ell_{\\lambda_\\ell}$ at the $\\ell$-th \nlevel of $G$ greedily, i.e., pick each next vertex such that the subtree corresponding to that \nvertex has largest weight among all remaining vertices in the level. \nAfter each selection of a vertex the greedy procedure can no longer \nselect any vertex in the corresponding subtree in subsequent iterations.\\footnote{\nFor $\\lambda=1$ this procedure produces a set of vertices, which comprise\na $\\frac{1}{2}$-approximation for the Firefighter problem, as it coincides\nwith the greedy algorithm of Hartnell and Li~\\cite{HartnellLi2000}.}\n\nNow, the tree $G'$ is constructed by deleting from $G$ any vertex\nthat is both not contained in any subtree $T_{u^\\ell_i}$, and not \ncontained in any path $P_{u^\\ell_i}$ for $\\ell\\in [L]$ and $i\\in [\\lambda_\\ell]$.\nIn other words, if $U\\subseteq V$ is the set of all leaves\nof $G$ that were disconnected from the root by the greedy\nalgorithm, then we consider the subtree of $G$ induced\nby the vertices $\\cup_{u\\in U}P_u$.\nFinally, the weights of vertices on the paths \n$P_{u^\\ell_i} \\setminus \\{u^\\ell_i\\}$ for $\\ell\\in [L]$ and $i\\in [\\lambda_\\ell]$ are reduced\nto zero. This concludes the construction of $G'=(V',E')$ and the new weight function $w'$. Denote\nby $D_\\ell = \\{u^\\ell_1,\\cdots, u^\\ell_{\\lambda_\\ell}\\}$ the set of vertices chosen by the\ngreedy procedure in level $\\ell$, and let $D=\\cup_{\\ell\\in [L]} D_{\\ell}$.\nObserve that by construction we have that each vertex\nwith non-zero weight is in the subtree of a vertex in $D$, i.e.,\n$$\nw'(V') = \\sum_{u\\in D} w'(T'_u).\n$$\nThe latter immediately implies point~\\ref{item:pruningLargeOpt}\nof Lemma~\\ref{lem:pruning} because the vertices $D$ can\nbe partitioned into $\\lambda$ many vertex sets that are\nbudget-feasible and can thus be protected in a Firefighter solution.\nHence an optimal solution to the Firefighter problem\non $G'$ covers at least a $\\frac{1}{\\lambda}$-fraction of the total\nweight of $G'$.\n\n\nIt remains to prove point~\\ref{item:pruningSmallLoss} of the Lemma.\nLet $S^* = S^*_1\\cup \\cdots \\cup S^*_L$ be the vertices protected in some optimal\nsolution in $G$, where $S^*_\\ell \\subseteq V_\\ell$ are the vertices protected in level $\\ell$ (and\nhence $|S^*_\\ell| \\leq B_\\ell$). \nWithout loss of generality, we assume that $S^*$ is redundancy-free.\nFor distinct vertices $u,v\\in V$ we say that $u$ \\emph{covers} $v$ if $v\\in T_u \\setminus \\{u\\}$.\n\nFor $\\ell \\in [L]$, let $I_\\ell = S^*_l \\cap D_\\ell$ be the set of vertices protected \nby the optimal solution that are also chosen by the greedy algorithm in level $\\ell$.\nFurthermore, let $J_\\ell \\subseteq S^*_\\ell$\nbe the set of vertices of the optimal solution that are \ncovered by vertices chosen by the greedy algorithm in earlier\niterations, i.e.,\n$J_\\ell = S^*_\\ell \\cap \\bigcup_{u\\in D_1\\cup\\cdots\\cup D_{\\ell -1}} T_u$. \nFinally, let $K_\\ell = S^*_\\ell \\setminus (I_\\ell \\cup J_\\ell)$ be all other\noptimal vertices in level $\\ell$. Clearly, $S^*_\\ell = I_\\ell \\cup J_\\ell \\cup K_\\ell$ \nis a partition of $S^*_\\ell$.\n\nConsider a vertex $u\\in K_\\ell$ for some $\\ell\\in [L]$. From the guarantee of the greedy \nalgorithm it holds that for every vertex $v\\in D_\\ell$ we have $w'(T_v) = w(T_v) \\geq w(T_u)$. \nThe same does not necessarily hold for covered vertices. \nOn the other hand, covered vertices\nare contained in $G'$ with their original weights. We exploit these two \nproperties to prove the existence of a solution in $G'$\nof almost the same weight as $S^*$.\n\nTo prove the existence of a good solution we construct\na solution $A = A_1 \\cup \\cdots \\cup A_L$ with $A_\\ell \\subseteq V_\\ell$ and $|A_\\ell| \\leq B_\\ell$\nrandomly, and prove a bound on its expected quality.\nWe process the levels of the tree $G'$ top-down to construct $A$ step\nby step.\nThis clearly does not compromise generality. Recall that we only need to prove the \nexistence of a good solution, and not compute it efficiently. We can hence assume the\nknowledge of $S^*$ in the construction of $A$. To this end assume that all levels\n$\\ell' < \\ell$ were already processed, and the corresponding sets $A_{\\ell'}$ were\nconstructed. The set $A_{\\ell}$ is constructed as follows:\n\n\\begin{enumerate}\n\\item Include in $A_\\ell$ all vertices in $I_\\ell$.\n\\item Include in $A_\\ell$ all vertices in $J_\\ell$ that are not \ncovered by vertices in $A_1\\cup \\cdots \\cup A_{\\ell-1}$ (vertices selected so far).\n\\item Include in $A_\\ell$ a \\emph{uniformly random subset} of $|K_\\ell|$ vertices\nfrom $D_\\ell \\setminus I_\\ell$.\n\\end{enumerate}\n\nIt is easy to verify that the latter algorithm returns a redundancy-free solution, as no two\nchosen vertices in $A$ lie on the same path to the root. Next, we show that the expected\nweight of vertices saved by $A$ is at least $(1-\\frac{1}{\\lambda})\\val(\\mathsf{OPT}(\\overline{\\mathcal{I}}))$, \nwhich will prove our claim, since then at least one solution has the desired quality.\n\nSince we only need a bound on the expectation we can focus on a single level $\\ell \\in [L]$ \nand show that the contribution of vertices in $A_\\ell$ is in expectation at least $1-\\frac{1}{\\lambda}$\ntimes the contribution of the vertices in $S^*_\\ell$. Observe that the vertices in $I_\\ell$ are\ncontained both in $S^*_\\ell$ and in $A_\\ell$, hence it suffices to show that the contribution\nof $A_\\ell \\setminus I_\\ell$ is at least $1-\\frac{1}{\\lambda}$ times the contribution \nof $S^*_\\ell \\setminus I_\\ell$, in expectation. Also, recall that every vertex in $D_\\ell$\ncontributes at least as much as any vertex in $K_\\ell$, by the greedy selection rule. It follows\nthat the $|K_\\ell|$ randomly selected vertices in $A_\\ell$ have at least as much contribution\nas the vertices in $K_\\ell$. Consequently, to prove the claim is suffices to bound the \nexpected contribution of vertices in $A_\\ell \\cap J_\\ell$ with respect to the contribution of\n$J_\\ell$. Since $A_\\ell \\cap J_\\ell \\subseteq J_\\ell$ it suffices to show that every vertex\n$u\\in J_\\ell$ is also present in $A_\\ell$ with probability at least $1-\\frac{1}{\\lambda}$.\n\nTo bound the latter probability we make use of the random choices in the construction\nof $A$ as follows. Let $\\ell' < \\ell$ be the level at which for some $w\\in D_{\\ell'}$ it \nholds that $u\\in T_w$. In other words, $\\ell'$ is the level that contains the ancestor \nof $u$ that was chosen by the greedy construction of $G'$. Now, since $S^*$ is redundancy-free,\nand by the way that $A$ is constructed, it holds that if $u\\not\\in A_\\ell$ \nthen $w\\in A_{\\ell'}$, namely if $u$ is covered, it can only be covered by the \nunique ancestor $w$ of $u$ that was chosen in the greedy construction of $G'$. Furthermore,\nin such a case the vertex $w$ was selected randomly in the third step of the $\\ell'$-th\niteration. Put differently, the probability that the vertex $u$ is covered \nis exactly the probability that its ancestor $w$ is chosen randomly to be part of $A_{\\ell'}$.\nSince these vertices are chosen to be a random subset of $|K_{\\ell'}|$ vertices from the set $D_{\\ell'}\\setminus I_{\\ell'}$,\nthis probability is at most \n$$\n\\frac{|K_{\\ell'}|}{|D_{\\ell'}| - |I_{\\ell'}|} =\n\\frac{|K_{\\ell'}|}{\\lambda B_{\\ell'} - |I_{\\ell'}|} \\leq \n\\frac{1}{\\lambda}, \n$$\nwhere the last inequality follows from $|K_{\\ell'}| + |I_{\\ell'}| \\leq B_{\\ell'}$.\nThis implies that $u\\in A_\\ell$ with probability of at least $1-\\frac{1}{\\lambda}$, as required\nand concludes the proof of the lemma.\n\n\n\\qed\n\n\n\n\n\n\\subsubsection*{Proof of Lemma~\\ref{lem:setQ}}\n\n\n\nWe construct the set $Q$ in two phases as follows. First we construct \na set $\\overline Q \\subseteq H$ of vertices fulfilling the first and the third properties, i.e.,\nit will satisfy $|\\overline Q| = O(\\frac{\\log N}{\\epsilon^3})$, as well as the property that\n$G[V\\setminus \\overline Q\\cup \\{r\\}]$ has connected components each of weight at most $\\eta$. Then,\nwe add to $\\overline Q$ all vertices of $H$ of degree at least three to arrive\nat the final set $Q$.\n\nIt will be convenient to define heavy vertices and heavy tree with respect to any \nsubtree $G'= (V', E')$ of $G$ which contains the root $r$. Concretely, we \ndefine $H_{G'} = \\{u\\in V'\\setminus \\{r\\} \\,\\mid\\, w(T'_u)\\geq \\eta\\}$ \nto be the set of $G'$-heavy vertices. The $G'$-heavy tree is the\nsubtree $G'[H_{G'} \\cup \\{r\\}]$ of $G'$. Observe that $H = H_G$ and that\n$H_{G'} \\subseteq H$ for every subtree $G'$ of $G$.\n\nTo construct $\\overline Q$ we process the tree $G$ in a bottom-up \nfashion starting with $\\overline Q = \\emptyset$. We will also remove\nparts of the tree in the end of every iteration. The first iteration \nstarts with $G' = G$. In every iteration that starts with tree $G'$, include in \n$\\overline Q$ an arbitrary leaf $u\\in H_{G'}$ of the heavy tree and remove $u$ and all vertices\nin its subtree from $G'$. The procedure ends when there is\neither no heavy vertex in $G'$ anymore, or when $r$ is the\nonly heavy vertex in $G'$.\n\nLet us verify that the claimed properties indeed hold. The fact that \n$|\\overline Q| = O(\\frac{\\log N}{\\epsilon^3})$ follows from the fact that at each iteration \nwe remove a $G'$-heavy vertex including all its subtree from the \ncurrent tree $G'$. This implies that the total weight of the tree $G'$\ndecreases by at least $\\eta$ in every iteration. Since we only include one \nvertex in every iteration we have\n$|\\overline Q| \\leq \\frac{w(V)}{\\eta} = O(\\frac{\\log N}{\\epsilon^3})$.\n\nThe third property follows from the fact that we always remove a leaf\nof the $G'$-heavy tree. Observe that the connected components of \n$G[V\\setminus (\\overline Q \\cup \\{r\\})]$ are contained in the subtrees\nwe disconnect in every iteration in the construction of $\\overline Q$.\nBy definition of $G'$-heavy leaves, in any such iteration where \na $G'$-heavy leaf $u$ is removed from the tree, these parts have weight \nat least $\\eta$, but any subtree rooted at any descendant of $u$ has\nweight strictly smaller than $\\eta$ (otherwise this descendant would\nbe $G'$-heavy as well, contradicting the assumption that it has a\n$G'$-heavy leaf $u$ as an ancestor). Now, since $u$ is included in $\\overline Q$,\nthe connected components are exactly these subtrees, so the property indeed holds.\n\nTo construct $Q$ and conclude the proof it remains to include in $\\overline Q$\nall remaining nodes of degree at least three in the heavy tree. The \nfact that also all leaves of the heavy tree are included in $Q$ is\nreadily implied by the construction of $\\overline Q$, so the second property \nholds for $Q$. Clearly, by removing more vertices from the heavy tree, the sizes\nof connected components only get smaller, so $Q$ also satisfies the third\ncondition, since $\\overline Q$ already did. Finally, the number of \nvertices of degree at least three in the heavy tree is strictly\nless than the number of its leaves, which is $O(\\frac{\\log N}{\\epsilon^3})$;\nfor otherwise a contradiction would occur since the tree would\nhave an average degree of at least $2$.\nThis implies that, in\ntotal, $|Q| = O(\\frac{\\log N}{\\epsilon^3})$,\nso the first property also holds.\n\nTo conclude the proof of the lemma it remains to note that the latter\nconstruction can be easily implemented in polynomial time.\n\n\\qed\n\n\n\n\\section{Missing details for $O(1)$-approximation\nfor RMFC}\\label{sec:proofsRMFC}\n\nThis section contains the missing proofs for our\n$12$-approximation for RMFC.\n\n\n\\subsection*{Proof of Theorem~\\ref{thm:bottomCover}}\n\n\n\nTo prove Theorem~\\ref{thm:bottomCover} we first show\nthe following result, based on which Theorem~\\ref{thm:bottomCover}\nfollows quite directly.\n\n\n\\begin{lemma}\\label{lem:sliceCover}\nLet $B\\in \\mathbb{R}_{\\geq 1}$, $\\eta\\in (0,1]$,\n$k \\in \\mathbb{Z}_{\\geq 1}$, and\n$\\ell_1 = \\lfloor \\log^{(k)} L \\rfloor$,\n$\\ell_2 = \\lfloor \\log^{(k-1)} L \\rfloor$.\nLet $x\\in P_B$ with\n$\\operatorname{supp}(x)\\subseteq V_{(\\ell_1,\\ell_2]}\n \\coloneqq V_{>\\ell_1} \\cap V_{\\leq \\ell_2}$,\nand we define $Y = \\{u\\in \\Gamma \\mid x(P_u) \\geq \\eta\\}$.\nThen one can efficiently compute a\nset $R\\subseteq V_{(\\ell_1,\\ell_2]}$ such\nthat\n\\smallskip\n\\begin{enumerate}[nosep, label=(\\roman*)]\n\\item\\label{item:scHitPath}\n$R\\cap P_u \\neq \\emptyset \\quad \\forall u\\in Y$, and\n\n\\item\\label{item:scBudgetOk}\n$\\chi^R\\in P_{\\bar{B}}$,\nwhere $\\bar{B} = \\frac{1}{\\eta} B + 1$.\n\\end{enumerate}\n\\end{lemma}\n\n\nWe first observe that Lemma~\\ref{lem:sliceCover} indeed\nimplies Theorem~\\ref{thm:bottomCover}.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:bottomCover}]\nFor $k=1,\\dots, q$, let\n$\\ell_1^k = \\lfloor \\log^{(k)} L\\rfloor$ and\n$\\ell_2^k = \\lfloor \\log^{(k-1)} L\\rfloor$, and we define\n$x^k\\in P_B$ by $x^k = x \\wedge \\chi^{V_{(\\ell_1^k, \\ell_2^k]}}$.\nHence, $x=\\sum_{k=1}^q x^k$.\nFor each $k\\in [q]$, we apply Lemma~\\ref{lem:sliceCover} to\n$x^k$ with $\\eta = \\frac{\\mu}{q}$ to obtain a set\n$R^k \\subseteq V_{(\\ell_1^k, \\ell_2^k]}$ satisfying\n\\begin{enumerate}[nosep, label=(\\roman*)]\n\\item $R^{k}\\cap P_u \\neq \\emptyset$ \\quad\n$\\forall u\\in Y^k=\\{u\\in \\Gamma \\mid x^k(P_u) \\geq \\eta\\}$, and\n\n\\item $\\chi^{R^k} \\in P_{\\bar{B}}$, where\n$\\bar{B} \\coloneqq \\frac{1}{\\eta} B + 1 = \\frac{q}{\\mu} B + 1\n  \\eqqcolon B'$.\n\\end{enumerate}\nWe claim that $R=\\cup_{k=1}^q R^k$ is a set satisfying\nthe conditions of Theorem~\\ref{thm:bottomCover}.\nThe set $R$ clearly satisfies $\\chi^R \\in P_{B'}$\nsince $\\chi^{R^k}\\in P_{B'}$ for $k\\in [q]$\nand the sets $R^k$ are on disjoint levels.\nFurthermore, for each $u\\in W=\\{v\\in \\Gamma \\mid x(P_v)\\geq \\mu\\}$\nwe indeed have $P_u\\cap R\\neq\\emptyset$ due to the following.\nSince $x=\\sum_{k=1}^q x^k$ and $x(P_u) \\geq \\mu$ there exists\nan index $j\\in [q]$ such that $x^j(P_u) \\geq \\eta = \\frac{\\mu}{q}$,\nand hence $P_u \\cap R \\supseteq P_u \\cap R^j \\neq \\emptyset$.\n\n\\end{proof}\n\n\n\n\nThus, it remains to prove Lemma~\\ref{lem:sliceCover}.\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:sliceCover}]\\leavevmode\n\nLet $\\tilde{B} = \\frac{1}{\\eta} B$.\nWe start by determining an optimal vertex solution $y$\nto the linear program $\\min\\{z(V\\setminus \\{r\\}) \\mid z\\in Q\\}$,\nwhere\n\\begin{equation*}\nQ = \\{z\\in P_{\\tilde{B}}\n  \\mid z(u) = 0\\;\\forall u\\in V\n    \\setminus (V_{(\\ell_1,\\ell_2]} \\cup \\{r\\}),\\;\\;\nz(P_u) \\geq 1 \\;\\forall u\\in Y\\}.\n\\end{equation*}\nNotice that $Q\\neq \\emptyset$\nsince $\\frac{1}{\\eta} x \\in Q$; hence, the above\nLP is feasible.\nFurthermore, notice that $y(P_u)\\leq 1$ for $u\\in \\Gamma$;\nfor otherwise, there is a vertex $v\\in \\operatorname{supp}(y)$ such that\n$y(P_v) > 1$, and hence $y - \\epsilon \\chi^{\\{v\\}}\\in Q$\nfor a small enough $\\epsilon >0$, violating that \n$y$ is an \\emph{optimal} vertex solution.\n\n\nLet $V^{\\mathcal{L}}$ be all $y$-loose vertices.\nWe will show that the set\n\\begin{equation*}\nR = V^{\\mathcal{L}} \\cup \\{u\\in V\\setminus \\{r\\} \\mid y(u)=1\\}\n\\end{equation*}\nfulfills the properties claimed by the lemma.\nClearly, $R\\subseteq V_{(\\ell_1,\\ell_2]}$ since\n$\\operatorname{supp}(y) \\subseteq V_{(\\ell_1,\\ell_2]}$.\n\nTo see that condition~\\ref{item:scHitPath}\nholds, let $u\\in Y$, and notice that we have $y(P_u)=1$.\nEither $|P_u \\cap \\operatorname{supp}(y)| =1$, in which case\nthe single vertex $v$ in $P_u\\cap \\operatorname{supp}(y)$ satisfies\n$y(u)=1$ and is thus contained in $R$; or $|P_u\\cap \\operatorname{supp}(y)| > 1$,\nin which case $P_u\\cap V^{\\mathcal{L}} \\neq \\emptyset$ which again\nimplies $R\\cap P_u \\neq \\emptyset$.\n\n\nTo show that $R$ satisfies~\\ref{item:scBudgetOk},\nwe have to show that $R$ does not exceed the budget\n$\\bar{B}\\cdot 2^\\ell = (\\frac{1}{\\eta}B + 1) 2^\\ell$ of any\nlevel $\\ell\\in \\{\\ell_1+1,\\dots, \\ell_2\\}$.\nWe have\n\\begin{align*}\n|R\\cap V_\\ell| \\leq y(V_\\ell) + |V^{\\mathcal{L}}|\n\\leq \\tilde{B} 2^\\ell + |V^{\\mathcal{L}}|\n= \\frac{1}{\\eta} B 2^\\ell + |V^{\\mathcal{L}}|,\n\\end{align*}\nwhere the second inequality follows from $y\\in Q$.\nTo complete the proof it suffices to show\n$|V^{\\mathcal{L}}| \\leq 2^\\ell$.\nThis follows by a sparsity reasoning analogous to\nLemma~\\ref{lem:sparsityFF} implying that the number\nof $y$-loose vertices is bounded by the number\nof tight budget constraints, and thus\n\\begin{equation}\\label{eq:budgetBoundsFirstStep}\n|V^{\\mathcal{L}}| \\leq \\ell_2 - \\ell_1 \\leq \\ell_2\n = \\lfloor \\log^{(k-1)} L \\rfloor.\n\\end{equation}\nFurthermore,\n\\begin{align*}\n2^\\ell &\\geq 2^{\\ell_1+1} = 2^{\\lfloor \\log^{(k)} L \\rfloor + 1}\n\\geq 2^{\\log^{(k)} L} = \\log^{(k-1)} L,\n\\end{align*}\nwhich, together with~\\eqref{eq:budgetBoundsFirstStep},\nimplies $|V^{\\mathcal{L}}| \\leq 2^\\ell$ and thus\ncompletes the proof.\n\n\\end{proof}\n\n\n\\subsection*{Proof of Theorem~\\ref{thm:bigBIsGood}}\n\nLet $(y,B)$ be an optimal solution to the RMFC\nrelaxation $\\min\\{B \\mid x\\in \\bar{P}_B\\}$\nand let $h=\\lfloor \\log L \\rfloor$.\nHence, $B\\leq B_\\mathsf{OPT}$.\nWe invoke Theorem~\\ref{thm:bottomCover} with respect\nto the vector $y\\wedge \\chi^{V_{>h}}$ and $\\mu=0.5$\nto obtain a set $R_1\\subseteq V_{>h}$ satisfying\n\\begin{enumerate}[nosep,label=(\\roman*)]\n\\item $R_1\\cap P_u\\neq \\emptyset\n\\quad\\forall u\\in W$,\nand\n\n\\item $\\chi^{R_1} \\in P_{2B+1}$,\n\\end{enumerate}\nwhere\n$W = \\{u\\in \\Gamma \\mid y(P_u\\cap V_{>h}) \\geq 0.5\\}$.\nHence, $R_1$ cuts off all leaves in $W$ from\nthe root by only protecting vertices on\nlevels $V_{> h}$ and using budget bounded by\n$2B+1 \\leq 3B \\leq 3 \\max\\{\\log L, B_\\mathsf{OPT}\\}$.\n\n\nWe now focus on the leaves $\\Gamma \\setminus W$,\nwhich we will cut off from the root by protecting\na vertex set $R_2 \\subseteq V_{\\leq h}$ feasible\nfor budget $3 \\max\\{\\log L, B_\\mathsf{OPT}\\}$.\nLet $(z,\\bar{B})$ be an optimal vertex\nsolution to the\nfollowing linear program\n\\begin{equation}\\label{eq:reoptTop}\n\\min\\left\\{\\bar{B} \\;\\middle\\vert\\;\nx\\in P_{\\bar{B}},\\; \nx(P_u) = 1 \\;\\forall u\\in \\Gamma\\setminus W\n\\right\\}.\n\\end{equation}\nFirst, notice that~\\eqref{eq:reoptTop} is feasible\nfor $\\bar{B}\\leq 2B$. This follows by observing\nthat the vector $q= 2(y\\wedge \\chi^{V_{\\leq h}})$\nsatisfies $q\\in P_{2 B}$ since $y\\in P_B$.\nMoreover, for $u\\in \\Gamma\\setminus W$,\nwe have\n\\begin{equation*}\nq(P_u) = 2 y(P_u \\cap V_{\\leq h})\n= 2 (1-y(P_u \\cap V_{> h})) > 1,\n\\end{equation*}\nwhere the last inequality follows from\n$y(P_u\\cap V_{>h}) < 0.5$ because\n$u\\in \\Gamma\\setminus W$.\nFinally, there exists a vector\n$q' < q$ such that\n$q'(P_u) =1$ for $u\\in \\Gamma\\setminus W$.\nThe vector $q'$ can be obtained from $q$ by\nsuccessively reducing values on vertices\n$v\\in \\operatorname{supp}(q)$ satisfying\n$q(P_v) > 1$.\nThis shows that $(q',2B)$ is a feasible\nsolution to~\\eqref{eq:reoptTop} and hence\n$\\bar{B} \\leq 2B$.\n\nConsider the set of all $z$-loose vertices\n$V^{\\mathcal{L}}=\\{u\\in \\operatorname{supp}(z) \\mid z(P_u)<1\\}$.\nWe define\n\\begin{equation*}\nR_2 = V^{\\mathcal{L}} \\cup \n\\{u\\in \\operatorname{supp}(z) \\mid z(u)=1\\}.\n\\end{equation*}\nNotice that for each $u\\in \\Gamma\\setminus W$,\nthe set $R_2$ contains a vertex on the\npath from $u$ to the root. Indeed, either\n$|\\operatorname{supp}(z)\\cap P_u|=1$ in which case there\nis a vertex $v\\in P_u$ with $z(v)=1$, which is\nthus contained in $R_2$, or $|\\operatorname{supp}(z)\\cap P_u|>1$\nin which case the vertex $v\\in \\operatorname{supp}(z)\\cap P_u$\nthat is closest to the root among all vertices in\n$\\operatorname{supp}(z)\\cap P_u$ is a $z$-loose vertex.\nHence, the set $R=R_1\\cup R_2$ cuts off all leaves\nfrom the root. It remains to show that it is\nfeasible for budget $3 \\max\\{\\log L, B_\\mathsf{OPT}\\}$.\n\nUsing an analogous sparsity reasoning as in\nLemma~\\ref{lem:sparsityFF}, we obtain that\n$|V^{\\mathcal{L}}|$ is bounded by the number\nof tight budget constraints, which is at most\n$h=\\lfloor \\log L \\rfloor \\leq \\log L$.\nHence, for any level $\\ell\\in [h]$, we have\n\\begin{align*}\n|R_2 \\cap V_\\ell| &\\leq |V^{\\mathcal{L}}| + z(V_\\ell) \\\\\n  &\\leq \\log L + 2^\\ell \\bar{B} && \\text{($(z,\\bar{B})$ feasible\nfor~\\eqref{eq:reoptTop})}\\\\\n  &\\leq \\log L + 2^\\ell \\cdot (2 B) && \\text{($\\bar{B}\\leq 2B$)}\\\\\n  &\\leq 2^\\ell \\cdot (3 \\max\\{\\log L, B_\\mathsf{OPT}\\}).\n     && \\text{($B\\leq B_\\mathsf{OPT}$)}\n\\end{align*}\nThus, both $R_1$ and $R_2$ are budget-feasible for\nbudget $3 \\max\\{\\log L, B_\\mathsf{OPT}\\}$, and since they\ncontain vertices on disjoint levels, $R=R_1\\cup R_2$\nis feasible for the same budget.\n\n\\qed\n\n\n\n\n\n\\subsection*{Proof of Lemma~\\ref{lem:enumWorks}}\n\nTo show that the running time of\n\\hyperlink{alg:enumRMFCtarget}%\n{$\\mathrm{Enum}(\\emptyset,\\emptyset,\\bar{\\gamma})$}\nis polynomial, we show that there is only a polynomial\nnumber of recursive calls to\n\\hyperlink{alg:enumRMFCtarget}%\n{$\\mathrm{Enum}(A,D,\\gamma)$}. Notice that the number\nof recursive calls done in one execution of\nstep~\\ref{item:enumRecCall} of the algorithm is equal\nto $2 |F_x|$.\nWe thus start by upper bounding $|F_x|$ for any solution\n$(x,B)$ to \\ref{eq:lpRMFCAD} with $B < \\log L$.\nConsider a vertex $f_u\\in F_x$, where\n$u\\in \\Gamma\\setminus W_x$.\nSince $u$ is a leaf not in $W_x$, we have\n$x(P_u \\cap V_{\\leq h}) > \\frac{1}{3}$, and\nthus\n\\begin{equation*}\nx(T_{f_u}\\cap V_{\\leq h}) > \\frac{1}{3}\n\\quad \\forall f_u \\in F_x.\n\\end{equation*}\nBecause no two vertices of $F_x$ lie on the same\nleaf-root path, the sets $T_{f_u} \\cap V_{\\leq h}$\nare all disjoint for different $f_u\\in F_x$,\nand hence\n\\begin{align*}\n\\frac{1}{3}|F_x| &< \\sum_{f \\in F_x} x(T_{f}\\cap V_{\\leq h})\\\\\n &\\leq x(V_{\\leq h})\n    && \\text{(disjointness of sets $T_{f}\\cap V_{\\leq h}$\n              for different $f \\in F_x$})\\\\\n &\\leq \\sum_{\\ell=1}^h 2^\\ell B\n    && \\text{($x$ satisfies budget constraints of~\\ref{eq:lpRMFCAD} )}\\\\\n &< 2^{h+1} B\\\\\n &< 2 (\\log L)^2.\n    && \\text{($h=\\lfloor \\log^{(2)} L \\rfloor$ and $B < \\log L$)}\n\\end{align*}\nSince the recursion depth is\n$\\bar{\\gamma}=2(\\log L)^2 \\log^{(2)} L$,\nthe number of recursive calls is bounded by\n\\begin{align*}\nO\\left((2 |F_x|)^{\\bar{\\gamma}}\\right) &= \n(\\log L)^{O((\\log L)^2 \\log^{(2)} L)}\n=2^{o(L)} = o(N),\n\\end{align*}\nthus showing that\n\\hyperlink{alg:enumRMFCtarget}%\n{$\\mathrm{Enum}(\\emptyset,\\emptyset,\\bar{\\gamma})$}\nruns in polynomial time.\n\nIt remains to show that \\hyperlink{alg:enumRMFCtarget}%\n{$\\mathrm{Enum}(\\emptyset,\\emptyset,\\bar{\\gamma})$} finds\na triple satisfying the conditions of Lemma~\\ref{lem:goodEnum}.\nFor this we identify a particular execution path of the\nrecursive procedure \n\\hyperlink{alg:enumRMFCtarget}%\n{$\\mathrm{Enum}(\\emptyset,\\emptyset,\\bar{\\gamma})$} that,\nat any point in the algorithm, will maintain a clean\npair $(A,D)$ that is compatible with $\\mathsf{OPT}$,\ni.e., $A\\subseteq \\mathsf{OPT}$ and $D\\cap \\mathsf{OPT} = \\emptyset$.\nAt the beginning of the algorithm we clearly have\ncompatibility with $\\mathsf{OPT}$ since $A=D=\\emptyset$.\nTo identify the execution path we are interested\nin, we highlight which recursive call we want to follow\ngiven that we are on the execution path.\nHence, consider a clean pair $(A,D)$\nthat is compatible with $\\mathsf{OPT}$ and assume we are\nwithin the execution of\n\\hyperlink{alg:enumRMFCtarget}%\n{$\\mathrm{Enum}(A,D,\\gamma)$}.\nLet $(x,B)$ be an optimal solution to~\\ref{eq:lpRMFCAD}.\nNotice that $B \\leq B_\\mathsf{OPT} \\leq \\log L$, because\n$(A,D)$ is compatible with $\\mathsf{OPT}$.\nIf $\\mathsf{OPT}\\cap Q_x=\\emptyset$, then $(A,D,x)$ fulfills the\nconditions of Lemma~\\ref{lem:goodEnum} and we are done.\nHence, assume $\\mathsf{OPT}\\cap Q_x \\neq \\emptyset$, and\nlet $f \\in F_x$ be such that\n$\\mathsf{OPT}\\cap T_{f}\\cap V_{\\leq h}\\neq \\emptyset$.\nIf $f \\in \\mathsf{OPT}$, then consider the execution path\ncontinuing with the call of \n\\hyperlink{alg:enumRMFCtarget}%\n{$\\mathrm{Enum}(A\\cup \\{f\\},D,\\gamma-1)$}; otherwise,\nif $f\\not\\in \\mathsf{OPT}$, we focus on the call of\n\\hyperlink{alg:enumRMFCtarget}%\n{$\\mathrm{Enum}(A,D\\cup \\{f\\},\\gamma-1)$}.\nNotice that compatibility with $\\mathsf{OPT}$ is maintained\nin both cases.\n\nTo show that the thus identified execution path of \n\\hyperlink{alg:enumRMFCtarget}%\n{$\\mathrm{Enum}(\\emptyset,\\emptyset,\\bar{\\gamma})$}\nindeed leads to a triple satisfying the conditions\nof Lemma~\\ref{lem:goodEnum}, we measure progress\nas follows. For any clean pair $(A,D)$\ncompatible with $\\mathsf{OPT}$, we define\na potential function $\\Phi(A,D)\\in \\mathbb{Z}_{\\geq 0}$\nin the following way.\nFor each $u\\in \\mathsf{OPT}\\cap V_{\\leq h}$,\nlet $d_u\\in \\mathbb{Z}_{\\geq 0}$\nbe the distance of $u$ to the first vertex in\n$A\\cup D \\cup \\{r\\}$ when following the unique\n$u$-$r$ path. We define\n$\\Phi(A,D)= \\sum_{u\\in \\mathsf{OPT} \\cap V_{\\leq h}} d_u$.\nNotice that as long as we have a triple $(A,D,x)$\non our execution path that does\nnot satisfy the conditions of Lemma~\\ref{lem:goodEnum},\nthen the next triple $(A',D',x')$ on our execution\npath satisfies $\\Phi(A',D') < \\Phi(A,D)$.\nHence, either we will encounter a triple on our\nexecution path satisfying\nthe conditions of Lemma~\\ref{lem:goodEnum}\nwhile still having a strictly positive potential,\nor we will encounter a triple $(A,D,x)$ compatible\nwith $\\mathsf{OPT}$ and $\\Phi(A,D)=0$, which implies\n$\\mathsf{OPT}\\cap V_{\\leq h} = A$,\nand we thus correctly guessed all vertices of\n$\\mathsf{OPT}\\cap V_{\\leq h}$ implying that\nthe conditions of Lemma~\\ref{lem:goodEnum}\nare satisfied for the triple $(A,D,x)$.\nSince $\\Phi(A,D)\\geq 0$ for any compatible clean\npair $(A,D)$, this implies that a triple\nsatisfying the conditions of Lemma~\\ref{lem:goodEnum}\nwill be encountered if the recursion depth $\\bar{\\gamma}$\nis at least $\\Phi(\\emptyset,\\emptyset)$.\nTo evaluate $\\Phi(\\emptyset,\\emptyset)$ we have to compute\nthe sum of the distances of all\nvertices $u\\in \\mathsf{OPT}\\cap V_{\\leq h}$\nto the root. The distance of $u$ to the root is at\nmost $h$ since $u\\in V_{\\leq h}$. Moreover, \n$|\\mathsf{OPT} \\cap V_{\\leq h}| < 2^{h+1} B_{\\mathsf{OPT}}$\ndue to the budget constraints. Hence,\n\\begin{align*}\n\\Phi(\\emptyset, \\emptyset)\n  &< h \\cdot 2^{h+1} \\cdot B_{\\mathsf{OPT}}\\\\\n  &\\leq 2 \\log^{(2)} L \\cdot (\\log L)^2\n    && \\text{($h=\\lfloor \\log^{(2)} L \\rfloor$ and $B_\\mathsf{OPT} \\leq \\log L$)}\\\\\n  &= \\bar{\\gamma},\n\\end{align*}\nimplying that a triple fulfilling the conditions of\nLemma~\\ref{lem:goodEnum} is encountered by\n\\hyperlink{alg:enumRMFCtarget}%\n{$\\mathrm{Enum}(\\emptyset,\\emptyset,\\bar{\\gamma})$}.\n\n\n\\qed\n\n\n\n\n\n\n\n\n\n\\section*{Acknowledgements}\nWe are grateful to Noy Rotbart for many stimulating discussions\nand for bringing several relevant references to our attention.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nCosmological observations have confirmed the big bang cosmology and determined the cosmological parameters precisely~\\cite{Ade:2015xua}. The matter contents of the Universe may be phenomenologically given by the standard model particles, the cosmological constant $\\Lambda$, and cold dark matter (CDM). However, the theoretical explanation of the origin of the extra ingredients, dark matter and dark energy, is still lacked. The theoretically expected value of the cosmological constant is too large to explain the present accelerating expansion. An alternative idea is that the acceleration is obtained by a potential of a scalar field instead of $\\Lambda$, and this idea is often called the quintessence model~\\cite{Caldwell:1997ii}. This scalar field could be originated from the gravity sector~\\cite{Fujii:2003pa}. A large class of scalar-tensor theories and $f(R)$ theories can be recast in the form of a theory of a canonical scalar field with a potential after the conformal transformation $\\tilde{g}_{\\mu\\nu}=A^2(\\phi) g_{\\mu\\nu}$ and the field redefinition $\\Phi=\\Phi(\\phi)$ where $\\Phi$ is the canonically normalized field. The metric $\\tilde{g}_{\\mu\\nu}$ is called the Jordan frame metric which the standard model particles are minimally coupled to whereas $g_{\\mu\\nu}$ is the Einstein frame metric in which the gravitational action is given by the Einstein-Hilbert action. In this case, the scalar field has the non-minimal coupling to the matter fields via the coupling function $A$.\n\n\nDark matter is also one of the biggest mystery of the modern cosmology. Although many dark matter candidates have been proposed in the context of the particle physics, any dark matter particles have not been discovered yet~\\cite{Agashe:2014kda,Ackermann:2015zua,Ahnen:2016qkx,Ackermann:2015lka,Khachatryan:2014rra,Conrad:2017pms}. The existence of dark matter is confirmed via only gravitational interactions. Hence, exploring dark matter candidate in the context of gravity is also a considerable approach. Not only dark energy but also dark matter could be explained by modifications of gravity. For instance, a natural extension of general relativity is a theory with a massive graviton (see \\cite{deRham:2014zqa} for a review). If a graviton obtains a mass, the massive graviton can be a dark matter candidate~\\cite{Dubovsky:2004ud,Pshirkov:2008nr,Aoki:2016zgp,Babichev:2016hir,Babichev:2016bxi,Aoki:2017cnz}.\n\n\nA viable dark matter scenario has to explain the present abundance of dark matter which usually leads to a constraint on a production scenario. However, a question arises: why are the energy densities of dark matter and baryon almost the same? If baryon and dark matter are produced by a common mechanism, almost the same abundance could be naturally obtained. On the other hand, if productions of the two are not related but independent, the coincidence might indicate that two energy densities are tuned to be the same order of the magnitude by a mechanism after the productions.\n\n\nIn the present paper, we shall combine two ideas of the modifications of gravity by using the proposal of \\cite{DeFelice:2017oym}: the non-minimal coupling of $\\phi$ and the existence of the massive graviton. We call this theory the chameleon bigravity theory which contains three types of gravitational degrees of freedom: the massless graviton, the massive graviton, and the chameleon field $\\phi$. We identify the massive graviton with dark matter. Since dark matter is originated from the gravity sector, the coupling between $\\phi$ and the dark matter may be given by a different way from the matter sector. We promote parameters in the graviton mass terms to functions of $\\phi$~\\cite{D'Amico:2012zv,Huang:2012pe}, giving rise to a new type of coupling between $\\phi$ and dark matter. In this case, as discussed in \\cite{DeFelice:2017oym}, the field value of $\\phi$ depends on the environment due to the non-minimal coupling as with the chameleon field \\cite{Khoury:2003aq,Khoury:2003rn}, which makes the graviton mass to depend on the environment.\n\n\nWe find that the ratio between energy densities of dark matter and baryon is dynamically adjusted to the observed value by the motion of $\\phi$ and then the ratio at the present is independent of the initial value. Hence, our model can explain the coincidence of the abundance of dark matter and baryon. Furthermore, if the potential of $\\phi$ is designed to be dark energy, the chameleon field $\\phi$ can give rise to the present acceleration of the universe. Both dark energy and dark matter are explained by the modifications of gravity in our model. \n\n\nThe paper is organized as follows. We introduce the chameleon bigravity theory in Sec.~\\ref{sec_bigravity}. In Sec.~\\ref{sec_Fri}, we show the Friedmann equation regarding the massive graviton is dark matter. We also point out the reason why the dark matter-baryon ratio can be naturally explained in the chameleon bigravity theory if we consider the massive graviton as dark matter. Some analytic solutions are given in Sec.~\\ref{sec_analytic} and numerical solutions are shown in Sec.~\\ref{sec_numerical}. These solutions reveal that the observed dark matter-baryon ratio is indeed dynamically obtained independently from the initial ratio. We summarize our results and give some remarks in Sec. \\ref{summary}. In Appendix, we detail the derivation of the Friedmann equation.\n\n\n\\section{Chameleon bigravity theory}\n\\label{sec_bigravity}\n\nWe consider the chameleon bigravity theory in which the mass of the massive graviton depends on the environment~\\cite{DeFelice:2017oym}. The action is given by\n\\begin{align}\nS&=\\int d^4x \\sqrt{-g} \\Biggl[  \\frac{M_g^2}{2}R[g]-\\frac{1}{2}\n K^2(\\phi) \ng^{\\mu\\nu}\\partial_{\\mu}\\phi\\partial_{\\nu}\\phi\n\\nn\n&\\qquad \\qquad \\qquad \\quad +M_g^2m^2 \\sum_{i=0}^4 \\beta_i (\\phi) U_i[s] \\Biggl]\n\\nn\n&+\\frac{M_f^2}{2}\\int d^4 x\\sqrt{-f}R[f]+S_{\\rm m}[\\tilde{g},\\psi]\\,, \\label{action}\n\\end{align}\nwhere $\\phi$ is the chameleon field and $S_{\\rm m}$ is the matter action. The functions $K(\\phi)$ and $\\beta_i(\\phi)$ are arbitrary functions of $\\phi$. The matter fields universally couple to the Jordan frame metric $\\tilde{g}_{\\mu\\nu}=A^2(\\phi)g_{\\mu\\nu}$ with a coupling function $A(\\phi)$. The potentials $U_i[s] \\, (i=0,\\cdots, 4)$ are the elementary symmetric polynomials of the eigenvalues of the matrix $s^{\\mu}{}_{\\nu}$ which is defined by the relation~\\cite{deRham:2010ik,deRham:2010kj,Hassan:2011zd}\n\\begin{align}\ns^{\\mu}{}_{\\alpha}s^{\\alpha}{}_{\\nu}=g^{\\mu\\alpha}f_{\\alpha\\nu}\\,.\n\\end{align}\nThe potential of $\\phi$ is not added explicitly since the couplings between $\\phi$ and the potentials $U_i$ yield the potential of $\\phi$ and thus an additional potential is redundant.\n\nNote that the field $\\phi$ is not a canonically normalized field. The canonical field $\\Phi$ is given by the relation\n\\begin{align}\nd \\Phi = K(\\phi) d\\phi \\,,\n\\end{align}\nby which the function $K$ does not appear explicitly in the action when we write down the theory in terms of $\\Phi$. Since $\\beta_i$ and $A$ are arbitrary functions, we can set $K=1$ by the redefinitions of $\\beta_i$ and $A$ without loss of generality. Nevertheless, we shall retain $K$ and discuss the general form of the action \\eqref{action}.\n\nIn general, the functions $\\beta_i(\\phi)$ can be chosen independently. In the present paper, however, we consider the simplest model such that $\\beta_i(\\phi)=-c_i f(\\phi)$ where $c_i$ are constant while $f(\\phi)$ is a function of $\\phi$. As we will see in next section, the graviton mass and the potential of $\\phi$ around the cosmological background are given by\n\\begin{align}\nm_T^2(\\phi)&:=\\frac{1+\\kappa}{\\kappa}m^2 f(\\phi)(c_1+2c_2+c_3)\n\\,, \\\\\nV_0(\\phi)&:=m^2M_p^2 f(\\phi)(c_0+3c_1+3c_2+c_3)\n\\,, \\label{bare_potential}\n\\end{align}\nwith $\\kappa=M_f^2\/M_g^2$ and $M_p^2=M_g^2+M_f^2$. In this case, both the potential form of $\\phi$ and the $\\phi$-dependence of the graviton mass are determined by $f(\\phi)$ only.\\footnote{Since we have absorbed the potential of $\\phi$ in the mass term of the graviton, $m_T^2M_p$ and $V_0$ seem to be a same order of magnitude. However, $m_T^2M_p^2$ and $V_0$ are not necessary to be the same order because they represent different physical quantities. Indeed, we will assume $V_0 \\ll m_T^2 M_p^2$.} Note that $V_0$ is the bare potential of $\\phi$. The effective potential of $\\phi$ is given by not only $V_0$ but also the amplitude of the massive graviton as well as the energy density of matter due to the non-minimal couplings (see Eq.~\\eqref{effective_potential}).\n\n\n\n\n\n\\section{Basic equations}\n\\label{sec_Fri}\nIn this section, we derive the basic equations to discuss the cosmological dynamics in the model \\eqref{action} supposing that the massive graviton is dark matter. We assume the coherent dark matter scenario in which dark matter is obtained from the coherent oscillation of the zero momentum mode massive gravitons~\\cite{Aoki:2017cnz}. Since the zero momentum mode of the graviton corresponds to the anisotropy of the spacetime, we study the Bianchi type I universe instead of the Friedmann-Lema{\\^i}tre-Robertson-Walker (FLRW) universe. The ansatz of the spacetime metrics are\n\\begin{align}\nds_g^2&=-dt^2 +a^2[ e^{4\\sigma_g} dx^2+e^{-2\\sigma_g}(dy^2+dz^2)]\\,, \\label{Bianchi_g} \\\\\nds_f^2&=\\xi^2\\left[ -c^2 dt^2+a^2\\{ e^{4\\sigma_f} dx^2+e^{-2\\sigma_f}(dy^2+dz^2)\\} \\right] \\,, \\label{Bianchi_f}\n\\end{align}\nwhere $\\{a,\\xi,c,\\sigma_g,\\sigma_f\\}$ are functions of the time $t$. We assume the matter field is a perfect fluid whose energy-momentum tensor is given by\n\\begin{align}\nT^{\\mu}{}_{\\nu}=A^4(\\phi)\\times {\\rm diag}[-\\rho(t),P(t),P(t),P(t)]\\,, \\label{Tmunu}\n\\end{align}\nwhere $\\rho$ and $P$ are the energy density and the pressure in the Jordan frame, respectively. The conservation law of the matter field is\n\\begin{align}\n\\dot{\\rho}+3\\frac{(Aa)^{\\cdot}}{Aa}(\\rho+P)=0\\,, \\label{conservation}\n\\end{align}\nwhere a dot is the derivative with respect to $t$.\n\nAs shown in \\cite{Maeda:2013bha,Aoki:2017cnz}, the small anisotropies $\\sigma_g$ and $\\sigma_f$ can be a dark matter component of the universe in the bimetric model without the chameleon field $\\phi$. We generalize their calculations to those in the present model \\eqref{action}. All equations under the ansatz \\eqref{Bianchi_g} and \\eqref{Bianchi_f} are summarized in Appendix. Here, we only show the Friedmann equation and the equations of motion of the massive graviton and the chameleon field because other equations are not important for the following discussion.\n\n\nWe assume the graviton mass $m_T$ is larger than the Hubble expansion rate $H:=\\dot{a}\/a$. After expanding the equations in terms of anisotropies and a small parameter $\\epsilon:=H\/m_T$, the Friedmann equation is given by\n\\begin{align}\n3M_p^2H^2= \\rho A^4+\\frac{1}{2}\n K^2 \n\\dot{\\phi}^2+V_0+\\frac{1}{2}\\dot{\\varphi}^2+\\frac{1}{2}m_T^2\\varphi^2 \\,, \\label{Fri_no_h}\n\\end{align}\nwhere $\\varphi$ is the massive graviton which is given by a combination of the anisotropies $\\sigma_g$ and $\\sigma_f$ (see Eqs.~\\eqref{graviton1} and \\eqref{graviton2}).\nThe equations of motion of the massive graviton $\\varphi(t)$ and the chameleon field $\\phi(t)$ are \n\\begin{align}\n\\ddot{\\varphi}+3H \\dot{\\varphi}+m_T^2(\\phi) \\varphi &=0 \\,, \\label{eq_varphi}\\\\\n K \\left( \\ddot{\\phi}+3H\\dot{\\phi} \\right)\n + \\dot{K} \\dot{\\phi} +  \\frac{\\partial V_{\\rm eff}}{ \\partial \\phi}&=0\\,, \\label{eq_cham}\n\\end{align}\nwhere the effective potential of the chameleon field is given by\n\\begin{align}\nV_{\\rm eff}:= V_0(\\phi)+\\frac{1}{2}m_T^2(\\phi)\\varphi^2 + \\frac{1}{4}A^4(\\phi) (\\rho-3p)\n\\,. \\label{effective_potential}\n\\end{align}\nNote that, although the bigravity theory contains the degree of freedom of the massless graviton (see Eq.~\\eqref{Friedmann_eq}), we neglect the contribution to the Friedmann equation from the massless graviton because the energy density of the massless graviton decreases faster than those of other fields. The effect of the massless graviton is not important for our discussions.\n\n\nWe notice that the basic equations \\eqref{Fri_no_h}, \\eqref{eq_varphi} and \\eqref{eq_cham} are exactly the same as the equations in the theory with two scalar fields given by the action\n\\begin{align}\nS=\\int d^4x \\sqrt{-g} \\Biggl[ &\\frac{M_p^2}{2} R[g]\n-\\frac{1}{2}K^2(\\phi) (\\partial \\phi)^2 -V_0(\\phi) \\nn\n&-\\frac{1}{2} (\\partial \\varphi)^2 -\\frac{1}{2}m_T^2(\\phi) \\varphi^2 \\Biggl]+S_m [\\tilde{g},\\psi]\n\\,. \\label{another_action}\n\\end{align}\nThe cosmological dynamics in \\eqref{action} with $H \\gg m_T$ can be reduced into that in \\eqref{another_action}. Our results obtained below can be straightforward generalized even in the case of \\eqref{another_action} up to the discussion about the cosmological dynamics. The action \\eqref{another_action} gives a toy model of the chameleon bigravity theory. However, the equivalence between \\eqref{action} and \\eqref{another_action} holds only for the background dynamics of the universe in $H\\gg m_T$. The equivalence between the two actions does not hold for small-scale perturbations around the cosmological background~\\cite{Aoki:2017cnz}. \n\n\nWe first consider a solution $\\phi=\\phi_{\\rm min}=$ constant which is realized when\n\\begin{align}\n\\frac{\\partial V_{\\rm eff}}{\\partial \\phi}\n=\\alpha_f \\left[ V_0 +\\frac{1}{2}m_T^2 \\varphi^2 \\right] +\\alpha_A (\\rho-3P) A^4=0\n\\,,\n\\label{phi=const}\n\\end{align}\nwhere\n\\begin{align}\n\\alpha_A :=\\frac{1}{K}\\frac{ d \\ln A}{d \\phi}=\\frac{ d \\ln A}{d \\Phi}\\,, \\quad\n\\alpha_f:=\\frac{1}{K}\\frac{d \\ln f}{d \\phi}=\\frac{ d \\ln f}{d \\Phi} \\,.\n\\end{align}\nThe equation \\eqref{phi=const} is not always compatible with $\\phi=$ constant since each term in \\eqref{phi=const} has different time dependence in general. Nonetheless, as we shall see below, they can be compatible with each other if $\\epsilon \\ll 1 $. In other words, a common constant value of $\\phi_{\\rm min}$ can be a solution all the way from the radiation dominant (RD) epoch to the matter dominant (MD) epoch of the universe. When the chameleon field is constant, the bare potential $V_0$ acts as a cosmological constant which has to be subdominant in the RD and the MD eras. The constant $\\phi$ implies that the graviton mass does not vary and thus we obtain\n\\begin{align}\n\\langle \\dot{\\varphi}^2 \\rangle_T= \\langle m_T^2 \\varphi^2 \\rangle_T \\propto a^{-3}\n\\,,\\label{eqn:scaling-mTconst}\n\\end{align}\nwhere $\\langle\\cdots \\rangle_T$ represents the time average over an oscillation period. The massive gravitons behave like a dark matter component of the universe. When we focus on the time scales much longer than $m_T^{-1}$, $m_T^2 \\varphi^2$ in Eq.~\\eqref{phi=const} can be replaced with $\\langle m_T^2 \\varphi^2 \\rangle_T$, which scales as \\eqref{eqn:scaling-mTconst}. Since $\\rho-3P$ also scales as $\\propto a^{-3}$ in the RD and the MD, the decaying laws of $\\rho-3P$ and $m_T^2 \\varphi^2$ in \\eqref{phi=const} are the same in this case. Hence, when the oscillation timescale of the massive graviton is much shorter than the timescale of the cosmic expansion, i.e., $\\epsilon \\ll 1 $, $\\phi=$ constant can be a solution all the way from the RD to the MD. The value of $\\phi_{\\rm min}$ is determined by simply solving Eq.~\\eqref{phi=const}.\n\n\nSupposing that the massive graviton is the dominant component of dark matter, Eq.~\\eqref{phi=const} in the RD and MD eras is replaced with\n\\begin{align}\n\\left( \\alpha_f \\rho_G+2\\alpha_A \\rho_{b} \\right) A^4=0\\,,\n\\end{align}\nwhere $\\rho_b$ is the baryon energy density and we have ignored $V_0$. The energy density of massive graviton in the Jordan frame is defined by\n\\begin{align}\n\\rho_G:=\\frac{1}{2}A^{-4}\\langle \\dot{\\varphi}^2+m_T^2\\varphi^2 \\rangle_T=A^{-4}m_T^2\\langle \\varphi^2 \\rangle_T\\,,\n\\end{align}\nwhich depends on the chameleon $\\phi$. Therefore, if $\\alpha_A$ and $\\alpha_f$ are assumed to be $\\alpha_A\/\\alpha_f \\simeq -5\/2$, the ratio between dark matter and baryon is automatically tuned to be the observational value. The dark matter-baryon ratio could be naturally explained without any fine-tuning of the productions of dark matter and baryon.\n\n\n\nNeedless to say, the initial value of $\\phi$ must not be at the bottom of the effective potential $(\\phi=\\phi_{\\rm min})$. We shall study the dynamics of $\\phi$ and discuss whether $\\phi$ approaches $\\phi_{\\rm min}$ before the MD era of the universe. \nAlthough we do not assume $\\phi$ is constant, we assume $\\phi$ does not rapidly move so that the graviton mass varies adiabatically\n\\begin{align}\n\\frac{\\dot{m}_T}{m_T^2} \\ll 1 \\,.\n\\label{adiabatic_condition}\n\\end{align}\nUnder the adiabatic condition  \\eqref{adiabatic_condition} we can take the adiabatic expansion for the massive graviton:\n\\begin{align}\n\\varphi=u(t)\\cos\\left[  \\int m_T[\\phi(t)] dt \\right]+\\cdots\\,,\n\\end{align}\nwith a slowly varying function $u(t)$. The adiabatic condition \\eqref{adiabatic_condition} is indeed viable for $\\epsilon \\ll 1$ since we will see the time dependence of $m_T$ is given by a power law of $a$ (see Eq.~\\eqref{time_dep_m} for example). The time average over an oscillation period yields $\\langle \\dot{\\varphi}^2 \\rangle_T=\\langle m_T^2 \\varphi^2 \\rangle_T=m_T^2 u^2\/2$. \n\n\nAfter taking the time average over an oscillation period under the adiabatic condition, the equations are reduced into\n\\begin{align}\n3M_p^2H^2= A^4 \\rho_r +A^4 \\rho_b +\\frac{1}{2}  K^2  \\dot{\\phi}^2 +V_0+ \\frac{1}{2}m_T^2 u^2 \\,, \\label{Fri}\n\\end{align}\nand\n\\begin{align}\n K \\left( \\ddot{\\phi}+3H \\dot{\\phi} \\right)\n +\\dot{K} \\dot{\\phi} +\\alpha_f V_0 \n& \\nn\n+\\frac{1}{4}\\alpha_f m_T^2 u^2+\\alpha_A A^4\\rho_b&=0\n\\,, \\label{eq_phi} \\\\\n4\\dot{u}+6H u+ \\alpha_f u K\\dot{\\phi}&=0\n\\,, \\label{eq_u}\n\\end{align} \nwhere $\\rho_r$ and $\\rho_b$ are the energy densities of radiation and baryon which decrease as $\\rho_r\\propto (aA)^{-4}$ and $\\rho_b \\propto (aA)^{-3}$ because of the conservation equation. The dynamics of the scale factor $a$, the chameleon field $\\phi$, and the amplitude of the massive graviton $u$ are determined by solving these three equations.\n\n\nBy using the density parameters, the Friedmann equation is rewritten as\n\\begin{align}\n1=\\Omega_r+\\Omega_b+\\Omega_{\\phi}+\\Omega_G \\,,\n\\end{align}\nwith\n\\begin{align}\n\\Omega_r&:=\\frac{A^4\\rho_r}{3M_p^2H^2} \n\\,, \\\\\n\\Omega_b&:=\\frac{A^4\\rho_b}{3M_p^2H^2} \n\\,, \\\\\n\\Omega_{\\phi}&:=\\frac{\\dot{\\phi}^2+2V_0}{6M_p^2H^2}\n\\,, \\\\\n\\Omega_G&:=\\frac{m_T^2 u^2}{6M_p^2H^2}\\,.\n\\end{align}\nWe also introduce the total equation of state parameter in the Einstein frame\n\\begin{align}\nw_E:=-1-\\frac{2\\dot{H}}{3H^2}\\,.\n\\end{align}\n\nThe above quantities are defined in the Einstein frame. Since the matter fields minimally couple with the Jordan frame metric, the observable universe is expressed by the Jordan frame metric. Hence, we also define the Hubble expansion rate and the effective equation of state parameter in the Jordan frame as\n\\begin{align}\nH_J&:=\\frac{(Aa)^{\\cdot}}{A^2a} \\,, \\\\\nw_{\\rm tot}&:=-1-\\frac{2\\dot{H}_J}{3AH_J^2}\\,.\n\\end{align}\n\n\n\n\n\n\n\n\n\n\\section{Analytic solutions}\n\\label{sec_analytic}\nIn this section we show some analytic solutions under the simplest case\n\\begin{align}\nK=1\\,, \\quad A=e^{\\beta\\phi\/M_p}\\,, \\quad\nf=e^{-\\lambda \\phi\/M_p}\\,, \\label{model_A}\n\\end{align}\nwith the dimensionless constants $\\beta$ and $\\lambda$. This model yields that the coupling strengths $\\alpha_A$ and $\\alpha_f$ are constant.\nWe consider four stages of the universe: the radiation dominant era, around the radiation-matter equality, the matter dominant era, and the accelerate expanding era. The analytic solutions are found in each stages of the universe as follows.\n\n\\subsection{Radiation dominant era}\nWe first consider the regime when the contributions to the Friedmann equation from baryon and dark matter are subdominant, that is, $\\Omega_b, \\Omega_G \\ll 1$. The Hubble expansion rate is then determined by the energy densities of radiation and $\\phi$. Since the effective potential of $\\phi$ are determined by the energy densities of baryon and dark matter, in this situation the potential force can be ignored compared with the Hubble friction term ($V_0$ is assumed to be always ignored during both radiation and matter dominations).\nThen, we obtain\n\\begin{align}\n\\dot{\\phi}\\propto a^{-3}\n\\,,\n\\end{align}\nwhich indicates that the field $\\phi$ loses its velocity due to the Hubble friction and then $\\phi$ becomes a constant $\\phi_i$. We can ignore $\\Omega_{\\phi}$ and then find the standard RD universe. At some fixed time deep in the radiation dominant era, we therefore set $\\phi=\\phi_i$ as the initial condition of $\\phi$. We shall then denote the initial values of the energy densities of baryon and the massive graviton as $\\rho_{b,i}$ and $\\rho_{G,i}$, respectively. \n\n\nNote that this constant initial value of $\\phi$ is not necessary to coincide with the potential minimum $\\phi=\\phi_{\\rm min}$, i.e., $\\phi_i \\neq \\phi_{\\rm min}$. The ratio $\\rho_{G,i}\/\\rho_{b,i}$ is not tuned to be five at this stage.\n\n\n\n\\subsection{Following-up era}\nWe then discuss the era just before radiation-matter equality in which we cannot ignore the potential force for $\\phi$. As discussed in the previous subsection, we find $\\phi=\\phi_i$ in the RD universe. When the potential force for $\\phi$ becomes relevant, the chameleon field $\\phi$ starts to evolve into the potential minimum $\\phi=\\phi_{\\rm min}$. Due to the motion of $\\phi$, the smaller one of $\\rho_G$ and $\\rho_b$ follows up the larger one. We obtain $\\rho_G\/\\rho_b=-2\\alpha_A\/\\alpha_f$ when the chameleon field reaches the minimum $\\phi_{\\rm min}$. We call this era of the universe the following-up era.\n\nIf the initial value $\\phi_i$ is close to the potential minimum $\\phi_{\\rm min}$, the dark matter-baryon ratio is already tuned to be almost the value $-2\\alpha_A\/\\alpha_f$, which we set to $\\sim 5$, and thus we do not need to discuss this case. We therefore study the case with $\\phi_i < \\phi_{\\rm min}$ and the case with $\\phi_i > \\phi_{\\rm min}$ (which correspond to $\\rho_{G,i} \\gg \\rho_{b,i}$ and $\\rho_{b,i} \\gg \\rho_{G,i}$, respectively). We shall discuss them in order.\n\n\n\n\\subsubsection{$\\rho_{G,i}\\gg \\rho_{b,i}$ before the equal time}\n\\label{DM>>b}\nIf dark matter (i.e., massive gravitons) is over-produced, the equations are reduced to\n\\begin{align}\n\\ddot{\\phi}+3H \\dot{\\phi}-\\frac{\\lambda}{4M_p} m_T^2 u^2 =0\\,,\n\\\\\n3M_p^2 H^2 =A^4 \\rho_r +\\frac{1}{2}\\dot{\\phi}^2+3m_T^2 u^2\n\\,,\n\\end{align}\nand \\eqref{eq_u}, where we have ignored the contributions from baryon. The system admits a scaling solution\n\\begin{align}\n\\phi&=\\frac{M_p}{\\lambda} \\ln t +{\\rm constant}\\,,\n\\nn\nu&\\propto t^{-1\/2}\n\\,,\n\\nn\na&\\propto t^{1\/2}\n\\,, \\label{scaling_DM}\n\\end{align}\nwhere the density parameters in the Einstein frame are given by\n\\begin{align}\n\\Omega_G=\\frac{4}{3\\lambda^2}\n\\,, \\quad\n\\Omega_{\\phi}=\\frac{2}{3 \\lambda^2}\n\\,, \\quad\n\\Omega_r=1-\\frac{2}{\\lambda^2}\n\\,.\n\\end{align}\nThe effective equation of state parameter in the Jordan frame is given by\n\\begin{align}\nw_{\\rm tot}=\\frac{\\lambda-2\\beta}{3(\\lambda+2\\beta)} \\,,\n\\end{align}\nand then $w_{\\rm tot} =-2\/9$ if $2\\beta=5\\lambda$. This solution exists only when $\\lambda^2>2$ since the density parameter has to be $0<\\Omega_r<1$. \n\n\nFor this scaling solution, the graviton mass decreases as\n\\begin{align}\nm_T^2 \\propto a^{-2}\\,, \\label{time_dep_m}\n\\end{align}\nwhich guarantees the adiabatic condition \\eqref{adiabatic_condition} when $\\epsilon \\ll 1$.\nThe energy density of massive gravitons in the Einstein frame decreases as\n\\begin{align}\nA^4 \\rho_G =\\frac{1}{2}m_T^2 u^2 \\propto a^{-4}\n\\,.\n\\end{align}\nOn the other hand, the energy density of baryon in the Einstein frame ``increases'' as\n\\begin{align}\nA^4 \\rho_b \\propto A a^{-3} \\propto a^{-3+2\\frac{\\beta}{\\lambda}}\n\\,,\n\\end{align}\n(For example, we obtain $A^4 \\rho_b \\propto a^2$ when $2\\beta = 5 \\lambda$). Therefore, even if baryon is negligible at initial, the baryon energy density grows and then it cannot be ignored when the energy density of baryon becomes comparable to that of dark matter.\n\nNote that the Jordan frame energy density of baryon, $\\rho_b$, always decays as $a_J^{-3}$ where $a_J=Aa$ is the scale factor of the Jordan frame metric. The quantity $A^4 \\rho_b$ is the energy density in the Einstein frame.\n\nIn the Einstein frame, the interpretation of the peculiar behavior of $A^4\\rho_G$ and $A^4\\rho_b$ is that the energy density of massive gravitons is converted to that of baryon through the motion of the chameleon field $\\phi$. Although we have considered the non-relativistic massive gravitons, the energy density of that in the Einstein frame behaves as radiation which implies that the field $\\phi$ removes the energy of massive gravitons (indeed, the graviton mass decreases due to the motion of $\\phi$). The removed energy is transferred into baryon via the non-minimal coupling.\n\nDuring the scaling solution, the massive graviton never dominates over radiation because both energy densities of the massive graviton and radiation obey the same decaying law $A^4\\rho_r, A^4\\rho_G \\propto a^4$. Hence, the field $\\phi$ can reach the bottom of the effective potential before the MD era. After reaching the bottom of the effective potential, the standard decaying laws for matters $A^4\\rho_r \\propto a^{-4}$ and $A^4\\rho_G, A^4 \\rho_b \\propto a^{-3}$ are recovered, then the usual dynamics of the universe is obtained with the observed dark matter-baryon ratio. \n\nWe note that the following-up of the baryon energy density can be realized even if the scaling solution does not exist $(\\lambda^2<2)$. The dynamics of this case is numerically studied in Sec.~\\ref{sec_numerical}.\n\n\n\n\\subsubsection{$\\rho_{b,i}\\gg \\rho_{G,i}$ before the equal time}\n\\label{b>>DM}\nIn this case,  the equations for the scale factor and $\\phi$ form a closed system given by\n\\begin{align}\n\\ddot{\\phi}+3H\\dot{\\phi}+\\frac{\\beta}{M_p} A^4 \\rho_b=0\\,,\n\\\\\n3M_p^2 H^2=A^4 \\rho_r+ A^4 \\rho_b+\\frac{1}{2}\\dot{\\phi}^2\n\\,. \n\\end{align}\nThe scaling solution is then found as\n\\begin{align}\n\\phi&=-\\frac{M_p}{2\\beta} \\ln t +{\\rm constant} \\,,\n\\nn\na &\\propto t^{1\/2}\n\\,,\n\\end{align}\nin which the density parameters are\n\\begin{align}\n\\Omega_b=\\frac{1}{3 \\beta^2}\n\\,, \\quad\n\\Omega_{\\phi}=\\frac{1}{6\\beta^2}\n\\,, \\quad\n\\Omega_r=1-\\frac{1}{2\\beta^2}\n\\,,\n\\end{align}\nwhere $\\beta$ has to satisfy $\\beta^2>1\/2$.\n\nDuring this scaling solution, the universe does not expand in the Jordan frame. Although the Einstein frame scale factor expands as the RD universe, $a\\propto t^{1\/2}$, the Jordan frame scale factor is given by\n\\begin{align}\na_J=aA={\\rm constant}\n\\,.\n\\end{align}\n\nThe solution for $u$ is found by substituting the scaling solution into \\eqref{eq_u}. We obtain\n\\begin{align}\nu\\propto a^{-\\frac{3}{2}+\\frac{\\lambda}{4\\beta}}\n\\,,\\quad\nm_T^2 \\propto a^{\\lambda\/\\beta}\n\\,,\n\\end{align}\nand then the energy density of massive graviton varies as\n\\begin{align}\nA^4 \\rho_G \\propto  a^{-3+\\lambda\/2\\beta}\\,.\n\\end{align}\nThe adiabatic condition \\eqref{adiabatic_condition} is guaranteed when $\\epsilon \\ll 1$.\nWhen $2\\beta \\simeq 5 \\lambda$, the graviton mass roughly increases as $m_T^2 \\propto a^{2\/5}$ and the energy density of massive gravitons in the Einstein frame decreases as $A^4\\rho_G \\propto a^{-14\/5}$. Therefore, even if the energy density of massive gravitons is significantly lower than that of baryon, the correct dark matter-baryon ratio is realized in time since the energy density of massive gravitons decreases slower than that of baryon. \n\n\n\n\\subsection{Matter dominant era}\nAfter $\\phi$ reaches the potential minimum $\\phi_{\\rm min}$, the chameleon field $\\phi$ does not move during the MD universe. As shown in \\cite{Aoki:2017cnz}, when $\\phi$ is constant, the massive graviton behaves as CDM and then the standard MD universe is obtained.\n\n\\subsection{Accelerating expanding era}\n\\label{sec_acc}\nAfter the MD era, the universe must show the accelerating expansion due to dark energy. Although one can introduce a new field to obtain the acceleration, we consider a minimal scenario such that the chameleon field itself is dark energy, i.e., the accelerating expansion is realized by the potential $V_0$. When $V_0$ becomes relevant to the dynamics of $\\phi$, the chameleon field again rolls down which leads to a decreasing of $m_T$. As a result, the energy density of massive gravitons rapidly decreases and then we can ignore the contributions from massive gravitons. The basic equation during the accelerating expansion is thus given by\n\\begin{align}\n3M_p^2 H^2=A^4 \\rho_b+\\frac{1}{2}\\dot{\\phi}^2 +V_0 \n\\,, \\label{Fri_DE} \\\\\n\\ddot{\\phi}+3H\\dot{\\phi}-\\frac{\\lambda}{M_p}V_0+\\frac{\\beta}{M_p}A^4 \\rho_b=0\n\\,, \\label{eq_phi_DE}\n\\end{align}\nwhich yield a scaling solution\n\\begin{align}\n\\phi&=\\frac{2M_p}{\\lambda} \\ln t +{\\rm constant}\n\\,, \\nn\na&\\propto t^{\\frac{2}{3}(1+\\beta\/\\lambda)} \\,,\n\\end{align}\nin which\n\\begin{align}\n\\Omega_b=\\frac{\\lambda^2+\\beta \\lambda -3}{(\\beta+\\lambda)^2}\n\\,, \\quad\n\\Omega_{\\phi}=\\frac{\\beta^2+\\beta \\lambda +3}{(\\beta+\\lambda)^2}\n\\,,\n\\end{align}\nand\n\\begin{align}\nw_{\\rm tot}=-\\frac{2\\beta}{4\\beta+\\lambda}\n\\,.\n\\end{align}\nThe scaling solution exists when\n\\begin{align}\n\\lambda(\\beta+\\lambda)>3\n\\,. \\label{inequality_DE}\n\\end{align}\nFor $2\\beta=5\\lambda$, we find $w_{\\rm tot}=-5\/11$ and the inequality \\eqref{inequality_DE} is reduced into $\\lambda^2 >6\/7$.\n\nThe amplitude of the massive graviton is given by\n\\begin{align}\nu \\propto t^{-\\frac{1}{2}(1+2\\beta\/\\lambda)}\n\\,,\n\\end{align}\nand then the density parameter of massive graviton decreases as\n\\begin{align}\n\\Omega_G\\propto t^{-1-2\\beta\/\\lambda}\n\\,.\n\\end{align}\nThe energy density of massive graviton gives just a negligible contribution during this scaling solution which guarantees the equations \\eqref{Fri_DE} and \\eqref{eq_phi_DE}. \n\n\nOn the other hand, when $\\lambda^2<6\/7$, the non-minimal coupling is small so that the field $\\phi$ can be approximated as a standard quintessence field. As a result, the acceleration is obtained by the slow-roll of $\\phi$ and then the dark energy dominant universe is realized.\n\n\n\n\n\n\n\n\n\n\\section{Cosmic evolutions}\n\\label{sec_numerical}\nIn this section, we numerically solve the equations \\eqref{Fri}-\\eqref{eq_u}. We discuss two cases, the over-produced case ($\\rho_{G,i} \\gg \\rho_{b,i}$) and the less-produced case ($\\rho_{G,i}\\ll \\rho_{b,i}$), in order.\n\n\n\\subsection{Over-produced case}\nFirst, we consider the over-produced case $\\rho_{G,i} \\gg \\rho_{b,i}$. We assume \\eqref{model_A} which we call Model A. A cosmological dynamics is shown in Fig.~\\ref{fig_modelA}. We set $\\rho_{G,i}\/\\rho_{b,i}=\\Omega_{G,i}\/\\Omega_{b,i}=200$ at the initial of the numerical calculation. Although dark matter is initially over-produced, the energy density of baryon follows up that of dark matter and then we obtain $\\rho_G\/\\rho_b\\simeq 5$ when $a_J=Aa\\sim 10^{-4}$ where we normalize the Jordan frame scale factor $a_J$ so that $\\Omega_{\\phi}|_{a_J=1}=0.7$. We note the following-up of $\\rho_b$ is obtained even if $\\lambda^2>2$ is not satisfied (In Fig.~\\ref{fig_modelA}, we set $\\lambda^2=(6\/5)^2<2$).\n\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=7cm,angle=0,clip]{Model_A.eps}\n\\caption{The evolution of the density parameters and the total equation of state parameters in terms of the Jordan frame scale factor $a_J=Aa$ which is normalized to be $\\Omega_{\\phi}|_{a_J=1}=0.7$. We set $\\beta=3$ and $\\lambda=2\\beta\/5=6\/5$ in Model A \\eqref{model_A}. We assume the initial ratio between dark matter and baryon as $\\rho_{G,i}\/\\rho_{b,i}=\\Omega_{G,i}\/\\Omega_{b,i}=200$ with $\\phi_i=0$.\n}\n\\label{fig_modelA}\n\\end{figure}\n\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=7cm,angle=0,clip]{phi.eps}\n\\caption{The evolution of the chameleon field $\\phi$ in Model A and Model B with $\\rho_{G,i}\/\\rho_{b,i}=200$.\n}\n\\label{fig_phi}\n\\end{figure}\n\nThe dynamics of the universe is precisely tested by the CMB observations after the decoupling time $a_J \\simeq 10^{-3}$. The evolutions of the total equation of state parameters are shown in Fig.~\\ref{fig_modelA}. The dynamics of the observable universe is represented by the Jordan frame quantity $w_{\\rm tot}$ because the visible matters couple with the Jordan frame metric. On the other hand, since the dark matter (i.e., massive gravitons) is originated from the the gravity sector, dark matter feels the dynamics of the Einstein frame whose equation of state parameter is denoted by $w_E$. Although the large deviation of dynamics from the standard cosmological one appears before the decoupling time $a_J \\lesssim 10^{-3}$, the standard dust dominant universe is recovered around the decoupling time.\n\n\nWhen we increase the values of $\\beta$ and $\\lambda$, the deviation from the standard evolution is amplified which is caused by the oscillation of $\\phi$ around $\\phi_{\\rm min}$ as shown in Fig,~\\ref{fig_phi}. Since the Jordan frame scale factor is given by $a_J=Aa=a e^{\\beta\\phi\/M_p}$, the oscillation of $\\phi$ yields the oscillation of $a_J$ which is amplified by increasing of $\\beta$.\n\n\nFig.~\\ref{fig_modelA} does not show the dark energy ``dominant'' universe even in the accelerating phase. Instead, the acceleration is realized by the scaling solution as explained in Sec.~\\ref{sec_analytic}. If this scaling solution can pass the observational constraints, it might give an answer for the other coincidence problem of dark energy: why the present dark energy density is almost same as that of matter? However, the cosmological dynamics after the decoupling time is strongly constrained by the observations. Thus, the dark energy model with the scaling solution should have a severe constraint (see \\cite{Amendola:1999er,Amendola:2003eq} for examples). Furthermore, the large coupling $\\alpha_A \\gtrsim M_p^{-1}$ leads to that the Compton wavelength of the chameleon field has to be less than Mpc to screen the fifth force in the Solar System~\\cite{Wang:2012kj}; however, the coupling functions \\eqref{model_A} require the Gpc scale Compton wavelength to give the current accelerating expansion.\n\n\n\nWe then provide a model in which the couplings $\\alpha_A$ and $\\alpha_f$ are initially large but they become small in time. This behavior is realized by the model\n\\begin{align}\nK^2=(1-\\phi^2\/M^2)^{-1} , \\, A=e^{\\beta\\phi\/M_p} ,\\,\nf=e^{-\\lambda \\phi\/M_p}, \\label{model_B}\n\\end{align}\nwhich we call Model B. The only difference from Model A is that $K$ is a function of $\\phi$. If the amplitude of the field $\\phi$ is small at initial $(\\phi \\ll M)$, Model B gives the same behavior as Model A. After $\\phi$ starts to roll and then $|\\phi| \\rightarrow M$, the kinetic function $K$ increases which causes the decreasing of the non-minimal couplings $\\alpha_A,\\alpha_f \\rightarrow 0$ (see Figs.~\\ref{fig_phi} and \\ref{fig_modelB}). \nNote that the field value is restricted in the range $-M<\\phi<M$ in Model B which gives a constraint $|\\phi_{\\rm min}|<M$ to obtain $\\rho_G\/\\rho_b\\simeq 5$.\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=7cm,angle=0,clip]{Model_B.eps}\n\\caption{The same figures as Fig.~\\ref{fig_modelA} and the evolution of $\\alpha_A$ and $\\alpha_f$ in Model B. We set $\\beta=3, \\lambda=6\/5$ and $M=M_p$. We assume the same initial condition as Fig.~\\ref{fig_modelA}.\n}\n\\label{fig_modelB}\n\\end{figure}\n\n\n\nA numerical solution of Model B is shown in Fig.~\\ref{fig_modelB}. The evolutions of $\\alpha_f$ and $\\alpha_A$ are shown in the bottom of Fig.~\\ref{fig_modelB}. As we expected, the couplings become weak in time. In particular, if $\\phi_{\\rm min} \\simeq M$ in Model B, the cosmological dynamics is quite similar to that in the $\\Lambda$CDM model after the decoupling time.\n\nThe evolutions in Model A and Model B are divided into four regimes: the radiation dominant era $(a_J \\lesssim 10^{-8})$, the following-up era $(10^{-8}\\lesssim a_J \\lesssim 10^{-3})$, the dust dominant era $(10^{-3}\\lesssim a_J \\gtrsim 10^{-1})$, and the accelerating era $(10^{-1}\\lesssim a_J)$. In Model A, the deviations from the $\\Lambda$CDM model appear in the following-up era and the accelerating era. On the other hand, the deviation appears only in the following-up era in Model B which is after the Big Bang Nucleosynthesis (BBN) $a_J\\simeq 10^{-8}$ and before the CMB last scattering surface $a_J\\simeq 10^{-3}$. Since the relation between the temperature and the Hubble expansion rate during the BBN era is essentially the same as in the standard cosmology, Model B is compatible with the standard BBN.  In Model B, evolution of perturbations at the CMB scales is also expected to be the same as in the standard cosmology. On the other hand, the non-standard evolution before the CMB last scattering surface may change the evolution of perturbations at smaller scales. These deviations may give observational constraints on our models or may help addressing some of the tensions between the standard $\\Lambda$CDM and observational data at small scales. \n\n\n\n\\subsection{Less-produced case}\nNext, we discuss the case of the less-produced dark matter $\\rho_{G,i}\\ll \\rho_{b,i}$. A numerical solution is shown in Fig.~\\ref{fig_modelA_less}. Although the energy density of dark matter is initially smaller than that of baryon, the correct abundance $(\\rho_G\/\\rho_b\\simeq 5)$ is obtained. Since the non-minimal coupling is not so large ($\\beta=1$ in the case of Fig.~\\ref{fig_modelA_less}), the universe evolves into the dark energy dominant universe in the future.\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=7cm,angle=0,clip]{Model_A_less.eps}\n\\caption{The evolution of the density parameters and the total equation of state parameter. We set $\\beta=1$ and $\\lambda=2\/5$ in Model A. We assume the initial ratio between dark matter and baryon as $\\rho_{G,i}\/\\rho_{b,i}=0.01$ with $\\phi_i=0$.\n}\n\\label{fig_modelA_less}\n\\end{figure}\n\n\n\nThe following-up era is realized when $\\Omega_b \\simeq 1\/3\\beta^2$ in the scaling solution in which the dynamics of the universe is deviated from the standard one. Hence, the small value of $\\beta$ yields that the following-up era is close to the decoupling time and then the deviation may give a large effect on the CMB physics. For instance, Fig.~\\ref{fig_modelA_less} indicates that the deviation still exists at $a_J \\simeq 10^{-3}$.\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=7cm,angle=0,clip]{Model_B_less.eps}\n\\caption{The same figures as Fig.~\\ref{fig_modelB} in the less-produced case. We set $\\beta=5, \\lambda=2$ and $M=M_p$, and assume $\\rho_{G,i}\/\\rho_{b,i}=0.01$ with $\\phi_i=0$.\n}\n\\label{fig_modelB_less}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=7cm,angle=0,clip]{phi_less.eps}\n\\caption{The evolution of the chameleon field $\\phi$ with $\\rho_{G,i}\/\\rho_{b,i}=0.01$ and $\\phi_i=0$. We set $\\beta=1, \\lambda=2\/5$ for Model A and $\\beta=5, \\lambda=2, M=M_p$ for Model B.\n}\n\\label{fig_phi_less}\n\\end{figure}\n\nIn Model B, the coupling strength is time-dependent. Hence, we can obtain a scenario in which the non-minimal couplings are initially large whereas the couplings turn to be weak after the ratio $\\rho_G\/\\rho_b$ is dynamically adjusted to the observed value as shown in Fig.~\\ref{fig_modelB_less}. In this case, the standard cosmological dynamics is recovered around $a_J\\simeq 10^{-4}$. Although the dynamics is deviated from the standard one in $10^{-6} \\lesssim a_J \\lesssim 10^{-4}$, it should have little effect on CMB.\n\nWhen the bare potential $V_0$ becomes relevant to $\\phi$, the field value increases as shown in Fig.~\\ref{fig_phi_less}. In Model B, this increasing leads to the increasing of the non-minimal couplings $\\alpha_A$ and $\\alpha_f$ in the future $a_J \\gg 1$. Needless to say, if the form of the bare potential $V_0$ is modified in order that $\\dot{\\phi}<0$ in the dark energy dominant era (for example, $\\lambda<0$), the coupling will be weak in the future as with Fig.~\\ref{fig_modelB}.\n\n\nAs a result, we can obtain a viable cosmological dynamics even in the less-produced case. Although the initial abundance of dark matter is much smaller than the observed value, the chameleon field provides the energy transfer from baryon to dark matter via the non-minimal couplings. The correct abundance of dark matter can be realized without the fine-tuning of the initial condition.\n\n\n\n\n\n\n\n\n\\section{Concluding remarks}\n\\label{summary}\nIn the present paper, we provide a cosmological scenario by which the observed dark matter-baryon ratio can be naturally explained.  We have added two new ingredients to the standard model: the massive graviton and the chameleon field corresponding to dark matter and dark energy, respectively. The matter fields are minimally coupled to the Jordan frame metric which leads to the non-minimal coupling of the matter fields to the chameleon field. On the other hand, the chameleon field may have a different coupling to dark matter since dark matter, the massive graviton, is originated from the gravity sector. We have assumed that the chameleon field has a non-minimal coupling to the massive graviton via the mass terms of the graviton. Two different non-minimal couplings of the chameleon field realize that the ratio $\\rho_G\/\\rho_b$ is dynamically relaxed to $\\sim 5$ without the tuning of the initial condition. \n\nWe have studied two simple models: Eq.~\\eqref{model_A} (Model A) and Eq.~\\eqref{model_B} (Model B).\\footnote{Although we chose exponential forms of $A$ and $f$, other types of the non-minimal coupling can be discussed. Even in this case, the dark matter-baryon ratio could be explained when $\\alpha_A\/\\alpha_f \\simeq -5\/2$.} In these models, we have an additional era in the universe before the radiation-matter equality which we have called the following-up era. Even if dark matter is initially over-produced or less-produced, the chameleon field transfers the energy of the larger one of dark matter and baryon into that of the smaller one via the non-minimal couplings. After the ratio $\\rho_G\/\\rho_b$ is tuned, the standard cosmological dynamics can be recovered.\n\nTo realize the following-up era, the non-minimal couplings should be at least comparable to the usual gravitational interaction, $|\\alpha_A|, |\\alpha_f| \\sim M_p^{-1}$. This ``large'' non-minimal couplings may lead to a deviation from the $\\Lambda$CDM cosmology. However, in Model B, the coupling strengths are time-dependent and then the present non-minimal couplings can be small as shown in Fig.~\\ref{fig_modelB} and Fig.~\\ref{fig_modelB_less}. \n\n\nThe chameleon bigravity theory \\eqref{action} was originally proposed in \\cite{DeFelice:2017oym} in order to avoid the Higuchi instability of cosmological solutions~\\cite{Higuchi:1986py,Higuchi:1989gz,Grisa:2009yy,Fasiello:2012rw,Fasiello:2013woa,Comelli:2012db,Comelli:2014bqa,DeFelice:2014nja,Aoki:2015xqa} and to circumvent the low cutoff scale of the standard bigravity theory. When the graviton mass is smaller than the Hubble expansion rate, the scalar mode of the massive graviton exhibits the ghost instability or the gradient instability, in general. However, the chameleon bigravity can yield a cosmological solution in which $m_T\/H$ is constant during the radiation dominant universe and then the homogeneous and isotropic solution does not suffer from the instability even in the early universe~\\cite{DeFelice:2017oym,DeFelice:2017gzc}. The cutoff scale of the gravity sector also becomes high in the early universe and thus this mechanism make it possible for us to apply the bigravity theory to the early universe. Since we have considered only the cases with $m_T \\gg H$, the Higuchi instability is not problematic for the present discussion and thus we have not discussed the stability issue of the homogeneous spacetimes \\eqref{Bianchi_g} and \\eqref{Bianchi_f}. Nevertheless, it would be interesting to study whether or not the chameleon bigravity also gives a stable homogeneous but anisotropic solution even in the early epoch of the universe.\n\n\nAs discussed in \\cite{Aoki:2016zgp,Aoki:2017cnz}, the massive graviton with a constant mass is a viable dark matter candidate in the wide range of the mass $10^{-23}{\\rm eV}\\lesssim m \\lesssim 10^7 (M_g\/M_f)^{2\/3} {\\rm eV}$. The lower bound is imposed so that dark matter halos form in dwarf galaxy scales, while the upper bound is given by the requirement that the lifetime of the massive graviton be longer than the age of the universe. In the chameleon bigravity theory studied in the present paper, by the same argument as before, we obtain the lower bound on the present value of $m_T$, i.e., $m_T|_{a_J = 1} \\gtrsim 10^{-23}{\\rm eV}$. The lifetime of the massive graviton also constrains the present value of the mass as follows. Since the abundance of dark matter is automatically tuned to the observed value  by the scaling solution, we need to discuss the lifetime of the massive graviton only after the following-up era in our scenario~\\footnote{Decay of massive graviton during and after the nucleosynthesis but before the end of the following-up era may have some impacts on the primordial abundances of the light elements. We leave studies of such effects to future publications.}. The graviton mass remains almost constant during this epoch. As a result, the requirement of a long enough lifetime gives the upper bound on the present value of the mass as $m_T|_{a_J=1} \\lesssim 10^7 (M_g\/M_f)^{2\/3} {\\rm eV}$.\n\n\nSince we have not discussed concrete observational constraints on our models, the observational viability of the models is an open question. Theoretically, our model can explain the dark matter-baryon ratio for any initial condition except for $\\rho_{G,i}=0$ or $\\rho_{b,i}=0$ because the following-up era can be the scaling solution and then it never ends unless the smaller one of $\\rho_g$ and $\\rho_b$ catches up the larger one. However, for example, $\\rho_{G,i}\/\\rho_{b,i} \\simeq 200$ should be an observational upper bound on the initial ratio in the over-produced case with $\\beta=3,\\lambda=6\/5$. If $\\rho_{G,i}\/\\rho_{b,i} \\gtrsim 200$, the dynamics of the universe at BBN is changed from the radiation dominant universe and then it may give an observational constraint. Furthermore, we should take into account local gravity constraints of the fifth force since the following-up of dark matter or baryon requires a large non-minimal coupling of the chameleon field. Although the current non-minimal coupling can be small in Model B, there still exists a small fifth force and then the local gravity experiments may give a constraint on our model as well.\\footnote{The massive graviton also yields a fifth force. We should also discuss the Vainshtein screening mechanism of the massive graviton. However, if the graviton mass is high enough (e.g., $m_T|_{a_J=1} \\gg 10^{-4}$ eV), the fifth force propagated by the massive graviton does not exist in the observed scales due to the Yukawa suppression.} We leave the details of the observational constraints of our models for a future work.\n\n\n\n\\section*{Acknowledgments}\nThe work of K.A. was supported in part by Grants-in-Aid from the Scientific Research Fund of the Japan Society for the Promotion of Science  (No. 15J05540). \nThe work of S.M. was supported in part by JSPS Grant-in-Aid for Scientific Research No. 17H02890, No. 17H06359,  and by World Premier International Research Center Initiative (WPI), MEXT, Japan. \n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\n\\par In recent years considerable progress has been made towards sensing and recovery of sparse signals, known as Compressive Sensing \\cite{CSCandes}\\cite{RobustUncertainty}. However, less attention has been paid to classifying sparse signals from compressive measurements, which can be stated as Compressive Classification.\n\\par The Compressive Classification is derived from the theory of statistical signal processing \\cite{KayV2}. In this compressive scenario, different sparse signals contaminated by gaussian noise are compressed by under-determined linear transformations, and then the compressed measurements are used for signal classification. Since the objective in classification is to minimize the error probability, or the probability of false classification of the classifiers, analysis of the influence of measurement matrices on the classification is different from that of estimating sparse signals.\n\\par To the best of our knowledge, the earliest analysis of the performance of Compressive Classification is from \\cite{Davenport06detectionand}\\cite{SPComp} by Davenport in 2006. In these publications, performance bounds on the probability of false classification of Compressive Classifiers using row-orthogonal random projection matrices were analyzed.\nBesides, there are other publications such as Ramin Zahedi \\textit{et al.} 's works \\cite{Ramin2010}\\cite{Zahedi201264}, in which Equiangular Tight Frames (ETFs, \\cite{Waldron2009}) were considered and proved to have best worst-case performance as the measurement matrices in Compressive Classification under the constraint that the measurement matrices are row-orthogonal.\n\\par\nThis paper is mainly concerned about the implementation of measurement matrices to improve the performance of Compressive Classification. We noticed that all these publications mentioned above (\\cite{Davenport06detectionand}\\cite{SPComp}\\cite{Ramin2010}\\cite{Zahedi201264}) and many other related most recent researches (\\cite{RaoICCASP2012}\\cite{PerfLimitsTranSP}) did their analysis under the constraint that the measurement matrices are row-orthogonal, \\textit{i.e.} the column vectors form an equi-norm tight frame. The reason why they use this constraint is for simplicity and avoiding the coloring of received signals between different hypotheses. While in this paper, we will prove that the transformation to equi-norm tight frames from arbitrary measurement matrices will reduce the probability of false classification of the commonly used Compressive Classifier, which coincides with the row-orthogonal constraint commonly used above in (\\cite{Davenport06detectionand}\\cite{SPComp}\\cite{Ramin2010}\\cite{Zahedi201264}). Although there are already proofs (\\cite{Ramin2010}\\cite{Zahedi201264}) showing that Equiangular Tight Frames (ETFs) have best worst-case performance among all tight frames, our job is different from theirs. Because constructing equi-norm tight frames is much simple and practical, so our results could provide a convenient approach to improve the performance of Compressive Classifiers -- The tightened, or row-orthogonalized measurement matrices could be better than before.\n\\par\nThe remainder of this paper is organized as follows:\nIn Section 2, we give the problem formulation of Compressive Classification and the error probability; in Section 3, we demonstrate our main result and proof of the influence of equi-norm tight frames on the error probability, and then some discussions and remarks; and in the last section numerical simulation results and discussions are provided for validation.\n\n\\section{Compressive Signal Classification}\n\\label{sec:compclassf}\n\\par\nDerived from the traditional theory of statistical signal detection and estimation \\cite{KayV2}, in the Compressive Classification scenario,  the sparse signals is compressed by under-determined linear transformation first, then these compressed measurements are used to do classification. As is stated in Davenport's works \\cite{Davenport06detectionand}\\cite{SPComp}, the model of the 2-ary compressive signal classification can be expressed as:\n\\begin{equation}\\label{CompressHypo}\n\\bm{y} =\n\\left \\{ \\begin{array}{ll}\n\t\\bm{\\Phi}(\\bm{s_1} + \\bm{n}) & \\text{Hypothesis  $H_1$} \\\\\n\t\\bm{\\Phi}(\\bm{s_2} + \\bm{n}) & \\text{Hypothesis  $H_2$} \\\\\n\n\n\n\\end{array} \\right.\n\\end{equation}\nin which $\\bm{s_i}\\in\\Lambda_k=\\{\\bm{s} \\in\\mathbb{R}^N,\\|\\bm{s}\\|_0\\leq k\\},i=1,2$ are k-sparse signals, i.e. the vectors with at most k nonzero elements, besides they are orthogonal to each other and have equal norms ($\\langle \\bm{s_i},\\bm{s_j}\\rangle = 0,\\|\\bm s_i\\|_2^2=\\|\\bm s_j\\|_2^2,i \\neq j$). $\\bm{y}\\in \\mathbb{R}^n$ is the received compressive measurements, $\\bm{n}\\thicksim \\mathcal{N} (0,\\sigma^2\\bm{I_N})$ is the Gaussian noise contaminating the sparse signals. And $\\bm{\\Phi}\\in \\mathbb{R}^{n\\times N},n<N$ is the under-determined measurement matrix, with full row rank and equi-norm columns, and satisfying Restricted Isometry Property (RIP) \\cite{Decoding}\\cite{SimpleProof}. It is best that we focus on the scenario of 2-ary Compressive Classification first, which means there are 2 hypotheses in (\\ref{CompressHypo}). More general results of multiple hypotheses (more than 2) can be easily deduced from the 2-ary scenario.\n\\par\nAccording to the works of Davenport \\cite{Davenport06detectionand}\\cite{SPComp}, Zahedi \\cite{Ramin2010} and Rao \\cite{RaoICCASP2012}, under the constraint of row-orthogonal ($\\bm{\\Phi\\Phi^T=I})$ to the measurement matrices, the following classifier can be used:\n\\begin{eqnarray}\nt_i&=&\\bm{y^T\\bm{\\Phi} s_i} - \\frac{1}{2}\\|\\bm{\\Phi s_i}\\|_2^2\\nonumber \\\\\n&=&\\langle\\bm{y},\\bm{\\Phi}\\bm{s_i}\\rangle-\\frac{1}{2}\\langle \\bm{\\Phi s_i},\\bm{\\Phi s_i} \\rangle,\\quad i=1,2 \\label{OMPDetct}.\n\\end{eqnarray}\nAnd the classification result is\n\\begin{equation}\\label{MatchFitler}\ni^*=\\arg \\max_{i} \\{t_i\\} \\quad i=1,2.\n\\end{equation}\n\\par\nThe classifier in (\\ref{OMPDetct}) is a most commonly used classifier, it is composed of a term of correlation and an minus term for normalization. The correlation part $\\bm{y^T\\bm{\\Phi} s_i}$ also plays an important role in iterations of the Orthogonal Matching Pursuit (OMP) algorithm \\cite{AnalOMP}. Since all the known analysis of Compressive Classifiers is restricted to row-orthogonal measurement matrices, our work is going to break this restriction. So the performance of the Compressive Classifier in (\\ref{OMPDetct}) with more general measurement matrices that may or may not be row-orthogonal will be analyzed as follows.\n\\begin{Theorem}\nIf we use Compressive Classifier (\\ref{OMPDetct}) and (\\ref{MatchFitler}) to classify (\\ref{CompressHypo}), the probability of false classification can be expressed as:\n\\begin{equation}\n\\mathcal{P}_E =\nQ(\\frac{\\|\\bm{\\Phi(s_1-s_2)}\\|_2^2}{2\\sigma\\|\\bm{\\Phi^T\\Phi(s_1-s_2)}\\|_2}) \\label{CompressDetect}.\n\\end{equation}\nHere $Q(x)=\\int_{x}^{\\infty} {\\frac{1}{\\sqrt{2\\pi}}\\exp{(-\\frac{t^2}{2})}\\mathrm{d}t}$, and $\\bm \\Phi \\in \\mathbb{R}^{n\\times N},n<N$ is the measurement matrix.\n\\end{Theorem}\n\\begin{proof}\nThe probability of false classification, or error probability is \\cite{KayV2}\n\\begin{eqnarray}\\label{ErrorProb}\n\\mathcal{P}_E &=& 1 -\n\\sum_{k=1}^2 \\mathcal{P}(t_k >t_j,j\\neq k|H_k)\\mathcal{P}(H_k)\n\\end{eqnarray}\\par\nIn general, the priori probability of different hypotheses can be assumed as uniformly distributed ($\\mathcal{P}(H_i)=\\mathcal{P}(H_j), i \\neq j$), and the conditional probability in (\\ref{ErrorProb}) is assumed to be equal \\cite{KayV2} for symmetry, that is\n\\begin{eqnarray}\n\\mathcal{P}(t_1 > t_2|H_1) =\n \\mathcal{P}(t_2 > t_1|H_2)\n\\end{eqnarray}\nthen the error probability is\n\\begin{eqnarray}\n\\mathcal{P}_E &=& 1 - \\mathcal{P}(t_1 > t_2|H_1\n.\n\\end{eqnarray}\nThus the probability can be analyzed with respect to the conditional probability $\\mathcal{P}( t_1> t_2|H_1)$ under the $H_1$ hypothesis.\nThe statistics (\\ref{OMPDetct}) under the $H_1$ hypothesis have the following joint distributions\n\\begin{eqnarray}\n\n\\left[ t_1 \\quad t_2 \\right]^T\n\t\t& \\thicksim & \\mathcal{N}(\\bm \\mu_{1,2},\\bm \\Sigma_{1,2})\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\bm \\mu_{1,2}=\\left[ \\begin{array}{c}\n\t\t\t \\frac{1}{2}\\|\\bm{\\Phi s_1}\\|_2^2 \\\\\n\t\t\t \\langle\\bm{\\Phi s_2},\\bm{\\Phi s_1}\\rangle-\\frac{1}{2}\\|\\bm{\\Phi s_2}\\|_2^2\n\t\t\\end{array} \\right] \\nonumber\n\\end{eqnarray}\n\\begin{equation}\n\\bm \\Sigma_{1,2} =\\sigma^2\\left[ \\begin{array}{cc}\n\t\t\t\t\t\t\t\t\t\\|\\bm{\\Phi^T\\Phi s_1}\\|_2^2 &\\langle \\bm{\\Phi^T\\Phi s_1},\\bm{\\Phi^T\\Phi s_2}\\rangle \\\\\n\t\t\t\t\t\t\t\t\t\\langle \\bm{\\Phi^1\\Phi s_2},\\bm{\\Phi^T\\Phi s_1}\\rangle &\\|\\bm{\\Phi^T\\Phi s_2}\\|_2^2\n\t\t\t\t\t\t\t\t\\end{array}\\right].\\nonumber\n\\end{equation}\nUsing the property of Gaussian distribution, the probability of false classification is then\n\\begin{eqnarray}\n\\mathcal{P}_E \n&=&Q(\\frac{\\|\\bm{\\Phi(s_1-s_2)}\\|_2^2}{2\\sigma\\|\\bm{\\Phi^T\\Phi(s_1-s_2)}\\|_2})\\nonumber\n\\end{eqnarray}\nand $Q(x)=\\int_{x}^{\\infty} {\\frac{1}{\\sqrt{2\\pi}}\\exp{(-\\frac{t^2}{2})}\\mathrm{d}t}$.\n\\end{proof}\n\\par As a matter of fact, if there is more than 2 hypotheses in the Compressive Classification problem (\\ref{CompressHypo}), the error probability of the Compressive Classifier (\\ref{OMPDetct}) and (\\ref{MatchFitler}) may not be so explicit as (\\ref{CompressDetect}) for the statistical correlation between different $t_i$'s in (\\ref{OMPDetct}). But similar techniques can be utilized and same results can be deduced, we will discuss these m-ary ($m>2$) hypotheses scenarios in the next section.\n\\par So in order to analyze the error probability (\\ref{CompressDetect}) of classifier (\\ref{OMPDetct}) without the constraint of row-orthogonality to measurement matrices and for all possible k-sparse signals $\\bm s_i,i=1,2$, we will have to focus on\n\\begin{equation}\n\\frac{\\|\\bm{\\Phi(s_1-s_2)}\\|_2^2}{\\|\\bm{\\Phi^T\\Phi(s_1-s_2)}\\|_2} \\label{MainTarget}\n\\end{equation}\nfor all k-sparse signals $\\bm{s_1,s_2}\\in\\Lambda_k=\\{\\bm{s} \\in\\mathbb{R}^N,\\|\\bm{s}\\|_0\\leq k\\}$. \\par\nIn the cases where measurement matrices satisfying row-orthogonality ($\\bm{\\Phi\\Phi^T}=\\bm I$), (\\ref{MainTarget}) is then reduced to\n\\begin{equation}\n\\frac{\\|\\bm{\\Phi(s_1-s_2)}\\|_2^2}{\\|\\bm{\\Phi^T\\Phi(s_1-s_2)}\\|_2} = \\|\\bm{\\Phi(s_1-s_2)}\\|_2.\n\\end{equation}\nAnd this is what Davenport \\cite{Davenport06detectionand}\\cite{SPComp} and Zahedi \\cite{Ramin2010}\\cite{Zahedi201264} analyzed in their publications.\n\n\\section{measurement Matrices and the Error Probability of Compressive Signal Classification}\n\\label{sec:main}\n\\par Although there has been plenty of works about the performance analysis of Compressive Classification, all these works have the same row-orthogonality presumption, but without a theoretical explanation. However, what we believe is that there exist other important reasons for the row-orthogonal condition to be necessary. Here is our main result of this paper:\n\\begin{Theorem}\nIn the Compressive Classification problem (\\ref{CompressHypo}), by tightening or row-orthogonalizing the measurement matrix $\\bm \\Phi \\in \\mathbb{R}^{n \\times N},n < N$, the error probability (\\ref{CompressDetect}) of the classifier (\\ref{OMPDetct}) will be reduced, which means\n\\begin{equation}\\label{MainIneq}\n\\frac{\\|\\bm{\\Phi(s_1-s_2)}\\|_2^2}{\\|\\bm{\\Phi^T\\Phi(s_1-s_2)}\\|_2} \\leq\n\\frac{\\|\\bm{\\hat \\Phi(s_1-s_2)}\\|_2^2}{\\|\\bm{\\hat \\Phi^T\\hat \\Phi(s_1-s_2)}\\|_2}\n\\end{equation}\nwhere $\\bm \\Phi \\in \\mathbb{R}^{n \\times N}$ is the arbitrary measurement matrix, and $\\bm {\\hat \\Phi} \\in \\mathbb{R}^{n \\times N}$ is the equi-norm tight frame measurement matrix row-orthogonalized from $\\bm \\Phi$.\n\\end{Theorem}\n\\begin{proof}\nAccording to Section 2, the error probability (\\ref{CompressDetect}) is determined by the following expression (\\ref{MainTarget}):\n\\begin{equation}\n\\frac{\\|\\bm{\\Phi(s_1-s_2)}\\|_2^2}{\\|\\bm{\\Phi^T\\Phi(s_1-s_2)}\\|_2}  \\nonumber\n\\end{equation}\nfor all k-sparse signals $\\bm s_1, \\bm s_2$, where $\\bm \\Phi$ is an arbitrary measurement matrix satisfying RIP. \\par\nAccording to the basic presumptions of $\\bm \\Phi$ in (\\ref{CompressHypo}), the arbitrary under-determined measurement matrix $\\bm \\Phi \\in \\mathbb{R}^{n \\times N}$, $n < N$ has full row rank, thus the singular value decomposition of $\\bm \\Phi$ is\n\\begin{equation}\\label{singular}\n\\bm \\Phi = \\bm{U \\left[ \\Sigma_n \\quad O\\right] V^T}\n\\end{equation}\nHere $\\bm \\Sigma_n \\in \\mathbb{R}^{n \\times n}$ is a diagonal matrix with each element $\\bm \\Phi$'s singular value $\\sigma_j \\neq 0$ $(1\\leq j \\leq n)$, and $\\bm U \\in \\mathbb{R}^{n \\times n}$, $\\bm V \\in \\mathbb{R}^{N \\times N}$ are orthogonal matrices composed of $\\bm \\Phi$'s left and right singular vectors. \\par\nIf an arbitrary equi-norm measurement matrix $\\bm \\Phi$ is transformed into an equi-norm tight frame $\\bm{\\hat \\Phi}$, we do orthogonalization to its row vectors, which is equivalent as:\n\\begin{equation}\\label{tighten}\n\\bm{\\hat \\Phi} = \\sqrt{c} \\cdot \\bm{U\\Sigma_n^{-1}U^T\\Phi} = \\sqrt{c}\\cdot \\bm{U \\left[ I_n \\quad O\\right] V^T}\n\\end{equation}\nWhere $\\bm U \\in \\mathbb{R}^{n \\times n}$, $\\bm V \\in \\mathbb{R}^{N \\times N}$ are $\\bm \\Phi$'s singular vector matrices. In a word, row-orthogonalization is equivalent to transforming all singular values of $\\bm \\Phi$ into equal ones. Thus $\\bm{\\hat \\Phi \\hat \\Phi^T = c \\cdot I_n}$, where $c>0$ is a certain constant for normalization.\\par\nThen\n\\begin{eqnarray}\\label{TF}\n\\frac{\\|\\bm{\\hat \\Phi(s_1-s_2)}\\|_2^2}{\\|\\bm{\\hat \\Phi^T\\hat \\Phi(s_1-s_2)}\\|_2}\n=\n\\| \\left[ \\begin{array}{cc}\n\t\t\t\t\t\t\t\t\t\\bm{I_n} &\\bm O\n\t\t\t\t\t\t\t\t\\end{array}\\right] \\bm V^T(\\bm{s_1-s_2})\\|_2.\n\\end{eqnarray}\nAnd for arbitrary measurement matrix $\\bm \\Phi$ that may not be row-orthogonal, we have\n\\begin{eqnarray}\n\\lefteqn{\\frac{\\|\\bm{\\Phi(s_1-s_2)}\\|_2^2}{\\|\\bm{\\Phi^T\\Phi(s_1-s_2)}\\|_2}} \\nonumber \\\\\n&=& \\frac{\\|\\left[ \\begin{array}{cc}\n\t\t\t\t\t\t\t\t\t\t\t\t\\bm \\Sigma_n & \\bm O \\end{array} \\right]\\bm{V^T(s_1-s_2)}\\|_2^2}\n\t\t\t{\\| \\left[ \\begin{array}{cc}\n\t\t\t\t\t\t\t\t\t\t\t\t\\bm \\Sigma_n^2 & \\bm O \\end{array} \\right]\\bm{V^T(s_1-s_2)}\\|_2} .\\label{singularfrac}\n\\end{eqnarray}\nIf we denote $\\bm{V^T(s_1-s_2)}$ by $\\bm u^{(1,2)}$, where $\\bm u^{(1,2)} = [u_1, u_2, \\cdots ,u_N]^T$. Then (\\ref{singularfrac}) becomes\n\\begin{eqnarray}\n\\frac{\\|\\left[ \\begin{array}{cc}\n\t\t\t\t\t\t\t\t\t\t\t\t\\bm \\Sigma_n & \\bm O \\end{array} \\right]\\bm{V^T(s_1-s_2)}\\|_2^2}\n\t\t\t{\\| \\left[ \\begin{array}{cc}\n\t\t\t\t\t\t\t\t\t\t\t\t\\bm \\Sigma_n^2 & \\bm O\\end{array} \\right]\\bm{V^T(s_1-s_2)}\\|_2}\n\t\t\t\t\t\t\t\t\t\t\t\t= \\frac{\\sum_{j=1}^n {\\sigma_j^2 u^{2}_j}}{\\sqrt{\\sum_{j=1}^n {\\sigma_j^4 u^{2}_j}}}\\nonumber \\\\\n\\leq \\sqrt{\\sum_{j=1}^n {u^{2}_j}}=\n\\| \\left[ \\begin{array}{cc}\n\t\t\t\t\t\t\t\t\t\\bm{I_n} &\\bm O\n\t\t\t\t\t\t\t\t\\end{array}\\right] \\bm V^T(\\bm{s_1-s_2})\\|_2 . \\label{CIneq}\n\\end{eqnarray}\nThe last inequality is derived from the Cauchy-Schwarz Inequality, combining (\\ref{singularfrac}), (\\ref{CIneq}) with (\\ref{TF}), then we have\n\\begin{equation}\n\\frac{\\|\\bm{\\Phi(s_1-s_2)}\\|_2^2}{\\|\\bm{\\Phi^T\\Phi(s_1-s_2)}\\|_2} \\leq\n\\frac{\\|\\bm{\\hat \\Phi(s_1-s_2)}\\|_2^2}{\\|\\bm{\\hat \\Phi^T\\hat \\Phi(s_1-s_2)}\\|_2}\n\\end{equation}\nwhich means that row-orthogonalization makes (\\ref{MainTarget}) larger and thus brings lower error probability.\nThe condition when the equality holds is that\n\\begin{eqnarray}\\label{condition}\n\\bm{\\left[\\Sigma_n^2\\quad O\\right]}\\bm{V^T}(\\bm{s_1-s_2})\n=c \\cdot \\bm{\\left[I_n\\quad O\\right]}\\bm{V^T}(\\bm{s_1-s_2})\n\\end{eqnarray}\nwhere $c>0$ is a certain constant.\\par\nIt is obvious that the equality in (\\ref{MainIneq}) holds for all k-sparse signals $\\bm s_1$ and $\\bm s_2$, if and only if $\\bm{\\Sigma_n^2=c \\cdot I_n}$, which means\n\\begin{equation}\n\\bm{\\Phi\\Phi^T} = c \\cdot \\bm I_n\\label{TightCond}\n\\end{equation}\n\\par So the result of (\\ref{MainIneq}) means that when arbitrary under-determined measurement matrices $\\bm \\Phi \\in \\mathbb{R}^{n \\times N},n < N$ are transformed into an equi-norm tight frame, i.e. row-ortho-gonalized, the equality in (\\ref{MainIneq}) will hold, then the value of (\\ref{MainTarget}) will increase, which means improvement of the performance of Compressive Classifier (\\ref{OMPDetct}).\n\\end{proof}\\par\nThe constant $c>0$ above is an amplitude constant for normalization and can take any value, with the equi-norm presumption of measurement matrices, the following corollary can be deduced:\n\\begin{Corollary}\nIf a matrix $\\bm \\Phi\\in \\mathbb{R}^{n \\times N}, n < N$ form an equi-norm tight frame, that is $\\bm{\\Phi\\Phi^T}=c\\cdot I_n$ and the column vectors satisfy $\\|\\bm \\phi_i\\|_2=\\|\\bm \\phi_j\\|_2=\\psi$, then $c = \\frac{N}{n}\\psi^2$.\n\\end{Corollary}\n\\begin{proof}\nIf $\\bm \\Phi$ has equal column norms and satisfies $\\bm{\\Phi\\Phi^T}=c\\cdot I_n$, then\n\\begin{eqnarray}\ntr(\\bm{\\Phi^T\\Phi})=N\\cdot \\psi ^2= tr(\\bm{\\Phi\\Phi^T})=n\\cdot c.\n\\end{eqnarray}\nAs a result, $c=\\frac{N}{n}\\psi^2$.\n\\end{proof}\\par\nIf we let $c = 1$, then we can get $\\|\\bm \\phi_i\\|_2=\\|\\bm \\phi_j\\|_2=\\sqrt{n\/N}$, which coincides with the results of \\cite{Ramin2010} and \\cite{Zahedi201264}.\n\n\\par Before the end of this section, some important discussions are believed to be necessary here.\n\\par Remark 1:\nFurther analysis of the result of Theorem 2 indicates that, when the measurement matrices of the commonly used Compressive Classifier (\\ref{OMPDetct}) are \"tightened\", i.e. row-orthogonalized, then the inequality (\\ref{MainIneq}) becomes equality, and the corresponding error probability will become\n\\begin{eqnarray}\\label{MFeqivalent}\n\\mathcal{P}_E(\\hat{\\bm \\Phi})=\n Q(\\frac{\\|\\bm{\\hat \\Phi(s_1-s_2)}\\|_2}{2 c^{-1\/2}\\cdot \\sigma})\\nonumber \\\\\n= Q(\\frac{\\|\\bm{P_{\\Phi^T} (s_1-s_2)}\\|_2}{2 c^{-1\/2}\\cdot \\sigma})\n\\end{eqnarray}\nwhere $\\bm{P_{\\Phi^T} = \\Phi^T ( \\Phi  \\Phi^T)^{-1} \\Phi}$. The last equality is derived from (\\ref{singular}) and (\\ref{tighten}), which equals, as a matter of fact, the error probability of the General Matched Filter Classifier \\cite{SPComp}\\cite{PHDThesis}:\n\\begin{eqnarray}\n\\hat t_i=\n\\langle\\bm{y},\\bm{(\\Phi \\Phi^T)^{-1}\\Phi}\\bm{s_i}\\rangle-\\frac{1}{2}\\|\\bm{P_{\\Phi^T} s_i} \\|_2^2,\\quad i=1,2. \\label{MFDetct}\n\\end{eqnarray}\nThus Theorem 2 and (\\ref{MFeqivalent}) indicates that equi-norm tight frames will improve the Compressive Classifier (\\ref{OMPDetct}) to the level of General Matched Filter Classifier (\\ref{MFDetct}) in the sense of error probability. Although it is obvious that row-orthogonality (\\ref{TightCond}) sufficiently ensures (\\ref{MFDetct}) to become equivalent as (\\ref{OMPDetct}), the necessity with row-orthogonality, or \"tightness\", of the equivalence between the Compressive Classifier (\\ref{OMPDetct}) and the General Matched Filter Classifier (\\ref{MFDetct}) is not so explicit but demonstrated by Theorem 2 and (\\ref{MFeqivalent}). \n\\par Besides, the error probability (\\ref{MFeqivalent}) coincides with Davenport's \\cite{Davenport06detectionand} and \\cite{SPComp} and Zahedi's \\cite{Ramin2010} and \\cite{Zahedi201264}, where they constrained row-orthogonality $\\bm{\\Phi \\Phi^T = c \\cdot I}$ and set $c = 1$. So Theorem 2 explains the benefits of using the row-orthogonal constraint to do Compressive Classification.\n Similar discussions about the improvement of equi-norm tight frames to oracle estimators can be found in \\cite{UniTightFrame}, which is another good support of the advantage of \"tight\".\n\\par Remark 2:\nAs is mentioned in last section, when there are more than 2 hypotheses, the m-ary ($m>2$) compressive classification problem model becomes:\n\\begin{equation}\\label{CompressHypo_mary}\n\\bm{y} =\n\\left \\{ \\begin{array}{ll}\n\t\\bm{\\Phi}(\\bm{s_1} + \\bm{n}) & \\text{Hypothesis  $H_1$} \\\\\n\t\\bm{\\Phi}(\\bm{s_2} + \\bm{n}) & \\text{Hypothesis  $H_2$} \\\\\n\t\\cdots & \\cdots \\\\\n\t\\bm{\\Phi}(\\bm{s_m} + \\bm{n}) & \\text{Hypothesis  $H_m$} \\\\\n\n\\end{array} \\right. .\n\\end{equation}\nUsing the same Compressive Classifier\n\\begin{eqnarray}\nt_i\n=\\langle\\bm{y},\\bm{\\Phi}\\bm{s_i}\\rangle-\\frac{1}{2}\\langle \\bm{\\Phi s_i},\\bm{\\Phi s_i} \\rangle,\\quad i=1,2,\\cdots ,m .\n\\end{eqnarray}\nThe corresponding error probability will be \n\\begin{eqnarray}\n\\mathcal{P}_E \n=1 - \\mathcal{P}( t_T> t_i,\\forall i \\neq T|H_T) ,\\quad T = 1,2,\\cdots,m.\n\\end{eqnarray}\nCombined with the Union Bound of probability theory, the error probability then satisfies\n\\begin{eqnarray}\n\\mathcal{P}_E \\leq\n\\sum_{i \\neq T}^m Q(\\frac{\\|\\bm{\\Phi(s_T-s_i)}\\|_2^2}{2\\sigma\\|\\bm{\\Phi^T\\Phi(s_T-s_i)}\\|_2}), T = 1,2,\\cdots,m .\\label{mErrorProb}\n\\end{eqnarray}\nThe error probability (\\ref{mErrorProb}) is similar to (\\ref{CompressDetect}) except for the inequality due to the use of union bound. In fact, it may be difficult to get any more accurate result than (\\ref{mErrorProb}), because of the statistical correlation between different $t_i$'s. It may not be persuasive to conclude the error probability's decrease brought by equi-norm tight frames, using the same proof in Theorem 2 in this m-ary scenario. because of the inequality in (\\ref{mErrorProb}); however, simulation results in the next section will demonstrate that equi-norm tight frames are still better in the m-mary ($m>2$) Compressive Classification scenario.\n\\par Remark 3: In comparison with the the work of Zahedi in \\cite{Ramin2010} and \\cite{Zahedi201264}, where Equiangular Tight Frames (ETFs) are proved to have the best worst-case performance among all tight frames (row-orthogonal constrained matrices), we just give the proof that for general under-determined measurement matrices, tightening can bring performance improvement for Compressive Classification. Our job is different from theirs, because all of Zahedi's analysis is based on the constraint that the measurement matrices are tight, or row-orthogonal, and the advantage of Equiangular Tight Frames (ETFs, \\cite{Waldron2009}) is that ETFs have the best worst-case (maximum of the minimum) performance among all tight frames of same dimensions, while our result shows that when arbitrary measurement matrices is \"tightened\", i.e. transformed into equi-norm tight frames, the performance of Compressive Classification will get improved. Nonetheless, the existence and construction of ETFs of some certain dimensions remains an open problem (\\cite{Waldron2009}), while doing row-orthogonalization for arbitrary matrices is very easy and practical. So our results provided a convenient approach to improve the performance of compressive classifiers.\n\\par\n\n\\section{Simulations}\n\\label{sec:simulation}\n\n\\begin{figure}\n\\centering\n\n\n\t\n\t\n\t\n\n\n\n\t\n\t\n\t\n\n\t\\begin{minipage}[htbp]{0.5\\textwidth}\n\t\\centering\n\t\t\\includegraphics[width=3.6in]{MonteCaro_2Hypo.eps}\n\t\t\\caption{Monte-Carlo simulation of 2-ary compressive classification error probability using non-tight frames and tight frames (for $k=1$ sparse signals)}\n\t\t\\label{figure1}\n\t\t\\end{minipage}\n\t\\begin{minipage}[htbp]{0.5\\textwidth}\n\t\\centering\n\t\t\\includegraphics[width=3.6in]{MonteCaro_2Hypok10.eps}\n\t\t\\caption{Monte-Carlo simulation of 2-ary compressive classification error probability using non-tight frames and tight frames (for $k=10$ sparse signals)}\n\t\t\\label{figure2}\n\t\t\\end{minipage}\n\t\n\t\\end{figure}\n\t\\begin{figure}\n\t\t\\begin{minipage}[htbp]{0.5\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=3.6in]{MonteCaro_10Hypo.eps}\n\t\t\\caption{Monte-Carlo simulation of 10-ary compressive classification error probability using non-tight frames and tight frames (for $k=1$ sparse signals)}\n\t\t\\label{figure3}\t\t\n\t\\end{minipage}\\\\\n\t\t\\begin{minipage}[htbp]{0.5\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=3.6in]{MonteCaro_10Hypok10.eps}\n\t\t\\caption{Monte-Carlo simulation of 10-ary compressive classification error probability using non-tight frames and tight frames (for $k=10$ sparse signals)}\n\t\t\\label{figure4}\t\t\n\t\\end{minipage\n\\end{figure}\n\\par In this section the main result of theorem 2 is verified by Monte-Carlo simulations. In the simulation some arbitrary $k=1$ and $k=10$ sparse signals are generated and classified using non-tight frames and tight frames. The Gaussian Random Matrices are chosen to be the non-tight frames, and the row-orthogonalized ones from those random matrices are chosen as the tight frames. Here we choose $N=500$, and the error probabilities of both 2-ary Compressive Classification and 10-ary Compressive Classification are demonstrated in Fig.\\ref{figure1} and Fig.\\ref{figure3} for $k=1$ sparse signals, and Fig.\\ref{figure2} and Fig.\\ref{figure4} for $k=10$ sparse signals, with the number of measurements $n$ ranging from 100 to 450 and signal to noise ratios $\\bm{\\|s_i\\|_2^2}\/\\sigma^2$ from 5 dB to 20 dB. Each error probability is calculated from average of 10000 independent experiments with tight or non-tight measurement matrices.\n\\par The simulation shows that equi-norm tight frames transformed from general Gaussian Random Matrices have better Compressive Classification performance than those non-tight Gaussian Random Matrices within $n$'s whole range, both for 2-ary classification and m-ary ($m>2$) classification scenarios, which is the benefit that \"tightening\" brings.\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\\par This paper deals with the performance improvement of a commonly used Compressive Classifier (\\ref{OMPDetct}). We prove that the transformation to equi-norm tight Frames from arbitrary measurement matrices will reduce the probability of false classification of the commonly used Compressive Classifier, thus improve the classification performance to the level of the General Matched Filter Classifier (\\ref{MFDetct}), which coincides with the row-orthogonal constraint commonly used before. Although there are other proofs that among all equi-norm tight frames the Equiangular Tight Frames (ETFs) achieve best worst-case classification performance, the existence and construction of ETFs of some dimensions is still an open problem.\nAs the construction of equi-norm tight frames from arbitrary matrices is much simple and practical, the conclusion of this paper can also provide a convenient approach to implement an improved measurement matrix for Compressive Classification. \n\n\n\n\n\n\\bibliographystyle{IEEEbib}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\n\nIn many computational settings, one wishes to transmit a $d$-dimensional real-valued vector.\nFor example, in distributed and federated learning scenarios, multiple participants (a.k.a. \\emph{clients}) in distributed SGD send gradients to a parameter server that averages them and updates the model parameters accordingly~\\cite{mcmahan2017communication}.\nIn these applications and others (e.g., traditional machine learning methods such K-Means and power iteration~\\cite{pmlr-v70-suresh17a} or other methods such as geometric monitoring~\\cite{icde2021}), sending \\emph{approximations} of vectors may suffice.  Moreover, the vectors' dimension $d$ is often large (e.g., in neural networks, $d$ can exceed a billion~\\cite{NIPS2012_6aca9700,shoeybi2019megatron,NEURIPS2019_093f65e0}), so sending compressed vectors is appealing.\n\nIndeed, recent works have studied how to send vector approximations using representations that use a small number of bits per entry (e.g.,~\\cite{pmlr-v70-suresh17a,ben2020send,wen2017terngrad,NIPS2017_6c340f25,konevcny2018randomized,caldas2018expanding}). \\new{Further, recent work has shown direct training time reduction from compressing the vectors to one bit per coordinate~\\cite{bai2021gradient}.}\nMost relevant to our work are solutions that address the distributed mean estimation problem. For example, \\cite{pmlr-v70-suresh17a} uses the randomized Hadamard transform followed by stochastic quantization (a.k.a. randomized rounding).\nWhen each of the $n$ clients transmits $O(d)$ bits, their Normalized Mean Squared Error (NMSE) is bounded by $O\\big(\\frac{\\log d}{n}\\big)$.\nThey also show a $O(\\frac{1}{n})$ bound with $O(d)$ bits via variable-length encoding, albeit at a higher computational cost.\nThe sampling method of~\\cite{konevcny2018randomized} yields an $O\\parentheses{\\frac{r\\cdot R}{n}}$ NMSE bound using $d(1+o(1))$ bits \\emph{in expectation}, where $r$ is each coordinate's representation length and $R$ is the normalized average variance of the sent vectors.\nRecently, researchers proposed to use Kashin's representation~\\cite{caldas2018expanding,lyubarskii2010uncertainty,iaab006}. Broadly speaking, it allows representing a $d$-dimensional vector in (a higher) dimension~$\\lambda\\cdot d$ for some $\\lambda>1$ using small coefficients. This results in an $O\\Big({\\frac{\\lambda^2}{(\\sqrt\\lambda -1)^4\\cdot n}}\\Big)$ NMSE bound, \\mbox{where each client transmits $\\lambda \\cdot d(1+o(1))$ bits~\\cite{iaab006}.}\n\\textcolor{black}{A recent work~\\cite{davies2021new} suggested an algorithm where if all clients' vectors have pairwise distances of at most $y\\in\\mathbb R$ (i.e., for any client pair $\\mathfrak c_{1},\\mathfrak c_{2}$, it holds that $\\norm{x_{(\\mathfrak c_1)}-x_{(\\mathfrak c_2)}}_2\\le y$), the resulting MSE is $O(y^2)$ (which is tight with respect to y) using $O(1)$ bits per coordinate on average. This solution provides a stronger MSE bound when vectors are sufficiently close \\mbox{(and thus $y$ is small) but does not improve the worst-case guarantee.}}\n\n\\ifdefined\n\\vbox{We step back and focus on approximating $d$-dimensional vectors \nusing $d(1+o(1))$ bits (e.g., one bit per dimension and a lower order overhead).  \nWe develop novel biased and unbiased compression techniques based on (uniform as well as structured) random rotations in high-dimensional spheres. Intuitively, after a rotation, the coordinates are identically distributed, allowing us to estimate each coordinate with respect to the resulting distribution. \nOur algorithms do not require expensive operations, such as variable-length encoding or computing the Kashin's representation, and are fast and easy to implement.\nWe obtain an $O\\parentheses{\\frac{1}{n}}$ NMSE bound using $d(1+o(1))$ bits, regardless of the coordinates' representation length, improving over previous works. \nEvaluation results indicate that this translates to a consistent \\mbox{improvement over the state of the art in different distributed and federated learning tasks.}}\n\\else\nWe step back and focus on approximating $d$-dimensional vectors \nusing $d(1+o(1))$ bits (e.g., one bit per dimension and a lower order overhead).  \nWe develop novel biased and unbiased compression techniques based on (uniform as well as structured) random rotations in high-dimensional spheres. Intuitively, after a rotation, the coordinates are identically distributed, allowing us to estimate each coordinate with respect to the resulting distribution. \nOur algorithms do not require expensive operations, such as variable-length encoding or computing the Kashin's representation, and are fast and easy to implement.\nWe obtain an $O\\parentheses{\\frac{1}{n}}$ NMSE bound using $d(1+o(1))$ bits, regardless of the coordinates' representation length, improving over previous works. \nEvaluation results indicate that this translates to a consistent \\mbox{improvement over the state of the art in different distributed and federated learning tasks.}\n\\fi\n\n\\section{Problem Formulation and Notation}\n\n\\T{1b~-~Vector Estimation.} \nWe start by formally defining the \\emph{1b~-~vector estimation} problem.\nA sender, called Buffy\\xspace, gets a real-valued vector $x\\in\\mathbb R^d$ and sends it using a $d(1+o(1))$ \nbits message (i.e., asymptotically \\emph{one bit per coordinate}).\nThe receiver, called Angel\\xspace, uses the message to derive an estimate $\\widehat x$ of the original vector $x$.\nWe are interested in the quantity $\\norm{x-\\widehat x}_2^2$, which is the sum of squared errors (SSE), and its expected value, the Mean Squared Error (MSE).  For ease of exposition, we hereafter assume that $x\\neq 0$ as this special case can be handled with one additional bit.\nOur goal is to minimize the \\emph{vector}-NMSE (denoted \\emph{vNMSE}), defined as the normalized \\mbox{MSE, i.e., $\\frac{\\mathbb E\\brackets{\\norm{x-\\widehat x}_2^2}}{\\norm{x}_2^2}$~.}\n\n\n\\T{1b~-~Distributed Mean Estimation.}\nThe above problem naturally generalizes to the \\emph{1b~-~Distributed Mean Estimation} problem. Here, we have a set of $n{\\,\\in\\,}\\mathbb N$ \\emph{clients} and a \\emph{server}. Each client $\\mathfrak c\\in\\set{1,\\ldots,n}$ has its own vector $x_{(\\mathfrak c)} {\\,\\in\\,}\\mathbb R^d$, which it sends using a $d(1{+}o(1))$-bits message to the server. \nThe server then produces an estimate $\\widehat {x_{\\text{avg}}}{\\,\\in\\,}\\mathbb R^d$ of the average $x_{\\text{avg}}=\\frac{1}{n}\\sum_{\\mathfrak  c=1}^n x_{(\\mathfrak c)}$ with the goal of minimizing {its \\emph{NMSE}, defined as the average estimate's MSE normalized by the average norm of the clients' original vectors, i.e., $\\frac{\\mathbb{E}\\brackets{\\norm{x_{\\text{avg}}-\\widehat {x_{\\text{avg}}}}_2^2}}{\\frac{1}{n}\\cdot\\sum_{\\mathfrak  c=1}^n\\norm{x_{(\\mathfrak  c)}}_2^2}$~.}\n\n\n\\T{Notation.} We use the following notation and definitions throughout the paper:\n\n\\textit{Subscripts.} $x_i$ denotes the $i$'th \\emph{coordinate} of the vector $x$, to distinguish it from client $\\mathfrak c$'s vector $x_{(\\mathfrak c)}$. \n\n\\textit{Binary-sign.} For a vector $x\\in \\mathbb R^d$, we denote its binary-sign function as $\\text{sign}(x)$, where $\\text{sign}(x)_i=1$ if $x_i\\ge 0$ and $\\text{sign}(x)_i=-1$ if $x_i < 0$. \n\n\\textit{Unit vector.} For any (non-zero) real-valued vector $x \\in \\mathbb R^d$, we denote its normalized vector by $\\breve{x} = \\frac{x}{\\norm{x}_2}$. That is, $\\breve{x}$ and $x$ has the same direction and it holds that $\\norm{\\breve{x}}_2=1$.\n\n\\textit{Rotation Matrix.} \nA matrix $R\\in\\mathbb R^{d\\times d}$ is a rotation matrix if $R^{T}  R=I$. The set of all rotation matrices is \\mbox{denoted as $\\mathcal O(d)$.\nIt follows that $\\forall R \\in \\mathcal{O}(d){:\\,}det(R) \\in \\set{-1,1}$ and $\\forall x \\in\\mathbb R^d{:\\,} \\norm{x}_2 {=} \\norm{Rx}_2$.}\n\n\\textit{Random Rotation.}\nA random rotation $\\mathcal{R}$ is a distribution over all random rotations in $\\mathcal O(d)$.\nFor ease of exposition, we abuse the notation and given $x\\in\\mathbb R^d$ denote the random rotation of $x$ by $\\mathcal R(x)=Rx$, where $R$ is drawn from $\\mathcal R$. Similarly, $\\mathcal R^{-1}(x)=R^{-1}x=R^Tx$ is the inverse rotation.\n\n\\textit{Rotation Property.}\nA quantity that determines the guarantees of our algorithms is~${\\ensuremath{\\mathcal L}\\xspace_{\\mathcal R, x}^d =\\frac{{\\norm{\\mathcal R(\\breve x)}_1^2}}{d}}$ (note the use of the $L_1$ norm).\nWe show that rotations with high $\\ensuremath{\\mathcal L}\\xspace_{\\mathcal R, x}^d$  values yield better estimates.\n\n\\T{Shared Randomness.}\nWe assume that Buffy\\xspace and Angel\\xspace have access to shared randomness, e.g., by agreeing on a common PRNG seed. Shared randomness is studied both in communication complexity (e.g., \\cite{newman1991private}) and in communication reduction in machine learning systems (e.g., \\cite{pmlr-v70-suresh17a,ben2020send}).\nIn our context, it means that \\mbox{Buffy\\xspace and Angel\\xspace can generate the same random rotations without communication.}\n\n\\section{The \\mbox{DRIVE}\\xspace Algorithm}\n\n\\ifdefined\nWe start by presenting \\mbox{DRIVE}\\xspace (Deterministically RoundIng randomly rotated VEctors), a novel 1b~-~Vector Estimation algorithm.\nFirst, we show how to minimize \\mbox{DRIVE}\\xspace's vNMSE. Then, we show how to make \\mbox{DRIVE}\\xspace unbiased, extending it to the 1b~-~Distributed \\mbox{Mean Estimation problem.}\n\nIn \\mbox{DRIVE}\\xspace, Buffy\\xspace uses shared randomness to sample a rotation matrix $R\\sim\\mathcal R$ and rotates the vector $x\\in\\mathbb R^d$ by computing $\\mathcal R(x)=R x$.\n\\else\nWe start by presenting \\mbox{DRIVE}\\xspace (Deterministically RoundIng randomly rotated VEctors), a novel 1b~-~Vector Estimation algorithm.\nLater, we extend \\mbox{DRIVE}\\xspace to the 1b~-~Distributed Mean Estimation problem.\nIn \\mbox{DRIVE}\\xspace, Buffy\\xspace uses shared randomness to sample a rotation matrix $R\\sim\\mathcal R$ and rotates the vector $x\\in\\mathbb R^d$ by computing $\\mathcal R(x)=R x$.\n\\fi\nBuffy\\xspace then calculates $S$, a scalar quantity we explain below. Buffy\\xspace then sends $\\big(S,\\text{sign}(\\mathcal R(x))\\big)$ to Angel\\xspace. \nAs we discuss later, sending $\\big(S,\\text{sign}(\\mathcal R(x))\\big)$ requires $d(1+o(1))$ bits.\nIn turn, Angel\\xspace computes $\\widehat {\\mathcal R(x)}=S\\cdot\\text{sign}(\\mathcal R(x)) \\in\\set{-S,+S}^d$.\nIt then uses the shared randomness to generate the same rotation matrix\nand employs the inverse {rotation, i.e., estimates $\\widehat x = \\mathcal R^{-1}(\\widehat {\\mathcal R(x)})$.\nThe pseudocode of \\mbox{DRIVE}\\xspace appears in Algorithm~\\ref{code:alg1}.}\n\n\nThe properties of \\mbox{DRIVE}\\xspace depend on the rotation $\\mathcal R$ and the \\emph{scale parameter} $S$.\nWe consider both uniform rotations, that provide stronger guarantees, and structured rotations that are orders of magnitude faster to compute.  \nAs for the scale $S = S({x,R})$, its exact formula determines the characteristics of \\mbox{DRIVE}\\xspace's estimate, e.g., having minimal vNMSE or being unbiased. \nThe latter allows us to apply \\mbox{DRIVE}\\xspace to the 1b~-~Distributed Mean Estimation (Section~\\ref{subsec:drive_dme}) and get an NMSE that decreases proportionally to the number of clients.  \n\n\\begin{algorithm}[t]\n\\caption{~\\mbox{DRIVE}\\xspace}\n\\label{code:alg1}\n\\begin{multicols}{2}\n\\begin{algorithmic}[1]\n  \\Statex \\hspace*{-4mm}\\textbf{Buffy\\xspace:}\n \n  \\State Compute $\\mathcal R(x),\\  S$.\\textcolor{white}{$\\big($}\n \n  \\State Send $\\big(S,\\text{sign}(\\mathcal R(x))\\big)$ to Angel\\xspace.\\textcolor{white}{$\\widehat {\\mathcal R(x)}$}\n\\end{algorithmic}\n\\columnbreak\n\\begin{algorithmic}[1]\n\\Statex \\hspace*{-4mm}\\textbf{Angel\\xspace:}\n\\State Compute $\\widehat {\\mathcal R(x)}=S\\cdot\\text{sign}\\big(\\mathcal R(x)\\big)$.\n\\State Estimate $\\widehat x = \\mathcal R^{-1}\\big(\\widehat {\\mathcal R(x)}\\big)$.\n\\end{algorithmic}\n\\end{multicols}\n\\end{algorithm}\nWe now prove a general result on the SSE of \\mbox{DRIVE}\\xspace that applies to any random rotation $\\mathcal R$ and any vector $x\\in\\mathbb R^d$. \nIn the following sections, we use this result to obtain the vNMSE \\mbox{when considering specific rotations and scaling methods as well as analyzing their guarantees.}\n\\begin{theorem}\\label{thm:theoreticalAlg}\nThe SSE of \\mbox{DRIVE}\\xspace is: $\\norm{x-\\widehat x}_2^2=\\norm{x}_2^2-2 \\cdot S \\cdot\\norm{\\mathcal R(x)}_1 + d \\cdot S^2$~.\n\\end{theorem}\n\\begin{proof}\nThe SSE in estimating $\\mathcal R(x)$ using $\\widehat {\\mathcal R(x)}$ equals that of estimating $x$ using $\\widehat x$. Therefore,\n\\ifdefined\n\\begin{multline}\n\\norm{x-\\widehat x}_2^2 = \\norm{\\mathcal R(x-\\widehat x)}_2^2 = \\norm{\\mathcal R(x)-\\mathcal R(\\widehat x)}_2^2 = \\norm{{\\mathcal R(x)}-\\widehat {\\mathcal R(x)}}_2^2\\\\\n= \\norm{\\mathcal R(x)}_2^2  - 2 \\angles{\\mathcal R(x),\\widehat {\\mathcal R(x)}} + {\\norm{\\widehat {\\mathcal R(x)}}_2^2} = \\norm{x}_2^2-2\\angles{\\mathcal R(x),\\widehat {\\mathcal R(x)}} + \\norm{\\widehat {\\mathcal R(x)}}_2^2.\\label{eq:thm_main}\n\\end{multline}\n\\else\n\\\\\n$\\hspace*{0.0cm}\\norm{x-\\widehat x}_2^2 = \\norm{\\mathcal R(x-\\widehat x)}_2^2 = \\norm{\\mathcal R(x)-\\mathcal R(\\widehat x)}_2^2 = \\norm{{\\mathcal R(x)}-\\widehat {\\mathcal R(x)}}_2^2$\\\\\n$\\hspace*{1.0cm}= \\norm{\\mathcal R(x)}_2^2  - 2 \\angles{\\mathcal R(x),\\widehat {\\mathcal R(x)}} + {\\norm{\\widehat {\\mathcal R(x)}}_2^2} = \\norm{x}_2^2-2\\angles{\\mathcal R(x),\\widehat {\\mathcal R(x)}} + \\norm{\\widehat {\\mathcal R(x)}}_2^2.\\refstepcounter{equation}\\hfill\\mbox{(\\theequation)}\\label{eq:thm_main}$\n\\fi\nNext, we have that,\n\\ifdefined\n\\begin{align}\n    \\angles{\\mathcal R(x),\\widehat {\\mathcal R(x)}} &= \\sum_{i=1}^d \\mathcal R(x)_i\\cdot \\widehat {\\mathcal R(x)}_i =  S \\cdot\\sum_{i=1}^d \\mathcal R(x)_i\\cdot \\text{sign}\\big(\\mathcal R(x)_i\\big) = S \\cdot\\norm{\\mathcal R(x)}_1\\ ,\\label{eq:thm_inner_product_1}\\\\\n\\norm{\\widehat {\\mathcal R(x)}}_2^2 &= \\sum_{i=1}^d \\widehat {\\mathcal R(x)}_i^2 = d\\cdot S^2~.\\label{eq:thm_inner_product_2}\n\\end{align}\n\\else\n\n$\\angles{\\mathcal R(x),\\widehat {\\mathcal R(x)}} = \\sum_{i=1}^d \\mathcal R(x)_i\\cdot \\widehat {\\mathcal R(x)}_i =  S \\cdot\\sum_{i=1}^d \\mathcal R(x)_i\\cdot \\text{sign}\\big(\\mathcal R(x)_i\\big) = S \\cdot\\norm{\\mathcal R(x)}_1\\ ,\\refstepcounter{equation}\\hfill\\mbox{(\\theequation)}\\label{eq:thm_inner_product_1}$\\\\\n$\\hspace*{0.85cm}\\norm{\\widehat {\\mathcal R(x)}}_2^2 = \\sum_{i=1}^d \\widehat {\\mathcal R(x)}_i^2 = d\\cdot S^2~.\\refstepcounter{equation}\\hfill\\mbox{(\\theequation)}\\label{eq:thm_inner_product_2}$\n\\fi\n\nSubstituting Eq.~\\eqref{eq:thm_inner_product_1} and Eq.~\\eqref{eq:thm_inner_product_2} in Eq.~\\eqref{eq:thm_main} yields the result.\n\\end{proof}\n\\section{\\mbox{DRIVE}\\xspace With a Uniform Random Rotation}\n\nWe first consider the thoroughly studied uniform random rotation (e.g., ~\\cite{mezzadri2006generate,wedderburn1975generating,heiberger1978generation,stewart1980efficient,tanner1982remark}), which we denote by $\\mathcal R_U$. The sampled matrix is denoted by $R_U\\sim\\mathcal R_U$, that is, $\\mathcal R_U(x)=R_U\\cdot x$.\nAn appealing property of a uniform random rotation is that, as we show later, it admits a scaling that results in a low constant vNMSE even with unbiased estimates.\n\n\n\\subsection{1b~-~Vector Estimation}\n\nUsing Theorem \\ref{thm:theoreticalAlg}, we obtain the following result. The result holds for any rotation, including $\\mathcal R_U$.\n\\begin{lemma}\\label{cor:biased_alg1_vNMSE}\nFor any $x\\in\\mathbb R^d$, \\mbox{DRIVE}\\xspace's SSE is minimized by $S=\\frac{\\norm{\\mathcal R(x)}_1}{d}$ (that is, $S=\\frac{\\norm{Rx}_1}{d}$ is determined after $R\\sim\\mathcal R$ is sampled). This yields a vNMSE of $\\frac{\\mathbb E\\brackets{\\norm{x-\\widehat x}_2^2}}{\\norm{x}_2^2}=1 - \\mathbb E\\brackets{\\ensuremath{\\mathcal L}\\xspace_{\\mathcal R,x}^d}$.\n\\end{lemma}\n\\begin{proof}\nBy Theorem \\ref{thm:theoreticalAlg}, to minimize the SSE we require\n\\ifdefined\n\\begin{align*}\n    \\hspace*{.4cm}\\frac{\\partial}{\\partial S}\\big({\\norm{x}_2^2-2 \\cdot S \\cdot\\norm{\\mathcal R(x)}_1 + d \\cdot S^2}\\big) = -2 \\cdot \\norm{\\mathcal R(x)}_1 + 2 \\cdot d \\cdot S = 0~,\n\\end{align*}\n\\else\n\\vspace*{1mm}\\\\\n$\\hspace*{0.9cm}\\frac{\\partial}{\\partial S}\\big({\\norm{x}_2^2-2 \\cdot S \\cdot\\norm{\\mathcal R(x)}_1 + d \\cdot S^2}\\big) = -2 \\cdot \\norm{\\mathcal R(x)}_1 + 2 \\cdot d \\cdot S = 0~,$\\\\\\vspace*{1mm}\n\\fi\nleading to $S=\\frac{\\norm{\\mathcal R(x)}_1}{d}$. Then, the SSE of \\mbox{DRIVE}\\xspace becomes:\n\\ifdefined\n{\n\\begin{multline*}\n\\norm{x-\\widehat x}_2^2 = \\norm{x}_2^2-2\\cdotS \\cdot\\norm{\\mathcal R(x)}_1+d\\cdot S^2 = \n\\norm{x}_2^2-2\\cdot\\frac{\\norm{\\mathcal R(x)}_1^2}{d} + d \\cdot \\frac{\\norm{\\mathcal R(x)}_1^2}{d^2}\n \\\\=\n\\norm{x}_2^2 - \\frac{\\norm{\\mathcal R(x)}_1^2}{d}\n=\n\\norm{x}_2^2 - \\frac{\\norm{x}_2^2 \\cdot \\norm{\\mathcal R(\\breve x)}_1^2}{d}\n= \\norm{x}_2^2 \\big(1-\\frac{\\norm{\\mathcal R(\\breve x)}_1^2}{d}\\,\\big)~.\n\\end{multline*}\n}\n\\else\n\\\\\n$\\hspace*{0.9cm}\\norm{x-\\widehat x}_2^2 = \\norm{x}_2^2-2\\cdotS \\cdot\\norm{\\mathcal R(x)}_1+d\\cdot S^2 = \n\\norm{x}_2^2-2\\cdot\\frac{\\norm{\\mathcal R(x)}_1^2}{d} + d \\cdot \\frac{\\norm{\\mathcal R(x)}_1^2}{d^2}$\\\\\n$\\hspace*{2.33cm}=\n\\norm{x}_2^2 - \\frac{\\norm{\\mathcal R(x)}_1^2}{d}\n=\n\\norm{x}_2^2 - \\frac{\\norm{x}_2^2 \\cdot \\norm{\\mathcal R(\\breve x)}_1^2}{d}\n= \\norm{x}_2^2 \\big(1-\\frac{\\norm{\\mathcal R(\\breve x)}_1^2}{d}\\,\\big)~.$\n\n\\fi\n{Thus, the normalized SSE is $\\frac{\\norm{x-\\widehat x}_2^2}{\\norm{x}_2^2}=1{-}\\ensuremath{\\mathcal L}\\xspace_{\\mathcal R,x}^d$. Taking expectation yields the result. \\qedhere}\n\\end{proof}\nInterestingly, for the uniform random rotation, $\\ensuremath{\\mathcal L}\\xspace_{\\mathcal R_U,x}^d$ follows the same distribution for all $x$. \nThis is because, by the definition of $\\mathcal R_U$, it holds that $\\mathcal R_U(\\breve x)$ is distributed uniformly over the unit sphere for any $x$.\nTherefore \\mbox{DRIVE}\\xspace's vNMSE depends only on the dimension $d$.\nWe next analyze the vNMSE attainable by the best possible $S$, as given in Lemma~\\ref{cor:biased_alg1_vNMSE},  when the algorithm uses $\\mathcal R_U$ and is not required to be unbiased. In particular, we state the following theorem whose\n\\ifdefined\n \\mbox{proof appears in Appendix \\ref{app:biased_drive_vNMSE}. }\n\\else\n proof appears in Appendix \\ref{app:biased_drive_vNMSE} \\mbox{(all appendices appear in the Supplementary Material and the extended paper version~\\cite{vargaftik2021drive}). }\n\\fi\n\\begin{restatable}{theorem}{biaseddrivevNMSE}\\label{thm:biased_drive_vNMSE}\nFor any $x \\in \\mathbb R^d$, the vNMSE of \\mbox{DRIVE}\\xspace with $S=\\frac{\\norm{\\mathcal R_U(x)}_1}{d}$ is\n$\\parentheses{1 - \\frac{2}{\\pi}} \\parentheses{ {1-\\frac{1}{d}}}$~.\n\\end{restatable}\n\n\n\n\\subsection{1b~-~Distributed Mean Estimation}\\label{subsec:drive_dme}\nAn appealing property of \\mbox{DRIVE}\\xspace with a uniform random rotation, established in this section, is that with a proper scaling parameter $S$, the estimate is unbiased. \nThat is, for any $x \\in \\mathbb R^d$, our scale guarantees that $\\mathbb E\\brackets{\\widehat x} = x$.\nUnbiasedness is useful when generalizing to the Distributed Mean Estimation problem. \nIntuitively, when $n$ clients send their vectors, any biased algorithm would result in an NMSE that may not decrease with respect to $n$. For example, if they have the same input vector, the bias would remain after averaging. Instead, an unbiased encoding algorithm has the property that when all clients \\mbox{act (e.g., use different PRNG seeds) independently, the NMSE decreases proportionally to $\\frac{1}{n}$.}\n\nAnother useful property of uniform random rotation is that its distribution is unchanged when composed with other rotations. We use it in \\mbox{the following theorem's proof, given in Appendix~\\ref{app:drive_is_unbiased}.}\n\n\\begin{restatable}{theorem}{driveisunbiased}\\label{theorem:drive_is_unbised}\nFor any $x \\in \\mathbb R^d$, set $S = \\frac{\\norm{x}_2^2}{\\norm{\\mathcal R_U(x)}_1}$. Then \\mbox{DRIVE}\\xspace satisfies $\\mathbb{E} [\\widehat x] = x$.\n\\end{restatable}\n\n\\mbox{Now, we proceed to obtain vNMSE guarantees for \\mbox{DRIVE}\\xspace's unbiased estimate.}\n\\begin{lemma}\\label{cor:alg1_unbiased_vNMSE}\nFor any $x\\in\\mathbb R^d$, \\mbox{DRIVE}\\xspace with $S = \\frac{\\norm{x}_2^2}{\\norm{\\mathcal R_U(x)}_1}$ has a vNMSE of \n$\\mathbb E\\brackets{{\\frac{1}{\\ensuremath{\\mathcal L}\\xspace_{\\mathcal R_U,x}^d}}} {-} 1$~.\n\\begin{proof}\nBy Theorem~\\ref{thm:theoreticalAlg}, the SSE of the algorithm satisfies:\n\\ifdefined\n{\n\\begin{equation*}\n\\begin{aligned}\n&\\norm{x}_2^2-2\\cdotS \\cdot\\norm{R_U \\cdot x}_1+d\\cdot S^2 = \n\\norm{x}_2^2-2\\cdot  \\frac{\\norm{x}_2^2}{\\norm{R_U \\cdot x}_1} \\cdot\\norm{R_U \\cdot x}_1+ d \\cdot \\parentheses{\\frac{\\norm{x}_2^2}{\\norm{R_U \\cdot x}_1}}^2\n \\\\\n&= d \\cdot \\parentheses{\\frac{\\norm{x}_2^2}{\\norm{x}_2\\norm{R_U \\cdot \\breve x}_1}}^2 {-} \\norm{x}_2^2 = d \\cdot \\frac{\\norm{x}_2^2}{\\norm{R_U \\cdot \\breve x}_1^2} {-} \\norm{x}_2^2 = \\norm{x}_2^2 \\cdot \\parentheses{\\parentheses{\\frac{d}{\\norm{R_U \\cdot \\breve x}_1^2}} {-} 1}~.\n\\end{aligned}    \n\\end{equation*}\n}\n\\else\n\\begin{equation*}\n\\small\n\\begin{aligned}\n&\\norm{x}_2^2-2\\cdotS \\cdot\\norm{R_U \\cdot x}_1+d\\cdot S^2 = \n\\norm{x}_2^2-2\\cdot  \\frac{\\norm{x}_2^2}{\\norm{R_U \\cdot x}_1} \\cdot\\norm{R_U \\cdot x}_1+ d \\cdot \\parentheses{\\frac{\\norm{x}_2^2}{\\norm{R_U \\cdot x}_1}}^2\n \\\\\n&= d \\cdot \\parentheses{\\frac{\\norm{x}_2^2}{\\norm{x}_2\\norm{R_U \\cdot \\breve x}_1}}^2 {-} \\norm{x}_2^2 = d \\cdot \\frac{\\norm{x}_2^2}{\\norm{R_U \\cdot \\breve x}_1^2} {-} \\norm{x}_2^2 = \\norm{x}_2^2 \\cdot \\parentheses{\\parentheses{\\frac{d}{\\norm{R_U \\cdot \\breve x}_1^2}} {-} 1}~.\n\\end{aligned}    \n\\end{equation*}\n\\fi\n\\mbox{Normalizing by $\\norm{x}_2^2$ and taking expectation over $R_U$ concludes the proof.}\n\\end{proof}\n\\end{lemma}\nOur goal is to derive an upper bound on the above expression and thus upper-bound the vNMSE. Most importantly, we show that even though the estimate is unbiased and we use only a single bit per coordinate, the vNMSE does not increase with the dimension and is bounded by a small constant. In particular, in Appendix \\ref{app:anbiased_drive_large_deviation}, we prove the following:\n\n\\begin{restatable}{theorem}{vNMSEofunbiaseddrive}\\label{thm:vNMSEofunbiaseddrive}\nFor any $x \\in\\ \\mathbb R^d$, the vNMSE of \\mbox{DRIVE}\\xspace with $S = \\frac{\\norm{x}_2^2}{\\norm{\\mathcal R_U(x)}_1}$\nsatisfies:\\\\\n\\mbox{$(i)$ For all $d \\ge 2$, it is at most $2.92$. $(ii)$ For all $d \\ge 135$, it is at most $\\frac{\\pi}{2} - 1 + \\sqrt{\\frac{{(6\\pi^3-12\\pi^2)}\\cdot\\ln d+1}{d}}$.}\n\\end{restatable}\n\nThis theorem yields strong bounds on the vNMSE. For example, the vNMSE is lower than $1$ for $d \\ge 4096$ and lower than $0.673$ for $d \\ge 10^5$. Finally, we obtain the following corollary,\n\\begin{corollary}\\label{cor:deviation_res_3}\nFor any $x\\in\\mathbb R^d$, the vNMSE tends to $\\frac{\\pi}{2} - 1 \\approx 0.571$ as $d\\to\\infty$~. \n\\end{corollary}\n\nRecall that \\mbox{DRIVE}\\xspace's above scale $S$ is a function of both $x$ and the sampled $R_U$. An alternative approach is to \\emph{deterministically} set $S$ to \n$\\frac{\\norm{x}_2^2}{\\mathbb E\\brackets{\\norm{\\mathcal R_U(x)}_1}}$.\nAs we prove in Appendix~\\ref{app:l1_expected_value}, the resulting scale is $\\frac{\\norm{x}_2\\cdot (d-1)\\cdot \\mathrm{B}(\\frac{1}{2},\\frac{d-1}{2})}{2d}$, where $\\mathrm{B}$ is the Beta function. \nInterestingly, this scale no longer depends on $x$ but only on its norm.\nIn the appendix, we also prove that the resulting vNMSE is bounded by $\\frac{\\pi}{2} - 1$ \\emph{for any $d$}. \nIn practice, we find that the benefit is marginal.\n\nFinally, with a vNMSE guarantee for the unbiased estimate by \\mbox{DRIVE}\\xspace, we obtain the following key result for the\n1b~-~Distributed Mean Estimation problem, whose proof appears in Appendix~\\ref{app:dme}.\nWe note that this result guarantees (e.g., see~\\cite{beznosikov2020biased}) that distributed SGD, where the participants' gradients are compressed with \\mbox{DRIVE}\\xspace, converges at the same asymptotic rate as without compression.\n\n\n\n\\begin{restatable}{theorem}{cordme}\\label{cor:dme}\nAssume $n$ clients, each with its own vector $x_{(\\mathfrak  c)}{\\,\\in\\,}\\mathbb R^d$. Let each client independently sample $R_{U,\\mathfrak  c}{\\,\\sim\\,}\\mathcal R_U$ and set its scale to $\\frac{\\norm{x_{(\\mathfrak c)}}_2^2}{\\norm{R_{U,\\mathfrak  c}\\cdot x_{(\\mathfrak  c)}}_1}$. Then, the server average estimate's NMSE satisfies:\n$\\frac{\\mathbb{E}\\brackets{\\norm{x_{\\text{avg}}-\\widehat {x_{\\text{avg}}}}_2^2}}{\\frac{1}{n}{\\cdot}\\sum_{\\mathfrak  c=1}^n\\norm{x_{(\\mathfrak  c)}}_2^2} {\\,=\\,} \\frac{\\mathit{vNMSE}}{n}$, {where \\textit{vNMSE} is given by Lemma~\\ref{cor:alg1_unbiased_vNMSE} and is bounded by Theorem~\\ref{thm:vNMSEofunbiaseddrive}.} \n\\end{restatable}\nTo the best of our knowledge, \\mbox{DRIVE}\\xspace is the first algorithm with a provable NMSE of $O(\\frac{1}{n})$ for the 1b~-~Distributed Mean Estimation problem (i.e., with $d(1+o(1))$ bits). \nIn practice, we use only $d+O(1)$ bits to implement \\mbox{DRIVE}\\xspace. We use the $d(1+o(1))$ notation to ensure compatibility with the theoretical results; see Appendix \\ref{app:MessageRepresentationLength} for a discussion.\n\n\n\n\\section{Reducing the vNMSE with \\mbox{DRIVE$^+$}\\xspace}\\label{sec:drive_plus}\n\nTo reduce the vNMSE further, we introduce the \\mbox{DRIVE$^+$}\\xspace algorithm.\nIn \\mbox{DRIVE$^+$}\\xspace, we also use a scale parameter, denoted $\\sc=\\sc(x, R)$ to differentiate it from the scale $S$ of \\mbox{DRIVE}\\xspace.\nHere, instead of reconstructing the rotated vector in a symmetric manner, i.e., $\\widehat {\\mathcal R(x)} \\in S \\cdot \\{-1,1\\}^d$, we have that $\\widehat {\\mathcal R(x)} \\in \\sc \\cdot \\{ c_1, c_2\\}^d$ where  $c_1,c_2$ are computed using K-Means clustering with $K=2$ over the $d$ entries of the rotated vector $\\mathcal R(x)$. \nThat is, $c_1,c_2$ are chosen to minimize the SSE over any choice of two values. \nThis does not increase the (asymptotic) time complexity over the random rotations considered in this paper as solving K-Means for the special case of one-dimensional data is deterministically solvable in $O(d \\log d)$ (e.g.,~\\cite{gronlund2017fast}).\nNotice that \\mbox{DRIVE$^+$}\\xspace still requires $d(1+o(1))$ bits as we communicate $\\parentheses{\\sc \\cdot c_1, \\sc \\cdot c_2}$ and a single bit per coordinate, indicating its nearest centroid.\nWe defer the pseudocode and analyses of \\mbox{DRIVE$^+$}\\xspace to Appendix~\\ref{app:drive_plus_app}. We show that with proper scaling, for both the 1b~-~Vector Estimation and 1b~-~Distributed Mean Estimation \\mbox{problems, \\mbox{DRIVE$^+$}\\xspace yields guarantees that are at least as strong as those of \\mbox{DRIVE}\\xspace.}\n\n\n\\section{\\mbox{DRIVE}\\xspace with a Structured Random Rotation}\\label{sec:Hadamard}\n\nUniform random rotation generation usually relies on QR factorization (e.g., see \\cite{pytorchqrfact}), which requires $O(d^3)$ time and $O(d^2)$ space.  \nTherefore, uniform random rotation can only be used in practice to rotate low-dimensional vectors. This is impractical for neural network architectures with many millions of parameters.\nTo that end, we continue to analyze \\mbox{DRIVE}\\xspace and \\mbox{DRIVE$^+$}\\xspace with the (randomized) Hadamard transform, a.k.a. \\emph{structured} random rotation~\\cite{pmlr-v70-suresh17a,ailon2009fast},  that admits a fast \\emph{in-place}, \n\\ifdefined\n\\mbox{parallelizable, $O(d\\log d)$ time implementation~\\cite{fino1976unified,uberHadamard}. We start with a few definitions.}\n\\else\n\\mbox{parallelizable, $O(d\\log d)$ time implementation~\\cite{fino1976unified,uberHadamard,openSource}. We start with a few definitions.}\n\\fi\n\n\\begin{definition}\nThe Walsh-Hadamard matrix (\\cite{horadam2012Hadamard}) $H_{2^k}\\in\\{+1,-1\\}^{2^{k}\\times 2^{k}}$ is recursively defined via: \\mbox{ \n$ \\small\nH_{2^k}{=} \\begin{pmatrix}\nH_{2^{k-1}} & H_{2^{k-1}} \\\\\nH_{2^{k-1}} & -H_{2^{k-1}}\n\\end{pmatrix} \n$ and $H_1 {=} \\begin{pmatrix} 1 \\end{pmatrix}$.\nAlso, $(\\frac{1}{\\sqrt d}H) \\cdot (\\frac{1}{\\sqrt d}H)^T {=} I$ and $\\mathit{det}(\\frac{1}{\\sqrt d}H) \\in [-1,1]$.} \n\\end{definition}\n\n\\ifdefined\n\\vbox{\n\\fi\n\\begin{definition}\nLet $R_H$ denote the rotation matrix $\\frac{HD}{\\sqrt d}\\in\\mathbb R^{d\\times d}$, where $H$ is a Walsh-Hadamard matrix and $D$ is a diagonal matrix whose diagonal entries are i.i.d. Rademacher random variables (i.e., taking values uniformly in $\\pm 1$). \nThen $\\mathcal R_H(x)=R_H \\cdot x =  \\frac{1}{\\sqrt d} H \\cdot (x_1 \\cdot D_{11}, \\dots, x_d \\cdot D_{dd})^T$ is the randomized Hadamard transform of $x$ and $\\mathcal R_H^{-1}(x)=R^T_H \\cdot x = \\frac{DH}{\\sqrt d} \\cdot x$ is the inverse transform\n\\end{definition}\n\\ifdefined\n}\n\\fi\n\n\n\n\\subsection{1b~-~Vector Estimation}\n\nRecall that the vNMSE of \\mbox{DRIVE}\\xspace, when minimized using $S=\\frac{\\norm{\\mathcal R (x)}_1}{d}$, is  $1 - \\mathbb E\\brackets{\\ensuremath{\\mathcal L}\\xspace_{\\mathcal R,x}^d}$ (see Lemma~\\ref{cor:biased_alg1_vNMSE}). We now bound this quantity of \\mbox{DRIVE}\\xspace with a structured random rotation.\n\\begin{lemma}\\label{lem:hadamard_biased}\nFor any dimension $d\\geq 2$ and vector $x \\in \\mathbb R^d$, the vNMSE of \\mbox{DRIVE}\\xspace with a structured random rotation and scale $S=\\frac{\\norm{\\mathcal R_H (x)}_1}{d}$ is: $1 - \\mathbb E\\brackets{\\ensuremath{\\mathcal L}\\xspace_{\\mathcal R_H,x}^d} \\le \\frac{1}{2}$.\\vspace{-1mm}\n\\end{lemma}\n\\begin{proof}\nObserve that for all $i$, \n$\\mathbb E\\brackets{\\abs{\\mathcal R_H(x)_i}} = \\mathbb E\\big[\\big|{\\sum_{j=1}^d \\frac{x_j}{\\sqrt{d}} H_{ij} D_{jj}}\\big|\\big]$. \nSince $\\set{H_{ij} D_{jj} \\mid j\\in[d]}$ are i.i.d. Rademacher random variables we can use the Khintchine inequality~\\cite{khintchine1923dyadische,szarek1976best} which implies that  $\\frac{1}{\\sqrt{2d}} \\cdot \\norm x_2\n\\le \\mathbb E\\brackets{\\abs{\\mathcal R_H(x)_i}} \\le \\frac{1}{\\sqrt{d}} \\cdot \\norm x_2$\n(see~\\cite{filmus2012khintchine,latala1994best} for simplified proofs).\nWe conclude that: \n\\ifdefined\n{\n\\begin{equation*}\n\\begin{aligned}\n\\mathbb E\\brackets{\\ensuremath{\\mathcal L}\\xspace_{\\mathcal R_H,x}^d} = \\frac{1}{d} \\cdot \\mathbb E\\brackets{ \\norm{\\mathcal R_H(\\breve x)}_1^2} \\ge \\frac{1}{d} \\cdot \\mathbb E\\brackets{ \\norm{\\mathcal R_H(\\breve x)}_1}^2 \\ge \\frac{1}{d} \\cdot \\Big(\\sum_{i=1}^d \\frac{1}{\\sqrt{2d}}\\Big)^2 = \\frac{1}{2}~.\n\\end{aligned}    \n\\end{equation*}\n}\n\\else\n\\\\\n\\hspace*{1.2cm}\n$\\mathbb E\\brackets{\\ensuremath{\\mathcal L}\\xspace_{\\mathcal R_H,x}^d} = \\frac{1}{d} \\cdot \\mathbb E\\brackets{ \\norm{\\mathcal R_H(\\breve x)}_1^2} \\ge \\frac{1}{d} \\cdot \\mathbb E\\brackets{ \\norm{\\mathcal R_H(\\breve x)}_1}^2 \\ge \\frac{1}{d} \\cdot (\\sum_{i=1}^d \\frac{1}{\\sqrt{2d}})^2 = \\frac{1}{2}~.$\\\\\n\\fi\nThis bound is sharp since for $d\\ge 2$ we have that \n$\\ensuremath{\\mathcal L}\\xspace_{\\mathcal R_H,x}^d = \\frac{1}{2}$ for $x=(\\frac{1}{\\sqrt 2}, \\frac{1}{\\sqrt 2},0,\\ldots,0)^T$~.\n\\end{proof}\n\\vspace{-1mm}\nObserve that unlike for the uniform random rotation, $\\mathbb{E}\\brackets{\\ensuremath{\\mathcal L}\\xspace_{\\mathcal R_H,x}^d}$ depends on $x$. We also note that this bound of $\\frac{1}{2}$ applies to \\mbox{DRIVE$^+$}\\xspace (with scale $\\sc=1$) as we show in Appendix~\\ref{app:1bDrive+}.\n\n\\subsection{1b~-~Distributed Mean Estimation}\\label{sec:dme_hadamard_subsec}\n\nFor an arbitrary $x \\in \\mathbb R^d$ and $\\mathcal R$, \nand in particular for $\\mathcal{R}_H$, \nthe estimates of \\mbox{DRIVE}\\xspace cannot be made unbiased.  For example,\nfor $x=(\\frac{2}{3}, \\frac{1}{3})^T$ we have that $\\text{sign}(\\mathcal R_H(x))=(D_{11},D_{11})^T$ and thus $\\widehat{\\mathcal R_H(x)} = S\\cdot (D_{11},D_{11})^T$. \nThis implies that $\\widehat x = \\mathcal R_H^{-1}(\\widehat{\\mathcal R_H(x)}) = \\frac{1}{\\sqrt 2} \\cdot D\\cdot H\\cdotS\\cdot (D_{11},D_{11})^T=\\sqrt 2 \\cdot S\\cdot D\\cdot (D_{11}, 0)^T =$ $\\sqrt 2 \\cdot S\\cdot (D_{11}^2,0)^T=(\\sqrt 2 \\cdot S ,0)^T$.  \\mbox{Therefore, $\\mathbb E[\\widehat x]\\neq x$ regardless of the scale.}\n\n\nNevertheless, we next provide evidence for why when the input vector is high dimensional and admits finite moments, a structured random rotation performs similarly to a uniform random rotation, yielding all the appealing aforementioned properties. Indeed, it is a common observation that the distribution of machine learning workloads and, in particular, neural network gradients are governed by such distributions (e.g., lognormal \\cite{chmiel2020neural} or normal \\cite{banner2018post,ye2020accelerating}).\n\nWe seek to show that at high dimensions, the distribution of $\\mathcal R_H(x)$ is sufficiently similar to that of $\\mathcal R_U(x)$. \nBy definition, the distribution of $\\mathcal R_U(x)$ is that of a uniformly at random distributed point on a sphere. \nPrevious studies of this distribution for high dimensions (e.g., \\cite{spruill2007asymptotic,diaconis1987dozen,rachev1991approximate,stam1982limit}) have shown that individual coordinates of $\\mathcal R_U(x)$ converge to the same normal distribution and that these coordinates are ``weakly'' dependent in the sense that the joint distribution of every $O(1)$-sized subset of coordinates is similar to that of independent normal variables for large $d$. \n\nWe hereafter assume that $x=(x_1, \\ldots, x_d)$, where the $x_i$s are i.i.d. and that $\\mathbb{E}[x_j^2]=\\sigma^2$ and $\\mathbb{E}[\\abs{x_j}^3]=\\rho < \\infty$ for all $j$. We show that $\\mathcal R_H (x)_i$ converges to the same normal distribution for all $i$.\nLet $F_{i,d}(x)$ be the cumulative distribution function (CDF) of $\\frac{1}{\\sigma} \\cdot \\mathcal R_H (x)_i$ and $\\Phi$ be the CDF of the standard normal \\mbox{distribution.\nThe following lemma, proven in Appendix~\\ref{app:proofOfBerryEsseen}, shows the convergence.}\n\\begin{restatable}{lemma}{proofOfBerryEsseen}\\label{lem:proofOfBerryEsseen}\n\\mbox{For all $i$, $\\mathcal R_H (x)_i$ converges to a normal variable: $\\sup_{x\\in \\mathbb R}\\abs{F_{i,d}(x)-\\Phi(x)} \\le \\frac{0.409\\cdot \\rho}{\\sigma^3 \\sqrt d}$.}\n\\end{restatable}\n\n\\ifdefined\n\\vbox{\n\\fi\nWith this result, we continue to lay out evidence for the ``weak dependency'' among the coordinates. We do so by calculating the moments of their joint distribution in increasing subset sizes showing that these moments converge to those of independent normal variables. Previous work has shown that a structured random rotation on vectors with specific distributions results in ``weakly dependent'' normal variables. This line of research~\\cite{rader1969new,thomas2013parallel,herendi1997fast} utilized the Hadamard transform for a different purpose. Their goal was to develop a computationally cheap method to generate independent normally distributed variables from simpler (e.g., uniform) distributions. We apply their analysis to our setting.\n\\ifdefined\n}\n\\fi\n\n\n\\mbox{We partially rely on the following observation that the Hadamard matrix satisfies.}\n\n\n\\begin{observation}\\label{obs:Hadamard_rows}(\\cite{rader1969new})\nThe Hadamard product (coordinate-wise product), $H_{\\angles{i}}\\circ H_{\\angles{\\ell}}$,\nof two rows $H_{\\angles{i}},H_{\\angles{\\ell}}$ in the Hadamard matrix yields another row at the matrix $H_{\\angles{i}}\\circ H_{\\angles{\\ell}} = H_{\\angles{1+(i-1) \\oplus (\\ell-1)}}$. Here, $(i-1) \\oplus (\\ell-1)$ is the bitwise {xor of the $(\\log d)$-sized binary representation of $(i-1)$ and $(\\ell-1)$}. \nIt follows that $\\sum_{j=1}^d H_{ij}H_{\\ell j}=\\sum_{j=1}^d (H_{\\angles{i}}\\circ H_{\\angles{\\ell}})_{{j}} = \\sum_{j=1}^d H_{1+(i-1) \\oplus (\\ell-1), j}$~.\n\\end{observation}\n\n\nWe now analyze the moments of the rotated variables, starting with the following observation. It follows from {the sign-symmetry of $D$ and matches the joint distribution of i.i.d. normal variables.}\n\\begin{observation}\nAll odd moments containing $\\mathcal R_H(x)$ entries are $0$. That is,\\\\ \n$\\hspace*{1.7cm}\\forall q\\in\\mathbb N, \\forall {i_1,\\ldots,i_{2q+1}}\\in\\set{1,\\ldots,d}:\\mathbb E\\brackets{\\mathcal R_H(x)_{i_1}\\cdot \\ldots\\cdot \\mathcal R_H(x)_{i_{2q+1}}}=0$.\n\\end{observation}\n\nTherefore, we need to examine only even moments. We start with showing that the second moments also match with the distribution of independent normal variables.\n\\begin{lemma}\nFor all $i\\neq \\ell$ it holds that $\\mathbb{E} \\brackets{(\\mathcal R_H \\cdot x)_i \\cdot (\\mathcal R_H \\cdot x)_\\ell} = 0$, whereas  $\\mathbb{E} \\brackets{(\\mathcal R_H \\cdot x)_i^2} = \\sigma^2$.\n\\end{lemma}\n\\begin{proof}\n{Since $\\set{D_{jj}\\mid j\\in\\set{1,\\ldots,d}}$ are sign-symmetric and i.i.d., }\n$\n\\mathbb{E} \\big[(\\mathcal R_H \\cdot x)_i \\cdot (\\mathcal R_H \\cdot x)_\\ell\\big] = \\mathbb{E} \\big[{\\frac{1}{d} (\\sum_{j=1}^d x_j H_{ij} D_{jj}) \\cdot (\\sum_{j=1}^d x_j H_{\\ell j} D_{jj})}\\big] = \\mathbb{E} \\brackets{x_j^2} \\cdot \\frac{1}{d} \\cdot \\sum_{j=1}^d H_{ij} H_{\\ell j}$.\nNotice that $\\sum_{j=1}^d H_{1j}{\\,=\\,}d$ \\mbox{ and $\\sum_{j=1}^d H_{ij}=0$ for all $i>1$. Thus, by \\Cref{obs:Hadamard_rows} we get $0$ if $i \\neq \\ell$ and $\\sigma^2$ otherwise.\\qedhere} \n\\end{proof}\nWe have established that the coordinates are pairwise uncorrelated. Similar but more involved analysis shows that the same trend continues under the assumption of the existence of $x$'s higher moments. In Appendix \\ref{app:Hadamard_4th_moments} we analyze the 4th moments showing that they indeed approach the 4th moments of independent normal variables with a rate of $\\frac{1}{d}$; the reader is referred to \\cite{rader1969new} for further intuition and higher moments analysis.\nWe therefore expect that using \\mbox{DRIVE}\\xspace and \\mbox{DRIVE$^+$}\\xspace with Hadamard transform will yield similar results to that of a uniform random rotation at high dimensions and when the input vectors respect the finite moments assumption.\n\n\\begin{figure}[]\n\\centering\n\\centerline{\\includegraphics[width=\\textwidth, trim=70 93 70 0, clip]{figures\/hadamard_vs_rr_4.pdf}}\n\\ifdefined\\else\\vspace*{-2mm}\\fi\n\\caption{Distributed mean estimation comparison: each data point is averaged over $10^4$ trials. In each trial, \\emph{the same} (randomly sampled) vector is sent by $n=10$ clients.}\n\\ifdefined\\else\\vspace*{-3mm}\\fi\n\\label{fig:theory_is_cool}\n\\end{figure}\n\nIn addition to the theoretical evidence, in Figure \\ref{fig:theory_is_cool}, we show experimental results comparing the measured NMSE for the 1b~-~Distributed Mean Estimation problem with $n=10$ clients (all given the \\emph{same} vector so biases do not cancel out) for \\mbox{DRIVE}\\xspace and \\mbox{DRIVE$^+$}\\xspace using both uniform and structured random rotations over three different distributions. The results indicate that all variants yield similar NMSEs in reasonable dimensions, \\mbox{in line with the theoretical guarantee of Corollary \\ref{cor:deviation_res_3} and Theorem~\\ref{cor:dme}.}\n\n\n\n\n\n\n\n\n\n\n\\section{Evaluation}\\label{sec:Evaluation}\n\nWe evaluate \\mbox{DRIVE}\\xspace and \\mbox{DRIVE$^+$}\\xspace, comparing them to standard and recent state-of-the-art techniques. We consider classic distributed learning tasks as well as federated learning tasks (e.g., where the data distribution is not i.i.d. and clients may change over time). All the distributed tasks are implemented over PyTorch \\cite{NIPS2019_9015} and all the federated tasks are implemented over TensorFlow Federated \\cite{tensorflowfed}.\nWe focus our comparison on vector quantization algorithms and recent sketching techniques and exclude sparsification methods (e.g.,~\\cite{konecy2017federated, WangSLCPW18, NEURIPS2019_d9fbed9d,NEURIPS2018_b440509a}) and methods that involve client-side memory since these can often work in conjunction with our algorithms.\n\n\\vbox{\n\\paragraph{\\textbf{{Datasets}.\\quad}}\\label{p:datasets}\nWe use MNIST~\\cite{lecun1998gradient,lecun2010mnist}, EMNIST~\\cite{cohen2017emnist}, CIFAR-10 and CIFAR-100 \\cite{krizhevsky2009learning} for image classification tasks; a next-character-prediction task using the Shakespeare dataset \\cite{shakespeare}; and a next-word-prediction \\mbox{task using the Stack Overflow dataset \\cite{stackoverflowdb}. Additional details appear in Appendix~\\ref{app:utilizedAssets}.}\n\n\\paragraph{\\textbf{{Algorithms}.\\quad}} \n\n\\addtocounter{footnote}{-1}\n\\new{Since our focus is on the distributed mean estimation problem and its federated and distributed learning applications, we run \\mbox{DRIVE}\\xspace and \\mbox{DRIVE$^+$}\\xspace with the unbiased scale quantities.\\footnotemark}\n}\n\n\nWe compare against several alternative algorithms: (1) \\emph{FedAvg} \\cite{mcmahan2017communication} that uses the full vectors (i.e., each coordinate is represented using a 32-bit float); (2) Hadamard transform followed by 1-bit stochastic quantization (SQ) \\cite{pmlr-v70-suresh17a,konevcny2018randomized}; (3) Kashin's representation followed by 1-bit stochastic quantization \\cite{caldas2018expanding}; (4) \\emph{TernGrad}~\\cite{wen2017terngrad}, which clips coordinates larger than 2.5 times the standard deviation, then performs 1-bit stochastic quantization on the absolute values and separately sends their signs and the maximum coordinate for scale (we note that TernGrad is a low-bit variant of a well-known algorithm called \\emph{QSGD}~\\cite{NIPS2017_6c340f25}, and we use TernGrad since we found it to perform better in our experiments);\\footnotetext{For \\mbox{DRIVE}\\xspace the scale is $S=\\frac{\\norm{x}_2^2}{\\norm{\\mathcal R(x)}_1}$ (see Theorem~\\ref{theorem:drive_is_unbised}). For \\mbox{DRIVE$^+$}\\xspace the scale is $\\sc=\\frac{\\norm{x}_2^2}{\\norm{c}_2^2}$, where $c\\in\\set{c_1,c_2}^d$ is the vector indicating the nearest centroid to each coordinate in $\\mathcal R(x)$ (see Section~\\ref{sec:drive_plus}).}\\footnote{\\new{When restricted to two quantization levels, TernGrad is identical to QSGD's max normalization variant with clipping (slightly better due to the ability to represent 0).}} and (5-6)~\\emph{Sketched-SGD}~\\cite{ivkin2019communication} and \\emph{FetchSGD}~\\cite{rothchild2020fetchsgd}, which are both count-sketch \\cite{charikar2002finding} based algorithms designed for distributed and federated learning, respectively.\n\n\n\n\n    \\begin{table}[]\n    \\resizebox{\\textwidth}{!}{%\n    \\begin{tabular}{r|l|l|l|l|l|l|}\n    \\cline{2-7}\n    \\multicolumn{1}{l|}{Dimension ($d$)} & \\multicolumn{1}{c|}{\\begin{tabular}[c]{@{}c@{}}Hadamard\\\\ + 1-bit SQ\\end{tabular}} & \\multicolumn{1}{c|}{\\begin{tabular}[c]{@{}c@{}}Kashin\\\\ + 1-bit SQ\\end{tabular}} & \\multicolumn{1}{c|}{\\begin{tabular}[c]{@{}c@{}}Drive \\\\ (Uniform)\\end{tabular}} & \\multicolumn{1}{c|}{\\begin{tabular}[c]{@{}c@{}}Drive$^+$\\\\ (Uniform)\\end{tabular}} & \\multicolumn{1}{c|}{\\begin{tabular}[c]{@{}c@{}}Drive\\\\  (Hadamard)\\end{tabular}} & \\multicolumn{1}{c|}{\\begin{tabular}[c]{@{}c@{}}Drive$^+$ \\\\ (Hadamard)\\end{tabular}} \\\\ \\hline\n    \\multicolumn{1}{|r|}{128}            & 0.5308, {\\textit{0.34}}     & 0.2550, \\textit{2.12}    & 0.0567, \\textit{40.4}             & \\textbf{0.0547}, \\textit{40.7}   &         0.0591,          \\textit{0.36}  & 0.0591,  \\textit{0.72}    \\\\ \\hline\n    \\multicolumn{1}{|r|}{8,192}          & 1.3338, {\\textit{0.57}}     & 0.3180, \\textit{3.42}    & \\textbf{0.0571}, \\textit{5088}    & \\textbf{0.0571}, \\textit{5101}   & \\textbf{0.0571},        {\\textit{0.60}}     & \\textbf{0.0571}, \\textit{1.06}    \\\\ \\hline\n    \\multicolumn{1}{|r|}{524,288}        & 2.1456, {\\textit{0.79}}     & 0.3178, \\textit{4.69}    & ---    & ---   & \\textbf{0.0571},         \\textit{0.82}      & \\textbf{0.0571}, \\textit{1.35}    \\\\ \\hline\n    \\multicolumn{1}{|r|}{33,554,432}     & 2.9332, {\\textit{27.1}}     & 0.3179, \\textit{332}     & ---    & ---   & \\textbf{0.0571},       {\\textit{27.2}}    & \\textbf{0.0571}, \\textit{37.8}     \\\\ \\hline\n    \\end{tabular}%\n    }\n    \\caption{Empirical NMSE and average per-vector encoding time (in milliseconds, on an RTX 3090 GPU) for distributed mean estimation with $n=10$ clients (same as in Figure~\\ref{fig:theory_is_cool}) and Lognormal(0,1) distribution. Each entry is a (NMSE, \\textit{time}) tuple and the most accurate result is highlighted in \\textbf{bold}.\n    }\\label{tbl:weAreFast}\n    \\ifdefined\\else\\vspace*{-2mm}\\fi\n    \\end{table} \n\nWe note that Hadamard with 1-bit stochastic quantization is our most fair comparison, as it uses the same number of bits as \\mbox{DRIVE$^+$}\\xspace (and slightly more than \\mbox{DRIVE}\\xspace) and has similar computational costs. This contrasts with Kashin's representation, where both the number of bits and the computational costs are higher. \nFor example, a standard TensorFlow Federated implementation (e.g., see ``\\textsc{class KashinHadamardEncodingStage}'' hyperparameters at \\cite{tensorflowfedkashincode}) uses a minimum of $1.17$ bits per coordinate, and three iterations of the algorithm resulting in five Hadamard transforms for each vector.\nAlso, note that TernGrad uses an extra bit per coordinate for sending the sign. Moreover, \\mbox{the clipping performed by TernGrad is a heuristic procedure, which is orthogonal to our work. }\n\n\nFor each task, we use a subset of datasets and the most relevant competition.\nDetailed configuration information and additional results appear in Appendix \\ref{appendix:additional_simulations}.\nWe first evaluate the vNMSE-Speed tradeoffs and then proceed to federated and distributed learning experiments.\n\n\n\n\n\n\n\n\\paragraph{\\textbf{vNMSE-Speed Tradeoff}.\\quad}\nAppearing in Table~\\ref{tbl:weAreFast}, the results show that our algorithms offer the lowest NMSE and that the gap increases with the dimension. \nAs expected, \\mbox{DRIVE}\\xspace and \\mbox{DRIVE$^+$}\\xspace with uniform rotation are more accurate for small dimensions but are significantly slower.\nSimilarly, \\mbox{DRIVE}\\xspace is as accurate as \\mbox{DRIVE$^+$}\\xspace, and both are significantly more accurate than Kashin (by a factor of 4.4$\\times$-5.5$\\times$) and Hadamard (9.3$\\times$-51$\\times$) with stochastic quantization. \nAdditionally, \\mbox{DRIVE}\\xspace is 5.7$\\times$-12$\\times$ faster than Kashin and about as fast as Hadamard.\nIn Appendix~\\ref{app:additional_speed_results} we discuss the result\n, give the complete experiment specification, and provide measurements on a commodity machine. \n\n\\new{We note that the above techniques, including \\mbox{DRIVE}\\xspace, are more computationally expensive than linear-time solutions like TernGrad.  Nevertheless, \\mbox{DRIVE}\\xspace's computational overhead becomes insignificant for modern learning tasks. For example, our measurements suggest that it can take 470~ms for computing the gradient on a ResNet18 architecture (for CIFAR100, batch size = 128, using NVIDIA GeForceGTX 1060 (6GB) GPU) while the encoding of \\mbox{DRIVE}\\xspace (Hadamard) takes 2.8~ms. That is, the overall computation time is only increased by 0.6\\% while the error reduces significantly. Taking the transmission and model update times into consideration would reduce the importance of the compression time further.}\n\n\n\\paragraph{\\textbf{{Federated Learning}.\\quad}}\nWe evaluate over four tasks: (1)~EMNIST over customized CNN architecture with two convolutional layers with ${\\approx}1.2M$ parameters \\cite{caldas2019leaf}; (2)~CIFAR-100 over \\mbox{ResNet-18}~\\cite{krizhevsky2009learning}\n; (3)~a~next-character-prediction task using the Shakespeare dataset \\cite{mcmahan2017communication}; (4)~a~next-word-prediction task using the Stack Overflow dataset \\cite{reddi2020adaptive}. Both (3) and (4) use LSTM recurrent models \\cite{Hochreiter1997LongSM} with\n${\\approx}820K$ and ${\\approx}4M$ parameters,\nrespectively. We use code, client partitioning, models, hyperparameters, and validation metrics from the federated learning benchmark of \\cite{reddi2020adaptive}. \n\n\\new{The results are depicted in Figure~\\ref{fig:federated_dnn}. \nWe observe that in all tasks, \\mbox{DRIVE}\\xspace and \\mbox{DRIVE$^+$}\\xspace have accuracy that is competitive with that of the baseline, FedAvg. In CIFAR-100, TernGrad and \\mbox{DRIVE}\\xspace provide the best accuracy. For the other tasks, \\mbox{DRIVE}\\xspace and \\mbox{DRIVE$^+$}\\xspace have the best accuracy, while the best alternative is either Kashin + 1-bit SQ or TernGrad, depending on the task. Hadamard + 1-bit SQ, which is the most similar to our algorithms (in terms of both bandwidth and compute), provides lower accuracy in all tasks. Additional details and hyperparameter \\mbox{configurations are presented in Appendix~\\ref{app:fl_details}.}}\n\n\n\\paragraph{\\textbf{{Distributed CNN Training}.\\quad}}\nWe evaluate distributed CNN training with 10 clients in two configurations: (1) CIFAR-10 dataset with ResNet-9; (2) CIFAR-100 with ResNet-18 \\cite{krizhevsky2009learning, he2016deep}. \n\\new{In both tasks, \\mbox{DRIVE}\\xspace and \\mbox{DRIVE$^+$}\\xspace have similar accuracy to FedAvg, closely followed by Kashin + 1-bit SQ. The other algorithms are less accurate, with Hadamard + 1-bit SQ being better than Sketched-SGD and TernGrad for both tasks.}\nAdditional details and {hyperparameter configurations are presented in Appendix \\ref{app:dl_details}. Figure~\\ref{fig:distributed-dnn} depicts the results.}\n\n\\begin{figure}[]\n\\centering\n\\centerline{\\includegraphics[width=\\textwidth, trim=10 45 0 0, clip]{figures\/Federated-DNN.pdf}}\n\\ifdefined  \n  \\vspace*{-2mm}\n\\fi\n\\caption{\n  Accuracy per round on various federated learning tasks. Smoothing is done using a rolling mean with a window size of 150. The second row zooms-in on the last 50 rounds.}\n\\label{fig:federated_dnn}\n\\ifdefined  \n  \\vspace*{-2mm}\n\\fi\n\\end{figure}\n\n\n\\begin{figure}[]\n\\centering\n\\centerline{\\includegraphics[width=\\textwidth, trim=0 45 0 0, clip]{figures\/Distributed-DNN.pdf}}\n\\ifdefined  \n  \\vspace*{-2mm}\n\\fi\n\\caption{\n \n  \\mbox{Accuracy per round on distributed learning tasks, with a zoom-in on the last 50 rounds.}\n}\n\\ifdefined  \n  \\vspace*{-4mm}\n\\fi\n\\label{fig:distributed-dnn}\n\\end{figure}\n\n\n\n\n\n\n\n\n\\paragraph{\\textbf{{Evaluation Summary}.\\quad}}\nOverall, it is evident that \\mbox{DRIVE}\\xspace and \\mbox{DRIVE$^+$}\\xspace consistently offer markedly favorable results in comparison to the alternatives in our setting. Kashin's representation appears to offer the best competition, albeit at somewhat higher computational complexity and bandwidth requirements. The lesser performance of the sketch-based techniques is attributed to the high noise of the sketch under such a low ($d(1+o(1))$ bits) communication requirement. This is because the \\mbox{number of counters they can use is too low, making too many coordinates map into each counter.}\n\\new{In Appendix~\\ref{subsection:power_iteration_appendix}, we also compare \\mbox{DRIVE}\\xspace and \\mbox{DRIVE$^+$}\\xspace to state of the art techniques over K-Means and Power Iteration tasks for 10, 100, and 1000 clients, yielding similar trends.}\n\n\n\\ifdefined\n\\begin{table}[]\n\\resizebox{\\textwidth}{!}{%\n\\renewcommand{\\arraystretch}{1.6}\n\\begin{tabular}{c|c|c|c|c|}\n\\cline{2-5}\n\\multirow{2}{*}{}                       & \\multirow{1}{*}{Scale} & \\multicolumn{3}{c|}{Rotation}                     \\\\ \\cline{3-5} Problem\n                                        &          $S$          & \\multicolumn{2}{c|}{Uniform} &   Hadamard                \\\\ \\hline\n\\multicolumn{1}{|l|}{1b - VE}                  &  $\\frac{\\norm{\\mathcal R(x)}_1}{d}$                  & \\multicolumn{2}{c|}{vNMSE $= \\parentheses{1 - \\frac{2}{\\pi}} \\parentheses{ {1-\\frac{1}{d}}}$} &   vNMSE $\\le \\frac{1}{2}$                 \\\\ \\hline\n\\multicolumn{1}{|l|}{\\multirow{2}{*}{1b - DME}} & \\multirow{2}{*}{$\\frac{\\norm{x}_2^2}{\\norm{\\mathcal R(x)}_1}$} & \\multicolumn{2}{c|}{$(i)$ $d \\ge 2 \\implies$ NMSE $\\le \\frac{1}{n} \\cdot 2.92$} & \\multirow{2}{*}{---} \\\\\n\\multicolumn{1}{|l|}{}                  &                   & \\multicolumn{2}{l|}{$(ii)$ \n$d \\ge 135 \\implies$ NMSE $\\le \\frac{1}{n} \\cdot \\left(\\frac{\\pi}{2} - 1 + \\sqrt{\\frac{{(6\\pi^3-12\\pi^2)}\\cdot\\ln d+1}{d}}\\right)$} &                   \\\\ \\hline\n\\end{tabular}%\n}\n\\caption{\\new{Summary of the proven error bounds for \\mbox{DRIVE}\\xspace.}}\n\\label{tab:summary_proven_bounds}\n\\end{table}\n\\else\n\\begin{table}[]\n\\resizebox{\\textwidth}{!}{%\n\\renewcommand{\\arraystretch}{1.6}\n\\begin{tabular}{c|c|c|c|c|}\n\\cline{2-5}\n\\multirow{2}{*}{}                       & \\multirow{1}{*}{Scale} & \\multicolumn{3}{c|}{Rotation}                     \\\\ \\cline{3-5} Problem\n                                        &          $S$          & \\multicolumn{2}{c|}{Uniform} &   Hadamard                \\\\ \\hline\n\\multicolumn{1}{|l|}{1b - VE}                  &  $\\frac{\\norm{\\mathcal R(x)}_1}{d}$                  & \\multicolumn{2}{c|}{vNMSE $= \\parentheses{1 - \\frac{2}{\\pi}} \\parentheses{ {1-\\frac{1}{d}}}$} &   vNMSE $\\le \\frac{1}{2}$                 \\\\ \\hline\n\\multicolumn{1}{|l|}{\\multirow{1}{*}{1b - DME}} & \\multirow{1}{*}{$\\frac{\\norm{x}_2^2}{\\norm{\\mathcal R(x)}_1}$} & \\multicolumn{2}{c|}{NMSE $\\le \\frac{1}{n} \\cdot 2.92$;\\quad $d \\ge 135 \\implies$ NMSE $\\le \\frac{1}{n} \\cdot \\left(\\frac{\\pi}{2} - 1 + \\sqrt{\\frac{{(6\\pi^3-12\\pi^2)}\\cdot\\ln d+1}{d}}\\right)$} & \\multirow{1}{*}{---}\\\\ \\hline\n\\end{tabular}%\n}\n\\caption{\\new{Summary of the proven error bounds for \\mbox{DRIVE}\\xspace.}}\n\\label{tab:summary_proven_bounds}\n\\end{table}\n\\fi\n\n\\section{\\new{Discussion}}\n\n\\ifdefined\nIn this section, we overview few limitations and future research directions for \\mbox{DRIVE}\\xspace.\n\\fi\n\n\\paragraph{\\textbf{{Proven Error Bounds}.\\quad}} We summarize the proven error bounds in Table \\ref{tab:summary_proven_bounds}. Since \\mbox{DRIVE}\\xspace (Hadamard) is generally not unbiased (as discussed in Section \\ref{sec:dme_hadamard_subsec}), we cannot establish a formal guarantee for the 1b - DME problem when using Hadamard. It is a challenging research question whether there exists \\mbox{other structured rotations with low computational complexity and stronger guarantees.}\n\n\\paragraph{\\textbf{{Input Distribution Assumption}.\\quad}} The distributed mean estimation analysis of our Hadamard-based variants is based on an assumption (Section \\ref{sec:dme_hadamard_subsec}) about the vector distributions. While machine learning workloads, and DNN gradients in particular (e.g.,~\\cite{chmiel2020neural,banner2018post,ye2020accelerating}), were observed to follow such distributions, this assumption may not hold for other applications.\n\nFor such cases, we note that \\mbox{DRIVE}\\xspace is compatible with the error feedback (EF) mechanism~\\cite{seide20141, karimireddy2019error} that ensured convergence and recovery of the convergence rate of non-compressed SGD. Specifically, as evident by Lemma \\ref{lem:hadamard_biased}, any scale $\\frac{\\norm{\\mathcal R (x)}_1}{d} \\le S \\le 2 \\cdot \\frac{\\norm{\\mathcal R (x)}_1}{d}$ is sufficient to respect the \\emph{compressor} (i.e., \\emph{bounded variance}) assumption. For completeness, in Appendix \\ref{app:exp:ef}, we perform EF experiments comparing \\mbox{DRIVE}\\xspace and \\mbox{DRIVE$^+$}\\xspace to other compression techniques that use EF.\n\n\n\\paragraph{\\textbf{{Varying Communication Budget}.\\quad}} \nUnlike some previous works, our algorithms' guarantees with more than one bit per coordinate are not established. It is thus an interesting future work to extend \\mbox{DRIVE}\\xspace to other communication budgets and understand what are the resulting guarantees. We refer the reader to~\\cite{vargaftik2021communication} for initial steps towards that direction. \n\n\n\\paragraph{\\textbf{{Entropy Encoding}.\\quad}}\n\nEntropy encoding methods (such as Huffman coding) can further compress vectors of values when the values are not uniformly distributed.  We have compared DRIVE against stochastic quantization methods using entropy encoding for the challenging setting for DRIVE where all vectors are the same (see Table \\ref{tbl:weAreFast} for further description).  The results appear in Appendix \\ref{subsec:app:ee}, where DRIVE still outperforms these methods.  We also note that, when computation allows and when using \\mbox{DRIVE}\\xspace with multiple bits per entry, \\mbox{DRIVE}\\xspace can also be enhanced by entropy encoding techniques.  We describe some initial results for this setting in~\\cite{vargaftik2021communication}.\n\n\n\n\\paragraph{\\textbf{{Structured Data}.\\quad}} When the data is highly sparse, skewed, or otherwise structured, one can leverage that for compression. We note that some techniques that exploit sparsity or structure can be use in conjunction with our techniques. For example, one may transmit only non-zero entries or Top-K entries \\mbox{while compressing these using \\mbox{DRIVE}\\xspace to reduce communication overhead even further.}\n\n\\paragraph{\\textbf{{Compatibility With Distributed All-Reduce Techniques}.\\quad}} Quantization techniques, including \\mbox{DRIVE}\\xspace, may introduce overheads in the context of All-Reduce (depending on the network architecture and communication patterns). In particular, if every node in a cluster uses a different rotation, \\mbox{DRIVE}\\xspace will not allow for efficient in-path aggregation without decoding the vectors. Further, the computational overhead of the receiver increases by a $\\log d$ factor as each vector has to be decoded separately before an average can be computed. It is an interesting future direction for \\mbox{DRIVE}\\xspace to understand how to minimize such potential overheads. For example, one can consider bucketizing co-located workers and apply DRIVE's quantization only for cross-rack traffic. \n\n\n\n\\section{Conclusions} \\label{sec:conclusions}\n{\nIn this paper, we studied the vector and distributed mean estimation problems. These problems are applicable to distributed and federated learning, where clients communicate real-valued vectors (e.g., gradients) to a server for averaging.\nTo the best of our knowledge, our algorithms are the first with a provable error of $O(\\frac{1}{n})$ for the 1b~-~Distributed Mean Estimation problem (i.e., with $d(1+o(1))$ bits). \nAs shown in~\\cite{iaab006}, any algorithm that uses $O(d)$ shared random bits (e.g., our Hadamard-based variant) has a vNMSE of $\\Omega(1)$, i.e., \\mbox{DRIVE}\\xspace and \\mbox{DRIVE$^+$}\\xspace are asymptotically optimal; additional discussion is given in Appendix~\\ref{app:lower_bounds}.\nOur experiments, carried over various tasks and datasets, indicate that our algorithms improve over the state of the art. \nAll the results presented in this paper are fully reproducible by our source code, \navailable at~\\cite{openSource}.\n}\n\n\n\n\n\\ifspacinglines\n\\fi\n\n\n\n\n\\begin{ack}\nMM was supported in part by NSF grants CCF-2101140, CCF-2107078, CCF-1563710, and DMS-2023528.  MM and RBB were supported in part by a gift to the Center for Research on Computation and Society at Harvard University. AP was supported in part by the Cyber Security Research Center at Ben-Gurion University of the Negev.\nWe thank Moshe Gabel, Mahmood Sharif, Yuval Filmus and, the anonymous reviewers for helpful comments and suggestions.\n\\end{ack}\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\nTo incorporate knowledge in real-world question-answering systems, knowledge base question answering (KBQA) utilizes a background knowledge base (KB) as the source of answers to factoid natural language questions. Leveraging the versatility of KB query languages like SPARQL \\citep{prud2008sparql}, many previous works \\citep{unger2012template, yahya2012natural} adopted a semantic parsing paradigm for KBQA, in which questions are converted to equivalent SPARQL queries and answers are given by executing the queries in KB. Regarding the intrinsic graph structure of SPARQLs, some works further reduced such procedure as generating the query graph of SPARQLs w.r.t. questions. However, these methods either require auxiliary tools (e.g. AMR in \\citealp{kapanipathi2021leveraging}, constituency tree in \\citealp{hu2021edg}, dependency tree in \\citealp{hu2017answering}) causing potential cascading errors, or rely on predefined templates \\citep{cui2019kbqa, athreya2021template} limiting their expressiveness and generalization abilities.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=1.0\\linewidth]{pics\/two_stage_2.pdf}\n    \\caption{Generating a query graph (bottom) by two stages to represent the SPARQL (right-top). At graph structure generation stage, node-extraction generates all graph nodes while graph-composition adds unlabeled edges between proper nodes. Then, the relation extraction stage decides the specific predicate of each edge.}\n    \\label{fig:two_stage}\n\\end{figure}\n\n\\begin{comment}\nTo address these, efforts were made on devising independent pipelines for query graph construction \\citep{lin2021deep}. As in Figure \\ref{fig:two_stage}, these pipelines usually involve a node extraction (NE) module to detect the mentions of all nodes in query graph and link entity mentions, a graph composition (GC) module to connect related nodes given by NE, and a relation extraction (RE) module deciding the KB predicate corresponding to each edge added in GC. Here, we observe strong causal effects between NE and GC, e.g. edges connected by GC are valid only between the node mentions extracted in NE, making GC decisions highly dependent on NE. To this regard, previous works \\citep{zhang2021namer, ravishankar2021two} that perform NE and GC separately without causal-modelling may fell short in deeply comprehending the correlated tasks and accurately generating query graphs.\n\\end{comment}\nTo address these, efforts were made on devising independent pipelines for query graph construction \\citep{lin2021deep}. As in Figure \\ref{fig:two_stage}, these pipelines usually involve a node extraction (NE) module to detect the mentions of all nodes in query graph and link entity mentions, a graph composition (GC) module to connect related nodes given by NE, and a relation extraction (RE) module deciding the KB predicate corresponding to each edge added in GC. In this framework, two drawbacks exist in previous works: 1) we observe strong causal effects between NE and GC, e.g. edges connected by GC are valid only between the node mentions extracted in NE, making GC decisions highly dependent on NE. To this regard, previous works \\citep{zhang2021namer, ravishankar2021two} that perform NE and GC separately without causal-modelling may fall short in deeply comprehending the correlated tasks and accurately generating query graphs. 2) GC is commonly modelled as a sequence-generation in prior methods, either through generative decoder \\citep{shen2019multi, chen2021outlining} or via stage-transition \\citep{yih2015semantic, hu2018state}. However, sequence-modelling generally undergoes sequence ambiguity and exposure bias \\citep{zhang-etal-2019-bridging} that harms model accuracy.\n\n\\begin{comment}\nIn this work, we formalize the generation of query graph in a two-staged manner as in Figure \\ref{fig:two_stage}. At the first stage, we adopt a novel causal-enhanced table-filling model to jointly complete NE and GC, resulting in a query graph structure representing the connectivity of all nodes. More specifically, inspired by \\citet{chen-etal-2020-exploring-logically}, we utilize a label transfer mechanism to facilitate the acquisition of causality between NE and GC. Further, unlike prior methods that generate graph structure iteratively either through generative decoder \\citep{shen2019multi, chen2021outlining} or via stage-transition \\citep{yih2015semantic, hu2018state}, we apply a table-filler to decode all edges simultaneously. In turn, it naturally circumvents the sequence ambiguity and exposure bias \\citep{zhang-etal-2019-bridging} of iterative decoding. For the second stage, we propose a beam-search-based relation extraction algorithm to determine the predicate that binds to each graph edge. Differ from prior works, we perform candidate predicate retrieval and ranking alternately for each edge, limiting the candidate scale linearly w.r.t. KB degree and making the algorithm scalable for large-scale KBs like DBpedia.\n\\end{comment}\n\nIn this work, we formalize the generation of query graph in a two-staged manner as in Figure \\ref{fig:two_stage}. At the first stage, we tackle the aforesaid weaknesses by a novel causal-enhanced table-filling model to jointly complete NE and GC, resulting in a query graph structure representing the connectivity of all nodes. More specifically, inspired by \\citet{chen-etal-2020-exploring-logically}, we utilize a label transfer mechanism to facilitate the acquisition of causality between NE and GC (which solves drawback 1 above). Further, we apply a table-filler to decode all edges simultaneously, which naturally circumvents the ambiguity and bias of iterative decoding (and solves drawback 2). For the second stage, we propose a beam-search-based relation extraction algorithm to determine the predicate that binds to each graph edge. Differ from prior works, we perform candidate predicate retrieval and ranking alternately for each edge, limiting the candidate scale linearly w.r.t. KB degree and making the algorithm scalable for large-scale KBs like DBpedia.\n\n\\begin{comment}\nIn short, the contributions of this paper are: we formalize the construction of query graph as two stages, for the graph structure generation stage, we devise a causal-enhanced table-filler to grasp intrinsic causal effects and avoid exposure bias; for the relation extraction stage, we present an efficient beam-search algorithm scalable for large KBs; our method outperforms previous state-of-the-arts on LC-QuAD 1.0, a prominent KBQA benchmark, by a large margin ($\\sim\\!\\!17\\%$), further experiments verifies the effectiveness of our approach.\n\\end{comment}\nIn short, the major contributions of this paper are: 1) to our knowledge, we are the first to model GC as a table-filling process, which prevents the ambiguity and bias in prior works; 2) we model the intrinsic causal effects in KBQA to grasp subtask correlations and improve pipeline integrity; 3) our method outperforms previous state-of-the-arts on LC-QuAD 1.0, a prominent KBQA benchmark, by a large margin ($\\sim\\!\\!17\\%$), further experiments verifies the effectiveness of our approach.\n\n\\section{Preliminaries}\n\\subsection{Problem Setting}\nWe solve KBQA in a semantic parsing way, given a question (left-top in Figure \\ref{fig:two_stage}), we generate a SPARQL query (right-top in Figure \\ref{fig:two_stage}) to represent its semantics and answer the question by executing the query in KB. By definition, SPARQL describes a query graph with each triple in its body referring to a graph edge; by matching the graph pattern in KB, certain KB entries binding to the query graph can be processed as query results (e.g. in Table \\ref{tab:trigger_words} for SELECT queries, all entries binding to the \"select\" node are results; for JUDGE queries, the existence of matched entries determines the boolean result). Hence, our task is further specified as constructing the query graph (bottom of Figure \\ref{fig:two_stage}) of a question to represent its corresponding SPARQL.\n\n\\begin{table}[t]\n\\centering\n\\begin{comment}\n\\resizebox{1.\\columnwidth}{!}{\n\\begin{tabular}{c c c}\n    \\toprule\n    \\bfseries Type & \\bfseries Example SPARQL & \\bfseries Trigger Words\\\\\n    \\cmidrule(lr){1-3}\n    JUDGE & ask \\{dbr:New\\_York a dbo:City\\} & did, is, ...\\\\\n    COUNT & select count(?x) \\{?x a dbo:City\\} & how many, ... \\\\\n    SELECT & select ?x \\{?x a dbo:City\\} & \/ \\\\\n    \\bottomrule\n\\end{tabular}}\n\\caption{Supported query types and their examples trigger words.}\n\\end{comment}\n\n\\resizebox{0.8\\columnwidth}{!}{\n\\begin{tabular}{c c}\n    \\toprule\n    \\bfseries Type & \\bfseries Example SPARQL\\\\\n    \\cmidrule(lr){1-2}\n    JUDGE & ask \\{dbr:New\\_York a dbo:City\\} \\\\\n    COUNT & select count(?x) \\{?x a dbo:City\\}  \\\\\n    SELECT & select ?x \\{?x a dbo:City\\} \\\\\n    \\bottomrule\n\\end{tabular}}\n\\caption{Supported query types.}\n\n\\label{tab:trigger_words}\n\\end{table}\n\n\\subsection{Methodology Overview}\nIllustrated by Figure \\ref{fig:two_stage}, we construct the query graph in two stages. In the graph structure generation stage (bottom-left in Figure \\ref{fig:two_stage}), we extract all graph nodes by finding the mention of each node in question and its tag among $\\{ variable,entity,type \\}$, e.g. the mention and tag for the node $?class$ is \"class\" and variable, respectively. Further, we link all non-variable nodes to KB entries, e.g. the $type$ node with mention \"person\" links to \\textit{dbo:person} in Figure \\ref{fig:two_stage}. Also, we decide the target (\"select\") node of the graph and add undirected edges between the nodes that are connected in the query graph, resulting in a graph structure representing the connectivity of all nodes.\n\nSince all edges above are undirected and unlabeled, we fill in the exact KB predicate of each edge in the relation extraction stage (bottom-right in Figure \\ref{fig:two_stage}) to construct a complete query graph. \n\nFinally, we compose a SPARQL w.r.t. the query graph as output. Note that the body of the SPARQL exactly corresponds to the query graph, so only the SPARQL header is yet undetermined. Like \\citealp{hu2021edg}, we collect frequent trigger words in the train data to classify questions into COUNT, JUDGE or SELECT queries as in Table \\ref{tab:trigger_words} (e.g. a question beginning with \"is\" triggers JUDGE). Thus, an entire SPARQL can now be formed. In the following sections, we expatiate our methodology for the two aforementioned stages.\n\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=0.999\\linewidth]{pics\/graph_structure_generation_4k_cut.png}\n    \\caption{Causal-enhanced table-filling model for graph structure generation. The label-to-node and table-to-edge correspondence is illustrated by the poker and fruit symbols respectively.}\n    \\label{fig:graph_structure_generation}\n\\end{figure*}\n\n\n\\section{Graph Structure Generation (GSG)}\n\\label{sec:gsg}\nThe overview of the model proposed for graph structure generation is illustrated by Figure \\ref{fig:graph_structure_generation}. As discussed in Section \\ref{sec:intro}, the model jointly deals with node extraction and graph composition via causal-modelling, which is detailed in this section below.\n\n\\subsection{Node Extraction (NE)}\nNode extraction discovers all nodes in the query graph, i.e. \\{\\textit{?person}, \\textit{?class}, \\textit{dbr:Swinhoe's\\_Crake}, \\textit{dbo:person}\\} in Figure \\ref{fig:two_stage}. We represent a node by its mention and tag, i.e. (\"person\", \\textit{variable}), (\"class\", \\textit{variable}), (\"Swinhoe's Crake\", \\textit{entity}) and (\"person\", \\textit{type}) for each node respectively.\n\nThis goal can naturally be achieved by multi-class sequence labeling. More specifically, let $\\mathbf{Q}\\in \\mathbb{N}^{n}$ be the question (token ids) with length $n$, we first encode it into hidden features $\\mathbf{H}_{rb}$ by a RoBERTa \\citep{liu2019roberta} encoder $E_{rb}: \\mathbb{N}^{n} \\rightarrow \\mathbb{R}^{n\\times h_{rb}}$ with hidden size $h_{rb}$:\n\\[ \\mathbf{H}_{rb} = E_{rb}(\\mathbf{Q}) \\in \\mathbb{R}^{n\\times h_{rb}} \\]\nThen, $\\mathbf{H}_{rb}$ is projected by a fully-connected-network (FCN) $E_{ne}: \\mathbb{R}^{n\\times h_{rb}} \\rightarrow \\mathbb{R}^{n\\times |L|}$ into $\\mathbf{Y}_{ne}$ in label space:\n\\[ \n\\mathbf{Y}_{ne} = E_{ne}(\\mathbf{H}_{rb}) \\in \\mathbb{R}^{n\\times |L|}\n\\]\n$L=\\{O\\}\\cup\\{B,I\\}\\times\\{V,E,T,VT\\}$ is the label set denoting the mention span of variables (V), entities (E), types (T), or overlapping variable and type (VT). Now, the label prediction of each token can be given by $\\mathbf{P}_{ne}=argmax(\\mathbf{Y}_{ne})$; also, given the gold token labels $\\mathbf{G}_{ne}\\in \\mathbb{N}^{n}$ (Figure \\ref{fig:graph_structure_generation} top), a model for NE can be trained by optimizing:\n\\[ \\ell_{ne} = -\\frac{1}{n} \\sum\\limits_{i=1}^n log(\\text{softmax}(\\mathbf{Y}_{ne})[i; \\mathbf{G}_{ne}[i]]) \\]\nWhere $[\\cdot ]$ denotes tensor indexing.\n\nAfter detecting all node mentions and tags, we link each non-variable node to KB entries by DBpedia Lookup and a mention-to-type dictionary built on train data to align the graph structure with KB. See Appendix \\ref{sec:appendix_linking} for more details in node linking.\n\n\\subsection{Graph Composition (GC)}\n\\label{sec:gc}\n\\begin{comment}\nAfter node extraction, all nodes in the query graph remain unconnected. To form the structure of the query graph, graph composition insert unlabeled and undirected edges between the nodes that are related in the query graph, leaving the specific predicate of each edge yet unresolved. Formerly, graph composition is commonly modelled as a sequence-generation process representing an edge by one or several sequence elements. To learn the generation of edge sequence, previous works train reward functions for stage-transition \\citep{yih2015semantic, hu2018state} or generative decoder models \\citep{shen2019multi, zhang2021namer}. Despite the strong expressiveness, modelling graph composition by a sequence usually suffers from two issues: 1) while the edge sequence is ordered, edges in the query graph are a set without order. For a graph with two edges $e_1$ and $e_2$, both sequence $e_1{\\text -}e_2$ and $e_2{\\text -}e_1$ correctly represents the edges in the graph, but they are distinct from the perspective of sequence-generation. As a result, the edge set itself becomes ambiguous for the sequence, which confuses the model when comprehending a sequence and potentially decelerates the convergence. 2) As discussed by \\citealp{zhang-etal-2019-bridging}, without extra augmentation, sequence-generation generally endures an exposure bias between training and inference, harming the model's accuracy when predicting. Hence, a robust model should address the issues above properly.\n\\end{comment}\nAfter node extraction, all nodes in the query graph remain unconnected. To form the structure of the query graph, graph composition inserts unlabeled and undirected edges between the nodes that are related in the query graph, leaving the specific predicate of each edge yet unresolved. Formerly, graph composition is commonly modelled as a edge-sequence-generation process via stage-transition \\citep{yih2015semantic, hu2018state} or generative decoders \\citep{shen2019multi, chen2021outlining}. Despite the strong expressiveness, modelling graph composition by a sequence usually suffers from two issues: 1) while the edge sequence is ordered, edges in the query graph are a set without order. For a graph with two edges $e_1$ and $e_2$, both sequence $e_1{\\text -}e_2$ and $e_2{\\text -}e_1$ correctly represents the edges in the graph, but they are distinct from the perspective of sequence-generation. As a result, the edge set itself becomes ambiguous for the sequence, which confuses the model when comprehending a sequence and potentially decelerates the convergence. 2) As discussed by \\citealp{zhang-etal-2019-bridging}, without extra augmentation, sequence-generation generally endures an exposure bias between training and inference, harming the model's accuracy when predicting. Hence, a robust model should address the issues above properly.\n\nHere, we model graph composition by a table-filling process to decide all edges simultaneously involving no sequence-generation, which naturally circumvents all issues above. Let $\\mathbf{H}_{gc}\\in \\mathbb{R}^{n\\times h_{gc}}$ be the hidden features for graph composition (the full definition of $\\mathbf{H}_{gc}$ with causal-modelling is given in Section \\ref{sec:causal_modelling}; without causal-modelling, we simply have $\\mathbf{H}_{gc}=\\mathbf{H}_{rb}$), we adopt a biaffine attention model \\citep{dozat2016deep, wang2021unire} to convert $\\mathbf{H}_{gc}$ into a table denoting the relationship between each token pair. More specifically, through two multi-layer-perceptrons (MLP) $E_{head}$ and $E_{tail}: \\mathbb{R}^{n\\times h_{gc}} \\rightarrow \\mathbb{R}^{n\\times h_{bi}}$, we first project $\\mathbf{H}_{gc}$ into head ($\\mathbf{H}_{head}$) and tail ($\\mathbf{H}_{tail}$) features:\n\\[ \\mathbf{H}_{\\{head,tail\\}} = E_{\\{head,tail\\}}(\\mathbf{H}_{gc}) \\in \\mathbb{R}^{n\\times h_{bi}} \\]\nThen, for $\\forall 1\\leq i,j\\leq n$, the biaffine attention is performed between the head features of the i\\textsuperscript{th} token $\\mathbf{h}_{head}^{(i)}$ and the tail features of the j\\textsuperscript{th} token $\\mathbf{h}_{tail}^{(j)}$, producing $\\mathbf{s}_{i,j}\\in \\mathbb{R}^2$ representing the probability that an edge exists between the i\\textsuperscript{th} and j\\textsuperscript{th} token:\n\\[ \\mathbf{s}_{i,j} = \\text{softmax}(\\text{Biaff}(\\mathbf{h}_{head}^{(i)}, \\mathbf{h}_{tail}^{(j)})) \\]\n\\[ \\text{Biaff}(\\mathbf{x},\\mathbf{y}) := \\mathbf{x}^T\\mathbf{U}_1\\mathbf{y} + \\mathbf{U}_2(\\mathbf{x}\\oplus \\mathbf{y}) + \\mathbf{b} \\]\nAs $\\mathbf{U}_1\\in\\mathbb{R}^{2\\times h_{bi}\\times h_{bi}}$, $\\mathbf{U}_2\\in\\mathbb{R}^{2\\times 2h_{bi}}$ and $\\mathbf{b}\\in\\mathbb{R}^2$ are trainable parameters, $\\oplus$ denotes concatenation. Combining all scores by $\\mathbf{Y}_{gc} = (\\mathbf{s}_{i,j})_{(1\\leq i,j\\leq n)}\\in \\mathbb{R}^{n\\times n\\times 2}$, we now have a table describing the edge existence likelihood between any two tokens.\n\nAt training, we first obtain the boolean gold table $\\mathbf{G}_{gc}\\in \\mathbb{B}^{n\\times n}$, for every connected node pair in the query graph, the element in $\\mathbf{G}_{gc}$ corresponding to any pair of tokens belonging to the mentions of the two nodes respectively is set to 1 (resulting in several rectangles of 1s). Also, we prefix the question with a special [CLS] token and connect it with the target node to represent the \"select\" edge; for ASK queries without target nodes, a [SEP] token is suffixed and connected with [CLS]. Note that since the graph structure is undirected, $\\mathbf{G}_{gc}$ is a symmetric matrix. An example of $\\mathbf{G}_{gc}$ can be found in Figure \\ref{fig:graph_structure_generation}. With $\\mathbf{G}_{gc}$, we can train the table-filler by $\\ell_{tb}$:\n\\[ \\ell_{tb} = -\\frac{1}{n^2} \\sum\\limits_{i=1}^n\\sum\\limits_{j=1}^n log(\\mathbf{Y}_{gc}[i;j;\\mathbf{G}_{gc}[i;j]]) \\]\nFollowing \\citealp{wang2021unire}, we also introduce $\\ell_{sym}$ to grasp the table symmetry. Finally, we optimize $\\ell_{gc} = \\ell_{tb} + \\ell_{sym}$ to train a model for GC.\n\\[ \\ell_{sym} = \\frac{1}{n^2} \\sum\\limits_{i=1}^n\\sum\\limits_{j=1}^n\\sum\\limits_{k=1}^2 |\\mathbf{Y}_{gc}[i;j;k]-\\mathbf{Y}_{gc}[j;i;k]| \\]\n\nAt inference, for each pair of nodes given by NE, we average the rectangle area in $\\mathbf{Y}_{gc}$ corresponding to the mentions of the node pair as its edge existence probability. The node pairs with a probability higher than 0.5 are connected. This threshold is selected intuitively to denote an edge is more likely to exist against to not exist, though we argue that the prediction is insensitive to any threshold in reasonable range (e.g. 0.3$\\sim$0.7).\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=.92\\linewidth]{pics\/causal_full.pdf}\n    \\caption{Modelling NE and GC with (a) and without (b) causality, as X, Y\\textsubscript{NE}, and Y\\textsubscript{GC} denotes question, NE predictions, and GC predictions. Model (c) learns the causal effects by a label transfer module.}\n    \\label{fig:causal_graphs}\n\\end{figure}\n\n\n\\subsection{Causal Modelling NE and GC}\n\\label{sec:causal_modelling}\nUp to now, NE and GC are treated as separate tasks that fail to model the intrinsic causal effects between them (e.g. edges in Y\\textsubscript{GC} only exist between the mentions detected in NE). Here, we model such causality by a mediation assumption in Figure \\ref{fig:causal_graphs}(b) denoting the causal dependence of GC on both question and NE prediction by edge X\\textrightarrow Y\\textsubscript{GC} and Y\\textsubscript{NE}\\textrightarrow Y\\textsubscript{GC} respectively. To grasp this causal graph, we devise a label transfer \\citep{chen-etal-2020-exploring-logically} module to enable the transfer of NE predictions to GC, i.e. representing Y\\textsubscript{NE}\\textrightarrow Y\\textsubscript{GC}, in Figure  \\ref{fig:causal_graphs}(c).\n\nIn detail, we sample NE predictions $\\widetilde{\\mathbf{Y}_{ne}}$ by gumbel softmax \\cite{nie2018relgan} with $\\boldsymbol{g}\\!\\!\\sim$Gumbel(0,1) and temperature $\\tau$.\n\\[ \\widetilde{\\mathbf{Y}_{ne}} = \\text{softmax}((\\mathbf{Y}_{ne}+\\boldsymbol{g})\/\\tau) \\in \\mathbb{R}^{n\\times |L|} \\]\n$\\widetilde{\\mathbf{Y}_{ne}}$ is then embedded by label embedding $\\mathbf{W}_{le}\\in \\mathbb{R}^{|L|\\times h_{le}}$ and concatenated with $\\mathbf{H}_{rb}$ to form $\\mathbf{H}_{gc}$ in Section \\ref{sec:gc} with $h_{gc}\\!=\\!h_{rb}+h_{le}$:\n\\[ \\mathbf{H}_{gc} = \\mathbf{H}_{rb} \\oplus (\\widetilde{\\mathbf{Y}_{ne}} \\mathbf{W}_{le})\\in \\mathbb{R}^{n\\times h_{gc}} \\]\nNow, by minimizing $\\ell_{gsg}\\!=\\!\\ell_{ne}\\!+\\!\\ell_{gc}$, a joint model for NE and GC can be obtained. In this model, GC receives NE labels to learn the causal effects from NE, while NE gets feedback through differentiable label transfer to further aid GC decision. In this sense, our model improves the integrity of graph structure generation compared with separately modelling each subtask or simple multitasking.\n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=.999\\linewidth]{pics\/re.pdf}\n    \\caption{Candidate retrieval and ranking framework for relation extraction.}\n    \\label{fig:re_overview}\n\\end{figure}\n\n\\section{Relation Extraction (RE)}\n\\label{sec:re}\nAs shown in Figure \\ref{fig:re_overview}, relation extraction (RE) conducts candidate retrieval and ranking in turn for each edge in graph structure $S$ to decide its predicate. For a question $q$, an edge $e$ connecting nodes $n_1$ and $n_2$ with mention $m_1,\\!m_2$ respectively, candidate retrieval recalls a set of predicates $P$ that can be bound to $e$. Note that unlike $e$, each predicate in $P$ is directional. Then, candidate ranking \\texttt{Rank}($P$,$q$,$m_1$,$m_2$) gives each predicate a score. This section details this procedure.\n\\paragraph{Candidate Ranking}\n\\label{sec:re_cand_rank}\nFor each $p_i\\in P$, we encode it together with $q,\\!m_1,\\!m_2$ by a RoBERTa encoder and pool them to $0\\!\\leq\\! s_i\\!\\leq\\! 1$ to score the predicate. If the direction of $p_i$ is $n_1\\!\\!\\rightarrow\\! n_2$, we join $q,m_1,m_2,p_i$ sequentially by [SEP] token as model input; otherwise (direction $n_2\\!\\!\\rightarrow\\! n_1$), the join order is $q,m_2,m_1,p_i$. By giving $s_i$ to each candidate, we can get the most proper predicates for $e$ by selecting those with highest scores. More details on training the ranking model can be found in Appendix \\ref{sec:appendix_re_ranking}.\n\\paragraph{Candidate Retrieval}\n\\citealp{zhang2021namer} proposed a straightforward way to retrieve candidates: if either $n_1$ or $n_2$ is a non-variable node, the predicates around that node in KB are viewed as candidates; otherwise, they trace $n_1$ or $n_2$ in other graph edges with non-variable nodes and view the predicates k-hop away from that node in KB as candidates (e.g. predicates 2-hop away from \\textit{dbo:person} are candidates for $?class\\text{-}?person$ in Figure \\ref{fig:re_overview}). We view this as the baseline in latter experiments.\n\n\\begin{algorithm}[t]\n\\caption{BeamSearchRE}\\label{alg:re_bs}\n\\LinesNumbered\n\\small\n\n\\SetKwFunction{FR}{Sample}\n\\SetKwFunction{FI}{inference}\n\\SetKwFunction{FG}{getNodePairsFromGraph}\n\\SetKwFunction{FC}{Retrieve}\n\\SetKwFunction{FCR}{Rank}\n\n\\begin{comment}\n\\KwIn{Question $q$, Query graph structure $S$, beam width $b$}\n\\KwOut{A beam of query graphs $\\mathcal{B}$}\n$\\mathcal{B}\\leftarrow\\{\\epsilon\\}$\\;\n$\\mathcal{S}_{undef}\\leftarrow\\mathcal{S}$\\;\n\\While{$\\mathcal{S}_{undef}\\ne\\varnothing$}{\n    $\\mathcal{B}'\\leftarrow\\{\\epsilon\\}$\\;\n    $e=(n_1,n_2)\\leftarrow$\\FR$(\\mathcal{S}_{undef})$\\;\n    \\For{$G\\in\\mathcal{B}$\n   \n    }{\n    $\\mathcal{P}\\leftarrow$\\FC$(G, n_1, n_2)$\\;\n   \n    $\\mathcal{C}=\\{(p_i,s_i)\\}\\leftarrow$\\FCR$(\\mathcal{P}, q, m_1, m_2)$\\;\n    $\\mathcal{B'}\\leftarrow\\mathcal{B'}\\cup \\{G\\}\\times\\mathcal{C}$\\;\n    }\n    $\\mathcal{B}\\leftarrow\\mathcal{B'}.\\text{topk}(b)$\\;\n    $\\mathcal{S}_{undef}.\\text{remove}(e)$\\;\n   \n   \n   \n   \n   \n   \n}\n\\end{comment}\n\n\\KwIn{Question $q$, Query graph structure $S$, beam width $b$}\n\\KwOut{A beam of query graphs $B$}\n$B\\leftarrow\\{\\{\\}\\}$;\\tcp*[h]{Start with an empth graph}\n\n$S_{pend}\\leftarrow S$;\\tcp*[h]{All edges are pending}\n\n\\While{$S_{pend}\\ne\\varnothing$}{\n    $B'\\leftarrow\\{\\}$\\;\n    \\tcp{Select a pending edge}\n    $e=(n_1,n_2)\\leftarrow$\\FR$(S_{pend})$\\;\n    \\For{$G\\in B$\n   \n    }{\n   \n    $P\\leftarrow$\\FC$(G, n_1, n_2)$\\;\n   \n    \\tcp{$n_1\/n_2$ has mention $m_1\/m_2$}\n    $C=\\{(p_i,s_i)\\}\\leftarrow$\\FCR$(P, q, m_1, m_2)$\\;\n   \n    \\tcp{Extend previous beams}\n    \\For{$(p_i,s_i)\\in C$}{\n        $B'\\leftarrow B'\\cup \\{G \\cup \\{(n_1, n_2, p_i, s_i)\\} \\}$;\n    }\n    }\n    $B\\leftarrow B'.\\text{topk}(b)$;\\tcp*[h]{Set up new beams}\n    \n    \\tcp{Mark \\textit{e} as determined}\n    $S_{pend}\\leftarrow S_{pend}\\setminus \\{e\\}$\\;\n   \n   \n   \n   \n   \n   \n}\n\\end{algorithm}\n\nHowever, this results in a candidate scale O($n^k$)\\footnote{n is the node degree in KB, k is the edge number in $S$}, making it unscalable to multi-hop queries (k$\\uparrow$) and large KBs (n$\\uparrow$). Here, we propose Algorithm \\ref{alg:re_bs} to limit the scale to O(n). We start by selecting an edge between $n_1^a$ and $n_1^b$ containing a non-variable node (e.g. edge \\textit{?class}-\\textit{dbr:Swinhoe's\\_Crake} in Figure \\ref{fig:re_overview}), retrieving all adjacent predicates of that node in KB and use \\texttt{Rank} to select the most proper predicate $p_1$ (e.g. dbp:named\\_by) of score $s_1$, this forms a subgraph $G$=\\{($n_1^a$,$n_1^b$,$p_1$)\\} with only one edge whose score is $s_1$. Then, we sample another edge between $n_2^a$ and $n_2^b$ (e.g. \\textit{?class}-\\textit{?person}) and retrieve its candidates $P$ based on $G$ (e.g. $G$ already entails \\textit{?class}=\\textit{dbr:bird}, so all neighbors of \\textit{dbr:bird} forms $P$), this process is denoted as \\texttt{Retrieve}($G,n_2^a,n_2^b$). Now, we use \\texttt{Rank} to select $p_2$ of score $s_2$ from $P$, add ($n_2^a$,$n_2^b$,$p_2$) to subgraph $G$ and update its score as $s_1*s_2$. Repeating this loop until all edges are bound with a predicate, we finally form a query graph.\n\nNote that for each edge, the candidate scale given by \\texttt{Retrieve} is O(n), since it is always among the neighbors of one or several KB nodes. Also, to improve the recall of query graphs, this process can trivially be extended as a beam search with each step maintaining a beam of subgraphs $B$, ordering each subgraph by $\\prod_i s_i$ as in Algorithm \\ref{alg:re_bs}.\n\n\\begin{table}[t]\n\\centering\n\\resizebox{.99\\columnwidth}{!}{\n\\begin{tabular}{c l c c c}\n    \\toprule\n    \\bfseries Type & \\bfseries Methods & \\bfseries P & \\bfseries R & \\bfseries F1\\\\\n    \\cmidrule(lr){1-5}\n    \\multirow{2}{*}{\\bfseries I} &\n    NSQA \\citep{kapanipathi2021leveraging} & .448 & .458 & .445 \\\\\n    & EDGQA \\citep{hu2021edg} & .505 & .560 & .531 \\\\\n    \\cmidrule(lr){1-5}\n    \\multirow{4}{*}{\\bfseries II} &\n    QAmp \\citep{vakulenko2019message} & .250 & .500 & .330 \\\\\n    & NAMER \\citep{zhang2021namer} & .438 & .438 & .435 \\\\\n    & STaG-QA \\citep{ravishankar2021two} & \\bfseries.745 & .548 & .536 \\\\\n    & \\bfseries Crake (ours) & .722 & \\bfseries.731 & \\bfseries.715 \\\\\n    \\bottomrule\n\\end{tabular}}\n\\caption{End-to-end performance on LC-QuAD 1.0 test set. I\/II stands for methods with\/without aux tools. We re-implement NAMER since its results on LC-QuAD is not provided; however, NAMER suffers from severe timeout issues on DBpedia to limit its performance, so we restrict each candidate query to run at most 45s in practice (which already requires $\\sim\\!$15h for a complete evaluation run).}\n\\label{tab:e2e_eval}\n\\end{table}\n\n\\section{Experiments}\n\\label{sec:exp}\n\\paragraph{Dataset} We adopt LC-QuAD 1.0 \\citep{trivedi2017lc}, a predominant open-domain English KBQA benchmark based on DBpedia \\citep{auer2007dbpedia} 2016-04, to test the performance of our system. We randomly sample 200 questions from train data as dev set and follow the raw test set, resulting in a 4800\/200\/1000 train\/dev\/test split. More details on the dataset can be found in Appendix \\ref{sec:appendix_dataset}. Like \\citealp{zhang2021namer}, we do not experiment on multiple datasets due to the high annotation cost involved, however, we conduct no dataset-specific optimizations in this work, so we consider the large improvements on LC-QuAD and detailed discussions sufficient to prove our effectiveness.\n\\paragraph{Annotation}\n\\label{sec:annotation}\nWe annotate the dataset with the mention of each node in query graph, e.g. the mention \"class\" and \"person\" for the node \\textit{?class} and \\textit{dbo:person} respectively in Figure \\ref{fig:two_stage}. With the annotation, we obtain the gold data ($G_{ne},\\!G_{gc}$) to train our models. Appendix \\ref{sec:appendix_annotation} details the annotation process. \n\n\\paragraph{Baselines}\nWe evaluate our method against existing works both with and without auxiliary tools. With aux tools, \\citealp{kapanipathi2021leveraging} constructs query graphs based on the AMR of questions; \\citealp{hu2021edg} designs rules on constituency tree to aid query graph formation. For independent pipelines without aux tools, \\citealp{vakulenko2019message} parses URI mentions from the question to match with KB via confidence score passing; \\citealp{ravishankar2021two} combines a generative graph-skeleton decoder with entity and relation detector to form a query; \\citealp{zhang2021namer} co-trains a pointer generator with the node extractor to build a query graph, it's worth to note that this work also requires the node-to-mention \\nameref{sec:annotation} for training.\n\n\\paragraph{Setup}\nWe utilize the RoBERTa-large released by huggingface \\citep{wolf-etal-2020-transformers} as our encoder. All experiments are averaged on two runs on an NVIDIA A40 GPU. For the GSG model, we train for at most 500 epochs (\\textasciitilde6 GPU-hours) and report the best checkpoint on dev set; for the RE model, we train for 20 epochs (\\textasciitilde16 GPU-hours) and report the final checkpoint. For hyperparameters, we find no apparent performance variance on dev set as long as the values are in reasonable range (e.g. $64\\!\\leq\\! h_{le}\\!\\leq\\!1024$, $1e\\text{-}6\\!\\leq\\! lr_{gsg}\\!\\leq\\!2e\\text{-}5$) so no further tuning is involved. See the full setting in Appendix \\ref{sec:appendix_hyper}.\n\n\\begin{table*}[t]\n\\centering\n\\resizebox{1.85\\columnwidth}{!}{\n\\begin{tabular}{l l c c c c c c c c}\n    \\toprule\n    \\multirow{2}{*}{\\bfseries Methods} &\n    \\multirow{2}{*}{\\bfseries Decoder Parameters} &\n    \\multicolumn{3}{c}{\\bfseries NE Accuracy} &\n    \\multicolumn{2}{c}{\\bfseries GSG Accuracy} &\n    \\multicolumn{3}{c}{\\bfseries End-to-end}\\\\\n    \\cmidrule(lr){3-5}\n    \\cmidrule(lr){6-7}\n    \\cmidrule(lr){8-10} & &\n    P & R & F1 &\n    EM & Actual &\n    P & R & F1\\\\\n    \\cmidrule(lr){1-10}\n    Seq2seq & 76.67M\\space($\\times1$) &  .895 & .901 & .897 & .695 & .768 & .653 & .674 & .654\\\\\n    TF  & 0.66M\\space\\space\\space($\\times1\/100$) & .895 & .901 & .897 & .728 & .795 & .655 & .674 & .657\\\\\n    TF+SMTL & 0.66M\\space\\space\\space($\\times1\/100$) & .901 & .904 & .902 & .735 & .805 & .665 & .684 & .667\\\\\n   \n    \\cmidrule(lr){1-10}\n    TF+Causal & 3.03M\\space\\space\\space($\\times1\/25$) & \\bfseries.909 & \\bfseries.914 & \\bfseries.911 & \\bfseries.755 & \\bfseries.828 & \\bfseries.677 & \\bfseries.696 & \\bfseries.680\\\\\n    \\bottomrule\n\\end{tabular}}\n\\caption{Experiments on table-filling and causal-modelling. Seq2seq and TF adopt a generative decoder and table-filler in GC respectively, while both deal with NE and GC by separate models. TF+SMTL (simple multitask learning) co-trains NE and GC by directly adding losses without modelling their intrinsic causal effects. TF+Causal denotes our full approach which models the causal effects between NE and GC by label transfer. We report the node-level P\/R\/F1 in NE, the exact-match (EM) and actual accuracy (that ignores variable mentions in judging accuracy) in GSG, and the overall answer-level P\/R\/F1 on LC-QuAD 1.0 dev set for comparison.}\n\\label{tab:ablation}\n\\end{table*}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=.999\\linewidth]{pics\/learning_curve.pdf}\n    \\caption{EM accuracy of GSG during training. See the meaning of each series in Table \\ref{tab:ablation}.}\n    \\label{fig:learn_curve}\n\\end{figure}\n\n\\subsection{End-to-end Evaluation}\nAs shown in Table \\ref{tab:e2e_eval}, our method, Crake, outperforms all former methods by a large $\\!\\sim\\!$17\\% margin on F1, becoming the new SoTA of LC-QuAD 1.0. Surpassing methods requiring aux tools (I) on all metrics, we present the effectiveness of independent pipelines (II) that avoid cascading errors. Also, we achieve consistent answer precision and recall to surpass other methods in II on F1, showing the superiority of our pipeline design, which is further discussed in the sections below.\n\n\\subsection{Effects of Tabel-Filling}\n\nAs explained in Section \\ref{sec:gc}, modelling GC as a sequence-generation causes a few issues that can be overcome by table-filling. Specifically, the sequence ambiguity confuses the learning process and requires large decoders to grasp the sequence generation policy, which may slow down the convergence. Besides, the exposure bias harms the decoding accuracy of the model at inference. This section, we try to verify such effects by experiments. To enable the comparison with sequence-generation, we construct a generative decoder as in \\citealp{zhang2021namer} as the baseline, which sequentially generates the connected node pairs in the graph structure to represent the edges. We train the generative model under the same settings (e.g. learning rate, warmup, epochs, etc.), resulting in the performance of \\texttt{Seq2seq} in Table \\ref{tab:ablation}.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=.93\\linewidth]{pics\/case_study_3.pdf}\n    \\caption{Case study on the effects of causal-modelling.}\n    \\label{fig:case_study}\n\\end{figure}\n\nComparing with the table-filling model (i.e. \\texttt{TF} in Table \\ref{tab:ablation}), \\texttt{Seq2seq} comes short in the accuracy of graph structure, indicating the negative effects of the exposure bias on predicting accuracy. Meanwhile, \\texttt{TF} requires only 1\/100 of \\texttt{Seq2seq}'s parameters to achieve comparable or better results, we attribute this to the removal of sequence ambiguity which frees the model from acquiring the complex and ambiguous scheme of sequence-generation. This speculation is further verified in Figure \\ref{fig:learn_curve}, in which \\texttt{TF} converges distinctly quicker than \\texttt{Seq2seq} since the simultaneous decision of all edges is well-defined and easier to learn. Thus, compared with sequence-modelling, handling GC via table-filling reduces model size and boosts training, which is essential for real-world applications.\n\n\\subsection{Effects of Causal-Modelling}\n\nWe propose a joint model to learn the NE-GC causalities in Section \\ref{sec:causal_modelling}, to discuss its effects, we compare it with two alternatives in Table \\ref{tab:ablation}: 1) using two separate models for NE and GC (\\texttt{TF} in Table \\ref{tab:ablation}) like \\citealp{ravishankar2021two}, 2) co-training NE and GC by sharing encoder and adding losses (\\texttt{TF+SMTL} in Table \\ref{tab:ablation}) like \\citealp{shen2019multi}. As shown, co-training consistently surpasses separate models by grasping the shared knowledge between NE and GC, nevertheless, our causal-modelling approach (\\texttt{TF+Causal}) further outperforms co-training. In detail, though \\texttt{TF+Causal} has similar results with \\texttt{TF+SMTL} in NE, it achieves better accuracy for overall GSG (NE+GC) and excels in end-to-end metrics. Therefore, we infer that causal-modelling improves the integrity of the GSG stage by expressing the internal causalities between its subtasks. To better understand this, we perform a case study in Figure \\ref{fig:case_study}, in which \\texttt{TF} fails to realize that \"skier\" also corresponds to a type node; in contrast, \\texttt{TF+SMTL} extract all nodes correctly by learning both NE and GC labels, but it still fails in generating a correct graph structure. Finally, \\texttt{TF+Causal} utilizes the VT tag of \"skier\" in NE predictions and correctly connects the II-IV edge in GC. Thus, Figure \\ref{fig:case_study} demonstrates the usage of causal effects to reach higher accuracy in GSG.\n\n\\subsection{Analysis on Beam-Search RE}\n\n\\begin{table}[t]\n\\centering\n\\resizebox{1.\\columnwidth}{!}{\n\\begin{tabular}{l c c c c c c}\n    \\toprule\n    \\multirow{2}{*}{\\bfseries Methods} & \n    \\multicolumn{3}{c}{\\bfseries Accuracy} &\n    \\multicolumn{3}{c}{\\bfseries Efficiency}\\\\\n    \\cmidrule(lr){2-4}\n    \\cmidrule(lr){5-7} & \\bfseries P & \\bfseries R & \\bfseries F1 & \\bfseries 1-hop & \\bfseries 2-hop & \\bfseries 3-hop\\\\\n    \\cmidrule(lr){1-7}\n    Baseline & .560 & .566 & .556 & \\bfseries0.12s & 42.4s & 84.2s\\\\\n    BeamSearch & \\bfseries.677 & \\bfseries.696 & \\bfseries.680 & \\bfseries0.12s & \\bfseries1.06s & \\bfseries2.72s\\\\\n    \\bottomrule\n\\end{tabular}}\n\\caption{Performance comparison between our beam-search RE algorithm and its baseline in Section \\ref{sec:re}. Accuracy refers to the answer-level P\/R\/F1, efficiency is measured by the average run time on 1\/2\/3-hop queries.}\n\\label{tab:re_ablation}\n\\end{table}\n\nIn this section, we compare our beam-search RE algorithm with its baseline. As stated in Section \\ref{sec:re}, by alternately performing retrieval and ranking on each edge (rather than retrieving the candidates of every edge before ranking), our approach lowers the scale of candidate predicates on multi-hop queries to get better efficiency, which is verified in Table \\ref{tab:re_ablation}. In detail, \\texttt{BeamSearch} costs substantially less time than \\texttt{Baseline} in 2 and 3-hop queries (note that for 1-hop queries, two methods reduce to a same process with similar time costs). Since \\texttt{BeamSearch} only operates on the neighbors of certain KB nodes, it avoids the retrieval of 2-hop neighbors, which requires considerable time on DBpedia, to improve efficiency. In addition, by pruning off useless candidates in \\texttt{Baseline}, \\texttt{BeamSearch} also achieves higher overall KBQA accuracy in Table \\ref{tab:re_ablation}.\nTherefore, Algorithm \\ref{alg:re_bs} transcends previous methods to reveal an efficient and accurate solution for ranking-based RE scalable to KB size and query complexity.\n\n\n\\section{Related Works}\n\\paragraph{KBQA via Semantic Parsing} A mainstream to solve KBQA is semantic parsing \\citep{yih2016value} which converts a question to a KB query to get answers. Due to the graph-like structure of KB queries, prior works construct query graphs to represent queries in semantic parsing. Among them, some works \\citep{zafar2018formal, chen2021formal} only focus on predicting the graph structure given node inputs. To perform end-to-end QA, \\citealp{hu2017answering} leverages the dependency parsing tree to match KB subgraphs for answers; \\citealp{kapanipathi2021leveraging} builds the query graph by transforming and linking the AMR \\citep{banarescu2012abstract} of the question; \\citealp{hu2021edg} uses the constituency tree to compose an entity description graph representing the query graph structure. Requiring aux tools or data structures, these works may be subjected to cascading errors. \\citealp{yih2015semantic} overcomes this by an independent stage-transition framework to generate the query graph, \\citealp{hu2018state} extends the transitions to express more complex graphs. Besides, \\citealp{zhang2021namer} adopts a pointer generator to decode graph structure, \\citealp{ravishankar2021two} generates the query skeleton by a seq2seq decoder. Unlike these methods that model the query graph as a sequence (by state-transition or generative decoder), we decode all edges at once via a table-filler in graph structure generation.\n\n\\paragraph{Modelling causal effects} Causality occurs in various deep-learning scenarios between multiple channels or subtasks, existing works models the causality for better performance. \\citealp{niu2021counterfactual} mitigates the false causal effects in VQA \\citep{antol2015vqa} to overcome language bias; \\citealp{zeng2020counterfactual} dispels the incorrect causalities from different input channels of NER by generating counterfacts. \\citealp{chen-etal-2020-exploring-logically} utilizes the inter-subtask causalities to improve multitask learning for JERE \\citep{li2014incremental}, ABSA \\citep{kirange2014aspect}, and LJP. Unlike them, we formulate and utilize the internal causal effects in KBQA.\n\n\n\\section{Conclusion}\nIn this work, we formalize the generation of query graphs in KBQA by two stages, namely graph structure generation (GSG) and relation extraction (RE). In GSG, we propose a table-filling model for graph composition to avoid the ambiguity and bias of sequence-modelling, meanwhile, we encode the inherent causal effects among GSG by a label-transfer block to improve the stage integrity. In RE, we introduce an effective beam-search algorithm to retrieve and rank predicates in order for each edge, which turns out to be scalable for large KBs and multi-hop queries. Consequently, our approach substantially surpasses previous state-of-the-arts in KBQA, revealing the effectiveness of our pipeline design. Detailed experiments also validate the effects of all our contributions.\n\n\n\\section{Limitation}\nAdmittedly, our approach endures certain limitations as discussed below.\n\\paragraph{Query Expressiveness} Like most semantic parsing systems, we fail to cover all the operations of SPARQL, limiting our capability to compose queries with complex \\texttt{filter} or property path. For the conciseness of our system, we only focus on constructing triples in the multi-hop query graph in this paper, while we plan to incorporate more functions into Crake in the future to improve the expressiveness of the system.\n\\paragraph{Annotation Cost} Training models with node mentions require expensive manual annotations, which is impractical for us to conduct on every popular KBQA dataset. As explained in Section \\ref{sec:exp}, without data-oriented optimization, we believe the significant gain presented adequate to verify our contributions. Further, we expect to extenuate such costs in two directions for the future: 1) some modules of our framework (e.g. NE) is generalizable to other English questions, gifting it the potential to be transferred to other datasets without re-training; 2) few-shot \\citep{wang2020generalizing} and active \\citep{aggarwal2014active} learning techniques aids the model to reach competitive performance with a small portion of annotated data, which can be explored in our framework to reduce annotation cost.\n\n\\section*{Acknowledgements}\n\nThis work was supported by National Key R\\&D Program of China (2020AAA0105200) and NSFC under grant U20A20174. The corresponding author of this work is Lei Zou (zoulei@pku.edu.cn). We would like to thank Zhen Niu and Sen Hu for their kind assistance on this work. We also appreciate anonymous reviewers for their valuable comments and advises.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Methodology}\n\\label{sect_methodology}\n\nIn order to facilitate our analysis, we start by making the following assumptions that are standard in the multi-armed bandits literature.\n\\begin{assumption} For all $u \\in [N]$, $i \\in [M]$ and $t \\in \\mathbb{N}$, the rewards $R_{t, u, i}$ are independent and $\\eta$-sub-Gaussian with mean $\\theta^*_{u, i} \\in [0, B]$.\n\n\\label{rew_assumptio}\n\\end{assumption}\n\nTo model the dependency between the mean rewards obtained from different user-item pairs, we employ the following assumption. We first present our algorithm and theoretical results under the general setting given by this assumption, and specialize for the setting of the collaborative filtering in following sections.\n\\begin{assumption}\n\\label{low_assum}\nThe mean reward matrix $\\vect{\\Theta}^*$ belongs to a known structure set $\\mathcal{L} \\subseteq \\mathbb{R}^{N \\times M}$.\n\\end{assumption}\n\nIn order to make use of initial historical data possibly available to the provider, we assume that the algorithm has access to an initial rough estimate $\\overline{\\vect{\\Theta}}$ that satisfies $\\|\\overline{\\vect{\\Theta}} - \\vect{\\Theta}^* \\|_\\text{F} \\leq G$. Such an estimate can be constructed using an off-the-shelf low-rank matrix completion algorithm on the initialization data. If such observations are not readily available at the time of initialization, they can be obtained by randomly sampling some of the user-item allocation pairs once. It is worth to note that one can also set $\\overline{\\vect{\\Theta}} = \\vect{0}$ and let $G$ be some number satisfying $\\|\\vect{\\Theta}^* \\|_\\text{F} \\leq G$.\n\n\\vspace{-3pt}\n\\begin{algorithm}\n\\caption{Structured Combinatorial Multi-Armed Bandit}\n\\begin{algorithmic}\n\\Require horizon $T$, initial estimate $\\overline{\\vect{\\Theta}} \\in \\mathbb{R}^{N \\times M}$ with $\\|\\overline{\\vect{\\Theta}} - \\vect{\\Theta}^* \\|_\\text{F} \\leq G$.\n\\For{$t = 1, 2, \\dots, T$}\n\\State Find the regularized least squares estimate $ \\widehat{\\vect{\\Theta}}_t = \\argmin_{\\vect{\\Theta} \\in \\mathcal{L}} \\left \\{ L_{2,t}(\\vect{\\Theta}) + \\gamma \\|\\vect{\\Theta} - \\overline{\\vect{\\Theta}} \\|_2^2 \\right \\}$\n\\State Construct the confidence set $\\mathcal{C}_t = \\{ \\vect{\\Theta} \\in \\mathcal{L} : \\|\\vect{\\Theta} - \\widehat{\\vect{\\Theta}}_t \\|_{2, E_t} \\leq \\sqrt{ \\beta_t^*(\\delta, \\alpha, \\gamma)}\\}$\n\\State Compute the action vector $\\vect{X}_t = \\argmax_{\\vect{X} \\in \\mathcal{X}_t} \\max_{\\vect{\\Theta} \\in \\mathcal{C}_t} \\; \\langle \\vect{X}, \\vect{\\Theta} \\rangle$\n\\State Play the arms $\\mathcal{A}_t$ according to $\\vect{X}_t$ \n\\State Observe $R_{t, u, i}$ for all $(u, i) \\in \\mathcal{A}_{t}$\n\\EndFor\n\\end{algorithmic}\n\\label{alg_low}\n\\end{algorithm}\n\\vspace{-3pt}\n\nOur method summarized in Algorithm \\ref{alg_low} follows the standard OFU (Optimism in Face of Uncertainty) principle \\cite{abbasi_2011}. It maintains a confidence set $\\mathcal{C}_t$ which contains the true parameter $\\vect{\\Theta}^*$ with high probability and chooses the allocation $\\vect{X}_t$ according to\n\\begin{equation}\n    \\vect{X}_t = \\argmax_{\\vect{X} \\in \\mathcal{X}_t} \\left \\{ \\max_{\\vect{\\Theta} \\in \\mathcal{C}_t} \\; \\langle \\vect{X}, \\vect{\\Theta} \\rangle \\right \\}\n\\label{low_oful}\n\\end{equation}\nTypically, the faster the confidence set $\\mathcal{C}_t$ shrinks, the lower regret we have. However, the main difficulty is to construct a series of $\\mathcal{C}_t$ that leverage the combinatorial observation model as well as the structure of the parameter so that we have low regret bounds. In this work, we consider constructing confidence sets that are centered around the regularized least square estimates. We let the cumulative squared prediciton error at time $t$ be\n\\begin{equation*}\n    L_{2,t}(\\vect{\\Theta}) = \\sum_{\\tau=1}^{t-1} \\sum_{(u, i) \\in \\mathcal{A}_\\tau} (\\theta_{ui}  - R_{\\tau, u, i})^2,\n\\end{equation*}\nand define the regularized least squares estimate at time $t$ as\n\\begin{equation}\n    \\widehat{\\vect{\\Theta}}_t = \\argmin_{\\vect{\\Theta} \\in \\mathcal{L}} \\left \\{ L_{2,t}(\\vect{\\Theta}) + \\gamma \\|\\vect{\\Theta} - \\overline{\\vect{\\Theta}} \\|_2^2 \\right \\}.\n    \\label{least_squares_estimate_low}\n\\end{equation}\nThen, the confidence sets take the form $\\mathcal{C}_t := \\{ \\vect{\\Theta} \\in \\mathcal{L} : \\|\\vect{\\Theta} - \\widehat{\\vect{\\Theta}}_t \\|_{2, E_t} \\leq \\sqrt{\\beta_t}\\}$ where $\\beta_t$ is an appropriately chosen confidence parameter, and the regularized empirical 2-norm $\\| \\cdot \\|_{2, E_t}$ is\n\\begin{equation*}\n    \\| \\vect{\\Delta} \\|_{2, E_t}^2 := \\sum_{u=1}^{N} \\sum_{i=1}^{M} (n_{t, u, i} + \\gamma) (\\Delta_{ui})^2,\n\\end{equation*}\nwhere $n_{t, u, i} := \\sum_{\\tau=1}^{t-1} \\mathds{1} \\{(u,i) \\in \\mathcal{A}_\\tau\\}$ is the number of times item $i$ has been allocated to user $u$ before time $t$ (excluding time $t$).\nHence, the empirical 2-norm is a measure of discrepancy that weighs the entries depending on how much they have been explored. Roughly speaking, since the confidence ellipsoid constructed using the 2-norm is wider in directions that are not yet well-explored, the OFU step described in \\ref{low_oful} is more inclined to make allocations that include the corresponding user-item pairs. In order to obtain low-regret guarantees for the allocations, the first step is to choose correct $\\beta_t$ parameter such that $\\mathcal{C}_t$ will contain the true parameter $\\vect{\\Theta}^*$ for all $t$ with high probability.\nIn order to take advantage of the structure of the arms, we let $\\mathcal{N}(\\mathcal{F}, \\alpha, \\| \\cdot \\|_{\\text{F}})$ denote the $\\alpha$-covering number of $\\mathcal{F}$ in the Frobenious-norm $\\| \\cdot \\|_{\\text{F}}$, and let\n\\begin{equation*}\n    \\beta_t^*(\\delta, \\alpha, \\gamma) := 8 \\eta^2 \\log \\left(\\mathcal{N}(\\mathcal{L}, \\alpha, \\| \\cdot \\|_{\\text{F}}) \/ \\delta \\right) + 2 \\alpha t NM \\left [ 8 B + \\sqrt{8 \\eta^2 \\log(4NM t^2\/\\delta)} \\right]  + 4 \\gamma G^2.\n\\end{equation*}\nThen, the following Lemma establishes that if we set $\\beta_t = \\beta_t^*(\\delta, \\alpha, \\gamma)$, the resulting confidence sets have the desired properties.\n\\begin{lemma} For any $\\delta > 0$, $\\alpha > 0$, $\\gamma > 0$, let $\\widehat{\\vect{\\Theta}}_t$ be the regularized least squares estimate given in \\ref{least_squares_estimate_low}. If the confidence sets are given as\n\\begin{equation}\n    \\mathcal{C}_t := \\{ \\vect{\\Theta} \\in \\mathcal{L} : \\|\\vect{\\Theta} - \\widehat{\\vect{\\Theta}}_t \\|_{2, E_t} \\leq \\sqrt{ \\beta_t^*(\\delta, \\alpha, \\gamma)}\\},\n    \\label{conf_sets_low}\n\\end{equation}\nthen with probability at least $1 - 2 \\delta$, $\\mathcal{C}_t \\ni \\vect{\\Theta}^*$, for all $t \\in \\mathbb{N}$.\n\\end{lemma}\n\nFinally, we show that if the structured combinatorial bandits algorithm follows the OFU allocations given in \\eqref{low_oful} while constructing the confidence sets according to \\eqref{conf_sets_low}, it obtains the following overall regret guarantee:\n\\begin{theorem}\\label{thm_alloc_regret}\nUnder Assumptions \\ref{rew_assumptio} and \\ref{low_assum}, for any $\\delta > 0$,  $\\alpha > 0$, $\\gamma \\geq 1$, with probability $1 - 2\\delta$, the cumulative regret of Algorithm \\ref{alg_low} is bounded by\n\\begin{equation*}\n    \\mathcal{R}(T, \\pi) \\leq \\sqrt{ 8 N M \\beta_T^* (\\delta, \\alpha, \\gamma) T \\log \\left(1 + T \/ \\gamma \\right) }.\n\\end{equation*}\n\\end{theorem}\n\n\\subsection{Low-Rank COMbinatorial Bandits (LR-COMB)}\n\n\\label{sect_lrcb}\n\nAs common in collaborative filtering settings, the correlation between users and arms can be captured through a matrix factorization model that leads to a low-rank mean reward matrix. \nEach user $u$ (item $i$) is associated with a feature vector $\\vect{p}_u$ ($\\vect{q}_i$) in a shared $R$-dimensional space (typically $R \\ll M, N$), and the mean reward of each user-item allocation pair is given by $\\theta_{ui}^* = \\vect{p}_u^\\textrm{T} \\vect{q}_i$. Consequently, the mean reward matrix satisfies the factorization $\\vect{\\Theta}^* = \\vect{P} \\vect{Q}^\\textrm{T}$ for some $\\vect{P} \\in \\mathbb{R}^{N \\times R}$ and $\\vect{Q} \\in \\mathbb{R}^{M \\times R}$. Based on this observation and the boundedness condition given in Assumption \\ref{rew_assumptio}, we can choose the structure set $\\mathcal{L}$ as\n\\begin{equation}\n    \\mathcal{L} = \\{ \\vect{\\Theta} \\in \\mathds{R}^{N \\times M} : \\text{rank}(\\vect{\\Theta}) \\leq R, \\theta_{ui} \\in [0, B], \\forall u,i\\}.\n    \\label{low_l}\n\\end{equation}\n\nThen, Lemma \\ref{lemma_covering} in the appendix shows that the covering number for $\\mathcal{L}$ given in equation \\eqref{low_l} is upper bounded by $\\log \\mathcal{N}(\\mathcal{L}, \\alpha, \\| \\cdot \\|_{\\text{F}}) \\leq (N + M + 1) R \\log ( 9B \\sqrt{NM} \/ \\alpha )$. Therefore, the regret guarantee for a setting with low-rank mean reward matrix becomes:\n\\begin{theorem}[Regret of LR-COMB] \\label{low_rank_regret_thm}\nUnder Assumption \\ref{rew_assumptio} and Assumption \\ref{low_assum} with $\\mathcal{L}$ given in \\eqref{low_l}, the Algorithm \\ref{alg_low} achieves cumulative regret\n\\begin{equation}\n    \\mathcal{R}(T, \\pi) = \\widetilde{\\mathcal{O}} \\left( \\sqrt{N M (N+M) RT} \\right),\n\\end{equation}\n\\end{theorem}\nwhere $\\widetilde{\\mathcal{O}}$ is the big-O notation, ignoring the poly-logarithmic factors of $N, M, T, R$. \n\nIn comparison, if we were to ignore the low-rank structure between the mean rewards obtained from user-item allocation pairs and apply the standard combinatorial bandit algorithms (e.g., CUCB \\cite{chen_2013}), we would suffer \\smash{$\\widetilde{\\mathcal{O}} ( N M \\sqrt{T} )$} regret \\cite{kveton_2015}. Since $R \\ll M, N$ in many applications of collaborative filtering, our algorithm significantly outperforms this naive approach. As common in the literature of combinatorial bandits, one possible approach to improve upon our theoretical analysis might be by assuming a problem setting where at most $K$ of the arms can be played in each round. However, our current analysis techniques do not allow us to incorporate and leverage such an assumption together with the low-rank structure of collaborative filtering. \n\n\\textbf{Implementation via Matrix Factorization: }\n\\label{section_mf}\nIn order to efficiently solve optimization problems  \\eqref{low_oful} and \\eqref{least_squares_estimate_low} in large scales, we take advantage of the matrix factorization model. As a result, we factorize $\\vect{\\Theta} = \\vect{P} \\vect{Q}^\\textrm{T}$ where $\\vect{P} \\in \\mathbb{R}^{N \\times R}$ and $\\vect{Q} \\in \\mathbb{R}^{M \\times R}$, and solve the problems by optimizing over $\\vect{P}$ and $\\vect{Q}$ rather than directly optimizing over $\\vect{\\Theta}$. Even if the problem \\eqref{least_squares_estimate_low} is not convex in the joint variable ($\\vect{P}$, $\\vect{Q}$), it is convex in $\\vect{P}$ for fixed $\\vect{Q}$ and it is convex in $\\vect{Q}$ for fixed $\\vect{P}$. Therefore, an alternating minimization algorithm becomes a feasible choice to find a reasonable solution for the least squares problem. Similarly, an alternating minimization approach is also useful to solve the problem \\eqref{low_oful}. We can fix an allocation $\\vect{X}$ and minimize over $\\vect{P}$ and $\\vect{Q}$. Then, for fixed $\\vect{P}$ and $\\vect{Q}$, the allocation $\\vect{X}$ is determined through the dual decomposition mechanism described in the section \\ref{sect_opt_allocations}. We call the resulting algorithm LR-COMB with Matrix Factorization and present it as Algorithm \\ref{alg_mf} in the Appendix. \n\n\\section{Implementation via Matrix Factorization}\n\nThe following algorithm describes an efficient implementation of our Low-Rank Combinatorial Bandit algorithm using matrix factorization. Note that converged $\\widehat{\\vect{\\Theta}}_t$ and $\\vect{X}_t$ are not necessarily the optimum solution for problems  \\eqref{low_oful} and \\eqref{least_squares_estimate_low} since the problems are not convex. However, the alternating optimization algorithm guarantees that, in each iteration, the objective value only decreases for \\eqref{least_squares_estimate_low}. Similarly, the objective value for \\eqref{low_oful} increases in each iteration of the alternating optimization.\n\n\\begin{algorithm}\n\\caption{LR-COMB with Matrix Factorization}\n\\begin{algorithmic}\n\\Require horizon $T$, initial estimate $\\overline{\\vect{\\Theta}} \\in \\mathbb{R}^d$ with $\\|\\overline{\\vect{\\Theta}} - \\vect{\\Theta}^* \\|_\\text{F} \\leq G$, parameters $\\delta, \\alpha > 0$, $\\gamma \\geq 1$.\n\\For{$t = 1, 2, \\dots, T$}\n\\State randomly initialize $\\widehat{\\vect{P}}$ and $\\widehat{\\vect{Q}}$\n\\While{convergence criterion not satisfied}\n    \\State $\\widehat{\\vect{P}} \\gets \\argmin_{\\vect{P} \\in \\mathbb{R}^{N \\times R}} \\left \\{ \\sum_{\\tau=1}^{t-1} \\sum_{(u, i) \\in \\mathcal{A}_\\tau} (\\vect{p}_u^\\textrm{T} \\vect{q}_i  - R_{\\tau, u, i})^2 + \\gamma \\|\\vect{P} \\vect{Q}^\\textrm{T} - \\overline{\\vect{\\Theta}} \\|_\\text{F}^2 \\right \\}$\n    \\State $\\widehat{\\vect{Q}} \\gets \\argmin_{\\vect{Q} \\in \\mathbb{R}^{M \\times R}} \\left \\{ \\sum_{\\tau=1}^{t-1} \\sum_{(u, i) \\in \\mathcal{A}_\\tau} (\\vect{p}_u^\\textrm{T} \\vect{q}_i  - R_{\\tau, u, i})^2 + \\gamma \\|\\vect{P} \\vect{Q}^\\textrm{T} - \\overline{\\vect{\\Theta}} \\|_\\text{F}^2 \\right \\}$\n\\EndWhile\n\\State $\\widehat{\\vect{\\Theta}}_t \\gets \\widehat{\\vect{P}} \\widehat{\\vect{Q}}^\\textrm{T}$\n\\State $\\vect{X} \\gets \\mathbb{1}_{N \\times M}$, $\\vect{P} \\gets \\widehat{\\vect{P}}$, $\\vect{Q} \\gets \\widehat{\\vect{Q}}$\n\\While{convergence criterion not satisfied}\n\\While{convergence criterion not satisfied}\n\\State $\\vect{P} \\gets \\argmax_{\\vect{P} \\in \\mathbb{R}^{N \\times R}} \\langle \\vect{X}, \\vect{P} \\vect{Q}^\\textrm{T} \\rangle$ s.t. $\\|\\vect{P} \\vect{Q}^\\textrm{T} - \\widehat{\\vect{\\Theta}}_t \\|_{2, E_t} \\leq \\sqrt{ \\beta_t^*(\\delta, \\alpha, \\gamma)}$\n\\State $\\vect{Q} \\gets \\argmax_{\\vect{Q} \\in \\mathbb{R}^{M \\times R}} \\langle \\vect{X}, \\vect{P} \\vect{Q}^\\textrm{T} \\rangle$ s.t. $\\|\\vect{P} \\vect{Q}^\\textrm{T} - \\widehat{\\vect{\\Theta}}_t \\|_{2, E_t} \\leq \\sqrt{ \\beta_t^*(\\delta, \\alpha, \\gamma)}$\n\\EndWhile\n\n\\State $\\vect{\\Theta} \\gets \\vect{P} \\vect{Q}^\\textrm{T}$\n\n\\While{convergence criterion not satisfied}\n    \\For{$u \\in [N]$} \n        \\State $\\vect{x}_u\\gets \\argmax_{\\vect{x}} \\left \\{ \\vect{x}^\\textrm{T} (\\vect{\\theta}_u - \\vect{\\lambda}) \\middle | \\vect{x} \\in \\{0, 1\\}^{M}, \\vect{x}^\\textrm{T} \\mathbb{1}_M \\leq d_{t, u} \\right \\} $\n    \\EndFor \n    \\State $\\vect{\\lambda} \\gets \\left[ \\vect{\\lambda} - \\alpha \\left( \\vect{c}_t - \\sum_{u = 1}^{N} \\vect{x}_u \\right) \\right]^{+} $\n\\EndWhile\n\\State $\\vect{X} \\gets [\\vect{x}_1, \\vect{x}_2, \\dots, \\vect{x}_N]^\\text{T}$\n\\EndWhile\n\\State $\\vect{X}_t \\gets \\vect{X}$\n\\State Play the arms $\\mathcal{A}_t$ according to $\\vect{X}_t$ \n\\State Observe $R_{t, u, i}$ for all $(u, i) \\in \\mathcal{A}_{t}$\n\\EndFor\n\\end{algorithmic}\n\\label{alg_mf}\n\\end{algorithm}\n\n\\section{Relaxation of Integer Program}\n\\label{appendix_num}\n\nA traditional linear integer program (IP) in matrix form is formulated as\n\\begin{align}\n\\begin{split}\n    \\max_{\\mathbf{x}} &\\; \\mathbf{t}^\\mathrm{T} \\mathbf{x}\\\\\n    \\text{s.t.} &\\; \\mathbf{A} \\mathbf{x} \\leq \\mathbf{b}\\\\\n    &\\; \\mathbf{x} \\in \\mathds{Z}_+^d\\\\\n    \\label{int_prog}\n\\end{split}\n\\end{align}\n\nThis problem can be relaxed to a linear program by dropping the integral constraints (setting $\\mathbf{x} \\in \\mathds{R}_+^d$). The integrality gap of an integer program is defined as the difference between the optimal values of the integer program in (IP) and its relaxed linear program. When the vector $\\mathbf{b}$ is integral and the matrix $\\mathbf{A}$ is totally unimodular (all entries are 1, 0, or -1 and every square sub-minor has determinant of +1 or -1) then the integrality gap is zero and the solution of the relaxed linear program is integer valued \\cite{bertsekas_1991}. \n\nHence, we can solve \\eqref{int_prog} by instead solving the following relaxed linear program:\n\\begin{align}\n\\begin{split}\n    \\max_{\\mathbf{x}} &\\; \\mathbf{t}^\\mathrm{T} \\mathbf{x}\\\\\n    \\text{s.t.} &\\; \\mathbf{A} \\mathbf{x} \\leq \\mathbf{b}\\\\\n    &\\; \\mathbf{x} \\in \\mathds{R}_+^d\\\\\n\\end{split}\n\\end{align}\n\nFor a matrix $\\vect{A}$ whose rows can be partitioned into two disjoint sets $\\mathcal{C}$  and $\\mathcal{D}$, the following four conditions together are sufficient for $\\vect{A}$ to be totally unimodular \\cite{heller_1957}:\n\\begin{enumerate}[nosep, labelindent= 2pt, align= left, labelsep=0.4em, leftmargin=*]\n    \\item Every entry in $\\vect{A}$ is $0$, $+1$, or $-1$.\n    \\item Every column of $\\vect{A}$ contains at most two non-zero entries.\n    \\item If two non-zero entries in a column of $\\vect{A}$ have the same sign, then the row of one is in $\\mathcal{C}$  , and the other in $\\mathcal{D}$ .\n    \\item If two non-zero entries in a column of $\\vect{A}$ have opposite signs, then the rows of both are in $\\mathcal{C}$  , or both in $\\mathcal{D}$ .\n\\end{enumerate}\n\nIn the setting of resource allocation, we can write problem \\eqref{integer_num} equivalently as problem \\eqref{int_prog} where $\\mathbf{x} = \\text{vec}(\\mathbf{X})$, $\\mathbf{t} = \\text{vec}(\\mathbf{\\Theta^*})$, $\\mathbf{A}$ and $\\mathbf{b}$ are given as\n\\begin{equation}\n    \\mathbf{A} = \n    \\begin{bmatrix}\n    \\mathds{1}_N^\\mathrm{T} \\otimes \\mathbf{I}_M\\\\\n    \\mathbf{I}_N \\otimes \\mathds{1}_M^\\mathrm{T}\\\\\n    \\end{bmatrix}\n    \\qquad \n    \\mathbf{b} = \n    \\begin{bmatrix}\n    \\vect{c}_t\\\\\n    \\vect{d}_t\\\\\n    \\end{bmatrix}\n\\label{def_Ab}\n\\end{equation}\n\nFor matrix $\\vect{A}$ given in \\eqref{def_Ab}, we can set $\\mathcal{C}$ to be the set of first $M$ rows corresponding to the capacity constraints, and $\\mathcal{D}$ to be the set of remaining rows corresponding to the demand constraints. Since this $\\vect{A}$ matrix satisfies the conditions of the proposition for sets $\\mathcal{C}$  and $\\mathcal{D}$, we obtain that $\\mathbf{A}$ is totally unimodular. Finally, since the vector $\\mathbf{b}$ is integral and the matrix $\\mathbf{A}$ is totally unimodular, the integrality gap is zero.\n\n\\section{Structured combinatorial multi-armed bandits}\n\\label{SCMAB}\n\nFor the ease of exposition, we present our proofs in the following setting with $d$ structured arms.\n\nWe consider a CMAB (Combinatorial Multi Armed Bandit) problem setting with $d$ arms associated with a set of independent random rewards $R_{t, i}$ for $i \\in [d]$ and $t \\in \\mathbb{N}$. Assume that the set of rewards $\\{R_{t, i} | t \\in \\mathbb{N}\\}$ associated with arm $i$ are $\\eta$-sub-Gaussian with mean $\\theta_i^* \\in [0, B]$. Let $\\vect{\\theta}^* = (\\theta_1, \\theta_2, \\dots, \\theta_d)$ be the vector of expectations of all arms and assume we know that it belongs to a structure set $\\mathcal{F} \\subseteq \\mathds{R}^{d}$. We further assume that we have access to an initial rough estimate $\\overline{\\vect{\\theta}}$ that satisfies $\\|\\overline{\\vect{\\theta}} - \\vect{\\theta}^* \\|_2 \\leq G$ (one can also set $\\overline{\\vect{\\theta}} = \\vect{0}$ and let $G$ be some number satisfying $\\|\\vect{\\theta}^* \\|_2 \\leq G$.).\n\nAt each round $t$, a subset of arms $\\mathcal{A}_t \\subseteq [d]$ are played and the individual outcomes of arms in $\\mathcal{A}_t$ are revealed.\nThe total reward at round $t$ is the sum of the rewards obtained from all arms in $\\mathcal{A}_t$. \nLetting $\\vect{e}_{i} \\in \\mathbb{R}^{d}$ denote the zero-one vector with a single one at the $i$-th entry, \ndefine the action vector for time $t$ as $\\vect{x}_{t}  = \\sum_{i \\in \\mathcal{A}_t} \\vect{e}_{i}$. \nConsequently, $\\vect{x}_{t}$ becomes a zero-one vector with ones at entries $\\mathcal{A}_t$ and zeros everywhere else. \nThe problem contains a constraint that any valid action $\\vect{x}_{t}$ must belong to a (time-varying) constraint set $\\mathcal{X}_{t} \\subseteq \\{0, 1\\}^{d}$. \n\nThe optimum allocation $\\vect{x}^*_t$ at time $t$ is given by $\\vect{x}^*_t \\in \\argmax_{\\vect{x} \\in \\mathcal{X}_t} \\; \\langle \\vect{x}, \\vect{\\theta}^* \\rangle$. We denote by $H_t$ the history $\\{\\vect{x}_{\\tau}, (R_{\\tau, i})_{i \\in \\mathcal{A}_{\\tau}}\\}_{\\tau = 1}^{t-1}$ of observations available when choosing the next action $\\vect{x}_{t}$. Let $\\pi$ be a policy which takes the action $\\vect{x}_{t}$ using the history $H_t$. Then, the $T$ period regret of a policy $\\pi$ is the random variable $ \\mathcal{R}(T, \\pi) =  \\sum_{t = 1}^{T} \\left[ \\langle \\vect{x}^*_t, \\vect{\\theta}^*\\rangle - \\langle \\vect{x}_{t}, \\vect{\\theta}^* \\rangle \\right] $.\n\n\\subsection{OFU for Structured Combinatorial Bandits}\n\nWe present our algorithm in the setting where the mean reward vector $\\vect{\\theta}^*$ belongs to a structure set $\\mathcal{F} \\subseteq \\mathds{R}^{d}$. Then, we analyze the algorithm to establish performance guarantees.\n\n\\begin{algorithm}\n\\caption{OFU for Structured Combinatorial Bandits}\\label{alg:cap}\n\\begin{algorithmic}\n\\Require horizon $T$, initial estimate $\\overline{\\vect{\\theta}} \\in \\mathbb{R}^d$ with $\\|\\overline{\\vect{\\theta}} - \\vect{\\theta}^* \\|_2 \\leq G$, parameters $\\delta, \\alpha > 0$, $\\gamma \\geq 1$.\n\\For{$t = 1, 2, \\dots, T$}\n\\State Find the least squares estimate $ \\widehat{\\vect{\\theta}}_t = \\argmin_{\\vect{\\theta} \\in \\mathcal{F}} \\left \\{ L_{2,t}(\\vect{\\theta}) + \\gamma \\|\\vect{\\theta} - \\overline{\\vect{\\theta}} \\|_2^2 \\right \\}$\n\\State Construct the confidence set $\\mathcal{C}_t = \\{ \\vect{\\theta} \\in \\mathcal{F} : \\|\\vect{\\theta} - \\widehat{\\vect{\\theta}}_t \\|_{2, E_t} \\leq \\sqrt{ \\beta_t^*(\\delta, \\alpha, \\gamma)}\\}$\n\\State Compute the action vector $\\vect{x}_t = \\argmax_{\\vect{x} \\in \\mathcal{X}_t} \\max_{\\vect{\\theta} \\in \\mathcal{C}_t} \\; \\langle \\vect{x}, \\vect{\\theta} \\rangle$\n\\State Play the arms $\\mathcal{A}_t$ according to $\\vect{x}_t$ \n\\State Observe $(R_{\\tau, i})_{i \\in \\mathcal{A}_{\\tau}}$\n\\EndFor\n\\end{algorithmic}\n\\label{alg_comb}\n\\end{algorithm}\n\nThe algorithm maintains a confidence set $\\mathcal{C}_t$ that contains the true parameter $\\vect{\\theta}^*$ with high probability and chooses the action $\\vect{x}_t$ according to\n\\begin{equation}\n    (\\vect{x}_t, \\widetilde{\\vect{\\theta}}_t) = \\argmax_{(\\vect{x}, \\vect{\\theta}) \\in \\mathcal{X}_t \\times \\mathcal{C}_t} \\; \\langle \\vect{x}, \\vect{\\theta} \\rangle\n\\end{equation}\n\nThe confidence sets that we construct are centered around the regualarized least square estimates defined next. We first let the cumulative squared prediciton error at time $t$ be\n\\begin{equation}\n    L_{2,t}(\\vect{\\theta}) = \\sum_{\\tau=1}^{t-1} \\sum_{i \\in \\mathcal{A}_\\tau} (\\theta_i  - R_{\\tau, i})^2\n\\end{equation}\nand define the regularized least squares estimate at time $t$ as\n\\begin{equation}\n    \\widehat{\\vect{\\theta}}_t = \\argmin_{\\vect{\\theta} \\in \\mathcal{F}} \\left \\{ L_{2,t}(\\vect{\\theta}) + \\gamma \\|\\vect{\\theta} - \\overline{\\vect{\\theta}} \\|_2^2 \\right \\}\n\\end{equation}\n\nThen, the confidence sets take the form $\\mathcal{C}_t := \\{ \\vect{\\theta} \\in \\mathcal{F} : \\|\\vect{\\theta} - \\widehat{\\vect{\\theta}}_t \\|_{2, E_t} \\leq \\sqrt{\\beta_t}\\}$ where $\\beta_t$ is an appropriately chosen confidence parameter, and the regularized empirical 2-norm $\\| \\cdot \\|_{2, E_t}$ is defined by \n\\begin{equation*}\n    \\| \\vect{\\Delta} \\|_{2, E_t}^2 := \\sum_{\\tau=1}^{t-1} \\sum_{i \\in \\mathcal{A}_\\tau} \\langle \\vect{\\Delta}, \\vect{e}_{i} \\rangle^2 + \\gamma \\| \\vect{\\Delta} \\|_{2}^2 =  \\sum_{i=1}^{d} (n_{t, i} + \\gamma) (\\Delta_{i})^2\n\\end{equation*}\nwhere $n_{t, i} := \\sum_{\\tau=1}^{t-1} \\mathds{1} \\{i \\in \\mathcal{A}_t\\}$ denotes the number of times arm $i$ has been pulled before time $t$ (excluding time $t$). For future reference, we also define the (non-regularized) empirical 2-norm $\\| \\cdot \\|_{2, \\widetilde{E}_t}$ by \n\\begin{equation*}\n    \\| \\vect{\\Delta} \\|^2_{2, \\widetilde{E}_t} := \\sum_{\\tau=1}^{t-1} \\sum_{i \\in \\mathcal{A}_\\tau} \\langle \\vect{\\Delta}, \\vect{e}_{i} \\rangle^2 =  \\sum_{i=1}^{d} (n_{t, i}) (\\Delta_{i})^2\n\\end{equation*}\n\nNote that the regularized empirical 2-norm is related to (non-regularized) empirical 2-norm as\n\\begin{equation*}\n    \\| \\vect{\\Delta} \\|^2_{2, E_t^2} = \\| \\vect{\\Delta} \\|^2_{2, \\widetilde{E}_t} + \\gamma \\| \\vect{\\Delta} \\|^2_{2}\n\\end{equation*}\n\nBy Lemma \\ref{lemma_lower_bound}, we establish that for any $\\vect{\\theta} \\in \\mathbb{R}^{d}$, \n\\begin{equation}\n    \\mathds{P} \\left( L_{2,t}(\\vect{\\theta}) \\geq L_{2,t}(\\vect{\\theta}^*) + \\frac{1}{2} \\|\\vect{\\theta}^* - \\vect{\\theta} \\|^2_{2, \\widetilde{E}_t} - 4 \\eta^2 \\log(1\/\\delta) \\quad ,\\forall t \\in \\mathbb{N} \\right) \\geq 1 - \\delta\n\\end{equation}\n\nHence, with high probability, $\\vect{\\theta}$ can achieve lower squared error than $\\vect{\\theta}^*$ only if the empirical deviation $\\|\\vect{\\theta}^* - \\vect{\\theta} \\|^2_{2, \\widetilde{E}_t}$ is less than $8 \\eta^2 \\log(1\/\\delta)$. \n\nIn order to make this property hold uniformly for all $\\vect{\\theta}$ in a subset $\\mathcal{C}_t$ of $\\mathcal{F}$, we discretize $\\mathcal{C}_t$ at some discretization scale $\\alpha$ and apply a union bound for this finite discretization set. Let $\\mathcal{N}(\\mathcal{F}, \\alpha, \\| \\cdot \\|_{2})$ denote the $\\alpha$-covering number of $\\mathcal{F}$ in the 2-norm $\\| \\cdot \\|_{2}$, and let \n\\begin{equation}\n    \\beta_t^*(\\delta, \\alpha, \\gamma) := 8 \\eta^2 \\log \\left(\\mathcal{N}(\\mathcal{F}, \\alpha, \\| \\cdot \\|_{2}) \/ \\delta \\right) + 2 \\alpha t \\sqrt{d} \\left [ 8 B + \\sqrt{8 \\eta^2 \\log(4d t^2\/\\delta)} \\right]  + 4 \\gamma G^2\n\\end{equation}\n\nThen, Lemma \\ref{lemma_confidence} shows that if we set $\\beta_t = \\beta_t^*(\\delta, \\alpha)$, the confidence sets $\\mathcal{C}_t$ contain the true parameter $\\vect{\\theta}^*$ for all $t$ with high probability. Following the construction of the confidence sets, the next step is to obtain the overall regret guarantee. As given in Corollary \\ref{corr_regret_order}, we find that the regret of Algorithm \\ref{alg_comb} satisfies\n\\begin{equation}\n    \\mathcal{R}(T, \\pi) = \\widetilde{\\mathcal{O}} \\left( \\sqrt{\\eta^2 d \\log \\left(\\mathcal{N}(\\mathcal{F}, T^{-1}, \\| \\cdot \\|_{2})\\right) T} \\right)\n\\end{equation}\n\n\\begin{remark}\nIn the literature of linear bandits, the typical observation model is such that each action $\\vect{x}_{t}$ results in a single reward feedback with mean $\\langle \\vect{x}_{t}, \\vect{\\theta}^* \\rangle$ and sub-Gaussianity parameter $d \\eta^2$ (since each arm has a $\\eta^2$-sub-Gaussian reward). Therefore, that observation model can only obtain $\\widetilde{\\mathcal{O}} ( \\sqrt{\\eta^2 d^2 \\log \\left(\\mathcal{N}(\\mathcal{F}, T^{-1}, \\| \\cdot \\|_{2})\\right) T} )$ regret guarantee. However, in our setting, the observations are sets of independent rewards $\\{R_{t, i}\\}_{i \\in \\mathcal{A}_t}$ where each element is $\\eta^2$-sub-Gaussian. Due to this richer nature of the observation model, we are able to achieve lower regret guarantees than only observing a single cumulative reward.\n\\end{remark}\n\n\\section{Proofs for Confidence Sets}\n\\label{pf_conf_sets}\n\n\\subsection{Martingale Exponential Inequalities}\n\nWe start with preliminary results on martingale exponential inequalities.\n\nConsider a sequence of random variables $(Z_n)_{n \\in \\mathds{N}}$ adapted to the filtration $(\\mathcal{H}_n)_{n \\in \\mathds{N}}$. Assume $\\mathds{E}[\\exp (\\lambda Z_i)]$ is finite for all $\\lambda$. Define the conditional mean $\\mu_i = \\mathds{E}[Z_i | \\mathcal{H}_{i-1}]$, and define the conditional cumulant generating function of the centered random variable $[Z_i - \\mu_i]$ by $\\psi_i(\\lambda) := \\log \\mathds{E}[ \\exp (\\lambda [Z_i - \\mu_i]) | \\mathcal{H}_{i-1}]$. Let \n\\begin{equation*}\n    M_n(\\lambda) = \\exp \\left \\{ \\sum_{i=1}^{n} \\lambda [Z_i - \\mu_i] - \\psi_i(\\lambda) \\right \\} \n\\end{equation*}\n\n\\begin{lemma}\n$(M_n (\\lambda))_{n \\in \\mathds{N}}$ is a martingale with respect to the filtration $(\\mathcal{H}_n)_{n \\in \\mathds{N}}$, and $\\mathds{E}[M_n (\\lambda)] = 1$.\n\\label{martingale}\n\\end{lemma}\n\n\\begin{proof}\nBy definition, we have\n\\begin{equation*}\n    \\mathds{E}[M_1 (\\lambda) | \\mathcal{H}_0] = \\mathds{E}[\\exp \\{ \\lambda [Z_1 - \\mu_1] - \\psi_1(\\lambda) \\} | \\mathcal{H}_0] = 1 \n\\end{equation*}\nThen, for any $n \\geq 2$,\n\\begin{align*}\n    \\mathds{E}[M_n (\\lambda) | \\mathcal{H}_{n-1}] &= \\mathds{E}[M_{n-1}(\\lambda) \\exp \\{ \\lambda [Z_n - \\mu_n] - \\psi_n(\\lambda) \\} | \\mathcal{H}_{n-1}] \\\\\n    &= M_{n-1}(\\lambda) \\mathds{E}[\\exp \\{ \\lambda [Z_n - \\mu_n] - \\psi_n(\\lambda) \\} | \\mathcal{H}_{n-1}]\\\\\n    &= M_{n-1}(\\lambda)\\\\\n\\end{align*}\nsince $M_{n-1}(\\lambda)$ is a measurable function of the filtration $\\mathcal{H}_{n-1}$.\n\\end{proof}\n\n\\begin{lemma}\nFor all $x \\geq 0$ and $\\lambda \\geq 0$, \n\\begin{equation*}\n\\mathds{P} \\left(\\sum_{i=1}^{n} \\lambda Z_i \\leq x + \\sum_{i=1}^{n} [\\lambda \\mu_i + \\psi_i(\\lambda)] \\quad ,\\forall t \\in \\mathds{N} \\right) \\geq 1 - e^{-x}\n\\end{equation*}\n\\label{exp_martingale}\n\\end{lemma}\n\n\\begin{proof}\nFor any $\\lambda$, $(M_n (\\lambda))_{n \\in \\mathds{N}}$ is a martingale with respect to $(\\mathcal{H}_n)_{n \\in \\mathds{N}}$ and $\\mathds{E}[M_n (\\lambda)] = 1$ by Lemma \\ref{martingale}. For arbitrary $x \\geq 0$, define $\\tau_x = \\inf \\{ n\\geq 0 | M_n(\\lambda) \\geq x \\}$ and note that $\\tau_x$ is a stopping time corresponding to the first time $M_n$ crosses the boundary $x$. Since $\\tau$ is a stopping time with respect to $(\\mathcal{H}_n)_{n \\in \\mathds{N}}$, we have $\\mathds{E}[M_{\\tau_x \\wedge n}(\\lambda)] = 1$. Then, by Markov's inequality\n\\begin{equation*}\n    x \\mathds{P} (M_{\\tau_x \\wedge n}(\\lambda) \\geq x) \\leq \\mathds{E}[M_{\\tau_x \\wedge n}(\\lambda)] = 1\n\\end{equation*}\n\nNoting that the event $\\{ M_{\\tau_x \\wedge n}(\\lambda) \\geq x \\} = \\bigcup_{k=1}^n \\{M_{k}(\\lambda) \\geq x\\} $, we have\n\\begin{equation*}\n\\mathds{P} \\left( \\bigcup_{k=1}^n \\{M_{k}(\\lambda) \\geq x\\} \\right) \\leq \\frac{1}{x}\n\\end{equation*}\n\nTaking the limit as $n \\to \\infty$, and applying monotone convergence theorem shows that $\\mathds{P} \\left( \\bigcup_{k=1}^\\infty \\{M_{k}(\\lambda) \\geq x\\} \\right) \\leq \\frac{1}{x}$ or $\\mathds{P} \\left( \\bigcup_{k=1}^\\infty \\{M_{k}(\\lambda) \\geq e^x\\} \\right) \\leq e^{-x}$. Then, by definition of $M_k (\\lambda)$, we conclude\n\\begin{equation*}\n    \\mathds{P} \\left( \\bigcup_{k=1}^\\infty \\left \\{ \\sum_{i=1}^{n} \\lambda [Z_i - \\mu_i] - \\psi_i(\\lambda) \\geq x \\right \\}  \\right) \\leq e^{-x}\n\\end{equation*}\n\n\\end{proof}\n\n\\subsection{Proofs for the construction of confidence sets}\n\n\\begin{lemma} For any $\\delta > 0$ and $\\vect{\\theta} \\in \\mathbb{R}^{d}$,\n\\begin{equation}\n    \\mathds{P} \\left( L_{2,t}(\\vect{\\theta}) \\geq L_{2,t}(\\vect{\\theta}^*) + \\frac{1}{2} \\|\\vect{\\theta}^* - \\vect{\\theta} \\|^2_{2, \\widetilde{E}_t} - 4 \\eta^2 \\log(1\/\\delta) \\quad ,\\forall t \\in \\mathbb{N} \\right) \\geq 1 - \\delta\n\\end{equation}\n\n\\label{lemma_lower_bound}\n\\end{lemma}\n\n\\begin{proof}\n\nLet $\\mathcal{H}_{t-1}$ be the $\\sigma$-algebra generated by $(H_t, \\mathcal{A}_t)$ and let $\\mathcal{H}_0 = \\sigma(\\emptyset, \\Omega)$. Then, define $\\epsilon_{t, i} := R_{t, i} - \\langle \\mathbf{X}_{t, i}, \\vect{\\theta}^*\\rangle$ for all $t \\in \\mathds{N}$ and $i \\in \\mathcal{A}_t$. By previous assumptions, $\\mathds{E} [\\epsilon_{t, i} | \\mathcal{H}_{t-1}] = 0$ and $\\mathds{E} [\\exp (\\lambda \\epsilon_{t, u}) | \\mathcal{H}_{t-1} ] \\leq \\exp \\left(\\frac{\\lambda^2 \\eta^2}{2} \\right)$ for all $t$.\n\nDefine $Z_{t, i} = (R_{t, i} - \\langle \\mathbf{X}_{t, i}, \\vect{\\theta}^*\\rangle)^2 - (R_{t, i} - \\langle \\mathbf{X}_{t, i}, \\vect{\\theta}\\rangle)^2$. Then, we have\n\\begin{align*}\nZ_{t, i} &= - (\\langle \\mathbf{X}_{t, i}, \\vect{\\theta}\\rangle - \\langle \\mathbf{X}_{t, i}, \\vect{\\theta}^*\\rangle)^2 + 2 \\epsilon_{t, i} (\\langle \\mathbf{X}_{t, i}, \\vect{\\theta}\\rangle - \\langle \\mathbf{X}_{t, i}, \\vect{\\theta}^*\\rangle)\\\\\n&= - \\langle \\mathbf{X}_{t, i}, \\vect{\\theta} - \\vect{\\theta}^*\\rangle^2 + 2 \\epsilon_{t, i} \\langle \\mathbf{X}_{t, i}, \\vect{\\theta} - \\vect{\\theta}^*\\rangle\n\\end{align*}\n\nTherefore, the conditional mean and conditional cumulant generating function satisfy\n\\begin{align*}\n    \\mu_{t, i} &:= \\mathds{E}[Z_{t, i} | \\mathcal{H}_{t-1}] = - \\langle \\mathbf{X}_{t, i}, \\vect{\\theta} - \\vect{\\theta}^*\\rangle^2\\\\\n    \\psi_t(\\lambda) &:= \\log \\mathds{E}[ \\exp (\\lambda [Z_{t, i} - \\mu_{t, i}]) | \\mathcal{H}_{t-1}]\\\\\n    &= \\log \\mathds{E}[ \\exp (2 \\lambda \\langle \\mathbf{X}_{t, i}, \\vect{\\theta} - \\vect{\\theta}^*\\rangle \\epsilon_{t, i} ) | \\mathcal{H}_{t-1}]\\\\\n    &\\leq \\frac{(2 \\lambda \\langle \\mathbf{X}_{t, i}, \\vect{\\theta} - \\vect{\\theta}^*\\rangle)^2 \\eta^2}{2}\n\\end{align*}\n\nApplying Lemma \\ref{exp_martingale} shows that for all $x \\geq 0$ and $\\lambda \\geq 0$,\n\\begin{equation*}\n\\mathds{P} \\left(\\sum_{\\tau=1}^{t-1} \\sum_{u \\in \\mathcal{A}_\\tau} Z_{\\tau, i} \\leq \\frac{x}{ \\lambda} + \\sum_{\\tau=1}^{t-1} \\sum_{u \\in \\mathcal{A}_\\tau} \\langle \\mathbf{X}_{\\tau, i}, \\vect{\\theta} - \\vect{\\theta}^*\\rangle^2 (2 \\lambda \\eta^2 - 1) \\quad ,\\forall t \\in \\mathds{N} \\right) \\geq 1 - e^{-x}\n\\end{equation*}\n\nNote that we have $\\sum_{\\tau=1}^{t-1} \\sum_{u \\in \\mathcal{A}_\\tau} Z_{\\tau, i} = L_{2,t}(\\vect{\\theta}^*) - L_{2,t}(\\vect{\\theta})$,\n\nand $\\sum_{\\tau=1}^{t-1} \\sum_{u \\in \\mathcal{A}_\\tau} \\langle \\mathbf{X}_{\\tau, i}, \\vect{\\theta} - \\vect{\\theta}^*\\rangle^2 = \\|\\vect{\\theta}^* - \\vect{\\theta} \\|^2_{2, \\widetilde{E}_t}$. \n\nThen, choosing $\\lambda = \\frac{1}{4 \\eta^2}$ and $x = \\log \\frac{1}{\\delta}$ gives\n\\begin{equation*}\n    \\mathds{P} \\left( L_{2,t}(\\vect{\\theta}) \\geq L_{2,t}(\\vect{\\theta}^*) + \\frac{1}{2} \\|\\vect{\\theta}^* - \\vect{\\theta} \\|^2_{2, \\widetilde{E}_t} - 4 \\eta^2 \\log(1\/\\delta) \\quad ,\\forall t \\in \\mathbb{N} \\right) \\geq 1 - \\delta\n\\end{equation*}\n\n\\end{proof}\n\n\\begin{lemma}\nIf $\\vect{\\theta}^{\\alpha} \\in \\mathcal{F}^{\\alpha}$ satisfies $\\|\\vect{\\theta} - \\vect{\\theta}^{\\alpha}\\|_{2} \\leq \\alpha$, then with probability at least $1 - \\delta$,\n\\begin{equation}\n    \\left |  \\frac{1}{2} \\|\\vect{\\theta}^* - \\vect{\\theta}^{\\alpha} \\|^2_{2, \\widetilde{E}_t} - \\frac{1}{2} \\|\\vect{\\theta}^* - \\vect{\\theta} \\|^2_{2, \\widetilde{E}_t} + L_{2,t}(\\vect{\\theta}) - L_{2,t}(\\vect{\\theta}^{\\alpha}) \\right | \\leq \\alpha t d \\left [ 8 B + \\sqrt{8 \\eta^2 \\log(4d t^2\/\\delta)} \\right]\n\\label{eqn_of_discr_lemma}\n\\end{equation}\n\\label{discr_lemma}\n\\end{lemma}\n\n\\begin{proof}\n\nSince any two $\\vect{\\theta}, \\vect{\\theta}^{\\alpha} \\in \\mathcal{F}$ satisfy $\\|\\vect{\\theta} - \\vect{\\theta}^{\\alpha}\\|_{2} \\leq \\sqrt{d} B$, it is enough to consider $\\alpha \\leq  \\sqrt{d} B$. We find \n\\begin{align*}\n    \\sum_{ i = 1 }^{d} | \\langle \\vect{\\theta}, \\vect{e}_{i} \\rangle^2 - \\langle \\vect{\\theta}^{\\alpha}, \\vect{e}_{i} \\rangle^2 | &\\leq \\max_{ \\|\\vect{\\Delta}\\|_2 \\leq \\alpha } \\left \\{\\sum_{ i = 1 }^{d} \\left | \\theta_{i}^2 - (\\theta_{i}+\\Delta_i)^2  \\right | \\right \\}\\\\\n    &= \\max_{ \\|\\vect{\\Delta}\\|_2 \\leq \\alpha } \\left \\{\\sum_{ i = 1 }^{d}\\left | 2 \\theta_{i} \\Delta_i + \\Delta_i^2  \\right | \\right \\}\\\\\n    &\\leq \\max_{ \\|\\vect{\\Delta}\\|_2 \\leq \\alpha } \\left \\{ 2 \\sum_{ i = 1 }^{d}\\left | \\theta_{i} \\Delta_i \\right | + \\sum_{ i = 1 }^{d} \\Delta_i^2 \\right \\}\\\\\n    &\\leq \\max_{ \\|\\vect{\\Delta}\\|_2 \\leq \\alpha }  \\left \\{ 2 B \\|\\vect{\\Delta}\\|_1 + \\|\\vect{\\Delta}\\|_2^2 \\right \\}\\\\\n    &\\leq 2 B \\sqrt{d} \\alpha + \\alpha^2\n\\end{align*}\nTherefore, it implies\n\\begin{align*}\n     \\sum_{ i = 1 }^{d} | \\langle \\vect{\\theta} - \\vect{\\theta}^*, \\vect{e}_{i} \\rangle^2 - \\langle \\vect{\\theta}^{\\alpha} - \\vect{\\theta}^*, \\vect{e}_{i} \\rangle^2 | &= \\sum_{ i = 1 }^{d} \\left| \\langle \\vect{\\theta}, \\vect{e}_{i} \\rangle^2 - \\langle \\vect{\\theta}^{\\alpha}, \\vect{e}_{i} \\rangle^2 + 2 \\langle \\vect{\\theta}^*, \\vect{e}_{i} \\rangle \\langle \\vect{\\theta}^{\\alpha} - \\vect{\\theta}, \\vect{e}_{i} \\rangle \\right| \\\\\n     &\\leq 2 B \\sqrt{d} \\alpha + \\alpha^2 + 2 B \\|\\vect{\\theta} - \\vect{\\theta}^{\\alpha}\\|_{1} \\\\\n     &\\leq 4 B \\sqrt{d} \\alpha + \\alpha^2\n\\end{align*}\n\nSimilarly, for any $t$, we have\n\\begin{align*}\n     \\sum_{ i = 1 }^{d} | \\left( R_{t,i} - \\langle \\vect{\\theta}, \\vect{e}_{i} \\rangle \\right)^2 -\\left( R_{t,i} - \\langle \\vect{\\theta}^{\\alpha}, \\vect{e}_{i} \\rangle \\right)^2 | &= \\sum_{ i = 1 }^{d} \\left| 2 R_{t,i} \\langle \\vect{\\theta}^{\\alpha} - \\vect{\\theta}, \\vect{e}_{i} \\rangle + \\langle \\vect{\\theta}, \\vect{e}_{i} \\rangle^2 - \\langle \\vect{\\theta}^{\\alpha}, \\vect{e}_{i} \\rangle^2 \\right| \\\\\n      &\\leq 2 \\sum_{ i = 1 }^{d} \\left| R_{t,i} \\right | \\left| \\langle \\vect{\\theta}^{\\alpha} - \\vect{\\theta}, \\vect{e}_{i} \\rangle \\right| + 2 B \\sqrt{d} \\alpha + \\alpha^2\\\\\n     &\\leq 2 \\| \\vect{\\theta}^{\\alpha} - \\vect{\\theta}\\|_{\\infty} \\sum_{ i = 1 }^{d} | R_{t,i} | + 2 B \\sqrt{d} \\alpha + \\alpha^2\\\\\n     &\\leq 2 \\alpha \\sum_{ i = 1 }^{d} | R_{t,i} | + 2 B \\sqrt{d} \\alpha + \\alpha^2\n\\end{align*}\n\nSumming over $t$ and noting that $\\mathcal{A}_t \\subseteq [d]$, the left hand side of \\eqref{eqn_of_discr_lemma} is bounded by\n\\begin{equation*}\n    \\sum_{\\tau = 1}^{t-1} \\left( \\frac{1}{2} \\left[ 4 B \\sqrt{d} \\alpha + \\alpha^2 \\right] + 2 \\alpha \\sum_{ i = 1 }^{d} | R_{t,i} | + 2 B \\sqrt{d} \\alpha + \\alpha^2  \\right) \\leq \\alpha \\sum_{\\tau = 1}^{t-1} \\left(6 B \\sqrt{d} + 2 \\sum_{ i = 1 }^{d} | R_{\\tau,i} | \\right)\n\\end{equation*}\n\nBecause $\\epsilon_{\\tau, i}$ is $\\eta$-sub-Gaussian, $\\mathds{P} \\left( |\\epsilon_{\\tau, i}| > \\sqrt{2 \\eta^2 \\log(2\/\\delta) }\\right) \\leq \\delta$. By a union bound, $\\mathds{P} \\left( \\exists \\tau, i \\text{ s.t. } |\\epsilon_{\\tau, i}| > \\sqrt{2 \\eta^2 \\log(4d \\tau^2\/\\delta) }\\right) \\leq \\frac{\\delta d}{2} \\sum_{\\tau = 1}^{\\infty} \\frac{1}{d \\tau^2} \\leq \\delta$. Since $| R_{\\tau,i} | \\leq B + | \\epsilon_{\\tau, i} |$, we have $| R_{\\tau,i} | \\leq B + \\sqrt{2 \\eta^2 \\log(4d \\tau^2\/\\delta)}$  with probability at least $1 - \\delta$. Consequently, the bound for the discretization error becomes\n\\begin{equation*}\n\\alpha t d \\left [ 8 B + 2 \\sqrt{2 \\eta^2 \\log(4d t^2\/\\delta)} \\right]\n\\end{equation*}\n\n\n\n\\end{proof}\n\n\n\\begin{lemma} For any $\\delta > 0$, $\\alpha > 0$ and $\\gamma > 0$, if\n\\begin{equation}\n    \\mathcal{C}_t = \\{ \\vect{\\theta} \\in \\mathcal{F} : \\|\\vect{\\theta} - \\widehat{\\vect{\\theta}}_t \\|_{2, E_t} \\leq \\sqrt{ \\beta_t^*(\\delta, \\alpha, \\gamma)}\\}\n\\end{equation}\nfor all $t \\in \\mathbb{N}$, then\n\\begin{equation}\n    \\mathds{P} \\left( \\vect{\\theta}^* \\in \\mathcal{C}_t \\quad ,\\forall t \\in \\mathbb{N} \\right) \\geq 1 - 2\\delta\n\\end{equation}\n\\label{lemma_confidence}\n\\end{lemma}\n\n\\begin{proof}\n\nLet $\\mathcal{F}^{\\alpha} \\subset \\mathcal{F}$ be an $\\alpha$-cover of $\\mathcal{F}$ in the 2-norm so that for any $\\vect{\\theta} \\in \\mathcal{F}$, there exists $\\vect{\\theta}^{\\alpha} \\in \\mathcal{F}^{\\alpha}$ such that $\\|\\vect{\\theta} - \\vect{\\theta}^{\\alpha}\\|_{2} \\leq \\alpha$. By a union bound applied to Lemma \\ref{lemma_lower_bound}, with probability at least $1 - \\delta$,\n\\begin{equation*}\n    L_{2,t}(\\vect{\\theta}^{\\alpha}) - L_{2,t}(\\vect{\\theta}^*) \\geq \\frac{1}{2} \\|\\vect{\\theta}^* - \\vect{\\theta}^{\\alpha} \\|^2_{2, \\widetilde{E}_t} - 4 \\eta^2 \\log(|\\mathcal{F}^{\\alpha}|\/\\delta) \\quad ,\\forall \\vect{\\theta}^{\\alpha} \\in \\mathcal{F}^{\\alpha}, t \\in \\mathbb{N}\n\\end{equation*}\n\nTherefore, with probability at least $1 - \\delta$, for all $\\vect{\\theta} \\in \\mathcal{F}, t \\in \\mathbb{N}$,\n\\begin{align*}\n    L_{2,t}(\\vect{\\theta}) - L_{2,t}(\\vect{\\theta}^*) \\geq & \\frac{1}{2} \\|\\vect{\\theta}^* - \\vect{\\theta} \\|^2_{2, \\widetilde{E}_t} - 4 \\eta^2 \\log(|\\mathcal{F}^{\\alpha}|\/\\delta) \\\\\n    &+ \\min_{\\vect{\\theta}^{\\alpha} \\in \\mathcal{F}^{\\alpha}} \\left \\{  \\frac{1}{2} \\|\\vect{\\theta}^* - \\vect{\\theta}^{\\alpha} \\|^2_{2, \\widetilde{E}_t} - \\frac{1}{2} \\|\\vect{\\theta}^* - \\vect{\\theta} \\|^2_{2, \\widetilde{E}_t} + L_{2,t}(\\vect{\\theta}) - L_{2,t}(\\vect{\\theta}^{\\alpha}) \\right \\}\n\\end{align*}\n\nBy Lemma \\ref{discr_lemma}, with probability at least $1 - 2 \\delta$,\n\\begin{equation*}\n    L_{2,t}(\\vect{\\theta}) - L_{2,t}(\\vect{\\theta}^*) \\geq \\frac{1}{2} \\|\\vect{\\theta}^* - \\vect{\\theta} \\|^2_{2, \\widetilde{E}_t} - D_t\n\\end{equation*}\nwhere $D_t := 4 \\eta^2 \\log(|\\mathcal{F}^{\\alpha}|\/\\delta) + \\alpha t d \\left [ 8 B + \\sqrt{8 \\eta^2 \\log(4d t^2\/\\delta)} \\right]$.\n\nAdding the regularization terms to both sides, we obtain\n\\begin{equation*}\n    L_{2,t}(\\vect{\\theta}) + \\gamma \\|\\vect{\\theta} - \\overline{\\vect{\\theta}} \\|_2^2 - L_{2,t}(\\vect{\\theta}^*) - \\gamma \\|\\vect{\\theta}^* - \\overline{\\vect{\\theta}} \\|_2^2 \\geq \\frac{1}{2} \\|\\vect{\\theta}^* - \\vect{\\theta} \\|^2_{2, \\widetilde{E}_t} + \\gamma \\|\\vect{\\theta} - \\overline{\\vect{\\theta}} \\|_2^2 - D_t - \\gamma  \\|\\vect{\\theta}^* - \\overline{\\vect{\\theta}} \\|_2^2 \n\\end{equation*}\n\nNote the definition of the least square estimate $\\widehat{\\vect{\\theta}}_t = \\argmin_{\\vect{\\theta} \\in \\mathcal{F}} \\left \\{ L_{2,t}(\\vect{\\theta}) + \\gamma \\|\\vect{\\theta} - \\overline{\\vect{\\theta}} \\|_2^2 \\right \\}$. By letting $\\vect{\\theta} = \\widehat{\\vect{\\theta}}_t$, the left hand side becomes non-positive, and hence \n\\begin{equation*}\n\\frac{1}{2} \\|\\vect{\\theta}^* - \\widehat{\\vect{\\theta}}_t \\|^2_{2, \\widetilde{E}_t} \\leq D_t + \\gamma \\left( \\|\\vect{\\theta}^* - \\overline{\\vect{\\theta}} \\|_2^2 - \\|\\widehat{\\vect{\\theta}}_t - \\overline{\\vect{\\theta}} \\|_2^2 \\right)\n\\end{equation*}\n\nThen, \n\\begin{equation*}\n\\frac{1}{2} \\|\\vect{\\theta}^* - \\widehat{\\vect{\\theta}}_t \\|^2_{2, \\widetilde{E}_t} + \\gamma \\left( \\|\\widehat{\\vect{\\theta}}_t - \\overline{\\vect{\\theta}} \\|_2^2 + \\|\\vect{\\theta}^* - \\overline{\\vect{\\theta}} \\|_2^2  \\right) \\leq D_t + 2 \\gamma \\|\\vect{\\theta}^* - \\overline{\\vect{\\theta}} \\|_2^2\n\\end{equation*}\n\nBy triangle inequality we have $ \\|\\widehat{\\vect{\\theta}}_t - \\overline{\\vect{\\theta}} \\|_2 + \\|\\vect{\\theta}^* - \\overline{\\vect{\\theta}} \\|_2  \\geq \\|\\vect{\\theta}^* - \\widehat{\\vect{\\theta}}_t \\|_2$. Taking squares on both sides, we obtain $ \\|\\widehat{\\vect{\\theta}}_t - \\overline{\\vect{\\theta}} \\|_2^2 + \\|\\vect{\\theta}^* - \\overline{\\vect{\\theta}} \\|_2^2  \\geq \\frac{1}{2} \\|\\vect{\\theta}^* - \\widehat{\\vect{\\theta}}_t \\|_2^2$. Then, noting that $\\| \\vect{\\Delta} \\|^2_{2, E_t^2} = \\| \\vect{\\Delta} \\|^2_{2, \\widetilde{E}_t} + \\gamma \\| \\vect{\\Delta} \\|^2_{2}$, we have\n\\begin{equation*}\n\\frac{1}{2} \\|\\vect{\\theta}^* - \\widehat{\\vect{\\theta}}_t \\|^2_{2, E_t} \\leq D_t + 2 \\gamma \\|\\vect{\\theta}^* - \\overline{\\vect{\\theta}} \\|_2^2\n\\end{equation*}\n\nLastly, using the inequality $\\|\\vect{\\theta}^* - \\overline{\\vect{\\theta}} \\|_2^2 \\leq G^2$,\n\\begin{equation*}\n\\|\\vect{\\theta}^* - \\widehat{\\vect{\\theta}}_t \\|^2_{2, E_t} \\leq 8 \\eta^2 \\log(|\\mathcal{F}^{\\alpha}|\/\\delta) + 2 \\alpha t d \\left [ 8 B + \\sqrt{8 \\eta^2 \\log(4d t^2\/\\delta)} \\right]  + 4 \\gamma G^2\n\\end{equation*}\n\nTaking the infimum over the size of $\\alpha$-covers, we obtain the final result.\n\n\\end{proof}\n\n\n\\section{Proofs for Regret Bounds}\n\\label{pf_regrets}\n\nThroughout this section we will use the shorthand $\\beta_t =  \\beta_t^*(\\delta, \\alpha, \\gamma)$.\n\nWe start with the definitions of weighted inner product and norms.\n\n\\begin{definition}\nFor a symmetric positive definite matrix $\\vect{W} \\in \\mathbb{R}^{d \\times d}$, define\n\\begin{itemize}\n    \\item $\\vect{W}$-inner product of two vectors $\\vect{x}, \\vect{y} \\in \\mathbb{R}^{d}$ as $\\langle \\vect{x}, \\vect{y} \\rangle_{\\vect{W}} := \\langle \\vect{W}  \\vect{x}, \\vect{y} \\rangle$\n    \\item $\\vect{W}$-norm of a vector $\\vect{x} \\in \\mathbb{R}^{d}$ as $\\|\\vect{x}\\|_{\\vect{W}} := \\sqrt{\\langle \\vect{x}, \\vect{x} \\rangle_{\\vect{W}}}$.\n\\end{itemize}\n\\end{definition}\n\nThen, the regularized empirical 2-norm of a vector $\\vect{z} \\in \\mathbb{R}^d$ can be written as \n\\begin{equation}\n    \\| \\vect{z} \\|_{2, E_t} = \\| \\vect{z} \\|_{\\vect{A}_t}\n\\end{equation}\nwhere $\\vect{A}_t$ is a diagonal matrix with diagonal entries $\\{ (n_{t, 1} + \\gamma), (n_{t, 2} + \\gamma), \\dots, (n_{t, d} + \\gamma) \\}$. \n\nRecall that $n_{t, i} = \\sum_{\\tau=1}^{t-1} \\mathds{1} \\{i \\in \\mathcal{A}_t\\}$ denotes the number of times arm $i$ has been pulled before time $t$ (excluding time $t$).\n\n\\begin{lemma}\nLet $x \\in \\mathcal{X}$ and $\\Theta \\in \\mathcal{F}_t$. Then,\n\\begin{equation}\n    |\\langle \\vect{\\theta} - \\widehat{\\vect{\\theta}}_t^{\\text{LS}}, \\vect{x} \\rangle| \\leq w \\sqrt{\\beta_t}\n\\end{equation}\nwhere $w = \\| \\vect{x} \\|_{\\vect{A}_t^{-1}}$ is the \"confidence width\" of an action $\\vect{x}$ at time $t$.\n\\label{lemma_conf_width}\n\\end{lemma}\n\n\\begin{proof} Let $\\vect{\\Delta} = \\vect{\\theta} - \\widehat{\\vect{\\theta}}_t^{\\text{LS}}$. Then,\n\\begin{align*}\n    |\\langle \\vect{\\Delta} , \\vect{x} \\rangle| &= |\\vect{\\Delta} ^\\textrm{T} \\vect{x} |\\\\\n    &= |\\vect{\\Delta}^\\textrm{T} \\vect{A}_t^{1\/2} \\vect{A}_t^{-1\/2} \\vect{x}|\\\\\n    &= |(\\vect{A}_t^{1\/2}\\vect{\\Delta})^\\textrm{T} \\vect{A}_t^{-1\/2} \\vect{x}|\\\\\n    &\\leq \\| \\vect{A}_t^{1\/2}\\vect{\\Delta} \\| \\|\\vect{A}_t^{-1\/2} \\vect{x}\\|\\\\\n    &= \\| \\vect{\\Delta} \\|_{\\vect{A}_t} \\| \\vect{x} \\|_{\\vect{A}_t^{-1}}\\\\\n    &= w \\| \\vect{\\Delta} \\|_{2, E_t}\\\\\n    &\\leq w \\sqrt{\\beta_t} \n\\end{align*}\n\\end{proof}\n\nDefine the widths of the allocations as\n\\begin{equation}\n    w_t := \\| \\vect{x}_t \\|_{\\vect{A}_t^{-1}} \\qquad \\text{and} \\qquad w_{t, i} := \\| \\vect{e}_i \\|_{\\vect{A}_t^{-1}}\n\\label{def_widths}\n\\end{equation}\n\n\\begin{lemma}\nFor any $t \\in \\mathbb{N}$, we have the identity\n\\begin{equation*}\n    w_t^2 = \\sum_{i \\in \\mathcal{A}_t} w_{t, i}^2\n\\end{equation*}\n\\label{lemma_width_identity}\n\\end{lemma}\n\\begin{proof}\n\\begin{align*}\n    w_t^2 &= \\langle \\vect{x}_t, \\vect{x}_t \\rangle_{\\vect{A}_t^{-1}}\\\\\n    &= \\left\\langle \\vect{A}_t^{-1} \\sum_{i \\in \\mathcal{A}_t} \\vect{e}_i, \\sum_{i \\in \\mathcal{A}_t} \\vect{e}_i \\right\\rangle \\\\\n    &= \\sum_{i \\in \\mathcal{A}_t} \\sum_{j \\in \\mathcal{A}_t} \\left\\langle \\vect{A}_t^{-1} \\vect{e}_i, \\vect{e}_j \\right\\rangle\\\\\n    &= \\sum_{i \\in \\mathcal{A}_t} \\left\\langle \\vect{A}_t^{-1} \\vect{e}_i, \\vect{e}_i \\right\\rangle\\\\\n    &= \\sum_{i \\in \\mathcal{A}_t} w_{t, i}^2\n\\end{align*}\nwhere the penultimate step follows because $\\left\\langle \\vect{A}_t^{-1} \\vect{e}_i, \\vect{e}_j \\right\\rangle = 0$ for $i \\neq j$.\n\\end{proof}\n\n\\begin{lemma}\nLet the regret at time $t$ be $r_t = \\langle \\vect{x}_t^*, \\vect{\\theta}^* \\rangle - \\langle \\vect{x}_t, \\vect{\\theta}^* \\rangle$. If $\\vect{\\theta}^* \\in \\mathcal{C}_t$, then\n\\begin{equation*}\n    r_t \\leq 2 w_t \\sqrt{\\beta_t}\n\\end{equation*}\n\\label{lemma_regret_ub}\n\\end{lemma}\n\n\\begin{proof}\nBy the choice of $(\\vect{x}_t, \\widetilde{\\vect{\\theta}}_t)$, we have\n\\begin{align*}\n    \\langle \\vect{x}_t, \\widetilde{\\vect{\\theta}}_t \\rangle = \\max_{(\\vect{x}, \\vect{\\theta}) \\in \\mathcal{X}_t \\times \\mathcal{C}_t} \\langle \\vect{x}, \\vect{\\theta} \\rangle \\geq \\langle \\vect{x}_t^*, \\vect{\\theta}^* \\rangle\n\\end{align*}\nwhere the inequality uses $\\vect{\\theta}^* \\in \\mathcal{C}_t$. Hence,\n\\begin{align*}\n    r_t &= \\langle \\vect{x}_t^*, \\vect{\\theta}^* \\rangle - \\langle \\vect{x}_t, \\vect{\\theta}^* \\rangle\\\\\n    &\\leq \\langle \\vect{x}_t, \\widetilde{\\vect{\\theta}}_t - \\vect{\\theta}^* \\rangle\\\\\n    &= \\langle \\vect{x}_t, \\widetilde{\\vect{\\theta}}_t - \\widehat{\\vect{\\theta}}_t^{\\text{LS}} \\rangle + \\langle \\vect{x}_t, \\widehat{\\vect{\\theta}}_t^{\\text{LS}} - \\vect{\\theta}^* \\rangle\\\\\n    &\\leq  2 w_t \\sqrt{\\beta_t}\n\\end{align*}\n\nwhere the last step follows from Lemma $\\ref{lemma_conf_width}$.\n\\end{proof}\n\nNext, we show that the confidence widths do not grow too fast.\n\n\\begin{lemma}\nFor every t,\n\\begin{equation}\n    \\log \\det \\vect{A}_{t+1} = d \\log \\gamma + \\sum_{\\tau = 1}^{t} \\sum_{i \\in \\mathcal{A}_\\tau} \\log(1 + w_{\\tau,i}^2) \n\\end{equation}\n\\label{lemma_logdet_multiply}\n\\end{lemma}\n\n\\begin{proof} By the definition of $\\vect{A}_t$, we have\n\\begin{align*}\n    \\det \\vect{A}_{t+1} &= \\det \\left( \\vect{A}_{t} + \\sum_{i \\in \\mathcal{A}_t} \\vect{e}_{i} \\vect{e}_{i}^\\textrm{T} \\right)\\\\\n    &= \\det \\left( \\vect{A}_{t}^{1\/2} \\left( \\vect{I} + \\vect{A}_{t}^{-1\/2}\\left( \\sum_{i \\in \\mathcal{A}_t} \\vect{e}_{i} \\vect{e}_{i}^\\textrm{T} \\right) \\vect{A}_{t}^{-1\/2} \\right) \\vect{A}_{t}^{1\/2} \\right) \\\\\n    &= \\det (\\vect{A}_{t}) \\det\\left( \\vect{I} + \\sum_{i \\in \\mathcal{A}_t} \\vect{A}_{t}^{-1\/2}  \\vect{e}_{i} \\vect{e}_{i}^\\textrm{T} \\vect{A}_{t}^{-1\/2}  \\right)\n\\end{align*}\n\nEach $\\vect{A}_{t}^{-1\/2}  \\vect{e}_{i} \\vect{e}_{i}^\\textrm{T} \\vect{A}_{t}^{-1\/2}$ term has zeros everywhere except one entry on the diagonal and that non-zero entry is equal to $w_{t, i}^2$. Furthermore, the location of the non-zero entry is different in for each term. Hence,\n\\begin{equation*}\n    \\det\\left( \\vect{I} + \\sum_{i \\in \\mathcal{A}_t} \\vect{A}_{t}^{-1\/2}  \\vect{e}_{i} \\vect{e}_{i}^\\textrm{T} \\vect{A}_{t}^{-1\/2}  \\right) = \\prod_{i \\in \\mathcal{A}_t} (1 + w_{t, i}^2)\n\\end{equation*}\n\nTherefore, we have\n\\begin{equation*}\n    \\log \\det \\vect{A}_{t+1} = \\log\\det \\vect{A}_{t} + \\sum_{i \\in \\mathcal{A}_t} \\log(1 + w_{t, i}^2)\n\\end{equation*}\nSince $\\vect{A}_1 = \\gamma \\vect{I}$, we have $\\log \\det \\vect{A}_1 = d \\log \\gamma$ and the result follows by induction.\n\\end{proof}\n\n\\begin{lemma}\nFor all t, $\\log \\det \\vect{A}_t \\leq d \\log (t + \\gamma - 1)$.\n\\label{lemma_logdet_upper}\n\\end{lemma}\n\\begin{proof}\nNoting that $\\vect{A}_t$ is a diagonal matrix with diagonals $(n_{t, i} + \\gamma)$,\n\\begin{align*}\n    \\tr \\vect{A}_t &= d \\gamma + \\sum_{i = 1}^{d} n_{t, i} \\\\\n    &= d \\gamma + d (t-1) \\\\\n    &= d(t + \\gamma - 1)\n\\end{align*}\nNow, recall that $\\tr \\vect{A}_t$ equals the sum of the eigenvalues of $\\vect{A}_t$. On the other hand, $\\det(\\vect{A}_t)$ equals the product of the eigenvalues. Since $\\vect{A}_t$ is positive definite, its eigenvalues are all positive. Subject to these constraints, $\\det(\\vect{A}_t)$ is maximized when all the eigenvalues are equal; the desired bound follows. \n\\end{proof}\n\n\\begin{lemma}\nLet $\\gamma \\geq 1$. Then, for all t, we have\n\\begin{equation*}\n    \\sum_{\\tau = 1}^{t} \\sum_{i \\in \\mathcal{A}_\\tau} w_{\\tau, i}^2 \\leq 2 d \\log \\left(1 + \\frac{t}{\\gamma} \\right)\n\\end{equation*}\n\\label{lemma_width_sum_ub}\n\\end{lemma}\n\n\\begin{proof}\nNote that $0 \\leq w_{\\tau, i}^2 \\leq 1$, if $\\gamma \\geq 1$. Using the inequality $y \\leq 2 \\log(1 + y)$ for $0 \\leq y \\leq 1$, we have\n\\begin{align*}\n    \\sum_{\\tau = 1}^{t} \\sum_{i \\in \\mathcal{A}_\\tau} w_{\\tau, i}^2 &\\leq 2 \\sum_{\\tau = 1}^{t} \\sum_{i \\in \\mathcal{A}_\\tau} \\log (1 + w_{\\tau, i}^2)\\\\\n    &= 2 \\log \\det \\vect{A}_{t+1} - 2 d \\log \\gamma\\\\\n    &\\leq 2 d \\log \\left(1 + \\frac{t}{\\gamma} \\right)\n\\end{align*}\nwhere the last two lines follow from Lemmas \\ref{lemma_logdet_multiply} and \\ref{lemma_logdet_upper} respectively.\n\\end{proof}\n\n\\begin{lemma}\nLet the instantaneous regret at time $t$ be $r_t = \\langle \\vect{x}_t^*, \\vect{\\theta}^* \\rangle - \\langle \\vect{x}_t, \\vect{\\theta}^* \\rangle$. If $\\gamma \\geq 1$ and $\\vect{\\theta}^* \\in \\mathcal{C}_t$ for all $t \\leq T$, then\n\\begin{equation*}\n    \\sum_{t = 1}^{T} r_t^2 \\leq 8 \\beta_T d \\log \\left(1 + \\frac{T}{\\gamma} \\right)\n\\end{equation*}\n\\label{lemma_square_regret}\n\\end{lemma}\n\n\\begin{proof}\n Assuming that $\\vect{\\theta}^* \\in \\mathcal{C}_t$ for all $t \\leq T$,\n\\begin{align*}\n    \\sum_{t = 1}^{T} r_t^2 &\\leq  \\sum_{t = 1}^{T} 4 w_t^2 \\beta_t\\\\\n    &\\leq 4 \\beta_T \\sum_{t = 1}^{T} w_t^2\\\\\n    &= 4 \\beta_T \\sum_{t = 1}^{T} \\sum_{i \\in \\mathcal{A}_t}  w_{t, i}^2 \\\\\n    &\\leq 8 \\beta_T d \\log \\left(1 + \\frac{T}{\\gamma} \\right)\n\\end{align*}\nwhere the first step follows from Lemma \\ref{lemma_regret_ub}, second step follows from the definition of $\\beta_t$, third step uses the identity given in Lemma \\ref{lemma_width_identity} and the last step is due to Lemma \\ref{lemma_width_sum_ub}.\n\\end{proof}\n\n\\begin{theorem}\nLet $\\gamma \\geq 1$. Then, with probability at least $1 - 2 \\delta$, the T period regret is bounded by\n\\begin{equation*}\n    \\mathcal{R}(T, \\pi) \\leq \\sqrt{ 8 d \\beta_T^* (\\delta, \\alpha, \\gamma) T \\log \\left(1 + \\frac{T}{\\gamma} \\right) }\n\\end{equation*}\nwhere\n\\begin{equation}\n    \\beta_T^*(\\delta, \\alpha, \\gamma) = 8 \\eta^2 \\log \\left(\\mathcal{N}(\\mathcal{F}, \\alpha, \\| \\cdot \\|_{2}) \/ \\delta \\right) + 2 \\alpha d T \\left(8 B + \\sqrt{8 \\eta^2 \\log(4 d T^2  \/ \\delta)} \\right) + 4 \\gamma G^2\n\\end{equation}\n\\label{thm_regret}\n\\end{theorem}\n\n\\begin{proof}\nAssuming that $\\vect{\\theta}^* \\in \\mathcal{C}_t$ for all $t \\leq T$,\n\\begin{align*}\n    \\mathcal{R}(T, \\pi) &= \\sum_{t = 1}^{T} r_t \\\\\n    &\\leq \\left( T \\sum_{t = 1}^{T} r_t^2 \\right)^{1\/2}\\\\\n    &\\leq \\left( 8 d \\beta_T T \\log \\left(1 + \\frac{T}{\\gamma} \\right) \\right)^{1\/2}\n\\end{align*}\nwhere the last step follows from Lemma \\ref{lemma_square_regret}. Then, by Lemma \\ref{lemma_confidence}, $\\vect{\\theta}^* \\in \\mathcal{C}_t$ for all $t \\leq T$ with probability at least $1 - 2 \\delta$. Therefore, the bound holds true with probability at least $1 - 2 \\delta$.\n\\end{proof}\n\n\\begin{corollary}\nLetting $\\delta = \\mathcal{O} ((dT)^{-1})$, $\\alpha = \\mathcal{O} ((dT)^{-1})$ and $\\gamma = 1$ results in a regret bound that satisfies\n\\begin{equation}\n    \\mathcal{R}(T, \\pi) = \\widetilde{\\mathcal{O}} \\left( \\sqrt{\\eta^2 d \\log \\left(\\mathcal{N}(\\mathcal{F}, T^{-1}, \\| \\cdot \\|_{2})\\right) T} \\right)\n\\end{equation}\n\\label{corr_regret_order}\n\\end{corollary}\n\n\\begin{proof}\nBy Theorem \\ref{thm_regret}, with probability $1$,\n\\begin{equation*}\n    \\mathcal{R}(T, \\pi) \\leq (1 - \\delta) \\sqrt{ 8 d \\beta_T^* (\\delta, \\alpha, \\gamma) T \\log \\left(1 + \\frac{T}{\\gamma} \\right) } + 2 \\delta BdT\n\\end{equation*}\n\nLetting $\\delta = \\mathcal{O} (T^{-1})$, $\\alpha = \\mathcal{O} = (T^{-1})$ and $\\gamma = 1$,\n\\begin{equation*}\n    \\mathcal{R}(T, \\pi) = \\widetilde{\\mathcal{O}} \\left( \\sqrt{d \\beta_T^* (T, T^{-1}, 1) T} \\right)\n\\end{equation*}\n\nNoting that $\\beta_T^* (T, T^{-1}, 1) = \\widetilde{\\mathcal{O}} \\left(\\eta^2 \\log \\left(\\mathcal{N}(\\mathcal{F}, T^{-1}, \\| \\cdot \\|_{2})\\right) \\right)$, the proof is complete.\n\n\\end{proof}\n\n\\section{Proofs for OFU-based Allocations}\n\\label{pf_allocation}\n\n\\subsection{Proof of Theorem \\ref{thm_alloc_regret}}\n\nIn the allocation problem, the mean reward of the arms are given in the matrix $\\vect{\\Theta} \\in \\mathbb{R}^{N \\times M}$. Consider setting $\\vect{\\theta} = \\text{vec}({\\vect{\\Theta}})$ as the mean reward vector for the problem described in Appendix Section \\ref{SCMAB}. Noting that $d = NM$ and $\\|\\cdot\\|_\\text{F} = \\|\\text{vec}(\\cdot)\\|_2$, the proof becomes a direct extension of Theorem \\ref{thm_regret}.\n\n\\subsection{Proof of Theorem  \\ref{low_rank_regret_thm}}\n\nThe proof is direct extension of Corollary \\ref{corr_regret_order} where the covering number is replaced with the following upper bound given for the choice of $\\mathcal{L}$ defined in equation \\eqref{low_l}. We provide the upper bound for the covering number of $\\mathcal{L}$ in the following lemma.\n\n\\begin{lemma}[Covering number for low-rank matrices]\nThe covering number of $\\mathcal{L}$ given in \\eqref{low_l} obeys\n\\begin{equation}\n    \\log \\mathcal{N}(\\mathcal{L}, \\alpha, \\| \\cdot \\|_{\\text{F}}) \\leq (N + M + 1) R \\log \\left( \\frac{9B \\sqrt{NM}}{\\alpha} \\right)\n\\end{equation}\n\\label{lemma_covering}\n\\end{lemma}\n\n\\begin{proof}\nThis proof is modified from \\cite{candes_2011}. Let $\\mathcal{S} = \\{ \\vect{\\Theta} \\in \\mathds{R}^{N \\times M} : \\text{rank}(\\vect{\\Theta}) \\leq R, \\|\\vect{\\Theta}\\|_\\text{F} \\leq 1\\}$. We will first show that there exists an $\\epsilon$-net $\\mathcal{S}^\\epsilon$ for the Frobenious norm obeying \n\\begin{equation*}\n    |\\mathcal{S}^\\epsilon| \\leq \\left(9 \/ \\epsilon \\right)^{(N+M+1)R}\n\\end{equation*} \n\nFor any $\\vect{\\Theta} \\in \\mathcal{S}$, singular value decomposition gives $\\vect{\\Theta} = \\vect{U} \\vect{\\Sigma} \\vect{V}^\\textrm{T}$, where $\\|\\vect{\\Sigma}\\|_{\\text{F}} \\leq 1$. We will construct an $\\epsilon$-net for $\\mathcal{S}$ by covering the set of permisible $\\vect{U}$, $\\vect{\\Sigma}$ and $\\vect{V}$. Let $\\mathcal{D}$ be the set of diagonal matrices with nonnegative diagonal entries and Frobenious norm less than or equal to one. We take $\\mathcal{D}^{\\epsilon\/3}$ be an $\\epsilon$\/3-net for $\\mathcal{D}$ with $|\\mathcal{D}^{\\epsilon\/3}| \\leq (9\/\\epsilon)^R$. Next, let $\\mathcal{O}_{N,R} = \\{\\vect{U} \\in \\mathbb{R}^{N \\times R} : \\vect{U}^\\textrm{T} \\vect{U} = \\vect{I} \\}$. To cover $\\mathcal{O}_{N,R}$, we use the $\\|\\cdot\\|_{1,2}$ norm defined as\n\\begin{equation*}\n    \\|\\vect{U}\\|_{1,2} = \\max_{i} \\|\\vect{u_i}\\|_{\\ell_2}\n\\end{equation*}\nwhere $\\vect{u_i}$ denotes the $i$th column of $\\vect{U}$. Let $\\mathcal{Q}_{N, R} = \\{\\vect{U} \\in \\mathbb{R}^{N, R} : \\|\\vect{U}\\|_{1,2} \\leq 1\\}$. It is easy to see that $\\mathcal{O}_{{N, R}} \\subset \\mathcal{Q}_{{N, R}}$ since the columns of an orthogonal matrix are unit normed. We see that there is an $\\epsilon\/3$-net $\\mathcal{O}_{{N, R}}^{\\epsilon\/3}$ for $\\mathcal{O}_{{N, R}}$ obeying $|\\mathcal{O}_{{N, R}}^{\\epsilon\/3}| \\leq (9\/\\epsilon)^{NR}$. Similarly, let $\\mathcal{P}_{M,R} = \\{\\vect{V} \\in \\mathbb{R}^{M \\times R} : \\vect{V}^\\textrm{T} \\vect{V} = \\vect{I} \\}$.  By the same argument, there is an $\\epsilon\/3$-net $\\mathcal{P}_{M, R}^{\\epsilon\/3}$ for $\\mathcal{P}_{M, R}$ obeying $|\\mathcal{P}_{{M, R}}^{\\epsilon\/3}| \\leq (9\/\\epsilon)^{MR}$. We now let $\\mathcal{S}^\\epsilon = \\{ \\bar{\\vect{U}} \\bar{\\vect{\\Sigma}} \\bar{\\vect{V}}^\\textrm{T} : \\bar{\\vect{U}} \\in \\mathcal{O}_{{N, R}}^{\\epsilon\/3}, \\bar{\\vect{V}} \\in \\mathcal{P}_{{M, R}}^{\\epsilon\/3}, \\bar{\\vect{\\Sigma}} \\in \\mathcal{D}^{\\epsilon\/3}\\}$, and remark $|\\mathcal{S}^\\epsilon| \\leq |\\mathcal{O}_{{N, R}}^{\\epsilon\/3}| |\\mathcal{P}_{{M, R}}^{\\epsilon\/3}| |\\mathcal{D}^{\\epsilon\/3}| \\leq (9\/\\epsilon)^{(N+M+1)R}$. It remains to show that for all $\\vect{\\Theta} \\in \\mathcal{S}$, there exists $\\bar{\\vect{\\Theta}} \\in \\mathcal{S}^\\epsilon$ with $\\|\\vect{\\Theta} - \\bar{\\vect{\\Theta}}\\|_\\text{F} \\leq \\epsilon$.\n\nFix $\\vect{\\Theta} \\in \\mathcal{S}$ and decompose it as $\\vect{\\Theta} = \\vect{U} \\vect{\\Sigma} \\vect{V}^\\textrm{T}$. Then, there exists $\\bar{\\vect{\\Theta}} = \\bar{\\vect{U}} \\bar{\\vect{\\Sigma}} \\bar{\\vect{V}}^\\textrm{T} \\in \\mathcal{S}^\\epsilon$ with $\\bar{\\vect{U}} \\in \\mathcal{O}_{{N, R}}^{\\epsilon\/3}, \\bar{\\vect{V}} \\in \\mathcal{P}_{{M, R}}^{\\epsilon\/3}, \\bar{\\vect{\\Sigma}} \\in \\mathcal{D}^{\\epsilon\/3}$ satisfying $\\|\\vect{U} - \\bar{\\vect{U}}\\|_{1, 2} \\leq \\epsilon\/3$, $\\|\\vect{V} - \\bar{\\vect{V}}\\|_{1, 2} \\leq \\epsilon\/3$ and $\\|\\vect{\\Sigma} - \\bar{\\vect{\\Sigma}}\\|_{\\text{F}} \\leq \\epsilon\/3$. This gives \n\\begin{align*}\n    \\|\\vect{\\Theta} - \\bar{\\vect{\\Theta}}\\|_{\\text{F}} &= \\|\\vect{U} \\vect{\\Sigma} \\vect{V}^\\textrm{T} - \\bar{\\vect{U}} \\bar{\\vect{\\Sigma}} \\bar{\\vect{V}}^\\textrm{T}\\|_{\\text{F}}\\\\\n    &= \\|\\vect{U} \\vect{\\Sigma} \\vect{V}^\\textrm{T} - \\bar{\\vect{U}} \\vect{\\Sigma} \\vect{V}^\\textrm{T} + \\bar{\\vect{U}} \\vect{\\Sigma} \\vect{V}^\\textrm{T} - \\bar{\\vect{U}} \\bar{\\vect{\\Sigma}} \\vect{V}^\\textrm{T} + \\bar{\\vect{U}} \\bar{\\vect{\\Sigma}} \\vect{V}^\\textrm{T} - \\bar{\\vect{U}} \\bar{\\vect{\\Sigma}} \\bar{\\vect{V}}^\\textrm{T}\\|_{\\text{F}} \\\\\n    &\\leq \\|(\\vect{U} - \\bar{\\vect{U}}) \\vect{\\Sigma} \\vect{V}^\\textrm{T}\\|_{\\text{F}} + \\|\\bar{\\vect{U}} (\\vect{\\Sigma} - \\bar{\\vect{\\Sigma}}) \\vect{V}^\\textrm{T}\\|_{\\text{F}} + \\|\\bar{\\vect{U}} \\bar{\\vect{\\Sigma}} (\\vect{V} - \\bar{\\vect{V}})^\\textrm{T} \\|_{\\text{F}}\n\\end{align*}\n\nFor the first term, since $\\vect{V}$ is an orthogonal matrix,\n\\begin{align*}\n    \\|(\\vect{U} - \\bar{\\vect{U}}) \\vect{\\Sigma} \\vect{V}^\\textrm{T}\\|^2_{\\text{F}} &= \\|(\\vect{U} - \\bar{\\vect{U}}) \\vect{\\Sigma} \\|^2_{\\text{F}} \\\\\n    &\\leq \\|\\Sigma\\|^2_{\\text{F}} \\|\\vect{U} - \\bar{\\vect{U}} \\|^2_{1, 2} \\leq (\\epsilon\/3)^2\n\\end{align*}\nBy the same argument, $\\|\\bar{\\vect{U}} \\bar{\\vect{\\Sigma}} (\\vect{V} - \\bar{\\vect{V}})^\\textrm{T} \\|_{\\text{F}} \\leq \\epsilon\/3$ as well. Lastly, $\\|\\bar{\\vect{U}} (\\vect{\\Sigma} - \\bar{\\vect{\\Sigma}}) \\vect{V}^\\textrm{T}\\|_{\\text{F}} = \\|\\vect{\\Sigma} - \\bar{\\vect{\\Sigma}}\\|_{\\text{F}} \\leq \\epsilon \/ 3$. Therefore, $\\|\\vect{\\Theta} - \\bar{\\vect{\\Theta}}\\|_{\\text{F}} \\leq \\epsilon$, showing that $\\mathcal{S}^\\epsilon$ is an $\\epsilon$-net for $\\mathcal{S}$ with respect to the Frobenious norm.\n\nNext, we will construct an $\\alpha$-net for $\\mathcal{L}$ given in equation \\ref{low_l}. Let $\\kappa = B\\sqrt{NM}$. We start by noting that for all $\\vect{\\Theta} \\in \\mathcal{L}$, the Frobenious norm obeys $\\|\\vect{\\Theta}\\|_\\text{F} \\leq \\kappa$. Then, define $\\vect{X} = \\frac{1}{\\kappa} \\vect{\\Theta} \\in \\mathcal{S}$ and $\\mathcal{L}^\\alpha := \\left\\{ \\kappa \\bar{\\vect{X}} : \\bar{\\vect{X}} \\in \\mathcal{S}^\\epsilon \\right\\}$. We previously showed that for any $\\vect{X} \\in \\mathcal{S}$,  there exists $\\bar{\\vect{X}} \\in \\mathcal{S}^\\epsilon$ such that $\\|\\vect{X} - \\bar{\\vect{X}}\\|_\\text{F} \\leq \\epsilon$. Therefore, for any $\\vect{\\Theta} \\in \\mathcal{L}$,  there exists $\\bar{\\vect{\\Theta}} = \\kappa \\bar{\\vect{X}} \\in \\mathcal{L}^\\alpha$ such that $\\|\\vect{\\Theta} - \\bar{\\vect{\\Theta}}\\|_\\text{F} \\leq \\kappa \\epsilon$. Setting $\\epsilon = \\alpha \/ \\kappa$, we obtain that $\\mathcal{L}^\\alpha$ is an $\\alpha$-net for $\\mathcal{L}$ with respect to the Frobenious norm. Finally, the size of $\\mathcal{L}^\\alpha$ obeys\n\\begin{equation*}\n    |\\mathcal{L}^\\alpha| = |\\mathcal{S}^{\\alpha \/ \\kappa}| \\leq \\left(9 \\kappa \/ \\alpha \\right)^{(N+M+1)R}\n\\end{equation*} \nThis completes the proof.\n\\end{proof}\n\n\\section{Additional Experimental Details}\n\\label{sect_additional_exp}\n\nAll experiments are implemented in Python and carried out on a machine with 2.3GHz 8-core Intel Core i9 CPU and 16GB of RAM. We solve the allocation integer program \\eqref{integer_num} using large-scale mixed integer programming (MIP) solver packages to have efficient computations.\n\n\\textbf{Parameter setup}:\n\\begin{itemize}[labelindent= 0pt, align= left, labelsep=0.4em, leftmargin=*]\n\\item In synthetic data, $\\vect{\\Theta}^*$ is scaled such that $B = 10$.\n\\item Standard deviation of the rewards: $\\eta = 1$.\n\\item In the static setting, $d_{t, u} = 1$ for all $u \\in [N]$.\n\\item In the dynamic setting, $d_{t, u} = 1$ with probability $0.2$, $0$ otherwise, independently for each $u \\in [N]$.\n\\item $C_\\text{max} = \\text{ceil} \\left( \\frac{3}{M} \\sum_{u = 1}^{N} d_{t, u} \\right)$.\n\\item $c_{t, i}$ are uniformly sampled over $\\{1, \\dots, C_\\text{max}\\}$ independently for each $t \\in [T]$ and $i \\in [M]$.\n\n\\end{itemize}\n\nTo complement our discussion on the importance of capacity-aware recommendations, Figure \\ref{fig_cap} shows the effects of having stricter capacity constraints. When $C_\\text{max}$ is low (the capacity constraints are stricter), we see that the performance gap between our proposed method and ICF2 is larger. Therefore, it is more crucial to employ capacity-aware mechanisms in settings with tighter capacity constraints. \n\n\\begin{figure}[ht]\n\\center\n\\includegraphics[width=0.4\\textwidth]{plots\/capacity_effect.jpeg}\n\\caption{Normalized cumulative reward obtained in $T = 300$ rounds for different choices of $C_{\\text{max}}$ (normalized by the cumulative reward of optimal allocations). Synthetic data in a static setting with $N = 400$, $M = 200$, $R = 10$. For each data point, the experiments are run on $20$ problem instances and means are reported together with error regions that indicate one standard deviation of uncertainty.}\n\\label{fig_cap}\n\\end{figure}\n\nIn Figures \\ref{exp_1}, \\ref{exp_2}, \\ref{exp_3} and \\ref{exp_4}, we provide detailed  results for different experimental settings described in Section \\ref{sect_exp}. Reward indicates the instantaneous reward obtained in each iteration, regret is the gap between the reward of the optimum allocation and the allocation achieved by the algorithm. Cumulative regret (defined in \\eqref{cumulative_regret}) is the cumulative sum of instantaneous regrets up to iteration $t$.  The average cumulative regret is obtained by normalizing the cumulative regret with $1 \/ t$.\n\n\\begin{figure}[ht]\n\\center\n\\includegraphics[width=\\textwidth]{plots\/experiment_1.jpeg}\n\\caption{Experimental results for synthetic data in a static setting with $N = 800$, $M = 400$, $R = 20$. The experiments are run on $10$ problem instances and means are reported together with error regions that indicate one standard deviation of uncertainty.}\n\\label{exp_1}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\center\n\\includegraphics[width=\\textwidth]{plots\/experiment_2.jpeg}\n\\caption{Experimental results for synthetic data in a dynamic setting with $N = 1000$, $M = 150$, $R = 20$, probability of activity $0.2$. The experiments are run on $10$ problem instances and means are reported together with error regions that indicate one standard deviation of uncertainty.}\n\\label{exp_2}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\center\n\\includegraphics[width=\\textwidth]{plots\/experiment_3.jpeg}\n\\caption{Experimental results for Restaurant-Customer data in a static setting. The experiments are run on $10$ problem instances and means are reported together with error regions that indicate one standard deviation of uncertainty.}\n\\label{exp_3}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\center\n\\includegraphics[width=\\textwidth]{plots\/experiment_4.jpeg}\n\\caption{Experimental results for MovieLens 100k data in a static setting. The experiments are run on $10$ problem instances and means are reported together with error regions that indicate one standard deviation of uncertainty.}\n\\label{exp_4}\n\\end{figure}\n\\section*{Checklist}\n\n\\begin{enumerate}\n\n\\item For all authors...\n\\begin{enumerate}\n  \\item Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?\n    \\answerYes{We pose the problem of making recommendations that will achieve optimal allocations in markets with constraints. Then, we formulate this problem as a structured combinatorial bandit and propose a no-regret algorithm.}\n  \\item Did you describe the limitations of your work?\n    \\answerYes{In Section \\ref{learning_opt_alloc}, we describe that we are studying the posed problem of interactive recommendations for optimum allocations under the assumptions that facilitate our theoretical analysis. In Section \\ref{sect_lrcb}, we indicate that one possible approach to improve upon our theoretical analysis might be by assuming a problem setting where at most some specific number of the arms can be played in each round.}\n  \\item Did you discuss any potential negative societal impacts of your work?\n    \\answerNA{}\n  \\item Have you read the ethics review guidelines and ensured that your paper conforms to them?\n    \\answerYes{}\n\\end{enumerate}\n\n\n\\item If you are including theoretical results...\n\\begin{enumerate}\n  \\item Did you state the full set of assumptions of all theoretical results?\n    \\answerYes{See Assumptions \\ref{rew_assumptio} and \\ref{low_assum} in Section \\ref{sect_methodology}.}\n        \\item Did you include complete proofs of all theoretical results?\n    \\answerYes{See Appendix \\ref{appendix_num}, \\ref{SCMAB}, \\ref{pf_conf_sets}, \\ref{pf_regrets} and \\ref{pf_allocation}.}\n\\end{enumerate}\n\n\\item If you ran experiments...\n\\begin{enumerate}\n  \\item Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?\n    \\answerYes{The code and data for the simulations are provided in the supplemental material.}\n  \\item Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?\n    \\answerYes{The hyperparameters of experimental settings are given in  Appendix \\ref{sect_additional_exp} as well as the captions of Figures \\ref{fig_regrets} and \\ref{fig_cap}. Further implementation details can be found in the code.}\n        \\item Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?\n    \\answerYes{See Figure \\ref{fig_regrets} and other figures in Appendix \\ref{sect_additional_exp}.}\n        \\item Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)?\n    \\answerYes{See Appendix \\ref{sect_additional_exp}.}\n\\end{enumerate}\n\n\\item If you are using existing assets (e.g., code, data, models) or curating\/releasing new assets...\n\\begin{enumerate}\n  \\item If your work uses existing assets, did you cite the creators?\n    \\answerYes{The datasets are cited appropriately.}\n  \\item Did you mention the license of the assets?\n    \\answerNA{}\n  \\item Did you include any new assets either in the supplemental material or as a URL?\n    \\answerYes{The code and data for the simulations are provided in the supplemental material.}\n  \\item Did you discuss whether and how consent was obtained from people whose data you're using\/curating?\n    \\answerNA{}\n  \\item Did you discuss whether the data you are using\/curating contains personally identifiable information or offensive content?\n    \\answerNA{}\n\\end{enumerate}\n\n\n\\item If you used crowdsourcing or conducted research with human subjects...\n\\begin{enumerate}\n  \\item Did you include the full text of instructions given to participants and screenshots, if applicable?\n    \\answerNA{}\n  \\item Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?\n    \\answerNA{}\n  \\item Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation?\n    \\answerNA{}\n\\end{enumerate}\n\n\n\\end{enumerate}\n\n\n\n\n\\section{Conclusion and future directions}\n\n\nIn this paper, we have studied the setting of interactive\nrecommendations that achieve socially optimal allocations under capacity constraints. We have formulated the problem as a low-rank combinatorial multi-armed bandit and proposed an algorithm that enjoys low regret. Building on the ideas founded in this work, we aim to pursue joint recommendation and pricing mechanisms that will achieve optimal allocations in the general problem setting with users actively reacting to the recommendations based on the prices determined by the provider. We believe that this is a practically-relevant question to be resolved because it allows for design of many interesting real-world recommendation applications for settings with associated markets. \n\n\\section{Discussion}\n\\label{sec:discussion}\n\nDescribe how recommendations have become an integral part of our socioeconomic life.\nDescribe how the current implementations miss the key factor of limited service capacities.\nUse examples such as restaurant suggestions, hospital suggestions, supply chain recommendations and business decisions.\nOur goal is to design a recommendation system that is aware of market conditions, service capacities, etc.\n\n\nEl Farol Bar, Kolkata Paisa Restaurant problem, etc (mainly statistical methods in this literature).\nLearning related works: Include the paper found by Efe related to such works, paper by Jordan, etc. \nShow what is missing in each one of them. \nFor example, pricing, or capacity constraints, or utilities, etc.\nKelly handles many of these aspects but lacks learning efficiciency.\nFor example, there is a huge communication overhead involved in signalling between the network and the users.\n\n\nWe consider the setting of online food delivery as an example setting to demonstrate our approach.\nConsider a platform that allocates diners to restaurants.\n(Doordash and UberEats are prominent examples of such platforms today.)\nDiners come to the platform seeking to place orders.\nThere are several restaurants registered with the platform ready to serve. \nBesides there are people who deliver the food from the restaurants to the diners.\nOther possible participants include the stakeholders, regulatory agencies, worker unions, etc.\nThis classification is useful to characterize the relevant attributes that are related to the different components in the network and how they connect with each other. \n\nHere is an example of the relevant attributes and their inter-connectedness.\nThe diners have preferences over the different restaurants.\nAlong with the food choice, the diners also care about the delay in their service, i.e.\nthe time it takes for them to recieve their order.\nPreference over restaurants and delays is an important attribute of the diners.\nThese preferences are their private knowledge which might be hard to learn with limited communication.\nBesides the diners might also be unwilling to report these truthfully.\nWe allow different diners to have different preferences, but they all belong to the same class of having preferences over restaurants and delays.\nOn the other hand, the restaurants have different serving capacities.\nThe number of diners that are being served \nby any given restaurant affects the delay of the diners served by that restaurant \nand in turn the experience of these diners.\n\n\nThere are several more aspects to this problem, for example, the availability of delivery people affects the delays.\nThe delivery people will have different preferences over where they wish to travel in order to deliver.\nFor example, at the end of the day a delivery person would prefer to deliver at a place closer to her home.\nThe restaurants can also have preferences over their idle times, for example, different restaurants would care differently about the lack of orders.\nFor example, restaurants that have online food delivery as their main source of revenue will care more whereas restaurants that rely more on on-foot traffic to their restaurants will care less.\nBesides, it is a dynamic setting since the availability of delivery people, the preferences of the agents in the network, and the serving capacities of the restaurants are all changing with time.\nWe will take a market-based approach to solve this problem.\n\n\nIn this paper, however, we will restrict our attention to serving the diners to the best extent possible given the constraints coming from the limited serving capacities of the restaurants and the availability of delivery persons.\nWe will only care about the preferences of the diners and treat the other components of the network as capacity constraints.\nWe will relegate the more general problem incorporating the preferences of the other components in the network for future papers.\n\n\\subsection{Notes from Intro}\n\n\n\n\nOur approach is based on integrating methods from network economics (Kelly decentralized network framework) with collaborative filtering (for example, matrix factorization or latent variable models) and bandits learning (for example, UCB or Thompson sampling). \nOur framework can be thought of as an extension of the market based methods in \nnetwork resource allocation settings with the additional feature of practical limitations in collecting the required information.\nThis requires striking a balance between opportunistically learning the unknown preferences and allocating the limited items.\n\n\nIntuition:\nIn market settings prices naturally adjust to the demand and the preferences of the agents so that \nwe allocate items to the agents that prefer them the most.\nKelly proposed a framework in the context of bandwidth allocation over the Internet.\nThe system consists of users with preferences and limited resources that can be used to serve the users.\nThe users preferences are modeled as utility functions over their outcomes,\nand the outcomes are a function of the resource allocation in the system.\nThe problem is formulated as a social welfare maximization one (or equivalently customer satisfaction maximization) subject to the capacity constraints in the system.\nOften, the system operator (platform) does not know the user preferences, only the users know their own preferences (or at least have an idea about it).\nOn the other hand the users do not know about the system capacity constraints and the system operator is best placed to collect this information.\n\n\n\nIn theory, the above framework should solve our problem.\nHowever, in the framework proposed by Kelly, \nwe would require the agents to respond constantly with their updated bids for different items.\nGiven how large the number of available items can be, \nit is impractical to get such responses from the agents.\nAssuming that the agents preferences are related to each other with a hidden low dimensional structure,\nit is natural to use methods from collaborative filtering  to resolve this problem.\n\n\n\\subsection{EXtending to dynamic settings}\n\nLet $[n] := \\{1, 2, \\dots, n\\}$ be the set of diners, with a typical diner denoted by $i$.\nLet $[m] := \\{1, 2, \\dots, m\\}$ be the set of restaurants, with a typical restaurant denoted by $r$.\nFor each diner, restaurant pair let $u_{i,r}:\\bbR^+ \\to \\bbR^+$ be the utility function, where $u_{i,r}(\\tau)$ denotes the utility diner $i$ gets from ordering from restaurant $r$ with a service delay of time $\\tau > 0$.\n\n\nAs in the serverless pricing paper, we will dicretize the delay times into several tiers or levels.\nLet $[l] := \\{1, 2, \\dots, l\\}$ be the set of delay tiers with a typical tier denoted by $t$.\nFor example, let tier $1$ correspond to a delay of at most $10$ mins, tier $2$ correspond to a delay or more than $10$ mins and less than $30$ mins, tier 3 correspond to a delay of more than $30$ mins and less than $60$ mins, and so on.\nNote that the intervals in the different tiers can be of different lengths.\nOur framework is general enough to model any of these.\nThe more the number of intervals, the higher precision we have over the utility functions, but it makes the computational problem harder leading to greater estimation and approximation errors.\nThus, the design of tiers is an important practical problem and will depend on the particular setting and tradeoff concerns.\nWe will assume that we have 6 tiers, each with a length of $10$ mins.\nThus, the maximum delay that we allow is $1$ hr beyond which we assume that the utility is $0$.\n\n(Include a diagram of utility function and values corresponding to tiers.)\n\n\nLet $U$ be an $n \\times m \\times l$ tensor, where $U(i,r,t)$ is the utility of diner $i$ if she receives her order from restaurant $r$ within the delay tier $t$.\nThis models the preferences of the diners.\nLet us now model the system constraints.\nWe will first have a stylized model for the system, but then relax it later.\nLet $\\mu_r$ denote the average service rate of restaurant $r$, i.e. if $\\mu_r = 1$ serving\/min then restaurant $r$ can cook 1 serving every min on average.\nGiven the current availability of delivery persons, let $\\delta_{i,r}$ denote the time it takes to deliver the order from restaurnat $r$ to diner $i$.\nLet us assume that this is also fixed for the moment, for example, say we use the average delay in delivery time.\n(The general RL framework will be stronger and be able to adapt to changing delivery times and restaurant serving rates. In that case we won't assume that we know these values a priori, but the RL framework will learn and predict these values in an online fashion.)\n\n\n\nI will now describe the dynamic aspect in this problem.\nWe envision the scheduler to operate periodically, say every $1$ min.\nLet us descritize the time into $1$ min intervals and starting at time $s = 0$.\nAt any instant, it maintains a queue of orders that are yet to be scheduled.\nTo elaborate, diners come to our platform and enter their order size. \nThe diners who are currently looking to order from our platform will be called active diners.\nLet $A(s) \\subset [n]$ denote the set of active diners at step $s$.\nEach diner $i$ once it comes to the platform, inputs its order size $J_i$, the amount of food she wishes to order.\nThus for each active diner, i.e. $i \\in A$, we know the order size $J_i$.\n\n\nOn the operation of the scheduler in step $s$ it comes up with a matching restaurant for each customer, a corresponding delay in service, and the corresponding cost for the order.\n(I will soon describe how the scheduler comes up with these allocations and pricing, but it is similar to the static scheduler in the serverless pricing paper.)\nEach active diner observes this suggestion and either accepts it or rejects it.\nIf the diner accepts the suggestion, she is scheduled for that order and removed from the queue and is not an active diner anymore.\n(I will use the queue and the set of active diners interchangeably, since they are identical in this framework.)\nIf the diner does not accept the suggestion, she is kept in the queue and will get a new suggestion in the next operation of the scheduler.\n(Possible considerations for practical reasons: we don't want to make the diners wait for $1$ min to come up with a new suggestion. We can consider providing multiple suggestions in each iteration to each diner, or a scheduler that operates at a faster time-scale.)\n\n\nLet the amount of food allocated to user $i$ from restaurant $r$ in tier $t$ by the scheduler be denoted by $x(i,r,t)$.\nThe static problem that the scheduler solves at every operation can be stated as follows:\n\n\\begin{align}\n\\text{\\bf\\underline{SYS}} \\nn \\\\\n\\maxi_{x(i,r,t)\\geq 0} ~~~~~& \\sum_{i=1}^n U(i,R_i,T_i)\\nn \\\\\n\\text{subject to} ~~~~~& \\sum_{i = 1}^N x(i,r,t) \\leq M(r,t), \\forall~ r \\in [m], t \\in [l], \\text{ and }\\label{eq: sys_capacity}\\\\\n& R_i, T_i \\in \\left\\{(r,t) \\in [m]\\times[l]: \\sum_{s = 1}^t x(i,r,s) \\geq J_i\\right\\} ~\\forall~ i \\in [n].\n\n \\label{eq: sys_T_def}\n\\end{align}\n\n\nLet\n\\[\n\tF(i,r,t) = \\frac{U(i,r,t)}{J_i}.\n\\]\nWe relax the above system problem as follows:\n\n\\begin{align}\n\\text{\\bf \\underline{SYS-LP}}\\nn\\\\\n\\maxi_{x(i,r,t)\\geq 0} ~~~~~&\\sum_{i=1}^N \\sum_{t=1}^T x(i,r,t)F(i,r,t)\\nn \\\\\n\\text{subject to} ~~~~~&\\sum_{t=1}^T \\sum_{r = 1}^m x(i,r,t) \\leq J_i, ~\\forall ~i \\in [N], \\label{tot_job_cons}\\\\\n&\\sum_{i = 1}^N x(i,r,t) \\leq M(r,t), ~\\forall~ r \\in [m], t \\in [l]. \\label{tot_resource_cons}\n\\end{align}\n\n\nThis is an LP problem and can be solved efficiently.\n(By performing sparcity analysis we should show that for most of the users their orders come from a single restaurant in a single tier. \nUse this to provide a bound for the relaxation.)\nFurther, we can use Kelly decomposition on this relaxed system problem to obtain a decentralized version for solving this problem.\nIt also gives rise to price discovery $p(i,r,t)$, which is the market equilibrium price for user $i$ to receive its food from restaurant $r$ within tier $t$.\n\n\n\\section{Experiments}\n\\label{sect_exp}\n\nIn this section, we demonstrate the efficiency of our proposed algorithm by conducting an experimental study over both synthetic and real-world datasets. The goal of our experimental evaluation is twofold: (i) evaluate our algorithm for making online recommendations and allocations in various market settings and (ii) understand the qualitative performance and intuition of our algorithm.\n\n\\textbf{Baseline algorithms:} We demonstrate the performance of our method by comparing it with baseline algorithms. To the best of our knowledge, there are no current approaches specifically designed to make interactive recommendations and allocations considering the capacity constraints. Therefore, based on currently available algorithms, we construct our baseline with methods that are designed for similar goals:\n\\begin{enumerate}[nosep, labelindent= 0pt, align= left, labelsep=0.4em, leftmargin=*]\n    \\item \\textbf{ACF:} (Allocations with Collaborative Filtering) It solves for the least squares problem \\eqref{least_squares_estimate_low} to estimate the mean rewards obtained from user-item allocation pairs. Then, makes the best allocation with respect to the estimated parameters at each round.\n    \\item \\textbf{CUCB:} It runs the Combinatorial-UCB algorithm \\cite{chen_2013} to decide on allocations without assuming any low-rank structure between the users and items. It views the user-item allocation pairs as arms that have no correlation in between. In every round, it pulls some subset of the arms according to the capacity and demand constraints.\n    \\item \\textbf{ICF:} It runs the Interactive Collaborative Filtering algorithm with linear UCB \\cite{zhao_2013} without considering the capacity constraints. For each user, the algorithm recommend the items that it estimates the users will obtain the most reward. Since this method does not consider the capacities, the recommendations do not necessarily satisfy the capacity constraints. Therefore, if an item is recommended to more users than its capacity, we assume that only a randomly chosen subset of the assigned users are able to get the item. The users that are not able to get the item do not send any reward feedback to the system.\n    \\item \\textbf{ICF2:} It is the same as ICF method described above, except that the algorithm observes a zero reward ($R_{t, u, i} = 0$) for all the user-item allocations that were not successfully achieved. As a result of the low rewards obtained from allocations that lead to capacity violations, the algorithm learns to avoid violating the capacities.\n\\end{enumerate}\n\n\n\\textbf{Experimental setup and datasets:} We use a synthetic dataset and two real world datasets to evaluate our approach. For the synthetic data, we generate an (approximately) low-rank random matrix $\\vect{\\Theta}^* \\in \\mathbb{R}^{N \\times M}$ with entries from $[0, B]$. For the real-world data, we consider the following publicly available data sets: Movielens 100k \\cite{harper_2015} which includes ratings from 943 users on 1682 movies and the RC (Restaurant and Consumer) dataset \\cite{blanca_2011} which includes ratings from 138 users on 130 restaurants. As the information of capacities are not given in the considered data, and to the best of our knowledge to any of the publicly available recommendation datasets, we consider instantiating random capacities for all items as described shortly. We consider settings with static and time-varying capacities\/demands. For the static case, we assume that all users request one item at all iterations, and the capacity of each each remains unchanged with time. In the dynamic setting, we allow both the demands $\\vect{d}_t$ and capacities $\\vect{c}_t$ vary with time $t$. At each allocation round, we consider that each entry of $\\vect{d}_t$ is independently sampled from a fixed probability distribution over $\\{0, 1\\}$. Therefore, while active users (with demand 1) are allocated at most one item, the inactive users (with demand 0) do not get allocated any item. Similarly, each entry of $\\vect{c}_t$ is independently sampled from a uniform distribution over  $\\{0, 1, \\dots, C_{\\text{max}}\\}$. At each round $t$, if user $u$ is allocated the item $i$, the system observes a reward with normal distribution $\\mathcal{N}(\\theta_{ui}^*, \\eta^2)$. \n \n\\textbf{Results:} We summarize our results in Figure \\ref{fig_regrets}. Further experimental details and results are left to Appendix \\ref{sect_additional_exp}. The observations can be summed up into following points: (1) LR-COMB (our proposed approach) is able to achieve lower regret than all other baseline methods in all experimental settings. (2) Even though the ACF method performs slightly better than LR-COMB in the initial rounds, it often gets stuck at high-regret allocations, and hence cannot achieve \\emph{no-regret}. It suffers from large regrets in the long-term because it tries to directly exploit the information it acquired so far without making any deliberate explorations. Therefore, we observe the significance of employing a bandit-based approach in achieving a no-regret algorithm. (3) Since CUCB does not leverage the low-rank structure of the parameters, it needs to sample and learn about each user-item allocation pair separately. Hence, it takes much longer for it to learn the optimum allocations. (4) Since ICF does not consider the capacities while making the allocations, it ends up incurring very large regrets. Even if it is able to identify the high-reward allocation pairs via collaborative filtering, the recommendations exceed the respective capacities of the items and we cannot obtain high rewards. (5) One possible ad-hoc approach to mitigate the issues with ICF is to use ICF2 which can indirectly capture the effects of the capacities since it receives zero rewards when the items are not successfully allocated. Nevertheless, ICF2 still does not directly use the knowledge of the capacities and hence it is still quite suboptimal. Even though it is able to show decent performance in static settings, its performance significantly degrades when the capacities dynamically change with time. \n\n\\vspace{-6 pt}\n\\begin{figure}[ht]\n\\center\n\\includegraphics[width=\\textwidth]{plots\/regrets_all_experiments.jpeg}\n\\caption{Instantaneous regret incurred in each round in different experimental settings. From left to right: (1) synthetic data in a static setting with $N = 800$, $M = 400$, $R = 20$, (2) synthetic data in a dynamic setting with $N = 1000$, $M = 150$, $R = 20$, probability of user activity $0.2$, (3) Restaurant-Customer data in a static setting, (4) Movielens 100k data in a static setting. In all settings, the experiments are run on $10$ problem instances and means are reported together with error regions that indicate one standard deviation of uncertainty.}\n\\label{fig_regrets}\n\\end{figure}\n\n\\section{Introduction}\n\n\n\n\nOnline recommendation systems have become an integral part of our socioeconomic life with the rapid increases in online services that help users discover options matching their preferences. \nDespite providing efficient ways to discover information about the preferences of users, they have played a largely complementary role to searching and browsing with little consideration of the accompanying \\emph{markets} within which recommended items are allocated to the users. Indeed, in many real-world scenarios, recommendations bring about the \\emph{allocation} of the corresponding items in a market that has possibly intrinsic constraints. \nIn particular, recommendations of candidate items that have associated notions of limited \\emph{capacities} naturally give rise to a market setting where users compete for the allocation of the recommended items.\n\nAllocation constraints are common in recommendation contexts.\nA few interesting examples include: (1) Point-of-Interest (PoI) recommendation systems (e.g., restaurants, theme parks, hotels), where the PoI can only accomodate limited number of visitors, (2) book recommendation systems employed by libraries, where the books recommended to the borrowers have limited copies, (3) route recommendation systems which aim to suggest the optimal road for travelling while avoiding traffic congestion, (4) course recommendation systems for universities, where each recommended course has limited number of seats. As similar systems become more ubiquitous and impactful in the broader aspects of daily life, there is a huge application drive and potential for delivering recommendations that respect the requirements of the market. Therefore, it is crucial to consider capacity-aware recommendation systems to maximize the user experience. \n\n\\textbf{Main Challenges: }We model the user preferences as rewards that users obtain by consuming different items, while the social welfare is the aggregate reward over the entire system comprising multiple users with heterogeneous preferences, and a provider who continually recommends items to the users and receives interactive reward feedback from them. The provider aims to maximize the social welfare while respecting the \\emph{time-varying} allocation constraints: indeed we consider system \\emph{dynamics} in terms of user demands and item capacities to be an important aspect of our problem. In the process of identifying the best match between users and target items, the provider encounters two challenges: The first challenge relates to the element of \\emph{recommendation} as the provider needs to make recommendations without exact knowledge of the user preferences ahead of time, and hence has to continue exploring user preferences while continually making recommendations. The second challenge relates to the \\emph{allocation} aspect of the problem induced by the market constraints. Note that even if matching the users with their most preferred items would result in high rewards, such an allocation may not respect the constraints of the market. For example, in a restaurant recommendation setting, if there is a hugely popular restaurant that most people love, a naive recommender would send many users to the same restaurant, causing overcrowding and considerable user dissatisfaction. \n\nThe key to overcoming the (first) challenge of making accurate recommendations is to learn the user preferences from the reward feedback. Since the preferences of different users for different items are highly correlated, it is natural to employ collaborative filtering techniques that have been widely applied in recommender systems \\cite{schafer_2007, bennett_2007, koren_2009, sarwar_01}. In order to learn the user preferences efficiently, previous works have established interactive collaborative filtering systems that query the users with well-chosen recommendations \\cite{kawale_2015, zhao_2013}. Typically, these works consider a setting where a single user arrives to the system at each round and the system makes a recommendation that will match the user's preferences. However, this assumption no longer holds in applications having  an associated market structure, as recommendations made to different users in the same time period must also respect the constraints of the market.\n\nThe common strategy to tackling the (second) allocation challenge is through pricing mechanisms that ensure social optimality. Such mechanisms have been studied in economics for two-sided (supply and demand) markets and are called Walrasian auctions \\cite{smith_1991}. In the networking literature, Kelly has also used similar mechanisms to do optimal bandwidth allocation over a network \\cite{kelly_97}. In pricing-based mechanisms, the users choose the items based on their preferences as well as the posted prices. The provider meanwhile successively adjusts these prices in response to the user's demand for the items, so that capacity constraints are satisfied in equilibrium. The equilibrium prices ensure that the limited number of items are allocated to users that are expected to obtain the largest reward. However, these mechanisms still require the users to know and evaluate their preferences for \\emph{all} possible items and respond constantly with their updated bids\/demands for each item. This is definitely not a scalable solution for the large-scale system (comprising large numbers of users and items) that we target. Furthermore, this framework assumes that users already know their preferences for all items, which is clearly not true in our setting, where users report their preferences through feedback \\emph{after} being targeted with their recommended items. For this reason, the provider must \\emph{learn} the user preferences in its quest to perform optimal capacity-constrained allocations.\n\n\\begin{wrapfigure}{r}{0.36 \\textwidth} \n\\vspace{-7pt}\n\\includegraphics[width= 0.36 \\textwidth]{plots\/recommendation-system.png}\n\\caption{The provider interactively learns the user preferences to achieve socially optimal capacity-constrained allocations.}\n\\label{system_diagram}\n\\vspace{-8pt}\n\\end{wrapfigure}\nHence, as depicted in Figure \\ref{system_diagram}, the goal of the provider is twofold: (1) to learn the user preferences and make recommendations that will guide the users to choose the items that they are likely to obtain high rewards, (2) to achieve allocations that will satisfy the capacity constraints. To achieve these goals, we envision developing the following market-aware recommendation mechanism for the provider. By recommending items, the provider helps the users to narrow down their options so that users can comprehend and evaluate their preferences among a smaller number of offered items. In addition, being aware of the market structure, the provider carefully determines the item prices that play the role of an intermediary for satisfying the constraints of the market. We believe that this is an important and practically-relevant question to be resolved because it allows for the analysis of many interesting real-world interactive recommendation settings with market constraints. In its full generality, this framework requires us to model the user decisions in a way that will capture the effects of the recommendations and prices that they are presented. In order to avoid the complications introduced by this modelling challenge and to obtain a profound understanding of fundamental aspects of the problem, we begin with focusing our attention on a central question whose solution will be key to making progress towards our longer-term goal of developing a complete framework.\n\n\nSpecifically, we focus our study on these essential aspects of the problem: recommending and allocating the items while interactively learning the user preferences, which to the best of our knowledge has not been addressed in the literature. In essence, we analyze a special case of the mechanism introduced above, by assuming that the provider makes recommendations such that the number of presented choices matches with the number of items the user is willing to consume, so that the users obtain all of the recommended items regardless of their prices. Then, the provider's task reduces to deciding on high-reward allocations while satisfying the constraints by allocating each item to at most certain number of users.\n\n\\textbf{Structured Combinatorial Multi-Armed Bandits: } The provider seeks to choose high-reward allocations subject to the constraints, while actively learning the user preferences by making queries that will give rise to the most informative responses. Therefore, it encounters the well-known \\textit{exploration-exploitation} dilemma. In essence, there exists a trade-off between two competing goals: maximizing social welfare using the historical feedback data, and gathering new information to improve the performance in the future. In the literature of interactive collaborative filtering, this dilemma is typically formulated as a multi-armed bandit problem where each arm corresponds to allocation of an item to a user \\cite{zhao_2013, barraza_2017, wang_2019}. When an item is allocated to a user, a random reward is obtained and the reward information is fed back to the provider to improve its future allocation strategies. However, in contrast to prior works, our setting further requires that a collection of actions taken for different users satisfy the constraints of the market. \n\nWe formulate our problem as a bandit problem with arms having correlated means, and call it Structured Combinatorial Bandit. Based on the standard OFU (Optimism in Face of Uncertainty) principle for linear bandits \\cite{dani_2008, abbasi_2011}, we devise a procedure that learns the mean reward values opportunistically so as to solve the system problem of optimal allocation with minimum regret. The estimation method benefits from both the combinatorial nature of the feedback and the dependencies induced by the low-rank structure of collaborative filtering setting. Moreover, using matrix factorization techniques, the algorithm is efficient even at scale in settings with a large number of users and items. As is standard with OFU-based methods, our algorithm maintains a confidence set of the mean rewards for all user-item pairs. If it has less data about some user-item allocation pair, the confidence set becomes wider in the corresponding direction. Then, due to optimism, the algorithm becomes more inclined to attempt the corresponding allocation pairs to explore and collect more information. \n\n\\textbf{Our contributions: } \n\\begin{itemize}[nosep, labelindent= 0pt, align= left, labelsep=0.4em, leftmargin=*]\n    \\item We formulate the problem of making recommendations that will facilitate socially optimal allocation of items with constraints. Our formulation further allows for the analysis of problem settings with dynamic (i.e., time-varying) item capacities and user demands.\n    \\item We pose the Structured Combinatorial Bandit problem under generic structural assumptions (not only low-rank) and propose an algorithm that achieves sublinear regret bounds in terms of parameters that depend on the problem-specific structure of the arms.\n    \\item For the recommendation setting, we specialize our results to low-rank structures and obtain a Low-Rank Combinatorial Bandit (LR-COMB) algorithm that achieves \\smash{$\\widetilde{\\mathcal{O}} ( \\sqrt{N M (N+M) RT} )$} regret in $T$ rounds for a problem with $N$ users, $M$ items and rank $R$ mean reward matrix.\n\\end{itemize}\n\\textbf{Experiments: } We run experiments both on synthetic and real-world datasets to show the efficacy of the proposed algorithms. Results show that proposed algorithm can obtain significant improvements over naive approaches in solving the problem of recommendation and allocation with constraints.\n\n\\textbf{Related work: }\n\n\n\\begin{itemize}[nosep, labelindent= 0pt, align= left, labelsep=0.4em, leftmargin=*]\n\n\\item \\textbf{Combinatorial Multi-Armed Bandits (CMAB) and Semi-Bandits: } The frameworks of CMAB \\cite{chen_2013, kveton_2015} and semi-bandits \\cite{audibert_2011} model multi-armed bandit problems where the player chooses a subset of arms in each round and observes individual outcomes of the played arms. However, they do not incorporate any structural assumptions about the rewards obtained from the arms. However, in a collaborative filtering setting like ours, the main promise is to leverage the intrinsic structure between different user-item pairs. To close this gap, we pose the problem of Structured Combinatorial Bandits and devise an algorithm that makes use of the structure of the arms as well. Additionally, CMAB framework assumes availability of an oracle that takes the means rewards for the arms and outputs the optimum subset of arms subject to the selection constraints. Due to the combinatorial nature of the problem, this oracle may not be readily available in general CMAB settings. In our case, due to the special structure of the capacity constraints, we can efficiently solve for the optimum allocations given the mean rewards of the allocation pairs.\n\n\\item \\textbf{Structured Linear Bandits: } Our formulation also shows parallelism with the frameworks of structured linear bandits \\cite{johnson_2016, combes_2017} and low-rank linear bandits \\cite{lu_2021}. However, it is distinct from them by having the additional ability to capture the combinatorial nature of the problem. In linear bandits, the player only observes the final total reward, but no outcome of any individual arm. Our setup differs from their case because the player (provider) is able to observe individual outcomes of all played arms. Due to this richer nature of the observation model, we can achieve lower regret guarantees than what is available in the literature of structured linear bandits.\n\n\\item \\textbf{Recommendation with Capacity Constraints: } There have been a few works using the notion of constrained resources to model and solve the problem of recommendation with capacity constraints \\cite{christakopoulou_2017, makhijani_2019}. However, these works only consider optimizing the recommendation accuracy subject to item usage constraints without any consideration of the interactive mechanisms that discover user preferences through recommendations.\n\n\\item \\textbf{Competing Bandits in Matching Markets: } One other related line of literature studies the stable matching problem in two-sided markets \\cite{liu_2020}. The model assumes that entities on each side of the market has preference orderings for the other side of the market and the allocations are driven by these preference orderings rather than the prices. In contrast to our work, these mechanisms necessitate at least the entities on one side of the market know their preferences over all the entities on the other side of the market. However, in many real-world settings of optimum recommendation and allocation, like the examples given above, the explicit preferences are not known ahead of time and can only be discovered through interactions. Furthermore, the matching markets only model one-to-one matches, meaning that they do not allow for the items to be allocated for multiple users.\n\\end{itemize}\n\\vspace{-2pt}\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\n\n\n\n\n\\section{Problem setting}\n\\label{sec:setting}\n\nWe use bold font for vectors $\\mathbf{x}$ and matrices $\\mathbf{X}$, and calligraphic font $\\mathcal{X}$ for sets. We denote by $[K]$ the set $\\{1,2, \\dots, K\\}$.\nFor a vector $\\mathbf{x}$, we denote its $i$-th entry by $x_i$ and for a matrix $\\mathbf{X}$, we denote its $(i,j)$-th entry by $x_{ij}$. We denote the Frobenius inner product of two matrices by $\\langle \\vect{A}, \\vect{B} \\rangle = \\tr (\\vect{A}^\\mathrm{T} \\vect{B})$, and the Frobenious norm of a matrix $\\vect{A}$ by $\\|\\vect{A}\\|_\\text{F}$.\n\nSuppose the \\emph{system} has $N$ \\emph{users} and $M$ \\emph{items} in record. \nThe items are \\emph{allocated} to the users in multiple \\emph{rounds} (or \\emph{periods}) denoted by $t \\in \\mathbb{N}$. Allocation of an item $i \\in [M]$ to a user $u \\in [N]$ results in a random \\emph{reward} that has a distribution unknown to the system provider. The expected reward obtained from allocating item $i$ to user $u$ is denoted by $\\theta^*_{ui}$ and these values are collected into the mean reward matrix $\\mathbf{\\Theta}^* \\in \\mathbb{R}^{N \\times M}$. \n\nWe assume that each item has (time-varying) capacity that corresponds to the maximum number of different users it can be allocated to. We denote the capacity of item $i \\in [M]$ by $c_{t,i}$, and collect these values into vectors $\\mathbf{c}_t \\in \\mathbb{R}^{M}$. Similarly, each user has a (time-varying) demand that corresponds to the maximum number of different items it can get allocated. We denote the demand of user $u \\in [N]$ by $d_{t,u}$, and collect these values into vectors $\\mathbf{d}_t \\in \\mathbb{R}^{N}$. Therefore, each item can only be allocated to at most $c_{t,i}$ different users, while each user can only get allocated at most $d_{t,u}$ different types of items in the period $t$. We shall call these the \\emph{allocation constraints}. One can consider the special case where $d_{t,u}$ parameters only take values from $\\{0, 1\\}$ so that each \\emph{active} user gets at most one allocation while the \\emph{inactive} users do not get any allocations.\n\nLet $\\mathbf{X}_{t}$ denote the \\emph{allocation matrix} for round $t$ where the $(u, i)$-th entry is one if user $u$ is allocated item $i$ at round $t$, and zero otherwise. Due to the allocation constraints, any valid $\\mathbf{X}_{t}$ must belong to the set of valid allocation matrices $\\mathcal{X}_t \\subseteq \\{0, 1\\}^{N \\times M}$ defined as:\n\\begin{equation*}\n    \\mathcal{X}_t = \\{ \\mathbf{X} \\in \\{0, 1\\}^{N \\times M} : \\mathbf{X} \\mathds{1}_M \\leq \\mathbf{d}_t \\text{ and } \\mathbf{X}^\\textrm{T} \\mathds{1}_N \\leq \\mathbf{c}_t\\}\n\\end{equation*}\n\\vspace{-1pt}\nwhere the inequalities are entry-wise and $\\mathds{1}_p$ denotes the all-ones vector of size $p$.\n\n\\subsection{Optimal allocations}\n\\label{sect_opt_allocations}\n\nGiven the knowledge about the mean reward matrix $\\vect{\\Theta}^*$, the optimal allocation $\\vect{X}^*_t$ at time $t$ can be obtained by solving the integer program:\n\\begin{equation}\n    \\vect{X}^*_t \\in \\argmax_{\\vect{X} \\in \\mathcal{X}_t} \\; \\langle \\vect{X}, \\vect{\\Theta}^* \\rangle\n    \\label{integer_num}\n\\end{equation}\nThis integer program can be relaxed to a linear program by dropping the integral constraints (setting $0 \\leq x_{ui} \\leq 1$). In Appendix \\ref{appendix_num}, we show that the integrality gap of this problem is zero. \\footnote{The integrality gap is the difference between optimal values of the integer program and its linear relaxation.} Hence, any integer solution found for the relaxed problem is also a solution for the allocation problem. \n\nWhen the provider does not have direct knowledge of the mean rewards associated with user-item allocation pairs, one standard approach is to employ pricing mechanisms \\cite{smith_1991, kelly_97}. \nThe idea is to apply dual decomposition on the (partial) Lagrangian function $ L( \\vect{X},  \\vect{\\lambda}) = \\langle \\vect{X}, \\vect{\\Theta}^* \\rangle + \\vect{\\lambda}^\\textrm{T} ( \\vect{c}_t - \\vect{X}^\\textrm{T} \\mathds{1}_N)$ where $\\vect{\\lambda} \\geq 0$ are the Lagrange multipliers (item prices) associated with the capacity constraints. Then, the allocation problem is decomposed into one problem for each user and one problem for the provider where the item prices mediates between the subsidiary problems. Each user calculates its demand by maximizing the corresponding component of the Lagrangian for a given set of prices. On the other side, the provider iteratively updates the prices based on users demands to achieve the optimal pricing. At the end of many consecutive updates from users and the provider, the equilibrium ensures that the limited items are allocated to users that are expected to obtain the largest reward. (See \\cite{palomar_2006} for further details.)\n\n\nHowever, as discussed in the introduction, this pricing mechanism has limitations in many real-world applications. Most importantly, it requires the user to solve a problem that involves the valuations even for the items that the user has no prior experience with. However, in many real-world scenarios, it is infeasible to request the users to choose among all the items in the system. Secondly, in the process of price discovery, the mechanism asks the users to repetitively respond to the prices by recomputing their demands. However, since it might take many iterations until convergence to the optimal pricing, asking the users to respond many times would be a burden for them. Furthermore, the final prices found by this iterative mechanism are only guaranteed to be optimal for the problem defined by the capacity $\\vect{c}_t$ and demand $\\vect{d}_t$ parameters at round $t$. If the capacities and demands vary with time, the optimal pricing and allocation for the next allocation round $t+1$ will be different and will be needed to be rediscovered.  \n\n\\subsection{Learning the optimal allocations}\n\n\\label{learning_opt_alloc}\n\nTo address the issues discussed in the previous section, we need mechanisms that can find the optimum allocations using fewer and simpler interactions. One resolution is to recommend a subset of items along with prices intelligently chosen by the provider. This way, the users will be able to easily evaluate their preference on the small number of recommended items and decide on their demand without requiring to consider all items in the system. The provider will decide on well-chosen offerings with correct prices so that it can satisfy the capacity constraints. However, as the provider does not have the complete knowledge of the user preferences, it needs to learn the unknown preference parameters $\\vect{\\Theta}^*$ from the user feedback so that it can determine better recommendations as well as the correct prices. Based on the examples of applications provided in the introduction, we believe that design of such system dynamics is a practically-relevant question to be resolved.\n\nAs a first step in this direction, we decide to restrict our attention to a setting that itself has interesting interactions between learning the user preferences and allocating the items. In order to facilitate our analysis, we consider that the number of choices presented to each user $u$ at round $t$ is limited exactly by the their demand $d_{t, u}$ and users are allocated with all of the items that they are recommended. Therefore, the problem essentially reduces to an allocation problem in which users get allocated a set of items directly by the provider instead of users choosing between the offerings. Then, after each round of allocation, users provide feedback about the items that they have been allocated so that the provider can enhance its performance in the following rounds. Hence, whilst the users get allocated sequentially, the predictions are constantly refined using the reward feedback.\n\nThe provider determines the allocations according to an \\textit{optimistic} estimate of the true mean reward matrix $\\vect{\\Theta}^*$. It solves the allocation problem \\eqref{integer_num} assuming that the estimated parameter is the underlying reward parameter and obtains an estimate for the optimum allocation at each round $t$. Even though these allocations can be suboptimal due to estimation errors, our analysis shows that the cumulative regret obtained from these sequential allocations can only grow sublinearly with the time horizon. Using this approach, we are coupling the general principle of optimism in the face of uncertainty (OFU) along with capacity aware resource allocation. In the experiments section, we show the importance of this connection by comparing our strategy with algorithms that only focus on one aspect of the problem: a non-OFU algorithm that only aims for achieving momentary performance and an OFU-based algorithm that is unaware of the capacities.\n\n\\begin{remark}\n\\label{remark_price}\nWhen the allocation problem \\eqref{integer_num} is solved with the estimated parameters, the Lagrange multipliers for the capacity constraints give estimates for the optimum prices of the items. As long as the user preferences are estimated well enough, these prices emerging from provider's problem are such that users who are aware of their preference for all items would still choose the recommended items. Hence, when the user preferences are learned, the mechanism is able to achieve high-reward allocations that complies with the user incentives under the optimal pricing.\n\\end{remark} \n\n\n\\subsection{Problem formulation}\n\nIn this section, we formulate the provider's problem and its objective. At each time period $t$, the provider chooses multiple user-item allocation pairs collected into a set $\\mathcal{A}_t \\subseteq [N] \\times [M]$. Then, the provider observes a random reward $R_{t, u, i}$ if user $u$ is allocated with item $i$ at round $t$. The total reward is the sum of rewards obtained from the system at all rounds during a time horizon $T$. The task is to repeatedly allocate the items to the users in multiple rounds so that the total expected reward of the system is as close to the reward of the optimal allocation as possible.\n\nLetting $\\mathbf{E}_{u, i} \\in \\mathbb{R}^{N \\times M}$ denote the zero-one matrix with a single one at the $(u, i)$ entry, we can write the indicator matrix for the allocation at time $t$ as $\\mathbf{X}_{t}  = \\sum_{(u, i) \\in \\mathcal{A}_t} \\mathbf{E}_{u, i}$. Consequently, $\\mathbf{X}_{t}$ becomes a zero-one matrix with ones at entries $\\mathcal{A}_t$ and zeros everywhere else. Note that there is a one-to-one relation between the matrix $\\mathbf{X}_{t}$ and the set $\\mathcal{A}_t$.\n\nWe denote by $H_t$ the history $\\{\\mathbf{X}_{\\tau}, (R_{\\tau, u, i})_{(u, i) \\in \\mathcal{A}_{\\tau}}\\}_{\\tau = 1}^{t-1}$ of observations available to the provider when choosing the next allocation $\\mathbf{X}_{t}$. The allocator employs a policy $\\pi = \\{ \\pi_t | t \\in \\mathbb{N}\\}$, which is a sequence of functions, each mapping the history $H_t$ to an action $\\mathbf{X}_{t}$. Then, the $T$ period cumulative regret of a policy $\\pi$ is the random variable\n\\vspace{-7pt}\n\\begin{equation*} \\label{cumulative_regret}\n    \\mathcal{R}(T, \\pi) =  \\sum_{t = 1}^{T} \\left[ \\langle \\mathbf{X}^*_t, \\mathbf{\\Theta}^*\\rangle - \\langle \\mathbf{X}_{t}, \\mathbf{\\Theta}^* \\rangle \\right]\n\\end{equation*}\n\\vspace{-4pt}\nwhere $\\vect{X}^*_t \\in \\argmax_{\\vect{X} \\in \\mathcal{X}_t} \\; \\langle \\vect{X}, \\vect{\\Theta}^* \\rangle$ denotes optimum allocation at time $t$.\n\n\\subsection{Combinatorial Multi-Armed Bandit Approach}\n\nOur first case of consideration is without any assumption on the structure of the matrix $R$. In this case, our problem maps to the setting of combinatorial multi-armed bandits. We can consider each allocation pair $(u, i)$ to be an arm with expected reward $R_{ui}$. In total, there are $N M$ arms and at each iteration we select of a subset of arms (a super arm in the context of CMAB). We are constrained to select a subset from $\\mathcal{S} \\in 2^{[NM]}$ where $\\mathcal{S}$ corresponds to the allocations that comply with problem constraints. \n\nLet $r_R(X)$ denote the expected reward of an allocation $X$ when the reward matrix is given by $R$. Then, \n\\begin{equation}\n    r_R(X) = \\sum_{u = 1}^{N} \\sum_{i = 1}^{M} R_{ui} x_{ui}\n\\end{equation}\n\nThen this reward structure satisfies the following properties:\n\n\\begin{enumerate}\n    \\item Monotonicity: If $R_{ui} \\leq R_{ui}'$ for all $u \\in [N], i \\in [M]$, then we have $r_R(X) \\leq r_{R'}(X)$\n    \\item Bounded smoothness: If $\\max_{(u,i) \\in [N]\\times [M]} |R_{ui} - R_{ui}'| \\leq \\Lambda$, then we have $|r_R(X) - r_{R'}(X)| \\leq N \\Lambda$.\n\\end{enumerate}\n\n\\subsection{Low-Rank Multi-Armed Bandit \\& Interactive Collaborative Filtering Approach}\n\nUsing the latent factor model (Hofmann and Puzicha 1999), the reward is assumed to be given by a product of user and item feature vectors $p_u$ and $q_i$:\n\\begin{equation}\n    r_{ui} = p_u^\\textrm{T}q_i + \\eta\n\\end{equation}\nwhere $\\eta$ is a zero-mean Gaussian observation noise with variance $\\sigma^2$.\n\n\\subsubsection{Distribution of User and Item Feature Vectors}\n\nWe adopt the PMF model (Salakhutdinov  and Mnih 2008) to build the distributions for the user and the item feature vectors, which are then used the generate the reward values. We define the prior distributions of the user and item feature vectors as Gaussian with variance $\\sigma_p^2$\nand $\\sigma_q^2$:\n\\begin{align}\n    p(p_u | \\sigma_p^2) &= \\mathcal{N}(p_u | 0, \\sigma_p^2 I)\\\\\n    p(q_i | \\sigma_q^2) &= \\mathcal{N}(q_i | 0, \\sigma_q^2 I)\n\\end{align}\n\nThen, the conditional distribution of the reward $r_{ui}$ given the user and item feature vectors follow a Gaussian distribution:\n\\begin{equation}\n    p(r_{ui} | p_u, q_i, \\sigma^2) =  \\mathcal{N}(p_u^\\textrm{T} q_i | 0, \\sigma^2)\n\\end{equation}\n\nBy obtaining a set of noisy rating observations $\\{u(k), i(k), r_{u(k) i(k)}\\}_{k = 1}^{K}$, we can obtain the posterior distributions for the user and item feature vectors. Here, we first focus on the conditional distribution user feature vectors, given the current item feature vectors:\n\\begin{align}\n    p(P | R, Q, \\sigma^2, \\sigma_p^2, \\sigma_q^2) &= p(P | R, Q, \\sigma^2, \\sigma_p^2)\\\\\n    &\\propto p(R | P, Q, \\sigma^2) \\cdot p(P | \\sigma_p^2)\n\\end{align}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\n  Automated systems that make fast decisions based on visual input, such as autonomous driving, drone control, or smart factories, rely on a very short response time to prevent damage or injury.\n  Low-latency network transmission enabled by recent development in networking technologies, such as 5G, allows edge devices with low computing power to offload expensive \\ac{DNN} inference of the vision task to a nearby server.\n \n \n  Figure~\\ref{fig:system} depicts a model scenario of an obstacle suddenly emerging in a trajectory of a self-driving car.\n  Considering the car's speed of 100 km\/h, if the end-to-end latency of the brake control system increased by 40 ms, for example, due to slow compression, the car would travel an additional 1.1 meters, potentially hitting the obstacle instead of stopping in front of it.\n\n \n \n\n  \\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/pdf\/system.pdf}\n   \n   \n    \\caption{The reaction time of an autonomous vehicle control system determines whether it can avoid hitting an obstacle.}\n    \\label{fig:system}\n   \n  \\end{figure}\n\n \n \n \n \n \n \n\n  Fast compression of source images is necessary to ensure low latencies over a transfer channel, and one way to decrease the latency is to reduce the codec complexity.\n  Removing coding features, however, typically results in decreased vision performance at the same bitrate.\n  Since \\acp{DNN} have the ability to learn from the input data, it is possible to retrain them on the compressed dataset (after decompressing it) to overcome the coding efficiency lost by pruning the coding features.\n  At the same time, pruning the existing codecs allows to reuse existing hardware support and retraining does not modify the neural network architecture since only the weights change.\n\n \n \n\n \n \n \n \n \n \n \n \n\n \n\n \n \n\n\n  Several types of low-complexity codecs exist.\n  Real-time texture compression can reach very high encoding speeds compared to other methods at the expense of rather low coding efficiency~\\cite{waveren2007, holub2013, zadnik2022}.\n  A ``mezzanine compression'' family of codecs is designed specifically to meet ultra-low latency requirements, with \\acs{JPEG}~XS~\\cite{descampe2021jpeg} as the newest standard in this family.\n  The recently standardized \\ac{HTJ2K}~\\cite{htj2k} simplifies the otherwise complex \\acs{JPEG}~2000~\\cite{skodras2001jpeg} with the goal of 10x throughput improvement.\n  However, it does not offer such a precise rate allocation as \\acs{JPEG}~XS.\n  Traditional hybrid video codecs, such as \\ac{HEVC}~\\cite{sullivan2012} or \\ac{VVC}~\\cite{bross2020} offer advanced coding features and great coding efficiency.\n  However, the additional complexity of, for example, the inter or intra prediction and advanced entropy coding can be prohibitive in resource-constrained devices.\n \n  \\Ac{JPEG}~\\cite{wallace1992} shares the core transform coding features with hybrid video codecs, but without the additional complexity.\n  At the same time, it can deliver sufficient quality for computer vision applications, as shown in the paper.\n\n  In this paper, we explore the idea of pruning the encoding configurations to reduce the encoding time and latency and compensating the lost vision performance by retraining the vision model with the compressed dataset.\n  As two case studies we chose reducing the configuration space of the otherwise very complex \\ac{ASTC}~\\cite{nystad2012} format and \\ac{JPEG}~XS that was designed specifically as a lightweight, low-latency codec.\n \n \n \n  Both codecs operate in a constant bitrate mode which is important for ensuring predictable latency.\n \n\n \n \n \n \n\n  We evaluate the effect of \\ac{ASTC} compression artifacts on the image classification accuracy of ShuffleNet~\\cite{ma2018shufflenet} V2 and both \\ac{ASTC} and \\ac{JPEG}~XS on the semantic segmentation of LR-ASPP-MobileNetV3~\\cite{howard2019}.\n  We compare both to \\ac{JPEG}, and in the case of the segmentation task also to \\ac{HTJ2K} and \\ac{JPEG}~2000.\n\n  The contributions of this paper are:\n  \\begin{itemize}\n      \\item We propose a lightweight \\ac{ASTC} encoder\\footnote{\\url{github.com\/cpc\/simple-texcomp}} that is approximately $2.3\\times$ faster than \\ac{JPEG} on a Samsung S10 smartphone.\n      \\item We study how pruning \\ac{JPEG}~XS encoding configurations impacts latency and computer vision performance.\n     \n     \n      \\item We demonstrate that the quality vs. latency tradeoff can be alleviated by retraining the classification and segmentation models with the compressed datasets.\n  \\end{itemize}\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{Background and Related Work}\n\n\\subsection{Adaptive Scalable Texture Compression}\n\n\\ac{ASTC} is the newest and most flexible texture compression format adopted as an OpenGL extension by the Khronos Group.\nLike other texture compression formats, it quantizes the input block's color space and represents its pixels as indices pointing at one of the quantized colors.\nCompared to the older BCn formats, \\ac{ASTC} supports many configuration options: scaling the input block size, partitioning, different color endpoint modes (CEM), endpoint and weight quantizations, dual-plane encoding, and \\ac{BISE}.\nAn important property of texture compression is a fixed compression ratio and random access: The individual pixels are randomly addressable from the compressed representation without decompressing.\n\nModern \\acp{GPU} have texture fetch units that can perform the decompression online during rendering with a negligible overhead which further enhances the low-latency potential of texture compression.\n\n\n\n\n\n\\subsection{JPEG XS}\n\n\\ac{JPEG}~XS is a wavelet-based mezzanine codec designed primarily for low complexity, low latency, high bandwidth, and high-quality video delivery.\nThe minimal coding unit of \\ac{JPEG}~XS is one precinct whose size can range from less than one pixel line up to several lines of the image.\n\nThe \\ac{JPEG}~XS rate allocation can predict the bitrate precisely, unlike \\ac{HTJ2K} where a precise rate allocation would require a significant additional complexity~\\cite{descampe2021jpeg}.\nThe latest version of the standard also supports direct Bayer data compression~\\cite{richter2021bayer} which can be used to bypass the traditional image processing pipeline at the sensor side and thus save latency.\n\nTo the best of our knowledge, no publicly available JPEG XS encoder currently exists.\nTherefore, we utilized the reference \\ac{JPEG}~XS reference software, version 1.4.0.4 (ISO\/IEC 21122-5:2020).\nIn the literature, \\cite{itakura2020} developed a JPEG XS codec capable of running at 60 \\ac{FPS} at 8K resolution on a 64-core AMD EPYC processor.\n\n\n\\subsection{Compression for Computer Vision}\n\\label{subs:cv}\n\n\nSome previous works optimize the perceptual model of \\ac{JPEG} for computer vision~\\cite{xie2019, liu2018}, leading to significant quality improvements.\n\\cite{brummer2020} optimized the global JPEG XS encoding parameters (gains and priorities) to better capture the characteristics of a computer vision target.\nOur approach of retraining the vision model with the compressed dataset is complementary to codec parameter optimizations.\n\n\\cite{zadnik2021} used retraining to recover object detection and semantic segmentation performance of BC1 and YCoCg-BC3~\\cite{waveren2007} texture compression.\nTo the best of our knowledge, no prior work implements a minimal-subset \\ac{ASTC} in the context of computer vision.\n\n\\cite{marie2022expert} proposed a modified loss function of a \\ac{DNN} to achieve more efficient restoration of classification accuracy lost to compression artifacts.\nThey achieved a minor but consistent gain of up to 0.79 \\ac{pp} validation accuracy compared to a simple retraining method used in this work.\n\nThe recent exploration of \\ac{VCM} by \\ac{MPEG} is an effort to develop a coding scheme with both machine and human perception in mind~\\cite{duan2020}.\nThe current development is being built on top of \\ac{VVC} which is a more complex format than what we target in this paper.\nFurthermore, our use case considers only the computer vision performance without the human in the loop.\nJPEG AI \\cite{jpegai2021} also explores compression for both human and computer vision targets, but focuses on utilizing learning-based coding methods.\n\nYet another approach to adapting compression for computer vision is ``feature compression'' which encodes intermediate neural features~\\cite{shao2020bottlenet}.\nFeature compression, however, requires computing some of the convolutional layers on the encoding device which contrasts with our approach of decreasing the encoding complexity.\n\n\\section{Implementation of Pruned Codecs}\n\n\\subsection{ASTC}\n\nDue to the \\ac{ASTC} complexity, exhaustively searching for encoding parameters is not feasible in real time, and such, heuristics must be used to prune the configuration space.\nIn our work, we reduce the configuration space to only one configuration: 5-bit color endpoint and 2-bit and weight quantization with a weight grid of $8\\times5$.\nThe selected configuration showed the lowest per-pixel distortion measured as \\ac{PSNR} on a sample dataset from adjacent configurations without requiring \\ac{BISE}.\n\nSince the only way to scale the \\ac{ASTC} bitrate is to modify the input block size, we implemented both $12\\times12$ and $8\\times8$ input block sizes, implying a \\ac{CR} of 27:1 and 12:1 ($0.\\overline{8}$ and 2.0 \\ac{bpp}), respectively.\n\nThe encoding of a block starts by selecting the endpoints with a small inset similar to~\\cite{waveren2007}.\nThen, ``ideal weights'' are selected by orthogonally projecting the input pixels onto the line defined by the endpoints.\nLastly, the ``ideal weights'' are bilinearly downsampled to the $8\\times5$ grid and quantized into two bits.\n\n\n\\subsection{JPEG XS}\n\n\nThe long encoding time of the reference encoder is caused by the rate allocation algorithm exhaustively computing the bit budget for each precinct at all quantization levels and using all possible coding methods.\nWe reduced the number of searched quantizations and coding methods to 13 and 5, respectively, without losing quality as the other combinations were unused in our tests.\n\nTo reduce the number of rate allocation passes further, we disabled the significance flag coding.\nSignificance flag coding detects a run of all-zero ``significance groups'' (groups of 8 adjacent coefficients) that can be encoded with a single flag and requires an additional rate allocation pass.\nDisabling this method brings the number of utilized coding methods from 5 to 3 and removes the need for ``refresh'' passes,  significantly reducing the encoding time.\nHowever, it also reduces the coding efficiency, which we try to recover by retraining with the compressed dataset.\nWe kept the coefficient prediction from a previous line, as disabling it would prevent the encoder meet the target bitrate.\n\n\n\\section{Experimental Setup}\n\n\\subsection{Implementation Details}\n\\label{subs:impl}\n\n\nThe \\ac{ASTC} encoder uses ARM NEON intrinsics to vectorize the most significant loops using 8-bit fixed-point representation.\nFor a fair runtime comparison with \\ac{JPEG}, we chose the \\ac{SIMD}-optimized \\texttt{libjpeg-turbo} library\\footnote{\\url{libjpeg-turbo.org} (version 2.1.1)} and developed a wrapper encoder application around the library.\nThe \\ac{JPEG} coding parameters were chosen to match the defaults of the \\texttt{cjpeg} command-line utility: YCbCr color space with 4:2:0 subsampling and no restart intervals.\nThe quality parameter (Q) 45 was selected so that the bitrate of a random sample of 10000 ImageNet images after encoding is the highest possible at or below the rate of \\ac{ASTC} $12\\times12$.\nBoth \\ac{ASTC} and \\ac{JPEG} were evaluated on a single core (A76) of a Samsung S10 smartphone.\n\n\n\nThe \\ac{JPEG}~XS encoder was compiled only for the x86 instruction set and evaluated on a single thread of Intel i7-8650U laptop CPU at a base frequency of 2.1 GHz with disabled frequency scaling.\nFor runtime comparison we chose two open source encoders: \\texttt{grok}\\footnote{\\url{github.com\/GrokImageCompression\/grok} (version 9.7.7)} for \\ac{JPEG}~2000 and \\texttt{OpenJPH}\\footnote{\\url{github.com\/aous72\/OpenJPH} (version 0.9.0)} for \\ac{HTJ2K}, both using irreversible \\ac{DWT}.\n\n\n\n\n\n\n\\subsection{Vision Tasks}\n\nWe evaluated the image classification accuracy of ShuffleNet V2\\footnote{\\url{pytorch.org\/hub\/pytorch_vision_shufflenet_v2}} in $0.5\\times$ and $1.0\\times$ sizes trained on the ImageNet dataset~\\cite{deng2009imagenet} with training hyperparameters derived from \\cite{zhang2018shufflenet} and \\cite{ma2018shufflenet}.\nWe also evaluated a semantic segmentation task with LR-ASPP-MobileNetV3\\footnote{\\url{github.com\/ekzhang\/fastseg} (commit 91238cd)} in both large and small versions trained on the Cityscapes dataset~\\cite{cordts2016}.\nThe Cityscapes images for training were cropped to $\\sqrt{2}$ of the original size in each dimension to avoid running out of \\ac{GPU} memory.\nBoth networks were retrained with the dataset compressed with \\ac{ASTC} to recover the vision performance lost by compression artifacts.\nUnfortunately, the \\ac{JPEG}~XS encoder was not able to encode some of the ImageNet images at the bitrate of $0.\\overline{8}$ \\ac{bpp}.\nTherefore, we evaluated only the segmentation task with this codec.\n\nAll encoders mentioned in the previous subsection were used for quality evaluations, along with \\texttt{astcenc}\\footnote{\\url{github.com\/ARM-software\/astc-encoder} (version 3.7)} at the fastest profile for quality evaluations on the Cityscapes dataset.\n\n\\ac{JPEG}-compressed images were used only for retraining the ShuffleNet V2 network.\n\\ac{JPEG}, \\ac{JPEG}~2000, and \\ac{HTJ2K} reach segmentation \\ac{mIoU} within 1.5\\% below the uncompressed result, and retraining is expected to bring the results on par with the uncompressed results, therefore, we did not retrain with these codecs.\n\n\\section{Results}\n\n\\subsection{Quality}\n\n\\paragraph*{Image Classification}\n\n\n\nTable~\\ref{tab:quality_shufflenet} summarizes the highest achieved classification accuracies of ShuffleNet V2 under different conditions.\nWhen the compressed data is used as an input to the network trained on uncompressed data (the ``orig.'' column), the \\ac{ASTC} compression degrades the accuracy by more than 15 \\ac{pp}, while the difference caused by \\ac{JPEG} compression is only 1.3 and 0.6 \\ac{pp}.\nHowever, when retrained with the compressed dataset (the ``retr.'' column), the accuracy decrease for \\ac{ASTC} with $12\\times12$ block size is only 4.9 and 5.0 \\ac{pp} for the $0.5\\times$ and $1.0\\times$ network sizes, respectively.\nThe \\ac{ASTC} $8\\times8$ achieves higher quality than $12\\times12$: only 2.3--1.8 \\ac{pp} accuracy decrease compared to the uncompressed result.\nRetraining with the \\ac{JPEG}-compressed dataset brings a 0.2 \\ac{pp} increase in the classification accuracy of the smaller network.\nThe results show that retraining the larger network does not improve the already high accuracy for \\ac{JPEG}.\n\n\\begin{table}[htbp]\n  \\caption{Validation top-1 accuracy of ShuffleNet V2 on ImageNet validation set with JPEG and the proposed ASTC compression with and without retraining.}\n  \\label{tab:quality_shufflenet}\n  \\centering\n  \\begin{threeparttable}\n      \\begin{tabular}{|l|r|rr|rr|}\n        \\hline\n            & & \\multicolumn{2}{c|}{ 0.5x } & \\multicolumn{2}{c|}{ 1.0x } \\\\\n        compression & bpp & orig. & retr. & orig. & retr. \\\\\n        \\hhline{|=|=|==|==|}\n        uncompressed &     24.0 & \\multicolumn{2}{c|}{54.4\\%} & \\multicolumn{2}{c|}{64.3\\%} \\\\\n        ASTC 12x12   &     0.89 & -16.8 & -4.9 & -15.1 & -5.0 \\\\\n        JPEG Q45     & \\raise.17ex\\hbox{$\\scriptstyle\\mathtt{\\sim}$} 0.89 &  -1.3 & -1.1 &  -0.6 & -0.7 \\\\\n        ASTC 8x8     &     2.00 &  -6.4 & -2.3 &  -6.6 & -1.8 \\\\\n        \\hline\n      \\end{tabular}\n  \\end{threeparttable}\n\\end{table}\n\n\\paragraph*{Semantic Segmentation}\n\nFigure~\\ref{fig:fastseg_eval} compares rate-distortion curves of multiple encoders according to three metrics: \\ac{PSNR}, \\ac{SSIM}, and validation \\ac{mIoU} of LR-ASPP-MobileNetV3 (large vesion) trained on an uncompressed dataset.\nThe small version of the model shows similar relations between the \\ac{mIoU} curves to the large version, therefore, we omitted it for brevity.\nThe plots show that despite significant \\ac{PSNR} and \\ac{SSIM} differences between \\ac{JPEG}~2000, \\ac{JPEG}, and \\ac{HTJ2K}, the \\ac{mIoU} difference is relatively small.\nDisabling significance flag coding of \\ac{JPEG}~XS (denoted as ``no-sf'') shows a consistent decrease of all metrics in both main and subline profiles.\nSimilarly, when compared to a full-featured \\texttt{astcenc} encoder at the fastest preset, our pruned implementation achieves significantly lower quality.\nBoth \\ac{SSIM} and \\ac{mIoU} metrics decrease rapidly with \\ac{JPEG}~XS at lower bitrates ($0.\\overline{8}$ and 1.0 \\ac{bpp}), especially the subline profile.\n\n\n\\begin{figure}[t]\n  \\centering\n  \\begin{tikzpicture}\n    \\tikzstyle{dataline} = [color=black, dashed, mark options=solid, mark=+]\n    \\begin{axis}\n      [\n        yshift           = -11cm,\n        width=0.95\\linewidth,\n        height=6cm,\n       \n       \n        xlabel           = {bpp},\n        xlabel near ticks,\n        ylabel           = {PSNR (dB)},\n       \n        ylabel near ticks,\n        grid             = both,\n        major grid style = {dotted,black!90},\n        minor grid style = {dotted,gray!60},\n        legend columns    = 3,\n        legend style      = { at = {(0.433, 1.05)}, anchor = south, font = \\scriptsize },\n        legend cell align = { left },\n        xmin = 0,\n        xmax = 4,\n        ymin = 30,\n        ymax = 55,\n        ytick distance = 10,\n        minor x tick num = 9,\n        minor y tick num = 9,\n      ]\n\n      \\addplot [dataline, black, mark=x]\n        table[x=bpp_mean, y=psnr_mean, col sep=comma] {data\/generated\/grok_jp2_irev.csv};\n     \n     \n      \\addplot [dataline, blue]\n        table[x=bpp_mean, y=psnr_mean, col sep=comma] {data\/generated\/openjph.csv};\n      \\addplot [dataline, green!75!black]\n        table[x=bpp_mean, y=psnr_mean, col sep=comma] {data\/generated\/turbojpeg.csv};\n      \\addplot [dataline, solid, black!40!purple!70!green]\n        table[x=bpp_mean, y=psnr_mean, col sep=comma] {data\/generated\/astc_u8.csv};\n      \\addplot [dataline, black!40!purple!70!green]\n        table[x=bpp_mean, y=psnr_mean, col sep=comma] {data\/generated\/astcenc_fastest.csv};\n      \\addplot [dataline, red]\n        table[x=bpp_mean, y=psnr_mean, col sep=comma] {data\/generated\/jxs_p3.csv};\n      \\addplot [dataline, orange!50!brown]\n        table[x=bpp_mean, y=psnr_mean, col sep=comma] {data\/generated\/jxs_p5.csv};\n      \\addplot [dataline, solid, red]\n        table[x=bpp_mean, y=psnr_mean, col sep=comma] {data\/generated\/jxs_nosf_p3.csv};\n      \\addplot [dataline, solid, orange!50!brown]\n        table[x=bpp_mean, y=psnr_mean, col sep=comma] {data\/generated\/jxs_nosf_p5.csv};\n\n      \\legend{ J2K (grok), HTJ2K (OpenJPH), JPEG (libjpeg-turbo), ASTC (ours),\n        ASTC (astcenc), JXS main, JXS subline, JXS main no-sf, JXS subline no-sf }\n\n    \\end{axis}\n    \\begin{axis}\n      [\n        yshift           = -16.6cm,\n        width=0.95\\linewidth,\n        height=6cm,\n        xlabel           = {bpp},\n        xlabel near ticks,\n        ylabel           = {SSIM (-)},\n       \n        ylabel near ticks,\n        grid             = both,\n        major grid style = {dotted,black!90},\n        minor grid style = {dotted,gray!60},\n        xmin = 0,\n        xmax = 4,\n        ymin = 0.89,\n        minor x tick num = 9,\n        minor y tick num = 4,\n      ]\n\n      \\addplot [dataline, black, mark=x]\n        table[x=bpp_mean, y=ssim_mean, col sep=comma] {data\/generated\/grok_jp2_irev.csv};\n     \n     \n      \\addplot [dataline, blue]\n        table[x=bpp_mean, y=ssim_mean, col sep=comma] {data\/generated\/openjph.csv};\n      \\addplot [dataline, green!75!black]\n        table[x=bpp_mean, y=ssim_mean, col sep=comma] {data\/generated\/turbojpeg.csv};\n      \\addplot [dataline, solid, black!40!purple!70!green]\n        table[x=bpp_mean, y=ssim_mean, col sep=comma] {data\/generated\/astc_u8.csv};\n      \\addplot [dataline, black!40!purple!70!green]\n        table[x=bpp_mean, y=ssim_mean, col sep=comma] {data\/generated\/astcenc_fastest.csv};\n      \\addplot [dataline, red]\n        table[x=bpp_mean, y=ssim_mean, col sep=comma] {data\/generated\/jxs_p3.csv};\n      \\addplot [dataline, orange!50!brown]\n        table[x=bpp_mean, y=ssim_mean, col sep=comma] {data\/generated\/jxs_p5.csv};\n      \\addplot [dataline, solid, red]\n        table[x=bpp_mean, y=ssim_mean, col sep=comma] {data\/generated\/jxs_nosf_p3.csv};\n      \\addplot [dataline, solid, orange!50!brown]\n        table[x=bpp_mean, y=ssim_mean, col sep=comma] {data\/generated\/jxs_nosf_p5.csv};\n\n    \\end{axis}\n    \\begin{axis}\n      [\n        yshift           = -22.2cm,\n        width=0.95\\linewidth,\n        height=6cm,\n       \n       \n        xlabel           = {bpp},\n        xlabel near ticks,\n        ylabel           = {mIoU (-)},\n       \n        ylabel near ticks,\n        grid              = both,\n        major grid style  = {dotted,black!90},\n        minor grid style  = {dotted,gray!60},\n        xmin              = 0,\n        xmax              = 4,\n        minor x tick num  = 9,\n        minor y tick num  = 9,\n      ]\n\n      \\addplot [dataline, black, mark=x]\n        table[x=bpp_mean, y=fastseg_large_miou, col sep=comma] {data\/generated\/grok_jp2_irev.csv};\n     \n     \n      \\addplot [dataline, blue]\n        table[x=bpp_mean, y=fastseg_large_miou, col sep=comma] {data\/generated\/openjph.csv};\n      \\addplot [dataline, green!75!black]\n        table[x=bpp_mean, y=fastseg_large_miou, col sep=comma] {data\/generated\/turbojpeg.csv};\n      \\addplot [dataline, solid, black!40!purple!70!green]\n        table[x=bpp_mean, y=fastseg_large_miou, col sep=comma] {data\/generated\/astc_u8.csv};\n      \\addplot [dataline, black!40!purple!70!green]\n        table[x=bpp_mean, y=fastseg_large_miou, col sep=comma] {data\/generated\/astcenc_fastest.csv};\n      \\addplot [dataline, red]\n        table[x=bpp_mean, y=fastseg_large_miou, col sep=comma] {data\/generated\/jxs_p3.csv};\n      \\addplot [dataline, orange!50!brown]\n        table[x=bpp_mean, y=fastseg_large_miou, col sep=comma] {data\/generated\/jxs_p5.csv};\n      \\addplot [dataline, solid, red]\n        table[x=bpp_mean, y=fastseg_large_miou, col sep=comma] {data\/generated\/jxs_nosf_p3.csv};\n      \\addplot [dataline, solid, orange!50!brown]\n        table[x=bpp_mean, y=fastseg_large_miou, col sep=comma] {data\/generated\/jxs_nosf_p5.csv};\n      \\addplot[mark=none, black, domain=0:4] {0.66539};\n\n      \\draw (0, 260) node [right] {\\scriptsize uncompressed};\n\n    \\end{axis}\n  \\end{tikzpicture}\n \n  \\caption{Mean intersection over union (mIoU) of a FastSeg large network (bottom), SSIM (middle) and PSNR (top) of a Cityscapes validation set compressed with different methods.}\n  \\label{fig:fastseg_eval}\n \n\\end{figure}\n\nTo recover the large \\ac{mIoU} degradation of \\ac{ASTC} and \\ac{JPEG}~XS at low bitrates, we retrained the network with the compressed datasets.\nTable~\\ref{tab:quality_mobilenet} summarizes the \\ac{mIoU} improvements of retraining LR-ASPP-MobileNetV3 in both small and large variants.\nFor \\ac{ASTC} $12\\times12$, retraining brought an improvement of 1.1 and 8.6 \\ac{pp} for the small and large networks, respectively.\nRetraining \\ac{JPEG}~XS in the main profile resulted in \\ac{mIoU} around 2.3--2.6 \\ac{pp} lower than the uncompressed result.\nThe subline profile of \\ac{JPEG}~XS shows a sharp decline in the vision performance without retraining.\nRetraining allows recovering most of the quality back.\nHowever, the results still do not reach the quality of \\ac{ASTC} $12\\times12$.\n\n\n\\begin{table}[htbp]\n  \\caption{Mean intersection over union (mIoU) of LR-ASPP-MobileNetV3 with Cityscapes validation set and the JPEG XS compression.}\n  \\label{tab:quality_mobilenet}\n  \\centering\n  \\begin{threeparttable}\n      \\begin{tabular}{|l|r|rr|rr|}\n        \\hline\n            & & \\multicolumn{2}{c|}{ small } & \\multicolumn{2}{c|}{ large } \\\\\n        compression              & bpp   & orig. & retr. & orig. & retr. \\\\\n        \\hhline{|=|=|==|==|}\n        uncompressed             & 24.0  & \\multicolumn{2}{c|}{61.2\\%} & \\multicolumn{2}{c|}{66.5\\%} \\\\\n        JPEG XS (main, sf)       & 0.89 & -4.9  & -2.3  & -6.5  & -2.6  \\\\\n        JPEG XS (main, no-sf)    & 0.89 & -5.8  & -2.7  & -7.4  & -2.3  \\\\\n        JPEG XS (subline, sf)    & 0.89 & -14.4 & -5.4  & -20.5 & -5.3  \\\\\n        JPEG XS (subline, no-sf) & 0.89 & -17.3 & -6.7  & -25.2 & -6.8  \\\\\n        ASTC 12x12               & 0.89 & -5.5  & -4.4  & -12.6 & -4.0  \\\\\n        ASTC 8x8                 & 2.00 & -3.5  & -2.8  & -6.4  & -1.7  \\\\\n        \\hline\n      \\end{tabular}\n  \\end{threeparttable}\n\\end{table}\n\nFigure~\\ref{fig:images} shows segmentations of two challenging scenes after the retrained model inference using compressed images to illustrate the effect of compression on the segmentation result.\nThe first image shows shape deformations caused by the \\ac{ASTC} and \\ac{JPEG}~XS main profile (``p3'') while the network trained with \\ac{JPEG}~XS subline profile (``p5'') fails to detect the people at all.\nIn the second image, all cases detect the person in the foreground but fail to detect some, or all, the people in the distance.\nIt should be emphasized that the MobileNet networks were trained with images containing approximately half of the pixels of the full-resolution images due to \\ac{GPU} memory limitations.\nTherefore, the examples do not correspond to the best predictions achievable with these networks.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/png\/retrain_comp.png}\n    \\caption{Visual comparison of LR-ASPP-MobileNetV3 (small version) segmentation of two Cityscapes images. From left to right: Original image (brightened), ground truth, segmentation of the model trained by datasets compressed by the ASTC ($12\\times12$), pruned JPEG XS (main profile) and pruned JPEG XS (subline profile) encoders.}\n    \\label{fig:images}\n\\end{figure}\n\n\\subsection{Runtime}\n\n\\paragraph*{ASTC Encoder}\n\nTable~\\ref{tab:runtime_arm} compares the average encoding time of a Cityscapes image (resolution $2048\\times1024$) of our \\ac{ASTC} encoder with the block size $12\\times12$ and $8\\times8$ to \\ac{JPEG} with the quality parameters Q 85 and 96.\nThe Q parameters were determined by the same procedure as in Subsection~\\ref{subs:impl} to ensure approximately the same bitrate as \\ac{ASTC} on a random subset of 100 images.\nFor the $12\\times12$ block size, the images were padded to a resolution divisible by a block size of 12 before \\ac{ASTC} encoding.\n\nThe results show our simple \\ac{ASTC} $12\\times12$ encoder is approximately $2.3\\times$ faster than the \\ac{JPEG} encoder based on \\texttt{libjpeg-turbo}.\nThe \\ac{JPEG} decoding was slightly slower than the encoding.\nWhile we did not conduct \\ac{ASTC} decoding measurements, in~\\cite{zadnik2021}, we measured BC1 and YCoCg-BC3 decoding time of an 8K frame as less than 1 ms on a desktop \\ac{GPU}.\n\\ac{ASTC} decoding is more complicated, but the decoding overhead is still expected to be close to negligible in comparison to the encoding.\n\n\\begin{table}[htbp]\n  \\caption{Encoding time of the proposed \\ac{ASTC} encoder and \\texttt{libjpeg-turbo} encoder and decoder (ARM A76 single-core).}\n  \\label{tab:runtime_arm}\n  \\centering\n  \\begin{threeparttable}\n      \\begin{tabular}{|l|rr|}\n        \\hline\n                              & \\multicolumn{2}{|c|}{bpp}  \\\\\n                              & \\tl0.89 & \\tl2.0 \\\\\n        \\hhline{|=|==|}\n        ASTC                  & 5.8     & 7.0    \\\\\n        JPEG (enc, libjpeg-turbo) & 13.3    & 16.7   \\\\\n        JPEG (dec, libjpeg-turbo) & 13.6    & 21.5   \\\\\n        \\hline\n      \\end{tabular}\n  \\end{threeparttable}\n\\end{table}\n\n\nFor comparison, we also measured the encoding and decoding time of \\ac{JPEG} at quality parameters 0 and 100 as 11 and 22 ms, and 7 and 32 ms, respectively.\nThese numbers establish the encoding speed bounds of this format.\n\nOn a single core of Intel i7-8650U laptop \\ac{CPU}, the AVX2-optimized \\texttt{astcenc} encoder compressed one Cityscapes image at approximately 44 and 61 ms (block sizes $12\\times12$ and $8\\times8$, respectively) at the fastest preset, suggesting that both \\ac{JPEG} and our pruned \\ac{ASTC} encoders are faster than a traditional \\ac{ASTC} encoder even with the latter evaluated on a more powerful \\ac{CPU}.\n\n\\paragraph*{JPEG XS Encoder}\n\n\nTable~\\ref{tab:runtime_x86} compares \\ac{JPEG}~XS with three encoders: \\ac{JPEG}, \\ac{JPEG}~2000, and \\ac{HTJ2K}.\nThe \\ac{HTJ2K} quantization was determined by a similar procedure as in Subsection~\\ref{subs:impl}.\nThe results show that by disabling the significance flag coding, the encoding time of one Cityscapes frame improved by 22--23\\%, and is only about 9--20\\% slower than \\ac{JPEG}~2000.\n\\ac{HTJ2K} encoding by OpenJPH was 3.8--$6.2\\times$ faster than \\ac{JPEG}~XS without the significance coding flags.\n\\ac{JPEG} by \\texttt{libjpeg-turbo} brought this difference further by almost another order of magnitude.\nKakadu \\ac{HTJ2K} is not publicly available, therefore, we used results published in~\\cite{taubman2019high} and extrapolated them to our setup.\nMore specifically, we scaled the result by the number of pixels from 4K to $2048\\times1024$, multiplied by 4 since the original result was obtained on a 4-core machine, and finally scaled to our frequency of 2.1 GHz from the original 3.4 GHz.\nBased on this rough estimation, it seems likely the \\ac{HTJ2K} encoder is capable of reaching encoding throughput close to \\ac{JPEG}.\n\n\\begin{table}[htbp]\n  \\caption{Encoding times of JPEG XS, JPEG, JPEG 2000, and HTJ2K encoders on a single core of i7-8650U CPU}\n  \\label{tab:runtime_x86}\n  \\centering\n  \\begin{threeparttable}\n      \\begin{tabular}{|l|rrr|}\n        \\hline\n                                              & \\multicolumn{3}{|c|}{bpp}   \\\\\n                                              & \\tl0.89 & \\tl2.0        & \\tl3.3 \\\\\n        \\hhline{|=|===|}\n        JPEG XS (main, sf)                    & 625     & 654           & 683    \\\\\n        JPEG XS (main, no-sf)                 & 477     & 503           & 531    \\\\\n        JPEG (libjpeg-turbo)                  & 11.3    & 13.5          & 16.0   \\\\\n        JPEG 2000 (grok)                      & 437     & 439           & 441    \\\\\n        HTJ2K (OpenJPH)                       & 73.7    & 105           & 138    \\\\\n        HTJ2K (Kakadu~\\cite{taubman2019high}) & -       & 14.4\\tnote{*} & -      \\\\\n        \\hline\n      \\end{tabular}\n      \\begin{tablenotes}[para]\n        \\item[*] extrapolated from the result in the publication\n      \\end{tablenotes}\n  \\end{threeparttable}\n\\end{table}\n\nTable~\\ref{tab:latency} shows \\ac{JPEG}~XS encoding time and latency at two different bitrates and three profiles: high, main, and subline.\nThe ``precinct'' column denotes the number of lines that form one presinct.\nThe ``enc'' column shows the total frame encoding time and the ``latency'' column shows the time until the first precinct is done encoding and thus represents the minimum theoretical achievable latency.\nWhile the overall frame encoding time does not differ dramatically between presets, the precinct size has a major impact on latency: The precinct of the high profile consisting of three lines shows a latency of 32--33\\% of the total encoding time, while the latency of a half-line precinct of the subline profile is three times smaller portion of the encoding time.\nThus, even without reaching a high throughput, it is possible to achieve latency almost an order of magnitude lower than the encoding time.\nIt should also be noted that the latency includes the wavelet transform over the whole frame and can be further reduced by pipelining it with the rest of the computation.\n\n\\begin{table}[htbp]\n  \\caption{Latency and throughput comparison of the pruned JPEG XS reference encoder without significance flag coding.}\n  \\label{tab:latency}\n  \\centering\n  \\begin{threeparttable}\n      \\begin{tabular}{|lll|rrr|}\n        \\hline\n                  & precinct & bpp  & enc  & \\multicolumn{2}{c|}{latency}      \\\\\n        profile   & (lines)  &      & (ms) & (ms) & (\\%) \\\\\n        \\hhline{|===|===|}\n          high    & 3        & 0.89 & 502  & 167  & 33\\%  \\\\\n          high    & 3        & 2.0  & 528  & 166  & 31\\%  \\\\\n          main    & 2        & 0.89 & 477  & 136  & 29\\%  \\\\\n          main    & 2        & 2.0  & 504  & 137  & 27\\%  \\\\\n          subline & 0.5      & 0.89 & 443  & 53   & 12\\%  \\\\\n          subline & 0.5      & 2.0  & 469  & 53   & 11\\%  \\\\\n        \\hline\n      \\end{tabular}\n  \\end{threeparttable}\n\\end{table}\n\n\\section{Discussion}\n\\label{sec:discussion}\n\n\n\n\nRetraining with the compressed dataset showed the largest improvements when the vision performance without retraining was very low, such as the classification with the \\ac{ASTC} $12\\times12$ and segmentation with the subline \\ac{JPEG}~XS encoders.\nOn the segmentation task, the overall quality decrease without retraining was smaller, because the images are less noisy and less prone to compression artifacts.\nThe vision performance could be further enhanced by improving the retraining process and optimizing encoding parameters for computer vision using one of the methods introduced in Subsection~\\ref{subs:cv}.\n\nThe lightweight \\ac{ASTC} encoder achieves a higher encoding speed than \\ac{JPEG}, making it the fastest encoder evaluated.\nOn the other hand, the pruned \\ac{JPEG}~XS reference encoder does not achieve sufficient runtime performance.\nHowever, its low complexity and results from literature~\\cite{itakura2020} suggest a fast implementation is possible.\nThe rate allocation can be further optimized by, for example, replacing the exhaustive search with a binary search, in combination with other heuristics.\nThe second most expensive operation in the reference encoder is the wavelet transform which we did not modify.\nIn the reference encoder, the wavelet transform is performed over the whole frame before the coding of individual precincts.\nHowever, it is possible to interleave the wavelet transform with the precinct coding as the latency of the wavelet transform ranges from a few pixels to 6 lines~\\cite{descampe2021jpeg}.\n\nTo put the results into a practical perspective, let us consider a scenario of compressing a Cityscapes image and sending it over a 500 Mbit\/s commercially available 5G network and an embedded transceiver capable of 2 Mbit\/s.\nAssuming a 1 ms latency budget for encoding and network transfer, the \\ac{ASTC} at the bitrate of $0.\\overline{8}$ would require a latency of 145 and 3.2 lines, respectively, assuming the encoding speed of 5.8 ms\/frame.\nWhile the first case allows partitioning the image into larger chunks, in the second case, as shown in Figure~\\ref{fig:fastseg_eval}, lowering the latency of \\ac{JPEG}~XS comes at a significant quality loss and thus necessitates the compensation by retraining.\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion}\n\nWe explored decreasing an image encoder complexity to achieve lower latency.\nNamely, we evaluated a lightweight implementation of an \\ac{ASTC} encoder and a pruned version of a \\ac{JPEG}~XS reference encoder.\n\nThe \\ac{ASTC} encoder outperforms \\ac{JPEG} in terms of encoding speed by approximately $2.3\\times$ at the same bitrate.\nWhen retrained with the dataset compressed with ASTC at the lowest bitrate of $0.\\overline{8}$, the classification accuracy was about 5 \\ac{pp}, and the segmentation \\ac{mIoU} 4.4--4.0 \\ac{pp} lower than the output of the networks trained and evaluated without any compression.\n\nThe pruned \\ac{JPEG}~XS reference encoder is not nearly as fast as \\ac{ASTC} and needs more optimizations to be usable for real-time tasks.\nNevertheless, we show that disabling significance flag coding decreases the number of required rate allocation passes, and boosts the encoding speed by 22--23\\% at the cost of only 0.4--0.3 \\ac{pp} of segmentation \\ac{mIoU} after retraining.\n\n\n\\ac{HTJ2K} and \\ac{JPEG} outperform both the tested codecs in terms of vision performance.\nHowever, \\ac{ASTC} still holds the advantage of the fastest coding speed, while \\ac{JPEG}~XS, if sufficiently optimized, is suitable for applications requiring ultra-low latencies.\nTo improve the quality, it is possible to apply computer vision--specific encoding parameter optimizations or improve the retraining process.\n\n\\section*{Acknowledgment}\n\nThe work was financially supported by the Tampere University ITC Graduate School. It was also supported by European Union's Horizon 2020 research and innovation program under Grant Agreement No 871738 (CPSoSaware) and in part by the Academy of Finland under Grant 325530.\n\n\n\n\n\n\\balance\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\label{sec:orgdeb2123}\n\nThose systems that we might classify as living, adaptive, or somehow intelligent all display a fundamental property: they resist or avoid perturbations that would result in their existence becoming unsustainable. This means that they must somehow be able to sense their current state of affairs (\\emph{perception}) and respond appropriately (\\emph{action}). In particular, an adaptive system should sense the relevant aspects of its current environmental state, and form expectations about the consequences of that state. In general, the interaction with the environment will be stochastic, and the statistically optimal method of `sensing' and prediction is Bayesian inference.\n\nTypically, however, the system has no direct access to the external state, only to sense data that indirectly have external causes. Moreover, sense data are often very high-dimensional, and predicting their consequences is underdetermined. As a result, it is common to assume that successful organisms are imbued with some kind of \\emph{generative model} of the process by which external causes generate their sense data. They can then use this model to infer those actions will bring (their beliefs about) their current state closer to those expectations: a process called \\emph{active inference}.\n\nSystems such as these are inherently open, and often their internal models and beliefs are supposed to be structured hierarchically---that is, compositionally. The processes of prediction and action sketched here are naturally bidirectional, and indeed our first contribution in the present work is to show that Bayesian inference is abstractly structured as a \\emph{category of optics} \\citep{Riley2018Categories,Clarke2020Profunctor}, the emerging canonical formalism for (open) bidirectionally structured compositional systems.\n\nThe compositional framework of \\emph{open games} \\citep{Bolt2019Bayesian,Ghani2016Compositional} builds on categories of optics to describe systems of motivated interacting agents, but it is substantially more general than needed for classical game theory: generalized open games naturally describe any bidirectionally structured open systems that can be associated with a measure of fitness. Consequently, such generalized open games provide a natural home for a compositional theory of interacting cybernetic systems, and using our notion of \\emph{Bayesian lens}, we characterize a number of canonical statistical models as \\emph{statistical games}.\n\nHowever, mere open games themselves supply no notion of \\emph{dynamics} mediating the interactions. We therefore introduce the concept of \\emph{dynamical realisation} of an open game (Definition \\ref{def:realisation}), as well as a coherence condition that ensures such a realisation behaves as we would expect from a cybernetic system (Definition \\ref{def:cyber-sys}). We use these concepts to show that two prominent frameworks for active inference instantiate such categories of cybernetic systems.\n\n\\paragraph{Acknowledgements} We thank the organizers of \\emph{Applied Category Theory 2020} for the opportunity to present this work, and the anonymous reviewers for helpful comments and questions. We also thank Bruno Gavranovi\u0107, Jules Hedges, and Neil Ghani for stimulating and insightful conversations, and credit Jules Hedges for observing the correct form of the Bayesian \\(\\mathsf{update}\\) map in discussions at SYCO 6.\n\n\\section{Bayesian Updates Compose Optically}\n\\label{sec:org69acce1}\n\nWe begin by proving that Bayesian updates compose according to the `lens' pattern \\citep{Foster2007Combinators} that sits at the heart of categories of open games and other `bidirectional' structures. We first show that Bayesian inversions are `vertical' maps in a fibred category of state-dependent channels. The Grothendieck construction of this structure gives a category of lenses. Open games are commonly defined using the more general `optics' pattern \\citep{Bolt2019Bayesian}, and so we also show that, under the Yoneda embedding, our category of lenses is equivalently a category of optics.\n\nThroughout the paper, we work in a general category of stochastic channels; abstractly, this corresponds to a \\emph{Markov category} \\citep{Fritz2019synthetic} or \\emph{copy-delete category} \\citep{Cho2017Disintegration}. Familiar examples of such categories include \\(\\mathcal{K}\\mspace{-2mu}\\ell(\\mathcal{D})\\), the Kleisli category of the finitely-supported distribution monad \\(\\mathcal{D}\\), and, for `continuous' probabiliy, \\(\\mathcal{K}\\mspace{-2mu}\\ell(\\mathcal{G})\\), the Kleisli category of the Giry monad. We will write \\(c^\\dag_\\pi := \\bdag{c}(\\pi)\\) to indicate the \\textbf{Bayesian inversion} of the channel \\(c\\) with respect to a state \\(\\pi\\). Then, given some \\(y \\in Y\\), \\(c^\\dag_\\pi (y)\\) is a new `posterior' distribution on X. We will call \\(c^\\dag_\\pi(y)\\) the \\textbf{Bayesian update} of \\(\\pi\\) along \\(c\\) given \\(y\\).\n\nFor a substantially expanded version of this section, including proofs and background exposition with precise definitions of Bayesian inversion, see the author's \\citep{Smithe2020Bayesian}. We will occasionally here refer to definitions or results in that paper.\n\n\\begin{defn}[State-indexed categories] \\label{def:stat-cat}\nLet \\((\\cat{C}, \\otimes, I)\\) be a monoidal category enriched in a Cartesian closed category \\(\\Cat{V}\\). Define the \\(\\cat{C}\\text{-state-indexed}\\) category \\(\\Fun{Stat}: \\cat{C}\\op \\to \\Cat{V{\\hbox{-}} Cat}\\) as follows. \n\\begin{align}\n\\Fun{Stat} \\;\\; : \\;\\; \\cat{C}\\op \\; & \\to \\; \\Cat{V{\\hbox{-}} Cat} \\nonumber \\\\\nX & \\mapsto \\Fun{Stat}(X) := \\quad \\begin{pmatrix*}[l]\n& \\Fun{Stat}(X)_0 & := \\quad \\;\\;\\; \\cat{C}_0 \\\\\n& \\Fun{Stat}(X)(A, B) & := \\quad \\;\\;\\; \\Cat{V}(\\cat{C}(I, X), \\cat{C}(A, B)) \\\\\n\\id_A \\: : & \\Fun{Stat}(x)(A, A) & := \\quad \n\\left\\{ \\begin{aligned}\n\\id_A : & \\; \\cat{C}(I, X)     \\to     \\cat{C}(A, A) \\\\\n        & \\quad\\;\\;\\: \\rho \\quad \\mapsto \\quad \\id_A\n\\end{aligned} \\right. \\label{eq:stat} \\\\\n\\end{pmatrix*} \\\\ \\nonumber \\\\\nf : \\cat{C}(Y, X) & \\mapsto \\begin{pmatrix*}[c]\n\\Fun{Stat}(f) \\; : & \\Fun{Stat}(X) & \\to & \\Fun{Stat}(Y) \\vspace*{0.5em} \\\\\n& \\Fun{Stat}(X)_0 & = & \\Fun{Stat}(Y)_0 \\vspace*{0.5em} \\\\\n& \\Cat{V}(\\cat{C}(I, X), \\cat{C}(A, B)) & \\to & \\Cat{V}(\\cat{C}(I, Y), \\cat{C}(A, B)) \\vspace*{0.125em} \\\\\n& \\alpha & \\mapsto & f^\\ast \\alpha : \\big( \\, \\sigma : \\cat{C}(I, Y) \\, \\big) \\mapsto \\big( \\, \\alpha(f \\klcirc \\sigma) : \\cat{C}(A, B) \\, \\big)\n\\end{pmatrix*} \\nonumber\n\\end{align}\nComposition in each fibre \\(\\Fun{Stat}(X)\\) is given by composition in \\(\\cat{C}\\); that is, by the left and right actions of the profunctor \\(\\Fun{Stat}(X)(-, =) : \\cat{C}\\op \\times \\cat{C} \\to \\Cat{V}\\). Explicitly, given \\(\\alpha : \\Cat{V}(\\cat{C}(I, X), \\cat{C}(A, B))\\) and \\(\\beta : \\Cat{V}(\\cat{C}(I, X), \\cat{C}(B, C))\\), their composite is \\(\\beta \\circ \\alpha : \\Cat{V}(\\cat{C}(I, X), \\cat{C}(A, C)) : = \\rho \\mapsto \\beta(\\rho) \\klcirc \\alpha(\\rho)\\). Since \\(\\Cat{V}\\) is Cartesian, there is a canonical copier \\(\\mathord{\\usebox\\sbcopier} : x \\mapsto (x, x)\\) on each object, so we can alternatively write \\((\\beta \\circ \\alpha)(\\rho) = \\big(\\beta(-) \\klcirc \\alpha(-)\\big) \\circ \\mathord{\\usebox\\sbcopier} \\circ \\rho\\). Note that we indicate composition in \\(\\cat{C}\\) by \\(\\klcirc\\) and composition in the fibres \\(\\Fun{Stat}(X)\\) by \\(\\circ\\).\n\\end{defn}\n\n\\begin{ex} \\label{ex:stat-meas}\nLet \\(\\Cat{V} = \\Cat{Meas}\\) be a `convenient' (\\emph{i.e.}, Cartesian closed) category of measurable spaces, such as the category of quasi-Borel spaces \\citep{Heunen2017Convenient}, let \\(\\mathcal{P} : \\Cat{Meas} \\to \\Cat{Meas}\\) be a probability monad defined on this category, and let \\(\\cat{C} = \\mathcal{K}\\mspace{-2mu}\\ell(\\mathcal{P})\\) be the Kleisli category of this monad. Its objects are the objects of \\(\\Cat{Meas}\\), and its hom-spaces \\(\\mathcal{K}\\mspace{-2mu}\\ell(\\mathcal{P})(A, B)\\) are the spaces \\(\\Cat{Meas}(A, \\mathcal{P} B)\\) \\citep{Fritz2019synthetic}. This \\(\\cat{C}\\) is a monoidal category of stochastic channels, whose monoidal unit \\(I\\) is the space with a single point. Consequently, states of \\(X\\) are just measures (distributions) in \\(\\mathcal{P} X\\). That is, \\(\\mathcal{K}\\mspace{-2mu}\\ell(\\mathcal{P})(I, X) \\cong \\Cat{Meas}(1, \\mathcal{P} X)\\). Instantiating  \\(\\Fun{Stat}\\) in this setting, we obtain:\n\\begin{align}\n\\Fun{Stat} \\;\\; : \\;\\; \\mathcal{K}\\mspace{-2mu}\\ell(\\mathcal{P})\\op \\; & \\to \\; \\Cat{V{\\hbox{-}} Cat} \\nonumber \\\\\nX & \\mapsto \\Fun{Stat}(X) := \\quad \\begin{pmatrix*}[l]\n& \\Fun{Stat}(X)_0 & := \\quad \\;\\;\\; \\Cat{Meas}_0 \\\\\n& \\Fun{Stat}(X)(A, B) & := \\quad \\;\\;\\; \\Cat{Meas}(\\mathcal{P} X, \\Cat{Meas}(A, \\mathcal{P} B)) \\\\\n\\id_A \\: : & \\Fun{Stat}(X)(A, A) & := \\quad\n\\left\\{ \\begin{aligned}\n\\id_A : & \\; \\mathcal{P} X     \\to     \\Cat{Meas}(A, \\mathcal{P} A) \\\\\n        & \\;\\;\\; \\rho \\;\\;\\, \\mapsto \\quad \\eta_A\n\\end{aligned} \\right. \\label{eq:stat-kl-d} \\\\\n\\end{pmatrix*} \\\\\nc : \\mathcal{K}\\mspace{-2mu}\\ell(\\mathcal{P})(Y, X) & \\mapsto \\Fun{Stat}(c) \\, := \\hfill\\nonumber\n\\end{align}\n\\begin{equation*}\n\\begin{pmatrix*}[c]\n\\Fun{Stat}(c) \\; : &\\Fun{Stat}(X) & \\to & \\Fun{Stat}(Y) \\vspace*{0.5em} \\\\\n& \\Fun{Stat}(X)_0 & = & \\Fun{Stat}(Y)_0 \\vspace*{0.5em} \\\\\n& \\begin{pmatrix*}[l]\n    d^\\dag : & \\mathcal{P} X & \\to \\mathcal{K}\\mspace{-2mu}\\ell(\\mathcal{P})(A, B) \\\\\n    & \\; \\pi & \\mapsto \\quad \\quad d^\\dag_\\pi\n  \\end{pmatrix*}\n  & \\mapsto &\n  \\begin{pmatrix*}\n    c^\\ast d^\\dag : \\mathcal{P} Y \\to \\mathcal{K}\\mspace{-2mu}\\ell(\\mathcal{P})(A, B) \\\\\n    \\rho \\quad \\mapsto \\quad d^\\dag_{c \\klcirc \\rho}\n  \\end{pmatrix*}\n\\end{pmatrix*} \\nonumber\n\\end{equation*}\nEach \\(\\Fun{Stat}(X)\\) is a category of stochastic channels with respect to measures on the space \\(X\\). We can write morphisms \\(d^\\dag : \\mathcal{P} X \\to \\mathcal{K}\\mspace{-2mu}\\ell(\\mathcal{P})(A, B)\\) in \\(\\Fun{Stat}(X)\\) as \\(d^\\dag_{(\\cdot)} : A \\xklto{(\\cdot)} B\\), and think of them  as generalized Bayesian inversions: given a measure \\(\\pi\\) on \\(X\\), we obtain a channel \\(d^\\dag_\\pi : A \\xklto{\\pi} B\\) with respect to \\(\\pi\\). Given a channel \\(c : Y \\mathoverlap{\\rightarrow}{\\smallklcirc\\,} X\\) in the base category of priors, we can pull \\(d^\\dag\\) back along \\(c\\), to obtain a \\(Y\\text{-dependent}\\) channel in \\(\\Fun{Stat}(Y)\\), \\(c^\\ast d^\\dag : \\mathcal{P} Y \\to \\mathcal{K}\\mspace{-2mu}\\ell(\\mathcal{P})(A, B)\\), which takes \\(\\rho : \\mathcal{P} Y\\) to the channel \\(d^\\dag_{c \\klcirc \\rho} : A \\xklto{c \\klcirc \\rho} B\\) defined by pushing \\(\\rho\\) through \\(c\\) and then applying \\(d^\\dag\\).\n\\end{ex}\n\n\\begin{rmk}\nNote that by taking \\(\\Cat{Meas}\\) to be Cartesian closed, we have \\(\\Cat{Meas}(\\mathcal{P} X, \\Cat{Meas}(A, \\mathcal{P} B)) \\cong \\Cat{Meas}(\\mathcal{P} X \\times A, \\mathcal{P} B)\\) for each \\(X\\), \\(A\\) and \\(B\\), and so a morphism \\(c^\\dag : \\mathcal{P} Y \\to \\mathcal{K}\\mspace{-2mu}\\ell(\\mathcal{P})(X, Y)\\) equivalently has the type \\(\\mathcal{P} Y \\times X \\to \\mathcal{P} Y\\). Paired with a channel \\(c : Y \\to \\mathcal{P} X\\), we have something like a Cartesian lens; and to compose such pairs, we can use the Grothendieck construction \\citep{nLab2020Grothendieck,Spivak2019Generalized}.\n\\end{rmk}\n\n\\begin{defn}[$\\Cat{GrLens}_{\\Fun{Stat}}$] \\label{def:stat-lens}\nInstantiating the category of Grothendieck \\(F\\text{-lenses } \\Cat{GrLens}_F\\) (see \\citep{Spivak2019Generalized})\nwith \\(F = \\Fun{Stat} : \\cat{C}\\op \\to \\Cat{V{\\hbox{-}} Cat}\\), we obtain the category \\(\\Cat{GrLens}_\\Fun{Stat}\\) whose objects are pairs \\((X, A)\\) of objects of \\(\\cat{C}\\) and whose morphisms \\((X, A) \\mathrel{\\ooalign{\\hfil$\\mapstochar\\mkern5mu$\\hfil\\cr$\\to$\\cr}} (Y, B)\\) are elements of the set\n\\begin{equation}\n\\Cat{GrLens}_\\Fun{Stat} \\big( (X, A), (Y, B) \\big) \\cong \\cat{C}(X, Y) \\times \\Cat{V} \\big( \\cat{C}(I, X), \\cat{C}(B, A) \\big) \\, .\n\\end{equation}\nThe identity \\(\\Fun{Stat}\\text{-lens}\\) on \\((Y, A)\\) is \\((\\id_Y, \\id_A)\\), where by abuse of notation \\(\\id_A : \\cat{C}(I, Y) \\to \\cat{C}(A, A)\\) is the constant map \\(\\id_A\\) defined in \\eqref{eq:stat} that takes any state on \\(Y\\) to the identity on \\(A\\). The sequential composite of \\((c, c^\\dag) : (X, A) \\mathrel{\\ooalign{\\hfil$\\mapstochar\\mkern5mu$\\hfil\\cr$\\to$\\cr}} (Y, B)\\) and \\((d, d^\\dag) : (Y, B) \\mathrel{\\ooalign{\\hfil$\\mapstochar\\mkern5mu$\\hfil\\cr$\\to$\\cr}} (Z, C)\\) is the \\(\\Fun{Stat}\\text{-lens } \\big( (d \\klcirc c), (c^\\dag \\circ c^\\ast d^\\dag) \\big) : (X, A) \\mathrel{\\ooalign{\\hfil$\\mapstochar\\mkern5mu$\\hfil\\cr$\\to$\\cr}} (Z, C)\\) with \\((d \\klcirc c) : \\cat{C}(X, Z)\\) and where \\((c^\\dag \\circ c^\\ast d^\\dag) : \\Cat{V}\\big(\\cat{C}(I, X), \\cat{C}(C, A)\\big)\\) takes a state \\(\\pi : \\cat{C}(I, X)\\) on \\(X\\) to the channel \\(c^\\dag_{\\pi} \\klcirc \\d^\\dag_{c \\klcirc \\pi}\\). If we think of the notation \\((\\cdot)^\\dag\\) as denoting the operation of forming the Bayesian inverse of a channel (in the case where \\(A = X\\), \\(B = Y\\) and \\(C = Z\\)), then the main result of this section is to show that \\((d \\klcirc c)^\\dag_\\pi \\overset{d \\klcirc c \\klcirc \\pi}{\\sim} c^\\dag_{\\pi} \\klcirc \\d^\\dag_{c \\klcirc \\pi}\\), where \\(\\overset{d \\klcirc c \\klcirc \\pi}{\\sim}\\) denotes \\((d \\klcirc c \\klcirc \\pi)\\text{-almost-equality}\\) \\citep[Definition 2.5]{Smithe2020Bayesian}.\n\\end{defn}\n\nIn order to give an optical form for \\(\\Cat{GrLens}_\\Fun{Stat}\\), we need to find two \\(\\cat{M}\\text{-actegories}\\) with a common category of actions \\(\\cat{M}\\). Let \\(\\hat{\\cat{C}}\\) and \\(\\check{\\cat{C}}\\) denote the categories \\(\\hat{\\cat{C}} := \\Cat{V{\\hbox{-}} Cat}(\\cat{C}\\op, \\Cat{V})\\) and \\(\\check{\\cat{C}} := \\Cat{V{\\hbox{-}} Cat}(\\cat{C}, \\Cat{V})\\) of presheaves and copresheaves on \\(\\cat{C}\\), and consider the following natural isomorphisms.\n\\begin{align}\n\\Cat{GrLens}_\\Fun{Stat} \\big( (X, A), (Y, B) \\big) & \\cong \\cat{C}(X, Y) \\times \\Cat{V} \\big( \\cat{C}(I, X), \\cat{C}(B, A) \\big) \\nonumber \\\\\n& \\cong \\int^{M \\, : \\, \\cat{C}} \\cat{C}(X, Y) \\times \\cat{C}(X, M) \\times \\Cat{V}\\big(\\cat{C}(I, M), \\cat{C}(B, A)\\big) \\nonumber \\\\\n& \\cong \\int^{\\hat{M} \\, : \\, \\hat{\\cat{C}}} \\cat{C}(X, Y) \\times \\hat{M}(X) \\times \\Cat{V}\\big(\\hat{M}(I), \\cat{C}(B, A)\\big) \\label{eq:stat-lens-coend}\n\\end{align}\nThe second isomorphism follows by Yoneda reduction \\citep{Loregian2015This,Roman2020Profunctor}, and the third follows by the Yoneda lemma. We take \\(\\cat{M}\\) to be \\(\\cat{M} := \\hat{\\cat{C}}\\), and define an action \\(\\odot\\) of \\(\\hat{\\cat{C}}\\) on \\(\\check{\\cat{C}}\\) as follows.\n\\begin{defn}[$\\odot$]\nWe give only the action on objects; the action on morphisms is analogous.\n\\begin{equation} \\label{eq:L-action}\n\\begin{aligned}\n\\odot : \\hat{\\cat{C}} & \\to \\Cat{V{\\hbox{-}} Cat}(\\check{\\cat{C}}, \\check{\\cat{C}}) \\\\\n\\hat{M} & \\mapsto\n  \\begin{pmatrix*}\n    \\hat{M} \\odot - & : & \\check{\\cat{C}} & \\to & \\check{\\cat{C}} \\\\\n    & & P & \\mapsto & \\Cat{V}\\big( \\hat{M}(I), P \\big)\n  \\end{pmatrix*}\n\\end{aligned}\n\\end{equation}\nFunctoriality of \\(\\odot\\) follows from the functoriality of copresheaves. \\qed\n\\end{defn}\n\n\\begin{prop} \\label{prop:stat-actegory}\n\\(\\odot\\) equips \\(\\check{\\cat{C}}\\) with a \\(\\hat{\\cat{C}}\\text{-actegory}\\) structure: unitor isomorphisms \\(\\lambda^{\\odot}_F : 1 \\odot F \\xto{\\sim} F\\) and associator isomorphisms \\(a^{\\odot}_{\\hat{M}, \\hat{N}, F} : (\\hat{M} \\times \\hat{N}) \\odot F \\xrightarrow{\\sim} \\hat{M} \\odot (\\hat{N} \\odot F)\\) for each \\(\\hat{M},\\hat{N}\\) in \\(\\check{\\cat{C}}\\), both natural in \\(F : \\Cat{V{\\hbox{-}} Cat}(\\cat{C}, \\Cat{V})\\).\n\\end{prop}\n\nWe are now in a position to define the category of abstract Bayesian lenses, and show that this category coincides with the category of \\(\\Fun{Stat}\\text{-lenses}\\).\n\\begin{defn}[Bayesian lenses]\nDenote by \\(\\Cat{BayesLens}\\) the category of optics \\(\\Cat{Optic}_{\\times, \\odot}\\) for the action of the Cartesian product on presheaf categories \\(\\times : \\hat{\\cat{C}} \\to \\Cat{V{\\hbox{-}} Cat}(\\hat{\\cat{C}}, \\hat{\\cat{C}})\\) and the action \\(\\odot : \\hat{\\cat{C}} \\to \\Cat{V{\\hbox{-}} Cat}(\\check{\\cat{C}}, \\check{\\cat{C}})\\) defined in \\eqref{eq:L-action}. Its objects \\((\\hat{X}, \\check{Y})\\) are pairs of a presheaf and a copresheaf on \\(\\cat{C}\\), and its morphisms \\((\\hat{X}, \\check{A}) \\mathrel{\\ooalign{\\hfil$\\mapstochar\\mkern5mu$\\hfil\\cr$\\to$\\cr}} (\\hat{Y}, \\check{B})\\) are abstract \\emph{Bayesian lenses}---elements of the type\n\\begin{equation}\n\\Cat{Optic}_{\\times, \\odot}\\Big((\\hat{X}, \\check{A}), (\\hat{Y}, \\check{B})\\Big)\n= \\int^{\\hat{M} \\, : \\, \\hat{\\cat{C}}} \\hat{\\cat{C}}(\\hat{X}, \\hat{M} \\times \\hat{Y}) \\times \\check{\\cat{C}}(\\hat{M} \\odot \\check{B}, \\check{A})\n\\end{equation}\nGiven \\(v : \\cat{C}(X, Y)\\) and \\(u : \\Cat{V}(\\cat{C}(I, X), \\cat{C}(B, A))\\), we denote the corresponding element of this type by \\(\\optar{v}{u}\\). A Bayesian lens \\((\\hat{X}, \\check{X}) \\mathrel{\\ooalign{\\hfil$\\mapstochar\\mkern5mu$\\hfil\\cr$\\to$\\cr}} (\\hat{Y}, \\check{Y})\\) is called a \\textbf{simple} Bayesian lens.\n\\end{defn}\n\n\\begin{prop} \\label{prop:bayeslens-are-lenses}\n\\(\\Cat{BayesLens}\\) is a category of lenses; a definition is given in \\citep[\u00a72.2.1]{Smithe2020Bayesian}.\n\\end{prop}\n\n\\begin{prop}[$\\Fun{Stat}\\text{-lenses}$ are Bayesian lenses] \\label{prop:stat-lens-bayeslens}\nLet \\(\\hat{(\\cdot)} : \\cat{C} \\hookrightarrow \\Cat{V{\\hbox{-}} Cat}(\\cat{C}\\op, \\Cat{V})\\) denote the Yoneda embedding and \\(\\check{(\\cdot)} : \\cat{C} \\hookrightarrow \\Cat{V{\\hbox{-}} Cat}(\\cat{C}, \\Cat{V})\\) the coYoneda embedding. Then\n\\begin{equation}\n\\Cat{Optic}_{\\times, \\odot}\\Big((\\hat{X}, \\check{A}), (\\hat{Y}, \\check{B})\\Big)\n\\cong\n\\Cat{GrLens}_\\Fun{Stat} \\Big( (X, A), (Y, B) \\Big)\n\\end{equation}\nso that \\(\\Cat{GrLens}_\\Fun{Stat}\\) is equivalent to the full subcategory of \\(\\Cat{Optic}_{\\times, \\odot}\\) on representable (co)presheaves.\n\\end{prop}\n\n\\begin{rmk}\nWe will often abuse notation by indicating representable objects in \\(\\Cat{BayesLens}\\) by their representations in \\(\\cat{C}\\). That is, we will write \\((X, A)\\) instead of \\((\\hat{X}, \\check{A})\\) where this would be unambiguous.\n\\end{rmk}\n\n\\begin{prop} \\label{prop:bayeslens-smc}\n\\(\\Cat{BayesLens}\\) is a symmetric monoidal category. The monoidal product \\(\\otimes\\) is inherited from \\(\\cat{C}\\); the unit object is the pair \\((I, I)\\) where \\(I\\) is the unit object in \\(\\cat{C}\\). For more details on the structure, see \\citep{Riley2018Categories} or \\citep{Moeller2018Monoidal}.\n\\end{prop}\n\n\\begin{defn}[Exact and approximate Bayesian lens]\nLet \\(\\optar{c}{c^\\dag} : (X, X) \\mathrel{\\ooalign{\\hfil$\\mapstochar\\mkern5mu$\\hfil\\cr$\\to$\\cr}} (Y, Y)\\) be a simple Bayesian lens. We say that \\(\\optar{c}{c^\\dag}\\) is \\textbf{exact} if \\(c\\) admits Bayesian inversion and, for each \\(\\pi : I \\mathoverlap{\\rightarrow}{\\smallklcirc\\,} X\\) such that \\(c \\klcirc \\pi\\) has non-empty support, \\(c^\\dag_\\pi\\) is the Bayesian inversion of \\(c\\) with respect to \\(\\pi\\). Simple Bayesian lenses that are not exact are said to be \\textbf{approximate}.\n\\end{defn}\n\n\\begin{lemma} \\label{lemma:optical-bayes}\nLet \\(\\optar{c}{c^\\dag}\\) and \\(\\optar{d}{d^\\dag}\\) be sequentially composable exact Bayesian lenses. Then the contravariant component of the composite lens \\(\\optar{d}{d^\\dag} \\lenscirc \\optar{c}{c^\\dag} \\cong \\optar{d \\klcirc c}{c^\\dag \\circ c^\\ast d^\\dag}\\) is, up to \\(d \\klcirc c \\klcirc \\pi \\text{-almost-}\\allowbreak\\text{equality}\\), the Bayesian inversion of \\(d \\klcirc c\\) with respect to any state \\(\\pi\\) on the domain of \\(c\\) such that \\(c \\klcirc \\pi\\) has non-empty support. That is to say, \\emph{Bayesian updates compose optically}: \\((d \\klcirc c)^\\dag_\\pi \\overset{d \\klcirc c \\klcirc \\pi}{\\sim} c^\\dag_\\pi \\klcirc d^\\dag_{c \\klcirc \\pi}\\).\n\\end{lemma}\n\n\\section{Open Games for General Optics}\n\\label{sec:org675467e}\n\\label{sec:games}\n\nIn this section, we supply mild generalizations of the structures underlying open games, building on those in \\citep{Bolt2019Bayesian}; at first, then, we consider games over arbitrary categories of optics \\(\\Cat{Optic}_{\\circR, \\circL}\\). Subsequently, we use games over Bayesian lenses (in the category of optics \\(\\Cat{BayesLens}\\) introduced above) to exemplify a number of canonical statistical concepts, such as maximum likelihood estimation and the variational autoencoder, and clarify their compositional structure using the notion of \\emph{optimization game} (Definition \\ref{def:opt-game}). Owing to space constraints, we omit most proofs in this section; they will appear in a full paper expanding the present abstract, and can be supplied at the request of the reader.\n\n\\begin{obs}\nIn the graphical calculus for the compact closed bicategory of profunctors \\(\\Cat{Prof}\\) \\citep{Roman2020Open}, the hom object \\(\\Cat{Optic}_{\\circR, \\circL}((X, A), (Y, B))\\) has the depiction\n\\[\n\\tikzfig{img\/optic-RL-XA-YB}\n\\]\nwhere the types on the wires are the 0-cells of \\(\\Cat{Prof}\\), the monoidal actions \\(\\circR\\) and \\(\\circL\\) are depicted as (co)monoids, and the states and effects are (co)representable functors on the objects \\(X,A,Y,B\\), treated as profunctors.\n\\end{obs}\n\n\\begin{defn}[Generalized context] \\label{def:ctx}\nThe context functor \\(C : \\Cat{Optic}_{\\circR, \\circL}\\op \\times \\Cat{Optic}_{\\circR, \\circL} \\to \\Set\\) takes the pair of optical objects \\(((X, A), (Y, B))\\) to the type with depiction\n\\[\n\\tikzfig{img\/context-RL-XA-YB}\n\\]\nThe triangles depict the (co)presheaves on the monoidal unit \\(I\\) in the underlying actegories. The action on morphisms (\\emph{i.e.}, optics) is by precomposition on the left and postcomposition on the right. Functoriality follows accordingly. \\qed\n\nWe can compose a context with an optic to obtain a `closed' system, as follows:\n\\[\n\\tikzfig{img\/closed-RL-XA-YB} \\mapsto \n\\tikzfig{img\/closed-RL-XA-YB-composed}\n\\]\n\\end{defn}\n\n\\begin{conjecture} \\label{conj:doubling}\nIt is easy to show that a context on \\(((X,A),(Y,B))\\) is equivalently a state \\((I, I) \\mathrel{\\ooalign{\\hfil$\\mapstochar\\mkern5mu$\\hfil\\cr$\\to$\\cr}} ((X,A),(Y,B))\\) in the monoidal category of `double lenses', \\(\\Cat{Lens}_{\\Cat{Optic}_{\\circR, \\circL}}\\) \\citep{Bolt2019Bayesian}. Rendering this graphically leads us to the following conjecture: categories of double optics are instances of the \\emph{doubling} or \\emph{CP} construction from categorical quantum mechanics (\\emph{cf.} \\citep{Coecke2015Categorical,Coecke2016Categorical}).\n\\end{conjecture}\n\n\\begin{prop} \\label{prop:ctx-nice}\nLet \\(\\cat{C}\\) and \\(\\cat{D}\\) be the (monoidal) actegories underlying \\(\\Cat{Optic}_{\\circR, \\circL}\\), and denote their respective monoidal units by \\(I_{\\cat{C}}\\) and \\(I_{\\cat{D}}\\). If these unit objects are terminal in their respective categories, then the contexts \\(C((X, A), (Y, B))\\) simplify to\n\\[\n\\tikzfig{img\/context-terminal}\n\\]\nwhere we have depicted the representable presheaf on \\(I_{\\cat{D}}\\) as \\(\\mathord{\\usebox\\sbground}\\) to indicate that \\(A\\) is just discarded. Consequently, in this case, a context is just an optic \\((I, B) \\mathrel{\\ooalign{\\hfil$\\mapstochar\\mkern5mu$\\hfil\\cr$\\to$\\cr}} (X, Y)\\).\n\n\\end{prop}\n\n\\begin{defn}[Generalized open game] \\label{def:open-game}\nLet \\((X, A)\\) and \\((Y, B)\\) be objects in any symmetric monoidal category of optics \\(\\Cat{Optic}_{\\circR, \\circL}\\). Let \\(\\Sigma\\) be a \\(\\cat{U}\\text{-category}\\), for any base of enrichment \\(\\cat{U}\\) such that \\(\\cat{U}\\text{-}\\Cat{Prof}\\) is compact closed. An \\textbf{open game} from \\((X, A)\\) to \\((Y, B)\\) with strategies in \\(\\Sigma\\), denoted \\(G : (X, A) \\xto{\\Sigma} (Y, B)\\), is given by:\n\\begin{enumerate}\n\\item a play function \\(P : \\Sigma_0 \\to \\Cat{Optic}_{\\circR, \\circL}((X, A), (Y, B))\\); and\n\\item a best response function \\(B : C((X, A), (Y, B)) \\to \\cat{U}\\text{-}\\Cat{Prof}(\\Sigma, \\Sigma)\\).\n\\end{enumerate}\nGiven a strategy \\(\\sigma : \\Sigma\\), we will often write \\(\\optar{v}{u}_\\sigma\\) or similar to denote its image under \\(P\\). A strategy is an \\textbf{equilibrium} in a context \\(\\optar{\\pi}{k}\\) if it is a fixed point of \\(B(\\optar{\\pi}{k})\\).\n\\end{defn}\n\nRoughly speaking, the `best responses' to a strategy \\(\\sigma\\) in a context is are those strategies \\(\\tau\\) such that choosing \\(\\tau\\) would result in performance at the game at least as good as choosing \\(\\sigma\\); equilibrium strategies are those for which such deviation would not improve performance.\n\n\\begin{rmk} \\label{rmk:b-relt}\n\nNote: whereas classic open games use a best-response relation, we categorify that here to a best-response \\emph{relator} (in the terminology of \\citep{Loregian2015This}; \\emph{i.e.}, a `proof-relevant' relation), so that we can describe the trajectories witnessing the computation of equilibria, rather than their mere existence.\n\\end{rmk}\n\n\\begin{prop} \\label{prop:cat-open-games}\nGeneralized open games over the symmetric monoidal category of optics \\(\\Cat{Optic}_{\\circR, \\circL}\\) with strategies enriched in \\(\\cat{U}\\) form a symmetric monoidal category denoted \\(\\Cat{Game}(\\cat{U}, \\circR, \\circL)\\).\n\\end{prop}\n\nSince our games are only a mild generalization of those of \\citep{Bolt2019Bayesian}, we refer the reader to \u00a73.10 of that paper for an idea of the proof of the foregoing proposition, which goes through analogously. The sequential composition of games is given by the sequential composition of optics, with the best response to the composite being the product of the best responses to the factors. Similarly, parallel composition is given by the monoidal product of optics, and the best response to the composite is again the product of the best responses to the factors.\n\nWe now consider some games over \\(\\Cat{BayesLens}\\) that supply the building blocks of the archetypal cybernetic systems to be considered in \\secref{sec:cyber-sys}. For now, we will take the strategies simply to be discrete categories (\\emph{i.e.}, sets), as in the standard formulation of open games. Consequently, we will take the codomain of the best response function to be \\(\\Set(\\Sigma, \\Set(\\Sigma, 2))\\), for each strategy type \\(\\Sigma\\). We assume the ambient category of stochastic channels is semicartesian, so that the monoidal unit is the terminal object.\n\n\\begin{rmk}\nAll the games we will consider henceforth will have play functions whose codomains restrict to the representable subcategory \\(\\Cat{GrLens}_\\Fun{Stat}\\) of \\(\\Cat{BayesLens}\\); in this work, we do not use the extra generality afforded by \\(\\Cat{BayesLens}\\), except insofar as it grants us the use of string diagrams in \\(\\Cat{Prof}\\), which we find helpful for reasoning intuitively about these systems. The generality of optics \\emph{is} however used in the `game-theoretic' games of \\citep{Bolt2019Bayesian}, and in future work we hope to relate the cybernetic systems of this paper to the game-theoretic setting of that earlier work.\n\\end{rmk}\n\n\\begin{rmk} \\label{rmk:atomic-games}\nAll the statistical games considered in this paper will be `atomic' in the sense of \\citep{Bolt2019Bayesian}: in particular, the best response functions we consider will be constant, meaning that, in any context, the set of best strategies does not depend on the `current' choice of strategy. Permitting such dependence will be important in future work, however, when we consider how cybernetic systems interact, and hence respond to each other.\n\\end{rmk}\n\n\\begin{ex} \\label{ex:ml-game}\nA Bayesian lens of the form \\((I, I) \\mathrel{\\ooalign{\\hfil$\\mapstochar\\mkern5mu$\\hfil\\cr$\\to$\\cr}} (X, X)\\) is fully specified by a state \\(\\pi : I \\mathoverlap{\\rightarrow}{\\smallklcirc\\,} X\\). A context for such a lens is given by a lens \\(\\optar{!}{k} : (I, X) \\mathrel{\\ooalign{\\hfil$\\mapstochar\\mkern5mu$\\hfil\\cr$\\to$\\cr}} (X, X)\\) where \\(! : I \\mathoverlap{\\rightarrow}{\\smallklcirc\\,} I\\) is the unique map and \\(k : X \\mathoverlap{\\rightarrow}{\\smallklcirc\\,} X\\) is any endochannel on \\(X\\). A \\textbf{maximum likelihood game} is any game whose play function has codomain in Bayesian lenses of this form \\((I, I) \\mathrel{\\ooalign{\\hfil$\\mapstochar\\mkern5mu$\\hfil\\cr$\\to$\\cr}} (X, X)\\) for any \\(X : \\cat{C}\\), and whose best response function is isomorphic to\n\\[\nB(\\optar{!}{k}) = \\optar{\\rho}{!}_\\sigma \\mapsto \\left\\{ \\optar{\\pi}{!}_\\tau \\middle| \\pi \\in \\underset{\\pi : I \\mathoverlap{\\rightarrow}{\\smallklcirc\\,} X}{\\arg\\max} \\E_{k \\klcirc \\pi} \\left[ \\pi \\right]  \\right\\}\n\\]\nwhere \\(\\E\\) is the canonical expectation operator (\\emph{i.e.} algebra evaluation) associated to states in \\(\\cat{C}\\), and where we have written \\(\\optar{\\rho}{!}_\\sigma\\) and \\(\\optar{\\pi}{!}_\\tau\\) to denote the images of the strategies \\(\\sigma\\) and \\(\\tau\\) under the play function. Intuitively, then, the best response is given by the strategy that maximises the likelihood of the state obtained from the context \\(k\\).\n\\end{ex}\n\n\\begin{rmk}\nIn what follows, we assume that the underlying category \\(\\cat{C}\\) of stochastic channels \\emph{admits density functions}. Informally, a density function for a stochastic channel \\(c : X \\mathoverlap{\\rightarrow}{\\smallklcirc\\,} Y\\) is a measurable function \\(p_c : Y \\times X \\to [0, 1]\\) whose values are the probabilities (or probability densities) \\(p_c(y | x)\\)  at each pair \\((y, x) : Y \\times X\\). We say that the value \\(p_c(y | x)\\) is the probability (or probability density) of \\(y\\) \\emph{given} \\(x\\). In a category such as \\(\\mathcal{K}\\mspace{-2mu}\\ell(\\mathcal{D}_{\\leq 1})\\), whose objects are sets and whose morphisms \\(X \\mathoverlap{\\rightarrow}{\\smallklcirc\\,} Y\\) are functions \\(X \\to \\mathcal{D}(Y + 1)\\), a density function for \\(c : X \\mathoverlap{\\rightarrow}{\\smallklcirc\\,} Y\\) is a morphism \\(Y \\otimes X \\mathoverlap{\\rightarrow}{\\smallklcirc\\,} I\\); note that in \\(\\mathcal{K}\\mspace{-2mu}\\ell(\\mathcal{D}_{\\leq 1})\\), \\(I\\) is not terminal. In the finitely-supported case, density functions are effectively equivalent to channels, but this is not the case in the continuous setting, where they are of most use. For more on this, see \\citep[\u00a72.1.4]{Smithe2020Bayesian}.\n\\end{rmk}\n\nA natural first generalization of maximum likelihood games takes us from states \\(I \\mathoverlap{\\rightarrow}{\\smallklcirc\\,} X\\) to channels \\(Z \\mathoverlap{\\rightarrow}{\\smallklcirc\\,} X\\); that is, from `elements' to `generalized elements' in the covariant (forwards) part of the lens. Unlike Bayesian lenses \\((I, I) \\mathrel{\\ooalign{\\hfil$\\mapstochar\\mkern5mu$\\hfil\\cr$\\to$\\cr}} (X, X)\\), lenses \\((Z, Z) \\mathrel{\\ooalign{\\hfil$\\mapstochar\\mkern5mu$\\hfil\\cr$\\to$\\cr}} (X, X)\\) admit nontrivial contravariant components, which we think of as generalized Bayesian inversions. Consequently, our first generalization is a notion of `Bayesian inference game'. A context \\(\\optar{\\pi}{k} : (I, X) \\mathrel{\\ooalign{\\hfil$\\mapstochar\\mkern5mu$\\hfil\\cr$\\to$\\cr}} (Z, X)\\) for a Bayesian lens \\((Z, Z) \\mathrel{\\ooalign{\\hfil$\\mapstochar\\mkern5mu$\\hfil\\cr$\\to$\\cr}} (X, X)\\) then constitutes a `prior' state \\(\\pi : I \\mathoverlap{\\rightarrow}{\\smallklcirc\\,} Z\\) and a `continuation' channel \\(k : X \\mathoverlap{\\rightarrow}{\\smallklcirc\\,} X\\) which together witness the closure of the otherwise open system.\n\n\\begin{ex} \\label{ex:simp-inf-game}\nFix a channel \\(c : Z \\mathoverlap{\\rightarrow}{\\smallklcirc\\,} X\\) with associated density function \\(p_c : X \\times Z \\to {\\mathbb{R}}_+\\) and a measure of divergence between states on \\(Z\\), \\(D : \\cat{C}(I, Z) \\times \\cat{C}(I, Z) \\to {\\mathbb{R}}\\). A corresponding (generalized) \\textbf{simple Bayesian inference game} is any game whose play function has codomain \\(\\Cat{BayesLens}((Z, Z), (X, X))\\) and whose best response function is isomorphic to\n\\begin{align*}\nB(\\optar{\\pi}{k}) = \\optar{d}{d'}_\\sigma & \\mapsto\n\\Bigg\\{ \\optar{c}{c'}_\\tau \\bigg| \nc' \\in \\underset{c' : \\Cat{V}(\\cat{C}(I, Z), \\, \\cat{C}(X, Z))}{\\arg\\min} \\E_{x \\sim k \\klcirc c \\klcirc \\pi} \\bigg[ \\E_{z \\sim c'_\\pi(x)} \\left[ - \\log p_c(x | z)  \\right]\n  + D(c'_\\pi(x), \\pi) \\bigg] \\Bigg\\} \\\\\n= \\optar{d}{d'}_\\sigma & \\mapsto\n\\Bigg\\{ \\optar{c}{c'}_\\tau \\bigg| \nc' \\in \\underset{c' : \\Cat{V}(\\cat{C}(I, Z), \\, \\cat{C}(X, Z))}{\\arg\\min} \\bigg( \\E_{z \\sim c'_\\pi \\klcirc k \\klcirc c \\klcirc \\pi} \\left[ - \\int_X \\log p_c(\\d k \\klcirc c \\klcirc \\pi | z)  \\right] \\\\\n& \\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\n  + D(c'_\\pi \\klcirc k \\klcirc c \\klcirc \\pi, \\pi) \\bigg) \\Bigg\\}\n\\end{align*}\nwhere \\(\\pi : I \\mathoverlap{\\rightarrow}{\\smallklcirc\\,} Z\\) and \\(k : X \\mathoverlap{\\rightarrow}{\\smallklcirc\\,} X\\), and where the notation \\(z \\sim \\pi\\) means ``\\(z\\) distributed according to the state \\(\\pi\\)''. Note that the second line follows from the first by linearity of expectation.\n\\end{ex}\n\n\\begin{prop}[{\\citep[Thm. 1]{Knoblauch2019Generalized}}]\nWhen \\(D\\) is chosen to be the Kullback-Leibler divergence \\(D_{KL}\\), minimizing the objective function defining a simple Bayesian inference game is equivalent to computing an (exact) Bayesian inversion.\n\\end{prop}\n\n\\begin{cor}\nGiven two Bayesian inference games \\(G : (Z, Z) \\mathrel{\\ooalign{\\hfil$\\mapstochar\\mkern5mu$\\hfil\\cr$\\to$\\cr}} (Y, Y)\\) and \\(H : (Y, Y) \\mathrel{\\ooalign{\\hfil$\\mapstochar\\mkern5mu$\\hfil\\cr$\\to$\\cr}} (X, X)\\), we can compose them sequentially to obtain a game \\(H \\lenscirc G : (Z, Z) \\mathrel{\\ooalign{\\hfil$\\mapstochar\\mkern5mu$\\hfil\\cr$\\to$\\cr}} (X, X)\\), which we will call a \\textbf{hierarchical Bayesian inference game}. It is then an immediate consequence of Lemma \\ref{lemma:optical-bayes} that, in any given context for which the forwards channels admit Bayesian inversion, the best response to the composite game \\(H \\lenscirc G\\) (that is, the optimal inversion of the composite channel) is given simply by (the composition of) the best responses to the factors \\(H\\) and \\(G\\). Consequently, Bayesian inference games are closed under composition.\n\\end{cor}\n\nSimilarly, given a channel \\(c : Z \\otimes Y \\mathoverlap{\\rightarrow}{\\smallklcirc\\,} X\\), we can consider the \\textbf{marginal Bayesian inference game} in which the objective is to compute the inversion of the channel onto just one of the factors \\(Z\\) or \\(Y\\) in the domain.\n\n\\begin{ex}[Variational autoencoder game] \\label{ex:vae-game}\nFix a family \\(\\mathcal{F} \\hookrightarrow \\cat{C}(Z, X)\\) of forward channels and a family \\(\\mathcal{P} \\hookrightarrow \\cat{C}(X, Z)\\) of backward channels such that each \\(c : \\mathcal{F}\\) admits a density function \\(p_c : X \\otimes Z \\to {\\mathbb{R}}_+\\) and each \\(d : \\mathcal{P}\\) admits a density function \\(q : Z \\otimes X \\to {\\mathbb{R}}_+\\); think of these families as determining parameterizations of the channels. We take our strategy type to be \\(\\Sigma = \\mathcal{F} \\times \\mathcal{P}\\). A \\textbf{simple variational autoencoder game} \\((Z,Z) \\xto{\\Sigma} (X,X)\\) is any game with play function \\(P : \\Sigma \\to \\Cat{BayesLens}((Z,Z), (X,X))\\) and whose best response function is isomorphic to\n\\begin{align*}\nB(\\optar{\\pi}{k}) = \\optar{d}{d'}_\\sigma \\mapsto\n\\Bigg\\{ \\optar{c}{c'}_\\tau \\Bigg| (c, c') \\in \\underset{\\substack{c \\in \\mathcal{F}, \\\\ c' \\in \\Cat{V}(\\cat{C}(I, Z), \\, \\mathcal{P})}}{\\arg\\min} \\E_{x \\sim k \\klcirc c \\klcirc \\pi} \\E_{z \\sim c'_\\pi(x)} \\left[ \\log \\frac{q(z|x)}{p_c(x|z)p_\\pi(z)} \\right]\n\\Bigg\\}\n\\end{align*}\nwhere \\(\\pi : I \\mathoverlap{\\rightarrow}{\\smallklcirc\\,} Z\\) admits a density function \\(p_\\pi : Z \\to {\\mathbb{R}}_+\\), \\(q : Z \\otimes X \\to {\\mathbb{R}}_+\\) is a density function associated to \\(c'_\\pi\\), and \\(k\\) has type \\(X \\mathoverlap{\\rightarrow}{\\smallklcirc\\,} X\\).\n\\end{ex}\n\n\\begin{prop} \\label{prop:vae-learns-model}\nA best response to a variational autoencoder game is a stochastic channel \\(c : \\mathcal{F}\\) that maximises the likelihood of the state observed through the continuation \\(k\\) under the assumption that the generative process is in \\(\\mathcal{F}\\), along with an inverse channel \\(c'_\\pi : \\mathcal{P}\\) that best approximates the exact Bayesian inverse \\(c^\\dag_\\pi\\) under the constraint of being in \\(\\mathcal{P}\\).\n\\end{prop}\n\n\\begin{prop} \\label{prop:vae-infers}\nVariational autoencoder games generalize inference games for the Kullback-Leibler divergence. More precisely, the objective function defining autoencoder games is of the same form as that defining inference games \\eqref{ex:simp-inf-game} when \\(D = D_{KL}\\).\n\\end{prop}\n\nThis prompts the following generalization:\n\n\\begin{ex}[Generalized autoencoder game] \\label{ex:ae-game}\nFix two families of channels \\(\\mathcal{F},\\mathcal{P}\\) and a strategy type \\(\\Sigma\\) as in Example \\ref{ex:vae-game}. Then a (generalized) \\textbf{simple autoencoder game} \\((Z,Z) \\xto{\\Sigma} (X,X)\\) is any game with play function \\(P : \\Sigma \\to \\Cat{BayesLens}((Z,Z), (X,X))\\) and whose best response function is isomorphic to\n\\begin{align*}\nB(\\optar{\\pi}{k}) = \\optar{d}{d'}_\\sigma \\mapsto\n\\Bigg\\{ \\optar{c}{c'}_\\tau \\Bigg| (c, c') \\in \\underset{\\substack{c \\in \\mathcal{F}, \\\\ c' \\in \\Cat{V}(\\cat{C}(I, Z), \\, \\mathcal{P})}}{\\arg\\min} \\bigg( & \\E_{z \\sim c'_\\pi \\klcirc k \\klcirc c \\klcirc \\pi} \\left[ - \\int_X \\log p_c(\\d k \\klcirc c \\klcirc \\pi | z) \\right] \\\\\n& \\qquad + D(c'_\\pi \\klcirc k \\klcirc c \\klcirc \\pi, \\pi) \\bigg)\n\\Bigg\\}\n\\end{align*}\nwhere \\(\\pi\\) and \\(k\\) have respective types \\(I \\mathoverlap{\\rightarrow}{\\smallklcirc\\,} Z\\) and \\(X \\mathoverlap{\\rightarrow}{\\smallklcirc\\,} X\\), and \\(D\\) is any measure of divergence between states.\n\nAs with Bayesian inference games, we can generalize simple autoencoder games to \\textbf{hierarchical} and \\textbf{marginal} autoencoder games via the corresponding sequential and parallel compositions.\n\\end{ex}\n\nThe foregoing games have been purely statistically formulated, without capturing the motivating feature of an open system as something in interaction with an external environment. Nonetheless, we can model a simple open system of hierarchical \\textbf{active inference} that receives stochastic inputs from an environment and emits actions stochastically into the environment, as follows.\n\\begin{ex}[Active inference game] \\label{ex:ai-game}\nLet \\(\\{S_i\\}_i\\) be set of spaces of sensory data indexed by hierarchical levels of abstraction \\(i\\) (for instance, the levels of abstraction might range from representations of whole objects to fine details about their texture); similarly, let \\(\\{A_i\\}_i\\) be a set of spaces of possible actions similarly hierarchically organized. Consider the marginal autoencoder games \\((S_{i+1} \\otimes A_{i}, S_{i+1}) \\mathrel{\\ooalign{\\hfil$\\mapstochar\\mkern5mu$\\hfil\\cr$\\to$\\cr}} (S_{i+1}, S_{i+1})\\) and \\((A_{i+1} \\otimes S_{i}, A_{i+1}) \\mathrel{\\ooalign{\\hfil$\\mapstochar\\mkern5mu$\\hfil\\cr$\\to$\\cr}} (A_{i+1}, A_{i+1})\\) coupled via the symmetric monoidal structure \\(\\otimes\\) of \\(\\cat{C}\\):\n\\[\n\\tikzfig{img\/optic-ai-S} \\qquad\n\\tikzfig{img\/optic-ai-A} \\; \\mapsto \\;\n\\tikzfig{img\/optic-ai}\n\\]\ngiving a composite game \\((S_{i+1} \\otimes A_{i+1}, S_{i+1} \\times A_{i+1}) \\mathrel{\\ooalign{\\hfil$\\mapstochar\\mkern5mu$\\hfil\\cr$\\to$\\cr}} (S_{i} \\otimes A_{i}, S_{i} \\times A_{i})\\). Recall from \\citep[\u00a7\u00a73.7-3.8]{Bolt2019Bayesian} that a composite game is given by the (sequential and parallel) composition of optics, with best-response given by the product of the best-responses of the factors.\n\nNote that the Bayesian posterior inferred by such a game has independent factors on \\(S_{i+1}\\) and \\(A_{i+1}\\). This is not merely a diagrammatic convenience, but coincides with a common `mean field' simplification in the modelling literature \\citep{Buckley2017free,Kingma2017Variational}. The dashed box is a functorial box \\citep{Mellies2006Functorial} depicting the Yoneda embedding; recall that optics in \\(\\Cat{BayesLens}\\) were defined over (co)presheaves, and so here we needed to lift the monoidal product on \\(\\cat{C}\\) into a diagram over its presheaf category \\(\\Cat{Cat}(\\cat{C}\\op, \\Set)\\).\n\nNext, compose these games along the hierarchy indexed by \\(i\\), to obtain a game \\((S_{N} \\otimes A_{N}, S_{N} \\times A_{N}) \\mathrel{\\ooalign{\\hfil$\\mapstochar\\mkern5mu$\\hfil\\cr$\\to$\\cr}} (S_{0} \\otimes A_{0}, S_{0} \\times A_{0})\\), such as an element of the following object:\n\\[\n\\tikzfig{img\/optic-ai-hierarchy}\n\\]\nGiven a context with a strong prior about expected sensory states and a continuation that responds to an action of type \\(A_0\\) by feeding back a state on \\(S_0\\), the best response can be shown to be that which selects actions that, under the current state, maximize the likelihood of obtaining the expected `goal' state \\citep{Buckley2017free,Friston2015Active}.\n\\end{ex}\n\n\\begin{rmk}\nWe have framed each of these statistical procedures as optimization problems not only to suggest a link to the utility-maximising agents of game theory, but also because it suggests the use of iterative methods to compute best responses; note that computational tractability is an important motivation in the proof of Proposition \\ref{prop:vae-learns-model}. \n\nThe question of providing such dynamical or, thinking of game composition as an algebra for building complex systems, `coalgebraic' semantics for (generalized) optimization games is the topic of the next section. We first formalize this notion.\n\\end{rmk}\n\n\\begin{defn} \\label{def:opt-game}\nAn \\textbf{optimization game} is any open game whose best response function can be defined by a function of the form \\(\\Sigma \\times C \\xto{\\pi} M \\xto{\\varphi} P\\), where \\(\\Sigma\\) is a strategy type, \\(C\\) a context type, \\(M\\) any space, and \\(P\\) a poset. We call \\(\\varphi\\) the \\textbf{fitness function}, and think of \\(\\pi\\) as projecting systems into a space whose points can be assigned a fitness. The best response function of an optimization game can then be defined by giving the subset of strategies contextually maximizing fitness, for each context \\(c : C\\).\n\\end{defn}\n\n\\section{Cybernetic Systems and Dynamical Realisation}\n\\label{sec:org4ff5161}\n\\label{sec:cyber-sys}\n\nIn this section, we begin to answer the question of precisely how the optimization games of the previous section may be realized in physical systems, such as brains or computers. More formally, this means we seek open dynamical systems whose input and output types correspond to the domain and codomain types of the foregoing games, such that there is a correspondence between the behaviours of the abstract games and their dynamical realisations, and such that the evolutions of the internal states of the dynamical systems correspond to strategic improvements in game-playing: by concentrating on optimization games, a natural measure of such improvement is encoded in the fitness function underlying the best-response relator.\n\nWe do not require that there is a correspondence between internal states of the realisations and strategies for the corresponding games, but we do require that the fitness functions extend to the the total state spaces of the closure of a realisation induced by the context. When there \\emph{is} a correspondence between internal states and strategies, we can take advantage of Definition \\ref{def:open-game} and interpret trajectories over the state space as trajectories over strategies witnessing the strategic improvement.\n\nWe begin by sketching categories of dynamical games, and then use these ideas to define preliminary notions of open cybernetic systems and categories thereof. We consider principally single systems whose underlying games are atomic (in the sense of Remark \\ref{rmk:atomic-games}), and leave the study of the behaviour of interacting cybernetic systems to future work. Once more, we omit proofs in this section; they will appear in a paper to follow.\n\n\\begin{defn}[Discrete-time dynamical system over $\\cat{C}$; after \\citep{Schultz2019Dynamical,Clarke2020Profunctor}] \\label{def:dds}\nA \\textbf{discrete-time dynamical system} over \\(\\cat{C}\\) with state space \\(S :\\cat{C}\\), input type \\(A : \\cat{C}\\) and output type \\(B : \\cat{C}\\) is a lens \\((S, S) \\mathrel{\\ooalign{\\hfil$\\mapstochar\\mkern5mu$\\hfil\\cr$\\to$\\cr}} (B, A)\\) over \\(\\cat{C}\\), \\emph{i.e.} in the following optical hom object:\n\\begin{align*}\n\\int^{M : \\cat{C}} \\Cat{Comon}(\\cat{C})(S, M \\otimes B) \\times \\cat{C}(M \\otimes A,S)\n\\cong \\Cat{Comon}(\\cat{C})(S, B) \\times \\cat{C}(S \\otimes A, S)\n\\end{align*}\nwhere the isomorphism follows by Yoneda reduction. Note that this requires that the `output' map of the dynamical system is a comonoid homomorphism in \\(\\cat{C}\\) and hence deterministic in a category of stochastic channels.\n\\end{defn}\n\n\\begin{defn}[Category of discrete-time dynamical systems] \\label{def:cat-dds}\nWe define a category \\(\\Cat{Dyn}_{\\cat{C}}\\) whose objects are the objects of \\(\\cat{C}\\) and whose morphisms, denoted \\(A \\xto{S} B\\), are discrete-time dynamical systems; the symbol above the arrow denotes the internal state space. Hom objects are given by\n\\[\n\\Cat{Dyn}_{\\cat{C}}(A, B) = \\sum_{S : \\cat{C}} \\Cat{Comon}(\\cat{C})(S, B) \\times \\cat{C}(S \\otimes A, S) \\, .\n\\]\nIdentity dynamical systems on each \\(A : \\cat{C}\\) are the `no-op' dynamical systems \\(A \\xto{A} A\\) given by identity optics \\(\\id_A : (A, A) \\mathrel{\\ooalign{\\hfil$\\mapstochar\\mkern5mu$\\hfil\\cr$\\to$\\cr}} (A, A)\\). Associativity and unitality of composition is inherited from the category of optics underlying Definition \\ref{def:dds}; a symmetric monoidal structure is similarly inherited. \\qed\n\\end{defn}\n\n\\begin{defn}[Lenses over dynamical systems; after \\citep{Riley2018Categories}] \\label{def:dyn-lens}\nThe category of (monoidal) lenses over \\(\\cat{C}\\text{-dynamical}\\) systems has as objects pairs \\((X, A)\\) of objects in \\(\\cat{C}\\) and as morphisms, \\textbf{dynamical lenses} \\((X, A) \\mathrel{\\ooalign{\\hfil$\\mapstochar\\mkern5mu$\\hfil\\cr$\\to$\\cr}} (Y, B)\\), elements of the type\n\\begin{gather*}\n\\tikzfig{img\/optic-dyn-XA-YB} \\\\\n\\int^{M : \\cat{C}} \\Cat{Dyn}_{\\cat{C}}(X, M \\otimes Y) \\times \\Cat{Dyn}_{\\cat{C}}(M \\otimes B, A) \\\\\n\\cong \\\\\n\\sum_{P,Q : \\cat{C}} \\int^{M : \\cat{C}} \\cat{C}(P \\otimes X, P) \\times \\Cat{Comon}(\\cat{C})(P, M \\otimes Y) \\times \\cat{C}(Q \\otimes M \\otimes B, Q) \\times \\Cat{Comon}(\\cat{C})(Q, A) \\\\\n\\sum_{P,Q : \\cat{C}} \\;\\; \\tikzfig{img\/optic-dyn-full-XA-YB} \\; .\n\\end{gather*}\nThat is, a dynamical lens is a pair of dynamical systems coupled along some `residual' type.\n\\end{defn}\n\n\\begin{rmk}\nAt this point we begin to run into sizes issues. However, for the purposes of this paper, we will simply assume that a satisfactory resolution of these matters is at hand; for instance, that there is a hierarchy of Grothendieck universes such that the coends over (`large') sums in the preceding definition constitute accessible objects.\n\\end{rmk}\n\nWe now expand the definition of context in the dynamical setting. We will see that a dynamical context is simply a closure of an open dynamical system: that is, a `larger' system into which a `smaller' open dynamical system can plug such that the composite is a closed (but still uninitialized) system.\n\n\\begin{prop} \\label{prop:dyn-ctx}\nIf \\(I\\) is terminal in \\(\\cat{C}\\), a context for a dynamical lens \\((X, A) \\mathrel{\\ooalign{\\hfil$\\mapstochar\\mkern5mu$\\hfil\\cr$\\to$\\cr}} (Y, B)\\) is an element of the following type, denoted \\(\\tilde{C}\\big((X, A), (Y, B)\\big)\\):\n\\[\n\\sum_{P,Q \\, : \\, \\cat{C}} \\;\\; \\tikzfig{img\/context-dyn-terminal}\n\\]\nInterpreting this diagram, a context for a dynamical lens \\((X, A) \\mathrel{\\ooalign{\\hfil$\\mapstochar\\mkern5mu$\\hfil\\cr$\\to$\\cr}} (Y, B)\\) amounts to an autonomous dynamical system with output type of the form \\(X \\otimes M\\) (for some residual type \\(M\\)), coupled along the residual \\(M\\) to an open dynamical system with input type \\(Y \\otimes M\\) and output type \\(B\\); and the \\(A\\) type is discarded. This is precisely what we should expect from a dynamical analogue of Proposition \\ref{prop:ctx-nice}.\n\\end{prop}\n\n\\begin{defn} \\label{def:dyn-game}\nA \\textbf{dynamical game} is just a generalized open game (\\ref{def:open-game}) over the category of dynamical lenses. We write \\((X, A) \\xto{\\tilde{\\Sigma}, S} (Y, B)\\) to indicate both the strategy type \\(\\tilde{\\Sigma}\\) and state space \\(S\\). Dynamical games form a symmetric monoidal category in the corresponding way. For notational clarity, we will write \\(\\tilde{G}\\) for a dynamical game, \\(\\tilde{P}\\) for its play function, and \\(\\tilde{B}\\) for its best response function.\n\\end{defn}\n\n\\begin{defn}[Dynamical realisation of an open game] \\label{def:realisation}\nLet \\(G : (X, A) \\xto{\\Sigma} (Y, B)\\) be an open game with \\(X, A, Y, B\\) all objects of some symmetric monoidal category \\(\\cat{C}\\). A \\textbf{dynamical realisation} of \\(G\\) is a choice of dynamical game \\(\\tilde{G} : (X, A) \\xto{\\tilde{\\Sigma}, S} (Y, B)\\) on the same objects, along with a function \\(\\interp{\\cdot} : C((X, A), (Y, B)) \\to \\tilde{C}((X, A), (Y, B))\\) lifting static contexts to dynamical contexts. Given a context \\(\\optar{\\pi}{k} : C((X, A), (Y, B))\\), we choose a representative \\(\\optar{\\interp{\\pi}}{\\interp{k}} \\cong \\interp{\\optar{\\pi}{k}} : \\tilde{C}((X, A), (Y, B))\\) for its realisation.\n\\end{defn}\n\nA `dynamical context' is an element of the type given in Proposition \\(\\ref{prop:dyn-ctx}\\): a context for a dynamical lens. A `static context' is simply a context for the `static' game that is being dynamically realized. At this stage, we impose no particular requirements on the context realisation function \\(\\interp{\\cdot}\\), except to say that in the intended semantics, \\(\\interp{\\optar{\\pi}{k}}\\) is a (coupled, open) dynamical system that constantly emits the state \\(\\pi\\) and (by some mechanism) realizes the channel \\(k\\). We call such a context \\emph{stationary} as neither \\(\\pi\\) nor \\(k\\) vary in time; future work will generalize the results of this section to \\emph{non-stationary} contexts.\n\n\\begin{defn}[Open cybernetic systems] \\label{def:cyber-sys}\nAn open \\textbf{cybernetic system} is defined by the data:\n\\begin{itemize}\n\\item an open optimization game (Def. \\ref{def:opt-game}) \\(G : (X, A) \\xto{\\Sigma} (Y, B)\\) with \\(X, A, Y, B\\) all objects of some symmetric monoidal category \\(\\cat{C}\\),\n\\item a fitness function \\(\\varphi_G : \\Sigma \\times C \\to M \\xto{\\varphi} F\\) where \\(C = C\\left((X, A), (Y, B)\\right)\\),\n\\item a dynamical realisation \\(\\big(\\tilde{G} : (X, A) \\xto{\\tilde{\\Sigma}, S} (Y, B), \\interp{\\cdot} : C((X, A), (Y, B)) \\to \\tilde{C}((X, A), (Y, B)) \\big)\\) of \\(G\\),\n\\end{itemize}\nsatisfying the following condition for each context \\(\\optar{\\pi}{k} : C((X, A), (Y, B))\\):\n\\begin{itemize}\n\\item there exists a dynamical strategy \\(\\tilde{\\sigma} : \\tilde{\\Sigma}\\), such that\n\\item writing \\(Z\\) for the total state space of the autonomous dynamical system \\(\\interp{\\optar{\\pi}{k}} \\lenscirc \\tilde{P}(\\tilde{\\sigma})\\) induced by the context, there exists a function \\(\\nu : Z \\to M\\) projecting \\(Z\\) into the `fitness landscape' \\(M\\), such that\n\\item there exists a fitness-maximising fixed point \\(\\zeta^\\ast : Z\\), in the sense that\n\\item for some equilibrium strategy of the static system \\(\\sigma^\\ast : \\text{fix } B(\\optar{\\pi}{k})\\), \\(\\varphi(\\nu(\\zeta^\\ast)) \\leq \\varphi_G(\\sigma^\\ast, \\optar{\\pi}{k})\\).\n\\end{itemize}\n\\noindent\nA \\textbf{category of open cybernetic systems} is a category of (generalized) open games such that each game is an open cybernetic system with dynamics realised in the same category \\(\\cat{C}\\), and such that the composite of games is a cybernetic system whose fitness-maximising fixed point projects onto fitness-maximising fixed points of each of the factors in their corresponding local contexts. (See \\citep[\u00a73.7]{Bolt2019Bayesian} for the definition of local context.)\n\\end{defn}\nThe idea here is that, by using the fitness function of the underlying optimization game, the cybernetic condition forces the behaviour of the dynamical realisation to coincide with the process of iteratively improving the strategies deployed by the system in playing the game. We summarize the condition in the diagram\n\\begin{equation*}\n\\begin{tikzcd}\n\\Sigma \\times C \\arrow[d, \"\\interp{\\cdot}\"'] \\arrow[r]  & M \\arrow[r, \"\\varphi\"] & F \\\\\n\\tilde{\\Sigma} \\times \\tilde{C} \\arrow[r, \"\\text{fix}\"] & Z \\arrow[u]            &  \n\\end{tikzcd}\n\\end{equation*}\nthough this is in general ill-defined: we do not require a function \\(\\interp{\\cdot} : \\Sigma \\to \\tilde{\\Sigma}\\), and nor do we require that the best response to \\(\\tilde{G}\\) coincides in any way with the best reponse to \\(G\\). Investigating such conditions is the subject of future work; for instance, we may be interested in nested cybernetic systems, such as characterize evolution by natural selection, and how their fitness functions constrain one another. For similar reasons, we are also interested in the case where the fitness function is itself non-stationary.\n\n\\begin{rmk} \\label{rmk:kubernetes}\nThe codomain category of the cybernetic realisation functor is in general much larger than the domain category of static games, and often it makes sense to consider dynamical games in this codomain category as if they were dynamical realisations of static games, even if in fact there is no static game to which they could correspond. For instance, adaptive systems in physical environments are in general not realisations of static games because their contexts are irreducibly dynamical and thus not the dynamical realisation of a static context; but over short time intervals, it can be productive to treat such systems as realisations of static games. In continuous time (not treated here), it is even possible to consider dynamical games that are indeed realisations of games that are static when represented in a smoothly varying coordinate system. The free-energy framework of Theorem \\ref{thm:cyber-fep} is an example of a category of cybernetic systems with a rich underlying category of dynamic games.\n\\end{rmk}\n\nA classic category of open cybernetic systems is found in the computational neuroscience literature, as summarized in the following theorem.\n\n\\begin{thm} \\label{thm:cyber-fep}\nConsider the subcategory of \\(\\Cat{BayesLens}\\) spanned by finite-dimensional Euclidean spaces, with morphisms generated (under sequential and parallel composition) by the (variational) autoencoder and inference games whose forwards and backwards channels emit Gaussian measures with high-precision. The (discrete-time) free-energy framework for action and perception \\citep{Buckley2017free} instantiates a category of open cybernetic systems realising games over this subcategory.\n\\end{thm}\n\n\\begin{rmk}\nTypical presentations of `active inference' under the free-energy principle are excessively complicated by the lack of attention paid to compositionality. Because the free-energy framework instantiates a \\emph{category} of open cybernetic systems, a radically simplified compositional presentation is possible. Such a presentation forms a companion to the present work.\n\\end{rmk}\n\n\\begin{cor} \\label{cor:bx-cortex}\nThe free-energy framework has been used to supply a computational explanation for the pervasive bidirectionality of cortical circuits in the mammalian brain \\citep{Bastos2012Canonical,Friston2010free}. A corollary of Theorem \\ref{thm:cyber-fep} is that this bidirectionality is furthermore justified by the abstract structure of Bayesian inference and its dynamical realisation: because Bayesian updates compose optically, a cybernetic system realising Bayesian inference compositionally must instantiate this structure. We note also that the parallel interacting bidirectional structure of the active inference game (Example \\ref{ex:ai-game}) is reproduced in the cortex.\n\\end{cor}\n\nThe free-energy framework realisation of autoencoder games is not unique; an alternative is found in machine learning.\n\n\\begin{thm} \\label{thm:cyber-vae}\nConsider the subcategory of \\(\\Cat{BayesLens}\\) spanned by finite-dimensional Euclidean spaces, with morphisms generated (under sequential and parallel composition) by the (variational) autoencoder and inference games whose forwards and backwards channels emit exponential-family measures. The deep (variational) autoencoder framework \\citep{Kingma2017Variational} instantiates a category of open cybernetic systems realising games over this subcategory.\n\\end{thm}\n\nIncreasingly, the variational autoencoder framework is used to model complete agents in machine learning, rather than merely dynamically realise static inference or learning problems. Indeed, thinking of the `free-energy framework' as a collection of cybernetic realisations of autoencoder and active-inference games, the demonstration of the following corollary of Theorem \\ref{thm:cyber-vae} is unsurprising:\n\n\\begin{cor} \\label{cor:deep-ai}\nThe ``deep active inference agent'' \\citep{Ueltzhoeffer2018Deep} is a cybernetic system realising an active inference game in the variational autoencoder framework.\n\\end{cor}\n\nWe have heretofore concentrated on `variational Bayesian' realisations of the games introduced in \\secref{sec:games}, as they most strikingly fit the language of optimization used there. But we expect any other family of approximate inference methods to supply a corresponding category of cybernetic systems. We thus make the following conjecture.\n\n\\begin{conjecture}\nConsider the subcategory of \\(\\Cat{BayesLens}\\) spanned by finite-dimensional smooth manifolds, with morphisms generated (under sequential and parallel composition) by the generalized autoencoder and inference games. We expect sampling algorithms, such as Markov chain Monte Carlo, to supply a corresponding category of open cybernetic systems of interest.\n\\end{conjecture}\n\nFinally, we provide further justification for Remark \\ref{rmk:b-relt}.\n\n\\begin{obs} \\label{prop:sigma-traj}\nConsider a variational autoencoder, realised as in Theorem \\ref{thm:cyber-vae}. By choosing the parameterizations \\(\\mathcal{F},\\mathcal{P}\\) of the forwards and backwards channels to coincide with the state spaces of their dynamical realisations, and the (static) play function \\(P\\) to take a parameter vector to the corresponding channel, the dynamical realisation induces a trajectory over the strategy space. Such trajectories organize into sheaf whose sections are trajectories of arbitrary length \\citep{Schultz2019Dynamical}, spans of which are again just (generalized) dynamical systems; these spans are equivalently profunctors \\citep{Benabou2000Distributors}. We can thus define a best-response function valued in profunctors whose elements are trajectories witnessing deviations of strategies to `better' strategies, and whose dynamical equilibria correspond precisely to the equilibria of the `static' best response function.\n\\end{obs}\n\n\\paragraph{On-going and Future Work}\n\\label{sec:org420ea85}\n\nThe structures sketched in this paper are merely first steps towards a categorical theory of cybernetics. In particular, since the first draft of this work was written, we have come to believe that the preliminary notions presented here of dynamical realisation, and by extension of open cybernetic system, are substantially less elegant than they could be. On-going work is focusing on this issue. We hope that a consequence of this refinement will be that the treatment of \\emph{interacting} cybernetic systems is simplified. In this new setting, we will also treat non-stationary systems in dynamical contexts and in continuous time, thereby supplying a general compositional treatment of (amongst other things) the `free-energy' framework.\n\nFinally, with respect to applications, we are interested in using these tools to realise game-theoretic games and to investigate the connections between repeated games and dynamical realisation. There are deep links with reinforcement learning to be explored, and we seek a setting for the study of nested and mutli-agent (`ecological') systems.\n\n\\bibliographystyle{eptcs}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nLet $(v_{t})_{t\\in\\mathbb Z}$ be a complex circularly symmetric Gaussian stationary process with zero mean and covariance function $(r_k)_{k\\in\\mathbb Z}$ with $r_k=\\mathbb{E}[v_{t+k}v^*_{t}]$ and $r_k\\to0$ as $k\\to\\infty$. We observe $N$ independent copies of $(v_t)_{t\\in\\mathbb Z}$ over the time window $t\\in\\{0,\\ldots,T-1\\}$, and stack the observations in a matrix  $V_T=[ v_{n,t} ]_{n,t = 0}^{N-1, T-1}$. This matrix can be written as $V_T=W_TR_T^{1\/2}$, where $W_T\\in\\mathbb{C}^{N\\times T}$ has independent $\\mathcal{CN}(0,1)$ (standard circularly symmetric complex Gaussian) entries and $R_T^{1\/2}$ is any square root of the Hermitian nonnegative definite Toeplitz $T\\times T$ matrix\n\\begin{equation*}\nR_T \\triangleq \\left[ r_{i-j} \\right]_{0\\leq i,j\\leq T-1} = \\begin{bmatrix}\nr_0 & r_{1} & \\ldots & r_{T-1} \\\\ \nr_{-1} & \\ddots & \\ddots & \\vdots \\\\ \n\\vdots  & \\ddots & \\ddots & r_{1}\\\\ \nr_{1-T} & \\ldots & r_{-1} & r_0\n\\end{bmatrix}.\n\\end{equation*}\nA classical problem in signal processing is to estimate $R_T$ from the observation of $V_T$. \nWith the growing importance of multi-antenna array processing, there has recently been a renewed interest for this estimation problem in the regime of large system dimensions, {\\it i.e.} for both $N$ and $T$ large. \n\nAt the core of the various estimation methods for $R_T$ are the biased and unbiased estimates $\\hat{r}_{k,T}^b$ and $\\hat{r}_{k,T}^u$ for $r_k$, respectively, defined by\n\\begin{align*}\n\t\\hat{r}_{k,T}^b &= \\frac{1}{NT}\\sum_{n=0}^{N-1}\\sum_{t=0}^{T-1} v_{n,t+k} v_{n,t}^* \\mathbbm{1}_{0 \\leq t+k \\leq T-1} \\\\\n\t\\hat{r}_{k,T}^u &= \\frac{1}{N(T-|k|)} \\sum_{n=0}^{N-1} \\sum_{t=0}^{T-1} v_{n,t+k} v_{n,t}^* \\mathbbm{1}_{0 \\leq t+k \\leq T-1}\n\\end{align*}\nwhere $\\mathbbm{1}_A$ is the indicator function on the set $A$.\nDepending on the relative rate of growth of $N$ and $T$, the matrices $\\widehat{R}_{T}^b = [\\hat{r}_{i-j,T}^b]_{0 \\leq i,j \\leq T-1}$ and $\\widehat{R}_{T}^u = [\\hat{r}_{i-j,T}^u]_{0 \\leq i,j \\leq T-1}$ may not satisfy $\\Vert R_T - \\widehat{R}_{T}^b \\Vert \\overset{\\rm a.s.}{\\longrightarrow} 0$ or $\\Vert R_T - \\widehat{R}_{T}^u \\Vert \\overset{\\rm a.s.}{\\longrightarrow} 0$. An important drawback of the biased entry-wise estimate lies in its inducing a general asymptotic bias in $\\widehat{R}_{T}^b$; as for the unbiased entry-wise estimate, it may induce too much inaccuracy in the top-right and bottom-left entries of $\\widehat{R}_{T}^u$. The estimation paradigm followed in the recent literature generally consists instead in building banded or tapered versions of $\\widehat{R}_{T}^b$ or $\\widehat{R}_{T}^u$ ({\\it i.e.} by weighting down or discarding a certain number of entries away from the diagonal), exploiting there the rate of decrease of $r_k$ as $k\\to\\infty$ \\cite{WuPour'09,BickLev'08a,LamFan'09,XiaoWu'12,CaiZhangZhou'10,CaiRenZhou'13}.\nSuch estimates use the fact that $\\Vert R_T-R_{\\gamma(T),T}\\Vert \\to 0$ with $R_{\\gamma,T} = [ [R_T]_{i,j} \\mathbbm{1}_{|i-j| \\leq \\gamma}]$ for some well-chosen functions $\\gamma(T)$ (usually satisfying $\\gamma(T)\\to \\infty$ and $\\gamma(T)\/T\\to 0$) and restrict the study to the consistent estimation of $R_{\\gamma(T),T}$. The aforementioned articles concentrate in particular on choices of functions $\\gamma(T)$ that ensure optimal rates of convergence of $\\Vert R_T-\\widehat{R}_{\\gamma(T),T}\\Vert$ for the banded or tapered estimate $\\widehat{R}_{\\gamma(T),T}$.\nThese procedures, although theoretically optimal, however suffer from several practical limitations. First, they assume the {\\it a priori} knowledge of the rate of decrease of $r_k$ (and restrict these rates to specific classes). Then, even if this were indeed known in practice, being asymptotic in nature, the results do not provide explicit rules for selecting $\\gamma(T)$ for practical finite values of $N$ and $T$. Finally, the operations of banding and tapering do not guarantee the positive definiteness of the resulting covariance estimate.\n\nIn the present article, we consider instead that the only constraint about $r_k$ is $\\sum_{k=-\\infty}^\\infty |r_k|<\\infty$ and estimate $R_T$ from the standard (non-banded and non-tapered) estimates $\\widehat{R}_T^b$ and $\\widehat{R}_T^u$. The consistence of these estimates, in general invalid, shall be enforced here by the choice $N,T\\to\\infty$ with $N\/T\\to c\\in(0,\\infty)$. This setting is more practical in applications as long as both the finite values $N$ and $T$ are sufficiently large and of similar order of magnitude.\nAnother context where a non banded Toeplitz rectification of the estimated \ncovariance matrix leads to a consistent estimate in the spectral norm is \nstudied in \\cite{val-loub-icassp14}. \n\nOur specific contribution lies in the establishment of concentration inequalities for the random variables $\\Vert R_T-\\widehat{R}_T^b\\Vert$ and $\\Vert R_T-\\widehat{R}_T^u\\Vert$. It is shown specifically that, for all $x>0$, $-\\log \\mathbb{P}[\\Vert R_T-\\widehat{R}_T^b \\Vert> x ]= O(T)$ and $-\\log \\mathbb{P}[\\Vert R_T-\\widehat{R}^u_T\\Vert > x ]= O(T\/ \\log T)$. Aside from the consistence in norm, this implies as a corollary that, as long as $\\limsup_T\\Vert R_T^{-1}\\Vert<\\infty$, for $T$ large enough, $\\widehat{R}^u_T$ is positive definite with outstanding probability ($\\widehat{R}^b_T$ is nonnegative definite by construction).\n\nFor application purposes, the results are then extended to the case where $V_T$ is changed into $V_T+P_T$ for a rank-one matrix $P_T$. Under some conditions on the right-eigenspaces of $P_T$, we show that the concentration inequalities hold identically. The application is that of a single source detection (modeled through $P_T$) by an array of $N$ sensors embedded in a temporally correlated noise (modeled by $V_T$). To proceed to detection, $R_T$ is estimated from $V_T+P_T$ as $\\widehat{R}_T^b$ or $\\widehat{R}_T^u$, which is used as a whitening matrix, before applying a generalized likelihood ratio test (GLRT) procedure on the whitened observation. Simulations corroborate the theoretical consistence of the test. \n\nThe remainder of the article is organized as follows. The concentration inequalities for both biased and unbiased estimates are exposed in Section~\\ref{unperturbed}. The generalization to the rank-one perturbation model is presented in Section~\\ref{sig+noise} and applied in the practical context of source detection in Section~\\ref{detect}.\n\n{\\it Notations:} The superscript $(\\cdot)^{\\sf H}$ denotes Hermitian transpose, $\\left\\| X \\right\\|$ stands for the spectral norm for a matrix and Euclidean norm for a vector, and $\\| \\cdot \\|_\\infty$ is the $\\sup$ norm of a function. The notations ${\\cal N}(a,\\sigma^2)$ and ${\\cal CN}(a,\\sigma^2)$ represent the real and complex circular Gaussian distributions with mean $a$ and variance $\\sigma^2$. For $x \\in \\mathbb{C}^{m}$, $D_x=\\diag (x)=\\diag (x_0, \\ldots, x_{m-1} )$ is the diagonal matrix having on its diagonal the elements of the vector $x$.\nFor $x=[x_{-(m-1)},\\ldots,x_{m-1}]^{\\sf T} \\in \\mathbb{C}^{2m+1}$, the matrix ${\\cal T}(x) \\in \\mathbb{C}^{m \\times m}$ is the Toeplitz matrix built from $x$ with entries $[{\\cal T}(x)]_{i,j}=x_{j-i}$.\nThe notations $\\Re(\\cdot)$ and $\\Im(\\cdot)$ stand for the real and the imaginary parts respectively.\n\n\n\\section{Performance of the covariance matrix estimators} \n\\label{unperturbed} \n\n\\subsection{Model, assumptions, and results}\n\\label{subsec-model} \n\nLet $(r_k)_{k\\in\\mathbb Z}$ be a doubly \ninfinite sequence of covariance coefficients. For any $T \\in \\mathbb N$, let \n$R_T = \\mathcal T( r_{-(T-1)},\\ldots, r_{T-1})$, a Hermitian nonnegative definite matrix. \nGiven $N = N(T) > 0$, consider the \nmatrix model \n\\begin{equation}\nV_T = [ v_{n,t} ]_{n,t = 0}^{N-1, T-1}  = W_T R_T^{1\/2}\n\\label{model1}\n\\end{equation}\nwhere $W_T = [ w_{n,t} ]_{n,t = 0}^{N-1, T-1}$ has independent ${\\cal CN}(0,1)$ entries. \nIt is clear that $r_k = \\mathbb{E} [v_{n,t+k} v_{n,t}^*]$  for any $t$, $k$, and $n \\in \\{0,\\ldots, N-1\\}$.\n\nIn the following, we shall make the two assumptions below.\n\\begin{assumption} \n\\label{ass-rk} \nThe covariance coefficients $r_k$ are absolutely summable and $r_0 \\neq 0$. \n\\end{assumption} \nWith this assumption, the covariance function \n\\[\n{\\boldsymbol\\Upsilon}(\\lambda) \\triangleq \n\\sum_{k=-\\infty}^\\infty r_k e^{-\\imath k\\lambda}, \\quad \n\\lambda \\in [0, 2\\pi) \n\\] \nis continuous on the interval $[0, 2\\pi]$. Since $\\| R_T \\| \\leq \\| \\boldsymbol\\Upsilon \\|_\\infty$ (see \\emph{e.g.} \\cite[Lemma 4.1]{Gray'06}), Assumption \\ref{ass-rk} \nimplies that $\\sup_T \\| R_T \\| < \\infty$.  \n\nWe assume the following asymptotic regime which will be simply denoted as ``$T\\rightarrow\\infty$'':\n\\begin{assumption} \n\\label{ass-regime} \n$T \\rightarrow \\infty$ and $N\/T \\rightarrow c > 0$.\n\\end{assumption} \n\n\nOur objective is to study the performance of two estimators of the \ncovariance function frequently considered in the literature. These estimators are defined as \n\\begin{align}\n\\label{est-b} \n\\hat{r}_{k,T}^b&= \n\\frac{1}{NT} \\sum_{n=0}^{N-1}\\sum_{t=0}^{T-1}\nv_{n,t+k} v_{n,t}^* \\mathbbm{1}_{0 \\leq t+k \\leq T-1} \\\\\n\\label{est-u} \n\\hat{r}_{k,T}^u&=\\frac{1}{N(T-|k|)}\n\\sum_{n=0}^{N-1}\\sum_{t=0}^{T-1} v_{n,t+k} v_{n,t}^* \n\\mathbbm{1}_{0 \\leq t+k \\leq T-1}.\n\\end{align}\nSince  $\\mathbb{E} \\hat{r}_{k,T}^b = ( 1 - |k|\/T ) r_k$ and $\\mathbb{E} \\hat{r}_{k,T}^u = r_k$,\nthe estimate $\\hat{r}_{k,T}^b$ is biased while $\\hat{r}_{k,T}^u$ is unbiased. \nLet also \n\\begin{align}\n\t\\label{est-Rb}\n\\widehat R^b_T &\\triangleq {\\cal T}\\left( \\hat{r}_{-(T-1),T}^b, \\ldots, \n\\hat{r}_{(T-1),T}^b \\right) \\\\\n\t\\label{est-Ru}\n\\widehat R^u_T &\\triangleq {\\cal T} \\left( \\hat{r}_{-(T-1),T}^u, \\ldots, \n\\hat{r}_{(T-1),T}^u \\right).\n\\end{align}\nA well known advantage of $\\widehat R^b_T$ over $\\widehat R^u_T$ as an estimate of $R_T$ is its structural nonnegative definiteness.\nIn this section, results on the spectral behavior of these matrices are provided under the form of concentration inequalities on $\\| \\widehat R^b_T - R_T \\|$ and $\\| \\widehat R^u_T - R_T \\|$: \n\\begin{theorem}\n\\label{th-biased} \nLet Assumptions~\\ref{ass-rk} and \\ref{ass-regime} hold true and let $\\widehat R^b_T$ be defined as in \\eqref{est-Rb}.\nThen, for any $x>0$,\n\\begin{equation*}\n\\mathbb{P} \\left[\\norme{\\widehat R_T^b - R_T} > x \\right] \\leq\n\\exp \\left( -cT \\left( \\frac{x}{\\| \\boldsymbol\\Upsilon \\|_\\infty} - \n \\log \\left( 1 + \\frac{x}{\\| \\boldsymbol\\Upsilon \\|_\\infty} \\right) + o(1) \\right)\n\\right)\n\\end{equation*}\nwhere $o(1)$ is with respect to $T$ and depends on $x$.\n\\end{theorem}\n\\begin{theorem}\n\\label{th-unbiased} \nLet Assumptions~\\ref{ass-rk} and \\ref{ass-regime} hold true and let $\\widehat R^u_T$ be defined as in \\eqref{est-Ru}.\nThen, for any $x>0$,\n\\begin{equation*}\n\\mathbb{P} \\left[\\norme{\\widehat R_T^u - R_T} > x \\right] \n\\leq \n\\exp \\left(- \\frac{cTx^2}{4\\norme{\\boldsymbol\\Upsilon}_{\\infty}^2\\log T} (1 + o(1)) \\right) \n\\end{equation*}\nwhere $o(1)$ is with respect to $T$ and depends on $x$.\n\\end{theorem}\n\nA consequence of these theorems, obtained by the Borel-Cantelli lemma, is that $\\| \\widehat R^b_T - R_T \\| \\to 0$ and \n$\\| \\widehat R^u_T - R_T \\| \\to 0$ almost surely as $T \\to \\infty$. \n\nThe slower rate of decrease of $T\/\\log(T)$ in the unbiased estimator exponent may be interpreted by the increased inaccuracy in the estimates of $r_k$ for values of $k$ close to $T-1$. \n\nWe now turn to the proofs of Theorems \\ref{th-biased} and \\ref{th-unbiased},\nstarting with some basic mathematical results that will be needed throughout \nthe proofs. \n\n\\subsection{Some basic mathematical facts} \n\n\\begin{lemma}\n\\label{lm-fq} \nFor $x,y \\in \\mathbb{C}^{m}$ and $A \\in \\mathbb{C}^{m\\times m}$,\n\\[\n\\left| x^{\\sf H} A x - y^{\\sf H} A y  \\right| \\leq \n\\norme{A} (\\norme{x}+\\norme{y})\\norme{x-y}.\n\\]\n\\end{lemma}\n\\begin{proof}\n\\begin{align*} \n\\left| x^{\\sf H} A x - y^{\\sf H} A y  \\right| &= \n\\left| x^{\\sf H} A x - y^{\\sf H} A x + y^{\\sf H} A x - y^{\\sf H} A y  \\right| \\\\\n&\\leq \\left| (x - y)^{\\sf H} A x \\right| + \\left| y^{\\sf H} A (x - y) \\right| \\\\\n&\\leq \\norme{A}(\\norme{x} + \\norme{y}) \\norme{x-y}.\n\\end{align*}\n\\end{proof}\n\n\\begin{lemma} \n\\label{chernoff} \nLet $X_0, \\ldots, X_{M-1}$ be independent $\\mathcal{CN}(0,1)$ random \nvariables. Then, for any $x > 0$, \n\\[\n\\mathbb{P} \\left[ \\frac1M \\sum_{m=0}^{M-1} (|X_m|^2 - 1) > x \\right] \n\\leq \\exp \\left( -M ( x - \\log(1+x) ) \\right) . \n\\] \n\\end{lemma}\n \n\\begin{proof} \nThis is a classical Chernoff bound. Indeed, given $\\xi \\in (0,1)$, we have\nby the Markov inequality \n\\begin{align*}\n\\mathbb{P} \\Bigl[ M^{-1} \\sum_{m=0}^{M-1} (|X_m|^2 - 1) > x \\Bigr] &= \\mathbb{P}\\left[ \\exp \\left( \\xi \\sum_{m=0}^{M-1} |X_m|^2 \\right) \n> \\exp \\xi M (x+1) \\right] \\\\\n&\\leq \\exp(-\\xi M (x+1)) \n\\mathbb{E} \\left[ \\exp \\left( \\xi \\sum_{m=0}^{M-1} |X_m|^2 \\right) \\right] \\\\\n&= \\exp \\left(- M \\left( \\xi(x+1) + \\log(1-\\xi) \\right) \\right) \n\\end{align*} \nsince $\\mathbb{E} \\left[\\exp (\\xi |X_m|^2) \\right] = 1\/(1-\\xi)$. The result follows upon \nminimizing this expression with respect to $\\xi$. \n\\end{proof}\n\n\\subsection{Biased estimator: proof of Theorem \\ref{th-biased}} \\label{biased}\nDefine\n\\begin{align*}\n\\widehat \\Upsilon^b_T(\\lambda) &\\triangleq  \n\\sum_{k=-(T-1)}^{T-1} \\hat{r}_{k,T}^b e^{\\imath k \\lambda} \n\\\\\n\\Upsilon_T(\\lambda) &\\triangleq  \n\\sum_{k=-(T-1)}^{T-1} r_k e^{\\imath k \\lambda}.\n\\end{align*} \nSince $\\widehat{R}_T^b-R_T$ is a Toeplitz matrix, from \\cite[Lemma 4.1]{Gray'06}, \n\\begin{equation*}\n\\norme{\\widehat{R}_T^b-R_T} \\leq \n\\underset{\\lambda\\in[0,2\\pi)}{\\text{sup}} \n\\left| \\widehat \\Upsilon_T^b(\\lambda) - \\Upsilon_T(\\lambda) \\right| \n\\leq \n\\underset{\\lambda\\in[0,2\\pi)}{\\text{sup}} \n\\left| \\widehat \\Upsilon_T^b(\\lambda) - \\mathbb{E} \\widehat \\Upsilon_T^b(\\lambda) \\right| +\n\\underset{\\lambda\\in[0,2\\pi)}{\\text{sup}} \n\\left| \\mathbb{E} \\widehat \\Upsilon_T^b(\\lambda) - \\Upsilon_T(\\lambda) \\right|. \n\\end{equation*}\nBy Kronecker's lemma (\\cite[Lemma 3.21]{Kallenberg'97}), the rightmost term at the right-hand side satisfies \n\\begin{equation}\n\\label{determ-biased} \n\\left| \\mathbb{E} \\widehat \\Upsilon_T^b(\\lambda) - \\Upsilon_T(\\lambda)\\right| \n\\leq  \\sum_{k=-(T-1)}^{T-1} \\frac{|k r_k|}{T}  \n\\xrightarrow[T\\to\\infty]{} 0. \n\\end{equation} \nIn order to deal with the term \n$\\sup_{\\lambda \\in [0,2\\pi)} | \\widehat \\Upsilon_T^b(\\lambda) - \\mathbb{E} \\widehat \\Upsilon_T^b(\\lambda) |$, \ntwo ingredients will be used. The first one is the \nfollowing lemma (proven in Appendix \\ref{anx-lm-qf}):\n\n\\begin{lemma}\n\\label{lemma_d_quad}\nThe following facts hold:\n\\begin{align*} \n\\widehat \\Upsilon_T^b(\\lambda) &= d_T(\\lambda)^{\\sf H}\\frac{V_T^{\\sf H}V_T}{N}d_T(\\lambda) \\\\\n\\mathbb{E} \\widehat \\Upsilon_T^b(\\lambda) &= d_T(\\lambda)^{\\sf H} R_T d_T(\\lambda)\n\\end{align*} \nwhere $d_T(\\lambda)=1\/\\sqrt{T}\\left[1, e^{- \\imath\\lambda}, \\ldots, \ne^{-\\imath(T-1)\\lambda} \\right]^{\\sf T}$.\n\\end{lemma}\n\nThe second ingredient is a Lipschitz property of the function \n$\\| d_T(\\lambda) - d_T(\\lambda') \\|$ seen as a function of $\\lambda$. \nFrom the inequality $|e^{-\\imath t\\lambda}-e^{-\\imath t\\lambda'}| \n\\leq t|\\lambda-\\lambda'|$, we indeed have\n\\begin{equation} \n\\label{lipschitz} \n\\| d_T(\\lambda) - d_T(\\lambda') \\| = \n\\sqrt{\\frac{1}{T} \\sum_{t=0}^{T-1} |e^{-\\imath t\\lambda}-\ne^{-\\imath t\\lambda'}|^2} \\leq \\frac{T|\\lambda-\\lambda'|}{\\sqrt{3}} . \n\\end{equation}\n\nNow, denoting by $\\lfloor \\cdot \\rfloor$ the floor function and choosing \n$\\beta > 2$, define ${\\cal I}=\\left\\{0, \\ldots, \\lfloor T^{\\beta} \\rfloor - 1 \\right\\}$.\nLet $\\lambda_i=2 \\pi \\frac{i}{\\lfloor T^{\\beta} \\rfloor }$, $i \\in {\\cal I}$, be a regular discretization of the \ninterval $[0, 2\\pi]$. \nWe write \n\\begin{align*}\n&\\underset{\\lambda \\in [0, 2\\pi)}{\\text{sup}} \\left| \\widehat \\Upsilon_T^b(\\lambda) - \\mathbb{E} \\widehat \\Upsilon_T^b(\\lambda) \\right| \n \\\\&\\leq \n\\underset{i \\in {\\cal I}}{\\text{max}} \n\\underset{\\lambda \\in [\\lambda_i,\\lambda_{i+1}]}{\\text{sup}} \\Bigl(\\left| \\widehat \\Upsilon_T^b(\\lambda) \n- \\widehat \\Upsilon_T^b(\\lambda_i) \\right| + \\left| \\widehat \\Upsilon_T^b(\\lambda_i) \n- \\mathbb{E} \\widehat \\Upsilon_T^b(\\lambda_i) \\right| + \\left| \\mathbb{E} \\widehat \\Upsilon_T^b(\\lambda_i) \n- \\mathbb{E} \\widehat \\Upsilon_T^b(\\lambda) \\right|\\Bigr)\\\\ &\\leq \n\\underset{i \\in {\\cal I}}{\\text{max}} \n\\underset{\\lambda \\in [\\lambda_i,\\lambda_{i+1}]}{\\text{sup}} \n\\left| \\widehat \\Upsilon_T^b(\\lambda) - \\widehat \\Upsilon_T^b(\\lambda_i) \\right| + \\underset{i \\in {\\cal I}}{\\text{max}} \\left| \\widehat \\Upsilon_T^b(\\lambda_i) \n- \\mathbb{E} \\widehat \\Upsilon_T^b(\\lambda_i) \\right| + \\underset{i \\in {\\cal I}}{\\text{max}} \n\\underset{\\lambda \\in [\\lambda_i,\\lambda_{i+1}]}{\\text{sup}} \n\\left| \\mathbb{E} \\widehat \\Upsilon_T^b(\\lambda_i) - \\mathbb{E} \\widehat \\Upsilon_T^b(\\lambda) \\right| \\\\ &\\triangleq \\chi_1 + \\chi_2 + \\chi_3 . \n\\end{align*} \nWith the help of Lemma \\ref{lemma_d_quad} and \\eqref{lipschitz}, \nwe shall provide concentration inequalities on the random terms $\\chi_1$ \nand $\\chi_2$ and a bound on the deterministic term $\\chi_3$. \nThis is the purpose of the three following lemmas. Herein and in the \nremainder, $C$ denotes a positive constant independent of $T$. This constant \ncan change from an expression to another. \n\n\\begin{lemma}\n\\label{chi1} \nThere exists a constant $C > 0$ such that for any $x > 0$ and any $T$ large\nenough, \n\\begin{equation*}\n\\mathbb{P} \\left[ \\chi_1 > x \\right]\n\\leq \\exp \\Biggl( - cT^2\\Biggl( \\frac{x T^{\\beta-2}}{C \\| \\boldsymbol\\Upsilon\\|_\\infty} \n- \\log \\frac{x T^{\\beta-2}}{C \\| \\boldsymbol\\Upsilon\\|_\\infty} - 1 \\Biggr) \\Biggr) . \n\\end{equation*}\n\\end{lemma} \n\n\\begin{proof}\nUsing Lemmas \\ref{lemma_d_quad} and \\ref{lm-fq} along with \n\\eqref{lipschitz}, we have \n\\begin{align*} \n\\left| \\widehat \\Upsilon_T^b(\\lambda) - \\widehat \\Upsilon_T^b(\\lambda_i) \\right|\n&= \\left| d_T(\\lambda)^{\\sf H}\\frac{V_T^{\\sf H} V_T}{N} d_T(\\lambda) \n        - d_T(\\lambda_i)^{\\sf H} \\frac{V_T^{\\sf H} V_T}{N}d_T(\\lambda_i) \\right| \\\\\n&\\leq 2 N^{-1} \\norme{d_T(\\lambda) - d_T(\\lambda_i)} \\| R_T\\| \\norme{W_T^{\\sf H} W_T} \\\\\n&\\leq C | \\lambda - \\lambda_i | \\| \\boldsymbol\\Upsilon\\|_\\infty \\norme{W_T^{\\sf H} W_T} . \n\\end{align*} \nFrom $\\| W_T^{\\sf H} W_T \\| \\leq \\tr(W_T^{\\sf H} W_T)$ and \nLemma~\\ref{chernoff}, assuming $T$ large enough so that $f(x,T) \\triangleq x T^{\\beta-1} \/ (C N \\| \\boldsymbol\\Upsilon\\|_\\infty)$ satisfies \n$f(x,T) \\geq 1$, we then obtain  \n\\begin{align*}\n\\mathbb{P} \\left[ \\chi_1 > x \\right] &\\leq \n\\mathbb{P} \\left[ \nC \\| \\boldsymbol\\Upsilon\\|_\\infty T^{-\\beta} \n\\sum_{t=0}^{T-1}\\sum_{n=0}^{N-1} |w_{n,t}|^2 > x \\right] \\\\\n&= \\mathbb{P} \\left[ \n\\frac{1}{NT} \\sum_{n,t} (| w_{n,t} |^2 -1 ) > f(x,T) - 1 \\right] \\\\\n&\\leq \\exp( - NT( f(x,T) - \\log f(x,T) - 1 ) ) . \n\\end{align*} \n\\end{proof}\n\n\\begin{lemma} \n\\label{chi2} \nThe following inequality holds\n\\begin{equation*}\n\\mathbb{P} \\left[ \\chi_2 > x \\right] \\leq\n2T^{\\beta} \\exp \\Biggl( - c T \\Biggl( \\frac{x}{\\|\\boldsymbol\\Upsilon\\|_\\infty} - \n\\log \\Bigl( 1 + \\frac{x}{\\|\\boldsymbol\\Upsilon\\|_\\infty} \\Bigr) \\Biggr) \\Biggr). \n\\end{equation*}\n\\end{lemma} \n\n\\begin{proof}\nFrom the union bound we obtain:\n\\begin{align*} \n\\mathbb{P} \\left[ \\chi_2 > x \\right]\n&\\leq \\sum_{i=0}^{\\lfloor T^{\\beta} \\rfloor - 1} \n\\mathbb{P} \\left[ \\left| \\widehat \\Upsilon_T^b(\\lambda_i) \n - \\mathbb{E} \\widehat \\Upsilon_T^b(\\lambda_i) \\right| > x \\right].\n\\end{align*} \nWe shall bound each term of the sum separately. Since \n\\begin{equation*}  \n\\mathbb{P} \\left[ \\left| \\widehat \\Upsilon_T^b(\\lambda_i) \n              - \\mathbb{E} \\widehat \\Upsilon_T^b(\\lambda_i) \\right| > x \\right] \n= \\mathbb{P} \\left[ \\widehat \\Upsilon_T^b(\\lambda_i) \n                 - \\mathbb{E} \\widehat \\Upsilon_T^b(\\lambda_i) > x \\right] +\n\\mathbb{P} \\left[ - \\left( \\widehat \\Upsilon_T^b(\\lambda_i) - \\mathbb{E} \\widehat \\Upsilon_T^b(\\lambda_i) \\right) > x \\right] \n\\end{equation*}\nit will be enough to deal with the first right-hand side term as the second \none is treated similarly.\nLet $\\eta_T(\\lambda_i) \\triangleq W_T q_T(\\lambda_i) = \n\\left[ \\eta_{0,T}(\\lambda_i), \\ldots, \\eta_{N-1,T}(\\lambda_i) \\right]^{\\sf T}$ \nwhere $q_T(\\lambda_i) \\triangleq R_T^{1\/2} d_T(\\lambda_i)$. Observe that \n$\\eta_{k,T}(\\lambda_i) \\sim \\mathcal{CN}(0,\\| q_T(\\lambda_i) \\|^2 I_N)$. We know from Lemma~\\ref{lemma_d_quad} that \n\\begin{equation}\n\\widehat \\Upsilon_T^b(\\lambda_i) - \\mathbb{E} \\widehat \\Upsilon_T^b(\\lambda_i) = \n\\frac 1N \\left( \\| \\eta_T(\\lambda_i) \\|^2 - \\mathbb{E} \\| \\eta_T(\\lambda_i) \\|^2 \\right).\n\\label{Epsilon_eta}\n\\end{equation}\nFrom (\\ref{Epsilon_eta}) and Lemma~\\ref{chernoff}, we therefore get  \n\\begin{equation*} \n\\mathbb{P}\\left[ \\widehat \\Upsilon_T^b(\\lambda_i) - \\mathbb{E} \\widehat \\Upsilon_T^b(\\lambda_i) > x \\right]\n\\leq \\exp \\Biggl( -N \\Biggl( \\frac{x}{\\| q_T(\\lambda_i) \\|^2} - \n \\log\\Bigl( 1 + \\frac{x}{\\| q_T(\\lambda_i) \\|^2} \\Bigr) \\Biggr) \\Biggr).  \n\\end{equation*} \nNoticing that $\\| q_T(\\lambda_i) \\|^2 \\leq \\|\\boldsymbol\\Upsilon\\|_\\infty$ and that the function $f(x) = x - \\log \\Bigl( 1 + x \\Bigr)$ is increasing for $x>0$, we get the result.\n\\end{proof} \n\nFinally, the bound for the deterministic term $\\chi_3$ is provided by the following lemma:\n\\begin{lemma}\n\\label{chi3} \n$\\displaystyle{ \n\\chi_3 \\leq C \\| \\boldsymbol\\Upsilon\\|_\\infty T^{-\\beta + 1}\n}$. \n\\end{lemma} \n\\begin{proof} \nFrom Lemmas \\ref{lemma_d_quad} and \\ref{lm-fq} along with \n\\eqref{lipschitz}, we obtain\n\\begin{align*} \n\\left| \\mathbb{E} \\widehat \\Upsilon_T^b(\\lambda) - \\mathbb{E} \\widehat \\Upsilon_T^b(\\lambda_i) \\right|\n&= \\left| d_T(\\lambda)^{\\sf H} R_T d_T(\\lambda) - d_T(\\lambda_i)^{\\sf H} R_T d_T(\\lambda_i) \\right| \\\\\n&\\leq 2 \\norme{R_T} \\norme{d_T(\\lambda) - d_T(\\lambda_i)} \\\\\n&\\leq C \\| \\boldsymbol\\Upsilon\\|_\\infty | \\lambda - \\lambda_i | T. \n\\end{align*} \nFrom $\\underset{i \\in {\\cal I}}{\\text{max}} \\underset{\\lambda \\in [\\lambda_i,\\lambda_{i+1}]}{\\text{sup}} \n| \\lambda - \\lambda_i | = \\lambda_{i+1} - \\lambda_i = T^{-\\beta}$ we get the result.\n\\end{proof}\n\nWe now complete the proof of Theorem \\ref{th-biased}. From \n\\eqref{determ-biased} and Lemma~\\ref{chi3}, we get\n\\begin{equation*}\n\\mathbb{P} \\left[\\norme{\\widehat R_T^b - R_T} > x \\right] = \n\\mathbb{P}\\left[ \\chi_1 + \\chi_2 > x + o(1) \\right]. \n\\end{equation*} \nGiven a parameter $\\epsilon_T \\in [0,1]$, we can write (with some slight notation abuse)\n\\begin{equation*}\n\\mathbb{P}\\left[ \\chi_1 + \\chi_2 > x + o(1) \\right] \\leq\n\\mathbb{P}\\left[ \\chi_1 > x \\epsilon_T \\right] + \\mathbb{P}\\left[\\chi_2 > x (1 - \\epsilon_T) + o(1) \\right].\n\\end{equation*}\nWith the results of Lemmas \\ref{chi1} and \\ref{chi2}, setting $\\epsilon_T=1\/T$, we get\n\\begin{align*}\n\\mathbb{P}\\left[ \\chi_1 + \\chi_2 > x + o(1) \\right] &\\leq\n\\mathbb{P}\\left[ \\chi_1 > \\frac{x}{T} \\right] + \\mathbb{P}\\left[ \\chi_2 > x (1 - \\frac{x}{T}) + o(1) \\right] \\\\\n&\\leq\n\\exp \\Bigl( - cT^2 \\Bigl( \\frac{x T^{\\beta-3}}{C \\| \\boldsymbol\\Upsilon\\|_\\infty} \n- \\log \\frac{x T^{\\beta-3}}{C \\| \\boldsymbol\\Upsilon\\|_\\infty} - 1 \\Bigr) \\Bigr) \\\\\n&+ \\exp \\Bigl( - cT \\Bigl( \\frac{x\\left( 1 - \\frac{1}{T} \\right)}{\\|\\boldsymbol\\Upsilon\\|_\\infty} - \n\\log \\Bigl( 1 + \\frac{x\\left( 1 - \\frac{1}{T} \\right)}{\\|\\boldsymbol\\Upsilon\\|_\\infty} \\Bigr) + o(1) \\Bigr) \\Bigr) \\\\\n&=\n\\exp \\Bigl( - cT \\Bigl( \\frac{x}{\\|\\boldsymbol\\Upsilon\\|_\\infty} - \n\\log \\Bigl( 1 + \\frac{x}{\\|\\boldsymbol\\Upsilon\\|_\\infty} \\Bigr) + o(1) \\Bigr) \\Bigr) \n\\end{align*}\nsince $\\beta>2$.\n\n\\subsection{Unbiased estimator: proof of Theorem \\ref{th-unbiased}}\\label{unbiased}\nThe proof follows basically the same main steps as for Theorem~\\ref{th-biased} \nwith an additional difficulty due to the scaling terms $1\/(T-|k|)$.\n\nDefining the function \n\\[ \n\\widehat \\Upsilon^u_T(\\lambda) \\triangleq  \n\\sum_{k=-(T-1)}^{T-1} \\hat{r}_{k,T}^u e^{ik \\lambda}\n\\]\nwe have\n\\begin{equation*} \n\\norme{\\widehat{R}_T^u-R_T} \\leq \n\\underset{\\lambda\\in[0,2\\pi)}{\\text{sup}} \n\\left| \\widehat \\Upsilon_T^u(\\lambda) - \\Upsilon_T(\\lambda) \\right| = \n\\underset{\\lambda\\in[0,2\\pi)}{\\text{sup}} \n\\left| \\widehat \\Upsilon_T^u(\\lambda) - \n    \\mathbb{E} \\widehat \\Upsilon_T^u(\\lambda) \\right|\n\\end{equation*} \nsince $\\Upsilon_T(\\lambda)=\\mathbb{E} \\widehat \\Upsilon_T^u(\\lambda)$, the estimates\n$\\hat{r}_{k,T}^u$ being unbiased. \n\nIn order to deal with the right-hand side of this expression, we need the \nfollowing analogue of Lemma~\\ref{lemma_d_quad}, borrowed from \n\\cite{val-loub-icassp14} and proven here in Appendix~\\ref{anx-lm-qf2}.\n\\begin{lemma}\n\\label{lemma_d_quad2}\nThe following fact holds:\n\\begin{align*}\n\\widehat \\Upsilon_T^u(\\lambda) &= d_T(\\lambda)^{\\sf H} \n\\left( \\frac{V_T^{\\sf H}V_T}{N} \\odot B_T \\right) d_T(\\lambda)\n\\end{align*} \nwhere $\\odot$ is the Hadamard product of matrices and where \n\\[\n\tB_T \\triangleq \\left[ \\frac{T}{T-|i-j|} \\right]_{0\\leq i,j\\leq T-1}.\n\\] \n\\end{lemma}\n\nIn order to make $\\widehat \\Upsilon_T^u(\\lambda)$ more tractable, we rely on the \nfollowing lemma which can be proven by direct calculation.\n\\begin{lemma}\n\\label{lemma_hadamard}\nLet $x$, $y \\in \\mathbb{C}^{m}$ and $A, B \\in C^{m \\times m}$. Then \n\\begin{equation*}\nx^{\\sf H}( A \\odot B ) y = \\tr (D_x^{\\sf H} A D_y B^{\\sf T}) \n\\end{equation*}\nwhere we recall $D_x = \\diag(x)$ and $D_y = \\diag(y)$.\n\\end{lemma}\n\nDenoting\n\\begin{align*}\nD_T(\\lambda) &\\triangleq \\diag (d_T(\\lambda)) = \\frac{1}{\\sqrt{T}} \\diag(1, e^{i\\lambda}, \\ldots, e^{i(T-1)\\lambda} ) \\\\\nQ_T(\\lambda) &\\triangleq R_T^{1\/2} D_T(\\lambda) B_T D_T(\\lambda)^{\\sf H} (R_T^{1\/2})^{\\sf H}\n\\end{align*}\nwe get from Lemmas~\\ref{lemma_d_quad2} and \\ref{lemma_hadamard} \n\\begin{align}\n\\label{Upsilon_sum} \n\\widehat \\Upsilon_T^u(\\lambda) &= \\frac1N\n\\tr(D_T(\\lambda)^{\\sf H} (R_T^{1\/2})^{\\sf H} W_T^{\\sf H} W_T R_T^{1\/2} D_T(\\lambda) B_T)\n\\nonumber\\\\ &= \\frac1N \\tr (W_T Q_T(\\lambda) W_T^{\\sf H}) \\nonumber \\\\\n&= \\frac1N\n\\sum_{n=0}^{N-1} w_n^{\\sf H} Q_T(\\lambda) w_n \n\\end{align} \nwhere $w_i^{\\sf H}$ is such that $W_T=[w_0^{\\sf H},\\ldots,w_{N-1}^{\\sf H}]$. \n\nCompared to the biased case, the main difficulty lies here in the fact \nthat the matrices $B_T\/T$ and $Q_T(\\lambda)$ have unbounded spectral norm as $T\\to\\infty$. \nThe following lemma, proven in Appendix~\\ref{prf-lm-B-Q}, provides some information on the spectral behavior of these\nmatrices that will be used subsequently.  \n\\begin{lemma}\n\\label{lm-B-Q} \nThe matrix $B_T$ satisfies \n\\begin{equation}\n\\label{norme_B} \n\\norme{B_T} \\leq \\sqrt{2} T( \\sqrt{\\log T} + C). \n\\end{equation} \nFor any $\\lambda \\in[0, 2\\pi)$, the eigenvalues \n$\\sigma_0, \\ldots, \\sigma_{T-1}$ of the matrix $Q(\\lambda)$ satisfy the \nfollowing inequalities: \n\\begin{eqnarray}\n\\sum_{t=0}^{T-1} \\sigma_t^2 &\\leq& 2 \\norme{\\boldsymbol\\Upsilon}_{\\infty}^2 \\log T + C \\label{sum_sigma2}\\\\ \n\\underset{t}{\\max} |\\sigma_t| &\\leq& \\sqrt{2} \\| \\boldsymbol \\Upsilon \\|_\\infty \n( \\log T )^{1\/2} + C \\label{sig_max} \\\\\n\\sum_{t=0}^{T-1} |\\sigma_t|^3 &\\leq& C ((\\log T)^{3\/2} +1)\\label{sum_sigma3}\n\\end{eqnarray}\nwhere the constant $C$ is independent of $\\lambda$.\n\\end{lemma} \n\nWe shall also need the following easily shown Lipschitz property of the \nfunction $\\norme{D_T(\\lambda) - D_T(\\lambda')}$:  \n\\begin{equation} \n\\label{lipschitz_D} \n\\| D_T(\\lambda) - D_T(\\lambda') \\| \\leq \\sqrt{T}|\\lambda - \\lambda'|. \n\\end{equation}\n\nWe now enter the core of the proof of Theorem~\\ref{th-unbiased}.  \nChoosing $\\beta>2$, let $\\lambda_i=2 \\pi \\frac{i}{\\lfloor T^{\\beta}\\rfloor}$, \n$i \\in {\\cal I}$, be a regular discretization of the interval $[0, 2\\pi]$ with \n${\\cal I}=\\left\\{0, \\ldots, \\lfloor T^{\\beta} \\rfloor - 1 \\right\\}$. We write \n\\begin{align*}\n\\underset{\\lambda \\in [0, 2\\pi)}{\\text{sup}} \n\\left| \\widehat \\Upsilon_T^u(\\lambda) - \\mathbb{E} \\widehat \\Upsilon_T^u(\\lambda) \\right|  \n&\\leq \\underset{i \\in {\\cal I}}{\\text{max}} \n\\underset{\\lambda \\in [\\lambda_i,\\lambda_{i+1}]}{\\text{sup}} \n\\left| \\widehat \\Upsilon_T^u(\\lambda)-\\widehat \\Upsilon_T^u(\\lambda_i) \\right| + \\underset{i \\in {\\cal I}}{\\text{max}} \\left| \\widehat \\Upsilon_T^u(\\lambda_i) - \\mathbb{E} \\widehat \\Upsilon_T^u(\\lambda_i) \\right|\\\\& + \\underset{i \\in {\\cal I}}{\\text{max}} \\underset{\\lambda \\in [\\lambda_i,\\lambda_{i+1}]}{\\text{sup}} \n\\left| \\mathbb{E} \\widehat \\Upsilon_T^u(\\lambda_i) - \\mathbb{E} \\widehat \\Upsilon_T^u(\\lambda) \\right| \\\\ \n&\\triangleq \\chi_1 + \\chi_2 + \\chi_3 . \n\\end{align*} \n\nOur task is now to provide concentration inequalities on the random terms \n$\\chi_1$ and $\\chi_2$ and a bound on the deterministic term $\\chi_3$.\n\n\\begin{lemma}\n\\label{chi1-u} \nThere exists a constant $C > 0$ such that, if $T$ is large enough, the following \ninequality holds:\n\\begin{equation*}\n\\mathbb{P} \\left[ \\chi_1 > x \\right]\n\\leq \\exp \\left( - cT^2 \\left( \\frac{x T^{\\beta-2}}{C \\sqrt{\\log T}} \n- \\log \\frac{x T^{\\beta-2}}{C \\sqrt{\\log T}} - 1 \\right) \\right). \n\\end{equation*} \n\\end{lemma} \n\n\\begin{proof}\nFrom Equation~\\eqref{Upsilon_sum}, we have\n\\begin{align} \n\\left| \\widehat \\Upsilon_T^u(\\lambda) - \\widehat \\Upsilon_T^u(\\lambda_i) \\right|\n& = \\frac1N \\left| \\sum_{n=0}^{N-1} w_n^{\\sf H}\\left( Q_T(\\lambda) - Q_T(\\lambda_i) \\right) w_n \\right| \\nonumber \\\\\n& \\leq \\frac1N \\sum_{n=0}^{N-1} \\left| w_n^{\\sf H} \\left( Q_T(\\lambda) - Q_T(\\lambda_i) \\right) w_n \\right| \\nonumber\\\\\n& \\leq \\frac1N \\norme{Q_T(\\lambda) - Q_T(\\lambda_i)} \\sum_{n=0}^{N-1} \\norme{w_n}^2 \\nonumber.\n\\end{align}\nThe norm above further develops as\n\\begin{align*} \n&\\norme{Q_T(\\lambda) - Q_T(\\lambda_i)}\\\\\n& \\leq \\norme{R_T} \\Vert D_T(\\lambda)B_TD_T(\\lambda)^{\\sf H}- \nD_T(\\lambda_i)B_TD_T(\\lambda)^{\\sf H} + D_T(\\lambda_i)B_TD_T(\\lambda)^{\\sf H} - \nD_T(\\lambda_i)B_TD_T(\\lambda_i)^{\\sf H} \\Vert \\\\\n&\\leq 2 \\norme{D_T(\\lambda)} \\norme{R_T} \\norme{B_T} \n\\norme{D_T(\\lambda) - D_T(\\lambda_i)}  \\leq C T ( \\sqrt{\\log T}  + 1)  \\left| \\lambda - \\lambda_i \\right| \n\\end{align*}\nwhere we used \\eqref{norme_B}, \\eqref{lipschitz_D}, and $\\norme{D_T(\\lambda)}=1\/\\sqrt{T}$.\nUp to a change in $C$, we can finally write $\\norme{Q_T(\\lambda) - Q_T(\\lambda_i)} \\leq \nC T^{1-\\beta} \\sqrt{\\log T}$. Assume that $f(x,T) \\triangleq xT^{\\beta-2}\/\n\\left( C \\sqrt{\\log T} \\right)$ \nsatisfies $f(x,T) > 1$ (always possible for every fixed $x$ by taking $T$ large). Then we get by Lemma~\\ref{chernoff} \n\\begin{align*} \n\\mathbb{P} \\left[ \\chi_1 > x \\right] & \\leq \\mathbb{P} \\left( C N^{-1} T^{1-\\beta} \n\\sqrt{\\log T} \\sum_{n,t} \\left|w_{n,t}\\right|^2 > x \\right) \\\\ \n&= \\mathbb{P} \\left( \\frac{1}{NT} \\sum_{n,t} (\\left|w_{n,t}\\right|^2 - 1) \n> f(x,T) - 1 \\right) \\\\\n&\\leq\n\\exp \\left( - NT \\left( f(x,T) - \\log \\left(f(x,T)\\right) - 1 \\right) \\right).\n\\end{align*}\n\\end{proof}\n\nThe most technical part of the proof is to control the term $\\chi_2$, which we handle hereafter.\n\n\\begin{lemma}\n\\label{chi2-u} \nThe following inequality holds:\n\\begin{equation*}\n\\mathbb{P} \\left[ \\chi_2 > x \\right]\n\\leq \\exp \\left(- \\frac{cx^2T}{4\\norme{\\boldsymbol\\Upsilon}_{\\infty}^2\\log T} (1 + o(1)) \\right).\n\\end{equation*}\n\\end{lemma}\n\n\\begin{proof}\nFrom the union bound we obtain:\n\\begin{equation} \\label{union_bound}\n\\mathbb{P} \\left[ \\chi_2 > x \\right]\n\\leq \\sum_{i=0}^{\\lfloor T^{\\beta} \\rfloor - 1} \n\\mathbb{P} \\left[ \\left| \\widehat \\Upsilon_T^u(\\lambda_i) \n - \\mathbb{E} \\widehat \\Upsilon_T^u(\\lambda_i) \\right| > x \\right].\n\\end{equation} \nEach term of the sum can be written\n\\begin{equation*}\n\\mathbb{P} \\left[ \\left| \\widehat \\Upsilon_T^u(\\lambda_i) - \\mathbb{E} \\widehat \\Upsilon_T^u(\\lambda_i) \\right| > x \\right] = \\mathbb{P} \\left[\\widehat \\Upsilon_T^u(\\lambda_i) - \\mathbb{E} \\widehat \\Upsilon_T^u(\\lambda_i) > x \\right] + \\mathbb{P} \\left[- \\left(\\widehat \\Upsilon_T^u(\\lambda_i) - \\mathbb{E} \\widehat \\Upsilon_T^u(\\lambda_i) \\right) > x \\right].\n\\end{equation*}\nWe will deal with the term $\\psi_i = \\mathbb{P} \\left[\\widehat \\Upsilon_T^u(\\lambda_i) - \\mathbb{E} \\widehat \\Upsilon_T^u(\\lambda_i) > x \\right]$, the term $\\mathbb{P} \\left[- \\left(\\widehat \\Upsilon_T^u(\\lambda_i) - \\mathbb{E} \\widehat \\Upsilon_T^u(\\lambda_i) \\right) > x \\right]$ being treated similarly.\nLet $Q_T(\\lambda_i) = U_T \\Sigma_T U_T^{\\sf H}$ be a spectral \nfactorization of the Hermitian matrix $Q_T(\\lambda_i)$ with \n$\\Sigma_T =\\diag (\\sigma_{0},\\ldots,\\sigma_{T-1})$. \nSince $U_T$ is unitary and $W_T$ has independent ${\\cal CN}(0,1)$ elements,\nwe get from Equation \\eqref{Upsilon_sum} \n\\begin{equation}\n\\label{Ups_u}\n\\widehat \\Upsilon_T^u(\\lambda_i) \\stackrel{\\cal L}{=}  \n\\frac{1}{N}\\sum_{n=0}^{N-1} w_n^{\\sf H} \\Sigma_T(\\lambda_i) w_n \n= \\frac{1}{N}\\sum_{n=0}^{N-1} \\sum_{t=0}^{T-1} |w_{n,t}|^2 \\sigma_t\n\\end{equation}\nwhere $\\stackrel{\\cal L}{=}$ denotes equality in law. \nSince $\\mathbb{E} \\left[ e^{a|X|^2} \\right]= 1\/(1-a)$ when $X \\sim {\\cal CN}(0, 1)$ \nand $0<a<1$, we have by Markov's inequality and from the independence of the\nvariables $|w_{n,t}|^2$\n\\begin{align}\n\\psi_i &= \\mathbb{P} \\left( \n\\frac{1}{N}\\sum_{n=0}^{N-1} \\sum_{t=0}^{T-1} |w_{n,t}|^2 \\sigma_t \n        - \\tr Q_T(\\lambda_i) > x \\right) \\nonumber \\\\ \n&\\leq \\mathbb{E}\\left[\\text{exp}\\Bigl( \n\\frac{\\tau}{N}\\sum_{n,t} |w_{n,t}|^2\\sigma_t \\Bigr) \\right]\n\\exp \\Bigl(-\\tau\\Bigl(x + \\sum_{t=0}^{T-1}\\sigma_t \\Bigr)\\Bigr) \\nonumber \\\\ \n&= \n\\exp \\Bigl( -\\tau \\Bigl(x+ \\sum_{t=0}^{T-1} \\sigma_t \\Bigr) \\Bigr) \n\\prod_{t=0}^{T-1} \\Bigl( 1- \\frac{\\sigma_t\\tau}{N} \\Bigr)^{-N} \n\\label{pro} \\\\\n&= \\exp\\Bigl(-\\tau \\Bigl(x+ \\sum_{t=0}^{T-1} \\sigma_t \\Bigr) \n  - N \\sum_{t=0}^{T-1}  \\log \\Bigl(1-\\frac{\\sigma_t\\tau}{N} \\Bigr) \\Bigr) \n\\nonumber\n\\end{align}\nfor any $\\tau$ such that $0 \\leq \\tau < \\underset{0\\leq t\\leq T-1}{\\text{min}}\\frac{N}{\\sigma_t}$.\nWriting $\\log(1-x) = - x - \\frac{x^2}{2} + R_3(x)$ with $\\left| R_3(x) \\right| \\leq \\frac{|x|^3}{3(1-\\epsilon)^3}$ when $|x|<\\epsilon<1$, we get\n\\begin{align}\n\\psi_i &\\leq\n\\exp \\Bigl(-\\tau x + N \\sum_{t=0}^{T-1}\\Bigl( \\frac{\\sigma_t^2\\tau^2}{2N^2} \n+  R_3 \\Bigl(\\frac{\\sigma_t\\tau}{N} \\Bigr) \\Bigr) \\Bigr) \\nonumber \\\\\n&\\leq \\exp \\Bigl(-N \\Bigl( \\frac{\\tau x}{N} - \\frac{\\tau^2}{2N^2} \\sum_{t=0}^{T-1} \\sigma_t^2 \\Bigr)  \\Bigr) \\exp  \\Bigl( N \\sum_{t=0}^{T-1} \\Bigl| R_3 \\Bigl(\\frac{\\sigma_t \\tau}{N} \\Bigr) \\Bigr|  \\Bigr) \\label{expr}.\n\\end{align}  \nWe shall manage this expression by using Lemma~\\ref{lm-B-Q}. In order to \ncontrol the term $\\exp(N \\sum |R_3(\\cdot)|)$, we make the choice\n\\begin{equation*}\n\\tau = \\frac{axT}{\\log T}\n\\end{equation*}\nwhere $a$ is a parameter of order one to be optimized later.\nFrom \\eqref{sig_max} we get \n$\\max_t\\frac{\\sigma_t \\tau}{N} = O \\left( (\\log T)^{-1\/2} \\right)$. Hence, \nfor all $T$ large, $\\tau < \\min_t \\frac{N}{\\sigma_t}$. Therefore, \\eqref{pro} \nis valid for this choice of $\\tau$ and for $T$ large. Moreover, for $\\epsilon$ \nfixed and $T$ large, $\\frac{\\sigma_t \\tau}{N} < \\epsilon <1$ so that for \nthese $T$\n\\begin{equation*}\nN \\sum_{t=0}^{T-1} \\Bigl| R_3 \\Bigl(\\frac{\\sigma_t\\tau}{N} \\Bigr) \\Bigr| \n\\leq \\frac{a^3T^3x^3}{3N^2(1-\\epsilon)^3 (\\log T)^3} \\sum_{t=0}^{T-1} |\\sigma_t|^3 \n= O \\left( {T}{(\\log T)^{-3\/2}}\\right) \n\\end{equation*}\nfrom (\\ref{sum_sigma3}).\nPlugging the expression of $\\tau$ in (\\ref{expr}), we get \n\\begin{equation*}\n\\psi_i \\leq \\exp \\Bigl(-N \\Bigl( \\frac{a T x^2}{(\\log T)N} - \\frac{a^2T^2x^2}{2N^2(\\log T)^2} \\sum_{t=0}^{T-1} \\sigma_t^2 \\Bigr)  \\Bigr) \\exp \\Bigl( C \\Bigl( {T}{(\\log T)^{-3\/2}} \\Bigr) \\Bigr) . \n\\end{equation*}\nUsing (\\ref{sum_sigma2}), we have\n\\begin{equation*}\n\\psi_i \\leq \\exp \\Bigl(-\\frac{x^2T}{\\log T} \n\\Bigl(a - \\frac{\\norme{\\boldsymbol\\Upsilon}_{\\infty}^2a^2T}{N} \\Bigr) \\Bigr) \n\\exp\\Bigl( \\frac{CT}{(\\log T)^{3\/2}} \\Bigr). \n\\end{equation*}\nThe right hand side term is minimized for $a=\\frac{N}{2T\\norme{\\boldsymbol\\Upsilon}_{\\infty}^2}$ which finally gives\n\\begin{equation*}\n\\psi_i \n\\leq \\exp \\Bigl(- \\frac{Nx^2}{4\\norme{\\boldsymbol\\Upsilon}_{\\infty}^2\\log T} (1 + o(1)) \\Bigr).\n\\end{equation*}\nCombining the above inequality with (\\ref{union_bound}) (which induces additional $o(1)$ terms in the argument of the exponential) concludes the lemma.\n\\end{proof}\n\n\\begin{lemma}\n\\label{chi3-u} \n$\\displaystyle{ \n\\chi_3 \\leq C T^{-\\beta+2} \\sqrt{\\log T}\n}$. \n\\end{lemma}\n\n\\begin{proof}\nFrom Lemma~\\ref{lemma_d_quad2}, $\\norme{R_T \\odot B_T} \\leq \\norme{R_T}\\norme{B_T}$ (see \\cite[Theorem 5.5.1]{HornJoh'91}), and \\eqref{lipschitz}, we get:\n\\begin{equation*}\n\\left| \\mathbb{E} \\widehat \\Upsilon_T^u(\\lambda_i) - \\mathbb{E} \\widehat \\Upsilon_T^u(\\lambda)  \\right| \n\\leq 2 \\norme{d_T(\\lambda) - d_T(\\lambda_i)} \\norme{R_T} \\norme{B_T}\n\\leq C T^2 \\left| \\lambda - \\lambda_{i} \\right| \\| \\boldsymbol\\Upsilon\\|_\\infty \\sqrt{\\log T}.\n\\end{equation*}\n\\end{proof}\nLemmas \\ref{chi1-u}--\\ref{chi3-u} show that $\\mathbb{P}[\\chi_2 > x]$ dominates the\nterm $\\mathbb{P}[\\chi_1 > x]$ and that the term $\\chi_3$ is vanishing. Mimicking the\nend of the proof of Theorem \\ref{th-biased}, we obtain Theorem \n\\ref{th-unbiased}. \n\nWe conclude this section by an empirical evaluation by Monte Carlo simulations of $\\mathbb{P}[\\Vert {\\widehat R_T - R_T}\\Vert > x]$ (curves labeled Biased and Unbiased), with $\\widehat R_T\\in\\{\\widehat R_T^b,\\widehat R_T^u\\}$, $T=2N$, $x=2$. This is shown in Figure~\\ref{det} against the theoretical exponential bounds of Theorems~\\ref{th-biased} and \\ref{th-unbiased} (curves labeled Biased theory and Unbiased theory). We observe that the rates obtained in Theorems~\\ref{th-biased} and \\ref{th-unbiased} are asymptotically close to optimal.\n\n\\begin{figure}[H]\n\\center\n\t\\begin{tikzpicture}[font=\\footnotesize]\n\t\t\\renewcommand{\\axisdefaulttryminticks}{2} \n\t\t\\pgfplotsset{every axis\/.append style={mark options=solid, mark size=2pt}}\n\t\t\\tikzstyle{every major grid}+=[style=densely dashed]       \n\t\t\t     \n\t\t\\pgfplotsset{every axis legend\/.append style={fill=white,cells={anchor=west},at={(0.99,0.02)},anchor=south east}}    \n\t\t\\tikzstyle{every axis y label}+=[yshift=-10pt] \n\t\t\\tikzstyle{every axis x label}+=[yshift=5pt]\n\t\t\\begin{axis}[\n\t\t\t\tgrid=major,\n\t\t\t\n\t\t\t\txlabel={$N$},\n\t\t\t\tylabel={$T^{-1} \\log \\left( \\mathbb P \\left[ \\norme {\\widehat R_T - R_T} > x \\right] \\right)$},\n\t\t\t    ytick={-0.2,-0.15,-0.1,-0.05,0},\n                yticklabels = {$-0.2$,$-0.15$,$-0.1$,$-0.05$,$0$},\n\t\t\t\n               \n\t\t\t\n\t\t\t\txmin=10,\n\t\t\t\txmax=40, \n\t\t\t\tymin=-0.2, \n\t\t\t\tymax=0,\n      width=0.7\\columnwidth,\n      height=0.5\\columnwidth\n\t\t\t]\n\n      \\addplot[smooth,black,densely dashed,line width=0.5pt,mark=star] plot coordinates{\n(10.000000,-0.052766)(12.000000,-0.051276)(14.000000,-0.050322)(16.000000,-0.049673)(18.000000,-0.049212)(20.000000,-0.048872)(22.000000,-0.048614)(24.000000,-0.048414)(26.000000,-0.048255)(28.000000,-0.048127)(30.000000,-0.048023)(32.000000,-0.047936)(34.000000,-0.047864)(36.000000,-0.047803)(38.000000,-0.047750)(40.000000,-0.047705)\n\n\n\n      };\n      \\addplot[smooth,black,line width=0.5pt,mark=star] plot coordinates{\n(10.000000,-0.178687)(12.000000,-0.154996)(14.000000,-0.137000)(16.000000,-0.122875)(18.000000,-0.112712)(20.000000,-0.1042589)(22.000000,-0.096377)(24.000000,-0.091108)(26.000000,-0.085960)(28.000000,-0.082679)(30.000000,-0.080274)(32.000000,-0.077053)(34.000000,-0.074633)(36.000000,-0.071135)(38.000000,-0.069311)(40.000000,-0.067182)\n\n\n\n\n      };\n      \\addplot[smooth,black,densely dashed,line width=0.5pt,mark=o] plot coordinates{\n(10.000000,-0.011823)(12.000000,-0.010787)(14.000000,-0.010070)(16.000000,-0.009540)(18.000000,-0.009129)(20.000000,-0.008799)(22.000000,-0.008526)(24.000000,-0.008295)(26.000000,-0.008097)(28.000000,-0.007924)(30.000000,-0.007771)(32.000000,-0.007635)(34.000000,-0.007512)(36.000000,-0.007401)(38.000000,-0.007300)(40.000000,-0.007206)\n\n\n\n      };\n      \n      \\addplot[smooth,black,line width=0.5pt,mark=o] plot coordinates{\n(10.000000,-0.063852)(12.000000,-0.050732)(14.000000,-0.041205)(16.000000,-0.034622)(18.000000,-0.029335)(20.000000,-0.025196)(22.000000,-0.021541)(24.000000,-0.018643)(26.000000,-0.015748)(28.000000,-0.013602)(30.000000,-0.012069)(32.000000,-0.010916)(34.000000,-0.010038)(36.000000,-0.007765)(38.000000,-0.007908)(40.000000,-0.007151)\n\n\n\n      };\n\n      \\legend{ {Biased theory},{Biased},{Unbiased theory},{Unbiased}}\n      \\end{axis}\n\t\\end{tikzpicture}\n\\caption{Error probability of the spectral norm for $x=2$, $c=0.5$, $[R_T]_{k,l}=a^{|k-l|}$ with $a=0.6$.}\n\\label{det}\n\\end{figure}\n\n\\section{Covariance matrix estimators for the \\\\ ``Signal plus Noise'' model} \n\\label{sig+noise} \n\n\\subsection{Model, assumptions, and results}\n\\label{subsec-model-perturbed} \n\nConsider now the following model:\n\\begin{equation}\\label{model2}\n\tY_T = [y_{n,t}]_{\\substack{0\\leq n\\leq N-1 \\\\0 \\leq t\\leq T-1}} =P_T+V_T\n\\end{equation}\nwhere the $N\\times T$ matrix $V_T$ is defined in \\eqref{model1} and where $P_T$ satisfies the \nfollowing assumption: \n\\begin{assumption} \n\\label{ass-signal} \n$P_T \\triangleq \\boldsymbol h_T \\boldsymbol s_T^{\\sf H} \\Gamma_T^{1\/2}$ where $\\boldsymbol h_T\\in \\mathbb{C}^N$ is a \ndeterministic vector such that\n$\\sup_T \\| \\boldsymbol h_T \\| < \\infty$, the vector \n$\\boldsymbol s_T = (s_0, \\ldots, s_{T-1})^{\\sf T} \\in \\mathbb{C}^T$ is a random vector\nindependent of $W_T$ with the distribution ${\\cal CN}(0, I_T)$, and \n$\\Gamma_T = [\\gamma_{ij} ]_{i,j=0}^{T-1}$ is Hermitian nonnegative such that $\\sup_T \\| \\Gamma_T \\| < \\infty$. \n\\end{assumption}\n\nWe have here a model for a rank-one signal corrupted with a Gaussian \nspatially white and temporally correlated noise with stationary temporal\ncorrelations. Observe that the signal can also be temporally correlated. \nOur purpose is still to estimate the noise correlation matrix\n$R_T$. To that end, we use one of the estimators \\eqref{est-b} or \\eqref{est-u} \nwith the difference that the samples $v_{n,t}$ are simply replaced with the \nsamples $y_{n,t}$. It turns out that these estimators are still consistent in \nspectral norm. Intuitively, $P_T$ does not break the \nconsistence of these estimators as it can be seen as a rank-one \nperturbation of the noise term $V_T$ in which the subspace spanned by \n$(\\Gamma^{1\/2})^{\\sf H} \\boldsymbol s_T$ is ``delocalized'' enough so as not to perturb \nmuch the estimators of $R_T$. In fact, we even have the following strong result.\n\\begin{theorem}\n\\label{th-perturb} \nLet $Y_T$ be defined as in \\eqref{model2} and let Assumptions~\\ref{ass-rk}--\\ref{ass-signal} hold. Define the\nestimates \n\\begin{align*}\n\\hat{r}_{k,T}^{bp}&= \n\\frac{1}{NT} \\sum_{n=0}^{N-1}\\sum_{t=0}^{T-1}\ny_{n,t+k} y_{n,t}^* \\mathbbm{1}_{0 \\leq t+k \\leq T-1} \\\\\n\\hat{r}_{k,T}^{up}&= \n\\frac{1}{N(T-|k|)} \\sum_{n=0}^{N-1}\\sum_{t=0}^{T-1}\ny_{n,t+k} y_{n,t}^* \\mathbbm{1}_{0 \\leq t+k \\leq T-1}\n\\end{align*}\nand let \n\\begin{align*}\n\\widehat R_T^{bp} &= {\\cal T}(\\hat{r}_{-(T-1),T}^{bp},\\ldots,\\hat{r}_{(T-1),T}^{bp}) \\\\\n\\widehat R_T^{up} &= {\\cal T}(\\hat{r}_{-(T-1),T}^{up},\\ldots,\\hat{r}_{(T-1),T}^{up}).\n\\end{align*}\nThen for any $x > 0$, \n\\begin{equation*} \n\\mathbb{P} \\left[\\norme{\\widehat R_T^{bp} - R_T} > x \\right]\n\\leq \n\\exp \\Bigl( -cT \\Bigl( \\frac{x}{\\| \\boldsymbol\\Upsilon \\|_\\infty} - \n \\log \\Bigl( 1 + \\frac{x}{\\| \\boldsymbol\\Upsilon \\|_\\infty} \\Bigr) + o(1) \\Bigr) \n\\Bigr) \n\\end{equation*} \nand \n\\begin{equation*} \n\\mathbb{P} \\left[\\norme{\\widehat R_T^{up} - R_T} > x \\right]\n\\leq \n\\exp \\Bigl(- \\frac{cTx^2}{4\\norme{\\boldsymbol\\Upsilon}_{\\infty}^2\\log T} (1 + o(1)) \n\\Bigr).  \n\\end{equation*} \n\\end{theorem}\n\nBefore proving this theorem, some remarks are in order.\n\\begin{remark}\nTheorem \\ref{th-perturb} generalizes without difficulty to the case where $P_T$ has a fixed rank $K>1$. This captures the situation of $K\\ll \\min(N,T)$ sources.\n\\end{remark} \n\\begin{remark}\n\tSimilar to the proofs of Theorems \\ref{th-biased} and \\ref{th-unbiased}, the proof of Theorem~\\ref{th-perturb} uses concentration inequalities for functionals of Gaussian random variables based on the moment generating function and the Chernoff bound. Exploiting instead McDiarmid's concentration inequality \\cite{ledoux}, it is possible to adapt Theorem~\\ref{th-perturb} to $\\boldsymbol s_T$ with bounded (instead of Gaussian) entries. This adaptation may account for discrete sources met in digital communication signals. \n\\end{remark} \n\n\\subsection{Main elements of the proof of Theorem \\ref{th-perturb}} \n\nWe restrict the proof to the more technical part that concerns $\\widehat R^{up}_T$. \nDefining\n\\begin{eqnarray*}\n\\widehat \\Upsilon_T^{up}(\\lambda) \\triangleq \\sum_{k=-(T-1)}^{T-1} \\hat{r}_{k,T}^{up} e^{ik \\lambda}\n\\end{eqnarray*}\nand recalling that $\\Upsilon_T(\\lambda) = \\sum_{k=-(T-1)}^{T-1} r_{k} e^{ik \\lambda}$, \nwe need to establish a concentration inequality on \\linebreak\n$\\mathbb{P} \\left[ \\sup_{\\lambda\\in[0,2\\pi)} | \\widehat \\Upsilon_T^{up}(\\lambda) - \n\\Upsilon_T(\\lambda) | > x \\right]$. For any $\\lambda\\in[0,2\\pi)$, the term \n$\\widehat \\Upsilon_T^{up}(\\lambda)$ can be written as \n(see Lemma~\\ref{lemma_d_quad2})  \n\\begin{align*}\n\\widehat \\Upsilon_T^{up}(\\lambda)&= \nd_T(\\lambda)^{\\sf H} \\left(\\frac{Y_T^{\\sf H}Y_T}{N} \\odot B_T \\right)\nd_T(\\lambda)  \\\\\n&= \nd_T(\\lambda)^{\\sf H} \\left(\\frac{V_T^{\\sf H}V_T}{N} \\odot B_T \\right)\nd_T(\\lambda) \\\\\n&+ \nd_T(\\lambda)^{\\sf H} \\left(\\frac{P_T^{\\sf H}V_T+V_T^{\\sf H}P_T}{N} \\odot B_T \n\\right) d_T(\\lambda) \\\\\n&+  d_T(\\lambda)^{\\sf H} \\left(\\frac{P_T^{\\sf H}P_T}{N} \\odot B_T \\right)\nd_T(\\lambda)  \\\\\n&\\triangleq \\widehat \\Upsilon_T^{u}(\\lambda) + \n\\widehat \\Upsilon_T^{cross}(\\lambda) + \\widehat \\Upsilon_T^{sig}(\\lambda) \n\\end{align*}  \nwhere $B_T$ is the matrix defined in the statement of \nLemma~\\ref{lemma_d_quad2}.\nWe know from the proof of Theorem~\\ref{th-unbiased} that \n\\begin{equation} \n\\label{noise-term} \n\\mathbb{P} \\left[\\sup_{\\lambda\\in[0,2\\pi)} \n| \\widehat \\Upsilon_T^{u}(\\lambda) - \\Upsilon_T(\\lambda) | > x \\right]\n\\leq \n\\exp \\left(- \\frac{cTx^2}{4\\norme{\\boldsymbol\\Upsilon}_{\\infty}^2\\log T} (1 + o(1)) \n\\right) .  \n\\end{equation} \nWe then need only handle the terms $\\widehat \\Upsilon_T^{cross}(\\lambda)$ and \n$\\widehat \\Upsilon_T^{sig}(\\lambda)$. \n\nWe start with a simple lemma. \n\\begin{lemma}\n\\label{lm-prod-gauss}\nLet $X$ and $Y$ be two independent ${\\cal N}(0,1)$ random variables. Then for \nany $\\tau \\in(-1,1)$, \n\\[\n\\mathbb{E}[\\exp(\\tau XY)] = (1 - \\tau^2)^{-1\/2}.\n\\]\n\\end{lemma}\n\\begin{proof}\n\\begin{align*}\n\\mathbb{E}[\\exp(\\tau XY)] &= \\frac{1}{2\\pi} \\int_{\\mathbb R^2} \ne^{\\tau xy} e^{-x^2\/2} e^{-y^2\/2} \\, dx\\, dy \\\\\n&= \\frac{1}{2\\pi} \\int_{\\mathbb R^2} \ne^{-(x-\\tau y)^2\/2} e^{-(1 - \\tau^2)y^2\/2} \\, dx\\, dy \\\\\n&= (1 - \\tau^2)^{-1\/2} .\n\\end{align*}\n\\end{proof} \nWith this result, we now have\n\\begin{lemma} \n\\label{cross-term}\nThere exists a constant $a > 0$ such that \n\\[\n\\mathbb{P}\\left[ \\sup_{\\lambda \\in[0,2\\pi)} | \\widehat \\Upsilon_T^{cross}(\\lambda) | \n> x \\right] \\leq \\exp\\Bigl( - \\frac{axT}{\\sqrt{\\log T}}(1 + o(1)) \\Bigr) . \n\\]\n\\end{lemma}\n\\begin{proof}\nWe only sketch the proof of this lemma. We show that for any \n$\\lambda \\in [0, 2\\pi]$, \n\\[\n\\mathbb{P}[ | \\widehat \\Upsilon_T^{cross}(\\lambda) | \n> x ] \\leq \\exp\\Bigl( - \\frac{axT}{\\sqrt{\\log T}} + C \\Bigr) \n\\]\nwhere $C$ does not depend on $\\lambda \\in [0,2\\pi]$. The lemma is then proven by a discretization\nargument of the interval $[0, 2\\pi]$ analogous to what was done in the \nproofs of Section~\\ref{unperturbed}.\nWe shall bound $\\mathbb{P}[ \\widehat \\Upsilon_T^{cross}(\\lambda) > x ]$, the term\n$\\mathbb{P}[ \\widehat \\Upsilon_T^{cross}(\\lambda) < - x ]$ being bounded similarly. \nFrom Lemma~\\ref{lemma_hadamard}, we get \n\\begin{align*}\n\\widehat \\Upsilon_T^{cross}(\\lambda) &= \n\\tr \\Bigl( D_T(\\lambda)^{\\sf H} \\frac{P^{\\sf H}_T V_T + V_T^{\\sf H} P_T}{N} \nD_T(\\lambda) B_T \\Bigr) \\\\\n&= \\tr \\frac{D_T(\\lambda)^{\\sf H} (\\Gamma_T^{1\/2})^{\\sf H} {\\boldsymbol s}_T \n{\\boldsymbol h}_T^{\\sf H} W_T R_T^{1\/2} D_T(\\lambda) B_T}{N} \\\\\n&\\phantom{=} + \n\\tr \\frac{D_T(\\lambda)^{\\sf H} (R_T^{1\/2})^{\\sf H} W_T^{\\sf H} {\\boldsymbol h}_T \n{\\boldsymbol s}_T^{\\sf H} \\Gamma_T^{1\/2} D_T(\\lambda) B_T}{N} \\\\\n&= \\frac 2N \\Re ( \\boldsymbol h_T^{\\sf H} W_T G_T(\\lambda) \\boldsymbol s_T ) \n\\end{align*} \nwhere $G_T(\\lambda) = R_T^{1\/2} D_T(\\lambda) B_T D_T(\\lambda)^{\\sf H} \n(\\Gamma_T^{1\/2})^{\\sf H}$. Let $G_T(\\lambda) = U_T \\Omega_T \n\\widetilde U_T^{\\sf H}$ be a singular value decomposition of $G_T(\\lambda)$ where \n$\\Omega = \\diag(\\omega_0, \\ldots, \\omega_{T-1})$. Observe that the vector \n$\\boldsymbol x_T \\triangleq W_T^{\\sf H} \\boldsymbol h_T = (x_0,\\ldots, x_{T-1})^{\\sf T}$ has \nthe distribution ${\\cal CN}(0, \\| \\boldsymbol h_T \\|^2 I_T)$. We can then write \n\\[\n\\widehat \\Upsilon_T^{cross}(\\lambda) \\stackrel{{\\cal L}}{=} \n\\frac 2N \\Re\\left( \\boldsymbol x_T^{\\sf H} \\Omega_T \\boldsymbol s_T \\right)\n= \\frac 2N \\sum_{t=0}^{T-1} \\omega_t ( \\Re x_t \\Re s_t + \\Im x_t \\Im s_t). \n\\]\nNotice that $\\{ \\Re x_t, \\Im x_t, \\Re s_t, \\Im s_t \\}_{t=0}^{T-1}$ are\nindependent with $\\Re x_t, \\Im x_t \\sim {\\cal N}(0, \\| \\boldsymbol h_T \\|^2\/2)$ and \n$\\Re s_t, \\Im s_t \\sim {\\cal N}(0, 1\/2)$. Letting $0 < \\tau < (\\sup_T \\| \\boldsymbol h_T \\|)^{-1}(\\sup_{\\lambda} \\| G_T(\\lambda) \\|)^{-1}$ and using Markov's inequality and Lemma~\\ref{lm-prod-gauss}, we get \n\\begin{align*} \n\\mathbb{P} \\left[ \\widehat \\Upsilon_T^{cross}(\\lambda) > x \\right] &= \n \\mathbb{P} \\left[ e^{N \\tau \\widehat \\Upsilon_T^{cross}(\\lambda)} > e^{N \\tau x} \\right] \n\\leq e^{-N \\tau x} \\mathbb{E} \\left[ e^{2\\tau \\sum_t \n\\omega_t ( \\Re x_t \\Re s_t + \\Im x_t \\Im s_t)} \\right] \\\\\n&= e^{-N\\tau x} \\prod_{t=0}^{T-1} \\left( 1 - \\tau^2 \\omega_t^2 \\| \\boldsymbol h_T \\|^2 \\right)^{-1} \n= \\exp \\left( -N \\tau x - \\sum_{t=0}^{T-1} \\log( 1 - \\tau^2 \\omega_t^2 \\| \\boldsymbol h_T\\|^2 ) \\right). \n\\end{align*}\nMimicking the proof of Lemma~\\ref{lm-B-Q}, we can establish that \n$\\sum_t \\omega_t^2 = O(\\log T)$ and $\\max_t \\omega_t = O(\\sqrt{\\log T})$ \nuniformly in $\\lambda \\in [0, 2\\pi]$. Set $\\tau = b \/ \\sqrt{\\log T}$ where\n$b > 0$ is small enough so that \n$\\sup_{T,\\lambda} (\\tau \\| \\boldsymbol h_T \\| \\, \\| G_T(\\lambda) \\|) < 1$. \nObserving that $\\log(1-x) = O(x)$ for $x$ small enough, we get \n\\[\n\\mathbb{P}[ \\widehat \\Upsilon_T^{cross}(\\lambda) > x ] \\leq \n\\exp \\bigl( -N bx\/\\sqrt{\\log T} + {\\cal E}(\\lambda, T) \\bigr)  \n\\]\nwhere $| {\\cal E}(\\lambda, T) | \\leq (C \/ \\log T) \\sum_t \\omega_t^2 \\leq C$. \nThis establishes Lemma~\\ref{cross-term}. \n\\end{proof}\n\n\\begin{lemma} \n\\label{signal-term}\nThere exists a constant $a > 0$ such that \n\\[\n\\mathbb{P}\\left[ \\sup_{\\lambda \\in[0,2\\pi)} | \\widehat \\Upsilon_T^{sig}(\\lambda) | \n> x \\right] \\leq \\exp\\Bigl( - \\frac{axT}{\\sqrt{\\log T}}(1 + o(1)) \\Bigr) . \n\\]\n\\end{lemma}\n\\begin{proof}\nBy Lemma~\\ref{lemma_hadamard},\n\\begin{align*} \n\\widehat \\Upsilon_T^{sig}(\\lambda) &= \nN^{-1} \\tr ( D_T^{\\sf H} P_T^{\\sf H} P_T D_T B_T ) \\\\\n&= \\frac{\\| \\boldsymbol h_T \\|^2}{N} \\boldsymbol s_T^{\\sf H} G_T(\\lambda) \\boldsymbol s_T \n\\end{align*} \nwhere $G_T(\\lambda) = \\Gamma_T^{1\/2} D_T(\\lambda) B_T D_T(\\lambda)^{\\sf H} \n(\\Gamma_T^{1\/2})^{\\sf H}$. By the spectral factorization \n$G_T(\\lambda) = U_T \\Sigma_T U_T^{\\sf H}$ with \n$\\Sigma_T = \\diag(\\sigma_0, \\ldots, \\sigma_{T-1})$, we get\n\\[\n\\widehat \\Upsilon_T^{sig}(\\lambda) \\stackrel{{\\cal L}}{=} \n\\frac{\\| \\boldsymbol h_T \\|^2}{N} \\sum_{t=0}^{T-1} \\sigma_t | s_t |^2 \n\\] \nand\n\\begin{align*} \n\\mathbb{P}[ \\widehat \\Upsilon_T^{sig}(\\lambda) > x ] &\\leq \ne^{-N\\tau x} \\mathbb{E} \\Bigl[ e^{\\tau \\|\\boldsymbol h_T\\|^2 \\sum_t \\sigma_t | s_t|^2}\\Bigr] \\\\\n&= \\exp\\Bigl( -N \\tau x - \\sum_{t=0}^{T-1} \n\\log(1 - \\sigma_t \\tau \\|\\boldsymbol h_T\\|^2) \\Bigr) \n\\end{align*} \nfor any $\\tau \\in (0, 1 \/ (\\|\\boldsymbol h_T\\|^2 \\sup_\\lambda \\| G_T(\\lambda)\\|))$. \nLet us show that \n\\[\n| \\tr G_T(\\lambda) | \\leq C \\sqrt{\\frac{\\log T + 1}{T}} . \n\\] \nIndeed, we have \n\\begin{align*}\n| \\tr G_T(\\lambda) | &= N^{-1} | \\tr D_T B_T D_T^{\\sf H} \\Gamma_T | = \\frac 1N \\left| \\sum_{k,\\ell=0}^{T-1} \\frac{e^{-\\imath (k-\\ell)\\lambda} \n\\gamma_{\\ell,k}}{T-|k-\\ell|} \\right| \\\\ \n&\\leq \\Bigl( \\frac 1N \\sum_{k,\\ell=0}^{T-1} |\\gamma_{k,\\ell}|^2 \\Bigr)^{1\/2} \n\\Bigl( \\frac 1N \\sum_{k,\\ell=0}^{T-1} \\frac{1}{(T-|k-\\ell|)^2} \\Bigr)^{1\/2} \\\\\n&= \\Bigl(\\frac{\\tr \\Gamma_T\\Gamma_T^{\\sf H}}{N} \\Bigr)^{1\/2} \n\\Bigl(\\frac 2N (\\log T + C) \\Bigr)^{1\/2} \\leq C \\sqrt{\\frac{\\log T + 1}{T}} .\n\\end{align*} \nMoreover, similar to the proof of Lemma~\\ref{lm-B-Q}, we can show that \n$\\sum_t\\sigma_t^2 = O(\\log T)$ and $\\max_t |\\sigma_t| = O(\\sqrt{\\log T})$ \nuniformly in $\\lambda$. Taking $\\tau = b \/ \\sqrt{\\log T}$ for \n$b > 0$ small enough, and recalling that $\\log(1-x) = 1 - x + O(x^2)$\nfor $x$ small enough, we get that \n\\[\n\\mathbb{P}[ \\widehat \\Upsilon_T^{sig}(\\lambda) > x ] \\leq \n\\exp\\Bigl( - \\frac{N bx}{\\sqrt{\\log T}} + \n\\frac{b \\| \\boldsymbol h_T \\|^2}{\\sqrt{\\log T}} \\tr G_T(\\lambda) + \n{\\cal E}(T,\\lambda) \\Bigr) \n\\]\nwhere $| {\\cal E}(T,\\lambda) | \\leq (C \/ \\log T) \\sum_t \\sigma_t^2 \\leq C$. \nWe therefore get \n\\[\n\\mathbb{P}[ \\widehat \\Upsilon_T^{sig}(\\lambda) > x ] \\leq \n\\exp\\Bigl( - \\frac{N bx}{\\sqrt{\\log T}} + C \\Bigr) \n\\]\nwhere $C$ is independent of $\\lambda$. Lemma~\\ref{signal-term} is then obtained\nby the discretization argument of the interval $[0,2\\pi]$.\n\\end{proof}\n\nGathering Inequality~\\eqref{noise-term} with Lemmas~\\ref{cross-term} and \n\\ref{signal-term}, we get the second inequality of the statement of \nTheorem~\\ref{th-perturb}. \n\n\n\\section{Application to source detection} \\label{detect}\n\nConsider a sensor network composed of $N$ sensors impinged by zero (hypothesis $H_0$) or one (hypothesis $H_1$) source signal. The stacked signal matrix $Y_T=[y_0,\\ldots,y_{T-1}]\\in\\mathbb{C}^{N\\times T}$ from time $t=0$ to $t=T-1$ is modeled as\n\\begin{eqnarray}\\label{model_det}\nY_T = \\left\\{\n    \\begin{array}{ll}\n        V_T & \\mbox{, $H_0$} \\\\\n        \\boldsymbol h_T \\boldsymbol s_T^{\\sf H} + V_T& \\mbox{, $H_1$}\n    \\end{array}\n\\right.\n\\end{eqnarray}\nwhere $s_T^{\\sf H}=[s_0^*,\\ldots,s_{T-1}^*]$ are (hypothetical) independent $\\mathcal{CN}(0,1)$ signals transmitted through the constant channel $\\boldsymbol h_T \\in \\mathbb{C}^{N}$, and $V_T=W_T R_T^{1\/2}\\in\\mathbb{C}^{N\\times T}$ models a stationary noise matrix as in \\eqref{model1}.\n\nAs opposed to standard procedures where preliminary pure noise data are available , we shall proceed here to an online signal detection test solely based on $Y_T$, by exploiting the consistence established in Theorem~\\ref{th-perturb}.\nThe approach consists precisely in estimating $R_T$ by $\\widehat{R}_T\\in\\{\\widehat{R}_T^{bp},\\widehat{R}_T^{up}\\}$, which is then used as a whitening matrix for $Y_T$. The binary hypothesis \\eqref{model_det} can then be equivalently written\n\\begin{eqnarray}\\label{model_w}\nY_T \\widehat{R}_T^{-1\/2}= \\left\\{\n    \\begin{array}{ll}\n        W_T {R}_T \\widehat{R}_T^{-1\/2}& \\mbox{, $H_0$}  \\\\\n        \\boldsymbol h_T \\boldsymbol s_T^{\\sf H} \\widehat{R}_T^{-1\/2} + W_T {R}_T \\widehat{R}_T^{-1\/2}& \\mbox{, $H_1$}.\n    \\end{array}\n\\right.\n\\end{eqnarray}\nSince $\\Vert R_T\\widehat{R}_T^{-1}-I_T\\Vert\\to 0$ almost surely (by Theorem~\\ref{th-perturb} as long as $\\inf_{\\lambda\\in[0,2\\pi)} {\\boldsymbol \\Upsilon}(\\lambda)>0$), for $T$ large, the decision on the hypotheses \\eqref{model_w} can be handled by the generalized likelihood ratio test (GLRT) \\cite{BianDebMai'11} by approximating $W_T {R}_T \\widehat{R}_T^{-1\/2}$ as a purely white noise. We then have the following result.\n\\begin{theorem}\\label{th_detection}\n\tLet $\\widehat{R}_T$ be any of $\\widehat{R}_T^{bp}$ or $\\widehat{R}_T^{up}$ strictly defined in Theorem~\\ref{th-perturb} for $Y_T$ now following model \\eqref{model_det}. Further assume $\\inf_{\\lambda\\in[0,2\\pi)} {\\boldsymbol \\Upsilon}(\\lambda)>0$ and define the test\n\\begin{equation}\\label{glrt_est}\n\\alpha = \\frac{N\\norme{Y_T \\widehat{R}_T^{-1}Y_T^{\\sf H}}}{\\tr \\left( Y_T \\widehat{R}_T^{-1} Y_T^{\\sf H} \\right)} ~ \\overset{H_0}{\\underset{H_1}{\\lessgtr}} ~ \\gamma\n\\end{equation}\nwhere $\\gamma\\in\\mathbb R^+$ satisfies $\\gamma>(1+\\sqrt c)^2$. Then, as $T \\to \\infty$,\n\\begin{align*}\n\t\\mathbb{P} \\left[ \\alpha \\geq \\gamma \\right] \\to \\left\\{ \\begin{array}{ll} 0 &,~H_0 \\\\ 1 &,~H_1. \\end{array}\\right.\n\\end{align*}\n\\end{theorem}\n\nRecall from \\cite{BianDebMai'11} that the decision threshold $(1+\\sqrt{c})^2$ corresponds to the almost sure limiting largest eigenvalue of $\\frac1T W_T W_T^{\\sf H}$, that is the right-edge of the support of the Mar\\v{c}enko--Pastur law. \n\nSimulations are performed hereafter to assess the performance of the test \\eqref{glrt_est} under several system settings. We take here ${\\boldsymbol h}_T$ to be the following steering vector ${\\boldsymbol h}_T=\\sqrt{p\/T}[1, \\ldots , e^{2i\\pi \\theta (T-1)}]$ with $\\theta = 10^\\circ$ and $p$ a power parameter. The matrix $R_T$ models an autoregressive process of order 1 with parameter $a$, {\\it i.e.} $[R_T]_{k,l}=a^{|k-l|}$. \n\nIn Figure~\\ref{det1}, the detection error $1-\\mathbb{P} [ \\alpha \\geq \\gamma|H_1]$ of the test \\eqref{glrt_est} for a false alarm rate (FAR) $\\mathbb{P} [ \\alpha \\geq \\gamma|H_0 ]=0.05$ under $\\widehat{R}_T=\\widehat{R}_T^{up}$ (Unbiased) or $\\widehat{R}_T=\\widehat{R}_T^{bp}$ (Biased) is compared against the estimator that assumes $R_T$ perfectly known (Oracle), {\\it i.e.} that sets $\\widehat{R}_T=R_T$ in \\eqref{glrt_est}, and against the GLRT test that wrongly assumes temporally white noise (White), {\\it i.e.} that sets $\\widehat{R}_T=I_T$ in \\eqref{glrt_est}. The source signal power is set to $p=1$, that is a signal-to-noise ratio (SNR) of $0$ dB, $N$ is varied from $10$ to $50$ and $T=N\/c$ for $c=0.5$ fixed. In the same setting as Figure~\\ref{det1}, the number of sensors is now fixed to $N=20$, $T=N\/c=40$ and the SNR (hence $p$) is varied from $-10$~dB to $4$~dB. The powers of the various tests are displayed in Figure~\\ref{det2} and compared to the detection methods which estimate $R_T$ from a pure noise sequence called Biased PN (pure noise) and Unbiased PN. The results of the proposed online method are close to that of Biase\/Unbiased PN, this last presenting the disadvantage to have at its disposal a pure noise sequence at the receiver.  \n\nBoth figures suggest a close match in performance between Oracle and Biased, while Unbiased shows weaker performance. The gap evidenced between Biased and Unbiased confirms the theoretical conclusions. \n\n\\begin{figure}[H]\n\\center\n  \\begin{tikzpicture}[font=\\footnotesize]\n    \\renewcommand{\\axisdefaulttryminticks}{2} \n    \\pgfplotsset{every axis\/.append style={mark options=solid, mark size=2pt}}\n    \\tikzstyle{every major grid}+=[style=densely dashed]       \n\\pgfplotsset{every axis legend\/.append style={fill=white,cells={anchor=west},at={(0.01,0.01)},anchor=south west}}    \n    \\tikzstyle{every axis y label}+=[yshift=-10pt] \n    \\tikzstyle{every axis x label}+=[yshift=5pt]\n    \\begin{semilogyaxis}[\n      grid=major,\n     \n      xlabel={$N$},\n      ylabel={$1-\\mathbb{P}[\\alpha>\\gamma|H_1]$},\n     \n     \n     \n      xmin=10,\n      xmax=50, \n      ymin=1e-4, \n      ymax=1,\n      width=0.7\\columnwidth,\n      height=0.5\\columnwidth\n      ]\n      \\addplot[smooth,black,line width=0.5pt,mark=star] plot coordinates{\n(10.000000,0.318300)(15.000000,0.164000)(20.000000,0.069100)(25.000000,0.028400)(30.000000,0.013300)(35.000000,0.006100)(40.000000,0.002900)(45.000000,0.001200)(50.000000,0.0005000)\n\n\n      };\n      \n      \\addplot[smooth,black,line width=0.5pt,mark=o] plot coordinates{\n(10.000000,0.848100)(15.000000,0.385600)(20.000000,0.168900)(25.000000,0.063900)(30.000000,0.028900)(35.000000,0.012800)(40.000000,0.005200)(45.000000,0.002100)(50.000000,0.000900)\n\n\n      };\n      \n      \\addplot[smooth,black,line width=0.5pt,mark=triangle] plot coordinates{\n(10.000000,0.973000)(15.000000,0.953000)(20.000000,0.954000)(25.000000,0.957000)(30.000000,0.958000)(35.000000,0.958000)(40.000000,0.940000)(45.000000,0.970000)(50.000000,0.949000)\n\n\n\n      };\n      \n      \\addplot[smooth,black,line width=0.5pt,mark=x] plot coordinates{\n(10.000000,0.200200)(15.000000,0.104000)(20.000000,0.043600)(25.000000,0.018700)(30.000000,0.008500)(35.000000,0.003300)(40.000000,0.0014500)(45.000000,0.000600)(50.000000,0.000200)\n\n\n      };\n\n\t\\legend{{Biased},{Unbiased},{White},{Oracle}}\n       \\end{semilogyaxis}\n       \\end{tikzpicture}\n       \\caption{Detection error versus $N$ with FAR$=0.05$, $p=1$, SNR$=0$ dB, $c=0.5$, and $a=0.6$.}\n\\label{det1}\n\\end{figure}\n\n\\begin{figure}[H]\n\\center\n  \\begin{tikzpicture}[font=\\footnotesize]\n    \\renewcommand{\\axisdefaulttryminticks}{2} \n    \\pgfplotsset{every axis\/.append style={mark options=solid, mark size=2pt}}\n    \\tikzstyle{every major grid}+=[style=densely dashed]       \n\\pgfplotsset{every axis legend\/.append style={fill=white,cells={anchor=west},at={(0.01,0.99)},anchor=north west}}    \n    \\tikzstyle{every axis y label}+=[yshift=-10pt] \n    \\tikzstyle{every axis x label}+=[yshift=5pt]\n    \\begin{axis}[\n      grid=major,\n     \n      xlabel={SNR (dB)},\n      ylabel={$\\mathbb{P}[\\alpha>\\gamma|H_1]$},\n     \n     \n  \t   \n      xmin=-10,\n      xmax=4, \n      ymin=0, \n      ymax=1,\n      width=0.7\\columnwidth,\n      height=0.5\\columnwidth\n      ]\n      \n            \\addplot[smooth,black,line width=0.5pt,mark=star] plot coordinates{\n(-15.000000,0.051200)(-14.000000,0.048100)(-13.000000,0.049100)(-12.000000,0.053100)(-11.000000,0.049800)(-10.000000,0.053200)(-9.000000,0.056100)(-8.000000,0.0593000)(-7.000000,0.067500)(-6.000000,0.073900)(-5.000000,0.110200)(-4.000000,0.190800)(-3.000000,0.330600)(-2.000000,0.538500)(-1.000000,0.766600)(0.000000,0.922000)(1.000000,0.985800)(2.000000,0.998600)(3.000000,1.000000)(4.000000,1.000000)(5.000000,1.000000)\n\n\n      };\n      \n      \\addplot[smooth,black,line width=0.5pt,mark=o] plot coordinates{\n(-15.000000,0.053900)(-14.000000,0.049300)(-13.000000,0.049500)(-12.000000,0.055100)(-11.000000,0.050800)(-10.000000,0.054000)(-9.000000,0.057800)(-8.000000,0.060100)(-7.000000,0.068400)(-6.000000,0.076300)(-5.000000,0.100400)(-4.000000,0.159500)(-3.000000,0.248400)(-2.000000,0.427000)(-1.000000,0.652600)(0.000000,0.854200)(1.000000,0.964900)(2.000000,0.995100)(3.000000,0.999500)(4.000000,1.000000)(5.000000,1.000000)\n\n\n      };\n      \n      \\addplot[smooth,black,densely dashed,line width=0.5pt,mark=star] plot coordinates{\n(-15.000000,0.047700)(-14.000000,0.047400)(-13.000000,0.044600)(-12.000000,0.049700)(-11.000000,0.049700)(-10.000000,0.051900)(-9.000000,0.059400)(-8.000000,0.063100)(-7.000000,0.073300)(-6.000000,0.099100)(-5.000000,0.149500)(-4.000000,0.240100)(-3.000000,0.399900)(-2.000000,0.613300)(-1.000000,0.816900)(0.000000,0.944300)(1.000000,0.991500)(2.000000,0.999200)(3.000000,1.000000)(4.000000,1.000000)(5.000000,1.000000)\n\n      };\n      \n      \\addplot[smooth,black,densely dashed,line width=0.5pt,mark=o] plot coordinates{\n(-15.000000,0.048300)(-14.000000,0.051100)(-13.000000,0.047100)(-12.000000,0.050300)(-11.000000,0.051800)(-10.000000,0.052900)(-9.000000,0.056300)(-8.000000,0.062600)(-7.000000,0.068100)(-6.000000,0.086500)(-5.000000,0.116600)(-4.000000,0.178600)(-3.000000,0.299600)(-2.000000,0.505700)(-1.000000,0.721500)(0.000000,0.897600)(1.000000,0.976000)(2.000000,0.996900)(3.000000,0.999900)(4.000000,0.999900)(5.000000,1.000000)\n\n      };\n      \n \n      \\addplot[smooth,black,line width=0.5pt,mark=x] plot coordinates{\n(-15.000000,0.048800)(-14.000000,0.050800)(-13.000000,0.048300)(-12.000000,0.048900)(-11.000000,0.055600)(-10.000000,0.058300)(-9.000000,0.063100)(-8.000000,0.066000)(-7.000000,0.084300)(-6.000000,0.110800)(-5.000000,0.169400)(-4.000000,0.280300)(-3.000000,0.446500)(-2.000000,0.669800)(-1.000000,0.858100)(0.000000,0.963000)(1.000000,0.994300)(2.000000,0.999400)(3.000000,1.000000)(4.000000,1.000000)(5.000000,1.000000)\n\n\n\n      };\n\n\t\\legend{{Biased},{Unbiased},{Biased PN},{Unbiased PN},{Oracle}}\n       \\end{axis}\n       \\end{tikzpicture}\n       \\caption{Power of detection tests versus SNR (dB) with FAR$=0.05$, $N=20$, $c=0.5$, and $a=0.6$.}\n\\label{det2}\n\\end{figure}\n\n\n\\begin{appendix}\n\\subsection{Proofs for Theorem~\\ref{th-biased}} \n\\subsubsection{Proof of Lemma \\ref{lemma_d_quad}} \n\\label{anx-lm-qf} \n\nDeveloping the quadratic forms given in the statement of the lemma, we get \n\\begin{align*} \nd_T(\\lambda)^{\\sf H}\\frac{V_T^{\\sf H}V_T}{N}d_T(\\lambda) &= \n\\frac{1}{NT}\\sum_{l,l'=0}^{T-1} e^{-\\imath(l'-l)\\lambda} [V_T^{\\sf H}V_T]_{l,l'} \\\\\n&= \\frac{1}{NT}\\sum_{l,l'=0}^{T-1} e^{-\\imath(l'-l)\\lambda} \n\\sum_{n=0}^{N-1} v^*_{n,l} v_{n,l'} \\\\ \n&= \\sum_{k=-(T-1)}^{T-1} e^{-\\imath k \\lambda} \n\\frac{1}{NT} \\sum_{n=0}^{N-1} \\sum_{t=0}^{T-1} v_{n,t}^* v_{n,t+k} \n\\mathbbm{1}_{0 \\leq t+k \\leq T-1}\\\\ \n&= \\sum_{k=-(T-1)}^{T-1} \\hat{r}_k^b e^{-\\imath k \\lambda}=\n\\widehat\\Upsilon_T^b(\\lambda), \n\\end{align*} \nand \n\\begin{align*} \n\\mathbb{E} \\left[ d_T(\\lambda)^{\\sf H} \\frac{V_T^{\\sf H}V_T}{N}d_T(\\lambda) \\right] \n&= d_T(\\lambda)^{\\sf H} (R_T^{1\/2})^{\\sf H} \n\\frac{\\mathbb{E}[ W_T^{\\sf H} W_T]}{N} R_T^{1\/2} d_T(\\lambda) \\\\ \n&= d_T(\\lambda)^{\\sf H} R_T d_T(\\lambda) .\n\\end{align*} \n\n\\subsection{Proofs for Theorem~\\ref{th-unbiased}}\n\\subsubsection{Proof of Lemma \\ref{lemma_d_quad2}} \n\\label{anx-lm-qf2} \nWe have \n\\begin{align*}\nd_T(\\lambda)^{\\sf H} \\left( \\frac{V_T^{\\sf H}V_T}{N} \\odot B_T \\right) d_T(\\lambda) &=\n\\frac{1}{NT}\\sum_{l,l'=-(T-1)}^{T-1}e^{i(l-l') \\lambda} [V_T^{\\sf H}V_T]_{l,l'}\\frac{T}{T-|l-l'|} \\nonumber \\\\\n&= \\sum_{k=-(T-1)}^{T-1}e^{ik\\lambda}\\frac{1}{N(T-|k|)}\\sum_{n=0}^{N-1}\\sum_{t=0}^{T-1}v_{n,t}^*v_{n,t+k}\\mathbbm{1}_{0 \\leq t+k \\leq T-1} \\\\\n&=\\sum_{k=-(T-1)}^{T-1} \\hat{r}_k^u e^{ik \\lambda} =\\widehat \\Upsilon_T^u(\\lambda).\n\\end{align*}\n\n\\subsubsection{Proof of Lemma \\ref{lm-B-Q}} \n\\label{prf-lm-B-Q} \nWe start by observing that \n\\begin{align*}\n\\tr B_T^2 &= \\sum_{i,j=0}^{T-1} \\left[ B_T \\right]_{i,j}^2 \n= \\sum_{i,j=0}^{T-1} \\left( \\frac{T}{T-|i-j|} \\right)^2 \n= 2\\sum_{i>j}^{T-1} \\left( \\frac{T}{T-|i-j|} \\right)^2 + T \\\\\n&= 2 \\sum_{k=1}^{T-1} \\left( \\frac{T}{T-k} \\right)^2 \\left( T - k \\right) + T \n= 2 T^2 \\sum_{k=1}^{T-1} \\frac{1}{T-k} + T \n= 2 T^2 \\left(\\log T + C \\right).\n\\end{align*}\nInequality \\eqref{norme_B} is then obtained upon noticing that \n$\\norme{B_T} \\leq \\sqrt{\\tr B_T^2}$. \n\nWe now show (\\ref{sum_sigma2}). \nUsing twice the inequality $\\tr (FG) \\leq \\norme{F} \\tr(G)$ when \n$F,G \\in \\mathbb{C}^{m \\times m}$ and $G$ is nonnegative definite \\cite{HornJoh'91}, \nwe get\n\\begin{align*}\n\\sum_{t=0}^{T-1} \\sigma_t^2(\\lambda_i) &= \\tr Q_T(\\lambda_i)^2  \n=  \\tr  R_T D_T(\\lambda_i)B_T D_T(\\lambda_i)^{\\sf H} R_T D_T(\\lambda_i) \nB_T D_T(\\lambda_i)^{\\sf H} \\\\\n&\\leq \\norme{R_T} \\tr R_T (D_T(\\lambda_i) B_T D_T(\\lambda_i)^{\\sf H})^2 \\\\\n&\\leq T^{-2} \\norme{R_T}^2 \\tr (B_T^2) \\leq 2\\norme{\\boldsymbol\\Upsilon}_{\\infty}^2 \\log T + C. \n\\end{align*}\n\nInequality \\eqref{sig_max} is immediate since $\\norme{Q_T}^2 \\leq \\tr Q_T^2$.\n\nAs regards \\eqref{sum_sigma3}, by the Cauchy--Schwarz inequality,\n\\begin{align*}\n\\sum_{t=0}^{T-1} |\\sigma_t^3(\\lambda_i)| &= \n\\sum_{t=0}^{T-1} \\sigma_t^2(\\lambda_i) |\\sigma_t(\\lambda_i)|  \n\\leq \\sqrt{\\sum_{t=0}^{T-1} \\sigma_t^4(\\lambda_i) \\sum_{t=0}^{T-1} \\sigma_t^2(\\lambda_i)} \\\\ \n&\\leq \\sqrt{\\left(\\sum_{t=0}^{T-1} \\sigma_t^2(\\lambda_i) \\right)^2 \\sum_{t=0}^{T-1} \\sigma_t^2(\\lambda_i)} = \\left( \\sum_{t=0}^{T-1} \\sigma_t^2(\\lambda_i) \\right)^{3\/2}\\\\\n&= C ( (\\log T)^{3\/2} +1 ).\n\\end{align*}\n\n\n\n\\end{appendix}\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{Intro:level1}Introduction}\nCoherent radiation can polarize the angular momentum distribution of an ensemble of atoms in various ways, creating different \npolarization moments, which modifies the way these atoms will interact with radiation. Carefully prepared spin polarized \natoms can make the absorption highly dependent on frequency \n(electromagnetically induced transparency~\\cite{Harris:1990}), causing large values of the dispersion, which, in turn, \nare useful for such interesting effects as slow light~\\cite{Hau:1999} and optical information storage~\\cite{Liu:2001}. \nElectric and magnetic fields, external or inherent in the radiation fields, may also influence the time evolution of \nthe spin polarization and cause measurable changes in absorption or fluorescence intensity and\/or polarization. \nThese effects are the basis of many magnetometry schemes~\\cite{Scully:1992,Budker:2007}, \nand must be taken into account in atomic clocks~\\cite{Knappe:2005} and when searching for \nfundamental symmetry violations~\\cite{Budker:2002} or exotic physics such as \nan electric dipole moment of the electron~\\cite{Regan:2002}. \nSufficiently strong laser radiation creates atomic polarization in excited as well as in the ground state~\\cite{Auzinsh:OptPolAt}\nThe polarization is destroyed when the Zeeman sublevel degeneracy is removed by a magnetic field. \nSince the ground state has a much longer lifetime, very narrow magneto-optical resonances can be created, which \nare related to the ground-state Hanle effect (see ~\\cite{Arimondo:1996} for a review). Such resonances were first \nobserved in cadmium in 1964~\\cite{Lehmann:1964}. \n\nThe theory of magneto-optical resonances has been understood for some time(see \\cite{Budker:2002,Alexandrov:2005,Auzinsh:OptPolAt} for a review), \nand bright (opposite sign) resonances have also been observed and explained~\\cite{Kazantsev:1984,Renzoni:2001a,Alnis:2001}; the challenge in describing experiments \nlies in choosing the effects to be included in the numerical calculations so as to find a balance between computation time and accuracy.\nThe optical Bloch equations (OBEs) for the density matrix have been used as early as 1978 to model magneto-optical \nresonances~\\cite{Picque:1978}. In order to achieve greater accuracy, later efforts to model signals took into account  \neffects such as Doppler broadening, the coherent properties of the laser radiation, \nand the mixing of magnetic sublevels in an external magnetic field to produce more and more \naccurate descriptions of experimental signals~\\cite{Auzinsh:2008}. Analytical models can also achieve excellent descriptions of \nexperimental signals at low laser powers in the regime of linear excitation where optical pumping plays a negligible \nrole~\\cite{Castagna:2011,Breschi:2012}. In recent \nyears, excellent agreement has been achieved by numerical calculations even when optical pumping plays a role. \nHowever, as soon as the laser radiation begins to saturate the \nabsorption transition, the model's accuracy suffers. The explanation has been that at high radiation intensities, \nit is no longer possible to model the \nrelaxation of atoms moving in and out of the beam with a single rate constant~\\cite{Auzinsh:1983, Auzinsh:2008}.  \nNevertheless, accurate numerical models of situations in an intense laser field are very desirable, because they \ncould arise in a number of experimental situations. Therefore, we have set out to model magneto-optical effects in the presence of \nintense laser radiation by taking better account of the fact that an atom experiences different laser intensity values as it \npasses through a beam. In practice, we solve the rate \nequations for the Zeeman coherences for different regions of the laser beam with a value of the Rabi frequency that more \nclosely approximates the real situation in that part of the beam.\nTo save computing time, stationary solutions to the rate equations for Zeeman sublevels and coherences are sought for each region~\\cite{Blushs:2004}. \nWith this simplification to take into account the motion of atoms through the beam, \nwe could now obtain accurate descriptions of experimental signals up to much higher intensities for reasonable computing times. \nMoreover, the model can be used to study the spatial distribution of the laser induced \nfluorescence within the laser beam. We performed such a study theoretically and experimentally using two overlapping lasers: \none spatially broad, intense pump laser, and a weaker, tightly focused, spatically narrow probe laser. \nThe qualitative agreement between experimental and theoretical fluorescence intensity profiles indicates \nthat the model is a useful tool for studying fluorescence dynamics\nas well as for modelling magneto-optical signals at high laser intensities. \n\n\n\\section{\\label{Theory:level1}Theory}\nThe theoretical model used here is a further development of previous efforts~\\cite{Auzinsh_crossing:2013}, which has been subjected \nto some initial testing in the specialized context of an extremely thin cell~\\cite{Auzinsh:2015}.\nThe description of coherent processes starts with the optical Bloch equation (OBE):\n\\begin{equation}\ni \\hbar \\frac{\\partial \\rho}{\\partial t} = \\left[\\hat{H},\\rho \\right]+ i \\hbar \\hat{R}\\rho,\n\\end{equation}\nwhere $\\rho$ is the density matrix describing the atomic state, $\\hat{H}$ is the Hamiltonian of the system, \nand $\\hat{R}$ is an operator that describes relaxation.  These equations are transformed into rate equations \nthat are solved under stationary conditions in order to obtain the Zeeman coherences in the \nground ($\\rho_{g_ig_j})$ and excited ($\\rho_{e_ie_j}$) states\\cite{Blushs:2004}. However, when the intensity distribution in the beam is not \nhomogeneous, more accurate results can be achieved by dividing the laser beam into concentric regions and solving the \nOBEs for each region separately while accounting for atoms that move into and out of each region as they fly through the \nbeam. Figure~\\ref{fig:dal1} illustrates the idea. \n\\begin{figure}\n\t\\includegraphics[width=0.45\\textwidth]{beam_division.pdf}\n\t\\caption{\\label{fig:dal1} Laser beam profile split into a number of concentric regions.}\n\\end{figure}\nThe top part of the figure shows the intensity profile of the laser beam, while \nthe bottom part of the figure shows a cross-section of the laser beam indicating the concentric regions. \n\nIn order to account for particles that leave one region and enter the next, an extra term must be added to the OBE:\n\\begin{equation}\n\\label{eq:transit}\n-i \\hbar\\hat{\\gamma_t} \\rho+i \\hbar\\hat{\\gamma_t} \\rho'.\n\\end{equation}\nIn this term, $\\rho'$ is the density matrix of the particles entering the region (identical to the density matrix of the \nprevious region), and $\\hat{\\gamma_t}$ is an operator that accounts for transit relaxation. This operator is \nessentially a diagonal matrix with elements $\\hat{\\gamma}_{t_{ij}}=(\\nicefrac{v_{yz}}{s_n})\\delta_{ij}$, where $v_{yz}$ \ncharacterizes the particle speed in the plane perpendicular to the beam and $s_n$ is the linear dimension of the region. \nTo simplify matters, we treat particle motion in only one direction and later average with particles that move in the other \ndirection. In that case, $\\rho'=\\rho^{n-1}$. \nThus, the rate equations for the density matrix $\\rho^n$ of the $n$textsuperscript{th} region become \n\\begin{align}\n\\label{eq:rate_region}\ni~\\hbar \\frac{\\partial\\rho^n}{\\partial t} &= \\left[ \\hat{H},\\rho^n\\right]+i ~\\hbar \\hat{R} \\rho^n -i~\\hbar\\hat{\\gamma_t}^n\\rho^n \\notag \\\\ \n& +i~\\hbar\\hat{\\gamma_t}^n\\rho^{n-1}-i~\\hbar\\hat{\\gamma_c}\\rho^n+i~\\hbar\\hat{\\gamma_c}\\rho^0.\n\\end{align}\nIn this equation the relaxation operator $\\hat{R}$ describes spontaneous relaxation only and $\\hat{\\gamma_c}$ is the collisional relaxation \nrate, which, however, becomes significant only at higher gas densities.  \n\nNext, the rotating wave approximation~\\cite{Allen:1975} is applied to the OBEs, which yield \nstochastic differential equations that can be simplified by means of the decorrelation \napproach~\\cite{Kampen:1976}. Since the measurable quantity is merely light intensity, \na formal statistical average is performed over the fluctuating phases of these stochastic equations, \nmaking use of the decorrelation approximation~\\cite{Blushs:2004}. As a result, the density matrix \nelements that correspond to optical coherences are eliminated and one is left with rate equations for the \nZeeman coherences:\n\\begin{align}\n\\label{eq:ground}\n\\frac{\\partial \\rho_{g_i,g_j}^n}{\\partial t} =& \\sum_{e_k,e_m}\\left(\\Xi_{g_ie_m}^n + (\\Xi_{e_kg_j}^n)^*\\right) d_{g_ie_k}^*d_{e_mg_j}\\rho_{e_ke_m}^n \\notag \\\\\n& - \\sum_{e_k,g_m}(\\Xi_{e_kg_j}^n)^*d_{g_ie_k}^*d_{e_kg_m}\\rho_{g_mg_j}^n \\notag \\\\ \n& - \\sum_{e_k,g_m}\\Xi_{g_ie_k}^n d_{g_me_k}^*d_{e_kg_j}\\rho_{g_ig_m}^n  \\\\\n& - i\\omega_{g_ig_j}\\rho_{g_ig_j}^n+\\sum_{e_ke_l}\\Gamma_{g_ig_j}^{e_ke_l}\\rho_{e_ke_l}^n-\\gamma_{t}\\rho_{g_ig_j}^n \\notag \\\\ \n& + \\gamma^{n}_{t}\\rho_{g_ig_j}^{n-1}-\\gamma^{n}_{c}\\rho_{g_ig_j}^n+\\gamma_c\\rho_{g_ig_j}^0\\notag\\\\\n\\label{eq:excited}\n\\frac{\\partial \\rho_{e_i,e_j}^n}{\\partial t} =& \\sum_{g_k,g_m}\\left((\\Xi_{e_ig_m}^n)^* + \\Xi_{g_ke_j}^n\\right) d_{e_ig_k}^*d_{g_me_j}\\rho_{g_kg_m}^n \\notag\\\\\n& - \\sum_{g_k,e_m}\\Xi_{g_ke_j}^nd_{e_ig_k}d_{g_ke_m}^*\\rho_{e_me_j}^n \\notag \\\\ \n& - \\sum_{g_k,e_m}(\\Xi_{e_ig_k}^n)^*d_{e_mg_k}d_{g_ke_j}^*\\rho_{e_ie_m}^n \\\\\n& - i\\omega_{e_ie_j}\\rho_{e_ie_j}^n-\\Gamma\\rho_{e_ie_j}^n-\\gamma^{n}_{t}\\rho_{e_ie_j}^n  \\notag \\\\\n& +\\gamma^{n}_{t}\\rho_{e_ie_j}^{n-1}-\\gamma_{c}\\rho_{e_ie_j}^n \\notag.\n\\end{align}\nIn both equations, the first term describes population increase and creation of coherence due to induced \ntransitions, the second and third terms describe population loss due to induced transitions, the fourth \nterm describes the destruction of Zeeman coherences  due to the splitting $\\omega{g_ig_j}$, \nrespectively, $\\omega_{e_ie_j}$ of the Zeeman sublevels in an external magnetic field, \nand the fifth term in Eq.~\\ref{eq:excited} describes spontaneous decay with $\\Gamma\\rho_{e_ie_j}^n$ giving the \nspontaneous rate of decay for the excited state. At the same time the fifth term in Eq.~\\ref{eq:ground} \ndescribes the transfer of population and coherences from the excited state matrix element $\\rho_{e_k e_l}$ to \nthe ground state density matrix element $\\rho_{g_i g_j}$ with rate $\\Gamma^{e_k e_l}_{g_i g_j}$. \nThese transfer rates are related to the rate of spontaneous decay $\\Gamma$ for the excited state. \nExplicit expressions for these $\\Gamma^{e_k e_i}_{g_i g_j}$ can be calculated from quantum angular \nmomentum theory and are given in~\\cite{Auzinsh:OptPolAt}. \nThe remaining terms have been described previously in the context of \nEqns.~\\ref{eq:transit} and~\\ref{eq:rate_region}. \nThe laser beam interaction is represented by the term\n\\begin{align}\n\\Xi_{g_ie_j}= \\frac{|\\bm\\varepsilon^n|^2}{\\frac{\\Gamma+\\Delta\\omega}{2}+i \\left(\\bar{\\omega}-\\mathbf{k}\\cdot \\mathbf{v}+\\omega_{g_ie_j}\\right)},\n\\end{align} \nwhere $|\\bm\\varepsilon^n|^2$ is the laser field's electric field strength in the $n$th region, $\\Gamma$ is the spontaneous \ndecay rate, $\\Delta\\omega$ is the laser beam's spectral width, $\\bar{\\omega}$ is the laser frequency, \n$k\\cdot v$ gives the Doppler shift, and $\\omega_{g_ie_j}$ is the difference in energy between levels \n$g_i$ and $e_j$. The system of linear equations can be solved for stationary conditions to \nobtain the density matrix $\\rho$. \n\nFrom the density matrix one can obtain the fluorescence intensity from each region for each velocity group $v$ and given \npolarization $\\bm\\varepsilon_f$ up to a constant factor of $\\tilde{I}_0$~\\cite{AuzFerb:OptPolMol, Barrat:1961, Dyakonov:1965}:\n\\begin{equation}\\label{eq:fluorescence}\n\tI_{n}(v,\\bm\\varepsilon_f) = \\tilde{I}_0\\sum\\limits_{g_i,e_j,e_k} d_{g_ie_j}^{\\ast(ob)}d_{e_kg_i}^{(ob)}\\rho_{e_je_k}.\n\\end{equation}\nFrom these quantities one can calculate the total fluorescence intensity for a given polarization $\\bm\\varepsilon_f$: \n\\begin{align}\n\tI(\\bm\\varepsilon_f) = \\sum_{n} \\sum_{v} f(v)\\Delta v \\frac{A_{n}}{A} I_{n}(v,\\bm\\varepsilon_f).\n\\end{align}\nHere the sum over $n$ represents the sum over the different beam regions of relative area \n$\\nicefrac{A_{n}}{A}$ as they are traversed by the \nparticle, $v$ is the particle velocity along the laser beam, and $f(v)\\Delta v$ gives the number of \natoms with velocity $v\\pm \\nicefrac{\\Delta v}{2}$. \n\nIn practice, we do not measure the electric field strength of the laser field, but the intensity $I=P\/A$, where $P$ is the \nlaser power and $A$ is the cross-sectional area of the beam. In the theoretical model it is more convenient to use the \nRabi frequency $\\Omega_R$, here defined as follows:\n\\begin{align}\n\\label{eq:Rabi}\n\\Omega_R = k_R \\frac{\\vert\\vert d \\vert\\vert \\cdot \\vert\\vert \\epsilon \\vert\\vert}{\\hbar} \\\n\t = k_R \\frac{\\vert\\vert d \\vert\\vert}{\\hbar} \\sqrt{\\frac{2 I}{\\epsilon_0 n c}},\n\\end{align}\nwhere $\\vert\\vert d \\vert\\vert$ is the reduced dipole matrix element for the transition in question, $\\epsilon_0$ is the \nvacuum permittivity, $n$ is the index of refraction of the medium, $c$ is the speed of light, and $k_R$ is a factor that \nwould be unity in an ideal case, but is adjusted to achieve the best fit between theory and experiment since the \nexperimental situation will always deviate from the ideal case in some way. \nWe assume that the laser beam's intensity distribution follows a Gaussian distribution. We define the average value of \n$\\Omega_R$ for the whole beam by taking the weighted average of a Gaussian distribution on the range [0,FWHM\/2], where \nFWHM is the full width at half maximum. Thus it follows that the Rabi frequency at the peak of the intensity distribution \n(see Fig.~\\ref{fig:dal1}) is $\\Omega_R=0.721\\Omega_{peak}$. From there the Rabi frequency of each region can be obtained \nby scaling by the value of the Gaussian distribution function. \n\n\n\n\n\n\n\\section{\\label{exp:level1}Experimental setup}\nThe theoretical model was tested with two experiments. The first experiment measured magneto-optical resonances on the $D_1$ line of \n$^{87}$Rb and is shown schematically in Fig.~\\ref{fig:exp_Rb87}. The experiment has been described elsewhere along with \ncomparison to an earlier version of the theoretical model that did not divide the laser beam into separate regions~\\cite{Auzinsh:2009}.\n\\begin{figure}\n\t\\includegraphics[width=0.45\\textwidth]{exp_rb87.pdf}\n\t\\caption{\\label{fig:exp_Rb87} (Color online) Basic experimental setup for measuring magneto-optical resonances. The inset \n\ton the left shows the level diagram of $^{87}$Rb~\\cite{Steck:rubidium87}. The other inset shows the geometrical orientation of the electric \n\tfield vector \\boldmath{E}, the magnetic field vector \\boldmath{B}, and laser propagation direction (Exc.) and  \n\tobservation direction (Obs.).}\n\\end{figure}\nThe laser was an extended cavity diode laser, whose frequency could be scanned by applying a voltage to a piezo crystal attached to the grating. \nNeutral density (ND) filters were used to regulate the laser intensity, and linear polarization was obtained using a \nGlan-Thomson polarizer. A set of three orthogonal Helmholtz coils scanned the magnetic field along the $z$ axis \nwhile compensating the ambient field in the other directions. A pyrex cell with a natural isotopic mixture \nof rubidium at room temperature was located at the center of the coils. The total laser induced fluorescence (LIF) \nin a selected direction (without frequency or polarization selection) was detected with \na photodiode (Thorlabs FDS-100) and data were acquired with a data acquisition card (National Instruments 6024E)\nor a digital oscilloscope (Agilent DSO5014). To generate the magnetic field scan with a rate of about 1~Hz, \na computer-controlled analog signal was applied to a bipolar power supply (Kepco BOP-50-8M). The laser \nfrequency was simultaneously scanned at a rate of about 10-20 MHz\/s, and it was measured by \na wavemeter (HighFinnesse WS-7). \nThe laser beam was characterized using a beam profiler (Thorlabs BP104-VIS). \n\nA second experimental setup was used to study the spatial profile of the fluorescence generated by atoms in a \nlaser beam at resonance. It is shown in Fig.~\\ref{fig:exp_setup}. Here two lasers were used to excite the \n$D_1$ and $D_2$ transitions of cesium. Both lasers were based on distributed feedback diodes from toptica \n(DL100-DFB). One of the lasers (Cs $D_2$) served as a pump laser with a spatially broad and intense beam, \nwhile the other (Cs $D_1$), \nspatially narrower beam probed the fluorescence dynamics within the pump beam. \nFigure~\\ref{fig:levels} shows the level scheme of the excited transitions. \nBoth lasers were stabilized with saturation absorption signals from cells shielded by three layers of mu-metal. \nMu-metal shields were used to avoid frequency drifts due to the magnetic field scan performed in the experiment and other \nmagnetic field fluctuations in the laboratory.\n\\begin{figure}\n\t\\includegraphics[width=0.45\\textwidth]{divu_staru_exp_sh_v3.pdf}\n\t\\caption{\\label{fig:exp_setup} (Color online) Experimental setup for the two-laser experiment. The lasers were stabilized by two Toptica Digilok modules \n\tlocked to error signals generated from saturated absorption spectroscopy measurements made in a separate, magnetically shielded cell.}\n\\end{figure}\n\\begin{figure}\n\t\\includegraphics[width=0.45\\textwidth]{Cs_limenu_shema_DiviStari.pdf}\n\t\\caption{\\label{fig:levels} Level scheme for the two-laser experiment. The bold, solid arrow represents the pump laser transition, \n\twhereas the arrows with dashed lines represent the scanning laser transitions. Other transitions are given as thin, solid lines.}\n\\end{figure}\n\nA bandpass filter (890~nm $\\pm$ 10~nm) was placed before the photodiode. \nTo reduce noise from the intense pump beam, the probe beam was modulated by placing a mechanical \nchopper near its focus, and the fluorescence signal was passed through a lock-in amplifier and recorded \non a digital oscilloscope (Yokogawa DL-6154).  \nThe probe laser was scanned through the pump laser beam profile using \na mirror mounted on a moving platform (Nanomax MAX301 from Thorlabs) with a scan range of 8 mm in one dimension. \nThe probing beam itself had a full width at half maximum (FWHM) \ndiameter of~\\SI{200}{\\micro\\metre} with typical laser power of~\\SI{100}{\\micro\\watt}. \nThe pump beam width was~\\SI{1.3}{\\milli\\metre} (FWHM) and its power was~\\SI{40}{\\milli\\watt}. This laser beam diameter was achieved by letting the \nlaser beam slowly diverge after passing the \nfocal point of a lens with focal length of~\\SI{1}{\\metre}. The pump laser beam diverged slowly enough to be effectively \nconstant within the vapor cell.\nThe probe beam was also focussed by the same lens to reach its focus point inside the cell. \n\n\n\n\n\n\n\\section{\\label{1laser:level1}Application of the model to magneto-optical signals obtained for high laser power densities}\nAs a first test for the numerical model with multiple regions inside the laser beam, we used the model to calculate the \nshapes of magneto-optical resonances for $^{87}$Rb in an optical cell. The experimental setup was described earlier \n(see Fig.~\\ref{fig:exp_Rb87}).   Figure~\\ref{fig:one_laser_exp}(a)--(c) \nshow experimental signals (markers) and theoretical calculations (curves) of magneto-optical signals in the \n$F_g=2\\longrightarrow F_e=1$ transition of the $D_1$ line of $^{87}$Rb. Three theoretical curves are shown: \ncurve N1 was calculated assuming a laser beam with a single average \nintensity; curve N20 was calculated using a laser beam divided into 20 concentric regions; curve N20MT was calculated \nin the same way as curve N20, but furthermore the results were averaged also over trajectories that did not pass through \nthe center. At the relatively low Rabi frequency of $\\Omega_R = 2.5$~MHz [Fig.~\\ref{fig:one_laser_exp}(a)] \nall calculated curves practically coincided and described well the experimental signals. The single region model \ntreats the beam as a cylindrical beam with an intensity of 2~mW\/cm$^2$, which is below the saturation intensity for \nthat transition of 4.5~mW\/cm$^2$~\\cite{Steck:rubidium87}. When the laser intensity was 20~mW\/cm$^2$ ($\\Omega_R = 8.0$~MHz), \nwell above the saturation intensity, model N1 is no longer adequate for describing the experimental signals \nand model N20MT works slightly better [Fig.~\\ref{fig:one_laser_exp}(b)]. \nIn particular, the resonance becomes sharper and sharper as the intensity increases, and models \nN20 and N20MT reproduce this sharpness. Even at an intensity around 200~mW\/cm$^2$ ($\\Omega_R = 25$~MHz),  \nthe models with 20 regions describe the shape of the experimental curve quite well, \nwhile model N1 describes the experimental results poorly in terms of width and overall shape [Fig.~\\ref{fig:one_laser_exp}(c)]. \n\n\\begin{figure}[htpb]\n\t\\includegraphics[width=0.45\\textwidth]{rb87_a.pdf}\n\t\\includegraphics[width=0.45\\textwidth]{rb87_b.pdf}\n\t\\includegraphics[width=0.45\\textwidth]{rb87_c.pdf}\n\t\\caption{(Color online) Magneto-optical resonances for the $F_g=2\\longrightarrow F_e=1$ transition of the $D_1$ line of $^{87}$Rb. \n\tFilled circles represent experimental measurements for (a) 28 $\\mu$W ($\\Omega_R$=2.5 MHz) \n\t(b) 280 $\\mu$W ($\\Omega_R$=8.0 MHz), and \n\t(c) 2800  $\\mu$W ($\\Omega_R$=25 MHz). \n\tCurve N1 (dashed) shows the results of a theoretical model that uses one Rabi frequency to model the entire beam profile. \n\tCurve N20 (dash-dotted)  shows the result of the calculation when the laser beam profile is divided into 20 concentric circles, \n\tand the optical Bloch equations are solved separately for each circle. \n\tCurve N20MT (solid) shows the results for a calculation with 20 concentric regions when trajectories are taken into account \n\tthat do not pass through the center of the beam.  \n\t}\n\t\\label{fig:one_laser_exp}\n\\end{figure}\n\n\n\\section{\\label{distribution:level1}Investigation of the spatial distribution of fluorescence in an intense laser beam}\n\\subsection{Theoretical investigation of the spatial dynamics of fluorescence in an extended beam}\nIn order to describe the magneto-optical signals in the previous sections, the fluorescence from all concentric beam regions \nin models N20 and N20MT was summed, since usually experiments measure only total fluorescence (or absorption), especially if \nthe beams are narrow. \nHowever, solving the optical Bloch equations separately for different concentric regions of the laser beam, it is possible \nto calculate the strength of the fluorescence as a function of distance from the center of the beam. With an appropriate \nexperimental technique, the distribution of fluorescence within a laser beam could also be measured. \n\nFigure~\\ref{fig:dynamics} shows the calculated fluorescence distribution as a function of position in the laser beam. \nAs atoms move through the beam in one direction, the intense laser radiation optically pumps the ground state. In a very \nintense beam, the ground state levels that can absorb light have emptied even before the atoms reach the center \n(solid, green curve). Since atoms are actually traversing the beam from all directions, the result is a fluorescence profile with a \nreduction in intensity near the center of the beam (dashed, red curve). \n\\begin{figure}[htpb]\n\t\\includegraphics[width=0.45\\textwidth]{f3.pdf}\n\t\\caption{(Color online) Fluorescence distribution as a function of position in the laser beam.\n\tDotted (blue) line---laser beam profile, solid (green) line---fluorescence from atoms moving in one direction;\n\tdash-dotted (red) line---the overall fluorescence as a function of position that results \n\tfrom averaging all beam trajectories. Results from theoretical calculations.}\n\t\\label{fig:dynamics}\n\\end{figure}\nThe effect of increasing the laser beam intensity (or Rabi frequency) can be seen in Fig.~\\ref{fig:dynamics_rabi}.\nAt a Rabi frequency of $\\Omega_R=0.6$ MHz, the fluorescence profile tracks the intensity profile of the laser beam \nexactly. When the Rabi frequency is increased ten times ($\\Omega_R=6.0$ MHz), which corresponds to an intensity \nincrease of 100, the fluorescence profile already appears somewhat deformed and wider than the actual laser beam profile. \nAt Rabi frequencies of $\\Omega_R=48.0$ MHz and greater, the fluorescence intensity at the center of the intense laser beam \nis weaker than towards the edges as a result of the ground state being depleted by the intense radiation \nbefore the atoms reach the center of the laser beam.\n\\begin{figure}[htpb]\n\t\\includegraphics[width=0.45\\textwidth]{f4.pdf}\n\t\\caption{(Color online) Fluorescence distribution as a function of position in the laser beam for various values of the Rabi frequency. \n\tResults from theoretical calculations. As the Rabi frequency increases, the distribution becomes broader.}\n\t\\label{fig:dynamics_rabi}\n\\end{figure}\n \n\\subsection{Experimental study of the spatial dynamics of excitation and fluorescence in an intense, extended beam}\n\nIn order to test our theoretical model of the spatial distribution of fluorescence from atoms in an intense, extended pumping beam, \nwe decided to record magneto-optical resonances  \nfrom various positions in the pumping beam. The experimental setup is shown in \nFig.~\\ref{fig:exp_setup}. To visualize these data, surface plots were generated where one horizontal \naxis represented the magnetic field and the other, the position of the probe beam relative to the pump beam axis. The \nheight of the surface represented the fluorescence intensity. In essense, the surface consists of a series of \nmagneto-optical resonances recorded for a series of positions of the probe beam axis relative the the pump beam axis. \nFig.~\\ref{fig:p44_s43} shows the results for experiments [(a)] and calculations [(b)] for which the pump beam was tuned to the \n$F_g=4\\longrightarrow F_e=4$ transition of the Cs $D_2$ line and the probe beam was tuned to the\n$F_g=4\\longrightarrow F_e=3$ transition of the Cs $D_1$ line. \n\\begin{figure*}\n\t\\includegraphics[width=0.45\\textwidth]{f7a.pdf}\n\t\\includegraphics[width=0.45\\textwidth]{f7b.pdf}\n\t\\caption{\\label{fig:p44_s43} (Color online) Magneto-optical resonances produced for various positions of the probing laser beam \n\t($F_g=4\\longrightarrow F_e=3$ transition of the $D_1$ line of cesium) with respect to the pump laser beam \n\t($F_g=4\\longrightarrow F_e=4$ transition of the $D_2$ line of cesium): \n\t(a) experimental results and (b) theoretical calculations.  }\n\\end{figure*}\nOne can see that the theoretical plot reproduces qualitatively all the features of the experimental measurement. \nSimilar agreement can be observed when the  probe beam was tuned to the $F_g=3\\longrightarrow F_e=4$ transition \nof the Cs $D_1$ line, as \nshown in Fig.~\\ref{fig:p44_s34}. \n\\vfill\n\\break\n\\begin{figure*}\n\t\\includegraphics[width=0.45\\textwidth]{f8a.pdf}\n\t\\includegraphics[width=0.45\\textwidth]{f8b.pdf}\n\t\\caption{\\label{fig:p44_s34} (Color online) Magneto-optical resonances produced for various positions of the probe laser beam \n\t($F_g=3\\longrightarrow F_e=4$ transition of the $D_1$ line of cesium) with respect to the pump laser beam \n\t($F_g=4\\longrightarrow F_e=4$ transition of the $D_2$ line of cesium): \n\t(a) experimental results and (b) theoretical calculations.   }\n\\end{figure*}\n\n\n\n\\section{\\label{Conclusions:level1}Conclusions}\nWe have set out to model magneto-optical signals more accurately at laser intensities significantly higher than the saturation \nintensity by dividing the laser beam into concentric circular regions and solving the rate equations for Zeeman coherences in each region while \ntaking into account the actual laser intensity in that region and the transport of atoms between regions. This approach was used to \nmodel magneto-optical resonances for the $F_g=2 \\longrightarrow F_e=1$ transitions of the $D_1$ line of $^{87}$Rb, comparing the \ncalculated curves to measured signals. \nWe have demonstrated that good agreement between theory and experiment can be achieved up to Rabi frequencies of at least 25~MHz, \nwhich corresponds to a laser intensity of 200 mW\/cm$^2$, or more than 40 times the saturation intensity of the transition.\nAs an additional check on the model, we have studied the spatial distribution of the fluorescence intensity within a laser beam theoretically \nand experimentally. The results indicated that at high laser power densities, the maximum fluorescence intensity is not produced \nin the center of the beam, because the atoms have been pumped free of absorbing levels prior to reaching the center. We compared\nexperimental and theoretical signals of magneto-optical resonance signals obtained by exciting cesium atoms with a \nnarrow, weak probe beam tuned to the $D_1$ transition at various locations inside a region illuminated by an intense pump beam \ntuned to the $D_2$ transition and obtained good qualitative agreement. \n\n \n\\begin{acknowledgments}\nWe gratefully acknowledge support from the Latvian Science Council Grant Nr. 119\/2012 and \nfrom the University of Latvia Academic Development Project Nr. AAP2015\/B013.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\nPrincipal component analysis (PCA) is  \nthe ``workhorse'' method\n  for dimensionality reduction and feature extraction. It finds well-documented applications, including  statistics, bioinformatics, genomics, quantitative finance, and engineering, to name just a few. The goal of PCA is to obtain low-dimensional representations for high-dimensional data, while preserving most of the high-dimensional data variance \\cite{1901pca}. \n\n\nYet, various practical scenarios involve \\emph{multiple} datasets, in which one is tasked with extracting the most discriminative information of one target dataset relative to others. \nFor instance, consider two gene-expression measurement datasets of volunteers from across different geographical  areas and genders: the first dataset collects gene-expression levels of cancer patients, considered here as the \n \\emph{target data}, \nwhile the second contains levels  from healthy individuals corresponding here to our \n \\emph{background data}. The  goal is to identify molecular subtypes of cancer within cancer patients.\nPerforming PCA on either the target data or the target together with background data is likely to yield principal components (PCs) that correspond to\nthe background information common to both datasets (e.g., the demographic patterns and genders) \\cite{1998background}, rather than the PCs uniquely describing the subtypes of cancer.\nAlbeit simple to comprehend and practically relevant, such discriminative data analytics has not been thoroughly addressed. \n\n\nGeneralizations of PCA include kernel (K) PCA \\cite{kpca,2017kpca}, graph PCA \\cite{proc2018gsk}, \n$\\ell_1$-PCA \\cite{2018l1pca}, \n robust PCA \\cite{jstsp2016shahid},\nmulti-dimensional scaling \\cite{mds}, \nlocally linear embedding \\cite{lle}, \nIsomap \\cite{2000isomap}, and Laplacian eigenmaps \\cite{2003eigenmap}. Linear discriminant analysis (LDA) is a\n\\emph{supervised} classifier of linearly projected reduced dimensionality data vectors. It is designed so that linearly projected training vectors (meaning \\emph{labeled} data) of the same class stay as close as possible, while projected data of different classes are positioned as far as possible \\cite{1933lda}. \nOther discriminative methods include \n\tre-constructive and discriminative subspaces \\cite{fidler2006combining},\n\t discriminative vanishing component analysis \\cite{hou2016discriminative}, \n\t and kernel LDA \\cite{mika1999fisher},\n\t  which similar to LDA rely on labeled data.\nSupervised PCA looks for orthogonal projection vectors so that the dependence of projected vectors from \n\tone dataset on the other dataset is maximized \\cite{barshan2011supervised}.\n\nMultiple-factor analysis, an extension of PCA to deal with multiple datasets, is implemented in two steps: S1) normalize each dataset by the largest eigenvalue of \n\tits sample covariance matrix; and, S2) perform PCA on the combined dataset of all normalized ones \\cite{abdi2013multiple}.\nOn the other hand, canonical correlation analysis is widely employed for analyzing multiple datasets \\cite{1936cca,2018cwsggcca,2018gmcca}, but its goal is to extract the shared low-dimensional structure. \nThe recent proposal called contrastive (c) PCA aims at extracting contrastive information between two datasets \\cite{2017cpca}, by searching for directions along which the target data variance is large while that of \n the background data  is small. Carried out using the singular value decomposition (SVD), cPCA can  reveal dataset-specific information often missed by standard PCA if the involved hyper-parameter is properly selected.  \nThough possible \n to automatically choose\nthe best hyper-parameter from a list of candidate values, performing SVD multiple\ntimes can be computationally cumbersome in large-scale feature extraction settings.\n\n\n  \n\n\nBuilding on but going beyond cPCA, this paper starts by developing a novel approach, termed discriminative (d) PCA, for discriminative analytics of \\emph{two} datasets. dPCA looks for linear projections (as in LDA) but of \\emph{unlabeled}\n data vectors, by \\textcolor{black}{maximizing the variance of projected  target data while minimizing that of background data. This  leads to a \\emph{ratio trace} maximization formulation,}\n and also justifies our chosen term \\emph{discriminative  PCA}. Under certain conditions, dPCA is proved to be least-squares (LS) optimal in the sense that it reveals PCs specific to the target data relative to background data. Different from \n cPCA, dPCA is parameter free, and it requires a single generalized eigendecomposition, lending itself favorably to large-scale discriminative data analytics. \n  However, real data vectors \n   often exhibit nonlinear correlations, rendering dPCA inadequate for complex practical setups. To this end, nonlinear dPCA is developed via kernel-based learning. Similarly, the solution of KdPCA can be provided analytically in terms of generalized eigenvalue decompositions. As the complexity of KdPCA grows only linearly with the dimensionality of data vectors, KdPCA is preferable over dPCA for discriminative analytics of high-dimensional data.\n\n\n\n\n\ndPCA is further extended to handle multiple (more than two) background datasets. Multi-background (M) dPCA is developed to extract low-dimensional discriminative structure unique to the target data but not to \\emph{multiple} sets of background data. This becomes possible by maximizing  \\textcolor{black}{the variance of projected \n\t target data while minimizing the sum of variances of all projected  \n\t  background data.}\nAt last, kernel (K) MdPCA is put forth to account for nonlinear data correlations.\n\n\n\\emph{Notation}: Bold uppercase (lowercase) letters denote matrices (column vectors).\nOperators $(\\cdot)^{\\top}$,\n$(\\cdot)^{-1}$, and ${\\rm Tr}(\\cdot)$ denote matrix transposition, \ninverse, and trace, respectively; \n$\\|\\mathbf{a}\\|_2 $ is the $\\ell_2$-norm of vector $\\mathbf{a}$;\n  ${\\rm diag}(\\{a_i\\}_{i=1}^m)$ is a diagonal matrix holding elements $\\{a_i\\}_{i=1}^m$ on its main diagonal; \n  $\\mathbf{0}$ denotes all-zero vectors or matrices;\n  and $\\mathbf{I}$ represents identity matrices of suitable dimensions.\n\n\n\\section{Preliminaries and Prior Art}\\label{sec:preli}\nConsider two datasets, namely a target dataset $\\{\\mathbf{x}_i\\in\\mathbb{R}^D\\}_{i=1}^m$ that we are interested in analyzing, and a background dataset $\\{\\mathbf{y}_j\\in\\mathbb{R}^D\\}_{j=1}^n$ that contains latent background-related vectors also present  in the target data. Generalization to multiple background datasets will be presented in Sec. \\ref{sec:mdpca}. \nAssume without loss of generality  that both datasets are centered; in other words, \n\\textcolor{black}{the sample mean $m^{-1}\\sum_{i=1}^m \\mathbf{x}_i$ $(n^{-1}\\sum_{j=1}^n \\mathbf{y}_j)$ has been subtracted from each $\\mathbf{x}_i$ $(\\mathbf{y}_j)$.}\nTo motivate our novel approaches \nin subsequent sections, some basics of PCA and cPCA are outlined next.\n\n\nStandard PCA handles a single dataset at a time. \nIt looks for low-dimensional representations $\\{\\boldsymbol{\\chi}_i\\in\\mathbb{R}^d \\}_{i=1}^m$ of $\\{\\mathbf{x}_i\\}_{i=1}^m$ with $d<D$ as linear projections  of $\\{\\mathbf{x}_i \\}_{i=1}^m$ by maximizing the variances of $\\{\\bm{\\chi}_i \\}_{i=1}^m$ \\cite{1901pca}. Specifically for $d=1$, (linear) PCA yields  ${\\chi}_i:=\\hat{\\mathbf{u}}^\\top\\mathbf{x}_i$, with the  vector $\\hat{\\mathbf{u}}\\in\\mathbb{R}^D$ found by\n\t\\begin{equation}\n\t\t\\label{eq:pca}\n\t\t\t\\hat{\\mathbf{u}}:=\\arg\n\t\t\t\\underset{\\mathbf{u}\\in\\mathbb{R}^D}{\\max}\n\t\t\t\\quad\\mathbf{u}^\\top\\mathbf{C}_{xx}\\mathbf{u}\\quad {\\rm s.\\,to}\\quad \\mathbf{u}^\\top\\mathbf{u}=1\t\n\t\t\\end{equation}\t\nwhere $\\mathbf{C}_{xx}:=(1\/m)\\sum_{i=1}^m\\mathbf{x}_i\\mathbf{x}_i^\\top\\in\\mathbb{R}^{D\\times D}$ is the sample covariance matrix of $\\{\\mathbf{x}_i \\}_{i=1}^m$. Solving \\eqref{eq:pca} yields $\\hat{\\mathbf{u}}$ as the \\textcolor{black}{normalized} eigenvector of $\\mathbf{C}_{xx}$ corresponding to the largest eigenvalue.\nThe resulting projections $\\{\\chi_i =\\hat{\\mathbf{u}}^\\top\\mathbf{x}_i \\}_{i=1}^m$ \n\\textcolor{black}{constitute}\n  the first principal \n  \\textcolor{black}{component (PC)} of the target data vectors. \nWhen $d>1$, PCA looks for  $\\{\\mathbf{u}_i\\in\\mathbb{R}^D \\}_{i=1}^d$, \nobtained from  the \\textcolor{black}{$d$ eigenvectors of $\\mathbf{C}_{xx}$ associated with the first $d$ largest eigenvalues sorted in a decreasing order}. As alluded to in Sec. \\ref{sec:intro}, PCA applied on $\\{\\mathbf{x}_i \\}_{i=1}^m$ only, or on the combined datasets\n $\\{\\{\\mathbf{x}_i\\}_{i=1}^m,\\,\\{\\mathbf{y}_j\\}_{j=1}^n \\}$ can generally not uncover the discriminative patterns or features of the target data relative to the background data.\n\n\n\n\nOn the other hand, the recent cPCA seeks a vector $\\mathbf{u}\\in\\mathbb{R}^D$ along which the target data exhibit large variations while the background\n data exhibit small variations, via solving \\cite{2017cpca}\n\t\\begin{subequations}\n\t\t\\label{eq:cpca}\n\t\t\\begin{align}\n\t\t\t\\underset{\\mathbf{u}\\in\\mathbb{R}^D}{\\max}\n\t\t\t\\quad&\\mathbf{u}^\\top\\mathbf{C}_{xx}\\mathbf{u}-\\alpha \\mathbf{u}^\\top\\mathbf{C}_{yy}\\mathbf{u}\\label{eq:cpcaobj}\\\\\n\t\t\t{\\rm s.\\,to}\\quad &\\mathbf{u}^\\top\\mathbf{u}=1\n\t\t\\end{align}\t\t\n\t\\end{subequations}\nwhere $\\mathbf{C}_{yy}:=(1\/n)\\sum_{j=1}^n\\mathbf{y}_j\\mathbf{y}_j^\\top\\in\\mathbb{R}^{D\\times D}$ denotes the sample covariance matrix of $\\{\\mathbf{y}_j \\}_{j=1}^n$, and the hyper-parameter $\\alpha\\ge 0$ trades off maximizing the target data variance (the first term in \\eqref{eq:cpcaobj}) for minimizing the background data variance (the second term). For a given $\\alpha$, the solution of \\eqref{eq:cpca} is given by the eigenvector of $\\mathbf{C}_{xx}-\\alpha\\mathbf{C}_{yy}$ associated with its largest eigenvalue, along which the obtained data projections constitute  the first contrastive (c) \\textcolor{black}{PC}. Nonetheless, there is no rule of thumb for choosing $\\alpha$.\n\\textcolor{black}{A spectral-clustering based algorithm was devised to automatically select $\\alpha$ from a list of candidate values \\cite{2017cpca}, but its brute-force search is computationally expensive to use in large-scale datasets.}\n\n\n\n\n\n\\section{Discriminative Principal Component Analysis} \\label{sec:dpca}\nUnlike PCA, LDA is a supervised classification method of linearly projected data at reduced dimensionality.   \nIt finds those linear projections  that reduce that variation in the same class and increase the separation between classes \\cite{1933lda}. This is accomplished by maximizing the ratio of the labeled data variance between classes to that within the classes. \n\nIn a related but unsupervised setup, \nconsider we are given a target dataset and a background dataset, and we are tasked with \\textcolor{black}{extracting   vectors that are meaningful in representing\n\t  $\\{\\mathbf{x}_i\\}_{i=1}^m$, but not $\\{\\mathbf{y}_j\\}_{j=1}^n$.}\n A meaningful \n approach would then be to maximize the ratio of the projected target data variance over that of the background data. Our  \\emph{discriminative (d) PCA} approach finds\n\t\t\\begin{equation}\t\\label{eq:dpca}\n\t\t\\textcolor{black}{\n\t\t\t\\hat{\\mathbf{u}}:=\\arg\n\t\t\t\\underset{\\mathbf{u}\\in\\mathbb{R}^{D}}{\\max}\n\t\t\t\\quad\\frac{\\mathbf{u}^\\top\\mathbf{C}_{xx}\\mathbf{u}}{ \\mathbf{u}^\\top\\mathbf{C}_{yy}\\mathbf{u}}}\n\t\t\\end{equation}\nWe will term the solution in \\eqref{eq:dpca}  discriminant subspace vector,\nand the projections $\\{\\hat{\\mathbf{u}}^\\top\\mathbf{x}_i\\}_{i=1}$  the first discriminative (d) \\textcolor{black}{PC}. \nNext, we discuss the solution in  \\eqref{eq:dpca}.\n\n\nUsing Lagrangian duality theory, the solution in \\eqref{eq:dpca} corresponds to the right eigenvector of $\\mathbf{C}_{yy}^{-1}\\mathbf{C}_{xx}$ associated with the largest eigenvalue. \nTo establish this,  note that \\eqref{eq:dpca} can be equivalently rewritten a\n\t\\begin{subequations}\n\\label{eq:dpcafm2}\n\\begin{align}\n\t\\hat{\\mathbf{u}}:=\\arg\n\t\t\\underset{\\mathbf{u}\\in\\mathbb{R}^{D}}{\\max}\\quad& \\mathbf{u}^\\top\\mathbf{C}_{xx}\\mathbf{u}\\label{eq:dpcafm2cos}\\\\\n\t{\\rm s.\\,to}\\quad& \\mathbf{u}^\\top\\mathbf{C}_{yy}\\mathbf{u}=1.\\label{eq:dpcafm2con}\n\\end{align}\n\t\\end{subequations}\n\tLetting $\\lambda$ denote the dual variable associated with the constraint \\eqref{eq:dpcafm2con}, the Lagrangian of \\eqref{eq:dpcafm2} becomes\n\\begin{equation}\\label{eq:lag}\n\\mathcal{L}(\\mathbf{u};\\,\\lambda)=\\mathbf{u}^\\top\\mathbf{C}_{xx}\\mathbf{u}+\\lambda\\left(1-\\mathbf{u}^\\top\\mathbf{C}_{yy}\\mathbf{u}\\right).\n\\end{equation}\nAt the optimum $(\\hat{\\mathbf{u}};\\,\\hat{\\lambda})$, \nthe KKT conditions confirm that\n\\begin{equation}\\label{eq:gep}\n\t\\mathbf{C}_{xx}\\hat{{\\mathbf{u}}}=\\hat{\\lambda}\\mathbf{C}_{yy}\\hat{\\mathbf{u}}.\n\\end{equation}\nThis is a generalized eigen-equation, whose solution $\\hat{\\mathbf{u}}$ is the generalized eigenvector of $(\\mathbf{C}_{xx},\\,\\mathbf{C}_{yy})$ corresponding to the generalized eigenvalue $\\hat{\\lambda}$. \nLeft-multiplying\n  \\eqref{eq:gep} by $\\hat{\\mathbf{u}}^\\top$ yields\n$\t\\hat{\\mathbf{u}}^\\top\\mathbf{C}_{xx}\\hat{\\mathbf{u}}=\\hat{\\lambda} \\hat{\\mathbf{u}}^\\top\\mathbf{C}_{yy}\\hat{\\mathbf{u}}\n$, corroborating that the optimal objective value of \\eqref{eq:dpcafm2cos} is attained when $\\hat{\\lambda}:=\\lambda_1$ is the largest generalized eigenvalue. Furthermore, \\eqref{eq:gep} can be solved efficiently using well-documented solvers that rely on e.g., Cholesky's factorization \\cite{saad1}. \n \n \n Supposing further that $\\mathbf{C}_{yy}$ is nonsingular\n   \\eqref{eq:gep}  yields \n  \\begin{equation}\n  \\label{eq:dpcasol}\n  \\mathbf{C}_{yy}^{-1}\\mathbf{C}_{xx}\\hat{\\mathbf{u}}=\\hat{\\lambda}\\hat{\\mathbf{u}}\n  \\end{equation}\nimplying that  $\\hat{\\mathbf{u}}$ in \\eqref{eq:dpcafm2} is the right eigenvector of $\\mathbf{C}_{yy}^{-1}\\mathbf{C}_{xx}$ corresponding to the largest eigenvalue    $\\hat{\\lambda}=\\lambda_1$.\n\n\n\nTo find multiple ($d\\ge 2$) subspace vectors, namely $\\{\\mathbf{u}_i\\in\\mathbb{R}^D\\}_{i=1}^d$ that form $\\mathbf{U}:=[\\mathbf{u}_1 \\, \\cdots \\, \\mathbf{u}_d]\\in\\mathbb{R}^{D\\times d}$, \\textcolor{black}{in \\eqref{eq:dpca} with  $\\mathbf{C}_{yy}$ being nonsingular,}  can be generalized as follows (cf. \\eqref{eq:dpca})\n\\vspace{4pt}\n\\textcolor{black}{\n\t\\begin{equation}\n\t\t\\label{eq:dpcam}\t\n\t\t\\hat{\\mathbf{U}}:=\\arg\t\\underset{\\mathbf{U}\\in\\mathbb{R}^{D\\times d}}{\\max}~\n\t\t{\\rm Tr}\\left[\\left(\\mathbf{U}^\\top\\mathbf{C}_{yy}\\mathbf{U}\\right )^{-1}\\mathbf{U}^\\top\\mathbf{C}_{xx}\\mathbf{U}\\right].\n\t\\end{equation}\n\t}\n\n\nClearly, \\eqref{eq:dpcam} is a \\emph{ratio trace} maximization problem; see e.g., \\cite{2014mati}, whose  solution\nis given in Thm. \\ref{the:dpca} (see a proof in \\cite[p. 448]{2013fukunaga}).\n\n\n\n\\begin{theorem}\n\t\\label{the:dpca}\n\tGiven centered data $\\{{\\mathbf{x}}_i\\in\\mathbb{R}^{D}\\}_{i=1}^m$ and $\\{{\\mathbf{y}}_j\\in\\mathbb{R}^{D}\\}_{j=1}^n$ with sample covariance matrices $\\mathbf{C}_{xx}:=(1\/m)\\sum_{i=1}^m\\mathbf{x}_i\\mathbf{x}_i^\\top$ and $\\mathbf{C}_{yy}:=(1\/n)\\sum_{j=1}^n\\mathbf{y}_j\\mathbf{y}_j^\\top\\succ\\mathbf{0}$, the $i$-th column of the dPCA optimal solution $\\hat{\\mathbf{U}}\\in\\mathbb{R}^{D\\times d}$ in \\eqref{eq:dpcam} is given by the right eigenvector of  $\\mathbf{C}_{yy}^{-1}\\mathbf{C}_{xx}$ associated with the $i$-th largest eigenvalue, where $i=1,\\,\\ldots,\\,d$.\n\\end{theorem}\n\n\n\nOur dPCA for \ndiscriminative analytics of two datasets is summarized in Alg. \\ref{alg:dpca}.\n\\textcolor{black}{Four} remarks are now in order.\n\n\\begin{remark}\nWithout background data, we have $\\mathbf{C}_{yy}=\\mathbf{I}$, and dPCA boils down to the standard PCA. \n\\end{remark}\n\n\n\\begin{remark}\n\tSeveral possible combinations of target and background datasets include:\n\t i) measurements from a healthy group $\\{\\mathbf{y}_j\\}$ and a diseased group $\\{\\mathbf{x}_i\\}$, where the former has similar population-level variation \n\t with the latter, but distinct variation \n\t due to subtypes of diseases; ii) before-treatment $\\{\\mathbf{y}_j\\}$ and after-treatment   $\\{\\mathbf{x}_i\\}$ datasets, in which the former contains additive \n\t measurement noise rather than the variation caused by treatment; and iii) signal-free $\\{\\mathbf{y}_j\\}$ and signal recordings $\\{\\mathbf{x}_i\\}$, where the former consists of only noise.\n\\end{remark}\n\n\\begin{remark}\\label{re:twoeq}\nConsider the eigenvalue decomposition \n$\\mathbf{C}_{yy}=\\mathbf{U}_y\\mathbf{\\Sigma}_{yy}\\mathbf{U}_y^\\top$. \nWith $\\mathbf{C}_{yy}^{1\/2}:=\\mathbf{\\Sigma}_{yy}^{1\/2}\\mathbf{U}_{y}^\\top$, and the definition \n  $\\mathbf{v}:=\\mathbf{C}_{yy}^{\\top\/2}\\mathbf{u}\\in\\mathbb{R}^D$,  \\eqref{eq:dpcafm2} can be expressed as\t\n\t\\begin{subequations}\n\t\t\\label{eq:v}\n\t\\begin{align}\n\t\t\\hat{\\mathbf{v}}:=\\arg\n\t\\max_{\\mathbf{v}\\in\\mathbb{R}^D}\\quad&\n\t\t\\mathbf{v}^\\top\\mathbf{C}^{-1\/2}_{yy}\\mathbf{C}_{xx}\\mathbf{C}^{-\\top\/2}_{yy}\\mathbf{v}\\\\\n\t\t{\\rm s.\\,to}\\quad &\\mathbf{v}^\\top\\mathbf{v}=1\t\n\t\t\\end{align}\n\t\t\\end{subequations}\nwhere $\\hat{\\mathbf{v}}$ corresponds to the leading eigenvector of $\\mathbf{C}_{yy}^{-1\/2}\\mathbf{C}_{xx} \\mathbf{C}_{yy}^{-\\top\/2}$. Subsequently,  $\\hat{\\mathbf{u}}$ in \\eqref{eq:dpcafm2} is recovered as $\\hat{\\mathbf{u}}=\\mathbf{C}_{yy}^{-\\top\/2}\\hat{\\mathbf{v}}$.\nThis indeed suggests that discriminative analytics of $\\{\\mathbf{x}_i\\}_{i=1}^m$ and $\\{\\mathbf{y}_j\\}_{j=1}^n$ using dPCA  \ncan be viewed as PCA of the `denoised' or `background-removed' data $\\{\\mathbf{C}^{-1\/2}_{yy}\\mathbf{x}_i \\}$,\nfollowed by an `inverse' transformation to \nmap the obtained subspace vector of the $\\{\\mathbf{C}^{-1\/2}_{yy}\\mathbf{x}_i \\}$ data  to $\\{\\mathbf{x}_i\\}$ that of the target \ndata.  \nIn this sense, $\\{\\mathbf{C}^{-1\/2}_{yy}\\mathbf{x}_i \\}$ can be seen as the data obtained after removing the dominant `background' subspace vectors from the target data. \n\n\n\n\t\n\n\n\n\\end{remark}\n\\begin{remark}\nInexpensive power or Lanczos iterations \\cite{saad1} can be employed to compute the principal eigenvectors in \\eqref{eq:dpcasol}.\n\\end{remark}\n\n\n\n\\begin{algorithm}[t]\n\t\\caption{Discriminative PCA.}\n\t\\label{alg:dpca}\n\t\\begin{algorithmic}[1]\n\t\t\\STATE {\\bfseries Input:}\n\t\tNonzero-mean target and background data $\\{\\accentset{\\circ}{\\mathbf{x}}_i\\}_{i=1}^m$ and $\\{\\accentset{\\circ}{\\mathbf{y}}_j\\}_{j=1}^n$; number of dPCs $d$.\n\t\t\\STATE {\\bfseries Exclude} the means from $\\{\\accentset{\\circ}{\\mathbf{x}}_i\\}$ and $\\{\\accentset{\\circ}{\\mathbf{y}}_j\\}$ to obtain centered data $\\{\\mathbf{x}_i\\}$, and $\\{\\mathbf{y}_j\\}$. Construct $\\mathbf{C}_{xx}$ and $\\mathbf{C}_{yy}$.\n\t\t\\STATE {\\bfseries Perform} \\label{step:4} eigendecomposition\n\t\ton $\\mathbf{C}_{yy}^{-1}\\mathbf{C}_{xx}$ to obtain the $d$ right eigenvectors $\\{\\hat{\\mathbf{u}}_i\\}_{i=1}^d$ associated with the $d$ largest eigenvalues.\n\t\t\\STATE {\\bfseries Output} $\\hat{\\mathbf{U}}=[\\hat{\\mathbf{u}}_1\\,\\cdots \\, \\hat{\\mathbf{u}}_d]$.\n\t\t\\vspace{-0pt}\n\t\\end{algorithmic}\n\\end{algorithm} \n\n\n\n\n\\textcolor{black}{Consider again \\eqref{eq:dpcafm2}. Based on Lagrange duality, when selecting $\\alpha=\\hat{\\lambda}$ in \\eqref{eq:cpca}, where $\\hat{\\lambda}$ is the largest eigenvalue of $\\mathbf{C}_{yy}^{-1}\\mathbf{C}_{xx}$, cPCA maximizing $\\textcolor{black}{{\\mathbf{u}}^\\top}(\\mathbf{C}_{xx}-\\hat{\\lambda}\\mathbf{C}_{yy}){\\mathbf{u}}$ is equivalent to $\\max_{\\mathbf{u}\\in\\mathbb{R}^{D}}\\mathcal{L}(\\mathbf{u};\\hat{\\lambda})=\\mathbf{u}^\\top(\\mathbf{C}_{xx}-\\hat{\\lambda}\\mathbf{C}_{yy})\\mathbf{u}+\\hat{\\lambda}$, which coincides with \\eqref{eq:lag} when $\\lambda=\\hat{\\lambda}$ at the optimum. \nThis suggests that the optimizers of cPCA and dPCA share the same direction when $\\alpha$ in cPCA is chosen to be the optimal dual variable $\\hat{\\lambda}$ of our dPCA in \\eqref{eq:dpcafm2}.\n\tThis equivalence between dPCA and cPCA with a proper $\\alpha$ can also be seen from \n\tthe following.\n\t\\begin{theorem}{\\cite[Theorem 2]{guo2003generalized}}\n\t\t\\label{the:cvsd}\n\t\tFor real symmetric matrices $\\mathbf{C}_{xx}\\succeq \\mathbf{0}$ and $\\mathbf{C}_{yy}\\succ\\mathbf{0}$, the following holds\n\t\t\t\\begin{equation*}\n\t\t\t\\check{\\lambda}=\\frac{\\check{\\mathbf{u}}^\\top\\mathbf{C}_{xx}\\check{\\mathbf{u}}}{\\check{\\mathbf{u}}^\\top\\mathbf{C}_{yy}\\check{\\mathbf{u}}}=\\underset{\\|\\mathbf{u}\\|_2=1}{\\max}\\frac{\\mathbf{u}^\\top\\mathbf{C}_{xx}\\mathbf{u}}{\\mathbf{u}^\\top\\mathbf{C}_{yy}\\mathbf{u}} \n\t\t\t\\end{equation*}\n\t\t\tif and only if\n\t\t\t\\begin{equation*}\n\t\t\t\\check{\\mathbf{u}}^\\top(\\mathbf{C}_{xx}-\\check{\\lambda}\\mathbf{C}_{yy})\\check{\\mathbf{u}}=\\underset{\\|\\mathbf{u}\\|_2=1}{\\max}\\mathbf{u}^\\top(\\mathbf{C}_{xx}-\\check{\\lambda}\\mathbf{C}_{yy})\\mathbf{u}.\n\t\t\t\\end{equation*}\n\t\\end{theorem}\n\t}\n\n\nTo gain further insight into the relationship between  dPCA and cPCA,\nsuppose that  $\\mathbf{C}_{xx}$ and $\\mathbf{C}_{yy}$ are simultaneously diagonalizable; that is, there exists an unitary matrix $\\mathbf{U}\\in\\mathbb{R}^{D\\times D}$ such that\n\\begin{equation*}\n\\mathbf{C}_{xx}:=\\mathbf{U}\\mathbf{\\Sigma}_{xx}\\mathbf{U}^\\top,\\quad {\\rm and}\\quad \\mathbf{C}_{yy}:=\\mathbf{U}\\mathbf{\\Sigma}_{yy}\\mathbf{U}^\\top\n\\end{equation*} \nwhere diagonal matrices $\\mathbf{\\Sigma}_{xx},\\,\\mathbf{\\Sigma}_{yy}\\succ \\mathbf{0}$ hold accordingly eigenvalues $\\{\\lambda_x^i\\}_{i=1}^D$ of $\\mathbf{C}_{xx}$ and $\\{\\lambda_y^i\\}_{i=1}^D$ of $\\mathbf{C}_{yy}$ on their main diagonals. \n\\textcolor{black}{Even if the two datasets may share some subspace vectors, $\\{\\lambda_x^i\\}_{i=1}^D$ and $\\{\\lambda_y^i\\}_{i=1}^D$ are in general not the same.}\nIt is easy to check that $\\mathbf{C}_{yy}^{-1}\\mathbf{C}_{xx}=\\mathbf{U}\\mathbf{\\Sigma}_{yy}^{-1}\\mathbf{\\Sigma}_{xx}\\mathbf{U}^\\top=\\mathbf{U} {\\rm diag}\\big(\\{\\frac{\\lambda_x^i}{\\lambda_y^i}\\}_{i=1}^D\\big)\\mathbf{U}^\\top$. Seeking the first $d$ latent subspace vectors is tantamount to taking the $d$ columns of $\\mathbf{U}$ that correspond to the $d$ largest values among $\\{\\frac{\\lambda_x^i}{\\lambda_y^i}\\}_{i=1}^D$.\nOn the other hand, cPCA for a fixed $\\alpha$, looks for the first $d$ latent subspace vectors of $\\mathbf{C}_{xx}-\\alpha\\mathbf{C}_{yy}=\\mathbf{U}(\\mathbf{\\Sigma}_{xx}-\\alpha{\\bm\\Sigma}_{yy})\\mathbf{U}^\\top=\\mathbf{U}{\\rm diag}\\big(\\{\\lambda_x^i-\\alpha\\lambda_y^i\\}_{i=1}^D\\big)\\mathbf{U}^\\top$, which amounts to taking the $d$ columns of $\\mathbf{U}$ associated with the $d$ largest values in $\\{\\lambda_x^i-\\alpha\\lambda_y^i\\}_{i=1}^D$. \n\\textcolor{black}{This further confirms that when $\\alpha$ is sufficiently large (small), cPCA returns the $d$ columns of $\\mathbf{U}$ associated with the $d$  largest $\\lambda_{y}^{i}$'s ($\\lambda_x^{i}$'s). When $\\alpha$ is not properly chosen, cPCA may fail to extract the most contrastive information from target data relative to background data. In contrast, this is not an issue is not present in dPCA simply because it has no tunable parameter.}\n\n\n\n\n\\section{Optimality of {d}PCA}\n\\label{sec:optim}\n\nIn this section, we show that dPCA is optimal when data obey a certain affine model. In a similar vein, PCA adopts  a factor analysis model  to express the non-centered background data $\\{\\accentset{\\circ}{\\mathbf{y}}_j\\in\\mathbb{R}^D\\}_{j=1}^n$ as\n\\begin{equation}\n\\accentset{\\circ}{\\mathbf{y}}_j=\\mathbf{m}_y+\n\\mathbf{U}_b\\bm{\\psi}_j+\\mathbf{e}_{y,j},\t\\quad j=1,\\,\\ldots,\\,n\\label{eq:y}\n\\end{equation}\nwhere $\\mathbf{m}_y\\in\\mathbb{R}^D$ denotes the unknown location (mean) vector; $\\mathbf{U}_b\\in\\mathbb{R}^{D\\times k}$ has orthonormal columns with $k<D$;\n $\\{\\bm{\\psi}_j\\in\\mathbb{R}^k\\}_{j=1}^n$ are some unknown coefficients with covariance matrix $\\bm{\\Sigma}_b:={\\rm diag}(\\lambda_{y,1},\\,\\lambda_{y,2},\\,\\ldots,\\,\\lambda_{y,k})\\in\\mathbb{R}^{k\\times k}$;\n and the modeling errors $\\{\\mathbf{e}_{y,j}\\in\\mathbb{R}^D\\}_{j=1}^n$ are assumed  zero-mean   with covariance matrix $\\mathbb{E}[\\mathbf{e}_{y,j}\\mathbf{e}_{y,j}^\\top]=\\mathbf{I}$. Adopting the least-squares (LS) criterion,\n the unknowns $\\mathbf{m}_y$, $\\mathbf{U}_b$, and $\\{\\bm{\\psi}_j\\}$ can be estimated by \\cite{1995pca} \n \\begin{equation*}\n\\underset{\\mathbf{m}_y,\\{\\bm{\\psi}_j\\}\\atop\n\\mathbf{U}_b\n}{\\min}\\quad \\sum_{j=1}^n\\left\\|\\accentset{\\circ}{\\mathbf{y}}_j-\\mathbf{m}_y-\\mathbf{U}_b\\bm{\\psi}_j\\right\\|_2^2\\quad{\\rm s.\\,to}\\quad  \\mathbf{U}_b^\\top\\mathbf{U}_b=\\mathbf{I}\n\\end{equation*}\nwe find at the optimum \n $\\hat{\\mathbf{m}}_y:=(1\/n)\\sum_{j=1}^n{\\accentset{\\circ}{\\mathbf{y}}}_j$, $\\big\\{\\hat{\\bm{\\psi}}_j:=\\hat{\\mathbf{U}}_b^\\top(\\accentset{\\circ}{\\mathbf{y}}_j-\\hat{\\mathbf{m}}_y)\\big\\}$, with $\\hat{\\mathbf{U}}_b$ columns given by the first $k$ leading eigenvectors of\n$\\mathbf{C}_{yy}=(1\/n)\\sum_{j=1}^n\\mathbf{y}_j\\mathbf{y}_j^\\top$, \nin which $\\mathbf{y}_j:={\\accentset{\\circ}{\\mathbf{y}}}_j-\\hat{\\mathbf{m}}_y$.  \nIt is clear that $\\mathbb{E}[\\mathbf{y}_j\\mathbf{y}_j^\\top]=\\mathbf{U}_b\\bm{\\Sigma}_{b}\\mathbf{U}_b^\\top+\\mathbf{I}$.\nLet matrix $\\mathbf{U}_n\\in\\mathbb{R}^{D\\times (D-k)}$ with orthonormal columns satisfying $\\mathbf{U}_n^\\top\\mathbf{U}_b=\\mathbf{0}$, and $\\mathbf{U}_y:=[\\mathbf{U}_b~\\mathbf{U}_n]\\in\\mathbb{R}^{D\\times D}$ with $\\bm{\\Sigma}_y:={\\rm diag}(\\{\\lambda_{y,i}\\}_{i=1}^D)$, where $\\{\\lambda_{y,k+\\ell}:=1\\}_{\\ell=1}^{D-k}$. Therefore, $\\mathbf{U}_b\\bm{\\Sigma}_{b}\\mathbf{U}_b^\\top+\\mathbf{I}=\\mathbf{U}_y\\bm{\\Sigma}_y\\mathbf{U}_y^\\top$. As $n\\to\\infty$, the strong law of large numbers asserts that $\\mathbf{C}_{yy}\\to\\mathbb{E}[\\mathbf{y}_j\\mathbf{y}_j^\\top]$; that is,  $\\mathbf{C}_{yy}=\\mathbf{U}_y\\bm{\\Sigma}_y\\mathbf{U}_y^\\top$ as $n\\to \\infty$.\n\n \nHere we assume that the target data  $\\{\\accentset{\\circ}{\\mathbf{x}}_i\\in\\mathbb{R}^D\\}_{i=1}^m$ share the  background related matrix $\\mathbf{U}_b$ with data $\\{\\accentset{\\circ}{\\mathbf{y}}_j\\}$, but also have $d$ extra  vectors \n specific to the target data relative to the background data. \n \\textcolor{black}{This assumption is well justified in realistic setups. In the example discussed in Sec. \\ref{sec:intro}, both patients'  and healthy persons' gene-expression data contain  common patterns corresponding to geographical and gender variances; while the patients' gene-expression data contain some specific latent subspace vectors corresponding to their diseases.}\n Focusing for simplicity  on $d=1$, we model $\\{\\accentset{\\circ}{\\mathbf{x}}_i\\}$ as\n\\begin{equation}\n\\accentset{\\circ}{\\mathbf{x}}_i=\\mathbf{m}_x+\\left[\\mathbf{U}_b~\\;\\mathbf{u}_s\\right]\\left[\\!\\!\\begin{array}{c}\\boldsymbol{\\chi}_{b,i}\\\\{\\chi}_{s,i}\\end{array}\\!\\!\\right]+\\mathbf{e}_{x,i},\\quad i=1,\\,\\ldots,\\,m\\label{eq:x}\n\\end{equation}\nwhere $\\mathbf{m}_x\\in\\mathbb{R}^D$ represents the location of $\\{\\accentset{\\circ}{\\mathbf{x}}_i\\}_{i=1}^m$; $\\{\\mathbf{e}_{x,i}\\}_{i=1}^m$ account for zero-mean modeling errors;  $\\mathbf{U}_x:=[\\mathbf{U}_b~\\mathbf{u}_s]\\in\\mathbb{R}^{D\\times (k+1)}$ collects orthonormal columns, where $\\mathbf{U}_b$ is the shared latent subspace vectors associated \n with background data, and $\\mathbf{u}_s\\in\\mathbb{R}^{D}$ is \n\\textcolor{black}{a latent subspace vector\n\t unique to the target data, but not to the background data.}\nSimply put, \nour goal is to extract this discriminative subspace $\\mathbf{u}_s$ given $\\{\\accentset{\\circ}{\\mathbf{x}}_i\\}_{i=1}^m$ and $\\{\\accentset{\\circ}{\\mathbf{y}}_j\\}_{j=1}^n$. \n\n\n\nSimilarly, given $\\{\\accentset{\\circ}{\\mathbf{x}}_i\\}$, the unknowns $\\mathbf{m}_x$, $\\mathbf{U}_x$, and\n $\\{\\bm{\\chi}_i:=[\\bm{\\chi}_{b,i}^\\top, \\,\\chi_{s,i}]^\\top\\}$\n can be estimated by \n  \t\\begin{equation*}\n \t\\underset{\\mathbf{m}_x,\\,\\{\\bm{\\chi}_i\\}\\atop\\mathbf{U}_x}{\\max}\n \t\\quad\\sum_{i=1}^m\\left\\|\\accentset{\\circ}{\\mathbf{x}}_i-\\mathbf{m}_x-\\mathbf{U}_x\\bm{\\chi}_i\\right\\|_2^2\n\t\\quad{\\rm s.\\,to}\\quad \\mathbf{U}_x^\\top\\mathbf{U}_x=\\mathbf{I} \n \t\\end{equation*}\n \tyielding $\\hat{\\mathbf{m}}_x:=(1\/m)\\sum_{i=1}^{m}\\accentset{\\circ}{\\mathbf{x}}_i$, $\\hat{\\bm{\\chi}}_i:=\\hat{\\mathbf{U}}_x^\\top\\mathbf{x}_i$ with $\\mathbf{x}_i:=\\accentset{\\circ}{\\mathbf{x}}_i-\\hat{\\mathbf{m}}_x$,\n where $\\hat{\\mathbf{U}}_x$ has columns the \\textcolor{black}{$(k+1)$ principal} eigenvectors of $\\mathbf{C}_{xx}=(1\/m)\\sum_{i=1}^{m}\\mathbf{x}_i\\mathbf{x}_i^\\top$.\nWhen $m\\to \\infty$, it holds that $\\mathbf{C}_{xx}\n=\\mathbf{U}_x \\bm{\\Sigma}_x \\mathbf{U}_x^\\top$, with $\\bm{\\Sigma}_x:=\\mathbb{E}[\\bm{\\chi}_i\\bm{\\chi}_i^\\top]={\\rm diag}(\\lambda_{x,1},\\,\\lambda_{x,2},\\,\\ldots,\\,\\lambda_{x,k+1})\\in\\mathbb{R}^{(k+1)\\times (k+1)}$. \n\n\n\n\nLet $\\bm{\\Sigma}_{x,k}\\in\\mathbb{R}^{k\\times k}$ denote the submatrix of $\\bm{\\Sigma}_x$ formed by its first $k$ rows and columns. When $m,\\,n\\to\\infty$ and $\\mathbf{C}_{yy}$ is nonsingular, one can express $\\mathbf{C}_{yy}^{-1}\\mathbf{C}_{xx}$ as  \n\\begin{align*}\n&\\mathbf{U}_y\\bm{\\Sigma}_y^{-1}\\mathbf{U}_y^\\top \\mathbf{U}_x\\bm{\\Sigma}_x\\mathbf{U}_x^\\top\\\\\n&=[\\mathbf{U}_b ~ \\mathbf{U}_n]\n\\left[\n\\!\\begin{array}\n{cc}\n\\bm{\\Sigma}_{b}^{-1} &\\mathbf{0}\\\\\n\\mathbf{0}  &\\mathbf{I}\n\\end{array}\\!\n\\right]\n\\left[\\!\n\\begin{array}\n{cc}\n\\mathbf{I}&\\mathbf{0}\\\\\n\\mathbf{0}&\\mathbf{U}_n^\\top \\mathbf{u}_s\n\\end{array}\n\\!\n\\right]\\\\\n&\\quad \\times\n\\left[\n\\!\\begin{array}\n{cc}\n\\bm{\\Sigma}_{x,k}&\\mathbf{0}\\\\\n\\mathbf{0}&\\lambda_{x,k+1}\n\\end{array}\n\\!\n\\right]\n\\left[\n\\!\n\\begin{array}\n{c}\n\\mathbf{U}_b^\\top\\\\\n\\mathbf{u}_s^\\top\n\\end{array}\n\\!\n\\right]\\\\\n&=[\\mathbf{U}_b ~ \\mathbf{U}_n]\n\\left[\\!\n\\begin{array}\n{cc}\n\\bm{\\Sigma}_{b}^{-1}\\bm{\\Sigma}_{x,k}&\\mathbf{0}\\\\\n\\mathbf{0}& \\lambda_{x,k+1}\\mathbf{U}_n^\\top \\mathbf{u}_s\n\\end{array}\n\\!\n\\right]\n\\left[\n\\!\n\\begin{array}\n{c}\n\\mathbf{U}_b^\\top\\\\\n\\mathbf{u}_s^\\top\n\\end{array}\n\\!\n\\right]\\\\\n&=\\mathbf{U}_b \\bm{\\Sigma}_{b}^{-1}\\bm{\\Sigma}_{x,k} \\mathbf{U}_b^\\top + \\lambda_{x,k+1}\\mathbf{U}_n\\mathbf{U}_n^\\top \\mathbf{u}_s\\mathbf{u}_s^\\top.\n\\end{align*}\nObserve that the first and second summands \nhave rank $k$ and $1$, respectively,\n  implying that $\\mathbf{C}_{yy}^{-1}\\mathbf{C}_{xx}$ has at most  rank $k+1$. If $\\mathbf{u}_{b,i}$ denotes the $i$-th column of $\\mathbf{U}_b$, that is orthogonal to $\\{\\mathbf{u}_{b,j}\\}_{j=1,j\\ne i}^k$ and $\\mathbf{u}_s$, right-multiplying $\\mathbf{C}_{yy}^{-1}\\mathbf{C}_{xx}$ by $\\mathbf{u}_{b,i}$ yields\n\\begin{equation*}\n\t\\mathbf{C}_{yy}^{-1}\\mathbf{C}_{xx}\\mathbf{u}_{b,i} = ({\\lambda_{x,i}}\/{\\lambda_{y,i}})\\,\\mathbf{u}_{b,i}\n\\end{equation*}\nfor $i=1,\\,\\ldots,\\,k$, which hints that\n $\\{\\mathbf{u}_{b,i}\\}_{i=1}^k$ are $k$ eigenvectors of $\\mathbf{C}_{yy}^{-1}\\mathbf{C}_{xx}$ associated with eigenvalues $\\{{\\lambda_{x,i}}\/{\\lambda_{y,i}}\\}_{i=1}^k$. \n Again, right-multiplying $\\mathbf{C}_{yy}^{-1}\\mathbf{C}_{xx}$ by $\\mathbf{u}_s$ gives rise to\n\\begin{equation}\\label{eq:yxus}\n\t\\mathbf{C}_{yy}^{-1}\\mathbf{C}_{xx}\\mathbf{u}_s =\\lambda_{x,k+1} \\mathbf{U}_n \\mathbf{U}_n^\\top \\mathbf{u}_s\\mathbf{u}_s^\\top \\mathbf{u}_s = \\lambda_{x,k+1} \\mathbf{U}_n \\mathbf{U}_n^\\top \\mathbf{u}_s.\n\\end{equation}\nTo proceed, we will leverage  the following three facts: i) $\\mathbf{u}_s$ is orthogonal to all columns of $\\mathbf{U}_b$; ii) columns of $\\mathbf{U}_n$ are orthogonal to those of $\\mathbf{U}_b$; and iii) $[\\mathbf{U}_b~\\mathbf{U}_n]$ has full rank. Based on i)-iii), \n it follows readily that $\\mathbf{u}_s$ can be uniquely expressed as a linear combination of columns of $\\mathbf{U}_n$; that is, $\\mathbf{u}_s:=\\sum_{i=1}^{D-k}p_i\\mathbf{u}_{n,i}$, where $\\{p_i\\}_{i=1}^{D-k}$ are some unknown coefficients, and $\\mathbf{u}_{n,i}$ denotes the $i$-th column of $\\mathbf{U}_n$. One can manipulate  $\\mathbf{U}_n \\mathbf{U}_n^\\top \\mathbf{u}_s$ in \\eqref{eq:yxus} as \n\\begin{align*}\n\t\\mathbf{U}_n \\mathbf{U}_n^\\top \\mathbf{u}_s&=[\\mathbf{u}_{n,1}~\\cdots~\\mathbf{u}_{n,D-k}]\n\t\\left[\\!\n\t\\begin{array}{c}\n\t\\mathbf{u}_{n,1}^\\top \\mathbf{u}_s\\\\\n    \\vdots\\\\\n\t\\mathbf{u}_{n,D-k}^\\top \\mathbf{u}_s\n\t\\end{array}\\!\n\t\\right]\\\\\n\t&=\\mathbf{u}_{n,1}\\mathbf{u}_{n,1}^\\top\\mathbf{u}_s+\\,\\cdots\\,+\\mathbf{u}_{n,D-k}\\mathbf{u}_{n,D-k}^\\top\\mathbf{u}_s\\\\\n\t&=p_1\\mathbf{u}_{n,1}+\\,\\cdots\\,+p_{D-k}\\mathbf{u}_{n,D-k}\\\\\n\t&=\\mathbf{u}_s\n\\end{align*}\nyielding $\\mathbf{C}_{yy}^{-1}\\mathbf{C}_{xx}\\mathbf{u}_s=\\lambda_{x,k+1}\\mathbf{u}_s$; that is, $\\mathbf{u}_s$ is the $(k+1)$-st eigenvector of $\\mathbf{C}_{yy}^{-1}\\mathbf{C}_{xx}$ corresponding to eigenvalue $\\lambda_{x,k+1}$.\n\n\n\nBefore moving on, we will make two assumptions.\n\n\\begin{assumption}\\label{asmp:model}\n\tBackground and target data  are generated according to the models \\eqref{eq:y} and \\eqref{eq:x}, respectively, with the background data sample covariance matrix being nonsingular.\n\\end{assumption}\n\\begin{assumption}\\label{asmp:unique}\n\tIt holds for all $i=1,\\,\\ldots,\\,k$ that ${\\lambda_{x,k+1}}\/{\\lambda_{y,k+1}}>{\\lambda_{x,i}}\/{\\lambda_{y,i}}$.\n\\end{assumption}\nAssumption \\ref{asmp:unique} essentially requires that $\\mathbf{u}_s$ is discriminative enough in the target data relative to the background data. \n\\textcolor{black}{After combining Assumption \\ref{asmp:unique} and the fact that $\\mathbf{u}_s$ is an eigenvector of $\\mathbf{C}_{yy}^{-1}\\mathbf{C}_{xx}$, it follows readily that the eigenvector of $\\mathbf{C}_{yy}^{-1}\\mathbf{C}_{xx}$ associated with the largest eigenvalue is $\\mathbf{u}_s$.}\nUnder these two assumptions, we establish the optimality of dPCA next. \n\\begin{theorem}\n\t\\label{the:optm}\n\tUnder Assumptions \\ref{asmp:model} and \\ref{asmp:unique} with $d=1$, as $m,\\,n\\to\\infty$, the solution of \\eqref{eq:dpca} recovers the subspace vector specific to target data relative to background data, namely $\\mathbf{u}_s$. \n\\end{theorem}\n\n \n\\section{Kernel dPCA} \\label{sec:kdpca}\n\nWith advances in data acquisition and data storage technologies, a sheer volume of possibly high-dimensional data are collected daily, that topologically lie on a  nonlinear manifold in general. This goes beyond the ability of the (linear) dPCA in Sec. \\ref{sec:dpca} due mainly to a couple of reasons: i) dPCA presumes a linear low-dimensional hyperplane to project the target data vectors; and ii) dPCA incurs computational complexity  $\\mathcal{O}(\\max (m,\\,n) D^2)$ that grows quadratically with the dimensionality of data vectors.\n To address these challenges, this section generalizes dPCA to account for nonlinear data relationships via kernel-based learning, and puts forth kernel (K) dPCA for nonlinear discriminative analytics. Specifically, KdPCA starts by \\textcolor{black}{mapping} both the target and background data vectors from the original data space to a higher-dimensional (possibly infinite-dimensional) feature space using a common nonlinear function, which is followed by performing linear dPCA on the \\textcolor{black}{transformed} data.\n\n\nConsider first  the dual version of dPCA, starting with  the $N:=m+n$ augmented data  $\\{\\mathbf{z}_i\\in\\mathbb{R}^D\\}_{i=1}^N$ as \n\\begin{equation*}\n\\label{eq:z}\n\\mathbf{z}_i:=\\left\\{\n\\begin{array}{ll}\n\\mathbf{x}_i, & 1\\le i \\le m\\\\\n\\mathbf{y}_{i-m}, & m< i \\le N\n\\end{array}\n\\right.\n\\end{equation*}\nand express the wanted\nsubspace vector $\\mathbf{u}\\in\\mathbb{R}^D$ in terms of\n $\\mathbf{Z}:=[\\mathbf{z}_1\\,\\cdots \\, \\mathbf{z}_N]\\in\\mathbb{R}^{D\\times N}$, yielding $\\mathbf{u}:=\\mathbf{Z}\\mathbf{a}$, where $\\mathbf{a}\\in\\mathbb{R}^N$ denotes  the dual vector. \\textcolor{black}{When ${\\rm min}(m\\,,n)\\gg D$, matrix $\\mathbf{Z}$ has full row rank in general. Thus, there always exists a vector $\\mathbf{a}$ so that $\\mathbf{u}=\\mathbf{Z}\\mathbf{a}$. Similar steps have also been used in obtaining dual versions of PCA and CCA \\cite{kpca,2004kernel}.} \nSubstituting $\\mathbf{u}=\\mathbf{Z}\\mathbf{a}$ into \\eqref{eq:dpca} leads to our dual dPCA \n\\begin{equation}\n\\textcolor{black}{\t\\label{eq:ddpca}\n\t\\underset{\\mathbf{a}\\in\\mathbb{R}^N}{\\max}\n\t\\quad \\frac{\\mathbf{a}^\\top \\mathbf{Z}^\\top \\mathbf{C}_{xx}\\mathbf{Z}\\mathbf{a}}{ \\mathbf{a}^\\top \\mathbf{Z}^\\top\\mathbf{C}_{yy}\\mathbf{Z}\\mathbf{a}}\n}\n\\end{equation}\nbased on which we will develop our KdPCA in the sequel.\n\n\n\n\nSimilar to deriving KPCA from dual PCA \\cite{kpca}, our approach is first to transform $\\{\\mathbf{z}_i\\}_{i=1}^N$ from $\\mathbb{R}^D$ to a high-dimensional space $\\mathbb{R}^L$ (possibly with $L=\\infty$) by some nonlinear mapping function $\\bm{\\phi}(\\cdot)$, followed by removing the sample means of $\\{\\bm{\\phi}(\\mathbf{x}_i)\\}$ and $\\{\\bm{\\phi}(\\mathbf{y}_j)\\}$ from the corresponding transformed data; and subsequently, implementing dPCA on the centered transformed datasets to obtain the low-dimensional \\textcolor{black}{kernel} dPCs. Specifically,  the sample covariance matrices of  $\\{\\bm{\\phi}(\\mathbf{x}_i)\\}_{i=1}^m$ and $\\{\\bm{\\phi}(\\mathbf{y}_j)\\}_{j=1}^n$ can be expressed as \n\\begin{align*}\t\n\t\\mathbf{C}_{xx}^{\\phi }&:=\\frac{1}{m}\\sum_{i=1}^m \\left(\\bm{\\phi}(\\mathbf{x}_i)-\\bm{\\mu}_{ x}\\right)\\left(\\bm{\\phi}(\\mathbf{x}_i)-\\bm{\\mu}_{ x}\\right)^\\top\\in\\mathbb{R}^{L\\times L}\\\\\n\t\\mathbf{C}_{yy}^{\\phi }&:=\\frac{1}{n}\\sum_{j=1}^n \\left(\\bm{\\phi}(\\mathbf{y}_j)-\\bm{\\mu}_{ y}\\right)\\left(\\bm{\\phi}(\\mathbf{y}_j)-\\bm{\\mu}_{ y}\\right)^\\top\\in\\mathbb{R}^{L\\times L}\n\\end{align*}\nwhere the $L$-dimensional vectors $\\bm{\\mu}_{x}:=(1\/m)\\sum_{i=1}^m\\bm{\\phi}(\\mathbf{x}_i)\n$ and $\\bm{\\mu}_{y}:=(1\/n)\\sum_{j=1}^n\\bm{\\phi}(\\mathbf{y}_j)\n$ are accordingly the sample means of $\\{\\bm{\\phi}(\\mathbf{x}_i)\\}$ and $\\{\\bm{\\phi}(\\mathbf{y}_j)\\}$.\nFor convenience, let\n $\\bm{\\Phi}(\\mathbf{Z}):=[\\bm{\\phi}(\\mathbf{x}_1)-\\bm{\\mu}_x,\\,\\cdots,\\,\\bm{\\phi}(\\mathbf{x}_m)-\\bm{\\mu}_x,\\,\\bm{\\phi}(\\mathbf{y}_1)-\\bm{\\mu}_y,\\,\\cdots,\\,\\bm{\\phi}(\\textcolor{black}{\\mathbf{y}_n})-\\bm{\\mu}_y]\\in\\mathbb{R}^{L\\times N}$.\n  Upon replacing $\\{\\mathbf{x}_i\\}$ and $\\{\\mathbf{y}_j\\}$ in \\eqref{eq:ddpca} with $\\{\\bm{\\phi}(\\mathbf{x}_i)-\\bm{\\mu}_x\\}$ and $\\{\\bm{\\phi}(\\mathbf{y}_j)-\\bm{\\mu}_y\\}$, respectively,  \\textcolor{black}{the kernel version of \\eqref{eq:ddpca} boils down to}\n\\textcolor{black}{\n\\begin{equation}\t\n\t\\label{eq:kdpca}\n\t\\underset{\\mathbf{a}\\in\\mathbb{R}^N}{\\max}\n\t\\quad\\frac{\\mathbf{a}^\\top \\bm{\\Phi}^\\top(\\mathbf{Z}) \\mathbf{C}_{xx}^{\\phi}\\bm{\\Phi}(\\mathbf{Z})\\mathbf{a}}{ \\mathbf{a}^\\top \\bm{\\Phi}^\\top(\\mathbf{Z})\\mathbf{C}_{y y}^{\\phi}\\bm{\\Phi}(\\mathbf{Z})\\mathbf{a}}.\n\\end{equation}}\n\n\nIn the sequel, \\eqref{eq:kdpca} will be further simplified by leveraging the so-termed `kernel trick' \\cite{RKHS}. \n\nTo start, define a kernel matrix $\\mathbf{K}_{xx}\\in\\mathbb{R}^{m\\times m}$ of  $\\{\\mathbf{x}_i\\}$ whose $(i,\\,j)$-th entry is $\\kappa(\\mathbf{x}_i,\\,\\mathbf{x}_j):=\\left<\\bm{\\phi}(\\mathbf{x}_i),\\,\\bm{\\phi}(\\mathbf{x}_j)\\right>$ for $i,\\,j=1,\\,\\ldots,\\,m$, where $\\kappa(\\cdot)$ represents some kernel function. Matrix $\\mathbf{K}_{yy}\\in\\mathbb{R}^{n\\times n}$ of $\\{\\mathbf{y}_j\\}$ is defined likewise. Further, the $(i,\\,j)$-th entry of matrix $\\mathbf{K}_{xy}\\in\\mathbb{R}^{m\\times n}$ is  $\\kappa(\\mathbf{x}_i,\\,\\mathbf{y}_j):=\\left<\\bm{\\phi}(\\mathbf{x}_i),\\,\\bm{\\phi}(\\mathbf{y}_j)\\right>$. Centering $\\mathbf{K}_{xx}$, $\\mathbf{K}_{yy}$, and $\\mathbf{K}_{xy}$ produces \n\\begin{align*}\n\\mathbf{K}_{xx}^c&:=\\mathbf{K}_{xx}-\\tfrac{1}{m}\\mathbf{1}_{m }\\mathbf{K}_{xx}-\\tfrac{1}{m}\\mathbf{K}_{xx}\\mathbf{1}_{m}+\\tfrac{1}{m^2}\\mathbf{1}_{m}\\mathbf{K}_{xx}\\mathbf{1}_{m}\\\\\n\\mathbf{K}_{yy}^c&:=\\mathbf{K}_{yy}-\\!\\tfrac{1}{n}\\mathbf{1}_{ n}\\mathbf{K}_{yy}-\\!\\tfrac{1}{n}\\mathbf{K}_{yy}\\mathbf{1}_{n}+\\!\\tfrac{1}{n^2}\\mathbf{1}_{n}\\mathbf{K}_{yy}\\mathbf{1}_{n}\\\\\n\\mathbf{K}_{xy}^c&:=\\mathbf{K}_{xy}-\\tfrac{1}{m}\\mathbf{1}_{ m}\\mathbf{K}_{xy}-\\tfrac{1}{n}\\mathbf{K}_{xy}\\mathbf{1}_{ n}+\\tfrac{1}{mn}\\mathbf{1}_{m}\\mathbf{K}_{xy}\\mathbf{1}_{n}\n\\end{align*}\nwith matrices $\\mathbf{1}_{m}\\in\\mathbb{R}^{m\\times m}$ and $\\mathbf{1}_n\\in\\mathbb{R}^{n\\times n}$ having all entries $1$. Based on those centered matrices, let\n\\begin{equation}\n\\label{eq:k}\n\\mathbf{K}:=\\left[\\begin{array}\n{cc}\n\\mathbf{K}_{xx}^c &\\mathbf{K}_{xy}^c\\\\\n(\\mathbf{K}_{xy}^c)^\\top  &\\mathbf{K}_{yy}^c\n\\end{array}\n\\right]\\in\\mathbb{R}^{N\\times N}.\n\\end{equation}\nDefine further  \n$\\mathbf{K}^x\\in\\mathbb{R}^{N\\times N}$ and $\\mathbf{K}^y\\in\\mathbb{R}^{N\\times N}$ with $(i,\\,j)$-th entries\n\\begin{subequations}\n\t\\label{eq:kxky}\n\t\\begin{align}\n\tK^x_{i,j}\\,:=\\left\\{\n\t\\begin{array}{ll}\n\tK_{i,j}\/m \\!&1\\le i \\le m\\\\\n\t~~~0 &m<i\\le N\n\t\\end{array}\n\t\\right.\\label{eq:kx}\\\\\n\tK^y_{i,j}\\,:=\\left\\{\n\t\\begin{array}{ll}\n\t~~~0 \\!&1\\le i \\le m\\\\\n\tK_{i,j}\/n &m<i\\le N\n\t\\end{array}\n\t\\right.\\label{eq:ky}\n\t\\end{align}\n\\end{subequations}\nwhere $K_{i,\\,j}$ stands for the $(i,\\,j)$-th entry of $\\mathbf{K}$.\n\n\n\\begin{algorithm}[t]\n\t\\caption{Kernel dPCA.}\n\t\\label{alg:kdpca}\n\t\\begin{algorithmic}[1]\n\t\t\\STATE {\\bfseries Input:}\n\t\tTarget data $\\{\\mathbf{x}_i\\}_{i=1}^m$ and background data $\\{\\mathbf{y}_j\\}_{j=1}^n$; number of dPCs $d$; kernel function $\\kappa(\\cdot)$; constant $\\epsilon$.\n\t\t\\STATE {\\bfseries Construct} $\\mathbf{K}$ using \\eqref{eq:k}. Build $\\mathbf{K}^x$ and $\\mathbf{K}^y$ via \\eqref{eq:kxky}.\n\t\t\\STATE {\\bfseries Solve} \\eqref{eq:kdpcasol} to obtain the first $d$  eigenvectors $\\{\\hat{\\mathbf{a}}_i\\}_{i=1}^d$.\n\t\t\\STATE {\\bfseries Output} $\\hat{\\mathbf{A}}:=[\\hat{\\mathbf{a}}_1\\,\\cdots\\,\\hat{\\mathbf{a}}_d]$.\n\t\t\\vspace{-0pt}\n\t\\end{algorithmic}\n\\end{algorithm} \n\n\n\\textcolor{black}{Using \\eqref{eq:k} and \\eqref{eq:kxky}, it can be easily verified that $\\mathbf{K}\\mathbf{K}^x=\\mathbf{K}{\\rm diag}(\\{\\iota_i^x\\}_{i=1}^N)\\mathbf{K}$, and $\\mathbf{K}\\mathbf{K}^y=\\mathbf{K}{\\rm diag}(\\{\\iota_i^y\\}_{i=1}^N)\\mathbf{K}$, where $\\{\\iota_i^x=1\\}_{i=1}^m$, $\\{\\iota_i^x=0\\}_{i=m+1}^N$, $\\{\\iota_i^y=0\\}_{i=1}^m$, and $\\{\\iota_i^y=1\\}_{i=m+1}^N$. That is, both  $\\mathbf{K}\\mathbf{K}^x$ and $\\mathbf{K}\\mathbf{K}^y$ are symmetric. }\nSubstituting \\eqref{eq:k} and \\eqref{eq:kxky} into \\eqref{eq:kdpca} yields \\textcolor{black}{(see details in the Appendix)}\n\\begin{equation}\t\\label{eq:kdpcafm2o}\n\t\\underset{\\mathbf{a}\\in\\mathbb{R}^N}{\\max}\n\t\\quad\\frac{\\mathbf{a}^\\top \\mathbf{K}\\mathbf{K}^x\\mathbf{a}}{ \\mathbf{a}^\\top \\mathbf{K}\\mathbf{K}^y\\mathbf{a}}.\n\\end{equation} \n\\textcolor{black}{Due to the rank-deficiency of $\\mathbf{K}\\mathbf{K}^y$ however, \\eqref{eq:kdpcafm2o} does not admit a meaningful solution. \n\tTo address this issue, following kPCA \\cite{kpca,2004kernel}, \n\ta positive constant $\\epsilon>0$ is added to the diagonal entries of $\\mathbf{K}\\mathbf{K}^y$. Hence, our KdPCA formulation for $d=1$ is given by}\n\t\\begin{equation}\t\\label{eq:kdpcafm2}\n\\textcolor{black}{\t\t\\hat{\\mathbf{a}}:=\\arg\t\n\t\\underset{\\mathbf{a}\\in\\mathbb{R}^N}{\\max}\n\t\\quad\\frac{\\mathbf{a}^\\top \\mathbf{K}\\mathbf{K}^x\\mathbf{a}}{ \\mathbf{a}^\\top \\left(\\mathbf{K}\\mathbf{K}^y+\\epsilon \\mathbf{I}\\right)\\mathbf{a}}.}\n\t\\end{equation}\nAlong the lines of  dPCA, the solution of KdPCA in \\eqref{eq:kdpcafm2} can be provided by \n\\begin{equation}\n\\label{eq:kdpcasol}\n\\textcolor{black}{\n\\left(\\mathbf{K}\\mathbf{K}^y+\\epsilon\\mathbf{I}\\right)^{-1}\\mathbf{K}\\mathbf{K}^x\\hat{\\mathbf{a}}=\\hat{\\lambda}\n\\hat{\\mathbf{a}}.}\n\\end{equation}\nThe optimizer $\\hat{\\mathbf{a}}$ coincides with the right eigenvector of $\\left(\\mathbf{K}\\mathbf{K}^y+\\epsilon\\mathbf{I}\\right)^{-1}\\mathbf{K}\\mathbf{K}^x$ corresponding to the largest eigenvalue $\\hat{\\lambda}=\\lambda_1$.\n\n When looking for $d$ dPCs, with $\\{\\mathbf{a}_i\\}_{i=1}^d$ collected as columns in $\\mathbf{A}:=[\\mathbf{a}_1\\,\\cdots \\,\\mathbf{a}_d]\\in\\mathbb{R}^{N\\times d}$, the KdPCA in  \\eqref{eq:kdpcafm2} can be generalized to $d\\ge 2$ as \n\t\\begin{equation*}\n\t\t\t\\hat{\\mathbf{A}}:=\\arg\n\t\t\\underset{\\mathbf{A}\\in\\mathbb{R}^{N\\times d}}{\\max}\n\t\t\\quad {\\rm Tr} \\left[ \\left(\\mathbf{A}^\\top (\\mathbf{K}\\mathbf{K}^y+\\epsilon \\mathbf{I}\\right)\\mathbf{A})^{-1} \\mathbf{A}^\\top \\mathbf{K}\\mathbf{K}^x\\mathbf{A}\\right]\n\t\\end{equation*}\nwhose columns correspond to the $d$ right eigenvectors of $\\left(\\mathbf{K}\\mathbf{K}^y+\\epsilon\\mathbf{I}\\right)^{-1}\\mathbf{K}\\mathbf{K}^x$\n associated with the $d$ largest  eigenvalues.\n   Having found  $\\hat{\\mathbf{A}}$, one can project the data $\\bm{\\Phi}(\\mathbf{Z})$ onto the obtained $d$ subspace vectors by $\\mathbf{K}\\hat{\\mathbf{A}}$. \n It is worth remarking that KdPCA can be performed in the high-dimensional feature space without explicitly forming and evaluating the nonlinear transformations. Indeed, this becomes possible by the `kernel trick' \\cite{RKHS}.   \n The main steps of KdPCA are given in Alg. \\ref{alg:kdpca}. \n \nTwo remarks are worth making at this point.\n \t\\begin{remark}\n \t\tWhen the kernel function required to form $\\mathbf{K}_{xx}$, $\\mathbf{K}_{yy}$, and $\\mathbf{K}_{xy}$ is not given, one may use the multi-kernel learning method to automatically choose the right kernel function(s); see for example, \\cite{mkl2004,zhang2017going,tsp2017zwrg}. Specifically, one can presume $\\mathbf{K}_{xx}:=\\sum_{i=1}^P\\delta_i\\mathbf{K}_{xx}^i$, $\\mathbf{K}_{yy}:=\\sum_{i=1}^P\\delta_i\\mathbf{K}_{yy}^i$, and $\\mathbf{K}_{xy}:=\\sum_{i=1}^P\\delta_i\\mathbf{K}_{xy}^i$ in \\eqref{eq:kdpcafm2}, where $\\mathbf{K}_{xx}^i\\in\\mathbb{R}^{m\\times m}$, $\\mathbf{K}_{yy}^i\\in\\mathbb{R}^{n\\times n}$, and $\\mathbf{K}_{xy}^i\\in\\mathbb{R}^{m\\times n}$ are formed using the kernel function $\\kappa_i(\\cdot)$; and $\\{\\kappa_i(\\cdot)\\}_{i=1}^P$ are a preselected dictionary of known kernels, but  $\\{\\delta_i\\}_{i=1}^P$ will be treated as unknowns to be learned along with $\\mathbf{A}$ in \\eqref{eq:kdpcafm2}.\n \t\\end{remark}\n \t\\begin{remark}\n \t\tIn the absence of  background data, upon setting $\\{\\bm{\\phi}(\\mathbf{y}_j)=\\mathbf{0}\\}$, and $\\epsilon=1$ in \\eqref{eq:kdpcafm2}, matrix $\\left(\\mathbf{K}\\mathbf{K}^y+\\epsilon\\mathbf{I}\\right)^{-1}\\mathbf{K}\\mathbf{K}^x$ reduces to\n \t\t\\begin{equation*}\n \t\t\\mathbf{M}:=\\left[\n \t\t\\begin{array}\n \t\t{cc}\n \t\t(\\mathbf{K}_{xx}^c)^2 &\\mathbf{0}\\\\\n \t\t\\mathbf{0}  &\\mathbf{0}\n \t\t\\end{array}\n \t\t\\right].\n \t\t\\end{equation*} After collecting the first $m$ entries of $\\hat{\\mathbf{a}}_i$ into $\\mathbf{w}_i\\in\\mathbb{R}^{m}$, \\eqref{eq:kdpcasol} suggests that $(\\mathbf{K}_{xx}^c)^2\\mathbf{w}_i=\\lambda_i\\mathbf{w}_i$, where $\\lambda_i$ denotes the $i$-th largest eigenvalue of $\\mathbf{M}$. \n \t\tClearly, $\\{\\mathbf{w}_i\\}_{i=1}^d$ can be viewed as the $d$ eigenvectors of $(\\mathbf{K}_{xx}^c)^2$  associated with their $d$ largest eigenvalues. Recall that KPCA finds the first $d$ principal eigenvectors of $\\mathbf{K}_{xx}^c$ \\cite{kpca}. Thus, KPCA is a special case of KdPCA, when no background data are employed.\n \t\\end{remark}\n \n\n\n\n\\section{Discriminative Analytics with\\\\ Multiple Background Datasets} \\label{sec:mdpca}\n\nSo far, we have presented discriminative analytics methods for two datasets. \nThis section presents their generalizations\nto cope with multiple (specifically, one target plus more than one background) datasets.  \nSuppose that, in addition to the zero-mean target dataset $\\{\\mathbf{x}_i\\in\\mathbb{R}^D\\}_{i=1}^m$, we are also given $M\\ge 2$ centered background datasets $\\{\\mathbf{y}_j^k\\}_{j=1}^{n_k}$ for $k=1,\\,\\ldots,\\,M$. \nThe $M$ sets of background data $\\{\\mathbf{y}_j^k\\}_{k=1}^M$  contain latent background subspace vectors that are also present  in $\\{\\mathbf{x}_i\\}$.\n\nLet $\\mathbf{C}_{xx}:=m^{-1}\\sum_{i=1}^{m}\\mathbf{x}_i\\mathbf{x}_i^\\top$ and $\\mathbf{C}_{yy}^k:=n_k^{-1}\\times $ $\n\\sum_{j=1}^{n_k}\\mathbf{y}_{j}^k(\\mathbf{y}^k_{j})^\\top$ be the corresponding sample covariance matrices. The goal here is to unveil  the latent subspace vectors \n\\textcolor{black}{that are significant in representing the target data, but not any of the background data.}\nBuilding on the dPCA in \\eqref{eq:dpcam} for a single background dataset, it is meaningful to seek directions that maximize the variance of target data, while minimizing those of all background data. Formally, \nwe pursue the following optimization, that we term multi-background (M) dPCA here, for discriminative analytics of multiple datasets\n\\begin{equation}\\label{eq:gdpca}\n\\underset{\\mathbf{U}\\in\\mathbb{R}^{D\\times d}}{\\max}\t\\quad {\\rm Tr}\\bigg[\\bigg(\\sum_{k=1}^M\\omega_k\\mathbf{U}^\\top\\mathbf{C}_{yy}^{k}\\mathbf{U}\\bigg)^{-1}\\mathbf{U}^\\top\\mathbf{C}_{xx}\\mathbf{U}\\bigg]\n\\end{equation}\nwhere $\\{\\omega_k\\ge 0\\}_{k=1}^M$ with $\\sum_{k=1}^M \\omega_k=1$\nweight the variances of the $M$ projected background datasets. \n\n\nUpon defining $\\mathbf{C}_{yy}:=\\sum_{k=1}^{M}\\omega_k\\mathbf{C}_{yy}^k\n$, it is straightforward to see that \\eqref{eq:gdpca} reduces to \\eqref{eq:dpcam}. Therefore, one readily deduces \n that the optimal ${\\mathbf{U}}$ in \\eqref{eq:gdpca} can be obtained by taking the $d$ right eigenvectors of $\\mathbf{C}_{yy}^{-1}\\mathbf{C}_{xx}$ that are associated with the $d$ largest eigenvalues. For implementation, the steps of MdPCA are presented in Alg. \\ref{alg:mdpca}.\n\n\\begin{algorithm}[t]\n\t\\caption{Multi-background dPCA.}\n\t\\label{alg:mdpca}\n\t\\begin{algorithmic}[1]\n\t\t\\STATE {\\bfseries Input:}\n\t\tTarget  data $\\{\\accentset{\\circ}{\\mathbf{x}}_i\\}_{i=1}^m$ and background data $\\{\\accentset{\\circ}{\\mathbf{y}}_{j}^k\\}_{j=1}^{n_k}$ for $k=1,\\,\\ldots,\\,M$; weight hyper-parameters $\\{\\omega_k\\}_{k=1}^M$; number of dPCs $d$.\n\t\t\\STATE {\\bfseries Remove} the means from $\\{\\accentset{\\circ}{\\mathbf{x}}_i\\}$ and $\\{\\accentset{\\circ}{\\mathbf{y}}_{j}^k\\}_{k=1}^{M}$ to obtain  $\\{\\mathbf{x}_i\\}$ and $\\{\\mathbf{y}_{j}^k\\}_{k=1}^{M}$. Form $\\mathbf{C}_{xx}$, $\\{\\mathbf{C}_{yy}^k\\}_{k=1}^M$, and $\\mathbf{C}_{yy}:=\\sum_{k=1}^{M}\\omega_k\\mathbf{C}^k_{yy}$.\n\t\t\\STATE {\\bfseries Perform} eigendecomposition\n\t\ton $\\mathbf{C}_{yy}^{-1}\\mathbf{C}_{xx}$  to obtain the first $d$ right eigenvectors $\\{\\hat{\\mathbf{u}}_i\\}_{i=1}^d$.\n\t\t\\STATE {\\bfseries Output} $\\hat{\\mathbf{U}}:=[\\hat{\\mathbf{u}}_1\\,\\cdots\\,\\hat{\\mathbf{u}}_d]$.\n\t\t\\vspace{-0pt}\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\n\\begin{remark}\n\t\\label{rmk:alpha}\t\n\tThe  parameters $\\{\\omega_k\\}_{k=1}^M$ can be decided using two possible methods:\n\t i) spectral-clustering \\cite{ng2002spectral} to select a few sets of $\\{\\omega_k\\}$ yielding \\textcolor{black}{the most  representative subspaces for projecting the target data across  $\\{\\omega_k\\}$;}\n\t or ii) optimizing  $\\{\\omega_k\\}_{k=1}^M$ jointly with $\\mathbf{U}$ in \\eqref{eq:gdpca}.\n\\end{remark}\n\n\n\n\nFor data  belonging to nonlinear manifolds, kernel (K) MdPCA will be developed next.\n With some nonlinear function  $\\phi(\\cdot)$, we obtain the transformed target data $\\{\\bm{\\phi}(\\mathbf{x}_i)\\in\\mathbb{R}^L\\}$ as well as background data $\\{\\bm{\\phi}(\\mathbf{y}_{j}^k)\\in\\mathbb{R}^L\\}$. Letting $\\bm{\\mu}_x\\in\\mathbb{R}^L$ and $\\bm{\\mu}_{y}^k:=(1\/n_k)\\sum_{j=1}^{n_k}\\bm{\\phi}(\\mathbf{y}_{j}^k)\\in\\mathbb{R}^L$ denote the means of $\\{\\bm{\\phi}(\\mathbf{x}_i)\\}$ and $\\{\\bm{\\phi}(\\mathbf{y}_{j}^k)\\}$, respectively, one can form the corresponding covariance matrices $\\mathbf{C}_{xx}^{\\phi}\\in\\mathbb{R}^{L\\times L}$, and \n\\begin{equation*}\n\t\\mathbf{C}_{yy}^{\\phi,k}:=\\frac{1}{n_k}\\sum_{j=1}^{n_k}\\left(\\bm{\\phi}(\\mathbf{y}_{j}^k)-\\bm{\\mu}_{y}^k\\right)\\left(\\bm{\\phi}(\\mathbf{y}_{j}^k)-\\bm{\\mu}_{y}^k\\right)^\\top\\in\\mathbb{R}^{L\\times L}\n\t\\end{equation*}\nfor $k=1,\\,\\ldots,\\, M$.\nDefine the aggregate vector $\\mathbf{b}_i\\in\\mathbb{R}^L$ \n\\begin{equation*}\n\\label{eq:b}\n\\mathbf{b}_i:=\\left\\{\n\\begin{array}{ll}\n\\bm{\\phi}(\\mathbf{x}_i)-\\bm{\\mu}_x, & 1\\le i \\le m\\\\\n\\bm{\\phi}(\\mathbf{y}_{i-m}^1)-\\bm{\\mu}_{y}^1, & m< i \\le m+n_1\\\\\n~~~\\vdots & \\\\\n\\bm{\\phi}(\\mathbf{y}_{i-\\textcolor{black}{(N-n_M)}}^M)-\\bm{\\mu}_y^M,& N-n_{M}< i\\le N\n\\end{array}\n\\right.\n\\end{equation*}\nwhere $N:=m+\\sum_{k=1}^M n_k$, for $i=1,\\,\\ldots,\\,N$, and collect vectors $\\{\\mathbf{b}_i\\}_{i=1}^N$ as columns to form $\\mathbf{B}:=[\\mathbf{b}_1\\,\\cdots\\,\\mathbf{b}_N]\\in\\mathbb{R}^{L\\times N}$.\nUpon assembling dual vectors $\\{\\mathbf{a}_i\\in\\mathbb{R}^N\\}_{i=1}^d$ to form $\\mathbf{A}:=[\\mathbf{a}_1\\,\\cdots \\, \\mathbf{a}_d]\\in\\mathbb{R}^{N\\times d}$,\nthe kernel version of \\eqref{eq:gdpca} can be obtained as\n\t\\begin{equation*}\n\n\\underset{\\mathbf{A}\\in\\mathbb{R}^{N\\times d}}{\\max} {\\rm Tr}\\bigg[\\bigg(\\mathbf{A}^\\top\\mathbf{B}^\\top\\sum_{k=1}^M\\omega_k\\mathbf{C}^{\\phi,k}_{yy}\\mathbf{B}\\mathbf{A}\\hspace{-0.1cm}\\bigg)^{-1}\\mathbf{A}^\\top\\mathbf{B}^\\top\\mathbf{C}^{\\phi} _{xx}\\mathbf{B}\\mathbf{A}\\bigg].\n\t\\end{equation*}\n\n\n\n\n\nConsider now  kernel matrices  $\\mathbf{K}_{xx}\\in\\mathbb{R}^{m\\times m}$ and $\\mathbf{K}_{kk}\\in\\mathbb{R}^{n_k\\times n_k}$, whose $(i,\\,j)$-th entries are   $\\kappa(\\mathbf{x}_i,\\,\\mathbf{x}_j)$ and $\\kappa(\\mathbf{y}_{i}^k,\\,\\mathbf{y}_{j}^k)$, respectively, for  $k=1,\\,\\ldots,\\,M$. \n Furthermore, matrices $\\mathbf{K}_{xk}\\in\\mathbb{R}^{m\\times n_k}$, and $\\mathbf{K}_{lk}\\in\\mathbb{R}^{n_l\\times n_k}$ are defined with  their corresponding $(i,\\,j)$-th elements  $\\kappa(\\mathbf{x}_{i},\\,\\mathbf{y}_{j}^k)$ and $\\kappa(\\mathbf{y}_{i}^{l},\\,\\mathbf{y}_{j}^k)$, for $l=1,\\,\\ldots,\\, k-1$ and  $k=1,\\,\\ldots,\\,M$.\nWe subsequently center those matrices to obtain $\\mathbf{K}_{xx}^c$ and \n\\begin{align*}\n\\mathbf{K}_{kk}^c&:=\\mathbf{K}_{kk}-\\tfrac{1}{n_k}\\mathbf{1}_{n_k}\\mathbf{K}_{kk}-\\tfrac{1}{n_k}\\mathbf{K}_{kk}\\mathbf{1}_{n_k}+\\tfrac{1}{n_k^2}\\mathbf{1}_{n_k}\\mathbf{K}_{kk}\\mathbf{1}_{n_k}\\\\\n\\mathbf{K}_{xk}^c&:=\\mathbf{K}_{xk}-\\tfrac{1}{m}\\mathbf{1}_{m }\\mathbf{K}_{xk}-\\tfrac{1}{n_k}\\mathbf{K}_{xk}\\mathbf{1}_{n_k}+\\tfrac{1}{mn_k}\\mathbf{1}_{m}\\mathbf{K}_{xk}\\mathbf{1}_{n_k}\\\\\n\\mathbf{K}_{lk}^c&:=\\mathbf{K}_{lk}-\\!\\tfrac{1}{n_l}\\mathbf{1}_{n_l}\\mathbf{K}_{lk}-\\!\\tfrac{1}{n_{k}}\\mathbf{K}_{lk}\\mathbf{1}_{n_k}+\\!\\tfrac{1}{n_ln_k}\\mathbf{1}_{n_l}\\mathbf{K}_{lk}\\mathbf{1}_{n_k}\n\\end{align*}\nwhere $\\mathbf{1}_{n_k}\\in\\mathbb{R}^{n_k\\times n_k}$ and $\\mathbf{1}_{n_{l}}\\in\\mathbb{R}^{n_{l}\\times n_{l}}$ are all-one matrices. With   $\\mathbf{K}^x$ as in \\eqref{eq:kx},  consider the $N\\times N$ matrix\n\\begin{equation}\n\\label{eq:km}\n\\mathbf{K}:=\\left[\\begin{array}\n{llll}\n\\mathbf{K}_{xx}^c  & \\mathbf{K}_{x1}^c & \\cdots & \\mathbf{K}_{xM}^c \\\\\n(\\mathbf{K}_{x1}^c)^\\top & \\mathbf{K}_{11}^c  & \\cdots & \\mathbf{K}_{1M}^c \\\\\n\\quad\\vdots           & \\quad\\vdots           & \\ddots & \\quad\\vdots \\\\\n(\\mathbf{K}_{xM}^c)^\\top & (\\mathbf{K}_{1M}^c)^\\top & \\cdots & \\mathbf{K}_{MM}^c\n\\end{array}\n\\right]\n\\end{equation}\nand $\\mathbf{K}^k\\in\\mathbb{R}^{N\\times N}$ with  $(i,\\,j)$-th entry\n\\begin{equation}\nK^k_{i,j}:=\\left\\{\\!\\!\\begin{array}{cl}\nK_{i,j}\/n_k, \\!&\\text{if}~ m+\\sum_{\\ell=1}^{n_{k-1}}n_{\\ell}< i \\le m+\\sum_{\\ell=1}^{n_{k}}n_{\\ell}\\\\\n0, &\\mbox{otherwise}\n\\end{array}\n\\right.\\label{eq:kk}\n\\end{equation}\nfor $k=1,\\,\\ldots,\\,M$. Adopting the regularization in \\eqref{eq:kdpcafm2}, our KMdPCA finds\n\t\\begin{equation*}\n\\hat{\\mathbf{A}}:=\\arg\\underset{\\mathbf{A}\\in\\mathbb{R}^{N\\times d}}{\\max}{\\rm Tr}\\bigg[\\bigg(\\mathbf{A}^\\top\\Big(\\mathbf{K}\\sum_{k=1}^M\\mathbf{K}^k+\\epsilon\\mathbf{I}\\Big)\\mathbf{A}\\bigg)^{-1}\\!\\!\\mathbf{A}^\\top\\mathbf{K}\\mathbf{K}^x\\mathbf{A}\\bigg]\n\t\\end{equation*}\nsimilar to (K)dPCA, whose solution comprises the right eigenvectors associated with the \\textcolor{black}{first $d$} largest  eigenvalues in\n\t\\begin{equation}\n\t\t\\label{eq:kmdpcasol}\n\t\t\\bigg(\\mathbf{K}\\sum_{k=1}^M\\mathbf{K}^k+\\epsilon\\mathbf{I}\\bigg)^{-1}\\mathbf{K}\\mathbf{K}^x\\hat{\\mathbf{a}}_i=\\hat{\\lambda}_i\\hat{\\mathbf{a}}_i.\n\t\\end{equation}\n\t\n\n\n\nFor implementation, KMdPCA is presented in Alg. \\ref{alg:mkdpca}.\n\\begin{algorithm}[t]\n\t\\caption{Kernel multi-background dPCA.}\n\t\\label{alg:mkdpca}\n\t\\begin{algorithmic}[1]\n\t\t\\STATE {\\bfseries Input:}\n\t\tTarget data $\\{\\mathbf{x}_i\\}_{i=1}^m$ and background data $\\{\\mathbf{y}_{j}^k\\}_{j=1}^{n_k}$ for $k=1,\\,\\ldots,\\,M$; number of dPCs $d$; kernel function $\\kappa(\\cdot)$; weight coefficients $\\{\\omega_k\\}_{k=1}^M$; constant $\\epsilon$.\n\t\t\\STATE {\\bfseries Construct} $\\mathbf{K}$ using \\eqref{eq:km}. Build $\\mathbf{K}^x$ and $\\{\\mathbf{K}^k\\}_{k=1}^M$ via \\eqref{eq:kx} and \\eqref{eq:kk}.\n\t\t\\STATE {\\bfseries Solve} \\eqref{eq:kmdpcasol} to obtain the first $d$  eigenvectors $\\{\\hat{\\mathbf{a}}_i\\}_{i=1}^d$.\n\t\t\\STATE {\\bfseries Output} $\\hat{\\mathbf{A}}:=[\\hat{\\mathbf{a}}_1\\,\\cdots\\,\\hat{\\mathbf{a}}_d]$. \n\t\t\\vspace{-0pt}\n\t\\end{algorithmic}\n\\end{algorithm} \n\n\n\n\\begin{remark}\nWe can verify that PCA, KPCA, dPCA, KdPCA, MdPCA, and KMdPCA incur computational complexities $\\mathcal{O}(mD^2)$, $\\mathcal{O}(m^2D)$, $\\mathcal{O}(\\max(m,n)D^2)$, $\\mathcal{O}(\\max(m^2,n^2)D)$, $\\mathcal{O}(\\max(m,\\bar{n} )D^2)$, and $\\mathcal{O}(\\max(m^2,\\bar{n}^2 )D)$, respectively, where $\\bar{n}:=\\max_k~\\{n_k\\}_{k=1}^M$. \nIt is also not difficult to check that the computational complexity of \n\tforming $\\mathbf{C}_{xx}$, $\\mathbf{C}_{yy}$,  $\\mathbf{C}_{yy}^{-1}$, and performing the eigendecomposition on $\\mathbf{C}_{yy}^{-1}\\mathbf{C}_{xx}$ is $\\mathcal{O}(mD^2)$, $\\mathcal{O}(nD^2)$, $\\mathcal{O}(D^3)$, and $\\mathcal{O}(D^3)$, respectively. As the number of data vectors ($m,\\,n$) is much larger than their dimensionality $D$, when performing dPCA in the primal domain, it follows readily that dPCA incurs  complexity $\\mathcal{O}(\\max(m,n)D^2)$. Similarly, the computational complexities of the other algorithms can be checked\nEvidently,  when $\\min(m,n)\\gg D $ or $\\min(m,\\underline{n})\\gg D$ with $\\underline{n}:=\\min_k\\,\\{n_k\\}_{k=1}^M$,\n\tdPCA and MdPCA are computationally more attractive than KdPCA and KMdPCA. On the other hand, KdPCA and KMdPCA become more appealing, when $D\\gg \\max(m,n)$ or $D\\gg \\max(m,\\bar{n})$.\n\tMoreover, the computational complexity of cPCA is $\\mathcal{O}(\\max (m,n)D^2L)$, where $L$ denotes the number of $\\alpha$'s candidates. Clearly, relative to dPCA, cPCA is computationally more expensive when $DL> \\max(m,n)$.\n\\end{remark}\n\n\n\n\n\\section{Numerical Tests}\\label{sec:simul}\n\nTo evaluate the performance of our proposed approaches for discriminative analytics, we carried out a number of  numerical tests using several synthetic and real-world datasets, a sample of which are reported in this section. \n\n\\subsection{dPCA tests}\n\n\n\n Semi-synthetic target \\textcolor{black}{$\\{\\accentset{\\circ}{\\mathbf{x}}_i\\in\\mathbb{R}^{784}\\}_{i=1}^{2,000}$} and background images \\textcolor{black}{$\\{\\accentset{\\circ}{\\mathbf{y}}_j\\in\\mathbb{R}^{784}\\}_{j=1}^{3,000}$} were obtained by superimposing images from the MNIST \\footnote{Downloaded from http:\/\/yann.lecun.com\/exdb\/mnist\/.} and CIFAR-10 \\cite{cifar10} datasets.\n Specifically, the target data $\\{\\mathbf{x}_i\\in\\mathbb{R}^{784}\\}_{i=1}^{2,000}$ were generated using $2,000$ handwritten digits 6 and 9 (1,000 for each) of size $28\\times 28$,\n  superimposed with $2,000$ frog images from the CIFAR-10 database \\cite{cifar10} followed by removing the sample mean from each data point; see Fig. \\ref{fig:targ}. The raw $32\\times 32$ frog images were converted into grayscale, and randomly cropped to $28\\times 28$. The zero-mean background data $\\{\\mathbf{y}_j\\in\\mathbb{R}^{784}\\}_{j=1}^{3,000}$ were constructed using $3,000$ cropped frog images, which were randomly chosen from the remaining frog images in the CIFAR-10 database.\n\n \\begin{figure}[t]\n \t\\centering \n \t\\includegraphics[scale=0.64]{targ.pdf} \n \t\\caption{\\small{Superimposed images.}}\n \t\\label{fig:targ}\n \\end{figure}\n \nThe dPCA Alg. \\ref{alg:dpca} was performed on $\\{\\accentset{\\circ}{\\mathbf{x}}_i\\}$ and  $\\{\\accentset{\\circ}{\\mathbf{y}}_j\\}$ with $d=2$. PCA was implemented on $\\{\\accentset{\\circ}{\\mathbf{x}}_i\\}$ only. The first two PCs and dPCs are presented in the left and right panels of Fig. \\ref{fig:digits}, respectively. Clearly, dPCA reveals the discriminative information of the target data describing digits $6$ and $9$ relative to the background data, enabling successful discovery of the digit $6$ and $9$ subgroups. On the contrary, PCA captures only the patterns that correspond to the generic background rather than those associated with  the digits $6$ and $9$. \n\\textcolor{black}{To further assess the performance of dPCA and PCA, K-means is carried out using the resulting low-dimensional representations of the target data. The clustering performance is evaluated in terms of two metrics: clustering error and scatter ratio. The clustering error is defined as the ratio of the number of incorrectly clustered data vectors over $m$.\n\tScatter ratio verifying cluster separation is defined as $S_t\/\\sum_{i=1}^2S_i$, where $S_t$ and $\\{S_i\\}_{i=1}^2$ denote the total scatter value and the within cluster scatter values, given by  $S_t:=\\sum_{j=1}^{2,000}\\|\\hat{\\mathbf{U}}^\\top\\mathbf{x}_j\\|_2^2$ and $\\{S_i:=\\sum_{j\\in \\mathcal{C}_i}\\|\\hat{\\mathbf{U}}^\\top\\mathbf{x}_j-\\hat{\\mathbf{U}}^\\top\\sum_{k\\in{\\mathcal{C}}_i}\\mathbf{x}_k\\|_2^2\\}_{i=1}^2$, respectively, with $\\mathcal{C}_i$ representing the set of data vectors belonging to cluster $i$. Table \\ref{tab:cluster} reports the clustering errors and scatter ratios of dPCA and PCA under  different $d$ values. Clearly, dPCA exhibits lower clustering error and higher scatter ratio.}\n\n\\begin{figure}[t]\n\t\\centering \n\t\\includegraphics[scale=0.53]{digits.pdf} \t\n\t\\caption{\\small{dPCA versus PCA on semi-synthetic images.}}\n\t\\label{fig:digits}\n\\end{figure}\n\n\n\n\n\\renewcommand{\\arraystretch}{1.5} \n\\begin{table}[tp]\t\n\\centering\n\t\\fontsize{8.5}{8}\\selectfont\n\t\t\\textcolor{black}{\\caption{\\textcolor{black}{Performance comparison between dPCA and PCA.}}\n\t\\label{tab:cluster}\n\t\\vspace{.8em}\n\t\\begin{tabular}{|c|c|c|c|c|}\n\t\t\\hline\n\t\t\\multirow{2}{*}{$d$}&\n\t\t\\multicolumn{2}{c|}{Clustering error}&\\multicolumn{2}{c|}{ Scatter ratio}\\cr\\cline{2-5}\n\t\t&dPCA&PCA&dPCA&PCA\\cr\n\t\t\\hline\n\t\t\\hline\n\t\t1&0.1660&0.4900&2.0368&1.0247\\cr\\hline\n\t\t2&0.1650&0.4905&1.8233&1.0209\\cr\\hline\n\t\t3&0.1660&0.4895&1.6719&1.1327\\cr\\hline\n\t\t4&0.1685&0.4885&1.4557&1.1190\\cr\\hline\n\t\t5&0.1660&0.4890&1.4182&1.1085\\cr\\hline\n\t\t10&0.1680&0.4885&1.2696&1.0865\\cr\\hline\n\t\t50&0.1700&0.4880&1.0730&1.0568\\cr\\hline\n\t   100&0.1655&0.4905&1.0411&1.0508\\cr\n\t\t\\hline\n\t\\end{tabular}}\n\\end{table}\n                                                                              \n\n\nReal protein expression data \\cite{mice}\n were also used to evaluate the ability of dPCA to discover subgroups in real-world conditions. Target data $\\{\\accentset{\\circ}{\\mathbf{x}}_i\\in\\mathbb{R}^{77}\\}_{i=1}^{267}$ contained $267$ data vectors, each collecting $77$ protein expression measurements of a mouse having Down Syndrome disease \\cite{mice}.  \nIn particular, the first $135$ data points $\\{\\accentset{\\circ}{\\mathbf{x}}_i\\}_{i=1}^{135}$ recorded protein expression measurements of $135$ mice with drug-memantine treatment, while the remaining $\\{\\accentset{\\circ}{\\mathbf{x}}_i\\}_{i=136}^{267}$ collected measurements of $134$ mice without such treatment. Background data  $\\{\\accentset{\\circ}{\\mathbf{y}}_j\\in\\mathbb{R}^{77}\\}_{j=1}^{135}$ on the other hand, comprised such measurements from $135$ healthy mice, which likely exhibited similar natural variations (due to e.g., age and sex) as the target mice, but without the differences that result from the Down Syndrome disease. \n\n\n\nWhen performing cPCA on $\\{\\accentset{\\circ}{\\mathbf{x}}_i\\}$ and $\\{\\accentset{\\circ}{\\mathbf{y}}_j\\}$, four $\\alpha$'s were selected from $15$ logarithmically-spaced values between $10^{-3}$ and $10^{3}$ via the spectral clustering method presented in \\cite{2017cpca}. \n\n\nExperimental results are reported in Fig. \\ref{fig:mice} with red circles and black diamonds representing sick mice with and without treatment, respectively. Evidently, when PCA is applied, the low-dimensional representations of the protein measurements from mice with and without treatment are distributed similarly. In contrast,  the low-dimensional representations cluster two groups of mice successfully when dPCA is employed. At the price of runtime (about $15$ times more than dPCA), cPCA with well {tuned} parameters ($\\alpha=3.5938$ and $27.8256$) can also separate the two groups.\n\n\n\n\\begin{figure}[t]\n\t\\centering \n\t\\includegraphics[scale=0.735]{mice.pdf} \n\t\\vspace{-4pt}\n\t\\caption{\\small{Discovering subgroups in mice protein expression data.}}\n\t\\label{fig:mice}\n\t\\vspace{-4pt}\n\\end{figure}\n\n\\subsection{KdPCA tests}\\label{sec:simu2}\n\n\n\n \\begin{figure}[t]\n \t\\centering \n \t\\includegraphics[scale=0.61]{kdpca_target.pdf} \n \t\\caption{\\small{Target data dimension distributions with $x_{i,j}$ representing the $j$-th entry of   $\\mathbf{x}_i$ for $j=1,\\, \\ldots,\\,4$ and $i=1,\\, \\ldots,\\,300$.}}\n \t\\label{fig:kdpca_target}\n \\end{figure}\n \n In this subsection, our KdPCA is evaluated \n using synthetic and real data.\nBy adopting the procedure described in \\cite[p. 546]{hastie2009elements}, we generated target data $\\{{\\mathbf{x}}_i:=[x_{i,1}\\, x_{i,2}\\,x_{i,3}\\,x_{i,4}]^\\top\\}_{i=1}^{300}$ and background data $\\{{\\mathbf{y}}_j\\in\\mathbb{R}^4\\}_{j=1}^{150}$. In detail, $\\{[x_{i,1}\\,x_{i,2}]^{\\top}\\}_{i=1}^{300}$ were sampled uniformly  from two circular concentric clusters with corresponding radii $1$ and $6$ shown in the left panel of Fig. \\ref{fig:kdpca_target}; and $\\{[x_{i,3}\\,x_{i,4}]^{\\top}\\}_{i=1}^{300}$ were uniformly drawn from a circle with radius $10$; see Fig. \\ref{fig:kdpca_target} (right panel) for illustration.\n The first and second two dimensions of $\\{{\\mathbf{y}}_j\\}_{j=1}^{150}$ were uniformly sampled from two concentric circles with corresponding radii of $4$ and $10$. All data points in $\\{{\\mathbf{x}}_i\\}$ and $\\{{\\mathbf{y}}_j\\}$ were corrupted with additive noise sampled independently from $\\mathcal{N}(\\mathbf{0},\\,0.1\\mathbf{I})$. \n To unveil the specific cluster structure of the target data relative to the background data,  \n  Alg. \\ref{alg:kdpca} was run with $\\epsilon=10^{-3}$ and using the degree-$2$ polynomial kernel $\\kappa(\\mathbf{z}_i,\\mathbf{z}_j)=(\\mathbf{z}_i^\\top\\mathbf{z}_j )^2$.   Competing alternatives including PCA, KPCA, cPCA, kernel (K) cPCA \\cite{2017cpca}, and dPCA were also implemented.  \n  Further, KPCA and KcPCA shared the kernel function with KdPCA. Three different values of $\\alpha$ were automatically chosen for cPCA \\cite{2017cpca}.\n  The parameter $\\alpha$ of KcPCA was set as $1$, $10$, and $100$.\n \n\nFigure \\ref{fig:kdpca_syn} depicts the first two dPCs, cPCs, and PCs of the aforementioned dimensionality reduction algorithms. Clearly, only KdPCA\nsuccessfully reveals the two unique clusters of $\\{{\\mathbf{x}}_i\\}$ relative to $\\{{\\mathbf{y}}_j\\}$. \n\n\\begin{figure}[t]\n\t\\centering \n\t\\includegraphics[scale=0.78]{kdpca_syn.pdf}  \n\t\t\t\\vspace{5pt}\n\t\\caption{\\small{Discovering subgroups in nonlinear synthetic data.}}\n\t\\label{fig:kdpca_syn}\n\\end{figure}\n\\begin{figure}[t]\n\t\\centering \n\t\\includegraphics[scale=0.74]{kdpca_real.pdf}  \n\t\\caption{\\small{Discovering subgroups in MHealth data.}}\n\t\\label{fig:kdpca_real}\n\\end{figure}\n\n\\begin{figure}[t]\n\t\\centering \n\t\\includegraphics[scale=0.71]{kdpca_real_test2.pdf}  \n\t\\caption{\\small{Distinguishing between waist bends forward and cycling.}}\n\t\\label{fig:kdpca_real_test2} \n\\end{figure}\n\n\\begin{figure}[t]\n\t\\centering \n\t\\includegraphics[scale=0.7]{kdpca_real_add.pdf}  \n\t\\caption{\\small{Face recognization by performing KdPCA.}}\n\t\\label{fig:kdpca_real_add} \n\\end{figure}\n\n \nKdPCA was tested \nin realistic settings using the real Mobile (M) Health data \\cite{mhealth}. This dataset consists of sensor (e.g., gyroscopes, accelerometers, and EKG) measurements from volunteers conducting a series of physical activities. In the first experiment, $200$ target data  $\\{{\\mathbf{x}}_i\\in\\mathbb{R}^{23}\\}_{i=1}^{200}$ were used, each of which recorded $23$ sensor measurements from one volunteer performing two different physical activities, namely laying down and having  frontal elevation of arms ($100$ data points correspond to each activity). Sensor measurements from the same volunteer standing still were utilized for the $100$ background data points $\\{{\\mathbf{y}}_j\\in\\mathbb{R}^{23}\\}_{j=1}^{100}$. \nFor KdPCA, KPCA, and KcPCA algorithms, the Gaussian kernel  \n with bandwidth $5$\nwas used. Three different values for the parameter $\\alpha$ in cPCA were automatically selected from a list of $40$ logarithmically-spaced values between $10^{-3}$ and $10^{3}$, whereas $\\alpha$ in KcPCA was set to $1$ \\cite{2017cpca}.\n\n\n\nThe first two dPCs, cPCs, and PCs of KdPCA, dPCA, KcPCA, cPCA, KPCA, and PCA are reported in Fig. \\ref{fig:kdpca_real}. It is self-evident that the two activities evolve into two separate clusters in the plots of KdPCA and KcPCA. On the contrary, due to the nonlinear data correlations, the other alternatives fail to distinguish the two activities.\n\n \nIn the second experiment, the target data were formed with sensor measurements of one volunteer executing waist bends forward and cycling. The background data were collected from the same volunteer standing still. The Gaussian kernel with bandwidth $40$ was used for KdPCA and KPCA, while the second-order polynomial kernel $\\kappa(\\mathbf{z}_i,\\,\\mathbf{z}_j)=(\\mathbf{z}_i^\\top\\mathbf{z}_j+3)^2$ was employed for KcPCA.  The first two dPCs, cPCs, and PCs of simulated schemes are depicted\n in Fig. \\ref{fig:kdpca_real_test2}. \nEvidently, KdPCA outperforms its competing alternatives in discovering the two physical activities of the target data.\n\n\\textcolor{black}{To test the scalability of our developed schemes,  the Extended Yale-B (EYB)  face image dataset \\cite{yaleb} was adopted to test \n\t the clustering performance of KdPCA, KcPCA, and KPCA. EYB  database contains frontal face images of $38$ individuals, each having about around $65$ color images of $192\\times 168$ ($32,256$) pixels. The color images of three individuals ($60$ images per individual) were converted into grayscale images and vectorized to obtain $180$ vectors of   size $32,256\\times 1$. The $120$ vectors from two individuals (clusters) comprised the target data, and the remaining $60$ vectors formed the background data. A Gaussian kernel with bandwidth $150$ was used for KdPCA, KcPCA, and KPCA. Figure \\ref{fig:kdpca_real_add} reports the first two dPCs, cPCs, and PCs of KdPCA, KcPCA (with 4 different values of $\\alpha$), and KPCA, with black circles and red stars representing the two different individuals from the target data. K-means is carried out using the resulting $2$-dimensional representations of the target data. The clustering errors of KdPCA, KcPCA with $\\lambda=1$, KcPCA with $\\lambda=10$, KcPCA with $\\lambda=50$, KcPCA with $\\lambda=100$, and KPCA are $0.1417$, $0.7$, $0.525$, $0.275$, $0.2833$, and  $0.4167$, respectively. Evidently, the face images of the two individuals can be better recognized with KdPCA than with other methods.}\n\n\\subsection{MdPCA tests}\n \n\t\nThe ability of the MdPCA Alg. \\ref{alg:mdpca} for discriminative dimensionality reduction  \nis examined here with two background datasets. \n For simplicity, the involved weights were set to $\\omega_1=\\omega_2=0.5$.\t\n \n In the first experiment,\n\ttwo clusters of $15$-dimensional data points were generated for the target data $\\{\\accentset{\\circ}{\\mathbf{x}}_i\\in\\mathbb{R}^{15}\\}_{i=1}^{300}$ ($150$ for each). \nSpecifically, the first $5$ dimensions of $\\{\\accentset{\\circ}{\\mathbf{x}}_i\\}_{i=1}^{150}$ and $\\{\\accentset{\\circ}{\\mathbf{x}}_i\\}_{i=151}^{300}$ were sampled from $\\mathcal{N}(\\mathbf{0},\\,\\mathbf{I})$ and $\\mathcal{N}(8\\mathbf{1},\\,2\\mathbf{I})$, respectively. The second and last $5$ dimensions of $\\{\\accentset{\\circ}{\\mathbf{x}}_i\\}_{i=1}^{300}$ were drawn accordingly from the normal distributions $\\mathcal{N}(\\mathbf{1},\\,10\\mathbf{I})$ and $\\mathcal{N}(\\mathbf{1},\\,20\\mathbf{I})$. The right top plot of Fig. \\ref{fig:mdpca_syn} shows that performing PCA cannot resolve the two clusters.\nThe first, second, and last $5$ dimensions of the first background dataset $\\{\\accentset{\\circ}{\\mathbf{y}}_{j}^1\\in\\mathbb{R}^{1}\\}_{j=15}^{150}$  were sampled from $\\mathcal{N}(\\mathbf{1},\\,2\\mathbf{I})$, $\\mathcal{N}(\\mathbf{1},\\,10\\mathbf{I})$, and $ \\mathcal{N}(\\mathbf{1},\\,2\\mathbf{I})$, respectively, while those of the second background dataset $\\{\\accentset{\\circ}{\\mathbf{y}}_{j}^2\\in\\mathbb{R}^{15}\\}_{j=1}^{150}$ were drawn from $\\mathcal{N}(\\mathbf{1},\\,2\\mathbf{I})$, $\\mathcal{N}(\\mathbf{1},\\,2\\mathbf{I})$, and $\\mathcal{N}(\\mathbf{1},\\,20\\mathbf{I})$.  \nThe two plots at the bottom of Fig. \\ref{fig:mdpca_syn} depict the first two dPCs of dPCA implemented with a single background dataset.  \nEvidently, MdPCA can discover the two clusters in the target data by leveraging \n the two background datasets.  \n\n\\begin{figure}[t]\n\t\\centering \n\t\\includegraphics[scale=0.66]{mdpca_syn.pdf}  \n\t\\vspace{-5pt}\n\t\\caption{\\small{Clustering structure by MdPCA using synthetic data.}}\n\t\\label{fig:mdpca_syn}\n\t\\vspace{-5pt}\n\\end{figure}\n\n\\begin{figure}[t]\n\t\\centering \n\t\\includegraphics[scale=0.69]{mdpca_semisyn.pdf}  \n\t\\vspace{-8pt}\n\t\\caption{\\small{Clustering structure by MdPCA using semi-synthetic data.}}\n\t\\label{fig:mdpca_semisyn}\n\t\\vspace{-10pt}\n\\end{figure}\n\n \nIn the second experiment, the\n  target data $\\{\\accentset{\\circ}{\\mathbf{x}}_{i}\\in\\mathbb{R}^{784}\\}_{i=1}^{400}$ were obtained using $400$ handwritten digits $6$ and $9$ ($200$ for each) of size $28\\times 28$ from the MNIST dataset superimposed with $400$ resized `girl' images from the CIFAR-100 dataset  \\cite{cifar10}. \nThe first $392$ dimensions of the first background dataset $\\{\\accentset{\\circ}{\\mathbf{y}}_{j}^1\\in\\mathbb{R}^{784}\\}_{j=1}^{200}$ and the last $392$ dimensions of the other background dataset $\\{\\accentset{\\circ}{\\mathbf{y}}_{j}^2\\in\\mathbb{R}^{784}\\}_{j=1}^{200}$ correspond to the first and last $392$ features of $200$ cropped girl images, respectively. The remaining dimensions of both background datasets were set zero.\n Figure \\ref{fig:mdpca_semisyn} presents the obtained (d)PCs of MdPCA, dPCA, and PCA, with red stars and black diamonds depicting digits $6$ and $9$, respectively.  \n PCA and dPCA based on a single background dataset (the bottom two plots in Fig. \\ref{fig:mdpca_semisyn}) reveal that the two clusters of data follow a similar distribution in the space spanned by the first two PCs. The separation between the two clusters becomes clear when the MdPCA is employed.\n \n\n \n \n\n\\subsection{KMdPCA tests}\n\\begin{figure}[t]\n\t\\centering \n\t\\includegraphics[scale=0.7]{kmdpca_syn.pdf} \n\t\\vspace{-10pt}\n\t\\caption{\\small{The first two dPCs obtained by  Alg. \\ref{alg:mkdpca}.}}\n\t\\label{fig:kmdpca}\n\t\\vspace{-13pt}\n\\end{figure}\n\n\nAlgorithm \\ref{alg:mkdpca} \\textcolor{black}{with $\\epsilon=10^{-4}$} is examined for dimensionality reduction using simulated data and compared against MdPCA, KdPCA, dPCA, and PCA.\nThe first two dimensions of the target data $\\{\\mathbf{x}_i\\in \\mathbb{R}^6\\}_{i=1}^{150}$ and $\\{\\mathbf{x}_i\\}_{i=151}^{300}$ were generated from two circular concentric clusters with respective radii  of $1$ and $6$.  The remaining four dimensions of the target data $\\{\\mathbf{x}_i\\}_{i=1}^{300}$ were sampled from   two concentric circles with radii of $20$ and $12$, respectively. Data $\\{\\mathbf{x}_i\\}_{i=1}^{150}$ and $\\{\\mathbf{x}_i\\}_{i=151}^{300}$ corresponded to two different clusters. \nThe first, second, and last two dimensions of one background dataset $\\{\\mathbf{y}_{j}^1\\in\\mathbb{R}^6\\}_{j=1}^{150}$ were sampled from three concentric circles with corresponding radii of $3$, $3$, and $12$. \nSimilarly,  three concentric circles with radii  $3$, $20$, and $3$ were used for generating the other background dataset $\\{\\mathbf{y}_{j}^2\\in\\mathbb{R}^6\\}_{j=1}^{150}$. Each datum in $\\{\\mathbf{x}_i\\}$, $\\{\\mathbf{y}_{j}^1\\}$, and $\\{\\mathbf{y}_{j}^2\\}$ was corrupted by additive noise $ \\mathcal{N}(\\mathbf{0}, \\,0.1\\mathbf{I})$. When running KMdPCA, the degree-$2$ polynomial kernel used in Sec. \\ref{sec:simu2} was adopted, and weights were set as $\\omega_1=\\omega_2=0.5$.\n\n\nFigure \\ref{fig:kmdpca} depicts the first two dPCs of KMdPCA, MdPCA, KdPCA and dPCA, as well as the first two PCs of (K)PCA.\nIt is evident that only KMdPCA is able to discover the two clusters in the target data.\n\n\n\n\n\n\n\n\n\n\n\n\\section{Concluding Summary}\\label{sec:concl}\nIn diverse practical setups, one is interested in extracting, visualizing, and leveraging the unique low-dimensional features of one dataset relative to a few others. This paper put forward a novel framework, that is termed discriminative (d) PCA, for performing discriminative analytics of multiple datasets. Both linear, kernel, and multi-background models were pursued.\n  In contrast with existing alternatives, dPCA is demonstrated to be optimal under certain   assumptions. \nFurthermore, dPCA is \\textcolor{black}{parameter free}, and requires only one generalized eigenvalue decomposition. \nExtensive  tests using both synthetic and real data corroborated the efficacy of our proposed approaches relative to relevant prior works. \n\nSeveral directions open up for future research: i) distributed and privacy-aware (MK)dPCA implementations to cope with large amounts of high-dimensional data; ii) robustifying (MK)dPCA to outliers; \n and iii) \n graph-aware (MK)dPCA generalizations exploiting additional priors of the data.\n\n\\section*{Acknowledgements}\nThe authors would like to thank Professor Mati Wax for pointing out an error in an early draft of this paper.\n\n\n\\bibliographystyle{IEEEtranS}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nMost of our experience (outside perturbation theory) with quantum mechanics concerns nonrelativistic quantum systems. This may be due to the fact that, as yet, no results on specific models of relativistic quantum field theory (rqft) in (physical) four space-time dimensions exist \\cite{GlJa}. In spite of that, it is still widely believed, and there are good reasons for that \\cite{Haag}, that rqft is the most fundamental physical theory. One of its most basic principles is \\textbf{microcausality} (\\cite{Haag}, \\cite{StrWight}), which is the local (i.e., in terms of local quantum fields) formulation of Einstein causality, the limitation of the velocity of propagation of signals by $c$, the velocity of light in the vacuum:\n$$\n[\\Phi(f),\\Phi(g)] = 0\n\\eqno{(1)}\n$$\nwhere the fields $\\Phi$ are regarded as (space-time) operator-valued distributions, when the supports of $f$ and $g$ are space-like to one another, see section 3.\n\nUnfortunately, in spite of its enormous success, nonrelativistic quantum mechanics (nrqm) is well-known to violate Einstein causality, which is not surprising, since nrqm is supposed to derive from the the non-relativistic limit $c \\to \\infty$ of rqft, mathematically speaking a rather singular limit, called a group contraction, from the Poincar\\'{e} to the Galilei group, first analysed by In\\\"{o}n\\\"{u} and Wigner \\cite{InWi}. For some systems, such as quantum spin systems, finite group velocity follows from the Lieb-Robinson bounds \\cite{LiR}, \\cite{NS}: these systems are, however, approximations to nonrelativistic many-body systems. Due to the crucial importance of Einstein causality for the foundations of physics (see \\cite{HMN}), it is important to understand in which precise sense nrqm is an \\textbf{approximation} of a causal theory, viz., rqft.\n\nIn classical physics, acausal behavior is well-known, e.g., in connection to the diffusion equation or heat conduction problems. In both cases, these equations may be viewed as an approximation of the telegraphy equation (see \\cite{Bar}, pg. 185 or \\cite{MorFes}, section 7.4), and the approximations are under mathematical control. What happens in quantum theory?\n\nFor imaginary times, the heat diffusion equation becomes the Schr\\\"{o}dinger equation, for a free particle of mass $m$ in infinite space ($\\hbar = 1$):\n$$\ni\\frac{\\partial}{\\partial t} \\Psi_{t} = - \\frac{1}{2m} \\triangle \\Psi_{t} \\mbox{ with } \\Psi \\in {\\cal H}=L^{2}(\\mathbf{R}^{3})\n\\eqno{(2.1)}\n$$\nThe Laplacean $\\triangle$ is a multiplication operator in momentum space, and the solution of (2.1) is (with $\\tilde{f}$ denoting Fourier transform of $f$),\n$$\n\\tilde{\\Psi_{t}}(\\vec{p}) = \\exp(-it\\frac{\\vec{p}^{2}}{2m}) \\tilde{\\Psi_{0}}(\\vec{p})\n\\eqno{(2.2)}\n$$\nAssuming that $\\Psi_{0}$ is a $C_{0}^{\\infty}(\\mathbf{R}^{3})$ function, i.e., smooth with compact support, it follows from (2.2) and the ''only if'' part of the Paley-Wiener theorem (see,e.g., \\cite{RSII}, Theorem IX-11) that, for any $t \\ne 0$, $\\Psi_{t}$ cannot be of compact support and is thus infinitely extended: one speaks of ''instantaneous spreading'' \\cite{GCH}. Of course, spreading is a general phenomenon in quantum physics, and this feature is demonstrated in a varied number of situations and in several possible ways, including an exact formula for the free propagator (see, e.g., \\cite{MWB}, chapter 2). The fact that the violation of Einstein causality is ''maximal'' was sharpened and made precise by Requardt \\cite{Req}, who showed that, for a class of one and n body non-relativistic systems, states localized at time zero in an arbitrarily small open set of $\\mathbf{R}^{n}$ are total after an arbitrarily small time.\n\nA very nice recent review of related questions was given by Yngvason in \\cite{Yng}. His theorem 1 transcribes a result proved by Perez and Wilde \\cite{PeWil} (see also \\cite{Yng} for additional related references), which shows that localization in terms of position operators is incompatible with causality in relativistic quantum physics. \n\nA different approach, which generalizes the argument after (2.2) by making a different use of analyticity, and also introduces the dichotomy mentioned in the abstract, was proposed by Hegerfeldt \\cite{GCH1}:\n\n\\textbf{Theorem 1} Let $H$ be a self-adjoint operator, bounded below, on a Hilbert space ${\\cal H}$. for given $\\Psi_{0} \\in {\\cal H}$, let $\\Psi_{t}, t \\in \\mathbf{R}$, be defined as\n$$\n\\Psi_{t} = \\exp(-iHt) \\Psi_{0}\n\\eqno{(3)}\n$$\nLet $A$ be a positive operator on ${\\cal H}$, $A \\ge 0$, and $p_{A}$ be defined by\n$$\np_{A}(t) \\equiv (\\Psi_{t}, A \\Psi_{t})\n\\eqno{(4)}\n$$\nThen, either\n$$\np_{A}(t) \\ne 0 \n\\eqno{(5)}\n$$\nfor almost all $t$  and the set of such $t$  is dense and open, or \n$$\np_{A}(t) \\equiv 0 \\mbox{ for all } t\n\\eqno{(6)}\n$$\n\nIf, now, the probability to find a particle inside a bounded region $V$ is given by the expectation value of an operator $N(V)$, such that\n$$\n0 \\le N(V) \\le \\mathbf{1}\n\\eqno{(7)}\n$$\n(e.g., $$N(V) = |\\chi_{V})(\\chi_{V}| \\eqno{(8)}$$ , where $\\chi_{V}$ is the characteristic function of $V$), and $\\mathbf{1}$ the identity operator,            it follows from theorem 1, with the choice\n$$\nA = \\mathbf{1} - N(V)\n\\eqno{(9)}\n$$\nthat, if at $t=0$ a particle is strictly localized in a bounded region $V_{0}$, then, unless it remains in $V_{0}$ for all times, it cannot be strictly localized in a bounded region $V$, however large, for any finite time interval thereafter, implying a violation of Einstein causality (see also \\cite{GCH1}, pg. 24, for further comments).\n\nOur main purpose in this review is to analyse the dichotomy (5)-(6) for nonrelativistic quantum systems (in case (6) we include some relativistic systems in the final discussion in the conclusion). We start with the option given by equation (6).\n\n\\section{Systems confined to a bounded region of space in quantum theory}\n\nOption (6) is found - with $A$  defined by (7)-(9) - in all systems restricted to lie in a finite region $V$ by \\textbf{boundaries}, with a Hamiltonian $H_{V}$ in theorem 1 self-adjoint and bounded below. This includes the electromagnetic field (Casimir effect, see the conclusion), but we now concentrate on nonrelativistic quantum systems. The simplest prototype of such is the free Hamiltonian $H_{V} = -\\frac{d^{2}}{dx^{2}}$, with $V=[0,L]$. The forthcoming theorem summarizes (and slightly extends) the rather detailed analysis in \\cite{GarKar}, using the results in \\cite{Robinson} (see the appendix of \\cite{GarKar} and references given there for the standard concepts used below). Our forthcoming conclusions differ, however, from \\cite{GarKar}. \n\n\\textbf{Theorem 2.1} In the following three cases, $H_{V}$ is self-adjoint and semi-bounded:\n\na1) $H_{V}^{\\sigma}$ on the domain \n\\begin{eqnarray*}\nD(H_{V}^{\\sigma}) = \\{\\mbox{ set of absolutely continuous (a.c.) functions } \\Psi \\mbox{ over } [0,L]\\\\\n\\mbox{ with a.c. first derivative } \\Psi^{'} \\mbox{ such that } \\Psi^{''} \\in L^{2}(0,L)\\}\n\\end{eqnarray*}\nand satisfying the boundary condition (b.c.)\n$$\n\\Psi^{'}(0)= \\sigma_{0} \\Psi(0) \\mbox{ and } \\Psi^{'}(L)= -\\sigma_{L} \\Psi(L)\n\\eqno{(10)}\n$$\nwhere $(\\sigma_{0},\\sigma_{L}) \\in (\\mathbf{R} \\times \\mathbf{R})$;\n\na2) $H_{V}^{\\infty}$ on $D(H_{V}^{\\infty})$, same as inside the brackets in a1), but with (10) replaced by\n$$\n\\Psi(0) = \\Psi(L) = 0\n\\eqno{(11)}\n$$\n\na3) $H_{V}^{\\theta}$, on $D(H_{V}^{\\theta})$, same as inside the brackets in a1), but with (10) replaced by \n$$\n\\Psi(0)= \\exp(i\\theta) \\Psi(L)\n\\eqno{(12)}\n$$\nwith $\\theta \\in \\mathbf{R}$. The case $\\sigma_{0}=0$ in a1) corresponds to Neumann b.c., $\\sigma_{0}>0$ to repulsive, $\\sigma_{0}<0$ to attractive boundaries (see \\cite{Robinson}, pg.17), with analogous statements for $\\sigma_{L}$. The case a2) corresponds to setting $\\sigma_{0} = -\\sigma_{L} = \\infty$ in (10), and is the case of impenetrable boundaries (Dirichlet b.c.). a3) is a generalization of periodic b.c.. We also have:\n\n\\textbf{Theorem 2.2} \n\na) In case a1), the momentum $p=-i\\frac{d}{dx}$ is not a symmetric operator;\n\nb) In case a2), $p$ defines a closed symmetric operator $p_{\\infty}$, and in case a3) it is a self-adjoint operator $p_{\\theta}$, which is a self-adjoint extension of $p_{\\infty}$, but for no $\\theta \\in \\mathbf{R}$ there are functions satisfying the Dirichlet b.c. (11) in the domain $D(p_{\\theta})$ of $p_{\\theta}$. Furthermore, in case a3),\n$$\nH_{V}^{\\theta} = p_{\\theta}^{2} = p_{\\theta}^{*} p_{\\theta}\n\\eqno{(13)}\n$$\n\nAn explicit proof of b.) may be found in \\cite{GarKar}, and a.) is straightforward. We see that in cases a1) and a2) the momentum is not well-defined (as a self-adjoint operator), while it is so in case a3), in which case the expected property (13) holds.\n\nWhat do we conclude fom theorems 2.1 and 2.2 (and their natural extensions to partial differential operators in higher dimensions, see \\cite{Robinson}, pg. 34)? As remarked by Robinson (\\cite{Robinson}, page 22), defining the probability current density $j(x)$ associated to the particle,\n$$\nj(x) = i(\\frac{d\\bar{\\Psi}}{dx} \\Psi(x)-\\bar{\\Psi}(x)\\frac{d\\Psi(x)}{dx})\n\\eqno{(14)}\n$$\nwe see that for a1),a2), $j(0)=0=j(L)$, while, in case a3), only $j(0)=j(L)$ holds. Thus, only in cases a1),a2) the particle flux both into and out of the system is zero, corresponding to an \\textbf{isolated} system, while a3) only means that all that flows in at $x=0$ flows out at $x=L$. This is the case with \\textbf{periodic} b.c. (a restriction of a3)), which requires for each $\\Psi$ that $\\Psi(x+L)=\\Psi(x)$ for all $x \\in [0,L]$. we thus call a3) generalised periodic b.c.: they allow a finite system to have a momentum operator \\cite{MaRo}, because, at the same time, they render the system ''infinite'' in a peculiar way, making it into a torus.  \n \nWe see, therefore, that the attempt to confine a quantum system in a bounded region of space by imposing on it b.c. originating from classical physics (a1),a2)) leads to physical inconsistencies, since the momentum is expected to exist as a local generator of space-translations (theorem 2.2). Generalized periodic b.c. (a3)) sometimes save the situation, for instance regarding thermodynamic quantities in statistical mechanics, which are expected (and often proven) not to depend on the boundary conditions \\cite{Ru}. For expectation values and correlation functions this  need not be the case. In addition, there are situations in rqft, such as the Casimir effect, for which periodic b.c. are definitely not adequate, as we shall comment in the conclusion.\n\n\\section{The problem of instantaneous spreading}\n\nWe now come to the option given by equation (5). Using theorem 1, Hegerfeldt (see \\cite{GCH} and references given there) proposed to analyse a two-atom model suggested by Fermi \\cite{Fermi} to check finite propagation speed in quantum electrodynamics, with $H$ the Hamiltonian, $A=A_{e_{B}}$, the probability that atom $B$, initially in the ground state, is excited by a photon resulting from the decay of atom $A$, initially in an excited state, and $\\Psi_{0}$ denoting the initial physical state of the system $A-B$. The conclusion is that $B$ is either immediately excited with nonzero probability or never. This conclusion was challenged by Buchholz and Yngvason \\cite{BYng} in a beautiful and subtle analysis, in which they concluded that there are no causality problems for the Fermi system in a full description of the system by rqft. One important point raised in \\cite{BYng} is that (4),(5) with $A$ positive is not an adequate criterion to investigate causality in rqft, as shown by the simple counterexample of the state $(\\Psi_{0}, \\cdot \\Psi_{0})$ equal to the vacuum state, for which $p_{A}(t)$ is \\textbf{always} nonzero for $A$ positive space-localized, by the Reeh-Schlieder theorem (see, e.g., \\cite{Ar}, pg. 101). It is perhaps worth noting that a non-perturbative rqft description of the two-atom system is not known, but the authors \\cite{BYng} relied on the general principles of rqft (\\cite{Haag}, \\cite{Ar}).\n\nIt follows from the above that instantaneous spreading would cease to be an obstacle to the physical consistency of nonrelativistic quantum mechanics if it could be shown that the latter is an approximation, in a suitable precise sense, to rqft. In this review we expand on a discussion by C. J\\\"{a}kel and one of us \\cite{JaWre} on this matter. In a not yet precise fashion (but see Lemma 2), one might propose as approximation criterion\n\n\\textbf{Proposal C} Nonrelativistic ground state expectation values are ''close'' to the corresponding relativistic vacuum expectation values when certain physical parameters are ''small''.\n\nFor an atom, e.g. hydrogen, in interaction with the electromagnetic field in its ground state, one such parameter is the ratio between the mean velocity of  the electron in the ground state and $c$, which is of order of the fine structure constant. It is clear that the Dirac atom, with a potential, is not a fully relativistic system, and therefore not a candidate to solve the Einstein causality problems in the manner proposed in \\cite{BYng}: thus, the well-known relativistic corrections \\cite{JJS} do not solve the causality issue as sketched above. Perturbative quantum electrodynamics, in spite of its great success, does not offer a solution either: for instance, the relativistic Lamb shift relies strongly on Bethe's nonrelativistic treatment (see, e.g., \\cite{JJS}, pg. 292). \n\nWe now attempt to make proposal C precise, and, at the same time, show some results relating relativistic and nonrelativistic systems which are not found in this form in the textbook literature. We take as the nonrelativistic systems, formulated in Fock space, the symmetric Fock space for Bosons, ${\\cal F}_{s}({\\cal H})$ which we simply denote by ${\\cal F}$ (see \\cite{MaRo} for a nice textbook presentation) and there the state $\\omega_{\\Psi_{0}}=(\\Psi_{0}, \\cdot \\Psi_{0})$. The observables will be functionals of the nonrelativistic free quantum fields at time zero:\n$$\n\\Phi(\\vec{x}) = \\phi(\\vec{x}) + \\phi^{*}(\\vec{x})\n\\eqno{(15.1)}\n$$\nwhere $*$ denotes hermitian conjugate and\n$$\n\\phi(\\vec{x}) = \\frac{1}{(2\\pi)^{3\/2}(2m_{0})^{1\/2}}\\int d\\vec{k} a(\\vec{k})\\exp(-i\\vec{k}\\cdot \\vec{x})\n\\eqno{(15.2)}\n$$\nand the canonically conjugate momenta\n$$\n\\Pi(\\vec{x}) = \\pi(\\vec{x}) + \\pi^{*}(\\vec{x})\n\\eqno{(16.1)}\n$$\nwhere\n$$\n\\pi(\\vec{x}) = -\\frac{i(2m_{0})^{1\/2}}{(2\\pi)^{3\/2}}\\int d\\vec{k} a(\\vec{k})\\exp(i\\vec{k}\\cdot \\vec{x}) \n\\eqno{(16.2)}\n$$\nAbove, $a, a^{*}$ are annihilation-creation operators satisfying\n$$\n[a(\\vec{k}),a^{*}(\\vec{l})] = \\delta(\\vec{k}-\\vec{l})\n\\eqno{(17)}\n$$\nIt is more adequate, both mathematically and physically (\\cite{MaRo},\\cite{RSII}) to use the smeared fields\n$$\n\\Phi(f) = \\int d\\vec{x} f(\\vec{x}) \\Phi(\\vec{x})\n\\eqno{(18)}\n$$\nand\n$$\n\\Pi(g) = \\int d\\vec{x} g(\\vec{x}) Pi(\\vec{x})\n\\eqno{(19)}\n$$\ni.e., to consider $\\Phi,\\Pi$ as operator-valued distributions, satisfyind the canonical commutation relations (CCR)\n$$\n[\\Phi(f),\\Pi(g)] = i(f,g)\n\\eqno{(20)}\n$$\non a suitable dense set (\\cite{RSII}, pg. 232), with\n$$\n(f,g) = \\int d\\vec{x} \\bar{f}(\\vec{x}) g(\\vec{x})\n\\eqno{(21)}\n$$\nfor $f,g \\in {\\cal S}(\\mathbf{R}^{3})$, the Schwarz space \\cite{RSII}. For the free relativistic quantum system, the corresponding state is again the no-particle state $\\omega_{\\Psi_{0}}$, the observables (functionals of) the relativistic free quantum fields\n$$\n\\Phi_{r}(\\vec{x}) = \\phi_{r}(\\vec{x}) + \\phi_{r}^{*}(\\vec{x})\n\\eqno{(22.1)}\n$$\nwhere\n$$ \n\\phi_{r}(\\vec{x}) = \\frac{c}{(2\\pi)^{3\/2}}\\int d\\vec{k}\\frac{1}{(2\\omega_{\\vec{k}}^{c})^{1\/2}} a(\\vec{k})\\exp(-i\\vec{k}\\cdot \\vec{x})\n\\eqno{(22.2)}\n$$\nand the canonically conjugate momentum\n$$\n\\Pi_{r}(\\vec{x}) = \\pi_{r}(\\vec{x}) + \\pi_{r}^{*}(\\vec{x})\n\\eqno{(23.1)}\n$$\nwith\n$$\n\\pi_{r}(\\vec{x}) = -\\frac{i}{(2\\pi)^{3\/2}c}\\int d\\vec{k}(2\\omega_{\\vec{k}}^{c})^{1\/2} a(\\vec{k})\\exp(i\\vec{k}\\cdot \\vec{x})\n\\eqno{(23.2)}\n$$\nIt is convenient to consider the CCR in the Weyl form\n$$\n\\exp(i\\Pi(f))\\exp(i\\Phi(g))=\\exp(i\\Phi(g))\\exp(i\\Pi(f))\\exp(-i(f,g))\n\\eqno{(24)}\n$$\nfor $f,g \\in {\\cal S}_{\\mathbf{R}}(\\mathbf{R}^{3})$, the Schwarz space of real-valued functions on $\\mathbf{R}^{3}$. Above,\n$$\n\\omega_{\\vec{k}}^{c} \\equiv (c^{2}\\vec{k}^{2}+m_{0}^{2}c^{4})^{1\/2}\n\\eqno{(25)}\n$$\nwith $m_{0}$ the ''bare mass'' of the particles. We write\n$$\na(f) = (2m_{0})^{1\/2}[\\Phi(f)+i \\Pi(f)]\n\\eqno{(26)}\n$$\nand similarly for $a^{*}(f),a_{r}(f),a_{r}^{*}(f)$. The zero-particle vector $\\Psi_{0} \\in {\\cal F}$ is such that\n$$\na(f)\\Psi_{0} = 0 \\mbox{ for all } f \\in {\\cal S}(\\mathbf{R}^{3})\n\\eqno{(27)}\n$$\nand similarly for $a_{r}(f)$. We assume that there exists a continuous unitary representation $U(\\vec{a},R)$ of the Euclidean group $\\vec{x} \\to R\\vec{x}+\\vec{a}$ on ${\\cal F}$ with $R$ a rotation and $\\vec{a}$ a translation, s.t.\n$$\nU(\\vec{a},R) a(f) U(\\vec{a},R)^{-1} = a(f_{\\vec{a},R})\n\\eqno{(28)}\n$$\nwith\n$$\nf_{\\vec{a},R}(\\vec{x})= f(R^{-1}(\\vec{x}-\\vec{a})  \n\\eqno{(29)}\n$$\nThe following lemma is fundamental:\n\n\\textbf{Lemma 1} The no-particle state $\\Psi_{0}$ is the unique state invariant under $U(\\vec{a},R)$.\n\n\\textbf{Proof} By (27),(28),\n$$\n0 = U(\\vec{a},R) a(f) \\Psi_{0} = a(f_{\\vec{a},R})U(\\vec{a},R) \\Psi_{0}\n\\eqno{(30)}\n$$\nSince, for all $f \\in {\\cal S}(\\mathbf{R}^{3})$, $U(\\vec{a},R) \\Psi_{0}$ is also a no-particle state by (30), it follows that\n$$\nU(\\vec{a},R) \\Psi_{0} = \\lambda(\\vec{a},R) \\Psi_{0}\n\\eqno{(31)}\n$$\nwith $|\\lambda|=1$, and the $\\lambda$ form a one-dimensional representation of the Euclidean group. Since the Euclidean group posesses only the trivial one-dimensional representation, we conclude that\n$$\nU(\\vec{a},R) \\Psi_{0} = \\Psi_{0}\n$$\ni.e., $\\Psi_{0}$ is necessarily a Euclidean invariant state (As Wightman observes \\cite{Wight}, this is \\textbf{not} assumed when one writes (28)!). In the case of the free (relativistic or nonrelativistic) field, the cluster property of the two-point function (a corollary of the Riemann-Lebesgue lemma, see, e.g., \\cite{MWB}, Lemma 3.8) implies, together with von Neumann's ergodic theorem (see, again, e.g., \\cite{MWB}, Theorem A.2), that $\\Psi_{0}$ is the unique state invariant under all space translations and, thus, the unique state invariant under all $U(\\vec{a},R)$. q.e.d.\n\nLemma 1 is the main ingredient of\n\n\\textbf{Theorem 3} The representations of the Weyl CCR (24) $(\\Phi,\\Pi)$ and $(\\Phi_{r},\\Pi_{r})$ are unitarily inequivalent.\n\n\\textbf{Proof} The proof of this theorem follows from (\\cite{RSII}, Theorem X.46), the inequivalence of the Weyl CCR for different masses $m_{1}$ and $m_{2}$, by identifying $\\Phi_{m_{1}}$ with $\\Phi$ and $\\Phi_{m_{2}}$ with $\\Phi_{r}$ (and similarly for the $\\Pi$). Let $G(R,\\vec{a})$ (resp. $G_{r}(R,\\vec{a})$) be the representatives of the Euclidean group leaving $(\\Phi,\\Pi)$ (resp.$(\\Phi_{r},\\Pi_{r})$) invariant. We assume that there exists a unitary map $T$ on ${\\cal F}$ which satisfies $$T \\exp(i\\Phi(f))T^{-1} = \\exp(i\\Phi_{r}(f)) \\eqno{(32)}$$ and $$T \\exp(i\\Pi(f))T^{-1} = \\exp(i\\Pi_{r}(f) \\eqno{(33)}$$. Exactly as in \\cite{RSII}, Theorem X.46, pg.234, this leads to \n$$\nTG(R,\\vec{a})T^{-1} = G_{r}(R,\\vec{a})\n\\eqno{(34)}\n$$\nfor all $(R,\\vec{a})$  in the Euclidean group. Applying (34) to $\\Psi_{0}$, we find\n$$\nT \\Psi_{0} = G_{r}(R,\\vec{a}) T \\Psi_{0}\n\\eqno{(35)}\n$$\nand, since, by lemma 1, $\\Psi_{0}$ is the unique vector in ${\\cal F}$ invariant under both $G(R,\\vec{a})$ and $G_{r}(R,\\vec{a})$, (35) yields\n$$\nT \\Psi_{0} = \\alpha \\Psi_{0}\n\\eqno{(36)}\n$$\nwhere $\\alpha$ is a phase. From (32), (33), and (36),\n\\begin{eqnarray*}\n(\\Psi_{0}, \\Phi(f) \\Phi(g) \\Psi_{0}) = (\\Psi_{0},  T \\exp(i\\Phi(f))T^{-1} T \\exp(i\\Phi(g))T^{-1}\\Psi_{0})\\\\\n= (\\Psi_{0},\\exp(i\\Phi_{r}(f))\\exp(i\\Phi_{r}(g))\\Psi_{0})\n\\end{eqnarray*}$$\\eqno{(37)}$$\nwhich implies that $\\Psi_{0}, \\Phi(f) \\Phi(g) \\Psi_{0})$ and $(\\Psi_{0},\\exp(i\\Phi_{r}(f))\\exp(i\\Phi_{r}(g))\\Psi_{0})$ are equal as tempered distributions on\n${\\cal S}(\\mathbf{R}^{3}) \\times {\\cal S}(\\mathbf{R}^{3})$. We have, from (15), (22),\n\\begin{eqnarray*}\n(\\Psi_{0}, \\Phi(\\vec{x})\\Phi(\\vec{y})\\Psi_{0}) = \\frac{1}{2m_{0}} \\delta(\\vec{x}-\\vec{y})=\\\\\n= \\frac{1}{(2m_{0})(2\\pi)^{3}} \\int d\\vec{k} \\exp(i\\vec{k} \\cdot (\\vec{x}-\\vec{y}))\n\\end{eqnarray*} $$\\eqno{(38)}$$\nwhile\n\\begin{eqnarray*}\n(\\Psi_{0}, \\Phi_{r}(\\vec{x}) \\Phi_{r}(\\vec{y}) \\Psi_{0}) = \\frac{1}{i}\\Delta_{+}(\\vec{x}-\\vec{y},m_{0}^{2})=\\\\\n=\\frac{1}{2(2\\pi)^{3}} \\int d\\vec{k} \\exp(i\\vec{k}\\cdot(\\vec{x}-\\vec{y}))\\frac{c^{2}}{\\omega_{\\vec{k}}^{c}}\n\\end{eqnarray*}$$\\eqno{(39)}$$\nand, by (25),\n$$\n\\frac{c^{2}}{\\omega_{\\vec{k}}^{c}}= \\frac{1}{m_{0}(1+\\frac{\\vec{k}^{2}}{m_{0}^{2}c^{2}})^{1\/2}}\n\\eqno{(40)}\n$$\nfrom which\n$$ \n\\frac{c^{2}}{\\omega_{\\vec{k}}^{c}} \\to \\frac{1}{m_{0}} \\mbox{ as } c \\to \\infty\n\\eqno{(41)}\n$$\nFor finite $c$, (38) and (39) do not, however, satisfy (37), leading to a contradiction. q.e.d.\n\nIn spite of the fact that the relativistic and nonrelativistic zero-time fields lead to inequivalent representations of the CCR due to the fact that the corresponding two-point functions are different for finite $c$, (41) shows that (39) tends to (38) as $c \\to \\infty$ and suggests that proposal C might be correct This is the content of the forthcoming lemma 2. We assume that we are given two ''wave-functions'' $f_{1},f_{2}$ such that\n$$\nf_{1}, f_{2} \\in {\\cal S}(\\mathbf{R}^{3})\n\\eqno{(42)}\n$$\n\\textbf{Lemma 2} Let (42) hold  and $\\epsilon $, $\\delta $ be chosen such that for $i=1,2$,\n\\begin{eqnarray*}\n\\int_{\\frac{|\\vec{k}|}{m_{0}c}>\\delta} d\\vec{k}|\\tilde{f_{i}}(\\vec{k})|^{2} < \\epsilon\\\\\n\\end{eqnarray*}\n$$\\eqno{(43)}$$\nThen\n\\begin{eqnarray*}\n2m_{0} \\Delta C \\equiv 2m_{0} |(\\Psi_{0}, \\Phi(f_{1})\\exp(-i(t_{1}-t_{2})H) \\Phi(f_{2})\\Psi_{0})-\\\\\n- (\\Psi_{0}, \\Phi_{r}(f_{1}) \\exp(-i(t_{1}-t_{2})H_{r}) \\Phi_{r}(f_{2})\\Psi_{0})| \\le\\\\\n\\le (2\\epsilon + \\delta^{2}\/2 + |t_{1}-t_{2}|\\frac{m_{0}c^{2}}{\\hbar} \\frac{\\delta^{4}}{8})\n\\end{eqnarray*} $$\\eqno{(44)}$$\nAbove,\n$$\nH \\equiv \\int d\\vec{k} \\frac{\\vec{k}^{2}}{2m_{0}} a^{*}(\\vec{k})a(\\vec{k})\n\\eqno{(45)}\n$$\n$$\nH_{r} \\equiv \\int d\\vec{k} (\\omega_{\\vec{k}}^{c}-m_{0}c^{2}) a^{*}(\\vec{k})a(\\vec{k}) \n\\eqno{(46)}\n$$\nWe also define the number operator\n$$\nN \\equiv \\int d\\vec{k} a^{*}(\\vec{k})a(\\vec{k})\n\\eqno{(47)}\n$$\n\n\\textbf{Remark} It is supposed that $\\delta$ is sufficiently small and is coupled to $\\epsilon$, so that both are small: a fine tuning is required in (43) and depends on the specific problem, but the requirement (43) is very natural and corresponds to the previously mentioned condition that the wavefunctions are ''small' beyond a certain critical momentum (in the ''relativistic'' region of momenta). In addition the time interval $|t_{1}-t_{2}|$ should be small in comparison with characteristic times related to the rest energy $\\frac{\\hbar}{m_{0}c^{2}}$ (here we reinserted $\\hbar$ for clarity). (45)-(47) may be understood as quadratic forms (see \\cite{RSII}, pg. 220). The quantity subtracted in (46) is the ''Zitterbewegungsterm'' \\cite{JJS}. Notice that the $2m_{0}$ factor in (44) cancels the product of two $(2m_{0})^{-1\/2}$ in each $\\Phi(f)$ in (15), or the corresponding relativistic term in the limit $c \\to \\infty$ by (41).\n\n\\textbf{Proof} We write, by (45), (46), (15) and (22), and setting $\\tau \\equiv t_{1}-t_{2}$,\n\\begin{eqnarray*}\n\\Delta C = | \\int d\\vec{k} (\\tilde{f}_{1}(\\vec{k})-\\tilde{f}_{2}(\\vec{k})) \\beta(\\vec{k},\\tau,c)|\\\\\n\\beta(\\vec{k},\\tau,c) \\equiv \\frac{1}{2m_{0}} \\exp(-i\\tau \\frac{\\vec{k}^{2}}{2m_{0}}-\\\\\n- \\frac{c^{2}}{\\omega_{\\vec{k}}^{c}} \\exp(-i\\tau(\\omega_{\\vec{k}}^{c}-m_{0}c^{2}))\n\\end{eqnarray*} $$\\eqno{(48)}$$\nWe split the integral defining $\\Delta C$ in (48) into one over $ I_{\\delta} \\equiv \\{\\vec{k} ;\\frac{|\\vec{k}|}{m_{0}c}>\\delta \\}$, and the other over the complementary region. We now insert the elementary inequalities valid inside $I_{\\delta}$: \n\\begin{eqnarray*}\n\\frac{1}{2m_{0}} - \\frac{c^{2}}{2\\omega_{\\vec{k}}^{c}} \\le \\frac{\\delta^{2}}{4m_{0}}\\\\\n|\\exp(-i\\tau \\frac{\\vec{k}^{2}}{2m_{0}}) -\\exp(-i\\tau(\\omega_{\\vec{k}}^{c}-m_{0}c^{2}))|\\\\\n\\le \\frac{m_{0}c^{2}|\\tau|\\delta^{4}}{8\\hbar}\\\\\n\\frac{c^{2}}{2\\omega_{\\vec{k}}^{c}} \\le \\frac{1}{2m_{0}}\n\\end{eqnarray*}\nas well as assumption (43) in the complement of $I_{\\delta}$, into (48), to obtain (44). q.e.d.\n\nLemma 2 shows that in the free field case, in spite of the nonequivalence of the relativistic and nonrelativistic representations shown in theorem 3, Einstein causality is saved, at least in an approximative sense. The real trouble starts with interactions. In that case, (37) implies, taking now for $\\Phi$ the interacting field, and $\\Psi_{0}$ the interacting vacuum $\\Omega_{0}$, that the two-point function of the interacting field must equal that of the free field of mass $m_{0}$ in the case of equivalence of representations. For a hermitian local scalar field for which the vacuum is cyclic, (37) (with $m_{0} > 0$) implies that $\\Phi$ is a free field of mass $m_{0}$ (Theorem 4.15 of \\cite{StrWight}). We know, however, at least for space dimensions less or equal to 2, interacting fields exist, the first one historically having been in one dimension \\cite{GlJa} the free scalar Boson field of mass $m_{0} >0$. Its Hamiltonian is\n$$\nH(g) = H_{0}+H_{I}(g)= \\int_{\\mathbf{R}}dk \\omega_{k}a^{*}(k)a(k)+\\int_{\\mathbf{R}}dx g(x):\\Phi_{r}(x)^{4}:\n\\eqno{(49)}\n$$\nwith $g \\in L^{2}(\\mathbf{R})$ a real valued function. $H(g)$ is a well-defined symmetric operator on a dense set in Fock space (see proposition pg. 227 of \\cite{RSII}; for self-adjointness see further in the same reference). The dots in (49) denote the so-called Wick product, which means that all creation operators in $\\Phi_{r}(x)^{4}$ are to be placed to the left of all annihilation operators (for further elementary discussion see \\cite{MaRo}, and a complete treatment \\cite{RSII}). In (49), the limit $g \\to \\lambda$ (with $\\lambda > 0$ a constant, interpreted as the coupling constant) exists in a well-defined sense \\cite{GliJa}). In the present case, the vacua $\\Omega_{0}$ and the no-particle state, which also belong to inequivalent representations, differ greatly. This may already be expected on the level of (49), because the ground state $\\Omega_{g}$ of $H(g)$ (whose existence was proved in \\cite{GliJa}) cannot be $\\Psi_{0}$ because of the vacuum polarizing term $H_{I}^{P}(g)$ in (49):\n$$\nH_{I}^{P}(g) = \\int dx g(x) \\phi_{r}^{*}(x)^{4}\n\\eqno{(50)}\n$$\nThose terms in (49) which commute with the number operator $N$ given by (47) are all equal to\n$$\nH_{I}^{C}(g) = 6 \\int dx g(x) \\phi_{r}^{*}(x)^{2} \\phi_{r}(x)^{2}        \n\\eqno{(51)}\n$$\nThe formal limit as $c \\to \\infty$ of the operator $H(g)-H_{I}^{C}(g)$ is not ''small'', for instance for (50) we get from (41)\n$$\nH_{I,\\infty}^{P}(g) = \\int dk_{1} \\cdots dk_{4} \\tilde{g}(k_{1}+\\cdots+k_{4})a^{*}(k_{1}) \\cdots a^{*}(k_{4})\n\\eqno{(52)}\n$$\nin the sense of quadratic forms. In the formal limit $c \\to \\infty$, $g \\to \\lambda$, (51) yields\n$$\nH_{I} = \\frac{3\\lambda}{2m_{0}^{2}} \\int dxdy a^{*}(x)a^{*}(y)\\delta(x-y)a(x)a(y)\n\\eqno{(53)}\n$$\nwith $a^{*}(x),a(x)$ defined by (26): together with $H_{0}$ in (49), this defines the Hamiltonian of a nonrelativistic system of Bosons with delta-function interactions (see \\cite{Do} for the precise definition in a segment with periodic b.c.).\n\nThe limit $g \\to \\lambda$, followed by $c \\to \\infty$, was controlled by Dimock \\cite{Dim} in a remarkable tour-de-force. He showed that the two-particle scattering amplitude of model (49) converges to that of model (53) (with the free Hamiltonian (45)). The proof in \\cite{Dim} does not, however, offer any hint as to how the contribution of all the terms in $H(g)-H_{I}^{C}(g)$ becomes irrelevant in that limit (W.F.W. thanks Prof. Dimock for a discussion about this topic). \n\nThe above-mentioned point is crucial, for the following reason. For quantum systems in general, it is essential to arrive at many-body systems with \\textbf{nonzero} density $\\rho$ in the thermodynamic limit, i.e., $N \\to \\infty$, $V \\to \\infty$, with $\\frac{N}{V}=\\rho >0$ (see \\cite{MaRo} for an overview of applications). The corresponding non-relativistic system has, in contrast to the situation considered in \\cite{Dim}, also an infinite number of degrees of freedom ($N \\to \\infty$). The situation has an analogy to the classical limit of quantum mechanical correlation functions considered by Hepp \\cite{HepC}, where two possible limits may be envisaged, one of them yielding quantum mechanical N-particle systems, the other one classical field theory. For free systems with nonzero density, non-Fock representations arise, both in the non-relativistic and in the relativistic cases \\cite{AW}, but it may be checked that lemma 2 continues to hold (for zero temperature). For interacting systems, however, $N$ is not a good quantum number, and, upon fixing it (at a large value proportional to the volume $V$), the relativistic system can only be close to the nonrelativistic one if the contribution of the terms  $H(g)-H_{I}^{C}(g$ becomes indeed irrelevant in the joint limit $g \\to \\lambda$ followed by $c \\to \\infty$. \n\nAs an example, we expect that the ground state energy per unit volume of the relativistic system (with a Hamiltonian for volume $V$ defined as in \\cite{GliJa}) tends, as $V \\to \\infty$ and $c \\to \\infty$, to the thermodynamic limit $e$ of the same quantity in the Lieb-Liniger model \\cite{LLi}, which is explicitly known to be $e(\\rho)= \\rho^{3}f(\\frac{\\lambda}{\\rho})$, with $f$ known explicitly as the unique solution of a Fredholm integral equation. Since $\\rho$ is not a parameter in the relativistic system, it is only when the above mentioned terms do not contribute (in the limit $g \\to \\lambda$, $c \\to \\infty$) that a similar fixing of the density becomes possible also for the relativistic system. This seems to be a deep mystery, whatever the way the problem is regarded.\n\nIn order to explain the last issue more completely, consider the l.h.s. of (44) in lemma 2. In the first term thereof, $\\Psi_{0}$ should be replaced by the ground state of the Hamiltonian $H+H_{I}$, where $H$ is given by (45) and $H_{I}$ by (53), and $\\Phi(f)$ replaced by a bounded function of the zero time nonrelativistic fields as in (24). Properly speaking, instead of smearing with a function $g$ one should consider the Hamiltonian restricted to a bounded region, e.g. a segment with, say, periodic b.c., and the thermodynamic limit taken, but we shall continue with the previous description for brevity. The second term on the left hand side of (44) should be replaced by\n$$\n\\lim_{g \\to \\lambda} (\\Omega_{g}, A_{r}(f_{1}) \\exp(-i(t_{1}-t_{2})H(g)) B_{r}(f_{2})\\Omega_{g})\n\\eqno{(54)}\n$$\nwhere $\\Omega_{g}$ is the ground state of $H(g)$ (shown to exist in \\cite{GliJa}), $A_{r}(f_{1}),B_{r}(f_{2})$ are bounded local functions of the fields (22), (23), i.e., with $f_{1},f_{2}$ with \\textbf{compact} support in the space variable. It was shown in \\cite{GliJa} that \n$$\n\\exists\\lim_{g \\to \\lambda} \\exp(i \\tau H(g)) A(f) \\exp(-i \\tau H(g))\n\\eqno{(55)}\n$$\nin the sense of the norm (in the C*-algebraic sense) for bounded local $A(f)$ (for these concepts, see \\cite{BRo2} or \\cite{Hug}). This limit, for both operators $A(f_{1}),B(f_{2})$ in (54), determines the observable content of (54), but it is clear that the whole of $H(g)$ will contribute to (55), in particular terms such as\n\\begin{eqnarray*}\nH_{I,c}^{P}(g) = \\int dk_{1} \\cdots dk_{4} \\prod_{i=1}^{4} \\frac{c}{(2\\omega_{k_{i}}^{c})^{1\/2}}\\\\\n\\tilde{g}(k_{1}+\\cdots+k_{4}) a^{*}(k_{1}) \\cdots a^{*}(k_{4})\n\\end{eqnarray*} $$\\eqno{(56)}$$ \nin $H(g)-H_{I}^{C}(g)$ will contribute, for a certain choice of observables in (55). Given that their formal limit as $c \\to \\infty$ does not vanish ((52)), it seems very unlikely that the limit, as $c \\to \\infty$, of (55) is independent of $H(g)-H_{I}^{C}$. In this connection, one may recall that the $S$-matrix, considered in \\cite{Dim} as an observable, is of a special kind, because it commutes with the free Hamiltonian $H_{0}$ in (49).\n\nThe basic ingredient of the proof of (55) \\cite{GliJa} is the fundamental property of microcausality (1) \\cite{Haag}\\cite{StrWight}. On the other hand, the form (49) of the Hamiltonian is dictated by the property of Lorentz covariance \\cite{GlJa}, proved for this model in \\cite{HeOs}.\n\n\\section{Conclusion}\n\nIn this review we discussed two aspects of the dynamics of non-relativistic quantum systems, unified by a dichotomy in Hegerfeldt's theorem 1. According to this theorem, there are exactly two options (5) and (6) for such systems.\n\nThe first aspect was related to option (6) in that theorem, viz. the attempt to isolate a quantum system from its surroundings by a set of boundary conditions, including those of Dirichlet and Neumann type (a1,a2). In general, this leads to physical inconsistencies, as reviewed in theorem 2.2 for non-relativistic systems, see also \\cite{GarKar}. We view these inconsistencies as consequences of trying to impose conditions deriving from classical physics to quantum systems, be they non-relativistic or relativistic. The latter case is well illustrated in Milonni's famous paper \\cite{Mil}, where he proved that, near a perfectly reflecting slab, the transverse vector potential and the electric field satisfy a set of equal-time CCR different from those holding for free fields. In \\cite{BohrRos}, Bohr and Rosenfeld showed, under the natural assumption that the fields are measured by observing the motion of quantum massive objects with which the fields interact, that the above-mentioned equal-time CCR follow. They are, therefore, very fundamental.  \n\nThis suggests that such idealized b.c. are unphysical: this fact was explicitly shown for Dirichlet or Neumann b.c. in the case of the Casimir effect \\cite{KNW}. The reason is that there are wild fluctuations of quantum fields over sharp surfaces \\cite{DeCa}. One promising direction to study this (as yet open) problem is to look at the electromagnetic field in the presence of dielectrics, instead of the ''infinitely thin'' conductor plates (see \\cite{Bim} and references given there). The Casimir problem (for conductors or dielectrics) is an example for which adoption of generalized periodic b.c. (a3) in theorems 2.1,2.2 is not a physically reasonable option: it is not compatible with the theory's classical limit.\n\nOur second topic concerned option (5) in theorem 1, the issue of instantaneous spreading for non-relativistic quantum systems. We concluded that there are serious obstacles on the way to rescue Einstein causality in the (natural) approximative sense of lemma 2, due to terms such as the vacuum polarizing term (56) in the interaction Hamiltonian (49), which are not ''small'' in the formal limit $c \\to \\infty$ by (52). Since the presence of such terms is dictated by such fundamental principles as Lorentz covariance and microcausality, the solution may not be simple.\n\nAlthough we used a special model for the sake of argument, any physical theory with vacuum polarization, such as quantum electrodynamics, is expected to be subject to analogous considerations. Notice that the remarks on the use of approximate theories in \\cite{Yng} do not apply here, because the problems we pose are not due to the approximative character of the theories, such as, e.g., various cutoffs in quantum-electrodynamics, but, as remarked in the previous paragraph, are due to an intrinsic property of relativistic quantum field theory, viz., vacuum polarization (or, more precisely, having a non-persistent vacuum).\n\nThe arguments we presented, however, are clearly no mathematical proof of a no-go theorem. One reason is that the limits $g \\to \\lambda$ and $c \\to \\infty$ do not necessarily commute: they may not. The problem is therefore open. It is hoped that a complete change of point of view may clarify the problem, but we conjecture that  $H(g)-H_{C}^{I}(g)$ will play a central role in the final solution.   \n\nProgress in both topics above would obviously be of great relevance for the foundations of quantum theory.   \n\n\\textbf{Acknowledgement} The idea of a part of this review arose at the meeting of operator algebras and quantum physics, satellite conference to the XVIII international congress of mathematical physics. We thank the organizers for making the participation of one of us (W. F. W.) possible, and Prof. J. Dimock for discussions there on matters related to section 3. We also thank Christian J\\\"{a}kel for critical remarks concerning possible changes of viewpoint, and for recalling some relevant references. W.F.W. also thanks J. Froehlich for calling his attention to the reference \\cite{Yng}.\n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nBiological systems, such as humans, use electrical signals as the medium of communication between their control centers (brains) and motor organs (arms, legs). While this is taken for granted by most people, those with severe physical impairments, such as quadriplegia, experience the breakdown of this communication system rendering them unable to perform the most basic physical movements. Modern technologies, such as BCIs, have attempted to ameliorate this through the use of brain signals as commands for assistive systems \\cite{perdikis2018cybathlon}. MI, a common paradigm for BCI control, requires the subject to simulate or imagine movement of the limbs on account of there being discernible differences in brain signals when moving different limbs \\cite{abiri2019comprehensive}. Due to it being non-invasive and cost-effective, EEG is the method of choice for collecting data for such systems \\cite{abiri2019comprehensive}.\n\nOne of the many recent developments in the application of EEG-driven BCIs is the Cybathlon competition held every four years under the auspices of Eidgen\u00f6ssische Technische Hochschule Z\u00fcrich (ETH Zurich) \\cite{riener2014cybathlon}. The competition involves physically challenged individuals completing routine tasks via assistive systems. One such task -- the BCI race -- has the participants (called pilots) control a virtual game character via brain signals only. Competing teams, who may hail from either academia or industry, are responsible for creating BCI systems and training their respective pilots. The goal of the Cybathlon is to push the state-of-the-art in BCI assistive systems, and accelerate its adoption in everyday lives of those who need it most.\n\nFor the 2020 edition of Cybathlon, a team from the Technische Universit\u00e4t M\u00fcnchen (TUM) called \"CyberTUM\" is amongst the competitors in the BCI race challenge. In order to achieve high scores in the competition, a major part of BCI development is the focus on robustness of the system i.e. minimizing the variability of the system for different sessions and environments. Lack of robustness, in fact, is an established concern in almost all BCI systems. Possible causes of the problem include nonstationarity of EEG signals (variance for the same subject) \\cite{wolpaw2002brain} \\cite{vidaurre2010towards}. An additional cause, as noted by participating teams in Cybathlon 2016, is the change in the subject's emotional state. During the race, As expected, a public event such as the BCI race, the pilots' stress stress levels increased. This is to be expected as a public event such as the BCI race can heighten stress. This change in the pilots' emotional state caused their respective BCI systems to perform sub-optimally. \n\nThe objective of this work is to mitigate this concern and develop MI systems that are robust to perturbations in the subject's emotional state, specifically to emotional arousal. In order to achieve this, we develop VR environments to induce high and low arousal in the subject before recording MI data. VR environments have been previously used along with EEG to prompt changes in emotional arousal \\cite{baumgartner2006neural}. Additionally, they have been used together with MI for treating Parkinson's disease \\cite{mirelman2013virtual}. To our knowledge, this is the first work where VR environments are used to increase robustness of MI-BCI systems. Subsequently, learning algorithms are trained, not only for MI but also for different arousal states. The idea is that during the BCI race, we first detect the pilot's emotional state of arousal, and choose the appropriate MI classifier. Due to COVID-19, many steps in the above mentioned outline had to be modified, the details of which are present as follows. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Related Work}\n\n\\subsection{Cybathlon 2016}\nThe inaugural Cybathlon competition was held in 2016. After the competition, the competing teams published their methods for training the participants, amongst which were Brain Tweakers (EPFL) \\cite{perdikis2018cybathlon} and Mirage91 (Graz University of Technology). One of the pilots of the former performed well in the qualifiers but poorly in the final, prompting the authors to cite psychological factors such as stress as the possible cause for the drop. A similar course of events was observed for the pilot of Mirage91, who after achieving an average runtime of 120 s in the days leading up to the Cybathlon, dropped to 196 s during the competition. The authors indicated that the pilot was showing signs of nervousness on competition day, with a heart beat of 132 beats per minute (bpm) prior to the race \\cite{statthaler2017cybathlon}. \n\nThe authors' hypothesis regarding the drop in their pilots' performances is supported by existing BCI literature \\cite{chaudhary2016brain} \\cite{lotte2013flaws} \\cite{hammer2012psychological} \\cite{jeunet2016standard}. Further support comes from evidence in affective science: It has been theorized that any event that causes an increase in emotional arousal can affect perception and memory in a manner which causes the retention of high-priority information and disregard of low-priority information \\cite{mather2012selective}.\n\n\n\\subsection{Emotional Valence and Arousal}\nEmotions are defined as complex psychological states, with three constituents: subjective experience, physiological and behavioral response \\cite{hockenbury2000discovering}. Following early attempts \\cite{wundt1897outlines}, more rigorous descriptions of emotions were made, the most widely accepted of which being the 'circumplex model' \\cite{russell1980circumplex}. It proposes that all emotions can be described as a combination of two properties: valence and arousal. These can be thought as orthogonal axes in two-dimensions. Neurologically, it entails that any emotional state is a result of two distinct and independent neural sub-systems \\cite{posner2005circumplex}. Figure \\ref{fig:circumplex} provides a visual representation of the circumplex model. As can be seen, emotions such as 'excited' are high on both the arousal and valence axes, while 'gloomy' is low in both arousal and valence.\n\n\\begin{figure}[tpb]\n  \\centering\n  \\includegraphics[width=0.4\\textwidth]{figures\/valence-arousal.png}\n  \\caption{The circumplex model of emotional classification. Figure courtesy of \\cite{Kim2016ImageRW}.} \\label{fig:circumplex}\n\\end{figure}\n\nAlternative descriptions, such as the 'vector model' \\cite{Bradley1992RememberingPP}, do not veer off sharply from the circumplex model; they too base emotional classification on both valence and arousal. Hence the circumplex model was used as the paradigm of emotional analysis for the duration of the project.\n\n\n\\subsection{Arousal, EEG and Motor Imagery}\\label{arousal-eeg-background}\nStates of high and low arousal can be inferred from EEG signals \\cite{pizzagalli2007electroencephalography}. This has been previously used to train learning systems for distinguishing between various arousal states \\cite{nagy2014predicting}. EEG bands pertinent to different states of arousal are alpha (8-14 Hz) -- related to a relaxed yet awakened state -- and gamma (36-44 Hz) -- a pattern associated with increased arousal and attention. The theta pattern (4-8 Hz), correlated with lethargy and sleepiness, is also useful for differentiating arousal. \n\nWith regards to motor imagery (MI), the most relevant EEG bands have been shown to be alpha (8-14 Hz) and beta (14-30 Hz) \\cite{graimann2010brain}, the latter of which is associated with high degrees of cognitive activity \\cite{pizzagalli2007electroencephalography}. \n\nMotor imagery data refers to data produced when the subject simulates limb movement. As movement of different limbs is sufficiently distinguishable, this can be used to perform control for various other tasks \\cite{padfield2019eeg}. To record EEG data for motor imagery, the 10-20 international system of electrode placement is used \\ref{fig:eeg-map}. Due to the cross-lateral nature of limb control in the human brain, movement of the right arm is recorded most faithfully by C3 and that of the left arm by C4 \\cite{graimann2010brain}.\n\n\\begin{figure}[htpb]\n  \\centering\n  \\includegraphics[width=0.3\\textwidth]{figures\/eeg-map.jpg}\n  \\caption{10-20 International system of EEG electrode placement. Electrodes C3 and C4 are most relevant for MI activity. Figure courtesy of \\cite{rojas2018study}.} \n  \\label{fig:eeg-map}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Methodology}\n\n\\subsection{Virtual Reality Environments}\nTraditional methods of inducing stress include the Sing-a-song stress test (SSST) \\cite{brouwer2014new} and the Trier social stress test (TSST) \\cite{kirschbaum1993trier}, while meditation has been shown to induce relaxation \\cite{sedlmeier2012psychological}\\cite{lumma2015meditation}. Emulating such environments faithfully in VR is sufficiently challenging, and may not be the most productive way to use VR to induce high\/low emotional arousal.\n\nPreviously, VR exposure therapy has been explored to alleviate various psychological disorders \\cite{krijn2004virtual}. One such example is using a VR height challenge -- placing the subject on higher ground in a virtual environment \\cite{diemer2016fear}. Not only does the challenge induce high emotional arousal in test subjects, but the control subjects -- the ones who are not acrophobic -- also exhibit the same physiological responses as the test group i.e. increased heart rate and skin conductance level \\cite{diemer2016fear}. Similarly, VR environments, particularly those with natural scenery e.g. a forest, have shown efficacy in reducing stress \\cite{anderson2017relaxation}\\cite{annerstedt2013inducing}. We thus developed two VR environments: one where the subject was placed on top of a skyscraper, called 'Height' while the second in a relaxing forest called 'Relaxation.' The environments were created using Unity 3D\\footnote{\\href{https:\/\/unity.com\/}{https:\/\/unity.com\/}}.\n\n\\begin{figure}[tbp]\n\\centering\n  \\begin{subfigure}[b]{0.4\\textwidth}\n    \\includegraphics[width=\\textwidth]{figures\/aroused.png}\n    \\label{fig:f1}\n  \\end{subfigure}\n  \\begin{subfigure}[b]{0.4\\textwidth}\n    \\includegraphics[width=\\textwidth]{figures\/relaxation.jpg}\n    \\label{fig:f2}\n  \\end{subfigure}\n  \\caption{Virtual reality environments for inducing arousal in subjects. On top is 'Height' designed to induce high arousal by placing the subject on the edge of a skyscraper. Below is 'Relaxation' intended to lower arousal via a natural, calming setting.}\n  \\label{fig:vr}\n\\end{figure}\n\n\n\\subsection{Dataset}\nAs this project was part of the CyberTUM team's participation in Cybathlon 2020, the original idea was to collect real data with the actual pilots who will be competing in the even proper. At the beginning of this work, however, no ethics approval had been acquired to run any experiments on the pilots. This was not detrimental to the project as a proof-of-concept could still be arrived at by collecting EEG data from volunteers within the CyberTUM team. The COVID-19 pandemic obstructed our means of collecting such data.\n\nIn the absence of our own motor-imagery and arousal data, we opted for the Graz 2b data set \\cite{leeb2008bci}. It belongs to a family of BCI datasets collected by the BCI Lab at Graz University of Technology. The dataset has been used previously in the BCI Competition IV \\cite{tangermann2012review}. EEG data is collected for 9 subjects doing a binary motor-imagery task (moving right and left hand on cue). The data is sampled at a frequency of 250 Hz with 3 EEG and 3 EOG channels. For our experiments, we use data from two subjects, B05 and B04, whom we refer to as subject 1 and 2 respectively henceforth.\n\n\n\\subsection{Subject Classification as Proxy for Arousal Classification}\\label{subject-classification}\nAs mentioned, we were unable to obtain our own EEG arousal data. To train the classifiers, we alternatively modified the experiment. Instead of using data with high\/low arousal emotional states as labels, we used different subjects as proxies for such states, making it a cross-subject classification task \\cite{del2014electroencephalogram} \\cite{riyad2019cross}. As EEG signals demonstrate significant variance between subjects, we can consider the data coming from subject A as that belonging to the emotional state of high arousal, and data from subject B as belonging to low arousal. With this approach, we can continue to train a classifier that would approximate the performance of one that is trained on actual arousal data, assuming the emotional states in this actual data are informative.\n\n\n\n\\subsection{Experimental Design}\nThe original scheme was to:\n\\begin{enumerate}\n    \\item Develop VR environments in line with existing literature that are known to induce stress (high arousal) and relaxation (low arousal) in subjects.\n    \\item Use electrodermal activity (EDA) activity to validate the efficacy of VR environments. EDA is a wide-used measure for emotional arousal, as skin conductance rises with rise in arousal \\cite{critchley2002electrodermal}.\n    \\item Record MI data alternating between states of low and high arousal for each session. Start with 60s of inducing high arousal via the \"Height\" environment, then record MI data for 45s. Repeat the same with \"Forest\" environment for relaxed state. Repeat this process for each trial. The MI data was to be recorded by using the common paradigm of showing the participant a cue on screen (typically left or right arrow) which would prompt them to imagine as if they were moving their left or right hand \\cite{ramoser2000optimal} \\cite{pfurtscheller1997eeg} \\cite{liu2017motor}.\n    \\item Train an arousal classifier. The aim of this classifier is to indicate the emotional state (high or low arousal) of the subject.\n    \\item Train separate MI classifiers for each emotional state. The goal is to optimize for accuracy, even if different types of pre-processing and classifier types were required for each state, unlike the arousal classifier which necessitates the same pre-processing steps.\n    \\item During deployment, first classify the emotional state using the arousal classifier, and based on its result, choose the appropriate MI classifier.\n\n\\end{enumerate}\nAs mentioned previously, due to numerous factors, many steps in the above formulation had to be either abandoned (2 and 3) or modified (4 and 5). The revised scheme, replaced steps 4-6 with the following:\n\n\\begin{enumerate}\n    \\item Train a cross-subject classifier replacing the arousal classifier. The task of this classifier is to take EEG as input from any of the two subjects, and classify the input as belonging to either subject 1 or 2. As the classifier is agnostic to the subject, the same pre-processing had to be done for each subject's data.\n    \\item Train separate MI classifiers for each subject instead of training for each emotional state.\n    \\item At test time, sample a run of a few data points (5 in our experiments), feeding them to the cross-subject classifier. Based on its mode (most frequent classification), select the appropriate MI classifier.\n\\end{enumerate}\n\n\n\\subsection{Learning algorithms}\\label{algos}\nWe experimented with a multitude of machine learning algorithms which are briefly described as follows.\n\n\\paragraph{Logistic regression}\nLogistic regression is a modification of linear regression for a binary classification task \\cite{kleinbaum2002logistic}. It predicts the probability of a class given the input, by first learning a weighted linear combination of input features and applying a logistic function to the result.\n\\begin{equation}\n    y = \\frac{1}{1+e^{-a}} \\quad where \\quad a = \\theta_0 + \\theta_1.x_1 + \\theta_2.x_2\n\\end{equation}\n\n\\paragraph{Linear discriminant analysis}\nLDA attempts to maximize inter-class variance while minimizing intra-class variance \\cite{balakrishnama1998linear} in the data. This results in a clustering of the data where it is easily separable. It is widely used in MI BCI \\cite{wang2006common} \\cite{wang2006common}.\n\n\\paragraph{Naive Bayes}\nA probabilistic classifier, naive bayes uses bayes' law to calculate the posterior probability of an event (class) given the prior and likelihood \\cite{murphy2006naive}. The posterior can then be updated with new evidence. It assumes that the features are independent, hence the term naive in its name.\n\\begin{equation}\n    P(y|x) = \\frac{P(y).P(x|y)}{P(x)}\n\\end{equation}\n\n\\paragraph{Ensemble model}\nThis is implemented as a voting classifier in gumpy. It uses a mix of classifiers such as nearest-neighbor, LDA and support vector machines (SVM) and uses the majority vote as the classification output. As such, it necessarily either equals or outperforms both Naive Bayes and LDA as it uses them in the ensemble.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Results}\n\n\\subsection{Artifact Removal}\n\\begin{figure}[!t]\n  \\begin{subfigure}[b]{0.5\\textwidth}\n    \\includegraphics[width=\\textwidth]{figures\/subject-5-1.png}\n    \\caption{Plotting ICA with EOG channels. A visual depiction of the first component (in red) of ICA being correlated with EOG.}\n    \\label{fig:f1}\n  \\end{subfigure}\n  \\begin{subfigure}[b]{0.5\\textwidth}\n    \\includegraphics[width=\\textwidth]{figures\/subject-5-2.png}\n    \\caption{Plotting ICA components against each other. The peak in the first component (blue) evidently due to an eye-blink. }\n    \\label{fig:f2}\n  \\end{subfigure}\n  \\caption{Artifact analysis using ICA for subject 1.}\n  \\label{ica}\n\\end{figure}\nThe data for subject 1 and 2 contained 324 and 399 trials (attempts at moving right or left hand) respectively. The standard approach to train MI classifiers is to analyze data and remove existing artifacts before extracting features from the data \\cite{uriguen2015eeg}. We first applied a Butterworth bandpass filter \\cite{daud2015butterworth} to extract frequencies within the range 2-60 Hz. We then analyze the data for artifacts. A common source of artifacts in MI data is noise from electrodes located in the forehead's proximity. This is in fact data collected from the Electrooculography (EOG) channels which detect movements such as eye blinks, which may show up in the MI data. Such noise can be detected by first performing independent component analysis (ICA) -- widely used in EEG preprocessing \\cite{ica} -- which tries to decompose a signal into constituent component under the assumption of statistical independence. We then see which of the resultant components correlates most with EOG channels, and filter it out \\cite{ica-eog}. An example of ICA on subject 1 can be seen in figure \\ref{ica}. We filter out the first component which seems to be picking up an eye blink. ICA on subject 2 did not improve the results.\n\n\n\\subsection{Feature Extraction}\n\\begin{figure*}[t]\n\\centering\n\\begin{subfigure}[b]{0.49\\linewidth}\n\\centering\n\\includegraphics[height=3cm]{figures\/pca-1.png}\n\\caption{PCA visualization of subject 1's feature vector.}\n\\end{subfigure} \n\\begin{subfigure}[b]{0.49\\linewidth}\n\\centering\n\\includegraphics[height=3cm]{figures\/pca-2.png}\n\\caption{PCA visualization of subject 2's feature vector.} \n\\end{subfigure}\n\\caption{Dimensionality reduction using PCA for feature space visualization of both subjects. Subject 2's features are more informative for the motor-imagery task compared to subject 1 which is also reflected in the training accuracy. Right hand movements are labeled red while left hand movements are blue.}\n\\label{fig:pca}\n\\end{figure*}\nSeveral methods were attempted to extract features. In principle, feature extraction in BCI takes two forms: frequency band selection and channel selection (also known as spatial filtering). In regards to the former, we've previously mentioned in \\ref{arousal-eeg-background} that alpha and beta bands have been shown to be most related to MI activity. Accordingly, we use these frequency bands as our features. In the same section we observed that channels C3 and C4 are the most relevant for MI, which we can use directly without any spatial filtering. For this, instead of using raw alpha and beta patterns, we opt for logarithmic sub-band powers of said patterns (see gumpy documentation\\footnote{\\href{http:\/\/gumpy.org\/}{http:\/\/gumpy.org\/}}). Each spectrum is divided into four sub-bands. An alternative approach for feature extraction in MI classification has been the use of the \"common spatial pattern (CSP)\" algorithm \\cite{Koles2005SpatialPU}. It tries to find optimal variances of subcomponents of a signal \\cite{csp} with respect to a given task. In our experiments, however, CSP performed poorly compared to logarithmic sub-band power of alpha and beta bands. The results when CSP was applied have thus been omitted from the report, but could be reproduced in the notebook (see section \\ref{documentation}). A visualization of the features using PCA for both subjects can be seen in figure \\ref{fig:pca}. As can be observed, the features for subject 2 are more conducive to discrimination of MI. This is also verified in the training results, where every classification algorithm achieved higher accuracy for subject 2 compared to subject 1.\n\n\n\\subsection{Training}\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=0.5\\textwidth]{figures\/training.pdf}\n  \\caption{Training scheme for both classifiers. MI classifiers are trained separately for each subject (labels corresponding to right and left hand) while Cross-subject classifier trained on features of both subjects.} \\label{fig:training}\n\\end{figure}\nAs mentioned previously, we train two types of classifiers: MI per subject classifier and cross-subject classifier. The entire training procedure is visually depicted in figure \\ref{fig:training}. After doing feature extraction, we first train an MI classifier for each subject with labels 0 and 1 (left and right hand movement respectively). Subsequently, we combine data of both subjects, labelling it 0 and 1 (subject 1 and subject 2 respectively) and train the cross-subject classifier. All classifiers described in \\ref{algos} are trained in each case, the results of which can be seen in table \\ref{results}.\n\n\\paragraph{MI classification}\nThe data for each subject was divided into an 80-20 split (training-test). The features were also standardized by rescaling to zero mean and unit standard deviation. Results for both subjects were satisfactory, although subject 1's data was harder to train on compared to subject 2. This can be observed by looking at the ranges of training accuracy for both subjects [55.84-70.12 vs. 91.25-95]\\%. Subject 2's classifiers achieved both a higher average accuracy as well as lower variance. LDA performed best for subject 1, while logistic regression achieved best results for subject 2. \n\n\n\\paragraph{Cross-subject classification}\nTraining for cross-subject classifiers followed the same procedure of feature extraction with the only difference being a re-labeling of the samples from limb movements to source subject. Once again, we split the data into 80-20 (train-test) portions, though this time the data is the combined samples from both subjects. For testing the classifiers, we split the test set further into sections containing five samples (trials) each. For each section, we take the mode (most frequent prediction) of the classifier which is considered the final result. For example, if our test data has 50 samples from each subject, we portion it into 20 sections (each subject with 10 sections). We then feed each section to the classifier and take the majority score for that section as the classifier's prediction. As can be seen, the ensemble model outperforms the rest of the algorithms by a considerable margin. In addition to this, we also created t-SNE embeddings of the features with 2 and 3 dimensions \\cite{tsne}. The results were not up to par and have thus been left out here (they can be reproduced via the notebook discussed in \\ref{documentation}). More details can be found in \\ref{discussion}.\n\n\n\n\n\\begin{table}[h]\n\\caption{Summary of results. Accuracy scores for MI (both subjects) as well as cross-subject (X-sub) using various classifiers. Best results in bold}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n\\textbf{Task}  &   \\multicolumn{4}{|c|}{\\textbf{Classifier}} \\\\\n\\cline{2-5} \n\\textbf{} &  \\textbf{Logistic Regression} & \\textbf{LDA} & \\textbf{Naive Bayes} & \\textbf{Ensemble}\\\\\n\\hline\nMI-sub 1    &   67.53\\%    &   \\textbf{70.12\\%}    &     55.84\\%     &   70.12\\% \\\\\n\\hline\nMI-sub 2    &   \\textbf{95\\%}    &   91.25\\%    &    91.25\\%     &   93.75\\% \\\\\n\\hline\nX-sub    &   58.65\\%    &   59.38\\%    &    59.38\\%     &   \\textbf{68.75\\%} \\\\\n\\hline\n\\end{tabular}\n\\label{results}\n\\end{center}\n\\end{table}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Discussion}\\label{discussion}\n\nThe results indicate that assuming different emotional states impart sufficient differences in EEG data, we can train classifiers that perform well above chance. Significant differences in the EEG signals between both subjects were observed during feature extraction and classification. This is not an uncommon phenomenon and has been documented in the literature \\cite{pizzagalli2007electroencephalography}. Blankertz et. al show that after testing on 80 subjects, the average classifier accuracy of a binary task was 74.4\\% with a spread of 16.5\\% \\cite{BLANKERTZ20101303}. Our findings buttress this as the best models for subject 1 and 2 achieved 75.38\\% and 95\\% accuracy respectively. This variability generally chalked up to differences in the subjects' abilities for implicit learning \\cite{kotchoubey2000learning}, performance in early neurofeedback sessions \\cite{Neumann1117} and attention spans \\cite{Daum94}. \n\nAccording to Tangermann et. al, the best results on data set 2b were achieved using filter-bank CSP as a pre-processing step followed by a naive bayes classifier during Competition IV \\cite{tangermann2012review}. In our testing, however, vanilla CSP for feature extraction was sub-optimal. Naive bayes was also found trailing behind other classifiers as seen in table \\ref{results}. We thus observe that vanilla CSP is not as performant as log band-power in our experiments, while we did not perform any experiments with filter-bank CSP.\n\nIn regards to cross-subject classification, appreciable results have been achieved by using ICA for feature extraction \\cite{tangkraingkij2009selecting} combined with a nearest-neighbor (NN) classifier. We verify the efficacy of ICA as a pre-processing step for feature extraction. Other approaches have shown PCA as an effective step for dimensionality reduction \\cite{palaniappan2005energy}. While we could not confirm this with PCA, using the more modern dimensionality reduction technique of t-SNE performed poorly in our experiments (tested using target dimensions 2 and 3). There is, however, recent evidence that using t-SNE in tandem with common dictionary learning may yield good results \\cite{nishimoto2020eeg}.\n\n\\subsection{Limitations and Future Outlook}\n\nA primary limitation of this work is the lack of testing on actual subjects. While the system ensures acceptable performance on an existing dataset, we can not conclude much about its usefulness in the real-world. To make such assertions with a certain degree of confidence, we need to evaluate how quickly we can switch between various MI classifiers based on the predictions of the emotion (cross-subject) classifier. This is also true for calibration time at the start of each session; while we use five trials during testing and get well above-chance results, comprehensive and systematic verification of the system is in order if it is to be of any practical use.\n\nIn addition to alpha patterns, gamma bands are correlated with increased arousal \\cite{pizzagalli2007electroencephalography}, which may have carried a strong supervision signal for the classifier. Had we acquired EEG data for aroused and relaxed states of a subject, an emphasis on gamma bands would have been warranted. As such, in the present case, as we did not have data corresponding to high and low arousal, gamma patterns were assumed not to be informative.\n\nFuture work may also look at training classifiers for more than two subjects. While two subjects suffice for the purposes of this study, as the original task was the discrimination between two emotional states of arousal, it may be worth exploring how the cross-subject classifier would scale to additional classes. This may be interpreted as having to classify not only emotional arousal but also valence (positive or negative) which may have important ethical implications.\n\nMost of the classifiers used in this project are classic algorithms, and were chosen for their still prevailing use in MI BCI. However, future work may also incorporate modern approaches such as deep neural networks for MI classification \\cite{Tabar_2016}. Deep learning could also be used to formulate our problem as that of multi-task learning for both arousal and MI classification \\cite{multitask}. In this manner we can replace training multiple classifiers with a single one which both classifies emotional arousal as well as motor imagery. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Interdisciplinary Work}\nThe nature of this project necessitated the undertaking of a multi-disciplinary approach, from understanding and systematizing human emotional arousal to developing algorithms for distinguishing both emotional states as well as motor function via EEG. Thus, this work borrows, incorporates and synthesizes elements from a number of disciplines including psychology (emotional arousal), neuroscience (EEG and motor-imagery), computer graphics (virtual reality environments) and artificial intelligence (machine learning for classification). Broadly, we can categorize psychology and neuroscience as brain sciences and computer graphics and artificial intelligence under the umbrella of informatics. Each of the two disciplines contributed unique methods and insights without which the project may not have come to fruition. The most valuable insight was the difficulty in training accurate machine learning algorithms for EEG. Although machine learning has become the dominant paradigm for classification tasks, this project demonstrates that pre-processing of data (via techniques such as ICA and log power-band) is at least as important to the success of the system as the classifier (the results for other feature extractors can be reproduced in the provided notebook), and even after pre-processing, we have no guarantees of robust performance. Another key insight was the extent to which EEG patterns vary between different people, pointing to the difficulty of transfer learning in this domain. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion}\nA major hurdle in the widespread and practical use of assistive systems based on MI-BCI is lack of reliability. While this can have many origins, an important source as identified by two Cybathlon teams in 2016 was related to shifts in the subject's state of emotional arousal. In this work, we present an end-to-end framework for inducing high\/low arousal in subjects, collecting EEG data and train learning algorithms for robust MI classification. While COVID-19 enforced certain constraints on data acquisition, we were still able to develop a proof-of-concept for how emotion-robust MI-BCI systems could be trained. Our results indicate that if the training signal contains sufficient information i.e. each emotional state has a distinct enough EEG signature, we can successfully train systems that are robust to variance in emotional arousal. A thorough study, however, needs to be conducted to determine the practicality of such a system with respect to variables such as classifier switching times and calibration periods.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Acknowledgments}\nThis project could not have been possible without the aid of Nicholas Berberich who provided constant and quality guidance on overall methodology, feature extraction and algorithms. Also worth gratitude are Matthijs Pals for his support in regards to MI data preprocessing and Svea Meyer for helping in the initial phase of the project as well as with explaining EEG terminology.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{FORWARD PHYSICS AND FORWARD INSTRUMENTATION AT THE CMS INTERACTION POINT}\n\nForward physics at the LHC covers a wide range of diverse physics subjects which have in \ncommon that particles produced at small polar angles $\\theta$ and hence large \nvalues of rapidity provide a defining characteristics. This article concentrates\non their physics interest in $pp$ collisions.\n\nAt the Large-Hadron-Collider (LHC), where proton-proton collisions occur at \ncenter-of-mass energies of 14 TeV, the maximal possible rapidity is \n$y_{max} = \\ln{\\frac{\\sqrt{s}}{m_{\\pi}}}\\sim 11.5$. The two multi-purpose detectors ATLAS and \nCMS at the LHC are designed primarily for efficient detection of processes with large\npolar angles and hence\nhigh transverse momentum $p_T$. The coverage in pseudorapidity \n$\\eta = - \\ln{[\\tan{( \\theta \/ 2 )} ] }$ of\ntheir main components extends down to about $|\\theta| = 1^\\circ$ from the beam axis\nor $|\\eta| = 5$.\n\nFor the CMS detector, several subdetectors with coverage beyond $|\\eta| =5$ are \ncurrently under construction (CASTOR and ZDC sampling calorimeters) or in the proposal \nstage (FP420 proton taggers and fast timing detectors). \n\nFuthermore, a salient feature of the forward instrumentation around the \ninteraction point\n(IP) of CMS is the presence of TOTEM~\\cite{TOTEM}. TOTEM is an approved experiment at \nthe LHC for measuring the $pp$ elastic cross section as a function of the four-momentum\ntransfer squared, $t$, and for measuring the total cross section with a precision of\napproximately 1\\%. The TOTEM experiment uses the same IP as CMS and supplements around \nthe CMS IP several tracking devices, located inside of the volume \nof the main CMS detector, plus near-beam proton taggers a distance up to $\\pm 220$~m. \nThe CMS and TOTEM collaborations have described the considerable physics potential of\njoint data taking in a report to the LHCC \\cite{opus}. \n\nThe kinematic coverage of the combined CMS and TOTEM apparatus is unprecedented at a\nhadron collider. It would be even further enhanced by complementing CMS with the\ndetectors of the FP420 proposal which would induce forward physics into the portfolio of\npossible discovery processes at the LHC~\\cite{fp420}.\n\nAn overview of the forward instrumentation up to $\\pm 220$~m from the CMS IP is given in \nFig.~\\ref{fig:overview}. There are two suites of calorimeters with tracking detectors in front.\nThe CMS Hadron Forward (HF) calorimeter with the TOTEM telescope T1 in front \ncovers the region $3 < |\\eta | < 5$, the CMS CASTOR calorimeter with the TOTEM telescope \nT2 in front covers $5.2 < |\\eta| < 6.6$. The CMS ZDC calorimeters will be \ninstalled at the end of the straight LHC beam-line section, at a distance of \n$\\pm 140$~m from the IP. Near-beam proton taggers will be installed by TOTEM at \n$\\pm 147$~m and $\\pm 220$~m from the IP.\nFurther near-beam proton taggers in combination with very fast timing detectors to be\ninstalled at $\\pm 420$~m from the IP are part of the FP420 proposal.\n\n\\begin{figure}\n\\hspace*{-0.5cm}\n\\includegraphics[scale=0.32, angle = -90]{cms_totem_detectors_new.eps}\n\\caption{Layout of the forward detectors around the CMS interaction point.}\n\\label{fig:overview}\n\\end{figure}\n\n\n\n\n\\section{PHYSICS WITH FORWARD DETECTORS}\n\nIn the following, we describe the physics interest of the CMS CASTOR and ZDC \ncalorimeters~\\cite{PTDR1} \nand the TOTEM T1 and T2 telescopes~\\cite{TOTEM}. Of particular interest are \nQCD measurements at values of Bjorken-$x$ as low as $x \\sim 10^{-6}$ and the resulting \nsensitivity to non-DGLAP dynamics, \nas well as forward particle and energy flow measurements. These can play an important \nrole in tuning the Monte Carlos description of underlying event and multiple interactions\nat the LHC and in constraining Monte Carlo generators used for cosmic ray studies.\n\n\\subsection{CMS CASTOR \\& ZDC calorimeters}\n\nThe two calorimeters are of interest for $pp$, $pA$ and $AA$ running at the LHC, \nwhere $A$ denotes a heavy ion. They are Cherenkov-light devices with electromagnetic \nand hadronic sections and will be present in \nthe first LHC $pp$ runs at luminosities where event pile-up should be low.\n\nThe CASTOR calorimeters are octagonal cylinders located at $\\sim 14$~m from the IP.\nThey are sampling calorimeters with tungsten plates as absorbers and fused silica quartz \nplates as active medium. The plates are inclined by $45^\\circ$ with respect to the \nbeam axis. Particles passing through the quartz emit Cherenkov\nphotons which are transmitted to photomultiplier tubes through aircore lightguides.\nThe electromagnetic section is 22 radiation lengths $X_0$ deep\nwith 2 tungsten-quartz sandwiches, the hadronic section consists of 12 tungsten-quartz\nsandwiches. The total depth is 10.3 interaction lengths $\\lambda_l$. The calorimeters\nare read out segmented azimuthally in 16 segments and logitudinally in 14 segments. \nThey do not have any segmentation in $\\eta$. The CASTOR coverage of \n$5.2 < |\\eta| < 6.6$ closes hermetically the total CMS calorimetric pseudorapidity range over\n13 units. \n\nCurrently, funding is available only for a CASTOR calorimeter on one side of the IP.\nConstruction is advanced, with concluding beamtests foreseen for this summer and \ninstallation in time for the 2009 LHC data taking. \n\nThe CMS Zero Degree Calorimeters, ZDC, are located inside the TAN absorbers \nat the ends of the straight section of \nthe LHC beamline, between the LHC beampipes, at $\\pm 140$~m distance on each side of the \nIP. They are very radiation-hard sampling calorimeters \nwith tungsten plates as absobers and as active medium quartz fibers read out via\naircore light guides and photomultiplier tubes.\nThe electromagnetic part, $19 X_0$ deep, is segmented into 5 units horizontally, the \nhadronic part into 4 units in depth. The total depth is 6.5 $\\lambda_l$. The ZDC \ncalorimeters have 100\\% acceptance for neutral particles with $|\\eta|>8.4$ and can measure\n50~GeV photons with an energy resolution of about 10\\%. \n\nThe ZDC calorimeters are already installed and will be operational already in 2008.\n\n\n\\subsection{TOTEM T1 \\& T2 telescopes}\n\nThe TOTEM T1 telescope consists of two arms symmetrically installed around the CMS IP \nin the endcaps of the\nCMS magnet, right in front of the CMS HF calorimeters and with $\\eta$ coverage similar to\nHF.\nEach arm consists of 5 planes of Cathod Strip Chambers (CSC) which measure\n3 projections per plane, resulting in a spatial resolution of 0.36~mm in the radial and\n0.62~mm in the azimuthal coordinate in test beam measurements.\n\nThe two arms of the TOTEM T2 telescope are mounted right in front of the CASTOR \ncalorimeters, with similar $\\eta$ coverage. Each arm consists of 10 planes of 20\nsemi-circular modules of Gas Electron Multipliers (GEMs). The detector read-out is\norganized in strips and pads, a resolution of $115~\\mu $m for the radial coordinate and\nof $16~\\mu$rad in azimuthal angle were reached in prototype test beam measurements.\n\n\n\\subsection{Proton-proton collisions at low $x_{Bj}$}\n\nIn order to arrive at parton-parton interactions at very low $x_{Bj}$ values, several \nsteps in the QCD cascade initiated by the partons from the \nproton may occur before the final hard interaction takes place. Low-$x_{Bj}$ QCD hence offers\nideal conditions for studying the QCD parton evolution dynamics. Measurements at the\nHERA $ep$ collider have explored low-$x_{Bj}$ dynamics down to values of a few $10^{-5}$.\nAt the LHC the minimum accessible $x$ decreases by a factor $\\sim 10$ for each\n2 units of rapidity. A process with a hard scale of $Q ~ 10$~GeV and within the \nacceptance of T2\/CASTOR ($\\eta = 6$) can occur at $x$ values as low as \n$10^{-6}$.\n\n\n\\begin{figure}[htb]\n\\includegraphics[scale =0.4]{pierre_m2vsx2_bw.eps}\n\\caption{Acceptance of the T2\/CASTOR detectors for Drell-Yan electrons, see text.}\n\\label{fig:DYcoverage}\n\\end{figure}\n\n\nForward particles at the LHC can be produced in collisions between two partons with\n$x_1 >> x_2$, in which case the hard interaction system is boosted forward.\nAn example is Drell-Yan production of $e^+ e^-$ pairs, \n$ q q \\rightarrow \\gamma^\\star \\rightarrow e^+ e^-$, a process that probes primarily \nthe quark content of the proton. Figure~\\ref{fig:DYcoverage} shows the distribution of the invariant\nmass $M$ of the $e^+ e^-$ system versus the $x_{Bj}$ of one of the quarks, where\n$x_2$ is chosen such that $x_1 >> x_2$. The solid curve shows the kinematic limit\n$M^{max} = \\sqrt{x_2 s}$. The dotted lines indicate the acceptance window for both\nelectrons to be detectable in T2\/CASTOR.\nThe black points correspond to any of the Drell-Yan events generated \nwith Pythia, the green\/light grey (blue\/dark grey ) ones refer to those events in which at least one (both)\nelectron lies within the T2\/CASTOR detector acceptance. For invariant masses of the $e^+ e^-$ \nsystem of $M> 10$~GeV, $x_{Bj}$ values down to $10^{-6}$ are accessible.\n\nThe rapid rise of the gluon density in the proton with decreasing values of $x_{Bj}$\nobserved by HERA in deep inelastic scattering cannot continue indefinitely without violating\nunitarity at some point. Hence, parton recombination within the proton must set in at low\nenough values of $x_{Bj}$ and leads to non-linear terms in the QCD gluon evolution. \nFigure~\\ref{fig:saturation} compares for \nDrell Yan processes with both electrons within the T2\/CASTOR detector acceptance the cross\nsection predicted by a PDF model without (CTEQ5L~\\cite{CTEQ}) and with (EHKQS~\\cite{EHKQS}) \nsaturation effects. A difference of a factor 2 is visible in the predictions. Further details \ncan be found in~\\cite{opus}.\n\n\n\n\\begin{figure}[htb]\n\\includegraphics[scale =0.3]{pierre_dsigmadx2dy_bw.eps}\n\\caption{Comparison of the cross section prediction of a model without (CTEQ5L) and \nwith (EHKQS) saturation for Drell-Yan events in which both electrons are detected in T2\/CASTOR.}\n\\label{fig:saturation}\n\\end{figure}\n\n\nComplementary information on the QCD evolution at low $x_{Bj}$ can be gained from\nforward jets. The DGLAP evolution~\\cite{DGLAP} \nassumes that parton emission in the cascade is strongly \nordered in transverse momentum while in the BFKL evolution~\\cite{BFKL}, \nno ordering in $k_t$ is assumed,\nbut strong ordering in $x$. At small $x_{Bj}$, the difference between the two approaches is\nexpected to be most pronounced for hard partons created at the beginning of the cascade, \nat pseudorapidities close to the proton, i.e. in the forward direction. Monte Carlo generator\nstudies indicate that the resulting excess of forward jets with high $p_T$, observed\nat HERA, might be measurable with T2\/CASTOR. Another observable sensitive to\nBFKL-like QCD evolution dynamics are dijets with large rapidity separation, which \nenhances the available phase space for BFKL-like parton radiation between the jets.\nLikewise dijets separated by a large rapidity gap are of interest since they indicate\na process in which no color flow occurs in the hard scatter but where, contrary to the \ntraditional picture of soft Pomeron exchange, also a high transverse momentum transfer \noccurs across the gap. \n\n\\subsection{Multiplicity \\& energy flow}\n\nThe forward detectors can be valuable tools for Monte Carlo tuning.\n\n\nThe hard scatter in hadron-hadron collisions takes place in a dynamic environment,\nrefered to as the ``underlying event'' (UE), where\nadditional soft or hard interactions between the partons and \ninitial and final state radiation occur. The effect of the UE can not be disentangled on an\nevent-by-event basis and needs to be included by means of tuning Monte Carlo multiplicities \nand energy flow predictions to data. The predictive power of these tunes obtained \nfrom Tevatron data is very limited, and ways need to be found to constrain the UE at LHC \nenergies with LHC data. As shown in~\\cite{Borras}, the forward detectors are sensitive\nto features of the UE which central detector information alone cannot constrain.\n\n\\begin{figure}[!b]\n\\includegraphics[scale =0.55]{cosmics_Eflow.epsi}\n\\caption{Energy flow as predicted by Monte Carlo generators used for the description of \ncosmic ray induced air showers~\\cite{opus}.}\n\\label{fig:cosmics}\n\\end{figure}\n\nAnother area with high uncertainties is modelling the interaction of primary cosmic rays in \nthe PeV energy range with the atmosphere. Their rate of occurance per year is too low for\nreliable quantitative analysis. The center-of-mass energy in $pp$ collisions at the LHC \ncorresponds to 100 PeV energy in a fixed target collision. Figure~\\ref{fig:cosmics} shows the \nenergy flow\nas function of pseudorapidity as predicted by different Monte Carlos in use in the cosmic ray\ncommunity. Clear differences in the predictions are visible in the acceptance region of\nT2\/CASTOR and ZDC.\n\n\n\n\\section{PHYSICS WITH A VETO ON FORWARD DETECTORS}\n\nEvents of the type $pp \\rightarrow pXp$ or $pp \\rightarrow Xp$, where no color exchange\ntakes place between the proton(s) and the system $X$, can be caused by $\\gamma$ exchange,\nor by diffractive interactions. In both cases, the absence of color flow between the\nproton(s) and the system $X$ results in a large gap in the rapidity distribution of the\nhadronic final state. Such a gap can be detected by requiring the absence of a signal in \nthe forward detectors. In the following, we discuss three exemplary processes which are \ncharacterized by a large rapidity gap in their hadronic final state.\n\n\\subsection{Diffraction with a hard scale}\n\nDiffraction, traditionally thought of as soft process and described in Regge theory, can also\noccur with a hard scale ($W$, dijets, heavy flavors) as\nhas been experimentally observed at UA8, HERA and Tevatron. In the presence of a hard scale,\ndiffractive processes can be described in perturbative QCD (pQCD) and their cross sections\ncan be factorized into that one of the hard scatter and a diffractive particle \ndistribution function (dPDF). In diffractive hadron-hadron scattering, rescattering between\nspectator particles breaks the factorization. The so-called rapidity gap survival \nprobability quantifies this effect~\\cite{survival}. A measure for it can be obtained by the ratio of\ndiffractive to inclusive processes with the same hard scale. At the Tevatron, the ratio \nis found to be ${\\cal O}(1 \\%)$~\\cite{tevatron}.\nTheoretical expectations for the LHC vary from a fraction of a percent to as much as \n30\\%~\\cite{predLHC}. \n\nSingle diffractive $W$ production, $pp \\rightarrow pX$, where $X$ includes a $W$, \nis an example for diffraction with a hard scale at the LHC and is in \nparticular sensitive to the quark component of the proton dPDF in an as-of-yet unmeasured \nregion. In the absence of event pile-up, a selection is possible based on the requirement\nthat there be no activity above noise level in the CMS forward calorimeters HF and CASTOR.\n\n\\begin{figure}\n\\includegraphics[scale=0.4]{nHFvsnCASTOROnlyMinus_nTrkMax_1_100pb_new.eps}\n\\caption{Number of towers with activity above noise level in HF versus in CASTOR for\nsingle diffractive $W$ production and for an integrated luminosity of 100~${\\rm pb}^{-1}$~\\cite{SDW}}\n\\label{fig:SDW}\n\\end{figure}\n\nFigure~\\ref{fig:SDW} shows the number of towers with activity above noise level in HF versus \nin CASTOR. The decay channel is $W \\rightarrow \\mu \\nu$ and a rapidity gap survical factor \nof 5\\% is assumed in the diffractive Monte Carlo sample (Pomwig). The number of events is\nnormalized to an integrated luminosity of 100~$\\rm pb^{-1}$ of single interactions (i.e. no\nevent pile-up). In the combined Pomwig + Pythia Monte Carlo sample, a clear excess in the \nbin [n(Castor), n(HF)] = [0,0] is visible, of ${\\cal O}(100)$ events. The ratio of diffraction \nto non-diffraction in the [0,0] bin of approximately 20 demonstrate the feasibility of \nobserving single diffractive $W$ production at the LHC.\n\nThe study assumes that CASTOR will be available only on one side. A second CASTOR in the \nopposite hemisphere and the use of T1, T2 will improve the observable excess \nfurther. \n\n\\subsection{Exclusive dilepton production}\n\nExclusive dimuon and dielectron production with no significant additional\nactivity in the CMS detector occurs with high cross section in\ngamma-mediated processes at the LHC, either as the pure QED process \n$\\gamma \\gamma \\rightarrow ll$ \nor in $\\Upsilon$ photoproduction~\\footnote{Photoproduction of $J\/psi$ mesons is also \npossible, but difficult to observe because of the trigger thresholds for leptons in CMS.}  \nA feasibility study to detect them with CMS was presented in this\nworkshop~\\cite{Hollar}. \n\nThe event selection is based on requiring that outside of the two leptons, no other \nsignificant activity is visible within the central CMS detector, neither in the calorimeter\nnor in the tracking system. In 100 $\\rm pb^{-1}$ of single interaction data, ${\\cal O} (700)$\nevents in the dimuon channels and ${\\cal O} (70)$ in the dielectron channel can be selected.\nEvents in which one of the  protons in the process does not stay intact but dissociates \nare the dominant source of background and are comparable in statistics to the signal. \nThis background can be significantly reduced by means of a veto condition on activity in CASTOR\nand ZDC, in a configuration with a ZDC on each side and a CASTOR on only one side of the IP\nby 2\/3.\n\nThe theoretically very precisely known cross section of the almost pure QED process \n$pp \\rightarrow pllp$ via $\\gamma$ exchange is an ideal calibration channel. With\n$100 \\rm pb^{-1}$ of data, an absolute luminosity calibration with 4\\% precision is feasible.\nFuthermore, exclusive dimuon production is an ideal alignment channel with high statistics \nfor the proposed proton taggers at 420~m from the IP. Upsilon photoproduction can constrain \nQCD models of diffraction, as discussed in the next section. \nThe $\\gamma \\gamma \\rightarrow e^+ e^-$ process has recently been observed at the Tevatron~\\cite{exclTevatron}.\n\n\n\\subsection{Upsilon photoproduction} \n\nAssuming the STARLIGHT~\\cite{starlight} \nMonte Carlo cross section prediction, the 1S, 2S and 3S resonances\nwill be clearly visible in $100 \\rm pb^{-1}$ of single interaction data. With their average\n$\\gamma p$ center-of-mass energy of $<W_{\\gamma p}> \\simeq 2400 \\rm GeV^2$ they will extend\nthe accessible range of the HERA measurement of the $W_{\\gamma p}$ dependence of \n$\\sigma (\\gamma p \\rightarrow \\Upsilon(1 S) p)$ by one order of magnitude. \n\n\n\\begin{figure}[htb]\n\\includegraphics[scale=0.4]{UpsilonSignalPAS.eps}\n\\caption{Invariant mass of exclusive dimuon production in the Upsilon mass region~\\cite{ExclDileptons}}\n\\label{fig:Upsilon}\n\\end{figure}\n\nBy means of the $p_T^2$ value of the $\\Upsilon$ as estimator of the transfered four-momentum \nsquared, $t$, at the proton vertex, it might be possible to measure the $t$ dependence of the \ncross section. This dependence is sensitive to the two-dimensional gluon distribution of the \nproton and would give access to the generalized parton distribution function (GPD) of the \nproton.\n\n\\section{PHYSICS WITH NEAR-BEAM PROTON TAGGERS}\n\nFor slightly off-momentum protons, the LHC beamline with its magnets is essentially a\nspectrometer. If a scattered proton is bent sufficiently, but little enough to remain within \nthe beam-pipe, they can be detected by means of detectors inserted into the beam-pipe and\napproaching the beam envelope as closely as possible. At ligh luminosity at the LHC,\nlarge rapidity gaps typical for diffractive events or events with $\\gamma$ exchange tend to be \nfilled in by particles from overlaid pile-up events. Hence tagging the outgoing scattered \nproton(s) becomes the only mean of detection at high luminosities.\n\n\\subsection{TOTEM and FP420 proton taggers}\n\nThe TOTEM proton taggers, located at $\\pm 147$~m and $\\pm 220$~m from the IP, each consist\nof Silicon strip detectors housed in movable Roman Pots~\\cite{TOTEM}. \nThe detector\ndesign is such that the beam can be approached up to a minimal distance of $10 \\sigma +$~0.5~mm.\nWith nominal LHC beam optics, scattered protons from the IP are within the acceptance of the \ntaggers at 220~m when for their fractional momentum loss $\\xi$ holds: $0.02 < \\xi < 0.2$.  \n\n\n\\begin{figure}\n\\includegraphics[scale=0.35, angle =-90]{fp420coverage.eps}\n\\caption{Acceptance in $x_L = 1 - \\xi$, where $\\xi$ is the fractional momentum loss of the \nscattered proton, of the TOTEM and FP420 proton taggers. The data points shown are from \nZEUS~\\cite{zeus}.}\n\\label{fig:xiCoverage}\n\\end{figure}\n\nIn order to achieve acceptance at smaller values of $\\xi$ with nominal LHC beam optics, \ndetectors have to be located further away from the IP. Proton taggers at $\\pm 420$~m from the\nIP have an acceptance of $0.002 < \\xi < 0.02$, complementing taggers at 220~m, as shown\nin Figure~\\ref{fig:xiCoverage}. \nThe proposal~\\cite{fp420} of the FP420 R\\&D collaboration foresees employing 3-D Silicon, an\nextremely radiation hard novel Silicon technology, for the proton taggers. Additional \nfast timing Cherenkov detectors will be capable of determining, within a resolution of a \nfew millimeters, whether the tagged proton came from the same vertex as the hard scatter visible\nin the central CMS detector. In order to comply with the space constraints of the location \nwithin the cryogenic region of the LHC, these detectors will be attached to a movable beam-pipe\nwith the help of which the detectors can approach the beam to within 3~mm.\n\nThe FP420 proposal is currently under scrutiny in CMS and ATLAS. If approved, installation could\nproceed in 2010, after the LHC start-up.\n\n\n\\subsection{Physics potential}\n\nForward proton tagging capabilities enhance the physics potential of CMS. They would\nrender possible a precise measurement of the mass and quantum numbers of the Higgs boson\nshould it be discovered by traditional searches. They also augment the CMS discovery reach\nfor Higgs production in the minimal supersymmetric extension (MSSM) of the Standard Model (SM)\nand for physics beyond the SM in $\\gamma p$ and $\\gamma \\gamma$ interactions.\n\nA case in point is the central exclusive production (CEP) process~\\cite{CEP}, \n$pp \\rightarrow p + \\phi + p$, where the plus sign denotes the absence of hadronic \nactivity between the outgoing protons, which survive the interaction intact, and the \nstate $\\phi$. The final state consists solely of the\nscattered protons, which may be detected in the forward proton taggers, and the decay \nproducts of $\\phi$ which can be detected in the central CMS detector. \nSelection rules force the produced state $\\phi$ to have $J^{CP} = n^{++}$ with $n =0, 2, ..$. \nThis process offers hence an experimentally very clean \nlaboratory for the discovery of any particle with these quantum numbers that couples \nstrongly to gluons. Additional advantages are the possibility to determine the mass of the state\n$\\phi$ with excellent resolution from the scattered protons alone, independent of its\ndecay products, and the possibility, unique at the LHC, to determine the quantum numbers of \n$\\phi$ directly from the azimuthal asymmetry between the scattered protons.\n\n\n\\begin{figure}[htb]\n\\includegraphics[angle=-90]{marek.eps}\n\\caption{Five $\\sigma$ discovery contours for central exclusive production of the \nheavier CP-even Higgs boson $H$~\\cite{Tasevsky}. See text for details.}\n\\label{fig:higgs}\n\\end{figure}\n\nIn the case of a SM Higgs boson with mass close to the current exclusion limit, which decays\npreferably into $b \\bar{b}$, CEP\nimproves the achievable signal-to-background ratio dramatically, to \n$\\cal{O}$(1)~\\cite{fp420,lightHiggs}. \nIn certain\nregions of the MSSM, generally known as ``LHC wedge region'', the heavy MSSM Higgs bosons would \nescape detection at the LHC. \nThere, the preferred search channels at the LHC are not available \nbecause the \nheavy Higgs bosons decouple from gauge bosons while their couplings to $b \\bar{b}$ and \n$\\tau \\bar{\\tau}$ are enhanced at high $\\tan{\\beta}$. Figure~\\ref{fig:higgs} depicts\nthe 5~$\\sigma$ discovery contour for the $H \\rightarrow b \\bar{b}$ channel in CEP in \nthe $M_A - \\tan{\\beta}$ plane of the MSSM within the $M_h^{max}$ benchmark scenario\nwith $\\mu = +200$~GeV and for different integrated luminosities. \nThe values of the mass of the heavier CP-even Higgs boson, $M_H$, are indicated by \ncontour lines. The dark region corresponds to the parameter region excluded by LEP. \n\nForward proton tagging will also give access to a rich QCD program on hard diffraction\nat high luminosities, where event pile-up is significant and makes undetectable the gaps \nin the hadronic final state otherwise typical of diffraction. Detailed studies with high\nstatistical precision will be possible on skewed, unintegrated gluon \ndensities; Generalized Parton Distributions which contain information on the correlations \nbetween partons in the proton; and the rapidity gap survival probability, a quantity closely \nlinked to soft rescattering effects and the features of the underlying event at the LHC.\n\nForward proton tagging also provides the possibility for precision studies of $\\gamma p$\nand $\\gamma \\gamma$ interactions at center-of-mass energies never reached before. Anomalous top\nproduction, anomalous gauge boson couplings, exclusive dilepton production and quarkonia \nproduction are possible topics, as was discussed in detail at this workshop.\n\n\n\\section{SUMMARY}\n\nForward physics in $pp$ collisions at the LHC covers a wide range of diverse physics subjects (low-$x_{Bj}$ QCD,\nhard diffraction, $\\gamma \\gamma$ and $\\gamma p$ interactions)\n which have in\ncommon that particles produced at large\nvalues of rapidity provide a defining characteristics. \nFor the CMS detector, several subdetectors with forward $\\eta$ coverage \nare currently under construction (CASTOR, ZDC) or in the proposal \nstage (FP420). The TOTEM experiment \nsupplements around the CMS IP several tracking devices and near-beam proton taggers at \ndistances up to $\\pm 220$~m. \nThe kinematic coverage of the combined CMS and TOTEM apparatus is unprecedented at a\nhadron collider. It would be even further enhanced by complementing CMS with the\ndetectors of the FP420 proposal which would add forward physics to the portfolio of\npossible discovery processes at the LHC.\n\n\n\\section{FORWARD PHYSICS AND FORWARD INSTRUMENTATION AT THE CMS INTERACTION POINT}\n\nForward physics at the LHC covers a wide range of diverse physics subjects which have in \ncommon that particles produced at small polar angles $\\theta$ and hence large \nvalues of rapidity provide a defining characteristics. This article concentrates\non their physics interest in $pp$ collisions.\n\nAt the Large-Hadron-Collider (LHC), where proton-proton collisions occur at \ncenter-of-mass energies of 14 TeV, the maximal possible rapidity is \n$y_{max} = \\ln{\\frac{\\sqrt{s}}{m_{\\pi}}}\\sim 11.5$. The two multi-purpose detectors ATLAS and \nCMS at the LHC are designed primarily for efficient detection of processes with large\npolar angles and hence\nhigh transverse momentum $p_T$. The coverage in pseudorapidity \n$\\eta = - \\ln{[\\tan{( \\theta \/ 2 )} ] }$ of\ntheir main components extends down to about $|\\theta| = 1^\\circ$ from the beam axis\nor $|\\eta| = 5$.\n\nFor the CMS detector, several subdetectors with coverage beyond $|\\eta| =5$ are \ncurrently under construction (CASTOR and ZDC sampling calorimeters) or in the proposal \nstage (FP420 proton taggers and fast timing detectors). \n\nFuthermore, a salient feature of the forward instrumentation around the \ninteraction point\n(IP) of CMS is the presence of TOTEM~\\cite{TOTEM}. TOTEM is an approved experiment at \nthe LHC for measuring the $pp$ elastic cross section as a function of the four-momentum\ntransfer squared, $t$, and for measuring the total cross section with a precision of\napproximately 1\\%. The TOTEM experiment uses the same IP as CMS and supplements around \nthe CMS IP several tracking devices, located inside of the volume \nof the main CMS detector, plus near-beam proton taggers a distance up to $\\pm 220$~m. \nThe CMS and TOTEM collaborations have described the considerable physics potential of\njoint data taking in a report to the LHCC \\cite{opus}. \n\nThe kinematic coverage of the combined CMS and TOTEM apparatus is unprecedented at a\nhadron collider. It would be even further enhanced by complementing CMS with the\ndetectors of the FP420 proposal which would induce forward physics into the portfolio of\npossible discovery processes at the LHC~\\cite{fp420}.\n\nAn overview of the forward instrumentation up to $\\pm 220$~m from the CMS IP is given in \nFig.~\\ref{fig:overview}. There are two suites of calorimeters with tracking detectors in front.\nThe CMS Hadron Forward (HF) calorimeter with the TOTEM telescope T1 in front \ncovers the region $3 < |\\eta | < 5$, the CMS CASTOR calorimeter with the TOTEM telescope \nT2 in front covers $5.2 < |\\eta| < 6.6$. The CMS ZDC calorimeters will be \ninstalled at the end of the straight LHC beam-line section, at a distance of \n$\\pm 140$~m from the IP. Near-beam proton taggers will be installed by TOTEM at \n$\\pm 147$~m and $\\pm 220$~m from the IP.\nFurther near-beam proton taggers in combination with very fast timing detectors to be\ninstalled at $\\pm 420$~m from the IP are part of the FP420 proposal.\n\n\\begin{figure}\n\\hspace*{-0.5cm}\n\\includegraphics[scale=0.32, angle = -90]{cms_totem_detectors_new.eps}\n\\caption{Layout of the forward detectors around the CMS interaction point.}\n\\label{fig:overview}\n\\end{figure}\n\n\n\n\n\\section{PHYSICS WITH FORWARD DETECTORS}\n\nIn the following, we describe the physics interest of the CMS CASTOR and ZDC \ncalorimeters~\\cite{PTDR1} \nand the TOTEM T1 and T2 telescopes~\\cite{TOTEM}. Of particular interest are \nQCD measurements at values of Bjorken-$x$ as low as $x \\sim 10^{-6}$ and the resulting \nsensitivity to non-DGLAP dynamics, \nas well as forward particle and energy flow measurements. These can play an important \nrole in tuning the Monte Carlos description of underlying event and multiple interactions\nat the LHC and in constraining Monte Carlo generators used for cosmic ray studies.\n\n\\subsection{CMS CASTOR \\& ZDC calorimeters}\n\nThe two calorimeters are of interest for $pp$, $pA$ and $AA$ running at the LHC, \nwhere $A$ denotes a heavy ion. They are Cherenkov-light devices with electromagnetic \nand hadronic sections and will be present in \nthe first LHC $pp$ runs at luminosities where event pile-up should be low.\n\nThe CASTOR calorimeters are octagonal cylinders located at $\\sim 14$~m from the IP.\nThey are sampling calorimeters with tungsten plates as absorbers and fused silica quartz \nplates as active medium. The plates are inclined by $45^\\circ$ with respect to the \nbeam axis. Particles passing through the quartz emit Cherenkov\nphotons which are transmitted to photomultiplier tubes through aircore lightguides.\nThe electromagnetic section is 22 radiation lengths $X_0$ deep\nwith 2 tungsten-quartz sandwiches, the hadronic section consists of 12 tungsten-quartz\nsandwiches. The total depth is 10.3 interaction lengths $\\lambda_l$. The calorimeters\nare read out segmented azimuthally in 16 segments and logitudinally in 14 segments. \nThey do not have any segmentation in $\\eta$. The CASTOR coverage of \n$5.2 < |\\eta| < 6.6$ closes hermetically the total CMS calorimetric pseudorapidity range over\n13 units. \n\nCurrently, funding is available only for a CASTOR calorimeter on one side of the IP.\nConstruction is advanced, with concluding beamtests foreseen for this summer and \ninstallation in time for the 2009 LHC data taking. \n\nThe CMS Zero Degree Calorimeters, ZDC, are located inside the TAN absorbers \nat the ends of the straight section of \nthe LHC beamline, between the LHC beampipes, at $\\pm 140$~m distance on each side of the \nIP. They are very radiation-hard sampling calorimeters \nwith tungsten plates as absobers and as active medium quartz fibers read out via\naircore light guides and photomultiplier tubes.\nThe electromagnetic part, $19 X_0$ deep, is segmented into 5 units horizontally, the \nhadronic part into 4 units in depth. The total depth is 6.5 $\\lambda_l$. The ZDC \ncalorimeters have 100\\% acceptance for neutral particles with $|\\eta|>8.4$ and can measure\n50~GeV photons with an energy resolution of about 10\\%. \n\nThe ZDC calorimeters are already installed and will be operational already in 2008.\n\n\n\\subsection{TOTEM T1 \\& T2 telescopes}\n\nThe TOTEM T1 telescope consists of two arms symmetrically installed around the CMS IP \nin the endcaps of the\nCMS magnet, right in front of the CMS HF calorimeters and with $\\eta$ coverage similar to\nHF.\nEach arm consists of 5 planes of Cathod Strip Chambers (CSC) which measure\n3 projections per plane, resulting in a spatial resolution of 0.36~mm in the radial and\n0.62~mm in the azimuthal coordinate in test beam measurements.\n\nThe two arms of the TOTEM T2 telescope are mounted right in front of the CASTOR \ncalorimeters, with similar $\\eta$ coverage. Each arm consists of 10 planes of 20\nsemi-circular modules of Gas Electron Multipliers (GEMs). The detector read-out is\norganized in strips and pads, a resolution of $115~\\mu $m for the radial coordinate and\nof $16~\\mu$rad in azimuthal angle were reached in prototype test beam measurements.\n\n\n\\subsection{Proton-proton collisions at low $x_{Bj}$}\n\nIn order to arrive at parton-parton interactions at very low $x_{Bj}$ values, several \nsteps in the QCD cascade initiated by the partons from the \nproton may occur before the final hard interaction takes place. Low-$x_{Bj}$ QCD hence offers\nideal conditions for studying the QCD parton evolution dynamics. Measurements at the\nHERA $ep$ collider have explored low-$x_{Bj}$ dynamics down to values of a few $10^{-5}$.\nAt the LHC the minimum accessible $x$ decreases by a factor $\\sim 10$ for each\n2 units of rapidity. A process with a hard scale of $Q ~ 10$~GeV and within the \nacceptance of T2\/CASTOR ($\\eta = 6$) can occur at $x$ values as low as \n$10^{-6}$.\n\n\n\\begin{figure}[htb]\n\\includegraphics[scale =0.4]{pierre_m2vsx2_bw.eps}\n\\caption{Acceptance of the T2\/CASTOR detectors for Drell-Yan electrons, see text.}\n\\label{fig:DYcoverage}\n\\end{figure}\n\n\nForward particles at the LHC can be produced in collisions between two partons with\n$x_1 >> x_2$, in which case the hard interaction system is boosted forward.\nAn example is Drell-Yan production of $e^+ e^-$ pairs, \n$ q q \\rightarrow \\gamma^\\star \\rightarrow e^+ e^-$, a process that probes primarily \nthe quark content of the proton. Figure~\\ref{fig:DYcoverage} shows the distribution of the invariant\nmass $M$ of the $e^+ e^-$ system versus the $x_{Bj}$ of one of the quarks, where\n$x_2$ is chosen such that $x_1 >> x_2$. The solid curve shows the kinematic limit\n$M^{max} = \\sqrt{x_2 s}$. The dotted lines indicate the acceptance window for both\nelectrons to be detectable in T2\/CASTOR.\nThe black points correspond to any of the Drell-Yan events generated \nwith Pythia, the green\/light grey (blue\/dark grey ) ones refer to those events in which at least one (both)\nelectron lies within the T2\/CASTOR detector acceptance. For invariant masses of the $e^+ e^-$ \nsystem of $M> 10$~GeV, $x_{Bj}$ values down to $10^{-6}$ are accessible.\n\nThe rapid rise of the gluon density in the proton with decreasing values of $x_{Bj}$\nobserved by HERA in deep inelastic scattering cannot continue indefinitely without violating\nunitarity at some point. Hence, parton recombination within the proton must set in at low\nenough values of $x_{Bj}$ and leads to non-linear terms in the QCD gluon evolution. \nFigure~\\ref{fig:saturation} compares for \nDrell Yan processes with both electrons within the T2\/CASTOR detector acceptance the cross\nsection predicted by a PDF model without (CTEQ5L~\\cite{CTEQ}) and with (EHKQS~\\cite{EHKQS}) \nsaturation effects. A difference of a factor 2 is visible in the predictions. Further details \ncan be found in~\\cite{opus}.\n\n\n\n\\begin{figure}[htb]\n\\includegraphics[scale =0.3]{pierre_dsigmadx2dy_bw.eps}\n\\caption{Comparison of the cross section prediction of a model without (CTEQ5L) and \nwith (EHKQS) saturation for Drell-Yan events in which both electrons are detected in T2\/CASTOR.}\n\\label{fig:saturation}\n\\end{figure}\n\n\nComplementary information on the QCD evolution at low $x_{Bj}$ can be gained from\nforward jets. The DGLAP evolution~\\cite{DGLAP} \nassumes that parton emission in the cascade is strongly \nordered in transverse momentum while in the BFKL evolution~\\cite{BFKL}, \nno ordering in $k_t$ is assumed,\nbut strong ordering in $x$. At small $x_{Bj}$, the difference between the two approaches is\nexpected to be most pronounced for hard partons created at the beginning of the cascade, \nat pseudorapidities close to the proton, i.e. in the forward direction. Monte Carlo generator\nstudies indicate that the resulting excess of forward jets with high $p_T$, observed\nat HERA, might be measurable with T2\/CASTOR. Another observable sensitive to\nBFKL-like QCD evolution dynamics are dijets with large rapidity separation, which \nenhances the available phase space for BFKL-like parton radiation between the jets.\nLikewise dijets separated by a large rapidity gap are of interest since they indicate\na process in which no color flow occurs in the hard scatter but where, contrary to the \ntraditional picture of soft Pomeron exchange, also a high transverse momentum transfer \noccurs across the gap. \n\n\\subsection{Multiplicity \\& energy flow}\n\nThe forward detectors can be valuable tools for Monte Carlo tuning.\n\n\nThe hard scatter in hadron-hadron collisions takes place in a dynamic environment,\nrefered to as the ``underlying event'' (UE), where\nadditional soft or hard interactions between the partons and \ninitial and final state radiation occur. The effect of the UE can not be disentangled on an\nevent-by-event basis and needs to be included by means of tuning Monte Carlo multiplicities \nand energy flow predictions to data. The predictive power of these tunes obtained \nfrom Tevatron data is very limited, and ways need to be found to constrain the UE at LHC \nenergies with LHC data. As shown in~\\cite{Borras}, the forward detectors are sensitive\nto features of the UE which central detector information alone cannot constrain.\n\n\\begin{figure}[!b]\n\\includegraphics[scale =0.55]{cosmics_Eflow.epsi}\n\\caption{Energy flow as predicted by Monte Carlo generators used for the description of \ncosmic ray induced air showers~\\cite{opus}.}\n\\label{fig:cosmics}\n\\end{figure}\n\nAnother area with high uncertainties is modelling the interaction of primary cosmic rays in \nthe PeV energy range with the atmosphere. Their rate of occurance per year is too low for\nreliable quantitative analysis. The center-of-mass energy in $pp$ collisions at the LHC \ncorresponds to 100 PeV energy in a fixed target collision. Figure~\\ref{fig:cosmics} shows the \nenergy flow\nas function of pseudorapidity as predicted by different Monte Carlos in use in the cosmic ray\ncommunity. Clear differences in the predictions are visible in the acceptance region of\nT2\/CASTOR and ZDC.\n\n\n\n\\section{PHYSICS WITH A VETO ON FORWARD DETECTORS}\n\nEvents of the type $pp \\rightarrow pXp$ or $pp \\rightarrow Xp$, where no color exchange\ntakes place between the proton(s) and the system $X$, can be caused by $\\gamma$ exchange,\nor by diffractive interactions. In both cases, the absence of color flow between the\nproton(s) and the system $X$ results in a large gap in the rapidity distribution of the\nhadronic final state. Such a gap can be detected by requiring the absence of a signal in \nthe forward detectors. In the following, we discuss three exemplary processes which are \ncharacterized by a large rapidity gap in their hadronic final state.\n\n\\subsection{Diffraction with a hard scale}\n\nDiffraction, traditionally thought of as soft process and described in Regge theory, can also\noccur with a hard scale ($W$, dijets, heavy flavors) as\nhas been experimentally observed at UA8, HERA and Tevatron. In the presence of a hard scale,\ndiffractive processes can be described in perturbative QCD (pQCD) and their cross sections\ncan be factorized into that one of the hard scatter and a diffractive particle \ndistribution function (dPDF). In diffractive hadron-hadron scattering, rescattering between\nspectator particles breaks the factorization. The so-called rapidity gap survival \nprobability quantifies this effect~\\cite{survival}. A measure for it can be obtained by the ratio of\ndiffractive to inclusive processes with the same hard scale. At the Tevatron, the ratio \nis found to be ${\\cal O}(1 \\%)$~\\cite{tevatron}.\nTheoretical expectations for the LHC vary from a fraction of a percent to as much as \n30\\%~\\cite{predLHC}. \n\nSingle diffractive $W$ production, $pp \\rightarrow pX$, where $X$ includes a $W$, \nis an example for diffraction with a hard scale at the LHC and is in \nparticular sensitive to the quark component of the proton dPDF in an as-of-yet unmeasured \nregion. In the absence of event pile-up, a selection is possible based on the requirement\nthat there be no activity above noise level in the CMS forward calorimeters HF and CASTOR.\n\n\\begin{figure}\n\\includegraphics[scale=0.4]{nHFvsnCASTOROnlyMinus_nTrkMax_1_100pb_new.eps}\n\\caption{Number of towers with activity above noise level in HF versus in CASTOR for\nsingle diffractive $W$ production and for an integrated luminosity of 100~${\\rm pb}^{-1}$~\\cite{SDW}}\n\\label{fig:SDW}\n\\end{figure}\n\nFigure~\\ref{fig:SDW} shows the number of towers with activity above noise level in HF versus \nin CASTOR. The decay channel is $W \\rightarrow \\mu \\nu$ and a rapidity gap survical factor \nof 5\\% is assumed in the diffractive Monte Carlo sample (Pomwig). The number of events is\nnormalized to an integrated luminosity of 100~$\\rm pb^{-1}$ of single interactions (i.e. no\nevent pile-up). In the combined Pomwig + Pythia Monte Carlo sample, a clear excess in the \nbin [n(Castor), n(HF)] = [0,0] is visible, of ${\\cal O}(100)$ events. The ratio of diffraction \nto non-diffraction in the [0,0] bin of approximately 20 demonstrate the feasibility of \nobserving single diffractive $W$ production at the LHC.\n\nThe study assumes that CASTOR will be available only on one side. A second CASTOR in the \nopposite hemisphere and the use of T1, T2 will improve the observable excess \nfurther. \n\n\\subsection{Exclusive dilepton production}\n\nExclusive dimuon and dielectron production with no significant additional\nactivity in the CMS detector occurs with high cross section in\ngamma-mediated processes at the LHC, either as the pure QED process \n$\\gamma \\gamma \\rightarrow ll$ \nor in $\\Upsilon$ photoproduction~\\footnote{Photoproduction of $J\/psi$ mesons is also \npossible, but difficult to observe because of the trigger thresholds for leptons in CMS.}  \nA feasibility study to detect them with CMS was presented in this\nworkshop~\\cite{Hollar}. \n\nThe event selection is based on requiring that outside of the two leptons, no other \nsignificant activity is visible within the central CMS detector, neither in the calorimeter\nnor in the tracking system. In 100 $\\rm pb^{-1}$ of single interaction data, ${\\cal O} (700)$\nevents in the dimuon channels and ${\\cal O} (70)$ in the dielectron channel can be selected.\nEvents in which one of the  protons in the process does not stay intact but dissociates \nare the dominant source of background and are comparable in statistics to the signal. \nThis background can be significantly reduced by means of a veto condition on activity in CASTOR\nand ZDC, in a configuration with a ZDC on each side and a CASTOR on only one side of the IP\nby 2\/3.\n\nThe theoretically very precisely known cross section of the almost pure QED process \n$pp \\rightarrow pllp$ via $\\gamma$ exchange is an ideal calibration channel. With\n$100 \\rm pb^{-1}$ of data, an absolute luminosity calibration with 4\\% precision is feasible.\nFuthermore, exclusive dimuon production is an ideal alignment channel with high statistics \nfor the proposed proton taggers at 420~m from the IP. Upsilon photoproduction can constrain \nQCD models of diffraction, as discussed in the next section. \nThe $\\gamma \\gamma \\rightarrow e^+ e^-$ process has recently been observed at the Tevatron~\\cite{exclTevatron}.\n\n\n\\subsection{Upsilon photoproduction} \n\nAssuming the STARLIGHT~\\cite{starlight} \nMonte Carlo cross section prediction, the 1S, 2S and 3S resonances\nwill be clearly visible in $100 \\rm pb^{-1}$ of single interaction data. With their average\n$\\gamma p$ center-of-mass energy of $<W_{\\gamma p}> \\simeq 2400 \\rm GeV^2$ they will extend\nthe accessible range of the HERA measurement of the $W_{\\gamma p}$ dependence of \n$\\sigma (\\gamma p \\rightarrow \\Upsilon(1 S) p)$ by one order of magnitude. \n\n\n\\begin{figure}[htb]\n\\includegraphics[scale=0.4]{UpsilonSignalPAS.eps}\n\\caption{Invariant mass of exclusive dimuon production in the Upsilon mass region~\\cite{ExclDileptons}}\n\\label{fig:Upsilon}\n\\end{figure}\n\nBy means of the $p_T^2$ value of the $\\Upsilon$ as estimator of the transfered four-momentum \nsquared, $t$, at the proton vertex, it might be possible to measure the $t$ dependence of the \ncross section. This dependence is sensitive to the two-dimensional gluon distribution of the \nproton and would give access to the generalized parton distribution function (GPD) of the \nproton.\n\n\\section{PHYSICS WITH NEAR-BEAM PROTON TAGGERS}\n\nFor slightly off-momentum protons, the LHC beamline with its magnets is essentially a\nspectrometer. If a scattered proton is bent sufficiently, but little enough to remain within \nthe beam-pipe, they can be detected by means of detectors inserted into the beam-pipe and\napproaching the beam envelope as closely as possible. At ligh luminosity at the LHC,\nlarge rapidity gaps typical for diffractive events or events with $\\gamma$ exchange tend to be \nfilled in by particles from overlaid pile-up events. Hence tagging the outgoing scattered \nproton(s) becomes the only mean of detection at high luminosities.\n\n\\subsection{TOTEM and FP420 proton taggers}\n\nThe TOTEM proton taggers, located at $\\pm 147$~m and $\\pm 220$~m from the IP, each consist\nof Silicon strip detectors housed in movable Roman Pots~\\cite{TOTEM}. \nThe detector\ndesign is such that the beam can be approached up to a minimal distance of $10 \\sigma +$~0.5~mm.\nWith nominal LHC beam optics, scattered protons from the IP are within the acceptance of the \ntaggers at 220~m when for their fractional momentum loss $\\xi$ holds: $0.02 < \\xi < 0.2$.  \n\n\n\\begin{figure}\n\\includegraphics[scale=0.35, angle =-90]{fp420coverage.eps}\n\\caption{Acceptance in $x_L = 1 - \\xi$, where $\\xi$ is the fractional momentum loss of the \nscattered proton, of the TOTEM and FP420 proton taggers. The data points shown are from \nZEUS~\\cite{zeus}.}\n\\label{fig:xiCoverage}\n\\end{figure}\n\nIn order to achieve acceptance at smaller values of $\\xi$ with nominal LHC beam optics, \ndetectors have to be located further away from the IP. Proton taggers at $\\pm 420$~m from the\nIP have an acceptance of $0.002 < \\xi < 0.02$, complementing taggers at 220~m, as shown\nin Figure~\\ref{fig:xiCoverage}. \nThe proposal~\\cite{fp420} of the FP420 R\\&D collaboration foresees employing 3-D Silicon, an\nextremely radiation hard novel Silicon technology, for the proton taggers. Additional \nfast timing Cherenkov detectors will be capable of determining, within a resolution of a \nfew millimeters, whether the tagged proton came from the same vertex as the hard scatter visible\nin the central CMS detector. In order to comply with the space constraints of the location \nwithin the cryogenic region of the LHC, these detectors will be attached to a movable beam-pipe\nwith the help of which the detectors can approach the beam to within 3~mm.\n\nThe FP420 proposal is currently under scrutiny in CMS and ATLAS. If approved, installation could\nproceed in 2010, after the LHC start-up.\n\n\n\\subsection{Physics potential}\n\nForward proton tagging capabilities enhance the physics potential of CMS. They would\nrender possible a precise measurement of the mass and quantum numbers of the Higgs boson\nshould it be discovered by traditional searches. They also augment the CMS discovery reach\nfor Higgs production in the minimal supersymmetric extension (MSSM) of the Standard Model (SM)\nand for physics beyond the SM in $\\gamma p$ and $\\gamma \\gamma$ interactions.\n\nA case in point is the central exclusive production (CEP) process~\\cite{CEP}, \n$pp \\rightarrow p + \\phi + p$, where the plus sign denotes the absence of hadronic \nactivity between the outgoing protons, which survive the interaction intact, and the \nstate $\\phi$. The final state consists solely of the\nscattered protons, which may be detected in the forward proton taggers, and the decay \nproducts of $\\phi$ which can be detected in the central CMS detector. \nSelection rules force the produced state $\\phi$ to have $J^{CP} = n^{++}$ with $n =0, 2, ..$. \nThis process offers hence an experimentally very clean \nlaboratory for the discovery of any particle with these quantum numbers that couples \nstrongly to gluons. Additional advantages are the possibility to determine the mass of the state\n$\\phi$ with excellent resolution from the scattered protons alone, independent of its\ndecay products, and the possibility, unique at the LHC, to determine the quantum numbers of \n$\\phi$ directly from the azimuthal asymmetry between the scattered protons.\n\n\n\\begin{figure}[!b]\n\\includegraphics[angle=-90]{marek.eps}\n\\caption{Five $\\sigma$ discovery contours for central exclusive production of the \nheavier CP-even Higgs boson $H$~\\cite{Tasevsky}. See text for details.}\n\\label{fig:higgs}\n\\end{figure}\n\nIn the case of a SM Higgs boson with mass close to the current exclusion limit, which decays\npreferably into $b \\bar{b}$, CEP\nimproves the achievable signal-to-background ratio dramatically, to \n$\\cal{O}$(1)~\\cite{fp420,lightHiggs}. \nIn certain\nregions of the MSSM, generally known as ``LHC wedge region'', the heavy MSSM Higgs bosons would \nescape detection at the LHC. \nThere, the preferred search channels at the LHC are not available \nbecause the \nheavy Higgs bosons decouple from gauge bosons while their couplings to $b \\bar{b}$ and \n$\\tau \\bar{\\tau}$ are enhanced at high $\\tan{\\beta}$. Figure~\\ref{fig:higgs} depicts\nthe 5~$\\sigma$ discovery contour for the $H \\rightarrow b \\bar{b}$ channel in CEP in \nthe $M_A - \\tan{\\beta}$ plane of the MSSM within the $M_h^{max}$ benchmark scenario\nwith $\\mu = +200$~GeV and for different integrated luminosities. \nThe values of the mass of the heavier CP-even Higgs boson, $M_H$, are indicated by \ncontour lines. The dark region corresponds to the parameter region excluded by LEP. \n\nForward proton tagging will also give access to a rich QCD program on hard diffraction\nat high luminosities, where event pile-up is significant and makes undetectable the gaps \nin the hadronic final state otherwise typical of diffraction. Detailed studies with high\nstatistical precision will be possible on skewed, unintegrated gluon \ndensities; Generalized Parton Distributions which contain information on the correlations \nbetween partons in the proton; and the rapidity gap survival probability, a quantity closely \nlinked to soft rescattering effects and the features of the underlying event at the LHC.\n\nForward proton tagging also provides the possibility for precision studies of $\\gamma p$\nand $\\gamma \\gamma$ interactions at center-of-mass energies never reached before. Anomalous top\nproduction, anomalous gauge boson couplings, exclusive dilepton production and quarkonia \nproduction are possible topics, as was discussed in detail at this workshop.\n\n\n\\section{SUMMARY}\n\nForward physics in $pp$ collisions at the LHC covers a wide range of diverse physics subjects (low-$x_{Bj}$ QCD,\nhard diffraction, $\\gamma \\gamma$ and $\\gamma p$ interactions)\n which have in\ncommon that particles produced at large\nvalues of rapidity provide a defining characteristics. \nFor the CMS detector, several subdetectors with forward $\\eta$ coverage \nare currently under construction (CASTOR, ZDC) or in the proposal \nstage (FP420). The TOTEM experiment \nsupplements around the CMS IP several tracking devices and near-beam proton taggers at \ndistances up to $\\pm 220$~m. \nThe kinematic coverage of the combined CMS and TOTEM apparatus is unprecedented at a\nhadron collider. It would be even further enhanced by complementing CMS with the\ndetectors of the FP420 proposal which would add forward physics to the portfolio of\npossible discovery processes at the LHC.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Acknowledgements}\nThis work was sponsored in part by the  National Science Foundation under contract ECCS-1800812. \nThis material is based upon work supported by the Google Cloud Research Credits program with the award GCP19980904.\nWe would like to sincerely thank Dr. Hassan Hijazi at Los Alamos National Laboratory for sharing a copy of his GravityX AC-OPF solver with us for comparison of our results. \n\\section{Background}\n\\label{sec:background}\n\n\\subsection{Homotopy Methods}\nThe Newton-Raphson (N-R) method is often used to solve the underlying non-linear equations in an AC-OPF formulation and its convergence can be sensitive to the choice of initial guess for the variables.\nIf the starting point is outside the basin of attraction for the problem, convergence can be very slow.\nA class of successive-relaxation methodologies, known as homotopy methods, was introduced to mitigate such issues.\nHomotopy is a numerical analysis method to solve a non-linear system of equations that traverses the solution space by deforming the non-linear equations from a trivial problem to the original.\nThe homotopy method initially defines a relaxation of the original problem which is trivial to solve, and proceeds to solve a sequence of deformations that ultimately leads back to the original problem.\nSuppose $\\mathcal{F}(x)=0$ is a set of non-linear equations that we aim to solve, we define a mapping to a trivial problem represented by $\\mathcal{G}(x)=0$.\nThe deformation from the trivial problem to the original non-linear problem is controlled by embedding a  scalar homotopy factor $\\nu \\in [0,1]$ into the non-linear equations, thereby defining a  sequence of problems given by (\\ref{eq:basic_homotopy}) .\n\\begin{equation}\n\\label{eq:basic_homotopy}\n    \\mathcal{H}(x, \\nu) = \\nu \\mathcal{G}(x) + (1-\\nu) \\mathcal{F}(x)=0,~ \\nu \\in [0,1]\n\\end{equation}\nHaving determined the solution, $x^0$ to the trivial problem, $\\mathcal{H}(x,1) = \\mathcal{G}(x) = 0$,  we can iteratively decrease $\\nu$, to move the system closer to the original problem, and use $x^0$ as our initial guess for the the next sub-problem.\nBy incrementally decreasing $\\nu$ and solving the updated sub-problems, we traverse the solution space from the trivial problem to the original problem.\nFor this method to be effective, each sub-problem's solution should lie within the basin of attraction of the solution of the previous sub-problem, in order to exploit the quadratic convergence of N-R.\nIt is often challenging to develop a general homotopy method that ensures a proper traversal of the solution space, where there exists a feasible solution for every sub-problem $\\mathcal{H}(x,\\nu)=0$ as it traverses the path from $\\nu: 1 \\rightarrow 0 $ \\cite{Allgower}. We present our homotopy method based on a circuit-inspired intuition that can intuitively ensure a feasible path. \n\n\\subsection{Homotopy Methods in Power System}\n\nA number of approaches have applied homotopy methods to solve power flow, AC-OPF, and other variants  in recent years \\cite{Murray-homotopy,Pandey-IMB, Park-homotopy,Network-Stepping,Pandey-Tx-Stepping}.\nIn \\cite{Pandey-Tx-Stepping}, the authors present a circuit-theoretic homotopy to robustly solve the power flow equations that embeds a homotopy factor in the equivalent circuits of the grid components to solve power flow.\nThese methods are extended in the Incremental Model Building (IMB) framework \\cite{Pandey-IMB} to solve the AC-OPF optimization problem.\nThe idea behind the IMB framework is to build the grid from ground-up using an embedded homotopy factor in AC-OPF equations.\nIMB defines a relaxed problem, $\\mathcal{G}(x)$, where the buses are almost completely shorted to one another, nearly all of the loads are removed at $\\nu=1$, and $\\nu$ is embedded in the generator limits so that the generator injections can be initially close to zero while remaining feasible with respect to the inequality constraints.\nAs a result, the relaxed network has very little current flow, so nearly all of the buses have voltage magnitude and angles close to that of the reference bus, and a flat-start initial point can reliably be used as a trivial solution of the homotopy sub-problem at $\\nu=1$.\nTo satisfy the requirement that a feasible solution exist for every sub-problem along the homotopy path, fictitious slack current sources are introduced at each bus for sub-problems $\\nu \\neq 0$, and their injections are heavily penalized \\cite{Jereminov-feasibility}.\n\nWhile IMB shows an approach to solve AC-OPF without good initial conditions, it does not include discrete control variables in the formulation.\nEven with a continuous relaxation to these variables, their introduction can significantly increase the nonlinearity of the network flow constraint equations and make the problem very challenging to solve.\nIn this work, we present a framework that builds on the IMB framework to include discrete control devices to solve the \\ACOPFD robustly.\n\n\\section{Conclusion}\n\\label{sec:conclusion}\nIn this paper we developed a two-stage homotopy algorithm to robustly solve real-world AC-OPF problems while incorporating discrete control devices. The proposed approach uses fundamentals from circuit-theory to design the mechanisms of the underlying homotopy methods and the approach does not depend on access to good initial conditions to solve the overall problem. To evaluate this approach, we ran a series of tests for four networks containing transformers and shunts with discrete settings.\nFor two of the cases, this method performed better than a state-of-the-art solver. Furthermore, we showed that by constructing different homotopy paths, we find different local optima solutions for the same problem, which had significantly different generation dispatch patterns. Therefore, we believe that the nonconvexity of the solution space of \\ACOPFDs problems warrant more investigation.\n\n\n\n\n\n\\section{AC-OPF with Discrete Variables Formulation \\ACOPFD}\n\\label{sec:formulation}\nThe paper solves the following AC-OPF with discrete control settings, which we will refer to as \\ACOPFD:\n\\begin{subequations}\n\\begin{gather}\n\\underset{x,\\xd}{\\text{minimize   }}\nf_0(x, \\xd) \\\\\n\\text{subject to: }\n g(x, \\xd) =0 \\\\%, \\; i = 1, \\ldots, m.\n h(x, \\xd) \\leq 0 \\\\\n \\xd^i \\in \\mathcal{D}^i, \\; i = 1, \\ldots, n_d\n\\end{gather}\n\\end{subequations}\nThe vector $x$ consists of the continuous-valued variables of the optimization, including complex bus voltages, real and reactive power injections by generators, continuous shunts settings, and substitution variables to track transformer and line flows.\nThe vector $\\xd\\in R^{n_d}$ represents the discrete control settings that are limited to a finite set of discrete values. In this paper, these include transformer taps $\\tau$  and phase-shifters $\\phi$ and discrete shunts $B^{sh}$ but the method is not restricted to just these. Each element of $\\xd$, $\\xd^i\\in\\xd$, has an associated integrality constraint (1d) restricting each device setting to be a finite value within the set, $\\mathcal{D}^i$. The objective function (1a) is chosen depending on the purpose of the AC-OPF study, but typically represents the economic cost of power generation.\nConstraint (1b) represents the AC network constraints, which can be formulated either to enforce net zero power-mismatch or current-mismatch at all nodes.\nConstraint (1c) contains the bus voltage magnitude limits, branch and transformer thermal limits, and real and reactive power generation limits.\n\n\n\n\\section{Implementation and Evaluation}\n\\label{sec:implementation}\nTo test the effectiveness of the proposed algorithm, we run the \\ACOPFDs solver on four networks based on cases used in the ARPA-E Grid Optimization Challenge 2 \\cite{go2} that contain transformers with adjustable tap ratios or phase shifts, and discrete switched shunt banks.\nThe cases were modified to remove additional features of the GO formulation in order to focus on the efficacy of our approach for incorporating discrete control devices, and to make all generation costs linear.\nThe details of cases used are shown in Table I, and the files have been made available in a public Github repository \\cite{cases-github}.\n\n\\begin{table}[h]\n\\label{tab:case-info}\n\\caption{Properties of \\ACOPFD ~Cases Tested}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\n\\textbf{Name} & \\textbf{Buses} & \\textbf{Generators} & \\textbf{Loads} & \\textbf{Lines} & \\textbf{Discrete Devices} \\\\ \\hline\nA & 3022 & 420 & 1625 & 3579 & 1384 \\\\ \\hline\nB & 6867 & 567 & 4618 & 7815 & 925 \\\\ \\hline\nC & 11152 & 1318 & 4661 & 16036 & 1030 \\\\ \\hline\nD & 16789 & 994 & 7846 & 23375 & 2722 \\\\ \\hline\n\\end{tabular}\n\\end{table}\nIn each evaluation, $\\xdbase$ values are set to the respective device settings listed in the .raw file, but to simulate running the \\ACOPFDs without any prior knowledge of settings, a copy of each case file is created where each transformer's initial setting is listed as its median available setting, and each switched shunt bank has all switches off (0 p.u.).\nThese cases with initial settings removed are denoted with a * superscript.\n\n\\subsection{Robustness and Scalability of IMB+D}\n\nTable \\ref{tab:big-results} shows a summary of results.\n$k_{adj}=0.1$ was applied equally across all devices in Stage 1, but normalized by the range of the individual device's settings ($\\xdupper^i - \\xdlower^i$)\nAs a comparison point, we also evaluated the same cases using the GravityX AC-OPF solver \\cite{Gravity}, a leading submission to the GO Challenge which utilizes the IPOPT non-linear optimization tool \\cite{IPOPT}.\n``\\% Adj'' indicates how many device settings in the solution differed from their initial value (the original .raw file values in the standard cases, and the simulated unknown in the * cases).\nFor each test, to evaluate the necessity of Stage II, a simple ``round and resolve'' approach is also attempted at the end of Stage I, in which the a feasible solution is sought immediately after rounding and fixing settings. \n\nThe proposed approach is able to find solutions for all the cases, even when initial device settings are removed.\nFor Cases A and D, evaluating from the simulated unknown settings actually produces a very slightly better objective value.\nGravityX produces a slightly better objective for both versions of cases B and C, but a less optimal solution for case A and does not converge for case D. \nObserve that Stage II is not strictly necessary for cases B, C, or D, but it is necessary for Case A to converge with discrete settings, which makes sense because in this case the tap steps are much further apart, so selecting discrete settings introduces a larger disturbance.\n\\begin{table}[]\n\\centering\n\\caption{Objective solutions and best $k_{adj}$ for tested cases}\n\\label{tab:big-results}\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline\n     & \\multicolumn{3}{c|}{IMB+D ($k_{adj}=0.1$)}                   & \\multicolumn{2}{c|}{GravityX+IP-OPT}                \\\\ \\hline\nCase  & Obj & \\% Adj & Need Stage II? & Obj & \\% Adj \\\\ \\hline\nA~ & 5.377e5 & 24.9\\% & Yes & 5.561e5 & 23.8\\% \\\\ \\hline\nA* & 5.357e5 & 25.6\\% & Yes & 5.561e5 & 17.6\\% \\\\ \\hline\nB~ & 1.216e5 & 90.8\\% & No & 1.215e5 & 74.5\\% \\\\ \\hline\nB* & 1.216e5 & 93.9\\% & No & 1.215e5 & 71.7\\% \\\\ \\hline\nC~ & 5.294e5 & 77.8 \\% & No & 5.292e5 & 80.29\\% \\\\ \\hline\nC* & 5.294e5 & 82.2\\% & No & 5.292e5 & 69.4\\% \\\\ \\hline\nD~ & 3.592e5 & 74.7\\% & No & No Solution & N\/A \\\\ \\hline\nD* & 3.591e5 & 89.2\\% & No & No Solution  & N\/A \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\\subsection{Impact of Different Homotopy Paths}\nTo construct different homotopy paths for the problem, we vary the Homotopy Stage I's adjustment penalty $k_{adj}$ parameter. We solve cases D and D* across a sweep of $k_{adj}$ values, and also at $k_{adj}=0$, meaning a penalty term is never used in Stage I.\nRecall that the term is completely removed at $\\nu_1 = 0$, so \\textit{the same final problem is being solved}, regardless of choice of $k_{adj}$. \nEssentially, all that differs is the homotopy path to the original \\ACOPFD problem. \n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=2.6in]{figures\/test_figure1.png}\n    \\caption{Changing $k_{adj}$ affects the homotopy path taken, and thus yields different solutions}\n   \n    \\label{fig:stage1-graphs}\n\\end{figure}\nFig. \\ref{fig:stage1-graphs} shows the effect of increasing $k_{adj}$ (i.e., varying the homotopy path) on both the final solution cost and percentage of adjustments made to devices in the solution.\nThis plot shows how it is possible to find multiple different local optima by parameterizing the penalty terms and essentially varying the homotopy path.\nFirst, the we obtain the best objective function with very low penalty factors.\nAdditionally, when knowledge of base settings is available, increasing the $k_{adj}$ reduces the number of adjustments.\nSurprisingly, however observe that for the D* simulations, increasing $k_{adj}$ does not have a large impact the number of discrete adjustments.\nWe hypothesize this is because some devices may require certain low or high settings for the network to be feasible, and so adjustments will be made regardless of the penalty value.\n\n\\subsection{Comparison of Local Solutions}\n\nTo further investigate the impact of different local solutions on the grid dispatch, we consider three sets of solution for the same exact problem as defined by Case A*: two solutions generated by IMB+D by taking separate homotopy paths through choice of $k_{adj}$, and one generated by GravityX.\nThe three solutions have generation dispatch costs in the range of \\$5.36e5-\\$5.84e5.\nHowever, more interesting insights can be gathered by looking at the actual dispatch of six large generators in Case A* for these three distinct local optima, which are shown in Fig. \\ref{fig:gen_dispatch}.\nWe notice that the dispatches for these generators vary significantly across the three dispatches.\nThis would imply that in the real world, the dispatch produced by an \\ACOPFDs study can have widely varying patterns dependent on the method (e.g. Gravity-X vs. IMB+D) used to solve the non-convex problem, or even the choice of homotopy path within IMB+D based on the value of $k_{adj}$.\nIn real world, this could make it harder for grid operators to justify the choice of one dispatch over another as the global minima may not be easily obtainable.\nToday, by running the convex DC-constraint based optimization with fixed settings, they are able to overcome the problem. \nHowever, due to the lack of granularity and accuracy, DC-based optimizations may be insufficient for future scenarios where adjustments to discrete devices are necessary in the optimization.\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=2.5in]{figures\/generator_dispatch.png}\n    \\caption{Changing the penalty scalar $k_{adj}$ on adjustments in Stage I causes a different homotopy path to be taken and can yield starkly different local optima in the final solution.}\n    \\label{fig:gen_dispatch}\n\\end{figure}\n\\section{Introduction}\n\\label{sec:introduction}\n\nA critical framework for modeling and optimizing the efficiency of today's power grid is based on the Alternating-Current Optimal Power Flow (AC-OPF) problem.\nIn the AC-OPF problem, a user-specified objective function, typically the cost of power generation, is optimized subject to network and device constraints.\nThese constraints include AC network constraints defined by Kirchhoff's Voltage and Current Laws, as well inequality constraints representing operational limits, such as bounds on voltages, power generation, and power transfer, to ensure reliable operation of the power system.\nIt has been estimated that improved methods to model and run the US power grid could improve dispatch efficiency in the US electricity system leading to savings between \\$6 billion and \\$19 billion per year \\cite{Cain}.\nMoreover, improved AC-OPF solution techniques can improve the reliability and resiliency of the grid, which is under increasing duress from extreme weather events, such as California's ongoing wildfires and the aftermath of the winter 2021 storm in Texas.\n\nTraditionally, the AC-OPF problem is a non-convex nonlinear problem with only continuous variables.\nHowever, many devices deployed in the grid today have controls with discrete-valued settings, and are likely to become more widespread as the modernization of the grid continues.\nDevices such as switched shunt banks and adjustable transformers can assist in balancing power flows in the system, meeting resilience-focused operational constraints, and locating a more optimal or resilient operating point than one located using fixed components settings.\nThe increased flexibility from these discrete devices also allows operators to avoid or delay costly upgrades to the network while increasing resiliency during extreme events. \nGiven the significant potential benefits, a recent Grid Optimization (GO) Competition (Challenge 2), organized by ARPA-E, sought new robust approaches to AC-OPF where adjustments to discrete devices like tap changers, phase shifters, and switched shunt banks are included in the variable set \\cite{go2}.\n\nAlthough there are clear benefits to inclusion of discrete control devices in an AC-OPF study, doing so directly results in a mixed-integer non-linear program (MINLP) that is significantly harder to solve.\nIncluding these discrete settings in the variable space could lead to searching over a combinatorially-large solution space, which would be intractable to solve in practical time. \nIf prior settings are at least known, then these can be used a starting point to begin the search for new settings.\nHowever, prior settings may be not known in some situations like planning or policy studies. For instance, when engineers evaluate the feasibility of 50\\% renewable penetration in a future U.S. Eastern Interconnection \\cite{miso50}. \n\nOne approach \\cite{liu-linearization} to include these discrete variables relies on sequential linearization of the optimization problem and then handling the discrete variables using mixed-integer linear problem (MILP) techniques.\nUnfortunately, the underlying network constraints can be highly non-linear with respect to certain control settings such as transformer tap ratios and phase shifts. Therefore these methods can suffer from a significant loss in model fidelity.\nAdditionally, linear relaxations to the OPF problem can lead to physically infeasible solutions, which are extremely undesirable for a grid dispatch \\cite{Baker-DCOPF}.\n\nThe simplest approach to this obstacle is a two-stage rounding technique \\cite{Tinney} \\cite{Papalexopoulos}.\nIn this method, the discrete control settings are initially treated as continuous-valued, and a relaxed formulation of the AC-OPF problem is solved; then, these variables are fixed to their nearest respective discrete values, and the optimization is solved again with these variables held constant.\nThe first challenge with this approach is that for realistically-sized networks, solving the relaxed problem with continuous valued transformer taps, phase-shifters and switched shunts is computationally challenging, especially when good initial conditions are unavailable.\nThe second challenge is that the rounding step to map the continuous-valued settings to their nearest respective discrete values can result in a physically infeasible solution, which can be difficult to avoid when the discrete values are spaced far apart or when very many control variables are rounded at once. \nDirectly rounding to the nearest discrete value also creates a discontinuous jump in the solution space, which is problematic for any Newton-method based solver that relies on first-order derivative continuity.\n\nA number of methods have been introduced to address the two challenges above. \nRecent work  \\cite{Coffrin} \\cite{Lavei} has pushed the state-of-the-art for solving AC-OPF for realistic networks, but these formulations did not include discrete variables. \nFor formulations that do include discrete control variables, several new approaches have been proposed \\cite{liu-penalty-function,Macfie-discrete-shunt,Capitanescu,Murray-discrete} to eliminate non-physical solutions and degradation of optimality due to rounding.\n\\cite{liu-penalty-function} proposes utilizing a penalty term in the objective function to push each relaxed control variable towards an available discrete value, so that the disturbance introduced by the rounding step is smaller.\nHowever, the use of penalty functions in optimizations with discrete variables can introduce stationary points and local minima \\cite{discrete-opt-overview}.\nAnother approach is to select subsets of relaxed discrete variables to round in an outer loop, while repeatedly solving the AC-OPF problem in between subsets until all of the settings have been fixed.\nIn \\cite{Macfie-discrete-shunt}, the authors present two methods for selecting which variables are rounded in each loop, and show these can reduce optimality degradation caused by rounding; however, unless only a single device is rounded at a time, this can introduce oscillations, since each rounding effectively adds a piecewise discontinuous function \\cite{Katzenelson}.\n\\cite{Capitanescu} uses sensitivities with respect to the discrete variables as metric to determine when to round settings, with the help of either a merit function or a MILP solver.\nIn \\cite{Murray-homotopy}, the authors point out that time constraints may make it impractical for grid operators to adjust a large number of control variable changes for a single dispatch.\nTo address this, they propose introducing a sparsity-inducing penalty term to the objective function along with a line-search of the discrete variable space to find a more limited number of control variable changes that can still improve optimality compared to holding all control variables constant.\nHowever, this method inherently assumes knowledge of good settings, which might not be available in some use cases for AC-OPF, such as planning studies.\n\nWe propose a two-stage homotopy algorithm for solving AC-OPF problems that explains how to incorporate discrete controls variables, can scale to large networks, and is robust to potential lack of knowledge of prior settings. \nOur approach builds off of the homotopy-based AC-OPF methodology presented in \\cite{Pandey-IMB}.\nIn the first stage,the discrete settings for the adjustable control devices are relaxed as continuous-valued variables and the optimization is solved using a robust homotopy technique.\nThe solution of the first stage is used to select discrete settings, and the respective variables are held constant thereafter. \nThen the errors induced by removing the relaxation are calculated and used in a second homotopy problem to locate a realistic solution.\n\nThe proposed novel approach can robustly determine a local optimum of any large real-world network with discrete controls without reliance on prior setting values. We also show that by choosing different homotopy paths, the proposed approach can obtain a variety of local minima solutions with significantly different generation dispatches.\nIn the results, we show that the method is not only novel in its approach but also more robust than another state-of-the-art optimization tool.\n\n\n\\section{Two Stage Homotopy Method for Solving \\ACOPFD}\n\\label{sec:methodology}\nWe propose a two-stage homotopy algorithm to solve the \\ACOPFDs problem described in Section \\ref{sec:formulation}.\nThe methods described are an extension of the IMB approach discussed above and we term the overall approach IMB+D\n\n\n\nThe first stage, Stage I, applies a relaxation to the \\ACOPFDs  problem in which we treat the discrete variables as continuous-valued by removing the integrality constraints. \nWe refer to the resulting relaxed optimization as \\ACOPFC, which is solved in Stage I.\nWe present a modeling framework for incorporating each adjustable device into the \\ACOPFC~ homotopy formulation so as to preserve the IMB concept of slowly ``turning on'' the grid as the sequence of sub-problems is traversed.\nAfter convergence of Stage I, in Stage II, the relaxed solution is used to select the nearest feasible discrete settings, and a second homotopy problem is defined to solve this problem. The local optima of this second stage yields an optimal solution for \\ACOPFD with the discrete value constraints satisfied. \n\n\n\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[width=2.8in]{figures\/Stage_One.png}\n    \\caption{A simple network with discrete devices in blue (a). In the completely relaxed network (b), the relaxation elements are shown in red. At the end of Stage I (c), a solution is found with all relaxations removed except continuous settings.}\n    \\label{fig:stageI_figure}\n\\end{figure}\n\\subsection{Homotopy Stage I: Solving the Relaxed \\ACOPFD}\n\\subsubsection{Embedding general network with homotopy}\nIn the constraints of an AC-OPF problem expressed using the current-voltage formulation, the current flows across lines are linear with respect to voltages, but the voltage and flow limit equations are quadratic, and the current injections from generators and loads are highly non-linear.\nEven without the introduction of additional control devices, Newton method's may fail to converge if the initial guess for variables is not within the basin of attraction of a solution \\cite{Murray-homotopy}. Therefore for large systems, without access to a reliable starting guess for Newton's method, we can ensure convergence using the IMB method presented in \\cite{Pandey-IMB}.\nBy starting with a deformed version of the network in IMB that has very little current flow, there exists a high voltage solution in which the bus voltages are close to one another, which is close to a flat start guess.\nTo define a deformed network and have a smooth trajectory of intermediate deformations back to the original network, in Stage I the homotopy factor $\\nu_1$ is embedded into the many of the parameters of the network's topology and devices: namely into high conductances in parallel to existing lines, the load factor, the generation limits (see Fig. \\ref{fig:stageI_figure}.b).\n\n\\subsubsection{Continuous relaxation and separation of discrete settings}\nIn Stage I, we apply a relaxation to the discrete control setting variables and split it into two components in order to leverage the robust methods of the IMB framework. \nFirst, a relaxation removing the integrality constraints (1d) is applied to replace the discrete variables $\\xd$ with a continuous vector, $\\xdcont$.\nAs a result, the constraints (1d) are replaced by:\n\\begin{equation}\n    \\xdlower^i \\leq \\xdcont^i \\leq \\xdupper^i \\text{, } i \\in 1...n_d\n\\end{equation}\nwhere $\\xdlower^i = \\min(\\mathcal{D}^i)$ and $\\xdupper^i = \\max(\\mathcal{D}^i)$, representing the minimum and maximum possible settings for a device.\n\nEven with this relaxation, adding devices that affect the flow of power across the network can significantly increase the nonlinearity of the constraint equations and the corresponding optimality conditions.\nFor example, introducing adjustable transformers causes the previously-linear transformer power flow model to be nonlinear with respect to voltage at its terminals, and adding phase shifters introduce trigonometric functions, and tap changers will introduce $\\frac{1}{\\tau}$ and $\\frac{1}{\\tau^2}$ terms.\nTo introduce these highly nonlinear models into the IMB framework, such that we preserve its initial trivial form where the entire grid is nearly shorted and a feasible homotopy path from $\\nu_1 = 1 \\rightarrow 0$, we design three measures.\n\nFirst, each relaxed setting variable $\\xdcont^i$ is separated into two components: a ``base'' value $\\xdbase^i$ and a continuous-valued ``adjustment'' variable $\\xdadjcont^i$:\n\\begin{equation}\n\\label{split}\n    \\xdcont^i = \\xdtriv^i + \\xdadjcont^i \n\\end{equation}\nSplitting $\\xdcont^i$ affects its initialization and bounding, but the total value still drives the respective device behavior. \nHowever, this step allows maintaining the solution space of the variable $\\xdadjcont$ around 0, which we have observed empirically improves the convergence in comparison to introducing $\\xdcont$ as variable directly;  we believe this could be due to improved search directions for the Newton's method as the partials are dependent on the value of control variables.\n\nSecond, to maintain the trivial shorted form of the IMB method at $\\nu_1 = 1$, we transform the discrete base setting $\\xdtriv^i$ to be homotopy dependent such that during early stages of homotopy it has almost no impact on the network solution. For example, a tap ratio of 1.0 p.u. on a transformer would have no affect on the system as a whole. We define this smooth deformation of the discrete base setting $\\xdtriv^i$ through embedding a homotopy factor $\\nu_1$:\n\\begin{equation}\n    \\xdbase^i = \\nu_1 \\xdtriv^i + (1-\\nu_1) \\xdbase_0^i\n\\end{equation}\n$\\xdtriv^i$ is chosen as a setting value that would ensure that the trivial solution for the $\\nu_1=1$ sub-problem is maintained. \n$\\xdbase_0^i$ must be chosen from within the feasible domain. A good prior setting can be used, but the median value can always be used if the user does not know one.\n\nLastly, we ensure the effective settings do not stray from their respective trivial values in the early sub-problems of Stage I by adding a homotopy-dependent penalty term to the objective function parameterized by a $\\nu_1$ and scaled by $k_{adj}$:\n\\begin{equation} \n    \\label{modified_objective_kadj}\n    f(x, \\xdadjcont) = f_0(x, \\xdadjcont) + \\nu_1 k_{adj}  \\sum_{i=1}^{N_d} |\\xdadjcont^i|^2\n\\end{equation}\nIncluding this term encourages minimizing $|\\xdcont^i|$ more strongly in the early stages of homotopy. However, as $\\nu_1$ is decreased, the penalty weight is reduced so that adjustment variables can move more freely as necessary to satisfy physical network constraints and decrease the primary objective function.\nNote that all penalty terms have been removed from the objective function when the \\ACOPFC ~is solved at $\\nu_1=0$, so the form of the final \\ACOPFC~ problem is independent of $k_{adj}$ value.\n\n\nTo address the likelihood of infeasible sub-problems along the homotopy path ($c(\\nu_1),  \\forall \\nu_1 \\in [0,1] $) where KCL cannot be satisfied without violating some variable limits, just as in IMB, homotopy dependent slack current injections (shown as red current sources in Fig. \\ref{fig:stageI_figure}.b) are defined at each node in the network to allow satisfaction of conservation of charge \\cite{Jereminov-feasibility}.\nThe magnitudes of the injection sources are penalized heavily in the objective function so that the sources only inject current if required for satisfaction of KCL, and the values are scaled by $\\nu_1$ so that the fictitious sources are removed entirely when $\\nu_1=0$.\n\n\\subsubsection{Solution of Stage I}\nFor each sub-problem defined in the homotopy path, the relaxed \\ACOPFC~is solved using the primal-dual interior point (PDIP) approach \\cite{Boyd}.\nThe Lagrangian for the sub-problem at any given $\\nu_1 \\in [0,1]$ is given by \n\\begin{equation}\n\\begin{aligned}\n    \\label{eq:lagrangian}\n    \\mathcal{L}^{\\nu_1}(\\xext, \\lambda,\\mu) = f_0^{\\nu_1}(\\xext) + \\lambda^T g^{\\nu_1}(\\xext) + \\mu^T h^{\\nu_1}(\\xext)\n\\end{aligned}\n\\end{equation}\nwhere $\\xext = [x, \\xdcont]^T$, $\\lambda$ is vector of dual variables for the equality constraints, and $\\mu$ is the vector of slack variables for the inequality constraints.\nA local minimizer, $\\theta^* = [\\xext^*, \\lambda^*, \\mu^*]$, is sought by using Newton's method to solve for the set of perturbed first order KKT conditions:\n\\begin{equation}\n\\label{eq:KKT}\n\\begin{split}\n&    \\mathcal{F}(\\theta)  = \\begin{bmatrix}\n\\nabla_{\\xext} f_0(\\xext) + \\nabla_{\\xext}^T g(\\xext) \\lambda + \\nabla_{\\xext}^T h(x_{ext}) \\mu \\\\\ng(\\xext) \\\\\n\\mu \\odot h(\\xext) + \\epsilon\n\\end{bmatrix} = 0 \\\\\n\\end{split}\n\\end{equation}\nwhere $\\odot$ is element-wise multiplication.\nIn order to facilitate convergence and primal-dual feasibility of the solution, heuristics based on diode-limiting from circuit simulation methods are applied \\cite{Pandey-Tx-Stepping}. \nThe located $\\theta^*$ is used as the initial guess for the next \\ACOPFC sub-problem once the homotopy parameter (and thus the network relaxation) has been updated. \nThis process is repeated until $\\nu_1=0$, at which point a solution has been found to \\ACOPFC (Fig. \\ref{fig:stageI_figure}.c).\n\n\\subsection{Homotopy Stage II: Discretization}\n\nAfter determining the optimal relaxed settings for the devices, we move onto the second stage of our approach, which solves for discrete value settings and the corresponding state of the grid using the relaxed solution from Stage I.\n\\subsubsection{Selection of discrete setting values}\nA practical discrete setting value must be chosen for each control device.\nFor each control variable, the nearest-neighbor discrete value to  is selected. The chosen setting is evaluated to estimate whether snapping the variable to this value might result in infeasibility that prevents convergence.\nThe sensitivities of bus voltages with respect to the setting are calculated and an approximate voltage after rounding is calculated using the sensitivity vector and a first-order Taylor approximation.\nWith predicted voltage values, we check inequality constraints affected by the setting's perturbation.\nIf the chosen rounded value resulted in an infeasibility or violation of bounds, then second closest available setting is chosen and checked.\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=2.8in]{figures\/Stage_Two.png}\n    \\caption{Process of using Stage I solution to discretize settings, formulate Stage II, and determine \\ACOPFD solution}\n    \\label{fig:Stage-Two-Networks}\n\\end{figure}\n\\subsubsection{Stage II Homotopy: Error Injections}\nAt the termination of the Stage I, we have a solution vector $\\theta^*$ that satisfies the perturbed KKT conditions for the continuous-valued OPF:\n\\begin{equation}\n    \\label{eq:KKT-sol-1}\n    \\mathcal{F}(\\theta^{*}) = 0\n\\end{equation}\nBy changing the setting variables from their converged continuous values to realistic setting values ($\\xdadjcont \\rightarrow \\xdadj$), the state vector is altered from $\\theta^*$ to $\\theta^\\prime$.\nBecause of the adjustments, evaluating the KKT conditions at $\\theta^\\prime$ will result in violations of the conditions, which we will refer to as residual vector $R$:\n\\begin{equation}\n    \\label{eq:KKT-sol-res}\n    \\mathcal{F}(\\theta^\\prime) = R\n\\end{equation}\nIn the case of primal variables, we can think of R as a set of independent current sources that compensates for the current mismatch at each node due to rounding discrete device settings.\nThis idea can be extended to dual variables as well, as the underlying equations and nonlinearities have the same form \\cite{Network-Stepping}.\nTherefore, in the network disturbed by the rounding step, the power flow constraints can be satisfied immediately after rounding by adding a current sources to each of these perturbed buses, with current injection values defined by the current mismatches caused by rounding (see Fig. \\ref{fig:Stage-Two-Networks}.b).\nIn a relaxed problem where we seek a solution to the same network but with the addition of $R$ to the respective equations, then we already know a solution to the problem: $\\theta^\\prime$.\n\\begin{equation}\n    \\mathcal{G}(\\theta^\\prime) = \\mathcal{F}(\\theta^\\prime) - R = 0\n\\end{equation}\nTherefore, we propose a second homotopy stage to find a feasible solution to the \\ACOPFDs problem after control variables have been rounded, in which the relaxed system of equations are KKT conditions for the \\ACOPFDs equations \\eqref{eq:KKT} after rounding but with $R$ added as error injections.\n\n\\begin{equation}\n\\begin{split}\n    \\mathcal{H}_2(x,\\nu_2) = (1-\\nu_2)\\mathcal{F}(\\theta) + \\nu_2(\\mathcal{F}(\\theta) - R) = 0\n\\end{split}\n\\end{equation}\nHere $\\nu_2$, the Stage II scalar homotopy factor, is used to gradually reduce the residual injections, tracing a continuous path to a feasible solution for $\\mathcal{F}(\\theta)$ with rounded values.\nThis avoids taking a discontinuous jump between solving \\ACOPFC and \\ACOPFD.\nWhen this homotopy problem is solved at $\\nu_2=0$, the error injections have been removed, and we have located a realistic, feasible solution to the \\ACOPFD.\n\n\\subsection{Generalization beyond IMB}\nWhile the two-stage approach in this paper is described as an extension of the larger IMB framework \\cite{Pandey-IMB} which assumes no knowledge of prior system settings, the two-stage homotopy algorithm can be applied to other approaches for solving AC-OPF without loss of generality. \nConsider the scenario in which good initial conditions for the general network are available but optimal settings for discrete variables ($\\xd^*$) are unknown.\nSuch a use-case may occur if a grid planner wishes to re-evaluate existing settings for discrete devices (e.g. aims to move from a feasible setting $(\\xd^k)$ to an optimal feasible setting $(\\xd^*)$), or to explore the effects of upgrading of fixed devices to adjustable devices). \nIn this situation, the grid planner may want to start from a feasible discrete setting but still explore whether a more optimal operating setting exists. In this scenario, we would still separate and relax each discrete setting $\\xd^i$ according to \\eqref{split}. \nBut here, we account for this knowledge of good setting in Stage I of the algorithm by defining base values using the previously known setting $\\xd^{k,i}$ such that $\\xdbase^{i} = \\xd^{k,i}$.\n\nTo ensure finding a solution at $\\nu_1=1$ is simple, the objective function is modified according to \\eqref{modified_objective_kadj}, but a very high penalty $k_{adj}$ value is used.\nNow at $\\nu_1=1$, with access to good initial conditions and feasible initial discrete settings, a trivial solution is obtained first.\nHowever, as we traverse the homotopy path from $\\nu_1 = 1 \\rightarrow 0$, the adjustment values for discrete settings take  according to the objective $f_0(x, \\xdadjcont)$ and eventually an optimal set of relaxed discrete settings are obtained at $\\nu_1 = 0$. \nTo find a feasible discrete setting, we perform Stage II as described in Section IV.B without modification. \n\n\t    \n\t\t\n\n\\section{Appendix: Additional Results}\nTable III contains a larger set of results obtained using the \\ACOPFDs approach. For all these simulations a $k_{adj}$ value of 0.1 was used. The case file can be found at \\cite{cases-github}. \n\\textit{Note: this table was omitted from the original PSCC submission due to space constraints.}\n\\bigbreak\n\n\\begin{table}[h]\n\\onecolumn\n\\begin{center}\n\\caption{Additional \\ACOPFDs results}\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline\n\\textbf{Case} & \\textbf{Buses} & \\textbf{Tap Changers} & \\textbf{Phase Shifter} & \\textbf{Switched Shunt} & \\textbf{Objective} & \\textbf{\\% Adj} \\\\ \\hline\nA  & 3022  & 981  & 4  & 399  & 5.377e5 & 24.9 \\% \\\\ \\hline\nA* & 3022  & 981  & 4  & 399  & 5.357e5 & 25.6 \\% \\\\ \\hline\nB  & 6867  & 759  & 5  & 161  & 1.216e5 & 90.8 \\% \\\\ \\hline\nB* & 6867  & 759  & 5  & 161  & 1.216e5 & 93.9 \\% \\\\ \\hline\nC  & 11152 & 530  & 0  & 500  & 5.294e5 & 77.8 \\% \\\\ \\hline\nC* & 11152 & 530  & 0  & 500  & 5.294e5 & 82.2 \\% \\\\ \\hline\nD  & 16789 & 997  & 2  & 1723 & 3.592e5 & 74.7 \\% \\\\ \\hline\nD* & 16789 & 997  & 2  & 1723 & 3.591e5 & 89.2 \\% \\\\ \\hline\nE  & 6549  & 1846 & 3  & 436  & 9.774e4 & 80.7 \\% \\\\ \\hline\nE* & 6549  & 1846 & 3  & 436  & 9.777e4 & 91.1 \\% \\\\ \\hline\nF  & 14393 & 0    & 5  & 724  & 2.317e4 & 33.7 \\% \\\\ \\hline\nF* & 14393 & 0    & 5  & 724  & 2.312e4 & 33.2 \\% \\\\ \\hline\nG  & 21849 & 997  & 2  & 1710 & 1.879e5 & 60.9 \\% \\\\ \\hline\nG* & 21849 & 997  & 2  & 1710 & 1.879e5 & 66.4 \\% \\\\ \\hline\nH  & 31156 & 12   & 36 & 2451 & 2.009e5 & 76.3 \\% \\\\ \\hline\nH* & 31156 & 12   & 36 & 2451 & 2.009e5 & 63.9 \\% \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\end{document}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nIntelligent reflecting surface (IRS) is an artificial planar array consisting of numerous reconfigurable passive elements with the capability of manipulating the impinging electromagnetic signals and offering anomalous reflections \\cite{Tan2018SRA,cuiTJ2017metasurface,Larsson2020Twocritical,Garcia2020JSACIRS_gap_scatter_reflect}.\nMany recent studies have indicated that IRS is a promising solution to build a programmable wireless environment via steering the incident signal in fully customizable ways  to enhance the spectral and energy efficiency of legacy systems \\cite{RenzoJSACposition,magzineWuqq,Liaskos2018magzineIRS,Renzo2019position}.\nMost contributions in this area focus on joint active and passive procoding design with various objectives and constraints \\cite{Wuq2019TWCprecoderIRS,HY2020TWCprecoderIRS,mux2020TWCprecoderIRS,ZhouG2020TSPIRSprecoderimperfectCE,LinS2021TWCprecoderIRS}.\nThe potential gains claimed by these works highly depend on the availability of accurate channel state information (CSI).\nHowever, channel estimation is a challenging task for the IRS-assisted system because there are no sensing elements or radio frequency  chains, and thus there is no baseband processing capability in the IRS.\n\n\n\n\n\n\\begin{figure}\n[!t]\n\\centering\n\\includegraphics[width=.8\\columnwidth]{model_v2.eps}\n\\caption{The IRS-assisted  multiuser MISO communication system.}\n\\label{IRS_system}\n\\end{figure}\n\n\\begin{figure}\n[!t]\n  \\centering\n  \\subfigure[Protocol for SU-MISO system]{\n    \\label{protocol:a}\n    \\includegraphics[width=.6\\columnwidth]{protocol_0.eps}}\n  \\subfigure[Directly extended protocol for MU-MISO system]{\n    \\label{protocol:b}\n    \\includegraphics[width=.6\\columnwidth]{protocol_1.eps}}\n  \\caption{The conventional uplink channel estimation protocols without exploiting the common-link structure.}\n  \\label{protocol_tran}\n\\end{figure}\n\nSome early-attempted works \\cite{Jensen2020SUCE,zhangruiJSACSUCE,YouxhSPLCESU} estimate the uplink cascaded IRS channel  for single user (SU) multiple-input-single-output (MISO) systems using the protocol shown in {\\figurename~\\ref{protocol:a}}.\nIn these works, the cascaded channel is equivalently represented as a traditional $M\\times N$ MIMO channel, where $M$ and $N$ is the base station (BS) array size and the IRS size, respectively, and the sensing matrix for channel reconstruction consists of the phase shifting vectors in consecutive training timeslots.\nMany works \\cite{Araujo2021JSAC_CE_PARAFAC,Mishra2019CEonoff,Elbir2020WCL_DL_CE,\nZhouzy2020decompositionCE,Kundu2021OJCSLMMSE_DFTGOOD,Alwazani2020OJCSLMMSE_DFT} directly extend the SU protocol to multi-user (MU) MISO systems, in which $K$ users transmit orthogonal pilot sequences in each training timeslot, as shown in {\\figurename~\\ref{protocol:b}}.\nBased on this protocol,\nthe on-off IRS state (amplitude) control strategy is proposed in \\cite{Araujo2021JSAC_CE_PARAFAC,Mishra2019CEonoff,Elbir2020WCL_DL_CE} to better decompose the MU cascaded channel coefficients   for easier channel estimation of the cascaded channel for each user.\n\nIt is pointed out by \\cite{Liuliang_CE2020TWC} that direct application of the SU protocol on an MU-MISO system fails to exploit the structural property and results in substantially larger pilot overheads.\nIntuitively, all the cascaded channels share a common BS-IRS link and it is possible to reduce the pilot overhead since the number of independent variables is $MN+NK$ instead of $MNK$.\nOne algorithm is proposed in \\cite{DaiLLFullD} with the idea of sequentially estimating the BS-IRS channel and the IRS-user channels.\nHowever, it requires that the BS can work at full-duplex mode.\nFor the cascaded channel estimation,\na new channel estimation protocol is proposed in \\cite{Liuliang_CE2020TWC}.\nSpecifically, the cascaded channel of one reference user is firstly estimated based on the SU protocol, and then other users' channels are estimated by only estimating the ratios of their channel coefficients to the reference channel, which can be referred to as the relative channels.\nThe overall training overhead is reduced from $NK$ to $K+N+\\lceil \\frac{N}{M}\\rceil(K-1)$.\nHowever, there is an error propagation issue associated with this scheme  since a low-accuracy estimation on the reference  channel may jeopardize  the estimations of the relative channels.\nMoreover, some IRS elements need to be switched off while estimating the relative channels for coefficients decomposition \\cite{Liuliang_CE2020TWC}.\n\nFor IRS design, the ``off'' state means no reflection (i.e., perfectly absorbing the incident signals),  and hence  it is difficult \\cite{Perfect_Absorption1,Perfect_Absorption2,Perfect_Absorption3} and also attracts additional implementation costs since this state is unnecessary for data transmission after the channel estimation.\nIn addition, switching off the IRS elements causes reflection power loss, which will lower the receive signal-to-noise ratio (SNR).\nSome recent works attempt to overcome this issue using ``always-ON'' training schemes.\nIn \\cite{Zhouzy2020decompositionCE,Kundu2021OJCSLMMSE_DFTGOOD,Alwazani2020OJCSLMMSE_DFT}, cascaded channel estimation algorithms based on tensor decomposition  are proposed for MU-MISO systems, without requiring selected IRS elements to be off using the protocol in {\\figurename~\\ref{protocol:b}}.\nIn particular, the training phase shifts are optimized to minimize the mean squared error (MSE), and it has been verified that the discrete-Fourier-transform (DFT)-based training phase shifting configuration is optimal in this scenario.\nHowever, the pilot overhead is $NK$ since the protocol in {\\figurename~\\ref{protocol:b}} does not utilize the\ncommon-link property.\nIn \\cite{double_IRS}, an always-ON training scheme is proposed, which extends the protocol in \\cite{Liuliang_CE2020TWC} to the double-IRS aided system.\nHowever, the number of BS antennas needs to be equal to or larger than the number of IRS elements (i.e., $M\\geq N$) to guarantee a full-rank measurement matrix to estimate the relative channels between the reference user and the other users.\\footnote{\nAccording to the property ${\\rm{rank}}({\\bf A} \\otimes{\\bf B})={\\rm{rank}}({\\bf A} ){\\rm{rank}}({\\bf B} )$, the rank of the measurement matrix in equation (40) of \\cite{double_IRS} cannot be larger than $(K-1)M$ while the targeted rank is $(K-1)N$.\n}\nThis assumption is quite restrictive as the number of elements of the IRS ($N$) is usually larger than the number of antennas at the BS.\n\n\n\nAnother critical problem is the feasibility issue when the channel statistical prior information is utilized to improve the channel estimation accuracy,\nalthough this is a common idea for conventional MIMO channel estimation.\nIn \\cite{Kundu2021OJCSLMMSE_DFTGOOD} and \\cite{Alwazani2020OJCSLMMSE_DFT}, statistical knowledge of the individual BS-IRS link and IRS-user links is required. However, these messages are not available in practice since\nnone of the existing algorithms, to the best of the authors' knowledge, can reconstruct the individual channel coefficients when $M<N$ due to the ambiguity issue, which introduces a random scaling on the estimated individual channels \\cite{Araujo2021JSAC_CE_PARAFAC,Hezqwcl2020IRS_CE_SU}.\nIn \\cite{Liuliang_CE2020TWC}, the linear minimum mean squared error (LMMSE) estimator is adopted to estimate the relative channels. However, the LMMSE estimator requires the covariance of the relative channels, which is also difficult since in general the distribution of the relative channels is heavy-tailed.\n\n\n\n\n\n\n\nIn this paper, we focus on the uplink cascaded IRS channel estimation for an MU-MISO system in sub-6 GHz bands.\nIn these frequency bands, the IRS design will be easier, but the channel propagation is not sparse.\\footnote{If the IRS operates at millimeter wave frequencies, some sparse channel estimation algorithms have been further proposed based on the protocol in {\\figurename~\\ref{protocol:b}} with less training timeslots   by exploiting the potential channel sparsity \\cite{Hezqwcl2020IRS_CE_SU,DaiLL2021CLpartII,Wangpl2020splCS_CE,Ardah2021splCS_TRICE,YuanXJ2020JSAConoffsparseCE,LClearning}.\n}\nIn addition, the number of IRS elements in the system can be larger than the number of BS antennas (i.e., $N>M$).\nWe propose a holistic solution to address the aforementioned issues.\nIn particular, a novel always-ON training protocol is designed; meanwhile the common-link structure is utilized to reduce the pilot overhead.\nFurthermore, an optimization-based cascaded channel estimation framework, which is flexible to utilize more practical channel statistical prior information, is proposed.\nThe following summarizes our key contributions.\n\\begin{itemize}\n\\item  {\\bf Always-ON Channel Estimation Protocol Exploiting the Common-Link Structure}:\n    We propose a novel channel estimation protocol without the need for on-off amplitude control to avoid the reflection power loss.\n    Meanwhile, the common-link structure is exploited and the pilot overhead is reduced to $K+N+\\lceil \\frac{N}{M}\\rceil(K-1)$.\n    \n\n    In addition, the proposed protocol is applicable with any number of  elements at the IRS (also including $N \\leq M$).\n    Further, it does not need a ``reference user'', and as such, the estimation performance is enhanced owing to the multiuser diversity.\n\n\n\\item  {\\bf Optimization-Based Cascaded Channel Estimation Framework}:\n    Since there is no on-off amplitude control, the cascaded channel coefficients are highly coupled.\n    In order to exploit the  common-link structure, we decompose the cascaded channel coefficients by the multiplication of the common-link variables and the user-specific variables, and then an optimization-based joint channel estimation problem is formulated based on the maximal a posterior probability (MAP) rule. The proposed optimization-based approach is flexible to incorporate different kinds of channel statistical prior setups. Specifically, we utilize the combined statistical information of the cascaded channels, which is a weaker requirement compared to statistical knowledge of the individual BS-IRS and IRS-user channels.\n    Then, a low-complexity alternating optimization algorithm is  proposed to achieve a local optimum solution.\n    Simulation results demonstrated that the optimization solution with proposed protocol achieves a more than $15$ dB gain compared to the benchmark.\n   \n\n\n\\item  {\\bf Training Phase Shifting Optimization for the Proposed Protocol}:\n    The phase shifting configuration can substantially enhance the channel estimation performance of the cascaded IRS channel because the phase shifting vectors are important components in the measurement matrix for channel reconstruction.\n    However, traditional solutions \\cite{Zhouzy2020decompositionCE,Kundu2021OJCSLMMSE_DFTGOOD,Alwazani2020OJCSLMMSE_DFT} of phase shifting optimization for SU cascaded IRS channel estimation cannot be directly applied to the MU case due to that  the cascaded channel coefficients are highly coupled when the common-link structure is exploited.\n    We propose a new formulation to optimize the phase shifting configuration, which maximizes the average reflection gain of the IRS.\n    Simulation results further verify the proposed configuration achieves a more than $3$ dB gain compared to the state-of-the-art  baselines.\n   \n\n\n\\end{itemize}\n\n\n\n\n\\section{System Model}\\label{system model}\n\n\n\\subsection{System Model of MU-MISO IRS Systems}\nThis paper investigates the uplink channel estimation in a narrow-band IRS-aided MU-MISO communication system that consists of one BS with $M$ antennas, one IRS with $N$ elements, and $K$ single-antenna users,\\footnote{\n{We adopt the single-antenna-user setup here for ease of presentation. The signal model can be directly extended to the setup when users have multiple antennas by transmitting orthogonal uplink pilot sequences in different antennas.\n}}\nas illustrated in {\\figurename~\\ref{IRS_system}}.\nLet ${\\bf h}_{{\\rm d},k} \\in {\\mathbb C}^{M \\times 1}$ denote the BS-user channel (a.k.a., the direct channel) for user $k$, ${\\bf G} \\in {\\mathbb C}^{N \\times M}$ denote the common BS-IRS channel, and ${\\bf h}_{{\\rm r},k} \\in {\\mathbb C}^{N \\times 1}$ denote the IRS-user channel for user $k$.\nWe assume  quasi-static block fading  for all the channels such that the channel coefficients remain constant within one channel coherence interval, and they are independent and identically distributed (i.i.d.) between coherence intervals.  Note that the quasi-static model considers the worse case scenarios where the temporal correlations between blocks are not exploited. In practice, the pilot overheads can be further reduced if one exploits temporal correlations of the channel blocks \\cite{kalman_filter_IRS2021TVT,kalman_filter_IRS2021chinacom}, but this is outside the scope of the paper.\n\nThe received baseband signal at the BS is given by\n\\begin{equation}\\label{equ:y_model1}\n\\begin{aligned}[b]\n{\\bf y}_{t}&=\\sum_{k=1}^K \\left({\\bf h}_{{\\rm d},k}+ {\\bf G}^{\\rm T} {\\bf \\Theta}_t  {\\bf h}_{{\\rm r},k} \\right) x_{k,t}\n+{ {\\bf z}_t}\n,\n\\end{aligned}\n\\end{equation}\nwhere $t$ is the time index, $x_{k,t}$ is the transmit pilot symbol from user $k$, ${\\bf z}_t \\sim {\\cal{CN}}({\\bm 0},\\sigma_0^2 {\\bf I}_{M}) $ is the {additive white Gaussian noise} (AWGN), and ${\\bf \\Theta}_t \\in {\\mathbb C}^{N \\times N}$ is the IRS reflection coefficient matrix.\nIt is known that ${\\bf \\Theta}_t$ is a diagonal matrix such that ${\\bf \\Theta}_t={\\rm diag}({\\bm \\theta}_t)$, where ${\\bm \\theta}_t=[e^{\\jmath \\varphi_{t,1}},e^{\\jmath \\varphi_{t,2}},\\cdots,e^{\\jmath \\varphi_{t,N}}]^{\\rm T}$ is the phase shifting vector from the IRS.\\footnote{\n{In practice, the reflection efficiency cannot be 1, which is known as the reflection loss. However, for ease of presentation, this loss can be absorbed into the path loss of ${\\bf G}$ since it is a constant value.\n}}\n\n\n\n\n\n\n\n\\subsection{IRS Cascaded Channel Model}\\label{sec:model_cascaded}\n\nDenote the cascaded channel related to the BS's $m$-th antenna and the $k$-th user by\n\\begin{equation}\\label{equ:cascaded_channel_vector}\n\\begin{aligned}[b]\n{\\bf h}_{{\\rm I},k,m}={\\rm{diag}}({\\bf h}_{{\\rm r},k}) {\\bf g}_m,\n\\end{aligned}\n\\end{equation}\nwhere ${\\bf g}_m$ is the $m$-th column in ${\\bf G}=[{\\bf g}_1,\\cdots,{\\bf g}_M]$.\nThe cascaded channel over all BS antennas is given by ${\\bf H}_{{\\rm I},k}=[{\\bf h}_{{\\rm I},k,1}^{\\rm T},\\cdots,{\\bf h}_{{\\rm I},k,M}^{\\rm T}]^{\\rm T}$, and we have\n\\begin{equation}\\label{equ:cascaded_channel}\n\\begin{aligned}[b]\n{\\bf H}_{{\\rm I},k}={\\bf G}^{\\rm T} {\\rm{diag}}({\\bf h}_{{\\rm r},k}).\n\\end{aligned}\n\\end{equation}\nSubstituting \\eqref{equ:cascaded_channel} into \\eqref{equ:y_model1}, the received signal is given by\n\\begin{equation}\\label{equ:y_model_vn}\n\\begin{aligned}[b]\n{\\bf y}_t&=\\sum_{k=1}^K \\left({\\bf h}_{{\\rm d},k}+ {\\bf H}_{{\\rm I},k} {\\bm{\\theta}}_t \\right) x_{k,t}\n+{\\bf z}_t\n.\n\\end{aligned}\n\\end{equation}\n\nIn \\cite{Araujo2021JSAC_CE_PARAFAC,Mishra2019CEonoff,Elbir2020WCL_DL_CE} and\n\\cite{Zhouzy2020decompositionCE,Kundu2021OJCSLMMSE_DFTGOOD,Alwazani2020OJCSLMMSE_DFT}, ${\\bf H}_{{\\rm I},k}$ for all $k$  are estimated\nwithout exploiting the implicit common link structure behind the ${\\bf H}_{{\\rm I},k}$ for all $k$. As a result, there are $MNK$ variables to be estimated and this poses a heavy penalty on the required pilot overheads in the MU-MISO system.\nOn the other hand, from \\eqref{equ:cascaded_channel}, we can see that the cascaded channels $\\left\\{{\\bf H}_{{\\rm I},1}, {\\bf H}_{{\\rm I},2},\\cdots,{\\bf H}_{{\\rm I},K}\\right\\}$ all share a common BS-IRS link $\\bf G$.\nSpecifically,\n ${\\bf H}_{{\\rm I},k}$ is the multiplication of the common $\\bf G$ and the user-specific ${\\rm{diag}}({\\bf h}_{{\\rm r},k})$.\nIn other words, the cascaded channels are not independent variables. In fact,  the common link structure $\\bf G$ should be exploited in the channel estimation.\nAs a result, the total number of independent variables\nis reduced to $MN+NK$.\n\n\nWe assume ${\\mathbb E}\\left[ {\\bf h}_{{\\rm I},k,m}\\right]={\\bf 0}$ for all $k$ and $m$.\\footnote{\n{The proposed algorithm in this paper is applicable for the case when an LoS link exists and ${\\mathbb E}\\left[ {\\bf h}_{{\\rm I},k,m}\\right]\\neq{\\bf 0}$. Simply substitute ${\\mathbb E}\\left[ {\\bf h}_{{\\rm I},k,m}\\right]$ into the prior distribution model in \\eqref{equ:prior}. Note that since the rank-1 LoS link usually is very strong and easier to be estimated, it will dominate the power of the channel coefficients, and the NMSE will be better than the NLoS scenario investigated in this paper.\n}}\nThe covariance of the cascaded channel ${\\bf h}_{{\\rm I},k,m}$ is given by\n\\begin{equation}\\label{equ:cascaded_channel_statisical}\n\\begin{aligned}[b]\n{\\bf C}_m^{(k)}={\\mathbb E}\\left[ {\\bf h}_{{\\rm I},k,m} {\\bf h}_{{\\rm I},k,m}^{\\rm H} \\right].\n\\end{aligned}\n\\end{equation}\nIn this paper, we focus on the case when ${\\bf C}_m^{(k)}$ is a full rank matrix for all $m$ and $k$, which is generally true in sub-6 GHz bands.\nWe  design a channel estimation algorithm and phase shifting configuration scheme by utilizing knowledge of  ${\\bf C}_m^{(k)}$.\nNote that in \\cite{Kundu2021OJCSLMMSE_DFTGOOD} and \\cite{Alwazani2020OJCSLMMSE_DFT}, the channel estimation algorithms for the cascaded channels require knowledge of the covariance of the IRS-BS link $\\bf G$ as well as the covariance of the IRS-user links ${\\bf h}_{{\\rm r},k}$. Also note that knowledge of the covariance of the cascaded channel ${\\bf C}_m^{(k)}$ is a weaker requirement compared to knowledge of the individual covariances $\\bf G$ and ${\\bf h}_{{\\rm r},k}$.\n\n\n\n\n\n\n\n\n\\section{Proposed Channel Estimation Protocol}\\label{dense_scheme}\n\n\n\n\n\\subsection{Overview of the Selected On-Off Channel Estimation Protocol}\\label{overview_onoff}\nThe selected on-off channel estimation protocol in \\cite{Liuliang_CE2020TWC} is illustrated in {\\figurename~\\ref{protocol:c}}, and consists of three stages.\nIn stage I, the BS-user channels are estimated by switching off all the IRS elements.\nIn stage II, a reference user is selected, which is indexed by user $1$, and its cascaded channel ${\\bf H}_{{\\rm I},1}$ is estimated using the algorithm for SU-MISO cases \\cite{zhangruiJSACSUCE}.\nIn stage III, the other $K-1$ users' cascaded channels are estimated by exploiting the common-link property.\n\n\\begin{figure}\n[!ht]\n\\centering\n\\includegraphics[width=.95\\columnwidth]{protocol_2.eps}\n\\caption{Selected on-off channel estimation protocol in \\cite{Liuliang_CE2020TWC}.}\n\\label{protocol:c}\n\\end{figure}\n\n\nWe focus on the estimation in stage III. Substituting ${\\bf H}_{{\\rm I},1}={\\bf G}^{\\rm T} {\\rm{diag}}({\\bf h}_{{\\rm r},1})$ into \\eqref{equ:y_model1}, the received signal is given by\n\\begin{equation}\\label{equ:y_model_LL}\n\\begin{aligned}[b]\n{\\bf y}_t&=\\sum_{k=1}^K {\\bf h}_{{\\rm d},k} x_{k,t}+\n{\\bf H}_{{\\rm I},1} {\\bm{\\theta}}_t x_{1,t}\\\\\n&\\qquad+\\sum_{k=2}^K {\\bf H}_{{\\rm I},1} {\\rm diag}({\\bm \\theta}_t)  {\\bf h}_{{\\rm u},k}  x_{k,t}\n+{ {\\bf z}_t}\n,\n\\end{aligned}\n\\end{equation}\nwhere ${\\bf h}_{{\\rm u},k} ={\\rm diag}({\\bf h}_{{\\rm r},1})^{-1}{\\bf h}_{{\\rm r},k}$ for all $k=2,3,\\cdots,K$, which are the user-specific variables to be estimated in this stage after exploiting ${\\bf H}_{{\\rm I},1}$ as the common-link variable.\nIn \\cite{Liuliang_CE2020TWC}, to estimate ${\\bf h}_{{\\rm u},k}$, only the $k$-th user sends $x_{k,t}=1$ and all the other users are inactive such that $x_{j,t}=0$ for all $j \\neq k$.\nThe received signal is given by\n\\begin{equation\n\\begin{aligned}[b]\n{\\bf y}_\n&={\\bf h}_{{\\rm d},k}+ {\\bf H}_{{{\\rm I},1}} {\\rm diag}({\\bm \\theta}_t)  {\\bf h}_{{\\rm u},k}\n+ {\\bf z}_t\n.\n\\end{aligned}\n\\end{equation}\nBy only switching on the first $M$ IRS elements with $[\\theta_{t,1},\\theta_{t,2},\\cdots,\\theta_{t,M}]^{\\rm T}={\\bf 1}$, the first $M$ coefficients in ${\\bf h}_{{\\rm u},k}$ can be estimated by\n\\begin{equation\n\\begin{aligned}[b]\n\\begin{bmatrix}{h}_{{\\rm u},k,1}\\\\ \\vdots \\\\ { h}_{{\\rm u},k,M}\\end{bmatrix}\n=\n{\n\\begin{bmatrix}\n    H_{{{\\rm I},1},11} & \\dots & H_{{{\\rm I},1},1M}\\\\\n    \\vdots  & \\ddots & \\vdots\\\\\n    H_{{{\\rm I},1},M1} & \\dots & H_{{{\\rm I},1},MM}\n\\end{bmatrix}\n}^{-1}\n({\\bf y}_i-{\\bf h}_{{\\rm d},k})\n.\n\\end{aligned}\n\\end{equation}\nNote that one may adopt the LMMSE estimator to achieve better performance if the covariance of ${\\bf h}_{{\\rm u},k} ={\\rm diag}({\\bf h}_{{\\rm r},1})^{-1}{\\bf h}_{{\\rm r},k}$ is available \\cite[Section V]{Liuliang_CE2020TWC}.\nIn the next timeslot, the next $M$ IRS elements are switched on with $[\\theta_{t,M+1},\\theta_{t,M+2},\\cdots,\\theta_{t,2M}]^{\\rm T}={\\bf 1}$ while the other elements are switched off to estimate the next $M$ coefficients in ${\\bf h}_{{\\rm u},k}$. The estimation continues in this way until all the coefficients in ${\\bf h}_{{\\rm u},k} $ are estimated, which finally costs $I=\\lceil \\frac{N}{M}\\rceil$ timeslots.\nThe overall pilot overhead of the protocol in \\cite{Liuliang_CE2020TWC} is $K+N+\\lceil \\frac{N}{M}\\rceil(K-1)$.\n\n\n\\subsection{Always-ON Channel Estimation Protocol}\\label{proposed_protocol}\nWe propose a novel always-ON channel estimation protocol without switching off\nselected IRS elements. The proposed protocol consists of two stages, as illustrated in {\\figurename~\\ref{protocol:d}}.\nIn particular, stage I contains $L_1+1$ timeslots, where $L_1 = \\lceil \\frac{N}{M}\\rceil$, and each timeslot contains $K$ samples. Stage II contains $L_2=N-L_1$ timeslots, and each timeslot contains only one sample.\nTherefore, we have $K(L_1+1)+L_2$ received samples in total. As defined in \\eqref{equ:y_model1}, the $t$-th received sample is given by\n\\begin{equation}\\label{equ:y_model2}\n\\begin{aligned}[b]\n{\\bf y}_{t}&=\\sum_{k=1}^K \\left({\\bf h}_{{\\rm d},k}+ {\\bf G}^{\\rm T} {\\rm diag} ({\\bm \\theta}_t)  {\\bf h}_{{\\rm r},k} \\right) x_{k,t}\n+{ {\\bf z}_t}\n,\n\\end{aligned}\n\\end{equation}\nwhere $t=1,2,\\cdots,K(L_1+1)+L_2$.\n\n\n\n\\begin{figure}\n[!ht]\n\\centering\n\\includegraphics[width=.8\\columnwidth]{protocol_3.eps}\n\\caption{Proposed always-ON protocol.}\n\\label{protocol:d}\n\\end{figure}\n\n\n\\subsubsection{Received Signal Samples in Each Training Timeslot}\nTo facilitate analysis, we introduce new notations on the received signal samples within one training timeslot indexed by $\\ell=0,1,\\cdots,N$.\n\\begin{itemize}\n\\item\n{{\\bf {Stage I}} (Users send orthogonal pilot sequences ${\\bf X}$)}:\nStage I consists of training timeslots $\\ell=0,1,\\cdots,L_1$.\nDenote by ${\\bf X} \\in {\\mathbb C}^{K \\times K}$,  where ${\\bf X}^{\\rm H} {\\bf X}=K{\\bf I}_K$, the orthogonal pilot sequences consisting of unit-modulus elements.\nIn the $\\ell$-th timeslot, $K$ users transmit ${\\bf X}$ by $K$ samples, and the IRS is configured by the phase  shifting vector ${\\bm \\theta}_\\ell$.\nThe $K$ received samples in the $\\ell$-th timeslot are ${\\bf y}_{K\\ell+1},{\\bf y}_{K\\ell+2},\\cdots,{\\bf y}_{K\\ell+K}$.\nLet ${{\\bf Y}}_{\\ell}=[{\\bf y}_{K\\ell+1},{\\bf y}_{K\\ell+2},\\cdots,{\\bf y}_{K\\ell+K}]$. We have\n\\begin{equation}\\label{equ:y_model_stage_I}\n\\begin{aligned}[b]\n{{\\bf Y}}_{\\ell}&=\\left({\\bf H}_{\\rm d} + {\\bf G}^{\\rm T} {\\rm diag} ({\\bm \\theta}_{\\ell}) {\\bf H}_{\\rm r} \\right)  {\\bm X}\n+{{\\bf Z}}_{\\ell}\n,\n\\end{aligned}\n\\end{equation}\nwhere\n\\begin{align}\n{\\bf H}_{\\rm d}&=[{\\bf h}_{{\\rm d},1},\\cdots,{\\bf h}_{{\\rm d},K}] \\in {\\mathbb C}^{M \\times K},\\\\\n{\\bf H}_{\\rm r}&=[{\\bf h}_{{\\rm r},1},\\cdots,{\\bf h}_{{\\rm r},K}] \\in {\\mathbb C}^{N \\times K}\n\\end{align}\nare the stacked channel coefficient matrices,\n$\\ell$ is the timeslot index,\nand ${{\\bf Z}}_{\\ell}=[{\\bf z}_{K\\ell+1},{\\bf z}_{K\\ell+2},\\cdots,{\\bf z}_{K\\ell+K}]$ denotes the noise.\n\n\n\\item {{\\bf {Stage II}} (Users send pilot $\\bar{\\bf x}$, which is the first column of $\\bf X$)}:\nStage II consists of training timeslots $\\ell=L_1+1,L_1+2,\\cdots,N$.\nWe denote the first column of $\\bf X$ by\n\\begin{equation}\\label{equ:barx}\n\\bar{\\bf x}=[{\\bar x}_{1},\\cdots,{\\bar x}_{K}]^{\\rm T}.\n\\end{equation}\nIn the $\\ell$-th timeslot, the users  transmit  $\\bar{\\bf x}$, while the IRS is configured by ${\\bm \\theta}_\\ell$.\nThe received signal in stage II is denoted by\n\\begin{equation}\\label{equ:y_model_stage_III}\n\\begin{aligned}[b]\n\\bar{\\bf y}_{\\ell}\n&= \\left({\\bf H}_{\\rm d} + {\\bf G}^{\\rm T} {\\rm diag} ({\\bm \\theta}_\\ell) {\\bf H}_{\\rm r} \\right)  \\bar{\\bf x}+{\\bf z}_{\\ell+(K-1)L_1+K}\n,\n\\end{aligned}\n\\end{equation}\nwhere $\\bar{\\bf y}_{\\ell}={\\bf y}_{\\ell+(K-1)L_1+K}$, and $\\ell=L_1+1,L_1+2,\\cdots,N$.\n\\end{itemize}\n\n$\\bar{\\bf y}_{\\ell}$ may extend to all the $N+1$ timeslots in the protocol by\n\\begin{equation}\\label{equ:bar_y}\n\\begin{aligned}[b]\n\\bar{\\bf y}_{\\ell}=\n\\begin{cases}\n{\\bf y}_{1+K \\ell}, \\; & {\\rm for} \\; \\ell=0,\\cdots,L_1,\\\\\n{\\bf y}_{\\ell+(K-1)L_1+K}, & {\\rm for} \\; \\ell=L_1+1,\\cdots,N.\n\\end{cases}\n\\end{aligned}\n\\end{equation}\n\nThe pilot overhead in stage I is $(\\lceil \\frac{N}{M}\\rceil+1) K$, and the overhead in stage II is $N-\\lceil \\frac{N}{M}\\rceil $. The overall pilot overhead of the proposed protocol is $K+N+\\lceil \\frac{N}{M}\\rceil(K-1)$.\nNote that the pilot overhead is significantly reduced by about $M$ times compared to the protocol in\n\\cite{Araujo2021JSAC_CE_PARAFAC,Mishra2019CEonoff,Elbir2020WCL_DL_CE} and\n\\cite{Zhouzy2020decompositionCE,Kundu2021OJCSLMMSE_DFTGOOD,Alwazani2020OJCSLMMSE_DFT}, which require $NK$ pilots.\n\n\n\\subsubsection{Signal Pre-processing}\nIn the proposed protocol, we set ${\\bm \\theta}_0=-{\\bm \\theta}_1$ to decouple the estimation on the BS-user channel\\footnote{\n The  BS-user channel can be estimated based on ${{\\bf Y}}_0$ and ${{\\bf Y}}_1$  by the linear minimum mean squared error  estimator, which is similar to the methods in existing works\n\\cite{Kundu2021OJCSLMMSE_DFTGOOD,Alwazani2020OJCSLMMSE_DFT,Liuliang_CE2020TWC} (See Appendix \\ref{app_Estimate_Hd}).} and  on the cascaded IRS channel.\nTo facilitate the estimation of the cascaded IRS channel, signal pre-processing to remove the BS-user channel  from the received signals  is performed.\n\n\\begin{itemize}\n\\item  \\emph{Pre-processing on ${{\\bf Y}}_\\ell$ for $\\ell=1,2,\\cdots,L_1$}:\nFor $\\ell=1$, we have\n\\begin{equation}\\label{equ:R1}\n\\begin{aligned}[b]\n{\\bf R}_{1} &= \\frac{1}{2}\\left({{\\bf Y}}_1-{{\\bf Y}}_0\\right) \\\\\n&={\\bf G}^{\\rm T} {\\rm diag} ({\\bm \\theta}_1) {\\bf H}_{\\rm r}{\\bf X}\n+\\tilde{\\bf Z}_1\n,\n\\end{aligned}\n\\end{equation}\nwhere the elements in $\\tilde{\\bf Z}_1$ follow i.i.d. ${\\cal{CN}}({ 0},\\frac{1}{2}\\sigma_0^2) $.\nFor $\\ell=2,3,\\cdots,L_1$, we have\n\\begin{equation}\\label{equ:R2L1}\n\\begin{aligned}[b]\n{\\bf R}_{\\ell} &= {{\\bf Y}}_{\\ell} -\\frac{1}{2}\\left({{\\bf Y}}_0+{{\\bf Y}}_1\\right)\\\\\n&={\\bf G}^{\\rm T} {\\rm diag} ({\\bm \\theta}_\\ell) {\\bf H}_{\\rm r} {\\bf X}\n+\\tilde{\\bf Z}_{\\ell}\n,\n\\end{aligned}\n\\end{equation}\nwhere the elements in $\\tilde{\\bf Z}_{\\ell}$ follow ${\\cal{CN}}({ 0},\\frac{3}{2}\\sigma_0^2) $.\nNote that the difference between \\eqref{equ:R1} and \\eqref{equ:R2L1} is that they have different noise variances.\n\n\\item \\emph{Pre-processing on $\\bar{\\bf y}_{\\ell}$ for $\\ell=1,2,\\cdots,N$}:\nIn the same manner, the BS-user channel is removed from $\\bar{\\bf y}_{\\ell}$\nfor $\\ell=1,2,\\cdots,N$:\n\\begin{equation}\\label{equ:rbar}\n\\begin{aligned}[b]\n\\bar{\\bf r}_{\\ell} &= \\bar{\\bf y}_{\\ell}-\\frac{1}{2}\\left(\\bar{\\bf y}_{0}+\\bar{\\bf y}_{1}\\right)\\\\\n&={\\bf G}^{\\rm T} {\\rm diag} ({\\bm \\theta}_\\ell) {\\bf H}_{\\rm r}\\bar{\\bf x}\n+\\bar{\\bf z}_{\\ell}\\\\\n&={\\bf G}^{\\rm T} {\\rm diag} ({\\bf H}_{\\rm r}\\bar{\\bf x})  {\\bm \\theta}_\\ell\n+\\bar{\\bf z}_{\\ell}\n,\n\\end{aligned}\n\\end{equation}\nwhere the elements in $\\bar{\\bf z}_{\\ell}$ follow ${\\cal{CN}}({ 0},\\frac{3}{2}\\sigma_0^2) $.\n\\end{itemize}\n\n\\begin{lemma}[Effectiveness of the proposed protocol]\\label{lemma0}\nThe cascaded channels ${\\bf H}_{{\\rm I},k}={\\bf G}^{\\rm T} {\\rm{diag}}({\\bf h}_{{\\rm r},k})$ ($k=1,2,\\cdots,K$) for all $K$ users can be perfectly recovered with probability one by adopting an orthogonal phase shifting configuration matrix ${\\bm \\Phi}=[{\\bm \\theta}_1,{\\bm \\theta}_2,\\cdots,{\\bm \\theta}_N]$ whose elements are all non-zero, if there is no noise, and\n${\\bf G}={\\bf F}_{\\rm R} {\\ddot{\\bf G}} {\\bf F}_{\\rm B}^{\\rm T}$ and ${\\bf h}_{{\\rm r},k}={\\bf F}_{\\rm R}{\\ddot{\\bf h}}_{{\\rm r},k}$ where ${\\bf F}_{\\rm B} \\in {\\mathbb C}^{M \\times M}$ and ${\\bf F}_{\\rm R} \\in {\\mathbb C}^{N \\times N}$ is respectively the angular domain basis for the BS antenna array and the IRS,\n${\\ddot{\\bf G}}=[{\\ddot{\\bf g}}_1,{\\ddot{\\bf g}}_2,\\cdots,{\\ddot{\\bf g}}_M]$, and ${\\ddot{\\bf g}}_m$ and ${\\ddot{\\bf h}}_{{\\rm r},k}$ for all $m$ and $k$ are pairwise  independent following zero-mean multivariate normal distributions.\n\\end{lemma}\n\n\\begin{IEEEproof}\nSee Appendix \\ref{proof_lemma0}.\n\\end{IEEEproof}\n\nCompared with \\cite{Liuliang_CE2020TWC} and \\cite{double_IRS}, the proposed protocol has two main differences, which provide the opportunity to keep all the IRS elements ON and to reduce the pilot overhead by utilizing the common-link structure at the same time.\nHere, we try to explain the intuition by supposing we adopt a similar estimation algorithm to those in \\cite{Liuliang_CE2020TWC} and \\cite{double_IRS}, i.e., first estimate the reference channel and then estimate the relative channels.\nFirstly, instead of requiring a specific reference user, we design a virtual reference channel ${\\bf H}_{\\rm v}={\\bf G}^{\\rm T} {\\rm diag} ({\\bf H}_{\\rm r}\\bar{\\bf x})$, which is fair for all users and can be reconstructed using the $N$ observations in \\eqref{equ:rbar}.\nSecondly, in our protocol, the relative channels can be estimated by using ${\\bf R}_{\\ell}$ in \\eqref{equ:R1} and \\eqref{equ:R2L1}\nresulting a  measurement matrix\n$[({\\bf H}_{\\rm v}{\\rm diag} ({\\bm \\theta}_1))^{\\rm T},\\cdots,({\\bf H}_{\\rm v}{\\rm diag} ({\\bm \\theta}_{L_1}))^{\\rm T}]^{\\rm T}$.\nOne can see that there are $L_1$ different ${\\bm \\theta}_{\\ell}$ in the measurement matrix instead of a fixed one as in \\cite{double_IRS}. When $L_1 \\geq \\lceil \\frac{N}{M}\\rceil$, it is possible that the rank of the  measurement matrix becomes $N$ with a proper phase shifting configuration to obtain a reasonable estimation.\\footnote{\nIt is seen that the  selected on-off protocol in \\cite{Liuliang_CE2020TWC} can be treated as a special case of our protocol by setting $\\bar{\\bf x}=[1,0,\\cdots,0]^{\\rm T}$ and selecting different parts of the IRS elements to be ON and OFF  to obtain $L_1$  different ${\\bm \\theta}_{\\ell}$, as introduced in Section \\ref{overview_onoff}.\n}\nNote that the above two-step channel estimation algorithm is only for explaining the intuition of the proposed approach. In the next section, we will propose an optimization-based cascaded channel estimation algorithm that may achieve more reliable  performance.\n\n\n\\section{Optimization-Based MU-Cascaded IRS Channel Estimation}\\label{sec:opt_est}\nIn this section, we propose an optimization-based channel estimation on the cascaded IRS channel ${\\bf h}_{{\\rm I},k,m}$ for all $m$ and $k$ based on the pre-processed observations $\\{{\\bf R}_{\\ell},\\bar{\\bf r}_{\\ell}\\}$ in \\eqref{equ:R1}, \\eqref{equ:R2L1} and \\eqref{equ:rbar}.\nWe  consider a general decomposition on the cascaded channel, which is friendly in utilizing the channel prior knowledge\nand the common-link structure across the multiple users.\nSpecifically, we adopt the MAP\napproach to estimate\n the cascaded channel ${\\bf h}_{{\\rm I},k,m}$ for all $m$ and $k$\ngiven the pre-processed observations $\\{{\\bf R}_{\\ell},\\bar{\\bf r}_{\\ell}\\}$.\nAn alternating optimization algorithm with efficient initialization is further proposed to achieve a local optimum  of the MAP problem.\n\n\n\n\n\\subsection{MAP Problem Formulation}\nAs shown in \\eqref{equ:cascaded_channel_vector}, the cascaded channel ${\\bf h}_{{\\rm I},k,m}$ can be decomposed by the common BS-IRS channel ${\\bf g}_m$ and the IRS-user channel ${\\bf h}_{{\\rm r},k}$ as follows:\n\\begin{equation}\\label{equ:cascaded_channel_vector_2v}\n\\begin{aligned}[b]\n{\\bf h}_{{\\rm I},k,m}={\\rm{diag}}({\\bf h}_{{\\rm r},k}) {\\bf g}_m.\n\\end{aligned}\n\\end{equation}\nHowever, the main challenge to estimate the individual ${\\bf h}_{{\\rm r},k}$ and ${\\bf g}_m$  is that  in the MAP formulation, prior distributions of ${\\bf g}_m$ and ${\\bf h}_{{\\rm r},k}$ will be needed, but\nit is difficult to  obtain individual  covariances of ${\\bf g}_m$ and ${\\bf h}_{{\\rm r},k}$ based on the covariance of the cascaded channel.\\footnote{\nOne possible way to estimate the covariance of the cascaded channel ${\\bf C}_m^{(k)}$ is using the\n the maximum likelihood estimator $\\hat{\\bf C}_m^{(k)}=\\frac{1}{J} \\sum_{j=1}^J \\left[ \\hat{\\bf h}_{{\\rm I},k,m}(j) \\hat{\\bf h}_{{\\rm I},k,m}(j)^{\\rm H} \\right]$, where $\\{\\hat{\\bf h}_{{\\rm I},k,m}(j)\\}$ are the estimated historical cascaded channels in the past $j=1,2,\\cdots,J$ transmission frames.\n  Note that similar covariances are also required by the LMMSE estimators for the selected on-off channel estimation protocol in \\cite{Liuliang_CE2020TWC} (See equations (72) and (86) in \\cite{Liuliang_CE2020TWC}).\n}\n\nTo address this issue, we consider\na more general auxiliary variable set for the cascaded channel decomposition:\n\\begin{equation}\\label{equ:set_opt}\n\\begin{aligned}[b]\n{\\cal A}=\\left\\{ \\left.\\left\\{{\\bf H}_{\\rm g},{\\bf H}_{\\rm u}\\right\\} \\right|\n{\\bf h}_{{\\rm I},k,m}={\\rm{diag}}({\\bf h}_{{\\rm u},k}) {\\bf h}_{{\\rm g},m}, \\forall k, \\forall m\n\\right\\},\n\\end{aligned}\n\\end{equation}\nwhere ${\\bf H}_{\\rm g}=[{\\bf h}_{{\\rm g},1},\\cdots,{\\bf h}_{{\\rm g},M}]$ is the common-link variable whose $m$-th column is ${\\bf h}_{{\\rm g},m}$, and ${\\bf H}_{\\rm u}=[{\\bf h}_{{\\rm u},1},\\cdots,{\\bf h}_{{\\rm u},K}]$ is the user-specific variable  whose $k$-th column is ${\\bf h}_{{\\rm u},k}$.\nOne may verify that $\\{{\\bf G}, {\\bf H}_{\\rm r}\\}\\in {\\cal A}$ according to \\eqref{equ:cascaded_channel_vector_2v}.\nBased on this, we formulate an optimization problem on  ${\\bf H}_{\\rm g}$  and ${\\bf H}_{\\rm u}$ using the MAP approach, which is given by\n\\begin{equation*\n\\begin{aligned}[b]\n&\n{\\mathcal{P}}{(\\text{A})}\n\\quad \\max_{ {\\bf H}_{\\rm g}, {\\bf H}_{\\rm u} } \\;\nf_{{\\rm A}}({\\bf H}_{\\rm g}, {\\bf H}_{\\rm u})\n,\n\\end{aligned}\n\\end{equation*}\nwhere the objective function is given by\\footnote{\nOne may substitute different prior setups to $p({\\bf h}_{{\\rm I},k,m} )$ in \\eqref{equ:prior}.\nIn addition, if channel prior knowledge is unavailable, one may simply remove $p({\\bf h}_{{\\rm I},k,m} )$ from \\eqref{equ:obj_f},\nand the optimization becomes the maximum likelihood approach.\n}\n\\begin{equation}\\label{equ:obj_f}\n\\begin{aligned}[b]\n&f_{{\\rm A}}({\\bf H}_{\\rm g}, {\\bf H}_{\\rm u})\n= \\sum_{m=1}^M \\sum_{k=1}^K \\ln p({\\bf h}_{{\\rm I},k,m} ) \\\\\n&\\quad+\\sum_{\\ell=1}^{L_1} \\ln p({\\tilde{\\bf R}}_{\\ell}| {\\bf H}_{\\rm g}, {\\bf H}_{\\rm u} )\n+\\sum_{\\ell=L_1+1}^{N} \\ln p( \\bar{\\bf r}_{\\ell} | {\\bf H}_{\\rm g}, {\\bf H}_{\\rm u} )\n,\n\\end{aligned}\n\\end{equation}\nand $\\tilde{\\bf R}_{\\ell}= {{\\bf R}}_{\\ell} {\\bf X}^{-1}$.\nNote that $f_{{\\rm A}}({\\bf H}_{\\rm g}, {\\bf H}_{\\rm u})$ in \\eqref{equ:obj_f} only requires the prior distribution of ${\\bf h}_{{\\rm I},k,m}$, which is given by\n\\begin{equation}\\label{equ:prior}\n\\begin{aligned}[b]\np({\\bf h}_{{\\rm I},k,m} )\\propto e^{\n- {\\bf h}_{{\\rm g},m}^{\\rm H} {\\rm{diag}}({\\bf h}_{{\\rm u},k})^{\\rm H} {{\\bf C}_m^{(k)}}^{-1} {\\rm{diag}}({\\bf h}_{{\\rm u},k}) {\\bf h}_{{\\rm g},m}\n},\n\\end{aligned}\n\\end{equation}\nwhere $\\propto$ denotes equality up to a\nscaling that is independent of the variables (i.e., ${\\bf h}_{{\\rm I},k,m}$ for \\eqref{equ:prior}).\nThe likelihood functions $p({\\tilde{\\bf R}}_{\\ell}| {\\bf H}_{\\rm g}, {\\bf H}_{\\rm u})$ and $p( \\bar{\\bf r}_{\\ell} | {\\bf H}_{\\rm g}, {\\bf H}_{\\rm u} )$ are given by\n\\begin{align}\np(\\tilde{{\\bf R}}_{\\ell}| {\\bf H}_{\\rm g}, {\\bf H}_{\\rm u} )\n&\\propto\ne^{-\\sigma_{\\ell}^{-2} \\left\\|\n\\tilde{{\\bf R}}_{\\ell}- {\\bf H}_{\\rm g}^{\\rm T} {\\rm diag} \\left({\\bm \\theta}_{\\ell}\\right) {\\bf H}_{{\\rm u}}\n\\right\\|^2_{\\rm F} }\n, \\label{equ:likelihood_R}\n\\\\\np( \\bar{\\bf r}_{\\ell} | {\\bf H}_{\\rm g}, {\\bf H}_{\\rm u} )\n&\\propto\ne^{-\\bar\\sigma_{\\ell}^{-2} \\left\\|\n\\bar{\\bf r}_{\\ell}-  {\\bf H}_{\\rm g}^{\\rm T} {\\rm diag} \\left({\\bm \\theta}_{\\ell} \\right) {\\bf H}_{{\\rm u}} \\bar{\\bf x}\n\\right\\|^2_2 }\n, \\label{equ:likelihood_r}\n\\end{align}\nwhere $\\sigma_{1}^2=\\frac{1}{2K}\\sigma_0^2$, $\\sigma_{\\ell}^2=\\frac{3}{2K}\\sigma_0^2$ for $\\ell=2,3,\\cdots,L_1$,\nand $\\bar\\sigma_{\\ell}^2=\\frac{3}{2}\\sigma_0^2$ for $\\ell=L_1+1,L_1+2,\\cdots,N$.\nFinally, after dropping all the irrelevant constant terms, the objective function is equivalently written as\n\\begin{equation}\\label{equ:obj_f_eq}\n\\begin{aligned}[b]\n&f_{{\\rm A}}({\\bf H}_{\\rm g}, {\\bf H}_{\\rm u})\n=-\\sum_{\\ell=1}^{L_1}\n{\\frac{1}{\\sigma_{\\ell}^{2}} \\left\\|\n\\tilde{{\\bf R}}_{\\ell}- {\\bf H}_{\\rm g}^{\\rm T} {\\rm diag} \\left({\\bm \\theta}_{\\ell}\\right) {\\bf H}_{{\\rm u}}\n\\right\\|^2_{\\rm F} }\\\\\n&\\quad-\\sum_{\\ell=L_1+1}^{N}\n{ \\frac{1}{\\bar\\sigma_{\\ell}^{2}} \\left\\|\n\\bar{\\bf r}_{\\ell}-{\\bf H}_{\\rm g}^{\\rm T} {\\rm diag} \\left({\\bm \\theta}_{\\ell} \\right) {\\bf H}_{{\\rm u}} \\bar{\\bf x}\n\\right\\|^2_2 }\n\\\\\n&\\quad-\\sum_{m=1}^M \\sum_{k=1}^K {\\bf h}_{{\\rm g},m}^{\\rm H} {\\rm{diag}}({\\bf h}_{{\\rm u},k})^{\\rm H} {{\\bf C}_m^{(k)}}^{-1} {\\rm{diag}}({\\bf h}_{{\\rm u},k}) {\\bf h}_{{\\rm g},m}  .\n\\end{aligned}\n\\end{equation}\nNote that ${\\mathcal{P}}{(\\text{A})}$ does not have a unique solution but all the solutions are equivalent for the purpose of estimation of the cascaded channel ${\\bf h}_{{\\rm I},k,m}$ for all $m$ and $k$.\n\\begin{lemma}[Equivalence of the solution of ${\\mathcal{P}}{(\\text{A})}$]\\label{eq_decomposite}\nLet ${\\bf H}_{\\rm g}^\\star$ and ${\\bf H}_{\\rm u}^\\star$ be an optimal solution of ${\\mathcal{P}}{(\\text{A})}$, then ${\\rm diag} \\left({\\bf a}\\right){\\bf H}_{\\rm g}^\\star$ and ${\\rm diag} \\left({\\bf a}\\right)^{-1}{\\bf H}_{\\rm u}^\\star$ is also an optimal solution of ${\\mathcal{P}}{(\\text{A})}$\nfor any coefficient ${\\bf a}=[a_1,a_2,\\cdots,a_N]^{\\rm T}$ with $|a_n|\\neq0$ for all $n=1,2,\\cdots,N$.\n\\end{lemma}\n\n\\begin{IEEEproof}\nSee Appendix \\ref{proof_lemma1}.\n\\end{IEEEproof}\n\n\nAs a result, there is ambiguity in estimating individual channels from solving ${\\mathcal{P}}{(\\text{A})}$. Nevertheless, the cascaded channel is unique regardless of the coefficient ${\\bf a}$.\n\n\n\n\n\\subsection{ Channel Estimation Algorithm based on Alternative Optimization}\\label{sec:opt_est_AO}\n\nSolving ${\\mathcal{P}}{(\\text{A})}$ is difficult  due to the optimization variables being coupled in the likelihood functions \\eqref{equ:likelihood_R} and \\eqref{equ:likelihood_r}.\nFortunately, we will show that ${\\mathcal{P}}{(\\text{A})}$ is actually bi-convex (see Lemma \\ref{fu_convex} and Lemma \\ref{fg_convex}), which can be solved by alternative optimization. In particular, we decompose ${\\mathcal{P}}{(\\text{A})}$ into two convex sub-problems, and\nthe optimal solutions for these sub-problems will be derived accordingly.\n\n\n\\subsubsection{Optimize ${\\bf H}_{\\rm u}$}\nWe investigate the optimization of ${\\bf H}_{\\rm u}$ while ${\\bf H}_{\\rm g}$ are fixed.\nAfter dropping all irrelevant terms, the sub-problem  is given by\n\\begin{align*}\n{\\mathcal{P}}{({\\text A}_{{\\rm u}})} \\quad \\min_{ {\\bf H}_{\\rm u} }\\; f_{{\\rm u}}({\\bf H}_{\\rm u})\n,\n\\end{align*}\nwhere\n\\begin{equation}\\label{equ:obj_f_hrk}\n\\begin{aligned}[b]\n&f_{{\\rm u}}({\\bf H}_{\\rm u})\n=\\sum_{\\ell=1}^{L_1}\n{\\frac{1}{\\sigma_{\\ell}^{2}} \\left\\|\n\\tilde{{\\bf R}}_{\\ell}- {\\bf D}_{\\ell} {\\bf H}_{{\\rm u}}\n\\right\\|^2_{\\rm F} }\\\\\n&\\quad+\\sum_{\\ell=L_1+1}^{N}\n{ \\frac{1}{\\bar\\sigma_{\\ell}^{2}} \\left\\|\n\\bar{\\bf r}_{\\ell}-{\\bf D}_{\\ell} {\\bf H}_{{\\rm u}} \\bar{\\bf x}\n\\right\\|^2_2 }\n+\\sum_{k=1}^K {\\bf h}_{{\\rm u},k}^{\\rm H} {\\bf C}_{{\\rm u},k} {\\bf h}_{{\\rm u},k}\n,\n\\end{aligned}\n\\end{equation}\nand\n\\begin{align}\n{\\bf D}_{\\ell}&={\\bf H}_{\\rm g}^{\\rm T} {\\rm diag} \\left({\\bm \\theta}_{\\ell}\\right), \\; {\\text{for}} \\; \\ell=1,2,\\cdots,N,\\\\\n{\\bf C}_{{\\rm u},k}&=\\sum_{m=1}^M {\\rm{diag}}({\\bf h}_{{\\rm g},m})^{\\rm H} {{\\bf C}_m^{(k)}}^{-1} {\\rm{diag}}({\\bf h}_{{\\rm g},m}), \\quad \\forall  k.\n\\end{align}\n\n\n\\begin{lemma}[Convexity of ${\\mathcal{P}}{({\\text A}_{{\\rm u}})}$]\\label{fu_convex}\nFor any fixed ${\\bf H}_{\\rm g}$, the objective function of ${\\mathcal{P}}{({\\text A}_{{\\rm u}})}$ is a convex quadratic function of the vectorization of ${\\bf H}_{\\rm u}$, which is denoted by ${\\rm{vec}}({\\bf H}_{\\rm u})$.\n\\end{lemma}\n\n\\begin{IEEEproof}\nSee Appendix \\ref{proof_lemmafu}.\n\\end{IEEEproof}\n\n\nBased on Lemma \\ref{fu_convex}, the optimal solution of ${\\mathcal{P}}{({\\text A}_{{\\rm u}})}$ is the root of the first order derivative of $f_{{\\rm u}}({\\rm{vec}}({\\bf H}_{\\rm u}))$,  which is given by\n\\begin{equation}\\label{equ:opt_hrk}\n\\begin{aligned}[b]\n{\\rm{vec}}({\\bf H}_{\\rm u}^\\star)\n&= {\\bm \\Lambda}_{{\\rm u}}^{-1} {\\bm \\nu}_{{\\rm u}}\n,\n\\end{aligned}\n\\end{equation}\nwhere\n\\begin{align}\n{\\bm \\Lambda}_{{\\rm u}}&=\n {\\bf C}_{{\\rm u}}\n+{\\bf I}_K \\otimes \\left(\\sum_{\\ell=1}^{L_1} \\frac{1}{\\sigma_{\\ell}^{2}} {\\bf D}_{\\ell}^{\\rm H} {\\bf D}_{\\ell}\\right) \\notag\\\\\n&\\quad + \\left(\\bar{\\bf x}^\\ast \\bar{\\bf x}^{\\rm T}\\right)\\otimes \\left(\\sum_{\\ell=L_1+1}^{N} \\frac{1}{\\bar\\sigma_{\\ell}^{2}} |\\bar{x}_k|^2 {\\bf D}_{\\ell}^{\\rm H} {\\bf D}_{\\ell}\\right)\n, \\label{equ:opt_hrk_e1}\\\\\n{\\bf C}_{{\\rm u}}&={\\rm{blkdiag}}({\\bf C}_{{\\rm u},1},{\\bf C}_{{\\rm u},2},\\cdots,{\\bf C}_{{\\rm u},K}),\n\\end{align}\nand\n\\begin{equation}\\label{equ:opt_hrk_e2}\n\\begin{aligned}[b]\n{\\bm \\nu}_{{\\rm u}}\n&={\\rm{vec}}\\left(\n\\sum_{\\ell=1}^{L_1} \\frac{1}{\\sigma_{\\ell}^{2}} {\\bf D}_{\\ell}^{\\rm H}\n\\tilde{{\\bf R}}_{\\ell}\n+ \\sum_{\\ell=L_1+1}^{N} \\frac{1}{\\bar\\sigma_{\\ell}^{2}}\n {\\bf D}_{\\ell}^{\\rm H} \\bar{\\bf r}_{\\ell} \\bar{\\bf x}^{\\rm H}\n\\right).\n\\end{aligned}\n\\end{equation}\n\n\n\n\n\\subsubsection{Optimize ${\\bf H}_{{\\rm g}}$}\nSimilarly, the sub-problem of optimizing ${\\bf H}_{{\\rm g}}$ is given by\n\\begin{align*}\n{\\mathcal{P}}{(\\text{A}_{{\\rm g}})} \\; \\min_{ {\\bf H}_{{\\rm g}} }\\; \\sum_{m=1}^M f_{{\\rm g},m}({\\bf h}_{{\\rm g},m})\n,\n\\end{align*}\nwhere\n\\begin{align}\n&f_{{\\rm g},m}({\\bf h}_{{\\rm g},m})\n=\n \\sum_{k=1}^K \\sum_{\\ell=1}^{L_1} \\frac{1}{\\sigma_{\\ell}^{2}} \\left\\|\n\\tilde{{r}}_{\\ell,m,k}-{\\bf h}_{{\\rm g},m}^{\\rm T}  {\\bf b}_{\\ell,k}\n\\right\\|^2 \\notag\\\\\n&+ \\sum_{\\ell=L_1+1}^{N}\\frac{1}{\\bar\\sigma_{\\ell}^{2}}   \\left\\|\n\\bar{r}_{\\ell,m}-\\sum_{k=1}^K \\bar{x}_k {\\bf h}_{{\\rm g},m}^{\\rm T} {\\bf b}_{\\ell,k}\n\\right\\|^2+{\\bf h}_{{\\rm g},m}^{\\rm H} {\\bf C}_{{\\rm g},m} {\\bf h}_{{\\rm g},m}\n, \\label{equ:obj_f_gm}\\\\\n&\\quad{\\bf b}_{\\ell,k}={\\rm diag}\\left({\\bm \\theta}_{\\ell}\\right) {\\bf h}_{{\\rm u},k}, \\; {\\text{for}} \\; \\ell=1,2,\\cdots,N,\\\\\n&\\quad{\\bf C}_{{\\rm g},m}=\\sum_{k=1}^K {\\rm{diag}}({\\bf h}_{{\\rm u},k})^{\\rm H} {{\\bf C}_m^{(k)}}^{-1} {\\rm{diag}}({\\bf h}_{{\\rm u},k}), \\quad \\forall  m,\n\\end{align}\n$\\tilde{{r}}_{\\ell,m,k}$ denotes the entry in the $m$-th row and $k$-th column of $\\tilde{\\bf R}_{\\ell}$,\nand $\\bar{r}_{\\ell,m}$ denotes the $m$-th entry in $\\bar{\\bf r}_{\\ell}$.\n\n\n\\begin{lemma}[Convexity of ${\\mathcal{P}}{({\\text A}_{{\\rm g}})}$]\\label{fg_convex}\nFor any fixed ${\\bf H}_{\\rm u}$, the objective function of ${\\mathcal{P}}{({\\text A}_{{\\rm g}})}$ is a convex quadratic function of $\\{{\\bf h}_{{\\rm g},1},{\\bf h}_{{\\rm g},2},\\cdots,{\\bf h}_{{\\rm g},M}\\}$.\n\\end{lemma}\n\n\\begin{IEEEproof}\nSee Appendix \\ref{proof_lemmafg}.\n\\end{IEEEproof}\n\n\n\nBased on lemma \\ref{fg_convex}, the optimal ${\\bf h}_{{\\rm g},m}$ is the\nroot of the first order derivative of $f_{{\\rm g},m}({\\bf h}_{{\\rm g},m})$, which is given by\n\\begin{equation}\\label{equ:opt_gm}\n\\begin{aligned}[b]\n{\\bf h}_{{\\rm g},m}^\\star\n&= {\\bm \\Lambda}_{{\\rm g},m}^{-1} {\\bm \\nu}_{{\\rm g},m}\n,\n\\end{aligned}\n\\end{equation}\nfor $m=1,2,\\cdots,M$, where\n\\begin{equation}\\label{equ:opt_gm_e1}\n\\begin{aligned}[b]\n{\\bm \\Lambda}_{{\\rm g},m}&={\\bf C}_{{\\rm g},m}^{-1}\n+\\sum_{k=1}^K\\sum_{\\ell=1}^{L_1} \\frac{1}{\\sigma_{\\ell}^{2}} {\\bf b}_{\\ell,k}^\\ast {\\bf b}_{\\ell,k}^{\\rm T}\\\\\n&\\;+ \\sum_{\\ell=L_1+1}^{N}\\frac{1}{\\bar\\sigma_{\\ell}^{2}}\n\\left(\\sum_{k =1}^K \\bar{x}_k {\\bf b}_{\\ell,k}^{\\rm T}\\right)^{\\rm H}\n\\left(\\sum_{k =1}^K \\bar{x}_k {\\bf b}_{\\ell,k}^{\\rm T}\\right)\n,\n\\end{aligned}\n\\end{equation}\nand\n\\begin{equation}\\label{equ:opt_gm_e2}\n\\begin{aligned}[b]\n&{\\bm \\nu}_{{\\rm g},m}=\\sum_{k=1}^K\\sum_{{\\ell}=1}^{L_1} \\frac{\\tilde{{r}}_{\\ell,m,k}}{\\sigma_{\\ell}^{2}} {\\bf b}_{\\ell,k}^\\ast\n+ \\sum_{\\ell=L_1+1}^{N}\\frac{\\bar{r}_{\\ell,m}}{\\bar\\sigma_{\\ell}^{2}}\n\\left(\\sum_{k =1}^K \\bar{x}_k^{\\ast} {\\bf b}_{\\ell,k}^{\\ast}\\right)\n.\n\\end{aligned}\n\\end{equation}\n\n\n\\subsubsection{Initial Estimation on ${\\bf H}_{{\\rm g}}$}\n The quality of the solution obtained by the alternative optimization  depends heavily on the initial point.\n Here, we  propose\nan efficient estimator for ${\\bf H}_{{\\rm g}}$  to initialize the proposed alternative optimization algorithm.\nIn particular, we construct a special  $\\left\\{{\\bf H}_{\\rm g},{\\bf H}_{\\rm u}\\right\\}$ pair whose elements are given by\n\\begin{align}\n{\\bf h}_{{\\rm g},m}&= {\\rm diag} ({\\bf H}_{\\rm r}\\bar{\\bf x}) {\\bf g}_m, \\label{equ:hg1}\\\\\n{\\bf h}_{{\\rm u},k}&={\\rm diag} ({\\bf H}_{\\rm r}\\bar{\\bf x})^{-1}{\\bf h}_{{\\rm r},k}. \\label{equ:hu1}\n\\end{align}\nSubstituting the above ${\\bf H}_{\\rm g}$  into \\eqref{equ:rbar},  we have\n\\begin{equation}\\label{equ:rbar_v2}\n\\begin{aligned}[b]\n\\bar{\\bf r}_{\\ell}\n&={\\bf H}_{\\rm g}^{\\rm T}  {\\bm \\theta}_\\ell\n+\\bar{\\bf z}_{\\ell}\n,\n\\end{aligned}\n\\end{equation}\nfor $\\ell=1,2,\\cdots,N$.\nTherefore, ${\\bf H}_{\\rm g}$ can be initialized by the least squares (LS) estimator as follows:\n\\begin{equation}\\label{equ:est_Hc}\n\\begin{aligned}[b]\n\\hat{\\bf H}_{\\rm g}=\\left( \\left[\\bar{\\bf r}_1,\\cdots,\\bar{\\bf r}_N \\right]\n\\left[{\\bm \\theta}_1,\\cdots,{\\bm \\theta}_N \\right]^{-1} \\right)^{\\rm T}.\n\\end{aligned}\n\\end{equation}\nSince $\\hat{\\bf H}_{\\rm g}$ in \\eqref{equ:est_Hc} is unbiased and it has exploited most of the available observations in all the $N+1$ training timeslots,\nit will give a good initial point.\n\n\n\\subsection{The Overall Proposed Algorithm}\nThe overall proposed cascaded IRS channel estimation algorithm  is summarized in Algorithm \\ref{alg:P1}.\nThe convergence of the proposed alternating optimization algorithm is analyzed in Lemma \\ref{convergence}.\n\n\\begin{algorithm}[!ht]\n\\caption{ Proposed alternating optimization algorithm for the  cascaded IRS channel estimation}\n\\label{alg:P1}\n\\begin{algorithmic}[1]\n\\STATE {Initialize ${\\bf H}_{{\\rm g}}$  by \\eqref{equ:est_Hc}.}\\\\\n\\REPEAT\n\\STATE  Update ${\\bf H}_{{\\rm u}}$ by \\eqref{equ:opt_hrk};\n\\STATE  Update ${\\bf h}_{{\\rm g},m}$ by \\eqref{equ:opt_gm}  for all $m$;\n\\UNTIL{ $f_{{\\rm A}}({\\bf H}_{\\rm g}, {\\bf H}_{\\rm u})$ in \\eqref{equ:obj_f_eq} converges;}\\\\\n\\STATE {Output the cascaded channel ${\\hat{\\bf h}}_{{\\rm I},k,m}={\\rm{diag}}({\\bf h}_{{\\rm u},k}) {\\bf h}_{{\\rm g},m}$ for all $k$ and $m$.}\n\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{lemma}[Convergence of the Proposed  Alternating Optimization Algorithm]\\label{convergence}\nThe objective function $f_{{\\rm A}}({\\bf H}_{\\rm g}, {\\bf H}_{\\rm u})$ is non-increasing in every step when ${\\bf H}_{{\\rm u}}$ or ${\\bf H}_{{\\rm g}}$ are updated, and the optimization  iterations in \\eqref{equ:opt_hrk} and \\eqref{equ:opt_gm}  converge to a local optimum\nof ${\\mathcal{P}}{(\\text{A})}$.\n\\end{lemma}\n\n\\begin{IEEEproof}\nAs shown in Section \\ref{sec:opt_est_AO}, the original problem is decomposed into two unconstrained minimization problems whose objectives are convex quadratic functions, and each subproblem has a unique optimal solution, which is derived in \\eqref{equ:opt_hrk} and \\eqref{equ:opt_gm}.\nTherefore, the  whole alternating optimization algorithm will converge to  a local optimum of the original problem ${\\mathcal{P}}{(\\text{A})}$ \\cite{BCD}.\n\\end{IEEEproof}\n\n\n\\emph{Remark}:\nThe complexity for updating ${\\bf H}_{{\\rm u}}$ by \\eqref{equ:opt_hrk} is ${\\cal O}(K^3 N^3+KMN^2)$.  The complexity for updating ${\\bf h}_{{\\rm g},m}$ by  \\eqref{equ:opt_gm} is ${\\cal O}(KN^3)$, and thus the complexity to update  ${\\bf H}_{{\\rm g}}$ is ${\\cal O}(KMN^3)$. Therefore, the overall complexity of the solution is ${\\cal O}(IK^3 N^3+IKMN^3)$, where $I$ denotes the number of iterations of the alternating optimization algorithm.\\footnote{We will show in simulation that the algorithm will converge quickly in about two or three iterations. In addition, the ${\\cal O}(N^3)$ complexity is costed by the matrix inversion operation. However, since the matrices required inversion operation are all Hermitian positive semi-definite matrices, they can be implemented very efficiently by advanced algorithms such as the Cholesky-decomposition-based algorithm \\cite{matrix_inverse}.  }\n\n\n\\section{Training Phase Shifting Configuration}\\label{sec:Phaseshift_cofig}\n\\subsection{ Motivation of the Phase Shifting Configuration}\nThe IRS steers the incident signal to different directions by configuring different phase shifting vectors ${\\bm \\theta}_{\\ell}$, as illustrated in {\\figurename~\\ref{theta_opt}}.\nAccording to the protocol, the cascaded channel is scanned by $N$ spatial directions in $N$ training timeslots, and a proper design on $\\{ {\\bm \\theta}_1,{\\bm \\theta}_2,\\cdots,{\\bm \\theta}_N \\}$ guarantees that the whole channel information in all directions is contained by the received signals such that good channel estimation performance can be achieved.\n\n\n\\begin{figure}\n[!ht]\n\\centering\n\\includegraphics[width=.9\\columnwidth]{phase_shift_opt.eps}\n\\caption{Illustration of the impact of different phase shiftings.}\n\\label{theta_opt}\n\\end{figure}\n\n\nIn the SU-MISO scenario, the overall received measurements after removing the pilots and the BS-user channels is given by\n\\begin{equation}\\label{equ:y_su}\n\\begin{aligned}[b]\n{\\bf Y}_{\\rm{SU}}&= {\\bf H}_{{\\rm I},1} {\\bm \\Phi}\n+{\\bf Z}\n,\n\\end{aligned}\n\\end{equation}\nwhere ${\\bm \\Phi}=[{\\bm \\theta}_1,{\\bm \\theta}_2,\\cdots,{\\bm \\theta}_N]$. The LS estimator may adopted as follows \\cite{Zhouzy2020decompositionCE}:\n\\begin{equation}\\label{equ:hatH_su}\n\\begin{aligned}[b]\n\\hat{\\bf H}_{{\\rm I},1}&= {\\bf Y}_{\\rm{SU}} {\\bm \\Phi}^{-1}\n,\n\\end{aligned}\n\\end{equation}\nwhere $\\hat{\\bf H}_{{\\rm I},1}$ denotes the estimated cascaded channel.\nThen ${\\bm \\Phi}$ is optimized by minimizing the MSE:\n\\begin{align*}\n\\min_{ {\\bm \\Phi} }\\; & {\\rm{tr}} \\left(\\left({\\bm \\Phi} {\\bm \\Phi}^{\\rm H}\\right)^{-1}\\right)\\\\\n{\\bf s.t.} \\;\n& |{\\bm \\Phi}_{i,j}|=1, \\quad \\forall i,j=1,2,\\cdots,N\n.\n\\end{align*}\nIt is proved in \\cite{Zhouzy2020decompositionCE} that the optimal value of the MSE is $1$, which can be achieved by the DFT matrix such that ${\\bm \\Phi}={\\bf F}$, where\n\\begin{equation}\\label{equ:DFT_F}\n\\begin{aligned}[b]\n{\\bf F}\n=\n\\begin{bmatrix}\n    1      &   1 &  1  & \\cdots & 1 \\\\\n    1 & e^{-\\jmath 2 \\pi \\frac{1}{N}} & e^{-\\jmath 2 \\pi \\frac{2}{N}} & \\cdots & e^{-\\jmath 2 \\pi \\frac{N-1}{N}} \\\\\n    1 & e^{-\\jmath 2 \\pi \\frac{2}{N}} & e^{-\\jmath 2 \\pi \\frac{4}{N}} & \\cdots & e^{-\\jmath 2 \\pi \\frac{2(N-1)}{N}} \\\\\n    \\vdots     &  \\vdots &  \\vdots  & \\ddots & \\vdots \\\\\n   1 & e^{-\\jmath 2 \\pi \\frac{N-1}{N}} & e^{-\\jmath 2 \\pi \\frac{2(N-1)}{N}} & \\cdots & e^{-\\jmath 2 \\pi \\frac{(N-1)^2}{N}}\n\\end{bmatrix}\n.\n\\end{aligned}\n\\end{equation}\n\nFor the protocol extended from the SU case \\cite{Zhouzy2020decompositionCE,Kundu2021OJCSLMMSE_DFTGOOD,Alwazani2020OJCSLMMSE_DFT} shown in {\\figurename~\\ref{protocol_tran}}, the transmit signals from users are the same in different timeslots.\nTherefore, the columns of ${\\bf F}$ may be permuted to any orders, and the MSE will remain the same.\nHowever, in our proposed protocol in {\\figurename~\\ref{protocol:d}}, the transmit signals are different in stage I and stage II.\nIn particular, the received signals in stage I contribute to the estimation on both the common-link variable and the user-specific variables in the cascaded channels, while the signals in stage II contribute to the common-link variable only.\nHence, the  phase shifting vectors $ {\\bm \\theta}_1, {\\bm \\theta}_2,\\cdots,{\\bm \\theta}_{L_1} $\nin stage I require additional design.\n\n\n\n\n\n\n\n\n\\subsection{Optimization Formulation of the Phase Shifting Configuration for MU-MISO IRS Systems}\n\nAs shown in \\eqref{equ:est_Hc}, the initial estimation on the common-link variable $\\hat{\\bf H}_{\\rm g}$ in \\eqref{equ:est_Hc}  is almost the same as the estimator for the SU case shown in \\eqref{equ:hatH_su}.\nTherefore, we still adopt the DFT-based phase shifting configuration for all the $N$ timeslots. Additionally, an additional steering direction ${\\bm{\\vartheta}} \\in {\\mathbb C}^{N \\times 1}$ is introduced for a more flexible design:\n\\begin{equation}\\label{equ:prop_phi}\n\\begin{aligned}[b]\n{\\bm \\Phi}&= {\\rm{diag}}({\\bm{\\vartheta}}){\\bf F}\n.\n\\end{aligned}\n\\end{equation}\nDenote by $\\vartheta_n$ the $n$-th element in ${\\bm{\\vartheta}}$. We have $|\\vartheta_n|=1$ for all $n=1,2,\\cdots,N$.\nOne can see that the value of ${\\rm{tr}} \\left(\\left({\\bm \\Phi} {\\bm \\Phi}^{\\rm H}\\right)^{-1}\\right)$ is kept at $1$ for any ${\\bm{\\vartheta}}$.\nDenote by ${\\bf f}_{\\ell}$ the $\\ell$-th column of $\\bf F$. The training phase shifting vector in the $\\ell$-th timeslot is given by\n\\begin{equation}\\label{equ:prop_theta_ell}\n\\begin{aligned}[b]\n{\\bm \\theta}_\\ell&= {\\rm{diag}}({\\bm{\\vartheta}}) {\\bf f}_{\\ell}\n,\n\\end{aligned}\n\\end{equation}\nwhere $\\ell=1,2,\\cdots,N$.\nThe remaining task is to design ${\\bm{\\vartheta}}$.\n\nHowever, it is difficult to design a straightforward objective function to optimize ${\\bm{\\vartheta}}$ since the MSE of the estimated ${\\bf H}_{\\rm u}$ by the proposed algorithm is complicated.\nConsidering that we have the knowledge of the covariances ${\\bf C}_m^{(k)}$ of the cascaded channels for all $m$ and $k$,\nthe  average received power of the effective IRS channel from user $k$ to the $m$-th BS antenna in timeslot $\\ell$  can be denoted by a function of ${\\bm{\\vartheta}}$, ${\\bf f}_{\\ell}$ and ${\\bf C}_m^{(k)}$:\n\\begin{equation}\\label{equ:channel_gain}\n\\begin{aligned}[b]\nQ_{\\ell,k,m} &= {\\mathbb E} \\left[ | {\\bf g}_m^{\\rm T} {\\rm{diag}}({\\bm \\theta}_\\ell) {\\bf h}_{{\\rm r},k} |^2  \\right]\\\\\n &= {\\mathbb E} \\left[ | {\\bf h}_{{\\rm I},k,m}^{\\rm T} {\\bm \\theta}_\\ell |^2  \\right]\\\\\n&= {\\mathbb E} \\left[   {\\bm \\theta}_\\ell^{\\rm H}\n\\left( {\\bf h}_{{\\rm I},k,m} {\\bf h}_{{\\rm I},k,m}^{\\rm H} \\right)^{\\ast}\n{\\bm \\theta}_\\ell  \\right]\\\\\n&= {\\bm \\theta}_\\ell^{\\rm H}\n\\left( {\\bf C}_m^{(k)} \\right)^{\\ast}\n{\\bm \\theta}_\\ell\\\\\n&={\\bm{\\vartheta}}^{\\rm H} {\\rm{diag}}({\\bf f}_{\\ell})^{\\rm H}\n\\left( {\\bf C}_m^{(k)} \\right)^{\\ast}\n{\\rm{diag}}({\\bf f}_{\\ell}){\\bm{\\vartheta}}\n.\n\\end{aligned}\n\\end{equation}\nSince ${\\bf f}_{\\ell}$ and ${\\bf C}_m^{(k)}$ are known variables, the summation of $Q_{\\ell,k,m}$ over antennas $m=1,2,\\cdots,M$, users $k=1,2,\\cdots,K$ and timeslots $\\ell=1,2,\\cdots,L_1$ is a function of ${\\bm{\\vartheta}}$, which is given by\n\\begin{equation}\\label{equ:channel_gain_all}\n\\begin{aligned}[b]\nf_{\\rm B}({\\bm{\\vartheta}})&=\\sum_{\\ell=1}^{L_1} \\sum_{k=1}^K \\sum_{m=1}^M Q_{\\ell,k,m} \\\\\n&=\\sum_{\\ell=1}^{L_1} \\sum_{k=1}^K \\sum_{m=1}^M\n{\\bm{\\vartheta}}^{\\rm H} {\\rm{diag}}({\\bf f}_{\\ell})^{\\rm H}\n\\left( {\\bf C}_m^{(k)} \\right)^{\\ast}\n{\\rm{diag}}({\\bf f}_{\\ell}){\\bm{\\vartheta}} \\\\\n&={\\bm{\\vartheta}}^{\\rm H} \\left(\\sum_{\\ell=1}^{L_1} \\sum_{k=1}^K \\sum_{m=1}^M\n {\\rm{diag}}({\\bf f}_{\\ell})^{\\rm H}\n\\left( {\\bf C}_m^{(k)} \\right)^{\\ast}\n{\\rm{diag}}({\\bf f}_{\\ell})\\right){\\bm{\\vartheta}} .\n\\end{aligned}\n\\end{equation}\nWe define\n\\begin{equation}\n\\begin{aligned}[b]\n{\\bf E}=\\left(\\sum_{\\ell=1}^{L_1} \\sum_{k=1}^K \\sum_{m=1}^M\n {\\rm{diag}}({\\bf f}_{\\ell})^{\\rm H}\n\\left( {\\bf C}_m^{(k)} \\right)^{\\ast}\n{\\rm{diag}}({\\bf f}_{\\ell})\\right).\n\\end{aligned}\n\\end{equation}\nThe optimization problem on ${\\bm{\\vartheta}}$ is formulated to maximize $f_{\\rm B}({\\bm{\\vartheta}})$:\n\\begin{equation*\n\\begin{aligned}[b]\n{\\mathcal{P}}{(\\text{B})}\\quad \\max_{ {\\bm{\\vartheta}} } \\; &\nf_{\\rm B}({\\bm{\\vartheta}})={\\bm{\\vartheta}}^{\\rm H} {\\bf E} {\\bm{\\vartheta}}\\\\\n{\\bf s.t.} \\;\n& |\\vartheta_n|=1, \\quad \\forall n=1,2,\\cdots,N.\n\\end{aligned}\n\\end{equation*}\n\n\n\\subsection{Solution for ${\\mathcal{P}}{(\\text{B})}$}\n${\\mathcal{P}}{(\\text{B})}$ is a non-convex problem due to the maximizing of a convex objective function and the unit-modulus constraints.\nWe solve ${\\mathcal{P}}{(\\text{B})}$ by the successive convex approximation (SCA) algorithm.\nIn particular, a surrogate problem, shown as follows, is iteratively solved:\n\\begin{equation*\n\\begin{aligned}[b]\n{\\mathcal{P}}{({\\text{B}}_i)}\\quad \\max_{ {\\bm{\\vartheta}} } \\; &\n{f}_{\\rm B}^{(i)} ({\\bm{\\vartheta}},\\bar{\\bm{\\vartheta}})\\\\\n{\\bf s.t.} \\;\n& |\\vartheta_n|=1, \\quad \\forall n=1,2,\\cdots,N,\n\\end{aligned}\n\\end{equation*}\nwhere $i$ is the iteration index, $\\bar{\\bm{\\vartheta}}$ is the solution of the surrogate problem in the $(i-1)$-th iteration, and ${f}_{\\rm B}^{(i)}({\\bm{\\vartheta}},\\bar{\\bm{\\vartheta}})$ is the first-order approximation of  ${f}_{\\rm B}({\\bm{\\vartheta}})$ at $\\bar{\\bm{\\vartheta}}$:\n\\begin{equation}\\label{equ:surrogate}\n\\begin{aligned}[b]\n{f}_{\\rm B}^{(i)} ({\\bm{\\vartheta}},\\bar{\\bm{\\vartheta}})\n&= 2 {\\rm Re} \\left\\{{\\bar{\\bm \\vartheta}}^{\\rm H} {\\bf E} {\\bm \\vartheta}\\right\\}-{\\bar{\\bm \\vartheta}}^{\\rm H} {\\bf E} {\\bar{\\bm \\vartheta}}.\n\\end{aligned}\n\\end{equation}\nOne can see that ${f}_{\\rm B}^{(i)}({\\bm{\\vartheta}},\\bar{\\bm{\\vartheta}})$ is a linear function of ${\\bm{\\vartheta}}$, and thus the optimal solution of ${\\mathcal{P}}{({\\text{B}}_i)}$ is given by\n\\begin{equation}\\label{equ:opt_theta}\n\\begin{aligned}[b]\n{\\bm{\\vartheta}}=e^{\\jmath \\angle ({\\bf E}{\\bar{\\bm \\vartheta}})}.\n\\end{aligned}\n\\end{equation}\nThe proof on the convergence of the SCA algorithm can be referred to in \\cite{SCA}.\n\n\n\n\\section{Numerical Examples}\\label{simulation}\n\\subsection{Simulation Setups}\nThis section evaluates the performance of the proposed cascaded channel estimation algorithm.\nIn particular, we consider the indoor femtocell network illustrated in {\\figurename~\\ref{indoor_8user}} in which $K=8$ users are randomly distributed in a 5 m $\\times$ 5 m square area and are served by one BS and one IRS.\nWe generate the channel coefficients according to the 3GPP ray-tracing model \\cite[Section 7.5]{3GPP} using the model parameters for the Indoor-Office scenario \\cite[Table 7.5-6]{3GPP}.\nThe system parameters for the simulations are summarized in Table \\ref{table_sim}, in which the path-loss is set according to the Indoor-Office pathloss model in \\cite[Table 7.4.1-1]{3GPP}.\n\n\n\\begin{figure}\n[!ht]\n\\centering\n\\includegraphics[width=.8\\columnwidth]{simulation_scena.eps}\n\\caption{The simulated IRS-aided $K$-user MISO communication scenario comprising of one $M$-antenna BS and one $N$-element IRS.}\n\\label{indoor_8user}\n\\end{figure}\n\n\\begin{table}[!ht]\n\\footnotesize\n\\renewcommand{\\arraystretch}{1.3}\n\\caption{Simulation Parameters}\n\\label{tablepm}\n\\centering\n\\begin{tabular}{c|c}\n\\hline\nParameters & Values \\\\\n\\hline\nCarrier frequency & 2.4 GHz\\\\\n\\hline\n Transmission bandwidth  & $200$ kHz\\\\\n\\hline\nNoise power spectral density & $-170$ dBm\/Hz\\\\\n\\hline\nPath-loss for BS-IRS and IRS-user links (dB)& $40 + 17.3 \\lg d$\\\\\n\\hline\nPath-loss for BS-user link (dB)& $30 + 31.9 \\lg d+\\zeta$\\\\\n\\hline\nPenetration loss $\\zeta$ due to obstacle  & 20 dB \\\\\n\\hline\n Reflection efficiency  of IRS & 0.8\\\\\n\\hline\nHeight of users & 1.5 m\\\\\n\\hline\nLocation of BS & (0, 0, 3m)\\\\\n\\hline\nLocation of IRS & (0, 10m, 3m)\\\\\n\\hline\n\\end{tabular}\n\\label{table_sim}\n\\end{table}\n\n\n\n\nIn the simulation, we consider two baseline schemes to benchmark the proposed scheme.\n\\begin{itemize}\n\\item {\\bf Baseline 1 [LMMSE using the protocol in {\\figurename~\\ref{protocol:b}} \\cite{Kundu2021OJCSLMMSE_DFTGOOD,Alwazani2020OJCSLMMSE_DFT}]}: This curve illustrates the performance of the LMMSE estimator proposed in \\cite{Kundu2021OJCSLMMSE_DFTGOOD} and \\cite{Alwazani2020OJCSLMMSE_DFT}. For simplicity, we assume that the BS-user channels have already been perfectly estimated by this scheme.\n    The protocol illustrated in {\\figurename~\\ref{protocol:b}} is adopted.\n    In additon, for fair comparison, the number of training timeslots is set as $\\lceil \\frac{N-1}{K}\\rceil+\\lceil \\frac{N}{M}\\rceil$ such that the total pilot overhead is just slightly higher than that of the proposed scheme.\n\\item {\\bf Baseline 2 [Bilinear alternating least squares (BALS) algorithm \\cite{Araujo2021JSAC_CE_PARAFAC}]}:\n In \\cite{Araujo2021JSAC_CE_PARAFAC}, an iterative algorithm is proposed to estimate ${\\bf G}$ and ${\\bf H}_{\\rm r}$ by utilizing the PARAFAC decomposition, which adopts the same channel estimation protocol as Baseline 1. Note that due to the ambiguity issue (see Lemma \\ref{eq_decomposite} or \\cite[Section IV]{Araujo2021JSAC_CE_PARAFAC}), the BALS also cannot exactly reconstruct ${\\bf G}$ and ${\\bf H}_{\\rm r}$, and the actually estimated variable is still the cascaded channel.\n\\item {\\bf Baseline 3 [MAP modification for the BALS in \\cite{Araujo2021JSAC_CE_PARAFAC}]}: In this baseline, we make a simple modification based on the MAP optimization in this paper to further enhance the performance of the BALS algorithm in \\cite{Araujo2021JSAC_CE_PARAFAC} by exploiting the prior knowledge of the cascaded channels.\n\\item {\\bf Baseline 4 [Selected On-off protocol \\cite{Liuliang_CE2020TWC}]}: This curve illustrates the performance of the estimation algorithm in \\cite{Liuliang_CE2020TWC} based on the  selected on-off channel estimation protocol shown in {\\figurename~\\ref{protocol:c}}. We assume that the covariances of ${\\bf h}_{{\\rm u},k} ={\\rm diag}({\\bf h}_{{\\rm r},1})^{-1}{\\bf h}_{{\\rm r},k}$ for all $k=1,2,\\cdots,8$ are available, and the LMMSE estimator in \\cite[Section V]{Liuliang_CE2020TWC} is adopted. In addition, we always select user $1$ as the reference user.\n\\end{itemize}\nNote that the proposed scheme and Baselines 3 and 4 have the same  pilot overhead, i.e., $K+N+\\lceil \\frac{N}{M}\\rceil(K-1)$.\nWe focus on the evaluation of the performance of cascaded channel estimation, and use the normalized MSE (NMSE) as the evaluation metric, which is given by\n\\begin{equation}\n{\\rm{NMSE}}=\\frac{\\sum_{k=1}^K\\sum_{m=1}^M{\\mathbb E} \\left[\\left|{\\bf h}_{{\\rm I},k,m}-{\\hat{\\bf h}}_{{\\rm I},k,m}\\right|_2^2\\right]}\n{\\sum_{k=1}^K\\sum_{m=1}^M {\\mathbb E} \\left[\\left|{\\bf h}_{{\\rm I},k,m}\\right|_2^2\\right]}\n.\n\\end{equation}\nIn addition, based on the proposed protocol, the BS-user channel estimation can be independent of the cascaded channel estimation by applying the signal pre-processing as shown in Section \\ref{proposed_protocol}. This signal pre-processing operation will provide a theoretical $3$ dB gain for the BS-user channel estimation compared to the conventional solution, which shuts down the IRS to estimate the BS-user channel  (see Appendix \\ref{app_Estimate_Hd}), and thus we do not compare the estimation performance for the BS-user channel in the simulations.\n\n\n\\begin{figure}\n[!t]\n  \\centering\n  \\subfigure[$P_{\\rm T}$ vs NMSE]{\n    \\label{nmse_vs_PT:a}\n    \\includegraphics[width=1\\columnwidth]{nmse_vs_pt_v2.eps}}\n \n  \\subfigure[Convergence behavior when $P_{\\rm T}=15$ dBm]{\n    \\label{nmse_vs_PT:b}\n    \\includegraphics[width=1\\columnwidth]{converge_v2.eps}}\n  \\caption{The NMSE versus transmit power when $M=8$ and $N=32$.}\n  \\label{nmse_vs_PT}\n\\end{figure}\n\n\\subsection{Simulation Results}\n{\\figurename~\\ref{nmse_vs_PT:a}} illustrates the NMSE of different schemes with respect to the transmit power of users, in which the BS adopts a $4\\times2$ uniform planar array (UPA), and the IRS adopts an $8\\times4$ UPA. Thus, we have $M=8$ and $N=32$.\nThe BALS algorithm in \\cite{Araujo2021JSAC_CE_PARAFAC} achieves the worst performance since it does not exploit the channel prior knowledge.\nThe performance of the LMMSE  using the traditional protocol in \\cite{Kundu2021OJCSLMMSE_DFTGOOD} and \\cite{Alwazani2020OJCSLMMSE_DFT} does not vary with the increase of $P_{\\rm T}$ since the main bottleneck is that the number of training timeslots is smaller than $N$.\nMoreover, the performance of BALS-MAP is better than that of LMMSE at a low SNR, but worsens as  $P_{\\rm T}$ increases since it will reduce to BALS when $P_{\\rm T}$ is infinite.\nBased on the above observations, we can draw a conclusion that the traditional channel estimation protocol shown in {\\figurename~\\ref{protocol:b}} is not effective for exploiting the common-link structure, and thus we do not consider  Baselines 2 and 3 in the remaining simulations.\nOn the other hand, it is seen that the proposed protocol with the optimization-based channel estimation algorithm achieves significant gain compared to all the baselines.\nIn addition, the phase shifting configuration by solving ${\\mathcal{P}}{(\\text{B})} $ achieves a more than 3 dB gain by steering the reflected signals in Stage I to the direction with a higher SNR compared to the random configuration baseline.\nNext, in {\\figurename~\\ref{nmse_vs_PT:b}}, we fix the transmit power $P_{\\rm T}$ at 15 dBm and show the convergence behaviors of the proposed Algorithm \\ref{alg:P1} for ${\\mathcal{P}}{(\\text{A})}$.\nOne can see that the proposed algorithm converges quickly.\nNote that although the solution without phase-shift optimization achieves a higher objective value, this does not imply it will have better performance since the objective functions of the two curves are different due to them adopting different ${\\bm{\\vartheta}}$ for the training phase shifting configuration.\n\n\\begin{figure}\n[!t]\n\\centering\n\\includegraphics[width=1\\columnwidth]{change_M_v2.eps}\n\\caption{NMSE versus $M$, when $P_{\\rm T}=20$ dBm.}\n\\label{M_vs_NMSE}\n\\end{figure}\n\nIn {\\figurename~\\ref{M_vs_NMSE}}, we simulate the performance of different BS antenna numbers $M$ when the BS adopts the uniform linear array and the IRS is still $8\\times4$ UPA. It is seen that the NMSE of all curves increases as $M$ increases since the ratio of the channel unknowns to the training observations decreases as $M$ increases. Moreover, the performance gain achieved by the phase shifting configuration increases as $M$ increases. This is because when $M$ increases, the number of training timeslots in Stage I of the proposed protocol decreases, and the probability that the random configuration scheme steers to the highest SNR direction becomes lower.\n\n\n\n\n\n\\begin{figure}\n[!t]\n\\centering\n\\includegraphics[width=1\\columnwidth]{change_N_v2.eps}\n\\caption{NMSE versus $N$, when $P_{\\rm T}=20$ dBm.}\n\\label{N_vs_NMSE}\n\\end{figure}\n\n\\begin{figure}\n[!t]\n\\centering\n\\includegraphics[width=1\\columnwidth]{CCDF_v2.eps}\n\\caption{The CCDFs for random user locations.}\n\\label{CCDF_location}\n\\end{figure}\n\nIn {\\figurename~\\ref{N_vs_NMSE}}, we simulate the NMSE of different schemes for different IRS sizes $N$. The BS is  $4\\times2$ UPA, and the IRS is $N_1 \\times 8$ UPA in which $N_1$ increases from $4$ to $10$. Note that as $N$ increases, the pilot overhead increases according to the proposed protocol but the ratio of the channel unknowns to the training observations is almost fixed.\nIt is seen that  the NMSE of Baselines 1 and 4 varies only a little, while the NMSE of the proposed scheme decreases as $N$ increases. This is because the channel becomes more correlated as $N$ becomes large, and the proposed scheme has a better capability of exploiting the channel prior knowledge.\n\nFinally, we investigate the impact of user locations on the estimation performance.\nIn particular, we fix $N=8 \\times 4$ and $M=4 \\times 2$, and generate $100$ snapshots for random user locations. For each snapshot, we further generate $1000$ channel realizations with independent small-scale fading to reduce the impact of other system parameters.\n{\\figurename~\\ref{CCDF_location}} plots the complementary cumulative distribution functions (CCDFs) of the NMSE for different snapshots.\nOne can see that the performance gains of the proposed scheme are irrespective of user locations.\nIn addition, we further increase $P_{\\rm T}=40$ dBm for Baseline 4 (i.e., the selected-on-off-protocol-based scheme \\cite{Liuliang_CE2020TWC}) such that it achieves a similar average NMSE to the proposed scheme with $P_{\\rm T}=20$ dBm. However, Baseline 4 achieves a much worse outage performance.\nThis is because the performance of the selected-on-off-protocol-based scheme \\cite{Liuliang_CE2020TWC} highly depends on the channel quality of the reference user, while the proposed scheme is much more robust since it does not require selecting one reference user.\n\n\n\n\\section{Conclusion}\\label{conclusion}\nIn this paper, we proposed a novel always-ON channel estimation protocol for uplink cascaded channel\nestimation in IRS-assisted MU-MISO systems.\nIn contrast to the existing schemes, the pilot overhead required by the proposed protocol is greatly reduced\nby exploiting the common-link structure.\nBased on the protocol, we formulated an optimization-based joint channel estimation problem that utilizes the combined\nstatistical information of the cascaded channels, and then we proposed an alternating optimization algorithm to solve the problem with the local optimum solution.\nIn addition, we optimized the phase shifting configuration in the proposed protocol, which may further enhance the channel estimation performance.\nThe simulation results demonstrated that the proposed protocol using the optimization based joint channel estimation algorithm achieves a more than $15$ dB gain compared to the benchmark. In addition, the proposed optimized phase shifting configuration  achieves a more than $3$ dB gain compared to the random configuration scheme.\n\n\n\n\\appendices\n\\section{Proof of Lemma \\ref{lemma0}}\\label{proof_lemma0}\nDefine the virtual reference channel by ${\\bf H}_{\\rm v}={\\bf G}^{\\rm T} {\\rm diag} ({\\bf H}_{\\rm r}\\bar{\\bf x})$. Based on \\eqref{equ:rbar}, we have\n\\begin{equation}\\label{equ:app0_Hv}\n\\begin{aligned}[b]\n\\left[\\bar{\\bf r}_1,\\cdots,\\bar{\\bf r}_N \\right]\n&={\\bf H}_{\\rm v}  {\\bm \\Phi}\n.\n\\end{aligned}\n\\end{equation}\nThus ${\\bf H}_{\\rm v}$ can be perfectly estimated by:\n\\begin{equation}\\label{equ:app0_Hv2}\n\\begin{aligned}[b]\n{\\bf H}_{\\rm v}= \\left[\\bar{\\bf r}_1,\\cdots,\\bar{\\bf r}_N \\right]\n{\\bm \\Phi}^{-1} .\n\\end{aligned}\n\\end{equation}\nWe further define the $K$ relative channels by ${\\bf h}_{{\\rm A},k}={\\rm diag} ({\\bf H}_{\\rm r}\\bar{\\bf x})^{-1} {\\bf h}_{{\\rm r},k}$, and ${\\bf H}_{\\rm A}=[{\\bf h}_{{\\rm A},1},{\\bf h}_{{\\rm A},2},\\cdots,{\\bf h}_{{\\rm A},K}]$.\nThen the cascaded channels become ${\\bf H}_{{\\rm I},k}= {\\bf H}_{\\rm v}{\\rm diag}({\\bf h}_{{\\rm A},k})$. Therefore, the remaining task is to perfectly estimate ${\\bf H}_{\\rm A}$.\n\nBased on the assumption on $\\bf G$ and ${\\bf H}_{\\rm r}$, we have\n${\\bf H}_{\\rm v}={\\bf F}_{\\rm B} {\\ddot{\\bf G}}^{\\rm T} {\\bf F}_{\\rm R}^{\\rm T} {\\rm diag} ({\\bf F}_{\\rm R} {\\ddot{\\bf H}}_{\\rm r}\\bar{\\bf x})$ where ${\\ddot{\\bf H}}_{\\rm r}=[{\\ddot{\\bf h}}_{{\\rm r},1},\\cdots,{\\ddot{\\bf h}}_{{\\rm r},K}]$.\nDefine ${\\bf V}=[{\\bf v}_{1},{\\bf v}_{2},\\cdots,{\\bf v}_{M}]$ which is given by\n\\begin{equation}\\label{equ:app0_V}\n\\begin{aligned}[b]\n{\\bf v}_m={\\rm diag} ({\\bf F}_{\\rm R} {\\ddot{\\bf H}}_{\\rm r}\\bar{\\bf x}){\\bf F}_{\\rm R} {\\ddot{\\bf g}}_m,\n\\end{aligned}\n\\end{equation}\nand thus ${\\bf H}_{\\rm v}={\\bf F}_{\\rm B} {\\bf V}^{\\rm T}$.\nUsing the independence of ${\\ddot{\\bf g}}_m$ and ${\\ddot{\\bf h}}_{{\\rm r},k}$, we have ${\\mathbb E}[{\\bf v}_i {\\bf v}_j^{\\rm H}]={\\bm 0}$. Since all ${\\bf v}_{m}$ ($m=1,2,\\cdots,M$) follow joint multivariate normal distribution, ${\\bf v}_{m}$ for all $m$ are pairwise  independent to each other.\nIn addition, since ${\\bf C}_m^{(k)}={\\mathbb E}\\left[ {\\bf h}_{{\\rm I},k,m} {\\bf h}_{{\\rm I},k,m}^{\\rm H} \\right]$ is full-rank for all $k$ and $m$, ${\\mathbb E}[{\\bf v}_m {\\bf v}_m^{\\rm H}]$ is also full-rank for all $m$ with properly-designed $\\bar{\\bf x}$.\n\nNext, using ${\\bf R}_{\\ell}$ in \\eqref{equ:R1} and \\eqref{equ:R2L1}, we have\n\\begin{equation}\\label{equ:app0_RL}\n\\begin{aligned}[b]\n\\tilde{\\bf R}_{\\ell} &= {\\bf F}_{\\rm B}^{-1} {\\bf R}_{\\ell} {\\bf X}^{-1}\\\\\n&={\\bf F}_{\\rm B}^{-1} {\\bf G}^{\\rm T} {\\rm diag} ({\\bm \\theta}_\\ell) {\\bf H}_{\\rm r}\\\\\n&={\\bf F}_{\\rm B}^{-1} {\\bf H}_{\\rm v} {\\rm diag} ({\\bm \\theta}_\\ell) {\\bf H}_{\\rm A}\\\\\n&={\\bf V}^{\\rm T} {\\rm diag} ({\\bm \\theta}_\\ell) {\\bf H}_{\\rm A}\n.\n\\end{aligned}\n\\end{equation}\nStacking all $\\tilde{\\bf R}_{\\ell}$, we have\n\\begin{equation}\\label{equ:app0_mesure}\n\\begin{aligned}[b]\n\\begin{bmatrix}\\tilde{\\bf R}_1\\\\ \\vdots \\\\ \\tilde{\\bf R}_{L_1}\\end{bmatrix}\n=\\begin{bmatrix}{\\bf V}^{\\rm T}{\\rm diag} ({\\bm \\theta}_1)\\\\ \\vdots \\\\ {\\bf V}^{\\rm T}{\\rm diag} ({\\bm \\theta}_{L_1})\\end{bmatrix}\n{\\bf H}_{\\rm A}.\n\\end{aligned}\n\\end{equation}\nDefine ${\\bm \\Psi}=[{\\rm diag} ({\\bm \\theta}_1){\\bf V},\\cdots,{\\rm diag} ({\\bm \\theta}_{L_1}){\\bf V}]^{\\rm T}$. Now we need to prove ${\\text{rank}}({\\bm \\Psi})={\\rm{min}}\\{N,ML_1\\}$ with probability one, and the whole proof is completed.\n\nBy permuting the columns of ${\\bm \\Psi}$, we have a new matrix\n$\\bar{\\bm \\Psi}=[{\\rm diag} ({\\bf v}_1)\\bar{\\bm \\Phi},\\cdots,{\\rm diag} ({\\bf v}_M)\\bar{\\bm \\Phi}]^{\\rm T}$\nwhere $\\bar{\\bm \\Phi}=[{\\bm \\theta}_1,\\cdots,{\\bm \\theta}_{L_1}]$.\nThen, it is equivalent to prove that ${\\text{rank}}(\\bar{\\bm \\Psi})={\\rm{min}}\\{N,ML_1\\}$ with probability one. We prove it by induction.\nDefine $\\bar{\\bm \\Psi}_m=[{\\rm diag} ({\\bf v}_1)\\bar{\\bm \\Phi},\\cdots,{\\rm diag} ({\\bf v}_m)\\bar{\\bm \\Phi}]^{\\rm T}$.\nSince $\\bar{\\bm \\Phi}$ is semi-orthogonal, ${\\text{rank}}(\\bar{\\bm \\Psi}_1)=L_1$ with probability one.\nLet ${\\text{rank}}(\\bar{\\bm \\Psi}_{m-1})={\\rm{min}}\\{N,(m-1)L_1\\}$ with probability one, and the rest task is to prove ${\\text{rank}}(\\bar{\\bm \\Psi}_m)={\\rm{min}}\\{N,mL_1\\}$ with probability one.\nWe prove it by contradiction.\nConsider the case when ${\\text{rank}}(\\bar{\\bm \\Psi}_{m-1})=(m-1)L_1$ which is smaller than $N$ but ${\\text{rank}}(\\bar{\\bm \\Psi}_m)<{\\rm{min}}\\{N,mL_1\\}$.\nSince ${\\rm diag} ({\\bf v}_m) {\\bm \\theta}_i$ and ${\\rm diag} ({\\bf v}_m) {\\bm \\theta}_j$ are orthogonal for $i \\neq j$, there shall exists $\\bf x$ which satisfies:\n\\begin{equation}\\label{equ:app0_x}\n\\begin{aligned}[b]\n[{\\rm diag} ({\\bf v}_1)\\bar{\\bm \\Phi},\\cdots,{\\rm diag} ({\\bf v}_{m-1})\\bar{\\bm \\Phi}]{\\bf x}={\\rm diag} ({\\bf v}_m) {\\bm \\theta}_\\ell,\n\\end{aligned}\n\\end{equation}\nfor some $1\\leq \\ell \\leq L_1$ such that ${\\text{rank}}(\\bar{\\bm \\Psi}_m)<{\\rm{min}}\\{N,mL_1\\}$ is true.\nHowever, since ${\\bf v}_m$ is independent to ${\\bf v}_1,{\\bf v}_2,\\cdots,{\\bf v}_{m-1}$ with full-rank covariance matrix, equation \\eqref{equ:app0_x} is inconsistent (which has no solution) with probability one, and the whole proof is finished.\n\n\\section{Estimation on the BS-User Channel}\\label{app_Estimate_Hd}\nBased on ${\\bm \\theta}_0=-{\\bm \\theta}_1$, we have\n\\begin{equation}\\label{equ:r_direct}\n\\begin{aligned}[b]\n{\\bf R}_{0} &= \\frac{1}{2}\\left({{\\bf Y}}_0+{{\\bf Y}}_1\\right) {\\bf X}^{-1}\\\\\n&={\\bf H}_{\\rm d}+\\tilde{\\bf Z}_0\n,\n\\end{aligned}\n\\end{equation}\nwhere $\\tilde{\\bf Z}_0$ is the noise matrix consisting of $MK$ i.i.d. complex Gaussian variables following ${\\cal{CN}}({ 0},\\frac{1}{2K}\\sigma_0^2) $.\nLet ${\\bf r}_{{0},k}$ be the $k$-th column of ${\\bf R}_{0}$.\nThe  BS-user direct channel for the $k$-th user can be estimated by the\nLMMSE\nestimator \\cite{Kay1993statisticalSP}:\n\\begin{equation}\\label{equ:hat_direct}\n\\begin{aligned}[b]\n\\hat{\\bf h}_{{\\rm d},k}={\\bf C}_{{\\rm d},k} \\left({\\bf C}_{{\\rm d},k}+ \\frac{\\sigma_0^2}{2K} {\\bf I}_M\\right)^{-1} {\\bf r}_{0,k},\n\\end{aligned}\n\\end{equation}\nwhere ${\\bf C}_{{\\rm d},k}={\\mathbb E}\\left[ {\\bf h}_{{\\rm d},k} {\\bf h}_{{\\rm d},k}^{\\rm H} \\right]$\nis the covariance matrix of the direct channel from the BS to the $k$-th user.\nNote that, as shown in \\eqref{equ:r_direct} and \\eqref{equ:hat_direct}, the proposed protocol may achieve a $3$ dB performance gain compared to the existing works, \\cite{Kundu2021OJCSLMMSE_DFTGOOD,Alwazani2020OJCSLMMSE_DFT,Liuliang_CE2020TWC}, on the estimation of the BS-user channels since it exploits doubled observation samples.\n\n\\section{Proof of Lemma \\ref{eq_decomposite}}\\label{proof_lemma1}\nThe objective function of ${\\mathcal{P}}{(\\text{A})}$ can be denoted by the function of the cascaded channel coefficients $\\{{\\bf h}_{{\\rm I},k,m}\\}$, as follows:\n\\begin{equation}\\label{equ:obj_f_A2}\n\\begin{aligned}[b]\n&f_{{\\rm A}}({\\bf H}_{\\rm g}, {\\bf H}_{\\rm u})\n=f_{{\\rm A}}(\\{{\\bf h}_{{\\rm I},k,m}\\})\\\\\n&=-\\sum_{\\ell=1}^{L_1} \\sum_{k=1}^K\n{\\frac{1}{\\sigma_{\\ell}^{2}} \\left\\|\n\\tilde{{\\bf r}}_{\\ell,k}- \\sum_{m=1}^M {\\bf h}_{{\\rm I},k,m}^{\\rm T} {\\bm \\theta}_{\\ell}\n\\right\\|^2 }\\\\\n&\\quad-\\sum_{\\ell=L_1+1}^{N}\n{ \\frac{1}{\\bar\\sigma_{\\ell}^{2}} \\left\\|\n\\bar{\\bf r}_{\\ell}-\\sum_{k=1}^K \\sum_{m=1}^M \\bar{x}_k {\\bf h}_{{\\rm I},k,m}^{\\rm T} {\\bm \\theta}_{\\ell}\n\\right\\|^2 }\n\\\\\n&\\quad-\\sum_{m=1}^M \\sum_{k=1}^K {\\bf h}_{{\\rm I},k,m}^{\\rm H} {{\\bf C}_m^{(k)}}^{-1} {\\bf h}_{{\\rm I},k,m}  .\n\\end{aligned}\n\\end{equation}\nTherefore, if one optimal solution $\\left\\{{\\bf H}_{\\rm g}^\\star,{\\bf H}_{\\rm u}^\\star \\right\\} \\in {\\cal A}$, any $\\left\\{{\\bf H}_{\\rm g},{\\bf H}_{\\rm u}\\right\\}$ pair in set ${\\cal A}$ is an optimal solution of ${\\mathcal{P}}{(\\text{A})}$, and the lemma is proved.\n\n\n\n\\section{Proof of Lemma \\ref{fu_convex}}\\label{proof_lemmafu}\nDenote $\\ddot{\\bf h}_{\\rm u}={\\rm{vec}}({\\bf H}_{\\rm u})$. The objective function $f_{{\\rm u}}$ in \\eqref{equ:obj_f_hrk} is given by\n\\begin{equation}\\label{equ:proof_fu_1}\n\\begin{aligned}[b]\n&f_{{\\rm u}}(\\ddot{\\bf h}_{\\rm u})\n=\\sum_{\\ell=1}^{L_1}\n{\\frac{1}{\\sigma_{\\ell}^{2}} \\left\\|\n{\\rm{vec}} (\\tilde{{\\bf R}}_{\\ell})- \\left({\\bf I}_K \\otimes {\\bf D}_{\\ell}\\right) \\ddot{\\bf h}_{\\rm u}\n\\right\\|^2_{2} }\\\\\n&\\quad+\\sum_{\\ell=L_1+1}^{N}\n{ \\frac{1}{\\bar\\sigma_{\\ell}^{2}} \\left\\|\n\\bar{\\bf r}_{\\ell}-\\left({\\bar{\\bf x}}^{\\rm T} \\otimes {\\bf D}_{\\ell}\\right) \\ddot{\\bf h}_{\\rm u}\n\\right\\|^2_2 }\n+ \\ddot{\\bf h}_{\\rm u}^{\\rm H} {\\bf C}_{{\\rm u}} \\ddot{\\bf h}_{\\rm u}\n,\n\\end{aligned}\n\\end{equation}\nwhere ${\\bf C}_{{\\rm u}}={\\rm{blkdiag}}({\\bf C}_{{\\rm u},1},{\\bf C}_{{\\rm u},2},\\cdots,{\\bf C}_{{\\rm u},K})$.\nThe second order derivative of $f_{{\\rm u}}(\\ddot{\\bf h}_{\\rm u})$ is given by\n\\begin{equation}\\label{equ:proof_fu_2}\n\\begin{aligned}[b]\n\\frac{ \\partial^2 f_{{\\rm u}}(\\ddot{\\bf h}_{\\rm u})}{\\partial \\ddot{\\bf h}_{\\rm u} \\partial \\ddot{\\bf h}_{\\rm u}^{\\rm H}}\n&=2{\\bf C}_{{\\rm u}}\n+2{\\bf I}_K \\otimes \\left(\\sum_{\\ell=1}^{L_1} \\frac{1}{\\sigma_{\\ell}^{2}} {\\bf D}_{\\ell}^{\\rm H} {\\bf D}_{\\ell}\\right) \\notag\\\\\n&\\quad + 2\\left(\\bar{\\bf x}^\\ast \\bar{\\bf x}^{\\rm T}\\right)\\otimes \\left(\\sum_{\\ell=L_1+1}^{N} \\frac{1}{\\bar\\sigma_{\\ell}^{2}} |\\bar{x}_k|^2 {\\bf D}_{\\ell}^{\\rm H} {\\bf D}_{\\ell}\\right)\n,\n\\end{aligned}\n\\end{equation}\nwhich is a  Hermitian positive semi-definite matrix. Thus, the lemma is proved.\n\n\\section{Proof of Lemma \\ref{fg_convex}}\\label{proof_lemmafg}\nThe second order derivative of $f_{{\\rm g},m}({\\bf h}_{{\\rm g},m})$ in \\eqref{equ:obj_f_gm} is given by\n\\begin{equation}\\label{equ:proof_fg_1}\n\\begin{aligned}[b]\n&\\frac{ \\partial^2 f_{{\\rm g},m}({\\bf h}_{{\\rm g},m})}{\\partial {\\bf h}_{{\\rm g},m} \\partial {\\bf h}_{{\\rm g},m}^{\\rm H}}\n=2{\\bf C}_{{\\rm g},m}^{-1}\n+2\\sum_{k=1}^K\\sum_{\\ell=1}^{L_1} \\frac{1}{\\sigma_{\\ell}^{2}} {\\bf b}_{\\ell,k}^\\ast {\\bf b}_{\\ell,k}^{\\rm T}\\\\\n&\\quad+ 2\\sum_{\\ell=L_1+1}^{N}\\frac{1}{\\bar\\sigma_{\\ell}^{2}}\n\\left(\\sum_{k =1}^K \\bar{x}_k {\\bf b}_{\\ell,k}^{\\rm T}\\right)^{\\rm H}\n\\left(\\sum_{k =1}^K \\bar{x}_k {\\bf b}_{\\ell,k}^{\\rm T}\\right)\n,\n\\end{aligned}\n\\end{equation}\nwhich is a  Hermitian positive semi-definite matrix. Thus, $f_{{\\rm g},m}({\\bf h}_{{\\rm g},m})$ is a convex quadratic function of ${\\bf h}_{{\\rm g},m}$. Then, the objective function $\\sum_{m=1}^M f_{{\\rm g},m}({\\bf h}_{{\\rm g},m})$ is a convex quadratic function of $\\{{\\bf h}_{{\\rm g},1},{\\bf h}_{{\\rm g},2},\\cdots,{\\bf h}_{{\\rm g},M}\\}$.\n\n\n\n\n\n\\bibliographystyle{IEEEtran\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n The stability of quantum motion in dynamical systems,\n measured by quantum Loschmidt echo \\cite{Peres84}, has attracted much attention\n in recent years.\n The echo is the overlap of the evolution of the\n same initial state under two Hamiltonians with slight difference in the classical limit,\n $ M(t) = |m(t) |^2 $, where\n \\begin{equation} m(t) = \\langle \\Psi_0|{\\rm exp}(iHt\/ \\hbar ) {\\rm exp}(-iH_0t \/ \\hbar) |\\Psi_0 \\rangle  \\label{mat} \\end{equation}\n is the fidelity amplitude.\n Here $H_0$ and $H$ are the unperturbed and perturbed Hamiltonians, respectively,\n $ H=H_0 + \\epsilon H_1 $,  with $\\epsilon $ a small quantity and $H_1$ a perturbation.\n This quantity $M(t)$ is called fidelity in the field of\n quantum information \\cite{nc-book}.\n\n\n Fidelity decay in quantum systems whose classical counterparts have\n strong chaos with exponential instability, has been studied well\n \\cite{JP01,JSB01,CLMPV02,JAB02,BC02,CT02,PZ02,WL02,VH03,STB03,WCL04,Vanicek04,WL05,WCLP05,GPSZ06}.\n Related to the perturbation strength, previous investigations show\n the existence of at least three regimes of fidelity decay:\n (i) In the perturbative regime in which the typical transition matrix element is smaller than the\n mean level spacing, the fidelity has a Gaussian decay.\n (ii) Above the perturbative regime, the fidelity has an exponential decay with a rate\n proportional to $\\epsilon^2$, usually called the Fermi-golden-rule (FGR) decay of fidelity.\n (iii) Above the FGR regime is the Lyapunov regime in which $M(t)$ has usually an\n approximate exponential decay with a perturbation-independent rate.\n\n\n Fidelity decay in regular systems with quasiperiodic motion in the\n classical limit has also attracted much attention\n \\cite{PZ02,JAB03,PZ03,SL03,Vanicek04,WH05,Comb05,HBSSR05,GPSZ06,WB06,pre07}.\n For single initial Gaussian wavepacket,  the fidelity has\n been found to have initial Gaussian decay followed by power law decay\\cite{PZ02,WH05,pre07}.\n\n\n Meanwhile, there exists a class of system which lies between the two classes of system mentioned above,\n namely, between chaotic systems with exponential instability and regular systems\n with quasiperiodic motion.\n One example of this class of system is the triangle map proposed by Casati and Prosen \\cite{triangle}.\n The map has linear instability with vanishing Lyapunov exponent,\n but can be ergodic and mixing with power-law decay of correlations.\n The classical Loschmidt echo in the triangle map has been studied recently\n and found behaving differently from that in systems with exponential instability\n and in systems with quasiperiodic motion \\cite{c-fid-tri}.\n This suggests that the decaying behavior of fidelity in the quantum triangle map may be\n different from that in the other two classes of system as well.\n In this paper, we present numerical results which confirm this expectation.\n\n\n Specifically, like in systems possessing strong chaos,\n in the triangle map three regimes of fidelity decay are found\n with respect to the perturbation strength: weak, intermediate and strong.\n However, in each of the three regimes, the decaying law(s) for the fidelity in the triangle map has\n been found different from that in systems possessing strong chaos.\n In section II, we recall properties of the classical triangle map\n and discuss its quantization.\n Section III is devoted to numerical investigations for the laws of fidelity decay\n in the three regimes of perturbation strength.\n Conclusions are given in section IV.\n\n\n \\section{Triangle map}\n\n\n \\begin{figure}\n \\includegraphics[width=\\columnwidth]{s0001-t1p7-N8.EPS}\n \\caption{ (color online).\n Averaged fidelity at weak perturbation, $\\sigma =10^{-4}$(solid curve),\n with average taken over 50 initial point sources chosen randomly, $N=2^{12}=4096$.\n The dashed-dotted straight line has a slope 1.7, showing that $\\log_{10}\\overline M(t)$ is approximately\n a function of $t^{1.7}$.\n For comparison, we also show two straight lines (dashed and dotted) with\n slopes 1 and 2, respectively.\n } \\label{fig-s0001-t1p7}\n \\end{figure}\n\n\n\n On the torus $(r,p) \\in {T}^2 = [-\\pi ,\\pi ) \\times [-\\pi ,\\pi )$,\n the triangle map is\n \\begin{eqnarray} \\nonumber p_{n+1} = p_n + \\alpha \\ \\text{sgn} (r_n)+ \\beta , \\hspace{1cm} (\\text{mod} 2\\pi )\n \\\\ r_{n+1} = r_n + p_{n+1} ,  \\hspace{1cm} (\\text{mod} 2\\pi ) \\label{map} \\end{eqnarray}\n where $\\text{sgn}(r) = \\pm 1 $ is the sign of $r$ for $r \\ne 0$ and\n $\\text{sgn}(r) =0$ for $r=0$ \\cite{triangle}.\n Rich behaviors have been found in the map:\n For rational $\\alpha \/\\pi$ and $\\beta \/\\pi$, the system is pseudointegrable.\n With the choice of $\\alpha =0$ and irrational $\\beta \/ \\pi$, it is ergodic but not mixing.\n Interestingly, for incommensurate irrational values of $\\alpha \/\\pi$ and $\\beta \/\\pi$,\n the dynamics is ergodic and mixing.\n In our numerical calculations, we take $\\alpha = \\pi^2 $ and $\\beta = (\\sqrt{5}  -1)\\pi \/2$,\n {for which $(\\beta \/ \\alpha )$ is an irrational number, the golden mean divided by $\\pi$,\n and the map is ergodic and mixing.}\n\n\n The triangle map (\\ref{map}) can be associated with the Hamiltonian\n \\begin{eqnarray} H = \\frac 12 \\widetilde p^2 + V(r) \\sum_{n=-\\infty }^{\\infty } \\delta (t-nT), \\label{H} \\end{eqnarray}\n where $ V(r) = - \\widetilde \\alpha |r| - \\widetilde \\beta r$ and $T$ is the period of kicking.\n It is easy to verify that the dynamics produced by this Hamiltonian gives the map (\\ref{map})\n with the replacement $p=T\\widetilde p, \\alpha = T\\widetilde \\alpha $, and $\\beta =T\\widetilde \\beta $.\n\n\n \\begin{figure}\n \\includegraphics[width=\\columnwidth]{s0001-001-N1.EPS}\n \\caption{ (color online).\n Averaged fidelity at three weak perturbation strengths, $\\sigma =10^{-4}$(thin solid curve), $10^{-3}$\n (dashed curve), and $10^{-2}$(thick solid curve),\n with average taken over 50 initial point sources chosen randomly, $N=2^{12}=4096$.\n The dashed-dotted straight line represents $M_1(t)$ in Eq.~(\\ref{ctgamma})\n with $\\gamma =1.7$ and $c$ as an adjusting parameter.\n Inset: Fidelity of $\\sigma =10^{-3}$ and $N=2^{n}$;\n the two curves are almost indistinguishable.\n } \\label{fig-s0001-tlog}\n \\end{figure}\n\n\n The classical map can be quantized by the method of quantization on torus\n \\cite{HB80-q-tori, FMR91,WB94,Haake}.\n Schr\\\"{o}dinger evolution under the Hamiltonian in Eq.~(\\ref{H})\n for one period of time is given by the Floquet operator\n \\begin{equation} \\label{U1} U = \\exp \\left [ -\\frac i2 ({\\hat{\\widetilde p}})^2T \\right ]\n \\exp [-i V( {\\hat r}) ], \\end{equation}\n where we set $\\hbar =1$ in Schr\\\"{o}dinger equation.\n In this quantization scheme, an effective Planck constant $\\hbar_{\\rm eff}=T$ is introduced.\n It has the following relation to the dimension $N$ of the Hilbert space,\n \\begin{equation} \\label{h} N h_{\\rm eff} =4\\pi^2, \\end{equation}\n hence, $\\hbar_{\\rm eff} = 2\\pi \/ N$.\n In what follows, for brevity, we will omit the subscript eff of $\\hbar_{\\rm eff}$.\n Eigenstates of $\\hat{r} $ and $\\hat p$ are discretized,\n $\\hat{r}|j\\rangle = j \\hbar |j\\rangle $ and $\\hat{p}|k\\rangle = k \\hbar |k\\rangle $,\n with  $j,k =-N\/2,-N\/2+1,\\ldots ,0,1, \\ldots , (N\/2)-1$.\n Then,\n {making use of the above discussed relations among $\\widetilde p, p, T, \\widetilde \\alpha , \\alpha ,\n \\widetilde \\beta , \\beta $, in particular, $T=\\hbar $},\n the Floquet operator in Eq.~(\\ref{U1}) can be written as\n \\begin{equation} \\label{U} U = \\exp \\left [ -\\frac {i}{2\\hbar} ({\\hat{ p}})^2 \\right ]\n \\exp \\left [ \\frac{i}{\\hbar} (\\alpha |\\hat r| +\\beta \\hat r) \\right ] . \\end{equation}\n In numerical computation, the time evolution\n $ |\\psi (t)\\rangle = U^t |\\psi_0\\rangle $ is calculated by the fast Fourier transform (FFT) method.\n\n\n The fidelity in Eq.~(\\ref{mat}) involves two slightly different Hamiltonians,\n unperturbed and perturbed.\n In this paper, for an unperturbed system with parameters $\\alpha $ and $\\beta $,\n the perturbed system is given by\n \\begin{equation} \\alpha  \\to \\alpha  + \\epsilon \\ \\ \\ \\ \\beta  \\to \\beta   . \\end{equation}\n Without the loss of generality, we assume $\\epsilon \\ge 0$.\n The parameter $\\sigma =(\\epsilon \/ \\hbar )$ can be used to characterize the strength of quantum\n perturbation.\n\n\n\n \\begin{figure}\n \\includegraphics[width=\\columnwidth]{mt-s001-01-N2.EPS}\n \\caption{ (color online).\n Variation of the averaged fidelity with $\\sigma t$ for  $\\sigma =0.01, 0.02$ and 0.1,\n with average taken over 100 initial point sources chosen randomly, $N=4096$.\n The solid straight line is drawn for a comparison with linear dependence on $\\sigma t$.\n For $\\sigma = 0.02$ and 0.1, $\\log_{10} \\overline M(t)$ is approximately a linear function of $ \\sigma t$,\n before it becomes close to the saturation value.\n Inset: The distribution $P(y)$ for the action difference $\\Delta S$ at $t=40$,\n where $y=(\\Delta S -\\langle \\Delta S \\rangle )\n \/ \\epsilon $ and $\\langle \\Delta S \\rangle$ is the average value of $\\Delta S$.\n It is calculated by taking randomly $10^7$ initial points in the phase space.\n $P(y)$ does not have a Gaussian shape.\n } \\label{fig-mt-s001-01}\n \\end{figure}\n\n\n \\section{Three regimes of fidelity decay}\n\n \\subsection{Weak perturbation regime}\n\n\n Let us first discuss weak perturbation.\n As mentioned in the introduction, in systems with strong chaos in the classical limit,\n the fidelity has a Gaussian decay under sufficiently weak perturbation.\n The Gaussian decay is derived by making use of the first order perturbation theory for eigensolutions\n of $H$ and $H_0$ and the random matrix\n theory for $\\Delta E_n \\equiv E_n-E^0_n$, where $E_n$ and $E^0_n$ are\n eigenenergies of $H$ and $H_0$, respectively.\n Numerical results in Ref.~\\cite{EKW05} show agreement of the spectral\n statistics in the triangle map with the prediction of random matrix theory,\n hence, at first sight, Gaussian decay might be expected for the fidelity decay\n in the weak perturbation regime of the triangle map.\n\n\n However, our numerical results show a non-Gaussian decay of fidelity for small perturbation.\n An example is given in Fig.~\\ref{fig-s0001-t1p7} for $\\sigma=10^{-4}$.\n To obtain relatively smooth curves for fidelity,\n average has been taken over 50 initial point sources (eigenstates of $\\hat r$) chosen randomly.\n This figure, plotted with $\\log_{10} \\left (-\\log_{10} \\overline M(t)\\right )$ versus $ \\log_{10} t $,\n shows clearly that $\\log_{10}\\overline M(t)$ is approximately proportional to $t^{1.7}$ (the\n dashed-dotted straight line), while is far from the Gaussian case of $t^2$ and the\n exponential case of $t$ represented by the dotted and dashed lines, respectively.\n\n\n Furthermore, we found that the averaged fidelity $\\overline M(t)$ can be fitted well by\n \\begin{equation} \\label{ctgamma} M_1(t) = \\exp (-c \\sigma^2 t^{\\gamma }) \\end{equation}\n with $\\gamma \\simeq 1.7$ and $c$ as a fitting parameter.\n In Fig.~\\ref{fig-s0001-tlog}, we show fidelity decay for three different values of $\\sigma $.\n With the horizontal axis scaling with $\\log_{10}\\sigma^2 t^{1.7 }$,\n the three curves corresponding to the three values of $\\sigma $\n are hardly distinguishable in their overlapping regions (except for long times).\n Note that, to show clearly the dashed-dotted straight line which represents\n $M_1(t)$ in Eq.~(\\ref{ctgamma}),\n we have deliberately adjusted a little the best-fitting value of $c$ such that the dashed-dotted line\n is a little above the curves of the fidelity.\n\n\n \\begin{figure}\n \\includegraphics[width=\\columnwidth]{s0p1-n11-n12-N3.EPS}\n \\caption{ (color online).\n Fidelity decay for $\\sigma =0.1$ and $N=2^{n}$,\n averaged over 100 initial point sources.\n } \\label{fig-s0p1-n11-n12}\n \\end{figure}\n\n\n\n In the inset of Fig.~\\ref{fig-s0001-tlog}, we show curves of fidelity for the same $\\sigma$\n but different values of $\\epsilon $ and $N$.\n The two curves are very close, supporting the assumption\n that $\\epsilon $ and $N$ appear in the form of the\n single variable $\\sigma $ as written on the right hand side of Eq.~(\\ref{ctgamma}).\n This dependence of $\\overline M(t)$ on the variable $\\sigma $ for sufficiently small $\\sigma $\n can be understood in a first-order perturbation treatment of fidelity,\n as shown in the following arguments.\n\n\n Let us consider a Hilbert space with sufficiently large dimension $N$\n and make use of arguments similar to those used in Ref.~\\cite{CT02}\n for deriving the Gaussian decay,\n but without assuming the applicability of the random matrix theory.\n It follows that, for times not very long, the averaged fidelity (averaged over initial states)\n is mainly determined by\n $\\langle \\exp (-i\\Delta \\omega_n t ) \\rangle$, where $\\Delta \\omega_n =\\omega_{n} -\\omega_n^0$\n and $\\langle \\ldots \\rangle $ indicates average over the quasi-spectrum.\n Here $\\omega_n^0$ is an eigen-frequency of the Floquet operator $U$ in Eq.~(\\ref{U})\n and $\\omega_n$ is the corresponding eigen-frequency of $(U e^{i\\sigma |r|})$.\n For large $N$, $\\langle \\exp (-i\\Delta \\omega_n t ) \\rangle$\n can be calculated by making use of the distribution of $\\Delta \\omega_n $.\n Since the two Floquet operators $U$ and $(U e^{i\\sigma |r|})$ differ by $e^{i\\sigma |r|}$,\n the distribution of $\\Delta \\omega_n $ is approximately a function of $\\sigma $.\n Then, $M(t)$ is approximately a function $\\sigma $.\n\n\n Finally, we give some remarks on the value of $\\gamma $.\n When $\\Delta \\omega_n$ has a Gaussian distribution, $\\overline M(t)$ has a Gaussian decay with $\\gamma =2$,\n as in the case of systems possessing strong chaos.\n In the triangle map, the non-Gaussian decay of fidelity discussed above implies\n that $\\Delta \\omega_n$ does not have a Gaussian distribution.\n Other types of distribution may predict values of $\\gamma $ different from 2, in particular,\n a L\\'{e}vy distribution would give $\\gamma <2$ in agreement with our numerical result.\n We also remark that the results here are not in confliction with numerical results of Ref.~\\cite{EKW05},\n in which only the statistics of $\\omega_n$ (not that of $\\Delta \\omega_n$) is found\n in agreement with the prediction of random matrix theory.\n\n\n \\subsection{Intermediate perturbation strength}\n\n\n \\begin{figure}\n \\includegraphics[width=\\columnwidth]{mt-s01-1-poi-st-N4.EPS}\n \\caption{ (color online).\n Averaged fidelity of $\\sigma$ from 0.1 to 1, with average taken over 1000 randomly chosen\n initial pointer sources, $N=2^{14}=16384$.\n For $\\sigma =0.2$ and above, the averaged fidelity obeys a decaying law which is different from\n that in Eq.~(\\ref{Mt-sigma-t}), in particular, it is not a function of $(\\sigma t)$.\n } \\label{fig-mt-s01-1-poi-st}\n \\end{figure}\n\n\n\n With increasing perturbation strength, exponential decay of $\\overline M(t)$ appears\n (see Fig.~\\ref{fig-mt-s001-01}).\n For $\\sigma $ from 0.02 to 0.1, after some initial times and before approaching\n its saturation value, the fidelity decays as\n \\begin{equation} M_2(t) = \\exp (- a \\sigma t), \\label{Mt-sigma-t} \\end{equation}\n with $a$ as a fitting parameter.\n Numerically, we found that $a \\approx 0.08$.\n The decay rate is proportional to $(\\sigma t)$, unlike in the FGR decay found in systems\n with strong chaos,\n \\begin{equation} M_{\\rm FGR}(t) \\sim \\exp (-2 \\sigma^2 K_E t), \\label{FGR} \\end{equation}\n where $K_E$ is the classical action diffusion constant \\cite{CT02}.\n The curves of $\\sigma =0.02$ and 0.1 in Fig.~\\ref{fig-mt-s001-01} are quite close,\n while that of $\\sigma =0.01$ has some deviation from the two.\n This implies that the $\\exp (- a \\sigma t)$ behavior of $\\overline M(t)$ appears\n between $\\sigma =0.01$ and 0.02.\n Note that vertical shifts have been made for the two curves of\n $\\sigma =0.02$ and 0.1  in Fig.~\\ref{fig-mt-s001-01} for better comparison.\n\n\n \\begin{figure}\n \\includegraphics[width=\\columnwidth]{lglgm-st2p5-s2-10-N5.EPS}\n \\caption{(color online).\n Averaged fidelity at strong perturbation,\n with average taken over 1000 randomly chosen initial Gaussian wavepackets,\n $N=2^{17}=131072$.\n $z=\\epsilon t^{2.5}\/\\hbar $ with $\\hbar $ fixed in this figure.\n The solid line represents a curve $\\exp (-c \\epsilon t^{2.5})$,\n where the fitting parameter $c$ is determined from comparison with the two curves\n of $\\sigma =2$ and 4 in the small-$z$ region.\n } \\label{fig-lglgm}\n \\end{figure}\n\n\n The origin of the non-FGR decay of fidelity\n in this regime of perturbation strength, may come from weak chaos.\n In fact, in another system which also possesses weak chaos in the classical limit,\n namely, the sawtooth map in some parameter regime, linear dependence of the decaying rate\n on $\\sigma$ has also been observed in the intermediate perturbation regime \\cite{WCL04,WL05,foot1}.\n In this regime of perturbation strength, the semiclassical theory predicts that,\n in the first order classical perturbation theory,\n the averaged fidelity is given by \\cite{WCL04}\n \\begin{eqnarray}  \\overline M(t) \\simeq \\left | \\int d\\Delta S e^{i\\Delta S\/ \\hbar }\n P(\\Delta S)\\right |^2,   \\label{Mp-ps} \\end{eqnarray}\n where\n $ \\Delta S( {\\bf p} _0 ,  {\\bf r} _0 ; t) = \\epsilon  \\int_0^t dt' H_1[( {\\bf r}  (t')]$\n is the action difference of two the classical trajectories starting at the same\n point $( {\\bf p} _0 ,  {\\bf r} _0)$ in the two systems,\n with $H_1$ evaluated along one of the two trajectories,\n and $P(\\Delta S)$ is the distribution of $ \\Delta S( {\\bf p} _0 ,  {\\bf r} _0 ; t)$.\n In systems possessing strong chaos, $P(\\Delta S)$ may have a Gaussian form, which implies\n the FGR decay for the fidelity.\n In the triangle map, $P(\\Delta S)$ is not a Gaussian distribution\n as shown in the inset of Fig.~\\ref{fig-mt-s001-01},\n hence, the fidelity does not have the FGR decay with a rate proportional to $\\sigma^2$.\n\n\n It is difficult to find an analytical expression for $P(\\Delta S)$,\n hence, we can not derive Eq.~(\\ref{Mt-sigma-t}) analytically.\n However, a qualitative understanding of the $(\\sigma t)$-dependence of $\\overline M(t)$ can be\n gained, as shown in the following arguments.\n Equation (\\ref{Mp-ps}) shows that the time-dependence of fidelity decay comes mainly from\n the dependence of $P(\\Delta S)$ on time.\n In the case of strong chaos, $\\Delta S$ behaves like a random walk, hence,\n $P(\\Delta S)$ has a Gaussian form with a width increasing as $\\sqrt t$ \\cite{CT02}.\n Since $\\Delta S \\propto \\epsilon $, the width of $P(\\Delta S)$ is a function of $(\\epsilon \\sqrt t)$;\n then, Eq.~(\\ref{Mp-ps}) gives the FGR decay of $\\overline M(t)$ which depends on $(\\sigma^2t)$.\n In the case of the triangle map, due to the linear instability of the map, it may happen\n that the width of $P(\\Delta S)$ increase linearly with $ t$\n in some situations when $t$ is not very long.\n This implies that the width of $P(\\Delta S)$ may be a function of the variable $(\\epsilon t)$.\n Then, it is possible for $\\overline M(t)$ to be approximately a function of $(\\sigma t)$.\n\n\n Equation (\\ref{Mp-ps}) predicts that, up to the first order classical perturbation theory,\n the dependence of $\\overline M(t)$ on $\\epsilon $ and $\\hbar $\n takes the single variable $\\sigma =\\epsilon \/ \\hbar $.\n Numerically we found that this is approximately correct, as shown in Fig.~\\ref{fig-s0p1-n11-n12}.\n Specifically, for fixed $\\sigma =0.1$, $\\overline M(t)$ of $N=2^{11}$ and of $N=2^{12}$ separate at about $t=15$.\n Indeed, for long times $t$, higher order contributions in the classical perturbation theory may\n need consideration and $\\overline M(t)$ may depend on $\\epsilon  $ and $\\hbar $ in a different way.\n For larger $N$, hence smaller $\\hbar $, the agreement becomes better,\n e.g., $\\overline M(t)$ of $N=2^{12}$ is closer to $N=2^{13}$ than to $N=2^{11}$.\n\n\n\n When $\\sigma $ goes beyond 0.1,\n the exponential decay of $\\overline M(t)$ expressed in Eq.~(\\ref{Mt-sigma-t}) disappears,\n in particular, the dependence of $\\overline M(t)$ on $\\sigma $ and $t$ does not take the form of $(\\sigma t)$\n (see Fig.~\\ref{fig-mt-s01-1-poi-st}).\n Meanwhile fluctuations of $\\overline M(t)$ becomes larger and larger with increasing $\\sigma $\n for initial point states.\n For example, Fig.~\\ref{fig-mt-s01-1-poi-st} shows that $\\overline M(t)$ of $\\sigma =1$\n has considerable fluctuations even after averaging over 1000 initial point sources.\n Taking initial Gaussian wavepackets, the fluctuations can be much suppressed.\n\n\n \\begin{figure}\n \\includegraphics[width=\\columnwidth]{mt-sgt1-tavg-N6.EPS}\n \\caption{ (color online).\n Averaged Fidelity for strong perturbation, from top to bottom, $\\sigma =2,4$ and 10.\n The average is taken over 1000 initial Gaussian wavepackets chosen randomly\n and over time from $t-2$ to $t+2$. $N=2^{17}$.\n The time axis is plotted in the logarithm scale.\n It shows that the long time decay of fidelity is slower than power law decay.\n } \\label{fig-mt-sgt1-tavg}\n \\end{figure}\n\n\n\n \\subsection{Strong perturbation regime}\n\n\n The triangle map has vanishing Lyapunov exponent, hence, its fidelity may not have\n the perturbation-independent decay\n which has been observed at strong perturbation in systems possessing exponential\n instability in the classical limit\n \\cite{JP01,BC02,STB03,WCLP05}.\n To understand fidelity decay in the triangle map, it is helpful to recall results\n about the classical fidelity given in \\cite{c-fid-tri}.\n In the classical triangle map, the classical fidelity decays as\n $M_{cl}(t) \\sim \\exp (-c \\epsilon t^{2.5})$\n for initial times when $M_{cl}(t)$ remains close to one,\n and has an exponential decay $\\exp (-c' \\epsilon^{2\/5}t)$ for longer times.\n The interesting feature is that the classical fidelity depends\n on the same scaling variable $\\tau \\equiv \\epsilon t^{2.5}$ in different time regions.\n\n\n In the weak and intermediate perturbation regimes discussed in the previous sections,\n the dependence of fidelity on $\\epsilon $ and $t$ does not take the form of the single\n variable $\\tau$.\n This is not strange, because the classical limit is achieved in the limit\n $\\hbar \\to 0$, which implies $\\sigma \\to \\infty$ for whatever small but fixed $\\epsilon $.\n Therefore, it is the strong perturbation regime in which\n the decaying behavior of fidelity may have some relevance to the classical fidelity.\n Numerical results presented below indeed support this expectation.\n\n\n \\begin{figure}\n \\includegraphics[width=\\columnwidth]{mt-sgt1-tavg-140-1k-N7.EPS}\n \\caption{ (color online).\n The same as in Fig.~\\ref{fig-mt-sgt1-tavg},\n with a different scale for the horizontal axis and\n for the time interval $140 < t < 1000$.\nFor $\\sigma =4$ and 10, $\\log_{10}\\overline M(t)$ form two lines for each $\\sigma$.\nThe three solid lines represent $\\log_{10} M_3(t)$ given by Eq.~(\\ref{loglogt}),\nwith $b=9.6,9.3$, and 8.3 from top to bottom.\n } \\label{fig-mt-sgt1-tavg-140-1k}\n \\end{figure}\n\n\n Figure \\ref{fig-lglgm} shows variation of the averaged fidelity with $\\log_{10}\\epsilon t^{2.5}$,\n with average taken over 1000 initial Gaussian wavepackets chosen randomly.\n The initial decay of the fidelity of $\\sigma =2$ and 4\n are quite close to the classical prediction $\\exp (-c \\epsilon t^{2.5})$.\n For longer times, the fidelity of $\\sigma $ from 2 to 10 (with $\\hbar $ fixed)\n is approximately a function of $\\tau$, the scaling variable predicted in the classical case,\n but, the decaying behavior of fidelity\n is not the same as that of the classical fidelity, i.e., not an exponential decay.\n We found that the dependence of $\\overline M(t)$ on $\\hbar$ does not take the form of $\\tau \/ \\hbar$,\n i.e., $\\overline M(t)$ is not a function of the single variable $(\\tau \/ \\hbar )$.\n\n\n For long times, the fidelity has large fluctuations even after averaging over 1000\n initial Gaussian wavepackets.\n The fluctuations can be much suppressed, when a further average is taken for time $t$ .\n Specifically, for each time $t$, we take average over $\\overline M(t')$ for $t'$ from $t-2$ to $t+2$.\n The results are given in Fig.~\\ref{fig-mt-sgt1-tavg},\n which shows that the long time decay of fidelity is slower than power law decay.\n To study the decaying behavior of the slower-than-power-law decay,\n we compare it with the function\n \\begin{equation}   M_3(t) = a(\\log_{10} t)^{-b},  \\label{loglogt} \\end{equation}\n with $a$ and $b$ as fitting parameters.\n In the time interval $140 < t < 1000$, the averaged fidelity can be fitted by this function,\n as shown in Fig.~\\ref{fig-mt-sgt1-tavg-140-1k}, where we plot $\\log_{10} M(t)$ versus\n $ \\log_{10} (\\log_{10} t)$.\n Further research work is needed to find analytical explanations for this slower-than-power-law decay\n of fidelity.\n\n\n\\vspace{1cm}\n\n \\section{Conclusions and Discussions}\n\n\n We present numerical results on fidelity decay in the triangle map with linear instability.\n Three regimes of fidelity decay has been found with respect to the perturbation strength:\n weak, intermediate and strong.\n At weak perturbation, the fidelity decays like $\\exp (-c \\sigma^2 t^{1.7})$.\n In the intermediate regime, the fidelity has an exponential decay\n which is approximately $\\exp (-c' \\sigma t)$.\n In the regime of strong perturbation, the fidelity is approximately a function of\n $\\epsilon t^{2.5}$\n and decays slower than power law decay for long times.\n\n\n These results show that the fidelity in the triangle map obeys decaying laws which are\n different from those in systems with strong chaos or with regular motion.\n The difference is closely related to the weak-chaos feature of the classical triangle map.\n In which way and to what extent does weak chaos influence the fidelity decay?\n This is still an open question.\n Indeed, common features of fidelity decay in systems with weak chaos, as well as\n their explanations, should be an interesting topic for future research work.\n In particular, one may note that stretch exponential decay of fidelity has also been observed\n for wave packets which initially reside in the border between chaotic\n and regular regions in mixed-type systems \\cite{WLT02}.\n\n\nACKNOWLEDGMENTS. The author is very grateful to G.~Casati and T.~Prosen\nfor valuable discussions and suggestions.\nThis work is partially supported by Natural Science Foundation of China Grant\nNo.~10775123 and the start-up funding of USTC.\n\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Acknowledgments}\n\nThis research is based upon work supported in part by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, under Award Number DE-SC0021398. This paper was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.\n\n\n\n\n\\section{Equivalence of Incremental and Terminal Information Gain in sOED} \n\\label{app:incre_terminal}\n\n\\begin{proof}[Proof of \\cref{prop:terminal_incremental}]\nUpon substituting \\cref{eq:terminal1,eq:terminal_info_gN} into  \\cref{eq:expected_utility}, the expected utility for a given deterministic policy $\\pi$ using the terminal formulation is\n\\begin{align}\n    U_T(\\pi)&=\\mathbb{E}_{y_0,...,y_{N-1}|\\pi,x_0}\\[\\int_{\\Theta} p(\\theta|I_N)\\ln{\\frac{p(\\theta|I_N)}{p(\\theta|I_0)}}\\,d\\theta \\] \\nonumber\\\\\n    &= \\mathbb{E}_{I_1,\\dots,I_N|\\pi,x_0}\\[\\int_{\\Theta} p(\\theta|I_N)\\ln{\\frac{p(\\theta|I_N)}{p(\\theta|I_0)}}\\,d\\theta \\]\n    \\label{e:app_UT}\n\\end{align}\nwhere recall $I_k=\\{ d_0,y_0,\\dots,d_{k-1},y_{k-1} \\}$ (and $I_0=\\emptyset$).\nSimilarly, substituting \\cref{eq:incremental1,eq:incremental2}, the expected utility for the same policy $\\pi$ using the incremental formulation is\n\\begin{align}\n    U_I(\\pi)&=\\mathbb{E}_{y_0,...,y_{N-1}|\\pi,x_0}\\[\\sum_{k=1}^N \\int_{\\Theta} p(\\theta|I_k)\\ln{\\frac{p(\\theta|I_k)}{p(\\theta|I_{k-1})}}\\,d\\theta \\] \\nonumber\\\\\n    &=\\mathbb{E}_{I_1,\\dots,I_N|\\pi,x_0}\\[\\sum_{k=1}^N \\int_{\\Theta} p(\\theta|I_k)\\ln{\\frac{p(\\theta|I_k)}{p(\\theta|I_{k-1})}}\\,d\\theta \\].\n    \\label{e:app_UI}\n\\end{align}\nIn both cases, \n$\\mathbb{E}_{y_0,\\dots,y_{N-1}|\\pi,x_0}$ can be equivalently replaced by $\\mathbb{E}_{I_1,\\dots,I_N|\\pi,x_0}$ since\n\\begin{align*}\n    \\mathbb{E}_{I_1,\\dots,I_N |\\pi,x_0} \\[\\cdots\\] &= \\mathbb{E}_{d_0,y_0,d_1,y_1,\\dots,d_{N-1},y_{N-1} | \\pi,x_0} \\[\\cdots\\] \\\\\n    &= \\mathbb{E}_{d_0|\\pi} \\mathbb{E}_{y_0,d_1,y_1,\\dots,d_{N-1},y_{N-1}|\\pi,x_0,d_0} \\[\\cdots\\] \\\\\n    &= \\mathbb{E}_{y_0,d_1,y_1,\\dots,d_{N-1},y_{N-1}|\\pi,x_0,\\mu_0(x_0)} \\[\\cdots\\] \\\\\n    &= \\mathbb{E}_{y_0,d_1,y_1,\\dots,d_{N-1},y_{N-1}|\\pi,x_0} \\[\\cdots\\] \\\\\n    &= \\mathbb{E}_{y_0|\\pi,x_0} \\mathbb{E}_{d_1|\\pi,x_0,y_0} \\mathbb{E}_{y_1,\\dots,d_{N-1},y_{N-1}|\\pi,x_0,y_0,d_1} \\[\\cdots\\] \\\\\n    &= \\mathbb{E}_{y_0|\\pi,x_0} \\mathbb{E}_{y_1,\\dots,d_{N-1},y_{N-1}|\\pi,x_0,y_0,\\mu_1(x_1)} \\[\\cdots\\] \\\\\n    &= \\mathbb{E}_{y_0|\\pi,x_0} \\mathbb{E}_{y_1,\\dots,d_{N-1},y_{N-1}|\\pi,x_0,y_0} \\[\\cdots\\] \\\\\n    &= \\mathbb{E}_{y_0|\\pi,x_0} \\mathbb{E}_{y_1|\\pi,x_0,y_0} \\mathbb{E}_{d_2,\\dots,d_{N-1},y_{N-1}|\\pi,x_0,y_0,y_1} \\[\\cdots\\] \\\\\n    & \\qquad\\vdots \\\\\n    &= \\mathbb{E}_{y_0|\\pi,x_0} \\mathbb{E}_{y_1|\\pi,x_0,y_0} \\cdots \\mathbb{E}_{y_{N-1}|\\pi,x_0,y_0,y_1,\\dots,y_{N-2},\\mu_{N-1}(x_{N-1})} \\[\\cdots\\] \\\\\n    &= \\mathbb{E}_{y_0|\\pi,x_0} \\mathbb{E}_{y_1|\\pi,x_0,y_0} \\cdots \\mathbb{E}_{y_{N-1}|\\pi,x_0,y_0,y_1,\\dots,y_{N-2}} \\[\\cdots\\] \\\\\n    &= \\mathbb{E}_{y_0,\\dots,y_{N-1}|\\pi,x_0} \\[\\cdots\\],\n\\end{align*}\nwhere the third equality is due to the deterministic policy (Dirac delta function) $d_0=\\mu_0(x_0)$, the fourth equality is due to \n$\\mu_0(x_0)$ being known if $\\pi$ and $x_0$ are given. The seventh equality is due to $\\mu_1(x_1)$ being known if $\\pi$ and $x_1$ are given, and $x_1$ is known if $x_0$, $d_0=\\mu_0(x_0)$ and $y_0$ are given, and $\\mu_0(x_0)$ is known if $\\pi$ and $x_0$ are given, so overall $\\mu_1(x_1)$ is known if $\\pi$, $x_0$ and $y_0$ are given.\nThe eighth to second-to-last equalities all apply the same reasoning recursively. The last equality brings the expression back to a conditional joint expectation. \n\nTaking the difference between \\cref{e:app_UT} and \\cref{e:app_UI}, we obtain\n\\begin{align*}\n    &U_I(\\pi) - U_T(\\pi)\\\\\n    &=\\mathbb{E}_{I_1,\\dots,I_N|\\pi,x_0}\\[\\sum_{k=1}^N \\int_{\\Theta} p(\\theta|I_k)\\ln{\\frac{p(\\theta|I_k)}{p(\\theta|I_{k-1})}}\\,d\\theta - \\int_{\\Theta} p(\\theta|I_N)\\ln{\\frac{p(\\theta|I_N)}{p(\\theta|I_0)}}\\,d\\theta \\]\\\\\n    &=\\int_{\\Theta}\\mathbb{E}_{I_1,\\dots,I_N|\\pi,x_0}\\[ \\sum_{k=1}^N p(\\theta|I_k)\\ln{\\frac{p(\\theta|I_k)}{p(\\theta|I_{k-1})}} - p(\\theta|I_N)\\ln{\\frac{p(\\theta|I_N)}{p(\\theta|I_0)}} \\]\\, d\\theta\\\\\n    &=\\int_{\\Theta}\\mathbb{E}_{I_1,\\dots,I_N|\\pi,x_0}\\[ \\sum_{k=1}^{N-1} p(\\theta|I_k)\\ln{\\frac{p(\\theta|I_k)}{p(\\theta|I_{k-1})}} + p(\\theta|I_N)\\ln{\\frac{p(\\theta|I_0)}{p(\\theta|I_{N-1})}} \\]\\,d\\theta\\\\\n    &=\\int_{\\Theta}\\mathbb{E}_{I_1,\\dots,I_{N-1}|\\pi,x_0} \\int_{I_N} p(I_N|I_{N-1},\\pi) \\[ \\sum_{k=1}^{N-1} p(\\theta|I_k)\\ln{\\frac{p(\\theta|I_k)}{p(\\theta|I_{k-1})}} + p(\\theta|I_N)\\ln{\\frac{p(\\theta|I_0)}{p(\\theta|I_{N-1})}} \\]\\,dI_N\\,d\\theta\\\\\n    &=\\int_{\\Theta}\\mathbb{E}_{I_1,\\dots,I_{N-1}|\\pi,x_0} \\[ \\sum_{k=1}^{N-1} p(\\theta|I_k)\\ln{\\frac{p(\\theta|I_k)}{p(\\theta|I_{k-1})}} + \\int_{I_N} p(\\theta,I_N|I_{N-1},\\pi)\\ln{\\frac{p(\\theta|I_0)}{p(\\theta|I_{N-1})}}\\,dI_N \\]\\,d\\theta\\\\\n    &=\\int_{\\Theta}\\mathbb{E}_{I_1,\\dots,I_{N-1}|\\pi,x_0} \\[ \\sum_{k=1}^{N-1} p(\\theta|I_k)\\ln{\\frac{p(\\theta|I_k)}{p(\\theta|I_{k-1})}} +  p(\\theta|I_{N-1})\\ln{\\frac{p(\\theta|I_0)}{p(\\theta|I_{N-1})}} \\]\\,d\\theta\\\\\n    &=\\int_{\\Theta}\\mathbb{E}_{I_1,\\dots,I_{N-1}|\\pi,x_0} \\[ \\sum_{k=1}^{N-2} p(\\theta|I_k)\\ln{\\frac{p(\\theta|I_k)}{p(\\theta|I_{k-1})}} +  p(\\theta|I_{N-1})\\ln{\\frac{p(\\theta|I_0)}{p(\\theta|I_{N-2})}} \\]\\,d\\theta\\\\\n    &=\\int_{\\Theta}\\mathbb{E}_{I_1,\\dots,I_{N-2}|\\pi,x_0} \\[ \\sum_{k=1}^{N-3} p(\\theta|I_k)\\ln{\\frac{p(\\theta|I_k)}{p(\\theta|I_{k-1})}} +  p(\\theta|I_{N-2})\\ln{\\frac{p(\\theta|I_0)}{p(\\theta|I_{N-3})}} \\]\\,d\\theta\\\\\n    &\\qquad \\vdots \\\\\n    &=\\int_{\\Theta}\\mathbb{E}_{I_1|\\pi,x_0} \\[ p(\\theta|I_{1})\\ln{\\frac{p(\\theta|I_0)}{p(\\theta|I_0)}} \\]\\,d\\theta\\\\\n    &=0,\n\\end{align*}\n\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\where the third equality takes the last term from the sigma-summation and combines it with the last term, the fourth equality expands the expectation and uses $p(I_N|I_1,\\ldots,I_{N-1},\\pi) = p(I_N|I_{N-1},\\pi)$, the fifth equality makes use of $p(\\theta|I_N)=p(\\theta|I_N,\\pi)$, and the seventh to second-to-last equalities repeat the same procedures recursively. \nHence, $U_T(\\pi)=U_I(\\pi)$.\n\\end{proof}\n\n\\section{Policy Gradient Expression}\n\\label{app:pg_derive}\n\nOur proof for \\cref{thm:PG} follows the proof given by \\cite{silver2014deterministic} for a general infinite-horizon  MDP. \nBefore presenting our proof, we first introduce a shorthand notation for writing the state transition probability:\n\\begin{align}\np(x_k \\rightarrow x_{k+1}|\\pi_w)=p(x_{k+1}|x_k,\\mu_{k,w}(x_k)).\n\\end{align}\nWhen taking an expectation over consecutive state transitions, we further use the simplifying notation\n\\begin{align}\n&\\int_{x_{k+1}}p(x_k \\rightarrow x_{k+1}|\\pi_w) \\int_{x_{k+2}} p(x_{k+1} \\rightarrow x_{k+2}|\\pi_w) \\nonumber\\\\\n&\\qquad \\cdots \\int_{x_{k+m}} p(x_{k+(m-1)} \\rightarrow x_{k+m}|\\pi_w) \\[\\cdots\\] \\,dx_{k+1} \\, dx_{k+2} \\cdots \\, dx_{k+m} \\nonumber\\\\ \n&= \\int_{x_{k+m}} p(x_k \\rightarrow x_{k+m}|\\pi_w) \\[\\cdots\\]  \\, dx_{k+m}\n\\\\ \n&= \\mathbb{E}_{x_{k+m} | \\pi_w, x_k} \\[\\cdots\\].\n\\end{align}\n\nTo avoid notation congestion, below we will also omit the subscript on $w$ and shorten  $\\mu_{k,w_k}(x_k)$ to $\\mu_{k,w}(x_k)$, with the understanding that $w$ takes the same subscript as the $\\mu$ function. \n\n\\begin{proof}[Proof of \\cref{thm:PG}]\n\nWe begin by recognizing that the gradient of expected utility in \\cref{eq:expected_utility_w} can be written using the V-function:\n\\begin{align}\n    \\nabla_w U(w) = \\nabla_w V^{\\pi_w}_0(x_0).\\label{e:gradU_derive}\n\\end{align}\nThe goal is then to derive the gradient expression for the V-functions. \n\nWe apply the definitions and recursive relations for the V- and Q-functions, and obtain a recursive relationship for the gradient of V-function:\n\\begin{align}\n    \\nabla_w V^{\\pi_w}_k(x_k) \n    &= \\nabla_w Q^{\\pi_w}_k(x_k,\\mu_{k,w}(x_k)) \n        \\nonumber\\\\\n    &= \\nabla_w \\Bigg[ \\int_{y_k} p(y_k|x_k,\\mu_{k,w}(x_k))g_k(x_k,\\mu_{k,w}(x_k),y_k)\\,dy_k \\nonumber\\\\\n    &\\qquad\\qquad + \\int_{x_{k+1}} p(x_{k+1}|x_k,\\mu_{k,w}(x_k)) V^{\\pi_w}_{k+1}(x_{k+1}) \\,dx_{k+1} \\Bigg] \\nonumber\\\\\n    &= \\nabla_w \\int_{y_k} p(y_k|x_k,\\mu_{k,w}(x_k))g_k(x_k,\\mu_{k,w}(x_k),y_k)\\,dy_k \\nonumber\\\\\n    &\\qquad\\qquad + \\nabla_w \\int_{x_{k+1}} p(x_{k+1}|x_k,\\mu_{k,w}(x_k)) V^{\\pi_w}_{k+1}(x_{k+1}) \\,dx_{k+1} \\nonumber\\\\\n    &= \\int_{y_k} \\nabla_w \\mu_{k,w}(x_k) \\nabla_{d_k} \\[ p(y_k|x_k,d_k) g_k(x_k,d_k,y_k) \\]\\Big|_{d_k=\\mu_{k,w}(x_k)} \\,dy_k \\nonumber\\\\\n    &\\qquad\\qquad + \\int_{x_{k+1}} \\Big[ p(x_{k+1}|x_k,\\mu_{k,w}(x_k)) \\nabla_w V^{\\pi_w}_{k+1}(x_{k+1}) \n    \\nonumber\\\\ \n    &\\qquad\\qquad +\\nabla_w \\mu_{k,w}(x_k) \\nabla_{d_k} p(x_{k+1}|x_k,d_k)\\Big|_{d_k=\\mu_{k,w}(x_k)} V^{\\pi_w}_{k+1}(x_{k+1}) \\Big] \\,dx_{k+1} \\nonumber\\\\\n    &= \\nabla_w \\mu_{k,w}(x_k) \\nabla_{d_k} \\Bigg[ \\int_{y_k} p(y_k|x_k,d_k) g_k(x_k,d_k,y_k) \\,dy_k \n    \\nonumber\\\\\n    &\\qquad\\qquad\\qquad\\qquad + \\int_{x_{k+1}}  p(x_{k+1}|x_k,d_k)V^{\\pi_w}_{k+1}(x_{k+1})dx_{k+1} \\Bigg]\\Bigg\\vert_{d_k=\\mu_{k,w}(x_k)} \\nonumber\\\\\n    &\\qquad\\qquad + \\int_{x_{k+1}} p(x_{k+1}|x_k,\\mu_{k,w}(x_k)) \\nabla_w V^{\\pi_w}_{k+1}(x_{k+1}) \\,dx_{k+1} \\nonumber\\\\\n    &= \\nabla_w \\mu_{k,w}(x_k) \\nabla_{d_k} Q^{\\pi_w}_{k}(x_k,d_k)\\Big|_{d_k=\\mu_{k,w}(x_k)} \n    \\label{e:gradV_recursive}\\\\\n    &\\qquad\\qquad + \\int_{x_{k+1}} p(x_k \\rightarrow x_{k+1}|\\pi_w) \\nabla_w V^{\\pi_w}_{k+1}(x_{k+1}) \\,dx_{k+1}. \\nonumber\n\\end{align}\nApplying the recursive formula \\cref{e:gradV_recursive} to itself repeatedly and expanding out the overall expression,\nwe obtain\n\\begin{align}\n    &\\nabla_w V^{\\pi_w}_k(x_k) \\nonumber\\\\\n    &= \\nabla_w \\mu_{k,w}(x_k) \\nabla_{d_k} Q^{\\pi_w}_k(x_k,d_k)\\Big|_{d_k=\\mu_{k,w}(x_k)} \\nonumber\\\\\n    &\\qquad + \\int_{x_{k+1}} p(x_k \\rightarrow x_{k+1}|\\pi_w) \\nabla_w \\mu_{k+1,w}(x_{k+1}) \\nabla_{d_{k+1}} Q^{\\pi_w}_{k+1}(x_{k+1},d_{k+1})\\Big|_{d_{k+1}=\\mu_{k+1,w}(x_{k+1})} \\,dx_{k+1} \\nonumber\\\\\n    &\\qquad + \\int_{x_{k+1}} p(x_k \\rightarrow x_{k+1}|\\pi_w) \\int_{x_{k+2}} p(x_{k+1} \\rightarrow x_{k+2}|\\pi_w) \\nabla_w V^{\\pi_w}_{k+2}(x_{k+2}) \\,dx_{k+2} \\,dx_{k+1} \\nonumber\\\\\n    &= \\nabla_w \\mu_{k,w}(x_k) \\nabla_{d_k} Q^{\\pi_w}_k(x_k,d_k)\\Big|_{d_k=\\mu_{k,w}(x_k)} \\nonumber\\\\\n    &\\qquad + \\int_{x_{k+1}} p(x_k \\rightarrow x_{k+1}|\\pi_w) \\nabla_w \\mu_{k+1,w}(x_{k+1}) \\nabla_{d_{k+1}} Q^{\\pi_w}_{k+1}(x_{k+1},d_{k+1})\\Big|_{d_{k+1}=\\mu_{k+1,w}(x_{k+1})} \\,dx_{k+1} \\nonumber\\\\\n    &\\qquad + \\int_{x_{k+2}} p(x_{k} \\rightarrow x_{k+2}|\\pi_w) \\nabla_w V^{\\pi_w}_{k+2}(x_{k+2}) \\,dx_{k+2} \\nonumber\\\\\n    &= \\nabla_w \\mu_{k,w}(x_k) \\nabla_{d_k} Q^{\\pi_w}_k(x_k,d_k)\\Big|_{d_k=\\mu_{k,w}(x_k)} \\nonumber\\\\\n    &\\qquad + \\int_{x_{k+1}} p(x_k \\rightarrow x_{k+1}|\\pi_w) \\nabla_w \\mu_{k+1,w}(x_{k+1}) \\nabla_{d_{k+1}} Q^{\\pi_w}_{k+1}(x_{k+1},d_{k+1})\\Big|_{d_{k+1}=\\mu_{k+1,w}(x_{k+1})} \\,dx_{k+1} \\nonumber\\\\\n    &\\qquad + \\int_{x_{k+2}} p(x_k \\rightarrow x_{k+2}|\\pi_w) \\nabla_w \\mu_{k+2,w}(x_{k+2}) \\nabla_{d_{k+2}} Q^{\\pi_w}_{k+2}(x_{k+2},d_{k+2})\\Big|_{d_{k+2}=\\mu_{k+2,w}(x_{k+2})} \\,dx_{k+2}\\nonumber\\\\\n    &\\hspace{2.5em}\\vdots \\nonumber\\\\\n    &\\qquad + \\int_{x_{N}} p(x_{k} \\rightarrow x_{N}|\\pi_w) \\nabla_w V^{\\pi_w}_{N}(x_{N}) \\,dx_{N} \\nonumber\\\\\n    &= \\sum_{l=k}^{N-1} \\int_{x_l} p(x_k \\rightarrow x_l|\\pi_w) \\nabla_w \\mu_{l,w}(x_l) \\nabla_{d_l} Q^{\\pi_w}_l(x_l,d_l)\\Big|_{d_l=\\mu_{l,w}(x_l)} \\,dx_l\\nonumber\\\\\n    &= \\sum_{l=k}^{N-1} \\mathbb{E}_{x_l| \\pi_w, x_k} \\[\\nabla_w \\mu_{l,w}(x_l) \\nabla_{d_l} Q^{\\pi_w}_l(x_l,d_l)\\Big|_{d_l=\\mu_{l,w}(x_l)}\\] \\,dx_l,\\label{e:gradV_final}\n\\end{align}\nwhere for the second-to-last equality, we absorb the first term into the sigma-notation by using\n\\begin{align*}\n& \\nabla_w \\mu_{{k},w}(x_{k}) \\nabla_{d_{k}} Q^{\\pi_w}_{k}(x_{k},d_{k})\\Big|_{d_{k}=\\mu_{k,w}(x_{k})} \\nonumber\\\\\n& \\qquad = \\int_{x_{k}} p(x_k | x_{k},\\mu_{k,w}(x_k)) \\nabla_w \\mu_{{k},w}(x_{k}) \\nabla_{d_{k}} Q^{\\pi_w}_{k}(x_{k},d_{k})\\Big|_{d_{k}=\\mu_{k,w}(x_{k})} \\,dx_{k}\n\\nonumber\\\\\n& \\qquad = \\int_{x_{k}} p(x_k \\rightarrow x_{k}|\\pi_w) \\nabla_w \\mu_{{k},w}(x_{k}) \\nabla_{d_{k}} Q^{\\pi_w}_{k}(x_{k},d_{k})\\Big|_{d_{k}=\\mu_{k,w}(x_{k})} \\,dx_{k},\n\\end{align*}\nand we eliminate the last term in the summation since\n$\\nabla_w V^{\\pi_w}_{N}(x_{N})=\\nabla_w g_{N}(x_{N})=0$.\n\nAt last, substituting \\cref{e:gradV_final} into \n\\cref{e:gradU_derive}, we obtain the policy gradient expression:\n\\begin{align}\n    \\nabla_w U(w) &= \\nabla_w V^{\\pi_w}_0(x_0) \\nonumber\\\\\n    &= \\sum_{l=0}^{N-1} \\mathbb{E}_{x_l|\\pi_w,x_0} \\[ \\nabla_w \\mu_{l,w}(x_l) \\nabla_{d_l} Q^{\\pi_w}_l(x_l,d_l)\\Big|_{d_l=\\mu_{l,w}(x_l)} \\]. \\nonumber\n    \\end{align}\nRenaming the iterator from $l$ to $k$ arrives at \\cref{eq:pg_theorem} in \\cref{thm:PG}, completing the proof.\n\\end{proof}\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\n\nThis paper presents a mathematical framework and computational methods to optimally design a finite number of sequential experiments (sOED); the code is available at \\url{https:\/\/github.com\/wgshen\/sOED}. \nWe formulate sOED as a finite-horizon POMDP. \nThis sOED form is provably optimal, incorporates both elements of feedback and lookahead, and generalizes the suboptimal batch (static) and greedy (myopic) design strategies. \nWe further structure the sOED problem in a fully Bayesian manner and with information-theoretic rewards (utilities), and prove the equivalence of incremental and terminal information gain setups. In particular, sOED can accommodate expensive nonlinear forward models with general non-Gaussian posteriors of continuous random variables. \n\n\n\n\nWe then introduce numerical methods for solving the sOED problem, which entails finding the optimal policy that maximizes the expected total reward.\nAt the core of our approach is PG, an actor-critic RL technique that parameterizes and learns both the policy and value functions in order to extract the gradient with respect to the policy parameters.\nWe derive and prove the PG expression for finite-horizon sOED, and propose an MC estimator. \nAccessing derivative information enables the use of gradient-based optimization algorithms to achieve  efficient policy search. \nSpecifically, we parameterize the policy and value functions as DNNs, and detail architecture choices that accommodate a nonparametric representation of the Bayesian posterior belief states. Further combined with a terminal information gain formulation, the Bayesian inference becomes embedded in the design sequence, allowing us to sidestep the need for explicitly and numerically computing the Bayesian posteriors at intermediate experiments.\n\n\n\n\nWe apply the overall PG-sOED method to two different examples.\nThe first is a linear-Gaussian problem that offers a closed form solution, serving as a benchmark. We validate the PG-sOED policy against the analytic optimal policy, and observe orders-of-magnitude speedups of PG-sOED over an ADP-sOED baseline.\nThe second entails a problem of contaminant source inversion in a convection-diffusion field. Through multiple sub-cases, we illustrate the advantages of PG-sOED over greedy and batch designs, and provide insights to the value of feedback and lookahead in the context of time-dependent convection-diffusion processes. \nThis demonstration also illustrates the ability of PG-sOED to accommodate expensive forward models with nonlinear physics and dynamics. \n\n\nThe main limitation of the current PG-sOED method is its inability \nto handle high-dimensional settings. While the nonparametric representation sidesteps the need to compute intermediate posteriors, Bayesian inference is ultimately required in order to estimate the KL divergence in the terminal reward.\nThus, an important direction of future work is to improve scalability for high-dimensional inference, to go beyond the current gridding method. This may be approached by employing more general and approximate inference methods such as MCMC, variational inference, approximate Bayesian computation, and transport maps,  perhaps in combination with dimension-reduction techniques.\n\nAnother fruitful area to explore is within advanced RL techniques\n(e.g., \\cite{mnih2015human,lillicrap2015continuous,mnih2013playing, \nschulman2017proximal}).\nFor example, replay buffer stores the experienced episodes, and training data can be sampled from this buffer to reduce sampling costs, control correlation among samples, and reach better convergence performance. \nOff-policy algorithms track two version of the policy network and Q-network---a behavior network for determining actions and a target network for learning---which have demonstrated improved sample efficiency.\nParameters of the policy and Q-networks may also be shared due to their similar features.\nFinally, adopting new utility measures, such as those reflecting goal-orientedness, robustness, and risk, would be of great interest to better capture the value of experiments and data in real-life and practical settings. \n\n\n\n\\section{Problem Formulation}\n\\label{sec:formulation}\n\n\n\\subsection{The Bayesian Paradigm}\n\nWe consider designing a finite\\footnote{In experimental design, the experiments are generally expensive and limited in number. Finite and small values of $N$ are therefore of interest. \nThis is in contrast to RL that often deals with infinite horizon.} number of $N$ experiments, indexed by  integers $k=0,1,\\ldots,N-1$.\nWhile the decision of how many experiments to perform (i.e. choice of $N$) is important, it is \nnot considered\nin this paper; instead, we assume $N$ is given and fixed.\nFurthermore, let \n$\\theta\\in \\mathbb{R}^{N_{\\theta}}$ denote the unknown model parameter we seek to \nlearn\nfrom the experiments, $d_k \\in \\mathcal{D}_k\\subseteq \\mathbb{R}^{N_d}$ the experimental design variable for the $k$th experiment (e.g., \nexperiment conditions),\n$y_k \\in \\mathbb{R}^{N_y}$ the noisy observation from the $k$th experiment (i.e. experiment measurements), and $N_{\\theta}$, $N_{d}$, and $N_{y}$ respectively the dimensions of parameter, design, and observation spaces. We further consider continuous $\\theta$, $d_k$, and $y_k$, although discrete or mixed settings can be accommodated as well.\nFor simplicity,\nwe also let $N_d$ and $N_y$ be constant across all experiments, but this is not a requirement.\n\nA Bayesian approach treats $\\theta$ as a random variable. \nAfter performing the $k$th \nexperiment, its conditional probability density function (PDF) is described by Bayes' rule:\n\\begin{align}\n    \\label{eq:bayes_rule}\n    p(\\theta|d_k,y_k,I_k) = \\frac{p(y_k|\\theta,d_k,I_k)p(\\theta|I_k)}{p(y_k|d_k,I_k)}\n\\end{align}\nwhere $I_k=\\{ d_0,y_0,\\dots,d_{k-1},y_{k-1} \\}$ (and $I_0=\\emptyset$) is the information set collecting the design and observation records from all experiments prior to the $k$th experiment, $p(\\theta|I_k)$ is the prior PDF for the $k$th experiment,\n$p(y_k|\\theta,d_k,I_k)$ is the likelihood function,\n$p(y_k|d_k,I_k)$ is the model evidence (or marginal likelihood, which is constant with respect to $\\theta$),\nand $p(\\theta|d_k,y_k,I_k)$ is the posterior PDF. The prior is then a representation of the uncertainty about $\\theta$ before \nthe $k$th experiment, and the posterior describes the updated uncertainty about $\\theta$ after having observed the outcome from the $k$th experiment.\nIn \\cref{eq:bayes_rule}, we also simplify the prior $p(\\theta|d_k,I_k)=p(\\theta|I_{k})$, invoking a reasonable assumption that knowing only the design for $k$th experiment (but without knowing its outcome) would not affect the prior. \nThe likelihood function carries the relation between the hidden parameter $\\theta$ and the observable $y_k$, through a forward model $G_k$ that governs the underlying process for the $k$th experiment (e.g., constrained via a system of partial differential equations (PDEs)). For example, a common likelihood form is \n\\begin{align}\n    y_k = G_k(\\theta, d_k; I_k) + \\epsilon_k,\n\\end{align}\nwhere $\\epsilon_k$ is a Gaussian random variable that describes the discrepancy between model prediction $G_k$ and observation $y_k$ due to, for instance, measurement noise. The inclusion of $I_k$ in $G_k$ signifies that model behavior may be affected by previous experiments. Each evaluation of the likelihood $p(y_k|\\theta,d_k,I_k) = p_{\\epsilon}(y_k-G_k(\\theta,d_k; I_k))$ thus involves a forward model solve, typically the most expensive part of the computation.\nLastly, \nthe posterior $p(\\theta|d_k,y_k,I_k)=p(\\theta|I_{k+1})$ becomes the prior for the $(k+1)$th experiment via the same form of \\cref{eq:bayes_rule}. Hence, Bayes' rule can be consistently and recursively applied for a sequence of multiple experiments. \n\n\\subsection{Sequential Optimal Experimental Design}\n\\label{sec:math_formulation}\n\nWe now present a general framework for sOED, posed as a POMDP.\nAn overview flowchart for sOED is presented in \\cref{fig:process} to accompany the definitions below.\n\n\\begin{figure}[htb]\n  \\centering\n  \\includegraphics[width=0.95\\linewidth]{Figures\/process.jpg}\n  \\caption{Flowchart of the process involved in a $N$-experiment sOED.}\n  \\label{fig:process}\n\\end{figure}\n\n\\textbf{State.} We introduce the state variable $x_k=[x_{k,b},x_{k,p}] \\in \\mathcal{X}_k$\nto be the state prior to designing and performing the $k$th experiment. \nHence, \n$x_0,\\ldots,x_{N-1}$ denote the respective states prior to each of the $N$ experiments, and $x_1,\\ldots,x_N$ denote the respective states after each of the $N$ experiments.\nThe state is an entity that summarizes past information needed for making experimental design decisions in the future. \nIt is very general and can contain different quantities deemed to be  decision-relevant. \nIn our case here, the state consists of a belief state $x_{k,b}$ reflecting our state of uncertainty about the hidden $\\theta$, and a physical state $x_{k,p}$ carrying other non-random variables pertinent to the design problem. \nSince $\\theta$ is not observable and can be only inferred from noisy and indirect observations $y_k$ through Bayes' rule in \\cref{eq:bayes_rule}, this setup can be viewed as a POMDP for $\\theta$ (or a MDP for $x_k$).\n\nConceptually, a \\emph{realization} of the belief state manifests as \nthe continuous posterior (conditional) random variable \n$(x_{k,b} = x'_{k,b}) = (\\theta|I_k=I_k')$, \nwhere the prime denotes realization. Such a random variable can be \nportrayed by, for example, its PDF, cumulative distribution function, or characteristic function\\footnote{\nIt is possible for $\\theta|I_k$'s with different $I_k'$'s to have the same PDF (or distribution or characteristic function), for example simply by exchanging the experiments. Hence, the mappings from $I_k$ to these portrayals (PDF, distribution, characteristic functions) are non-injective. This may be problematic when considering transition probabilities of the belief state, but avoided if we keep to our root definition of belief state based on $I_k$, which remains unique.}.\nAttempting to directly represent these infinite-dimensional quantities  in practice would require some finite-dimensional approximation or discretization.\nAlternatively, one can adopt a nonparametric approach and track \n$I_k$\n(from a given initial $x_0$),\nwhich then yields a \nrepresentation of $x_{k}$ (both $x_{k,b}$ and $x_{k,p}$) without any approximation\\footnote{$I_k$ collects the complete history of experiments and their observations, therefore is a sufficient statistic for $x_k$ by definition. Hence, if $I_k$ is known, then the full state $x_k$ is equivalently represented. \nAll of these are conditioned on a given initial $x_0$ (which includes the prior on $\\theta$), but for simplicity we will omit this conditioning when writing the PDFs in this paper, with the understanding that it is always implied. }\nbut its dimension grows with $k$. However, the dimension is always bounded since the maximum number of experiments considered is finite (i.e. $k < N$).\nIn any case, the belief state space is uncountably infinite since $\\theta$ is a continuous random variable (i.e. the possible posteriors that can be realized is uncountably infinite).\nWe will further detail our numerical representation of the belief state in \\cref{sec:policy_net} and \\cref{sec:numerical_belief_state}.\n\n\\textbf{Design (action) and policy.} Sequential experimental design involves building policies mapping from the state space to the design space, $\\pi = \\{\\mu_k : \\mathcal{X}_k \\mapsto \\mathcal{D}_k, k=0,\\ldots,N-1\n\\}$, such that the design for the $k$th experiment is determined by the state via $d_k=\\mu_k(x_k)$. Thus, sequential design is inherently adaptive, computing designs based on the current state which depends on the previous experiments and their outcomes.\nWe focus on deterministic policies in this study, where policy functions $\\mu_k$ produce deterministic outputs. \n\n\\textbf{System dynamics (transition function).} The system dynamics, denoted by $x_{k+1}=\\mathcal{F}_k(x_k,d_k,y_k)$, describes the transition from state $x_k$ to state $x_{k+1}$ after carrying out the $k$th experiment with design $d_k$ and observation $y_k$. For the belief state, the prior $x_{k,b}$ can be updated to the posterior $x_{k+1,b}$ via Bayes' rule in \\cref{eq:bayes_rule}. The physical state, if present, evolves based on the relevant physical process.\nWhile the system dynamics described in \\cref{eq:bayes_rule} appears deterministic given a specific realization of $d_k$ and $y_k$, it is a stochastic transition since the observation $y_k$ is random. In particular, there exists an underlying transition probability\n\\begin{align}\np(x_{k+1}|x_{k},d_{k})=p(y_k|x_k,d_k)=p(I_{k+1}|d_{k},I_{k}) =p(y_{k}|d_{k},I_{k}) = \n\\int_{\\Theta} p(y_k|\\theta,d_k, I_k)p(\\theta|I_{k})\\,d\\theta,\n\\label{eq:transition}\n\\end{align}\nwhere we \nsimplify the prior with $p(\\theta|d_k,I_k)=p(\\theta|I_{k})$. \nThis transition probability is intractable and does not have a closed form. However, we are able to generate samples of the next state by sampling from the prior and likelihood, as suggested by the last equality in \\cref{eq:transition}. Hence, we have a model-based (via a sampling model) setup.\n\n\n\\textbf{Utility (reward).} We denote $g_k(x_k,d_k,y_k) \\in \\mathbb{R}$ to be the immediate reward from performing an experiment. Most generally, this quantity can depend on the state, design, and observation. For example, it may simply be the (negative) cost of the $k$th experiment.\nSimilarly, we define a terminal reward $g_N(x_N) \\in \\mathbb{R}$ containing any additional reward measure that reflects the benefit of reaching certain final state, and that can only be computed after the entire set of experiments is completed. We will provide a specific example of reward functions pertaining to information measures in \\cref{sec:information_gain}.\n\n\n\\textbf{sOED problem statement.} The sOED problem seeks the policy that solves the following optimization problem: \nfrom a given initial state $x_0$, \\begin{align}\n    \\label{eq:optimal_policy}\n    \\pi^\\ast = \\operatornamewithlimits{arg\\,max}_{\\pi=\\{\\mu_0,\\ldots,\\mu_{N-1}\\}}& \\qquad U(\\pi)\\\\\n    \\text{s.t.}& \n    \\qquad d_k = \\mu_k(x_k) \\in \\mathcal{D}_k, \\nonumber\\\\\n    &\\qquad x_{k+1}=\\mathcal{F}_k(x_k,d_k,y_k),\n    \\hspace{3em} \\text{for}\\quad k=0,\\dots,N-1, \\nonumber\n\\end{align}\nwhere\n\\begin{align}\n    \\label{eq:expected_utility}\n    U(\\pi) = \\mathbb{E}_{y_0,...,y_{N-1}|\\pi,x_0}\\[\\sum_{k=0}^{N-1}g_k(x_k,d_k,y_k)+g_N(x_N)\\]\n\\end{align}\nis the expected total utility functional. \nWhile here $x_0$ is fixed, this formulation can easily be adjusted to accommodate stochastic $x_0$ as well, by including $x_0$ as a part of $I_k$ and taking another expectation over $x_0$ in \\cref{eq:expected_utility}. \n\nOverall, our sOED problem corresponds to a model-based planning problem of RL. It is challenging for several reasons: \n\\begin{itemize}\n\\item finite horizon, where the policy functions $\\mu_k$ are different for each $k$ and need to be tracked and solved for separately; \n\\item partially and indirectly observed hidden $\\theta$ whose belief state space is uncountably infinite and also infinite-dimensional or nonparametric; \n\\item deterministic policy;\n\\item continuous design (action) and observation spaces; \n\\item transition probability intractable to compute, and transition can only be sampled;\n\\item each belief state transition involves a Bayesian inference, \nrequiring many \nforward model evaluations; \n\\item reward functions are information measures for continuous random variables (discussed below), which are difficult to estimate.\n\\end{itemize}\n\n\\subsection{Information Measures as Experimental Design Rewards}\n\\label{sec:information_gain}\n\nWe wish to adopt reward functions \nthat reflect the degree of success for the experiments, \nnot only\nthe experiment costs. Determining such an appropriate quantity depends on the experimental goals, e.g., to achieve inference, prediction, model discrimination, etc. One popular choice corresponding to the goal of parameter inference is to maximize a measure of information gained on $\\theta$. \nLindley's seminal paper~\\cite{Lindley1956} proposes to use the mutual information between the parameter and observation as the expected utility, and Ginebra~\\cite{Ginebra2007} provides more general criteria for proper measure of information gained from an experiment. \nFrom the former, mutual information is equal to the expected KL\ndivergence from the prior to the posterior. The KL divergence provides an intuitive interpretation as it quantifies the farness between the prior and the posterior distributions, and thus a larger divergence corresponds to a greater degree of belief update---and hence information gain---resulting from the experiment and its observation.\n\nIn this paper, we follow Lindley's approach and demonstrate the use of KL divergence as sOED rewards,\nand present two reasonable sequential design formulations that are in fact equivalent. The first, call it the \\emph{terminal formulation}, involves clumping the information gain from all $N$ experiments in the terminal reward (for clarity, we omit all other reward contributions common to the two formulations, although it would be trivial to show the equivalence for those cases too): \n\\begin{align}\n g_k(x_k, d_k, y_k) &= 0, \\qquad k=0,\\ldots,N-1 \\label{eq:terminal1}\\\\\n g_N(x_N) &= D_{\\mathrm{KL}}\\( p(\\cdot|I_N)\\,||\\,p(\\cdot|I_0) \\) \\nonumber\\\\ &= \\int_{\\Theta} p(\\theta|I_N) \\ln\\[\\frac{p(\\theta|I_N)}{p(\\theta|I_0)}\\]\\,d\\theta.\\label{eq:terminal_info_gN}\n\\end{align}\nThe second, call it the \\emph{incremental formulation}, entails the use of incremental information gain from each experiment in their respective immediate rewards:\n\\begin{align}\n    g_k(x_k, d_k, y_k) &= D_{\\mathrm{KL}}\\( p(\\cdot|I_{k+1})\\,||\\,p(\\cdot|I_k) \\) \\nonumber\\\\&= \\int_{\\Theta} p(\\theta|I_{k+1}) \\ln\\[\\frac{p(\\theta|I_{k+1})}{p(\\theta|I_k)}\\]\\,d\\theta, \\qquad k=0,\\ldots,N-1\\label{eq:incremental1}\\\\\n    g_N(x_N) &= 0. \\label{eq:incremental2}\n\\end{align}\n\n\\begin{theorem} \n\\label{prop:terminal_incremental}\nLet $U_T(\\pi)$ be the sOED expected utility defined in \\cref{eq:expected_utility} subject to the constraints in \\cref{eq:optimal_policy} for a given policy $\\pi$ while using the terminal formulation \\cref{eq:terminal1,eq:terminal_info_gN}. Let $U_I(\\pi)$ be the same except using  the incremental formulation \\cref{eq:incremental1,eq:incremental2}. Then $U_T(\\pi)=U_I(\\pi)$. \n\\end{theorem}\n\nA proof is provided in \\cref{app:incre_terminal}.\nAs a result, the two formulations correspond to the same sOED problem. \n\n\\subsection{Generalization of Suboptimal Experimental Design Strategies}\n\\label{sec:subopt_design}\n\n\nWe also make the connection between sOED to \nthe commonly used batch design and greedy sequential design.\nWe illustrate below that both batch and greedy designs are,\nin general, suboptimal with respect to the expected utility \\cref{eq:expected_utility}. Thus, sOED generalizes these design strategies.\n\nBatch OED designs all $N$ experiments together prior to performing any of those experiments. Consequently, it is non-adaptive, and cannot make use of  new information acquired from any of the $N$ experiments to help adjust the design of other experiments. Mathematically, batch design seeks static design values (instead of a policy) over the joint design space $\\mathcal{D}:=\\mathcal{D}_0 \\times \\mathcal{D}_1 \\times \\cdots \\times \\mathcal{D}_{N-1}$:\n\\begin{align}\n    (d_0^{\\mathrm{ba}},\\dots,d_{N-1}^{\\mathrm{ba}}) = \\operatornamewithlimits{arg\\,max}_{(d_0,\\dots,d_{N-1}) \\in \\mathcal{D}} \\mathbb{E}_{y_0,\\dots,y_{N-1}|d_0,\\dots,d_{N-1},x_0}\\[ \\sum_{k=0}^{N-1}g_k(x_k,d_k,y_k) + g_N(x_N) \\],\\label{eq:batch}\n\\end{align}\nsubject to the system dynamics. In other words, the design $d_k$ is chosen independent of $x_k$ (for $k > 0$).\nThe suboptimality of batch design becomes clear once realizing \\cref{eq:batch} is equivalent to the sOED formulation in \\cref{eq:optimal_policy} but restricting all $\\mu_k$ to be only constant functions. Thus, $U(\\pi^{\\ast}) \\geq U(\\pi^{\\mathrm{ba}}=d^{\\mathrm{ba}})$. \n\nGreedy design is also a type of sequential experimental design and produces a policy. It optimizes only for the immediate reward at each experiment:\n\\begin{align}\n\\mu_k^{\\mathrm{gr}} = \\operatornamewithlimits{arg\\,max}_{\\mu_k} \\mathbb{E}_{y_k|x_k,\\mu_k(x_k)}\\[ g_k(x_k,\\mu_k(x_k),y_k) \\], \\qquad k=0,\\dots,N-1,\\label{eq:greedy}\n\\end{align}\nwithout needing to subject to the system dynamics since the policy functions $\\mu_k^{\\mathrm{gr}}$ are decoupled. $U(\\pi^{\\ast}) \\geq U(\\pi^{\\mathrm{gr}})$ follows trivially.\nAs a more specific example when using information measure utilities described in \\cref{sec:information_gain}, greedy design would only make sense under the incremental formulation (\\cref{eq:incremental1,eq:incremental2}).\nThen, together with \\cref{prop:terminal_incremental}, we have \n$U_{T}(\\pi^{\\ast})=U_{I}(\\pi^{\\ast}) \\geq U_{I}(\\pi^{\\mathrm{gr}})$.\n\n\n\n\n\n\\section{Introduction}\n\\label{sec:intro}\n\n\n\n\n\nExperiments are indispensable for scientific research. Carefully designed experiments can provide substantial savings for these often expensive data-acquisition opportunities. However,  designs based on heuristics are usually not optimal, especially for complex systems with high dimensionality, nonlinear responses and dynamics, multiphysics, and uncertain and noisy environments. \nOptimal experimental design (OED), while leveraging a\ncriteria based on a forward model that simulates the experiment process, systematically quantifies and maximizes the value of experiments. \n\n\n\n\nOED for linear models \n\\cite{Fedorov1972,Atkinson2007} \nuses criteria based on the information matrix derived from the model, which can be calculated analytically. Different operations on this matrix form the core of the well-known alphabetical designs, such as the $A$- (trace), $D$- (determinant), and $E$-optimal (largest eigenvalue) designs. \nBayesian OED further incorporates the notion of prior and posterior distributions that reflect the uncertainty update as a result of the experiment data \n\\cite{Berger1985, Chaloner1995}.\nIn particular, the Bayesian $D$-optimal criterion generalizes to the nonlinear setting under an information-theoretic perspective \\cite{Lindley1956}, and is equivalent to the expected Kullback\u2013Leibler (KL) divergence from the prior to the posterior. \nHowever, these OED criteria are generally intractable to compute for nonlinear models\nand must be\napproximated~\\cite{Box1959,Ford1989,Chaloner1995,Muller2005,Ryan2016}. With advances in computing power and a need to tackle bigger and more complex systems in engineering and science, there is a growing interest, urgency, and opportunity for computational development \nof nonlinear OED methods~\\cite{Ryan2003,Terejanu2012,Huan2013,Long2015,Weaver2016,Alexanderian2016,Tsilifis2017,Overstall2017,Beck2018,Kleinegesse2019,Foster2019,Wu2020}. \n\n\nWhen designing multiple experiments, commonly used \napproaches are often suboptimal. The first is \\emph{batch} (or static) design: it rigidly designs all experiments together \\emph{a priori} using the aforementioned linear or nonlinear OED method, and does not offer any opportunity to adapt when new information becomes available (i.e. no feedback). \nThe second is \\emph{greedy} (or myopic) design \n\\cite{Box1992, Dror2008, Cavagnaro2010, Solonen2012, Drovandi2013, Drovandi2014, Kim2014,Hainy2016,Kleinegesse2021}:\nit plans only for the \\emph{next} experiment, \nupdates with its observation, and repeats the design process. While greedy design has feedback, it lacks consideration for future effects and consequences (i.e. no lookahead). Hence, greedy design does not see the big picture or plan for the future. It is easy to relate, even from everyday experience (e.g., driving a car, planning a project), that a lack of feedback (for adaptation) and lookahead (for foresight) can lead to suboptimal decision-making with \nundesirable  consequences. \n\n\n\n\n\n\n\nA provably optimal formulation of sequential experimental design---we refer to as sequential OED (sOED)~\\cite{Muller2007,VonToussaint2011,Huan2015,Huan2016}---needs both elements of feedback and lookahead, and \ngeneralizes the batch and greedy designs. The main features\nof sOED are twofold. First, sOED works with design \\emph{policies} (i.e. functions that can adaptively suggest what experiment to perform depending on the current situation) in contrast to\nstatic design values. Second, sOED always designs for all remaining experiments, thus capturing the effect on the entire future horizon when each design decision is made. \nFormally, the sOED problem can be formulated as a {partially observable Markov decision process} (POMDP). Under this agent-based view, the experimenter (agent) selects the experimental design (action) following a policy, and observes the experiment measurements (observation) in order to maximize the total utility (reward) that depends on the unknown model parameters (hidden state). \nA belief state \ncan be further formed\nbased on the Bayesian posterior that describes the uncertainty \nof the hidden state, thereby turning the POMDP into a \nbelief Markov decision process (MDP) \\cite{littman1995learning}. \n\nThe sOED problem targeted in our paper presents an atypical and challenging POMDP: finite horizon, continuous random variables, uncountably infinite belief state space, deterministic policy, continuous designs and observations, sampling-only transitions that each involves a Bayesian inference, and information measures as rewards. Thus, while there exists an extensive POMDP literature (e.g.,~\\cite{cassandra1994acting, littman1995efficient, cassandra1998survey, kurniawati2016online, igl2018deep}), off-the-shelf methods cannot be directly applied to this sOED problem. \nAt the same time, attempts for sOED have been sparse, with examples~\\cite{Carlin1998,Gautier2000,Pronzato2002, Brockwell2003, Christen2003, Murphy2003, Wathen2006} \nfocusing on discrete settings \nand with special problem and solution forms,\nand do not use an information criteria or do not adopt a Bayesian framework. \nMore recent efforts for Bayesian sOED~\\cite{Huan2015,Huan2016} employ approximate dynamic programming (ADP) and transport maps, and illustrate the advantages of sOED over batch and greedy designs. However, this ADP-sOED method remains computationally expensive.\n\n\n\n\n\n\n\n\n\n\nIn this paper, we create new methods to solve the sOED problem in a computationally efficient manner, by drawing the state-of-the-art from reinforcement learning (RL) \\cite{watkins1992q, sutton2000policy, szepesvari2010algorithms, mnih2015human, schulman2015trust, silver2016mastering, silver2017mastering, li2017deep, sutton2018reinforcement}.\nRL approaches are often categorized as value-based (learn value functions only)\n\\cite{watkins1992q,mnih2015human,wang2016dueling,van2016deep}, policy-based (learn policy only)\n\\cite{willianms1988toward, williams1992simple}, or actor-critic (learn policy and value functions together) \\cite{konda2000actor, peters2008natural, silver2014deterministic, lillicrap2015continuous}. \nADP-sOED~\\cite{Huan2015,Huan2016} is thus value-based, where the policy is only implicitly expressed via the learnt value functions. Consequently, each policy evaluation involves optimizing the value functions on-the-fly, a costly calculation especially for continuous action spaces. \nBoth policy-based and actor-critic methods are more efficient in this respect. \nActor-critic methods have further been observed to produce lower solution variance and faster convergence \\cite{sutton2018reinforcement}. \n\nWe adopt an actor-critic approach in this work. \nRepresenting and learning the policy explicitly further enables the use of policy gradient (PG) techniques \\cite{sutton2000policy, kakade2001natural, degris2012off, silver2014deterministic, lillicrap2015continuous, schulman2015trust, mnih2016asynchronous, schulman2017proximal, lowe2017multi, liu2017stein, barth2018distributed} that estimate the gradient with respect to policy parameters, and in turn permits the use of gradient-based optimization algorithms.\nInspired by deep deterministic policy gradient (DDPG)~\\cite{lillicrap2015continuous}, we further employ deep neural networks (DNNs) to parameterize and approximate the policy and value functions. The use of DNNs can take advantage of potentially large number of episode samples generated from the transition simulations, and compute gradients efficiently through back-propagation. \nNevertheless, care needs be taken to design the DNNs and their hyperparameters in order to \nobtain stable and rapid convergence to a good sOED policy, which we will describe in the paper. \n\nThe main contributions of our paper are as follows.\n\\begin{itemize}\n\\item We formulate the sOED problem as a finite-horizon POMDP under a Bayesian setting for continuous random variables, and illustrate its generalization over the batch and greedy designs.\n\\item We present the PG-based sOED (that we call PG-sOED) algorithm, proving the key gradient expression and proposing its Monte Carlo estimator. We further present the DNN architectures for the policy and value functions, and detail the numerical setup of the overall method.\n\\item We demonstrate the speed and optimality advantages of PG-sOED over ADP-sOED, batch, and greedy designs, on a benchmark and a problem of contaminant source inversion in a convection-diffusion field that involves an expensive forward model. \n\\item We make available our PG-sOED code at \\url{https:\/\/github.com\/wgshen\/sOED}. \n\\end{itemize}\n\nThis paper is organized as follows. \\Cref{sec:formulation} introduces the components needed in an sOED problem, culminating with the sOED problem statement. \\Cref{sec:method} describes the details of the entire PG-sOED method.\n\\Cref{sec:results} presents numerical examples, a linear-Gaussian benchmark and a problem of contaminant source inversion in a convection-diffusion field, to validate PG-sOED and demonstrate its advantages over other baselines.\nFinally, \\cref{sec:conclusions} concludes the paper and provides an outlook for future work.\n\n\n\n\n\\section{Policy Gradient for Sequential Optimal Experimental Design}\n\\label{sec:method}\n\nWe approach the sOED problem by directly parameterizing the policy functions and representing them explicitly. We then develop gradient expression with respect to the policy parameters, so to enable gradient-based optimization for numerically identifying optimal or near-optimal policies. Such approach is known as the PG\nmethod (e.g., \\cite{silver2014deterministic, lillicrap2015continuous}).\nIn addition to the policy, we also parameterize and learn the value functions,\nthus arriving at an actor-critic form. \nPG contrasts with previous  ADP-sOED efforts~\\cite{Huan2015,Huan2016} that \napproximate only the value functions. In those works, the policy is represented implicitly, and requires solving a (stochastic) optimization problem each time the policy is evaluated. This renders both the offline training and online policy usage computationally expensive. As we will demonstrate, PG sidesteps this requirement.\n\n\nIn the following, we first derive the exact PG expression in \\cref{ss:PG_exact}. We then present numerical methods in \\cref{ss:PG_numerical} to estimate this exact PG expression. In particular, this requires adopting a parameterization of the policy functions; we will present the use of DNNs to achieve this parameterization. Once the policy parameterization is established, we can then compute the PG with respect to the parameters, and optimize them using a gradient ascent procedure.\n\n\\subsection{Derivation of the Policy Gradient}\n\\label{ss:PG_exact}\n\nThe PG approach to sOED (PG-sOED) involves parameterizing each policy function $\\mu_{k}$  with parameters $w_k$ ($k=0,\\ldots,N-1$), which we denote by the shorthand form $\\mu_{k,w_k}$. In turn, the policy $\\pi$ is parameterized by $w=\\{w_k, \\forall k\\} \\in \\mathbb{R}^{N_w}$ \nand denoted by $\\pi_{w}$, where $N_w$ is the dimension of the overall policy parameter vector. The sOED problem statement from \\cref{eq:optimal_policy,eq:expected_utility} then updates to: from a given initial state $x_0$,\n\\begin{align}\n    \\label{eq:PG_sOED}\n    w^{\\ast} = \\operatornamewithlimits{arg\\,max}_{w}& \\qquad U(w)\\\\\n    \\text{s.t.}& \n    \\qquad d_k = \\mu_{k,w_k}(x_k) \\in \\mathcal{D}_k, \\nonumber\\\\\n    &\\qquad x_{k+1}=\\mathcal{F}_k(x_k,d_k,y_k), \n    \\hspace{3em} \\text{for}\\quad k=0,\\dots,N-1, \\nonumber\n\\end{align}\nwhere\n\\begin{align}\n    \\label{eq:expected_utility_w}\n    U(w) = \\mathbb{E}_{y_0,...,y_{N-1}|\\pi_w,x_0}\\[\\sum_{k=0}^{N-1}g_k(x_k,d_k,y_k)+g_N(x_N)\\].\n\\end{align}\nWe now aim to derive the gradient $\\nabla_{w} U(w)$.\n\nBefore presenting the gradient expression, we need to introduce the value functions. \nThe \\emph{state-value function} (or \\emph{V-function}) following policy $\\pi_{w}$ and at the $k$th experiment is\n\\begin{align}\nV_k^{\\pi_{w}}(x_k)&=\\mathbb{E}_{y_k,\\dots,y_{N-1}|\\pi_{w},x_k}\\[\\sum_{t=k}^{N-1} g_t(x_t,\\mu_{t,w_t}(x_t),y_t) + g_N(x_N)\\] \\\\\n &= \\mathbb{E}_{y_k|\\pi_w,x_k} \\[ g_k(x_k,\\mu_{k,w_k}(x_k),y_k) + V^{\\pi_w}_{k+1}(x_{k+1}) \\] \\\\\n V_N^{\\pi_{w}}(x_N) &= g_N(x_N)\n\\end{align}\nfor $k=0,\\ldots,N-1$, where $x_{k+1}=\\mathcal{F}_k(x_k,\\mu_{k,w_k}(x_k),y_k)$.\nThe V-function is the expected cumulative remaining reward starting from a given state $x_k$ and following policy $\\pi_{w}$ for all remaining experiments. \nThe \\emph{action-value function} (or \\emph{Q-function}) following policy $\\pi_{w}$ and at the $k$th experiment is\n\\begin{align}\n\\label{eq:action_bellman}\nQ_k^{\\pi_{w}}(x_k,d_k)&=\\mathbb{E}_{y_k,\\dots,y_{N-1}|\\pi_{w},x_k,d_k}\\[g_k(x_k,d_k,y_k) + \\sum_{t=k+1}^{N-1} g_t(x_t,\\mu_{t,w_t}(x_t),y_t) + g_N(x_N)\\]\n\\\\\n&=\\mathbb{E}_{y_k|x_k,d_k} \\[ g_k(x_k,d_k,y_k) + Q^{\\pi_w}_{k+1}(x_{k+1},\\mu_{k+1,w_{k+1}}(x_{k+1}))\\]\n\\label{eq:action_bellman2}\n\\\\\nQ_{N}^{\\pi_{w}}(x_N,\\cdot) &= g_N(x_N).\n\\end{align}\nfor $k=0,\\ldots,N-1$, where $x_{k+1}=\\mathcal{F}_k(x_k,d_k,y_k)$. \nThe Q-function is the expected cumulative remaining reward for performing the $k$th experiment at the given design $d_k$ from a given state $x_k$ and thereafter following policy $\\pi_{w}$. The two functions are related via\n\\begin{align}\nV_k^{\\pi_{w}}(x_k)=Q_k^{\\pi_{w}}(x_k,\\mu_{k,w_k}(x_k)).\n\\end{align}\n\n\n\n\n\\begin{theorem}\n\\label{thm:PG}\nThe gradient of the expected utility in \\cref{eq:expected_utility_w} with respect to the policy parameters (i.e. the policy gradient) is \n\\begin{align}\n    \\nabla_w U(w) = \\sum_{k=0}^{N-1} \\mathbb{E}_{x_k|\\pi_w,x_0} \\[ \\nabla_w \\mu_{k,w_k}(x_k) \\nabla_{d_k} Q^{\\pi_w}_k(x_k,d_k)\\Big|_{d_k=\\mu_{k,w_k}(x_k)} \\].\\label{eq:pg_theorem}\n\\end{align}\n\\end{theorem}\nWe provide a proof in \\cref{app:pg_derive}, which follows the proof in \\cite{silver2014deterministic} for a general infinite-horizon MDP. \n\n\\subsection{Numerical Estimation of the Policy Gradient}\n\\label{ss:PG_numerical}\n\n\nThe PG \\cref{eq:pg_theorem} generally cannot be evaluated in closed form, and needs to be approximated numerically. We propose a Monte Carlo (MC) estimator:\n\\begin{align}\n    \\label{eq:policy_grad}\n    \\nabla_w U(w) \\approx \\frac{1}{M} \\sum_{i=1}^M \\sum_{k=0}^{N-1} \\nabla_w \\mu_{k,w_k}(x^{(i)}_k) \\nabla_{d^{(i)}_k} Q^{\\pi_w}_k(x^{(i)}_k,d^{(i)}_k)\\Big|_{d^{(i)}_k=\\mu_{k,w_k}(x^{(i)}_k)}\n\\end{align}\nwhere superscript indicates the $i$th episode (i.e. trajectory instance) generated from MC sampling. Note that the \\emph{sampling} only requires a given policy and does not need any Q-function. Specifically, for the $i$th episode, we first sample a hypothetical ``true'' $\\theta^{(i)}$ from the prior belief state $x_{0,b}$ and freeze it for the remainder of this episode---that is, all subsequent $y_k^{(i)}$ will be generated from this $\\theta^{(i)}$.\nWe then compute $d_k^{(i)}$ from the current policy $\\pi_w$, sample $y_k^{(i)}$ from the likelihood $p(y_k|\\theta^{(i)},d_k^{(i)},I_k^{(i)})$, and repeat for all experiments $k=0,\\dots,N-1$. The same procedure is then repeated for all episodes $i=1,\\dots,M$. The choice of $M$ can be selected based on indicators such as MC standard error, ratio of noise level compared to gradient magnitude, or the validation expected utility from sOED policies produced under different $M$. \nWhile we propose to employ a fixed sample $\\theta^{(i)}$ for the entire $i$th episode, one may also choose to resample $\\theta_k^{(i)}$ at each stage $k$ from the updated posterior belief state $x_{k,b}^{(i)}$.\nThese two approaches are in fact equivalent, since from factoring out the expectations we have\n\\begin{align}\n    \\label{eq:equivalency_sample_theta}\n    U(w) &= \\mathbb{E}_{y_0,...,y_{N-1}|\\pi_w,x_0}\\[\\sum_{k=0}^{N-1}g_k(x_k,d_k,y_k)+g_N(x_N)\\] \\nonumber \\\\\n    &= \\mathbb{E}_{\\theta|x_{0,b}} \\mathbb{E}_{y_0|\\pi_w,\\theta,x_0} \\mathbb{E}_{y_1|\\pi_w,\\theta,x_0,y_0} \\cdots \\nonumber\\\\\n    &\\qquad \\qquad \\qquad \\cdots \\mathbb{E}_{y_{N-1}|\\pi_w,\\theta,x_0,y_0,\\dots,y_{N-2}} \\[\\sum_{k=0}^{N-1}g_k(x_k,d_k,y_k)+g_N(x_N)\\] \\\\\n    &= \\mathbb{E}_{\\theta_0|x_{0,b}} \\mathbb{E}_{y_0|\\pi_w,\\theta_0,x_0} \\mathbb{E}_{\\theta_1|x_{1,b}} \\mathbb{E}_{y_1|\\pi_w,\\theta_1,x_{1}} \\cdots \\nonumber\\\\\n    & \\qquad \\qquad \\qquad \\cdots\\mathbb{E}_{\\theta_{N-1}|x_{N-1,b}} \\mathbb{E}_{y_{N-1}|\\pi_w,\\theta_{N-1},x_{N-1}} \\[\\sum_{k=0}^{N-1}g_k(x_k,d_k,y_k)+g_N(x_N)\\],\n\\end{align}\nwhere the second equality corresponds to the episode-fixed $\\theta^{(i)}$, and the last equality corresponds to the resampling of $\\theta_k^{(i)}$. The former, however, is computationally easier, since it does not require working with the intermediate posteriors.\n\nFrom \\cref{eq:policy_grad}, the MC estimator for PG entails computing the gradients $\\nabla_w \\mu_{k,w_k}(x^{(i)}_k)$ and $\\nabla_{d^{(i)}_k} Q^{\\pi_w}_k(x^{(i)}_k,d^{(i)}_k)$. While the former can be obtained through the parameterization of the policy functions, the latter typically requires parameterization of the Q-functions as well. We thus parameterize both the policy and Q-functions, arriving at an actor-critic method. Furthermore, we adopt the approaches from Deep Q-Network (DQN)~\\cite{mnih2015human} and \nDDPG~\\cite{lillicrap2015continuous}, and use DNNs to approximate the policy and Q-functions. We present these details next. \n\n\n\n\\subsubsection{Policy Network}\n\\label{sec:policy_net}\n\nConceptually, we would need to construct individual DNNs $\\mu_{k,w_k}$ to approximate $\\mu_{k} : \\mathcal{X}_k \\mapsto \\mathcal{D}_k$ for each $k$. Instead, we choose to combine them together \ninto a single function $\\mu_{w}(k, x_k)$, which then requires only a single DNN for the entire policy at the cost of a higher input dimension. Subsequently, the $\\nabla_w \\mu_{k,w_k}(x^{(i)}_k)=\\nabla_w \\mu_{w}(k,x^{(i)}_k)$ term from \\cref{eq:policy_grad} can be obtained via back-propagation. Below, we discuss the architecture design of such a DNN, with particular focus on its input layer.\n\n\nFor the first input component, i.e. the stage index $k$,\ninstead of passing in the integer directly we opt to use one-hot encoding that takes the form of a unit vector:\n\\begin{align}\n    k \\qquad \\longrightarrow \\qquad e_{k+1}=[0,\\dots,0,\\underbrace{1}_{(k+1)\\rm{th}},0,\\dots,0]^T.\n\\end{align}\nWe choose one-hot encoding because the stage index\nis an ordered categorical variable instead of a quantitative variable (i.e. it has notion of ordering but no notion of metric). Furthermore, these unit vectors are always orthogonal, which we observed to offer good overall numerical performance of the policy network. The tradeoff is that the dimension of representing $k$ is increased from 1 to $N$.\n\nFor the second component, i.e. the state $x_k$ (including both $x_{k,b}$ and $x_{k,p}$), we represent it in a nonparametric manner as discussed in \\cref{sec:math_formulation}:\n\\begin{align}\nx_k \\qquad \\longrightarrow \\qquad I_k=\\{d_0,y_0,\\dots,d_{k-1},y_{k-1}\\}.\n\\end{align}\nTo accommodate states up to stage $(N-1)$ (i.e. $x_{N-1}$), we use a fixed total dimension of $(N-1)(N_d+N_y)$ for this representation, where for $k < (N-1)$ the entries for $\\{d_l, y_l \\,|\\, l \\geq k\\}$ (experiments that have not happened yet) are padded with zeros (see \\cref{eq:NN_input}). \nIn addition to providing a \nstate representation without any approximation, another major advantage of such nonparametric form can be seen under the terminal formulation in \\cref{eq:terminal_info_gN}, where now none of the intermediate belief states (i.e. $x_{k,b}$ for $k<N$) needs to be computed since the policy network can directly take input of $I_k$. \nAs a result, only a single final Bayesian inference conditioned on all experiments and all observations needs be performed at the end of each episode. The number of Bayesian inference calculations is greatly reduced. \n\n\nPutting together the two input components, the overall input layer for the policy network $\\mu_{w}(k,x_k)$, when evaluating at $(k,x_k)$, has the form\n\\begin{align}\n    \\label{eq:NN_input}\n    [ \\underbrace{e_{k+1}}_{N},\\overbrace{d_0}^{N_d},\\dots,d_{k-1},\\underbrace{0,\\dots,0}_{N_d(N-1-k)},\\overbrace{y_0}^{N_y},\\dots,y_{k-1},\\underbrace{0,\\dots,0}_{N_y(N-1-k)} ] ^ T,\n\\end{align}\nwhere we also indicate the zero-paddings for the entries corresponding to future experiments $l \\geq k$. \nThe overall input layer has a total dimension of $N + (N-1)(N_d+N_y)$, which is linear in $N$, $N_d$, and $N_y$. \nThe remainder of the policy network is relatively straightforward. The output layer is an $N_d$-dimensional vector representing $d_k$, and the network architecture can be chosen by the user. \nWe have experimented with dense layers, and experience suggests 2-3 hidden layers often achieve good performance for our numerical cases. More systematic hyperparameter tuning for DNNs can also be employed to optimize the architecture, but not pursued in this paper. \n\nWe end the introduction of the policy network by emphasizing that $\\mu_{w}(k,x_k)$ is not trained in a supervised learning manner from training data; instead, it is updated iteratively via PG en route to maximizing $U(w)$.\n\n\n\\subsubsection{Q-Network}\n\\label{sec:Q_net}\n\n\nWhile seeking $Q^{\\pi_w}_{k,\\eta_k}$ (parameterized by $\\eta_k$) that approximates $Q^{\\pi_w}_k : \\mathcal{X}_k \\times \\mathcal{D}_k \\mapsto \\mathbb{R}$ for $k=0,\\dots,N-1$,\nwe also combine them into a single function $Q^{\\pi_w}_{\\eta}(k,x_k,d_k)$ in a similar manner as the policy network; we call $Q^{\\pi_w}_{\\eta}$ the Q-network.\nSubsequently, the $\\nabla_{d^{(i)}_k} Q^{\\pi_w}_k(x^{(i)}_k,d^{(i)}_k)$\nterm from \\cref{eq:policy_grad} can be approximated by \n$\\nabla_{d^{(i)}_k} Q^{\\pi_w}_{\\eta}(k,x^{(i)}_k,d^{(i)}_k)$, which can now also be obtained via back-propagation. \nThe input layer then takes the same form as the policy network, except we augment extra entries for $d_k$ as well. The overall input dimension is then $N+(N-1)(N_d+N_y)+N_d$. The network output is a scalar. \n\n\nThe Q-network is trained in a supervised learning manner from the MC episodes generated for \\cref{eq:policy_grad}, by finding $\\eta$ that minimizes the following loss function built based on\n\\cref{eq:action_bellman2}:\n\\begin{align}\n    \\label{eq:value_loss}\n    \\mathcal{L}(\\eta) = \\frac{1}{M} \\sum_{i=1}^M \\sum_{k=0}^{N-1} \\[ Q^{\\pi_w}_{\\eta}(k,x^{(i)}_k,d^{(i)}_k) - \\(g_k(x^{(i)}_k,d^{(i)}_k,y^{(i)}_k) + Q^{\\pi_w}_{k+1}(x^{(i)}_{k+1},d^{(i)}_{k+1})\\) \\]^2\n\\end{align}\nwhere $d^{(i)}_k = \\mu_{w}(k,x^{(i)}_k)$ \nand $Q^{\\pi_w}_N(x^{(i)}_N,\\cdot) = g_N(x^{(i)}_N)$. It is worth noting that $Q^{\\pi_w}_{k+1}(x^{(i)}_{k+1},d^{(i)}_{k+1})$ \ndoes not depend on $\\eta$, but in practice is often approximated by  $Q^{\\pi_w}_{\\eta}(k+1,x^{(i)}_{k+1},d^{(i)}_{k+1})$ ({for $k=0,\\dots,N-2$})\\footnote{The use of an approximate Q-value in the next (i.e. $k+1$) stage rather than expanding further with $g_{k+1}$, $g_{k+2}$, etc. renders this a \\emph{one-step lookahead} approach. This is not to be confused with greedy or myopic design, which does not include any future value term.}. When minimizing the loss, the gradient contribution with respect to $\\eta$ from this term is therefore ignored. \n\n\n\n\\subsubsection{Evaluation of Kullback-Leibler Rewards}\n\\label{sec:numerical_belief_state}\n\nA final step needed to construct the Q-network following \\cref{eq:value_loss} (and in turn, the policy network) is the ability to evaluate our immediate and terminal rewards $g_k$ and $g_N$. \nHaving established the equivalence of terminal and incremental formulations in \\cref{sec:information_gain}, we focus on the former since it requires fewer KL divergence calculations with only the KL in $g_N$ needed at the end of each episode. With the nonparametric state representation using $I_k$ (\\cref{sec:policy_net}), we do not need to explicitly update the Bayesian posteriors throughout the intermediate experiments. Instead, we only need a single Bayesian inference to obtain $p(\\theta|I_N)$, and use it to estimate the KL divergence \\cref{eq:terminal_info_gN}. \n\nIn general, the posteriors will be of non-standard distributions and the KL divergence \nmust be approximated numerically. \nFor the small $N_{\\theta}$ (e.g., $\\leq 4$) examples presented in this paper, we use a grid discretization of the $\\theta$-space and estimate its posterior PDF pointwise; in this work, we always use a uniform grid with 50 nodes in each dimension for evaluation, and 20 nodes for training only in the higher-dimensional $N_{\\theta}=4$ case to speed up the computation (spot-checking between 20 and 50 nodes indicated their results to be close). \nHowever, the exponential growth of grid points with $N_{\\theta}$ would require higher dimensional problems to seek alternative methods, such as Markov chain Monte Carlo (MCMC) with kernel density estimation or likelihood-free ratio estimation \\cite{thomas2021likelihood}, variational inference \\cite{blei2017variational} and transport maps \\cite{Huan2015}. These will be important future directions of our research.\n\n\n\n\\subsubsection{Exploration Versus Exploitation}\n\nThe tradeoff between exploration and exploitation is an important consideration in RL, especially in an uncertain environment. \nExploration searches unexplored or under-explored regions that may contain good policies (i.e. global search), and invests for long-term performance. Exploitation focuses on region deemed promising based on our current knowledge (i.e. local search), thus targets short-term performance. Insufficient exploration may strand the policy search in a local optimum, and insufficient exploitation may lack \nconvergence. A mixed strategy to balance \nexploration and exploitation is prudent \\cite{burnetas1997optimal,li2017deep}, such as through the commonly used epsilon-greedy technique \\cite{sutton2018reinforcement} and many other advanced methods. \n\n\nIn this work, we inject exploration by adding a perturbation to our deterministic policy. \nWe employ this exploration \\emph{only} when generating the MC episodes in \\cref{eq:policy_grad} for estimating the PG during training, and nowhere else (including testing). Thus we view this exploration as an aid solely to the training data generation, and our policy remains deterministic. \nWhen this exploration perturbation is used, the design becomes:\n\\begin{align}\n    d_k = \\mu_k(x_k) + \\epsilon_{\\rm{explore}},\n\\end{align}\nwhere $\\epsilon_{\\rm{explore}}\\sim\\mathcal{N}(0,\\mathbb{I}_{N_d}\\sigma_{\\rm{explore}}^2)$.\nThe perturbed $d_k$ should also be truncated by any design constraints to remain within $\\mathcal{D}_k$. The value of $\\sigma_{\\rm{explore}}$ reflects the degree of exploration versus exploitation, and should be selected based on the problem context. For example, a reasonable approach is to set a large $\\sigma_{\\rm{explore}}$ early in the algorithm and reduce it gradually. More advanced techniques have been proposed to reach a better exploration, for instance, by adding noise to the policy network parameters instead of the design variable \\cite{plappert2017parameter,fortunato2017noisy}; however, these strategies are beyond the scope of our paper.\n\n\\subsection{Simplification to Batch and Greedy Designs}\n\\label{sec:PG-based_batch_greedy}\n\nIn \\cref{sec:subopt_design}, we illustrated batch and greedy designs to be suboptimal cases of the sOED. Similarly, we can reduce the PG-sOED formulation, with very minor modifications, to arrive at PG-based batch and greedy designs. \n\nFor batch design, we simply replace the input layer of the policy network to contain only the one-hot encoding terms $e_{k+1}$. The resulting policy structure thus only maps from the stage index $k$ to a design, and $d_k$ does not depend on the state. \nFor greedy design, we \nuse the incremental formulation\nand retain only the immediate reward term. The Q-function then becomes $Q^{\\pi}_k(x_k,d_k)=\\mathbb{E}_{y_k|x_k,d_k}[g_k(x_k,d_k,y_k)]$, and the loss in \\cref{eq:value_loss} simplifies to\n\\begin{align*}\n        {\\mathcal{L}^{\\mathrm{gr}}}(\\eta) = \\frac{1}{M} \\sum_{i=1}^M \\sum_{k=0}^{N-1} \\[ Q^{\\pi_w}_{\\eta}(k,x^{(i)}_k,d^{(i)}_k) - g_k(x^{(i)}_k,d^{(i)}_k,y^{(i)}_k) \\]^2.\n\\end{align*}\n\n\n\n\n\n\n\n\n\n\\subsection{Pseudocode for the Overall Algorithm}\n\nWe present the detailed algorithm for PG-sOED in \\cref{alg:PG-sOED}. \nWe re-emphasize that the exploration perturbation is only used in generating the MC episodes on line 5, but not used anywhere else including when evaluating the policy after it is constructed. \nFurthermore, with our DNN parameterization of the policy and Q-functions (\\cref{sec:policy_net,sec:Q_net}), $w$ and $\\eta$ are their respective DNN parameter weights. Hence, DNNs computations are encountered in \\cref{alg:PG-sOED} wherever $w$ and $\\eta$ appear.\n\n\n\\begin{algorithm}\n\\caption{The PG-sOED algorithm.}\n\\label{alg:PG-sOED}\n\\begin{algorithmic}[1]\n\\STATE{Define all components in \\cref{sec:math_formulation};}\n\\STATE{Set initial state $x_0$, policy updates $L$, MC sample size $M$, policy and Q-network architectures, learning rate $\\alpha$ for policy update, exploration scale $\\sigma_{\\rm{explore}}$;\n}\n\\STATE{Initialize policy and Q-network parameters $w$ and $\\eta$;}\n\\FOR{$l=1,\\dots,L$}\n\\STATE{Simulate $M$ episodes: sample $\\theta\\sim x_{0,b}$, and then for $k=0,\\dots,N-1$ sample $d_k=\\mu_{w}(k,x_k)+\\epsilon_{\\rm{explore}}$} and $y_k\\sim p(y_k|\\theta,d_k,I_k)$;\n\\STATE{Store the full information vectors from all episodes $\\{I_N^{(i)}\\}_{i=1}^M$}, from which the intermediate $\\{I_1^{(i)}, I_2^{(i)},\\ldots,I_{N-1}^{(i)}\\}$ can also be formed trivially;\n\\STATE{Compute and store immediate and terminal rewards for all episodes $\\{g_k^{(i)}\\}_{i=1}^M$, $k=0,\\dots,N$;}\n\\STATE{Update $\\eta$ by minimizing the loss in \\cref{eq:value_loss};}\n\\STATE{Update $w$ by gradient ascent: $w = w + \\alpha \\nabla_wU(w)$, where $\\nabla_wU(w)$ is estimated through \\cref{eq:policy_grad};}\n\\STATE{(Optional) Reduce $\\alpha$ and $\\sigma_{\\rm{explore}}$;}\n\\ENDFOR\n\\STATE{Return optimized policy $\\pi_{w}$;}\n\\end{algorithmic}\n\\end{algorithm}\n\n\\section{Numerical Results}\n\\label{sec:results}\n\nWe present two examples to illustrate different aspects of PG-sOED. The first is a linear-Gaussian problem (\\cref{sec:ex_linGau}) that offers a closed form solution due to its conjugate prior form. This problem serves as a benchmark, where we validate the optimal policy and expected utility obtained by PG-sOED against the analytic solution. We also illustrate the superior computational speed of PG-sOED over an existing ADP-sOED baseline.\nThe second example entails a problem of contaminant source inversion in a convection-diffusion field (\\cref{sec:ex_sourceInv}). It further divides into three cases: case 1 compares PG-sOED to  greedy design, and cases 2 and 3 compare PG-sOED to both greedy and batch designs. This example thus demonstrates the advantages of PG-sOED over greedy and batch designs, and its ability to accommodate expensive forward models with nonlinear physics and dynamics. \n\n\\subsection{Linear-Gaussian Benchmark}\n\\label{sec:ex_linGau}\n\\subsubsection{Problem Setup}\nWe adopt the linear-Gaussian problem from~\\cite{Huan2016} as a benchmark case for validating PG-sOED. The forward model is linear in $\\theta$, and corrupted with an additive Gaussian observation noise $\\epsilon_k \\sim \\mathcal{N}(0,1^2)$:\n\\begin{align}\n    y_k = G(\\theta,d_k) + \\epsilon_k = \\theta d_k + \\epsilon_k.\n\\end{align}\nWe design $N=2$ experiments, with prior $\\theta\\sim\\mathcal{N}(0,3^2)$, and design constrained in $d_k \\in [0.1,3]$. The resulting conjugate form renders all subsequent posteriors to be analytically Gaussian, thus allowing us to compute the optimal policy in closed form. \nThere is no  physical state for this problem. \nThe stage rewards and terminal reward are chosen to be\n\\begin{align}\n    g_k(x_k,d_k,y_k) &= 0, \\quad k=0,1 \\\\\n    g_N(x_N) &= D_{\\mathrm{KL}}\\( p(\\cdot|I_N)\\,||\\,p(\\cdot|I_0) \\) - 2\\( \\ln{\\sigma_N^2} - \\ln{2} \\)^2\n\\end{align}\nwhere $\\sigma_N^2$ represents the variance of the final belief state, and the additive penalty in the terminal reward is purposefully inserted \nto make the design problem more challenging.\nWe solve this sOED problem both by ADP-sOED \\cite{Huan2016} and PG-sOED. ADP-sOED adopts the same setting in \\cite{Huan2016}. For PG-sOED, we set \n$L=100$, \n$M=1000$, \n$\\alpha=0.15$, \nand $\\sigma_{\\rm{explore}}=0.2$ {that also decreases by\na factor of $0.95$ per policy update. Both the policy network and Q-network contain two hidden layers with ReLU activation, and each hidden layer has 80 nodes.} In particular, we have observed even a low $M=10$ yielded similar performance, but $M=1000$ is used here to further reduce the effect of MC error in the demonstration. \n\n\\subsubsection{Results}\n\nDue to the conjugate form, we can obtain all posteriors in closed form, and find the (non-unique) optimal policies analytically~\\cite{Huan2015,Huan2016}. To evaluate the policies found by ADP-sOED and PG-sOED, we sample $10^4$ episodes using their final policies and compute their total rewards. \nADP-sOED yields a mean total reward of \n$0.775\\pm0.006$ and PG-sOED also \n$0.775\\pm0.006$, where the $\\pm$ is the MC standard error. Both match extremely well with the analytic result $U(\\pi^\\ast)\\approx0.783$, where the discrepancy (due to MC sampling {and grid discretization of the posterior}) is within two standard errors. These results thus support that both ADP-sOED and PG-sOED have found the optimal policy. \n \n \n\n\n\n\\Cref{fig:LinGau_Cnvg_reward,fig:LinGau_Cnvg_residual} present the convergence history for the expected utility and residual ($|U(\\pi^\\ast)-U(w)|$) as a function of the PG-sOED iterations. \nThe convergence is rapid, reaching over 3 orders of magnitude reduction of the residual \nwithin 30 iterations. \nThe much lower initial expected utility (around $-8.5$) also indicates that a random policy (from random initialization) performs much worse than the optimized policy.\n\n\\begin{figure}[htbp]\n  \\centering\n  \\subfloat[Reward history]{\\label{fig:LinGau_Cnvg_reward}\\includegraphics[width=0.49\\linewidth]{Figures\/LinGau_Cnvg_PG.pdf}}\n  \\subfloat[Residual history $|U(\\pi^\\ast)-U(\\pi)|$]{\\label{fig:LinGau_Cnvg_residual}\\includegraphics[width=0.49\\linewidth]{Figures\/LinGau_residual_PG.pdf}}\n  \\caption{Convergence history of PG-sOED.}\n  \\label{fig:LinGau_Cnvg}\n  \n\\end{figure}\n\n\n\\cref{tab:ADPvsPG} compares the computational costs between ADP-sOED and PG-sOED for this linear-Gaussian problem, obtained using a single 2.6 GHz CPU on a MacBook Pro laptop.\nThe timing figures are from 30 gradient ascent updates for PG-sOED in the training stage, and 1 policy update (minimum needed) for ADP-sOED. \nPG-sOED produces orders-of-magnitude speedups compared to ADP-sOED. Noteworthy is the extremely low testing time (i.e. using the policy online during the experimental campaign after the policy has been constructed offline), achieving $0.0002$ seconds per experiment (4 seconds per $10^4$ episodes with $N=2$ experiments per episode). \nThe advantage is due to ADP-sOED being a value-based approach where each policy evaluation needs to solve a (stochastic) optimization problem which in turns requires many forward model evaluations; in contrast, PG-sOED only requires a single forward pass of its policy-network free of any forward model runs. The online speed make the PG-sOED an excellent candidate for situations where fast, possibly real-time design responses are needed.\n\n\\begin{table}[hbt]\n    \\centering\n    \\caption{Comparison of computational costs between ADP-sOED and PG-sOED.\n        }\n    \\begin{tabular}{|c| c c |c|}\n    \\hline\n        & \\textbf{Training time (s)} & \\textbf{Forward model evaluations} & \\textbf{Testing time (s)} \\\\\n    \\hline\n    ADP-sOED & 837     & $5.3 \\times 10^8$     & 24,396     \\\\\n    PG-sOED & 24     & $3.1 \\times 10^6$     & 4     \\\\\n    \\hline\n    \\end{tabular}\n    \\label{tab:ADPvsPG}\n\\end{table}\n\n\n\\subsection{Contaminant Source Inversion in Convection-Diffusion Field}\n\\label{sec:ex_sourceInv}\n\n\\subsubsection{Problem Setup}\n\nThe next example entails mobile sensor design in a convection-diffusion field (e.g., of a chemical contaminant plume) to take concentration measurements in order to infer the plume source location. We model the contaminant field in a two-dimensional square domain, where the contaminant concentration $G$ at spatial location $z=[z_x,z_y]$ and time $t$ is constrained by the convection-diffusion PDE:\n\\begin{align}\n    \\frac{\\partial G(z,t;\\theta)}{\\partial t} = \\nabla^2 G-{u}(t) \\cdot \\nabla G + S(z,t;\\theta), \\qquad z\\in [z_L,z_R]^2, \\quad t>0,\n    \\label{e:PDE}\n\\end{align}\nwhere ${u}=[u_x,u_y] \\in \\mathbb{R}^2$\nis a time-dependent convection velocity, and $\\theta =[\\theta_x,\\theta_y, \\theta_h, \\theta_s] \\in \\mathbb{R}^4$ \n{is the source parameter} residing within the source function\n\\begin{align}\n    S(z,t;\\theta) = \\frac{\\theta_s}{2\\pi \\theta_h^2} \\exp\\( -\\frac{(\\theta_x-z_x)^2 + (\\theta_y-z_y)^2}{2 \\theta_h^2} \\)\n\\end{align}\nwith $\\theta_x$ and $\\theta_y$ denoting the source location, and $\\theta_h$ and $\\theta_s$ denoting the source width and source strength. \nThe initial condition is $G(z,0;\\theta)=0$, and homogeneous Neumann boundary condition \nis imposed for all sides of the square domain. \nWe solve the PDE numerically using second-order finite volume method on a uniform grid of size $\\Delta z_x = \\Delta z_y = 0.01$ and {a second-order fractional step method for} time-marching with stepsize $\\Delta t = 5.0 \\times 10^{-4}$. \nFor example, \\cref{fig:Source_exp} illustrates the solution $G$ for such a convection-diffusion scenario where the convection speed increases over time. \n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[width=1.0\\linewidth]{Figures\/Source_example.png}\n    \\caption{Sample numerical solution of the concentration field $G$ at different time snapshots. The solution is solved in a wider computational domain $[-1,2]^2$ but displayed here in a region of interest $[0,1]^2$. In this case, $\\theta=[0.210,0.203,0.05,2]$ and the convection grows over time with $u_x=u_y=10t\/0.2$. Hence, isotropic diffusion dominates early on and the plume stretches towards the convection direction with time. \n    }\n  \\label{fig:Source_exp}\n\\end{figure}\n\nFor the design problem, we have a vehicle with sensing equipment for measuring the contaminant concentration $G$ that can be relocated at fixed time intervals. We seek to determine where we should relocate this vehicle such that its measurements can lead to the best inference of the source parameter $\\theta$.\nWe consider $N$ measurement opportunities respectively at time $t_k$ for $k=0,\\dots,N-1$. The vehicle starts with initial belief state $x_{0,b}=(\\theta|I_0)$ (i.e. prior on $\\theta$) and initial physical state $x_{0,p}$ (i.e. vehicle location). The design variable is the displacement of the vehicle from the current location and constrained $d_k \\in [d_L, d_R]^2$ to reflect the range of vehicle movement. The physical state is then updated via\n\\begin{align}\n    x_{k+1,p} = x_{k,p} + d_k.\n\\end{align}\nAt the new physical location, a noisy measurement of the contaminant concentration is obtained in the form\n\\begin{align}\n    y_k = G(z=x_{k+1,p},t_k;\\theta) + \\epsilon_k \\( 1+\\vert G(x_{k+1,p},t_k;\\theta) \\vert \\)\n\\end{align}\nwhere \n$\\epsilon_k \\sim \\mathcal{N}(0,\\sigma_\\epsilon^2)$, thus the observation noise is affected by the signal magnitude. \nOnce the new measurement is acquired, the belief state is updated from $x_{k,b}=(\\theta|I_k)$ to $x_{k+1,b}=(\\theta|I_{k+1})$ through Bayes' rule. The reward functions are \n\\begin{align}\n    g_k(x_k,d_k,y_k) &= - c_q f_c(d_k), \\quad k=0,\\dots,N-1 \\\\\n    g_N(x_N) &= \n                        D_{\\mathrm{KL}}\\( p(\\cdot|I_N)\\,||\\,p(\\cdot|I_0) \\).\n\\end{align} \nIn particular, the immediate reward reflects a cost on the vehicle movement considering the effect of the wind, {where $f_c(d_k)$ denotes the cost function on sensor movement.}\n\nWe further set up 3 cases for this source inversion problem with their settings summarized in \\cref{tab:Source_setup}. \nCase 1 is diffusion-only and cases 2 and 3 are convection-diffusion. {In cases 1 and 2, $N=2$ experiments will be designed, and the source width $\\theta_h$ and source strength $\\theta_s$ are known. In case 3, $N=4$ experiments will be designed, and $\\theta_h$ and $\\theta_s$ are treated as unknown parameters}.\nFor case 1, contaminant source is off ($s=0$) initially and activated ($s=2$) at $t=0.16$. The first experiment ($t_0=0.15$) thus takes place when there is no contaminant and only the second ($t_1=0.32$) encounters the plume. We anticipate no immediate gain from the first experiment but it may be used to set up a better second experiment via lookahead. We use case 1 to highlight the difference between sOED and greedy design. \nCase 2, by including time-dependent convection, emphasizes the value of feedback adaptation to environmental change and dynamics. We use case 2 to compare sOED with both greedy and batch designs. \n{Case 3 features a longer horizon and a higher dimensional parameter space, and with a penalty on the sensor movement. We use case 3 to demonstrate the usage of sOED in more complicated problems.}\nAll batch and greedy designs are implemented following \\cref{sec:PG-based_batch_greedy}. \nFor PG-sOED, we set $L=300$, $M=1000$, $\\alpha=0.01$ {with the Adam optimizer \\cite{kingma2014adam}}, and $\\sigma_{\\rm{explore}} = 0.05$. From experimentation with different $M$'s, we have found $M=1000$ to be suitable here while lower settings such as $M=100$ starts to {produce noisier results}.\nFinally, $10^4$ episodes are simulated for evaluating the performance of different design policies.  \n\n\n\\begin{table}[hbt]\n    \\centering\n    \\caption{Setup for the three cases in the contaminant source inversion problem.}\n    \\begin{tabular}{|c|c|c|c|}\n    \\hline\n    & \\textbf{Case 1} & \\textbf{Case 2} & \\textbf{Case 3}\\\\\n    \\hline\n    Number of experiments & \\multicolumn{2}{c|}{$N=2$} & $N=4$ \\\\\n    \\cline{2-4}\n    Prior of $\\theta_x$ and $\\theta_y$ & \\multicolumn{3}{c|}{$\\theta_x,\\theta_y \\sim \\mathcal{U}([0,1])$} \\\\\n    \\cline{2-4}\n    Prior of $\\theta_h$ & \\multicolumn{2}{c|}{$\\theta_h=0.05$} & $\\theta_h \\sim \\mathcal{U}([0.02, 0.1])$ \\\\\n    \\cline{2-4}\n    Prior of $\\theta_s$ & $\\theta_s= \\left\\{ \\begin{array}{c}\n     0\\,\\, \\mathrm{if} \\,\\, t<0.16 \\\\ 2\\,\\, \\mathrm{if} \\,\\, t\\geq 0.16 \\end{array}\n     \\right.$ & $\\theta_s=2$ & $\\theta_s \\sim \\mathcal{U}([0, 5])$ \\\\\n    \\cline{2-4}\n    Initial physical state & \\multicolumn{3}{c|}{$x_{0,p} = [0.5,0.5]$} \\\\\n    \\cline{2-4}\n    Design constraint & \\multicolumn{2}{c|}{$d_k \\in [-0.25,0.25]^2$} & $x_{k,p} \\in [0,1]^2$ \\\\\n    \\cline{2-4}\n    Computational domain  & $z_L=0$, $z_R=1$ & \\multicolumn{2}{c|}{$z_L=-1$, $z_R=2$} \\\\\n    \\cline{2-4}\n    Experiment time & $t_0=0.15$, $t_1=0.32$ & $t_0=0.05$, $t_1=0.2$ & $t_k=0.05(k+1)$ \\\\\n    \\cline{2-4}\n    Velocity field & $u_x=u_y=0$ & \\multicolumn{2}{c|}{$u_x=u_y=10t\/0.2$} \\\\\n    Noise scale & $\\sigma_\\epsilon = 0.1$ & \\multicolumn{2}{c|}{$\\sigma_\\epsilon = 0.05$} \\\\\\cline{2-4}\n    Cost function $f_c(d_k)$ & \\multicolumn{2}{c|}{$\\Vert d_k \\Vert ^2$} & $\\Vert d_k \\Vert - \\frac{\\sqrt{2}}{40} d_k \\cdot u(t_k)$ \\\\\n    \\cline{2-4}\n    Cost coefficient & $c_q=0.5$ & $c_q=0$ & $c_q=0.2$ \\\\\n    \\hline\n    \\end{tabular}\n    \\label{tab:Source_setup}\n\\end{table}\n\n\\subsubsection{Surrogate Model}\n\nSolving the forward model \\cref{e:PDE} using finite volume is still computationally viable for PG-sOED, but expensive. \nOne strategy to accelerate the computation is to employ surrogate models to replace the original forward model. \nWe use DNNs to construct surrogate models of for $G(z,t_k;\\theta)$ for $k=0,\\dots,N-1$. \nWe use the following architecture for each DNN: a 4-dimensional input layer taking $z$ and $\\theta$; five hidden layers with 40, 80, 40, 20, and 10 nodes; and a scalar output $G$. \nA dataset is generated by solving for $G$ on $2000$ samples of $\\theta$ drawn from its prior distribution.\nThese concentration values are then first restricted to only the domain that is reacheable by the vehicle (due to the design constraint), then shuffled across $\\theta$ and split 80\\% for training and 20\\% for testing. We achieve low test mean-squared-errors of around \n$10^{-6}$\nfor surrogate models $G(z,t_k;\\theta)$ for all three cases.\n\\Cref{fig:Source_surrogate} provides an example comparing the concentration contours from $t=0.05$ and $t=0.2$ of case 2 using the DNN surrogates (left column) and finite volume (right column). They appear nearly identical. \nMore importantly, the surrogate models provide a significant  speedup over the finite volume solver by a factor of $10^5$. \n\\begin{figure}[htb]\n  \\centering\n  \\subfloat[$t=0.05$]{\\label{fig:Source_surrogate_1}\\includegraphics[width=0.48\\linewidth]{Figures\/Source_surrogate_1.png}}\\hspace{1em}\n  \\subfloat[$t=0.2$]{\\label{fig:Source_surrogate_2}\\includegraphics[width=0.48\\linewidth]{Figures\/Source_surrogate_2.png}}\n  \\caption{Sample comparison of the concentration field $G$ at $t=0.05$ and $t=0.2$ of case 2 using the DNN surrogates (left column) and finite volume (right column). They appear nearly identical. \n    }\n  \\label{fig:Source_surrogate}\n\\end{figure}\n\n\\subsubsection{Case 1}\n\nCase 1 is diffusion-only. \nBefore presenting the sOED results, we first offer some intuition about high-value designs from a simpler one-experiment design. \n\\Cref{fig:Diffusion_bestloc_util} illustrates the expected utility surface versus sensor location for \na \\emph{single} experiment measuring concentration at\n$t=0.32$. The key insight is that high-value experiments are at the corners of the domain. \nThis can be explained by the isotropic nature of diffusion process that carries information about distance but not direction, thereby leading to posterior probabilities concentrating around regions that resemble an arc of a circle (\\cref{fig:Diffusion_1sensor_post}). Combined with the square domain geometry and Neumann boundary conditions, the ``covered area'' of high-probability posterior is smallest (i.e. least uncertain), averaged over all possible $\\theta$ source locations, when the measurements are made at the corners of the domain.\n\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[width=0.7\\linewidth]{Figures\/Diffusion_bestloc_util.pdf}\n  \\caption{Case 1. Expected utility for one-experiment design at $t=0.32$. The best design locations are at the corners.}\n  \\label{fig:Diffusion_bestloc_util}\n\\end{figure}\n\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[width=1\\linewidth]{Figures\/Diffusion_bestloc_post.pdf}\n  \\caption{Case 1. Posterior PDF contours for the one-experiment design under different design locations (red dot) and a sample source location (purple star). The posteriors exhibit shapes resemble an arc of a circle, due to the isotropic nature of diffusion and the domain geometry. \n    }\n  \\label{fig:Diffusion_1sensor_post}\n\\end{figure}\n\n\nWith the insight that corners are good design locations,\nunderstanding the behavior of PG-sOED\nbecomes easier. \n\\Cref{fig:Diffusion_vsgreedy_soed1} displays the posterior contours after 1 and 2 experiments (i.e. $p(\\theta|I_1)$ and $p(\\theta|I_2)$; also recall the prior $p(\\theta|I_0)$ is uniform) of an episode instance when following PG-sOED; \\cref{fig:Diffusion_vsgreedy_greedy1} displays those for greedy design. In each plot, the purple star represents the true source location for that episode, the red dot represents the physical state (vehicle location), and the red line segment tracks the vehicle displacement (design) from the preceding experiment. \n\nIn PG-sOED (\\cref{fig:Diffusion_vsgreedy_soed1}), the first design moves the vehicle towards a corner despite the source is off at $t_0$ and that no concentration signal is obtained, incurring a negative reward $g_0=-0.040$ due to the movement penalty. The greedy design realizes the source is off and remains at the initial location (center), keeping its reward at $g_0=0$.\nAt this point, it would appear greedy is performing better. The source then becomes active in the second experiment at $t_1$, and both PG-sOED and greedy shift the vehicle towards a corner. However, PG-sOED is able to arrive much closer to the corner and obtains a more informative measurement compared to greedy design, since PG-sOED has already made a head start in the first experiment. Therefore, PG-sOED is able to look ahead and take into account future outcomes. With an initial ``sacrifice'' of seemingly fruitless first experiment, PG-sOED is able to better position the vehicle for a much more lucrative second experiment, such that the expected \\emph{total} reward is maximized ($\\sum_{k=0}^2 g_k=2.941$ for PG-sOED versus $\\sum_\n  {k=0}^2 g_k=2.022$ for greedy). \n\nWe further generate $10^4$ episodes under different \n$\\theta$ samples drawn from the prior, and collect their realized total rewards in\n\\cref{fig:Diffusion_hist}. Indeed, the mean total reward for PG-sOED is $0.615 \\pm 0.007$, higher than greedy design's $0.552 \\pm 0.005$. Note that while PG-sOED has more low-reward episodes {corresponding to when the true source location is far away from the top right corner and when incurring high movement penalty in the first stage}, it also has more high-reward episodes {corresponding to when the true source location is near the top right corner}. Overall, PG-sOED achieves a greater mean value than greedy design. \n\n\\begin{figure}[htbp]\n  \\centering\n  \\subfloat[PG-sOED, $g_0=-0.040,\\sum_{k=0}^2 g_k=2.941 $]{\\label{fig:Diffusion_vsgreedy_soed1}\\includegraphics[width=0.48\\linewidth]{Figures\/Diffusion_vsgreedy_soed1.pdf}}\\hspace{1em}\n  \\subfloat[Greedy, $g_0=-0.000, \\sum_\n  {k=0}^2 g_k=2.022$]{\\label{fig:Diffusion_vsgreedy_greedy1}\\includegraphics[width=0.48\\linewidth]{Figures\/Diffusion_vsgreedy_greedy1.pdf}}\n  \\caption{Case 1. An episode instance obtained by PG-sOED and greedy design. The purple star represents the true $\\theta$, red dot represents the physical state (vehicle location), red line segment tracks the vehicle displacement (design) from the preceding experiment, and contours plot the posterior PDF. }\n  \\label{fig:Diffusion_vsgreedy}\n\\end{figure}\n\n\n\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[width=0.7\\linewidth]{Figures\/Diffusion_soed_greedy_hist.pdf}\n  \\caption{Case 1. Histograms of total rewards from $10^4$ episodes generated using PG-sOED and greedy designs. \n  The mean total reward for PG-sOED is $0.615 \\pm 0.007$, higher than greedy design's $0.552 \\pm 0.005$.\n  }\n  \\label{fig:Diffusion_hist}\n\\end{figure}\n\n\n\n\\subsubsection{Case 2}\nCase 2 incorporates convection in addition to diffusion, and disables movement penalty for the vehicle (i.e. $c_q=0$).\nIn \\cref{fig:Source_xp}, we plot the physical states $x_{1,p}$ and $x_{2,p}$ (i.e. vehicle locations after the first and second experiments) from $10^4$ episodes sampled from PG-sOED, greedy, and batch designs. \nWe observe both PG-sOED and batch design initially move the vehicle towards the top right corner and then turn back; greedy design roughly moves in the opposite direction. Notably, $d_1$ (design for the second experiment) for batch design is always the same, in contrast to PG-sOED and greedy that are adaptive. \nThe behavior of the different policies can be better understood through \\cref{fig:Source_util}, which shows the contours of expected utility versus sensor location if performing only a \\emph{single} experiment at $t_0$ or $t_1$, respectively. \nIn \\cref{fig:Source_util_1}, we find the global maximum to be around $(0.3,0.3)$, which explains the initial movement of greedy design towards the bottom left.\nHowever, \\cref{fig:Source_util_2} reveals that the top right region becomes more informative at $t_1$. Physically, this makes sense since the convection velocity grows over time towards the top-right direction, and more information can be gathered if we ``catch'' the flow at a downstream position. \nThis explains why PG-sOED and batch design both move towards the top right even in the first experiment since both of those designs can see the more informative second experiment (except that batch design cannot adapt). \n\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[width=0.8\\linewidth]{Figures\/Source_Xp.png}\n  \\caption{Case 2. Vehicle locations of episodes obtained from PG-sOED, greedy, and batch designs.}\n  \\label{fig:Source_xp}\n\\end{figure}\n\n\\begin{figure}[htbp]\n  \\centering\n  \\subfloat[   $t=0.05$]{\\label{fig:Source_util_1}\\includegraphics[width=0.48\\linewidth]{Figures\/Source_util_1.pdf}}\n  \\subfloat[   $t=0.2$]{\\label{fig:Source_util_2}\\includegraphics[width=0.48\\linewidth]{Figures\/Source_util_2.pdf}}\n  \\caption{Case 2. Expected utility versus sensor location if conducting a single experiment at $t=0.05$ or $t=0.2$.\n    }\n  \\label{fig:Source_util}\n\\end{figure}\n\n\n\nBack to the two-experiment design, \\cref{fig:Source_hist} summarizes the total rewards from all $10^4$ episodes, with PG-sOED having the highest mean value at\n$1.344 \\pm 0.008$ followed by batch design's $1.264 \\pm 0.007$ and greedy design's $1.178 \\pm 0.010$.\nThe advantage of PG-sOED is greater over greedy and less over batch, suggesting a more prominent role of lookahead.\nFrom the histograms, greedy design has many low-reward episodes, corresponding to scenarios when the true source location is in the upper-right region. At the same time, greedy also has a similar distribution of high-reward episodes as sOED because it is able to adapt. In contrast, batch design does not have many low-reward episodes since it moves towards the upper-right in the first experiment. However, it also has fewer high-reward episodes compared to sOED because it is unable to adapt.\n\n\\begin{figure}[htbp]\n  \\centering\n  \\subfloat[sOED versus greedy]{\\label{fig:Source_soed_greedy_hist}\\includegraphics[width=0.48\\linewidth]{Figures\/Source_soed_greedy_hist.pdf}}\n  \\subfloat[sOED versus batch]{\\label{fig:Source_soed_batch_hist}\\includegraphics[width=0.48\\linewidth]{Figures\/Source_soed_batch_hist.pdf}}\n  \\caption{Case 2. Histograms of total rewards from $10^4$ episodes generated using PG-sOED, greedy, and batch designs. The mean total reward for PG-sOED is $1.344 \\pm 0.008$, higher than greedy design's $1.178 \\pm 0.010$ and batch design's $1.264 \\pm 0.007$.\n  }\n  \\label{fig:Source_hist}\n\\end{figure}\n\nLastly, we provide examples of posteriors resulting from sample episodes. \\Cref{fig:Source_vsgreedy} presents scenarios where PG-sOED visibly achieves a ``narrower'' posterior compared to greedy and batch designs, which is reflected quantitatively through the higher total reward. However, there are also scenarios \nwhere PG-sOED achieves a lower total reward, such as shown in \\cref{fig:Source_vsbatch}. It is the \\emph{expected} utility averaged over all possible scenarios that PG-sOED maximizes.\n\\begin{figure}[htbp]\n  \\centering\n  \\subfloat[PG-sOED, $\\theta=(0.7,0.9)$, total reward $=2.020$]{\\label{fig:Source_vsgreedy_soed1}\\includegraphics[width=0.48\\linewidth]{Figures\/Source_vsgreedy_soed1.pdf}} \n  \\subfloat[Greedy, $\\theta=(0.7,0.9)$, total reward $=0.493$]{\\label{fig:Source_vsgreedy_greedy1}\\includegraphics[width=0.48\\linewidth]{Figures\/Source_vsgreedy_greedy1.pdf}}\\\\\n  \\subfloat[PG-sOED, $\\theta=(0.65,0.7)$, total reward $=4.014$]{\\label{fig:Source_vsbatch_soed}\\includegraphics[width=0.48\\linewidth]{Figures\/Source_vsbatch_soed.pdf}} \n  \\subfloat[Batch, $\\theta=(0.65,0.7)$, total reward $=2.647$]{\\label{fig:Source_vsbatch_batch}\\includegraphics[width=0.48\\linewidth]{Figures\/Source_vsbatch_batch.pdf}}\n\\\\  \\caption{Case 2. \n  Episode instances obtained by PG-sOED, greedy, and batch designs, where PG-sOED outperforms both greedy and batch designs. The purple star represents the true $\\theta$, red dot represents the physical state (vehicle location), red line segment tracks the vehicle displacement (design) from the preceding experiment, and contours plot the posterior PDF. The inset zooms in on the high-probability posterior region.\n  }\n  \\label{fig:Source_vsgreedy}\n\\end{figure}\n\n\\begin{figure}[htbp]\n  \\centering\n  \\subfloat[PG-sOED, $\\theta=(0.1,0.4)$, total reward $=1.076$]{\\label{fig:Source_vsgreedy_soed2}\\includegraphics[width=0.48\\linewidth]{Figures\/Source_vsgreedy_soed2.pdf}} \n  \\subfloat[Greedy, $\\theta=(0.1,0.4)$, total reward $=1.687$]{\\label{fig:Source_vsgreedy_greedy2}\\includegraphics[width=0.48\\linewidth]{Figures\/Source_vsgreedy_greedy2.pdf}}\n  \\caption{Case 2. \n  Episode instances obtained by PG-sOED and greedy design, where greedy design outperforms PG-sOED. The purple star represents the true $\\theta$, red dot represents the physical state (vehicle location), red line segment tracks the vehicle displacement (design) from the preceding experiment, and contours plot the posterior PDF.}\n  \\label{fig:Source_vsbatch}\n\\end{figure}\n\n\\subsubsection{Case 3}\n\nCase 3 provides demonstration on a problem with longer design horizon ($N=4$ experiments) and a higher dimensional parameter space ($N_\\theta=4$) that now also includes the source strength $\\theta_s$ and width $\\theta_h$. \n\nFrom \\cref{fig:Source3_hist}, we see that PG-sOED's mean total reward ($3.435 \\pm 0.016$) outperforms both greedy ($3.057 \\pm 0.015$) and batch ($2.856 \\pm 0.012$) designs.\nIn particular, PG-sOED features a prominent bimodal distribution of the total rewards, but also a heavier tail to the right leading to an overall greater total reward \\emph{in expectation} compared to greedy and batch designs. \nIn \\cref{fig:Source3_xp}, we plot the physical states $x_{k+1,p}$ (i.e. vehicle locations after the $k$-th experiments) from $10^4$ episodes sampled from PG-sOED, greedy, and batch designs. As expected, batch design always produces identical movement paths since it is non-adaptive, while the PG-sOED and greedy designs can spread into the entire domain. Notably, we do not observe movement to the left and bottom regions even after all four experiments. This is because moving towards the bottom left is against the wind direction and requires a higher movement cost, and the bottom left does not offer measurements as informative as the top right since the convection direction is towards the top right.\n\n\\begin{figure}[htbp]\n  \\centering\n  \\subfloat[PG-sOED versus greedy]{\\label{fig:Source3_soed_greedy_hist}\\includegraphics[width=0.48\\linewidth]{Figures\/Source3_soed_greedy_hist.pdf}}\n  \\subfloat[PG-sOED versus batch]{\\label{fig:Source3_soed_batch_hist}\\includegraphics[width=0.48\\linewidth]{Figures\/Source3_soed_batch_hist.pdf}}\n  \\caption{Case 3. Histograms of total rewards from $10^4$ episodes generated using PG-sOED, greedy, and batch designs. The mean total reward for PG-sOED is $3.435 \\pm 0.016$, higher than greedy design's $3.057 \\pm 0.015$ and batch design's $2.856 \\pm 0.012$.\n}\n  \\label{fig:Source3_hist}\n\\end{figure}\n\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[width=0.85\\linewidth]{Figures\/Source3_Xp.png}\n  \\caption{Case 3. Vehicle locations of episodes obtained from PG-sOED, greedy, and batch designs (rows) plotted for experiments 1--4 (columns).}\n  \\label{fig:Source3_xp}\n\\end{figure}\n\n\\cref{fig:Source3_contour} shows the marginal posterior PDF contours from episode instances obtained by PG-sOED, greedy and batch designs. Interestingly, both PG-sOED and batch design move towards the bottom right corner in the first experiment, leading to a marginal posterior in $\\theta_x$ and $\\theta_y$ with a half circle shape that provides a better setup for the next movement. In contrast, the marginal posterior of greedy design leads to a full circle shape, which does not narrow down the direction as much. Consequently, PG-sOED outperforms greedy design. However, this initial advantage in batch design is overtaken by its lack of adaptation in the subsequent experiments, leading to a lower expected total reward than greedy design. We also find the posterior mode \nfor $\\theta_h$ to be noticeably shifted from its true data-generating value, and the uncertainty remains high even after four experiments. \nThis can be explained by the low sensitivity of the concentration to the range of source widths considered, making $\\theta_h$ more difficult to identify.\nHowever, the mode of the full four-dimensional joint posterior (not plotted) is still close to the true value.\n\n\\begin{figure}[htbp]\n  \\centering\n  \\subfloat[PG-sOED, $\\theta=(0.6,0.7,0.06,0.2)$, total reward $=3.428$]{\\label{fig:Source3_contour_soed}\\includegraphics[width=0.8\\linewidth]{Figures\/Source3_contour_soed.pdf}} \\\\\n  \\subfloat[Greedy, $\\theta=(0.6,0.7,0.06,0.2)$, total reward $=2.614$]{\\label{fig:Source3_contour_greedy}\\includegraphics[width=0.8\\linewidth]{Figures\/Source3_contour_greedy.pdf}} \\\\\n  \\subfloat[Batch, $\\theta=(0.6,0.7,0.06,0.2)$, total reward $=1.984$]{\\label{fig:Source3_contour_batch}\\includegraphics[width=0.8\\linewidth]{Figures\/Source3_contour_batch.pdf}} \n  \\caption{Case 3. \n  Episode instances obtained by PG-sOED, greedy and batch designs. The purple star represents the true $\\theta$, red dot represents the physical state (vehicle location), red line segment tracks the vehicle displacement (design) from the preceding experiment, and contours plot the marginal posterior PDF.}\n  \\label{fig:Source3_contour}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nThe production of top quarks at high-energy colliders is a process of utmost importance, both in\ntesting the validity of the Standard Model~(SM) and in the quest for new physics.\nWithin the SM, the main source of top-quark events in hadronic collisions is\ntop-quark pair ($t {\\bar t}$) production.\nThe large data set delivered by the CERN LHC\nenables precise measurements of the $t {\\bar t}$ production cross section as a\nfunction of the $t {\\bar t}$ kinematics (see e.g. Refs.~\\cite{Khachatryan:2015oqa,Khachatryan:2015fwh,Aad:2015hna,Aad:2015mbv,Khachatryan:2016gxp,Aaboud:2016iot,Aaboud:2016syx,Sirunyan:2017mzl,Sirunyan:2018wem,Sirunyan:2019zvx}), which can be compared with the SM predictions.\nAt the same time these studies have a wider relevance, since $t {\\bar t}$ production is a crucial\nbackground in many new-physics searches.\n\nNext-to-leading-order (NLO) QCD corrections \nfor this production process were obtained thirty years ago\n\\cite{Nason:1987xz, Beenakker:1988bq, Beenakker:1990maa, Nason:1989zy,Mangano:1991jk}.\nBeyond the on-shell approximation of $t {\\bar t}$ production, first NLO QCD studies were carried out \nwithin the narrow-width approximation~\\cite{Melnikov:2009dn,Bernreuther:2010ny,Campbell:2012uf}.\nSuch NLO studies were later\nperformed by considering\nthe complete $W^+W^-b\\bar b$ final states, with off-shell\nleptonic~\\cite{Bevilacqua:2010qb,Denner:2010jp,Denner:2012yc,Heinrich:2013qaa} and\nsemi-leptonic~\\cite{Denner:2017kzu} decays.\nIn the leptonic channel the case of massive bottom quarks was investigated in\nRefs.~\\cite{Cascioli:2013wga,Frederix:2013gra}.\nNLO QCD results for off-shell $t {\\bar t}$ production in association with an additional jet were obtained\nin Refs.~\\cite{Bevilacqua:2015qha,Bevilacqua:2016jfk}.\nNLO electroweak~(EW) corrections for on-shell $t {\\bar t}$ production were studied in\nRefs.~\\cite{Kuhn:2006vh,Bernreuther:2006vg,Kuhn:2013zoa,Bernreuther:2010ny,Hollik:2011ps,Pagani:2016caq},\nand a merged calculation for $t {\\bar t}+0,1\\,$jets including EW corrections was presented in\nRef.~\\cite{Gutschow:2018tuk}.\nFor the leptonic decay channel the complete NLO EW corrections to the production of the\nsix-particle final state are known~\\cite{Denner:2016jyo}.\n\nThe calculation of the next-to-next-to-leading-order (NNLO) QCD \ncorrections to the $t {\\bar t}$ total cross section was completed a few years ago\n\\cite{Baernreuther:2012ws, Czakon:2012zr, Czakon:2012pz, Czakon:2013goa}.\nNNLO results for some differential distributions were presented in\nRefs.~\\cite{Czakon:2015owf,Czakon:2016ckf,Czakon:2017dip}.\nThis calculation was recently combined with NLO EW corrections~\\cite{Czakon:2017wor}.\nThe $t {\\bar t}$ charge asymmetry is known at NLO \\cite{Kuhn:1998kw} and NNLO \\cite{Czakon:2014xsa} in QCD,\nand also including NLO EW corrections \\cite{Czakon:2017lgo}.\nFirst NNLO QCD results including top-quark decays are starting to appear \\cite{Behring:2019iiv}.\n\nIn the present paper we deal with on-shell $t {\\bar t}$ production in NNLO QCD.\nThe calculation of the $t {\\bar t}$ production cross section at this perturbative order requires \ntree-level contributions with two additional\nfinal-state partons,\none-loop contributions with one additional parton and \npurely virtual contributions.\nThe required tree-level and one-loop scattering amplitudes\nare known. They enter the NLO calculation of \nthe associated production of a $t {\\bar t}$ pair and one jet~\\cite{Dittmaier:2007wz,Dittmaier:2008uj},\nbut in the case of NNLO $t {\\bar t}$ production they\nneed to be accurately evaluated also in the infrared-singular regions where the jet becomes unresolved.\nThe purely virtual contributions entail the square of one-loop scattering amplitudes and the two-loop\nscattering amplitudes.\nThe squared one-loop amplitudes are known~\\cite{Korner:2008bn,Anastasiou:2008vd,Kniehl:2008fd}.\nThe complete computation of the two-loop amplitudes has been carried out\nnumerically~\\cite{Czakon:2008zk,Baernreuther:2013caa}.\nPartial results for these amplitudes are available in analytic\nform~\\cite{Bonciani:2008az,Bonciani:2009nb,Bonciani:2010mn,Bonciani:2013ywa}.\nRecent progress in the computation of non-planar\ntwo-loop master integrals~\\cite{Becchetti:2019tjy,DiVita:2019lpl} indicates that\nthe analytic calculation can be completed in the near future.\n\nThe implementation of the various scattering amplitudes in a (fully differential)\nNNLO calculation is definitely a non-trivial task because of the presence of infrared~(IR) \ndivergences at intermediate stages of the calculation. \nVarious methods have been proposed and used to overcome these difficulties \nat the NNLO level (the interested reader can consult the list of references in\nRef.~\\cite{Bendavid:2018nar}).\n\nUsing the antenna subtraction method~\\cite{GehrmannDeRidder:2005cm, Abelof:2011jv},\npartial results for $t {\\bar t}$ production in the $q{\\bar q}$ partonic channel\nwere obtained by considering the complete fermionic contributions and evaluating the\nremaining contributions\nin the leading-colour approximation~\\cite{Abelof:2014fza, Abelof:2014jna,Abelof:2015lna}.\nThe complete NNLO computation of\nRefs.~\\cite{Baernreuther:2012ws, Czakon:2012zr, Czakon:2012pz, Czakon:2013goa,Czakon:2014xsa,Czakon:2015owf,Czakon:2016ckf,Czakon:2017dip}\nwas performed by using the {\\sc Stripper} method~\\cite{Czakon:2010td,Czakon:2011ve,Czakon:2014oma}.\n\n\nIn a recent paper~\\cite{Catani:2019iny} we have presented a new calculation of the inclusive\n$t {\\bar t}$ production cross section in NNLO QCD.\nThis calculation completes a previous computation\nthat was limited to the flavour off-diagonal partonic channels~\\cite{Bonciani:2015sha}.\nThe calculation uses the $q_T$ subtraction formalism~\\cite{Catani:2007vq} to handle and cancel\nIR-singular contributions\nin real and virtual corrections,\nand it is now completely integrated into the {\\sc Matrix} framework~\\cite{Grazzini:2017mhc}.\nThis allows us to perform fast and efficient computations of fiducial cross sections and\n(multi-)differential kinematical distributions for the production of on-shell top quarks.\n\nIn the present paper we extend the results of Ref.~\\cite{Catani:2019iny} in various respects.\nWe present NNLO QCD predictions for several differential distributions in the transverse momenta\nand rapidities of the top quarks, as well as in the invariant mass and the rapidity of the\n$t {\\bar t}$ system, and we discuss the results obtained by using different scale choices. \nWe compare these results with CMS measurements in the lepton+jets channel at the centre-of-mass energy\n\\mbox{$\\sqrt{s}=13$~TeV}~\\cite{Sirunyan:2018wem}.\nWe then consider double-differential distributions and compare our results with the corresponding\nmeasurements by CMS~\\cite{Sirunyan:2018wem}.\n\n\nThe paper is organized as follows. In Section~\\ref{sec:matrix} we illustrate the framework\nin which the calculation is performed. In Section~\\ref{sec:resu}\nwe present results for single-differential and double-differential distributions,\nand we compare them with the experimental measurements.\nFinally, in Section~\\ref{sec:summary} we present our conclusions.\nIn Appendix~\\ref{sec:validation} we present a quantitative comparison of our NNLO differential results with those available\nin the literature.\n\n\n\\section[Calculation within the {\\sc Matrix} framework]{Calculation within the M{\\normalsize ATRIX} framework}\n\\label{sec:matrix}\n\nOur fully differential NNLO computation of $t {\\bar t}$ production is carried out within\nthe \\Matrix{}~\\cite{Grazzini:2017mhc} framework.\n\\Matrix{} features a completely automated implementation of the $q_T$ subtraction formalism~\\cite{Catani:2007vq}\nto compute NNLO corrections,\nand it is thus applicable to the production of an arbitrary set of colourless final-state particles\nin hadronic collisions~\\cite{Catani:2013tia},\nas long as the two-loop virtual corrections to the corresponding leading-order~(LO) process are provided.\nWith appropriate modifications of the NNLO subtraction counterterm\nand the explicit computation of additional soft contributions (see below),\n\\Matrix{} can now also deal with the production of heavy-quark pairs.\n\nAccording to the $q_T$ subtraction method, the NNLO\ndifferential cross section $d{\\sigma}^{t{\\bar t}}_{\\rm NNLO}$ for \nthe production process $pp\\to t{\\bar t}+X$ can be written as\n\\begin{equation}\n\\label{eq:main}\nd{\\sigma}^{t{\\bar t}}_{\\rm NNLO}={\\cal H}^{t{\\bar t}}_{\\rm NNLO}\\otimes d{\\sigma}^{t{\\bar t}}_{\\rm LO}\n+\\left[ d{\\sigma}^{t{\\bar t}+\\rm{jet}}_{\\rm NLO}-\nd{\\sigma}^{t{\\bar t}, \\, CT}_{\\rm NNLO}\\right],\n\\end{equation}\nwhere $d{\\sigma}^{t{\\bar t}+\\rm{jet}}_{\\rm NLO}$ is the $t {\\bar t}$+jet cross section at NLO accuracy.\n\nThe square bracket term of Eq.~(\\ref{eq:main}) is IR finite in the limit\nin which the transverse momentum of the $t {\\bar t}$ pair, $q_T$, vanishes.\nHowever, the individual contributions\n$d{\\sigma}^{t{\\bar t}+\\rm{jet}}_{\\rm NLO}$ and\n$d{\\sigma}^{t{\\bar t}, \\, CT}_{\\rm NNLO}$ are separately divergent.\nThe contribution $d{\\sigma}^{t{\\bar t}+\\rm{jet}}_{\\rm NLO}$ can be evaluated with any available NLO method to handle and cancel IR divergences.\nThe IR subtraction counterterm $d{\\sigma}^{t{\\bar t}, \\,CT}_{\\rm NNLO}$\nis obtained from the NNLO perturbative expansion \n(see e.g.\\ Refs.~\\cite{Bozzi:2005wk,Bozzi:2007pn,Bonciani:2015sha})\nof the resummation formula\nof the logarithmically-enhanced\ncontributions to the $q_T$ distribution\nof the $t {\\bar t}$ pair~\\cite{Zhu:2012ts,Li:2013mia,Catani:2014qha}:\nthe explicit form of $d{\\sigma}^{t{\\bar t}, \\,CT}_{\\rm NNLO}$ is fully known.\n\nTo complete the NNLO calculation, the second-order functions\n${\\cal H}^{t{\\bar t}}_{\\rm NNLO}$ in Eq.~(\\ref{eq:main}) are needed.\nThese functions embody\nprocess-independent and process-dependent contributions. The process-independent contributions to \n${\\cal H}^{t{\\bar t}}_{\\rm NNLO}$ are analogous to those entering Higgs\nboson~\\cite{Catani:2007vq} and vector-boson~\\cite{Catani:2009sm} production,\nand they are explicitly known~\\cite{Catani:2011kr,Catani:2012qa,Catani:2013tia,Gehrmann:2012ze,Gehrmann:2014yya}.\nSince in $t {\\bar t}$ production both the $gg$ and the $q{\\bar q}$ partonic channels contribute\nat the same perturbative order,\nall these process-independent contributions are required.\nIn the flavour off-diagonal channels \nthe process-dependent contributions to ${\\cal H}^{t{\\bar t}}_{\\rm NNLO}$\ninvolve only amplitudes of the partonic\nprocesses \\mbox{$q{\\bar q} \\to t{\\bar t}$} and \\mbox{$gg\\to t{\\bar t}$} up to the one-loop level,\nand the explicit results on the NLO {\\em azimuthal-correlation} terms\nin the transverse-momentum resummation formalism~\\cite{Catani:2014qha}.\nThe computation of ${\\cal H}^{t{\\bar t}}_{\\rm NNLO}$ in the flavour diagonal $q{\\bar q}$ and $gg$\nchannels additionally requires the two-loop amplitudes\nfor \\mbox{$q{\\bar q}\\to t{\\bar t}$} and \\mbox{$gg\\to t{\\bar t}$},\nand the evaluation of new contributions of purely {\\it soft} origin.\nThe two-loop amplitudes are available in a numerical form~\\cite{Baernreuther:2013caa},\nand the corresponding grids have been implemented into {\\sc Matrix} through a suitable\ninterpolation routine.\nThe computation of the additional soft contributions\nhas been completed by some of us~\\cite{inprep}\\footnote{An independent computation of\nthese soft contributions is presented in Ref.~\\cite{Angeles-Martinez:2018mqh}.},\nand it has been implemented into {\\sc Matrix} as well.\n\nThe core of the \\Matrix{} framework is the Monte Carlo program \\Munich{}\\footnote{\\Munich{} is the \nabbreviation of ``MUlti-chaNnel Integrator at Swiss~(CH) precision'' --- an automated parton-level\nNLO generator by S.~Kallweit.}, which includes a fully automated implementation of the\ndipole-subtraction method for massless~\\cite{Catani:1996jh,Catani:1996vz}\nand massive~\\cite{Catani:2002hc} partons,\nand an efficient phase-space integration.\nAll the required (spin- and colour-correlated) tree-level and one-loop (squared) amplitudes\nare obtained by using {\\sc OpenLoops~2}~\\cite{Cascioli:2011va,openloops2},\nexcept for the four-parton tree-level colour correlations that are based on an analytic implementation.\n{\\sc OpenLoops~2} relies on its new on-the-fly tensor reduction~\\cite{Buccioni:2017yxi} that guarantees\nstability all over the phase space, especially in the IR-singular regions,\nwhile scalar integrals from {\\sc Collier}~\\cite{Denner:2014gla,Denner:2016kdg} are used.\nTo the purpose of validating our results for the real--virtual corrections,\nwe have also used the independent matrix-element generator {\\sc Recola}~\\cite{Actis:2016mpe,Denner:2017wsf}, which employs tensor reduction and scalar integrals from {\\sc Collier}, and we find complete agreement.\n\nThe subtraction in the square brackets of Eq.~(\\ref{eq:main}) is not local, but the cross section \nis formally finite in the limit \\mbox{$q_T \\to 0$}. In practice, a technical cut on $q_T$\nis introduced to render $d{\\sigma}^{t {\\bar t}+\\mathrm{jet}}_{\\mathrm{(N)LO}}$ and \n$d{\\sigma}^{\\mathrm{CT}}_{\\mathrm{(N)NLO}}$ separately finite.\nTherefore, in our actual implementation, the $q_T$ subtraction method is\nvery similar to a phase-space slicing method. \nIt turns out that a cut, $r_{\\mathrm{cut}}$, on the \ndimensionless quantity \\mbox{$r=q_T\/m_{t{\\bar t}}$} ($m_{t{\\bar t}}$\ndenotes the invariant mass of the $t {\\bar t}$ pair) \nis more convenient from a practical point of view. \nThe absence of any residual logarithmic\ndependence on $r_{\\mathrm{cut}}$ is a strong evidence of the correctness of the \ncomputation, since any mismatch between the contributions would result in a divergence \nof the cross section in the limit \\mbox{$r_{\\mathrm{cut}}\\to0$}.\nThe remaining power-suppressed contributions vanish in that limit, and they can be controlled by\nmonitoring the $r_{\\mathrm{cut}}$ dependence of the cross section.\n\n\nThe $r_\\text{cut}\\to 0$ extrapolation for the total cross section is carried out by using the approach\nintroduced in Ref.~\\cite{Grazzini:2017mhc}.\nA quadratic least $\\chi^2$ fit to the $r_\\text{cut}$ dependent results is performed and repeated\nby varying the upper bound of the $r_\\text{cut}$ interval.\nFinally, the result with the lowest $\\chi^2\/$degrees-of-freedom value is taken as the best fit,\nwhile the remaining results are used to estimate the extrapolation uncertainty.\nIn addition to this analysis at the level of the total cross section,\nwe have performed a similar bin-wise extrapolation in the computation of differential cross sections.\nWe find that the results are in good agreement with those obtained by directly\nusing a sufficiently low value of $r_\\text{cut}$ (\\mbox{$r_{\\rm cut} \\lesssim 0.15\\%$}).\n\n\n\\section{Results}\n\\label{sec:resu}\n\nTo present our quantitative results, we consider $pp$ collisions at \\mbox{$\\sqrt{s}=13$~TeV},\nand we fix the pole mass ${m_{t}}$ of the top quark to the value \\mbox{${m_{t}}=173.3$~GeV}.\nWe consider $n_F=5$ massless quark flavours, and we use the corresponding NNPDF31~\\cite{Ball:2017nwa}\nsets of parton distribution functions~(PDFs) with \\mbox{$\\as({m_Z})=0.118$}.\nIn particular, N$^n$LO (with $n = 0,1,2$) predictions are obtained by using PDFs\nat the corresponding perturbative order and the evolution of $\\as$ at \\mbox{$(n + 1)$}-loop order,\nas provided by the PDF set.\n\nQCD scale uncertainties are estimated through the customary procedure of independently varying\nthe renormalization ($\\mu_R$) and factorization ($\\mu_F$) scales \nby a factor of two\naround their common central value $\\mu_0$ with the constraint \\mbox{$0.5\\leq \\mu_F\/\\mu_R\\leq 2$},\ni.e.\\ we use the standard 7-point scale variation.\n\nSetting $\\mu_0={m_{t}}$, the total cross sections and their corresponding scale uncertainties read\n\\begin{equation}\n\\sigma_\\text{LO}^{t {\\bar t}} = 478.9(1)^{+29.6\\%}_{-21.4\\%}\\text{~pb}\\,,\\quad\n\\sigma_\\text{NLO}^{t {\\bar t}} = 726.9(1)^{+11.7\\%}_{-11.9\\%}\\text{~pb}\\,,\\quad\n\\sigma_\\text{NNLO}^{t {\\bar t}} = 794.0(8)^{+3.5\\%}_{-5.7\\%}\\text{~pb}\\,.\n\\label{eq:sigmatot}\n\\end{equation}\nWe note that the LO and NLO results are not fully consistent within the corresponding uncertainties,\nindicating that, at least at LO, scale variations cannot be trusted as perturbative uncertainties.\nSimilar features are shared by various other hard-scattering processes at hadron colliders.\nIn contrast, the NLO and NNLO predictions are consistent, suggesting that scale variations\ncan be used to estimate the size of perturbative contributions beyond NNLO. \n\n\nThe characteristic hard-scattering scale that controls the perturbative QCD behaviour\nof the total cross section $\\sigma^{t {\\bar t}}$ is ${m_{t}}$. In our calculation of\n$\\sigma^{t {\\bar t}}$, as reported in Eq.~(\\ref{eq:sigmatot}), we have used QCD scales ($\\mu_R$ and\n$\\mu_F$) at values of the order of ${m_{t}}$. \nDifferential cross sections are \ncontrolled by corresponding characteristic hard scales, and we use QCD scales of that order\nin our computation of these observables.\nThe characteristic hard scale specifically depends on the differential cross section under consideration. \n\nHaving at our disposal a fully differential calculation we can also use {\\it dynamical}\nQCD scales. By dynamical we mean hard scales that refer to multi-differential\ncross sections\neventually integrated over the phase space to obtain the specific \ndifferential cross section under consideration. \nThe use of a dynamical scale produces practical simplifications\nsince it allows us to compute several observables (e.g., differential cross sections)\nsimultaneously, without changing the QCD scales on an observable-dependent basis.\nIn practice, we use dynamical scales that are expected to be ``effectively similar''\nto characteristic hard scales. Moreover, the study of dynamical scales\nis of interest independently of how we use them.\n\nThe default dynamical scale that we use throughout the paper is set to the \ncentral value \\mbox{$\\mu_0=H_{T}\/2$}, where $H_{T}$ is the sum of the transverse masses\nof the top and antitop quarks,\n\\begin{equation}\n  H_{T}=m_{T,t}+m_{T,{\\bar t}}\\,,\n\\end{equation}\nwith\n\\begin{equation}\nm_{T,t({\\bar t})}=\\sqrt{{m^2_{t}}+p^2_{T,t({\\bar t})}}\\, ,\n\\end{equation}\nand $p_{T,t}$ and $p_{T,{\\bar t}}$ are the transverse momenta of the top and the antitop quark, respectively.\nWe present differential cross sections that are obtained by using \n\\mbox{$\\mu_0=H_{T}\/2$} and values of $\\mu_0$ of the order of the characteristic hard scale\nfor that cross section.  We also show results obtained by using central scales that are\nlowered by a factor of $1\/2$. A reduced central scale, such as $H_{T}\/4$, was considered\nin the studies of Ref.~\\cite{Czakon:2016dgf} on the basis of features of\nfastest perturbative convergence of some observables,\nand it was also already suggested in Ref.~\\cite{Denner:2012yc}.\n\nWe have chosen the dynamical scale $H_{T}$ since it is expected to be parametrically\nof the same order as the characteristic hard scale of the observables that we examine\nin this paper. This {\\it a priori} expectation is based on the kinematical features of\nthese observables and on the general dynamical features of $t {\\bar t}$ production.\nIn the following paragraph we briefly comment about this. Independently of the expectation,\nthroughout the paper we comment on the actual quantitative results that we obtain by\nusing different QCD scales.\n\nOwing to dynamics, the typical size of both $p_{T,t}$ and $p_{T,{\\bar t}}$ is\nof the order of ${m_{t}}$ (see, e.g., Figs.~\\ref{fig:pt1}--\\ref{fig:pth} and \n\\ref{fig:pth_yth}). Therefore, in the case of observables that are inclusive over\n$p_{T,t}$ and $p_{T,{\\bar t}}$, such as the total cross section and the pair rapidity\ndistribution in Fig.~\\ref{fig:ytt}, $H_{T}\/2$ turns out to be of the same order as \n${m_{t}}$, which is the characteristic hard scale for these observables. Analogously,\nsince \\mbox{$p_{T,t} \\sim p_{T,{\\bar t}}$}, $H_{T}\/2$ turns out to be of the same order\nas the transverse masses, which are the characteristic hard scales for the differential cross sections\nin Figs.~\\ref{fig:pt1}--\\ref{fig:pth} and \\ref{fig:pth_yth}.\nThe invariant mass $m_{t{\\bar t}}$ of the $t {\\bar t}$ pair is the characteristic hard scale in the case\nof the differential cross sections in Figs.~\\ref{fig:mtt}, \\ref{fig:ytt_mtt} and \\ref{fig:mtt_pth}.\nThe invariant mass is of the same order as $H_{T}$ with the exception of the kinematical subregions\nwhere the transverse momentum $p_{T,t{\\bar t}}$\nof the pair or the rapidity separation \\mbox{$|y_t - y_{\\bar t}|$} between\nthe top and the antitop quark are large. However, these subregions are dynamically suppressed,\nand therefore they give a minor contribution to the inclusive\n(over $p_{T,t{\\bar t}}$ and \\mbox{$|y_t - y_{\\bar t}|$}) cross sections\nin Figs.~\\ref{fig:mtt}, \\ref{fig:ytt_mtt} and \\ref{fig:mtt_pth}.\n\n\nOur numerical results for differential cross sections are compared with the measurements of the\nCMS collaboration~\\cite{Sirunyan:2018wem} (the data correspond to an integrated luminosity of $35.8~{\\rm fb}^{-1}$) in the lepton+jets channel at parton level.\nThe extrapolation from particle to parton level is carried out by the CMS collaboration in the inclusive phase space,\nand therefore no kinematical cuts are applied to obtain our theoretical predictions.\nTo perform the comparison, our results are multiplied by the factor $0.292$,\nwhich corresponds to the value $0.438$~\\cite{Tanabashi:2018oca}\nof the semileptonic decay fraction of the $t {\\bar t}$ pair, multiplied by a factor of $2\/3$\nsince Ref.~\\cite{Sirunyan:2018wem} considers only the decay into electrons and muons.\n\nIn Ref.~\\cite{Sirunyan:2018wem} the CMS data for single- and double-differential distributions\nare compared to theoretical results obtained with the NLO Monte Carlo event generators\n{\\sc POWHEG}~\\cite{Nason:2004rx,Frixione:2007vw,Alioli:2010xd},\ninterfaced either to {\\sc PYTHIA8}~\\cite{Sjostrand:2007gs} or to {\\sc HERWIG++}~\\cite{Bahr:2008pv},\nand {\\sc MG5\\_aMC@NLO}~\\cite{Alwall:2014hca} interfaced to {\\sc PYTHIA8}~\\cite{Sjostrand:2007gs}\n(using the {\\sc FxFx} method \\cite{Frederix:2012ps} to deal with multijet merging).\nIn addition, some of the measured parton-level single-differential distributions,\nnamely the transverse-momentum and rapidity distributions of the leptonically and hadronically\ndecaying top quark and the invariant-mass and rapidity distribution of the $t {\\bar t}$ pair,\nare also compared to the NNLO QCD+NLO EW results of Ref.~\\cite{Czakon:2017wor}.\nNone of the double-differential distributions in Ref.~\\cite{Sirunyan:2018wem} are compared to theoretical results beyond NLO QCD.\n\n\n\n\\subsection{Single-differential distributions}\n\\label{sec:single}\n\nIn this section we present LO, NLO and NNLO results for a selection of single-differential distributions\nand compare them with the CMS measurements from Ref.~\\cite{Sirunyan:2018wem}.\nAt each perturbative order the scale-uncertainty bands in the figures\nare computed as explained at the beginning of Section~\\ref{sec:resu}. \n\nWe start the presentation by considering the transverse-momentum distributions of the top and antitop quarks.\nFor each event we classify the transverse momenta\naccording to their maximum and minimum values, $p_{T,t_\\text{high}}$ and $p_{T,t_\\text{low}}$.\n\n\n\\begin{figure}[t]\n\\includegraphics[height=0.30\\textheight]{plots\/LHC13.CMSsetup\/plotCMS13_HT_2_pT_t1.pdf}\n\\includegraphics[height=0.30\\textheight]{plots\/LHC13.CMSsetup\/plotCMS13_mT_t1_pT_t1.pdf}\n\\includegraphics[height=0.30\\textheight]{plots\/LHC13.CMSsetup\/plotCMS13_mT_t1_2_pT_t1.pdf}\n\\vspace{-4ex}\n\\caption{\\label{fig:pt1}\n  Single-differential cross sections as a function of $p_{T,t_\\text{high}}$.\n  CMS data~\\cite{Sirunyan:2018wem} and LO, NLO and NNLO results\n  for central scales equal to $H_{T}\/2$ (left), $m_{T,t_\\text{high}}$ (central) and $m_{T,t_\\text{high}}\/2$ (right).\n}\n\\end{figure}\n\\begin{figure}[t]\n\\includegraphics[height=0.30\\textheight]{plots\/LHC13.CMSsetup\/plotCMS13_HT_2_pT_t2.pdf}\n\\includegraphics[height=0.30\\textheight]{plots\/LHC13.CMSsetup\/plotCMS13_mT_t2_pT_t2.pdf}\n\\includegraphics[height=0.30\\textheight]{plots\/LHC13.CMSsetup\/plotCMS13_mT_t2_2_pT_t2.pdf}\n\\vspace{-4ex}\n\\caption{\\label{fig:pt2}\n  Single-differential cross sections as a function of  $p_{T,t_\\text{low}}$.\n  CMS data~\\cite{Sirunyan:2018wem} and LO, NLO and NNLO results\n  for central scales equal to $H_{T}\/2$ (left), $m_{T,t_\\text{low}}$ (central) and $m_{T,t_\\text{low}}\/2$ (right).\n}\n\\end{figure}\n\nIn Figs.~\\ref{fig:pt1} and \\ref{fig:pt2}~(left)\nwe show these distributions\\footnote{NNLO results for these distributions have been recently presented in Ref.~\\cite{Czakon:2019bcq}.}\ncomputed at our reference scale \\mbox{$\\mu_0=H_T\/2$}.\nThe characteristic hard scale of a transverse-momentum distribution is the transverse mass $m_T$.\nTherefore, in Fig.~\\ref{fig:pt1} (central and right) we also report the $p_{T,t_\\text{high}}$ distribution for\ncentral scales \\mbox{$\\mu_0=m_{T,t_\\text{high}}$} and \\mbox{$\\mu_0=m_{T,t_\\text{high}}\/2$}, respectively,\nwhile in Fig.~\\ref{fig:pt2} (central and right) we consider $p_{T,t_\\text{low}}$\nfor \\mbox{$\\mu_0=m_{T,t_\\text{low}}$} and \\mbox{$\\mu_0=m_{T,t_\\text{low}}\/2$}.\nThe $p_{T,t_\\text{high}}$ distribution is peaked at \\mbox{$p_{T,t_\\text{high}}\\sim 100$~GeV},\nwhile the $p_{T,t_\\text{low}}$ distribution is peaked at \\mbox{$p_{T,t_\\text{low}}\\sim 60$~GeV}.\n\nWe first discuss the $p_{T,t_\\text{high}}$ distribution and focus on the \\mbox{$p_{T,t_\\text{high}}\\to 0$} region.\nIf $p_{T,t_\\text{high}}$ is small, both top quarks are forced to have small transverse momenta.\nAs a consequence, this kinematical region corresponds to a small transverse momentum of\nthe top-quark pair, $p_{T,t{\\bar t}}$.\nThe small-$p_{T,t{\\bar t}}$ region exhibits Sudakov-type divergences \\cite{Zhu:2012ts,Li:2013mia,Catani:2014qha,Catani:2018mei} at fixed order in perturbation theory,\nfrom the strong unbalance of real and virtual contributions due to soft-collinear emissions.\nIn the computation of $p_{T,t_\\text{high}}$, the unphysical fixed-order behaviour of $p_{T,t{\\bar t}}$ is smeared\ndue to the integration over $p_{T,t_\\text{low}}$, and the Sudakov-type perturbative divergences disappear, by leaving (possibly large) residual effects.\nThe amount of smearing is controlled by the shape of the $p_{T,t_\\text{high}}$ distribution in the low-$p_T$ region\nat LO, which affects the unbalance between real and virtual contributions.\nThe steeply rising LO distribution at low $p_T$ strongly suppresses real radiation,\nand the NLO radiative corrections to $p_{T,t_\\text{high}}$ tend to be large and negative\nas $p_{T,t_\\text{high}}$ decreases (a large and positive effect occurs at NNLO, and so forth).\nAccurate theoretical predictions of the detailed shape of the $p_{T,t_\\text{high}}$ distribution at small $p_T$ require studies of all-order resummation effects of Sudakov type.\nHowever, in the case of large $p_T$ bins (as is the case in Fig.~\\ref{fig:pt1}), reliable predictions can be obtained by considering perturbation theory\nat a sufficiently high order.\n\nComparing the results in Fig.~\\ref{fig:pt1} for the scales $\\mu_0=H_T\/2$ (left) and $\\mu_0=m_{T,t_\\text{high}}$ (central)\nwe see that they are rather similar, and that the NNLO prediction agrees with the data.\nThe scale \\mbox{$\\mu_0=m_{T,t_\\text{high}}\/2$} also leads to good agreement with the data,\nbut the corresponding NNLO uncertainty band is significantly narrower,\nespecially in the intermediate region of transverse momenta,\nwhich is not observed for the corresponding band at NLO.\nThis behaviour, namely the drastic shrinking of the scale-uncertainty band from NLO to NNLO,\nmight indicate that for this choice scale variations cannot be trusted\nas an estimate of the perturbative uncertainties at NNLO.\nWe also note that in the intermediate and large $p_T$ region the result obtained by using \\mbox{$\\mu_0=m_{T,t_\\text{high}}\/2$} coincides\nwith the upper bound of the corresponding NNLO band  (i.e., the point $\\mu_0=m_{T,t_\\text{high}}\/2$ corresponds to a local maximum of the NNLO cross section as a function of the scales).\n\nWe now discuss the $p_{T,t_\\text{low}}$ distribution. In the region \\mbox{$p_{T,t_\\text{low}}\\to 0$},\nfor LO kinematics\nboth the top and the antitop quark are required to have small $p_T$.\nAt NLO, real corrections open up a phase-space region where one top quark has a small $p_T$ and\nthe other one has a relatively large $p_T$,\nthereby leading to large positive radiative contributions.\nThe perturbative instability affecting the \\mbox{$p_{T,t_\\text{high}}\\to 0$} behaviour is now spread\nover the entire region of transverse momenta since, contrary to the low-$p_{T,t_\\text{high}}$ region,\nsmall values of $p_{T,t_\\text{low}}$ do not constrain $p_{T,t{\\bar t}}$ to be small.\nThe choices \\mbox{$\\mu_0=H_T\/2$} and \\mbox{$\\mu_0=m_{T,t_\\text{low}}$} lead to rather similar results.\nIn both cases, at low and large $p_{T,t_\\text{low}}$ NLO and NNLO bands overlap,\nwhereas they do not in the intermediate region where\nthe NLO band shrinks, showing that NLO perturbative uncertainties are underestimated.\nWe note that the scale \\mbox{$\\mu_0=m_{T,t_\\text{low}}\/2$} makes the perturbative convergence worse and the scale uncertainties\nlarger at both NLO and NNLO.\n\n\\begin{figure}\n\\includegraphics[height=0.30\\textheight]{plots\/LHC13.CMSsetup\/plotCMS13_HT_2_pT_th.pdf}\n\\includegraphics[height=0.30\\textheight]{plots\/LHC13.CMSsetup\/plotCMS13_mT_avt_pT_th.pdf}\n\\includegraphics[height=0.30\\textheight]{plots\/LHC13.CMSsetup\/plotCMS13_mT_avt_2_pT_th.pdf}\n\\vspace{-4ex}\n\\caption{\\label{fig:pth}\n  Single-differential cross sections as a function of $p_{T,t_\\text{had}}$.\n  CMS data~\\cite{Sirunyan:2018wem} and LO, NLO and NNLO results\n  for central scales equal to $H_{T}\/2$ (left), $m_{T,t_{\\rm av}}$ (central) and $m_{T,t_{\\rm av}}\/2$ (right).\n}\n\\end{figure}\n\n\nWe next consider the distribution in the transverse momentum of the hadronically decaying top or antitop quark, $p_{T,t_\\text{had}}$.\nSince our calculation refers to stable top quarks, a prediction for the $p_{T,t_\\text{had}}$ distribution\ncan be obtained by computing the transverse-momentum spectra of the top and the antitop quark,\nand taking their average afterwards.%\n\\footnote{This is also the definition used in the data\/theory comparison performed\nby the CMS collaboration \\cite{Sirunyan:2018wem}.}\nDiscussing our theoretical predictions, we refer to this as the $p_{T,t_{\\mathrm{av}}}$ distribution.\nThe corresponding LO, NLO and NNLO results are depicted in Fig.~\\ref{fig:pth}\nfor three different scale choices.\nWe show the predictions for our default choice \\mbox{$\\mu_0=H_T\/2$} on the left.\nIn the predictions for our natural scale choices, \nthe top (antitop) $p_T$ distributions required to compute the average are evaluated\nat the corresponding transverse mass, $\\mu_0=m_{T,t({\\bar t})}$ (central) and $\\mu_0=m_{T,t({\\bar t})}\/2$ (right).\nWe denote these scale choices as $m_{T,t_{\\rm av}}$ and $m_{T,t_{\\rm av}}\/2$, respectively.\nThe $p_{T,t_\\text{had}}$ distribution has a maximum at \\mbox{$p_{T,t_\\text{had}} \\sim 80~{\\rm GeV}$}.\nThe LO and NLO scale-uncertainty bands do not overlap, except for \\mbox{$\\mu_0=m_{T,t_{\\rm av}}\/2$}.\nThis is consistent with what happens for the corresponding total cross sections.\nThe NLO and NNLO bands do overlap in the entire $p_{T,t_\\text{had}}$ range,\nsuggesting a good convergence of the perturbative expansion.\nIn Fig.~\\ref{fig:pth} we also observe that the scale choices $\\mu_0=H_T\/2$ (left) and $\\mu_0=m_{T,t_{\\rm av}}$ (central)\ngive rather similar results.\nOn the contrary, the choice \\mbox{$\\mu_0=m_{T,t_{\\rm av}}\/2$} suggests a faster convergence\nof the perturbative expansion \\cite{Czakon:2016dgf}. \nHowever, we also note that with this scale choice the NLO scale dependence is similar\nto what we obtain with $\\mu_0=H_T\/2$ and $\\mu_0=m_{T,t_{\\rm av}}$,\nwhereas the NNLO scale dependence is significantly smaller than at NLO,\nthereby suggesting a possible underestimation of the perturbative uncertainty at NNLO.\nWe note that \\mbox{$\\mu_0=m_{T,t_{\\rm av}}\/2$} is also the scale used\nfor the NNLO QCD+NLO EW prediction~\\cite{Czakon:2017wor}\nto which the CMS data are compared in Ref.~\\cite{Sirunyan:2018wem}.\nThe data show that the measured $p_{T,t_\\text{had}}$ distribution is slightly softer than the NNLO prediction.\nThis is noticed also by the CMS collaboration~\\cite{Sirunyan:2018wem} and in previous comparisons between NNLO results and LHC measurements \\cite{Khachatryan:2015oqa,Khachatryan:2015fwh,Aad:2015hna,Aad:2015mbv,Khachatryan:2016gxp,Aaboud:2016iot,Aaboud:2016syx,Sirunyan:2017mzl}.\nHowever Fig.~\\ref{fig:pth} shows that the NNLO result and the data are consistent within the respective uncertainties.\nOur predictions for the $p_T$ spectrum of the leptonically decaying top quark are, of course,\nidentical to those for $p_{T,t_\\text{had}}$, and they are not shown here.\nThe comparison with the data shows similar features.\n\n\n\nWe add a few comments on the perturbative behaviour of the $p_{T,t_\\text{high}}$, $p_{T,t_\\text{low}}$ and $p_{T,t_{\\mathrm{av}}}$ distributions\npresented in Figs.~\\ref{fig:pt1}, \\ref{fig:pt2} and \\ref{fig:pth}.\nThe three distributions are identical at LO,\nbut their behaviour beyond LO is clearly very different.\nAs we can see from Fig.~\\ref{fig:pth}, the shape of the $p_{T,t_{\\mathrm{av}}}$ distribution is almost unchanged\nwith respect to the LO prediction.\nThis feature is somehow expected, since the transverse-momentum spectrum of the top (antitop) quark\nat higher orders is affected by recoiling hard multijet radiation,\nwhich leads to a partly softer spectrum only at quite high $p_{T,t_{\\mathrm{av}}}$\n(beyond the $p_{T,t_\\text{had}}$ range in Fig.~\\ref{fig:pth}),\nwhere hard multijet radiation is kinematically suppressed.\n\nThe physical shape of the $p_{T,t_\\text{high}}$ and $p_{T,t_\\text{low}}$ distributions is expected\nto be different from $p_{T,t_{\\mathrm{av}}}$ at low and intermediate values of $p_T$.\nThere we roughly have \\mbox{$p_{T,t_\\text{high}} - p_{T,t_\\text{low}} \\sim p_{T,t{\\bar t}}$}.\nThe $p_{T,t{\\bar t}}$ distribution, which is confined to \\mbox{$p_{T,t{\\bar t}} = 0$} at LO,\nhas an average value of about 50~GeV (which is already achieved at NLO),\nand it is localized in the small-$p_{T,t{\\bar t}}$ region with a peak around 10~GeV~\\cite{Zhu:2012ts,Catani:2018mei}.\nWe thus expect that the physical shape of $p_{T,t_\\text{high}}$ ($p_{T,t_\\text{low}}$) is harder (softer) than its LO counterpart,\nwith shape distortions of few tens of GeV as given by the size of $p_{T,t{\\bar t}}$.\nIndeed, this is what we can observe from the comparison between the data and the LO prediction\nat small and intermediate values of $p_T$ in Figs.~\\ref{fig:pt1} and~\\ref{fig:pt2}.\nThis shape distortion has a smaller effect at high values of both $p_{T,t_\\text{high}}$ and $p_{T,t_\\text{low}}$.\n\nIn view of these physical expectations, it is not surprising that the shape\nof the $p_{T,t_\\text{high}}$ and $p_{T,t_\\text{low}}$ distributions is strongly affected by beyond-LO contributions.\nAs discussed before, their fixed-order perturbative features are a smoothened version\nof the corresponding features of the $p_{T,t{\\bar t}}$ distribution \\cite{Catani:2018mei},\nthe smoother behaviour being due to the smearing that is produced by the integration\nof $p_{T,t{\\bar t}}$ over the respective unobserved $p_T$.\n\n\n\\begin{figure}\n\\includegraphics[height=0.30\\textheight]{plots\/LHC13.CMSsetup\/plotCMS13_HT_2_m_ttx.pdf}\n\\includegraphics[height=0.30\\textheight]{plots\/LHC13.CMSsetup\/plotCMS13_m_ttx_2_m_ttx.pdf}\n\\includegraphics[height=0.30\\textheight]{plots\/LHC13.CMSsetup\/plotCMS13_HT_4_m_ttx.pdf}\n\\vspace{-4ex}\n\\caption{\\label{fig:mtt}\n  Single-differential cross sections as a function of $m_{t{\\bar t}}$.\n  CMS data~\\cite{Sirunyan:2018wem} and LO, NLO and NNLO results\n  for central scales equal to $H_{T}\/2$ (left), $m_{t{\\bar t}}\/2$ (central) and $H_{T}\/4$ (right).\n}\n\\end{figure}\n\nThe invariant-mass distribution of the top-quark pair is reported in Fig.~\\ref{fig:mtt}.\nThe distribution is peaked at \\mbox{$m_{t{\\bar t}} \\sim 400~{\\rm GeV}$}.\nThe characteristic hard-scattering scale for this distribution is of the order of $m_{t{\\bar t}}$ itself.\nWe use our default scale choice\n\\mbox{$\\mu_0=H_T\/2$}~(left), and two other central values, namely \\mbox{$\\mu_0=m_{t{\\bar t}}\/2$}~(central) and \\mbox{$\\mu_0=H_T\/4$}~(right).\n\n\nWe first comment on the convergence of the perturbative series for the three scale choices.\nIn the cases \\mbox{$\\mu_0=H_T\/2$} and \\mbox{$\\mu_0=m_{t{\\bar t}}\/2$} we see that LO and NLO bands\ndo not overlap, analogously to what was previously observed for the $p_{T,t_\\text{had}}$ distribution and for the total cross section in Eq.~(\\ref{eq:sigmatot}).\nIn the case \\mbox{$\\mu_0=H_T\/2$}, the NNLO corrections enhance the NLO result\nby about $10\\%$ in the peak region,\nand their effect slightly increases with $m_{t{\\bar t}}$ up to about $15\\%$ in the highest-$m_{t{\\bar t}}$ bin.\nUsing \\mbox{$\\mu_0=m_{t{\\bar t}}\/2$}, the NNLO effect is similar in the peak region,\nbut it increases to about $20\\%$ at high $m_{t{\\bar t}}$. The NLO and NNLO bands do overlap using both\n\\mbox{$\\mu_0=H_T\/2$} and \\mbox{$\\mu_0=m_{t{\\bar t}}\/2$}.\nAs observed in Ref.~\\cite{Czakon:2016dgf}, the choice \\mbox{$\\mu_0=H_T\/4$} leads to a faster convergence\nof the perturbative series.\nHowever, in the region where $m_{t{\\bar t}} > 360~{\\rm GeV}$ we see that the size of the scale-variation band is very much reduced in going from the NLO to the NNLO result.\nThis behaviour suggests that the central scale $\\mu_0=H_T\/4$ is (accidentally) quite close to a region of (local) minimal sensitivity \\cite{Stevenson:1981vj} of the scale dependence of the NNLO result. In view of this feature, we think that the NNLO scale variation band\nwith $\\mu_0=H_T\/4$ likely underestimates the perturbative uncertainty in this $m_{t{\\bar t}}$ region and, especially, in the intermediate mass range $400~{\\rm GeV}\\ltapm_{t{\\bar t}} \\ltap 1~{\\rm TeV}$.\n\n\nWe now comment on the comparison with the data. The first bin, \\mbox{$300\\,{\\rm GeV}<m_{t{\\bar t}}< 360\\,{\\rm GeV}$},\ndeserves a separate discussion.\nThe experimental result in this bin is significantly above the theoretical predictions,\nindependently of the scale choice.\nThis disagreement may have various origins.\nWe first note that the result of our calculation for on-shell top quarks is non-vanishing only in the limited region where\n\\mbox{$2{m_{t}}=346.6~{\\rm GeV} <m_{t{\\bar t}}< 360~{\\rm GeV}$}.\nIn this region the NLO and NNLO effects are largest, regardless of the scale choice.\nMoreover, this region is particularly sensitive to the value of the top-quark mass.\nIf the top-quark mass is smaller\\footnote{Incidentally, we note that in Ref.~\\cite{Sirunyan:2019zvx}\nthe CMS collaboration uses NLO QCD results obtained with NNPDF31 PDFs to extract the value\n${m_{t}}=170.81 \\pm 0.68$~GeV from a fit of a measurement of the triple-differential cross section\nas a funtion of $m_{t{\\bar t}}$, $y_{t{\\bar t}}$ and the multiplicity of additional jets.}\nby a couple of {\\rm GeV}, the predicted cross section\nin this bin becomes larger, without significantly affecting the NNLO result in each of the higher-$m_{t{\\bar t}}$ bins.\nAnother possible reason for the discrepancy may be the unfolding procedure~\\cite{Sirunyan:2018wem}\nthat is used to convert the data from particle to parton level.\nWe expect such unfolding procedure to be particularly delicate in the threshold region:\nboth the actual value of the top-quark mass used in the extrapolation\nand off-shell effects may have a significant impact.\nFurther perturbative QCD effects beyond NNLO may also be relevant,\nbut they are not expected to be large enough to explain the observed discrepancy with respect to the data.\nOverall, excluding the first bin, the NNLO result for the central scale \\mbox{$\\mu_0=m_{t{\\bar t}}\/2$} leads to\nthe best agreement with the data. The result for \\mbox{$\\mu_0=H_T\/2$} is also consistent\nwith the data within uncertainties.\n\n\n\\begin{figure}\n\\includegraphics[height=0.30\\textheight]{plots\/LHC13.CMSsetup\/plotCMS13_HT_2_absy_ttx.pdf}\n\\includegraphics[height=0.30\\textheight]{plots\/LHC13.CMSsetup\/plotCMS13_m_t_absy_ttx.pdf}\n\\includegraphics[height=0.30\\textheight]{plots\/LHC13.CMSsetup\/plotCMS13_HT_4_absy_ttx.pdf}\n\\vspace{-4ex}\n\\caption{\\label{fig:ytt}\n  Single-differential cross sections as a function of  $y_{t{\\bar t}}$.\n  CMS data~\\cite{Sirunyan:2018wem} and LO, NLO and NNLO results\n  for central scales equal to $H_{T}\/2$ (left), ${m_{t}}$ (central) and $H_{T}\/4$ (right).\n}\n\\end{figure}\n\n\nWe finally consider the distribution in the rapidity of the top-quark pair, $y_{t{\\bar t}}$.\nA natural scale choice for this distribution is the top-quark mass,\nas in the case of the computation of the total cross section.\nIn Fig.~\\ref{fig:ytt} we report the rapidity distribution of the $t {\\bar t}$ system for our default choice\n\\mbox{$\\mu_0=H_T\/2$} (left), for \\mbox{$\\mu_0={m_{t}}$} (central) and for \\mbox{$\\mu_0=H_T\/4$} (right).\nWe first observe that the choices \\mbox{$\\mu_0=H_T\/2$} and \\mbox{$\\mu_0={m_{t}}$} lead to rather similar results.\nThe LO and NLO bands marginally overlap, and the impact of NNLO corrections in the central rapidity region\nis about $10\\%$, consistently with what we find for the total cross section.\nAs previously observed for the $m_{t{\\bar t}}$ distribution, the choice \\mbox{$\\mu_0=H_T\/4$} leads to a faster convergence\nof the perturbative expansion \\cite{Czakon:2016dgf},\nand to a strong reduction of scale uncertainties at NNLO.\nThe data nicely agree with the NNLO predictions for all the three scale choices.\n\nWe conclude this section with a few comments on the effects of EW corrections\non the distributions considered so far.\nThe effect of the complete EW corrections to top-quark pair production was studied in\nRef.~\\cite{Pagani:2016caq}, and NNLO QCD+NLO EW predictions for some differential observables\nwere presented in Ref.~\\cite{Czakon:2017wor}.\nIn the case of the $p_{T,t_\\text{had}}$ distribution, the EW corrections are negative and lead\nto percent-level effects at low transverse momenta, which increase up to about $-5\\%$\nin the highest-$p_T$ bins in Fig.~\\ref{fig:pth}.\nSuch effects are of the order of (or smaller than)\nthe residual perturbative uncertainties of the NNLO result,\nand significantly smaller than the current experimental uncertainties.\nAn analogous conclusion can be drawn for the invariant-mass distribution,\nwhich receives EW corrections of about $+2\\%$ in the small-$m_{t{\\bar t}}$ region\nand $-2\\%$ in the highest-$m_{t{\\bar t}}$ region in Fig.~\\ref{fig:mtt}.\nThe impact of EW corrections on the rapidity distributions is typically of ${\\cal O}(1\\%)$ or smaller.\nTherefore, we conclude that the impact of EW corrections is not expected to significantly affect\nour comparison with the data. Similar conclusions can be drawn for the double-differential distributions\nthat we present in the following.\nMore precise data, with higher reach in transverse momenta, will definitely call for the inclusion\nof EW corrections~\\cite{Czakon:2017wor}.\n\n\n\\subsection{Double-differential distributions}\n\\label{sec:double}\n\nIn this section we present results for double-differential distributions and compare them with the\nCMS measurements from Ref.~\\cite{Sirunyan:2018wem}.\nWe consider three double-differential distributions, namely\n\\begin{itemize}\n\\item the $y_{t{\\bar t}}$ distribution in $m_{t{\\bar t}}$ intervals~(see Fig.~\\ref{fig:ytt_mtt});\n\\item the $p_{T,t_\\text{had}}$ distribution in $y_{t_\\text{had}}$ intervals~(see Fig.~\\ref{fig:pth_yth});\n\\item the $m_{t{\\bar t}}$ distribution in $p_{T,t_\\text{had}}$ intervals~(see Fig.~\\ref{fig:mtt_pth}).\n\\end{itemize}\nWe present results for the default scale choice \\mbox{$\\mu_0=H_T\/2$}\nand \nat an additional scale of the order of the characteristic hard scale of the corresponding double-differential distribution. \n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[height=0.30\\textheight]{plots\/LHC13.CMSsetup\/wideCMS13_HT_2_m_ttx-absy_ttx.pdf}\\\\\n\\includegraphics[height=0.30\\textheight]{plots\/LHC13.CMSsetup\/wideCMS13_m_ttx_2_m_ttx-absy_ttx.pdf}\n\\end{center}\n\\vspace{-4ex}\n\\caption{\\label{fig:ytt_mtt}\n  Double-differential cross sections as a function of $y_{t{\\bar t}}$ in four $m_{t{\\bar t}}$ intervals.\n  CMS data~\\cite{Sirunyan:2018wem} and LO, NLO and NNLO results\n  for central scales $H_{T}\/2$ (upper) and $m_{t{\\bar t}}\/2$ (lower).\n}\n\\end{figure}\n\nThe $y_{t{\\bar t}}$ distribution in four $m_{t{\\bar t}}$ intervals is presented in Fig.~\\ref{fig:ytt_mtt}.\nThe central scales are \\mbox{$\\mu_0=H_T\/2$}~(upper) and \\mbox{$\\mu_0=m_{t{\\bar t}}\/2$}~(lower), and they lead to similar results, consistently with those for the single-differential cross sections in Figs.~\\ref{fig:mtt} and \\ref{fig:ytt}.\nThe impact of the radiative corrections is relatively uniform in $y_{t{\\bar t}}$ and in $m_{t{\\bar t}}$.\nThe scale \\mbox{$\\mu_0=m_{t{\\bar t}}\/2$} leads to slightly larger radiative corrections in the highest-$m_{t{\\bar t}}$ interval.\nIn the first $m_{t{\\bar t}}$ interval, the NNLO prediction slightly undershoots the data,\nconsistently with what is observed in the low-$m_{t{\\bar t}}$ region in Fig.~\\ref{fig:mtt}.\nWe note that, at variance with the first $m_{t{\\bar t}}$ bin in Fig.~\\ref{fig:mtt},\nhere the first $m_{t{\\bar t}}$ interval extends up to $450$ GeV,\nthereby leading to a better agreement between the NNLO prediction and the data.\nAt high values of $m_{t{\\bar t}}$, the scale \\mbox{$\\mu_0=m_{t{\\bar t}}\/2$} leads to a slightly better agreement\nwith the data.\nIn the highest-$y_{t{\\bar t}}$ bin, for both central scales, the experimental result\nis below the NNLO prediction in all invariant-mass intervals except the first one.\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[height=0.30\\textheight]{plots\/LHC13.CMSsetup\/wideCMS13_HT_2_absy_th-pT_th.pdf}\\\\\n\\includegraphics[height=0.30\\textheight]{plots\/LHC13.CMSsetup\/wideCMS13_mT_avt_absy_th-pT_th.pdf}\n\\end{center}\n\\vspace{-4ex}\n\\caption{\\label{fig:pth_yth}\n  Double-differential cross sections as a function of $p_{T,t_\\text{had}}$ in four $y_{t_\\text{had}}$ intervals.\n  CMS data~\\cite{Sirunyan:2018wem} and LO, NLO and NNLO results\n  for central scales $H_{T}\/2$ (upper) and $m_{T,t_{\\rm av}}$ (lower).\n}\n\\end{figure}\n\nIn Fig.~\\ref{fig:pth_yth} we present the $p_{T,t_\\text{had}}$ distribution in $y_{t_\\text{had}}$ intervals,\nwhere $y_{t_\\text{had}}$ is the rapidity of the hadronically decaying top or antitop\nquark with transverse momentum $p_{T,t_\\text{had}}$.\nThe QCD results use the scales \\mbox{$\\mu_0=H_T\/2$}~(upper) and \\mbox{$\\mu_0=m_{T,t_{\\rm av}}$}~(lower) as a natural scale,\nanalogously to the $p_{T,t_\\text{had}}$ distribution in Fig.~\\ref{fig:pth}.\nWe observe that the two scale choices lead to very similar results.\nThe impact of the radiative corrections is rather uniform in the considered $y_{t_\\text{had}}$ intervals.\nDiscussing the single-differential $p_{T,t_\\text{had}}$ spectrum in Fig.~\\ref{fig:pth},\nwe observed that the NNLO prediction is slightly harder than the data.\nIn Fig.~\\ref{fig:pth_yth} we observe that this feature holds in all the $y_{t_\\text{had}}$ intervals.\nNonetheless, the overall agreement between the NNLO prediction and the data is good.\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[height=0.30\\textheight]{plots\/LHC13.CMSsetup\/wideCMS13_HT_2_pT_th-m_ttx.pdf}\\\\\n\\includegraphics[height=0.30\\textheight]{plots\/LHC13.CMSsetup\/wideCMS13_m_ttx_2_pT_th-m_ttx.pdf}\n\\end{center}\n\\vspace{-4ex}\n\\caption{\\label{fig:mtt_pth}\n  Double-differential cross sections as a function of $m_{t{\\bar t}}$ in four $p_{T,t_\\text{had}}$ intervals.\n  CMS data~\\cite{Sirunyan:2018wem} and LO, NLO and NNLO results\n  for central scales $H_{T}\/2$ (upper) and $m_{t{\\bar t}}\/2$ (lower).\n}\n\\end{figure}\n\nWe finally discuss the $m_{t{\\bar t}}$ distribution in $p_{T,t_\\text{had}}$ intervals, which is shown in Fig.~\\ref{fig:mtt_pth}.\nWe use the central scales \\mbox{$\\mu_0=H_T\/2$}~(upper) and \\mbox{$\\mu_0=m_{t{\\bar t}}\/2$}~(lower).\nIn each $p_{T,t_\\text{had}}$ interval we have $p_{T,{\\rm min}} < p_{T,t_\\text{had}} < p_{T,{\\rm max}}$.\nThe introduction of kinematical cuts on $p_{T,t_\\text{had}}$ significantly affects the $m_{t{\\bar t}}$ distribution.\nIn the presence of a cut, $p_{T,{\\rm max}}$, on the maximum $p_T$ of the hadronically decaying top quark,\nhigh values of $m_{t{\\bar t}}$ can be reached by either increasing the transverse momentum of the top quark\ndecaying to a leptonic final state, or increasing the rapidity separation \\mbox{$|y_t - y_{\\bar t}|$}.\nThese kinematical regions are both dynamically suppressed for heavy-quark production\nand, therefore, $p_{T,{\\rm max}}$ produces a faster suppression at high invariant masses.\nThis effect can be observed by comparing the $m_{t{\\bar t}}$ single-differential distribution in Fig.~\\ref{fig:mtt} with the present double-differential distribution.\n\nThe presence of a lower cut, $p_{T,{\\rm min}}$, on the minimum $p_{T,t_\\text{had}}$ has an even more drastic effect on the $m_{t{\\bar t}}$ distribution,\nas it imposes the constraint \\mbox{$m_{t{\\bar t}}> 2\\sqrt{{m^2_{t}}+p_{T,{\\rm min}}^2} \\equiv 2 m_{T,{\\rm min}}$} for LO kinematics.\nBelow this unphysical threshold the LO result vanishes,\nand the NLO and NNLO results are effectively LO and NLO predictions, respectively.\nAs a consequence, they suffer from larger theoretical uncertainties,\nwhich is reflected by the stronger scale dependence.\nAbove this threshold, the LO distribution sharply increases up to a kinematical peak\nclose to $2m_{T,{\\rm min}}$.\nOwing to this LO behaviour, soft-collinear radiation produces shape instabilities~\\cite{Catani:1997xc}\nin this $m_{t{\\bar t}}$ region at each subsequent perturbative order.\nThe qualitative behaviour of these shape instabilities is completely analogous to that observed and discussed in Ref.~\\cite{Catani:2018krb} (see Figs.~10 and 20 and related comments therein) in the case of diphoton production\nin the presence of $p_T$ cuts.\nThese perturbative instabilities are localized in a very narrow region\naround the LO threshold, and therefore their effect is smeared if a sufficiently large bin size is considered, as is the case for the differential distribution in Fig.~\\ref{fig:mtt_pth}.\n\nThe comparison with the data shows that in the first two $p_{T,t_\\text{had}}$ intervals\nthe NNLO prediction undershoots the data in the first $m_{t{\\bar t}}$ bin.\nThis discrepancy at low-$m_{t{\\bar t}}$ is not resolved in the two highest-$p_{T,t_\\text{had}}$ intervals,\nsince the larger bin size ($300\\text{ GeV} < m_{t{\\bar t}} < 430\\text{ GeV}$) renders the distribution less sensitive to the behaviour close to the physical threshold.\nThese observations are consistent with the expectations\nfrom the behaviour of the single-differential distribution in Fig.~\\ref{fig:mtt} at low $m_{t{\\bar t}}$.\nExcluding the narrower bins at low $m_{t{\\bar t}}$, the NNLO prediction in Fig.~\\ref{fig:mtt_pth} is in very good agreement with the experimental measurements.\nThe results with \\mbox{$\\mu_0=H_T\/2$}~(upper) and \\mbox{$\\mu_0=m_{t{\\bar t}}\/2$}~(lower)\nturn out to be rather similar, consistently with\nour general expectation at the beginning of Section~\\ref{sec:resu}.\nThe exception is the highest-$p_{T,t_\\text{had}}$ interval, where the scale \\mbox{$\\mu_0=m_{t{\\bar t}}\/2$} leads\nto a quite large NLO scale dependence at low $m_{t{\\bar t}}$, which is drastically reduced at NNLO.\nIn this region of low $m_{t{\\bar t}}$ and high $p_{T,t_\\text{had}}$, the invariant mass $m_{t{\\bar t}}$ is not the characteristic hard scale anymore:\nthe scale choice \\mbox{$\\mu_0=m_{t{\\bar t}}\/2$} is not expected to be optimal,\nand the scale \\mbox{$\\mu_0=H_T\/2$} turns out to be more appropriate.\n\n\\section{Summary and outlook}\n\\label{sec:summary}\n\nIn this paper we have presented a new fully differential NNLO calculation\nof top-quark pair production at hadron colliders.\nThe calculation is carried out by using the $q_T$ subtraction formalism\nto handle IR divergences from real and virtual contributions, and it is implemented in the \\Matrix{} framework.\nOur code enables fast and efficient calculations of fiducial cross sections\nand multi-differential distributions.\n\nWe have computed several single- and double-differential distributions of the top quarks,\nand we have compared our results with recent measurements performed by the CMS collaboration\nin the lepton+jets decay channel.\nWe have considered several values of the renormalization and factorization scales to compute each of the distributions.\nWe have used natural scales (i.e. ${m_{t}}$, $m_{t{\\bar t}}\/2$ and the relevant transverse masses $m_T$)\nof the order of the characteristic hard scale of the computed distribution,\nand we have shown that the corresponding results are similar to what is obtained \nwith the overall choice $\\mu_0 = H_T\/2$.\nWe find that both the natural scales and $\\mu_0 = H_T\/2$ lead\nto a reasonable perturbative behaviour for all the distributions that we have examined.\nThe NNLO corrections substantially reduce the uncertainties of the theoretical predictions,\nand they improve the overall agreement with the experimental measurements.\nThe largest deviation between data and the NNLO result\noccurs close to the $m_{t{\\bar t}}$ threshold in single- and double-differential distributions.\nThis discrepancy could be related to a variety of effects,\nincluding issues in the extrapolation of the data from particle to parton level,\nwhich is expected to be delicate in such threshold region.\nA lower value of the top-quark mass also has a significant impact close to the threshold.\n\nThe code that is used to perform these calculations is going to become public in a future \\Matrix{} release,\nproviding a fast and flexible tool to compute (multi-)differential distributions\nwith arbitrary cuts on the top-quark kinematical variables.\nThe inclusion of NLO EW corrections and of top-quark decays is left to future work.\n\n\\vspace*{2ex}\n\\noindent {\\bf Acknowledgements}\n\n\\noindent We are very grateful to Hayk Sargsyan for his contribution at early stages of this work.\nWe are also indebted to Federico Buccioni, Jean-Nicolas Lang, Jonas Lindert and Stefano Pozzorini for their ongoing support with {\\sc OpenLoops~2}.\nWe wish to thank Ben Kilminster and Florencia Canelli for useful discussions.\nThis work is supported in part by the Swiss National Science Foundation (SNF) under contract 200020-169041. The work of SK is supported by the ERC Starting Grant 714788 REINVENT. \n\n\n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe mechanism underlying the explosive deaths of massive stars remains poorly understood.  \\citet{Colg66} first \nsuggested that neutrino energy deposition plays a central role in powering core-collapse supernovae (CCSNe).  \nEver since, much of the effort in CCSN theory has focused on building increasingly sophisticated \nneutrino radiation hydrodynamical models, with the hopes of reproducing the properties of \nCCSNe, including the kinetic energies, debris morphologies, nucleosynthetic yields, and the remnant mass, \nspin, and velocity distributions.  Despite this effort, the best models still fall short of accounting \nfor any of these properties, much less all of them simultaneously.  Perhaps even more alarming, \nthe various groups involved in CCSN modeling often reach qualitatively different conclusions with \ncalculations that are ostensibly quite similar, vis-\\'{a}-vis whether an explosion even occurs.  \n\nIn \\citet{Mull12,Mull12_sasi}, results are reported from 2D axisymmetric modeling with the \\textsc{Vertex-CoCoNuT} code, \nwhich uses a conformally-flat spacetime approximation of general relativity \\citep{Mull10}.  They find \nexplosions for $8.1$-$\\,{\\rm M}_\\odot$, $11.2$-$\\,{\\rm M}_\\odot$, $15$-$\\,{\\rm M}_\\odot$, and $27$-$\\,{\\rm M}_\\odot$ progenitors, but, when \nreported, the explosion energies are $\\sim$10 times smaller than the canonical $10^{51}\\, {\\rm erg}$ energy \nof typical CCSNe.  Similar findings were presented for a variety of other progenitors (Janka et al., Nuclear Astrophysics Workshop, Ringberg Castle, 2014).  Recently, \\citet{Hank13} reported results from \na three-dimensional simulation with the \\textsc{Prometheus-Vertex} code of the same $27$-$\\,{\\rm M}_\\odot$ progenitor \nconsidered in \\citet{Mull12_sasi} and found no explosion.  This negative result has been recapitulated for all\nother 3D simulations performed recently by this group \\citep{Tamb14}, despite their having seen explosions in the corresponding 2D simulations.    \n\n\\citet{Suwa10} reports an explosion of a $13$-$\\,{\\rm M}_\\odot$ progenitor in a 2D simulation and \\citet{Taki12} finds explosions of an $11.2$-$\\,{\\rm M}_\\odot$ progenitor in both 2D and 3D.  These models neglected the heavy lepton neutrinos, which were recently incorporated with an approximate leakage scheme \\citep{Taki13}.  In all cases, the $\\nu_e$ and $\\bar{\\nu}_e$ transport was computed using the isotropic diffusion source approximation (IDSA) \\citep{Lieb09}, a crude approximation meant to enable multi-D simulations at minimal cost.  While interesting, their results are difficult to interpret in the context of the viability of the neutrino mechanism, as the authors acknowledge.\n\nMeanwhile, \\citet{Brue13} report results of \n2D axisymmetric modeling with their \\textsc{Chimera} code.  They consider $12$-$\\,{\\rm M}_\\odot$, $15$-$\\,{\\rm M}_\\odot$, \n$20$-$\\,{\\rm M}_\\odot$, and $25$-$\\,{\\rm M}_\\odot$ progenitors from \\citet{Woos07} and find explosions in all cases, \ncuriously at almost the same post-bounce time.  They also report energies that are somewhat larger than \nthe those reported in \\citet{Mull12}, but that still fall short of the $10^{51}\\, {\\rm erg}$ mark.  Janka at el. (Nuclear Astrophysics Workshop, Ringberg Castle, 2014) recently reported 2D models of the same four progenitors, and found significantly different results, with, for example, the $12$-$\\,{\\rm M}_\\odot$ model not yet exploding more than $700\\,{\\rm ms}$ after bounce.\n\nImportantly, all of the studies discussed above relied on the so-called \n``ray-by-ray-plus'' approximation of neutrino transport, which replaces the real transport problem \nwith a series of independent spherically-symmetric transport solves.  This is a crude approximation \nthat introduces large variations in angle and time in the neutrino fluxes and the associated neutrino energy deposition \nso crucial for the neutrino-driven mechanism.  This simplification has yet to be clearly justified, \nand may be producing qualitatively incorrect results, particularly in 2D.  \n\nThe only calculations ever performed \nwhich allow for multidimensional transport were the VULCAN\/2D results reported in \\citet{Burr06}, \\citet{Burr07}, \\citet{Ott08}, and \\citet{Bran11}, \nand none of these calculations showed a revival of the stalled shock in 2D by the delayed-neutrino mechanism.\nThe calculations of \\citet{Ott08} and \\citet{Bran11} were multi-angle as well.\nHowever, these calculations were performed without $\\mathcal{O}(v\/c)$ transport effects \\citep{Hube07}.\nWe are, therefore, motivated in this paper to perform new 2D multi-group radiation hydrodynamics calculations with\na new code with both multi-D transport (avoiding the simplifications of the ray-by-ray approach) and \nthe velocity-dependent terms to determine whether these earlier results were artifacts of the neglect \nof $\\mathcal{O}(v\/c)$ terms, and for comparison with the 2D results of other groups.\nTo accomplish this, we have developed the CASTRO radiation hydrodynamics code. CASTRO contains a \nmulti-group flux-limited neutrino transport solver, is time-dependent and multidimensional, \nand treats three neutrino species ($\\nu_e$, $\\bar{\\nu}_e$, $\\nu_x$, where the $\\nu_x$ species \nincludes the $\\mu$ and $\\tau$ neutrinos and their antiparticles), including all relevant $\\mathcal{O}(v\/c)$ terms.\nWe find that none of our new 2D calculations, employing the same progenitors as \\citet{Brue13}, explode by the delayed-neutrino mechanism.\nWith this paper, we describe our results and speculate on the reasons for the different outcomes we find.\nSince all other groups are using the ray-by-ray approach, we suggest that one reason for the different outcomes\nmay be in the handling of multi-D transport.  In 2D, the axial sloshing motions, often not seen in 3D \\citep{Burr12}, may be reinforcing the errors in the ray-by-ray approach\nand leading to a qualitatively incorrect outcome.  In 3D, these axial sloshing effects are often absent, and the ray-by-ray approach\nmay be less anomalous (due to the greater sphericity of the hydrodynamics), so the lack\nof explosions seen by the Garching group in 3D, when they observe explosions for the same progenitors in 2D, remains\npuzzling.\n\n\n\n\\section{Numerics and Setup}\n\nWe use the CASTRO code to carry out our CCSN simulations \\citep{Almg10,Zhan11,Zhan13}.  \nCASTRO is a second-order, Eulerian, compressible, Godunov-type, radiation hydrodynamics \ncode that uses block-structured adaptive mesh refinement to simultaneously refine \nin both space and time.  Simulations can be performed in 1D (spherical), 2D (cylindrical), and 3D (Cartesian).  The hydrodynamic \nupdates use piecewise-parabolic reconstruction with higher-order limiters to preserve \naccuracy at smooth extrema, an approximate Riemann solver detailed in \\citet{Almg10}, and incorporates \nfull corner coupling in the directionally unsplit integration.  In this work, we make \nuse of the multi-group flux-limited diffusion (MGFLD) neutrino transport solver detailed \nin \\citet{Zhan13}.  \n\nCurrently, multi-dimensional multi-angle\ntransport is not feasible and will require the exascale.\nOur comoving frame multi-group flux-limited diffusion (MGFLD) formulation includes $\\mathcal{O}(v\/c)$ terms that can\nresult (correctly) in significant differences in the dynamic diffusion limit where\nradiation transport is dominated by motion of the fluid \\citep{Cast04}.\nThe approach used in CASTRO\nsplits the system into three parts: 1) a part that couples the radiation\nand fluid in a hyperbolic subsystem with a piecewise parabolic\nmethod (PPM) with characteristic tracing and full corner\ncoupling \\citep{Mill02}, 2) another part that is a system of coupled\nparabolic equations that evolves radiation diffusion over all the\ngroups (along with $Y_e$ in the neutrino case) and source-sink\nterms, and 3) a final part that performs frequency-space advection.\nThe hyperbolic subsystem is solved explicitly with a high-order\nGodunov scheme as part of the hydrodynamic component of the algorithm,\nwhereas the parabolic part is solved implicitly with a first-order\nbackward Euler method.  Frequency-space advection is performed using\na standard approach based on the method of lines.  The frequency-space advection has its\nown CFL condition for stability and, if necessary, subcycling in time is employed in order to satisfy the\nfrequency-space CFL condition. The primary computational expense of\nthe radiation is in the solution of linear systems as part of the\niteration over energy groups.  We rely on the {\\sl hypre} library\n\\citep{Falg02} for solving these systems on large parallel machines.\n\nCASTRO uses a hybrid parallelization strategy based on MPI + OpenMP\nusing the BoxLib framework \\citep{Almg10}. The basic strategy\nis to distribute grids within the AMR hierarchy to computational nodes.\nThis provides a natural coarse-grained approach to distributing the\ncomputational work.  A dynamic load balancing technique is needed to\nadjust the load.  Although the code supports both a heuristic knapsack\nalgorithm and a space-filling curve algorithm for load balancing, the\ndata-locality properties make the space-filling curve the method of\nchoice for problems with radiation. The main advantages of the CASTRO code\nare the efficiency due to the use of AMR, the temporal sub-cycling, and\nthe accuracy due to the coupling of radiation force into the Riemann solver.\n\nCASTRO incorporates 1) a complicated set of opacity tables for the\nvarious neutrino species that includes weak-magnetism,\nion-ion correlation, and ion form-factor corrections,\n2) various extant nuclear equations of state (including the\nShen [default] and various Lattimer and Swesty equations of state), 3) inelastic\nneutrino-electron and neutrino-nucleon scattering, and 4) a\ntemporal sub-cycling algorithm that accelerates computation\nby many factors over traditional codes.  CASTRO\ntreats three neutrino species ($\\nu_e$, $\\bar{\\nu}_e$, $\\nu_x$, where \nthe $\\nu_x$ species includes the $\\mu$ and $\\tau$ neutrinos and their antiparticles).  In this work, \nfor computational expediency, we neglect in the 2D simulations energy-group coupling processes such as \ninelastic scattering, and justify this choice in Section~\\ref{1D} through the comparison of model shock radius \nevolutions in 1D, with and without them.  We also adopt the Shen equation of state \\citep{Shen98a,Shen98b}. \n\nWe follow the collapse, bounce, and subsequent evolution of four progenitors in 1D and 2D.  The progenitors are nonrotating, solar metallicity models with zero-age main sequence masses of $12$-$\\,{\\rm M}_\\odot$, $15$-$\\,{\\rm M}_\\odot$, $20$-$\\,{\\rm M}_\\odot$, and $25$-$\\,{\\rm M}_\\odot$ \\citep{Woos07}.  These are the same progenitors recently considered by \\citet{Brue13} and Janka et al. (Nuclear Astrophysics Workshop, Ringberg Castle, 2014).  Our numerical grid has $0.5\\,{\\rm km}$ resolution in the inner $\\sim$$128\\,{\\rm km}$, $1\\,{\\rm km}$ resolution out to $\\sim$$270\\,{\\rm km}$, never worse than $4\\,{\\rm km}$ resolution anywhere beneath the shock, and extends out to $5120\\,{\\rm km}$.\n\n\\section{Results of Two-Dimensional Simulations}\n\\label{results}\n\n\\subsection{Global Structure}\n\\label{global}\nFigure~\\ref{fig:snapshots} shows representative snapshots of the radial velocity and entropy from our $12$-$\\,{\\rm M}_\\odot$ and $25$-$\\,{\\rm M}_\\odot$ models.  At $200\\,{\\rm ms}$ after bounce, both models are nearly spherical and show weak convective activity with a characteristic angular scale corresponding to $\\ell\\sim10$ ($\\ell$ is spherical harmonic degree) as is clearly seen in the radial velocities.  By $400\\,{\\rm ms}$ after bounce, both models have become more aspherical, indicating stronger convective activity that is at larger angular scales (smaller $\\ell$).  The snapshots at $600\\,{\\rm ms}$ after bounce look qualitatively similar to those at $400\\,{\\rm ms}$ after bounce, showing large scale convective motions including relatively low entropy accretion streams and higher entropy buoyant plumes.\n\nFigure~\\ref{fig:rshock} shows the evolutions of the average shock radii for our 1D and 2D models.  Not surprisingly, our 1D models do not explode, consistent with nearly all spherically symmetric CCSN calculations.  In 2D, despite evolving the models to $\\gtrsim$$600\\,{\\rm ms}$ after bounce, we do not find explosions.  Nevertheless, there are interesting differences between 1D and 2D models.  Perhaps most importantly, the 2D stalled shock radii are consistently larger by a factor 1.3--2, as found in essentially all 1D-2D comparisons \\citep[e.g.][]{Mull12}.  This is often attributed to aspherical instabilities, namely convection and the standing accretion shock instability (SASI), both increasing the neutrino heating efficiency in the gain region and providing turbulent pressure support \\citep[e.g.][]{Hera92,Burr95,Jank96}.  In our models, we see convective activity develop immediately as the bounce shock moves out.  This brief phase of ``prompt convection,'' seeded by perturbations introduced by our aspherical grid, drives the shock to large radii much faster than in the corresponding 1D models.  By $\\sim$$20\\,{\\rm ms}$ after bounce, all four 2D models have shock radii of $\\sim$$200\\,{\\rm km}$ and these shocks stall $\\sim$$60\\,{\\rm ms}$ earlier than the 1D models, which never reach radii much beyond $\\sim$$150\\,{\\rm km}$.  Between 300 and 400$\\,{\\rm ms}$, the $20$-$\\,{\\rm M}_\\odot$ and $25$-$\\,{\\rm M}_\\odot$ models show jumps in the average shock radii, both in 1D and 2D.  These jumps correspond to the accretion of the Si\/O interface in these models, wherein $\\dot{M}$ drops suddenly and the shock responds by moving outward.  In 1-D, the jump is only about $10\\%$ of the pre-interface shock radius, but the effect is 2--3 times larger in 2D.  Apparently and interestingly, the sensitivity to the accretion of shelves increases with shock radius. \n\nThe shock radii are only one aspect of the global structure that determines if and when explosions commence.  The gain radius, $R_g(\\theta)$, is defined as the radius above which the net rate of energy deposition in the gas, integrated over all neutrino energies and summed over species, is positive.  The gain region, bounded by the gain and shock radii, contains the mass where neutrino energy deposition, occurring at a rate\n\\begin{equation}\\label{eq:heating_rate}\n(\\mathcal{H} - \\mathcal{C})_{\\rm gain} = \\sum_s \\int d\\Omega \\int_{R_g(\\theta)}^{R_s(\\theta)} r^2 dr \\int_{\\varepsilon_{s}^{\\rm min}}^{\\varepsilon_{s}^{\\rm max}} (c\\kappa_{s,\\varepsilon} E_{s,\\varepsilon} - j_{s,\\varepsilon}) d\\varepsilon\\,,\n\\end{equation}\n is thought to lead to shock revival.  In Eq.~\\ref{eq:heating_rate}, $s\\in\\left\\{\\nu_e,\\bar{\\nu}_e,\\nu_x\\right\\}$, $R_s(\\theta)$ is the angle-dependent shock radius, $\\varepsilon$ is neutrino energy, $\\kappa$ is the absorption cross section, $E$ is the neutrino energy density, and $j$ is the volume emissivity.  As important is the cooling region where accreted material cools and settles onto the proto-neutron star, undermining the pressure support crucial to reviving the stalled shock.  We define a net cooling rate analogous to the heating rate above as\n\\begin{equation}\n(\\mathcal{H} - \\mathcal{C})_{\\rm cool} = \\sum_s \\int d\\Omega \\int_{R_{\\nu_s}(\\theta)}^{R_g(\\theta)} r^2 dr \\int_{\\varepsilon_{\\nu_s}^{\\rm min}}^{\\varepsilon_{\\nu_s}^{\\rm max}} (c\\kappa_{s,\\varepsilon} E_{s,\\varepsilon} - j_{s,\\varepsilon}) d\\varepsilon\\,,\n\\end{equation}\nwhere $R_{\\nu_s}$ is the angle-, species-, and energy-dependent neutrinosphere radius defined implicitly by\n\\begin{equation}\n\\int_{R_{\\nu_s}}^\\infty \\kappa dr = \\frac{2}{3}\\,.\n\\end{equation}\nThis definition roughly accounts for all the optically-thin cooling in the region, but does not include diffusive flux from beneath the $\\tau=2\/3$ surface.  We define an effective inner cooling radius as\n\\begin{equation}\nR_{\\rm in} = \\frac{\\sum_s \\sum_\\varepsilon |w_s| R_{\\nu_s}}{\\sum_s \\sum_\\varepsilon |w_s|}\\,,\n\\end{equation}\nwhere\n\\begin{equation}\nw_s = \\int_{R_{\\nu_s}}^\\infty (c\\kappa E - j) r^2 dr\n\\end{equation}\nis an integral of the net neutrino energy deposition along a column (at fixed $\\theta$) and should be understood to be angle-, species-, and energy-dependent.  This inner cooling radius is related to the neutrinosphere radii of the dominant cooling agents.  The region between $R_{\\rm in}$ and $R_g$ is cooling rapidly by optically-thin neutrino emission, predominantly through the $\\nu_e$ and $\\bar{\\nu}_e$ species, and represents at least one component of the accretion luminosity.  Figure~\\ref{fig:radii} shows the evolutions of the shock, gain, and inner radii for our four 2D models.  After a short-lived initial transient, the gain and inner radii generally recede in all four models, reflecting the contraction of the inner proto-neutron star.\n\n\\subsection{Diagnostic Quantities}\n\\label{diagnostics}\nThe evolutions of certain diagnostic quantities have proven useful in distilling insight from the complicated multi-D dynamics of pre-explosion supernova cores.  The heating efficiency, $\\epsilon_h$, is defined as the ratio of the net heating in the gain region to the sum $L_{\\nu_e}+L_{\\bar{\\nu}_e}$.  We also define an analogous cooling efficiency, $\\epsilon_c\\equiv-(\\mathcal{H}-\\mathcal{C})_{\\rm cool}\/(L_{\\nu_e}+L_{\\bar{\\nu}_e})$.  Both efficiencies are shown in Fig.~\\ref{fig:efficiencies}.  The heating efficiencies initially rise slowly and show broad peaks around $200\\,{\\rm ms}$ after bounce.  The peak values range from 6\\% to 9\\%, monotonically increasing with progenitor mass.  At low optical depths, the heating efficiency is equivalent to an effective optical depth, giving a heating rate $(L_{\\nu_e}+L_{\\bar{\\nu}_e}) (1-\\exp{(-\\tau_{\\rm eff})})\\approx (L_{\\nu_e}+L_{\\bar{\\nu}_e}) \\tau_{\\rm eff}$.  The basic character of the heating efficiencies can be understood by a simple estimate of $\\tau_{\\rm eff}$.  First, write $\\tau_{\\rm eff}\\sim\\langle \\rho\\rangle_g \\sigma \\Delta R$, where $\\langle \\rho\\rangle_g$ is the average density in the gain region, $\\sigma$ is a characteristic neutrino absorption cross section per unit mass, and $\\Delta R=\\langle R_s\\rangle - \\langle R_g\\rangle$.  The average density is easily computed from the gain mass and shock and gain radii.  We write the cross section as $\\sigma\\approx\\sigma_0 (\\varepsilon\/\\varepsilon_0)^2\\sim\\sigma_0 (R_0\/R_{\\rm in})^2$, where the 0-subscript signifies some characteristic value.  This last scaling is justified by the approximately inverse relationship between root-mean-square neutrino energy and inner cooling radius.  Finally, we arrive at\n\\begin{equation}\\label{eq:tau}\n\\tau_{\\rm eff}\\sim \\frac{3\\sigma_0 M_g}{4\\pi} \\frac{R_s-R_g}{R_s^3-R_g^3} \\left(\\frac{R_0}{R_{\\rm in}}\\right)^2\\,.\n\\end{equation}\nThis estimate reproduces the heating efficiencies reasonably well, accounting, for example, for the slow decline in the $12$-$\\,{\\rm M}_\\odot$ model, the nearly constant value for the $15$-$\\,{\\rm M}_\\odot$ model, and the rise in the latter half of the $20$-$\\,{\\rm M}_\\odot$ and $25$-$\\,{\\rm M}_\\odot$ models.  The virtue of this estimate is that it shows how the heating efficiencies depend on the global structure of the solutions.\n\nThe cooling efficiency $\\epsilon_c$ represents the fraction of the $\\nu_e$ and $\\bar{\\nu}_e$ luminosities arising from optically thin cooling.  Since some of the accreted material advects into optically thick regions before cooling appreciably, the net cooling rate $(\\mathcal{H}-\\mathcal{C})_{\\rm cool}$ entering into the definition of $\\epsilon_c$ is not equal to the total accretion luminosity, which must be $\\approx GM_{\\rm pns}\\dot{M}_{\\rm pns}\/R_{\\rm pns}$ (pns$\\equiv$proto-neutron star), at least in an integral-averaged sense.  Rather, $\\epsilon_c$ represents a response function to sudden changes in $\\dot{M}$.  For example, if the accretion rate drops due to the accretion of a composition interface, the immediate response in the luminosity (delayed somewhat by the advection time between the shock and cooling region) will be to drop fractionally by approximately $\\epsilon_c \\dot{M}\/\\dot{M}_0$, where $\\dot{M}_0$ and $\\dot{M}$ are the pre- and post-interface accretion rates, respectively.  The noticeable shelves accreted in the $12$-$\\,{\\rm M}_\\odot$, $20$-$\\,{\\rm M}_\\odot$, and $25$-$\\,{\\rm M}_\\odot$ models confirm this basic picture.  Apparently, a low cooling efficiency may make a model prone to explosion when shelves are accreted since, though $\\dot{M}$ can drop appreciably, the luminosity responds only weakly, leaving the model closer to (or perhaps beyond) the critical luminosity for explosion \\citep{Burr13}.  Evidently, this effect is not sufficient to drive explosions in our models.\n\nA variety of ``explosion conditions'' have been proposed in the literature.  The most popular considers the ratio of the timescale for advection through the gain region to the heating timescale.  Define the advection timescale as $\\tau_{\\rm adv} = M_{\\rm gain}\/\\dot{M}$ and the heating timescale as\n\\begin{equation}\n\\tau_{\\rm heat} = \\frac{\\int_{\\rm gain} (\\rho e - \\rho e_0) dV}{(\\mathcal{H}-\\mathcal{C})_{\\rm gain}}\\,,\n\\end{equation}\nwhere $e_0$ is approximately the zero-point of the EOS, i.e. $e_0=e(\\rho,Y_e,T_{\\rm min})$ where $T_{\\rm min}$ is the lowest temperature of the tabulated EOS ($10^{-2}$ MeV).  The left panel of Fig.~\\ref{fig:crit_panel} shows the evolutions of this ratio for all four models.  Surprisingly, the curves all show the same basic structure, peaking around $200\\,{\\rm ms}$ after bounce before slowly declining.  Superimposed on these slowly varying curves, the $20$-$\\,{\\rm M}_\\odot$ and $25$-$\\,{\\rm M}_\\odot$ models show narrow peaks around $300\\,{\\rm ms}$ after bounce, corresponding to the accretion of the Si\/O interfaces in these models when $\\dot{M}$ drops abruptly by a factor of two.\n\nAnother interesting dimensionless quantity that arises naturally in discussing critical conditions for explosion \\citep[see, e.g.,][]{Dole13} is the ratio of heating in the gain region to the accretion power $\\mathcal{H}_g R_s\/(G M \\dot{M})$, shown in the right panel of Fig.~\\ref{fig:crit_panel}.\nComparing the left and right panels of Fig.~\\ref{fig:crit_panel}, it is clear that these dimensionless ratios have very similar behaviors.  Writing the heating in the gain region $\\mathcal{H}_g = h_g M_g$, where $h_g$ is the specific heating rate and $M_g$ is the mass in the gain region, and noting that the zero-point-subtracted specific internal energy is of order $G M\/R_s$, it is easy to see that $\\mathcal{H}_g R_s\/(G M \\dot{M})\\sim \\tau_{\\rm adv}\/\\tau_{\\rm heat}$, so the similarities between the left and right panels of Fig.~\\ref{fig:crit_panel} are not surprising.  Similarly, the ``antesonic'' condition, proposed by \\citet{Pejc12}, is easily related to these other conditions.  Apparently, $\\tau_{\\rm adv}\/\\tau_{\\rm heat} \\sim \\mathcal{H}_g R_s\/(G M \\dot{M}) \\propto c_s^2\/v_{\\rm esc}^2$.\n\nWhile all three ratios are interesting diagnostics and can clearly indicate when explosions are underway, none seem to have a well-determined critical threshold above which explosions are inevitable.  In particular, the value of one is not necessarily associated with transitioning to explosion.  The critical values likely depend in detail on how these quantities are defined, and may be below or above one, but reasonable definitions seem to at least give critical values of order one.  Of course, since we do not obtain explosions, we are unable to determine these critical values for our models and definitions.\n\n\\section{Comparisons with One-Dimensional Results}\n\\label{1D}\n\nMultidimensional effects seem to be crucial if the neutrino mechanism succeeds in producing explosions.  A variety of effects may be important.  The post-shock turbulent flow, driven by convective and\/or SASI activity, leads to longer dwell times in the gain region on average, exposing the accreted material to more net heating \\citep{Murp08,Dole13}.  The turbulent flow itself gives rise to Reynolds stresses that can help support the shock \\citep{Murp13}.  Convection in the proto-neutron star can advect neutrinos closer to their neutrinosphere, shortening their escape time and boosting the diffusive luminosity from the core \\citep{Dess06,Mull14}.  Even the efficiency of cooling through the accretion luminosity can vary between dimensions.  We have already explored the former two effects in parametrized setups \\citep{Dole13,Murp13}, so here we focus on characterizing how the neutrino luminosities and average energies differ between 1D and 2D models.\n\nFigure~\\ref{fig:le_1d2d} shows the evolutions of the $\\nu_e$ and $\\bar{\\nu}_e$ luminosities and root-mean-square neutrino energies, as measured in the laboratory frame at $500\\,{\\rm km}$ and defined by\n\\begin{equation}\n\\varepsilon_{\\rm rms} = \\sqrt{\\frac{\\int_{\\varepsilon_{\\rm min}}^{\\varepsilon_{\\rm max}} \\varepsilon^2 F_\\varepsilon d\\varepsilon}{\\int_{\\varepsilon_{\\rm min}}^{\\varepsilon_{\\rm max}} F_\\varepsilon d\\varepsilon}}\\, ,\n\\end{equation}\nfor all four progenitor models.  Some differences between 1D and 2D seem to be model-independent.  For example, the rms energies are systematically higher in 1D than in 2D by $\\sim$5--10\\%, consistent with results obtained by other groups \\citep{Bura06}.  The luminosities, on the other hand, do not show such a generic difference.  Around $100\\,{\\rm ms}$ post-bounce, the $20$-$\\,{\\rm M}_\\odot$ and $25$-$\\,{\\rm M}_\\odot$ models show deficits of up to $\\sim$$20\\%$ in $\\nu_e$ and $\\bar{\\nu}_e$ luminosities in 2D relative to 1D.  Beyond $\\sim$$200\\,{\\rm ms}$ post bounce, all the 2D models tend to have higher $\\nu_e$ and $\\bar{\\nu}_e$ luminosities by $\\sim$$5\\%$, likely an effect of Ledoux convection in the proto-neutron star, driven by the destabilizing lepton gradient \\citep{Bura06,Dess06}.\n\nThe idea of a critical luminosity for explosion at a given mass accretion rate was first discussed in \\citet{Burr93}.  \\citet{Murp08} later showed, in the context of parametrized modeling, that the critical luminosity is lower in 2D than in 1D, a result confirmed by other studies \\citep{Hank13,Dole13,Couc13_2d3d}.  Unfortunately, the critical curves that emerged from these studies are not easily adapted to self-consistent radiation hydrodynamic models.  One reason is that the critical luminosity almost certainly depends on quantities other than $L_{\\nu_e}$ and $\\dot{M}$.  For example, it may depend on the proto-neutron star mass and radius and\/or the shock radius, as suggested in the discussion of the dimensionless quantities in Section~\\ref{diagnostics}.  It also depends on the structure of the flow beneath the gain layer, which can be quite different between parametrized and self-consistent models.  Nevertheless, the lower critical luminosity in 2D very likely remains in self-consistent models.\n\nIn Fig.~\\ref{fig:crit}, we show the evolutions in the $L_{\\nu_e}$-$\\dot{M}$ plane for all four progenitors in both 1D and 2D.  All of the models show relatively flat curves until late times, indicating that the neutrino luminosities remain roughly constant while the accretion rate drops.  It is during this phase that a crossing of critical luminosity curve (and subsequent explosion) seem most likely, though this does not occur in our models.  As in Fig.~\\ref{fig:le_1d2d}, the 2D models tend to show higher $\\nu_e$ luminosities at late times relative to 1D.  Naively, this might suggest that 2D models are easier to explode not only because the critical curve is lower, but also because 2D models tend to have higher luminosities at a given mass accretion rate.  Of course, this argument neglects other important differences between 1D and 2D including the higher rms neutrino energies in 1D, for example.  Rewriting Eq.~\\ref{eq:tau} for the effective optical depth (or, equivalently, the heating efficiency) with $M_g=\\dot{M} \\tau_{\\rm adv}$ and $R_0\/R_{\\rm in}=\\varepsilon\/\\varepsilon_0$, we arrive at\n\\begin{equation}\n\\frac{\\tau_{\\rm eff}^{\\rm 2D}}{\\tau_{\\rm eff}^{\\rm 1D}} \\sim  \\left(\\frac{\\tau_{\\rm adv}^{\\rm 2D}}{\\tau_{\\rm adv}^{\\rm 1D}}\\right) \\left[ \\left(\\frac{R_s-R_g}{R_s^3-R_g^3}\\right)^{\\rm 2D} \\left(\\frac{R_s^3-R_g^3}{R_s-R_g}\\right)^{\\rm 1D} \\right] \\left(\\frac{\\varepsilon_{\\rm rms}^{\\rm 2D}}{\\varepsilon_{\\rm rms}^{\\rm 1D}}\\right)^2\\;.\n\\end{equation}\nThe first term on the right hand side is the ratio of the advection timescales, which favors 2D.  The second term (in square brackets) favors 1D since the gain volume is typically much larger in 2D.  As we show in Fig.~\\ref{fig:le_1d2d}, the last term given by the square of the ratio of rms neutrino energies favors 1D.  Evidently, though some effects in 2D tend toward lower heating efficiencies, these are overwhelmed by the different advection timescales, leading to higher heating efficiencies in 2D.  Now consider the diagnostic given by the ratio of heating in the gain region to accretion power, which Fig.~\\ref{fig:crit_panel} shows is intimately related to the ratio of advection to heating timescales.  It is easy to show that the 2D to 1D ratio of this diagnostic is\n\\begin{equation}\n\\left(\\frac{H R_s}{G M \\dot{M}}\\right)^{\\rm 2D}\\left\/\\left(\\frac{H R_s}{G M \\dot{M}}\\right)^{\\rm 1D}\\right. \\approx \\left(\\frac{L_{\\nu_e}^{\\rm 2D}}{L_{\\nu_e}^{\\rm 1D}}\\right) \\left(\\frac{\\tau_{\\rm eff}^{\\rm 2D}}{\\tau_{\\rm eff}^{\\rm 1D}}\\right) \\left(\\frac{R_s^{\\rm 2D}}{R_s^{\\rm 1D}}\\right)\\;.\n\\end{equation}\nAt least after the first 100--200$\\,{\\rm ms}$, all of these terms favor 2D, making 2D models much easier to explode than corresponding 1D models.\n\nIn the 1D and 2D models discussed thus far, we have ignored inelastic scattering processes.  Figure~\\ref{fig:inelastic} shows results from two 1D simulations of the $15$-$\\,{\\rm M}_\\odot$ progenitor that differ only in whether inelastic scattering on electrons is included.  The most marked difference is in the rms energy of the $\\nu_x$ species, which is lower with inelastic scattering included by about $5\\,{\\rm MeV}$.  In principle, depositing $5\\,{\\rm MeV}$ per $\\nu_x$ neutrino represents a significant source of heating, but this energy deposition occurs mainly at large optical depths for the $\\nu_e$ and $\\bar{\\nu}_e$ species, so the net effect is substantially muted.  For example, the $\\nu_e$ and $\\bar{\\nu}_e$ luminosities are higher by only 3--4\\% and the rms neutrino energies are higher by only $\\sim$$0.5\\%$ in the model with inelastic scattering.  The $\\nu_x$ luminosity with inelastic scattering adjusts so that the total neutrino luminosity, summed over all species, is nearly identical to the model without inelastic scattering, with fractional differences typically less than $0.1\\%$.  Thus, though inelastic scattering on electrons does lead to small changes in the luminosities and average energies of each species, the net effect on the structure as a whole is quite small.  For example, the shock radii shown in Fig.~\\ref{fig:inelastic} are almost identical, differing by at most $2\\%$ and typically less than $1\\%$.\n\n\n\\section{Angular and Temporal Variations in the Neutrino Sector}\n\\label{angle}\n\n\\subsection{A Criticism of ``Ray-by-ray''}\nOne of the motivations for carrying out this study is to compare our MGFLD neutrino transport results with those obtained with the \\textit{de facto} standard ``ray-by-ray'' formulation currently used by all other groups carrying out radiation hydrodynamic simulations of core-collapse supernovae.  \\citet{Ott08} carried out both MGFLD and full multi-angle transport calculations and found that, though MGFLD is only approximate in the semi-transparent and free streaming regimes, the two methods actually agree quite well in the supernova context.  No such comparison with ``ray-by-ray'' has yet been discussed \\citep[but see][for a comparison of multi-angle and ``ray-by-ray'' transport results based on time-independent snapshots of multidimensional simulations]{Sumi14}.  Here, we take a small step towards this comparison, which captures one of our main criticisms of the technique --- the unphysically high degree of angular and temporal correlation between the matter and neutrino radiation fields.\n\nAn argument often made in the core-collapse community is that the spatio-temporal variations in the flow effectively average out the error made in employing the ``ray-by-ray'' approach \\citep{Bura06a,Mezz14}.  However, both \\citet{Bura06a} and \\citet{Mezz14} acknowledge that such averaging can only be approximate and that artifacts may remain in ``ray-by-ray'' calculations.  For example, \\citet{Bura06a}, in an attempt to address this concern, post-processed their ``ray-by-ray'' models by recomputing the heating in the gain region using an angularly averaged radiation field.  When integrated over the volume of the gain region and also time-averaged, they found good agreement between the ``ray-by-ray'' and angularly-averaged ``ray-by-ray'' heating rates \\citep[see also][]{Sumi14}.  Beyond this zeroth order comparison, however, larger differences begin to emerge.  For example, \\citet{Bura06a} find that the heating of downflows and high-entropy bubbles, when analyzed separately, show differences larger than 10\\% in the time-averaged heating rates as computed by ``ray-by-ray'' and angularly-averaged calculations.  This finding strongly reinforces our concern about the artificially high degree of correlation between matter and neutrino sectors.  Importantly, the most crucial question with regards to ``ray-by-ray,'' how it effects the time-dependent hydrodynamic response of the system, remains open.  We do not attempt to address that here and therefore make no claim that we have proven that ``ray-by-ray'' calculations are wholly wrong.  Instead, we aim only to show some ways in which ``ray-by-ray'' calculations fall short, and to suggest that these shortcomings may make the current crop of supernova modeling efforts in the community difficult to reconcile.\n\nTo that end, we consider a snapshot from our 2D simulation of the $12$-$\\,{\\rm M}_\\odot$ model $604\\,{\\rm ms}$ after bounce (i.e., at the end of the run).  We extract the energy integrated fluxes of the electron neutrinos at $250\\,{\\rm km}$ as computed by CASTRO's MGFLD solver.  With this snapshot, we do two more transport calculations, ignoring velocity- and time-dependence and scattering.  The first calculation gives the full angle- and energy-dependent specific intensities at discrete latitudes at $250\\,{\\rm km}$.  We then integrate over energy and take the first moment to get the fluxes.  Finally, we compute the angle- and energy-dependent specific intensities under the ``ray-by-ray'' assumption, then integrate over energy and take the first moment to recover the ``ray-by-ray'' fluxes.  We describe our techniques for the multi-angle and ``ray-by-ray'' calculations in Appendix~\\ref{appendix}.  The simulation snapshot and the results of these transport calculations are shown in Fig.~\\ref{fig:rbr_comp}.  The MGFLD and full multi-angle transport results agree quite well in the general character of the angular variation, despite the neglect of velocity- and time-dependence and of scattering in the simplified multi-angle calculation.  The variation is smooth and has a small amplitude.  Though the low-$\\ell$ modes have similar amplitudes compared with the other two schemes, the ``ray-by-ray'' results show much more intermediate- to high-$\\ell$ power, manifest as wild variations that close inspection reveals are highly correlated with structures in the flow.  The peak-to-peak variation in the ``ray-by-ray'' result is about a factor of five larger than for MGFLD and about a factor of four larger than for multi-angle transport.  While this may already be a concern, the most serious failing of the ``ray-by-ray'' approach is that these large variations are tightly correlated with hydrodynamic structures along each ray.  The neutrino radiation field is produced by an integral over many sources at depth, an effect which can only be truly captured by doing full multi-angle transport and seems reasonably well approximated by MGFLD, but which a ``ray-by-ray'' approach can never reliably reproduce.  The ``ray-by-ray'' flux depends only on the profile along a given radial ray, introducing large variations which should be washed away by the integral character of transport and which are highly correlated with the hydrodynamic variations of the flow.  We note that our choice of measuring the fluxes at $250\\,{\\rm km}$ was motivated by the desire to include transport through the entire gain region, but choosing a radius far from the neutrinosphere tends to emphasize the differences between the schemes.  However, one must measure the fluxes very close to the neutrinosphere for the methods to show comparable fluctuation, and even then the ``ray-by-ray'' scheme exaggerates the variation, though to a lesser degree than when viewed at $250\\,{\\rm km}$.\n\n\n\\subsection{Angular Variations}\n\\label{angular_variations}\nAs has been discussed previously, the angular variations of the neutrino radiation field are muted relative to variations in the matter sector \\citep{Bran11}.  In Fig.~\\ref{fig:shock_lum_corr}, we show the normalized standard deviations of the shock radii and $\\nu_e$-fluxes (at $500\\,{\\rm km}$) as a function of time.  We define the standard deviation ($\\sigma$) normalized by the mean ($\\mu$) of quantity $Q$ as \n\\begin{equation}\n\\frac{\\sigma}{\\mu} = \\frac{1}{\\langle Q\\rangle}\\left(\\frac{\\int_0^\\pi (Q(\\theta)-\\langle Q(\\theta)\\rangle)^2 \\sin\\theta d\\theta}{\\int_0^\\pi \\sin\\theta d\\theta}\\right)^{1\/2}\\;,\n\\end{equation}\nwhere the angle brackets indicate solid-angle averaging.  Typically, the fractional angular variations in the shock radii are about an order-of-magnitude larger than the corresponding variations in the fluxes.  We also show the cross-correlations of the shock radii and $\\nu_e$-fluxes, defined by\n\\begin{equation}\nCorr(R_s,F_{\\nu_e}) = \\frac{1}{2}\\int_0^\\pi \\frac{(R_s - \\langle R_s \\rangle)(F_{\\nu_e}-\\langle F_{\\nu_e}\\rangle)}{\\sigma_{R_s} \\sigma_{F_{\\nu_e}}} \\sin\\theta d\\theta\\;,\n\\end{equation}\nwhere $\\sigma$ is the standard deviation, computed as above, but without the normalization by the mean.  All the models show a positive correlation on average, as indicated by the gray dashed lines in the figure.  The $20$-$\\,{\\rm M}_\\odot$ and $25$-$\\,{\\rm M}_\\odot$ models show a period $\\sim$$100\\,{\\rm ms}$ long of consistently high correlation.  Interestingly, these phases follow immediately after the accretion of the significant Si\/O interfaces in these models, where the accretion rates drop by a factor $\\approx$2.  Importantly, however, though the shock and $\\nu_e$-fluxes are correlated in all models, the amplitude of the $\\nu_e$-flux variation is much smaller than one would find with a ``ray-by-ray'' calculation, so the degree to which these variations couple may be quite different than in ``ray-by-ray'' calculations \\citep{Burr13}.\n\n\\citet{Tamb14} recently reported systematic asymmetries in the net lepton-number emission in several 3D models and dubbed this feature LESA (Lepton-number Emission Self-sustained Asymmetry).  To investigate whether we see a similar phenomenon, we decompose the lepton-number flux $F^n_{\\nu_e} - F^n_{\\bar{\\nu}_e}$ of our $12$-$\\,{\\rm M}_\\odot$ model ($n$ indicates number instead of energy flux) into spherical harmonics and focus on the evolution of the normalized dipolar component $a_1\/a_0$.  In the model described in \\citet{Tamb14}, deviations from spherical symmetry in the lepton-number flux are intimately related to corresponding deviations in the shock structure.  In the bottom panel of Fig.~\\ref{fig:lesa}, we show the evolutions of $a_1\/a_0$ for the lepton-number flux and the shock surface.  For ease of comparison, we have multiplied $a_1\/a_0$ for the lepton-number flux by five.  We see no sign of a strong correlation between the two quantities.  Indeed, the normalized cross-correlation as a function of time lag shows a broad feature around zero offset, with a modest peak value of $\\sim$0.3 (1 indicates perfectly correlated signals) at $-21\\,{\\rm ms}$, comparable to the advection timescale.  Importantly, the magnitude of the asymmetry is also about an order-of-magnitude smaller than shown in \\citet{Tamb14}.  In short, we find no evidence for LESA in our models, but they are limited to 2D and more work is needed before a final judgment on the existence of LESA can be made.  On the other hand, in our 2D simulations we do find that the dipolar component of the \\textit{sum} $F_{\\nu_e}+F_{\\bar{\\nu}_e}$ (and also the sum of number fluxes) is highly correlated with the dipolar component of the shock, as shown in the top panel of Fig.~\\ref{fig:lesa}.  In this panel, $a_1\/a_0$ of $F_{\\nu_e}+F_{\\bar{\\nu}_e}$ is shown, multiplied by ten for ease of comparison with $a_1\/a_0$ of the shock.\n\n\\subsection{Temporal Variations}\n\\label{temporal}\nAn interesting diagnostic of the temporal variations in the neutrino sector is the power spectrum of the signal expected to be detected from a galactic supernova.  Following \\citet{Lund10}, we compute the event rate expected for the IceCube detector at 256 observer orientations for our $25$-$\\,{\\rm M}_\\odot$ model, assumed to be at a distance of 10 kpc.  We compute the power spectra of these signals and then solid-angle average these to produce the power spectrum shown in Fig.~\\ref{fig:nu_pspec}.  Comparing with Fig.~5 of \\citet{Lund10}, we see remarkable agreement between our power spectrum and the ``north hemispheric average'' reported in that work.  Interestingly, their power spectrum produced by averaging the spectra from each ray, in a manner similar to what we have done, shows significantly more power, particularly at high frequencies.  Their hemispheric averaging is meant to mask the inherent error in their ``ray-by-ray'' transport by mimicking the angular integral character of transport.  Evidently, our MGFLD transport naturally produces power spectra that agree quite nicely with their hemisphere-averaged results.  By contrast, the ``ray-by-ray'' technique clearly leads to significant overestimates of the variability in the quantity $L_{\\bar{\\nu}_e} \\varepsilon_{\\rm rms}^2$, particularly at high frequencies.  This quantity bears directly on the energy deposition in the gain region, and so directly on the viability of the neutrino-driven supernova mechanism.\n\nOne interesting aspect of the power spectrum that was not addressed by \\citet{Lund10} (or the follow-on references \\citealt{Lund12} and \\citealt{Tamb13}) is its exponential shape.  The dashed line in Fig.~\\ref{fig:nu_pspec} is a fit of the power spectrum above $50\\,{\\rm Hz}$ with $P(f)=P_0 \\exp (-4\\pi f\\tau)$, where $P_0$ and $\\tau$ are free parameters.  The resulting fit seems to describe our results quite well, and appears to be consistent with what others have found.  At least one way to produce an exponential power spectrum is with Lorentzian pulses of the form\n\\begin{equation}\nL(t) \\propto \\frac{\\gamma}{(t-t_0)^2 + \\gamma^2}\\;,\n\\end{equation}\nwhere $t_0$ is the time of the pulse and $\\gamma$ is the half-width at half-maximum.  In Fig.~\\ref{fig:lorentz}, we show a fake signal, produced as a constant plus 500 Lorentzian pulses\\footnote{The number of pulses only effects the normalization, not the e-folding timescale of the exponential.} at randomly chosen times and with randomly chosen amplitudes.  The power spectrum, shown below the signal, has an exponential dependence on frequency, with an e-folding timescale of $4\\pi\\gamma$.  So, one interpretation of the exponential power spectrum of our modeled IceCube signal is that the signal is comprised of many approximately Lorentzian pulses with a typical full-width at half-maximum timescale, extracted from our fit, of $2\\tau=2.3\\,{\\rm ms}$.  This represents very rapid variability.  A simplistic picture of how this timescale may emerge is of blobs cooling as they advect through the cooling region.  In this picture, we adopt the simple scaling $L_{\\rm blob} \\varepsilon_{\\rm rms}^2\\sim T^8$ and estimate the timescale $2\\tau\\sim(8 v_r d\\ln T\/dr)^{-1}$, with $v_r$ the radial velocity.  Using the velocity and temperature gradient in the cooling region shown in Fig.~\\ref{fig:rbr_comp}, our estimate yields $2\\tau\\sim2.5\\,{\\rm ms}$, in surprisingly good agreement with the measured value of $2.3\\,{\\rm ms}$.\n\n\n\n\\section{Summary and Conclusions}\n\\label{sum}\n\nUsing our new multi-group, multi-dimensional radiation hydrodynamics code CASTRO, which incorporates all terms to $\\mathcal{O}(v\/c)$ in the transport and does not make the ray-by-ray approximation employed by all other groups now modeling core-collapse supernovae, we have simulated in two spatial dimensions the dynamics of four progenitor massive star models. One goal was to determine, using a different code, whether the outcome of our previous simulations using the VULCAN\/2D methodology \\citep{Burr06,Burr07,Ott08} depended upon the absence of the $\\mathcal{O}(v\/c)$ terms in VULCAN\/2D. We have determined that the results are qualitatively the same and, as when employing VULCAN\/2D, we do not see explosions by the neutrino heating mechanism after $\\sim$600 milliseconds after bounce. Both codes perform two-dimensional transport, though using a multi-group flux-limited (MGFLD) formulation.  This conclusion concerning the overall outcome of these models (i.e., explosion in 2D, driven by neutrino heating) is in contrast with the results of \\citet{Brue13} and Janka et al. (Nuclear Astrophysics Workshop, Ringberg Castle, 2014), who also do not agree one with the other, but who do obtain neutrino-driven explosions in some or all of their 2D simulations.\n\nOne is left to ponder the reasons for these remaining differences in the community of researchers engaged in detailed simulations of the core-collapse supernova phenomenon. We have demonstrated that the ray-by-ray approach does not reproduce the correct angular and temporal neutrino field variations, though no one has yet performed the head-to-head ray-by-ray versus correct transport comparisons needed to definitively clarify the impact of the ray-by-ray approximation. We speculate, however, that the combination of the ray-by-ray approach with the artificiality of the axial sloshing effects manifest in 2D simulations may be the reason the groups using ray-by-ray obtain explosions in 2D (when they do).\n\nWhile the ray-by-ray approximation is clearly suspect, there are other differences that may prove to play an important role in producing the range of findings in the community.  One might suspect that differences in the neutrino interaction physics may play an important role, but our experimentation indicates that the numerous hydrodynamic, thermal, and radiative feedbacks in the core-collapse problem mute the effects of even large changes in the neutrino-matter cross sections and associated emissivities on the dynamic evolution after collapse.  In 1D test calculations we have performed in which the $\\nu_{e}$$-$neutron absorption cross section was changed by a factor of two (both increased and decreased), the resulting stalled shock radii were the same to within a few percent.  Some recent calculations suggest there may be some sensitivity to the choice of equation of state (EOS), with calculations using the Lattimer and Swesty EOS tending to explode more easily than those using the Shen EOS \\citep{Janka12,Suwa13,Couch13}.  Since both the present study and the VULCAN\/2D studies used the Shen EOS and failed to explode, it may prove illuminating to repeat some of these calculations with the Lattimer and Swesty EOS.  The effects of general relativity (GR) and the differing fidelity with which they are included in calculations may also contribute \\citep[e.g.][]{Mull12}, but note that GR seems not to be generally requisite for explosions, as demonstrated by the 2D $27$-$\\,{\\rm M}_\\odot$ models reported in \\citet{Mull12_sasi} that included GR and in \\citet{Hank13} that used a monopolar gravity approximation with mock GR corrections, nevertheless transitioning to explosion at nearly the same post-bounce time.  The marked difference between 1D and 2D in the early evolution of the shock radius in our models, which we attribute to a vigorous burst of prompt convection seeded by perturbations from our aspherical grid, may also be a concern, but we would expect the memory of this defect to be lost within a few dynamical times ($<100\\,{\\rm ms}$) as the system dynamically relaxes to a quasi-steady configuration.  Differences in the transport algorithms (apart from the ray-by-ray versus multi-D transport issue) could be to blame, and code-to-code comparisons are called for.  This was one early motivation for embarking upon this study with CASTRO---to see whether the outcomes were different from those we obtained using VULCAN\/2D. But, more inter-group comparisons, not just intra-group comparisons, are needed.\n\nThe fact that the 3D simulations of the Garching group are not exploding when they were in 2D \\citep{Hank13} should be a wake-up call to the community to determine the origins of these differences between the various simulation groups and between 2D and 3D.  As we have suggested, the use of the ray-by-ray approach is dubious, and since its artificial character is more manifest in 2D we suspect that it is part of the problem. However, this does not explain the current conundrum in 3D---something else may be amiss. It could be that the progenitor models are to blame and a new generation of such models, performed in 3D at the terminal stages of a massive star's life are needed \\citep{Meak11}. It could be that rotation, even the modest rotation expected from the pulsar injection constraint \\citep{Emme89}, could, by the resultant centrifugal support and consequent expected boost in the stalled shock radius, convert duds into explosions.  This is the simplest solution, and one is reminded that the exploding model of \\citet{Mare09} was rotating. Both large-scale and turbulent magnetic fields could play a role, through the associated stress, but also due to enhanced angular momentum transport from the core to the mantle \\citep[e.g.][]{Sawa14}.  However, without very rapid rotation, which might be associated with the rare hypernovae \\citep{Burr07mhd} that serve as a bridge to the collapsar model of long-soft gamma-ray bursts, there would not seem to be enough extra free energy to power explosion generically.  Perturbations of the progenitor cores that collapse have never been properly included into supernova theory, and might be a fruitful line of investigation \\citep{Couc13}. Such perturbations seed the instabilities long identified with more robust dynamics and the viability of the delayed neutrino mechanism.\n\nWhatever the solution to this recalcitrant problem, advances in the numerical arts seem destined to play a central role.  Approximations have been made by all groups to accommodate the limitations of the available computer resources, leaving one to wonder whether such compromises have corrupted the results.  One would hope that simple, compelling reasoning, and physical insight could in the end lead to a solution.  This has happened before in astrophysics. However, the complexity of the dynamics, the fact that the explosion energy is a small fraction of the available energy, and the circumstance that the central ``engine'' is shrouded in mystery by the profound opacity of the stellar envelope, and, hence, is itself (almost) inaccessible to direct observation or measurement, may mitigate against a breakthrough unaided by computation.\n\n\n\\acknowledgments\n\nThe authors acknowledge conversations and collaborations with Jeremiah Murphy, \nChristian Ott, Stan Woosley, Ann Almgren, John Bell, and Louis Howell.\nThe development of the CASTRO code was supported by the Scientific Discovery through\nAdvanced Computing (SciDAC) program of the DOE, under grant number DE-FG02-08ER41544,\nthe NSF under the subaward no. ND201387 to the Joint Institute for Nuclear Astrophysics (JINA, NSF PHY-0822648),\nand the NSF PetaApps program, under award OCI-0905046 via a subaward\nno. 44592 from Louisiana State University to Princeton University.\nThe authors employed computational resources provided by the TIGRESS\nhigh performance computer center at Princeton University, which is jointly supported by the Princeton\nInstitute for Computational Science and Engineering (PICSciE) and the Princeton University Office of\nInformation Technology; by the National Energy Research Scientific Computing Center\n(NERSC), which is supported by the Office of Science of the US Department of\nEnergy under contract DE-AC03-76SF00098; and on the Kraken supercomputer,\nhosted at NICS and provided by the National Science Foundation through\nthe TeraGrid Advanced Support Program under grant number TG-AST100001.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\n\nNonlinear parabolic equations of the form \n\\begin{equation} \\label{eq:pde}\n\\partial u\/\\partial t = \\nabla\\cdot \\bigl(D(u,\\nabla u)\\nabla u\\bigr)\\quad\\text{on }\\Omega\\times (0,T),\n\\end{equation}\nequipped with suitable boundary and initial conditions, are frequently encountered in applications. If the diffusion constant $D(u,\\nabla u)$ vanishes for some values of $u$ and $\\nabla u$, i.e., the equation is degenerate, one obtains a quite different dynamics compared to the linear case. The two main nonlinear features are finite speed of propagation and the absence of parabolic smoothening of the solution. Concrete applications can, e.g., be found when modelling gas flow through porous media, phase transitions and population dynamics. A survey of such applications is given in \\cite[Section~1.3 and Chapter~2]{Vasquez.2007}. In order to keep the presentation as clear-cut as possible, we will mostly ignore the presence of lower-order advection and reactions terms. \n\nApproximating the solution of a partial differential equation typically results in large-scale computations, which require the usage of parallel and distributed hardware. One possibility to design numerical schemes that make use of such hardware is to decompose the equation's domain into a family of subdomains. The domain decomposition method then consists of an iterative procedure where, in every step, the equation is solved independently on each subdomain and the resulting solutions are thereafter communicated to the adjacent subdomains. This independence of the decomposed equations and the absence of global communication enables the parallel and distributed implementation of domain decomposition methods. For linear parabolic equations the common procedure is to first discretize the equation in time by a standard implicit integrator. Then an elliptic equation on $\\Omega$ is obtained in every time step, which is iteratively solved by a domain decomposition based discretization. We refer to the monographs~\\cite{Mathew.2008,QuarteroniValli.1999,ToselliWidlund.2005} for an in-depth treatment of this approach.  Another possibility is to apply the domain decomposition method to the full space-time domain $\\Omega\\times (0,T)$, which leads to an iterative procedure over parabolic problems that can be parallelized both in space and time; see, e.g., \\cite{Gander.1999,GanderHalpern.2007,GiladiKeller.2002}. \n\nWhen considering nonlinear parabolic problems one finds that there are hardly any results concerning the analysis of domain decomposition based schemes. Two exceptions are the papers~\\cite{KimEtal.2000,Lapin.1991}, where domain decomposition schemes are analyzed for non-degenerate quasilinear parabolic equations and the degenerate two-phase Stefan problem, respectively. The lack of results in the context of degenerate equations is rather surprising from a practical point of view, as the equations' finite speed of propagation is ideal for applying domain decomposition strategies. For example, a solution that is initially zero in parts of the domain $\\Omega$ will in each time step only propagate to a small number of neighboring subdomains, which limits the computational work considerably. However, from a theoretical perspective the lack of convergence results is less surprising. The issue is that the standard domain decomposition schemes all link together the equations on the subdomains via boundary conditions. As the solutions of degenerate parabolic equations typically lack higher-order regularity, making sense of such boundary linking is, at the very least, challenging.\n\n\\begin{figure}\n \\centering \n \\includegraphics[scale=0.6]{Fig1.pdf}\n \\includegraphics[scale=0.6]{Fig2.pdf}\n\\caption{Examples of overlapping domain decompositions $\\{ \\Omega_{\\ell} \\}_{\\ell =1}^{s}$ of a domain $\\Omega\\subset \\mathbb{R}^{2}$,\nwith $s=4$ subdomains (left) and $s=2$ subdomains that are further decomposed into families of pairwise disjoint sets (right), respectively.}\n\\label{fig:dom}\n\\end{figure}\n\nIn order to remedy this, we propose to directly introduce the domain decomposition in the time integrator via an operator splitting procedure. More precisely, let~$\\{ \\Omega_{\\ell} \\}_{\\ell =1}^{s}$ be an overlapping decomposition of the spatial domain $\\Omega$, as exemplified in Figure~\\ref{fig:dom}. On these subdomains we introduce the partition of unity $\\{ \\chi_{\\ell} \\}_{\\ell =1}^{s}$ and the operator decomposition, or splitting, \n\\begin{equation} \\label{eq:opdecomp}\nfu= \\nabla\\cdot \\bigl(D(u,\\nabla u)\\nabla u\\bigr)= \\sum_{\\ell=1}^{s}\\nabla\\cdot \\bigl(\\chi_{\\ell} D(u,\\nabla u)\\nabla u\\bigr) = \\sum_{\\ell=1}^{s} f_{\\ell}u.\n\\end{equation}\nTwo possible (formally) first-order integrators are then the sum splitting \n\\begin{equation}\\label{schemes:sumpre}\n\\left\\{\n\\begin{aligned}\n&v_{\\ell}  =u_{n}+ sh f_{\\ell} v_{\\ell}, \\quad \\ell=1,\\ldots, s,\\\\\n& u_{n+1} = \\sfrac1s\\sum_{\\ell=1}^{s} v_{\\ell},\n\\end{aligned}\n\\right.\n\\end{equation}\nwhich represents a ``quick and dirty'' scheme that is straightforward to parallelize, and the Lie splitting \n\\begin{equation}\\label{schemes:liepre}\n\\left\\{\n\\begin{aligned}\n& v_{0}     =u_{n},\\\\\n& v_{\\ell}   =v_{\\ell-1}+ h f_{\\ell} v_{\\ell}, \\quad \\ell=1,\\ldots, s,\\\\\n& u_{n+1} = v_{s},\n\\end{aligned}\n\\right.\n\\end{equation}\nwhich is usually more accurate but requires a further partitioning of the subdomains $\\Omega_\\ell$ in order to enable parallelization, as illustrated in Figure~\\ref{fig:dom}. In contrast to the earlier domain decomposition based schemes, where an iterative procedure is required with possibly many instances of boundary communications, one time step of either splitting scheme only needs the solution of $s$ elliptic equations together with the communication of the data related to the overlaps. Similar splitting schemes have, e.g., been considered in the papers \\cite{Arraras.2015,Hansen.2016,Mathew.1998,Vabishchevich.2013} when applied to linear, and to some extent semilinear, parabolic problems. However, there does not seem to be any analysis applicable to degenerate, or even quasilinear, parabolic equations in the literature.\n\nHence, the goal of this paper is twofold. First, we aim to derive a new energetic, or variational, framework that allows a proper interpretation of the operator decomposition~\\eqref{eq:opdecomp} for two commonly occurring families of degenerate parabolic equations. These are the $p$-Laplace type evolutions, where the prototypical example is given by $D(u,\\nabla u)=|\\nabla u|^{p-2}$, and the porous medium type equations, where $D(u,\\nabla u)=(p-1)|u|^{p-2}$ in the simplest case. For the porous medium application we will use the strategic reformulation \n\\begin{equation*} \nfu=\\Delta \\alpha (u)=\\sum_{\\ell=1}^{s}\\Delta \\bigl(\\chi_{\\ell}\\alpha (u)\\bigr)=\\sum_{\\ell=1}^{s} f_{\\ell}u\n\\end{equation*}\nof the decomposition~\\eqref{eq:opdecomp}, in order to enable an energetic interpretation. \n\nSecondly, we will strive to obtain a general convergence analysis for the domain decomposition based time integrators, including the sum and Lie splitting schemes. The main idea of the convergence analysis is to introduce the nonlinear Friedrich extensions of the operators $f$ and $f_{\\ell}$, via our new abstract energetic framework, and then to employ a Lax-type result from the nonlinear semigroup theory~\\cite{BrezisPazy.1972}.\n\n\\section{Function spaces}\\label{sec:func}\n\nThroughout the analysis $\\Omega \\subset \\mathbb{R}^d$, $d\\geq 1$, will be an open, connected and bounded set and the parameter $p\\in (1,\\infty)$ is fixed. Next, let $\\{ \\Omega_{\\ell} \\}_{\\ell =1}^{s}$ be a family of overlapping subsets of $\\Omega$ such that $\\bigcup_{\\ell =1}^s \\Omega_{\\ell} = \\Omega$ holds. Here, each $\\Omega_{\\ell}$ is either an open connected set, or a union of pairwise disjoint open, connected sets $\\Omega_{\\ell,k}$ such that $\\bigcup_{k=1}^{r}\\Omega_{\\ell,k} = \\Omega_{\\ell}$. On $\\{ \\Omega_{\\ell} \\}_{\\ell =1}^{s}$ we introduce the partition of unity $\\{ \\chi_{\\ell} \\}_{\\ell =1}^{s}\\subset C^{\\infty}(\\Omega)$ such that\n\\begin{align*}\n\\chi_{\\ell} (x)>0\\text{ for all }x\\in\\Omega_{\\ell},\\quad \\chi_{\\ell} (x) = 0\\text{ for all }x\\in\\Omega\\setminus\\Omega_{\\ell}\\quad \\text{and} \\quad \\sum_{\\ell =1}^{s} \\chi_{\\ell}= 1.\n\\end{align*} \nFor details on the construction of explicit domain decompositions $\\{ \\Omega_{\\ell} \\}_{\\ell =1}^{s}$ and partitions of unity $\\{ \\chi_{\\ell} \\}_{\\ell =1}^{s}$ we refer to \\cite[Section~3.2]{Arraras.2015} and \\cite[Section~4.1]{Mathew.1998}.\n\nThe related weighted Lebesgue space $L^p(\\Omega_{\\ell},\\chi_{\\ell})$ can now be defined as the set of all measurable functions $u$ on $\\Omega_{\\ell}$ such that the norm\n\\begin{align*}\n\\|u\\|^p_{L^p(\\Omega_{\\ell},\\chi_{\\ell})} = \\int_{\\Omega_{\\ell}}\\chi_{\\ell} |u|^p \\,\\mathrm{d}x\n\\end{align*}\nis finite. The space $L^p(\\Omega_{\\ell},\\chi_{\\ell})$ is a reflexive Banach space, which follows by observing that the map $G : L^p(\\Omega_{\\ell},\\chi_{\\ell}) \\to L^p(\\Omega_{\\ell}):u\\mapsto \\chi_{\\ell}^{\\nicefrac 1p} u$ is an isometric isomorphism \\cite[Chapter~1]{DrabekEtAl.1997}. We will also make frequent use of the product space $L^p(\\Omega_{\\ell},\\chi_{\\ell})^k$, equipped with the norm\n\\begin{align*}\n\\|(u_1,\\ldots,u_{k})\\|_{L^p(\\Omega_{\\ell},\\chi_{\\ell})^{k}}^{p}= \\int_{\\Omega_{\\ell}}\\chi_{\\ell} |(u_1,\\ldots,u_{k})|^p \\,\\mathrm{d}x,\n\\end{align*}\nwhich is again a reflexive Banach space~\\cite[Theorem 1.23]{AdamsFournier.2003}. \n\nNext, let $\\left(H, \\ska{\\cdot}{\\cdot}_{H}\\right)$ be a real Hilbert space and denote the space of distributions on \n$\\Omega$  by $\\mathcal{D}'(\\Omega)$. For a given $k\\geq 1$ we introduce the linear operator\n\\begin{align*}\n\\delta:  H \\to \\mathcal{D}'(\\Omega)^k,\n\\end{align*}\nwhich is assumed to be continuous in the following fashion.\n\\begin{ass}\\label{ass:1}\nIf $\\lim_{n\\to\\infty} u_{n}=u$ in $H$ then, \nfor $j=1,\\ldots,k$,\n\\begin{equation*}\n\\lim_{n\\to\\infty} (\\delta u_n )_j(\\phi) = (\\delta u)_{j}(\\phi)\\quad \\text{in } \\mathbb{R}\\quad \\text{for all } \\phi \\in C_0^{\\infty}(\\Omega).\n\\end{equation*}\n\\end{ass}\nAs the regularity of the weights $\\chi_{\\ell}$ implies that $\\chi_{\\ell}\\phi \\in C^{\\infty}_0(\\Omega_\\ell)$ for all $\\phi \\in C^{\\infty}_0(\\Omega)$, we can define the product $\\chi_{\\ell}\\delta u$ by \n\\begin{align*}\n(\\chi_{\\ell}\\delta u)_{j}(\\phi)=(\\delta u)_{j}(\\chi_{\\ell}\\phi)\\quad \\text{for all } \\phi \\in C^{\\infty}_0(\\Omega).\n\\end{align*}\nWith this in place we can introduce our energetic spaces $V$ and $V_{\\ell}$ as subspaces of $H$ given by\n\\begin{align*}\nV &= \\Bigl\\{ u \\in H:\\text{ there exists a } v_j\\in L^p(\\Omega) \\text{ such that }\\\\\n& \\qquad \\qquad  (\\delta u )_j(\\phi) = \\int_{\\Omega} v_j \\phi \\,\\mathrm{d}x\\quad \\text{for all } \\phi \\in C^{\\infty}_0(\\Omega),\\ j=1,\\dots,k  \\Bigr\\}\n\\end{align*}\nand\n\\begin{align*}\nV_{\\ell} &= \\Bigl\\{ u \\in H:\\text{ there exists a } v_j\\in L^p(\\Omega_{\\ell},\\chi_{\\ell})\\text{ such that }\\\\\n& \\qquad \\qquad (\\chi_{\\ell}\\delta u )_j(\\phi) = \\int_{\\Omega_{\\ell}} v_j \\chi_{\\ell} \\phi \\,\\mathrm{d}x\\quad\\text{for all } \\phi \\in C^{\\infty}_0(\\Omega),\\ j=1,\\dots,k   \\Bigr\\}, \n\\end{align*}\nrespectively. On the energetic spaces we consider the operators\n\\begin{align*}\n\\delta_{p}: V \\subseteq H \\to L^p(\\Omega)^k \\quad \\text{ and } \\quad \n\\delta_{p, \\ell}: V_{\\ell} \\subseteq H \\to L^p(\\Omega_{\\ell},\\chi_{\\ell})^k,\n\\end{align*}\nwhere $\\delta_{p}$ maps $u\\in V$ to the corresponding $L^p(\\Omega)$ functions that $\\delta u$ can be represented by, and $\\delta_{p,\\ell}$ maps $u\\in V_{\\ell}$ to the corresponding $L^p(\\Omega_{\\ell},\\chi_{\\ell})$ functions that $\\chi_{\\ell}\\delta u$ can be represented by, respectively.\n\n\\begin{lemma}\\label{lem:Vintersec}\n$V = \\bigcap_{\\ell =1}^s V_{\\ell}$.\n\\end{lemma}\n\n\\begin{proof}\nFor an arbitrary $u \\in V$ it follows, for $\\ell = 1,\\dots,s$,  that \n\\begin{align*}\n(\\chi_{\\ell}\\delta u)_j (\\phi) = (\\delta u)_j (\\chi_{\\ell}\\phi) = \\int_{\\Omega} (\\delta_p u)_j \\chi_{\\ell}\\phi \\,\\mathrm{d}x\n\\end{align*}\nfor every $\\phi \\in C_0^{\\infty}(\\Omega)$ and $j = 1,\\dots,k$. As $(\\delta_p u)_j|_{\\Omega_{\\ell}} \\in L^p(\\Omega_{\\ell}) \\subseteq L^p(\\Omega_{\\ell},\\chi_{\\ell} )$, we have a representation of $(\\delta u)_j$ in $L^p(\\Omega_{\\ell},\\chi_{\\ell} )$, i.e., $u \\in V_{\\ell}$ for every $\\ell = 1,\\dots, s$. Hence, $V \\subseteq \\bigcap_{\\ell =1}^s V_{\\ell}$. \n\t\t\nNext, assume that $u\\in\\bigcap_{\\ell =1}^s V_{\\ell}$. Then we can write \n\\begin{align*}\n(\\delta u)_j (\\phi) = (\\delta u)_j \\bigl(\\sum_{\\ell =1}^{s} \\chi_{\\ell} \\phi \\bigr) \n\t                   = \\sum_{\\ell =1}^{s} (\\delta u)_j \\left(  \\chi_{\\ell} \\phi \\right) \n\t                   = \\sum_{\\ell =1}^{s} \\int_{\\Omega_{\\ell} }(\\delta_{p,\\ell} u)_j   \\chi_{\\ell} \\phi \\,\\mathrm{d}x\n\\end{align*}\nfor every $\\phi \\in C_0^{\\infty}(\\Omega)$ and $j =1,\\dots,k$. Let $w_{\\ell,j}$ be the zero extension of $(\\delta_{p,\\ell} u)_j$ to the whole of $\\Omega$. We can then define the measurable function \n$v_{j}$ on $\\Omega$ as $v_j = \\sum_{\\ell =1}^{s} \\chi_{\\ell} w_{\\ell,j}$, which satisfies\n\\begin{align*}\n(\\delta u)_j (\\phi)= \\int_{\\Omega} v_j  \\phi \\,\\mathrm{d}x\\quad\\text{for all }\\phi \\in C_0^{\\infty}(\\Omega).\n\\end{align*}\nFurthermore, the $L^{p}(\\Omega)$ norm of $v_{j}$ can be bounded by\n\\begin{align*}\n\\|v_j\\|_{L^p(\\Omega)} \n\t\\leq \\sum_{\\ell=1}^{s} \\bigl(\\int_{\\Omega_{\\ell} } \\chi_{\\ell}^p \\left|(\\delta_{p,\\ell} u)_j\\right|^p \\,\\mathrm{d}x\\bigr)^{\\nicefrac{1}{p}}\n\t\\leq \\sum_{\\ell=1}^{s}  \\|\\chi_{\\ell}\\|_{L^\\infty(\\Omega_{\\ell})}^{\\nicefrac{(p-1)}{p}} \\left\\|(\\delta_{p,\\ell} u)_j\\right\\|_{L^p(\\Omega_{\\ell}, \\chi_{\\ell})}.\n\\end{align*}\nThis yields that $(\\delta_p u)_j = v_j \\in L^p(\\Omega)$ for $j =1,\\dots,k$, i.e., $u\\in V$ and we thereby have the identification $V=\\bigcap_{\\ell =1}^s V_{\\ell}$. \\qed\n\\end{proof}\n\n\\begin{lemma}\nIf Assumption~\\ref{ass:1} holds, then the operators $\\delta_p$ and $\\delta_{p,\\ell}$, $\\ell=1,\\dots,s$, are linear and closed.\n\\end{lemma}\n\n\\begin{proof}\nThe linearity of the operators is clear, since $\\delta$ is a linear operator.  \nLet the sequence $\\seq{u}\\subset V_{\\ell}$ satisfy\n\\begin{align*}\n\\lim_{n\\to\\infty} u_{n}=u\\quad\\text{in }H \\quad \\text{and} \\quad\n\\lim_{n\\to\\infty} \\delta_{p,\\ell} u_n=v\\quad\\text{in }L^p(\\Omega_{\\ell},\\chi_{\\ell})^k.\n\\end{align*} \nAssumption~\\ref{ass:1} then yields that \n\\begin{align*}\n(\\chi_{\\ell} \\delta u)_j (\\phi) = \\lim_{n\\to \\infty} (\\delta u_n)_j (\\chi_{\\ell}\\phi)\t\n= \\lim_{n\\to \\infty } \\int_{\\Omega_{\\ell}} (\\delta_{p,\\ell} u_n)_j \\chi_{\\ell} \\phi\t \\,\\mathrm{d}x \n= \\int_{\\Omega_{\\ell}} v_j \\chi_{\\ell} \t\\phi \\,\\mathrm{d}x\n\\end{align*}\nfor every $\\phi \\in C_0^{\\infty}(\\Omega)$ and $j=1,\\dots,k$. Hence, $(\\chi_{\\ell}\\delta u)_j$ can be represented by the \n$L^p(\\Omega_{\\ell},\\chi_{\\ell})$ function $v_j$, i.e., $\\delta_{p,\\ell} u = v$ holds and the operator $\\delta_{p, \\ell}$ is therefore closed. The closedness of $\\delta_{p}$ follows by the same line of reasoning. \n\\qed\n\\end{proof}\n\nOn the energetic spaces $V$ and $V_{\\ell}$, $\\ell =1,\\dots,s$, we define the norms\n\\begin{align*}\n\\|\\cdot\\|_{V}= \\|\\cdot\\|_H + \\|\\delta_{p}\\cdot\\|_{L^p(\\Omega)^k}\\quad\\text{and}\\quad\n\\|\\cdot\\|_{V_{\\ell}}= \\|\\cdot\\|_H + \\| \\delta_{p, \\ell}\\cdot \\|_{L^p(\\Omega_{\\ell},\\chi_{\\ell})^k},\n\\end{align*}\nrespectively. \n\n\\begin{lemma}\nIf Assumption~\\ref{ass:1} holds, then the spaces $(V, \\|\\cdot\\|_V)$ and $(V_{\\ell}, \\|\\cdot\\|_{V_{\\ell}})$, $\\ell =1,\\dots,s$, are reflexive Banach spaces.\n\\end{lemma}\n\n\\begin{proof}\nConsider the reflexive Banach space $X=H\\times L^p(\\Omega_{\\ell},\\chi_{\\ell})^k$,\nequipped with the norm $\\|(u_{1},u_{2})\\|_{X}=\\|u_{1}\\|_{H}+\\|u_{2}\\|_{L^p(\\Omega_{\\ell},\\chi_{\\ell})^k}$,\nand introduce the linear and isometric operator\n\\begin{align*}\nG:V_{\\ell}\\to X:u\\mapsto (u,\\delta_{p,\\ell}u).\n\\end{align*}\nThe graph of the closed operator $\\delta_{p,\\ell}$ coincides with the image $G(V_{\\ell})$, which makes $G(V_{\\ell})$ a closed linear subset of $X$. Here, $(G(V_{\\ell}),\\|\\cdot\\|_{X})$ is a reflexive Banach space \\cite[Theorem 1.22]{AdamsFournier.2003} and, as $G$ is isometric, it is isometrically isomorphic to $(V_{\\ell},\\|\\cdot\\|_{V_{\\ell}})$. Hence, the latter is also a reflexive Banach space. The same line of argumentation yields that $V$ is a reflexive Banach space.\n\\qed\n\\end{proof}\n\nHereafter, we will assume the following.\n\\begin{ass}\\label{ass:2}\nThe set $V$ is dense in $H$.\n\\end{ass}\nUnder this assumption it also holds that $V_{\\ell}$ is a dense subsets of $H$. By the construction of the energetic norms, one then obtains that the reflexive Banach spaces $(V, \\|\\cdot\\|_V)$ and $(V_{\\ell}, \\|\\cdot\\|_{V_{\\ell}})$ are densely and continuously embedded in $H$ and we have the following Gelfand triplets\n\\begin{align*}\nV \\overset{d}{\\hookrightarrow} H \\cong H^* \\overset{d}{\\hookrightarrow}  V^*\n\\quad \\text{and}\\quad \nV_{\\ell} \\overset{d}{\\hookrightarrow} H \\cong H^* \\overset{d}{\\hookrightarrow} V^*_{\\ell}.\n\\end{align*}\nHere, the density of  $H^*$ in $V^*$ and $V^*_{\\ell}$, respectively, follows, e.g., by~\\cite[Bemerkung I.5.14]{GGZ.1974}. For future reference, we denote the dual pairing between a Banach space $X$ and its dual $X^*$ by $\\dualX{\\cdot}{\\cdot}$, \nand the Riesz isomorphism from $H$ to $H^{*}$ by\n\\begin{align*}\n\\gamma: H \\to H^*: u\\mapsto \\ska{u}{\\cdot}_{H}.\n\\end{align*}\nHere, the Riesz isomorphism satisfies the relations\n\\begin{align*}\n\\dualV{\\gamma u}{v}=\\ska{u}{v}_{H}\\quad\\text{and}\n\\quad\\dualVi{\\gamma u}{v_{\\ell}}=\\ska{u}{v_{\\ell}}_{H}\n\\end{align*}\nfor all $u\\in H$, $v\\in V$ and $v_{\\ell}\\in V_{\\ell}$. \n\n\\begin{remark}\\label{rem:hm1}\nThroughout the derivation of the energetic framework we have assumed that the partition of unity $\\{ \\chi_{\\ell} \\}_{\\ell =1}^{s}$ consists of elements in $C^{\\infty}(\\Omega)$. This is somewhat restrictive from a numerical point of view, but this regularity is required if nothing else is known about the operator $\\delta: H \\to \\mathcal{D}'(\\Omega)^k$. Fortunately, in concrete examples; see Sections~\\ref{sec:pLap} and~\\ref{sec:por}, one commonly has that $\\delta(H)\\subseteq H^{-1}(\\Omega)^k$. If we then choose a partition of unity $\\{ \\chi_{\\ell} \\}_{\\ell =1}^{s}$ in $W^{1,\\infty}(\\Omega)$, we have the property that $\\chi_{\\ell}\\phi\\in H^{1}_{0}(\\Omega)$  for every $\\phi\\in H^{1}_{0}(\\Omega)$, and we can once more derive the above energetic setting by testing with functions $\\phi$ in $H^{1}_{0}(\\Omega)$, instead of in $C^{\\infty}_{0}(\\Omega)$.\n\\end{remark}\n\n\\section{Energetic extensions of the vector fields}\\label{sec:enform}\n\nWith the function spaces in place, we are now able to define the general energetic extensions of our vector fields.\n\\begin{ass}\\label{ass:3}\nFor a fixed $p\\in(1,\\infty)$, let $\\alpha: \\Omega \\times \\mathbb{R}^k \\to \\mathbb{R}^k$ fulfill the properties below.\n\\begin{itemize}\n\\item[$\\alpha_{1})$] The map $\\alpha: \\Omega \\times \\mathbb{R}^k \\to \\mathbb{R}^k$ fulfills the Carath\\'{e}odory condition, i.e., $z \\mapsto \\alpha(x,z)$ is continuous for a.e.\\ $x\\in \\Omega$ and $x \\mapsto \\alpha(x,z)$ is measurable for every $z\\in \\mathbb{R}^k$.\n\\item[$\\alpha_{2})$] The growth condition $|\\alpha(x,z)| \\leq c_1 |z|^{p-1} +c_2(x) $ holds for a.e.\\ $x\\in \\Omega$ and every $z\\in \\mathbb{R}^k$, where $c_1>0$ and $c_2\\in L^{\\nicefrac{p}{(p-1)}}(\\Omega)$ is nonnegative.\n\\item[$\\alpha_{3})$] The map $\\alpha$ is monotone, i.e., for every $z,\\tilde{z} \\in \\mathbb{R}^k$ and a.e.\\ $x\\in \\Omega$ the inequality $(\\alpha(x,z) - \\alpha(x,\\tilde{z}))\\cdot(z - \\tilde{z}) \\geq 0 $ holds. \n\\item[$\\alpha_{4})$] The map $\\alpha$ is coercive, i.e., there exists $c_3>0$ and $c_4\\in L^1(\\Omega)$ such that for every $z\\in \\mathbb{R}^k$ and a.e.\\ $x\\in \\Omega$ the condition $\\alpha(x,z) \\cdot z \\geq c_3 |z|^p - c_4(x)$ holds.\n\\end{itemize}\n\\end{ass}\nCompare with~\\cite[Section 26.3]{Zeidler.1989}.\n\nWe introduce the full energetic operator $F : V \\to V^*$ as\n\\begin{align*}\n\\dualV{Fu}{v} = \\int_{\\Omega} \\alpha(\\delta_p u ) \\cdot \\delta_p v \\,\\mathrm{d}x\\quad\\text{for }u,v\\in V.\n\\end{align*}\nThe operator $F$ is well defined, as $\\delta_p v \\in L^p(\\Omega)^k$ for $v\\in V$ and by ($\\alpha_{2}$) \nwe obtain that $\\alpha(\\delta_p v) \\in  L^{\\nicefrac{p}{(p-1)}}(\\Omega)^k \\cong \\left(L^{p}(\\Omega)^k\\right)^*$.\nFurthermore, we define the decomposed energetic operators $F_{\\ell} : V_{\\ell} \\to V^*_{\\ell}$, $\\ell =1,\\dots, s$, by\n\\begin{align*}\n\\dualVi{F_{\\ell} u}{v} = \\int_{\\Omega_{\\ell}} \\chi_{\\ell} \\alpha(\\delta_{p, \\ell} u ) \\cdot \\delta_{p, \\ell}v \\,\\mathrm{d}x\\quad\\text{for all }u,v\\in V_{\\ell}.\n\\end{align*}\nThese operators are well defined, as\n\\begin{align*}\n|\\dualVi{F_{\\ell} u& }{v}| \n\\leq \\int_{\\Omega_{\\ell}} \\chi_{\\ell} (c_{1}|\\delta_{p, \\ell} u |^{p-1} +c_{2}) |\\delta_{p, \\ell}v|\\,\\mathrm{d}x\\\\\n&\\leq \\bigl(c_{1}\\bigl(\\int_{\\Omega_{\\ell}} \\chi_{\\ell}|\\delta_{p, \\ell}u|^p\\,\\mathrm{d}x\\bigr)^{\\nicefrac{(p-1)}{p}}\n                +\\bigl(\\int_{\\Omega_{\\ell}} \\chi_{\\ell}c_{2}^{\\nicefrac{p}{(p-1)}}\\,\\mathrm{d}x\\bigr)^{\\nicefrac{(p-1)}{p}}\\bigr)\n\\bigl(\\int_{\\Omega_{\\ell}} \\chi_{\\ell} |\\delta_{p, \\ell}v|^p \\,\\mathrm{d}x\\bigr)^{\\nicefrac{1}{p}}\n\\end{align*}\nis finite for every $u,v\\in V_{\\ell}$, due to ($\\alpha_{2}$). This family of operators is a decomposition of $F$, as it fulfills\n\\begin{align*}\n\\dualV{Fu}{v}=\\sum_{\\ell =1}^{s} \\dualVi{F_{\\ell}u}{v}\\quad\\text{for all }u,v\\in V.\n\\end{align*}\nWe can now derive the basic properties of the energetic operators.\n\n\\begin{lemma}\\label{lem:energy}\nIf the Assumptions \\ref{ass:1}--\\ref{ass:3} hold and $h>0$ , then the operators $\\gamma + hF: V \\to V^*$ and \n$\\gamma+ hF_{\\ell}: V_{\\ell} \\to V^*_{\\ell}$, $\\ell=1,\\ldots,s$, are strictly monotone, hemicontinuous and coercive.\n\\end{lemma}\n\n\\begin{proof} \nWe will only derive the properties for $\\gamma + hF_{\\ell}$, as the same argumentation holds for $\\gamma + hF$. The strict monotonicity of the operator follows using ($\\alpha_{3}$), as\n\\begin{align*}\n\\dualVi{(\\gamma+ hF_{\\ell})u - &(\\gamma + hF_{\\ell})v}{u-v}= \\\\\n                          & \\ska{u-v}{u-v}_{H} + h\\int_{\\Omega_{\\ell}} \\chi_{\\ell} \n                             \\bigl(\\alpha(\\delta_{p, \\ell} u) - \\alpha(\\delta_{p, \\ell} v)  \\bigr) \\cdot \\delta_{p, \\ell} (u-v) \\,\\mathrm{d}x> 0\n\\end{align*}\nholds for all $u,v\\in V_{\\ell}$ with $u\\neq v$.\n\nNext, we prove that $F_{\\ell}$ is hemicontinuous, i.e., $t \\mapsto \\dualVi{F_{\\ell}(u +tv)}{w}$ is continuous on $[0,1]$ for $u,v,w\\in V_{\\ell}$. Consider a sequence $\\seq{t}$ in $[0,1]$ with limit $t$ and introduce\n\\begin{align*}\ng(t,x)= \\chi_{\\ell}(x) \\alpha\\bigr(x,(\\delta_{p, \\ell} u +t\\delta_{p, \\ell}v) (x)\\bigr) \\cdot \\delta_{p, \\ell} w(x).\n\\end{align*}\nAs $\\lim_{n\\to \\infty } g(t_{n},x)=g(t,x)$ holds for almost every $x\\in \\Omega_{\\ell}$, due to ($\\alpha_{1}$), and\n\\begin{align*}\n|g(t,x)|\\leq \\chi_{\\ell}(x) \\bigl( c_{1} \\bigl(|\\delta_{p, \\ell} u(x)|+|\\delta_{p, \\ell} v(x)|\\bigr)^{p-1}+c_{2}(x)\\bigr)\n|\\delta_{p, \\ell} w(x)|,\n\\end{align*}\nwhere the right-hand side is an $L^1(\\Omega_{\\ell})$ element, we obtain that\n\\begin{align*}\n\\lim_{n\\to \\infty} \\dualVi{F_{\\ell}(u +t_nv)}{w}\t\n    &= \\lim_{n\\to \\infty}\\int_{\\Omega_{\\ell}}\\chi_{\\ell} \\alpha(\\delta_{p, \\ell} (u +t_nv)  ) \\cdot \\delta_{p, \\ell} w \\,\\mathrm{d}x\\\\\n    &= \\dualVi{F_{\\ell}(u +tv)}{w},\n\\end{align*}\nby the dominated convergence theorem. This implies that $F_{\\ell}$ is hemicontinuous, \nand the same trivially holds for $\\gamma + hF_{\\ell}$.\n\t\nLast, we prove the coercivity of $\\gamma + hF_{\\ell}$. By assumption ($\\alpha_{4}$), we have\\begin{align*}\n\\dualVi{(\\gamma + hF_{\\ell}) u }{u} \n\t&= \\ska{u}{u}_{H} + h \\int_{\\Omega_{\\ell}} \\chi_{\\ell} \\alpha(\\delta_{p, \\ell} u  ) \\cdot \\delta_{p, \\ell} u \\,\\mathrm{d}x \\\\\n\t&\\geq  \\|u\\|_H^2 + h \\int_{\\Omega_{\\ell}} \\chi_{\\ell}(c_3|\\delta_{p, \\ell} u |^p -c_{4})\\,\\mathrm{d}x\\\\\n\t&\\geq\\|u\\|_H^2 + c_3h  \\|\\delta_{p, \\ell} u\\|_{L^p(\\Omega_{\\ell},\\chi_{\\ell})^k}^p - h\\|\\chi_{\\ell}\\|_{L^\\infty(\\Omega_{\\ell})}\\|c_4\\|_{L^1(\\Omega_{\\ell})}\n\\end{align*} \nfor every $u \\in V_{\\ell}$. Hence, we have the limit\n\\begin{align*}\n\\frac{\\dualVi{(\\gamma + hF_{\\ell}) u }{u}}{\\|u\\|_{V_{\\ell}}}\\geq  \n\\min(1,c_{3}h)\\frac{\\|u\\|_H^2 + \\|\\delta_{p, \\ell} u\\|_{L^p(\\Omega_{\\ell},\\chi_{\\ell})^k}^p}{\\|u\\|_H \n+ \\|\\delta_{p, \\ell} u\\|_{L^p(\\Omega_{\\ell},\\chi_{\\ell})^k} } \n- \\frac{c(\\chi_{\\ell},c_{4})}{\\|u\\|_{V_{\\ell}}}\\to \\infty,\n\\end{align*}\nas $\\|u\\|_{V_{\\ell}} \\to \\infty$, which implies the coercivity of $\\gamma + hF_{\\ell}$. \\qed\n\\end{proof}\n\n\\begin{corollary}\\label{cor:energy}\nIf the Assumptions \\ref{ass:1}--\\ref{ass:3} hold and $h>0$ , then the operators $\\gamma + hF: V \\to V^*$ and \n$\\gamma+ hF_{\\ell}: V_{\\ell} \\to V^*_{\\ell}$, $\\ell=1,\\ldots,s$, are all bijective.\n\\end{corollary}\n\n\\begin{proof}\nAs  $\\gamma + hF: V \\to V^*$ and $\\gamma+ hF_{\\ell}: V_{\\ell} \\to V^*_{\\ell}$ are all, by Lemma~\\ref{lem:energy}, \nstrictly monotone, hemicontinuous and coercive, their bijectivity follows by the Browder--Minty theorem; \nsee, e.g., \\cite[Theorem 26.A]{Zeidler.1989}.\\qed\n\\end{proof}\n\n\\section{Friedrich extensions of the vector fields}\\label{sec:frform}\n\nThe energetic setting is too general for the convergence analysis that we have in mind. We therefore\nintroduce the nonlinear Friedrich extensions of our vector fields, i.e., we restrict the domains of the energetic operators such that they become (unbounded) operators on the pivot space $H$. More precisely, we define the Friedrich extension $f: {D}(f) \\subseteq H \\to H$ of the full vector field by\n\\begin{align*} \n{D} (f) = \\{u \\in V : F u \\in H^* \\}\\quad\\text{and}\n\\quad f u =  -\\gamma^{-1} Fu\\quad\\text{for } u\\in{D} (f).\n\\end{align*}\nAnalogously, we introduce the Friedrich extensions $f_{\\ell}: {D}(f_{\\ell}) \\subseteq H \\to H$, $\\ell =1,\\dots,s$, of the decomposed vector fields by\n\\begin{align*} \n{D} (f_{\\ell}) = \\{u \\in V_{\\ell} : F_{\\ell} u \\in H^* \\}\\quad\\text{and}\n\\quad f_{\\ell} u = -\\gamma^{-1} F_{\\ell}u\\quad\\text{for  } u\\in{D} (f_{\\ell}).\n\\end{align*}\n\n\\begin{lemma} \\label{lem:friedrich}\nIf the Assumptions \\ref{ass:1}--\\ref{ass:3} hold, then the operators $f:{D}(f) \\subseteq H \\to H$ and $f_{\\ell}:{D}(f_{\\ell}) \\subseteq H \\to H$, \n$\\ell =1,\\dots,s$, are all maximal dissipative.\n\\end{lemma}\n\n\\begin{proof}\nBy ($\\alpha_{3}$) of Assumption~\\ref{ass:3}, we have that\n\\begin{align*}\n\\ska{f_{\\ell}u - f_{\\ell}v}{u - v}_{H} & = -\\dualVi{F_{\\ell}u-F_{\\ell}v}{u-v}\\\\\n&=-\\int_{\\Omega_{\\ell}} \\chi_{\\ell}\\bigl(\\alpha(\\delta_{p,\\ell} u)- \\alpha(\\delta_{p,\\ell} v)\\bigr)\n\\cdot \\delta_{p,\\ell} (u - v)\\,\\mathrm{d}x \\leq 0\n\\end{align*}\nfor all $u,v\\in{D}(f_{\\ell})$, i.e., $f_{\\ell}$ is dissipative. Next, for given $h>0$ and $v\\in H$ one has, in virtue of Corollary~\\ref{cor:energy}, that \nthere exists a unique $u\\in V_{\\ell}$ such that $(\\gamma+hF_{\\ell})u=\\gamma v$, or equivalently\n\\begin{align*}\nF_{\\ell} u = -\\sfrac1h\\, \\gamma (u-v)\\in H^{*}.\n\\end{align*}\nHence, $u\\in{D}(f_{\\ell})$ and $(I-hf)u=v$ in $H$, i.e., ${R}(I-hf_{\\ell})=H$\nand $f_{\\ell}$ is therefore maximal. The same argumentation also yields that $f$ is maximal dissipative.\n\\qed\n\\end{proof}\n\nBefore we continue with our analysis we recapitulate a few properties of a general maximal dissipative \noperator $g:{D}(g)\\subseteq H\\to H$. The resolvent\n\\begin{align*}\n(I-hg)^{-1}:H\\to{D}(g)\\subseteq H\n\\end{align*}\nis well defined, for every $h>0$, and nonexpansive, i.e.,\n\\begin{align*}\n\\|(I-hg)^{-1}u-(I-hg)^{-1}v\\|_{H}\\leq\\|u-v\\|_{H}\\quad\\text{for all }u,v\\in H.\n\\end{align*}\nThe latter follows directly by the definition of dissipativity. Furthermore, the resolvent \nand the related Yosida approximation $g(I - hg)^{-1}$ satisfies the following.\n\n\\begin{lemma} \\label{lem:yosida}\nIf $g:{D}(g)\\subseteq H\\to H$ is maximal dissipative, then \n\\begin{align*}\n\\lim_{h\\to 0} (I - hg)^{-1}u=u\\quad\\text{and}\\quad\\lim_{h\\to 0} g(I - hg)^{-1}v= gv\n\\end{align*}\nin $H$ for every $u\\in \\overline{{D}(g)}$ and $v\\in{D}(g)$, respectively.\n\\end{lemma}\n\nThe proof of Lemma~\\ref{lem:yosida} can, e.g., be found in \\cite[Proposition II. 3.6]{Barbu.1976} or \n\\cite[Proposition 11.3]{Deimling.1985}. Next, we will relate the full vector field $f$ with its decomposition $\\sum_{\\ell =1}^{s} f_{\\ell}$.\n\n\\begin{lemma} \\label{lem:frieddom}\nIf the Assumptions \\ref{ass:1}--\\ref{ass:3} hold, then $\\bigcap_{\\ell =1}^s {D}(f_{\\ell}) \\subseteq {D}(f)$ and\n$fu = \\sum_{\\ell =1}^{s} f_{\\ell}u$ for every $u\\in\\bigcap_{\\ell =1}^s {D}(f_{\\ell})$.\n\\end{lemma}\n\n\\begin{proof}\nChoose a $u\\in\\bigcap_{\\ell =1}^s {D}(f_{\\ell})$, then $u\\in\\bigcap_{\\ell =1}^s V_{\\ell} = V$ and the sum $z=\\sum_{\\ell=1}^{s} f_{\\ell}u\\in H$ satisfies the relation\n\\begin{align*}\n\\ska{-z}{v}_{H}=\\sum_{\\ell =1}^{s} \\dualVi{F_{\\ell}u}{v}=\\dualV{Fu}{v}\n\\end{align*}\nfor all $v\\in V$. Hence, $Fu\\in H ^{*}$, which yields that $u\\in{D}(f)$ and $fu=-\\gamma^{-1} Fu =z$.\n\\qed\n\\end{proof}\n\nUnfortunately, the set ${D}(f)$ is in general not equal to $\\bigcap_{\\ell =1}^s {D}(f_{\\ell})$, as $u\\in {D}(f)$ does not necessarily imply that $F_{\\ell} u \\in H^*$ for every $\\ell =1,\\dots,s$. This issue is well known and we will encounter it when decomposing the $p$-Laplacian; compare with Section~\\ref{sec:pLap}. We will therefore assume that the mild regularity property below holds.\n\\begin{ass}\\label{ass:f1}\n$V \\subseteq  {R}\\bigl( I - h f |_{\\bigcap_{\\ell =1}^s {D}(f_{\\ell})} \\bigr)\\quad$for all $h>0$.\n\\end{ass}\nUnder this assumption one has the following identification, which is sufficient for our convergence analysis.\n\n\\begin{lemma}\\label{lem:close}\nIf the Assumptions \\ref{ass:1}--\\ref{ass:f1} hold, then the closure of $f|_{\\bigcap_{\\ell =1}^s{D}(f_{\\ell})}$ is $f$, i.e., \n\\begin{align*}\n\\overline{\\graph\\bigl( f|_{\\bigcap_{\\ell =1}^s {D}(f_{\\ell})} \\bigr)} = \\graph(f).\n\\end{align*}\n\\end{lemma}\n\n\\begin{proof}\nBy  Lemma~\\ref{lem:frieddom} and the fact that the maximal dissipative operator $f$ is closed~\\cite[Proposition II.3.4]{Barbu.1976}, \nwe obtain that\n\\begin{align*}\n\t\\overline{ \\graph\\bigl(f|_{\\bigcap_{\\ell =1}^s {D}(f_{\\ell})} \\bigr)} \n\t\\subseteq \\overline{ \\graph(f)}=\\graph(f).\n\\end{align*}\nNext, choose an arbitrary $(u,fu) \\in \\graph(f)$. Since\n\\begin{align*}\nu \\in {D}(f) \\subseteq V \\subseteq  {R}\\bigl( I - h f |_{\\bigcap_{\\ell =1}^s {D}(f_{\\ell})} \\bigr),\n\\end{align*}\nwe can define $v_h \\in  \\bigcap_{\\ell =1}^s{D}(f_{\\ell})$ via\n\\begin{align*}\nv_h = (I - hf)^{-1} u = \\bigl(I - h f |_{\\bigcap_{\\ell =1}^s {D}(f_{\\ell})} \\bigr)^{-1} u\n\\end{align*}\nfor every $h>0$.\nBy Lemma~\\ref{lem:yosida}, we have the limits\n\\begin{align*}\n\\lim_{h\\to 0} v_h = u\\quad \\text{ and }\\quad \\lim_{h\\to 0} f v_h = \\lim_{h\\to 0} f(I-hf)^{-1} u  =fu\\quad \\text{ in } H.\n\\end{align*}\nHence, the set $\\graph\\bigl( f|_{\\bigcap_{\\ell =1}^s {D}(f_{\\ell})} \\bigr)$ is dense in \n$\\graph(f)$, i.e., its closure in $H\\times H$ is equal to $\\graph(f)$. \\qed\n\\end{proof}\n\n\\section{Abstract evolution equations and their approximations}\\label{sec:approx}\n\nWith the Friedrich formulation of our full vector field $f:{D}(f) \\subseteq H \\to H$, the parabolic equations \nall take the form of an abstract evolution equations, i.e., \n\\begin{equation}\\label{eq:evolv}\n\\dot{u}=fu,\\quad u(0)=\\eta,\n\\end{equation}\non $H$. Furthermore, with the decomposition $f=\\sum_{\\ell =1}^s f_{\\ell}$, the splitting schemes \n\\eqref{schemes:sumpre} and \\eqref{schemes:liepre} are given by the operators\n\\begin{align*}\nS_{h}=\\sfrac1s\\,\\sum_{\\ell=1}^{s}\\bigl(I-hsf_{\\ell}\\bigr)^{-1} : H \\to H\\quad\\text{and}\\quad\nP_{h}= \\prod_{\\ell=1}^{s}\\bigl(I-hf_{\\ell}\\bigr)^{-1}: H\\to H,\n\\end{align*}\nrespectively. Here, $S^{n}_{h}\\eta$ and $P^{n}_{h}\\eta$ are both approximations of the exact solution $u$ at time $t=nh$.\n\nAs the resolvent of a maximal dissipative operator is well defined and nonexpansive on $H$, it is a natural starting point for a solution concept. To this end, consider the operator family $\\{\\ee{tf}\\}_{t\\geq 0}$ defined by \n\\begin{equation*}\n\\ee{tf}\\eta=\\lim_{n\\to\\infty}\\bigl(I-\\sfrac tn\\,f\\bigr)^{-n}\\eta,\n\\end{equation*}\nwhere the limit is well defined in $H$ for every $\\eta\\in\\overline{{D}(f)}$ and $t\\geq 0$; see \\cite[Theorem~I]{CrandallLiggett.1971}. \nThe operator family $\\{\\ee{tf}\\}_{t\\geq 0}$ is in fact a (nonlinear) semigroup and each $\\ee{tf}:\\overline{{D}(f)}\\to\\overline{{D}(f)}$ is a nonexpansive operator on $H$. The unique mild solution of the evolution equation \\eqref{eq:evolv} is then given by the function $u:t\\mapsto \\ee{tf}\\eta$, which is continuous on bounded time intervals.  An extensive exposition of the nonlinear semigroup theory can, e.g., be found in \\cite{Barbu.1976}.\n\nThere is a discrepancy between the domain of the solution operator, i.e., ${D}(\\ee{tf})=\\overline{{D}(f)}$, and \nthe fact that the operators $S_{h}$ and $P_{h}$ are not necessarily invariant over it. In order to\navoid several technicalities induced by this, we will assume the following.\n\\begin{ass}\\label{ass:f2}\nThe domain ${D}(f)$  is dense in $H$.\n\\end{ass}\nAs $f$ is the closure of $f|_{\\bigcap_{\\ell =1}^s {D}(f_{\\ell})}$, one has the inclusions\n\\begin{equation*}\n{D}(f|_{\\bigcap_{\\ell =1}^s {D}(f_{\\ell})})\\subseteq D(f)\\subseteq\\overline{{D}(f|_{\\bigcap_{\\ell =1}^s {D}(f_{\\ell})})},\n\\end{equation*}\nwhich implies that $\\overline{{D}(f)}=\\overline{{D}(f|_{\\bigcap_{\\ell =1}^s {D}(f_{\\ell})})}$.\nHence, ${D}(f|_{\\bigcap_{\\ell =1}^s {D}(f_{\\ell})})$ is also dense in $H$  when Assumption~\\ref{ass:f2} holds.\n\nWe can now formulate the following simplified version of the Lax-type convergence \nresult given in~\\cite[Corollary~4.3]{BrezisPazy.1972}.\n\n\\begin{lemma}\\label{lem:BrePaz}\nConsider an operator family $\\{G_{h}\\}_{h>0}$, where each operator $G_{h}:H\\to H$ is nonexpansive on $H$\nand the operator family is consistent, i.e., \n\\begin{align*}\n\\lim_{h\\to 0} \\sfrac 1h\\,(G_{h}-I)u= fu\\quad\\text{in } H \\quad\\text{for all }u\\in \\cap_{\\ell =1}^s {D}(f_{\\ell}). \n\\end{align*}\nIf the Assumptions~\\ref{ass:1}--\\ref{ass:f2} hold, then \n\\begin{align*}\n\\lim_{n\\to\\infty}\\sup_{t\\in(0,\\,T)}\\bigl\\| G^{n}_{t\/n}\\eta-\\ee{tf}\\eta\\bigr\\|_{H}=0\n\\end{align*}\nfor every $\\eta\\in H$ and $T<\\infty$.\n\\end{lemma}\n\n\\begin{theorem}\\label{thm:sum}\nIf the Assumptions~\\ref{ass:1}--\\ref{ass:f2} hold, then the sum splitting~\\eqref{schemes:sumpre} \nis convergent in $H$, uniformly on bounded time intervals, to the mild solution of the abstract evolution \nequation~\\eqref{eq:evolv}, i.e., \n\\begin{align*}\n\\lim_{n\\to\\infty}\\sup_{t\\in(0,\\,T)}\\bigl\\| S^{n}_{t\/n}\\eta-\\ee{tf}\\eta\\bigr\\|_{H}=0\n\\end{align*}\nfor every $\\eta\\in H$ and $T<\\infty$.\n\\end{theorem}\n\n\\begin{proof}\nAs each resolvent $(I-hsf_{\\ell})^{-1}$ is nonexpansive on $H$ for all values of $hs>0$, one has the bound\n\\begin{align*}\n\\|S_{h}u-S_{h}v\\|_{H}\\leq \\sfrac 1s\\, \\sum_{\\ell=1}^{s} \\|(I-hsf_{\\ell})^{-1}u-(I-hsf_{\\ell})^{-1}v\\|_{H}\\leq \\|u-v\\|_{H},\n\\end{align*}\nand $S_{h}$ is therefore nonexpansive on $H$. To validate the consistency of $\\{S_{h}\\}_{h>0}$, we first observe that\n\\begin{align*}\n\\sfrac 1h\\,\\bigl((I-hsf_{\\ell})^{-1}-I\\bigr)=sf_{\\ell}(I-hsf_{\\ell})^{-1}.\n\\end{align*}\nThe consistency can then be formulated in terms of the Yosida approximation, i.e.,\nfor every $u\\in \\cap_{\\ell =1}^s {D}(f_{\\ell})$ one has the limit\n\\begin{align*}\n\\sfrac1h\\,(S_{h}-I)u = \\sum_{\\ell=1}^{s} \\sfrac{1}{hs}\\,\\bigl((I-hsf_{\\ell})^{-1}-I\\bigr)u \n= \\sum_{\\ell=1}^{s} f_{\\ell}(I-hsf_{\\ell})^{-1}u\\to \\sum_{\\ell=1}^{s} f_{\\ell}u=fu\n\\end{align*}\nin $H$, as $h\\to 0$; compare with Lemma~\\ref{lem:yosida}. The desired convergence is then proven \nas the hypotheses of Lemma~\\ref{lem:BrePaz} hold. \\qed\n\\end{proof}\n\n\\begin{theorem}\\label{thm:lie}\nIf the Assumptions~\\ref{ass:1}--\\ref{ass:f2} hold, then the Lie splitting~\\eqref{schemes:liepre} \nis convergent in $H$, uniformly on bounded time intervals, to the mild solution of the abstract evolution \nequation~\\eqref{eq:evolv}, i.e., \n\\begin{align*}\n\\lim_{n\\to\\infty}\\sup_{t\\in(0,\\,T)}\\bigl\\| P^{n}_{t\/n}\\eta-\\ee{tf}\\eta\\bigr\\|_{H}=0\n\\end{align*}\nfor every $\\eta\\in H$ and $T<\\infty$.\n\\end{theorem}\n\n\\begin{proof}\nWe once more prove convergence by validating the hypotheses of Lemma~\\ref{lem:BrePaz}. The nonexpansivity\nof the operator $P_{h}$ on $H$ follows trivially as every resolvent $(I-hf_{\\ell})^{-1}$ has the same property. In order to validate the consistency of $\\{P_{h}\\}_{h>0}$, let $u\\in \\cap_{\\ell =1}^s {D}(f_{\\ell})$ and consider the telescopic expansion\n\\begin{equation}\\label{eq:Pconist}\n\\sfrac1h\\,(P_{h}-I)u = \\sum_{\\ell=1}^{s} \\sfrac1h\\,\\bigl((I-hf_{\\ell})^{-1}-I\\bigr)u_{\\ell,h}= \n\\sum_{\\ell=1}^{s} f_{\\ell}(I-hf_{\\ell})^{-1}u_{\\ell,h},\n\\end{equation}\nwhere $u_{1,h}=u$ and \n\\begin{equation*}\nu_{\\ell,h}=(I-hf_{\\ell-1})^{-1}\\ldots(I-hf_{1})^{-1}u\\quad \\text{for }\\ell=2,\\ldots,s.\n\\end{equation*}\nAs the arguments of the Yosida approximations in~\\eqref{eq:Pconist} are $h$ dependent, we can not directly use Lemma~\\ref{lem:yosida}. Instead, we assume for the time being that the limit \n\\begin{equation}\\label{eq:uhlimit}\n\\lim_{h\\to 0} \\sfrac1h(u-u_{\\ell,h}) =z_{\\ell},\\quad \\text{in }H,\n\\end{equation}\nexists. By introducing the maximal dissipative operator \n\\begin{align*}\ne_{\\ell}: {D}(f_{\\ell}) \\subseteq H \\to H : u\\mapsto f_{\\ell}u-z_{\\ell},\n\\end{align*}\nwhich satisfies $(I-hf_{\\ell})^{-1}u_{\\ell,h}=(I-he_{\\ell})^{-1}(u_{\\ell,h}+hz_{\\ell})$, we have the reformulation\n\\begin{align*}\nf_{\\ell}(I-hf_{\\ell})^{-1}u_{\\ell,h} &= \\sfrac1h\\,(I-he_{\\ell})^{-1}(u_{\\ell,h}+hz_{\\ell})-\\sfrac1h\\,(I-he_{\\ell})^{-1}u\\\\\n&\\qquad+\\sfrac1h\\,\\bigl((I-he_{\\ell})^{-1}-I\\bigr)u+\\sfrac1h\\,(u-u_{\\ell,h}).\n\\end{align*}\nBy Lemma~\\ref{lem:yosida} and the nonexpansivity of $(I-he_{\\ell})^{-1}$, one then obtains the limit \n\\begin{align*}\n\\|f_{\\ell}&(I-hf_{\\ell})^{-1}u_{\\ell,h} -f_{\\ell}u\\|_{H}\\\\\n&\\leq \\|\\sfrac1h\\,(I-he_{\\ell})^{-1}(u_{\\ell,h}+hz_{\\ell})-\\sfrac1h\\,(I-he_{\\ell})^{-1}u\\|_{H}\\\\\n&\\qquad+\\|\\sfrac1h\\,\\bigl((I-he_{\\ell})^{-1}-I\\bigr)u-e_{\\ell} u\\|_{H} + \\|\\sfrac1h\\,(u-u_{\\ell,h}) + e_{\\ell} u - f_{\\ell} u\\|_{H}\\\\\n&\\leq \\|-\\sfrac1h\\,(u-u_{\\ell,h})+z_{\\ell}\\|_{H}+\\|e_{\\ell}(I-he_{\\ell})^{-1}u-e_{\\ell}u\\|_{H}\\\\\n&\\qquad+ \\|\\sfrac1h\\,(u-u_{\\ell,h})-z_{\\ell}\\|_{H}\\to 0,\\quad\\text{as }h\\to 0.\n\\end{align*}\nHence, if \\eqref{eq:uhlimit} exists then $ \\lim_{h\\to 0}f_{\\ell}(I-hf_{\\ell})^{-1}u_{\\ell,h}=f_{\\ell}u$. Furthermore,\nif \\eqref{eq:uhlimit} exists for every $\\ell=1,\\ldots,s$, then $\\lim_{h\\to 0}1\/h\\,(P_{h}-I)u=fu$ in $H$.\n\nThe limit \\eqref{eq:uhlimit} obviously exists for $\\ell=1$. If it exists for $\\ell=k$ then it also exists for $\\ell=k+1$, as\n\\begin{align*}\n\\sfrac1h\\,(u-u_{k+1,h})&=\\sfrac1h\\,(u-u_{k,h})-\\sfrac1h\\,\\bigl((I-hf_{k})^{-1}-I\\bigr)u_{k,h}\\\\\n&=\\sfrac1h\\,(u-u_{k,h})-f_{k}(I-hf_{k})^{-1}u_{k,h}\\to z_{k}-f_{k}u\n\\end{align*}\nin $H$, as $h\\to 0$. By induction, the limit~\\eqref{eq:uhlimit} exists for every $\\ell=1,\\ldots,s$, and $\\{P_{h}\\}_{h>0}$ is therefore consistent.\\qed\n\\end{proof}\n\n\\begin{remark}\nThe results can be extended to perturbed equations $\\dot{u}=(f+g)u$, e.g., arising if a lower-order advection or reaction term is added to the diffusion process. Here, $g$ and $f+g$ are both assumed to satisfy a shifted dissipativity condition of the form\n\\begin{align*}\n\\ska{gu - gv}{u - v}_{H} \\leq M[g] \\|u-v\\|_{H}^{2}\\quad\\text{for all }u,v\\in{D}(g),\n\\end{align*}\nwith $M$ being a nonnegative constant, and the range condition ${R}(I-hg)=H$ for $h\\in(0,1\/M)$.\nThis is, e.g., satisfied when $g:H\\to H$ is Lipschitz continuous. \nMore elaborate perturbation examples are given in \\cite[Section~II.3.2]{Barbu.1976}.\nFor these perturbed evolution equations, one has convergence for the modified splitting schemes, with a single step given by\n\\begin{align*}\n\\tilde{S}_{h}= (I-hg)^{-1}S_{h}\\quad\\text{and}\\quad \\tilde{P}_{h}= (I-hg)^{-1}P_{h},\n\\end{align*}\nrespectively. If $g:H\\to H$ is in addition Lipschitz continuous, then convergence is also obtained for the semi-implicit schemes\n\\begin{align*}\n\\hat{S}_{h}= (I+hg)S_{h}\\quad\\text{and}\\quad \\hat{P}_{h}= (I+hg)P_{h}.\n\\end{align*}\nThe convergence of the modified schemes follow just as for the proof of Theorem~\\ref{thm:lie} together with \nthe fact that \\cite[Corollary~4.3]{BrezisPazy.1972} is valid for operators $G_{h}$ that have Lipschitz constants of the form $1+Ch$.\n\\end{remark}\n\n\\section{Parabolic equations of p-Laplace type}\\label{sec:pLap}\n\nAs a first problem class we consider the parabolic equations of $p$-Laplace type with \nhomogeneous Neumann boundary conditions, i.e., \n\\begin{equation}\\label{eq:pLap}\n\\begin{cases}\n\\partial u\/ \\partial t = \\nabla \\cdot \\alpha(\\nabla u)  &\\text{in } \\Omega\\times (0,T),\\\\\n\\alpha(\\nabla u)\\cdot n =0 &\\text{on } \\partial \\Omega\\times (0,T),\\\\\nu(0) =\\eta  &\\text{in } \\Omega.\n\\end{cases}\n\\end{equation}\nThe domain $\\Omega\\subset \\mathbb{R}^d$ is assumed to have a locally Lipschitz boundary $ \\partial \\Omega$, and the map $\\alpha:\\Omega\\times\\mathbb{R}^{d}\\to \\mathbb{R}^{d}$ satisfies Assumption~\\ref{ass:3} for a given $p\\geq 2$. The classical $p$-Laplacian is then given by \n\\begin{equation*}\n\\alpha(x,z)=|z|^{p-2}z.\n\\end{equation*}\nAfter multiplication with $v$ and a subsequent integration by parts, the variational form of~\\eqref{eq:pLap} and its decomposition is formally given by\n\\begin{equation}\\label{eq:pLaplvar}\n(\\partial u\/ \\partial t,v)_{L^2(\\Omega)} =  - \\int_{\\Omega}\\alpha (\\nabla u)\\cdot \\nabla v \\,\\mathrm{d}x = \n- \\sum_{\\ell=1}^{s}  \\int_{\\Omega_{\\ell}}\\chi_{\\ell} \\alpha (\\nabla u)\\cdot \\nabla v \\,\\mathrm{d}x.\n\\end{equation}\nHere, we have introduce a domain decomposition $\\{\\Omega_{\\ell}\\}_{\\ell =1}^s$, where $\\bigcup_{\\ell =1}^s \\Omega_{\\ell}=\\Omega$, together with a partition of unity $\\{\\chi_{\\ell}\\}_{\\ell=1}^{s}$ chosen in $W^{1,\\infty}(\\Omega)$; compare with Remark~\\ref{rem:hm1}. \n\nIn order to fit the variational form into the abstract setting of Sections~\\ref{sec:enform}, we choose the pivot space $H = L^2(\\Omega)$ and the operator $\\delta$ as the distributional gradient\n\\begin{align*}\n\\delta: L^2(\\Omega)  \\to \\mathcal{D}'(\\Omega)^d: u \\mapsto \\nabla u.\n\\end{align*}\nThis choice of $\\delta$ fulfills the continuity Assumption~\\ref{ass:1}, since for a convergent sequence $\\seq{u}\\subset L^2(\\Omega)$ and an arbitrary $\\phi \\in C^{\\infty}_0(\\Omega)$ one can write\n\\begin{align*}\n\\lim_{n\\to \\infty} (D_j u_n) (\\phi)  = -\\lim_{n\\to \\infty} \\int_{\\Omega} u_n D_j \\phi \\,\\mathrm{d}x = - \\int_{\\Omega} u D_j \\phi \\,\\mathrm{d}x = (D_j u) (\\phi) \n\\end{align*} \nfor every$j=1,\\dots,d$, where $D_j$ is the $j$-th partial derivative in a distributional sense.\nThe space $V$ is then\n\\begin{align*}\nV = \\bigl\\{ u\\in L^2(\\Omega): \\nabla u \\in L^p(\\Omega)^d \\bigr\\}.\n\\end{align*}\nA bootstrap argument using the Sobolev embedding theorem yields the identification $V = W^{1,p}(\\Omega)$. Since $W^{1,p}(\\Omega)$ is dense in $L^2(\\Omega)$, Assumption~\\ref{ass:2} is also fulfilled. \n\nWith these choices, $\\delta_{p}u$ is simply the weak gradient of $u\\inW^{1,p}(\\Omega)$ and we obtain the standard energetic form $F :V \\to V^*$ of $p$-Laplace type vector fields, i.e., \n\\begin{align*}\n\\dualV{Fu}{v}  = \\int_{\\Omega}\\alpha (\\nabla u)\\cdot \\nabla v \\,\\mathrm{d}x.\n\\end{align*}\nThe domain of the corresponding Friedrich extension can be written as\n\\begin{align*}\n{D}(f) &= \\Bigl\\{u \\in W^{1,p}(\\Omega):\\text{ there exists a }z\\in L^2(\\Omega) \\text{ such that }\\\\\n& \\qquad \\qquad -\\int_{\\Omega}\\alpha (\\nabla u)\\cdot \\nabla v \\,\\mathrm{d}x=\\int_{\\Omega} zv\\,\\mathrm{d}x \\quad\\text{for all } v \\in W^{1,p}(\\Omega) \\Bigr\\}, \n\\end{align*}\nand $fu$ is given by the weak divergence of $\\alpha(\\nabla u)$. The same characterization can be made for $F_{\\ell}$ and $f_{\\ell}$, respectively. Applying Lemma~\\ref{lem:friedrich} the operators $f$ and $f_{\\ell}$, $\\ell =1,\\dots,s$, are maximal dissipative and Lemma~\\ref{lem:frieddom} yields that\n\\begin{align*}\n\\bigcap_{\\ell =1}^{s} {D}(f_{\\ell}) \\subseteq {D}(f)\\quad \\text{ and }\\quad\nfu = \\sum_{\\ell =1}^{s} f_{\\ell}u\\quad\\text{for }u\\in\\bigcap_{\\ell =1}^{s} {D}(f_{\\ell}).\n\\end{align*}\nValidation of Assumption~\\ref{ass:f2} requires further structure of the map $\\alpha$. For the classical $p$-Laplacian the related $\\alpha$ is continuously differentiable and $\\alpha(0)=0$, which implies that $C_0^{\\infty}(\\Omega)$ is a subset of ${D}(f)$. Hence, ${D}(f)$ is dense in $L^2(\\Omega)$ and Assumption~\\ref{ass:f2} is valid in this context. Finally, if Assumption~\\ref{ass:f1} holds then the convergence results from Section \\ref{sec:approx} can directly be applied. \n\n\\begin{figure}\n \\centering \n \\includegraphics[scale=0.7]{Fig3.pdf}\n \\caption{An example of a domain decomposition $\\{\\Omega_{\\ell}\\}_{\\ell =1}^3$ that fulfills \\eqref{ass:dom}.}\n \\label{fig:pLaplace}\n\\end{figure}\n\nApart from the special cases when $d=1$ or $p=2$, the domains ${D}(f)$ of $p$-Laplace type vector fields can not be expected to coincide with $\\bigcap_{\\ell =1}^{s} {D}(f_{\\ell})$. The issue is that for an element $u \\in {D}(f)$ one has\n\\begin{align*}\nf_{\\ell}u=\\nabla\\cdot \\bigl(\\chi_{\\ell} \\alpha(\\nabla u) \\bigr)=  \\nabla \\chi_{\\ell} \\cdot\\alpha(\\nabla u) \n+ \\chi_{\\ell} \\nabla\\cdot \\alpha(\\nabla u),\n\\end{align*}\nwhere the function $\\alpha(\\nabla u)$ only lies in $L^{\\nicefrac{p}{(p-1)}}(\\Omega)^{d}$, with $p>2$. The term $f_{\\ell}u$ is therefore, in general, not an $L^{2}(\\Omega)$ function. In order to give a possible setting for which Assumption~\\ref{ass:f1} is valid, we assume that the domain decomposition $\\{\\Omega_{\\ell}\\}_{\\ell =1}^s$ is chosen such that\n\\begin{equation}\\label{ass:dom}\n\\text{closure}(\\bigcup_{\\ell =1}^{s-1}\\Omega_{\\ell})\\setminus\\partial\\Omega = \\emptyset.\n\\end{equation}\nThat is, the subdomain $\\Omega_{s}$ separates the boundary $\\partial\\Omega$ from the other subdomains; as illustrated in Figure~\\ref{fig:pLaplace}.\n\n\\begin{lemma} \nConsider a domain decomposition $\\{\\Omega_{\\ell}\\}_{\\ell =1}^s$ that satisfies \\eqref{ass:dom} and with subdomains $\\Omega_{\\ell}$, $\\ell=1,\\ldots,s-1$, that all have the segment property. If $p\\geq 2$ in addition satisfies $p>(d+1)\/2$ and the map $\\alpha$ fulfills Assumption~\\ref{ass:3}($\\alpha_{2}$) with $c_{2}\\in L^{2}(\\Omega)$, then the Friedrich extension $f$ of a $p$-Laplace type vector field and its decomposition into the operators $f_{\\ell}$ fulfill Assumption~\\ref{ass:f1}.\n\\end{lemma}\n\n\\begin{proof}\nFor an arbitrary $g \\in V=W^{1,p}(\\Omega)$ there exists a unique $u\\in {D}(f)$ such that\n$u-hfu = g$ and Assumption~\\ref{ass:f1} is then valid if $u\\in\\bigcup_{\\ell =1}^s{D}(f_{\\ell})$. To prove this, we first observe that  $fu=\\nabla \\cdot \\alpha(\\nabla u) = (u-g)\/h \\inW^{1,p}(\\Omega)$ and $W^{1,p}(\\Omega) \\hookrightarrow L^r(\\Omega)$ for some $r > dp\/(p-1)$, as $p \\geq 2$ and $p > (d+1)\/2$. Hence, \\cite[Theorem 2 and Remarks pp.~829--830]{diBenedetto.1983} implies that $\\nabla u$  is locally H\\\"older continuous on $\\Omega$ and we obtain that\n\\begin{align*}\n\\alpha(\\nabla u)|_{\\Omega_{int}}\\in L^{2}(\\Omega_{int})^{d}\n\\end{align*}\nfor every open domain $\\Omega_{int}$ such that $\\overline{\\Omega}_{int}\\subset\\Omega$. \n\nAs $u\\in{D}(f)$, we have the integration by parts\n\\begin{align} \\label{eq:intbypar}\n-\\int_{\\Omega} \\alpha (\\nabla u)\\cdot \\nabla w\\,\\mathrm{d}x\n= \\int_{\\Omega}  \\nabla\\cdot\\alpha (\\nabla u) w\\,\\mathrm{d}x\n\\end{align} \nfor every $w\\inW^{1,p}(\\Omega)$. Due to the  extra interior regularity of $\\alpha(\\nabla u)$ we can, e.g., extend~\\eqref{eq:intbypar} to \nall $w=w_{1}+w_{2}$, where $w_{1}\\in W^{1,p}(\\Omega)$ and $w_{2}\\in H^{1}(\\Omega)$ is a.e.\\ zero on $\\Omega\\setminus\\Omega_{int}$\nfor some open subdomain $\\Omega_{int}$ that has the segment property and fulfills $\\overline{\\Omega}_{int}\\subset\\Omega$.\nThe latter implies that $w_{2}$ is the zero extension of $w_{2}|_{\\Omega_{int}}\\in H^{1}_0 (\\Omega_{int})$; see, e.g., \\cite[Theorem 5.29]{AdamsFournier.2003}.  \n\nNext, let $v\\in V_{\\ell}\\subset L^{2} (\\Omega)$, for $\\ell=1,\\ldots,s$, and consider $\\chi_{\\ell} v \\in L^2(\\Omega)$. Here, \n\\begin{align*}\nD_{j}(\\chi_{\\ell} v) (\\phi) &= D_{j}( v) (\\chi_{\\ell}\\phi) + \\int_{\\Omega} (D_{j}\\chi_{\\ell}) v\\phi\\,\\mathrm{d}x= \\int_{\\Omega_{\\ell}} \\bigl(\\chi_{\\ell} (\\delta_{p,\\ell}v)_{j}+(D_{j}\\chi_{\\ell})v\\bigr)\\phi \\,\\mathrm{d}x\n\\end{align*}\nfor every $\\phi \\in C^\\infty_0 (\\Omega)$, i.e.,  $\\chi_{\\ell} v \\in H^1(\\Omega)$ and $\\chi_{\\ell} v = 0$ a.e.\\ on $\\Omega\\setminus \\Omega_{\\ell}$. If $\\ell<s$ then $\\chi_{\\ell} v|_{\\Omega_{\\ell}}\\in H^1_{0}(\\Omega_{\\ell})$. \n\nFor $\\ell=1,\\ldots,s-1$, we can test with $w=\\chi_{\\ell}v$ and integrate by parts~\\eqref{eq:intbypar}. Writing out $\\nabla(\\chi_{\\ell}v)$ and rearranging the terms gives us \n\\begin{align*} \n-\\int_{\\Omega_{\\ell}} \\chi_{\\ell}\\alpha (\\nabla u)\\cdot \\delta_{p,\\ell}v\\,\\mathrm{d}x =\n \\int_{\\Omega} \\bigl(\\chi_{\\ell}\\nabla\\cdot \\alpha (\\nabla u)+\\nabla\\chi_{\\ell}\\cdot  \\alpha (\\nabla u)\\bigr)v\\,\\mathrm{d}x,\n\\end{align*}\ni.e., $u\\in\\bigcap_{\\ell=1}^{s-1} {D}(f_{\\ell})$, as the integrand on the right-hand side is in $L^{2}(\\Omega)$.\n\nIt remains to prove that $u$ lies in ${D}(f_{s})$.  As the closure of $\\bigcup_{\\ell =1}^{s-1}\\Omega_{\\ell}$ does not intersect the outer boundary $\\partial\\Omega$, we can choose an open subset $\\Omega_{out}\\subset\\Omega_{s}$ such that $\\chi_{s}=1$ on \n$\\Omega_{out}$, its boundary $\\partial \\Omega_{out}$ is locally Lipschitz continuous and $\\partial\\Omega\\subset\\partial\\Omega_{s}$.\nLet $v\\in V_{s}$, then $\\chi_{s}\\delta_{p,s}v=\\nabla v$ a.e.\\ on $\\Omega_{out}$ and $\\chi_{s}v|_{\\Omega_{out}}=v|_{\\Omega_{out}}\\in W^{1,p}(\\Omega_{out})$. The local Lipschitz continuity of  $\\partial \\Omega_{out}$ implies, e.g., via \\cite[Theorem 5.24]{AdamsFournier.2003}, that there exists an extension $w_{1}\\in W^{1,p}(\\Omega)$ such that $w_{1}=\\chi_{s}v$ a.e.\\ on $\\Omega_{out}$.  Furthermore, $w_{2}=\\chi_{s}v-w_{1}\\in H^{1}(\\Omega)$ is zero a.e. on  $\\Omega_{out}$, i.e., it is a zero extension of an $H^{1}_0 (\\Omega_{int})$ function on some subdomain $\\Omega_{int}$, with $\\partial\\Omega_{int}\\subset \\Omega_{out}$. For every $v\\in V_{s}$ we therefore have a partitioning of the form $w=\\chi_{s}v=w_{1}+w_{2}$ and the integration by parts \\eqref{eq:intbypar} is well defined for $\\ell=s$. By the same argumentation as for $\\ell<s$, one obtains that $u$ lies in ${D}(f_{s})$.\n\\qed\n\\end{proof}\n\n\\begin{remark}\nFrom a numerical perspective the construction~\\eqref{ass:dom} with a separating subdomain $\\Omega_{s}$ is suboptimal for general time dependent PDEs, as it may increase the amount of communication in the implementation of scheme. However, as discussed in Section~\\ref{sec:intro}, we are foremost interested in the approximation of solutions with compact support in $\\Omega$. Hence, for sufficiently short time intervals $(0,T)$ there is obviously no communication related to $\\Omega_{s}$; as exemplified in Figure~\\ref{fig:pLaplace}.\n\\end{remark}\n\n\\section{Parabolic equations of porous medium type}\\label{sec:por}\n\nA second problem class that fits into our abstract setting is the parabolic equations of porous medium type with homogeneous Dirichlet boundary conditions, i.e., \n\\begin{equation}\\label{eq:pme}\n\\begin{cases}\n\\partial u\/ \\partial t = \\Delta \\alpha(u)  &\\text{in } \\Omega\\times (0,T),\\\\\n\\alpha(u) =0 &\\text{on } \\partial \\Omega\\times (0,T),\\\\\nu(0) =\\eta  &\\text{in } \\Omega.\n\\end{cases}\n\\end{equation}\nHere, the domain $\\Omega\\subset \\mathbb{R}^d$ is assumed to have a locally Lipschitz boundary $ \\partial \\Omega$, and the map $\\alpha:\\Omega\\times\\mathbb{R}\\to\\mathbb{R}$ fulfills Assumption~\\ref{ass:3} for a given $p$ that satisfies \n\\begin{align*}\np\\in (1,\\infty)\\quad\\text{if }d\\leq 2,\\quad\\text{and}\\quad p\\in [2d\/(d+2), \\infty)\\quad\\text{if }d>2.\n\\end{align*}\nThis restriction on $p$ is made in order to assure the embedding\n\\begin{equation}\\label{eq:H1inLq}\nH_0^1(\\Omega)\\overset{d}{\\hookrightarrow} L^{\\nicefrac{p}{(p-1)}}(\\Omega),\n\\end{equation}\nwhich is central in our forthcoming analysis. The standard porous medium equation is then given by\n\\begin{align*}\n\\alpha(x,z)=|z|^{p-2}z,\\quad\\text{with }p\\geq 2,\n\\end{align*}\nand the fast diffusion equation is obtained for the same $\\alpha$, but with $1<p<2$; see~\\cite{Vasquez.2007}. The two-phase Stefan problem \\cite[Section~5.10]{Friedman.1988} follows by choosing \n\\begin{align*}\n\\alpha (x,z)  = \n\\begin{cases}\na(z+1)  &\\text{ for } z \\leq -1\\\\\n0  &\\text{ for } z \\in (-1,1)\\\\\nb(z-1)  &\\text{ for } z \\geq 1\\, ,\n\\end{cases}\n\\end{align*}\nwhere $a,b>0$, and Assumption~\\ref{ass:3} is then valid for $p=2$. \n\nAfter multiplying \\eqref{eq:pme} by $w$, where $-\\Delta w = v$ in $\\Omega$ and $w=0$ on $\\partial\\Omega$, and integrating by parts twice, the variational form of~\\eqref{eq:pme} and its decomposition is formally \n\\begin{equation}\\label{eq:pmevar}\n\\int_{\\Omega}\\frac{\\partial u}{\\partial t}\\,(-\\Delta)^{-1}v\\,\\mathrm{d}x = \n-\\int_{\\Omega}\\alpha (u) v \\,\\mathrm{d}x = \n- \\sum_{\\ell=1}^{s} \\int_{\\Omega_{\\ell}}\\chi_{\\ell} \\alpha (u) v \\,\\mathrm{d}x.\n\\end{equation}\nAbove, we have once more introduced a domain decomposition $\\{\\Omega_{\\ell}\\}_{\\ell =1}^s$ of $\\Omega$ together with a partition of unity $\\{\\chi_{\\ell}\\}_{\\ell=1}^{s}$. \n\nWith the proper interpretation, the left-hand side of \\eqref{eq:pmevar} is given by the inner product on\n$H^{-1}(\\Omega)$; compare with \\cite[Bemerkung III.1.13]{GGZ.1974}. The formal variational formulation~\\eqref{eq:pmevar} therefore leads us to choosing the pivot space $H=H^{-1}(\\Omega)$ and the operator\n\\begin{align*}\n\\delta:H^{-1}(\\Omega)  \\to \\mathcal{D}'(\\Omega): u \\mapsto u.\n\\end{align*}\nThe operator $\\delta$ obviously fulfills the continuity Assumption~\\ref{ass:1}. The space $V$ is now\n\\begin{align*}\nV &=  \\Bigl\\{ u \\in H^{-1}(\\Omega) :\\text{ there exists a } v\\in L^p(\\Omega) \\text{ such that }\\\\\n&  \\qquad\\qquad \\dualHH{u}{\\phi} = \\int_{\\Omega} v \\phi \\,\\mathrm{d}x\\quad\\text{for all } \\phi \\in H^{1}_0(\\Omega)\\Bigr\\}\n= \\bigl(L^{\\nicefrac{p}{(p-1)}}(\\Omega)\\bigr)^*,\n\\end{align*}\nand as before $\\delta_p u = v$, where $v$ is the unique function stated in the definition of $V$. By the embedding~\\eqref{eq:H1inLq} and~\\cite[Bemerkung I.5.14]{GGZ.1974}, we obtain that \n\\begin{align*}\n\\bigl(L^{\\nicefrac{p}{(p-1)}}(\\Omega)\\bigr)^*\\overset{d}{\\hookrightarrow} H^{-1}(\\Omega),\n\\end{align*} \ni.e., Assumption~\\ref{ass:2} is fulfilled. With these choices, we have the energetic form $F :V \\to V^*$ given by\n\\begin{align*}\n\\dualV{Fu}{v}  = \\int_{\\Omega}\\alpha (\\delta_{p} u)\\delta_{p} v \\,\\mathrm{d}x.\n\\end{align*}\n\nIn order to characterize the Friedrich operator $f$, we introduce the Dirichlet Laplacian $-\\Delta:H^{1}_{0}(\\Omega)\\to H^{-1}(\\Omega)$, where\n\\begin{align*}\n\\dualHH{-\\Delta u}{v} = \\int_{\\Omega} \\nabla u\\cdot\\nabla v\\,\\mathrm{d}x\\quad\\text{for all } u,v\\in H^{1}_{0}(\\Omega).\n\\end{align*}\nAs $-\\Delta$ is the Riesz isomorphism from $H^{1}_{0}(\\Omega)$ to $H^{-1}(\\Omega)$,\nthe inner product on $H^{-1}(\\Omega)$ satisfies\n\\begin{align*}\n\\ska{u}{v}_{H^{-1}(\\Omega)} \n&= \\sfrac14\\bigl(\\|u+v\\|_{H^{-1}(\\Omega)}-\\|u-v\\|_{H^{-1}(\\Omega)}\\bigr)\\\\\n&=\\sfrac14\\bigl(\\|(-\\Delta)^{-1}(u+v)\\|_{H^{1}_{0}(\\Omega)}-\\|(-\\Delta)^{-1}(u-v)\\|_{H^1_{0}(\\Omega)}\\bigr)\\\\\n&=\\ska{(-\\Delta)^{-1}u}{(-\\Delta)^{-1}v}_{H^{1}_{0}(\\Omega)}\\\\\n&=\\dualHH{u}{(-\\Delta)^{-1}v}\n\\end{align*}\nfor all $u,v\\in H^{-1}(\\Omega)$; compare with \\cite{EmmrichSiska.2012}. Next, for $u\\in{D}(f)$ there exists a $z\\in H^{-1}(\\Omega)$ such that\n\\begin{align*}\n-\\int_{\\Omega}\\alpha (\\delta_{p} u)\\,\\delta_{p} v \\,\\mathrm{d}x=(z,v)_{H^{-1}(\\Omega)}=\\dualHH{v}{(-\\Delta)^{-1}z}\n\\end{align*}\nfor all $v \\in \\bigl(L^{\\nicefrac{p}{(p-1)}}(\\Omega)\\bigr)^*$, or equivalently \n\\begin{align*}\n-\\int_{\\Omega}\\alpha (\\delta_{p} u)\\,w \\,\\mathrm{d}x= \\int_{\\Omega} w \\,(-\\Delta)^{-1} z\\,\\mathrm{d}x\n\\quad\\text{ for all } w \\in L^{p}(\\Omega).\n\\end{align*}\nHence, $-\\alpha (\\delta_{p} u)=(-\\Delta)^{-1}z\\in H^{1}_{0}(\\Omega)$; see, e.g., \\cite[Lemma~3.31]{AdamsFournier.2003}, and we obtain the characterization \n\\begin{align*}\n{D}(f)=\\bigl\\{u\\in \\bigl(L^{\\nicefrac{p}{(p-1)}}(\\Omega)\\bigr)^*:\\alpha (\\delta_{p} u)\\in H^{1}_{0}(\\Omega)\\bigr\\},\n\\end{align*}\nand $fu=\\Delta\\alpha (\\delta_{p} u)$ for $u\\in{D}(f)$.\n\nAnalogously to Section~\\ref{sec:pLap}, we have ${R}(\\delta)= H^{-1}(\\Omega) \\subset \\mathcal{D}'(\\Omega)$ and we can therefore allow a partition of unity $\\{\\chi_{\\ell}\\}$ in $W^{1,\\infty}(\\Omega)$. The spaces $V_{ \\ell}$, $\\ell =1,\\dots,s$, are then\n\\begin{align*}\nV_{\\ell}&= \\big\\{ u \\in H^{-1}(\\Omega): \\text{ there exists a } v \\in L^p(\\Omega_{\\ell},\\chi_{\\ell}) \\text{ such that }\\\\\n& \\qquad\\qquad  \\dualHH{\\chi_{\\ell} u}{\\phi} = \\int_{\\Omega_{\\ell}} \\chi_{\\ell} v \\phi \\,\\mathrm{d}x\\quad\\text{for every } \\phi \\in H^{1}_0(\\Omega) \\big\\}.\n\\end{align*}\nAgain, we write $\\delta_{p, \\ell} u$ for the unique $L^p(\\Omega_{\\ell},\\chi_{\\ell})$ function $v$ from this definition.\n\nAfter introducing $F_{\\ell}$ and $f_{\\ell}$, as described in Sections~\\ref{sec:enform}, we have by Lemmas~\\ref{lem:friedrich} and~\\ref{lem:frieddom} that the operators $f$ and $f_{\\ell}$, $\\ell =1,\\dots,s$, are maximal dissipative and\n\\begin{align*}\nfu = \\sum_{\\ell =1}^{s} f_{\\ell}u\\quad\\text{for }u\\in\\bigcap_{\\ell =1}^{s} {D}(f_{\\ell})\\subseteq {D}(f).\n\\end{align*}\nInstead of Assumption~\\ref{ass:f1} we can prove the stronger condition \n\\begin{align*}\n\\bigcap_{\\ell =1}^s {D}(f_{\\ell}) = {D}(f).\n\\end{align*}\nTo prove the equality take an arbitrary $u\\in {D}(f)$. Since $\\alpha(\\delta_p u )\\in H_0^1(\\Omega)$, we also have\nthat $\\chi_{\\ell}\\alpha(\\delta_p u ) \\in H_0^1(\\Omega)$ for every weight function $\\chi_{\\ell} \\in W^{1,\\infty}(\\Omega)$ and\n\\begin{align*} \n-\\int_{\\Omega_{\\ell}} \\chi_{\\ell}\\alpha (\\delta_{p} u)\\, \\delta_{p,\\ell}v\\,\\mathrm{d}x &=\\dualHH{v}{-\\chi_{\\ell}\\alpha (\\delta_{p} u)}\\\\\n&= \\ska{\\Delta\\bigl(\\chi_{\\ell}\\alpha (\\delta_{p} u)\\bigr)}{v}_{H^{-1}(\\Omega)}\\quad \\text{ for all } v\\in V_{\\ell}.\n\\end{align*}\nThat is, $u$ also lies in ${D}(f_{\\ell})$ for $\\ell =1,\\dots,s$.\n\nAssumption~\\ref{ass:f2} requires some further regularity of the map $\\alpha$ and the validation that $\\alpha (\\delta_{p} u)$ vanishes on the boundary $\\partial\\Omega$. For the porous medium equation and the two-phase Stefan problem one has that $\\alpha(\\phi)\\in H^{1}_{0}(\\Omega)$ for every $\\phi\\in C_0^{\\infty}(\\Omega)$. The set of functionals of the form $v\\mapsto\\int_{\\Omega}u v\\,\\mathrm{d}x$, where $u\\in C_0^{\\infty}(\\Omega)$ and $v\\in H^{1}_{0}(\\Omega)$, is therefore a subset of ${D}(f)$. It is also a dense subset of $H^{-1}(\\Omega)$, as $C_0^{\\infty}(\\Omega)$ is dense in $L^{2}(\\Omega)$ and $L^{2}(\\Omega)^*$ is dense in $H^{-1}(\\Omega)$. Hence, Assumption~\\ref{ass:f2} is valid for these two prototypical examples, and the convergence results of Section~\\ref{sec:approx} hold.\n\n\\begin{remark}\nThe variational setting of porous medium type equations, with $H^{-1}(\\Omega)$ as pivot space, is by no means standard. However, it enables a clear-cut way of introducing the related Friedrich operator. The variational setting has, e.g., been proposed in \\cite[Bemerkung I.5.14]{GGZ.1974}. It has also been employed in \\cite{EmmrichSiska.2012} when proving convergence of finite element\/implicit Euler approximations for the porous medium equation, on its very weak form. Note that the standard approach to prove that $\\Delta\\alpha$ is a maximal dissipative operator on $H^{-1}(\\Omega)$ is to directly observe that it is the gradient of a convex function; see \\cite[Example 3]{Brezis.1971}.\n\\end{remark}\n\n\\begin{acknowledgements}\nPart of this study was conducted during Hansen's guest research stay at the Institut f\\\"{u}r Mathematik, TU Berlin. Hansen would like to thank Etienne Emmrich for enabling this inspiring stay. \n\\end{acknowledgements}\n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\@startsection {section}{1}{\\z@}%\n                               {-3.5ex \\@plus -1ex \\@minus -.2ex\n                               {2.3ex \\@plus.2ex}%\n                               {\\normalfont\\large\\bfseries}}\n\\renewcommand\\subsection{\\@startsection{subsection}{2}{\\z@}%\n                                 {-3.25ex\\@plus -1ex \\@minus -.2ex}%\n                                 {1.5ex \\@plus .2ex}%\n                                 {\\normalfont\\bfseries}}\n\\makeatother\n\n\\newcommand\\bl{\\color{blue}}  \n\n\n\n\\begin{document}\n\n\\begin{titlepage}\n \n\\begin{center}\n{\\Large \\bf Holographic Quenches and\n\\vspace{3mm} Fermionic Spectral Functions}\n\n\\vskip 10mm\n\n{\\large N.~Callebaut$^{a,b}$, B.~Craps$^{a,c}$, F.~Galli$^d$, D.~C.~Thompson$^a$, \\\\\n\\vspace{3mm}\nJ.~Vanhoof$^a$, J.~Zaanen$^e$, Hongbao~Zhang$^a$}\n\\vskip 7mm\n\n$^a$ Theoretische Natuurkunde, Vrije Universiteit Brussel, and \\\\\n\\hspace*{0.15cm}  International Solvay Institutes, \nPleinlaan 2, B-1050 Brussels, Belgium \\\\\n$^b$ Ghent University, Department of Physics and Astronomy,\\\\ Krijgslaan 281-S9, 9000 Gent, Belgium \\\\\n$^c$ Laboratoire de Physique Th\\'eorique, Ecole Normale Sup\\'erieure,\\\\ 24 rue Lhomond, F-75231 Paris Cedex 05, France \\\\\n$^d$ Instituut voor Theoretische Fysica, KU Leuven,\\\\ \nCelestijnenlaan 200D, B-3001 Leuven, Belgium.\\\\\n$^e$ Instituut-Lorentz for Theoretical Physics, Universiteit Leiden,\\\\ PO Box 9506, NL-2300 RA Leiden, The Netherlands\n\n\\vskip 10mm\n \n{\\small\\noindent  {\\tt ncalleba.Callebaut@ugent.be, Ben.Craps@vub.ac.be, federico@itf.fys.kuleuven.be, Daniel.Thompson@vub.ac.be, Joris.Vanhoof@vub.ac.be, Jan@lorentz.leidenuniv.nl, Hongbao.Zhang@vub.ac.be}}\n \n\\end{center}\n\n\\vskip 10mm\n\n\\begin{center}\n{\\bf ABSTRACT}\n\\vspace{3mm}\n\\end{center}\nUsing holographic methods we investigate the behaviour of fermionic spectral functions of strongly coupled $2+1$ dimensional field theories as both temperature and chemical potential are quenched.  \n\\vfill \n\\end{titlepage}\n\n\n\n\\section{Introduction}\n\nArguably two of the most fundamental problems in quantum field theory, or equivalently quantum many body physics, are non-equilibrium unitary time evolution and finite density, respectively.  With the exception of integrable systems in 1+1D (see, for instance, \\cite{Calabrese:2006rx}) there exists  no direct  field theoretical method that can deal with far from equilibrium dynamics in strongly interacting systems with  infinite   degrees of freedom.  Even at equilibrium, for strongly interacting fermionic systems at finite density, the fermion sign problem obscures the view of non-perturbative physics.\n\nThese problems, associated with understanding non-Fermi liquid physics, have been pushed to center stage\nin condensed matter physics since the discovery of the strange metals in   high $T_c$ superconductors and related systems \\cite{Anderson,VLSAR,VNS}. The study of non-equilibrium physics of quantum many body systems is at present rapidly evolving in the laboratory because of the new techniques becoming available in the field of cold atoms, high energy physics (heavy ion collisions)   and the ``ultrafast\" experiments on condensed matter systems. \n\nThe AdS\/CFT correspondence is unique in its capacity to deal both with finite density and non-equilibrium in a mathematically controlled way, albeit within \nthe limitations of the ``large $N$\" limit and the restrictions on the non-equilibrium physics that can be addressed easily.  To illustrate this power, we present \nhere an extreme example of a time-dependent ``experiment'' involving strongly interacting fermion matter. \n\nConsider a strongly interacting \nrelativistic quantum critical state formed from fermions and other degrees of freedom in 2+1D at zero density and zero temperature. \nThis can be pictured as graphene right at the  quantum phase transition to the Mott insulator \\cite{Herbut}.  At large negative times its fermion\nspectral functions, which can be measured by (inverse) photoemission, will be of the ``branch cut\" form dictated by Lorentz and scale invariance: \n\\begin{equation}\n A (\\omega, k) \\sim \\text{Im} \\left[(  \\sqrt{k^2 - \\omega^2})^{2\\Delta-d}\\right]  \\ .\n\\end{equation}\nHere,  $d$ is the spacetime dimension and throughout this work we will consider  $d=3$.  We are typically interested in the regime  $2\\Delta - d < 0$, noticing that the scaling dimension is bounded from below by the unitarity limit for the fermionic CFT operators, $\\Delta = (d-1)\/2$. \n\nSubsequently, this system is prepared in a highly excited state by a sudden ``quench\" at a given instant, but now at some finite {\\em charge density} which is uniform in space. The main purpose of this paper is to compute time-dependent spectral properties of this system using holography. Specifically, we compute its time-dependent spectral function and show that it suggests the gradual build-up of a Fermi surface. We also highlight some caveats, commenting in particular on the relation with quantities potentially measurable using time-resolved ARPES experiments. \n\n\n\\section{Time-dependent fermion spectral functions}\n\nIn equilibrium, the spectral function of an operator  is defined as ($-2$ times) the imaginary part of its corresponding retarded Green's function in frequency space. However, out of equilibrium such a definition is ambiguous;  the retarded Green's function $G_R(\\vec{x}_1, t_1; \\vec{x}_2, t_2)$ may no longer depend on just the relative time interval $  t =  t_2  - t_1$ but on $t_1$ and $t_2$ separately and  a conventional Fourier transform to frequency space can not be taken. \nIn \\cite{Balasubramanian:2012tu}, a  generalised notion of a spectral function was proposed and computed in a far-from-equilibrium AdS\/CFT context\n(see \\cite{CaronHuot:2011dr,Chesler:2011ds} and \\cite{Banerjee:2012uq,Mukhopadhyay:2012hv,Steineder:2013ana} for studies of this and related quantities far from equilibrium and closer to equilibrium, respectively).  \nIntroducing both the relative time $t$ and the average time, $T=(t_1+t_2)\/2$, \n   we consider the Fourier transform of the  Green's  function with respect to  $t$ whilst keeping $T$ fixed, \n   \\begin{equation}\\label{eq:Wignertrans}\n G(T, \\omega, k) = \\int dt~e^{i \\omega t} G(T,t, k)  \\ .\n  \\end{equation}\nWe assume  that  homogeneity   is preserved such that we can Fourier transform to spatial momenta with no change. Then a notion of a time-dependent spectral function is defined by \n  \\def{\\rm{ Im}}{{\\rm{ Im}}}\n  \\begin{equation}\\label{eq:SpectralDef}\n  A(T,\\omega, k) = - 2 \\, {\\rm{ Im}}  \\, G_R(T, \\omega, k) \\ . \n  \\end{equation}\nIn passing we note that this recipe for taking a Fourier transform has a rather well established analogue as the Wigner distribution \\cite{Wigner:1932eb} used in phase space approaches to Quantum Mechanics.\n\nSuppose that  a system starts in equilibrium, and then undergoes an instantaneous quench at time $t_0=0$, eventually relaxing to find a new equilibrium. \nAt $T=-\\infty$ the spectral function \\eqref{eq:SpectralDef} will match the conventional equilibrium spectral function of the initial configuration and at  $T=+\\infty$ it will match that of the system at its new equilibrium.  \nFor intermediate values of average time,  \\eqref{eq:SpectralDef} can generically display rather wild oscillations and indeed need not remain positive for positive frequencies (see, for instance, \\cite{Balasubramanian:2012tu}). This feature is similar to the ``negative probabilities'' displayed by the Wigner distribution and may be countered by coarse-graining with a Gaussian filter.  \n \nOne consequence of the inherent non-locality of the definition \\eqref{eq:SpectralDef} is that for $T<0$ the spectral function is already influenced by the effects of the quench; this does not represent non-causality, it is simply because for a large enough relative time, the interval centered around $T$ will necessarily extend through the time of the quench.   \nOne could choose an alternate definition removing this feature, for instance by considering the retarded Green's function as a function of final time $t_2$  and relative time $t=t_2-t_1$, and Fourier transforming with respect to the relative time $t$ at fixed final time. (Note that the retarded Green's function vanishes for $t<0$.)\n\n \n\n\\section{Holographic setup}  \n\nHow can a charged quench be described in AdS\/CFT? In AdS\/CFT the field theoretical problem in $d$ dimensions is dualized in an equivalent gravitational problem living in a $d+1$ dimensional bulk, described by classical gravity in the large $N$ limit of  the  gauge theory. When the bulk is characterised by an Anti-de-Sitter (AdS) geometry it describes the vacuum of a conformal field theory on\nthe boundary at zero temperature and  density. To study the non-equilibrium  physics associated with an instantaneous quench in the zero density theory, a standard simple approach (see, for instance, \\cite{Bhattacharyya:2009uu,Balasubramanian:2010ce,Wu:2012rib,Hubeny:2007xt,AbajoArrastia:2010yt,Albash:2010mv,Liu:2013iza, Balasubramanian:2011at,Allais:2011ys,Callan:2012ip,Hubeny:2013hz})  is to inject at the AdS boundary a shell of light-like ``dust\" (or other matter that effectively behaves like dust) in  AdS$_{d+1}$. This corresponds in the field theory with the sudden creation of an excited state whose time evolution is subsequently determined by the equivalent gravitational evolution in the bulk.   A time-dependent metric describing such a process is available in closed form; it is the AdS version of a metric derived by Vaidya in the 1930's.   After some time this in-falling shell of dust will form a black brane, which in turn encodes a thermal equilibrium state in the field theory: this is the holographic description of the anticipated thermalization of the field theory in the long time limit. \n\nThis in-falling shell paradigm of holographic thermalisation was recently extended to Einstein-Maxwell gravity, where one can consider the  injection of a shell of dust which is {\\em electrically charged} \\cite{Galante:2012pv,Caceres:2012em}.  Fig.~\\ref{fig:Penrose} illustrates the Penrose diagram for the corresponding Reissner-Nordstr\\\"om-AdS-Vaidya spacetime.\n\\begin{figure}[ht] \n\\centering \n\\includegraphics[width=0.28 \\textwidth]{penroseRN.pdf}\n\\caption{ Penrose diagram for Vaidya AdS Reissner-Nordstr{\\\"o}m (RN) spacetime. }\\label{fig:Penrose}\n\\end{figure} \nAlthough the (scale-dependent) equilibration time becomes dependent on whether the chemical potential is large or small \ncompared to temperature, the gross evolution in the bulk is similar as in the zero density case: after some time a charged Reissner-Nordstr{\\\"o}m (RN) black brane will form, describing a holographic strange metal at finite temperature.\n\nIt has been known for a while that, for sufficiently relevant scaling dimensions of the fermion operators of the zero density CFT,  Fermi liquid like \nquasiparticles form  at finite density \\cite{Liu:2009dm,Cubrovic:2009ye,Faulkner:2009wj}, characterised by a propagator  \n \\begin{equation}\\label{fermion2pt}\nG(\\omega,k)=\\frac{1}{\\omega-v_F(k-k_F)+\\Sigma(\\omega,k)}\n\\end{equation}\nwith self energy $\\Sigma \\sim \\omega^{2\\nu_{k=k_F}}$ where $\\nu_k$ is associated with the scaling dimension in the AdS$_2\\times \\Rbar^2$\nnear horizon geometry. We deliberately choose parameters such that $2 \\nu_{k_F} > 1$, so that sharp quasiparticles are formed in the finite density \nsystem.\n\nIt is however by now well understood that, when backreaction from bulk fermions is taken into account, the RN black hole is quantum mechanically unstable to the formation of an electron star \\cite{Hartnoll:2009ns,Hartnoll:2010gu,Cubrovic:2010bf,Cubrovic:2011xm,Allais:2013lha} (see \\cite{Hartnoll:2011fn} for a review). This also corresponds to a thermodynamic instability, in the sense that for fixed temperature and chemical potential, the electron star solution has lower free energy than the RN black brane. While it would be interesting to study this effect in the present fixed-energy (as opposed to fixed-temperature) context, in the present paper we choose to work in the limit in which we treat the bulk classically (which can be thought of as ignoring perturbative and non-perturbative $1\/N$ corrections).\n \nIn our $d=3$ case, the metric and the bulk spacetime gauge fields are  given  in Eddington-Finkelstein coordinates    by \n\\begin{equation}\n\\begin{aligned}\n&ds^2 = \\frac{1}{z^2} \\left(-f(v,z) dv^2 -  2 dv dz  + d\\vec{x}^2 \\right) \\ , \\\\\n&A = g(v,z) dv  \\, , \n\\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation}\n\\begin{aligned}\nf(v,z) &=1+  \\Theta(v)\\left( - ( 1 + Q^{2}) z^3 +  Q^2 z^4 \\right)  \\ , \\\\\n\\quad g(v,z) &= \\Theta(v) \\mu (1- z)  \\ . \n\\end{aligned}\n\\end{equation}\nAt the  spacetime boundary $z=0$, the bulk lightcone time $v$ coincides with the field theory time.\nAs in \\cite{Liu:2009dm} we work with dimensionless quantities obtained by rescaling such that the horizon is fixed at $z=1$. The temperature of the final black hole produced is  $\\mathcal T=\\frac{1}{4\\pi} \\left(3 - Q^2 \\right)$  and the parameter $\\mu= g_F Q$ can be identified with the chemical potential of the theory where $g_F$ is the $U(1)$ coupling which we shall set to unity.\n\nTo determine fermionic spectral functions, we use the same strategy that was employed in \\cite{Balasubramanian:2012tu} for scalar spectral functions. The retarded propagator of an operator is determined by computing its expectation value following a perturbation by a delta-function source \\cite{Iqbal:2008by}. The expectation value and the source are identified with certain coefficients in a near-boundary expansion of the dual bulk field. The bulk field is obtained by solving its equation of motion subject to boundary conditions determined by the delta-function source and initial conditions that set the field to zero outside the future lightcone of the source.  \n\nTo implement this procedure for a fermionic operator of conformal dimension $\\Delta$ and $U(1)$ charge $q$, we consider the Dirac equation for a bulk fermionic field of mass $m$, \n\\begin{equation}\n(\\slashed{D} - m)\\Psi \\equiv \\gamma^{\\mu}\\partial_{\\mu}\\Psi  + \\frac{1}{4} \\gamma^{\\mu} \\omega_{\\mu AB}\\Gamma^{AB}\\Psi - i q \\gamma^{\\mu}A_{\\mu} \\Psi  - m \\Psi = 0  \\ , \n\\end{equation} \nin which $\\gamma^\\mu$ are the curved space gamma matrices of the $AdS$ spacetime, $\\omega_{\\mu AB}$ the spin connection, $\\Gamma^A$ the constant bulk gamma matrices in the local tangent frame and  $\\Gamma^{AB} = \\frac{1}{2} [\\Gamma^A,\\Gamma^B]$ the antisymmetric generators of Lorentz transformations. \n\nThe conformal dimension is related to the mass of the bulk field by $(\\Delta-3\/2)^{2}=m^{2}$ or $\\Delta_{\\pm}=3\/2\\pm m$. One should keep in mind that the unitarity bound on fermionic CFT operators is $\\Delta>(d-1)\/2=1$. This implies that in the mass range $m<-1\/2$ only the choice $\\Delta=\\Delta_{-}$ is allowed and in the range $1\/2<m$ only $\\Delta=\\Delta_{+}$. In the mass range $-1\/2<m<1\/2$, both $\\Delta=\\Delta_{-}$ and $\\Delta=\\Delta_{+}$ are allowed. These two possibilities are related by switching the roles of the source and the expectation value in the identification with coefficients of the bulk field. Throughout this paper, we are assuming that the conformal dimension of the operator is \n\\begin{equation}\n\\Delta=\\Delta_{+}=\\frac32+m.\n\\end{equation} \nWe will consider operators of  conformal dimension $\\Delta \\leq 3\/2$ (with the main focus of the analysis on the case $m=0$).  Spectral functions of fermionic operators in this conformal dimension range have been studied at equilibrium in the RN geometry  in \\cite{Liu:2009dm,Cubrovic:2009ye,Faulkner:2009wj}.    \n\nTo proceed, we will choose the spatial momentum to be aligned in the  $x^{1}$ direction and define\n\\begin{equation} \\label{eq:spinorresc}\n\\Psi = z^{m+ \\frac{1}{2}} e^{i k x^{1}  }   \\left(  \\begin{array}{c} y_{+} (v,z) \\\\  z_{+} (v,z) \\\\ y_{-} (v,z) \\\\ z_{-} (v,z)\\end{array} \\right) \\ . \n\\end{equation}\nWith a  judicious choice of gamma matrices, \n\\begin{equation}\n\\Gamma^{v}=\\left( \\begin{array}{cc}  0& i \\sigma_{2} \\\\ i\\sigma_{2} & 0  \\end{array}\\right) \\ , \\quad\n\\Gamma^{z}=\\left( \\begin{array}{cc}  \\mathbb{I}_{2} & 0 \\\\ 0 & -\\mathbb{I}_{2} \\end{array}\\right) \\ , \\quad\n\\Gamma^{1}=\\left( \\begin{array}{cc}  0&  \\sigma_{1} \\\\   \\sigma_{1} & 0  \\end{array}\\right) \\ , \\quad\n\\Gamma^{2}=\\left( \\begin{array}{cc}  0&  \\sigma_{3} \\\\   \\sigma_{3} & 0  \\end{array}\\right) \\ , \\quad\n\\end{equation}\nand frame fields,\n \\begin{equation}\n e^{v} = \\frac{1}{z}\\left(d z +\\frac{1}{2} ( f(v,z)+1) dv \\right)\\ , \\quad e^{z} = \\frac{1}{z}\\left(d z +\\frac{1}{2} (f(v,z)-1) dv \\right)\\ , \\quad    e^{x^{i}  } = \\frac{1}{z}dx^{i}\\ ,\n  \\end{equation}\none finds the equations of motion for $y_{+}$ and $z_{-}$ decouple from those of $y_{-}$ and $z_{+}$  \\cite{Liu:2009dm}.  It suits our purpose to define the combinations  $\\alpha = y_+ + z_-$ and $\\beta= y_+- z_-$,\nin terms of which the equations of motion become \n\\begin{equation}\n\\begin{aligned} \n0&=  (-1 + m) \\alpha  - (m + i k z) \\beta + z \\partial_{z}\\alpha  \\ ,    \\\\\n0&=  -2 (m - i k z) \\alpha + \\left( z \\partial_{z}f +  2 f (m-1 )+4 i q z g \\right) \\beta  + 2 z f \\partial_{z}\\beta - 4 z \\partial_{v}\\beta \\ .  \\label{eq:eqms}\n\\end{aligned}\n\\end{equation}  \nSimilar equations for $y_{-}$ and $z_{+}$ can be obtained by substituting $k \\leftrightarrow - k$ in these.    \n\n\nSince we are working in a mixed representation, in which only spatial momenta are Fourier transformed at first, the prescription for finding the  bulk-to-boundary retarded Green's  function is to simply   consider the causal response to a delta-function source on the boundary. Then, using the standard holographic dictionary identifying the source and vev of the boundary CFT operator with the asymptotic behaviour of the bulk field \\cite{Henningson:1998cd,Iqbal:2009fd}, one can extract the boundary retarded Green's function $G_R$. Whilst $G_R$ is a $2\\times2$ matrix, in this basis of gamma matrices it is diagonalised.  \n\nSpecifically, $z_{-}$ and $y_{+}$  have the asymptotic expansion\n\\begin{equation}\n\\begin{aligned}\nz_{-} &\\approx z^{1-2m} (A + O(z)) + z^{2} (B + O(z)) \\, ,\\\\\ny_{+} &\\approx z^{2-2m} (C + O(z)) + z ~(D + O(z))\\, ,\n\\end{aligned}\n\\end{equation}\nwhere $A$ is identified as the source and $D$ is identified as the expectation value. $C$ and $B$ are local functions of $A$ and $D$ (and of their derivatives), respectively, as determined by the asymptotic Dirac equation.\\footnote{See for instance \\cite{Liu:2009dm}. Notice however the different rescaling and chirality conventions.}\n\nTo invoke a delta-function source at a time $t_1$ on the boundary one thus enforces \n\\begin{equation}\n\\lim_{z\\to0} z^{2m -1} z_{-} (v,z)=\\lim_{z\\to 0} z^{2m-1}\\frac{\\alpha(v,z) -\\beta(v,z)}{2} = \\delta(v-t_1) \\ ,\n\\end{equation}\nand to ensure that the propagator is causal (i.e., the retarded propagator) we impose that \n\\begin{equation}\n\\beta(v<t_1, z ) = 0 \\ . \n\\end{equation}\nBy virtue of the Dirac equation, this is sufficient to ensure that also $\\alpha(v<t_1,z)= 0$. With these conditions one can then determine the appropriate solution to \\eqref{eq:eqms}. \n\nFor $t_2 > t_1$ the source has no support and the expectation value of the boundary operator can be easily extracted. The upper component of $G_R$ can  be calculated as  \n\\begin{equation} \nG_{11}(t_2, t_1)  = -i \\lim_{z\\to0} z^{-1} y_{+}(v_2,z) = -i \\lim_{z\\to0} z^{-1} \\frac{\\alpha(v_2,z) +\\beta(v_2,z)}{2}  \\ ,  \n\\end{equation} \nwith a similar  definition for the lower component $G_{22}$ coming from the analogous equations for  $y_{-}$ and $z_{+}$. \n\n\n\\section{Numerical solution strategy}\n\nTo solve the equations of motions  \\eqref{eq:eqms}  we can distinguish three cases: a) when the relative time interval  $t=t_2-t_1$ occurs entirely before the quench; b) when it extends across the quench and c) when it occurs entirely after the quench.  \n\nIn case a) one needs only to consider the pure AdS region of the Vaidya spacetime for which exact analytic expressions are known for the bulk-to-boundary retarded Green's function.  \n\n In case b)  the same analytic expressions may be used for the AdS portion of the spacetime and in particular on the shell of null dust at $v=0$.  We then   employ numerical methods to propagate these analytic expressions forwards from the shell at $v=0$ to fill out the remainder of the spacetime outside the horizon. \nIn particular, knowing $\\beta$ on a fixed $v$ slice, the first equation in \\eqref{eq:eqms} with suitable boundary conditions can be integrated to obtain the bulk profile  for $\\alpha$.  Given the bulk profiles for $\\alpha$  and $\\beta$ on a fixed $v$ slice the second equation in \\eqref{eq:eqms} is used to evolve $\\beta$ to a subsequent time. \nTo perform the integration in the radial, $z$, direction we employ a  Chebyshev pseudospectral method, combined with an explicit $4^{th}$ order Runge-Kutta method in the $v$ direction.   For numerical purposes, it is also necessary to impose a cut-off at late $v_{cut}$ in our numerical integration.  To generate the plots in the next section we took  $N=27$ Chebyshev points, a Runge-Kutta time step $\\delta v= 0.05$ and a cut-off in the Runge-Kutta integration $v_{cut}=60$.   Details of the convergence and accuracy of these methods are provided in appendix A where we show that for the chosen values of parameters the results for the spectral function are convergent to within the resolution of plots shown.\n \nIn case c), although the relevant geometry is the AdS-RN black hole, one does not have access to analytic expressions for the retarded propagator in the mixed representation $G_R(t, k)$.  We found a sensible approach in this case to  first calculate $G_R(\\omega, k)$ in Fourier space  as in  \\cite{Liu:2009dm}  and to then perform an inverse Fourier transform to find the desired Green's function in the mixed representation.     \n\nThe next step to obtain our time-dependent spectral function is to perform the Wigner transformation to Fourier space as described  in \\eqref{eq:Wignertrans}. Depending on the conformal dimension of the operator,   $G_R(T,t, k)$ can exhibit power law divergences as $t\\rightarrow 0$, which need to be appropriately regulated.  To address  this problem one can take  \n the ratio of $G_R(T,t, k)$ with another analytically known and appropriately regulated function ${\\cal A}(t)$  such that the ratio is finite as $t\\rightarrow 0$.  If the Fourier transform of  ${\\cal A}(t)$ is also known analytically, then a simple application of the convolution theorem can be used to establish  $G_R(T, \\omega, k)$.   In this work, Fourier transformations are implemented numerically using discrete Fourier routines.  To avoid artefacts associated to the late time cut-off $v_{cut}$ of our numerical evolution, e.g.\\ Wilbraham-Gibbs or ringing phenomena,  we include a Lanczos $\\sigma$ factor in our Fourier transformation \\cite{Lanczos}. \n\n\n\\section{Results}\n\nFig.~\\ref{fig:DensityPlots} shows the spectral sum (the trace of the spectral function matrix),\n\\begin{equation}\nA(T, \\omega, k)=-2\\left[ {\\rm{ Im}}(G_{11}(T, \\omega, k))+ {\\rm{ Im}}(G_{22}(T, \\omega, k))\\right] \\ , \n\\end{equation}\nin the $\\{\\omega, k\\}$ plane for a fermionic operator of conformal dimension $\\Delta = 3\/2$ (dual to a bulk field of mass $m=0$)  when we quench the system to a near extremal final state:  the chemical potential is quenched from $\\mu =0 $ to $\\mu = 1.7$ and the temperature is kept close to zero (recall at zero temperature $Q=\\sqrt{3} \\approx 1.732$).\nA small non-zero temperature has the advantage of stabilising the numerics, while being closer to what one would obtain in a real experimental situation. \n \\begin{figure}[ht]\n \\begin{center}\n\\includegraphics[width=0.32 \\textwidth]{densityplotM0Q1-7TavMinusInfty.pdf} \n \\includegraphics[width= 0.32 \\textwidth]{densityplotM0Q1-7Tav-5.pdf}  \n\\includegraphics[width=0.32 \\textwidth]{densityplotM0Q1-7Tav-2-5.pdf}  \\\\\n \\includegraphics[width=0.32 \\textwidth]{densityplotM0Q1-7Tav0.pdf} \n \\includegraphics[width=0.32\\textwidth]{densityplotM0Q1-7Tav1-5withMu.pdf}\n\\includegraphics[width=0.32\\textwidth]{densityplotM0Q1-7TavPlusInfty.pdf}\n  \\end{center}\n\\caption{False colour density plots  for spectral sum $A= -2 \\left[ {\\rm{ Im}}(G_{11})+ {\\rm{ Im}}(G_{22})\\right]$  for bulk field of mass $m=0$ when quenched to $Q=1.7$,   at   (from  top to bottom) $T=\\{-\\infty,\n-5,-2.5,  0,   1.5,+\\infty\n\\}$. For reference we plot, in red dashed lines, lightcones centered around $\\omega =0$.} \n  \\label{fig:DensityPlots}\n\\end{figure} \n\nThe top central panel of Fig.~\\ref{fig:DensityPlots} shows the result at an early average time $T=-5$ and we see that whilst the density   is roughly symmetric around the light cone centered at $\\omega=0$ (indicated in the figure by the dashed red lines), a number of oscillations have already begun to influence the profile. In the top right panel, at $T=-2.5$, the period of the oscillations has roughly doubled, as can be seen more clearly in  Fig.~\\ref{fig:SpectralSection}.\n\nIn the  bottom left panel, at $T=0$, the oscillations have essentially disappeared, and the spectral function has developed a clear asymmetry between positive and negative frequency branches.  In the bottom central panel, at $T=1.5$, it is now clear that the asymptotic behaviour of the peak of the spectral function has shifted down still further, characteristic of a system at finite density.\n\nTaking Fig.~\\ref{fig:DensityPlots} at face value, after a transient regime at short times, characterised by oscillations  of the ``critical Dirac cones\", it appears that at longer times a Fermi energy and Fermi surface develop. Observing that the asymptotic ``Dirac cones'' gradually move down, it is tempting to define a notion of time-dependent effective Fermi energy, or equivalently time-dependent effective chemical potential, $\\mu_{eff}(T)$, which measures how much the asymptotic cones have moved down. (This is indicated by the green dashed lightcone in the bottom central panel of Fig.\\ \\ref{fig:DensityPlots}.)\\footnote{At a practical level we numerically calculate the location of the maximum value $\\omega_{\\star}$ of the spectral sum at a fixed large $k$ and use this to define  $\\mu_{eff}(T)= k-\\omega_{\\star}$. However, since the peaks of the spectral functions have a finite width and a profile that depends on both $Q$ and $\\Delta$, in what follows to make true comparisons we always take the ratio of this quantity with the same quantity calculated in the final equilibrium state.}\n  \\begin{figure}[h]\n  \\begin{center}  \n      \\includegraphics[width= 0.45\\textwidth]{peakprofilewposM0Q1-7Tav-5.pdf} \\qquad\n      \\includegraphics[width= 0.45\\textwidth]{peakprofilewposM0Q1-7Tav-2-5.pdf} \n \\end{center}\n  \\caption{Detail of section of  spectral sum along  $\\omega = k-2$ at average times   $T=-5$ (left) and $T = -2.5$ (right) highlighting an approximate halving of the frequency of oscillations.}\n  \\label{fig:SpectralSection}\n\\end{figure} \n\nFig.~\\ref{fig:omega0Plots} \n\\begin{figure}[h]\n  \\begin{center}\n    \\includegraphics[width=0.5\\textwidth]{omega0plotsv2.pdf}\n       \\end{center}\n\\caption{Plots of spectral sum at $\\omega = -10^{-3}$ for field of mass $m=0$ when quenched to $Q=1.7$ for $T= \\{-5,-2.5,-0.5,0,0.25, 0.5, 1.5,2.5,5, 10,15 \\}$. The higher $T$, the higher the maximal value.  Also shown is a line at $k_F \\approx 0.92$, the value of the Fermi momenta established in \\cite{Liu:2009dm}. }   \\label{fig:omega0Plots}\n\\end{figure} \nshows the accumulation of a peak at zero frequency for a number of average times.  We see for very early times (the curves with lowest peaks in Fig.~\\ref{fig:omega0Plots}) that  the spectral function attains its maximum very close to zero momenta, as would be expected at zero density,  and exhibits clear oscillations including negative spectral weight.  As the average time evolves one finds that these oscillations die out, the location of the maximum at zero frequency migrates to larger values of $k$ and the value at the maximum increases.  The edge at larger $k$ sharpens and a clear peak develops with the location of the peak moving to where ultimately a sharp spike at $k=k_F \\approx 0.92$ \\cite{Liu:2009dm} will occur. \n\n\nIn Fig.~\\ref{fig:chempot} \n\\begin{figure}[h]\n  \\begin{center}\n    \\includegraphics[width=0.5\\textwidth]{effchemv2.pdf}\n       \\end{center}\n\\caption{Plots of the relative effective chemical potential $\\Delta \\mu(T) = \\mu_{eff}(T)\/\\mu_{eff}(+\\infty)$ as a function of average time, $T$, for $m=0$ quenches with $Q = 0.5$ (red circles), $1$ (orange squares), $1.25$ (cyan diamonds), $1.7$  (blue triangles).  }   \\label{fig:chempot}\n\\end{figure} \nand Fig.~\\ref{fig:Deltachempot} \n\\begin{figure}[h]\n  \\begin{center}\n    \\includegraphics[width=0.5\\textwidth]{deltacomparev2.pdf}\n       \\end{center}\n\\caption{Plots of the relative effective chemical potential as a function of average time for  quenches with $Q =1$  for different conformal dimensions $\\Delta = 1.5$  (red circles), $\\Delta = 1.35$ (orange squares), $\\Delta = 1.25$ (cyan diamonds), $\\Delta = 1.15$  (blue triangles) and  $\\Delta = 1.01$  (green inverted triangles). }  \\label{fig:Deltachempot}\n\\end{figure} \nwe show how this effective chemical potential $\\mu_{eff}(T) $  evolves as a  function  of time for various choices of the net charge density and fermion scaling dimension.  At late times, the system seems to settle in an equilibrium (quasi) Fermi liquid with a Fermi surface that obeys the Luttinger Volume theorem (which states that  the volume enclosed by the Fermi surface is proportional to the fermion charge density) at a finite temperature, chosen to be small compared to the chemical potential in the examples we show.  \nIn Fig.~\\ref{fig:chempot} we examine this effective chemical potential as a function of time for quenches with different values of $Q$ and find several distinct features. As might be anticipated, at $T=0$ the effective chemical potential appears to attain half its final value.  For $T>0$ there is a universality in the approach to the final value of chemical potential, while for $T<0$ the onset of chemical potential gain occurs later for larger $Q$.  \nFor charges $Q$ of order 1, i.e., comparable to the extremal value $\\sqrt3$, the unit of time in  Fig.~\\ref{fig:chempot} is of order the inverse final chemical potential. \nFig.~\\ref{fig:chempot} shows that the effective chemical potential is built up on a time scale of the same order.\nIn Fig.~\\ref{fig:Deltachempot} we examine this effective chemical potential for different values of the fermionic operator conformal dimension (bulk mass) and find a final clear feature: the smaller $\\Delta$, the quicker chemical potential is acquired.\n\nThe effective chemical potential can be viewed as a new non-local probe of thermalization. Probes considered in previous work include equal-time two-point functions, Wilson loops and entanglement entropy \\cite{Balasubramanian:2010ce,Galante:2012pv,Caceres:2012em}, in which case the thermalization time depends on the spatial extent of the probe. If one chose probes of size set by the final chemical potential, one would find thermalization times of the same order as in Fig.~\\ref{fig:chempot}.\n\n\n\\section{Caveats and further discussion}\n\nIn interpreting our results, several caveats need to be taken into account. A first caveat, which we have already mentioned, is that our ``quasi\" Fermi liquid realised at long times is in fact a false vacuum artefact associated with the large $N$ limit. \nAs we already discussed, it is understood that the true equilibrium state is a Fermi liquid which is dual to the electron star in the bulk. The dynamical ``uncollapse'' of the black brane in the electron star requires a quantised description of the geometry: in classical gravity this cannot happen.  However, as long as $N$ is not too small we expect a separation of time scales such that the further relaxation from the RN ``quasi'' Fermi liquid to the electron star Fermi liquid will take place at a much later time. We leave this problem of the dynamical formation of the electron star due to quantised geometry as a challenge for holography in general.\n\nNext, while our notions of time-dependent spectral function and of time-dependent effective Fermi energy are well-defined and reduce to the appropriate equilibrium concepts in static situations, we should ask to what extent these quantities are measurable in a lab, and exactly which physical information they carry.\n\nIn equilibrium, spectral functions can be experimentally determined using Angle-Resolved Photoemission Spectroscopy (ARPES). Actually, electrons can only be emitted from occupied states, so what ARPES measures is really the product of the spectral function and the Fermi factor. Technically, this product equals the ``lesser Green function'' in Fourier space, often denoted $G^<(\\omega, k)$ or $D^<(\\omega, k)$; see, for instance, \\cite{Bellac}. Because of the fluctuation-dissipation theorem, knowledge of the spectral function suffices to determine the lesser Green function, and vice versa (at least for nonzero temperature).\n\nAway from equilibrium, time-resolved ARPES is the tool of choice. As shown in \\cite{Freericks}, the lesser Green function (which now depends on two moments of time) still carries the relevant information. In quite close analogy to \\eqref{eq:Wignertrans}, it should be Fourier transformed with respect to the relative time coordinate, with the additional complication that the experimental setup introduces time windows for both moments of time. For slowly varying systems, at least some version of the fluctuation-dissipation theorem remains valid \\cite{Keller_Jarrell, Mukhopadhyay:2012hv}. Far from equilibrium, however, the retarded and lesser Green functions are not straightforwardly related  (see, for instance, \\cite{CaronHuot:2011dr,Chesler:2011ds, Balasubramanian:2012tu}), making it much less clear how ARPES is related to time-dependent spectral functions. While in current versions of time-resolved ARPES the fluctuation-dissipation relation is a good approximation, this would not be the case for an idealized ARPES experiment preformed on our system, which is quenched instantaneously.\n\nOne may wonder about the meaning of the oscillations and of the negative regions in our time-dependent spectral functions. Our present understanding is that these are due to simple interference between the pre-quench and post-quench periods, as was the case for the quenched harmonic oscillator discussed in \\cite{Balasubramanian:2012tu}. One piece of evidence supporting this statement is that the wavelength of the oscillations in $\\omega$ seems to be set by the time $T$ to the quench (see Fig.~\\ref{fig:SpectralSection}). This presumably makes it hard to relate this quantity to practical experiments. Given a system prepared in an out-of-equilibrium state, what one would really like to probe is how it behaves {\\em after} it has been prepared in that state. But in our setup, the retarded Green function with both moments of time after the quench behaves exactly as if the system were in equilibrium (as discussed in \\cite{Bhattacharyya:2009uu} for a thermal quench). So if one defined a time-dependent spectral function using only those ``post-quench'' data (by introducing appropriate time windows), the oscillations and negative regions would disappear. The reason for the equilibrium behaviour is that the post-quench retarded propagator is only sensitive to the bulk geometry outside the shell, which agrees with that of a charged black brane.\n\nNevertheless, our system right after the quench is far from equilibrium: as also mentioned in \\cite{Bhattacharyya:2009uu} (and worked out in detail in \\cite{Bhattacharyya:2009uu, Wu:2012rib, Balasubramanian:2010ce, Hubeny:2007xt,AbajoArrastia:2010yt,Albash:2010mv,Liu:2013iza, Balasubramanian:2011at,Allais:2011ys,Callan:2012ip,Hubeny:2013hz}), spatially nonlocal observables do not immediately take their equilibrium values. Therefore, time-resolved ARPES, which does not probe the retarded propagator but the lesser Green function, should not give equilibrium results either (at least in principle: in practice it may not be easy to have good enough time resolution to see deviations from equilibrium). From this point of view (linear response after the quench being trivial, but time-resolved ARPES not), it would be interesting to compute the lesser Green function for our system. While it is in principle known how to do this (see, for instance, \\cite{Herzog:2002pc,Skenderis:2008dg}), we leave this computation to future work.\n\n\n\\section*{Acknowledgements}\nWe thank D.~Dudal, N.~Iqbal,  V.~Ker\\\"anen and E.~Keski-Vakkuri for useful discussions, M.~Heller for helpful lectures on numerical methods, and Z.~Cao and Y.~Tian for stimulating discussions on the numerical strategy.\nB.C.\\ and J.Z.\\ thank the organizers of the 2014 Amsterdam String Workshop for hospitality during the final stages of this work. This work was supported in part by the Belgian Federal Science Policy Office through the Interuniversity Attraction Pole P7\/37, by FWO-Vlaanderen through projects G020714N and G.0651.11, by the Vrije Universiteit Brussel through the Strategic Research Program ``High-Energy Physics'' and by the European Science Foundation Holograv Network. J.Z.\\ acknowledges support of a grant from the John Templeton Foundation. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation. J.V.\\ is Aspirant FWO. D.T.\\ and F.G.\\ are FWO postdocs.\n\n\n\n\\begin{appendix}\n\\section{Numerical convergence}\nWe provide some details illustrating the convergence of the numerical methods used.  There are three key variables that influence the numerical accuracy: {\\it i}) $N$, the number of nodes used in the pseudospectral method to integrate in the radial direction; {\\it ii}) $\\delta v$, the time-stepping used in the Runge-Kutta integration in the $v$ direction; {\\it iii}) $v_{cut}$, the final relative time used for the numerical integration.  The convergence for each  of these is illustrated in Figs.~\\ref{fig:convcheb}--\\ref{fig:convcut} showing the imaginary part of the Fourier transformed Green's function $G_R(\\omega, k)$ for the case of $Q=1.7$, $T=-2.5$ at a fixed $k=0.9$. These values are chosen since they highlight the most challenging parts to obtain numerically. Setting $Q=1.7$, as was used in the bulk of this paper, means that we are at  low temperature and hence $G_{R}(t,k)$ is not thermally suppressed.  The choice  $k=0.9$ corresponds to focusing on the region of momenta space of most interest, where we might anticipate long lived excitations. With $T=-2.5$  we capture the difficult rapidly fluctuating parts of the spectral function arising from transient behaviour in the quench regime. \n\\begin{figure}[ht]\n  \\begin{center}\n    \\includegraphics[width=0.48   \\textwidth]{ChebconvTavm2p5.pdf}\n       \\end{center} \n\\caption{Convergence as number of Chebyshev nodes is varied  (we fix the Runge-Kutta step  $\\delta v=0.05$ and $v_{cut}=60$).  Very good convergence, at the resolution of this plot, is achieved after $N=25$ points. The most sensitive point is the absolute height of the transient negative peak, but even this is well converged at $N=30$.  In the paper we used $N=27$ to balance resource use and accuracy. } \\label{fig:convcheb}\n\\end{figure} \n \\begin{figure}[ht]\n  \\begin{center}\n    \\includegraphics[width=0.48\\textwidth]{RKconvTavm2p5.pdf}\n       \\end{center}\n\\caption{Convergence as Runge-Kutta time step $\\delta v= \\frac{1}{5}2^{-n}$  is varied (we fix $N=30$ Chebyshev nodes and $v_{cut}=60$).  Very good convergence, at the resolution of this plot the different curves are virtually indistinguishable, is achieved with time stepping $\\delta v = 0.05$ as was adopted in the paper. }\\label{fig:convRK}\n\\end{figure} \n  \\begin{figure}[ht]\n  \\begin{center}\n    \\includegraphics[width=0.48\\textwidth]{TcutconvTavm2p5.pdf}\n       \\end{center}\n\\caption{Convergence as the final integration time $v_{cut}$ is varied (we fix $N=30$ Chebyshev nodes and the Runge-Kutta step $\\delta v=0.05$).   Here the results are more sensitive but are well converged for $v_{cut} = 45$ (the green, blue and cyan lines  are essentially coincident).  We used $v_{cut}=60$ in the paper. }\\label{fig:convcut}\n\\end{figure} \n\\end{appendix}\n \n \n \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:introduction}}\n\\IEEEPARstart{P}{ath} planning is a method to find an optimal route from the starting point to the target point. It has been widely used in various fields such as robotics \\cite{yu2020path,kumar2020comparison,wohlke2021hierarchies}, drone \\cite{jeauneau2018path,qie2019joint,wang2020multi,wang2020mobile,hayat2020multi,bayerlein2021multi}, military service \\cite{lee2020autonomous,hu2022autonomous}, and self-driving car \\cite{wang2019obstacle,tammvee2021human}. Recently, reinforcement learning (RL) has been mainly studied for the path planning \\cite{lee2020autonomous,wang2020mobile,yao2020path,qiao2020hierarchical,bayerlein2021multi,lin2021collision,cao2021confidence,wohlke2021hierarchies,hu2022autonomous}. To get an optimal solution, it is essential to give enough reward for an agent to reach the goal and to set up a specific environment. Several studies on learning the RL model have proposed to make an agent be robust in a complicated or an unknown environment for the path planning \\cite{li2019deep,yang2020efficient,hu2020voronoi}. However, existing studies have defined one single goal before the learning. That is, the agent's ability to search the path when completed learning can be limited. To make the agent reach a number of goals in a dynamic environment, learning a controllable agent is needed. One of the recent approaches for controlling the agent has a limitation in that the agent can only learn the behavior from trajectories that have been directly experienced \\cite{lee2022learning}. Therefore, the agent can only be under control in the area visited by the agent. In this paper, I focus on learning a fully controllable agent in the path planning using a goal-conditioned RL. Especially, I apply, to the goal-conditioned RL, a bi-directional memory editing and a sub-goals dedicated network to improve the ability to search the path of the agent. \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.40]{fig1.pdf}\n\\caption{Examples of training and test environment in this study. The experiment was set as a simple version of the two-dimensional grid environment. The starting point and the target point were fixed at the location in the training environment. The policy of the agent is to search the path from the starting point to the target point. (a), The training environment was set for the agent to easily get an optimal route. (b), The test environment was set for the agent to get an optimal route in a difficult way. Several sub-goals including the final goal were given to the agent.  }\n\\label{fig1}\n\\vskip -0.2in\n\\end{figure}\n\nIn the goal-conditioned RL, the agent learns the sub-goals, part of the trajectory of the agent \\cite{lee2020weakly,okudo2021subgoal,zhang2021world,chane2021goal,yang2022rethinking}. By learning the sub-goals, eventually, the agent can reach the final goal. This method showed good performance mainly in robotics. However, previous studies have focused on reaching a complicated single goal. In addition, the multi-goal RL model, in which the agent can perform many goals, has been proposed \\cite{bai2019guided,zhao2019maximum}. However, the multi-goals should also be defined before the learning. Unlike these studies, in this present study, I propose an RL framework in which the agent can perform a number of sub-goals in various scenarios. An important point here is that the sub-goals are not defined in advance. In other words, the agent completed the learning reaches the goals, which have never been visited by the agent in the learning.\n\n\nFig \\ref{fig1} shows the examples of the training and the test environments of this study. The training environment has been simply set for the agent to reach the final goal. But the test environments have been set in a difficult way. The agent should go through the sub-goals and reach the target point, even if the starting point and the target point are set in reverse. The difficult missions were also given to the agent such as a round trip. In the test environments, an important thing is that the agent has to reach the goal never been visited in the learning. To perform the goals, the agent must be fully controlled by the sub-goals that are customized by the user. \n\n\n Meanwhile, memory editing occurs in our daily lives. \\cite{bernstein2009tell,schacter2011memory,fernandez2015benefits,phelps2019memory}. The memory editing helps us to get away from mental illnesses such as trauma \\cite{phelps2019memory}. We can also get precise information by editing untidy memories \\cite{fernandez2015benefits}. A recent study used the concept of the memory editing on goal-conditioned RL for the agent to reach sub-goals so that the user can control the agent \\cite{lee2022learning}. However, the agent cannot move the sub-goals in difficult environments. This is because the memory of the RL model is edited only based on one direction of the paths that are moved by the agent. The first purpose of the RL is to achieve the final goal. Despite the sub-goals, the agent can ignore the sub-goals if the sub-goals get in the way of the final goal. That is, the agent does not need to go round and round to reach the final goal. In this study, I developed the concept of the memory editing for the fully controllable agent in the path planning. \n \nLet's assume that we walk to the destination (See Fig \\ref{fig0}). From the route that we walked, we can learn how we reach the destination. We can remember intermediate stops in the route and know that the total path is the overall set, which consists of the intermediate stops. Further, if we recall our memories backward, we can also find the route back to the starting point. For example, in the Fig \\ref{fig0}, when we want to return to the starting point, we know which action we have to perform. However, it is difficult to find an inverse action in a dynamic RL environment. Thus, an inverse module to predict the inverse action is necessary to obtain the exact knowledge to come back to the starting point. \n\nBased on the processes, in which we recall our memories and obtain knowledge, I utilize a bi-directional memory reminiscence and editing (bi-directional memory editing) to obtain various trajectories. Using these trajectories, the agent can learn various behaviors. Furthermore, I use the dedicated network for learning the sub-goals to improve learning efficiency. Finally, I present a reward shaping for the shorter path of the agent. Using these techniques, the agent can achieve various sub-tasks as well as the final goal in the path planning environment. The fully controllable agent can be useful in the environments where we have to consider a number of variables and we have to assume various scenarios. The main contributions of this article are as follows:\n\n\\begin{itemize}\n\\item Using the bi-directional memory editing, we can obtain various trajectories, and can learn the agent to perform various tasks, based on the trajectories. The agent can be fully controlled so that the agent can reach any point in the path planning environment. \\\\\n\\item I employ the sub-goals dedicated network to improve the efficiency of learning the sub-goals. To distinguish the network from the policy network, the agent can focus on performing the various sub-goals.  \\\\\n\\item I propose the reward shaping for the shorter path of the agent. In the path planning, it is important for the agent not only to reach the final goal but also to reach it within a limited time. By applying the reward shaping in the bi-directional memory editing, the agent can reach the final goal within a shorter time. \\\\\n\\item To the best of our knowledge, this study is the first RL methodology for the path planning in that the agent is fully under control. Therefore, the agent achieves the user-defined sub-goals such as a round trip in a difficult test environment. Moreover, the agent can move to the point that has never been reached by the agent in the training. By using this methodology, we can suppose and conduct various scenarios in the path planning. \n\\end{itemize}\n\n\nThe rest of the paper proceeds as follows. Section 2 presents the background of this study. In Section 3, learning a fully controllable agent is proposed. To do this, I introduce the bi-directional memory editing and the sub-goals dedicated network. Furthermore, I propose the reward shaping for the shorter path of the agent. Section 4 provides the results of the experiment. I confirmed whether the agent can successfully perform the sub-goals in various scenarios. Section 5 concludes this paper and calls for future research.\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.6]{fig0.pdf}\n\\caption{ Illustration of the concept of the bi-directional memory editing. After we arrive at the destination, if we recall our memory, we can find out the route for returning to the starting point. Further, we can also obtain knowledge about which action to perform to reach the starting point.    }\n\\label{fig0}\n\\end{figure}\n\n\n\n\n\n\\section{Background}\n\\subsection{Path planning}\n\n\nTo get an optimal solution to reach the target point from the starting point, traditionally, optimization methods have been used in the path planning. Several studies using an A* \\cite{yan2018path,chen2020improved}, a genetic algorithm \\cite{zhou2020trajectory}, and a particle swarm optimization \\cite{huang2018uav,chen2021three, wang2022improved} have been proposed. Combining two optimization methods also has been studied \\cite{jamshidi2020analysis}. \n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=\\textwidth]{fig2.png}\n\\caption{Illustration of the proposed method. The red line indicates the route visited by the agent. The blue line indicates the inverse direction of the route. The inverse module predicts the action when $s_t$ and $s_{t+1}$ are given, and the predicted action is used to collect the reverse trajectories. In the bi-directional memory editing, various sub-goals are generated and stored in the separated replay memory $(D_s)$. The transition samples from $D_s$ are trained by the sub-goals dedicated network.}\n\\label{fig2}\n\\end{figure*}\n\nRecently, with the development of deep learning, studies on the path planning using the RL have mainly been proposed \\cite{lei2018dynamic,lee2020autonomous,liu2021novel,wang2020mobile,yao2020path,yan2020towards,wang2020multi,qiao2020hierarchical,bayerlein2021multi,lin2021collision,cao2021confidence,wohlke2021hierarchies,hu2022autonomous}. They have supposed the specific scenario and set an environment to apply the agent in the path planning. Especially, they have applied their study to robotics \\cite{wang2020mobile,wohlke2021hierarchies,lin2021collision}, \ndrone \\cite{jeauneau2018path, qie2019joint ,yan2020towards, wang2020multi,hayat2020multi,bayerlein2021multi }, and ship \\cite{chen2019knowledge, guo2020autonomous}. Also, they have focused on the one single goal of the agent to reach the target point, avoiding obstacles. After learning is completed, the agent could reach the goal; however, the agent cannot be under control in the previous studies. That is, the agent can only perform the predefined goal. \n\nIn addition, learning user-defined sub-goals has been proposed \\cite{lee2022learning}. However, the agent was partially under control. The agent could not perform the round trip and only moved the area that was visited by the agent. In this study, I focus on learning the fully controllable agent so that the agent can perform various trips, as shown in Fig \\ref{fig1}.b. \n\n\n\n\\subsection{Goal-conditioned RL}\n\nHindsight experience replay (HER) used the sub-goals and the pseudo rewards to make the RL model converge to the final policy \\cite{andrychowicz2017hindsight}. The reason why learning the sub-goals improves the performance of the RL is that the sub-goals are located on the route to reaching the final goal. Several studies developed the HER for an exploration of the agent \\cite{nguyen2019hindsight, fang2019curriculum,lai2020hindsight }. \n\nThe goal-conditional RL models excel to reach the goal through the intermediate sub-goals and show excellent performance on the robotics problem \\cite{nasiriany2019planning,zhao2019maximum,ghosh2019learning,nair2018visual,eysenbach2019search,bai2019guided,eysenbach2020c,lee2020weakly,okudo2021subgoal,zhang2021world,chane2021goal,yang2022rethinking}. They have focused on searching for meaningful sub-goals and improving the performance of the main policy network (high-level policy network). To reach the desired policy, it is important to induce the agent to reach the landmark of the sub-goals \\cite{nair2018visual,nasiriany2019planning,bai2019guided,zhang2021world,kim2021landmark} and to improve sample efficiency \\cite{nachum2018data,eysenbach2019search,gurtler2021hierarchical}. The existing studies with learning the sub-goals are similar to this study in terms of generating the sub-goals and relabeling the rewards. \n\n\n\nHowever, in this study, the sample efficiency and searching for the valid sub-goals are not essential. By performing the bi-directional memory editing, we can get trajectories two times more than when bi-directional memory editing is not performed. Then, enough sub-goals can be collected for learning the sub-goals. Also, it does not matter the time when the sub-goals dedicated network is trained, which is separated from the original policy network. Whether the sub-goals are trained after learning the policy network or during learning the policy network, if sufficient trajectories are gathered, the agent can learn various behaviors and sub-goals. The main purpose of this study is to make the agent under control for performing various tasks which are not defined in the training environment. After learning is completed, the agent can achieve various user-defined sub-goals as well as the final goal.  \n\nI consider a discounted, finite-horizon, goal-conditioned Markov decision process (MDP) defined as a tuple ($\\mathcal{S}, \\mathcal{G}, \\mathcal{A}, p,R, \\gamma, H$), where $\\mathcal{S}$ is the set of state, $\\mathcal{G}$ is the set of goals, $\\mathcal{A}$ is set of actions, $p(s_{t+1}|s_{t},a_{t})$ is a dynamics function, $R$ is the reward function, $\\gamma \\in [0,1)$ is the discounted factor, and $H$ is the horizon. In the goal-conditioned RL, the agent learns to maximize the expected discounted cumulative reward $\\mathbb{E}[ \\sum_{t=1}^\\infty R(s_t,g,t)]$, where $g$ is the goal. The objective is to obtain a policy $\\pi(a_t|s_t,g,t)$.\n\n\n\n\n\n\n\\section{Proposed method}\nFig \\ref{fig2} shows the illustration of a summary of the proposed method. For the fully controllable agent in the path planning, I introduce three simple techniques. First, I propose a bi-directional memory editing to generate various behaviors and sub-goals of the agent. Here, to secure reverse trajectories, an inverse module to predict actions is used. Second, to improve the efficiency of learning, I utilize the sub-goals dedicated network separated from the policy network. Finally, I present a reward shaping for the shorter path of the agent.\n\n\\subsection{Bi-directional memory editing}\nThe memory editing is performed to generate sub-goals of the agent. The sub-goals are generated from the trajectories of the agent, and additional rewards are given to the agent. As the agent begins to recognize the sub-goals, the agent can achieve the sub-goals and greedily reach the final goal in the previous studies.\n\nIn the path planning, it is important to allow the agent to visit a wider area such that the agent can visit various locations and learn the optimal route to reach the goal. Thus, if we can get various trajectories more than the trajectories that are actually visited by the agent, it can bring greater benefit to learning the agent's ability for searching the path. In addition to a forward route, the reverse route from the goal to the starting point can be a useful ingredient for the agent to learn various behaviors and sub-goals. To do this, I employ the reverse trajectory to generate various sub-goals and to learn the robust RL model by performing the bi-directional memory editing. \n\nFirst, a forward memory editing is performed, which is described in line 21-24 in Algorithm 1. The sub-goals ($g$) are generated from the forward route, and the state of the sub-goals ($s_{t+1} \\| g$) are made. After that, a backward memory editing is performed, which is described in line 25-29 in Algorithm 1. Reverse transition is $\\{(s_{t+1},a_{t+1}',r_t,s_t)\\}$ whereas original transition is $\\{(s_t,a_t,r_t,s_{t+1})\\}$. Here, it is difficult to obtain $a_{t+1}'$. The reason is that it is not simple to find an action to derive the $s_t$, when $s_{t+1}$ is given. I propose to use the inverse module to obtain $\\hat{a}_{t+1}$ by predicting $a_{t+1}'$ when $s_t$ and $s_{t+1}$ are given. Like the forward memory editing, the sub-goals ($g$) and the state of the sub-goals ($s_{t+1} \\| g$) are generated. Finally, the edited memories are stored in the replay memory for the sub-goals ($\\mathcal{D_s}$).\n\nUsing the bi-directional memory editing, we can obtain two routes from the one single route moved by the agent. Beyond being two times the trajectories, it means that the agent can learn various relationships between the actions and the sub-goals. In the path planning, the agent can almost only learn the route that is visited by the agent. For instance, as shown in Fig \\ref{fig1}.(a), the agent can almost learn the leftward and upward directions because of the location of the goal. However, using the bi-directional memory editing, the agent can learn all directions in various locations. Therefore, by learning a number of sub-goals and behaviors, the agent can reach the goals that were never visited in the training.\n\n\n\n\n\n\\begin{algorithm}[t]\n\\begin{algorithmic}[1]\n\\caption{Learning the sub-goals using bi-directional memory editing }\\label{alg:algorithm1}\n\n\\State Initialize policy network parameters $\\theta_{p}$\n\\State Initialize sub-goals dedicated network parameters $\\theta_{g}$\n\\State Initialize inverse module parameters $\\theta_{iv}$\n\\State Initialize replay buffer for original goal $\\mathcal{D} \\leftarrow \\emptyset$\n\\State Initialize replay buffer for sub-goals $\\mathcal{D_s} \\leftarrow \\emptyset$\n\\Procedure{Learning the sub-goals}{}\n\\For{episode = 1, M}\n\\State  \\verb|\\\\| Simulation stage.\n\\For{each step}\n\\State Execute an action $s_t,a_t,e_t,s_{t+1} \\approx \\pi_{\\theta}(a_t \\mid s_t)$\n\\State Store transition $\\mathcal{E}\\leftarrow \\mathcal{E} \\cup \\{(s_t,a_t,r_t)\\}$\n\\State Optimize inverse module $\\theta_{iv}$ \n\\EndFor\n\n\n\\If{ $s_{t+1}$ is terminal}\n\\State Compute returns $R_t= \\Sigma^\\infty_{k}\\gamma^{k-t}{r}_k$ in $\\mathcal{E}$\n\\State $\\mathcal{D}\\leftarrow \\mathcal{D}\\cup \\{(s_t,a_t,R_t)\\}$\n\\State Clear episode buffer $\\mathcal{E} \\leftarrow \\emptyset$\n\\EndIf\n\\State\n\n\\State   \\verb|\\\\|Bi-directional memory editing\n\\State Generate sub-goals $g$ \n\\State Generate state of sub-goals $s_{t} \\| g$\n \\algorithmiccomment{$\\|$ denotes concatenation}\n\\State Set additional rewards $r_t'$ \n\\State $\\mathcal{D_s}\\leftarrow \\mathcal{D_s}\\cup \\{s_t \\| g,a_t,r_t')\\}$\n\\State Predict the inverse action $\\hat{a_{t+1}} \\leftarrow \\theta_{iv}(s_{t+1},s_t)$ \n\\State Generate reverse transition $\\{(s_{t+1},\\hat{a}_{t+1},r_t,s_{t})\\}$\n\\State Generate sub-goals $g$ \n\\State Generate state of sub-goals $s_{t+1} \\| g$\n\\State Set additional rewards $r_t''$ \n\\State $s_t \\leftarrow s_{t+1}$,$a_t  \\leftarrow \\hat{a}_{t+1}$,$r_t  \\leftarrow r_t''$\n\\State $\\mathcal{D_s}\\leftarrow \\mathcal{D_s}\\cup \\{s_t \\| g,a_t,r_t')\\}$\n\n\\State\n\\State   \\verb|\\\\|Learning stage.\n\\For{k= 1, N}\n\\State  Sample a minibatch $\\{(s,a,R)\\}$ from $\\mathcal{D}$\n\\State \\algorithmiccomment{Optimize policy network  $\\theta_{p}$ }\n\\EndFor\n\\For{k= 1, P}\n\\State  Sample a minibatch $\\{(s \\| g,a,r')\\}$ from $\\mathcal{D_s}$\n\\State \\algorithmiccomment{Optimize sub-goals dedicated network $\\theta_{g}$ }\n\n\\EndFor\n\\EndFor\n\\EndProcedure\n\n\n\\end{algorithmic}\n\\label{algo1}\n\\end{algorithm}\n\n\n\n\n\n\\subsection{The sub-goals dedicated network}\nUsing the bi-directional memory editing, the agent can learn various behaviors and sub-goals. However, the agent at the middle of the route can be confused about where the agent has to go. Because the agent is forced to learn how to go in both directions at one point due to the bi-directionally edited memories. For example, in the Fig \\ref{fig0}, if the agent is located near the tree, the agent learns to move both upwards and leftwards. If the agent is trained using the policy network, the agent can be confused about where the agent has to go if the agent is located near the tree. Because the purpose of the policy network is only to train the agent to reach the final goal. Actually, the agent has been hovering around the middle point of the environment in the experiment. Therefore, I employ the network for the sub-goals separately from the network for the final goal. \n\nThe sub-goals dedicated network only learns the sub-goals, and the original policy network only learns the final goal. To do this, I also employ the replay memory for the sub-goals ($\\mathcal{D_s}$) in addition to replay memory for the final goal ($\\mathcal{D}$). Moreover, using the sub-goals dedicated network can improve the sample efficiency. In general, the capacity of the replay memory is limited and the replay memory is updated with the last transition of the agent. Thus, the agent mainly learns the recent transitions and gradually reaches the goal. However, as previously mentioned, using the bi-directional memory editing, we can obtain a number of sub-goals and behavior of the agent. Therefore, using the separated network and the replay memory, we can learn the agent to reach the sub-goals whenever, during or after the policy network learning.\n\nAlthough the agent of the policy network cannot reach the final policy, the agent of the sub-goals dedicated network can reach the various sub-goals as well as the final goal. The users can fully control the agent by collecting various sub-goals from the bi-directional memory editing and by learning the agent on the sub-goals dedicated network. \n\n\n\n\n\\subsection{Reward shaping for the shorter path}\nIn the path planning, one of the important factors is to make the agent reach the destination within a specific period. To improve the agent's ability, it is necessary to give enough rewards according to the steps to reach the target point. However, in this study, I focus on learning the sub-goals, and additional rewards are given to the agent to reach the sub-goals, regardless of the shortest path. Furthermore, I want to confirm that the agent can reach various sub-points in the environment. Thus, I assume that the environment of this study is a sparse reward environment so that the agent does not need to reach the target point in the shortest path. Rather, due to the exploration bonus, it is likely for the agent to delay the one's arrival. Therefore, we propose the reward shaping when the bi-directional editing is performed for the shorter path of the agent in the path planning.\nWhen the bi-directional editing is performed, the additional rewards ($r_t'$) are given with the corresponding sub-goals. Here, the rewards are reshaped as follows: \n\\begin{eqnarray}\n{r_{tg}} & = & r_{t}' + (dist_{short} \u2013 dist_{s_t})\n\\end{eqnarray}\n: where $dist_{short}$ indicates the number of the shortest steps that can be possibly reached by the agent from $s_t$ to the current sub-goal ($g_s$), and $dist_{st}$ indicates the number of steps of the path that is moved by the agent from $s_t$ to the current sub-goal ($g_s$). That is, in each sub-goal, the agent gets a penalty by the number of steps to reach the sub-goal point. If the agent reaches the sub-goal in the shortest path, the agent does not receive any penalty. Otherwise, the agent receives a penalty depending on the number of steps to reach the sub-goal points. \n\n\n\n\n\\subsection{Learning a fully controllable agent in the path planning}\nFig \\ref{fig2} shows the summary of the proposed method. The inverse module predicts the action of when $s_t$ and $s_{t+1}$ are given in the bi-directional memory editing. The sub-goals from two trajectories are generated. At this time, the reward shaping for the shorter path is performed. Then, the transitions are stored in the replay memory for the sub-goals ($\\mathcal{D_s}$). The sub-goals dedicated network is trained independently with the policy network, whereas the policy network is only trained for the final policy. After learning is completed, in the various scenarios, the agent gets the sub-goals that are defined by the users and tried to achieve the sub-goals as well as the final goal. \n\nAlgorithm 1 shows the procedure in the proposed method in detail. The simulation stage is similar to the other RL methods except for the inverse module to obtain $\\hat{a}_{t+1}$ by predict $a_{t+1}'$ when $s_t$ and $s_{t+1}$ are given. In the bi-directional memory editing, the reverse transition $\\{(s_{t+1},\\hat{a}_{t+1},r_t,s_{t})\\}$ is obtained using the inverse module and various sub-goals are generated and stored in the memory $(D_s)$. In the learning stage, optimize the policy network $(\\theta_p)$ for the final goal and optimize the sub-goals dedicated network  $(\\theta_g)$  for the sub-goals. In fact, it does not matter to learn the goals dedicated network separately after learning the policy network is completed, if the number of the edited memories is enough. \n\n\n\n\n\n\n\n\n\n\n\\section{Experiments}\n\\subsection{Experimental setting}\nIn the experiments, I wanted to confirm that the agent can be fully controlled by the sub-goals and that the agent can achieve various sub-goals, which were never visited by the agent in the learning. Thus, the training environment was constructed in a simpler way, while the test environment in a difficult way. I assumed a number of scenarios to test the agent's ability of the path searching. Furthermore, I wanted to show the effect of the reward shaping for the shorter path.\n\n  Fig \\ref{fig1} shows the first environment. The goal of the agent was set to reach the target point in a simple two-dimensional (2D) grid environment. The reward was 0 except for the agent reaching the target point (+30). The RL model was performed for a total of 10,000 episodes. In the test environment, I set a total of 26 scenarios. In each scenario, several sub-goals were given to the agent. At first, a sub-goal nearest the current location of the agent was given to the agent. Then, if the agent reaches the sub-goal, the next sub-goal nearest the current location was given. Also, if the agent could not reach the given sub-goal, the next sub-goal nearest the current location was given.\n In various scenarios, I observed whether the agent reaches various sub-goals and successfully performs difficult missions such as a round trip. Moreover, I compared with and without the reward shaping for the shorter path in the proposed method. In each scenario, I calculated the number of steps to reach the target point.\n \n\n\n\n\n  Fig \\ref{fig3} shows the second environment. The environment is the 'key-door domain'. The environment has a total of 4 stages and the agent should go through the bonus point (key) to clear the stage (door). Even though the agent reaches the target point (door), if the agent could not pass the bonus point (key), the agent cannot jump up to the next stage. The reward was set +10 for a bonus point, -10 for a penalty point, and +100 for the goal, when the agent goes through these points, respectively. This environment was difficult because of the condition that the agent must pass the bonus point to clear each stage and that the environment was defined as the sparse reward setting. The RL model performed for a total of 50,000 episodes. In the test environment, I set the two scenarios. In each stage, two sub-goals, defined by the user, the bonus point, and the goal, were given to the agent as the sub-goals in an order.\n\n\n\n \\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.35]{fig3.png}\n\\caption{The key-door domain environment. In each stage, the factors (a starting point, a bonus point, a wall, and a target point) were set differently. The agent must go through the bonus point to clear the stage. Even if the agent reaches the target point if the agent did not go through the bonus point, the agent cannot pass by the next stage.}\n\\label{fig3}\n\\end{figure}\n\n\\subsection{Base architecture of the RL model}\n\nSelf-imitation learning (SIL) is an on-policy algorithm to exploit valuable past decisions from the replay memory \\cite{oh2018self}. In the learning stage, the transitions are sampled, and they are trained by the policy network. At this time, if the transitions of the past are not valuable compared to the current value, the transitions are not exploited. That is, the SIL imitates valuable behaviors of the agent in the past. The authors combined the SIL and the on-policy RL model \\cite{mnih2016asynchronous,schulman2017proximal} and proposed the following off-policy actor-critic loss:\n\n\\begin{eqnarray}\n\\mathcal{L}&=&\\mathbb{E}_{s,a,R\\in \\mathcal{D}}[\\mathcal{L}_{policy} + \\beta\\mathcal{L}_{value}] , \\label{eq:sil1} \\\\\n\\mathcal{L}_{policy}&=&-log\\pi_{\\theta}(a|s)(R-V_\\theta(S))_+  , \\label{eq:sil2} \\\\\n\\mathcal{L}_{value}& =&{{1}\\over{2}} \\parallel(R-V_\\theta(S))_+\\parallel ^2 , \\label{eq:sil3}\n\\end{eqnarray}\nwhere $(\\cdot)_+=max(\\cdot,0)$; $\\pi_\\theta$ and $V_{\\theta}(s)$ are the policy network and the value network, respectively, parameterized by $\\theta$. The value loss is controlled by $\\beta \\in \\mathbb{R}^+$. From the $(\\cdot)_+$ operator in the loss, the transition, in which the current value is larger than the past return, is trained by the policy network and the value network.\n\nThe reason why that the SIL is used in the study is that the exploitation of valuable transition is needed. In fact, in the study, the off-policy RL model is necessary to utilize the replay memory \\cite{mnih2015human,schaul2015prioritized,van2016deep}. However, in the path planning, to reach the target point, various routes can be obtained. Accordingly, I utilize the off-policy actor-critic RL model to get an effect of both the on-policy and the off-policy. The final RL architecture in this study is the combination of the SIL and the actor-critic network (ASIL).\n\nIn addition, I utilized the random network distillation (RND), which is widely used as an exploration bonus method \\cite{burda2018exploration}. The RND uses two networks: a fixed and random initialized network (target network), and a predictor network trained using the output of the target network. The exploration bonus is given as a difference between the outputs of the two networks. If the agent visits a specific point continually, the exploration bonus is gradually decreased. Otherwise, if the agent visits a novel space, a large exploration bonus will be given to the agent.\n\n\n\n\n\n\n\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.45]{fig4.pdf}\n\\caption{The visualization of the path of the agent for the first environment. A color change from blue to red indicates that the agent has visited more often. \\textbf{a}, The path of the agent of the policy network without learning the sub-goals. The agent easily reached the target point. \\textbf{b}, The path of the agent of policy network with learning the sub-goals using the bi-directional memory editing. The agent was confused by the sub-goals, so that the agent could not reach the final goal. }\n\\label{fig4}\n\\end{figure}\n\n\n\\subsection{Simple 2D grid environment}\nFig \\ref{fig4} shows the visualization of the path that is moved by the agent of the policy network without (a) and with (b) learning the sub-goals. A color change from blue to red means that the agent has visited more often. When the policy network did not learn the sub-goals, the agent of the policy network easily reached the target point, and the agent mostly moved only between the starting point and the target point. That is, the agent mostly moved the left and top areas in the environment because of the location of the target point.\n\n\n\\begin{figure*}[h]\n\\centering\n\\includegraphics[scale=0.45]{fig6.png}\n\\caption{\\textbf{a}, The visualization of the path of the agent of the policy network with learning the sub-goals. The agent could not learn the sub-goals so the agent had a tendency to just move to the right. \\textbf{b},  The visualization of the path of the agent of the sub-dedicated network with the forward directional memory editing. The agent was trained using a one-directional route. Therefore, the agent tried leftward, even though the given sub-goals were located on the right of the agent. }\n\\label{fig6}\n\\end{figure*}\n\n\n\\begin{figure*}[h]\n\\centering\n\\includegraphics[scale=0.42]{fig5.png}\n\\caption{The visualization of the path of the agent when the sub-goals were given in the test environment according to the reward shaping. The top and the bottom of the figure indicate the route of the agent without and with the reward shaping, respectively. (a), The agent successfully went through the sub-goals and reached the final goal. (b), In a round-trip environment, the agent also passed all the sub-goals and came back to the starting point. With the reward shaping (bottom), the agent arrived at each sub-goal point and the final goal faster than without the reward shaping (top). }\n\\label{fig5}\n\\end{figure*}\n\nIn addition, if the policy network was trained on the sub-goals using the bi-directional memory editing, the agent of the policy network could not reach the target point, as shown in Fig \\ref{fig4}.b. At a specific point, if the agent is trained to go in various directions by the sub-goals, the agent of the policy network was confused about where it has to go. Then, the agent would fail to reach the final goal. In the experiment, the agent just moved to the right area in the environment. This case was repeated every time the network was trained. Notably, in the experiment, the agent of the policy network without learning the sub-goals always could reach the final goal. In the test environment, the agent of the policy network, which learns the sub-goals using  bi-directional editing, also failed to reach the sub-goals, as shown in Fig \\ref{fig6}.a. The agent even left the environment as soon as the agent departed, as shown in Fig \\ref{fig6}.a.(left). This is because the agent of the policy network mainly moved to the right area in the environment due to the confusion. \n\n\nWhen the sub-goals dedicated network was trained using the forward directional memory editing only, the agent failed to reach the sub-goals, as shown in Fig \\ref{fig6}.b. It was observed that the agent tried to move in the left direction. The reason is that the agent of the policy network mainly moved to the leftward and the upward direction, and the agent is trained using the one-directional trajectories. \n\nFig \\ref{fig5} shows the result of learning the sub-goals in the test environments without (top) and with (bottom) the reward shaping for the shorter path. The agent was trained from the sub-goals dedicated network using the trajectories, which were collected from the agent of the policy network. In the test environment, I set the sub-goals difficultly, even though all the points including the starting point and the target point were set inversely. The agent successfully reached all the sub-goals and the target points in the experiment, as shown in Fig \\ref{fig5}.a. That is, the agent was able to be fully controlled by the sub-goals, in various scenarios. The agent reached the sub-goals that had never been visited by the agent of the policy network. Interestingly, in extremely hard environments (round trip tasks), as shown in Fig \\ref{fig5}.b, the agent departed from the starting point and went through the sub-goals, and then the agent turned halfway point and came back to the starting point. \n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[scale=0.55]{fig7.png}\n\\caption{The visualization of the path of the agent without and with the reward shaping for the shorter path in 20 scenarios. The number, located in the upper of each figure, indicates the number of steps to reach the target point. The agent of the sub-goals dedicated network arrived at the target point in all scenarios. With a visual inspection, the path of the agent with the reward shaping was shorter than without the reward shaping. In the comparison of the number of steps, it can be seen that the agent with the reward shaping reaches the target point within a short time than the agent without the reward shaping. }\n\\label{fig7}\n\\end{figure*}\n\nIn addition, with the reward shaping for the shorter path, the agent reached the target point faster than without the reward shaping, as shown in Fig \\ref{fig5}. With a visual inspection, we can observe that the agent with the reward shaping reached the sub-goals located diagonally with a shorter distance from the current location. Moreover, the difference between the number of steps to reach the target point with and without the reward shaping was significant. \n\n\n\n\n\n\n\n\n\nTo strongly confirm the performance difference between without and with the reward shaping of the proposed method, I additionally constructed 20 scenarios. Fig \\ref{fig7} shows the route of the agent with and without the reward shaping in each scenario. The number in each figure indicates the number of steps to reach the target point. In all scenarios, the agent went through the sub-goals and reached the final goal. It can be seen that the agent was able to be fully under control by the sub-goals. Moreover, except for three scenarios, the scenario 14, 15, and 16, the agent with the reward shaping reached the target point much faster in all scenarios. This characteristic was salient in complex environments such as the scenario 1 and 2. The average of the steps with the reward shaping was 338.25 and the average of the steps without the reward shaping was 411.45. It was confirmed that the rewards shaping can shorten the number of steps to arrive at the destination by 21.6.\n\n\n\n\n\\subsection{Key-door domain}\nThe agent of the (ASIL + RND) reached the final stage within 50,000 episodes only one time out of 10 trials in the 'key-door domain' environment. It was a very difficult environment to clear. This was because of the condition to clear each stage and the sparseness of the reward. I assumed the two scenarios in the test environment. Two sub-goals were imposed on the agent differently in each stage. The bonus point and the goal point are also given as the sub-goals. The agent was enforced to reach the sub-goals at first and after that, the agent was encouraged to reach the bonus point and the target point. \n\n\\begin{figure*}[h]\n\\centering\n\\includegraphics[width=\\textwidth]{fig8.png}\n\\caption{The visualization of the path of the agent in the key-door domain environment. \\textbf{a}, The path of the agent of the policy network in each stage, when the agent cleared all stages. The agent of the policy network cleared only one time out of 10 trials in total. \\textbf{b}, The path of the agent of the policy network in each stage, when the agent did not clear all stages. \\textbf{c} $\\sim$ \\textbf{d}, The path of the agent of the sub-goals dedicated network, when the sub-goals were given. The sub-goals dedicated network was used, when the agent of the policy network failed to clear all stages.}\n\\label{fig8}\n\\end{figure*}\n\n\nFig \\ref{fig8}.a shows the visualization of the path of the agent, success case of learning the desired policy. If the policy network converges to the desired policy, the agent of the network showed a clear path to the goal of Stage 4. However, in the experiment, the agent of the policy network almost failed to reach the final stage in the training environment within 50,000 episodes. Fig \\ref{fig8}.b shows the visualization of the path of the agent, when the agent of the policy network failed to learn the desired policy. The agent could reach Stage 4, but the agent failed to clear the stage. In contrast, the agent of the sub-goals dedicated network was able to reach the goal in the final stage, going through the sub-goals and the bonus point in the two scenarios as shown in Fig \\ref{fig8}.c $\\sim$ d, even though the agent of the policy network failed, as shown in Fig \\ref{fig8}.b. This result means that learning the sub-goals can improve the ability of the agent to reach the final goal, just like in previous studies. Unlike the previous study, I set the mission for the agent's performance to the sub-goals dedicated network in order to apply  in the path planning problem, in which the ability to move in various directions is necessary. Indeed, in the experiments, the last 300 episodes of the policy network were enough to learn the sub-goals dedicated network so that the agent could be under control. We do not need to collect a number of trajectories to learn the agent in the path planning, if we utilize the proposed method. \n\n\n\n\n\n\n\nHowever, the agent of the sub-goals dedicated network did not show the shortest path whereas the agent of the policy network showed the almost shortest path. Furthermore, like a previous study \\cite{lee2022learning}, sometimes, the agent that learned the sub-goals was confused when the bonus point was near the current location of the agent. The reason is that the agent was trained to go through the bonus point, at first, to clear each stage. That is, the agent was enforced to move to the sub-goals in learning the sub-goals, and the agent was also encouraged to move to the bonus point and the target point. When the agent was near the bonus point, the agent could get a larger reward, although the agent did not reach the sub-goals. The agent was partially able to be controllable in a complex environment with various variables that have a great deal of influence on the agent. These remaining issues call for further studies. \n\n\n\n\n\n\\section{Conclusion}\nIn this paper, I propose a novel RL framework within which the agent can be under control in the path planning environment so that the agent can reach various sub-goals. The agent that completed the learning can perform the difficult missions such as a round trip, even the agent can reach an unknown area. Therefore, the bi-directional memory editing and the sub-goals dedicated network were presented on the goal-conditioned RL. From the bi-directional memory editing, we can obtain various sub-goals and behaviors of the agent such that the agent can be more robust in the test environment. In addition, using the sub-goals dedicated network, the agent can perform several behaviors that are directed by different sub-goals at one point. It was confirmed that the agent can be fully controlled and can achieve various sub-goals that are customized by the users in the test environment. Furthermore, the proposed reward shaping for the shorter path can improve the ability of the path planning. \n\nHowever, in a complex environment with various variables such as the key-door domain, the agent was confused about whether to select the sub-goals or the bonus point. Although a fully controllable agent is useful in the path planning, various variables such as an obstacle and the limited number of steps in an environmental setting should be considered. Further, the reward shaping cannot guarantee the optimal path. Future studies are required for a fully controllable agent for the path planning. I expect that this study would be applied and studied in a variety of domains with a fully controllable agent in various scenarios. \n\n\n\n\n\\ifCLASSOPTIONcompsoc\n \n  \\section*{Acknowledgments}\n\\else\n \n  \\section*{Acknowledgment}\n\\fi\n\n\n\n\n\\ifCLASSOPTIONcaptionsoff\n  \\newpage\n\\fi\n\n\n\n\n\n\n\\bibliographystyle{IEEEtran}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section*{References}}\n\n\\makeatletter \n\\renewcommand\\@biblabel[1]{#1.} \n\\makeatother\n\n\n\\begin{document}\n\n\\title{Span Labeling Approach for Vietnamese and Chinese Word Segmentation}\n\\titlerunning{Span Labeling Approach for Vietnamese and Chinese Word Segmentation}\n\n\\author{Duc-Vu Nguyen\\inst{1,3}\\and Linh-Bao Vo\\inst{2,3} \\and Dang Van Thin\\inst{1,3} \\and Ngan Luu-Thuy Nguyen\\inst{2,3}}\n\n\\authorrunning{D.-V. Nguyen et al.}\n\n\\institute{Multimedia Communications Laboratory, University of Information Technology,\\\\Ho Chi Minh City, Vietnam \\\\ \\email{\\{vund, thindv\\}@uit.edu.vn}\\\\ \\and University of Information Technology, Ho Chi Minh City, Vietnam\\\\ \\email{18520503@gm.uit.edu.vn, ngannlt@uit.edu.vn} \\and Vietnam National University, Ho Chi Minh City, Vietnam}\n\n\\maketitle\n\n\\begin{abstract}\nIn this paper, we propose a span labeling approach to model n-gram information for Vietnamese word segmentation, namely \\textsc{SpanSeg}. We compare the span labeling approach with the conditional random field by using encoders with the same architecture. Since Vietnamese and Chinese have similar linguistic phenomena, we evaluated the proposed method on the Vietnamese treebank benchmark dataset and five Chinese benchmark datasets. Through our experimental results, the proposed approach \\textsc{SpanSeg} achieves higher performance than the sequence tagging approach with the state-of-the-art F-score of 98.31\\% on the Vietnamese treebank benchmark, when they both apply the contextual pre-trained language model XLM-RoBERTa and the predicted word boundary information. Besides, we do fine-tuning experiments for the span labeling approach on BERT and ZEN pre-trained language model for Chinese with fewer parameters, faster inference time, and competitive or higher F-scores than the previous state-of-the-art approach, word segmentation with word-hood memory networks, on five Chinese benchmarks.\n\\keywords{Natural Language Processing \\and Word Segmentation \\and Vietnamese \\and Chinese.}\n\\end{abstract}\n\n\n\\section{Introduction}\nWord segmentation is the first essential task for both Vietnamese and Chinese. The input of Vietnamese word segmentation (VWS) is the sequence of syllables delimited by space. In contrast, the input of Chinese word segmentation (CWS) is the sequence of characters without explicit delimiter. The use of a Vietnamese syllable is similar to a Chinese character. Despite deep learning dealing with natural language processing tasks without the word segmentation phase, the research on word segmentation is still necessary regarding the linguistic aspect. Since Vietnamese and Chinese have similar linguistic phenomena such as overlapping ambiguity in VWS \\cite{hongphuong08} and in CWS \\cite{sun-etal-1998-chinese}, therefore the research about VWS and CWS is a challenging problem.\n\nMany previous approaches for VWS have been proposed. For instance, in the early stage of VWS, \\citet{diend1} supposed VWS as a stochastic transduction problem. Therefore, they represented the input sentence as an unweighted Finite-State Acceptor. As a consequence, \\citet{hongphuong08} proposed the ambiguity resolver using a bi-gram language model as a component in their model for VWS. After that, \\citet{nguyenetal2006} used conditional random fields (CRFs) and support vector machines (SVMs) for VWS. Recently, \\citet{phongnt1} utilized rules based on the predicted word boundary and threshold for the classifier in the post-processing stage to control overlapping ambiguities for VWS. Besides, \\citet{datnq1} proposes a method for auto-learning rule based on the predicted word boundary for VWS. Furthermore, \\citet{nguyen-2019-neural} proposed the joint neural network model for Vietnamese word segmentation, part-of-speech tagging, and dependency parsing. Lastly, \\citet{nguyen-etal-2019-ws-svm} proposed feature extraction to deal with overlapping ambiguity and capturing word containing suffixes.\n\nFrom our observation, the number of research and approaches for CWS is greater than VWS. The research \\cite{tseng-etal-2005-conditional,song-etal-2006-chinese,chen-etal-2015-long,zhang-etal-2016-transition-based, ma-etal-2018-state, higashiyama-etal-2019-incorporating} treated CWS as a character-based sequence labeling task. The contextual feature extractions were proved helpful in CWS \\cite{higashiyama-etal-2019-incorporating}. After that, neural networks were powerful for CWS \\cite{chen-etal-2015-long, ma-etal-2018-state, higashiyama-etal-2019-incorporating}. The measuring word-hood for n-grams was an effective method for non-neural network model \\cite{sun-etal-1998-chinese} and neural network model \\cite{tian-etal-2020-improving-chinese}. Besides, the multi-criteria learning from many different datasets is a strong method \\cite{chen-etal-2017-adversarial, qiu-etal-2020-concise, ke-etal-2021-pre}. Remarkably, \\citet{tian-etal-2020-improving-chinese} incorporated the word-hood for n-gram into neural network model effectively.\n\nWe have an observation that most of the approaches for VWS and CWS treated word segmentation as a token-based sequence tagging problem, where the token is a syllable in VWS and character in CWS. Secondly, the intersection of VWS and CWS approaches leverages the context to model n-gram of token information, such as measuring the word-hood of the n-gram in CWS. All of the previous approaches in CWS incorporate the word-hood information as a module of their models. Therefore, our research hypothesizes whether we can model a simple model that can simulate measuring word-hood operation.\n\nFrom our observation and hypothesis, we get the inspiration of span representation in constituency parsing \\cite{stern-etal-2017-minimal} to propose our \\textsc{SpanSeg} model for VWS and CWS. The main idea of our \\textsc{SpanSeg} is to model all n-grams in the input sentence and score them. Modeling an n-gram is equivalent to find the probability of a span being a word. Via experimental results, the proposed approach \\textsc{SpanSeg} achieves higher performance than the sequence tagging approach when both utilize contextual pre-trained language model XLM-RoBERTa and predicted word boundary information on the Vietnamese treebank benchmark with the state-of-the-art F-score of 98.31\\%. Additionally, we do fine-tuning experiments for the span labeling method on BERT and ZEN pre-trained language model for Chinese with fewer parameters, faster inference time, and competitive or higher F-scores than the previous state-of-the-art approach, word segmentation with word-hood memory networks, on five Chinese benchmarks.\n\n\\section{The Proposed Framework}\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=\\textwidth]{fig1.pdf}\n\\caption{\\label{figmodel}The architecture of \\textsc{SpanSeg} for VWS. The input sentence is ``\\vi{h\u1ecdc sinh h\u1ecdc sinh h\u1ecdc}'' (\\textit{student learn biology}) including five syllables \\{``\\vi{h\u1ecdc}'', ``\\vi{sinh}'', ``\\vi{h\u1ecdc}'', ``\\vi{sinh}'', and ``\\vi{h\u1ecdc}''\\}. The gold-standard segmentation for the input sentence is ``\\vi{h\u1ecdc\\_sinh h\u1ecdc sinh\\_h\u1ecdc}'' including three words \\{``\\vi{h\u1ecdc\\_sinh}'', ``\\vi{h\u1ecdc}'', and ``\\vi{sinh\\_h\u1ecdc}''\\}. The initial BIES (Begin, Inside, End, or Singleton) word boundary tags (differing from gold-standard segmentation) were predicted by an off-the-shelf toolkit RDRsegmenter \\cite{datnq1}.}\n\\end{figure}\n\nDiffering from previous studies, we regard word segmentation as a span labeling task. The architecture of our proposed model, namely \\textsc{SpanSeg}, is illustrated in Figure~\\ref{figmodel}, where the general span labeling paradigm is at the top of the figure. This paper is the first work approach to word segmentation as a span labeling task to the best of our knowledge. Before presenting the details of \\textsc{SpanSeg}, we take a first look at problem representation of \\textsc{SpanSeg}. In Figure~\\ref{figmodel}, we consider the input sentence in the form of the index (integer type) and syllable (string type) as an array \\{$0$: ``\\vi{h\u1ecdc}'', $1$: ``\\vi{sinh}'', $2$: ``\\vi{h\u1ecdc}'', $3$: ``\\vi{sinh}'', $4$: ``\\vi{h\u1ecdc}''\\}. With this consideration, the gold-standard segmentation ``\\vi{h\u1ecdc\\_sinh h\u1ecdc sinh\\_h\u1ecdc}'' (\\textit{student learn biology}) (including three words \\{``\\vi{h\u1ecdc\\_sinh}'', ``\\vi{h\u1ecdc}'', and ``\\vi{sinh\\_h\u1ecdc}''\\}) is presented by three spans $(0,2)$ (``\\vi{h\u1ecdc\\_sinh}''), $(2, 3)$ (``\\vi{h\u1ecdc}''), and $(3, 5)$ (``\\vi{sinh\\_h\u1ecdc}''). By approaching word segmentation as a span labeling task, we have three positive samples (three circles filled with gray color) for the input sentence in Figure~\\ref{figmodel}, whereas other circles filled with white color with solid border are negative samples for the input sentence in Figure~\\ref{figmodel}. Also, in Figure~\\ref{figmodel}, we note that all circles with dashed border (e.g, spans $(0,0)$, $(1,1)$, $\\dots$, $(n,n)$, where $n$ is the length of the input sentence) are skipped in \\textsc{SpanSeg} because they do not represent spans.\n\nAfter presenting \\textsc{SpanSeg}, in the rest of this section, we firstly introduce problem representation of word segmentation as a span labeling task (in subsection~\\ref{subproblem}). Secondly, we introduce the proposed span post-processing algorithm for word segmentation (in subsection~\\ref{spanpost}). The first and second subsections are two important points of our research. Thirdly, we describe the span scoring module (in subsection~\\ref{scoringmodule}). In the last two subsections, we provide the architecture encoder for VWS and CWS. We describe the model \\textsc{SpanSeg} for VWS (in subsection~\\ref{spansegvi}). Lastly, we describe \\textsc{SpanSeg} for CWS (in subsection~\\ref{spansegzh}).\n\n\\subsection{Word segmentation as span labeling task for Vietnamese and Chinese}\n\\label{subproblem}\nThe input sentence of word segmentation task is a sequence of tokens $\\mathcal{X} = x_1 x_2 \\dots x_{n}$ with the length of $n$. The token $x_i$ is a syllable or character toward Vietnamese or Chinese, respectively. Given the input $\\mathcal{X}$, the output of word segmentation is a sequence of words $\\mathcal{W} = w_1 w_2 \\dots w_{m}$ with the length of $m$, where $1 \\leq m \\leq n$. We have a property that the word $w_j$ is constituted by one token or consecutive tokens. So, we use the sequence of tokens $x_i x_{i+1} \\dots x_{i+k-1}$ for denoting the word $w_j$ be constituted by $k$ consecutive tokens beginning at token $x_i$, where $1 \\leq k \\leq n$ (concretely, $k = 1$ representing single words and $2 \\leq k \\leq n$ representing compound words for both Vietnamese and Chinese). Inspired by the work of \\citet{stern-etal-2017-minimal} for constituency parsing, we use the span $(i - 1, i - 1 + k)$ to represent the word constituted by $k$ consecutive tokens $x_i x_{i+1} \\dots x_{i+k-1}$ beginning at token $x_i$. Therefore, the goal of the span labeling task for both VWS and CWS is to find the list of spans $\\hat{\\mathcal{S}}$ such that every token $x_i$ is spanned, and there is no overlapping between every two spans. Formally, the word segmentation model as span labeling task for both VWS and CWS can be formalized as:\n\\begin{IEEEeqnarray}{rCl}\n\\hat{\\mathcal{S}} = \\textsc{SpanPostProcessing}(\\hat{\\mathcal{Y}})\n\\end{IEEEeqnarray}\nwhere \\textsc{SpanPostProcessing}($\\cdot$) simply is a algorithm for producing the word segmentation boundary satisfying non-overlapping between every two spans. The $\\hat{\\mathcal{Y}}$ is the set of predicted spans as following:\n\\begin{IEEEeqnarray}{rCl}\n\\hat{\\mathcal{Y}} = \\{(l, r) | 0 \\leq l \\leq n - 1 ~\\text{and}~l < r \\leq n~\\text{and}~\\textsc{Scorer}(\\mathcal{X}, l, r) > 0.5\\}\n\\end{IEEEeqnarray}\nwhere $n$ is the length of the input sentence. The \\textsc{Scorer}($\\cdot$) is the scoring module for the span $(l, r)$ of sentence $\\mathcal{X}$. The output of \\textsc{Scorer}($\\cdot$) has a value in the range of 0 to 1. In our research, we choose the sigmoid function as an activation function at the last layer of \\textsc{Scorer}($\\cdot$) module. Lastly, the word segmentation as a span labeling task is the binary classification problem. We use the binary cross-entropy loss for the cost function as following:\n\\begin{align}\nJ(\\theta) = -\\frac{1}{|\\mathcal{D}|}\\sum_{\\mathcal{X, S} \\in \\mathcal{D}} \\biggl(\\frac{1}{(n(n+1))\/2}&\\sum_{l=0}^{n-1} \\sum_{r=l+1}^{n} \\left[(l,r) \\in \\mathcal{S}\\right]\\log\\big(\\textsc{Scorer}(\\mathcal{X}, l, r)\\big) \\nonumber\\\\\n{}+{}&\\left[(l,r) \\notin \\mathcal{S}\\right]\\log\\big(1 - \\textsc{Scorer}(\\mathcal{X}, l, r)\\big) \\biggr)\n\\end{align}\nwhere $\\mathcal{D}$ is the training set and $|\\mathcal{D}|$ is the size of training set. For each pair ($\\mathcal{X, S}$) in training set $\\mathcal{D}$, we compute binary cross-entropy loss for all spans $(l, r)$, where $0 \\leq l \\leq n - 1 ~\\text{and}~l < r \\leq n$, and $n$ is the length of sentence $\\mathcal{X}$. The term $\\left[(l,r) \\in \\mathcal{S}\\right]$ has the value of $1$ if span $(l, r)$ belongs to the list $\\mathcal{S}$ of sentence $\\mathcal{X}$ and conversely, of 0. Similarly, the term $\\left[(l,r) \\notin \\mathcal{S}\\right]$ has the value of $1$ if span $(l, r)$ does not belong to the list $\\mathcal{S}$ of sentence $\\mathcal{X}$ and conversely, of 0. Lastly, we make a note that in our training and prediction progress, we will discard spans with length greater than 7 for both Vietnamese and Chinese (7 is maximum n-gram length following \\cite{diao-etal-2020-zen} for Chinese, so we decide to choose 7 for Vietnamese according to the statistics in the work of \\cite{nguyen-etal-2019-ws-svm}).\n\n\n\\subsection{Post-Processing Algorithm For Predicted Spans}\n\\label{spanpost}\n\nIn the previous subsection~\\ref{subproblem}, we presented word segmentation as a span labeling task for Vietnamese and Chinese. In this subsection, we present our proposed post-processing algorithm for predicted spans from the span labeling problem. However, we found that in the predicted spans set $\\hat{\\mathcal{Y}}$ there exists overlapping between some two spans. We deal with the overlapping ambiguity by choosing the spans with the highest score and removing the rest. The overlapping ambiguity phenomenon occurs when our \\textsc{SpanSeg} predicts compound words. It occurs in our \\textsc{SpanSeg} and other word segmenters on Vietnamese \\cite{hongphuong08} and Chinese \\cite{sun-etal-1998-chinese}.\n\nApart from overlapping ambiguity, our \\textsc{SpanSeg} faces the missing word boundary problem. That problem can be caused by originally predicted spans or as a result of solving overlapping ambiguity. We choose the missing word boundary based on all predicted spans $(i - 1, i - 1 + k)$ with $k = 1$ for single words to deal with the missing word boundary problem. To sum up, our proposed post-processing algorithm for predicted spans from the span labeling problem, namely \\textsc{SpanPostProcessing}, deals with overlapping ambiguity and missing spans from predicted spans. The detail of our \\textsc{SpanPostProcessing} is presented in Algorithm~\\ref{alg}.\n\n\\begin{algorithm}[!ht]\n  \\caption{\\textsc{SpanPostProcessing}}\\label{alg}\n  \\begin{algorithmic}[1]\n    \\Require\n      \\Statex The input sentence $\\mathcal{X}$ with the length of $n$;\n      \\Statex The scoring module \\textsc{Scorer}($\\cdot$) for any span $(l, r)$ in $\\mathcal{X}$, where $0 \\leq l \\leq n - 1 ~\\text{and}~l < r \\leq n$;\n      \\Statex The set of predicted spans $\\hat{\\mathcal{Y}}$, sorted in ascending order.\n    \\Ensure\n      \\Statex The list of valid predicted spans $\\hat{\\mathcal{S}}$, satisfying non-overlapping between every two spans.\n    \\Statex\n    \\State $\\hat{\\mathcal{S}}_\\text{novlp} = [(0, 0)]$ \\Comment{The list of predicted spans without overlapping ambiguity.}\n    \\State $\\hat{\\mathcal{S}} = []$ \\Comment{The final list of valid predicted spans.}\n    \\For{$\\hat{y}$ \\textbf{in} $\\hat{\\mathcal{Y}}$} \\Comment{The $\\hat{y}[0]$ is the left boundary and $\\hat{y}[1]$ is the right boundary of each span $\\hat{y}$.}\n        \\If{$\\hat{\\mathcal{S}}_\\text{novlp}[\\text{-}1][1] < \\hat{y}[0]$} \\Comment{Check for missing boundary.}\n        \\State $\\hat{\\mathcal{S}}_\\text{novlp}$.\\textbf{append}$\\big((\\hat{\\mathcal{S}}_\\text{novlp}[\\text{-}1][1], \\hat{y}[0])\\big)$ \\Comment{Add the missing span to $\\hat{\\mathcal{S}}_\\text{novlp}$}\n        \\EndIf\n        \\If{$\\hat{\\mathcal{S}}_\\text{novlp}[\\text{-}1][0] \\leq \\hat{y}[0] < \\hat{\\mathcal{S}}_\\text{novlp}[\\text{-}1][1]$}  \\Comment{Check for overlapping ambiguity.}\n        \\If{$\\textsc{Scorer}(\\mathcal{X}, \\hat{\\mathcal{S}}_\\text{novlp}[\\text{-}1][0], \\hat{\\mathcal{S}}_\\text{novlp}[\\text{-}1][1]) < \\textsc{Scorer}(\\mathcal{X}, \\hat{y}[0], \\hat{y}[1])$}\n            \\State $\\hat{\\mathcal{S}}_\\text{novlp}$.\\textbf{pop}()  \\Comment{Remove the span causing overlapping with the lower score than $\\hat{y}$.}\n            \\State $\\hat{\\mathcal{S}}_\\text{novlp}$.\\textbf{append}$\\big((\\hat{y}[0], \\hat{y}[1])\\big)$ \\Comment{Add the span $\\hat{y}$ to $\\hat{\\mathcal{S}}_\\text{novlp}$.}\n        \\EndIf\n        \\Else{}\n        \\State $\\hat{\\mathcal{S}}_\\text{novlp}$.\\textbf{append}$\\big((\\hat{y}[0], \\hat{y}[1])\\big)$ \\Comment{Add the span $\\hat{y}$ to $\\hat{\\mathcal{S}}_\\text{novlp}$.}\n        \\EndIf\n    \\EndFor\n    \\If{$\\hat{\\mathcal{S}}_\\text{novlp}[\\text{-}1][1] < n$} \\Comment{Check for missing boundary.}   \n        \\State $\\hat{\\mathcal{S}}_\\text{novlp}$.\\textbf{append}$\\big((\\hat{\\mathcal{S}}_\\text{novlp}[\\text{-}1][1], n)\\big)$ \\Comment{Add the missing span to $\\hat{\\mathcal{S}}_\\text{novlp}$}\n    \\EndIf\n    \\For{$i, \\hat{y}$ \\textbf{in} \\textbf{enumerate}($\\hat{\\mathcal{S}}_\\text{novlp}$)} \\Comment{The $\\hat{y}[0]$ is the left boundary and $\\hat{y}[1]$ is the right boundary of each span $\\hat{y}$, and $i$ is the index of $\\hat{y}$ in list $\\hat{\\mathcal{S}}_\\text{novlp}$.}\n      \\If{$0 < i$ \\textbf{and} $\\hat{\\mathcal{S}}_\\text{novlp}[i-1][1] < \\hat{y}[0]$} \\Comment{Check for missing boundary.}\n      \\State \\textit{missed\\_boundaries} $= \\big[\\hat{\\mathcal{S}}_\\text{novlp}[i-1][1]\\big]$\n        \\For{\\textit{bound} \\textbf{in} \\textbf{range}$\\big(\\hat{\\mathcal{S}}_\\text{novlp}[i-1][1], \\hat{y}[0]\\big)$}\n          \\If{$\\textsc{Scorer}(\\mathcal{X}, \\textit{bound}, \\textit{bound} + 1) > 0.5$}\\Comment{Check for single word.}\n            \\State \\textit{missed\\_boundaries}.\\textbf{append}($\\textit{bound} + 1$)\n          \\EndIf\n        \\EndFor\n        \\State \\textit{missed\\_boundaries}.\\textbf{append}($\\hat{y}[0]$)\n        \\For{\\textit{j} \\textbf{in} \\textbf{range}$\\big(\\textbf{len}(\\textit{missed\\_boundaries}) - 1\\big)$}\n          \\State $\\hat{\\mathcal{S}}$.\\textbf{append}$\\big((\\textit{missed\\_boundaries}[j], \\textit{missed\\_boundaries}[j+1])\\big)$\\Comment{Add the missing span to $\\hat{\\mathcal{S}}$}\n        \\EndFor\n      \\EndIf\n      \\State $\\hat{\\mathcal{S}}$.\\textbf{append}$\\big(\\hat{y}[0], \\hat{y}[1]\\big)$ \\Comment{Add the non-overlapping span to $\\hat{\\mathcal{S}}$}\n    \\EndFor\n  \\end{algorithmic}\n\\end{algorithm}\n\n\n\n\\subsection{Span Scoring Module}\n\\label{scoringmodule}\nIn two previous subsections~\\ref{subproblem} and ~\\ref{spanpost}, we presented two critical points of our research. There we mentioned the \\textsc{Scorer}($\\cdot$) module many times. In this section, we present \\textsc{Scorer}($\\cdot$) module. It is based on the familiar module that name Biaffine \\cite{dozat2017deep}. While \\citet{ijcai2020-560} experimenting with the Biaffine module for constituency parsing, we use the Biaffine module for span labeling word segmentation. The Biaffine module is used in \\cite{dozat2017deep} to capture the directed relation between two words in a sentence for dependency parsing. In the constituency parsing problem, \\citet{ijcai2020-560} used the Biaffine module to find the representation of phrases. Our research uses the Biaffine module to model the representation of n-gram for the word segmentation task.\n\nAs we can see in Figure~\\ref{figmodel}, each token $x_i$ in the input sentence has two context-aware word representations including left and right boundary representations except the begin (``<s>'') and end (``<\/s>'') tokens. In case we use the BiLSTM (Bidirectional Long Short Term Memory) encoder, the left boundary representation of token $x_i$ is the concatenation of the hidden state forward vector $\\textbf{f}_{i-1}$ and the hidden state backward vector $\\textbf{b}_{i}$ and the right boundary representation of token $x_i$ is the concatenation of the hidden state forward vector $\\textbf{f}_{i}$ and the hidden state backward vector $\\textbf{b}_{i+1}$, following \\citet{stern-etal-2017-minimal}. In case we use BERT \\cite{devlin-etal-2019-bert} or ZEN \\cite{diao-etal-2020-zen} encoder, we chunk the last hidden state vector into two vectors with the same size as forward and backward vectors of the BiLSTM encoder. Even though we use the BiLSTM, BERT, or ZEN encoder, we always have the left and right boundary representation for each token $x_i$ in the input sentence. Therefore, in Figure~\\ref{figmodel}, we see that the right boundary representation $\\textbf{f}_{i}\\oplus \\textbf{b}_{i+1}$ of token $x_i$ is the left boundary representation of token $\\textbf{x}_{i+1}$. As the work of \\citet{ijcai2020-560}, we use two MLPs to make the difference between the right boundary representation of token $x_i$ and the left boundary representation of token $x_{i+1}$. To sum up, we have the left $\\textbf{r}_i^{\\text{left}}$ and right $\\textbf{r}_i^{\\text{right}}$ boundary representations of token $x_{i}$ as following:\n\\begin{IEEEeqnarray}{rCl}\n\\textbf{r}_i^{\\text{left}} &=& \\text{MLP}^{\\text{left}}(\\textbf{f}_{i-1}\\oplus \\textbf{b}_{i})\\\\\n\\textbf{r}_i^{\\text{right}} &=& \\text{MLP}^{\\text{right}}(\\textbf{f}_{i}\\oplus \\textbf{b}_{i+1})\n\\end{IEEEeqnarray}\nFinally, inspired by \\citet{ijcai2020-560}, given the input sentence $\\mathcal{X}$, the span scoring module \\textsc{Scorer}($\\cdot$) for span $(l, r)$ in our \\textsc{SpanSeg} model is computed by using a biaffine operation over the left boundary representation of token $x_l$ and the right boundary representation of token $x_r$ as following:\n\\begin{IEEEeqnarray}{rCl}\n\\textsc{Scorer}(\\mathcal{X}, l, r) = \\text{sigmoid}\\bigg( \\begin{bmatrix}\\textbf{r}_l^{\\text{left}} \\\\ 1\\end{bmatrix}^{\\text{T}}\\textbf{W}\\textbf{r}_r^{\\text{right}}\\bigg)\n\\end{IEEEeqnarray}\nwhere $\\textbf{W} \\in \\mathbb{R}^{d\\times d}$. To sum up, the \\textsc{Scorer}($\\mathcal{X}, l, r$) gives us a score to predict whether a span $(l, r)$ is a word.\n\n\\subsection{Encoder and input representation for VWS}\n\\label{spansegvi}\n\\begin{sloppypar}In three previous subsection~\\ref{subproblem}, \\ref{spanpost}, and \\ref{scoringmodule}, we describe three mutual parts of the \\textsc{SpanSeg} model for Vietnamese and Chinese. In this subsection, we present the encoder and the input representation for VWS of the \\textsc{SpanSeg} model. Firstly, the default configuration of \\textsc{SpanSeg} for the input representation of token $x_i$ is composed as following:\n\\begin{align}\n\\textbf{default\\_embedding}_i =  \\big(&\\textbf{static\\_syl\\_embedding}_{i} \\nonumber\\\\\n& + \\textbf{dynamic\\_syl\\_embedding}_{i}\\big) \\oplus \\textbf{char\\_embedding}_i\n\\end{align}\nwhere the symbol $\\oplus$ denotes the concatenation operation. The $\\textbf{static\\_syl\\_embedding}_{i}$ is extracted from the pre-trained Vietnamese syllable embedding with the dimension of 100 provided by \\citet{nguyen-etal-2017-word}. So, the dimension of vector $\\textbf{dynamic\\_syl\\_embedding}_{i}$ also is 100. We initialize randomly and update the value of $\\textbf{dynamic\\_syl\\_embedding}_{i}$ in the training progress. We do not update the value of $\\textbf{static\\_syl\\_embedding}_{i}$ during training model. Besides, we also use a character embedding for the input representation by using BiLSTM network for sequence of characters in token $x_i$ to obtain $\\textbf{char\\_embedding}_i$.\\end{sloppypar}\n\nThe default configuration does not utilize the Vietnamese predicted word boundary information as many previous works on VWS did. Following the work of \\citet{nguyen-2019-neural}, we additionally use the boundary BIES tag embedding for the input representation of token $x_i$. Therefore, the second configuration of \\textsc{SpanSeg}, namely \\textsc{SpanSeg} (TAG) is presented as following:\n\\begin{align}\n\\textbf{default\\_tag\\_embedding}_i = \\textbf{default\\_embedding}_i \\oplus \\textbf{bies\\_tag\\_embedding}_i\n\\end{align}\nwhere the value of $\\textbf{bies\\_tag\\_embedding}_i$ (with the dimension of 100) is initialized randomly and updated; and the boundary BIES tag is predicted by the off-the-shelf toolkit RDRsegmenter \\cite{datnq1}.\n\nRecently, many contextual pre-trained language models were proposed inspired by the work of \\citet{devlin-etal-2019-bert}. However, our research utilizes contextual pre-trained multilingual language model XLM-Roberta (XLM-R) \\cite{conneau-etal-2020-unsupervised} with the \\textit{base} architecture for VWS since there is no contextual pre-trained monolingual language model for Vietnamese at this time. So, the third configuration of \\textsc{SpanSeg}, namely \\textsc{SpanSeg} (XLM-R), is presented as following:\n\\begin{align}\n\\textbf{default\\_xlmr\\_embedding}_i = \\textbf{default\\_embedding}_i \\oplus \\textbf{xlmr\\_embedding}_i\n\\end{align}\nwhere the $\\textbf{xlmr\\_embedding}_i$ is the projected vector from the hidden state of \\textit{the last four layers} of the XLM-R model. The dimension of $\\textbf{xlmr\\_embedding}_i$ is 100. We do not update parameters of the XLM-R model during the training process.\n\nLastly, we make the fourth configuration for \\textsc{SpanSeg}, namely \\textsc{SpanSeg} (TAG + XLM-R). This configuration aims to combine all syllables, characters, predicted word boundaries, and contextual information for VWS.\n\\begin{align}\n\\textbf{default\\_tag\\_xlmr\\_embedding}_i =~& \\textbf{default\\_embedding}_i \\nonumber\\\\\n & \\oplus \\textbf{bies\\_tag\\_embedding}_i \\oplus \\textbf{xlmr\\_embedding}_i\n\\end{align}\n\nAfter we have the input representation for each token $x_i$ of the input sentence $\\mathcal{X}$, we feed them into the BiLSTM network to obtain the forward $f_i$ and backward $b_i$ vectors. The forward $f_i$ and backward $b_i$ vectors is used in the \\textsc{Scorer}($\\cdot$) module in subsection~\\ref{scoringmodule}.\n\n\n\\subsection{Encoder and input representation for CWS}\n\\label{spansegzh}\nTo make a fair comparison to the state-of-the-art model for CWS, we used the same encoder as the work of \\citet{tian-etal-2020-improving-chinese}. Following the work \\cite{tian-etal-2020-improving-chinese}, we choose two BERT \\cite{devlin-etal-2019-bert} and ZEN \\cite{diao-etal-2020-zen} encoders with the \\textit{base} architecture. The BERT and ZEN are two famous encoders utilizing contextual information for Chinese language processing, in which the ZEN encoder enhances n-gram of characters information. For each character $x_i$ in the input sentence $\\mathcal{X}$, we chunk the hidden state vector of \\textit{the last layer} of BERT or ZEN into two vectors with the same size as the forward $f_i$ and backward $b_i$ vectors in the BiLSTM network. Finally, the forward $f_i$ and backward $b_i$ vectors are used in the \\textsc{Scorer}($\\cdot$) module in subsection~\\ref{scoringmodule}. We update the parameters of BERT and ZEN in training progress following the work of \\citet{tian-etal-2020-improving-chinese}.\n\n\\section{Experimental Settings}\n\\subsection{Datasets}\n\nThe largest VWS benchmark dataset\\footnote{The details of VTB dataset are presented at \\url{https:\/\/vlsp.org.vn\/vlsp2013\/eval\/ws-pos}.} is a part of the Vietnamese treebank (VTB) project \\cite{nguyen-etal-2009-building}. We use the same split as the work of \\citet{datnq1}. The summary of the VTB dataset for the word segmentation task is provided in Table~\\ref{vtbstat}.\n\n\n\\begin{table}[ht]\n\\caption{\\label{vtbstat}Statistics of the Vietnamese treebank dataset for word segmentation. We provide the number of sentences, characters, syllables, words, character types, syllable types, word types. We also compute the out-of-vocabulary (OOV) rate as the percentage of unseen words in the development and test set.}\n\\centering\n\\resizebox{0.535\\textwidth}{!}{%\n\\begin{tabular}{l|r|r|r}\n\\hline\n\\multirow{2}{*}{} & \\multicolumn{3}{c}{\\textbf{VTB}} \\\\ \\cline{2-4} \n & \\multicolumn{1}{c|}{Train} & \\multicolumn{1}{c|}{Dev} & \\multicolumn{1}{c}{Test} \\\\ \\hline\n\\# sentences & 74,889 & 500 & 2,120 \\\\ \\hline\n\\# characters & 6,779,116 & 55,476 & 307,932 \\\\ \\hline\n\\# syllables & 2,176,398 & 17,429 & 96,560 \\\\ \\hline\n\\# words & 1,722,271 & 13,165 & 66,346 \\\\ \\hline\n\\# character types & 155 & 117 & 121 \\\\ \\hline\n\\# syllable types & 17,840 & 1,785 & 2,025 \\\\ \\hline\n\\# word types & 41,355 & 2,227 & 3,730 \\\\ \\hline\nOOV Rate & - & 2.2 & 1.6 \\\\ \\hline\n\\end{tabular}%\n}\n\\end{table}\n\n\nFor evaluating our \\textsc{SpanSeg} on CWS, we employ five benchmark datasets including MSR, PKU, AS, CityU (from SIGHAN 2005 Bakeoff \\cite{emerson-2005-second}), and CTB6 \\cite{xue_xia_chiou_palmer_2005}. We convert traditional Chinese characters in AS, and CityU into simplified ones following previous studies \\cite{chen-etal-2015-long, qiu-etal-2020-concise, tian-etal-2020-improving-chinese}. We follow the official training\/test data split of MSR, PKU, AS, and CityU, in which we randomly extract 10\\% of the training dataset for development as many previous works. For CTB6, we the same split as the work of \\citet{tian-etal-2020-improving-chinese}. For pre-processing phase of all CWS dataset in our research, we inherit the process\\footnote{\\url{https:\/\/github.com\/SVAIGBA\/WMSeg}} of \\citet{tian-etal-2020-improving-chinese}. The summary of five Chinese benchmark datasets for the word segmentation task is presented in Table~\\ref{zhstat}.\n\n\n\\begin{table}[H]\n\\caption{\\label{zhstat}Statistics of five Chinese benchmark dataset for word segmentation. We provide the number of sentences, characters, words, character types, word types. We also compute the out-of-vocabulary (OOV) rate as the percentage of unseen words in the test set.}\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r|r}\n\\hline\n\\multirow{2}{*}{} & \\multicolumn{2}{c|}{\\textbf{MSR}} & \\multicolumn{2}{c|}{\\textbf{PKU}} & \\multicolumn{2}{c|}{\\textbf{AS}} & \\multicolumn{2}{c|}{\\textbf{\\textsc{CityU}}} & \\multicolumn{3}{c}{\\textbf{CTB6}} \\\\ \\cline{2-12} \n & \\multicolumn{1}{c|}{Train} & \\multicolumn{1}{c|}{Test} & \\multicolumn{1}{c|}{Train} & \\multicolumn{1}{c|}{Test} & \\multicolumn{1}{c|}{Train} & \\multicolumn{1}{c|}{Test} & \\multicolumn{1}{c|}{Train} & \\multicolumn{1}{c|}{Test} & \\multicolumn{1}{c|}{Train} & \\multicolumn{1}{c|}{Dev} & \\multicolumn{1}{c}{Test} \\\\ \\hline\n\\# sentences & 86,918 & 3,985 & 19,054 & 1,944 & 708,953 & 14,429 & 53,019 & 1,492 & 23,420 & 2,079 & 2,796 \\\\ \\hline\n\\# characters & 4,050,469 & 184,355 & 1,826,448 & 172,733 & 8,368,050 & 197,681 & 2,403,354 & 67,689 & 1,055,583 & 100,316 & 134,149 \\\\ \\hline\n\\# words & 2,368,391 & 106,873 & 1,109,947 & 104,372 & 5,449,581 & 122,610 & 1,455,630 & 40,936 & 641,368 & 59,955 & 81,578 \\\\ \\hline\n\\# character types & 5,140 & 2,838 & 4,675 & 2,918 & 5,948 & 3,578 & 4,806 & 2,642 & 4,243 & 2,648 & 2,917 \\\\ \\hline\n\\# word types & 88,104 & 12,923 & 55,303 & 13,148 & 140,009 & 18,757 & 68,928 & 8,989 & 42,246 & 9,811 & 12,278 \\\\ \\hline\nOOV Rate & - & 2.7 & - & 5.8 & - & 4.3 & - & 7.2 & - & 5.4 & 5.6 \\\\ \\hline\n\\end{tabular}%\n}\n\\end{table}\n\n\n\\subsection{Model Implementation}\n\\subsubsection{The detail of \\textsc{SpanSeg} for Vietnamese}\nFor the encoder mentioned in the subsection~\\ref{spansegvi}, the number of layers of BiLSTM is $3$, and the hidden size of BiLSTM is 400. The size of MLPs mentioned in the subsection~\\ref{scoringmodule} is 500. The dropout rate for embedding, BiLSTM, and MLPs is 0.33. We inherit hyper-parameters from the work of \\cite{dozat2017deep}. We trained all models up to 100 with the early stopping strategy with patience epochs of 20. We used AdamW optimizer \\cite{loshchilov2019decoupled} with the default configuration and learning rate of $\\text{10}^{\\text{-3}}$. The batch size for training and evaluating is up to 5000.\n\n\\subsubsection{The detail of \\textsc{SpanSeg} for Chinese}\nFor the encoder mentioned in the subsection~\\ref{spansegzh}, we do fine-tuning experiments based on BERT \\cite{devlin-etal-2019-bert} and ZEN \\cite{diao-etal-2020-zen} encoders. The size of MLPs mentioned in the subsection~\\ref{scoringmodule} is 500. The dropout rate for BERT and ZEN is 0.1. We trained all models up to 30 with the early stopping strategy with patience epochs of 5. We used AdamW optimizer \\cite{loshchilov2019decoupled} with the default configuration and learning rate of $\\text{10}^{\\text{-5}}$. The batch size for training and evaluating is 16.\n\n\n\\section{Results and Analysis}\n\n\n\\subsection{Main Results}\n\nFor VWS, we also implement the BiLSTM-CRF model with the same backbone and hyper-parameters as our \\textsc{SpanSeg}. The overall results are presented in Table~\\ref{mainvtb}. On the default configuration, our \\textsc{SpanSeg} gives a higher result than BiLSTM-CRF with the F-score of 97.76\\%. On the configuration with pre-trained XLM-R, our \\textsc{SpanSeg} (XLM-R) gives a higher result than BiLSTM-CRF (XLM-R) with the F-score of 97.95\\%. On the configuration with predicted boundary BIES tag from off-the-shelf toolkit RDRsegmenter \\cite{datnq1}, the BiLSTM-CRF (TAG) gives a higher result than our \\textsc{SpanSeg} (TAG) with the F-score of 98.10\\%. Finally, on the configuration with a combination of all features, our \\textsc{SpanSeg} (TAG+XLM-R) gives a higher result than BiLSTM-CRF (TAG+XLM-R) with the F-score of 98.31\\%, which is also the state-of-the-art performance on VTB. We can see that the contextual information is essential for \\textsc{SpanSeg} since \\textsc{SpanSeg} models the left and right boundary of a word rather than the between to consecutive tokens.\n\n\n\\begin{table}[ht]\n\\caption{\\label{mainvtb}Performance (F-score) comparison between \\textsc{SpanSeg} (with different configurations) and previous state-of-the-art models on the test set of VTB dataset.}\n\\centering\n\\resizebox{0.65\\textwidth}{!}{%\n\\begin{tabular}{l|c|c|c|c}\n\\hline\n\\multirow{2}{*}{} & \\multicolumn{4}{c}{\\textbf{VTB}} \\\\ \\cline{2-5} \n & P & R & F & $\\text{R}_\\text{OOV}$ \\\\ \\hline\nvnTokenizer \\cite{hongphuong08} & 96.98 &  97.69 &  97.33 & - \\\\ \\hline\nJVnSegmenter-Maxent \\cite{nguyenetal2006} & 96.60 &  97.40 &  97.00 & -\\\\ \\hline\nJVnSegmenter-CRFs \\cite{nguyenetal2006} &  96.63 &  97.49 & 97.06 & - \\\\ \\hline\nDongDu \\cite{luu2012} & 96.35 & 97.46 & 96.90 & - \\\\ \\hline\nUETsegmenter \\cite{phongnt1}  & 97.51 & 98.23 & 97.87 & - \\\\ \\hline\nRDRsegmenter \\cite{datnq1} & 97.46 & 98.35 & 97.90 & - \\\\ \\hline\nUITsegmenter \\cite{nguyen-etal-2019-ws-svm} & 97.81 & \\textbf{98.57} & 98.19 & - \\\\ \\hline\nBiLSTM-CRF & 97.42 & 97.84 & 97.63 & 72.47 \\\\\n\\textsc{SpanSeg} & 97.58 & 97.94 & 97.76 & \\textbf{74.65} \\\\\nBiLSTM-CRF (XLM-R) & 97.69 & 97.99 & 97.84 & 72.66 \\\\\n\\textsc{SpanSeg} (XLM-R) & 97.75 & 98.16 & 97.95 & 70.01 \\\\\nBiLSTM-CRF (TAG) & 97.91 & 98.28 & 98.10 & 69.16 \\\\\n\\textsc{SpanSeg} (TAG) & 97.67 & 98.28 & 97.97 & 65.94 \\\\\nBiLSTM-CRF (TAG+XLM-R) & 97.94 & 98.44 & 98.19 & 68.87 \\\\\n\\textsc{SpanSeg} (TAG+XLM-R) & \\textbf{98.21} & 98.41 & \\textbf{98.31} & 72.28 \\\\ \\hline\n\\end{tabular}%\n}\n\\end{table}\n\nFor CWS, we presented the performances of our \\textsc{SpanSeg} in Table~\\ref{zhmain}. We do not compare our method with previous studies approaching multi-criteria learning since simply the training data is different. Our research focuses on the comparison between our \\textsc{SpanSeg} and sequence tagging approaches. Firstly, we can see that our \\textsc{SpanSeg} (BERT) achieves higher results than state-of-the-art methods \\textsc{WMSeg} (BERT-CRF) \\cite{tian-etal-2020-improving-chinese} on four datasets including MSR (98.31\\%), PKU (96.56\\%), AS (96.62\\%), and CTB6 (97.26\\%) except CityU (97.74\\%). Our \\textsc{SpanSeg} (ZEN) do not achieve the stable performance as \\textsc{SpanSeg} (BERT). The potential reason for this problem is that both ZEN \\cite{diao-etal-2020-zen} encoder and our \\textsc{SpanSeg} try to model n-gram of Chinese characters causing inconsistency.\n\n\\begin{table}[ht]\n\\caption{\\label{zhmain}Performance (F-score) comparison between \\textsc{SpanSeg} (BERT and ZEN) and previous state-of-the-art models on the test set of five Chinese benchmark datasets. The symbol [$\\bigstar$] denotes the methods learning from data annotated through different segmentation criteria, which means that the labeled training data are different from the rest.}\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{l|c|c|c|c|c|c|c|c|c|c}\n\\hline\n\\multirow{2}{*}{} & \\multicolumn{2}{c|}{\\textbf{MSR}} & \\multicolumn{2}{c|}{\\textbf{PKU}} & \\multicolumn{2}{c|}{\\textbf{AS}} & \\multicolumn{2}{c|}{\\textbf{\\textsc{CityU}}} & \\multicolumn{2}{c}{\\textbf{CTB6}} \\\\ \\cline{2-11} \n & F & $\\text{R}_\\text{OOV}$ & F & $\\text{R}_\\text{OOV}$ & F & $\\text{R}_\\text{OOV}$ & F & $\\text{R}_\\text{OOV}$ & F & $\\text{R}_\\text{OOV}$ \\\\ \\hline\n\\citet{chen-etal-2015-long} & 97.40 & - & 96.50 & - & - & - & - & - & 96.00 & - \\\\\n\\citet{xu-sun-2016-dependency} & 96.30 & - & 96.10 & - & - & - & - & - & 95.80 & - \\\\\n\\citet{zhang-etal-2016-transition-based} & 97.70 & - & 95.70 & - & - & - & - & - & 95.95 & - \\\\\n\\citet{chen-etal-2017-adversarial} [$\\bigstar$] & 96.04 & 71.60 & 94.32 & 72.64 & 94.75 & 75.34 & 95.55 & 81.40 & - & - \\\\\n\\citet{wang-xu-2017-convolutional} & 98.00 & - & 96.50 & - & - & - & - & - & - & - \\\\\n\\citet{zhou-etal-2017-word} & 97.80 & - & 96.00 & - & - & - & - & - & 96.20 & - \\\\\n\\citet{ma-etal-2018-state} & 98.10 & 80.00 & 96.10 & 78.80 & 96.20 & 70.70 & 97.20 & 87.50 & 96.70 & 85.40 \\\\\n\\citet{GongCGQ19} & 97.78 & 64.20 & 96.15 & 69.88 & 95.22 & 77.33 & 96.22 & 73.58 & - & - \\\\\n\\citet{higashiyama-etal-2019-incorporating} & 97.80 & - & - & - & - & - & - & - & 96.40 & - \\\\\n\\citet{qiu-etal-2020-concise} [$\\bigstar$] & 98.05 & 78.92 & 96.41 & 78.91 & 96.44 & 76.39 & 96.91 & 86.91 & - & - \\\\\\hline\n\\textsc{WMSeg (BERT-CRF)} \\cite{tian-etal-2020-improving-chinese} & 98.28 & \\textbf{86.67} & 96.51 & \\textbf{86.76} & 96.58 & 78.48 & 97.80 & 87.57 & 97.16 & 88.00 \\\\\n\\textsc{WMSeg (ZEN-CRF)} \\cite{tian-etal-2020-improving-chinese} & \\textbf{98.40} & 84.87 & 96.53 & 85.36 & \\textbf{96.62} & \\textbf{79.64} & 97.93 & \\textbf{90.15} & \\textbf{97.25} & \\textbf{88.46} \\\\ \\hline\n\\textsc{MetaSeg} \\cite{ke-etal-2021-pre} [$\\bigstar$] & 98.50 & - & 96.92 & - & 97.01 & - & 98.20 & - & 97.89 & - \\\\ \\hline\n\\textsc{SpanSeg (BERT)} & 98.31 & 85.32 & \\textbf{96.56} & 85.53 & \\textbf{96.62} & 79.36 & 97.74 & 87.45 & \\textbf{97.25} & 87.91 \\\\\n\\textsc{SpanSeg (ZEN)} & 98.35 & 85.66 & 96.35 & 83.66 & 96.52 & 78.43 & \\textbf{97.96} & 90.11 & 97.17 & 87.76 \\\\\n\\hline\n\\end{tabular}%\n}\n\\end{table}\n\n\\begin{table}[ht]\n\\caption{\\label{timetab}Statistics of model size (MB) and inference time (minute) of \\textsc{WMSeg} \\cite{tian-etal-2020-improving-chinese} and our \\textsc{SpanSeg} dealing with the training set of the AS dataset on Chinese. We use the same batch size as the work of \\citet{tian-etal-2020-improving-chinese}. The inference time is done by using Tesla P100-PCIE GPU with memory size of 16,280 MiB via Google Colaboratory.}\n\\centering\n\\resizebox{0.65\\textwidth}{!}{%\n\\begin{tabular}{l|c|r|c|r}\n\\hline\n\\multicolumn{1}{r|}{\\multirow{2}{*}{}} & \\multicolumn{2}{c|}{\\textbf{BERT Encoder}} & \\multicolumn{2}{c}{\\textbf{ZEN Encoder}} \\\\ \\cline{2-5} \n\\multicolumn{1}{r|}{} & \\textbf{\\textsc{WMSeg}} & \\multicolumn{1}{c|}{\\textbf{\\textsc{SpanSeg}}} & \\textbf{\\textsc{WMSeg}} & \\multicolumn{1}{c}{\\textbf{\\textsc{SpanSeg}}} \\\\ \\hline\nSize (MB) & \\multicolumn{1}{r|}{704} & 397 & \\multicolumn{1}{r|}{1,150} & 872 \\\\ \\hline\nInference Time (minute) & \\multicolumn{1}{r|}{28} & 15 & \\multicolumn{1}{r|}{46} & 32 \\\\ \\hline\n\\end{tabular}%\n}\n\\end{table}\n\nLastly, we test the \\textsc{WMSeg} and our \\textsc{SpanSeg} when dealing with the largest benchmark dataset AS on Chinese to discuss the size of the model and the inference time. The statistics are presented in Table~\\ref{timetab}, showing that our \\textsc{SpanSeg} has the smaller size and faster inference time than \\textsc{WMSeg}. The statistics can be explained by \\textsc{WMSeg} \\cite{tian-etal-2020-improving-chinese} containing word-hood memory networks to encode both n-grams and the word-hood information, while our \\textsc{SpanSeg} encodes n-grams information via span representation.\n\n\n\n\n\\subsection{Analysis}\n\n\\begin{table}[H]\n\\caption{\\label{viambi}Error statistics of the overlapping ambiguity problem involving three consecutive tokens on VWS dataset. The symbols \\cmark~and \\xmark~denote predicting correctly and incorrectly, respectively.}\n\\centering\n\\resizebox{0.6\\textwidth}{!}{%\n\\begin{tabular}{c|c|r|r|r|r}\n\\hline\n\\multicolumn{1}{l|}{\\multirow{2}{*}{\\textbf{BiLSTM-CRF}}} & \\multicolumn{1}{l|}{\\multirow{2}{*}{\\textbf{\\textsc{SpanSeg}}}} & \\multicolumn{4}{c}{\\textbf{Configuration}} \\\\ \\cline{3-6} \n\\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{c|}{\\textbf{Defalut}} & \\multicolumn{1}{c|}{\\textbf{XLM-R}} & \\multicolumn{1}{c|}{\\textbf{TAG}} & \\multicolumn{1}{c}{\\textbf{TAG+XLM-R}} \\\\ \\hline\n\\xmark & \\xmark & 15 & 5 & 19 & 7 \\\\ \\hline\n\\cmark & \\xmark & \\textbf{7} & \\textbf{0} & 4 & 0 \\\\ \\hline\n\\xmark & \\cmark & \\textbf{7} & \\textbf{0} & \\textbf{18} & \\textbf{1} \\\\ \\hline\n\\end{tabular}%\n}\n\\end{table}\n\nTo explore how our \\textsc{SpanSeg} learns to predict VWS and CWS, we select the statistics of the overlapping ambiguity problem involving three consecutive tokens. The first case is that given the gold standard tags ``\\textbf{\\color{blue}\\underline{B E} S}'', the prediction is incorrect if its tags ``\\textbf{\\color{red}S \\underline{B E}}'', and is correct if its tags ``\\textbf{\\color{blue}\\underline{B E} S}''. The second case is that given the gold standard tags ``\\textbf{\\color{blue}S \\underline{B E}}'', the prediction is incorrect if its tags ``\\textbf{\\color{red}\\underline{B E} S}'', and is correct if its tags ``\\textbf{\\color{blue}S \\underline{B E}}''. Notably, we do not count the case that is not one in two cases we describe. We present the error statistics for Vietnamese in Table~\\ref{viambi}. We can see that the contextual information from XLM-R helps both BiLSTM-CRF and our \\textsc{SpanSeg} in reducing ambiguity. However, according to Table~\\ref{mainvtb}, the predicted word boundary information helps both BiLSTM-CRF and our \\textsc{SpanSeg} in increasing overall performance but causes the overlapping ambiguity problem. Our \\textsc{SpanSeg} (TAG) solves overlapping ambiguity better than BiLSTM-CRF (TAG) when utilizing predicted word boundary information. Lastly, we also provide error statistics for Chinese in Table~\\ref{zhambi}. We can see that overlapping ambiguity is the crucial problem for both \\textsc{WMSeg} \\cite{tian-etal-2020-improving-chinese} and our \\textsc{SpanSeg} on MSR, PKU, and AS datasets.\n\n\\begin{table}[H]\n\\caption{\\label{zhambi}Error statistics of the overlapping ambiguity problem involving three consecutive tokens on five Chinese benchmark datasets. The symbols \\cmark~and \\xmark~denote predicting correctly and incorrectly, respectively.}\n\\centering\n\\resizebox{0.6\\textwidth}{!}{%\n\\begin{tabular}{c|c|r|r|r|r|r}\n\\hline\n\\textbf{\\textsc{\\textbf{WMSeg}}} \\cite{tian-etal-2020-improving-chinese} & \\textbf{\\textsc{\\textbf{SpanSeg}}} & \\multicolumn{1}{c|}{\\textbf{MSR}} & \\multicolumn{1}{c|}{\\textbf{PKU}} & \\multicolumn{1}{c|}{\\textbf{AS}} & \\multicolumn{1}{c|}{\\textbf{\\textsc{CityU}}} & \\multicolumn{1}{c}{\\textbf{CTB6}} \\\\ \\hline\n\\xmark & \\xmark & 14 & 13 & 12 & 2 & 3 \\\\ \\hline\n\\cmark & \\xmark & \\textbf{2} & \\textbf{2} & 2 & \\textbf{1} & \\textbf{2} \\\\ \\hline\n\\xmark & \\cmark & \\textbf{2} & 1 & \\textbf{5} & 0 & 0 \\\\ \\hline\n\\end{tabular}%\n}\n\\end{table}\n\n\n\n\\section{Conclusion}\nThis paper proposes a span labeling approach, namely \\textsc{SpanSeg}, for VWS. Straightforwardly, our approach encodes the n-gram information by using span representations. We evaluate our \\textsc{SpanSeg} on the Vietnamese treebank dataset for the word segmentation task with the predicted word boundary information and the contextual pre-trained embedding from the XLM-RoBerta model. The experimental results on VWS show that our \\textsc{SpanSeg} is better than BiLSTM-CRF when utilizing the predicted word boundary and contextual information with the state-of-the-art F-score of 98.31\\%. We also evaluate our \\textsc{SpanSeg} on five Chinese benchmark datasets to verify our approach. Our \\textsc{SpanSeg} achieves competitive or higher F-scores through experimental results, fewer parameters, and faster inference time than the previous state-of-the-art method, \\textsc{WMSeg}. Lastly, we also show that overlapping ambiguity is a complex problem for VWS and CWS. Via the error analysis on the Vietnamese treebank dataset, we found that utilizing the predicted word boundary information causes overlapping ambiguity; however, our \\textsc{SpanSeg} is better than BiLSTM-CRF in this case. Finally, our \\textsc{SpanSeg} will be made available to the open-source community for further research and development.\n\n\\bibliographystyle{splncs04nat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n \\label{sec:intro}\n \n With the discovery of 1I\/2017 U1 (`Oumuamua) we now have our first glimpse of an interstellar object \\citep{meech2017}.  With a velocity at infinity of around 26~km\/s, and an inclination of 122$^{\\circ}$ that precludes a close encounter with one of the Solar system planets\\footnote{Orbital parameters taken from the JPL small body database, \\url{ssd.jpl.nasa.gov\/sbdb.cgi}}, 1I\/`Oumuamua is securely of interstellar origin.  Furthermore, \\citet{mamajek2017} showed that the trajectory of 1I\/`Oumuamua prior to encountering the Solar system is not consistent with a recent ejection from a nearby star and that its velocity relative to the galactic background is close to the local standard of rest.  This suggests that 1I\/`Oumuamua was ejected at low speed from its parent system and that it has been wandering interstellar space for a long time since.\n \n The existence of 1I\/`Oumuamua can also be used to place constraints on the mass of material typically ejected by planetary systems, \\citep[e.g.][]{laughlin2017, raymond2017}, albeit that with only a single object such estimates are subject to large uncertainties.  In addition while the orbital characteristics of 1I\/`Oumuamua are consistent with our expectations for an interstellar object, its physical characteristics are rather more surprising, in particular its lack of observable activity \\citep[e.g.][]{jewitt2017} and highly elongated shape \\citep[e.g.][]{bolin2017, meech2017}.\n \n \\citet{cuk2017} recently noted that binary and multiple star systems could be a major source of interstellar bodies and suggested that 1I\/`Oumuamua might have originated as a fragment of a much larger body that was tidally disrupted during ejection from its parent system.\n \n In this work we quantitatively examine this scenario, showing that tidal disruptions are unlikely, but that tight binary systems can nevertheless eject an amount of rocky material comparable to the predominantly icy material thrown out by single and wide binary star systems.  We begin in Section~\\ref{sec:comp} by summarising the current state of knowledge of the composition of 1I\/`Oumuamua, in particular examining whether it is a rocky or icy object since this has important implications for its origin.  In Section~\\ref{sec:ejection} we outline our rationale for expecting that binary systems may be a major source of ejected material and then describe our methods in Section~\\ref{sec:method}.  We present the results of our analysis in Section~\\ref{sec:results} and discuss their implications, before summarising in Section~\\ref{sec:conc}.\n \n \\section{The composition and nature of 1I\/`Oumuamua}\n \\label{sec:comp}\n \n Three clues to the composition of 1I\/`Oumuamua are its spectral reflectance profile, lack of a coma and elongated shape.\n A number of groups have obtained photometric colours across a variety of bands in the visible and near infrared \\citep[e.g.][]{bannister2017, bolin2017, jewitt2017, masiero2017, meech2017, ye2017}.  These observations reveal an object with a relatively constant shallow red slope to its reflectance in the visible and near infrared.  It is redder than some inner Solar system asteroid classes and does not appear to demonstrate the broad absorption longwards of 0.75~$\\mu$m seen in many asteroid classes.  However it is also significantly less red than many Kuiper belt objects.  Rather it seems fairly similar to D-type asteroids or the nuclei of long period comets.\n \n Despite its spectral similarity to volatile rich objects,\n 1I\/`Oumuamua has shown no detectable level of activity.  \\citet{jewitt2017} place an upper limit on the rate of mass loss due to activity at $\\sim 2\\times 10^{-4}$~kg~s$^{-1}$, limiting the area of exposed water ice on the surface to $<$1~m$^2$.  As they point out however, this does not preclude the existence of water ice within the interior of 1I\/`Oumuamua since regolith can be an extremely good insulator (with thermal diffusivities as low as $\\sim10^{-8}$~m$^2$~s$^{-1}$) and as such the thermal impulse during its transit of the inner Solar system may only have penetrated around 0.5-1~m below the surface at the present time. \n \n One might propose that since 1I\/`Oumuamua has spent a very long time outside the envelope of a stellar magnetosphere, exposed to the background galactic radiation environment, it might have developed a thick volatile depleted layer \\citep[e.g.][]{fitzsimmons2018}. We find this unlikely however, as the nuclei of long period comets spend the majority of their time in the outer reaches of the Oort cloud, experiencing the same interstellar radiation field that 1I\/`Oumuamua would have done, yet they become active within the inner solar system, an issue also acknowledged by \\citet{fitzsimmons2018}. Additionally, detailed calculations have shown that interstellar radiation is only capable of altering the upper few centimetres, even after over 10$^9$~years beyond the Solar heliopause, where exposure is highest to the most chemically significant radiation \\citep{cooper2003}. This penetration depth would be sufficient to change the colour of the surface, but probably not sufficient to generate an insulating layer that could keep volatiles from degassing during passage through the inner Solar system.\n \n As such we conclude that the lack of activity from 1I\/`Oumuamua is unlikely to be related to radiation exposure during its long sojourn in interstellar space.  Instead we argue that the lack of activity is related to its origin: either it is a truly volatile poor body, or it underwent sufficient processing within its parent system to generate a thick ($\\ga$0.5~m) insulating crust.  Both of these options imply that 1I\/`Oumuamua spent a significant period of time within the inner parts of its parent system prior to being ejected.\n \n \\section{The case for binary systems as a major source of ejected material}\n \\label{sec:ejection}\n \n If we consider a star with a companion and a population of small bodies, simple analysis shows that for ejection to dominate over accretion in encounters between the small bodies and the companion, the escape velocity of the companion should exceed the Keplerian orbital velocity at its semi-major axis \\citep[e.g.][]{wyatt2017}.  This is independent of whether the companion is a planet or a star.  In the Solar system Jupiter, Saturn, Uranus and Neptune all satisfy this criterion, however as discussed by \\citet{wyatt2017} the timescale for ejection is also important.  For Neptune and Uranus, while ejection dominates over accretion, bodies can take 100s of millions of years to be ejected, long enough for galactic tides to perturb the bodies such that they enter the Oort cloud.  Moreover, as we discussed in Section~\\ref{sec:comp} 1I\/`Oumuamua appears to be rocky or devolatilised, suggesting it was ejected from inside the ice line.\n \n For a solar-mass star, efficient ejection inside typical ice line distances of a few AU within the first few Myr of the life of the system requires that the companion has a mass greater than that of Saturn.  However, radial velocity surveys show that the occurrence rate of giant planets at orbital periods of 100-400 days is low---approximately 3\\% \\citep{santerne2016} rising to around 10\\% for orbital periods of $<$10 years \\citep{mayor2011}.  As such, we expect that at most 10\\% of Sun-like single stars will host a planet capable of efficiently ejecting material interior to the ice line.  \\citet{laughlin2017} and \\cite{raymond2017} thus argue that if 1I\/`Oumuamua is indeed rocky, then typical extrasolar asteroid belts must be unusually massive.\\footnote{While material can still be ejected from our own inner Solar system as a result of the terrestrial planets passing material out to Jupiter, the timescales for doing so are longer ($\\sim$10-100~Myr), which means that the total mass can be significantly depleted by collisional processing prior to ejection \\citep{jackson2012, shannon2015}.}  Similarly, recent results from micro-lensing surveys \\citep[e.g.][]{suzuki2016, mroz2017} suggest that giant planets at larger separations are also not common.\n \n While giant planets are relatively uncommon, tight binary systems are abundant \\citep{duchene2013}, and are extremely efficient at ejecting material \\citep{smullen2016}. They may therefore represent a dominant source of interstellar small bodies. We now examine this hypothesis in detail.\n \n \\section{Method}\n \\label{sec:method}\n \n We want to be able to examine the ejection of small bodies from binary systems, and compare this with single stars, in a way that is consistent across spectral classes. As summarized by \\citet{duchene2013}, the properties of binary systems, in particular their separation and their multiplicity frequency, are dependent on the mass of the primary, though they note that these dependences are subject to a fair degree of uncertainty and are the subject of active investigation.  As such, we need to take into account the relative abundance of stars of different masses to be able to build a complete picture of the binary population. To do this we construct a population synthesis model as detailed below.\n \n Physically, our picture is one of planetesimals migrating inwards during the early phases of planet formation, in the presence of a protoplanetary disk.  \\cite{holman1999} showed that any material in circumbinary orbit migrating inward will become unstable on short timescales once it passes a stability boundary $a_{\\rm c, out}$, for which they provide an empirical fit to results from N-body simulations (their equation 3).  This critical distance is a function of the binary mass ratio and eccentricity and ranges from around 2 to 4 times the binary separation.  We thus envisage planetesimals migrating in and then being ejected once they pass $a_{\\rm c, out}$.\n \n  \\begin{table*}\n     \\centering\n     \\caption{Binary fractions and separation distributions used in the population synthesis model.  Chosen to match \\citet{duchene2013}}\n     \\label{table:bindists}\n     \\begin{tabular}{r|c|l}\n        Mass ($M_{\\odot}$) & Binary fraction (\\%) & Separation distribution \\\\\n         $0.1 \\leq M_1 < 0.6$ & 26 & log-normal, mean = 5.30~AU, $\\sigma_{\\log_{10} a}$ = 0.867\\\\\n         $0.6 \\leq M_1 < 1.4$ & 44 & log-normal, mean = 44.7~AU, $\\sigma_{\\log_{10} a}$ = 1.53\\\\\n         $1.4 \\leq M_1 < 6.5$ & 50 & log-normal, mean = 0.141~AU, $\\sigma_{\\log_{10} a}$ = 0.50, + log-normal, mean = 316~AU, $\\sigma_{\\log_{10} a}$ = 1.65\\\\\n         $M_1 \\geq 6.5$       & 70 & log-normal, mean = 0.178~AU,  $\\sigma_{\\log_{10} a}$ = 0.36, + log-uniform, min = 0.50~AU, max = 10$^6$\n     \\end{tabular}\n \\end{table*}\n \n \n \\subsection{Population synthesis model}\n \\label{sec:method:popsynth}\n \n Our population synthesis model proceeds as follows for the construction of a single system:\n \n \\begin{enumerate}\n     \\item Sample the system mass, $M_{\\rm sys}$, from the system initial mass function of \\citet{chabrier2003}.\n     \\item Sample the mass ratio, $\\mu = M_2\/(M_1+M_2)$ uniformly in the range [0, 0.5], as suggested by \\citet{duchene2013}, and use this to calculate $M_1$ and $M_2$.\n     \\item Determine which of the \\citet{duchene2013} mass bins  $M_1$ falls into and use the appropriate multiplicity fraction to determine if the system is actually binary\\footnote{Hierarchical triples would act similarly to binaries in our models, so for simplicity we ignore multiple systems.}. If the system is not binary set $\\mu=0$, $M_1=M_{\\rm sys}$ and return to step (i) and begin again for the next system.\n     \\item Sample the eccentricity uniformly in the range [0, 0.9]\\footnote{We cap the eccentricity at 0.9 to avoid computationally expensive N-body simulations.} and the binary separation from the distributions listed in Table~\\ref{table:bindists}, taken to approximately match \\citet{duchene2013}.\n     \\item Determine the radii and luminosities of the stars by interpolating on the isochrones generated by MIST (the MESA Isochrones and Stellar Tracks, \\citealt{dotter2016, choi2016}) for a system age of 1~Myr.\\footnote{We take 1~Myr to be a representative age of a young system in which substantial disk migration may be ongoing.}\n \\end{enumerate}\n \n Our population synthesis model ensures that binary systems of different masses have the correct weighting and also simultaneously constructs a single star comparison population.  Integrated over all stellar masses the model produces binary systems at a rate of 30\\% by number, while the higher multiplicity rates for more massive stars means that binaries constitute 41\\% of all mass.\n \n We assume that disk dynamics are broadly the same across all systems and that disk masses are a roughly constant fraction of the total stellar mass, such that the mass of material that is ejected by each binary is a constant fraction of the system mass.  As such we will weight systems by their total mass for all of our analysis in the later sections.\n \n In examining our binary systems we should also take into account that binaries with very wide separations are unlikely to have substantial material in circumbinary orbit, but rather can be expected to behave like single stars.  Exactly where to draw the boundary between a wide binary in which we expect the stars to have their own disks that are largely independent and close binaries in which we expect a dominant circumbinary disk is not clear.  We choose to set the boundary as the point at which the outermost stable orbit around the larger star, $a_{\\rm c, in}$ (as determined by Eq.~1 of \\citealt{holman1999}), is greater than 10~AU.  Our results are not sensitive to the precise choice of this cut-off however, as discussed in Section~\\ref{sec:results}. Given these assumptions, we estimate a mass fraction of 26\\% in circumbinaries, with the remainder in singles and wide binaries.\n \n One could envisage there being some intermediate separation binaries in which a circumprimary disk extends to $a_{\\rm c, in}$ such that material in the outer regions of the disk can be ejected due to viscous spreading across the $a_{\\rm c, in}$.  The mass available for ejection in such a system is fairly limited however, being sourced from the lower density outer regions of the disk, whereas inward drift can potentially make a large fraction of the mass of a circumbinary disk available for ejection.  As such we focus here on circumbinary systems.\n \n \\subsection{$N$-body simulations}\n \\label{sec:method:nbody}\n \n While an object that migrates inward past $a_{\\rm c, out}$ will become unstable on short timescales, this does not determine the fate of the object, which can either be ejected or accreted onto one of the stars.  In addition we also want to examine the distribution of close encounter distances prior to ejection to assess the possible role of tidal disruptions as suggested by \\citet{cuk2017}.  Since the unstable region inside $a_{\\rm c, out}$ also applies to gas, we expect that the disk will have a central cavity where the gas density is low. The fate of bodies migrating into this central unstable region can thus be determined using simple $N$-body integrations.\n \n We conduct a set of 2000 $N$-body simulations using the high-order adaptive-timestep integrator {\\sc \\tt IAS15}\\xspace in the {\\sc \\tt REBOUND}\\xspace integration package \\citep{ReinSpiegel2015,ReinLiu2012}.  As described in Section~\\ref{sec:method:popsynth} the mass ratio is uniformly distributed on~[0,~0.5] and the eccentricity is uniformly distributed on~[0,~0.9]. We then initialize particles on co-planar and initially circular orbits at orbital distances beyond the instability limit from \\citet{holman1999}. Finally, we apply an inward drag force that (in an orbit-averaged sense) exponentially decreases the semimajor axis on a timescale 1000 times longer than the orbital period using the \\texttt{modify\\_orbits\\_forces} routine in {\\sc \\tt REBOUNDx}\\xspace, and integrate for $10^4$ binary orbits. We track ejections by flagging particles that go beyond 100 binary orbit separations.\n \n Since Newtonian gravity is scale invariant, we express masses in terms of the primary mass and distances in units of the binary separation. Collisions introduce the scale of the stellar radii into the problem, so we first run a universal set of 2000 integrations of point particles, and dimensionalize after the fact, drawing stellar masses, radii, and binary separations as described above.  We then look for collisions with the stars in this post-processing step.\n \n \\section{Results and discussion}\n \\label{sec:results}\n \n Comparing the distribution of collisional and ejection outcomes for our $N$-body simulations we find that the fraction of particles ejected is near unity, reaching a minimum of $95\\%$ for O and B stars that have the largest collisional cross-sections. As such, it is a good approximation to assume that all material that migrates to within the critical semi-major axis will be ejected.\n \n We now generate a synthetic population of 10$^7$ systems using our population synthesis models.  As noted in Section~\\ref{sec:method:popsynth} we assume that the mass of material that is ejected by each binary is a constant fraction of the system mass.  For now we remain agnostic as to what this ejection fraction is and label it as $M_{\\rm ej}^{\\rm bin}$.  We discuss possible values for $M_{\\rm ej}^{\\rm bin}$ and their implications in Section~\\ref{sec:results:interstellarpop}.\n \n As we discussed above, we believe that 1I\/`Oumuamua is either rocky or has a substantial devolatilised crust, implying that it spent a considerable time inside the ice line in its parent system.  Since bodies are rapidly ejected once they move within $a_{\\rm c, out}$ we are interested specifically in the subset of circumbinary systems where the ice line is outside $a_{\\rm c, out}$.\n \n We determine the ice line distance, $a_{\\rm ice}$, as the distance at which the radiative equilibrium temperature of a blackbody is 150~K, assuming the total luminosity of the two stars is located at the centre of mass of the system.  While the presence of a massive protoplanetary disk can potentially have a large effect on the ice line location through optical depth preventing stellar radiation reaching the interior and through viscous heating, we note that the region interior to $a_{\\rm c, out}$ is expected to have a low gas density such that these effects can be ignored in that region.  As such, while the precise location of the ice line might vary if it is outside $a_{\\rm c, out}$ the fraction of systems with $a_{\\rm ice} > a_{\\rm c, out}$ determined in this way should provide a reasonable estimate.\n \n Since we are allowing $a_{\\rm ice}$ to be close to $a_{\\rm c, out}$ it is worth considering how long a body might need to be inside $a_{\\rm ice}$ to develop a substantial devolatilised layer.  For Solar system comets mass loss rates are typically a few $10^{-7}$~kg~m$^{-2}$~s$^{-1}$ just inside the ice line \\citep{ahearn2012}, corresponding to an erosion rate of $\\sim$0.01~m~yr$^{-1}$.  These mass loss rates rise very rapidly as orbital distance decreases however, reaching $\\sim10^{-5}$-$10^{-4}$~kg~m$^{-2}$~s$^{-1}$ (1-10~m~yr$^{-1}$) by 1~AU where Solar intensity is twice as high as at the ice line.  As such we might expect to require a few centuries to perhaps a few millenia of continuous activity to generate a sufficiently thick devolatilised crust, which is very plausible for a 100~m cometary body migrating in via gas drag from close to $a_{\\rm c, out}$.\n \n  \\begin{figure}\n     \\centering\n     \\includegraphics[width=\\columnwidth]{cbaiacmwcomp3.png}\n     \\caption{Histogram of $a_{\\rm ice}\/a_{\\rm c, out}$ for all close binaries, weighted by the mass of the system.  The total distribution is shown in black, while the orange, green, blue and purple curves show the contributions from stars of different masses.  The grey shading shows the effect of changing the wide binary cut-off to $a_{\\rm c, in} >$ 2.5, 5, 20, or 40~AU.  All curves are normalised to the total mass of close binaries for a wide binary cut-off of $a_{\\rm c, in} >$ 10~AU.  The dashed line indicates $a_{\\rm ice}\/a_{\\rm c, out}=1$.}\n     \\label{fig:aiac}\n \\end{figure}\n \n In Fig.~\\ref{fig:aiac} we then show the distribution of $a_{\\rm ice}\/a_{\\rm c, out}$, weighted by the mass of the system.  With our assumption that the mass of material ejected relative to the system mass is the same for all binaries the fraction with $a_{\\rm ice}\/a_{\\rm c, out} > 1$ in this distribution then tells us the fraction of interstellar material ejected by binaries that will be rocky or devolatilised.  We find that this fraction is 36\\%, such that the ratio of icy to rocky\/devolatilised material is roughly 2:1.  We also show the contributions from stars of different masses in Fig.~\\ref{fig:aiac}.  This shows that the population of icy interstellar material predominantly originates from low mass stars while the population of rocky\/devolatilised material is dominated by intermediate mass stars.  The fraction of systems with $a_{\\rm ice}\/a_{\\rm c, out} > 1$ is relatively insensitive to the choice we made in Section~\\ref{sec:method:popsynth} regarding where to place the wide binary cut-off, as shown by the grey shading in Fig.~\\ref{fig:aiac}.\n \n \\subsection{The population of interstellar bodies}\n \\label{sec:results:interstellarpop}\n \n The total population of interstellar bodies will be the combination of those ejected by binary systems and those ejected by single star systems.  We previously defined the fractional mass ejected by binary systems as $M_{\\rm ej}^{\\rm bin}$.  For single star systems (and wide binaries) we assume that ejection of significant masses of material is limited to those stars that host giant planets, and that those systems eject a fraction of their mass equal to $M_{\\rm ej}^{\\rm sin, gp}$.  Averaged over all singles the fractional mass ejected is then $f_{\\rm giant}M_{\\rm ej}^{\\rm sin, gp}$, where $f_{\\rm giant}$ is the fraction of singles that host giant planets.  The total fractional mass averaged over all stars is then\n \\begin{equation}\n     M_{\\rm interstellar} = f_{\\rm bin} M_{\\rm ej}^{\\rm bin} + (1 - f_{\\rm bin}) f_{\\rm giant}M_{\\rm ej}^{\\rm sin, gp},\n     \\label{eq:minter1}\n \\end{equation}\n where $f_{\\rm bin}$ is the mass fraction of stars that are binaries.  In Section~\\ref{sec:method:popsynth} we found that $f_{\\rm bin}$ = 26\\%, while in Section~\\ref{sec:ejection} we argued that $f_{\\rm giant}$ is no more than 10\\%.  Using these values Equation~\\ref{eq:minter1} becomes\n  \\begin{equation}\n     M_{\\rm interstellar} = 0.26 M_{\\rm ej}^{\\rm bin} + 0.074 M_{\\rm ej}^{\\rm sin, gp}.\n     \\label{eq:minter2}\n \\end{equation}\n We are also interested in the fraction of all interstellar bodies that are rocky or devolatilised rather than icy, $R_{\\rm interstellar}$.  We thus divide the ejected masses into rocky and icy components such that\n\\begin{equation}\n   \\label{eq:minter3}\n   \\begin{split}\n        R_{\\rm interstellar} & = \\frac{M_{\\rm interstellar, rock}}{M_{\\rm interstellar}} \\\\\n            & = \\frac{0.26 R_{\\rm bin}  M_{\\rm ej}^{\\rm bin} + 0.074 R_{\\rm sin} M_{\\rm ej}^{\\rm sin, gp}}{0.26 M_{\\rm ej}^{\\rm bin} + 0.074 M_{\\rm ej}^{\\rm sin, gp}},\n   \\end{split}\n \\end{equation}\n where $R_{\\rm bin}$ and $R_{\\rm sin}$ are the rock\/ice fractions for binaries and single systems respectively.\n \n The Nice model for the early evolution of the outer Solar system suggests that the Solar system began with an outer planetesimal disk of $\\sim$30~$M_{\\oplus}$, the majority of which was ejected \\citep[e.g.][]{gomes2005, levison2008}.  In addition perhaps 1~$M_{\\oplus}$ was ejected from the inner Solar system \\citep[e.g.][]{shannon2015}.  If other systems that host giant planets behave in a similar way to the Solar system then, taking a median value we can expect that $M_{\\rm ej}^{\\rm sin, gp} \\sim 30 M_{\\oplus}\/M_{\\odot}$ and that $R_{\\rm sin} \\sim 0.033$.\n \n We must now estimate $M_{\\rm ej}^{\\rm bin}$.  In doing so we first note that from our $N$-body simulations we expect binaries to eject essentially all material that migrates to within $a_{\\rm c, out}$.  Moreover since the ejection mechanism only relies on the central binary itself, material can be ejected from very early in the life of the system, whereas in a single system ejection can only begin once a giant planet has formed.  If most systems begin with disks close to the gravitational instability limit of $\\sim$0.1~$M_{\\rm sys}$ and a typical gas to dust ratio of around 100 this implies a total mass of solids of $\\sim$300~$M_{\\oplus}\/M_{\\odot}$.  If we estimate that around 10\\% of this material migrates to within the critical stability radius this implies that every binary system ejects as much material as a single star system that hosts giant planets.  Using $M_{\\rm ej}^{\\rm bin} \\sim 30 M_{\\oplus}\/M_{\\odot}$ as our fiducial value we then find that $M_{\\rm interstellar} \\sim 10 M_{\\oplus}\/M_{\\odot}$.  Since we expect $R_{\\rm bin}$ = 0.36 this leads to $R_{\\rm interstellar} \\sim 0.29$.  With this estimate more than three quarters of interstellar bodies originate from binary stars, while for rocky objects the fraction would be even higher.\n \n \\subsection{Tidal disruptions vs extreme heating}\n \\label{sec:results:disruption}\n \n \\citet{cuk2017} suggested that 1I\/`Oumuamua might have originated as a tidal disruption event of a planet. We can test this hypothesis with our $N$-body simulations. For massive stars, the Roche limit (inside which tidal disruptions will occur) is smaller than the stellar radius. To provide the best-case scenario for tidal disruptions, we re-calculate the stellar collision outcomes using stellar radii for the Zero Age Main Sequence, when the radii are at a minimum.  Treating a gravity-dominated planet as a strengthless body, we compute the Roche radius as $R_{\\rm Roche} = 1.26 (\\rho_*\/\\rho_{p})^{(1\/3)} R_*$, where we assume a lower bound for the density of the planet of $\\rho_{p}$ = 3000~kg~m$^{-3}$. We find that none of our 2000 simulations results in a close encounter within the Roche radius.  Indeed the majority of closest approach distances are far outside the Roche radius.  This implies that the frequency of tidal disruptions is $\\lesssim 10^{-3}$ times the ejection rate. Note that there is no contribution from higher mass stars since they always have their Roche radius inside the star.\n \n \\begin{figure}\n     \\centering\n     \\includegraphics[width=\\columnwidth]{danclosetemps2.png}\n     \\caption{Mass weighted distribution of peak temperatures reached by particles ejected in our $N$-body simulations.}\n     \\label{fig:peaktemps}\n \\end{figure}\n \n While we have shown that tidal disruptions are rare,  material that is ejected can potentially be heated to high temperatures during a close approach.  In Fig.~\\ref{fig:peaktemps} we show the mass weighted distribution of peak temperatures experienced by particles in our $N$-body simulations before they are ejected.  Around 10\\% of bodies experience peak temperatures in excess of 1800~K, sufficient to melt rock.  These are drawn from systems independent of their ice line distances, though we note that systems with higher temperatures at the stability boundary $a_{\\rm c, out}$ are more likely to experience extreme heating. \n It would be interesting to consider in future work how extreme heating may modify the shape and surface layers of 1I\/`Oumuamua.  We do note that the extreme elongation would be easier to maintain for an annealed, monolithic body than for a rubble pile.  The fact that such heating predominantly occurs around high-mass stars also further motivates understanding the planet formation process in these extreme regimes.\n \n  \n Since submission of this work, the revised manuscript of \\citet{cuk2017} points out that planets on circumprimary (S type), rather than the circumbinary (P type) orbits considered here may be tidally disrupted more effectively. As he points out, a dynamical instability of several planets around a single star would be required to scatter the doomed planet into a region of phase space where it can chaotically transfer between stars and suffer a close enough encounter for tidal disruption. It would be valuable to extend the analysis of \\citet{cuk2017} to quantify the fraction of instabilities that could plausibly deliver planets into this chaotic intermediate regime instead of directly ejecting them. \n\n Alternatively, planets in such an S type configuration could have their eccentricities secularly driven to high enough values by the binary companion \\citep{naoz2012, petrovich2015} for tidal disruption, though this slow secular driving should be quenched by the tidal forces themselves before the planet is able to come close enough to the star for breakup \\citep[e.g.][]{wu2011, liu2014}.\n \n \\section{Conclusions}\n \\label{sec:conc}\n \n We have shown that the population of interstellar objects can be dominated by planetesimals ejected during planet formation in circumbinary systems.  Even if a typical circumbinary only ejects as much material as the Solar system we would still expect close binaries to be the source of more than three quarters of interstellar bodies due to the relatively low abundance of single star systems with giant planets like the Solar system.  Whereas in the Solar system the ejected material is overwhelmingly icy, we expect that around 36\\% of binaries may predominantly eject material that is rocky or substantially devolatilised, leading to similar expectations for the abundance of rocky\/devolatilised bodies in the interstellar population.\n \n Within the close binary population the dominant source of rocky\/devolatilised material are intermediate mass stars, A-stars and late B-stars.  As such, we suggest that the apparently rocky or devolatilised appearance of 1I\/`Oumuamua indicates that it likely originated in such an intermediate mass binary system.\n \n We find that tidal disruptions during ejection from binary systems, as suggested by \\citet{cuk2017} are rare.  While close encounter distances that result in strong tidal effects are highly unusual around 10\\% of bodies may experience extremely high peak temperatures during ejection.  How these extreme temperatures could influence the shape and surface layers of a body like 1I\/`Oumuamua would be an interesting topic for future work, and that such heating predominantly occurs around high-mass stars also further motivates understanding the planet formation process in these extreme regimes.\n \n \n \\section*{Acknowledgements}\n\nAPJ thanks Andrew Shannon and Casey Lisse for helpful discussions.\nAPJ gratefully acknowledges funding through NASA grant NNX16AI31G (Stop hitting yourself). \nDT thanks Greg Laughlin for insightful discussions.\nHR gratefully acknowledges funding through NSERC Discovery Grant RGPIN-2014-04553.\nThe authors thank the referee for a rapid review and helpful comments that have improved the manuscript.\n\n{\\footnotesize\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\IEEEPARstart{H}{yperspectral} analysis has received an increasing attention because of its high spectral resolution, which enables a variety of objects to be identified and classified \\cite{Plaza2009}. \\textit{Mixed pixels} caused by the presence of multiple objects within a single pixel adversely affect the performance of hyperspectral analysis~\\cite{Kesha2002}. To address this problem, a wide variety of spectral unmixing methods have been developed over the last decades~\\cite{Biouc2012,Dobig2014,Heyle2014,Uezat2017}. Spectral unmixing methods aim at decomposing a mixed spectrum into a collection of reference spectra (known as \\textit{endmembers}) characterizing the macroscopic materials present in the scene, and their respective proportions (known as \\textit{abundances}) in each image pixel~\\cite{Kesha2002}. Despite the large number of developed spectral unmixing methods, there are still major challenges for accurate estimates of endmember signatures and abundances~\\cite{Biouc2012}. Among these challenges, endmember variability may lead to large amounts of errors in abundance estimates~\\cite{Zare2014}. It results from the fact that each endmember can rarely be represented by a unique spectral signature. Conversely, it is subject to so-called spectral variability, e.g., caused by variations in the acquisition process, the intensity of illumination or other physical characteristics of the materials~\\cite{Murph2012,Uezat2016a}. Taking this endmember variability into account during the spectral unmixing process is one of keys for successful application of spectral unmixing~\\cite{Somer2011}.\n\nThe methods that incorporate endmember variability can be categorized into two main approaches (see \\cite{Somer2011,Zare2014,Drume2016b} for recent overviews). The first approach relies on the definition of a set of multiple spectral signatures, referred to as endmember bundles, to characterize each endmember class. Endmember bundles can be collected from field campaign  or can be extracted from data itself using endmember bundle extraction methods~\\cite{Bates2000,Somer2012,Uezat2016a}. The advantage of this approach is that the method can use \\textit{a priori} information representing endmember variability. Endmember bundles can be validated by experts in order to provide accurate representation about endmember variability~\\cite{Bates2000}. Although traditional methods incorporating endmember bundles (e.g.~\\cite{Rober1998}) are known to be computationally expensive, more  efficient methods have been recently developed and have shown great potential~\\cite{Vegan2014,Goena2013,Uezat2016b,Meyer2016}. However, as pointed out in~\\cite{Uezat2016}, it is unlikely that the endmember bundles completely represent endmember variability present in an image. Such incomplete endmember bundles may lead to poor estimates of abundances. The second approach uses physical or statistical descriptions of the endmember variability. More precisely, these methods describes the endmember variability thanks to a statistical distribution~\\cite{Halim2015} or by incorporating additional variability terms in the mixing model~\\cite{Drume2016,Thouv2016,Hong2017}. The advantage of this approach results from the adaptive learning of the endmember variability. Indeed, state-of-the-art methods such as those recently introduced in \\cite{Drume2016,Thouv2016,Hong2017} enable endmember spectra to spatially vary within each pixel in order to describe endmember variability. This is important since the endmember spectra to be used for the abundance estimation can be different between pixels. However, estimating endmember variability without \\textit{a priori} knowledge is a challenging task, especially when large amounts of endmember variability are present in an image. In addition, the statistical distribution or additional terms used in these methods may be overly simplified to represent endmember variability.\n\nBoth approaches demonstrate benefits and drawbacks. A natural question arises: is it possible to combine the strong advantages of both approaches to robustly represent endmember variability? This paper addresses the question and introduces a novel spectral unmixing method that bridges the gap between the aforementioned two approaches with the help of a double sparsity-based method inspired by \\cite{Rubin2010}. Specifically, the proposed method aims at adaptively recovering endmember spectra within each pixel to describe endmember variability while incorporating available \\textit{a priori} information. The proposed method is closely related to the existing methods. Thus, the main contributions of this paper are threefold: 1) to propose a novel spectral unmixing method that incorporates endmember bundles and generates adaptive endmember spectra within each pixel, 2) to give a systematic review of related work and show the relationship between the proposed method and existing methods, 3) to provide comparison between the proposed method and other sparsity-based methods.\n\nThis paper is organized as follows. Section \\ref{section2} describes related works and existing methods, while highlighting their inherent drawbacks. In Section \\ref{section3}, a novel mixing model that incorporates endmember variability is proposed and its relationships with existing methods are discussed. Section \\ref{sec:algo} introduces an associated unmixing algorithm designed to recover the endmember classes, adaptive bundles and abundances. Section \\ref{section4} and Section \\ref{section5} show experimental results obtained from simulated data and real hyperspectral images. Finally, conclusions are drawn in Section \\ref{section6}.\n\n\\section{Related works and issues raised by existing methods}\\label{section2}\n\\subsection{Conventional (variability-free) linear mixing model}\nLet $\\mathbf{y}_i \\in \\mathbb{R}^{L \\times 1}$ denote the $L$-spectrum measured at the $i$th pixel of an hyperspectral image. According to the linear mixing model (LMM), the observed spectrum of the $i$th pixel $\\mathbf{y}_i$ is approximated by a weighted linear combination of endmember spectra and abundance fractions\n\\begin{eqnarray}\n\\label{eq:LMM}\n  \\mathbf{y}_i=\\mathbf{M}\\mathbf{a}_i+\\mathbf{n}_i\n\\end{eqnarray}\nwhere $\\mathbf{M} \\in \\mathbb{R}^{L \\times K}$ is the matrix of the spectral signatures associated with the $K$ endmember classes, $\\mathbf{a}_i = \\left[a_{1i},\\ldots,a_{Ki}\\right]^T \\in \\mathbb{R}^{K \\times 1}$ is the abundance fractions of the pixel and $\\mathbf{n}_i \\in \\mathbb{R}^{L \\times 1}$ represents noise and modeling error. LMM is generally accompanied by abundance non-negativity constraint (ANC) and the abundance sum-to-one constraint (ASC)\n\\begin{eqnarray}\n    \\forall k, \\forall i, \\   a_{ki}\\geq 0,\\quad \\text{and} \\quad  \\forall i,\\  \\sum_{k=1}^{K}a_{ki}=1\n\\end{eqnarray}\nwhere $a_{ki}$ is the abundance fraction of the $k$th class in the $i$th pixel. This model implicitly relies on two assumptions: \\emph{i}) each endmember class is described by a unique spectrum and \\emph{ii}) the endmember matrix $\\mathbf{M}$ is fixed and commonly used to unmix all the pixels of a given image. In other words, LMM does not account for spectral variability. However, as discussed previously, this is likely unrealistic because spectral variability is naturally observed in hyperspectral images, e.g., because of variations in illumination or  physical intrinsic characteristics of materials \\cite{Uezat2016}.\n\n\\subsection{Linear mixing models incorporating endmember bundles}\nMultiple endmember spectral mixture analysis (MESMA) \\cite{Rober1998} allows the variability of the endmember spectrum representative of each class and a varying number of endmember classes present within each pixel. Although MESMA has been widely used for a variety of applications \\cite{Rober1998,Denni2003}, it owns several major limitations: \\emph{i}) it is highly computationally expensive because MESMA needs to test a large number of combinations of endmember spectra \\cite{Tits2013}, \\emph{ii}) MESMA tends to select an overestimated number of endmember classes because it uses the reconstruction error to select the appropriate combination of endmember spectra \\cite{Demar2012} and \\emph{iii}) the performance of MESMA may significantly decrease when endmember spectra (or \\emph{bundles}) within each class do not completely represent the spectral variability \\cite{Uezat2016}.\n\nTo overcome these limitations, recent works have proposed a new class of methods that incorporate all endmember bundles  defined as \\cite{Goena2013,Vegan2014}\n\\begin{eqnarray}\n\\label{eq:bundles}\n  \\mathbf{E}=\\begin{bmatrix}\\ \\mathbf{E}_1 \\ | \\ \\mathbf{E}_2\\  |\\cdots|\\  \\mathbf{E}_K\\ \\end{bmatrix}\n\\end{eqnarray}\nwhere $\\mathbf{E}_k \\in \\mathbb{R}^{L \\times N_k}$ represents a set of endmember spectra (aka bundle) characterizing the $k$th class, $N_k$ is the number of endmember spectra in the $k$th class and $N$ is the total number of endmember spectra of all classes with $N= \\sum_{k=1}^K N_k$. Generalizing LMM in \\eqref{eq:LMM}, those methods firstly model a given observed pixel spectrum with respect to all spectra in endmember bundles and corresponding multiple abundances\n\\begin{eqnarray}\\label{eq:4}\n  \\mathbf{y}_i=\\mathbf{E}\\mathbf{r}_i+\\mathbf{n}_i\n\\end{eqnarray}\nwhere $\\mathbf{r}_i \\in \\mathbb{R}^{N \\times 1}$ is multiple abundance fractions corresponding to each spectrum of the endmember bundles $\\mathbf{E}$. As for LMM, ASC or ANC can also be imposed to $\\mathbf{r}_i$. As a second step, multiple abundance fractions $\\mathbf{r}_i$ are summed within each class to generate a single abundance fraction for each class\n\\begin{eqnarray}\\label{summed}\n  \\mathbf{a}_i=\\mathbf{G}^T\\mathbf{r}_i\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n  \\mathbf{G} = \\begin{bmatrix} \\mathbf{1}_{N_1} & \\mathbf{0}_{N_1} & \\dotsb & \\mathbf{0}_{N_1}\n    \\\\  \\mathbf{0}_{N_2} & \\mathbf{1}_{N_2} & \\mathbf{\\dotsb} & \\mathbf{0}_{N_2}\n\\\\ \\mathbf{\\vdots} & \\mathbf{\\vdots} & \\mathbf{\\ddots} & \\mathbf{\\vdots}\n\\\\ \\mathbf{0}_{N_K} & \\mathbf{0}_{N_K} & \\mathbf{\\dotsb} & \\mathbf{1}_{N_K} \\end{bmatrix}\n\\end{eqnarray}\nwhere $\\mathbf{1}_{N_k} \\in \\mathbb{R}^{N_k \\times 1}$ is a column vector of ones and $\\mathbf{0}_{N_k} \\in \\mathbb{R}^{N_k \\times 1}$ represent a $N_k$-dimensional vector whose components are zeros. While these two steps are conducted separately in \\cite{Goena2013,Vegan2014}, they can be also considered jointly  within a multi-task Gaussian process framework \\cite{Uezat2016,Uezat2016b}. Even if these methods have been shown to be effective, a large number of endmember spectra within each class may be redundant. In such case, following a model selection inspiration, Veganzones \\emph{et al.} introduce a complementary sparsity regularization on the multiple abundance vectors \\cite{Vegan2014}\n\\begin{eqnarray}\n\\begin{aligned}\n\t& \\min\\limits_{\\mathbf{r}_i}\\frac{1}{2}\\Vert\\mathbf{E}\\mathbf{r}_i-\\mathbf{y}_i\\Vert^2_2+\\lambda_r\\Vert\\mathbf{r}_i\\Vert_1 \\\\\n    & \\text{s.t. }   \\forall i, \\quad  \\mathbf{r}_{i} \\succeq 0\n\\end{aligned}\n\\end{eqnarray}\nwhere $\\succeq$ represents the element-wise comparison, $\\Vert\\cdot\\Vert_2$ is the $\\ell_2$-norm, $\\Vert\\cdot\\Vert_1$ is the $\\ell_1$-norm which is known to promote sparsity. Once the multiple abundance vector $\\mathbf{r}_i$ has been estimated, it is normalized in order to reduce the effects of multiplicative factors and satisfy ASC. Following the same approach, further sparsity can be imposed using $\\ell_p$-norm \\cite{Sigur2014} or reweighted $\\ell_1$-approaches \\cite{Zheng2016,He2017}. Overall, this sparsity property allows the selection of a smaller number of endmember spectra. However, it may not lead to the selection of a smaller number of endmember classes. Conversely, to promote sparsity on the number of endmember classes, one strategy consists in formulating the unmixing problem through a sparse group lasso \\cite{Iorda2011a}\n\\begin{eqnarray}\n\\begin{aligned}\n\t& \\min\\limits_{\\mathbf{r}_i}\\left\\lbrace \\frac{1}{2}\\Vert\\sum_{k=1}^{K}\\mathbf{E}_k(\\mathbf{g}_k\\odot\\mathbf{r}_i)-\\mathbf{y}_i\\Vert^2_2 \\right. \\\\\n    & \\left. +\\lambda_g\\sum_{k=1}^{K}\\Vert\\mathbf{g}_k\\odot\\mathbf{r}_i\\Vert_2 + \\lambda_r\\Vert\\mathbf{r}_i\\Vert_1 \\right\\rbrace\\\\\n    & \\text{s.t. }   \\forall i, \\quad  \\mathbf{r}_{i} \\succeq 0\n\\end{aligned}\n\\end{eqnarray}\nwhere $\\mathbf{g}_k$ is the $k$th column of $\\mathbf{G}$, $\\odot$ is the element-wise product and thus $\\mathbf{g}_k\\odot\\mathbf{r}_i$ extracts the elements in $\\mathbf{r}_i$ belonging to the $k$th class. This approach has the great advantage of promoting sparsity in both the number of endmember spectra and the number of endmember classes. Another strategy relies on the concept of ``social sparsity'' that can exploit the structure of endmember bundles more explicitly \\cite{Meyer2016}. The method assumes that $\\mathbf{r}_i$ can be partitioned into $K$ groups representing each endmember class, leading to the optimization problem\n\\begin{eqnarray}\n\\begin{aligned}\n\t& \\min\\limits_{\\mathbf{r}_i} \\left\\lbrace \\frac{1}{2}\\Vert\\mathbf{E}\\mathbf{r}_i-\\mathbf{y}_i\\Vert^2_2+\\lambda_r\\left(\\sum_{k=1}^{K}\\Vert \\mathbf{g}_k\\odot\\mathbf{r}_i\\Vert_p^q\\right)^{\\frac{1}{q}} \\right\\rbrace\\\\\n    & \\text{s.t. }  \\forall i,\\forall n, \\quad  \\mathbf{r}_{i} \\succeq 0, \\quad \\sum_{n=1}^{N}r_{ni}=1\n\\end{aligned}\n\\end{eqnarray}\nwhere $\\Vert\\cdot\\Vert_p$ is the $\\ell_p$-norm and $r_{ni}$ is the $n$th multiple abundance fraction of the $i$th pixel.  Finally, abundances associated each endmember class can be obtained by summing the multiple abundances within each class as in \\eqref{summed}. This method can be considered as a generalized model since, by adjusting the values of $(p,q)$, it boils down to the group lasso, the elitist lasso or the fractional case \\cite{Meyer2016}.\n\nHowever, all aforementioned sparsity-based methods still suffer from the following limitations:\n\\begin{enumerate}\n  \\item \\emph{Physically unrealistic abundance fractions}: They explicitly generate unrealistic multiple abundances corresponding to each spectrum in endmember bundles.\n  \\item \\emph{Lack of adaptability to describe endmember variability}: ASC imposed on $\\mathbf{r}_i$ does not allow a consistant description of the endmembers within each pixel. In addition, it cannot capture adaptive and hierarchical structure of endmember spectra for each pixel.\n\\end{enumerate}\nTo overcome these two shorcomings, the present paper capitalizes on this abundant literature to design a new multiple endember mixing model introduced in the next section.\n\n\\section{Multiple endmember mixing models}\\label{section3}\n\\subsection{MEMM}\nThe proposed model relies on 3 main ingredients, namely endmember bundles, bundling coefficients and abundances. According to this model, each endmember bundle is mixed to provide a suitable and adaptive endmember spectrum used to unmix a given pixel. The proposed multiple endmember mixing model (MEMM) is defined as\n\\begin{eqnarray}\n\\label{eq:MEMM_model}\n  \\mathbf{y}_i=\\mathbf{E}\\mathbf{B}_i\\mathbf{a}_i+\\mathbf{n}_i\n\\end{eqnarray}\nwhere $\\mathbf{B}_i \\in \\mathbb{R}^{N \\times K}$ gathers so-called bundling coefficients of the $i$th pixel which decompose the endmember signatures according to the endmember bundles for the considered pixel. To enforce the bundle structure, the bundling coefficients $\\mathbf{B}_i$ associated with the pixel is defined as the following block-diagonal matrix\n\\begin{eqnarray}\n  \\mathbf{B}_i = \\begin{bmatrix} \\mathbf{b}_{1i} & \\mathbf{0}_{N_1} & \\dotsb & \\mathbf{0}_{N_1}\n    \\\\  \\mathbf{0}_{N_2} & \\mathbf{b}_{2i} & \\mathbf{\\dotsb} & \\mathbf{0}_{N_2}\n\\\\ \\mathbf{\\vdots} & \\mathbf{\\vdots} & \\mathbf{\\ddots} & \\mathbf{\\vdots}\n\\\\ \\mathbf{0}_{N_K} & \\mathbf{0}_{N_K} & \\mathbf{\\dotsb} & \\mathbf{b}_{Ki} \\end{bmatrix}\n\\end{eqnarray}\nwhere $\\mathbf{b}_{ki} \\in \\mathbb{R}^{N_k \\times 1}$ is the bundling coefficients for the $k$th class at the $i$th pixel. Each bundling coefficient must be nonnegative and the bundling vector $\\mathbf{b}_{ki}$ is expected to be sparse. Indeed multiple endmember spectra within each class are usually redundant and only a few endmember spectra within each class should be enough to unmix a pixel. This property can be induced by considering the following bundling constraints\n\\begin{equation}\n\\label{eq:B_constraints}\n    \\forall i, \\ \\mathbf{B}_i \\succeq 0 \\quad \\text{and} \\quad  \\Vert\\mathbf{B}_i\\Vert_0=\\sum_{k=1}^{K}\\Vert\\mathbf{b}_{ki}\\Vert_0 \\leq s\n\\end{equation}\nwhere $\\Vert\\cdot\\Vert_0$ is the $\\ell_0$-norm that counts the number of nonzero elements and $s$ is the maximum number of nonzero elements in $\\mathbf{B}_i$, i.e., the maximum number of endmembers to be used within each class to describe the pixel. The abundance non-negativity constraint (ANC) and the abundance sum-to-one constraint (ASC) are usually imposed. In addition, in this work, complementary sparsity is imposed on each abundance vector, i.e.,\n\\begin{equation}\n\\label{eq:A_constraints_MEMM}\n  \\forall k,\\forall i,\\  a_{ki}\\geq 0,\\quad \\text{and} \\quad \\forall i,\\ \\sum_{k=1}^{K}a_{ki}=1, \\ \\ \\Vert\\mathbf{a}_i\\Vert_0 \\leq v\n\\end{equation}\nwhere $v$ is the number of endmember classes to be used to decompose the image pixel.\n\n\\subsection{MEMMs}\\label{subsec:MEMMs}\nThe sparsity constraint \\eqref{eq:B_constraints} applied to $\\mathbf{B}$ can be slightly modified to obtain another meaningful set of constraints\n\\begin{equation}\n\\label{eq:B_constraints_MEMMs}\n    \\forall i,\\ \\mathbf{B}_i \\succeq 0 \\quad \\text{and} \\quad \\forall k, \\forall i,\\ \\Vert\\mathbf{b}_{ki}\\Vert_0 \\leq 1.\n\\end{equation}\nThe resulting model, referred to as MEMM$_s$ in what follows, is designed to generate at most one scaled endmember spectrum for each class.\n\n\n   \n   \n\n\\subsection{Relationships between MEMM and existing models}\n\\subsubsection{MEMM$_s$ and MESMA}\nWhen the sparsity constraint on the abundances $\\Vert\\mathbf{a}_i\\Vert_0 \\leq v$ in \\eqref{eq:A_constraints_MEMM} is not considered, the optimization problem associated with the MEMM$_s$ model described in paragraph \\ref{subsec:MEMMs} is equivalent to MESMA and sparse MESMA \\cite{Chen2016}. Unlike MESMA that considers the reconstruction error to determine the optimal combination of endmember classes within each pixel, MEMM$_s$ incorporates the sparsity constraint to select the optimal combination. This prevents a larger number of endmember classes to be selected for each pixel.\n\n\\subsubsection{MEMM and pixel-wise endmember variability models}\nBy denoting $\\mathbf{\\tilde{M}}_i=\\mathbf{E}\\mathbf{B}_i$ the equivalent endmember matrix associated with the $i$th pixel, MEMM models the observed pixel spectra as\n\\begin{eqnarray}\n\t\\mathbf{y}_i=\\mathbf{\\tilde{M}}_i\\mathbf{a}_i+\\mathbf{n}_i\n\\end{eqnarray}\nwhere $\\mathbf{\\tilde{M}}_i$ can be interpreted as a set of $K$ spatially varying endmember spectra. This approach has been also adpoted in recent works to incorporate endmember variability as additive factors \\cite{Thouv2016}, multiplicative factors \\cite{Drume2016,Imbir2017} or a combination of additive and multiplicative factors \\cite{Hong2017}. In particular, when $N_1=\\ldots,N_k=1$ in \\eqref{eq:bundles}, the endmember bundles $\\mathbf{E}_1,\\ldots, \\mathbf{E}_K$ are reduced to unique endmember spectra characterizing each class. The associated bundling coefficient matrix $\\mathbf{B} = \\mathrm{diag}[b_1,\\ldots,b_K]$ is diagonal where each coefficient $b_k$ scales the corresponding endmember spectrum $\\mathbf{E}_k$ ($k=1,\\ldots,K$). Thus, MEMM generalizes the recently introduced  extended linear mixing model \\cite{Drume2016}.\n\nHowever, MEMM is different from the aforementioned methods since it resorts to \\textit{a priori} information (i.e., endmember bundles) to model the endmember variability. More precisely, MEMM describes the admissible variability within an endmember class as the convex cone spanned by the corresponding bundles. As a consequence, \\emph{per se}, MEMM offers an adaptive description of the spectral variability even when pre-defined endmember bundles do not completely capture this variability within each class.\n\n\\subsubsection{MEMM and sparsity-based unmixing methods}\nBy setting $\\mathbf{r}_i=\\mathbf{B}_i\\mathbf{a}_i$, MEMM in \\eqref{eq:MEMM_model} can be rewritten as\n\\begin{eqnarray}\n\t\\mathbf{y}_i=\\mathbf{E}\\mathbf{r}_i+\\mathbf{n}_i\n\\end{eqnarray}\nsimilarly to the existing models discussed in Section \\ref{section2}. The main difference is that MEMM enables the multiple abundances $\\mathbf{r}_i$ to be decomposed into bundling coefficients $\\mathbf{B}_i$ within each class and abundances $\\mathbf{a}_i$, resulting into a bi-layer description of the abundances. This hierarchical decomposition has been also adopted by unmixing methods based on multilayer nonnegative matrix factorization (MLNMF) \\cite{Rajab2015}. However, each layer induced by MEMM (i.e., bundling matrix and abundance vector) has a clear and meaningful role. Moreover, when the existing methods impose ASC onto $\\mathbf{r}_i$, they assume that mixed spectra belong to the simplex spanned by the endmember bundles. However, this is a limited assumption since observed spectra may be outside the simplex, e.g., when affected by variations in illumination. Conversely, MEMM imposes ASC only onto the abundances $\\mathbf{a}_i$ of endmember classes and enables the bundling coefficients $\\mathbf{B}_i$ to scale the endmember signature, e.g.,  to capture variability induced by varying illumination (see experiments in Section \\ref{section6}). In addition,\nMEMM complements this bi-layer hierarchy with a twofold structured, physically-motivated sparsity imposed on the multiple abundance vector $\\mathbf{r}_i$. This bilevel sparsity has the significant advantage of reducing overfitting and one may expect a significant improvement of stability and interpretability of the abundance estimates.\n\nFinally, the intrinsic structure of MEMM is also similar to the recently developed methods based on robust constrained matrix factorization \\cite{Akhta2017} and kernel archetypoid analysis \\cite{Zhao2017}. These methods model a set $\\mathbf{Y} = \\left[\\mathbf{y}_1,\\ldots,\\mathbf{y}_P\\right] \\in \\mathbb{R}^{L \\times P}$ of $P$ pixel spectra as\n\\begin{eqnarray}\n\t\\mathbf{Y}=\\mathbf{Y}\\mathbf{C}\\mathbf{A}+\\mathbf{N}\n\\end{eqnarray}\nwhere $\\mathbf{C} \\in \\mathbb{R}^{P \\times K}$ is a matrix gathering a set of coefficients, $\\mathbf{A} = \\left[\\mathbf{a}_1,\\ldots,\\mathbf{a}_P\\right] \\in \\mathbb{R}^{K \\times P}$ is the abundance matrix and $\\mathbf{N} = \\left[\\mathbf{n}_1,\\ldots,\\mathbf{n}_P\\right] \\in \\mathbb{R}^{L \\times N}$ is the error and noise matrix. Sparsity (induced by ASC or the use of $\\ell_0$-pseudonorm) and nonnegativity constraints are imposed onto each column of $\\mathbf{C}$ and $\\mathbf{Y}\\mathbf{C}$ can be interpreted as synthetic endmember spectra. These methods use the subset of whole image pixels to generate synthetic endmember spectra that are fixed within each image. On the other hand, MEMM uses the subset of endmember bundles to generate synthetic endmember spectra that may be different for each pixel.\n\n\\section{MEMM-based unmixing algorithm}\\label{sec:algo}\nUnmixing according to the proposed MEMM can be formulated as the minimization problem\n\\begin{eqnarray}\\label{eq_sep}\n\\begin{aligned}\n\t& \\min\\limits_{\\mathbf{B}_i,\\mathbf{a}_i}\\frac{1}{2}\\Vert\\mathbf{E}\\mathbf{B}_i\\mathbf{a}_i-\\mathbf{y}_i\\Vert^2_2 \\\\\n    \\text{s.t.}\\   \\forall k,\\forall i, & \\ a_{ki} \\geq 0,\\quad  \\sum_{k=1}^{K}a_{ki}=1,\\quad \\Vert\\mathbf{a}_i\\Vert_0 \\leq v, \\\\\n    &  \\mathbf{B}_i \\succeq 0,\\quad \\Vert\\mathbf{B}_i\\Vert_0 \\leq s.\n\\end{aligned}\n\\end{eqnarray}\nThis minimization problem is similar to the double sparsity-inducing method proposed in \\cite{Rubin2010}. Using an alternative formulation, the minimization problem can be written as the following non-convex minimization problem:\n\\begin{equation}\n\t\\min\\limits_{\\mathbf{B}_i,\\mathbf{a}_i} \\mathcal{J}\\left(\\mathbf{B}_i,\\mathbf{a}_i\\right)=\\left\\lbrace f(\\mathbf{B}_i,\\mathbf{a}_i) + h(\\mathbf{B}_i)+g(\\mathbf{a}_i) \\right\\rbrace\n\\end{equation}\nwith\n\\begin{eqnarray}\n     f(\\mathbf{B}_i,\\mathbf{a}_i)&=\\frac{1}{2}\\Vert\\mathbf{E}\\mathbf{B}_i\\mathbf{a}_i-\\mathbf{y}_i\\Vert^2_2\\\\\n     h(\\mathbf{B}_i)&=\\iota_{\\mathbb{R}_+}(\\mathbf{B}_i) + \\lambda_b\\Vert\\mathbf{B}_i\\Vert_0\\\\\n     g(\\mathbf{a}_i)&=\\iota_{\\mathbb{S}}(\\mathbf{a}_i) + \\lambda_a\\Vert\\mathbf{a}_i\\Vert_0\n\\end{eqnarray}\nwhere $\\lambda_a$ and $\\lambda_b$ are parameters which control the balance between the data fitting term and the sparse regularizations, $\\iota_{\\mathcal{C}}(\\mathbf{x})$ is the indicator function on the set $C$ (i.e., $\\iota_{\\mathcal{C}}(\\mathbf{x})=0$ when $\\mathbf{x} \\in \\mathcal{C}$ whereas $\\iota_{\\mathcal{C}}(\\mathbf{x})=\\infty$ when $\\mathbf{x} \\notin \\mathcal{C}$), and $\\mathbb{S}$ is the simplex defined by the ASC and ANC. Solving this optimization problem is challenging since the  regularization functions $h$ and $g$ are nonconvex and nonsmooth. However, it can be tackled thanks to the proximal alternating linearized minimization (PALM)  \\cite{Bolte2014}. With guarantees to converge to a critical point, PALM iteratively updates the parameters $\\mathbf{a}_i$ and $\\mathbf{B}_i$ by alternatively minimizing the objective function with respect to (w.r.t.) these parameters, i.e., by solving the following proximal problems\n\\begin{eqnarray}\n\\begin{aligned}\n\t\\mathbf{B}_i^{(t+1)} \\in & \\min\\limits_{\\mathbf{B}_i} \\left\\lbrace h(\\mathbf{B}_i) + \\langle \\mathbf{B}_i-\\mathbf{B}_i^{(t)},\\nabla_{\\mathbf{B}_i}f(\\mathbf{B}_i^{(t)},\\mathbf{a}_i^{(t)})  \\rangle \\right.\\\\\n    &\\left. + \\frac{c_t}{2}\\Vert\\mathbf{B}_i-\\mathbf{B}_i^{(t)}\\Vert_2^2 \\right\\rbrace\\\\\n    \\mathbf{a}_i^{(t+1)} \\in & \\min\\limits_{\\mathbf{a}_i} \\bigg\\{ g(\\mathbf{a}_i) + \\langle \\mathbf{a}_i-\\mathbf{a}_i^{(t)},\\nabla_{\\mathbf{a}_i}f(\\mathbf{B}_i^{(t+1)},\\mathbf{a}_i^{(t)}) \\rangle \\}. \\\\\n    &\\left. + \\frac{d_t}{2}\\Vert\\mathbf{a}_i-\\mathbf{a}_i^{(t)}\\Vert_2^2 \\right\\rbrace\n\\end{aligned}\n\\end{eqnarray}\nThe pseudocode for MEMM is shown in Algorithm~\\ref{algorithm_memm} and these two steps are described in what follows.\n\n\\subsection{Optimization w.r.t. $\\mathbf{B}_i$}\nTo optimize only w.r.t. the diagonal entries in $\\mathbf{B}_i$, the objective function can be rewritten with the following decomposition\n\\begin{eqnarray}\n\\begin{aligned}\n\tf(\\mathbf{b}_i,\\mathbf{a}_i)& =\\frac{1}{2}\\Vert\\mathbf{U}_i\\mathbf{b}_i-\\mathbf{y}_i\\Vert^2_2\\\\\n    h(\\mathbf{b}_i)& =\\iota_{\\mathbb{R}_+}(\\mathbf{b}_i) + \\lambda_b\\Vert\\mathbf{b}_i\\Vert_0\n\\end{aligned}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray*}\n\\begin{aligned}\n\t\\mathbf{U}_i&=\\left[\\mathbf{E}_1 \\odot a_{1i}| \\cdots| \\mathbf{E}_K\\odot a_{Ki}\\right]\\\\\n    \\mathbf{b}_i &= \\left[ \\mathbf{b}_{1i}^T, \\mathbf{b}_{2i}^T, \\cdots, \\mathbf{b}_{Ki}^T\\right]^T.\n\\end{aligned}\n\\end{eqnarray*}\nThis leads to the following updating rule\n\\begin{eqnarray*}\n\t\\min\\limits_{\\mathbf{b}_i} \\left\\lbrace h(\\mathbf{b}_i) + \\frac{c_{t}}{2}\\Vert\\mathbf{b}_i-(\\mathbf{b}_i^{(t)}-\\frac{1}{c_{t}}\\nabla_{\\mathbf{b}}f(\\mathbf{b}_i^{(t)},\\mathbf{a}_i^{(t)}))\\Vert_2^2 \\right\\rbrace\n\\end{eqnarray*}\nwhere $\\nabla_{\\mathbf{b}_i}f(\\mathbf{b}_i^{(t)},\\mathbf{a}_i^{(t)})=\\mathbf{U}_i^T\\left(\\mathbf{U}_i \\mathbf{b}_i - \\mathbf{y}_i\\right)$.\n\nUsing similar computations as in \\cite{Bolte2014}, this can be conducted as\n\\begin{eqnarray}\n\t\\begin{aligned}\n\t\\mathbf{b}_i^{(t+1)} & \\in \\text{prox}_{c_t\/\\lambda_b}^h(\\mathbf{b}_i^{(t)}-\\frac{1}{c_t}\\nabla_{\\mathbf{b}_i}f(\\mathbf{b}_i^{(t)},\\mathbf{a}_i^{(t)}))\\\\\n  \n    \\end{aligned}\n\\end{eqnarray}\nwhere $c_{t}=\\gamma_m\\Vert\\mathbf{U}_i^T\\mathbf{U}_i\\Vert_{\\mathrm{F}}$ represents a step size for each iteration. The proximal operator associated with $f$ can be computed using the approach \\cite{Bolte2014}. Finally, the bundling matrix $\\mathbf{B}_i$ can be reconstructed as $\\mathbf{B}_i=\\text{blkdiag}(\\mathbf{b}_i)$ where $\\text{blkdiag}(\\cdot)$ generates the block diagonal matrix $\\mathbf{B}_i$ from the vector $\\mathbf{b}_i$.\\\\\n\n\\subsection{Optimization with respect to $\\mathbf{a}_i$}\nTo optimize w.r.t. $\\mathbf{a}_i$, the objective function can be rewritten using the decomposition\n\\begin{eqnarray}\n\t f(\\mathbf{B}_i,\\mathbf{a}_i)=&\\frac{1}{2}\\Vert\\mathbf{\\tilde{M}}_i\\mathbf{a}_i-\\mathbf{y}_i\\Vert^2_2\\\\\n    g(\\mathbf{a}_i)=&\\iota_{\\mathbb{S}}(\\mathbf{a}_i) + \\lambda_a\\Vert\\mathbf{a}_i\\Vert_0\n\\end{eqnarray}\nwhere $\\mathbf{\\tilde{M}}_i=\\mathbf{E}\\mathbf{B}_i$. Thus, updating the abundance vector can be formulated as\n\\begin{eqnarray*}\n\t\\min\\limits_{\\mathbf{a}_i} \\left\\lbrace g(\\mathbf{a}_i) + \\frac{d_{t}}{2}\\left\\|\\mathbf{a}_i-\\left(\\mathbf{a}_i^{(t)}-\\frac{1}{d_{t}}\\nabla_{\\mathbf{a}_i}f(\\mathbf{B}_i^{(t+1)},\\mathbf{a}_i^{(t)})\\right)\\right\\|^2 \\right\\rbrace\n\\end{eqnarray*}\nwhere $\\nabla_{\\mathbf{a}_i}f(\\mathbf{B}_i^{(t+1)},\\mathbf{a}_i^{(t)})=\\mathbf{\\tilde{M}}_i^T\\left(\\mathbf{\\tilde{M}}_i \\mathbf{a}_i - \\mathbf{y}_i\\right)$. Using the proximal operator, this can be written as\n\\begin{eqnarray}\n\t\\begin{aligned}\n\t\\mathbf{a}_i^{(t+1)} & \\in \\text{prox}_{d_{t}\/\\lambda_a}^g\\left(\\mathbf{a}_i^{(t)}-\\frac{1}{d_{t}}\\nabla_{\\mathbf{a}_i}f(\\mathbf{B}_i^{(t+1)},\\mathbf{a}_i^{(t)})\\right)\n    \\end{aligned}\n\\end{eqnarray}\nwhere $d_{t}=\\gamma_a\\Vert\\mathbf{\\tilde{M}}_i^T\\mathbf{\\tilde{M}}_i\\Vert_{\\mathrm{F}}$ represents a step size for each iteration. Moreover the proximal mapping associated with $g$  can be  performed using the method developed in \\cite{Anast2013}.\n\n\\begin{algorithm}\n\\caption{Algorithm for MEMM-based unmixing}\\label{algorithm_memm}\n\\begin{algorithmic}[1]\n\\State $\\mathbf{Input}: \\mathbf{y}_i,\\mathbf{E}$\n\\State \\textbf{Initialization}: $\\mathbf{a}_i^{(0)}$ and $\\mathbf{B}_i^{(0)}$.\n\\State Set $\\mathbf{r}_i^{(0)}$ using an unmixing method (e.g. FCLS).\n\\State $\\mathbf{a}_i^{(0)}=\\mathbf{G}^T\\mathbf{r}_i^{(0)}$\n\\State $\\forall_k, \\mathbf{b}_{ik}^{(0)}=(\\mathbf{g}_k\\odot\\mathbf{r}_i^{(0)}) \\oslash a_{ki}^{(0)}$\n\\State \\textbf{Main procedure}:\n\\While{ the stopping criterion is not satisfied}\n\\State $\\mathbf{b}_i^{(t+1)} \\leftarrow  \\text{prox}_{c_t\/\\lambda_b}^h(\\mathbf{b}_i^{(t)}-\\frac{1}{c_{t}}\\nabla_{\\mathbf{b}_i}f(\\mathbf{b}_i^{(t)},\\mathbf{a}_i^{(t)}))$\n\\State $\\mathbf{B}_i^{(t+1)}=\\text{blkdiag}(\\mathbf{b}_i^{(t+1)})$\n\\State $\\mathbf{a}_i^{(t+1)} \\leftarrow  \\text{prox}_{d_{t}\/\\lambda_a}^g(\\mathbf{a}_i^{(t)}-\\frac{1}{d_{t}}\\nabla_{\\mathbf{a}_i}f(\\mathbf{B}_i^{(t+1)},\\mathbf{a}_i^{(t)}))$\n\\EndWhile\n\\State $\\mathbf{Output}: \\mathbf{a}_i^{(t+1)},\\mathbf{B}_i^{(t+1)}$\n\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{Initialization and stopping rule}\nMEMM requires initial estimates $\\mathbf{a}_i^{0}$ and $\\mathbf{B}_i^{0}$ of the abundance vector and bundling matrix, respectively. To do so, first, MEMM estimates a multiple abundance vector $\\mathbf{r}_i^{(0)}$ using a state-of-the-art LMM-based unmixing method  (e.g., FCLS, see line 3). Then an initial estimate of the single abundance vector $\\mathbf{a}_i^{(0)}$ is computed according to \\eqref{summed} (see line 4). Finally, the bundling matrix $\\mathbf{B}_i^{(0)}$ is arbitrarily initialized as the corresponding scaling factor (see line 5, where $\\oslash$ stands for the element-wise division). Once initial estimates have been obtained, $\\mathbf{a}_i^{(t+1)}$ and $\\mathbf{b}^{(t+1)}$ are iteratively updated in lines 8--10. The algorithm stops when the difference between updated and previous values of the objective function $f(\\mathbf{B}_i,\\mathbf{a}_i)$ is smaller than a predetermined threshold.\n\n\n\\begin{figure*}\n        \\centering\n        \\includegraphics[width=\\linewidth]{endm_b_sim}\n        \\caption{Synthetically generated endmember bundles.}\\label{fig:endm_b_sim}\n\\end{figure*}\n\n\\section{Experiments using simulated data}\\label{section4}\nFirst, the relevance of the proposed MEMM and its variant MEMM$_{s}$ has been evaluated thanks to experiments conducted on simulated datasets.\n\n\\subsection{Generation of bundles}\nFirst, $K=10$ spectra were selected from the USGS spectral library. The $10$ spectra were chosen so that the minimum angle between any two spectra was larger than $5^\\circ$. This pruning prevented synthetic endmember bundles to overlap each other. Second, endmember bundles $\\mathbf{E}_k$ ($k=1,\\ldots,K$)  were designed by randomly generated $N_k=30$ endmember spectra for each endmember bundle using the approach proposed in \\cite{Thouv2016a}. These bundles, depicted in Fig. \\ref{fig:endm_b_sim},  were used for generating the following two simulated data.\n\n\\subsubsection{Simulated dataset 1 (SIM1)}\nThe first dataset, referred to as SIM1 in what follows, was generated using MESMA. A mixed spectrum was generated using the following 5 steps. First, the number of endmember classes $K$ was randomly determined in the set ($\\{1,\\ldots,5\\}$). Second, a random combination of endmember classes was selected. Third, one spectrum within each selected endmember class was randomly chosen. Forth, the abundances of the selected endmember classes were randomly generated using a Dirichlet distribution to jointly ensure ANC and ASC. Finally, a mixed spectrum was generated by a linear combination of endmember spectra of the selected endmember classes and the randomly generated abundances. A set of $P=100$ mixed spectra were generated in this study. Different amounts of additive Gaussian noise with corresponding signal-to-noise ratios of $50$dB, $40$dB and $30$dB were considered to the mixed spectra.\n\n\\subsubsection{Simulated dataset 2 (SIM2)}\nSIM2 was generated using MEMM. A mixed spectrum was generated similarly as done in SIM1. The main difference is the bundling coefficients in MEMM. In order to generate the bundling coefficients, the number of spectra $N_k$ was randomly chosen in the set ($\\{1,\\ldots,5\\}$). The bundling coefficients of the randomly selected spectra were generated from a Dirichlet distribution. A mixed spectrum was generated by a linear combination of spectra, the bundling coefficients and the abundances. As for SIM1, $P=100$ mixed spectra were generated and different amounts of Gaussian noise were also added to SIM2.\n\n\n\\subsection{Compared methods}\nMEMM and MEMM$_s$ were compared with other 5 methods that incorporate endmember bundles and promote sparsity: fully constrained least squares (FCLS) \\cite{Biouc2010}, sparse unmixing by variable splitting and augmented Lagrangian (SUnSAL) \\cite{Iorda2011}, alternating angle minimization (AAM) \\cite{Heyle2016a} and methods based on group lasso and elitist lasso \\cite{Meyer2016}. Note that FCLS and SUnSAL can incorporate endmember bundles by considering the model in (\\ref{eq:4}). FCLS and SUnSAL were chosen for comparing the proposed methods with most widely used unmixing methods that promote sparsity. AAM was selected for comparison because it is a latest variant of MESMA and it is more computationally efficient than MESMA while achieving good performance. Group lasso and elitist lasso were also included for comparison because they can incorporate the structure of endmember bundles as well as MEMM. MEMM, MEMM$_s$ and the methods based on group lasso and elitist lasso require initial estimates of abundances. In order to fairly compare methods, this study used abundances estimated by FCLS for the initial estimates required for the methods. SUnSAL, the methods based on group lasso or elitist lasso and MEMM$_s$ require a parameter $\\lambda_r$ or $\\lambda_a$ controlling sparsity regularization. MEMM requires two parameters $\\lambda_b$ and $\\lambda_a$. In order to fairly compare the methods, these parameters were empirically determined in the set $(0.0001, 0.001, 0.01, 0.1, 1, 5)$ for each simulated data so that the selected values produced the highest SRE$_a$. Parameter sensitivity of the proposed method is reported in the supplementary document \\cite{Uezat2018sup}. Finally, computational time were also discussed.\n\n\\subsection{Performance criteria}\nThe main objective of this experiment was to assess the algorithm performance while selecting a combination of endmember classes and spectra, and estimating abundances corresponding to endmember classes and spectra. Three criteria were chosen for quantitative validation of the methods. To evaluate the quality of the reconstruction, one defines the signal-to-reconstruction error (SRE) per endmember class and per endmember spectrum as \\cite{Iorda2014a}\n\\begin{equation}\n\\begin{aligned}\n\\text{SRE}_a \\equiv \\mathbb{E}\\left[\\Vert \\mathbf{a} \\Vert_2^2 \\right]\/\\mathbb{E}\\left[\\Vert \\mathbf{a}-\\hat{\\mathbf{a}} \\Vert_2^2 \\right]\\\\\n\\text{SRE}_r \\equiv \\mathbb{E}\\left[\\Vert \\mathbf{r} \\Vert_2^2 \\right]\/\\mathbb{E}\\left[\\Vert \\mathbf{r}-\\hat{\\mathbf{r}} \\Vert_2^2 \\right]\\\\\n\\end{aligned}\n\\end{equation}\nwhere $\\mathbf{a}$ and $\\hat{\\mathbf{a}}$ are the actual and estimated abundance vectors of all pixels, $\\mathbf{r}$ and $\\hat{\\mathbf{r}}$ are the actual and estimated multiple abundance vectors of all pixels.\n\nSecond, the number of nonzero abundances were used to evaluate the sparsity level per endmember class and per endmember spectrum recovered by the methods, i.e.,  \\cite{Iorda2014a,Chen2016}\n\\begin{equation}\n\\begin{aligned}\n\\text{SL}_a \\equiv \\frac{1}{P}\\sum_{i=1}^{P} \\Vert \\hat{\\mathbf{a}}_i \\Vert_0\\\\\n\\text{SL}_r \\equiv \\frac{1}{P}\\sum_{i=1}^{P} \\Vert \\hat{\\mathbf{r}}_i \\Vert_0.\n\\end{aligned}\n\\end{equation}\nAs in \\cite{Iorda2014a}, abundances smaller than $10^{-4}$ were considered as \\textit{zero abundances}.\n\nFinally, to validate the performance in selecting a relevant combination of endmember classes or spectra, one defines the distance between the two actual and estimated supports (DIST) \\cite{Elad2010,Chen2016}\n\\begin{equation}\n\\begin{aligned}\n\\text{DIST}_a \\equiv \\frac{1}{P}\\sum_{i=1}^{P} \\frac{\\max\\left(\\vert \\mathcal{S}_{i}^a \\vert, \\vert \\hat{\\mathcal{S}}_{i}^a \\vert\\right) - \\vert \\mathcal{S}_{i}^a \\cap \\hat{\\mathcal{S}}_{i}^a\\vert}{\\max\\left(\\vert \\mathcal{S}_{i}^a \\vert, \\vert \\hat{\\mathcal{S}}_{i}^a \\vert\\right)}\\\\\n\\text{DIST}_r \\equiv \\frac{1}{P}\\sum_{i=1}^{P} \\frac{\\max\\left(\\vert \\mathcal{S}_{i}^r \\vert, \\vert \\hat{\\mathcal{S}}_{i}^r \\vert\\right) - \\vert \\mathcal{S}_{i}^r \\cap \\hat{\\mathcal{S}}_{i}^r\\vert}{\\max\\left(\\vert \\mathcal{S}_{i}^r \\vert, \\vert \\hat{\\mathcal{S}}_{i}^r \\vert\\right)}\\\\\n\\end{aligned}\n\\end{equation}\nwhere $\\mathcal{S}$ and $\\hat{\\mathcal{S}}$ are true and estimated support sets (i.e., indexes of nonzero values), $\\vert \\mathcal{S} \\vert$ represents the total number of elements in the set $\\mathcal{S}$ and $\\cap$ stands for the intersection operator. The figures of merit DIST$_a$ and DIST$_r$ evaluated the distance between two supports of endmember classes and the distance between two supports of endmember spectra, respectively.\n\n\n\n\\begin{table}[h!]\n    \\caption{SRE per endmember class ($\\text{SRE}_a$).}\\label{table:sre_c}\n\\resizebox{\\columnwidth}{!}{\n\\begin{tabular}{c||c||ccccccc}\t\t\t\t\t\t\t\t\t\n\\toprule\t\t\t\t\t\t\t\t\t\n\t&SNR\t&FCLS\t&AAM\t&SUnSAL\t&Group lasso\t&Elitist lasso\t&MEMM$_s$\t&MEMM\t\\\\\n\\midrule\t\t\t\t\t\t\t\t\t\n\\multirow{3}{*}{SIM1}\t&30dB\t&18.0507\t&18.7656\t&18.9285\t&\\textbf{19.4215}\t&19.2333\t &14.5739\t&18.6904\t\\\\\n\t&40dB\t&22.9074\t&\\textbf{25.8112}\t&23.0496\t&23.589\t&23.303\t&14.8229\t&23.3532\t\\\\\n\t&50dB\t&27.358\t&\\textbf{33.8279}\t&27.5013\t&27.4184\t&26.8979\t&14.1579\t&28.0017\t \\\\\n\\midrule\t\t\t\t\t\t\t\t\t\n\\multirow{3}{*}{SIM2}\t&30dB\t&16.1865\t&15.9501\t&16.5182\t&\\textbf{18.0736}\t&16.4315\t &12.1614\t&16.4785\t\\\\\n\t&40dB\t&21.6942\t&19.4906\t&21.6671\t&22.0263\t&21.3809\t&13.6682\t &\\textbf{22.1381}\t\\\\\n\t&50dB\t&25.6617\t&22.9952\t&25.5168\t&25.9215\t&24.7702\t&14.9405\t &\\textbf{26.5442}\t\\\\\n\\bottomrule\t\t\t\t\t\t\t\t\t\n\\end{tabular}\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n}\n \\end{table}\n\n\\begin{table}[h!]\n    \\caption{SRE per endmember spectrum ($\\text{SRE}_r$).}\\label{table:sre_s}\n\\resizebox{\\columnwidth}{!}{\n\\begin{tabular}{c||c||ccccccc}\t\t\t\t\t\t\t\t\t\n\\toprule\t\t\t\t\t\t\t\t\t\n\t&SNR\t&FCLS\t&AAM\t&SUnSAL\t&Group lasso\t&Elitist lasso\t&MEMM$_s$\t&MEMM\t\\\\\n\\midrule\t\t\t\t\t\t\t\t\t\n\\multirow{3}{*}{SIM1}\t&30dB\t&1.1752\t&\\textbf{2.2035}\t&1.7234\t&2.0398\t&1.5188\t&-0.4570\t &0.7692\t\\\\\n\t&40dB\t&2.8611\t&\\textbf{4.6569}\t&3.0262\t&3.0357\t&2.9957\t&0.5171\t&2.7653\t\\\\\n\t&50dB\t&3.6312\t&\\textbf{12.7796}\t&3.7692\t&3.7591\t&3.7576\t&0.1700\t&3.5721\t\\\\\n\\midrule\t\t\t\t\t\t\t\t\t\n\\multirow{3}{*}{SIM2}\t&30dB\t&1.0084\t&-0.2324\t&1.4405\t&\\textbf{1.8935}\t&1.3043\t&-2.0312\t &0.4313\t\\\\\n\t&40dB\t&2.3323\t&1.4226\t&2.4976\t&\\textbf{2.5741}\t&2.48\t&-2.1441\t&2.2462\t\\\\\n\t&50dB\t&3.2175\t&3.3104\t&3.3273\t&3.3266\t&\\textbf{3.3666}\t&-3.0151\t&3.2304\t\\\\\n\\bottomrule\t\t\t\t\t\t\t\t\t\n\\end{tabular}\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n}\n \\end{table}\n\n\n\\subsection{Results}\nSRE per class was calculated for each method  and reported in Table \\ref{table:sre_c}. For SIM1 with $50$dB, AAM performed best among all methods. The performance of AAM, however, was degraded as SNR became lower. For data with $30$dB, the results derived from AAM were  worse than those derived from SUnSAL, elitist lasso and group lasso. In addition, AAM performed poorly compared with other methods in SIM2. This showed that the MESMA-based approach (AAM) was less effective when given endmember bundles did not completely represent endmember variability present in the data and SNR of the data was low ($<40$dB). This finding was also observed in \\cite{Uezat2016}. MEMM produced better results for SIM2 with $50$dB  than the other sparsity-methods and produced comparable results with $40$dB and $30$dB. MEMM$_s$ performed poorly, compared with all methods. SRE per spectrum was also calculated from each method  and reported in Table \\ref{table:sre_s}. Compared with SRE per class, SRE per spectrum was very low for all methods. This showed that the exact recovery of multiple abundances $\\mathbf{r}_i$ was challenging under conditions where a large number of endmember spectra were present within each class.\n\n\n\\begin{table}[h!]\n    \\caption{Sparsity level per endmember class ($\\text{SL}_a$). Reference: $\\text{SL}_a=2.01$ in SIM1 and $\\text{SL}_a=2.27$ in SIM2.)}\\label{table:nonneg_c}\n\\resizebox{\\columnwidth}{!}{\n\\begin{tabular}{c||c||ccccccc}\t\t\t\n\\toprule\t\t\t\t\t\t\t\t\t\n\t&SNR\t&FCLS\t&AAM\t&SUNSAL\t&Group lasso\t&Elitist lasso\t&MEMM$_s$\t&MEMM\t\\\\\n\\midrule\t\t\t\t\t\t\t\t\t\n\\multirow{3}{*}{SIM1}\t&30dB\t&4.8\t&4.25\t&3.32\t&4.59\t&6.83\t&3.05\t&\\textbf{1.84}\t\\\\\n\t&40dB\t&4.62\t&4.29\t&3.82\t&4.54\t&5.64\t&3.64\t&\\textbf{2.14}\t\\\\\n\t&50dB\t&4.34\t&4.05\t&4.24\t&4.58\t&5\t&3.55\t&\\textbf{1.99}\t\\\\\n\\midrule\t\t\t\t\t\t\t\t\t\n\\multirow{3}{*}{SIM2}\t&30dB\t&5.02\t&4.45\t&3.56\t&4.72\t&6.63\t&4.27\t&\\textbf{2.03}\t\\\\\n\t&40dB\t&5.21\t&4.59\t&4.16\t&5.02\t&5.6\t&4.25\t&\\textbf{2.5}\t\\\\\n\t&50dB\t&4.97\t&4.49\t&4.87\t&4.72\t&5.54\t&3.9\t&\\textbf{2.28}\t\\\\\n\\bottomrule\t\t\t\t\t\t\t\t\t\n\\end{tabular}\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n}\n \\end{table}\n\n\\begin{table}[h!]\n    \\caption{Sparsity level per endmember class ($\\text{SL}_r$). Reference: $\\text{SL}_r=2.01$ in SIM1 and $\\text{SL}_r=2.39$ in SIM2.}\\label{table:nonneg_s}\n\\resizebox{\\columnwidth}{!}{\n\\begin{tabular}{c||c||ccccccc}\t\t\t\t\t\t\t\t\n\\toprule\t\t\t\t\t\t\t\t\t\n\t&SNR\t&FCLS\t&AAM\t&SUNSAL\t&Group lasso\t&Elitist lasso\t&MEMM$_s$\t&MEMM\t\\\\\n\\midrule\t\t\n\\multirow{3}{*}{SIM1}\t&30dB\t&10.83\t&4.25\t&10.52\t&43.12\t&11.28\t&\\textbf{3.04}\t&5.07\t\\\\\n\t&40dB\t&14.25\t&4.29\t&12.39\t&29.88\t&15.81\t&\\textbf{3.55}\t&8.72\t\\\\\n\t&50dB\t&18.66\t&4.05\t&18.07\t&23.57\t&21.31\t&\\textbf{3.45}\t&11.13\t\\\\\n\\midrule\t\t\t\t\t\t\t\t\t\n\\multirow{3}{*}{SIM2}\t&30dB\t&11.98\t&4.45\t&13.08\t&66.64\t&13.16\t&\\textbf{4.22}\t&5.26\t\\\\\n\t&40dB\t&16.79\t&4.59\t&15.6\t&45.99\t&19.67\t&\\textbf{4.15}\t&10.88\t\\\\\n\t&50dB\t&25.34\t&4.49\t&24.84\t&65.97\t&29.76\t&\\textbf{3.78}\t&19.38\t\\\\\n\\bottomrule\t\t\t\t\t\t\t\t\t\n\\end{tabular}\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n}\n \\end{table}\n\nThe average number of nonzero abundances in $\\mathbf{a}_i$ and $\\mathbf{r}_i$ were shown in Table \\ref{table:nonneg_c} and Table \\ref{table:nonneg_s}, respectively. For the number of nonzero abundances per class, MEMM performed best among all methods for both SIM1 and SIM2. This showed that the sparse constraint used in MEMM successfully led to the selection of smaller numbers of endmember classes. For the number of nonzero abundances per spectrum, AAM and MEMM$_s$ produced the smaller number of nonzero abundances than other methods. This was because these methods selected at most one spectrum within each class and enforced greater sparsity when selecting endmember spectra.\n\n\n\\begin{table}[h!]\n    \\caption{Distance between actual and estimated supports per endmember class ($\\text{DIST}_a$).}\\label{table:dist_c}\n\\resizebox{\\columnwidth}{!}{\n\\begin{tabular}{c||c||ccccccc}\t\t\t\t\t\t\t\t\t\n\\toprule\t\t\t\t\t\t\t\t\t\n\t&SNR\t&FCLS\t&AAM\t&SUNSAL\t&Group lasso\t&Elitist lasso\t&MEMM$_s$\t&MEMM\t\\\\\n\\midrule\t\t\t\t\t\t\t\t\t\n\\multirow{3}{*}{SIM1}\t&30dB\t&0.5893\t&0.5489\t&0.3939\t&0.5866\t&0.6967\t&0.3491\t&\\textbf{0.1195}\t \\\\\n\t&40dB\t&0.5689\t&0.5163\t&0.4616\t&0.5594\t&0.6368\t&0.4498\t&\\textbf{0.1265}\t\\\\\n\t&50dB\t&0.5137\t&0.4828\t&0.5035\t&0.5479\t&0.5795\t&0.4180\t&\\textbf{0.0758}\t\\\\\n\\midrule\t\t\t\t\t\t\t\t\t\n\\multirow{3}{*}{SIM2}\t&30dB\t&0.5725\t&0.5156\t&0.3775\t&0.5487\t&0.6538\t&0.5025\t&\\textbf{0.1843}\t \\\\\n\t&40dB\t&0.5789\t&0.4843\t&0.4521\t&0.5534\t&0.5952\t&0.4938\t&\\textbf{0.162}\t\\\\\n\t&50dB\t&0.5345\t&0.4696\t&0.5247\t&0.5185\t&0.5825\t&0.4179\t&\\textbf{0.1033}\t\\\\\n\\bottomrule\t\t\t\t\t\t\t\t\t\n\\end{tabular}\t\t\t\t\t\t\t\n}\n \\end{table}\n\n\\begin{table}[h!]\n    \\caption{Distance between actual and estimated supports per endmember spectrum ($\\text{DIST}_r$).}\\label{table:dist_s}\n\\resizebox{\\columnwidth}{!}{\n\\begin{tabular}{c||c||ccccccc}\t\t\t\t\t\t\t\t\t\n\\toprule\t\t\t\t\t\t\t\t\t\n\t&SNR\t&FCLS\t&AAM\t&SUNSAL\t&Group lasso\t&Elitist lasso\t&MEMM$_s$\t&MEMM\t\\\\\n\\midrule\t\t\t\t\t\t\t\t\t\n\\multirow{3}{*}{SIM1}\t&30dB\t&0.9167\t&0.767\t&0.853\t&0.94\t&0.8963\t&\\textbf{0.7493}\t&0.8115\t \\\\\n\t&40dB\t&0.8914\t&\\textbf{0.6707}\t&0.8587\t&0.9137\t&0.8881\t&0.7351\t&0.774\t\\\\\n\t&50dB\t&0.8649\t&\\textbf{0.5622}\t&0.8578\t&0.8971\t&0.8911\t&0.6586\t&0.7431\t\\\\\n\\midrule\t\t\t\t\t\t\t\t\t\n\\multirow{3}{*}{SIM2}\t&30dB\t&0.9039\t&0.8426\t&0.8476\t&0.9332\t&0.8915\t&0.8729\t&\\textbf{0.8142}\t \\\\\n\t&40dB\t&0.8791\t&\\textbf{0.7672}\t&0.8525\t&0.9167\t&0.883\t&0.8175\t&0.7993\t\\\\\n\t&50dB\t&0.8738\t&\\textbf{0.6949}\t&0.8696\t&0.9172\t&0.8816\t&0.7277\t&0.8095\t\\\\\n\\bottomrule\t\t\t\t\t\t\t\t\t\n\\end{tabular}\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n}\n \\end{table}\n\nThe errors in the support sets were shown in Table \\ref{table:dist_c} and Table \\ref{table:dist_s}. When estimating the support set per class, MEMM outperformed other methods for both SIM1 and SIM2. This showed that the double sparsity imposed by MEMM also led to the appropriate selection of the combination of endmember classes for each pixel. The performance of MEMM, however, was degraded when estimating the support set per spectrum. Other methods also performed poorly for the support set per spectrum. This shows that when multiple highly correlated endmember spectra are present within each class, the existing methods experience difficulty to select an optimal combination of endmember spectra.\n\n\n\\begin{table}[h!]\n    \\caption{Computational time for unmixing $P=100$ pixels using endmember bundles of $N=300$ endmember spectra.}\\label{table:time}\n\\resizebox{\\columnwidth}{!}{\n\\begin{tabular}{c|ccccccc}\t\t\t\t\t\t\t\t\n\\toprule\t\t\t\t\t\t\t\t\nData\t&FCLS\t&AAM\t&SUNSAL\t&Group lasso\t&Elitist lasso\t&MEMM$_s$\t&MEMM\t\\\\\n\\midrule\t\t\t\t\t\t\t\t\nSIM1\t&0.3006\t&517.7361\t&\\textbf{0.155}\t&0.9517\t&1.2694\t&0.4846\t&2.6625\t\\\\\nSIM2\t&\\textbf{0.1966}\t&549.1244\t&0.2616\t&0.8426\t&1.2122\t&2.2311\t&2.5705\t\\\\\n\\bottomrule\t\t\t\t\t\t\t\t\n\\end{tabular}\t\t\t\t\t\t\t\t\n}\n \\end{table}\n\nFinally, the computational times of all methods were shown in Table \\ref{table:time}. MEMM was more computationally expensive than FCLS, SUnSAL, Group lasso and Elitist lasso. The proposed methods, however, were computationally cheaper than AAM because they did not need to test a large number of the combinations of endmember spectra.\n\n\\begin{figure}[h!]\n        \\begin{subfigure}[b]{0.5\\columnwidth}\n                \\centering\n                \\includegraphics[width=.9\\linewidth]{rgb_1}\n                \\caption{}\n                \\label{fig:real_rgb_1}\n        \\end{subfigure}%\n        \\begin{subfigure}[b]{0.5\\columnwidth}\n                \\centering\n                \\includegraphics[width=.9\\linewidth]{rgb_2}\n                \\caption{}\n                \\label{fig:real_rbg_2}\n        \\end{subfigure}\n        \\caption{Real hyperspectral images: (a) MUESLI image and (b) AVIRIS image.}\\label{fig:rgb}\n\\end{figure}\n\n\\section{Experiments using real data}\\label{section5}\n\\subsection{Description of real hyperspectral images}\n\nThe methods were finally compared on two real hyperspectral images. The first hyperspectral image was acquired in June 2016, over the city of Saint-Andr{\\'e}, France, during the MUESLI airborne acquisition campaign. The image was composed of $415$ spectral bands. The spectral bands affected by noise (between $1.34-1.55\\mu$m and $1.80-1.98\\mu$m) were removed, leading to $L=345$ spectral bands.  In the image scene, spatially discrete objects were present. In this study, each spatially discrete region was assumed to be composed of a single endmember class. Endmember bundles were extracted from each region. As a consequence, large amounts of endmember variability were expected to be present within each spatially discrete region associated with a particular class and mixed pixels were expected to be located in the boundary of these spatially discrete regions. Moreover, the scene of interest was composed of two flight lines under significantly different illumination conditions, as shown in Fig. \\ref{fig:real_rgb_1}. Thus, this image (referred to as MUESLI image) was used to evaluated whether the methods could accurately estimate abundances when large amounts of endmember variability were present. From this image, $K=6$ endmember bundles composed of a total of $N=180$ spectral signatures representing spatially discrete objects were extracted using the n-Dimensional Visualizer provided by the ENVI software. These bundles are represented in Fig. \\ref{fig:endm_b_real1}(top). Some endmember classes present in the studied area were affected by different illumination conditions. Unlike simulated data, ground truth was not available. Estimated abundances were qualitatively validated by visual inspection of the abundance maps.\n\\begin{figure*}\n        \\centering\n        \\includegraphics[width=\\linewidth]{endm_b_real1}\n        \\caption{First row: endmember bundles used for unmixing the MUESLI image. Second row: endmember bundles generated by MEMM.}\\label{fig:endm_b_real1}\n\\end{figure*}\n\nThe second image was acquired over Moffett Field, CA, USA, by the Airborne Visible\/Infrared Imaging Spectrometer (AVIRIS). The image, depicted in Fig. \\ref{fig:real_rbg_2}, initially comprised $224$ spectral bands. After the noisy spectral bands were removed, $L=178$ spectral bands remained. The area of interest, composed of a lake and a vegetated coastal area, was considered in many previous studies, e.g., to assess the performance of unmixing methods. Thus, this second image (referred to as AVIRIS image) was used to test whether the proposed method could perform at least as well as the existing methods to analyze this widely used image scene. As for the AVIRIS image, $K=3$ endmember bundles were extracted and qualitative validation was conducted because of the lack of ground truth. The extracted bundles are depicted in Fig. \\ref{fig:endm_b_real2}(left).\n\n\\begin{figure}\n        \\centering\n        \\includegraphics[width=.7\\linewidth]{endm_b_real2}\n        \\caption{First column: endmember bundles used for unmixing the AVIRIS image. Second column: endmember bundles generated by MEMM.}\\label{fig:endm_b_real2}\n\\end{figure}\n\n\\subsection{Results}\nFor both images, the proposed method was compared with FCLS, SUnSAL, AAM, the methods based on group lasso and elitist lasso. The parameters ($\\lambda_r$, $\\lambda_b$ and $\\lambda_a$) required for the methods were empirically determined by qualitatively evaluating abundances derived from different values of the parameters. Finally, synthetic endmember bundles generated by MEMM were compared with endmember bundles initially used for unmixing.\n\n\\begin{figure*}\n        \\centering\n        \\includegraphics[width=\\linewidth]{abun_all_real1}\n        \\caption{MUESLI image: estimated abundance maps. From top to bottom: building, road, shrub, crop land~1, crop land~2 and grass.}\\label{fig:abun_all_real1}\n\\end{figure*}\n\nAbundance maps estimated by the $7$ methods on the MUESLI image are depicted in Fig. \\ref{fig:abun_all_real1}. These maps show that the abundances estimated by MEMM were more consistent at the boundary affected by the different illumination conditions. This showed that MEMM was more robust to the different illumination conditions than concurrent methods. The abundances estimated by MEMM were also high for each endmember class and showed less noisy. This suggested that MEMM also promoted more sparsity than other methods.\n\n\\begin{figure*}\n        \\centering\n        \\includegraphics[width=\\linewidth]{abun_all_real2}\n        \\caption{AVIRIS image: abundance maps. From top to bottom: vegetation, soil and water.}\\label{fig:abun_all_real2}\n\\end{figure*}\n\nFig. \\ref{fig:abun_all_real2} shows the abundance maps estimated for the AVIRIS image. All methods except elitist lasso generated similar abundances. Abundances estimated by elitist lasso were different because it was designed to use a larger number endmember classes for unmixing each pixel. MEMM produced similar abundances when compared with FCLS, group lasso. This showed that MEMM could perform at least as well as other sparsity-based methods to unmix this well-studied test site. MEMM$_s$, however, generated more noisy abundance maps. This showed that the initial estimates of abundances used in MEMM$_s$ did not lead to an optimal combination of endmember spectra and the optimal abundances.\n\nFinally, the endmember bundles recovered by MEMM were compared with endmember bundles initially used to unmix both hyperspectral images, see Fig. \\ref{fig:endm_b_real1} and Fig. \\ref{fig:endm_b_real2}. In Fig. \\ref{fig:endm_b_real1}, the synthetic endmember bundles filled the gaps that were present in the original endmember bundles. The extended endmember bundles showed more detailed spectral variability within each class in terms of both spectrum amplitudes and shapes also in Fig. \\ref{fig:endm_b_real2}. This enabled MEMM to generate adaptive endmember spectra within each pixel and estimate more accurate abundances even when initial endmember bundles did not completely represent endmember variability.\n\n\\section{Conclusion}\\label{section6}\nThis paper proposed a multiple endmember mixing model that bridges the gap between endmember bundle-based method and data driven-based methods. MEMM appeared to be superior to the existing methods as follows: \\emph{i}) it incorporated endmember bundles to generate adaptive endmember spectra for each pixel, \\emph{ii}) it had explicit physical meaning and generated hierarchical structure of endmember spectra, \\emph{iii}) it imposed double sparsity for the selection of both endmember classes and endmember spectra. MEMM were tested and compared to the state-of-the-art methods using simulated data and real hyperspectral images. MEMM showed comparable results for estimating abundances while it outperformed other methods in terms of selecting a set of endmember classes within each pixel. This paper deeply focused on sparsity constraints for both bundling coefficients and abundances. However, other constraints (e.g., spatial constraints) can be easily incorporated in the proposed unmixing framework. Future work will consist of reducing the computational complexity of the proposed method.\n\n\n\n\n\n\n\n\n\n\\section*{Acknowledgment}\nThe authors would like to thank Prof. Jose M. Bioucas-Dias, Dr. Rob Heylen and Dr. Travis R. Meyer for sharing the MATLAB codes of the SUnSAL, AAM and goup\/elitist lasso unmixing algorithms.\n\n\n\\ifCLASSOPTIONcaptionsoff\n  \\newpage\n\\fi\n\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSQL is the {\\em de facto} relational database query language that stands still  \\cite{Atzeni:2013:RMD:2503792.2503808} despite the advent of new trends as Big Data, NoSQL, RDF stores and others.\nIt builds upon the Codd's \\cite{Codd70,Codd72} seminal relational data model accompanied by an algebra and calculus to operate on data.\nFormer proposals such as \\cite{Grefen1994} better provide a formal framework for current SQL implementations.\nAs a query language, SQL can be well understood from Codd's tuple relational calculus but also from logic programming (in particular, \\cite{Ullman88} includes equivalences between relational operations and logic rules).\nHowever, among other features beyond the original relational model, SQL provides the notion of temporary view defined in {\\tt WITH} clauses (as described in Section \\ref{with}), whose definition is available only to the query in which it occurs \\cite{Silberschatz6th}.\nThis is no longer representable either in relational formal languages or directly in logic programming.\\footnote{Obviously, logic programming implementations provide general-purpose languages as Prolog that can emulate a temporary definition.}\n\nHere is when intuitionistic logic programming may come at help to providing first-class citizen semantics:\nApproaches as \\cite{DBLP:journals\/jlp\/Gabbay85,DBLP:journals\/jlp\/McCarty88,MillerModulesJLP,Bonner90hypotheticaldatalog,Hodas94} fit into this logic, an extension of logic programming including in particular embedded implications.\nAdding negation to intuitionistic logic programming might develop paradoxes which are circumvented in \\cite{bonner90adding} by dealing with two kind of implications: for rules ($\\leftarrow$) and for goals ($\\Leftarrow$, i.e., an embedded implication).\nWhereas in the formula $A \\leftarrow B$, the atom $B$ is ''executed\" for proving $A$, in the formula $A \\Leftarrow B$, the atom $B$ is ``assumed\" to be true for proving $A$.\nHypothetical Datalog \\cite{Bonner89hypotheticaldatalog,Bonner90hypotheticaldatalog,bonner90adding,bonner89expressing} incorporated this logic and has been a proposal thoroughly studied from semantic and complexity point-of-views.\nThe work \\cite{sae13c-ictai13} (recalled in Section \\ref{hypothetical-datalog}) presented an extended (w.r.t. \\cite{Bonner89hypotheticaldatalog,Bonner90hypotheticaldatalog,bonner90adding,bonner89expressing}) intuitionistic setting along with an implementation in the deductive system DES \\cite{saenzDES}, in which a rule is accepted in the antecedent of an embedded implication, and not only facts as in \\cite{bonner90adding}.\n\nDriven from the need for supporting a broader subset of SQL in this system, we show how to take advantage of the intuitionistic embedded implication to model {\\tt WITH} SQL queries in a logic setting, an application which has not been proposed to the best of our knowledge so far.\nThus, as shown in Section \\ref{translating}, it is possible to have such SQL queries translated into Datalog, and can be therefore processed by a deductive engine. \nBut Hypothetical Datalog is powerful enough to even apply the same technique to model assumptions in SQL queries, with the (non-standard) clause \\mytt{ASSUME}.\nThis clause enables both positive and negative assumptions on data, as shown in Section \\ref{assume}, which are useful for modelling ``what-if'' scenarios.\nFinally, we present the deductive system DES at work with examples of \\mytt{WITH} and \\mytt{ASSUME} queries in Section \\ref{examples}.\nOur approach is also useful for connecting with external relational database systems which cannot process these clauses.\nDES then behaves as a front-end capable of processing either novel or unsupported features in such systems.\n\n\\section{The SQL {\\tt WITH} Clause}\n\\label{with}\n\nTypically, complex SQL queries are broken-down for applying the {\\em divide-and-conquer} principle as well as for enhancing readability and maintenance.\nIntroducing intermediate views with {\\tt CREATE VIEW} statements is in the order of the day, but this might neither be recommendable (making these views observable for other users) nor possible (only certain users with administration permissions are allowed to create views). \nThe {\\tt WITH} clause provides a form of encapsulation in SQL by locally defining those broken-down views, making their realms to pertain to the context of a given query.\nNext, the syntax of a query $Q$ including this clause is recalled:\n\n\\vspace{2mm}\n\\begin{tabular}{ll}\n\\mytt{WITH} &\\mytt{{\\em R}$_1$ AS {\\em SQL}$_1$,}\\\\\n            &\\mytt{\\ldots,} \\\\\n            &\\mytt{{\\em R}$_n$ AS {\\em SQL}$_n$}\\\\\n\\mytt{{\\em SQL}}\n\\end{tabular}\n\n\\noindent where each \\mytt{{\\em R}$_i$} is a temporary view name defined by the SQL statement \\mytt{{\\em SQL}$_i$}, and which can be referenced only in \\mytt{{\\em SQL}}, the ultimate query that builds the outcome of the query $Q$.\nThis query can be understood as a relation with name \\mytt{{\\em R}} and defined with the DDL statement \\mytt{CREATE VIEW {\\em R} AS} $Q$. \n\nWith respect to the semantics of an SQL query, we recall and adapt the notation in \\cite{cgs12a-flops2012} which in turn is based on \\cite{Grefen1994}.\nA {\\em table instance} is a multiset of facts (following logic programming instead of relational databases).\nA {\\em database instance} $\\Delta$ of a database\nschema is a set of table instances, one for each defined table ({\\em extensional} relation) in the database.\nThe notation $\\Delta(T)$ represents the instance of a table $T$ in $\\Delta$.\nEach query or view ({\\em intensional} relation) $R$ is defined as a multiset of facts,\nand $\\Phi_R$ represents the ERA (Extended Relational Algebra) expression associated to an SQL query or view $R$, as explained in \\cite{Molina08}.\nAn intensional relation usually depends on previously defined relations, and sometimes it will be useful to write  $\\Phi_R(R_1,\\dots,R_n)$ indicating that $R$ depends on $R_1, \\dots, R_n$.\nHere, we assume that each extensional relation in a database instance has attached type information for each one of its named arguments.\nAs well, each intensional relation argument receives its type via inferencing and arbitrary names if not provided in its definition.\nTables are denoted by their names, that is, $\\Phi_T = T$ if $T$ is a table.\n\t\\begin{definition}  \\label{def:SQLERA} {\\em\n\tThe \\emph{computed answer} of an ERA expression $\\Phi_R$ with respect to some schema instance $\\Delta$ is denoted by $\\evalSQL{\\Phi_R}{\\Delta}$, where:\n\t\\begin{itemize}\n\t\t\\item If $R$ is an extensional relation, $\\evalSQL{\\Phi_R}{\\Delta} =\\, \\Delta(R)$.\n\t\t\\item If $R$ is an intensional relation and $R_1, \\dots, R_n$ the relations defined in $R$, then\n\t\t$\\evalSQL{\\Phi_R}{\\Delta} =\\, \\Phi_R(\\evalSQL{\\Phi_{R_1}}{\\Delta}, \\dots, \\evalSQL{\\Phi_{R_n}}{\\Delta})$. \\hfill $_\\square$\n\t\\end{itemize}\n}\n    \\end{definition}\n\nQueries are executed by SQL systems. \nThe answer for a query $Q$ and a database instance $\\Delta$ in an implementation is represented by $\\ans_\\Delta$($Q$). The notation $\\ans_\\Delta$($R$) abbreviates $\\ans_\\Delta$(\\mytt{SELECT * FROM $R$}). \nIn particular, we assume the existence of \\emph{correct} SQL implementations.\n\n\t\\begin{definition} \\label{def:correctSQL} {\\em \n\tA \\emph{correct} SQL implementation verifies that $\\ans_\\Delta$($Q$) = $\\evalSQL{\\Phi_Q}{\\Delta}$ for every query $Q$.\\hfill $_\\square$\n}\n\t\\end{definition}\n\n\n\n\n\n\\section{Hypothetical Datalog}\n\\label{hypothetical-datalog}\n\nHypothetical Datalog is an extension of function-free Horn logic\n\\cite{bonner90adding}.\nFollowing \\cite{sae13c-ictai13}, the syntax of the logic is first order and includes a universe of constant symbols, a set of variables and a set of predicate symbols.\nFor concrete symbols, we write variables starting with either an upper-case letter or an underscore, and the rest of symbols starting with lower-case.\nRemoving function symbols from the logic is a condition for finiteness of answers, a natural requirement of relational database users.\nAs in Horn-logic, a rule has the form $A \\leftarrow \\phi$, where $A$ is an atom and $\\phi$ is a conjunction of goals.\nSince we consider a hypothetical system,\na goal can also take the form $R_1 \\land \\ldots \\land R_n \\Rightarrow G$, a construction known as an {\\em embedded implication}.\nThe following definition captures the syntax of the language, where $vars(T)$ is the set of variables occurring in $T$:\n\n\\begin{definition} {\\em \n\t\\quad \\\\\n\n\t$R := A \\mid A \\leftarrow G_1 \\land \\ldots \\land G_n$ \\\\\n\n\t$G := A \\mid \\neg G \\mid R_1 \\land \\ldots \\land R_n \\Rightarrow G$\\\\\n\n\t\\noindent where $R$ and $R_i$ stand for rules, $G$ and $G_i$ for goals, $A$ for an atom  (possibly containing variables and constants, but no compound terms), and $\\bigcup vars(R_i) \\cap vars(R) = \\emptyset$, and $vars(R_i)$ and $vars(G)$ are disjoint.\\hfill $_\\square$\n}\n\\end{definition}\n\nDisjoint conditions ensure that assumed rules do not depend on actual substitutions along inference, i.e., assumed rules take the form they have in the program.\n\nSemantics is built with a stratified inference system which can be consulted in \\cite{sae13c-ictai13}.\nHere we recall the inference rule for the embedded implication:\\footnote{Each rule in this inference system is read as: If the formulas above the line can be inferred, then those below the line can also be inferred.}\nFor any goal $\\phi$ and database instance $\\Delta$:\n\n\\begin{center}\n\\medskip\n$\\infer[]{\\Delta \\vdash R_1 \\land \\ldots \\land R_n \\Rightarrow \\phi}{\\Delta \\cup \\{R_1, \\ldots, R_n\\} \\vdash \\phi}$\n\\medskip\n\\end{center}\n\nThis means that for proving the conclusion $\\phi$, rules $R_i$, together with the current database instance $\\Delta$ can be used in subsequent inference steps.\nThe {\\em unified stratified semantics} defined in \\cite{sae13c-ictai13} builds a set of axioms $\\mathcal{E}$ that provides a means to assign a meaning to a goal as: \n$solve(\\phi,\\mathcal{E}) = \\{\\Delta \\vdash id:\\psi \\in \\mathcal{E} ~ \\mbox{such that} ~ \\phi\\theta = \\psi\\}$, where $\\theta$ is a substitution and each axiom in $\\mathcal{E}$ is mapped to the database $\\Delta$ it was deduced for, and the inferred fact $\\psi$ is labelled with its data source (for supporting duplicates).\nWe use $\\Delta(\\mathcal{E})$ to denote the multiset of facts $\\psi$ so that $\\Delta \\vdash id:\\psi \\in \\mathcal{E}$ for any $id$.\nSo, this inference rule captures what SQL {\\tt WITH} statements need if translated to Hypothetical Datalog, because each $R_i$ can represent each temporary view definition, as will be shown in the next section.\n\n\n\\section{Translating SQL into Datalog}\n\\label{translating}\n\nWe consider standard SQL as found in many textbooks (e.g., \\cite{Silberschatz6th}), but also allowing \\verb|FROM|-less statements, i.e., providing a single-row output constructed with the comma-separated expressions after the \\verb|SELECT| keyword (Oracle, for instance, resorts to feed the row from the \\verb|dual| table to express the same feature).\nHere, we define a function \\mytt{{\\em SQL\\_to\\_DL}} that takes a relation name  and an SQL statement as input and returns a multiset of Datalog rules providing the same meaning as the SQL relation for a corresponding predicate with the same name as the relation.\nThe following (incomplete) definition for this function includes only a couple of the basic cases, where others can be easily developed from \\cite{Ullman88}.\nFrom here on, set-related operators and symbols refer to multisets, as SQL relations can contain duplicates.\n\n\t \n\\medskip\n\\noindent {\\em \\% Basic SELECT statement} \\\\\n\\noindent \\mytt{{\\em SQL\\_to\\_DL}}($r$, \\mytt{SELECT A$_1$,\\ldots,A$_n$ FROM $Rel$ WHERE $Cond$}) =\\\\\n\\indent \\{ $r(\\overline{X_i}) \\leftarrow DLRel(\\overline{X_i}), DLCond(\\overline{X_j}) ~ \\} \\bigcup RelRules \\bigcup CondRules$, \\\\\nwhere \\mytt{{\\em SQLREL\\_to\\_DL}}$(Rel)=(DLRel(\\overline{X_i}), RelRules)$, and \\\\\n\\indent \\mytt{{\\em SQLCOND\\_to\\_DL}}$(Cond)=(DLCond(\\overline{X_j}), CondRules)$\n\\medskip\n\n\\noindent {\\em \\% Duplicate-preserving union} \\\\\n\\noindent \\mytt{{\\em SQL\\_to\\_DL}}($r$, \\mytt{{\\em SQL}}$_1$ \\mytt{UNION ALL} \\mytt{{\\em SQL}}$_2$) = \\\\\n\\indent \n\\mytt{{\\em SQL\\_to\\_DL}}($r$, \\mytt{{\\em SQL}}$_1$) $\\bigcup$\n\\mytt{{\\em SQL\\_to\\_DL}}($r$, \\mytt{{\\em SQL}}$_2$) \n\\medskip\n\n\nHere, each \\mytt{A}$_i$ is an argument name present in the relation $Rel$ with corresponding logic variable $X_i$. \n$Rel$ is constructed with either a single defined relation (table or view), or a join of relations, or an SQL statement.\nFunction \\mytt{{\\em SQLREL\\_to\\_DL}} (resp., \\mytt{{\\em SQLCOND\\_to\\_DL}}) takes an SQL relation (resp. condition) and returns a goal and, possibly, additional rules which result from the translation.\nVariables $\\overline{X_j}$ come as a result of the translation of the condition $DLCond$ to a goal. \nAs well, some basic cases are presented next for these functions, where $GoalName$ is an arbitrary, fresh new goal name:\n\n\\medskip\n\\noindent {\\em \\% Extensional\/Intensional Relation Name} \\\\\n\\noindent \\mytt{{\\em SQLREL\\_to\\_DL}}$(RelName)=(RelName(\\overline{X_i}), \\{ \\})$ \\\\\n\\noindent where $\\overline{X_i}$ are the $n$ variables corresponding to the $n$-degree relation $RelName$.\n\\medskip\n\n\\noindent {\\em \\% SQL Statement} \\\\\n\\noindent \\mytt{{\\em SQLREL\\_to\\_DL}}$(SQL)=(GoalName(\\overline{X_i})$, \\mytt{{\\em SQL\\_to\\_DL}}$(GoalName, SQL)$)\\\\\n\\noindent where $\\overline{X_i}$ are the $n$ variables corresponding to the $n$-degree statement $SQL$.\n\\medskip\n\n\\noindent {\\em \\% NOT IN Condition} \\\\\n\\noindent \\mytt{{\\em SQLCOND\\_to\\_DL}}(\\mytt{A$_i$ NOT IN} $Rel)=(not ~ DLRel(\\overline{X_j}), RelRules)$\\\\\nwhere \\mytt{{\\em SQLREL\\_to\\_DL}}$(Rel)=(DLRel(\\overline{X_j}), RelRules)$, and $X_i \\in \\overline{X_j}$ is the corresponding variable to argument \\mytt{A}$_i$.\n\\medskip\n\nCompleting this function by including the \\mytt{WITH} statement is straightforward because every temporary view can be represented by a predicate resulting from the translation of the temporary view definition into Datalog rules.\nAssuming such predicates as the antecedent of an embedded implication can be used to augment the (local, temporary) database for interpreting the meaning of the translated SQL outcome query:\n\n\\medskip\n\\noindent\n\t{\\mytt{{\\em SQL\\_to\\_DL}}($r$, \\mytt{WITH $r_1$ AS {\\em SQL}$_1$, \\ldots, $r_n$ AS {\\em SQL}$_n$ {\\em SQL}}) =}\\\\\n\\indent\t\\{ $r(\\overline{X_i}) \\leftarrow$ \\\\\n\\indent\t\\hspace*{1.7mm}\n\t$\\land($\\mytt{{\\em SQL\\_to\\_DL}}($r_1$,\\mytt{{\\em SQL}}$_1$)) $\\land$ \\ldots \\\\\n\\indent\t\\hspace*{1.7mm}\n\t$\\land($\\mytt{{\\em SQL\\_to\\_DL}}($r_n$,\\mytt{{\\em SQL}}$_n$)) $\\Rightarrow$ \n\t$s(\\overline{X_i})$ \\} $\\bigcup$ \\texttt{{\\em SQL\\_to\\_DL}}($s$,\\texttt{{\\em SQL}})  \n\\medskip\n\n\\noindent where $\\land(Bag)$ denotes $B_1 \\land \\cdots \\land B_m$ ($B_i \\in Bag$).\n\nThe following theorem establishes the semantic equivalence of an SQL relation and its counterpart Datalog translation.\n\n\\begin{theorem} {\\em\n\tThe semantics of an SQL $n$-degree relation $r$ defined by the query $Q$ on a database instance $\\Delta$ coincides with the meaning of a goal $r(\\overline{X_i})$, $1 \\leq i \\leq n$, for $\\Delta' = \\Delta \\bigcup$ \\mytt{{\\em SQL\\_to\\_DL}}($r$,$Q$), that is:\n\t$\\ans_\\Delta(Q) = \\Delta(solve(r(\\overline{X_i}),\\mathcal{E}))$, where $\\mathcal{E}$ is the unified stratified semantics for $\\Delta'$.}  \\hfill $_\\square$\n\\end{theorem}\n\n\\section{Beyond the WITH Clause: Expressing Assumptions}\n\\label{assume}\n\nAs a novel feature, hypothetical SQL queries (absent in the standard) were introduced (inspired in \\cite{nss-flops08}) in DES version 2.6 for solving ``what-if'' scenarios.\nSyntax for such queries is:\n\n{\\fontsize{11}{11}\n\t\\vspace{3mm}\n\t\\noindent {\\tt ASSUME {\\em SQL}$_1$ IN {\\tt {\\em Rel}}$_1$, ..., {\\em SQL}$_n$ IN {\\tt {\\em Rel}}$_n$}\n\t{\\tt {\\em SQL};}\n\t\\vspace{3mm}\n}\n\n\\noindent which makes to {\\em assume} the result of {\\tt {\\em SQL}}$_i$ in {\\tt {\\em Rel}}$_i$ when processing {\\tt {\\em SQL}}.\nThis means that the semantics of each {\\tt {\\em Rel}}$_i$ is either overloaded (if the relation already exists) or otherwise defined with the facts of {\\tt {\\em SQL}}$_i$.\nImplementing this resorted to {\\em globally} define each {\\tt {\\em Rel}}$_i$, which is not the expected behaviour as its definition must be local to {\\tt {\\em SQL}}.\nRoughly, solving an \\mytt{ASSUME} query resorted to overload the meanings of each {\\tt {\\em Rel}}$_i$ (by inserting the required facts) before computing {\\tt {\\em SQL}} and, after solving, to restore them (by deleting the same facts).\nThis also precluded nested assumptions, and such statements were allowed only as top-level queries but not as part of query definitions.\nFor instance, if it would be allowed, the following query would be incorrectly computed in that scenario:\n\n{\\fontsize{11}{11}\n\t\\begin{verbatim}\nASSUME SELECT 1 IN r(a), \n       (ASSUME SELECT 2 IN r(a) SELECT * FROM r) IN s \nSELECT * FROM r,s;\n\t\\end{verbatim}\n}\n\n\\noindent because the meaning of \\mytt{r} in the context of \\mytt{SELECT * FROM r,s} would be overloaded with both \\mytt{\\{(1)\\}} and \\mytt{\\{(2)\\}}, instead of just with \\mytt{\\{(1)\\}}.\n\nApplying hypothetical reasoning in this case solves this issue, allowing us not only to use nested assumptions in both top-level queries and views, but also to take advantage of negative assumptions.\nA negative assumption allows to {\\em remove} facts from the meaning of a relation, which broadens the applicability of queries in decision-support scenarios.\nTo specify negative assumptions, \\mytt{NOT IN} is used instead of just \\mytt{IN}.\nHypothetical Datalog in \\cite{Sae15c} introduces the notion of {\\em restricted predicate} to handle negative assumptions in embedded implications. \nA restricted predicate includes at least a restricting rule whose head is an atom preceded by a minus sign.\nIts meaning is the set of facts deduced from regular rules minus the set of facts deduced from restricting rules.\nSo, a negative assumption is modelled with a restricting rule in the antecedent of an embedded implication, so that the translation from SQL to Hypothetical Datalog for \\mytt{ASSUME} statements becomes:\n\n\\medskip\n\\noindent\n\\begin{tabular}{m{0.1cm}m{0.1cm}l}\n\t\\multicolumn{3}{l}{\\mytt{{\\em SQL\\_to\\_DL}}($r$, \\mytt{ASSUME {\\em SQL}$_1$ [NOT] IN $r_1$,\\ldots,{\\em SQL}$_n$ [NOT] IN $r_n$} \\mytt{{\\em SQL}}) =}\\\\\n\t& \\{ & $r(\\overline{X_i}) \\leftarrow$ \\\\\n\t& & $\\land$(\\mytt{{\\em SQL\\_to\\_DL}}($r_1$, \\mytt{{\\em SQL}}$_1$)$[$\\mytt{[-]}$r_1$\/$r_1]$) $\\land$\n\t\\ldots \\\\\n\t& & $\\land$(\\mytt{{\\em SQL\\_to\\_DL}}($r_n$, \\mytt{{\\em SQL}}$_n$)$[$\\mytt{[-]}$r_n$\/$r_n]$) \\texttt{=>} \n\t$s(\\overline{X_i})$ \\} \\\\\n\t& & $\\cup$ $\\texttt{{\\em SQL\\_to\\_DL}}(s,\\texttt{{\\em SQL}})$\n\\end{tabular}\n\\medskip\n\n\\noindent where $A[B\/C]$ represents the application of the syntactic substitution $C$ by $B$ in all the rule heads in $A$, \\mytt{[{\\em T}]} represents that \\mytt{{\\em T}} is optional but it must occur if the corresponding $i$-th entry also occurs (i.e., if \\mytt{NOT} occurs in the assumption for $r_i$, then \\mytt{-} also occurs in the corresponding substitution).\n\n\\section{Playing with the System}\n\\label{examples}\n\nTranslating an SQL query to Datalog in a practical system involves more features that the ones briefly suggested before and are out of the scope of this extended abstract.\nFor example, the {\\tt SELECT} list can include expressions and scalar SQL statements, nested statements can be correlated, aggregate functions and grouping can be used, and so on.\nAlso, from a capacity point-of-view, a needed stage in the translation is folding\/unfolding of rules \\cite{prolog} to simplify the Datalog program resulting from the translation.\nThis is quite relevant because deducing the meaning (either complete or restricted to a given call) of the involved relations along query solving is needed, therefore significantly augmenting their space and time requirements.\nNext, we introduce a couple of examples of this translation with the system DES \\cite{saenzDES}, which in particular supports such features and inputs from several query languages, including Datalog and SQL.\nHere, we resort to the actual textual syntax of Datalog rules in this system, which follows the syntax of Prolog.\n\n\\begin{wrapfigure}{r}{45mm}\n\t\\vspace{-35pt}\n\t\\begin{center}\n\t\t\\begin{tabular}{cc}\n\t\t\t\\begin{minipage}[c]{2cm}\n\t\t\t\t\\begin{tabular}{|c|}\n\t\t\t\t\t\\hline \n\t\t\t\t\t\\mytt{student}\\\\\n\t\t\t\t\t\\hline \n\t\t\t\t\t\\mytt{(adam)} \\\\ \n\t\t\t\t\t\\mytt{(bob)} \\\\ \n\t\t\t\t\t\\mytt{(pete)} \\\\ \n\t\t\t\t\t\\mytt{(scott)} \\\\ \n\t\t\t\t\t\\hline \n\t\t\t\t\\end{tabular}\n\t\t\t\\end{minipage}\n\t\t\t& \n\t\t\t\\begin{tabular}{|c|}\n\t\t\t\t\\hline \n\t\t\t\t\\mytt{take}\\\\\n\t\t\t\t\\hline \n\t\t\t\t\\mytt{(adam,db)} \\\\ \n\t\t\t\t\\mytt{(pete,db)} \\\\ \n\t\t\t\t\\mytt{(pete,lp)} \\\\ \n\t\t\t\t\\mytt{(scott,lp)} \\\\ \n\t\t\t\t\\hline \n\t\t\t\\end{tabular}\t\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\vspace{-25pt}\n\n\n\n\n\\end{wrapfigure}\nLet us consider a database containing the relations \\mytt{student(name)} and \\mytt{take(name, title)}.\nThe first one states names of students and the second one the course (\\mytt{title}) each student (\\mytt{name}) is enrolled in. \nTypes can be specified either with a Datalog assertion (as \\mytt{:-type(student(name:string))} for the first case) or a DDL SQL statement (as \\mytt{create table take(name string, title string)} for the second one, where a foreign key \\mytt{take.name} $\\rightarrow$ \\mytt{student.name} could be stated as well). \nWe consider the database instance depicted in the tables above.\n\n\nThe next SQL statement (looking for students that have not been already enrolled in a course)\nis translated as follows in a system session with DES 4.0:\n\n{\\fontsize{11}{11}\n\t\\begin{verbatim}\n\tDES> select * from student where name not in \n        (select name from take)\n\tInfo: SQL statement compiled to:\n\t  answer(A) :- student(A), not take(A,_B).\n\tanswer(student.name:string) ->\n\t{ answer(bob) }\n\tInfo: 1 tuple computed.          \n\t\\end{verbatim}\n}\n\nThis example shows a few of things.\nFirst, as a query $Q$ is allowed at the system prompt, the call to the translation function becomes \\mytt{{\\em SQL\\_to\\_DL}}(\\mytt{answer},$Q$), i.e., the outcome relation is automatically renamed to the reserved keyword \\mytt{answer}.\nSecond, the outcome schema \\mytt{answer(student.name:string)} shows that the single output argument comes from the argument \\mytt{name} of the relation \\mytt{student}, with type \\mytt{string}.\nThird, following the definition of the translation function, this query should be translated into:\n\n{\\fontsize{11}{11}\n\t\\begin{verbatim}\n\tanswer(A) :- student(A), goal1(A).\n\tgoal1(A) :- not take(A,_B)\n\t\\end{verbatim}\n}\n\nBut folding\/unfolding simplifies this as it was displayed in the system session.\nThese translations have been displayed because it was specified so by issuing the command \\mytt{\/show\\_compilations on}. \nFinally, non-relevant variables to a rule outcome are underscored (otherwise, they would be signalled as anonymous).\nThis is important in this case to identify as safe the rule in which this underscored variable occurs.\nClassical safety \\cite{Ullman88} would tag the rule \\mytt{answer(A) :- student(A), not take(A,\\_B)} as unsafe, but an equivalent set of safe rules can be found: \\mytt{answer(A) :- student(A), not goal1(A)} and \\mytt{goal1(A) :- take(A,\\_B)}.\nUnderscored variables are a means to encapsulate this form of safety, which is identified by the system and processed correspondingly.\nIn general, there can be several rules for \\mytt{answer} (e.g., when a \\mytt{UNION} is involved) and others on which this predicate depends on.\n\nAs an example of a \\mytt{WITH} query, the following statement defines the relation \\mytt{grad} intended to retrieve the eligible students for graduation (those that took both \\mytt{db} and \\mytt{lp} in this tiny example):\n\n{\\fontsize{11}{11}\n\t\\begin{verbatim}\n\tDES> with grad(name) as \n\t       (select student.name\n\t        from student, take t1, take t2\n\t        where student.name=t1.name\n\t          and t1.name=t2.name \n\t          and t1.title='db' and t2.title='lp') \n\t     select * from grad;\n\tInfo: SQL statement compiled to:\n\t  answer(A) :-\n\t  (grad(B) :- student(B), take(B,db), take(B,lp))\n\t  =>\n\t  grad(A).\n\tanswer(grad.name:varchar(30)) ->\n\t{ answer(pete) }\n\tInfo: 1 tuple computed.          \n\t\\end{verbatim}\n}\n\nAs an example of an \\mytt{ASSUME} query, we reuse the \\mytt{grad} definition above, assume that \\mytt{adam} is not an eligible student, and that  \\mytt{adam} and \\mytt{scott} took \\mytt{lp} and \\mytt{db} respectively:\n\n{\\fontsize{11}{11}\n\t\\begin{verbatim}\nDES> assume \n       (select 'adam') not in student, \n       (select 'adam','lp' union all select 'scott','db')\n         in take, \n       (select student.name from student, take t1, take t2\n         where student.name=t1.name and t1.name=t2.name and\n               t1.title='lp' and t2.title='db') in grad(name) \n     select * from grad;\nInfo: SQL statement compiled to:\n  answer(A) :-\n  -student(adam) \/\\ take(adam,lp) \/\\  take(scott,db) \/\\\n  (grad(B) :- student(B), take(B,lp), take(B,db))\n  =>\n  grad(A).\nanswer(grad.name:varchar(30)) ->\n{ answer(pete), answer(scott) }\nInfo: 2 tuples computed.          \n\\end{verbatim}\n}\n\nHere, the assumption on \\mytt{student} is negative and is compiled to a restricting fact. \nThe second one is compiled to a couple of facts because of the union.\nThe last one is the same as the previous example.\nThe SQL statement after the assumptions simply leads to the goal \\mytt{grad(A)}, for which even when \\mytt{adam} took the courses to graduate, he was removed as an eligible student and therefore from the answer.\n\nIf the extensional relations \\mytt{student} and \\mytt{take} are already defined in an external relational database (as, e.g., MySQL or PostgreSQL), they can be made available to DES via an ODBC connection (with the command \\mytt{\/open\\_db}), and queried as if they were local \\cite{Sae14a}.\nThis way, DES behaves as a front-end for both straight calls to native (i.e., supported by the external relational system) SQL queries and non-native queries (as those including \\mytt{ASSUME}).\nFor non-native statements, prepending the command \\mytt{\/des} to the query makes DES to handle such queries which are unsupported in the external database.\nFor example:\n\n{\\fontsize{11}{11}\n\t\\begin{verbatim}\n\tDES> \/open_db postgresql \n\tDES> \/des assume ...\n\tDES> \/open_db mysql \n\tDES> \/des with ...\n\t\\end{verbatim}\n}\n  \n\\noindent obtaining the same answers as before for the same queries (both omitted here in the ellipses).\nNote that, in particular, \\mytt{WITH} is unsupported in both MySQL and MS Access.\n\nEven when \\mytt{WITH} is supported in several relational database systems, they are somewhat restricted because, referring to the syntax in Section \\ref{with}, \\mytt{{\\em SQL}} cannot contain a \\mytt{WITH} clause, whereas we do allow for it.\n\n\n\n\n\n\n\n\\section{Conclusions}\nThis work has presented a proposal to take advantage of intuitionistic logic programming to model both temporary definitions (with the \\mytt{WITH} clause) and assumptions (with the \\mytt{ASSUME} clause) in SQL.\nIts motivation lies in providing a clean semantics that makes assumptions to behave as first-class citizen in the object language.\nThe deductive database system DES was used as a test bed to experiment with assumptions, translating SQL queries into Hypothetical Datalog.\nFurther, this system can be used as a front-end to relational systems lacking features as the \\mytt{WITH} clause.\nThe most related work is \\cite{ANSS13d}, which includes assumptions in SQL with a tailored semantics, and generates SQL scripts implementing fixpoint computations.\nWith respect to the intuitionistic formal framework, our work is based on \\cite{Bonner89hypotheticaldatalog,Bonner90hypotheticaldatalog,bonner90adding,bonner89expressing} and adapted to assume rules and deal with duplicates in \\cite{sae13c-ictai13}.\nHowever, it is not powerful enough to include embedded universal quantifiers in premises as in \\cite{bonner89expressing}, which provides the ability to create new constant symbols hypothetically along inference.\nThough this is not directly applicable to the current work, it is indeed an interesting subject to explore by considering that domains can be finitely constrained in practical applications, as with foreign keys.\n\n\\section*{Acknowledgements}\nThanks to the anonymous referees for their suggestions to improve this work, which has been partially supported by the Spanish MINECO project CAVI-ART (TIN2013-44742-C4-3-R), Madrid regional project N-GREENS Software-CM (S2013\/ICE-2731)  and UCM grant GR3\/14-910502.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nMean-field approximations of stochastic processes are crucial in many areas of science, where a large system is described by a stochastic process and its expected behaviour is approximated by a simpler mean-field model using a system of differential equations. The stochastic processes of particular interest to us are binary network processes, where each node of a large network can be in one of two states and the state of a node changes depending on the states of the neighbouring nodes.  The theory of their mathematical modelling can be found in several books and review papers \\cite{Barratetal, Danon, Nekovee, Newmanetal}. The two well-known examples we analyze in Section 4 are epidemic and rumour spreading.\n\n\nFor concreteness we consider a continuous time Markov process, $X(t)$, with states $k\\in\\{0,1,\\ldots,N\\}$ and let $p_k(t)=P[X(t)=k]$ denote the discrete probability distribution of $X(t)$.  The time evolution of $p_k(t)$ is described by a linear system of differential equations, called the master equations. There is a well established theory for solving linear systems by expressing their solution as $\\exp(At)$, where $A$ is the transition rate matrix of the system. It is also well-known that the matrix exponential is hard to compute for large matrices, hence methods have been developed exploiting any special structure of the matrix. Recently, an extremely powerful method has been worked out for tridiagonal matrices by Smith and Shahrezaei \\cite{SmithShahrezaei}. Since one-step processes have tridiagonal transition matrices, the time dependence of the probabilities $p_k(t)$ can be computed very efficiently. Besides computational methods there are theoretical approaches to approximate, estimate and characterize qualitatively the time dependence of the probability distribution $p_k$ and its moments (mainly its expected value). The typical approach is to introduce a low dimensional system of non-linear ODEs called the mean-field equations. These are not only faster to solve computationally but also allow for analysis giving a better qualitative understanding of the system.\n\nThe accuracy of mean-field approximations is not obvious. The problem of rigorously linking exact stochastic models to mean-field approximations goes back to the early work of Kurtz \\cite{Kurtz1970} (see \\cite{EthierKurtz} for a more recent reference). He studied density-dependent Markov processes and proved their stochastic convergence to the deterministic mean-field model. It can be shown that the difference between the solution of the mean-field equation and the expected value of the process is of order $1\/N$ as $N$ tends to infinity \\cite{BaKiSiSi}.\n\nHowever, it is natural to look for actual lower and upper bounds that can be used for finite $N$ (in contrast to the previous asymptotic results). It is known that in many cases, the mean-field model yields an upper bound on the expected value of the process. The first lower bound is by Armbruster and Beck \\cite{ArBe} where a system of two ODEs proves a lower bound on the expected value in the case of a susceptible-infected-susceptible (SIS) epidemic model on a complete graph. The aim of this paper is to extend these results to a wider class of Markov chains, including several which approximate network processes.\n\nWe develop bounds for \\emph{one-step processes}, also known as \\emph{birth-death processes}, that form an important class of continuous time Markov processes, for which the state space is $\\{ 0,1, \\ldots , N\\}$ and it is assumed that transition from state $k$ is possible only to states $k-1$ at rate $c_k$ and to state $k+1$ at rate $a_k$. The master equations are\n\\begin{equation}\n    p_k'= a_{k-1}p_{k-1}-(a_{k}+c_{k})p_k+c_{k+1}p_{k+1},\\quad k=0,\\ldots,  N , \\label{eqKolm}\n\\end{equation}\nwhere of course $a_{-1}=a_N=c_0=c_{N+1}=0$.\nThe bounds we develop are the solutions of simple ODEs and will hold for all $N$.  We will illustrate with numerical examples that the upper and lower bounds are remarkably close to each other.\n\nIn Section 2 we set up the problem, give the mean-field equation and the approximating system, state the main result, and state the tools used in the proof.  Section 3 proves the main result.  In Section 4 we illustrate the performance of the bounds on two examples: an SIS epidemic with and without airborne infection, and a voter-like model. Section 5 concludes and discusses directions for future research.\n\n\n\n\\section{Formulation of the main result}\n\n\\subsection{Differential equations for the moments}\nWe first establish the differential equations for the moments of the process $X(t)$. We focus on the fraction $X\/N$ since we are interested in situations with large $N$ and density dependent coefficients. By definition, the $n$-th moment of $(X\/N)$ is\n\\begin{equation}\\label{ynODE}\ny_n(t) := \\mathbb{E}[(X(t)\/N)^n]=\\sum_{k=0}^N (k\/N)^n p_k(t)\\quad (n=0,1,\\dots).\n\\end{equation}\nOf course $y_0=1$.\nThe following equations for $y_n'$ can be derived using Lemma 2 in \\cite{BaKiSiSi}, using the Kolmogorov backward equations, or taking the time derivative of \\eqref{ynODE} and substituting \\eqref{eqKolm}.\n\\[\ny'_n = \\frac{1}{N^n} \\sum_{k=0} ^N \\left( a_k ( (k+1)^n-k^n) + c_k ( (k-1)^n-k^n) \\right) p_k .\n\\]\nWe are interested in the case when $a_k\/N$ and $c_k\/N$ are polynomials of $k\/N$. Then, $y'_n$ can be expressed in terms of the coefficients of these polynomials. In fact, the following result was proved in \\cite{BaKiSiSi}.\n\n\\begin{lemma}\\label{momlem}\nLet\n\\begin{equation}\\label{akck}\n\\frac{a_k}N=A(k\/N) \\quad\\text{and}\\quad\\frac{c_k}N=C(k\/N)\n\\end{equation}\nwith polynomials $A(x)=\\sum_{j=0}^mA_jx^j$ and $C(x)=\\sum_{j=0}^mC_jx^j$ such that $A(1)=0$ and $C(0)=0$.\nThen\n\\begin{align}\\label{ysystem1}\ny_1'&=\\sum_{j=0}^mD_jy_j,\\\\\\label{ysystem2}\ny_n'&=n\\sum_{j=0}^mD_jy_{n+j-1}+\\frac1NR_n\\quad (n=2,3,\\dots)\n\\end{align}\nwhere $D_j=A_j-C_j$, $0\\leq R_n\\leq \\frac{n(n-1)}2 c$, and $c=\\sum_{j=0}^m (|A_j|+|C_j|)$.\n\\end{lemma}\n\n\\subsection{Mean-field equation and an approximating system}\nOur mean-field equation\n\\begin{equation}\\label{meanf}\ny'=\\sum_{j=0}^mD_jy^j,\\quad y(0)=y_1(0),\n\\end{equation}\nis motivated by the approximations $\\frac1NR_n\\approx0$ and $y_n\\approx y_1^n$ for large $N$,\nwhere the second approximation essentially assumes $X\/N$ is deterministic. We are interested whether the solution $y$ of the mean-field equation converges uniformly on $[0,T]$ to $y_1$ as $N\\to\\infty$. Recently in \\cite{ArBe}, the following special case was proved in an elementary way.\n\\begin{thm}\\label{arbethm}\nLet $m=2$, $D_0=0$, $D_1=\\tau-\\gamma$, and $D_2=-\\tau$ with positive constants $\\tau$ and $\\gamma$.\nIf $y(0)=y_1(0)=u\\in [0,1]$ is fixed, then for any fixed $T>0$,\n\\[y_1(t)\\leq y(t)\\quad \\text{ for } t\\in[0,T],\\]\nand\n\\[\\lim_{N\\to\\infty}|y(t)-y_1(t)|=0\\quad \\text{uniformly in } t\\in [0,T].\\]\n\\end{thm}\nIn the theorem above, the constants $\\tau$ and $\\gamma$ correspond to the infection and recovery rates in an SIS model of disease spread (see Subsection \\ref{sissub}). Our aim is to generalize Theorem \\ref{arbethm} to a broader class of coefficients $D_j$ and to provide a lower bound for $y_1$. In the proof of our main result we will not only compare $y$ with $y_1$ but also $y^n$ with $y_n$.  Thus using \\eqref{meanf},\n\\[(y^n)'=ny^{n-1}y'=n\\sum_{j=0}^mD_jy^{n+j-1}=n\\sum_{j=0}^mD_j(y^n)^{\\frac{n+j-1}n}.\\]\nHence, the powers of $y$ satisfy the initial value problem below:\n\\begin{align}\\label{psystem1}\ny'&=\\sum_{j=0}^mD_jy^j,\\quad y(0)=y_1(0),\\\\\\label{psystem2}\n(y^n)'&=n\\sum_{j=0}^mD_j(y^n)^{\\frac{n+j-1}n},\\quad y^n(0)=y_1^n(0)\\quad (n=2,3,\\dots).\n\\end{align}\n(The equation for $y'$ is separated from $(y^n)'$ for $n\\geq 2$ because it will have a different role.)\nThis system in combination with system \\eqref{ysystem1}--\\eqref{ysystem2} for $y_n'$ motivates the following initial value problem:\n\\begin{align}\\label{zsystem1}\nz_1'&=\\sum_{j=0}^mD_jz_j,\\quad z_1(0)=y_1(0),\\\\\\label{zsystem2}\nz_n'&=n\\sum_{j=0}^mD_jz_n^{\\frac{n+j-1}n}+\\frac{n(n-1)}{2N}c,\\quad z_n(0)=y_1^n(0)\\quad (n=2,\\dots,m),\n\\end{align}\nwhere we let $z_0=1$.\n\n\\subsection{Main result}\nWe are now ready to state our main result.\n\\begin{thm}\\label{main}\nAssume that\n\\begin{equation}\\label{sign}\nD_0\\geq0, D_1\\in\\mathbb{R}\\text{ and } D_j\\leq0 \\text{ for }j\\geq 2,\n\\end{equation}\nand let $y_1(0)=u\\in(0,1]$ be fixed.\nThen for the solutions $y$ of \\eqref{meanf} and $z_1$ of \\eqref{zsystem1}--\\eqref{zsystem2}, it holds that\n\\begin{align*}\nz_1(t)\\leq y_1(t)\\leq y(t)& \\text{ for }t\\geq0,\\\\\nz_1(t)^n \\leq y_n(t)\\leq z_n(t)& \\text{ for }t\\geq0,\\ 2\\leq n\\leq m,\n\\end{align*}\nand for every $T>0$ there exists a constant $C_T>0$ such that\n\\[\nz_1(t)-y(t)\\leq \\frac{C_T}N \\text{ in }[0,T].\n\\]\n\\end{thm}\n\nThe proof is based on some familiar inequalities which we recall in the next subsection.\n\n\\subsection{Tools of the proof}\n\nThe following comparison results are standard in the theory of ODEs, see \\cite{Hale}.\n\n\\begin{lemma}[Comparison]\\label{complem}\nSuppose that $f(t,x)$ is continuous in $x$;\n\\begin{itemize}\n\\item the initial value problem $x_2'(t)=f(t,x_2(t))$, $x_2(0)=x_0$ has a unique solution for $t\\in[0,T]$;\n\\item $x_1'(t)\\leq f(t,x_1(t))$ for $t\\in[0,T]$; and $x_1(0)\\leq x_0$.\n\\end{itemize}\nThen $x_1(t)\\leq x_2(t)$ for $t\\in[0,T]$.\n\\end{lemma}\n\n\\begin{lemma}[Peano's inequality]\nSuppose that $f_1,f_2\\colon[0,T]\\times[a,b]\\to\\mathbb{R}$ are Lipschitz continuous functions in their second variable with Lipschitz constant $L$ and $|f_1(t,x)-f_2(t,x)|\\leq M$ in $[0,T]\\times[a,b]$ with some constant $M$. If\n\\begin{itemize}\n\\item $x_1'(t)=f_1(t,x_1(t)), x_2'(t)=f_2(t,x_2(t))$ for $t\\in(0,T]$ and\n\\item $x_1(0)=x_2(0)$,\n\\end{itemize}\nthen\n\\[|x_1(t)-x_2(t)|\\leq \\frac{M}L\\left(e^{Lt}-1\\right) \\quad (t\\in[0,T]).\\]\n\\end{lemma}\n\nThe classical Jensen's inequality and the definition of the expected value yields the probabilistic version of Jensen's inequality, see \\cite{Ross}.\n\n\\begin{lemma}[Jensen's inequality]\nIf $X$ is a random variable and $\\varphi\\colon\\mathbb{R}\\to\\mathbb{R}$ is a convex function, then\n\\[\\varphi(\\mathbb{E}[X])\\leq\\mathbb{E}[\\varphi(X)].\\]\nFor concave $\\varphi$, the reverse inequality holds.\n\\end{lemma}\n\n\n\n\n\\section{Proof of the main result}\n\n\n\\begin{proof}[Proof of Theorem \\ref{main}]\nThe proof will be carried out in multiple steps. First, we derive bounds for $y_n$ and $z_n$ independent of $N$ which will guarantee uniform Lipschitz constants with respect to $N$ in steps three and four. Second, we show that $y_1\\leq y$. In the third step, we show that $y_n \\leq z_n$ for $n\\geq 2$ and use Peano's inequality to deduce from equations \\eqref{psystem2}, \\eqref{ysystem2} and \\eqref{zsystem2} that there is some constant $C(n,T)$ depending on $n$ and $T$ such that\n\\[|y^n(t)-z_n(t)|\\leq \\frac{C(n,T)}N \\text{ for } t\\in[0,T],\\ n=2,\\ldots,m.\\]\nFinally with these estimates at hand, we obtain in the same manner $z_1\\leq y_1$ and\n\\[|y(t)-z_1(t)|\\leq \\frac{C_T}N \\text{ for } t\\in[0,T],\\]\nwith some suitable constant $C_T$.\n\n\\textbf{Step 1: A priori bounds independent of $N$.}\nWe first focus on a lower bound for $y_n$.\nThe definition of $y_n$ implies $0\\leq y_\\ell\\leq y_n$ for all $n\\leq \\ell$. Therefore,\n\\[y_1'\\geq D_0+y_1\\sum_{j=1}^mD_j\\geq y_1 D,\\ y_1(0)=u,\\]\nwhere the second inequality is due to \\eqref{sign} and we define $D=\\sum_{j=1}^mD_j$.  Applying Lemma \\ref{complem} we obtain\n\\[y_1(t)\\geq ue^{tD}.\\]\nNow due to Jensen's inequality,\n\\begin{equation}\\label{Jensen}\ny_1^n=\\mathbb{E}[X\/N]^n\\leq \\mathbb{E}[(X\/N)^n]=y_n\\text{ for } n\\geq 1.\n\\end{equation}\nDefining $\\delta_1(T)=u^m\\min\\{1,e^{TmD}\\}>0$, this leads to our lower bound,\n\\begin{equation}\\label{also}\ny_n(t) \\geq y_1^n(t) \\geq u^n e^{tnD}\\geq \\delta_1(T)\\text{ for } 1\\leq n\\leq m,t\\in[0,T].\n\\end{equation}\n\nNext we focus on an upper bound for $z_n$ for $n\\geq 2$.  We bound the definition \\eqref{zsystem2} by assuming $z_n\\geq 1$. Then using the sign condition \\eqref{sign}, $z_n^{\\frac{n-1}{n}}\\leq z_n$, and $N\\geq 1$,\n\\[z'_n \\leq n(D_0+|D_1|+c n)z_n,\\quad z_n(0)=u^n \\text{ for } n\\geq 2.\\]\nApplying Lemma \\ref{complem} to this differential inequality, we obtain an upper bound,\n\\begin{equation}\\label{zbound}\nz_n(t) \\leq u^n e^{n(D_0+|D_1|+cn)t}\\leq \\delta_2(T)\\text{ for } 2\\leq n\\leq m,t\\in[0,T],\n\\end{equation}\nwhere $\\delta_2(T)=\\max\\{1,u^2 e^{m(D_0+|D_1|+cm)T}\\}$.\n\n\\textbf{Step 2: Comparison of $y$ and $y_1$.}\nSubstituting \\eqref{Jensen} into \\eqref{ysystem1} with regard to the sign condition \\eqref{sign} yields\n\\[y_1'\\leq \\sum_{j=0}^m D_jy_1^j.\\]\nApplying Lemma \\ref{complem} to the above inequality and to \\eqref{meanf} yields for $t\\in[0,T]$,\n\\begin{equation}\\label{y1lessy}\ny_1(t)\\leq y(t).\n\\end{equation}\n\n\\textbf{Step 3: Comparison of $y_n$, $y^n$, and $z_n$.}\nApplying Jensen's inequality again,\n\\begin{gather}\\label{jen1}\ny_{n+j-1}=\\mathbb{E}[(X\/N)^{n+j-1}]\\geq \\mathbb{E}[(X\/N)^n]^{\\frac{n+j-1}n}=y_n^{\\frac{n+j-1}n}\\text{ for } j\\geq 1,\\\\\ny_{n-1}=\\mathbb{E}[(X\/N)^{n-1}]\\leq \\mathbb{E}[(X\/N)^n]^{\\frac{n-1}n}=y_n^{\\frac{n-1}{n}}.\\label{jen2}\n\\end{gather}\nWe also have the trivial estimate\n\\begin{equation}\\label{triv}\n0\\leq y_n\\leq1.\n\\end{equation}\n\nPutting the estimates \\eqref{jen1}, \\eqref{jen2} and \\eqref{triv} into the differential equation \\eqref{ysystem2} of the $n$-th moment for $n\\geq 2$ and taking care of the sign condition \\eqref{sign} we obtain\n\\begin{equation}\\label{ynest}\ny_n'\\leq n\\sum_{j=0}^mD_jy_n^{\\frac{n+j-1}n}+\\frac{n(n-1)}{2N}c,\\quad y_n(0)=y_1^n(0)=u^n.\n\\end{equation}\nBy introducing the function $g_n(x)=n\\sum_{j=0}^mD_jx^{\\frac{n+j-1}n}+\\frac{n(n-1)}{2N}c$, \\eqref{ynest} can be written as $y_n'\\leq g_n(y_n)$, and \\eqref{zsystem2} has the form $z_n'=g_n(z_n)$.\nNow, $g_n(x)$ is Lipschitz continuous except at 0. Thus we can apply Lemma \\ref{complem} while $z_n\\geq\\delta_1(T)$ holds, to obtain\n\\begin{equation}\\label{ysubnzn}\ny_n(t)\\leq z_n(t) \\text{ for } 2\\leq n\\leq m,t\\in [0,T].\n\\end{equation}\nFortunately, \\eqref{also} ensures $z_n(t)\\geq y_n(t)\\geq \\delta_1(T)$ for $t\\leq T$ and $2\\leq n\\leq m$.\n\nUsing the same steps and \\eqref{psystem2} we can also show that\n\\begin{equation}\\label{ynzn}\ny^n(t)\\leq z_n(t) \\text{ for } 2\\leq n\\leq m,t\\in [0,T].\n\\end{equation}\nNow,\n\\begin{equation}\\label{zybound}\n\\delta_2(T)\\geq z_n(t)\\geq y^n(t) \\geq y_1^n(t) \\geq \\delta_1(T)\n\\text{ for } 2\\leq n\\leq m,t\\in [0,T],\n\\end{equation}\nwhere the inequalities are from \\eqref{zbound}, \\eqref{ynzn}, \\eqref{y1lessy}, and \\eqref{also}, respectively.\n\nThis lets us use Peano's inequality to estimate the difference of the solution $y^n$ of \\eqref{psystem2} and $z_n$ of \\eqref{zsystem2} since $z_n'=g_n(z_n)$, $(y^n)'=g_n(y^n)-\\frac{n(n-1)}{2N}c$, and \\eqref{zybound} ensures a Lipschitz constant independent of $N$. Therefore, there is some constant $C(n,T)$ depending on $n$ and $T$ such that\n\\begin{equation}\\label{est}\n|y^n(t)-z_n(t)|\\leq \\frac{C(n,T)}N \\text{ for } 2\\leq n\\leq m,t\\in[0,T].\n\\end{equation}\n\n\\textbf{Step 4: Comparison of $y_1$ and $z_1$.}\nNow, with estimates \\eqref{ysubnzn} and \\eqref{est} in our hand we turn to equations \\eqref{psystem1}, \\eqref{ysystem1}, \\eqref{zsystem1} to obtain estimates on their solutions analogously to the preceding part of the proof. First, by substitution of estimates \\eqref{ysubnzn} into \\eqref{ysystem1} with regard to the sign condition \\eqref{sign} it follows that\n\\begin{equation}\\label{y1est}\ny_1'\\geq D_0+D_1y_1+\\sum_{j=2}^mD_jz_j,\\quad y(0)=y_1(0)=u.\n\\end{equation}\nConsidering functions $z_n$ ($n=2,\\dots,m$) fixed, then \\eqref{y1est} and \\eqref{zsystem1} have the form $y_1'(t)\\geq g_1(t,y_1(t))$ and $z_1'(t)=g_1(t,z_1(t))$ where $g_1(t,x)=D_0+D_1x+\\sum_{j=2}^mD_jz_j(t)$. Thus, Lemma \\ref{complem} implies\n\\[y_1\\geq z_1\\]\nand this leads to the lower bound for the $n$-th moment as $y_n\\geq z_1^n$ by using \\eqref{also}.\n\nNow we apply Peano's inequality to estimate the difference of the solution $y$ of \\eqref{psystem1} and $z_1$ of \\eqref{zsystem1}. Indeed, $y'(t)=h_1(t,y(t))$ and $z_1'(t)=g_1(t,z_1(t))$ where $h_1(t,x)=D_0+D_1x+\\sum_{j=2}^mD_jy^j(t)$. Therefore,\n\\[h_1(t,x)-g_1(t,x)=\\sum_{j=2}^m D_j(y^j(t)-z_j(t)).\\]\nThus by \\eqref{est},\n\\[| h_1(t,x)-g_1(t,x)|\\leq \\frac{m \\cdot \\max_{2\\leq n\\leq m} |D_n|C(n,T)}N \\text{ for } t\\in[0,T].\\]\nThen Peano's inequality implies that there is some constant $C$ depending on $m$ and $T$ such that\n\\[\n|y(t)-z_1(t)|\\leq \\frac{C}N \\text{ for } t\\in[0,T].\n\\]\nThe proof is now complete.\n\\end{proof}\n\n\n\n\n\n\n\n\\section{Examples}\n\nHere we show numerically the lower and upper bounds in the case of two network processes.\n\n\\subsection{SIS epidemic propagation with and without airborne infection}\\label{sissub}\n\nConsider SIS epidemic propagation on a regular random graph with $N$ nodes.  We allow for multiple routes of infection so that the infection spreads not only via the contact network, but also due to external forcing, such as airborne infection. The state space of the corresponding one-step process is $\\{ 0,1, \\ldots , N\\}$, where $k$ denotes the state with $k$ infected nodes. In fact, the position of the infected nodes also affects the spreading process, hence the state space is larger, and therefore our model is only an approximation of the real infection propagation process. The validity of this approximation is discussed in detail in \\cite{akckcikk}. We note that in the case of a complete graph, the model is exact. Starting from state\n$k$ the system can move either to state $k+1$ or to $k-1$, since at a given instant only one node can change its state. When the system moves from state $k$ to $k+1$ then a susceptible node becomes infected. The rate of external infection of a susceptible node is denoted by $\\beta$. The rate of internal infection is proportional to the number of infected neighbours, which is $d\\frac{k}{N}$ in average, where $d$ denotes the degree of each node in the network and $\\frac{k}{N}$ is the proportion of the infected nodes. Since there are $N-k$ susceptible nodes and each of them has $d\\frac{k}{N}$ infected neighbours the total number of $SI$ edges is $d(N-k)\\frac{k}{N}$. Then the rate of transition from state $k$ to state $k+1$ is obtained by adding the rates of the two infection processes\n$$\na_k=\\tau d(N-k)\\frac{k}{N} +\\beta (N-k) .\n$$\nwhere $\\tau$ is the infection rate. The rate of transition from state $k$ to $k-1$ is $c_k=\\gamma k$, because any of the $k$ infected nodes can recover with recovery rate $\\gamma$. Using these coefficients, $a_k$ and $c_k$, the spreading process can be described by equation \\eqref{eqKolm}. The coefficients can be given in the form \\eqref{akck} by choosing the functions $A(x)=\\tau d x (1-x)+\\beta (1-x) $ and $C(x)=\\gamma x$. The coefficients of these polynomials are $A_0=\\beta $, $A_1=\\tau d -\\beta$, $A_2=-\\tau d $, $C_0=0$, $C_1=\\gamma$ and $C_2=0$. Thus the coefficients\n$$\nD_0=\\beta, \\quad D_1=\\tau d -\\beta - \\gamma, \\quad D_2= -\\tau d\n$$\nsatisfy the sign condition \\eqref{sign}. According to \\eqref{meanf}, the mean-field equation takes the form\n\\begin{equation}\ny'=\\beta+(\\tau d -\\beta - \\gamma)y-\\tau d y^2 \\label{meanfSISa}\n\\end{equation}\nsubject to the initial condition $y(0)=i\/N$, where $i$ is the number of initially infected nodes.\nSystem \\eqref{zsystem1}-\\eqref{zsystem2} can be written as\n\\begin{align}\\label{zsystem1SISa}\nz_1'&=\\beta+(\\tau d -\\beta - \\gamma)z_1-\\tau d z_2,\\\\\\label{zsystem2SISa}\nz_2'&=2\\beta z_2^{1\/2}+2(\\tau d -\\beta - \\gamma)z_2 -2\\tau d z_2^{3\/2} + \\frac{c}{N},\n\\end{align}\nwhere $c=\\beta+|\\tau d-\\beta|+\\tau d +\\gamma$. The initial condition is $z_1=i\/N$, $z_2=(i\/N)^2$. The mean-field equation \\eqref{meanfSISa} and the system \\eqref{zsystem1SISa}--\\eqref{zsystem2SISa} can be easily solved with an ODE solver.\n\nThe solutions without airborne infection, $\\beta=0$, are shown in Figure \\ref{fig:SIS} for $N=10^6$ and $N=10^7$. It is important to note that for such large values of $N$ the master equation \\eqref{eqKolm} cannot be solved numerically, but we know that the expected value $y_1(t)= \\sum_{k=0}^N  \\frac{k}{N}  p_k(t)$ is between the two curves given in the Figure. We can also see that for small times the two bounds are nearly identical, i.e., we get the expected value with high accuracy. As time increases the bounds move apart, moreover the length of time interval, where the two bounds give the expected value accurately increases with $N$.\n\nThe solutions with airborne infection, $\\beta>0$, are shown in Figure \\ref{fig:SISair} for $N=100$ together with the expected value $y_1(t)= \\sum_{k=0}^N  \\frac{k}{N} p_k(t)$ obtained by solving the master equation \\eqref{eqKolm} for $p_k$. One can see that the expected value is between the two bounds, in fact, it is hardly to distinguish from the solution of the mean-field equation, therefore the stationary part of the curves are enlarged in the inset. Note that the performance of the bounds is much better than in the case without airborne infection. Here we get much closer bounds even for a small value of $N$.\n\n\\subsection{A voter-like model}\n\nConsider again a regular random network where each node can be in one of two states, 0 or 1, representing two opinions propagating along the edges of the network (see \\cite{HolleyLiggett}). If a node is in state 0 and has $j$ neighbours in state 1, then its state will change to 1 with probability $j\\tau \\Delta t$ in a small time interval $\\Delta t$. This describes a node switching to opinion 1. The opposite case can also happen, that is a node in state 1 can become a node with opinion 0 with a probability $j\\gamma \\Delta t$ in a small time interval $\\Delta t$, if it has $j$ neighbours in state 0. The parameters $\\tau$ and $\\gamma$ characterize the strengths of the two opinions. Voter models are related to the famous Ising spin model in physics where the atomic spin, $\\pm1$, in a domain is affected by the spin in neighboring domains. The state space of the corresponding one-step process is $\\{ 0,1, \\ldots , N\\}$, where $k$ denotes the state, in which there are $k$ nodes with opinion 1. Starting from state $k$ the system can move either to state $k+1$ or to $k-1$, since at a given instant only one node can change its opinion. When the system moves from state $k$ to $k+1$ then a node with opinion 0 is ``invaded'' and becomes a node with opinion 1. The rate of this transition is proportional to the number of neighbours with opinion 1, which is $d\\frac{k}{N}$ in average, where $d$ is the degree of each node in the network and $\\frac{k}{N}$ is the proportion of the nodes with opinion 1. Since there are $N-k$ nodes with opinion 0 and each of them has $d\\frac{k}{N}$ neighbours with the opposite opinion the total number of edges connecting nodes with two different opinions is $d(N-k)\\frac{k}{N}$. Hence the rate of transition from state $k$ to state $k+1$ is\n$$\na_k=\\tau d(N-k)\\frac{k}{N}.\n$$\nSimilar reasoning leads to the rate of transition from state $k$ to $k-1$ as\n$$\nc_k=\\gamma dk\\frac{N-k}{N}.\n$$\nUsing these coefficients, $a_k$ and $c_k$, the spreading process can be described by equation \\eqref{eqKolm}. The coefficients can be given in the form \\eqref{akck} by choosing the functions $A(x)=\\tau d x (1-x)$ and $C(x)=\\gamma d x(1-x)$. The coefficients of these polynomials are $A_0=0$, $A_1=\\tau d $, $A_2=-\\tau d $, $C_0=0$, $C_1=\\gamma d$ and $C_2=-\\gamma d$. Thus the coefficients\n$$\nD_0=0, \\quad D_1=\\tau d - \\gamma d, \\quad D_2= \\gamma d-\\tau d\n$$\nsatisfy the sign condition \\eqref{sign} if $\\gamma < \\tau$. According to \\eqref{meanf} the mean-field equation takes the form\n\\begin{equation}\ny'=(\\tau d - \\gamma d)(y- y^2) \\label{meanfvot}\n\\end{equation}\nsubject to the initial condition $y(0)=i\/N$, where $i$ is the number of nodes with opinion 1 at time 0.\nSystem \\eqref{zsystem1}-\\eqref{zsystem2} can be written as\n\\begin{align}\\label{zsystem1vot}\nz_1'&=(\\tau d - \\gamma d)z_1-(\\tau d - \\gamma d) z_2,\\\\\\label{zsystem2vot}\nz_2'&=2(\\tau d - \\gamma d)z_2 -2(\\tau d - \\gamma d) z_2^{3\/2} + \\frac{c}{N},\n\\end{align}\nwhere $c=2\\tau d +2\\gamma d $. The initial condition is $z_1= i\/N$, $z_2= (i\/N)^2$. The mean-field equation \\eqref{meanfvot} and system \\eqref{zsystem1vot}--\\eqref{zsystem2vot} can be easily solved with an ODE solver. The solutions are shown in Figure \\ref{fig:vot} both for $D_2<0$ (left panel) and for $D_2>0$ (right panel). For $D_2<0$ the lower bound performs well only for large values of $N$. The master equation \\eqref{eqKolm} cannot be solved numerically  for such large values of $N$. Hence the expected value $y_1$ is not shown in the left panel. For $D_2>0$, i.e., when $\\gamma > \\tau$, the role of $y$ and $z_1$ is exchanged. The solution $y$ of the mean-field equation becomes the lower bound and $z_1$ becomes the upper bound. The function $y$ is hardly distinguishable from the expected value $y_1$, the inset shows that $y$ is really a lower bound.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.4]{figSIS}\n\\caption{SIS epidemic without airborne infection. The solution $y$ of the mean-field equation \\eqref{meanfSISa} (continuous curve) and the first coordinate $z_1$ of the solution of system \\eqref{zsystem1SISa}--\\eqref{zsystem2SISa} for $N=10^6$ (dashed curve) and for $N=10^7$ (dashed-dotted curve). The parameter values are $\\gamma=1$, $\\tau=0.1$, $d=30$, and $\\beta=0$. }\\label{fig:SIS}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.4]{figSISair}\n\\caption{SIS epidemic with airborne infection. The solution $y$ of the mean-field equation \\eqref{meanfSISa} (continuous curve), the first coordinate $z_1$ of the solution of system \\eqref{zsystem1SISa}--\\eqref{zsystem2SISa} (dashed curve) and the expected value $y_1$ obtained from the solution of the master equation (dashed-dotted curve). The stationary part of the curves are enlarged in the inset. The parameter values are $N=100$, $\\gamma=1$, $\\tau=0.05$, $d=20$ and $\\beta=1$. }\\label{fig:SISair}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.4]{figvot_a}\n\\includegraphics[scale=0.4]{figvot_b}\n\\caption{Voter-like model. The solution $y$ of the mean-field equation \\eqref{meanfvot} (continuous curve), the first coordinate $z_1$ of the solution of system \\eqref{zsystem1vot}--\\eqref{zsystem2vot} (dashed curve) and the expected value $y_1$ obtained from the solution of the master equation (dashed-dotted curve). Left panel: the case $D_2<0$, the values of $N$ are shown in the figure, the other parameter values are $\\gamma =0.1$, $\\tau=0.2$, $d=10$. Right panel: the case $D_2>0$, the parameter values are $N=200$, $\\gamma =0.2$, $\\tau=0.1$, $d=10$. The stationary part of the curves are enlarged in the inset. }\\label{fig:vot}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.4]{figSIS_qCS}\n\\caption{Potential lower bounds of SIS model without airborne infection. The solution $y$ of the mean-field equation \\eqref{meanfSISa} (continuous curve), the first coordinate $z_1$ of the solution of system \\eqref{zsystem1SISq}--\\eqref{zsystem2SISq} (dashed curve) for different values of $q$ shown in the figure, $z_1$ of the solution of system \\eqref{zsystem1SISCS}--\\eqref{zsystem2SISCS} (dotted curve) and the expected value $y_1$ obtained from the solution of the master equation (dashed-dotted curve). The parameter values are $N=100$, $\\gamma =1$, $\\tau=0.1$, $d=30$, and $\\beta=0$. }\\label{fig:SISq}\n\\end{center}\n\\end{figure}\n\n\n\\section{Discussion}\n\nWe started from the master equation of a one-step process, assumed that the coefficients are density dependent, and the functions $A$ and $C$ are polynomials. Then under certain sign condition on the coefficients of the polynomials we proved that the mean-field equation yields an upper bound for the expected value of the process. We constructed an auxiliary system for the artificially defined functions $z_j$, and proved that $z_1$ is a lower bound. We showed several examples, where the upper and lower bounds are close to each other, hence the method can be used to approximate the expected value without solving the large system of master equations or using simulation, which only gives probabilistic guarantees.\n\nTwo avenues for future research are relaxing the sign condition \\eqref{sign} and improving the lower bound.\nIt is easy to see that in the case $D_j \\geq 0$ for $j\\geq 2$ and following the argument in Step 2 that the mean field equation yields a lower bound. Also in the voter-like model, a violation of the sign condition (i.e., when $\\gamma$, the rate of switching to opinion 0, is greater than $\\tau$, the rate of switching to opinion 1) can be dealt with by switching the labels of the opinions or equivalently replacing $x$ by $1-x$.  Perhaps more general violations of the sign condition can be dealt with by considering the convexity or concavity of the entire polynomials $A(x)$ and $C(x)$ instead of by the signs of their coefficients.\n\nThe second avenue for future research is improving the lower bound. Unlike the case of SIS disease propagation with airborne infection, $\\beta>0$, where the upper and lower bounds are very close to each other even for small system sizes, say $N=100$, we can see that for the case of regular SIS epidemic without airborne infection, $\\beta=0$, and the voter-like model, that the lower bound may be quite far from the expected value for moderately large $N$, say $N=10^5$.\n\nThe reason that the lower bound veers off to 0 is that $z_2$ converges to a positive steady state and then the derivative of $z_1$ becomes negative after some time. This problem can be overcome by altering the differential equation of $z_1$. The term $D_2z_2$ could be changed to a term that contains also $z_1$ in order to prevent $z_1$ from becoming negative. The new term could be introduced by exploiting the fact that $z_j$ approximates the $j$-th moment $y_j$ and this function can be approximated by $y^j$. This suggests that $z_2$ can be approximated by $z_1^2$, or in other words, $z_1$ approximates $\\sqrt{z_2}$. In order to tune the approximation we introduce the artificial parameter $q\\in [0,2]$ and change the term  $D_2z_2$ to $D_2z_2^{q\/2} z_1^{2-q}$. For $q=0$ we obtain the mean-field upper bound and for $q=2$ we get back the original lower bound. In the quadratic case the modified differential equations for the lower bound of the SIS epidemic \\eqref{zsystem1SISa}--\\eqref{zsystem2SISa}\nwithout airborne infection, $\\beta=0$, are\n\\begin{align}\\label{zsystem1SISq}\nz_1'&=D_0 + D_1 z_1 + D_2z_2^{q\/2} z_1^{2-q} \\\\\n\\label{zsystem2SISq}\nz_2'&=2D_0 z_2^{1\/2} + 2D_1 z_2 +2 D_2 z_2^{3\/2} +\\frac{c}{N}.\n\\end{align}\nIn Figure \\ref{fig:SISq} the solution of this system is shown for different values of $q$ together with the solution of the mean-field equation and the expected value. The curve for $q=0.5$ is close to the mean-field upper bound (i.e., $q=0$) and at least appears to be a valid lower bound for $y_1$.\n\nAn alternate approach for an improved lower bound modifies the differential equation for $z_2$ \\eqref{zsystem2} by replacing the $z_2^{3\/2}$ term by $z_2^2\/z_1$ resulting in the system\n\\begin{align}\\label{zsystem1SISCS}\nz_1'&=D_0 + D_1 z_1 + D_2z_2 \\\\\n\\label{zsystem2SISCS}\nz_2'&=2D_0 z_2^{1\/2} + 2D_1 z_2 +2 D_2 z_2^2\/z_1 +\\frac{c}{N}.\n\\end{align}\nBy Jensen's inequality, $y_2^{3\/2}\\leq y_3$.\nThus assuming $z_2\\approx y_2$, the $z_2^{3\/2}$ term can be interpreted as a lower bound for a $y_3$ term.  Now we motivate the $z_2^2\/z_1$ term. Using the Cauchy-Schwarz inequality, $y_2^2\\leq y_3y_1$, leads to a lower bound $y_2^2\/y_1\\leq y_3$ that appears to be tighter than $y_2^{3\/2}$.  Assuming $z_1\\approx y_1$ and $z_2\\approx y_2$ then motivates using $z_2^2\/z_1$ instead of $z_2^{3\/2}$.  From Figure \\ref{fig:SISq} it appears to be a valid lower bound for $y_1$ and closer to the mean-field upper bound than the original lower bound.\nOf course these potential lower bounds \\eqref{zsystem1SISq}--\\eqref{zsystem2SISq} and \\eqref{zsystem1SISCS}--\\eqref{zsystem2SISCS} are only justified by a numerical experiment (Figure \\ref{fig:SISq}) and some intuition.  It is an open question whether these are provable lower bounds or not.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nThe results of this article strengthen the connection between invariants of Legendrian knots in standard contact $\\mathbb {R}^3$ and the $2$-variable Kauffman polynomial. Relations between the $2$-variable knot polynomials (HOMFLY-PT and Kauffman) and Legendrian knot theory were first realized in the work of Fuchs and Tabachnikov \\cite{FT} who observed, based on results of Bennequin \\cite{Be}, Franks--Morton--Williams \\cite{FW}, and Rudolph \\cite{Ru}, that these polynomials provide upper bounds on the Thurston--Bennequin number of a Legendrian knot. At that time, it was still unknown whether Legendrian knots in $\\mathbb {R}^3$ were determined up to Legendrian isotopy by their Thurston--Bennequin number, rotation number, and underlying smooth knot type (the so-called ``classical invariants'' of Legendrian knots). This question was soon resolved with the introduction of several non-classical invariants in the late 90's and early 2000's including the Legendrian contact homology algebra which is a differential graded algebra (DGA) coming from $J$-holomorphic curve theory that was constructed by Chekanov in \\cite{Chekanov02} and discovered independently by Eliashberg and Hofer, and combinatorial invariants introduced by Chekanov and Pushkar \\cite{ChP} defined by counting certain decompositions of front diagrams called normal rulings.\\footnote{Around the same time, generating family homology invariants capable of distinguishing Legendrian links with the same classical invariants were introduced by Traynor~\\cite{Tr}.} Interestingly, normal rulings were discovered independently by Fuchs in connection with augmentations of the Le\\-gendrian contact homology DGA. Moreover, Fuchs again pointed toward a connection between Legendrian invariants and topological knot invariants by conjecturing in \\cite{Fuchs} that a Legendrian knot should have a normal ruling if and only if the Kauffman polynomial estimate for the Thurston--Bennequin number is sharp. This conjecture was resolved affirmatively in \\cite{R1} by interpreting Chekanov and Pushkar's combinatorial invariants as polynomials, and showing that the ungraded ruling polynomial, $R^1_K(z)$, of a Legendrian knot $K \\subset \\mathbb {R}^3$ arises as a specialization\n\\begin{gather} \\label{eq:specialization}\nR^1_K(z) = F_K(a,z)|_{a^{-1}=0}\n\\end{gather}\nof the framed version of the Kauffman polynomial $F_K \\in \\mathbb {Z}\\big[a^{\\pm1}, z^{\\pm1}\\big]$; the specialization has the property that it is non-zero if and only if the Kauffman polynomial estimate for $\\operatorname{tb}(K)$ is sharp. An analogous result also established in~\\cite{R1} holds for the $2$-graded ruling polynomial, $R^2_K(z)$, and the HOMFLY-PT polynomial.\n\nInitially, the Legendrian invariance of the ruling polynomials, based on establishing bijections between rulings during bifurcations of the front diagram occurring in a Legendrian isotopy, was somewhat mysterious from the point of view of symplectic topology. Building on the earlier works \\cite{Fuchs, FI,Henry,NgSab, Sab}, Henry and the second author showed in~\\cite{HenryRu} that the ruling polynomials are in fact determined by the Legendrian contact homology DGA, $(\\mathcal{A}, \\partial)$, since their specializations at $z = q^{1\/2} -q^{-1\/2}$ with $q$ a prime power agree with normalized counts of augmentations of $(\\mathcal{A}, \\partial)$ to the finite field $\\mathbb{F}_q$, i.e., DGA representations from $(\\mathcal{A},\\partial)$ to $(\\mathbb{F}_q,0)$. Thus, in the ungraded case (\\ref{eq:specialization}) shows that counts of ungraded augmentations are actually topological (depending only on the underlying framed knot type of $K$), as they arise from a specialization of the Kauffman polynomial. In this article we extend this result by relating counts of higher dimensional (ungraded) representations of $(\\mathcal{A}, \\partial)$ with the $n$-colored Kauffman polynomials.\n\nTo give a statement of our main result, for $n \\geq 1$, let $\\operatorname{Rep}_1\\big(K,\\mathbb{F}_q^n\\big)$ denote the {\\it ungraded total $n$-dimensional representation number} of $K$ as defined in~\\cite{LeRu}; see Definition \\ref{def:total}. Let $F_{n,K}(a,q)$ denote the {\\it $n$-colored Kauffman polynomial} (for framed knots); see Definition~\\ref{def:Kauffman}. In Section~\\ref{sec:ncolored}, we define an {\\it ungraded $n$-colored ruling polynomial} denoted~$R^1_{n,K}(z)$.\n\n\\begin{Theorem} \\label{thm:main} For any Legendrian knot $K$ in $\\mathbb {R}^3$ with its standard contact structure and any $n \\geq 1$, there is a well-defined specialization $F_{n,K}(a,q)|_{a^{-1}=0}$, and we have\n\\[\n\\operatorname{Rep}_1\\big(K,\\mathbb{F}_q^n\\big)=R^1_{n,K}(z)=F_{n,K}(a,q)|_{a^{-1}=0}.\n\\]\n\\end{Theorem}\n\nAs an immediate consequence, we get:\n\n\\begin{Corollary}\nThe ungraded total $n$-dimensional representation number $\\operatorname{Rep}_1\\big(K,\\mathbb{F}_q^n\\big)$ depends only on the underlying framed knot type of $K$.\n\\end{Corollary}\nThe corollary is a significant strengthening of a result from~\\cite{NgR2012} that the existence of an ungraded representation of~$(\\mathcal{A}, \\partial)$ on $\\mathbb{F}_2^n$ depends only on the Thurston--Bennequin number and topological knot type of~$K$. Precisely how much of the Legendrian contact homology DGA is determined by the framed knot type of~$K$ remains an interesting question. See~\\cite{NgR2012} and~\\cite{Ng1} for some open conjectures along this line.\n\nA previous article \\cite{LeRu} establishes analogous results in the case of $2$-graded representations and the HOMFLY-PT polynomial, and in fact establishes the equality $\\operatorname{Rep}_m\\big(K,\\mathbb{F}_q^n\\big)=R^m_{n,K}(z)$ between the $m$-graded total representation numbers and $m$-graded colored ruling polynomials for all $m \\in \\mathbb {Z}_{\\geq 0}$ {\\it except for the ungraded case where $m=1$}.  The $m=1$ case is more involved for a number of reasons. In the following we briefly review the argument from~\\cite{LeRu} and then outline our approach to Theorem~\\ref{thm:main}.\n\nFor $m \\neq 1$, the $n$-colored ruling polynomial is defined as a linear combination of satellite ruling polynomials of the form\n\\begin{gather} \\label{eq:mgraded}\nR^m_{n,K}(q) = \\frac{1}{c_n}\\sum_{\\beta \\in S_n} q^{\\lambda(\\beta)\/2} R^m_{S(K,\\beta)}(z)\\big|_{z = q^{1\/2}-q^{-1\/2}}, \\qquad m \\neq 1,\n\\end{gather}\nwhere \\looseness=1 $S(K,\\beta)$ is the Legendrian satellite of $K$ with a Legendrian positive permutation braid associated to $\\beta \\in S_n$. The same linear combination of HOMFLY-PT polynomials defines the $n$-colored HOMFLY-PT polynomial. In~\\cite{LeRu}, the total $n$-dimensional representation number is recovered from~(\\ref{eq:mgraded}) via a bijection between $m$-graded augmentations of $S(K,\\beta)$ and $n$-dimensional representations of the DGA of $K$ mapping a distinguished invertible generator into $B_\\beta \\subset {\\rm GL}(n, \\mathbb{F})$ where ${\\rm GL}(n,\\mathbb{F}) = \\sqcup_{\\beta \\in S_n}B_\\beta$ is the Bruhat decomposition. Thus, summing over all $\\beta \\in S_n$ corresponds to considering all $n$-dimensional representations of the DGA of~$K$ on~$\\mathbb{F}^n$.\n\nWhen $m=1$, the above bijection becomes modified in an interesting way, as augmentations of $S(K,\\beta)$ now correspond to (ungraded) representations of $(\\mathcal{A}, \\partial)$ on differential vector spaces of the form $\\big(\\mathbb{F}^n,d\\big)$ where $d$ varies over all (ungraded) upper triangular differential on $\\mathbb{F}^n$. (When $m \\neq 1$, $d =0$ is automatic for grading reasons.) The total $n$-dimensional representation number, $\\operatorname{Rep}_1\\big(K,\\mathbb{F}_q^n\\big)$, only counts representations with $d=0$, so the definition of $R^1_{n,K}$ needs to be changed to only take into account normal rulings corresponding to representations with $d=0$. This is done by replacing each $R_{S(K,\\beta)}^1$ in~(\\ref{eq:mgraded}) with the corresponding {\\it reduced ruling polynomial}~$\\widetilde{R}^1_{S(K,\\beta)}$ as introduced in~\\cite{NgR2012} that only counts normal rulings of $S(K,\\beta)$ that never pair two strands of the satellite that correspond to a single strand of $K$. Up to a technical point about the use of different diagrams in~\\cite{LeRu} and~\\cite{HenryRu} that the bulk of Section~\\ref{sec:5} is spent addressing, this leads to the equality $\\operatorname{Rep}_1\\big(K,\\mathbb{F}_q^n\\big)=R^1_{n,K}(z)$.\n\nEstablishing that $R^1_{n,K}(z)=F_{n,K}(a,q)|_{a^{-1}=0}$ requires a much more involved argument than for the case of the colored HOMFLY-PT polynomial and $R^2_{n,K}(z)$ (where the result is immediate from~\\cite{R1} and the definition). The $n$-colored Kauffman polynomial is defined by satelliting $K$ with the symmetrizer in the BMW algebra, $\\mathcal{Y}_n \\in \\operatorname{BMW}_n$. In addition to a sum over permutation braids as in the HOMFLY-PT case, $\\mathcal{Y}_n$ also has terms of a less explicit nature (though, see~\\cite{D}) involving tangle diagrams in $[0,1]\\times \\mathbb {R}$ with turn-backs, i.e., components that have both endpoints on the same boundary component of $[0,1]\\times \\mathbb {R}$. To relate $R^1_{n,K}$ and $F_{n,K}$, we use the combinatorics of normal rulings to find an inductive characterization of $R^1_{n,K}$ in terms of ordinary ruling polynomials rather than reduced ruling polynomials, and then compare this with an inductive characterization of $\\mathcal{Y}_n$ due to Heckenberger and Sch{\\\"u}ler~\\cite{Heck}.\n\nThe remainder of the article is organized as follows. In Section \\ref{sec:ncolored}, we define the ungraded $n$-colored ruling polynomial and establish an inductive characterization of it in Theorem \\ref{thm:main1}. In Section \\ref{sec:Kauffman}, we recall the definition of the colored Kauffman polynomial and prove the second equality of Theorem \\ref{thm:main} (see Theorem~\\ref{thm:main2}). Section \\ref{sec:5} reviews definitions of representation numbers from \\cite{LeRu} and then establishes the first equality of Theorem~\\ref{thm:main} (see Theorem~\\ref{thm:Rep}). In Section \\ref{sec:multicomp}, we close the article with a brief discussion of a modification of Theorem \\ref{thm:main} for the case of multi-component Legendrian links.\n\n\\section[The $n$-colored ungraded ruling polynomial]{The $\\boldsymbol{n}$-colored ungraded ruling polynomial} \\label{sec:ncolored}\n\nIn this section, after a brief review of ruling polynomials and Legendrian satellites, we define the ungraded $n$-colored ruling polynomial $R^1_{n,K}$ as a linear combination of {\\it reduced} ruling polynomials indexed by permutations $\\beta \\in S_n$. Reduced rulings, considered earlier in~\\cite{NgR2012}, form a restricted class of normal rulings of satellite links, so that it is not immediately clear how to describe $R^1_{n,K}$ in terms of ordinary ruling polynomials. For this purpose, we work in a Legendrian version of the $n$-stranded BMW algebra, $\\operatorname{BMW}_n^\\mathfrak{Leg}$, and inductively construct elements $L_n \\in \\operatorname{BMW}_n^\\mathfrak{Leg}$ that can be used to produce $R^1_{n,K}$ via (non-reduced) satellite ruling polynomials.\n\n\\subsection{Legendrian fronts and ruling polynomials}\nIn this article we consider Legendrian links and tangles in a $1$-jet space $J^1M$ where $M$ is one of~$\\mathbb {R}$,~$S^1$, or~$[0,1]$. In all cases, we can view $J^1M = T^*M\\times \\mathbb {R}$ as $M \\times \\mathbb {R}^2$, and using coordinates $(x,y,z)$ with $x \\in M$ and $y,z \\in \\mathbb {R}$ the contact form is ${\\rm d}z - y\\,{\\rm d}x$. Legendrian curves can be viewed via their {\\it front projection} $\\pi_{xz}\\colon J^1M \\rightarrow M \\times \\mathbb {R}$, $(x,y,z) \\mapsto (x,z)$ which is a collection of curves having cusp singularities and transverse double points but no vertical tangencies. The original Legendrian is recovered via $y= \\frac{{\\rm d}z}{{\\rm d}x}$, so in front diagrams (implicitly) the over-strand at a crossing is the strand with lesser slope (as the $y$-axis is oriented away from the viewer). Legendrian links have a {\\it contact framing} which is the framing given by the upward unit normal vector to the contact planes.\n\n\\begin{figure}[t] \\centering\n \\includegraphics{images\/normality.png}\n \\caption{Each closed curve of a normal ruling consists of a pair of companion paths with monotonically increasing $x$-coordinate beginning and ending at a common left and right cusp of~$\\pi_{xz}(L)$. At switches, paths from two different closed curves of~$\\rho$ meet and both turn a corner at a crossing. The normality condition requires that near switches the switching paths and their companion paths match one of the pictured configurations. At crossings that are not switches paths from two different closed curves cross transversally.} \\label{fig:normality}\n\\end{figure}\n\n\\begin{figure}[h!] \\centering\n \\begin{align*}\n \\raisebox{-.5cm}{\\includegraphics[scale=.7]{images\/mruling1.png}}-\\raisebox{-.5cm}{\\includegraphics[scale=.7]{images\/mruling2.png}} &=z\\left( \\;\\raisebox{-.5cm}{\\includegraphics[scale=.7]{images\/mruling3.png}}-\\; \\raisebox{-.5cm}{\\includegraphics[scale=.7]{images\/mruling4.png}}\\right), \\tag{R1} \\\\\n \\raisebox{-.5cm}{\\includegraphics[scale=.7]{images\/mruling5.png}}& =\\raisebox{-.5cm}{\\includegraphics[scale=.7]{images\/mruling6.png}}=0, \\tag{R2} \\\\\n \\raisebox{-.5cm}{\\includegraphics[scale=.7]{images\/unknot.png}} \\;\\sqcup K &=z^{-1}K. \\tag{R3}\n \\end{align*}\\vspace{-0.7cm}\n \\caption{The ungraded ruling polynomial skein relations.}\\label{fig:psuedo-gen}\n\\end{figure}\n\nRecall that for a Legendrian link $K \\subset J^1\\mathbb {R}$ a {\\it normal ruling} $\\rho$ of $K$ is a decomposition of the front diagram of $K$ into a collection of simple closed curves with corners at a left and right cusp and at {\\it switches} (adhering to the normality condition, see Fig.~\\ref{fig:normality}). For each $x=x_0$ where the front projection of $K$ does not have crossings or cusps, a normal ruling $\\rho$ divides the strands of~$K$ at $x=x_0$ into pairs, so that $\\rho$ can be viewed as a sequence of pairings of strands of~$K$. For a~more detailed discussion of normal rulings see for instance \\cite{Fuchs,NgR2012, R1, Sab}. The {\\it ungraded ruling polynomial} of $L$ (also called the $1$-graded ruling polynomial) is defined as\n\\[\nR_K^1(z):=\\sum_{\\rho\\in\\Gamma(K)}z^{j(\\rho)} \\in \\mathbb {Z}\\big[z^{\\pm1}\\big],\n\\] where the sum is over all normal rulings of $K$ and $j(\\rho)=\\#\\text{switches}-\\#\\text{right cusps}$. For Legendrian links in $J^1\\mathbb {R}$, the ungraded ruling polynomial of $K$ satisfies and is uniquely characterized by the skein relations in Fig.~\\ref{fig:psuedo-gen} and the normalization $R^1_{\\includegraphics[scale=.2]{images\/Unknot2}} = z^{-1}$. (See~\\cite{R1}.) The relations in Fig.~\\ref{fig:psuedo-gen} imply two additional relations that we will make use of, cf.\\ \\cite[Section~6]{R2}.\n\\begin{gather} \\label{eq:fish}\n \\text{fishtail relation:} \\ \\raisebox{-.35cm}{\\includegraphics[scale=.5]{images\/lfish.png}} =\\raisebox{-.35cm}{\\includegraphics[scale=.5]{images\/rfish.png}}=0,\n \\vspace{.5cm}\\\\\n \\text{double-crossing relation:} \\ \\raisebox{-.35cm}{\\includegraphics[scale=.5]{images\/dblesigma.png}}= \\raisebox{-.35cm}{\\includegraphics[scale=.5]{images\/id2.png}}+ z\\cdot\\raisebox{-.35cm}{\\includegraphics[scale=.5]{images\/cross.png}}- z\\cdot\\raisebox{-.35cm}{\\includegraphics[scale=.5]{images\/cusps.png}}.\\nonumber\n\\end{gather}\n\\begin{Remark} The double crossing relation can be used to show that the first relation of Fig.~\\ref{fig:psuedo-gen} also hold with right cusps. Moreover, the third relation from Fig.~\\ref{fig:psuedo-gen} is implied by the first two as long as $K \\neq \\varnothing$.\n\\end{Remark}\n\n\\subsection{A Legendrian BMW algebra}\n\nA {\\it Legendrian $n$-tangle} is a properly embedded Legendrian $\\alpha \\subset J^1[0,1]$ (i.e., compact with $\\partial \\alpha \\subset \\partial J^1[0,1]$) whose front projection agrees with the collection of horizontal lines $z=i$, $1\\leq i \\leq n$, near $\\partial J^1[0,1]$.  Legendrian isotopies of $n$-tangles are required to remain fixed in a~neighborhood of $\\partial J^1[0,1]$. At $x=0$ and $x=1$ we enumerate the endpoints and strands of a Legendrian tangle from $1$ to $n$ with {\\it descending} $z$-coordinate. For any permutation $\\beta \\in S_n$, there is a corresponding {\\it positive permutation braid} that is a Legendrian $n$-tangle, also denoted $\\beta \\subset J^1[0,1]$, that connects endpoint $i$ at $x=0$ to endpoint $\\beta(i)$ at $x=1$ for $1 \\leq i \\leq n$. Up to Legendrian isotopy, $\\beta$ is uniquely characterized by requiring that\n\\begin{itemize}\\itemsep=0pt\n\\item[(i)] the front projection does not have cusps and,\n\\item[(ii)] for $i<j$, the front projection has no crossings (resp.\\ exactly one crossing) between the strands with endpoints $i$ and $j$ at $x=0$ if $\\beta(i) < \\beta(j)$ (resp.\\ $\\beta(i)> \\beta(j)$).\n\\end{itemize}\nThe number of crossings in such a front diagram for $\\beta$ is called the {\\it length} of $\\beta$ and will be deno\\-ted~$\\lambda(\\beta)$. For Legendrian $n$-tangles $\\alpha,\\beta \\subset J^1[0,1]$, we define their multiplication $\\alpha\\cdot\\beta \\subset J^1[0,1]$ by stacking $\\beta$ to the {\\it left} of $\\alpha$ (as in composition of permutations). Diagrammatically:\n\\[\n\\labellist\n\\small\n\\pinlabel $\\alpha$ at 45 38\n\\endlabellist\n\\includegraphics[scale=.5]{images\/Tangle1} \\,\\, \\raisebox{.6cm}{$\\cdot$} \\,\\,\n\\labellist\n\\small\n\\pinlabel $\\beta$ at 45 38\n\\endlabellist\n\\includegraphics[scale=.5]{images\/Tangle1}\n\\quad\\raisebox{.6cm}{$=$} \\quad\n\\labellist\n\\small\n\\pinlabel $\\beta$ at 45 38\n\\pinlabel $\\alpha$ at 120 38\n\\endlabellist\n\\includegraphics[scale=.5]{images\/Tangle2}\n\\]\n\n\\begin{Definition} Let $\\mathcal{R}$ be a coefficient ring containing $\\mathbb {Z}[z^{\\pm1}]$ as a subring. Define the {\\it Legendrian BMW algebra}, $\\operatorname{BMW}_n^\\mathfrak{Leg}$, as an $\\mathcal{R}$-module to be the quotient $\\mathcal{R} \\mathfrak{Leg}_n\/\\mathcal{S}$ where $\\mathcal{R} \\mathfrak{Leg}_n$ is the free $\\mathcal{R}$-module generated by Legendrian isotopy classes of Legendrian $n$-tangles in $J^1[0,1]$, and $\\mathcal{S}$ is the $\\mathcal{R}$-submodule generated by the ruling polynomial skein relations from Fig.~\\ref{fig:psuedo-gen}. Multiplication of $n$-tangles induces an $\\mathcal{R}$-bilinear product on $\\operatorname{BMW}_n^\\mathfrak{Leg}$.\n\\end{Definition}\n\nIn the remainder of the article, we fix the coefficient ring $\\mathcal{R}$ to be $\\mathbb {Z}\\big[s^{\\pm1}\\big]$ localized to include denominators of the form $s^n-s^{-n}$ for $n\\geq 1$ where $z = s- s^{-1}$. In Section~\\ref{sec:5}, we will work with the alternate variable $s= q^{1\/2}$.\n\n\\begin{figure} \\centering\n $\\sigma_i:=\\raisebox{-.75cm}{\\includegraphics[scale=.5]{images\/sigmai.png}}\\hspace{.75cm} \\hspace{.75cm} e_i:=\\raisebox{-.75cm}{\\includegraphics[scale=.5]{images\/ei.png}}$\n \\caption{Crossing and hook elements in $\\operatorname{BMW}_n^\\mathfrak{Leg}$.} \\label{fig:sigma}\n\\end{figure}\n\nFig.~\\ref{fig:sigma} indicates crossing and hook elements, $\\sigma_i, e_i \\in \\operatorname{BMW}_n^\\mathfrak{Leg}$, for $1 \\leq i < n$. Note that the fishtail and double-crossing relations imply\n\\begin{gather}\n \\sigma_i e_i = e_i \\sigma_i =0, \\qquad \\mbox{and} \\label{eq:fish2} \\\\\n \\sigma_i^2 = 1 + z \\sigma_i - z e_i, \\qquad \\mbox{for $1 \\leq i <n$.} \\label{eq:double}\n\\end{gather}\n\n\\begin{Remark} \\label{rem:BMW}\\samepage\\quad\n\\begin{enumerate}\\itemsep=0pt\n\\item Occasionally, we will multiply an element $\\alpha \\in \\operatorname{BMW}_i^\\mathfrak{Leg}$ by an element of $\\operatorname{BMW}_j^\\mathfrak{Leg}$ with $i<j$. Unless indicated otherwise, we do so by extending $i$-tangles to $j$-tangles by placing $j-i$ horizontal strands below $\\alpha$.\n\n\\item We do not give a complete set of generators and relations for $\\operatorname{BMW}_n^\\mathfrak{Leg}$, as the main role of $\\operatorname{BMW}_n^\\mathfrak{Leg}$ in this article is to provide a convenient setting for ruling polynomial calculations. We leave finding a presentation for $\\operatorname{BMW}_n^\\mathfrak{Leg}$ as an open problem.\n\\end{enumerate}\n\\end{Remark}\n\n\\subsection{Legendrian satellites and reduced ruling polynomials} \\label{sec:sat} We will be considering rulings of satellites. Given a connected Legendrian knot $K \\subset J^1\\mathbb {R}$ and a Legendrian $n$-tangle $L \\subset J^1[0,1]$, a~front diagrammatic description of the {\\it Legendrian satellite} $S(K,L) \\subset J^1\\mathbb {R}$ is the following: Form the $n$-copy of~$K$, i.e., take $n$ copies of $K$ each shifted up a~small amount in the $z$-direction, then insert a rescaled version of the $n$-tangle $L$ into the $n$-copy at a small rectangular neighborhood $J \\cong [0,1] \\times [-\\epsilon, \\epsilon]$ of part of a strand of $K$ that is oriented from left-to-right. We refer to $J$ as the {\\it $J^1[0,1]$-part} of the satellite. See Fig.~\\ref{fig:Sat}. It can be shown that the Legendrian isotopy type of~$S(K,L)$ depends only on~$K$ and the closure of $L$ in $J^1S^1$, cf.~\\cite{NgTraynor}. (See also Section~\\ref{sec:satellite} for additional discussion.)\n\n\\begin{figure} \\centering\n\t\t\\labellist\n\\small\n\\pinlabel $K$ [t] at 36 30\n\\pinlabel $\\beta$ [t] at 170 44\n\\pinlabel $S(K,\\beta)$ [t] at 352 0\n\\endlabellist\n\n \\includegraphics{images\/Sat}\n\t\t\n\\caption{The Legendrian satellite $S(K,\\beta)$ where $\\beta = \\sigma_1 \\sigma_2 \\subset J^1[0,1]$ is the positive permutation braid associated to the $3$-cycle $(1 \\, 2 \\,3)$. The $J^1[0,1]$-part of the satellite is indicated by the dotted rectangular box.} \\label{fig:Sat}\n\\end{figure}\n\nIn \\cite{NgR2012} a variant of the ruling polynomial was introduced for satellites using reduced normal rulings. Let $L \\subset J^1[0,1]$ be a Legendrian $n$-tangle. A normal ruling $\\rho$ of $S(K,L)$ is said to be {\\it reduced} if, outside of the $J^1[0,1]$-part of $S(K,L)$, parallel strands of $S(K,L)$ corresponding to a~single strand of~$K$ are not paired by $\\rho$. See Fig.~\\ref{fig:reduced}.\n\\begin{figure}[h] \\centering\n \\includegraphics{images\/reducedtrefoil.png}\n \\caption{A reduced normal ruling (left) and a non-reduced normal ruling (right) of $S(K,\\beta)$ where $K$ is a right-handed trefoil and $\\beta=1$.} \\label{fig:reduced}\n\\end{figure}\n We denote the set of reduced normal rulings of $S(K,L)$ by $\\widetilde{\\Gamma}(K,L)$ and define the {\\it reduced ruling polynomial} of $S(K,L)$ as $\\widetilde{R}_{S(K,L)}(z):=\\sum\\limits_{\\rho\\in\\widetilde{\\Gamma}(K,L)}z^{j(\\rho)}$.\n\n\\begin{Remark} \\label{rem:thin}If the reduced condition holds for a normal ruling $\\rho$ of $S(K,L)$ for the parallel strands of $S(K,L)$ corresponding to a single point $k_0 \\in K$ outside of the $J^1[0,1]$-part of $S(K,L)$, then $\\rho$ will be reduced. Indeed, the involution of parallel strands of $S(K,L)$ at $k_0 \\in K$ that arises from restriction the pairing of $\\rho$ (strands of $S(K,L)$ at $k_0$ that are paired with strands in a part of the satellite away from~$k_0$ are fixed points of the involution) is called the ``thin part'' of~$\\rho$ at~$k_0$ in~\\cite{NgR2012}, and Lemmas~3.3 and~3.4 from~\\cite{NgR2012} show that the thin part is independent of~$k_0$.\n\\end{Remark}\n\nWe are now prepared to make the following key definition.\n\n\\begin{Definition} \\label{def:nruling}For any (connected) Legendrian knot $K\\subset J^1\\mathbb {R}$ we define the {\\it ungraded $n$-colored ruling polynomial} as\n\\begin{gather*}\n R_{n,K}^1(s):=\\frac{1}{c_n}\\sum_{\\beta\\in S_n}s^{\\lambda(\\beta)}\\widetilde{R}_{S(K,\\beta)}(z),\n\\end{gather*}\nwhere the sum is over positive permutation braids, $z=s-s^{-1}$, $\\lambda(\\beta)$ is the length of $\\beta$, and $c_n=s^{n(n-1)\/2}\\prod\\limits_{i=1}^n\\frac{s^i-s^{-i}}{s-s^{-1}}$.\n \\end{Definition}\nNote that $R_{n,K}^1$ belongs to coefficient ring $\\mathcal{R}$ defined above.\n\n\\begin{Remark}It is proved in \\cite{NgR2012} that with $L$ fixed the reduced ruling polynomial $\\widetilde{R}_{S(K,L)}(z)$ is a Legendrian isotopy invariant of~$K$. Alternatively, the Legendrian isotopy invariance of $R_{n,K}^1$ also follows from Theorem~\\ref{thm:main1} below and invariance of the ordinary ruling polynomials.\n\\end{Remark}\n\n\\subsection[Inductive characterization of $R_{n,K}^1$]{Inductive characterization of $\\boldsymbol{R_{n,K}^1}$} \\label{sec:Ln}\n\nIt is convenient to extend the concept of satellite ruling polynomials slightly by defining $R^1_{S(K,\\eta)}$ for $\\eta \\in \\operatorname{BMW}_n^{\\mathfrak{Leg}}$ represented as an $\\mathcal{R}$-linear combination of $n$-tangles $\\sum\\limits_{i=1}^k r_i \\alpha_i$ to be\n\\begin{gather} \\label{eq:BMWsatellite}\nR^1_{S(K,\\eta)} = \\sum_{i=1}^kr_i R^1_{S(K,\\alpha_i)}.\n\\end{gather}\n(This is well-defined since the front diagram of each $\\alpha_i$ appears as a~subset of $\\pi_{xz}(S(K,\\alpha_i))$ and~$R^1$ satisfies the skein relations that define $\\operatorname{BMW}_n^\\mathfrak{Leg}$.) The goal in the remainder of this section is to give such a characterization of the ungraded $n$-colored ruling polynomials via appropriate elements $L_n \\in \\operatorname{BMW}_n^\\mathfrak{Leg}$.\n\nLet\n\\[\n\\gamma_n=1+\\sum_{j=1}^{n-1}s^j\\sigma_{n-1}\\sigma_{n-2}\\cdots \\sigma_{n-j} \\in \\operatorname{BMW}^\\mathfrak{Leg}_n,\n\\]\n where $\\sigma_i$ is as in Fig.~\\ref{fig:sigma} and consider the elements of $\\operatorname{BMW}^\\mathfrak{Leg}_n$\n defined inductively for $n\\geq1$ by $L_1 =1$ and\n \\[L_n=L_{n-1}\\beta_n, \\qquad n \\geq 2,\\]\nwhere\n\\[ \\beta_n:=\\left(1-z\\sum_{k=2}^n\\alpha_{k,n}\\right)\\gamma_n.\\]\nHere, $\\alpha_{k,n}\\in \\operatorname{BMW}^\\mathfrak{Leg}_n$ is defined inductively on the {\\it bottom} $k$ strands by $\\alpha_{2,n} = e_{n-1}$ and\n \\begin{gather} \\alpha_{k,n}=\\raisebox{-.25cm}{\\includegraphics{images\/alpha1}}-z\\cdot\\raisebox{-.25cm}{\\includegraphics{images\/alpha2}}-z\\cdot\\raisebox{-.25cm}{\\includegraphics{images\/alpha3}}+z^2\\cdot\\raisebox{-.25cm}{\\includegraphics{images\/alpha4}} \\label{alpha}\\end{gather}\n\t\twith the identity tangle appearing as the top $n-k$ strands of $\\alpha_{k,n}$. For example, using (\\ref{eq:fish2}) we have $L_2=\\mathbf{1}+s\\sigma_1-ze_1$.\n\t\tThe double subscript notation on $\\alpha_{k,n}$ is to emphasize that $\\alpha_{k,n}$\n\t\tinvolves the bottom $k$ strands, $n-k+1, n-k+2, \\ldots, n$, out of $n$ total strands.\n\t\t(By contrast, when multiplying $L_{n-1} \\beta_n$ an extra strand is placed at the bottom of $L_{n-1}$; see Remark~\\ref{rem:BMW}.)\n\nIn Section~\\ref{sec:Kauffman}, we will also make use of a non-inductive characterization of the~$\\alpha_{k,n}$. Given a~front diagram~$D$, let $\\operatorname{cr}(D)$ denote the set of crossings appearing in $D$. We define the {\\it resolution of $D$ with respect to $X\\subset \\operatorname{cr}(D)$} to be the tangle $r_X(D)$ obtained from resolving all crossings of~$X$ as depicted in Fig.~\\ref{fig:res}.\n\n \\begin{figure}[h] \\centering\n \\includegraphics[scale=.4]{images\/xres.png}\\raisebox{.9cm}{$\\;\\longrightarrow$} \\includegraphics[scale=.4]{images\/res.png}\n \\caption{The resolution of a crossing in $X\\subset \\operatorname{cr}(D)$.} \\label{fig:res}\n \\end{figure}\n \\begin{Lemma} For $1<k\\leq n$, \\[\\raisebox{.6cm}{$\\alpha_{k,n}=\\sum\\limits_{X\\subset \\operatorname{cr}(C_{k,n})} (-z)^{\\vert X \\vert} r_X(C_{k,n})$, \\quad where $C_{k,n}=$ }\\includegraphics[scale=.65]{images\/Ck.png}\\]\\label{lemma:alpha}\n \\end{Lemma}\n\t\t\n\\begin{proof}This is a straightforward induction on~$k$.\n\\end{proof}\n\n\\begin{Theorem}\\label{thm:main1}For any Legendrian $K\\subset J^1\\mathbb {R}$, $R_{n,K}^1=\\frac{1}{c_n}R^1_{S(K,L_n)}$.\n\\end{Theorem}\n\nThe proof will be given below after Lemma~\\ref{lemma:THMB}.\n\nGiven $K\\subset J^1\\mathbb {R}$, and $\\eta$ a Legendrian $n$-tangle, recall that the $J^1[0,1]$-part of $S(K,\\eta)$ refers to a rectangular region, $J\\cong [0,1] \\times [-\\epsilon,\\epsilon]$, where the front diagram of $\\eta$ appears. We will call a normal ruling~$\\rho$ of~$S(K,\\eta)$ {\\it $k$-reduced} if {\\it at the right boundary of~$J$} none of the top~$k$ parallel strands of $\\eta$ within $S(K,\\eta)$ are paired with one another. We denote by $\\widetilde{\\Gamma^k}(K,\\eta)$, the set of $k$-reduced normal rulings of~$S(K,\\eta)$. Similarly, given a~location $*$ within $J$ corresponding to an $x$-value in $J^1[0,1]$ without double points of $\\eta$ a~normal ruling $\\rho$ of $S(K,\\eta)$ is said to be {\\it $(k,m)$-paired at $*$} if $\\rho$ is $(m-1)$-reduced (at the right boundary of~$J$) and $\\rho$ pairs strand $k$ of~$\\eta$ with strand~$m$ of $\\eta$ at $*$ (where strands are numbered from top to bottom as they appear in~$\\eta$ at~$*$). The set of $(k,m)$-paired rulings of $S(K,\\eta)$ is denoted by $\\Gamma_{*}^{\\tau(k,m)}(K,\\eta)$. The {\\it $k$-reduced ruling polynomial} and {\\it $(k,m)$-paired ruling polynomial} (at $*$) are defined respectively by \\[ \\widetilde{R}^{(k)}_{(K,\\eta)}(z)=\\sum_{\\rho\\in\\widetilde{\\Gamma^k}(K,\\eta)}z^{j(\\rho)} \\qquad \\text{and} \\qquad R^{\\tau(k,m)}_{(*;K,\\eta)}(z)=\\sum_{\\rho\\in\\Gamma_{*}^{\\tau(k,m)}(K,\\eta)}z^{j(\\rho)}.\\]\nEither polynomial can be extended by linearity to allow $\\eta$ to be an $\\mathcal{R}$-linear combination of Legendrian $n$-tangles.\n\n\\begin{Remark}\\label{rem:reduced}If $\\eta$ is a $n$-stranded positive braid, then any $n$-reduced normal ruling is reduced. This is because if parallel strands of the satellite $S(K,\\eta)$ corresponding to a single strand of $K$ are not paired at the right side of the $J^1[0,1]$-part of the satellite, then such parallel strands cannot be paired anywhere outside of the $J^1[0,1]$-part. See Remark~\\ref{rem:thin}.\n\\end{Remark}\n\n\\begin{Lemma}\\label{lemma:THMB} For any $1<k \\leq m \\leq n$, let $\\beta \\in S_{m-1}$ be a positive permutation braid extended to an Legendrian $n$-tangle by placing the $n-m+1$ stranded identity tangle below $\\beta$, and let $\\nu$ be a linear combination of Legendrian $n$-tangles. Then, we have\n\\[z\\widetilde{R}^{(m-1)}_{(K,\\beta\\alpha_{k,m}\\nu)}(z)=R^{\\tau(m-k+1,m)}_{(*;K,\\beta\\nu)}(z),\\]\nwhere $*$ is the location between $\\beta$ and $\\nu$ $($immediately to the left diagrammatically of~$\\beta)$.\n\\end{Lemma}\n\n\\begin{proof}The proof is by induction on $k$ with $m$ and $n$ fixed. The base case is clear since $\\alpha_{2,m}=e_{m-1}$.\n\nSuppose the statement is true for~$k$. For clarity of this proof, we emphasize the location of our paired rulings within our notation, and abbreviate $\\alpha_{k,m}$ as~$\\alpha_k$. Using~(\\ref{alpha}) and the inductive hypothesis, we compute\n\\begin{gather*}\n z\\widetilde{R}^{(m-1)}_{(K,\\beta\\alpha_{k+1}\\nu)}(z)\\\\\n =z\\widetilde{R}^{(m-1)}_{(K,\\beta\\sigma_{m-k}\\alpha_{k}\\sigma_{m-k}\\nu)}(z) -z^2\\widetilde{R}^{(m-1)}_{(K,\\beta\\sigma_{m-k}\\alpha_{k}\\nu)}(z) -z^2\\widetilde{R}^{(m-1)}_{(K,\\beta\\alpha_{k}\\sigma_{m-k}\\nu)}(z)+z^3\\widetilde{R}^{(m-1)}_{(K,\\beta\\alpha_{k}\\nu)}(z)\\\\\n =R^{\\tau(m-k+1,m)}_{(*;K,\\beta\\sigma_{m-k}*\\sigma_{m-k}\\nu)}(z) -zR^{\\tau(m-k+1,m)}_{(*;K,\\beta\\sigma_{m-k}*\\nu)}(z) -zR^{\\tau(m-k+1,m)}_{(*;K,\\beta*\\sigma_{m-k}\\nu)}(z)+z^2R^{\\tau(m-k+1,m)}_{(*;K,\\beta*\\nu)}(z) \\hypertarget{star}{}\\\\\n \\overset{\\star}{=}R^{\\tau(m-k,m)}_{(*;K,\\beta*\\nu)}(z).\n\\end{gather*}\nWe now establish equality $\\star$. For the diagrams $D$ that follow, when considering $(k,m)$-paired rulings, we indicate the location $*$ with a green vertical segment having endpoints on the paired strands~$k$ and~$m$. When we encode such information in $D$ we suppress the paired notation $R^{\\tau(k,m)}_{(*;K,D)}$ as $R^\\tau_{K,D}$ (and similarly for sets of rulings: $\\Gamma_{*}^{\\tau(k,m)}(K,D)$ becomes $\\Gamma^{\\tau}(K,D)$). For instance,\n\\[\nR^{\\tau(m-k,m)}_{(*;K,\\beta*\\nu)}(z) =\n R^\\tau_{\\left(K,\\; \\raisebox{-.25cm}{\\includegraphics[scale=.6]{images\/paired1.png}}\\right)}(z),\n\\]\nwhere the diagram only pictures strands with numberings in the range $m-k$ to $m$ at $*$. Observe the bijection\n\\begin{gather}\\label{eq:color}\n \\Gamma^\\tau\\left(K, \\raisebox{-.25cm}{\\includegraphics[scale=.6]{images\/paired1.png}}\\right)=\\left\\{\\rho \\in \\Gamma^\\tau\\left(K, \\raisebox{-.25cm}{\\includegraphics[scale=.6]{images\/paired2.png}}\\right) \\,\\bigg|\\, \\mbox{$r_1$ and $r_2$ are not switches of $\\rho$}\\right\\},\n\\end{gather}\n where $r_1$ and $r_2$ denote the left and right crossings of the diagram on the right. In addition, we have the following (separate) bijections:\n\\begin{gather}\\label{eq:bijectionsec2}\n\\begin{array}{@{}lrcl}\n &\\rho&\\mapsto &\\Phi(\\rho),\\\\\n \\Big\\{ \\rho\\in\\Gamma^\\tau\\left(K,\\raisebox{-.25cm}{\\includegraphics[scale=.65]{images\/paired3.png}}\\right) \\,\\big|\\,& \\mbox{ $r_1$ is a switch} \\Big\\} &{\\longleftrightarrow}\n & \\Gamma^\\tau\\left(K,\\raisebox{-.25cm}{\\includegraphics[scale=.65]{images\/paired5.png}}\\right), \\vspace{1mm}\\\\\n \\Big\\{\\rho\\in\\Gamma^\\tau\\left(K,\\raisebox{-.25cm}{\\includegraphics[scale=.65]{images\/paired3.png}}\\right) \\, \\big|\\, & \\mbox{$r_2$ is a switch}\\Big\\} &\n \\longleftrightarrow &\n \\Gamma^\\tau\\left(K,\\raisebox{-.25cm}{\\includegraphics[scale=.65]{images\/paired4.png}}\\right), \\vspace{1mm}\\\\\n \\Big\\{\\rho\\in\\Gamma^\\tau\\left(K,\\raisebox{-.25cm}{\\includegraphics[scale=.65]{images\/paired3.png}}\\right) \\,\\big|\\, & \\mbox{$r_1$ and $r_2$ are switches }\\Big\\} & \\longleftrightarrow & \\Gamma^\\tau\\left(K,\\raisebox{-.25cm}{\\includegraphics[scale=.65]{images\/paired6.png}}\\right) ,\n \\end{array}\n\\end{gather}\nwhere $\\Phi(\\rho)$ is the unique ruling that agrees with $\\rho$ outside of a neighborhood of the resolved crossing(s). (Any ruling of a diagram $D$ with switches at a set of crossings~$X$ gives rise to a~ruling of $r_{X}(D)$.) Each $\\Phi$ is clearly injective. To verify surjectivity, observe that any ruling $\\rho'$ in one of the sets on the right side of~(\\ref{eq:bijectionsec2}) indeed comes from a ruling on the left side by replacing the resolved crossing(s) with switches. Here, it is crucial to note that the normality condition is automatically satisfied. If the normality condition was not satisfied, then $\\rho'$ must pair strand $m-k$ with a strand numbered in the range $m-k+2$ to $m-1$ at the location~$*$ indicated by the vertical segment. Since $\\beta \\in S_{m-1}$, it follows that at least~2 of the top $m-1$ strands are paired at the right side of the $J^1[0,1]$-part of the satellite, contradicting that $\\rho'$ is $(m-1)$-reduced. Finally, the inclusion-exclusion principle and the above bijections~\\eqref{eq:color} and~\\eqref{eq:bijectionsec2} imply\n\\begin{gather*}\n R^\\tau_{\\left(K, \\;\n \\raisebox{-.25cm}{\\includegraphics[scale=.54]{images\/paired1.png}}\\right)}(z)\\\\\n =R^\\tau_{\\left(K,\\; \\raisebox{-.25cm}{\\includegraphics[scale=.54]{images\/paired3.png}}\\right)}(z)\n -zR^\\tau_{\\left(K, \\; \\raisebox{-.25cm}{\\includegraphics[scale=.54]{images\/paired4.png}}\\right)}(z)\n -zR^\\tau_{\\left(K,\\; \\raisebox{-.25cm}{\\includegraphics[scale=.54]{images\/paired5.png}}\\right)}(z)\n +z^2R^\\tau_{\\left(K,\\; \\raisebox{-.25cm}{\\includegraphics[scale=.54]{images\/paired6.png}}\\right)}(z).\n\\end{gather*}\n(The factors of $z$ and $z^2$ appear since the first two bijections in (\\ref{eq:bijectionsec2}) decrease the number of switches by $1$, while the last bijection decreases the number by 2.) This establishes \\hyperlink{star}{$\\star$}, so we are done.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:main1}] With $n\\geq 1$ fixed we prove the following statement by induction on~$m$. The theorem follows from the special case where $m=n$ and $\\mu = 1$ (in view of Remark~\\ref{rem:reduced}).\n\n{\\bf Inductive statement:} For any (linear combination of) Legendrian $n$-tangle(s) $\\mu \\subset J^1[0,1]$\n and any $1 \\leq m \\leq n$,\n\\[\nR_{S(K,L_{m}\\mu)}(z)=\\sum_{\\beta\\in S_{m}}s^{\\lambda(\\beta)}\\widetilde{R}^{(m)}_{(K,\\beta\\mu)}(z)\n\\]\n(with the products $L_m\\mu$ and $\\beta\\mu$ formed as in Remark~\\ref{rem:BMW}).\n\nThe base case of $m=1$ is immediate since any normal ruling is $1$-reduced. For the inductive step, with $m\\geq 2$ and the statement assumed for $m-1$ we compute\n\\begin{align*}\n \\notag R_{S(K,L_m\\mu)}(z)&=R_{S(K,L_{m-1}(1-z\\sum\\limits_{k=2}^m\\alpha_{k,m})\\gamma_m\\mu)}(z)\\\\\n \\notag &=R_{S(K,L_{m-1}\\gamma_m\\mu)}-z\\sum_{k=2}^m R_{S(K,L_{m-1}\\alpha_{k,m}\\gamma_m\\mu)}(z)\\\\\n \\notag &=\\sum_{\\beta\\in S_{m-1}}s^{\\lambda(\\beta)}\\left(\\widetilde{R}^{(m-1)}_{(K,\\beta\\gamma_m\\mu)}(z) -z\\sum_{k=2}^m\\widetilde{R}^{(m-1)}_{(K,\\beta\\alpha_{k,m}\\gamma_m\\mu)}(z)\\right)&&(\\text{inductive hyp})\\\\\n &=\\sum_{\\beta\\in S_{m-1}}s^{\\lambda(\\beta)}\\left(\\widetilde{R}^{(m-1)}_{(K,\\beta\\gamma_m\\mu)}(z) -\\sum_{k=2}^mR^{\\tau(m-k+1,m)}_{(*;K,\\beta*\\gamma_m\\mu)}(z)\\right). &&(\\text{Lemma~\\ref{lemma:THMB}})\n\t\\end{align*}\nNote that since there are no crossings in $\\beta$ involving the $m$-th strand, an $(m-1)$-reduced ruling is $m$-reduced unless at $*$ strand $m$ is paired\n with one of the strands numbered $1, \\ldots, m-1$ in the $J^1[0,1]$-part of the satellite, i.e., a strand numbered $m-k+1$ with $2 \\leq k \\leq m$. Thus, continuing the above computation, we have\n\\begin{equation*}\n \\sum_{\\beta\\in S_{m-1}}s^{\\lambda(\\beta)}\\left(\\widetilde{R}^{(m-1)}_{(K,\\beta\\gamma_m\\mu)}(z)-\\sum_{k=2}^mR^{\\tau(m-k+1,m)}_{(*;K,\\beta*\\gamma_m\\mu)}(z)\\right)=\\sum_{\\beta\\in S_{m-1}}s^{\\lambda(\\beta)}\\widetilde{R}^{(m)}_{(K,\\beta\\gamma_m\\mu)}.\n\\end{equation*}\nNext note that any $\\beta\\in S_m$ can be written uniquely as $\\beta=\\beta'\\sigma_{m-1}\\cdots \\sigma_{m-j}$ with $\\beta'\\in S_{m-1}$ and $0 \\leq j \\leq m-1$. (Here, $m-j = \\beta^{-1}(m)$.) Moreover, such a factorization realizes $\\beta$ as a~positive permutation braid, so that $\\lambda(\\beta)=\\lambda(\\beta')+j$. Using the definition of~$\\gamma_m$, we see that\n\\[\\sum_{\\beta\\in S_{m-1}}s^{\\lambda(\\beta)}\\widetilde{R}^{(m)}_{(K,\\beta\\gamma_m\\mu)}=\\sum_{\\beta\\in S_m}s^{\\lambda(\\beta)}\\widetilde{R}^{(m)}_{(K,\\beta\\mu)}.\\] This completes the inductive step.\n\\end{proof}\n\n\\section{Relation to the Kauffman polynomial} \\label{sec:Kauffman}\n We begin this section with a review of the definition of the $n$-colored Kauffman polynomial~$F_{n,K}$ via satelliting with the symmetrizer $\\mathcal{Y}_n \\in \\operatorname{BMW}_n$ in the BMW algebra. Then, using an inductive characterization of $\\mathcal{Y}_n$ from~\\cite{Heck}, combined with the earlier characterization of $R^1_{n,K}$ we show how to recover $R^1_{n,K}$ as a specialization $F_{n,K}\\big\\vert_{a^{-1}=0}$. This is accomplished by relating the Legendrian~BMW element $L_n \\in \\operatorname{BMW}^\\mathfrak{Leg}_{n}$ from Theorem~\\ref{thm:main1} to~$\\mathcal{Y}_n$ in a certain sub-quotient of~$\\operatorname{BMW}_n$ defined in terms of Legendrian tangles. Along the way we pause to make a conjecture relating ungraded ruling polynomial skein modules with Kauffman skein modules in general.\n\n\\subsection{The Kauffman polynomial and the BMW algebra} Recall that the (framed) Kauffman polynomial (Dubrovnik version) is a regular isotopy invariant that assigns a Laurent polynomial to framed links $L\\subset\\mathbb {R}^3$ characterized by the skein relations (shown with blackboard framing)\n\\begin{gather*}\n \\raisebox{-.5cm}{\\includegraphics[scale=.6]{images\/Kauf1.png}} -\\raisebox{-.5cm}{\\includegraphics[scale=.6]{images\/Kauf2.png}}=z\\left(\\; \\raisebox{-.5cm}{\\includegraphics[scale=.6]{images\/Kauf4.png}}- \\raisebox{-.5cm}{\\includegraphics[scale=.6]{images\/Kauf3.png}}\\;\\right), \\tag{F1} \\\\\n \\raisebox{-.5cm}{\\includegraphics[scale=.6]{images\/Kauf5.png}}=a^{-1}\\raisebox{-.5cm}{\\includegraphics[scale=.6]{images\/Kauf.png}} \\hspace{1cm} \\raisebox{-.5cm}{\\includegraphics[scale=.6]{images\/Kauf6.png}} =a\\raisebox{-.5cm}{\\includegraphics[scale=.6]{images\/Kauf.png}}\\,, \\tag{F2} \\\\\n \\raisebox{-.5cm}{\\includegraphics[scale=.6]{images\/Kauf7.png}} \\;\\sqcup L =\\left(\\frac{a-a^{-1}}{z}+1\\right)L, \\tag{F3}\n\\end{gather*}\nwhere $F_L:=1$ if $L$ is the empty link.\\footnote{This is equivalent to choosing the normalization $F_U=\\frac{a-a^{-1}}{z}+1$, where $U$ is the $0$-framed unknot.}\n\nThe ordinary BMW algebra, $\\operatorname{BMW}_n$ (see \\cite{BW, Mu}) can be defined as the Kauffman skein module for framed tangles in $[0,1]\\times \\mathbb {R}^2 \\cong J^1[0,1]$ with $n$ boundary points on each component. In more detail, let $\\mathfrak{Fr}_n$ denote the set of isotopy classes of {\\it framed $n$-tangles} where we require that near $\\partial J^1[0,1]$ framed $n$-tangles agree with the horizontal lines $z=i$, $y =0$, $1\\leq i \\leq n$ with framing vector $\\frac{\\partial}{\\partial y}$; this requirement is maintained during isotopies. Working over the field of rational functions $\\mathbb{F} = \\mathbb {Z}(a,s)$ with $z=s-s^{-1}$, $\\operatorname{BMW}_n= \\mathbb{F}\\mathfrak{Fr}_n\/\\mathcal{T}$ where $\\mathcal{T}$ is the $\\mathbb{F}$-submodule generated by the Kauffman polynomial skein relations. Multiplication in $\\operatorname{BMW}_n$ is as in the Legendrian case, i.e., the product $\\alpha \\cdot \\beta$ is $\\beta$ stacked to the left of~$\\alpha$.\n\nFor viewing framed $n$-tangles diagrammatically, we continue to use the $xz$-plane for projections, and we require the framing to be globally given by $\\frac{\\partial}{\\partial y}$, that is, perpendicular to the projection plane and away from the viewer. This framing becomes isotopic to the blackboard framing when $n$-tangles are closed to become links in $J^1S^1$ (by identifying the left and right boundaries of $J^1[0,1]$). To view a Legendrian $n$-tangle $L \\subset J^1[0,1]$ as a framed $n$-tangle $\\operatorname{fr}(L) \\in \\mathfrak{Fr}_n$ we form a diagram for $\\operatorname{fr}(L)$ from the front diagram of $L$ by smoothing left cusps and adding a small loop with a negative crossing at right cusps. See Fig.~\\ref{fig:framed}. This has the property that the closure of $\\operatorname{fr}(L)$ in $J^1S^1$ is framed isotopic to the Legendrian closure of~$L$ with its contact framing.\n\n\\begin{figure}\n \\centering\n\t\t\\labellist\n\\small\n\\pinlabel $L$ [t] at 64 0\n\\pinlabel $\\operatorname{fr}(L)$ [t] at 250 0\n\\endlabellist\n\n \\includegraphics{images\/FramedL}\n\t\t\n\t\t\\quad\n\t\t\n \\caption{The framed $n$-tangle $\\operatorname{fr}(L)$ associated to a Legendrian $n$-tangle $L \\subset J^1[0,1]$.}\n \\label{fig:framed}\n\\end{figure}\n\nThe topological satellite operation produces from a framed knot $K \\subset J^1\\mathbb {R}$ and a framed $n$-tangle $L \\subset J^1[0,1]$ a satellite link $S(K,L) \\subset J^1\\mathbb {R}$. A diagram for $S(K,L)$ arises from taking a blackboard framed diagram for $K$ and placing a blackboard framed diagram for the closure of $L$ in an (immersed) annular neighborhood of $K$ in the projection plane. When $K$ and $L$ are Legendrian, the framed knot type of $S(K,L)$ agrees with that of the previously defined Legendrian satellite (with contact framing).\n For $K \\subset J^1\\mathbb {R}$ (a framed link) and $\\eta \\in \\operatorname{BMW}_n$, the Kauffman polynomial of the satellite $F_{S(K,\\eta)}$ is defined by linearity as in (\\ref{eq:BMWsatellite}).\n\n\\subsection[Symmetrizer in $\\operatorname{BMW}_n$ and the $n$-colored Kauffman polynomial]{Symmetrizer in $\\boldsymbol{\\operatorname{BMW}_n}$ and the $\\boldsymbol{n}$-colored Kauffman polynomial}\n\n The $n$-colored Kauffman polynomial is defined by satelliting with the {\\it symmetrizer} $\\mathcal{Y}_n \\in \\operatorname{BMW}_n$. (More general, colored Kauffman polynomials, where the coloring is by a partition $\\lambda$, can be defined using other idempotents in $\\operatorname{BMW}_n$ and are related to quantum invariants of type~$B$,~$C$, and~$D$; see, e.g.,~\\cite{BlBe}.) The symmetrizer $\\mathcal{Y}_n$ is characterized as the unique non-zero element of $\\operatorname{BMW}_n$ that is idempotent, i.e., has $\\mathcal{Y}_n^2 = \\mathcal{Y}_n$, and satisfies\n\\begin{gather}\n \\mathcal{Y}_n\\sigma_i =s\\mathcal{Y}_n, \\qquad \\text{for }1\\leq i <n. \\quad \\quad \\quad (\\mbox{crossing absorbing property})\\label{eats}\n\\end{gather}\nThe following inductive formula for $\\mathcal{Y}_n$ is due to Heckenberger and Sch{\\\"u}ler in \\cite{Heck}. Consider $Y_n \\in \\operatorname{BMW}_n$ defined by\n \\begin{gather}\nY_1 = 1, \\notag\\\\\n\\label{inductiveform}\n Y_n=Y_{n-1}\\gamma_n+Y_{n-1}\\frac{z}{1-s^{2n-3}a} \\left(\\sum_{i=1}^{n-1}as^{2n-2i-1}\\left(\\sum_{j=0}^{i-1}s^{j}D_{i-j,i+1}\\right)\\right),\n\\end{gather}\nwhere\n \\[D_{i,j}:=\\begin{cases} {\\includegraphics[scale=.8]{images\/DIJ}}\\,, &\\raisebox{.7cm}{ $i<j$,}\\\\\n0,&\\;i\\geq j.\n\\end{cases}\n\\] The symmetrizer is then obtained as the normalization\n$\\mathcal{Y}_n=\\frac{Y_n}{c_n}$ (with $c_n$ as in Definition~\\ref{def:nruling}). For example, $\\mathcal{Y}_2=\\frac{1}{c_2}\\big({\\rm id}+s\\sigma_1+\\frac{sza}{1-sa}e_1\\big)$. Note that $Y_n$ is quasi-idempotent (i.e., $Y_n^2 = c_n Y_n$) and also has the crossing absorbing property~(\\ref{eats}).\n\n\\begin{Remark} \\looseness=-1 Our notations $\\sigma_i$, $s$, $a$, $e_i$ translate into the notations from \\cite{Heck} as $g_i$, $q$, $r$, $r^{-1} e_i$. (The~$r^{-1}$ is because our $e_i$ is a Legendrian hook, so that $\\operatorname{fr}(e_i)$ has an extra loop that produces the $r^{-1}$ factor.) Our $D_{i,j}$ is such that $a \\cdot D_{i-j, i+1}$ is the $j$-th term in $d^+_{n,i}$ from~\\cite{Heck}, the factor of $a$ again arising from our use of Legendrian tangles in representing the quasi-idempotent~$Y_n$. Note that \\cite[Proposition~1]{Heck} gives an inductive formula for $\\mathcal{Y}_n$ (notated there as~$S_n$) rather than~$Y_n$, and the denominator $q^n [\\![ n ]\\!]$ that appears there is accounted for by our factor of $\\frac{1}{c_n}$ relating~$Y_n$ and~$\\mathcal{Y}_n$.\n\\end{Remark}\n\n\\begin{Definition} \\label{def:Kauffman}\nGiven a framed knot $K\\subset\\mathbb {R}^3$, we define the {\\it $n$-colored Kauffman polynomial} of~$K$ by\n\\[F_{n,K}(a,s)=F_{S(K,\\mathcal{Y}_n)}(a,z)\\vert_{z=s-s^{-1}},\\]\n where $\\mathcal{Y}_n$ is the symmetrizer in $\\operatorname{BMW}_n$.\n\\end{Definition}\n\n\\subsection{Ruling polynomials via specializations of the Kauffman polynomial}\nFor a Legendrian link in $L \\subset J^1\\mathbb {R}$ the Thurston--Bennequin number satisfies the inequality $\\operatorname{tb}(L) \\leq -\\deg_a \\widehat{F}_L(a,z)$, where $\\widehat{F}_L = a^{-w(L)} F_L$ is the framing independent version of the Kauffman polynomial obtained by normalizing $F_L$ using the writhe of a diagram for~$L$. (See~\\cite{FT, Ng3, T}.) This is equivalent to the inequality\n\\begin{gather} \\label{eq:Estimate}\n\\deg_a F_L(a,z) \\leq 0.\n\\end{gather}\n Thus, when $L$ is Legendrian $F_L \\in \\mathbb {Z}\\big[a^{\\pm 1}, z^{\\pm}\\big]$ does not contain positive powers of~$a$, and a~specialization $F_L(a,z)|_{a^{-1}=0}$ arises from simply setting $a^{-1}=0$.\n\n\\begin{Theorem}[\\cite{R1}]\\label{thm:rulings and the Kauffman poly}\nFor any Legendrian link $L\\subset J^1\\mathbb {R}$, the ungraded ruling polynomial is the specialization \\[R^1_L(z)=F_L(a,z)|_{a^{-1}=0}.\\]\n\\end{Theorem}\nWe will want to perform a similar specialization on elements of $\\operatorname{BMW}_n$ whose coefficients in $\\mathbb{F} = \\mathbb {Z}(a,s)$ may not be Laurent polynomials in $a$.\n To do so, note that the notion of degree in $a$ extends to arbitrary non-zero rational functions via\n $\\deg_a\\colon \\mathbb {Z}(a,s) \\setminus \\{0\\} \\rightarrow \\mathbb {Z}$,\n\\[\n \\deg_a\\left(\\frac{f}{g}\\right) = \\deg_a f - \\deg_a g, \\qquad \\mbox{for $f,g \\in \\mathbb {Z}[a,s]$}.\n\\]\nIn addition, we use the convention that $\\deg_a0:=-\\infty$. Moreover, on the subring $\\mathbb{F}^- := \\{F\\in \\mathbb{F} \\,|  \\allowbreak \\deg_a F\\leq 0 \\}$ we can define a specialization $\\vert_{a^{-1}=0}\\colon \\mathbb{F}^-\\rightarrow \\mathbb{Z}(s)$, by\n\\[\n\\frac{f}{g}\\bigg\\vert_{a^{-1}=0}:=\\begin{cases}0, & \\deg_a(f\/g)<0,\\\\\n\\dfrac{c_a(f)}{c_a(g)},&\\deg_a(f\/g)= 0,\\end{cases}\n\\]\nwhere $c_a(f) \\in \\mathbb {Z}[s]$ denotes the leading coefficient in $a$ of $f$.\n\\begin{Proposition} The specialization $\\vert_{a^{-1}=0}\\colon \\mathbb{F}^-\\rightarrow \\mathbb{Z}(s)$ is a well-defined, unital ring homomorphism.\n \\end{Proposition}\n\\begin{proof}Suppose $\\frac{f_1}{g_1}=\\frac{f_2}{g_2}$ in $\\mathbb {F}^-$. Then $f_1g_2=g_1f_2$ and so $c_a(f_1)c_a(g_2)=c_a(g_1)c_a(f_2)$, since $c_a(fg)=c_a(f)c_a(g)$ is true for polynomials $f$ and $g$. It follows that $\\frac{f_1}{g_1}\\big\\vert_{a^{-1}=0}=\\frac{f_2}{g_2}\\big\\vert_{a^{-1}=0}$ and so the specialization $\\big\\vert_{a^{-1}=0}$ is well-defined. Note that $1\\vert_{a^{-1}=0}=1$ follows from the definition. Now suppose $\\frac{f_1}{g_1},\\frac{f_2}{g_2}\\in\\mathbb {F}^-$. If $\\deg_a({f_1}\/{g_1})=\\deg_a({f_2}\/{g_2})=0$, then $\\deg_a(f_1g_2)=\\deg_a(g_1f_2)=\\deg_a(g_1g_2)$. In which case, \\begin{align*}\n\\left(\\frac{f_1}{g_2}+\\frac{f_2}{g_2}\\right)\\bigg\\vert_{a^{-1}=0} & =\\frac{f_1g_2+g_1f_2}{g_1g_2}\\bigg\\vert_{a^{-1}=0}\\\\\n& =\\frac{c_a(f_1)c_a(g_2)+c_a(g_1)c_a(f_2)}{c_a(g_1)c_a(g_2)} =\\frac{f_1}{g_1}\\bigg\\vert_{a^{-1}=0}+\\frac{f_2}{g_2}\\bigg\\vert_{a^{-1}=0}.\n\\end{align*} If $\\deg_a({f_1}\/{g_1}),  \\deg_a({f_2}\/{g_2})<0$, then\n\\[\n\\deg_a(f_1g_2+g_1f_2)\\leq \\max\\{\\deg_a(f_1g_2),\\deg_a(g_1f_2)\\}<\\deg_a(g_1g_2)\n\\] and so $\\big(\\frac{f_1}{g_1}+\\frac{f_2}{g_2}\\big)\\big\\vert_{a^{-1}=0}= 0 = \\frac{f_1}{g_1}\\big\\vert_{a^{-1}=0}+\\frac{f_2}{g_2}\\big\\vert_{a^{-1}=0}$ holds by definition. Lastly, if $\\deg_a(f_1\/g_1)<\\deg_a(f_2\/g_2)=0$, then $\\deg_a(f_1g_2+g_1f_2)=\\deg_a(g_1f_2)=\\deg(g_1g_2)$ and \\[\\left(\\frac{f_1}{g_2}+\\frac{f_2}{g_2}\\right)\\bigg\\vert_{a^{-1}=0}=\\frac{f_1g_2+g_1f_2}{g_1g_2}\\bigg\\vert_{a^{-1}=0}=\\frac{c_a(g_1)c_a(f_2)}{c_a(g_1)c_a(g_2)}=\\frac{f_1}{g_1}\\bigg\\vert_{a^{-1}=0}+\\frac{f_2}{g_2}\\bigg\\vert_{a^{-1}=0}.\\] It remains to show $\\vert_{a^{-1}=0}$ preserves multiplication. In the case where\n $\\deg_a(f_1\/g_1)<0$, it follows that $\\deg_a(f_1f_2)<\\deg_a(g_1g_2)$ and so\n \\[\\left(\\frac{f_1f_2}{g_1g_2}\\right)\\bigg\\vert_{a^{-1}=0}=0 =\\frac{f_1}{g_1}\\bigg\\vert_{a^{-1}=0}\\cdot\\frac{f_2}{g_2}\\bigg\\vert_{a^{-1}=0}.\\]\n The remaining case where $\\deg_a(f_1\/g_1)=\\deg_a(f_1\/g_1)=0$ immediately follows from the fact that $c_a$ preserves multiplication for then\n\\begin{gather*}\n\\left(\\frac{f_1f_2}{g_1g_2}\\right)\\bigg\\vert_{a^{-1}=0}=\\frac{c_a(f_1f_2)}{c_a(g_1g_2)} =\\frac{c_a(f_1)c_a(f_2)}{c_a(g_1)c_a(g_2)} =\\frac{f_1}{g_1}\\bigg\\vert_{a^{-1}=0}\\cdot\\frac{f_2}{g_2}\\bigg\\vert_{a^{-1}=0}.\\tag*{\\qed}\n\\end{gather*}\\renewcommand{\\qed}{}\n\\end{proof}\n\nIn the following we will make a connection between the Legendrian BMW algebra from Section~\\ref{sec:ncolored} and $\\operatorname{BMW}_n$. Towards this end, define\n\\[\n\\operatorname{BMW}^-_n= \\operatorname{Span}_{\\mathbb{F}^-}\\mathfrak{Leg}_n \\subset \\operatorname{BMW}_n\n\\]\nto be the $\\mathbb{F}^-$-submodule generated by (framed isotopy classes of) Legendrian $n$-tangles. Next, let $\\mathfrak{m} \\subset \\mathbb{F}^-$ denote the maximal ideal $\\mathfrak{m} = \\{F \\in \\mathbb{F}^-\\,|\\, \\deg_a F < 0\\}$, that is the kernel of $\\vert_{a^{-1}=0}$, and consider the quotient $\\mathbb{F}^-$-module\n\\[\n\\operatorname{BMW}^\\infty_n = \\operatorname{BMW}^-_n\/\\mathfrak{m} \\cdot \\operatorname{BMW}^-_n,\n\\]\nwith the projection map notated as\n\\[\n\\operatorname{BMW}^-_n \\rightarrow \\operatorname{BMW}^\\infty_n, \\qquad y \\mapsto y\\vert_{a^{-1}=0}.\n\\]\n\nNote that the estimate~(\\ref{eq:Estimate}) shows that for any fixed Legendrian knot $K \\subset J^1\\mathbb {R}$ the $\\mathbb{F}$-module homomorphism $\\operatorname{BMW}_n \\rightarrow \\mathbb{F}$, $L \\mapsto F_{S(K,L)}(a,s)$ maps $\\operatorname{BMW}_n^-$ to $\\mathbb{F}^-$. Moreover, the specialization $\\operatorname{BMW}^-_n \\rightarrow \\mathbb{Z}(s)$, $L \\mapsto F_{S(K,L)}(a,s)\\big\\vert_{a^{-1}=0}$ induces a well defined map \\mbox{$\\operatorname{BMW}^\\infty_n \\rightarrow \\mathbb{Z}(s)$}.\n\\begin{Proposition} \\label{prop:algebra}\nThere is an $\\mathcal{R}$-algebra homomorphism $\\varphi\\colon \\operatorname{BMW}_n^{\\mathfrak{Leg}} \\rightarrow \\operatorname{BMW}_{n}^\\infty$ induced by the map $\\mathfrak{Leg}_n \\rightarrow \\mathfrak{Fr}_n$. Moreover, we have a commutative diagram\n\\[\n\\xymatrix{ \\operatorname{BMW}^-_n \\ar[rr]^{F_{S(K,\\cdot)}} \\ar[d]_{|_{a^{-1}=0}} & & \\mathbb{F}^- \\ar[rr]^{|_{a^{-1}=0}} & & \\mathbb{Z}(s). \\\\ \\operatorname{BMW}^\\infty_n \\ar[rrrru] & & & & \\\\ \\operatorname{BMW}^\\mathfrak{Leg}_n \\ar[u]^{\\varphi} \\ar[rrrruu]_{R^1_{S(K,\\cdot)}} & & & &}\n\\]\n\\end{Proposition}\n\\begin{proof} To see that $\\varphi$ is well-defined, note that the ruling polynomial relation (R1) holds in $\\operatorname{BMW}_n$ since it is implied by the Kauffman relation (F1). Moreover, using the (F2), we see that when $L \\subset J^1[0,1]$ has a zig-zag, and $L' \\subset J^1[0,1]$ is the Legendrian obtained by removing the zig-zag from $L$, we have $L = a^{-1}L'$ in $\\operatorname{BMW}^-$, and this implies that $[L]=0$ in $\\operatorname{BMW}^\\infty$ as required by (R2). Finally, to verify (R3) when $K \\in \\mathfrak{Leg}_n$ we can compute in $\\operatorname{BMW}^\\infty_n$\n\\begin{align*}\nK \\sqcup \\raisebox{-.3cm}{\\includegraphics[scale=.5]{images\/unknot}} & = a^{-1} K\\sqcup \\raisebox{-.3cm}{\\includegraphics[scale=.4]{images\/Kauf7}} = a^{-1}\\left(\\frac{a-a^{-1}}{z} +1\\right) K \\\\\n& = z^{-1}K + \\big(a^{-1}-a^{-2}\/z\\big) K = z^{-1}K,\n\\end{align*}\nsince $\\big(a^{-1}-a^{-2}\/z\\big) \\in \\mathfrak{m}$.\n\nThe upper triangle of the diagram is commutative by definitions, and the commutativity of the lower triangle follows from Theorem~\\ref{thm:rulings and the Kauffman poly}.\n\\end{proof}\n\n\\begin{Conjecture}When $\\operatorname{BMW}^\\mathfrak{Leg}_n$ is defined over $\\mathbb{Z}(s)$, the map $\\varphi$ is an algebra isomorphism.\n\\end{Conjecture}\n\n\\begin{Remark} A similar conjecture can be made involving the (suitably defined) $1$-graded ruling polynomial skein module and the Kauffman skein module of any contact $3$-manifold, $M$.\n \\end{Remark}\n\nRecall the element $L_n \\in \\operatorname{BMW}^\\mathfrak{Leg}_n$ from Section~\\ref{sec:Ln}.\n\\begin{Proposition} \\label{prop:BMW}\nFor any $n\\geq 1$, we have\n\\begin{enumerate}\\itemsep=0pt\n\\item[$1)$] $\\mathcal{Y}_n \\in \\operatorname{BMW}^-_n$, and\n\\item[$2)$] $\\varphi\\big(\\frac{1}{c_n} L_n\\big) = \\mathcal{Y}_n\\big\\vert_{a^{-1}=0}$ holds in $\\operatorname{BMW}^\\infty_n$.\n\\end{enumerate}\n\\end{Proposition}\nWe prove (1) now; the proof of~(2) is deferred until Section~\\ref{sec:prop}.\n\\begin{proof}[Proof of (1)] Note that all the tangles involved in the inductive characterization of $Y_n$ from (\\ref{inductiveform}) are Legendrian with coefficients in~$\\mathbb{F}^-$, and the normalizing factor $\\frac{1}{c_n}$ also belongs to~$\\mathbb{F}^-$.\n\\end{proof}\n\nThe following is the second equality in the statement of Theorem~\\ref{thm:main} from the introduction.\n\\begin{Theorem} \\label{thm:main2}\nFor any Legendrian knot $K \\subset J^1\\mathbb {R}$, the $n$-colored Kauffman polynomial has $F_{n,K} \\in \\mathbb{F}^-$ and satisfies\n\\[\nF_{n,K}|_{a^{-1}=0} = R^1_{n,K}.\n\\]\n\\end{Theorem}\n\\begin{proof}Since $\\mathcal{Y}_n \\in \\operatorname{BMW}^-_n$ we get that $F_{n,K} = F_{S(K,\\mathcal{Y}_n)} \\in \\mathbb{F}^-$, and since $\\varphi\\big(\\frac{1}{c_n}L_n\\big) = [\\mathcal{Y}_n]$ the commutativity of the diagram in Proposition~\\ref{prop:algebra} together with Proposition~\\ref{thm:main1}\n shows that\n\\begin{gather*}\nF_{n,K}|_{a^{-1}=0} = F_{S(K,\\mathcal{Y}_n)}|_{a^{-1}=0} = R^1_{S(K, \\frac{1}{c_n} L_n)} = R^1_{n,K}.\\tag*{\\qed}\n\\end{gather*}\\renewcommand{\\qed}{}\n\\end{proof}\n\n\\begin{Corollary}For each $n\\geq2$, $\\varphi\\big(\\frac{1}{c_n}L_n\\big)$ is a central idempotent in $\\operatorname{BMW}_n^\\infty$.\n\\end{Corollary}\n\\begin{proof}This follows from Proposition~\\ref{prop:BMW} since $\\mathcal{Y}_n$ has this property already in $\\operatorname{BMW}_n$. (See, e.g.,~\\cite{BlBe, Heck}.)\n\\end{proof}\n\n\\subsection{Establishing (2) of Proposition \\ref{prop:BMW}} \\label{sec:prop}\n\nWe now embark on showing $\\varphi\\big(\\frac{1}{c_n}L_n\\big)=\\mathcal{Y}_n\\vert_{a^{-1}=0}$. Throughout this section we will work in $\\operatorname{BMW}^{\\infty}_n$, but we will simplify notation by writing $L_n$ for $\\varphi(L_n)$. We begin with some preparatory lemmas that provide formulas for $L_n$ that are closer to the inductive formula for $Y_n$ from (\\ref{inductiveform}). Lemma~\\ref{lemma:beta} uses the hypothesis that $L_{n-1}$ has the crossing absorbing property. This assumption is later verified to be true (see Proposition~\\ref{prop:almostmainthm}).\n\n\\begin{Lemma}\\label{lemma:beta}Let $n\\geq2$ and assume that, in $\\operatorname{BMW}^\\infty_n$, $L_{n-1}$ has the crossing absorbing pro\\-per\\-ty~\\eqref{eats}. Then, in $\\operatorname{BMW}^\\infty_n$ we have\n \\begin{equation*}\n {L_{n-1}\\beta_n}=L_{n-1}\\left(1-z\\sum_{k=2}^n s^{2-k}\\left[\\sum_{j=1}^{k-2}-zs^{2+j-k} D_{n-j,n}+ D_{n-k+1,n}\\right]\\right)\\gamma_n.\n\\end{equation*}\n\\end{Lemma}\n\n\\begin{proof}We use Lemma \\ref{lemma:alpha}. For each $2\\leq k\\leq n$, let $\\operatorname{cr}_\\ell(C_{k,n})$ (respectively $\\operatorname{cr}_r(C_{k,n})$) denote the set of crossings appearing in the left (resp.\\ right) half of $C_{k,n}$, and abbreviate $C_k:=C_{k,n}$. Then,\n\\begin{align*}\n {L_{n-1}\\beta_n}&=L_{n-1}\\left(1-z\\sum_{k=2}^n \\sum_{X\\subset \\operatorname{cr}(C_k)}(-z)^{|X|}r_X(C_k)\\right)\\gamma_n\\\\\n &=L_{n-1}\\left(1-z\\sum_{k=2}^n \\sum_{X\\subset \\operatorname{cr}_\\ell(C_k)}\\sum_{Y\\subset \\operatorname{cr}_r(C_k)}(-z)^{|X|+|Y|}r_{X\\bigsqcup Y}(C_k)\\right)\\gamma_n\\\\\n &=L_{n-1}\\left(1-z\\sum_{k=2}^n \\sum_{X\\subset \\operatorname{cr}_\\ell(C_k)}(-z)^{|X|}\\sum_{Y\\subset \\operatorname{cr}_r(C_k)}(-z)^{|Y|}r_{X\\bigsqcup Y}(C_k)\\right)\\gamma_n \\, .\n\\end{align*}\nSince $L_{n-1}$ absorbs crossings $L_{n-1}r_{X \\bigsqcup Y}(C_k)=L_{n-1}s^{k-2-|Y|}r_X(D_{n-k+1,n})$; see Fig.~\\ref{fig:Absorbing}. Furthermore, because there are $\\binom{k-2}{|Y|}$ subsets of $\\operatorname{cr}_r(C_k)$ having $|Y|$ crossings, summing over $|Y|$ one obtains\n\\begin{gather*}\n L_{n-1}\\sum_{Y\\subset \\operatorname{cr}_r(C_k)}(-z)^{|Y|}r_{X\\bigsqcup Y}(C_k) =L_{n-1}\\sum_{j=0}^{k-2}\\binom{k-2}{j}(-z)^j s^{k-2-j}r_X(D_{n-k+1,n})\\\\\n\\qquad{} =L_{n-1}(s-z)^{k-2}r_X(D_{n-k+1,n})\n=L_{n-1}s^{2-k}r_X(D_{n-k+1,n}).\n\\end{gather*}\nTherefore, \\[L_{n-1}\\beta_n=L_{n-1}\\left(1-z\\sum_{k=2}^ns^{2-k}\\sum_{X\\subset \\operatorname{cr}_\\ell(C_k)}(-z)^{|X|}r_X(D_{n-k+1,n})\\right)\\gamma_n.\\]\n\n\\begin{figure}\n \\centering\n\t\t\\labellist\n \\small\n\\pinlabel $n$ [r] at 0 0\n\\pinlabel $n-k+1$ [r] at 0 112\n\\pinlabel $L_{n-1}$ [b] at 168 56\n\\pinlabel $L_{n-1}$ [b] at 448 56\n\\pinlabel $=\\,s^{k-2-|Y|}$ [b] at 236 48\n\\endlabellist\n\n \\quad \\includegraphics[scale=.8]{images\/Absorbing}\n\t\t\n \\caption{An illustration of the identity $L_{n-1}r_{X \\bigsqcup Y}(C_k)=s^{k-2-|Y|}L_{n-1}r_X(D_{n-k+1,n})$, with the resolved crossings from $X \\bigsqcup Y$ indicated in dotted ovals. The number of crossings in the right half of $r_{X\\sqcup Y}(C_{k,n})$ is $k-2-|Y|$.} \\label{fig:Absorbing}\n\\end{figure}\n\nIt remains to establish the innermost sum satisfies\n\\begin{gather} \\label{eq:Xsum}\nL_{n-1}\\sum_{X\\subset \\operatorname{cr}_\\ell(C_k)}(-z)^{|X|}r_X(D_{n-k+1,n}) = L_{n-1}\\left[\\sum_{j=1}^{k-2}-zs^{2+j-k} D_{n-j,n}+ D_{n-k+1,n}\\right].\n\\end{gather}\nWhen we perform the resolution by subsets of $\\operatorname{cr}_\\ell(C_k)$ we will not be able to feed all of the remaining crossings into $L_{n-1}$ (in fact, when $X$ is the empty set there are no crossings that we can push into $L_{n-1}$). We remedy this by partitioning the nonempty subsets of $\\operatorname{cr}_\\ell(C_k)$. Label the crossings in $\\operatorname{cr}_\\ell(C_k)$ by ascending $z$-coordinate as $c_1, \\ldots, c_{k-2}$, and define $\\chi_j=\\{X\\subset \\operatorname{cr}_\\ell(C_k)\\colon \\allowbreak j=\\min\\{i\\,|\\,c_i \\in X\\}\\}$, for $1\\leq j\\leq k-2$. Let $X\\in\\chi_j$ be given. By definition $c_j\\in X$ is the lowest crossing that is resolved in $r_X(D_{n-k+1,n})$. Isotopy allows us to push the $k-2-j-(|X|-1)$ remaining crossings lying above $c_j$ into $L_{n-1}$. Hence, using $i=|X|-1$ and that the requirement that $X \\in \\chi_j$ leaves $i$ choices from the $k-2-j$ crossings above $c_j$ to determine $X$, we have\n \\begin{align*}\n L_{n-1}\\sum_{X\\in\\chi_j}(-z)^{|X|}r_X(D_{n-k+1,n})&=L_{n-1}\\sum_{i=0}^{k-2-j}\\left(\\begin{array}{c} k-2-j \\\\ i \\end{array} \\right)(-z)^{i+1}s^{(k-2-j)-i}D_{n-j,n} \\\\\n\t\t& = -zL_{n-1}(-z+s)^{k-2-j} D_{n-j,n} \\quad \\mbox{(binomial theorem)} \\\\\n\t\t &=-zL_{n-1}s^{2+j-k}D_{n-j,n}.\n \\end{align*}\nSumming over all $j$, and adding the term $D_{n-k+1,n}$ for the case when $X$ is empty, establishes~(\\ref{eq:Xsum}) and completes the proof.\n\\end{proof}\n\\begin{Lemma}\\label{lemma:mult}For any $1\\leq i\\leq n$, in $\\operatorname{BMW}^\\infty_n$ we have\n$D_{i,n}\\gamma_n=\\sum\\limits_{r=0}^{n-i-1}s^rD_{i,n-r}$.\n\\end{Lemma}\n\\begin{proof}\nThis is just applying type II Reidemeister moves to see that\n\\[\nD_{i,n}s^r\\sigma_{n-1}\\sigma_{n-2}\\cdots\\sigma_{n-r} =  \\begin{cases} s^r D_{i,n-r}, &   r < n-i, \\\\ 0, & r \\geq n-i,\\end{cases}\n\\]\n since the fishtail relation (\\ref{eq:fish}) may be applied when $r \\geq n-i$. See Fig.~\\ref{fig:DIJprod}.\n\\end{proof}\n\n\\begin{figure}[h]\n \\centering\n \\raisebox{-1.25cm}{\\includegraphics[scale=.9]{images\/DIJprod1.png}}\\quad=\\quad\\raisebox{-1.25cm}{\\includegraphics{images\/DIJprod2.png}}\n\n \\caption{An Illustration of the Type II Reidemeister moves used in Lemma \\ref{lemma:mult}.}\n\t\t \\label{fig:DIJprod}\n\\end{figure}\nProposition \\ref{prop:BMW} (2) follows from the following.\n\\begin{Proposition} \\label{prop:almostmainthm}\nFor all $n\\geq1$, $L_n$ has the following properties in $\\operatorname{BMW}^\\infty_n$:\n\\begin{itemize}\\itemsep=0pt\n \\item[$(i)$] $L_n=Y_n\\vert_{a^{-1}=0}$,\n \\item[$(ii)$] $L_n$ has the crossing absorbing property~\\eqref{eats}.\n\\end{itemize}\n\\end{Proposition}\n\\begin{proof} The proof is by induction on $n$. In the base case $n=1$, (i) is immediate from definitions and (ii) is vacuous. Assume the result for $n-1$. Note that it suffices to establish (i) since the fact that $Y_n$ has the crossing absorbing property and $\\vert_{a^{-1}=0}$ is a $\\mathbb{Z}(s)$-algebra homomorphism would then allow us to verify (ii) via\n\\[\nL_n\\sigma_i=(Y_n\\vert_{a^{-1}=0})\\sigma_i=(Y_n\\sigma_i)\\vert_{a^{-1}=0}=(sY_n)\\vert_{a^{-1}=0}=sL_n.\n\\]\n Showing $L_n=Y_n\\vert_{a^{-1}=0}$ is the following a computation (with Lemmas \\ref{lemma:beta} and \\ref{lemma:mult} used at the 2nd and 3rd equality):\n\\begin{align*}\n L_n&=L_{n-1}\\beta_n =L_{n-1}\\left(1-z\\sum_{k=2}^n s^{2-k}\\left[\\sum_{j=1}^{k-2}-zs^{2+j-k} D_{n-j,n}+ D_{n-k+1,n}\\right]\\gamma_n\\right)\\\\\n &=L_{n-1}\\left(\\gamma_n-z\\sum_{k=2}^n s^{2-k}\\left[\\sum_{j=1}^{k-2}-zs^{2+j-k}\\sum_{r=0}^{j-1}s^rD_{n-j,n-r} + \\sum_{r=0}^{k-2}s^rD_{n-k+1,n-r}\\right]\\right)\\\\\n &=L_{n-1}\\left(\\gamma_n-z\\sum_{k=2}^n s^{2-k}\\left[\\sum_{r=0}^{k-3}\\sum_{j=r+1}^{k-2}-zs^{2+j-k+r}D_{n-j,n-r} + \\sum_{r=0}^{k-2}s^rD_{n-k+1,n-r}\\right]\\right)\\\\\n &=L_{n-1}\\left(\\gamma_n-z\\left[\\sum_{k=2}^n \\sum_{r=0}^{k-3}\\sum_{j=r+1}^{k-2}-zs^{4+j-2k+r}D_{n-j,n-r} + \\sum_{k=2}^n\\sum_{r=0}^{k-2}s^{2-k+r}D_{n-k+1,n-r}\\right]\\right)\\\\\n\t &=L_{n-1}\\left(\\gamma_n-z\\left[\\sum_{r=0}^{n-3} \\sum_{k=r+3}^{n}\\sum_{j=r+1}^{k-2}-zs^{4+j-2k+r}D_{n-j,n-r} \\right.\\right.\\\\\n& \\left.\\left. \\quad{} + \\sum_{r=0}^{n-2}\\sum_{k=r+2}^{n}s^{2-k+r}D_{n-k+1,n-r}\\right]\\right)\\\\\n &=L_{n-1}\\left(\\gamma_n-zD_{1,2}-z\\sum_{r=0}^{n-3}s^r\\left[ \\sum_{k=r+3}^{n}\\sum_{j=r+1}^{k-2}-zs^{4+j-2k}D_{n-j,n-r}\\right.\\right.\\\\\n& \\left.\\left. \\quad{} +\\sum_{j=r+1}^{n-1}s^{2-j-1}D_{n-j,n-r}\\right]\\right)\\\\\n &=L_{n-1}\\left(\\gamma_n-zD_{1,2}-z\\sum_{r=0}^{n-3}s^r\\left[ \\sum_{j=r+1}^{n-2}s^j D_{n-j,n-r}\\left(\\sum_{k=j+2}^{n}-zs^{4-2k}\\right)\\right.\\right.\\\\\n& \\left.\\left. \\quad{}\n +\\sum_{j=r+1}^{n-1}s^{2-j-1} D_{n-j,n-r}\\right]\\right)\\\\\n &=L_{n-1}\\left(\\gamma_n-z D_{1,2}-z\\sum_{r=0}^{n-3}s^r\\left[ \\sum_{j=r+1}^{n-2}s^j D_{n-j,n-r}\\left(z\\frac{s^{4-2n}-s^{2-2j}}{s^2-1}\\right)\\right.\\right.\\\\\n& \\left.\\left. \\quad{} +\\sum_{j=r+1}^{n-1}s^{2-j-1} D_{n-j,n-r}\\right]\\right)\\\\\n &=L_{n-1}\\left(\\gamma_n-z D_{1,2}-z\\sum_{r=0}^{n-3}s^r\\left[ \\sum_{j=r+1}^{n-2}\\left(s^{j-2n+3}-s^{1-j}\\right) D_{n-j,n-r}\\right.\\right.\\\\\n& \\left.\\left. \\quad{}+\\sum_{j=r+1}^{n-1}s^{2-j-1} D_{n-j,n-r}\\right]\\right)\\\\\n &=L_{n-1}\\left(\\gamma_n-z D_{1,2}-z\\sum_{r=0}^{n-3}s^r\\Bigg[s^{2-n} D_{1,n-r}\n  \\right.\\\\\n& \\left.\\left. \\quad{} +\\sum_{j=r+1}^{n-2}\\left(s^{j-2n+3}-s^{1-j}+s^{2-j-1}\\right) D_{n-j,n-r}\\right]\\right)\\\\\n &=L_{n-1}\\left(\\gamma_n-z D_{1,2}-z\\sum_{r=0}^{n-3}s^r\\left[s^{2-n} D_{1,n-r} +\\sum_{j=r+1}^{n-2}\\left(s^{j-2n+3}\\right) D_{n-j,n-r}\\right]\\right)\\\\\n &=L_{n-1}\\left(\\gamma_n-z D_{1,2}-z\\sum_{r=0}^{n-3}s^r\\sum_{j=r+1}^{n-1}s^{j-2n+3} D_{n-j,n-r}\\right)\\\\\n &=L_{n-1}\\left(\\gamma_n-\\frac{z}{s^{2n-3}}\\left(s^{2n-3} D_{1,2}+\\sum_{r=0}^{n-3}s^r\\sum_{j=r+1}^{n-1}s^{j} D_{n-j,n-r}\\right)\\right)\\\\\n &=L_{n-1}\\left(\\gamma_n-\\frac{z}{s^{2n-3}}\\left(s^{2n-3} D_{1,2}+\\sum_{i=2}^{n-1}s^{n-i-1}\\sum_{j=n-i}^{n-1}s^j D_{n-j,i+1}\\right)\\right)\\\\\n &=L_{n-1}\\left(\\gamma_n-\\frac{z}{s^{2n-3}}\\left(s^{2n-3} D_{1,2}+\\sum_{i=2}^{n-1}s^{n-i-1}\\sum_{j=0}^{i-1}s^{n-i+j} D_{i-j,i+1}\\right)\\right)\\\\\n &=L_{n-1}\\left(\\gamma_n-\\frac{z}{s^{2n-3}}\\left(s^{2n-3} D_{1,2}+\\sum_{i=2}^{n-1}s^{2n-2i-1}\\sum_{j=0}^{i-1}s^{j} D_{i-j,i+1}\\right)\\right)\\\\\n &=L_{n-1}\\left(\\gamma_n-\\frac{z}{s^{2n-3}}\\left(\\sum_{i=1}^{n-1}s^{2n-2i-1}\\sum_{j=0}^{i-1}s^{j} D_{i-j,i+1}\\right)\\right) \\\\\n\t\t& = Y_{n-1}\\big\\vert_{a^{-1}=0} \\cdot \\left(\\gamma_n+\\frac{z}{1-s^{2n-3}a}\\left(\\sum_{i=1}^{n-1}as^{2n-2i-1} \\sum_{j=0}^{i-1}s^{j} D_{i-j,i+1}\\right) \\right)\\bigg\\vert_{a^{-1}=0} = Y_n\\big\\vert_{a^{-1}=0}.\n\\end{align*}\nAt the last equality, we used \\eqref{inductiveform}.\n\\end{proof}\n\n\\section[The $n$-colored ruling polynomial and representation numbers]{The $\\boldsymbol{n}$-colored ruling polynomial and representation numbers} \\label{sec:5}\n\nIn this section, we show that the $1$-graded $n$-colored ruling polynomial agrees with the $1$-graded, total $n$-dimensional representation number defined in~\\cite{LeRu}; see Theorem~\\ref{thm:Rep}. After a brief review of Legendrian contact homology and relevant material from~\\cite{LeRu}, the remainder of the section contains the proof of Theorem~\\ref{thm:Rep}.\n\n\\subsection{Review of the Legendrian contact homology DGA} We assume familiarity with the Legendrian contact homology differential graded algebra, (abbrv. LCH DGA), aka.\\ the Chekanov--Eliashberg algebra, in the setting of Legendrian links in $J^1M$ with $M = \\mathbb {R}$ or $S^1$, and refer the reader to any of \\cite{Chekanov02, EN, ENS, Ng1, NgR2012} for this background material. We continue to use coordinates $(x,y,z) \\in J^1M = T^*M \\times \\mathbb {R} = M \\times \\mathbb {R}^2$, and to view projections to $S^1\\times \\mathbb {R}$ in $[0,1]\\times \\mathbb {R}$ with the left and right boundary identified.\n The Reeb vector field is $\\frac{\\partial}{\\partial z}$, so Reeb chords of $K$ are in bijection with double points of the {\\it Lagrangian projection} aka. the {\\it $xy$-diagram} of $K$ which is the projection to $T^*M$.\n Representation numbers are defined in \\cite{LeRu} using the fully non-commutative version of the LCH DGA associated to a Legendrian knot or link $K$ equipped with a collection of base points, $*_1, \\ldots, *_\\ell$, with the requirement that every component of $K$ has at least one base point. The resulting DGA, notated $(\\mathcal{A}(K), \\partial)$ or $(\\mathcal{A}(K, *_1, \\ldots, *_\\ell), \\partial)$ when the choice of base points should be emphasized, is an associative, non-commutative algebra with identity generated over $\\mathbb {Z}$ by\n\\begin{itemize}\\itemsep=0pt\n\\item[(i)] the Reeb chords of $K$, denoted $b_1, \\ldots, b_r$, and\n\\item[(ii)] invertible generators $t_1^{\\pm1}, \\ldots, t_\\ell^{\\pm1}$ corresponding to the base points $*_1, \\ldots, *_\\ell$.\n\\end{itemize}\nThere are no relations other than $t_it_i^{-1} = t_i^{-1}t_i=1$. The differential $\\partial$ vanishes on the $t_i$; for a Reeb chord, $a$, the differential $\\partial a$ is defined via a signed count of rigid holomorphic disks in~$T^*M$ with boundary on the Lagrangian projection of $K$ and having a single positive boundary puncture at $a$ and an arbitrary number of negative boundary punctures. Each such a disk $u$ contributes a term $\\pm w(u)$ to $\\partial a$ where~$w(u)$ is the product of base point generators and negative punctures as they appear in counter-clockwise order along the boundary of the domain of~$u$, starting from the positive puncture at~$a$. Occurrences of~$t_i$ appear with exponent~$\\pm1$ according to the oriented intersection number of $\\partial u$ with $*_i$. See Fig.~\\ref{fig:PointedDisk}. For associating $\\pm1$ signs to disks, we use the conventions as in \\cite{HenryRu, LeRu}. Most results of this section concern DGA representations defined over a field of characteristic~$2$, and in this case it suffices to work with the version of~$\\mathcal{A}(K)$ defined over $\\mathbb {Z}\/2$ where the~$\\pm1$ signs become irrelevant.\n\n\\begin{figure}\n\\labellist\n\\small\n\\pinlabel $a$ [l] at 186 102\n\\pinlabel $+$ [r] at 166 102\n\\pinlabel $b_1$ [b] at 136 180\n\\pinlabel $b_2$ [b] at 24 182\n\\pinlabel $b_3$ [tl] at 84 20\n\\pinlabel $-$ [t] at 132 152\n\\pinlabel $-$ [t] at 28 162\n\\pinlabel $-$ [b] at 86 34\n\\endlabellist\n\\centerline{\\includegraphics[scale=.6]{images\/PointedDisk}}\n\n\\caption{A holomorphic disk contributing the term $\\partial a = \\pm b_1 b_2 t^{-1} b_3 + \\cdots$\n to the differential of $\\mathcal{A}(K)$.}\n\\label{fig:PointedDisk}\n\\end{figure}\n\n\\subsection{1-graded representation numbers} In~\\cite{LeRu}, Legendrian invariant $m$-graded representation numbers are defined for any non-negative integer $m\\geq 0$ by considering DGA homomorphisms that are only required to preserve grading mod $m$. In the current article, we are concerned only with $1$-graded representations, which we will refer to as {\\it ungraded} representations since the grading condition becomes vacuous when $m=1$. We review definitions from~\\cite{LeRu} in the ungraded setting.\n\nLet $V$ be a vector space over a field, $\\mathbb{F}$, with $\\operatorname{char}(\\mathbb{F}) = 2$, and let $d\\colon V \\rightarrow V$ be an ungraded differential on~$V$, i.e., $d$ is just a linear map satisfying $d^2=0$.\n Then, $d$ induces a differential on the endomorphism algebra\n\\[\n\\delta\\colon \\ \\operatorname{End}(V) \\rightarrow \\operatorname{End}(V), \\qquad \\delta(T) = d \\circ T + T \\circ d\n\\]\nmaking $(\\operatorname{End}(V),\\delta)$ into an ungraded DGA, i.e.,\n $\\delta$ satisfies $\\delta^2=0$ and $\\delta(T_1T_2) = \\delta(T_1)T_2 + T_1\\delta(T_2)$. An {\\it ungraded representation} of a DGA, $(\\mathcal{A}, \\partial)$, on $(V,d)$ is an ungraded DGA homomorphism\n\\[\nf\\colon \\ (\\mathcal{A}, \\partial) \\rightarrow (\\operatorname{End}(V), \\delta),\n\\]\ni.e., a ring homomorphism satisfying $f(1)=1$ and $f \\circ \\partial = \\delta \\circ f$.\nIn the $1$-dimensional case where $V = \\mathbb{F}$ and $d=0$, an ungraded representation on $(\\mathbb{F},0)$ is also called an {\\it ungraded augmentation}.\n\nFor $\\mathcal{A} = \\mathcal{A}(K)$, we use the notation $\\overline{\\operatorname{Rep}}_1(K, (V, d))$ for the set of all ungraded representations of $(\\mathcal{A},\\partial)$ on $(V,d)$ and $\\overline{\\operatorname{Aug}}_1(K, \\mathbb{F})$ for the set of augmentations to $\\mathbb{F}$.\n In the case where $K$ is connected with base points $*_1, \\ldots, *_\\ell$ appearing in order starting at $*_1$ and following the orientation of $K$, given a subset $T \\subset {\\rm GL}(V)$ we will use the notation\n $\\overline{\\operatorname{Rep}}_1(K,(V,d),T)$ to denote the set of those $f \\in \\overline{\\operatorname{Rep}}_1(K, (V, d))$ such that $f(t_1\\cdots t_\\ell) \\in T$. In particular, $\\overline{\\operatorname{Rep}}_1(K, (V, d)) =\\overline{\\operatorname{Rep}}_1(K,(V,d), {\\rm GL}(V))$.\n\n\\begin{Definition} \\label{def:total} Let $\\mathbb{F}_q$ denote the finite field of order $q$ with $q$ a power of $2$. The ($1$-graded) {\\it total $n$-dimensional representation number} of $K$ is defined by\n\\begin{align*}\n\\operatorname{Rep}_1\\big(K, \\mathbb{F}^n_q\\big) & := \\big|\\operatorname{End}\\big(\\mathbb{F}^n_q\\big)\\big|^{-{\\rm rb}(K)\/2}  |{\\rm GL}(n, \\mathbb{F}_q) |^{-\\ell} \\big| \\overline{\\operatorname{Rep}}_1\\big(K, \\big(\\mathbb{F}^n_q, 0\\big)\\big) \\big| \\\\\n & = \\big(q^{n^2}\\big)^{-{\\rm rb}(K)\/2}   \\left(q^{n(n-1)\/2}   \\prod_{m=1}^n\\big(q^m-1\\big)\\right)^{-\\ell} \\big| \\overline{\\operatorname{Rep}}_1\\big(K, \\big(\\mathbb{F}^n_q, 0\\big)\\big) \\big|,\n\\end{align*}\n where ${\\rm rb}(K)$ is the number of Reeb chords of $K$ and $\\ell$ is the number of basepoints.\n\\end{Definition}\n\nThat $\\operatorname{Rep}_1\\big(K, \\mathbb{F}^n_q\\big)$ only depends on the Legendrian isotopy type of $K$ is established in \\cite[Proposition~3.10]{LeRu}. The following theorem shows that the $1$-graded $n$-colored ruling polynomial and the total $n$-dimensional representation numbers of~$K$ are equivalent Legendrian invariants.\n\n\\begin{Theorem} \\label{thm:Rep}\nLet $K \\subset J^1 \\mathbb {R}$ be a $($connected$)$ Legendrian knot and $\\mathbb{F}_q$ a finite field of order $q$ with characteristic~$2$. Then,\n\\[\nR^1_{n,K}(q) = \\operatorname{Rep}_1(K, \\mathbb{F}_q),\n\\]\nwhere $R^1_{n,K}(q) = R^1_{n,K}(s)\\big\\vert_{s=q^{1\/2}}$.\n\\end{Theorem}\n\nThe proof of Theorem \\ref{thm:Rep} rests on the following relation between representation counts and reduced ruling polynomials which we will establish in Sections~\\ref{sec:strat}--\\ref{sec:augsat} using extensions of results from~\\cite{HenryRu} and~\\cite{LeRu}. In \\cite[Section~4]{LeRu}, a ``path subset''\\footnote{The {\\it path subset} $B_\\beta$ is the subset of ${\\rm GL}(n,\\mathbb{F}_q)$ arising from specializing the path matrix $P^{xy}_\\beta$ using arbitrary ring homomorphisms from $\\mathcal{A}(\\beta)$ to $\\mathbb{F}_q$. The {\\it path matrix} $P^{xy}_\\beta$ is a matrix whose entries belong to $\\mathcal{A}(\\beta)$ and record certain left-to-right paths through the $xy$-diagram of $\\beta$ that reflect the possible behavior of boundaries of holomorphic disks bordering $\\beta$ from above. See \\cite[Section~4.1]{LeRu}. Note that~\\cite{LeRu} uses the opposite convention for composing Legendrian $n$-tangles, with $\\alpha \\cdot \\beta$ defined as $\\alpha$ stacked to the {\\it left} of~$\\beta$. Consequently, for a permutation $\\beta \\in S_n$, the permutation braid $\\beta \\subset J^1S^1$ used in the present article corresponds to $\\beta^{-1} \\subset J^1S^1$ in~\\cite{LeRu}. As a~result, the notations $B_\\beta$ and $P^{xy}_\\beta$ used here correspond to $B_{\\beta^{-1}}$ and $P_{\\beta^{-1}}^{xy}$ in~\\cite{LeRu}.} $B_\\beta \\subset {\\rm GL}(n, \\mathbb{F}_q)$ is associated to a reduced\\footnote{A positive permutation braid $\\beta\\subset J^1S^1$ is {\\it reduced} if its front diagram corresponds to a reduced braid word, i.e., one where the product $\\sigma_i\\sigma_{i+1}\\sigma_i$ does not appear for any~$i$.} positive permutation braid, $\\beta \\in S_n$.\n\n\\begin{Lemma} \\label{lem:count}\nLet $\\beta \\in S_n$ be an $n$-stranded reduced positive permutation braid, and suppose that the $($connected$)$ Legendrian knot $K \\subset J^1\\mathbb {R}$ has its front diagram in plat position\\footnote{A front diagram is in {\\it plat position} if all left cusps appear at a common $x$-coordinate at the far left of the diagram and all right cusps appear at a common $x$-coordinate at the far right of the diagram.} and is equipped with $\\ell$ base points where~$\\ell$ is the number of right cusps of~$K$. Then,\n\\begin{gather*}\n\\big|\\overline{\\operatorname{Rep}}_1\\big(K, \\big(\\mathbb{F}_q^n,0\\big), B_\\beta\\big)\\big| = |{\\rm GL}(n, \\mathbb{F}_q)|^{\\ell-1}  q^{n(n-1)\/2}  (q-1)^n   q^{n^2   {\\rm rb}(K)\/2}   q^{\\lambda(\\beta)\/2}\\widetilde{R}_{S(K, \\beta)}(z),\n\\end{gather*}\nwhere ${\\rm rb}(K)$ is the number of Reeb chords of $K$, $\\lambda(\\beta)$ is the length of $\\beta$, i.e., the number of crossings of $\\beta$, and $z=q^{1\/2}-q^{-1\/2}$.\n\\end{Lemma}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:Rep}] Since both sides of the equation are Legendrian isotopy invariant, we may assume $K$ is in plat position.\nIn \\cite[Proposition 4.14]{LeRu}, it is shown that ${\\rm GL}(n, \\mathbb{F}_q) = \\sqcup_{\\beta \\in S_n} B_\\beta$. (Actually, this coincides with the Bruhat decomposition of ${\\rm GL}(n, \\mathbb{F}_q)$.) Thus,\n\\[\n\\overline{\\operatorname{Rep}}_1\\big(K, \\big(\\mathbb{F}_q^n,0\\big), {\\rm GL}(n, \\mathbb{F}_q)\\big) = \\sqcup_{\\beta \\in S_n} \\overline{\\operatorname{Rep}}_1\\big(K, \\big(\\mathbb{F}_q^n,0\\big),B_\\beta\\big),\n\\]\n and using Lemma \\ref{lem:count} and Definition \\ref{def:nruling} we compute\n\\begin{align*}\n\\operatorname{Rep}_1\\big(K, \\mathbb{F}_q^n\\big) & = \\big(q^{n^2}\\big)^{-{\\rm rb}(K)\/2}  |{\\rm GL}(n)|^{-\\ell}  \\sum_{\\beta \\in S_n} \\overline{\\operatorname{Rep}}_1\\big(K, \\big(\\mathbb{F}_q^n,0\\big),B_\\beta\\big) \\\\\n & = |{\\rm GL}(n)|^{-1}   q^{n(n-1)\/2}(q-1)^n \\sum_{\\beta \\in S_n}q^{\\lambda(\\beta)\/2} \\widetilde{R}_{S(K, \\beta)}(z) \\\\\n& = \\left( q^{n(n-1)\/2}  \\prod_{m=1}^n\\big(q^m-1\\big)\\right)^{-1}  q^{n(n-1)\/2}(q-1)^n \\sum_{\\beta \\in S_n} q^{\\lambda(\\beta)\/2} \\widetilde{R}_{S(K, \\beta)}(z) \\\\\n& = \\left(\\prod_{m=1}^n\\left(\\frac{q^m-1}{q-1} \\right) \\right)^{-1} \\sum_{\\beta \\in S_n} q^{\\lambda(\\beta)\/2} \\widetilde{R}_{S(K, \\beta)}(z) \\\\\n& = \\frac{1}{c_n} \\sum_{\\beta \\in S_n} q^{\\lambda(\\beta)\/2} \\widetilde{R}_{S(K, \\beta)}(z) = R^1_{n,K}(q).\n\\end{align*}\n(The notation $c_n$ is as in Definition \\ref{def:nruling} with $s = q^{1\/2}$.)\n\\end{proof}\n\n\\subsection{Strategy of the proof of Lemma \\ref{lem:count}} \\label{sec:strat}\n\nThe relation between counts of representations on $\\big(\\mathbb{F}_q^n,0\\big)$ and reduced ruling polynomials stated in Lemma \\ref{lem:count} is based on refinements of two results:\n\\begin{enumerate}\\itemsep=0pt\n\\item In \\cite[Theorem~6.1]{LeRu}, it is shown that for any $n$-stranded reduced positive permutation braid $\\beta \\in S_n$,\n (using $1$ base point on $K$ and a particular $xy$-diagram of $S(K,\\beta)$) when $\\operatorname{char}(\\mathbb{F}) = 2$ there is a bijection\n\\begin{gather} \\label{eq:LeRu}\n\\bigsqcup_{d}\\overline{\\operatorname{Rep}}_1\\big(K, \\big(\\mathbb{F}^n,d\\big),B_\\beta\\big) \\quad \\longleftrightarrow \\quad \\overline{\\operatorname{Aug}}_1(S(K, \\beta), \\mathbb{F}),\n\\end{gather}\nwhere the union is over all strictly upper triangular differentials on $\\mathbb{F}^n$.\n\n\\item In  \\cite[Theorem 3.2]{HenryRu}, for any Legendrian $K' \\subset J^1\\mathbb {R}$ (using a particular $xy$-diagram of~$K'$) the set of augmentations of~$K'$ is decomposed into pieces\n\\begin{gather} \\label{eq:HenryRu}\n\\operatorname{Aug}_1(K', \\mathbb{F}) = \\bigsqcup_{\\rho} (\\mathbb{F}^*)^{a(\\rho)} \\times \\mathbb{F}^{b(\\rho)},\n\\end{gather}\nwhere the disjoint union is indexed by all normal rulings of $K'$ and the exponents~$a(\\rho)$ and $b(\\rho)$ are specified by the combinatorics of~$\\rho$. Theorem~1.1 of~\\cite{HenryRu} then applies the decomposition to relate the Legendrian invariant augmentation numbers with the ruling polynomial.\n\\end{enumerate}\nThe idea behind the proof of Lemma~\\ref{lem:count} is then to check that the subset of $\\operatorname{Aug}_1(S(K,\\beta), \\mathbb{F})$ corresponding under~(\\ref{eq:LeRu}) to $\\overline{\\operatorname{Rep}}_1\\big(K, \\big(\\mathbb{F}^n, 0\\big), B_\\beta\\big)$, i.e., those representations with $d=0$, is the part of the disjoint union~(\\ref{eq:HenryRu}) with $K' = S(K,\\beta)$ that is indexed by {\\it reduced} normal rulings of~$S(K, \\beta)$. This is roughly what we shall do. However, complications arise as different (but Legendrian isotopic) $xy$-diagrams for the satellite $S(K,\\beta)$ having different (but stable tame isomorphic) DGAs are used in~(\\ref{eq:LeRu}) and~(\\ref{eq:HenryRu}). As a result, we need to also keep track of the way the set $\\operatorname{Aug}_1(S(K, \\beta), \\mathbb{F})$ changes when transitioning between these different diagrams for $S(K,\\beta)$. This contributes to the factor appearing in front of~$\\widetilde{R}_{S(K,\\beta)}$ in the statement of Lemma~\\ref{lem:count}.\n\n\\begin{Remark}\n In the case of $m$-graded representations with $m \\neq 1$, $d=0$ is the only term in the disjoint union (\\ref{eq:LeRu}) for grading reasons.\n\\end{Remark}\n\n\\subsection[Four diagrams for the satellite, $S(K, \\beta)$]{Four diagrams for the satellite, $\\boldsymbol{S(K, \\beta)}$} \\label{sec:satellite}\n\nIn establishing Lemma~\\ref{lem:count} we will make use of four different (but Legendrian isotopic) versions of the satellite $S(K,\\beta)$. The four $xy$-diagrams are denoted $S^1_{xy}(K,\\beta)$, $S^2_{xy}(K, \\beta)$, $S^1_{xz}(K,\\beta)$, $S^2_{xz}(K,\\beta)$ and will be defined momentarily; see Fig.~\\ref{fig:SKbxy}. As a~preliminary, we consider $xz$-diag\\-rams (front projection) and $xy$-diagrams (Lagrangian projection) for the companion \\mbox{$K \\subset J^1\\mathbb {R}$} and pattern $\\beta \\subset J^1S^1$.\n\n\\subsubsection[Diagrams for $K$ and $\\beta$]{Diagrams for $\\boldsymbol{K}$ and $\\boldsymbol{\\beta}$}\n\nFor the companion knot $K \\subset J^1\\mathbb {R}$, apply Ng's resolution procedure (see~\\cite{Ng1}) so that the $xy$-diagram for $K$ is related to the front projection of~$K$ by placing the strand with smaller slope on top at crossings and adding an extra loop at right cusps. Enumerate the Reeb chords of~$K$ by $a_1, \\ldots, a_m$, and $c_1, \\ldots, c_\\ell$ where the $a_i$ correspond to crossings of the front projection of $K$ and the $c_i$ are the extra crossings near right cusps that arise from the resolution procedure. Choose an initial base point~$*$ of~$K$ not located on any of the loops at right cusps and at a point where the front diagram of~$K$ is oriented left-to-right. For convenience, we assume the $c_1, \\ldots, c_\\ell$ are enumerated in the order they appear when following along~$K$ according to its orientation, starting at~$*$.\n\nFor the braid $\\beta \\subset J^1S^1$, we form an $xy$-diagram as indicated in Fig.~\\ref{fig:Dips} having $\\ell$ dips in addition to the original crossings coming from the front projection of $\\beta$. This is done by applying the resolution procedure to the front diagram of $\\beta$ as in \\cite[Section~2.2]{NgTraynor} (this amounts to adding a dip to the right of the crossings of $\\beta$), and then adding $\\ell-1$ extra dips to the $xy$-diagram. In addition, a collection of $n$ basepoints, $*_1, \\ldots, *_n$, are placed immediately to the left of the crossings of $\\beta$, one on each strand. Note that the addition of the dips can be accomplished by Legendrian isotopy as indicated in \\cite[Section 3.1]{Sab}, and we have used a minor variation on the resolution procedure of \\cite[Section 2.2]{NgTraynor} so that the dip arising from the resolution procedure has the same form as the others. Numbering the dips $1,\\ldots, \\ell$ as they appear from left to right in $T^*S^1$ (viewed as $[0,1]\\times \\mathbb {R}$ with boundaries identified) the $k$-th dip consists of two groups of crossings $x_{i,j}^k$ and $y_{i,j}^k$ for $1 \\leq i < j \\leq n$ that appear in the left and right half of the dip respectively. For $1 \\leq k \\leq \\ell$, we collect these crossings as the entries of strictly upper triangular matrices denoted $X_k$ and $Y_k$. The crossings that correspond to $xz$-crossings of $\\beta$ are labeled $p_1, \\ldots, p_\\lambda$, and DGA generators associated to the basepoints $*_1,\\ldots, *_n$ will be labelled $s_1, \\ldots, s_n$.\n\n\\begin{figure}\n\\labellist\n\\small\n\\pinlabel $z$ [l] at 4 26\n\\pinlabel $x$ [l] at 28 2\n\\pinlabel $y$ [l] at 250 26\n\\pinlabel $x$ [l] at 274 2\n\\pinlabel $X_1$ at 392 24\n\\pinlabel $Y_1$ at 452 24\n\\pinlabel $X_2$ at 542 24\n\\pinlabel $Y_2$ at 602 24\n\\pinlabel $\\leadsto$ at 200 96\n\\pinlabel $*_3$ at 264 82\n\\pinlabel $*_2$ at 264 98\n\\pinlabel $*_1$ at 264 112\n\n\\endlabellist\n\\centerline{\\includegraphics[scale=.7]{images\/Dips}}\n\n\\caption{A Lagrangian diagram for the positive braid $\\beta = \\sigma_2 \\sigma_1\\sigma_2$ with $\\ell =2$ dips.}\n\\label{fig:Dips}\n\\end{figure}\n\nIn constructing the diagrams $S^1_{xy}(K,\\beta)$ and $S^2_{xy}(K,\\beta)$ and comparing their DGAs, it is useful to be able to move the locations of dips around via a Legendrian isotopy. This may be accomplished by converting an isotopy $\\phi_t\\colon S^1 \\rightarrow S^1$, $0\\leq t \\leq 1$, to an ambient contact isotopy\n\\begin{gather} \\label{eq:contact}\n\\Phi_t\\colon J^1S^1 \\rightarrow J^1S^1,\\qquad \\Phi_t(x,y,z) = \\left(\\phi_t(x), \\frac{y}{\\phi'_t(x)}, z\\right), \\qquad 0\\leq t \\leq 1.\n\\end{gather}\nIn particular, given any cyclically ordered collection of points $x_1, \\ldots, x_\\ell \\in S^1$, after a Legendrian isotopy the $X_k$ and $Y_k$ crossings can be arranged to appear above an arbitrarily small neighborhood of $x_k$ in $S^1$ for $1\\leq k \\leq \\ell$.\n\n\\subsubsection{Diagrams for Legendrian satellites}\n\nThe Legendrian satellite $S(K,\\beta) \\subset J^1\\mathbb {R}$ is formed by scaling the $y$ and $z$ coordinates of $J^1S^1$ so that $\\beta$ sits in a small neighborhood, $N_0 \\subset J^1S^1$, of the $0$-section and then applying a~contactomorphism $\\Psi\\colon N_0 \\rightarrow N(K)$ of $N_0$ onto a Weinstein tubular neighborhood of~$K$. Two general methods for forming $xy$-diagrams of satellites are as follows:\n\n\\begin{itemize}\\itemsep=0pt\n\\item {\\it The $xy$-method.} Use the orientations of $K$ and $\\mathbb {R}^2$ to identify an (immersed) annular neighborhood, $N_{xy}$, of the $xy$-diagram of $K$ with $S^1 \\times (-\\epsilon,\\epsilon)$. Then, (after scaling the $y$-coordinate appropriately) place an $xy$-diagram for $\\beta$ into $N_{xy}$. As $\\Psi$ can be chosen to preserve the Reeb vector field (see \\cite{Geiges}), this indeed produces an $xy$-diagram for $S(K,\\beta)$.\n\n\\item {\\it The $xz$-method.} First form an $xz$-diagram for $S(K,\\beta)$ as in Section \\ref{sec:sat}: start with the $n$-copy of $K$ (this is $n$-parallel copies of $K$ shifted a small amount in the $z$-direction), then insert the $xz$-projection of $\\beta$\n at the location of the initial base point of $K$.\n Finally, produce an $xy$-diagram for $S(K,\\beta)$ by applying Ng's resolution procedure.\n\\end{itemize}\n\n{\\bf Definition of $\\boldsymbol{S^1_{xy}(K,\\beta)}$:} To form $S^1_{xy}(K,\\beta)$ expand $*$ to a cluster of base points $*_1, \\ldots, *_\\ell$ all appearing in order (according to the orientation of $K$) in a small neighborhood of~$*$. Then, form the satellite with the $xy$-diagram of $\\beta$ described above in such a way that the crossings $p_1, \\ldots, p_\\lambda$ from $\\beta$ and the crossings from $X_1$, $Y_1$ appear in a neighborhood of~$*_1$, and the crossings from $X_k, Y_k$ appear in a neighborhood of $*_k$ for $2 \\leq k \\leq \\ell$.\n\n{\\bf Definition of $\\boldsymbol{S^2_{xy}(K,\\beta)}$:} This $xy$-diagram is formed from $S^1_{xy}(K,\\beta)$ by using a Legendrian isotopy of $\\beta$ (constructed from an isotopy of~$S^1$ as in~(\\ref{eq:contact})) to relocate each collection of~$X_i$ crossings to a neighborhood of a right cusp of~$K$ (so that exactly one $X_i$ appears near each right cusp); relocate each group of $Y_i$ crossings to a neighborhood of a left cusp; and leave the crossings $p_1, \\ldots, p_\\lambda$ of $\\beta$ (and the basepoints $*_1, \\ldots, *_n$) in place at the location of the initial base point $*$ for~$K$. (Recall that~$\\ell$ is both the number of dips and the number of right cusps of~$K$.)\n\n{\\bf Definition of $\\boldsymbol{S^1_{xz}(K,\\beta)}$ and $\\boldsymbol{S^2_{xz}(K,\\beta)}$:} Both $S^1_{xz}(K,\\beta)$ and $S^2_{xz}(K,\\beta)$ are formed using the $xz$-method. The only difference between the two is the placement of base points. For~$S^1_{xz}(K,\\beta)$, base points $*_1, \\ldots, *_n$ appear, one on each parallel strand of the $n$-copy, just before~$\\beta$ (with respect to the orientation of $K$). For $S^2_{xz}(K,\\beta)$, we have base points $*_1, \\ldots, *_c$, one on each component of $S(K,\\beta)$ placed on a loop near some right cusp of the component.\n\nSee Fig.~\\ref{fig:SKbxy}.\n\n\\begin{figure}\n\\labellist\n\\small\n\\pinlabel $S^1_{xy}(K,\\beta)$ [t] at 130 0\n\\pinlabel $S^2_{xy}(K,\\beta)$ [t] at 474 0\n\n\\endlabellist\n\\centerline{\\includegraphics[scale=.6]{images\/SKbxy}}\n\n\\quad\n\n\\quad\n\n\\labellist\n\\small\n\\pinlabel $S^1_{xz}(K,\\beta)$ [t] at 126 0\n\\pinlabel $S^2_{xz}(K,\\beta)$ [t] at 460 0\n\\endlabellist\n\n\\centerline{\\includegraphics[scale=.6]{images\/SKbxz}}\n\n\\vspace{2mm}\n\n\\caption{The Lagrangian ($xy$)-diagrams $S^1_{xy}(K,\\beta)$, $S^2_{xy}(K,\\beta)$, $S^1_{xz}(K,\\beta)$, and $S^2_{xz}(K,\\beta)$ where $K$ is a Legendrian trefoil and $\\beta = \\sigma_1 \\in S_2$.}\n\\label{fig:SKbxy}\n\\end{figure}\n\n\\subsubsection{DGA generators} We set notations for the generators of the DGAs arising from the various $xy$-diagrams of $S(K,\\beta)$ that have been defined. Many generators are indexed with a pair of subscripts $i$,~$j$. Such a~subscript indicates that at the overstrand of the crossing belongs to the $i$-th copy of $K$ and the understrand belongs to the $j$-th copy of $K$. Here,\n outside of an arc $A \\subset K$ where the $p_1, \\ldots, p_\\lambda$ crossings of $\\beta$ appear, $S(K,\\beta)$ consists of $n$ copies of $K\\setminus A$, which we label from $1$ to $n$ according to the descending order of their $y$-coordinates at the boundary of $A$.\n\n{\\bf Generators of $\\boldsymbol{\\mathcal{A}(S^1_{xy}(K,\\beta))}$ and $\\boldsymbol{\\mathcal{A}(S^2_{xy}(K,\\beta))}$:} The generating sets for these DGAs are in bijection. Both contain the DGA of $\\beta$ as a sub-DGA, and this accounts for generators of the form $p_1, \\ldots, p_\\lambda$, $x_{i,j}^k$, $y^k_{i,j}$ for $1\\leq k \\leq \\ell$ and $1\\leq i < j \\leq n$ as well as invertible generators $s_1, \\ldots, s_n$ associated to the base points $*_1, \\ldots, *_n$ on $\\beta$. In addition, for each of the Reeb chords $a_1, \\ldots, a_m$, and $c_1,\\ldots, c_\\ell$ of $K$ there are $n^2$ Reeb chords for $S^1_{xy}(K,\\beta)$ and $S^2_{xy}(K,\\beta)$ that we denote by $a^{k_1}_{i,j}$ and $c^{k_2}_{i,j}$, $1\\leq k_1 \\leq m$, $1\\leq k_2 \\leq \\ell$, $1 \\leq i,j \\leq n$.\n\n{\\bf Generators of $\\boldsymbol{\\mathcal{A}(S^1_{xz}(K,\\beta))}$ and $\\boldsymbol{\\mathcal{A}(S^2_{xz}(K,\\beta))}$:} Both of these diagram have Reeb chords\n\\begin{itemize}\\itemsep=0pt\n\\item $p_1, \\ldots, p_\\lambda$ from the $xz$-crossings of $\\beta$;\n\\item $a^{k_1}_{i,j}$, $1 \\leq k_1 \\leq m$, $1 \\leq i,j \\leq n$, from the $xz$-crossings of~$K$;\n\\item $y^{k_2}_{i,j}$, $1 \\leq k_2 \\leq \\ell$, $1 \\leq i<j \\leq n$, from crossings near left cusps;\n\\item $c^{k_2}_{i,j}$, $1 \\leq k_2 \\leq \\ell$, $1 \\leq j \\leq i \\leq n$, from crossings near right cusps.\n\\end{itemize}\nThe invertible generators coming from base points will be denoted as $s_1, \\ldots, s_n$ for $S^1_{xz}(K,\\beta)$ and as $r_1, \\ldots, r_c$ for $S^2_{xz}(K,\\beta)$.\n\nIn particular, note that for any of the four diagrams for $S(K,\\beta)$ there is a collection of generators of the form $y^{k}_{i,j}$ which we will refer to as {\\it $Y$-generators}.\n\\begin{Definition} Let $K'$ denote one of $S^1_{xy}(K,\\beta)$, $S^2_{xy}(K,\\beta)$, $S^1_{xz}(K,\\beta)$, or $S^2_{xz}(K,\\beta)$. We denote by\n\\[\n\\overline{\\operatorname{Aug}}_1(K', \\mathbb{F})_{Y=0}\n\\]\nthe set of augmentations of $\\mathcal{A}(K')$ to $\\mathbb{F}$ that map all $Y$-generators to~$0$.\n\\end{Definition}\n\nThe subsets $\\overline{\\operatorname{Aug}}_1(K', \\mathbb{F})_{Y=0} \\subset \\overline{\\operatorname{Aug}}_1(K', \\mathbb{F})$ play an important role in the proof Lemma \\ref{lem:count} as they correspond to representations with $d=0$ as well as to reduced normal rulings.\n\n\\begin{Remark}Computations of the differential for similar DGAs of $1$-dimensional Legendrian satellites are presented in detail in several places, and it is not difficult to extend these computations to give complete formulas for differentials in the present setting. See especially~\\cite{LeRu} for satellites formed via the $xy$-method and~\\cite{NgR2012} and~\\cite{NRSSZ} for satellites formed via the $xz$-method. Rather than giving a complete description of differentials here, we state several partial formulas as they become useful in the following proofs.\n\\end{Remark}\n\n\\subsection{Representations and augmentations of the satellite} \\label{sec:augsat}\n\nWe will use the following mild variation of \\cite[Theorem 6.1]{LeRu} to transition between $n$-dimensional representations and augmentations of satellites.\nAs in the construction of $S^1_{xy}(K,\\beta)$, expand the initial base point $*$ of $K$ into a cluster of base points $*_1, \\ldots, *_\\ell$ and consider the DGA $(\\mathcal{A}(K), \\partial)$ with invertible generators $t_1, \\ldots, t_\\ell$.\n\n\\begin{Proposition} \\label{prop:LeRu} Let $\\beta$ be a reduced positive permutation braid, and construct $S^1_{xy}(K, \\beta)$ as above.\nThere is a bijection\n\\[\n\\big\\{ f \\in \\overline{\\operatorname{Rep}}_1\\big(K,\\big(\\mathbb{F}^n,0\\big)\\big) \\,|\\, f(t_1) \\in B_\\beta \\mbox{ and } f(t_i) \\in N_+ \\mbox{ for $i \\geq 2$}\\big\\} \\quad \\leftrightarrow \\quad \\overline{{\\rm Aug}}_1(S^1_{xy}(K, \\beta), \\mathbb{F})_{Y=0},\n\\]\nwhere $B_\\beta$ is the path subset of $\\beta$ and $N_+ \\subset {\\rm GL}(n, \\mathbb{F})$ is the group of upper triangular matrices with~$1$'s on the diagonal.\n\\end{Proposition}\n\n\\begin{proof}The proof is similar to \\cite[Theorem~6.1]{LeRu} which implies the case of only~$1$ base point. We sketch the argument and highlight the modifications to the proof for the case of more than one basepoint. To avoid considering signs, we only treat the case where $\\operatorname{char}(\\mathbb{F}) = 2$ (which is the only case needed for Lemma~\\ref{lem:count}). This allows us to work with the LCH DGA defined over~$\\mathbb {Z}\/2$ rather than over~$\\mathbb {Z}$ (since any representation with $\\operatorname{char}(\\mathbb{F})=2$ factors through the change of coefficients map from the DGA over~$\\mathbb {Z}$ to the DGA over~$\\mathbb {Z}\/2$).\n\n To relate the differential on $D\\colon \\mathcal{A}\\big(S^1_{xy}(K,\\beta)\\big) \\rightarrow \\mathcal{A}\\big(S^1_{xy}(K,\\beta)\\big)$ to the differential $\\partial$ on $\\mathcal{A}(K)$ consider a $\\mathbb {Z}\/2$-algebra homomorphism\n\\[\n\\Phi\\colon \\ \\mathcal{A}(K) \\rightarrow \\operatorname{Mat}\\big(n, \\mathcal{A}\\big(S^1_{xy}(K,\\beta)\\big)\\big)\n\\]\nthat sends Reeb chords $a_k$ or $c_k$ to the corresponding $n\\times n$ matrices of Reeb chords $A_k = \\big(a^k_{i,j}\\big)$ or $C_k = \\big(c^k_{i,j}\\big)$ and satisfies\n\\begin{gather*}\n\\Phi(t_1) = P^{xy}_{\\beta},\n\\qquad \\Phi(t_k) = (I+X_k), \\qquad \\mbox{for $2\\leq k \\leq \\ell$},\n\\end{gather*}\nwhere $P^{xy}_{\\beta}$ and $P^{xz}_{\\beta}$ denote the $xy$- and $xz$-path matrices of $\\beta$ as defined in \\cite[Section~4.1]{LeRu}. Over~$\\mathbb {Z}\/2$, with the differential $D$ on $\\mathcal{A}\\big(S^1_{xy}(K,\\beta)\\big)$ extended entry-by-entry to \\linebreak $\\operatorname{Mat}\\big(n, \\mathcal{A}\\big(S^1_{xy}(K,\\beta)\\big)\\big)$ we have the identities,\n\\begin{gather}\nD Y_k = Y_k^2, \\qquad 1 \\leq k \\leq \\ell, \\qquad \\mbox{and} \\label{eq:Y1} \\\\\nD \\circ \\Phi(x) = \\Phi \\circ \\partial(x) + O(Y), \\qquad \\mbox{for any generator $x \\in\\mathcal{A}(K)$}, \\label{eq:Y2}\n\\end{gather}\nwhere $O(Y)$ denotes a term belonging to the $2$-sided ideal generated by the $Y$-generators. This is essentially as in \\cite[Section~5]{LeRu}; see especially Proposition~5.2 and Corollary~5.3 of~\\cite{LeRu}. For generalizing to the case of more than one base point, note that, for $1 \\leq k \\leq \\ell$, $\\Phi(t_k)$ is the left-to-right $xy$-path matrix for the part of the $xy$-diagram of~$\\beta$ that sits in a~neighborhood of the basepoint~$*_k$ on~$K$. As a result, the entries of $\\Phi(t_k)$ (resp.\\ $\\Phi\\big(t_k^{-1}\\big)$)\n record the possibly negative punctures that boundaries of ``thick disks'' of~$S^1_{xy}(K,\\beta)$ can have when they pass through the location of~$*_k$ on~$K$ in a way that agrees (resp.\\ disagrees) with the orientation of~$K$; see \\cite[Sections~4.1 and~5.2]{LeRu}.\n\nNow, the bijection from the statement of the proposition arises from associating to an augmentation $\\epsilon \\in \\overline{{\\rm Aug}}_1\\big(S^1_{xy}(K, \\beta), \\mathbb{F}\\big)_{Y=0}$ the matrix representation $f\\colon  \\mathcal{A}(K) \\rightarrow \\operatorname{Mat}(n,\\mathbb{F})$ given by $f = \\epsilon \\circ \\Phi$. From here~(\\ref{eq:Y1}) and~(\\ref{eq:Y2}) can then be used to show that under the assumption that $\\epsilon(Y)=0$, the augmentation equation $\\epsilon \\circ D = 0$ is equivalent to the representation equation $f \\circ \\partial = 0$, cf.\\ \\cite[Theorem~6.1]{LeRu}. (Note that the hypothesis that $\\beta$ is a reduced positive permutation braid is used as in~\\cite{LeRu} to see that the equation $\\epsilon(D \\Phi(t_1)) =0 $ is equivalent to having $\\epsilon \\circ D(x) =0$ for all generators of the form $p_i$ or $x^1_{i,j}$.)\n \\end{proof}\n\nWe are now prepared to give the proof of Lemma~\\ref{lem:count}.\n\n\\begin{proof}[Proof of Lemma \\ref{lem:count}]\\quad\n\n {\\bf Step 1.} Establish that\n\\begin{gather*}\n\\big|\\overline{\\operatorname{Rep}}_1\\big(K, \\big(\\mathbb{F}_q^n,0\\big), B_\\beta\\big)\\big| = \\frac{|{\\rm GL}(n)|^{\\ell-1}}{(q^{n(n-1)\/2})^{\\ell-1}} \\cdot \\big| \\overline{{\\rm Aug}}_1\\big(S^{1}_{xy}(K, \\beta), \\mathbb{F}_q\\big)_{Y=0}\\big|.\n\\end{gather*}\n\nKeeping in mind that $\\overline{\\operatorname{Rep}}_1(K, (\\mathbb{F}_q^n,0), B_\\beta)$ is defined in terms of the DGA $\\mathcal{A}(K, *_1, \\ldots, *_\\ell)$ of~$K$ equipped with $\\ell$ base points $*_1, \\ldots, *_\\ell$, let $\\overline{\\operatorname{Rep}}_1\\big(\\mathcal{A}(K, *), \\big(\\mathbb{F}_q^n,0\\big), B_\\beta\\big)$ denote the corresponding set of representations with respect to the DGA of~$K$ equipped with only the one initial base point~$*$, i.e., the set of ungraded representations $f\\colon (\\mathcal{A}(K,*), \\partial) \\rightarrow (\\operatorname{Mat}(n, \\mathbb{F}_q),0)$ ha\\-ving \\mbox{$f(t) \\in B_\\beta$}. The differential of a Reeb chord~$b$ of $K$ in $\\mathcal{A}(K, *_1, \\ldots, *_\\ell)$ is obtained from its differential in~$\\mathcal{A}(K,*)$ by replacing all occurrences of~$t$ by the product $t_1\\cdots t_\\ell$. Consequently,\n\\[\n\\big|\\overline{\\operatorname{Rep}}_1\\big(K, \\big(\\mathbb{F}_q^n,0\\big), B_\\beta\\big)\\big| = |{\\rm GL}(n)|^{\\ell -1} \\cdot \\big|\\overline{\\operatorname{Rep}}_1\\big(\\mathcal{A}(K, *), \\big(\\mathbb{F}_q^n,0\\big), B_\\beta\\big)\\big|,\n\\]\nwhere the factor $|{\\rm GL}(n)|^{\\ell-1}$ arises as the number of ways to factor a given matrix $f(t) \\in B_\\beta$ into a product $f(t_1) \\cdots f(t_\\ell)$ with $f(t_i) \\in {\\rm GL}(n)$ for $1 \\leq i \\leq \\ell$. On the other hand, using Proposition~\\ref{prop:LeRu} gives\n\\[\n\\big| \\overline{{\\rm Aug}}_1(S^{1}_{xy}(K, \\beta), \\mathbb{F}_q)_{Y=0}\\big| = \\big(q^{n(n-1)\/2}\\big)^{\\ell-1} \\big|\\overline{\\operatorname{Rep}}_1\\big(\\mathcal{A}(K, *), \\big(\\mathbb{F}_q^n,0\\big), B_\\beta\\big)\\big|,\n\\]\nwhere in this case the factor $\\big(q^{n(n-1)\/2}\\big)^{\\ell-1}$ arises as the number of ways to factor a given matrix $f(t) \\in B_\\beta$ into a product $f(t_1) \\cdots f(t_\\ell)$ with $f(t_1) \\in B_\\beta$ and $f(t_i) \\in N_+$ for $2 \\leq i \\leq \\ell$. Here, we use the connection with the Bruhat decomposition from \\cite[Section~4.3]{LeRu} so that~$B_\\beta$ has the form $BS_{\\beta} B$ where $S_\\beta$ is a permutation matrix and~$B$ is the group of invertible upper triangular matrices. Therefore, given $f(t) \\in B_\\beta$ and strictly upper-triangular matrices \\mbox{$f(t_2), \\ldots, f(t_{\\ell}) \\in N_+$} there is a unique element $f(t_1) \\in B_\\beta$ so that $f(t) = f(t_1) f(t_2)\\cdots f(t_{\\ell})$.\n\n{\\bf Step 2.} Establish that\n\\begin{gather*}\n\\big| \\overline{{\\rm Aug}}_1\\big(S^{1}_{xy}(K, \\beta), \\mathbb{F}_q\\big)_{Y=0}\\big| = \\big| \\overline{{\\rm Aug}}_1\\big(S^{2}_{xy}(K, \\beta), \\mathbb{F}_q\\big)_{Y=0}\\big|.\n\\end{gather*}\n\nA Legendrian isotopy of $\\beta$ as in (\\ref{eq:contact}) can be used to produce a Legendrian isotopy of $S(K,\\beta)$ that moves the $X_k$ and $Y_k$ crossings around the annular neighborhood $N_{xy}$ of the Lagrangian projections of~$K$. In particular, this procedure leads to a Legendrian isotopy $\\Lambda_t$, $0 \\leq t \\leq 1$, from $S^1_{xy}(K,\\beta)$ to $S^2_{xy}(K,\\beta)$. Note that the crossings $p_1, \\ldots, p_\\lambda$ and the base points $*_1, \\ldots, *_n$ can be assumed to remain in place during the isotopy. Moreover, since the $xy$-diagram of $\\beta$ has no vertical tangencies, by scaling the $y$- and $z$- coordinates by an appropriately small factor, it can be assumed that, for all $0\\leq t \\leq 1$, the $xy$-diagram of $\\Lambda_t$ does not have self-tangencies. As a result, the Reeb chords of $\\Lambda_t$ appear in continuous $1$-parameter families parametrized by $t \\in [0,1]$, and (identifying the corresponding generators of all $\\mathcal{A}(\\Lambda_t)$)\n the differential remains constant except for a finite number of handleslide disk bifurcations.\n These occur when an~$X_k$ or~$Y_k$ crossing of $\\beta$ passes over or under another strand of the satellite as in Move~I of \\cite[Section~6]{ENS}. During the move, there are three Reeb chords, one that is a crossing of $\\beta$ and two more of the form $a^k_{i,j}$ or $c^k_{i,j}$, that all come together at a triple point. Label the three Reeb chords as $x$, $y$, $z$ so that their lengths (i.e., the difference of $z$-coordinates at endpoints) satisfy $h(x) > h(y) >h(z)$, and note that $z$ is the crossing from $\\beta$. The DGAs before and after the triple point move are related by a DGA isomorphism $\\phi\\colon (\\mathcal{A}, \\partial) \\rightarrow (\\mathcal{A}', \\partial')$ that maps~$x$ to an element of the form $x \\pm yz$ or $x\\pm zy$ and fixes all other generators. In particular, $\\phi$ restricts to the identity on the sub-algebra generated by the $Y$-generators.\nComposing all of the DGA isomorphisms from handleslide disks, we see that there is a DGA isomorphism $\\varphi\\colon \\big(\\mathcal{A}\\big(S^1_{xy}(K, \\beta)\\big), \\partial_1\\big) \\rightarrow \\big(\\mathcal{A}\\big(S^2_{xy}(K, \\beta)\\big), \\partial_2\\big)$ that restricts to the identity on all $Y$-generators. As a result, $\\varphi^*\\colon \\overline{{\\rm Aug}}_1\\big(S^{2}_{xy}(K, \\beta), \\mathbb{F}_q\\big) \\rightarrow \\overline{{\\rm Aug}}_1\\big(S^{1}_{xy}(K, \\beta), \\mathbb{F}_q\\big)$, $\\varphi^*\\epsilon = \\epsilon \\circ \\varphi$ induces the required bijection between $\\overline{{\\rm Aug}}_1\\big(S^{1}_{xy}(K, \\beta), \\mathbb{F}_q\\big)_{Y=0}$ and $\\overline{{\\rm Aug}}_1\\big(S^{2}_{xy}(K, \\beta), \\mathbb{F}_q\\big)_{Y=0}$.\n\n{\\bf Step 3.} Establish that\n \\begin{gather} \\label{eq:Step3}\n\\big| \\overline{{\\rm Aug}}_1\\big(S^{2}_{xy}(K, \\beta), \\mathbb{F}_q\\big)_{Y=0}\\big| = \\big(q^{n(n-1)\/2}\\big)^\\ell\\big| \\overline{{\\rm Aug}}_1\\big(S^{1}_{xz}(K, \\beta), \\mathbb{F}_q\\big)_{Y=0}\\big|.\n\\end{gather}\n\nThe generators of $S^1_{xz}(K,\\beta)$ are identified with a subset of the generators of $S^2_{xy}(K,\\beta)$, and this leads to an algebra inclusion $i\\colon \\mathcal{A}\\big(S^1_{xz}(K,\\beta)\\big) \\rightarrow \\mathcal{A}\\big(S^2_{xy}(K,\\beta)\\big)$. In fact, it is not hard to check that~$i$ is a DGA homomorphism; see \\cite[Proposition~4.23]{NRSSZ} for a detailed explanation in the case where $\\beta$ is the identity braid. The difference between the two DGAs is that $\\mathcal{A}\\big(S^2_{xy}(K,\\beta)\\big)$ has additional generators of the form $c^k_{i,j}$ and $x^k_{i,j}$ with $1 \\leq k \\leq \\ell$, $1 \\leq i<j \\leq n$ that $\\mathcal{A}\\big(S^1_{xz}(K,\\beta)\\big)$ does not have. Equation~(\\ref{eq:Step3}) then follows from:\n\n{\\bf Claim:} Given any $\\epsilon \\in \\overline{{\\rm Aug}}_1\\big(S^{1}_{xz}(K, \\beta), \\mathbb{F}_q\\big)_{Y=0}$ and arbitrary values $\\epsilon'\\big(c^k_{i,j}\\big) \\in \\mathbb{F}_q$, there exists a unique augmentation $\\epsilon' \\in \\overline{{\\rm Aug}}_1\\big(S^{2}_{xy}(K, \\beta), \\mathbb{F}_q\\big)_{Y=0}$ extending these values and restricting to $\\epsilon$ on $\\mathcal{A}\\big(S^{1}_{xz}(K, \\beta)\\big)$.\n\nGiven $\\epsilon$ and $\\epsilon'\\big(c^k_{i,j}\\big)$ we need to show that there are unique values $\\epsilon'\\big(x^k_{i,j}\\big)$ for which the equations\n\\[\n\\epsilon' \\circ \\partial\\big(c^k_{i,j}\\big) = 0 \\qquad \\mbox{and} \\qquad \\epsilon' \\circ \\partial\\big(x^k_{i,j}\\big) =0, \\qquad 1 \\leq k \\leq \\ell, \\  1 \\leq i<j\\leq n\n\\]\nhold. Since $\\partial x^k_{i,j}$ belongs to the $2$-sided ideal generated by the $Y$-generators, and $\\epsilon$ vanishes on all $Y$-generators, the equations $\\epsilon' \\circ \\partial\\big(x^k_{i,j}\\big) =0$ are satisfied. Note that (again computing over~$\\mathbb {Z}\/2$)\n\\[\n\\partial C_k = (I+X_k)^{\\pm 1} + W_k,\n\\]\nwhere $W_k$ denotes a matrix with entries in the subalgebra generated by $\\mathcal{A}_{xz}^1(S(K,\\beta))$ and by the~$c^k_{i,j}$ generators. The map $\\mathbb{F}_q^{n(n-1)\/2} \\rightarrow \\mathbb{F}_q^{n(n-1)\/2}$ that sends a collection of values $\\big(\\epsilon'\\big(x^k_{i,j}\\big)\\big)_{i<j}$ to the above diagonal part of $\\epsilon'\\big((I+X_k)^{\\pm 1}\\big)$ is a bijection. Hence, there is a unique way to choose $\\epsilon'\\big(x^k_{i,j}\\big)$ so that the equations $\\epsilon \\circ\\partial c^k_{i,j}=0$, $1 \\leq i<j\\neq n$ hold (since this is equivalent to having $\\epsilon'\\big((I+X_k)^{\\pm1}\\big) = \\epsilon'(W_k)$ hold on the upper triangular entries.)\n\n{\\bf Step 4.} Establish that\n \\begin{gather} \\label{eq:Step4}\n\\big| \\overline{{\\rm Aug}}_1\\big(S^{1}_{xz}(K, \\beta), \\mathbb{F}_q\\big)_{Y=0}\\big| = (q-1)^{n-c}\\big| \\overline{{\\rm Aug}}_1\\big(S^{2}_{xz}(K, \\beta), \\mathbb{F}_q\\big)_{Y=0}\\big|.\n\\end{gather}\n\nFirst, note that when the locations of basepoints are moved around the number of $Y=0$ augmentations does not change. (This is because when a basepoint~$s_i$ moves through an $xy$-crossing~$a$, the DGAs before and afterward are related by an isomorphism $\\phi$ that maps~$a$ to an element of the form $s_i^{\\pm1} a$ or $a s_i^{\\pm1}$ (depending on orientation of $K$ and whether $s_i$ passes the upper or lower endpoint of~$a$) and fixes all other generators; see \\cite[Theorem~2.20]{NgR2012}. In particular, the induced bijection between augmentation sets, $\\phi^*$, preserves the $Y=0$ subsets (since $\\epsilon(a) = 0$ if and only if $\\epsilon\\big(t_i^{\\pm1}a\\big) =0$).\n\nTo understand the effect of changing the number of basepoints, suppose that $K_1$ and $K_2$ are two $xy$-diagrams, identical except that on $K_1$ a collection of base points with corresponding generators $u_1, \\ldots, u_s$ appears near the location of a single base point $u$ of $K_2$. Arguing as in Step 1, shows that there is a DGA map $\\phi\\colon \\mathcal{A}(K_2) \\rightarrow \\mathcal{A}(K_1)$ with $\\phi(u) = u_1\\cdots u_s$ and $\\phi(x) = x$ for all other generators, and moreover $\\phi^*\\colon \\overline{{\\rm Aug}}_1(K_1, \\mathbb{F}_q) \\rightarrow \\overline{{\\rm Aug}}_1(K_2, \\mathbb{F}_q)$ is surjective and $(q-1)^{s-1}$-to-$1$. (Here, $(q-1)^{s-1}$ represents the number of ways to factor an element of~$\\mathbb{F}_q^*$ into a product of~$s$ elements in $\\mathbb{F}_q^*$.) Since $\\phi$ is the identity on Reeb chords, when applied to~$S(K,\\beta)$, $\\phi^*$~restricts to a~surjective, $(q-1)^{s-1}$-to-$1$ map between the $Y=0$ augmentation sets. Starting with $S^1_{xz}(K,\\beta)$ and applying this procedure repeatedly with the collection of base points on each component (keeping in mind that $S^1_{xz}(K,\\beta)$ has $n$ base points while $S^2_{xz}(K,\\beta)$ has $c$) leads to the formula~(\\ref{eq:Step4}).\n\n{\\bf Step 5.} Establish a decomposition of the form\n\\[\n \\overline{{\\rm Aug}}_1\\big(S^{2}_{xz}(K, \\beta), \\mathbb{F}_q\\big)_{Y=0} = \\bigsqcup_\\rho W_\\rho,\n\\]\nwhere the disjoint union is over reduced normal rulings, $\\rho$, and\n\\[\n|W_\\rho| = (q-1)^{j(\\rho)+c} \\cdot q^{\\frac{1}{2}(-j(\\rho) +{\\rm rb}(S(K,\\beta)))}\n\\]\nwith $c$ the number of components of $S(K,\\beta)$ and ${\\rm rb}(S(K,\\beta))$ the number of Reeb chords of~$S_{xz}^2(K,\\beta)$.\n\nA decomposition of the entire augmentation variety $\\overline{{\\rm Aug}}_1\\big(S^{2}_{xz}(K, \\beta), \\mathbb{F}_q\\big)$ (without imposing the $Y=0$ condition) as $\\bigsqcup W_\\rho$ with the disjoint union over all normal rulings (without the reduced condition) is established in Theorem~3.4 of~\\cite{HenryRu}. The statement of that theorem implies that $|W_\\rho| = (q-1)^{j(\\rho)-c}q^{b(\\rho)}$ where $b(\\rho)$ is the number of ``returns''\\footnote{Given a normal ruling $\\rho$ for a Legendrian link $K'$, the $xz$-crossings of $K'$ that are not switches are either {\\it departures} or {\\it returns}. At a departure (resp.\\ return) the normality condition holds to the left (resp.\\ right) of the crossing but not to the right (resp.\\ left) of the crossing. See \\cite[Section~3]{NgSab}.} of $\\rho$ plus the number of right cusps of $\\rho$, and Lemma~5 of~\\cite{NgSab} shows that $b(\\rho) = \\frac{1}{2} ( -j(\\rho) + {\\rm rb}(S(K,\\beta)))$. Thus, it suffices to show that an augmentation $\\epsilon \\in \\overline{{\\rm Aug}}_1\\big(S^{2}_{xz}(K, \\beta), \\mathbb{F}_q\\big)$ belongs to $W_\\rho$ with $\\rho$ {\\it reduced} if and only if the $Y=0$ condition is satisfied.\n\nA summary of the construction of the decomposition $\\overline{{\\rm Aug}}_1\\big(S^{2}_{xz}(K, \\beta), \\mathbb{F}_q\\big)=\\bigsqcup W_\\rho$ is as follows. Let $K' \\subset J^1\\mathbb {R}$ be a Legendrian in plat position whose DGA is computed from resolving the front projection of $K'$ and positioning a single base point on each component of $K'$ in the loop near some chosen right cusp. The DGA of $S^2_{xz}(K,\\beta)$ is of this required type. For such a~$K'$, the article~\\cite{HenryRu} considers objects called Morse complex sequences (MCSs) that consist of a sequence of chain complexes and formal handleslide marks (which are vertical segments on the front diagram of $K'$) subject to several axioms motivated by Morse theory. Section~5 of \\cite{HenryRu} gives a bijection between $\\overline{{\\rm Aug}}_1(K', \\mathbb{F}_q)$ and the set of ``$A$-form''\\footnote{An MCS is in {\\it $A$-form} if its handleslides only appear in specified locations on the front diagram of $K'$ to the left of crossings and right cusps. See \\cite[Section~5]{HenryRu}.} MCSs for $K'$, denoted here $\\operatorname{MCS}^A(K')$, where an augmentation $\\epsilon$ corresponds to an $A$-form MCS with one handleslide mark just to the left of every crossing or right cusp, $x$, with $\\epsilon(x) \\neq 0$ with the handleslide coefficient determined by the value $\\epsilon(x)$. In Section~4.1 of~\\cite{HenryRu}, another class of MCSs called ``$SR$-form'' MCSs are considered; we will denote the set of all $SR$-form MCSs for~$K'$ as~$\\operatorname{MCS}^{SR}(K')$. Each $SR$-form MCS, $\\mathcal{C}$, has an associated normal ruling~$\\rho$ of~$K'$, and all handleslide marks of $\\mathcal{C}$ appear in collections of a~standard form near switches, returns, and right cusps of~$\\rho$. In particular, at every switch of~$\\rho$,~$\\mathcal{C}$ must have a collection of handleslides with non-zero coefficients; returns and right cusps may or may not have handleslides. In Section~6 of~\\cite{HenryRu} a bijection $\\Psi\\colon \\operatorname{MCS}^A(K') \\rightarrow \\operatorname{MCS}^{SR}(K')$ and its inverse $\\Phi=\\Psi^{-1}$ are constructed. By definition, $\\epsilon \\in W_\\rho$ if the corresponding $A$-form MCS, $\\mathcal{C}_\\epsilon$, is such that the $SR$-form MCS $\\Psi(\\mathcal{C}_\\epsilon)$ has associated normal ruling~$\\rho$.\n\nNote that for an $A$-form or $SR$-form MCS every handleslide has some associated $xz$-crossing or right cusp. (For an $SR$-form MCS with normal ruling $\\rho$, associated crossings can only be switches or returns of $\\rho$ and a single crossing may have more than one handleslide associated to it.) Let $\\operatorname{MCS}^A(S(K,\\beta))_{Y=0}$ and $\\operatorname{MCS}^{SR}(S(K,\\beta))_{Y=0}$ denote those $A$-form and $SR$-form MCSs for $S^2_{xz}(K,\\beta)$ that do not have any handleslide marks associated to $xz$-crossings corresponding to the $Y$-generators. (These are the groups of $n(n-1)\/2$ crossings that appear in $S^2_{xz}(K,\\beta)$ near the location of left cusps of~$K$.) Step~5 is then completed by the following.\n\n\\begin{Lemma}The above constructions from {\\rm \\cite{HenryRu}} restrict to bijections\n\\[\n \\overline{{\\rm Aug}}_1\\big(S^{2}_{xz}(K, \\beta), \\mathbb{F}_q\\big)_{Y=0} \\leftrightarrow \\operatorname{MCS}^A(S(K,\\beta))_{Y=0} \\leftrightarrow \\operatorname{MCS}^{SR}(S(K,\\beta))_{Y=0}\n\\]\nand an $SR$-form MCS $\\mathcal{C}$ belongs to $\\operatorname{MCS}^{SR}(S(K,\\beta))_{Y=0}$ if and only if the associated normal ruling $\\rho$ is reduced.\n\\end{Lemma}\n\\begin{proof}The first bijection is clear since an $A$-form MCS has no handleslides at $Y$ crossings if and only if the corresponding augmentation vanishes on $Y$-generators. Turning to the second bijection, given an $A$-form or $SR$-form MCS, $\\mathcal{C}$, let $i(\\mathcal{C})$ denote the first (from left to right) $xz$-crossing of $S^2_{xy}(K,\\beta)$ that has a handleslide associated to it. The defining requirement for both $\\operatorname{MCS}^A(S(K,\\beta))_{Y=0}$ and $\\operatorname{MCS}^{SR}(S(K,\\beta))_{Y=0}$ is equivalent to $i(\\mathcal{C})$ not being one of the $Y$-crossings. (The $Y$-crossings all appear to the left of the other $xz$-crossings of $S^2_{xz}(K,\\beta)$ since $K$ is in plat position.) Examining the definition of $\\Psi$ and $\\Phi$ in Section 6 of \\cite{HenryRu}, it is straightforward to see that $i(\\Psi(\\mathcal{C})) = i(\\mathcal{C})$ and $i(\\Phi(\\mathcal{C})) = i(\\mathcal{C})$, so that $\\Psi$ and $\\Phi$ restrict to provide the bijection $\\operatorname{MCS}^A(S(K,\\beta))_{Y=0} \\leftrightarrow \\operatorname{MCS}^{SR}(S(K,\\beta))_{Y=0}$.\n\nFor the final statement of the lemma, note that a normal ruling of $S^2_{xz}(K,\\beta)$ is reduced if and only if it has no switches at $Y$-crossings. See \\cite[Lemma~3.2]{NgR2012}. Moreover, if a $Y$-crossing is a~return then there must be another $Y$-crossing somewhere to its left that is a switch. (If there are no switches in the $Y$-crossings then, it is easy to see that all $Y$-crossings are departures.) Thus, for $\\mathcal{C}$ in $SR$-form, $i(\\mathcal{C})$ is a $Y$-crossing if and only if the corresponding ruling is not reduced.\n\\end{proof}\n\n{\\bf Step 6.} Completion of the proof.\n\nCombining the identities from Steps 1--5, we have\n\\begin{align*}\n\\big|\\overline{\\operatorname{Rep}}_1\\big(K, \\big(\\mathbb{F}_q^n,0\\big), B_\\beta\\big)\\big| & = |{\\rm GL}(n)|^{\\ell-1}  q^{n(n-1)\/2} (q-1)^{n-c}  \\sum_{\\rho} (q-1)^{j(\\rho)+c}q^{\\frac{1}{2}(-j(\\rho) + {\\rm rb}(S(K,\\beta)))} \\\\\n& = |{\\rm GL}(n)|^{\\ell-1}  q^{n(n-1)\/2} (q-1)^{n}   q^{\\frac{1}{2}{\\rm rb}(S(K,\\beta))}  \\sum_{\\rho} (q^{1\/2}-q^{-1\/2})^{j(\\rho)} \\\\\n& = |{\\rm GL}(n)|^{\\ell-1}  q^{n(n-1)\/2} (q-1)^{n}   q^{n^2 {\\rm rb}(K)\/2} q^{\\lambda(\\beta)\/2} \\widetilde{R}_{S(K,\\beta)}(z),\n\\end{align*}\nwhere the summations are over all {\\it reduced} normal rulings and at the last equality we used that the number of Reeb chords of $S^2_{xz}(K,\\beta)$ is $n^2 \\cdot {\\rm rb}(K)+\\lambda(\\beta)$.\n\\end{proof}\n\n\\section{The multi-component case} \\label{sec:multicomp}\n\nFor simplicity, we have restricted the focus of this article to the case where $K$ is a connected Legendrian knot. We close with a discussion of an appropriate modification of Theorem~\\ref{thm:main} for the case of Legendrian links with multiple components.\n\nWhen $K = \\sqcup_{i =1}^c K_i$ is a Legendrian link with $c$ components, for $\\vec{n} = (n_1, \\ldots, n_c)$ with $n_i\\geq 1$ one can consider the $\\vec{n}$-colored Kauffman polynomial defined by satelliting each $K_i$ with the symmetrizer from the $n_i$-stranded BMW algebra in a multi-linear manner. With the $\\vec{n}$-colored $1$-graded ruling polynomial defined by the analogous modification,\n\\[\nR^1_{\\vec{n},K}(q) = \\left(\\prod_{i=1}^c\\frac{1}{c_{n_i}} \\right)\\sum_{\\vec{\\beta} \\in S_{n_1} \\times \\cdots \\times S_{n_c}} \\left(\\prod_{i=1}^c q^{\\lambda(\\beta_i)\/2} \\right) \\widetilde{R}_{S(K,\\vec{\\beta})}(z)\\vert_{z = q^{1\/2}-q^{-1\/2}},\n\\]\nthe second equality of Theorem \\ref{thm:main}, modified to read $R^1_{\\vec{n},K}(z)=F_{\\vec{n},K}(a,q)|_{a^{-1}=0}$, follows by a~mild variation on the arguments of Sections~\\ref{sec:ncolored} and~\\ref{sec:Kauffman}.\n\nTo obtain an additional equality with a total representation number, one should work with the so-called {\\it composable algebra} version of the LCH DGA, $(\\mathcal{A}_{{\\rm comp}}, \\partial)$, cf.~\\cite{BEE, EENS}. The underlying algebra $\\mathcal{A}_{{\\rm comp}}$ has generators $b_1, \\ldots, b_r$ and $t_1^{\\pm1}, \\ldots, t_{\\ell}^{\\pm1}$ from Reeb chords and basepoints as well as idempotent generators $e_1, \\ldots, e_c$ corresponding to the components of $K$. Moreover, $\\mathcal{A}_{{\\rm comp}}$ has relations\n\\begin{alignat*}{3}\n& e_ie_j = \\delta_{i,j}, \\qquad & & \\displaystyle \\sum_{i=1}^c e_i = 1,& \\\\\n& e_i b_k = \\delta_{i,u(k)} b_k, & & b_ke_i = \\delta_{i, l(k)} b_k, & \\\\\n & e_i t_k = t_k e_i = \\delta_{i, s(k)} t_k, \\qquad & & t_kt_k^{-1} = t_k^{-1}t_k = e_{s(k)}, &\n\\end{alignat*}\nwhere the upper (resp.\\ lower) endpoint of the Reeb chord~$b_k$ is on the component $K_{u(k)}$ (resp.\\ $K_{l(k)}$) and the basepoint $t_k$ sits on the~$K_{s(k)}$ component. Note that an algebra representation $f\\colon \\mathcal{A}_{{\\rm comp}} \\rightarrow \\operatorname{End}(V)$ is equivalent to a~collection of vector spaces $V_1, \\ldots, V_c$ together with linear maps\n\\[\nf(b_k)\\colon \\ V_{l(k)} \\rightarrow V_{u(k)}\n\\]\nassigned to Reeb chords, and invertible linear maps\n\\[\nf(t_k)\\colon \\ V_{s(k)} \\rightarrow V_{s(k)}\n\\]\nassigned to base points. Here, the $V_i$ are determined from $V$ via $V_i = f(e_{i}) V$. (This is as in the correspondence between quiver representations and representations of the corresponding path algebra, see, e.g., \\cite[Section~1.2]{Brion}. Except for the relation $t_kt_k^{-1} = t_k^{-1}t_k = e_{s(k)}$, the composable algebra $\\mathcal{A}_{{\\rm comp}}$ is precisely the path algebra associated to the quiver with vertices indexed by components of $K$ and edges corresponding to Reeb chords and base points.) The DGA differential $\\partial$ on $\\mathcal{A}_{{\\rm comp}}$ is defined to satisfy $\\partial e_i = \\partial t_k= 0$ and by the usual holomorphic disk count on Reeb chords.\n\nFor $\\vec{n}= (n_1, \\ldots, n_c)$, with $n_i \\geq 1$ as above, we denote by $\\overline{\\operatorname{Rep}}_1\\big(K, \\big(\\mathbb{F}^{\\vec{n}}_q, 0\\big)\\big)$\nthe set of DGA representations $f\\colon (\\mathcal{A}_{\\rm comp}, \\partial) \\rightarrow \\big(\\operatorname{End}\\big(\\oplus_{i} \\mathbb{F}_q^{n_i}\\big), 0\\big)$ that,\nwhen viewed as quiver representations, assign the collection of vector spaces $\\mathbb{F}_q^{n_1}, \\ldots, \\mathbb{F}_q^{n_c}$ to the components of~$K$, i.e., $f(e_i)$ is the projection to the $\\mathbb{F}_{q}^{n_i}$ component. Note that to obtain a Legendrian isotopy invariant we need to adjust the normalizing factor from~\\cite{LeRu} used in Definition~\\ref{def:total}. This is done by defining the ($1$-graded) {\\it total $\\vec{n}$-dimensional representation number} of~$K$ to be\n\\begin{align*}\n\\operatorname{Rep}_1\\big(K, \\mathbb{F}^{\\vec{n}}_q\\big) & := \\left(\\prod_{i,j} \\big|\\operatorname{Hom}_{\\mathbb{F}_q}\\big(\\mathbb{F}^{n_j}_q, \\mathbb{F}^{n_i}_q\\big)\\big|^{-{\\rm rb}_{i,j}(K)\/2}\\right)\n\\\\\n & \\quad {}\\times \\left( \\prod_i \\big|{\\rm GL}\\big(n, \\mathbb{F}^{n_i}_q\\big)\\big|^{-\\ell_i} \\right)  \\big| \\overline{\\operatorname{Rep}}_1\\big(K, \\big(\\mathbb{F}^{\\vec{n}}_q, 0\\big)\\big)\\big| \\\\\n & = \\left(\\prod_{i,j} \\big(q^{n_in_j}\\big)^{-{\\rm rb}_{i,j}(K)\/2}\\right) \\\\\n & \\quad {} \\times \\left(\\prod_{i} \\left(q^{n_i(n_i-1)\/2}   \\prod_{m=1}^{n_i}\\big(q^m-1\\big)\\right)^{-\\ell_i} \\right)  \\big| \\overline{\\operatorname{Rep}}_1\\big(K, \\big(\\mathbb{F}^{\\vec{n}}_q, 0\\big)\\big) \\big|,\n\\end{align*}\n where ${\\rm rb}_{i,j}(K)$ is the number of Reeb chords, $b_k$, with $u(k) =i$ and $l(k) = j$ and $\\ell_i$ is the number of basepoints on the $K_i$ component of $K$.\n\nWith the composable algebra used for $K$, a suitable modification of Theorem~6.1 from~\\cite{LeRu} relating augmentations of the satellites $S\\big(K, \\vec{\\beta}\\big)$ with higher dimensional representations of $(\\mathcal{A}_{{\\rm comp}}(K), \\partial)$ continues to hold. [The key point being that when components $K_i$ and $K_j$ are satellited with~$n_i$ and~$n_j$ stranded braids respectively, each Reeb chord, $b_k$, with $u(k) = i$ and $l(k) =j$ corresponds to an $n_i \\times n_j$ matrix of Reeb chords in the satellite $S(K, \\vec{\\beta})$.] From this starting point, the analog of the first equality of Theorem~\\ref{thm:main}, $\\operatorname{Rep}_1\\big(K, \\mathbb{F}^{\\vec{n}}_q\\big)= R^1_{\\vec{n},K}(z)$, can be deduced as in Section~\\ref{sec:5}.\n\n\\subsection*{Acknowledgements}\nThis article is dedicated to Dmitry Fuchs to whom the second author is grateful for his generosity and support through years in grad school and beyond. Thank you Dmitry! DR acknowledges support from Simons Foundation grant~\\#429536.\n\n\\pdfbookmark[1]{References}{ref}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nThe Polonyi problem~\\cite{PLB131-59,PRD49-779} is one of the most serious\nproblems in models based on the $N=1$ supergravity~\\cite{NPB212-413}. In a\nwide class of supergravity models, the Polonyi field $\\phi$, which is\na scalar field related to the supersymmetry (SUSY) breaking,\nhas a mass $m_\\phi$ of the order of the gravitino mass. During\ninflation, $\\phi$ takes an amplitude of the order of the gravitational\nscale $M\\equiv M_{\\rm pl}\/\\sqrt{8\\pi}\\simeq 2.4\\times\n10^{18} {\\rm GeV} $. After the inflation, the condensation of $\\phi$ starts to\noscillate when the expansion rate of the universe, $H$, becomes\ncomparable to $m_\\phi$ and finally decays into particles in the\nobservable sector. Since the interactions of the Polonyi field are\nsuppressed by powers of $M^{-1}$, the decay rate of the Polonyi field,\n$\\Gamma_\\phi$, is very small as\n\\begin{equation}\n    \\Gamma_\\phi \\sim N\\frac{m_\\phi^3}{M_{\\rm pl}^2},\n    \\label{decay_rate}\n\\end{equation}\nwhere $N$ is the number of the decay modes. (In the following\ncalculations, we take $N=100$.)  Therefore, the Polonyi field is\nexpected to decay when the temperature of the universe becomes very\nlow. The reheating temperature $T_R$ due to the decay of the Polonyi\nfield is given by\n\\begin{equation}\n    \\label{rtemp}\n    T_R \\sim 1 {\\rm MeV} \n    \\left(\\frac{m_{\\phi}}{10{\\rm TeV}}\\right)^{3\/2}.\n\\end{equation}\nThis causes serious cosmological difficulties; namely the Polonyi\nfield may destroy the great success of the big-bang nucleosynthesis\n(BBN), and the entropy production due to its decay may dilute the\nbaryon number of the universe too much.\n\nIn the previous works~\\cite{PLB174-176,PLB342-105} it has been pointed\nout that the Polonyi problem can be solved if the gravitino mass\n$m_{3\/2}$ (which is the same order of the Polonyi mass) is larger than\n$O(10) {\\rm TeV} $ in order to hasten the decay of the Polonyi field. Thus,\nit is desirable to raise the gravitino mass while keeping the\neffective SUSY breaking scale in the observable sector at\n$O(100) {\\rm GeV} $. In no-scale type supergravity models, such mass\nhierarchy is realized~\\cite{PRD50-2356} and hence the no-scale type\nsupergravity model with $m_{3\/2} \\mathop{}_{\\textstyle \\sim}^{\\textstyle >}  O(10) {\\rm TeV} $ has been suggested as\nan attractive solution to the Polonyi problem.\n\nIn reference~\\cite{PLB342-105}, however, it has been also pointed out\nthat the mass density of the lightest superparticle (LSP) produced by\nthe decay of the Polonyi field may overclose our universe if LSP is\nstable. As we will see below, the mass density of LSP increases as the\nreheating temperature $T_R$ due to the decay of the Polonyi field\ndecreases.  Therefore, a lowerbound on $T_R$ is derived requiring that\nthe present mass density of LSP should not exceed the critical density\nof the universe $\\rho_c$. In this letter, we obtain the lowerbound on\n$T_R$ in the framework of the minimal SUSY SU(5) model with no-scale\ntype boundary conditions on the SUSY breaking parameters.\n\n\n\n\\section{The Model}\n\\label{sec:model}\n\nBefore starting cosmological arguments, let us first describe our\nbasic assumptions. We consider the minimal SUSY SU(5) model with\nno-scale type boundary conditions. This model has three types of Higgs\nfield; $H({\\bf 5})$ and $\\bar{H}({\\bf 5^*})$ which contain flavor\nHiggses $H_f$ and $\\bar{H}_f$, and $\\Sigma ({\\bf 24})$ whose\ncondensation breaks the SU(5) group into the gauge group of the\nminimal SUSY standard model (MSSM), $\\rm SU(3)_C\\times SU(2)_L\\times\nU(1)_Y$. For the Higgs sector, the superpotential is given by\n\\begin{equation}\n    W = \\frac{1}{3} \\lambda {\\rm tr} \\Sigma^3 \n    + \\frac{1}{2} M_\\Sigma {\\rm tr} \\Sigma^2\n    + \\kappa \\bar{H} \\Sigma H\n    + M_H \\bar{H} H,\n    \\label{W_su5}\n\\end{equation}\nwhere $\\lambda$ and $\\kappa$ are dimensionless constants, while\n$M_\\Sigma$ and $M_H$ are mass parameters which are of the order of the\ngrand unified theory (GUT) scale $M_{\\rm GUT} (\\sim\n10^{16} {\\rm GeV} )$. Furthermore, the model also has the soft SUSY breaking\nterms;\n\\begin{equation}\n    {\\cal L}_{\\rm soft} = \n      - \\frac{1}{3} \\lambda A_\\Sigma {\\rm tr} \\Sigma^3 \n      - \\frac{1}{2} M_\\Sigma B_\\Sigma {\\rm tr} \\Sigma^2\n      - \\kappa A_H \\bar{H} \\Sigma H\n      - M_H B_H \\bar{H} H +h.c.,\n    \\label{L_soft_su5}\n\\end{equation}\nwhere $A_\\Sigma$, $B_\\Sigma$, $A_H$ and $B_H$ are SUSY breaking\nparameters.  Minimising the Higgs potential, we find the following\nstationary point;\n\\begin{equation}\n    \\vev{\\Sigma} = \n    \\frac{1}{\\lambda}  \\left \\{  \n    M_\\Sigma + 2  \\left (  A_\\Sigma - B_\\Sigma  \\right ) \n    + O \\left (  \\frac{A_\\Sigma}{M_\\Sigma}, \\frac{B_\\Sigma}{M_\\Sigma}  \\right )  \n     \\right \\}  \\times {\\rm diag}(2,2,2,-3,-3),\n    \\label{vac_su5}\n\\end{equation}\nwhere the SU(5) is broken down to $\\rm SU(3)_C\\times SU(2)_L\\times\nU(1)_Y$. Regarding this stationary point as the vacuum, we obtain MSSM\nas the effective theory below the GUT scale $M_{\\rm GUT}$. Here, the\nmasslessness of the flavor Higgses $H_f$ and $\\bar{H}_f$ is achieved\nby a fine tuning among several parameters; $M_H - 3 \\kappa\nM_\\Sigma\/\\lambda \\simeq \\mu_H$, where $\\mu_H$ is the SUSY-invariant\nHiggs mass in MSSM.\n\nIn the present model, the parameters in MSSM at the electroweak scale\nis obtained by solving renormalization group equations (RGEs). Our\nmethod is as follows. The boundary conditions on the parameters in the\nminimal SUSY SU(5) model are given at the gravitational scale $M$.\nSince we assume the no-scale type supergravity models, all the SUSY\nbreaking parameters except for the gaugino mass vanish at the\ngravitational scale. From the gravitational scale to the GUT scale,\nthe parameters follow the renormalization group flow derived from RGEs\nin the minimal SUSY SU(5) model. Then we determine the parameters in\nMSSM at the GUT scale through an appropriate matching condition\nbetween the parameters in the SUSY SU(5) model and those in MSSM.\nFinally, we use RGEs in MSSM from the GUT scale to the electroweak\nscale in order to obtain the low energy parameters.\n\nAs for the matching condition, we have a comment. In the stationary\npoint (\\ref{vac_su5}), the mixing soft mass term of the two flavor\nHiggs bosons, $m_{12}^2\\bar{H}_fH_f$, is generated at the tree level,\nwhere $m_{12}^2$ is given by\n\\begin{equation}\n    m_{12}^2 (M_{\\rm GUT}) \\simeq \n       \\left [  \\frac{6\\kappa}{\\lambda} \n      (A_\\Sigma - B_\\Sigma)(A_H - B_\\Sigma) \n      -\\mu_H B_H.\n       \\right ] _{\\mu =M_{\\rm GUT}}\n      \\label{m12^2}\n\\end{equation}\nSince the mixing mass term depends on unknown parameters, $\\lambda$\nand $\\kappa$ in eq.(\\ref{W_su5}), we regard $m_{12}^2$ as a free\nparameter taking account of the uncertainty of $\\lambda$ and $\\kappa$\nin our analysis.  Then, the low energy parameters are essentially\ndetermined by the gauge and Yukawa coupling constants and the\nfollowing three parameters; the supersymmetric Higgs mass $\\mu_H$, the\nmixing mass of the two flavor Higgs bosons $m_{12}^2$, and the unified\ngaugino mass.\\footnote\n{In fact, parameters in MSSM slightly depend on the parameters in the\nSUSY GUT such as $\\lambda$, $\\kappa$ and so on. In our numerical\ncalculation, we ignore the effects of these parameters on the \nrenormalization group flow.}\nHowever, it is more convenient to express these parameters by other\nphysical ones. In fact, one combination of them is constrained so that\nthe flavor Higgs bosons have correct vacuum expectation values;\n$\\langle H_f\\rangle^2+\\langle\\bar{H}_f\\rangle^2\\simeq (174 {\\rm GeV} )^2$. As\nthe other two physical parameters, we use the mass of LSP, $m_{\\rm\nLSP}$, and the vacuum angle\n$\\tan\\beta\\equiv\\langle{H_f}\\rangle\/\\langle{\\bar{H}_f}\\rangle$. Thus,\nonce we fix $m_{\\rm LSP}$ and $\\tan\\beta$, we can determine all the\nparameters in MSSM.\\footnote\n{Yukawa coupling constants are determined so that the fermions have\ncorrect masses. The gauge coupling constants are also fixed so that\ntheir correct values at the electroweak scale are reproduced.}\n\nFollowing the above procedure, we solve the RGEs numerically, and\nobtain the low energy parameters in MSSM. Then, we determine the mass\nspectrum and the mixing angles for all superparticles. One remarkable\nthing is that {\\it LSP almost consists of bino $\\tilde{B}$ which is\nthe superpartner of the gauge field for $U(1)_Y$} if we require that\nLSP is neutral. Therefore, in our model, the LSP mass $m_{\\rm LSP}$ is\nessentially equivalent to the bino mass. This fact simplifies the\nfollowing analysis very much.\n\n\\section{Density of LSP}\n\\label{sec:density}\n\nNow we are in a position to discuss the mass density of LSP produced\nby the decay of the Polonyi field. The decay of the Polonyi field\nproduces a large number of superparticles, which promptly decay into\nLSPs. The number density of LSP produced by the decay, $n_{{\\rm\nLSP},i}$, is of the same order of that of the Polonyi field\n$n_\\phi\\equiv\\rho_\\phi \/m_\\phi$ (with $\\rho_\\phi$ being the energy\ndensity of the Polonyi field). Just after the decay of the Polonyi\nfield, the yield variable for LSP, $Y_{\\rm LSP}$, which is defined by\nthe ratio of the number density of LSP to the entropy density $s$, is\ngiven by\n\\begin{eqnarray}\n    m_{\\rm LSP} Y_{\\rm LSP} & \\simeq & \\frac{\\rho_\\phi}{s}\n    \\simeq \\frac{m_{\\rm LSP}\\rho_{{\\rm LSP},i}}{m_\\phi s}\n    \\sim \\left(\\frac{m_{\\rm LSP}T_R}{m_\\phi}\\right)\n    \\nonumber \\\\\n    & \\sim & 10^{-5} {\\rm GeV}  \\left(\\frac{m_{\\rm LSP}}{100 {\\rm GeV} }\\right)\n    \\left(\\frac{T_R}{1 {\\rm MeV} }\\right)\\left(\\frac{10 {\\rm TeV} }{m_\\phi}\\right),\n    \\label{mY_LSP}\n\\end{eqnarray}\nwhere $\\rho_{{\\rm LSP},i}$ is the mass density of LSP just after the\ndecay of the Polonyi field. If LSP is stable and the pair annihilation\nof LSP is not effective, $Y_{\\rm LSP}$ is conserved until today.\nComparing the ratio given in eq.(\\ref{mY_LSP}) with the ratio of the\ncritical density $\\rho_c$ to the present entropy density $s_{0}$,\n\\begin{equation}\n    \\frac{\\rho_c}{s_{0}} \\simeq 3.6 \\times 10^{-9}h^2~ {\\rm GeV} ,\n    \\label{critical}\n\\end{equation}\nwhere $h$ is the Hubble constant in units of 100km\/sec\/Mpc, we see\nthat LSP overcloses the universe in the wide parameter region for\n$m_{\\rm LSP}, m_{\\phi}$ and $T_R$ which we are concerned with.\n\nIf the pair annihilation of LSP takes place effectively, its abundance\nis reduced to \n\\begin{equation}\n    \\frac{n_{\\rm LSP}}{s} \\simeq \n    \\left. \\frac{H}{s\\langle\\sigma_{\\rm ann}v_{\\rm rel}\\rangle} \n       \\right | _{T=T_R},\n      \\label{abundance_LSP}\n\\end{equation}\nwhere $\\sigma_{\\rm ann}$ is the annihilation cross section, $v_{\\rm\nrel}$ is the relative velocity, and $\\langle\\cdots\\rangle$ represents\nthe average over the phase space distribution of LSP. From\neq.(\\ref{abundance_LSP}), we obtain a lowerbound on the annihilation\ncross section,\n\\begin{equation}\n    \\langle\\sigma_{\\rm ann}v_{\\rm rel}\\rangle  \\mathop{}_{\\textstyle \\sim}^{\\textstyle >} \n    3\\times 10^{-8}h^{-2} {\\rm GeV} ^{-2}\n    \\left( \\frac{m_{\\rm LSP}}{100 {\\rm GeV} }\\right)\n    \\left( \\frac{100 {\\rm MeV} }{T_R}\\right),\n    \\label{sv_limit}\n\\end{equation}\nin order that the mass density of LSP does not overclose the universe. \n\nComparing this bound with the annihilation cross section of LSP, we\nderive a bound on the reheating temperature by the decay of the\nPolonyi field. Since LSP is most dominated by bino, it annihilates\ninto fermion pairs. The annihilation cross section is given\nby~\\cite{PLB230-78}\n\\begin{equation}\n    \\langle\\sigma_{\\rm ann}v_{\\rm rel}\\rangle \n    = a + b\\langle v^2\\rangle,\n    \\label{sigma*v}\n\\end{equation}\nwhere $\\langle v^2\\rangle$ is the average velocity of LSP,\nand\n\\begin{eqnarray}\n    a  & \\simeq &\n    \\frac{32\\pi\\alpha_1^2}{27} \n    \\frac{m_t^2}{(m_{\\tilde{t}_R}^2 + m_{\\rm LSP}^2 - m_t^2)^2}\n     \\left (  1 - \\frac{m_t^2}{m_{\\rm LSP}^2}  \\right ) ^{1\/2} \n    \\theta (m_{\\rm LSP}-m_t),\n    \\label{s-wave} \\\\\n    b &\\simeq& \\frac{8\\pi\\alpha_1^2}{3} \\sum_{m_f\\leq T} Y_f^4  \\left \\{ \n    \\frac{m_{\\rm LSP}^2}{(m_{\\rm LSP}^2+m_{\\tilde{f}}^2)^2}\n    - \\frac{2m_{\\rm LSP}^4}{(m_{\\rm LSP}^2+m_{\\tilde{f}}^2)^3}\n    + \\frac{2m_{\\rm LSP}^6}{(m_{\\rm LSP}^2+m_{\\tilde{f}}^2)^4}  \\right \\} .\n    \\label{p-wave}\n\\end{eqnarray}\nHere, $\\alpha_1^2\\equiv g_1^2\/4\\pi\\simeq 0.01$ represents the fine\nstructure constant for U(1)$_{\\rm Y}$, $m_t$ the top-quark mass, $Y_f$\nthe hypercharge of the fermion $f$, and $m_{\\tilde{f}}$ the mass of\nthe sfermion $\\tilde{f}$.  Notice that $a$ and $b$ terms correspond to\n$s$- and $p$-wave contributions, respectively. Taking\n$m_{\\tilde{f}}\\sim m_{\\rm LSP}\\sim 100 {\\rm GeV} $, the annihilation cross\nsection given in eq.(\\ref{sigma*v}) is at most $3\\times\n10^{-8} {\\rm GeV} ^{-2}$. Using this result in the inequality\n(\\ref{sv_limit}), we can see that the reheating temperature must be\nhigher than about 100MeV even if $\\langle v^2\\rangle\\sim 1$. If the\naverage velocity is smaller than 1, the constraint becomes more\nstringent, as we will see below.\n\n\n\n\\section{Thermalization of LSP}\n\\label{sec:thermalization}\n\n\nIn order to obtain the precise lowerbound on the reheating temperature\n$T_R$, we have to know $\\langle v^2\\rangle$, as well as the mass\nspectrum of the superparticles on which the annihilation cross section\ndepends. First, let us discuss the averaged velocity of LSP, $\\langle\nv^2\\rangle$. Since LSP is mostly the bino, it loses its energy by\nscattering off the background fermions. In the model with the no-scale\ntype boundary conditions, right-handed sleptons become the lightest\namong the sfermions, and hence LSP loses its energy mainly by\nscattering off the background electron (and $\\mu$ and $\\tau$, if the\ntemperature is higher than their masses). If LSP is relativistic, the\ncross section for this process, $\\sigma_{\\rm scatt}$, is estimated as\n\\begin{equation}\n    \\langle \\sigma_{\\rm scatt} v_{\\rm rel} \\rangle \n    \\simeq 128\\pi \\alpha_1^2\n    \\frac{E_{\\rm LSP}^2 T_R^2}{m_{\\tilde{e}_R}^4 m_{\\rm LSP}^2},\n    \\label{sigma_scatt}\n\\end{equation}\nwhere $E_{\\rm LSP}$ is the energy of LSP, and $m_{\\tilde{e}_R}$ the\nmass of the right-handed selectron.\\footnote%\n{This cross section is applied for $E_{\\rm LSP} T_{R} \\ll\nm_{\\tilde{e}_R}^2$.}\nThe energy loss rate for\nthe relativistic LSP, $\\Gamma_{\\rm scatt}^{\\rm R}$, is given by\n\\begin{equation}\n    \\label{loss-rate}\n    \\Gamma_{\\rm scatt}^{\\rm R} \\simeq \n     n_e \\langle \\sigma_{\\rm scatt} v_{\\rm rel} \\rangle \n     \\frac{\\Delta E_{\\rm LSP}}{E_{\\rm LSP}},\n\\end{equation}\nwhere $n_e$ represents the number density of the background electron\nand $\\Delta E_{\\rm LSP}$ is the averaged  energy loss of LSP in \none scattering which is  given by\n\\begin{equation}\n    \\label{energy-loss}\n    \\Delta E_{\\rm LSP} \\simeq  \n    -12E_{\\rm LSP}\\left(\\frac{T_R E_{\\rm LSP}}{m_{\\rm LSP}^2}\\right).\n\\end{equation}\nTaking the ratio of the energy loss  rate $\\Gamma_{\\rm\nscatt}^{\\rm R}$ to the expansion rate $H$ of the universe, we find\n\\begin{equation}\n    \\left. \\frac{\\Gamma_{\\rm scatt}^{\\rm R}}{H}\n       \\right | _{E_{\\rm LSP}\\gg m_{\\rm LSP}}\n      \\simeq \n      2\\times10^3 \\left( \\frac{E_{\\rm LSP}}{10^2 {\\rm GeV} } \\right)^3\n      \\left( \\frac{T_R}{100 {\\rm MeV} } \\right)^4\n      \\left( \\frac{100 {\\rm GeV} }{m_{\\tilde{e}_R}} \\right)^4\n      \\left( \\frac{100 {\\rm GeV} }{m_{\\rm LSP}} \\right)^4.\n\\end{equation}\nThus, if $T_R  \\mathop{}_{\\textstyle \\sim}^{\\textstyle >} $ a few $\\times$ 10MeV, the energetic LSP loses its\nenergy through the scattering off thermal electrons efficiently for\n$m_{\\tilde{e}_R}\\sim m_{\\rm LSP}\\simeq O(100\/GEV)$, and becomes a\nnon-relativistic particle.\n\n\nThe non-relativistic LSP further loses its energy by scattering off\nbackground electrons. The averaged loss of the kinetic energy for the\nnon-relativistic LSP in one scattering process, $\\Delta \\epsilon_{\\rm\nLSP}$, is given\nby\\footnote{%\nNaively, it is expected that $\\Delta \\epsilon_{\\rm LSP}$ is $\\sim\nT_R$. However this order of the energy loss is cancelled out when the\naverage is taken over angles of the incident particles and  the actual\nenergy loss is much smaller than the naive expectation.}\n\\begin{equation}\n    \\Delta \\epsilon_{\\rm LSP} \\simeq\n    -\\frac{20\\epsilon_{\\rm LSP} T_R}{m_{\\rm LSP}}\n    \\left( 1 -\\frac{T_R}{\\epsilon_{\\rm LSP}}\\right),\n    \\label{Delta_E}\n\\end{equation}\nwhere $\\epsilon_{\\rm LSP}\\equiv E_{\\rm LSP}-m_{\\rm LSP}$ is the\nkinetic energy of LSP. As one can see in eq.(\\ref{Delta_E}), the LSP\nwhich has a kinetic energy larger than $\\sim T_R$ tends to lose its\nenergy through the scattering process, while LSP receives energy from\nthe thermal bath if its energy is smaller than $\\sim T_R$. Thus, if\nthe scattering processes take place effectively, the averaged kinetic\nenergy of LSP becomes $\\sim T_R$, {\\it i.e.} LSP goes into the\nkinetic equilibrium.\n\nThe energy loss rate $\\Gamma_{\\rm scatt}^{\\rm NR}$ for the \nnon-relativistic LSP is given by\n\\begin{equation}\n    \\Gamma_{\\rm scatt}^{\\rm NR} \\simeq \n    n_e \\langle\\sigma_{\\rm scatt} v_{\\rm rel}\\rangle \\times\n    \\frac{20T_R}{m_{\\rm LSP}}\n    \\simeq \n    \\frac{5760\\alpha_1^2}{\\pi}\n    \\frac{T_R^6}{m_{\\rm LSP}m_{\\tilde{e}_R}^4}.\n    \\label{gamma_scatt}\n\\end{equation}\nThe LSP goes into the kinetic equilibrium if the scattering rate\n$\\Gamma_{\\rm scatt}^{\\rm NR}$ is larger than the expansion rate of the\nuniverse. Taking $m_{\\tilde{e}_R}\\sim m_{\\rm LSP}$, the ratio of\n$\\Gamma_{\\rm scatt}^{\\rm NR}$ to the expansion rate of the universe,\n$H$, is given by\n\\begin{equation}\n    \\left. \\frac{\\Gamma_{\\rm scatt}^{\\rm NR}}{H}  \\right | _{E_{\\rm LSP}\n      \\sim m_{\\rm LSP}}\n      \\simeq 4 \\times 10^{3} ~\n       \\left (  \\frac{m_{\\rm LSP}}{100 {\\rm GeV} }  \\right ) ^{-5}\n       \\left (  \\frac{T_R}{100 {\\rm MeV} }  \\right ) ^{4}.\n\\end{equation}\nThus, if the reheating temperature is higher than about 10MeV,\nproduced LSPs go into kinetic equilibrium as far as $m_{\\rm LSP} \\sim\nO(100)$GeV.  Furthermore, as we discussed in the previous section,\nthe reheating temperature should be higher than at least 100MeV in\norder to decrease the mass density of LSP sufficiently. Thus, we\nconclude that the produced LSPs go into kinetic equilibrium if we\nrequire that the mass density of the relic LSP should not overclose\nthe universe.\\footnote\n{By the numerical calculation we have checked, in fact, that the\nscattering rate given in eq.(\\ref{gamma_scatt}) is always larger than\nthe expansion rate of the universe when the relic LSP does not\noverclose the universe. (See fig.2.)}\nIn this case, the averaged velocity is given by\n\\begin{equation}\n    \\langle v^2\\rangle \\simeq \\frac{3T_R}{m_{\\rm LSP}}.\n\\end{equation}\n>From this we easily see that the LSP abundance given in\neq.(\\ref{abundance_LSP}) decreases as the reheating temperature gets\nhigher. Thus, we obtain the lowerbound on the reheating temperature.\n\n\n\\section{Results}\n\\label{sec:results}\n\nOnce we know the averaged velocity $\\langle v^2\\rangle$, we can\ncalculate the annihilation cross section of LSP, and get the\nlowerbound on the reheating temperature after the decay of the Polonyi\nfield. In this letter, we first solve RGEs based on the minimal SU(5)\nmodel with the no-scale boundary conditions, and determine the mass\nspectrum of the superparticles. We only investigate the parameter\nspace which is not excluded by the experimental or theoretical\nconstraints. The constraints which we use are as follows:\n\\begin{itemize}\n\\item Higgs bosons $H_f$ and $\\bar{H}_f$ have correct vacuum\nexpectation values.\n\\item Perturbative picture is valid below the gravitational scale.\n\\item LSP is neutral.\n\\item Sfermions (especially, charged sleptons) have masses larger than\nthe experimental lower limits~\\cite{PDG}.\n\\item The branching ratio for $Z$-boson decaying into neutralinos is \nnot too large~\\cite{PLB350-109}.\n\\end{itemize}\nThen, with the obtained mass spectrum of superparticles, we calculate\nthe annihilation cross section and determine the lowerbound on the\nreheating temperature from the following equation;\n\\begin{equation}\n    \\left. \\frac{H}{s\\langle\\sigma_{\\rm ann}v_{\\rm rel}\\rangle} \n       \\right | _{T=T_R} \\leq\n      \\frac{\\rho_c}{s_0} \\simeq 3.6h^2 \\times 10^{-9} {\\rm GeV} .\n\\end{equation}\n\nIn fig.~1, we show the lowerbound on the reheating temperature in the\n$\\tan\\beta$ vs. $m_{\\rm LSP}$ plane. In the figures, large or small\n$\\tan\\beta$'s are not allowed since the Yukawa coupling constant for\nthe top quark or bottom quark blows up below the gravitational scale\nfor such $\\tan\\beta$'s. Furthermore, there also exists a lowerbound on\nthe LSP mass. In the case where $\\tan\\beta \\mathop{}_{\\textstyle \\sim}^{\\textstyle <}  20$, charged sfermions\nbecome lighter than the experimental limit if the LSP mass becomes\nlighter than $\\sim 50 {\\rm GeV} $.  On the other hand, for the large\n$\\tan\\beta$ case, unless the bino mass is sufficiently large, the\nlightest charged slepton becomes LSP. (Remember that the dominant\ncomponent of LSP is bino.)  Thus, the lowerbound on $m_{\\rm LSP}$ is\nobtained. As we can see, the reheating temperature should be larger\nthan about 100MeV, even for the case where $m_{\\rm LSP}\\sim 50 {\\rm GeV} $.\nThe constraint becomes more stringent as $m_{\\rm LSP}$ increases,\nsince the masses of the superparticles which mediate the annihilation\nof LSP becomes larger as the LSP mass increases.\n\nIf we translate the lowerbound on the reheating temperature into that\nof the Polonyi mass $m_\\phi$, we obtain $m_\\phi \\mathop{}_{\\textstyle \\sim}^{\\textstyle >}  100 {\\rm TeV} $ (see\neq.(\\ref{rtemp})). We can also see that the lowerbound is almost\nindependent of $\\tan\\beta$. In fig.~2, We show the lowerbound on $T_R$\nas a function of the LSP mass for $\\tan\\beta =10$, and $\\mu_H >0$.\n\nHere, we should comment on the accidental case where the annihilation\nprocess hits the Higgs pole in the $s$-channel. If the LSP mass is\njust half of the lightest Higgs boson mass, the LSP annihilation cross\nsection is enhanced since LSP has small but nonvanishing fraction of\nhiggsino component. If the parameters are well tuned, such a situation\ncan be realized and the lowerbound of $T_R$ decreases to $O(10) {\\rm MeV} $.\nHowever, we consider that such a scenario are very unnatural since a\nprecise adjustment of the\nparameters is required in order to hit the Higgs pole.\\footnote\n{In the case where the annihilation process hits the pole of heavier\nHiggs bosons, the cross section is not enhanced so much, since the\nwidths of the heavier Higgs bosons are quite large.}\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\n\nIn this letter, we have obtained the lowerbound on the reheating\ntemperature due to the decay of the Polonyi field in a framework of\nthe no-scale type supergravity model. As a result, we have seen that\nthe Polonyi mass should be larger than about 100TeV which may raise a\nnew fine-tuning problem~\\cite{PLB173-303}.\n\nWe have assumed the minimal SUSY GUT model in the present\nanalysis. However, the main conclusion is not changed as far as LSP\nis mostly the bino, because the minimum value of the lowerbound ($T_R\n\\simeq 100$MeV) is obtained when the mass of the selectron takes the\nexperimentally allowed lower limit.\n\nWe have assumed that LSP is stable so far. However, if we introduce\n$R$-parity violation, LSP becomes unstable and the allowed $T_R$ is as\nlow as a few MeV. It is also the case if we assume a very light LSP\nsuch as a neutral higgsino~\\cite{PLB131-59} or an\naxino~\\cite{NPB358-447} whose masses are less than about 100MeV.\n\n\\section*{Acknowledgement}\n\nTwo of the authors (T.M. and T.Y.) would like to thank M.~Yamaguchi for\nuseful discussions in the early stage of this work.\n\n\\newpage\n\\newcommand{\\Journal}[4]{{\\sl #1} {\\bf #2} {(#3)} {#4}}\n\\newcommand{Ap. J.}{Ap. J.}\n\\newcommand{Can. J. Phys.}{Can. J. Phys.}\n\\newcommand{Nuovo Cimento}{Nuovo Cimento}\n\\newcommand{Nucl. Phys.}{Nucl. Phys.}\n\\newcommand{Phys. Lett.}{Phys. Lett.}\n\\newcommand{Phys. Rev.}{Phys. Rev.}\n\\newcommand{Phys. Rep.}{Phys. Rep.}\n\\newcommand{Phys. Rev. Lett.}{Phys. Rev. Lett.}\n\\newcommand{Prog. Theor. Phys.}{Prog. Theor. Phys.}\n\\newcommand{Sov. J. Nucl. Phys.}{Sov. J. Nucl. Phys.}\n\\newcommand{Z. Phys.}{Z. Phys.}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction\\label{ch5:int}}\n\nThis work is aimed at the question of what provides the\ndominant power source of ultraluminous infrared galaxies, that\nis galaxies with luminosities in the wavelength range\n8--1000 \\hbox{\\micron~} in excess of $10^{12}$ \\hbox{L$_\\odot$}~($H_0$=75 km\ns$^{-1}$ Mpc$^{-1}$).  This question is of interest because\nit is possible that the power source of these galaxies is\nqualitatively different from the dominant power source of most\ninfrared galaxies of lower luminosity.  The observational\napproach of this work is the use of moderate resolution\n8--13 \\hbox{\\micron~} spectroscopy.  \n\nIn very broad strokes, the context of the debate over the\nquestion addressed here can be summarized as \nan attempt, on the one hand, to explain ultraluminous infrared\ngalaxies as more powerful versions of starburst galaxies,\nthat is as super starbursts (Joseph \\&\nWright~1985), where massive star\nformation provides the power source, and on the other hand,\nto demonstrate that ultraluminous infrared galaxies are\nthe precursors of quasars (Sanders et al. 1988) and are\npowered by active galactic nuclei (AGNs).\n\nThe question has remained open to serious debate for\nessentially one reason: dust.  Infrared galaxies by their\nnature radiate the bulk of their energy through the \ndegradation of the optical and UV photons emitted by the\nprimary power source into infrared photons emitted by dust\nheated to a temperature of $\\sim$50 K.  This has the\neffect of erasing most of the commonly accepted\nfeatures used to distinguish the characteristics of\nstarbursts and AGNs.\n\nOptical and, to a lesser degree, near-infrared studies are\nparticularly susceptible to this difficulty since spectroscopic signatures of AGNs in ultraluminous\ninfrared galaxies should only betray sources that are\nrelatively unobscured and therefore might not contribute\nsubstantially to the heating of the dust that is the final source\nof most of the observed emission.  However, Veilleux et\nal. (1995) have shown in a large sample of infrared galaxies\nthat the fraction of infrared galaxies with AGN-like optical\nspectra increases as a function of increasing luminosity, providing\nimportant circumstantial evidence that AGN-like activity is\nassociated with ultraluminous infrared galaxies.  The strength\nof this evidence might be considered to be weakened, however, by\nthe finding of Goldader et al. (1995) that the ratio of\nBr$\\gamma$ to infrared luminosity decreases as a function of\nluminosity, suggesting that dust obscuration plays an even\ngreater role in ultraluminous infrared galaxies than in\nluminous infrared galaxies.  This suggests the perverse\npossibility that energetically dominant starbursts in\nultraluminous infrared galaxies are so obscured in the optical\nthat only incidental unobscured AGNs are detected.\n\nRadio observations are unaffected by dust obscuration and can\nprovide high spatial resolution.  However, the primarily\nsynchrotron radiation that is detected is only indirectly linked\nto the power source of infrared galaxies.  In starbursts, it is\nthought that the radio emission arises from electrons\naccelerated in supernovae remnants, while in AGN, relativistic\njets are strong radio emitters. Condon et al. (1991; hereafter\nCHYT) have employed the high spatial resolution provided by the\nVery Large Array (VLA) to show that the radio sizes of a large\nfraction of ultraluminous infrared galaxies are comparable to\nthe minimum size required for a blackbody that reproduces the\n25, 60, and 100 \\hbox{\\micron~} flux densities of these sources.  From this\nthey infer that starbursts rather than AGNs may power these\nsources.  At even higher spatial resolution provided by very\nlong baseline interferometry (VLBI), Lonsdale, Smith \\& Lonsdale\n(1993) find evidence for radio structure on the scale of\nmilliarcseconds, some components of which have brightness\ntemperatures well in excess of what could be expected in\nstarburst-related emission.  They deduce from this that AGNs are\npresent in five of eight of the ultraluminous infrared galaxies\nin their sample\\footnote{Smith, Lonsdale \\& Lonsdale (1998)\nsuggest that radio supernova remnants may be responsible for\nsome of the structure they detect}.\n\nIn this work, the dust emission itself is examined for\nspectral clues that can reveal the nature of the power\nsource in infrared galaxies.  In a sample of 60 galaxies,\nRoche et al. (1991; hereafter RASW) have shown that their\n8--13 \\hbox{\\micron~} spectra can be classified into three types:\nthose with prominent polycyclic aromatic hydrocarbon (PAH)\nfeatures generally show \\ion{H}{2} region-like optical\nspectra, those with flat featureless spectra have Seyfert\n1 or quasar-like optical spectra, and those rare galaxies\nwith silicate absorption features usually have Seyfert 2 optical\nspectra.  Dudley \\& Wynn-Williams (1997) have extended\nthe work on galaxies with very deep silicate features to\nshow that the size of the power source can be constrained\nto be smaller than a few parsecs, implying that such sources\nare powered by deeply embedded AGNs.  Here, the 8--13 \\hbox{\\micron~}\nspectra of a well-defined sample of 25 luminous and\nultraluminous infrared galaxies are examined in this\ncontext. \n\nIt should be noted that all of the sources for which new\nspectral observations are reported here can be found in one or\nmore of four {\\em IRAS}-based catalogs: the Bright Galaxy Survey\n(Soifer et al. 1989), the Extended Bright Galaxy Survey\n(Sanders et al. 1995), the Two Jansky Survey (Strauss et\nal. 1992), or the Extended 12 \\hbox{\\micron~} Survey (Rush, Makan \\&\nSpinogio 1993).\n\nObservations and data reduction procedures are described in\nSection 2.  Section 3 presents the new spectral observations.  The\nclassification of the spectra is discussed in\nSection 4.1.  The justification for the classification of the ultraluminous galaxies that fall within the later subsample\nis discussed in some detail in Section 4.2.  In Section 4.3 a subsample\nof galaxies is defined and an apparent trend with luminosity\nwithin it is examined.  The possibility of an\nanti-correlation between the equivalent width of the 11.3\n\\hbox{\\micron~} PAH feature and the ratio of infrared light to molecular\ngas mass is discussed in Section 4.4, and conclusions are\ndrawn in Section 5.\n\n\n\\section{Observations and Data Reduction\\protect\\label{ch5:odr}}\n\n\n\nThe new observations reported here were obtained over the\ncourse of 5 years, from 1991 through 1995, using the 10 and\n20 \\hbox{\\micron~} Cooled Grating Spectrograph (CGS3) mounted at the\nCassegrain focus of the United Kingdom Infrared Telescope\n(UKIRT) on Mauna Kea in Hawaii.  Observations in 1991 were\nobtained by G. Wynn-Williams and J. Goldader, in 1992 by\nG. Wynn-Williams and C. Dudley, and in 1993--1995 by\nC. Dudley. For parts of each observing run a UKIRT\nscientist (J. Davies, T. Geballe, or G. Wright) was present\nat the summit and participated in the observations.  CGS3 has\nbeen described by Cohen \\& Davies (1995).  All but one of\nthe spectra reported here were obtained using the low-resolution spectroscopy mode with $\\lambda\/\\Delta\\lambda\n\\sim 60$ in a series of beam-switched, chopped observations\nwith a chop frequency of $\\sim$10 Hz and a beam switch\nfrequency of $\\sim$0.5 Hz.  To fully sample each\nspectrum, two interlacing grating positions were observed\nwith changes between the two positions occurring after\n10--16 beam-switched pairs.  Table 1 gives a log of\nobservations for the galaxies.  In Table 1, column 1 is\nthe object name, columns 2 and 3 give the aperture centre\nin RA and DEC, column 4 gives the position reference,\ncolumns 5 and 6 give the object redshift and reference,\ncolumn 7 gives the date of observation, column 8 gives the\ncircular aperture size used in arcseconds (full width at\n10 per cent power), column 9 gives the chop throw in arcseconds,\nand column 10 gives the chop direction.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nObservations were calibrated in wavelength by means of\nhigh order spectral measurements of a Kr lamp in the\nmanner discussed by Hanner, Brooke \\& Tokunaga (1995).\nA master calibration spectrum was first selected and\ncalibrated in wavelength by flattening a 2$\\times$\nover-sampled Kr lamp spectrum (4 grating positions) with\nthe spectrum of a soldering iron replacing the Kr lamp in\nthe beam.  Kr lamp spectra from each night's observations\nwere then cross correlated against the unflattened master\nspectrum to give a wavelength calibration for each night.\nTypical shifts from the nominal CGS3 wavelength\ncalibration were less than a resolution element, and were\nnever more than 2 resolution elements.\n\nCorrections for instrumental response, atmospheric\ntransmission, and flux calibration were performed by dividing\nsource spectra by the spectrum of a standard star with a\nknown 10 \\hbox{\\micron~} flux density and an assumed blackbody\ntemperature based on its spectral type, and then multiplying\nby the Planck curve of that temperature scaled to match the\nstandard star in flux density.  Observations of standard\nstars were performed with the same aperture, chop throw,\nposition angle and frequency as the galaxy observations they\ncalibrate. The standard star observations were made to match\nin airmass so that no additional airmass corrections would be\nrequired.  For the standard stars, the number of beam-switched pairs between grating shifts was typically 3--5. The\nstars have been chosen, where possible, to have a spectral\ntype of K0 or earlier to avoid the effects of SiO absorption,\nH$^-$ opacity, and circumstellar dust emission that can\neffect the spectral shape of late-type stars.  Table 2 gives\nthe HR number, 10.1 \\hbox{\\micron~} flux density, photometric reference,\nand assumed blackbody temperature of each standard star, in\ncolumns 4, 5, 6, and 7, respectively, for each of the\nobservations given in Table 1.  In column 2 of Table 2,\nalternative names are given for galaxies that are found in\nmultiple catalogs, while column 3 gives the date of each\nobservation as in Table 1.  It should be noted that an error of $\\sim$1000 K in the assumed blackbody temperature of the standard\nstars leads to a $\\sim$2 per cent error in flux calibration at\neither end of the passband 8--13 \\hbox{\\micron~} which is negligible for\nthe purposes of this work, while for those stars of late\nspectral type, deviations from the assumed blackbody can be\n$\\sim$10 per cent in the SiO band (Cohen et al. 1995).  Systematic\nuncertainties in overall flux calibration of 15--20 per cent can\narise due to telescope pointing errors with the stronger\neffects for the smaller apertures.  For wavelengths shortward\nof 8.2 \\hbox{$\\mu$m}, strong systematic effects can occur as a\nresult of a variable water vapor column in the Earth's\natmosphere.\n\n\n\n\n\nAll spectra were regridded to a common wavelength scale using a\nGaussian filter of $\\sigma$ = 0.064 \\hbox{\\micron~}, which lead to a slight\ncorrelation between neighboring data points.  Those spectra of the\nsame source obtained on two or more nights were combined,\nweighted by integration time.  Data not falling in the ranges\n7.7--9.1 and 10.0--13.4 \\hbox{\\micron~} have been excluded due to poor\natmospheric transmission.\n\n\nSpectra of Wynn-Williams et al.'s (1991) Arp 299B1 were obtained on 1994 February 6 and 7\nusing CGS3 in high-resolution mode ($\\lambda\/\\Delta\\lambda\n\\sim 250$) using a 3\\farcs26 full width at 10 per cent power\ncircular aperture.  The spectra obtained on each night were\nthe result of two sets of observations each consisting of two\ninterlacing grating positions but with different wavelength\nranges to cover the spectral range 10--13.2 \\hbox{$\\mu$m}.  The chop\nthrow was 20\\arcsec, and the chop and beam switch PA were\n45$^\\circ$.  About 30 per cent of the first night's observations were\naffected by clouds; these data were rejected by inspecting the\nchange in the dispersion of the total signal.  The Kr lamp was\nobserved on each night to provide wavelength calibration.  The\nstrongest lines from each night agreed in wavelength to 3 per cent of\na resolution element so that the Kr spectra of the two nights\nwere combined, and a difference in wavelength of 30 per cent of a\nresolution element was measured between the expected (based on\nthe nominal calibration of CGS3) and the measured wavelengths\nof 5 Kr lines.  An additive correction of 0.02 \\hbox{\\micron~} was\napplied to the observations, this being the interpolation of\nthe 2.117 and 2.191 \\hbox{\\micron~} Kr lines measured at 6th order at the\nexpected position of the galaxy [\\ion{Ne}{2}] line.  Flux\ncalibration was performed using HR 4067 with a 10.1 \\hbox{\\micron~} flux\ndensity of 103 Jy and an assumed temperature of 4000 K on the\nfirst night, and HR 5340 using a 10.1 \\hbox{\\micron~} flux density of 738\nJy and an assumed temperature of 4400 K on the second night.\nNo significant difference was found between the two nights'\nobservations, so they were combined according to their\nstatistical weights. The difference in airmass between the\nsource and standard observations was $\\sim$0.5 but monitoring\nof BS 4518 on the first night at airmasses similar to the\nsource observations showed that the level of spectroscopic\nvariation over the observed wavelength range as a function of\nairmass was negligible compared to the final signal-to-noise\nratio.  The overall photometric calibration is estimated to be\ngood to about 25 per cent while the typical signal-to-noise ratio of\nan independent data point is about 10.\n\n\n\n\n\\section{Results\\label{ch5:r}}\n\n\n\n\n\nThe intensity and equivalent width of the 11.3 \\hbox{\\micron~} PAH\nfeature have been measured directly for all of the sources by\nestimating the continuum in the rest wavelength (see Table 1) ranges\n10.5--10.9 and 11.7--12.1 \\hbox{\\micron~} and subtracting the linear\ninterpolation of these two estimates from the spectra to\nmeasure the residual flux in the range 10.9--11.7 \\hbox{$\\mu$m}.\nUncertainties in the feature intensities represent the\ncombined uncertainties in the residual flux in this range but\nnot the uncertainties in continuum subtraction.  For the\nequivalent width estimates, uncertainties in the continuum\nlevel have been included.  In most cases the continuum\nuncertainties are dominant in the equivalent width estimates,\nleading to indeterminate results in the case of limits, but in\na few cases the uncertainty in the residual is the dominant\ncomponent, and upper limits to these equivalent widths can be\nset.\n\nSince model fitting can lead to substantially cleaner\nestimates of feature fluxes (of order the signal-to-noise\nratio in the peak spectral element being fit)\nestimates based on the model that is applied to these \ndata to test for the presence of a silicate absorption\nfeature are also given.  The model consists of a\npower-law continuum and an additive PAH spectrum\nrepresented by the spectrum of the Orion Bar taken from\nRoche (1989).  The fitting of this model was accomplished\nusing a version of Bevington's (1969) program CURVEFIT\nsupplied with IDL ver. 4.0 using three free parameters,\nthe power-law continuum scaling and index ($f_\\nu =\n\\beta\\times\\nu^{-\\alpha}$), and the scaling of the Orion Bar\nspectrum.  The data were fit in the rest wavelength ranges\n10.0--10.9, 11.0--11.4, and 11.7--12.6 \\hbox{\\micron~} using initial\nguesses based on linear fits to the logarithm of data\nbinned in the ranges 10.0--10.5, 10.5--10.9, and\n11.7--12.6 \\hbox{\\micron~} for the power-law scaling and index, and\nbased on the continuum subtracted estimates of the flux in\nthe 11.3 \\hbox{\\micron~} feature.  The weighting of the data was\nstatistical.  As discussed recently by Goldader et\nal. (1995), the use of a power-law as a fitting function\ntends to throw the bulk of the errors in the fit\nparameters into the power-law scaling, so that the formal\nerrors of the remaining parameters are underestimated.\nFurther, the model assumptions introduce further\nsystematic uncertainties.  In particular, by fitting the\ndata in the range 11.2--12.6 \\hbox{$\\mu$m}, it is assumed that\nthe ratio of 11.3 \\hbox{\\micron~} feature emission with respect to the\nunderlying plateau emission that extends to 12.9 \\hbox{\\micron~} is\nthe same as that found at the position of peak of the 11.3\n\\hbox{\\micron~} emission feature in the Orion Bar.  This ratio,\nhowever, is know to vary spatially within the Orion Bar (Roche\net al. 1989), and also within at least one external galaxy\n(Dudley \\& Wynn-Williams 1999).  The consequences\nof this uncertainty are largest in spectra where the PAH\nemission is most powerful relative to the continuum.  Given\nthat the galaxy redshifts make it impossible to estimate the\ncontinuum beyond 13.1 \\hbox{\\micron~} (rest frame) in $\\sim$50 per cent of the\nsources, it has not been possible to quantify the importance of\nthis effect in these data, regardless of obvious\nsignal-to-noise issues.  Due to these multiple and tangled\nuncertainties, both results of direct measurements and model\nfits with formal errors are presented, but it must be\ncautioned that the model parameter errors are seriously\nunderestimated.  In particular, model 3 $\\sigma$ upper limits\nto the 11.3 \\hbox{\\micron~} intensity are roughly the same as measured 1\n$\\sigma$ upper limits to the 11.3 \\hbox{\\micron~} intensity.  The new\nspectra and model fits are presented in Fig. 1.\n\n\n\n\n\n\n\n\n\nTable 3 presents the model power-law index in column 2,\nthe measured (upper) and model (lower) 11.3 \\hbox{\\micron~} feature\nintensity in column 3, the 11.3 \\hbox{\\micron~} feature equivalent width in\ncolumn 4, the ratio of the flux density of the source in\nthe rest wavelength range 8.0--8.4 \\hbox{\\micron~} to the model flux\ndensity in the same range where the errors represent the\nerrors in the estimates of the source flux densities only.\nMeasurements of fine structure lines are given in columns\n6 and 7.  For the measurements of fine structure lines the\ncontinuum was estimated by eye, and for detections of the\n12.8 \\hbox{\\micron~} [\\ion{Ne}{2}] line it must be cautioned that a\nsignificant amount of the intensity could be due to a\nblended PAH emission feature at 12.7 \\hbox{\\micron~} except in the\ncase of Arp 299B1, where the spectral resolution is\nsufficient to distinguish the two features.  Errors for\ndetections and limits are 1 $\\sigma$ and 3 $\\sigma$,\nrespectively, except in the case of the model 11.3 \\hbox{\\micron~}\nequivalent width, which is reported in all cases to give a\nsense of the model continuum uncertainties.  In the case\nof Arp 299B1, the fine structure [\\ion{S}{4}] and\n[\\ion{Ne}{2}] results are derived from the high-resolution\nspectrum.  The redshift of the [\\ion{Ne}{2}] line (2900\n\\hbox{km s$^{-1}$}) agrees within 1.4 $\\sigma$ (1 $\\sigma$ = 170 \\hbox{km s$^{-1}$}\nbased on Gaussian fitting) with the CO redshift measured\nby Sargent \\& Scoville (1991), while much more precise\nagreement between unpublished IRTF CSHELL \\hbox{Br$\\gamma$}~ data and the CO\nobservations is known, so that some\nconfidence in [\\ion{Ne}{2}] identification is warranted. It\nshould be noted\nthat [\\ion{Ne}{2}] is not detected in the low resolution\n8--13 \\hbox{\\micron~} spectrum of Arp 299B1 but that the 3 $\\sigma$\nupper limit for this feature is 2.1 $\\times 10^{-15} $ W\nm$^{-2}$, which is consistent with the intensity measured\nat high resolution.  Further, the ratio of I(11.3) to\nI([\\ion{Ne}{2}]) (1.7) is not unusual for galaxies for\nwhich the ratio has previously been measured (RASW).\n\n\n\n\n\n\n\n\\addtocounter{table}{1}\n\n\n\n\n\n\n\n\nAs is apparent in Table 3, while a wide range of power-law\nindices are necessary to describe the long-wavelength\ncontinuum, the ratio of 8.2 \\hbox{\\micron~} flux to model 8.2 \\hbox{\\micron~}\nflux is not significantly different from unity for a large\nfraction of the galaxies even in cases where the power-law\nindex is large and a minimum near 9.7 \\hbox{\\micron~} is pronounced.\nAs a result, it is not generally necessary to invoke silicate\nabsorption to explain the observed spectra.  \n\n\n\n\\section{Discussion\\label{ch5:d}}\n\n\\subsection{Spectral classification\\label{ch5:d1}}\n\nRASW have found that a three-type\nclassification scheme is sufficient to classify the 8--13\n\\hbox{\\micron~} spectra of most galaxies.  Their scheme consists of\nflat-featureless, silicate absorption or emission, and PAH-type (their UIR)\nspectra.  Further, there is a general correspondence\nbetween the 8--13 \\hbox{\\micron~} spectrum of a galaxy and its optical\nspectral classification based on optical emission lines.  The\ncorrespondence is that those galaxies with flat 8--13 \\hbox{\\micron~}\nspectra are often broad-line AGNs, those with silicate\nabsorption are frequently narrow-line AGNs, and those with PAH\nfeatures typically have optical spectra with line ratios\nsimilar to Galactic\n\\ion{H}{2} regions.  \n\nParenthetically, the identification of the mid-infrared\nemission feature analysed here is not entirely secure.  Other\nlaboratory analogs, including quenched carbonaceous composites\n(Sakata \\& Wada~1989), hydrogenated amorphous\ncarbon (Duley~1989), and anthracite (Ellis et\nal.~1994), have been proposed in addition to\nthe PAH model (Puget \\& L\\'eger~1989) that is\nadopted here for ease of expression.  However, the precise\nidentification of the chemical source of these features is not\nparticularly important for the present work.  What is\nimportant is that the strength of one feature may be predicted\nto some extent from observations of another.  In particular\nthe strength of the 7.7 \\hbox{\\micron~} feature may be predicted from\nobservations of the 11.3 \\hbox{\\micron~} feature to within about a factor of 2\n(Zavagno, Cox \\& Baluteau 1992).\n\nThe model fitting performed at the long-wavelength end of the\nspectra can be used effectively to test for the presence of\nstrong silicate absorption.  Two cases might occur.  First, a\nPAH spectrum might undergo external extinction.  In this case,\ntwo effects will contribute to the model under predicting the\nobserved flux in the shorter wavelength half of a spectrum.  The\nfirst effect is that the 11.3 \\hbox{\\micron~} feature will undergo more\nsevere extinction than the region near 8 \\hbox{\\micron~} due to the\nspectral shape the silicate component of the interstellar\nextinction curve.  This has the effect of reducing the predicted\nPAH strength.  The second effect is that the slope of the model\ncontinuum will have a larger spectral index than would otherwise\nbe the case if no silicate absorption were present.  The model\nwould therefore under-predict the continuum underlying the 7.7\nand 8.6 \\hbox{\\micron~} feature complex.  Both of these effects will cause\nthe model to under-predict the shorter wavelength observations.\nSecond, a PAH spectrum may undergo no particular extinction,\nhowever, a second silicate-absorbed continuum component may also\ncontribute to the spectrum.  In this case, the model should\npredict the PAH component well.  However, if the silicate\nabsorption is strong, the model spectral index will be increased\ndue to the additional component, and the continuum in the short\nwavelength half will be underestimated so that the model will\nalso under-predict the observed flux.  \n\nThe RASW classification scheme is applied to the present data\nin the following way: the ratio of the observed to model flux\ndensity between 8.0 and 8.4 is used as a guide to classification\nsuch that if this ratio is 3 $\\sigma$ above 2.5, then the\nsource is classified as silicate.  This cut may somewhat\nunderestimate the number of galaxies in which silicate\nabsorption is present compared to the approach taken in\nDudley \\& Wynn-Williams (1999), where the cut is taken at\n2.0. For the purposes of definite classification, under the\nworking hypothesis that all infrared galaxies are PAH-type\nunless convincingly shown to be otherwise, this more\nconservative value is appropriate.  In cases where the 11.3\n\\hbox{\\micron~} feature is not formally detected but only allowed (dot\ndashed curves in Fig. 1), and where a minimum is present\nnear 9.7 \\hbox{$\\mu$m}, a question mark is appended to the PAH\ndesignation; however, whether or not there is a formal\ndetection of the 11.3 \\hbox{\\micron~} feature in sources with obviously\nflat spectra, the classification is given as flat.  A mixed\nclassification of PAH and silicate is given where PAH\nemission can explain $\\sim$30 per cent of the observed emission\nbetween 8.0 and 8.4 \\hbox{$\\mu$m}.  Under this condition, the\nclassification of Arp 220 by RASW as both PAH and silicate\nwould be revised to just silicate.  Arp 220 is discussed\nin detail in Section 4.2.5.\n\n\n\n\n\n\n\n\n\\addtocounter{table}{1}\n\n\nTable 4 gives, for each source in column 1, the spectral\nclassification based on the present data and on optical\nobservations, along with 12--100 \\hbox{\\micron~} photometry, the infrared\nluminosity, and the star formation efficiency.  In some cases\nthe optical spectral classification has been inferred from the\ndata of Veilleux et al. (1995), if some diagnostics were\nmissing. It is clear from Table 4 that the general\ncorrespondence between optical spectroscopy and 8--13 \\hbox{\\micron~}\nspectroscopy is quite similar to that found by RASW.  Neglecting\nLINER-type galaxies, four exceptions where AGNs are identified\noptically but which are found to have PAH-type spectra in this\nwork are Mkn 1034, IRAS 04154+1755, IRAS 05189$-$2524, and Zw\n475.056.\n\nMkn 1034 is a known Seyfert 1 galaxy; however its 8--13 \\hbox{\\micron~}\nspectrum can be fit with a PAH plus continuum model.  This may\nbe consistent with the finding of Sopp \\& Alexander (1992) that\nonly half of the 6-cm flux density measured in a 14\\farcs5 beam\nis also contained within a 1\\farcs5 beam, suggesting that\nspatially extended star formation plays a role in this source.\nA similar explanation may be invoked for the presence of PAH\nemission in the spectra of the Seyfert 2 galaxies IRAS\n04154+1755 and Zw 475.056.  Crawford et al. (1996) find IRAS\n04154+1755 to be resolved at 6 cm using both 4-arcsec and\n0.5-arcsec beams.  In the map of Zw 475.056 presented by CHYT,\nlow-level radio emission extends over 1.5-arcsec.  IRAS\n05189$-$2524 is discussed in detail in the next section; however\nit should be noted here that the presence of exceptions, in this\nsample, to the the correspondence between optical and 8--13 \\hbox{\\micron~}\nspectral classification found by RASW should not be taken as\nevidence that the correspondence is merely coincidental.  The\nsources presented by RASW were sometimes selected on the basis\nof optical spectroscopy to examine the question of what kind of\nspectrum they would present at 8--13 \\hbox{$\\mu$m}.  Hence, many of\ntheir sources were archetypical examples of their optical class.\nIn the present sample, selection is based solely on infrared\nproperties, so that mixed cases might well be expected.\n\nIt is probably not coincidental that these four sources are\namong the sources in Table 3. with the smaller values of\n$\\alpha$ suggesting that the AGN in these sources may contribute\nsomewhat to the 8--13 \\hbox{\\micron~} continuum; strong 12 and 25 \\hbox{\\micron~}\nemission is known to select for AGNs that have unambiguous\noptical spectra (de Grijp et al. 1985; Spinoglio \\& Malkan, see also\nthe AGN spectral energy distributions presented by Sanders et\nal. 1989).\n\n\\subsection{Comments on individual ultraluminous infrared galaxies}\n\n\n\n\nThe observed properties of the ultraluminous infrared galaxies\npresented here show a surprising range.  A closer look at these\nproperties will be discussed in this section.  \n\n\\subsubsection{A phenomenological description of IRAS 05189$-$2524}\n\nOne of the greatest surprises in these data is the case of\nIRAS 05189$-$2524, which might have been expected to show a flat\nor silicate absorbed 8--13 micron spectrum based on its\noptical AGN-like spectrum (Sanders et al. 1988).  What is\nobserved however is a spectrum that is well fit by a PAH plus\npower-law continuum model (see Fig. 1).  Additionally, the\nstrength of the 11.3 \\hbox{\\micron~} emission relative to the 8--1000\n\\hbox{\\micron~} emission is quite similar to known starburst galaxies so that\nit seems possible that star formation might be the primary\npower source of the galaxy.  Here, a tentative decomposition\nof its spectral energy distribution is proposed.  In Fig.\n2, a comparison is made between IRAS 05189$-$2524 and two\nknown AGNs to suggest that while the 1--5 \\hbox{\\micron~} emission in IRAS\n05189$-$2524 may be due to hot dust heated by an AGN, the 8--100\n\\hbox{\\micron~} emission may be better explained as arising from star\nformation traced by the observed PAH emission.  In Fig. 2\nthe power-law extrapolations of the dust continuum emission\n(solid lines) found in the 8--13 \\hbox{\\micron~} spectra are projected to 5\nand 25 \\hbox{$\\mu$m}.  For the two sources where the 8--13 micron\nemission is thought to be due to AGN-heated dust (Mkn 231 and\nIRAS 13349+2438) (Roche et al. 1983; Beichman et al. 1986),\nthese extrapolations are reasonable predictors of the observed\nflux densities at both 5 and 25 \\hbox{$\\mu$m}.  For IRAS 05189$-$2524\nthe extrapolation of the continuum to 25 \\hbox{\\micron~} is in reasonable\nagreement with the observed flux density; however the\nextrapolation to 5 \\hbox{\\micron~} fails to predict the 4--5 \\hbox{\\micron~}\nphotometry of IRAS 05189$-$2524 (see double arrow), suggesting\nthat an unrelated power source is responsible for this\nemission.  Given the apparent similarity in the shape of 1--5\n\\hbox{\\micron~} emission in both IRAS 05189$-$2524 and IRAS 13349+2438, it\nis possible that the spectral energy distribution of IRAS\n05189$-$2524 is the result of a starburst producing the majority\nof the 8--100 \\hbox{\\micron~} emission and an AGN with a spectral energy\ndistribution similar to that of IRAS 13349+2438 responsible\nfor the bulk of the shorter wavelength emission but\ncontributing little to the 8--100 \\hbox{\\micron~} emission.  Such a\ndecomposition could reconcile the apparently contradictory\nobservations of both an AGN-like optical spectrum and the\npresent PAH type 8--13 \\hbox{\\micron~} spectrum.  A decomposition of this\nkind could also be consistent with the suggestion of Goldader\net al. (1995) that the 2 \\hbox{\\micron~} continuum is dominated by AGN\n[heated dust] emission based on the small equivalent width of the observed\n\\hbox{Br$\\gamma$}~ emission.  That hot dust typically plays a lesser\nrole in starburst galaxies at 2 \\hbox{\\micron~} may also be deduced\nfrom the frequent detection of stellar continuum with probably\nlittle veiling of the CO absorption features, while in IRAS\n05189$-$2524 CO absorption is not detected (Goldader et\nal. 1995).  Another piece of evidence that supports the view that\nthe AGN in IRAS 05189$-$2524 may not be energetically dominant\nis the failure to detect a radio core of extreme brightness\ntemperature in VLBI observations (Lonsdale, et al.\n1993).\n\nThe VLA radio map of CHYT shows the radio continuum in this\nsource to be extended on a scale of 160 pc, which is\ncomparable to the minimum blackbody diameter of 280 pc based on\nthe observed 60 and 100 \\hbox{\\micron~} flux densities and assuming optically thick\ndust emission.  It is therefore plausible that the far-infrared\nemission arises from optically thick warm dense clouds embedded\nin the starburst with a low volume filling factor but a column\ncovering factor of order unity. The mid-infrared continuum and\nPAH emission present in this spectrum might arise from the\nsurfaces of these clouds and may be optically thin in the sense\nthat the emission arises from cloud surfaces facing our line of\nsight, and from a layer that is only 1--2 $A_V$ thick on any\ngiven line of sight into the starburst.  \n\nGiven the possibility\nof a decomposition of the 1--100 \\hbox{\\micron~} spectral energy\ndistribution into a dominant starburst component and an AGN\ncomponent important only at shorter wavelengths, the\nplausibility of the infrared emission being spatially associated\nwith the radio continuum, and most importantly, the presence of\n11.3 \\hbox{\\micron~} PAH emission in the 8--13 \\hbox{\\micron~} spectrum taken together with the\nadequate fit of the model given in Fig. 1, the 8--1000 \\hbox{\\micron~}\nemission from IRAS 05189$-$2524 is attributed to a starburst for\nthe purposes of this work.\n\n\\subsubsection{Digression on newly available {\\em ISO} data}\n\nThe model applied to the data presented in Fig. 1 for the\npurpose of spectral classification is not required in cases\nwhere the strength of the 7.7 \\hbox{\\micron~} PAH feature has been\nobserved as is now the case for four of the ultraluminous\ninfrared galaxies in this combined sample as a result of {\\em\nInfrared Space Observatory (ISO)} observations (Genzel et\nal. 1998).  With the additional information it is possible to\ncompare observations with a variety of scenarios to better\nunderstand the physical situation with greater confidence than\npossible when only ground-based data are available.  In Fig. 3,\nthree possibilities are considered: (1) For the top spectrum\nin each panel, a PAH plus power-law continuum model such as\nthat employed in Fig. 1 is fit not just in the spectral region\nsurrounding the 11.3 \\hbox{\\micron~} feature, but to the entire 6--13 \\hbox{\\micron~}\nrange.  (2) For the middle spectrum in each panel, a PAH plus\npower-law continuum model is subjected to extinction by cold\ndust that includes the silicate absorption feature, as adopted\nby Dudley \\& Wynn-Williams (1997), and again fit to the entire\n6--13 \\hbox{\\micron~} range.  (3) For the bottom spectrum in each panel, a\nPAH plus power-law continuum model is additively combined with\na silicate-absorbed continuum source, and again fit over the entire\n6--13 \\hbox{\\micron~} range.  The model fits presented in Fig. 3 have been\ncarried out using CURVEFIT with equal weighting.  In the first\nmodel, the index of the power-law was specified either from\nTable 3 (IRAS 17208$-$0014 and Mkn 273) or as 12.8 and 4.9 for\nArp 220 and Mkn 231, respectively.  It should be noted that\nthree of the sources exhibit a large dynamic range, and in the\nfollowing discussion some attention will be drawn to the poor\nfit at low flux density levels, which is also an expected systematic\neffect of the fitting procedure, since differences between\nsmall numbers are small.  It may also be noted that the\nmismatches are apparent for a large number of neighboring data\npoints.\n\n\n\\begin{figure}\n\\end{figure}\n\n\n\n\n\\subsubsection{IRAS 17208$-$0014}\n\nIn IRAS 17208$-$0014, all models give fairly adequate fits\nto the data, and deciding between them depends mainly on\nother observational evidence.  The strongest evidence that\nsupports a preference for the fit shown for the top spectrum\nis that this source is extended on a scale $\\sim$1.7 kpc in\nthe radio (Condon et al. 1996), although higher spatial\nresolution observations reveal the presence of a sub-kpc\nradio continuum source associated with the OH maser emission\n(Martin et al. 1989).  The failure of Martin et al. (1989)\nto detect more extended emission may be due to the\nrestricted set of base lines they employed.  When this\nevidence is taken along with the optical spectral\nclassification of Veilleux et al. (1995) in which the line\nratios in this source are similar to those in \\ion{H}{2}\nregions, there is no strong reason to suspect that anything\nother than star formation is responsible for the observed\nmid-infrared spectrum.  Nor is there any reason to suspect a\nhigh degree of extinction; IRAS 17208$-$0014 is among the\nsources presented by Genzel et al. (1998) for which the\ncontinuum underlying the 12.8-\\hbox{\\micron~} [\\ion{Ne}{2}] feature in\ntheir SWS data can be comfotably plotted on the scale of the\nISOPHOT-S data in their fig. 2, suggesting a lack of strong\nsilicate absorption (see also Fig. 1).  The fit to the upper\nspectrum might be improved somewhat by increasing the\nstrength of the 7.7 and 6.2 \\hbox{\\micron~} features by a factor of 2,\na procedure which is probably allowed for the 7.7 \\hbox{\\micron~}\nfeature at least since the Orion Bar spectrum used as a\ntemplate here is known to have a somewhat weaker than usual\n7.7 \\hbox{\\micron~} feature relative to the 11.3 \\hbox{\\micron~} feature (Zavagno\net al. 1992).\n\n\\subsubsection{Mkn 273}\n\nThe fit to the top spectrum of Fig. 3 for Mkn 273 gives, as\nmight have been anticipated from Fig. 1, a poor representation\nof the data.  The fitted strength of the 11.3 \\hbox{\\micron~} feature is\nclearly too large, which is roughly the converse of the result\nin Fig. 1.  Thus, it may be concluded that a simple PAH plus\npower-law continuum model does not describe these data.  The fit\nshown for the middle spectrum is better, in that it requires\nless PAH emission over all and the strength of the 11.3 \\hbox{\\micron~}\nfeature is not over-predicted; however the continuum around the\n9.7 \\hbox{\\micron~} silicate minimum is not well reproduced.  This\nsuggests that an unaccounted-for emission component is required\nto produce this emission.  The fit to the lower spectrum\novercomes this difficulty by superimposing an unobscured PAH plus power-law\ncontinuum such as that which fit IRAS 17208$-$0014 (top\nspectrum) upon a continuum source that suffers strong silicate\nabsorption.  Since Mkn 273 shows clear signs of nuclear\nactivity (Armus, Heckman \\& Miley 1989; Veilleux et al. 1995),\na silicate-absorbed continuum source might be produced by an\nedge-on view of a disk model such as those proposed by Pier \\&\nKrolik (1992) or Efstathiou \\& Rowan-Robinson (1995), which are\nnatural elaborations of the AGN unification model proposed by\nAntonucci \\& Miller (1985).  The continuum level evident in the\nSWS observations of 12.8-\\hbox{\\micron~} [\\ion{Ne}{2}] in Genzel et\nal. (1998) is consistent with the new CGS3 data presented here;\nand therefore consistent with silicate absorption playing a\nrole in this source.  If the fit to the bottom spectrum is accepted,\nthen the AGN component could well be responsible for the\ngreater fraction of the 60 and 100 \\hbox{\\micron~} emission.\n\n\\subsubsection{Arp 220}\n\nPrior to the new {\\em ISO} observations the analysis of the\nArp 220 spectrum would have proceeded very much along the\nlines that have just been given for Mkn 273 (Smith et\nal. 1989), with the proviso that Arp 220 has less obvious\nsigns of activity in its nucleus so that disk-like models\nfor the silicate-absorbed continuum emission may be less\nhelpful in advancing our understanding, and deeply embedded\nmodels perhaps more descriptive (Dudley \\& Wynn-Williams\n1997).  Applying the model used in Fig. 1 to the data of\nSmith et al. (1989) severely under-predicts the observed 8.2\n\\hbox{\\micron~} flux density.  However, as can be seen in Fig. 3 the\nnew {\\em ISO} data can apparently be fit by a simple PAH\nplus power-law continuum model due to the apparent\ndiscrepancy between the observations of the 11.3 \\hbox{\\micron~}\nfeature reported by Smith et al. (1989) and the {\\em ISO}\ndata of Genzel et al. (1998).  It should be noted that there\nare no continuum points longward of the 11.3 \\hbox{\\micron~} feature in\nthe new {\\em ISO} data, so that determining its strength\nfrom those data is difficult.  It seems safer at this point\nto accept the ground-based data as a guide in this spectral\nregion and thus hold this model in abeyance, pending further\nobservational evidence.  The fit to the middle spectrum\nis essentially the same as that given in fig. 1 of Smith et\nal. (1989) and suffers from the same difficulty noted by\nSmith et al. (1989), namely that the minimum near 9.7 \\hbox{\\micron~}\nis poorly fit.  The fit to the lower plot is similar to that\nshown in Smith et al.'s (1989) fig. 2 or as adopted by Dudley \\&\nWynn-Williams (1997).  The continuum in the 12.8-\\hbox{\\micron~}\n[\\ion{Ne}{2}] SWS data of Genzel et al. (1998) is consistent\nwith the Smith et al. (1989) data but is not apparently\nconsistent with the ISOCAM-CVF spectrum presented by Elbaz\net al. (1998).\n\nFor Arp 220, therefore, the new {\\em ISO} data presented by\nGenzel et al. (1998) seems to allow the possibility that\nunextinguished PAH emission plus a simple power-law\ncontinuum describes this spectrum as a result of the\napparent strength of the 11.3 \\hbox{\\micron~} feature in the new data.\nIt has been suggested that differing spectral resolution\n(roughly a factor of 2) between the Smith et al. (1989) and\nGenzel et al. (1998) data may account for this difference,\nhowever this could only work and if the 11.3 \\hbox{\\micron~} feature\nwere unresolved in both spectra, and that does not appear to\nbe the case.  Gauging the over all PAH emission from the 6.2\n\\hbox{\\micron~} feature which displays continuum on either side, the\nintensity of which is found to be well correlated with the\nthat of the 7.7 \\hbox{\\micron~} feature in Galactic sources (Cohen\net al. 1989), it is clear the PAH emission relative to the\nover-all 8--1000 \\hbox{\\micron~} emission is exceedingly weak, really\nno stronger than what might be allowed in Mkn 231 (see\nnext), if the apparent 6.2 \\hbox{\\micron~} feature in the {\\em ISO}\ndata for Mkn 231 is real.  In contrast, again using the 6.2\n\\hbox{\\micron~} PAH feature as a guide, the strength of the PAH\nemission relative to the 8--1000 \\hbox{\\micron~} emission in Mkn 273 is\ntwice as strong, and in IRAS 17208$-$0014, five times as\nstrong. The Arp 220 spectrum might possibly be produced by a\nstarburst that reprocesses more that 90 per cent of its\nluminosity not once but many times so that the observed PAH\nemission is produced only in a thin outer shell or in a\nsmall number of narrow chinks. Such a model would be even\nmore extreme than what is argued above as plausible for IRAS\n05189$-$2524 in a starburst context.  However, given the\ndisagreement between the two spectra of this source near\n11.3 \\hbox{\\micron~} and in consideration of the arguments of Dudley \\&\nWynn-Williams (1997), a model that has both unobscured PAH\nemission and a silicate-absorbed continuum due to a deeply\nembedded AGN continues to seem more natural. This\ninterpretation also seems to account simultaneously for both\nthe 158-\\hbox{\\micron~} [CII] (Luhman et al. 1998) and 2-10 keV X-ray\n(Iwasawa 1999) deficits in this source if the ``AGNs''\n(Soifer et al. 1999) are sufficiently obscured.  It should\nbe noted that in this preferred model, only a fraction of\nthe flux density at 7.7 \\hbox{\\micron~} can be attributed to the 7.7\n\\hbox{\\micron~} PAH feature, in accordance with either the 12.8 of 14.3\n\\hbox{\\micron~} continuum seen in the SWS data presented by Genzel et\nal. (1998).  As suggested by Dudley \\& Wynn-Williams (1997),\nspatially resolved spectroscopic observations may help to\nclarify this issue.\n\n\\subsubsection{Mkn 231}\n\nFor Mkn 231 any PAH contribution to the spectrum must be\nminimal compared to a continuum contribution (Roche et\nal. 1983), as can also be seen from the fit to the upper\nspectrum in Fig. 3.  This source had been fit adequately\nwith a model consisting of a continuum source suffering\nsilicate absorption by Roche et al. (1983), and the present\nmodels are {\\it not} an improvement.  It would be of\ninterest to know if disk radiative transfer models such as\nthose proposed by Pier and Krolik (1992) would fair as well\nas the cold absorber model of Roche et al. (1983) in the\ndetailed fit to the silicate absorption feature given their\nlikely success in fitting the overall infrared spectral\nenergy distribution of Mkn 231.  The 11.3 \\hbox{\\micron~} emission\nfeature is not detected in this source and it is not clear\nthat the single point that is spectrally coincident with the\n6.2 \\hbox{\\micron~} feature in the {\\em ISO} data should be considered\nas a real indication of the presence of PAH emission.  Some\nPAH emission is not unexpected in this source given the\npresence of star-forming knots near the nucleus of this\nsource that might account for $\\sim$10 per cent of its\ninfrared luminosity (Surace et al. 1998).  The estimate of\nthe 7.7 \\hbox{\\micron~} PAH strength given by Genzel et al. (1998)\nrelies on a continuum point at 10.9 \\hbox{\\micron~} which lies in the\nsilicate absorption feature.  Of the sources measured in\nthis way by Genzel et al. (1998) Mkn 231 perhaps provides\nthe clearest example of how this method is prone to\noverestimate the PAH strength when silicate absorption is\npresent.  The 12.8 or 14.3 \\hbox{\\micron~} continuum in their fig. 1\nprovide a rough guide as to when there may be a problem in\nthis respect.  For Mkn 231, the 12.8 \\hbox{\\micron~} continuum of\nGenzel et al. (1998) is high in comparison with the data of\nRoche et al. (1983) but it is possible that it would agree\nwith the extrapolation of the long-wavelength ISOPHOT-S data\nwhich also apparently fall significantly above the Roche et\nal. (1983) data as seen in Fig. 3.  It is not clear if these\ndifferences should be attributed to aperture effects, or\nother systematics, or if they reflect intrinsic source\nvariability over 1.4 decade; Roche et al. (1983) found good\nagreement with the narrowband photometric observations of\nReike (1976) made over 0.4 decade earlier.\n\n\n\\subsection{A 60 \\protect\\hbox{$\\mu$m}~ flux-limited subsample\\label{ch5:d2}}\n\n\n\nAmong galaxies with \\hbox{$L_{\\rm IR}$} $\\geq 1.6\\times10^{11}$ \\hbox{L$_\\odot$}, a nearly\ncomplete (all but three sources have 8--13 \\hbox{\\micron~} spectra)\nflux-limited (at 60 \\hbox{$\\mu$m}) sample of 25 sources may now be\ndefined on the combined Bright Galaxy Survey (Soifer et\nal. 1989) and the Extended Bright Galaxy Survey (Sanders et\nal. 1995) (combined BGS) for $F_\\nu$(60) $\\geq$ 11.25 Jy\n(roughly twice the flux limit of  combined BGS).  The only other\nrestriction on this subsample is that the sources be observable\nusing UKIRT, i.e. $-25^\\circ$ $\\leq \\delta \\leq $ 60$^\\circ$.\nTable 5 gives the ultraluminous and luminous samples with the\nsource name in column 1, the 8--13 spectral classification in\ncolumn 2, the reference for the 8--13 \\hbox{\\micron~} spectrum in column 3,\nthe $\\log$ of the 8--1000 \\hbox{\\micron~} luminosity in column 4, and the\n$\\log$ of the ratio of the 12 \\hbox{\\micron~} flux density to the 60 \\hbox{\\micron~}\nflux density in column 5. The ultraluminous sample has 5 members\nand the luminous comparison sample has 20 members.  The\nultraluminous sample is {\\em unlikely} to span all the possible\ncharacteristics of ultraluminous infrared galaxies and therefore\nis not fully representative.  There are two observations that\nconfirm this: (1) if Arp 220 were scaled to the 60 \\hbox{\\micron~} 11.25 Jy\nflux limit used to define this sample, it would not have been\nobservable using CGS3 or other ground-based spectrographs\navailable when these data were collected, and (2) the ratio\ngiven in column 5 of Table 5 for the two brightest ultraluminous\ngalaxies (Arp 220 and Mkn 231) bracket the same ratio for all\nthe galaxies in the sample with $1.6\\times10^{11} \\leq$ \\hbox{$L_{\\rm IR}$}\n$\\leq 10^{12}$ \\hbox{L$_\\odot$}~ except NGC 1068 and NGC 7469.  An analysis\nover the entire Bright Galaxy Survey shows that the dispersion\nof this ratio is indeed larger for ultraluminous infrared\ngalaxies than for luminous infrared galaxies by a factor of 2\n(Soifer \\& Neugebauer 1991).  Thus, conclusions based on the\npresent flux-limited sample of ultraluminous galaxies should be\nviewed with caution not simply because of the small number of\ngalaxies in this sample (counting statistics on a bimodal parent population), but because the\ncharacteristics of the parent population are not well\nrepresented (under-sampling).\n\n\n\nIt is possible that the last of these two difficulties\ncould be somewhat ameliorated by the following\nconsiderations.  Ultraluminous galaxies that fall outside\nthe range of 12 on 60 \\hbox{\\micron~} flux density ratios defined by\nArp 220 and Mkn 231 may be assigned to the AGN-dominated\nbin in the context of exploring the AGN vs. starburst\nhypotheses.  For ultraluminous galaxies with 12\/60 \\hbox{\\micron~}\ncolors redder than Arp 220, PAH emission cannot make a\ncontribution to the 12 \\hbox{\\micron~} flux density that is\nsufficiently large to explain the 60 \\hbox{\\micron~} flux density. \nIt is therefore likely that such sources will have deep\nsilicate absorption features, and by the arguments\npresented by Dudley \\& Wynn-Williams (1997), the size of\ntheir power source should be too small to be due to a\nstarburst.  For galaxies with 12\/60 \\hbox{\\micron~} colors bluer than\nMkn 231, 60 \\hbox{\\micron~} selected samples will overlap 25 or 12\n\\hbox{\\micron~} selected samples that are known to select for AGNs\n(de Grijp et al. 1985; Spinoglio \\& Malkan 1989).  However,\nreliance on these notions would be ill advised given the\ninternal evidence in the present data showing that PAH\nemission can be important in ultraluminous galaxies with\n12 to 60 \\hbox{\\micron~} flux density ratios similar to the two\nextremes, namely, IRAS 17208$-$0014 and IRAS 05189$-$2524.\n\n\n\n\n\n\nInspection of Table 5 shows that 3 out of 5 of the ultraluminous\ngalaxies are powered predominantly by AGNs although if judgment\nis reserved for Arp 220, this fraction would be 2 out of 5. The\ntrend is thus consistent with about half of ultraluminous\ngalaxies being powered by AGNs.  On the other hand, the fraction\nof the more numerous luminous galaxies in this sample that are\npowered predominantly by AGNs is at most 0.25 and could be as\nsmall as 0.05 or smaller if the relative contributions of the\nAGN and starburst in NGC 1068 are approximately equal (Telesco\net al. 1984).  This trend supports the evidence from optical\nspectroscopy that AGNs are present in greater numbers in\nultraluminous infrared galaxies (Veilleux et al. 1995).  It goes\nfurther than optical or radio studies because it is possible to\ndemonstrate with greater certainty which type of power source\n(starburst or AGN) dominates the infrared luminosity.  Further,\nit is apparent that for luminosities less than 10$^{12}$ \\hbox{L$_\\odot$},\nstar formation is far and away the dominant power source, a\nresult that confirms work at many wavelengths.  This analysis\nlends some support to the proposal of Sanders et al. (1988) that\nthere is an evolutionary connection between ultraluminous\ninfrared galaxies and quasars, while at the same time\ndemonstrating that massive star formation can indeed be the\ndominant power source of some ultraluminous infrared galaxies.\nHowever, it must be conceded that the evidence presented\nconstitutes only a trend. The precise proportion of\nAGN-dominated sources among 60 \\hbox{\\micron~} selected ultraluminous\ninfrared galaxies cannot be said to have been established here,\nand thus no statistically meaningful difference between luminous\nand ultraluminous galaxies has truly been shown.  It is\nof interest that the trend does continue with the next step down\nin 60 \\hbox{\\micron~} flux density.  Adding IRAS 12112+0305 and IRAS\n08572+3915 to the present sample of five ultraluminous galaxies adds\none apparently starburst-dominated (based on the apparently\nnormal ratio of the 6.2 \\hbox{\\micron~} PAH feature emission to overall infrared\nemission in the IRAS 12112+0305 spectrum of Genzel et\nal. 1998\\footnote{A crude attempt to correct for the\ncontribution of the silicate absorption edge to the 7.7 \\hbox{\\micron~} PAH\nfeature by fitting a power law to continuum points at 5.9 and\n14.3 \\hbox{\\micron~} suggests a deficit in the strength of the 7.7 \\hbox{\\micron~} feature for\nIRAS 12112+0305.})\nand one AGN-dominated object (Dudley \\& Wynn-Williams 1997).\n\n\\subsection{Anti-correlation between 11.3 \\hbox{\\micron~} eqivalent width and\n$L_{\\rm IR}$\/$M({\\rm H}_2)$\\label{ch5:d3}}\n\n\nIn Fig. 4  a possible anti-correlation between\nthe equivalent width (EW) of the 11.3 \\hbox{\\micron~} feature and the global ratio of $L_{\\rm IR}$\/$M({\\rm H}_2)$ is presented for a number of galaxies.  The\nEW data are based either on the\nmodel fits given in Table 3 or the previously published\nvalues of RASW reproduced in Table 6. The ratios of\ninfrared luminosity to H$_2$ mass are taken from the\nliterature (Tables 4 and 6).  The solid\nline in Fig. 4 indicates inverse proportionality.\n\n\n\n\n\nThe ratio ${L_{\\rm IR} \\over {\\rm M(H}_2)}$ is often taken to be\na measure of star formation efficiency.  When it is large, then\na large fraction of the available gas is involved in star\nformation; when it is small, conditions may be such that the gas\nexists in a more quiescent state.  However, for a very large\nratio of ${L_{\\rm IR} \\over {\\rm M(H}_2)}$, Sanders et\nal. (1991) argued that even a very high star formation\nefficiency cannot account for the infrared luminosity, on the\ngrounds that star formation would too quickly consume the\navailable gas mass, so that AGN-like activity might be the more\nlikely primary power source of the infrared luminosity.  Both\n${L_{\\rm IR} \\over {\\rm M(H}_2)}$ and EW(11.3) are independent\nof distance in so far as EW(11.3) is independent of aperture\nsize.\n\nA tendency for EW(11.3) to be anti-correlated with the star\nformation efficiency might arise if two conditions were met.\nFirst, if the intensity of the 11.3 \\hbox{\\micron~} feature is a roughly\nconstant fraction of the overall infrared emission (proportional\nto $L_{\\rm IR}$) then this will not contribute to the\nanti-correlation.  Second, if the strength of the continuum at\n11.3 \\hbox{\\micron~} increases with decreasing availability of dust\n(inversely proportional to M(H$_2$)) to reprocess the light\nemitted by massive stars so that a larger fraction of the dust\ngrain population is heated to the warm temperatures required for\nthe dust grains to emit at 11.3 \\hbox{\\micron~} then this could be the\nsource of anti-correlation.  Such an explanation requires\nfurther elaboration since the process by which the 11.3 \\hbox{\\micron~}\ncontinuum is produced is likely, in general, to be a local\nphenomenon rather than a global one.\n\nAn elaboration that may be in keeping with recent {\\em HST}\nobservations (Watson et al. 1996; Surace et al. 1998;\nSatyapal et al. 1997) is that as the star formation\nefficiency increases, the proportion of massive stars formed\nin dense clusters may increase.  If this is the case, then\nthe time-scale for clearing the interstellar medium from\naround these clusters to a distance where dust grains would\nno longer be heated sufficiently to contribute to the 11.3\n\\hbox{\\micron~} continuum may be longer that the time-scale for\nclearing the interstellar medium around the groups of\nmassive stars that might form if the star formation\nefficiency is lower.  Two effects may contribute to this.\nFirst, the distance at which a grain could be heated to a\ntemperature high enough to contribute to the 11.3 \\hbox{\\micron~}\ncontinuum would increase as the square root of the number of\nmassive stars in a cluster so that if the velocity of gas\nand dust being cleared from clusters of massive stars is not\ntoo different from the velocity of gas and dust cleared from\ngroups of massive stars, the dust remains heated for a\nlonger period of time.  Second, dense clusters might be\ntemporarily gravitationally bound and behave ballistically\nwith respect to the dissipative gas. They may sometimes move\ninto a fresh reservoir of gas, heating the associated dust\nto higher temperatures.\n\nIn this view, the additional contribution to the 11.3 \\hbox{\\micron~}\ncontinuum emission at higher star formation efficiency would\nbe provided by possibly equilibrium temperature grains\nheated by a combination of direct and Ly$\\alpha$ heating\nrather than small grains heated by a single photon.  To\nzeroth order, this is a sensible proposition, since one\nexpects (at least within photo-dissociation regions) that\nthe PAH emission and the spike-heated grain emission to have\nroughly fixed relative contributions at 11.3 \\hbox{\\micron~} due to the\nrough similarity of the emission time scales for the two\nemission processes, so that the apparent anti-correlation\nwould not necessarily arise.\n\nA number of observations will be required to examine this\nsituation.  Most important are higher signal-to-noise\nobservations that will allow real measurements of EW(11.3)\nfor a number of the sources included in Fig. 4.  These are\nrequired to decide if the anti-correlation is has any merit.\nAnother avenue of investigation would be to make higher\nspatial resolution observations of nearby starbursts where\nclusters of massive stars are observed to exist to establish\nwhether or not EW(11.3) does decrease in the vicinity of\nsome clusters (see Dudley \\& Wynn-Williams 1999 for a\npreliminary attempt).  Finally, high spatial resolution 2\n\\hbox{\\micron~} adaptive optics imaging of a number of starbursts with\na range of star formation efficiency can help to determine\nif the fraction of massive stars that exist in dense\nclusters varies with the star formation efficency parameter\nas conjectured here.\n\n\\section{Conclusions\\label{ch5:c}}\n\nNew 8--13 \\hbox{\\micron~} spectroscopy of 27 infrared galaxies has been\npresented.  The spectra have been fit with a two-component model\nconsisting of a power-law continuum underlying a PAH emission\nspectrum.  These fits are based on the spectral data between 10 and\n13 \\hbox{\\micron~} and are used to predict the flux\nemitted between 8.0 and 8.4 \\hbox{$\\mu$m}.  It is\nfound that for the majority of the observed galaxies, the\nmodel predicts the observed 8.0--8.4 \\hbox{\\micron~} emission to\nwithin an acceptable factor of 2, suggesting that the\nmajority of the sources are powered by starbursts.\n\nOne source, Mkn 273, has been found to show very clear evidence for\nsilicate absorption.\n\nA flux-limited 60 \\hbox{\\micron~} selected subsample that combines the\npresent spectroscopic results and those of RASW had been\ndefined, consisting of 25 galaxies.  In this subsample the fraction of ultraluminous\ninfrared galaxies powered by AGN is $\\sim$50 per cent in\ncontrast to 5--25\\% among the luminous infrared galaxies.  \nOwing to the small sample size for the ultraluminous infrared\ngalaxies, this difference in AGN fraction can be considered only\na trend.\n\nA possible anti-correlation is identified between the\nequivalent width of the 11.3 \\hbox{\\micron~} PAH feature and the star\nformation efficiency in galaxies where both have been\nmeasured.  It is suggested that variation in the continuum\ndust temperature as a function of star formation efficiency\ncould produce this apparent effect.\n\n\n\n\\noindent{ACKNOWLEDGMENTS}\n\nThe author would like to thank the UKIRT staff: Joel Aikock,\nTim Caroll, Dolores Walthers, and Thor Wold, for assistance at\nthe telescope, and John Davies, Tom Geballe, and Gillian Wright\nfor fruitful discussions and calibration data in advance of\npublication.  UKIRT is operated by the Joint Astronomy Centre\non behalf of the U.~K.~Particle Physics and Astronomy Research\nCouncil.  Brian Rush, Reinhart Genzel, Vassilis Chamanadaris,\nJoe Hora, and D.-C. Kim all kindly provided data in advance of\npublication. A series of exchanges with Dieter Lutz has help to\nclarify certain issues, as has a discussion with Gary\nNeugebauer, Keith Matthews and Eichi Egami.  Eric Becklin has\nprovided thoughtful comments at several stages of this work.\nMany members of the Institute for Astronomy have benefited this\nwork through discussion and sharing of work in progress.  A few\nare Gareth Wynn-Williams (dissertation committee chair), Dave Sanders, Bob Joseph, Klaus\nHoddap, Josh Barnes, Alan Tokunaga, Jeff Goldader, Jason\nSurace, and Masatoshi Imanishi; thanks are due to these and\nothers.  Louise Good proofread the manuscript prior to\nsubmission.  This work has been partially supported by NSF\ngrants ASTR-8919563, NASA grant NAGW-3938, and the Frank Orrall\nfund.  The NASA Extragalactic Database (NED) has been a\nconstant aid.  It is run by the Jet Propulsion Laboratory,\nCalifornia Institute of Technology, under contract with NASA.\nThis work has also made use of NASA's Astrophysics Data System\nAbstract Service.\n\n\\newpage\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\n\nUnderstanding the interplay of rare-earth local-moment magnetism\nand $3d$ transition metal itinerant magnetism is of fundamental\nphysical interest\\cite{Campbell72} and may help in the design of\nmore efficient permanent magnets.\nThe tetragonal $R$Fe$_4$Al$_8$ ($R=$rare earth) compounds\n(Fig.\\ \\ref{Fig1}) are well suited for studying this interplay\nbecause of simple symmetry conditions and because the interaction\nbetween the two magnetic sublattices is rather weak: Fe moments\nappear to order without a corresponding $R$ $4f$ order.\nConsequently, numerous studies have been performed on\n$R$Fe$_4$Al$_8$ (R148) over the last 30 years and interest in\nthese compounds has remained\nhigh.\\cite{Buschow78,Felner78,Fujiwara87,Duong01,Duong02,Langridge99,Paixao00,Paixao01,Schobinger98,Schobinger99}\n\n\\begin{figure}[tb]\n\\includegraphics[width=0.98\\linewidth]{pixGd148struc.eps}\n\\caption{(Color online) The crystal structure of $R$Fe$_4$Al$_8$.\nThin lines outline the tetragonal (I$4\/$mmm) unit cell, thick\nlines denote the Fe ``cage'' around the rare earth $R$. }\n\\label{Fig1}\n\\end{figure}\n\n\nNeutron scattering studies performed on R148 with various $R$\nindicate that Fe moments, between which the coupling is strongest,\norder between $100$ and $200\\, {\\rm K}$, generally in cycloidal\nstructures with propagation vectors parallel to [110] and moments\nconfined to the $a$$-$$b$\nplane.\\cite{Paixao00,Paixao01,Schobinger98,Schobinger99} For\nmagnetic rare earths $R$, ordering of the $R$ moments, with much\nweaker coupling and at much lower temperature $T$, has been\nreported. However, the critical temperature associated with this\nordering is very poorly defined. For example in Dy148 at\n$\\sim$$50\\,{\\rm K}$ the magnetic dc susceptibility rises faster\nthan expected for Curie-Weiss behavior, but there are no sharp\nfeatures.\\cite{Paixao00} Furthermore, the rare earth moments were\nfound to follow the Fe-moment\nmodulation.\\cite{Paixao00,Schobinger98} This seems to imply that\nthe Fe$-R$ moment interaction is stronger than the $R$$-$$R$\nmoment interaction.\n\nOf all rare earth elements, Gd has two highly interesting specific\nfeatures associated with its high-spin and zero-angular-momentum\n$4f$ state: 1) high spin results in strong magnetic coupling\nbetween the localized $4f$ and the conduction electrons, implying\nlarge magnetic interactions and the largest de Gennes factor of\nall rare earths; 2) zero angular momentum implies a spherical $4f$\ncharge-cloud and no magneto-crystalline anisotropy (MCA) resulting\nfrom crystal-electric-field effects $-$ the direct interplay of\nmagnetic interactions can be studied without\ncrystal-electric-field effects. In view of this, it is somewhat\nsurprising that GdFe$_4$Al$_8$ (Gd148) has been much less\nstudied\\cite{Fujiwara87,Duong01} than R148 with other rare earths.\nIn particular, apart from M{\\\"{o}}ssbauer\nspectroscopy,\\cite{Buschow78} there have been no microscopic\nstudies of Gd148.\n\nHere, we report on the flux-growth of single crystals of the\nGdFe$_4$Al$_8$ phase, and on the characterization of the crystals\nby magnetization, electrical transport, specific heat, and x-ray\ndiffraction. We also report on first synchrotron x-ray resonant\nmagnetic scattering (XRMS) data. We provide evidence for two\nconsecutive phase transitions at low temperature $T$ in addition\nto the N\\'{e}el transition at $155\\, {\\rm K}$, and show that the\ntwo, low-$T$, transitions are associated with the ordering of the\nGd $4f$ moments, resulting in complex magnetic structures\ninvolving both ferro- and antiferromagnetic components. In\ncontrast to other R148 compounds, the $4f$-moment ordering is\nassociated with a propagation vector that is distinct from that\nassociated with the Fe moment order, which at low $T$ has the same\nmodulation as in the vicinity of the {N{\\'{e}}el} temperature. We\nalso present the magnetic phase diagram for two field directions\nand discuss the implications of changes in the Fe stoichiometry.\n\nThe paper is organized into seven sections. In Sec.\\ \\ref{exp} we\ndescribe the flux growth of the single crystalline samples used in\nthe study, and the experimental procedures. In Sec.\\ \\ref{lowH}\nzero and low field electrical transport, magnetization, and\nspecific-heat data are presented, and we evaluate the influence of\nthe iron stoichiometry on the physical properties. In Sec.\\\n\\ref{highH} we present electrical transport and magnetization data\nmeasured in fields for two field directions, and in Sec.\\\n\\ref{xrms} the results of a first XRMS experiment are described.\nFinally, we discuss the low-temperature magnetic phase diagram and\npossible magnetic structures in Sec.\\ \\ref{disc}, before\nsummarizing our main conclusions in Sec.\\ \\ref{conc}.\n\n\\section{Experimental}\n\\label{exp}\n\nWhereas all previous single crystals of $R$Fe$_4$Al$_8$ material\nwere obtained using the Czochralsky-method, we have grown single\ncrystals of Gd148 (and other R148) with a self-flux\nmethod.\\cite{Sol1,Sol2,Sol3} One of the problems with studies on\nsingle crystals of R148 is a width of formation often observed,\ninvolving some Fe atoms occupying nominal Al positions and vice\nversa; this can lead to striking differences in the magnetic\nstructure and phase diagram as compared to the stoichiometric\nmaterial (see, e.g., Refs.\\ \\onlinecite{Paixao01,Waerenborgh00}).\nA flux-growth procedure may allow better control of the\nstoichiometry of the crystals by varying the starting composition.\nUnfortunately, information about the ternary phase diagram\n$R$-Fe-Al is very limited. We used differential thermal analysis\n(DTA; see Ref.\\ \\onlinecite{Janssen05} for a review of the use of\nDTA in the flux growth of crystals) to establish i) that\nGdFe$_4$Al$_8$ is congruently melting and ii) selected solidus\ntemperatures in the ternary around the GdFe$_4$Al$_8$\nstoichiometry. For the crystals used in this study, we selected\nstarting compositions between Gd148 and the also congruently\nmelting Fe$_2$Al$_5$ (Gd$_3$Fe$_{16}$Al$_{34}$ and\nGd$_2$Fe$_{14}$Al$_{31}$), which should not lead to large\nimbalances in the Fe to Al ratio.\n\nElements in ratios corresponding to the starting composition were\nfirst arc-melted together. The resulting ingot was placed in an\nalumina crucible, which was wrapped in Ta foil (to prevent any\nresidual oxygen in the Ar from reacting with the sample) and\nplaced in a vertical tube furnace in flowing Ar. Crystals were\ngrown by heating to $1475^{\\circ}{\\rm C}$ and then slowly\n($2^{\\circ}{\\rm C} \/ {\\rm h}$) cooling to $1180^{\\circ}{\\rm C}$,\nat which point the furnace was turned off. The flux was removed\nfrom the crystals in a second step by heating to $1200^{\\circ}{\\rm\nC}$, keeping the temperature for $30\\,{\\rm min}$ and then\ndecanting, following procedures described in Refs.\\\n\\onlinecite{Sol1,Sol2,Sol3,Janssen05}.\n\nFor both starting compositions, we obtained crystals of the Gd148\nphase, as identified via powder x-ray diffraction. Crystals\ntypically grow prismatically with (110) facets and the long\ndirection parallel to [001], as determined with Laue scattering.\nWe obtained crystals of dimensions up to about $10\\times 2\\times 2\n{\\rm mm}^3$. We determined, via single crystal\nx-ray-diffraction-structure refinement and electron-microprobe\nanalysis (employing single crystals of GdFe$_2$ as standards), the\ncrystals to be slightly iron deficient, but detected no Al on Fe\nsites or vice versa (measured compositions were between\nGdFe$_{3.88(5)}$Al$_8$ and GdFe$_{3.96(1)}$Al$_8$; deviations of\nthe Al stoichiometry from $8$ were always less than 2 standard\ndeviations and are not listed).\n\nWe note that the x-ray-diffraction-structure refinement cannot\ndistinguish between a Fe deficiency due to vacancies and a\n(larger) Fe deficiency due to Al on Fe sites, both scenarios can\ngive an equal electron density. However, assuming an Al\/Fe mixture\nto retain the full occupancy of the Fe site leads (for one of the\ninvestigated crystals) to a GdFe$_{3.92(1)}$Al$_{8.08(1)}$\ncomposition and, thus, to an excess of Al. Since the\nelectron-microprobe analysis results indicated no significant\nvariations in the $1\/8$ ratio of Gd to Al, we did the structure\nrefinements under the assumption that the electron density on the\nFe sites is caused by vacancies, although some Al\/Fe substitution\ncould never be excluded. Whatever scenario is chosen, none of our\nconclusions drawn in this article are affected by the issue of\nwhether or not some Al atoms are present on the Fe site, and for\ndefiniteness subsequently we will assume that the vacancy scenario\nis the correct one. The composition of our samples can thus be\ndescribed with the empirical formula GdFe$_{4-\\delta}$Al$_8$, with\n$\\delta \\approx 0.04-0.12$. Surface scans in the\nelectron-microprobe analysis gave no indications of compositional\nvariations within the same crystal.\n\nSince the exact iron stoichiometry varies slightly from crystal to\ncrystal (even within the same growth batch) and this was found to\nhave a significant influence on the magnetic properties (see Sec.\\\n\\ref{lowH} below) most of the measurements presented below were\nperformed on the same crystal, sample I, which has a refined\ncomposition GdFe$_{3.96(1)}$Al$_8$ ($a=8.7699(9)\\, $\\AA,\n$c=5.0440(6)\\, $\\AA). A bar cut from the crystal by a wire saw was\nconnected with contacts for 4-point electrical-transport\nmeasurements with the current density $j\\|$[110] (sample Ia).\nOther samples were connected with contacts for electrical\ntransport measurements with either $j\\|$[110] or $j\\|$[001]. On a\nsecond bar (sample Ib, which had a mass of $2.22\\,{\\rm mg}$) cut\nfrom crystal I we measured the longitudinal magnetization and the\nzero-field specific heat. In order to investigate the relation\nbetween compositional variations and variations in the transition\ntemperatures, we carried out additional magnetization measurements\non several other crystals, the composition of which was determined\nby x-ray diffraction and electron microprobe. The additional\ncrystals are labelled II to VII in the order of their appearance\nin the text. The above measurements were performed with commercial\n(Quantum Design) laboratory equipment. For the high-field\nmeasurements, the sample orientation was fixed with a\ntwo-component glue (X60 from {W{\\\"{a}}getechnik} GmbH, Darmstadt,\nGermany).\n\nSample Ib was also used in the XRMS experiment, performed on the\n6ID-B undulator beamline in the MUCAT sector at the Advanced\nPhoton Source, Argonne National Laboratory. The incident energy of\nx-rays was tuned to the Gd $L_{\\rm II}$ edge ($E=7.934\\,{\\rm\nkeV}$), using a liquid nitrogen cooled, double crystal Si (111)\nmonochromator and a bent mirror. Sample Ib was mounted on a copper\nrod on the cold finger of a closed cycle displex refrigerator,\nsuch that a natural (110) facet was exposed to the x-ray beam.\nThermal transfer was enhanced by embedding the sample in copper\npaste. The sample, oriented so that the scattering plane of the\nexperiment was coincident with the $a-b$ plane, was encapsulated\nin a Be dome with He exchange gas to further enhance thermal\ntransfer. The incident radiation was linearly polarized\nperpendicular to the scattering plane ($\\sigma$ polarized). In\nthis geometry, only the component of the magnetic moment that is\nin the scattering plane contributes to the resonant magnetic\nscattering arising from electric dipole transitions ($E1$) from\n$2p$ to $5d$ states. Furthermore, the dipole resonant magnetic\nscattering rotates the linear polarization into the scattering\nplane ($\\pi$ polarization). In contrast, charge scattering does\nnot change the polarization of the incident beam\n($\\sigma\\rightarrow\\sigma$ scattering). Pyrolytic graphite PG\n(006) was used as a polarization analyzer, selecting primarily\n$\\pi$ polarized radiation. For $E=7.934\\,{\\rm keV}$, the\npolarization analyzer used reduces the detected intensity\nresulting from $\\sigma\\rightarrow\\sigma$ charge scattering by\nabout $99.9\\%$, whereas the $\\sigma\\rightarrow\\pi$ resonant\nmagnetic scattering is passed with little loss. Thus, the\npolarization analysis suppresses the charge scattering relative to\nthe magnetic scattering signal.\n\n\n\\label{res}\n\\section{Low field measurements and influence of Fe content}\n\\label{lowH}\n\\begin{figure}[tb]\n\\includegraphics[width=0.98\\linewidth]{xlHTP.eps}\n\\caption{(Color online) Zero-field resistivity $\\rho$ (along\n[110], sample Ia) and inverse dc susceptibility $\\chi ^{-1}$ vs\ntemperature $T$ (sample Ib). } \\label{Fig2}\n\\end{figure}\n\nIn this section, we will first present zero-field transport and\nlow field magnetization measurements on our main crystal (samples\nIa\/Ib). Then, we will briefly discuss sample-to-sample differences\nin the characteristic temperatures, which we think are related to\nFe deficiency. Finally, we will present specificheat measurements\non sample Ib indicating that the Gd $4f$ moments are ordered only\nat low temperatures. The main implications of the measurements\nwill be discussed later in Sec.\\ \\ref{disc}.\n\nIn Fig.\\ \\ref{Fig2}, we show, for sample Ia, the $T$ dependence of\nthe resistivity $\\rho$ ($j\\|$[110]). Since the sample is rather\nsmall, the estimated uncertainty in the absolute value of the\nresistivity is about 15\\%. The antiferromagnetic ordering at $T_N\n\\simeq 155\\, {\\rm K}$ is visible as a sharp kink in $\\rho\n(T)$.\\cite{note_Palasyuk04} The inverse dc magnetic susceptibility\n$\\chi^{-1}$ ($M\/H$) of sample Ib, cut from the same crystal as\nsample Ia, is also presented in Fig.\\ \\ref{Fig2}. The dashed and\ndotted curves shown are determined from the magnetization $M(T)$\nin a field of $\\mu_0 \\,H=0.1\\,{\\rm T}$ applied parallel to [110]\nand parallel to [001]. Below this field $M$ vs $H$ is linear for\n$T \\gtrsim 40\\, {\\rm K}$. Since $\\chi$ is slightly anisotropic, we\nalso calculated the polycrystalline average $\\chi_{\\rm\npoly}=(2\\chi_{110}+\\chi_{001})\/3$, shown as squares. In\n$\\chi^{-1}$ for both directions the antiferromagnetic ordering is\nmanifest by a change in slope, but it is much less sharp than the\ncorresponding signature in the resistivity. A weak signature of\nthe {N{\\'{e}}el} transition may be expected if, as we will discuss\nbelow, only the Fe moments order at $T_N$. In agreement with\nmeasurements on polycrystalline samples,\\cite{Buschow78} $\\chi$\nfollows Curie-Weiss behavior both above and, over a limited $T$\nrange, below $T_N$. We found (on the calculated polycrystalline\naverage) above $T_N$ a Weiss temperature of $-165\\,{\\rm K}$ and an\neffective moment of $11.4 \\, \\mu_B \\, \/ {\\rm f.u.}$. Assuming\ncontributions by the magnetic atoms that are additive in the Curie\nconstant and a contribution by Gd of $7.9 \\, \\mu_B \\, \/ {\\rm Gd}$,\nthe free moment per Fe atom in the paramagnetic state is $4.1 \\,\n\\mu_B \\, \/ {\\rm Fe}$. Below $T_N$ (range $70-115\\,{\\rm K}$) we\nfound a Weiss temperature of $17\\,{\\rm K}$ and an effective moment\nof $7.4 \\, \\mu_B \\, \/ {\\rm f.u.}$. The latter is closer to the Gd\nfree-ion value ($7.9 \\, \\mu_B \\, \/ {\\rm f.u.}$) than the one\nreported by Buschow {\\em et al.}\\cite{Buschow78} ($6.2 \\, \\mu_B \\,\n\/ {\\rm f.u.}$).\n\n\\begin{figure}[t!b]\n\\includegraphics[width=0.98\\linewidth]{xlRcomps.eps}\n\\caption{(Color online) Low-temperature normalized resistance\n(panel a) and derivative of high temperature resistance (panel b)\nvs temperature $T$ of various crystals: Ia (thick line) and II\n(dotted line) for $j\\|$[110], III (open circles) and IV (full\ncircles) for $j\\|$[001].} \\label{Fig3}\n\\end{figure}\n\nThe resistivity curve shows additional features just below $30\\,\n{\\rm K}$, magnified in Fig.\\ \\ref{Fig3}a) (thick line): The broad\npeak in $\\rho (T)$ at $\\sim \\! \\! 28\\, {\\rm K}$ with a drop at\n$\\sim \\! \\! 26.5\\,{\\rm K}$ ($T_2$) indicates a phase transition.\nFurthermore, with decreasing $T$, at $\\sim \\! \\! 21\\, {\\rm K}$\n($T_1$) there is a sudden increase in $\\rho$ by $\\sim 10\\%$. This\nfeature is hysteretic in temperature, suggesting that the feature\nis associated with a first-order transition.\\cite{note_Fujiwara}\nNote that the hysteresis visible at $T_2$ is opposite to what is\nexpected for a first-order transition (i.e.,\nsuperheating\/supercooling). It is most likely a remnant of the\nhysteresis associated with the $T_1$ transition. At $T_1$, the\nform of the hysteresis is consistent with\nsuperheating\/supercooling.\n\nAlso shown in Fig.\\ \\ref{Fig3}a) are low-$T$ zero-field\nresistivity curves for three additional samples, one also with\n$j\\|$[110] and two with $j\\|$[001]. Whereas the feature of a\nsudden increase of $\\rho$ with decreasing $T$ is visible in all\ncurves, the other feature at slightly higher temperatures is only\nidentifiable when $j\\|$[110]. Based on the results from sample Ia\nonly, it may be tempting to associate the increase in $\\rho$ (with\ndecreasing $T$) with the opening of a superzone\ngap.\\cite{Mackintosh62} However, the presence of this increase\nwith the same order of magnitude for both $j\\|$[110] and\n$j\\|$[001] makes a superzone gap scenario less likely.\n\nIt is also clear from Fig.\\ \\ref{Fig3}a) that there are\nsignificant variations of the temperatures $T_1$ corresponding to\nthis feature. For comparison, Fig.\\ \\ref{Fig3}b) shows the\nresistivity derivative ${\\rm d} \\rho \/ {\\rm d} T$ for the same\nsamples at higher $T$, in order to make the {N{\\'{e}}el}\ntransition more visible, which manifests itself differently for\nthe two current density directions. The curves indicate a sharp,\nwell defined $T_N$ for each sample and comparing Figs.\\\n\\ref{Fig3}a) and \\ref{Fig3}b) suggests a correlation between\nhigher $T_1$ and lower $T_N$.\n\n\n\\begin{figure}[tb]\n\\includegraphics[width=0.98\\linewidth]{xlMTmicroprobe.eps}\n\\caption{(Color online) Low field ($H\\|[001]$) Magnetization of\nvarious crystals with Fe stoichiometry (as described in Sec.\\\n\\ref{exp}) determined by electron microprobe or\nx-ray-diffraction-structure refinement.} \\label{Fig4}\n\\end{figure}\n\nFigure \\ref{Fig4} displays low field ($H\\|$[001]) magnetization\nmeasurements performed on three samples with different Fe\nstoichiometry as determined by x-ray diffraction and electron\nmicroprobe: samples Ib (GdFe$_{3.96(1)}$Al$_8$, thick line), V\n(GdFe$_{3.94(6)}$Al$_8$, full squares), and VI\n(GdFe$_{3.88(5)}$Al$_8$, dashed line). The data were taken for\nboth decreasing and increasing $T$ (field cooled), and\nnormalized\\cite{note_normalizedM} to the maximum in magnetization.\nA comparison with Fig.\\ \\ref{Fig3}a) indicates that the rise in\n$M$ with $T$ is associated with $T_1$, the subsequent decrease\nwith $T_2$. A decrease in the Fe content seems to systematically\nshift $T_1$ and $T_2$ to higher temperatures. We note that for\ncrystal I $M(T)$ (Fig.\\ \\ref{Fig4}) indicates a higher $T_2$ than\n$\\rho (T)$ [Fig.\\ \\ref{Fig3}a)]. In principle, this might be due\nto a different Fe content of the two pieces (Ia and Ib) cut from\nthe crystal and used for these measurements. However, we do not\nbelieve that this is the case, because the specific-heat peak\nassociated with $T_2$ (see below; measured on Ib) corresponds well\nto $T_2$ deduced from $\\rho (T)$ on sample Ia.\n\nCombining the above data it seems that decreasing the Fe content\nresults in a decrease of $T_N$ and an increase of $T_1$ and $T_2$.\nIn order to confirm this, a study involving samples that are much\nmore Fe deficient, would be highly desirable.\n\n\\begin{figure}[tb]\n\\includegraphics[width=0.98\\linewidth]{nxlHCalt.eps}\n\\caption{(Color online) %\nSpecific heat $C_p$ measured on sample Ib in zero field\n($\\blacksquare$) and Gd $4f$ moment contribution to the specific\nheat ($\\circ$), obtained by subtracting the specific heat of\nYFe$_4$Al$_8$ (full line, from Ref.\\ \\onlinecite{Hagmusa98}).\nInset: Entropy of the Gd $4f$ moments obtained by integrating the\n$4f$ moment contribution to $C_p \/ T$. The two lines shown were\ncalculated using two extrapolations of the specific heat to $T=0$,\nindicated by dashed lines in the main panel (see text for details\nof the extrapolations). The dotted line in the inset indicates the\nfull Gd $4f$ entropy.} \\label{Fig5}\n\\end{figure}\n\n\nFigure \\ref{Fig5} shows the measured specific heat $C_p$ of sample\nIb (closed squares) after subtraction of the addenda contribution\n(the specific heat of sample platform and grease measured before\nmounting the sample). Also shown is the mass-scaled specific heat\nof YFe$_4$Al$_8$ from Ref.\\ \\onlinecite{Hagmusa98} (full line),\nand the difference between the two specific heats (open circles).\nSince the necessary mass-scaling is small, the remaining specific\nheat is close to the magnetic contribution associated with the Gd\nsublattice. The broad, asymmetric, peak at $\\sim \\! \\! 26.5 \\,\n{\\rm K}$ corresponds to the drop in $\\rho$ at $T_2$ [determined on\nsample Ia cut from the same crystal as sample Ib, see Fig.\\\n\\ref{Fig3}a) dashed curve]. The broadness of the specific-heat\npeak may suggest a degree of inhomogeneity of the Fe distribution\nin the sample, although the electron-microprobe analysis provided\nno indications of inhomogeneities within crystals.\n\n\nNo feature in the specific heat is visible around $T_1$, the small\ndiscontinuity at $20\\,{\\rm K}$ indicated by the data was found to\nbe an artifact due to the change of the heat pulse intensity\napplied by the system. We checked the specific-heat raw data, and\ncan exclude any latent heat restricted to a temperature region\nsmaller than the spacing of measurement points. We conclude that\nthe latent heat of the $T_1$ transition has to be small, i.e., the\nphases above and below $T_1$ have similar entropies. The\nalternative explanation of the specific heat, that $T_1$ is not a\nfirst-order transition, seems unlikely in light of not only the\npresence, but also the shape of thermal hysteresis observed in\n$\\rho (T)$, which is particularly clear when a magnetic field is\napplied in the [110] direction (see Sec.\\ \\ref{highH} below).\n\nThe corresponding magnetic entropy $S$, obtained by integration,\nis shown in the inset of Fig.\\ \\ref{Fig5}. The two curves shown\nare calculated with two different extrapolations of $C_p$ to zero\ntemperature (dashed lines in the main panel). One extrapolation is\nobtained by connecting the $C_p$ data point at the lowest $T$\nlinearly to $C_p=0$ at zero temperature, the other by connecting\nthe $C_p\/T$ data at the lowest $T$ linearly to $C_p\/T=0$ at zero\ntemperature. Given typically observed specific heats of Gd\ncompounds,\\cite{note_DuongGdFe2Ge2} the actual specific heat at\nlow temperature will most likely be between these two\nextrapolations, and thus the two curves shown in the inset of\nFig.\\ \\ref{Fig5} may be considered lower and upper limits of the\nGd $4f$ entropy (neglecting additional experimental\nuncertainties). At $30\\, {\\rm K}$ the magnetic entropy already\nreaches 80\\% (78\\% and 86\\% for the two low-$T$ extrapolations\nmade) of the full entropy of Gd $4f$ moments (${\\rm R}\\,\\ln 8$,\ndashed line). Above $45\\, {\\rm K}$ it hardly varies anymore,\nhaving reached $>90\\%$ and $>97\\%$ of the full $4f$ entropy for\nthe two low-$T$ extrapolations made. This strongly suggests that\nthe Gd $4f$ moments are not ordered above $T_2$, and the\ntransition at $T_2$ corresponds to the ordering of the $4f$\nmoments.\n\nBy careful measurement and analysis of the low-field thermodynamic\nand transport data of solution grown single crystals of Gd148 we\nhave shown that there are sharp features associated with two\nmagnetic phase transitions. These transitions appear to be\nprimarily related to the ordering of the Gd sublattice. In\naddition we have been able to establish a clear correlation\nbetween small variations in the Fe stoichiometry and $T_N$, $T_1$,\nand $T_2$. Having done this, we will now focus on the sample that\nis closest to full stoichiometry (crystal I) and try to more fully\ndelineate and understand the field and temperature dependence and\nthe structure of the magnetic phases in this sample.\n\n\n\\section{High-field measurements}\n\\label{highH}\n\nApplying a magnetic field often helps to clarify the nature of\nmagnetic phases observed in zero (or low) field measurements. We\ntherefore measured the field dependent electrical resistivity and\nthe magnetization on several samples. Shown in this section are\nelectrical transport (subsection \\ref{highHrho}) and magnetization\n(subsection \\ref{highHM}) data measured on samples Ia\/Ib at low\ntemperatures in fields applied in-plane and out-of-plane. The\nfeatures in resistivity corresponding to the two low-temperature\ntransitions are much sharper when an in-plane field is applied and\nobserved features suggest complex domain effects. The\nmagnetization data are used to estimate the in-plane and\nout-of-plane components of the spontaneous ferromagnetic moment.\nThe phase diagrams for $H\\|$[110] and $H\\|$[001], which can be\nconstructed from these transport and magnetization data, will be\npresented and discussed in Sec.\\ \\ref{disc}.\n\n\\subsection{Electrical transport}\n\\label{highHrho}\n\\begin{figure}[tb]\n\\includegraphics[width=0.98\\linewidth]{xldomains.eps}\n\\caption{(Color online) resistivity $\\rho$ vs $T$ in fields of\n$3.5\\,{\\rm T}$ applied in-plane parallel or perpendicular to the\ncurrent density $j\\|[110]$ (sample Ia). Arrows indicate the $T$\ndirection in which the measurements were performed. The open\ncircle indicates the value of the $T_1$ transition in $3.5\\,{\\rm\nT}$, averaged for $T$ increasing and decreasing.} \\label{Fig6}\n\\end{figure}\n\nFigure \\ref{Fig6} shows the resistivity $\\rho$ ($j\\|$[110]) of\nsample Ia when a field of $3.5\\,{\\rm T}$ is applied either $\\|j$\n(longitudinal resistivity) or $\\|[1\\overline{1}0]$ (i.e.\\\ntransverse resistivity, $\\perp$$j$, but in a crystallographically\nequivalent in-plane direction). For both field directions, the\nmagnitude of the jump at $T_1$ is enhanced and the hysteresis in\ntemperature greatly increased, making the first-order nature of\nthe transition more apparent. The average\n($T\\uparrow$,$T\\downarrow$) temperature of the $T_1$ feature,\nhowever, is only weakly influenced by an in-plane field. In\ncontrast to this, the $T_2$ feature is shifted to higher $T$ by\n$H\\|$[110], suggesting a ferromagnetic nature of the phase at\n$T<T_2$. Furthermore, the fields remove the peak in $\\rho$ just\nabove the drop at $T_2$, and cause the feature to exhibit\nhysteresis. This different response to $H\\|$[110] confirms that\nthere are indeed two independent magnetic transitions $T_1$ and\n$T_2$, dividing the ordered state below $T_N$ into three magnetic\nphases, a low-temperature phase (LTP), an intermediate-temperature\nphase (ITP), and a high-temperature phase (HTP).\n\n\n\\begin{figure}[tb]\n\\includegraphics[width=0.98\\linewidth]{xlRTgraph.eps}\n\\caption{(Color online) a) Resistivity $\\rho$ vs $T$ (sample Ia,\n$j\\|$[110], $T$ increasing) in fields applied parallel to\n$j\\|$[110]. From top to bottom 0 (thick line), 0.1, 0.2, 0.4, 0.5,\n0.8, 1 (thick line), 1.5, 2, 2.5, $3.5\\,{\\rm T}$. b) As panel a),\nbut in fields applied parallel to [001] ($j\\|$[110]). From top to\nbottom 0 (thick line), 0.1, 0.2, 0.3, 0.4, 0.5, 0.7, 0.8, 0.9, 1,\n2, $5.5\\,{\\rm T}$.} \\label{Fig7}\n\\end{figure}\n\nWhereas there are similarities in $\\rho (T)$ for the two field\ndirections considered, there are also very pronounced differences,\nparticularly for the $T_1$ feature, below which the field cooled\n$\\rho(T)$ is much higher for $H$$\\perp$$j$ than for $H\\|j$. A\ntransverse magnetoresistance higher than the longitudinal one\nmight arise from a bending of the electron trajectory by the\nLorentz force (see, e.g., Ref.\\ \\onlinecite{Harris95}). However,\nthe zero-field-cooled $\\rho (T)$ below $33\\,{\\rm K}$ (black dotted\nline) is the same (within error bars) for $H\\|j$ and $H$$\\perp j$.\nIndeed, as we will see (Fig.\\ \\ref{Fig8} below), at $5\\,{\\rm K}$\n{\\em after cooling in zero field} the resistivity is only very\nweakly field dependent for an in-plane field (both for transverse\nand longitudinal configurations).\n\n\\begin{figure}[tb]\n\\includegraphics[width=0.98\\linewidth]{nxlRHgraph.eps}\n\\caption{(Color online) Resistivity vs $H\\|j\\|$[110] at selected\n$T$ (sample Ia).} \\label{Fig8}\n\\end{figure}\n\nThis indicates that the difference in the $\\rho (T)$ curves of\nFig.\\ \\ref{Fig6} originates from the magnetism. The\nmagnetoresistance then may arise either from spin-disorder\nscattering or scattering of domain walls\\cite{Singh76} and the\ndifference can be due to a different configuration of domains. In\nprinciple, this configuration (domain population, average size of\ndomains, etc.) may be different for $H\\|[1\\overline{1}0]$\n(transverse magnetoresistance) and $H\\|[110]$ (longitudinal\nmagnetoresistance) due to different demagnetizing effects (the\nsample was cut in a rod such that the [110] direction is a bout\nthree times longer than the $[1\\overline{1}0]$ direction). In any\ncase, the fact that the resistivity is almost field-independent\n(for an in-plane field) at $5\\,{\\rm K}$ indicates that the\nrelevant (for resistivity) domain configuration in the LTP is very\nhard to change by applying a magnetic field, whereas a field\napplied in the ITP prepares a certain LTP domain configuration\nupon cooling below $T_1$. We note that application of a field in\nthe ITP also seems to lead to small, but systematic, changes in\nthe resistivity values in the HTP, persisting up to $\\sim$$T_N$.\n\nFigure \\ref{Fig7}a) shows the evolution of $\\rho (T)$ [for clarity\nonly the data measured upon warming, after the lowest temperature\nhad been reached by cooling with the same field applied\n(field-cooled-warming measurement protocol) are displayed] with\nincreasing $H\\|j$ (again sample Ia, $j\\|$[110]). It seems clear\nthat the $T_2$ transition systematically shifts to higher\ntemperatures with increasing $H$. The $T_1$ transition is far less\nfield dependent (for $H\\|$[110]), and as discussed above, quite\nhysteretic. Furthermore, the resistivity in the ITP becomes lower\nin higher $H$, developing into a well-defined weakly\ntemperature-dependent valley above $\\sim$$2\\,{\\rm T}$. This\ndevelopment is also visible in the $\\rho (H)$ dependence at\nconstant $T$ (Fig.\\ \\ref{Fig8}). There is only a minimal field\ndependence of $\\rho$ at $5\\,{\\rm K}$ in the LTP. In contrast to\nthis, a pronounced drop in low fields at $25\\,{\\rm K}$ in the ITP\nis shifted to higher fields by increasing $T$ and seems to\ntransform into the drop associated with the ITP-HTP transition. In\nthe $\\rho (T)$ curves of Fig.\\ \\ref{Fig7}a), for intermediate\nfield ranges, there are indications for an additional feature in\nthe ITP-HTP transition. The details of this feature have to be\nclarified in future work.\n\nThe evolution of $\\rho (T)$ with increasing $H\\|$[001] for the\nsame sample and current density direction is shown in Fig.\\\n\\ref{Fig7}b). The data shown were again obtained using the\nfield-cooled-warming measurement protocol, but we note that\napplication of $H\\|$[001] (unlike $H\\|$[110]) quickly suppresses\nany hysteresis, and broadens the transitions. Apart from a\nbroadening, the $T_2$ transition does not seem to be strongly\naffected. In contrast to this, $H\\|$[001] systematically shifts\nthe $T_1$ transition to lower temperatures and above\n$\\sim$$1\\,{\\rm T}$ there are no longer any indications for a\ntransition to the LTP. This indicates that the LTP is suppressed\nby $H\\|$[001].\n\n\\subsection{Magnetization}\n\\label{highHM}\n\n\n\\begin{figure}[tb]\n\\includegraphics[width=0.98\\linewidth]{xlMH.eps}\n\\caption{(Color online) Magnetization $M$ as a function of field\n$H$ at selected $T$, $H\\|[110]$ (blue) and $H\\|[001]$ (red),\nsample Ib. Thin arrows indicate the $H$ direction of the\nmeasurements, thick arrows indicate the criterion taken for the\nestimation of the spontaneous ferromagnetic moment.} \\label{Fig9}\n\\end{figure}\n\n\nIn order to investigate the magnetic nature of these phases, we\nmeasured the magnetization $M$ as a function of $T$ and $H$\n($\\|$[110] and $\\|$[001]). In Fig.\\ \\ref{Fig9}, we display for\nsample Ib $M(H)$ loops at selected $T$. For $H\\|$[110], at\n$40\\,{\\rm K}$, $M(H)$ shows almost linear behavior, but at\n$2\\,{\\rm K}$ $M(H)$ has a behavior typical for soft ferromagnets:\nsaturation in a field which is of the order of the estimated\ndemagnetizing field, but no hysteresis or remanence. However, the\nsaturation value of $4.5 \\mu_B\/{\\rm f.u.}$ at $2\\, {\\rm K}$ is\nmuch lower than the full Gd moment of $7 \\mu_B\/{\\rm f.u.}$. At\n$26\\,{\\rm K}$, pronounced hysteresis is visible in low $H$. This\nis ubiquitous for $M(H)$ in the ITP and correlates with the\nhysteresis in the drop in $\\rho$ visible in Fig.\\ \\ref{Fig8}a).\nThe hysteresis may suggest a first-order metamagnetic transition.\nIn higher fields $M(H)$ is nearly constant, resembling the\nbehavior at $2\\,{\\rm K}$. In low fields $M(H)$ indicates a\nspontaneous ferromagnetic moment, unlike at $2\\,{\\rm K}$, at\n$26\\,{\\rm K}$ we did observe remanence. The $T$ dependence of the\nspontaneous ferromagnetic moment, as estimated from Fig.\\\n\\ref{Fig9} taking into account demagnetizing effects (we took the\nmagnetization at the first kink in low $H$, indicated by thick\narrows in Fig.\\ \\ref{Fig9}), is shown in Fig.\\ \\ref{Fig9N} (black\nsquares). The LTP-ITP transition in $M(H)$, $H\\|$[110] is\naccompanied by a drop of the spontaneous moment to $\\sim \\! \\! 1\\,\n\\mu_B$, and by the appearance of hysteresis.\n\nFor $H\\|$[001] at $26\\,{\\rm K}$ (ITP), the form of $M(H)$ (Fig.\\\n\\ref{Fig9} red curves) is consistent with a spontaneous\nferromagnetic moment of about $2\\, \\mu_B$. For the LTP consider\n$M(H)$ at $2\\,{\\rm K}$. Below $0.5\\, {\\rm T}$, $M(H)$ is linear,\nabove it rises more strongly before slowly starting to saturate.\nThe field $H_1$ where $M(H)$ starts to increase faster than linear\nis lower at higher $T$, extrapolating to zero at $T_1$. For\nsamples with a higher $T_1$, $H_1$ at low temperatures is\nproportionally higher (not shown). Whereas the feature at $H_1$ as\ndefined above is rather sharp, the slow saturation of $M$ above\n$H_1$ may be due to a small inhomogeneity of the Fe content in the\nsample, as discussed in Sec.\\ \\ref{lowH}.\n\n\\begin{figure}[tb]\n\\includegraphics[width=0.98\\linewidth]{xlMsat.eps}\n\\caption{(Color online) Spontaneous ferromagnetic moment $M_s$,\nestimated from the $M(H)$ curves in Fig.\\ \\ref{Fig9} vs\ntemperature $T$, $H\\|[110]$ (black squares) and $H\\|[001]$ (red\ncircles), sample Ib. The square root of the sum of the squares for\nboth field directions is shown as open triangles.} \\label{Fig9N}\n\\end{figure}\n\n\nFor $H\\|$[001], the saturation value of $M \\lesssim 5\\, \\mu_B\/{\\rm\nf.u.}$ is higher than for $H\\|$[110], but still far below the full\nGd $4f$ moment. Note that the difference to $7\\, \\mu_B\/{\\rm f.u.}$\nis much too large to be explained by a negative contribution of\nthe Gd conduction electrons. The low saturated moment implies that\nthe magnetization does not correspond to the magnetization of a\nsimple ferromagnet. For $H\\|$[001], the temperature dependence of\nthe estimated spontaneous ferromagnetic moment is shown in Fig.\\\n\\ref{Fig9N} as red circles. It is similar to $M(T)$ in low $H$\n(c.f.\\ Fig.\\ \\ref{Fig4}).\n\nThe linear $M(H)$ behavior (below $0.5 \\, {\\rm T}$ at $2\\,{\\rm\nK}$) may be due in-plane ferromagnetic moments resisting a\nreorientation by an out-of-plane field. The LTP-ITP transition\ncould correspond to a reorientation of ferromagnetically aligned\n$4f$ moments (as suggested in Ref.\\ \\onlinecite{Fujiwara87}).\nFigure \\ref{Fig9N} indicates that in the LTP the ferromagnetically\naligned moments are confined to the $a$$-$$b$ plane, whereas in\nthe ITP the moments are mainly out-of-plane, pointing into a\ndirection with an angle of about $25^{\\circ}$ to [001]. A crude\nestimate of the magnitude of the spontaneous moment is given by\n$(M_{s,110}^2 + M_{s,001}^2)^{1\/2}$, shown in Fig.\\ \\ref{Fig9N} as\nopen triangles. Apart from one point near the LTP-ITP transition\n(where the estimation of $M_s$ is less clear than in other\nregions) the temperature dependence of $(M_{s,110}^2 +\nM_{s,001}^2)^{1\/2}$ is consistent with the LTP-ITP transition not\naffecting the magnitude of the ferromagnetic component, but only\nit's direction. Finally, we note that, like the low-field\nmagnetization (Fig.\\ \\ref{Fig4} full line), the estimated\nspontaneous moment extrapolates to zero around $30-35\\,{\\rm K}$,\ni.e., slightly higher than $T_2$ as indicated by resistivity and\nthe specific heat maximum. This roughly correlates with the {\\em\nonset} of the $4f$ contribution to the specific heat (Fig.\\\n\\ref{Fig5}), which is not sharp at all. We therefore think it is\nnot an indication of an additional transition, rather it may be\ndue to an inhomogeneous Fe distribution or due to significant\nprecursor-effects (e.g.\\ fluctuations).\n\nThe electrical-transport and magnetization data presented in this\nsection will allow us to construct $H$$-$$T$ phase diagrams for\n$H\\|$[110] and $H\\|$[001]. We will do this in Sec.\\ \\ref{disc}.\nFurthermore, we have established that the Gd $4f$ moment order\nsetting in at $T_2$ has a large ferromagnetic component, and the\nfirst-order (see Fig.\\ \\ref{Fig6}) transition at $T_1$ involves a\nreorientation of the ferromagnetic component into the $a$$-$$b$\nplane for $T<T_1$. In order to elucidate the underlying magnetic\nstructure we will now focus on scattering data.\n\n\\section{X-ray magnetic resonant scattering}\n\\label{xrms}\n\nIn order to prove the long-range nature of the magnetic order, and\nto learn more about the magnetic structure than the information\nthat is obtainable from thermodynamic and transport measurements,\nscattering techniques can be applied, typically neutron\nscattering. For Gd148, neutron scattering is less feasible because\nof the large neutron absorption cross-section of the natural\nisotope of Gd. An alternative technique, which relies on intense\nsynchrotron radiation and exploits a large increase in the\nscattered intensity due to resonant processes when the energy of\nthe radiation is close to an absorption edge of a magnetically\nordered element, is x-ray resonant magnetic scattering\n(XRMS).\\cite{Gibbs88,Hannon88,XRMSreview} To gain further insight\ninto the magnetism of Gd148, we therefore performed a first XRMS\nexperiment on the (together with piece Ia from the same crystal)\nbest characterized sample Ib.\n\nBased on the neutron scattering and XRMS results on R148 with\nother rare\nearths,\\cite{Langridge99,Paixao00,Paixao01,Schobinger98,Schobinger99}\nas well as on the thermodynamic and transport results presented in\nearlier sections, we expected at temperatures between $T_2$ and\n$T_N$ a modulation of Fe moments with propagation along the [110]\ndirection to be present, with a feasible resonant enhancement of\nthe intensity scattered at the corresponding satellite reflections\nat the Gd $L_{\\rm II}$ edge. Figure \\ref{Fig11}a) shows reciprocal\nspace scans along [$h$,$h$,$0$] with $\\sigma \\rightarrow \\pi$\npolarization analysis and the energy tuned to the Gd $L_{\\rm II}$\nedge ($7.934\\,{\\rm keV}$) at $37$ and $160\\,{\\rm K}$. At $37\\,{\\rm\nK}$, the charge reflection at (4,4,0) is accompanied by small\nsatellites at ($4\\pm \\tau$,$4\\pm \\tau$,$0$) with $\\tau=0.086$. The\nsatellite reflections are not present at $160\\,{\\rm K}$,\nindicating that they are due to the magnetic ordering at the\n{N{\\'{e}}el} temperature (c.f.\\ Fig.\\ \\ref{Fig2}). We also found\nsatellite reflections at ($3\\pm \\tau$,$3\\pm \\tau$,$0$), ($2\\pm\n\\tau$,$2\\pm \\tau$,$0$), and ($1\\pm \\tau$,$1\\pm \\tau$,$0$), with\n$0.06\\leq \\tau \\leq 0.135$, depending on $T$. The full width at\nhalf maximum (FWHM) of the satellite reflections is approximately\ntwice the FWHM of the charge reflections. The corresponding\nsmaller correlation length of the magnetic order may be an\nindication of the presence of small domains. We roughly estimate\nthe average size of the domains (in [110] direction) to be\n$\\sim$$0.17\\,\\mu{\\rm m}$ or about $140$ unit cells. Rearrangements\nof these domains might be the cause of the small changes in\nresistivity values in the HTP (see the discussion of Fig.\\\n\\ref{Fig6} in Sec.\\ \\ref{highHrho}).\n\n\\begin{figure}[t!b]\n\\includegraphics[width=0.98\\linewidth]{xlnewXRMS.eps}\n\\caption{(Color online) a) Reciprocal space scans along\n[$h$,$h$,0] at $37$ and $160\\,{\\rm K}$ (sample Ib, Gd $L_{\\rm II}$\nedge). The data in the range $h=3.96$ to $4.04$ have been divided\nby a factor of $10000$.\nb) Energy scans of the intensity of the ($4-\\tau$,$4-\\tau$,$0$)\nsatellite reflections at $9$, $37$, and $130\\,{\\rm K}$. Note that\nintensities in panel a) are not comparable to intensities in panel\nb).} \\label{Fig11}\n\\end{figure}\n\n\nThe intensity of the ($4\\pm \\tau$,$4\\pm \\tau$,$0$) satellite\nreflections is about four orders of magnitude lower than the\nintensity of the (4,4,0) charge reflection, despite the use of the\npolarization analysis, which suppresses the charge reflection\nintensity by three orders of magnitude. Due to the small intensity\nof the satellite reflections and their vicinity to the charge\nreflections, there is considerable intensity due to the charge\ntail at the position of the satellite reflections. A slight\nasymmetry in the charge tail leads to a larger charge contribution\nto the scattered intensity at the ($4+\\tau$,$4+\\tau$,$0$)\nreflection, and we therefore further analyzed only the\n($4-\\tau$,$4-\\tau$,$0$) satellite reflection.\n\n\n\nFigure \\ref{Fig11}b) shows, at selected $T$, energy scans over the\nGd $L_{\\rm II}$ edge for the satellite reflections at ($4-\\tau ,\n4-\\tau , 0$). The intensity of the resonance is largest at\n$37\\,{\\rm K}$. The lower intensity at $130\\,{\\rm K}$ is due to the\nvicinity of the {N{\\'{e}}el} transition, the lower intensity at\n$9\\,{\\rm K}$ will be discussed below in Sec.\\ \\ref{disc}. The\nposition and the (amplitude-normalized) peak shape of the\nresonance is $T$ independent within our resolution and consistent\nwith dipolar ($E1$)\ntransitions.\\cite{Langridge99,Gibbs88,Hannon88,XRMSreview} We note\nthat a resonant enhancement at the Gd $L_{\\rm II}$ edge was\nobserved only at the ($4\\pm\\tau , 4\\pm\\tau , 0$) satellite\nreflections, there was no enhancement at the (4,4,0) charge\nreflection, consistent with resonant magnetic scattering.\nReciprocal space scans and the resonant enhancement at the Gd\n$L_{\\rm II}$ edge are consistent with a magnetic modulation with\npropagation vector ($\\tau$,$\\tau$,0) associated with Gd.\n\nThe specific-heat measurement discussed in Sec.\\ \\ref{lowH}\nimplies that in the HTP the Gd $4f$ moments are not ordered. Since\nXRMS is element-sensitive, this might seem in contradiction to our\nobservation of a resonant magnetic signal at the Gd $L_{\\rm II}$\nedge at $37$ and $130\\, {\\rm K}$ (Fig.\\ \\ref{Fig11}). However,\nXRMS ($E1$) of the Gd $L$ edges is sensitive to the polarization\nof the Gd $5d$ bands, not (directly) to the $4f$\nmoments.\\cite{Langridge99,Gibbs88,Hannon88,XRMSreview} A Gd $5d$\nband polarization can also be induced by the Fe $3d$ moments. For\nthis situation, a very weak intensity of the corresponding\nmagnetic satellites is expected.\\cite{Everitt95} Compared to other\nGd compounds,\\cite{Granado04,Good05,Tan05,Kim05branch} the\nintensity of the satellite reflections we observed is indeed very\nweak $-$ more than three orders of magnitude lower. This low\nintensity might be due to the ordered moments pointing into a\ndirection close to [001], since in our scattering geometry we are\nnot sensitive to moments in [001] direction. However, changing the\norientation of the sample such that [001] is in the scattering\nplane\\cite{Langridge99,Tan05} did not significantly change the\nintensity, suggesting that its low value is not related to the\ndirection of magnetic moments. The behavior is similar to the\nsituation in Dy148 at high temperatures.\\cite{Langridge99}\n\nAt this point it may be useful to briefly comment on an XRMS\nsignal coming directly from the Fe moments. This would of course\nbe very important to confirm the above considerations.\nUnfortunately, the energies of the $L$ absorption edges of $3d$\ntransition metals do not allow diffraction experiments for regular\ncrystalline materials.\\cite{Tonnerre95} Resonances at the $K$ edge\nmay be used instead, but the x-ray resonant process at transition\nmetal $K$ edges is neither well understood nor efficient. Reported\nresonant enhancements at transition metal $K$ edges of magnetic\nscattering intensities were typically very small and often broad\n(see, e.g., Refs.\\\n\\onlinecite{Hill97,Stunault99,Hill00,Neubeck01,Paolasini02}),\ncompared to the substantial enhancements often observed at rare\nearth $L$ edges (see for example Refs.\\\n\\onlinecite{Granado04,Good05,Tan05,Kim05branch}).\n\nAn additional problem in our case is the large charge scattering\nbackground due to the vicinity of the magnetic satellite\nreflections to the charge reflections. This is more problematic\nfor Fe $K$ edge measurements, because at the Fe $K$ edge, the\nangle of the analyzer crystal for the polarization analysis is\nfurther away from the ideal $90^{\\circ}$ than in the case of the\nGd $L_{\\rm II}$ edge, resulting in a significantly higher charge\nscattering background at the Fe $K$ edge. Performing energy scans\nin the range of the Fe $K$ edge ($7.11\\,{\\rm keV}$) we did not\nobserve a clear resonant enhancement. In view of the mentioned\ndifficulties, this does not, however, imply that the Fe $3d$\nmoments are not ordered. Rather, we conclude that the direct\nobservation of an XRMS signal from the Fe moments will need a\nseparate experiment with specialized equipment and enough time to\ngather high statistics.\n\nAlthough we did not observe a clear resonant enhancement at the Fe\n$K$ edge, based on the above discussion of the intensity of the\nXRMS at the Gd $L_{\\rm II}$ edge and the comparison with other\nR148 compounds, we conclude that the ($\\tau ,\\tau , 0$)\npropagation vector describes the order of the Fe moments. A Fe\nmoment modulation along [110] is common for R148 compounds, but in\nR148 with other rare earths $\\tau$ is typically larger ($0.127 <\n\\tau < 0.243$) and varies less with\n$T$.\\cite{Paixao00,Paixao01,Schobinger98,Schobinger99}\n\nAt $9\\,{\\rm K}$, the satellite reflections are present as well\n[Fig.\\ \\ref{Fig11}b)], with the propagation vector in the same\ndirection. This demonstrates that the Fe moments maintain their\nmodulation from the HTP, where the Gd $4f$ moments are not\nordered, in the LTP, where the Gd $4f$ moments {\\em are} ordered.\nThis is similar to the situation in other R148 compounds. However,\nthe low intensity of the ($4-\\tau , 4-\\tau , 0$) satellite\nreflections at low $T$ indicate that for Gd148, unlike the\nsituation in other R148\ncompounds,\\cite{Langridge99,Paixao00,Schobinger98} the rare-earth\n$4f$ moments do {\\em not} follow the Fe moment modulation. In\norder to find possible modulations involving the Gd $4f$ moments,\nwe searched at $9\\,{\\rm K}$ also for magnetic satellite\nreflections in other high-symmetry directions. However, full scans\nalong [110], [100], and [001] (for scattering geometries mentioned\nabove) did not turn up any reflections with intensity high enough\nto correspond to a modulation of a significant part of the $4f$\nmoments.\n\nFurther XRMS experiments, including tracking the observed magnetic\nsatellite reflections in detail through temperature and applied\nfield, and comparing the resonances at absorption edges other than\nGd $L_{\\rm II}$ are planned.\n\n\\section{Discussion and conclusions}\n\\label{disc}\n\nIn the following we will discuss our results, starting with the\nmagnetic ordering at the {N{\\'{e}}el} temperature and the nature\nof the resulting phase. We will then discuss the $H-T$ phase\ndiagrams (for two field directions) at low temperature and the\nmagnetic order in these phases.\n\nOur resistivity and magnetization data indicate that\nGdFe$_4$Al$_8$ orders magnetically at $T_N \\approx 155\\,{\\rm K}$\n(see Sec.\\ \\ref{lowH}). Even small ($1-3\\%$) iron deficiency\nsignificantly lowers $T_N$, indicating that magnetic interactions\ninvolving the Fe $3d$ moments drive the transition. In the x-ray\nresonant magnetic scattering (XRMS) experiment (Sec.\\ \\ref{xrms}),\nthe order below $T_N$ was identified as an incommensurate\nantiferromagnetic order with propagation in [110] direction.\nAlthough corresponding satellite reflections were measured on the\nGd $L_{\\rm II}$ edge, the $T$ dependence of entropy associated\nwith the Gd $4f$ moments, estimated from specific-heat data (see\nFig.\\ \\ref{Fig5} in Sec.\\ \\ref{lowH}), indicates that at high\ntemperature, the Gd $4f$ moments do not participate in this\nantiferromagnetic order, which is thus driven by Fe-Fe moment\ninteractions. This can explain the weak intensity of the measured\nsatellite reflections. Antiferromagnetic order propagating in\n[110] direction, at high $T$ confined to Fe $3d$ moments, seems to\nbe rather typical for $R$Fe$_4$Al$_8$ compounds.\n\nIn addition to the {N{\\'{e}}el} transition, we found two more\nphase transitions at low temperature ($T_1$ and $T_2$), dividing\nthe ordered state into three phases: LTP (low-temperature phase),\nITP (intermediate-temperature phase), and HTP (high-temperature\nphase). Due to the proximity of the two low-$T$ transitions it\nproved to be very useful to apply magnetic fields in two\ndirections and thereby more clearly separate these phases.\n\n\\begin{figure}[tb]\n\\includegraphics[width=0.98\\linewidth]{xlnewphd.eps}\n\\caption{(Color online) Low-temperature phase diagrams of\nGdFe$_{3.96(1)}$Al$_8$ (sample Ia\/b) for $H\\|$[110] (a) and\n$H\\|$[001] (b). Triangles in a) are from resistivity measurements\non sample Ia, the other symbols from magnetization measurements on\nsample Ib (see text for details). The three magnetic phases LTP\n(low-temperature phase), ITP (intermediate-temperature phase), and\nHTP (high-temperature phase), as well as the zero-field transition\ntemperatures $T_1$ and $T_2$ are indicated in both panels. A\npossible fourth magnetic phase existing in fields applied parallel\nto [110] (HFP) is indicated in panel a). Underlaid is a contour\nplot of the magnetization $M$ (sample Ib).} \\label{Fig10}\n\\end{figure}\n\nLow-temperature phase diagrams established from resistivity and\nand magnetization measurements (see Sec.\\ \\ref{highH}) on crystal\nI (samples Ia and Ib) are presented in Fig.\\ \\ref{Fig10} [for\n$H\\|$[110] in panel a) and for $H\\|$[001] in panel b)]. The phase\ndiagrams are underlaid with a contour-plot of the magnetization\n[data determined from $M(H)$ measurements as displayed in Fig.\\\n\\ref{Fig9}, for $H$ increasing].\n\nFor $H\\|$[110] [Fig.\\ \\ref{Fig10}a)] the phase boundaries\n($\\bigtriangleup$,$\\bigtriangledown$) were determined from $\\rho\n(T)$ measurements [steepest slope, c.f.\\ Fig.\\ \\ref{Fig7}a); shown\nis the average between points determined on curves measured with\n$T$ increasing and decreasing]. In $M(T,H)$ the ITP-HTP transition\ncorresponds to, upon decreasing $T$, a steep rise in $M$ and\nsaturation. For definiteness, we used the extrema in ${\\rm\nd}M\/{\\rm d}H$ [${\\rm d}(MT)\/{\\rm d}T$], plotted in Fig.\\\n\\ref{Fig10}a) as $\\circ$ ($\\Box$). In case of hysteresis we again\ntook the average between $T$ or $H$ increasing and decreasing.\nAbove $0.5\\,{\\rm T}$, this tracks the ITP-HTP transition deduced\nfrom $\\rho (T)$ well, but in lower $H$ the smallness of the\nmagnetization component in [110] direction leads the transition\nline established by these criteria two lower $T$, in fact\nextrapolating to $T_1$ for $H=0$. The $M(H)$ curve measured at\n$26\\,{\\rm K}$ (Fig.\\ \\ref{Fig9}) and the $\\rho (H)$ curve measured\nat $25\\,{\\rm K}$ (Fig.\\ \\ref{Fig8}) {\\em do} suggest an additional\nmetamagnetic transition in the temperature region of the ITP. This\ntransition is what is determined from the criteria above below\n$0.5\\,{\\rm T}$, starting from $T_1$ in zero field and apparently\nmerging with the boundary to the HTP at about $28\\,{\\rm K}$ in\n$0.5\\,{\\rm T}$.\n\nThis seems to separate the ITP as deduced from resistivity\nmeasurements into two phases, we call the tentative high field\nphase HFP. In the ({\\em longitudinal}, $H\\|$[110]) magnetization\nthe HFP is indistinguishable from the LTP, and thus it could be\nthat the HFP is better described as a ``modified LTP'' phase\nrather than as a ``modified ITP'' one. However, the electrical\ntransport data (see, e.g., Fig.\\ \\ref{Fig6}) clearly indicate a\nfirst-order transition separating LTP and HFP. This transition\nexhibits a wide hysteresis in temperature when $H\\|$[110] is\nsubstantial (see Fig.\\ \\ref{Fig6}) and likely involves complex\ntransformation of domains present in the LTP and in the HFP (see\nthe brief discussion in Sec.\\ \\ref{highHrho}; note that the HFP\nwas not yet introduced there) warranting further investigation.\nThe connection of the LTP-HFP transition with pronounced domain\neffects might indicate that domains are a significant force\ndriving this transition, but presently this is tentative at best.\nThe large hysteresis associated with the LTP-HFP transition\ncomplicates the determination of the equilibrium transition line.\nEstimating the equilibrium transition by taking the average of the\ntransitions for $T$ increasing and decreasing [for $3.5\\,{\\rm T}$\nthis is indicated by an open circle in Fig.\\ \\ref{Fig6}, in the\nphase diagram, the transition is indicated by open down triangles]\nsuggests that $H\\|$[110] favors neither the LTP nor the HFP\nstrongly. Similarly, the transition between the ITP and the HTP is\nhard to pin down from $M(T)$ measurements with $H\\|$[110] because\nof the smallness of the component along [110] of the\nmagnetization. We observed no second ``extremum in $[{\\rm d}(MT) \/\n{\\rm d}T]$'', using this criterion only leads to the transition\nbetween the HFP and the ITP, rather than to the one between the\nITP and the HTP. For a field applied along [110], resistivity vs\ntemperature measurements clearly indicate two transitions, whereas\n(longitudinal) magnetization vs temperature measurements indicate\nonly one. Only when examining the spontaneous magnetization along\n[110] (Fig.\\ \\ref{Fig9N}) is the presence of two separate\ntransitions clearly indicated.\n\nFor $H\\|$[001] [Fig.\\ \\ref{Fig10}b)], three distinct regions in\nthe contour-plot of the magnetization are readily identified with\nLTP (low $M$), ITP (high $M$) and HTP (low $M$). The transition\nlines between these phases are qualitatively consistent with what\nmay be deduced from the resistivity measurements presented in\nFig.\\ \\ref{Fig7}b). In contrast to the phase diagram for\n$H\\|$[110], there are no indications for the presence of\nadditional phases for $H\\|$[001]. Since $H\\|$[001] significantly\nbroadens the features in $\\rho (T)$, we rely here on magnetization\nmeasurements (see Figs.\\ \\ref{Fig4} and \\ref{Fig9}) only to draw\nthe transition lines in Fig.\\ \\ref{Fig10}b). To fix the ITP-HTP\ntransition, we used the same procedure as for $H\\|$[110]\n(squares). For $H\\|$[001] too, the $H$ dependence of the\ntransition and a saturation consistent (considering demagnetizing\neffects) with a spontaneous moment indicate a ferromagnetic\ncomponent in the ITP (see also Fig.\\ \\ref{Fig9N}). The LTP-ITP\ntransition is determined from the $M(H)$ loops (c.f.\\ Fig.\\\n\\ref{Fig9}), taking the field value where $M$ starts to increase\nfaster than linear as the criterion for the transition field. This\ntransition is displayed by yellow circles in Fig.\\ \\ref{Fig10}b).\nFor $H\\rightarrow 0$ the transition line extrapolates to $T_1$,\nreaffirming it's identity as the LTP-ITP transition. It can be\nseen that $H\\|$[001] quickly suppresses the LTP.\n\nBased on the specific-heat data, we can conclude that the HTP to\nITP transition corresponds to the ordering of the Gd $4f$ moments.\nSince the magnetization data indicate the presence of a\nspontaneous ferromagnetic moment in the ITP, this Gd $4f$ ordering\nis either ferromagnetic or has a ferromagnetic component. The\nincrease of the transition temperature $T_2$ upon application of a\nmagnetic field (Fig.\\ \\ref{Fig10}) is consistent with this. Thus,\nGd moments do not simply follow the Fe moment order from higher\ntemperatures [this is in contrast to, e.g., DyFe$_4$Al$_8$ (Ref.\\\n\\onlinecite{Paixao00})].\n\nThe transition from the LTP to the ITP seems to be of first-order\nas indicated by the observation of thermal hysteresis in the\nresistivity (see Fig.\\ \\ref{Fig3}) as well as the magnetization\n(see Fig.\\ \\ref{Fig9}). Surprisingly though, no feature could be\nidentified in the specific heat corresponding to this transition\nand any latent heat would have to be very small. The LTP-ITP\ntransition is associated with a lock-in of the observed\nspontaneous ferromagnetic component (Fig.\\ \\ref{Fig9N}) into the\n$a$$-$$b$ plane in the LTP, but there are no clear indications\nthat its magnitude is changed by the transition. The appearance of\nan additional antiferromagnetic modulation is possible, at least\nit is not clear why a pure lock-in transition of a ferromagnetic\ncomponent should result in a sizeable increase of the resistivity\n(both for $j\\|$[110] and $j\\|$[001]).\n\nThe presence of the HFP in $H\\|$[110] complicates the situation.\nIn fact, there are clear indications  (meta-stability) for the\nfirst-order nature of the transitions at all boundaries of the\nHFP. In contrast to this, there are no indications for\nmeta-stability in the ITP-HTP transition. Meta-stability seems to\nbe present in the LTP-ITP transition, but it is much weaker than\nin any transition from or to the HFP. We point out that the\nfirst-order transition lines LTP-HFP and HFP-HTP both seem to\nextrapolate to $T_1$ in zero field, and therefore the observed\nsmaller meta-stability of the LTP-ITP transition might well be\nconnected with the HFP phase. This is a strong incentive for\nfurther investigations of the HFP.\n\nWe now attempt to draw a picture accounting for the full magnetic\nmoment of both Gd and Fe. The full ordered moment of Gd is\n$7\\,{\\mu_B}\/{\\rm Gd}$. Since the valence of Gd is fixed and there\nare no crystal-electric-field effects, this is the same for\ndifferent compounds, with only small corrections, up to $\\sim\n0.5\\,\\mu_B$, due to contributions by itinerant electrons (such as\nthe Gd $5d$ electrons). The full moment of Fe can vary from\ncompound to compound.\n\nWe know from the specific heat and XRMS measurements that in the\nHTP Gd moments (the Gd $4f$ moments at least) are not ordered, and\nFe moments are antiferromagnetically ordered with propagation\nalong [110]. Magnetization measurements further show that there is\nno significant ferromagnetic component present [see the $40\\,{\\rm\nK}$ $M(H)$ loops in Fig.\\ \\ref{Fig9}]. We recall that, according\nto the x-ray-diffraction-structure refinement and specific-heat\nmeasurements, the Gd lattice is fully occupied and that at low\ntemperatures the Gd $4f$ moments are well ordered. The appearance\nof a spontaneous ferromagnetic moment below the temperature $T_2$\nwhere the Gd moments start to order (compare Figs.\\ \\ref{Fig5} and\n\\ref{Fig9N}) indicates that the ordering of the Gd moments is\nferromagnetic, but the saturation moments for both field\ndirections are far below the $7\\,{\\mu_B}\/{\\rm f.u.}$ (see Fig.\\\n\\ref{Fig9}) expected for a full ferromagnetic alignment of Gd\nmoments. What could be the reason for this?\n\nOne possibility is that the Gd $4f$ moments are not fully\nferromagnetically ordered, but rather also have an\nantiferromagnetic component. However, since we did not find any\nsatellite reflections with high enough intensity to account for\nthe necessary fraction of the full Gd $4f$ moment in any of the\nhigh symmetry directions (see Sec.\\ \\ref{xrms}, note that the\nintensity of the ($4-\\tau,4-\\tau,0$) reflection is much too low)\nthis scenario is not very likely.\n\nFrom high-field measurements on finely ground free powder\nparticles of Gd148 (likely, at least roughly, corresponding to\n$H\\|$[001], ITP), Duong {\\em et al.}\\cite{Duong01} suggested for\n$H\\rightarrow 0$ a ferrimagnetic structure with Gd moments\nparallel, Fe moments antiparallel to the field, with an\nantiferromagnetic Fe moment order re-established only in high\nfields. On the one hand, such a simple ferrimagnetic arrangement\nof Gd and Fe moments is inconsistent with our observation in the\nXRMS experiment of the presence of the ($4-\\tau,4-\\tau,0$)\nsatellite reflection characteristic for the antiferromagnetic Fe\nmodulation even at $9\\,{\\rm K}$ [Fig.\\ \\ref{Fig11}b)]. On the\nother hand, a related idea would be consistent with both the XRMS\nand the magnetization measurements: upon the ferromagnetic\nordering of the Gd $4f$ moments, the Fe moments {\\em could} pick\nup a ferromagnetic component, leading to a canted\nantiferromagnetic structure. Such a canted magnetic structure with\na large ferromagnetic component may also explain why the intensity\nof the ($4-\\tau,4-\\tau,0$) satellite reflection is about three\ntimes smaller at $9\\,{\\rm K}$ (in the LTP) than at $37\\,{\\rm K}$\n(in the HTP). Furthermore, it would also allow Fe to influence the\nlow-$T$ transitions, as indicated by the influence of Fe\ndeficiencies on all characteristic temperatures (see Sec.\\\n\\ref{lowH}). The results of a recent dichroism experiment (on a\ncrystal ground to powder), which will be discussed in detail\nelsewhere, are consistent with such a scenario as well.\n\nThis picture covers, at least for the LTP and the HTP, all the\ndata presented in the paper, but without more detailed scattering\ndata it remains speculative. Particularly concerning the ITP and\nHFP, the absence of scattering data makes detailed conclusions\nelusive.\n\nSince the direction of the spontaneous ferromagnetic moment is\ndifferent for the ITP and the LTP, magnetocrystalline anisotropy\n(MCA) is likely one of the forces driving this transition. In rare\nearth compounds, MCA typically arises from crystal-electric-field\neffects. As we mentioned in Sec.\\ \\ref{intro}, this source of\nanisotropy is absent for Gd. Significant MCA that was nevertheless\nobserved for both elemental Gd and Gd compounds has been ascribed\nto a combination of dipole-dipole interactions and (mainly)\nanisotropy in the $5d$ bands.\\cite{ColarietiTosti03,KimPhD} The\ncorresponding magnetic-anisotropy energies are much lower than the\nmagnetic-anisotropy energies resulting from crystal-electric-field\neffects in other rare earths. A weak magnetic-anisotropy energy\nmight also, at least partly, explain why no latent heat could be\nobserved at $T_1$. In Gd148, anisotropy in the Fe $3d$ bands might\nbe an additional source of MCA.\\cite{note_Lu148} In any case, the\ndifferent spontaneous moment direction in the ITP likely results\nin a different structure of the Fe moments. The apparent shift of\n$T_1$ upon changing the Fe stoichiometry may be taken as an\nindication of the involvement of Fe moments in the LTP-ITP\ntransition. Before going any further in the discussion of the\ndriving forces (there is likely a delicate balance between\nmultiple energy scales) of the transitions between LTP, ITP, and\nHFP, the magnetic structures of these phases need to be solved.\nWith this aim in mind, additional scattering experiments are\nplanned.\n\nRegardless of the details of the magnetic structure at low\ntemperature, it is important to point out that i) on the one hand,\nthe Fe moments have at least a component antiferromagnetically\nmodulated in [110] direction ii) The Gd $4f$ moments, on the other\nhand can {\\em not} have a component ordered with the same\npropagation as the Fe $3d$ moments, because then the intensity of\nthe reflections measured at $9\\,{\\rm K}$ would have to be orders\nof magnitude higher (see Sec.\\ \\ref{xrms}). This is strikingly\ndifferent from the behavior for example in Dy148, where the rare\nearth moments order at lower temperature, but then follow the Fe\nmoment modulation, and it implies a {\\em co-existence of two\ndistinct orders} associated with Gd $4f$ and Fe $3d$ moments.\nWhereas compounds in which one magnetic sublattice orders at\ntemperatures much lower than the other magnetic sublattice are not\nvery rare (indeed this was well known to be the case for R148\ncompounds in general), we are aware of only one example,\nPrBa$_2$Cu$_3$O$_{6.92}$,\\cite{Hill00} in which the magnetic\nsublattice ordering at lower $T$ has no component with the same\nmodulation that the other sublattice has at all $T$.\n\nThe at low temperatures persisting antiferromagnetic modulation of\nthe Fe moments coexisting with an order of the Gd moments with a\nlarge ferromagnetic component and without a component having the\nFe moment modulation is the central point of our paper. It may\nindicate a modified relation between the strength of the Gd-Gd and\nGd-Fe moment interactions in Gd148 as compared to other R148\ncompounds. As we mentioned, apart from the stronger couplings\nbetween $4f$ and conduction electrons in general, the most\npeculiar property of Gd is the absence of crystal-electric-field\neffects.\n\nIt seems remarkable that in the absence of the MCA due to crystal\nelectric field effects, the magnetic behavior in Gd148 is {\\em\nmore} complex than in other R148. Likely, strong\ncrystal-electric-field effects play a central role in the\nmagnetism in R148 (except for Gd148). E.g., in Dy148,\ncrystal-electric-field effects press the $4f$ moments into the\ncrystallographic plane, making an order following the Fe-moment\nmodulation more probable, since the Fe moments are already\nin-plane. Our results on Gd148, in contrast, indicate a complex\ninterplay of local $4f$ and itinerant $3d$ moments when\ncrystal-electric-field effects are absent.\n\n\\section{Summary}\n\\label{conc}\n\nWe presented an extensive set of data including magnetization,\nelectrical transport, specific heat, and x-ray resonant magnetic\nscattering, measured on flux-grown single crystals of\nGdFe$_4$Al$_8$. We found that two transitions at $T_1 \\sim 21\\,\n{\\rm K}$ and $T_2 \\sim 27\\, {\\rm K}$ at low temperature divide the\nordered state below $T_N \\sim 155\\, {\\rm K}$, where Fe moments\norder antiferromagnetically with modulation along [110], into\nthree phases. The corresponding phase diagrams for fields applied\nin [110] and [001] direction were presented.\n\nGd $4f$ moments order, mainly ferromagnetically, below $T_2$.\nAbove $T_1$ the ferromagnetic component of the moments points into\na direction close to [001], whereas below $T_1$ the ferromagnetic\ncomponent is locked into the $a$$-$$b$ plane. The Gd $4f$ moment\norder is distinct from the Fe-moment order At low temperature $T$:\nthe Fe moments still have at least a component modulated along\n[110], but the Gd $4f$ moments apparently do not have such a\ncomponent. We proposed a canted antiferromagnetic structure of the\nFe moments at low $T$, which can cover all the low-$T$ data\npresented in the paper, but to some extent still is speculative.\nAbove  The transition at $T_1$ likely involves a delicate balance\nof multiple energy scales associated with both Gd and Fe.\n\nThe complex magnetism in GdFe$_4$Al$_8$ as compared to\n$R$Fe$_4$Al$_8$ compounds with other rare earths $R$ likely is\nrelated to a modified ratio of coupling strengths and to the\nabsence of crystal-electric-field effects. In order to further\nelucidate the structure of the magnetic phases of GdFe$_4$Al$_8$\nadditional scattering experiments are planned.\n\n\\begin{acknowledgments}\n\\label{ack} We thank S.~L. {Bud'ko}, P.~Ryan, and R.~W. McCallum\nfor useful discussions, P.~Ryan also for technical assistance\nduring the synchrotron experiment. Ames Laboratory is operated for\nthe U.S. Department of Energy by Iowa State University under\nContract No.\\ W-7405-Eng-82. This work was supported by the\nDirector for Energy Research, Office of Basic Energy Sciences.\nSynchrotron work was performed at the MuCAT sector at the Advanced\nPhoton Source supported by the U.S. DOE, BES, and OS under\nContract No.\\ W-31-109-Eng-38.\n\\end{acknowledgments}\n\n\\newcommand{\\noopsort}[1]{} \\newcommand{\\printfirst}[2]{#1}\n  \\newcommand{\\singleletter}[1]{#1} \\newcommand{\\switchargs}[2]{#2#1}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{intro}\nIn perturbative QCD the two well-known parton evolution equations DGLAP \\cite{63,64,65} and BFKL \\cite{66,67} serve as the basic tools for prediction of parton distribution functions (PDFs) on their respective kinematic domains. The DGLAP approach is valid for the large interaction scale $Q^2$ since it sums higher order $\\alpha_s$ contributions enhanced by the leading logarithmic powers of $\\ln Q^2$. On the other hand, BFKL evolution deals with the small-$x$ and semi-hard $Q^2$ kinematic region by summing the leading logarithmic contributions of $(\\alpha_s \\ln 1\/x)^n$. However, both the linear evolutions turn out to be inadequate to comply with the unitary bound or the conventional Froissart bound \\cite{68}. To restore the unitarity, a higher order pQCD correction is required to shadow the rapid parton growth which eventually leads to the saturation phenomena at very small-$x$. In this respect, several nonlinear equations have been proposed in recent years to entertain this shadowing correction in DGLAP and BFKL equation.\n\\par In pQCD interactions, the origin of the shadowing correction is primarily considered as the gluon recombination ($g+g\\rightarrow  g$) which is simply the inverse process of gluon splitting ($g\\rightarrow g+g$). The modification of gluon recombination to the original DGLAP equation was first performed in GLR-MQ equation \\cite{16,75,86,87} proposed by Gribov-Levin-Ryskin and Mueller-Qui. It is considered that the interferant cut diagrams of the recombination amplitude are the origin  for the negative shadowing effects in the gluon recombination processes \\cite{59,69}. The GLR-MQ equation was originally derived using AGK-cutting rules \\cite{71} to compute the contributions from these interference processes. However, GLR-MQ does not possess any antishadowing correction term since in this equation momentum conservation is violated in the gluon recombination processes. The shadowing and antishadowing effects are expected to happen in a complementary fashion which in fact ensure the momentum conservation during the process of gluon recombination \\cite{70}. In consequence of gluon recombination, a part of gluon momentum lost due to shadowing is believed to be compensated in terms of new gluons with comparatively larger $x$. This causes an enhanced density of larger momentum gluons responsible for the antishadowing effect. In this respect, to take care of momentum conservation and antishadowing effects in gluon evolution, a modified DGLAP equation was proposed \\cite{69} to replace the GLR-MQ equation. In the derivation of the modified DGLAP equation, the author has used TOPT-cutting rules based on time ordered perturbation theory (TOPT) to calculate the contribution of the gluon recombination, instead of using AGK-cutting rules unlike in the derivation of GLR-MQ equation. This equation naturally comes with two separate nonlinear terms, quadratic in gluon distribution function which are identified as the positive antishadowing and negative shadowing components in the gluon evolution. This suggests that negative shadowing effects are suppressed by antishadowing effects in the gluon evolution process.\n\\par   However, above mentioned modified DGLAP approach cannot predict the evolution of unintegrated gluon distribution because DGLAP is based on collinear factorization scheme while the later deals with $k_T$-factorization. In this respect, a new unitarized BFKL equation was proposed by Ruan, Shen, Yang and Zhu \\cite{1} incorporating both shadowing and antishadowing correction. This modified BFKL (MD-BFKL) equation allows one to study antishadowing effects in unintegrated gluon distribution platform.\n\\par However, there are several other BFKL based nonlinear evolution equations that consider corrections from gluon fusion. One of the most widely studied models is Balitsky-Kovchegov (BK) equation \\cite{10,11} which is originally derived in terms of scattering amplitude. It can be explained by the dipole model in which the nonlinear term comes from the dipole splitting while the screening effect origins from the double scattering of the dipole on the target \\cite{85}. The solution of the BK equation suggests an intense shadowing leading to the so-called saturation of the scattering amplitude showing a complete flat spectrum. However, like GLR-MQ equation the BK equation is also irrelevant to the gluon antishadowing. On the other hand, besides inclusion of antishadowing correction term, the nonlinear terms in MD-BFKL equation hold a simple form which is quadratic in unintegrated gluon distribution as well as running coupling constant with some constant coefficients. These unique features of MD-BFKL equation motivate our current studies on small-$x$ physics.\n\\par A numerical solution of MD-BFKL equation suggests a sizable impact of antishadowing effect on gluon evolution particularly in the pre-asymptotic regime \\cite{1}. In the literature \\cite{73} the phenomenology of the MD-BFKL equation is extended to study the implicit nuclear shadowing and antishadowing effects. In our current work, we try to construct a modified evolution equation by implanting the so-called kinematic constraint or consistency constraint on the MD-BFKL equation. The kinematic constraint (see Fig.~\\ref{2x} ) is implemented in different forms:\n\\begin{align}\n\t\\label{1.1}\n\tq_T^2&<\\frac{k_T^2}{z} \\text{     } \\text{ LDC \\cite{6,74}},\\\\\n\t\\label{1.2}\n\tq_T^2&<\\frac{(1-z)k_T^2}{z}\\text{  \\cite{3}},\\\\\n\t\\label{1.3}\n\tk_T^{'2}&<\\frac{k_T^2}{z}\\text{      }\\text{  BFKL \\cite{3}}.\n\\end{align}\nThis constraint arises as a consequence of BFKL multi-Regge kinematics which suggests the exchanged gluon virtuality is dominated by transverse components while the longitudinal components of the gluon momentum are required to be small i.e. $k^{'2}\\approx k_T^{'2}$. The kinematic constraint gives an implicit cutoff on $k_T^{'2}$ as depicted by \\eqref{1.3}. The inequality \\eqref{1.3} can be considered as a special case of \\eqref{1.1} recalling the fact that for a given value of $k_T^2$, a high $q_T^2$ implies an equally high $k_T^{'2}$ \\cite{3}. Although there are other constraints coming from energy-momentum conservation \\cite{7}, this bound is considerably tighter than the later \\cite{3}. Besides the introduction of this cutoff on the upper limit of integration found to preserve the scale invariance of the BFKL equation.\n\\par Our primary goal of this work is to investigate the nonlinear effects on gluon evolution in terms of MD-BFKL equation supplemented by the kinematic constraint. Our studies are particularly focused on the near saturation region where shadowing effects are dominant over antishadowing effects. The content of the paper is organized as follows. In Sect. \\ref{analytic} we present our kinematic constraint improved MD-BFKL equation starting from the construction of the equation and a brief discussion on some of its main features (Sect. \\ref{analytic1}). We suggest an analytical solution to this equation and sketch a particular solution in terms of $x$ and $k_T^2$ evolution followed by three-dimensional realization of gluon evolution in $x\\text{-}k_T^2$ phase space (Sect. \\ref{analytic2}). We have also extracted collinear gluon distribution $xg(x,Q^2)$ from unintegrated gluon distribution $f(x,k_T^2)$ and compared our prediction with that of global parameterization groups NNPDF 3.1sx and CT 14. In Sect. \\ref{critical} we suggest a differential geometric approach towards finding the equation of the critical boundary between high and low gluon density regime. In this section, a phenomenological insight of the geometrical scaling is discussed as well.\n\\par In Sect. \\ref{hera} we explore the phenomenological implication of our solution for unintegrated gluon distribution towards prediction of DIS structure function $F_2$ and longitudinal structure function $F_L$ at HERA. We show the comparisons of reduced cross sections (Sect. \\ref{hera1}) as well as virtual photon-proton cross sections in the transition region (Sect. \\ref{hera2}) with HERA H1 and ZEUS data. Finally, in Sect. \\ref{con} we summarize and outline our conclusion as well as possible future prospects.\n\n\n\n\n\\section{Nonlinear effects in kinematic constraint improved MD-BFKL equation}\\label{analytic}\n\n\\subsection{Construction of kinematic constraint improved MD-BFKL}\\label{analytic1}\nThe modified BFKL (MD-BFKL) equation reads \\cite{1},\n\\begin{equation}\n\t\\label{1}\n\t\\begin{split}\n\t\t&-x\\frac{\\partial f \\left(x, k_T^2\\right)}{\\partial x}=\\\\&\\frac{\\alpha_s N_c k_T^2}{\\pi}\\int_{k_{T_{\\min }}^{'2}}^{\\infty}\\frac{dk_T^{'2}}{k_T^{'2}}\\left[\\frac{f (x, k_T^{'2})-f \\left(x, k_T^2\\right)}{|k_T^{'2}-k_T^2|} + \\frac{f \\left(x, k_T^2\\right)}{\\sqrt{k_T^4+4k_T^{'4}}}\\right]\\\\\n\t\t&-\\frac{36\\alpha_s^2}{\\pi k_T^2 R^2}\\frac{N_c^2}{N_c^2-1}f^2(x,k_T^2)+\\frac{18\\alpha_s^2}{\\pi k_T^2 R^2}\\frac{N_c^2}{N_c^2-1}f^2(\\frac{x}{2},k_T^2),\n\t\\end{split}\n\\end{equation}\nwhere $f \\left(x, k_T^2\\right)$ denotes the gluon distribution unintegrated over the transverse momentum of gluon $k_T$ and $k_{T_{\\min }}^{'2}$ is the infrared cutoff of the evolution. The first part of \\eqref{1} linear in $f(x,k_T^{'2})$ (or $f(x,k_T^2)$) is the BFKL kernel at leading logarithmic of $1\/x$ (LLx) accuracy. A diagrammatic representation for probing gluon content of the proton at high photon virtuality $Q^2$ is sketched in Fig.~\\ref{2x} (a). The BFKL kernel corresponds to the sum of gluon ladder diagram Fig.~\\ref{2x}(b) generated by squaring the amplitude of Fig.~\\ref{2x}(a). The 1st term in the BFKL kernel, involving $f(x,k_T^{'2})$   corresponds to diagrams with real gluon emission while the second term takes care of diagrams with virtual corrections. Note that the apparent singularity that is observed at $k_T^{'2} =k_T^2$ cancels between real and virtual contributions. \n\n\\par The quadratic terms in \\eqref{1} viz.  \n$\\frac{\\partial f_{\\text{shad}} }{\\partial \\ln{1\/x}}=-\\frac{36\\alpha_s^2}{\\pi k_T^2 R^2}\\frac{N_c^2}{N_c^2-1}f^2$\nand\n$\\frac{\\partial f_{\\text{antishad}} }{\\partial \\ln{1\/x}}$ $=\\frac{18\\alpha_s^2}{\\pi k_T^2 R^2}\\frac{N_c^2}{N_c^2-1}f^2$ depict shadowing and antishadowing correction to the original BFKL equation respectively. The factor $\\pi R^2$ represents the transverse area populated by gluons. The radius parameter $R$ arises in the QCD cut diagram coupling 4 gluons to 2 gluons (see Fig.~\\ref{3x}), the value of which depends on how exactly the gluon ladders couple to hadron. If the gluon ladder couples to two different constituents of hadron, $R$ is characterized by hadronic radius i.e. $R\\approx R_H=5 \\text{ GeV}^{-1}$ whereas if we consider the possibility of ladder coupling to the same parton then the appropriate choice of $R$ is the radius of valence quark i.e. $R=2\\text{ GeV}^{-1}$. In the latter case $(R=2\\text{ GeV}^{-1})$ we have \"hot spots\" \\cite{75,72}.\n\\begin{figure*}[tbp]\n\t\\centering\n\t\\includegraphics[width=.21\\textwidth,clip]{GluonLadderII.eps}\\hspace{5mm}\n\t\\includegraphics[width=.23\\textwidth,clip]{Structure_box.eps}\\hspace{5mm}\n\t\\includegraphics[width=.15\\textwidth,clip]{f011.eps}\\hspace{5mm}\n\t\\includegraphics[width=.15\\textwidth,clip]{f026.eps}\n\t\n\n\t\\caption{\\label{2x} From left: (a) chain of sequential gluon emission which forms the basis of BFKL equation. On squaring the amplitude of (a) the ladder diagram (b) is generated which when summed gives the BFKL kernel. (c) and (d) are two simple examples of the inhomogeneous driving term of \\eqref{4} which correspond to the shaded region of gluon-virtual photon coupling in (b). }\n\\end{figure*}\n\n\\par The interpretation of the real emission term in the BFKL kernel is that we have a splitting $k^{'}\\rightarrow k + q$ inside hadron resulting an infinite chain of reggeized gluons labeled as $k_1,k_2,k_3,\\text{...}k_n$ (Fig.~\\ref{2x}(a)). In high energy limit, the longitudinal components of the gluon momentum are strongly ordered while there is no ordering on the transverse components of the gluon momentum i.e.\n\t\\begin{align}\n\t\t\\label{2.2a}\n\t\t&x_1\\ll x_2\\ll x_3\\text{.....}\\ll x_{n-1}\\ll x_n,\\\\\n\t\t\\label{2.2b}\n\t\t&k_{1 T}\\sim k_{2 T}\\sim k_{3 T}\\sim \\text{...}.\\sim k_{n-1 T}\\sim k_{\\text{nT}}.\n\t\\end{align}\nIn addition, the small-$x$ regime where the BFKL is valid, the gluon virtuality along the chain must be dominated by the transverse components of the gluon momentum, \n\\begin{align}\n\t\\label{2.3}\n\tk^2=2 k^+ k^--k_T^2\\approx k_T^2.\n\\end{align}\\\\\nThe above kinematics corresponding to \\eqref{2.2a},\\eqref{2.2b} and \\eqref{2.3} is referred as multi-Regge kinematics. The inequality \\eqref{2.2a} implies $z=\\frac{x}{\\frac{x}{z}}=\\frac{x_{n-1}}{x_n}\\ll1$ and since transverse momenta are of the same order: $k_T\\simeq k_T{}^{'}$, depict a cutoff \n\\begin{equation}\n\t\\label{2.4}\n\tk_T{}^{'}{}^2< \\frac{k_T{}^2}{z},\n\\end{equation}\nwhich is so-called  kinematic constraint or consistency constraint for real gluon emission \\cite{3,4,5,6}. The BFKL kernel in \\eqref{1} is at leading logarithmic $1\/x$ (LLx) accuracy. However, higher order corrections to the BFKL equation are already been evaluated up to NLLx accuracy, which turns out to be quite large \\cite{2}. Implementation of the constraint \\eqref{2.4} on the evolution of BFKL based equations makes the theory more realistic in the sense that this ensures the participation of only the LLx part of higher order correction in the evolution.  This, in fact, portrays the importance NLLx correction in BFKL kernel.\n\n\\par Recalling that BFKL equation can be written as an integral equation of  $f(x,k_T^2)$ \\cite{15,16}, the kinematic  constraint \\eqref{2.4} can be imposed onto the real emission part of the BFKL kernel as follows\n\\begin{equation}\n\t\\label{2.5}\n\t\\begin{split}\n\t\t\\int_{x}^{1}\\frac{dz}{z}\\int _{k_{T_{\\min }}^{'2}}\\frac{dk_T^{'2}}{k_T^{'2}}\\bigg[&\\frac{\\Theta \\left(\\frac{k_T^2}{z}-k_T^{'2}\\right)f\\left(\\frac{x}{z},k_T^{'2}\\right)-f\\left(\\frac{x}{z},k_T^2\\right)}{\\left| k_T^{'}{}^2-k_T^2\\right| }\\\\\n\t\t&+\\frac{f\\left(\\frac{x}{z},k_T^2\\right)}{\\sqrt{k_T^4+4k_T^{'4}}}\\bigg],\n\t\\end{split}\n\\end{equation}\nwhere the heaviside function $\\Theta \\left(\\frac{k^2}{z}-k^{'}{}^2\\right)$ in \\eqref{2.5} serves the purpose of the kinematic cutoff. The upper limit of the integration over $k_T^{'2}$ is implicit in $\\Theta$. The BFKL kernel gains one more degree of freedom after implementation of kinematic constraint for real emission. Now expressing  the MD-BFKL equation \\eqref{1} in terms of the KC improved BFKL kernel \\eqref{2.5} yields\n\\begin{equation}\n\t\\label{4}\n\t\\begin{split}\n\t\t&f\\left(x,k_T^2\\right)=f^{(0)}\\left(x,k_T^2\\right)+\\\\&\\frac{\\alpha _s k_T^2 N_c }{\\pi }{\\int _x^1\\frac{dz}{z}\\int _{k_{T_{\\min }}^{'2}}}\\frac{dk_T^{'2}}{k_T^{'2}}\\bigg[\\frac{\\Theta \\left(\\frac{k_T^2}{z}-k_T^{'2}\\right)f\\left(\\frac{x}{z},k_T^{'2}\\right)-f\\left(\\frac{x}{z},k_T^2\\right)}{\\left| k_T^{'2}-k_T^2\\right| }\\\\\n\t\t&+\\frac{f\\left(\\frac{x}{z},k_T^2\\right)}{\\sqrt{k_T^4+4k_T^{'4}}}\\bigg]-\\frac{36 \\alpha _s^2}{\\pi  k_T^2 R^2}\\frac{N_c^2}{N_c^2-1}\\int _x^1\\frac{dz}{z}f^2\\left(x,k_T^2\\right)\\\\\n\t\t&+\\frac{18 \\alpha _s^2}{\\pi  k_T^2 R^2}\\frac{N_c^2}{N_c^2-1}\\int _x^1\\frac{dz}{z}f^2\\left(\\frac{x}{2},k_T^2\\right).\n\t\\end{split}\n\\end{equation}\nThe above equation \\eqref{4} is the integral form of our KC improved MD-BFKL equation. The inhomogeneous driving term $f^{(0)}(x,k_T^2)$ depicts gluon-proton coupling corresponding to the shaded region in Fig.~\\ref{2x}(b). The Fig.~\\ref{2x}(c, d) represents two simple possible contributions to $f^{0}(x,k_T^2)$ which indicates radiation of gluons from valence quark.\n\n\\par To sketch an integro-differential form of KC improved MD-BFKL equation out of \\eqref{4} we differentiate \\eqref{4} w.r.t. $\\ln(1\/x)$, then using properties of $\\Theta$ and Dirac-$\\delta$ function, to be specific  $\\Theta '(t)=\\delta  (t)$ and $f( t)\\delta  (t-a)=f(a)\\delta  (t-a)$ and doing some algebra one can show\n\n\\begin{equation}\n\t\\label{5}\n\t\\begin{split}\n\t\t\\frac{\\partial}{\\partial \\ln\\frac{1}{x}}\\int_{x}^{1}\\frac{\\text{d}z}{z}\\Theta \\left(\\frac{k_T^2}{z}-k_T^{'2}\\right)&f(x,k_T^{'2})\\longrightarrow          \\Theta(k_T^2-k_T^{'2})f(x,k_T^{'2})\\\\\n\t\t&+\\Theta(k_T^{'2}-k_T^{2})f\\left(\\frac{k_T^{'2}}{k_T^{2}}x,k_T^{'2}\\right).\n\t\\end{split}\n\\end{equation}\nAbove prescription allows one to express \\eqref{4} in the following integro-differential form:\n\\begin{align}\n\t\\label{2.8}\n\t\\begin{split}\n\t\t&-x\\frac{\\partial f\\left(x,k_T^2\\right)}{\\partial x}=\\frac{\\alpha _s k_T^2 N_c }{\\pi }{\\int _{k_{T_{\\min }}^{'2}}}\\frac{dk_T^{'2}}{k_T^{'2}}\\\\&\\times\\bigg[\\frac{\\Theta \\left(k_T^2-k_T^{'2}\\right)f\\left(x,k_T^{'2}\\right)+\\Theta (k_T^{'2}-k_T^{2})f\\left(\\frac{k_T^{'2}}{k_T^{2}}x,k_T^2\\right)}{\\left| k_T^{'2}-k_T^2\\right| }\\\\&-\\frac{f(x,k_T^2)}{|k_T^{'2}-k_T^2|}+\\frac{f\\left(x,k_T^2\\right)}{\\sqrt{k_T^4+4k_T^{'4}}}\\bigg]-\\frac{36 \\alpha _s^2}{\\pi  k_T^2 R^2}\\frac{N_c^2}{N_c^2-1}f^2\\left(x,k_T^2\\right)\\\\&+\\frac{18 \\alpha _s^2}{\\pi  k_T^2 R^2}\\frac{N_c^2}{N_c^2-1}f^2\\left(\\frac{x}{2},k_T^2\\right).\n\t\\end{split}\n\\end{align}\nIn derivation of \\eqref{2.8} we have neglected the term $x\\frac{\\partial f^{(0)}}{\\partial x}$. This is justified in the sense that $x\\frac{\\partial f^{(0)}}{\\partial x}$ is much less singular than $x\\frac{\\partial f}{\\partial x}$ at small-$x$. Moreover if we consider $f^{(0)}$ is particularly contributed by the diagrams shown in Fig.~\\ref{2x} (c, d) then $f^{(0)}$ would be independent of $x$ (or, $\\frac{\\partial f^{(0)}}{\\partial x}=0$ ) at small-$x$ limit \\cite{6,15}.\n\\begin{figure}[tbp]\n\t\\centering\n\t\\includegraphics[width=.27\\textwidth,clip]{Shadow.eps}\t\n\n\t\\caption{\\label{3x} QCD cut diagram representing nonlinear terms in \\eqref{1} with $4\\rightarrow 2$ gluon recombination kernel (dashed box). The dashed box includes all possible pQCD diagrams which couple 4 gluons to 2 gluons.}\n\\end{figure}\n\n\\par To simplify the distribution function $f(\\frac{k_T^{'2}}{k_T^{'2}}x,k_T^{2})$ corresponding to real emission term in \\eqref{2.8} we have incorporated Regge like behavior of gluon distribution. Before the advent of pQCD, Regge theory is been considered as a very successful theory regarding the phenomenological analysis of total hadronic and  photoproduction cross sections. These non-perturbative processes can economically described by two Regge contributions namely pomerons with intercept slightly above unity (${\\alpha_{P}(0)\\approx 1.08}$) and Reggons with intercept $\\alpha_{R}(0)\\approx 0.5$. The Regge behavior corresponding to sea quark and anti-quark distributions is given by $q_{sea}\\sim x^{-\\alpha_{P}}$ whereas that of valence quark distribution is given by $q_{v}\\sim x^{-\\alpha_{R}}$. On the other hand, in pQCD also, particularly the small-$x$ region is considered to have higher possibility towards exploring Regge limit of pQCD. Note that the BFKL dynamics itself is based on the concept of pomeranchuk theorem or pomeron: the Regge-pole carrying the quantum-numbers of the vacuum. However, the BFKL pomeron (also called hard pomeron) should be contrasted with non perturbative description of pomeron (soft pomeron) in the sense that they pose relatively different magnitude of intercept. For BFKL-pomeron the intercept is\n\\begin{equation*}\n\t\\alpha_{P}^{BFKL}(0) = 1 + \\lambda_{\\text{BFKL}},\n\\end{equation*}\nwhere  $\\lambda_{\\text{BFKL}}= \\frac{3 \\alpha_s}{\\pi} 28  \\zeta(3)$, $\\zeta$ being Reiman zeta function \\cite{9}. The typical value of $\\lambda_{\\text{BFKL}}$ for $\\alpha_s$=0.2 is $\\sim 0.5$ implies $\\alpha_{P}^{BFKL}(0)\\approx1.5$ which is potentially large in magnitude compared to soft pomeron $(\\alpha_{P}(0)=1.08)$.\n\\par In pQCD the small-$x$ behavior of the parton distribution is supposed to be controlled by intercept of appropriate Regge trajectory. Regge model provides parametrizations of DIS distribution functions, $f_i(x,Q^2)=A_i(Q^2)x^{-\\lambda_i}$ (i=$\\sum$ (singlet structure function)and g (gluon distribution)), where$\\lambda_i$ is the pomeron intercept minus one $(\\alpha_{P}(0)-1)$.  At small-$x$, the leading order calculations in \\(\\text{ln}(1\/x)\\) with fixed \\(\\alpha _s\\) predicts a steep power law behavior of \\(\\textit{f}\\left(x,k^2\\right)\\sim x^{-\\lambda_\\text{BFKL}}\\) \\cite{7,8}. This motivates us to consider a simple form of Regge factorization as follows,\n\\begin{equation}\n\t\\label{6}\n\t\\begin{split}\n\t\tf\\left(\\frac{k_T^{'2}}{k_T^2}x,k_T^2\\right)&\\simeq x^{-\\lambda_{\\text{BFKL}}}\\left(\\frac{k_T^{2}}{k_T^{'2}}\\right)^{\\lambda_{\\text{BFKL}}} H(k_T^2)\\\\\n\t\t&= \\left(\\frac{k_T^2}{k_T^{'2}}\\right)^{\\lambda }f\\left(x,k_T^2\\right),\n\t\\end{split}\n\\end{equation}\nwhere we have dropped the subscript on $\\lambda$ and in the rest of the text we will follow this notation. Similar way we can express \n\\begin{equation}\n\t\\label{2.10}\n\t\\begin{split}\n\t\tf\\left(\\frac{x}{2},k^2\\right)&\\simeq 2^{\\lambda }f\\left(x,k^2\\right).\n\t\\end{split}\n\\end{equation}\nThis type of Regge like form considered in \\eqref{6} and \\eqref{2.10} is supported by various literatures \\cite{76,77,78,79}. But note that Regge factorization can't be taken as good ansatz for the entire kinematic region of $x$ and $k_T^2$ \\cite{80}. This type of Regge behavior is considered to be valid only in the vicinity of the saturation scale, $Q_s^2$ where scattering amplitude depends only on a single dimensionless variable, $Q^2\/Q_s^2$. The Regge theory is applicable if the quantity invariant mass, $W$ $(=\\sqrt{Q^2(1-x)\/x})$ is much greater than all other variables. Therefore, we expect it to be valid if $x$ is enough small, for any value of $Q^2$.\n\n\n\\par Now recalling $\\Theta(t)=1-\\Theta(-t)$ and substituting \\eqref{6} and \\eqref{2.10} in \\eqref{2.8} we get\n\\begin{equation}\n\t\\label{7}\n\t\\begin{split}\n\t\t&-x\\frac{\\partial f\\left(x,k_T^2\\right)}{\\partial x}=\\\\&\\frac{\\alpha _s N_c k_T^2 }{\\pi }\\int _{k_{T_{\\min }}^{'2}}\\frac{dk'^2}{k_T^{'}{}^2}\\bigg[\\Theta \\left(k_T^2-k_T^{'2}\\right)\\frac{1-\\left(\\frac{k_T^2}{k_T^{'2}}\\right)^{\\lambda }}{\\left| k_T^{'2}-k_T^2\\right| }f\\left(x,k_T^{'2}\\right)\\\\\n\t\t&+\\left(\\frac{k_T^2}{k_T^{'2}}\\right)^{\\lambda }\\frac{f\\left(x,k_T^2\\right)}{\\left| k_T^{'2}-k^2\\right|}-\\frac{f\\left(x,k_T^2\\right)}{\\left| k_T^{'2}-k^2\\right|} +\\frac{f\\left(x,k_T^2\\right)}{\\sqrt{k_T^4+4k_T^{'4}}}\\bigg]\\\\\n\t\t&-\\frac{36 \\alpha _s^2}{\\pi  k_T^2 R^2}\\frac{N_c^2}{N_c^2-1}(1-2^{2 \\lambda -1})f^2\\left(x,k_T^2\\right).\n\t\\end{split}\n\\end{equation}\nThe above equation \\eqref{7} is our kinematic constraint improved MD-BFKL equation in LLx accuracy.\n\n\\par Before going into in depth phenomenological analysis and consistency of the equation towards experimental result, we want to highlight two important features of the evolution equation \\eqref{7}. As we follow, in the region far below saturation limit, for the canonical choice of $\\lambda\\sim0.5$, the quadratic term in \\eqref{7} tends to vanish since $(1-2^{2\\lambda-1}) \\rightarrow0$. This indicates that below saturation region both the contribution from shadowing correction $\\frac{\\partial f_{\\text{shad}}}{\\partial \\ln{\\frac{1}{x}}}$ and antishadowing correction $\\frac{\\partial f_{\\text{antishad}}}{\\partial \\ln{\\frac{1}{x}}}$ coexists but seem to balance each other for which nonlinear effect becomes negligible. Therefore, in the below saturation regime, we can revert  \\eqref{7} back to the original BFKL equation choosing $\\lambda = 0.5$.\n\nOn the other hand, in the vicinity of saturation limit, our interpretation of gluon distribution for the nonlinear term has to be reviewed. In this region, the gluon distribution becomes flat which makes our Regge factorization for the nonlinear term in  \\eqref{2.10} invalid. Rather essentially, we should take the approximation\n\\begin{equation*}\n\tf\\left(\\frac{x}{2},k_T^2\\right) \\simeq f(x,k_T^2),\n\\end{equation*} \nwhich is supposed to justify our understanding of the saturation region. Taking this approximation in \\eqref{2.8} one can arrive at\n\\begin{equation}\n\t\\label{8}\n\t\\begin{split}\n\t\t&-x\\frac{\\partial f\\left(x,k_T^2\\right)}{\\partial x}\\\\\n\t\t&=\\frac{N_c \\alpha _sk_T^2 }{\\pi }\\int _{k_{T_{\\min }}^{'2}}\\frac{dk'^2}{k_T^{'}{}^2}\\bigg[\\Theta \\left(k_T^2-k_T^{'2}\\right)\\frac{1-\\left(\\frac{k_T^2}{k_T^{'2}}\\right)^{\\lambda }}{\\left| k_T^{'2}-k_T^2\\right| }f\\left(x,k_T^{'2}\\right)\\\\\n\t\t&+\\left(\\frac{k_T^2}{k_T^{'2}}\\right)^{\\lambda }\\frac{f\\left(x,k_T^{'2}\\right)}{\\left| k_T^{'2}-k^2\\right|}-\\frac{f\\left(x,k_T^2\\right)}{\\left| k_T^{'2}-k^2\\right|} +\\frac{f\\left(x,k_T^2\\right)}{\\sqrt{k_T^4+4k_T^{'4}}}\\bigg]\\\\\n\t\t&-\\frac{18 \\alpha _s^2}{\\pi  k_T^2 R^2}\\frac{N_c^2}{N_c^2-1}f^2\\left(x,k_T^2\\right).\n\t\\end{split}\n\\end{equation}\nEquation \\eqref{8} serves the evolution of gluon near the saturation limit. Here the $\\lambda$ dependence of the nonlinear term has been completely wiped up, unlike \\eqref{7}. In this regime, the shadowing contribution becomes twice as that of antishadowing effect forecasting a net shadowing effect in the evolution. \n\n\n\n\n\n\n\n\\subsection{Analytical solution of KC improved MD-BFKL}\\label{analytic2}\n\nIn this section, we present an analytical solution of our KC improved MD-BFKL equation, particularly in the saturation region where shadowing effect is the dominant one. Since our calculations are limited to fixed strong coupling $\\alpha_s$, so do fix $\\lambda$, therefore,  the solution for both \\eqref{7} and \\eqref{8} will exhibit the same form  only differ by the coefficients of their respective quadratic terms.\n\n\\par Recalling that there is no ordering for transverse momenta $k_T^{'}\\backsimeq k_T$ in BFKL multi-Regge kinematics, this allows us to write the gluon distribution in Taylor series,\n\\begin{equation}\n\t\\label{9}\n\tf\\left(x,k_T^{'2}\\right)=f\\left(x,k_T^2\\right)+\\frac{\\partial f\\left(x,k_T^2\\right)}{\\partial k_T^2}\\left(k_T^{'2}-k_T^2\\right) +  \\EuScript{O}\\left(k_T^{'2}-k_T^2\\right),\n\\end{equation}\nwhere  $\\EuScript{O}\\left(k_T^{'2}-k_T^2\\right)$ denotes the higher order terms. Above series is a convergent series in $\\left(k_T^{'2}-k_T^2\\right)$ as no ordering of the transverse momenta in BFKL kinematics $k_T^{'}- k_T\\backsimeq 0$ implies the higher order terms  $\\EuScript{O}\\left(k_T^{'2}-k_T^2\\right)\\rightarrow 0$, thereby ensures the higher order terms to become insignificant and can be neglected. Thus this assumption would hold good as long as no ordering condition of the transverse momenta in BFKL kinematics is concerned.  This type of series expansion of distribution function is well supported in the literature \\cite{12}.\n\\par Now neglecting these higher order terms  we can express \\eqref{8} as\n\\begin{equation}\n\t\\label{10}\n\t\\begin{split}\n\t\t-x\\frac{\\partial f\\left(x,k_T^2\\right)}{\\partial x}=&\\xi(k_T^2) \\frac{\\partial f\\left(x,k_T^2\\right)}{\\partial k_T^2}\n\t\t+\\zeta(k_T^2) f\\left(x,k_T^2\\right)\\\\&-\\Delta(k_T^2) f^2\\left(x,k_T^2\\right),\n\t\\end{split}\n\\end{equation}\nwhere\n\\begin{equation}\n\t\\label{11}\n\t\\begin{split}\n\t\t\\xi(k_T^2)=\\frac{\\alpha _s N_c }{\\pi }k_T^2\\bigg[&\\int _{k_{T_{\\min }}^{'2}}^{k_T^2}\\frac{dk_T^{'2}}{k_T^{'2}}\\frac{k_T^{'2}-k_T^2}{\\left| k_T^{'2}-k_T^2\\right| }\\\\&+\\int _{k_T^{2}}^\\infty\\frac{dk_T^{'2}}{k_T^{'2}}\\left(\\frac{k_T^2}{k_T^{'2}}\\right)^{\\lambda }\\frac{k_T^{'2}-k_T^2}{\\left| k_T^{'2}-k_T^2\\right| }\\bigg],\\\\\n\t\\end{split}\n\\end{equation}\n\n\\begin{equation}\n\t\\label{12}\n\t\\begin{split}\n\t\t\\zeta(k_T^2)&=\\frac{\\alpha _s N_c }{\\pi }k_T^2\\bigg[-\\int _{k_T^2}^{\\infty }\\frac{dk_T^{'2}}{k_T^{'2}}\\frac{1}{\\left| k_T^{'2}-k_T^2\\right| }\\\\&+\\int _{k_T^2}^\\infty\\frac{dk_T^{'2}}{k_T^{'2}}\\left(\\frac{k_T^2}{k_T^{'2}}\\right)^{\\lambda }\\frac{1}{\\left| k_T^{'2}-k_T^2\\right| }+\\int _{k_{T_{\\min }}^{'2}}^{\\infty }\\frac{dk_T^{'2}}{k_T^{'2}}\\frac{1}{\\sqrt{k_T^4+4k_T^{'4}}}\\bigg],\n\t\\end{split}\n\\end{equation}\nand $\\Delta(k_T^2)=\\frac{18 \\alpha _s^2}{\\pi  k_T^2 R^2}\\frac{N_c^2}{N_c^2-1}$.\n\\par Note that the integrals in \\eqref{12}, $I_1=\\int _{k_T^2}^{\\infty }\\frac{dk_T^{'2}}{k_T^{'2}}\\frac{1}{\\left| k_T^{'2}-k_T^2\\right| }$ and $I_2=\\int _{k_T^2}\\frac{dk_T^{'2}}{k_T^{'2}}\\left(\\frac{k_T^2}{k_T^{'2}}\\right)^{\\lambda }\\frac{1}{\\left| k_T^{'2}-k_T^2\\right| }$ are improper since both blow-up at the lower limit $k_T^2$.\nTo get rid of the singularity in $I_1$ we have performed some angular integral prescription. To be specific we have used the standard trigonometric integral\n\n\\begin{equation}\n\t\\label{13}\n\t\\int_{0}^{2\\pi}\\frac{\\text{d}\\theta}{p+q\\cos\\theta}=\\frac{2\\pi}{\\sqrt{p^2+q^2}},\n\\end{equation} \nvalid for $p+q>0$. Replacing the variable $k_T^{'} \\rightarrow k_T^{'} +k_T$ one can write\n\\begin{equation}\n\t\\label{14}\n\t\\int\\frac{\\text{d}^2k_T^{'}}{(k_T^{'}-k_T)^2}\\frac{1}{(k_T^{'}-k_T)^2+k_T^{'2}}=\\int\\frac{\\text{d}^2k_T^{'}}{k_T^{'2}}\\frac{1}{k_T^{'2}+(k_T^{'}+k_T)^2}.\n\\end{equation}\nNow using \\eqref{13} in the r.h.s. of \\eqref{14} we get\n\\begin{equation}\n\t\\label{15}\n\t\\begin{split}\n\t\t\\int\\frac{\\text{d}k_T^{'}  k_T^{'} \\text{d}\\theta}{k_T^{'2}(2k_T^{'2}+k_T^2+2k_T^{'}k_T\\cos\\theta)}&=2\\pi\\int\\frac{\\text{d}k_T^{'} k_T^{'}}{k_T^{'2} \\sqrt{4k_T^{'4}+k_T^4}}\\\\&=\\pi\\int\\frac{\\text{d}k_T^{'2}}{k_T^{'2} \\sqrt{4k_T^{'4}+k_T^4}}.\n\t\\end{split}\n\\end{equation}\nSimilarly using \\eqref{13} in the l.h.s. of \\eqref{14} we get\n\n\\begin{equation}\n\t\\label{16}\n\t\\begin{split}\n\t\t\\int\\frac{\\text{d}^2k_T^{'}}{(k_T^{'}-k_T)^2}\\frac{1}{(k_T^{'}-k_T)^2+k_T^{'2}} &=\\pi \\int \\frac{dk_T^{'2}}{k_T^{'2}}\\frac{1}{\\left| k_T^{'2}-k_T^2\\right| }\\\\& -\\pi\\int\\frac{\\text{d}k_T^{'2}}{k_T^{'2} \\sqrt{4k_T^{'4}+k_T^4}}.\n\t\\end{split}\n\\end{equation}\nEquating \\eqref{15} and \\eqref{16} we obtain\n\n\\begin{equation}\n\t\\label{17}\n\t\\int\\frac{dk_T^{'2}}{k_T^{'2}}\\frac{1}{\\left| k_T^{'2}-k_T^2\\right| }= 2 \\int\\frac{\\text{d}k_T^{'2} }{k_T^{'2} \\sqrt{4k_T^{'4}+k_T^4}},\n\\end{equation}\nwhich turns out to be well-behaved for both integration limit for $I_1$. On the other hand, in the limit $k_{T_{\\text{min}}}^{'2} \\leq k_T^{'2}\\leq k_T^2$ i.e. for  not too large $k_T^{'2}$, the contribution from the longitudinal component to the gluon virtuality $k^{'2}$ becomes negligible which in turn preserves the no ordering condition of transverse momentum of BFKL kinematics i.e. $k_T^{'2}\\approx k_T^2$ very strictly. Therefore, in this limit $k_{T_{\\text{min}}}^{'2} \\leq k_T^{'2}\\leq k_T^2$, it is justified to implant a factor $\\left(k_T^{2}\/k_T^{'2}\\right)^\\lambda$ inside the integrals which will in fact make our calculation simpler without altering the underlying physics. Moreover, this will fix the infrared divergence problem of the second improper integral $I_2$ by allowing us to evaluate the integral down to $k_{T_\\text{min}}^{'2}$.\n\\par Now considering these approximations in \\eqref{11} and \\eqref{12} and putting $I_1$ in \\eqref{12} we obtain\n\\begin{align}\n\t\\label{18}\n\t\\begin{split}\n\t\t\\xi(k_T^2)=&\\frac{\\alpha _s N_c k_T^2}{\\pi }\\bigg[\\int _{k_{T_{\\min }}^{'2}}^{k_T^2}\\frac{dk_T^{'2}}{k_T^{'2}}\\left(\\frac{k_T^2}{k_T^{'2}}\\right)^\\lambda\\frac{k_T^{'2}-k_T^2}{\\left| k_T^{'2}-k_T^2\\right| }\\\\&+\\int _{k_T^{2}}^{\\infty}\\frac{dk_T^{'2}}{k_T^{'2}}\\left(\\frac{k_T^2}{k_T^{'2}}\\right)^{\\lambda }\\frac{k_T^{'2}-k_T^2}{\\left| k_T^{'2}-k_T^2\\right| }\\bigg]\\\\=&\\frac{\\alpha_s N_c k_T^2}{\\pi}\\frac{1}{\\lambda}\\left(2-k_T^{2\\lambda}\\right)\\approx-\\frac{\\alpha_s N_c}{\\pi\\lambda}(k_T^2)^{\\lambda+1},\n\t\\end{split}\n\\end{align}\n\n\\begin{align}\n\t\\label{19}\n\t\\begin{split}\n\t\t\\zeta(k_T^2)=&\\frac{\\alpha _s N_c }{\\pi }k_T^2\\bigg[\\int _{k_{T_\\text{min}}^{'2}}^{\\infty }\\left(\\frac{k_T^2}{k_T^{'2}}\\right)^\\lambda\\frac{dk_T^{'2}}{k_T^{'2}}\\frac{1}{\\left| k_T^{'2}-k_T^2\\right| }\n\t\t\\\\&-\\int _{k_{T_{\\min }}^{'2}}^{\\infty }\\frac{dk_T^{'2}}{k_T^{'2}}\\frac{1}{\\sqrt{k_T^4+4k_T^{'4}}}\\bigg]\\\\\n\t\t=&\\frac{\\alpha _s N_c }{\\pi }\\bigg[\\frac{k_T^{2\\lambda}}{\\lambda}- 2^{2-\\frac{\\lambda }{2}}  \\lambda^{-1}k_T^{2\\lambda}\\bigg(1-\\frac{\\sqrt{k_T^4+4}}{k_T^2}\\bigg)^{\\lambda \/2} \\, _2F_1\\\\&-\\lambda\\ln \\left(\\frac{k_T^2}{2}+\\sqrt{1+\\frac{k_T^4}{4}}\\right)\\bigg]\\approx \\frac{\\alpha _s N_c }{\\pi }\\left(\\epsilon+\\frac{(k_T^2)^\\lambda }{\\lambda}\\right),\n\t\\end{split}\n\\end{align}\nwhere $\\, _2F_1$=$\\, _2\\text{F}_1\\left(-\\frac{\\lambda }{2},\\frac{\\lambda }{2};1-\\frac{\\lambda }{2};\\frac{k_T^2+\\sqrt{k_T^4+4}}{2 k_T^2}\\right)$ is a standard hypergeometric function. In the above calculations we have taken infrared cutoff $k_{T_\\text{min}}^{'2}= 1 \\text{ GeV}^2$ since for unified BFKL-DGLAP framework  this provided a very consistent result towards HERA DIS  data for proton structure function $F_2$ \\cite{14}.  In \\eqref{19} $(k_T^2)^\\lambda\/\\lambda$ is the only dominant term since other terms are significantly small.  The contribution from the logarithmic term in \\eqref{19} is negligible in comparison to the net contribution for all $k_T^2$. However, phenomenological studies shows the term involving hypergeometric function in \\eqref{19} becomes irrelevant towards change in $k_T^2$ i.e. it possess a constant value $(\\approx -4.79)$ (see Fig.~\\ref{reg}(a)). This constant contribution is insignificant to the net contribution for $\\zeta(k_T^2)$ if $k_T^2$ is enough high. But for small $k_T^2$ this contribution can not be neglected since at this range, the $k_T^{2\\lambda}\/\\lambda$ contribution  itself is small. In consequence, this constant contribution can be treated as a small perturbation $\\epsilon$ to the dominant term $k_T^{2\\lambda}\/\\lambda$ . We have performed phenomenological determination of the constant perturbation parameter $\\epsilon$ using standard non-linear regression method (see Fig.~\\ref{reg}(b)).\n\n\\begin{figure*}[tbp]\n\t\\centering\n\t\\includegraphics[width=.38\\textwidth,clip]{epsilon.eps}\\hspace{1cm}\n\t\\includegraphics[width=.38\\textwidth,clip]{N11.eps}\n\t\\caption{\\label{reg} (a) $k_T^2$ dependence of the small perturbation $\\tilde{\\epsilon}=|\\epsilon|$ corresponding to the term with the hypergeometric function in \\eqref{9} (left). (b) A phenomenological calculation of $\\epsilon$ using standard non-linear regression (right). Comparison between $\\tilde{\\zeta}=\\frac{\\pi}{\\alpha_sN_c}\\zeta$ (green dotted line) and $\\epsilon+2\\lambda^{-1}k_T^{2\\lambda}$ (red dotted line) is shown.}\n\\end{figure*}\n\n\nNow we are set to solve our original equation \\eqref{10} which is indeed a 1st order  semilinear (nonlinear) PDE. Our analytical approach of solving the same involves two steps: first we will express the nonlinear PDE in terms of a linear PDE then we will solve the linear PDE via. change of coordinates.\n\\par Substitution of $f(x, k_T^2)$  by its inverse function $\\omega(x,k_T^2)$ i.e. $f=\\omega^{-1}$ in \\eqref{10} yields\n\\begin{equation}\n\t\\label{20}\n\t-x\\frac{\\partial \\omega }{\\partial x}=\\xi \\frac{\\partial \\omega }{\\partial k_T^2}-\\zeta  \\omega +\\Delta,\n\\end{equation}\nwhich is in fact a linear PDE in $\\frac{\\partial \\omega}{\\partial x}$, $\\frac{\\partial \\omega}{\\partial k_T^2}$\nand $\\omega$. Now we construct a new set of  co-ordinate  $\\sigma\\equiv\\sigma(x, k_T^2)$ and $\\eta\\equiv\\eta(x,k_T^2)$ such that it transforms \\eqref{20} into an ODE. To be specific we define this transformation in  such a way that it is one to one for all $(x,k_T^2)$ in some set of points D in $x$-$k_T^2$ plane. This will allow us to solve \\eqref{20} for $x$ and $k_T^2$ as functions of $\\sigma$ and $\\eta$. The only requirement is that we should ensure  the Jacobian of the transformation does not vanish  i.e.\n$\\text{J}=\\left|\n\\begin{array}{cc}\n\\sigma_x & \\eta_x  \\\\\n\\sigma_{k_T^2} & \\eta_{k_T^2}  \n\\end{array} \\right| \\neq 0$ in D.\\\\\n\\par Next we want to recast \\eqref{20} in $(\\sigma,\\eta)$ plane computing the derivatives via chain rule:\n\\begin{equation}\n\t\\label{21}\n\t\\begin{split}\n\t\t&\\omega_x=\\omega_\\sigma\\sigma_x+\\omega_\\eta\\eta_x, \\\\\n\t\t&\\omega_{k_T^2}=\\omega_\\sigma\\sigma_{k_T^2}+\\omega_\\eta\\eta_{k_T^2}.\n\t\\end{split}\n\\end{equation}\nSubstitution of \\eqref{21} into \\eqref{20} yields\n\\begin{equation}\n\t\\label{22}\n\t-(x \\sigma_x+\\xi\\sigma_{k_T^2})\\omega_\\sigma-(x\\eta_x+\\xi\\eta_{k_T^2})\\omega_\\eta+\\zeta\\omega-\\Delta=0.\n\\end{equation}\nSince we want above equation to be expressed as an ODE, we require either,\n\\begin{equation*}\n\tx \\sigma_x+\\xi\\sigma_{k_T^2}=0 \\text{    or  } x\\eta_x+\\xi\\eta_{k_T^2}=0.\n\\end{equation*}\nIf we consider $x \\sigma_x+\\xi\\sigma_{k_T^2}=0$, the solution $\\sigma$ is constant along the curves that satisfy\n\\begin{equation}\n\t\\label{23}\n\t\\frac{\\text{d}k_T^2}{\\text{d}x}=\\frac{\\xi}{x} \\implies \\ln (x C_1)=\\int\\frac{\\text{d} k_T^2}{\\xi}\n\t\\implies C_1=\\frac{e^{-\\frac{k_T^{-2\\lambda}}{l \\lambda}}}{x},\n\\end{equation}\nwhere $l=-\\frac{\\alpha_s N_c}{\\pi \\lambda}$ and $C_1$ is the constant of integration. Now we can choose the new coordinates as $\\sigma =\\frac{e^{-\\frac{k_T^{-2\\lambda}}{l \\lambda}}}{x}$ and $\\eta=x$. Here $\\text{J}=\\left|\n\\begin{array}{cc}\n\\sigma_x & \\eta_x  \\\\\n\\sigma_{k_T^2} & \\eta_{k_T^2}  \n\\end{array} \\right| \\neq 0$ as required.\n\\par We rewrite \\eqref{21} as\n\\begin{align}\n\t\\label{24}\n\t\\begin{split}\n\t\t&\\omega_x=\\omega_\\sigma\\sigma_x+\\omega_\\eta\\eta_x=-\\frac{e^{-\\frac{k_T^{-2\\lambda}}{l \\lambda}}}{x^2}\\omega_\\sigma+\\omega_\\eta \\text{,}\\\\\n\t\t&\\omega_{k_T^2}=\\omega_\\sigma\\sigma_{k_T^2}+\\omega_\\eta\\eta_{k_T^2}=\\frac{(k_T^2)^{-1-\\lambda}e^{-\\frac{k_T^{-2\\lambda}}{l \\lambda}}}{lx}.\n\t\\end{split}\n\\end{align}\nNow putting \\eqref{24} in \\eqref{22} we get\n\\begin{align}\n\t\\label{25}\n\t-\\eta \\omega_\\eta+\\zeta\\omega-\\Delta=0.\n\\end{align}\nEquation \\eqref{25} is an ODE and it can be solved using standard ODE solving techniques. Now solving \\eqref{25} and then transforming it to the original coordinates $(\\sigma,\\eta)\\rightarrow (x,k_T^2)$ we get the general solution of the KC improved MD-BFKL equation, \n\\begin{widetext}\n\t\\begin{align}\n\t\\begin{split}\n\t\\label{26}\n\t&f(x,k_T^2)=\\omega^{-1}(x,k_T^2)=\\frac{k_T^{-2\\frac{n}{l}} (-1)^{\\frac{n}{\\lambda  l}} l^{-\\frac{n}{\\lambda  l}} \\lambda ^{-\\frac{n}{\\lambda  l}}}{\\text{G}\\left(\\frac{e^{-\\frac{k_T^{-2\\lambda}}{l \\lambda}}}{x}\\right) x^m+\\frac{18\\alpha_s ^2 (-1)^{1\/\\lambda } \\lambda ^{1\/\\lambda } N_c^2 l^{1\/\\lambda } e^{-\\frac{m k_T^{-2\\lambda }}{\\lambda  l}} m^{-\\frac{\\lambda  l+l+n}{\\lambda  l}} \\Gamma \\left(\\frac{l+n}{l \\lambda }+1,-\\frac{k_T^{-2\\lambda } m}{l \\lambda }\\right)}{\\pi  R^2 \\left(N_c^2-1\\right)}},\n\t\\end{split}\n\t\\end{align}\n\\end{widetext}\nwhere  $\\Gamma \\left(\\frac{l+n}{l \\lambda }+1,-\\frac{k_T^{-2\\lambda } m}{l \\lambda }\\right)$ is a standard Gamma function and $\\text{G}\\left(\\frac{e^{-\\frac{k^{-2\\lambda}}{l \\lambda}}}{x}\\right)$ is an arbitrary continuously differentiable function. The parameters $m=\\frac{\\epsilon\\alpha_sN_c }{\\pi}$ and $n=\\frac{\\alpha_sN_c}{\\pi\\lambda}$ are coming from \\eqref{19}. In the following section, we try to get particular solutions for the PDE applying some initial boundary condition and present an analysis of the phenomenological aspects of \\eqref{26}.\n\n\n\n\\subsubsection{$x$ and $k_T^2$ evolution}\n\nIn this section, we study the small-$x$ dependence of gluon distribution by picking an appropriate input distribution at some high $x$, as well as the $k_T^2$ dependence of gluon distribution setting input distribution at some low $k_T^2$.\n\\par Let us rewrite \\eqref{26} rearranging a bit \n\\begin{widetext}\n\\begin{align}\n\t\\label{27}\n\t\\begin{split}\n\t\t\\text{G}\\left(\\frac{e^{-\\frac{k_T^{-2\\lambda}}{l \\lambda}}}{x}\\right)=\\frac{1}{N_c^2-1}\\Bigg[&x^{-m}N_C^2 \\bigg(\\frac{18\\alpha_s ^2 (-1)^{\\frac{1}{\\lambda }+1} \\lambda ^{1\/\\lambda }  l^{1\/\\lambda } e^{-\\frac{m k_T^{-2\\lambda }}{\\lambda  l}} m^{-\\frac{\\lambda  l+l+n}{\\lambda  l}} \\Gamma \\left(\\frac{l+n}{l \\lambda }+1,-\\frac{k_T^{-2\\lambda } m}{l \\lambda }\\right)}{\\pi  R^2}\\\\\n\t\t&+\\frac{1}{f(x,k_T^2)} k_T^{-2\\frac{n}{l}} (-1)^{\\frac{n}{\\lambda  l}} l^{-\\frac{n}{\\lambda  l}} \\lambda ^{-\\frac{n}{\\lambda  l}}\\bigg)+\\frac{1}{f(x,k_T^2)}x^{-m} k_T^{-2\\frac{n}{l}} (-1)^{\\frac{n}{\\lambda  l}+1} l^{-\\frac{n}{\\lambda  l}} \\lambda ^{-\\frac{n}{\\lambda  l}}\\Bigg],\n\t\\end{split}\n\\end{align}\n\\end{widetext}First we will try to evaluate the functional form of the arbitrary differentiable function G applying some initial boundary condition on it. For $k_T^2$ evolution we set the initial distribution at $(x,k_0^2)$ where $x$ is fixed throughout the evolution. At $(x,k_0^2)$  let the argument of G be denoted by\n$t=\\frac{e^{-\\frac{k_0^{-2\\lambda}}{l \\lambda}}}{x}$.\n\nNow writing \\eqref{27} for $(x,k_0^2)$ and putting $x=\\frac{e^{-\\frac{k_0^{-2\\lambda}}{l \\lambda}}}{t}$ we get\n\\begin{widetext}\n\\begin{equation}\n\t\\label{28}\n\t\\begin{split}\n\t\t\\text{G}\\left(t\\right)=\\frac{1}{N_c^2-1}\\Bigg[&\\frac{e^{\\frac{mk_0^{-2\\lambda}}{l \\lambda}}}{t^{-m}}N_c^2 \\bigg(\\frac{18\\alpha_s ^2 (-1)^{\\frac{1}{\\lambda }+1} \\lambda ^{1\/\\lambda }  l^{1\/\\lambda } e^{-\\frac{m k_0^{-2\\lambda }}{\\lambda  l}} m^{-\\frac{\\lambda  l+l+n}{\\lambda  l}} \\Gamma \\left(\\frac{l+n}{l \\lambda }+1,-\\frac{k_0^{-2\\lambda } m}{l \\lambda }\\right)}{\\pi  R^2}\\\\\n\t\t&+\\frac{1}{F_0} k_0^{-2\\frac{n}{l}} (-1)^{\\frac{n}{\\lambda  l}} l^{-\\frac{n}{\\lambda  l}} \\lambda ^{-\\frac{n}{\\lambda  l}}\\bigg)+\\frac{1}{F_0}\\frac{e^{\\frac{mk_0^{-2\\lambda}}{l \\lambda}}}{t^{-m}} k_0^{-2\\frac{n}{l}} (-1)^{\\frac{n}{\\lambda  l}+1} l^{-\\frac{n}{\\lambda  l}} \\lambda ^{-\\frac{n}{\\lambda  l}}\\Bigg],\n\t\\end{split}\n\\end{equation}\n\\end{widetext}\nwhere $F_0$ is the unintegrated gluon distribution at $(x,k_0^2)$.\nEquation \\eqref{28} is the functional form of G. Replacing $t$ with any other argument will give us the value of G at that particular argument. We put $t=\\frac{e^{-\\frac{k_T^{-2\\lambda}}{l \\lambda}}}{x}$ in \\eqref{28} which gives us the l.h.s. of \\eqref{27} and then we solve  \\eqref{27} for $f(x,k_T^2)$. The solution for $k_T^2$ evolution turns out to be\n\\begin{widetext}\n\\begin{equation}\n\t\\label{29}\n\t\\begin{split}\n\t\tf(x,k_T^2)=\\dfrac{(-1)^{\\frac{n}{\\lambda  l}} \\big(N_c^2-1\\big) k_T^{-2\\frac{n}{l}} k_0^{2n\/l}   \\big( e^{\\frac{k_T^{-2\\lambda }-k_0^{-2\\lambda }}{\\lambda  l}}\\big)^m\\text{A}[x,k_T^2]}{(-1)^{\\frac{1}{\\lambda }+1}N_c^2 \\text{B}[ k_0^2 ] +(-1)^{\\frac{n}{\\lambda  l}}N_c^2 F_0 \\text{A}[x,k_T^2] +(-1)^{1\/\\lambda } N_c^2 \\text{B}[k_T^2]+ (-1)^{\\frac{n}{\\lambda  l}+1}F_0 \\text{A}[x,k_T^2]},\n\t\\end{split}\n\\end{equation}\n\\end{widetext}\nwhere\n\\begin{align*}\n\t\\begin{split}\n\t\t&\\text{A}[x,k_T^2]=\\pi  R^2 x^m m^{\\frac{\\lambda  l+l+n}{\\lambda  l}} e^{\\frac{m \\big(k_T^{-2\\lambda }+k_0^{-2\\lambda }\\big)}{\\lambda  l}}; \\\\\n\t\t&\\text{B}[i]=18 \\alpha_s ^2 x^m l^{\\frac{l+n}{\\lambda  l}} \\lambda ^{\\frac{l+n}{\\lambda  l}} k_0^{2n\/l} \\Gamma\\left. \\left(\\frac{l+n}{l \\lambda }+1,-\\frac{m (i)^{-\\lambda }}{l \\lambda }\\right)\\right\\vert_{i=k_T^2 , k_0^2}.\n\t\\end{split}\n\\end{align*}\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[trim=25 5 21.5 0,width=.46\\textwidth,clip]{x-4.eps}\\hspace{5mm}\n\t\\includegraphics[trim=25 5 21.5 0,width=.46\\textwidth,clip]{x-5.eps}\n\t\n\n\t\\caption{\\label{k1}$k_T^2$ evolution of unintegrated gluon distribution $f(x,k_T^2)$. Our results of KC improved MD-BFKL are shown for conventional $R=5 \\text{ GeV}^{-1}$ (gluons are distributed throughout the nucleus) and $R=2 \\text{ GeV}^{-1}$(at \"hot-spots\" within proton disk). Prediction from modified BK equation is plotted for comparison.}\n\\end{figure*}\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[trim=25 5 21.5 0,width=.46\\textwidth,clip]{x-3.eps}\\hspace{5mm}\n\t\\includegraphics[trim=25 5 21.5 0,width=.46\\textwidth,clip]{x-6.eps}\n\t\n\n\t\\caption{\\label{k2}$Q^2$ evolution of collinear gluon distribution $xg(x,Q^2)$. Our results of KC improved MD-BFKL are shown for conventional $R=5 \\text{ GeV}^{-1}$ (gluons are distributed throughout the nucleus) and $R=2 \\text{ GeV}^{-1}$(at \"hot-spots\" within proton disk). Theoretical prediction is compared with global datasets NNPDF 3.1sx and CT 14.}\n\\end{figure*}\nSimilarly setting the initial distribution at $(x_0,k_T^2)$, we obtain the solution for x-evolution as follows\n\\begin{widetext}\n\\begin{equation}\n\t\\label{30}\n\tf(x,k_T^2)=\\dfrac{(-1)^{\\frac{3}{\\lambda }+\\frac{2 n}{\\lambda  l}}\\left(N_c^2-1\\right) (\\frac{x_0}{x})^m k_T^{-2\\frac{n}{l}}  \\left(-\\frac{k_T^{-2\\lambda }}{\\lambda  l}-\\ln \\frac{x}{x_0}\\right)^{-\\frac{n}{\\lambda l} }\\text{C}[x,k_T^2]}\n\t{\\Bigg[\\splitdfrac{(-1)^{\\frac{3}{\\lambda }+\\frac{3 n}{\\lambda  l}}N_c^2 F_0^{'} \\text{C}[x,k_T^2] +\\frac{ (-1)^{\\frac{2}{\\lambda }+\\frac{n}{\\lambda  l}} }{k_T^2} \\Gamma_1 \\text{D}[x,k_T^2]}{+\n\t\t\t(-1)^{\\frac{2}{\\lambda }+\\frac{n}{\\lambda  l}+1}  k_T^{2\\left(\\frac{n}{l}-\\lambda  \\left(\\frac{1}{\\lambda }+\\frac{n}{\\lambda  l}\\right)\\right)} \\Gamma_2 \\text{D}[x,k_T^2]+(-1)^{\\frac{3}{\\lambda }+\\frac{3 n}{\\lambda  l}+1}\\pi  R^2 F_0^{'} x^m \\text{C}[x,k_T^2]}\\Bigg]},\n\\end{equation}\n\\end{widetext}\nwhere, $F_0^{'}$ is the initial gluon distribution at $(x_0,k_T^2)$,\n\\begin{equation*}\n\t\\begin{split}\n\t\t&\\Gamma_1=\\Gamma\\left(\\frac{l+n}{l \\lambda }+1,-\\frac{k_T^{-2\\lambda } m}{l \\lambda }\\right),\\text{     }\n\t\t\\\\&\\Gamma_2=\\Gamma\\left(\\frac{l+n}{l \\lambda }+1,m \\left(-\\frac{k_T^{-2\\lambda }}{l \\lambda }-\\ln \\frac{x}{x_0}\\right)\\right),\\\\\n\t\t&\\text{C}[x,k_T^2]=\\pi  R^2 x^ml^{-\\frac{1}{\\lambda }-\\frac{n}{\\lambda  l}} \\lambda ^{-\\frac{1}{\\lambda }-\\frac{n}{\\lambda  l}} k_T^{2\\left(\\frac{n}{l}-\\lambda  \\left(\\frac{1}{\\lambda }+\\frac{n}{\\lambda  l}\\right)\\right)}\\\\\n\t\t&\\times m^{\\frac{2}{\\lambda }+\\frac{2 n}{\\lambda  l}+1} \\left(-\\frac{k_T^{-2\\lambda }}{\\lambda  l}-\\ln \\frac{x}{x_0}\\right)^{\\frac{1}{\\lambda }+\\frac{n}{\\lambda  l}},\\\\\n\t\t&\\text{D}[x,k_T^2]=18 \\alpha_s ^2 N_c^2 x_0^m  l^{-\\frac{n}{\\lambda  l}} \\lambda ^{-\\frac{n}{\\lambda  l}} e^{-\\frac{m k_T^{-2\\lambda }}{\\lambda  l}}\\\\\n\t\t&\\times m^{\\frac{1}{\\lambda }+\\frac{n}{\\lambda  l}} \\left(-\\frac{k_T^{-2\\lambda }}{\\lambda  l}-\\ln \\frac{x}{x_0}\\right)^{1\/\\lambda }.\n\t\\end{split}\n\\end{equation*}\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[trim=25 5 21.5 0,width=.46\\textwidth,clip]{k_5.eps}\\hspace{5mm}\n\t\\includegraphics[trim=25 5 21.5 0,width=.46\\textwidth,clip]{k_50.eps}\n\t\n\n\t\\caption{\\label{x1}$x$ evolution of unintegrated gluon distribution $f(x,k_T^2)$. Our results of KC improved MD-BFKL are shown for conventional $R=5 \\text{ GeV}^{-1}$ (gluons are distributed throughout the nucleus) and $R=2 \\text{ GeV}^{-1}$(at \"hot-spots\" within proton disk). Prediction from modified BK equation is plotted for comparison.}\n\\end{figure*}\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[trim=25 5 21.5 0,width=.46\\textwidth,clip]{k_35.eps}\\hspace{5mm}\n\t\\includegraphics[trim=25 5 21.5 0,width=.46\\textwidth,clip]{k_100.eps}\n\t\n\n\t\\caption{\\label{x2}$x$ evolution of collinear gluon distribution $xg(x,k_T^2)$. Our results of KC improved MD-BFKL are shown for conventional $R=5 \\text{ GeV}^{-1}$ (gluons are distributed throughout the nucleus) and $R=2 \\text{ GeV}^{-1}$(at \"hot-spots\" within proton disk). Theoretical prediction is compared with global datasets NNPDF 3.1sx and CT 14.}\n\\end{figure*}\n\nWe have plotted the solution for both $x$ evolution \\eqref{30} and $k_T^2$ evolution \\eqref{29} in Fig.~\\ref{k1} and Fig.~\\ref{x1}. Our prediction of unintegrated gluon distribution $f(x,k_T^2)$ is contrasted with that of modified BK equation \\cite{5},\n\n\\begin{equation}\n\\label{bk}\n\\begin{split}\n&-x\\frac{\\partial f \\left(x, k_T^2\\right)}{\\partial x}=\\\\&\\frac{\\alpha_s N_c k_T^2}{\\pi}\\int_{k_{T_{\\min }}^{'2}}^{\\infty}\\frac{dk_T^{'2}}{k_T^{'2}}\\left[\\frac{f (x, k_T^{'2})-f \\left(x, k_T^2\\right)}{|k_T^{'2}-k_T^2|} + \\frac{f \\left(x, k_T^2\\right)}{\\sqrt{k_T^4+4k_T^{'4}}}\\right]\\\\\n&-\\alpha_{s}\\left(1-k_T^2\\frac{\\text{d}}{\\text{d}k_T^2}\\right)^2\\frac{k_T^2}{R^2}\\left[\\int_{k_T^2}^{\\infty}\\frac{\\text{d}k_T^{'2}}{k_T^{'2}}\\ln \\left(\\frac{k_T^{'2}}{k_T^2}\\right)f(x,k_T^2)\\right]\n\\end{split}\n\\end{equation}\nwhich is BFKL equation supplemented by the negative nonlinear term, derived in approximation of infinite and uniform target. In \\cite{5} the perturbative parton saturation is studied to a vast extent, including modification of \\eqref{bk} in terms of kinematic constraint, DGLAP, $\\text{P}_{gg}$ splitting function and running coupling constant, then solving the same numerically. We have also extracted collinear gluon distribution from unintegrated gluon distribution using the standard relation,\n\\begin{equation}\n\\label{f2g}\nxg(x,Q^2)=\\int_{0}^{Q^2}\\frac{\\text{d}k_T^2}{k_T^2}f(x,k_T^2)\n\\end{equation}\nsketched in Fig.~\\ref{k2} and Fig.~\\ref{x2}. Our predicted collinear gluon density is compared with that of LHAPDF global parameterization groups NNPDF 3.1sx \\cite{36} and CT 14 \\cite{53}. Both of the LHAPDF datasets  include HERA as well as recent LHC data with high precision PDF sensitive measurements.\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[trim=25 0 21.5 0,width=.47\\textwidth,clip]{compar_x.eps}\n\t\\hspace{1cm}\n\t\\includegraphics[trim=25 0 21 0,width=.46\\textwidth,clip]{compar_k.eps}\n\t\n\n\t\\caption{\\label{3a} (a) Comparison between linear BFKL, MD-BFKL without KC and MD-BFKL with KC for $k_T^2$ evolution at $x=10^{-4}$ and $10^{-6}$ (b) Comparison between linear BFKL, MD-BFKL without KC and MD-BFKL with KC for $x$ evolution at $k_T^2=5\\text{ GeV}^2$ and $50\\text{ GeV}^2$. Results are shown for $R= 5 \\text{ GeV}^{-1}$. }\n\\end{figure*} \n\\par Our prediction of both unintegrated and collinear gluon distribution is obtained for  two different  form of shadowing: conventional $R= 5 \\text{ GeV}^{-1}$ (order of proton radius) where gluons are spread throughout the nucleus and extreme $R= 2 \\text{ GeV}^{-1}$ where gluons are expected to concentrated in hotspots within the proton disk,  recalling that  $\\pi R^2$ is the transverse area within which gluons are concentrated inside proton. From Fig.~\\ref{k1}-\\ref{x2} it is clear that shadowing correction are more prominent when gluons are concentrated in hotspots within proton. \n\n\n\\par The $k_T^2$ (and $Q^2$) evolution in Fig.~\\ref{k1}-\\ref{k2} is studied for the kinematic range $1 \\text{ GeV}^2\\leq k_T^2(\\text{or }Q^2)\\leq 10^3 \\text{ GeV}^2$ corresponding to four different values of $x$ as indicated in the figure. Our evolution for both $f(x,k_T^2)$ and $xg(x,Q^2)$ shows a similar growth as modified BK as well as datasets respectively for all $x$. It is also observed that the growth of $f(x,k_T^2)$ (or $xg(x,Q^2)$) is almost linear for the entire kinematic range of $k_T^2$ (or $Q^2$). This is expected since the net shadowing term in \\eqref{8} has $1\/k_T^2$ dependence, which suppresses the contribution from the shadowing term at large $k_T^2$.\n\n\n\\par The $x$ evolution of $f(x,k_T^2)$ and $xg(x,Q^2)$ is shown in Fig.~\\ref{x1}-\\ref{x2} for  two $k_T^2$ values viz. $5 \\text{ GeV}^2$, $50 \\text{ GeV}^2$ and two $Q^2$ values viz. $35 \\text{ GeV}^2$, $100 \\text{ GeV}^2$. The input is taken at higher $x$ value $x=10^{-2}$ and then evolved down to smaller $x$ value upto $x=10^{-6}$ thereby setting the kinematic range of evolution $10^{-6}\\leq x\\leq10^{-2}$. We observed that the singular $x^{-\\lambda}$ growth of the gluon is eventually suppressed by the net shadowing effect. KC improved MD-BFKL seem to poses a more intense shadowing then modified BK. However, it is hard to establish the existence of shadowing for $x\\geq 10^{-3}$. The obvious distinction between the two form of shadowing $R=  \\text{5 GeV}^{-1}$ (conventional) and $R= 2\\text{ GeV}^{-1}$ (\"hotspot\") is also observed towards small-$x$ $(x\\leq 10^{-3})$.  Interestingly towards very  small-$x$ $(x\\leq 10^{-5})$, at small $k_T^2$ (or $Q^2$) values (viz. $k_T^2$=$5\\text{ GeV}^2$, $Q^2$=$35 \\text{ GeV}^2$) the gluon distribution becomes almost irrelevant of change in $x$. This could be a strong hint for the possible saturation phenomena in the small-$x$ high density regime.\n\n\\par In Fig.~\\ref{3a}(a) we have shown a comparison between the linear BFKL equation, MD-BFKL without kinematic constraint and MD-BFKL with kinematic constraint for $x$ evolution. The three solutions are compared for two different values of $k_T^2$ i.e. $ 5 \\text{ GeV}^2$ and $50 \\text{ GeV}^2$. Similarly in Fig.~\\ref{3a}(b) we have shown similar comparison but for $k_T^2$ evolution for two different values of $x$ i.e. $x=10^{-4}$ and $x=10^{-6}$. In Fig.~\\ref{3a}  the singular $x^{-\\lambda}$  behavior of $f(x,k_T^2)$ is distinct for unshadowed linear BFKL equation. On the other hand, the deviation of the two different forms of the MD-BFKL equation from linear BFKL equation reflects the underlying shadowing correction. It is also observed that shadowing effect is more intense in MD-BFKL with KC than MD-BFKL without KC.\n\n\n\\subsubsection{Complete solution of KC improved MD-BFKL}\nIn this section we implant a functional form of the input distribution (more likely a dynamic one) on the general solution of our KC improved MD-BFKL equation \\eqref{26} and try to obtain a parametric form of the solution. The underlying motivation towards doing so is that this allows us to evolve our solution for both $x$ and $k_T^2$ simultaneously in $x\\text{-}k_T^2$ phase space which help us to portray a three-dimensional realization of the gluon evolution. \n\\par Recall the well-known solution of the linear BFKL equation \\cite{9}\n\\begin{equation}\n\t\\label{32}\n\tf(x,k_T^2)=\\beta \\frac{x^{-\\lambda}\\sqrt{k_T^2}}{\\sqrt{\\ln\\frac{1}{x}}}\\exp \\left(-\\frac{\\ln^2(k_T^2\/k_s^2)}{2\\Omega\\ln (1\/x)}\\right),\n\\end{equation}\nwhere $\\lambda= \\frac{3 \\alpha_s}{\\pi} 28  \\zeta(3)$, $\\zeta$ being Reimann zeta function and $\\Omega=32.1 \\alpha_s$. The nonperturbative parameter $k_s^2= 1 \\text{ GeV}^2$ and the normalization constant $\\beta\\sim 0.01$ \\cite{1}. Since far below saturation region both linear and nonlinear equation should give the same solution, therefore we can take \\eqref{32} as the input distribution for our general solution of KC improved MD-BFKL equation.\n\nFirst we try to find the functional form of the arbitrary differentiable function $\\text{G}\\left(\\frac{e^{-\\frac{k_T^{-2\\lambda}}{l \\lambda}}}{x}\\right)$ present in the general solution \\eqref{26} applying the initial distribution \\eqref{32}.\nLet us rewrite \\eqref{27} which is the rearranged form of the general solution \\eqref{26}\n\n\n\\begin{widetext}\n\\begin{equation}\n\t\\label{33}\n\t\\begin{split}\n\t\t\\text{G}\\left(\\frac{e^{-\\frac{k_T^{-2\\lambda}}{l \\lambda}}}{x}\\right)=&\\frac{1}{N_c^2-1}\\Bigg[x^{-m}N_c^2 \\bigg(\\frac{18\\alpha_s ^2 (-1)^{\\frac{1}{\\lambda }+1} \\lambda ^{1\/\\lambda }  l^{1\/\\lambda } e^{-\\frac{m k_T^{-2\\lambda }}{\\lambda  l}} m^{-\\frac{\\lambda  l+l+n}{\\lambda  l}} \\Gamma \\left(\\frac{l+n}{l \\lambda }+1,-\\frac{k_T^{-2\\lambda } m}{l \\lambda }\\right)}{\\pi  R^2}\\\\\n\t\t&+\\frac{1}{f(x,k_T^2)} k_T^{-2\\frac{n}{l}} (-1)^{\\frac{n}{\\lambda  l}} l^{-\\frac{n}{\\lambda  l}} \\lambda ^{-\\frac{n}{\\lambda  l}}\\bigg)+\\frac{1}{f(x,k_T^2)}x^{-m} k_T^{-2\\frac{n}{l}} (-1)^{\\frac{n}{\\lambda  l}+1} l^{-\\frac{n}{\\lambda  l}} \\lambda ^{-\\frac{n}{\\lambda  l}}\\Bigg].\n\t\\end{split}\n\\end{equation}\n\\end{widetext}\nSetting initial parameters at $(x_0,k_T^2)$ we get initial distribution \\eqref{32} as\n\\begin{equation}\n\t\\label{34}\n\tf(x_0,k_T^2)=\\beta \\frac{x_0^{-\\lambda}\\sqrt{k_T^2}}{\\sqrt{\\ln\\frac{1}{x_0}}}\\exp \\left(-\\frac{\\ln^2(k_T^2\/k_s^2)}{2\\Omega\\ln (1\/x_0)}\\right).\n\\end{equation}\n\nWe denote the argument of G at $(x_0,k_T^2)$ as  $\\tau=\\frac{e^{-\\frac{k_T^{-2\\lambda}}{l \\lambda}}}{x_0}$ which implies $k_T^2=( -l\\lambda  \\ln (\\tau  x_0))^{-1\/\\lambda }$. Now writing \\eqref{33} for $(x_0,k_T^2)$  we obtain\n\\begin{widetext}\n\\begin{equation}\n\t\\label{35}\n\t\\begin{split}\n\t\t\\text{G}(\\tau)=&x^{-m} k_T^{-\\frac{n}{l}} (-l\\lambda)^{\\frac{n}{\\lambda  l}}  \\Bigg(\\frac{x^{\\lambda } \\sqrt{\\ln \\frac{1}{x}} (\\lambda  (-l) \\ln (\\tau  x_0))^{1\/\\lambda } \\exp \\left(\\frac{\\ln ^2\\left((\\lambda  (-l) \\ln (\\tau  x_0))^{-1\/\\lambda }\\right)}{2 \\Omega  \\ln \\frac{1}{x}}\\right)}{\\beta }\\\\\n\t\t&-\\frac{18\\alpha_s ^2  N_c^2 \\left(k_T^{-\\lambda }\\right)^{1\/\\lambda } \\left(e^{-\\frac{k_T^{-\\lambda }}{\\lambda  l}}\\right)^m \\left(-\\frac{m k_T^{-\\lambda }}{\\lambda  l}\\right)^{-\\frac{l+n}{\\lambda  l}} \\Gamma \\left(\\frac{l+n}{l \\lambda }+1,-\\frac{k_T^{-\\lambda } m}{l \\lambda }\\right)}{\\pi  m R^2 \\left(N_c^2-1\\right)}\\Bigg),\n\t\\end{split}\n\\end{equation}\n\\end{widetext}\nwhere, $k_T^2=( -l\\lambda  \\ln (\\tau  x_0))^{-1\/\\lambda }$. Note that \\eqref{35} is the functional form of G. We substitute $\\tau=\\frac{e^{-\\frac{k_T^{-2\\lambda}}{l \\lambda}}}{x}$ in \\eqref{35} which gives us the l.h.s. of \\eqref{33} and then solve \\eqref{33} for $f(x,k_T^2)$,\n\\begin{widetext}\n\\begin{equation}\n\t\\label{36}\n\t\\begin{split}\n\t\tf(x,k_T^2)=\\frac{\\beta  m q^{n\/l} \\left(-\\frac{m k_T^{-2\\lambda }}{\\lambda  l}\\right)^{\\frac{l+n}{\\lambda  l}} \\left(-\\frac{m q^{-\\lambda }}{\\lambda  l}\\right)^{\\frac{l+n}{\\lambda  l}}}\n\t\t{\\left(-\\frac{m q^{-\\lambda }}{\\lambda  l}\\right)^{\\frac{l+n}{\\lambda  l}} \\left(\\tilde{\\Delta}  q^{n\/l} \\tilde{A}[ k_T^2 ] +\\chi\\right)-\\tilde{\\Delta} \\beta   k_T^{2n\/l} \\tilde{A}(q) \\left(-\\frac{m k_T^{-2\\lambda }}{\\lambda  l}\\right)^{\\frac{l+n}{\\lambda  l}}},\n\t\\end{split}\n\\end{equation}\n\n\n\nwhere,\n\\begin{equation*}\n\t\\begin{split}\n\t\t&\\chi=m x^{\\lambda } \\sqrt{\\ln \\frac{1}{x}} k_T^{2n\/l} e^{\\frac{\\ln ^2(q)}{2 \\Omega  \\ln \\frac{1}{x}}} \\left(-\\frac{m k_T^{-2\\lambda }}{\\lambda  l}\\right)^{\\frac{l+n}{\\lambda  l}} q^{-1},\\text{    } \\text{   }\\tilde{A}[i]= i^{-1}\\beta  \\left(e^{-\\frac{q^{-\\lambda }}{\\lambda  l}}\\right)^m  \\Gamma \\left[i\\right], \\text{    }\\text{    } \\text{   }\n\t\tq=\\left(-l\\lambda \\ln \\frac{x_0 e^{-\\frac{k_T^{-2\\lambda }}{\\lambda  l}}}{x}\\right)^{-1\/\\lambda },\\\\\n\t\t&\\Gamma \\left[i\\right]=\\Gamma\\left(1+\\frac{l+n}{l\\lambda},\\frac{m (i)^{-\\lambda}}{l\\lambda}\\right), \\text{  }\\tilde{\\Delta}=\\frac{18 \\alpha _s^2}{\\pi  R^2}\\frac{N_c^2}{N_c^2-1}.\n\t\\end{split}\n\\end{equation*}\n\\end{widetext}\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=.40\\textwidth,clip]{MD-BFKL3D1.eps}\n\t\\caption{\\label{5x} Three dimensional representation of (a) TMDlib HERA data fit: PB-NLO-HERAI+II+2018 (left)   (b) our solution for KC improved MD-BFKL in $x$-$k_T^2$ phase space (right).}\n\\end{figure}\n\n\n\nFor simplicity let us denote r.h.s. of \\eqref{36} by $\\gamma$ i.e.\n\n\\begin{equation}\n\t\\label{37}\n\tf(x,k_T^2) = \\gamma(x,k_T^2).\n\\end{equation}\nAt $x=x_0$ and $k_T^2=k_0^2$\n\\begin{equation}\n\t\\label{38}\n\tf(x_0,k_0^2) = \\gamma(x_0,k_0^2).\n\\end{equation}\nNow dividing \\eqref{37} by \\eqref{38} we get\n\n\\begin{equation}\n\t\\label{39}\n\tf(x,k_T^2) = f(x_0,k_0^2)\\frac{\\gamma(x,k_T^2)}{\\gamma(x_0,k_0^2)}.\n\\end{equation}\nFrom \\eqref{32} we have the input distribution\n\\begin{equation}\n\t\\label{40}\n\tf(x_0,k_0^2)=\\beta \\frac{x_0^{-\\lambda}\\sqrt{k_0^2}}{\\sqrt{\\ln\\frac{1}{x_0}}}\\exp \\left(-\\frac{\\ln^2(k_0^2\/k_s^2)}{2\\Omega\\ln (1\/x_0)}\\right).\n\\end{equation}\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[width=.29\\textwidth,clip]{lambda04.eps}\\hspace{1cm}\n\t\\includegraphics[width=.29\\textwidth,clip]{lambda06.eps}\n\t\\caption{\\label{6y} Density plots showing $\\lambda$ sensitivity of unintegrated gluon distribution $f(x,k_T^2)$, sketched for two canonical choice of $\\lambda$ viz.  $\\lambda=0.4$ (left) and  $\\lambda=0.6$ (right). }\n\\end{figure*}\nNow substituting $f(x,k_0^2)$ from \\eqref{40} into \\eqref{39} we obtain\n\\begin{equation}\n\t\\label{41}\n\t\\begin{split}\n\t\t&f(x,k_T^2)=\\\\&\\frac{(\\ln \\frac{1}{x_0})^{-\\frac{1}{2}}q^{n\/l} k_0^{-2\\frac{n}{l}+4\\frac{l+n}{ l}-2}\\beta  x_0^{-\\lambda }  e^{-\\frac{\\ln ^2(k_0^2)}{2 \\Omega  \\ln \\frac{1}{x_0}}} \\left(k_T^2q\\right)^{-\\frac{l+n}{l}} \\tilde{\\phi}}\n\t\t{\\left(-\\frac{m q^{-\\lambda }}{\\lambda  l}\\right)^{\\frac{l+n}{\\lambda  l}} \\left(\\tilde{\\Delta}  q^{n\/l} \\tilde{A}[k_T^2] +\\chi\\right)-\\tilde{\\Delta} k_T^{2n\/l} \\tilde{A}[q] \\left(-\\frac{m k_T^{-2\\lambda }}{\\lambda  l}\\right)^{\\frac{l+n}{\\lambda  l}}},\n\t\\end{split}\n\\end{equation}\nwhere\n\\begin{equation*}\n\t\\begin{split}\n\t\t\\delta=&m x_0^{\\lambda } \\sqrt{\\ln \\frac{1}{x_0}} k_0^{2n\/l-2} e^{\\frac{\\ln ^2(k_0^2)}{2 \\Omega  \\ln \\frac{1}{x_0}}}   \\left(-\\frac{m k_0^{-2\\lambda }}{\\lambda  l}\\right)^{\\frac{l+n}{\\lambda  l}}, \\\\\n\t\t\\tilde{\\phi}=&\\left(-\\frac{m k_0^{-2\\lambda }}{\\lambda  l}\\right)^{\\frac{l+n}{\\lambda  l}} \\left(\\tilde{\\Delta}k_0^{2n\/l} \\tilde{A}[k_0^2] +\\delta[x,k_T^2]\\right)\\\\\n\t\t&-\\tilde{\\Delta} k_0^{2n\/l}\\tilde{A}[k_0^2]\\left(-\\frac{m k_0^{-2\\lambda }}{\\lambda  l}\\right)^{\\frac{l+n}{\\lambda  l}}.\t\n\t\\end{split}\n\\end{equation*}\n\n\\begin{figure*}[tbp]\n\t\\centering\n\t\\includegraphics[width=.29\\textwidth,clip]{R5.eps}\\hspace{1cm}\n\t\\includegraphics[width=.29\\textwidth,clip]{R2.eps}\n\t\\caption{\\label{6xx} Density plots showing $R$ sensitivity of unintegrated gluon distribution $f(x,k_T^2)$, sketched for two form of shadowing: conventional $R=5 \\text{ GeV}^{-1}$ (left) and extreme $R=2 \\text{ GeV}^{-1}$ (right).} \n\\end{figure*}\n\nEquation \\eqref{41} serves as the parametric form of the solution for KC improved MD-BFKL equation. The input distribution is inclusive in the solution \\eqref{41} itself, therefore, we do not have to depend on the experimental data fits for input distribution unlike we did in the previous section of $x$ and $k_T^2$ evolution. This parametric form of the solution actually helps us to explore the three-dimensional insight of the gluon evolution in $x$-$k_T^2$ phase space. However, using \\eqref{41} one may also study $x$ and $k_T^2$ evolution separately by setting $x$ fixed for $k_T^2$ evolution or $k_T^2$ fixed for $x$ evolution.\n\n\n\\par In Fig.~\\ref{5x} we have shown our solution of KC improved MD-BFKL equation \\eqref{41} in three dimension. The kinematic region for our study is set to be $10^{-6}\\leq x \\leq10^{-2}$ and $1 \\text{ GeV}^2\\leq k_T^2 \\leq10^3\\text{ GeV}^2$. In the 3D surface, the suppression in the rise of the gluon distribution towards small $x$ due to shadowing correction is visible. However, a linear rise of the surface in the $k_T^2$ direction can be seen, attributed to the $1\/k_T^2$ factor in the nonlinear term, which offset the effect of shadowing at large $x$.    \n\n\\par In Fig.~\\ref{6y} we have shown density plots of $f(x,k_T^2)$ in $x\\text{-}k_T^2$ domain to examine the sensitivity of $f(x,k_T^2)$ towards the parameter $\\lambda$. Density plot allows us to visualize and distinguish the kinematic regions with high\/low $f(x,k_T^2)$ in $x\\text{-}k_T^2$ plane which is more informative then any ordinary 3D plot. In Fig.~\\ref{6y} the plots are sketched for two canonical choices of $\\lambda$ viz. $\\lambda=0.4$ and $0.6$ corresponding to two $\\alpha_s$ values 0.15 and 0.23.\nOur solution seems to be very sensitive towards a small change in $\\lambda$. An apparent 50\\% change in $\\lambda$ (0.4 to 0.6) suggests approximately around one order of magnitude rise in gluon distribution $f(x,k_T^2)$ for an approximate limit of $x$ and $k_T^2$: $10^{-6}\\leq x\\leq 10^{-5}$ and $50 \\text{ GeV}^2 \\leq k_T^2\\leq 100 \\text{ GeV}^2$. It is also observed that the range of high $k_T^2$ and very small-$x$ is the high gluon distribution $f(x,k_T^2)$ region where gluons are mostly populated. In Fig.~\\ref{6xx} we have shown $R$ sensitivity of our solution for two choices of the shadowing parameter $R$ viz. $R= 5 \\text{ GeV}^{-1}$ and $2 \\text{ GeV}^{-1}$. A satisfactory shadowing effect is observed  from the comparison of the two plots. The extreme form of shadowing $(R=2 \\text{ GeV}^{-1})$ is found to suppress atleast $10\\text{-}20 \\%$ magnitude of gluon density than the conventional shadowing $(R=5 \\text{ GeV}^{-1})$ in the high $k_T^2$ and small-$x$ region.\n\n\\section{Equation of Critical line and prediction of saturation scale: a differential geometric approach}\\label{critical}\n\\begin{figure*}[tbp]\n\t\\centering\n\t\\includegraphics[width=.4\\textwidth,clip]{Saturation_scale.eps}\\hspace{1.5cm}\n\t\\includegraphics[width=.4\\textwidth,clip]{critical.eps}\n\t\\caption{\\label{6x} (a) Contour plot obtained by solving \\eqref{35x} (left). Solid lines show contours of constant gradients along the curve. (b)Diagram showing critical boundary line separating high and low gluon density region (right). Blue dashed line is obtained from \\eqref{38x} while red dashed line is corresponding to GBW model \\cite{21}.  }\n\\end{figure*}\nThe so-called saturation physics allows one to study the high parton density region in the small coupling regime. The transition from the linear region to saturation region is characterized by the saturation scale. The saturation momentum scale $Q_s$ is the threshold transverse momentum for which non-linearity becomes visible. The boundary in $(x, Q^2)$ or $(x,k_T^2)$ plane along which saturation sets in is characterized by the critical line. An important feature of our analytical solution to the  KC improved MD-BFKL equation is the finding of the equation of the critical line which is supposed to mark the boundary between dilute and dense partonic system in $x\\text{-}k_T^2$ phase space. \n\\par Although the gluon saturation can be achieved only when $Q_s\\sim Q$ (or $k_T^2$), the observables already become sensitive to $Q_s$ during the approach to saturation regime. This property is known as geometrical scaling in DIS inclusive event which means that instead of depending on $k_T^2$ and $x$ separately, the gluon distribution depends on a single dimensionless variable $\\frac{k_T^2}{Q_s^2}$ i.e.\n\\begin{equation}\n\t\\label{31s}\n\t\\phi(x,k_T^2)\\equiv\\phi\\left(\\frac{k_T^2}{Q_s^2}\\right).\n\\end{equation}\nIn recent years, this geometrical scaling property of DIS observables is studied very extensively for various frameworks \\cite{42,56, 57, 58,84}. In \\cite{56} an analysis of the saturation scale has been performed in the platform of resummed NLLx BFKL where the saturation scale was calculated via the relation\n\\begin{equation}\n\t\\label{32s}\n\t-\\frac{d\\omega(\\gamma_c)}{d\\gamma_c}=\\frac{\\omega_s(\\gamma_c)}{1-\\gamma_c},\n\\end{equation}\nwhich has been repeatedly derived in several literature \\cite{59,60,61}. In \\cite{42} the saturation scale $Q_s$ was obtained from the numerical solution of a nonlinear equation by finding the maximum of the momentum distribution of the gluon. Another approach for determination of $Q_s$ can be found in \\cite{5} where a parameter $\\beta$ is defined as the relative difference between the solutions to the linear and nonlinear equation,\n\\begin{equation}\n\t\\label{33s}\n\t\\beta=\\frac{|f^\\text{lin}(x,Q_s^2(x,\\beta))-f^\\text{lin}(x,Q_s^2(x,\\beta))|}{f^\\text{lin}(x,Q_s^2(x,\\beta))}.\n\\end{equation} \nThe crucial parameter $\\beta$ actually depicts the percentage deviation that the non linear solution shows from the linear one and it lies in the order of $0.1 \\text{-} 0.5$ (or $10\\text{-}50 \\%$ deviation).\n\n\n\\par Our approach towards studying geometrical scaling and critical line is primarily based on the basic understanding from differential geometry, in particular gradient of a function which is considered as the direction of steepest ascent of that function. Each component of the gradient gives us the rate of change of the function with respect to some standard basis i.e. it gives us an idea about how fast our function grows or decays or saturates with respect to the change of the variables. One important advantage of choosing gradient is that it is a two dimensional object since it does not possess any component along $f(x,k_T^2)$ axis in $\\mathbb{R}^3$. This ensures that it does not have any direct dependence on the magnitude of gluon density $f(x,k_T^2)$, rather it depends on the rate at which $f(x,k_T^2)$ changes with respect to $x$ and $k_T^2$ change. This property of gradient actually helps us in distinguishing out the saturation region and linear region although the distribution function $f(x,k_T^2)$ has large variation in order of magnitude for different regions in $x\\text{-}k_T^2$ phase space.\n\\par Recall that towards small-$x$, gluon evolution is suppressed due to shadowing effect as sketch in Fig.~\\ref{5x}  which motivates us to evaluate the gradient of $f(x,k_T^2)$ particularly along $x$ basis. For simplicity we consider an unit vector $\\vec{\\nu}$ along $\\tilde{\\eta} \\text{ }( = 1\/x)$ basis, we can project the gradient $\\nabla f(\\tilde{\\eta},k_T^2)$ along $\\vec{\\nu}$ via dot product $\\nabla f(\\tilde{\\eta},k_T^2)$.$\\vec{\\nu}$. This scalar quantity can also be interpreted as the directional derivative along the direction $\\vec{\\nu}$,\n\\begin{equation}\n\t\\label{34x}\n\tg\\equiv \\nabla_{\\vec{\\nu}} f(\\tilde{\\eta},k_T^2) = \\nabla f(\\tilde{\\eta},k_T^2).\\vec{\\nu}.\n\\end{equation}\nTaking the Euclidean norm yields\n\\begin{equation}\n\t\\label{35x}\n\tg = \\pm|\\nabla_{\\vec{\\nu}} f(\\tilde{\\eta},k_T^2)|,\n\\end{equation}\nwhere the negative (-) sign is for the descending function (or negative slope).\n\\par We obtained a family of contours (or level curves)  $\\tilde{\\eta}(k_T^2)$ in $\\tilde{\\eta}\\text{-}k_T^2$ plane solving \\eqref{35x} as shown in Fig.~\\ref{6x}(a). Each contour depicts a constant gradient $g$ along the curves. The set of contours can also be identified as some set of possible saturation scales. As sketch in Fig.~\\ref{6x}(a) the $\\tilde{\\eta}\\text{-}k_T^2$ plane is divided into two regions: low gluon density region where the spacing between two consecutive contours is very small and high gluon density region where the spacing becomes very large compared to previous. This distinction in contour spacing in the two regions comes from the fact that the gradient changes very fast until the saturation is reached and then after reaching the saturation boundary gradient changes very slowly or almost freezes for further increase in $\\tilde{\\eta}(k_T^2)$ as can be seen in Fig.~\\ref{6x}(a). In high gluon density region, the contour curves tend to $\\tilde{\\eta}(k_T^2)\\rightarrow \\infty $ towards high $k_T^2$. For low gluon density region, shadowing effects are negligible and the contours become almost parallel straight lines. \n\n\n\\par Let us try to find the equation of critical line which divides the two regions in $\\tilde{\\eta}\\text{-}k_T^2$ space. The level curves of the function $g=\\pm| \\nabla_{\\vec{\\nu}} f(\\tilde{\\eta},k_T^2)| $ are two dimensional curves that can be obtained by setting $g=k$ where $k$ is a constant $(k\\in \\mathbb{R})$. Therefore, the equations of the level curves are given by \n\\begin{equation}\n\t\\label{36x}\n\t| \\nabla_{\\vec{\\nu}} f(\\tilde{\\eta},k_T^2)|=\\pm k.\n\\end{equation}\nNow for a known initial saturation momentum scale $Q_{s0}(1\/x_0)$ we can predict equation of  the critical line using  \\eqref{35x} and \\eqref{36x}. The equation of the critical line is the equation for the level curve of the function $g=\\pm| \\nabla_{\\vec{\\nu}} f(\\tilde{\\eta},k_T^2)| $ that passes through the point $(Q_{s0}(\\tilde{\\eta}_0),\\tilde{\\eta}_0)$. First we find the value of $k$ by plugging the point $(Q_{s0}(\\tilde{\\eta}_0),\\tilde{\\eta}_0)$ into \\eqref{36x}\n\\begin{equation}\n\t\\label{37x}\n\tk_{s0}=\\pm| \\nabla_{\\vec{\\nu}} f(Q_{s0}^2(\\tilde{\\eta}_0))|.\n\\end{equation}\nNow the level curve passing through $(Q_{s0}(\\tilde{\\eta}_0),\\tilde{\\eta}_0)$ is obtained by setting\n\\begin{equation}\n\t\\label{38x}\n\t| \\nabla_{\\vec{\\nu}} f(\\tilde{\\eta},Q_s^2)|=| \\nabla_{\\vec{\\nu}} f(Q_{s0}^2(\\tilde{\\eta}_0))|,\n\\end{equation}\nwhich is the equation of the critical boundary. The knowledge of an appropriate initial saturation scale $Q_{s0}(1\/x_0)$ allows one to separate out the linear and saturation region using \\eqref{38x}. In Fig.~\\ref{6x}(b) we have sketched a possible critical line obtained from \\eqref{38x} for the choice of initial saturation scale $Q_{s0}^2(\\eta_0)\\backsimeq 2.8 \\text{ GeV}^{2}$ at $\\eta_0=10^6$ (or $x_0=10^{-6}$) which is taken from the calculation from the original saturation model by Golec-Biernat and Wusthoff \\cite{21, 5}. A rough agreement between our prediction and that of GBW model is observed in Fig.~\\ref{6x}(b). However, $Q_s$ given by \\eqref{38x} is found to have weaker $x$ dependence than the one from GBW model $Q_s^{'2}$. The saturation scale has direct dependence on partons per unit transverse area. Smaller $x$ suggests larger parton density giving rise to a larger saturation momentum scale, $Q_s^2$. In other words the saturation scale $Q_s$ depends on $x$ in such a way that with decreasing $x$ one has to probe  smaller distances or higher $Q^2$ in order to resolve the dense partonic structure of the proton which is clear from our analysis.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{KC improved MD-BFKL prediction of HERA DIS data}\\label{hera}\n\\subsection{DIS structure functions and reduced cross section}\\label{hera1}\nIn this section we present a quantitative prediction of proton structure functions $F_2(x,Q^2)$ and longitudinal structure function $F_L(x,Q^2)$ as an outcome of our solution to the KC improved MD-BFKL equation. At small-$x$, the sea quark distribution is driven by gluons via. $g\\rightarrow q\\bar{q}$ process. This component from sea quark distribution can be calculated in perturbative QCD. The relevant diagram for this QCD process is shown in Fig.~\\ref {d1} and the contribution to the transverse and longitudinal components of the structure functions can be written using the $k_T$-factorization theorem \\cite{25,26}\n\n\\begin{equation}\n\t\\label{42}\n\tF_\\text{i}(x,Q^2)=\\int_x^1\\frac{dx^{'}}{x^{'}}\\int\\frac{dk_T^2}{k_T^4}f\\left(\\frac{x}{x^{'}},k_T^2\\right)F_\\text{i}^{(0)}(x^{'},k_T^2,Q^2),\n\\end{equation}\nwhere i= T, L and $\\frac{x}{x^{'}}$ is the fractional momentum carried by gluon which splits into $q\\bar{q}$ pair. $F_\\text{i}^{(0)}$ includes both the quark box and cross box contribution which comes from virtual gluon-virtual photon subprocess leading to $q\\bar{q}$ production (Fig.~\\ref{d1} ). The gluon distribution $f(\\frac{x}{x^{'}},k_T^2)$ in \\eqref{42} represents the sum of the gluon ladder diagrams in the lower part of the box as shown in Fig.~\\ref {d1} is given by BFKL equation \\cite{2}. Here we will study the effect of gluon shadowing using the  solution $f(x,k_T^2)$ of KC improved MD-BFKL in \\eqref{42}.\n\n\n\\par The explicit expression for quark box contribution $F_\\text{i}^{0}$ can be obtained by writing four momentum in terms of the convenient light like momenta $p$ and  $q^{'}\\equiv q +xp$, where $x=\\frac{Q^2}{2p.q}$ and $Q^2=-q^2$ like as usual (see Fig.~\\ref{d1} ). Now we can decompose quark and gluon momentum in terms of sudakov variables\n\\begin{equation}\n\t\\begin{split}\n\t\t&\\kappa=\\alpha p-\\beta q^{'}+\\kappa_T,\\\\\n\t\t&k= a p+bq^{'}+k_T.\n\t\\end{split}\n\\end{equation}\nThe integration should be performed over the box diagram subject to quark mass-shell constraints \\cite{8}\n\n\\begin{align}\n\t\\begin{split}\n\t\t&(\\alpha -x)2p.q(1-\\beta)-\\kappa_T^2=m_q^2,\\\\\n\t\t&(a-\\alpha)2p.q\\beta-(\\kappa_T-k_T)^2=m_q^2,\n\t\\end{split}\n\\end{align}\nwhich leads to the box contribution \\cite{29}\n\\begin{align}\n\t\\label{43}\n\t\\begin{split}\n\t\t&\\tilde{F}_T^{(0)}(k_T^2,Q^2)=2 \\sum_q e_q^2\\frac{Q^2}{4\\pi^2}\\int_{0}^{1}d\\beta\\int d^2\\kappa_T\\alpha_s(\\kappa_T)\\\\&\\times\\big\\{[\\beta^2+(1-\\beta)^2]\\bigg[ \\frac{\\kappa_T^2}{L_1^2}-\\frac{\\kappa_T.(\\kappa_T-k_T)}{L_1L_2}\\bigg]+\\frac{m_q}{L_1^2}-\\frac{m_q^2}{L_1L_2} \\big\\},\n\t\t\\end{split}\n\t\\end{align}\n\t\\begin{align}\n\t\\label{44}\n\t\\begin{split}\n\t\t\\tilde{F}_L^{(0)}(k_T^2,Q^2)=&2 \\sum_q e_q^2\\frac{Q^4}{\\pi^2}\\int_{0}^{1}d\\beta\\int d^2\\kappa_T\\alpha_s(\\kappa_T)\\\\&\\times\\beta^2(1-\\beta)^2\\bigg( \\frac{1}{L_1^2}-\\frac{1}{L_1L_2}\\bigg),\n\t\t\\end{split}\n\\end{align}\nwhere the denominators $L_\\text{i}$ are\n\\begin{equation*}\n\t\\begin{split}\n\t\t&L_1=\\kappa_T^2+\\beta(1-\\beta)Q^2+m_q^2,\\\\\n\t\t&L_2=(\\kappa_T-k_T)^2+\\beta(1-\\beta)Q^2+m_q^2.\n\t\\end{split}\n\\end{equation*}\n\\begin{figure*}[tbp]\n\t\\centering\n\t\\includegraphics[width=.23\\textwidth,clip]{Structure_box1.eps}\n\t\\hspace{2cm}\n\t\\includegraphics[width=.23\\textwidth,clip]{Structure_Cross1.eps}\n\t\n\n\t\\caption{\\label{d1} Diagrammatic representation of the factorization formula \\eqref{42} where gluon couples to virtual photon through the (a) quark box (left) and (b) crossed box (right) diagrams. }\n\\end{figure*}\n\n\nNote that $\\tilde{F}_\\text{i}^{(0)}(k_T^2,Q^2)\\equiv \\int_x^1\\frac{dx^{'}}{x^{'}}F_\\text{i}^{(0)}(x^{'},k_T^2,Q^2)$ i.e. the $x^{'}$ integration of \\eqref{42} is implicit in the $d^2k_T^{'}$ and $d\\beta$ integration in \\eqref{43} and \\eqref{44}. Now plugging \\eqref{43} and \\eqref{44} in the $k_T$-factorization formula \\eqref{42} we get\n\\begin{align}\n\t\\label{45}\n\t\\begin{split}\n\t\tF_T(x,Q^2)=&2 \\sum_q e_q^2\\frac{Q^2}{4\\pi^2}\\int_{k_0^2}^\\infty\\frac{dk_T^2}{k_T^4} \\int_{0}^{1}d\\beta\\int d^2\\kappa_T\\alpha_s(\\kappa_T)\\\\&\\times\\big\\{[\\beta^2+(1-\\beta)^2]\\bigg[ \\frac{\\kappa_T^2}{L_1^2}-\\frac{\\kappa_T.(\\kappa_T-k_T)}{L_1L_2}\\bigg]\\\\\n\t\t&+\\frac{m_q}{L_1^2}-\\frac{m_q^2}{L_1L_2} \\big\\}f(\\frac{x}{x^{'}},k_T^2),\n\t\t\\end{split}\n\t\\end{align}\n\\begin{align}\n\\begin{split}\t\n\t\\label{46}\n\t\tF_L(k_T^2,Q^2)=&2 \\sum_q e_q^2\\frac{Q^4}{\\pi^2}\\int_{k_0^2}^\\infty\\frac{dk_T^2}{k_T^4}\\int_{0}^{1}d\\beta\\int d^2\\kappa_T\\alpha_s(\\kappa_T)\\\\&\\times\\beta^2(1-\\beta)^2\\bigg( \\frac{1}{L_1^2}-\\frac{1}{L_1L_2}\\bigg)f(\\frac{x}{x^{'}},k_T^2).\n\t\\end{split}\n\\end{align}\n\\begin{figure*}[tbp]\n\t\\label{}\n\t\\centering\n\t\\includegraphics[trim=21 0 21.5 0,width=.45\\textwidth,clip]{F2_5.eps}\\hspace{5mm}\n\t\\includegraphics[trim=21 0 21.5 0,width=.45\\textwidth,clip]{F2_15.eps}\\\\\n\t\\includegraphics[trim=21 0 21.5 0,width=.45\\textwidth,clip]{F2_35.eps}\\hspace{5mm}\n\t\\includegraphics[trim=21 0 21.5 0,width=.45\\textwidth,clip]{F2_45.eps}\n\t\n\t\n\t\n\n\t\\caption{\\label{7x} Prediction for proton structure function F$_2$ obtained for two choices of shadowing: conventional ($R= 5 \\text{ GeV}^{-1}$) and \"hotspot\" ($R= 2 \\text{ GeV}^{-1}$). Data are taken from HERA (H1 \\cite{32} and ZEUS \\cite{33}) as well as fixed target experiment (NMC \\cite{31} and BCDMS \\cite{34}). Data-sets from global parametrization groups NNPDF3.1sx \\cite{36} and PDF4LHC 15 \\cite{35} is also included. The background contribution is given by \\eqref{49} with $F_2^{BG}(x_0=0.1)\\approx$ 0.316, 0.384, 0.391 and 0.406 corresponding to four $Q^2$ values viz. $5 \\text{ GeV}^2, 15 \\text{ GeV}^2, 35 \\text{ GeV}^2$ and $45 \\text{ GeV}^2$. Separate plots of $F_2^{\\text{npert.}}$ vs. $F_2^{\\text{npert.+pert.}}$ is also sketched.\n\t}\n\\end{figure*}\nEquations \\eqref{45} and \\eqref{46} serve as the basic tool for calculating structure functions at small-$x$ provided that the gluon distribution $f(x,k_T^2)$ is known. An analytical approach towards the calculation of $F_2$ and $F_L$ can be found in literature \\cite{27,28} for fixed coupling case using linear BFKL equation. In the literature \\cite{9}, structure functions are calculated taking gluon distribution $f(x,k_T^2)$ from the numerical solution of unitarized BFKL equation for running coupling consideration. It has been seen that the analytical approach \\cite{28} considerably overestimates the actual (numerical) solution \\cite{9}. This is because of the fact that analytical approach neglects terms down only by powers of $\\ln (1\/z)^{-1}$ as well as it does not accommodate running of strong coupling. However, our approach is a semi-analytical one, in the sense that we take gluon distribution from our analytical solution while further integrations are performed numerically. This indeed allows us to check the feasibility of our analytical solution in describing DIS data. \n\\par In \\eqref{42}  $m_q$ denotes the quark mass and it is taken to be $m_q=1.28 \\text{ GeV}$ for charm quark while  massless ($m_q=0$) for light quarks (u, d and s). Our phenomenology is limited for light quark therefore putting $m=0$ in \\eqref{43} and replacing the quark transverse momentum by $\\kappa_T=\\kappa_T^{'}+(1-\\lambda)k_T$ we obtain \\cite{28}\n\\begin{equation}\n\t\\label{47}\n\t\\begin{split}\n\t\t&\\tilde{F}_T^{(0)}(k_T^2,Q^2)=\\\\& \\sum_q e_q^2\\frac{\\alpha_s}{\\pi}Q^2k_T^2\\int_{0}^{1}d\\beta\\int_{0}^{1}d\\lambda\\int_0^{\\infty} d\\kappa_T^{'2}[\\beta^2+(1-\\beta)^2]\\\\&\\times\\lambda\\frac{(2\\lambda-1)\\kappa_T^{'2}+(1-\\lambda)[\\lambda(1-\\lambda)k_T^2+\\beta(1-\\beta)Q^2]}{[\\kappa_T^{'2}+\\lambda(1-\\lambda)k_T^2+\\beta(1-\\beta)Q^2]^3}.\n\t\\end{split}\n\\end{equation}\nAfter integrating over $\\kappa_T^{'2}$ one can arrive at\n\\begin{equation}\n\t\\label{48}\n\t\\begin{split}\n\t\t\\tilde{F}_T^{(0)}(k_T^2,Q^2)=& \\sum_q e_q^2\\frac{\\alpha_s}{\\pi}Q^2k_T^2\\int_{0}^{1}d\\beta\\int_{0}^{1}d\\lambda\\\\&\\times\\frac{[\\lambda^2+(1-\\lambda)^2][\\beta^2+(1-\\beta)^2]}{\\lambda(1-\\lambda)k_T^2+\\beta(1-\\beta)Q^2}.\n\t\\end{split}\n\\end{equation}\nEquation \\eqref{47} and \\eqref{48}   are written in terms of Feynman integral which actually eliminates the azimuthal dependence and reduces the two fold integral $d^2\\kappa_T$ of \\eqref{43} to single integral $\\pi d\\kappa_T^{'2}$ . From \\eqref{48} it is clear that $\\tilde{F}_T^{(0)}(k_T^2,Q^2)$ or $F(x^{'},k_T^2,Q^2)$ possess the dimension of $k_T^2$. Therefore, $F(x^{'},$$k_T^2,$ $Q^2)$ or more conveniently $F(x^{'},k_T^2,Q^2)\/k_T^2$ may be considered as the structure function of an off mass shell gluon of approximate virtuality $k_T^2$. In \\cite{27} differential structure functions have been studied for fixed coupling and it is found that the ratio between longitudinal $F_L$ and transverse structure function $F_T$ is 2:9 for fixed coupling approximation. We have considered this ratio directly in our calculation of longitudinal structure function. Finally, we have taken an assumption $f(x\/x^{'}.k_T^2)\\rightarrow f(x,k_T^2)$ i.e. we ignore the $x^{'}$ dependence of $f(x\/x^{'}.k_T^2)$ which is reasonable in LLx accuracy since \n\\begin{equation*}\n\t(\\ln\\frac{x}{x^{'}})^n=(\\ln x)^n[1+O(1\/\\ln x)].\n\\end{equation*}\nThe advantage of taking this assumption is that we do not have to impose the possible constraint \\cite{8} coming from $x\/x^{'}<1$ on the region of integration. In principle, the factorization formula \\eqref{42} require to be run down to $k_T^2=0$. The integral itself is infrared finite as both the functions $f(\\frac{x}{x^{'}},k_T^2)$ and $F_i^{(0)}(x^{'},k_T^2,Q^2)$ vanish at $k_T^2=0$. However, BFKL dynamics is based on perturbative QCD which is not expected to hold the nonperturbative small $k_T^2$ physics. On the other hand, for small $k_T^2$ the gluon distribution vanishes linearly with the decrease in $k_T^2$ on account of gauge invariance \\cite{9} making the contribution small. Therefore, we have neglected this small contribution from small $k_T^2$ region in our calculations of unintegrated gluon distribution $f(x,k_T^2)$.\n\n\\begin{figure*}[tbp]\n\t\\label{}\n\t\\centering\n\t\\includegraphics[trim=21 0 21.5 0,width=.45\\textwidth,clip]{FL_5.eps}\\hspace{5mm}\n\t\\includegraphics[trim=21 0 21.5 0,width=.45\\textwidth,clip]{FL_15.eps}\\\\\n\t\\includegraphics[trim=21 0 21.5 0,width=.45\\textwidth,clip]{FL_35.eps}\\hspace{5mm}\n\t\\includegraphics[trim=21 0 21.5 0,width=.45\\textwidth,clip]{FL_45.eps}\n\t\n\t\n\t\n\n\t\\caption{\\label{77x} Prediction for proton structure function $F_L$ obtained for two choices of shadowing: conventional ($R= 5 \\text{ GeV}^{-1}$) and \"hotspot\" ($R= 2 \\text{ GeV}^{-1}$). Data are taken from HERA H1 \\cite{49} and ZEUS \\cite{33}. Data-sets from global parametrization group PDF4LHC 15 \\cite{35} is also included. The background contribution is given by \\eqref{49} with $F_L^{BG}(x_0=0.1)\\approx$ 0.057, 0.063, 0.071 and 0.073 corresponding to four $Q^2$ values viz. 5 $\\text{ GeV}^2$, 15 $\\text{ GeV}^2$, 35 $\\text{ GeV}^2$ and 45 $\\text{ GeV}^2$. Separate plots of $F_2^{\\text{npert.}}$ vs. $F_2^{\\text{npert.+pert.}}$ is also sketched. }\n\\end{figure*}\n\\begin{figure}[tbp]\n\t\\centering\n\t\\includegraphics[width=.48\\textwidth,clip]{RQ1.eps}\n\t\\caption{\\label{Rx} pQCD prediction from KC improved MD-BFKL for the ratio $R(Q^2)$ in the kinematic range $2 \\text{ GeV}^2\\leq Q^2\\leq 100 \\text{ GeV}^2$. The data \\cite{49} are from H1  ($\\sqrt{s}= 252\\text{ GeV}$) and ZEUS ($\\sqrt{s}= 225\\text{ GeV}$) experiment. The theoretical calculation are for $\\sqrt{s}= 225\\text{ GeV}$ analogous to c.m.s. of ZEUS .\n\t}\t\n\\end{figure}\n\n\\par Before proceeding to the realistic estimation of the structure functions, we add on the \"background\" some non-BFKL contribution, $F_\\text{i}^{\\text{BG}}$ to $F_\\text{i}$  since the above prediction of structure function is not enough for describing DIS data \\cite{9}. This is because \\eqref{45} and \\eqref{46} represent only the LLx gluon contribution and they are not the only contributions to the DIS structure functions. Although gluonic contribution is dominant at small-$x$, towards higher $x$ their effect becomes weak and we cannot neglect other non-BFKL contribution. For instance, we assume that $F_\\text{i}^{\\text{BG}}$ evolves like $x^{-0.08}$ motivated from soft pomeron intercept $\\alpha_{P}(0)=1.08$ \\cite{30}. To be precise we use \n\\begin{equation}\n\t\\label{49}\n\tF_\\text{i}^{\\text{BG}}(x,Q^2)=F_\\text{i}(x_0,Q^2)\\left(\\frac{x}{x_0}\\right)^{-0.08}.\n\\end{equation}\nAn intelligent way of calculating this non-BFKL contribution is to choose $x_0$ at some high $x$ and then take $F_\\text{i}^{\\text{BG}}(x_0,Q^2)$ from data \\cite{31} which is also listed in the figure caption. The recent HERA DIS data taken for comparison with our result can be found in \\cite{32,49} from H1 collaboration and in \\cite{33} from ZEUS collaboration. In the text \\cite{32} from H1 collaboration inclusive neutral current $e^{\\pm}p$ scattering cross section data collected during the years (2003-2007) is presented. The beam energies $E_p$ of corresponding H1 experiment run are $920$, $575$ and $460 \\text{ GeV}^2$. Corresponding $F_L$ data of H1 is taken from \\cite{49} where measurement are performed at c.m.s. energies $\\sqrt{s}=$ 225 and 252 $\\text{GeV}$. On the other hand, in \\cite{33} from ZEUS collaboration reduced cross sections for $ep$ scattering for different c.m.s. energies viz. $318$, $251$ and $225\\text{ GeV}$ is presented. The fixed target data from NMC \\cite{31} and BCDMS \\cite{34} collaboration which exist for $x>10^{-2}$ is also shown in the Figure \\ref{7x}. Finally, we have included data from global parameterization groups viz. NNPDF3.1sx \\cite{36} and PDF4LHC15 \\cite{35} for comparison. \n\n\n\\begin{figure*}[tbp]\n\t\\centering\n\t\\includegraphics[width=.32\\textwidth,clip]{cros2.eps}\n\t\\includegraphics[width=.32\\textwidth,clip]{cros5.eps}\n\t\\includegraphics[width=.32\\textwidth,clip]{cros85.eps}\\\\\n\t\\includegraphics[width=.32\\textwidth,clip]{cros12.eps}\t\n\t\\includegraphics[width=.32\\textwidth,clip]{cros25.eps}\n\t\\includegraphics[width=.32\\textwidth,clip]{cros45.eps}\n\n\t\\caption{\\label{8x} Theoretical prediction from KC improved MD-BFKL for reduced cross section $\\sigma_r (x,Q^2)$. The data are from H1 \\cite{40,83} ($\\sqrt{s}\\approx$ 318, 300 and 252 $\\text{GeV}^2$) and ZEUS \\cite{33} ($\\sqrt{s}\\approx$ 318 $\\text{GeV}^2$). Theoretical calculations are for $\\sqrt{s}= 318 \\text{ GeV}^2$.}\n\\end{figure*}\n\\par The $x$ dependence of the structure functions $F_L$ and $F_2$ is shown in Fig.~\\ref{7x} and Fig.~\\ref{77x} for the four $Q^2$ values $5 \\text{ GeV}^2$, $15 \\text{ GeV}^2$, $35 \\text{ GeV}^2$ and $45 \\text{ GeV}^2$. The nonperturbative contribution, $F_{2\/L}^{\\text{npert.}}$ (or $F_{2\/L}^{\\text{BG}}$) given by \\eqref{49} contrasted with total (nonpert. +pert.) contribution,  $F_{2\/L}^{\\text{npert.+pert.}}$. QCD prediction shows a satisfactory agreement with DIS data for both structure functions $F_2$ and $F_L$. It is clear that perturbative contribution is insignificant towards large $x$  ($\\geq 10^{-2}$), while for small $x$ around $x\\leq 10^{-3}$ this contribution becomes visibly important. Interestingly both data and theory for $F_2$ structure function seem to preserve the $x^{-\\lambda}$ singular behavior rather than showing taming due to net shadowing effect in the asymptotic limit $x\\rightarrow0$. It is difficult to observe the existence of any sizable shadowing effect for $x>10^{-3}$. However, for $x<10^{-3}$, recalling that the shadowing term is proportional to $1\/R^2$ it is seen that for $R=2\\text{ GeV}^{-1}$ (gluons in hotspots) the rise of structure function becomes slower than that of $R=5\\text{ GeV}^{-1}$ ($R=R_H$, hadron radius) as expected. But even the extreme form of the gluon shadowing ($R=2\\text{ GeV}^{-1}$) suppresses $F_2$ only by about $10\\%$ or less at $10^{-3}$. It just depicts that shadowing has the negligible impact on structure functions even towards smaller $x$. This is because of the fact that in this low $x$ regime gluons are expected to drive the sea quark distribution via $g\\rightarrow q\\bar{q}$. Therefore, a similar sea quark contribution to the structure function $F_2$ in addition to the gluon contribution can be expected in low $x$ regime. On the other hand from Fig.~\\ref{77x} it is clear that for $x>10^{-3}$, the size of the longitudinal structure function is negligibly small while for very small-$x$ regime ($x<10^{-3}$) , $F_L$  grows eventually. This is in accord with our expectation since measurement of $F_L$ directly probes the gluonic content of the proton which is dominant in small-$x$ regime. \n\n\n\\par The proton structure functions $F_2$ and $F_L$ are of complementary in nature. These are related to the $\\gamma^{*}p$ cross sections of longitudinally and transversely polarized photons $\\sigma_L$ and $\\sigma_T$ as $F_L\\propto\\sigma_L$ and $F_2\\propto (\\sigma_L+\\sigma_T)$.  Since $\\sigma_L$ and $\\sigma_T$ are positive, that imposes a restriction on $F_L$ and $F_2$ i.e. $0\\leq F_L\\leq F_2$. To circumvent the need of a proper relationship between the structure functions and  polarized cross sections to study $\\gamma^*p$ cross section one can define a ratio between the structure functions or the equivalent cross section ratio\n\\begin{equation}\n\t\\label{R}\n\tR=\\frac{\\sigma_L}{\\sigma_T}=\\frac{F_L}{F_2-F_L}.\n\\end{equation}\nThe advantage behind this formulation is that this is independent of any normalization factors.\n\n\\par In Fig \\ref{Rx} a phenomenological comparison between data and theory is shown to illustrate $Q^2$ dependence of the ratio $R(x,Q^2)$. The HERA data taken for comparison can be found in \\cite{49} where $ep$ cross section data measured for two center of mass energies $\\sqrt{s}=225$ and $252\\text{ GeV}$ is enlisted. From Fig.~\\ref{Rx} it  is clear that the HERA data is in general agreement with the QCD expectation. Available H1 data for very low $Q^2$ $(Q^2\\leq 5\\text{ GeV})$ is also sketched in Fig.~\\ref{Rx}. We have excluded the very low $Q^2$ region in our calculation of $R$ since we are bounded to stick in the perturbative region. However, a more realistic study in this transition region is performed in Sect.~\\ref{hera2}. \n\n\\par Now we try to predict reduced cross section $\\sigma_r$ for $ep$ scattering process from the knowledge of structure functions that we have discussed above. The inclusive deep-inelastic differential cross section for $ep$ scattering can be represented in terms of three structure functions $F_2$, $F_L$ and $xF_3$. The structure functions have direct dependence with DIS kinematic variables, $x$ and $Q^2$ only, while the cross section has additional dependence with inelasticity $y=Q^2\/s x$. The inclusive cross section for neutral current $ep$ scattering is given by\n\\begin{equation}\n\t\\label{50}\n\t\\begin{split}\n\t\t\\frac{d^2 \\sigma^{e^{\\pm}p}}{dx dQ^2}=\\frac{2\\pi\\alpha^2}{x Q^4}Y_{+}\\left(F_2-\\frac{y^2}{Y_+}F_L\\mp\\frac{Y_{-}}{Y_{+}}xF_3\\right),\n\t\\end{split}\n\\end{equation}\nwhere $Y_{\\pm}=1\\pm(1-y)^2$. The cross section for exchanged virtual boson $(\\text{Z} \\text{ or } \\gamma^{*})$ is related to $F_2$ and $F_3$ in which contribution from both longitudinal and transverse boson polarization state exists. On the other hand, only the longitudinal polarized virtual boson exchange processes contribute to $F_L$ which has a significant impact on higher order QCD though it vanishes at lowest order. At small momentum transfer $Q^2$ $(\\text{i.e. } Q^2\\ll M_Z^2\\approx800\\text{ GeV}^2)$, interaction of massless photon is dominant over the exchange of heavy Z boson. Thus, the contribution from Z boson exchange and the influence of the term $xF_3$ is negligible at low and moderate $Q^2$. Therefore, in this range of $Q^2$, in one photon exchange approximation, the differential cross section formula \\eqref{50} can be written as \n\n\\begin{equation}\n\t\\label{51}\n\t\\sigma_r\\equiv\\frac{d^2 \\sigma^{e^{\\pm}p}}{dx dQ^2}\\frac{x Q^4}{2\\pi\\alpha^2}\\frac{1}{Y_{+}}=\\left(F_2-\\frac{y^2}{Y_+}F_L\\right),\n\\end{equation}\nwhere $\\sigma_r$ is the reduced cross section. Note that \\eqref{51} is symmetric under charge exchange i.e. identical for both $e^{+}p$ and $e^{-}p$ processes unlike \\eqref{50}. Additionally, it is independent  of incoming electron  helicity state. Thus, at low and moderate $Q^2$ $(\\leq m_Z^2)$ which is our region of study, the knowledge of $F_2$ and $F_L$ is enough to predict the reduced cross section. We can also express reduced cross section in terms of the ratio $R(x,Q^2)$ defined in \\eqref{R} replacing $F_L$ by $F_L=\\frac{R}{1+R}F_2$ which yields\n\\begin{equation}\n\t\\label{51x}\n\t\\sigma_r=F_2(x,Q^2)\\left[1-\\frac{y^2}{Y_+}\\frac{R}{1+R}\\right].\n\\end{equation}\n\n\\par The $x$ dependence of $e^{\\pm}p$ reduced cross section $\\sigma_r$ calculated for center of mass energy $\\sqrt{s}=318\\text{ GeV}$ is shown in Fig.~\\ref{8x}. Our theoretical expectation is compared with HERA H1 measurement \\cite{40,83} and ZEUS  ($\\sqrt{s}=318\\text{ GeV}$) \\cite{33}. The available H1 data for low $Q^2$ ($\\leq12\\text{ GeV}^2$) measured at $\\sqrt{s}=$ 318 $\\text{GeV}$ (SVX) and 300 $\\text{GeV}$ (NVX-BST) is taken from \\cite{40}, while the same for high $Q^2$ ($>12\\text{ GeV}^2$) is taken from \\cite{83}.\nThe cross section measurement of SVX is found to be slightly higher than that of NVX-BST as expected because of the increase in center of mass energy. Both theory and data agree well for our phenomenology range $Q^2\\leq 100\\text{ GeV}^2$. The distinction in $\\sigma_r$ due to the two forms of shadowing $R=5\\text{ GeV}^{-1}$ and $R=2\\text{ GeV}^{-1}$ is more prominent for smaller $Q^2$ values. For each $Q^2$, starting at some high $x$ the reduced cross section first increases as $x\\rightarrow0$ and then an abrupt fall in cross section can be observed in both theory and data at very small-$x$ region ($x<10^{-4}$) . For all $Q^2$, this region of $x$ corresponds to the highest inelasticity $y\\approx0.65$ ($y=Q^2\/sx$) and thus characteristic turn over of cross section at $y\\approx0.65$ can be attributed to the influence of $F_L$.  In simple words, towards very small-$x$ (or high $y$) the monotonic rise of $F_2$ is suppressed by the contribution from longitudinal structure function $F_L$ thereby causing an overall fall in the cross section. For low inelasticity $y<0.65$, the contribution from the longitudinal structure function is small on the other hand, structure function $F_2$ exhibits a steady increase as $x\\rightarrow 0$. Therefore, in the region where $x$ is not so small, the growth of the cross section is found to be power like as expected. \n\n\n\\subsection{Virtual photon-proton cross section prediction in transition region}\\label{hera2}\n\\par Traditionally the photon-proton interaction is classified into two separate processes depending upon the photon virtuality $Q^2$ viz. photoproduction (at low $Q^2$) and deep inelastic scattering (at high $Q^2$). DIS is considered  as a basic tool for exploring pQCD where the point like virtual photon directly probes the partonic contents of the proton. On the other hand, photoproduction is completely nonperturbative phenomena defined in the limit of vanishing $Q^2$ where real (or quasi-real) photons interact with the proton more likely a hadron-hadron collision. The experiment at HERA collider provides a unique opportunity to study both the processes photoproduction and DIS on their respective kinematic domains. The $Q^2$ dependence of the proton structure functions are well described by pQCD over a wide range of $x$ and $Q^2$ \\cite{37,38} in accordance with HERA data. However, for $Q^2\\lesssim 2\\text{ GeV}^2$ (photo production region) the pQCD breakdowns since the higher order contributions to the perturbative expansion becomes very large. In this region, data can be only described by non-perturbative phenomenological models \\cite{39}. Our present study is especially focused on the transition region $(2\\text{ GeV}^2\\leq Q^2\\leq10\\text{ GeV}^2)$ from photoproduction to deep inelastic scattering. To measure the photon-proton cross section in the transition region two dedicated runs were performed in the years 1999 (Nominal vertex \"NVX'99\") and 2000 (Shifted Vertex \"SVX'00\") by H1 experiment at HERA. The published data can be found in \\cite{40}.\n\n\n\n\\par Recall the neutral current $ep$ double differential cross section formula \\eqref{51} defined in the region $Q^2\\ll M_Z^2$. In this region massive boson $(M_Z)$ exchange is neglected and only one photon exchange is considered, thereby the role of incoming electron reduces to be a source of virtual photon interacting proton. Thus, we can recast the formula \\eqref{51} in terms of photon-proton reaction. In fact the structure function $F_2$ and $F_L$ in \\eqref{51} related to the longitudinally and transversely polarized photon-proton scattering cross sections $\\sigma_L$ and $\\sigma_T$ by the relations\n\\begin{align}\n\t\\label{52}\n\t&F_L = \\frac{Q^2}{4\\pi^2\\alpha}(1-x)\\sigma_L,\\\\\n\t\\label{53}\n\t&F_2 = \\frac{Q^2}{4\\pi\\alpha}(1-x)(\\sigma_L+\\sigma_T),\n\\end{align}\nwhich hold good at low $x$. Considering \\eqref{52} and \\eqref{53} the reduce cross section in \\eqref{51} can be written as\n\\begin{equation}\n\t\\label{54}\n\t\\sigma_r = \\frac{Q^2(1-x)}{4\\pi^2\\alpha}\\sigma_{\\gamma^*p}^{\\text{eff}},\n\\end{equation}\nwhere \\begin{align}\n\t\\label{55}\n\t\\sigma_{\\gamma^*p}^{\\text{eff}}=\\sigma_T+(1-\\frac{y^2}{Y_+})\\sigma_L\n\\end{align}\nis the effective virtual photon-proton cross section. Note that the expression for  $\\sigma_{\\gamma^*p}^{\\text{eff}}$ is similar to the total cross section $\\sigma_{\\gamma^*p}^{\\text{tot}}$ which is linear combination of $\\sigma_L$ and $\\sigma_T$ i.e.  \n$\\sigma_{\\gamma^*p}^{\\text{tot}}=\\sigma_L+\\sigma_T$. In fact the total cross section $\\sigma_{\\gamma^{*}p}^{\\text{tot}}$ and $\\sigma_{\\gamma^*p}^{\\text{eff}}$ can be regarded as the same quantity at low inelasticity $y$ i.e. $\\sigma_{\\gamma^*p}^{\\text{eff}} \\xrightarrow{y\\rightarrow0}\\sigma_{\\gamma^*p}^{\\text{tot}}$ which differ only in the region of high y.\n\n\\begin{figure}[tbp]\n\t\\centering\n\t\\includegraphics[width=.4\\textwidth,clip]{W_cross.eps}\n\t\\caption{\\label{7xx} QCD prediction from KC improved MD-BFKL for virtual photon-proton cross section $\\sigma_{\\gamma^{*}p}^\\text{eff}$ as a function of $Q^2$ at different values of $W$. The cross section for different $W$ values are multiplied by factor multiple of 2 indicated in the figure. The included data are from  H1 \\cite{40} ($\\sqrt{s}= 318\\text{ GeV}$) and ZEUS  \\cite{81} ($\\sqrt{s}= 300\\text{ GeV}$). }\t\n\\end{figure}\n\\par Figure \\ref{7xx} shows the measurement of the virtual photon-proton cross section $\\sigma_{\\gamma^{*}p}^\\text{eff}$ as a function of $Q^2$ corresponding to different values of $W$.  The total cross section $\\sigma_{\\gamma^*p}^{\\text{tot}}$ is often expressed as a function of $Q^2$ and invariant mass $W$. The standard relation between $W$, $x$ and $Q^2$ is $W=\\sqrt{Q^2(1-x)\/x}$. Since $Q^2\\approx syx$, for small-$x$ we can have an approximate relationship between $W$ and $y$ i.e. $W^2\\simeq sy$ which we have used in our calculations. HERA measurement for $\\sigma^{\\text{eff}}_{\\gamma^{*}p}$ from H1 \\cite{40} ($\\sqrt{s}= 318\\text{ GeV}$) and ZEUS  \\cite{81} ($\\sqrt{s}= 300\\text{ GeV}$) are included for comparison with our theoretical prediction. We have chosen  $\\sqrt{s}= 318\\text{ GeV}$ for our calculation analogous to H1 measurement. The precision of the data is such that their errors are hardly visible.  Both the HERA data and theoretically measured cross sections for different values of $W$ are multiplied by the factors multiple of 2 as indicated in the figure. For $Q^2\\gtrsim3 \\text{ GeV}^2$, an excellent agreement between the theory and HERA data can be observed for the wide range of $W$, while for $Q^2<3 \\text{ GeV}^2$, the slop of the QCD prediction seems unsatisfactory. This indicates the inadequacy of perturbative QCD at very low $Q^2$ ($<3 \\text{ GeV}^2$) and roughly provides a lower bound of $Q^2$ to our theory.\n\n\n\n\n\n\n\n\n\n\\section{Conclusion}\\label{con}\n\n\\par In conclusion, we have presented a phenomenological study on the behavior of unintegrated gluon distribution at small-$x$ and moderate $k_T^2$ region particularly $10^{-6}\\leq x\\leq 10^{-2}$ and $2 \\text{ GeV}^2\\leq k_T^2\\leq 1000 \\text{ GeV}^2 $ which is also the accessible kinematic range to experiments performed at HERA $ep$ collider. In the beginning, we have improved the MD-BFKL equation supplementing so-called kinematic constraint on it. Then we have solved this unitarized BFKL equation analytically in order to study $x$ and $k_T^2$ dependence of unintegrated gluon distribution function. Our prediction of gluon distribution is contrasted with that of modified BK equation as well as global datasets NNPDF 3.1sx and CT 14. The $x$ evolution of gluon distribution shows the singular $x^{-\\lambda}$ type behavior of gluon evolution tamed by shadowing correction. We found that for intense shadowing $(R= 2 \\text{ GeV}^{-1})$, towards smaller x, certainly from $x=10^{-4}$, the gluon distribution emerges from rapid BFKL growth which indicates the dominance of gluon shadowing. While in case of conventional shadowing $(R= 5 \\text{ GeV}^{-1}) $ an appreciable modification of BFKL behavior can only be seen from $x= 10^{-5}$. The $k_T^2$ dependence of unintegrated gluon distribution is also studied. Although obvious suppression due to net shadowing correction is seen in our studies, however no significant saturation phenomenon is observed. The reason is the nonlinear contribution term is suppressed by the factor $1\/k_T^2$ at large values of $k_T^2$.\n\n\\par We have obtained a more general solution of KC improved MD-BFKL equation implementing a pre-defined input distribution on it. This has allowed us to visualize the gluon evolution in three dimensions $\\mathbb{R}^3$ into $x\\text{-}k_T^2$ phase space. We have also shown the sensitiveness of unintegrated gluon distribution towards $R$ and $\\lambda$ using density plots in $x\\text{-}k_T^2$ plane.\n\n\\par An important achievement of obtaining an analytical solution in this work is its implication on qualitative studies on geometrical scaling which is presented in Sect. \\ref{critical}. Starting from a basic concept of differential geometry and knowledge of level curves we have managed to obtain an equation of the critical boundary which is supposed to separate low and high gluon density regions.\n\\par In Sect. \\ref{hera} we have studied the small-$x$ dependence of the structure function $F_2$ and $F_L$ obtained via $k_T$-factorization formula $F_i=f\\otimes F_i^{(0)}$. Here the unintegrated gluon distribution $f(x,k_T^2)$ is taken from our analytical solution of KC improved MD-BFKL equation setting boundary condition at $x_0=0.01$. Surprisingly the quantitative size of the shadowing correction to $F_2$ and $F_L$ is found to be very small and the structure functions seem to hold the singular $x^{-\\lambda}$ behavior. Even at intense shadowing condition $(R= 2 \\text{ GeV}^{-1})$ the $F_2$ structure function is found to be suppressed only by $10\\%$. In addition we have measured $e^{\\pm}p$ reduced cross section as well as equivalent $\\gamma^*p$ longitudinal to transverse cross section ratio $R(x,Q^2)$ from the knowledge of $F_2$ and $F_L$. Our results are compared with recent high precision HERA measurements. The comparison reveals a good agreement between our theory and DIS data.\n\\par In Sect. \\ref{hera2} we have examined the virtual photon-proton effective cross section, particularly in the transition region from photoproduction to deep inelastic scattering. The quantity $\\sigma_{\\gamma^{*}p}^\\text{eff}$ serves the role of the total cross section if small inelasticity $y$ is concerned. We have compared our predicted $\\sigma_{\\gamma^{*}p}^\\text{eff}$ with the HERA data of two dedicated runs \"NVX'99\" and \"SVX'00\" by H1 as well as ZEUS for the transition region. Our theoretical prediction shows well-consistency with HERA data particularly upto $Q^2\\sim3 \\text{ GeV}^2$ in the transition region.\n\n\n\\par In summary, there are several attractive features of our present study. First, we have able to predict a wide range of physical quantities, starting right from our solution for unintegrated gluon density. Secondly, all analysis is performed in terms of relatively small numbers of parameters. Moreover, the idea developed in this work for studying geometrical scaling can be implemented in any other framework. Finally, a very significant feature of this analysis is that we have considered two extreme possibilities of shadowing which can be distinguished by experimental data. In the end, we conclude that, as per feasibility towards HERA DIS data is concerned, the KC improved MD-BFKL equation could be a reliable framework for exploring high energy physics over a wide range of $x$ and $k_T^2$ which is also relevant for LHC probe and future collider phenomenology. \n\n\n\\section*{Acknowledgments}\nWe thank Prof.~Johannes Bluemlein, DESY (Hamburg, Germany) and Prof.~Dieter Schildknecht, Bielefeld university (Germany)  for their valuable comments. Two of us ( P.P. and M.L.) acknowledge Department of Science and Technology (DST), India (grant DST\/INSPIRE Fellowship \/2017 \/IF160770) and Council of Scientific and Industrial Research (CSIR), New Delhi (grant 09\/796(0064)2016-EMR-I) respectively for the financial assistantship. \n\n\n\n\n\\bibliographystyle{spphys}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nWe consider a composite quantum system consisting of two subsystems $A$ and $B$ of Hilbert space dimensions $m$ and $n$, respectively. The Hilbert space $\\mathcal{H}_{A+B}$ of the composite system is given by the tensor product of the Hilbert spaces of the subsystems, $\\mathcal{H}_{A+B}=\\mathcal{H}_{A}\\otimes\\mathcal{H}_{B}$. The random pure state (as opposed to the mixed state) of the composite system is written as a linear combination of the random coefficients $x_{i,j}$ and the complete basis $\\left\\{\\Ket{i^{A}}\\right\\}$ and $\\left\\{\\Ket{j^{B}}\\right\\}$ of $\\mathcal{H}_{A}$ and $\\mathcal{H}_{B}$, $\\Ket{\\psi}=\\sum_{i=1}^{m}\\sum_{j=1}^{n}x_{i,j}\\Ket{i^{A}}\\otimes\\Ket{j^{B}}$. The corresponding density matrix $\\rho=\\Ket{\\psi}\\Bra{\\psi}$ has the natural constraint $\\tr(\\rho)=1$. This implies that the $m\\times n$ random coefficient matrix $\\mathbf{X}=(x_{i,j})$ satisfies\n\\begin{equation}\\label{eq:pcv}\n\\tr\\left(\\mathbf{XX}^{\\dag}\\right)=1.\n\\end{equation}\nWithout loss of generality, it is assumed that $m\\leq n$. The reduced density matrix $\\rho_{A}$ of the smaller subsystem $A$ admits the Schmidt decomposition $\\rho_{A}=\\sum_{i=1}^{m}\\lambda_{i}\\Ket{\\phi_{i}^{A}}\\Bra{\\phi_{i}^{A}}$,\nwhere $\\lambda_{i}$ is the $i$-th largest eigenvalue of $\\mathbf{XX}^{\\dag}$. The conservation of probability~(\\ref{eq:pcv}) now implies the constraint $\\sum_{i=1}^{m}\\lambda_{i}=1$. The probability measure of the random coefficient matrix $\\mathbf{X}$ is the Haar measure, where the entries are uniformly distributed over all the possible values satisfying the constraint~(\\ref{eq:pcv}). The resulting eigenvalue density of $\\mathbf{XX}^{\\dag}$ is (see, e.g., Ref.~\\onlinecite{Page1993}),\n\\begin{equation}\\label{eq:fte}\nf\\left(\\bm{\\lambda}\\right)=\\frac{\\Gamma(mn)}{c}~\\delta\\left(1-\\sum_{i=1}^{m}\\lambda_{i}\\right)\\prod_{1\\leq i<j\\leq m}\\left(\\lambda_{i}-\\lambda_{j}\\right)^{2}\\prod_{i=1}^{m}\\lambda_{i}^{n-m},\n\\end{equation}\nwhere $\\delta(\\cdot)$ is the Dirac delta function and the constant\n\\begin{equation}\\label{eq:con}\nc=\\prod_{i=1}^{m}\\Gamma(n-i+1)\\Gamma(i).\n\\end{equation}\nThe random matrix ensemble~(\\ref{eq:fte}) is also known as the (unitary) fixed-trace ensemble. The considered bipartite quantum system is a fundamental model that describes the interaction between a physical object and its environment. For example~\\cite{Page1993}, the subsystem $A$ is the black hole and the subsystem $B$ is the associated radiation field. In another example~\\cite{Majumdar}, the subsystem $A$ is a set of spins and the subsystem $B$ represents the environment of a heat bath.\n\nQuantum entanglement is the most fundamental feature in quantum mechanics. Quantum states that are highly entangled contain more information about different parts of the composite system. The degree of entanglement is measured by the entanglement entropy, which is a function of the eigenvalues of $\\mathbf{XX}^{\\dag}$. The function should monotonically increase from the separable state ($\\lambda_{1}=1$, $\\lambda_{2}=\\dots=\\lambda_{m}=0$) to the maximally-entangled state ($\\lambda_{1}=\\lambda_{2}=\\dots\\lambda_{m}=1\/m$). The most well-known entanglement entropy is the von Neumann entropy\n\\begin{equation}\\label{eq:vN}\nS=-\\sum_{i=1}^{m}\\lambda_{i}\\ln\\lambda_{i},\n\\end{equation}\nwhich achieves the separable state and maximally-entangled state when $S=0$ and when $S=\\ln{m}$, respectively. A one-parameter generalization of the von Neumann entropy is Tsallis entropy~\\cite{Tsallis98}\n\\begin{equation}\\label{eq:T}\nT=\\frac{1}{q-1}\\left(1-\\sum_{i=1}^{m}\\lambda_{i}^{q}\\right),~~~~q\\in\\mathbb{R}\\backslash\\{0\\},\n\\end{equation}\nwhich, by l'H\\^{o}pital's rule, reduces to the von Neumann entropy~(\\ref{eq:vN}) when the non-zero real parameter $q$ approaches $1$. The Tsallis entropy~(\\ref{eq:T}) achieves the separable state and maximally-entangled state when $T=0$ and $T=\\left(m^{q-1}-1\\right)\/(q-1)m^{q-1}$, respectively. The Tsallis entropy enjoys certain advantage in describing the entanglement. In particular, it overcomes the inability of the von Neumann entropy to model systems with long-range interactions~\\cite{Malacarne02}. The Tsallis entropy has the unique nonadditivity (also known as nonextensivity) property, whose physical relevance to quantum systems has been increasingly identified~\\cite{Gell-Mann}. It also has a definite concavity for any $q$, i.e., being convex for $q<0$ and concave for $q>0$.\n\nStatistical behavior of entanglement entropies can be understood from their moments. In particular, one is interested in higher moments beyond mean values (first moments) that govern the fluctuation of the entropies. In the literature, the first moment of von Neumann entropy $\\mathbb{E}_{f}\\!\\left[S\\right]$ (the subscript $f$ emphasizes that the expectation is taken over the fixed-trace ensemble~(\\ref{eq:fte})) was conjectured by Page~\\cite{Page1993}. Page's conjecture was proved independently by Foong-Kanno~\\cite{Foong1994}, S\\'{a}nchez-Ruiz~\\cite{Ruiz1995}, Sen~\\cite{Sen1996}, and Adachi-Toda-Kubotani~\\cite{Adachi2009}. Recently, an expression for the variance of von Neumann entropy $\\mathbb{V}\\!_{f}\\!\\left[S\\right]$ was conjectured by Vivo-Pato-Oshanin~\\cite{VPO16}, which was subsequently proved by the author~\\cite{Wei17}. For the Tsallis entropy, the first moment $\\mathbb{E}_{f}\\!\\left[T\\right]$ was derived by Malacarne-Mendes-Lenzi~\\cite{Malacarne02}. The task of the present work is to study the variance of Tsallis entropy $\\mathbb{V}\\!_{f}\\!\\left[T\\right]$.\n\nThe paper is organized as follows. In Sec.~\\ref{sec:var} we derive an exact variance formula of Tsallis entropy valid for $q>-1$ in terms of finite sums of terminating hypergeometric functions. As a byproduct of the result, we provide in the Appendix another proof to the recently proved~\\cite{Wei17} Vivo-Pato-Oshanin's conjecture~\\cite{VPO16}. In Sec.~\\ref{sec:spe} the derived variance formula is further simplified to explicit expressions in the special cases of quadratic entropy ($q=2$) and small subsystem dimensions ($m=2,3$). We summary the main results and point out possible approaches to study the higher moments in Sec.~\\ref{sec:dis}.\n\n\\section{Exact Variance of Tsallis Entropy}\\label{sec:var}\nSimilar to the case of von Neumann entropy~\\cite{Page1993,Wei17}, the starting point of the calculation is to convert the moments defined over the fixed-traced ensemble~(\\ref{eq:fte}) to the well-studied Laguerre ensemble, whose correlation functions are explicitly known. Before discussing the moments conversion approach, we first set up necessary definitions relevant to the Laguerre ensemble. By construction~(\\ref{eq:pcv}), the random coefficient matrix $\\mathbf{X}$ is naturally related to a Wishart matrix $\\mathbf{YY}^{\\dag}$ as\n\\begin{equation}\\label{eq:wf}\n\\mathbf{XX}^{\\dag}=\\frac{\\mathbf{YY}^{\\dag}}{\\tr\\left(\\mathbf{YY}^{\\dag}\\right)},\n\\end{equation}\nwhere $\\mathbf{Y}$ is an $m\\times n$ ($m\\leq n$) matrix of independently and identically distributed complex Gaussian entries (complex Ginibre matrix). The density of the eigenvalues $0<\\theta_{m}<\\dots<\\theta_{1}<\\infty$ of $\\mathbf{YY}^{\\dag}$ equals~\\cite{Forrester}\n\\begin{equation}\\label{eq:we}\ng\\left(\\bm{\\theta}\\right)=\\frac{1}{c}\\prod_{1\\leq i<j\\leq m}\\left(\\theta_{i}-\\theta_{j}\\right)^{2}\\prod_{i=1}^{m}\\theta_{i}^{n-m}\\e^{-\\theta_{i}},\n\\end{equation}\nwhere $c$ is the same as in~(\\ref{eq:con}) and the above ensemble is known as the Laguerre ensemble. The trace of the Wishart matrix\n\\begin{equation}\\label{eq:tr}\nr=\\tr\\left(\\mathbf{YY}^{\\dag}\\right)=\\sum_{i=1}^{m}\\theta_{i}\n\\end{equation}\nfollows a gamma distribution with the density~\\cite{VPO16}\n\\begin{equation}\\label{eq:r}\nh_{mn}(r)=\\frac{1}{\\Gamma(mn)}\\e^{-r}r^{mn-1},~~~~r\\in[0,\\infty).\n\\end{equation}\nThe relation~(\\ref{eq:wf}) induces the change of variables\n\\begin{equation}\\label{eq:cv}\n\\lambda_{i}=\\frac{\\theta_{i}}{r},~~~~i=1,\\ldots,m,\n\\end{equation}\nthat leads to a well-known relation (see, e.g., Ref.~\\onlinecite{Page1993}) among the densities~(\\ref{eq:fte}),~(\\ref{eq:we}), and~(\\ref{eq:r}) as\n\\begin{equation}\\label{eq:relation}\nf\\left(\\bm{\\lambda}\\right)h_{mn}(r)\\dd r\\prod_{i=1}^{m}\\dd\\lambda_{i}=g\\left(\\bm{\\theta}\\right)\\prod_{i=1}^{m}\\dd\\theta_{i}.\n\\end{equation}\nThis implies that $r$ is independent of each $\\lambda_{i}$, $i=1,\\ldots,m$, since their densities factorize.\n\nFor the von Neumann entropy~(\\ref{eq:vN}), the relation~(\\ref{eq:relation}) has been exploited to convert the first two moments~\\cite{Page1993,Wei17} from the fixed-trace ensemble~(\\ref{eq:fte}) to the Laguerre ensemble~(\\ref{eq:we}). The moments conversion was an essential starting point in proving the conjectures of Page~\\cite{Page1993,Ruiz1995} and Vivo-Pato-Oshanin~\\cite{Wei17}. We now show that the moments conversion approach can be also applied to study the Tsallis entropy. We first define\n\\begin{equation}\\label{eq:L}\nL=\\sum_{i=1}^{m}\\theta_{i}^{q}\n\\end{equation}\nas the induced\\footnote{For convenience of the discussion, we define the induced entropy, which may not have physical meaning of an entropy.} Tsallis entropy of the Laguerre ensemble~(\\ref{eq:we}). Using the change of variables~(\\ref{eq:cv}), the $k$-th power of Tsallis entropy~(\\ref{eq:T}) can be written as\n\\begin{equation}\nT^{k}=\\frac{1}{(q-1)^{k}}\\left(1-\\frac{L}{r^{q}}\\right)^{k}=\\frac{1}{(q-1)^{k}}\\sum_{i=0}^{k}(-1)^{i}\\binom{k}{i}\\frac{L^{i}}{r^{qi}}\n\\end{equation}\nand thus we have\n\\begin{equation}\\label{eq:fg0}\n\\mathbb{E}_{f}\\!\\left[T^{k}\\right]=\\frac{1}{(q-1)^{k}}\\sum_{i=0}^{k}(-1)^{i}\\binom{k}{i}\\mathbb{E}_{f}\\!\\left[\\frac{L^{i}}{r^{qi}}\\right].\n\\end{equation}\nThe expectation on the left hand side is computed as\n\\begin{eqnarray}\n\\mathbb{E}_{f}\\!\\left[\\frac{L^{i}}{r^{qi}}\\right]&=&\\int_{\\bm{\\lambda}}\\frac{L^{i}}{r^{qi}}f\\left(\\bm{\\lambda}\\right)\\prod_{i=1}^{m}\\dd\\lambda_{i}\\\\\n&=&\\int_{\\bm{\\lambda}}\\frac{L^{i}}{r^{qi}}f\\left(\\bm{\\lambda}\\right)\\prod_{i=1}^{m}\\dd\\lambda_{i}\\int_{r}h_{mn+qi}(r)\\dd r\\label{eq:fg1}\\\\\n&=&\\frac{\\Gamma(mn)}{\\Gamma(mn+qi)}\\int_{\\bm{\\lambda}}\\int_{r}L^{i}f\\left(\\bm{\\lambda}\\right)h_{mn}(r)\\dd r\\prod_{i=1}^{m}\\dd\\lambda_{i}\\label{eq:fg2}\\\\\n&=&\\frac{\\Gamma(mn)}{\\Gamma(mn+qi)}\\mathbb{E}_{g}\\!\\left[L^{i}\\right]\\label{eq:fg3},\n\\end{eqnarray}\nwhere the multiplication of an appropriate constant $1=\\int_{r}h_{mn+qi}(r)\\dd r$ in~(\\ref{eq:fg1}) along with the fact that $r^{-qi}h_{mn+qi}(r)=\\Gamma(mn)h_{mn}(r)\/\\Gamma(mn+qi)$ lead to~(\\ref{eq:fg2}) and the last equality~(\\ref{eq:fg3}) is established by the change of measures~(\\ref{eq:relation}). Inserting~(\\ref{eq:fg3}) into~(\\ref{eq:fg0}), the $k$-th moment of Tsallis entropy~(\\ref{eq:T}) is written as a sum involving the first $k$ moments of the induced Tsallis entropy~(\\ref{eq:L}) as\n\\begin{equation}\\label{eq:TL}\n\\mathbb{E}_{f}\\!\\left[T^{k}\\right]=\\frac{\\Gamma(mn)}{(q-1)^{k}}\\sum_{i=0}^{k}\\binom{k}{i}\\frac{(-1)^{i}}{\\Gamma(mn+qi)}\\mathbb{E}_{g}\\!\\left[L^{i}\\right].\n\\end{equation}\nWith the above relation~(\\ref{eq:TL}), the computation of moments over the less tractable correlation functions of the fixed-trace ensemble~(\\ref{eq:fte}) is now converted the one over the Laguerre ensemble~(\\ref{eq:we}), which will be calculated explicitly. In particular, computing the variance $\\mathbb{V}\\!_{f}\\!\\left[T\\right]=\\mathbb{E}_{f}\\!\\left[T^2\\right]-\\mathbb{E}_{f}^{2}\\!\\left[T\\right]$ requires the moments relation~(\\ref{eq:TL}) for $k=1$,\n\\begin{equation}\\label{eq:TL1}\n\\mathbb{E}_{f}\\!\\left[T\\right]=\\frac{1}{q-1}\\left(1-\\frac{\\Gamma(mn)}{\\Gamma(mn+q)}\\mathbb{E}_{g}\\!\\left[L\\right]\\right)\n\\end{equation}\nand $k=2$,\n\\begin{equation}\\label{eq:TL2}\n\\mathbb{E}_{f}\\!\\left[T^{2}\\right]=\\frac{1}{(q-1)^{2}}\\left(1-\\frac{2\\Gamma(mn)}{\\Gamma(mn+q)}\\mathbb{E}_{g}\\!\\left[L\\right]+\\frac{\\Gamma(mn)}{\\Gamma(mn+2q)}\\mathbb{E}_{g}\\!\\left[L^{2}\\right]\\right),\n\\end{equation}\nwhere the first moment relation~(\\ref{eq:TL1}) has also appeared in Ref.~\\onlinecite{Malacarne02}. It is seen from~(\\ref{eq:TL2}) that the essential task now is to compute $\\mathbb{E}_{g}\\!\\left[L\\right]$ and $\\mathbb{E}_{g}\\!\\left[L^{2}\\right]$. Before proceeding to the calculation, we point out that in the limit $q\\to1$ the derived second moments relation~(\\ref{eq:TL2}) leads to a new proof to the recently proved variance formula of von Neumann entropy~\\cite{Wei17} with details provided in the Appendix.\n\nThe computation of $\\mathbb{E}_{g}\\!\\left[L\\right]$ and $\\mathbb{E}_{g}\\!\\left[L^{2}\\right]$ involves one and two arbitrary eigenvalue densities, denoted respectively by $g_{1}(x_{1})$ and $g_{2}(x_{1},x_{2})$, of the Laguerre ensemble as\n\\begin{eqnarray}\n\\mathbb{E}_{g}\\!\\left[L\\right]&=&m\\int_{0}^{\\infty}\\!\\!x_{1}^{q}~g_{1}(x_{1})\\dd x_{1},\\label{eq:L1}\\\\\n\\mathbb{E}_{g}\\!\\left[L^{2}\\right]&=&m\\int_{0}^{\\infty}\\!\\!x_{1}^{2q}~g_{1}(x_{1})\\dd x_{1}+m(m-1)\\int_{0}^{\\infty}\\!\\!\\int_{0}^{\\infty}\\!\\!x_{1}^{q}x_{2}^{q}~g_{2}\\left(x_{1},x_{2}\\right)\\dd x_{1}\\dd x_{2}.\\label{eq:L2}\n\\end{eqnarray}\nIn general, the joint density of $N$ arbitrary eigenvalues $g_{N}(x_{1},\\dots,x_{N})$ is related to the $N$-point correlation function\n\\begin{equation}\\label{eq:cf}\nX_{N}\\left(x_{1},\\dots,x_{N}\\right)=\\det\\left(K\\left(x_{i},x_{j}\\right)\\right)_{i,j=1}^{N}\n\\end{equation}\nas~\\cite{Forrester} $g_{N}(x_{1},\\dots,x_{N})=X_{N}\\left(x_{1},\\dots,x_{N}\\right)(m-N)!\/m!$, where $\\det(\\cdot)$ is the matrix determinant and the symmetric function $K(x_{i},x_{j})$ is the correlation kernel. In particular, we have\n\\begin{eqnarray}\ng_{1}(x_{1})&=&\\frac{1}{m}K(x_{1},x_{1}),\\label{eq:x}\\\\\ng_{2}(x_{1},x_{2})&=&\\frac{1}{m(m-1)}\\left(K(x_{1},x_{1})K(x_{2},x_{2})-K^{2}(x_{1},x_{2})\\right),\\label{eq:xy}\n\\end{eqnarray}\nand the correlation kernel $K(x_{i},x_{j})$ of the Laguerre ensemble can be explicitly written as~\\cite{Forrester}\n\\begin{equation}\\label{eq:ker}\nK(x_{i},x_{j})=\\sqrt{\\e^{-x_{i}-x_{j}}(x_{i}x_{j})^{n-m}}\\sum_{k=0}^{m-1}\\frac{C_{k}(x_{i})C_{k}(x_{j})}{k!(n-m+k)!},\n\\end{equation}\nwhere\n\\begin{equation}\nC_{k}(x)=(-1)^{k}k!L_{k}^{(n-m)}(x)\n\\end{equation}\nwith\n\\begin{equation}\nL_{k}^{(n-m)}(x)=\\sum_{i=0}^{k}(-1)^{i}\\binom{n-m+k}{k-i}\\frac{x^i}{i!}\n\\end{equation}\nbeing the (generalized) Laguerre polynomial of degree $k$. The Laguerre polynomials satisfy the orthogonality relation~\\cite{Forrester}\n\\begin{equation}\\label{eq:oc}\n\\int_{0}^{\\infty}\\!\\!x^{n-m}\\e^{-x}L_{k}^{(n-m)}(x)L_{l}^{(n-m)}(x)\\dd{x}=\\frac{(n-m+k)!}{k!}\\delta_{kl},\n\\end{equation}\nwhere $\\delta_{kl}$ is the Kronecker delta function. It is known that the one-point correlation function admits a more convenient representation as~\\cite{Ruiz1995,Forrester}\n\\begin{equation}\\label{eq:one}\nX_{1}(x)=K(x,x)=\\frac{m!}{(n-1)!}x^{n-m}\\e^{-x}\\left(\\left(L_{m-1}^{(n-m+1)}(x)\\right)^{2}-L_{m-2}^{(n-m+1)}(x)L_{m}^{(n-m+1)}(x)\\right).\n\\end{equation}\nWe also need the following integral identity, due to Schr{\\\"{o}}dinger~\\cite{Schrodinger1926}, that generalizes the integral~(\\ref{eq:oc}) to\n\\begin{eqnarray}\\label{eq:Swm}\nA_{s,t}^{(\\alpha,\\beta)}(q)&=&\\int_{0}^{\\infty}\\!\\!x^{q}\\e^{-x}L_{s}^{(\\alpha)}(x)L_{t}^{(\\beta)}(x)\\dd{x}\\nonumber\\\\\n&=&(-1)^{s+t}\\sum_{k=0}^{\\min(s,t)}\\binom{q-\\alpha}{s-k}\\binom{q-\\beta}{t-k}\\frac{\\Gamma(k+q+1)}{k!},~~~~q>-1.\n\\end{eqnarray}\n\n\nWith the above preparation, we now proceed to the calculation of $\\mathbb{E}_{g}\\!\\left[L\\right]$ and $\\mathbb{E}_{g}\\!\\left[L^{2}\\right]$. Inserting~(\\ref{eq:x}) and~(\\ref{eq:one}) into~(\\ref{eq:L1}), and defining further\n\\begin{equation}\n\\mathcal{A}_{s,t}=A_{s,t}^{(n-m+1,n-m+1)}(n-m+q),\n\\end{equation}\none obtains by using~(\\ref{eq:Swm}) that\n\\begin{eqnarray}\n\\mathbb{E}_{g}\\!\\left[L\\right]&=&\\frac{m!}{(n-1)!}\\left(\\mathcal{A}_{m-1,m-1}-\\mathcal{A}_{m-2,m}\\right)\\\\\n&=&\\frac{m!}{(n-1)!}\\Bigg(\\sum_{k=0}^{m-1}\\binom{q-1}{m-k-1}^{2}\\frac{\\Gamma(n-m+q+k+1)}{k!}-\\nonumber\\\\\n&&\\sum_{k=0}^{m-2}\\binom{q-1}{m-k-2}\\binom{q-1}{m-k}\\frac{\\Gamma(n-m+q+k+1)}{k!}\\Bigg),\\nonumber\n\\end{eqnarray}\nwhich is valid for $q>-1$. The first moment expression in the above form has been obtained in Ref.~\\onlinecite{Malacarne02} and we continue to show that it can be compactly written as a terminating hypergeometric function of unit argument. Indeed, since\n\\begin{equation}\n\\binom{q-1}{-1}\\binom{q-1}{1}\\frac{\\Gamma(n+q)}{(m-1)!}=0,~~~~q>-1\\backslash\\{0\\},\n\\end{equation}\nwe have\n\\begin{eqnarray}\n\\mathbb{E}_{g}\\!\\left[L\\right]&=&\\frac{m!}{(n-1)!}\\sum_{k=0}^{m-1}\\frac{\\Gamma(n-m+q+k+1)}{k!}\\left(\\binom{q-1}{m-k-1}^{2}-\\binom{q-1}{m-k-2}\\binom{q-1}{m-k}\\right)\\\\\n&=&\\frac{m!\\Gamma^{2}(q)}{(n-1)!}\\sum_{k=0}^{m-1}\\frac{\\Gamma(n+q-k)}{(m-k-1)!}\\left(\\frac{1}{k!^{2}\\Gamma^{2}(q-k)}-\\frac{1}{(k-1)!(k+1)!\\Gamma(q-k+1)\\Gamma(q-k-1)}\\right)\\nonumber\\\\\n&=&\\frac{m!\\Gamma(q+1)\\Gamma(q)}{(n-1)!}\\sum_{k=0}^{m-1}\\frac{\\Gamma(n+q-k)}{(m-k-1)!\\Gamma(q-k+1)\\Gamma(q-k)k!(k+1)!}\\\\\n&=&\\frac{m!\\Gamma(q+1)\\Gamma(q)}{(n-1)!}\\frac{\\Gamma(n+q)}{(m-1)!\\Gamma(q+1)\\Gamma(q)}\\sum_{k=0}^{m-1}\\frac{(1-m)_{k}(-q)_{k}(1-q)_{k}}{(1-n-q)_{k}(2)_{k}}\\frac{1}{k!}\\label{eq:Lf2}\\\\\n&=&\\frac{m\\Gamma(n+q)}{(n-1)!}~\\!\\pFq[4]{3}{2}{1-m,-q,1-q}{1-n-q,2}{1},~~~~q>-1\\backslash\\{0\\},\\label{eq:Lf3}\n\\end{eqnarray}\nwhere the second equality follows from the change of variable $k\\to m-1-k$, (\\ref{eq:Lf2}) is obtained by repeated use of the identity\n\\begin{equation}\\label{eq:s2h}\n\\Gamma(m-k)=\\frac{(-1)^{k}}{(1-m)_{k}}\\Gamma(m)\n\\end{equation}\nwith $(a)_{n}=\\Gamma(a+n)\/\\Gamma(a)$ being the Pochhammer's symbol, and~(\\ref{eq:Lf3}) is obtained by the series definition of hypergeometric function\n\\begin{equation}\n\\pFq[4]{p}{q}{a_{1},\\dots,a_{p}}{b_{1},\\dots,b_{q}}{z}=\\sum_{k=0}^{\\infty}\\frac{(a_{1})_{k}\\dots(a_{p})_{k}}{(b_{1})_{k}\\dots(b_{q})_{k}}\\frac{z^{k}}{k!}\n\\end{equation}\nthat reduces to a finite sum if one of the parameters $a_{i}$ is a negative integer. Inserting~(\\ref{eq:Lf3}) into~(\\ref{eq:TL1}), we arrive at a compact expression for the first moment of Tsallis entropy as\n\\begin{equation}\n\\mathbb{E}_{f}\\!\\left[T\\right]=\\frac{1}{q-1}\\left(1-\\frac{m(mn-1)!\\Gamma(n+q)}{(n-1)!\\Gamma(mn+q)}~\\!\\pFq[4]{3}{2}{1-m,-q,1-q}{1-n-q,2}{1}\\right),~~~~q>-1\\backslash\\{0\\}.\n\\end{equation}\n\nWe now calculate $\\mathbb{E}_{g}\\!\\left[L^{2}\\right]$. Inserting~(\\ref{eq:x}) and~(\\ref{eq:xy}) into~(\\ref{eq:L2}), one has\n\\begin{equation}\\label{eq:L2s}\n\\mathbb{E}_{g}\\!\\left[L^{2}\\right]=I_{1}-I_{2}+\\frac{m^{2}\\Gamma^{2}(n+q)}{(n-1)!^{2}}~\\!\\pFq[4]{3}{2}{1-m,-q,1-q}{1-n-q,2}{1}^{2},\n\\end{equation}\nwhere\n\\begin{eqnarray}\nI_{1}&=&\\int_{0}^{\\infty}\\!\\!x_{1}^{2q}~K(x_{1},x_{1})\\dd x_{1},\\\\\nI_{2}&=&\\int_{0}^{\\infty}\\!\\!\\int_{0}^{\\infty}\\!\\!x_{1}^{q}x_{2}^{q}~K^{2}\\left(x_{1},x_{2}\\right)\\dd x_{1}\\dd x_{2},\\label{eq:iI2}\n\\end{eqnarray}\nand we have used the result~(\\ref{eq:Lf3}) with the fact that\n\\begin{equation}\n\\int_{0}^{\\infty}\\!\\!x^{q}K(x,x)\\dd{x}=\\mathbb{E}_{g}\\!\\left[L\\right].\n\\end{equation}\nThe integral $I_{1}$ can be read off from the steps that led to~(\\ref{eq:Lf3}) by replacing $q$ with $2q$ as\n\\begin{equation}\\label{eq:I1}\nI_{1}=\\frac{m\\Gamma(n+2q)}{(n-1)!}\\pFq[4]{3}{2}{1-m,-2q,1-2q}{1-n-2q,2}{1}.\n\\end{equation}\nInserting~(\\ref{eq:ker}) into~(\\ref{eq:iI2}), and defining further (cf.~(\\ref{eq:Swm}))\n\\begin{equation}\n\\mathbb{A}_{s,t}=A_{s,t}^{(n-m,n-m)}(n-m+q),\n\\end{equation}\nthe integral $I_{2}$ is written as\n\\begin{equation}\\label{eq:I2}\nI_{2}=\\sum_{k=0}^{m-1}\\frac{k!^{2}\\mathbb{A}_{k,k}^{2}}{(n-m+k)!^2}+2\\sum_{j=1}^{m-1}\\sum_{i=0}^{j-1}\\frac{i!j!\\mathbb{A}_{i,j}^{2}}{(n-m+i)!(n-m+j)!},\n\\end{equation}\nwhere by using~(\\ref{eq:Swm}) and~(\\ref{eq:s2h}) we obtain\n\\begin{eqnarray}\n\\mathbb{A}_{i,j}&=&\\Gamma^{2}(q+1)\\sum_{k=0}^{i}\\frac{\\Gamma(n-m+q+k+1)}{\\Gamma(q-i+k+1)\\Gamma(q-j+k+1)(i-k)!(j-k)!k!}\\\\\n&=&\\frac{\\Gamma(n-m+q+1)}{\\Gamma(q-i+1)\\Gamma(q-j+1)i!j!}\\sum_{k=0}^{i}\\frac{(-i)_{k}(-j)_{k}(n-m+q+1)_{k}}{(q-i+1)_{k}(q-j+1)_{k}}\\frac{1}{k!}\\label{eq:1I2s}\\\\\n&=&\\frac{\\Gamma(n-m+q+1)}{\\Gamma(q-i+1)\\Gamma(q-j+1)i!j!}\\pFq[4]{3}{2}{-i,-j,n-m+q+1}{q-i+1,q-j+1}{1},\\label{eq:1I2}\n\\end{eqnarray}\nand similarly\n\\begin{equation}\\label{eq:2I2}\n\\mathbb{A}_{k,k}=\\frac{\\Gamma(n-m+q+1)}{\\Gamma^{2}(q-k+1)k!^{2}}\\pFq[4]{3}{2}{-k,-k,n-m+q+1}{q-k+1,q-k+1}{1}.\n\\end{equation}\nFinally, by inserting~(\\ref{eq:I1}), (\\ref{eq:I2}), (\\ref{eq:1I2}), and (\\ref{eq:2I2}) into~(\\ref{eq:L2s}) we arrive at\n\\begin{eqnarray}\\label{eq:L2F}\n\\mathbb{E}_{g}\\!\\left[L^{2}\\right]&=&\\frac{m^{2}\\Gamma^{2}(n+q)}{(n-1)!^{2}}~\\!\\pFq[4]{3}{2}{1-m,-q,1-q}{1-n-q,2}{1}^{2}+\\frac{m\\Gamma(n+2q)}{(n-1)!}\\pFq[4]{3}{2}{1-m,-2q,1-2q}{1-n-2q,2}{1}\\nonumber\\\\\n&&-\\Gamma^{4}(q+1)\\Gamma^{2}(n-m+q+1)\\left(\\sum_{i=0}^{m-1}L^{2}(i,i)+2\\sum_{j=1}^{m-1}\\sum_{i=0}^{j-1}L^{2}(i,j)\\!\\right),~q>-1\\backslash\\{0\\},\n\\end{eqnarray}\nwhere the symmetric function $L(i,j)=L(j,i)$ is\n\\begin{equation}\nL(i,j)=\\frac{\\pFq[4]{3}{2}{-i,-j,n-m+q+1}{q-i+1,q-j+1}{1}}{\\Gamma(q-i+1)\\Gamma(q-j+1)\\sqrt{i!j!(n-m+i)!(n-m+j)!}}.\n\\end{equation}\nWith the derived first two moments~(\\ref{eq:Lf3}),~(\\ref{eq:L2F}) and the relations~(\\ref{eq:TL1}),~(\\ref{eq:TL2}), an exact variance formula of Tsallis entropy is obtained.\n\n\\section{Special Cases}\\label{sec:spe}\nThough the derived results~(\\ref{eq:Lf3}),~(\\ref{eq:L2F}) may not be further simplified for an arbitrary $m$, $n$, and $q$, we will show that explicit variance expressions can be obtained in some special cases of practical relevance.\n\n\\subsection{Quadratic Entropy $q=2$}\nIn the special case $q=2$, the Tsallis entropy~(\\ref{eq:T}) reduces to the quadratic entropy\n\\begin{equation}\\label{eq:T2}\nT=1-\\sum_{i=1}^{m}\\lambda_{i}^{2},\n\\end{equation}\nwhich was first considered in physics by Fermi~\\cite{Malacarne02}. The quadratic entropy~(\\ref{eq:T2}) is the only entropy among all possible $q$ values that satisfies the information invariance and continuity criterion~\\cite{Brukner09}.\n\nBy the series representations~(\\ref{eq:Lf2}) and~(\\ref{eq:1I2s}), the first two moments in the case $q=2$ are directly computed as\n\\begin{eqnarray}\n\\mathbb{E}_{g}\\!\\left[L\\right]&=&mn(m+n),\\\\\n\\mathbb{E}_{g}\\!\\left[L^{2}\\right]&=&mn\\left(mn^{3}+2m^{2}n^{2}+4n^{2}+m^{3}n+10mn+4m^{2}+2\\right).\n\\end{eqnarray}\nBy~(\\ref{eq:TL1}) and~(\\ref{eq:TL2}), we immediately have\n\\begin{eqnarray}\n\\!\\!\\!\\!\\!\\!\\!\\mathbb{E}_{f}\\!\\left[T\\right]&=&\\frac{mn-m-n+1}{mn+1},\\label{eq:ETq2}\\\\\n\\!\\!\\!\\!\\!\\!\\!\\mathbb{E}_{f}\\!\\left[T^{2}\\right]&=&\\frac{(m-1)(n-1)}{(mn+1)(mn+2)(mn+3)}\\left(m^{2}n^{2}-mn^{2}-m^{2}n+5mn-4n-4m+8\\right),\n\\end{eqnarray}\nwhich lead to the variance of Tsallis entropy for $q=2$ as\n\\begin{equation}\\label{eq:q2}\n\\mathbb{V}\\!_{f}\\!\\left[T\\right]=\\frac{2\\left(m^{2}-1\\right)\\left(n^{2}-1\\right)}{\\left(mn+1\\right)^2\\left(mn+2\\right)\\left(mn+3\\right)}.\n\\end{equation}\nFinally, we note that explicit variance expressions for other positive integer values of $q$ can be similarly obtained.\n\n\\subsection{Subsystems of Dimensions $m=2$ and $m=3$}\nWe now consider the cases when dimensions $m$ of the smaller subsystems are small. This is a relevant scenario for subsystems consisting of, for example, only a few entangled particles~\\cite{Majumdar}. For $m=2$ with any $n$ and $q$, the series representations~(\\ref{eq:Lf2}) and~(\\ref{eq:1I2s}) directly lead to the results\n\\begin{eqnarray}\n\\mathbb{E}_{g}\\!\\left[L\\right]&=&\\frac{q^{2}+q+2n-2}{(n-1)!}\\Gamma(q+n-1),\\\\\n\\mathbb{E}_{g}\\!\\left[L^{2}\\right]&=&\\frac{2}{(n-1)!}\\left(\\frac{\\Gamma(q+n-1)\\Gamma(q+n)}{(n-2)!}+\\left(2q^{2}+q+n-1\\right)\\Gamma(2q+n-1)\\right).\\label{eq:m2}\n\\end{eqnarray}\nIn the same manner, for $m=3$ with any $n$ and $q$ we obtain\n\\begin{eqnarray}\n\\mathbb{E}_{g}\\!\\left[L\\right]&=&\\frac{6n\\left(q^{2}+q-3\\right)+(q-2)(q-1)(q+2)(q+3)+6n^{2}}{2(n-1)!}\\Gamma(q+n-2),\\\\\n\\mathbb{E}_{g}\\!\\left[L^{2}\\right]&=&\\frac{6n(q^2+q-3)+q^4+4q^3-7q^2-10q+12+6n^2}{(n-1)!(n-2)!}\\Gamma(q+n-2)\\Gamma(q+n-1)+\\nonumber\\\\\n&&\\frac{3n(4q^2+2q-3)+8q^4+8q^3-14q^2-8q+6+3n^2}{(n-1)!}\\Gamma(2q+n-2).\\label{eq:m3}\n\\end{eqnarray}\nThe corresponding variances are obtained by keeping in mind the relations~(\\ref{eq:TL1}) and~(\\ref{eq:TL2}). For $m\\geq4$, explicit variance expressions can be similarly calculated. However, it does not seem promising to find an explicit variance formula valid for any $m$, $n$, and $q$.\n\n\\section{Summary and Perspectives on Higher Moments}\\label{sec:dis}\nWe studied the exact variance of Tsallis entropy, which is a one parameter ($q$) generalization of the von Neumann entropy. The main result is an exact variance expression~(\\ref{eq:L2F}) valid for $q>-1$ as finite sums of terminating hypergeometric functions. For $q=1$, we find a short proof to the variance formula of the degenerate case of von Neumann entropy in the Appendix. For other special cases of practical importance of $q=2$, $m=2$, and $m=3$, explicit variance expressions have been obtained in~(\\ref{eq:q2}),~(\\ref{eq:m2}), and~(\\ref{eq:m3}), respectively.\n\nWe end this paper with some perspectives on the higher moments of Tsallis entropy. In principle, the higher moments can be calculated by integrating over the correlation kernel~(\\ref{eq:ker}) as demonstrated for the first two moments. In practice, the calculation becomes progressively complicated as the order of moments increases. Here, we outline an alternative path that may systematically lead to the moments of any order in a recursive manner.\nWe focus on the induced Tsallis entropy $L$ as defined in~(\\ref{eq:L}) since the moments conversion is available~(\\ref{eq:TL}). The starting point is the generating function of $L$\n\\begin{eqnarray}\n\\tau_{m}(t,q)=\\mathbb{E}_{g}\\!\\left[\\e^{tL}\\right]&=&\\frac{1}{c}\\int_{0}^{\\infty}\\!\\!\\cdots\\!\\!\\int_{0}^{\\infty}\\!\\prod_{1\\leq i<j\\leq m}\\left(\\theta_{i}-\\theta_{j}\\right)^{2}\\prod_{i=1}^{m}\\theta_{i}^{n-m}\\e^{-\\theta_{i}+t\\theta^{q}_{i}}\\dd\\theta_{i}\\label{eq:gT}\\\\\n&=&\\frac{1}{c}\\det\\left(\\int_{0}^{\\infty}x^{i+j+n-m-2}\\e^{-x+tx^{q}}\\dd x\\right)_{i,j=1}^{m}\\label{eq:gT1},\n\\end{eqnarray}\nwhich is a two-parameter ($t$ and $q$) deformation of the Laguerre ensemble~(\\ref{eq:we}). Compared to the weight function $w(x)=x^{n-m}\\e^{-x}$ of the Laguerre ensemble, the deformation induces a new weight function\n\\begin{equation}\\label{eq:nw}\nw(x)=x^{n-m}\\e^{-x+tx^{q}},\n\\end{equation}\nwhich generalizes the Toda deformation~\\cite{Ismail} $w(x)=x^{n-m}\\e^{-x+tx}$ with the parameter $q$. The basic idea to systematically produce the moments is to find some differential and difference equations of the generating function $\\tau_{m}(t,q)$. The theory of integrable systems~\\cite{Forrester} may provide the possibility to obtain differential equations for the Hankel determinant~(\\ref{eq:gT1}) with respect to continuous variables $t$ and $q$ as well as difference equations with respect to the discrete variable $m$. In particular, when $q$ is a positive integer the deformation~(\\ref{eq:nw}) is known as multi-time Toda deformation~\\cite{Ismail}, where much of the integrable structure is known~\\cite{Ismail}.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nRecently, \\citet{piatti2021b}  obtained a rotating disc for \nSmall Magellanic Cloud (SMC) star clusters using available radial velocities, \nderived proper motions, and positions of 25 clusters with ages between $\\sim$ 1 \nand 10 Gyr old. He computed the three rotational velocity components \n\\citep{vdmareletal2002} of each cluster for 5.3$\\times$10$^8$ possible combinations \nof values of right ascension and declination of its centre, of heliocentric distance of \nits centre, of radial velocity and proper motions in right ascension and declination, of \ninclination of the disc, of position angle of the line-of-nodes, of rotational velocity, \nand of velocity dispersions along the three independent axes of motion.\nThe resulting  rotating disc, represented by a combination of parameter values that\nbest resembled the observed clusters' movements, turned out to be very\nsimilar to that best-fitted by \\citet{zivicketal2020} for field SMC giants \n\\citep[see, also][]{niederhoferetal2021,deleoetal2020,gaiaetal2020b}. This means\nthat both galaxy components are kinematically synchronized at some level for\ndistances smaller than $\\sim$ 2 kpc from the SMC centre.\n\n\\newpage\n\nThe SMC clusters' motions derived by \\citet{piatti2021b} show also some\nnoticeable scatter around the obtained representative rotating disc, which\nis not caused by the considerable range of the clusters' ages used. \nInstead, \\citet{piatti2021b} found that the velocity dispersion plays an important role\nin dealing with such a scatter around the mean rotation curve. Particularly,\nhe found that velocity dispersions of 50, 20, and 25 km\/s along the right\nascension, declination, and line-of-sight axes, respectively, reasonably\nrepresent the observed velocity dispersion. This finding reveals that a rotating\ndisc is not enough to describe the SMC clusters' kinematics and therefore,\nthere exists clusters with motions that depart from that of  the rotation on a plane\naround the SMC centre.  According to \\citet{piatti2021b}, the  rotating disc is \nroughly  a tilted edge-on disc  (disc inclination = (70.0$\\pm$10.0)$\\degr$, \nposition angle of the line-of-nodes = (200.0$\\pm$30.0)$\\degr$), with a stretching along the direction perpendicular\nto it, that nearly coincides with that of the Magellanic Clouds connecting bridge.\n\nIt has been shown that the SMC is under tidal effects by the interaction with the\nLarge Magellanic Cloud \\citep[LMC,][]{pieresetal2017,zivicketal2018,mackeyetal2018,deleoetal2020,niederhoferetal2021,omkumaretal2021}. \\citet{beslaetal2012} showed that the observed irregular morphology \nand internal kinematics of the Magellanic System (in gas and stars) are naturally \nexplained by interactions between the LMC and SMC, rather than gravitational \ninteractions with the Milky Way. They examined the gas and stellar kinematic centres \nof the LMC; the warped LMC old stellar disc and bar; the gaseous arms stripped out \nof the LMC by the SMC in the direction of the Magellanic Bridge; the stellar debris \nfrom the SMC seen in the LMC disc field; etc., to strongly reinforce the suspicions of \n\\citet{dVF1972} that the interaction with the Milky Way is not responsible for the \nLMC\/SMC's morphology. Moreover, these conclusions provide further support that the Magellanic Clouds are completing their first infall to the Milky Way. Hence, a question arises unavoidably: Are SMC tidal structures\nkinematically different from that of the SMC rotating disc? or in other words, do \nkinematically perturbed clusters have any kinematic signatures that differentiate them from \nbound clusters? \\citet{diasetal2021} observed clusters located in a branch of the\nMagellanic Bridge, placed toward the northeastern outskirts of the SMC; in the SMC\nmain body; and in the so-called Counter-Brigde, that is  nearly superimpose in the sky \nto that  Magellanic Bridge branch. They estimated heliocentric distances, radial \nvelocities, and proper motions for a sample of 7 clusters that can be useful, once\nthey are converted into space velocity components, to probe the existence of any \ndistinguishable kinematic features between clusters belonging to these three different\ngalaxy structures, two of them (Magellanic Bridge and Counter-Bridge) originated by \ntidal interaction  with the LMC \\citep{db2012,beslaetal2016}. \n\nIn this work, we show that there exist different space residual velocities  for\nclusters that are spatially distributed throughout SMC tidal structures \nand those bound to the SMC main body. This outcome allowed us to propose\na simple kinematic criterion to assess at first glance whether a SMC cluster is\nbound to the galaxy or belong to any  kinematically perturbed stellar component. In Section\n2 we describe the gathered data employed in this work, while in Section\n3 we analyze and discuss the kinematic characteristics of SMC clusters distributed\nat projected distances in the sky between $\\sim$ 1 kpc and 4 kpc from the SMC \ncentre.\n\n\\section{Data gathering}\n\nWe used the $v_1$, $v_2$, and $v_3$ space velocity components according to\n\\citet{vdmareletal2002} of 25 SMC clusters analyzed by \\citet{piatti2021b} and\nthose of the 7 clusters studied by \\citet{diasetal2021}. As far as we are aware,\nthis is the largest sample of SMC clusters with derived space velocity components.\nThey were calculated from measured radial velocities and proper motions, and the\ntransformation equations (9), (13), and (21) in \n\\citet{vdmareletal2002}\\citep[see, also][]{piattietal2019}. We started from the\nhypothesis of, if a cluster has been perturbed by the LMC, its circular motion should \nchange, and in the process of escaping from the SMC, its velocity behavior \nwill be measurably different. In order to avoid projection effects of the clusters' motions\nspace velocity components are suitable; inspecting proper motions or radial\nvelocities could be misleading.\n\nWe also searched the literature for accurate heliocentric distances of the\ncluster sample. Unfortunately, only the 7 clusters studied by \\citet{diasetal2021}\n(B\\,168, BS\\,188, BS\\,196, HW\\,56, HW\\,85, IC\\,1708, L1) and 12 clusters\nanalyzed by \\citet{piatti2021b} (K\\,1, K\\,3, K\\,4, K\\,8, K\\,44, L\\,1, L\\,110, L\\,113, \nNGC\\,121, NGC\\,339, NGC\\,361, NGC\\,419) have available distances. Their\nvalues and corresponding uncertainties were taken from \\citet{cetal01,glattetal2008a,diasetal2016,paetal14,mvetal2021}; and \n\\citet{diasetal2021}. We also computed the position angles of the clusters\nfrom north to east.  Table~\\ref{fig1} lists the values of position angles and\ndistances adopted, alongside with their respective references.\n\nFrom a careful reading of the references used to compile SMC clusters' distances, it \nturns out to be that the estimation of distances of SMC clusters has not been\ntargeted by any observational campaign yet. On the contrary, there is a very limited \nnumber of SMC clusters with distance measurements. Most of them have been obtained \nfrom Padova's isochrone fitting using very sophisticated statistical methods \n\\citep[see, e.g.][]{diasetal2016,diasetal2021} and\/or considering spectroscopic \nmetallicities \\citep[see, e.g.][]{glattetal2008a,paetal14,diasetal2021}.\nThe compiled distances are the most valuable treasure available up-to-date to carry out \nthe analysis described in Section\\,3, whose results must be interpreted as those\nbased  on the available cluster sample. The outcomes found in \nthis work are pioneers to uncover an SMC picture that allows us to question about the \nnature and consequences of the interactions between both Magellanic Clouds. Precisely, \nthis pioneering approach  anticipates further studies that will be hopefully carried out from observational  data obtained from the new generation telescopes.\n\nTable~\\ref{tab:tab1} indicates the method employed to estimate the adopted distances, namely:\ntheoretical isochrone fits to the cluster colour-magnitude diagrams, magnitudes of the cluster red giant clumps, \nor the Luminosity-Period relationship for $\\delta$ Sct stars. We adopted here as a reference distance\nscale that of the Padova theoretical isochrones \\citep{betal12}, because $\\sim$ 65$\\%$ of the cluster\nsample have distance estimates that rely on those isochrones. Using 7 clusters in our sample with distances\nestimated from fitting of Teramo \\citep{pietrinfernietal2004} and Dartmouth \\citep{dotteretal2007} isochrones, we found\na mean difference in the distance modulus of (Teramo - Dartmouth) of 0.009$\\pm$0.021 mag, and for\n4 clusters in common in our sample with distances from Padova and Teramo isochrone fits, a difference in the distance modulus of -0.008$\\pm$0.020 mag.\n We note that a difference of 0.01 mag in the distance\nmodulus implies a distance difference of 0.27 kpc at a distance of 60 kpc. Therefore, we assumed that\nthere is no offsets between distances obtained from these sets of theoretical isochrones that could affect\n the subsequent analysis. As for distances derived from estimation of the magnitude of the Red Clump, \n we found 4 clusters \\citep[K\\,3, L\\,1, NGC\\,121, 339;][]{cetal01} with distances estimated using the Red Clump \n method and  Padova isochrone fits (see Table~\\ref{tab:tab1}) and the mean difference turned our to be \n (Red Clump - Padova iso) = -2.5$\\pm$1.2 kpc. Therefore, we corrected the distances of K\\,44, L\\,113 and\n NGC\\,361 by adding 2.5 kpc to the mean values listed in Table~\\ref{tab:tab1}. As for the distance of\n NGC\\,419, we found a mean difference of 0.1 kpc with respect to the value estimated from Padova\n isochrone fits \\citep{glattetal2008a}.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig1.png}\n\\caption{Spatial distribution of SMC clusters compiled in \\citet{bicaetal2020} \n(gray dots).  Coloured circles and stars correspond to clusters with \nresidual velocities smaller and larger than 60 km\/s.}\n\\label{fig1}\n\\end{figure}\n\n\\begin{table*}\n\\caption{Properties of SMC star clusters.}\n\\label{tab:tab1}\n\\begin{tabular}{@{}lccccc}\\hline\nID & PA (deg) & $\\Delta$$V$ (km\/s) & $D$ (kpc)  & Method & Ref.\\\\\\hline\nBruck168 & 50.8\t& 29.2$\\pm$49.4\t&61.9$\\pm$2.1\t &Padova iso\t&6\\\\\nBS188 \t&67.9\t& 115.7$\\pm$40.5\t& 52.7$\\pm$3.0 &Padova iso\t&6\\\\\nBS196 \t&56.9\t& 80.6$\\pm$35.7\t& 50.1$\\pm$2.0 &Padova iso\t&6\\\\\nHW56 \t&30.3\t& 63.8$\\pm$40.0\t& 53.5$\\pm$1.2 &Padova iso\t&6\\\\\nHW85 \t&65.6\t& 112.1$\\pm$44.0\t& 54.0$\\pm$1.6 &Padova iso\t&6\\\\\nIC1708 \t&54.0\t& 145.5$\\pm$48.5\t& 65.2$\\pm$1.5 &Padova iso\t&6\\\\\nK1\t\t&247.2\t& 73.1$\\pm$34.2\t& 53.7$\\pm$2.4 &Teramo \\& Dartmouth iso\t&4\\\\\nK3 \t\t&270.9\t& 14.6$\\pm$27.0\t& 60.6$\\pm$1.1 &Padova, Teramo \\& Dartmouth iso\t&2\\\\\nK4 \t\t&248.0\t& 24.4$\\pm$37.2\t& 54.9$\\pm$2.3 &Teramo \\& Dartmouth iso\t&4\\\\\nK8 \t\t&255.0\t& 99.2$\\pm$24.0\t& 69.8$\\pm$2.3 &Padova iso\t&3\\\\\nK44 \t\t&149.5\t& 48.0$\\pm$32.2\t& 62.2$\\pm$2.7 &Red Clump\t&1\\\\\nL1 \t\t&259.5\t& 129.2$\\pm$20.7\t& 56.9$\\pm$1.0 &Padova, Teramo \\& Dartmouth iso\t&2\\\\\nL100\t\t&61.4\t& 78.1$\\pm$36.3\t& 58.6$\\pm$0.7 &Padova iso \t&6\\\\\nL110 \t&90.8\t& 58.2$\\pm$32.7\t& 47.9$\\pm$2.3 &Teramo \\& Dartmouth\t&4\\\\\nL113 \t&102.7\t& 98.8$\\pm$33.2\t&50.5$\\pm$1.7\t & Red Clump\t&1\\\\\nNGC121 \t&302.2\t& 63.6$\\pm$25.9\t& 64.9$\\pm$1.2 &Padova, Teramo \\& Dartmouth iso\t&2\\\\\nNGC339 \t&168.9\t& 42.0$\\pm$29.7\t& 57.6$\\pm$4.1 &Padova, Teramo \\& Dartmouth iso\t&2\\\\\nNGC361 \t&31.3\t& 44.2$\\pm$16.9\t& 55.8$\\pm$1.7 &Red Clump\t&1\\\\\nNGC419 \t&92.7\t& 44.6$\\pm$21.2\t& 56.2$\\pm$1.3 &$\\delta$ Sct L-P relation\t&5\\\\\n\\hline\n\\end{tabular}\n\n\\noindent Ref.: (1) \\citet{cetal01}; (2) \\citet{glattetal2008a}; \n(3) \\citet{diasetal2016};  \n(4) \\citet{paetal14};  \\\\\n(5) \\citet{mvetal2021}; (6) \\citet{diasetal2021}. \n\\end{table*}\n\n\n\\section{Analysis and discussion}\n\nThe rotating disc geometry obtained by \\citet[][see his Fig. 4]{piatti2021b} can be \ndescribed as a 3D vector with components $v_1$, $v_2$, and $v_3$ that\nare functions of the position angle (PA). Therefore, the differences between each \nvelocity component of a cluster and the respective ones for the rotational curve \nat the cluster's PA ($\\Delta$$v_i$, i=1,2,3) represent the components of the \nresidual velocity vector with  respect to the SMC  rotating disc. By adding them\nin quadrature we obtained the magnitude of such a vector $\\Delta$$V$ =\n($\\Sigma$$_{i=1}^{i=3}$ $\\Delta$$v_i$$^2$)$^{1\/2}$. Figure\\,1 shows the\nspatial distribution of the 32 clusters with calculated $v_1$, $v_2$, and $v_3$\nvelocity components by adopting the parameters of the disc obtained\nby \\citet[][see his Table\\,2]{piatti2021b}. As can be seen, clusters span a wide\nrange of  $\\Delta$$V$  values, that we refer as the residual velocity vector.\n\nBy using as residual velocity vector\ncomponents the values of the dispersion velocity found by\n\\citet{piatti2021b} (i.e., $\\Delta$$v_1$=25 km\/s, $\\Delta$$v_2$=50 km\/s, \n$\\Delta$$v_3$=20 km\/s), we derived the\ncorresponding residual velocity vector magnitude, $\\Delta$$V$ = 60 km\/s. \nWe adopted this value to differentiate clusters bound to the SMC\nrotating disc ($\\Delta$$V$ $<$ 60 km\/s) from those scattered by the tidal\ninteraction with the LMC ($\\Delta$$V$ $>$ 60 km\/s). With \nthis  residual velocity cut, we found 13 clusters with a kinematics that very \nwell match that of a  rotating disc, and others 19 whose orbital motions we \ninterpreted have been affected by the effects of tidal forces. Figure~\\ref{fig1}\ndepicts their spatial distribution. Clusters with $\\Delta$$V$ $>$ 60 km\/s are mainly \nlocated in SMC regions projected in the sky that have some previous hints for \ntidal structures arisen from the interaction between both Magellanic Clouds, \nnamely: a northern branch of the Magellanic Bridge and the Counter-Bridge,\nwhich are superimposed at the northeastern outskirts of the SMC; the main\nMagellanic Bridge, located toward the east from the SMC centre; the West\nHalo \\citep{db2012,diasetal2016} placed at the southwestern end of the SMC disc.\nThis finding confirms our suspicions that  kinematically perturbed clusters -- those with orbital \nmotions different from those on a  rotating disc -- accelerated by the LMC \ngravitational field. Likewise, there are some clusters with $\\Delta$$V$ $>$ 60 \nkm\/s that appear projected toward the SMC main body. As far as we are\naware, this is the first observational evidence of the distinctive hot kinematics\nof clusters that depart from the SMC cold disc kinematics. Note that the \nrotation velocity of the SMC disc is 25.0$\\pm$5.0 km\/s \\citep{piatti2021b,zivicketal2020}.\n\nTwo additional features can be glimpsed at Fig.~\\ref{fig1}. On the one hand,\nsome clusters with $\\Delta$$V$ $<$ 60 km\/s are also projected on\nthe above mentioned SMC tidal structures, which could imply that\nSMC disc clusters may be projected on galaxy tidal features, in very\ngood agreement with the disc extension out to  $\\sim$ 4 kpc \n\\citep[see Fig. 1 in][]{diasetal2021}. Therefore, the sole projected cluster position \nis not enough to straightforwardly identify any clusters as\nbeing tidally  perturbed.  We do not rule out the possible existence of cold \nkinematics clusters in the SMC tidal structures. On the other hand, some clusters \nwith $\\Delta$$V$ $>$ 60 km\/s are seen projected on the SMC main body, which \ncould witness the presence of runaway clusters traveling throughout the SMC disc. \n\nIn order to disentangle whether there is a possible mixture of  kinematically perturbed\nclusters \nwith $\\Delta$$V$ smaller and larger than 60 km\/s, which would reveal that\nthey would not be easily recognized from their  residual velocity \nvectors, their heliocentric distances are necessary. Unfortunately, there is a small\nnumber of clusters with measured distances (see Section 2), which points to the \nneed of  observational campaigns in order to build a realistic spatial distribution \nof them \\citep{piatti2021a}, and consequently, to trace more precisely the 3D \nSMC cluster population structure. Previous star cluster studies adopted, in general, \na mean \nSMC distance when dealing with cluster colour-magnitudes diagrams\n\\citep[][and references therein]{p12b,p14c,petal2016}. This is\nbecause changes in the distance modulus by an amount equivalent to the average \ndepth of the SMC \\citep{cetal01,ripepietal2017,graczyketal2020}, so that clusters can be \nplaced in front or behind the SMC, leads to a smaller age difference than that \nresulting from the  isochrones bracketing the observed star cluster features in the \ncolour-magnitude diagram. \n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig2.png}\n\\caption{Spatial distribution of SMC clusters compiled in \\citet{bicaetal2020} \n(gray dots). Circles and stars correspond to clusters with \n residual velocities smaller and larger than 60 km\/s, colour-coded\naccording to their heliocentric distances.}\n\\label{fig2}\n\\end{figure}\n\nWith available heliocentric distances for 19 clusters\nin our sample, we built Fig.~\\ref{fig2}, which confirms that there is a wide\ncluster distance range among those projected on SMC tidal structures,\nin comparison with clusters projected toward the SMC main body. In the\nnortheastern outermost region, there are clusters located in from of the SMC \n(they belong to the northern branch of the Magellanic Bridge); in the SMC body \nitself; and behind the galaxy (they are part of the Counter-Bridge)\n\\citep{diasetal2021}. These clusters have very different $\\Delta$$V$ values as \njudged by the different symbols used to represent them (circles and stars as in\nFig.~\\ref{fig1}). Similarly, the line-of-sight toward the West Halo also\npresents clusters placed in front of the SMC; at the SMC distance; and \nbehind it. This is the first evidence from clusters's distances that the West\nHalo is a perturbed structure of the SMC, consisting of  leading and trailing tails\n\\citep{db2012,diasetal2016}. Nevertheless,  in order to determine the extension \nof both tails, further distance estimates for a larger sample of  clusters projected \non this sky region are desirable. We also showed that SMC disc clusters \n($\\Delta$$V$ $<$ 60 km\/s) are spread throughout the whole covered area and \nlocated at distances compatible with them being part of the SMC main body \n\\citep[56 $\\la$ $D$ (kpc) $\\la$ 62,][and references therein]{piatti2021b}.\n\n\n\\begin{figure*}\n\\includegraphics[width=\\textwidth]{fig3.png}\n\\caption{Residual velocities $\\Delta$$V$ versus heliocentric distance\nof SMC clusters. The uncertainties in $\\Delta$$V$ were calculated using\nthe measured errors in proper motion and radial velocity, and those from the \nsolution for the 3D movement of the SMC centre obtained by \\citet{piatti2021b}, \npropagated through the transformation equations and added in quadrature. \nSolid and dashed lines represent the SMC centre obtained by \\citet{piatti2021b} \nand a readily visible SMC main body depth, respectively.}\n\\label{fig3}\n\\end{figure*}\n\nFigure~\\ref{fig3} reveals a striking distribution of SMC clusters bound\nto the SMC main body and those pertaining to SMC tidal structures. According to\nthe observed distances of the Magellanic Bridge \n\\citep[$D$  $<$ 55 kpc,][]{wagnerkaiser17,jacyszyndobrzenieckaetal2020}; of the\nCounter-Bridge \\citep[$D$ $>$ 65 kpc,][]{diasetal2021}; and the present West\nHalo depth (52 $\\la$ $D$ (kpc) $\\la$ 70) derived from 5 clusters at PA between 230$\\degr$ \nand 270$\\degr$, there is a suggestive  residual velocity level at $\\sim$ 50 km\/s\nthat broadly separates  SMC main body clusters from those in tidally induced\nfeatures. Clusters that belong to these last structures show  residual velocities\n$\\Delta$$V$ $>$ 50km\/s, in very good agreement with the  residual velocity \ncut calculated above (60 km\/s) from the SMC dispersion velocities obtained by\n\\citet{piatti2021b}. SMC main body clusters span a wide range of\nresidual velocities, but most of them show $\\Delta$$V$ $<$ 50 km\/s, meaning\nthat they have motions similar to the SMC  rotating disc kinematics \\citep{zivicketal2020}.\nThis outcome shows that the kinematics of SMC tidal\nstructures differentiates from that of its disc. \n\nAmong clusters with heliocentric distances smaller than 55 kpc, there are\nfour that belong to the northern branch of the Magellanic Bridge (PA $<$ 90$\\degr$);\nanother one belongs to the main branch of the Magellanic Bridge (PA $\\approx$ \n1$\\degr$); and one is part of the leading West Halo. All these clusters have \n$\\Delta$$V$ $>$ 50. km\/s. As clusters placed behind the SMC ($D$ $>$ 65 kpc)\nare considered, there is one that pertains to the Counter-Bridge, and another \nto the trailing West Halo. They also have $\\Delta$$V$ $>$ 50. km\/s. \nNGC\\,121 is the oldest known SMC cluster \\citep[10.5 Gyr,][]{p11b}; its location does \nnot coincide with any tidally induced structure \n(((RA-RA$_0$)$\\times$cos(Dec),(Dec-Dec$_0$)) $\\approx$ (-2.0 kpc, 1.3 kpc)),\nsee Fig.~\\ref{fig2}), but the cluster is at $D$ = (64.9$\\pm$1.2) kpc and has\n$\\Delta$$V$ = 64 km\/s. It is within the kinematic regime of tidally perturbed clusters.\nThe lack of ancient globular clusters in the SMC has been one of the most intriguing enigmas in our understanding of the SMC formation and evolution. \\citet{carpinteroetal2013} modeled the dynamical LMC-SMC interaction and their\nrespective star cluster populations aiming at exploring whether the lack of\nold SMC globular clusters can be the result of such galaxy tidal interactions.\nThey found that clusters that originally belonged to the SMC are more likely to be\nfound in the outskirts of the LMC or ejected into the intergalactic medium.\nIn this context, NGC\\,121 could be identified as one of those SMC old escaping clusters.\n\nThere are two clusters located at distances compatible with the SMC main body\ndepth whose  residual velocity vectors ($\\Delta$$V$ $>$ 50 km\/s) suggest that\nthey do not belong to the SMC  rotating disc. These clusters are projected\ntoward the northern branch of the Magellanic Cloud and the West Halo, \nrespectively, so that they could either indicate that at the very onset of both tidal \nstructures clusters' kinematics change or that they are runaways clusters.\nWe note that other runaway clusters have been identified in the LMC \\citep{piattietal2018a}.\nThe remaining eight clusters with distance estimates in our sample are SMC\nmain body objects with an SMC  rotating disc kinematics ($\\Delta$$V$ $<$ 50 km\/s).\n\n\n\\begin{figure*}\n\\includegraphics[width=\\textwidth]{fig4.png}\n\\caption{Projected spatial distributions of SMC clusters, with the corresponding\nvelocity vectors. The black arrow  represents a velocity of 1$\\pm$km\/s.\nThe projection of the Magellanic Bridge is also schematically\nrepresented by black lines \\citep{wagnerkaiser17,jacyszyndobrzenieckaetal2020,bicaetal2020}. }  \n\\label{fig4}\n\\end{figure*}\n\nFor completeness purposes, Fig.~\\ref{fig4} illustrates the residual velocity \ncomponents projected on the Cartesians planes \\citep{vdmareletal2002}.\nThey witness the stretching experienced by the star cluster population due to\ntidal effects: those on the northern branch of the Magellanic Bridge roughly moving \ntoward the  LMC, while the one observed in the Counter-Bridge traveling in an\noposite direction; a picture that is also verified among clusters placed in the closer \nand farther regions of the West Halo.\n\n In what follows, we mention two different scenarios that could be considered\nto speculate on possible interpretations of the nature and consequences of\nthe tidal interaction between the LMC and the SMC. They are not exhaustive, but\nsimply introductory, based on recent results.\n\nThe gaps and spurs seen in the spatial distribution of Magellanic Bridge's clusters\n\\citep[see Fig. 3 in][]{bicaetal2020} resemble those observed in tidal tails or streams\naround Milky Way globular clusters \n\\citep[e.g., Pal\\,5, GD-1 streams:][]{gd2006a,gd2006b}, which have become the \nmost sensitive tracers of the nature and distribution of dark matter in the\nMilky Way \\citep{bonacaetal2019}. \nThese tidal tails resulted to be exceedingly cold, \nwith stellar velocity dispersions on the order of 2 km\/s or less, and mean motions\nsimilar to their parent globular clusters \\citep{pricewhelanetal2019}. Moreover, the\ndiscovery of tidal tails around Milky Way globular clusters using astrometric\ninformation usually relies on the assumption that stars in tidal tails share the mean\nproper motions and radial velocities of globular clusters' stars \\citep[e.g.,][]{carballobello2019,sollima2020,shippetal2020}.\nThe presence of tidal stream substructures extending to hundreds of parsecs around \nglobular clusters also suggest that at least some globular clusters continue to be \nembedded in dark matter halos \\citep{olszewskietal2009,kuzmaetal2016}.\n\\citet{diteodoroetal2019} studied high-resolution H\\,I data in the SMC and found that\nout to $\\sim$ 4 kpc from its centre, the galaxy exhibits a rotating disc kinematics\nthat possibly flattens outwards, despite of the strong gravitational interaction with \nthe LMC. In order to reproduce the observed rotation curved, they argued that\na dominant dark matter halo is required. Recently, \\citet{koposovetal2018} \ndiscovered a mildly elliptical ultra-faint dwarf galaxy (Hydrus) located in the \nMagellanic Bridge with a velocity dispersion of 2.7$\\pm$0.5 km\/s, which indicates \nthat the system is dark matter dominated. \n\nTo the light of our knowledge about the kinematics of globular clusters' tidal\ntails, the Magellanic Bridge and West Halo do not behave similarly with\nrespect to the SMC. We found that they show motions\nlargely departed from that of the SMC main body (see Fig~\\ref{fig3}).\nTherefore, the present observational evidence would be supporting either the \nnotion of a tidal interaction pattern between the LMC and the SMC not comparable \nto that known for globular clusters and the Milky Way, or that the SMC\nis not a dark matter dominated dwarf galaxy. We think that Fig.~\\ref{fig3} is\na suitable starting point to be considered for further investigations on such a \ndebatable conundrum.\n\n\n\\section{Data availability}\n\nData used in this work are available upon request to the author.\n\n\n\\section*{Acknowledgements}\nI thank the referee for the thorough reading of the manuscript and\ntimely suggestions to improve it. \n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe recent discovery of extended ultraviolet (XUV) disks (e.g.,\n\\cite{Th05}, \\cite{Gil05}), reflecting ongoing star formation beyond\nthe optical radii and traditional star formation thresholds of\nlate-type galaxies, has provided an intriguing look at disk\ngrowth in progress at $z$$\\sim$$0$. A separate line of research on\n``blue-sequence E\/S0s,'' morphologically defined E\/S0 galaxies on the\nblue sequence in color vs. stellar mass space, may provide a direct look\nat disk \\emph{regrowth} in progress at $z$$\\sim$$0$ (\\cite{KGB}), a\nprocess predicted by hierarchical models (e.g., \\cite{SM},\n\\cite{Gv07}). Blue-sequence E\/S0s increase dramatically in abundance\nbelow the gas-richness threshold mass at \\break$M$$_{*}$ $\\sim$ $5 \\times\n10^{9}\\, M_{\\odot}$, the regime in which neutral-atomic-gas\/stellar mass\nratios $\\gtrsim$1 become common (\\cite{K04}, \\cite{KW}, see \\cite{KGB}\nregarding corrected mass scale). As shown in \\cite{KGB} and\n\\cite{We10a}, many blue-sequence E\/S0s display the global gas\nreservoirs and specific star formation rates necessary for significant\nstellar disk growth on relatively short timescales. As a result, if\nthe XUV-disk phenomenon is associated with disk building in general,\nwe might expect to observe XUV disks preferentially among\n\\emph{blue-sequence} E\/S0s.\n\nAlthough the first XUV-disk studies, such as \\cite{Th07}, emphasized\nlate-type galaxies, XUV disks around E\/S0s are increasingly being\nfound as well (e.g., \\cite{D09}, \\cite{CH}, \\cite{Th10},\n\\cite{SR10}). However, the question remains: are XUV disks in E\/S0s\nrelated to the probable active disk-builders, the blue-sequence E\/S0\npopulation? We address this question by examining the frequency of XUV\ndisks in E\/S0s as a function of both sequence and mass, finding a\nsurprising but illuminating lack of association between Type~1 XUV\ndisks and E\/S0s at the low-mass, gas-rich end of the blue sequence.\n\nOur sample of 31 E\/S0s encompasses all of the Nearby Field Galaxy\nSurvey (NFGS, \\cite{Jan}) blue-sequence E\/S0s and the majority of NFGS\nred-sequence E\/S0s with \\break $M$$_{*}$~$\\lesssim$~$4 \\times 10^{10}\\,\nM_{\\odot}$ (Fig.\\ \\ref{RBfig}), where many E\/S0s have substantial gas,\nand relatively undisturbed blue-sequence E\/S0s with the potential for\ndisk regrowth are observed (\\cite{KGB}). In addition to these 25 NFGS\nE\/S0s, we include 6 blue-sequence E\/S0s from the ``HyperLeda+'' sample\nof \\cite{KGB}.\n\nIn \\cite{Th07}, Type~1 XUV disks are defined as displaying more than\none structured UV-bright emission complex beyond a centralized\nsurface-brightness contour corresponding to the expected star\nformation threshold (equated to an NUV surface brightness of 27.35 AB\nmag\\,arcsec$^{-2}$ in \\cite{Th07}, roughly matching typical H$\\alpha$\nand HI thresholds; we label the corresponding radius $R_{\\rm\n\\it{UVSF}}$). In addition, this definition requires that the XUV\nemission take on a different morphology from any underlying optical\nemission.\n\nWe classify our sample galaxies based on the Type~1 XUV-disk\ndefinition using \\emph{GALEX} NUV images with a minimum exposure time\nof 1500~s. For comparison of UV and optical morphologies, we employ\nDSS-II red images (http:\/\/archive.stsci.edu\/dss\/). These\nclassifications supersede the preliminary, purely visual (made without\nreference to $R_{\\rm \\it{UVSF}}$) classifications of \\cite{Me09}. The\nType~1 XUV disks in our E\/S0s can extend to several times $R_{25}$ as\nhas been found in late types (e.g., \\cite{Th05}, \\cite{Gil05},\n\\cite{Th07}, see also \\cite{ZC}). We find radial extents (to the last\nmeasured NUV point) between $\\sim$0.8 and 3 times $R_{25}$, with an\naverage of $\\sim$1.6.\n\n\\begin{figure}\n\\resizebox{0.95\\columnwidth}{!}{\\includegraphics{Full_Redblue_lyonconf.eps} }\n\\caption{E\/S0 sample in color-stellar mass space. Small grey symbols\nindicate galaxies in the NFGS, the parent sample for all but six of\nour E\/S0s. The dashed line divides the red and blue sequences, and the\nvertical line marks the gas-richness threshold mass (\\cite{KGB}). The\n30 E\/S0s from \\emph{GALEX} program GI3-0046 are denoted by open\ncircles, and CGCG~065-002 from program GI5-042 is denoted by an open\ntriangle. Type~1 XUV-disk galaxies in this sample are marked with\nasterisks.}\n\\label{RBfig}\n\\end{figure}\n\nCompared to XUV disks in late-type galaxies, our E\/S0 XUV disks tend\nto be redder. However, using outer-disk FUV$-$NUV colors and assuming\nthe observed UV emission comes from young stars, we still estimate\n$<$1 Gyr ages for our E\/S0 XUV disks from simple stellar population\n(SSP) models (although a mixed old plus young population is presumably\npresent, we defer multiple-component stellar population modeling to a\nfuture paper). Comparing with UV model colors of \\cite{B03} for an\ninstantaneous starburst with $Z=0.02$ (as in \\cite{Th07} Figure 1),\nour average Type~1 XUV-disk FUV$-$NUV color of $\\sim$1.6 corresponds\nto a stellar population with an approximate age of 500 Myr.\n\nAlthough the UV upturn, i.e., UV emission associated with old stellar\npopulations (\\cite{OC99}), is a plausible source for excess UV flux\nin E\/S0s, the FUV$-$$K$ colors we measure, as well as independent\nevidence for actual or potential star formation (e.g., H$\\alpha$ or HI\ndetections in all but two of our XUV-disk galaxies), largely support an\nassociation of our XUV disks with recent star formation.\n\nType~1 XUV disks occur in 13\/31 or 42$^{+11}_{-10}$\\% of our E\/S0\nsample. This frequency is approximately double the 20\\% Type~1\nXUV-disk frequency found in \\cite{Th07} for a mostly late-type galaxy\nsample. We find XUV disks in both red- and blue-sequence E\/S0s over a\nlarge range in stellar mass (Fig.\\ \\ref{RBfig}). The widespread nature\nof the XUV-disk phenome-non seems to suggest an association with\nevolutionary processes affecting the galaxy population broadly, such\nas gas accretion and\/or minor satellite interactions.\n\nWe also observe a complex mass and sequence dependence of XUV-disk\nincidence: the relative abundance of red- \\emph{vs.} blue-sequence\nXUV-disk galaxies \\emph{seems to reverse} across the gas-richness\nthreshold mass. On the red sequence, the XUV-disk frequencies are low\nand high (33$^{+41}_{-28}$\\% and 71$^{+18}_{-26}$\\%) above and below\nthe threshold mass, respectively. On the blue-sequence, the\ncorresponding frequencies are high and \\emph{low} (50$\\pm$20\\% and\n18$^{+19}_{-12}$\\%). For low-mass E\/S0s, the association we observe\nbetween \\emph{red} color and a high frequency of Type 1 XUV disks\nsuggests that these disks typically experience inefficient or\nweak star formation instead of the pronounced star formation expected\non the \\emph{blue} sequence in this gas-rich regime. This link between\nType~1 XUV disks and weak\/inefficient star formation is supported by\nprevious studies (e.g., \\cite{Th07}, \\cite{B08}).\n \nBased on these results, observing Type 1 XUV disks in E\/S0s may not\nprovide the best indication of \\emph{significant} disk\nregrowth. Instead, relatively \\emph{low} XUV-disk incidence, as is\nfound for both low-mass blue-sequence E\/S0s and normal late-types\n(\\cite{Th07}), may be characteristic of populations with active disk\nbuilding in progress. Our findings highlight the need for a more\ngeneral UV-bright disk definition, which will be addressed in future\nwork.\n\\vspace{1.2 mm}\n\nWe thank S. Jogee for her role in acquiring the \\emph{Spitzer} data\nand M. Haynes for the early release of \\emph{GALEX} imaging of\nNGC~3773, an archival target in our sample. We also thank L. Wei,\nM. Norris, C. Clemens, and A. Leroy for useful discussions. AJM\nacknowledges support from the NASA Harriett G. Jenkins Pre-doctoral\nFellowship Program. This work uses observations made with the NASA\nGalaxy Evolution Explorer and \\emph{Spitzer} Space\nTelescope. \\emph{GALEX} is operated for NASA by Caltech under NASA\ncontract NAS5-98034. We acknowledge support from the \\emph{GALEX}\nGuest Investigator program under NASA grants NNX07AT33G and\nNNX09AF69G. \\emph{Spitzer} is operated by the Jet Propulsion\nLaboratory, Caltech under a contract with NASA. Support for this work\nwas also provided by NASA through an award issued by JPL\/Caltech.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\nThe basis of the classical algebraic geometry is the Galois\ncorrespondence between \\textit{P}-closed ideals in\n$P[x_1,x_2,\\ldots,x_n]$ and algebraic sets in the affine space\n$P^n$. Having noted that a point $a=(a_1,a_2,\\ldots,a_n)$ unique\ndetermines a homomorphism $x\\in\\textrm{Hom}(P[X])$\n($s(x_i)=a_i$), B. Plotkin in [11-13] makes foundation of the\nuniversal algebraic geometry, i.e., algebraic geometry for\narbitrary variety $\\Theta$ of universal algebras. Let us fix a\nvariety $\\Theta$ and an algebra $H\\in\\Theta$. Let $W=W(X)$ be a\nfree algebra in $\\Theta$ with finite set of generators. The set\n$\\textrm{Hom}(W,H)$ we consider as affine space of points over\n$H$. Then arbitrary congruence $T$ in $W$ determines the set of\npoints in affine space $\\textrm{Hom}(W,H)$:\n$$T'_H=A=\\{\\mu\\in\\textrm{Hom}|T\\subset\\textrm{Ker}\\mu\\}.$$\nA set of points $A\\subset\\textrm{Hom}(W,H)$ determines congruence\nof W\n$$A'=A'_W=T=\\bigcap_{\\mu\\in A}\\textrm{Ker}\\mu.$$\nWe call the set of points $A$ such that $A=T'$ for some $T$ an\nalgebraic set in $\\textrm{Hom}(W,H)$. A relation $T$ with $T=A'$\nfor some $T$ is a congruence in $W$. We call such a congruence an\n$H$-closed one.\n\nIt was found [3, 4, 11-14] that many problems in the universal\nalgebraic geometry, such as geometric equivalence, geometric\nsimilarity, isomorphism and equivalence of categories of algebraic\nsets and varieties depend on structure of ${\\rm Aut}(\\Theta^0)$\nand Aut(End($W$)), where $\\Theta^0$ is the category of algebras\n$W=W(X)$ with the finite $X$ that are free in $\\Theta$, and $W$ is\na free algebra in $\\Theta$. Structure of ${\\rm Aut(Com}-P))^0$\n(i.e, the classical case) was described in [2]. Here was\nintroduced variant of the concept of quasiinner automorphism -\nbasic  concept for description of Aut(End($W$)) and\n$\\textrm{Aut}(\\Theta^0)$. Later groups $\\textrm{Aut}(\\Theta^0)$\nwere described for categories of free Lie algebras and free\nassociative algebras [1, 5, 8, 9]and for categories of free\nmodules and free Lie modules [5, 6]. Some other definition of\nquasiinner automorphism was used in mentioned papers. Also are\ndescribed ${\\rm Aut}(\\Theta^0)$ for varieties of groups,\nsemigroups and some another varieties. For some varieties is\ndescribed Aut(End($W$)), but these problems are more difficult.\nIn particular, this problem was not solved even in the classical\ncase of free commutative algebras, i.e. the structure of\nAut(End$P[x_1,\\ldots ,x_n]$)) is not describe if $n>2$. In all\nsolved cases groups Aut(End($W$)) are generated by semi-inner\nautomorphisms and mirror automorphism. In [3] this problem in\nclassical case was solved, if the field $P$ is algebraically\nclosed.\n\nI want to note that the proof in [3] does not use the description\nof the group ${\\rm Aut}(P[X])$). It is very important, because\nstructure of the group ${\\rm Aut}(P[x_1,\\ldots ,x_n])$ is not\ndescribed, when $n>2$, but counterexamples show that structures of\n${\\rm Aut}(P[x_1,\\ldots ,x_n])$ and ${\\rm Aut}(P[x,y]$) are\nprincipally different. Now in the presented paper the group\nAut(End($W_n$)) is described for commutative and associative\nalgebras over arbitrary infinite field.\n\n\n\\section{Definitions}\n\nLet us recall some definitions for variety of commutative algebras\n$\\textrm{Com}-P$ (see \\textrm{[4,13]} for general case).\n\n\\begin{ddd}\n\nLet $P[X]=P[x_1,\\ldots ,x_n]$ be a free commutative algebra over\na field $P$ with a finite set of generators and\n$\\tau\\in\\textrm{Aut}(\\textrm{End}(P[X]))$. It is known\n$\\textrm{[4]}$ that there exists a bijection $\\mu:P[X]\\mapsto\nP[X]$ such that for every $s\\in \\textrm{End}(P[X])$\n$$ s^{\\tau}=\\mu s\\mu^{-1}.$$\nSuch representation of $\\tau$ is called a representation of $\\tau$\nas quasiinner automorphism generated by $\\mu$.\n\n\\end{ddd}\nOf course, arbitrary set of substitutions does not generate an\nautomorphism, But there are two kinds of bijections $\\mu$ in\ncommutative case and three in the associative case that\ngenerate an automorphism.\\\\\n\n1. $\\mu=\\overline{\\alpha}$, where $\\overline{\\alpha}$ is the\nnatural extension of an automorphism $\\alpha \\in\n\\textrm{Aut}(P[X])$ on $W$, i.e. $\\overline{\\alpha}\\left( \\sum\na_ku_k\\right)=\\sum \\alpha(a_k)u_k$.\n We shall write $\\overline{\\alpha}=\\alpha$ and call it \"automorphism\\hspace{2pt} of\n field\".\\\\\n\n 2. $\\mu=\\eta\\in \\textrm{Aut}(P[X])$.\\\\\n\n\\begin{ddd}\n\nThe automorphism $\\tau$ generated by an automorphism\n$\\mu\\in\\textrm{Aut}(P[X])$ is called inner automorphism.\n\n\\end{ddd}\n\n\\begin{ddd}\n\nNote, that every $\\alpha \\in \\textrm{Aut}(P)$ belongs to\nnormalizer of subgroup  $\\textrm{Aut}(P[X]))$ in the group of all\nbijections of $P[X]$. So, every product of elements of\n$\\textrm{Aut}(P)$ and $\\textrm{Aut}(P[X])$ my be represented in\nthe form $\\mu=\\alpha\\eta$, $\\alpha \\in \\textrm{Aut}P$,\n$\\eta\\in\\textrm{Aut}(P[X])$. Automorphism\n$\\tau\\in\\textrm{Aut}(\\textrm{End}(P[X])$ generated by $\\mu$ is\ncalled semiinner.\n\n\\end{ddd}\n\n3. Now we describe the third type of $\\mu$ for associative\nalgebras. Let $S=S(X)$ be a free semi-group. For every\n$u=x_{i_1}x_{i_2}\\ldots x_{i_n}$ in $S$, take\n$\\overline{u}=x_{i_n}\\ldots x_{i_2}x_{i_1}$. Then $u\\rightarrow\n\\overline{u}$ is an antiautomorphism\\hspace{2pt} of the semigroup S.\n\n\\begin{ddd}\n\nLet $W=W(X)$ be a free associative algebra. For every its element\n$\\omega=\\lambda_0+\\lambda_1u_1+\\cdots+\\lambda_ku_k$ denote\n$\\beta(\\omega)=\\overline{\\omega}=\\lambda_0+\\lambda_1\\overline{u_1}+\\cdots+\n\\lambda_k\\overline{u_k}$. The transition $\\beta:W(X)\\rightarrow\nW(X)$ is an antiautomorphism of the algebra $W$, and it generates\nthe automorphism $\\delta\\in \\textrm{Aut}(\\textrm{End}(W(X))$,\n$s^\\delta =\\beta s \\beta^{-1}$. We call $\\beta$ mirror\nantiautomorphism of $W$ and $\\delta$ mirror automorphism of\n$\\textrm{Aut}(\\Theta^0)$.\n\n\\end{ddd}\n\nHere $\\delta$ is not inner and is not semi-inner, but is\nquasi-inner. Note that\n\n\\begin{enumerate}\n\n\\item $\\delta$ belongs to the normalizer of subgroup\n$\\textrm{Sinn}(\\textrm{End}(W(X))$ in\n$\\textrm{Aut}(\\textrm{End}(W(X))$ that consists of all semi-inner\nautomorphisms;\n\n\\item every antiautomorphism of $W$ is a product of the mirror\nantiautomorphism $\\beta$ and an automorphism of $W$.\n\n\\end{enumerate}\n\nLet now $W(X)$ be a free finitely generated algebra in arbitrary\nvariety $\\Theta$.\n\n\\begin{ddd}\n\nBijection $\\mu : W(X)\\mapsto W(X)$ is called central, if for every\n$s\\in \\textrm{End}(W(X))$\n$$\\mu s=s\\mu.$$\nAlgebra $W(X)$ is called central, if every its central bijection\nis identical.\n\n\\end{ddd}\n\n\\begin{ttt}\n\nLet $W(X)$ be a free central algebra and $\\mu$ some bijection of\n$W(X)$ generating an automorphism\n$\\tau\\in\\textrm{Aut}(\\textrm{End}(W(X)))$. Then $\\mu$ transforms\nevery base of algebra $W(X)$ to a base of algebra.\n\n\\end{ttt}\n\\textbf{Proof.} [see 4]\\\\[6pt]\n\nFor free algebras in a variety $\\Theta$ of algebras over field $P$\n(Com-P, Ass-P, and others) we shall correct the definition of\ncentral algebra. Clearly, in every such algebra $W(X)$ we have a\nbijection $\\mu(u)=au+b$, where $a,b\\in P,$ $a\\neq 0$ (linear\nbijection). This bijection commutate with every endomorphism of\nalgebra $W(X)$ and, obviously, transforms every base of $W(X)$ to\na base.\n\n\\begin{ddd}\n\nFinitely generated algebra $W(X)$ over field $P$ is called almost\ncentral, if every its central bijection is linear.\n\n\\end{ddd}\n\n\n\\begin{ttt}\n\nFree finitely generated commutative, associative are almost\ncentral.\n\n\\end{ttt}\n\\textbf{Proof.} Let $\\mu$ be a central bijection of $W(X)$,\n$\\mu(x_i)=r_i(x_1,x_2,\\ldots,x_n)\\in W(X)$ , and $s\\in\n\\textrm{End}(W(X)$, $s(x_1)=x_1$, $s(x_i)=0$ for $i=1$. Then\n$$\\mu s(x_1)=\\mu (x_1)=r_1(x_1,x_2,\\ldots,x_n),$$\n$$s\\mu(x_1)=s(r_1(x_1,x_2,\\ldots,x_n))=r_1(x_1,0,\\ldots,0)=r(x_1).$$\nSo $r_1(x_1,x_2,\\ldots,x_n)=r(x_1)$ is a polynomial of one\nvariable.\n\nFor arbitrary $u\\in W(X)$ take $s\\in \\textrm{End}(W(X))$\n$s(x_1)=u$, $s(x_i)=x_i$ for $i>1$. Then\n$$\\mu(u)=\\mu s(x_1)=s\\mu(x_1)=s(r(x_1))=r(u),$$\nand so $\\mu(u)=r(u)=a_0+a_1u+\\cdots+a_ku^k$. Because $\\mu$ is\nbijection, so $r$ is linear, Q.E.D..\\\\[3pt]\n\\textbf{Corollary.} \\textit{From the theorems 1 and 2 we have that\nin mentioned algebras every bijection $\\mu$, which generate an\nautomorphism of $\\textrm{End}(W(X))$  transform every base of\nalgebra $W(X)$ to a base of this algebra.}\n\n\\section{Linearity of automorphism generating bijection}\n\nLet $W=W(X)$ be a free commutative, associative or Lee algebra\nover the field $P$ with a finite set of generators, and $\\tau$ be\nan automorphism of $\\textrm{End}(W(X))$ generated by bijection\n$\\mu$. We call $s\\in \\textrm{End}(W(X))$ a constant endomorphism\nif $\\textrm{Im}(s)=P$. Clearly, if $s$ is constant then $s^{\\tau}$\nalso is constant. Denote the set of all constant endomorphisms by\n$Const$.\n\nLet $a\\in P$ and $s$ be a constant endomorphism such that\n$s(x_1)=a$. Then $\\mu(a)=\\mu(s(x_1))=s^{\\tau}(\\mu(x_1))\\in P$. So\na restriction of $\\mu$ on $P$ is a bijection: $\\mu:P\\mapsto P$.\n\nLet $\\mu(0)=a$ and $\\mu(1)=b$. We shall consider the linear\nbijection $l:W(X)\\mapsto W(X)$, $l(u)=cu+d$ such that $l(a)=0$ and\n$l(b)=1$. Since linear bijection generates identical automorphism\nof the semigroup $\\textrm{End}(W(X))$, so bijections $\\mu$ and\n$\\mu'=l\\mu$ generate equal automorphisms. Note that $\\mu'(0)=0$\nand $\\mu'(1)=1$. We denote $\\mu=\\mu'$ and assume that $\\mu(0)=0$\nand $\\mu(1)=1$.\n\nSince $\\mu$ transforms base of algebra $W(X)$ to base, so the\nendomorphism $\\eta:$ $\\eta(x_i)=\\mu(x_i)$ is an automorphism of\n$W(X)$. Now we shall consider the bijection $\\mu'=\\eta^{-1}\\mu$.\nWe have for $\\mu'$ that $\\mu'(x_i)=x_i$.\n\nNow we shall consider the Galois correspondence between ideals in\n$W(X)$ and algebraic sets in affine space over $W(X)$. We recall\nthat every ideal $T$ in $W(X)$ defines an algebraic set of points\nin affine space over $W(X)$.\n$$T'=T'_{W(X)}=A=\\{\\eta\\in\\textrm{End}(W(X))|T\\subset\\textrm{Ker}\\eta\\}.$$\nA set of points $A\\subset\\textrm{End}(W(X))$ determines ideal in\n$W(X)$:\n$$A'=A'_W=\\bigcap_{\\eta\\in A}\\textrm{Ker}\\hspace{4pt}\\eta.$$\n\nAn element $u\\in W(X)$ is called basic, if it belongs to some base\nof algebra $W(X)$.\n\n\\begin{lll} If $u$ is an basic element of W(X), then\n$$\\left\\langle u\\right\\rangle'' =\\left\\langle u\\right\\rangle$$\n\\end{lll}\n\\textbf{Proof.} Clearly,\n$$A=\\left\\langle u_1\\right\\rangle'=\\{\\eta\\in\\textrm{End}W|\\hspace{4pt}\\eta(u_1)=0\\}$$\nand\n$$A'=\\left\\langle u_1\\right\\rangle''=\\bigcap_{\\eta\\in\nA}\\textrm{Ker}\\hspace{4pt}\\eta=\\bigcap_{\\eta(u_1)=0}\\textrm{Ker}\\hspace{4pt}\\eta.$$\nLet $f(u_1,u_2,\\ldots,u_n)\\in\\left\\langle u_1\\right\\rangle''$ and\n$\\eta:\\hspace{4pt}\\eta(u_1)=0,\\hspace{4pt}\\eta(u_i)=u_i$ for\n$i>1$. Then $0=\\eta(f(u_1,u_2,\\ldots,u_n))=f(0,u_2,\\ldots,u_n)$,\nso $f\\in\\left\\langle u_1\\right\\rangle$, Q.E.D..\n\n\\begin{lll}\nLet $\\mu$ be an automorphism generating bijection of $W$ such that\n$\\mu(0)=0,\\hspace{2pt}\\mu(1)=1$ and $\\mu(x_i)=x_i$. Then\n$$\\left\\langle\\mu(\\alpha\nx_1)\\right\\rangle=\\left\\langle\\mu(x_1)\\right\\rangle=\\left\\langle\nx_1\\right\\rangle$$\n\\end{lll}\n\\textbf{Proof.} At first note that for any endomorphism\n$s\\in\\textrm{End}(W)$\n$$s(x_1)=0\\Leftrightarrow s^{\\tau}(x_1)=0.$$\nReally, if $s(x_1)=0$, then $s^{\\tau}(x_1)=s^{\\tau}\\mu(x_1)=\\mu\ns(x_1)=0$; and if $s^{\\tau}(x_1)=0$, then\n$0=s^{\\tau}(x_1)=s^{\\tau}\\mu(x_1)=\\mu s(x_1)=0$, i.e. $s(x_1)=0$.\n\nLet $\\mu(\\alpha x_1)=r_1(x_1,\\ldots,x_n)$. Take\n$s\\in\\textrm{End}(W)$ such that $s^{\\tau}(x_1)=0$ and\n$s^{\\tau}(x_i)=x_i$ for $i>1$. Then $0=\\mu(0)=\\mu s(\\alpha\nx_1)=s^{\\tau}\\mu(\\alpha\nx_1)=s^{\\tau}r_1(x_1,\\ldots,x_n)=r_1(0,x_1,\\ldots,x_n)$, i.e.\n$\\mu(\\alpha x_1)\\in\\left\\langle x_1\\right\\rangle''=\\left\\langle\nx_1\\right\\rangle$.\n\nNow let us prove, that $x_1\\in\\left\\langle\\mu(\\alpha\nx_1)\\right\\rangle$. Note that\n$$\\langle\\mu(\\alpha x_1)\\rangle=\\langle\\mu(\\alpha\nx_1)\\rangle''=\\bigcap_{\\eta^{\\tau}:\\hspace{2pt}\\eta^{\\tau}(\\mu(\\alpha\nx_1))=0}\\textrm{Ker}\\hspace{2pt}\\eta^{\\tau}.$$ So we must prove\nthat, if $\\eta^{\\tau}(\\mu(\\alpha x_1))=0$ then\n$\\eta^{\\tau}(x_1)=0$. Really,\n$$0=\\eta^{\\tau}(\\mu(\\alpha x_1))=\\mu\\eta(\\alpha\nx_1)=\\mu(\\alpha\\eta(x_1))\\Rightarrow\\alpha\\eta(x_1)=0\\Rightarrow\\eta^{\\tau}(x_1)=0.$$\nQ.E.D.\n\n\\begin{lll}\nLet $\\mu$ be an automorphism generating bijection of $W$ such that\n$\\mu(0)=0,\\hspace{2pt}\\mu(1)=1$ and $\\mu(x_i)=x_i$.Then for every\n$a\\in P$ and $u\\in P[X]$\n$$\\mu(au)=\\mu(a)\\mu(u).$$\n\\end{lll}\n\\textbf{Proof.} Since $\\langle\\mu(ax_1)\\rangle=\\langle\nx_1\\rangle$, so $\\mu(ax_1)=bx_1=b\\mu(x_1)$, where $w\\in P[X]$.\nSubstitution $s(x_1)=1$ gives that $b=\\mu(a)$, i.e.\n$\\mu(ax_1)=\\mu(a)\\mu(x_1)$.\n\nIn associative case we consider only the case, where the field\n$P$ is of characteristic 0. I must note, that $\\mu(ax_1)$ together\nwith $x_2,\\ldots.x_n$ is a base of algebra, and so\n$\\mu(ax_1)=bx_1+f(x_2,\\ldots,x_n)$ (see [7]). Because\n$\\mu(ax_1)\\in\\langle x_1\\rangle$, so $\\mu(ax_1)=bx_1$.\n\n\\begin{lll} With the preceding notation\n$$\\mu(u+v)=\\mu(u)+\\mu(v).$$\n\\end{lll}\n\\textbf{Proof.} Let\n$\\mu(x_1+x_2)-\\mu(x_1)-\\mu(x_2)=f(x_1,\\ldots,x_n)=g(\\mu(x_1),\\ldots,\\mu(x_n))$,\nthen $\\mu(u_1+u_2)-\\mu(u_1)-\\mu(u_2)=g(\\mu(u_1),\\ldots,\\mu(u_n)$.\nSubstitution $s:\\hspace{4pt}s(u_1)=u_1,s(u_2)=u_2,s(u_i)=0$ for\n$i>2$ gives identity\n$$\\mu(u_1+u_2)-\\mu(u_1)-\\mu(u_n)=\\overline{g}(\\mu(u_1),\\mu(u_2)).$$\nClearly, $\\overline{g}$ is homogenous of degree 1 of $u_1$ and\n$u_2$. Therefore $\\overline{g}(u_1,u_2)=au_1+bu_2$ and\n$$\\mu(u+v)=\\mu(u)+\\mu(v)+a\\mu(u)+b\\mu(v).$$\nSubstitutions $u=0$ and $v=0$ gives that $a=b=0$, Q.E.D..\\\\[6pt]\n\\textbf{Corollary.} Note that restriction $\\mu$ on $P$ is an\nautomorphism of $P$ and denote it $\\eta$. Let\n$\\mu'=\\eta^{-1}\\mu$, then $\\mu'$ is identical on $P$. So for\n$\\mu=\\mu'$ we have identity\n\\begin{equation}\n\\mu(au)=a\\mu(u),\n\\end{equation}\nfor every $a\\in P$ and $u\\in W(X)$.\n\n\\section{Description of  $\\textrm{Aut}(\\textrm{End}(W(X)))$}\n\nLet consider at first the commutative  case.\n\n\\begin{lll}\nLet $\\mu$ be a bijection of $P[X]$ generating an automorphism\n$\\tau\\in \\textrm{Aut}(\\textrm{End}(P[X]))$ such that $\\mu$ is\nidentical on $P$ and $\\mu(x_i)=x_i$. Then for every $u,v\\in P[X]$\nit holds that\n$$\\mu(u\\cdot v)=\\mu(u)\\cdot\\mu(v)$$\n\\end{lll}\n\\textbf{Proof.} Let $s$ be a constant endomorphism of $P[X])$.Then\n$s^{\\tau}$ also is constant and so\n$$s^{\\tau}(\\mu(x_1x_2)-\\mu(x_1)\\mu(x_2))=s^{\\tau}\\mu(x_1x_2)-s^{\\tau}\\mu(x_1)\\cdot\ns^{\\tau}\\mu(x_2)=$$\n$$\\mu s(x_1x_2)-\\mu s(x_1)\\cdot \\mu\ns(x_2)=\\mu(a_1a_2)-\\mu(a_1)\\cdot\\mu(a_2)=0,$$\n$$\\mu(x_1x_2)-\\mu(x_1)\\mu(x_2)\\in\\bigcap_{s^{\\tau}\\in\nConst}\\textrm{Ker}\\hspace{2pt}s^{\\tau}=0,$$ Q.E.D..\n\nNow we can formulate the main result for commutative case.\n\n\\begin{ttt}\nEvery automorphism of $\\textrm{End}(P[X])$ is semiinner.\n\\end{ttt}\n\\textbf{Proof.} It follows from lemma 3.4, lemma 4.1 and identity\n(1).\\\\[6pt]\n\nNow let consider associative case.\n\n\\begin{lll}\nLet $\\mu$ be a bijection of $W(X)$ generating an automorphism\n$\\tau\\in \\textrm{Aut}(\\textrm{End}(W(X)))$ such that $\\mu'$ is\nidentical on $P$ and $\\mu(x_i)=x_i$. Then for every $u,v\\in W(X)$\nit holds that\n$$\\mu(u\\cdot v)=\\mu(u)\\cdot\\mu(v),$$\nor for every $u,v\\in W(X)$ it holds that\n$$\\mu(u\\cdot v)=\\mu(v)\\cdot\\mu(u),$$\n\\end{lll}\n\\textbf{Proof.} Let\n$\\mu(x_1x_2)-\\mu(x_1)\\mu(x_1)=f(x_1,\\ldots,x_n)=g(\\mu(x_1),\\ldots,\\mu(x_n))=\\overline{g}(\\mu(x_1),\\mu(x_2)$.\nNote that $\\overline{g}$ is 1-homogenous of $x_1$ and 1-homogenous\nof $x_2$. Following the proof of lemma 3.4 we get identity\n$$\\mu(uv)=a\\mu(u)\\mu(v)+b\\mu(v)\\mu(u).$$\nSubstitution $v=1$ gives equality $a+b=1$. We shall denote\n$\\mu(u)=\\overline{u}$ and calculate $\\mu(xxy)$ .\n$$\\mu(xxy)=a\\overline{xx}\\cdot\\overline{y}+b\\overline{y}\\cdot\\overline{xx}=$$\n$$a(a+b)\\overline{x}\\cdot\\overline{x}\\cdot\\overline{y}+b(a+b)\\overline{y}\\cdot\\overline{x}\\cdot\\overline{x}=$$\n$$a\\overline{x}\\cdot\\overline{x}\\cdot\\overline{y}+b\\overline{y}\\cdot\\overline{x}\\cdot\\overline{x},$$\n$$\\mu(xxy)=a\\overline{x}\\cdot\\overline{xy}+b\\overline{xy}\\cdot\\overline{x}=$$\n$$a^2\\overline{x}\\cdot\\overline{x}\\cdot\\overline{y}+2ab\\overline{x}\\cdot\\overline{y}\\cdot\\overline{x}+b^2\\overline{y}\\cdot\\overline{x}\\cdot\\overline{xy}.$$\nSo $a=a^2,\\hspace{4pt}b=b^2,\\hspace{4pt}ab=0,\\hspace{4pt}a+b=1$\nand we have two solutions $a=1,b=0$ or $a=0,b=1$, Q.E.D..\n\n\nNow we can formulate the main result for associative algebras.\n\n\\begin{ttt}\nThe group $\\textrm{Aut}(\\textrm{End}(W(X)))$ is generated by\nsemiinner automorphisms and mirror automorphism.\n\\end{ttt}\n\\textbf{Proof.} It follows from lemma 3.4, lemma 4.2 and identity\n(1).\\\\[6pt]\n\n\\noindent{\\Large\\textbf{Acknowledgments}}\\\\[12pt]\nThe author is happy to thank professor B. Plotkin for stimulating\ndiscussions of the results.\\\\[30pt]\n\n\\noindent{\\Large\\textbf{References}}\n\\begin{enumerate}\n\n\\item A. Berzins, {\\it The automorphisms of \\,{\\rm End}{}$K[x]$}, Proc.\nLatvian Acad. Sci., Section B, 2003, vol. 57, no. 3\/4, pp. 78-81\n\n\\item A. Berzins, {\\it Geometric equivalence of algebras}, Int. J.\nAlg. Comput., 2001, vol. 11, no. 4, pp. 447-456.\n\n\\item A. Berzins, {\\it The group of automorphisms of semigroup\nEnd($P[X]$},(to appear)\n\n\\item A. Berzins, B. Plotkin and E. Plotkin, {\\it Algebraic\ngeometry in Varieties with the Given Algebra of constants}, J.\nMath. Sci., New York, 2000, vol. 102, no. 3, pp. 4039-4070\n\n \\item R. Lipyanski, B. Plotkin {\\it Automorphisms of categories\n of free modules and free Lie algebras}, ArXiv math.RA\/0502212 (10\n Feb 2005) pp. 14.\n\n \\item Y. Katsov, R. Lipyanski, B. Plotkin, {\\it Automorphisms of categories\n of free modules, free semimodules  and free Lie modules}, (to appear)\n\n\\item L. Makar-Limanov  {\\it Algebraically closed skew field}, Journal of Algebra, Vol.\n93, 1985, pp. 117-135.\n\n\\item G. Mashevitzky, {\\it Automorphisms\nof categories of free associative algebras.}, Preprint.\n\n\\item G. Mashevitzky, B. Plotkin, E. Plotkin, {\\it Automorphisms\nof categories of free Lie algebras.}, Journal of Algebra, Vol.\n282, 2004, pp. 490-512.\n\n\\item B. Plotkin, {\\it Algebraic logic, varieties of algebras and\nalgebraic varieties}, Proc. Int. Alg. Conf., St. Petersburg, 1995,\nWalter de Gruyter, New York, London, 1996.\n\n\\item B. Plotkin, {\\it Varieties of algebras and algebraic\nvarieties}, Israel J. of Mathematics, 1996, vol. 96, no. 2, pp.\n511-522.\n\n\\item B. Plotkin, {\\it Varieties of algebras and algebraic\nvarieties. Categories of algebraic varieties}, Sib. Adv. Math.,\n1997, vol. 7, no. 2, pp. 64-97.\n\n\\item B. Plotkin, {\\it Algebras with the same (algebraic)\ngeometry}, Proc. Steklov Inst. Math, Vol. 242, 2003, pp. 165-196.\n\n\\item B. Plotkin, G Zhitomirski, {\\it On automorphisms of\ncategories of universal algebras}, ArXiv math.CT\/0411408 (18 Nov\n2004). pp. 24.\n\n\\end{enumerate}\n\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nEquivalence principle acts as the guiding lighthouse to understand how matter fields behave in a curved spacetime background. Unfortunately, there exists no such principle which helps to answer the opposite, viz. how matter fields curve the spacetime \\cite{gravitation,LL2,Eddington:1924,MTW,Poisson,Wald,Hawking:1973uf}. Lack of such principle resulted in the vast landscape of various alternative gravity theories among which general relativity\\ remains the most useful one. Despite this large variety of alternative gravitational theories one can do better by imposing some physical requirements on the systems of interest. In particular, the restriction to the class of theories having utmost second order derivatives of the metric is a judicious one, since this helps to overcome the well known Ostrogradski instability by evading the existence of any ghost modes in the theory \\cite{Chen:2012au,Motohashi:2016ftl,Klein:2016aiq}. Surprisingly enough the above criteria turns out to be a very interesting one, as it singles out a \nvery specific class of unique gravitational theories, known in the literature as the Lanczos-Lovelock  models of gravity \\cite{Lanczos:1938sf,Lanczos:1932zz,Lovelock:1971yv,Padmanabhan:2011ex,\nPadmanabhan:2013xyr,Chakraborty:2014joa}. In particular, all these models satisfy the criteria that Bianchi derivative of the field tensor identically vanishes \\cite{Dadhich:2008df,Camanho:2015hea}. One can add to this list a large number of interesting properties, a few among them are --- (a) The action functional for Lanczos-Lovelock  gravity can be broken into a bulk part and a surface contribution, which is closely related to the Wald entropy associated with the black hole horizon \\cite{Padmanabhan:2004fq,Padmanabhan:2007en,Kothawala:2008in}; (b) The field equations in Lanczos-Lovelock  gravity can be expressed as a thermodynamic identity on an arbitrary null surface \\cite{Paranjape:2006ca,Akbar:2006er,Cai:2005ra,Ge:2007yu,Gong:2007md,Wu:2007se,\nKothawala:2009kc,Chakraborty:2010wn,Chakraborty:2015aja,Chakraborty:2015wma,\nHansen:2016gud,Dey:2016zka,Chakraborty:2016dwb}; (c) The difference between a suitably defined surface and bulk degrees of freedom in the context of Lanczos-Lovelock  gravity can be interpreted as the evolution of the spacetime \\cite{Padmanabhan:2009kr,Padmanabhan:2013nxa,Chakraborty:2014rga}; (d) The surface and bulk terms in the Lanczos-Lovelock  gravity are related by a holographic relation between them \\cite{Mukhopadhyay:2006vu,Dieks:2015jja}. \n\nIn the standard textbook treatment, one first introduces the gravitational Lagrangian $L$, which is $\\sqrt{-g}R$ for general relativity, while the Lagrangian is a polynomial in the Riemann tensor for general Lanczos-Lovelock  gravity. Variation of the action functional due to arbitrary variation of the metric, with appropriate boundary conditions \\cite{Parattu:2015gga,Parattu:2016trq,Chakraborty:2016yna,Chakraborty:2017zep} leads to the corresponding field equations for gravity, namely the rule by which matter tells the spacetime how to curve (see also \\cite{Deser:2009fq}). However, as emphasized earlier there is no physical principle which guides us to the precise mathematical structure of the gravitational action, what is more the field equations so obtained equates matter which is intrinsically \\emph{quantum} with spacetime geometry describing gravity \\emph{classically}. Despite this conceptual discomfort, there exist two additional representations of the gravitational dynamics, which are intrinsically observer dependent. The first one uses time like observers with four velocity $u^{a}$ and equates two quantities that this observer measures in the geometric as well as in the matter sector, leading to a scalar equation,\n\\begin{align}\\label{Eq_01}\nG_{ab}u^{a}u^{b}= \\kappa T_{ab}u^{a}u^{b}\n\\end{align}\nHere $\\kappa$ stands for the gravitational constant in four dimensions and enforcing this equation for all observers with four velocity $u_{i}$ leads to the Einstein's equations $G_{ab}=\\kappa T_{ab}$. The other approach uses null vectors $\\ell _{a}$ (i.e., $\\ell ^{2}=0$) and leads to the following scalar equation,\n\\begin{align}\\label{Eq_02}\nR_{ab}\\ell ^{a}\\ell ^{b}=\\kappa T_{ab}\\ell ^{a}\\ell ^{b}\n\\end{align}\nIn this situation as well validity of the above expression for all null vectors $\\ell _{a}$, along with two times contracted Bianchi identity and covariant conservation of matter energy-momentum tensor amounts to furnish the ten components of the Einstein's equations. Following \\cite{Padmanabhan:2016eld}, in this article, we would like to derive \\ref{Eq_01} and \\ref{Eq_02} from a geometrical point of view for Lanczos-Lovelock  theories of gravity. We would also argue why the route taken in this work is a natural one compared to the standard derivation of Einstein's equations from the gravitational action. We have organized the paper as follows: In \\ref{Einstein} we briefly review the derivation of Einstein's equations, which will be generalized in \\ref{Lovelock} to Lanczos-Lovelock  theories of gravity. Finally we provide the concluding remarks before presenting an derivation regarding the curvature tensor associated with the Lanczos-Lovelock  theory of gravity in the appendix.\n\\section{Einstein's equations following Newton's path}\\label{Einstein}\n\nTo set the stage for Lanczos-Lovelock  theories of gravity we would like to briefly review the corresponding situation in general relativity, which will mainly follow from \\cite{Padmanabhan:2016eld}. In Newtonian theory one describes gravity using the gravitational potential $\\phi(\\mathbf{x},t)$. Given the matter density $\\rho(\\mathbf{x},t)$, the dynamics of the gravitational field is being determined through the Poisson's equation $\\nabla ^{2}\\phi=(\\kappa \/2)\\rho$. When one invokes principle of equivalence, the potential $\\phi$ is readily identified with components of the metric, in particular, $g_{00}=-(1+2\\phi)$. One then interprets the Poisson's equation in metric language as $-\\nabla ^{2}g_{00}=\\kappa T_{00}$, where matter energy density is interpreted as the time-time component of the divergence free, symmetric, matter energy momentum tensor $T_{ab}$. In order to have a relativistic generalization one naively makes the following replacement $\\nabla ^{2}\\rightarrow \\square$. This immediately suggests to look for second derivatives of the metric which is a second rank tensor as well as divergence free, paving the way to the Einstein tensor. On the other hand, for the right hand side the natural choice being the matter energy momentum tensor $T_{ab}$, one ends up equating the Einstein's tensor with the matter energy momentum tensor resulting in the Einstein's equations (for a more detailed discussion, see \\cite{Padmanabhan:2016eld}).  \n\nHowever, we could have just followed Newton's path and look for relativistic generalization of Poisson's equation itself, possibly leading to a scalar equation describing gravity in general relativity. As a first step one must realize that the right hand side of Poisson's equation is intrinsically observer dependent, as it is the energy density of some matter field measured by an observer with some four velocity $u_{i}$. Given the symmetric and conserved matter energy momentum tensor $T_{ab}$, the matter energy density $\\rho$ as measured by an observer with four velocity $u_{a}$ is $T_{ab}u^{a}u^{b}$. Hence the same observer dependence must continue to exist in general relativity\\ and the right hand side of the desired general relativistic equation for gravity should be $T_{ab}u^{a}u^{b}$ \\cite{Padmanabhan:2016eld}. \n\nTo derive the left hand side, i.e., analog of $\\nabla ^{2}\\phi$ we note that this requires to derive a scalar object which involves two spatial derivatives of the metric (since this is what is responsible for the $\\nabla ^{2}\\phi$ term in non-relativistic limit) and necessarily depends on the four-velocity $u^{i}$. The only tensor depending on two derivative of the metric that can be constructed corresponds to the curvature tensor $R_{abcd}$. In order to obtain double spatial derivatives acting on the metric one has to project all the components of the curvature tensor on the plane orthogonal to $u_{a}$ using $h^{a}_{b}=\\delta ^{a}_{b}+u^{a}u_{b}$ and hence construct a scalar thereof. To keep generality, instead of looking for a timelike vector $u_{i}$, we will concentrate on a particular vector $\\ell ^{a}$, with norm $\\ell _{a}\\ell ^{a}\\equiv \\ell ^{2}$. Then one can immediately introduce the following tensor,\n\\begin{equation}\\label{Eq_03}\n\\mathbb{P}^{a}_{b}=\\delta ^{a}_{b}-\\frac{1}{\\ell ^{2}}\\ell ^{a}\\ell _{b};\\qquad \\mathbb{P}^{a}_{b}\\mathbb{P}^{b}_{c}=\\left(\\delta ^{a}_{b}-\\frac{1}{\\ell ^{2}}\\ell ^{a}\\ell _{b}\\right)\\left(\\delta ^{b}_{c}-\\frac{1}{\\ell ^{2}}\\ell ^{b}\\ell _{c}\\right)=\\delta ^{a}_{c}-\\frac{1}{\\ell ^{2}}\\ell ^{a}\\ell _{c}=\\mathbb{P}^{a}_{c}\n\\end{equation}\nfor which the final property ensures that it is a projector. Then one can define a projected Riemann curvature with respect to the vector $\\ell ^{a}$ by projecting all the indices of the Riemann curvature tensor, leading to\\footnote{By the very definition, the Riemann tensor $R_{abcd}$ has a generic form $\\partial _{b}\\partial _{c}g_{ad}$ in local inertial frame. Thus the above projection ensures that $g_{ab}$ has only double spatial derivatives when $\\ell _{a}$ is a timelike vector. All time derivatives appearing in $\\mathsf{R}_{abcd}$ are single in nature and hence vanishes in the local inertial frame all together. This being the prime motivation of introduction of this projected Riemann tensor. The same can be ascertained from the Gauss-Codazzi equation as well, see \\cite{gravitation,Poisson}.}\n\\begin{equation}\n\\mathsf{R}_{mnrs}=\\mathbb{P}^{a}_{m}\\mathbb{P}^{b}_{n}\\mathbb{P}^{c}_{r}\\mathbb{P}^{d}_{s}R_{abcd}\n\\end{equation}\nThe scalar constructed out of this projected Riemann tensor becomes,\n\\begin{equation}\n\\mathsf{R}=\\mathbb{P}_{mr}\\mathbb{P}_{ns}R^{mnrs}=R-\\frac{2}{\\ell ^{2}}R_{ab}\\ell ^{a}\\ell ^{b}\n\\end{equation}\nNote that we have thrown away another term involving $R_{abcd}\\ell ^{a}\\ell ^{b}\\ell ^{c}\\ell ^{d}$, since due to antisymmetry properties of the Riemann tensor the above quantity identically vanishes. Furthermore, the above scalar \\emph{by} construction has only two derivatives of the metric and is completely spatial in the instantaneous rest frame of the time-like observer. Hence $\\mathsf{R}$ is the relativistic generalization of $\\nabla ^{2}\\phi$ and thus must be equal to the corresponding matter energy density, which in this case corresponds to, $-(2\\kappa\/\\ell ^{2})T_{ab}\\ell ^{a}\\ell ^{b}$, where $T_{ab}$ is the matter energy momentum tensor. Thus finally one obtains the following equation,\n\\begin{equation}\nR_{ab}\\ell ^{a}\\ell ^{b}-\\frac{1}{2}R\\ell ^{2}=\\kappa T_{ab}\\ell ^{a}\\ell ^{b}\n\\end{equation}\nSurprisingly, through the above exercise we have achieved two results in one go. Firstly, for $\\ell ^{2}=-1$, i.e., for unit normalized timelike vectors we get back the equation $G_{ab}u^{a}u^{b}=\\kappa T_{ab}u^{a}u^{b}$. Then requiring these equations to hold for all timelike observers will lead to $G_{ab}=\\kappa T_{ab}$. At this stage it will be worthwhile to mention that a similar approach as the above one, was taken in \\cite{Straumann:2013} to derive the Einstein's equations. However the key difference between the approach in \\cite{Straumann:2013} and in this work being use of the projection tensor $\\mathbb{P}^{a}_{b}$ judiciously. For our approach (also see \\cite{Padmanabhan:2016eld}) the use of projection tensor is of utmost importance in contrast to \\cite{Straumann:2013}.\n\nFinally the limit $\\ell ^{2}\\rightarrow 0$ would lead to the null version of the above equation, which reads, $R_{ab}\\ell ^{a}\\ell ^{b}=\\kappa T_{ab}\\ell ^{a}\\ell ^{b}$. In this case besides demanding the validity of the above equation for all null vectors, it is important to use contracted Bianchi identity as well as covariant conservation of energy-momentum tensor, leading to: $G_{ab}+\\Lambda g_{ab}=\\kappa T_{ab}$. Note that in the second situation a cosmological constant has automatically come into existence.\n\\section{Newton leads way to Lovelock}\\label{Lovelock}\n\nIn view of the above result, one immediately asks for the corresponding situation in Lovelock gravity: Can field equations of Lovelock gravity be derived following the above procedure? This is what we will explore in this section and shall show that one can indeed derive the field equations for Lovelock gravity following an equivalent procedure. Before jumping into the details let us mention some structural aspects of Lovelock gravity. Briefly speaking, Lovelock gravity corresponds to a class of gravitational Lagrangians which are polynomial in the Riemann curvature tensor yielding field equations which are second order in the metric. The analytical form of such a polynomial (also called a pure Lovelock term) of order $m$ involves $m$ Riemann curvature tensors contracted appropriately, such that\n\\begin{align}\nL^{(m)}=\\frac{1}{2^{m}}\\delta ^{a_{1}b_{1}\\cdots a_{m}b_{m}}_{c_{1}d_{1}\\cdots c_{m}d_{m}}R^{c_{1}d_{1}}_{a_{1}b_{1}}\\cdots R^{c_{m}d_{m}}_{a_{m}b_{m}}\\equiv \\frac{1}{m}P^{ab~(m)}_{cd}R^{cd}_{ab}\n\\end{align}\nThe above relation defines the tensor $P^{ab~(m)}_{cd}$ associated with the $m$th order Lanczos-Lovelock  gravity, having all the symmetries of the Riemann tensor with the following algebraic structure\n\\begin{align}\\label{Entropy_Tensor}\nP^{ab~(m)}_{cd}=\\frac{m}{2^{m}}\\delta ^{aba_{2}b_{2}\\cdots a_{m}b_{m}}_{cdc_{2}d_{2}\\cdots c_{m}d_{m}}R^{c_{2}d_{2}}_{a_{2}b_{2}}\\cdots R^{c_{m}d_{m}}_{a_{m}b_{m}}\n\\end{align}\nThe tensor $P^{ab~(m)}_{cd}$ satisfies an additional criteria $\\nabla _{a}P^{ab~(m)}_{cd}=0$, which ensures that the field equations derived from this Lagrangian are of second order. In what follows we will exclusively concentrate on the $m$th order Lanczos-Lovelock  gravity and hence shall remove the symbol ``$(m)$'' from the superscripts of various geometrical expressions.\n\nOne can now start from the Poisson's equation and fix the right hand side to be $-T_{ab}\\ell ^{a}\\ell ^{b}\/\\ell ^{2}$, which is the matter energy density associated with the vector field $\\ell ^{a}$. In order to get the left hand side we must construct an appropriate curvature tensor suited for the Lovelock gravity and project it using the projector $\\mathbb{P}^{a}_{b}$ introduced in \\ref{Eq_03} before constructing a scalar out of it. There are two possible choices for such a curvature tensor among which we will discuss the simpler one in the following, while the complicated one is deferred to \\ref{App_01}. The curvature tensor described here was first introduced in \\cite{Dadhich:2008df,Camanho:2015hea} and is defined as follows,\n\\begin{align}\n\\mathtt{R}^{ab}_{cd}=\\frac{1}{2}\\left(P^{ab}_{mn}R^{mn}_{cd}+P^{mn}_{cd}R_{mn}^{ab}\\right)-\\frac{(m-1)}{(d-1)(d-2)}\\left(\\delta ^{a}_{c}\\delta ^{b}_{d}-\\delta ^{a}_{d}\\delta ^{b}_{c}\\right)L\n\\end{align}\nwhere, $L$ is the $m$th order Lovelock polynomial and $P^{ab}_{cd}=\\partial L\/\\partial R^{cd}_{ab}$. \nSince the $m$th order Lanczos-Lovelock  Lagrangian depends on $m$ powers of Riemann, the above definition for $P^{ab}_{cd}$ ensures that it depends on $(m-1)$ powers of Riemann tensor and hence exactly coincides with \\ref{Entropy_Tensor}. Furthermore, the tensor $P^{ab}_{cd}$ can be easily generalized to the full Lovelock polynomial by just adding over different $m$ values, but we will concentrate on a single term in the full Lanczos-Lovelock  Lagrangian. Also note that the above defined `Lovelock' Riemann tensor has all the symmetries of the original `Einstein' Riemann tensor $R^{ab}_{cd}$. Further it satisfies the contracted Bianchi identity, which will be sufficient for our purpose. Given the above $m$th order `Lovelock' Riemann, one can project it in the plane orthogonal to $\\ell _{a}$ and obtain the following scalar\n\\begin{equation}\n\\mathbb{R}=\\mathtt{R}^{ab}_{cd}\\mathbb{P}^{c}_{a}\\mathbb{P}^{d}_{b}\n\\end{equation}\nExplicit evaluation of the above scalar can be performed keeping in mind that the tensor $\\mathtt{R}^{ab}_{cd}$ is antisymmetric under exchange of the indices $(a,b)$ and $(c,d)$ respectively, leading to\n\\begin{align}\\label{Eq_Love_F_02}\n\\mathbb{R}&=\\mathtt{R}^{ab}_{cd}\\delta ^{c}_{a}\\delta ^{d}_{b}-\\frac{2}{\\ell ^{2}}\\mathtt{R}^{ab}_{cd}\\ell^{c}\\ell_{a}\\delta ^{d}_{b}\n\\nonumber\n\\\\\n&=P^{ab}_{cd}R^{cd}_{ab}-\\frac{(m-1)}{(d-1)(d-2)}(d^{2}-d)L\n-\\frac{2}{\\ell ^{2}}\\left\\lbrace P^{ab}_{mn}R^{mn}_{cb}-\\frac{(m-1)}{(d-1)(d-2)}\\left(d\\delta ^{a}_{c}-\\delta ^{a}_{c}\\right)L \\right\\rbrace \\ell_{a}\\ell^{c}\n\\nonumber\n\\\\\n&=mL-\\frac{d(m-1)}{d-2}L-\\frac{2}{\\ell ^{2}}P^{ab}_{mn}R^{mn}_{cb}\\ell_{a}\\ell^{c}+2\\frac{m-1}{d-2}L=-\\frac{2}{\\ell ^{2}}P^{ab}_{mn}R^{mn}_{cb}\\ell _{a}\\ell ^{c}+L\n\\end{align}\nThus equating \\ref{Eq_Love_F_02} to the matter energy density through $-(2\\kappa\/\\ell ^{2})T_{ab}\\ell ^{a}\\ell ^{b}$ we finally obtain,\n\\begin{align}\nP^{a}_{bmn}R_{c}^{bmn}\\ell _{a}\\ell ^{c}-\\frac{1}{2}L\\ell ^{2}=\\kappa T_{ab}\\ell ^{a}\\ell ^{b}\n\\end{align}\nFor unit normalized time-like vectors, $\\ell _{a}=u_{a}$ and $\\ell ^{2}=-1$, leading to $E_{ab}u^{a}u^{b}=\\kappa T_{ab}u^{a}u^{b}$, where $E_{ab}\\equiv P_{a}^{~pqr}R_{bpqr}-(1\/2)Lg_{ab}$ is the analogue of Einstein tensor in Lovelock gravity. On the other hand, for the null vectors we arrive at $\\mathcal{R}_{ab}\\ell ^{a}\\ell ^{b}=\\kappa T_{ab}\\ell ^{a}\\ell ^{b}$, where the tensor $\\mathcal{R}_{ab}\\equiv P_{a}^{~pqr}R_{bpqr}$ is the analogue of Ricci tensor in Lovelock gravity. Hence even in the case of Lovelock gravity, if one assumes that $E_{ab}u^{a}u^{b}=\\kappa T_{ab}u^{a}u^{b}$ holds for all timelike observers, the field equations for Lovelock gravity: $E_{ab}=\\kappa T_{ab}$ follows. While in the null case, besides demanding the validity of $E_{ab}\\ell^{a}\\ell^{b}=\\kappa T_{ab}\\ell^{a}\\ell^{b}$ for all null vectors, one have to use the Bianchi identity associated with Lovelock theories \\cite{Padmanabhan:2011ex} as well as covariant conservation of matter energy-momentum tensor to arrive at the Lovelock field equations:  $E_{ab}+\\Lambda g_{ab}=\\kappa T_{ab}$, which inherits the cosmological constant as well. Note that the above result has been derived in the context of $m$th order Lanczos-Lovelock  gravity, which can be generalized to the general Lanczos-Lovelock  Lagrangian in a straightforward manner. This results into: $\\{\\sum _{m}E_{ab}^{(m)}\\}+\\{\\sum _{m}\\Lambda ^{(m)}\\}g_{ab}=\\kappa T_{ab}$. Interestingly, the cosmological constant besides being generated as an integration constant of the field equations also gets contribution from the $m=0$ term in the Lanczos-Lovelock  gravity. Therefore one may choose this term in the Lagrangian appropriately to arrive at the present small value of the cosmological constant. Therefore we conclude that the derivation of field equations for gravity can always be achieved starting from the Poisson's equation and subsequently projecting a suitable curvature tensor, whether it is Einstein gravity or Lovelock. \n\\section{Concluding Remarks}\\label{Conclusion}\n\nBy equivalence principle gravity manifests itself by curving the spacetime, which the material particles follow. In particular one can invoke special relativity in locally freely falling frame and hence write down the laws of motion in curvilinear coordinates, thus describing motion in curved spacetime. The notion of locally freely falling observer brings in intrinsic observer dependence in the theory and introduces observers for whom a local spacetime region is causally inaccessible, known as a local Rindler observer. Remarkably the local vacuum state of a test quantum field (as fit for local inertial observers) will appear as thermal to the local Rindler observer \\cite{Davies:1974th,Unruh:1976db} (however also see \\cite{Lochan:2016cxt}). If any matter field (characterized by matter energy momentum tensor $T^{a}_{b}$) crosses the local Rindler horizon, it will appear to be thermalized by the Rindler observer (since the matter will take infinite time to reach the horizon) and the corresponding heat density is being given by $T^{b}_{a}\\ell ^{a}\\ell _{b}$ (for a perfect fluid the above quantity is given by $\\rho+p$, which by Gibbs-Duhem relation is the matter heat density), where $\\ell _{a}$ is the null normal to the horizon. Note that the above heat density for matter is invariant under the transformation $T^{a}_{b}\\rightarrow T^{a}_{b}+(\\textrm{constant})\\delta ^{a}_{b}$.\n\nAt this stage, one can ask a natural question, ``what about heat density of gravity?''. Surprisingly, one can answer the same in the above setting. Considering a general null surface it turns out that one can interpret $R^{a}_{b}\\ell ^{b}\\ell _{a}$ as the heat density of the spacetime. This originates from the fact that one can have a one to one correspondence between $R_{ab}\\ell ^{a}\\ell ^{b}$ and the viscous dissipation term $2\\eta \\sigma _{ab}\\sigma ^{ab}+\\zeta \\theta ^{2}$, where $\\sigma_{ab}$ is the shear of the null congruence $\\ell _{a}$ and $\\theta$ is its expansion, with $\\eta$ and $\\zeta$ being shear and bulk viscous coefficients. Thus the term $R_{ab}\\ell ^{a}\\ell ^{b}$ is related to heating of the spacetime \\cite{Chakraborty:2015hna,Padmanabhan:2010rp,Kolekar:2011gw,Bhattacharya:2017mrg,Bhattacharya:2015qkt}. Given this thermodynamic backdrop, it is clear that the Einstein's equations when written as \\ref{Eq_02} not only yields the geometrical input of the gravitational theory but is also \nphysically well-motivated since the equality of \\ref{Eq_02} can be thought of as an equilibrium \nsituation, where the heat produced by gravity is being compensated by that of matter. \n\nThus the equation $2G_{ab}\\ell ^{a}\\ell ^{b}=T_{ab}\\ell ^{a}\\ell ^{b}$ not only arises more naturally from the relativistic generalization of Newton's law but the usefulness of writing Einstein's equations in this manner stems from the fact that one might interpret both sides of these equations \\emph{independently} and the equations themselves follow due to a balancing act performed by spacetime itself \\cite{Padmanabhan:2014jta,Padmanabhan:2015lla,Padmanabhan:2015pza,\nChakraborty:2015hna,Padmanabhan:2016eld,Wang:2015cna,Navia:2017ltw}. We would like to reiterate that the field equations derived in this context is purely geometrical and follows Newton's path. One first realizes that energy density associated with any material body is intrinsically observer dependent and surprisingly one can construct a tensor (again dependent on observer) which contains spatial derivatives of the metric alone. Keeping this as a curved spacetime generalization of Newton's law one uniquely arrives at Einstein's equations when one make use of all observers (or all the null surfaces). This shows that the most natural generalization of Newton's law to curved spacetime is $R_{ab}\\ell ^{a}\\ell ^{b}=8\\pi T_{ab}\\ell ^{a}\\ell ^{b}$ (of course, leading to Einstein's equations, but at a secondary level) bolstering the claim that gravity is intrinsically a thermodynamic phenomenon.\n\\section*{Acknowledgements}\n\nS.C. gratefully acknowledges the help of T. Padmanabhan for suggesting this project and also for helpful discussions throughout the same. He also thanks Naresh Dadhich, Kinjalk Lochan and Krishnamohan Parattu for various fruitful discussions. A part of this work was completed while the author was visiting Albert Einstein Institute in Golm, Germany and the author gratefully acknowledges the warm hospitality there. Research of S.C. is supported by the SERB-NPDF grant (PDF\/2016\/001589) from DST, Government of India.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nThe abundance of heavy elements in the H~II regions of galaxies\nreflects the past history of star formation and the effects of\ninflows and outflows of gas. A characterization of the evolution\nof chemical abundances for galaxies of different masses is\ntherefore essential to a complete model of galaxy formation that\nincludes the physics of baryons \\citep{delucia04,finlator07}.\nImportant observational constraints for such models come from\ndetermining the scaling relations at different redshifts among\ngalaxy luminosity, stellar mass, and metallicity, which, for\nstar-forming galaxies, typically consists of the oxygen abundance.\nHowever, one of the key challenges is to take the observationally\nmeasured quantities, i.e. strong, rest-frame optical emission-line\nratios, and connect them with the physical quantity of interest,\ni.e. oxygen abundance.\n\n\n\n\\begin{deluxetable*}{lcccccccc}\n\\tablewidth{0pt} \\tabletypesize{\\footnotesize}\n\\tablecaption{Galaxies Observed with Keck~II\nNIRSPEC\\label{tab:obs}}\n\\tablehead{\n\\colhead{~~~~~~~~~~DEEP ID~~~~~~~~~~} &\n\\colhead{~~~~~R.A. (J2000)~~~~~} &\n\\colhead{~~~~~Dec. (J2000)~~~~~} &\n\\colhead{~~~~~$z_{{\\rm H}\\alpha}$~~~~~} &\n\\colhead{~~~~~$B$~~~~~} &\n\\colhead{~~~~~$R$~~~~~} &\n\\colhead{~~~~~$I$~~~~~} &\n\\colhead{~~~~~$M_B$~~~~~} &\n\\colhead{~~~~~$U-B$~~~~~}\n}\n\\startdata\n    42044579 \\dotfill   &    02 30 43.46   &   00 42 43.60 & 1.0180 &   23.22   &    22.97   &    22.40  &  -21.20 &   0.54 \\\\\n    22046630 \\dotfill   &    16 50 13.83   &   35 02 01.78 & 1.0225 &   23.64   &    23.02   &    22.31  &  -21.37 &   0.69 \\\\\n    22046748 \\dotfill   &    16 50 14.55   &   35 02 04.31 & 1.0241 &   24.43   &    23.76   &    22.90  &  -20.86 &   0.86 \\\\\n    42044575 \\dotfill   &    02 30 44.85   &   00 42 51.33 & 1.0490 &   23.08   &    22.94   &    22.56  &  -21.06 &   0.34 \\\\\n    42010638 \\dotfill   &    02 29 08.74   &   00 23 28.40 & 1.3877 &   22.93   &    22.85   &    22.54  &  -22.12 &   0.49 \\\\\n    42010637 \\dotfill   &    02 29 08.74   &   00 23 32.87 & 1.3882 &   24.20   &    23.98   &    23.72  &  -20.87 &   0.44 \\\\\n    42021412 \\dotfill   &    02 30 44.55   &   00 30 50.73 & 1.3962 &   24.07   &    23.74   &    23.12  &  -21.91 &   0.78 \\\\\n    42021652 \\dotfill   &    02 30 44.70   &   00 30 46.19 & 1.3984 &   22.97   &    22.24   &    21.32  &  -24.01 &   1.01 \\\\\n\\enddata\n\\tablecomments{Units of right ascension are hours, minutes, and seconds, and units of declination are degrees, arcminutes and arcseconds.}\n\\end{deluxetable*}\n\n\nIn the local universe, \\citet{tremonti04} have used a sample of\n$\\sim$53,000 emission-line galaxies from the Sloan Digital Sky\nSurvey (SDSS) to investigate the luminosity-metallicity ($L-Z$)\nand mass-metallicity ($M-Z$) relationships. For this sample,\nmetallicities were estimated on the observed spectra of several\nstrong emission lines, including [O II] $\\lambda\\lambda$3726,\n3729, H$\\beta$, [O III]$ \\lambda\\lambda$5007, 4959, H$\\alpha$, [N\nII] $\\lambda\\lambda$6548, 6584, and [S II] $\\lambda\\lambda$6717,\n6731. At increasing redshifts, as the strong rest-frame optical\nemission lines shift into the near-IR, metallicities are typically\nbased on smaller subsets of strong emission lines through the use\nof empirically calibrated abundance indicators\n\\citep[e.g.][]{pp04,pagel79}. Much progress has been made recently\nin assembling large samples of star-forming galaxies with\nabundance measurements at both intermediate redshift\n\\citep{kewley02,savaglio05} and at $z>2$ \\citep{erb06a}. However,\nwe have only begun to gather chemical abundance measurements for\ngalaxies at $z \\sim 1-2$ \\citep[][hereafter Paper\nI,]{maier06,shapley05}. In this work, we continue our efforts to\nfill in the gap of chemical abundance measurements during this\nimportant redshift range, which hosts the emergence of the Hubble\nsequence of disk and elliptical galaxies \\citep{dickinson00}, and\nthe buildup of a significant fraction of the stellar mass in the\nuniverse \\citep{drory05,dickinson03} prior to the decline in\nglobal star formation rate (SFR) density \\citep{madau96}.\n\n\n\n\n\nChemical abundances for high-redshift galaxies are commonly\nestimated using locally calibrated empirical indicators. Yet it is\ncrucial to recognize the fact that a considerable fraction of the\n$z \\sim 1$ and $2$ galaxies with measurements of multiple\nrest-frame optical emission lines do not follow the local\nexcitation sequence described by nearby H~II regions and\nstar-forming galaxies in the diagnostic diagram featuring the [O\nIII] $\\lambda$5007\/H$\\beta$ and [N II] $\\lambda$6584\/H$\\alpha$\nemission-line ratios \\citep[Paper I;][]{erb06a}. On average, the\ndistant galaxies lie offset towards higher [O III]\n$\\lambda$5007\/H$\\beta$ and [N II] $\\lambda$6584\/H$\\alpha$,\nrelative to local galaxies. As discussed in Paper I and\n\\citet{groves06}, several causes may account for this offset, in\nterms of the prevailing physical conditions in the H~II regions of\nhigh-redshift galaxies. The relevant conditions are H~II region\nelectron density, hardness of the ionizing spectrum, ionization\nparameter, the effects of shock excitation, and contributions from\nan active galactic nucleus (AGN). It is still unclear which of\nthese are most important for determining the emergent spectra of\nhigh- redshift galaxies. Understanding this offset in\nemission-line ratios is important, not only because it provides\nevidence that physical conditions in the high redshift universe\nare different from the local ones, but also because the\napplication of an empirically calibrated abundance indicator to a\nset of H~II regions or star-forming galaxies rests on the\nassumption that these objects are similar, on average, to those on\nwhich the calibration is based.\n\nIn this sense, the current work is also motivated by the\ninterpretation of the offset in emission-line ratios among distant\ngalaxies, and an assessment of the reliability of using local\nabundance calibrations for high-redshift star-forming galaxies.\nInstead of focusing on high-redshift objects, another approach is\nto study the properties in a class of nearby objects, which\nexhibit similar offset behavior on the emission-line diagnostic\ndiagram. Unravelling the relations between the physical conditions\nand unusual diagnostic line ratios for such objects aids the\nunderstanding of high-redshift galaxies. The SDSS, with its rich\nset of photometric and spectroscopic information, provides an\nideal local comparison sample.\n\n\nIn this paper we expand on the analysis presented in Paper I, with\nan enlarged sample of DEEP2 star-forming galaxies observed with\nNIRSPEC on the Keck II telescope. The larger number of DEEP2\nobjects with near-IR observations enforces the conclusions drawn\nin the previous work. Furthermore, our detailed study of nearby\nSDSS objects with similar emission-line diagnostic ratios leads to\na clearer physical explanation of the observed properties of the\nDEEP2 galaxies. The DEEP2 sample, near-IR spectroscopic\nobservations, data reduction, and measurements are described in \\S\n2. We present the oxygen abundances derived from measurements of\n[O III], H$\\beta$, H$\\alpha$, and [N II] emission lines in both\nindividual as well as composite spectra in \\S 3. The\nmass-metallicity relationship and its evolution through cosmic\ntime are discussed in \\S 4. In \\S 5 we investigate differences in\n$z \\sim 1.0-1.5$ H II region physical conditions with respect to\nlocal samples by examining nearby SDSS galaxies with similar\nemission-line diagnostic ratios. Finally, we summarize our main\nconclusions in \\S 6. A cosmology with $\\Omega_m = 0.3$,\n$\\Omega_{\\Lambda} = 0.7$, and $h = 0.7$ is assumed throughout.\n\n\\section{DEEP2 Sample, Observations, and Data Reduction}\n\n\\subsection{DEEP2 Target Sample and Near-IR Spectroscopy}\n\nThe high-redshift galaxies presented in this paper are drawn from\nthe DEEP2 Galaxy Redshift Survey \\citep[hereafter\nDEEP2;][]{davis03,faber05}, which contains $>30,000$ galaxies with\nhigh-confidence redshifts at $0.7 \\leq z \\leq 1.5$ down to a\nlimiting magnitude of $R_{AB}=24.1$. The motivation for our\nfollow-up near-infrared spectroscopic program, along with detailed\ndescriptions of the sample selection, optical and near-IR\nphotometry, and spectroscopy are presented in Paper I. Only a\nbrief overview is given here.\n\n\n\nThe new sample contains four galaxies at $z \\sim 1.0$ and four\ngalaxies at $z \\sim 1.4$, which, in combination with the pilot\nprogram presented in Paper I, leads to a sample of 20 galaxies in\ntotal. These galaxies are located in fields 2, 3, and 4 of the\nDEEP2 survey, at 16, 23, and 2 hr right ascension, respectively.\nTo probe chemical abundances and H~II region physical conditions,\nobservations of several strong H~II region emission lines are\nrequired, ideally at least [O II], H$\\beta$, [O III], H$\\alpha$,\nand [N II]. At $z \\geq 0.85$, however, the only strong H~II region\nemission feature contained in the DEEP2 DEIMOS spectroscopic data\nis the [O II] doublet.  Therefore, near-IR spectroscopy is needed\nto measure longer wavelength H~II region emission lines at $z \\geq\n1$. We target two narrow redshift windows within the larger DEEP2\nredshift distribution: $0.96 \\leq z \\leq 1.05$ and $1.36 \\leq z\n\\leq 1.50$, within which it is possible to measure the full set of\nH$\\beta$, [O III], H$\\alpha$, and [N II], in spite of the bright\nsky lines and strong atmospheric absorption in the near-IR.\n\n\\begin{figure*}\n\\epsscale{.8} \\plotone{f1.eps} \\caption{DEEP2 color-magnitude and\nmagnitude-stellar mass diagrams. The rest-frame $(U - B)$ vs.\n$M_B$ color-magnitude diagrams ({\\it top}) and the $M_B$ vs.\nstellar mass diagrams ({\\it bottom}) are for DEEP2 galaxies at\n$0.96 \\le z \\le 1.05$ ({\\it left}) and $1.36 \\le z \\le 1.50$ ({\\it\nright}). In each plot, DEEP2 galaxies from both the pilot sample\npresented in Paper I as well as objects with new NIRSPEC\nobservations are indicated with red squares. As shown here, all\nNIRSPEC targets were drawn from the ``blue cloud'' of the color\nbimodality. \\label{fig:colormag}.}\n\\end{figure*}\n\n\nThe absolute $B$ magnitude, $M_B$, and stellar mass estimates are\ngiven in Tables \\ref{tab:obs} and \\ref{tab:emi} for the new\nobjects and plotted as red squares in the lower panels of Figure\n\\ref{fig:colormag}, together with the data for the pilot sample.\nFor all the objects, we use the $M_B$ values from\n\\citet{willmer06} based on optical data alone and confirm their\ngood agreement with estimates based on fits to the $BRIK_S$ SEDs\nthat span through rest-frame $J$ ($I$) band for the objects at $z\n\\sim 1.0$ ($z \\sim 1.4$). Stellar masses for the objects in our\nsample are derived with $K_S$-band photometry, following the\nprocedure outlined in \\citet{bundy05}, which assumes a\n\\citet{chabrier03} stellar initial mass function (IMF). As\ndiscussed in detail in Paper I, the \\citet{bundy05} stellar mass\nmodelling technique agrees well with that used by\n\\citet{kauffmann03a} for SDSS galaxies, based on both spectral\nfeatures and broadband photometry. Stellar masses for galaxies in\nthe pilot sample have been updated to reflect both the most\ncurrent DEEP2 near-IR photometric catalog, and population\nsynthesis models limiting the stellar population age to be younger\nthan the age of the Universe at $z\\sim 1$. Thus, in a few cases,\nthe stellar masses differ slightly from those presented in Table\n(2) of Paper I. As shown in the lower panels of Figure\n(\\ref{fig:colormag}), the $z \\sim 1.4$ galaxies in our sample span\nthe full range of absolute $B$ luminosities in the DEEP2 survey,\nfrom $M_{B} \\sim$ -20 to -23, while the smaller set of $z \\sim\n1.0$ galaxies happens to cover the faint end of the luminosity\nfunction. Galaxies in both redshift intervals cover more than an\norder of magnitude in stellar mass and therefore should be able to\nprobe an interesting dynamic range. Since our goal was to study\nthe emission-line properties of galaxies, all objects in our\nsample lie in the blue component of the observed $(U-B)$ color\nbimodality in the DEEP2 survey, as shown in the upper panels of\nFigure \\ref{fig:colormag}.\n\n\n\n\\begin{deluxetable*}{lcccccccccr}\n\\tabletypesize{\\scriptsize}\n\\tablecaption{Emission Lines and Physical Quantities.\\label{tab:emi}}\n\\tablewidth{0pt}\n\\tablehead{\n     &  &  &  &  &  &\n    \\multicolumn{2}{c}{12 + log(O\/H)} &  &  &  \\\\\n    \\cline{7-8} \\\\\n    \\colhead{DEEP ID} & \\colhead{z$_{{\\rm H}\\alpha}$} & \\colhead{F$_{{\\rm H}\\beta}$\\tablenotemark{a}} &\n    \\colhead{F$_{{\\rm [O III]}\\lambda5007}$\\tablenotemark{a}} & \\colhead{F$_{{\\rm H}\\alpha}$\\tablenotemark{a}} &\n    \\colhead{F$_{{\\rm [N II]}\\lambda6584}$\\tablenotemark{a}} & \\colhead{N2\\tablenotemark{b}} &\n    \\colhead{O3N2\\tablenotemark{c}} & \\colhead{$L_{H\\alpha}$\\tablenotemark{d}} &\n    \\colhead{SFR$_{{\\rm H}\\alpha}$\\tablenotemark{e}}  &\n    \\colhead{log$(M_{\\ast}\/M_{\\odot})$\\tablenotemark{f}}\n    }\n\\startdata\n42044579 & 1.0180 & 2.4$\\pm$0.4 &  5.4$\\pm$0.3 & 12.1$\\pm$0.3 & 2.8$\\pm$0.3 & 8.54$\\pm$0.18 & 8.41$\\pm$0.14 & 0.7 &  3  & 10.26$\\pm$0.15 \\\\\n22046630 & 1.0225 & 3.7$\\pm$0.5 &  8.7$\\pm$0.4 & 15.2$\\pm$0.3 & 1.8$\\pm$0.3 & 8.37$\\pm$0.18 & 8.32$\\pm$0.14 & 0.8 &  3  & 10.29$\\pm$0.16 \\\\\n22046748 & 1.0241 & 2.2$\\pm$0.4 &  6.4$\\pm$0.3 &  9.6$\\pm$0.2 & 2.3$\\pm$0.2 & 8.55$\\pm$0.18 & 8.39$\\pm$0.14 & 0.5 &  2  & 10.28$\\pm$0.10 \\\\\n42044575 & 1.0490 & 7.7$\\pm$0.7 & 22.9$\\pm$0.3 & 23.6$\\pm$0.3 & 3.6$\\pm$0.3 & 8.43$\\pm$0.18 & 8.32$\\pm$0.14 & 1.4 &  6  &  9.74$\\pm$0.06 \\\\\n42010638 & 1.3877 & 3.8$\\pm$0.9 & 17.3$\\pm$0.4 & 16.6$\\pm$0.5 & 2.8$\\pm$0.5 & 8.46$\\pm$0.19 & 8.27$\\pm$0.15 & 2.0 &  9  & 10.21$\\pm$0.06 \\\\\n42010637 & 1.3882 & 3.3$\\pm$0.9 &  3.7$\\pm$0.3 &  6.4$\\pm$0.4 & 2.9$\\pm$0.4 & 8.70$\\pm$0.18 & 8.60$\\pm$0.15 & 0.8 &  4  & 9.96$\\pm$0.12 \\\\\n42021412 & 1.3962 & 7.1$\\pm$0.8 &  $>$ 2.0     & 12.7$\\pm$0.3 & 2.9$\\pm$0.3 & 8.54$\\pm$0.18 &  $<$ 8.70     & 1.5 &  7  & 10.89$\\pm$0.13 \\\\\n42021652\\tablenotemark{g} & 1.3984 & 4.6$\\pm$0.5 &  9.7$\\pm$0.5 &  9.5$\\pm$0.3 & 1.2$\\pm$0.3 & 8.38$\\pm$0.19 & 8.33$\\pm$0.15 & 1.1 & 5 & ... \\\\\n\\enddata\n\\tablenotetext{a}{Emission-line flux and random error in units of $10^{-17}$ ergs s$^{-1}$ cm$^{-2}$.} \n\\tablenotetext{b}{Oxygen abundance deduced from the N2 relationship presented in \\citet{pp04}.} \n\\tablenotetext{c}{Oxygen abundance deduced from the O3N2 relationship presented in \\citet{pp04}.}\n\\tablenotetext{d}{H$\\alpha$ luminosity in units of $10^{42}$ ergs s$^{-1}$.} \n\\tablenotetext{e}{Star formation rate in units of $M_{\\odot}$ yr$^{-1}$, calculated from $L_{{\\rm H}\\alpha}$ using the calibration of \\citet{kennicutt98}, and divided by a factor of 1.8 to convert to a \\citet{chabrier03} IMF from the Salpeter IMF assumed by \\citet{kennicutt98}. Note that SFRs have not been corrected for dust extinction or aperture effects, which may amount to a factor of $2$ difference \\citep{erb06c}.}\n\\tablenotetext{f}{Stellar mass and uncertainty estimated using the methods described in Bundy et al. (2005), and assuming a \\citet{chabrier03} IMF.} \\tablenotetext{g}{This object has a double morphology. The separation between the two components is about 0.9$^{''}$, which corresponds to $\\sim$8 kpc at $z = 1.3984$. We measured line fluxes for the emission-line-dominated component, but do not have a robust estimate of the corresponding stellar mass. In the DEEP2 photometry the two components were counted as one source, and the resulting stellar-mass estimate has contribution from both components. We therefore do not include this stellar mass estimate in our sample.}\n\\end{deluxetable*}\n\n\nThe near-IR spectra were obtained on 2005 September 17 and 18 with\nthe NIRSPEC spectrograph \\citep{mclean98} on the Keck~II\ntelescope. Over the range of redshifts of the galaxies presented\nhere, two filter setups are required to measure the full set of\nH$\\beta$, [O III], H$\\alpha$, and [N II]. For objects at $z \\sim\n1.4$, the NIRSPEC 5 filter (similar to $H$ band) is used to\nobserve H$\\alpha$ and [N II], whereas the NIRSPEC 3 filter\n(similar to $J$ band) is used for H$\\beta$ and [O III]. For\nobjects at $z \\sim 1.0$, the NIRSPEC 3 filter is used to observe\nH$\\alpha$ and [N II], whereas the NIRSPEC 1 filter ($\\Delta\\lambda\n=$ 0.95-1.10 $\\mu$m) is used for H$\\beta$ and [O III]. All targets\nwere observed for 3$\\times$900 s in each filter with a\n0.76$^{''}$$\\times$42$^{''}$ long slit. The spectral resolution\ndetermined from sky lines is $\\sim10 {\\rm \\AA}$ for all four\nNIRSPEC filters used here. Photometric conditions and seeing were\nvariable throughout both nights, with seeing ranging from\n0.5$^{''}$ to 0.7$^{''}$ in the near-IR. In order to enhance the\nlong-slit observing efficiency, we targeted two galaxies\nsimultaneously by placing them both on the slit.\n\nWe observed a total of 10 DEEP2 galaxies, successfully measuring\nthe full set of H$\\beta$, [O III], H$\\alpha$, and [N II] for eight\nout of 10. For the remaining pair, we only detected H$\\alpha$ in\nthe $H$ band, but no H$\\beta$ nor [O III] in the $J$-band\nexposures, in which the background in between sky lines was\ncharacterized by a significantly higher level of continuum than\nusual. This anomalous background is likely due to an increased\ncontribution from clouds, which may have affected both $H$- and\n$J$-band observations of the pair. Since a clear measurement was\nnot obtained for these two objects, due to variable weather\nconditions, we exclude them from our study. The object, 42021652,\nhas a double morphology, with one component dominated by emission\nlines with a weak continuum, and another component dominated by\nstrong continuum, with only weak emission lines at roughly the\nsame redshift. The separation between the two components on the\nsky is $\\sim$0.9$^{''}$, which corresponds to $\\sim$8 kpc at $z =\n1.3984$, perhaps indicative of a merger event. This interpretation\nis supported by the small velocity difference of $\\Delta v\\sim\n125$~km s$^{-1}$ between the two components. We measure line\nfluxes for the component dominated by emission lines, since it\nprovides a more robust estimate of line ratios. Deblended optical\nand near-infrared magnitudes would be required to obtain robust\nstellar masses for the individual components. However, in the\nDEEP2 photometry the two components were counted as one source\nsince they are too close to be deblended and the stellar mass has\ncontribution from both of them. For now, we only include flux\nmeasurements of the emission-line component for the\ndiagnostic-line-ratio analysis but do not include this object in\nthe mass-metallicity studies. A summary of the observations\nincluding target coordinates, redshifts, and optical and near-IR\nphotometry is given in Table \\ref{tab:obs}.\n\n\n\n\n\\subsection{Data Reduction and Optimal Background Subtraction}\n\nData reduction was performed with a similar procedure to the one\ndescribed in Paper I and \\citet{erb03}, with the exception of an\nimproved background subtraction method applied to the\ntwo-dimensional galaxy spectral images \\citep[][private\ncommunication]{kelson03,becker06}.\nIn the custom NIRSPEC long-slit reduction package written by D. G.\nBecker (2006, private communication), optimal background\nsubtraction is performed on the unrectified science frames. First,\na transformation is calculated between CCD ($x$, $y$) coordinates\nand those of slit position and wavelength, using the\nwavelength-dependent traces of bright standard stars and the\nspatially dependent curves of bright sky lines. Then a\ntwo-dimensional model of the sky background is constructed as a\nfunction of slit position and wavelength, using a low-order\npolynomial in the slit-position dimension, and a $b$-spline\nfunction in the wavelength dimension. This two-dimensional model\nis iteratively fit in the differenced frame of adjacent science\nexposures and subtracted from the unrectified data. After\nbackground subtraction, cosmic rays were removed from each\nexposure, which was then rotated, cut out along the slit, and\nrectified. Finally, all background-subtracted, rectified exposures\nof a given science target were combined in two dimensions. This\nnew approach to reducing NIRSPEC spectra results in fewer\nartifacts around bright sky lines and cosmic rays, which are\ncommonly introduced when rectification is performed before sky\nsubtraction and cosmic-ray zapping.\nOne-dimensional spectra, along with error spectra, were then\nextracted and flux- calibrated using A-star observations,\naccording to the procedure described in Paper I and \\citet{erb03}.\n\n\n\n\\subsection{Measurements and Physical Quantities}\n\n\n\\begin{figure*}\n\\epsscale{1} \\plotone{f2.eps} \\caption{NIRSPEC spectra of DEEP2\ngalaxies in our new sample at $z\\sim 1.0$. H$\\beta$ and [O III]\nare observed in the NIRSPEC 1 filter, with H$\\alpha$ and [N II] in\nthe NIRSPEC 3 filter (similar to the $J$ band). The 1 $\\sigma$\nerror spectra are shown as dotted lines, offset vertically by $-5\n\\times 10^{-18}$ ergs s$^{-1}$ cm$^{-2}$ \\AA$^{-1}$ for\nclarity.\\label{fig:spec1}} \\epsscale{1.}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\epsscale{1.} \\plotone{f3.eps} \\caption{NIRSPEC spectra of DEEP2\ngalaxies in our new sample at $z\\sim 1.4$. H$\\beta$ and [O III]\nare observed in the NIRSPEC 3 filter, with H$\\alpha$ and [N II] in\nthe NIRSPEC 5 filter (similar to the $H$ band). The 1 $\\sigma$\nerror spectra are shown as dotted lines, offset vertically by $-5\n\\times 10^{-18}$ ergs s$^{-1}$ cm$^{-2}$ \\AA$^{-1}$ for\nclarity.\\label{fig:spec2}} \\epsscale{1.}\n\\end{figure*}\n\nOne-dimensional, flux-calibrated NIRSPEC spectra along with the 1\n$\\sigma$ error spectra of galaxies in our new sample are shown in\nFigures \\ref{fig:spec1} and \\ref{fig:spec2}. Emission-line fluxes\nand uncertainties measured from the one-dimensional spectra are\ngiven in Table \\ref{tab:emi}. H$\\alpha$ and [N II] $\\lambda$6584\nemission-line fluxes were determined by first fitting a Gaussian\nprofile to the H$\\alpha$ feature to obtain the redshift and FWHM,\nand using these values to constrain the fit to the [N II] emission\nline. This method is based on the assumption that the H$\\alpha$\nand [N II] lines have exactly the same redshift and FWHM, with the\nH$\\alpha$ line having a higher signal-to-noise ratio (S\/N). For\nmost of the objects in our sample, the [N II] $\\lambda$6548 line\nwas too faint to measure. [O III] $\\lambda$5007 and H$\\beta$\nfluxes were determined with independent fits. In most cases,\nredshifts from H$\\alpha$, [O III] $\\lambda$5007, and H$\\beta$\nagree to within $\\Delta z = 0.0004$ ($\\Delta v = 50-60$ km\ns$^{-1}$ at $z=1.0-1.4$). For the object 42021412, [O III]\n$\\lambda$5007 lies on top of a bright sky line, and only a lower\nlimit is given.\n\n\nSFRs inferred from H$\\alpha$ luminosities using the calibration of\n\\citet{kennicutt98} are shown in Table (\\ref{tab:emi}). The\nresults have been converted from the Salpeter IMF used by\n\\citet{kennicutt98} to a \\citet{chabrier03} IMF by dividing the\nresults by a factor of $1.8$. In our whole sample of 20 galaxies,\nthe H$\\alpha$ fluxes range from $5.6 \\times 10^{-17}$ to $2.4\n\\times 10^{-16}$ erg s$^{-1}$ cm$^{-2}$. The mean H$\\alpha$ flux\nfor the sample at $z \\sim 1.0$ ($z \\sim 1.4$) is $1.3 \\times\n10^{-16}$ ($1.2 \\times 10^{-16}$), corresponding to a star\nformation rate of 3 (6) $M_{\\odot}$ yr$^{-1}$, uncorrected for\ndust extinction or aperture effects, which may amount to a factor\nof $2$ difference \\citep{erb06c}. Note that these characteristic\nH$\\alpha$ star formation rates, after being corrected for aperture\neffects, would be significantly higher than those of local\ngalaxies in the SDSS sample of \\citet{tremonti04} and the $\\langle\nz \\rangle = 0.4$ TKRS subsample of \\citet{kobulnicky04}, even\nbefore correction for dust extinction. We also note that the mean\nspecific SFR for both $z\\sim 1.0$ and $z \\sim 1.4$ samples is\n$\\log((SFR\/M_{\\star})\\mbox{ yr}^{-1})=-9.7$.\n\nAs discussed in Paper I, absolute line flux measurements suffer\nfrom several sources of systematic error, which can amount to at\nleast a $\\sim25\\%$ uncertainty \\citep{erb03}. This level of\nuncertainty is present even under photometric conditions, which\nmay not have applied through the full extent of our observations.\nFor the remainder of the discussion we therefore focus on the\nmeasured line {\\it ratios}, [N II] $\\lambda$6584\/H$\\alpha$ and [O\nIII] $\\lambda$5007\/H$\\beta$, which are not only unaffected by\nuncertainties in flux calibration and other systematics but also\nrelatively free from the effects of dust extinction, due to the\nclose wavelength spacing of the lines in each ratio. Hereafter, we\nuse ``[N II]\/H$\\alpha$\" to refer to the measured emission-line\nflux ratio between [N II] $\\lambda$6584 and H$\\alpha$, and ``[O\nIII]\/H$\\beta$\" for that between [O III] $\\lambda$5007 and\nH$\\beta$.\n\n\n\\section{The Oxygen Abundance}\n\nH II region metallicity is an important probe of galaxy formation\nand evolution, as it represents the integrated products of past\nstar formation, modulated by the inflow and outflow of gas. Oxygen\nabundance is often used as a proxy for metallicity since oxygen\nmakes up about half of the metal content of the interstellar\nmedium and exhibits strong emission lines from multiple ionization\nstates in the rest-frame optical that are easy to measure. For\ncomparison, we use the solar oxygen abundance expressed as 12 +\nlog(O\/H) = 8.66 \\citep{allende02,asplund04}.\n\nThe most robust way to estimate the oxygen abundance is the\nso-called direct $T_e$ method, based on the measurement of the\ntemperature-sensitive ratio of auroral and nebular emission lines.\nHowever, in distant galaxies the auroral lines are almost always\nundetectable \\citep[but see][]{hoyos05,kakazu07} since they become\nextremely weak at metallicities above $\\sim$0.5 solar. Even at\nlower metallicities, the auroral lines are typically beyond the\nreach of the low S\/N typical of the spectra of distant galaxies.\nFor distant star-forming galaxies, therefore, measuring strong\nemission-line ratios is the only viable way of obtaining the H II\nregion gas-phase oxygen abundance \\citep{kobulnicky99,pettini01}.\n\nGiven our NIRSPEC data set, and our desire to avoid the systematic\nuncertainties entailed in adding [O II] line fluxes obtained with\nthe DEIMOS spectrograph (without real-time flux-calibration) to [O\nIII] fluxes obtained with NIRSPEC, we focus on two strong-line\nratios as indicators for the oxygen abundance: N2$\\equiv$ log([N\nII]\/H$\\alpha$) and O3N2$\\equiv$ log\\{([O III]\/H$\\beta$)\/([N\nII]\/H$\\alpha$)\\}. These indicators have been calibrated by\n\\citet{pp04} using local H II regions, most of which have direct\n$T_e$ abundance determinations. The sensitivity of the indicators\nto oxygen abundance, as well as their limitations, have been\ndiscussed in \\citet{pp04} and Paper~I.  Absolute estimates of\nmetal abundances are quite uncertain, as abundances determined\nwith different indicators or with different calibrations of the\nsame indicator may have substantial biases or discrepancies\n\\citep[e.g.][]{kennicutt03,kobulnicky04}. We therefore emphasize\nrelative abundances determined with the same method, using the\nsame calibration.\n\nThe N2 indicator, pointed out by several works\n\\citep{storchi94,raimann00,denicolo02}, is related to the oxygen\nabundance via\n\\begin{equation}\n{\\rm 12 + log(O\/H) = 8.90 + 0.57 \\times} {\\rm N2},\n\\end{equation}\nwhich is valid for ${\\rm7.50 < 12 + log(O\/H) < 8.75}$, with a 1\n$\\sigma$ scatter of $\\pm0.18$ dex \\citep{pp04}. It has been used\nby \\citet{erb06a} to estimate oxygen abundances for UV-selected $z\n\\sim 2$ galaxies. Note that the N2 indicator is not sensitive to\nincreasing oxygen abundance above roughly solar metallicity, as\nshown with photoionization models \\citep{kewley02}. Thus, for a\nsubset of 12 galaxies in our sample with measurements of the full\nset of H$\\beta$, [O III], H$\\alpha$, and [N II], we also use the\nO3N2 indicator introduced by \\citet{alloin79}, which is expected\nto be particularly useful at solar and super-solar metallicities\nwhere [N II] saturates but the strength of [O III] continues to\ndecrease with increasing metallicity. Using the same calibration\nsample, \\citet{pp04} show that O3N2 is related to the oxygen\nabundance via\n\\begin{equation}\n{\\rm 12 + log(O\/H) = 8.73 - 0.32 \\times} {\\rm O3N2},\n\\end{equation}\nwhich is valid for ${\\rm8.12 < 12 + log(O\/H) < 9.05}$, with a 1\n$\\sigma$ scatter of $\\pm0.14$ dex. Table \\ref{tab:emi} lists\noxygen abundances derived using these two indicators. The errors\non the oxygen abundances are dominated by the systematic\nuncertainties in the calibrations of the indicators.\n\n\n\n\\subsection{Composite Spectra}\n\n\n\n\\begin{figure*}\n\\epsscale{1.} \\plotone{f4.eps} \\caption{Composite NIRSPEC spectra\nof the 18 $z\\sim 1.0-1.5$ galaxies with both stellar mass\nestimation as well as [N II] and H$\\alpha$ measurement in our\nsample, divided separately at $z \\sim 1.0$ and $z \\sim 1.4$. The 1\n$\\sigma$ error spectra are shown as dotted lines, offset\nvertically by $-3 \\times 10^{-18}$ ergs s$^{-1}$ cm$^{-2}$\n\\AA$^{-1}$ for clarity. The spectra are labeled with the mean\nstellar mass from each bin, and the H$\\alpha$, [N II] and [S II]\nlines are marked by dashed lines. Note that the spectra near the\ndensity-sensitive [S II] lines are very noisy, due to the large\ndispersion of the flux per count from near-IR standard star\ncalibration, caused by low efficiency near the filter\nedge.\\label{fig:compspec:n2}}\n\\end{figure*}\n\n\\begin{figure*}\n\\epsscale{.95} \\plotone{f5.eps} \\caption{Composite NIRSPEC spectra\nof the 12 $z\\sim 1-1.5$ galaxies with both stellar mass estimation\nas well as all-four-line measurement in our sample, divided\nseparately at $z \\sim 1.0$ and $z \\sim 1.4$. The 1 $\\sigma$ error\nspectra are shown as dotted lines, offset vertically by $-3 \\times\n10^{-18}$ ergs s$^{-1}$ cm$^{-2}$ \\AA$^{-1}$ for clarity. The\nspectra are labeled with the mean stellar mass from each bin, and\nthe H$\\beta$, [O III], H$\\alpha$, [N II] and [S II] lines are\nmarked by dashed lines.\\label{fig:compspec:o3n2}}\n\\end{figure*}\n\n\n\n\n\nRelative, average abundances determined from composite spectra can\nbe more accurately determined than those from individual spectra.\nAs discussed by \\citet{erb06a}, making a composite spectrum not\nonly reduces the uncertainties associated with the strong-line\ncalibration by a factor $N^{1\/2}$, where $N$ is the number of\nobjects included in the composite spectrum, but also enhances the\nspectrum S\/N since sky lines generally lie at different\nwavelengths for spectra at different redshifts. In addition, one\nof our goals is to determine the average properties of subgroups\nof galaxies in our sample. For the subset of 18 galaxies in our\nsample with H$\\alpha$ and [N II] measurements, as well as stellar\nmass estimates, we divide the sample into four bins by stellar\nmass, with two bins at $z \\sim 1.0$ and two bins at $z \\sim 1.4$.\nFor the subset of 12 galaxies with measurements of not only\nH$\\alpha$ and [N II], but also H$\\beta$ and [O III], we also\ndivide the sample into four bins by stellar mass with two bins\neach at $z \\sim 1.0$ and at $z \\sim 1.4$.\n\nTo make the composite spectra, we first shift the individual\none-dimensional flux-calibrated spectra into the rest frame and\nthen combine them by generating the median spectrum, which\npreserves the relative fluxes of the emission features\n\\citep{vanden01}. We use N2 and N2+O3 composite spectra to refer\nto the composites with H$\\alpha$ and [N II], and those with all\nfour lines, respectively. The N2 and N2+O3 composite spectra,\nlabelled with mean stellar mass in each bin, are shown in Figures\n\\ref{fig:compspec:n2} and \\ref{fig:compspec:o3n2}. The\ncorresponding emission-line flux ratios along with uncertainties\nmeasured from the composite spectra, as well as the inferred\noxygen abundances, are listed in Tables \\ref{tab:comp:n2} and\n\\ref{tab:comp:o3n2}. The listed errors in 12 + log(O\/H) include\nthe uncertainties from the propagation of emission-line flux\nmeasurements, as well as the systematic scatter from the\nstrong-line calibration.\nAs shown in paper I, the systematic discrepancies between the N2-\nand O3N2-based abundances are mainly due to the fact that, on\naverage, DEEP2 galaxies are offset from the excitation sequence\nformed by local H II regions and star-forming galaxies. We discuss\nthis issue in detail in \\S \\ref{sec:offset}.\n\n\n\n\n\\section{The Mass-Metallicity Relation}\\label{sec:mz_relation}\n\n\\begin{figure*}\n\\epsscale{1.} \\plotone{f6.eps} \\caption{Mass-metallicity relation\nobserved at $z \\sim 1.0$ and $z \\sim 1.4$. In the left panel,\nmetallicities are inferred from the N2 indicator, while the right\npanel shows metallicities estimated from the O3N2 indicator. In\nboth plots, open (filled) diamonds and squares are used for N2\n(N2+O3) composite spectra at $z \\sim 1.0$ and $z \\sim 1.4$,\nrespectively (see section 3.1). Errors show the uncertainties\npropagated from the strong-line ratio measurement, while the\nsystematic uncertainties from the strong-line method calibration\n\\citep{pp04} are shown in the lower right corner of each plot as\ndotted (solid) error bars for N2 (N2+O3) composite spectra. The\nsystematic uncertainties from the calibration are reduced by a\nfactor $N^{1\/2}$, where $N$ is the number of individual spectra\nincluded in each composite. For stellar masses, the horizontal\nbars give the mass range in each bin. Data points for individual\nobjects are shown as filled triangles and circles for $z \\sim 1.0$\nand $z \\sim 1.4$, respectively. Associated error bars are listed\nin Table 2 in both this paper and in Paper~I. For comparison,\nmetallicities as a function of stellar mass are also shown, for\nboth local SDSS galaxies ({\\it grey contours and dots}) and the\n\\citet{erb06a} $z \\sim 2$ sample ({\\it grey open circles}, {\\it\nleft panel}). Note that here and throughout, we use contours and\ndots to show SDSS objects. On each such applicable plot, SDSS data\npoints were mapped onto 10 evenly spaced levels according to\nnumber density, where objects on the lowest level are denoted by\ndots while other levels are presented by contours. Solar\nmetallicity is marked with a horizontal dashed line. It can be\nseen that the N2 indicator saturates near the solar abundance.\n\\label{fig:massz}}\n\\end{figure*}\n\n\nThe redshift evolution of the luminosity-metallicity and\nmass-metallicity relations provides important constraints on\nmodels of galaxy evolution. A correlation between gas-phase\nmetallicity and stellar mass can be explained by either the\ntendency of lower mass galaxies to have larger gas fractions and\nlower star formation efficiencies\n\\citep{mcgaugh97,bell00,kobulnicky03}, or the preferential loss of\nmetals from galaxies with shallow potential wells by\ngalactic-scale winds \\citep{larson74}. In the local universe,\nstrong correlations between rest-frame optical luminosity and the\ndegree of chemical enrichment have been observed in both\nstar-forming and early-type galaxies\n\\citep{garnett87,brodie91,tremonti04}. The correlation has also\nbeen observed in intermediate- and high- redshift samples\n\\citep{kobulnicky03,lilly03,kobulnicky04,erb06a}, although caution\nmust be taken when comparing samples with metallicities determined\nfrom different methods. Physically, the correlation between\nstellar mass and metallicity is more fundamental than that between\nluminosity and metallicity \\citep[Paper I;][]{tremonti04,erb06a}.\nWe therefore focus on the mass-metallicity relation in the\nfollowing discussion.\n\n\n\n\n\n\n\nThe left panel of Figure \\ref{fig:massz} shows the average\nmetallicity of the galaxies in each mass bin determined from the\nN2 composite spectra plotted against their average stellar mass at\n$z \\sim 1.0$ ({\\it open diamonds}) and at $z \\sim 1.4$ ({\\it open\nsquares}). Although our sample is still small, we do see evidence\nfor mass-metallicity relations at both $z \\sim 1.0$ and $z \\sim\n1.4$. These trends are also present when we examine the\nmetallicities and stellar masses for individual objects, which are\nplotted in the figure as well. For comparison, the local SDSS\ngalaxies discussed by \\citet{tremonti04} are denoted by contours\nand dots\\footnote{Note that here and throughout, we use a\ncombination of contours and dots to indicate SDSS objects. On each\nsuch applicable plot, SDSS data points were mapped onto 10 evenly\nspaced levels according to surface density, where objects on the\nlowest level are denoted by dots while other levels are presented\nby contours.} and the $z \\sim 2$ \\citet{erb06a} sample as open\ncircles. Metallicities for SDSS galaxies were calculated using the\nsame strong-line indicator that was applied to the DEEP2 galaxies\nand not the Bayesian O\/H estimate from \\citet{tremonti04}.\n\nAt fixed stellar mass, the metallicities of our $z \\sim 1.0-1.5$\nsample as a whole are lower than those of local galaxies yet\nhigher than those of the $z \\sim 2$ sample. However, there is\nevidence for a reverse trend between the subsets of our sample at\n$z \\sim 1.0$ and at $z \\sim 1.4$. In Paper I, this difference was\nattributed to the fact that the $z \\sim 1.0$ sample was on average\nfainter and less massive than the $z\\sim 1.4$ sample. With a\nlarger sample, however, we find that the higher mass $z \\sim 1.0$\nbin does have lower metallicity than the lower mass $z \\sim 1.4$\nbin. Differences in outflow or inflow rate of unenriched gas at $z\n\\sim 1.0$ and at $z \\sim 1.4$ could give rise to this trend.\nHowever, the interval in cosmic time between $z\\sim 1.4$ and $1.0$\nis small enough that typical gas inflow rates at fixed stellar\nmass, and the corresponding star formation and outflow rates, will\nnot significantly evolve. Therefore, this explanation is not a\nlikely cause of the reverse trend. A different average degree of\ndust reddening at $z \\sim 1.0$ and at $z \\sim 1.4$ is also not a\nlikely cause, since the N2 indicator is based on emission lines\nwith very close spacing in wavelength. On the other hand, if the\n$z \\sim 1.0$ galaxies have systematically different physical\nconditions or less significant contributions from AGN activity\nrelative to the $z \\sim 1.4$ objects, metallicities estimated with\nthe same calibration would be systematically biased between the\ntwo samples in such a way to produce the observed trend. As\ndiscussed in \\S \\ref{sec:offset}, we propose that the most likely\ncause for the reverse trend is this difference in H II region\nphysical conditions. Since the systematic uncertainty from\nstrong-line calibration is large, and our sample is still too\nsmall to draw any solid conclusion, it will become feasible to\nclarify this issue only when a statistically large enough sample\nis assembled, and both the high- and low- mass ends are spanned at\n$z \\sim 1.0$ as well as at $z \\sim 1.4$.\n\n\nWe also plot the mass-metallicity relation from the N2+O3\ncomposite spectra in Figure \\ref{fig:massz}, where the left panel\nshows metallicities determined from the N2 indicator, while the\nright panel shows those determined from the O3N2 indicator, at\nboth $z \\sim 1.0$ ({\\it filled diamonds}) and $z \\sim 1.4$ ({\\it\nfilled squares}). The O3N2-based abundances are systematically\nlower than those based on N2. As suggested in Paper I and\ndiscussed in detail in \\S \\ref{sec:offset}, these systematic\ndiscrepancies between N2- and O3N2-based abundances are due to the\nfact that DEEP2 galaxies depart from the local H~II region\nexcitation sequence. In addition, the reverse trend between $z\n\\sim 1.0$ and $1.4$ in metallicity estimated from O3N2 is much\nless significant than the one in metallicity estimated from N2. We\nreturn to this issue as well in \\S \\ref{sec:offset}. Despite these\ndiscrepancies, the overall correlation between average stellar\nmass and metallicity observed among the N2 composite spectra is\nstill present for the N2+O3 spectra. This is evidence that the\ncorrelation is insensitive to the spectrum of any particular\nobject, as it is robust to analyses using different binning\nschemes. At the lower mass end ($M_{\\star} \\sim 8\\times10^{9}$\n$M_{\\sun}$), the average metallicity of $z \\sim 1.0-1.5$ galaxies\nbased on N2 is at least 0.22 dex lower than the local typical\nvalue. Since the N2 indicator saturates near solar abundance, as\ndiscussed in \\citet{erb06a}, it is difficult to determine the true\nmetallicity offset between two samples at different redshifts\nusing this indicator. We can also determine the offset from the\nO3N2-based abundances, particularly near the solar abundance.\nBased on O3N2 abundances, the metallicity offset between our DEEP2\nobjects and the local SDSS sample is at least 0.21 dex at the\nhigh-mass end ($M_{\\star} \\sim 5\\times10^{10}$ $M_{\\sun}$).\nHowever, as we discuss in \\S \\ref{sec:offset}, these estimates\nbased on strong-line indicators may still be subject to systematic\nuncertainties from using the calibration of H II regions with\nsignificantly different physical properties.\n\n\n\n\\begin{deluxetable}{lccccc}\n\\tabletypesize{\\scriptsize}\n\\tablecaption{Oxygen Abundances from N2 composite spectra.\\label{tab:comp:n2}}\n\\tablewidth{0pt}\n\\tablehead\n{\n\\colhead{~~~~~Bin~~~~~} & \n\\colhead{N\\tablenotemark{a}} & \n\\colhead{$\\langle {\\rm z}_{{\\rm H}\\alpha} \\rangle$\\tablenotemark{b}} & \n\\colhead{log$(M_{\\ast}\/M_{\\odot})$\\tablenotemark{c}} & \n\\colhead{N2\\tablenotemark{d}} &\n\\colhead{12 + log(O\/H)\\tablenotemark{e}}\n}\n\\startdata\n1\\dotfill & 3 & 1.0375 & $9.88^{+0.10}_{-0.13}$   & -0.96$\\pm$0.03 & 8.35$\\pm$0.11 \\\\\n2\\dotfill & 4 & 1.0207 & $10.37^{+0.05}_{-0.06}$  & -0.74$\\pm$0.02 & 8.48$\\pm$0.09 \\\\\n3\\dotfill & 5 & 1.3930 & $10.10^{+0.06}_{-0.06}$  & -0.63$\\pm$0.02 & 8.54$\\pm$0.09 \\\\\n4\\dotfill & 6 & 1.3903 & $10.83^{+0.05}_{-0.05}$  & -0.49$\\pm$0.02 & 8.62$\\pm$0.08 \\\\\n\\enddata\n\\tablenotetext{a}{Number of objects contained in each bin.}\n\\tablenotetext{b}{Mean redshift for each bin.}\n\\tablenotetext{c}{Mean stellar mass and uncertainty from error propagation.}\n\\tablenotetext{d}{N2 $\\equiv$ log([N II]$\\lambda6584$\/H$\\alpha$).}\n\\tablenotetext{e}{Oxygen abundance deduced from the N2 relationship presented in \\citet{pp04}.}\n\\end{deluxetable}\n\n\n\\begin{deluxetable*}{lccccccc}\n\\centering\n\\tabletypesize{\\scriptsize}\n\\tablecaption{Oxygen Abundances from N2+O3 composite spectra.\\label{tab:comp:o3n2}}\n\\tablewidth{0pt}\n\\tablehead\n{\n\\colhead{~~~~~~~~~~Bin~~~~~~~~~~} & \n\\colhead{~~~N~~~} & \n\\colhead{~~~$\\langle {\\rm z}_{{\\rm H}\\alpha} \\rangle$~~~} &\n\\colhead{~~~log$(M_{\\ast}\/M_{\\odot})$~~~}  & \n\\colhead{~~~N2~~~} & \n\\colhead{~~~O3N2\\tablenotemark{a}~~~}  &\n\\colhead{~~~[12 + log(O\/H)]$_{N2}$\\tablenotemark{b}~~~} & \n\\colhead{~~~[12 + log(O\/H)]$_{O3N2}$\\tablenotemark{c}~~~}\n}\n\\startdata\n1\\dotfill & 3 & 1.0372 &  $9.98^{+0.09}_{-0.11}$  & -0.89$\\pm$0.03 &  1.37$\\pm$0.03 & 8.39$\\pm$0.10 & 8.29$\\pm$0.08 \\\\\n2\\dotfill & 3 & 1.0216 &  $10.40^{+0.06}_{-0.06}$ & -0.79$\\pm$0.02 &  1.17$\\pm$0.06 & 8.45$\\pm$0.10 & 8.35$\\pm$0.08 \\\\\n3\\dotfill & 3 & 1.3923 &  $10.06^{+0.06}_{-0.06}$ & -0.61$\\pm$0.04 &  1.09$\\pm$0.08 & 8.55$\\pm$0.11 & 8.38$\\pm$0.08 \\\\\n4\\dotfill & 3 & 1.3858 &  $10.63^{+0.07}_{-0.09}$ & -0.51$\\pm$0.02 &       $>$ 0.62 & 8.61$\\pm$0.10 & $<$ 8.53 \\\\\n\\enddata\n\\tablenotetext{a}{O3N2 $\\equiv$ log$\\{$([O III]$\\lambda$5007\/H$\\beta$)\/([N II] $\\lambda$6584\/H$\\alpha$)$\\}$.}\n\\tablenotetext{b}{Oxygen abundance deduced from the N2 relationship presented in \\citet{pp04}.} \n\\tablenotetext{c}{Oxygen abundance deduced from the O3N2 relationship presented in \\citet{pp04}.}\n\\end{deluxetable*}\n\n\n\n\\section{The Offset in Diagnostic Line Ratios of High-Redshift Galaxies}\\label{sec:offset}\n\n\\subsection{Emission-Line Diagnostics}\n\n\\begin{figure*}\n  \\centering\n    \\includegraphics[width=80mm]{f7a.eps}\n    \\includegraphics[width=80mm]{f7b.eps}\n    \\caption{H II region log[N II]\/H$\\alpha$ vs. log[O III]\/H$\\beta$ diagram.\n    On the left, the $z\\sim$1.0-1.5 DEEP2 galaxies are shown as indicated on\n    the plot. The right panel shows the average diagnostic line ratios from\n    the composite spectra of DEEP2 objects. Note the bin with the highest\n    [N II]\/H$\\alpha$ value only has a lower limit on [O III]\/H$\\beta$ because\n    it includes the object 42021412. SDSS emission-line galaxies that satisfy\n    the criteria described in \\S \\ref{subsec:sdsspreselect} are shown\n    as grey contours and dots. The dashed line is an empirical demarcation\n    from Ka03 based on the SDSS galaxies, whereas the dotted\n    line is the theoretical limit for star-forming galaxies from Ke01.\n    Nearby star-forming galaxies and H II regions form a well-defined excitation\n    sequence of photoionization by massive stars, below and to the left of these\n    curves. The $z \\sim 1.0-1.5$ DEEP2 objects on average are offset from this\n    excitation sequence, with objects at $z\\sim$1.4 more offset than those at $z\\sim$1.0.}\n    \\label{fig:subfig:bpt}\n\\end{figure*}\n\n\\begin{figure*}\n  \\centering\n    \\includegraphics[width=160mm]{f8.eps}\n    \\caption{H II region diagnostic diagram: log[N II]\/H$\\alpha$ vs.\n    log[O III]\/H$\\beta$. SDSS Main, Offset-AGN, Offset-ambiguous,\n    and Offset-SF samples are shown as grey contours with dots, blue triangles,\n    green crosses, and red circles, respectively (see \\S \\ref{subsec:sdsspreselect}\n    for more information on sample selection rules). The $z\\sim1.0-1.5$\n    DEEP2 objects are also shown for a comparison using the same legends\n    as in the left panel of Figure \\ref{fig:subfig:bpt}, and the dotted\n    and dashed lines have the same meanings as those in Figure \\ref{fig:subfig:bpt}.}\n    \\label{fig:subfig:bptn2}\n\\end{figure*}\n\n\nThere is evidence that physical conditions in the H II regions of\nhigh-redshift galaxies hosting intense star formation are\ndifferent from those of local H II regions (Paper I). The most\ncommon method for probing H~II region physical conditions, and\ndiscriminating between star-forming galaxies and AGNs, is based on\nthe positions of objects in the \\citet*{bpt81} empirical\ndiagnostic diagrams (hereafter BPT diagrams). These plots feature\nthe optical line ratios [N II]\/H$\\alpha$, [O I]\/H$\\alpha$, [S\nII]\/H$\\alpha$, and [O III]\/H$\\beta$, and have been updated by many\nauthors, including \\citet{osterbrock85}, \\citet{veilleux87},\n\\citet[][hereafter Ke01]{kewley01a}, and \\citet[][hereafter\nKa03]{kauffmann03b}. A considerable fraction of high-redshift\nstar-forming galaxies at both z $\\sim$ 1 (paper I) and z $\\geq$ 2\n\\citep{erb06a} do not follow the local excitation sequence formed\nby nearby H II regions and star-forming galaxies on the\nemission-line diagnostic diagram of [N II]\/H$\\alpha$ versus [O\nIII]\/H$\\beta$; on average, they lie offset upward and to the\nright. These differences must be taken into account when applying\nempirically calibrated abundance indicators to galaxy samples at\ndifferent redshifts. Several possible causes for the offset have\nbeen discussed in Paper I, including differences in the ionizing\nspectrum, ionization parameter, electron density, and the effects\nof shocks and AGNs.\n\n\nIn Figure \\ref{fig:subfig:bpt}, [O III]\/H$\\beta$ and [N\nII]\/H$\\alpha$ ratios are plotted in the left panel for the 13\ngalaxies in our sample with the full set of emission lines, and\nthe average [O III]\/H$\\beta$ and [N II]\/H$\\alpha$ ratios from the\nN2+O3 composite spectra are also shown in the right panel for bins\nat $z \\sim 1.0$ and $1.4$, respectively. A subset of emission-line\nobjects from SDSS are also shown as grey contours and dots for\ncomparison. The dotted curve is from Ke01, representing a\ntheoretical upper limit on the location of star-forming galaxies\nin the diagnostic diagram. The dashed curve is from Ka03 and\nserves as an empirical discriminator between star-forming galaxies\nand AGNs. On average, the H$\\alpha$ flux of galaxies below this\ncurve should have $<1\\%$ contribution from AGNs\n\\citep{brinchmann04}.\nThe effect observed in Paper I is still present in our larger\nsample, in the sense that the $z \\sim 1.0-1.4$ sample is, on\naverage, significantly offset from the excitation sequence formed\nby star-forming galaxies from SDSS. Furthermore, the average\noffset for the $z\\sim 1.4$ objects is larger than for those at\n$z\\sim 1.0$. A similar, if not stronger, effect is observed in\nstar-forming galaxies at $z \\sim 2$ \\citep{erb06a}. As discussed\nin Paper I, unaccounted-for stellar Balmer absorption is not the\nexplanation for the offsets in emission-line ratios, since the\ncorrections would shift the DEEP2 galaxies by no more than 0.1 dex\ndownward and by an insignificant amount in [N II]\/H$\\alpha$.\n\nIsolating the causes of the offset of the high-redshift samples\nfrom the local excitation sequence on the diagnostic diagram will\nprovide important insight into the physical conditions in distant\nstar-forming galaxies. These conditions also comprise an essential\nsystematic factor determining the emergent strong emission-line\nratios and associated calibration of chemical abundances.\nUnfortunately, we do not currently have a large high-redshift\nsample with available additional spectral features, stellar\npopulation parameters, and morphological information, which is\nneeded for a direct study of how the diagnostic line ratios vary\nas functions of galaxy properties. However, as seen in Figure\n\\ref{fig:subfig:bpt}, while only a tiny fraction of the SDSS\ngalaxies reside in the region of BPT parameter space inhabited by\nour most extremely offset high-redshift galaxies, in between the\nstar-forming excitation sequence and the AGN branch, the sheer\nsize of the SDSS sample still results in a set of $\\sim 100$ such\nlocal objects. These objects can serve as possible local\ncounterparts for our DEEP2 objects, with the added benefit of high\nS\/N photometric, spectroscopic, and morphological information from\nSDSS. We will make use of this detailed information to understand\nthe cause of the local galaxies' offset in the BPT diagram, and,\nby extension, the likely cause of the offset among the\nhigh-redshift galaxies.\n\nIn the following sections, we compare in detail these anomalous\nSDSS objects against more typical SDSS star-forming galaxies.\nAccordingly, we analyze possible causes for their offset in terms\nof different physical conditions in the ionized regions, which\ninclude H~II region electron density, hardness of the ionizing\nspectrum, ionization parameter, the effects of shock excitation,\nand contributions from an AGN. We further try to unravel possible\nconnections between physical conditions of these anomalous SDSS\nobjects and their host galaxy properties and use them to interpret\nour observations of high-redshift star-forming galaxies.\n\n\n\\subsection{Local Counterparts: SDSS Main and Offset Samples}\\label{subsec:sdsspreselect}\n\n\\begin{figure*}\n  \\centering\n    \\includegraphics[width=160mm]{f9.eps}\n    \\caption{H II region diagnostic diagrams. Descriptions for SDSS objects\nare the same as those in Figure \\ref{fig:subfig:bptn2}. On the [O\nI]\/H$\\alpha$ and [S II]\/H$\\alpha$ diagrams, the dotted (blue\nsolid) lines are empirical curves separating star-forming galaxies\n(Seyferts) and AGN (LINERs) from \\citet{kewley06}. The log[N\nII]\/[O II] vs. log[O III]\/H$\\beta$ diagram is useful in separating\nionization parameter and abundance but becomes insensitive to\nionization parameter above solar metallicity \\citep{dopita00}. The\n[N II]\/[O II] flux ratio has been corrected for dust extinction,\ngiven the wide spacing in wavelength between [N II] and [O II].}\n    \\label{fig:subfig:agnbpt}\n\\end{figure*}\n\n\nWithin this work we use the SDSS Data Release 4\n\\citep[DR4;][]{adelman06} spectroscopic galaxy sample, which\ncontains $u$-, $g$-, $r$-, $i$-, and $z$-band photometry and\nspectroscopy of 567,486 objects. As described in\n\\citet{tremonti04}, emission-line fluxes of these galaxies were\nmeasured from the stellar-continuum subtracted spectra with the\nlatest high spectral resolution population synthesis models by\n\\citet{bc03}.\n\nBefore being further divided into groups with different\nemission-line diagnostic ratios, our sample was selected from the\n567,486 galaxy DR4 sample according to the following criteria:\n\n(1) Redshifts in the range $0.005 < z < 0.25$. (2) Signal-to-noise\nratio (S\/N) $>$ 3 in the strong emission-lines [O II]\n$\\lambda\\lambda$3726, 3729, H$\\beta$, [O III]\n$\\lambda\\lambda$4959, 5007, H$\\alpha$, [N II] $\\lambda$6584, and\n[S II] $\\lambda\\lambda$6717, 6731; and S\/N $>$ 1 in the weak\nemission-line [O I] $\\lambda$6300. In practice, $\\sim 99$\\% of the\nobjects also have [O I] $\\lambda$6300 S\/N$>$3. (3) The fiber\naperture covers at least 20\\% of the total $g$-band photons. (4)\nStellar population parameter estimates are available in the\ncatalog derived using methods described by \\citet{kauffmann03a}.\n\nThe first criterion on redshift range is the same as that adopted\nby \\citet{tremonti04}, which enables a fair comparison with the\nmass-metallicity relation from their work. The second criterion\nselects galaxies with well-measured emission-line fluxes, required\nby our study based on emission-line diagnostic ratios. The S\/N\nvalues were obtained using errors of the emission-line fluxes,\nwhich were first taken from the emission-line flux catalog on the\nMPA DR4 Web site\\footnote{See\nhttp:\/\/www.mpa-garching.mpg.de\/SDSS\/} and then scaled by the\nrecommended factors (see the Web site for more details about the\nemission-line flux error). The third criterion is required to\navoid significant aperture effects on the flux ratios\n\\citep{kewley05,tremonti04}. Lower aperture fraction can cause\nsignificant discrepancies between aperture and global parameter\nestimates. The fourth criterion enables a comparison of stellar\npopulation properties with diagnostic emission-line ratios.\nRoughly 30\\% of the initial set of DR4 galaxies were ruled out\nbecause they do not have stellar population parameter estimates;\nan additional $\\sim$20\\% was cut out according to the redshift and\naperture-coverage constraints. Rules on the S\/N in both strong and\nweak emission lines removed another $\\sim$45\\% of the objects. In\nall, these selection criteria leave us with $\\sim$31,000\nemission-line objects, shown as grey contours and dots in Figure\n\\ref{fig:subfig:bpt}, which is about 5\\% of the 567,486 galaxy DR4\nsample.\n\n\n\nWith the exception of one of the 13 DEEP2 objects, which lies on\nthe part of the [N II]\/H$\\alpha$ versus [O III]\/H$\\beta$ diagram\npopulated by many H II\/AGN composites, the high-redshift galaxies\non average fall between the excitation sequence of typical SDSS\nstar-forming galaxies and the AGN branch. Although many of the\nDEEP2 objects have a less extreme offset relative to typical SDSS\nstar-forming galaxies than those of the sample of \\citet{erb06a},\nand the most extreme $z\\sim 1.4$ objects in Paper~I, there are\nstill several that are significantly offset from the main\nsequence. We proceed by studying properties of the SDSS galaxies\nwith similar [N II]\/H$\\alpha$ and [O III]\/H$\\beta$ values to those\nof the significantly offset DEEP2 objects, and comparing these\nunusual objects with typical SDSS star-forming galaxies along the\nexcitation sequence. In this sense we further divide the SDSS\nsample into the ``Main'', and ``Offset'' subgroups, as shown in\nFigure \\ref{fig:subfig:bptn2}.\n\nObjects in the Main sample are selected to lie below the Ka03\nempirical curve, whereas Offset objects are located in between the\nKa03 and the Ke01 curves. The Offset sample is also constrained to\nhave log([N II]\/H$\\alpha$) $\\leq$ -0.44 and log([O III]\/H$\\beta$)\n$\\geq$ 0.26, according to the two emission-line ratios of the\nDEEP2 object with the second largest [N II]\/H$\\alpha$ value. We do\nnot use the galaxy with the largest [N II]\/H$\\alpha$ value in our\nDEEP2 sample for the selection criteria of the two line ratios,\nbecause the number density in the composite region increases very\nrapidly as the AGN branch is approached, yet most of our DEEP2\ngalaxies clearly do not fall in that regime. Also, it is worth\npointing out that, while they are significantly displaced from the\nlocal excitation sequence, more than half of the DEEP2 objects\nactually lie below the Ka03 curve. Indeed, the SDSS Offset sample\nis constructed according to our DEEP2 objects that are offset the\nmost, in order to create a strong contrast with the Main sample.\nIt is also representative of the $z\\sim 2$ galaxies observed by\n\\citet{erb06a}. As we describe in \\S 5.6, the conclusions drawn\nfrom this extreme sample are corroborated by the work of\n\\citet{brinchmann07}, where objects with less extreme offsets are\nconsidered and therefore should be valid for typical objects in\nour DEEP2 sample. Finally, the Offset sample only covers a certain\nrange of stellar masses, so we further select the Main control\nsample according to the same stellar-mass range in order to ensure\na fair comparison. This leaves us with $\\sim$21,000 objects for\nthe Main sample and 101 objects for the Offset sample.\n\nFigure \\ref{fig:subfig:agnbpt} displays how objects in the Offset\nsample are distributed in the additional BPT diagrams featuring [O\nI]\/H$\\alpha$ and [S II]\/H$\\alpha$. These plots indicate that\nOffset objects selected solely on the basis of their [O\nIII]\/H$\\beta$ and [N II]\/H$\\alpha$ ratios span a diverse range of\nproperties in other physical parameter spaces. High [O\nI]\/H$\\alpha$ and [S II]\/H$\\alpha$ both occur when there is a hard\nionizing radiation field, including significant contribution from\nX-ray photons. In this case, there is an extended, partially\nionized zone, where H I and H II coexist, and [O I] and [S II] are\ndominant forms of O and S. The extended zone of partially ionized\nH does not exist in H II regions photoionized by OB stars\n\\citep{evans85,veilleux87}. High [O I]\/H$\\alpha$ and [S\nII]\/H$\\alpha$ are also produced in gas that has been heated by\nfast, radiative shocks, which also produce partially ionized\nshock-precursor regions \\citep{dopita95,dopita96}. Material in\nsupernova remnants provides an example of shocked gas. While it is\nsensitive to shocks, [S II] is more susceptible to collisional\nde-excitation than [O I], given its critical density\n(2$\\times10^3$ cm$^{-3}$), as opposed to that of [O I]\n(2$\\times10^6$ cm$^{-3}$), and therefore might be suppressed in\nregions of high electron densities \\citep{dopita97,kewley01b}.\nAnother point worth mentioning is that [O I]\/H$\\alpha$ should\nreveal larger differences in regions ionized by hard spectrum as\nopposed to that of stars, than [S II]\/H$\\alpha$ does. This is\nbecause the ionization potential of [O I] matches that of H I, so\n[O I] enhancement should reflect increased presence of partially\nionized zone. However, [S II] exists in both completely and\npartially ionized zones. So the contrast between AGN- and\nstellar-ionized regions for [S II]\/H$\\alpha$ is not as great as\nthat for [O I]\/H$\\alpha$.\n\nIn summary, the various locations of the Offset objects on the [O\nI]\/H$\\alpha$ and [S II]\/H$\\alpha$ diagrams indicate different\nlevels of contribution from AGN and shock excitations to the\nemerging spectra. We therefore further divide the Offset sample on\nthe [O I]\/H$\\alpha$ and [S II]\/H$\\alpha$ diagrams, according to\nthe theoretical scheme for classifying starburst galaxies and AGNs\n\\citep{kewley01a,kewley06}. It is worth noting here that all\nobjects in the Offset sample have S\/N$>$3 in both [S II] and [O I]\nemission lines, so this division should not be compromised by\nspurious measurements. While there may still be a low-level\n($<10$\\%) AGN contribution to the spectra of objects classified as\nstarbursts by this scheme, it serves as a rough guide to the range\nof properties in the Offset sample. This classification leaves us\nwith (1) ``Offset-AGN'' (43), where objects lie above both the\nKe01 curves of the two diagrams; (2) ``Offset-ambiguous'' (33),\nwhere objects lie above one Ke01 curve and under the other one;\nand (3) ``Offset-SF'' (25), where objects lie below both the Ke01\ncurves. The division of Offset-AGN and Offset-SF is only based on\nhardness of the ionizing spectrum and the contribution of shock\nexcitation. We show in Section \\ref{sec:sub:comparison} that these\ntwo offset samples have different host galaxy properties.\n\n\n\\subsection{H II Region Emission-Line Diagnostic Ratio, Physical Conditions, and Galaxy Properties}\n\n\\begin{figure*}\n  \\centering\n    \\includegraphics[width=160mm]{f10a.eps}\n    \\includegraphics[width=160mm]{f10b.eps}\n    \\includegraphics[width=160mm]{f10c.eps}\n    \\includegraphics[width=160mm]{f10d.eps}\n    \\caption{Distribution of row (1): redshift and metallicities\n    based on various strong-line indicators; row (2): ionization-parameter\n    indicator log($O_{32}$), electron density indicator [S II] $\\lambda6717$\/[S II] $\\lambda6731$,\n    D$_n(4000)$ and H$\\delta_{A}$; row (3): stellar mass,\n    fiber-corrected specific star formation rate, $r$-band Petrosian\n    concentration index $C = R_{90}\/R_{50}$, and color-magnitude diagram; row\n    (4): surface mass density log($\\mu_{\\star}$), SFR surface density\n    log($\\dot{\\Sigma}_{\\star}$), $R_{50}(z)$, which is the radius enclosing\n    50\\% of the Petrosian $z$-band luminosity of a galaxy, and H$\\alpha$\n    emission equivalent widths, for the SDSS Main ({\\it black dotted line}),\n    Offset-AGN ({\\it blue dashed line}), and Offset-SF samples ({\\it red solid line}; see \\S \\ref{subsec:sdsspreselect}\n    for more information). For clarity, the distribution functions are\n    normalized to three different levels, with the maxima being 1.0,\n    0.7, and 0.5 for the SDSS Main, Offset-AGN, and Offset-SF samples, respectively.}\n    \\label{fig:subfig:dis}\n\\end{figure*}\n\n\nIn order to determine the origin of the observed difference in\nemission-line ratios between the Main and Offset samples, we\nconsider a large set of galaxy properties. Measurements of\nemission lines as well as photometric, physical, and environmental\nproperties of stellar population are available for our SDSS\nsamples. In this section, we describe the galaxy properties of\ninterest, while in the following section, a comparison of the Main\nand Offset sample distributions in these properties is presented.\n\nFirst, using emission-line ratios, we examine the main factors\ncontrolling the emission-line spectrum in an H~II region, i.e.,\nthe gas phase metallicity, the shape or hardness of the ionizing\nradiation spectrum, and the geometrical distribution of gas with\nrespect to the ionizing sources, which can be represented by the\nmean ionization parameter and electron density \\citep{dopita00}.\nSeveral H~II region emission-line ratios can be used as\ndiagnostics of these factors.\n\nApart from the N2 and O3N2 indicators, we also use the ratio N2O2\n$\\equiv$ log([N II] $\\lambda$6584\/[O II] $\\lambda\\lambda$3726,\n3729), suggested by \\citet{vanzee98}, as an abundance diagnostic\nfor SDSS objects, because N2O2 is virtually independent of\nionization parameter and also strongly sensitive to metallicity.\nThis indicator is monotonic between 0.1 and over 3.0 times solar\nmetallicity \\citep{dopita00,kewley02}. Concerns about reddening\ncorrection and reliable calibration over such a large wavelength\nbaseline have hampered the use of this ratio. As shown in\n\\citet{kewley02}, however, the use of classical reddening curves\nand standard calibration are quite sufficient to allow this [N\nII]\/[O II] diagnostic to be used as a reliable abundance\nindicator. The \\citet{bresolin07} calibration of N2O2 is used to\ninfer oxygen abundance. We also use the $R_{23} \\equiv$ log\\{([O\nII] $\\lambda$3727 + [O III] $\\lambda\\lambda$4959, 5007)\/H$\\beta$\\}\nparameter, introduced by \\citet{pagel79}, as an abundance\nindicator, to make comparison with several works\n\\citep{lilly03,savaglio05}, although this indicator has some\nwell-documented drawbacks \\citep[e.g.][]{kobulnicky99,kewley02}.\n\n$O_{32} \\equiv$ log([O III] $\\lambda$5007\/[O II]\n$\\lambda\\lambda$3726, 3729) is used as an indicator of the\nionization parameter \\citep{mcgaugh91,dopita00}. However, $O_{32}$\nis sensitive to both ionization parameter and abundance; both low\nionization parameter and high abundances produce low values of\n$O_{32}$ \\citep{dopita00,kewley02}. Thus, when $O_{32}$ is used as\na diagnostic of the relative ionization parameters of different\nsamples, the metallicities of these samples must also be taken\ninto account for a fair comparison. A diagnostic plot, for\nexample, N2O2 versus $O_{32}$, can be used to separate ionization\nparameter and abundance. The ratio [S III]\/[S II] provides another\ndiagnostic of the ionization parameter, which is independent of\nmetallicity except at very high values of the ratio\n\\citep{kewley02}. Unfortunately, the SDSS spectra do not cover [S\nIII] $\\lambda\\lambda$9069, 9532 from SDSS, so it is not included\nhere.\n\nFinally, [S II] $\\lambda6717$\/[S II] $\\lambda6731$ is adopted as\nan electron-density indicator. We do not use the density-sensitive\nratio, [O II] $\\lambda3726$\/[O II] $\\lambda3729$, since this\ndoublet is typically blended in SDSS spectra.\n\nWe note that emission-line ratios, such as $O_{32}$, N2O2, and\n$R_{23}$ have large separations in wavelength and thus need to be\ncorrected for dust reddening. The Balmer decrement method with the\nreddening curve of \\citet{calzetti00} is used to correct these\nquantities for dust extinction, assuming an intrinsic ratio of\n$H\\alpha\/H\\beta=2.78$, appropriate for $T_e=10,000$ K.\n\n\nNext, we turn to the question of the stellar populations and\nstructural and environmental properties of the Main and the Offset\nsample host galaxies. To examine stellar ages, we adopt the narrow\ndefinition of the 4000 ${\\rm \\AA}$ break denoted as D$_n(4000)$\n\\citep{balogh99}, which is small for young stellar populations and\nlarge for old, metal-rich galaxies \\citep{kauffmann03a}. We use\nthe H$\\delta_{A}$ index as an indicator of recent starburst\nactivities \\citep[e.g.][]{kauffmann03a,worthey97}. Strong\nH$\\delta$ absorption lines arise in galaxies that experienced a\nburst of star formation that ended $\\sim$0.1-1 Gyr ago. For galaxy\nmorphology, we use the standard concentration parameter defined as\nthe ratio $C = R_{90}(r)\/R_{50}(r)$, where $R_{90}(r)$ and\n$R_{50}(r)$ are the radii enclosing 90\\% and 50\\% of the Petrosian\n$r$-band luminosity of the galaxy. For galaxy size, we study\n$R_{50}(z)$, the radius enclosing 50\\% of the Petrosian $z$-band\nluminosity of a galaxy. We also examine the surface mass density\n$\\mu_{\\star}$, defined as $M_{\\star}\/[2\\pi R^2_{50}(z)]$\n\\citep{kauffmann03c}, the SFR surface density\n$\\dot{\\Sigma}_{\\star}\\equiv {\\rm SFR}\/[2\\pi R^2_{50}(r)]$, where\nwe use a radius defined in the $r$ band rather than the $z$ band,\nas it is more appropriate for H$\\alpha$ luminosities. Additional\nparameters include the fiber-corrected specific star formation\nrate SFR\/M$_{\\star}$, H$\\alpha$ equivalent width (EW), and the\ncolor-magnitude diagram $(g-i)^{0.1}$ versus $M_r$, where\n$(g-i)^{0.1}$ denotes the $(g-i)$ color $k$-corrected to $z =\n0.1$, and $M_r$ stands for the $k$-corrected $r$-band absolute\nmagnitude \\citep{kauffmann03a}. In order to search for any\nenvironmental dependence of the Offset sample properties, we count\nthe number of spectroscopically observed galaxies that are located\nwithin 2 Mpc in projected radius and $\\pm$500 km s$^{-1}$ in\nvelocity difference from each object in our SDSS samples. Here we\nfollow the procedures described in \\citet{kauffmann04},\nconstructing a volume-limited tracer sample using the SDSS DR4\ndata.\n\n\n\\subsection{Comparison with Typical SDSS Star-forming Galaxies}\\label{sec:sub:comparison}\n\n\\begin{figure*}\n\\epsscale{1.} \\plotone{f11.eps} \\caption{Ionization-parameter\nindicator $O_{32}$, as a function of various parameters.\nDescriptions are the same as those in Figure\n\\ref{fig:subfig:agnbpt}.\\label{o32_dep}}\n\\end{figure*}\n\n\nIn this section, we compare the properties of Offset and Main\nsample objects, in terms of their H~II region physical conditions,\nstellar populations, structural parameters, and environments. In\nparticular, we find striking differences in H~II region ionization\nparameter, electron density, galaxy size, and star formation rate\nsurface density. Figure \\ref{fig:subfig:dis} shows relative\ndistributions of diagnostic-line ratios and galaxy properties of\nthe SDSS Main, Offset-AGN, and Offset-SF samples. For clarity, the\ndistribution functions are normalized to three different levels,\nwith the maxima being 1.0, 0.7 and 0.5 for the SDSS Main,\nOffset-AGN, and Offset-SF samples, respectively.\n\nFirst, we consider H~II region physical conditions. Compared to\nthe Main sample, both Offset samples on average have larger\nionization parameters. The Offset-SF and Offset-AGN samples have\n$O_{32} = 0.0$ and -0.2, respectively, as opposed to -0.7 for the\nMain sample. Here and throughout, we refer to the median values of\nthe distributions. Since $O_{32}$ also depends on metallicity, we\nneed to make a fair comparison for the ionization parameter at a\nfixed metallicity. Figure \\ref{o32_dep} shows $O_{32}$ as a\nfunction of several emission-line diagnostic ratios and galaxy\nproperties. In the lower four panels, we can see that, regardless\nof which metallicity indicator is used, the Offset-SF sample has\nbigger $O_{32}$ values than the Main sample at a fixed strong-line\nindicator value. Therefore, the Offset-SF sample has larger\naverage ionization parameter than Main sample objects having\nsimilar metallicities, independent of the metallicity indicator.\nCompared with the Main sample, all indicators except $R_{23}$\nsuggest that the Offset-AGN has higher ionization parameters at\nfixed metallicities.\n\nThe Offset-SF sample also has significantly larger electron\ndensities, on average, than the Main sample ([S II]\n$\\lambda6717$\/[S II] $\\lambda6731$ of $\\sim$1.23 compared to\n$\\sim$1.40, corresponding to electron densities of $\\sim$208\ncm$^{-3}$ compared to $\\sim$47 cm$^{-3}$), whereas the Offset-AGN\nsample has only slightly higher electron densities than those of\nthe Main sample ([S II] $\\lambda6717$\/[S II] $\\lambda6731$ of\n$\\sim$1.37, corresponding to electron densities of $\\sim$67\ncm$^{-3}$). As discussed in Paper I, for fixed ionization\nparameter, metallicity, and input ionizing spectrum, the\nphotoionization models presented in \\citet{kewley01a} display a\ndependence on electron density, in the sense that model grids with\nhigher electron density have an upper envelope in the space of [O\nIII]\/H$\\beta$ versus [N II]\/H$\\alpha$ that is offset upward and to\nthe right, relative to model grids with lower electron density.\nThis theoretical shift in the BPT diagram due to increased\nelectron density is qualitatively reflected in the properties of\nOffset-SF objects.\n\nNext, we consider photometric and spectroscopic stellar population\nproperties. The Offset-SF sample is similar to the Main sample in\nterms of median color [$(g-i)^{0.1}$ $\\sim$ 0.64 compared to\n$\\sim$0.66], stellar age [the same value of D$_n$(4000) = 1.24],\nand fiber-corrected specific star formation rates (the same value\nof $\\log((SFR\/M_{\\star})\\mbox{ yr}^{-1}) = -9.7$), whereas the\nOffset-AGN sample has redder colors [$(g-i)^{0.1}$ of $\\sim$0.86],\nolder stellar ages [D$_n$(4000) = 1.50], and lower specific star\nformation rates ($\\log((SFR\/M_{\\star})\\mbox{ yr}^{-1}) = -10.0$).\nAlso, the Offset-SF sample has a larger H$\\alpha$ EW than the Main\nsample (62 ${\\rm \\AA}$ compared to 40 ${\\rm \\AA}$), whereas the\nOffset-AGN sample has a much smaller H$\\alpha$ EW (15${\\rm \\AA}$).\nThe Offset-SF sample has a smaller burst fraction in stellar mass\nthan the Main sample (H$\\delta_{A}$ of 4.3 compared to 5.5), while\nthe discrepancy between the Offset-AGN and Main samples is even\nlarger (H$\\delta_{A}$ of 3.3 compared to 5.5).\n\nIn terms of galaxy structure, the Offset-SF sample has larger\nconcentration ($C$ = 2.58 compared to 2.36), smaller half-light\nradii [$R_{50}(z)$ of 1.2 kpc compared to 1.9 kpc], higher surface\nstellar mass density [log($\\mu_{\\star}$) of 8.78 M$_{\\odot}$\nkpc$^{-2}$ compared to 8.61 M$_{\\odot}$ kpc$^{-2}$], and higher\nSFR surface density [log($\\dot{\\Sigma}_{\\star}$) of -0.86\nM$_{\\odot}$ yr$^{-1}$ kpc$^{-2}$ compared to -1.12 M$_{\\odot}$\nyr$^{-1}$ kpc$^{-2}$] than the Main sample, whereas the Offset-AGN\nhas larger concentration ($C$ of 2.66), moderately smaller sizes\n[$R_{50}(z)$ of 1.5 kpc], moderately higher surface stellar mass\ndensity [log($\\mu_{\\star}$) of 8.68 M$_{\\odot}$ kpc$^{-2}$], and\nsimilar SFR surface density [log($\\dot{\\Sigma}_{\\star}$) of -1.12\nM$_{\\odot}$ yr$^{-1}$ kpc$^{-2}$].\n\nIn terms of environment, we find that 90\\%-95\\% of objects in the\nMain, Offset-SF, and Offset-AGN samples have zero or one neighbor,\nthe lowest density bin in \\citet{kauffmann04}. Furthermore, the\ndistributions of environments for the Offset samples are similar\nto that of the Main sample. Therefore, it appears that the vast\nmajority of galaxies considered here reside in the lowest density\nenvironment defined by \\citet{kauffmann04}. This result is not\nsurprising, as our SDSS samples all contain emission-line\ngalaxies, which are more likely to be found in low-density\nenvironments \\citep{kauffmann04}. Currently we have too few Offset\nobjects to draw any solid conclusion about the question on galaxy\nenvironmental dependence.\n\n\n\n\n\\citet{groves06} suggest that the offset on the BPT diagram seen\nin some high-redshift galaxies is mostly caused by contribution\nfrom AGNs, as they found properties of their candidate\nlow-metallicity H II\/AGN composites similar to those of the host\ngalaxies of AGNs \\citep{kauffmann03b,heckman04}; these objects\nhave on average significantly higher stellar masses, older stellar\npopulations, and redder colors than the sample of pure\nstar-forming galaxies used for comparison. \\citet{groves06} arrive\nat this conclusion after constructing their offset sample in a\nregion bounded by the Ka03 and the Ke01 curves in the [N\nII]\/H$\\alpha$ versus [O III]\/H$\\beta$ diagram, and the Seyfert\nbranch [log([O III]\/H$\\beta$) $\\geq$ 3log([N II]\/H$\\alpha$)]. When\nwe reproduced their offset sample, we found the median values of\nlog([N II]\/H$\\alpha$) $\\sim$ -0.286 and log([O III]\/H$\\beta$)\n$\\sim$ -0.024, whereas the corresponding values for our DEEP2\nobjects are -0.638 and 0.344. As the density of objects increases\nrapidly towards increasing [N II]\/H$\\alpha$, the offset sample\ndefinition of \\citet{groves06} is weighted heavily towards H\nII\/AGN composites and not necessarily representative of the\naverage [N II]\/H$\\alpha$ or [O III]\/H$\\beta$ of high-redshift\ngalaxies. Furthermore, it is not clear that their ``typical''\ncomparison sample, which was defined as all galaxies within\n$\\pm$0.05 dex of [-0.55, 0.10] in the [N II]\/H$\\alpha$ versus [O\nIII]\/H$\\beta$ diagram, represents a fair one, as they have not\ncontrolled for any galaxy property. Given the strong correlations\namong galaxy properties, it is crucial to compare galaxies that\nare similar in at least some basic parameters.\nHere we want to emphasize that in order to make a controlled\ncomparison, the Main sample was selected to have similar range and\nmedian of stellar masses as that of the Offset sample. The Main\nsample also has a similar median [N II]\/H$\\alpha$ value. With this\ncontrolled comparison of typical and Offset SDSS objects, we find\nthat the unusual objects on the BPT diagram are populated not only\nby likely AGN hosts (Offset-AGN) but also by the Offset-SF sample,\nmost of which do not resemble typical AGN-host galaxies.\nFurthermore, there is no segregation of the two offset samples in\nthe [N II]\/H$\\alpha$ versus [O III]\/H$\\beta$ diagram.\n\nAs suggested in \\citet{groves06}, another possible test of the\npresence of an AGN would require the detection of either He II\n$\\lambda$4686 or [Ne V] $\\lambda$3426 lines. The [Ne V]\n$\\lambda$3426 line is not redshifted into the SDSS band for the\nmajority of our objects: we have only four such objects in\nOffset-AGN, of which the spectra near [Ne V] $\\lambda$3426 are too\nnoisy to provide any meaningful constraints. The He II\n$\\lambda$4686 line is expected to be very weak: a 20\\% AGN\ncontribution to H$\\beta$ implies He II\/H$\\beta$ = 0.05. Note that\nbesides the AGN photoionization, possible mechanisms for producing\nHe II emission also include hot stellar ionizing continua and\nshock excitation \\citep{garnett91}. Another important goal is to\ninfer metallicities of the Offset-SF objects using a method that\nis independent of strong-line indicators, since these objects have\nabnormal strong-line ratios. A more calibration-independent way to\naddress metallicity is through the classic $T_e$ method, which\nrelies on measuring weak auroral lines. Both the test of the\npresence of AGNs, and the determination of metallicity with the\ndirect $T_e$ method, require the measurement of very weak lines.\nThese measurements become feasible through the use of composite\nspectra.\n\n\n\n\n\n\n\n\n\\subsection{SDSS Composite Spectra}\n\n\\begin{figure*}\n  \\centering\n    \\includegraphics[width=150mm]{f12a.eps}\n    \\includegraphics[width=150mm]{f12b.eps}\n    \\includegraphics[width=150mm]{f12c.eps}\n    \\caption{SDSS composite spectra. For clarity, only the portion\n    of the spectrum containing the weak emission lines of interest\n    is presented here. The original composite and stellar-continuum-subtracted\n    spectra for samples (A) Main, (B) Offset-AGN, and (C)\n    Offset-SF\n    are shown as black solid curves. Stellar-continuum fits using\n    \\citet{bc03} population synthesis models are plotted as red dotted\n    curves. The [O III] $\\lambda$4363 and He II $\\lambda$4686 lines are marked by dotted lines.}\n    \\label{fig:sdsscompspec}\n\\end{figure*}\n\n\nIn order to measure He II $\\lambda$4686 as another test of\npossible AGN contribution to the ionizing spectrum, and [O III]\n$\\lambda$4363 to infer the oxygen abundance using the $T_e$\nmethod, we construct composite spectra for both Offset-SF and\nOffset-AGN objects. For comparison, we also make composite spectra\nfor the Main sample. We adopt the same method of making composite\nspectra used for our DEEP2 objects, employing median scaling to\npreserve the relative fluxes of the emission features. The\nvariance spectrum is combined in the same way and the error for\nthe corresponding composite spectrum is the square root of the\ncomposite variance spectrum divided by the number of objects.\n\nThe stellar continuum was modelled over the rest-frame wavelength\nrange of 3800 - 7500 ${\\rm \\AA}$ and then subtracted from the\ncomposite spectra, with a method similar to the one described in\n\\citet{tremonti04} and \\citet{brinchmann04}. This technique\n\\citep[][private communication]{charlot07} relies on the\n\\citet{bc03} stellar population synthesis model, but with two\nmajor improvements. First, the number of the metallicity grids was\nexpanded from three to four; second, and most importantly, the\nstellar continuum fitting was carried out using both the STELIB\n\\citep{leborgne03} and the MILES \\citep{sanchez06} libraries of\nobserved stellar spectra. Models using the MILES library fit the\ndata better than those with the STELIB one, in terms of the\nagreement around Balmer lines. It is especially essential to\nobtain a reasonable stellar continuum fit near H$\\gamma$, which is\nvery close in wavelength to the weak [O III] $\\lambda$4363 auroral\nline. Models using the STELIB library tend to over-estimate\nstellar continua, especially around Balmer lines. We therefore\nadopt the stellar continuum models based on the MILES library.\n\nThe resultant continuum-subtracted spectra, along with the\noriginal composite spectra and the best-fit model of the stellar\ncontinua, are shown in Figure (\\ref{fig:sdsscompspec}). We only\npresent portions of the entire composite spectra to enable\nscrutiny of the relevant weak lines ([O III] $\\lambda$4363 and He\nII $\\lambda$4686) and our capability of making a reasonably good\nstellar continuum fit. The corresponding error spectra are\nestimated by combining the measurement error for the composite\nspectra and the systematic uncertainties from the stellar\ncontinuum fitting. The systematic error from the stellar continuum\nfitting is estimated using the rms of a portion of the\ncontinuum-subtracted spectrum, which is relatively free of\nemission-line features. This method of estimating continuum fit\nerror is motivated by the fact that the rms would be zero if the\nfitting was ideally good and if there were no emission-line\nfeatures. At all of the wavelengths considered, the systematic\nerror is significantly larger than the measurement error for the\ncomposite spectra and therefore dominates the total error budget.\nEmission-line fluxes and flux ratios with the total uncertainties\nare given in Table \\ref{tab:sdsscomp}.\n\nThere are a few caveats that must be mentioned in the analysis of\ncomposite spectra described here. First, we are averaging over\ntens of thousands of galaxies for the SDSS Main sample and dozens\nof galaxies for the Offset-SF and Offset-AGN samples; second, for\neach single galaxy, we are dealing with integrated spectra\ncontaining contributions from multiple H II regions, which may\nshow a range of metallicities, ionization parameters, ionizing\nspectra, and electron densities. Also, even though we select the\nsamples to have fiber apertures covering more than 20\\% of the\ntotal $g$-band photons, because of the presence of radial\ngradients in galaxy properties, uncertainties still remain since\nthe spectrum is weighted towards the nucleus. Despite these\ncaveats, however, we want to emphasize that the analysis of\ncomposite integrated fibre-based spectra is still meaningful in\nterms of determining average properties and the relative\ndifferences among samples.\n\n\n\n\n\\begin{deluxetable*}{lccccccc}\n\\tabletypesize{\\scriptsize}\n\\tablecaption{Emission-line flux measurements of SDSS composite spectra.\\label{tab:sdsscomp}}\n\\tablewidth{0pt}\n\\tablehead\n{\n\\colhead{~~~~~~~~~Sample~~~~~~~~~} & \n\\colhead{F$_{{\\rm [O II]}\\lambda\\lambda3726, 3729}$\\tablenotemark{a}} & \n\\colhead{F$_{{\\rm [O III]}\\lambda4363}$\\tablenotemark{a}} & \n\\colhead{F$_{{\\rm He II}\\lambda4686}$\\tablenotemark{a}} & \n\\colhead{F$_{{\\rm H}\\beta}$\\tablenotemark{a}} &\n\\colhead{F$_{{\\rm [O III]}\\lambda4959}$\\tablenotemark{a}} & \n\\colhead{F$_{{\\rm [O III]}\\lambda5007}$\\tablenotemark{a}} &\n\\colhead{F$_{{\\rm [O I]}\\lambda6300}$\\tablenotemark{a}}\n}\n\\startdata\nMain\\dotfill       & 327.3$\\pm$0.3 & 0.6$\\pm$0.2 &  1.4$\\pm$0.2 & 153.5$\\pm$0.2 &  45.2$\\pm$0.1 & 137.0$\\pm$0.3 & 19.1$\\pm$0.2 \\\\\nOffset-AGN\\dotfill & 271.9$\\pm$0.8 & 2.8$\\pm$0.5 &  4.9$\\pm$0.5 &  90.4$\\pm$0.5 &  86.4$\\pm$0.2 & 261.7$\\pm$0.6 & 24.3$\\pm$0.6 \\\\\nOffset-SF\\dotfill  & 629.1$\\pm$1.0 & 7.1$\\pm$0.6 & 12.5$\\pm$0.7 & 321.0$\\pm$0.7 & 266.6$\\pm$0.3 & 807.4$\\pm$0.8 & 38.8$\\pm$0.7 \\\\\n\\hline\n~~~~~~~~~Sample~~~~~~~~~ &  F$_{{\\rm H}\\alpha}$\\tablenotemark{a} & F$_{{\\rm [S II]}\\lambda6717}$\\tablenotemark{a} & F$_{{\\rm [S II]}\\lambda6731}$\\tablenotemark{a} & $\\frac{{\\rm F_{[O III] \\lambda \\lambda 4959, 5007}}}{{\\rm F_{[O III] \\lambda 4363}}}$\\tablenotemark{b}  & $\\frac{{\\rm F_{He II \\lambda 4686}}}{{\\rm F_{H\\beta}}}$ & $\\frac{{\\rm F_{[O I] \\lambda 6300}}}{{\\rm F_{H\\alpha}}}$ & $\\frac{{\\rm F_{[S II]\\lambda\\lambda6717+6731}}}{{\\rm F_{H\\alpha}}}$ \\\\\n\\hline\n          &   &    &    &   &     &   &    \\\\\nMain\\dotfill         &  633.6$\\pm$0.3 & 117.6$\\pm$0.3  & 83.9$\\pm$0.3  & 249$\\pm$82 &  0.009$\\pm$0.001  & 0.0301$\\pm$0.0004 & 0.3180$\\pm$0.0006  \\\\\nOffset-AGN\\dotfill   &  337.3$\\pm$0.7 &  80.6$\\pm$0.6  & 58.5$\\pm$0.6  & 110$\\pm$19 &  0.054$\\pm$0.006  & 0.0720$\\pm$0.0018 & 0.4124$\\pm$0.0028   \\\\\nOffset-SF\\dotfill    & 1272.8$\\pm$0.9 & 170.6$\\pm$0.8  & 137.3$\\pm$0.8 & 129$\\pm$11 &  0.039$\\pm$0.002  & 0.0305$\\pm$0.0006 & 0.2419$\\pm$0.0009  \\\\\n\\enddata\n\\tablenotetext{a}{Emission-line flux and 1 $\\sigma$ error in units of $10^{-17}$ ergs s$^{-1}$ cm$^{-2}$ from SDSS composite spectra.}\n\\tablenotetext{b}{Dust-reddening corrected using Balmer decrement method.}\n\\end{deluxetable*}\n\n\n\n\\subsubsection{AGN Contribution}\n\nThe composite spectra we construct can be used to address the\nlevel at which AGN ionization contributes to the anomalous\nemission-line ratios, discovered in both nearby SDSS Offset\nobjects and high-redshift DEEP2 galaxies. Based on their [S\nII]\/H$\\alpha$ and [O I]\/H$\\alpha$ ratios, we expect the Offset-AGN\nobjects to have a non-negligible contribution to their ionization\nfrom AGNs and\/or shock heating. Based on the composite spectrum,\nwe can also probe the weaker He II line, which is another\nindicator of AGN activity.\n\nFrom the He II\/H$\\beta$ ratio (0.054$\\pm$0.006), we can see that\nthe Offset-AGN sample may have up to $\\sim$20\\% AGN contribution\nto the Balmer emission lines \\citep{groves06}. The Offset-SF\nsample, on the other hand, shows a lower level of possible AGN\ncontamination (He II\/H$\\beta$ of 0.039$\\pm$0.002). This\ndistinction is also suggested by the different host-galaxy\nproperties of the two samples. The Offset-AGN objects with similar\nstellar masses to the Main sample have redder colors, older\nstellar populations, lower starburst stellar mass fraction,\nsmaller H$\\alpha$ EWs, and smaller specific star formation rates,\nwhich are similar to host galaxies of AGN \\citep{kauffmann03b},\nwhereas the Offset-SF sample is similar to the Main sample in\nterms of these galaxy properties. Furthermore, by construction,\nthe Offset-SF objects have low [S II]\/H$\\alpha$ and [O\nI]\/H$\\alpha$ in the range favored for typical star-forming\ngalaxies.\n\nWe note, however, that the Offset-SF sample still has a higher\nvalue of He II\/H$\\beta$ than the Main sample (0.009$\\pm$0.001).\nThere might be some mixing with the Offset-AGN objects because of\nuncertainties in emission-line diagnostic ratios. Indeed, three\nobjects in the Offset-SF sample have cross-identifications with\n$ROSAT$ sources, and two of them have optical spectra indicating\nbroad Balmer lines. However, $\\sim$90\\% of the Offset-SF sample\nshow no evidence of AGNs in terms of $ROSAT$-source\ncross-identifications and broad Balmer lines. In addition, we have\nexamined the He II\/H$\\beta$ ratios from individual galaxy spectra\nin the Offset-SF sample, finding that five objects (including the\nthree with $ROSAT$ cross-identifications) of the 25 have He\nII\/H$\\beta$ $\\sim$ 0.05-0.10, while the remaining 20 objects all\nhave He II\/H$\\beta$ $\\lesssim$ 0.01, similar to the typical He\nII\/H$\\beta$ of the Main sample. Even the five objects with He\nII\/H$\\beta$ $\\sim$ 0.05-0.10 could have other possible ionizing\nsources including hot stars and shocks \\citep{garnett91}. It is\nworth mentioning that nondetection in $ROSAT$ or broad Balmer\nlines cannot fully exclude the possibility of AGN contamination,\nas the majority of obscured AGNs are completely absorbed in the\nsoft X-ray range and do not show broad Balmer lines. However, if\nthe dominant cause of the Offset-SF on the BPT diagram for our\nOffset-SF sample was mildly obscured AGN contamination, namely, if\ntheir optical spectra were contaminated by AGN ionization at a\nlower level than the Offset-AGN sample, then we would expect them\nto exhibit intermediate emission-line diagnostic ratios and\nhost-galaxy properties, relative to the Main and the Offset-AGN\nsamples. This is not the case. Instead, the Offset-SF objects have\neven higher ionization parameters and electron densities than the\nOffset-AGN sample, relative to the Main sample. In addition, as\ndiscussed in \\S \\ref{sec:sub:comparison}, Offset-SF objects have\nhost-galaxy properties different from those of typical weak AGNs\nin SDSS \\citep{kauffmann03b}. Therefore, it appears that AGN\nexcitation does not provide the dominant cause of the anomalous\nemission-line ratios for most of the Offset-SF sample.\n\n\\subsubsection{Electron Temperature and Oxygen Abundance}\\label{sec:sub:te}\n\n\\begin{figure*}\n\\epsscale{1.} \\plotone{f13.eps} \\caption{Mass-metallicity relation\nfor the SDSS Main ({\\it grey contours}) and Offset-SF ({\\it red\nopen circles}) samples, based on various strong-line indicators.\nCalibrations are from \\citet[][for $R_{23}$]{zaritsky94},\n\\citet[][for N2 and O3N2]{pp04}, and \\citet[][for\nN2O2]{bresolin07}, respectively. \\label{fig:sdss:massz}}\n\\end{figure*}\n\n\n\n\n\\begin{deluxetable*}{lcccccc}\n\\centering\n\\tabletypesize{\\scriptsize}\n\\tablecaption{Physical Quantities from SDSS composite spectra.\\label{tab:sdssabundance}}\n\\tablewidth{0pt}\n\\tablehead\n{\n\\colhead{~~~~~~~~~~Sample~~~~~~~~~~} & \n\\colhead{~~~~~N$_e$\\tablenotemark{a}~~~~~} &\n\\colhead{~~~~~T$_e$[O III]\\tablenotemark{b}~~~~~} &\n\\colhead{~~~~~T$_e$[O II]\\tablenotemark{b}~~~~~} & \n\\colhead{~~~~~O$^+$\/H$^+$\\tablenotemark{c}~~~~~} &\n\\colhead{~~~~~O$^{++}$\/H$^+$\\tablenotemark{c}~~~~~} & \n\\colhead{~~~~~12 + log(O\/H)\\tablenotemark{d}~~~~~}\n}\n\\startdata\nMain\\dotfill      &  28$\\pm$ 5 & 0.950$^{+0.124}_{-0.072}$ &  0.965$^{+0.087}_{-0.050}$ & 13.2$^{+3.7}_{-4.1}$ &  3.7$\\pm$1.3 & 8.23$^{+0.11}_{-0.17}$ \\\\\nOffset-SF\\dotfill & 190$\\pm$12 & 1.170$^{+0.039}_{-0.033}$ &  1.119$^{+0.027}_{-0.023}$ &  6.5$\\pm$0.6         &  5.2$\\pm$0.5 & 8.07$\\pm$0.04    \\\\\n\\enddata\n\\tablenotetext{a}{Electron density and 1 $\\sigma$ error in units of cm$^{-3}$ derived from $({\\rm [S \\,\\,II]} \\lambda 6717\/{\\rm [S \\,\\,II]} \\lambda 6731$).}\n\\tablenotetext{b}{Electron temperature and 1 $\\sigma$ error in units of $10^4$ K.}\n\\tablenotetext{c}{Ionic abundance and 1 $\\sigma$ error from collisionally excited lines in units of $10^{-5}$.}\n\\tablenotetext{d}{Total oxygen abundance and 1 $\\sigma$ error assuming O\/H = (O$^{+}$ + O$^{++}$)\/H$^{+}$.}\n\\end{deluxetable*}\n\n\n\n\n\nBy making composite spectra, and hence enhancing the S\/N of weak\nauroral lines, we can determine metallicities with a method\nindependent of strong-line indicators. We have already discussed\nthe fact that the Offset-SF objects have larger ionization\nparameters, as indicated by their higher $O_{32}$ values. However,\nit is still unclear whether the Offset-SF sample has comparable\nmetallicity with the Main one, on average, as suggested by N2 and\nN2O2, or if it has $\\sim$0.3 dex lower metallicity, as suggested\nby O3N2 and $R_{23}$. Settling this question is important for\nmultiple reasons.\n\nFirst, gas-phase metallicity and electron temperature, themselves,\nserve as key parameters of H~II regions. Second, understanding the\nrelative metallicities of the Main and the Offset-SF samples on\naverage can help us determine the difference in average ionization\nparameter quantitatively. If the Offset-SF sample has comparable\nmetallicities with the Main sample, their ionization-parameter\ndifference would be very large; on the other hand, if the\nOffset-SF sample has much lower metallicities than the Main one\ndoes, their ionization-parameter difference would be much smaller,\nsince both lower metallicity and higher ionization parameter can\ncause large $O_{32}$. Finally, if the Offset-SF objects actually\nhave much lower metallicities than the Main objects, then they\nwill not follow the local mass-metallicity relation\n\\citep{tremonti04}, given their comparable stellar masses.\n\n\nIn order to determine oxygen abundances using the direct method,\nestimates of both the electron density and temperature are\nrequired. Based on the ratio [S II] $\\lambda$6717\/[S II]\n$\\lambda$6731, we establish that on average all of our objects are\nin the low-density regime. Then using the five-level atom program\n{\\tt nebular} implemented in IRAF\/STSDAS \\citep{shaw95}, we derive\nthe electron temperature $T$[O III] from the ratio of auroral line\n[O III] $\\lambda$4363 to nebular lines [O III] $\\lambda\n\\lambda$4959, 5007 \\citep{agn2} and correct for dust extinction\nusing the Balmer decrement method. As for $T$[O II], we adopt a\nsimple scaling relation between the temperatures in different\nionization zones of an H II region predicted by the\nphotoionization models of Garnett (1992):\n\\begin{equation}\nT{\\rm [O \\,\\,II]} = 0.70T{\\rm [O\\,\\, III]} + 3000 \\,\\,{\\rm K},\n\\end{equation}\nwhich is applicable in a wide range of $T$[O III] (2000-18,000 K).\nIonic abundances O$^{++}$\/H$^{+}$ and O$^{+}$\/H$^{+}$ are then\ndetermined using programs in the {\\tt nebular} package, and,\nfinally, we obtain the total gas phase oxygen abundance assuming\nO\/H = (O$^{+}$ + O$^{++}$)\/H$^{+}$. The results are summarized in\nTable \\ref{tab:sdssabundance}.\n\n\nFirst, the Offset-SF sample has $\\sim$2200 K higher electron\ntemperatures than the Main one does on average. These objects are\nunusual in the sense that they have higher electron densities and\nelectron temperatures, hence ambient interstellar pressures\nassuming pressure equilibrium between H II regions and ambient\ngas, and larger ionization parameters. It is not clear what is at\nthe root of the special H II region physical conditions, in terms\nof galaxy properties, or larger scale environments. However, in\nthe simple theoretical estimates for local starburst galaxies, the\nISM pressure scales roughly linearly with the SFR surface density\n\\citep[e.g.,][]{thompson05}. Indeed, the Offset-SF sample has\nseveral prominently related properties, including higher\nconcentration ($r$-band Petrosian concentration index\n$R_{90}\/R_{50}$ of $\\sim$2.6 compared to $\\sim$2.4), smaller\nhalf-light radii [$R_{50}(z)$ of 1.2 kpc compared to 1.9 kpc], and\nmost notably, higher SFR surface density\n[log($\\dot{\\Sigma}_{\\star}$) of -0.86 M$_{\\odot}$ yr$^{-1}$\nkpc$^{-2}$ compared to -1.12 M$_{\\odot}$ yr$^{-1}$ kpc$^{-2}$)],\nas shown in Figure \\ref{fig:subfig:dis}, compared to typical\nstar-forming galaxies. This higher SFR surface density may\ntherefore account for the higher interstellar pressure seen in the\nH II regions of Offset-SF objects.\n\n\n\nIt is also possible to relate electron temperature, electron\ndensity, and ionization parameter in a single H II region, through\nthe classical picture of \\citet{stromgren39}. With higher ambient\ninterstellar pressure due to higher electron densities and\ntemperatures, H II regions are surrounded by denser molecular gas\ndust and therefore have smaller radii under the stall condition.\nIn an idealized case of a fully ionized spherical H II region with\npure hydrogen, the radius $R$ of the ionized region is given by $R\n= [3Q\/(4\\pi N_e^2 \\alpha_B)]^{1\/3}$, where $Q$ is the rate of\nemission of hydrogen ionizing photons, and $\\alpha_B$ is the case\nB recombination coefficient \\citep{agn2}. The ionization parameter\n$U$ defined by $[F\/4\\pi c R^2 N_{{\\rm H II}}]$, where $F$ is the\nflux of ionizing photons, $c$ is the speed of light, and $N_{{\\rm\nH II}}$ is the particle density in H II region, is then\nproportional to $N_e^{1\/3}$ for a given central hot star.\nTherefore, in this simple picture, H II regions with higher\nelectron densities and temperatures will have higher ionization\nparameters. Accordingly, the higher electron densities and\ntemperatures inferred from the integrated spectra of the Offset-SF\nobjects may result in the observed higher ionization parameters as\nwell.\n\nSecond, the Offset-SF sample has $\\sim $0.16 dex lower\nmetallicities than the Main sample. Note that the absolute\nabundance values inferred using the $T_e$ method are significantly\nlower than those based on any strong-line indicators. This\ndifference may stem from temperature fluctuations in H~II regions\nand the strong $T_e$ dependence of line emissivities, from which\nthe net effect is that the electron temperature derived from\nauroral lines tends to overestimate the real temperature and hence\nsystematically underestimate the abundance \\citep{garnett92}.\nTherefore, we only focus on relative differences based on the same\nmethod between the Offset-SF and Main samples. Given that the\nOffset-SF and Main samples are characterized by similar average\nstellar masses, this difference in average metallicity suggests\nthat Offset-SF objects do not follow the local mass-metallicity\nrelation, in addition to being more compact and sustaining higher\ninterstellar pressures than the Main sample.\n\nTo investigate this question further, we examine several versions\nof the mass-metallicity relation for the Offset-SF and the Main\nobjects, based on different strong-line metallicity indicators.\nFigure \\ref{fig:sdss:massz} includes these mass-metallicity\ncorrelations for both the Main and the Offset-SF samples,\ndemonstrating that one would draw different conclusions about the\nrelationship between the Offset-SF and the Main samples, depending\non which strong-line indicator was used. Between the Main and\nOffset-SF samples, the relative average metallicity differences at\nfixed mass based on $R_{23}$, N2, O3N2, and N2O2 are $\\sim$0.28,\n0.00, 0.18, and 0.00 dex, respectively. This ambiguity could be\nunderstood as the result of adopting the same calibration for\nsamples with different physical conditions, of which the most\nimportant are ionization parameter and electron density.\n\nThe discrepancy between conclusions drawn from the strong-line\nindicators and the $T_e$ method can be understood using the\nphotoionization models of \\citet{kewley02}. For example, according\nto $R_{23}$, the average metallicity difference between the\nOffset-SF and Main samples is $\\sim$0.28 dex, whereas this\nmetallicity difference based on the $T_e$ method is $\\sim$0.16\ndex. This $\\sim$0.12 dex discrepancy can be explained by examining\nthe theoretical grids in the $R_{23}$ versus 12 + log(O\/H) space,\ngiven by photoionization models with different values of electron\ndensity and ionization parameter. If the incorrect assumption is\nadopted that both the Main and Offset-SF samples have electron\ndensities $N_e \\sim 10$ cm$^{-3}$ and ionization parameters\n$q=3\\times10^{7}\\mbox{cm s}^{-1}$ -- values appropriate for the\nMain but not Offset-SF sample --  we obtain that the Offset-SF\nsample is $\\sim 0.25$~dex lower in oxygen abundance than the Main\nsample. On the other hand, if an electron density $N_e \\sim 350$\ncm$^{-3}$ and ionization parameter $q=5\\times10^{7}\\mbox{cm\ns}^{-1}$ are assumed for the Offset-SF sample --  more appropriate\nbased on their inferred H~II region physical conditions -- the\ndifference in estimated Main and Offset-SF oxygen abundance is\nonly $\\sim$0.10 dex. This difference is $\\sim$0.15 dex smaller\nthan the one inferred with the incorrect assumption that the\nOffset-SF and Main samples have the same electron densities and\nionization parameters. The actual difference in electron density\nbetween the two samples is less extreme ($N_e \\sim 190$ cm$^{-3}$\nof the Offset-SF compared to $N_e \\sim 28$ cm$^{-3}$ of the Main),\nso this discrepancy may be smaller, as observed ($\\sim$0.12 dex).\nAnalogous effects occur for the N2 and O3N2 indicators.\n\n\n\n\nIn summary, the metallicities of the Offset-SF objects based on\nstrong-line indicators can be either overestimated or\nunderestimated, depending on the relative shifts of the\ntheoretical grids caused by varying physical conditions, and an\nincorrect assumption of the values of ionization parameter and\nelectron density. For the SDSS Offset-SF sample, the metallicities\nbased on strong-line indicators can either be systematically too\nlarge by $\\sim$0.16 dex (N2, N2O2), or too low by $\\sim$0.12 dex\n($R_{23}$). The bias of O3N2 is little (too large by $\\sim$0.02\ndex) for our SDSS Offset-SF objects, although it may not always be\nnegligible given other ranges of physical conditions in terms of\nmetallicity, ionization parameter, and electron density. The\nreason for the Offset-SF objects' anomalous positions on the BPT\ndiagram is the same: H~II regions with distinct ionization\nparameters and electron densities fall on different surfaces in\nthe diagnostic-line parameter space.\n\n\n\nOne caveat that should be mentioned here is the fact that we are\ninferring average biases over the stellar mass and metallicity\nranges spanned by the Offset-SF sample. Another caveat is that\nresults of the SDSS data are all based on integrated fiber spectra\nand hence limited in terms of spatial information. We plan to\nfurther study the spatial dependence of H~II region emission lines\nin these offset objects using long-slit spectrographs in the\nfuture.\n\n\n\n\n\\subsection{Implications for DEEP2 Objects}\n\nWe have analyzed emission-line diagnostic ratios, physical\nconditions and galaxy properties of SDSS objects with similar [N\nII]\/H$\\alpha$ and [O III]\/H$\\beta$ values to those of DEEP2\ngalaxies with the most extreme offset in our sample, and $z \\sim\n2$ star-forming galaxies \\citep{erb06a}. There are two major\ncauses for their offset: one is different H II region physical\nconditions characterized by higher electron density and\ntemperature, and hence larger ionization parameter, compared to\ntypical SDSS star-forming galaxies. These physical conditions of\nthe SDSS Offset-SF objects are also connected to their host galaxy\nproperties, particularly the higher SFR surface density. The other\npossible cause for the offset is contribution from AGNs and\/or\nshock excitation. We cannot rule out either of these two\npossibilities, or a combination of both, to explain the offset of\nDEEP2 objects on the diagnostic diagram.\n\nAs for the question of H II region physical conditions, it is\npossible to determine electron densities for our DEEP2 objects\nusing the [O II] doublet contained in the DEIMOS spectra. Note\nthat we will not directly compare electron densities of the $z\n\\sim 1.0-1.5$ objects to those of SDSS local galaxies, because\nthey were estimated using different density-sensitive doublets,\nand therefore the comparison might suffer from the associated\nsystematic uncertainties. However, when we divide all of the\nobjects in our DEEP2 sample that have reliable [O II]-doublet\nmeasurements into two groups, according to the empirical Ka03\ncurve in the [N II]\/H$\\alpha$ versus [O III]\/H$\\beta$ diagram, we\nfind that for the group below the Kauffmann curve, the median\nvalue of electron density inferred from [O II] $\\lambda$3726\/[O\nII] $\\lambda$3729 is $\\sim$23 cm$^{-3}$, whereas for the group\nabove the Kauffmann curve, the median value is $\\sim$159\ncm$^{-3}$. This difference in electron density as a function of\nposition on the BPT diagram is qualitatively consistent with what\nwe have found for SDSS objects (Table (\\ref{tab:sdssabundance});\nelectron density N$_e$ [cm$^{-3}$] of 190$\\pm$12 for the Offset-SF\ncompared to 28$\\pm$5 for the Main), lending independent support\nfor our method of using Offset-SF objects as local analogs for the\nH~II regions in our high-redshift sample. In principle, we could\nexamine the ionization parameter indicator, $O_{32}$, for our\nDEEP2 objects, using [O II] from DEIMOS and [O III] from NIRSPEC\nspectra. However, at this point, the comparison does not seem\nuseful, as it would suffer from large systematic uncertainties,\ndue to the manner in which the optical and near-IR spectra were\ncollected and flux-calibrated.\n\nA related question is whether objects that are more offset in the\n[N II]\/H$\\alpha$ versus [O III]\/H$\\beta$ diagram have higher\naverage SFR surface density than objects with less offset within\nour DEEP2 $z \\sim 1-1.5$ sample. The spatial extent of H$\\alpha$\nemission contains information about galaxy sizes. However, we can\nonly measure the extent along the slit, which does not necessarily\nreflect the actual galaxy size. In addition, the size estimated\nfrom the spatial extent of H$\\alpha$ emission is subject to\nuncertainties due to the variation in the seeing FWHM, which is\ncomparable to the H$\\alpha$ size itself. As we do not have\nreliable estimates for the sizes of our DEEP2 objects, the above\nquestion can be finally answered only when the galaxy\nmorphological information is robustly gathered. Furthermore, the\nSFRs of our DEEP2 objects have not been corrected for dust\nextinction or aperture effects, which may amount to factor of 2\ndifferences \\citep{erb06c}. Limited by the associated systematic\nuncertainties, we are not able to compare the SFR surface\ndensities of our DEEP2 objects directly with those of the SDSS\nsamples. However, after correcting the SFR-associated systematic\nuncertainties for their $z \\sim 2$ sample, \\citet{erb06b} found a\nmean log($\\dot{\\Sigma}_{\\star}$) $\\approx$ 0.46, significantly\nhigher than that of SDSS local star-forming galaxies. In addition,\nobservational evidence exists that the galaxy size at fixed\nmass\/luminosity decreases with increasing redshift out to z $\\sim$\n3 \\citep{trujillo06,dahlen07}. At the same time, galaxies with\nstar formation rates significantly higher than those found among\ntypical SDSS emission-line galaxies are more commonly found at\nhigher redshifts \\citep[e.g.,][]{dahlen07,reddy07}. Therefore,\ngalaxies with high SFR surface densities are more prevalent in the\nhigh-redshift universe. For such objects, a noticeable difference\nin H~II region physical conditions is expected.\n\n\nAs for the question of AGN contribution, the DEEP2 object 42010637\nclearly falls on the part where many objects have contribution\nfrom AGNs in their ionizing spectra. However, it does not have\nmulti-wavelength information that can either confirm or rule out\nthe AGN excitation. In addition, all of the $z\\sim 2$ objects with\n[O III]\/H$\\beta$ and [N II]\/H$\\alpha$ measurements are offset by\nan amount similar to our most extremely offset $z\\sim 1.0-1.5$\nobjects but show no evidence of AGN contamination in at least\ntheir rest-frame UV spectra \\citep{erb06a}. The lack of such\nevidence in the rest-frame UV rules out the presence of at least a\nfairly unobscured AGN. Note that \\citet{daddi07} find that roughly\n20-30\\% of star-forming galaxies with $M\\sim 10^{10}$-$10^{11}\nM_{\\odot}$ at $z\\sim 2$ display a mid-IR excess, as evidence for\nhosting an obscured AGN. These authors show that this fraction\nincreases with stellar mass, reaching $\\sim$50-60\\% for an extreme\nmass range of $M > 4\\times10^{10} M_{\\odot}$. Our DEEP2 $z\\sim\n1.0-1.5$ sample covers a range of $M \\sim 5\\times10^9$- $10^{11}\nM_{\\odot}$, not all of which are in the ``massive'' range of\n\\citet{daddi07}, and only two of 20 are in the extreme range. If\nthe mid-IR excess, hence AGN contamination, found by\n\\citet{daddi07} is also applicable to our DEEP2 $z\\sim 1.0-1.5$\nobjects, the fraction of objects containing potential AGN\ncontamination would be at most $\\sim 20$\\%-30\\%. However, nine (or\n10, one with a lower limit) of the 13 objects in our sample with\nall four line measurements show evidence for being offset in the\n[O III]\/H$\\beta$ versus [N II]\/H$\\alpha$ BPT diagram (i.e., more\nthan 20-30\\%).  More generally, the associated space densities of\nboth the blue DEEP2 galaxies \\citep{coil07} and UV-selected $z\n\\sim 2$ objects for which we have NIRSPEC spectra\n\\citep{adelberger05,reddy07} are significantly higher than that of\nthe obscured AGN population featured in \\citet{daddi07}.\nTherefore, as one of the possible causes for the offset on the BPT\ndiagram, AGN contamination cannot account for nor is consistent\nwith all the rest-frame optical emission-line measurements\npresented here.\n\nWe will address the AGN-contamination issue in the future by\nlooking at more DEEP2 objects for which multi-wavelength\ninformation is available and obtaining spatially resolved spectra\nwith an integral field unit assisted by adaptive optics to isolate\nthe contribution from the nucleus. Also, the measurements of\nflux-calibrated emission lines including [O II] $\\lambda$3727, [O\nIII] $\\lambda$4363, H$\\beta$, [O III] $\\lambda$5007, [O I]\n$\\lambda$6300, [N II] $\\lambda$6584, H$\\alpha$ and [S II]\n$\\lambda\\lambda$6717, 6731, as well as host-galaxy morphological\ninformation are required, in order to finally settle the causes of\nthe offset in the [N II]\/H$\\alpha$ versus [O III]\/H$\\beta$ diagram\nfor high-redshift star-forming galaxies. [O III] $\\lambda$4363 may\nbe difficult to measure for all but the most metal-poor objects or\nin deep $z \\sim 1$ composite spectra, and intensity limits on [O\nI] $\\lambda$6300 may also be difficult to obtain. Yet measurements\nof the stronger emission lines for a statistical sample of objects\nat $z \\geq 1$ will be feasible with the next generation of\nground-based multi object near-IR spectrographs.\n\n\nIn an independent study, \\citet{brinchmann07} have also analyzed\nthe possible causes for the high-redshift galaxies' offset in the\nBPT diagram, using theoretical models of nebular emission from\nstar-forming galaxies \\citep{charlot01} and the SDSS DR4 data.\nThey have found a relationship in SDSS galaxies between their\nlocation in the BPT diagram and their excess specific SFRs and\nlarger H$\\alpha$ EWs relative to galaxies of similar mass. We note\nthat they have only examined SDSS star-forming galaxies below the\nKa03 curve, whereas our SDSS Offset samples have been selected to\nbe above this curve, in order to probe exactly the regime where\nthe most offset objects in our DEEP2 $z \\sim 1-1.5$ sample and the\n\\citet{erb06a} $z - 2$ star-forming galaxies reside.\n\\citet{brinchmann07} have inferred that an elevated ionization\nparameter $U$ is at the root of the excess specific SFRs of the\nmore offset objects within their pure star-forming galaxy sample\nand further speculated that higher electron densities and escape\nfractions of hydrogen ionizing photons might be the factors\nresponsible for the systematically higher values of $U$ in the H\nII regions of high-redshift galaxies. Using a different technique\nand sample of galaxies, we have reached a similar conclusion about\nthe higher ionization parameter and larger H$\\alpha$ EWs in our\nSDSS Offset-SF samples. In addition, we have also uncovered that\nthe higher electron density and temperature, hence higher\ninterstellar ambient pressure, is at the root of the higher\nionization parameter. We have further shown that these unusual H\nII region physical conditions are well connected to the higher SFR\nsurface density of host galaxies. The trend of higher electron\ndensity with increasing BPT diagram offset found within our DEEP2\n$z \\sim 1-1.5$ sample, and the observational evidence that\ngalaxies with high SFR surface densities are more prevalent at\nhigh redshifts, lend further support to our conclusions drawn\nbased on the SDSS local emission-line galaxies.\n\n\nThese differences discovered in H II region physical conditions,\nwhich may commonly apply to $z \\sim 1-1.5$ star-forming galaxies,\nmust be taken into account when strong-line abundance indicators\nare used to study the evolution of galaxy metallicity with\nredshift. The resulting systematic bias in inferred oxygen\nabundance can be estimated quantitatively either via detailed\nphotoionization models, given the difference in H II region\nphysical conditions inferred from the relevant density- and\nionization parameter-sensitive line ratios, or through empirical\ncomparisons, as we have illustrated for the SDSS Main and\nOffset-SF samples.\n\nWe have shown in \\S \\ref{sec:sub:te} that, for the SDSS Offset-SF\nobjects, the metallicities based on various strong-line indicators\ncan either be systematically too large by $\\sim$0.16 dex (N2,\nN2O2) or too low by $\\sim$0.12 dex ($R_{23}$). For N2 in\nparticular, the inferred metallicities of the SDSS Offset-SF\nobjects can be systematically too large by $\\sim$0.16 dex, which\nis already comparable to the inherent scatter in the N2\ncalibration \\citep[1 $\\sigma$ dispersion of 0.18 dex;][]{pp04}.\nThis systematic bias may be even larger if the actual difference\nin H II region physical conditions is more extreme. In addition,\nwe note that this systematic uncertainty stemming from the\ndifference in H II region physical conditions has different\neffects from that of the inherent scatter in the calibration, when\nthe strong-line relation calibrated with local H II regions is\napplied to high-redshift star-forming galaxies. As there is\nevidence that the average ionization parameter and electron\ndensity in high-redshift star-forming galaxies are systematically\nhigher than the local typical values, there will be a systematic\n``bias,'' instead of a scatter as a result. For example, the\nN2-based metallicities for the DEEP2 $z\\sim 1.4$ sample may be\nsystematically overestimated by as much as $\\sim$ 0.16 dex.\nFurthermore, as discussed in \\S 4, the fact, that our DEEP2 $z\\sim\n1.4$ sample is more offset from the local excitation sequence than\nthe $z\\sim 1.0$ sample, may result in N2-based metallicities that\nare more significantly overestimated as well. This effect may\ncause the apparent reverse trend of average O\/H with redshift\nwithin the DEEP2 sample, such that the $z\\sim 1.4$ sample is\ndescribed by apparently higher metallicities at fixed stellar mass\nthan the one at $z\\sim 1.0$, despite the general trend of\nincreasing O\/H towards lower redshift. The reverse trend within\nthe DEEP2 sample based on O3N2 is less significant than that based\non N2, which is consistent with the fact that the O3N2-based\nmetallicity bias due to the offset in the BPT diagram is much less\nthan that of N2.\n\nIn addition, we can quantify the potential bias for the sample of\nstar-forming galaxies at $z\\sim2$ presented by \\citet{erb06a}. If\nthe four $z\\sim2$ objects with the full set of H$\\beta$, [O III]\n$\\lambda 5007$, H$\\alpha$, and [N II] $\\lambda 6584$ measurements\nare representative of the larger population of star-forming\ngalaxies in \\citet{erb06a} in terms of diagnostic line ratios,\nsuch that the $z\\sim 2$ star-forming galaxies are even more\nsignificantly offset than the DEEP2 $z\\sim 1.4$ sample on average,\nthen their metallicities based on N2 would also be overestimated\nby $\\sim$0.16 dex. Therefore, the true metallicity offset between\nthe  \\citet{erb06a} $z\\sim2$ sample and local SDSS objects may be\nas much as $\\sim$0.16 dex larger than it appears now in Figure\n(\\ref{fig:massz}a). Accounting for the systematic differences in\nconverting strong emission-line ratios to oxygen abundances is\ntherefore a crucial component of comparing galaxy metallicities at\ndifferent redshifts.\n\n\\section{Summary}\n\nWe have compiled a sample of 20 star-forming galaxies at $1.0 < z\n< 1.5$ drawn from the blue cloud of the color bimodality observed\nin the DEEP2 survey, to study the correlation between stellar mass\nand metallicity, across a dynamical range of 2 orders of magnitude\nin stellar mass, as well as H~II region physical conditions at\nthis redshift range. In order to gain some insights on the causes\nof the offset in the BPT diagram observed in high-redshift\nstar-forming galaxies, we have examined the H II region diagnostic\nline ratios and host galaxy properties of the small fraction of\nSDSS galaxies that have similar diagnostic ratios to those of the\nDEEP2 sample. Our main results are summarized as follows:\n\\begin{enumerate}\n\n\\item[1.] There is a correlation between stellar mass and\ngas-phase oxygen abundance in DEEP2 star-forming galaxies at $z\n\\sim 1.0$ and at $z \\sim 1.4$. We have found that the zero point\nof the $M-Z$ relationship evolves with redshift, in the sense that\ngalaxies at fixed stellar mass become more metal-rich at lower\nredshift, by comparing the $1.0 < z < 1.5$ sample with UV-selected\n$z \\sim 2$ and SDSS local star-forming galaxies. At the low-mass\nend ($M_{\\star} \\sim 8\\times10^9 M_{\\odot}$), the relation at $1.0\n< z < 1.5$ is offset by $\\sim$0.2 (0.35) dex from the local\nmass-metallicity relation according to the N2 (O3N2) indicator.\nThe N2-based offset could be larger by as much as $\\sim$0.16 dex,\nwhen the systematic bias due to difference in H II region physical\nconditions between $1.0 < z < 1.5$ and the local universe is taken\ninto account. At the high-mass end ($M_{\\star} \\sim 5\\times10^{10}\nM_{\\odot}$), the metallicity offset between the DEEP2 $1.0 < z <\n1.5$ sample and the local SDSS sample is at least $\\sim 0.2$ dex,\naccording to the O3N2 indicator.\n\n\n\n\\item[2.] As observed previously for a very small sample of\nhigh-redshift galaxies, on average our new DEEP2 sample at $1.0 <\nz < 1.5$ is offset from the excitation sequence formed by nearby\nH~II regions and SDSS emission-line galaxies. By examining the\nsmall fraction of SDSS galaxies that have similar diagnostic\nratios to those of the DEEP2 sample, we have found two likely\ncauses for the anomalous emission-line ratios. One is the\ncontribution from AGN and\/or shock excitation at the level of\n$\\sim 20$\\%. The other is the difference in H~II region physical\nconditions, characterized by significantly larger ionization\nparameters, as a result of higher electron densities and\ntemperatures, and hence higher interstellar ambient pressure, than\nthe typical values of local star-forming galaxies with similar\nstellar mass. These unusual physical conditions are possibly\nconnected to the host-galaxy properties, most importantly smaller\nsizes and higher star-formation rate surface densities. Our\nconclusion drawn from analyzing the SDSS data has been further\nverified by the fact that the DEEP2 objects more offset from the\nlocal excitation sequence in the BPT diagram also have higher\nelectron densities than those closer to the local sequence. We\ncannot rule out either the contribution from AGN and\/or shock\nexcitation, or the difference in H~II region physical conditions,\nfor the unusual emission-line diagnostic ratios of high-redshift\nstar-forming galaxies.\n\n\\item[3.] We have quantified the effects of different H II region\nphysical conditions on the strong-line metallicity calibrations.\nThe direct electron temperature method was used to estimate the\n``true'' metallicity difference between offset SDSS objects with\nanomalous line ratios and more typical objects of similar stellar\nmass. Strong-line indicators were also used to estimate this\ndifference. A comparison of these results reveals potential biases\nin the strong-line indicators. According to our test, the\nmetallicities based on strong-line indicators can either be\nsystematically too large by $\\sim$0.16 dex (N2, N2O2), or too low\nby $\\sim$0.12 dex ($R_{23}$), for objects with similar H~II region\nphysical conditions to those observed in high-redshift galaxies.\nThe bias of O3N2 is much less significant (too large by $\\sim$0.02\ndex) for offset SDSS objects with anomalous line ratios, although\nit may not always be negligible given other ranges of physical\nconditions in terms of metallicity, ionization parameter, and\nelectron density.\n\\end{enumerate}\n\n\n\nThe difference in H~II region physical conditions, which may\ncommonly apply to $z\\sim 1.0-1.5$ star-forming galaxies, must be\ntaken into account when strong-line abundance indicators are used\nto study the evolution of galaxy metallicity with redshift. There\nare at least two methods to remove the systematic bias from the\neffect of significantly different H~II region physical conditions\non the strong-line abundance calibrations. One is to gather the\nabundance information with direct $T_e$ method for a sample of\nhigh-redshift H~II regions as the calibration sample, which may be\nhard to achieve, as auroral lines are difficult to measure except\nfor very metal-poor objects or in deep, composite spectra. The\nother, which is currently feasible yet relies on photoionization\nmodels, is to quantify the biases in strong-line indicators when\ncertain physical conditions are present and then compensate the\nbiases when inferring abundances from strong-line indicators for\nhigh-redshift galaxies.\n\nIn this study we have presented evidence that high-redshift\nstar-forming galaxies possess distinct H~II region physical\nproperties, as characterized by on average larger ionization\nparameters, higher electron densities, and temperatures, which are\npossibly connected to their relatively smaller sizes and higher\nSFR surface densities. These conditions may be quite common during\nthe epoch at $z \\geq 1$ when at least 50\\% of the local stellar\nmass density was formed \\citep{bundy06,drory05}. Therefore, they\nshould be characterized in more detail for a full understanding of\nthe star formation history of the universe as well as the buildup\nof heavy elements in galaxies. The next generation of ground-based\nnear-IR multi-object spectrographs will play a key role in\nassembling rest-frame optical emission-line measurements for large\nsamples of high-redshift galaxies, enabling the detailed study of\nstar-forming galaxies in the early universe.\n\n\\acknowledgments We are indebted to the DEEP2 team, whose\nsignificant efforts in establishing such a tremendous\nspectroscopic sample at $z\\sim1$ made this project possible. We\nalso thank Kevin Bundy for his assistance with estimating stellar\nmasses, and Bruce Draine and Jenny Greene for helpful discussions.\nA. E. S. acknowledges support from the David and Lucile Packard\nFoundation and the Alfred P. Sloan Foundation. A. L. C. is\nsupported by NASA through Hubble Fellowship grant HF-01182.01-A\nawarded by the Space Telescope Science Institute, which is\noperated by the Association of Universities for Research in\nAstronomy, Inc., for NASA, under contract NAS 5-26555. C. P. M. is\nsupported in part by NSF grant AST 04-07351. Funding for the DEEP2\nsurvey has been provided by NSF grants AST95-09298, AST-0071048,\nAST-0071198, AST-0507428, and AST-0507483 as well as NASA LTSA\ngrant NNG04GC89G. Funding for the Sloan Digital Sky Survey (SDSS)\nhas been provided by the Alfred P. Sloan Foundation, the\nParticipating Institutions, the National Aeronautics and Space\nAdministration, the National Science Foundation, the U.S.\nDepartment of Energy, the Japanese Monbukagakusho, and the Max\nPlanck Society. The SDSS Web site is http:\/\/www.sdss.org\/. We wish\nto extend special thanks to those of Hawaiian ancestry on whose\nsacred mountain we are privileged to be guests.  Without their\ngenerous hospitality, most of the observations presented herein\nwould not have been possible.\n\n\n\n\\bibliographystyle{apj}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn recent years, ubiquitous computer vision applications are pervading our lives. Smartphones, self-driving terrestrial and aerial vehicles, Visual Sensor Networks (VSNs) are capable of acquiring visual data and performing complex analysis tasks. In particular, VSNs are expected to play a major role in the advent of the \\emph{Internet-of-Things} paradigm. Such computer vision tasks usually exploit a concise yet effective representation of the acquired visual content, rather than being based on the pixel-level content. In this context, local features represent an effective solution that is being successfully exploited for a number of tasks such as content-based retrieval, object tracking, image registration, etc. Local feature extraction algorithms usually consist of two distinct components. First, a keypoint detector aims at identifying salient regions (e.g. corners, blobs) within a given image. Second, a descriptor assigns each identified keypoint a descriptor, in the form of a set of values, based on the local characteristics of the image patch surrounding such keypoint. Such information is further processed in order to extract a semantic representation of the acquired content, e.g., by identifying and tracking objects, recognizing faces, monitoring the environment and recognizing events. \n\nAs regards visual feature extraction algorithms, SIFT~\\cite{DBLP:journals:ijcv:Lowe04} is widely considered as the state-of-the-art for a large number of tasks. It consists in a keypoint detector based on the Difference-of-Gaussians (DoG) algorithm, and in a scale- and rotation-invariant real-valued descriptor, based on local intensity gradients. Besides, SURF~\\cite{DBLP:conf:eccv:BayTG06} is partially inspired by SIFT and aims at achieving a similar level of accuracy at a lower computational cost. More recently, several low-complexity algorithms have been proposed, with the objective of alleviating the computational burden required by both traditional keypoint detectors and descriptors. For example, FAST~\\cite{rosten_2005_tracking} and AGAST~\\cite{mair2010_agast} are computationally efficient detectors capable of identifying stable corners. As for descriptors, binary-valued features are emerging as an efficient alternative to traditional real-valued features. BRIEF~\\cite{DBLP:conf\/eccv\/CalonderLSF10}, BRISK~\\cite{DBLP:conf\/iccv\/LeuteneggerCS11}, FREAK~\\cite{DBLP:conf\/cvpr\/AlahiOV12} and BAMBOO~\\cite{baroffio:2014:ICIP:BAMBOO} are instances of such category. For each identified keypoint, they compute a descriptor vector in the form of a sequence of binary values, each of which is obtained by comparing the (smoothed) intensities of a pair of pixels sampled around the keypoint. \nIn some cases, ad-hoc software-based implementations are available for specific hardware architectures~\\cite{baroffio:2014:ICIP:BRISKOLA}. \n\nLocal feature detection in video sequences has been addressed in the past literature, with the goal of identifying keypoints that are stable across time. For example, Shi and Tomasi~\\cite{Shi94goodfeatures} propose a widely adopted detector suitable for tracking applications. Zhang et. al propose a complex video-retrieval system based on color, shape and texture features extracted from the key-frames of a video~\\cite{Zhang1997643}. More recently, Zha et al. propose a method to extract spatio-temporal features from video content~\\cite{2009:Zha:ACCV}. Besides being a key to tasks such as object tracking, event identification and video calibration, temporally stable features improve the efficiency of coding architectures tailored to features extracted from video content~\\cite{BaroffioRCTT:TIP, BaroffioRCTT:ICIP2013, Makar:2014:TIP}.\nMore recently, Girod et al.~\\cite{journals\/ijsc\/MakarTCCG13} propose a feature detection and coding algorithm inspired by traditional motion estimation methods. Such algorithm selects a set of features corresponding to canonical image patches whose content is stable across frames, leading to a significant reduction of the transmission bitrate thanks to ad-hoc coding primitives. Although such algorithm represents a good solution for applications that require the efficient transmission of local features for further processing, it might not be the best in terms of computational complexity. Considering low-power devices, computationally intensive operations might significantly reduce the detection frame rate, possibly impairing performance of time-critical tasks or introducing undue delay. \nIn this paper, we introduce a fast detection algorithm based on BRISK~\\cite{DBLP:conf\/iccv\/LeuteneggerCS11} and tailored to the context of video sequences, aimed at reducing the computational complexity and thus enabling high frame rates, without significantly affecting performance in terms of accuracy.\n\nThe rest of this paper is organized as follows. Section~\\ref{sec:BRISK} introduces the main concepts behind BRISK. Section~\\ref{sec:algorithm} illustrates the proposed fast detection architecture. Section~\\ref{sec:experiments} defines the experimental setup and presents results. Finally, conclusions are drawn in Section~\\ref{sec:conclusions}.\n\n\n\n\n\\section{Binary Robust Invariant Scalable Keypoints (BRISK)}\\label{sec:BRISK}\n\nLeutenegger et al.~\\cite{DBLP:conf\/iccv\/LeuteneggerCS11} propose the Binary Robust Invariant Scalable Keypoints (BRISK) algorithm as a computationally efficient alternative to traditional local feature detectors and descriptors. The algorithm consists in two main steps: i) a keypoint detector, that identifies salient points in a scale-space and ii) a keypoint descriptor, that assigns each keypoint a rotation- and scale- invariant binary descriptor. Each element of such descriptor is obtained by comparing the intensities of a given pair of pixels sampled within the neighborhood of the keypoint at hand. \n\nThe BRISK detector is a scale-invariant version of the lightweight FAST~\\cite{rosten_2005_tracking} corner detector, based on the Accelerated Segment Test (AST). Such a test classifies a candidate point $p$ (with intensity $I_p$) as a keypoint if $n$ contiguous pixels in the Bresenham circle of radius 3 around $p$ are all brighter than $I_p + t$, or all darker than $I_p - t$, with $t$ a predefined threshold. Thus, the highest the threshold, the lowest the number of keypoints which are detected and vice-versa.\n\nScale-invariance is achieved in BRISK by building a scale-space pyramid consisting of a pre-determined number of octaves and intra-octaves, obtained by progressively downsampling the original image.\nThe FAST detector is applied separately to each layer of the scale-space pyramid, in order to identify potential regions of interest having different sizes. Then, non-maxima suppression is applied in a 3x3 scale-space neighborhood, retaining only features corresponding to local maxima. Finally, a three-step interpolation process is applied in order to refine the correct position of the keypoint with sub-pixel and sub-scale precision.\n\n\n\n\n\n\n\\section{Fast video feature extraction}\\label{sec:algorithm}\nLet $\\mathcal{I}_n$ denote the $n$-th frame of a video sequence of size $N_x \\times N_y$, \nwhich is processed to extract a set of local features $\\mathcal{D}_n$. First, a keypoint detector is applied to identify a set of interest points. Then, a descriptor is applied on the (rotated) patches surrounding each keypoint. Hence, each element of $d_{n,i} \\in \\mathcal{D}_n$ is a visual feature, which consists of two components: i) a 4-dimensional vector $\\mathbf{p}_{n,i} = [x, y, \\sigma, \\theta]^T$, indicating the position $(x,y)$, the scale $\\sigma$ of the detected keypoint, and the orientation angle $\\theta$ of the image patch; ii) a $P$-dimensional \\emph{binary} vector $\\mathbf{\\descr}_{n,i} \\in \\{0,1\\}^P$, which represents the descriptor associated to the keypoint $\\mathbf{p}_{n,i}$.\n\nTraditionally, local feature extraction algorithms have been designed to efficiently extract and describe salient points within a single frame. Considering video sequences, a straightforward approach consists in applying a feature extraction algorithm separately to each frame of the video sequence at hand. \nHowever, such a method is inefficient from a computational point of view, as the temporal redundancy between contiguous frame is not taken into consideration. The main idea behind our approach is to apply a keypoint detection algorithm only on some regions of each frame. To this end, for each frame $\\mathcal{I}_n$, a binary \\emph{Detection Mask} $\\mathcal{M}_n \\in \\{0, 1\\}^{N_x \\times N_y}$ having the same size of the input image is computed, exploiting the information extracted from previous frames. Such mask defines the regions of the frame where a keypoint detector has to be applied. That is, considering an image pixel $\\mathcal{I}_n(x,y)$, a keypoint detector is applied to such a pixel if the corresponding mask element $\\mathcal{M}_n(x,y)$ is equal to 1. Furthermore, we assume that if a region of the $n$-th frame is not subject to keypoint detection , the keypoints that are present in such an area in the previous frame, i.e. $\\mathcal{I}_{n-1}$, are still valid. Hence, such keypoints are propagated to the current set of features. That is,\n\n\\begin{equation}\n\t\\mathcal{D}_n = \\{d_{n,i} : \\mathcal{M}_n(\\mathbf{p}_{n,i}) = 1 \\;\\cup\\; d_{n-1, j} : \\mathcal{M}_n(\\mathbf{p}_{n-1, j}) = 0\\}\n\\end{equation} \n\nNote that the algorithm used to compute the \\emph{Detection Mask} needs to be computationally efficient, so that the savings achievable by skipping detection in some parts of the frame are not offset by this extra cost. In the following, two efficient algorithms for obtaining a \\emph{Detection Mask} are proposed: \\emph{Intensity Difference Detection Mask} and \\emph{Keypoint Binning Detection Mask}.\n\n\\vspace{-3 mm}\n\n\\subsection{Intensity Difference Detection Mask}\\label{sec:IDDM}\n\nThe key tenet is to apply the detector only to \nthose regions that change significantly across the frames of the video. In order to identify such regions and build the \\emph{Detection Mask}, we exploit the scale-space pyramid built by the BRISK detector, thus incurring in no extra cost. \nConsidering frame $\\mathcal{I}_n$ and $\\mathcal{O}$ detection octaves, pyramid layers $\\mathcal{L}_{n,o}, o = 1, \\dots, \\mathcal{O}$ are obtained by progressively smoothing and half-sampling the original image, as illustrated in Section~\\ref{sec:BRISK}. Then, considering two contiguous frames $\\mathcal{I}_{n-1}$ and $\\mathcal{I}_{n}$ and octave $o$, a subsampled version of the \\emph{Detection Mask} is obtained as follows:\n\n\n\\begin{equation}\n\t\\mathcal{M}'_{n,o}(k, l) =\n\t\t\\begin{cases} 1 &\\mbox{if } \\lvert \\mathcal{L}_{n, o}(k,l) - \\mathcal{L}_{n-1, o}(k,l) \\rvert \\le \\mathcal{T}_{I} \\\\ \n\t\t\t0 & \\mbox{if } \\lvert \\mathcal{L}_{n, o}(k,l) - \\mathcal{L}_{n-1, o}(k,l) \\rvert > \\mathcal{T}_{I},\n\t\t\\end{cases}\n\\end{equation}\n\nwhere $\\mathcal{T}_{I}$ is an arbitrarily chosen threshold and $(k,l)$ the coordinates of the pixels in\nthe intermediate representation $\\mathcal{M}'_{n,o}$. \nFinally, the intermediate representation $\\mathcal{M}'_{n,o}$ resulting from the previous operation needs to be upsampled in order to obtain the final mask  $\\mathcal{M}_n \\in \\{0, 1\\}^{N_x \\times N_y}$. Masks can then be applied to detection in different fashions: i) exploiting the mask obtained resorting to each scale-space layer $o = 1, \\dots, \\mathcal{O}$ in order to detect keypoint at the corresponding layer $o$; ii) use a single detection mask for all the scale-space layers. \n\n\\vspace{-3 mm}\n\n\\subsection{Keypoint Binning Detection Mask}\\label{sec:KBDM}\n\nConsidering two contiguous frames of a video sequence, the amount of features identified in a given area are often correlated~\\cite{Eriksson:2014:ICASSP}. To exploit such information, \nthe detector is applied to a region of the input image only if the number of features extracted in the co-located region in the previous frame is greater than a threshold. Specifically, in order to obtain a \\emph{Detection Mask} for the $n-$th frame, a spatial binning process is applied to the features extracted from frame $\\mathcal{I}_{n-1}$. To this end, we define a grid consisting of $\\mathcal{N}_{r} \\times \\mathcal{N}_{c}$ spatial bins $\\mathcal{B}_{i, j}, i = 0, \\dots, \\mathcal{N}_{r}, j = 0, \\dots, \\mathcal{N}_{c}$. Thus, each bin refers to a rectangular area of $S_x \\times S_y$ pixels, where $S_x = \\nicefrac[]{N_x}{\\mathcal{N}_{c}}$ and $S_y = \\nicefrac[]{N_y}{\\mathcal{N}_{r}}$. Then, a two-dimensional spatial histogram of keypoints is created by assigning each feature to the corresponding bin as follows:\n\n\\begin{equation}\n\t\\mathcal{M}''_{n}(k, l) = \\lvert d_{n-1, i} \\in \\mathcal{D}_{n-1} \\rvert : \\lfloor \\nicefrac{x_{n-1, i}}{S_x}\\rfloor = k , \\lfloor \\nicefrac{y_{n-1, i}}{S_y}\\rfloor = l,\n\n\\end{equation}\n\nwhere $(x_{n-1, i}, y_{n-1, i})$ represents the location of feature $d_{n-1, i}$ and $|\\cdot|$ the number of elements in a set.\nThen, a binary subsampled version of the \\emph{Detection Mask} is obtained by thresholding such histogram, employing a tunable threshold $\\mathcal{T}_{H}$:\n\n\\begin{equation}\n\t\\mathcal{M}'_{n}(k, l) =\n\t\t\\begin{cases} 1 &\\mbox{if } \\mathcal{M}''_{n}(k, l) \\ge \\mathcal{T}_{H} \\\\ \n\t\t\t0 & \\mbox{if } \\mathcal{M}''_{n}(k, l) < \\mathcal{T}_{H},\n\t\t\\end{cases}\n\\end{equation}\n\nFinally, the \\emph{Detection Mask} $\\mathcal{M}_n$ having size $N_x \\times N_y$ pixels is obtained by upsampling the intermediate representation $\\mathcal{M}'_{n}$. Such a detection mask is applied to all scale-space octaves.\n\n\n\n\n\n\n\n\n\n\\section{Experiments}\\label{sec:experiments}\n\\textbf{Dataset: }\nWe evaluated the proposed algorithms with respect to three different test scenarios. First, we exploited the \\emph{Stanford MAR dataset}~\\cite{Makar:2014:TIP}, containing the four VGA size, 200 frames long video sequences \\emph{Alicia Keys}, \\emph{Fogelberg}, \\emph{Anne Murray} and \\emph{Reba}. Each sequence contains a CD cover recorded with a hand-held mobile phone, under different imaging conditions such as illumination, zoom, perspective, rotation, glare, etc. Furthermore, for each sequence, the dataset contains the ground truth information, in the form of a still image of the corresponding CD cover, having a size of $500 \\times 500$ pixels. \n\nAs a second test, we evaluated the approaches resorting to the \\emph{Rome Landmark Dataset}. Such dataset includes a set of 10 query video sequences, each capturing a different landmark in the city of Rome with a camera embedded in a mobile device~\\cite{romelandmark}. The frame rate of such sequences is equal to 24fps, whereas the resolution ranges from 480x360 pixels (4:3) to 640x360 pixels (16:9). The first 50 frames of each video were used as query. On average, each query video corresponds to 9 relevant images representing the same physical object under different conditions and with heterogeneous qualities and resolutions. Then, distractor images randomly sampled from the \\emph{MIRFLICKR-1M} dataset~\\cite{huiskes08}, so that the final database contains 10k images.\n\nFinally, we tested our method on the \\emph{Stanford MAR multiple object} video set~\\cite{Makar:2014:TIP}. Such a set is made up of 4 video sequences, each consisting of 200 frames at 640x480 resolution. Each video is recorded with a handheld camera and portrays three different objects, one at a time.\n\n\\textbf{Methods: \nWe tested the two detection methods presented in Section~\\ref{sec:algorithm}, that is, \\emph{Intensity Difference Detecion Mask} and \\emph{Keypoint Binning Detection Mask}. In both cases, we employed the original BRISK implementation from the authors\\footnote{http:\/\/www.asl.ethz.ch\/people\/lestefan\/personal\/BRISK}, setting the number of octaves to 4 and the detection threshold to 55 and 70 for the \\emph{Stanford MAR dataset} and the \\emph{Rome landmark dataset}, respectively.\nAs regards \\emph{Intensity Difference Detection Mask}, we built the mask testing several different configurations. \nWe tested our algorithm with the 4 layers corresponding to each scale-space octaves.  \nSince the performance was similar when using different layers, we resorted to the top-layer, i.e., the one with the lowest spatial resolution and processing cost. \nBoth \\emph{Intensity Difference Detection Mask} and \\emph{Keypoint Binning Detection Mask} require a threshold to be set in order to obtain the final detection mask. We tested several different configurations, each representing a tradeoff between computational efficiency and task accuracy. \n\nWe compared our algorithms with a \\emph{Temporally Coherent Detector} based on non-canonical patch matching~\\cite{Makar:2014:TIP}, which also exploits temporal redundancy in the detected keypoints.\nSuch algorithm aims at propagating stable keypoints across frames, exploiting a pixel-level representation of local features. In details, a traditional keypoint detector is applied to the first frame of a Group Of Pictures of size $\\Delta$. Given an identified keypoint, a non-canonical square image patch is extracted from the neighborhood of such a point. Then, considering the following frame, we searched for a matching patch in a window surrounding such a keypoint. Two patches are assumed to be a match if the Sum of Absolute Differences (SAD) between their pixels is below a given threshold $\\mathcal{T}_{BM}$. Finally, keypoints for which a match is found are propagated to the next frame, and their position is determined by the aforementioned block matching procedure. \nIn our tests, according to the prescriptions of~\\cite{Makar:2014:TIP}, we employed patches of $16 \\times 16$ pixels and we set $\\Delta = 10$ and $\\mathcal{T}_{BM} = 1800$. Furthermore, to make the procedure faster, we implemented a coarse-to-fine matching algorithm, where the first step consists in a spiral search algorithm with a precision of $2$ in a search window of $24 \\times 24$ pixels, whereas the second step in a spiral search algorithm with quarter-pixel precision in a search window of $1.75 \\times 1.75$ pixels. Finally, to further speed-up the process, we set an early termination SAD threshold $\\mathcal{T}_{ET} = 1000$. \nThis detector was originally proposed with the goal of maximizing coding efficiency, when patches around the detected keypoints need to be compressed and transmitted. To this end, this method can also adopt more sophisticated matching strategies, e.g., based on affine warping. However, in this paper we consider an implementation based on block matching to minimize the computational complexity. \n\n\n\\textbf{Evaluation methods and measures: }In the case of the \\emph{Stanford MAR dataset}, for a given video sequence, we extracted a set of features for each frame.\nThen, the set of features extracted from a frame is matched with the ones extracted from the ground truth frame. A radius match algorithm is used, where the matching threshold is set to $\\mathcal{T}_{M} = 0.18*512 \\simeq 102$. Finally, geometric coherence of matches is enforced resorting to the RANSAC algorithm. Finally, the number of \\emph{Matches-Post-Ransac} (MPR) is employed as the accuracy measure. \n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[trim=0cm 0cm 0cm 0cm, clip=true,width=0.43\\textwidth]{figures\/AliciaKeys_55_40_MPSvsTime_BLOCK_MATCHING.pdf}\n\t\\caption{Accuracy, measured as the number of matches post-RANSAC (MPR), and computational time for each frame of the \\emph{Alicia Keys} test sequence.}\n\t\\label{fig:MPR_over_time_Alicia}\n\t\\vspace{-3 mm}\n\\end{figure}\n\nIn the case of the \\emph{Rome Landmark dataset}, the accuracy of the task was evaluated according to the \\emph{Mean Average Precision} (MAP). Given an input query sequence $q$, for each frame $\\mathcal{I}_{q,n}$ it is possible to define the \\emph{Average Precision} as \n\\begin{equation}\n\tAP_{q,n} = \\frac{\\sum_{k=1}^Z P_{q,n}(k)r_{q,n}(k)}{R_{q,n}},\n\\end{equation}\nwhere $P_{q,n}(k)$ is the precision (i.e., the fraction of relevant documents retrieved) considering the top-$k$ results in the ranked list of database images; $r_{q,n}(k)$ is an indicator function, which is equal to 1 if the item at rank $k$ is relevant for the query, and zero otherwise; $R_{q,n}$ is the total number of relevant document for frame $\\mathcal{I}_{q,n}$ of the query sequence $q$ and $Z$ is the total number of documents in the list.\nThe overall accuracy for the query sequence $q$ is evaluated according to\n\n\\begin{equation}\n\tAP_q = \\frac{\\sum_{n = 1}^N AP_{q,n}}{N}, \n\\end{equation}\n\nwhere $N$ is the total number of frames of the query video $q$. \n\nFinally, the \\emph{Mean Average Precision} is obtained as\n\\begin{equation}\\label{eq:MAP}\nMAP = \\frac{\\sum_{q = 1}^Q AP_q}{Q}, \n\\end{equation}\nthat is, the mean of the $MAP_q$ measure over all the query sequences.\n\nIn the case of the \\emph{Stanford MAR multiple object} video set, the accuracy is measured according to a combined detection and tracking precision metric. In particular, for each frame, the goal is to correctly detect the portrayed database object and to identify its position within the frame. Each frame of the video sequences is matched against all the database object. Radius match and geometric verification steps are performed as in the case of~\\emph{Stanford MAR dataset} scenario. The matching object is the one with the highest number of matches-post-RANSAC. The bounding box for the identified object is obtained by projecting the database object corners according to the homography computed with the RANSAC algorithm at the previous step. Each frame is deemed as correct if the correct object is detected, and if the estimated position is consistent with the ground-truth information. As to the latter, the estimated object position is deemed correct if the displacement between the estimated centroid and the ground truth one is lower than a threshold. We set the value of such a threshold to 10 pixels. \n\nWe evaluated the complexity of the feature extraction methods by means of the required CPU time. We performed our tests on a laptop equipped with a 2.5GHz Intel Core i5 processor and 10 GB of RAM. \n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[trim=0.5cm 0cm 0cm 0cm, clip=true,width=0.40\\textwidth]{figures\/StanfordMAR_MPSvsTime2_new.pdf}\n\t\\caption{Energy-Accuracy curves for \\emph{Stanford MAR} dataset.}\n\t\\label{fig:energyaccuracy} \n\t\\vspace{-3 mm}\n\\end{figure}\n\n\\textbf{Results: } As an illustrative example, Figure~\\ref{fig:MPR_over_time_Alicia} shows the results obtained for the \\emph{Alicia Keys} sequence. The charts also report the results obtained when detection is performed independently on a frame-by-frame basis (full detection) to serve as a reference. \nWe observe that the method using the \\emph{Intensity Difference Detection Mask} (threshold 20) achieves an accuracy level similar to that of full detection ($MPR = 55$ vs. $56$), at a reduced computational time (20.5 ms vs. 24.5 ms). As for \\emph{Temporally Coherent Detector}, it leads to a significant loss in terms of accuracy ($MPR = 41$), while being quite computationally intensive (72 ms on average). While accuracy could be further improved by resorting to matching based on affine warping, this would further increase its complexity. This confirms the fact that this detector was originally designed with the goal of maximizing coding efficiency rather than computational cost. Since this is confirmed also on other test sequences, we do not report additional results for this detector. \n\nIt is interesting to observe the energy-accuracy trade-off that can be achieved by varying the threshold used by the algorithms based on detection masks. To this end,  Figure~\\ref{fig:energyaccuracy} compares the performance of \\emph{Intensity Difference Detection Mask} and \\emph{Keypoint Binning Detection Mask} with that of full detection, averaging the results on the \\emph{Stanford MAR dataset}. The two methods based on a detection mask performs on a par, reducing the required computational time by 30\\% while losing as few as 4 matches. \n\nFurthermore, we tested our approach based on a \\emph{Detection Mask} on the \\emph{Rome Landmark Dataset}. \nFigure~\\ref{fig:RomeLandmarkMAP} compares the results of \\emph{Intensity Difference Detection Mask} with that of full detection, showing that computational time can be reduced by about 35\\% without affecting task accuracy. Furthermore, the feature extraction process can be speeded un by 3 times at the cost of 0.03\\% lower \\emph{Mean Average Precision}.\n\nFinally, the results of our fast detection algorithms on the \\emph{Stanford MAR multiple object} video set are reported in Figure~\\ref{fig:StanfordMultiple}. The computational time can be be reduced up to 40\\% without significantly impairing object detection and tracking performance. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[trim=0cm 0cm 0cm 0cm, clip=true,width=0.40\\textwidth]{figures\/RomeLandmarkMAPvsTime.pdf}\n\t\\caption{Energy-Accuracy curve for the \\emph{Rome Landmark} dataset, when using the \\emph{Intensity Difference Detection Mask} in order to reduce the detection area and with different values for the thresholding parameter. The computational time for each frame can be reduced from 28ms to 18ms, without significantly affecting the accuracy of the task.}\n\t\\label{fig:RomeLandmarkMAP} \n\t\\vspace{-1 mm}\n\\end{figure}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[trim=0cm 0cm 0cm 0cm, clip=true,width=0.40\\textwidth]{figures\/StanfordMultiple_1_thr10.pdf}\n\t\\caption{Energy-Accuracy curves for the \\emph{Stanford MAR} multiple object sequences.}\n\t\\label{fig:StanfordMultiple} \n\t\\vspace{-4 mm}\n\\end{figure}\n\n\n\\section{Conclusions}\\label{sec:conclusions}\n\nIn this paper we presented a method for fast keypoint detection in video sequences based on \\emph{Detection Masks}. Results show that the proposed approach allows for a reduction in terms of computational complexity of up to 35\\% without significantly impair task performance.\nIn our future investigation we plan to further improve the \\emph{Detection Mask} building process, by introducing more sophisticated yet computationally efficient solutions. \n\n\n\n\\balance\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nMany real-world networks can be formulated as graphs for modeling different relationships among nodes, such as social networks, chemical molecule structures and citation networks. Recently, there have been various attempts to extend Convolutional Neural Network (CNNs) and pooling methods to graph-structured data. These methods are named as Graph Neural Networks (GNNs) and have been successfully applied to different graph related tasks containing graph classification, node classification and link prediction \\cite{zhang2018deep,zhou2018graph}.\nGraph classification aims to extract accurate information from graph-structured data for classification. However, most existing GNNs based graph classification methods overlook that it's complicated and time consuming to collect or label the graph data. Learning with few labeled graph data is still a challenge for the practical applications of graph classification. \n\\begin{figure*}[!t]\\small\n\n\\centering\n\\subfigure[\\scriptsize{Graph spectral measures method \\cite{Chauhan2020FEW-SHOT}.}]{\n\\label{fig:gmsdemo}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{pic\/gsm.PNG}\n}\n\\subfigure[\\scriptsize{Our method.}]{\n\\label{fig:ourdemo}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{pic\/assumption1.PNG}\n}\n\\caption{Comparison of different methods. (a) The method from the global structure of dataset. The most representative graph of each class is viewed as class-prototype graph. The super-classes are clustered from training classes. (b) Our method from the view of local structure. $\\Theta$ is meta-learner's parameters and $\\theta_{1},\\theta_{2}$ are task specific parameters derived from meta-learner's fast adaptation. The components in dotted boxes have similar triangle structure. We assume the similarity can be discovered by a well initialized meta-learner within a few adaptation steps.}\n\\label{fig:first_image}\n\\end{figure*} \n\nFew-shot learning, aiming to label the query data when given only a few labeled support data, is a natural way to alleviate the problem. There are many papers discussing few-shot learning with meta-learning \\cite{FinnAL17},  data augmentation \\cite{NIPS2018_7504} or regularization \\cite{Yin2020Meta-Learning}, but most of them don't consider the graph data. \nFurthermore, there have been several methods for few-shot node classification \\cite{SatorrasICLR2018graph,Kim_2019_CVPR,meta_gnn} and few-shot link prediction \\cite{NIPS2017_7266,recommendation,MeLU,adCold-Start}, but they only focus on node-level embedding. Recently, Chauhan et al. \\cite{Chauhan2020FEW-SHOT} proposed few-shot graph classification based on graph spectral measures and got satisfactory performance. \nFrom the global structure of dataset, they \\textit{bridge} the test classes and training classes by assuming that the test classes belong to the same set of super-classes clustered from training classes. However, the above methods based on graph spectral measures might have some limitations for the following reasons: (1) the label spaces of training classes and test classes usually do not overlap in few-shot settings; (2) the bridging methods above may diminish the model to capture the local structure of test data. \n\n\nFrom the perspective of the graph's local structure, we observe that the graphs of training classes and test classes have similar sub-structures. For example, different social networks usually have similar groups; different protein molecules usually have similar spike proteins. We assume these similarities can be discovered by a well initialized meta-learner within a few adaptation steps. \nTherefore, we consider fast adaptation by a meta-learner from learned graph classification tasks to new tasks. Figure \\ref{fig:first_image} illustrates the existing method and our assumption. \n\nCurrently, GNNs have reliable ability to capture local structures over graphs by convolutional operations and pooling operations, but lack of fast adaptation mechanism when dealing with never seen graph classes. Inspired by Model Agnostic Meta-Learning (MAML, \\cite{FinnAL17}), which has attracted great attention because of its fast adaptation mechanism, we leverage GNNs as graph embedding backbone and meta-learning as a training paradigm to rapidly capture task-specific knowledge in graph classification tasks and transfer them to new tasks.\n\nHowever, directly applying MAML for fast adaptation is suboptimal due to the following reasons: (1) MAML requires painstaking hyperparameter searches to stabilize training and achieve high generalization \\cite{antoniou2018how}; (2) unlike images, graphs have arbitrary node size and sub-structure, which brings uncertainty for adaptation. There have some variants of MAML trying to overcome these problems by incorporating an online hyperparameter adaptation \\cite{behl2019alpha}, reducing optimization difficulty \\cite{nichol2018firstorder} or increasing context parameters for adaptation \\cite{zintgraf2019fast}, but they don't consider the structure of graph data. In this paper, we design a novel component named as adaptive step controller to learn optimal adaptation step for meta-learner to improve its learning robustness and generalization. The controller evaluates the meta-learner and decides when to stop adaptation by two kinds of inputs: (1) graphs' embedding quality, which is viewed as a meta-feature and indicated with Average Node Information (ANI, the average amount of node information in a batch of graphs); (2) meta-learner's training state, which is indicated with training loss of classification. \n\nWe formulate our framework as \\textbf{A}daptive \\textbf{S}tep MAML (AS-MAML). To the best of our knowledge, we are the first to consider the few-shot graph classification problem from the view of graph's local structure and propose a fast adaptation mechanism on graphs via meta-learning. Our contributions are summarized as follows:\n\\begin{itemize}\n\\item We propose a novel GNNs based graph meta-learner, which captures the features efficiently of sub-structures on unseen graphs by fast adaptation mechanism.\n\\item We design a novel controller for meta-learner. Driven by Reinforcement Learning (RL, \\cite{rl}), the controller provide optimal adaptation step for the meta-learner via graph's embedding quality and training loss. Our ablation experiments show its effectiveness to improve learning robustness and generalization.\n\\item We perform quantitative analysis and provide a generalization guarantee of key algorithms via a graph-dependent upper bound.  \n\n\\item We evaluate our framework's performance against different baselines on four graph datasets and achieve state-of-the-art performance in almost all the tasks. We also evaluate the transferability of popular graph embedding modules on our few-shot graph classification tasks.\n\\end{itemize}\n\\section{Related Works}\n\\subsection{Graph Classification}\n\nIn graph classification tasks, each full graph is assigned a class label. There exist several branches for graph classification. The first is graph kernel methods which design kernels for the sub-structures exploration and exploitation of graph data. The typical kernels include Shortest-path Kernel \\cite{borgwardt2005shortest}, Graphlet Kernel \\cite{shervashidze2009efficient} and Weisfeiler-Lehman Kernel \\cite{shervashidze2011weisfeiler}.\n\nAs the main branch in recent years, GNNs have been successfully applied to graph classification. GNNs focus on node representations, which are iteratively computed by message passing from the features of their neighbor nodes using a differentiable aggregation operation.  GCN \\cite{kipf2016semi} proposed Graph Convolutional Neural Network (termed as  GCN) and got satisfying results based on directly feature aggregation from neighborhood nodes. GAT \\cite{GAT} imported attention mechanism for graph convolutional operations. GraphSAGE \\cite{SAGE} proposed an inductive framework which leverages node features to generate node embeddings efficiently for unseen nodes. In our framework, we use these classical methods to update nodes of graphs, while other methods like Graph Isomorphism Network (GIN)   \\cite{xu2018how} are also applicable. \n\nIn the meantime, inspired by pooling in CNNs, a bunch of researchers concentrates on efficient pooling methods for accurate graph summary and computation efficiency.  Beyond pooling layers in CNNs, graph pooling layers can enable GNNs to reason and get global representation from adjacent nodes. More and more evidence shows that graph pooling promotes the graph classification performance  \\cite{lee2019self,diehl2019edge,unet}. SAGPool \\cite{lee2019self} implemented self-attention pooling on graphs considering both node features and graph topology. EdgePool \\cite{diehl2019edge} implemented a localized and sparse pooling transform backed by the notion of edge contraction. Graph U-nets \\cite{unet} implemented novel graph pooling and unpooling operations. Based on these operations, they developed a new model containing graph encoder and graph decoder and got satisfactory performance on graph classification tasks.\n\n\\subsection{Few-Shot Learning and Meta-Learning}\nFew-shot classification aims to learn a model under the circumstances of low sample resources and is usually powered by meta-learning in recent years. Meta-learning was also known as learning to learn, with a meta-learner observing various task learning processes and summarizing meta-knowledge to accelerate the learning efficiency of new tasks.  Baxter et al.  \\cite{BaxterBiasLearning} proposed a model to learn inductive bias from the perspective of bias learning, and they analytically showed that the number of examples required of each task decreases as the number of task rises.\n\nRecent meta-learning related works can be classified into three categories: optimization (or gradients) based methods, metric learning based methods and memory based methods. Optimization based methods aim to train a model to learn optimization \\cite{RaviICLR2017,LiICML2018}, learn a good initialization \\cite{FinnAL17} for rapid adaptation, or train parameter generator for task-specific classifier \\cite{RusuICLR2019}. Metric learning based methods aim to learn a feature space shared with new tasks \\cite{VinyalsBLKW16,SnellSZ17}. Moreover, memory based methods learn new tasks by reminiscence mechanism in virtue of physical memory \\cite{SantoroBBWL16}.\n\nFurthermore, almost all the previous few-shot learning methods are devised for image data, where images are prone to be represented in Euclidean space. Because we all have the idea that CNNs based models can perform efficient transfer in Euclidean space by feature reuse \\cite{Raghu2020Rapid}, in virtue of that different images usually share common edge features and corner features.  Graph data such as social networks, which are appropriate to be formed into non-Euclidean space instead of Euclidean space. Few-shot learning in non-Euclidean space is addressed in our work.\n\\subsection{GNNs and its Generalization on Graph data}\nWe have seen several works of few-shot node classification promoting performance via GNNs \\cite{SatorrasICLR2018graph, Kim_2019_CVPR, liu2019GPN, yang2020dpgn,Yao2020Automated,NIPS2019_8389,yao2019graph,Gidaris_2019_CVPR,Liu2019PrototypePN}, but they just leverage the message passing mechanism of GNNs to enhance the performance on node classification, without involving the generalization of GNNs themselves and compatibility with graph classification task.  For graph classification, Knyazev et al.  \\cite{NIPS2019_8673} focus on the ability of attention GNNs to generalize to larger, more complex or noisy graphs. Lee et al. \\cite{LeeTransfer} imported domain transfer method by transferring the intrinsic geometric information learned in the source domain to the target. Hu et al. \\cite{Hu*2020Strategies} systematically studied the effectiveness of pre-training strategies on multiple graph datasets. Based on graph spectral measures, Chauhan et al. \\cite{Chauhan2020FEW-SHOT} proposed few-shot graph classification using the notion of super-graph by two steps: (1) they define the \\(p\\)-th Wasserstein distance to measure the spectral distance among graphs and select the most representative graph as prototype graph for each class; (2) by clustering the prototype graphs based on spectral distance, they clustered the prototype graph again into a super-graph consisting of super-classes. Therefore, they assume that the test classes belong to the same set of super-classes clustered from the training classes. We loosen the assumption and emphasize fast adaptation to boost few-shot graph classification. \n\n\\section{Problem Setup}\n\\begin{figure*}[!ht]\n  \\centering\n  \\includegraphics[width=0.70\\textwidth]{pic\/frameworkYZ.pdf}\n  \\caption{Diagram of the AS-MAML framework's learning process in a single episode on the 2-way-1-shot graph classification task. The yellow arrows show meta-learner's $T$ step adaptations on support graphs. The blue dash arrows show $T$ step evaluations (Accuracies) on the query graphs. The orange dash arrows show the backpropagation (BP) according to $T$-th loss on query graphs. The step controller receives ANIs and classification losses on support graphs of each step. After that, the controller outputs the adaptation step $T$. Finally, the controller receives accuracies on query graphs as rewards and updates its own parameters. }\n  \\label{fig:fm}\n\\end{figure*}\n\nWe form the few-shot problem as N-way-K-shot graph classification. Firstly, given graph data $\\mathcal{G}=\\left\\{({G}_{1},\\mathbf{y}_{1}), ({G}_{2},\\mathbf{y}_{2}), \\cdots, ({G}_{n},\\mathbf{y}_{n})\\right\\}$, where ${G}_{i}= \\left(\\mathcal{V}_{i}, \\mathcal{E}_{i}, \\mathbf{X}_{i}\\right)$. We use ${n}_{i}$ to denote the number of node set $ \\mathcal{V}_{i}$. So each graph ${G}_{i}$ has an adjacent matrix  $\\mathbf{A}_{i} \\in \\mathbb{R}^{n_{i} \\times n_{i}}$ and a node attribute matrix $\\mathbf{X}_{i} \\in \\mathbb{R}^{n_{i} \\times d}$, where $d$ is the dimension of node attribute. Secondly, according to label $\\mathbf{y}$, we split $\\mathcal{G}$ into $\\{(\\mathcal{G}^{train},\\mathbf{y}^{train})\\}$ and $\\{(\\mathcal{G}^{test},\\mathbf{y}^{test})\\}$ as training set and test set respectively. Notice that $\\mathbf{y}^{train}$ and $\\mathbf{y}^{test}$ must have no common classes. We use episodic training method, which means at the training stage we sample a task $\\mathcal{T}$ each time, and each task contains support data $D^{train}_{sup}={ \\{ (G_{i}^{train},\\mathbf{y}_{i}^{train}) \\} }_{i=1}^{s}$ and query data $D^{train}_{que}={ \\{ (G_{i}^{train},\\mathbf{y}_{i}^{train}) \\} }_{i=1}^{q}$,\nwhere $s$ and $q$ are the number of support data and query data respectively. Given labeled support data, our goal is to predict the labels of query data. Please note that in a single task, support data and query data share the same class space. If $s={N \\times K}$, which means that support data contain N classes and K labeled samples per class, we name the problem as N-way-K-shot graph classification. At test stage when performing classification tasks on unseen classes, we firstly fine tune the meta-learner on the support data of test classes $D^{test}_{sup}={ \\{ (G_{i}^{test},\\mathbf{y}_{i}^{test}) \\} }_{i=1}^{s}$ , then we report classification performance on  $D^{test}_{que}={ \\{ (G_{i}^{test},\\mathbf{y}_{i}^{test}) \\} }_{i=1}^{q}$.\n\\section{Proposed Framework}\n\nOverall, our few-shot graph classification framework consists of GNNs based meta-learner and a step controller to decide the adaptation steps of meta-learner. We use MAML to implement a fast adaptation mechanism for meta-learner because of its model agnostic property. Du et al. \\cite{recommendation} proposed an RL based step controller to guide meta-learner for link prediction. We argue that classification loss is suboptimal to be viewed as rewards for overcoming overfitting. Therefore, we adopt a novel step controller to accelerate training and overcome overfitting. Our step controller is also driven by RL but learns the optimal adaptation step by using ANIs and losses as inputs and classification accuracy as rewards. Figure \\ref{fig:fm} demonstrates the training process of our framework.\n\n\\subsection{Graph Embedding Backbone }\nWe explain our proposed framework with typical graph convolutional modules and pooling modules as embedding backbone, due to that novel graph convolutional modules or pooling modules are out of concern for this paper. The first step to represent a graph is to embed the nodes it contains. We investigate several embedding methods such as GCN, GAT, GraphSAGE and GIN. Here we focus on GraphSAGE as following reasons: (1) GraphSAGE has more flexible aggregators in few-shot learning scenarios; (2) Errica et at. \\cite{Errica2020A} set GraphSAGE as a strong baseline when compared to GIN for graph classification task. Hence we use mean aggregator of GraphSAGE as follows:   \n\\begin{equation}\n\\mathbf{h}_{v}^{l} = \\sigma\\left(\\mathbf{W} \\cdot \\operatorname{mean}\\left(\\left\\{\\mathbf{h}_{v}^{l-1}\\right\\} \\cup\\left\\{\\mathbf{h}_{u}^{l-1}, \\forall u \\in \\mathcal{N}(v)\\right\\}\\right)\\right.,\n\\end{equation}\nwhere $\\mathbf{h}_{v}^{l}$ is the $l$-th layer representation of node $v$, $\\sigma$ is the sigmoid function, $\\mathbf{W}$ is the parameters and $\\mathcal{N}(v)$ contains the neighborhoods of $v$.  Please note that this mean aggregator just belongs to the group of typical aggregators we use in experiments. We will provide concrete analysis for other aggregators in Section \\ref{section:bound}  and Section \\ref{section:exp}.\n\nAfter that, we discuss existing pooling operations. Under the circumstances of few-shot learning, the meta-learner urgently needs a flexible pooling strategy with learning capability to strengthen its generalization. Here, we focus on self-attention pooling (SAGPool) \\cite{lee2019self} as our pooling layer thanks to its flexible attention parameters. The main step of SAGPool is to calculate the attention score matrix of graph ${G}_{i}$ as follows: \n\\begin{equation}\n\\mathbf{S}_{i}=\\sigma\\left(\\mathbf{\\tilde{D}}_{i}^{-\\frac{1}{2}} \\mathbf{\\tilde{A}}_{i} \\mathbf{\\tilde{D}}_{i}^{-\\frac{1}{2}} \\mathbf{X}_{i} \\mathbf{\\Theta}_{a t t}\\right),\n\\end{equation}\nwhere the $\\mathbf{S}_{i} \\in \\mathbb{R}^{n_{i} \\times 1}$  indicates the self-attention score, $n_{i}$ is node number of the graph. $\\sigma$ is the activation function (e.g., tanh), $\\mathbf{\\tilde{A}}_{i} \\in \\mathbb{R}^{n_{i} \\times n_{i}}$ is the adjacency matrix with self-connections, $\\mathbf{\\tilde{D}}_{i} \\in \\mathbb{R}^{n_{i} \\times n_{i}}$ is the diagonal degree matrix of $\\mathbf{\\tilde{A}}_{i} $, $\\mathbf{X}_{i} \\in \\mathbb{R}^{n_{i} \\times d}$ is $n$ input features with dimension $d$, and $\\mathbf{\\Theta}_{a t t} \\in \\mathbb{R}^{d \\times 1}$ is the learnable parameters of pooling layer.  Based on the attention score, we select top $c<n_{i}$ nodes that have larger scores with keeping their origin edges unchanged.\n\nTo get fixed representation dimension for each graph, we need Read-Out operation to form each graph embedding vector into identical dimension.  Following Zhang et al. \\cite{zhang2019hierarchical}, we use the concatenation of mean-pooling and max-pooling for each level of graph embeddings of ${G}_{i}$ as follows:\n\n\\begin{equation}\n\\mathbf{r}_{i}^{l}=\\mathcal{R}\\left(\\mathbf{H}_{i}^{l}\\right)=\\sigma\\left(\\frac{1}{n_{i}^{l}} \\sum_{p=1} \\mathbf{H}_{i}^{l}(p,:) \\| \\max _{q=1}^{d} \\mathbf{H}_{i}^{l}(:, q)\\right),\n\\end{equation}\nwhere $\\mathbf{r}_{i}^{l} \\in \\mathbb{R}^{2 d}$ is the $l$-th layer embedding, $n_{i}^{l}$ is the node number in $l$-th layer, $\\mathbf{H}_{i}^{l}$ denotes $l$-th layer hidden representation matrix , $\\|$ is concatenation operation, $p$ and $q$ are row number and column number respectively, $d$ is feature dimension, and $\\sigma$ is the activation function (e.g., Rectified Linear Unit, ReLU \\cite{ReLU}).\n\nFollowing the graph embedding backbone, we compute  the final graph embedding of  ${G}_{i}$ as \n\\begin{equation}\n\\mathbf{z}_{i}= \\mathbf{r}_{i}^{1}+\\mathbf{r}_{i}^{2}+\\cdots+\\mathbf{r}_{i}^{L}\n\\end{equation}\n and put it into Multi-Layer Perceptron (MLP) classifier to perform classification using cross-entropy loss.\n \\subsection{Meta-Learner for Fast Adaptation}\nWe use $\\mathbf{\\theta}_{e}$ and $\\mathbf{\\theta}_{c}$ to denote the parameters of graph embedding modules and MLP classifier respectively, where $\\mathbf{\\theta}_{e}$ contains the parameters of node embedding layers and pooling layers. \nTo achieve the fast adaptation of $\\mathbf{\\theta}_{e}$ and $\\mathbf{\\theta}_{c}$, we put them into a nested loop framework to create a GNNs based meta-learner.  Specifically, our meta-learner is optimized from two procedures. One of the procedures is called the outer loop aiming to get optimal initialization for new classification tasks, and one is called the inner loop to implement fast adaptation based on a suitable initialization. Algorithm \\ref{alg:algorithm1} elaborates on how to train a graph meta-learner at the training state. First, we sample support data $D^{train}_{sup}$ and query data $D^{train}_{que}$ in an episode. Then we perform adaptation operation by updating $\\mathbf{\\theta}_{e}$ and $\\mathbf{\\theta}_{c}$ for $T$ steps on $D^{train}_{sup}$. Lines 7 to 8 in Algorithm \\ref{alg:algorithm1} demonstrate the adaptation of meta-learner. After adaptation, demonstrated by line 13, we use the losses on $D^{train}_{que}$ to perform backpropagation and update $\\mathbf{\\theta}_{e}$ as well as $\\mathbf{\\theta}_{c}$. Similarly, at the test stage, the meta-learner will perform adaptation on labeled support graphs $D^{test}_{sup}$ and predict the label of query graphs $D^{test}_{que}$. \n\n\\begin{algorithm}[tb]\n\\caption{Training Stage of AS-MAML}\n\\label{alg:algorithm1}\n\\begin{flushleft}\n\\textbf{Input}: Task distribution $p(\\mathcal{T})$ over $\\{(G^{train},\\mathbf{y}^{train})\\}$\\\\\n\\textbf{Parameter}: Graph embedding parameters $\\theta_{e}$, classifier parameters $\\theta_{c}$, step controller parameters $\\theta_{s} $, learning rate $ \\alpha_{1}, \\alpha_{2}, \\alpha_{3}$ \\\\\n\\textbf{Output}: The trained parameters $\\theta_{e},\\theta_{c},\\theta_{s}$\n\n\\end{flushleft}\n\n\\begin{algorithmic}[1]\n\\STATE Randomly initialize $\\theta_{e} $, $\\theta_{c} $, $\\theta_{s} $\n\\WHILE{not convergence}\n\\STATE Sample task $\\mathcal{T}_{i}$ with support graphs $D^{train}_{sup}$ and query graphs $D^{train}_{que}$\n\\STATE Get adaptation step $T$ via  Equation \\ref{calT}\n\\STATE Set fast adaptation parameters: $\\theta^{\\prime}=\\theta=\\{\\theta_{e},\\theta_{c}\\}$\n\\FOR {$t = 0 \\to T$ }\n\\STATE Evaluate  $\\nabla_{\\theta^{\\prime}} \\mathcal{L}_{\\mathcal{T}_{i}}\\left(f_{\\theta^{\\prime}}\\right)$ on $D^{train}_{sup}$ by classification loss $\\mathbf{L}^{(t)}$.\n\\STATE Update $\\theta^{\\prime} $ : $\\theta^{\\prime} \\leftarrow \\theta^{\\prime}- \\alpha_{1} \\nabla_{\\theta^{\\prime}} \\mathcal{L}_{\\mathcal{T}_{i}}\\left(f_{\\theta^{\\prime}}\\right)$\n\\STATE Calculate ANI $\\mathbf{M}^{(t)}$ via Equation \\ref{eq:ANIs}\n\\STATE Calculate stop probability $p(t)$ via Equation \\ref{stop_prb}\n\\STATE Calculate reward ${Q^{(t)}}$ on $D^{train}_{que}$  by Equation \\ref{eq:Q}\n\\ENDFOR\n\\STATE  $\\theta \\leftarrow \\theta-\\alpha_{2} \\nabla_{\\theta}  \\mathcal{L}_{\\mathcal{T}_{i}}\\left(f_{\\theta^{\\prime}}\\right)$ on $D^{train}_{que}$\n\\FOR {$t = 0 \\to T$ }\n\\STATE $\\theta_{s} \\leftarrow \\theta_{s} +  \\alpha_{3}  Q^{(t)} \\nabla_{ \\theta_{s}} \\ln p(t)$\n\\ENDFOR\n\\ENDWHILE\n\\end{algorithmic}\n\\end{algorithm}\n\\subsection{Adaptation Controller}\nFinding optimal combinations of learning rates and step size is difficult for MAML \\cite{antoniou2018how}. Besides, arbitrary graph size and structure bring difficulty for ascertaining optimal step size manually. As an empirical solution to alleviate these problems, we design an RL based controller to decide optimal step size for the adaptation of meta-learner when given other hyper-parameters. Therefore, our controller must roughly know when to stop adaptation according to the embedding quality and training state (denoted by loss). We focus on Average Node Information (ANI) to indicate the embedding quality. Intuitively, if a node can be well reconstructed by its neighborhoods, it has less information for the graph classification. Similarly, the rising of batch graphs' ANI indicates that the pooling module has learned to select the most informative nodes.  Hou et al. \\cite{Hou2020Measuring} proposed similar concept called \\textit{Feature Smoothness} measuring node information over graphs.\nHere we adopt another practical method defined by \\cite{zhang2019hierarchical},  where they compute node information as the Manhattan distance between the node representation itself and the one constructed from its neighbors. Inspired by their work, we define the ANI of a single graph ${G}_{i}$ as follows:\n    \\begin{equation}\n    \\label{eq:ANI}\n{ANI}_{i}^{l}= \\frac{1}{n_{i}^{l}}\\sum_{j=1} \\left\\| \\left[ \\left(\\mathbf{I}_{i}^{l}-\\left(\\mathbf{D}_{i}^{l}\\right)^{-1} \\mathbf{A}_{i}^{l}\\right) \\mathbf{H}_{i}^{l}  \\right]_{j} \\right\\|_{1},\n\\end{equation}\nwhere $l$ denotes the embedding layer of the graph, $n_{i}^{l}$ denotes the number of node, $j$ denotes the row index of matrix or $j$-th node, ${\\| \\cdot \\|}_{1}$ denotes the L1 norm of row vector,  $\\mathbf{A}_{i}^{l}$ denotes the adjacency matrix, $\\mathbf{D}_{i}^{l}$ is the degree matrix of $\\mathbf{A}_{i}^{l}$, $\\mathbf{H}_{i}^{l}$ denotes $l$-th layer hidden representation matrix. In our work, we only use the last layer of graph embedding (i.e., $\\mathbf{H}_{i}^{L}$). And unless specifically stated, we use scalar value ANI to denote the average node information of the batch graphs:\n \\begin{equation}\n \\label{eq:ANIs}\nANI=1\/n*\\sum_{i=1}^{n} ANI_{i}^{L},\n\\end{equation}\nwhere $n$ denotes the number of batch graphs, $L$ denotes the L-th layer of graph embedding.\n\nNext we set the number of initial step as $T_{i}$, the ANIs in $T_{i}$ steps as $\\mathbf{M} \\in \\mathbb{R}^{T_{i} \\times 1}$ and denote classification losses as $\\mathbf{L} \\in \\mathbb{R}^{T_{i} \\times 1}$. Then we compute stop  probability $p^{(t)}$ at step $t$ as follows:\n\\begin{equation}\n\\label{stop_prb}\n\\boldsymbol{h}^{\\left(t\\right)}=\\operatorname{LSTM}\\left(\\left[\\mathbf{L}^{(t)},\\mathbf{M}^{(t)}\\right], \\boldsymbol{h}^{(t-1)} \\right),  \\\\\np^{(t)}=\\sigma\\left(\\mathbf{W}\\mathbf{h}^{(t)}+\\mathbf{b}\\right),\n\\end{equation}\nwhere $\\mathbf{W}$ and $\\mathbf{b}$ are the parameters of a MLP module, $\\sigma$ is sigmoid function and $h^{(t)}$ is the output of LSTM module. Note that the adaptation will not stop until $T_{i}$ steps at current task regardless of $p^{(t)}$. We set $p^{(t)}$ as a prior for the next task:\n\\begin{equation}\n\\label{calT}\nT_{i+1} =  \\left \\lfloor \\frac{1}{p^{(T_{i})}} \\right \\rfloor ,\n\\end{equation}where $ \\lfloor \\rfloor$ is round down operation. We can observe that we compute $T$ of next task according to the ANIs and losses, where both of them are produced by current task. The reason behind it is that if we stop adaptation according to $p^{(t)}$ in current task, it will output  larger variance of step and bring instability for optimization.  \nBesides, we get controller's rewards as following:\n\n\\begin{equation}\n\\label{eq:Q}\nQ^{(t)}=\\sum_{t=1}^{T} r^{(t)}=\\sum_{t=1}^{T}(e_{T}-e_{t} - \\eta*t),\n\\end{equation}\nwhere $T$ is total steps and $e_{t}$ is the classification accuracy on query data at step $t$, and $\\eta*t$ denotes the penalty item.  Then we update our controller by policy gradients, which is a typical method in RL: \n\\begin{equation}\n\\theta_{s} = \\theta_{s} +  \\alpha_{3}  Q^{(t)}\\nabla_{ \\theta_{s}} \\ln p(t),\n\\end{equation}where $\\ln p(t)$ is the log of  stop probability p(t) $\\cdot$, $\\nabla_{ \\theta_{s}}$ is the gradients over $\\theta_{s}$ and $\\alpha_{3}$ is learning rate. \n\n \n \n \\section{Graph structure-dependent bound for meta-learning}\n \\label{section:bound}\nIn this section, we theoretically analyze the proposed framework with \nunnecessary details omitted. Driven by graph data, we will build a generalization error bound in meta-learning scenario considering the locality of graph structure (e.g., the degrees of nodes in a graph). Before getting closer to analysis, we first give the key results: (1) The bound is dependent on the number of graphs in the support example of test classes and the number of graphs of training classes.  (2) The bound is dependent on the locality structure of graphs between training classes and test classes. The first result is common in meta-learning scenarios, while the second result is derived from the structure of graphs.\n\nFrom the perspective of representation learning and probability distribution, our framework try to minimize the distance of latent distributions between $\\mathcal{G}^{train} $ and $\\mathcal {G}^{test} $ directly.  Based on this premise, we use the integral probability metric (IPM, \\cite{IPM}) as a general distance metric to produce the upper bound. IPM has been employed in the analysis of transfer learning \\cite{10.5555\/2999325.2999503}, non-parametric estimation \\cite{NPE} and generative models \\cite{zhang2018on}. Generally, IPM can be formalized as:\n\n\\begin{equation}\n\\gamma_{\\mathcal{H}}(\\mathbb{P}, \\mathbb{Q}):=\\sup _{h \\in \\mathcal{H}}\\left|\\mathbb{E}_{\\mathbb{P}} h(z)-\\mathbb{E}_{\\mathbb{Q}} h(z)\\right|,\n\\end{equation}\nwhere $\\mathbb{P}$ and $\\mathbb{Q}$ denote two different probability distributions, respectively,  $\\mathcal{H}$ denotes the collection of real-valued functions (e.g., square loss function and margin loss function ). \n\nWe set the empirical distribution of $\\mathcal{G}^{train} $ as  $\\hat{\\mathbb{P}}$.  With a slight abuse of notation, in a single task at the test stage, we set the empirical distribution of support graph $D^{test}_{sup}$ as $\\hat{\\mathbb{Q}}$, and set the expected distribution query data $D^{test}_{que}$ as $\\mathbb{Q}$. Then in the adaptation process at test stage, we actually want to  minimize the following bound:\n\n\\begin{equation}\n\\gamma_{\\mathcal{H}}(\\hat{\\mathbb{P}}, {\\mathbb{Q}}) = \\sup _{h \\in \\mathcal{H}}\\left|\\mathbb{E}_{\\hat{\\mathbb{P}}} h({G})-\\mathbb{E}_{{\\mathbb{Q}}} h({G})\\right|,\n\\label{equation:PhatQ}\n\\end{equation}\nwhere $h(G)$ is the classification loss of meta-learner after the adaptation on $D^{test}_{sup}$.  \nThen with the help of IPM, $\\gamma_{\\mathcal{H}}(\\hat{\\mathbb{P}}, {\\mathbb{Q}}) $can be bounded by following theorem \\cite{cai2020weighted}:\n\\begin{theorem}\nLet $\\mathcal{H}$ denote a class of functions whose members map from ${G}_{i}$ to [a, b], and suppose that the training data test data are independent, and that the data instances of each are i.i.d. within a sample. Let $\\epsilon > 0 $. Then with probability at least $1 - \\epsilon$ over the draws of the training and query samples, \n\\begin{equation}\n\\gamma_{\\mathcal{H}}\\left(\\hat{\\mathbb{P}}, \\mathbb{Q}\\right) \\leq \\gamma_{\\mathcal{H}}\\left(\\hat{\\mathbb{P}}, \\hat{\\mathbb{Q}}\\right)+2 \\mathcal{R}\\left( \\mathcal{H} | \\{{G}_{i}\\}^{m}_{i=1} \\right)+3 \\sqrt{\\frac{(b-a)^{2} \\log (2 \/ \\epsilon)}{2m}},\n\\end{equation}\nwhere $m$ is the number of support graphs $D^{test}_{sup}$,  $\\mathcal{R}\\left( \\mathcal{H} |\\{{G}_{i}\\}^{m}_{i=1} \\right)$ denotes the empirical Rademacher complexity \\cite{Radermacher_com} of the function class $\\mathcal{H}$  w.r.t. the support graphs.\n\\label{theorem1}\n\\end{theorem}\n\nNow we provide the proof of Theorem \\ref{theorem1} referencing Cai et al. \\cite{cai2020weighted}. First, IPM is interrelated with  \\textit{uniform deviation with empirical Rademacher complexity} \\cite{Radermacher_com}:\n\\begin{theorem}\nLet $\\mathcal{H}$ denote a class of functions whose members map from ${z}_{i}$ to [a, b], and suppose that $\\{{z}_{i}\\}^{m}_{i=1}$ is sampled from a i.i.d distribution $\\mathbb{P}$. Then for $\\epsilon > 0 $. Then with probability at least $1 - \\epsilon$ over the sample, \n\\begin{equation}\n\\sup _{h \\in \\mathcal{H}}\\left|\\mathbb{E}_{\\mathbb{\\hat{P}}} h(z)-\\mathbb{E}_{\\mathbb{P}} h(z)\\right| \\leq 2 \\mathcal{R}\\left( \\mathcal{H} | \\{{z}_{i}\\}^{m}_{i=1} \\right)+3 \\sqrt{\\frac{(b-a)^{2} \\log (2 \/ \\epsilon)}{2m}},\n\\label{theorem2}\n\\end{equation}where $\\mathbb{\\hat{P}}$  represents the empirical distribution of the sample, $ \\mathcal{R}\\left( \\mathcal{H} | \\{{z}_{i}\\}^{m}_{i=1} \\right)$ denotes the empirical Rademacher complexity of the function class $\\mathcal{H}$  w.r.t. the sample. \n\\end{theorem}\n\n\\begin{proof}\nThen Equation \\ref{equation:PhatQ} is unfolded as:\n\\begin{equation}\n\\begin{split}\n\\gamma_{\\mathcal{H}}(\\hat{\\mathbb{P}}, {\\mathbb{Q}}) &= \\sup _{h \\in \\mathcal{H}}\\left|\\mathbb{E}_{\\hat{\\mathbb{P}}} h({G})  -\\mathbb{E}_{{\\mathbb{Q}}} h({G})\\right|  \\\\\n&= \\sup _{h \\in \\mathcal{H}}\\left|\\mathbb{E}_{\\hat{\\mathbb{P}}} h({G}) + \\mathbb{E}_{{\\hat{\\mathbb{Q}}}} h({G}) -\\mathbb{E}_{\\hat{\\mathbb{Q}}} h({G}) -\\mathbb{E}_{{\\mathbb{Q}}} h({G})\\right| \\\\\n& \\leq \\sup _{h \\in \\mathcal{H}}\\left[ \\left|\\mathbb{E}_{\\hat{\\mathbb{P}}} h({G}) - \\mathbb{E}_{{\\hat{\\mathbb{Q}}}} h({G}) \\right| + \\left|\n\\mathbb{E}_{\\hat{\\mathbb{Q}}} h({G}) -\n\\mathbb{E}_{{\\mathbb{Q}}} h({G}) \\right| \\right] \\\\\n& \\leq \\sup _{h \\in \\mathcal{H}} \\left|\\mathbb{E}_{\\hat{\\mathbb{P}}} h({G}) - \\mathbb{E}_{{\\hat{\\mathbb{Q}}}} h({G}) \\right| + \n\\sup _{h \\in \\mathcal{H}} \\left| \\mathbb{E}_{\\hat{\\mathbb{Q}}} h({G}) -\n\\mathbb{E}_{{\\mathbb{Q}}} h({G}) \\right|  \\\\\n&=\\gamma_{\\mathcal{H}}\\left(\\hat{\\mathbb{P}}, \\hat{\\mathbb{Q}}\\right) + \\sup _{h \\in \\mathcal{H}} \\left| \\mathbb{E}_{\\hat{\\mathbb{Q}}} h({G}) -\n\\mathbb{E}_{{\\mathbb{Q}}} h({G}) \\right|\n\\end{split}\n\\label{equation:proof}\n\\end{equation}When the last term of Formula \\ref{equation:proof} substituted by Theorem \\ref{theorem2},  Theorem \\ref{theorem1} is proved.\n\\end{proof}\n\nAfter giving the proof of Theorem \\ref{theorem1}, $\\gamma_{\\mathcal{H}}\\left(\\hat{\\mathbb{P}}, \\hat{\\mathbb{Q}}\\right)$ can be optimized by the adaptation of meta-learner on the test stage.  As the final step, we focus on the evaluation of $\\mathcal{R}\\left( \\mathcal{H} |\\{{G}_{i}\\}^{m}_{i=1} \\right)$. The diversity of graph structure brings difficulty for directly calculating this term. We adopt the method proposed by K. Garg et al. \\cite{garg2020generalization},  who gave an upper bound for $\\mathcal{R}\\left( \\mathcal{H} |\\{{G}_{i}\\}^{m}_{i=1} \\right)$, by transforming each graph to the corresponding collection of local computation trees. Because when updating every node embedding, the node to be updated can be viewed as a root node of a tree, with its neighbor nodes as children.  Given this hypothesis, they derived a strict upper bound mainly based on the degree of nodes, where the degree was used to denote the local complexity of graph. Furthermore, the local complexity of graph is made full use by our model. In the experiment of Section \\ref{comparisonGSM}, our model performs better over datasets which have \nclear local structure.\n\n\n\n\\section{Experiments}\nIn the experiments, we focus on two aspects: (1)How does the framework perform on few-shot graph classification tasks? (2) How does the controller work when training meta-learner? In this section, we will introduce experiment datasets, comparison with baselines and details of implementation. Finally, we demonstrate the effectiveness of key modules by ablation study and detail analysis.\n\\subsection{Datasets}\nWe select four public graph datasets including COIL-DEL, R52, Letter-High and  TRIANGLES. These datasets are publicly available \\footnote{https:\/\/ls11-www.cs.tu-dortmund.de\/staff\/morris\/graphkerneldatasets} \\footnote{https:\/\/www.cs.umb.edu\/~smimarog\/textmining\/datasets\/}. The statistics are summarized in Table \\ref{tab:1}. The visualization of each datasets are shown in Figure \\ref{fig:data}. In the work proposed by \\cite{Chauhan2020FEW-SHOT}  for few-shot graph classification, they  focus on two datasets containing Letter-High and  TRIANGLES. We use two additional datasets where Graph-R52 was built from text classification dataset and  COIL-DEL was built from images. \n\\paragraph{COIL-DEL} COIL-DEL is built on images, and each graph is constructed by applying corner detection and Delaunay triangulation to corresponding image \\cite{COIL}. \n\\paragraph{R52} R52 is a text dataset in which each text is viewed as a graph. We transformed it into a graph dataset as follows: if two words appear together in a specified sliding window, they have an undirected edge in the graph. We keep classes with more than 20 samples and finally get 28 classes. We name the new dataset as Graph-R52 for clarity.\n\\paragraph{Letter-High} Each graph represents distorted letter prototype drawings with representing lines by undirected edges and ending points of lines by nodes \\cite{COIL}. More specifically, Letter-High contains 15 classes from English alphabets: A, E, F, H, I, K, L, M, N, T, V, W, X, Y, Z. \n\\paragraph{TRIANGLES} The dataset was proposed for the task of triangle counting, where the model is required to give the number of triangles of each graph. TRIANGLES contains 10 different graph classes numbered from 1 to 10 corresponding to the number of triangles in graphs of each class. In our experiments, we use the partition of \\cite{Chauhan2020FEW-SHOT},  where they remove oversize graph samples so the total sample size of TRIANGLES is reduced from 45000 to 2000.\n\\begin{table}[!tbp]\n\\centering\n\\begin{tabular}{ccccccc}  \n\\toprule\nDatasets   & $|G|$     & Avg.$|\\mathcal{V}|$     & Avg.$|\\mathcal{E}|$ & $C_{0}$ & $C_{1}$ & $C_{2}$ \\\\\n\\midrule\nCOIL-DEL   & 3900  & 21.54  & 54.24  & 60  & 16  & 20    \\\\\nGraph-R52        & 8214   & 30.92    & 165.78    & 18  & 5  & 5   \\\\\nLetter-High      & 2250   & 4.67     & 4.50      & 11  & 0  & 4   \\\\\nTRIANGLES        & 45000  & 20.85    & 35.50     & 7   & 0  & 3   \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{Statistics of datasets. For each dataset, we show total graph number $|G|$, average node number Avg.$|\\mathcal{V}|$, Average edge number Avg.$|\\mathcal{E}|$ and class number for training ($C_{0}$), validation ($C_{1}$) and test ($C_{2}$). }\n\\label{tab:1}\n\\end{table}\n\\begin{figure}[!t]\\small\n\\centering\n\\subfigure[\\scriptsize{COIL-DEL}]{\n\\label{fig:example1}\n\\centering\n\\includegraphics[width=0.22\\textwidth]{pic\/DEL.png}\n}\n\\subfigure[\\scriptsize{Graph-R52}]{\n\\label{fig:example2}\n\\centering\n\\includegraphics[width=0.22\\textwidth]{pic\/r52.png}\n}\n\\subfigure[\\scriptsize{TRIANGLES}]{\n\\label{fig:example3}\n\\centering\n\\includegraphics[width=0.22\\textwidth]{pic\/tri15.png}\n}\n\\subfigure[\\scriptsize{Letter-High}]{\n\\label{fig:example4}\n\\centering\n\\includegraphics[width=0.22\\textwidth]{pic\/lh1.png}\n}\n\\caption{Visualization of typical instances of the used datasets, where their different graph size and graph structures bring challenge for graph classification models.}\n\\label{fig:data}\n\\end{figure} \n\n\n\\begin{table*}[!htbp]\n\\centering\n\\begin{tabular}{llcccc} \n\\toprule\n\\multicolumn{1}{c}{\\multirow{2}{*}{Categories}}  & \\multicolumn{1}{c}{\\multirow{2}{*}{Baselines}} & \\multicolumn{2}{c}{\\multirow{1}{*}{COIL-DEL}} & \\multicolumn{2}{c}{\\multirow{1}{*}{Graph-R52}} \\\\ \n\\cline{3-6}\n & &5-way-5-shot &5-way-10-shot &2-way-5-shot &2-way-10-shot \\\\\n\\toprule[0.5pt]\n\\multirow{3}{*}{Kernels} & GRAPHLET & $47.47\\pm1.06$ & $49.04\\pm0.98$ & $56.52\\pm1.46$ & $57.16\\pm1.47$ \\\\\n                         & SP        & $38.33\\pm0.62$ & $42.18\\pm0.69$ & $74.38\\pm1.50$ & \\textcolor{blue}{76.96 $\\pm$ 1.34} \\\\\n                         & WL        & $43.05\\pm1.25$ & $52.28\\pm1.47$ & \\textbf{76.90 $\\pm$ 1.48} & $75.91\\pm1.46$ \\\\\n\\midrule\n\n\\multirow{1}{*}{Finetuning} & finetuning & $68.21\\pm1.29$ & $72.38\\pm1.40$ & $71.87\\pm2.04$ & $72.39\\pm1.88$ \\\\\n\\midrule\n\\multirow{9}{*}{GNNs-Pro}\n                        & GCN, TopKPool & $78.01\\pm1.83$ & $78.98\\pm1.53$ & $69.98\\pm1.53$ & $70.19\\pm1.37$ \\\\\n                         & GCN, EdgePool     & $76.21\\pm1.54$ & $79.43\\pm1.58$ & $67.24\\pm1.34$ & $67.72\\pm1.59$ \\\\\n                         & GCN, SAGPool          & $76.58\\pm1.19$ & $79.16\\pm1.06$ & $69.88\\pm1.40$ & $70.46\\pm1.47$ \\\\\n                         & GraphSAGE, TopKPool    & $69.80\\pm1.25$ & $74.18\\pm1.73$ & $70.43\\pm1.76$ & $70.52\\pm1.83$ \\\\\n                         \n                         & GraphSAGE, EdgePool        & $80.08\\pm1.26$ & $80.96\\pm1.26$ & $68.13\\pm1.59$ & $70.72\\pm1.58$ \\\\\n                         \n                         & GraphSAGE, SAGPool & $79.30\\pm1.12$ & $80.91\\pm1.62$ & $68.10\\pm1.40$ & $70.49\\pm1.32$ \\\\\n                         \n                         & GAT, TopKPool          & $76.37\\pm1.10$ & $77.29\\pm1.40$ & $71.99\\pm1.51$ & $73.31\\pm1.44$ \\\\\n                         \n                         & GAT, EdgePool        & \\textcolor{blue}{81.00 $\\pm$ 1.22} & \\textcolor{blue}{83.57 $\\pm$ 0.99} & $66.49\\pm1.32$ & $70.49\\pm1.17$ \\\\\n    \n                         & GAT, SAGPool          & $72.54\\pm1.07$ & $73.99\\pm1.00$ & $67.78\\pm1.52$ & $74.10\\pm1.57$ \\\\\n                        \n\\midrule\n\n\\multirow{2}{*}{Ours}   &AS-MAML (wo\/AS)  & $79.54 \\pm 1.48$ & $81.24\\pm1.27$ & $74.12\\pm1.39$ & $76.05\\pm1.17$ \\\\\n                        & AS-MAML (w\/AS)   & \\textbf{81.55 $\\pm$ 1.39} & \\textbf{84.75 $\\pm$ 1.30} & \\textcolor{blue}{75.33 $\\pm$ 1.19} & \\textbf{78.33 $\\pm$ 1.17} \\\\\n                                  \n\\bottomrule\n\\end{tabular}\n\\caption{Accuracies with a standard deviation of baseline methods and our framework. We tested 200 and 500 N-way-K-shot tasks on COIL-DEL and  Graph-R52, respectively. The \\textbf{bold black} numbers denote the best results we get, and the  \\textcolor{blue} {blue} numbers denote the second best results. AS-MAML (wo\/AS) denotes our framework without Adaptive Step (AS) which is controlled by our step controller, and AS-MAML (w\/AS) denotes the whole framework we proposed.}\n\\label{tab:2}\n\\end{table*}\n\n\\subsection{Baselines}\nWe adopt four groups of baselines made up of  Graph Kernel, Finetuning, GNNs-Prototypical-Classifier (GNNs-Pro) and Graph Spectral Measures (GSM) \\cite{Chauhan2020FEW-SHOT} . For Graph Kernel baselines, we perform  N-way-K-shot graph classification over the test set directly because there are no parameters to transfer. The baselines of the last three groups train a GNNs based graph classifier by performing classification over $C_0$ training classes (see Table \\ref{tab:1}). On the test stage, they perform N-way-K-Shot classification. \n\n\\paragraph{Graph Kernel.} This group of methods firstly measure the similarity between labeled support data and query data on the test stage. After that, the similarity matrix was put into a Prototypical Classifier, which has none of parameters \\cite{SnellSZ17}, to get predicted labels of query data. We choose typical graph kernel algorithms including Shortest-path  Kernel (SP) \\cite{borgwardt2005shortest}, Graphlet Kernel \\cite{shervashidze2009efficient} and Weisfeiler-Lehman Kernel (WL)  \\cite{shervashidze2011weisfeiler}. \n\n\n\\paragraph{Finetuning.} In this baseline, we train a naive graph classifier consisting of GraphSAGE, SAGPool and MLP classifier. On the test stage, we change the output dimension of the last layer of the classifier and fine-tuning the layer's parameters, while keeping other modules unchanged. \n\n\\paragraph{GNNs-Pro.}  We train a graph classifier following Finetuning. On the test stage, we replace the MLP classifier with Prototypical Classifier. We choose GCN \\cite{kipf2016semi}, GraphSAGE \\cite{SAGE} and GAT \\cite{GAT} as graph convolutional modules, and  Self-attention Pooling (SAGPool) \\cite{lee2019self}, TopK Pooling (TopKPool) \\cite{unet} and Edge Pooling (EdgePool) \\cite{diehl2019edge} as graph pooling modules. \n\n\n\\paragraph{GSM} Chauhan et al. \\cite{Chauhan2020FEW-SHOT} proposed the GSM based method customized for few-shot graph classification. On the training stage, they compute prototype graphs from each class, then they cluster the prototype graphs to produce super-classes. After that, they predict the origin class and super-class of each graph. On the test stage, they only update the classifier based on the classification of origin classes.\n\n\\subsection{Experimental Details}\nTo ensure a fair comparison, we use three convolutional layers followed by corresponding pooling layers for the GNNs based baselines and our proposed framework. We set the same node dimension as 128 for all GNNs based baselines. For the adaptation step, we set the minimum and maximum step by 4 and 15. We implement GNNs based baselines and our framework with  PyTorch Geometric (PyG \\footnote{ https:\/\/github.com\/rusty1s\/pytorch\\_geometric } )  and graph kernel baselines based on GraKel {\\footnote{https:\/\/github.com\/ysig\/GraKeL}}.  We use SGD optimizer with 1e-5 for weight decay and versatile learning rates 0.0001, 0.001,  0.0001 for $\\alpha_{1}$, $\\alpha_{2}$ and $\\alpha_{3}$, respectively.\n\n\n\n\\subsection{Comparison with Graph Kernel, Finetuning and GNNs-Pro}\n \\label{section:exp}\nTo evaluate the performance of our framework, we performed 5-way-5-shot and 5-way-10-shot graph classification on COIL-DEL dataset. On Graph-R52 dataset, we performed 2-way-5-shot and 2-way-10-shot graph classification.  The results are reported in Table \\ref{tab:2}.  \n Our framework utilizes GraphSAGE and SAGPool as graph embedding backbone. So firstly we compare our framework with the finetuning baseline built on GraphSAGE and SAGPool. We found our framework is superior to the finetuning baseline with a large margin, which indicates that the meta-leaner works well with a fast adaptation mechanism.  Moreover, under 5-way-10-shot setting in GraphSAGE-SAGPool baseline, our framework achieves about 3.84\\%  improvement on the COIL-DEL dataset.\n\nSurprisingly, traditional graph kernel based baselines achieve competitive performance on Graph-R52 dataset. The reasons are two-fold: (1)  our graphs from texts contain many well defined sub-graphs built by text themes and their neighbor words, and this pattern gives graph kernels a favorable position; (2) the parameters of kernel methods are far less than GNNs based methods. So they are not prone to be overfitting, which is GNNs' common problem for the few-shot task. However,  finding an appropriate kernel is difficult (e.g., From Table \\ref{tab:2}, SP behaves badly compared to WL and GRAPHLET on the COIL-DEL dataset).\n\n\\begin{figure*}[!t]\\small\n\\centering\n\\subfigure[\\scriptsize{GSM on TRIANGLES}]{\n\\label{fig:exa1}\n\\centering\n\\includegraphics[width=0.21\\textwidth]{pic\/GSM_3w5shot_150_sample_per_class_.png}\n}\n\\subfigure[\\scriptsize{ AS-MAML on TRIANGLES}]{\n\\label{fig:exa}\n\\centering\n\\includegraphics[width=0.21\\textwidth]{pic\/tri_mine.png}\n}\n\\subfigure[\\scriptsize{GSM on Letter-High}]{\n\\label{fig:exm3}\n\\centering\n\\includegraphics[width=0.21\\textwidth]{pic\/gsm_lh.png}\n}\n\\subfigure[\\scriptsize{AS-MAML on Letter-High}]{\n\\label{fig:exa4}\n\\centering\n\\includegraphics[width=0.21\\textwidth]{pic\/lh_mine3.png}\n}\n\\caption{T-SNE visualization of unseen classes under 5-way-5-shot scenarios. We randomly sample 400 examples per dataset and visualized the representations before they were put into classifying layer. We can see more clear distribution boundary using our method.}\n\\label{fig:tsne}\n\\end{figure*} \n\\begin{table}[!tpbh]\n\\centering\n\\begin{tabular}{cccc} \n\\toprule\n\\multicolumn{1}{c}{\\multirow{2}{*}{Methods}} & \\multicolumn{1}{c}{\\multirow{2}{*}{Shots}} & \\multicolumn{2}{c}{\\multirow{1}{*}{Datasets}} \\\\ \n\n\\cline{3-4}\n& &TRIANGLES &Letter-High  \\\\ \n\\toprule[0.6pt]\n\n\\multirow{2}{*}{GSM}   &5-shot  &  $71.40\\pm4.34$ & $69.91\\pm5.90$   \\\\\n\n                       &10-shot &  $75.60\\pm3.67$ & $73.28\\pm3.46$  \\\\\n\\midrule                      \n\\multirow{2}{*}{Ours}  &5-shot  &  $86.47\\pm0.74$ & $76.29\\pm0.89$   \\\\\n\n                       &10-shot &  $87.26\\pm0.69$ & $77.87\\pm0.75$  \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{Accuracies evaluated from GSM and AS-MAML we proposed. For AS-MAML, we test 200 N-way-K-shot tasks on both datasets. For GSM, we use the best results in their paper.}\n\\label{tab:3}\n\\end{table}\n\n\\begin{figure*}[!htp]\n\\centering\n\\subfigure[\\scriptsize{Test accuracies}]{\n\\label{fig:com1}\n\\centering\n\\includegraphics[width=0.22\\textwidth]{pic\/ada_vs_maml_test_acc1.pdf}\n}\n\\subfigure[\\scriptsize{ ANI \\& training accuracies}]{\n\\label{fig:com2}\n\\centering\n\\includegraphics[width=0.22\\textwidth]{pic\/train_ANI_acc1.pdf}\n}\n\\subfigure[\\scriptsize{Adaptation steps }]{\n\\label{fig:com3}\n\\centering\n\\includegraphics[width=0.22\\textwidth]{pic\/step_stair.pdf}\n}\n\n\\caption{Illustrations of the learning process under the 5-way-10-shot setting on  COIL-DEL dataset. (a) Test accuracies calculated from AS-MAML (wo\/AS) and AS-MAML (w\/AS) respectively. The adaptation step of AS-MAML (wo\/AS) is 6, and other hyper-parameters are same as AS-MAML (w\/AS). The initial values are reported after $0$-th training epoch. (b) The normalized ANIs and training accuracies in the first 50 epochs. Both of them are extracted from support graphs of the training set. (c) The variations of the adaptation step on the training stage. the value at epoch 0 is the initial adaptation step, and then we calculate an average for every 5 epochs. }\n\\label{fig:train_process}\n\\end{figure*} \n\n\\subsection{Comparison with GSM Based Method}\n\\label{comparisonGSM}\nThe GSM based method proposed by Chauhan et al. \\cite{Chauhan2020FEW-SHOT} did not adopt the episodic training paradigm, which is a key idea in our paper, so the method is inappropriate to be trained by N-way-K-shot graph classification on COIL-DEL and Graph-R52 dataset. For a fair comparison, we evaluate our framework on TRIANGLES and Letter-High, which are typical datasets used in their paper. As their partition, we randomly split out 20\\% examples from the training set to perform validation. Following their test configuration, we perform 3-way-K-shot and 4-way-K-shot classification on TRIANGLES and Letter-High respectively. The comparison of performances is shown in Table \\ref{tab:3}. From the table, we conclude that our framework outperforms theirs with a large margin. The reason behind it is that they assume the test classes belong to the same set of super-classes built from the training classes. However, the label spaces of training classes and test classes usually do not overlap in few-shot settings. We observe that the graphs of training classes and test classes have similar sub-structures, which can be discovered by a well initialized meta-learner within a few adaptation steps. As mentioned before, different classes in TRIANGLES have similar triangle structure. Therefore our framework gets the most obvious improvement on this dataset. Besides, the visualization (Figure \\ref{fig:tsne}) of latent representation on unseen classes demonstrates the effectiveness of our method.\n\n\n\\subsection{Ablation Study and Detail Analysis} \nIn this section, we show the effect of the controller module by ablation study. First of all,  without the adaptive step (AS), we evaluate the performance of our framework by just putting GraphSAGE, SAGPool into meta-learner. From Table \\ref{tab:2}, we found that under 2-way-10-shot setting on Graph-R52, AS-MAML (wo\/AS) brings about 3.06\\% improvement compared with finetuning baseline. Furthermore, the step controller brings about 2.28\\% improvement under 2-way-10-shot setting on Graph-R52 and 3.51\\% improvement under 5-way-5-shot setting on COIL-DEL.\n\nWe did a deeper analysis for ANI and give more details of the step controller module under 5-way-10-shot setting on  COIL-DEL dataset. \nFigure \\ref{fig:com1} shows the effect on the test set after adding the adaptive step (AS).\nThe scatter diagram (Figure \\ref{fig:com2}) shows that ANIs have a positive correlation with classification accuracies, which means larger ANI indicates better graph embedding for the MLP classifier module. The advantage of ANI against classification accuracy is that a larger ANI implies more discriminative graph embedding modules, while a better classification accuracy may mean that the MLP classifier module is overfitted on poor graph embedding modules. Finally, Figure \\ref{fig:com3} shows the variations of the adaptation step produced by the controller. At the beginning, the controller receives larger loss and smaller ANI, so it gives more adaptation steps to meta-learner for encouraging exploration. When the meta-learner has been trained well, the controller receives smaller loss and larger ANI, so it outputs smaller step size to alleviate overfitting. \n\n\n\n\\section{Conclusion and Future Works}\nModeling real-world data into graphs is getting more attention in recent years. In this paper, we focus on few-shot graph classification and propose a novel framework named AS-MAML. To control the meta-learner's adaptation step, we proposed a novel step controller in a RL way by using ANI to demonstrate embedding quality. Beyond that, ANI is calculated by unsupervised way like estimating Mutual Information on graphs \\cite{2018deep}. Exploring and utilizing them for graph representation learning is an interesting future work.  Moreover, we expect better graph embedding methods to improve the performance of our framework, including GIN and its variants. We also expect our work can be expanded to more challenging graph classification tasks like skeleton based action recognition, protein classification and subgraph analysis of social networks.\n\n\n\\bibliographystyle{ACM-Reference-Format}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}}\n\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.92\\linewidth]{figures\/motivation.pdf}\n\t\\vspace{-0.5em}\n\t\\caption{Two indispensable characteristics of an ideal VQA model. (a) \\textbf{Visual-explainable ability}: The model can not only predict correct answer (\\textit{e}.\\textit{g}., ``\\texttt{surfing}\"), but also rely on the right visual reference regions when making this prediction. (b) \\textbf{Question-sensitive ability}: The model should be sensitive to the linguistic variations. For example, after replacing the critical word ``\\texttt{luggage}\" with ``\\texttt{bus}\", the predicted answers of the two questions should be different. LMH~\\cite{clark2019don} is a SOTA VQA model.\n\t}\n\t\\vspace{-1em}\n\t\\label{fig:1}\n\\end{figure}\n\n\n\n\n\\IEEEPARstart{V}{isual} Question Answering (VQA), \\textit{i}.\\textit{e}., answering natural language questions about the given visual content, is one of the essential abilities of advanced AI agents. With the release of multiple large-scale VQA datasets (\\textit{e}.\\textit{g}., VQA v1~\\cite{antol2015vqa}, VQA v2~\\cite{goyal2017making}, CLEVR~\\cite{johnson2017clevr},  and GQA~\\cite{hudson2019gqa}), VQA has received unprecedented attentions and hundreds of VQA models have been developed. However, since the inevitable annotation artifacts in the real image datasets, today's VQA models always over-rely on superficial linguistic correlations between the questions and answers (\\textit{a}.\\textit{k}.\\textit{a}., language biases)~\\cite{agrawal2016analyzing, zhang2016yin, johnson2017clevr, goyal2017making}. For example, a model naively answering ``\\texttt{2}\" for all ``\\texttt{How many X?}\" questions can still get satisfactory performance regardless of ``\\texttt{X}\". To disentangle the bias factors and clearly monitor the progress of VQA research, several diagnostic benchmarks have been proposed, such as VQA-CP~\\cite{agrawal2018don} and GQA-OOD~\\cite{kervadec2021roses}. These benchmarks deliberately keep different question-answer distributions in training and test sets. Moreover, the performance of many state-of-the-art VQA models~\\cite{andreas2016neural, fukui2016multimodal, yang2016stacked, anderson2018bottom, tan2019lxmert} all consistently drop significantly on these diagnostic benchmarks compared to their counterparts (\\textit{i}.\\textit{e}., VQA v2\/v1 and GQA).\n\n\n\\begin{figure*}[tbp]\n\t\\centering\n\t\\includegraphics[width=0.9\\linewidth]{figures\/CST.pdf}\n\t\\caption{Comparisons on three different VQA frameworks, and the weights of all VQA models are shared in each framework. \\textbf{(a)} The most common VQA framework, \\textit{i}.\\textit{e}., a plain VQA model with XE loss using original samples. \\textbf{(b)} The framework of previous CSS work~\\cite{chen2020counterfactual}: Using CSS to generate counterfactual samples, and training the VQA model using XE loss for both original samples and counterfactual samples. \\textbf{(c)} The framework of the proposed CSST: Besides the XE training objectives, we use CSS to further generate positive-negative sample pairs, and train the VQA model with an extra contrastive loss. For conciseness, we only show one (type of) counterfactual sample and negative sample in each figure, and we skip the answer sets for all samples in each figure.}\n\t\\label{fig:2}\n\\end{figure*}\n\n\nCurrently, the most prevalent solutions to mitigate the language bias issues are \\textbf{ensemble-based} methods: they introduce an auxiliary question-only model to regularize the training of targeted VQA model. Specifically, these methods can further be categorized into two sub-types: 1) \\emph{Adversary-based models}~\\cite{ramakrishnan2018overcoming, grand2019adversarial, belinkov2019don}: They train two models (\\textit{i}.\\textit{e}., the question-only and targeted VQA model) in an adversarial manner~\\cite{goodfellow2014generative, chen2018zero}, \\textit{i}.\\textit{e}., minimizing the loss of VQA model while maximizing the loss of question-only model. Since these two models are typically designed to share the same question encoder, these adversary-based methods aim to reduce the language biases by learning bias-neutral question representations. Unfortunately, this adversarial training scheme brings significant noise into gradient calculations, which results in an unstable training process~\\cite{grand2019adversarial}. 2) \\emph{Fusion-based models}~\\cite{cadene2019rubi, clark2019don, mahabadi2019simple, niu2021counterfactual, han2021greedy}: They late fuse the predicted answer distributions of the two models, and derive training gradients based on fused answer distribution. The design philosophy of these fusion-based models, is to let the targeted VQA model focuses more on samples, which can't be easily answered correctly by the question-only model.\n\n\nAlthough the ensemble-based methods have dominated the performance on these diagnostic benchmarks, it is worth noting that current methods fail to equip themselves with two indispensable characteristics of an ideal VQA model: 1) \\textbf{Visual-explainable ability}: The VQA model should rely on the right visual regions when making decisions, \\textit{i}.\\textit{e}., right for the right reasons~\\cite{ross2017right}. As shown in Fig.~\\ref{fig:1} (a), although both two models can predict the correct answer ``\\texttt{surfing}\", they actually refer to different visual reference regions when making their respective answer predictions. 2) \\textbf{Question-sensitive ability}: The VQA model should be sensitive to the linguistic variations in questions. As shown in Fig.~\\ref{fig:1} (b), for two questions with similar sentence structure (\\textit{e}.\\textit{g}.,  only replacing the word ``\\texttt{luggage}\" with ``\\texttt{bus}\"), if the meanings of the two questions are different, the model should perceive the discrepancy and make corresponding predictions.\n\n\\begin{figure}[tbp]\n\t\\centering\n\t\\includegraphics[width=0.98\\linewidth]{figures\/V-CSS_Q-CSS.pdf}\n\t\\vspace{-0.5em}\n\t\\caption{(a): A training sample from VQA-CP. (b): The synthesized training sample by V-CSS. It masks ciritcal objects (\\textit{e}.\\textit{g}., \\inlineimage{figures\/tie.jpg}) in image and assigns different GT answers (\\textit{e}.\\textit{g}., ``\\texttt{not green}\"). (c): The synthesized training sample by Q-CSS. It replaces critical words (\\textit{e}.\\textit{g}., ``\\texttt{tie}\") with special token ``[\\texttt{MASK}]\" in the question and assigns different GT answers (\\textit{e}.\\textit{g}., ``\\texttt{not green}\").}\n\t\\label{fig:3}\n\\end{figure}\n\n\nIn this paper, we propose a novel model-agnostic \\emph{Counterfactual Samples Synthesizing and Training} (CSST) strategy. CSST serves as a plug-and-play component to improve VQA models' visual-explainable and question-sensitive abilities, even for complex ensemble-based models. Specifically, CSST is composed of two parts: Counterfactual Samples Synthesizing (CSS) and Counterfactual Samples Training (CST).\n\nFor each original VQA training sample, CSS can generate a corresponding counterfactual sample. As shown in Fig.~\\ref{fig:3}, CSS consists of two  different types of sample synthesizing mechanisms: \\emph{V-CSS} and \\emph{Q-CSS}. For V-CSS, it synthesizes counterfactual images by masking critical objects in the original image. By ``critical\", we mean that these objects are important for answering a certain question. For example, object~\\inlineimage{figures\/tie.jpg}~(\\textit{i}.\\textit{e}., green tie) is a critical object for question ``\\texttt{What color is the man's tie?}\". Then, the counterfactual image and original question compose a new image-question (VQ) pair. For Q-CSS, it synthesizes counterfactual questions by replacing critical words in original question with a special token ``[\\texttt{MASK}]\". Similarly, the counterfactual question and original image compose a new VQ pair. Meanwhile, to avoid expensive manual annotations, we design a dynamic answer assigning mechanism to approximate ground-truth answers for all synthesized VQ pairs (\\textit{e}.\\textit{g}., ``\\texttt{not green}\").\n\nBased on our general sample synthesizing mechanism CSS, we also propose a new CST training strategy for VQA, which helps the VQA models to focus on the critical objects and words, \\textit{i}.\\textit{e}., improving VQA models' visual-explainable and question-sensitive abilities. Specifically, the CST consists of two different training objectives: XE loss and contrastive loss (cf. Fig.~\\ref{fig:2} (c)). For XE training, we use CSS to generate a counterfactual sample for each original sample, and feed both two samples into a same VQA model. After training with these complementary sample pairs simultaneously, the VQA models are forced to focus on masked critical objects and words. For contrastive training, we sample (or generate) a set of positive\/negative samples for each original sample, and propose two variants of \\emph{supervised contrastive loss}~\\cite{khosla2020supervised} for optimization. After training with the contrastive objective, the VQA models can further distinguish the original samples and superficially similar counterfactual ones. \n\nFor more effective contrastive training, we further design a novel positive and negative sample selection mechanism based on CSS. For each original training sample, we regard all samples with the same question type and ground-truth answers as \\emph{positive} samples, and utilize CSS to generate corresponding counterfactual samples of each positive sample as \\emph{negative} samples. Since CSS can effectively find the critical objects in images or words in questions, these positive and negative samples are superficially similar but semantically different. Compared to directly generating negative samples from the original sample~\\cite{liang2020learning}, our strategy can significantly improve the diversity of visual content at contrastive training, which is empirically important for contrastive training.\n\n\n\n\nWe evaluated CSST on two challenging datasets: VQA-CP and GQA-OOD. Extensive ablations have demonstrated the effectiveness of CSST. CSST can be seamlessly incorporated into any VQA architecture, which not only improves their both visual-explainable and question-sensitive abilities, but also consistently boosts their performance. Particularly, by building on top of model LMH+SAR~\\cite{clark2019don,si2021check}, we achieve new record-breaking results on both benchmarks.\n\nThis paper is a substantial and systematic extension of our previous CVPR version~\\cite{chen2020counterfactual} with several improvements: 1) We design two variants of supervised contrastive loss for VQA. 2) We extend CSS into contrastive training (\\textit{i}.\\textit{e}., CSST), and design a novel positive and negative sample selection mechanism based on CSS. 3) We make two improvements on CSS, which achieves better performance than the original version. 4) We achieve new state-of-the-art performance on all benchmarks, and conduct more experiments to further verify the generalization and superiority of CSST, including more datasets, backbones, ablations, and visualizations.\n\n\n\\section{Related Work}\n\n\\noindent\\textbf{Visual Question Answering (VQA).} VQA, \\textit{i}.\\textit{e}., understanding both the visual content and natural language question, and making the answer prediction, is an important research task in both computer vision and natural language processing. Benefits from the deep learning techniques and large-scale VQA datasets~\\cite{antol2015vqa,goyal2017making,johnson2017clevr,hudson2019gqa}, VQA has realized impressive progress and achieved good results in real images. With the advance in large-scale multimodal representation pretraining, today's VQA performance is mainly dominated by pretrained multimodal BERT models~\\cite{tan2019lxmert,su2020vl,li2020oscar,chen2020uniter}.\n\n\n\\noindent\\textbf{Language Biases in VQA.}\nDespite VQA is a typical multimodal task, a large body of research~\\cite{jabri2016revisiting, agrawal2016analyzing, zhang2016yin, goyal2017making} has shown the existence of language biases in VQA. There are two main solutions to reduce language biases:\n\n\\textit{1. Balancing Datasets to Reduce Biases.} The most straightforward solution is to create more balanced datasets. For example, Zhang~\\textit{et}.\\textit{al}.~\\cite{zhang2016yin} and Goyal~\\textit{et}.\\textit{al}.~\\cite{goyal2017making} collected complementary images with opposite answers for all questions. Although these ``balanced\" datasets have reduced biases to some extent, the statistical biases from questions still can be leveraged~\\cite{agrawal2018don}. As shown in VQA-CP, the performance of numerous models drop significantly compared to these ``balanced\" datasets. In this paper, CSST follows the same spirit of dataset balancing and trains VQA models with more complementary samples. Especially, we don't need any extra manual annotations.\n\n\\textit{2. Designing Models to Reduce Biases.} Another solution is to design specific debiasing models. So far, the most effective debiasing models for VQA are ensemble-based methods~\\cite{ramakrishnan2018overcoming, grand2019adversarial, belinkov2019don, cadene2019rubi, clark2019don, mahabadi2019simple, han2021greedy}. In this paper, we propose a novel CSST strategy, which can be seamlessly incorporated into the ensemble-based models to further reduce the biases.\n\n\\noindent\\textbf{Visual-Explainable Ability in VQA Models.}\nTo improve visual-explainable ability, early works~\\cite{qiao2018exploring, liu2017attention, zhang2019interpretable} directly apply human attention as supervision to guide the models' attention maps. However, since the existence of strong biases, even with appropriate attention maps, the remaining layers of network may still disregard the visual signal~\\cite{selvaraju2019taking}. Thus, some recent works~\\cite{selvaraju2019taking, wu2019self} utilize Grad-CAM~\\cite{selvaraju2017grad} to obtain the private contribution of each object to correct answers, and encourage the rank of all object contributions to be consistent with human annotations. Unfortunately, these models have two inherent drawbacks: 1) They need extra human annotations. 2) The training is not end-to-end.\n\n\\noindent\\textbf{Question-Sensitive Ability in VQA Models.}\nIf VQA systems really ``understand\" the question, they should be sensitive to the linguistic variations in question. Surprisingly, to the best of our knowledge, there is only a few works~\\cite{shah2019cycle} have studied the influence of linguistic variations in VQA. Specifically, it designs a cycle-consistent loss between two dual tasks, and utilizes sampled noises to generate diverse questions. However, Shah~\\textit{et}.\\textit{al}.~\\cite{shah2019cycle} only consider the robustness to different rephrasings of the same question. In contrast, we also encourage models to perceive the difference of questions when changing some critical words.\n\n\\noindent\\textbf{Data Augmentation in VQA.} Some concurrent works and following works after our CSS~\\cite{chen2020counterfactual} also utilize data augmentation to improve VQA performance. Specifically, there are two directions: 1) \\textit{Images Augmentation.} Almost all other image augmentation works~\\cite{agarwal2019towards,pan2019question,gokhale2020mutant} resort to GAN~\\cite{goodfellow2014generative} to generate corresponding counterfactual images, which is notorious for unstable training. Meanwhile, photo-realistic image generation itself is still an open challenge. 2) \\textit{Questions Augmentation.} The most straightforward question augmentation strategy in VQA~\\cite{kant2020contrast} is back-translation~\\cite{edunov2018understanding}, which translating a sentence from one language to another and then translating it back using a pair of translation models.\n\nIn contrast, in this paper, our proposed CSST only masks critical objects or words, which is easier and more adoptable\n\n\n\\noindent\\textbf{Contrastive Training in VQA.} Contrastive learning techniques have achieved unprecedented success in vision community, especially for self-supervised representation learning~\\cite{oord2018representation, he2020momentum, chen2020simple}. The core idea of contrastive learning is to maximize the mutual information between the input samples and positive samples, and minimize the one between negative samples. Currently, there are several VQA works~\\cite{liang2020learning, kant2020contrast, zhu2020overcoming} also utilize contrastive training to distinguish the original training samples and superficially similar counterparts. Different from all existing works, we equip our CSS mechanism into positive and negative selection for contrastive training, which not only increases the sample diversity, but also meets the semantic requirements.\n\n\n\n\\section{Approach}\n\nWe follow the common formulation and regard the VQA task as a multi-label classification problem. Without loss of generality, given a dataset $\\mathcal{D} = \\{I_i, Q_i, a_i \\}^N_i$ consisting of triplets of images $I_i \\in \\mathcal{I}$, questions $Q_i \\in \\mathcal{Q}$ and answer sets $a_i \\in \\{a_i^j | a_i^j \\in \\mathcal{A}\\}$, the VQA task learns a mapping $f_{vqa}: \\mathcal{I} \\times \\mathcal{Q} \\rightarrow [0, 1]^{|\\mathcal{A}|}$, which produces an answer distribution given image-question pair. $\\mathcal{I}$, $\\mathcal{Q}$, and $\\mathcal{A}$ denote the set of images questions, and answers, respectively. For each ground-truth answer $a_i^j \\in a_i$, it has a soft target score $t_i^j$, and we use the same ground-truth soft target scores as in prior works~\\cite{anderson2018bottom}. For simplicity, we omit subscript $i$ in the following sections.\n\nIn this section, we first introduce the UpDn baseline~\\cite{anderson2018bottom} and a typical ensemble-based framework in Sec.~\\ref{sec:3.1}. Then, we introduce the details of Counterfactual Samples Synthesizing (CSS) in Sec.~\\ref{sec:3.2}. Last, we introduce the details of Counterfactual Samples Training (CST) in Sec.~\\ref{sec:3.3}.\n\n\n\\subsection{Preliminaries} \\label{sec:3.1}\n\n\\noindent\\textbf{Bottom-Up Top-Down (UpDn) Model~\\cite{anderson2018bottom}.}\nFor each image $I$, the UpDn uses an image encoder $e_v$ to output a set of object features: $\\bm{V} = \\{\\bm{v}_1, ..., \\bm{v}_{n_v}\\}$, where $\\bm{v}_i$ is $i$-th object feature. For each question $Q$, the UpDn uses a question encoder $e_q$ to output a set of word features: $\\bm{Q} = \\{\\bm{w}_1, ..., \\bm{w}_{n_q}\\}$, where $\\bm{w}_j$ is $j$-th word feature. Then, both $\\bm{V}$ and $\\bm{Q}$ are fed into the model $f_{vqa}$ to predict answer distributions:\n\\begin{equation} \\label{eq:p_vqa}\nP_{vqa}(\\bm{a}|I, Q) = f_{vqa}(\\bm{V}, \\bm{Q}).\n\\end{equation}\nTypically, model $f_{vqa}$ contains a soft attention mechanism, and it is trained with a (binary) cross-entropy (XE) loss.\n\n\\begin{algorithm}[t]\n\t\\caption{Ensemble-based Model (fusion-based)}\\label{alg:VQA}\n    \\textcolor{gray}{\\textbf{Inputs:} original training sample ($I$, $Q$, $a$), $update\\_cond$ is a control condition for parameter updates. \\\\\n    \\textbf{Outputs:} images features $\\bm{V}$, question features $\\bm{Q}$, and predicted answer distribution $P_{vqa}(\\bm{a})$.}\n\t\\begin{algorithmic}[1]\n\t\t\\Function {\\textbf{VQA}}{$I, Q, a, update\\_cond$}\n\t\t\\State $ \\bm{V} \\leftarrow e_v(I) $\n\t\t\\State $ \\bm{Q} \\leftarrow e_q(Q) $\n\t\t\\State $ P_{vqa}(\\bm{a}) \\leftarrow f_{vqa}(\\bm{V}, \\bm{Q}) $\n\t\t\\If{$update\\_cond$}\n\t\t\\State $ P_{q}(\\bm{a}) \\leftarrow f_{q}(\\bm{Q})$      \\textcolor{gray}{\\Comment{question-only model}}\n\t\t\\State $ \\hat{P}_{vqa}(\\bm{a}) \\leftarrow M\\left(P_{vqa}(\\bm{a}), P_{q}(\\bm{a})\\right)  $\n\t\t\\State $ L_{\\text{XE}} \\leftarrow \\text{XE}(\\hat{P}_{vqa}(\\bm{a}), a)$ \n\t\t\\textcolor{gray}{\\Comment{update parameters}}\n\t\t\\EndIf \n\t\t\\State \\textbf{return} $\\bm{V}, \\bm{Q}, P_{vqa}(\\bm{a})$\n\t\t\\EndFunction\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\n\\noindent\\textbf{Ensemble-Based Models.} As we discussed in Sec.~\\ref{sec:intro}, the ensemble-based models can be grouped into two sub-types: \\emph{adversary-based} and \\emph{fusion-based}. Since adversary-based models~\\cite{ramakrishnan2018overcoming, grand2019adversarial, belinkov2019don} always suffer severe unstable training and relatively worse performance, in this section, we only introduce the typical fusion-based framework~\\cite{cadene2019rubi, clark2019don, mahabadi2019simple}. As shown in Algorithm~\\ref{alg:VQA}, they introduce an auxiliary question-only model $f_q$ which takes $\\bm{Q}$ as input and predicts answers:\n\\begin{equation}\nP_{q}(\\bm{a}|Q) = f_{q}(\\bm{Q}).\n\\end{equation}\n\nThen, they combine these two answer distributions and obtain a new answer distribution $\\hat{P}_{vqa}(\\bm{a})$ by a function $M$:\n\\begin{equation} \\label{eq:eq3}\n\\hat{P}_{vqa}(\\bm{a}|I, Q) = M \\left( P_{vqa}(\\bm{a}|I, Q), P_{q}(\\bm{a}|Q) \\right).\n\\end{equation}\n\nIn the training stage, the XE loss is computed based on the fused answer distribution $\\hat{P}_{vqa}(\\bm{a})$ and the training gradients are backpropagated through $f_{vqa}$ and $f_q$. In the test stage, only the model $f_{vqa}$ is used as plain VQA models.\n\n\\subsection{Counterfactual Samples Synthesizing (CSS)} \\label{sec:3.2}\n\n\nThe overall structure of the CSS training scheme is shown in Algorithm~\\ref{alg:css}. Specifically, for any \\textbf{VQA} model, given an original training sample $(I, Q, a)$, CSS can generate two types of counterfactual samples: $(I^-, Q, a^-)$ by V-CSS or $(I, Q^-, a^-)$ by Q-CSS. In the following, we mainly introduce the details of V-CSS and Q-CSS. As shown in Algorithm~\\ref{alg:css}, for each specific training sample, we only use one certain synthesizing mechanism, and $\\delta$ is the trade-off weight (see Fig.~\\ref{fig:5} (c) for more details about the influence of different $\\delta$).\n\n\n\\subsubsection{V-CSS} \\label{sec:v-css}\n\nWe sequentially introduce all steps of V-CSS following its execution path (\\textit{i}.\\textit{e}., line 5 to 8 in Algorithm~\\ref{alg:css}), which consists of four main steps: initial objects selection (\\textsc{IO\\_Sel}), object local contributions calculation, critical objects selection (\\textsc{CO\\_Sel}), and dynamic soft answer assigning (\\textsc{DSA\\_Ass}).\n\n\n\\textbf{1. Initial Objects Selection (\\textsc{IO\\_Sel}).} In general, for any specific QA pair $(Q, a)$, only a few objects in image $I$ are related. To narrow the scope of critical objects selection, we first construct a smaller object set $\\mathcal{I}$, and assume all objects in $\\mathcal{I}$ are possibly important in answering this specific question. Since we lack manual annotations about the critical objects for each sample, we followed~\\cite{wu2019self} to extract objects which are highly related to the QA. Specifically, we first assign POS tags to each word in the QA using the spaCy POS tagger~\\cite{honnibal2017spacy} and extract nouns in QA. Then, we calculate the cosine similarity between the GloVe~\\cite{pennington2014glove} embeddings of object categories\\footnote{For the VQA-CP dataset, we followed prior works~\\cite{wu2019self} to utilize these object category annotations, and the influence of different sizes of initial objects are also illustrated in Fig.~\\ref{fig:5} (a). For the GQA-OOD dataset, for fair comparisons with others, we skip this step for all experiments.} and the extracted nouns, the similarity scores between all objects in $I$ and the QA are denoted as $\\mathcal{SIM}$. We select $|\\mathcal{I}|$ objects with the highest $\\mathcal{SIM}$ scores as the initial object set $\\mathcal{I}$.\n\n\\textbf{2. Object Local Contributions Calculation.} After obtaining the initial object set $\\mathcal{I}$, we start to calculate the local contribution of each object to the predicted probability of ground-truth answer. Following recent works~\\cite{jain2019attention, selvaraju2019taking, wu2019self} which utilize modified Grad-CAM~\\cite{selvaraju2017grad} to derive the local contribution of each participant, we calculate the contribution of $i$-th object feature to the ground-truth answer $a$ as:\n\\begin{equation} \\label{eq:object_gradcam}\ns(a, \\bm{v}_i) = \\mathcal{S}(P_{vqa}(a), \\bm{v}_i) \\coloneqq \\left(\\nabla_{\\bm{v}_i} P_{vqa}(a) \\right)^T\\mathbf{1},\n\\end{equation}\nwhere $P_{vqa}(a)$ is the predicted answer probability of the ground-truth answer a,  $\\bm{v}_i$ is $i$-th object feature, and $\\mathbf{1}$ is an all-ones vector. Obviously, if the score $s(a, \\bm{v}_i)$ is higher, the contributions of object $\\bm{v}_i$ to answer $a$ is larger.\n\n\n\\begin{algorithm}[tbp]\n\t\\caption{Counterfactual Samples Synthesizing (CSS)}\\label{alg:css}\n\t\\textcolor{gray}{\\textbf{Inputs:} original training sample ($I$, $Q$, $a$), and a typical VQA model \\textbf{VQA}.  \\\\\n    \\textbf{Outputs:} counterfactual training sample ($I^-$, $Q$, $a^-$) from V-CSS or  ($I$, $Q^-$, $a^-$) from Q-CSS.}\n\t\\begin{algorithmic}[1]\n\t\t\\Function {\\textbf{CSS}}{$I, Q, a$}\n\t\t\\State $ \\bm{V}, \\bm{Q}, P_{vqa}(\\bm{a}) \\leftarrow \\textbf{\\text{VQA}}(I, Q, a, \\text{False})$\n\t\t\\State $ cond \\sim U[0, 1]$\n\t\t\\If {$cond \\geq \\delta $}\n\t\t\\textcolor{gray}{\\Comment{execute V-CSS}}\n\t\t\\State $ \\mathcal{I} \\leftarrow  \\textsc{IO\\_Sel}(I, Q) $\n\t\t\\State $ s(a, \\bm{v}_i) \\leftarrow \\mathcal{S}(P_{vqa}(a), \\bm{v}_i)$\n\t\t\\State $ I^+, I^- \\leftarrow \\textsc{CO\\_Sel}(\\mathcal{I}, \\{s(a, \\bm{v}_i) \\}) $\n\t\t\\State $ a^- \\leftarrow \\textsc{DSA\\_Ass}(I^+, Q, \\textbf{\\text{VQA}}, a) $\n\t\t\\State \\textbf{return} ($I^-$, $Q$, $a^-$)\n\t\t\\Else \n\t\t\\textcolor{gray}{\\Comment{execute Q-CSS}}\n\t\t\\State $ s(a, \\bm{w}_i) \\leftarrow \\mathcal{S}(P_{vqa}(a), \\bm{w}_i) $\n\t\t\\State $ Q^+, Q^- \\leftarrow \\textsc{CW\\_Sel}(\\{s(a, \\bm{w}_i)\\})$\n\t\t\\State $ a^- \\leftarrow \\textsc{DSA\\_Ass}(I, Q^+, \\textbf{\\text{VQA}}, a) $\n        \\State \\textbf{return} ($I$, $Q^-$, $a^-$)\n\t\t\\EndIf\n\t\t\\EndFunction\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\n\n\\textbf{3. Critical Objects Selection (\\textsc{CO\\_Sel}).} After obtaining the private contribution scores $s(a, \\bm{v}_i)$ for all objects in $\\mathcal{I}$, we select the top-K objects with highest scores as the critical object set $I^+$. The K is a dynamic number for each image, which is the smallest number meets Eq.~\\eqref{eq:topk_objects}:\n\\begin{equation} \\label{eq:topk_objects}\n\\sum_{\\bm{v}_i \\in I^+} \\exp(s(a, \\bm{v}_i)) \/ \\sum_{\\bm{v}_j \\in \\mathcal{I}} \\exp(s(a, \\bm{v}_j))  > \\eta,\n\\end{equation}\nwhere $\\eta$ is a constant, we set $\\eta=0.65$ in all experiments (See Fig.~\\ref{fig:5} for more details about the influence of dynamic K setting). Since the visual features $\\bm{V} = \\{\\bm{v}_i\\}$ are extracted from a pretrained Faster R-CNN, \\textit{i}.\\textit{e}., they always have a lot of repetitions due to overlapped image regions. To avoid visual clue leakage from the other object visual features~\\cite{lu202012}, we also regard the object that overlaps any critical objects by 60\\% IoU or more as critical objects. Then, the counterfactual input $I^-$ is the absolute complement of set $I^+$ in set $I$, \\textit{i}.\\textit{e}., $I^- = I \\backslash I^+$. We show an example of $I$, $I^+$ and $I^-$ in Fig.~\\ref{fig:4}. \n\n\n\\textbf{4. Dynamic Soft Answer Assigning (\\textsc{DSA\\_Ass}).} Given the counterfactual visual input $I^-$ and original question $Q$, we compose a new VQ pair ($I^-, Q$). To assign ground-truth answers for the VQ pair  ($I^-, Q$), we design a new dynamic soft answer assigning (\\textsc{DSA\\_Ass}) mechanism. The details of \\textsc{DSA\\_Ass} are shown in Algorithm~\\ref{alg:daass}. Specifically, we first feed another VQ pair ($I^+, Q$) into the \\textbf{VQA} model, and obtain the predicted answer distribution $P^+_{vqa}(\\bm{a})$. We directly use the same set of ground-truth answer categories as the pseudo ground-truth for the corresponding counterfactual sample. Meanwhile, we re-assign the ground-truth probability for each answer category of $a^-$, \\textit{e}.\\textit{g}., the $i$-th answer category $a^{i-}$ is set to $P^{gt}(a^{i-}) = 1 - P_{vqa}^+(a^{i-})$. In an extreme case, if the model predicts all ground-truth answers correctly for the VQ pair ($I^+, Q$) with 100\\% probabilities, then $a^-$ is a $\\emptyset$, \\textit{i}.\\textit{e}., zero for all answer candidates. The basic motivation is that if the current model can predict ground-truth answers for ($I^+, Q$) (\\textit{i}.\\textit{e}., $I^+$ contains critical objects and $I^-$ not), the ground-truth for ($I^-, Q$) should not contain original ground-truth answers anymore, \\textit{e}.\\textit{g}., ``\\texttt{not green}\" in Fig~\\ref{fig:2}. Similarly, the higher confidence for ($I^+, Q$), the less probability for this answer as the ground-truth for ($I^-, Q$).\n\n\\begin{algorithm}[tbp]\n\t\\caption{Dynamic Soft Answer Assigning (\\textsc{DSA\\_{Ass}})}\\label{alg:daass}\n\t\\textcolor{gray}{\\textbf{Inputs:} VQ pair ($I^+$, $Q^+$), a typical VQA model \\textbf{VQA}, and original ground-truth answer set $a$.  \\\\\n    \\textbf{Outputs:} automatically assigned pseudo gt answer set $a^-$, and corresponding ground-truth soft target scores $\\{t^{i-}\\}$.}\n\t\\begin{algorithmic}[1]\n\t\t\\Function {$\\textsc{DA\\_Ass}$}{$I^+, Q^+, \\textbf{\\text{VQA}}, a$}\n\t\t\\State  $ \\_, \\_, P_{vqa}^+(\\bm{a}) \\leftarrow \\textbf{\\text{VQA}}(I^+, Q^+, a, \\text{False}) $\n\t\t\\State $ a^- \\coloneqq a$\n\t\t\\textcolor{gray}{\\Comment{same gt answer set}}\n\t\t\\For{$a^{i-} ~\\text{in}~ a^-$}\n\t\t\\State $t^{i-} \\coloneqq 1 - P_{vqa}^+(a^{i-})$\n\t\t\\textcolor{gray}{\\Comment{soft answer}}\n\t\t\\EndFor\n\t\t\\State \\textbf{return} $a^-$\n\t\t\\EndFunction\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{figure}[tbp]\n\t\\centering\n\t\\includegraphics[width=0.95\\linewidth]{figures\/IQ+-.pdf}\n\t\\vspace{-1em}\n\t\\caption{An informal illustration example of the $I^+$, $I^-$, $Q^+$, and $Q^-$ in CSS. For $I^+$ and $I^-$, they are two mutual exclusive object sets. For $Q^+$ and $Q^-$, we show the example when word ``\\texttt{kite}\" is selected as critical word. For clarity, we omit some objects. \n\t}\n\t\\label{fig:4}\n\\end{figure}\n\n\n\\subsubsection{Q-CSS}\nAll steps in Q-CSS are similar to V-CSS. Following its execution path (\\textit{i}.\\textit{e}., line 11 to 13 in Algorithm~\\ref{alg:css}), it consists of word local contribution calculation, critical words selection (\\textsc{CW\\_Sel}), and dynamic soft answer assigning (\\textsc{DSA\\_Ass}).\n\n\\textbf{1. Word Local Contribution Calculation.} Similar with the V-CSS (cf. Eq.~\\eqref{eq:object_gradcam}), we calculate the contribution of $i$-th word feature to the ground-truth answer $a$ as:\n\\begin{equation} \\label{eq:word_gradcam}\ns(a, \\bm{w}_i) = \\mathcal{S}(P_{vqa}(a), \\bm{w}_i) \\coloneqq (\\nabla_{\\bm{w}_i} P_{vqa}(a))^T\\mathbf{1}.\n\\end{equation}\n\n\\textbf{2. Critical Words Selection (\\textsc{CW\\_Sel}).}  In this step, we first extract question-type\\footnote{We slightly abuse ``question-type\" here. For VQA-CP, the question-type denotes the default question-type annotations in the original dataset. For GQA-OOD, the question-type denotes the annotated local groups, which can be easily mapped back to each question. \\label{qtype}} words for each question $Q$ (\\textit{e}.\\textit{g}., ``\\texttt{what color}\" in Fig.~\\ref{fig:4} is the question-type of question ``\\texttt{What color is the kite?}\"). Then, we select top-K words with the highest scores from the remaining sentence (except question-type words) as critical words. The counterfactual question $Q^-$ is the sentence by replacing all critical words in $Q$ with a special token ``[\\texttt{MASK}]\". Meanwhile, the $Q^+$ is the sentence by replacing all other words (except question-type and critical words) with ``[\\texttt{MASK}]\". We show an example of $Q$, $Q^+$, and $Q^-$ in Fig.~\\ref{fig:4}.\n\n\\textbf{3. Dynamic Soft Answer Assigning (\\textsc{DSA\\_Ass}.)} This step is identical to the \\textsc{DSA\\_Ass} in V-CSS, \\textit{i}.\\textit{e}., Algorithm~\\ref{alg:daass}. For Q-CSS, the input for \\textsc{DSA\\_Ass} is the VQ pair $(I, Q^+)$.\n\n\\subsubsection{Highlights of Two Improvements on CSS~\\cite{chen2020counterfactual}} \\label{sec:3.2.3}\nCompared to the initial CSS in the conference version~\\cite{chen2020counterfactual}, we make two important modifications and improvements:\n\\begin{enumerate}\n    \\item \\textbf{Object Overlapping in \\textsc{CO\\_Sel}.} Different from the words features in questions, which are relatively ``independent\", object features from the same image always have a lot of repetitions due to their overlapped image regions. We avoid this visual clue leakage by considering the IoU overlaps between all objects. This strategy helps models rely more on the calculated critical objects. \n    \n    \\item \\textbf{``\\emph{Soft}\" Answer Assigning in \\textsc{DSA\\_Ass}.} In initial CSS work~\\cite{chen2020counterfactual}, we select top-K predictions based on $P^+_{vqa}$ (cf. Algorithm~\\ref{alg:daass}), and assign ground-truth answers based on these top predictions. Compared to this ``hard\" answer selection, \\textsc{DSA\\_Ass} is more robust to the selection of hyperparameter $K$, and is more sensitive to perceive the changes of answer predictions (\\textit{i}.\\textit{e}., $P^+_{vqa}$).\n\n\\end{enumerate}\n\nTo distinguish these two types of CSS, we denote the current improved version as CSS$^+$ in the following sections (See results in Sec.~\\ref{sec:exp} for comparisons between CSS \\& CSS$^+$).\n\n\n\\subsection{Counterfactual Samples Training (CST)} \\label{sec:3.3}\n\nTo further benefit from the counterfactual samples generated by CSS, we propose a CST strategy for model training (cf. Algorithm~\\ref{alg:cst}). Specifically, it consists of a cross-entropy (XE) training loss and a contrastive (CR) training loss.\n \n\\subsubsection{XE Training on Counterfactual Samples}\n\nFor XE training, we directly follow other state-of-the-art VQA models (\\textit{e}.\\textit{g}., UpDn~\\cite{anderson2018bottom}) and use binary cross-entropy loss as training objective, \\textit{i}.\\textit{e}., \n\\begin{equation}\n    \\begin{split}\n        L_{\\text{XE}} & = - \\sum^M_i \\left[t^i \\log \\sigma(\\hat{P}_{vqa}(\\bm{a}|I, Q)) + \\right. \\\\\n        & \\left. + (1 - t^i) \\log (1 - \\sigma(\\hat{P}_{vqa}(\\bm{a}|I, Q))) \\right],\n    \\end{split}\n\\end{equation}\nwhere $\\sigma$ denotes the sigmoid activation function, and $t^i$ is the soft target score of $i$-th ground-truth answer for this training sample (cf. line 5 in Algorithm~\\ref{alg:daass}). Different from the most prevalent VQA framework which only uses original samples in the XE training (cf. Fig.~\\ref{fig:2} (a)), in our XE training, we feed both original training samples and their corresponding counterfactual samples into the same \\textbf{VQA} model (\\textit{i}.\\textit{e}., line 3 to 5 in Algorithm~\\ref{alg:cst}).\n\n\n\\begin{algorithm}[tbp]\n    \\textcolor{gray}{\\textbf{Inputs:} original training sample ($I$, $Q$, $a$), and a typical VQA model \\textbf{VQA}, and \\textbf{CSS}. \n   \n    }\n\t\\caption{Counterfactual Samples Training (CST)}\\label{alg:cst}\n\t\\begin{algorithmic}[1]\n\t\t\\Function {\\textbf{CST}}{$I, Q, a$} \n        \\State \\textcolor{gray}{\\# XE training} \n        \\State $(I^-, Q, a^-) \\leftarrow \\textbf{\\text{CSS}}(I, Q, a)$\n\t\t\\textcolor{gray}{\\Comment{V-CSS for example}}\n        \\State $ \\_, \\_, P_a \\leftarrow \\textbf{\\text{VQA}}(I, Q, a, \\text{True})$\n        \\textcolor{gray}{\\Comment{original sples.}}\n        \\State $ \\_, \\_, \\_ \\leftarrow \\textbf{\\text{VQA}}(I^-, Q, a^-, \\text{True})$\n        \\textcolor{gray}{\\Comment{counterfact. sples.}}\n        \\\\\n        \\State \\textcolor{gray}{\\# samples selection} \n\t\t\\State $(I_p, Q_p, a) \\leftarrow \\textsc{POS\\_Sel}(I, Q, a) $\n\t\t\\textcolor{gray}{\\Comment{positive sples.}}\n\t\t\\State $\\{(I_n^i, Q_n^i, \\_)\\} \\leftarrow \\textsc{NEG\\_Sel}(I, Q, a) $ \n        \\textcolor{gray}{\\Comment{negative sples.}}\n        \\\\\n        \\State \\textcolor{gray}{\\# contrastive training}\n        \\State $\\_, \\_, P_p \\leftarrow \\textbf{\\text{VQA}} (I_p, Q_p, _, \\text{False})$\n        \\textcolor{gray}{\\Comment{for pos. sples.}}\n        \\State $\\_, \\_, P^i_n \\leftarrow \\textbf{\\text{VQA}} (I_n^i, Q_n^i, _, \\text{False})$\n        \\textcolor{gray}{\\Comment{for all neg. sples.}}\n        \\State $L_{\\text{CR}} \\leftarrow \\text{CR}(P_a, P_p, \\{P_n^i\\})$\n        \\textcolor{gray}{\\Comment{CR training}}\n\t\t\\EndFunction\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\n\n\\begin{figure*}[htbp]\n\t\\centering\n\t\\includegraphics[width=0.95\\linewidth]{figures\/ablative_studies_v2.pdf}\n\t\\caption{\\textbf{Ablations}. Accuracies (\\%) on VQA-CP v2 test set of different hyperparameters settings of V-CSS$^+$ or Q-CSS$^+$. (a) The results of different size of $\\mathcal{I}$ and critical objects in V-CSS$^+$. All results come from model LMH+V-CSS$^+$. (b) The results of different size of critical words in Q-CSS$^+$. All results come from model LMH+Q-CSS$^+$. (c) The results of different proportion $\\delta$ between V-CSS$^+$ and Q-CSS$^+$. All results come from model LMH+V-CSS$^+$+Q-CSS$^+$.}\n\t\\label{fig:5}\n\\end{figure*}\n\n\n\\subsubsection{Contrastive Training on Counterfactual Samples}\nFor contrastive training, we propose a novel and effective positive and negative samples selection mechanism based on our CSS, and two variants of contrastive loss for VQA.\n\n\\noindent\\textbf{Positive \\& Negative Samples Selection Strategies.} Without loss of generality, we regard each original training sample as the \\emph{anchor} sample, and we regard all samples with the same question-type\\footref{qtype} and ground-truth answer categories as the candidate positive set\\footnote{For those samples do not have any sample meets the two requirements, we directly regard the original sample as the candidate positive set, and the proportions are quite small, \\textit{e}.\\textit{g}., $\\approx$ 1\\% for VQA-CP v2.}. In each training step, we randomly sample one sample from the candidate set as the \\emph{positive} sample (\\textit{i}.\\textit{e}., \\textsc{POS\\_Sel} in Algorithm~\\ref{alg:cst}). For each positive sample, we compose four different types of \\emph{negative} samples, which can be categorized into two groups based on their sampling strategies. The first group of negative samples is the counterfactual samples of the positive samples corresponding to V-CSS and Q-CSS, respectively. The second group of negative samples is randomly sampled samples: including 1) a random sampled sample with the same question-type\\footref{qtype} but different ground-truth answer; and 2) a composed sample by replacing the original image with a randomly sampled image from the same batch. All four types of negative sample selection strategies constitute the \\textsc{NEG\\_Sel} step in Algorithm~\\ref{alg:cst}. Detailed quantitative results of different samples are reported in TABLE~\\ref{tab:pos&neg}.\n\n\\emph{Advantages over Existing Positive\/Negative Sampling Solutions.} Currently, there are two types of positive and negative sample selection strategies. The first type also builts on top of our CSS~\\cite{liang2020learning}. For each sample $(I, Q)$, they directly use the $(I^-, Q)$ and $(I, Q^-)$ in our \\textbf{CSS} (cf. Algorithm~\\ref{alg:css}) as negative samples. The second type is directly randomly replaced an image or question~\\cite{kant2020contrast} to compose negative samples (\\textit{e}.\\textit{g}., $(I_{\\text{rand}}, Q)$ or $(I, Q_{\\text{rand}})$). Instead, we benefit from the ability of CSS, and generate counterfactual samples for positive samples (our first group), \\textit{i}.\\textit{e}., negative samples consists of $(I_{pos}^-, Q_{pos})$ and $(I_{pos}, Q_{pos}^-)$. Compared to the first solution, our strategy significantly increases the \\emph{diversity} of visual contents and questions in positive and negative samples, \\textit{i}.\\textit{e}., in each training epoch, we can use total different counterfactual samples as negative samples. Compared to the second solution, our strategy based on CSS can help models to focus more on fine-grained differences (the masked critical objects and words) under the contrastive training. \n\n\n\\noindent\\textbf{Two Variants of Contrastive Loss for VQA.} In this paper, we propose two different types of contrastive loss for VQA: a global version which calculates the similarity (or distance) between anchor samples and positive\/negative samples on the whole answer probabilities (dubbed as CR-G), and a local version which calculates the distance based on only ground-truth answer probabilities (dubbed as CR-L).\n\n\\emph{Global Contrastive Loss (CR-G).} Given the predicted answer distributions (before sigmoid activation function) of anchor sample, positive sample, and negative samples (denoted as $P_a$, $P_p$ and $\\{P_n^i\\}$ respectively), the CR-G loss is:\n\\begin{equation}\n    L_{\\text{CR-G}} = - \\log \\left( \n    \\frac{\\mathrm{e}^{s(P_a, P_p)\/\\tau}}{\\mathrm{e}^{s(P_a, P_p)\/\\tau} + \\sum_i\\mathrm{e}^{s(P_a, P_n^i)\/\\tau}} \n    \\right),\n\\end{equation}\nwhere $s(P_i, P_j)$ is the cosine similarity of answer distributions $P_i$ and $P_j$, and $\\tau$ is the hyperparameter temperature.\n\n\\emph{Local Contrastive Loss (CR-L).} Instead of calculating similarity based on the whole answer distributions, there is another alternative choice which only focuses on probabilities of ground-truth answers. Thus, the CR-L loss is:\n\\begin{equation}\n    L_{\\text{CR-L}} = - \\log  \\left( \n    \\frac{\\mathrm{e}^{\\tilde{P}_a(m)\/\\tau}}{\\mathrm{e}^{\\tilde{P}_a(m)\/\\tau} + \\sum_i \\tilde{P}_n^i(m) \\times \\mathrm{e}^{\\tilde{P}_n^i(m)\/\\tau} } \n\\right),\n\\end{equation}\nwhere $m$ is the answer index of ground-truth answer with the highest target score, and $\\tilde{P}_*(\\cdot)$ is the answer probabilities after sigmoid activation function (\\textit{i}.\\textit{e}., $\\tilde{P}_*(\\cdot) = \\sigma(P_*(\\cdot))$). In addition, we add $\\tilde{P}_n^i(m)$ as weights which helps models to focus more on hard negative samples with larger $\\tilde{P}_n^i(m)$.\n\n\\emph{Differences between CR-G and CR-L.} Since CR-L only optimize models along a single ground-truth answer direction, it can directly suppress models output over-high scores for negative samples on original ground-truth answers, which is important for vanilla VQA models (\\textit{e}.\\textit{g}.. UpDn). But for ensemble-based VQA models (\\textit{e}.\\textit{g}., LMH), since we overlook the accuracy of fused answer distributions (cf. $\\hat{P}_{vqa}(\\bm{a}$) in Eq.~\\eqref{eq:eq3}), it would be better to use CR-G, which calculates the sample similarity over the global answer distributions.\n\n\n\n\\begin{table*}[tbp]\n    \\caption{Accuracies (\\%) of different VQA architectures on VQA-CP v2 test set and GQA-OOD testdev set. CSST-G and CSST-L denote the CSST with CR-G and CR-L losses, respectively. $^*$ denotes the results from our reimplementation using official codes. Q-CSS$^+$\/V-CSS$^+$\/CSS$^+$ denote the results of this improved version of CSS, and Q-CSS\/V-CSS\/CSS denote the results from~\\cite{chen2020counterfactual}.}\n\t\\vspace{-1em}\n\n\t\\subfloat[Ablation studies of different CSST variants and components on UpDn~\\cite{anderson2018bottom} base model.]{\n\t\t\\tablestyle{3.5pt}{1.05}\\begin{tabular}{|l|x{23}x{23}x{23}x{23}|}\n\t\t\t\\hline\n\t\t\t\\multirow{2}{*}{Models} & \\multicolumn{4}{c|}{VQA-CP v2} \\\\\n\t\t\t& All & Y\/N & Num & Other  \\\\\n\t\t\t\\hline\n\t\t\t\\textbf{UpDn}~\\cite{anderson2018bottom}  & 39.74 & 42.27 & 11.93  & 46.05 \\\\\n\t\t\tBaseline$^*$ & 39.68 & 41.93 & 12.68 & 45.91 \\\\\n\t\t\t~+Q-CSS & 40.05 & 42.16 & 12.30 & 46.56 \\\\\n\t\t\t~+V-CSS & 40.98 & 43.12 & 12.28 & 46.86 \\\\\n\t\t\t~+CSS & 41.16 & 43.96 & 12.78 & 47.48 \\\\\n\t\t\t~+CSS$^+$ & \\cellcolor{mygray-bg}{40.84} & \\cellcolor{mygray-bg}{43.09}  & \\cellcolor{mygray-bg}{12.74}  & \\cellcolor{mygray-bg}{47.37} \\\\\n\t\t\t~+CSST-G & \\cellcolor{mygray-bg}{41.68} & \\cellcolor{mygray-bg}{45.70}  & \\cellcolor{mygray-bg}{14.01}  & \\cellcolor{mygray-bg}{47.16} \\\\\n\t\t\t~+CSST-L & \\cellcolor{mygray-bg}{\\textbf{56.55}} & \\cellcolor{mygray-bg}{\\textbf{80.45}}  & \\cellcolor{mygray-bg}{\\textbf{36.29}}  & \\cellcolor{mygray-bg}{\\textbf{49.58}}  \\\\\n\t\t    \\hline\n\t\t\t\\multirow{2}{*}{Models} & \\multicolumn{4}{c|}{GQA-OOD} \\\\\n\t\t\t& Acc-A & Acc-T & Acc-H & $\\Delta$$\\downarrow$  \\\\\n\t\t\t\\hline\n\t\t\t\\textbf{UpDn}~\\cite{anderson2018bottom}  & 46.4 & 42.1 & 49.1 & 16.6 \\\\\n\t\t\tBaseline$^*$ & 46.96 & 42.90 & 49.45 & 15.3 \\\\\n\t\t\t~+Q-CSS$^+$ & \\cellcolor{mygray-bg}{46.42} & \\cellcolor{mygray-bg}{43.37}  & \\cellcolor{mygray-bg}{48.30}  & \\cellcolor{mygray-bg}{11.4} \\\\\n\t\t\t~+V-CSS$^+$ & \\cellcolor{mygray-bg}{46.64} & \\cellcolor{mygray-bg}{43.56}  & \\cellcolor{mygray-bg}{48.53}  & \\cellcolor{mygray-bg}{11.4} \\\\\n\t\t\t~+CSS$^+$ & \\cellcolor{mygray-bg}{46.17} & \\cellcolor{mygray-bg}{\\textbf{44.40}}  & \\cellcolor{mygray-bg}{47.26}  & \\cellcolor{mygray-bg}{\\textbf{6.4}} \\\\\n\t\t\t~+CSST-G & \\cellcolor{mygray-bg}{\\textbf{47.78}} & \\cellcolor{mygray-bg}{44.21}  & \\cellcolor{mygray-bg}{\\textbf{49.97}}  & \\cellcolor{mygray-bg}{13.0} \\\\\n\t\t\t~+CSST-L & \\cellcolor{mygray-bg}{47.25} & \\cellcolor{mygray-bg}{43.37}  & \\cellcolor{mygray-bg}{49.62}  & \\cellcolor{mygray-bg}{14.4} \\\\\n\t\t\t\\hline\n\t\\end{tabular}}\\hspace{2mm}\n\n\t\\subfloat[Ablation studies of different CSST variants and components on RUBi~\\cite{cadene2019rubi} base model.]{\n\t\t\\tablestyle{3.5pt}{1.05}\\begin{tabular}{|l|x{23}x{23}x{23}x{23}|}\n\t\t\t\\hline\n\t\t\t\\multirow{2}{*}{Models} & \\multicolumn{4}{c|}{VQA-CP v2} \\\\\n\t\t\t& All & Y\/N & Num & Other  \\\\\n\t\t\t\\hline\n\t\t\t\\textbf{RUBi}~\\cite{cadene2019rubi} & 44.23 & --- & ---  & ---  \\\\\n\t\t\tBaseline$^*$ & 45.23 & 64.85 & 11.83 & 44.11 \\\\\n\t\t\t~+Q-CSS & 46.31 & 68.70 & 12.15 & 43.95 \\\\\n\t\t\t~+V-CSS & 46.00 & 62.08 & 11.84 & \\textbf{46.95} \\\\\n\t\t\t~+CSS & 46.67 & 67.26 & 11.62 & 45.13 \\\\\n\t\t\t~+CSS$^+$ & \\cellcolor{mygray-bg}{47.46} & \\cellcolor{mygray-bg}{\\textbf{69.42}}  & \\cellcolor{mygray-bg}{12.18}  & \\cellcolor{mygray-bg}{45.63} \\\\\n\t\t\t~+CSST-G & \\cellcolor{mygray-bg}{47.06} & \\cellcolor{mygray-bg}{65.91}  & \\cellcolor{mygray-bg}{12.61}  & \\cellcolor{mygray-bg}{46.64} \\\\\n\t\t\t~+CSST-L & \\cellcolor{mygray-bg}{\\textbf{47.65}} & \\cellcolor{mygray-bg}{62.06}  & \\cellcolor{mygray-bg}{\\textbf{33.46}} & \\cellcolor{mygray-bg}{43.98} \\\\\n\t\t    \\hline\n\t\t\t\\multirow{2}{*}{Models} & \\multicolumn{4}{c|}{GQA-OOD} \\\\\n\t\t\t& Acc-A & Acc-T & Acc-H & $\\Delta$$\\downarrow$  \\\\\n\t\t\t\\hline\n\t\t\t\\textbf{RUBi}~\\cite{cadene2019rubi}  & 38.8 & 35.7 & 40.8 & 14.3 \\\\\n\t\t\tBaseline$^*$ & 45.85 & 43.37 & 47.37 & \\textbf{9.2} \\\\\n\t\t\t~+Q-CSS$^+$ & \\cellcolor{mygray-bg}{47.32} & \\cellcolor{mygray-bg}{43.18} & \\cellcolor{mygray-bg}{49.86} & \\cellcolor{mygray-bg}{15.5} \\\\\n\t\t    ~+V-CSS$^+$ & \\cellcolor{mygray-bg}{45.46} & \\cellcolor{mygray-bg}{41.86} & \\cellcolor{mygray-bg}{47.66} & \\cellcolor{mygray-bg}{13.9} \\\\\n\t\t\t~+CSS$^+$ & \\cellcolor{mygray-bg}{46.75} & \\cellcolor{mygray-bg}{42.90}  & \\cellcolor{mygray-bg}{49.11}  & \\cellcolor{mygray-bg}{14.5} \\\\\n\t\t\t~+CSST-G & \\cellcolor{mygray-bg}{47.64} & \\cellcolor{mygray-bg}{43.18}  & \\cellcolor{mygray-bg}{50.38}  & \\cellcolor{mygray-bg}{16.7} \\\\\n\t\t\t~+CSST-L & \\cellcolor{mygray-bg}{\\textbf{48.39}} & \\cellcolor{mygray-bg}{\\textbf{44.31}}  & \\cellcolor{mygray-bg}{\\textbf{50.89}}  & \\cellcolor{mygray-bg}{14.8} \\\\\n\t\t\t\\hline\n\t\\end{tabular}}\\hspace{2mm}\n\n\t\\subfloat[Ablation studies of different CSST variants and components on LMH~\\cite{clark2019don} base model.]{\n\t\t\\tablestyle{3.5pt}{1.05}\\begin{tabular}{|l|x{23}x{23}x{23}x{23}|}\n\t\t\t\\hline\n\t\t\t\\multirow{2}{*}{Models} & \\multicolumn{4}{c|}{VQA-CP v2} \\\\\n\t\t\t& All & Y\/N & Num & Other  \\\\\n\t\t\t\\hline\n\t\t\t\\textbf{LMH}~\\cite{clark2019don}  & 52.05 & --- & ---  & ---  \\\\\n\t\t\tBaseline$^*$ & 52.45 & 69.81 & 44.46 & 45.54 \\\\\n\t\t\t~+Q-CSS & 56.66 & 80.82 & 45.83 & 46.98 \\\\\n\t\t\t~+V-CSS & 58.23 & 80.53 & 52.48 & 48.13 \\\\\n\t\t\t~+CSS & 58.95 & 84.37 & 49.42 & 48.21 \\\\\n\t\t\t~+CSS$^+$ & \\cellcolor{mygray-bg}{59.54} & \\cellcolor{mygray-bg}{83.37}  & \\cellcolor{mygray-bg}{52.57}  & \\cellcolor{mygray-bg}{48.97} \\\\\n\t\t\t~+CSST-G & \\cellcolor{mygray-bg}{\\textbf{61.66}} & \\cellcolor{mygray-bg}{\\textbf{90.20}}  & \\cellcolor{mygray-bg}{\\textbf{54.42}}  & \\cellcolor{mygray-bg}{48.69} \\\\\n\t\t\t~+CSST-L & \\cellcolor{mygray-bg}{55.38} & \\cellcolor{mygray-bg}{68.01}  & \\cellcolor{mygray-bg}{52.89}  & \\cellcolor{mygray-bg}{\\textbf{49.45}} \\\\\n\t\t    \\hline\n\t\t\t\\multirow{2}{*}{Models} & \\multicolumn{4}{c|}{GQA-OOD} \\\\\n\t\t\t& Acc-A & Acc-T & Acc-H & $\\Delta$$\\downarrow$  \\\\\n\t\t\t\\hline\n\t\t\t\\textbf{LMH}~\\cite{clark2019don} & --- & --- & --- & --- \\\\\n\t\t\tBaseline$^*$ & 43.96 & 40.73 & 45.93 & 12.8 \\\\\n\t\t\t~+Q-CSS$^+$ & \\cellcolor{mygray-bg}{43.45} & \\cellcolor{mygray-bg}{41.20} & \\cellcolor{mygray-bg}{44.84} & \\cellcolor{mygray-bg}{\\textbf{8.8}} \\\\\n\t\t\t~+V-CSS$^+$ & \\cellcolor{mygray-bg}{42.63} & \\cellcolor{mygray-bg}{41.02} & \\cellcolor{mygray-bg}{43.62} & \\cellcolor{mygray-bg}{6.3} \\\\\n\t\t\t~+CSS$^+$ & \\cellcolor{mygray-bg}{44.24} & \\cellcolor{mygray-bg}{41.20}  & \\cellcolor{mygray-bg}{46.11}  & \\cellcolor{mygray-bg}{11.9} \\\\\n\t\t\t~+CSST-G & \\cellcolor{mygray-bg}{45.42} & \\cellcolor{mygray-bg}{42.90}  & \\cellcolor{mygray-bg}{46.97}  & \\cellcolor{mygray-bg}{9.5} \\\\\n\t\t\t~+CSST-L & \\cellcolor{mygray-bg}{\\textbf{47.93}} & \\cellcolor{mygray-bg}{\\textbf{44.31}}  & \\cellcolor{mygray-bg}{\\textbf{50.14}}  & \\cellcolor{mygray-bg}{13.2} \\\\\n\t\t\t\\hline\n\t\\end{tabular}}\\hspace{3mm}\n\t\\vspace{-1em}\n\t\\label{tab:boost_models}\n\\end{table*}\n\n\n\\addtolength{\\tabcolsep}{-1pt}\n\\begin{table}[tbp]\n\t\\small\n\t\\caption{Ablation studies of the influence of different positive and negative samples selection strategies on VQA-CP v2. ``CSS\" denotes the model using only two negative samples from V-CSS$^+$ and Q-CSS$^+$ in CR training. ``Rand\" denotes the model using only the last two randomly composed negative samples in CR training. $^*$ denotes the results from our reimplementation.}\n\t\\vspace{-1em}\n\t\\begin{center}\n\t\t\\scalebox{0.98}{\n\t\t\t\\begin{tabular}{| l | c c | c c c c|}\n\t\t\t\t\\hline\n\t\t\t\tModels & CSS & Rand & All & Y\/N & Num & Other  \\\\\n\t\t\t\t\\hline\n\t\t\t\tBaseline (LMH) & & & 52.45 & 69.81 & 44.46 & 45.54 \\\\\n\t\t\t\tLMH-CSS & & & 58.95 & 84.37 & 49.42 & 48.21 \\\\\n\t\t\t\t~+CL~\\cite{liang2020learning} & & & 59.18 & 86.99 & 49.89 & 47.16 \\\\\n\t\t\t\tLMH-CSS$^+$ & & & 59.54 & 83.37 & 52.57 & 48.97  \\\\\n\t\t\t\t~+CL$^*$~\\cite{liang2020learning} & & & 59.78 & 83.85 & 53.12 & \\textbf{49.00} \\\\\n\t\t\t\t~+CR & & \\boldcheckmark & \\cellcolor{mygray-bg}{60.74} & \\cellcolor{mygray-bg}{89.49} & \\cellcolor{mygray-bg}{52.05} & \\cellcolor{mygray-bg}{48.06} \\\\\n\t\t\t\t~+CR & \\boldcheckmark & & \\cellcolor{mygray-bg}{61.13} & \\cellcolor{mygray-bg}{90.09} & \\cellcolor{mygray-bg}{53.48} &  \\cellcolor{mygray-bg}{48.06} \\\\\n\t\t\t\t~+CR (CSST-G) & \\boldcheckmark & \\boldcheckmark & \\cellcolor{mygray-bg}{\\textbf{61.66}} & \\cellcolor{mygray-bg}{\\textbf{90.20}} & \\cellcolor{mygray-bg}{\\textbf{54.42}} & \\cellcolor{mygray-bg}{48.69} \\\\\n\t\t\t\t\\hline\n\t\t\t\\end{tabular}\n\t\t}\n\t\\end{center}\n\t\\label{tab:pos&neg}\n\\end{table}\n\\addtolength{\\tabcolsep}{1pt}\n\n\n\\section{Experiments} \\label{sec:exp}\n\n\n\\subsection{Datasets and Evaluation Metrics}\n\\noindent\\textbf{Datasets.} We evaluated CSST on two challenging diagnostic VQA benchmarks: 1) \\textbf{VQA-CP}~\\cite{agrawal2018don}. It is a re-organization of the training and test sets of widely-used VQA v1~\\cite{antol2015vqa} and VQA v2~\\cite{goyal2017making}, where the answer distribution of each question type in the training set is made explicitly different from the one in the test set. There are two VQA-CP datasets: VQA-CP v2 and VQA-CP v1. We followed the official splits~\\cite{agrawal2018don} for both two datasets. We also reported our results on the VQA v2 validation set for more complete comparisons. 2) \\textbf{GQA-OOD}~\\cite{kervadec2021roses}. It is a fine-grained re-organization of GQA~\\cite{hudson2019gqa}. They share the same training set, but GQA-OOD introduces fine-grained shifts into both the validation and test sets. \n\n\\noindent\\textbf{Evaluation Metrics.}\nFor model accuracies on VQA-CP, we followed standard  VQA evaluation metric~\\cite{antol2015vqa}. Similarly, we reported accuracy on all test samples (All) and three different categories separately: Yes\/No (Y\/N), number counting (Num), and other (Other) categories. For model accuracies on GQA-OOD, there are four metrics: 1) \\textbf{Acc-A} (all): overall accuracy over all test samples. 2) \\textbf{Acc-T} (tail): the accuracy on OOD samples (\\textit{i}.\\textit{e}., samples of the tail of the answer class distribution). 3) \\textbf{Acc-H} (head): the accuracy on in-domain samples (\\textit{i}.\\textit{e}., samples of the head of each local group). 4) $\\mathbf{\\Delta}$ = (Acc-H - Acc-T) \/ Acc-T: the error prediction imbalance between frequent (in-domain) and rare (OOD) answers.\n\n\n\\subsection{Experimental Settings} \\label{sec:expsetting}\n\\noindent\\textbf{Data Preprocessing Details.}\nFor fair comparisons, we did all the same data preprocessing steps with the widely-used UpDn model~\\cite{anderson2018bottom} using the publicly available reimplementation\\footnote{\\href{https:\/\/github.com\/hengyuan-hu\/bottom-up-attention-vqa}{https:\/\/github.com\/hengyuan-hu\/bottom-up-attention-vqa}\\label{updn_github}}. Specifically, for image feature extraction, we used Faster R-CNN~\\cite{ren2015faster} pretrained on VG to extract the top-$K$ objects. Following the convention of prior works, we set $K=36$ for VQA-CP and VQA, and $K=100$ for GQA-OOD. For question feature extraction, there are slight differences for the models with different backbones. For the UpDn-based models, we set the maximum length of questions as 14, and the word embeddings are initialized with GloVe ~\\cite{pennington2014glove} vectors with the dimension of 300. A single-layer GRU is used to obtain question embedding vectors with the dimension of 1,024. For the LXMERT-based models, we used the official tokenizer in LXMERT to segment each question into word tokens. A pre-trained LXMERT model is used to obtain question features with the dimension of 768.\n\n\\noindent\\textbf{Training Details and Hyperparameters.}\nWe trained UpDn-based models or LXMERT-based models for 30 epochs with batch size 512 on all datasets. We used the Adamax algorithm as the optimizer following the public reimplementation\\footref{updn_github}. All parameters were initialized from scratch and the random seed was set to 0. Loss weights of $L_\\text{XE}$, $L_\\text{CR-G}$, and $L_\\text{CR-L}$ were set to 1, 1, and 8 in most of experiments. $\\tau$ in the contrastive loss was set to 1 in all experiments. Results about the influence of hyperparameters are reported in Sec.~\\ref{sec:exp_hyper}\n\n\n\n\\noindent\\textbf{Adapting Ensemble-based Methods to GQA-OOD Benchmark.} In original GQA-OOD paper~\\cite{kervadec2021roses}, the authors claim none of the typical ensemble-based models (\\textit{e}.\\textit{g}., RUBi~\\cite{cadene2019rubi} and LMH~\\cite{clark2019don}) can improve the Acc-T performance on GQA-OOD, and they even deteriorate Acc-H performance. In this paper, we argue that it is not suitable to apply these debiasing methods on all samples, because ``biases\" only reside in \\emph{imbalanced local groups}~\\cite{kervadec2021roses}. Therefore, in our experiments, we only utilized these debiasing methods on imbalanced local groups. Since the local groups of samples are different in the training and test sets, in the training stage, we only selected the imbalanced local groups from the training set based on the same Shannon entropy and threshold, \\textit{i}.\\textit{e}., we don't use any extra information about test set in advance.\n\n\n\\noindent\\textbf{Adapting LXMERT Backbone for VQA.} LXMERT~\\cite{tan2019lxmert} itself is a multimodal BERT model, which can be easily used for VQA. Similar to other multimodal BERT models, after pretraining, it typically utilizes the output of the [CLS] token as a multimodal fused feature, and trains a linear classifier for answer prediction. Unfortunately, this setting is slightly different from the UpDn backbone, whose inputs are both visual and question features. To benefit from the pretrained weights and seamlessly equipped LXMERT into other existing ensemble-based methods with UpDn backbone, in this paper, we utilized the outputs of visual and language embedding tokens as the visual and question features, respectively (\\textit{i}.\\textit{e}., replacing original inputs of ensemble-based methods). The pretrained weights of LXMERT keep fixed. (See results of models with LXMERT in following sections.)\n\n\n\\noindent\\textbf{Adapting SAR~\\cite{si2021check} strategy to LMH-CSST.} In our experiments, we also equipped the model-agnostic SAR~\\cite{si2021check} to our LMH-CSST to further improve performance. Specifically, for the Select And Rerank (SAR) module, we used SAR to refer to SAR+LMH (\\textit{i}.\\textit{e}., incorporate LMH into SAR). We chose LMH-CSST as Candidate Answer Selector (CAS), and used top-20 answers as candidates. We utilized the ``R $\\to$ C\" strategy to combine question and answer into a synthetic dense caption. We trained SAR for 10 epochs, and batch size was set to 32. For Question Type Discriminator, we selected 1 or 2 candidates for Y\/N questions and 12 candidate answers for non-Y\/N questions when testing on VQA-CP. Meanwhile, Since we lack manual question type annotations for samples on GQA-OOD, we removed Question Type Discriminator when testing on GQA-OOD, and selected 12 candidates for all questions. We refer readers to SAR ~\\cite{si2021check} for more details.\n\n\n\n\n\\begin{table*}\n\t\\small\n\t\\caption[]{Accuracies (\\%) on VQA-CP v2 test set and VQA v2 val set of state-of-the-art models. The gap (other) represents the accuracy difference between the other category of VQA v2 and VQA-CP v2. ``Pre.\" represents ``Pretraining\", \\textit{i}.\\textit{e}., these models use extra datasets at their pretraining stage (\\textit{e}.\\textit{g}., LXMERT backbone~\\cite{tan2019lxmert} or SAR~\\cite{si2021check}). ``Ann.\" represents ``Annotation\", \\textit{i}.\\textit{e}., these models rely on extra manual anntations. $^\\dagger$ represents the \\emph{ensemble-based} methods, $^*$ indicates the results from our reimplementation using official released codes, $^\\ddag$ denotes these models use the same adapting strategy for LXMERT backbone mentioned in Sec.~\\ref{sec:expsetting}.} \n\t\\vspace{-1em}\n\t\\begin{center}\n\t\t\\scalebox{0.96}{\n\t\t\t\\begin{tabular}{| l | c | c | c | c c c c | c c c c| c |}\n\t\t\t\t\\hline\n\t\t\t\t\\multirow{2}{*}{Models}  & \\multirow{2}{*}{Base} & \\multirow{2}{*}{Pre.} & \\multirow{2}{*}{Ann.} & \\multicolumn{4}{c|}{VQA-CP v2 test $\\uparrow$} & \\multicolumn{4}{c|}{VQA v2 val $\\uparrow$} & Gap $\\downarrow$ \\\\\n\t\t\t\t& & & & All & Yes\/No & Num & Other & All & Yes\/No & Num & Other & Other \\\\\n\t\t\t    \\hline\n\t\t\t    UpDn~\\cite{anderson2018bottom}$_{\\textit{CVPR'18}}$ & UpDn &  &  & 39.74 & 42.27 & 11.93 & 46.05 & 63.48 & 81.18 & 42.14 & 55.66 & 9.61 \\\\\n\t\t\t\tAReg$^\\dagger$~\\cite{ramakrishnan2018overcoming}$_{\\textit{NeurIPS'18}}$ & UpDn & & & 41.17 & 65.49 & 15.48 & 35.48 & 62.75 & 79.84 & 42.35 & 55.16 & 19.58 \\\\\n\t\t\t\tMuRel~\\cite{cadene2019murel}$_{\\textit{CVPR'19}}$ & UpDn & & & 39.54 & 42.85 & 13.17 & 45.04 & --- & --- & --- & --- &  --- \\\\\n\t\t\t\tGRL$^\\dagger$~\\cite{grand2019adversarial}$_{\\textit{ACL'19}}$ & UpDn & & & 42.33 & 59.74 & 14.78 & 40.76 & 51.92 & --- & --- & --- & ---  \\\\\n\t\t\t\tRUBi$^\\dagger$$^*$~\\cite{cadene2019rubi}$_{\\textit{NeurIPS'19}}$ & UpDn & & & 45.23 & 64.85 & 11.83 & 44.11 & 50.56 & 49.45 & 41.02 & 53.95 & 9.84  \\\\\n\t\t\t\tSCR~\\cite{wu2019self}$_{\\textit{NeurIPS'19}}$ & UpDn & & & 48.47 & 70.41 & 10.42 & 47.29 & 62.30 & 77.40 & 40.90 & 56.50 & 9.21 \\\\\n\t\t\t\tLMH$^\\dagger$$^*$~\\cite{clark2019don}$_{\\textit{EMNLP'19}}$ & UpDn & & & 52.45 & 69.81 & 44.46 & 45.54 & 61.64 & 77.85 & 40.03 & 55.04 & 9.50 \\\\\n\t\t\t\tCVL~\\cite{abbasnejad2020counterfactual}$_{\\textit{CVPR'20}}$ & UpDn & & & 42.12 & 45.72 & 12.45 & 48.34 &  --- & --- & ---  & --- & --- \\\\\n\t\t\t\tUnshuffling~\\cite{teney2020unshuffling}$_{\\textit{arXiv'20}}$ & UpDn & & & 42.39 & 47.72 & 14.43 & 47.24 & 61.08 & 78.32 & 42.16 & 52.81 & 5.57 \\\\\n\t\t\t\tRandImg~\\cite{teney2020value}$_{\\textit{NeurIPS'20}}$ & UpDn & & & 55.37 & 83.89 & 41.60 & 44.20 & 57.24 & 76.53 & 33.87 & 48.57 & 4.37 \\\\\n\t\t\t\tSSL~\\cite{zhu2020overcoming}$_{\\textit{IJCAI'20}}$ & UpDn & & & 57.59 & 86.53 & 29.87 & 50.03 & 63.73 &  --- & --- & --- & --- \\\\\n\t\t\t\tCSS+CL$^\\dagger$~\\cite{liang2020learning}$_{\\textit{EMNLP'20}}$ & UpDn & & & 59.18 & 86.99 & 49.89 & 47.16 & 57.29 & 67.27 & 38.40 & 54.71 & 7.55 \\\\\n\t\t\t\tCF-VQA$^\\dagger$~\\cite{niu2021counterfactual}$_{\\textit{CVPR'21}}$ & UpDn & & & 53.55 & \\textbf{91.15} & 13.03 & 44.97 & 63.54 & 82.51 & 43.96 & 54.30 & 9.33 \\\\\n\t\t\t\tGGE-DQ$^\\dagger$~\\cite{han2021greedy}$_{\\textit{ICCV'21}}$ & UpDn & & & 57.32 & 87.04 & 27.75 & 49.59 & 59.11 & 73.27 & 39.99 & 54.39 & 4.80 \\\\\n\t\t\t\tLMH+SAR$^\\dagger$$^*$~\\cite{si2021check}$_{\\textit{ACL'21}}$ & UpDn & \\boldcheckmark & & 62.51 & 76.40 & \\textbf{59.40} & 56.09 & 65.79 & 77.26 & 52.71 & 60.52 & 4.43\\\\\n\t\t\t\t\\textbf{LMH-CSS}$_{\\textit{CVPR'20}}$ & UpDn &  &  & \\cellcolor{mygray-bg}{58.95} & \\cellcolor{mygray-bg}{84.37} & \\cellcolor{mygray-bg}{49.42} & \\cellcolor{mygray-bg}{48.21} & \\cellcolor{mygray-bg}{59.91} & \\cellcolor{mygray-bg}{73.25} & \\cellcolor{mygray-bg}{39.77} & \\cellcolor{mygray-bg}{55.11} & \\cellcolor{mygray-bg}{6.90} \\\\\n\t\t\t\t\\textbf{LMH-CSS$^+$} & UpDn &  & & \\cellcolor{mygray-bg}{59.54} & \\cellcolor{mygray-bg}{83.37}  & \\cellcolor{mygray-bg}{52.57}  & \\cellcolor{mygray-bg}{48.97}  & \\cellcolor{mygray-bg}{59.96} & \\cellcolor{mygray-bg}{73.69}  & \\cellcolor{mygray-bg}{40.18}  & \\cellcolor{mygray-bg}{54.77}  & \\cellcolor{mygray-bg}{5.80} \\\\\n\t\t\t\t\\textbf{LMH-CSST} & UpDn &  & & \\cellcolor{mygray-bg}{61.66} & \\cellcolor{mygray-bg}{90.20}  & \\cellcolor{mygray-bg}{54.42}  & \\cellcolor{mygray-bg}{48.69}  & \\cellcolor{mygray-bg}{62.37} & \\cellcolor{mygray-bg}{80.05}  & \\cellcolor{mygray-bg}{39.24}  & \\cellcolor{mygray-bg}{55.04}  & \\cellcolor{mygray-bg}{6.35} \\\\\n\t\t\t\t\\textbf{LMH-CSST+SAR} & UpDn & \\boldcheckmark & & \\cellcolor{mygray-bg}{\\textbf{66.49}} & \\cellcolor{mygray-bg}{86.97}  & \\cellcolor{mygray-bg}{57.95}  & \\cellcolor{mygray-bg}{\\textbf{58.53}} & \\cellcolor{mygray-bg}{\\textbf{69.31}} & \\cellcolor{mygray-bg}{\\textbf{85.85}} & \\cellcolor{mygray-bg}{\\textbf{52.87}} & \\cellcolor{mygray-bg}{\\textbf{61.08}} & \\cellcolor{mygray-bg}{\\textbf{2.55}} \\\\\n\t\t\t\t\\hline\n\t\t\t\n\t\t\t\tUpDn$^{*\\ddag}$~\\cite{anderson2018bottom}$_{\\textit{CVPR'18}}$ & LXMERT & \\boldcheckmark & &  44.14 & 43.12  & 17.07 & 51.66  & 67.69 & 83.83 & 50.80 & 59.89 & 8.23 \\\\\n\t\t\t\tLMH$^{*\\dagger\\ddag}$~\\cite{clark2019don}$_{\\textit{EMNLP'19}}$ & LXMERT & \\boldcheckmark & & 59.66 & 73.41 & 57.72 & 52.99 & 59.57 & 64.17  & 47.51 & 59.27 & 6.28 \\\\\n\t\t\t    \\textbf{LMH-CSS}$^+$$^\\ddag$ & LXMERT & \\boldcheckmark & & \\cellcolor{mygray-bg}{63.63} & \\cellcolor{mygray-bg}{84.70}  & \\cellcolor{mygray-bg}{62.12}  & \\cellcolor{mygray-bg}{53.00}  & \\cellcolor{mygray-bg}{58.01} & \\cellcolor{mygray-bg}{60.37}  & \\cellcolor{mygray-bg}{47.02}  & \\cellcolor{mygray-bg}{59.14} & \\cellcolor{mygray-bg}{6.14} \\\\\n\t\t\t\t\\textbf{LMH-CSST}$^\\ddag$ & LXMERT & \\boldcheckmark & & \\cellcolor{mygray-bg}{65.71} & \\cellcolor{mygray-bg}{\\textbf{90.10}}  & \\cellcolor{mygray-bg}{\\textbf{63.70}}  & \\cellcolor{mygray-bg}{53.48} & \\cellcolor{mygray-bg}{65.71} & \\cellcolor{mygray-bg}{80.61}  & \\cellcolor{mygray-bg}{48.26}  & \\cellcolor{mygray-bg}{58.99} & \\cellcolor{mygray-bg}{5.51} \\\\\n\t\t\t\t\\textbf{LMH-CSST+SAR}$^\\ddag$ & LXMERT & \\boldcheckmark & & \\cellcolor{mygray-bg}{\\textbf{67.49}} & \\cellcolor{mygray-bg}{88.06}  & \\cellcolor{mygray-bg}{56.57}  & \\cellcolor{mygray-bg}{\\textbf{59.71}}  & \\cellcolor{mygray-bg}{\\textbf{69.32}} & \\cellcolor{mygray-bg}{\\textbf{84.11}}  & \\cellcolor{mygray-bg}{\\textbf{54.90}}  & \\cellcolor{mygray-bg}{\\textbf{61.88}} & \\cellcolor{mygray-bg}{\\textbf{2.17}} \\\\\n\t\t\t\t\\hline\n                MUTANT~\\cite{gokhale2020mutant}$_{\\textit{EMNLP'20}}$ & UpDn & & \\boldcheckmark & 61.72 & 88.90 & 49.68 & 50.78 & 62.56 & 82.07 & 42.52 & 53.28 & 2.50 \\\\\n\t\t\t\tMUTANT~\\cite{gokhale2020mutant}$_{\\textit{EMNLP'20}}$ & LXMERT & \\boldcheckmark & \\boldcheckmark & 69.52 & 93.15 & 67.17 & 57.78 & 70.24 & 89.01 & 54.21 & 59.96 & 2.18 \\\\\n\t\t\t\t\\hline\n\t\t\t\\end{tabular}\n\t\t}\n\t\\end{center}\n\t\\vspace{-1em}\n\t\\label{tab:SOTA_v2}\n\\end{table*}\n\n\n\n\\addtolength{\\tabcolsep}{-2pt}\n\\begin{table}[tbp]\n\t\\small\n\t\\caption{Accuracies (\\%) on VQA-CP v1 test set of state-of-the-art models. $^\\dagger$ denotes ensemble-based methods. $^*$ indicates the results from our reimplementation. $^\\ddag$ denotes models use the adapting strategy for LXMERT backbone mentioned in Sec.~\\ref{sec:expsetting}.}\n\t\\vspace{-1em}\n\t\\begin{center}\n\t\t\\scalebox{0.98}{\n\t\t\t\\begin{tabular}{| l | c | c c c c|}\n\t\t\t\t\\hline\n\t\t\t\tModels & Base & All & Y\/N & Num & Other  \\\\\n\t\t\t\t\\hline\n\t\t\t\tGVQA~\\cite{agrawal2018don}$_{\\textit{CVPR'18}}$ & SAN & 39.23 & 64.72 & 11.87 & 24.86 \\\\\n\t\t\t\tUpDn~\\cite{anderson2018bottom}$_{\\textit{CVPR'18}}$ & UpDn & 39.74 & 42.27 & 11.93 & 46.05 \\\\\n\t\t\t\tAReg$^\\dagger$~\\cite{ramakrishnan2018overcoming}$_{\\textit{NeurIPS'18}}$ & UpDn & 41.17 & 65.49 & 15.48 & 35.48 \\\\\n\t\t\t\tGRL$^\\dagger$~\\cite{grand2019adversarial}$_{\\textit{ACL'19}}$ & UpDn & 45.69 & 77.64 & 13.21 & 26.97 \\\\\n\t\t\t\tRUBi$^\\dagger$$^*$~\\cite{cadene2019rubi}$_{\\textit{NeurIPS'19}}$ & UpDn & 50.90 & 80.83 & 13.84 & 36.02 \\\\\n\t\t\t\tLMH$^\\dagger$$^*$~\\cite{clark2019don}$_{\\textit{EMNLP'19}}$ & UpDn & 55.27 & 76.47 & 26.66 & 45.68 \\\\\n\t\t\t\t\\textbf{LMH-CSS}$_{\\textit{CVPR'20}}$ & UpDn & 60.95 & 85.60 & 40.57 & 44.62  \\\\\n\t\t\t\t\\textbf{LMH-CSS$^+$} & UpDn & \\cellcolor{mygray-bg}{61.66} & \\cellcolor{mygray-bg}{84.97} & \\cellcolor{mygray-bg}{41.65}  & \\cellcolor{mygray-bg}{46.51}  \\\\\n\t\t\t\t\\textbf{LMH-CSST} & UpDn & \\cellcolor{mygray-bg}{63.16} & \\cellcolor{mygray-bg}{90.58} & \\cellcolor{mygray-bg}{39.16}  & \\cellcolor{mygray-bg}{45.53}  \\\\\n\t\t\t\t\\textbf{LMH-CSST+SAR} & UpDn & \\cellcolor{mygray-bg}{\\textbf{69.57}} & \\cellcolor{mygray-bg}{\\textbf{90.81}}  & \\cellcolor{mygray-bg}{\\textbf{49.29}} & \\cellcolor{mygray-bg}{\\textbf{56.59}}   \\\\\n\t\t\t\t\\hline\n\t\t\t\t\\textbf{LMH-CSS$^+$}$^\\ddag$ & LXMERT & \\cellcolor{mygray-bg}{66.13} & \\cellcolor{mygray-bg}{88.89} & \\cellcolor{mygray-bg}{42.74}  & \\cellcolor{mygray-bg}{52.88}  \\\\\n\t\t\t\t\\textbf{LMH-CSST}$^\\ddag$ & LXMERT & \\cellcolor{mygray-bg}{67.27} & \\cellcolor{mygray-bg}{92.04} & \\cellcolor{mygray-bg}{43.23}  & \\cellcolor{mygray-bg}{52.30}  \\\\\n\t\t\t\t\\textbf{LMH-CSST+SAR}$^\\ddag$ & LXMERT & \\cellcolor{mygray-bg}{\\textbf{69.75}} & \\cellcolor{mygray-bg}{\\textbf{92.27}} & \\cellcolor{mygray-bg}{\\textbf{44.05}}  & \\cellcolor{mygray-bg}{\\textbf{57.69}}  \\\\\n\t\t\t\t\\hline\n\t\t\t\\end{tabular}\n\t\t}\n\t\\end{center}\n\t\\label{tab:SOTA_v1}\n\\end{table}\n\\addtolength{\\tabcolsep}{2pt}\n\n\n\\addtolength{\\tabcolsep}{-2pt} \n\\begin{table}[tbp]\n\t\\small\n\t\\caption{Accuracies (\\%) on GQA-OOD testdev set of state-of-the-art models. $^*$ indicates the results from our reimplementation using official released codes. $^\\ddag$ denotes models use the adapting strategy for LXMERT backbone mentioned in Sec.~\\ref{sec:expsetting}.}\n\t\\vspace{-1em}\n\t\\begin{center}\n\t\t\\scalebox{0.98}{\n\t\t\t\\begin{tabular}{| l | c | c c c c|}\n\t\t\t\t\\hline\n\t\t\t\tModels & Base & Acc-A & Acc-T & Acc-H & $\\Delta$$\\downarrow$  \\\\\n\t\t\t\t\\hline\n\t\t\t\n\t\t\t\n\t\t\t\tMCAN~\\cite{yu2019deep}$_{\\textit{CVPR'19}}$ & MCAN & 50.8 & 46.5 & 53.4 & 14.8 \\\\\n\t\t\t\tBAN4~\\cite{kim2018bilinear}$_{\\textit{NeurIPS'18}}$ & BAN4 & 50.2 & 47.2 & 51.9 & 9.9 \\\\\n\t\t\t\n\t\t\t\n\t\t\t\t\n\t\t\t\tUpDn$^*$$^\\ddag$~\\cite{anderson2018bottom}$_{\\textit{CVPR'18}}$ & LXMERT & 49.61 & 46.10 & 51.76 & 12.3 \\\\\n\t\t\t\tLMH$^*$$^\\ddag$~\\cite{clark2019don}$_{\\textit{EMNLP'19}}$ & LXMERT & 47.78 & 45.44 & 49.22 & \\textbf{8.3} \\\\\n\t\t\t\t\\textbf{LMH-CSS$^+$}$^\\ddag$ & LXMERT & \\cellcolor{mygray-bg}{49.21} & \\cellcolor{mygray-bg}{46.28} & \\cellcolor{mygray-bg}{51.01}  & \\cellcolor{mygray-bg}{10.2} \\\\\n\t\t\t\t\\textbf{LMH-CSST}$^\\ddag$ & LXMERT & \\cellcolor{mygray-bg}{49.75} & \\cellcolor{mygray-bg}{46.38}  & \\cellcolor{mygray-bg}{51.82}  & \\cellcolor{mygray-bg}{11.7} \\\\\n\t\t\t\t\\textbf{LMH-CSST+SAR}$^\\ddag$ & LXMERT & \\cellcolor{mygray-bg}{\\textbf{51.36}} & \\cellcolor{mygray-bg}{\\textbf{48.07}}  & \\cellcolor{mygray-bg}{\\textbf{53.38}}  & \\cellcolor{mygray-bg}{11.0} \\\\\n\t\t\t\t\\hline\\hline\n\t\t\t    UpDn$^*$~\\cite{anderson2018bottom}$_{\\textit{CVPR'18}}$ & UpDn & 46.96 & 42.90 &  49.45 & 15.3 \\\\\n\t\t\t\tRUBi$^*$~\\cite{cadene2019rubi}$_{\\textit{NeurIPS'19}}$  & UpDn & 45.85 & 43.37 & 47.37 & \\textbf{9.2} \\\\\n\t\t\t\tLMH$^*$~\\cite{clark2019don}$_{\\textit{EMNLP'19}}$ & UpDn  & 43.96 & 40.73 & 45.93 & 12.8 \\\\\n\t\t\t\t\\textbf{LMH-CSS$^+$} & UpDn & \\cellcolor{mygray-bg}{44.24} & \\cellcolor{mygray-bg}{41.20}  & \\cellcolor{mygray-bg}{46.11}  & \\cellcolor{mygray-bg}{11.9} \\\\\n\t\t\t\t\\textbf{LMH-CSST} & UpDn & \\cellcolor{mygray-bg}{45.42} & \\cellcolor{mygray-bg}{42.90}  & \\cellcolor{mygray-bg}{46.97}  & \\cellcolor{mygray-bg}{9.5}   \\\\\n\t\t\t\t\\textbf{LMH-CSST+SAR} & UpDn & \\cellcolor{mygray-bg}{\\textbf{51.07}} & \\cellcolor{mygray-bg}{\\textbf{47.98}}  & \\cellcolor{mygray-bg}{\\textbf{52.97}}  & \\cellcolor{mygray-bg}{10.4}  \\\\\n\t\t\t\t\\hline\n\t\t\t\\end{tabular}\n\t\t}\n\t\\end{center}\n\t\\label{tab:SOTA_GQA}\n\\end{table}\n\\addtolength{\\tabcolsep}{2pt}\n\n\n\n\\subsection{Ablative Studies }\n\\subsubsection{Influence of Different Hyperparameters of CSS} \\label{sec:exp_hyper}\nWe run a number of ablation studies to analyze the influence of different hyperparameters of CSS$^+$ (\\textit{i}.\\textit{e}., V-CSS$^+$ and Q-CSS$^+$\\footnote{V-CSS$^+$ and Q-CSS$^+$ denotes the improved version of V-CSS and Q-CSS, respectively (cf. Sec.~\\ref{sec:3.2.3}).}). Specifically, we conducted all ablations by building on top of a typical ensemble-based VQA model LMH~\\cite{clark2019don}. To disentangle the influence of our contrastive training, we only use the XE loss as training objective and both original samples and counterfactual samples as the inputs (cf. Fig.~\\ref{fig:2} (b), similar with~\\cite{chen2020counterfactual}). All results are illustrated in Fig.~\\ref{fig:5}.\n\n\\noindent\\textbf{Size of $\\mathcal{I}$ in V-CSS$^+$.} The influence of different size of $\\mathcal{I}$ is shown in Fig.~\\ref{fig:5} (a). We can observe that the model's performance gradually decreases with the increase of $|\\mathcal{I}|$. \n\n\\noindent\\textbf{Size of critical objects in V-CSS$^+$.} The influence of masking different numbers of critical objects is shown in Fig.~\\ref{fig:5} (a). We compared dynamic K (cf. Eq.~\\eqref{eq:topk_objects}) with some fixed constants (\\textit{e}.\\textit{g}., 1, 3, 5). From the results, we can observe that the model with dynamic K achieves the best performance.\n\n\n\\begin{table*}[!t]\n    \\centering\n\t\\caption{Quantitative results about the evaluation of the VQA models' visual-explainable and question-sensitive abilities.}\n    \\vspace{-0.5em}\n\n\t\\subfloat[Accuracies (\\%) on VQA-CP v2 test set.]{\n\t\t\\tablestyle{2.5pt}{1.05}\\begin{tabular}{l|x{22}x{22}x{22}x{22}}\n\t\t\t\\hline\n\t\t\tModels & All & Y\/N & Num & Other  \\\\\n\t\t\t\\hline\n\t\t\t\\multicolumn{5}{l}{\\emph{Models with UpDn backbone}} \\\\\n\t\t\t\\hline\n\t\t\tSCR & 48.47 & 70.41 & 10.42 & 47.29 \\\\\n\t\t\tLMH &52.45 & 69.81 & 44.46 & 45.54  \\\\\n\t\t\t~+SCR & \\multicolumn{4}{c}{continued decrease} \\\\\n \t\t\t~+\\textbf{CSS$^+$} & \\cellcolor{mygray-bg}{59.54} & \\cellcolor{mygray-bg}{83.37}  & \\cellcolor{mygray-bg}{52.57}  & \\cellcolor{mygray-bg}{\\textbf{48.97}} \\\\\n\t\t\t~+\\textbf{CSST} & \\cellcolor{mygray-bg}{\\textbf{61.66}} & \\cellcolor{mygray-bg}{\\textbf{90.20}}  & \\cellcolor{mygray-bg}{\\textbf{54.42}}  & \\cellcolor{mygray-bg}{48.69} \\\\\n\t\t\t\\hline\n\t\t\t\\multicolumn{5}{l}{\\emph{Models with LXMERT backbone}} \\\\\n\t\t\t\\hline\n\t\t\tLMH & 59.66 & 73.41 & 57.72 & 52.99 \\\\\n\t\t\t~+\\textbf{CSS$^+$} & \\cellcolor{mygray-bg}{63.63} & \\cellcolor{mygray-bg}{84.70}  & \\cellcolor{mygray-bg}{62.12}  & \\cellcolor{mygray-bg}{53.00} \\\\\n\t\t\t~+\\textbf{CSST} & \\cellcolor{mygray-bg}{\\textbf{65.71}} & \\cellcolor{mygray-bg}{\\textbf{90.10}}  & \\cellcolor{mygray-bg}{\\textbf{63.70}}  & \\cellcolor{mygray-bg}{\\textbf{53.48}} \\\\\n\t\t\t\\hline\n\t\\end{tabular}}\\hspace{3mm}\n\n\t\\subfloat[$\\mathcal{AI}$ score (\\%) on VQA-CP v2 test set.]{\n\t\t\\tablestyle{2.5pt}{1.05}\\begin{tabular}{l|x{22}x{22}x{22}x{22}}\n\t\t\t\\hline\n\t\t\tModels & Top-1 & Top-2 & Top-3  \\\\\n\t\t\t\\hline\n\t\t\t\\multicolumn{4}{l}{\\emph{Models with UpDn backbone}} \\\\\n\t\t\t\\hline\n\t\t\tSCR & 15.55 & 13.40 & 11.97 \\\\\n\t\t\tLMH & 17.81 & 15.19 & 13.52 \\\\\n\t\t\t~+V-CSS$^+$ & \\cellcolor{mygray-bg}{\\textbf{20.66}} & \\cellcolor{mygray-bg}{\\textbf{16.71}}  & \\cellcolor{mygray-bg}{14.34} \\\\\n\t\t\t ~+\\textbf{CSS$^+$} & \\cellcolor{mygray-bg}{18.81} & \\cellcolor{mygray-bg}{15.73}  & \\cellcolor{mygray-bg}{13.84} \\\\\n\t\t\t~+\\textbf{CSST} & \\cellcolor{mygray-bg}{19.54} & \\cellcolor{mygray-bg}{16.37}  & \\cellcolor{mygray-bg}{\\textbf{14.41}} \\\\\n\t\t\t\\hline\n\t\t\t\\multicolumn{4}{l}{\\emph{Models with LXMERT backbone}} \\\\\n\t\t\t\\hline\t\t\t\n\t\t\tLMH & 17.16 & 15.06 & 13.60\\\\\n\t\t\t~+\\textbf{CSS$^+$} & \\cellcolor{mygray-bg}{18.90} & \\cellcolor{mygray-bg}{16.28}  & \\cellcolor{mygray-bg}{14.43} \\\\\n\t\t\t~+\\textbf{CSST} & \\cellcolor{mygray-bg}{\\textbf{22.68}} & \\cellcolor{mygray-bg}{\\textbf{18.99}}  & \\cellcolor{mygray-bg}{\\textbf{16.48}} \\\\\n\t\t\t\\hline\n\t\\end{tabular}}\\hspace{3mm}\n\n\t\\subfloat[\\textbf{Left}: $CS(k)$ (\\%) on VQA-CP-Rephrasing; \\textbf{Right}: $\\mathcal{CI}$ score (\\%) on VQA-CP v2 test set.]{\n\t\t\\tablestyle{2.5pt}{1.05}\\begin{tabular}{l|x{22}x{22}x{22}x{22}|x{22}}\n\t\t\t\\hline\n\t\t\tModels & K=1 & K=2 & K=3 & K=4  & $\\mathcal{CI}$ \\\\\n\t\t\t\\hline\n\t\t\t\\multicolumn{6}{l}{\\emph{Models with UpDn backbone}} \\\\\n\t\t\t\\hline\n\t\t\tUpDn & 49.94 & 38.80 & 31.55 & 28.08 & 33.70 \\\\\n\t\t\tLMH & 51.68 & 39.84 & 33.38 & 29.11 & 40.26 \\\\\n\t\t\t~+Q-CSS$^+$ & \\cellcolor{mygray-bg}{55.69} & \\cellcolor{mygray-bg}{43.95}  & \\cellcolor{mygray-bg}{37.22}  & \\cellcolor{mygray-bg}{32.68} & \\cellcolor{mygray-bg}{57.08}  \\\\\n\t\t\t~+\\textbf{CSS$^+$} & \\cellcolor{mygray-bg}{56.31} & \\cellcolor{mygray-bg}{44.08}  & \\cellcolor{mygray-bg}{37.00}  & \\cellcolor{mygray-bg}{32.29} & \\cellcolor{mygray-bg}{51.87} \\\\\n\t\t\t~+\\textbf{CSST} & \\cellcolor{mygray-bg}{\\textbf{59.97}} & \\cellcolor{mygray-bg}{\\textbf{49.49}}  & \\cellcolor{mygray-bg}{\\textbf{43.37}}  & \\cellcolor{mygray-bg}{\\textbf{39.24}} & \\cellcolor{mygray-bg}{\\textbf{58.45}} \\\\\n\t\t\t\\hline\n\t\t\t\\multicolumn{6}{l}{\\emph{Models with LXMERT backbone}} \\\\\n\t\t\t\\hline\t\t\t\n\t\t\tLMH & 61.21 & 50.33 & 43.94 & 39.52 & 55.42\\\\\n\t\t\t~+\\textbf{CSS$^+$} & \\cellcolor{mygray-bg}{61.75} & \\cellcolor{mygray-bg}{50.62}  & \\cellcolor{mygray-bg}{43.92}  & \\cellcolor{mygray-bg}{39.27} & \\cellcolor{mygray-bg}{55.44} \\\\\n\t\t\t~+\\textbf{CSST} & \\cellcolor{mygray-bg}{\\textbf{63.87}} & \\cellcolor{mygray-bg}{\\textbf{53.72}}  & \\cellcolor{mygray-bg}{\\textbf{47.70}}  & \\cellcolor{mygray-bg}{\\textbf{43.58}} & \\cellcolor{mygray-bg}{\\textbf{58.22}} \\\\\n\t\t\t\\hline\n\t\\end{tabular}}\\hspace{3mm}\n\t\\label{tab:vq_ablatives}\n\\end{table*}\n\n\n\\noindent\\textbf{Size of critical words in Q-CSS$^+$.} The influence of replacing different sizes of critical words is shown in Fig.~\\ref{fig:5} (b). From the results, we can observe that replacing only one word (\\textit{i}.\\textit{e}., top-1) achieves the best performance.\n\n\\noindent\\textbf{Proportion $\\delta$ between V-CSS$^+$ and Q-CSS$^+$.} The influence of different $\\delta$ is shown in Fig.~\\ref{fig:5} (c). From the results, we can observe that the performance is best when $\\delta = 0.5$.\n\nIn all the following experiments (including experiments on different benchmarks, models with different backbones, and models with contrastive training), we used the same best hyperparameter settings for both V-CSS$^+$ and Q-CSS$^+$.\n\n\n\\subsubsection{Architecture Generalization of CSST}\n\n\\noindent\\textbf{Settings.} Since our CSST is a model-agnostic training strategy, which can be seamlessly incorporated into any different VQA architecture. To evaluate the effectiveness of CSST to boost the debiasing performance of different backbones, we incorporated the CSST into multiple architectures including: UpDn~\\cite{anderson2018bottom}, RUBi~\\cite{cadene2019rubi}, LMH~\\cite{clark2019don}. Especially, RUBi and LMH are ensemble-based methods. For VQA-CP v2, we followed the same settings as prior works. For GQA-OOD, we used the adapting strategies mentioned in Sec.~\\ref{sec:expsetting}. All results are shown in TABLE~\\ref{tab:boost_models}. For more clear comparisons, we used superscript $^+$ to distinguish this improved version of CSS (Q-CSS\/V-CSS) with their respective initial counterparts~\\cite{chen2020counterfactual}.\n\n\\noindent\\textbf{Results.} Compared to these baseline models, our CSST (\\textit{i}.\\textit{e}., both CSST-G and CSST-L) can consistently improve the performance for all architectures. For different architectures, the behaviors of CSST-G and CSST-L are slightly different: CSST-G is more suitable for complex ensemble-based models (\\textit{e}.\\textit{g}., 9.21\\% absolute gains in LMH on VQA-CP v2), and CSST-L is more suitable for vanilla VQA models (\\textit{e}.\\textit{g}., 16.87\\% absolute gains in UpDN on VQA-CP v2). Meanwhile, when both two types of CSS are used (\\textit{i}.\\textit{e}., Q-CSS$^+$ and V-CSS$^+$), models often achieve better performance than single type CSS. Furthermore, the model with CSST (\\textit{i}.\\textit{e}., CSST-G or CSST-L) always achieves the best performance.\n\n\n\\subsubsection{Influence of Positive \\& Negative Sample Selection}\n\n\\noindent\\textbf{Settings.} To demonstrate the effectiveness of our proposed samples selection strategy, we compared CSST with a strong baseline CL~\\cite{liang2020learning}, which has shown effectiveness under our CSS mechanism. Meanwhile, to further show the importance of \\emph{counterfactual samples} in the sample selection, we separate these four negative samples into two groups: 1) ``CSS\": the first two negative samples from V-CSS$^+$ and Q-CSS$^+$; 2) ``Random\" (Rand): the last two randomly composed negative samples (cf. \\textsc{NEG\\_Sel} in Algorithm~\\ref{alg:cst}). All results are reported in TABLE~\\ref{tab:pos&neg}.\n\n\\noindent\\nonumber\\textbf{Results.} Compared to the CL baseline, model with CSST can achieve much better performance (\\textit{e}.\\textit{g}., 61.66\\% vs. 59.78\\%), which shows the importance of training sample diversity in contrastive training. Compared to the model with ``Rand\" negative samples, model with ``CSS\" negative samples also achieve better performance (\\textit{e}.\\textit{g}., 61.13\\% vs. 60.74\\%), which shows the importance of fine-grained differences in the samples for contrastive training. Meanwhile, when all negative samples are used, the model achieves the best performance.\n\n\n\\begin{figure*}[htbp]\n\t\\centering\n\t\\includegraphics[width=0.85\\linewidth]{figures\/visualization_img.pdf}\n\t\\caption{Examples of visual-explainable ability from VQA-CP v2 test set. The \\textcolor{green}{\\textbf{green}} boxes denote their scores $s(\\hat{a}, \\bm{v})$ \\textgreater $0$, \\textit{i}.\\textit{e}., positive contributions to final predictions. The \\textcolor{red}{\\textbf{red}} boxes denote their scores $s(\\hat{a}, \\bm{v})$ \\textless $0$, \\textit{i}.\\textit{e}., negative contributions to final predictions. Only objects which are highly related to the QA pair are shown (\\textit{i}.\\textit{e}., $\\mathcal{SIM} \\geq 0.6$). LMH-CSS is the model with only XE loss~\\cite{chen2020counterfactual}}\n\t\\label{fig:6}\n\\end{figure*}\n\n\\begin{figure*}[htbp]\n\t\\centering\n\t\\includegraphics[width=0.85\\linewidth]{figures\/visualization_words.pdf}\n\t\\vspace{-1.0em}\n\t\\caption{Visualization exmaples of question-sensitive ability from VQA-CP v2 test set. Different shades of green color in the question denotes the relative values of $s(\\hat{a}, \\bm{w})$, \\textit{i}.\\textit{e}., the word with darker green denotes the word has larger contribution to final predictions.}\n\t\\label{fig:7}\n\\end{figure*}\n\n\\subsection{Comparisons with State-of-the-Arts Models} \\label{sec:SOTA}\n\\subsubsection{Performance on VQA-CP v2 and VQA v2}\n\\noindent\\textbf{Settings.} We incorporated the CSST-G into model LMH~\\cite{clark2019don}, which is dubbed as \\textbf{LMH-CSST}, and compared it with the state-of-the-art models on both VQA-CP v2 and VQA v2. According to the model framework, we group them into: 1) AReg~\\cite{ramakrishnan2018overcoming}, GRL~\\cite{grand2019adversarial}, RUBi~\\cite{cadene2019rubi}, LMH~\\cite{clark2019don}, CSS+CL~\\cite{liang2020learning}, CF-VQA~\\cite{niu2021counterfactual}, GGE-DQ~\\cite{han2021greedy}, and LMH+SAR~\\cite{si2021check}. They are all ensemble-based models. 2) UpDn~\\cite{anderson2018bottom}, MuRel~\\cite{cadene2019murel}, SCR~\\cite{wu2019self}, CVL~\\cite{abbasnejad2020counterfactual}, Unshuffling~\\cite{teney2020unshuffling}, RandImg~\\cite{teney2020value}, and SSL~\\cite{zhu2020overcoming}. These models are vanilla VQA models. For complete comparisons, we reported results on two backbones: UpDn~\\cite{anderson2018bottom} and LXMERT~\\cite{tan2019lxmert}. For the UpDn backbone, we followed same settings as prior works~\\cite{clark2019don}. For the LXMERT backbone, we used adapting strategies mentioned in Sec.~\\ref{sec:expsetting}. Since SAR~\\cite{si2021check} is another model-agnostic model, which is orthogonal to our contributions, \\textit{i}.\\textit{e}., we can also equip LMH-CSST with SAR (dubbed as \\textbf{LMH-CSST+SAR}) to further boost performance. Moreover, following suggestions from~\\cite{teney2020value}, we also reported the performance gap on the \\emph{Other} category.\n\n\n\\noindent\\textbf{Results.} The results are reported in TABLE~\\ref{tab:SOTA_v2}. When trained and tested on the VQA-CP v2 dataset (\\textit{i}.\\textit{e}., the left side of TABLE~\\ref{tab:SOTA_v2}), LMH-CSST+SAR achieves a new SOTA performance on both UpDn and LXMERT backbones, with 66.49\\% and 67.49\\% accuracies, respectively. Compared to all models without extra pretraining or annotations (\\textit{i}.\\textit{e}., ``Pre.\" and ``Ann.\" in TABLE~\\ref{tab:SOTA_v2}), LMH-CSST still achieves SOTA performance with 61.66\\% accuracies. Particularly, CSST improves the performance of LMH with 9.21\\% (61.66\\% vs. 52.45\\%) and 6.05\\% (65.71\\% vs. 59.66\\%) absolute gains on UpDn and LXMERT backbones, respectively. When trained and tested on the VQA v2 dataset (\\textit{i}.\\textit{e}., the right side of TABLE~\\ref{tab:SOTA_v2}), CSST consistently improves the performance of LMH with 0.73\\% (62.37\\% vs. 61.64\\%) and 6.14\\% (65.71\\% vs. 59.57\\%) absolute gains on UnDn and LXMERT backbones, respectively. Different from previous SOTA models that suffer severe performance drops between VQA-CP v2 and VQA v2 (\\textit{e}.\\textit{g}., 9.50\\% and 6.28\\% on LMH), LMH-CSST+SAR can significantly decrease the performance gap into 2.55\\% and 2.17\\%, which demonstrates that the effectiveness of CSST to reduce language biases.\n\n\n\\subsubsection{Performance on VQA-CP v1}\n\n\\noindent\\textbf{Settings.} We also compared the LMH-CSST with state-of-the-art models on the VQA-CP v1. Similarly, we group these models into: 1) AReg~\\cite{ramakrishnan2018overcoming}, GRL~\\cite{grand2019adversarial}, RUBi~\\cite{cadene2019rubi} and LMH~\\cite{clark2019don} are all ensemble-based models. 2) GVQA~\\cite{agrawal2018don} and UpDn are vanilla VQA models, and GVQA is with SAN~\\cite{yang2016stacked} backbone. All settings are same as the ones on VQA-CP v2.\n\n\\noindent\\textbf{Results.} The results are reported in TABLE~\\ref{tab:SOTA_v1}. Compared to baseline models, both LMH-CSST and LMH-CSST+SAR achieve new state-of-the-art performance on two different backbones over all metrics. Particularly, the CSST improves the performance of LMH with a 7.89\\% (63.16\\% vs. 55.27\\%) absolution performance gains on UpDn backbones.\n\n\n\n\\subsubsection{Performance on GQA-OOD}\n\\noindent\\textbf{Settings.} We further compared LMH-CSST with state-of-the-art models on GQA-OOD. Similarly, we group these models into 1) RUBi~\\cite{cadene2019rubi} and LMH~\\cite{clark2019don} are ensemble-based models. 2) MCAN~\\cite{yu2019deep}, BAN4~\\cite{kim2018bilinear}, and UpDn are vanilla VQA models. All settings are same as experiments on VQA-CP. To adapt these ensemble-based methods to GQA-OOD, we used the adapting strategy mentioned in Sec.~\\ref{sec:expsetting}.\n\n\\noindent\\textbf{Results.} The results are reported in TABLE~\\ref{tab:SOTA_GQA}. For fair comparisons, the baselines (UpDn and LMH) with the LXMERT backbones were reimplemented by using the same adapting strategy. Compared to baselines, LMH-CSST+SAR achieves new state-of-the-art performance on most of the metrics on both two backbones. Especially for the most important metrics Acc-T~\\cite{kervadec2021roses}, our achieve the best performance over others (\\textit{e}.\\textit{g}., 47.98\\% vs. 43.37\\% and 48.07\\% vs. 47.2\\%).\n\n\n\n\n\n\n\n\n\n\n\\subsection{Improving Visual-Explainable Ability}\n\nWe will validate the effectiveness of CSST to improve visual-explainable ability by answering the following questions: \\textbf{Q1}: Can existing visual-explainable models be incorporated into the ensemble-based framework? \\textbf{Q2}: How does CSST improve the model's visual-explainable ability?\n\n\n\\subsubsection{CSST vs. SCR (Q1)}\n\n\\noindent\\textbf{Settings.} We equipped the existing state-of-the-art visual-explainable model SCR~\\cite{wu2019self} into the LMH framework, and compared it with CSST. Results are reported in TABLE~\\ref{tab:vq_ablatives} (a).\n\n\\noindent\\textbf{Results.} Since the training of all SOTA visual-explainable models (\\textit{e}.\\textit{g}., SCR~\\cite{wu2019self}, HINT~\\cite{selvaraju2019taking}) are not end-to-end, for fair comparisons, we used a well-trained LMH (\\textit{i}.\\textit{e}., 52.45\\% accuracies on VQA-CP v2) as the initial model. However, we observe that its performance continues to decrease from the start, which shows that the existing visual-explainable models can not be easily incorporated into the ensemble-based framework. In contrast, the proposed CSST can consistently improve performance on different backbones.\n\n\n\\subsubsection{Evaluations of Visual-Explainable Ability (Q2)}\n\n\\noindent\\textbf{Settings.} We evaluated the effectiveness of CSS to improve the visual-explainable ability on both quantitative and qualitative results. For quantitative results, since we lack human annotations about the critical objects for each question, we regard the $\\mathcal{SIM}$ scores (cf. \\textsc{IO\\_Sel} in Sec.~\\ref{sec:v-css}) as pseudo ground-truths. Thus, we design a new metric \\emph{Average Importance} ($\\mathcal{AI}$): the average $\\mathcal{SIM}$ score of the top-K objects with highest $|s(a, \\bm{v})|$. We formally define $\\mathcal{AI}$ as:\n\\begin{equation} \n\\label{eq:AI}\n\\small\n\\mathcal{AI} = \\frac{\\sum_{(I, Q)} \\left[ \\mathbf{1}(a = \\hat{a}) \\cdot \\sum_{k}\\mathcal{SIM}_k^{(I, Q)} \\right]}{\\sum_{(I, Q)} 1}\n\\end{equation}\nwhere $k$ is the index of top-K objects, $\\mathcal{SIM}_k^{(I, Q)}$ is the $\\mathcal{SIM}$ score of k-th object for the sample $(I, Q)$, and $\\mathbf{1}$ is an indicator function. The results are shown in TABLE~\\ref{tab:vq_ablatives} (b). We further demonstrate some qualitative results in Fig.~\\ref{fig:6}.\n\n\\noindent\\textbf{Results.} From TABLE~\\ref{tab:vq_ablatives} (b), we can observe that CSST dramatically improves $\\mathcal{AI}$ scores, which means the influential objects for predictions are more related to the QA pairs. From Fig.~\\ref{fig:6}, we can find that both CSS (LMH vs. LMH-CSS) and CST (LMH-CSS vs. LMH-CSST) help the model to make predictions based on critical objects (\\textit{i}.\\textit{e}., green boxes) and suppress the influence of irrelevant objects (red boxes).\n\n\n\n\\subsection{Improving Question-Sensitive Ability}\nWe will validate the effectiveness of our CSST to improve the question-sensitive ability by answering the following questions: \\textbf{Q3}: Does CSST help to improve the robustness to diverse rephrasings of questions? \\textbf{Q4}: How does CSST improve the model's question-sensitive abilities?\n\n\\subsubsection{Robustness to Rephrasings of Questions (Q3)}\n\\noindent\\textbf{Settings.} As discussed in previous work~\\cite{shah2019cycle}, being robust to diverse rephrasing of questions is one of key behaviors of a question-sensitive model. To more accurately evaluate the robustness, we re-splited the existing dataset VQA-Rephrasings~\\cite{shah2019cycle} with the same splits as VQA-CP, and denoted it as VQA-CP-Rephrasings. For evaluation, we used the standard metric \\emph{Consensus Score} $CS(k)$. Results are reported in TABLE~\\ref{tab:vq_ablatives} (c) (left). We refer readers to~\\cite{shah2019cycle} for more details about the VQA-Rephrasings and metric $CS(k)$.\n\n\\noindent\\textbf{Results.} From TABLE~\\ref{tab:vq_ablatives} (c), we can observe that Q-CSS$^+$ dramatically improves the robustness to diverse rephrasings of questions. Furthermore, V-CSS$^+$ can help to further improve the robustness, \\textit{i}.\\textit{e}., CSS$^+$ achieves better performance.\n\n\n\\subsubsection{Evaluations of Question-Sensitive Ability (Q4)}\n\n\\noindent\\textbf{Settings.} We evaluated the effectiveness of CSST to improve the question-sensitive ability on quantitative and qualitative results. For quantitative results, since there is no standard evaluation metric, we design a new metric \\emph{Confidence Improvement} ($\\mathcal{CI}$): Given a test sample $(I, Q, a)$, we remove a critical noun in question $Q$, and obtain a new test sample $(I, Q^*, a)$\\footnote{Test set is released at: \\href{https:\/\/github.com\/yanxinzju\/CSS-VQA}{https:\/\/github.com\/yanxinzju\/CSS-VQA}.}. Then we fed both two samples into the evaluated model, and calculated the confidence decreases of the ground-truth answer. We formally define $\\mathcal{CI}$ as:\n\\begin{equation}\n\\small\n\\mathcal{CI} = \\frac{\\sum_{(I, Q)}  \\mathbf{1}(P_{vqa}(a | I, Q) > P_{vqa}(a | I, Q^*)) \\cdot \\mathbf{1}(a = \\hat{a}) }{\\sum_{(I, Q)} 1}\n\\end{equation}\nwhere $\\hat{a}$ is the predicted answer for sample $(I, Q)$, $\\mathbf{1}$ is an indicator function. The results are reported in TABLE~\\ref{tab:vq_ablatives} (c). We further demonstrate some qualitative results in Fig.~\\ref{fig:7}.\n\n\\noindent\\textbf{Results.} From TABLE~\\ref{tab:vq_ablatives} (c), we can observe that CSST helps the model to benefit more from the critical words, \\textit{i}.\\textit{e}., removing critical words results in more confidence drops for the ground-truth answers. From Fig.~\\ref{fig:7}, we can find that both CSS (LMH vs. LMH-CSS) and CST (LMH-CSS vs. LMH-CSST) help the model to make predictions based on critical words (\\textit{e}.\\textit{g}., ``\nbus\" or ``objects\"), \\textit{i}.\\textit{e}., forcing the model to understand the whole questions before making predictions.\n\n\n\\section{Conclusions and Future Works}\nIn this paper, we proposed a model-agnostic Counterfactual Samples Synthesizing and Training (CSST) strategy to improve the VQA model's visual-explainable and question-sensitive abilities. The CSST is composed of two main components: CSS and CST, where CSS generates counterfactual training samples by masking critical objects or words, and CST consists of a XE training loss and a contrastive training loss. The CSST (\\textit{i}.\\textit{e}., CSS and CST) can consistently boost the performance of different VQA models. We validate the effectiveness of CSST through extensive comparative and ablative studies on three benchmarks and multiple different backbones. Moving forward, we are going to 1) extend CSST to other visual-language tasks that suffer severe language biases or other types of biases; 2) design a specific VQA backbone that benefits from the design of CSST.\n\n\n\n\n\n\n\n\n\n\n\n\n\\ifCLASSOPTIONcaptionsoff\n  \\newpage\n\\fi\n\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\n\n\n\n\n\n\n\n\n\n\n\n\nDeep neural networks (DNNs) have made remarkable progress in almost all areas of AI research in recent years. \nDespite their powerful ability, the development of modern neural network architectures is usually accompanied by an increase in computational budget, memory usage, and energy consumption \\cite{simonyan2014very,szegedy2015going,he2016deep,huang2017densely,tan2019efficientnet,dosovitskiy2020image,wang2020deep,liu2022convnet}. \nOn the other hand, common edge-computing platforms such as cell phones, mini-robots, AR glasses, and autopilot systems can only provide limited computing power and battery life.  \nConsequently, there is an immense technical gap between the evolutionary trend of modern neural networks and edge applications.\nOnly co-designed optimization from both sides can maximize the potential of deep neural networks on the edge.\n\n\\begin{figure}[]\n\\captionsetup[subfigure]{justification=centering, font=tiny}\n\\begin{center}\n\\hspace*{-10pt} \n\\begin{subfigure}[t]{.36\\textwidth}\n  \\centering\n  \\includegraphics[width=\\linewidth, scale=0.5]{comparison_sota.pdf}\n  \\caption{SOTA Comparison on ImageNet}\n  \\label{fig:sota comparison}\n\\end{subfigure}\n\\hspace{-1pt}\n\\vspace{-10pt}\n\\begin{subfigure}[t]{.12\\textwidth}\n    \\centering\n    \\includegraphics[width=17mm, height=52mm]{Motivation_Visualization.pdf}\n    \\centering\n    \\caption{Loss Landscape}\n    \\label{fig:loss landscape motivation}\n\\end{subfigure}\n\\end{center}\n\\vskip -2pt \n\\caption{(a) Comparison with SOTA binary neural networks on ILSVRC-2012 ImageNet in terms of accuracy and model param. \nGoogle's PokeBNN relies on a huge batch size of 8192, and ours is 512, in line with most previous works.\n(b) 3D loss landscape visualization \\cite{li2018visualizing} of full-precision ResNet-18 \\cite{he2016deep}, BinaryNet-18 \\cite{hubara2016binarized} and our BNext-18.\nThe smoothness of BNext-18 is particularly close to the full-precision counterpart.}\n\\label{fig:teaser}\n\\vskip -8pt \n\\end{figure}\n\nMotivated by this, researchers have developed various techniques to compress and accelerate modern neural networks (e.g., Network Pruning \\cite{han2015deep}, Knowledge Distillation\\cite{hinton2015distilling}, Compact Architecture Design \\cite{howard2017mobilenets}, low-bit Quantization \\cite{Jacob_2018_CVPR,banner2019post}). \nAmong them, Binary Neural Networks (BNNs) have received the attention of researchers owing to their vast potential on edge devices \\cite{courbariaux2016binarized,rastegari2016xnor, liu2018birealnet,real2binICLR20,liu2020reactnet, zhang2021pokebnn}.\nBy restricting the value range to 1-bit (e.g., ${\\pm1}$), the binary quantization technique can compress memory requirements by ${32\\times}$ and achieve a theoretical speedup of ${58\\times}$ on the CPU by replacing the floating-point dot products with XNOR and bit-counting operations \\cite{courbariaux2016binarized,rastegari2016xnor}. \nDespite the apparent efficiency advantages, binary neural networks have long suffered from optimization difficulties and accuracy degradation problems. \nPrevious works have made great efforts to narrow the accuracy gap to full-precision models such as ResNet-18 and ResNet-50 \\cite{courbariaux2016binarized,rastegari2016xnor,liu2018birealnet,real2binICLR20,liu2020reactnet,zhang2021pokebnn}. \nHowever, the latest BNNs are still far behind the latest full-precision counterparts with \\raisebox{-0.55ex}{\\~{}}10\\% lower accuracy \\cite{dosovitskiy2020image, liu2022convnet, liu2021swin}, due to their extreme degradation of representation capacity (${3.4 \\times 10^{38}}$ lower) \\cite{tu2022adabin} and increased optimization difficulty. \nTo solve the problem of non-differentiability in backpropagation, BNN training relies on gradient estimation techniques, e.g., the Straight-Through Estimator \\cite{courbariaux2016binarized}. \nBNNs also have a much coarser loss landscape where the optimization roadmap to the global minimum is more rugged than for full-precision networks (see Fig.~\\ref{fig:loss landscape motivation}).\nThe aforementioned problems complicate the optimization of BNNs, which then require specific architecture designs and optimization methodology for highly accurate results.\nTherefore, the question of \\emph{how a binary neural network can achieve the crucial accuracy level (e.g., 80\\%) on ILSVRC-2012 ImageNet} is still an open question.\n\n\nIn this paper, we build BNext, the first BNN with 80\\%+ accuracy on the ILSVRC-2012 ImageNet dataset. \nTo enhance the representative capacity, we first construct a novel binary processing unit with an adaptive information re-coupling structure called Info-RCP. \nPrevious works \\cite{rastegari2016xnor, real2binICLR20, zhang2021pokebnn} utilize channel-wise scaling to adjust the distribution after each binary convolution. \nAlthough this alleviates the information loss caused by binarization, these traditional Real2Binary-style designs \\cite{real2binICLR20} ignore the semantic gaps before and after binary convolution, which causes sub-optimal information coupling. \nInstead, we first reduce this distribution gap by scaling and shifting the binary convolution outputs with an extra BatchNorm and a PReLU layer. \nThen, we reshape the output distribution of this unit using a lightweight attention branch, which combines the information before and after this binary unit as the inputs. \nSubsequently, the basic BNext block is constructed by stacking multiple Info-RCP modules.\nBased on our analysis of classical binary architectures and motivated by the regularized architecture design of the recent vision transformer \\cite{han2021demystifying, liu2021swin}, we propose a novel element-wise attention (ELM-Attention) module to enhance the binary basic block further. \nMore specifically, each basic block utilizes an element-wise bypass using multiplication to dynamically calibrate the output of the first Info-RCP module in the basic block. \nFinally, we build the BNext family with different model sizes by stacking the basic block using different stage-design strategies. \nAs shown in Fig.~\\ref{fig:loss landscape motivation}, the loss landscape of BNext-18 is much smoother and more complete compared to a traditional binary ResNet-18 and is pretty close to a full-precision ResNet-18. \n\nThe optimization pipeline design plays a crucial role in high-accuracy BNN optimization.  \nCompared to 32-bit DNNs, BNNs own a coarser loss landscape, which adds more obstacles for optimizing towards the global minimum.\nIt thus requires more fine-grained and flexible gradient information to escape from sub-optimal convergence.\nTo this end, Knowledge Distillation (KD) is a commonly used technique in previous works \\cite{real2binICLR20,liu2020reactnet,liu2021adam}.\nHowever, when we attempt to push the accuracy level of BNNs from the current 70\\% to 80\\% level, we observe that BNNs are more prone to counter-intuitive overfitting using the standard KD techniques.\nTherefore, improving the teacher model selection and KD pipeline design is essential.\nWe thus propose a new metric, \\emph{Knowledge Complexity}, as a simple yet effective indicator for teacher model selection.\nWe further use the selected teachers (e.g., \\cite{tan2019efficientnet,liu2022convnet}), including a principal teacher model and an assistant teacher group.\nWe adaptively pick the best suitable assistant teacher at different training stages to improve the student's accuracy and avoid overfitting.\nThe assistant teachers serve as regularizers for the knowledge from high-confidence prediction. \nFurthermore, modern DNNs \\cite{dosovitskiy2020image, liu2022convnet, liu2021swin} usually rely on well-designed data augmentation pipelines. \nTo the best of our knowledge, this is the first work that thoroughly verifies recently proposed data augmentation techniques on BNNs and provides practical guidance based on extensive experiments. \nOverall, as shown in Fig.~\\ref{fig:sota comparison}, BNext outperforms all previous BNNs by a large margin and is the first binary neural network to reach the 80\\% accuracy level on ImageNet.\n\n\n\nOur main contributions can be summarized as follows:\n\\begin{enumerate}\n    \\item\n   \n    \\emph{BNext}, a novel binary architecture with a smoother loss landscape than existing popular designs which makes it easier to optimize.\n    \n    \\item\n   \n    The \\emph{Diversified Consecutive Knowledge Distillation} technique, which alleviates the counter-intuitive overfitting problem in highly accurate BNNs. \n    \n    \\item\n   \n    A modern training pipeline for BNNs based on a fair and comprehensive verification of up-to-date data augmentation techniques.\n\n    \\item\n   \n    The first BNN to achieve $>$\\textbf{80\\%} top-1 accuracy on ImageNet, which reshapes our vision of BNNs' potential.\n\\end{enumerate}\n\n\n\n\n\n\\section{Related Work}\n\\label{sec:related work}\n\n\\textbf{BNN Optimization.} Binary neural networks are difficult to optimize.\nCompared to their 32-bit counterparts, BNNs are almost universally non-differentiable.\nBinaryNet \\cite{courbariaux2016binarized} solved this challenge and proved the feasibility of optimizing BNNs by using the straight-through-estimation (STE) technique.\nLater works (e.g. \\cite{liu2018birealnet, lin2020rotated, wang2021gradient, guo2021boolnet}) have attempted to improve the optimization using different variants of STE for approximating the gradient of sign function.\n\\cite{NEURIPS2019_9ca8c9b0} revisited the functional role of latent weights in BNNs and proposed a specialized optimizer BOP to flip the binary states.\nReal2BinaryNet \\cite{real2binICLR20} applies knowledge distillation in three stages to optimize BNNs.\nSimilar works (\\cite{liu2020reactnet, zhang2021pokebnn}) simplified the KD pipeline to a two-stage process.\nMoreover, Liu \\etal \\cite{liu2020reactnet} explore how training strategies (e.g., optimizers, weight decay) to aid BNN optimization.\nIn this paper, we provide the first in-depth analysis of the relationship between BNN optimization and the corresponding architecture design.\nThe analysis helps us to understand why BNNs are hard to optimize and how to construct a highly optimization-friendly architecture. \n\n\\textbf{Knowledge Distillation.} Knowledge distillation \\cite{hinton2015distilling} is used to ``distill'' dark knowledge from a pre-trained teacher model to a  student model.\nIn more recent works, researchers have turned their attention to a deeper understanding of this technique.\nFor example, it was found that there is a surprisingly large discrepancy between the predictive distributions of the teacher and the student, and a good fit for the teacher paradoxically does not generalize well for the student \\cite{stanton2021does}.\nBeyer \\etal \\cite{beyer2022knowledge} show that certain implicit design decisions drastically affect the effectiveness of distillation. \nPark \\etal \\cite{park2021learning} improves the distillation process by building student-friendly teachers and implicitly reveal the presence of a knowledge mismatch between teacher and student. \nMeanwhile, modern BNN optimization \\cite{real2binICLR20,liu2020reactnet,zhang2021pokebnn} heavily relies on knowledge distillation techniques \\cite{hinton2015distilling, shen2020meal}. \nThe soft label from a pre-trained teacher provides a more fine-grained supervision signal than a one-hot label. \nTherefore, the BNN research community is encouraged to explore the latest advances in KD technologies and incorporate new insights into BNN research.\n\n\n\\textbf{Modern Neural Network Optimization.} \nThe success of modern deep neural networks \\cite{tan2019efficientnet,dosovitskiy2020image, liu2021swin,liu2022convnet} for image classification relies not only on architecture design ideas, such as the attention mechanism but also on a bag of up-to-date optimization tricks. \nVarious data augmentation strategies such as MixUp \\cite{zhang2017mixup}, CutMix \\cite{yun2019cutmix}, RandAugmentation \\cite{cubuk2020randaugment} and Augmentation Repetition \\cite{hoffer2020augment} are evaluated to be beneficial for the generalization of regular 32-bit networks \\cite{ridnik2022solving}. \nHowever, for highly accurate BNNs it is still an open question whether regular strategies can be applied or new BNNs-specific augmentation strategies should be used. \nThis work gives the first fair and comprehensive empirical study on modern optimization tricks for BNNs to check their feasibility in large-scale image classification tasks. \nThe detailed ablation study demonstrates the necessity of revisiting BNN-specific augmentation strategies and reveals possible solutions.\n\n\n\n\n\n\n\\begin{figure*}[]\n\\captionsetup[subfigure]{justification=centering, font=tiny}\n\\begin{center}\n\\begin{subfigure}[t]{.13\\textwidth}\n    \\centering\n    \\includegraphics[width=22mm, height=22mm]{visualization_resnet18.pdf}\n    \\centering\n    \\caption{ResNet-18}\n    \\label{fig:loss landscape resnet18}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[t]{.13\\textwidth}\n    \\centering\n    \\includegraphics[width=22mm, height=22mm]{visualization_bresnet18.pdf}\n    \\centering\n    \\caption{BinaryNet-18}\n    \\label{fig:loss landscape binarynet-18}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[t]{.13\\textwidth}\n    \\centering\n    \\includegraphics[width=22mm, height=22mm]{visualization_bresnet_double_shortcut.pdf}\n    \\centering\n    \\caption{BiRealNet-18}\n    \\label{fig:loss landscape birealnet-18}\n\\end{subfigure}\n\\hfil\n\\begin{subfigure}[t]{.13\\textwidth}\n    \\centering\n    \\includegraphics[width=22mm, height=22mm]{visualization_real2binary.pdf}\n    \\centering\n    \\caption{Real2BinaryNet-18}\n    \\label{fig:loss landscape real2binarynet-18}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[t]{.13\\textwidth}\n    \\centering\n    \\includegraphics[width=22mm, height=22mm]{visualization_bnext_mix2binary.pdf}\n    \\centering\n    \\caption{BiRealNet-18 + Info-RCP (ours)}\n    \\label{fig:loss landscape birealnet-18 + info-rcp}\n\\end{subfigure}\n\\hfil\n\\begin{subfigure}[t]{.13\\textwidth}\n    \\centering\n    \\includegraphics[width=22mm, height=22mm]{visualization_bnext_elm_attention.pdf}\n    \\centering\n    \\caption{BiRealNet-18 + ELM-Attention (ours)}\n    \\label{fig:loss landscape birealnet-18 + ELM Attention}\n\\end{subfigure}\n\\hfil\n\\begin{subfigure}[t]{.13\\textwidth}\n    \\centering\n    \\includegraphics[width=22mm, height=22mm]{visualization_bnext_full.pdf}\n    \\centering\n    \\caption{BNext (ours)}\n    \\label{fig:loss landscape bnext}\n\\end{subfigure}\n\\hfill\n\\end{center}\n\\caption{The loss landscape of popular BNNs (contour line view) on CIFAR10 dataset. we follow the setting of \\cite{li2018visualizing}.  ``Info-RCP'' and ``ELM-Attention'' are the module designs in this paper. The corresponding 3D views are available in the supplementary material.}\n\\label{fig:loss_landscape_visualization}\n\\end{figure*}\n\n\\begin{figure}[]\n\\captionsetup[subfigure]{justification=centering, font=tiny}\n    {\\begin{subfigure}[c]{.28\\columnwidth}\n        \\includegraphics[height=26mm,clip=false]{Conv_Block_XNOR-Net.pdf}\n        \\caption{XNOR Net}\n        \\label{fig:xnor conv block}\n    \\end{subfigure}}\n    {\\begin{subfigure}[c]{.32\\columnwidth}\n       \n        \\includegraphics[ height=26mm,clip=false]{Conv_Block_Real2BinaryNet.pdf}\n        \\caption{Real2BinaryNet}\n        \\label{fig:real2binary conv block}\n    \\end{subfigure}}\n    {\\begin{subfigure}[c]{.38\\columnwidth}\n       \n        \\includegraphics[height=26mm,clip=false]{Infor-RCP_Block.pdf}\n        \\caption{Info-RCP (ours)}\n        \\label{fig:infor-rcp}\n    \\end{subfigure}}\n\\caption{Binary convolution module design comparison among XNOR-Net \\cite{rastegari2016xnor}, Real2BinaryNet \\cite{real2binICLR20} and Ours.}\n\\label{fig:Conv Block Compare}\n\\end{figure}\n\\section{BNext Architecture Design}\n\\label{sec:macro architecture design}\nIn this section, we introduce the motivation and details of BNext architecture design. \nIn section \\ref{subsection:Visualizing Optimization Bottleneck of BNNs}, we visualize the loss landscape of the most popular binary neural networks and provide our observations and insights.\nWe then present the two core modules of BNext architecture design and the structure of the BNext model family in section \\ref{Section:Building Optimization-Friendly BNNs}.\n\n\n\n\n\\subsection{Visualizing the Optimization Bottleneck}\n\\label{subsection:Visualizing Optimization Bottleneck of BNNs}\nBinary neural networks are lightweight neural networks constructed from stacked 1-bit convolutions. \nIn forward propagation, both the inputs $x_r^t$ and the proxy weights $w_r^t$ in each 1-bit convolution are binarized into ${\\pm{1}}$ using the sign function before the dot product computation. \nIn backward propagation, the sign function is approximated using the gradient of hardtanh function \\cite{hubara2016binarized}, so that we can compute the gradient of $w_b^t$ and optimize the weights $w_r^t$.\nThe gradient of $w_b^t$ is accumulated in the proxy weights $w_r^t$ during each iteration  \\cite{courbariaux2016binarized}. This process can be mathematically formulated as follows:\n\\begin{equation}\n\\label{func. binarization}\n\\small\nx_b^t, w_b^t = \\mathrm{binarize}\\left( x_r^t, w_r^t \\right) =\\begin{cases}\n\t+1&\t\tif\\ x_r^t, w_r^t\\ge 0,\\\\\n\t-1&\t\t\\mathrm{otherwise.}\\\\\n\\end{cases}\n\\end{equation}\n\\begin{equation}\nw_{r}^{t+1}=w_{r}^{t}-\\alpha \\cdot \\frac{\\partial \\ell ^t}{\\partial w_{r}^{t}},\\ \\ \\ \\frac{\\partial \\ell ^t}{\\partial w_{r}^{t}}=\\frac{\\partial \\ell ^t}{\\partial w_{b}^{t}}\\cdot \\frac{\\partial w_{b}^{t}}{\\partial w_{r}^{t}},\n\\end{equation}\nwhere ${\\frac{\\partial w_{b}^{t}}{\\partial w_{r}^{t}}=1_{||w_{r}^{t}||\\le c},}$ represents the Straight-Through-Estimation (STE) strategy. \n$r$, $b$, and $t$ denote real-valued variable, binary variable, and iteration number, respectively. \nDespite STE partially solving the non-differentiable problem, the sparse binary representation and gradient mismatch problem make BNNs suffer from serious accuracy degradation \\cite{courbariaux2016binarized,rastegari2016xnor}. \nVarious subsequent designs \\cite{rastegari2016xnor,liu2018birealnet,real2binICLR20,bethge2020meliusnet,liu2020reactnet} have tried to solve the two core bottlenecks of BNNs: capacity degradation and optimization difficulty.  \nTo understand the optimization bottleneck in BNNs, we first focus on the most popular BNN architectures. \nSpecifically, previous researchers have proposed several performance-beneficial structures in BNNs, such as double residual connection \\cite{liu2018birealnet}, attention as introduced in Real2BinaryNet \\cite{real2binICLR20, zhang2021pokebnn} and adaptive distribution reshaping \\cite{liu2020reactnet}.\nHowever, the shared insight behind these techniques is still unclear. \n\n\n\nLiu \\etal \\cite{liu2021adam} studied the impact of training strategies and optimizers on BNNs by visualizing the loss landscape.\nInspired by this idea, we further use it as an indicator to analyze the most popular binary architectures, including BinaryNet \\cite{courbariaux2016binarized}, BiRealNet \\cite{liu2018birealnet}, Real2BinaryNet \\cite{real2binICLR20}, and their full-precision counterpart ResNet-18 \\cite{he2016deep}. \nWe plotted the corresponding 2D loss landscapes (contour line view) for each network, as shown in Fig.~\\ref{fig:loss_landscape_visualization}. \nBy comparing Fig.~\\ref{fig:loss landscape resnet18} and Fig.~\\ref{fig:loss landscape binarynet-18}, we can see that directly binarizing the network \\cite{courbariaux2016binarized} makes the loss landscape surface extremely discontinuous and rugged, the contour lines are unstructured and are unaligned in Fig.~\\ref{fig:loss landscape binarynet-18}. \nThe highly rugged landscape surface makes BNNs easier to converge into sub-optimal minima and more sensitive to inappropriate optimization processes. \nIn Fig.~\\ref{fig:loss landscape birealnet-18}, the double skip-connection design proposed in BiRealNet \\cite{liu2018birealnet} alleviates the information sparsity problem to some extent and allows a more continuous information flow through bypass connections. \nTherefore, we can observe more structured and complete contours, although they still appear rugged.\nReal2BinaryNet \\cite{real2binICLR20} suggests reshaping the output of binary convolution in a data-driven manner by using a trainable attention module, which reduces the information bottleneck.\nCompared to Fig.~\\ref{fig:loss landscape binarynet-18} and \\ref{fig:loss landscape birealnet-18}, Real2BinaryNet further flattens the area around the global minimum and the gap between each contour line in Fig.~\\ref{fig:loss landscape real2binarynet-18} is more uniform, which means more robust to the initialization and optimization. \n\nInspired by these findings in the visualization, we formulate the following three hypotheses:\n(1) The activation binarization restricts the feature patterns available for forward propagation before each 1-bit convolution layer. \n(2) Properly reshaping the distribution after a binary convolution is essential for the feature modeling between adjacent 1-bit processing units. \n(3) An effective shortcut design enhances the information density in forward and backward propagation of BNNs. \nAlthough a lot of previous works are based on similar optimization ideas \\cite{rastegari2016xnor, liu2018birealnet, real2binICLR20, liu2020reactnet, zhang2021pokebnn}, as seen from the loss landscape, their optimization effects can still be improved compared to the full-precision backbone Fig.~\\ref{fig:loss landscape resnet18}. \nThis conclusion motivates us to develop a better architectural design.\n\n\n\\begin{figure*}[]\n\\captionsetup[subfigure]{justification=centering}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{Pipeline.pdf}\n\\end{center}\n\\caption{The architecture of BNext. ``Processor'' is the basic binary convolution with improved Info-RCP design, ``BN'' represents batch normalization. ``C'' indicates the base width (channel number). ``N'' and ``M'' are the number of basic blocks in each stage respectively.}\n\\label{fig:architecture}\n\\end{figure*}\n\n\\subsection{Building Optimization-Friendly BNNs}\n\\label{Section:Building Optimization-Friendly BNNs}\nThis section introduces the core building components and their design ideas of our BNext network design.\n\n\\textbf{Binary Convolution with Information Recoupling} \nTo facilitate the information propagation and alleviate possible bottlenecks between adjacent 1-bit processing units, we construct a new binary convolution module with adaptive information reshaping and coupling, as shown in Fig.~\\ref{fig:infor-rcp}. \nDuring the forward pass, the output distribution of a binary convolution is adjusted with a subsequent Batch Norm and a PReLU layer. \nNext, we explicitly calibrate the information flow through the shift and scale design.\nWe first use a shortcut connection to fuse the input and output of the main branch [BConv-BN-PReLU] by element-wise additioning. \nThe output is subsequently fed into a lightweight gap-aware Squeeze-and-Expand (SE) \\cite{hu2018squeeze} branch, which is used to scale the main branch output. \nSpecifically, both features before and after the main branch are considered as the input, which enables SE to perceive more information flow changes and to adaptively adjust the output distribution through learning.\nThis design is referred to as Information-Recoupling (Info-RCP) module in the rest of the paper.\nCompared to XNOR-Net and Real2Binary-Net (Fig.~\\ref{fig:xnor conv block} and \\ref{fig:real2binary conv block}), our design offers a significantly more robust and flexible post-convolution reshaping mechanism.\nOur ablation study (Table \\ref{tab:ablation on module designs}) verifies the effectiveness of Info-RCP. \nMeanwhile, we find that the loss landscape also becomes more uniform and smooth (Fig.~\\ref{fig:loss landscape birealnet-18 + info-rcp}), indicating that Info-RCP can effectively facilitate optimization.\n\n\\textbf{Basic Block Design with ELM-Attention}\nMotivated by the analysis in Sec. \\ref{subsection:Visualizing Optimization Bottleneck of BNNs} and the recent regularized architecture design ideas in the vision transformer \\cite{vaswani2017attention,dosovitskiy2020image, han2021demystifying}, we investigate a novel basic block design with an enhanced bypass structure. \nAs shown in Fig.~\\ref{fig:architecture}, we stack multiple Info-RCP modules (Processor) to build the fundamental building block. \nEach Processor is followed by a BatchNorm layer. \nThen we use continuous residual connections to enclose each basic Processor [Info-RCP-BN], which relieves the information bottlenecks in forward propagation. \nFurthermore, we utilize an element-wise bypass multiplication to dynamically calibrate the output of the first 3$\\times$3 Processor. \nThe element-wise multiplication is helpful for the forward feature fusion and propagation in each block, which can substantially improve the loss landscape, as shown in Fig.~\\ref{fig:loss landscape birealnet-18 + ELM Attention}.\nFinally, by combining both Info-RCP and ELM-Attension we obtain a significantly better loss landscape, as shown in Fig.~\\ref{fig:loss landscape bnext}.\nIts smoothness is already very close to the full-precision backbone.\n\n\n\\textbf{BNext Family Construction} \nWe build the BNext family by stacking basic blocks under different stage design strategies.\nFor a fair comparison with existing works, we apply the proposed block designs to the popular backbone MobileNetV1 and get the BNext variants by changing the block width and depth in each stage. \nThe overall BNext family architectures are shown in Fig.~\\ref{fig:architecture}.  \nSpecifically, we build four BNext variants with increasing capacity to test the scalability of BNext architecture.\n\n\nTable \\ref{tab:architecture information} shows the configuration of BNext networks.\n$C$ indicates the width of input layer, $N$ and $M$ denote the stage depth ratios, as shown in Fig.~\\ref{fig:architecture}. \n\\begin{table}[]\n\\centering\n\\caption{BNext architecture overview. ``Q'' represents the quantization bit-width of weights and activations (W\/A) in Fig. \\ref{fig:architecture} for the first layer, the SE branch and the output layer, respectively.\n}\n\\setlength\\tabcolsep{4pt}\n\\footnotesize\n\\begin{tabular}{cccc}\n\\hline\nModels  & Stage Ratios (N,M)  & Base Width (C) & Q (W\/A) \\\\ \\hline\nBNext-T & 1:1:3:1       & 32         & 8\/8-4\/8-8\/8     \\\\ \\hline\nBNext-S & 1:1:3:1       & 48         & 8\/8-4\/8-8\/8     \\\\ \\hline\nBNext-M & 2:2:4:2       & 48         & 8\/8-4\/8-8\/8     \\\\ \\hline\nBNext-L & 2:2:8:2       & 64         & 8\/8-4\/8-8\/8     \\\\ \\hline\n\\end{tabular}%\n\\label{tab:architecture information}\n\\end{table}\n\n\\section{Knowledge Distillation Strategy}\n\n\\subsection{Knowledge Complexity for Teacher Selection}\nKnowledge Distillation (KD) is an essential optimization technique for BNNs \\cite{real2binICLR20, liu2020reactnet,liu2021adam,zhang2021pokebnn}.\nTo achieve the target of 80\\% accuracy on ImageNet, we thus need to assess which teacher models provide the best supervision for BNNs. \nThe only study that has studied the impact of teacher selection on high-accuracy BNN optimization is PokeBNN \\cite{zhang2021pokebnn}, which uses a high precision VIT teacher instead of a ResNet-50 \\cite{dosovitskiy2020image}. \nHowever, they report that the student model does not gain accuracy from a more accurate teacher, reflecting that just considering the teacher's accuracy is insufficient.\nIn our work, we did a more comprehensive investigation.\nWe observed different degrees of overfitting on BNext-T and BNext-L if we used stronger full-precision teachers and the standard KD.\nDue to space limitations, we provide more details of the counter-intuitive overfitting in the Appendix (Section \\ref{appendix: Counter Intuitive Overfitting}). \nTo solve this issue, we propose a simple yet effective metric, \\emph{Knowledge Complexity} (KC) incorporating accuracy and model compactness for teacher selection, and a diversified consecutive KD method.\n\\begin{equation}\n    \\mathcal{KC}(T) = \\log{\\left(1+\\frac{\\mathrm{Params}(T)}{\\mathrm{Accuracy}(T) \\cdot 10^6 + 10^{-8}}\\right)},\n\\label{func. knowledge complexity}       \n\\end{equation}\nwhere $T$ is the selected teacher model, $Params$ is the number of parameters and $Accuray$ the teacher's test accuracy. \nGenerally speaking, KC defines the cost of teacher parameter-space complexity to the dark knowledge effectiveness. \nTo the best of our knowledge, this is the first work to investigate this problem for BNNs and propose a practical solution.\n\nThe \\emph{Knowledge Complexity} analysis of three popular DNN families and the corresponding teacher and student top-1 accuracy are shown in Fig.~\\ref{fig:knowledge complexity}.\nBNext-T has been used as the student in this study.\nAs common sense, we need a stronger teacher to pull the upper limit of students' accuracy.\nHowever, as indicated by Fig.~\\ref{fig:knowledge complexity}, a teacher with both high accuracy and high KC will readily cause the binary student to have the overfitting problem (significant drop in students' top-1 acc.).  \nA possible reason is that training a binary student with a too-complex teacher at the initial training stage does not regard the tremendous discrepancy between the predicted distributions of the teacher and the student.\nSimilar observations have also been obtained in \\cite{park2021prune}, where a pruned teacher works better than the original one.\nTherefore, we suggest prioritizing teachers with lower KC for BNNs' KD training.\n\n\n\n\n\n\\subsection{Diversified Consecutive KD}\n\\label{subsec: knowledge distillation optimization}\nWe develop a diversified consecutive KD algorithm (Alg. \\ref{Algorithm: Curriculum Knowledge Distillation}) to boost the performance of KD training further. \nThis method consists of a gap-sensitive knowledge ensemble and a knowledge-boosting strategy.   \n\n\\begin{figure}[]\n\\captionsetup[subfigure]{justification=centering}\n\\begin{center}\n\\includegraphics[width=0.243\\textwidth]{teacher_train_top1_kc_alt.pdf}\n\\hfill\n\\includegraphics[width=0.227\\textwidth]{student_test_top1_kc_alt.pdf}\n\\end{center}\n\\vskip -12pt \n\\caption{Counterintuitive overfitting effects and the knowledge-complexity impacts under the selection of different teachers. The two y-axes indicate the teacher (left) and student (right) test Top-1 accuracy, evaluated on ImageNet with the input resolution 224\u00d7224. More details are available in supplementary materials.}\n\\vskip -8pt \n\\label{fig:knowledge complexity}\n\\end{figure}\n\n\\begin{algorithm}\n\\caption{Diversified Consecutive KD Algorithm}\n\\begin{algorithmic}\n\\Require A binary student $S_{\\theta}$, a strong teacher $T_{\\theta}^{s}$ and a group of assistant teachers \\{$T_{\\theta,1}^{w}, ..., T_{\\theta,n}^{w}$\\}. Training dataset $D$. Total epochs $N$. Initial assistant teacher index $m=1$. AMP quantization $Q$, provided by PyTorch. \n\n\\For{${i}$ in epochs}: \\Comment{start training}\n    \\For{$j$, $x$ in enumerate($D$)}:\n        \\State 1.forward  $S_{\\theta}(x), Q(T_{\\theta}^{s})(x), Q(T_{\\theta,m}^{w})(x)$,\n        \\State 2.calculate $\\mathcal{L}(x, T_{\\theta}^{s}, T_{\\theta}^{w})$, Func. \\ref{func. capacity aware KD}, \n        \\State 3.backward propagation and updates, Func. \\ref{func. binarization}.\n    \\EndFor\n    \\If{$i$\\%$\\left(\\frac{N}{n}\\right)$=0}: \\Comment{switch assistant teacher $T_{\\theta,m}^{w}$}\n        \\State m+=1,  \\Comment{increase knowledge effectiveness}\n    \\EndIf\n\\EndFor             \\Comment{end training}\n\\end{algorithmic}\n\\label{Algorithm: Curriculum Knowledge Distillation}\n\\end{algorithm}\n\\textbf{A Gap-Sensitive Knowledge Ensemble}\nOur core design principle is to use the knowledge ensemble to increase the diversity of teacher knowledge. \nMoreover, the knowledge ensemble is not static but dynamically adapted regarding the difficulty of knowledge.\nTo guarantee a consistent accuracy upper bound, we use the same strong teacher ${T_{\\theta}^s}$ during training.\nWe obtain additional guidance by adding another less confident assistant teacher ${T_{\\theta}^w}$ to enhance the overall knowledge diversity.\nThe binary student ${S_{\\theta}}$ consistently learns from two teachers.\nThe loss functions are mathematically formulated as follows:\n\\begin{multline}\n    \\mathcal{L}(x)=\\sigma(x){KL}_{S_{\\theta}, T_{\\theta}^s}(x) + (1-\\sigma(x)){KL}_{S_{\\theta}, T_{\\theta}^w}(x), \\\\\n    \\sigma(x) = \\frac{e^{\\mathcal{L}_{CE}(S_\\theta, T_\\theta^s)}}{\\sum_i e^{\\mathcal{L}_{CE}({S_\\theta, T_\\theta^i))}}}, \\ \\ \\ \\mathcal{L}_{CE}^i = -\\sum_{x}S_\\theta(x)\\log(T_{\\theta}^i(x)),\n   \n\\label{func. capacity aware KD}    \n\\end{multline}\nwhere $CE$ and $KL$ indicate the Cross-Entropy and KL-Divergence loss \\cite{shen2020meal}, respectively. \nWe calculate the distance between student prediction and each teacher prediction for every forward pass. \nWe then weigh each knowledge entry $KL_{S_{\\theta}, T_{\\theta}^i}$ using a Softmax term $\\sigma(x)$. \nConsequently, our dynamic weighing and fusion method constrains the capacity-limited binary student ${S_{\\theta}}$ to pay more attention to the more complex knowledge. \n\n\\textbf{Knowledge-Boosting Strategy} \n\\label{subsec: Knowledge-Boosting Strategy}\nTo further decrease the knowledge discrepancy in BNext-M\/L optimization, we develop a knowledge-boosting strategy.\nAs aforementioned, the strong teacher ${T_{\\theta}^s}$ is consistent throughout the training, but the assistant teacher is selected from a pre-defined candidate group.\nThe assistant teachers will join the KD process one by one in increasing order of accuracy and knowledge-complexity and have the same length of active time.\nIn this way, we use adapted assistants at different stages of training to adjust teacher knowledge, gaining better diversity. \nBinary students can thus receive more diverse supervision information, which have more robust regularization and can effectively solve the strong teacher overfitting problem shown in Fig.~\\ref{fig:knowledge complexity}.\nThis is an important reason why BNext-M\/L can achieve such high accuracy.\nThe teacher selection rule is based on knowledge complexity statistics. \nA detailed description of the assistant teacher group settings can be found in the supplementary material.\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\\begin{table*}[]\n\\centering\n\\caption{Performance comparison with modern DNN designs and SOTA binary designs on ImageNet.\n``W\/A'' indicates the bit-width of convolutions except for the input layer.\n``QOPs'' is the INT-4\/8 operations in total and ``$\\dagger$\" indicates results with post-training quantization.\nIn most cases, BNext performs better than models of similar (or larger) size - marked with (a) through (e).\n}\n\\setlength\\tabcolsep{5pt}\n\\footnotesize\n\\begin{tabular}{@{}rlcccc@{}p{24pt}@{}rlrccccc@{}}\n\\cline{2-6} \\cline{9-15}\n~ & \\thead{Regular\\\\ Designs}     & \\thead{W\/A}   & \\thead{OPs\\\\($10^{8}$)}   & \\thead{\\#Param\\\\(MB)}     & \\thead{Top-1\\\\(\\%)} &~&\n~ & \\thead{Binary\\\\Designs}       & \\thead{BOPs\\\\$(10^{9})$}      & \\thead{QOPs\\\\$(10^{6})$}       &\\thead{FLOPs\\\\$(10^{8})$}      & \\thead{OPs\\\\$(10^{8})$}       & \\thead{\\#Param\\\\(MB)} & \\thead{Top-1\\\\(\\%)} \\\\ \\cline{2-6} \\cline{9-15}\n(a) & LQ-Net-18 \\cite{zhang2018lq}       & 2\/2   & -               & \\phantom{00}8.4 & 64.9          &~&  ~  & BNN \\cite{hubara2016binarized}           & 1.70      & -                & 1.20        & 1.47      & \\phantom{0}4.2       & 42.2          \\\\\n ~  & LQ-Net-18 \\cite{zhang2018lq}       & 4\/4   & -               & \\phantom{0}16.8 & 69.3          &~&  ~  & XNOR-Net \\cite{rastegari2016xnor}        & 1.70      & -                & 1.20        & 1.47      & \\phantom{0}4.2       & 51.2          \\\\\n(b) & MobileNetV2 \\cite{bai2019}         & 8\/8   & \\phantom{0}0.8  & \\phantom{00}3.5 & 68.3          &~&  ~  & XNOR-Net++ \\cite{bulat2019xnor}          & 1.70      & -                & 1.20        & 1.47      & \\phantom{0}4.2       & 57.1          \\\\\n(d) & ResNet-50 \\cite{bai2019}           & 8\/8   & \\phantom{0}4.8  & \\phantom{0}24.6 & 74.7          &~& (a) & Bi-RealNet-18 \\cite{liu2018birealnet}    & 1.68      & -                & 1.39        & 1.65      & \\phantom{0}4.2       & 56.4          \\\\\n ~  & ResNet-18 \\cite{he2016deep}        & 32\/32 & \\phantom{}18.0  & \\phantom{0}44.6 & 69.6          &~& (b) & Bi-RealNet-34 \\cite{liu2018birealnet}    & 3.53      & -                & 1.39        & 1.94      & \\phantom{0}5.1       & 62.2          \\\\\n(c) & MobileNetV3 \\cite{howard2019searching} & 32\/32 & \\phantom{0}2.2 & \\phantom{0}21.6 & 75.2          &~&  ~  & Bi-RealNet-152 \\cite{liu2020bi}          & 10.7      & -                & 4.48        & 6.15      & -                    & 64.5          \\\\\n~ & ResNet-50 \\cite{he2016deep}        & 32\/32 & \\phantom{}38.0  & \\phantom{0}97.5 & 76.0          &~& (b) & MeliusNet-29 \\cite{bethge2020meliusnet}  & 5.47      & -                & 1.29        & 2.14      & \\phantom{0}5.1       & 65.8          \\\\\n(e) & DeiT-S \\cite{touvron2021training}  & 32\/32 & \\phantom{}46.0  & \\phantom{0}83.9 & 79.7          &~&  ~  & MeliusNet-42 \\cite{bethge2020meliusnet}  & 9.69      & -                & 1.74        & 3.25      & \\phantom{}10.1       & 69.2          \\\\\n(e) & RegNetY-4G \\cite{radosavovic2020designing} & 32\/32 & \\phantom{}40.0  & \\phantom{0}80.1 & 80.0          &~&  ~  & MeliusNet-59 \\cite{bethge2020meliusnet}  & 18.30     & -                & 2.45        & 5.30      & \\phantom{}17.4       & 71.0          \\\\\n~ & Swin-T \\cite{liu2021swin}          & 32\/32 & \\phantom{}45.0  & \\phantom{}106.0 & 81.3         &~& (a) & Real2Binary-Net \\cite{real2binICLR20}    & 1.67      & -                & 1.56        & 1.82      & \\phantom{0}5.1       & 65.4          \\\\\n ~  & ConvNext-T \\cite{liu2022convnet}   & 32\/32 & \\phantom{}45.0  & \\phantom{}114.3 & 82.5      &~& (a) & ReActNet-BiR18 \\cite{ReActNetGithub}     & 1.68      & -                & 1.63        & 1.89      & \\phantom{0}4.2       & 65.9          \\\\    \\cline{2-6}\n ~  &                                    &       &                 &                 &               &~& (b) & ReActNet-A \\cite{liu2018birealnet}       & 4.82      & -                & 0.12        & 0.87      & \\phantom{0}7.4       & 69.4          \\\\\n ~  &                                    &       &                 &                 &               &~& (b) & ReActNet-Adam \\cite{Liu2019radam}        & 4.82      & -                & 0.12        & 0.87      & \\phantom{0}7.4       & 70.5          \\\\ \\cline{2-6}\n ~  &                                    &       &                 &                 &               &~& (c) & PokeBNN 1.75x \\cite{zhang2021pokebnn}    & 11.03     & \\phantom{0}21.3  & -           & 1.74      & \\phantom{}16.3       & 76.8          \\\\ \n ~  & \\mth{Our Design\\\\ w\/o Post-Quant.} & \\mth{W\/A} & \\mth{OPs\\\\($10^{8}$)} & \\mth{\\#Param\\\\(MB)} & \\mth{Top-1\\\\(\\%)} \n                                                                                                     &~& (d) & PokeBNN 2.0x \\cite{zhang2021pokebnn}     & 14.14     & \\phantom{0}25.5  & -           & 2.27      & \\phantom{}20.7       & 77.2          \\\\ \\cline{2-6} \\cline{9-15} \n ~  & BNext-18 (ours)                    & 1\/1   & \\phantom{0}1.64  & \\phantom{00}5.4 & 68.4         &~& (a) & BNext-18 $\\dagger$ (ours)                & 1.68      & \\phantom{}135.2  & -      & 0.43      & \\phantom{0}2.2       & \\textbf{67.9} \\\\\n ~  & BNext-T (ours)                     & 1\/1   & \\phantom{0}0.88  & \\phantom{0}13.3 & 72.4         &~& (b) & BNext-T $\\dagger$ (ours)                 & 4.82      & \\phantom{0}13.4  & -      & 0.77      & \\phantom{0}5.3       & \\textbf{72.0} \\\\\n ~  & BNext-S (ours)                     & 1\/1   & \\phantom{0}1.90  & \\phantom{0}26.7 & 76.1            &~& (c) & BNext-S $\\dagger$ (ours)                 & 10.84     & \\phantom{0}21.1  & -      & 1.72      & \\phantom{}11.1       & \\textbf{75.8}             \\\\ \n ~  & BNext-M (ours)                     & 1\/1   & \\phantom{0}3.38  & \\phantom{0}46.5 & 78.3         &~& (d) & BNext-M $\\dagger$ (ours)                 & 20.09     & \\phantom{0}24.3  & -      & 3.17      & \\phantom{}20.4       & \\textbf{77.9} \\\\ \n ~  & BNext-L (ours)                     & 1\/1   & \\phantom{0}8.54  & \\phantom{}106.1 & 80.6         &~& (e) & BNext-L $\\dagger$ (ours)                 & 52.15     & \\phantom{0}39.4  & -      & 8.19      & \\phantom{}47.6       & \\textbf{80.4} \\\\ \\cline{2-6} \\cline{9-15}\n\\end{tabular}\n\\label{Table:ImageNet SOTA Comparison}\n\\end{table*}\n\n\\section{Experiments}\nIn this section, we evaluate our model and methods on the ILSVRC12 ImageNet \\cite{deng2009imagenet} and the CIFAR dataset.\n\n\\subsection{Experimental Setups}\n\\label{subsec:Experimental Settings and Datasets}\nWe use the AdamW \\cite{loshchilov2017decoupled} optimizer ($\\beta_{1} {=} 0.99$, $\\beta_{2} {=} 0.999$) with an initial learning rate of $10^{-3}$ and weight decays of $10^{-3}$ and $10^{-8}$ for non-binary and binary parameters, respectively. \nWe use 5 epochs of warmup for the learning rate and then reduce it with a Cosine Scheduler \\cite{paszke2019pytorch}. \nWe set the backward gradient clipping range of the STE (Func.~\\ref{func. binarization}) as [-1.5, 1.5]. \nWe use hard binarization (Func.~\\ref{func. binarization}) for activation and progressive weight binarization as proposed in \\cite{guo2021boolnet}.\nTwo different backbones, ResNet \\cite{he2016deep} and MobileNetV1 \\cite{howard2017mobilenets} are used for technique evaluations.\nThe model is trained on 8 Nvidia DGX-A100 GPUs. \n\n\\textbf{ImageNet:} We train for an input resolution of 224x224 with a batch size of 512 for 512 epochs.\nRandAugment (7, 0.5) \\cite{cubuk2020randaugment} is used for enhanced data augmentation.\nWe use the Diversified Consecutive KD process described in Section.~\\ref{subsec: knowledge distillation optimization} for BNext optimization with\nConvNext-Tiny \\cite{liu2022convnet} as the strong teacher and [EfficientNet-(B0-B2-B4)\\cite{tan2019efficientnet}, ConvNext-Tiny] as the candidate assistant teacher group\n\n\\textbf{CIFAR:} We train the model with a batch size of 128 for 256 epochs.\nWe use the standard data augmentations random crop, random horizontal flip and normalization \\cite{paszke2019pytorch}, and use cross entropy loss \\cite{paszke2019pytorch} for optimization.\nResults on CIFAR are the average of five runs. \n\n\n\\subsection{Performance Evaluation}\n\\textbf{ImageNet:} BNext achieves a new state-of-the-art top-1 accuracy for BNNs on ImageNet classification (see Table \\ref{Table:ImageNet SOTA Comparison}).\nWhile most existing BNNs \\cite{courbariaux2016binarized,rastegari2016xnor,liu2018birealnet,bethge2020meliusnet,real2binICLR20,liu2020reactnet, tu2022adabin}\nare still less accurate than a 32-bit ResNet-18 (69.74\\% top-1 accuracy) \\cite{he2016deep}, BNext-XL pushes the upper boundary of BNNs to a crucial accuracy level of 80.57\\%. \nWith this result it surpasses most existing works by 10\\% accuracy and achieves a result close to SOTA 32-bit designs, such as ConvNext \\cite{liu2022convnet}, Swin Transformer \\cite{liu2021swin} and RegNetY-4G \\cite{radosavovic2020designing}. \nCompared to the previous SOTA design PokeBNN, our BNext-M$\\dagger$ achieves 0.7\\% higher accuracy with similar model size. \nThe BNext architecture also relies on lower optimization requirements than PokeBNN family. \nPokeBNN is trained with a huge batch size of 8192 for 720 epochs \\cite{zhang2021pokebnn} on 64 TPU-v3 chips, while BNext only uses a batch size of 512 for 512 epochs. \nCompared to the popular ReActNet \\cite{liu2020reactnet} design, BNext-Tiny$\\dagger$ achieves 2.6\\% higher accuracy with 10M less operations. \nTo further verify the effectiveness of the BNext design, we combine the proposed techniques with a ResNet-18 backbone.\nWe use standard data augmentation and only ResNet-34 as a teacher for a fair comparison. \nThe BNext-18 achieves 68.4\\% top-1 accuracy on ImageNet dataset, only 1.2\\% lower than the original ResNet-18 \\cite{he2016deep}. \nMeanwhile, the performance is higher than all the existing designs such as BNN \\cite{courbariaux2016binarized}, XNOR-Net \\cite{rastegari2016xnor}, BiReal-Net \\cite{liu2018birealnet}, Real2BinaryNet \\cite{real2binICLR20}, and ReActNet-BIR18 \\cite{liu2018birealnet}. For the sake of fair comparison, our calculation of OPs  excludes  negligible floating-point counts as \\cite{liu2018birealnet,bethge2020meliusnet,liu2020reactnet,liu2021adam,zhang2021pokebnn}.           \n\n\\textbf{CIFAR:} We further explore the generalization ability of BNext (with a ResNet-18 backbone) on the smaller dataset CIFAR10 (see Table.~\\ref{exp:comparison cifar10}).\nCompared to the latest designs such as AdaBNN \\cite{tu2022adabin}, ReCU \\cite{xu2021recu} and RBNN \\cite{lin2020rotated}, BNext achieves the best performance.\nIt surpasses one of the most recent works AdaBNN by 0.5\\% and closes the gap to 32-bit ResNet to 0.8\\%.\nThe complete evaluation on CIFAR100 is provided in the supplementary material.  \n\n\\begin{table}[]\n\\centering\n\\small\n\\caption{A comparison of state-of-the-art BNNs on CIFAR10.\n\\resizebox{\\columnwidth}{!}{%\n\\setlength\\tabcolsep{3pt}\n\\begin{tabular}{c|c|c|c}\n\\hline\nMethod      & W\/A (Bitwidth)   & Top-1 Acc {[}\\%{]} & Top-5 Acc {[}\\%{]}\\\\ \\hline\nBaseline (ResNet-18)    & 32\/32 & 94.8  & -                \\\\\nRAD  \\cite{ding2019regularizing}       & 1\/1   & 90.5    & -            \\\\\nIR-Net \\cite{qin2020forward}      & 1\/1   & 91.5    & -            \\\\\nRBNN  \\cite{lin2020rotated}      & 1\/1   & 92.2     & -           \\\\\nReCU  \\cite{xu2021recu}       & 1\/1   & 92.8    & -            \\\\\nAdaBNN \\cite{tu2022adabin}     & 1\/1   & 93.1     & -           \\\\\nBNext (ResNet-18) (ours) & 1\/1   & \\textbf{93.6 ($\\pm0.12$)}    & 98.8 ($\\pm0.03$)            \\\\ \\hline\n\\end{tabular}\n}\n\\label{exp:comparison cifar10}\n\\end{table}\n\n\\begin{table}[]\n\\centering\n\\setlength\\tabcolsep{7pt}\n\\footnotesize\n\\caption{The ablation study of Info-RCP and ELM-Attention on ImageNet, showing the top-1\/top-5 accuracy.}\n\\label{tab:ablation on module designs}\n\\begin{tabular}{c|cc}\n\\hline\nMethod                & w\/o ELM-Attention  & w\/ ELM-Attention   \\\\ \\hline\nw\/o Info-Recoupling        & 62.43\/83.75           & 64.61\/85.49                \\\\ \nw\/ Info-Recoupling    & 64.91\/85.52           & 65.02\/85.73                \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\n\\subsection{Ablation Study}\n\\label{Ablation Study}\nWe evaluate the impacts of the proposed techniques in this paper in a detailed ablation study on the proposed module designs,  optimization schemes, data augmentation strategies and post quantization impacts.\nWe use BNext-Tiny for all experiments with only standard data augmentation and cross-entropy loss for model training (unless stated otherwise).  \n\n\\textbf{Module Designs.}\nWe conduct an ablation study on the proposed Info-RCP and ELM-Attention modules in Table.~\\ref{tab:ablation on module designs}.\nThe baseline model without both structures only achieves 62.43\\% on ImageNet.\nAdding only ELM-Attention to BNext-Tiny increases the accuracy to 64.91\\%.\nAdding only Info-RCP to BNext-Tiny increases the accuracy to 64.61\\%.\nAdding both designs, achieves the highest accuracy of 65.02\\%.\nThus, both designs enhance the representational capacity, which confirms the expectations raised by the loss landscape comparison in Fig.~\\ref{fig:loss_landscape_visualization}.\n\n\n\\textbf{Optimization Schemes.}\nWe evaluate the effectiveness of all optimization schemes used in our experiments in Table \\ref{exp:ablation training schemes}.\nThe traditional knowledge distillation strategy wth ResNet-101 \\cite{liu2020reactnet} as teacher is utilized as backbone. Replacing the teacher with $KC(x)$ (Func. \\ref{func. knowledge complexity}) based teacher selection (Efficient-B0) pushes the accuracy by 0.57\\%.\nThis reveals the importance of teacher selection in binary network optimization.\nUsing a gap-sensitive ensemble (Func. \\ref{func. capacity aware KD}) of Efficient-B0 and Efficient-B2 gives an extra 0.05\\% improvement. A further 0.08\\% accuracy gain is achieved by adding Rand Augmentation \\cite{cubuk2020randaugment}. Extending the training epochs to 512 pushes the accuracy to 72.40\\%.\n\\begin{table}[]\n\\centering\n\\caption{\nThe ablation study on the effects of adding the different training strategies on ImageNet.}\n\\setlength\\tabcolsep{4pt}\n\\footnotesize\n\\begin{tabular}{c|ccc}\n\\hline\nMethod                                                   & Epochs &  Top-1 {[}\\%{]} & Top-5 {[}\\%{]} \\\\ \\hline\n\\multicolumn{1}{c|}{+ KD }        & 128      & 70.76        & 89.64        \\\\\n\\multicolumn{1}{c|}{+ KC-Based $T(x)$ Selection} & 128      & 71.33 (+0.57)        & 90.02 (+0.38)       \\\\\n\\multicolumn{1}{c|}{+ Gap-Sensitive KD} & 128  & 71.38 (+0.05) & 90.04 (+0.02) \\\\\n\\multicolumn{1}{c|}{+ Rand Augmentation \\cite{cubuk2020randaugment}}   & 128  & 71.46 (+0.08)  & 90.06 (+0.02) \\\\ \n\\multicolumn{1}{c|}{+ Long Training}   & 512  & 72.40 (+0.94)  & 90.64 (+0.58) \\\\ \n\\hline\n\\end{tabular}%\n\\label{exp:ablation training schemes}\n\\end{table}\n\n\\begin{table}[]\n\\centering\n\\caption{The ablation study on popular data augmentation strategies on ImageNet. BNext-Tiny is selected as the baseline.}\n\\setlength\\tabcolsep{2pt}\n\\footnotesize\n\\begin{tabular}{c|ccc}\n\\hline\n\\multicolumn{1}{c}{Data Augmentation}     & Epochs & Top-1 (\\%) & Top-5 (\\%) \\\\ \\hline\n\\multicolumn{1}{c|}{Baseline}                      & 128 & 65.02 & 85.73 \\\\\n\\multicolumn{1}{c|}{Mixup \\cite{zhang2017mixup}}              & 128 & 65.83 \\phantom{0}(+0.81) & 86.16 \\phantom{0}(+0.43) \\\\\n\\multicolumn{1}{c|}{Cutmix \\cite{yun2019cutmix}}             & 128 & 64.51 \\phantom{0}(-0.51)  & 85.32 \\phantom{0}(-0.41)\\\\\n\\multicolumn{1}{c|}{Repeat Augment \\cite{hoffer2020augment}}        & 128 & 53.74 (-11.28)& 75.65 (-10.08) \\\\\n\\multicolumn{1}{c|}{Rand Augment \\cite{cubuk2020randaugment}} & 128 & \\textbf{67.51 \\phantom{0}(+2.49)} & \\textbf{87.28 \\phantom{0}(+1.55)} \\\\\n\\multicolumn{1}{c|}{Rand Augment \\cite{cubuk2020randaugment}, Mixup \\cite{zhang2017mixup}} & 121    & 66.51 \\phantom{0}(+1.49) & 86.77 \\phantom{0}(+1.04)\\\\ \\hline\n\\end{tabular}%\n\\label{exp:ablation data augmentation}\n\\end{table}\n\n\\textbf{Data Augmentations.}\nWe verify the effectiveness of existing popular data augmentations such as Mixup \\cite{zhang2017mixup}, Cutmix \\cite{yun2019cutmix}, Augment-Repeat \\cite{hoffer2020augment} and Rand Augmentation \\cite{cubuk2020randaugment}. The results are shown in Table. \\ref{exp:ablation data augmentation}.\nMost of the popular choices  \\cite{dosovitskiy2020image,liu2022convnet, liu2021swin} are harmful for the BNext model optimization.\nFor example, the Augment-Repeat \\cite{hoffer2020augment} decreases the Top-1 accuracy by 10.08\\%.\nBoth Rand Augmentation and Mixup improve the generalization separately, but combining them produces sub-optimal results.\nApplying only Rand Augmentation achieves the best improvement of 2.49\\%.\n\\begin{table}[]\n\\centering\n\\caption{Impact of post-training quantization on the first input layer, the last layer, and the SE branch in BNext Design. The best and second-best quantization results are highlighted.}\n\\setlength\\tabcolsep{3pt}\n\\footnotesize\n\\begin{tabular}{c|cccccc}\n\\hline\n\\multicolumn{1}{c|}{I-S-E-O (W\/A)}           & BNext-18  & BNext-T & BNext-S & BNext-M & BNext-L  \\\\ \\hline\n\\multicolumn{1}{c|}{32\/32}             & 68.37          & 72.36       & 76.06     &  78.27        & 80.57    \\\\ \\hline\n\\multicolumn{1}{c|}{8\/8-8\/8-8\/8-8\/8}   & \\textbf{68.38}          & \\textbf{72.36}       &   \\textbf{76.05}      & \\textbf{78.11}        & \\textbf{80.47}    \\\\ \n\\multicolumn{1}{c|}{8\/8-4\/8-4\/8-8\/8}   & \\textbf{67.94}          &   \\textbf{72.04}      &  \\textbf{75.75}       & \\textbf{77.97}        & \\textbf{80.37}    \\\\\n\\multicolumn{1}{c|}{8\/8-4\/8-4\/4-8\/8}   & 67.91          & 71.95        &   75.61       & 77.93        & 80.26   \\\\\n\\multicolumn{1}{c|}{8\/8-4\/4-4\/4-8\/8}   & 67.29          & 71.72       &  75.50       & 77.79        & 79.52   \\\\\n\\hline\n\\end{tabular}\n\\label{table: ablation quantization}\n\\end{table}\n\n\\textbf{Quantization Impacts.} \nWe apply post-training quantization to the input convolution layer, the output fully-connected layer and the layers in each SE branch, Table. \\ref{table: ablation quantization}. \nWithout quantization, BNext-L reaches the top-1 accuracy of 80.57\\%.\nApplying 8-bit quantization to weights and activations of all layers only marginally reduces accuracy to 80.47\\%.\nFurther, decreasing the bit-width of weights in SE branch to 4 bits (with 8-bit activations) achieves the final BNext-L model whose classification performance 80.37\\% Top-1 accuracy is still higher than the 80\\% mark. \n\n\n\n\n\n\n\n\n\\section{Conclusion}\nIn this paper, we proposed the first binary neural network architecture which achieves 80.57\\% Top-1 accuracy on ImageNet, BNext.\nThis is achieved by enhancing the optimization process with the novel Info-RCP and ELM-Attention modules.\nThe smoother loss landscape and enhanced optimization allows better fitting to the train data and even shows slight overfitting, which was not possible with previous BNN architectures.\nTo increase the generalization and reduce any overfitting, we propose \\emph{Diversified Consecutive KD} and discover several interesting phenomena in the process.\nThe highly accurate results on ImageNet demonstrate that BNext is sufficiently effective as a strong feature extractor for representation learning.\n\n\n\n\n\n{\\small\n\\bibliographystyle{ieee_fullname}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nRadio-Frequency (RF) waves can be utilized for transmission of both information and power simultaneously. As one of the primary works in the information theory literature, Varshney studied this problem in \\cite{Varshney_2008}, in which he characterized the capacity-power function for a point-to-point discrete memoryless channel (DMC). He showed the existence of tradeoff between the information rate and the delivered power for some channels, such as, point-to-point binary channels and amplitude constraint Gaussian channels. Recent results in the literature have also revealed that in many scenarios, there is a tradeoff between information rate and delivered power. Just to name a few, frequency-selective channel \\cite{Grover_Sahai_2010}, MIMO broadcasting \\cite{Zhang_Keong_2013}, interference channel \\cite{Park_Clerckx_2013,Park_Clerckx_2014}, relaying \\cite{Nasir_Zhou_Durrani_Kennedy_2013,Huang_Clerckx_2016_2}.\n\nOne of the major efforts in a Simultaneous Wireless Information and Power Transfer (SWIPT) architecture is to increase the Direct-Current (DC) power at the output of the harvester without increasing transmit power. The harvester, known as rectenna, is composed of a rectifier\\footnote{In the literature, the rectifier is usually considered as a diode, which is the main source of nonlinearity induced in the system.} followed by a low-pass filter. In \\cite{Trotter_Griffin_Durgin_2009,Clerckx_Bayguzina_2016}, it is shown that the RF-to-DC conversion efficiency is a function of rectenna's structure, as well as its input waveform. Accordingly, in order to maximize rectenna's DC power output, a systematic waveform design is crucial to make the best use of an available RF spectrum. In \\cite{Clerckx_Bayguzina_2016}, an analytical model for rectenna's output is introduced via the Taylor expansion of the diode characteristic function. As one of the main conclusions, it is shown that the rectifier's nonlinearity is key to design efficient wireless powered systems.\n\nThe design of an efficient SWIPT architecture fundamentally relies on designing an efficient Wireless Power Transfer (WPT) structure as an important building\nblock of SWIPT. The SWIPT literature has so far focused on the linear model of the rectifier, e.g., \\cite{Grover_Sahai_2010,Zhang_Keong_2013,Park_Clerckx_2013,Park_Clerckx_2014,Nasir_Zhou_Durrani_Kennedy_2013,Huang_Clerckx_2016_2}, whereas, it is expected that considering nonlinearity effect changes the SWIPT design, signalling and architecture significantly. Indeed, in \\cite{Clerckx_2016,Clerckx_2016_Proc}, the design of SWIPT waveforms is studied accounting for rectenna's nonlinearity with a power splitter at the receiver. It is shown that superposing deterministic multisines (for power transfer purposes) with Orthogonal Frequency Division Multiplexing (OFDM) symbols modulated with Circularly Symmetric Complex Gaussian (CSCG) zero-mean inputs (for information purposes) enlarges the Rate-Power (R-P) region, compared to merely zero-mean inputs. This highlights the potential and benefits of departing from conventional CSCG inputs in SWIPT.\n\nLeveraging the aforementioned observations, we provide a step closer at identifying the fundamental limits of SWIPT accounting for the nonlinearity of rectenna. In this paper, we study a flat-fading Additive White Gaussian Noise (AWGN) channel for SWIPT. Taking the advantage of the approximation for rectenna's nonlinear output introduced in \\cite{Clerckx_Bayguzina_2016}, we obtain the general form of the delivered power in terms of system baseband parameters. Assuming that the receiver jointly extracts information and harvests power from the received RF signal,\\footnote{We note that, leveraging the results in thermodynamics of computing, it is demonstrated that energy need not be dissipated in the decoding process. This is due to the reason that to perform a mathematical work, energy is not required \\cite[Ch. 5]{Feynman_1998}. In particular, decoders that are reversible computational devices would not dissipate any energy \\cite{Landauer_1987} and electronic circuits that are almost thermodynamically reversible have been built \\cite{Frank_thesis}. Motivated by this, we also assume that at the receiver, the decoder is able to jointly harvest power and extract information from the received RF signal.} it is shown that the delivered power at the receiver is dependent on the first to fourth moment statistics of the channel input distribution. Considering the optimization problem of maximizing R-P region, we obtain an achievable scheme as an inner bound for the general problem.  The scheme is based on constraining the channel inputs to independent and identically distributed (i.i.d.) distributions that are determined by their first and second moment statistics. For the studied inner bound, we show that there is a tradeoff between the delivered power at the receiver and the rate of the received information. This result is highlighted in contrast to the scenario, in which a linear model is considered for the power harvester at the receiver. It can be easily verified that under an assumption of linear model for the power harvester, the goal of maximum rate and maximum energy are aligned in the flat-fading channel. Additionally, we show that the maximum rate-power (for the studied inner bound) is achieved when the channel input distributions is Gaussian with mean zero, however, with different (asymmetric) power allocations to the real and imaginary dimensions.\n\n\\textit{Organization}: In Section \\ref{Sec_System_Model}, we introduce the system model. In Section \\ref{Sec_Delivered_power}, the delivered power at the receiver is obtained in terms of system baseband parameters accounting the approximation for nonlinearity of rectenna. In Section \\ref{Sec_Problem_statement}, we introduce the problem considered in this paper, and accordingly, in Section \\ref{Sec_Main_Result}, we obtain an achievable scheme as an inner bound for the general optimization problem. In Section \\ref{Sec_Conclusion}, we conclude the paper.\n\n\\textit{Notation}: Throughout this paper, the standard CSCG distribution is denoted by $\\mathcal{CN}(0,1)$. Complex conjugate of a complex number $c$ is denoted by $c^{*}$. For a random process $X(t)$, corresponding random variable at time index $n$ is represented by $X_n$. The operators $\\mathbb{E}[\\cdot]$ and $\\mathcal{E}[\\cdot]$ denote the expectation over statistical randomness and the average over time, respectively. $\\Re\\{\\cdot\\}$ and $\\Im\\{\\cdot\\}$ are real and imaginary operators, respectively. We use the notations $\\mathrm{sinc}(t)=\\frac{\\sin(\\pi t)}{\\pi t}$ and $s_l=\\text{sinc}(l+1\/2)$ for integer $l$. We also define $\\delta_l$ as\n\\begin{align}\n\\delta_l=\\left\\{\\begin{array}{ll}\n  1 & l=0 \\\\\n  0 & l\\neq 0\n\\end{array}\\right..\n\\end{align}\n\n\\section{System Model}\\label{Sec_System_Model}\nConsidering a point-to-point flat-fading AWGN channel, in the following, we explain the operation of the transmitter and the receiver.\n\n\\subsection{Transmitter}\nAt the transmitter, the signal $X(t)$ is produced as\n\\begin{align}\nX(t)=\\sum_{n}X_n\\text{sinc}(f_wt-n),\n\\end{align}\nwhere $X_n$ is an information-power symbol at time index $n$, modelled as a random variable, which is produced in an i.i.d. fashion and $X(t)$ is with bandwidth $[-f_w\/2,f_w\/2]$. Next, the signal $X(t)$ is upconverted to the carrier frequency $f_c$ and is sent over the channel.\n\n\\subsection{Receiver}\nThe filtered received RF waveform at the receiver is modelled as\n\\begin{align}\n  Y_{\\text{rf}}(t) &=\\sqrt{2}\\Re\\left\\{Y(t)e^{j2\\pi f_ct}\\right\\},\n\\end{align}\nwhere $Y(t)$ is the baseband equivalent of the channel output with bandwidth $[-f_w\/2,f_w\/2]$. We assume that $f_c>2f_w$.\n\n\\textit{Power}: At the receiver, the power of the RF signal $Y_{\\text{rf}}(t) $ is delivered through the rectenna. In the following, we leverage the approximation for rectenna's output introduced in \\cite{Clerckx_Bayguzina_2016}\\footnote{According to \\cite{Clerckx_Bayguzina_2016}, due to the presence of a diode in rectenna's structure, its output current is an exponential function, which is approximated by expanding its Taylor series. The approximation used here, is the fourth moment truncation of Taylor series, in which the first and third moments are zero with respect to the time averaging.}. Accordingly, the delivered power (denoted by $P_{\\text{del}}$) is modelled as\\footnote{According to \\cite{Clerckx_Bayguzina_2016}, rectenna's output in (\\ref{eqn:1}) is in the form of current with unit Ampere. However, since power is proportional to current, with abuse of notation, we refer to the term in (\\ref{eqn:1}) as power.}\n\\begin{align}\\label{eqn:1}\nP_{\\text{del}}=\\mathbb{E}\\mathcal{E}[k_2Y_{\\text{rf}}(t)^2 + k_4 Y_{\\text{rf}}(t)^4],\n\\end{align}\nwhere $k_2$ and $k_4$ are constants. Note that, in the linear model for the delivered power $P_{\\text{del}}$, in (\\ref{eqn:1}), we have only the second moment of the received RF signal $Y_{\\text{rf}}(t)$. Validating through circuit simulations in \\cite{Clerckx_Bayguzina_2016}, it is shown that the linear model is inaccurate and inefficient from a signal design perspective.\n\n\\textit{Information}: The signal $Y_{\\text{rf}}(t) $ is downconverted producing the baseband signal $Y(t)$ given as \\footnote{We model the baseband equivalent channel impulse response as $H(\\tau,t)=\\sum_{i}a_i^b(t)\\delta(\\tau-\\tau_i(t))+W(t)$ where $\\alpha_i^b(t)$, $\\tau_i(t)$ are the channel coefficient and delay of path $i$.}\n\\begin{align}\nY(t)=\\sum_{i}a_i^b(t)X(t-\\tau_i(t))+W(t).\n\\end{align}\nNext, $Y(t)$ is sampled with a sampling frequency $f_w$ producing $Y_{m}=Y(m\/f_w)$ given as\n\\begin{align}\\label{eqn:20}\nY_m&=X_m \\sum_{i}a_i^b(m\/f_w)+W_m,\n\\end{align}\nwhere in (\\ref{eqn:20}), we used $\\tau_i(m\/f_w)f_w\\approx 0$ because the channel is flat-fading. $W_m$ and $X_m$ represent samples of the additive noise $W(t)$ and the signal $X(t)$ at time $t=m\/f_w$, respectively.\n\nWe model $W_m$ as an i.i.d. and CSCG random variable with variance $\\sigma_w^2$, i.e., $W_m\\sim \\mathcal{CN}(0,\\sigma_w^2)$. We assume that both the transmitter and the receiver know the Channel gain, namely, $h(t)=\\sum_{i}a_i^b(t)$ at times $t=m\/2f_w$ for integer $m$, which is assumed to be fixed over all the transmissions. Throughout the paper, since the transmitted symbols $X_m$ and the noise $W_m$ are i.i.d., we drop the index $m$ for $X_m,~W_m$. We also define $h=\\sum_i a_i^b((2m)\/(2f_w))$ and $\\tilde{h}=\\sum_i a_i^b((2m+1)\/(2f_w))$. Note that $h$ and $\\tilde{h}$ are assumed to be fixed, however, we assume they are not necessarily equal. Therefore, (\\ref{eqn:20}) reads\n\\begin{align}\\label{eqn:34}\nY=h X + W.\n\\end{align}\nNote that in (\\ref{eqn:34}), only even samples of the channel, i.e., $h$ are involved.\n\n\\section{Delivered power}\\label{Sec_Delivered_power}\nIn this section, we study the power delivered at the receiver. Note that most of the communication processes, such as, coding\/ decoding, modulation\/ demodulation, etc, is done at the baseband. Therefore, from a communication system design point of view, it is most preferable to have baseband equivalent presentation of the system. Henceforth, in the following Proposition, we derive the delivered power $P_{\\text{del}}$ at the receiver in terms of system baseband parameters.\n\n\\begin{prop}\\label{Prop1}\nAssuming the channel input distributions are i.i.d., the delivered power $P_{\\text{del}}$ at the receiver, can be expressed in terms of system baseband parameters as\n\\begin{align}\\label{eqn:22}\nP_{\\text{del}}=\\alpha Q+\\tilde{\\alpha}\\tilde{Q}+(\\beta +\\tilde{\\beta})P+\\gamma,\n\\end{align}\nwhere $\\tilde{Q}$ is given by\n\\begin{align}\\nonumber\n\\tilde{Q}&=\\frac{1}{3}\\big(Q_{r}+Q_{i}+2(\\mu_{r}T_{r}+\\mu_{i}T_{i})\\\\\n&+6P_{r}P_{i}+6P_{r}(P_{r}-\\mu_{r}^{2})+6P_{i}(P_{i}-\\mu_{i}^{2})\\big),\n\\end{align}\nwhere the parameters $\\alpha,~\\tilde{\\alpha},~\\beta,~\\tilde{\\beta}$ and $\\gamma$ are given as\n\\begin{align}\\label{eqn:32}\n\\alpha&=\\frac{3k_4}{4f_w}|h|^4,\\\\\n\\tilde{\\alpha}&=\\frac{3k_4}{4f_w}|\\tilde{h}|^4,\\\\\n \\beta&=\\frac{1}{f_w}\\left(k_2+6k_4\\sigma_w^2\\right)|h|^2, \\\\\n \\tilde{\\beta}&=\\frac{1}{f_w}\\left(k_2+6k_4\\sigma_w^2\\right)|\\tilde{h}|^2, \\\\\\label{eqn:33}\n \\gamma&=\\frac{1}{f_w}(k_2\\sigma_w^2+3k_4\\sigma_w^4),\n\\end{align}\nand $Q=\\mathbb{E}[|X|^4]$, $T=\\mathbb{E}[|X|^3]$, $P=\\mathbb{E}[|X|^2]$, $\\mu=\\mathbb{E}[X]$. Similarly, $Q_r=\\mathbb{E}[\\Re\\{X\\}^4]$, $T_r=\\mathbb{E}[\\Re\\{X\\}^3]$, $P_r=\\mathbb{E}[\\Re\\{X\\}^2]$, $\\mu_r=\\mathbb{E}[\\Re\\{X\\}]$ and $Q_i=\\mathbb{E}[\\Im\\{X\\}^4]$, $T_i=\\mathbb{E}[\\Im\\{X\\}^3]$, $P_i=\\mathbb{E}[\\Im\\{X\\}^2]$, $\\mu_i=\\mathbb{E}[\\Im\\{X\\}]$.\n\\end{prop}\n\\textit{Proof}: See Appendix \\ref{app:1}.\n\n\\begin{rem}\nWe note that obtaining a closed form expression for the delivered power $P_{\\text{del}}$ at the receiver when the channel inputs are not i.i.d. is cumbersome. This is due to the fact that the fourth moment of the received signal $Y_{\\text{rf}}(t)$ creates dependencies of the statistics of the present channel input on the statistics of the channel inputs on other time indices (see e.g., eq. (\\ref{eqn:9}) and eq. (\\ref{eqn:30}) in Appendix \\ref{app:1}).\n\\end{rem}\n\n\\section{Problem statement}\\label{Sec_Problem_statement}\n\nWe aim at maximizing the rate of the received information, as well as the amount of power delivered at the receiver. Accordingly, the optimization problem we consider, is the maximization of mutual information between the channel input $X$ and the channel output $Y$ under a given power constraint at the transmitter and a minimum delivered power constraint at the receiver. Hence, for the optimization problem, we have\n\\begin{equation}\\label{eqn:13}\n\\begin{aligned}\n& \\underset{ p_{X}(x)}{\\text{sup}}\n& & I\\left(X;Y\\right) \\\\\n& \\text{s.t.}\n& & \\left\\{\\begin{array}{l}\n      P\\leq P_{a} \\\\\n      P_{\\text{del}}\\geq P_d\n    \\end{array}\\right.,\n\\end{aligned}\n\\end{equation}\nwhere $\\sup$ is taken over all input distributions $p_{X}(x)$ satisfying the constraints in (\\ref{eqn:13}). $P_{a}$ is the available power budget at the transmitter and $P_d$ is the minimum amount of power that is to be delivered to the receiver.\n\n\\begin{rem}\nWe note that, for the problem in (\\ref{eqn:13}), if the second constraint (the minimum delivered power at the receiver) is represented via a linear model, i.e., $\\mathbb{E}[|Y|^2]\\geq P_d$, the maximum is achieved using a CSCG input distribution. It can also be verified easily that there is no tradeoff between the received information rate and delivered power at the receiver.\n\\end{rem}\n\n\\section{Main Result}\\label{Sec_Main_Result}\nIn this section, we obtain an inner bound for the problem in (\\ref{eqn:13}) by constraining the input distributions to those that are determined by their first and second moment statistics\\footnote{This assumption is justified due to the fact that in practice, most of the modulation schemes are i.i.d. and are fully characterized by the knowledge of the first and second moment statistics only.}. We show that for the considered scenario, there is a tradeoff between the rate of the transmitted information, namely $I(X;Y)$ and delivered power $P_{\\text{dc}}$ at the receiver and accordingly, we characterize the tradeoff.\n\n\\begin{prop}\\label{Prop2}\nWhen a channel input distribution $p_{X}(x)$ is completely determined by its first and second moment statistics, the supremum in (\\ref{eqn:13}) is achieved by a zero mean Gaussian distribution as the channel input, i.e., $\\Re\\{X\\}\\sim\\mathcal{N} (0,P_r)$, and $\\Im\\{X\\}\\sim \\mathcal{N}(0,P_i)$, where $P=P_r+P_i=P_a$. Furthermore, let $P_{\\text{dc,max}}=3(\\alpha+\\tilde{\\alpha}) {P_{a}}^2+(\\beta+\\tilde{\\beta}) P_{a} +\\gamma$ and $P_{\\text{dc,min}}=2(\\alpha+\\tilde{\\alpha}) {P_{a}}^2+(\\beta+\\tilde{\\beta}) P_{a} +\\gamma$ be the maximum and minimum delivered power at the receiver, respectively. For $P_d=P_{\\text{dc,max}}$, the maximum in (\\ref{eqn:13}) is attained by $P_i=0,P_r=P_{a}$ or $P_i=P_a,P_r=0$. For $P_d=P_{\\text{dc,min}}$, the maximum in (\\ref{eqn:13}) is attained by $P_i=P_a\/2,P_r=P_a\/2$. For $P_{\\text{dc,min}}<P_d< P_{\\text{dc,max}}$, the optimal power allocation that attains the maximum rate is given by $P_i^*$ and $P_r^*=P_a-P_i^*$, where $P_i^*$ is chosen such that the following equation is satisfied\n\\begin{align}\\label{eqn:31}\n\\alpha Q+\\tilde{\\alpha} \\tilde{Q}+(\\beta+\\tilde{\\beta}) P_a+\\gamma=P_d.\n\\end{align}\n\nFor $P_d< P_{\\text{dc,min}}$, the optimal power allocation is attained by $P_i^*=P_r^*=P_{a}\/2$ and the delivered power is still $P_{\\text{dc,min}}$.\n\n\\end{prop}\n\\textit{Proof}: See Appendix \\ref{app:2}.\n\n\nNote that in (\\ref{eqn:31}), for a complex zero-mean Gaussian distributed channel input with $P_r$ and $P_i$ as the variances of real and imaginary dimensions, respectively, we have\n\\begin{align}\nQ=\\tilde{Q}=3(P_i^2+P_r^2)+2P_iP_r.\n\\end{align}\n\n\\begin{rem}\nFrom (\\ref{eqn:22}), it is seen that the delivered power $P_{\\text{del}}$ at the receiver depends on the second moment statistics $P_r,P_i$, as well as the fourth moment statistics $Q_r,Q_i$ of the channel input $X$. This is due to the presence of the fourth moment of the received signal $Y_{\\text{rf}}$ in modelling the rectenna's output. Accordingly, the rate is minimized (corresponding to $P_{\\text{dc,max}}$) when the available power at the transmitter is fully allocated to one of the real or imaginary dimensions. This is because allocating power to one dimension, leads to a higher fourth order moment. On the other hand, the maximum rate is achieved (corresponding to $P_{\\text{dc,min}}$) when the available power is equally distributed between the real and the imaginary dimensions. This is elaborated in Figure  \\ref{Figure:1}.\n\\end{rem}\n\n\\begin{rem}\nAs mentioned earlier, maximization of both the delivered power $P_{\\text{del}}$ and the rate $I(X;Y)$ are aligned under the linear modelling for the delivered power. Therefore, regardless of the channel condition, the best power allocation is always $P_i=P_r=P\/2$. However, accounting the fourth moment of the received signal $Y_{\\text{rf}}$ in (\\ref{eqn:1}), the receiver chooses the proper power allocation such that the constraints in (\\ref{eqn:13}) are satisfied (if $P_d\\leq P_{\\text{dc,max}}$). We also note that since we have assumed the channel is fixed for the whole transmission, therefore, the transmitter keeps using the same power allocation for the whole transmission.\n\nIt is also noted that at each time index the rate of the information $I(X;Y)$ is affected only through $h$, whereas, the delivered power $P_{\\text{del}}$ is affected through both $h$ and $\\tilde{h}$. This is illustrated in Figure \\ref{Figure:1} by representing three different regions by varying the channel coefficients.\n\\end{rem}\n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[scale=0.3]{fig_sim.eps}\n\\caption{Delivered power $P_{\\text{del}}$ and achievable information rate $I(X;Y)$ for the problem in (\\ref{eqn:13}) for i.i.d. channel inputs that are determined by their first and second moment statistics $(P_{a}=1,~ \\sigma_w^2=10^{-4},~f_w=1,~k_2=0.17,~k_4=19.145$). The points $A,~B,~C,~D$ correspond to $(P_r,P_i)=(0,1),~=(0.03,0.97),~=(0.2,0.8),~=(0.5,0.5)$, respectively, where the values for $P_r$ and $P_i$ can be interchanged without affecting the resulting points. The values for $k_2,~k_4$ are adapted from \\cite{Clerckx_2016}.} \\label{Figure:1}\n\\par\\end{centering}\n\\vspace{0mm}\n\\end{figure}\n\n\\section{Conclusion}\\label{Sec_Conclusion}\nIn this paper, we studied SWIPT over a point-to-point complex AWGN channel in the presence of a nonlinear power harvester at the receiver. Assuming that the channel state information is available at both the transmitter and the receiver, we studied the problem of maximizing rate of the transmitted information as well as delivered power at the receiver. Assuming that the channel inputs are i.i.d. and are fully characterized by the knowledge of their first and second moment statistics, we derived an inner bound for the optimal R-P region. We showed that for the obtained inner bound, there is a trade off (due to the nonlinearity of the power harvester at the receiver) between the rate of transmitted information and delivered power. Accordingly, we characterized the inner bound, which demonstrates that the optimal channel input is still a zero mean Gaussian distribution, however, with asymmetric power allocations to the real and imaginary dimensions.\n\nAmong open problems that are left for future research, we mention here the optimal input distribution for the problem in (\\ref{eqn:13}). Another interesting problem is the extension of the problem studied in this paper to the frequency-selective AWGN channel.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Conclusion}\n\nIn this paper, we present the SignLanguageGAN, a novel approach for photo-realistic synthesis of sign language videos. \nIn addition to previous work on SLP we extended our GAN with a human semantic parser.\nThis allows our system to learn region-level spatial layouts in order to guide the generator to produce more realistic appearing videos on a large variety of signers.\nThe introduction of a periodic lambda for weighting the discriminator is worth to investigate further since it allows to create images with less artifacts in long training sessions.\n\n\n\n\n\n\\section{Evaluation}\n\n\n\\subsection{Setup}\n\n\nWe evaluate our approach on the MS-ASL~\\cite{MSASL} dataset, the most diverse sign language dataset in terms of number of signers and scene conditions. We compare our generated frames with ground truth images and determine the MSE, PSNR and SSIM to assess their quality. Note, unfortunately, we cannot compare our results directly to Text2Sign~\\cite{Text2Sign}, since the SMILE dataset~\\cite{SMILE} is not publicly available.\n\n\\begin{table}[h]\n  \\centering\n  \\begin{tabular}{ c | c | c | c }\n    \\toprule\n    {Architecture}     & {PSNR}          & {MSE}            & {SSIM}         \\\\ \\midrule\n    pix2pix            & 20.546          & 783.376          & 0.786          \\\\\n    \\addlinespace[0.5em]\n    Soft-Gated Warping & \\textbf{22.033} & \\textbf{622.683} & \\textbf{0.813} \\\\\n  \\end{tabular}\n  \\caption{pix2pix and Soft-Gated Warping architectures tested on 50 videos of the MS-ASL dataset after three epochs of training.}\n  \\label{tab:architectures}\n\\end{table}\n\n\n\\subsection{Multi-Signer Generation Challenges}\nIn order to evaluate the Soft-Gated Warping extension in our architecture we have compared it with a network arrangement using pix2pix~\\cite{Pix2Pix} instead. Table~\\ref{tab:architectures} shows that the human semantic parser extension leads to improved results for all evaluated metrics using 50 randomly selected videos from the MS-ASL dataset. We also have compared these multi-signer MS-ASL results to results on homogenious \\emph{Geb\u00e4rdenlernen}\\footnote{\\url{https:\/\/gebaerdenlernen.de\/index.php?article_id=72}} videos, which show a single-signer in the same scene performing german sign language glosses. In this way, we wanted to assess how the number of signers and the conditions of the scene affect SLP quality. The results show that single signer setups (with PSNR=25.634, MSE=194.174, SSIM=0.882) can easily outperform multi-signer datasets (with PSNR=22.033, MSE=622.683, SSIM=0.813). That underlines our assumption that our pose-guided approach would perform well in real-world scenarios.\n\n\n\n\n\nSince SignLanguageGAN is the first SLP approach to use the MS-ASL dataset it is difficult to make a meaningful comparison. Nevertheless, we use a subset of the MS-ASL dataset with 184 videos consisting of a total of 16601 frames as a benchmark\nresulting in a PSNR of 26.044, MSE of 260.363 and SSIM of 0.893 after 300 epochs of training.\nWe compare our results to the results for three individual signers from Text2Sign~\\cite[p. 901 Table 3]{Text2Sign}, even though our test setup is significantly different from the Text2Sign setup. First, we did not have access to the SMILE dataset, which makes the comparison much more difficult and also prevents us from using the pix2pix HD~\\cite{Pix2PixHD} architecture. Second, Text2Sign performs a fine-tuning for the three signers.\nSince pix2pix HD performs better and SLP for only three signers is not as difficult as for more than 200 we are not surprised that Text2Sign results, with an average PSNR of 25.604, MSE of 166,614 and SSIM of 0.942 are better than ours in terms of pure numbers. Nevertheless, we show that our architecture performs well on extremely diverse data that also contains many low quality samples and without special fine-tuning for individual signers. Figure~\\ref{fig:results_grid_1} shows five examples of our results.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{figure}[h]\n  \\includegraphics[width=0.48\\textwidth]{images\/results_grid_1.jpg}\n  \\caption{Five examples of our produced sign language glosses. First line shows the input image, second the image produced by our SignlanguageGAN, third the ground truth.}\n  \\label{fig:results_grid_1}\n\\end{figure}\n\\section{Introduction}\nThe WHO states that 430 million people in the world are deaf or suffer from significant hearing loss. Affected people often use one of the many different sign languages to communicate. In both directions of translation, the first approaches to barrier-free communication between signers and non-signers have emerged in recent years~\\cite{bangham2000virtual, RWTH2014, camgoz1}. Our approach refers to the generation of personalized sign videos that can be synthesized in a pose-guided pipeline. Our network architecture is based on the findings of the Text2Sign~\\cite{Text2Sign} paper, which presented the first non-graphical avatar approach to continuous gesture video generation. However, we only consider here the generation of glosses in videos. Our sign language GAN generates a video over the space of all signs from the MS-ASL dataset~\\cite{MSASL} from a set of input-target pairs.\nOur goal is to improve the approach by Stoll et al. and enhance the quality of the generated videos to support a variety of signers and different scene conditions.\nHence, we apply our approach to the challenging MS-ASL dataset. In this way, we hope to drive the development of real-world sign language applications with personalized generated signers.\n\n\n\nWhen training on highly variable data, such as the MS-ASL dataset, we have observed that our approach first finds a stable equilibrium, but after some time the discriminator loss starts to increase, which means that it is too easily fooled (see Figure~\\ref{fig:dynamic_vs_static}). It seems, however, that with a dataset as diverse as MS-ASL, not enough uniform criteria can be learned to judge the authenticity of an image when the generator has exceeded a certain threshold. If the discriminator is too weak, the L1 loss of the generator will prevail and the resulting images will be blurred. We have evaluated different strategies to stabilize the training and found that a dynamic lambda, a value that periodically weights the generator error against the discriminator, reactivates the training and leads to an improvement in image quality as shown in Figure~\\ref{fig:static_vs_dynamic_result_variance}.\n\n\n\n\n\\section{Method}\n\\label{sec:method}\n\\subsection{Approach \\& Architecture}\nOur \\emph{SignLanguageGAN} pipeline combines the approaches from Soft-Gated Warping-GAN~\\citep{WarpingGan} and Text2Sign~\\cite{Text2Sign} and is displayed in Figure~\\ref{fig:pipeline}. At the first stage the human semantic parser (A1), a U-Net~\\citep{UNet} autoencoder trained on the CIHP dataset, a subset of the LIP dataset~\\citep{LIP} creates an abstraction of the input frame. It has been shown that a semantic translation can be beneficial for spatial transformation of a person in an image \\citep{WarpingGan, Guha}. The human semantic parser provides such a mapping from limb categories to colour values. This initial parsing is now transformed by a pix2pix network~\\citep{Pix2Pix} (B) using the initial parsing and the OpenPose~\\citep{OpenPose1} generated target pose (A2) representing the desired gloss. Finally, the result is the target parsing, a semantic color map of the person in the input frame who takes the pose represented by the OpenPose skeleton.\nThe second stage forms the actual adversarial model. Starting with a generator (C) that was expanded by a second convolutional encoder network. The latent spaces of both encoders are concatenated and a decoder learns to reproduce the target image from them. The encoder-decoder with skip connections is derived from U-Net architecture~\\citep{UNet}, which was utilized in pix2pix. However, unlike in the original U-Net architecture, down-sampling is performed from two separate networks, while up-sampling is executed by a single one which merges the information of its counterparts. In its setup, the first encoder processes the generated target parsing from the first stage together with the OpenPose generated target pose. Encoder two, on the other hand, receives the input frame together with the respective initial parsing. Both encoders learn their individual mapping into a low dimensional representation. From the concatenation of these two latent spaces the decoder learns how to create a new image of the person seen in the initial frame performing the desired sign language gloss. This output is passed to the SignLanguageGAN discriminator (D) for the adversarial training. The discriminator determines whether the image originates from the generator or if a real sample from the dataset was used. In particular, it is important that the discriminator receives all information for assessment that was processed in the generation.\n\n\n\\subsection{Training with a Dynamic Lambda}\nWhile our architecture is a composite of several technologies, the training process differs in certain aspects from the frameworks used. We apply a least squares objective function in the discriminator as Mao et al. show that the L2 loss reduces vanishing gradients due to its sensitivity~\\citep{LSGAN}.\nThis effect is particularly noticeable with highly heterogeneous data such as the MS-ASL~\\citep{MSASL} collection.\nAnother challenge in working with such dynamic data became apparent as the training has advanced. The equilibrium between discriminator and generator becomes increasingly unstable, presumably because the discriminator has more and more difficulty learning uniform criteria to determine. This results in images with more artifacts and lower variance $\\sigma$, as the generator contribution becomes stronger, for which we choose a L1 loss in accordance with pix2pix. $\\sigma$ influences contrast and structure in SSIM. To break this trend, we apply a dynamic weighting between generator and discriminator to cyclically amplify the impact of the discriminator. For this we use the lambda parameter, which in pix2pix is a constant value. Several approaches have been evaluated to adjust lambda periodically or adaptively. However, strategies that interfered too much with the balance of the GAN were counterproductive. It have become evident that a steady fluctuation yielded the best results (see Figure~\\ref{fig:dynamic_vs_static}).\nCompared to a static weighting the dynamic lambda actively manipulates the loss leading to more updates for the generator.\nThose updates encourage the generator to increase the output quality in order to fool the discriminator.\nWith the aim of having a high generator contribution at the beginning of the training, we use a cosine period with an amplitude span of 1.5 epochs $\\approx \\pi$ and a value range of 50 to 150. Our objective function consists of the conditional least squares discriminator objective $V(D,G)$ and the $L1$ loss with the dynamic term where $i$ and $j$ determine the amplitude range.\n\n$$\\lambda = i + \\frac{cos([0, \\pi]) + 1}{2} \\cdot j$$\n$$\\min_{D}\\min_{G}L_{SgnlGAN} = V(D,G)+L_{1}(G) \\cdot \\lambda$$\n\nBy periodically manipulating the loss we observe an overall better performing generator with fewer artifacts and higher variance. This leads to a slight improvement in the SSIM and improved image results as shown in Figure~\\ref{fig:static_vs_dynamic_result_variance}(a,b).\n\n\n\n\n\n\n\n\n\\begin{figure}[h]\n  \\includegraphics[width=0.48\\textwidth]{images\/dynamicvsstatic_lambda.png}\n  \\caption{SignLanguageGAN discriminator losses for 20 epochs of training with dynamic lambda (blue) and static lambda (red).}\n  \\label{fig:dynamic_vs_static}\n\\end{figure}\n\n\\begin{figure}[h]\n  \\includegraphics[width=0.48\\textwidth]{images\/dynamic_vs_static_result_variance.pdf}\n  \\caption{(a) (Top:) Static lambda during training. (Bottom:) Dynamic lambda during training. (b) Curve shows increased variance (in blue) when dynamic weighting is applied which leads to higher SSIM.}\n  \\label{fig:static_vs_dynamic_result_variance}\n\\end{figure}\n\n\\section{Related Work}\nSign language related tasks can be divided into three subcategories\n\\emph{Sign Language Recognition (SLR)}~\\citep{AlphabetRecognition, DeepASL, GestureRecognitionASL}, \\emph{Sign Language Translation (SLT)}~\\citep{camgoz1, camgoz2} and~\\emph{Sign Language Production (SLP)}~\\citep{Text2Sign}. Our approach relates to SLP tasks to produce glosses and signs of performed sign language.\nSign language datasets mostly consist of annotated videos. Recent approaches like \"Word Level American Sign Language\" (WLASL)~\\citep{WLASL} or \"Microsoft - American Sign Language\" (MS-ASL)~\\citep{MSASL} contain annotated high quality videos showing many signers performing various glosses. For our approach we use the MS-ASL dataset as it contains over 25000 videos of more than 200 signers with very different scene conditions, which comes closest to video data in the wild. Other datasets like SMILE~\\citep{SMILE} used by~\\cite{Text2Sign} or~\\citep{how2sign} even contain multi-sensor data but fewer signers and unfortunately have not been published yet.\nThe use of GANs to tackle SLP problems is still rather new.\nAn important contribution from 2016 is the pix2pix~\\citep{Pix2Pix} framework which uses a U-Net~\\citep{UNet} generator with skip-connections.\nIt is a key component in our SLP approach as well. Synthesizing sign language videos share some challenges with tasks of Warping-GANs. They aim to transfer a person into previously unseen poses. There have been some remarkable developments in this area in the last years \\citep{WarpingGan, Guha, PoseGuided, Liquid}.\nIn early 2020, Stoll et al. released a comprehensive approach to translate spoken language into sign language videos. The pipeline called \\emph{Text2Sign}~\\citep{Text2Sign} solves the task in two major stages. First the spoken glosses are translated into a sequence of motion graphs by an encoder-decoder architecture (Text2Pose). These poses are then passed to a sign language production GAN (Pose2Video), which is also based on a pix2pix architecture and an extension \\citep{Pix2PixHD} of it. Our work picks up on the results of the \\emph{Pose2Video} part and further develops the approach. Later in 2020, more GAN applications for SLP have been released \\citep{multichannelSLP, SignSynth, caneverybody, WiGAN}. Which is an indication of the great interest in the issue and show improved results. Contrary to all this work, our approach concentrates on the synthesis of sign language in more real life conditions using the MS-ASL dataset.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAs aptly pointed out by \\citet{nicosia2013characteristic} ``Networks are the fabric of complex systems''. Concepts like organized complexity (or middle way \\cite{laughlin2000middle}, mesoscopic systems \\cite{imry1997introduction}) revolve around the existence of features shared by systems made of many interacting parts, which can be suitably described in terms of networks.\nIn 1948 \\citet{weaver1991science} defined the notions of simplicity and complexity in Science by a three categories classification: (1) \\textit{Organized Simplicity}. The paradigm of this kind of problems is exemplified by Newton's law of universal gravitation; (2) \\textit{Disorganized Complexity}. In these cases, systems are made by a very large number of interacting elements. Even if these systems cannot be studied at the single-element level, nevertheless they allow for a very efficient statistical treatment (e.g., in terms of statistical mechanics).\nThere is a third kind of complexity that cannot be studied this way and that Weaver identified with biological systems: \\textit{Organized Complexity}.\n\nBoth the arguments and terminology used by Weaver were largely coincident with Laughlin \\textit{et al.}, identifying ``The Middle Way'' \\cite{laughlin2000middle} as the real frontier of basic Science in the XXI century.\nThe natural entities in which the organized complexity approach was more fruitful were protein molecules.\nNot only they were exactly at the boundary between simple and complex physics \\cite{frauenfelder1994biomolecules} but they allowed for both very accurate experiments and for a very refined structural description (e.g., via X-ray crystallography).\nThe situation nowadays continues to be unchanged with respect to Laughlin \\textit{et al.} \\cite{laughlin2000middle}: protein science is still the most explored field in the realm of organized complexity \\cite{vijayabaskar2010interaction,bagler2005network,bode2007network,yan2014construction,doi:10.1021\/cr3002356,orozco2014theoretical,banavar2007physics}.\n\nThe resurgent interest in graph-theoretical methods for the analysis complex systems \\cite{dehmer2013quantitative,ecoli_graph_complexity_arxiv,giuliani2014network,boccaletti+latora+moreno+chavez+hwang2006,csermely2013structure,doi:10.1021\/cr3002356,bashan2012network,mirshahvalad2014dynamics,dorogovtsev2008critical,song2006origins,rosvall2014memory,barthelemy2011spatial,holme2012temporal,newman2005power,csermely2012disordered,csermely2013structuredynamics,newman2010networks,song2005self} set forth by the works by Barabasi, Strogatz and other pioneers in the first decade of XXI century, allowed for a rich set of measures providing a multifaceted description of complex networks \\cite{Han20121958,newman2005power,ecoli_graph_complexity_arxiv,escolano2012heat,rossi2013characterizing,PhysRevE.80.045102,escolano2012heat,doi:10.1021\/ci300321f,duardo2013modeling,riera2012new}.\nIt is sufficient that a given problem is formalized by means of a vertex-and-edge representation -- with vertices being the parts and edges their pairwise relations -- to allow facing the problem according with the terms of organized complexity \\cite{giuliani2014network}.\nProtein molecules are at the forefront of complex network way to organized complexity.\nIn fact, it is possible to represent a protein molecule in terms of protein contact network (i.e., a network having amino acid residues as vertices and edges representing physical contacts between them) by considering its native 3D structure \\cite{yan2014construction,estrada2010universality}.\nIt is worth noting that the graph-based representation is at the core of chemical thought: the structural formulas are graphs \\cite{trinajstiac1983chemical}.\n\nIn this work, we face a data-driven quest for mesoscopic organization principles of complex biological systems by analyzing different complex networks: protein contact networks, metabolic networks, and genetic networks, together with simulated archetypal networks whose wiring scheme is generated by mathematical rules.\nAll considered networks are characterized in terms of two collections of numerical features.\nThe first collection is based on classical topological descriptors, such as the modularity and statistics of the shortest paths. The second one exploits the discrete heat kernel (HK), elaborated using the eigendecomposition of the (normalized) graph Laplacian \\cite{Xiao:2009:GCH:1563046.1563099}.\nWith a first preliminary analysis, we show that the different classes of networks are discriminated by a suitable embedding of the numeric features. This is reasonably expected given the substantially different natures of the analyzed networks, but by no means can be considered as a trivial result.\nAs a matter of fact, the distinction in terms of metabolic, genetic, and protein contact networks is based on network functions, and the demonstration of a link between functional and structural properties of the corresponding graph representation is a prerequisite for the soundness of the proposed strategy of analysis.\nAn important result is that the two herein considered network characterizations resulted to be strongly correlated with each other, so giving a proof-of-concept of the reliability and interpretability of the adopted network descriptions.\nFrom this first analysis, it also emerged that protein contact networks display unique properties that do not allow for a straightforward classification in any of the considered classical archetypal networks, stressing the need for the search of new generative models for protein contact networks.\nThe second, and more important, claim of this paper is that the ensemble HK analysis allows us to derive results in agreement with known chemico-physical properties of protein molecules.\nOur analysis demonstrates that a (simulated) diffusion process in protein contact networks proceeds slower than normal diffusion, i.e., we observe subdiffusion. Notably, a two-regime diffusion emerged from the analysis of the heat trace decay: a fast and a slow regime. The fast regime is driven by short cuts putting in contact amino acid residues far-away along the sequence.\nSubdiffusion is ubiquitous in Nature \\cite{0034-4885-76-4-046602} and it is a well-studied peculiarity describing energy flow \\cite{doi:10.1146\/annurev.physchem.59.032607.093606,lervik2010heat,li2014anisotropic,sangha2009proteins} and vibration dynamics \\cite{yu2003anomalous,reuveni2010anomalies,granek2011proteins,neusius2008subdiffusion,reuveni2012dynamic} in protein molecules.\nInterestingly, here we are able to observe the same phenomenon by exploiting only algebraic properties of the considered graph representations of an ensemble of varying-size proteins.\nThe demonstration of the ability of such a network formalization to explain a unique chemico-physical property of protein molecules, as heat diffusion, is a proof-of-concept of the usefulness of graph theory in chemical systems modeling.\n\nThere is sufficient agreement on the fact that proteins, considering their native structure, are highly modular and fractal networks \\cite{PhysRevE.79.020901,banerji2011fractal,ecoli_graph_complexity_arxiv,di2012proteins,PhysRevE.71.011912,delvenne2010stability}; yet they are characterized also by short paths connecting distant regions of the molecules responsible for the fast-track transport of energy and protein allosteric properties \\cite{doi:10.1146\/annurev.physchem.59.032607.093606}.\nThe modular character of protein contact networks has a direct impact on diffusion processes simulated via graph-theoretical methods.\nIn fact, well-defined modular organizations slow down diffusion processes in networks \\cite{gallos2007scaling}.\nAdditionally, theoretical and experimental results regarding diffusion on porous and fractal media predict fractional scaling exponents for the mean squared displacement (a well-known measure of diffusion), which give rise to the so-called anomalous diffusion \\cite{ben2000diffusion,nakayama1994dynamical}.\nTo conclude, another interesting fact deduced from our analysis is that, at odds with the other networks, modularity of protein graphs increases with the size of the network.\nThis observation is fairly interesting since path efficiency and modular properties are two conflicting features in networks \\cite{gallos2012small,bullmore2012economy}, which however seem to be suitably optimized in proteins.\n\nThe remainder of this paper is organized as follows.\nIn Sec. \\ref{sec:material_and_methods} we first introduce the data that we considered in our study (Sec. \\ref{sec:datasets}) and successively we describe the two adopted network characterizations (Sec. \\ref{sec:characteristics}).\nIn Sec. \\ref{sec:results} we present the experimental results.\nIn Sec. \\ref{sec:pca} we discuss the results in terms Principal Component Analysis (PCA) of the considered network characterizations in terms of numerical features; in Sec. \\ref{sec:cca} we demonstrate the important statistical agreement among the two characterizations.\nFinally, in Sec. \\ref{sec:scaling} we present the principal results of this paper, in which we show the two-regime diffusion of proteins.\nSec. \\ref{sec:conclusions} concludes the paper; Appendix \\ref{sec:heat_kernel} provides the HK technical details and Appendix \\ref{sec:heat_kernel_spectrum} offers a justification for considering ensemble statistics in the analysis of the HT decay.\n\n\n\n\\section{Material and Methods}\n\\label{sec:material_and_methods}\n\n\\subsection{The Considered Dataset}\n\\label{sec:datasets}\n\nOur dataset consists of 323 connected networks (simple graphs).\nWe considered 100 randomly selected E.Coli protein contact networks (PCN) from the dataset recently elaborated by us \\cite{ecoli_graph__arxiv,ecoli_graph_complexity_arxiv} -- in the literature PCN are also referred to as protein contact maps, although here we consider the two denominations as equivalent. Such proteins have been obtained by integrating the Niwa \\textit{et al.} \\cite{niwa2009} E.Coli data with the available information of the respective native structures gathered from the Protein Data Bank repository \\cite{pdb}.\nThe selected proteins contain from 300 to 1000 amino acid residues. As prescribed by \\citet{doi:10.1021\/cr3002356}, undirected edges are added among any two alpha carbon atoms within the $[4, 8]$ \\AA{} range; the lower 4 \\AA{} filter allows to get rid of trivial contacts that do not have important topological properties.\nThen, we considered 43 metabolic networks (MN) describing organisms belonging to all three domains of life. Vertices of such networks are the substrates and product of the chemical reactions, while the edges are the reactions catalyzed by the enzymes.\nAs demonstrated in Ref. \\cite{jeong2000large}, those large metabolic networks exhibit a typical scale-free topology.\nThe size of the 43 metabolic networks ranges from 300 to 1500 vertices.\nWe also considered a collection of 50 realistic gene regulatory networks (GEN) with a number of vertices varying from 200 to 1100 genes\/vertices. The GEN are generated with the SysGenSIM software \\cite{pinna2011simulating}, a MATLAB\\texttrademark  toolbox for the simulation of systems genetics datasets.\nArtificial networks and data by SysGenSIM have already been officially employed for the verification of gene network inference algorithms, such as in the DREAM5 Systems Genetics challenge \\cite{dream5}; they have also been used as benchmarks for the development of state-of-the-art reverse-engineering algorithms \\cite{pinna2010knockouts}.\nThe considered GEN networks have been generated with the Exponential Input Power-law Output (EIPO) model, i.e., they are built by (i) sampling the number of ingoing and outgoing edges for each vertex from, respectively, an exponential and a power law distribution, and then by (ii) connecting the vertices accordingly.\nThese artificial EIPO networks exhibit two well-known structural characteristics of real gene networks: the modularity \\cite{barabasi2004network}, and the vertex in-degree and out-degree distributions fitting, respectively, an exponential and a power-law curve \\cite{guelzim2002topological}.\nBesides the adopted EIPO topology, we considered an average vertex degree varying from 4 to 8: the average degree has been sampled in such a wide range due to the uncertainty in the size of the interactome in typical gene regulatory networks. Apparently, the complexity of biological organisms better correlates with the number of interactions between genes than with the number of genes. Therefore, the average number of edges in gene regulatory networks varies according to the complexity of the represented organism \\cite{stumpf2008estimating}; it makes then sense to study gene regulatory networks with a different number of interactions\/edges.\n\nTo obtain suitable references with the aim of helping us in discussing the results, we considered 130 additional networks of varying size belonging to well-known classes of graphs -- such networks play the role of ``probes'' in our dataset.\nNotably, we considered 10 Erd\\H{o}s-R\\'{e}nyi (ER) graphs generated with probability $p=\\log(n)\/n$; 10 Barab\\'{a}si-Albert (BA) scale-free networks \\cite{barabasi1999emergence} with a six-edges preferential attachment scheme; and 10 random regular graphs (REG) with degree equal to six.\nTo cover all network sizes, such probe networks are generated with a number of vertices ranging from 200 to 1100.\nFinally, we also considered the synthetic counterpart of the 100 real proteins (denoted as PCN-S in the following). \nSuch synthetic proteins have been generated by considering the same number of vertices and edges of the real proteins.\nThe generation mechanism of the topology follows the three-rule scheme proposed by \\citet{bartoli2007effect}, to simulate the folded configuration of the protein backbone by a probability of contact decreasing with the sequence distance. The only exception is for the rule involving edges of the backbone structure.\nIn fact, to be consistent with the architecture of our real proteins (we considered edges among residues within 4--8 \\AA{}), in PCN-S we added edges only among consecutive residues in the sequence having distance 2. It is worth pointing out that such a generation mechanism gives rise to networks with typical small world topology \\cite{bartoli2007effect}.\n\n\n\\subsection{Characterization of the Graph Topology}\n\\label{sec:characteristics}\n\nIn the following, we provide an essential description of the two graph characterizations used in this paper. Full details are omitted for the sake of brevity; we include references to the literature.\nIn practice, each characterization is meant to offer a description of the original graphs as vectors of numerical features.\nThe first characterization employs ``classical'' topological descriptors (TD), which include statistics of the degrees\/shortest paths and also elaborations of the graph spectrum.\nIn particular we consider the number of vertices (V) and edges (E) as basic descriptors of the size of the network; the modularity (MOD) \\cite{newman2006modularity,blondel2008fast} for quantifying the presence of a global community\/cluster structure (please note that we consider as feature the value associated to the partition with maximum modularity); the average closeness centrality (ACC), average shortest path (ASP), average degree centrality (ADC), and average clustering coefficient (ACL) \\cite{costa2007characterization}; the energy (EN) and Laplacian energy (LEN) of the spectrum  \\cite{gutman2006laplacian}; two invariant features from the heat kernel -- see later for details; the ambiguity (A) \\cite{Livi_ga_2013}, which expresses the degree of irregularity of the topology; and finally the entropy of a stationary Markovian random walk (H) \\cite{Dehmer201157}.\n\nIn the second characterization we exploit the HK only, whose technical details are reported in Appendix \\ref{sec:heat_kernel}.\nIn this respect, we consider here three sets of features: those extracted from the heat trace (HT), the heat content (HC), and the series of invariants (\\ref{eq:heat_coeff}) associated to the HC, which are called heat content invariants (HCI).\nPlease note that HT and HC are time-dependent characteristics, while HCI is not. Therefore, in the characterization exploiting the HK only, we consider several time instants for HT and HC; in the TD characterization, instead, we consider only one time instant for the HT -- corresponding to a ``transient'' regime of the diffusion -- and only the first HCI coefficient. Further details are progressively provided in the following sections.\n\n\n\n\\section{Results}\n\\label{sec:results}\n\nThe results of the computational experiments are organized in the following subsections. In Sec. \\ref{sec:pca} we show and discuss the PCA performed over the two aforementioned characterizations of the considered networks.\nIn Sec. \\ref{sec:cca} we discuss the canonical correlation analysis (CCA) calculated among these two characterizations.\nFinally, in Sec. \\ref{sec:scaling} we present the results obtained in terms of scaling and diffusion dynamics.\n\n\n\\subsection{Principal Component Analysis}\n\\label{sec:pca}\n\nNumeric data have been standardized using the component-wise mean and standard deviation.\nTD and HK fields were submitted to two separated PCA, projecting hence the networks in the spaces spanned by respective principal components (PC).\nThe emerging of local models specific for each type of network, and the correlation of the components extracted on the entire data set, is thus a proof of the presence of different architectures characterizing the considered classes of networks.\nMoreover, the mutual correlation between PCs of TD and HK -- assessed in Sec. \\ref{sec:cca} by a CCA -- gave us a demonstration of the consistency and interpretability of the considered network descriptions.\n\n\n\\subsubsection{Analysis of Topological Descriptors}\n\\label{sec:pca_td}\n\nFig. \\ref{fig:PCA_Topo_Descriptors} shows the PCA of the topological descriptors (PCA-TD).\nThe first three PCs are sufficient to explain more than 90\\% of the variance ($\\simeq 91\\%$).\nAs it is possible to observe in Fig. \\ref{fig:topo_pc1-2}, PC1--PC2 offer a very clear discrimination among the different classes of networks.\nThe separability persists also by considering PC1--PC3, while however we observe that GEN lose compactness and overlap with MN. By considering PC2--PC3, instead, PCN overlap with REG.\nHowever, the overall picture emerging from PCA-TD clearly points to the possibility of distinguishing among the network types.\n\nLet us now interpret such PCs. Tab. \\ref{tab:factors} shows the loadings of the first three factors of PCA-TD.\nThe first factor (FACTOR-1) is primarily characterized by MOD, ACC, and ASP. As MOD increases (the community structure becomes more evident) preferential paths connecting different regions of the network increase as well. In fact, ACC and ASP are, respectively, negatively and positively correlated with MOD.\nIt is worth mentioning that ACC and ASP offer a somewhat opposite view of the same feature, i.e., the efficiency of the paths in the networks.\nAs the global community structure emerges (captured by MOD), also the local clustering structure (ACL) increases as well, although ACL is less loaded on FACTOR-1.\nIn addition it is worth noting the agreement among the randomness (H) and the modularity: predictability of stationary random walks is affected by the presence of network modules\/communities.\n\nFACTOR-2 positively correlates the number of vertices (V) with LEN, which clearly points to the correlation among the network size in terms of number of vertices and the global architecture.\nThe meaning of this factor will appear more clear in Sec. \\ref{sec:scaling}, where we will discuss the scaling of the number of vertices with MOD and the invariant characteristics of the HK.\n\nFinally, the third factor (FACTOR-3) could be interpreted as the ``redundancy'' of the network wiring substrate.\nIn fact, descriptors heavily loaded on FACTOR-3 are those more directly related to the adjacency matrix--edges.\nThe ambiguity (A) decreases as the number of edges increases. This means that adding redundancy to the network (i.e., alternative paths) affects the regularity of the topology.\n\nIt is immediate to recognize how the different types of networks are characterized by local linear models in the globally orthogonal PC spaces.\nThese linear models correspond to different scaling relations with network size -- discussed later in Sec. \\ref{sec:scaling}.\n\nA simple look at Fig. \\ref{fig:PCA_Topo_Descriptors} allows to catch the singular position of PCN on the extreme right of the most informative PC1--PC2 space, so confirming  the peculiar character of PCN  with respect to classical network architectures. Moreover it is worth noting that the artificial polymer networks -- PCN-S -- are the most similar to PCN, although it is not possible to appreciate any overlap. This fact suggests that proteins are not just ``coiled strings'' as hypothesized in Ref. \\cite{bartoli2007effect}.\nIn addition to the features coming from the folding of a continuous backbone, PCN have other peculiar characteristics.\n\\begin{figure*}[ht!]\n\\centering\n\n\\subfigure[PC1--PC2]{\n\\includegraphics[viewport=0 0 343 245,scale=0.62,keepaspectratio=true]{.\/TD_PCA1-2.pdf}\n\\label{fig:topo_pc1-2}}\n~\n\\subfigure[PC1--PC3]{\n\\includegraphics[viewport=0 0 343 245,scale=0.62,keepaspectratio=true]{.\/TD_PCA1-3.pdf}\n\\label{fig:topo_pc1-3}}\n\n\\subfigure[PC2--PC3]{\n\\includegraphics[viewport=0 0 343 245,scale=0.62,keepaspectratio=true]{.\/TD_PCA2-3.pdf}\n\\label{fig:topo_pc2-3}}\n~\n\\subfigure[PC1--PC2--PC3]{\n\\includegraphics[viewport=0 0 321 204,scale=0.65,keepaspectratio=true]{.\/TD_PCA1-2-3.pdf}\n\\label{fig:topo_pc1-2-3}}\n\n\\caption{Embedding considering the first three PCs of PCA-TD.}\n\\label{fig:PCA_Topo_Descriptors}\n\\end{figure*}\n\\begin{table}[thp!]\\scriptsize\n\\begin{center}\n\\caption{Loadings of the first three factors of PCA-TD. Relevant correlations are in bold.}\n\\label{tab:factors}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\\textbf{DESCRIPTOR} & \\textbf{FACTOR-1} & \\textbf{FACTOR-2} & \\textbf{FACTOR-3} \\\\\n\\hline\nV & -0.0441 & \\textbf{0.9953} & 0.0513 \\\\\nE & -0.2053 & 0.5095 & \\textbf{0.8082} \\\\\nMOD & \\textbf{0.9591} & -0.1383 & -0.1036 \\\\\nADC & -0.2294 & 0.0428 & \\textbf{0.9375} \\\\\nACC & \\textbf{-0.9918} & 0.0353 & 0.0637 \\\\\nASP & \\textbf{0.9281} & -0.0279 & 0.0315 \\\\\nACL & 0.6716 & -0.3588 & -0.0010 \\\\\nEN & -0.0166 & 0.6830 & \\textbf{0.7268} \\\\\nLEN & -0.3944 & \\textbf{0.8407} & 0.1712 \\\\\nHT ($t=5$) & 0.6696 & 0.6486 & -0.1007 \\\\\nHCI ($m=1$) & 0.4914 & -0.6172 & 0.4639 \\\\\nH & 0.6906 & -0.2774 & 0.4918 \\\\\nA & -0.3229 & 0.1878 & \\textbf{-0.7584} \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\subsubsection{Analysis of the Heat Kernel}\n\\label{sec:anal_hk}\n\nWe consider three types of invariant features elaborated from the HK: HT, HC, and HCI.\nFor the PCA of HT and HC we take into account ten time instants going from $t=0$ to $t=9$; this choice will be justified later in Sec. \\ref{sec:scaling}; for the PCA of HCI we consider the first ten coefficients $q_m$ of the series in Eq. \\ref{eq:heat_coeff} -- this choice is motivated by the fact that for higher-order coefficients the values become numerically unstable.\nIn all cases, the first three PCs are sufficient to explain more than 90\\% of the variance of the original data, and so they are retained for the embedding.\n\nIn Fig. \\ref{fig:PCA_HT} it is shown the PCA of the HT representation (PCA-HT).\nFrom PC1--PC2 and PC2--PC3 of PCA-HT it is possible to understand that PCN are clearly recognizable by considering the HT, while however GEN depict a not-so-coherent pattern (this is valid for all three PCs).\nIn Fig. \\ref{fig:PCA_HC}, instead, we show the PCA of the HC representation (PCA-HC). We remind to the reader that HT and HC are correlated, since HC considers the information provided by both eigenvalues and eigenvectors of the normalized Laplacian, and not just the eigenvalues as in the HT case.\nIn fact, from PCA-HC it is possible to note that all networks denote a clear distinguishability; considering either PC1--PC2 and PC2--PC3 almost all networks seems to denote a very peculiar configuration in the PCA space.\n\nFinally, in Fig. \\ref{fig:PCA_HCI} we show the PCA of the HCI representation (PCA-HCI).\nThe PCs of PCA-HCI allow us noting how PCN, REG, and ER denote a very compact configuration in the PCA-HCI space, while GEN, BA, and MN present a more sparse distribution.\nThis fact might be interpreted by observing that such two groups differentiate among networks having a clear scale-free topology (second group) and those that are not scale-free (first group).\nInterestingly, PCN-S seem to lie in-between those two groups.\n\\begin{figure*}[ht!]\n\\centering\n\n\\subfigure[PC1--PC2.]{\n\\includegraphics[viewport=0 0 343 245,scale=0.62,keepaspectratio=true]{.\/HT_PCA1-2.pdf}\n\\label{fig:HT_PC1-PC2}}\n~\n\\subfigure[PC1--PC3.]{\n\\includegraphics[viewport=0 0 343 245,scale=0.62,keepaspectratio=true]{.\/HT_PCA1-3.pdf}\n\\label{fig:HT_PC1-PC3}}\n\n\\subfigure[PC2--PC3.]{\n\\includegraphics[viewport=0 0 343 245,scale=0.62,keepaspectratio=true]{.\/HT_PCA2-3.pdf}\n\\label{fig:HT_PC2-PC3}}\n~\n\\subfigure[PC1--PC2--PC3.]{\n\\includegraphics[viewport=0 0 318 195,scale=0.65,keepaspectratio=true]{.\/HT_PCA1-2-3.pdf}\n\\label{fig:HT_PC1-PC2-PC3}}\n\n\\caption{Embedding of the first three PCs of PCA-HT considering $t=0, 1, ..., 9$.}\n\\label{fig:PCA_HT}\n\\end{figure*}\n\\begin{figure*}[ht!]\n\\centering\n\n\\subfigure[PC1--PC2.]{\n\\includegraphics[viewport=0 0 343 245,scale=0.62,keepaspectratio=true]{.\/HC_PCA1-2.pdf}\n\\label{fig:HC_PC1-PC2}}\n~\n\\subfigure[PC1--PC3.]{\n\\includegraphics[viewport=0 0 343 245,scale=0.62,keepaspectratio=true]{.\/HC_PCA1-3.pdf}\n\\label{fig:HC_PC1-PC3}}\n\n\\subfigure[PC2--PC3.]{\n\\includegraphics[viewport=0 0 343 245,scale=0.62,keepaspectratio=true]{.\/HC_PCA2-3.pdf}\n\\label{fig:HC_PC2-PC3}}\n~\n\\subfigure[PC1--PC2--PC3.]{\n\\includegraphics[viewport=0 0 318 199,scale=0.65,keepaspectratio=true]{.\/HC_PCA1-2-3.pdf}\n\\label{fig:HC_PC1-PC2-PC3}}\n\n\\caption{Embedding of the first three PCs of PCA-HC considering $t=0, 1, ..., 9$.}\n\\label{fig:PCA_HC}\n\\end{figure*}\n\\begin{figure*}[ht!]\n\\centering\n\n\\subfigure[PC1--PC2.]{\n\\includegraphics[viewport=0 0 346 245,scale=0.62,keepaspectratio=true]{.\/HCI_PCA1-2.pdf}\n\\label{fig:HCI_PC1-PC2}}\n~\n\\subfigure[PC1--PC3.]{\n\\includegraphics[viewport=0 0 346 245,scale=0.62,keepaspectratio=true]{.\/HCI_PCA1-3.pdf}\n\\label{fig:HCI_PC1-PC3}}\n\n\\subfigure[PC2--PC3.]{\n\\includegraphics[viewport=0 0 346 245,scale=0.62,keepaspectratio=true]{.\/HCI_PCA2-3.pdf}\n\\label{fig:HCI_PC2-PC3}}\n~\n\\subfigure[PC1--PC2--PC3.]{\n\\includegraphics[viewport=0 0 318 195,scale=0.65,keepaspectratio=true]{.\/HCI_PCA1-2-3.pdf}\n\\label{fig:HCI_PC1-PC2-PC3}}\n\n\\caption{Embedding of the first three PCs of PCA-HCI considering the first ten HCI coefficients.}\n\\label{fig:PCA_HCI}\n\\end{figure*}\n\n\n\n\\subsection{Canonical Correlation Analysis of the PCA Representations}\n\\label{sec:cca}\n\nHere we discuss the canonical correlation analysis (CCA) calculated among the various PCA discussed in Sec. \\ref{sec:pca}.\nFor the CCA, we always consider the first three PCs of each representation.\nIn Tab. \\ref{tab:cca} are reported the pairwise correlation values among the most important canonical variates.\nThere is a strong agreement among all considered PCA representations.\nSince part of the information from the HK is present also in TD, we have considered also a PCA representation of TD that does not include such information -- in the table it is indicated as ``PCA-TD\\_NO-HK''.\nInterestingly, removing the information of the HK from the TD does not alter the scored correlation, so giving a demonstration of the strong coherence between TD and HK based representations of the considered networks.\nThis result suggests the possibility to interpret the three HK based representations in terms of the more understandable linear correlation structure discussed in Sec. \\ref{sec:pca_td}.\n\\begin{table}[thp!]\\scriptsize\n\\begin{center}\n\\caption{Canonical correlation coefficients between the first canonical variates relative to different principal component spaces.}\n\\label{tab:cca}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n& \\textbf{PCA-HT} & \\textbf{PCA-HC} & \\textbf{PCA-HCI} \\\\\n\\hline\n\\textbf{PCA-TD} & 0.993 & 0.992 & 0.961 \\\\\n\\textbf{PCA-TD\\_NO-HK} & 0.988 & 0.986 & 0.946 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\subsection{Scaling and Heat Diffusion Analysis}\n\\label{sec:scaling}\n\nIn this section we first study the results in terms of scaling of MOD, HT, HC, and HCI w.r.t. the number of vertices of the considered networks.\nSuccessively, we provide an analysis of the characteristic diffusion patterns emerged from the focused study of the HK.\n\nFig. \\ref{fig:scaling_modularity} shows the scaling of the modularity with the size (number of vertices) of the networks.\nAs already noted in Tab. \\ref{tab:factors}, V and MOD do not appear to be globally correlated.\nIn fact, PCN and PCN-S are the only architectures that show an increasing trend, while the others appear to be almost uncorrelated. We note an exception for ER that tend toward a negative correlation; please note that in the ER case, analytical results are available \\cite{guimera2004modularity}.\nIt is worth explaining the particular pattern of GEN, which does not show a clear trend. In Fig. \\ref{fig:scaling_modularity_ge} we show the scaling for GEN by considering the different average degrees used for the EIPO model, where we can observe that each average degree gives rise to a definite trend of MOD.\nPlease note that the linear fitting lines in Fig. \\ref{fig:scaling_modularity} are introduced only to help the reader visually, and are not meant to provide a model describing the MOD trend asymptotically.\n\\begin{figure*}[ht!]\n\\centering\n\n\\subfigure[Scaling of modularity.]{\n\\includegraphics[viewport=0 0 350 245,scale=0.62,keepaspectratio=true]{.\/scaling_modularity.pdf}\n\\label{fig:scaling_modularity_all}}\n~\n\\subfigure[Modularity of GEN with different average degree.]{\n\\includegraphics[viewport=0 0 339 245,scale=0.62,keepaspectratio=true]{.\/scaling_modularity_ge.pdf}\n\\label{fig:scaling_modularity_ge}}\n\n\\caption{Scaling of modularity over network size. Please note that the linear fitting lines are introduced only to visually guide the reader, and are not meant to provide a model describing the MOD trend asymptotically.}\n\\label{fig:scaling_modularity}\n\\end{figure*}\n\nFigs. \\ref{fig:scaling_HT}, \\ref{fig:scaling_HT_slopes}, and \\ref{fig:scaling_HC} show the scaling of all considered HK invariants.\nInitially we consider only three relevant time instants for HT, i.e., $t=1, 5, 9$, which are depicted, respectively, in Figs. \\ref{fig:scaling_HT_t1}, \\ref{fig:scaling_HT_t5}, and \\ref{fig:scaling_HT_t9}.\nIt is possible to note that, as expected, at $t=1$ all networks show a similar increasing linear trend w.r.t. the network size.\nAs the time instant increases, instead, PCN show a positive slope at least one order of magnitude greater than the others.\nAt first, this fact might be attributed solely to the intrinsic high modularity characterizing the protein structures.\nTo this end, in Fig. \\ref{fig:correlations} we globally correlated MOD with HT over time -- the time $t$ here has a fine-grained sampling going from 0.1 to 100 with an increment step of 0.1. In the same plot, we show also the partial correlation obtained when considering the number of vertices as the control variable (indicated as ``MOD--HT(t) \/ V'' in the figure).\nThe linear correlation trend shows that initially the two quantities are fairly anti-correlated, while they soon become very correlated, reaching the maximum correlation ($\\simeq 0.88$) around the time instant $t=10$. Successively, the correlation decreases with a smooth trend.\nThe partial correlation demonstrates that the initial negative correlation is due to the network size effect; correlations are positive when the size is removed.\nThis variability in the correlation points out the fact that, although the information provided by HT is fairly consistent with the one provided by MOD, they are by no means equivalent. Notably, HT offers a richer type of information that we will further exploit in the following in terms of diffusion.\nIn addition, this particular trend justifies the selection of the first ten time (integer-valued) instants for the calculations of PCA-HT, PCA-HC, and PCA-HCI performed in Sec. \\ref{sec:anal_hk}.\n\\begin{figure*}[ht!]\n\\centering\n\n\\subfigure[Scaling of HT for $t=1$.]{\n\\includegraphics[viewport=0 0 350 245,scale=0.62,keepaspectratio=true]{.\/scaling_HT_t1.pdf}\n\\label{fig:scaling_HT_t1}}\n~\n\\subfigure[Scaling of HT for $t=5$.]{\n\\includegraphics[viewport=0 0 350 245,scale=0.62,keepaspectratio=true]{.\/scaling_HT_t5.pdf}\n\\label{fig:scaling_HT_t5}}\n\n\\subfigure[Scaling of HT for $t=9$.]{\n\\includegraphics[viewport=0 0 350 245,scale=0.62,keepaspectratio=true]{.\/scaling_HT_t9.pdf}\n\\label{fig:scaling_HT_t9}}\n~\n\\subfigure[Linear correlations among MOD and $\\mathrm{HT}(t)$.]{\n\\includegraphics[viewport=0 0 349 241,scale=0.62,keepaspectratio=true]{.\/correlations.pdf}\n\\label{fig:correlations}}\n\n\\caption{Scaling of HT over networks size.}\n\\label{fig:scaling_HT}\n\\end{figure*}\n\nNow we elaborate over the diffusion dynamics of the considered network ensembles by analyzing the log-log plots shown in Fig. \\ref{fig:scaling_HT_slopes}.\nLet us denote the time-dependent linear best-fitting (as those in Figs. \\ref{fig:scaling_HT_t1}, \\ref{fig:scaling_HT_t5}, and \\ref{fig:scaling_HT_t9}) for the HT over the networks of a specific class, $\\mathcal{C}$, as a function of the network size $n$, with the following expression:\n\\begin{equation}\n\\label{eq:linearfit_ht}\n\\text{HT}_\\mathcal{C}(n; t) \\simeq \\alpha_{\\mathcal{C}}(t)\\cdot n,\n\\end{equation}\nwhere $n$ is the number of vertices and $\\alpha_{\\mathcal{C}}(t)\\geq 0$ is the time-dependent slope.\nAs discussed in Appendix \\ref{sec:heat_kernel_spectrum}, by fitting linearly the HT we implicitly hypothesize the possibility to consistently describe each class of networks with an ensemble, $\\mathcal{C}$, characterized by a unique probability density function of the normalized Laplacian eigenvalues (see Ref. \\cite{mitrovic2009spectral} for a related theoretical study on a similar topic). This assumption is also justified by the results of PCA-HT reported in Fig. \\ref{fig:PCA_HT}, which show good agreement among networks of the same class.\nAs a consequence, the linear best-fitting (\\ref{eq:linearfit_ht}) allows to consider a statistic over an entire homogeneous class of networks, instead of focusing on each isolated network dynamics separately.\nIt is straightforward to realize that $\\text{HT}_\\mathcal{C}(n; t) = n$ for $t=0$, i.e., $\\alpha_{\\mathcal{C}}(0)=1$.\nAs $t$ grows, $\\alpha_{\\mathcal{C}}(t)$ becomes always smaller, with a rate that is related to the characteristic HT decay of the ensemble.\nIn Fig. \\ref{fig:scaling_HT_slopes_time} we show the linear best-fitting slopes of HT, $\\alpha_{\\mathcal{C}}(t)$, as a function of time -- note that $t$ always varies from 0 to 100 with an increment step of 0.1.\nWhile one expects to observe trends consistent with an exponential decay (see definition of HT in Eq. \\ref{eq:heat_trace}), it is possible to recognize a different asymptotic trend for the PCN ensemble. For the sake of a better visualization, in Figs. \\ref{fig:scaling_HT_slopes_time__P}, \\ref{fig:scaling_HT_slopes_time__M}, and \\ref{fig:scaling_HT_slopes_time__G} we report the same plot but isolating, respectively, PCN, MN, and GEN; other networks are omitted for brevity.\nFig. \\ref{fig:scaling_HT_slopes_time__P} depicts what we might consider a change of functional form for the PCN trend at some point in time (i.e., starting around $t\\simeq 5$).\nThis change of regime in the diffusion lasts few time instants, then the trend switches from exponential to power law like. This is not noted for the other networks that, instead, remain consistent with an exponential decay -- they are actually expressed as sums of exponentials.\nIn practice, from a certain $\\widetilde{t}>5$, the diffusion in PCN seems to be consistent with a power law, $\\alpha_{\\mathcal{C}}(\\widetilde{t}) \\sim \\widetilde{t}^{-\\beta}$, where in our case the characteristic exponent is $\\beta\\simeq 1.1$.\nIt is worth mentioning that we achieved the very same results for PCN constructed without considering the lower filter at 4 \\AA{}, that is, by considering all contacts within 8 \\AA{} (data not shown).\n\nSimilar anomalies of functional form have been observed in the (cumulative) distribution of many experimental time series, especially in those related to financial markets \\cite{kwapien2012physical}.\nThis phenomenon might happen when the functional form is consistent with one of the $q$-exponentials family, which originated in the field of non-extensive statistical mechanics \\cite{tsallis2001nonextensive}.\nIn the case of PCN, this behavior is the signature of a crucial physical property of proteins, i.e., the energy flow.\nEnergy flow in proteins mimics the transport in a three-dimensional percolation cluster \\cite{doi:10.1146\/annurev.physchem.59.032607.093606}: energy flows readily between connected sites of the cluster and only slowly between non connected sites.\nThis experimentally validated double regime seems to be captured by the HT decay trend shown in Fig. \\ref{fig:scaling_HT_slopes_time__P}. \nWe stress that this result is elaborated from the herein exploited minimalistic PCN model, so confirming the relevance of graph-based representations in protein science.\n\\begin{figure*}[ht!]\n\\centering\n\n\\subfigure[Linear fitting slopes of HT over time.]{\n\\includegraphics[viewport=0 0 340 247,scale=0.6,keepaspectratio=true]{.\/scaling_HT_slopes_time_log_log.pdf}\n\\label{fig:scaling_HT_slopes_time}}\n~\n\\subfigure[Linear fitting slopes of HT over time (PCN only).]{\n\\includegraphics[viewport=0 0 340 247,scale=0.6,keepaspectratio=true]{.\/scaling_HT_slopes_time_log_log_PROTONLY.pdf}\n\\label{fig:scaling_HT_slopes_time__P}}\n\n\\subfigure[Linear fitting slopes of HT over time (MN only).]{\n\\includegraphics[viewport=0 0 340 247,scale=0.6,keepaspectratio=true]{.\/scaling_HT_slopes_time_log_log_MNONLY.pdf}\n\\label{fig:scaling_HT_slopes_time__M}}\n~\n\\subfigure[Linear fitting slopes of HT over time (GEN only).]{\n\\includegraphics[viewport=0 0 340 247,scale=0.6,keepaspectratio=true]{.\/scaling_HT_slopes_time_log_log_GEONLY.pdf}\n\\label{fig:scaling_HT_slopes_time__G}}\n\n\\caption{Scaling of HT linear best fitting slopes over time (time is sampled in 1000 equally-spaced points between 0 and 100).}\n\\label{fig:scaling_HT_slopes}\n\\end{figure*}\n\nNow let us consider the results for the HC (Figs. \\ref{fig:scaling_HC_t1}, \\ref{fig:scaling_HC_t5}, and \\ref{fig:scaling_HC_t9}). Those three figures depict the scaling of the HC over the network size, considering the information of the entire HM. Notably, PCN and PCN-S are the only network types showing a consistent linear scaling with the size for all time instants. Other networks are not well-described by a linear fit as the time increases.\nFinally, in Fig. \\ref{fig:scaling_HCI} we show the scaling of the first HCI coefficient with the vertices (please note that for $m=1$, Eq. \\ref{eq:hci} yields negative values). Of notable interest is the fact that PCN denote a nearly constant trend.\nThis means that, since the HCIs are time-independent features synthetically describing the HC information, PCN denote a similar characteristic in this respect, as in fact HC scaling in Fig. \\ref{fig:scaling_HC} is consistently preserved over time.\n\nIn Fig. \\ref{fig:HM_time} we offer a visual representation of the heat diffusion pattern over time that is observable through the entire HM.\nWe considered two exemplar networks of exactly the same size: the ``JW0058'' protein and the synthetic counterpart belonging to PCN-S that we denote here as ``JW0058-SYNTH''.\nAs discussed before, PCN are characterized by a highly modular and fractal structure, while the considered synthetic counterpart exhibits a typical small world topology.\nAccordingly, by comparing the diffusion occurring on the two networks over time, it is possible to recognize significantly different patterns that were not noted in the scalings of Fig. \\ref{fig:scaling_HC}.\nOf course, initially ($t=1$) the heat is mostly concentrated in the vertices, which results in a very intense trace.\nAs the time increases, the diffusion pattern for the real protein is more evident and also persistent.\nThis is in agreement with recent laboratory experiments \\cite{doi:10.1146\/annurev.physchem.59.032607.093606,lervik2010heat}, which demonstrated that diffusion in proteins proceeds slower than normal diffusion.\nConversely, the diffusion for JW0058-SYNTH is in general faster since in fact the trace vanishes quickly.\nIn graph-theoretical terms, this means that the spectral gap of PCN dominates Eq. \\ref{eq:heat_matrix} as $t$ becomes large.\nThis result suggests us that, in future research studies, it could be interesting to devote focused attention to the properties of the spectral density of the ensembles.\nAnalogue results have been obtained by considering the other network types; we do not show them for the sake of brevity.\n\\begin{figure*}[ht!]\n\\centering\n\n\\subfigure[Scaling of HC for $t=1$.]{\n\\includegraphics[viewport=0 0 351 245,scale=0.62,keepaspectratio=true]{.\/scaling_HC_t1.pdf}\n\\label{fig:scaling_HC_t1}}\n~\n\\subfigure[Scaling of HC for $t=5$.]{\n\\includegraphics[viewport=0 0 351 245,scale=0.62,keepaspectratio=true]{.\/scaling_HC_t5.pdf}\n\\label{fig:scaling_HC_t5}}\n\n\\subfigure[Scaling of HC for $t=9$.]{\n\\includegraphics[viewport=0 0 351 245,scale=0.62,keepaspectratio=true]{.\/scaling_HC_t9.pdf}\n\\label{fig:scaling_HC_t9}}\n~\n\\subfigure[Scaling of HCI (first coefficient).]{\n\\includegraphics[viewport=0 0 351 245,scale=0.62,keepaspectratio=true]{.\/scaling_HCI.pdf}\n\\label{fig:scaling_HCI}}\n\n\\caption{Scaling of HC and HCI over network size.}\n\\label{fig:scaling_HC}\n\\end{figure*}\n\\begin{figure*}[ht!]\n\\centering\n\n\\subfigure[HM for JW0058 at $t=1$.]{\n\\includegraphics[viewport=0 0 1043 634,scale=0.205,keepaspectratio=true]{.\/HM_JW0058_t1.pdf}\n\\label{fig:HM_JW0058_t1}}\n~\n\\subfigure[HM for JW0058-SYNTH at $t=1$.]{\n\\includegraphics[viewport=0 0 1037 615,scale=0.21,keepaspectratio=true]{.\/HM_JW0058_SYNTH_t1.pdf}\n\\label{fig:HM_TY_t1}}\n\n\\subfigure[HM for JW0058 at $t=5$.]{\n\\includegraphics[viewport=0 0 1037 615,scale=0.21,keepaspectratio=true]{.\/HM_JW0058_t5.pdf}\n\\label{fig:HM_JW0058_t5}}\n~\n\\subfigure[HM for JW0058-SYNTH at $t=5$.]{\n\\includegraphics[viewport=0 0 1043 634,scale=0.205,keepaspectratio=true]{.\/HM_JW0058_SYNTH_t5.pdf}\n\\label{fig:HM_TY_t5}}\n\n\\subfigure[HM for JW0058 at $t=9$.]{\n\\includegraphics[viewport=0 0 1037 615,scale=0.21,keepaspectratio=true]{.\/HM_JW0058_t9.pdf}\n\\label{fig:HM_JW0058_t9}}\n~\n\\subfigure[HM for JW0058-SYNTH at $t=9$.]{\n\\includegraphics[viewport=0 0 1037 615,scale=0.21,keepaspectratio=true]{.\/HM_JW0058_SYNTH_t9.pdf}\n\\label{fig:HM_TY_t9}}\n\n\\subfigure[HM for JW0058 at $t=15$.]{\n\\includegraphics[viewport=0 0 1037 615,scale=0.21,keepaspectratio=true]{.\/HM_JW0058_t15.pdf}\n\\label{fig:HM_JW0058_t15}}\n~\n\\subfigure[HM for JW0058-SYNTH at $t=15$.]{\n\\includegraphics[viewport=0 0 1037 615,scale=0.21,keepaspectratio=true]{.\/HM_JW0058_SYNTH_t15.pdf}\n\\label{fig:HM_TY_t15}}\n\n\\caption{HM diffusion pattern over time for protein JW0058 and its synthetic counterpart.}\n\\label{fig:HM_time}\n\\end{figure*}\n\n\n\\clearpage\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nIn this paper we have investigated the structure of three types of complex networks: protein contact networks, metabolic networks, and gene regulatory networks, together with simulated archetypal models acting as probes.\nWe biased the study on protein contact networks, highlighting their peculiar structure with respect to the other networks.\nOur analysis focused on ensemble statistics, that is, we analyzed the features elaborated by considering several instances of varying size of such networks.\nWe considered two main network characterizations: the first one based on classical topological descriptors, while the second one exploited several invariants extracted from the discrete heat kernel.\nWe found strong statistical agreement among those two representations, which allowed for a consistent interpretation of the results in terms of principal component analysis.\nOur major result was the demonstration of a double regime characterizing a (simulated) diffusion process in the considered protein contact networks.\nAs shown by laboratory experiments, both energy flow and vibration dynamics in proteins exhibit subdiffusive properties, i.e., slower-than-normal diffusion \\cite{doi:10.1146\/annurev.physchem.59.032607.093606}.\nThe notable difference in the diffusion pattern between real proteins and the herein considered simulated polymers (whose contact networks have the same local structure of the corresponding real proteins), points to a peculiar mesoscopic organization of proteins going beyond the pure backbone folding.\nThe observed correlations between MOD and HT indicates this principle in the presence of well-characterized domains.\nThe novelty of our results is that we were able to demonstrate such a well-known property of proteins by exploiting graph-based representations and computational tools only.\nThe fact that the observed properties emerged with no explicit reference to chemico-physical characterization of proteins, relying hence on pure topological properties, suggests the existence of general universal mesoscopic principles fulfilling the hopes expressed by Laughlin \\textit{et al.} \\cite{laughlin2000middle}.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nRepeated studies have illustrated that neural networks can be trained to solve inverse problems in imaging, including problems such as image reconstruction in MRI, inpainting, superresolution, deblurring, and more. Recent reviews and tutorials on this topic \\cite{arridge2019solving,ongie2020deep} have described various approaches to this problem. \\change{For concreteness, we focus on the case of \\emph{linear} inverse problems in imaging.} In the general framework of interest, an unknown\n$n$-pixel image (in vectorized form) $x \\in \\mathbb{R}^n$ (or $\\mathbb{C}^n$) is observed via $m$ noisy \\change{linear} measurements \n$y\\in\\mathbb{R}^m$ (or $\\mathbb{C}^m$) according to the model\n\\begin{equation}\\label{eq:invprob0}\ny=A_0x+\\varepsilon,\n\\end{equation}\nwhere the matrix $A_0 \\in \\mathbb{R}^{m\\times n}$ (or $\\mathbb{C}^{m\\times n})$ is the {\\em forward model} and $\\varepsilon$ represents a vector of noise. The goal is to recover $x$ from $y$.\n\n\\begin{figure*}[tb]\n\\centering\n\\setlength{\\tabcolsep}{0pt}\n\\begin{tabular}{cccc}\n\\subfloat{\\includegraphics[width = 0.245\\textwidth]{figures\/revisedbigfig\/mri\/nomask.png}}\\hspace{0.1em} &\n\\subfloat{\\includegraphics[width = 0.245\\textwidth]{figures\/revisedbigfig\/mri\/initial.png}}\\hspace{0.1em}  &\n\\subfloat{\\includegraphics[width = 0.245\\textwidth]{figures\/revisedbigfig\/mri\/newmodel.png}} \\hspace{0.1em} &\n\\subfloat{\\includegraphics[width = 0.245\\textwidth]{figures\/revisedbigfig\/mri\/rnr.png}} \\\\\n\\subfloat{\\includegraphics[width = 0.245\\textwidth]{figures\/newbigfig\/motionblur\/ground_truth}}\\hspace{0.1em}  &\n\\subfloat{\\includegraphics[width = 0.245\\textwidth]{figures\/newbigfig\/motionblur\/original_30.679604}}\\hspace{0.1em}  &\n\\subfloat{\\includegraphics[width = 0.245\\textwidth]{figures\/newbigfig\/motionblur\/newmodel_21.416325}}\\hspace{0.1em}  &\n\\subfloat{\\includegraphics[width = 0.245\\textwidth]{figures\/newbigfig\/motionblur\/PnP_29.681627}} \\\\\n\\small (a) Ground truth & \\small (b) No model drift & \\small (c) Model drift  & \\begin{minipage}{0.25\\linewidth}\\small (d) Model drift w\/model adaptation \\end{minipage}\n\\end{tabular} \\\\\n\\caption{Small perturbations in measurements for deep learning-based image reconstruction operators can lead to both subtle and obvious artifacts in reconstructions across problems and domains. In the top row, we present results for undersampled MRI reconstruction of knee images, and the second row illustrates deblurring images of human faces.\n(a) Ground truth image. (b) No model drift. Training and test data correspond to same model, $A_0$, yielding accurate reconstruction  via learned model. (c) Model drift but no model adaptation. Training assumes model $A_0$ but at test time we have model $A_1$. Reconstruction using trained network {\\em without model adaptation} gives significant distortions. (d) Model drift and model adaptation. Training assumes model $A_0$ but at test time we have model $A_1$. Reconstruction using model adaptation prevents distortions and compares favorably to the setting without model drift. The MRI example demonstrates our Reuse and Regularize method (Alg. \\ref{alg:rnr}), and the deblurring example demonstrates our Parameterize and Perturb method (Alg. \\ref{alg:pnp}). Experimental details are in Section~\\ref{sec:exp}.}\n\\label{fig:fig1}\n\\end{figure*}\n\nIn this paper, we focus on the setting in which the forward model $A_0$ is known and used during training. Past work has illustrated that leveraging knowledge of $A_0$ during training can reduce the sample complexity \\cite{gilton2019neumann}. This paradigm is particularly common in applications such as medical imaging, where $A_0$ represents a model of the imaging system. For instance, in magnetic resonance imaging (MRI), $A_0$ reflects which k-space measurements are collected. \n\n\n\n\nUnfortunately, these methods can be surprisingly  fragile in the face of {\\em model drift}, which occurs when, at test time, we are provided samples of the form \n\\begin{equation}\n\\label{eq:invprob1}\ny = A_1 x+\\varepsilon'\n\\end{equation}\nfor some new forward model $A_1 \\neq A_0$  \\change{and\/or a change in the noise distribution (\\emph{i.e.},~ the noise $\\varepsilon'$ is distributed differently than $\\varepsilon$).} That is, \nassume we have trained a solver that is a function of both the original forward model $A_0$ and a learned neural network. \nOne might try to reconstruct $x$\nfrom $y$ using this solver, but it will perform poorly because it is using a misspecified model ($A_0$ instead of $A_1$). Alternatively, we might attempt to use the same general solver where we replace $A_0$ with $A_1$ but leave the learned component intact. In this case, the estimate $x$ computed from $y$ may also be poor, as illustrated in \\cite{antun2020instabilities} and \\cite{hussein2020correction}. The situation is complicated even further if we do not have a precise model of $A_1$ at test time. \n\nThese are real challenges in practice. For example, in MRI reconstruction there is substantial variation in the forward model depending on the type of acquisition -- e.g., Cartesian versus non-Cartesian k-space sampling trajectories, different undersampling factors, different number of coils and coil sensitivity maps, magnetic field inhomogeneity maps, and other calibration parameters \\cite{fessler} -- all which need to be accounted for during training and testing. A network trained for one of these forward models may need to be retrained from scratch in order to perform well on even a slightly different setting (e.g., from six-fold to four-fold undersampling of k-space). Furthermore, training a new network from scratch may not always be feasible after deployment due to a lack of access to ground truth images. This could be either due to privacy concerns of sharing patient data between hospitals and researchers, or because acquiring ground truth images is difficult for the new inverse problem. \n\n\nThis leads us to formulate the problem of {\\em model adaptation}: given a reconstruction network trained on measurements from one forward model adapt\/retrain\/modify the network to reconstruct images from measurements reflecting a new forward model. We consider a few  variants of this problem: (a) the new forward model $A_1$ is known, along with one or more unlabeled training samples $y_i$ reflecting $A_1$, and (b) $A_1$ is unknown or only partially known, and we only have one or more unlabeled training samples reflecting $A_1$. These training samples are unlabeled in the sense that they are not paired with ``ground truth'' images used to generate the $y_i$'s. \nOur proposed model adaptation methods allow \na reconstruction network to be trained for a known forward model and then adapted to a related forward model without access to ground truth images, and without knowing the exact parameters of the new forward model.\n\n\nModel drift as stated above is a particular form of \\emph{distribution drift}, in which the distribution of $Y|X=x$ changes between training and deployment and we know $Y$ has a linear dependence on $X$ before and after the drift (even if we do not know the parameters of those linear relationships, represented as $A_0$ and $A_1$). \\change{That is, if we assume $\\varepsilon \\sim {\\cal N}(0,\\sigma^2 I)$, then the training distribution is $Y|X=x \\sim {\\cal N}(A_0 x,\\sigma^2 I)$ and the distribution at deployment is $Y|X=x \\sim {\\cal N}(A_1 x,\\sigma^2 I)$. In general, distribution drift challenges may be addressed using transfer learning \\cite{pan2009survey,weiss2016survey,tan2018survey} and domain adaptation \\cite{muandet2013domain,li2017deeper,wang2018deep}. One of the methods we explore in the body of the paper, Parameterize and Perturb, shares several features with transfer learning methodology. However, since in our setting we have a specific form of distribution drift, it is possible to design more targeted methods with better performance, as illustrated by\nexisting specialized methods for image inpainting \\cite{fawzi2016image}, as well as our general-purpose Reuse and Regularize method (detailed below).}\n\\subsection{Related Work}\n\nA broad collection of recent works, as surveyed by \\cite{arridge2019solving} and \\cite{ongie2020deep}, have explored using machine learning methods to help solve inverse problems in imaging. The current paper is motivated in part by experiments presented in \\cite{antun2020instabilities}, which show that deep neural networks trained to solve inverse problems are prone to several types of instabilities. \nSpecifically, they \nshowed that model drift in the form of slight changes in the forward model (even ``beneficial'' ones, like increasing the number of k-space samples in MRI) often have detrimental impacts on reconstruction performance. While \\cite{antun2020instabilities}\nis mostly empirical in nature, a follow-up mathematical study \\cite{gottschling2020troublesome} provides theoretical support to this finding, implying that instability arises naturally from training standard deep learning inverse solvers. However, recent work also shows that the instabilities observed in in \\cite{antun2020instabilities} can be mitigated to some extent by adding noise to measurements during training, though such techniques are not sufficient to resolve artifacts arising from substantial model drift.\nTo address a subset of these issues, \\cite{raj2020improving} and \\cite{lunz2018adversarial} propose  adversarial training frameworks that increases the robustness of inverse problem solvers. However, \\cite{raj2020improving} and \\cite{lunz2018adversarial} focus on robustness to adversarial perturbations in the measurements for a fixed forward model, and do not address \\change{a global change in the forward model}, which is the focus of this work.\n\nSimilar to this work, a recent paper \\cite{jong} has proposed domain adaptation techniques to transfer a reconstruction network from one inverse problem setting to another, e.g., adapting a network trained for CT reconstruction to perform MRI reconstruction. However, the focus of that approach is on adapting to {\\em changes in the image distribution}, whereas our approaches focus on {\\em changes to the forward model} assuming the image distribution is unchanged. Additionally, to our knowledge, no existing domain adaptation approaches consider the scenario where the new forward model depends on unknown calibration parameters, as we do in this work. \n\nAnother line of work explores learned methods for image reconstruction with automatic parameter tuning; see \\cite{wei2020tuning} and references therein. However, this work focuses on learning regularization and optimization parameters, not parameters of a drifting forward model. Also, \\cite{wu2019learning} describes a unrolling approach to learning a forward model in an imaging context, but with the goal of designing a forward model that optimizes reconstruction quality, rather than estimating a correction to the forward model from measurements. Some recent studies have used pre-trained generative models to solve inverse problems with unknown calibration parameters \\cite{anirudh2018unsupervised}; this line of work can be viewed as an extension of compressed sensing with general models framework introduced in \\cite{bora2017compressed}.\n\n\\section{Problem Formulation}\nHere we formalize the problem of \\emph{model adaptation} as introduced above.\n\nSuppose we have access to an estimator $\\widehat{x}  = f_0(y)$ that has been designed\/trained to solve the inverse problem\n\\begin{equation}\\label{eq:invprob0a}\n    y = A_0x + \\varepsilon,\\quad x \\sim P_X,~\\varepsilon\\sim P_{N_0}\\tag{P0}\n\\end{equation}\nwhere $A_0$ is a known (linear) forward model, $P_X$ denotes the distribution of images $x$ and $P_{N_0}$ denotes the distribution of the noise $\\varepsilon$. We assume the trained estimator ``solves'' the inverse problem in the sense that it produces an estimate $\\hat x = f_0(y)$ such that the mean-squared error (MSE) $\\mathbb{E}_{x,\\varepsilon}[\\|\\hat x - x\\|^2]$ is small. \n\nNow assume that the forward model has changed from $A_0$ to a new model $A_1$ and\/or the noise distribution has changed from $P_{N_0}$ to a new noise distribution $P_{N_1}$, resulting in the new inverse problem\n\\begin{equation}\\label{eq:invprob1a}\n    y = A_1 x + \\varepsilon',\\quad x \\sim P_X,~\\varepsilon'\\sim P_{N_1}.\\tag{P1}\n\\end{equation}\nWe consider both the case where $A_1$ is \\emph{known} (\\emph{i.e.},~ we have an accurate estimate of $A_1$) and the case \\change{where $A_1$ is partially unknown, in the sense that it belongs to a class of parameterized forward models, \\emph{i.e.},~ $A_1 \\in \\{A(\\sigma): \\sigma \\in \\mathbb{R}^q\\}$, where the parameters $\\sigma \\in \\mathbb{R}^q$ are unknown.}\n\nThe goal of \\emph{model adaptation} is to adapt\/retrain\/modify the estimator $\\widehat{x} = f_0(y)$ that was designed to solve the original inverse problem (P0) to solve the new inverse problem (P1). We will consider two variants of this problem:\n\\change{\n\\begin{itemize}\n    \\item \\emph{Model adaptation without calibration data:} In this setting, we assume access to only one measurement vector $y$ generated according to \\eqref{eq:invprob1a}.\n    \\item \\emph{Model adaptation with calibration data:} In this setting we assume access to a new set of measurement vectors $\\{y_i\\}_{i=1}^N$ generated according to \\eqref{eq:invprob1a}, but without access to the paired ground truth images (i.e., the corresponding $x_i$'s).\n\\end{itemize}\n}\n\nWhile the above discussion centers around a general estimator $\\widehat{x} = f_0(y)$, we are particularly interested in estimators that combine a trained trained deep neural network component depending on a vector of weights\/parameters $\\theta_0$, along with the original forward model $A_0$; we will call such an estimator a \\emph{reconstruction network}. Specifically, we assume that the forward model $A_0$ (or other derived quantities, such as its transpose $A_0^\n\\top$, psuedo-inverse $A_0^\\dagger$, etc.) is embedded in the reconstruction network, either in an initialization layer and\/or in multiple subsequent layers. This is the case for networks based on unrolling of iterative algorithms (see, for example, \\cite{gregor2010learning, sun2016deep, mardani2018neural,ongie2020deep,arridge2019solving}, and references therein), in which $A_0$ appears repeatedly in the network in ``data-consistency'' layers that approximately re-project the intermediate outputs of the network onto the set of data constraints $\\{x\\in \\mathbb{R}^n : A_0 x = y\\}$.\nIn general, we will\nassume the reconstruction network can be parametrized as $f_0(\\cdot) =  f(\\cdot;\\theta_0,A_0)$ where  $\\theta_0\\in\\mathbb{R}^p$ is the vector of pre-trained neural network weights\/parameters and $A_0$ is the original forward model.\n\n\\change{Finally, to simplify the presentation, we will assume an additive white Gaussian noise model for both (P0) and (P1), \\emph{i.e.},~ $P_{N_0} = \\mathcal{N}(0,\\sigma_0^2 I)$ and $P_{N_1} = \\mathcal{N}(0,\\sigma_1^2 I)$ with known variances $\\sigma_0^2$ and $\\sigma_1^2$. In this case the negative log-likelihood of $x$ given $y$ under measurement model (P1) is $\\frac{1}{2\\sigma_1^2}\\|A_1x-y\\|_2^2$, which justifies our use of quadratic data-consistency terms in the development below.}\n\n\\begin{figure}[ht]\n\\centering\n\\begin{subfigure}{.8\\linewidth}\n\n    \\caption{Original reconstruction network.}\n    \\centering\n  \\includegraphics[width=.6\\linewidth]{figures\/fig2_orig.png}  \n\\end{subfigure} \\\\[1em]\n\\begin{subfigure}{.8\\linewidth}\n    \\caption{Model adaptation by Parametrize and Perturb (P\\&P).}\n    \\centering\n  \\includegraphics[width=.6\\linewidth]{figures\/fig2_pnp.png}  \n\\end{subfigure} \\\\[1em]\n\\begin{subfigure}{0.8\\linewidth}\n  \\caption{Model adaptation by Reuse \\& Regularize (R\\&R).}\n  \\centering\n  \\includegraphics[width=0.8\\linewidth]{figures\/fig2_rnr.png}  \n\\end{subfigure}\n\\caption{Three basic paradigms of reconstruction under ``model drift''. (a) If the training data is generated using the model $y = A_0 x + \\varepsilon$, this can be used to learn a reconstruction network $f(y;\\theta_0,A_0)$ which is parameterized by weights or parameters $\\theta_0$ and may also explicitly depend on forward model $A_0$. (b) {\\bf Parametrize and Perturb (P\\&P):} If at test time we are presented with data corresponding to the model $y = A_1 x + \\varepsilon'$, we may not only use the new forward model $A_1$ but also learn a perturbation $\\delta$ to the original network parameters $\\theta_0$ to compensate for the model drift.\n(c) {\\bf Reuse and Regularize (R\\&R):} Alternatively to P\\&P, we may reuse the pre-trained network $f_0$ as an implicit regularizer in an iterative model-based reconstruction scheme. The proposed scheme alternates between applying $f_0\\circ A_0$, which denoises and\/or removes artifacts, and a data-consistency step (denoted by $\\DC_{A_1}$ above) that enforces the estimated image $\\widehat{x}$ satisfies $A_1 \\widehat{x} \\approx y$.\n}\n\\label{fig:proposedmethods}\n\\end{figure}\n\n\\subsection{The feasibility of model adaptation}\n\nTo compute an accurate reconstruction under the original forward model, $A_0$, the learned solver must reconstruct components of the image that lie in the null space $N(A_0)$: for superresolution, these are high-frequency details lost during downsampling, and in inpainting, these are the pixels removed by $A_0$. \n\\change{The neural network trained as a component of $f_0$ implicitly represents a mapping from image components in the range of $A_0$ to components in $N(A_0)$.} \n\n\\change{Reconstructing under a different forward model, $A_1$, requires reconstructing different components of the image in the null space $N(A_1)$. The general intuition behind model adaptation is that if $A_0$ and $A_1$ are similar, then the mapping represented by $f_0$ can inform the new mapping that we need to learn from image components in the range of $A_1$ to components in $N(A_1)$. For example, in an inpainting setting, the learned network not only represents the missing pixels, but it also represents some function of the observed pixels that are relevant to filling in the missing pixels. Thus if $A_1$ has a similar null space (\\emph{e.g.},~ an offset in the collection of missing pixels), it is reasonable to expect that the original network has learned to represent some information about image components in the null space of $A_1$ but not in the null space of $A_0$. As the null spaces of $A_0$ and $A_1$ get further apart, model adaptation becomes less effective. This is similar to the widely-noted behavior of transfer learning, where transfer learning efficacy depends on the similarity of the training and target distributions. This intuition is supported by our empirical results, which illustrate that when $A_0$ and $A_1$ correspond to different blur kernels or perturbed k-space sampling patterns in MRI, the learned mapping $f_0$ does contain information about image components in the null space of $A_1$ that can be leveraged to improve reconstruction accuracy, even without additional training samples drawn using the model $A_1$.}\n\n\\section{Proposed Approaches}\n\\label{sec:proposed}\nWe propose two distinct model adaptation approaches, {\\em Parameterize \\& Perturb (P\\&P)} and {\\em Reuse \\& Regularize (R\\&R)}, as detailed below.\n\n\\subsection{Parametrize and Perturb: A transfer learning approach}\n\nLet $f_0$ be a reconstruction network trained to solve inverse problem (P0). Suppose we can explicitly parameterize $f_0$ both in terms of the trained weights\/parameters $\\theta_0$ and the original forward model $A_0$, \\emph{i.e.},~ we may write $f_0(\\cdot) = f(\\cdot\\,;\\theta,A_0)$. Given a new measurement vector $y$ under the measurement model (P1), a ``naive'' approach to model adaptation is to simply to replace substitute the new forward model $A_1$ for $A_0$ in this parametrization, and estimate the image as $f(y;\\theta_0,A_1)$. However, as illustrated in Figure \\ref{fig:fig1}, this can lead to artifacts in the reconstruction due to model mismatch.\n\nInstead, we propose estimating the image as $ f(y;\\theta_1,A_1)$ where $\\theta_1$ is a perturbed set of of network parameters obtained by solving the optimization problem:\n\\begin{equation}\\label{eq:A1}\n\\min_{\\theta} \\|y - A_1 f(y;\\theta,A_1)\\|_2^2 + \\mu \\|\\theta-\\theta_0\\|_2^2.\n\\end{equation}\nwhere $\\mu > 0$ is a tunable parameter.\nThe first term enforces data consistency, i.e., the estimated image $\\widehat{x}$ should satisfy $y \\approx A_1 \\widehat{x}$, while the second term $\\|\\theta-\\theta_0\\|_2^2$ ensures the retrained parameters $\\theta$ stay close to the original network parameters $\\theta_0$. This term is necessary to avoid degenerate solutions, \\change{which we demonstrate in the Supplement. Our use of a proximity term of this form is also inspired in part by its success in other transfer learning applications (see, \\emph{e.g.},~ \\cite{xuhong2018explicit}).}\n\nIf the forward model $A_1$ is also unknown, we propose optimizing for it as well in the above formulation, which gives:\n\\begin{equation}\\label{eq:A2}\n\\min_{\\theta,A \\in {\\cal A}} \\|y - A f(y;\\theta,A)\\|_2^2 + \\mu \\|\\theta-\\theta_0\\|_2^2.\n\\end{equation}\nwhere $\\mathcal{A}$ denotes a constraint set.\nWe assume the forward model is parameterized such that the constraint set is given by $\\mathcal{A} = \\{A(\\sigma) : \\sigma \\in \\mathbb{R}^q\\}$, where $A(\\sigma)$ denotes a class of of forward models parametrized by a vector $\\sigma \\in \\mathbb{R}^q$ with $q \\ll m\\cdot n$ (e.g., in a blind deconvolution setting, $A(\\sigma)$ corresponds convolution with an unknown kernel $\\sigma$). In particular, we propose optimizing over the parameters $\\sigma$, which is possible with first-order methods such as stochastic gradient descent, provided the map $\\sigma \\mapsto A_\\sigma$ is first-order differentiable.\n\n{\\centering\n\\begin{minipage}{.6\\linewidth}\n\\centering\n\\begin{algorithm}[H]\n    \\caption{Parameterize \\& Perturb (P\\&P)}\n    \\label{alg:pnp}\n    \\centering\n    \\begin{algorithmic}[1]\n        \\Require{Original forward model $A_0$, new forward model $A_1$, pre-trained reconstruction network $f_0(\\cdot) = f(\\cdot; \\theta_0,A_0)$,  regularization parameter $\\mu$, new measurements $y$.}\n            \\State Modify the reconstruction network $f_0$ by internally changing $A_0$ to $A_1$, to obtain the estimate $f(y;\\theta_0, A_1)$\n            \\State Fine-tune the network weights as  $\\theta_1 = \\theta_0+\\delta$ where $\\delta$ is a perturbation learned by solving \\eqref{eq:A1}\n    \\end{algorithmic}    \n\\end{algorithm}\n\\end{minipage}\n\\par\n}\\vspace{1em}\n\n\nThe preceding discussion focused on the case of reconstructing single measurement vector $y$ at test time, \\emph{i.e.},~ model adaptation without calibration data. Additionally, we consider a P\\&P\\ approach in the case where we have access to calibration data $y_1,...,y_N$ generated according to (P1). In this case we propose retraining the network\nby minimizing the sum of data-consistency terms over the calibration set:\n\\begin{equation}\\label{eq:A3}\n(\\theta_1,A_1) = \\argmin_{\\theta,A \\in {\\cal A}} \\frac{1}{N}\\sum_{i=1}^N\\|y_i - A f(y_i;\\theta,A)\\|_2^2 + \\lambda \\|\\theta-\\theta_0\\|_2^2.\n\\end{equation}\nAt deployment, we propose using the retrained  network $\\widehat{x} = f(y;\\theta_1,A_1)$ as our estimator.\n\n\\change{\nIt is worth noting that the P\\&P\\ model adaptation technique presented above bears similarities to the deep image prior (DIP) approach to solving inverse problems as introduced in \\cite{dip}.  However, P\\&P\\ differs from DIP in two key aspects: First, in DIP the reconstruction network is initialized with random weights, whereas in P\\&P\\ we start with a network whose initial weights $\\theta_0$ are obtained by training to solve the initial inverse problem (P0). Second, we explicitly enforce proximity to the initial weights to prevent overfitting to the data, and do not rely on early stopping heuristics as is the case DIP. The P\\&P\\ approach also shares similarities to the ``fine-tuning'' step proposed in the $\\Sigma$-net MRI reconstruction framework \\cite{hammernik2019sigmanet}, where a loss similar to \\eqref{eq:A1} is minimized to enforce data consistency at test time. However, different from P\\&P, the fine-tuning approach in \\cite{hammernik2019sigmanet} regularizes the reconstruction by minimizing the loss between initial reconstruction and the new network output in the SSIM metric. As demonstrated in Figure \\ref{fig:fig1}, this initial reconstruction can have severe artifacts in certain settings due to model mismatch, in which case enforcing proximity in image space to an initial reconstruction is less justified.\n}\n\n\n\\subsection{Reuse \\& Regularize: Model adaptation without retraining} \n\\change{\nOne drawback of the P\\&P\\ approach is that it requires fine-tuning the network for each input $y$, which is computationally expensive relative to a feed-forward reconstruction approach. Additionally, the P\\&P\\ approach is somewhat indirect, relying only on the inductive bias of the network architecture and its original parameter configuration to impart a regularization effect for the new inverse problem (P1). Here we propose a different model adaptation approach that does not require retraining the original reconstruction network, and explicitly makes use of the fact that the original network is designed to solve (P0).\n\nSuppose we are given a reconstruction network $f_0(y)$\ntrained to solve (P0). The key idea we exploit is that the composition of $f_0$  with the original forward model $A_0$, should act as an \\emph{auto-encoder}, \\emph{i.e.},~ if we define the map $g:\\mathbb{R}^d\\rightarrow\\mathbb{R}^d$ by $g(x) = f_0(A_0x)$ then by design we should have $g(x)\\approx x$ for any image $x$ sampled from the image distribution $P_X$. See Figure \\ref{fig:autoencoder} for an illustration in the case of undersampled MRI reconstruction.\n\n\nGiven this fact, one simple approach to reconstructing a measurement vector $y$ under (P1) is to start from an initial guess, \\emph{e.g.},~ the least squares solution $x ^{(0)}= A_1^\\dagger y$, and attempt to find a fixed-point of $g(\\cdot)$ by iterating:\n\\begin{equation}\\label{eq:fp}\nx^{(k+1)} = g(x^{(k)}),~~k=0,1,2,...\n\\end{equation}\nHowever, this approach only uses knowledge of the new forward model $A_1$ in the initialization step. Also, unless we can guarantee the map $g(\\cdot)$ is non-expansive (i.e., its Jacobian is 1-Lipschitz), these iterations could diverge.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.7\\columnwidth]{figures\/autoencoder.pdf}\n    \\caption{Illustration of the auto-encoding property of the map $f_0 \\circ A_0$ as used in the proposed R\\&R\\ model adaptation approach, illustrated in an undersampled MRI reconstruction setting.}\n    \\label{fig:autoencoder}\n\\end{figure}\n\nInstead, building off the intuition that $g$ acts as an auto-encoder, we propose using $g$ as a regularizer in an iterative model-based reconstruction scheme. In particular, we adopt a regularization-by-denoising (RED) approach, which allows one to convert an arbitrary denoiser\/de-artifacting map into a regularizer  \\cite{RED}. The RED approach is motivated by the following cost function:\n\\begin{equation}\\label{eq:RED}\n    \\min_{x} \\frac{1}{2}\\|A_1x - y\\|_2^2 + \\lambda\\rho(x)\n\\end{equation}\nwhere the function $\\rho(x) := x^\\top(x-g(x))$ can be interpreted as a regularizer induced by the map $g(x)$ and $\\lambda > 0$ is a regularization parameter. Under appropriate conditions on the function $g$, one can show $\\nabla \\rho(x) = x-g(x)$. This fact is used in \\cite{RED} to derive a variety of iterative algorithms based on first-order methods (see also \\cite{reehorst2018regularization} for further analysis of RED, including convergence guarantees).\n\n{\\centering\n\\begin{minipage}{.6\\linewidth}\n\\centering\n\\begin{algorithm}[H]\n    \\caption{Reuse \\& Regularize (R\\&R)}\n    \\label{alg:rnr}\n    \\centering\n    \\begin{algorithmic}[1]\n        \\Require{Pre-trained reconstruction network $f_0(\\cdot)$, original forward model $A_0$, new forward model $A_1$, regularization parameter $\\lambda>0$, max iterations $K$, new measurements $y$.}\n            \\State $x\\gets A_1^\\dagger y$ \\Comment{\\emph{least-squares initialization}}\n            \\For{$k=1,2,...,K$}\n                \\State $z \\gets f_0(A_0x)$ \\Comment{\\emph{regularize using pre-trained network}}\n                \\State $x \\gets (A_1^\\top A_1 + \\lambda I)^{-1}(A_1^\\top y + \\lambda z)$ \n                \\Comment{\\emph{data consistency}}\n            \\EndFor\n            \\State \\textbf{return} $x$ \n       \n    \\end{algorithmic}    \n\\end{algorithm}\n\\end{minipage}\n\\par\n}\\vspace{1em}\n\nFor simplicity, we focus on a RED approach with proximal gradient descent (see, e.g., \\cite{parikh2014proximal}) as the base algorithm with stepsize $\\tau > 0$. This results in an alternating scheme:\n\\begin{align*}\n    z^{(k)} & = (1-\\tau)x^{(k)} + \\tau g(x^{(k)})\\\\\n    x^{(k+1)} & = \\argmin_{x} \\frac{1}{2\\lambda}\\|A_1x - y\\|_2^2 + \\frac{1}{2\\tau} \\|x- z^{(k)}\\|_2^2\n\\end{align*}\nThe $x$-update above has the closed-form expression\n\\begin{equation}\n    x^{(k+1)} = \\left(A_1^\\top A_1 + \\tfrac{\\lambda}{\\tau} I\\right)^{-1}\\left(A_1^\\top y + \\tfrac{\\lambda}{\\tau} z^{(k)}\\right) \\label{eq:xup}\n\\end{equation}\nFor simplicity of implementation and to reduce the number of tuning parameters, we fix the stepsize to $\\tau = 1$ in all our experiments. We summarize these steps in Algorithm \\ref{alg:rnr}.\n\nNote that in the limit as $\\lambda\\rightarrow \\infty$, Algorithm \\ref{alg:rnr} reduces to the fixed-point scheme \\eqref{eq:fp}, and in the limit as $\\lambda \\rightarrow 0$ Algorithm \\ref{alg:rnr} will return the initialization $x = A_1^\\dagger y$. In general, the output from Algorithm \\ref{alg:rnr} will interpolate between these two extremes: $x$ will be an approximate fixed point of $g$ and will approximately satisfy data consistency, \\emph{i.e.},~ $y \\approx A_1 x$.\n\n\nFor certain types of forward models the $x$-update in \\eqref{eq:xup} can be computed efficiently (e.g., if $A_1$ corresponds to a 2-D discrete convolution with circular boundary conditions, then $A_1^\\top A_1$ diagonalizes under the 2-D discrete Fourier transform). However, in general, the matrix inverse $(A_1^\\top A_1 + \\lambda I)^{-1}$ may be expensive to apply. Therefore, in practice we propose approximating \\eqref{eq:xup} with a fixed number of conjugate gradient iterations.\n\nA notable aspect of the R\\&R\\ approach is that it has potential to improve the accuracy of network-based reconstructions \\emph{even in the absence of model drift}, \\emph{i.e.},~ even if $A_1 = A_0$. This is because data-consistency is not guaranteed by certain reconstruction networks (e.g., U-Nets). However, we are less likely to see a benefit in the case where the reconstruction network already incorporates data-consistency layers, such as networks inspired by unrolling iterative optimization algorithms. We explore this aspect empirically in Section \\ref{sec:mrisamplingexp} in the context of MRI reconstruction.\n\n{\\centering\n\\begin{minipage}{.6\\linewidth}\n\\centering\n\\begin{algorithm}[H]\n    \\caption{Reuse \\& Regularize with fine-tuning (R\\&R$+$)}\n    \\label{alg:rnrp}\n    \\centering\n    \\begin{algorithmic}[1]\n        \\Require{Pretrained reconstruction network $f_0(\\cdot) = f(\\cdot; \\theta_0)$, original forward model $A_0$, new forward model $A_1$, regularization parameter $\\lambda$, new measurements $y$.}\n            \\State Construct an estimator $\\widehat{x}(y;\\theta_0)$ by unrolling $K$ iterations of Algorithm \\ref{alg:rnr}\n            \\State Fine-tune the network weights as  $\\theta_1 = \\theta_0+\\delta$ where $\\delta$ is a perturbation learned by approximately minimizing the cost \\eqref{eq:rnrpcost} via SGD.\n            \\State \\textbf{return} $x = \\widehat{x}(y;\\theta_1)$\n    \\end{algorithmic}    \n\\end{algorithm}\n\\end{minipage}\n\\par\n}\\vspace{1em}\n\nThe R\\&R\\ approach can also be extended the case where the new forward model $A_1$ depends on unknown parameters. First, we define an estimator $\\hat{x}(y;A_1)$ by unrolling of a fixed number of iterates of Algorithm \\ref{alg:rnr}, \\emph{i.e.},~ we take $\\hat{x}(y;A_1) = x^{(K)}$ where $x^{(K)}$ is the $K$th iterate of Algorithm \\ref{alg:rnr} with input $y$ for some small fixed value of $K$ (e.g., $K=5$). Supposing $A_1$ belongs to a parameterized class of forward models $A(\\sigma)$, i.e., $A_1 = A(\\sigma_1)$ for some set of parameters $\\sigma_1$, we propose estimating $\\sigma_1$ by minimizing data-consistency of the estimator:\n\\begin{equation}\\label{eq:B2}\n\\tilde{\\sigma}_1 = \\argmin_\\sigma  \\|A(\\sigma)\\hat{x}(y;A(\\sigma)) - y\\|_2^2\n\\end{equation}\nThe resulting image estimate is then taken to be $\\hat{x}(y;\\tilde{A}_1)$ where $\\tilde{A}_1 = A(\\tilde{\\sigma}_1)$.\n\nFinally, we also consider a combination of the P\\&P\\ and R\\&R\\ approaches where we additionally fine-tune the weights $\\theta_0$ of the reconstruction network $f_0$ embedded in the unrolled R\\&R\\ estimator.  Writing this estimator as $\\widehat{x}(y;\\theta_0)$, similar to the P\\&P\\ approach we propose ``fine-tuning'' the weights $\\theta_0$ by approximately minimizing the cost function\n\\begin{equation}\\label{eq:rnrpcost}\n\\min_\\theta \\|A_1 \\widehat{x}(y;\\theta) -y\\|_2^2,\n\\end{equation}\nto obtain the updated network parameters $\\theta_1 = \\theta + \\delta$ where $\\delta$ is some small perturbation. The estimated image is then given by $x = \\widehat{x}(y;\\theta_1)$. We call this approach R\\&R$+$. Empirically we see consistent improvement in reconstruction accuracy from R\\&R$+$ over R\\&R\\ without any fine-tuning (see Figures \\ref{fig:megafigmotblur} and \\ref{fig:megafigmri}). However, this comes at the additional computational cost of having to retrain the reconstruction network parameters at test time.\n}\n\n\n\\section{Experiments}\\label{sec:exp}\n\nIn this section we empirically demonstrate our approach to model adaptation on three types of inverse problems with two example reconstruction network architectures. We have chosen these comparison points for their simplicity and to illustrate the broad applicability of our proposed approaches. In particular, our approaches to model adaptation are not tied to a specific architectural design.\n\n\\subsection{Methods and datasets used}\n\nWe demonstrate our approaches on three inverse problems: motion deblurring, superresolution, and undersampled single-coil MRI reconstruction. \n\nFor motion deblurring, our initial model $A_0$ corresponds to a $10^\\circ$ motion blur with a $7\\times7$ kernel, and $A_1$ is a \\change{$20^\\circ$} motion blur with a $7\\times7$ kernel, with angle given with respect to the horizontal axis. In superresolution, our initial model is a bilinear downsampling with rate $2\\times$, and $A_1$ corresponds to $2\\times$ bicubic downsampling. \n\nMRI reconstruction is performed with a \\change{$6\\times$} undersampling of k-space in the phase encoding direction for both $A_0$ and $A_1$. The sampling maps are shown in Fig \\ref{fig:mrimasks}.\n\n\\begin{figure}[ht]\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[width=0.25\\columnwidth, trim= 20 20 20 20, clip]{figures\/A0_mask.png} & \\includegraphics[width=0.25\\columnwidth,trim= 20 20 20 20, clip]{figures\/A1_mask.png}\\\\[0.1em]\n\\begin{minipage}{0.25\\columnwidth}\n(a) Original k-space sampling pattern ($A_0$)\n\\end{minipage} &\n\\begin{minipage}{0.25\\columnwidth}\n(b) Resampled k-space sampling pattern ($A_1$)\n\\end{minipage}\n\\end{tabular}\n\\caption{Visualization of k-space masks used for MRI experiments. Each mask represents a \\change{6-fold Cartesian undersampling} with 4\\% of the center k-space lines fully sampled, and the remaining lines sampled according to a Gaussian variable density scheme. \\change{The $A_1$ mask contains the same center lines, but the higher frequency k-space lines are sampled separately.}}\\label{fig:mrimasks}\n\\end{figure}\n\nWe use two datasets in our experiments. First, for motion deblurring and superresolution, we train and test on 128x128-pixel aligned photos of human faces from the CelebA dataset \\cite{liu2015faceattributes}. \n\nThe data used in the undersampled MRI experiments were obtained from the NYU fastMRI Initiative \\cite{zbontar2018fastMRI}. The primary goal of the fastMRI dataset is to test whether machine learning can aid in the reconstruction of medical images. We trained and tested on a subset of the single-coil knee dataset\\change{, which consist of simulated single-coil measurements. In all tests, we use complex-valued data, which interfaces with our deep networks by treating the real and imaginary parts of the images as separate channels. We measure reconstruction accuracy with respect to the center 320$\\times$320 pixels of the complex IFFT of the fully-sampled k-space data. For the purpose of visualization, we display only the magnitude images in the following sections.}\n\nLearning rates and regularization parameters (\\emph{i.e.},~ $\\mu$ in Algorithm \\ref{alg:pnp} and $\\lambda$ in Algorithm \\ref{alg:rnr})\nwere tuned via cross-validation on a hold out validation set of 512 images for CelebA, and 64 MR images for fastMRI. Batch sizes were fixed in advance to be 128 for the motion blur and superresolution settings, and 8 for the MRI setting. Hyperparameters were tuned via grid search on a log scale. \\change{For R\\&R, we use $K=5$ iterations in the main loop of Algorithm \\ref{alg:rnr}. During training, we add Gaussian noise with $\\sigma=0.01$ to all measurements, as suggested by \\cite{genzel2020solving} to improve robustness.} \n\nWe compare the performance of two reconstruction network architectures across all datasets. First, we utilize the U-Net architecture \\cite{ronneberger2015u}. Our U-Net implementation takes as input the adjoint of the measurements under the forward model $A_0^\\top y$ or $A_1^\\top y$, which is then passed through several CNN layers before obtaining a reconstructed image $\\widehat{x}$.\n\nWe also utilize the MoDL architecture \\cite{aggarwal2018modl}, a learned architecture designed for solving inverse problems with known forward models. MoDL is an iterative or ``unrolled'' architecture, which alternates between a data-consistency step and a trained CNN denoiser, with weights tied across unrolled iterations. We use a U-Net architecture as the denoiser in our implementation of MoDL, ensuring that the overall number of parameters (except for a learned scaling factor in MoDL) is the same in both architectures.\n\n\\change{To compare to deep learning-based approaches which do not require training on particular forward models, we compare to the Image-Adaptive GAN (IAGAN) \\cite{hussein2020image} and to Regularization by Denoising (RED) \\cite{RED}. IAGAN leverages a pretrained GAN to reconstruct arbitrary linear measurements by fitting the latent code input to the GAN, while also tuning the GAN parameters in a way similar to our proposed P\\&P\\ approach and the Deep Image Prior approach \\cite{dip}.\n\nRED requires only a pretrained denoiser, which we implement by pretraining a set of residual U-Net denoisers on the fastMRI and CelebA training sets, with a variety of different Gaussian noise levels. Specifically, we train 15 denoisers for each problem setting, with $\\sigma$ ranging from $10^{-4}$ to $10^1$ on a logarithmic scale. All results shown are tuned on the validation set to ensure the optimal denoisers are used.\n\nWe also compare to a penalized least squares approach with total variation regularization \\cite{rudin1992nonlinear}, a classical approach that does not use any learned elements. While more complex regularizers are possible, total variation (TV) is used because of its status as a simple, widely-used conventional baseline.}\n\n\n\\subsection{Parametrizing forward models}\n\nBoth of our proposed model adaptation methods permit the new forward model to be unknown during training, provided it has a known parametrization. In this case, the parameters describing the forward model are learned along with the reconstruction. Here we describe the parametrizations of the forward models that are used.\n\nFor the deblurring task, the unknown blur kernel is parametrized as a 7x7 blur kernel, initialized with the weights used for the ground-truth kernel during the initial stage of training. Practically, this is identical to a standard convolutional layer with a fixed initialization and only one learned kernel.\n\nA similar approach is used for superresolution. The forward model can be efficiently represented by strided convolution, and the adjoint is represented by a standard \"convolution transpose\" layer, again with the weights initialized to match the forward operator in the initial pre-training phase.\n\n\\change{In the case of MRI, we use two choices of $A_1$, depending on whether we assume $A_1$ is fully known or not. In the case $A_1$ is fully known, we utilize another $6\\times$ undersampled k-space mask, but with resampled high-frequency lines. We display the original and new k-space sampling masks in Figure \\ref{fig:mrimasks}. To illustrate the utility of our approach under miscalibration of the forward model in an MRI reconstruction setting, we also consider a unknown random perturbation of the original k-space lines, which we attempt to learn during reconstruction. The vertical k-space lines are still fully sampled, as are the center 4$\\%$ of frequencies, but all high frequency lines are perturbed uniformly at random with a continuous value from -2 to 2. We wish to emphasize that this experiment is not meant to reflect clinical practice, since such miscalibration of k-space sampling locations is not typically encountered in anatomical imaging with Cartesian k-space sampling trajectories. However, we include this experiment simply to illustrate that our approach could be extended to unknown parametric changes in the forward model in an MR reconstruction setting.\n}\n\n\\subsection{Main results}\n\n\n\\begin{table*}\t\n\\centering\n\t\\adjustbox{max width=\\columnwidth}{\n\t\\begin{tabular}{cc|c|c|c|c|c|c|}\n\t\t\\hline\n         & & \\multicolumn{6}{c|}{Baselines} \\\\\n         & & \\multirow{2}{*}{TV} & \\multirow{2}{*}{RED} & & Train w\/$A_0$ & Train w\/$A_0$ & Train w\/$A_1$ \\\\\n\t\t & & &  & & Test w\/$A_0$ & Test w\/$A_1$ & Test w\/$A_1$ \\\\ \\hline \\hline\n\t \\multicolumn{2}{c|}{\\multirow{2}{*}{Blur}} & \\multirow{2}{*}{27.61}  & \\multirow{2}{*}{30.23} & U-Net & 34.15   & 25.42  & 33.98 \\\\\n\t  & & & & MoDL & 36.25  & 23.91 & 36.13 \\\\ \\hline\n\t \\multicolumn{2}{c|}{\\multirow{2}{*}{SR}} & \\multirow{2}{*}{28.33} & \\multirow{2}{*}{28.59} &  U-Net & 30.74  & 26.3 & 31.22 \\\\\n\t  & & & & MoDL & 31.32  & 22.27 & 31.98 \\\\ \\hline\n\t \\multicolumn{2}{c|}{\\multirow{2}{*}{MRI}} & \\multirow{2}{*}{25.09} & \\multirow{2}{*}{27.76} & U-Net & 31.51  & 27.47 & 32.33 \\\\\n\t\t & & & & MoDL & 31.88 & 22.82 & 31.79 \\\\ \\hline \\hline\n\t\t \\multicolumn{8}{c}{Proposed Model Adaptation Methods} \\\\\n\t &\t & \\multicolumn{3}{c|}{Known $A_1$} &  \\multicolumn{3}{c}{Unknown $A_1$} \\\\\n\t &\t & P\\&P (Alg. \\ref{alg:pnp}) & R\\&R (Alg. \\ref{alg:rnr}) & R\\&R+ (Alg. \\ref{alg:rnrp}) & P\\&P (Alg. \\ref{alg:pnp}) & R\\&R (Alg. \\ref{alg:rnr}) & R\\&R+ (Alg. \\ref{alg:rnrp}) \\\\ \\hline\n\t\t \\multirow{2}{*}{Blur}  & U-Net & 33.01 & 32.11 & \\bf  33.50 & 29.18  & 27.67 & 30.05 \\\\\n\t\t & MoDL & 30.08 & 33.82 & \\bf 34.73 & 29.89  & 27.81 & 27.94 \\\\ \\hline\n\t\t \\multirow{2}{*}{SR}  & U-Net & 28.00 & 29.95 & \\bf  29.99  & 27.77  & 26.98 & 29.35 \\\\\n\t\t & MoDL & 24.59 & 28.18 & \\bf  29.83 & 23.14 & 24.93 & 25.29 \\\\ \\hline\n\t\t \\multirow{2}{*}{MRI}  & U-Net & 29.07 & 29.71 & \\bf  31.43 & 28.92  & 28.06 & 29.54 \\\\\n\t\t & MoDL & 30.63 & 30.25 & \\bf 31.44 & 26.64 & 23.46 & 27.67 \\\\\n\t\\end{tabular}\n\t}\n\t\\vspace{0.5em}\n\t\\caption{\\change{Comparison of performance of various baseline methods for inverse problems across a variety of datasets and forward models. The metric presented is the mean PSNR. SSIM values can be found in Table \\ref{table:ssimtable}}}\n\t\\label{table:mainadaptationtable}\n\\end{table*}\n\nIn Table \\ref{table:mainadaptationtable} we present our main results. We present sample reconstructions for the deblurring problem and MRI reconstruction problem in Figs. \\ref{fig:megafigmotblur} and \\ref{fig:megafigmri}. For reference, the ground truth, inputs to the networks, a total variation regularized reconstruction, and a RED reconstruction are presented in Figs. \\ref{fig:deblurinitialmethods} and \\ref{fig:mriinitialmethods}. We also provide in the Appendix a table of SSIM values as well as the full version of Table \\ref{table:mainadaptationtable}, which contains the standard deviations of PSNR.\n\nWhile the magnitude of the improvements vary across domains and problems, we find that retraining the network with the proposed model adaptation techniques significantly improve performance by several dBs in the new setting. This effect is particularly striking in the case of MRI reconstruction with MoDL, where the ``naive'' approach of replacing $A_0$ with $A_1$ in the network gives catastrophic results (a roughly 9 dB drop in reconstruction PSNR), while the proposed model adaptation approaches give reconstruction PSNRs within 1-2 dB of the baseline approach of training and testing with the same forward model in the case where $A_1$ is known.\n\n\\begin{figure}\n\\centering\n\\adjustbox{max width=0.5\\columnwidth}{\n\\renewcommand*{\\arraystretch}{0}\n\\begin{tabular}{c@{}c@{}c@{}c@{}}\n  \\small Ground & \\small Blurred & \\small TV-Regularized & \\small RED  \\\\\n  \\small Truth &                    & \\small Reconstruction & \\\\ \\vspace{2pt} && \\\\\n \\includegraphics[align=c, width = 0.15\\columnwidth]{figures\/megafig\/truth\/celeba\/48_9.032992.png} &\n\\includegraphics[align=c,width = 0.15\\columnwidth]{figures\/megafig\/corrupted\/motionblur\/48_20.794127.png} &\n \\includegraphics[align=c, width = 0.15\\columnwidth]{figures\/megafig\/tv\/48_9.032992_blur_tv.png} &\n\\includegraphics[align=c, width = 0.15\\columnwidth]{figures\/revisionfigs\/blur\/48_red_32.89262294769287_0.9620666.png} \\\\ \n \\includegraphics[align=c, width = 0.15\\columnwidth]{figures\/megafig\/truth\/celeba\/48_9.032992_res.png}\n& \\includegraphics[align=c, width = 0.15\\columnwidth]{figures\/megafig\/corrupted\/motionblur\/48_20.794127_res.png} &\n\\includegraphics[align=c, width = 0.15\\columnwidth]{figures\/megafig\/tv\/48_9.032992_blur_tv_res.png} &\n\\includegraphics[align=c, width = 0.15\\columnwidth]{figures\/revisionfigs\/blur\/48_red_err__32.89196014404297_0.9620606.png}\n\\end{tabular}\n}\n\\caption{Comparison figures for the deblurring methods in Figure \\ref{fig:megafigmotblur}. We present the ground truth, the blurred image (with Gaussian noise with $\\sigma=0.01$ added), a total variation (TV) regularized reconstruction, \\change{and a comparison to Regularization by Denoising (RED), a model-agnostic method leveraging a deep denoiser}. Below each of the above is the residual image, multiplied by 5$\\times$ for ease of visualization.}\n\\label{fig:deblurinitialmethods}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\adjustbox{max width=0.5\\columnwidth}{\n\\renewcommand*{\\arraystretch}{0}\n\\begin{tabular}{@{}c@{}c@{}c@{}c@{}}\n \\small Ground & \\small IFFT & \\small TV-Regularized & \\small RED \\\\\n \\small Truth & \\small Reconstruction & \\small Reconstruction & \\\\ \\vspace{1pt} && \\\\%\\vspace{2pt} \\\\\n \\includegraphics[align=c,width = 0.15\\columnwidth]{figures\/revisionfigs\/mri\/111_truth.png} &\n\\includegraphics[align=c,width = 0.15\\columnwidth]{figures\/revisionfigs\/mri\/111_ifft.png} &\n\\includegraphics[align=c,width = 0.15\\columnwidth]{figures\/revisionfigs\/mri\/111_tv.png} &\n\\includegraphics[align=c,width = 0.15\\columnwidth]{figures\/revisionfigs\/mri\/111_red.png} \\\\\n \\includegraphics[align=c,width = 0.15\\columnwidth]{figures\/megafig\/truth\/mri\/14_48.477177_res.png} &\n\\includegraphics[align=c,width = 0.15\\columnwidth]{figures\/revisionfigs\/mri\/111_ifft_err.png} &\n\\includegraphics[align=c,width = 0.15\\columnwidth]{figures\/revisionfigs\/mri\/111_tv_err.png} &\n\\includegraphics[align=c,width = 0.15\\columnwidth]{figures\/revisionfigs\/mri\/111_red_err.png}\n\\end{tabular}\n}\n\\caption{Comparison figures for the MRI reconstruction methods in Figure \\ref{fig:megafigmri}. We present the IFFT with all k-space data maintained, the na\u00efve IFFT reconstruction after k-space masking, a total variation (TV) regularized reconstruction (with PSNR 27.3 dB), and a RED reconstruction (with PSNR 28.4 dB). We also present the residuals relative to the fully-sampled IFFT, multiplied by 5$\\times$ for ease of visualization.}\n\n\\label{fig:mriinitialmethods}\n\\end{figure}\n\n\n\n\\begin{figure*}\n\\centering\n\\adjustbox{max width=\\columnwidth}{\n\\renewcommand*{\\arraystretch}{0}\n\\begin{tabular}{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}\n& \\small Train w\/$A_0$  & \\small Train w\/$A_0$ & \\small P\\&P & \\small R\\&R & \\small R\\&R+ & \\small P\\&P & \\small R\\&R & \\small R\\&R+ \\\\\n & \\small Test w\/$A_0$ & \\small Test w\/$A_1$ & \\small Known $A_1$ & \\small Known $A_1$ & \\small Known $A_1$ & \\small Unknown $A_1$ & \\small Unknown $A_1$ & \\small Unknown $A_1$ \\\\ \\vspace{2pt} \\\\\n\\begin{minipage}{0.08\\linewidth} U-Net \\end{minipage} &\n\\includegraphics[align=c, width = 0.12\\linewidth]{figures\/revisionfigs\/blur\/unet\/48_orig__34.860546588897705_0.9174336.png} & \n\\includegraphics[align=c, width = 0.12\\linewidth]{figures\/revisionfigs\/blur\/unet\/48_noadapt__24.471983909606934_0.8112663.png} & \n\\includegraphics[align=c, width = 0.12\\linewidth]{figures\/revisionfigs\/blur\/unet\/48_pnp__32.33306646347046_0.9213726.png} &\n\\includegraphics[align=c, width = 0.12\\linewidth]{figures\/revisionfigs\/blur\/unet\/48_rnr_32.935969829559326_0.97213316.png} & \n\\includegraphics[align=c, width = 0.12\\linewidth]{figures\/revisionfigs\/blur\/unet\/48_rnrp_33.25258016586304_0.9752408.png} & \n\\includegraphics[align=c, width = 0.12\\linewidth]{figures\/revisionfigs\/blur\/unet\/48_pnp1__26.703567504882812_0.85022515.png} & \n\\includegraphics[align=c, width = 0.12\\linewidth]{figures\/revisionfigs\/blur\/unet\/48_rnr1_0.9337511_27.55173921585083_0.9337511.png} &\n\\includegraphics[align=c, width = 0.12\\linewidth]{figures\/revisionfigs\/blur\/unet\/48_rnrp1_0.95398253_29.146764278411865_0.95398253.png} \\\\\n\\vspace{2pt}\\begin{minipage}{0.08\\linewidth} U-Net \\\\ Residual \\end{minipage} &\n\\includegraphics[align=c, width = 0.12\\linewidth]{figures\/revisionfigs\/blur\/unet\/48_orig_err__34.860546588897705_0.9174336.png} & \n\\includegraphics[align=c, width = 0.12\\linewidth]{figures\/revisionfigs\/blur\/unet\/48_noadapt_err__24.471983909606934_0.8112663.png} & \n\\includegraphics[align=c, width = 0.12\\linewidth]{figures\/revisionfigs\/blur\/unet\/48_pnp_err__32.33306646347046_0.9213726.png} &\n\\includegraphics[align=c, width = 0.12\\linewidth]{figures\/revisionfigs\/blur\/unet\/48_rnr_err__32.935969829559326_0.97213316.png} & \n\\includegraphics[align=c, width = 0.12\\linewidth]{figures\/revisionfigs\/blur\/unet\/48_rnrp_err__33.25258016586304_0.9752408.png} & \n\\includegraphics[align=c, width = 0.12\\linewidth]{figures\/revisionfigs\/blur\/unet\/48_pnp1_err__26.703567504882812_0.85022515.png} &\n\\includegraphics[align=c, width = 0.12\\linewidth]{figures\/revisionfigs\/blur\/unet\/48_rnr1_err__27.55173921585083_0.9337511.png} & \n\\includegraphics[align=c, width = 0.12\\linewidth]{figures\/revisionfigs\/blur\/unet\/48_rnrp1_err__29.146764278411865_0.95398253.png} \\\\\n\\begin{minipage}{0.08\\linewidth} MoDL \\end{minipage} &\n\\includegraphics[align=c, width = 0.12\\linewidth]{figures\/revisionfigs\/blur\/modl\/48_orig__36.96568489074707_0.91479874.png} & \n\\includegraphics[align=c, width = 0.12\\linewidth]{figures\/revisionfigs\/blur\/modl\/48_noadapt__23.280608654022217_0.69428253.png} & \n\\includegraphics[align=c, width = 0.12\\linewidth]{figures\/revisionfigs\/blur\/modl\/48_pnp__35.34075736999512_0.9167714.png} &\n\\includegraphics[align=c, width = 0.12\\linewidth]{figures\/revisionfigs\/blur\/modl\/48_rnr__34.358670711517334_0.97277963.png} & \n\\includegraphics[align=c, width = 0.12\\linewidth]{figures\/revisionfigs\/blur\/modl\/48_rnrp__34.52039957046509_0.97645557.png} & \n\\includegraphics[align=c, width = 0.12\\linewidth]{figures\/revisionfigs\/blur\/modl\/48_pnp1__34.72598075866699_0.91442525.png} & \n\\includegraphics[align=c, width = 0.12\\linewidth]{figures\/revisionfigs\/blur\/modl\/48_rnr1__27.185657024383545_0.92795646.png} & \n\\includegraphics[align=c, width = 0.12\\linewidth]{figures\/revisionfigs\/blur\/modl\/48_rnrp1__29.811813831329346_0.9518878.png} \\\\\n\\begin{minipage}{0.08\\linewidth} MoDL \\\\ Residual \\end{minipage} &\n\\includegraphics[align=c, width = 0.12\\linewidth]{figures\/revisionfigs\/blur\/modl\/48_orig_err__36.96568489074707_0.91479874.png} & \n\\includegraphics[align=c, width = 0.12\\linewidth]{figures\/revisionfigs\/blur\/modl\/48_noadapt_err__23.280608654022217_0.69428253.png} & \n\\includegraphics[align=c, width = 0.12\\linewidth]{figures\/revisionfigs\/blur\/modl\/48_pnp_err__35.34075736999512_0.9167714.png} &\n\\includegraphics[align=c, width = 0.12\\linewidth]{figures\/revisionfigs\/blur\/modl\/48_rnr_err__34.358670711517334_0.97277963.png} & \n\\includegraphics[align=c, width = 0.12\\linewidth]{figures\/revisionfigs\/blur\/modl\/48_rnrp_err__34.52039957046509_0.97645557.png} & \n\\includegraphics[align=c, width = 0.12\\linewidth]{figures\/revisionfigs\/blur\/modl\/48_pnp1_err__34.72598075866699_0.91442525.png} & \n\\includegraphics[align=c, width = 0.12\\linewidth]{figures\/revisionfigs\/blur\/modl\/48_rnr1_err__27.185657024383545_0.92795646.png} & \n\\includegraphics[align=c, width = 0.12\\linewidth]{figures\/revisionfigs\/blur\/modl\/48_rnrp1_err__29.811813831329346_0.9518878.png}\n\\end{tabular}\n}\n\\caption{Visual examples of reconstruction quality for the motion deblurring inverse problem solved by U-Net and MoDL, as well as the associated residuals. Each residual is multiplied by 5 for ease of inspection. The initial forward model $A_0$ is a 7x7 motion blur with angle $10^\\circ$, and the $A_1$ model has a 7x7 motion blur kernel with angle $20^\\circ$. The analogous figure for the superresolution problem, and further examples, are available in the Supplement. Best viewed electronically.}\n\\label{fig:megafigmotblur}\n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\\renewcommand*{\\arraystretch}{0}\n\\begin{tabular}{@{}c@{}c@{}c@{}c@{}c@{}c@{}}\n & \\small Train w\/$A_0$  & \\small Train w\/$A_0$ & \\small P\\&P & \\small R\\&R & \\small R\\&R+ \\\\\n& \\small Test w\/$A_0$ & \\small Test w\/$A_1$ & & & \\\\\n\\begin{minipage}{0.08\\linewidth} \\small U-Net \\end{minipage} &\n\\includegraphics[align=c,width = 0.11\\linewidth]{figures\/revisionfigs\/mri\/unet\/111_orig__31.326417922973633_0.81668746.png} & \n\\includegraphics[align=c,width = 0.11\\linewidth]{figures\/revisionfigs\/mri\/unet\/111_noadapt__27.354295253753662_0.79132897.png} & \n\\includegraphics[align=c,width = 0.11\\linewidth]{figures\/revisionfigs\/mri\/unet\/111_pnp__30.4063081741333_0.81077015.png} &\n\\includegraphics[align=c,width = 0.11\\linewidth]{figures\/revisionfigs\/mri\/unet\/111_rnr__29.316372871398926_0.797895.png} &\n\\includegraphics[align=c,width = 0.11\\linewidth]{figures\/revisionfigs\/mri\/unet\/111_rnrp_31.570298671722412_0.8103058.png} \\\\\n\\vspace{2pt}\\begin{minipage}{0.08\\linewidth}\\small U-Net \\\\ Residual \\end{minipage} & \n\\includegraphics[align=c,width = 0.11\\linewidth]{figures\/revisionfigs\/mri\/unet\/111_orig_err__31.326417922973633_0.81668746.png} & \n\\includegraphics[align=c,width = 0.11\\linewidth]{figures\/revisionfigs\/mri\/unet\/111_noadapt_err__27.354295253753662_0.79132897.png} & \n\\includegraphics[align=c,width = 0.11\\linewidth]{figures\/revisionfigs\/mri\/unet\/111_pnp_err__30.406277179718018_0.81077003.png} &\n\\includegraphics[align=c,width = 0.11\\linewidth]{figures\/revisionfigs\/mri\/unet\/111_rnr_err__29.316372871398926_0.797895.png} &\n\\includegraphics[align=c,width = 0.11\\linewidth]{figures\/revisionfigs\/mri\/unet\/111_rnrp_err__31.42012119293213_0.8079765.png} \\\\ \n\\vspace{2pt} \\begin{minipage}{0.08\\linewidth} \\small  MoDL \\end{minipage} &\n\\includegraphics[align=c,width = 0.11\\linewidth]{figures\/revisionfigs\/mri\/modl\/111_orig__33.28503370285034_0.8331989.png} & \n\\includegraphics[align=c,width = 0.11\\linewidth]{figures\/revisionfigs\/mri\/modl\/111_noadapt__23.445076942443848_0.6545583.png} & \n\\includegraphics[align=c,width = 0.11\\linewidth]{figures\/revisionfigs\/mri\/modl\/111_pnp__31.869094371795654_0.82581556.png} &\n\\includegraphics[align=c,width = 0.11\\linewidth,align=c]{figures\/revisionfigs\/mri\/modl\/111_rnr__31.056983470916748_0.82304215.png} &\n\\includegraphics[align=c,width = 0.11\\linewidth,align=c]{figures\/revisionfigs\/mri\/modl\/111_rnrp__32.294108867645264_0.8277144.png} \\\\\n\\vspace{2pt}\\begin{minipage}{0.08\\linewidth}\\small MoDL \\\\ Residual \\end{minipage} & \n\\includegraphics[align=c,width = 0.11\\linewidth]{figures\/revisionfigs\/mri\/modl\/111_orig_err__33.28503370285034_0.8331989.png} & \n\\includegraphics[align=c,width = 0.11\\linewidth]{figures\/revisionfigs\/mri\/modl\/111_noadapt_err__23.445076942443848_0.6545583.png} & \n\\includegraphics[align=c,width = 0.11\\linewidth]{figures\/revisionfigs\/mri\/modl\/111_pnp_err__31.869094371795654_0.82581556.png} &\n\\includegraphics[align=c,width = 0.11\\linewidth,align=c]{figures\/revisionfigs\/mri\/modl\/111_rnr_err__31.056983470916748_0.82304215.png} &\n\\includegraphics[align=c,width = 0.11\\linewidth,align=c]{figures\/revisionfigs\/mri\/modl\/111_rnrp_err__32.294108867645264_0.8277144.png}\n\\end{tabular}\n\\caption{Visual examples of different reconstruction approaches for the MRI inverse problem under model drift, along with associated residuals. All residual images are scaled by 5x for ease of inspection. Best viewed electronically.}\n\\label{fig:megafigmri}\n\\end{figure*}\n\n\n\\subsection{Learning multiple forward models}\n\n\\definecolor{mygreen}{HTML}{01B051}\n\\definecolor{myblue}{HTML}{5B9BD5}\n\\definecolor{myred}{HTML}{FF0001}\n\\definecolor{myorange}{HTML}{ED7D31}\n\\definecolor{myyellow}{HTML}{FFC000}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\linewidth]{figures\/multimodelexp.pdf}\n\\caption{Na\\\"ively learning to deblur with a single network and multiple blur kernels sacrifices performance on all blurs. \nIn \\textcolor{mygreen}{\\bf green}, the test-time accuracy of a network trained to deblur multiple blurs, tested on a known kernel. In \\textcolor{myorange}{\\bf orange}, the same network, but tested on a new blur that was not used during training. In \\textcolor{black}{\\bf black}, our proposed P\\&P approach (Alg. \\ref{alg:pnp}), with a known model, and in \\textcolor{myyellow}{\\bf yellow} the same with a learned forward model. \\textcolor{myblue}{\\bf Blue} and \\textcolor{myred}{\\bf red} show the performance of our \\change{R\\&R} approach (Alg. \\ref{alg:rnr}), with and without a known forward model.}\n\\label{fig:multimodelexp}\n\\end{figure}\n\nIn this section we explore an alternative approach to model adaptation. In this setting, we assume that a set of candidate forward models are known during training time. During test time, a single forward model is used for measurement, but the test-time forward model is not known during training. This case represents the setting where the forward model might be parametrized, and so a reasonable approach may be to train the learned network using a number of different forward models to improve robustness.\n\nIn simple settings, training on multiple models might be reasonable. However, when the forward model parameterization is high-dimensional, learning to invert all possible forward models may be difficult. \n\nWe demonstrate this setting with a deblurring example, in which the same network is trained using a number of blur kernels. The blur kernels are the same kernels used for comparisons in \\cite{hradivs2015convolutional}. For consistency, we resize all 50 blur kernels to 7x7, and normalize the weights to sum to 1. We compare reconstruction accuracy when the ground truth blur kernel is included in the set of kernels used for training, as well as when the reconstruction network has never seen data blurred with the testing kernel.\n\nThe results are shown in Fig \\ref{fig:multimodelexp}. Experimentally, we find that training on multiple blur kernels simultaneously incurs a  performance penalty as the number of blur kernels used in training increases. In this setting, where the forward model has many degrees of freedom and data is limited, attempting to learn to solve all models simultaneously is worse than transferring a single learned model, even in the absence of further ground truth data for calibration.\n\n\\change{\\subsection{Adapting to variable sampling rates in single-coil MRI}\\label{sec:mrisamplingexp}\n\nA particular concern raised in \\cite{antun2020instabilities} is related to the stability of a learned solver with respect to the level of undersampling at measurement time. In particular, the authors of that work observe that an image reconstruction system trained to recover images sampled at a particular rate would experience a degradation in reconstruction accuracy for \\emph{higher} sampling rates than the one the system was trained on.\n\nIn Fig. \\ref{fig:samplerate} we explore this problem in the MRI setting using a U-Net as the reconstruction method, and demonstrate that our R\\&R\\ method can adapt to this setting as well. By using R\\&R\\ during inference, the learned network was trained at a $6\\times$ accceleration acquisition setting, but was safely deployed for other accelerations \\emph{without significant degradation in reconstruction quality}, and \\emph{comparing favorably to networks trained explicitly for other sampling rates}.\n\nFor comparison purposes, we also train a U-Net using multiple sampling masks. During training, the multiple-model solver is trained to reconstruct MRIs that are measured using the five different sampling patterns demonstrated in Fig. \\ref{fig:samplerate}. We present the mean PSNR on the test set in Table \\ref{table:samplingratetable}, along with the mean test PSNR for applying R\\&R\\ to the multiple-model solver, assuming at test time that the sampling pattern is known. Reconstructions from the multiple-model solver can be found in the Supplement. We observe that training with multiple models means that at test time all models produce reasonable reconstructions, but at the cost of reconstruction quality compared to networks trained for single sampling patterns.\n\nIn this experiment, we also observe an interesting side-effect of R\\&R: when R\\&R\\ is used to ``adapt'' to a forward model $A_0$ that the original network was trained on, we tend to see an improvement in reconstruction quality. This effect is most pronounced for the U-Net trained to reconstruct multiple sampling patterns, but is also true for the ``dedicated'' solver, demonstrated in Fig. \\ref{fig:samplerate}.\n}\n\n\\begin{figure*}[ht]\n  \\centering\n \n  \\includegraphics[width=\\linewidth]{figures\/revisionfigs\/accelfig.png}\n\\caption{\\change{Using the R\\&R\\ model adaptation approach permits using a U-Net trained for $6\\times$ acceleration on MRI reconstruction across a range of acceleration parameters. The various k-space sampling patterns used in these experiments are shown in the top row. Without adaptation (second row), the reconstruction quality decreases when changing the acceleration factor, \\emph{even when more k-space measurements are taken}, as originally observed in \\cite{antun2020instabilities}. The R\\&R\\ reconstructions (third row) compare favorably to the performance of networks trained on each particular k-space sampling pattern (bottom row). The PSNR of each image is presented in dB in yellow on each image.}}\n\\label{fig:samplerate}\n\\end{figure*}\n\n\\begin{table}\t\n\\centering\n\t\\begin{tabular}{c|c|c|c|c|c}\n\t\t\\hline\n\t & 2$\\times$ & 4$\\times$ & 6$\\times$ & 8$\\times$ & 12$\\times$ \\\\ \\hline\n\t\t Single-Model & 35.74 & 32.53 & 31.51 & \\bf 30.69 & \\bf 29.48 \\\\\n\t\t 6$\\times$ No adaptation & 27.02 & 30.20 & 31.51 & 27.76 & 26.15 \\\\\n\t\t 6$\\times$ R\\&R & 35.11 & \\bf 32.61 & \\bf 31.73 & 29.40 & 27.34 \\\\\n\t\t Multi-Model & 33.99 & 31.62 & 30.48 & 29.25 & 28.35 \\\\\n\t\t Multi-Model R\\&R & \\bf 35.80 & 32.35 & 30.81 & 29.60 & 28.61 \\\\\n\t\\end{tabular}\n\t\\vspace{0.5em}\n\t\\caption{\\change{Comparison of reconstruction PSNR for a variety of MRI acceleration factors for several different approaches. ``Multi-model'' refers to a U-Net trained for reconstruction on all shown sampling rates, whereas each column of the ``Single-Model'' results represents a network trained for that particular sampling pattern. $6\\times$ refers to the network shown in other experiments. Training a single-sample model consistently performs well for that particular forward model, but at the cost of lower performance on other accelerations, even for higher sampling rates. The multi-model approach sacrifices performance on any one forward model, but most of the difference can be removed by augmenting the multi-model network with our R\\&R\\ method.}}\n\t\\label{table:samplingratetable}\n\\end{table}\n\n\\change{\\subsection{Model adaptation under variable model overlap}\\label{sec:exp:nullspaceshift}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\linewidth]{figures\/revisionfigs\/samplingfig.png}\n\\caption{\\change{Comparison of the mean PSNR for the R\\&R\\ method and no adaptation for single-sample MRI reconstruction vs. the number of frequencies that differ between $A_0$ and $A_1$. The shaded areas represent the standard deviation of mean test PSNR over 10 runs, since frequencies are replaced randomly.}}\n\\label{fig:nullspaceoverlap}\n\\end{figure}\n\nIn this section we explore how varying the distance between the forward models $A_0$ and $A_1$ affects reconstruction quality, and how our proposed R\\&R\\ method deals with different amounts of overlap between $A_0$ and $A_1$. The forward model under investigation is $6\\times$ single-coil MRI reconstruction.\n\nTo explore variable levels of model drift in the single-coil MRI reconstruction case, we vary which k-space frequencies are sampled in a Cartesian pattern. Specifically, we construct a list of ``non-sampled'' frequencies and a list of ``sampled'' frequencies under $A_0$. We create $A_1$ by swapping $n$ ``sampled'' frequencies for $n$ frequencies in the original ``non-sampled'' list, to ensure that the new $A_1$ contains exactly $n$ frequencies that were not sampled under $A_0$. We do not swap the 4$\\%$ center frequencies in any test.\n\nIn Fig. \\ref{fig:nullspaceoverlap} we plot $n$ vs the mean PSNR over 10 separate instantiations of the above experiment for a no-adaptation approach as well as our R\\&R\\ method. We run 10 separate instances since the frequencies that are swapped, as well as what frequencies they are swapped to, is random, introducing some variance to the process. We also visually represent the maximum and minimum PSNR across all instances with shading.\n\nNote that the setup in this subsection is different than the experiments used in Fig. \\ref{fig:samplecompfig}, Table \\ref{table:psnrtable}, or Fig. \\ref{fig:megafigmri}. In those experiments, $A_1$ resamples using a Gaussian distribution, biasing the frequencies towards central frequencies, and does not check for overlaps. Since $A_0$ is also biased toward central frequencies, and all frequencies are uniformly swapped, the $A_1$'s in this section may be ``harder'' to reconstruct from than the original $A_1$. In this experiment, the number of changed frequencies acts as a proxy for the difference between $N(A_0)$ and $N(A_1)$, the null spaces of $A_0$ and $A_1$. This experiment also therefore illustrates the extent to which $f_0$ contains information about image components in the null space of $A_1$ for different $A_1$, since if $f_0$ did not contain any further information about the null space of $A_1$, we would expect the R\\&R\\ curve in Fig. \\ref{fig:nullspaceoverlap} to overlap with the ``No Adaptation'' curve.\n}\n\n\\subsection{Sample complexity}\\label{sec:exp:samplecomp}\n\n\\change{Our other experiments assume that model adaptation is performed at the level of individual test samples. However, in the case where we have access to a \\emph{calibration set} of measurements under the new forward model $A_1$ that we can leverage to retrain the network using the P\\&P\\ approach.}\n\nIn the transfer learning setting, a key concern is the size of the transfer learning set necessary to achieve high-quality results. In this section we compare the performance of P\\&P across different calibration set sizes.\n\nIn Fig. \\ref{fig:samplecompfig} we explore the effect of the number of  samples observed under the new forward model on the adapted model. We observe that even without knowing the forward model, a single calibration sample is sufficient to give improvement over the ``naive'' method that replaces $A_0$ with $A_1$ without further retraining.\nWhen the forward model is known during calibration and testing, a single example image can result in a 2 dB improvement in PSNR.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\linewidth]{figures\/calibrationsamplecomp.png}\n\\caption{Performance of the P\\&P model adaptation approach for motion deblurring as a function of the number of calibration samples (blurred images) under the new forward model. Both of our approaches outperform a naive approach (``No Adaptation''), even without exact knowledge of the new forward model.}\n\\label{fig:samplecompfig}\n\\end{figure}\n\n\\subsection{Model-blind reconstruction with generative networks}\\label{sec:exp:gan}\n\nRecent work \\cite{bora2017compressed, anirudh2018unsupervised, basioti2020image, hussein2020image} has explored solving inverse problems using generative networks, which permit reconstruction under arbitrary forward models assuming an expressive enough generative network. In particular, \\cite{anirudh2018unsupervised} and \\cite{basioti2020image} consider the case where the forward model is either partially or entirely unknown, and hence may be learned by parameterizing and jointly optimizing over both the forward model and the latent code for the generative network.\n\n\\change{In Fig. \\ref{fig:ganfig} we provide an illustration of reconstructions obtained by the method of \\cite{hussein2020image}, compared to our proposed R\\&R\\ approach. In our demonstration, as in \\cite{hussein2020image}, the generative network under consideration is a pretrained Boundary Equilibrium GAN (BEGAN) \\cite{berthelot2017began}.\nThe reconstruction quality is higher when a model-specific network is used, especially when examining fine details and textures.}\n\nIn the absence of $(x_i, y_i)$ pairs, a generative approach may be reasonable. However, learning the data manifold in its entirety requires a great deal of data at minimum, along with a sufficiently large and well-tuned generator. The authors of \\cite{yeh2017semantic} also note this fundamental limitation: for smaller or simpler applications, learning a high-quality GAN is straightforward, but for more complex applications it is difficult to train GAN models that are sufficiently accurate to rely on for high-quality reconstructions.\n\n\\begin{figure}\n\\centering\n\\renewcommand*{\\arraystretch}{0}\n\\begin{tabular}{@{}c@{}c@{}c@{}}\n\\small Truth & \\small \\change{R\\&R$+$} & \\small IAGAN \\\\ \\vspace{2pt} \\\\\n\\subfloat{\\includegraphics[width = 0.15\\linewidth, height = 0.15\\linewidth]{figures\/ganfigs\/truth\/3_10.818476.png}} \\hfill &\n\\subfloat{\\includegraphics[width = 0.15\\linewidth]{figures\/ganfigs\/rnrsup\/3_rnr_0.94_36.72.png}} &  \n\\subfloat{\\includegraphics[width = 0.15\\linewidth]{figures\/ganfigs\/gansuperres\/3_iagan_0.84_27.62.png}} \\\\\n\\subfloat{\\includegraphics[width = 0.15\\linewidth]{figures\/ganfigs\/truth\/5_8.551126.png}} &\n\\subfloat{\\includegraphics[width = 0.15\\linewidth]{figures\/ganfigs\/rnrsup\/5_rnr_0.88_31.34.png}} &  \n\\subfloat{\\includegraphics[width = 0.15\\linewidth]{figures\/ganfigs\/gansuperres\/5_iagan_0.84_23.96.png}} \\\\\n\\subfloat{\\includegraphics[width = 0.15\\linewidth]{figures\/ganfigs\/truth\/26_8.849044.png}} & \n\\subfloat{\\includegraphics[width = 0.15\\linewidth]{figures\/ganfigs\/rnrsup\/26_rnr_0.95_31.64.png}} & \n\\subfloat{\\includegraphics[width = 0.15\\linewidth]{figures\/ganfigs\/gansuperres\/26_iagan_0.87_26.77.png}}\\\\\n\\end{tabular}\n\\caption{Comparison of model adaptation (\\change{R\\&R$+$}) with a model-blind GAN-based reconstruction approach \\change{(IAGAN \\cite{hussein2020image})} for 2$\\times$ super-resolution. While a GAN-based approach only requires learning a single generative network for all forward models, our results suggest that a network trained for a specific forward model with the same number of training samples gives better reconstructions. Best viewed electronically.}\n\\label{fig:ganfig}\n\\end{figure}\n\n\\section{Discussion and Conclusion}\n\nThis paper explores solutions to the fragility of learned inverse problem solvers in the face of model drift. We demonstrate across a range of simple, practical applications that using a learned image reconstruction network in settings even slightly different than they were trained in results in significant reconstruction errors, both subtle and obvious. We propose two model adaptation procedures: the first is based on a transfer learning approach that attempts to learn a perturbation to the pre-trained network parameters, which we call Parametrize and Perturb (P\\&P); \\change{the second reuses the network as an implicitly defined regularizer in an iterative model-based reconstruction scheme, which we call Reuse and Regularize (R\\&R). We also look at a hybrid approach combining these techniques we call R\\&R+.}\n\nWe show that our model adaptation techniques enable retraining\/reuse of learned solvers under a change in the forward model, even when the change in forward model is not known. In addition, we demonstrate that just learning to invert a variety of forward models at once is not necessarily the solution to the problem of model drift: directly training on many forward models empirically appears to cause reconstruction quality to fall across all learned models. We also show that our approach is superior to one that requires learning a model of the entire image space via a generative model.\n\n\\change{The proposed P\\&P, R\\&R, and R\\&R+ model adaptation approaches each have different trade-offs, and may be useful in different scenarios. In general, we observe that R\\&R+\\ produces superior reconstructions over R\\&R\\ and P\\&P, but incurs significant computation and time costs associated with network retraining specified in  \\eqref{eq:rnrpcost}. The P\\&P\\ approach also incurs similar costs associated with network retraining. However, when a calibration set is available (as in Section \\ref{sec:exp:samplecomp}), the P\\&P\\ approach only needs to be retrained once, and computation cost at deployment matches the original solver. However, we observe two significant benefits of the R\\&R\\ approach. First, empirically we observe that only few iterations of R\\&R\\ (see Algorithm \\ref{alg:rnr}) tend to be required to give accurate results (namely, $5$ iterations in all our experiments), which increases computational cost by only a constant factor relative to the original reconstruction network. In addition, in the R\\&R\\ approach only one new parameter is introduced, in contrast to several parameters related to the optimization required for P\\&P\\ and R\\&R+. Finally, our experiments suggest that the improvement offered by R\\&R+ tends to be marginal relative to the improvement seen by going from no adaptation to R\\&R. Therefore, in situations where reconstruction time is crucial, model adaptation by R\\&R\\ may be preferred over R\\&R+.}\n\n\\change{One surprising benefit of the R\\&R\\ approach is that even in the absence of model drift (i.e., $A_0=A_1$) the reconstruction accuracy improves relative to the output from the reconstruction network. This is because R\\&R\\ iteratively modifies the output of the network to enforce data-consistency at test time. This may potentially resolve the issue raised in \\cite{sidky2020cnns} about whether learned image reconstruction networks are truly ``solving'' a given inverse problem, \\emph{i.e.},~ give a well-defined inverse map of the measurement model. However, to show this would require a much more detailed analysis of the estimator defined by the R\\&R\\ approach that is beyond the scope of this work.}\n\n\n\nAdapting learned inverse problem solvers to function under new forward models is just one step towards robustifying these powerful approaches. Our approach for unknown $A_1$ assumes an explicit parametrization of the forward model, but such a parametrization is not always straightforward or realistic. How best to adapt to complex changes in the forward model that are not easily parametrized is an important open question for future work; \\change{see \\cite{lunz2021learned} for one recent approach to learning non-parametric (and potentially non-linear) changes to forward models in an iterative reconstruction framework.}\n\n\\change{While this work focused on linear inverse problem, many of the principles introduced in this work extend also to non-linear inverse problems. For example, the R\\&R\\ approach, which is based on the regularization-by-denoising technique (RED), is readily adapted to non-linear problems amenable to a RED approach, which includes phase retrieval \\cite{metzler2018prdeep} among others.}\n\n\\change{Our empirical evidence suggests that successful model adaptation is possible provided the nullspace (or approximate nullspaces) of $A_0$ and $A_1$ are close in some sense. However, in settings where nullspaces of $A_0$ and $A_1$ are far apart, model adaptation may lead to  artifacts or hallucinated details in the reconstructions. In order to understand these limitations of model adaptation, recent methods introduced to quantify hallucinations induced by neural-network based reconstructions may prove to be useful \\cite{bhadra2020hallucinations}.}\n\n\n\nFinally, while we focused our attention on model drift, an important open problem is how to adapt to simultaneous model and data distribution drift, and the extent to which these effects can be treated independently. We hope to address these questions in future work.\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\n\nPartially identified models have been receiving extensive attentions in recent years due to their broad applications in econometrics. Partial identification of a structural parameter arises when the data available and the constraints coming from economic theory only allow to place the parameter inside a proper subset of the parameter space. Due to the limitation of the data generating process, the data cannot provide any information within the set where the structural parameter is partially identified (called \\textit{identified set}).\n\nThis paper aims at developing a semi-parametric Bayesian inference for partially identified models. A Bayesian approach may be appealing for several reasons. First, Bayesian procedures often conduct inference through Bayesian credible sets (BCS's), which are often relatively easy to construct thanks to the use of Markov Chain Monte Carlo (MCMC) methods. This is particularly useful when we are concerned about marginalizing the BCS to a low-dimensional space.  In some situations,   we are interested only in a projection of the identified region for  the subset inference.   We demonstrate that     our proposed approach provides tractable computational tools for projecting a high-dimensional identified region to a low-dimensional space. This   has important implications for practical implementations.\n\n\n\nSecondly, our Bayesian procedures also have comprehensive frequentist validations. In particular, our constructed   BCS of the identified set also has a correct asymptotic frequentist coverage probability. We construct credible sets  based on the support function; the latter completely characterizes convex and closed identified sets. We also show the Bernstein-von Mises theorem for the posterior distribution of the support function.  At the best of our knowledge this has not been studied in the literature yet.  This powerful result in turn allows us to establish the (asymptotic) equivalence between BCS's and frequentist confidence sets (FCS's) for the identified set. The literature on partial identification distinguishes between credible\/confidence sets for the partially identified parameter and for the identified set. Credible sets for the identified set play an important role not only when the target of interest is the partially identified parameter but even when the identified set is itself the object of interest. While focusing on the study of BCS for the identified set, we also extend  Moon and Schorfheide (2010)'s analysis for the partially identified parameter to a semi-parametric setup,  which is relevant in more general \\textit{moment inequality models} where the likelihood function may be unknown. Moreover,  if we admit the existence of a true value of the structural parameter and the identified set,  the corresponding posterior distributions concentrate asymptotically in a neighborhood of the true value (set). This property is known as the \\textit{posterior consistency}. It is important because it guarantees that, with a sufficiently large amount of data, we can recover the truth accurately with large probabilities.\n\n\n\n\nThird, putting a prior on the partially identified parameter can be viewed  as a way of incorporating researchers' beliefs. A Bayesian approach conveniently combines the information from both the observed data and other sources of prior information. The prior information is, for instance, information coming from historical data, information based on experience or on previous survey data. In some applications this information is largely available, \\textit{e.g.} in macroeconomics, central banking and finance.   We stress that the prior information will not affect the boundary of the identified set, but will only play a role in determining which areas inside the identified set are a priori ``more likely'' than others.   On the other hand, when specifying a prior distribution on the partially identified parameter is either  difficult or conflicting with the philosophy of partial identification, a  researcher can still use our procedure and either specify a uniform prior or just construct the BCS for the identified set for inference.   The latter is due to an important feature   of our procedure  that the Bayesian analysis of the identified set does not  require to specify a prior on the partially identified parameter.   Therefore,  we accommodate both situations where a researcher does  have prior information as well as situations where she does not.\n\n\n\n\nFrom the posterior perspective, the Bayesian  partial identification produces a posterior distribution of the partially identified parameter whose support will asymptotically concentrate around the true identified set.  When informative  priors are available,  the shape of the posterior density may  not be flat   inside the identified set,  and  will ground on the prior distribution even asymptotically. Therefore, the asymptotic behavior of the posterior distribution is different from that of the traditional point identified case where (in the latter case) the information from the prior is often washed away by the data asymptotically. Thus, the Bayesian approach to partial identification links conclusions and inferences to various information sources -- data, prior, experience, etc.-- in a transparent way.\n\nFinally, when the identified set depends on a point identified nuisance parameter, say $\\phi$, and this is integrated out with respect to its posterior, then the prior information on the partially identified parameter is completely revised by the data. Hence,  the proposed procedure also learns about the partially identified parameter based on the whole posterior distribution of $\\phi$, which is potentially  useful  in finite samples. Consequently, there is a strong  motivation for us to conduct a comprehensive Bayesian study for the partially identified econometric models.\\\\\n\n\n\nThere are in general two approaches in the literature on Bayesian partial identification. The first approach specifies a parametric likelihood function and assumes it is known  up to a finite-dimensional parameter. This approach has been used frequently in the literature, see e.g., Moon and Schorfheide (2012), Poirier (1998),  Bollinger and  Hasselt (2009), Norets and Tang (2012) among many others. In many applications, however, econometric models usually only identify a set of moment inequalities instead of the full likelihood function. Examples are: interval-censored data, interval instrumental regression, asset pricing (Chernozhukov et al. 2008),  incomplete structural models (Menzel 2011), etc. Assuming a parametric form of the likelihood function is  ad-hoc in these applications. Once the likelihood is mis-specified, the posterior can be misleading. The second approach starts from a set of moment inequalities, and uses a moment-condition-based likelihood such as the limited information likelihood (Kim 2002) and the exponential tilted empirical likelihood (Schennach 2005). Further references may be found in Liao and Jiang (2010),  Chernozhukov and Hong (2003) and Wan (2011). This approach avoids assuming the knowledge of the true likelihood function. However, it only studies the structural parameter, and it is hard to construct posteriors and credible sets for the identified set. Moreover, it does not have a Bayesian probabilistic interpretation.\n\n\nThis paper proposes a pure Bayesian procedure without assuming a parametric form of the true likelihood function. We place a nonparametric prior on the likelihood and obtain the marginal posterior distribution for the partially identified parameter  and  the identified set. A similar Bayesian procedure was recently used in Florens and Simoni (2011). As a result, our procedure is semi-parametric Bayesian and  can make inference about both the partially identified parameter and its identified set easily. It only requires a set of moment conditions and then it can be completely nonparametric on the data generating process.  This is an appealing feature in general moment inequality models.   On the other hand, if the likelihood function is known,   our procedure continues to work and this paper is still well-motivated. In fact, many contributions of this paper, e.g., Bayesian inference of the  support function, construction of BCS for the identified set, subset inferences, etc., are relevant and original also for the case with a known likelihood.\n\n\n\n\n\n\n\n\n\n\n\n\nThere is a growing literature on Bayesian partially identified models. Besides those mentioned above, the list also includes Gelfand and Sahu (1999), Neath and Samaniego (1997), Gustafson (2012), Epstein and Seo (2011), Stoye (2012), Kitagawa (2012), Kline (2011), etc. There is also an extensive literature that analyzes partially identified models from a frequentist point of view. A partial list includes Andrews and Guggenberger (2009), Andrews and Soares (2010), Andrews and Shi (2013), Beresteanu, Molchanov and Molinari (2011), Bugni (2010), Canay (2010), Chernozhukov, Hong and Tamer (2007),  Chiburis (2009),  Imbens and Manski (2004), Romano and Shaikh (2010),  Rosen (2008),  Stoye (2009),  among others. See Tamer (2010) for a review.\n\n\nWhen the identified set is closed and convex, the support function becomes one of the useful tools to characterize its properties.  The literature on this perspective has been growing rapidly, see for example,   Bontemps, Magnac and Maurin (2012),  Beresteanu and Molinari (2008),   Beresteanu et al. (2012),   Kaido and Santos (2013),  Kaido (2012) and Chandrasekhar et al. (2012). This   paper is also closely related to the asymptotic  nonparametric Bayesian literature: Wu and Ghosal (2008), Ghosh and Ramamoorthi (2003), Ghosal and van der Vaart (2001),  Shen and Wasserman (2001),  Ghosal et al. (1999), Amewou-Atisso et al. (2003),   Walker et al. (2007),   van der Vaart and van Zanten (2008), Bickel and Kleijn (2012),  Jiang (2007), Choi and  Ramamoorthi (2008), Castillo (2008),  Freedman (1999),  Rivoirard  and Rousseau (2012),  among others.\n\n\n\n\nThe paper is organized as follows. Section \\ref{highlight} outlines our main results and contributions. Section \\ref{s_general_setup} sets up the model and discusses the prior specification on the underlying likelihood function. Section \\ref{s_nonparametric_prior} studies the (marginal) posterior distribution of the structural parameter. Section \\ref{ssf} studies the posterior of the support function in moment inequality models. In particular, the Bernstein-von Mises theorem and a linear representation for the support function are obtained. Section \\ref{sbcs} constructs the Bayesian credible sets for both the structural parameter and its identified set. In addition, the frequentist coverages of these credible sets are studied.  Section \\ref{s_Projection_Subset_Inference} addresses the subset inference when the target of interest is only a component of the full parameter. Section \\ref{s_posterior_consistency_set} shows the posterior consistency for the identified set and provides the concentration rate. Section \\ref{s_further_illustration_Uniformity} addresses the uniformity. In particular, it discusses the case when point identification is actually achieved.  Section \\ref{s_Financial_asset_pricing} applies the support function approach to a financial asset pricing study.\nFinally, Section \\ref{s_Conclusion} concludes with further discussions. All the proofs are given in the appendix to this paper and in a supplementary appendix.\n\n\n\n\n\n\\section{Highlights of Our Contributions}\\label{highlight}\n\n\n\nThis section provides a global vision of our main contributions of this paper. Formal setup of the model starts from Section 3.\n\n\n\\subsubsection*{Semi-parametric Bayesian partial identification}\n\n\n\nWe focus on semi-parametric models where the true likelihood function may be unknown, which is more relevant in moment inequality models. Then there are three types of parameters in the Bayesian setup: $\\theta$, which is the partially identified structural parameter;  $\\phi$,   a   point-identified parameter that characterizes the identified set, and the  unknown likelihood  $F$.  The identified set can be written as $\\Theta(\\phi)$. According to the Bayesian philosophy, we treat the identified set as random, and construct its posterior distribution.\n\nWithout assuming any parametric form for the likelihood, we place a nonparametric prior $\\pi(F)$ on it. The posteriors of $\\phi$ and of the identified set can then be constructed via the posterior of $F$.  Such a semi-parametric posterior requires only a set of moment inequalities, and therefore is robust to the likelihood specification.  Moreover, to make inference about the partially identified $\\theta,$ we place a conditional prior $\\pi(\\theta|\\phi)$ supported only on $\\Theta(\\phi)$.  Note that Bayesian inference for the identified set may be carried out based on the posterior of $\\Theta(\\phi)$ which does not depend on $\\pi(\\theta|\\phi)$. So the  prior specification for $\\theta$ plays a role only in the inference about $\\theta$.\n\n\n\n\n\n\nFor these posteriors, we show that asymptotically $p(\\theta|Data)$ will be supported  within an arbitrarily small neighborhood of the true identified set, and the  posterior of $\\Theta(\\phi)$ also concentrates around the true set in the Hausdorff distance. These are  the notion of \\emph{posterior consistency} under partial identification.\n\n\n\\subsubsection*{Support function}\n\n\nTo make inference on $\\Theta(\\phi)$ we can take advantage of the fact that when $\\Theta(\\phi)$ is closed and convex it is completely characterized by its \\textit{support function} $S_{\\phi}(\\cdot)$ defined as:\n$$\nS_{\\phi}(\\nu)=\\sup_{\\theta\\in\\Theta(\\phi)}\\theta^T\\nu\n$$\nwhere $\\nu\\in\\mathbb{S}^{\\dim(\\theta)}$, the unit sphere. Therefore, inference on $\\Theta(\\phi)$ may be conveniently carried out through inference on its support function. The posterior distribution of $S_{\\phi}(\\cdot)$ is also determined by that of $\\phi$. We show that in a general moment inequality model, the support function has an asymptotic linear representation in a neighborhood of the true value of $\\phi$, which  potentially extends the inference in Bontemps et al. (2012) to nonlinear models.\n \nOur paper also establishes  the Bernstein-von Mises theorem for the support function, that is, the posterior distribution of  $S_{\\phi}(\\cdot)$  converges weakly to a Gaussian process. \nWe also calculate the support function for a number of interesting examples, including interval censored data, missing data, interval instrumental regression and asset pricing models.\n\n\n\\subsubsection*{Two-sided Bayesian credible sets for the identified set}\n\nWe construct two types of Bayesian credible sets (BCS's): one for the identified set $\\Theta(\\phi)$ and the other for the partially identified parameter $\\theta$. In particular, the BCS for the  {identified set} is constructed based on the support function, is two-sided, and has an asymptotically correct frequentist coverage probability.  Specifically,  we find sets $\\Theta(\\hat\\phi )^{-q_{\\tau}\/\\sqrt{n}}$ and $\\Theta(\\hat\\phi )^{q_{\\tau}\/\\sqrt{n}}$,  satisfying:  for  level $1-\\tau$ where $\\tau\\in (0,1)$,\n\nBayesian coverage:\n\\begin{equation}\\label{eq2.1add}\nP(\\Theta(\\hat\\phi )^{-q_{\\tau}\/\\sqrt{n}}\\subset\\Theta(\\phi)\\subset\\Theta(\\hat\\phi )^{q_{\\tau}\/\\sqrt{n}}|Data)=1-\\tau;\n\\end{equation}\n\nFrequentist coverage:\n\\begin{equation}\\label{eq2.2add}\nP_{D_n}(\\Theta(\\hat\\phi )^{-q_{\\tau}\/\\sqrt{n}}\\subset\\Theta(\\phi_0)\\subset\\Theta(\\hat\\phi )^{q_{\\tau}\/\\sqrt{n}})\\geq1-\\tau,\n\\end{equation}\nwhere   $P_{D_n}$ denotes the sampling probability, and $\\Theta(\\phi_0)$ is the true identified set.    In (\\ref{eq2.1add}) the random set is $\\Theta(\\phi)$ while in (\\ref{eq2.2add}) the random sets are $\\Theta(\\hat\\phi )^{-q_{\\tau}\/\\sqrt{n}}$ and $\\Theta(\\hat\\phi )^{q_{\\tau}\/\\sqrt{n}}$.  One of the important features is that the BCS for the identified set does not require specifying a prior on the partially identified parameter.  The notation $\\Theta(\\hat\\phi )^{-q_{\\tau}\/\\sqrt{n}}$, $\\Theta(\\hat\\phi )^{q_{\\tau}\/\\sqrt{n}}$, $\\hat\\phi$ and $q_{\\tau}$ are to be formally defined in the paper. Therefore, the constructed two-sided BCS can also be used as frequentist confidence set for the identified set.\n\n\n\nFurthermore, we find that in the semi-parametric Bayesian model, Moon and Schorfheide (2012)'s conclusion about the BCS for the partially identified parameter $\\theta$ still holds:  it is smaller than frequentist confidence sets in large samples.\nHence, while the BCS for the partially identified parameter does not have a correct frequentist coverage, the asymptotic equivalence between BCS and FCS for the identified set holds. Intuitively, this is because the prior information still plays an important role in the posterior of the partially identified parameter even asymptotically; on the other hand, as the identified set is ``point identified\", whose BCS is independent of the prior on $\\theta$, then its prior information is ``washed away'' asymptotically.  Thus, the proposed inference for the identified set and the support function is asymptotically robust to their prior specification.\n\n\n\\subsubsection*{Projection and subset inference}\n\nWe show that with our approach it is easy to project\n(marginalize) onto low-dimensional subspaces for subset inferences. This computation is fast.\nSuppose the dimension of $\\theta$ is relatively large, but we are interested in only a few components of $\\theta$,   and aim to make inference about these components and their marginal identified set. In our approach, constructing the identified set and BCS for the marginal components   simply requires the marginalization of a joint distribution and can be carried out efficiently thanks to the use of MCMC methods. It is also computationally convenient to calculate the BCS for the marginal identified set. Hence, the proposed   procedure has large potentiality in many empirical applications.\n\n\\subsubsection*{Uniformity}\n\nThe proposed  Bayesian inference for the identified set is valid uniformly over a class of data generating process. In particular, using specific examples, we illustrate that as the identified set shrinks to a singleton, so that point identification is (nearly) achieved, our Bayesian inference for the identified set carries over.\n\n\n\\subsubsection*{Applications}\n\nWe develop a detailed application of Bayesian partial identification to financial asset pricing, which is an example where the identified set is of direct interest. Estimation and inference for the support function as well as for the identified set are conducted. Moreover, throughout the paper, we study in detail other typical examples including the interval censoring, interval regression and missing data problems.\n\n\n\n\n\n\n\n\n\n\n\n\\section{General Setup of Bayesian Partially Identified Model}\\label{s_general_setup}\n\n\\subsection{The Model}\nEconometric models often involve a structural parameter $\\theta\\in\\Theta$ that is only partially identified by the data generating process (DGP) on a non-singleton set, which we call \\textit{identified set}. The model also contains two parameters that are point identified by the DGP:  a finite-dimensional parameter $\\phi\\in\\Phi\\subset\\mathbb{R}^{d_{\\phi}}$ and the distribution function $F$ of the observed data, which is infinite-dimensional. Here, $\\Phi$ denotes the parameter space for $\\phi$ and $d_{\\phi}$ its dimension. The point identified parameter $\\phi$ often arises naturally as it characterizes the data distribution. In most of partially identified models, the identified set is also characterized by $\\phi$, hence we denote it by $\\Theta(\\phi)$ to indicate that once $\\phi$ is determined, so is the identified set. Let $d=\\dim(\\theta)$ and $\\Theta\\subset\\mathbb{R}^{d}$ denote the parameter space for $\\theta$; we assume $\\Theta(\\phi)\\subseteq\\Theta$.\n\nIn a parametric Bayesian partially identified model as in  Poirier (1998), Gustafson (2012) and Moon and Schorfheide (2012), $F$ is linked with a known likelihood function to $\\phi$. However, as in the usual point identified models, in some applications  assuming a known likelihood function may suffer from a model specification problem, and may lead to misleading conclusions. Instead, econometric applications often  involve only a set of moment conditions as in (\\ref{eq2.1}) below. This gives rise  to the \\textit{moment inequality models}.  A parametric form of the likelihood function and of $F$ can be unavailable in these models. A robust approach is to proceed without assuming a parametric form for the likelihood function, but to put a prior on $(\\theta,\\phi,F)$ instead.   This yields the semi-parametric Bayesian setup.\n\n\nWe  specify a nonparametric prior  on data's cumulated distribution function (CDF) $F$, which can deduce a prior for $\\phi $ through a transformation $\\phi=\\phi(F)$, as $\\phi$ often is a functional of $F$.   Moreover, the prior  on the identified set $\\Theta(\\phi)$   is determined through that of $\\phi$.\n Due to the identification feature, for any given $\\phi\\in\\Phi$, we specify a conditional prior $\\pi(\\theta|\\phi)$ such that\n$$\n\\pi(\\theta\\in\\Theta(\\phi)|\\phi)=1.\n$$\nBy construction, this prior for $\\theta$ puts all its mass on $\\Theta(\\phi)$ for any $\\phi\\in\\Phi$.  So it takes the form:\n    \\begin{displaymath}\n      \\pi(\\theta|\\phi) \\propto I_{\\theta\\in\\Theta(\\phi)}g(\\theta),\n    \\end{displaymath}\nwhere $g(\\cdot)$ is some probability density function and $I_{\\theta\\in\\Theta(\\phi)}$ is the indicator function of $\\Theta(\\phi)$. In Section \\ref{app_prior_theta} we discuss the philosophy of specifying the prior on $\\theta$.\n\n\n\n\n\nOur analysis focuses on the situation where $\\Theta(\\phi)$ is a closed and convex set for each $\\phi$. Therefore, $\\Theta(\\phi)$ can be uniquely characterized by its \\emph{support function}. For any fixed $\\phi$, the support function for $\\Theta(\\phi)$ is a function $S_{\\phi}(\\cdot): \\mathbb{S}^d\\rightarrow\\mathbb{R}$ such that\n    \\begin{displaymath}\n      S_{\\phi}(\\nu)= \\sup_{\\theta\\in\\Theta(\\phi)}\\theta^T\\nu.\n    \\end{displaymath}\nwhere $\\mathbb{S}^d$ denotes the unit sphere in $\\mathbb{R}^d$. The support function plays a central role in convex analysis since it determines all the characteristics of a convex set. Hence, it is one of the essential objects for our Bayesian inference. In a similar way as for $\\Theta(\\phi)$, we put a prior on $S_{\\phi}(\\cdot)$ via the prior on $\\phi$.\n\n\nSuppose $p(\\phi|D_n)$ denotes the posterior of $\\phi$, given the data $D_n$ and a prior $\\pi(\\phi)$. It is readily seen  that (see e.g., Poirier 1998) the joint posterior of $(\\theta, \\phi)$ is given by\n$$\np(\\theta, \\phi|D_n)\\propto\\pi(\\theta|\\phi)p(\\phi|D_n).\n$$\nBy integrating out $\\phi$, we obtain the marginal posterior for $\\theta$.  On the other hand,  the posteriors of $\\Theta(\\phi)$ and $S_{\\phi}(\\cdot)$ are also determined through the marginal posterior  $p(\\phi|D_n)$.   This also highlights an important feature of this paper: our results  on $\\Theta(\\phi)$ and the support function  do not require placing a prior on the partially identified parameter $\\theta$,  because  as far as $p(\\phi|D_n)$ is concerned, the prior for $\\theta$ is not needed at all.  Furthermore, as the identified set and support function are ``point identified'', their posteriors are asymptotically robust to the prior specifications on $\\phi$.\n\n\n\n\n\n\n\nLet us present a few examples that have received much attention in partially identified econometric models literature. In the rest of the paper, we denote by $X$ the observable random variable for which we have $n$ i.i.d. observations $D_n=\\{X_i\\}_{i=1}^n$. Let $(\\mathcal{X},\\mathcal{B}_{x},F)$ denote a probability space in which $X$ takes values and $\\mathcal{F}$ denote the parameter space of $F$.\n\n\n\n\n\n\n \\begin{comment}\n  These moment restrictions can be written as moment inequalities through the use of a function $\\Psi: \\Theta \\times \\Phi \\rightarrow \\mathbb{R}^{m}$ as\n\n    \\begin{equation}\\label{eq_moment_inequalities}\n      \\Psi(\\theta,\\phi) \\leq 0.\n    \\end{equation}\n\n\\noindent By denoting $\\Psi_{i}(\\theta,\\phi)$ the $i$-th component of $\\Psi(\\theta,\\phi)$, then (\\ref{eq_moment_inequalities}) is equivalent to $\\Psi_{i}(\\theta,\\phi)\\leq 0$ for all $1\\leq i\\leq m$. A common feature of many settings we are considering is that the parameter $\\theta$ does not characterize the sampling distribution $F$ and the relationship between $(F,\\phi)$ and $\\theta$ is only described by the moment restrictions in (\\ref{eq_moment_inequalities}).\\\\\n\\indent For a given value of $\\phi$ we denote by $\\Theta_{0}(\\phi)$ the set of all $\\theta\\in \\Theta$ that satisfy the moment restrictions:\n\n    \\begin{displaymath}\n      \\Theta_{0}(\\phi) \\equiv \\left\\{\\theta\\in\\Theta; \\; \\Psi(\\theta,\\phi) \\leq 0\\right\\}.\n    \\end{displaymath}\n\n\\noindent We refer to this set as the \\emph{identified set}. The structural parameter $\\theta$ is \\emph{point-identified} if $\\forall\\phi\\in\\Phi$, $\\Theta_{0}(\\phi)$ is a singleton; otherwise, we say that $\\theta$ is \\emph{partially identified}. In this paper, we focus on the partially identified case where $\\Theta_{0}(\\phi)$ is a nonsingleton set.\\\\\n\n\n\\noindent where $<\\cdot,\\cdot>$ denotes the inner product in $\\mathbb{R}^{d}$. Since the support function is homogeneous of degree $1$, we restrict its domain to the unit sphere $\\mathbb{S}^{d}$ in $\\mathbb{R}^{d}$ as it is common in the literature. Therefore, $S_{\\phi}(\\cdot):\\mathbb{S}^{d}\\rightarrow \\mathbb{R}$.\n\n\\end{comment}\n\n\\begin{exm}[Interval censored data]\\label{ex2.1}\n  Let $(Y,Y_{1},Y_{2})$ be a $3$-dimensional random vector such that $Y\\in[Y_{1},Y_{2}]$ with probability one. The random variables $Y_{1}$ and $Y_{2}$ are observed while $Y$ is unobservable (see, e.g., Moon and Schorfheide 2012).  We denote:  $\\theta=E(Y)$ and $\\phi=(\\phi_{1},\\phi_{2})' \\equiv (E(Y_{1}),E(Y_{2}))'$. Therefore, we have the following\nidentified set for $\\theta$: $\\Theta(\\phi)=[\\phi_1, \\phi_2]$. The support function for $\\Theta(\\phi)$ is easy to derive:\n    \\begin{displaymath}\n  \n   S_{\\phi}(1)=\\phi_2, \\quad S_{\\phi}(-1)=-\\phi_1.\n    \\end{displaymath}\n    The non-parametric prior specification on the likelihood is to be discussed in Section \\ref{sss_nonparametric_prior}.\n$\\square$\n\n\n \n    \n  \n\n\\end{exm}\n\n\\begin{exm}[Interval regression model]\\label{ex2.2}\n The regression model with interval censoring has been studied by, for example, Haile and Tamer (2003). Let $(Y,Y_{1},Y_{2})$ be a $3$-dimensional random vector such that $Y\\in[Y_{1},Y_{2}]$ with probability one. The random variables $Y_{1}$ and $Y_{2}$ are observed while $Y$ is unobservable. Assume that\n$$\nY=x^T\\theta+\\epsilon\n$$\nwhere $x$ is a vector of observable regressors. In addition, assume there is a $d$-dimensional vector of nonnegative exogenous variables $Z$ such that $E(Z\\epsilon)=0$. Here $Z$ can be either a vector of instrumental variables when $X$ is endogenous, or a nonnegative transformation of $x$ when $x$ is exogenous. It follows that\n\\begin{equation}\\label{eq2.1}E(ZY_1)\\leq E(ZY)=E(Zx^T)\\theta\\leq E(ZY_2).\n\\end{equation}\nWe denote $\\phi=(\\phi_1, \\phi_2, \\phi_3)$ where $(\\phi_1^T,\\phi_3^T)=(E(ZY_1)^T, E(ZY_2)^T)$ and $\\phi_2=E(Zx^T).$ Then the identified set for $\\theta$ is given by $\\Theta(\\phi)=\\{\\theta\\in\\Theta: \\phi_1\\leq \\phi_2\\theta\\leq \\phi_3\\}.$ Suppose $\\phi_2^{-1}$ exists. The support function for   $\\Theta(\\phi)$ is given by (denote $(x)_i$ as the $i$th component of $x$)\\footnote{See Appendix \\ref{s_appendix_support_function_interval_regression} in the supplementary material for detailed derivations of the support function in this example. Similar results but in a slightly different form are  presented in Bontemps et al. (2012).}:\n$$\nS_{\\phi}(\\nu)=\\nu^T\\phi_2^{-1}\\left(\\frac{\\phi_1+\\phi_3}{2}\\right)+\\alpha_{\\nu}^T\\left(\\frac{\\phi_3-\\phi_1}{2}\\right),\\quad \\nu\\in\\mathbb{S}^d\n$$\nwhere  $\\alpha_{\\nu}=( |(\\nu^T\\phi_2^{-1})_1|,..., |(\\nu^T\\phi_2^{-1})_d|)^T.$\n \n\n\n\n\n $\\square$\n\\end{exm}\n\n\\begin{exm}[Missing data]\\label{ex2.3}\n  Consider a bivariate random vector $(Y,M)$ where $M$ is a binary random variable which takes the value $M=0$ when $Y$ is missing and $1$ otherwise. Here   $Y$ represents whether a treatment is successful ($Y=1$) or not ($Y=0$).  The parameter of interest is the probability  $\\theta=P(Y=1)$.  This problem without the missing-at-random assumption has been extensively studied in the literature, see for example, Manski  and Tamer (2002), Manski (2003), etc.  Since $P(Y=1|M=0)$ cannot be recovered from the data,  the empirical evidence partially identifies $\\theta$ and $\\theta$ is characterized by the following moment restrictions:\n    \\begin{displaymath}\n      P(Y=1|M=1)P(M=1) \\leq \\theta \\leq  P(Y=1|M=1)P(M=1)+ P(M=0).\n    \\end{displaymath}\n Here, $\\phi=(P(M=1), P(Y=1|M=1))=(\\phi_1,\\phi_2)$. The identified set is $\\Theta(\\phi)=[\\phi_1\\phi_2,\\phi_1\\phi_2+1-\\phi_1]$,\n and its support function is:\n$S_{\\phi}(1) = \\phi_1\\phi_2+1-\\phi_1$, $S_{\\phi}(-1)=-\\phi_1\\phi_2$.\n\n\\end{exm}\n\n\\subsection{Nonparametric prior scheme for $(\\phi, F)$}\\label{sss_nonparametric_prior}\n\n\n\nWhen the model only specifies a set of moment inequalities, we can place a non-parametric prior on the likelihood function through $F$, e.g., a Dirichlet process prior.   Since $\\phi$ is point identified, we assume it can be rewritten as a measurable function of $F$ as  $\\phi = \\phi(F)$.  The prior distribution for $\\phi$ is then deduced   from that of $F$ via $\\phi(F)$. The Bayesian experiment is (we use the notation ``$\\sim$\" to mean ``distributed as\")\n$$\n      X|F \\sim F,\\qquad F\\sim \\pi(F), \\qquad \\theta|\\phi \\sim \\pi(\\theta|\\phi(F))\n$$\nFor instance, in the interval censored data example \\ref{ex2.1},  let $F$ be the joint CDF of $(Y_1, Y_2)$, then $(\\phi_1, \\phi_2)=\\phi(F)=(E(Y_1|F), E(Y_2|F))$, and the identified set is modeled as  $\\Theta(\\phi)=[\\phi_1(F), \\phi_2(F)]$, which is a set-valued function of $F$.\n\n\nThe prior distribution $\\pi(F)$ is a  distribution on $\\mathcal{F}$. Examples of such a prior  include Dirichlet process priors (Ferguson 1973) and Polya tree (Lavine 1992). The case where $\\pi(F)$ is a Dirichlet process prior in partially identified models is  proposed by \\cite{FlorensSimoni2011}.\n\n\n\nLet $p(F|D_n)$ denote the marginal posterior of $F$ which, by abuse of notation, can be written $p(F|D_n)\\propto\\pi(F)\\prod_{i=1}^nF(X_i)$. The posterior distributions of $\\phi$,    $\\Theta(\\phi)$, and the support function $S_{\\phi}(\\cdot)$ are deduced from the posterior of $F$, but do not depend on the prior on $\\theta$. Moreover, it can be shown that $p(\\theta|\\phi(F), D_n)=\\pi(\\theta|\\phi(F)).$ Then, for any measurable set $B\\subset\\Theta$, the marginal posterior probability of $\\theta$ is given by, averaging over $F$:\n    \\begin{eqnarray*}\\label{eq2.3}\n      P(\\theta\\in B|D_n) & = & \\int_{\\mathcal{F}}P(\\theta\\in B |\\phi(F), D_n)p(F|D_n)dF\\nonumber\\\\\n      & = & \\int_{\\mathcal{F}}\\pi(\\theta\\in B|\\phi(F))p(F|D_n)dF = E[\\pi(\\theta\\in B|\\phi(F))|D_n]\n    \\end{eqnarray*}\nwhere the conditional expectation is taken with respect to the posterior of $F$. The above posterior is easy to calculate via simulation when $F$ has a Dirichlet process prior.\n\n\n\nAn alternative prior scheme for $(\\phi, F)$ consists in putting a prior   on $\\phi$ directly. This  is particularly useful when there is informative prior information for $\\phi$. It models the  unknown likelihood function semi-parametrically through reformulating $F$ as $F=F_{\\phi, \\eta}$ where $\\eta$ is an infinite-dimensional nuisance parameter (often a density function) that is \\textit{a priori} independent of $\\phi$. The prior on $(\\phi, F)$ is then deduced from the prior on $(\\phi,\\eta)$.  We describe this alternative semi-parametric prior in Appendix \\ref{sss_semiparametric_prior}.\n\n\n\n\n\n\n\n\n\n\\section{Bayesian Inference for $\\theta$}\\label{s_nonparametric_prior}\n\n\\subsection{Putting priors on partially identified $\\theta$}\\label{app_prior_theta}\n\nIn this section we briefly discuss the meaning of the prior $\\pi(\\theta|\\phi)$. As  stated in  Tamer (2010): ``(Partial identification) links conclusions drawn from various\nempirical models to sets of assumptions made in a transparent way.\nIt allows researchers to examine the informational content of their\nassumptions and their impacts on the inferences made.''\n\nBy imposing a  prior  on the partially identified parameter $\\theta$, we reflect how prior beliefs and\/or assumptions can impact the associated statistical inference. To illustrate the rationale of imposing such a prior, let us consider the missing data example (Example \\ref{ex2.3}).    Writing $\\alpha=P(Y=1|M=0)$, we then link $\\theta$ with $\\alpha$ by\n\\begin{equation}\\label{eq2.2}\n\\theta=\\phi_2\\phi_1+\\alpha(1-\\phi_1).\n\\end{equation}\nAs $\\phi$ is point identified, statistical inferences about $\\theta$ therefore relies on the  treatment of $\\alpha$. On the other hand, various ways of dealing with $\\alpha$ reflect various researchers' prior beliefs, which also correspond to the ``informational content of their assumptions\".\n\nFrom a Bayesian point of view, this is fulfilled by putting a distribution on $\\alpha$, as a prior $\\pi(\\alpha)$ supported on $[0,1]$ (possibly also depending on $\\phi$).  The traditional exogeneity assumption such as missing-at-random, in this case, corresponds to a point mass prior on $\\alpha=\\phi_2=P(Y=1|M=1)$.   The more concentrating is the prior, the stronger are the assumptions we impose on the missing mechanism.  Such a prior distribution can also come from a previous study based on a different dataset that contain information about $\\alpha$ where only summarizing statistics are available instead of the complete data set.  Above all, when no informative knowledge about $\\alpha$ is available, a uniform prior on $[0,1]$ is imposed for $\\alpha$, which reduces to Manski's bounds approach.\n\nGiven the imposed distribution $\\pi(\\alpha)$ that reflects researchers' assumptions or beliefs about the missing mechanism, we can deduce a conditional prior for $\\theta$ through (\\ref{eq2.2}) given $\\phi=(\\phi_1,\\phi_2)$.\nAs a result, putting a prior on the partially identified parameter can be viewed as a way of incorporating researchers' assumptions on the missing mechanisms. This varies from the  traditional exogeneity approach to the  most  robust bounds approach, which also bridges point identification and partial identification.\n\n\n\\subsection{Posterior Consistency for $\\theta$}\nThe shape of the posterior of a partially identified parameter    still relies upon its prior distribution asymptotically, which distinguishes from the asymptotic posterior behavior in the classical point identified case.  On the other hand, the support of the prior distribution of $\\theta$  is revised after data are observed and eventually converges towards the true identified set asymptotically. The latter corresponds to the frequentist consistency of the posterior distribution for partially identified parameters. Posterior consistency  is one of the benchmarks  of a Bayesian procedure under consideration, which ensures that with a sufficiently large amount of data, it is nearly possible to discover the true identified set.\n\n\n\n\n\nWe assume there is a true value of $\\phi$, denoted by $\\phi_0$, which induces a true identified set $\\Theta(\\phi_0)$, and a true $F$, denoted by $F_{0}$. Our goal is to achieve the frequentist \\textit{posterior consistency} for the partially identified parameter:   for any $\\epsilon > 0$ there is $\\tau \\in (0,1]$ such that\n$$\nP(\\theta\\in\\Theta(\\phi_0)^{\\epsilon}|D_n)\\rightarrow^p 1,  \\text{ and }\\quad P(\\theta\\in\\Theta(\\phi_0)^{-\\epsilon}|D_n)\\rightarrow^p (1 - \\tau).\n$$\nHere  $   \\Theta(\\phi)^{\\epsilon} $ and $ \\Theta(\\phi)^{-\\epsilon} $ are the $\\epsilon$-envelope and $\\epsilon$-contraction of $\\Theta(\\phi)$, respectively:\n    \\begin{equation}\\label{eq_e_envelope}\n      \\Theta(\\phi)^{\\epsilon} = \\{\\theta\\in\\Theta: d(\\theta, \\Theta(\\phi))\\leq\\epsilon\\},\\quad  \\Theta(\\phi)^{-\\epsilon} = \\{\\theta\\in\\Theta(\\phi): d(\\theta, \\Theta\\backslash\\Theta(\\phi))\\geq\\epsilon\\},\n    \\end{equation}\n  with $\\Theta\\backslash\\Theta(\\phi) =\\{\\theta\\in\\Theta; \\theta\\notin \\Theta(\\phi)\\}$ and $d(\\theta,\\Theta(\\phi))=\\inf_{x\\in\\Theta(\\phi)}\\|\\theta-x\\|$. Note that this result still carries over  when  $\\theta$ is point identified, in which  case $\\Theta(\\phi)^{\\epsilon}$ is an $\\epsilon$-ball around $\\theta$, $\\Theta(\\phi)^{-\\epsilon}$ is empty, and $\\tau=1.$\n\n\n\n\n \n \n\nThe likelihood function is endowed with a prior through either the nonparametric prior $\\pi(F)$ as described in Section \\ref{sss_nonparametric_prior} or the semi-parametric prior $\\pi(\\phi)$ as described in Appendix \\ref{sss_semiparametric_prior}. We assume that the priors $\\pi(F)$ and $\\pi(\\phi)$ specified for $F$ and $\\phi$ are such that the corresponding posterior distribution of $p(\\phi|D_n)$ is consistent.\n\n\\begin{assum}\\label{ass3.1}\n  At least one of the following holds:\n\n    \\begin{itemize}\n      \\item[(i)] The measurable function $\\phi(F):\\mathcal{F}\\rightarrow\\Phi$ is continuous. The prior $\\pi(F)$ is such that the posterior $p(F|D_{n})$ satisfies:\n\n        \\begin{displaymath}\n          \\int_{\\mathcal{F}} m(F)p(F|D_n)dF \\rightarrow^p  \\int_{\\mathcal{F}} m(F)\\delta_{F_0}(dF)\n        \\end{displaymath}\n\n        for any bounded and continuous function $m(\\cdot)$ on $\\mathcal{F}$ where $\\delta_{F_0}$ is the Dirac function at the true distribution function $F_0$;\n\n      \\item[(ii)] The prior $\\pi(\\phi)$ is such that the posterior $p(\\phi|D_{n})$ satisfies:\n\n        \\begin{displaymath}\n          \\int_{\\Phi} m(\\phi)p(\\phi|D_n)d\\phi \\rightarrow^p  \\int_{\\Phi} m(\\phi)\\delta_{\\phi_0}(d\\phi)\n        \\end{displaymath}\n\n        for any bounded and continuous function $m(\\cdot)$ on $\\Phi$ where $\\delta_{\\phi_0}$ is the Dirac function at the true  $\\phi_0$.\n\n    \\end{itemize}\n\\end{assum}\n\nAssumptions \\ref{ass3.1} \\textit{(i)} and \\textit{(ii)} refer to the nonparametric and semi-parametric prior scheme respectively, and are verified by many nonparametric and semi-parametric priors. Examples are: Dirichlet process priors, Polya Tree process priors, Gaussian process priors, etc.  For instance, when $\\pi(F)$ is the Dirichlet process prior, the second part of Assumption \\ref{ass3.1} \\textit{(i)} was proved in Ghosh and Ramamoorthi (2003, Theorem 3.2.7) while the condition that $\\phi(F)$ is continuous in $F$ is verified in many examples relevant for applications. For instance, in Example \\ref{ex2.1}, $\\phi(F)=(E(Y_1|F),E(Y_2|F))^T$ and in Example \\ref{ex2.2}, $\\phi(F)=(E(ZY_1|F), E(ZX^T|F), E(ZY_2|F))$, which are all bounded linear functionals of $F$.\nWe refer to \\cite{GhoshRamamoorthi2003} for examples and sufficient conditions for this assumption.\n\n\\begin{assum}[Prior for $\\phi$]\\label{ass3.2} For any $\\epsilon>0$ there are  measurable sets $A_2\\subset A_1\\subset\\Phi$ such that  $0<\\pi(\\phi\\in A_i)\\leq 1$,  $i=1,2$ and \\\\\n(i)  for all $\\phi\\in A_1$, $\\Theta(\\phi_0)^{\\epsilon}\\cap \\Theta(\\phi)\\neq\\emptyset$; for all $\\phi\\notin A_1$, $\\Theta(\\phi_0)^{\\epsilon}\\cap \\Theta(\\phi)=\\emptyset$,\\\\\n(ii) for all $\\phi\\in A_2$, $\\Theta(\\phi_0)^{-\\epsilon}\\cap \\Theta(\\phi)\\neq\\emptyset$; for all $\\phi\\notin A_2$, $\\Theta(\\phi_0)^{-\\epsilon}\\cap \\Theta(\\phi)=\\emptyset$.\n\\end{assum}\n \n\n\n\n\nAssumption \\ref{ass3.2} is satisfied as long as the identified set $\\Theta(\\phi)$ is bounded and the prior of $\\phi$ spreads over a large support of the parameter space. This assumption allows us to prove the posterior consistency without assuming the prior $\\pi(\\theta|\\phi)$ to be a continuous function of $\\phi$, and therefore priors like $I_{\\phi_1<\\theta<\\phi_2}$ in the interval censoring data example are allowed. Under this assumption\nthe conditional prior probability of the $\\epsilon$-envelope of the true identified set can be approximated by a continuous function, that is, there is a sequence of bounded and continuous functions $h_m(\\phi)$ such that (see lemma \\ref{LemB.1} in the appendix) almost surely in $\\phi$:\n$$\n\\pi (\\theta\\in\\Theta(\\phi_0)^{\\epsilon}|\\phi) = \\lim_{m\\rightarrow\\infty}h_m(\\phi).\n$$\n\\noindent A similar approximation holds for the conditional prior of the $\\epsilon$-contraction $\\pi(\\theta\\in\\Theta(\\phi_0)^{-\\epsilon}|\\phi)$.\n\n\n\\begin{assum}[Prior for $\\theta$]\\label{ass3.3}\nFor any $\\epsilon>0$, and $\\phi\\in\\Phi$,  $\\pi(\\theta\\in\\Theta(\\phi)^{-\\epsilon}|\\phi)<1.$\n\\end{assum}\n\n In the special case when $\\theta$ is point identified ($\\Theta(\\phi)$ is a singleton), the $\\epsilon$-contraction  is empty and thus $\\pi(\\theta\\in\\Theta(\\phi)^{-\\epsilon}|\\phi)=0.$\n\n\nAssumption \\ref{ass3.3}  is an assumption on the prior for $\\theta$, which means the identified set should be \\textit{sharp} with respect to the prior information. Roughly speaking, the support of the prior should not be a proper subset of any $\\epsilon$-contraction of the identified set $\\Theta(\\phi)$. If otherwise the prior information restricts $\\theta$ to be inside a strict subset of $\\Theta(\\phi)$ so that Assumption \\ref{ass3.3} is violated, then that prior information should be taken into account in order to shrink $\\Theta(\\phi)$ to a sharper set. In that case, the posterior will asymptotically concentrate around a set that is smaller than the   set identified by the data alone. Remark that assumption \\ref{ass3.3} is not needed for the first part of Theorem \\ref{th3.1} below.\n\n\n\n\n The following theorem gives the posterior consistency for partially identified parameters.\n\\begin{thm}\\label{th3.1}\n \n  Under Assumptions \\ref{ass3.1} and \\ref{ass3.2}, for any $\\epsilon>0$,\n\n    \\begin{displaymath}\n      P(\\theta\\in \\Theta(\\phi_{0})^{\\epsilon}|D_n)\\rightarrow^p 1.    \\end{displaymath}\n      If Assumption \\ref{ass3.3} is further satisfied, then  there is $  \\tau \\in (0,1]$ such that\n      $$\n  P(\\theta\\in\\Theta(\\phi_0)^{-\\epsilon}|D_n)\\rightarrow^p (1 - \\tau).\n      $$\n\\end{thm}\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{Bayesian Inference of Support Function}\\label{ssf}\nOur analysis focuses on identified sets which are closed and convex. These sets are completely determined by their support functions, and efficient estimation of support functions may lead to optimality of estimation and inference of the identified set. As a result, much of the new development in the partially identified literature focuses on the support function, e.g., Kaido and Santos (2013),  Kaido (2012),  Beresteanu and Molinari (2008), Bontemps et al. (2012).\n\nThis section develops Bayesian analysis for the support function $S_{\\phi}(\\nu)$ of the identified set $\\Theta(\\phi)$. We consider a more specific partially identified model: the \\textit{moment inequality model} which is described in section \\ref{ss_5.1} below. Bayesian inference for the support function has two main interests. First, it provides an alternative way to characterize and perform estimation of the identified set $\\Theta(\\phi)$, which in many cases is relatively easy for computations and simulations. Second, it allows us to construct a two-sided BCS for $\\Theta(\\phi)$ that is also asymptotically equivalent to a frequentist confidence set. In this section we first develop a  local linearization in $\\phi$ of the support function.  As the support function itself is ``point identified'', we prove that its  posterior satisfies the Bernstein-von Mises theorem. This result is \\textit{per se} of particular interest in the nonparametric Bayesian literature.\n\n\n\n\\subsection{Moment Inequality Model}\\label{ss_5.1}\nThe \\textit{moment inequality model} assumes that $\\theta$ satisfies $k$ moment restrictions:\n\\begin{equation}\\label{eq4.1}\n\\Psi(\\theta, \\phi)\\leq 0,\\quad \\Psi(\\theta, \\phi)=(\\Psi_1(\\theta,\\phi),..., \\Psi_k(\\theta,\\phi))^T\n\\end{equation}\nwhere $\\Psi: \\Theta\\times\\Phi\\rightarrow\\mathbb{R}^k$ is a known function of $(\\theta, \\phi)$.  The identified set can be characterized as:\n    \\begin{equation}\\label{eq4.2old}\n      \\Theta(\\phi)=\\{\\theta\\in\\Theta: \\Psi(\\theta, \\phi)\\leq 0\\}.\n    \\end{equation}\nSince most of the partially identified models can be characterized as moment inequality models, model (\\ref{eq4.1})-(\\ref{eq4.2old}) has received extensive attention in the  literature.\n\nWe assume  each component of $\\Psi(\\theta,\\phi)$ to be a convex function of $\\theta$ for every $\\phi\\in\\Phi$,  as stated in the next assumption.\n\n\\begin{assum}\\label{ass5.1}\n  $\\Psi(\\theta,\\phi)$ is continuous in $(\\theta,\\phi)$ and convex in $\\theta$ for every $\\phi\\in\\Phi$.\n\\end{assum}\n\nLet us consider the support function $S_{\\phi}(\\cdot):\\mathbb{S}^{d}\\rightarrow\\mathbb{R}$ of the identified set $\\Theta(\\phi)$. We restrict its domain to the unit sphere $\\mathbb{S}^{d}$ in $\\mathbb{R}^{d}$ since $S_{\\phi}(\\nu)$ is positively homogeneous in $\\nu$. Under Assumption \\ref{ass5.1} the support function is the optimal value of an ordinary convex program:\n\\begin{displaymath}\n  S_{\\phi}(\\nu) = \\sup_{\\theta\\in\\Theta}\\{\\nu^T\\theta;\\:  \\Psi(\\theta, \\phi)\\leq 0\\}.\n\\end{displaymath}\nTherefore, it also admits a Lagrangian representation (see Rockafellar 1970, chapter 28):\n\\begin{equation}\\label{eq5.1}\n  S_{\\phi}(\\nu) = \\sup_{\\theta\\in\\Theta}\\{\\nu^T\\theta - \\lambda(\\nu,\\phi)^T\\Psi(\\theta,\\phi)\\},\n\\end{equation}\n\n\\noindent where $\\lambda(\\nu,\\phi): \\mathbb{S}^{d}\\times\\mathbb{R}^{d_{\\phi}}\\rightarrow\\mathbb{R}_{+}^{k}$ is a $k$-vector of Lagrange multipliers.\\\\% Note that $d_{\\phi}$ is the dimension of $\\phi$.\\\\\n\\indent We denote by $\\Psi_{S}(\\theta,\\phi_{0})$ the $k_{S}$-subvector of $\\Psi(\\theta,\\phi_{0})$ containing the constraints that are strictly convex functions of $\\theta$ and by $\\Psi_{L}(\\theta,\\phi_{0})$ the $k_{L}$ constraints that are linear in $\\theta$.  So $k_S + k_L = k$. The corresponding Lagrange multipliers are denoted by $\\lambda_{S}(\\nu,\\phi_{0})$ and $\\lambda_{L}(\\nu,\\phi_{0})$, respectively, for $\\nu\\in\\mathbb{S}^{d}$. Moreover, define $\\Xi(\\nu,\\phi) = \\arg\\max_{\\theta \\in\\Theta}\\{\\nu^T\\theta; \\: \\Psi(\\theta,\\phi)\\leq 0\\}$  as the \\textit{support set} of $\\Theta(\\phi)$. Then, by definition,\n$$\n\\nu^T\\theta=S_{\\phi}(\\nu),\\quad \\forall\\theta\\in\\Xi(\\nu,\\phi).\n$$\nWe also denote by $\\nabla_{\\phi}\\Psi(\\theta,\\phi)$ the $k\\times d_{\\phi}$ matrix of partial derivatives of $\\Psi$ with respect to $\\phi$, and by $\\nabla_{\\theta}\\Psi_{i}(\\theta,\\phi)$ the $d$-vector of partial derivatives of $\\Psi_{i}$ with respect to $\\theta$ for each $i\\leq k$. In addition, let\n$$Act(\\theta,\\phi)\\equiv\\{i\\leq k;\\: \\Psi_i (\\theta,\\phi)=0\\}$$\nbe the set of the inequality active constraint indices. For some $\\delta>0$, let $B(\\phi_{0}, \\delta)=\\{\\phi\\in\\Phi; \\: \\|\\phi - \\phi_{0}\\|\\leq \\delta\\}$.\n\nWe assume the following:\n\n\\begin{assum}\\label{ass5.2}\n  The true value $\\phi_{0}$ is in the interior of $\\Phi$, and $\\Theta$ is convex and compact.\n\\end{assum}\n\n\n\n\\begin{assum}\\label{ass5.3} There is some $\\delta>0$ such that for all $\\phi \\in B(\\phi_{0}, \\delta)$, we have:\\\\\n(i)  the  matrix $\\nabla_{\\phi}\\Psi(\\theta,\\phi)$ exists and is continuous in $(\\theta,\\phi)$;\\\\\n(ii) the set $\\Theta(\\phi)$ is non empty;\\\\\n(iii)  there exists a $\\theta\\in\\Theta$ such that $\\Psi(\\theta,\\phi)<0$;\\\\\n(iv) $\\Theta(\\phi)$ belongs to the interior of $\\Theta$;\\\\\n(v) for every $i\\in Act(\\theta,\\phi_{0})$, with $\\theta\\in\\Theta(\\phi_{0})$, the vector $\\nabla_{\\theta}\\Psi_{i}(\\theta,\\phi)$ exists and is continuous in $(\\theta,\\phi)$ for every $\\phi\\in B(\\phi_{0},\\delta)$ and $\\theta\\in\\Theta(\\phi)$.\n\\end{assum}\n\nAssumption \\ref{ass5.3} \\textit{(iii)} is the Slater's condition which is a sufficient condition for strong duality to hold. It implies Assumption \\ref{ass5.3} \\textit{(ii)}. However, we keep both conditions because in order to establish some technical results we only need condition \\textit{(ii)} which is weaker.\n\n\n\nThe next assumption concerns the inequality active constraints. Assumption \\ref{ass5.4} requires that the active inequality constraints gradients $\\nabla_{\\theta}\\Psi_{i}(\\theta,\\phi_{0})$ be linearly independent. This assumption guarantees that a $\\theta$ which solves the optimization problem (\\ref{eq5.1}) with $\\phi = \\phi_{0}$ satisfies the Kuhn-Tucker conditions. Alternative assumptions that are weaker than Assumption \\ref{ass5.4} could be used, but the advantage of Assumption \\ref{ass5.4} is that it is easy to check.\n\n\\begin{assum}\\label{ass5.4}\n   \n   \n   \n   For any $  \\theta\\in\\Theta(\\phi_{0})$,  the gradient vectors $\\{\\nabla_{\\theta}\\Psi_{i}(\\theta,\\phi_{0})\\}_{i\\in Act(\\theta,\\phi_{0})}$ are linearly independent\n   \n\\end{assum}\n\n The following assumption is key for our analysis, and is sufficient for the differentiability of the support function at $\\phi_0$:\n\\begin{assum}\\label{ass5.6}\nAt least one of the following holds:\n    \\begin{itemize}\n      \\item[\\textit{(i)}] For the ball $B(\\phi_0,\\delta)$ in Assumption \\ref{ass5.3},  for every $(\\nu,\\phi)\\in\\mathbb{S}^{d}\\times B(\\phi_{0},\\delta)$, $\\Xi(\\nu,\\phi)$ is a singleton;\n      \\item[\\textit{(ii)}] There are linear constraints in $\\Psi(\\theta,\\phi_0)$, which are also separable in $\\theta$, that is, $k_L>0$ and $\\Psi_{L}(\\theta,\\phi_{0}) = A_{1}\\theta + A_{2}(\\phi_{0})$ for some function $A_{2}: \\Phi\\rightarrow \\mathbb{R}^{k_{L}}$ (not necessarily linear) and some $(k_L\\times d)$-matrix $A_{1}$.\n    \\end{itemize}\n\\end{assum}\n\n\n Assumption \\ref{ass5.6} is particularly important for the linearization of the support function that we develop in Section \\ref{ss_5.2}. In fact, if one of the two parts of Assumption \\ref{ass5.6} holds then the support function is differentiable at $\\phi$ for every $(\\nu,\\phi)\\in\\mathbb{S}^{d}\\times B(\\phi_{0}, \\delta)$,  and we have a closed form for its derivative. This assumption also plays one of the key roles in the  study of asymptotic efficiency by Kaido and Santos (2013).\n\nThe last set of assumptions   will be used to prove the Bernstein-von Mises theorem for $S_{\\phi}(\\cdot)$ and allows to strengthen the result of Theorem \\ref{Lem5.2} below. The first three assumptions are (local) Lipschitz equi-continuity assumptions.\n\n\\begin{assum}\\label{ass5.5}\n  For the ball $B(\\phi_{0},\\delta)$ in Assumption \\ref{ass5.3}, for some $K>0$ and $\\forall \\phi_{1},\\phi_{2}\\in B(\\phi_{0},\\delta)$:\n    \\begin{itemize}\n    \\item[(i)] $\\sup_{\\nu\\in\\mathbb{S}^d}\\|\\lambda(\\nu,\\phi_{1}) - \\lambda(\\nu,\\phi_{2})\\| \\leq K\\|\\phi_{1} - \\phi_{2}\\|$;\n    \\item[(ii)] $\\sup_{\\theta\\in\\Theta}\\|\\nabla_{\\phi}\\Psi(\\theta,\\phi_{1}) - \\nabla_{\\phi}\\Psi(\\theta,\\phi_{2})\\| \\leq K\\|\\phi_{1} - \\phi_{2}\\|$;\n    \\item[(iii)] $\\|\\nabla_{\\phi}\\Psi(\\theta_1,\\phi_{0}) - \\nabla_{\\phi}\\Psi(\\theta_2,\\phi_{0})\\| \\leq K\\|\\theta_{1} - \\theta_{2}\\|$, for every $\\theta_1, \\theta_2\\in \\Theta$;\n    \\item[(iv)] If $\\Xi(\\nu,\\phi_0)$ is a singleton for any $\\nu$ in  some compact subset $W\\subseteq \\mathbb{S}^d$, and if the correspondence $(\\nu,\\phi) \\mapsto \\Xi(\\nu,\\phi)$ is upper hemicontinous on $\\mathbb{S}^d\\times B(\\phi_0,\\delta)$ then there exists $\\varepsilon =O(\\delta)$ such that $\\Xi(\\nu,\\phi_1) \\subseteq \\Xi^{\\varepsilon}(\\nu,\\phi_0)$.\n  \\end{itemize}\n\\end{assum}\nHere $\\|\\nabla_{\\phi}\\Psi(\\theta,\\phi)\\|$ denotes the Frobenius norm of the matrix. The above conditions are not stringent.  In particular,  condition \\textit{(iv)} is easy to understand when $\\Xi(p,\\phi)$ is a singleton, that is, when the optimization problem for the support function has a unique solution, for each $\\phi\\in B(\\phi_{0},\\delta)$. Then $\\Xi(\\nu,\\phi_1)$ and $\\Xi(\\nu,\\phi_0)$ are singletons that are close to each other, and $\\Xi(\\nu,\\phi_0)^{\\varepsilon}$ is a small ball around $\\Xi(\\nu,\\phi_0)$.\n\n\n\\indent We show in the following example that Assumptions \\ref{ass5.1}-\\ref{ass5.5}  are easily satisfied.\n\n\\begin{exm}[Interval censored data - \\textit{continued}]\n  The setup is the same as in Example \\ref{ex2.1}. Assumption \\ref{ass5.2} is verified if $Y_{1}$ and $Y_{2}$ are two random variables with finite first moments $\\phi_{0,1}$ and $\\phi_{0,2}$, respectively. Moreover, $\\Psi(\\theta,\\phi) = (\\phi_{1} - \\theta, \\theta - \\phi_{2})^T$, $\\phi = (\\phi_{1},\\phi_{2})^T$,\n\\begin{displaymath}\n  \\nabla_{\\phi}\\Psi(\\theta,\\phi) = \\left(\\begin{array}\n  {cc} 1 & 0\\\\ 0 & -1\n\\end{array}\\right)\n\\end{displaymath}\nso that Assumptions \\ref{ass5.1}, \\ref{ass5.2} and \\ref{ass5.3} \\textit{(i)}-\\textit{(ii)} are trivially satisfied. Assumption \\ref{ass5.3} \\textit{(iii)} holds for every $\\theta$ inside $(\\phi_{1},\\phi_{2})$; Assumption \\ref{ass5.3} \\textit{(iv)} is satisfied if $\\phi_{1}$ and $\\phi_{2}$ are bounded. To see that Assumptions \\ref{ass5.3} \\textit{(v)} and \\ref{ass5.4} are satisfied note that $\\forall \\theta < \\phi_{0,1}$ we have $Act(\\theta,\\phi_{0}) = \\{1\\}$, $\\forall \\theta > \\phi_{0,2}$ we have $Act(\\theta,\\phi_{0}) = \\{2\\}$ while $\\forall \\theta \\in[\\phi_{0,1}, \\phi_{0,2}]$ we have $Act(\\theta,\\phi_{0}) = \\emptyset$. Assumption \\ref{ass5.6} \\textit{(i)} and \\textit{(ii)} are both satisfied since the support set takes the values $\\Xi(1,\\phi)=\\phi_2$ and $\\Xi(-1,\\phi)=-\\phi_1$ and the constraints in $\\Psi(\\theta,\\phi_{0})$ are both linear with $A_{1} = (-1,1)^T$ and $A_{2}(\\phi_{0}) = \\nabla_{\\phi}\\Psi(\\theta,\\phi_{0})\\phi_{0}$.\\\\\n\\indent  Assumptions \\ref{ass5.5} \\textit{(ii)}-\\textit{(iii)} are naturally satisfied because $\\nabla_{\\phi} \\Phi(\\theta, \\phi)$ does not depend on $(\\theta, \\phi)$.  The Lagrange multiplier is $\\lambda(\\nu,\\phi) = (-\\nu I(\\nu<0), \\nu I(\\nu\\geq 0))^{T}$ so that Assumption \\ref{ass5.5} \\textit{(i)} is satisfied since the norm is equal to $0$. Finally, the support set $\\Xi(\\nu,\\phi) = \\phi_{1}I(\\nu<0) + \\phi_{2}I(\\nu\\geq 0)$ is a singleton for every $\\phi\\in B(\\phi_{0}, \\delta)$ and $\\Xi(\\nu,\\phi_{0})^{\\varepsilon} = \\{\\theta\\in\\Theta;\\|\\theta - \\theta_{*}\\| \\leq \\varepsilon \\}$ where $\\theta_{*} = \\Xi(\\nu,\\phi_{0}) = \\phi_{0,1}I(\\nu<0) + \\phi_{0,2}I(\\nu\\geq 0)$. Therefore, $\\|\\Xi(\\nu,\\phi) - \\theta_{*}\\| \\leq \\delta$ and Assumption \\ref{ass5.5} \\textit{(iv)} holds with $\\varepsilon = \\delta$. $\\square$\n\\end{exm}\n\n\\begin{comment}\n\\begin{exm}[Interval regression model - \\textit{continued}]\n  Consider Example \\ref{ex2.2}. Assumption \\ref{ass5.2} is verified if the random vector $(Y_{1},Y_{2},X^T,Z^T)$ is such that there exists positive constants $c_1,c_2$ and $c_3$ such that $\\|E(ZY_1)\\|<c_1$, $\\|E(ZY_2)\\|<c_2$ and $\\|E(ZX^T)\\|<c_3$ where $E$ is taken with respect to the true distribution. Assume $\\dim(Z) = \\dim(X) \\equiv d$ and $rank(E(ZX^T)) = d$, then $\\Psi(\\theta,\\phi) = ([\\phi_{2}\\theta - \\phi_3]^T,[\\phi_{1} - \\phi_2\\theta]^T)^T$, $\\phi = (\\phi_{1},\\phi_{2},\\phi_3)$,\n\\begin{displaymath}\n  \\nabla_{\\phi}\\Psi(\\theta,\\phi) = \\left(\\begin{array}\n  {ccc} 0_d & (\\theta'\\otimes I_d)&-I_d\\\\I_d & -(\\theta'\\otimes I_d) & 0_d\n\\end{array}\\right)\n\\end{displaymath}\nwhere $0_d$ (resp. $I_d$) denotes a $d$-square matrix of zeros (resp. of ones). This verifies Assumptions \\ref{ass5.1} and \\ref{ass5.3}. Remark that $\\Psi(\\theta,\\phi)$ is a $2d$-vector and $\\nabla_{\\phi}\\Psi(\\theta,\\phi)$ is a $(2d\\times d(2+d))$-matrix. Assumption \\ref{ass5.5} holds with $k_S = 0$. Assumption \\ref{ass5.6} \\textit{(i)} does not hold in general for non-degenerate $p$ and for $d > 1$; assumption \\ref{ass5.6}  \\textit{(ii)} holds if we assume that $\\phi_{2}$ is a known parameter.\\\\\nUnder the assumption that $\\dim(Z) = \\dim(X) \\equiv d$ and $rank(E(ZX^T)) = d$ we are able to rewrite the identified set in an equivalent form.\n\\begin{lem}\\label{Lem5.1}\n  Suppose $\\phi_2^{-1}$ exists, then\n  $$\\Theta(\\phi)=\\left\\{\\theta\\in\\Theta: \\theta=\\phi_2^{-1}(\\frac{\\phi_1+\\phi_3}{2}+u), u\\in(-\\frac{\\phi_3-\\phi_1}{2},\\frac{\\phi_3-\\phi_1}{2})\\right\\}.$$\n\\end{lem}\n\nNow we are ready to calculate the support function for $\\Theta(\\phi)$.\n\\begin{thm}\\label{th5.1}\n  Suppose $\\phi_2^{-1}$ exists. The support function for $\\Theta(\\phi)$ is given by:\n  $$\n  S_{\\phi}(p)=p^T\\phi_2^{-1}(\\frac{\\phi_1+\\phi_3}{2})+\\alpha_p^T(\\frac{\\phi_3-\\phi_1}{2}),\n  $$\n  where $d=\\dim(\\theta)$, $\\text{sgn}(x)=I(x>0)-I(x<0)$,\n  $$\n  \\alpha_p=\\begin{pmatrix}\n    (p^T\\phi_2^{-1})_1\\text{sgn} (p^T\\phi_2^{-1})_1\\\\\n  \\vdots\\\\\n  (p^T\\phi_2^{-1})_d\\text{sgn} (p^T\\phi_2^{-1})_d\n  \\end{pmatrix}.\n  $$\n\\end{thm}\n$\\square$\n\\end{exm}\n\n\n\\end{comment}\n\n\n\n\\subsection{Asymptotic Analysis}\\label{ss_5.2}\nThe support function of the closed and convex set $\\Theta(\\phi)$\nadmits directional derivatives in $\\phi$, see e.g. \\cite{MilgromSegal2002}. Moreover, if Assumption \\ref{ass5.6} holds for a particular value $(\\nu,\\phi)$, then $S_{\\phi}(\\nu)$ is differentiable at $\\phi$ and its derivative is equal to the left and right directional derivatives. The next theorem exploits this fact  and states that the support function can be locally approximated by a linear function of $\\phi$.  \n\n\n\\begin{thm}\\label{Lem5.2}\n  If Assumptions \\ref{ass5.1}-\\ref{ass5.6} hold with $\\delta = r_{n}$ for some $r_n=o(1)$,  then there is  $N\\in\\mathbb{N}$ such that for every $n\\geq N$,  there exist: (i) a real function $f(\\phi_{1},\\phi_{2})$ defined for every $\\phi_1, \\phi_2\\in B(\\phi_{0},r_{n})$, \n  (ii) a Lagrange multiplier  function $\\lambda(\\nu,\\phi_0): \\mathbb{S}^{d}\\times\\mathbb{R}^{d_{\\phi}}\\rightarrow\\mathbb{R}_{+}^{k}$, and (iii)   a Borel measurable mapping  $\\theta_{*}(\\nu):\\mathbb{S}^{d}\\rightarrow \\Theta$  satisfying $\\theta_{*}(\\nu)\\in\\Xi(\\nu,\\phi_{0})$  for all $\\nu\\in\\mathbb{S}^{d}$,  such that for every $\\phi_1, \\phi_2\\in B(\\phi_{0},r_{n})$:\n\n  \\begin{displaymath}\n    \\sup_{\\nu\\in\\mathbb{S}^{d}}\\left|\\left(S_{\\phi_1}(\\nu) - S_{\\phi_2}(\\nu)\\right) - \\lambda(\\nu,\\phi_{0})^T\\nabla_{\\phi}\\Psi(\\theta_{*}(\\nu),\\phi_{0})[\\phi_1-\\phi_2]\\right| = f(\\phi_{1},\\phi_{2})\n  \\end{displaymath}\n\n  \\noindent and $\\frac{f(\\phi_{1},\\phi_{2})}{\\|\\phi_{1} - \\phi_{2}\\|} \\rightarrow 0$ uniformly in $\\phi_1, \\phi_2\\in B(\\phi_{0},r_{n})$ as $n\\rightarrow \\infty$.\n\\end{thm}\n\nWe remark that the functions $\\lambda$ and $\\theta_*$ do not depend on the specific choice of $\\phi_1$ and $\\phi_2$ inside $B(\\phi_0,r_n)$, but only on $\\nu$ and the true value $\\phi_0$.   The expansion can also be viewed as  stochastic when $\\phi_1, \\phi_2$ are interpreted as random variables associated with the posterior distribution $P(\\phi|D_{n})$. This interpretation  is particularly useful to understand Theorems \\ref{th5.2} and \\ref{th5.3}.\n\n\n\nWith the approximation given in the theorem we are now ready to state posterior consistency (with concentration rate) and asymptotic normality of the posterior distribution of $S_{\\phi}(\\nu)$. The posterior consistency of the support function is also based upon the posterior concentration rate for $\\phi$. In a semi-parametric Bayesian model where $\\phi$ is point identified, the posterior of $\\phi$ achieves a near-parametric concentration rate under proper prior conditions. Since our goal is to study the posterior of $S_{\\phi}(\\nu)$, we state a high-level assumption on the posterior of $\\phi$ as follows, instead of deriving it from more general conditions.\n\n\\begin{assum}\\label{ass4.1}The marginal posterior of $\\phi$ is such that, for some $C>0$,\n$$P(\\|\\phi-\\phi_0\\|\\leq Cn^{-1\/2}(\\log n)^{1\/2}|D_n)\\rightarrow^p1.$$\n\\end{assum}\n\nThis assumption is\na standard result in semi\/non-parametric Bayesian literature. If we place a nonparametric prior on $F$ as described in Section \\ref{sss_nonparametric_prior}, the notation used in this assumption is a shorthand for $$P(\\|\\phi(F)-\\phi(F_0)\\|\\leq Cn^{-1\/2}(\\log n)^{1\/2}|D_n)\\rightarrow^p1.$$ When the likelihood function is unknown, a formal derivation of this assumption for a semi-parametric prior of $(\\phi, F)$ will be presented in Appendix \\ref{s_appendix_posterior_concentration_phi}.\n\n\n\nThe next theorem gives the contraction rate for the posterior of the support function.\n\n\n\\begin{thm}\\label{th5.2}\n  Under Assumption \\ref{ass4.1} and the Assumptions of Theorem \\ref{Lem5.2} with $r_n=\\sqrt{(\\log n)\/n}$, for some $C>0$,\n    \\begin{equation}\\label{eq5.2}\n      P\\left(\\left.\\sup_{\\nu\\in\\mathbb{S}^{d}}|S_{\\phi}(\\nu) - S_{\\phi_{0}}(\\nu)|<C(\\log n)^{1\/2}n^{-1\/2} \\right|D_{n}\\right) \\rightarrow^{p} 1.\n    \\end{equation}\n\\end{thm}\n\n\\begin{remark}\nThe above result holds for both nonparametric and semi-parametric prior on $(\\phi,F)$. The concentration rate, as given in the theorem, is nearly  parametric: $\\sqrt{\\frac{\\log n}{n}}$ and is the same as the rate in assumption \\ref{ass4.1}. Thus, when the posterior for $\\phi$ contracts at the rate $n^{-1\/2}$,  the same holds for the posterior of the support function. The posterior probability in the theorem is now converging to zero, instead of being smaller than an arbitrarily small constant. This often gives rise to the term $\\sqrt{\\log n}$, which  arises commonly in the posterior concentration rate literature (e.g., Ghosal et al. 2000, Shen and Wasserman 2001). The same rate of convergence in the frequentist perspective has been achieved by Chernozhukov et al. (2007),  Beresteanu and Molinari  (2008),  Kaido and Santos (2013), among others, when estimating the identified set.\n\\end{remark}\n\n\n\nWe now state a Bernstein-von Mises (BvM) theorem for the support function. This theorem is valid under the assumption that a BvM  theorem holds for the posterior distribution of the   identified parameter $\\phi$. We denote by $\\|\\cdot\\|_{TV}$ the total variation distance, that is, for two probability measures $P$ and $Q$, $$\\|P-Q\\|_{TV} \\equiv \\sup_{B}|P(B) - Q(B)|$$ where $B$ is an element of the $\\sigma$-algebra on which $P$ and $Q$ are defined.\n\n\\begin{assum}\\label{ass5.9}\n  Let $P_{\\sqrt{n}(\\phi - \\phi_{0})|D_{n}}$ denote the posterior distribution of $\\sqrt{n}(\\phi - \\phi_{0})$. We assume\n  %\n    \\begin{displaymath}\n      \\|P_{\\sqrt{n}(\\phi - \\phi_{0})|D_{n}} - \\mathcal{N}({\\Delta}_{n,\\phi_{0}}, I_0^{-1})\\|_{TV} \\rightarrow^p 0\n    \\end{displaymath}\n\n  \\noindent where $\\mathcal{N}$ denotes the $d_{\\phi}$-dimensional normal distribution, ${\\Delta}_{n,\\phi_{0}}\\equiv n^{-1\/2}\\sum_{i=1}^{n}I_0^{-1}l_{\\phi_{0}}(X_{i})$, $l_{\\phi_{0}}$ is the semi-parametric efficient score function of the model and $I_0$ denotes the semi-parametric efficient  information matrix.\n\\end{assum}\n\nAs we focus on the partial identification features, we state the above assumption as a high-level condition instead of proving it.  We refer to Bickel and Kleijn (2012) and  Rivoirard and Rousseau (2012) for primitive conditions of  this assumption in semi-parametric models. Remark that $l_{\\phi_{0}}$ and $I_0$ also depend  on the true  likelihood function. The \\textit{semi-parametric efficient score function} and the \\textit{semi-parametric efficient information} contribute to the stochastic local asymptotic normality (LAN, Le Cam 1986) expansion of the integrated likelihood, which is necessary in order to get the BvM result in Assumption \\ref{ass5.9}. A precise definition of $l_{\\phi_{0}}$ and $I_0$ may be found in \\cite{vanderVaart2002} (Definition 2.15). In particular, they become the usual score function (first derivative of the likelihood) and Fisher's information matrix when the true likelihood is fully parametric.\n\nBelow we denote $P_{\\sqrt{n}(S_{\\phi}(\\nu) - S_{\\phi_{0}}(\\nu))|D_{n}} $ as the posterior distribution of $\\sqrt{n}(S_{\\phi}(\\nu) - S_{\\phi_{0}}(\\nu)).$\n\\begin{thm}\\label{th5.3}\n  Let Assumption \\ref{ass5.9} hold. If the assumptions of Theorem \\ref{th5.2} and Assumption \\ref{ass5.5} hold with $\\delta = r_{n} = \\sqrt{(\\log n)\/n}$, then for any $\\nu\\in\\mathbb{S}^{d}$,\n  %\n    \\begin{displaymath}\n      \\|P_{\\sqrt{n}(S_{\\phi}(\\nu) - S_{\\phi_{0}}(\\nu))|D_{n}} - \\mathcal{N}(\\bar{\\Delta}_{n,\\phi_{0}}(\\nu), \\bar{I}_{0}^{-1}(\\nu))\\|_{TV} \\rightarrow^p 0,\n    \\end{displaymath}\n  \\noindent where $\\bar{\\Delta}_{n,\\phi_{0}}(\\nu)= \\lambda(\\nu,\\phi_{0})^T\\nabla_{\\phi}\\Psi(\\theta_{*}(\\nu),\\phi_{0}){\\Delta}_{n,\\phi_{0}}$ and $$\\bar{I}_{0}^{-1}(\\nu) = \\lambda(\\nu,\\phi_{0})^T\\nabla_{\\phi}\\Psi(\\theta_{*}(\\nu),\\phi_{0}) I_0^{-1} \\nabla_{\\phi}\\Psi(\\theta_{*}(\\nu),\\phi_{0})^{T}\\lambda(\\nu,\\phi_{0}).$$\n\\end{thm}\n\n The asymptotic mean and covariance matrix can be easily estimated by replacing $\\phi_{0}$ by any consistent estimator $\\hat{\\phi}$. Thus, $\\theta_{*}(\\nu)$ will be replaced by an element $\\hat{\\theta}_{*}(\\nu)\\in\\Xi(\\nu,\\hat{\\phi})$ and an estimate of $\\lambda(\\nu,\\phi_{0})$ will be obtained by numerically solving the ordinary convex program in (\\ref{eq5.1}) with $\\phi_{0}$ replaced by $\\hat{\\phi}$.\n\n\\begin{remark} The posterior asymptotic variance of the support function $\\bar{I}_{0}^{-1}$ is the same as that of the frequentist estimator obtained by Kaido and Santos (2013, Theorem 3.2). Both are derived based on a linear expansion of the support function.  This implies that the  Bayesian estimation of the  support function   is also asymptotically semi-parametric efficient in the frequentist sense. On the other hand,  there is also a major difference between our results and theirs because when studying the posterior distributions, we do not have an empirical process as Kaido and Santos (2013) do. This requires us to develop a different strategy to prove the linear expansion given in Theorem \\ref{Lem5.2} as well as the asymptotic normalities given in Theorems  \\ref{th5.3} and \\ref{t5.4} below.  This also achieves a more strengthened result because the expansion   in Theorem \\ref{Lem5.2} is  uniformly valid in a neighborhood of $\\phi_0$.  \n\n\n \n\n\\end{remark}\n\n\n\\begin{remark}\nThe support function $S_{\\phi}(\\cdot)$ is a stochastic process with realizations in   $\\mathcal{C}(\\mathbb{S}^{d})$, the space of bounded continuous functions on $\\mathbb{S}^{d}$.  Despite of the pointwise convergence in Theorem  \\ref{th5.3} for each fixed $\\nu$, however, the posterior distribution of the process $\\sqrt{n}(S_{\\phi}(\\cdot) - S_{\\phi_{0}}(\\cdot))$ does not converge to a Gaussian measure on  $\\mathcal{C}(\\mathbb{S}^{d})$ in the total variation distance. Roughly speaking, the convergence in total variation would require the existence of a Gaussian measure $\\mathbb{G}(\\cdot)$ on $\\mathcal{C}(\\mathbb{S}^{d})$ such that uniformly in all Borel measurable sets $B$ of $\\mathcal{C}(\\mathbb{S}^{d})$,\n  \\begin{equation}\\label{e5.3}\n    |P_{\\sqrt{n}(S_{\\phi}(\\cdot) - S_{\\phi_{0}}(\\cdot))|D_{n}}(B) - \\mathbb{G}(B)| \\rightarrow^{p} 0,\n  \\end{equation}\nwhere $P_{\\sqrt{n}(S_{\\phi}(\\cdot) - S_{\\phi_{0}}(\\cdot))|D_{n}}$ denotes the posterior distribution of the centered support function. However, in general (\\ref{e5.3}) does not hold   uniformly in all the Borel sets $B$. Such a negative result can be made rigorous, and is generally known, see e.g., Freedman (1999) or Leahu (2011).\n$\\square$\n \\end{remark}\n\nOn the positive side,  a \\emph{weak} Bernstein-von Mises theorem holds  with respect to the weak topology. More precisely, let $\\mathbb{G}$ be a Gaussian measure on $\\mathcal{C}(\\mathbb{S}^{d})$ with mean function $\\bar{\\Delta}_{n,\\phi_{0}}(\\cdot)= \\lambda(\\cdot,\\phi_{0})^T\\nabla_{\\phi}\\Psi(\\theta_{*}(\\cdot),\\phi_{0}){\\Delta}_{n,\\phi_{0}}$ and covariance operator with kernel\n  $$\\bar{I}_{0}^{-1}(\\nu_{1},\\nu_{2}) = \\lambda(\\nu_{1},\\phi_{0})^T\\nabla_{\\phi}\\Psi(\\theta_{*}(\\nu_{1}),\\phi_{0}) I_0^{-1} \\nabla_{\\phi}\\Psi(\\theta_{*}(\\nu_{2}),\\phi_{0})^{T}\\lambda(\\nu_{2},\\phi_{0}),\\qquad \\forall \\nu_{1},\\nu_{2}\\in\\mathbb{S}^{d}.$$  We then have the following theorem. For a set $B$,  denote by $\\partial B$ the boundary set of $B$.\n\n  \\begin{thm}\\label{t5.4} Let $\\mathcal{B}$ be the class of Borel measurable sets in $\\mathcal{C}(\\mathbb{S}^{d})$ such that $\\mathbb{G}(\\partial B)=0$.   Under the assumptions of Theorem \\ref{th5.3},\n    \\begin{equation}\\label{eq_weak_convergence_posterior_equivalent}\n      \\sup_{B\\in\\mathcal{B}}\\left|P_{\\sqrt{n}(S_{\\phi}(\\cdot) - S_{\\phi_{0}}(\\cdot))|D_{n}}(B) - \\mathbb{G}(B)\\right| \\rightarrow^{p} 0.\n    \\end{equation}\n    Let  `$\\Rightarrow$' denote weak convergence on the class of probability measures on $\\mathcal{C}(\\mathbb{S}^{d}).$\n    Then equivalently,\n        \\begin{equation}\\label{eq_weak_convergence_posterior}\n      P_{\\sqrt{n}(S_{\\phi}(\\cdot) - S_{\\phi_{0}}(\\cdot))|D_{n}} \\Rightarrow \\mathbb{G}(\\cdot).\n    \\end{equation}\n  \\end{thm}\n\n\n\n\n\\subsection{Models with moment equalities}\\label{ss_5.3}\nOur  analysis carries over when the model contains both moment equalities and inequalities if the moment equality functions are affine functions. This case is more general than the previous one. Suppose that the identified set writes as\n\n    \\begin{eqnarray}\\label{eq_5.3}\n      \\Theta(\\phi) = \\{\\theta\\in\\Theta; && \\Psi_{i}(\\theta,\\phi)\\leq 0,\\: i=1,\\ldots,k_1 \\quad\\textrm{and }\\nonumber\\\\\n      && a_{i}^{T}\\theta + b_{i}(\\phi) = 0, \\: i = k_1 + 1,\\ldots, k_1 + k_2\\}\n    \\end{eqnarray}\n\n\\noindent where $a_{i}$ is a $(d\\times 1)$-vector and $b_{i}$ is a continuous real-valued function of $\\phi$ for all $i$. Let $k_1$ denote the number of moment inequalities, $k_2$ denote the number of moment equalities, and  $k=k_1 + k_2$. We then define $\\Psi(\\theta,\\phi)$ as the $(k\\times 1)$ vector whose first $k_1$ components are the functions $\\Psi_{i}(\\theta,\\phi)$, $i=1,\\ldots,k_1$ and the last $k_2$ components are the functions $a_{i}^{T}\\theta + b_{i}(\\phi)$, $i=k_1 + 1, \\ldots, k_1 + k_2$. \n\n  The set $\\Theta(\\phi)$ is closed and convex (with an empty interior) since it is the intersection of a closed and convex set with closed hyperplanes. In this case, the support function   still has a  Lagrangian representation  as:\n    \\begin{displaymath}\n      S_{\\phi}(\\nu) = \\sup_{\\theta\\in\\Theta}\\{\\nu^{T}\\theta - \\lambda(\\nu,\\phi)^{T}\\Psi(\\theta,\\phi)\\},\n    \\end{displaymath}\n\n\\noindent where $\\lambda(\\nu,\\phi):\\mathbb{S}^{d}\\times\\mathbb{R}^{d_{\\phi}}\\rightarrow \\mathbb{R}_{+}^{k_1}\\times\\mathbb{R}^{k_2}$ is a $k$-vector of Lagrange multipliers (see Rockafellar 1970, chapter 28).\nAssumptions \\ref{ass5.1}-\\ref{ass5.6} remain unchanged except for Assumption \\ref{ass5.3} \\textit{(iii)}, which is replaced by:\n\n    \\setcounter{assum}{2}\n    \\begin{assum}  \n      \\textit{(iii)} There is some $\\delta >0$ such that for all $\\phi\\in B(\\phi_{0},\\delta)$ there exists a $\\theta\\in\\Theta$ such that $\\Psi_{i}(\\theta,\\phi)<0$, $\\forall i=1,\\ldots, k_1$.\n    \\end{assum}\n\nThe results of Section \\ref{ss_5.2} are still valid with minor modifications in the proofs. We detail these modifications in Appendix \\ref{ss_E.4}.\n\n\n\\section{Bayesian Credible Sets }\\label{sbcs}\nInferences can be carried out through finite-sample Bayesian credible sets (BCS's).   We study  two kinds of BCS's: credible sets for $\\theta$ and credible sets for the identified set $\\Theta(\\phi)$.\n\n\n\n\\subsection{Credible set for $\\Theta(\\phi)$}\n\n\\subsubsection{Two-sided  BCS}\nWe focus on the case when the identified set is convex and closed, and aim at constructing two-sided credible sets $A_1$ and $A_2$ such that\n$$\nP(A_1\\subset\\Theta(\\phi)\\subset A_2|D_n)\\geq 1-\\tau\n$$\n for $\\tau\\in (0,1)$, where the probability is taken with respect to the posterior of $\\phi$.\nOur construction is based on the support function. To illustrate why support function can help, for a set $\\Theta(\\phi)$ recall its $\\epsilon$-envelope: $\\Theta(\\phi)^{\\epsilon}=\\{\\theta\\in\\Theta: d(\\theta, \\Theta(\\phi))\\leq\\epsilon\\}$ and its $\\epsilon$-contraction: $\\Theta(\\phi)^{-\\epsilon}=\\{\\theta\\in\\Theta(\\phi): d(\\theta, \\Theta\\backslash\\Theta(\\phi))\\geq\\epsilon\\}$ where $\\epsilon\\geq 0$. \nLet $\\hat{\\phi}$ be a Bayesian estimator for $\\phi_0$,   which can be, e.g., the posterior mean or mode.      We have, for any $c_n\\geq0$,\n$$\nP(\\Theta(\\hat\\phi )^{-c_n}\\subset\\Theta(\\phi)\\subset\\Theta(\\hat\\phi )^{c_n}|D_n)=P(\\sup_{\\|\\nu\\|=1}|S_{\\phi}(\\nu)-S_{\\hat\\phi}(\\nu)|\\leq c_n|D_n).\n$$\nNote that the right hand side of the above equation depends on the posterior of the support function.  Let $q_{\\tau}$ be the $1-\\tau$ quantile of the posterior of $$J(\\phi)=\\sqrt{n}\\sup_{\\|\\nu\\|=1}|S_{\\phi}(\\nu)-S_{\\hat\\phi }(\\nu)|$$ so that\n\\begin{equation}\\label{eq6.2}\nP\\left(J(\\phi)\\leq q_{\\tau}\\bigg|D_n\\right)=1-\\tau.\n\\end{equation}\nThe posterior of $J(\\phi)$ is determined by that of $\\phi$. Hence $q_{\\tau}$ can be simulated efficiently from the MCMC draws from $p(\\phi|D_n)$. Immediately, we have the following theorem:\n\n\\begin{thm}\\label{th6.2}Suppose  for any $\\tau\\in(0,1),$  $q_{\\tau}$ is defined as  in (\\ref{eq6.2}),  then for every sampling sequence $D_n,$\n$$\nP(\\Theta(\\hat\\phi )^{-q_{\\tau}\/\\sqrt{n}}\\subset\\Theta(\\phi)\\subset\\Theta(\\hat\\phi )^{q_{\\tau}\/\\sqrt{n}}|D_n)= 1-\\tau.\n$$\nIn particular, $\\Theta(\\hat\\phi )^{-q_{\\tau}\/\\sqrt{n}}$ is allowed to be an empty set.\n\\end{thm}\n\n\n\n\\begin{remark}\\label{rem_6.3}\nIt is straightforward to construct the one-sided BCS for $\\Theta(\\phi)$ using the described procedure. For example, let $\\tilde{q}_{\\tau}$   be such that $$P( \\sqrt{n}\\sup_{\\|\\nu\\|=1}(S_{\\phi}(\\nu)-S_{\\hat\\phi }(\\nu))\\leq \\tilde{q}_{\\tau}|D_n) = 1 - \\tau.$$  Then, $P(\\Theta(\\phi)\\subset \\Theta(\\hat\\phi )^{\\tilde{q}_\\tau\/\\sqrt{n}}|D_n)=1-\\tau$ for every sampling sequence $D_n.$\n\\end{remark}\n\n\n\n\n\n\n\n\n\\subsubsection{Frequentist coverage probability of BCS for $\\Theta(\\phi)$}\n\nThe constructed two-sided BCS for the identified set has desired frequentist properties, which follows from the Bernstein-von Mises Theorem   (see Theorem \\ref{th5.3}) of the support function.\nThe frequentist coverage probability for a  general (two-sided) multi-dimensional BCS has been largely unknown in the literature before.\nThe analysis relies on the following assumption, which requires the asymptotic normality and semi-parametric efficiency of the consistent estimator $\\hat \\phi$. Under  mild conditions, it  holds for many regular estimators such as the posterior mean, mode and the maximum likelihood estimator.\n\n\n\\begin{assum}\\label{ass6.2}\nThe consistent estimator $\\hat\\phi $ satisfies $$\n\\sqrt{n}(\\hat\\phi -\\phi_0)\\rightarrow^d \\mathcal{N}(0, I_0^{-1})\n$$\nwhere $I_0$ denotes the semi-parametric efficient information matrix as in Assumption \\ref{ass5.9}.\n\\end{assum}\n\n\n\n\n\n\n\\begin{thm} \\label{th6.3} Consider the moment inequality model in (\\ref{eq4.1})-(\\ref{eq4.2old}). If the Assumptions of Theorem \\ref{th5.3} and Assumption \\ref{ass6.2} hold,  then the constructed two-sided Bayesian credible set has asymptotically correct frequentist coverage probability,  that is, for any $\\tau\\in(0,1)$,\n$$\nP_{D_n}(\\Theta(\\hat\\phi )^{-q_{\\tau}\/\\sqrt{n}}\\subset\\Theta(\\phi_0)\\subset\\Theta(\\hat\\phi )^{q_{\\tau}\/\\sqrt{n}})\\geq 1-\\tau+o_p(1).\\footnote{The result presented here is understood as: There is a random sequence $\\Delta(D_n)$ that depends on $D_n$  such that $\\Delta(D_n)=o_p(1)$, and for any sampling sequence $D_n$, we have $P_{D_n}(\\Theta(\\hat\\phi )^{-q_{\\tau}\/\\sqrt{n}}\\subset\\Theta(\\phi_0)\\subset\\Theta(\\hat\\phi )^{q_{\\tau}\/\\sqrt{n}})\\geq 1-\\tau+\\Delta(D_n)$. Similar interpretation applies to (\\ref{eq6.3add}).}\n$$\nwhere $P_{D_n}(.)$ denote the probability measure based on the sampling distribution, fixing $(\\theta, \\phi)=(\\theta_0,\\phi_0)$.\n\\end{thm}\n\n\n\nNote that in   Theorem \\ref{th6.2},   the random set is $\\Theta(\\phi)$, while in Theorem \\ref{th6.3} the random sets are  $\\Theta(\\hat\\phi )^{-q_{\\tau}\/\\sqrt{n}}$ and $\\Theta(\\hat\\phi )^{q_{\\tau}\/\\sqrt{n}}$. \n The rationale of this theorem is that, because the identified set itself is ``point identified\", its prior does not depend on that of $\\theta$ and is   dominated by the data asymptotically.\n  \n\\begin{remark}\nNote that $q_{\\tau}$ depends only on the posterior of $\\phi$.  Hence Theorem \\ref{th6.3} does not rely on the  prior of $\\theta$, and shows asymptotic robustness to the prior of $\\phi$. It    also holds when $\\Theta(\\phi)$ becomes a singleton, and in that case the lower-side $\\Theta(\\hat\\phi )^{-q_{\\tau}\/\\sqrt{n}}$ is empty. Therefore the point identified case is also nested. We shall discuss the uniformity issue in more detail in Section \\ref{s_further_illustration_Uniformity}.\n\\end{remark}\n\n \n\n\n\nSimilarly, we can show that the one-sided BCS as constructed in Remark \\ref{rem_6.3} above has asymptotically correct coverage probability too.  For example, for $\\tilde{q}_{\\tau}$  such that \\\\ $P(\\sqrt{n}\\sup_{\\|\\nu\\|=1}(S_{\\phi}(\\nu)-S_{\\hat\\phi }(\\nu))\\leq \\tilde{q}_{\\tau}|D_n)=1-\\tau$,  then\n\\begin{equation}\\label{eq6.3add}\nP_{D_n}(\\Theta(\\phi_0)\\subset\\Theta(\\hat\\phi )^{\\tilde{q}_{\\tau}\/\\sqrt{n}})\\geq 1-\\tau+o_p(1).\n\\end{equation}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Credible set for $\\theta$}\nWe now construct the Bayesian credible set for $\\theta$. A BCS for $\\theta$ at level $1-\\tau$ is a set BCS$(\\tau)$ such that\n\\begin{equation*}\\label{eq6.1}\nP(\\theta\\in \\text{BCS}(\\tau)|D_n)=1-\\tau\n\\end{equation*}\nfor $\\tau\\in(0,1)$.  One of the popular choices of the credible set is the highest-probability-density (HPD) set, which has been widely used in empirical studies and also used in the Bayesian partially identified literature by e.g., Moon and Schorfheide (2012) and Norets and Tang (2012).\n\nThe BCS  can be compared with the frequentist confidence set (FCS).  A frequentist confidence set FCS$(\\tau)$ for $\\theta_0$  satisfies\n$$\n\\lim_{n\\rightarrow\\infty}\\inf_{\\phi\\in\\Phi}\\inf_{\\theta_{0}\\in\\Theta(\\phi)}P_{D_n}(\\theta_{0}\\in \\text{FCS}(\\tau))\\geq 1-\\tau.\n$$\nThere have been various procedures in the literature to construct a FCS$(\\tau)$ that satisfies the above inequality. One of the important  FCS's  is   based on a consistent estimator $\\hat{\\phi}$ of $\\phi_0$ such that $\\Theta(\\hat{\\phi})\\subset\\text{FCS}({\\tau})$. By using a known likelihood function, Moon and Schorfheide (2012) compared the BCS with this type of FCS and showed that the BCS and FCS are asymptotically different. As Theorem \\ref{t6.1} below shows, such a comparison still carries over under the more robust semi-parametric Bayesian setup.  The following assumption is needed.\n\\begin{assum}\\label{ass6.1}\n(i) The frequentist FCS($\\tau$) is such that, there is $\\hat{\\phi}$ with $\\|\\hat{\\phi}-\\phi_0\\|=o_p(1)$ satisfying $\\Theta(\\hat{\\phi})\\subset \\text{FCS}(\\tau)$.\\\\\n(ii) $\\pi(\\theta\\in\\Theta(\\phi)|\\phi)=1$ for all $\\phi\\in\\Phi$; $\\sup_{(\\theta,\\phi)\\in\\Theta\\times\\Phi}\\pi(\\theta|\\phi)<\\infty$.\n\\end{assum}\n Many frequentist FCS's satisfy condition \\textit{(i)}, see, e.g., Imbens and Manski (2004), Chernozhukov et al. (2007), Rosen (2008), Andrews and Soares (2010), etc. Condition \\textit{(ii)} is easy to verify since $\\Theta\\times\\Phi$ is compact. When for every $\\phi\\in\\Phi$, $\\Theta(\\phi)$ is not a singleton, examples of $\\pi(\\theta|\\phi)$ satisfying assumption \\ref{ass6.1} \\textit{(ii)} include: the uniform prior with density $$\\pi(\\theta|\\phi)=\\mu(\\Theta(\\phi))^{-1}I_{\\theta\\in\\Theta(\\phi)},$$ where $\\mu(\\cdot)$ denotes the Lebesgue measure; and the truncated normal prior with density\n$$\\pi(\\theta|\\phi)=\\left[\\int_{\\Theta(\\phi)}h(x;\\lambda,\\Sigma)dx\\right]^{-1}h(\\theta; \\lambda,\\Sigma)I_{\\theta\\in\\Theta(\\phi)},$$\nwhere $h(x;\\lambda,\\Sigma)$ is the density function of a multinormal distribution $\\mathcal{N}(\\lambda,\\Sigma)$.\n\n\\begin{thm}\\label{t6.1}\nUnder Assumption \\ref{ass6.1} and  the assumptions of Theorem \\ref{th5.2},   $\\forall\\tau\\in(0,1)$,\\\\\n(i)\n$$\nP(\\theta\\in\\text{FCS}(\\tau)|D_n)\\rightarrow^p1,\n$$\n(ii) $$\nP(\\theta\\in\\text{FCS}(\\tau), \\theta\\notin\\text{BCS}(\\tau)|D_n)\\rightarrow^p\\tau.\n$$\n\\end{thm}\n\\begin{remark}\n\n\nTheorem \\ref{t6.1} \\textit{(i)} shows that the posterior probability that $\\theta$ lies inside the frequentist confidence set is arbitrarily close to one, as $n\\rightarrow\\infty$. This indicates that the  posterior will asymptotically concentrate inside the FCS. On the other hand, by  \\textit{(ii)},  there is  a non-negligible probability that  FCS is strictly larger than BCS.  The prior information on $\\theta$ still plays a non-negligible role in the posterior as the sample size increases.\n\nOur prior condition in Assumption \\ref{ass6.1} \\textit{(ii)} implies that Theorem \\ref{t6.1} only focuses on partial identification. It can be restrictive in the point identified case. Because our prior is such that $\\pi(\\theta\\in\\Theta(\\phi)|\\phi)=1$ for each $\\phi$, when $\\Theta(\\phi)$ is a singleton $\\pi(\\theta|\\phi)$ becomes a Dirac function and $\\sup_{\\theta,\\phi}\\pi(\\theta|\\phi)<\\infty$  cannot be expected to hold in this case. On the other hand, Assumption \\ref{ass6.1} \\textit{(ii)} does cover many partially identified models of interest, and it is a prior assumption that has been used frequently elsewhere in the literature, e.g., Moon and Schorfheide (2012) and Gustafson (2012).\n\\end{remark}\n\n\n\n\\section{Projection and Subset Inference}\\label{s_Projection_Subset_Inference}\nOne of the important features of the proposed   procedure is that it is relatively easy to marginalize onto low-dimensional subspaces, and the computation is fast. Suppose the dimension of $\\theta$ is relatively large, but we are interested in only one component of $\\theta$, say $\\theta_1$. Then projections aim at constructing the BCS's for $\\theta_1$ and for its identified set  $\\widetilde{\\Theta}(\\phi)_1$.\n\nWe illustrate this by using the interval regression example (Example \\ref{ex2.2}). Suppose the full parameter $\\theta$ is high-dimensional. Let $W_1=ZY_1$, $W_2=ZY_2$ and $V=Zx^T$. Here $\\phi_1=EW_1$,  $\\phi_2=EV$ and $\\phi_3=EW_2$. Let $\\phi=(\\phi_1^T,vec(\\phi_2)^T,\\phi_3^T)$, and $e=(1,0,...,0)^T$. The identified set for $\\theta_1$ can be expressed using the support function $S_{\\phi}(\\cdot)$:\n$$\n\\widetilde{\\Theta}(\\phi)_1=\\{\\theta_1: \\exists  \\omega=(\\theta_2,...,\\theta_{d})\\text{ such that } (\\theta_1,\\omega)\\in\\Theta(\\phi)\\}=[-S_{\\phi}(-e), S_{\\phi}(e)],\n$$\nwhere the exact expression for $S_{\\phi}(\\cdot)$ is given in Appendix C.1 in the supplementary material. We place a Dirichlet process prior $ \\mathcal{D}ir(\\nu_{0},Q_{0})$ on the joint CDF of $(W_1, W_2, V)$. By the stick-breaking representation (see Sethuraman 1994) the deduced   posterior distribution of $\\phi$ is the distribution of the following quantity:\n\\begin{equation}\\label{e6.4}\n       \\phi|D_{n}= \\rho\\sum_{i=1}^{n}\\beta_{i}D_{n,i} + (1 - \\rho)\\sum_{j=1}^{\\infty}\\alpha_{j}\\xi_{j}\n\\end{equation}\n where $D_{n,i}$ is the $i$th observation of the vector $(W_1^T, vec(V)^T, W_2^T)$, $\\rho$ is drawn from a Beta distribution $\\mathcal{B}e(n, \\nu_{0})$ independently of the other quantities, $(\\beta_{1},\\ldots,\\beta_{n})$ is drawn from a Dirichlet distribution of parameters $(1,\\ldots,1)$ on the simplex $S_{n-1}$ of dimension $(n-1)$, $\\xi_{j}\\sim \\:iid\\: Q_{0}$ and $\\{\\alpha_{k}\\}_{k\\geq 1}$ are computed as $\\alpha_{k} = v_{k}\\prod_{l=1}^{k}(1 - v_{l})$ where $\\{v_{l}\\}_{l\\geq 1}$ are independent drawings from a beta distribution $\\mathcal{B}e(1,\\nu_{0})$ and $\\{v_{j}\\}_{j\\geq1}$ are independent of $\\{\\xi_{j}\\}_{j\\geq 1}$. In practice, we can set a  truncation $K$, so the infinite sum in the posterior representation in (\\ref{e6.4}) is replaced with a truncated sum $(1-\\rho)\\sum_{j=1}^K\\alpha_j\\xi_j.$ In addition, $(\\alpha_1,...\\alpha_K)$ are normalized so that $\\sum_{j=1}^K\\alpha_j=1.$\n\n We can place a uniform prior for $\\theta$, and draw $\\{\\theta^{(i)},\\phi^{(i)}\\}_{i=1}^B$ from the posterior  $(\\theta, \\phi)|D_n$. Then $\\{\\theta_1^{(i)}\\}_{i=1}^B$ are the draws from the marginal  posterior of $\\theta_1.$  Let $\\theta_1^{(\\tau\/2)}$ and $\\theta_1^{(1-\\tau\/2)}$ be the $\\tau\/2$th and  ${(1-\\tau\/2)}$th sample quantiles of $\\{\\theta_1^{(i)}\\}_{i=1}^B$. Then $[\\theta_1^{(\\tau\/2)}, \\theta_1^{(1-\\tau\/2)}]$ is the BCS($\\tau$) of $\\theta_1$. Moreover, let $q_{\\tau}$ be the $(1-\\tau)$th quantile of the posterior of\n$$\nJ(\\phi)=\\sqrt{n}\\max\\{S_{\\phi}(e)-S_{\\hat\\phi }(e), S_{\\phi}(-e)-S_{\\hat\\phi }(-e)\\},\n$$\nwhich can be approximated by the $(1-\\tau)$th sample quantile of $\\{J(\\phi^{(i)})\\}_{i=1}^B$.\nWe then construct the BCS$(\\tau)$ for $\\widetilde{\\Theta}(\\phi)_1$ as\n$\n[-S_{\\hat\\phi }(-e)-\\frac{q_{\\tau}}{\\sqrt{n}},  S_{\\hat\\phi }(e)+\\frac{q_{\\tau}}{\\sqrt{n}}].\n$\n\n\n\nWe present a simple numerical result for illustration, where $\\theta_{01}=1$, but the total dimension is high: $\\dim(\\theta_0)=10.$ Let $W_1\\sim \\mathcal{N}(0,0.5I)$ and $W_2\\sim \\mathcal{N}(5, I)$.  Set $\\nu_0=3$ and the base measure $Q_0=\\mathcal{N}(0, I)$.  $B=100$ posterior draws are sampled.  While the finite sample performance is very robust to the choice of the truncation $K$,  we choose $K$ following the guidance of Ishwaran and James (2002), who obtained an approximation error of order $n\\exp(-(K-1)\/\\nu_0)$ for truncations. Hence, in the simulation with $n=500, \\nu_0=3$, the choice $K=50$ gives  an error of order $4\\times 10^{-5}$. Table \\ref{table1} summarizes the true identified set $\\widetilde\\Theta(\\phi_0)_1$ for $\\theta_1$, and the averaged BCS$(0.1)$ for both $\\theta_1$ and the projected set $\\widetilde{\\Theta}(\\phi)_1$ over 50 replications. Results based on various choices of $(n, B, K)$ are reported.\n\n\n\n\\begin{table}[htdp]\n\\caption{90\\% Bayesian credible sets  marginalized to the subset for $\\theta_1$}\n\\begin{center}\n\n\\begin{tabular}{cccccc}\n\\hline\n$n$ & $K$ & $B$ &   $\\widetilde\\Theta(\\phi_0)_1$& BCS for $\\theta_1$ & BCS for $\\widetilde{\\Theta}(\\phi)_1$ \\\\\n\\hline\n500 & 50 & 100 & [0,    1.667] & [0.007,    1.667] & [-0.174,    1.844] \\\\\n & 50 & 500 &  & [-0.008    1.666] & [-0.174,    1.837] \\\\\n & 100 & 100 &  & [-0.012    1.662] & [-0.181,    1.832] \\\\\n & 100 & 500 &  & [-0.000    1.667] & [-0.169,    1.840] \\\\\n \\hline\n1000 & 50 & 100 &  [0,    1.667] & [0.023,   1.641] & [-0.121,    1.789] \\\\\n & 50 & 500 &  & [0.011,    1.641] & [-0.126,  1.786] \\\\\n & 100 & 100 &  & [0.036,    1.636] & [-0.120,    1.781] \\\\\n & 100 & 500 &  & [0.025,   1.641] & [-0.121,    1.786] \\\\\n\\hline\n\\end{tabular}\n\n\n\n\\label{table1}\n  \\end{center}\n\\end{table}\n\n\n\n\nWhen computing the BCS for $\\widetilde{\\Theta}(\\phi)_1$,  it is also interesting to compare the computation time      with that of the high-dimensional projection based on the criterion function approach as in, e.g. Chernozhukov et al. (2007), Andrews and Soares (2010) etc, because they have the same asymptotic frequentist coverages as ours.  For the moment inequalities $\\Psi(\\theta,\\phi)=(\\phi_2\\theta-\\phi_3,\\phi_1-\\phi_2\\theta)\\leq0$ we employ the criterion function and construct a confidence set FCS as in Chernozhukov et al. (2007):  $$Q_n(\\theta)=\\sum_{j}\\max(\\Psi_j(\\theta,\\hat\\phi),0)^2w_j,\\quad \\text{FCS}(\\tau)=\\{\\theta: \\sqrt{n}Q_n(\\theta)\\leq c_{\\tau}\\},\n$$  with $w_j=1$ and $\\hat\\phi$ the sample mean estimator of $\\phi$. The critical value $c_{\\tau}$ is obtained via the bootstrap procedure proposed by Bugni (2010), which  requires solving a constrained optimization problem. We  use     the ``fmincon\" toolbox in Matlab for the numerical optimization\\footnote{We use the Matlab code of Bugni (2010), downloaded from the online supplement of Econometrica. The optimization is solved constrained on an estimated identified set, which involves an additional parameter $t_n$. We set $t_n=\\log(n).$}, and then project  FCS$(\\tau)$ onto the subspace for $\\theta_1$ to get the marginal confidence set FCS$_1(\\tau)$. The projection is done through the following steps: generate $\\{\\theta_j^*\\}_{j=1}^M$ uniformly from $\\Theta$. Let $\\theta_{j,1}^*$ be the first component of $\\theta_j^*$ and\n\\begin{equation}\\label{eq7.2}\nL(\\tau)=\\min\\{\\theta_{j,1}^*:  \\theta_{j}^*\\in\\text{FCS}(\\tau), j=1,...,M\\},\\quad U(\\tau)=\\max\\{\\theta_{j,1}^*:  \\theta_{j}^*\\in\\text{FCS}(\\tau), j=1,...,M\\}.\n\\end{equation}\nThen $[L(\\tau), U(\\tau)]$ forms a projected frequentist confidence interval for $\\widetilde\\Theta(\\phi_0)_1.$  In the simulation, we set a small parameter space $\\Theta=\\otimes_{i=1}^{10}[-2, 2]$ in order to calculate $L(\\tau)$ and $U(\\tau)$ efficiently.\n\nTable  \\ref{table2add} compares the computation times necessary to obtain the projected sets using our proposed BCS and using the criterion function approach. Reported is the averaged time for one computation  over 50 replications, using the same simulated model.  We see that the proposed BCS projection computes much faster.\n\n\n\\begin{table}[htdp]\n\\caption{Computation times (in seconds) for the projected BCS and criterion-function-FCS }\n\\begin{center}\n\n\\begin{tabular}{cc|ccc|ccc}\n\\hline\n &  &  & BCS   &  & & FCS& \\\\\n &  &  & $K$        &  & &$M$ \\\\\n$n$ & $B$ & 50 & 100 & 500 & 30 & 50 & 100 \\\\\n\\hline\n &  &  &         &  & & \\\\\n500 & 50 & 0.065 & 0.073 & 0.129 & 9.169 & 10.214 & 10.955 \\\\\n & 100 & 0.128 & 0.142 & 0.248 & 18.963 & 18.893 & 19.496 \\\\\n & 200 & 0.244 & 0.273 & 0.479 & 37.067 & 36.599 & 37.288 \\\\\n  &  &  &         &  & & \\\\\n1000 & 50 & 0.072 & 0.079 & 0.136 & 14.123 & 14.248 & 15.837 \\\\\n & 100 & 0.137 & 0.155 & 0.259 & 26 & 27.029 & 28.442 \\\\\n & 200 & 0.269 & 0.295 & 0.549 & 54.027 & 52.661 & 54.240 \\\\\n\\hline\n\\end{tabular}\n\n\\label{table2add}\n  \\end{center}\n \\small \\textit{Proposed BCS and criterion-function-based-FCS are compared. $B$ is the number of either posterior draws (for BCS) or Bootstrap draws (for FCS)   to  compute the critical values; $K$ is the truncation number to approximate the Dirichlet process posterior; $M$ is used in (\\ref{eq7.2}) for  the projected FCS. Computations are conducted using a 2.3 GHz Mac with Intel Core i7 CPU.}\n\n\\end{table}\n\n\n\n\n\n\n\n\n\n\n\\section{Posterior consistency for $\\Theta(\\phi)$}\\label{s_posterior_consistency_set}\n\n The estimation accuracy of the identified set is often measured, in the literature, by the Hausdorff distance. Specifically, for a point $ a$ and a set $A$, let\n$\nd(a,A)=\\inf_{x\\in A}\\|a-x\\|,\n$\nwhere $\\|\\cdot\\|$ denotes the Euclidean norm. The Hausdorff distance between sets $A$ and $B$ is defined as\n\\begin{equation*}\nd_H(A, B)= \\max\\left\\{\\sup_{a\\in A}d(a, B), \\sup_{b\\in B}d(b, A)\\right\\}=\\max\\left\\{\\sup_{a\\in A}\\inf_{b\\in B}\\|a-b\\|, \\sup_{b\\in B}\\inf_{a\\in A}\\|b-a\\|\\right\\}.\n\\end{equation*}\n  This section aims at deriving a rate $r_n=o(1)$ such that for some constant $C>0,$\n$$\nP(d_H(\\Theta(\\phi), \\Theta(\\phi_0))<Cr_n|D_n)\\rightarrow^p1.\n$$\nThe above result is based upon two important features: (1) the posterior concentration rate for $\\phi$ which is stated in assumption \\ref{ass4.1}, and (2) the continuity of the Hausdorff distance with respect to $\\phi$.\n\n\n\nThe continuity of $d_{H}(\\Theta(\\phi),\\Theta(\\phi_{0}))$ with respect to $\\phi$ in multi-dimensional models is hard to verify. Hence, instead of assuming the continuity directly, we place a less demanding assumption which implicitly implies the continuity but is relatively easier to verify. With this aim, we consider the \\textit{moment inequality model} described in equations (\\ref{eq4.1}) - (\\ref{eq4.2old}) and place the following assumptions.\n\n\\begin{assum}\\label{ass8.1}\nThe parameter space $\\Theta\\times\\Phi$ is compact.\n\\end{assum}\n\n\\begin{assum} \\label{ass4.3}\n$\\{\\Psi(\\theta,\\cdot): \\theta\\in\\Theta\\}$ is Lipschitz equi-continuous on $\\Phi$, that is, for some $K>0$, $\\forall \\phi_1, \\phi_2\\in \\Phi$,\n$$\n\\sup_{\\theta\\in\\Theta}\\|\\Psi(\\theta, \\phi_1)-\\Psi(\\theta, \\phi_2)\\|\\leq K\\|\\phi_1-\\phi_2\\|.\n$$\n\\end{assum}\nGiven the compactness of $\\Theta$, this assumption is satisfied by many interesting examples of  moment inequality models.\n\n\n\\begin{assum} \\label{ass4.4}\nThere exists a closed neighborhood $U(\\phi_0)$ of $\\phi_0$, such that for any $a_n=O(1)$, and any $\\phi\\in U(\\phi_0)$, there exists $C>0$ that might depend on $\\phi$, so that\n$$\n\\inf_{\\theta: d(\\theta, \\Theta(\\phi))\\geq Ca_n}\\max_{i\\leq k} \\Psi_i(\\theta,\\phi)>a_n.\n$$\n\\end{assum}\n\nIntuitively, when $\\theta$ is bounded away from $\\Theta(\\phi)$ (up to a rate $a_n$), at least one of the moment inequalities is violated, which means $\\max_{i\\leq k}\\Psi_i(\\theta,\\phi)>0$. This assumption quantifies how much $\\max_{i\\leq k}\\Psi_i(\\theta,\\phi)$ will depart from zero. This is a sufficient condition for the partial identification condition (4.5) in Chernozhukov, Hong and Tamer (2007). If we define $$\nQ(\\theta,\\phi)=\\|\\max(\\Psi(\\theta,\\phi),0)\\|= \\left[\\sum_{i=1}^k(\\max(\\Psi_i(\\theta,\\phi), 0))^2\\right]^{1\/2}\n$$\nthen $Q(\\theta,\\phi)=0$ if and only if $\\theta\\in\\Theta(\\phi)$. The partial identification condition in Chernozhukov et al. (2007, condition (4.5)) assumes that  there exists $K>0$ so that for all $\\theta$,\n\\begin{equation}\\label{eq4.2}\nQ(\\theta,\\phi)\\geq  Kd(\\theta,\\Theta(\\phi)),\n\\end{equation}\nwhich says that $Q$ should be bounded below by a number proportional to the distance of $\\theta$ from the\nidentified set if $\\theta$ is bounded away from the identified set.  Assumption \\ref{ass4.4} is a sufficient condition for (\\ref{eq4.2}).\n\\begin{exm}[Interval censored data - \\textit{continued}] In the interval censoring data example, $\\Psi(\\theta, \\phi)=(\\theta-\\phi_2, \\phi_1-\\theta)^T$ and for any $\\phi=(\\phi_1, \\phi_2)$ and $\\tilde{\\phi}=(\\tilde{\\phi}_1, \\tilde{\\phi}_2)$ we have:\n$\\|\\Psi(\\theta, \\phi)-\\Psi(\\theta, \\tilde{\\phi})\\|=\\|\\phi-\\tilde{\\phi}\\|$. This verifies Assumption \\ref{ass4.3}. Moreover,   for any $\\theta$ such that $d(\\theta,\\Theta(\\phi))\\geq a_n$, either $\\theta\\leq\\phi_1-a_n$ or $\\theta\\geq\\phi_2+a_n.$ If $\\theta\\leq \\phi_1-a_n$, then $\\Psi_2(\\theta,\\phi)=\\phi_1-\\theta\\geq a_n$; if $\\theta\\geq \\phi_2+a_n$, then $\\Psi_1(\\theta,\\phi)=\\theta-\\phi_2\\geq a_n.$ This verifies Assumption \\ref{ass4.4}. $\\square$\n\\end{exm}\n\nThe following theorem shows the concentration rate for the posterior of the identified set.\n\\begin{thm}\\label{t4.1}\nUnder Assumptions \\ref{ass4.1}, \\ref{ass8.1}-\\ref{ass4.4}, for some $C>0$,\n\\begin{equation}\\label{eq_t4.1}\n  P(d_H(\\Theta(\\phi), \\Theta(\\phi_0))>C\\sqrt{\\frac{\\log n}{n}}|D_n)\\rightarrow^p0.\n\\end{equation}\n\\end{thm}\n\\begin{remark}\nThe convergence in Hausdorff distance can be implied by that of the support function for convex and close sets (e.g., Beresteanu and Molinari 2008).  Therefore, (\\ref{eq_t4.1}) is another statement of result (\\ref{eq5.2}). However, they are obtained under different assumptions   and  (\\ref{eq_t4.1}) is obtained directly from the perspective of the posterior of the identified set.\n\\end{remark}\n\\begin{remark}\nRecently, Kitagawa (2012) obtained the posterior consistency for $\\Theta(\\phi)$ in the one-dimensional case:\n$\nP(d_H(\\Theta(\\phi), \\Theta(\\phi_0))>\\epsilon|D_n)\\rightarrow0\n$\nfor almost every sampling sequence of $D_n.$ This result was obtained for the case where  $\\Theta(\\phi)$ is a connected interval and $d_H(\\Theta(\\phi), \\Theta(\\phi_0))$ is assumed to be a continuous map of $\\phi$. In multi-dimensional cases where $\\Theta(\\phi)$ is a more general convex set, however, verifying the continuity of  $d_H(\\Theta(\\phi), \\Theta(\\phi_0))$  is much more technically involved, due to the challenge of computing the Hausdorff distance in multi-dimensional manifolds. In contrast, our Lipschitz equi-continuity condition in Assumption \\ref{ass4.3} and Assumption \\ref{ass4.4} are  much easier to verify in specific examples, as they depend on the moment conditions directly.\n\\end{remark}\n\n\\begin{comment}\n\n\\section{Set-Optimality  based on Bayesian Decision Theory}\\label{sopt}\nOptimality is an important problem for partially identified models. While there are some recent results on statistical inference (e.g., Canay 2010), the optimality  issue has not been well addressed in the literature from an estimation point of view.   Kaido and Santos (2013) used a semi-parametric efficiency  criterion of estimating the identified set based on  the Hausdorff distance.  We propose a new optimality criterion based on the Bayesian decision theory for the set estimation.  In the literature, Kitagawa (2012) proposed a ``gamma-minimax\"  analysis for the partially identified parameter, which does not possess a closed form.\n\n In the  Bayesian decision theory, using the Hausdorff distance as the loss function is considered to be too strong. Instead, we define a new loss function as follows: for any estimator $\\Omega$ of the identified set, define the loss function\n$$\nL(\\Omega, \\phi)=\\mu(\\Theta(\\phi)\\Delta\\Omega), \\quad \\Theta(\\phi)\\Delta\\Omega=(\\Theta(\\phi)\\cap\\Omega^c)\\cup(\\Theta(\\phi)^c\\cap\\Omega),\n$$\nwhere $\\mu(\\cdot)$ denotes the Lebesgue measure. Note that $\\Theta(\\phi)\\Delta\\Omega$ is the \\textit{symmetric difference} between $\\Omega$ and $\\Theta(\\phi)$. Like the Hausdorff distance, it is also  commonly used to measure the accuracy of set approximations in many literature, for instance,   density  level set estimation (e.g., Gayraud and Rousseau 2005, Rigollet and Vert 2009), convex geometry (Bianchi et al. 2012), etc.\n\nDefine  a set estimator of the identified set:\n$$\n\\widehat\\Omega=\\{x\\in\\Theta: P(x\\in\\Theta(\\phi)|D_n)\\geq 0.5\\}.\n$$\nwhere $P(x\\in\\Theta(\\phi)|D_n)$ is the probability measure taken with respect to the posterior of $\\Theta(\\phi)$, for a given $x\\in\\Theta$. The following theorem shows the optimality of $\\widehat\\Omega$ under  the posterior expected loss, defined as $E[L(\\Omega, \\phi)|D_n]=\\int \\mu(\\Theta(\\phi)\\Delta\\Omega)p(\\phi|D_n)d\\phi$ for each set $\\Omega$.\n\n\\begin{thm}\\label{t51}For any   set $\\Omega\\subset\\Theta$ and every sampling sequence $D_n$,\n$$\nE[L(\\widehat \\Omega, \\phi)|D_n]\\leq E[L(\\Omega, \\phi)|D_n].\n$$\n\\end{thm}\n\n\n\nHence $\\widehat\\Omega$ is an optimal estimator of the identified set in the sense that it minimizes the Bayesian risk, which is the posterior expected loss. Therefore we can also call it the \\textit{Bayes rule} from a decision-making point of view.  As often the case in the decision theory, the Bayes rule is also \\textit{admissible}:  for any set   estimator $\\Omega$ that depends on the data $D_n$, let\n$$\nR(\\Omega,\\phi)=E_{D_n}[L(\\Omega,\\phi)|\\phi],\n$$\nwhere the expectation is taken with respect to the  true sampling distribution, given any fixed $\\phi$. We have,\n\n\\begin{cor} \\label{c51} Suppose the prior $\\pi(\\phi)$ is such that $\\pi(\\phi\\in B)>0$ for any open ball $B\\subset\\Phi$. For any $\\Omega\\subset\\Theta$, assume $R(\\Omega, \\phi)$ is continuous in $\\phi$. Then  either one of the following holds:\\\\\n(i) there is $\\phi^*\\in\\Phi$ such that $R(\\Omega,\\phi^*)>R(\\widehat\\Omega, \\phi^*)$;\\\\\n(ii) for any $\\phi\\in\\Phi$,  $R(\\Omega, \\phi)\\geq R(\\widehat\\Omega, \\phi)$.\n\n\\end{cor}\n\n\n \\end{comment}\n\n\n\n\n\n\n\n\n\n\\section{Further Illustrations and  Uniformity}\\label{s_further_illustration_Uniformity}\n\\subsection{Missing data: coverage probabilities and prior sensitivity}\\label{smd}\nThis subsection illustrates the coverages of the proposed BCS in the missing data problem (example \\ref{ex2.3}),  previously discussed  by Manski (2003).   Let $Y$ be a binary variable, indicating whether a treatment is successful ($Y=1$) or not ($Y=0$).  However, $Y$ is observed subject to missing. We write $M=0$ if $Y$ is missing, and $M=1$ otherwise. Hence, we observe $(M, MY)$. The parameter of interest is $\\theta = P(Y=1)$. The identified parameters are denoted by\n$$\n\\phi_1=P(M=1),\\quad \\phi_2=P(Y=1|M=1).\n$$ Let $\\phi_0=(\\phi_{10},\\phi_{20})$ be the true value of $\\phi=(\\phi_1,\\phi_2)$.\nThen, without further assumption on $P(Y=1|M=0)$, $\\theta$ is only partially identified on $\n\\Theta(\\phi)=[\\phi_1\\phi_2,\\phi_1\\phi_2+1-\\phi_1]\n$.\nThe support function is easy to calculate and is\n$$S_{\\phi}(1)=\\phi_1\\phi_2+1-\\phi_1\\quad S_{\\phi}(-1)=-\\phi_1\\phi_2.$$ Suppose we observe i.i.d. data  $\\{(M_i, Y_iM_i)\\}_{i= 1}^{n}$, and define $\\sum_{i=1}^nM_i=n_1$ and $\\sum_{i=1}^nY_iM_i=n_2$. In this example, the true likelihood function\n$\nl_n(\\phi) \\propto \\phi_1^{n_1}(1-\\phi_1)^{n-n_1}\\phi_2^{n_2}(1-\\phi_2)^{n_1-n_2}\n$ is known.\n\n\nWe place independent beta priors, Beta($\\alpha_1,\\beta_1$) and Beta$(\\alpha_2,\\beta_2)$, on $(\\phi_1,\\phi_2)$. The uniform distribution is a special case of Beta prior. Then the posterior of $(\\phi_1,\\phi_2)$ is a product of Beta($\\alpha_1+n_1,\\beta_1+n-n_1$) and Beta($\\alpha_2+n_2,\\beta_2+n_1-n_2$). If in addition, we have  prior information on $\\theta$ and place a prior $\\pi(\\theta|\\phi)$ supported on $\\Theta(\\phi)$, then by integrating out $\\phi$, we immediately obtain the marginal posterior of $\\theta.$\n\n\nWe now present the two-sided BCS for $\\Theta(\\phi)$ obtained by using the support function of $\\Theta(\\phi)$. The estimator $\\hat\\phi$ is taken as the posterior mode: $\n\\hat{\\phi}_{1}= (n_1+\\alpha_1-1)\/(n+\\alpha_1+\\beta_1-2),$ and $\\hat{\\phi}_{2}=(n_2+\\alpha_2-1)\/(n_1+\\alpha_2+\\beta_2-2)$.\nThen\n$$J(\\phi)=\\sqrt{n}\\max\\left\\{|\\phi_1\\phi_2-\\phi_1-\\hat{\\phi}_{1}\\hat{\\phi}_{2}+\\hat{\\phi}_{1}|, |\\phi_1\\phi_2-\\hat{\\phi}_{1}\\hat{\\phi}_{2}|\\right\\}.\n$$\nLet $q_{\\tau}$ be  the $1-\\tau$ quantile of the posterior of $J(\\phi)$, which can be obtained by simulating from the Beta distributions. The lower and upper $1-\\tau$ level BCS's  for $\\Theta(\\phi)$ are $\\Theta(\\hat\\phi )^{-q_{\\tau}\/\\sqrt{n}}\\subset \\Theta(\\phi)\\subset \\Theta(\\hat\\phi )^{q_{\\tau}\/\\sqrt{n}}$ where\n$$\n\\Theta(\\hat\\phi )^{-q_{\\tau}\/\\sqrt{n}}=[\\hat{\\phi}_{1}\\hat{\\phi}_{2}+q_{\\tau}\/\\sqrt{n}, \\hat{\\phi}_{1}\\hat{\\phi}_{2}+1-\\hat{\\phi}_{1}-q_{\\tau}\/\\sqrt{n}]\n$$\n\\noindent and\n$$\n\\Theta(\\hat\\phi )^{q_{\\tau}\/\\sqrt{n}}=[\\hat{\\phi}_{1}\\hat{\\phi}_{2}-q_{\\tau}\/\\sqrt{n}, \\hat{\\phi}_{1}\\hat{\\phi}_{2}+1-\\hat{\\phi}_{1}+q_{\\tau}\/\\sqrt{n}],\n$$\nwhich are also two-sided asymptotic $1-\\tau$ frequentist confidence intervals of the true $\\Theta(\\phi_{0}).$\n\nHere we present a simple simulated example, where the true $\\phi_0=(0.7,0.5)$. This implies the true identified interval to be $[0.35, 0.65]$ and about thirty percent of the simulated data are ``missing\". We   set  $\\alpha_1=\\alpha_2, \\beta_1=\\beta_2$ in the prior. In addition, $B=1,000$ posterior draws $\\{\\phi^i\\}_{i=1}^B$ are sampled from  the posterior Beta distribution. For each of them, compute $J(\\phi^{i})$ and set $q_{0.05}$ as the 95\\% upper quantile of   $\\{J(\\phi^i)\\}_{i=1}^B$ to obtain the critical value of the BCS and construct the two-sided BCS for the identified set. Each simulation is repeated for 500 times to calculate the coverage frequency of the true identified interval.  Table \\ref{table2} presents the results. We see that the coverage probability for the two-sided is close to the desired 95\\% when sample size increases. In addition, the marginal coverages of the lower and upper sets are close to 97.5\\% when sample size is relatively large.\n\n\nMoreover, Figure 1 plots the five conjugate prior  specifications  used in this study: flat prior, reverse $J$-shaped with a right tail, $J$-shaped with a left tail,  $U$-shaped, and uni-mode. These priors reflect different types of prior beliefs: the first prior is used if a researcher has no informative prior information, the second one (resp. third one) is used if one strongly believes that the probability of missing is low (resp. high), the fourth prior is used when one   thinks  that the probability of missing is either very high or very low, and the last prior corresponds to a  symmetric prior belief centered at fifty percent. So Table \\ref{table2} also provides   simple sensitivity analysis of prior specification. The results demonstrate robustness of the coverage probabilities to conjugate prior specification.\n\n\n\n\\begin{table}[htdp]\n\\caption{Frequentist coverage probability of  BCS and prior sensitivity for missing data}\n\\begin{center}\n\n\\begin{tabular}{c|cc|ccc}\n\\hline\n$n$ & $\\alpha$ & $\\beta$ & Lower & Upper & Two-sided \\\\\n\\hline\n50 & 1 & 1 & 0.978 & 0.944 & 0.924 \\\\\n & 1 & 0.1 & 0.964 & 0.944 & 0.912 \\\\\n & 0.1 & 1 & 0.952 & 0.958 & 0.916 \\\\\n & 0.1 & 0.1 & 0.974 & 0.958 & 0.938 \\\\\n   &2&2&0.958 &  0.970& 0.932\\\\\n \\hline\n100 & 1 & 1 & 0.982 & 0.96 & 0.948 \\\\\n & 1 & 0.1 & 0.978 & 0.968 & 0.950 \\\\\n & 0.1 & 1 & 0.968 & 0.968 & 0.948 \\\\\n &0.1  & 0.1 & 0.972 & 0.972 & 0.944 \\\\\n  &2&2&0.956 &  0.978& 0.944\\\\\n \\hline\n500 & 1 & 1 & 0.970 & 0.974 & 0.950 \\\\\n &  1& 0.1 & 0.978 & 0.978 & 0.958 \\\\\n & 0.1 & 1 & 0.974 & 0.972 & 0.948 \\\\\n & 0.1 & 0.1 & 0.972 & 0.974 & 0.950 \\\\\n &2&2& 0.976 & 0.978& 0.956\\\\\n\\hline\n\\end{tabular}\n\n\n\\label{table2}\n\\small\n\n\\it Lower, Upper and Two-sided represent the  frequencies of the events $\\Theta(\\hat\\phi )^{-q_{\\tau}\/\\sqrt{n}}\\subset\\Theta(\\phi_0)$, $\\Theta(\\phi_0)\\subset \\Theta(\\hat\\phi )^{q_{\\tau}\/\\sqrt{n}}$, and $\\Theta(\\hat\\phi )^{-q_{\\tau}\/\\sqrt{n}}\\subset\\Theta(\\phi_0)\\subset \\Theta(\\hat\\phi )^{q_{\\tau}\/\\sqrt{n}}$ over 500 replicates. The coverage probability for the two-sided BCS is set to 95\\%.\n\n\\end{center}\n\\end{table}\n\n\n\n \\begin{figure}[htbp]\n\\begin{center}\n\\caption{Conjugate priors   for sensitivity analysis in the missing data problem}\n\\includegraphics[width=12cm]{prior.eps}\n\\label{figureadd}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\n\\subsection{Uniformity: from partial identification to point identification}\\label{s_from_partial_identification_point_Identification}\n\n\nWe have been focusing on partially identified models, and the inference results achieved are for a fixed data generating process. It is interesting to see whether they  still hold uniformly over a class of data generating processes, including the case   when point identification is nearly achieved. This is important because in many cases it is possible that we actually have point identification and, in that event, $\\Theta(\\phi)$ degenerates to a singleton. For example, in the interval censored model, when  $EY_1=EY_2$,  $\\theta=EY$ is point identified.\n\nWhen point identification is indeed achieved, the frequentist coverage probability of the upper-sided BCS $\\Theta(\\phi)\\subset\\Theta(\\hat\\phi )^{q_{\\tau}\/\\sqrt{n}}$ and the asymptotic normality for the posterior of the support function    still hold   because they are  generally guaranteed by the semi-parametric Bernstein-von Mises theorem for $\\phi$ when $\\Theta(\\phi)$ is a singleton (e.g., Rivoirard and Rousseau 2012, Bickel and Kleijn 2011). But the low-side BCS for $\\Theta(\\phi)$ will be empty with a  positive probability.\nTheorem \\ref{t6.1}, however, does not hold anymore  when $\\theta$ is point identified, as we discussed previously.\nWe further illustrate the uniformity in two examples.\n\n\\begin{exm}[Interval censored data - \\textit{continued}] We  show that the   the upper  BCS for the identified set has a uniformly correct frequentist asymptotic coverage probability.\nTo simplify our illustration, we assume $Y_1$ and $Y_2$ are independent and follow $\\mathcal{N}(\\phi_{10}, 1)$ and $\\mathcal{N}(\\phi_{20},1)$, respectively. A sequence  of different $\\phi_0$ are  considered which includes the case $\\phi_{10}-\\phi_{20}=o(1)$. When $\\phi_{10}=\\phi_{20}$, however,  we suppose $Y_1, Y_2$ are sampled independently.  Suppose   econometricians   place independent standard normal priors on $\\phi_1$ and $\\phi_2$,   then the posteriors  are independent, given by $\\phi_i|D_n\\sim \\mathcal{N}(\\bar{Y}_i\\frac{n}{1+n}, \\frac{1}{1+n}), i=1,2,\n$\nand  $\\hat{\\phi} = \\frac{n}{1+n}(\\bar{Y}_1,\\bar{Y}_2)$ is the posterior mode of the joint distribution.\nThe support function is $S_{\\phi}(1)=\\phi_2$, $S_{\\phi}(-1)=-\\phi_1$. Let $$J(\\phi)=\\sqrt{n}\\sup_{\\|\\nu\\|=1}(S_{\\phi}(\\nu)-S_{\\hat\\phi }(\\nu))=\\sqrt{n}\\max\\left\\{\\phi_2-\\frac{n}{1+n}\\bar{Y}_2, \\frac{n}{1+n}\\bar{Y}_1-\\phi_1\\right\\},$$ and let $\\tilde{q}_{\\tau}$ be the $1-\\tau$ quantile of the posterior of $J(\\phi)$.   We now show that the frequentist coverage of  BCS($\\tau$)$=[\\bar{Y}_1\\frac{n}{1+n}-\\frac{\\tilde{q}_{\\tau}}{\\sqrt{n}}, \\bar{Y}_2\\frac{n}{1+n}+\\frac{\\tilde{q}_{\\tau}}{\\sqrt{n}}]$ is  valid uniformly for $\\phi_0=(\\phi_{10},\\phi_{20})\\in\\Phi$, that is,\n\\begin{equation}\\label{e71}\n\\liminf_{n\\rightarrow\\infty}\\inf_{(\\phi_{01},\\phi_{02})\\in\\Phi}P_{D_n}(\\Theta(\\phi_0)\\subset \\text{BCS}(\\tau))= 1-\\tau.\n\\end{equation}\nWe can simplify $J(\\phi)$ to be $\\sqrt{n\/(1+n)}\\max\\{Z_1, Z_2\\}$ where $Z_i\\sim \\mathcal{N}(0,1)$, $i=1,2$ and $Z_1$ and $Z_2$ are independent. This implies, for the standard normal's CDF $H(\\cdot)$,\n$$\n1 - \\tau = P(J(\\phi)\\leq\\tilde{q}_{\\tau}|D_n)=P\\left(\\max\\{Z_1,Z_2\\}\\leq\\tilde{q}_{\\tau}\\sqrt{\\frac{1+n}{n}}\\right)\n= H\\left(\\tilde{q}_{\\tau}\\sqrt{\\frac{1+n}{n}}\\right)^2.\n$$\nHence, $H\\left(\\tilde{q}_{\\tau}\\sqrt{(1+n)\/n}\\right)^2 = H(\\tilde{q}_{\\tau})^2 + o(1)$ and so $H(\\tilde{q}_{\\tau})^2\\rightarrow1-\\tau.$ The event $\\{\\Theta(\\phi_0)\\subset \\textrm{BCS}(\\tau)\\}$ is equivalent to $(\\bar{Y}_1-\\phi_{10})\\frac{n}{1+n}\\leq \\frac{\\tilde{q}_{\\tau}}{\\sqrt{n}}+\\frac{\\phi_{10}}{1+n}$ and $(\\bar{Y}_2-\\phi_{20})\\frac{n}{1+n}\\geq \\frac{\\phi_{20}}{1+n}- \\frac{\\tilde{q}_{\\tau}}{\\sqrt{n}}$. Hence,\n\\begin{eqnarray*}\n\\inf_{\\phi_0\\in\\Phi}P_{D_n}(\\Theta(\\phi_0)\\subset\\text{BCS}(\\tau))&=&\\inf_{\\phi_0\\in\\Phi}H\\left((\\frac{\\tilde{q}_{\\tau}}{\\sqrt{n}}+\\frac{\\phi_{10}}{1+n})\\frac{1+n}{\\sqrt{n}}\\right)\nH\\left((\\frac{\\tilde{q}_{\\tau}}{\\sqrt{n}}-\\frac{\\phi_{20}}{1+n})\\frac{1+n}{\\sqrt{n}}\\right)\\cr\n&=&H(\\tilde q_{\\tau})^2+o(1)\\rightarrow1-\\tau.\n\\end{eqnarray*}\n  This gives (\\ref{e71}).\n\n  On the other hand, if $\\phi_{20}-\\phi_{10}=o(1)$, the lower BCS for $\\Theta(\\phi)$ is empty with a large probability. To see this, for any fixed $q$,  the lower BCS is $A=[\\bar{Y}_1\\frac{n}{1+n}+\\frac{q}{\\sqrt{n}}, \\bar{Y}_2\\frac{n}{1+n}-\\frac{q}{\\sqrt{n}}]$. Let $\\Delta_n=\\phi_{20}-\\phi_{10}$, then\n$\nP(A=\\emptyset)=P((\\bar{Y}_2-\\bar{Y}_1)\\frac{n}{1+n}<\\frac{2q}{\\sqrt{n}})=H(\\sqrt{2}q-\\sqrt{\\frac{n}{2}}\\Delta_n)+o(1).\n$\nSuppose $\\sqrt{n}\\Delta_n=o(1)$, then $P(A=\\emptyset)\\rightarrow H(\\sqrt{2}q)$.\nThis  probability is very large  for many reasonable cut-off $q$. For example, if $q=1.96$, $H(\\sqrt{2}q)=0.997.$\n\nFor a numerical illustration,  set $\\phi_{10}=1, \\phi_{20}=1+\\Delta_n$ for a sequence of  small  $\\Delta_n$ that decreases to zero, and calculate the frequency that $\\Theta(\\phi_0)\\subset$ BCS$(0.05)$ and $A=\\emptyset.$  The model is nearly point identified, and point identification is achieved when $\\Delta_n=0.$ Results are reported in Table \\ref{t2add}.\n\n\n\n\\end{exm}\n\n\n\n\n\\begin{exm}[Missing data example - \\textit{continued}]\nConsider again the missing data example in Section \\ref{smd}, where now the true $\\phi_{10}$ is $\\phi_{10}=1-\\Delta_n$ with $\\Delta_n\\rightarrow0,$ that is, the probability of missing is close to zero. So the model is close to point identification.  However, suppose we still place   priors on $\\phi_1$ and $\\phi_2$ and $\\Theta(\\phi)=[\\phi_1\\phi_2, \\phi_1\\phi_2+1-\\phi_1]$ as before.  Our result shows that\n\\begin{equation}\\label{eq8.2}\nP_{D_n}(\\Theta(\\phi_0)\\subset\\Theta(\\hat\\phi )^{\\tilde{q}_{\\tau}\/\\sqrt{n}})\\rightarrow 1-\\tau\n\\end{equation}\n when $\\tilde{q}_{\\tau}$ is the $1-\\tau$ quantile of the posterior of\n $$\n \\sqrt{n}\\max\\left\\{\\phi_1\\phi_2-\\phi_1-\\hat{\\phi}_{1}\\hat{\\phi}_{2}+\\hat{\\phi}_{1},  \\hat{\\phi}_{1}\\hat{\\phi}_{2}-\\phi_1\\phi_2\\right\\}.\n $$\nIt can also be shown that the coverage (\\ref{eq8.2})   holds uniformly for $\\phi_0$ inside a compact parameter space.\nIt is also easy to see that, if $\\Delta_n=o(n^{-1\/2})$, then for any $\\tau\\in(0,1)$, the lower BCS$(\\tau)$ is empty with probability approaching one.  \n\n\n\nWe illustrate the above discussions using a simulated example, where $\\phi_{10}=1-\\Delta_n$ for a sequence of   small $\\Delta_n$.   We use the uniform priors  and compute the frequency of the events that $\\Theta(\\phi_0)\\subset\\Theta(\\hat\\phi )^{\\tilde q_{0.05}\/\\sqrt{n}}$ and that the lower BCS is empty. We set $\\phi_{20}=0.5$ so that $\\hat{\\phi}_{2}$ has the maximum possible variance. Therefore, our simulation also demonstrates how sensitive  the coverage frequency is to the variance of the point identified estimator. The frequency of coverage over 500 replications are summarized in Table \\ref{t2add} below.\n\n\\begin{table}[htdp]\n\\caption{Frequency of BCS(0.05) coverages for near point identification}\n\\begin{center}\n\n\n\\begin{tabular}{ccc|cccc}\n\\hline\n& &  &  & $\\Delta_n$ \\\\\n&$n$ & event & 0.1 & 0.05 & 0.01 & 0 \\\\\n\\hline\n &&&&&&\\\\\nInterval censoring&50 &  Lower BCS$=\\emptyset$ & 0.966 & 0.94 & 0.972 & 0.956 \\\\\n && $\\Theta(\\phi_0)\\subset\\Theta(\\hat\\phi )^{\\tilde q\/\\sqrt{n}}$ & 0.952 & 0.964 & 0.952 & 0.944 \\\\\n  &&&&&&\\\\\n&100 &   Lower BCS$=\\emptyset$ & 0.964 & 0.96 & 0.962 & 0.962 \\\\\n &&$\\Theta(\\phi_0)\\subset\\Theta(\\hat\\phi )^{\\tilde q\/\\sqrt{n}}$ & 0.962 & 0.966 & 0.958 & 0.95 \\\\\n\\hline\n& &&&&&\\\\\nMissing data &50 &  Lower BCS$=\\emptyset$ & 0.998 & 1 & 1 & 1 \\\\\n && $\\Theta(\\phi_0)\\subset\\Theta(\\hat\\phi )^{\\tilde q\/\\sqrt{n}}$& 0.95 & 0.952 & 0.942 & 0.944 \\\\\n& &&&&&\\\\\n&100 &  Lower BCS$=\\emptyset$ & 0.99 & 1 & 1 & 1 \\\\\n && $\\Theta(\\phi_0)\\subset\\Theta(\\hat\\phi )^{\\tilde q\/\\sqrt{n}}$& 0.952 & 0.956 & 0.958 & 0.952 \\\\\n\\hline\n  \\end{tabular}\n\n\n  \\label{t2add}\n\\small\n\n\\it\nThe frequencies (over 500 replications) that the lower BCS is empty and that the upper BCS covers the true identified set are summarized. The length of the true identified set is $\\Delta_n$. The model achieves point identification when $\\Delta_n=0$.\n\n\\end{center}\n\\end{table}\n\n\nWe see that the upper BCS with 95\\% credible level has the coverage probability for the true identified set close to 0.95.\n Also, the lower BCS is empty almost all the times.\n\n \\end{exm}\n\n\n\n\n\\section{Financial Asset Pricing}\\label{s_Financial_asset_pricing}\nWe develop a detailed application in financial asset pricing model, where the identified set is of direct interest.\n\n\\subsection{The model}\nAsset pricing models state that the equilibrium price $P_{t}^{i}$ of a financial asset $i$ is equal to\n$$\n      P_{t}^{i} =  E [M_{t+1}P_{t+1}^{i}|\\mathcal{I}_{t}],\\qquad i=1,\\ldots,N\n$$\n\n\\noindent where $P_{t+1}^{i}$ denotes the price of asset $i$ at the period $(t+1)$, $M_{t+1}$ is the stochastic discount factor (SDF, hereafter) and $\\mathcal{I}_{t}$ denotes the information set at time $t$. In vectorial form this rewrites as\n   \\begin{equation}\\label{eq9.1}\n      \\iota =  E [M_{t+1}R_{t+1}|\\mathcal{I}_{t}]\n    \\end{equation}\n\n\\noindent where $\\iota$ is the $N$-dimensional vector of ones and $R_{t+1}$ is the $N$-dimensional vector of gross asset returns at time $(t+1)$: $R_{t+1} = (r_{1,t+1},\\ldots, r_{N,t+1})^{T}$ with $r_{i,t+1} = P_{t+1}^{i}\/P_{t}^{i}$. This model can be interpreted as a model of the SDF $M_{t+1}$ and may be used to detect the SDFs that are compatible with asset return data. Hansen and Jagannathan (1991) have obtained a lower bound on the volatility of SDFs that could be compatible with a given SDF-mean value and a given set of asset return data. Therefore, the set of SDFs $M_{t+1}$ that can price existing assets generally form a proper set.\\\\\n\\indent Let $m$ and $\\Sigma$ denote, respectively, the vector of unconditional mean returns and covariance matrix of returns of the $N$ risky assets, that is, $m =  E (R_{t+1})$ and $\\Sigma =  E (R_{t+1} - m)(R_{t+1} - m)^{T}$. Denote $\\mu =  E (M_{t+1})$ and $\\sigma^{2} = Var(M_{t+1})$, which are partially identified. We assume that $m$, $\\Sigma$, $\\mu$ and $\\sigma^{2}$ do not vary with $t$.  Hansen and Jagannathan (1991) showed that given $(m,\\Sigma)$, which are point identified by the observed $\\{R_{t+1}\\}$, if the SDF $M_{t+1}$ satisfies (\\ref{eq9.1}), then its variance $\\sigma^2$ should be no smaller than:\n     \\begin{eqnarray}\\label{eq7.1}\n      \\sigma_{\\phi}^{2}(\\mu) & = & (\\iota - \\mu m)^{T}\\Sigma^{-1} (\\iota - \\mu m) \\equiv \\phi_{1}\\mu^{2} - 2\\phi_{2}\\mu + \\phi_{3}\\cr\n      & & \\nonumber\\\\\n      \\textrm{with } \\phi_{1} & = & m^{T}\\Sigma^{-1}m, \\qquad \\phi_{2} = m^{T}\\Sigma^{-1}\\iota, \\qquad \\phi_{3} = \\iota^{T}\\Sigma^{-1}\\iota.\n    \\end{eqnarray}\n\n\\noindent Therefore, an SDF correctly prices an asset only if, for given $(m,\\Sigma)$, its mean $\\mu$ and variance $\\sigma^{2}$ are such that $\\sigma^{2}\\geq \\sigma_{\\phi}^{2}(\\mu)$, and in this case the SDF is called \\textit{admissible}. Inadmissible SDFs do not satisfy  model (\\ref{eq9.1}).\n\nDefine the set of admissible SDF's means and variances:\n    \\begin{equation*}\n      \\Theta(\\phi) = \\left\\{(\\mu,\\sigma^{2})\\in \\Theta; \\: \\sigma_{\\phi}^{2}(\\mu) - \\sigma^{2} \\leq 0\\right\\}\n    \\end{equation*}\n\n\\noindent where $\\phi = (\\phi_{1},\\phi_{2},\\phi_{3})^{T}$ and $\\Theta\\subset \\mathbb{R}_{+}\\times\\mathbb{R}_{+}$ is a compact set that we can choose based   on experiences. Usually, we can fix upper bounds $\\bar{\\mu}>0$ and $\\bar{\\sigma}>0$ as large as we want and take $\\Theta = [0,\\bar{\\mu}] \\times [0,\\bar{\\sigma}^{2}]$. In practice, $\\bar{\\mu}$ and $\\bar{\\sigma}$ must be chosen sufficiently large such that $\\Theta(\\phi)$ is non-empty. We also point out that to be consistent with our developed theory, the parameter space is chosen to be compact. Thus, the space for $\\sigma^2$ includes zero. In practice, one can require $\\sigma^2\\geq \\epsilon$ for a sufficiently small $\\epsilon>0.$ For simplicity, we keep the current parameter space for $\\sigma^2$, which is also used sometimes in the literature. Making inference on $\\Theta(\\phi)$ allows to check whether a family of SDF (and then a given utility function) prices a financial asset correctly or not. Frequentist inference for this set is carried out in Chernozhukov, Kocatulum and Menzel (2012).\n\n Using our previous notation,  we define $\\theta = (\\mu,\\sigma^{2})$ and\n    \\begin{displaymath}\n      \\Psi(\\theta,\\phi) = \\sigma_{\\phi}^2(\\mu) - \\sigma^{2},\n    \\end{displaymath}\nwhich gives a moment inequality model.\n\n\\subsection{Support function}\n\nIn this case $\\Psi(\\theta,\\phi)$ is convex in $\\theta$. More precisely, $\\Psi(\\theta,\\phi)$ is linear in $\\sigma^{2}$ and strictly convex in $\\mu$ (because $\\Sigma$ is positive definite so  $\\phi_{1}>0$). Assumptions \\ref{ass5.1}- \\ref{ass5.5}  are easy to verify except for Assumptions \\ref{ass5.6} and \\ref{ass5.5}\\textit{(i)} and \\textit{(iv)}. However, it can be shown that the support function is differentiable at $\\phi_{0}$ without Assumption \\ref{ass5.6} being satisfied. So, our Bayesian analysis on the support function of $\\Theta(\\phi)$ still goes through. Assumption \\ref{ass5.5} \\textit{(i)} and \\textit{(iv)} must be checked case by case (that is, for every region of values of $\\nu$) since $\\lambda(\\nu,\\phi)$ takes a different expression in each case, see Appendix \\ref{ss_support_set_HJ} in the supplementary material.\n\n\n\n\nWe can rewrite the support function to be $S_{\\phi}(\\nu)=\\Xi(\\nu,\\phi)^T\\nu$, where\n    \\begin{eqnarray*}\n      \\Xi(\\nu,\\phi) & = & \\arg\\max_{\\theta\\in\\Theta}\\left\\{\\nu^T\\theta; \\:\\Psi(\\theta,\\phi)\\leq 0\\right\\}\\\\\n      & = & \\arg\\max_{0\\leq\\mu<\\bar{\\mu},\\:0<\\sigma^{2}<\\bar{\\sigma}^{2}}\\left\\{\\nu_{1}\\mu + \\nu_{2}\\sigma^{2} - \\lambda(\\nu,\\phi)(\\phi_{1}\\mu^{2} - 2\\phi_{2}\\mu + \\phi_{3} - \\sigma^{2})\\right\\}\n     \n    \\end{eqnarray*}\n\n\\noindent where $\\nu=(\\nu_{1},\\nu_{2})$. The support function and $\\Xi(\\nu,\\phi)$ have explicit expressions, but they are very long and complicated. Thus, we present them in Appendix \\ref{ss_support_set_HJ}\n\n\\begin{comment}\n\\begin{thm}\n  Let $p\\in\\mathbb{S}^{2}$ and denote $p=(p_{1},p_{2})$. Then we have:\n\n     \\begin{enumerate}\n      \\item $p_{2}<0$ and $p_{1}>0$:\n        \\begin{displaymath}\n          \\Xi(p,\\phi) = \\left\\{\\begin{array}\n            {cc} \\left(\\frac{\\phi_{2}}{\\phi_{1}} - \\frac{p_{1}}{2p_{2}\\phi_{1}},\\frac{p_{1}^{2}}{4p_{2}^{2}\\phi_{1}} + \\frac{\\phi_{1}\\phi_{3} - \\phi_{2}^{2}}{\\phi_{1}}\\right) & \\textrm{ if } \\bar{\\mu} \\geq \\frac{\\phi_{2}}{\\phi_{1}} - \\frac{p_{1}}{2p_{2}\\phi_{1}} \\textrm{ and }\\bar{\\sigma}^{2}\\geq \\frac{p_{1}^{2}}{4p_{2}^{2}\\phi_{1}} + \\frac{\\phi_{1}\\phi_{3} - \\phi_{2}^{2}}{\\phi_{1}}\\\\\n            &\\\\\n            \\left(\\bar{\\mu}, \\phi_{1}\\bar{\\mu}^{2} - 2\\phi_{2}\\bar{\\mu} + \\phi_{3}\\right)& \\textrm{ if } \\bar{\\mu} \\leq \\frac{\\phi_{2}}{\\phi_{1}} - \\frac{p_{1}}{2p_{2}\\phi_{1}} \\textrm{ and }\\bar{\\sigma}^{2}\\geq \\phi_{1}\\bar{\\mu}^{2} - 2\\phi_{2}\\bar{\\mu} + \\phi_{3}\\\\\n            & \\\\\n            \\left(\\frac{\\phi_{2} + \\sqrt{\\phi_{2}^{2} - \\phi_{1}(\\phi_{3} - \\bar{\\sigma}^{2})}}{\\phi_{1}},\\bar{\\sigma}^{2}\\right)& \\textrm{ if } \\bar{\\mu} \\geq \\frac{\\phi_{2} + \\sqrt{\\phi_{2}^{2} - \\phi_{1}(\\phi_{3} - \\bar{\\sigma}^{2})}}{\\phi_{1}} \\textrm{ and }\\bar{\\sigma}^{2}\\leq \\frac{p_{1}^{2}}{4p_{2}^{2}\\phi_{1}} + \\frac{\\phi_{1}\\phi_{3} - \\phi_{2}^{2}}{\\phi_{1}}\n          \\end{array}\\right.\n        \\end{displaymath}\n\n      \\item $p_{2}<0$ and $p_{1}<0$:\n        \\begin{displaymath}\n          \\Xi(p,\\phi) = \\left\\{\\begin{array}\n            {cc} (0,\\phi_{3}) & \\textrm{ if } 2p_{2}\\phi_{2}\\leq p_{1}<0 \\textrm{ and }\\bar{\\sigma}^{2}\\geq \\phi_{3}\\\\\n            &\\\\\n            \\left(\\frac{\\phi_{2} + \\sqrt{\\phi_{2}^{2} - \\phi_{1}(\\phi_{3} - \\bar{\\sigma}^{2})}}{\\phi_{1}},\\bar{\\sigma}^{2}\\right)& \\textrm{ if } 2p_{2}\\phi_{2}\\leq p_{1}<0,\\;\\bar{\\mu} \\geq \\frac{\\phi_{2} + \\sqrt{\\phi_{2}^{2} - \\phi_{1}(\\phi_{3} - \\bar{\\sigma}^{2})}}{\\phi_{1}}\\\\\n            & \\textrm{ and }\\bar{\\sigma}^{2}\\leq \\phi_{3}\\\\\n            \\textrm{ same as in 1. } & \\textrm{ if } p_{1}<2p_{2}\\phi_{2}<0\n           \n           \n           \n           \n           \n          \\end{array}\\right.\n        \\end{displaymath}\n\n      \\item $p_{2}>0$ and $p_{1}>0$:\n        \\begin{displaymath}\n          \\Xi(p,\\phi) = \\left\\{\\begin{array}\n            {cc} (\\bar{\\mu},\\bar{\\sigma}^{2}) & \\textrm{ if } \\bar{\\sigma}^{2}\\geq \\phi_{1}\\bar{\\mu}^{2} - 2\\phi_{2}\\bar{\\mu} + \\phi_{3}\\\\\n            \\left(\\frac{\\phi_{2} + \\sqrt{\\phi_{2}^{2} - \\phi_{1}(\\phi_{3} - \\bar{\\sigma}^{2})}}{\\phi_{1}},\\bar{\\sigma}^{2}\\right) & \\textrm{ if } \\bar{\\sigma}^{2} < \\phi_{1}\\bar{\\mu}^{2} - 2\\phi_{2}\\bar{\\mu} + \\phi_{3} \\textrm{ and } \\bar{\\mu}> \\frac{\\phi_{2}}{\\phi_{1}}\n          \\end{array}\\right.\n        \\end{displaymath}\n\n      \\item $p_{2}>0$ and $p_{1}<0$:\n        \\begin{displaymath}\n          \\Xi(p,\\phi) = \\left\\{\\begin{array}\n            {cc} (0,\\bar{\\sigma}^{2}) & \\textrm{ if } \\bar{\\sigma}^{2}\\geq \\phi_{3}\\\\\n            \\left(\\mu_{min},\\bar{\\sigma}^{2}\\right) & \\textrm{ otherwise}\n          \\end{array}\\right.\n        \\end{displaymath}\n      \\noindent where $\\mu_{\\min} = \\min\\{\\mu\\in[0,\\bar{\\mu}];\\:\\phi_{1}\\mu^{2} - 2\\phi_{2}\\mu + \\phi_{3} \\leq \\bar{\\sigma}^{2}\\}$.\n    \\end{enumerate}\nThe support function is given by\n$$\nS_{\\phi}(p)=p^T\\Xi(p,\\phi), \\quad\\| p\\|=1.\n$$\n\n    \\end{thm}\n\n\\noindent Thus, $\\Xi(p,\\phi)$ is a singleton and the support function can be easily obtained. We remark that in cases 3 and 4 the constraint given by the Hansen-Jagannathan bound is no longer binding, so that the corresponding Lagrange multiplier is $0$.\n\n\\end{comment}\n\\subsection{Dirichlet process prior}\n Let $F$ denote a probability distribution. The Bayesian model is $R_t|F\\sim F$ and $  (m, \\Sigma)=(m(F), \\Sigma(F))$, where\n    $$\nm(F)=\\int r F(dr),\\quad \\Sigma(F)=\\int rr^TF(dr)-\\int rF(dr)\\int rF(dr)^T.\n    $$\n    Let us impose a Dirichlet process prior for $F$, with parameter $v_0$ and base probability measure $Q_0$ on $\\mathbb{R}^N$. By Sethuraman (1994)'s decomposition, the Dirichet process prior induces a prior for $(m, \\Sigma)$ as:\n    $m=\\sum_{j=1}^{\\infty}\\alpha_j\\xi_j$,  and $\\Sigma=\\sum_{j=1}^{\\infty}\\alpha_j\\xi_j\\xi_j^T-\\sum_{i=1}^{\\infty}\\alpha_i\\xi_i\\sum_{j=1}^{\\infty}\\alpha_j\\xi_j^T$\n    where $\\xi_j$ are independently sampled from $Q_0$;  $\\alpha_j=u_j\\prod_{l=1}^j(1-u_l)$ with  $\\{u_i\\}_{i=1}^n$ drawn from Beta$(1,v_0)$. These priors then induce a prior for $\\phi.$ The posterior distribution for $(m,\\Sigma)$ can be calculated explicitly:\n    \\begin{eqnarray*}\n \\Sigma|D_n&\\sim&(1-\\gamma)\\sum_{j=1}^{\\infty}\\alpha_j\\xi_j\\xi_j^T+\\gamma\\sum_{t=1}^n\\beta_tR_tR_t^n\\cr\n &&-\\left( (1-\\gamma)\\sum_{j=1}^{\\infty}\\alpha_j\\xi_j+\\gamma\\sum_{t=1}^n\\beta_tR_t\\right)\\left( (1-\\gamma)\\sum_{j=1}^{\\infty}\\alpha_j\\xi_j+\\gamma\\sum_{t=1}^n\\beta_tR_t\\right)^T,\n \\end{eqnarray*}\n $$\n m|D_n\\sim (1-\\gamma)\\sum_{j=1}^{\\infty}\\alpha_j\\xi_j+\\gamma\\sum_{t=1}^n\\beta_tR_t,\\quad \\gamma\\sim\\text{Beta}(n,v_0),\\quad \\{\\beta_j\\}_{j=1}^n\\sim Dir(1,...,1).\n $$\nWe can set a  truncation $K>0$, so the infinite sums in the posterior representation  are  replaced with a truncated sum.     We can then simulate the posterior for $\\phi$  based on the distributions of $\\Sigma|D_n$, $m|D_n$ and (\\ref{eq7.1}).\n\n\\subsection{Simulation}\nWe present a simple simulated example. The returns $R_t$ are generated from  a 2-factor model:\n$R_t=\\Lambda f_t+u_t+2\\iota$, where $\\Lambda$ is a $N\\times 2$ matrix of factor loadings. The error terms $\\{u_{it}\\}_{i\\leq N, t\\leq n}$ are i.i.d. uniform U$[-2,2]$.  Besides, the components of $\\Lambda$ are standard normal, and the factors are also uniform U$[-2,2]$.  The true $m=ER_t=2\\iota$, $\\Sigma=\\Lambda\\Lambda^{T}+I_N$.\n\n\n\nWe set $N=5, n=200$. When   the posterior is calculated,  the DGP's distributions and the factor model structure are treated unknown, and we  apply the nonparametric Dirichlet Process prior on the CDF of $R_t-m$, with parameter $v_0=3$, and based measure $Q_0=\\mathcal{N}(0,1)$. We use a uniform prior for $(\\sigma^2,\\mu)$, and obtain the posterior distributions for $(m,\\Sigma, \\phi_1,\\phi_2,\\phi_3,\\sigma^2,\\mu)$.  More concretely, the prior is assumed to be:\n$$\n\\pi(\\sigma^2,\\mu|\\phi)=\\pi(\\sigma^2|\\phi,\\mu)\\pi(\\mu);\\quad \\pi( \\sigma^2|\\phi, \\mu)\\sim U[\\sigma^2_{\\phi}(\\mu), \\bar{\\sigma}^2], \\pi(\\mu)\\sim U[0,\\bar{\\mu}],\n$$\nwhere $\\mu$ and $\\phi$ are a priori independent. We sample 1,000 draws from the posterior of $(\\phi, \\sigma^2,\\mu)$.  Each time we first draw $(m,\\Sigma)$ from their marginal posterior distributions, based on which obtain the posterior draw of $\\phi$ from (\\ref{eq7.1}). In addition, draw  $\\mu$ uniformly from $[0,\\bar{\\mu}]$, and finally $\\sigma^2$ uniformly from $[\\sigma^2_{\\phi}(\\mu), \\bar{\\sigma}^2]$, where $\\sigma^2_{\\phi}(\\mu)$ is calculated based on the drawn $\\phi$ and $\\mu$.\n\n\n\nThe posterior mean $(\\hat\\phi_1,\\hat\\phi_2,\\hat\\phi_3)$ of $\\phi$ is calculated, based on which we calculate a Bayesian estimate of the boundary of the identified set (we set $\\bar{\\mu}=1.4$ and $\\bar{\\sigma}^2=6$):\n$$\n\\partial\\Theta(\\hat\\phi)=\\{\\mu\\in[0,\\bar{\\mu}], \\sigma^2\\in[0,\\bar{\\sigma}^2]: \\sigma^2=\\hat\\phi_1\\mu^2-2\\hat\\phi_2\\mu+\\hat\\phi_3\\},\n$$\nwhich is helpful to compute the BCS for the identified set. In addition, we estimate the support function $S_{\\phi}(\\nu)$ using  the posterior mean of $\\phi$.\n  In   Figure \\ref{f1}, we plot the Bayesian estimates of the support function for two cases: $\\nu_2\\in [0,1]$, $\\nu_1=\\sqrt{1-\\nu_2^2}$, and $\\nu_2\\in [-1,0]$, $\\nu_1=-\\sqrt{1-\\nu_2^2}$.\n\n \\begin{figure}[htbp]\n\\begin{center}\n\\caption{ Posterior estimates of support function. Left panel is for $\\nu_2\\in [0,1]$, $\\nu_1=\\sqrt{1-\\nu_2^2}$; right panel is for $\\nu_2\\in [-1,0]$, $\\nu_1=-\\sqrt{1-\\nu_2^2}$}\n\\includegraphics[width=8cm]{su1.eps}\n\\includegraphics[width=8cm]{su2.eps}\n\\label{f1}\n\\end{center}\n\\end{figure}\n\n\n\n Using the  support function, we  calculate the 95\\% posterior quantile $q_{\\tau}$ for $J(\\phi)$, based on which we construct the  BCS $\\Theta(\\hat\\phi )^{q_{\\tau}\/\\sqrt{n}}$ for the identified set. The boundary of $\\Theta(\\hat\\phi )^{q_{\\tau}\/\\sqrt{n}}$ is given by\n $$\n\\partial \\Theta(\\hat\\phi )^{q_{\\tau}\/\\sqrt{n}}= \\left\\{\\mu\\in[0,\\bar{\\mu}], \\sigma^2\\in[0,\\bar{\\sigma}^2]: \\inf_{z}\\sqrt{|z-\\mu|^2+|\\sigma^2_{\\hat\\phi }(z)-\\sigma^2|^2}=q_{\\tau}\/\\sqrt{n}\\right\\}.\n $$\nIn Figure \\ref{f2}, we plot the posterior draws of $(\\mu, \\sigma^2)$, $\\partial\\Theta(\\hat\\phi), \\partial \\Theta(\\hat\\phi )^{q_{\\tau}\/\\sqrt{n}}$ and the boundary of the true identified set. The scatter plot of posterior draws, the estimated boundaries and the BCS show that the true identified set is well estimated.\n\n\n\n\n\n\n  \\begin{figure}[htbp]\n\\begin{center}\n\\caption{1,000 posterior draws of $(\\mu,\\sigma^2)$. Solid line is the boundary of the true identified set; dashed line represents the estimated boundary using the posterior mean; dotted line gives the 95\\% BCS of the identified set. Plots are obtained based on a single set of simulated data. The BCS also covers a part of negative values for $\\sigma^2$. In practice, we can truncate it to ensure it is always positive.}\n\\includegraphics[width=7cm]{f2.eps}\n\\label{f2}\n\\end{center}\n\\end{figure}\n\n\n\\begin{comment}\n \\begin{table}[htdp]\n\\caption{Posterior estimates of $\\phi$}\n\\begin{center}\n\\begin{tabular}{c|c|c|c}\n\\hline\n&true&posterior mean & posterior mode\\\\\n\\hline\n$\\phi_1$&19.01&13.11& 12.18\\\\\n$\\phi_2$&9.51&6.67& 6.23\\\\\n$\\phi_3$&4.75&3.4& 3.21\\\\\n\\hline\n\\end{tabular}\n\\label{table3}\n\\small\n\n\\end{center}\n\\end{table}\n\n\n\\end{comment}\n\n\n\n\\section{Conclusion}\\label{s_Conclusion}\n\nWe propose a semi-parametric Bayesian procedure for inference about partially identified models.  Bayesian approaches are appealing in many aspects.  Classical Bayesian approach in this literature has been assuming a known  likelihood function. However, in many applications econometric models   only identify a set of moment inequalities, and therefore assuming a known likelihood function suffers from the risk of misspecification, and may result in inconsistent estimations of the identified set. On the other hand, Bayesian approaches that use moment-condition-based likelihoods such as the limited information and exponential tilted empirical likelihood, though guarantee the consistency, lack of probabilistic interpretations, and do not provide an easy way to make inference about the identified set.  Our approach, on the contrary, only requires a set of moment conditions but still possesses a pure Bayesian interpretation. Importantly, we can conveniently analyze both the partially identified parameter and its identified set. Moreover, we shed light on many appealing features of our proposed approach such as the computational efficiency for subset inference and projections.\n\n\nOur analysis primarily focuses on identified sets which are closed and convex. These sets are completely characterized by their support function. By imposing a prior on the support function, we construct its posterior distribution. It is shown that the support function for a very general moment inequality model admits a linear expansion, and its posterior is consistent. The Bernstein-von Mises theorem for the posterior of the support function is proven.\n\n\n\nSome researcher may argue that frequentist and Bayesian approaches are asymptotically equivalent because the prior is ``washed away\" as the sample size increases. So why bother with Bayesian asymptotics? Interestingly,   this is no longer the case under partial identification.  As was shown by Moon and Schorfheide (2012), when making inference about the partially identified parameter, the BCS can be strictly smaller than the FCS.   Therefore, the two fundamental statistical methodologies provide different inferences that are not necessarily equivalent even asymptotically.  This paper completes such a comparison. We also establish the large-sample correspondence of the two approaches for the identified set. It is also illustrated that the results hold in the uniform sense, in particular when point identification is nearly achieved.\n\n\n\n\n\n\n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{}\n\nMuch of our knowledge on quantum nature of atomic nuclei \ncomes from studies of nuclear reactions \nin which an energetic beam of one nuclear species \ncollides with a target made of another. \nAmong various collision processes, \nthe nucleon knockout reaction \nhas become recognized as one of the most sensitive tools \nfor spectroscopic studies, \nespecially for nuclei away from the stability line, \nwhich include even those beyond the drip line. \nThe knockout residue produced by removing a nucleon (or nucleons) \nfrom a fast moving beam particle, \nwhich impinges on a light target fixed in the laboratory, \nis observed in inverse kinematics \nby a detector placed in forward hemisphere efficiently. \nThe removed nucleon(s) will be selected democratically from the valence space, \nallowing states with unique, often rarely accessible configurations \nto be populated in this process. \nThe final state in the residue is identified \nby tagging de-excitation \n$\\gamma$ rays~\\cite{Navin08,Aumann00,Maddalena01} \n(see also references in Ref.~\\cite{Gade08b}) \nand by observing decay neutrons and constructing the invariant \nmass~\\cite{Simon99,Hoffman08,Hoffman09,Kondo10,Cao12,Lunderberg12,Aksyutina13,Kohley13}. \nFor one-nucleon knockout case, \nthe momentum spread of the residue reflects the Fermi motion of the nucleon \nsuddenly removed, \nand is sensitive to the orbital angular momentum (the $l$ value) \nof the struck nucleon. \nFurthermore, \nthe cross sections leading to individual final states \nrelate to the occupancy of single-particle orbits, \nproviding a link to details of the nuclear structure. \n\nThe present study aims at exploring unbound states in $^{16}{\\rm C}$ \nthrough an application of the one-neutron knockout technique \nto a $^{17}{\\rm C}$ beam \nimpinged on a proton target, \nfor which simple reaction mechanisms are expected. \nFocus is placed in a search of lowest-lying cross shell transitions, \nthe location of which reflects the shell gap between $p$ and $sd$ orbits. \nThe neutron-rich carbon (C) isotopes \nhave attracted attention as they often exhibit unique features: \nnone of the odd mass C (heavier than $^{13}{\\rm C}$) \nhas the ground-state spin parity of $J^{\\pi}_{\\rm g.s.}$=5\/2$^+$, \nthe values which are expected from a naive shell model. \nThere has been a debate about a reduced E2 transition strength \n(small proton collectivity) for the $2^+_1$ state \nin $^{16}{\\rm C}$~\\cite{Imai04,Elekes04,Ong08,Wiedeking08,Wuosmaa10,Petri12}. \nThere is evidence for neutron halo formation for \n$^{15}{\\rm C}$~\\cite{Sauvan00}, \n$^{19}{\\rm C}$~\\cite{Nakamura99}, and \n$^{22}{\\rm C}$~\\cite{Tanaka10,Kobayashi12}. \nFor some, if not all, of these features, \nnuclear deformation may play a key role, \nwhich occurs in this mass region \ndue to near degeneracy of the $\\nu d_{5\/2}$-$\\nu s_{1\/2}$ orbits: \nneutrons in these orbits can gain energy by breaking spherical symmetry \n(the Jahn-Teller effect)~\\cite{Hamamoto07}. \nThe effect of nuclear deformation \nwill further be signified by large quadrupole transition \nstrengths~\\cite{Elekes05,Ysatou08} \nand by a reduction of the major \n$p$-$sd$ shell gap~\\cite{Talmi60,Suzuki97,Wiedeking05}. \nA recent $\\beta$-delayed neutron \nemission study of $^{17}{\\rm B}$~\\cite{Ueno13} \nhas reported low-lying negative parity states in $^{17}{\\rm C}$, \namong which the lowest one was the $J^{\\pi}$=1\/2$^-$ state \nat the excitation energy of $E_x$=2.71(2) MeV. \nThe energy of this state, \nreflecting the $p$-$sd$ shell gap, \nturned out to be \nlower than those of neighboring odd mass C isotopes:  \n$E_x$=3.10 MeV for the $1\/2^-_1$ state in $^{15}{\\rm C}$ \nand $E_x$=3.09 MeV for the $1\/2^+_1$ state \nin $^{13}{\\rm C}$~\\cite{Ajzenberg91}. \nThis \nmight indicate an onset of the $p$-$sd$ shell gap quenching \ntowards heavier C isotopes. \nTo examine this picture in more detail \nit is worthwhile to accumulate data \non cross shell transitions in neighboring isotopes. \nThis Letter reports on new relevant spectroscopic information \non $^{16}{\\rm C}$ in its unbound $E_x$ region. \nBesides, \nbased on the parallel momentum distribution \nof the core fragment populated in a final state, \nit is demonstrated that the width of the distribution \nprovides a good measure of the $l$ value (and thus the parity) \nof the state populated; \nfor neutron knockout involving a proton target, \nthis has previously been shown \nbased on the transverse momentum distributions \nin the $^1{\\rm H}(^{18}{\\rm C},$$^{17}{\\rm C}^*)$~\\cite{Kondo09} \nand $^1{\\rm H}(^{14}{\\rm Be},$$^{13}{\\rm Be}^*)$~\\cite{Kondo10} reactions \nwith the aid of elaborate reaction mechanism calculations. \n\nDespite relative proximity to stability, \ninformation on energy levels of $^{16}\\textrm{C}$, \nparticularly that above the neutron threshold \n(the neutron separation energy of $^{16}{\\rm C}$ \nis $S_n$=4.250(4) MeV~\\cite{Audi03}), \nhas been limited. \nThis is partially due to ineffectiveness of $\\beta$ decay \nfor this particular nucleus, \nas recognized by the absence of a parent nucleus \n($^{16}{\\textrm B}$ is particle unstable). \nEarly spectroscopic studies on $^{16}\\textrm{C}$ \nutilized binary reactions involving transfers of neutrons. \nThe $^{14}\\textrm{C}(t,p)^{16}\\textrm{C}$ two-neutron transfer \nstudies~\\cite{Fortune77,Balamuth77,Fortune78,Sercely78} \nhave investigated levels below 7 MeV, \nincluding six bound states and an unbound state at 6.11 MeV. \nSince the ground state of $^{14}\\textrm{C}$ \nis characterized by neutron $p$-shell closure, \nthe states populated \nmostly involved configurations with two $sd$-shell neutrons, $(1s0d)^2$. \nThe $^{13}\\textrm{C}(^{12}\\textrm{C},$$^9\\textrm{C})^{16}\\textrm{C}$ \nthree-neutron transfer study~\\cite{Bohlen03} \nhas reported 14 more states up to $E_x$=17.4 MeV, \nincluding states with more complex configurations. \nDue to kinematical matching~\\cite{Bohlen03} \nstates with high angular momenta \nwere favorably populated. \nCombining information \nfrom the recent $^{15}\\textrm{C}(d,p)^{16}\\textrm{C}$ reaction study \nusing a radioactive $^{15}\\textrm{C}$ beam~\\cite{Wuosmaa10}, \nsound $J^{\\pi}$ assignments \nhave been available for bound states. \nFor unbound states \nonly the 8.92-MeV level \nhas received a firm assignment of $5^-$~\\cite{Bohlen03}. \nTwo earlier one-neutron knockout studies on $^{17}{\\rm C}$ using Be targets \nfocused on transitions leading to bound final states \nin $^{16}{\\rm C}$ by means of \nin-beam $\\gamma$-ray spectroscopy~\\cite{Maddalena01,Rodriguez-Tajes12}. \nThey provided information \nnot only on excited states of $^{16}{\\rm C}$ \nbut also on ground state properties of $^{17}{\\rm C}$, \ne.g., the spin parity, \n$J^{\\pi}_{\\rm g.s.}(^{17}{\\rm C})$=$3\/2^+$, and \nno halo formation in spite of the remarkable low neutron separation energy \nof $S_n$=0.727(18) MeV~\\cite{Audi03} \ndue to a high angular momentum of $l$=2 for the valence neutron. \n\nThe experiment was performed at the RIPS facility~\\cite{Kubo92} of RIKEN. \nDetails of the setup \nare provided in Refs.~\\cite{Ysatou08,Tshoo12}, \nand a preliminary report of this work has been presented \nin Ref.~\\cite{Hwang13}. \nThe $^{17}\\textrm{C}$ beam \nwas produced from a 110-MeV\/nucleon $^{22}\\textrm{Ne}$ beam \nwhich impinged on a 6-mm-thick Be target. \nThe typical $^{17}\\textrm{C}$ beam intensity \nwas 10.2 kcps with a momentum spread of $\\Delta P\/P$=$\\pm 0.1\\%$. \nThe beam profile \nwas monitored by a set of parallel-plate avalanche counters (PPACs) \nplaced upstream of the experimental target. \nThe target \nwas pure liquid hydrogen~\\cite{Ryuto05} contained in a cylindrical cell: \n3 cm in diameter, 120$\\pm$2 mg\/cm$^2$ in thickness, \nand having 6-$\\mu$m-thick Havar foils for the entrance and exit windows. \nThe average energy of $^{17}\\textrm{C}$ at the middle of the target \nwas 70 MeV\/nucleon. \nThe target was surrounded by a NaI(Tl) scintillator array \nused to detect $\\gamma$ rays from charged fragments. \nThe fragment was bent by a dipole magnet behind the target, \nand was detected by a plastic counter hodoscope \nplaced downstream of the magnet. \nThe $\\Delta E$ and time-of-flight (TOF) information in the hodoscope \nwas used to identify the $Z$ number of the fragment. \nThe trajectory \nwas reconstructed by a set of multi-wire drift chambers (MWDCs) \nbefore and after the magnet, \nwhich, together with TOF, \ngave mass identification. \nNeutrons were detected by two walls \nof plastic scintillator arrays placed 4.6 and 5.8 m downstream from the target. \nThe neutron detection efficiency \nwas $24.1\\pm0.8\\%$ for a threshold setting of 4 MeVee. \nThe relative energy ($E_{\\rm rel}$) of the final system \nwas calculated from momentum vectors of the charged fragment and the neutron. \nIn deducing the fragment vector, \ninformation on the impact point on target in transverse directions \n(determined by the upstream tracking detectors) \nwas taken into account \ntogether with hit information within the MWDC placed behind the target. \nNeutron coincidence events \nwere classified in terms of $E_{\\rm rel}$ \nand the Fermi momentum of the struck neutron $k_3$. \nIn the sudden approximation, \nthe latter \ncorresponds to the transferred momentum \nto the knockout residue ($^{16}{\\rm C}$). \nThe detector acceptance \nwas evaluated by a Monte Carlo simulation \nas a function of $E_{\\rm rel}$ and $k_3$. \n\nFigure~\\ref{fig:spectrum_fit} \nshows relative energy dependence of cross sections for the \n(a) $^1\\textrm{H}(^{17}\\textrm{C},$$^{15}\\textrm{C}$+$n)$ and \n(b) $^1\\textrm{H}(^{17}\\textrm{C},$$^{15}\\textrm{C}\n(5\/2^+; 0.74\\ {\\rm MeV})$+$n)$ reactions at 70 MeV\/nucleon. \nBackground contributions measured by an empty target were subtracted. \nError bars are statistical ones. \nShown in the inset of Fig.~\\ref{fig:spectrum_fit} (b) \nis the \"Doppler-corrected\" energy spectrum for $\\gamma$ rays \nemitted from the decay product nucleus $^{15}{\\rm C}$. \nA peak around $E_{\\gamma}$=0.8 MeV \narises from the decay of the first $5\/2^+$ level at 0.74 MeV \n(the only bound excited state) \nto $^{15}{\\rm C}_{\\rm g.s.}$. \nThe $5\/2^+$ state \nis an isomeric state having a half-life of 2.61$\\pm$0.07 ns~\\cite{Ajzenberg91}. \nThis \nlong life time \ncaused the emission point \nof the de-excitation $\\gamma$ ray \nto be distributed along the path of the fast-moving decay product. \nThe Doppler correction for the $\\gamma$ ray energy \nwas made by assuming that the decay occurs at 40.7 cm downstream of the target \n(an average decay point expected from the average beam energy \nand the known mean life for the isomeric state) \nin both data reduction and simulation by the \\textsc{geant} code~\\cite{GEANT}. \nThe latter \nwas done fully taking into account realistic geometry of the experiment. \nA higher energy tail for the photo-electric peak \nis due to this incomplete Doppler correction procedure. \nThe simulated response, however, \nreproduced the data well \nas shown by the green solid line. \nThe photo-peak efficiency over $E_{\\gamma}$=0.60--1.12 MeV \n($\\gamma$-ray window) \nwas estimated to be 5.1(3)\\% by the \\textsc{geant} simulation. \nFigure~\\ref{fig:spectrum_fit} (b) \nwas obtained by gating on the $5\/2^+$ $\\gamma$ peak \nwith the above window \nand by correcting for the detection efficiency. \nThe background component \nwas subtracted \nby assuming (i) \nthat the background portion is the same \nas that in the $\\gamma$-ray spectrum \nin the inset of Fig.~\\ref{fig:spectrum_fit} (b): \nthe portion of the area beneath the dotted line over the $\\gamma$-ray window, \nwhich amounts to 46\\%, \nand (ii) \nthat the background shape is \ncharacterized by \nthe inclusive spectrum in Fig.~\\ref{fig:spectrum_fit} (a). \nA peak \nis clearly seen at $E_{\\rm rel}$=0.46 MeV \nin Fig.~\\ref{fig:spectrum_fit} (a). \nThis is also evident in Fig.~\\ref{fig:spectrum_fit} (b), \nindicating that \nthis peak feeds the $5\/2^+$ state in $^{15}{\\rm C}$ after emitting a neutron. \nBesides, another resonance \nis visible at $E_{\\rm rel}$$\\approx$1.3 MeV \nin Fig.~\\ref{fig:spectrum_fit} (b). \nThese were observed for the first time. \n\n\\begin{figure}[t]\n\\resizebox{0.90\\columnwidth}{!}{%\n\\includegraphics[angle=-90]{spectrum_sub.eps}%\n}\n\\caption{(Color online.) \nRelative energy spectra \nfor the \n(a) $^1\\textrm{H}(^{17}\\textrm{C},$$^{15}\\textrm{C}$+$n)$ and \n(b) $^1\\textrm{H}(^{17}\\textrm{C},$$^{15}\\textrm{C}\n(5\/2^+; 0.74\\ {\\rm MeV})$+$n)$ \none-neutron knockout reactions at 70 MeV\/nucleon. \nShown in the inset of panel (b) \nis the Doppler-corrected energy spectrum of $\\gamma$ rays \nemitted from $^{15}{\\rm C}$. \nNeutron coincidence is required for this spectrum. \nGreen solid lines represent the results of the fit; \ndotted lines assumed background; \nred dashed lines extracted individual resonances. \nA decay scheme for states populated is shown in panel (a). \n\\label{fig:spectrum_fit}}\n\\end{figure}\n\nThe relative energy spectrum of Fig.~\\ref{fig:spectrum_fit} (a) \nwas used in a fitting analysis \nto extract the parameters for the resonances: \n$E_{\\rm rel}$ and the populating cross section $\\sigma^{\\rm exp}_{-1n}$. \nTheir responses (dashed lines) \nwere generated by a Monte Carlo simulation \nwhich takes into account the detector resolution, beam profile, \nCoulomb multiple scattering, \nand range difference inside the target. \nThe relative energy resolution \nwas estimated to scale as $\\Delta E_{\\rm rel}$=$0.17\\sqrt{E_{\\rm rel}}$ MeV \n(in rms). \nFor the $E_{\\rm rel}$=0.46-MeV state, \na finite line width ${\\mathit \\Gamma}$ for an $l$=1 neutron emission \nwas considered by adopting \na single Breit-Wigner function~\\cite{Lane-Thomas58,Ysatou08} \n(for other states, \nnot as well isolated in the present data as this state, \nonly the instrumental resolution was taken into account). \nIn the analysis an excess strength \nwas recognized at $E_{\\rm rel}$$\\approx$1.9 MeV. \nThis may correspond to the nearest known state \nat $E_x$=6.11 MeV~\\cite{Fortune77,Sercely78} \nif the decay product nucleus $^{15}{\\rm C}$ is populated in the ground state \n(in Fig.~\\ref{fig:spectrum_fit} (a) the response of this strength \nwas created by assuming $E_{\\rm rel}$=1.86 MeV: \nthe difference between $E_x$=6.11 MeV and $S_n$ for $^{16}{\\rm C}$). \nA similar fitting analysis using the $\\gamma$-ray coincidence spectrum \nin Fig.~\\ref{fig:spectrum_fit} (b), however, \ndid not exclude the possibility that this component \nis absent from this spectrum. \nIt is quoted that the upper limit on the fraction of $^{15}{\\rm C}$ fragments \nfrom the decay of the $E_{\\rm rel}$=1.86-MeV resonance, \nthat were in the 0.74-MeV state \nis 20\\% of the strength found in the spectrum \nin Fig.~\\ref{fig:spectrum_fit} (a). \nThe solid line in Fig.~\\ref{fig:spectrum_fit} (a) \nshows the result of the fit to the total inclusive spectrum. \nThe background (dotted line) coming from transitions to the continuum and \nfrom detecting neutrons \nnot associated with the decay of excited states in $^{16}{\\rm C}$, \ne.g., neutrons emitted from excited $^{17}{\\rm C}$ nuclei \nthat are created by inelastic scattering processes, \nwas simulated by a function \n$a(E_{\\rm rel})^b\\exp(-cE_{\\rm rel})$ \nwith $a$, $b$, and $c$ free parameters. \nThe results of the fit \nare summarized in Table~\\ref{tbl:sigma_1n}. \n$E_x$ \nwas calculated by $E_x$=$E_{\\rm rel}$+$S_n$+$E^*$, \nwith $E^*$ the excitation energy of $^{15}{\\rm C}$. \nThe errors include statistical ones \nand those due to the choice of the background shape \n(the latter, \nestimated by further assuming thermal emission of a neutron ($b$=1\/2) \nand neutron evaporation ($b$=1) for the background shape, \nturned out to be a dominant source of the error: \nthey were 80, 90, and 70\\% of the errors quoted in Table~\\ref{tbl:sigma_1n} \nfor $E_{\\rm rel}$, ${\\mathit \\Gamma}$, and $\\sigma^{\\rm exp}_{-1n}$, respectively, \nfor the $E_{\\rm rel}$=0.46-MeV state, \nwhile they \nwere 90, 20, and 70\\% in the (upper bounds of) errors \nof $\\sigma^{\\rm exp}_{-1n}$ for the $E_{\\rm rel}$=1.86-MeV state \nand of $E_{\\rm rel}$ and $\\sigma^{\\rm exp}_{-1n}$ \nfor the $E_{\\rm rel}$=1.29-MeV state, respectively). \nFor $\\sigma^{\\rm exp}_{-1n}$, \nuncertainties originating from the target thickness and \nneutron detection efficiency \nare included. \nThe same fit was repeated \nfor the $\\gamma$-ray coincidence spectrum of Fig.~\\ref{fig:spectrum_fit} (b) \nusing the responses \nfor the $E_{\\rm rel}$=0.46 and 1.29-MeV states, \nobtained from the fit to the inclusive spectrum \nin Fig.~\\ref{fig:spectrum_fit} (a). \nTo quantify the character of the latter state, \nas a state built on the $^{15}{\\rm C}^*$(0.74 MeV) excited core, \nthe fit was repeated by changing the strength from the original one. \nBy finding the fractional value at which the $\\chi^2$ of the fit \nalters from the minimum by one unit, \na lower limit of 32\\% was deduced. \nThe extraction of the $l$ value of the knocked-out neutron \nfrom a differential quantity \nfor the $E_{\\rm rel}$=0.46-MeV state \nis explained later. \n\n\nTo allow discussion in terms of nuclear structure, \nreaction model calculations based on the Glauber \napproximation~\\cite{Hansen03} \nwere performed. \nThe one-neutron removal cross section $\\sigma^{\\rm th}_{-1n}$ \nis expressed for a given final state with $J^{\\pi}$ as \n\\begin{equation}\n\\sigma^{\\rm th}_{-1n}=\\sum_{nlj}\\left( \\frac{A}{A-1}\\right)^{N}\nC^2S(J^{\\pi},nlj)\n\\sigma_{\\rm sp}(nlj,S_n^{\\rm eff}), \n\\end{equation}\nwhere $A$ is the projectile mass, $N$ the major oscillator quantum number, \n$C^2S$ the spectroscopic factor, \nand $\\sigma_{\\rm sp}$ the single-particle cross section. \nThe quantum numbers of the removed neutron are denoted by $nlj$. \n$S_n^{\\rm eff}$ is the effective separation energy given by \nthe sum of $S_n$ of the projectile \nand $E_x$ of the residue. \n$\\sigma_{\\rm sp}$ \nwas calculated by the code \\textsc{csc\\_gm}~\\cite{Abu-Ibrahim03} \ntaking into account \nboth stripping and diffractive processes \n(effective nature of the nucleon-nucleon ($NN$) profile function used \nresulted in small non-zero stripping cross sections)~\\cite{Hansen03}. \nThe elastic $S$ matrix \nfor the collision \nof the residue (core) with the proton target \nwas calculated by folding the finite-range Gaussian $NN$ \nprofile function~\\cite{Abu-Ibrahim08} \nwith the point proton and neutron densities of the core \nobtained from \nthe Hartree-Fock (HF) calculation using the SkX interaction~\\cite{Brown98}. \nThe $S$ matrix for describing the scattering of \nthe valence neutron with the target proton \nwas given by $S(b)$=1$-{\\mathit \\Gamma}_{pn}(b)$, \nhere $b$ is the impact parameter of the colliding nucleons, \nand ${\\mathit \\Gamma}_{pn}$ \nthe profile function for proton-neutron scattering. \nThe parameters chosen for the profile function \nare those \ndescribing the $NN$ total and elastic cross sections \nconsistently~\\cite{Abu-Ibrahim08}.\nThe neutron-residue relative motion \nwas calculated in a Woods-Saxon potential. \nThe depth was adjusted \nso as to reproduce $S_n^{\\rm eff}$, \nfor a diffuseness $a_0$=$0.7$ fm and a reduced radius $r_0$ specifically chosen \nto be consistent with the HF calculation~\\cite{Terry04,Gade08}: \n$r_0$ generates a single-particle \nwave function with a rms neutron-core separation \nof $r_{\\rm sp}$=$[A\/(A-1)]^{1\/2}r_{\\rm HF}$ at the HF-predicted binding energy, \nwhere $r_{\\rm HF}$ is the HF rms radius of each orbit. \nThe spin-orbit potential \nhad the same $a_0$ and $r_0$ as the central one with a strength of $-12$ MeV \nin the notation of Ref.~\\cite{Bertulani06}. \nThe HF radius for the $p_{1\/2}$ ($p_{3\/2}$) orbit of $^{17}\\textrm{C}$, \nfor example, \nis \n2.966 (2.779) fm; \nthis translates into $r_{\\rm sp}$=3.057 (2.865) fm, \nwhich is reproduced by taking $r_0$=$1.263$ (1.234) fm. \nThe $C^2S$ values \nwere obtained by the shell-model code \\textsc{nushell}~\\cite{nushell} \nusing the WBT interaction~\\cite{Warburton92} in the $spsdpf$ model space. \nThe calculated results \nfor relevant states \nare given in Table~\\ref{tbl:sigma_1n}. \n$\\sigma^{\\rm th}_{-1n}$ \nincludes contributions from both stripping ($\\sigma_{\\rm str}$) and \ndiffractive ($\\sigma_{\\rm diff}$) mechanisms. \nDue to inert nature of the proton, \nthe latter dominates the knockout processes. \n\n\\begin{table}[t]\n\\caption{States populated by \nthe $^1{\\rm H}$($^{17}{\\rm C}$,$^{16}{\\rm C}$) reaction. \nTheoretical cross sections were obtained by using the Glauber-model code \n\\textsc{csc\\_gm}~\\cite{Abu-Ibrahim03} and the shell-model spectroscopic factors \ncalculated with the WBT interaction~\\cite{Warburton92}. \nCalculations \nused $S_n^{\\rm eff}$ involving experimental $E_x$ values. \n\\label{tbl:sigma_1n}} \n\\begin{ruledtabular}\n\\begin{tabular}{ccccccccccc}\n\\multicolumn{4}{l}{Experiment}  & &\n\\multicolumn{3}{l}{Theory} \\\\\n\\cline{1-5} \n\\cline{6-10}\n$E_{\\rm rel}$   & \n$E_x$   & \n${\\mathit \\Gamma}$ & \n$l$ &\n$\\sigma^{\\rm exp}_{-1n}$ & &\n$E_x$   & \n$\\sigma_{\\rm str}$  & \n$\\sigma_{\\rm diff}$  & \n$\\sigma^{\\rm th}_{-1n}$  & \n$J^{\\pi}$     \\\\ \n(MeV)  & (MeV) & (MeV) & ($\\hbar$) & (mb)  &  & \n(MeV)  & (mb)  & (mb)  & (mb)  & \n    \\\\ \\hline\n0.463(3)\\protect\\footnotemark[1] & 5.45(1) & 0.03(1) & 1 & 10.6(6) & & \n5.57 & 1.38 & 14.23 & 15.61 & \\multicolumn{1}{c}{$2^-_1$} \\\\\n1.86\\protect\\footnotemark[2]  & \n6.11 & \n--- & --- & \n$2.0^{+0.4}_{-0.8}$ \n& &  \n5.75  & 0.05 & 0.53 &  0.58 & ($3^-_1$) \\\\\n      &                              &     &     &           & &  \n7.60  & 0.09 & 1.37 &  1.46 & ($2^+_3$) \\\\\n      &                              &     &     &           & &  \n8.81  & 0.04 & 0.31 &  0.35 & ($4^+_2$) \\\\\n1.29(2)\\protect\\footnotemark[1]  & 6.28(2) & --- & --- & \n$2.5^{+0.2}_{-1.9}$\n& &  \n6.55  &  0.61  & 5.43  & 6.04 & ($1^-_2$) \\\\\n                                 &         &     &     &          & &  \n6.63  &  0.28  & 2.57  & 2.85 & ($2^-_2$)\n\\end{tabular}\n\\footnotetext[1]{Observed in coincidence with the 0.74-MeV \n$\\gamma$ ray from $^{15}{\\rm C}$. }\n\\footnotetext[2]{Derived from the energy $E_x$=6.11 MeV \nin Ref.~\\cite{Fortune77} \nby assuming the $^{15}{\\rm C}$ core is in the ground state. }\n\\end{ruledtabular}\n\\end{table}\n\nThe state observed at $E_x$=5.45 MeV \nwas found to be well explained by the $2^-_1$ shell-model state \nin both position and cross section, \nmaking an assignment of $2^-$ appropriate. \nThe $2^-_1$ state \nexhibited the highest cross section of unidentified shell-model states \nin the energy region of interest. \nThe summed $\\sigma^{\\rm th}_{-1n}$ for predicted $2^-$ and $1^-$ states \nbelow 8 MeV, \nwhere major strengths are concentrated, \nare 20.9 \n(15.5, 2.8, and 2.6 mb at $E_x$=5.57, 6.63, and 7.23 MeV, respectively) \nand 11.5 mb (see below for composition), respectively. \nTheir ratio is near to the statistical ratio of 5:3 \nexpected for a doublet with spins $J$=2 and 1, \nallowing an interpretation that the $2^-$ and $1^-$ states \nare formed by coupling a hole in the $\\nu p_{1\/2}$ orbit \nto three neutrons with $J$=3\/2 in the $sd$ orbits \n(note that $J^{\\pi}_{\\rm g.s.}(^{17}{\\rm C})$=$3\/2^+$). \nThe predicted $1^-$ strength is distributed \namong states at $E_x$=5.79, 6.55, and 6.98 MeV \nwith $\\sigma^{\\rm th}_{-1n}$=0.6, 6.0, and 4.9 mb, respectively. \nThe fragmentation of the strength \nand the general trend in the Glauber model \nto overestimate the cross section~\\cite{Gade08b,Gade08} \nwould exclude \nan assignment of $1^-$ for the 5.45-MeV state \nwith $\\sigma^{\\rm exp}_{-1n}$=10.6(6) mb. \n\nFigure~\\ref{fig:pl} \nshows the laboratory parallel momentum ($p_{||}$) distribution \nleading to the 5.45-MeV state. \nThis was obtained by subdividing, in terms of $p_{||}$, \nthe inclusive spectrum \nand repeating the fitting procedure described above. \nThe errors are statistical ones. \nAlso plotted in Fig.~\\ref{fig:pl} \nare the $p_{||}$ distributions calculated with \\textsc{csc\\_gm} \nfor varying $l$ values. \nAn experimental resolution of 43(1) MeV\/$c$ in rms is convoluted. \nFactors relevant to stripping mechanisms are dropped, \nand the curves \nrepresent the Fourier transform of the single-particle wave functions. \nThe full width at half maximum (FWHM) of the experimental distribution \nfor the 5.45-MeV state \nwas determined by a fit using a Gaussian \nto be 210(11) MeV\/$c$ after unfolding the resolution. \nIn the fit, \na low-energy tail ($p_{||} <$ 5.72 GeV\/$c$), \nwhich often suffers from higher-order effects~\\cite{Tostevin02}, \nwas \neliminated. \nThe fit curve (not shown) is similar to the $l$=1 curve (solid line). \nThe width \nagrees well with 233 MeV\/$c$ FWHM \ncalculated \nfor $p$-wave knockout, \nwhereas for $s$- (dotted line) and $d$-wave (dashed line) knockout, \nwidths of 121 and 377 MeV\/$c$ FWHM were respectively predicted, \nincompatible with the measurement. \nThis observation \nagrees to the expected character \nof the 5.45-MeV, $2^-$ state as having a neutron hole in the $p$ orbit, \nillustrating the robust feature of the $p_{||}$ distribution \nas an $l$ identifier. \n\nThe large populating cross sections observed for the 6.11-MeV state \nin the $^{14}{\\rm C}(t,p)^{16}{\\rm C}$ reactions~\\cite{Fortune77,Sercely78} \nhave suggested that this state is either of the natural parity \n$2^+$, $3^-$, or $4^+$ states. \nThe knockout cross sections calculated for the relevant \n$2_3^+$, $3_1^-$, and $4_2^+$ shell-model states, \ntogether with their shell-model energies, \nare compared to the data in Table~\\ref{tbl:sigma_1n}. \nThe present data turned out not to provide a strong constraint \non the $J^{\\pi}$ values for this state, \nalthough in terms of comparisons in both $E_x$ and $\\sigma_{-1n}$ \nthey seem to prefer the assignment of $2^+$ or $3^-$. \nThe 6.28-MeV state exhibited the same decay pattern as the strongest \n5.45-MeV $2^-$ state \nwith a sizable cross section. \nThe $1^-_2$ and $2^-_2$ states \npredicted at 6.55 and 6.63 MeV, respectively, \nhad large populating cross sections and are candidates for this state. \n\n\n\\begin{figure}[t]\n\\resizebox{0.90\\columnwidth}{!}{%\n\\includegraphics[angle=0]{pl_lab.eps}%\n}\n\\caption{Laboratory $p_{||}$ distribution of $^{16}{\\rm C}$ populated \nin the 5.45-MeV state after knockout from $^{17}{\\rm C}$ (open circles). \nThe dotted, solid, and dashed lines \nare the Fourier transform of single-particle wave functions \nof orbitals with $l$=0, 1, and 2, respectively. \n\\label{fig:pl}}\n\\end{figure}\n\nThe presently observed $2^-$, 5.45-MeV state in $^{16}{\\rm C}$ \nbelongs to a member of the lowest-lying states \nhaving \nan opposite parity to the ground state. \nThe location of such states provides a measure \nof the $p$-$sd$ shell gap and \nit is well explained by the shell model using the WBT interaction \nacross the C isotopes, $^{11-15}{\\rm C}$. \nTo illustrate the latter, \ntheir energies are compared \nto the shell-model values \nin Fig.~\\ref{fig:wbt_cross_shell_states}. \nIn a recent study of $\\beta$-delayed neutron emission \nof $^{17}{\\rm B}$~\\cite{Ueno13}, \nthree low-lying negative parity states \nwere newly identified in just one-neutron heavier nucleus $^{17}{\\rm C}$. \nThe WBT interaction turned out to fail \nin predicting their location by about 1 MeV \n(theory predicts lower values, see also Fig.~\\ref{fig:wbt_cross_shell_states}), \nand several possible mechanisms, \nsuch as reduction in pairing energy for neutrons in the $sd$ orbits, \nwere discussed. \nThe present study adds a case \nin which the shell model with the WBT interaction \npredicts the location of the lowest-lying cross shell transition properly \n(see also Fig.~\\ref{fig:wbt_cross_shell_states}), \nshowing \nthat this interaction \ndescribes the $p$-$sd$ shell gap in $^{16}{\\rm C}$ adequately. \nTo pin down the source of the discrepancy \nbetween theory and experiment \non the position of the cross shell transition in $^{17}{\\rm C}$, \nas discussed in Ref.~\\cite{Ueno13}, \nand to better understand the dynamical evolution of single-particle orbits \nand relevant residual interactions \naway from stability, \nfurther spectroscopic studies on such states \nin heavier C as well as neighboring isotopes are of help. \n\n\\begin{figure}[t]\n\\resizebox{0.90\\columnwidth}{!}{%\n\\includegraphics[angle=0]{wbt_cross_shell_states.eps}%\n}\n\\caption{(Color online.) \nThe migration of energies of (known) lowest-lying states in C isotopes, \nwhose parities are opposite to those of their respective ground states, \nin comparison to shell-model values \nobtained by using the WBT interaction~\\cite{Warburton92}. \nData for $^{11-15}{\\rm C}$ (filled circles) \nare from Refs.~\\cite{Ajzenberg91,Ajzenberg90}. \nThe data point for $^{16}{\\rm C}$ (red filled square) \nis from the present study, \nwhile that for $^{17}{\\rm C}$ (open diamond) is from Ref.~\\cite{Ueno13}. \nThe shell-model calculations \nwere performed within the 2$\\hbar \\omega$ and 0$\\hbar \\omega$ bases \n(for both positive and negative parity states) \nfor $^{11-15}{\\rm C}$ and $^{16,17}{\\rm C}$, respectively. \n\\label{fig:wbt_cross_shell_states}}\n\\end{figure}\n\n\nIn summary, \none-neutron knockout from $^{17}{\\rm C}$ on a proton target \nwas exploited in populating two new states at 5.45(1) and 6.28(2) MeV, \nand a previously known state at 6.11 MeV in $^{16}{\\rm C}$. \nThe energy spectrum \nwas constructed utilizing the invariant mass method \ninvolving a decay neutron and a $^{15}{\\rm C}$ fragment. \nDe-excitation $\\gamma$ rays from the latter \nwere measured to correctly locate the resonances. \nFor the 5.45-MeV state, \nan attempt was made to deduce the orbital angular momentum \nof the knocked-out neutron \nfrom the parallel momentum distribution \nassociated with the unbound knockout residue. \nThis, together with a comparison in terms of the measured and calculated \nknockout cross sections, \nhas led to a spin-parity assignment of $2^-$ for this state. \nPossible spins and parities have been suggested for the other states, \nbringing about an advanced understanding \nof the level scheme of $^{16}{\\rm C}$. \nThe energy of the first $2^-$ state was adequately reproduced by the \nstandard shell-model calculation using the WBT interaction \nwithout invoking modifications to the residual interaction. \n\n\n\n\n\n\n\n\\begin{acknowledgments}\nThis work was supported in part by \nthe Grant-in-Aid for Scientific Research (15740145) of MEXT Japan \nand the NRF grant (R32-2008-000-10155-0 (WCU), 2010-0027136) of MEST Korea. \n\\end{acknowledgments}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAveraging techniques have been widely used in the construction and the stability analysis of solutions to time varying differential equations \\cite{bogoliubov1961, sanders2007averaging, sarychev2001lie, bullo2002averaging, vela2003ageneral,maggia2020higher}. The rigorous application of the method of averaging begins by writing the system on the form:\n\\begin{align}\n    \\label{eq:voc_form}\\frac{d\\bm{\\zeta}}{d\\tau} & = \\textbf{f}_0(\\bm{\\zeta},\\tau) + \\varepsilon\\,\\textbf{f}_1(\\bm{\\zeta},\\tau,\\varepsilon)\n\\end{align}\npossibly via coordinate changes and time scaling, where the vector field $\\textbf{f}_1$ is periodic in $\\tau$ and $\\varepsilon$ is a small parameter. When $\\textbf{f}_0=0$, the system is said to be on the averaging canonical form and the averaging theorem can be directly applied \\cite[Chapter 2]{sanders2007averaging}. If $\\textbf{f}_0\\neq 0$, the Variation of Constants (VOC) formula may be used to force the system on the averaging canonical form \\cite[Section 1.6]{sanders2007averaging},\\cite[Section 9.1]{bullo2004geometric}. However, even when the vector field $\\textbf{f}_0$ is linear and time invariant, i.e. $\\textbf{f}_0(\\bm{\\zeta},\\tau) = \\textbf{A}\\bm{\\zeta}$ for some matrix $\\textbf{A}$, the VOC formula is practically useful only when the eigenvalues of the matrix $\\textbf{A}$ are purely imaginary. This is due to the fact that the pullback of the vector field $\\textbf{f}_1$ under the flow of $\\textbf{f}_0$ will contain exponentially growing terms (see the discussion in \\cite[Section 1.7]{sanders2007averaging}). \n\nIn this manuscript, we analyze a class of singularly perturbed high-frequency, high-amplitude oscillatory systems described by equation (\\ref{eq:orig_sys}) which naturally arises in extremum seeking applications \\cite{krstic2000stability, durr2013lie}, and can be put on the form (\\ref{eq:voc_form}). Yet, the VOC formula is not useful in analyzing this class of systems for the reasons outlined in the previous paragraph. Moreover, recent results in the literature such as the singularly perturbed Lie Bracket Approximation framework \\cite{durr2015singularly, durr2017extremum} do not capture the stability properties as we illustrate below.\n\nTo analyze the behavior of this class of systems, we combine the higher order averaging theorem \\cite{sanders2007averaging} with singular perturbation techniques \\cite{khalil2002nonlinear} in a way that accounts for the interaction between the fast periodic time scale and the singularly perturbed part of the system. Furthermore, we propose a novel 3D source seeking algorithm for rigid bodies with a non-collocated sensor. The proposed algorithm is inspired by the chemotactic strategy of sea urchins sperm cells for seeking the egg in 3D \\cite{friedrich2007chemotaxis,abdelgalil2021sperm}, and it utilizes the special structure of the matrix group SO(3). We prove the practical stability of the proposed algorithm using the singularly perturbed averaging results we state here. \n\\iffalse 0\\section{Motivational Example}\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figs\/motivational_example_numerical_results.pdf}\n    \\caption{Numerical results for the motivation example. The initial condition is taken as $(-2,2,2,2)$, and the parameters are $\\beta=1\/2$, $\\omega=4\\pi$ and $\\mu=1\/(4\\pi)$.}\n    \\label{fig:motivation_exmp}\n\\end{figure}\nWe begin the discussion with a simple example to illustrate the motivation behind our results. Consider the system:\n\\begin{gather}\n    \\label{eq:mot_ex_1}\\frac{dx_1}{d\\tau}= \\mu \\beta x_1 + \\mu \\sqrt{2\\omega}\\sin(x_1^2+ x_4-x_3 +\\omega \\mu \\tau)\\\\\n    \\label{eq:mot_ex_2}\\frac{dx_2}{d\\tau}= \\mu \\beta x_2 + \\mu \\sqrt{2\\omega}\\cos(x_1^2+ x_4-x_3 +\\omega \\mu \\tau)\\\\\n    \\label{eq:mot_ex_3}\\frac{dx_3}{d\\tau}= x_4 - x_3, \\qquad     \\frac{dx_4}{d\\tau}= x_2^2 -x_4\n\\end{gather}\nwith an initial condition $(x_{1,0},x_{2,0},x_{3,0},x_{4,0})\\in\\mathbb{R}^4$, where $\\beta$ is a fixed positive parameter, and $\\mu$ and $\\omega$ are positive control parameters. Observe that the system (\\ref{eq:mot_ex_1})-(\\ref{eq:mot_ex_3}) has the same form as the class of systems considered in \\cite{durr2015singularly}. If we attempt to apply the results in \\cite{durr2015singularly}, we obtain the quasi-steady state $(x_3,x_4)=\\bm{\\varphi}_0(x_1,x_2)=(x_2^2,x_2^2)$ for the singularly perturbed part of the system. Hence, the reduced order system according to \\cite{durr2015singularly} becomes:\n\\begin{gather}\n    \\label{eq:mot_ex_reduced_1}\\frac{dx_1}{d\\tau}= \\beta x_1+\\sqrt{2\\omega} \\sin(x_1^2+\\omega\\tau)\\\\\n    \\label{eq:mot_ex_reduced_2}\\frac{dx_2}{d\\tau}= \\beta x_2+\\sqrt{2\\omega} \\cos(x_1^2+\\omega\\tau)\\\\\n    \\label{eq:mot_ex_reduced_3}x_3 = x_2^2, \\qquad    x_4= x_2^2\n\\end{gather}\nIt is clear that the system (\\ref{eq:mot_ex_reduced_1})-(\\ref{eq:mot_ex_reduced_3}) has no practically asymptotically stable subsets for any value of $\\omega$, and so the results in \\cite{durr2015singularly} are not useful here. However, numerical simulations shown in Fig.\\ref{fig:motivation_exmp} demonstrate that the system exhibits practical stability-like behavior, which is not captured by the results in \\cite{durr2015singularly}.\n\\fi\n\\section{Singularly Perturbed Second Order Averaging}\nIn this section, we state the main theorem of our work. Consider the interconnection of systems on the form:\n\\begin{align}\n    \\label{eq:orig_sys}\n    \\begin{aligned}\n        \\dot{\\textbf{x}}&= \\textbf{f}(\\textbf{x},\\textbf{y},t,\\omega) + \\frac{1}{\\sqrt{\\omega}}\\textbf{f}_3(\\textbf{x},\\textbf{y},t,\\omega),  & \\textbf{x}(t_0) &= \\textbf{x}_0\\\\ \\dot{\\textbf{y}}&=\\textbf{g}(\\textbf{x},\\textbf{y},t,\\omega)+\\frac{1}{\\sqrt{\\omega}}\\textbf{g}_3(\\textbf{x},\\textbf{y},t,\\omega),  & \\textbf{y}(t_0) &= \\textbf{y}_0\n    \\end{aligned}\n\\end{align}\nwhere $\\textbf{x},\\textbf{x}_0\\in\\mathbb{R}^n,\\,\\textbf{y},\\textbf{y}_0\\in\\mathbb{R}^m,\\,t,t_0\\in\\mathbb{R}$, $\\omega\\in(0,\\infty)$, and the maps $\\textbf{f},\\,\\textbf{g}$ are given by:\n\\begin{gather*}\n    \\textbf{f}(\\textbf{x},\\textbf{y},t,\\omega) = \\sum_{i\\in\\{1,2\\}}\\omega^{1-\\frac{i}{2}}\\, \\textbf{f}_i(\\textbf{x},\\textbf{y},\\omega t)\\\\\n    \\textbf{g}(\\textbf{x},\\textbf{y},t,\\omega)= \\omega \\textbf{A}\\,(\\textbf{y}-\\bm{\\varphi}_0(\\textbf{x})) +\\sum_{i\\in\\{1,2\\}}\\omega^{1-\\frac{i}{2}}\\, \\textbf{g}_i(\\textbf{x},\\textbf{y},\\omega t)\n\\end{gather*}\nWe adopt the following assumptions on the regularity of the right-hand side of equation (\\ref{eq:orig_sys}):\n\\begin{asmp}\\thlabel{asmp:A}\nSuppose that for $i\\in\\{1,2\\}$:\n    \\begin{enumerate}\n        \\item \\label{asmp:item_A1}  ${ \\textbf{f}_i(\\cdot,\\cdot,\\tau)\\in\\mathcal{C}^{3-i}(\\mathbb{R}^{n+m};\\mathbb{R}^n)}$, ${ \\textbf{f}_i\\in\\mathcal{C}^0(\\mathbb{R}^{n+m+1};\\mathbb{R}^n)}$,\n        \\item $\\textbf{g}_i(\\cdot,\\cdot,\\tau)\\in\\mathcal{C}^{3-i}(\\mathbb{R}^{n+m};\\mathbb{R}^n)$, ${ \\textbf{g}_i\\in\\mathcal{C}^0(\\mathbb{R}^{n+m+1};\\mathbb{R}^n)}$,\n        \\item \\label{asmp:item_A2} $\\exists T>0$ s.t. ${ \\textbf{f}_i(\\cdot,\\cdot,\\tau+T) = \\textbf{f}_i(\\cdot,\\cdot,,\\tau)}$, and $ \\textbf{g}_i(\\cdot,\\cdot,\\tau+T)$ $= \\textbf{g}_i(\\cdot,\\cdot,,\\tau)$, $\\,\\forall \\tau \\in\\mathbb{R}$,\n        \\item $\\textbf{f}_3$ and $\\textbf{g}_3$ are locally Lipschitz continuous in $\\textbf{x},\\textbf{y}$ and jointly continuous in all of their arguments,\n        \\item ${ \\int_{0}^{T}\\textbf{f}_1(\\cdot,\\cdot,s)ds} = 0$, $\\bm{\\varphi}_0(\\cdot)\\in\\mathcal{C}^{3}(\\mathbb{R}^n;\\mathbb{R}^m)$, and the matrix $ \\textbf{A}$ is Hurwitz.\n    \\end{enumerate}\n\\end{asmp}\n\\begin{rem}\nWe restrict our treatment here to the periodic case for simplicity. The extension to the case when the vector fields are quasi-periodic is straightforward but the computations are more involved.\n\\end{rem} \nNext, consider the reduced order system:\n\\begin{gather} \\label{eq:reduced_system}\n    \\dot{\\tilde{\\textbf{x}}}= \\sqrt{\\omega}\\,\\tilde{\\textbf{f}}_1(\\tilde{\\textbf{x}},\\omega t )+ \\tilde{\\textbf{f}}_2(\\tilde{\\textbf{x}},\\omega t),\\qquad\\qquad\n    \\tilde{\\textbf{y}}= \\bm{\\varphi}_0(\\tilde{\\textbf{x}})\n\\end{gather}\nwhere the time-varying vector fields $\\tilde{\\textbf{f}}_i$ are defined by:\n\\begin{gather}\n    \\label{eq:red_avg_sys_dets_1}\\tilde{\\textbf{f}}_1(\\textbf{x},\\tau)= \\textbf{f}_1(\\textbf{x},\\bm{\\varphi}_0(\\textbf{x}),\\tau) \\\\\n    \\tilde{\\textbf{f}}_2(\\textbf{x},\\tau)= \\textbf{f}_2(\\textbf{x},\\bm{\\varphi}_0(\\textbf{x}),\\tau) + \\textbf{C}(\\textbf{x},\\bm{\\varphi}_0(\\textbf{x}),\\tau)\\bm{\\varphi}_1(\\textbf{x},\\tau)\\\\\n    \\textbf{C}(\\textbf{x},\\textbf{y},\\tau)= \\partial_\\textbf{w} \\textbf{f}_1(\\textbf{x},\\textbf{w},\\tau)|_{\\textbf{w}=\\textbf{y}} \\\\\n    \\bm{\\varphi}_1(\\textbf{x},\\tau)= \\left(\\text{Id}-\\text{e}^{T \\textbf{A}}\\right)^{-1}\\textstyle\\int_{0}^T \\text{e}^{ (T-s)\\textbf{A}}\\,\\textbf{b}_1(\\textbf{x},s+\\tau)ds \\\\\n    \\label{eq:red_avg_sys_dets_5}\\textbf{b}_1(\\textbf{x},\\tau)= \\textbf{g}_1(\\textbf{x},\\bm{\\varphi}_0(\\textbf{x}),\\tau)-\\partial_{\\textbf{x}}\\bm{\\varphi}_0(\\textbf{x})\\, \\textbf{f}_1(\\textbf{x},\\bm{\\varphi}_0(\\textbf{x}),\\tau)\n\\end{gather}\nand $\\text{Id}$ is the identity matrix. In a companion paper \\cite{abdelgalil2022recursive}, see also \\cite{murdock1983some}, we showed that the higher order averaging theorem may be applied to the reduced order system (\\ref{eq:reduced_system}) to obtain the reduced order averaged system:\n\\begin{gather}\\label{eq:reduced_avg_system}\n    \\dot{\\bar{\\textbf{x}}}= \\bar{\\textbf{f}}(\\bar{\\textbf{x}}), \\qquad\\qquad \\bar{\\textbf{y}} = \\bm{\\varphi}_0(\\textbf{x})\n\\end{gather}\nwhere the vector field $\\bar{\\textbf{f}}(\\cdot)$ is given by:\n\\begin{equation}\n\\begin{aligned}\n    \\bar{\\textbf{f}}(\\textbf{x})= \\frac{1}{T}\\int_0^{T}\\bigg(\\tilde{\\textbf{f}}_2(\\textbf{x},\\tau_1)+\\frac{1}{2}\\Big[ \\int_0^{\\tau_1}&\\tilde{\\textbf{f}}_1(\\textbf{x}, \\tau_2) d\\tau_2,\\\\\n    &\\tilde{\\textbf{f}}_1(\\textbf{x},\\tau_1)\\Big]\\bigg)\\,d\\tau_1\n\\end{aligned}\n\\end{equation}\nUnder \\thref{asmp:A}, we have the following theorem concerning the relation between the stability of the system (\\ref{eq:orig_sys}) and the reduced order averaged system (\\ref{eq:reduced_avg_system}):\n\\begin{thm}\\thlabel{thm:A}\n    Let \\thref{asmp:A} be satisfied, and suppose that a compact subset $\\mathcal{S}\\subset\\mathbb{R}^n$ is globally uniformly asymptotically stable for the reduced order averaged system (\\ref{eq:reduced_avg_system}). Then, $\\mathcal{S}$ is singularly semi-globally practically uniformly asymptotically stable for the original system (\\ref{eq:orig_sys}).\n\\end{thm}\n\\begin{rem}\n    We note that our definition of singular semi-global practical uniform asymptotic stability, which can be found in the appendix A, is slightly different from that in \\cite{durr2015singularly}. The proof of \\thref{thm:A} proceeds by establishing each part of \\thref{defn:A} similar to \\cite{moreau2000practical,durr2015singularly}, relying on \\thref{prop:A} which we can be found along with a proof sketch in the appendix B. We omit the proof of the theorem from this manuscript to be included in the extended version. \n\\end{rem}\n\n\\iffalse 0\nGoing back to the motivational example, we see that when $\\mu=1\/\\omega$, the system can be written on the form:\n\\begin{gather}\n    \\label{eq:mot_ex_sol_1}\\dot{x}_1= \\beta x_1 + \\sqrt{2\\omega}\\sin(x_1^2+ x_4-x_3 +\\omega t)\\\\\n    \\label{eq:mot_ex_sol_2}\\dot{x}_2= \\beta x_2 + \\sqrt{2\\omega}\\cos(x_1^2+ x_4-x_3 +\\omega t)\\\\\n    \\label{eq:mot_ex_sol_3}\\dot{x}_3=\\omega\\,(x_4 - x_3), \\qquad     \\dot{x}_4= \\omega\\,(x_2^2 -x_4)\n\\end{gather}\nIf we apply our results, leaving the details of the computations as an easy exercise for the reader, we obtain the reduced order system:\n\\begin{align}\n\\dot{\\tilde{x}}_1&= \\beta \\tilde{x}_1  + \\sqrt{2\\omega} \\sin(\\tilde{x}_1^2+\\omega t)+ \\tilde{x}_2 \\sin(2\\tilde{x}_1^2+2\\omega t)\\\\\n\\dot{\\tilde{x}}_2&=\\alpha \\tilde{x}_2  + \\sqrt{2\\omega} \\cos(\\tilde{x}_1^2+\\omega t)+ \\tilde{x}_2 \\cos(2\\tilde{x}_1^2+2\\omega t)\n\\end{align}\nwhere $\\alpha = \\beta-1$. By applying the averaging procedure, we obtain the reduced order averaged system:\n\\begin{align}\n\\dot{\\bar{x}}_1&= (\\beta-2) \\bar{x}_1 \\\\\n\\dot{\\bar{x}}_2&= (\\beta-1) \\bar{x}_2 \n\\end{align}\nIt is clear that the reduced order averaged system has a globally uniformly asymptotically stable equilibrium point at the origin when $\\beta<1$. Hence, we conclude from \\thref{thm:A} that the origin $x_1=x_2=0$ is a singularly semi-globally practically uniformly asymptotically stable subset for the original system (\\ref{eq:mot_ex_1})-(\\ref{eq:mot_ex_3}). \n\\fi\n\\section{3D Source Seeking}\nSource seeking is the problem of locating a target that emits a scalar measurable signal, typically without global positioning information \\cite{cochran20093,krstic2008extremum}. Interestingly, microorganisms are routinely faced with the source seeking problem. In particular, sea urchin sperm cells seek the egg by swimming up the gradient of the concentration field  of a chemical secreted by the egg \\cite{abdelgalil2021sperm,friedrich2007chemotaxis}. The sperm cells do so by swimming in helical paths that dynamically align with the gradient. In this section, we propose a bio-inspired 3D source seeking algorithm for rigid bodies with a non-collocated signal strength sensor that partially mimics the strategy of sperm cells for seeking the egg.\n\nThe kinematics of a rigid body in 3D space are given by:\n\\begin{align}\n    \\label{eq:kin}\\dot{\\textbf{p}} &= \\textbf{R}\\textbf{v}, & \\dot{\\textbf{R}} &= \\textbf{R}\\widehat{\\bm{\\Omega}}\n\\end{align}\nwhere $\\textbf{p}$ denotes the position of a designated point on the body with respect to a fixed frame of reference, $\\textbf{R}$ relates the body frame to the fixed frame, and $\\textbf{v}$ and $\\bm{\\Omega}$ are the linear and angular velocities in body coordinates, respectively. The map $\\widehat{\\bullet}:\\mathbb{R}^3\\rightarrow\\mathbb{R}^{3\\times 3}$ takes a vector $\\bm{\\Omega}=\\left[\\Omega_1,\\Omega_2,\\Omega_3\\right]^\\intercal \\in \\mathbb{R}^3$ to the corresponding skew symmetric matrix. \n\nWe assume a vehicle model in which the linear and angular velocity vectors are given in body coordinates by:\n\\begin{align}\n    \\textbf{v}&= v\\textbf{e}_1, & \\bm{\\Omega} &= \\Omega_\\parallel\\textbf{e}_1 + \\Omega_\\perp \\textbf{e}_3\n\\end{align} \nwhere $\\textbf{e}_i$ for $i\\in\\{1,2,3\\}$ are the standard unit vectors. \\begin{rem}\nThis model is a natural extension of the unicycle model to the 3D setting. It is well known that this system is controllable using depth one Lie brackets \\cite{leonard1993averaging}.\n\\end{rem}\n\nLet $c:\\mathbb{R}^3\\rightarrow\\mathbb{R}$ be the signal strength field emitted by the source, and consider the case of a non-collocated signal strength sensor that is mounted at $\\textbf{p}_s$, where:\n\\begin{align}\\label{eq:sens_loc}\n    \\textbf{p}_s&= \\textbf{p} + r \\textbf{R}\\textbf{e}_2\n\\end{align}\n\\begin{asmp}\\thlabel{asmp:B}\nSuppose that the signal strength field $c\\in \\mathcal{C}^3(\\mathbb{R}^3;\\mathbb{R})$ is radially unbounded, $\\exists!\\textbf{p}^*\\in \\mathbb{R}^3$ such that $\\nabla c(\\textbf{p})= 0 \\iff \\textbf{p}=\\textbf{p}^*$, and it satisfies $c(\\textbf{p}^*)-c(\\textbf{p})\\leq \\kappa \\lVert \\nabla c(\\textbf{p})\\lVert^2, \\, \\forall \\textbf{p}\\in\\mathbb{R}^n$ and $\\kappa>0$.\\\\\n\\end{asmp}\n\nNow, consider the following control law:\n\\begin{gather}\n\\label{eq:ctrl_law_1}v = \\sqrt{4\\omega}\\cos(2\\omega t - c(\\textbf{p}_s))\\\\\n    \\label{eq:ctrl_law_2}\\dot{\\textbf{y}} = \\omega\\,\\textbf{A}\\,\\textbf{y} + \\omega\\,\\textbf{B}\\, c(\\textbf{p}_s) \\\\\n    \\Omega_\\perp  = \\sqrt{\\omega}\\,\\textbf{C}\\, \\textbf{y}, \\qquad\n    \\label{eq:ctrl_law_3}\\Omega_\\parallel = \\omega\n   \n\\end{gather}\nwhere $\\textbf{y}\\in\\mathbb{R}^2$, and $\\textbf{A},\\textbf{B},\\textbf{C}$ are given by:\n\\begin{align}\n    \\label{eq:flter_mats}\\textbf{A} &= \\left[\\begin{array}{cc} -1 & 1\\\\ 0 & -1 \\end{array}\\right], & \\textbf{B}&= \\left[\\begin{array}{c} 0\\\\ 1\\end{array}\\right], & \\textbf{C}&= \\left[\\begin{array}{cc} -1 & 1\\end{array}\\right]\n\\end{align}\nThe static part of this controller, i.e. equation (\\ref{eq:ctrl_law_1}) is a 1D extremum seeking control law \\cite{scheinker2014extremum}. Note that other choices of this control law are possible \\cite{grushkovskaya2018class}. \nThe dynamic part of this controller, i.e. the equations (\\ref{eq:ctrl_law_2})-(\\ref{eq:ctrl_law_3}), is a narrow band-pass filter centered around the frequency $\\omega$. The motivation to consider this setup is that in the presence of noise, a narrow band-pass filter centered around the dither frequency $\\omega$ optimally extracts the gradient information in the measured output while attenuating noise. In addition, assume that the distance $r$ specifying the offset of the sensor from the center of the frame is such that $r=1\/\\sqrt{\\omega}$. This assumption may seem artificial at first glance, though its implication is clear; we require that as the frequency of oscillation $\\omega$ tends to $\\infty$, the distance $r$ from the center of the vehicle is small enough so as not to amplify unwanted nonlinearities in the signal strength field. Alternatively, one may consider this assumption as a ``distinguished limit'' \\cite{kevorkian2012multiple} for the perturbation calculation in the presence of the two parameters $\\omega$ and $r$. Under these assumptions, we have the following proposition:\n\\begin{prop}\nLet \\thref{asmp:B} be satisfied, and let $r=1\/\\sqrt{\\omega}$. Then, the compact subset $\\mathcal{S}=\\{\\textbf{p}^*\\}\\times \\text{SO}(3)$ is singularly semi-globally practically uniformly asymptotically stable for the system defined by equations (\\ref{eq:kin})-(\\ref{eq:flter_mats}).\n\\end{prop}\n\\begin{proof}\nLet $\\textbf{R}_0 =\\text{exp}\\left(\\omega t \\,\\widehat{\\textbf{e}}_1 \\right),\\,\\textbf{Q} = \\textbf{R}\\textbf{R}_0^\\intercal$, and compute:\n\\begin{align}\n    \\dot{\\textbf{Q}} &= \\dot{\\textbf{R}}\\textbf{R}_0^\\intercal +  \\textbf{R}\\dot{\\textbf{R}}_0^\\intercal = \\Omega_\\perp\\textbf{R}\\widehat{\\textbf{e}}_3\\textbf{R}_0^\\intercal \\\\\n    &=  \\Omega_\\perp\\textbf{Q}\\textbf{R}_0 \\widehat{\\textbf{e}}_3\\textbf{R}_0^\\intercal = \\Omega_\\perp\\textbf{Q}\\widehat{\\textbf{R}_0 \\textbf{e}_3}\n\\end{align}\nLet $\\bm{\\Lambda}(\\textbf{y},\\omega t)= \\textbf{C}\\, \\textbf{y}\\,\\textbf{R}_0\\textbf{e}_3$ and observe that:\n\\begin{gather}\n    \\dot{\\textbf{p}}= \\textbf{R}\\textbf{v} = \\textbf{R}\\textbf{R}_0^\\intercal \\textbf{R}_0 \\textbf{v} = v \\,\\textbf{Q} \\textbf{R}_0 \\textbf{e}_1 = v\\,\\textbf{Q} \\textbf{e}_1 \\\\\n    \\dot{\\textbf{Q}}= \\sqrt{\\omega}\\,\\textbf{Q} \\widehat{\\bm{\\Lambda}}(\\textbf{y},\\omega t)\n\\end{gather}\nTo simplify the presentation, we embed $\\text{SO}(3)$ into $\\mathbb{R}^9$ by partitioning the matrix $\\textbf{Q}=[\\textbf{q}_1,\\, \\textbf{q}_2,\\, \\textbf{q}_3 ]$, and defining the state vector $\\textbf{q}=[\\textbf{q}_1^\\intercal,\\, \\textbf{q}_2^\\intercal,\\, \\textbf{q}_3^\\intercal ]^\\intercal$. Restrict the initial conditions for $\\textbf{q}$ to lie on the compact submanifold $\\mathcal{M} = \\left\\{\\textbf{q}_i\\in\\mathbb{R}^{3}:\\, \\textbf{q}_i^\\intercal\\textbf{q}_j=\\delta_{ij},~\\textbf{q}_i\\times\\textbf{q}_j = \\epsilon_{ijk}\\textbf{q}_k  \\right\\}$, \nwhere $\\delta_{ij}$ is the Kronecker symbol and $\\epsilon_{ijk}$ is the Levi-Civita symbol.\nOn $\\mathbb{R}^3\\times\\mathcal{M}\\times\\mathbb{R}^2$, the system is governed by:\n\\begin{align}\n    \\label{eq:src_seek_sys_1} \\dot{\\textbf{p}} &= \\sqrt{4\\omega}\\cos(2\\omega t - c(\\textbf{p}_s))\\,\\textbf{q}_1\\\\\n    \\label{eq:src_seek_sys_2}\\dot{\\textbf{q}}_i &= \\sqrt{\\omega}\\sum\\limits_{j,k=1}^3\\Lambda_j(\\textbf{y},\\omega t)\\epsilon_{ijk}\\textbf{q}_k \\\\\n    \\label{eq:src_seek_sys_3}\\dot{\\textbf{y}} &=\\omega \\left(\\textbf{A}\\,\\textbf{y} + \\,\\textbf{B}\\, c(\\textbf{p}_s) \\right)\n\\end{align}\nFor more details on this embedding, see the companion paper \\cite{abdelgalil2022recursive}. The signal strength field can be expanded as a series in $1\/\\sqrt{\\omega}$ using Taylor's theorem:\n\\begin{gather*}\n    c(\\textbf{p}_s)=\\,c(\\textbf{p})+\\frac{1}{\\sqrt{\\omega}} \\nabla c(\\textbf{p})^\\intercal (\\cos(\\omega t) \\textbf{q}_2 + \\sin(\\omega t) \\textbf{q}_3) \\\\\n    +\\frac{1}{2\\omega }\\nabla^2 c(\\textbf{p})[(\\cos(\\omega t) \\textbf{q}_2 + \\sin(\\omega t) \\textbf{q}_3)] \\\\\n    + \\frac{1}{\\omega \\sqrt{\\omega}}\\bm{\\rho}(\\textbf{p},\\textbf{p}_s,\\omega t,1\/\\sqrt{\\omega})\n\\end{gather*}\nwhere the remainder $\\bm{\\rho}$ is Lipschitz continuous in all of its arguments, and $\\nabla^2c(\\textbf{p})[\\textbf{w}]=\\textbf{w}^\\intercal \\nabla^2c(\\textbf{p}) \\textbf{w}$. \nNow, observe that the system governed by the equations (\\ref{eq:src_seek_sys_1})-(\\ref{eq:src_seek_sys_3}) belongs to the class of systems described by (\\ref{eq:orig_sys}). Hence, we may employ \\thref{thm:A} in analyzing the stability of the system. In order to proceed, the reduced order averaged system needs to be computed. Due to space constraints, we leave the computations as an exercise for the interested reader in the light of equations (\\ref{eq:red_avg_sys_dets_1})-(\\ref{eq:red_avg_sys_dets_5}), and we provide only the end result of the computation:\n\\noindent\\begin{minipage}{.5\\linewidth}\n\\begin{gather*}\n    \\dot{\\bar{\\textbf{p}}} = \\vphantom{\\frac{1}{4}}\\bar{\\textbf{q}}_1\\bar{\\textbf{q}}_1^\\intercal \\nabla c(\\bar{\\textbf{p}}) \\\\\n    \\dot{\\bar{\\textbf{q}}}_2 = -\\frac{1}{4}\\bar{\\textbf{q}}_1\\bar{\\textbf{q}}_3^\\intercal \\nabla c(\\bar{\\textbf{p}}),\n\\end{gather*}\n\\end{minipage}%\n\\begin{minipage}{.5\\linewidth}\n\\begin{gather*}\n    \\dot{\\bar{\\textbf{q}}}_1= \\frac{1}{4}(\\bar{\\textbf{q}}_2\\bar{\\textbf{q}}_2^\\intercal + \\bar{\\textbf{q}}_3\\bar{\\textbf{q}}_3^\\intercal)\\nabla c(\\bar{\\textbf{p}}) \\\\\n    \\dot{\\bar{\\textbf{q}}}_3 = -\\frac{1}{4}\\bar{\\textbf{q}}_1\\bar{\\textbf{q}}_2^\\intercal \\nabla c(\\bar{\\textbf{p}})\n\\end{gather*}\n\\end{minipage}\\vspace*{0.1in} \\\\\nEquivalently, this system can be written as:\n\\begin{gather}\\label{eq:exmp_red_avg_sys_1}\n    \\dot{\\bar{\\textbf{p}}} = \\bar{\\textbf{Q}}\\textbf{e}_1\\textbf{e}_1^\\intercal \\bar{\\textbf{Q}}^\\intercal \\nabla c(\\bar{\\textbf{p}}) \\\\\n    \\label{eq:exmp_red_avg_sys_2}\\dot{\\bar{\\textbf{Q}}} = \\bar{\\textbf{Q}} \\widehat{\\bar{\\bm{\\Lambda}}}(\\textbf{p},\\textbf{Q})\n\\end{gather}\nwhere the average angular velocity vector $\\bar{\\bm{\\Lambda}}$ is given by:\n\\begin{align}\n    \\label{eq:avg_rot_vel} \\bar{\\bm{\\Lambda}}(\\textbf{p},\\textbf{Q})&= \\frac{1}{4}\\textbf{Q}^\\intercal \\nabla c(\\textbf{p})\\times\\textbf{e}_1\n\\end{align}\nWe claim that the compact subset $\\mathcal{S}$ is globally uniformly asymptotically stable for the reduced order averaged system (\\ref{eq:exmp_red_avg_sys_1})-(\\ref{eq:exmp_red_avg_sys_2}). To prove this claim, we use the negative of the signal strength field as a Lyapunov function $V_c(\\textbf{p})=c(\\textbf{p}^*)-c(\\textbf{p})$. Observe that the system (\\ref{eq:exmp_red_avg_sys_1})-(\\ref{eq:exmp_red_avg_sys_2}) is autonomous, and so the function $V_c$ is indeed a Lyapunov function for the compact subset $\\mathcal{S}$ due to \\thref{asmp:B} \\cite{khalil2002nonlinear}. We proceed to compute the derivative of $V_c$:\n\\begin{gather}\n    \\dot{V}_c = -\\nabla c(\\bar{\\textbf{p}})^\\intercal \\bar{\\textbf{Q}}\\textbf{e}_1\\textbf{e}_1^\\intercal \\bar{\\textbf{Q}}^\\intercal \\nabla c(\\bar{\\textbf{p}})\\leq 0\n\\end{gather}\nNow, consider the subset $\\mathcal{N}=\\{(\\textbf{p},\\textbf{Q})\\in\\mathbb{R}^3\\times \\text{SO}(3): \\dot{V}_c=0\\}$, and observe that $\\mathcal{S}\\subset \\mathcal{N}$, and that $\\mathcal{S}$ is an invariant subset of the reduced order averaged system (\\ref{eq:exmp_red_avg_sys_1})-(\\ref{eq:exmp_red_avg_sys_2}). Suppose that a trajectory $(\\bar{\\textbf{p}}(t),\\bar{\\textbf{Q}}(t))$ of the system (\\ref{eq:exmp_red_avg_sys_1})-(\\ref{eq:exmp_red_avg_sys_2}) exists such that $(\\bar{\\textbf{p}}(t),\\bar{\\textbf{Q}}(t))\\in \\mathcal{N}\\backslash\\mathcal{S},\\, \\forall t\\in I$, where $I$ is the maximal interval of existence and uniqueness of the trajectory. Such a trajectory must satisfy:\n\\begin{align}\n    \\nabla c(\\bar{\\textbf{p}}(t))^\\intercal \\bar{\\textbf{Q}}(t)\\textbf{e}_1 = 0,\\quad \\forall t\\in I\n\\end{align}\nThe differentiability of the trajectories allows us to compute the derivative of this identity and obtain that:\n\\begin{align}\n    \\frac{d}{dt}\\big(\\nabla c(\\bar{\\textbf{p}}(t))^\\intercal \\bar{\\textbf{Q}}(t)\\textbf{e}_1\\big) = 0,\\quad \\forall t\\in I\n\\end{align}\nwhich simplifies to:\n\\begin{align}\n    \\label{eq:second_identity}\\nabla c(\\bar{\\textbf{p}}(t))^\\intercal \\bar{\\textbf{Q}}(t)(\\bar{\\bm{\\Lambda}}(\\bar{\\textbf{p}}(t),\\bar{\\textbf{Q}}(t))\\times \\textbf{e}_1) = 0\n\\end{align}\nRecalling equation (\\ref{eq:avg_rot_vel}), we see that:\n\\begin{gather}\n    \\bar{\\bm{\\Lambda}}(\\bar{\\textbf{p}}(t),\\bar{\\textbf{Q}}(t))\\times \\textbf{e}_1 = \\frac{1}{4}(\\text{Id}-\\textbf{e}_1\\textbf{e}_1^\\intercal)\\bar{\\textbf{Q}}(t)^\\intercal \\nabla c(\\bar{\\textbf{p}}(t))\\\\\n    = \\frac{1}{4}\\bar{\\textbf{Q}}(t)^\\intercal \\nabla c(\\bar{\\textbf{p}}(t))\n\\end{gather}\nHence, the equation (\\ref{eq:second_identity}) necessitates that:\n\\begin{align}\n    \\lVert \\nabla c(\\bar{\\textbf{p}}(t)) \\lVert^2 = 0,\\quad \\forall t\\in I\n\\end{align}\nwhich is clearly in contradiction with \\thref{asmp:B}. Accordingly, it follows from LaSalle's Invariance principle \\cite[Corollary 4.2 to Theorem 4.4]{khalil2002nonlinear} that the compact subset $\\mathcal{S}$ is globally uniformly asymptotically stable for the system (\\ref{eq:exmp_red_avg_sys_1})-(\\ref{eq:exmp_red_avg_sys_2}). Hence, we conclude by \\thref{thm:A} that the subset $\\mathcal{S}$ is singularly semi-globally practically uniformly asymptotically stable for the original system defined by (\\ref{eq:kin})-(\\ref{eq:flter_mats}). \n\\end{proof}\n\\begin{figure*}[t]\n    \\centering \\includegraphics[width=1\\textwidth]{figs\/signal_strength_and_pos.pdf}\n    \\caption{Numerical results: the history of the signal strength at the vehicle center and the position coordinates (left), and the 3D spatial trajectory (right)}\n    \\label{fig:num_res}\n\\end{figure*}\n\\begin{rem}\nIf we attempt to apply the framework of singularly pertubed Lie Bracket Approximation introduced in \\cite{durr2015singularly} to the system (\\ref{eq:src_seek_sys_1})-(\\ref{eq:src_seek_sys_3}), then the quasi-steady state of the system will be $\\textbf{y} = [c(\\textbf{p}_s),c(\\textbf{p}_s)]^\\intercal$. Hence, according to \\cite{durr2015singularly}, the reduced order system is:\n\\begin{gather}\n    \\dot{\\textbf{p}} = \\sqrt{4\\omega}\\cos(2\\omega t - c(\\textbf{p}_s))\\,\\textbf{q}_1,\\qquad \n    \\dot{\\textbf{q}}_i =0\n\\end{gather}\nwhich yields the Lie Bracket system:\n\\begin{gather}\n    \\dot{\\bar{\\textbf{p}}} = \\bar{\\textbf{Q}}\\textbf{e}_1\\textbf{e}_1^\\intercal \\bar{\\textbf{Q}}^\\intercal \\nabla c(\\bar{\\textbf{p}}), \\qquad \n    \\dot{\\bar{\\textbf{Q}}} =0\n\\end{gather}\nIt is clear that the compact subset $\\mathcal{S}$ is not asymptotically stable for the Lie Bracket system, and so the framework in \\cite{durr2015singularly} does not capture the stability of this system. \\end{rem}\n\\section{Numerical Simulations}\nWe demonstrate our results by providing a numerical example. Consider the signal strength field given by $c(\\textbf{p})= -\\log\\left(1+\\textbf{p}^\\intercal\\textbf{p}\/2\\right)$, which represents a stationary source located at the origin. We take the initial conditions as $\\textbf{p}(0)=[6,~2,~-2]^\\intercal$ and $\\textbf{R}(0)=\\textbf{I}_{3\\times3}$, and the frequency as $\\omega=4\\pi$. The numerical simulations are shown in Fig.(\\ref{fig:num_res}). Observe that the behavior near the source is nontrivial, i.e. there is a limit cycle. However, in the limit $\\omega\\rightarrow \\infty$ and $r=O(1\/\\sqrt{\\omega})$, this complex behavior does not appear in the reduced order averaged system.\n\n\\section{Conclusion}\nIn this manuscript, we analyzed a class of singularly perturbed high-amplitude, high-frequency oscillatory systems that naturally arises in extremum seeking applications and stabilization by oscillatory controls. We combined singular perturbation with the higher order averaging theorem in order to capture the stability properties of this class of systems. As an application, we proposed a novel 3D source seeking algorithm for rigid bodies with a non-collocated sensor inspired by the chemotaxis of sea urchin sperm cells.\n\\section*{Acknowledgement} The authors like to acknowledge the support of the NSF Grant CMMI-1846308. \\\\\n\\appendices\n\\renewcommand{\\thelem}{\\thesection.\\arabic{lem}} \n\\renewcommand{\\thecor}{\\thesection.\\arabic{cor}}\n\\renewcommand{\\thedefn}{\\thesection.\\arabic{defn}} \n\\renewcommand{\\theprop}{\\thesection.\\arabic{prop}} \n\\section{Definitions}\n\\begin{defn}\\thlabel{defn:A}\n The set $\\mathcal{S}$ is said to be \\textit{singularly semi-globally practically uniformly asymptotically stable} for system (\\ref{eq:orig_sys}) if the following is satisfied:\n\\begin{enumerate}\n    \\item $\\forall \\epsilon_x,\\epsilon_z\\in(0,\\infty)$ there exists $\\delta_x,\\delta_z\\in(0,\\infty)$ and $\\omega^*\\in(0,\\infty)$ such that $\\forall \\omega\\in(\\omega^*,\\infty)$, $\\forall t_0\\in\\mathbb{R}$, and $\\forall t\\in[0,\\infty)$ we have: \n\\begin{align*}\n\\begin{drcases}\n     \\hfill \\textbf{x}_0\\in \\mathcal{U}_{\\delta_x}^{\\mathcal{S}}  \\hfill \\\\\n     \\textbf{y}_0-\\bm{\\varphi}_0(\\textbf{x}_0)\\in \\mathcal{U}_{\\delta_z}^{0}\n\\end{drcases} &\\implies \\begin{cases} \n     \\hfill\\textbf{x}(t)\\in \\mathcal{U}_{\\epsilon_x}^{\\mathcal{S}}\\hfill \\\\\n     \\textbf{y}(t)-\\bm{\\varphi}_0(\\textbf{x}(t))\\in \\mathcal{U}_{\\epsilon_z}^{0} \n\\end{cases}\n\\end{align*}\n\\item $\\forall \\epsilon_x,\\epsilon_z\\in(0,\\infty)$ and all $\\delta_x,\\delta_z\\in(0,\\infty)$, there exists a time $T_f\\in(0,\\infty)$ and $\\omega^*\\in(0,\\infty)$ such that $\\forall \\omega\\in(\\omega^*,\\infty)$, $\\forall t_0\\in\\mathbb{R}$, $\\forall t_1\\in[T_f,\\infty)$, and $\\forall t_2\\in[T_f\/\\omega,\\infty)$ we have: \n\\begin{align*}\n\\begin{drcases}\n     \\hfill \\textbf{x}_0\\in \\mathcal{U}_{\\delta_x}^{\\mathcal{S}}  \\hfill\\\\\n     \\textbf{y}_0-\\bm{\\varphi}_0(\\textbf{x}_0)\\in \\mathcal{U}_{\\delta_z}^{0}\n\\end{drcases} &\\implies \\begin{cases} \n     \\hfill \\textbf{x}(t_1)\\in \\mathcal{U}_{\\epsilon_x}^{\\mathcal{S}} \\hfill \\\\\n     \\textbf{y}(t_2)-\\bm{\\varphi}_0(\\textbf{x}(t_2))\\in \\mathcal{U}_{\\epsilon_z}^{0}\n\\end{cases}\n\\end{align*}\n\\item $\\forall \\delta_x,\\delta_z\\in(0,\\infty)$ there exists $\\epsilon_x,\\epsilon_z\\in(0,\\infty)$ and $\\omega^*\\in(0,\\infty)$ such that $\\forall \\omega\\in(\\omega^*,\\infty)$, $\\forall t_0\\in\\mathbb{R}$, and $\\forall t\\in[0,\\infty)$ we have: \n\\begin{align*}\n\\begin{drcases}\n     \\hfill \\textbf{x}_0\\in \\mathcal{U}_{\\delta_x}^{\\mathcal{S}}  \\hfill\\\\\n     \\textbf{y}_0-\\bm{\\varphi}_0(\\textbf{x}_0)\\in \\mathcal{U}_{\\delta_z}^{0}\n\\end{drcases} &\\implies \\begin{cases} \n     \\hfill \\textbf{x}(t)\\in \\mathcal{U}_{\\epsilon_x}^{\\mathcal{S}}\\hfill \\\\\n     \\textbf{y}(t)-\\bm{\\varphi}_0(\\textbf{x}(t))\\in \\mathcal{U}_{\\epsilon_z}^{0}\n\\end{cases}\\\\\n\\end{align*}\n\\end{enumerate}\n\\end{defn}\nObserve that our definition of singular semi-global practical uniform asymptotic stability is different from the definitions introduced in \\cite{durr2015singularly} due to the absence of a second parameter. Intuitively, when two or more parameters are involved, the so called `distinguished limit' is a standard technique in perturbation theory that simplifies the interaction between the limiting behavior of the two parameters on the trajectories of differential equations \\cite[Chapter 2]{kevorkian2012multiple}. Our definitions are motivated by this concept of distinguished limits. \n\\section{Trajectory Approximation}\nFor the purpose of brevity, we state here some notations that may enhance the readability of the proof. Whenever a Lipschitz property of a map $\\textbf{f}$ over a subset $\\mathcal{K}$ is employed, the corresponding Lipschitz constant is labelled as $L_{\\textbf{f},\\mathcal{K}}$. Similarly, when a uniform bound is employed, it is labelled as $B_{\\textbf{f},\\mathcal{K}}$. Sometimes we use $M_{\\textbf{f},\\mathcal{K}}$ as a generic constant when a mix of the two properties is used. Finally, we may omit mentioning the map in the constant label when it is too long or when it is clear from the context. \n\nUnder \\thref{asmp:A}, we have a trajectory approximation result between the original system (\\ref{eq:orig_sys}) and the reduced order averaged system (\\ref{eq:reduced_avg_system}): \n\\begin{prop}\\thlabel{prop:A}\nLet \\thref{asmp:A} be satisfied, and suppose that a compact subset $\\mathcal{S}\\subset\\mathbb{R}^n$ is globally uniformly asymptotically stable for the averaged reduced order system (\\ref{eq:reduced_avg_system}). Then, there exist constants $\\lambda >0 $ and $\\gamma > 0$ such that for every bounded subset $ \\mathcal{B}_{\\textbf{x}}\\times \\mathcal{B}_{\\textbf{z}}\\subset\\mathbb{R}^n\\times\\mathbb{R}^m$, $\\forall t_f\\in(0,\\infty)$, and $\\forall D\\in(0,\\infty)$, there exists $\\omega^*\\in(0,\\infty)$ such that $\\forall \\omega\\in(\\omega^*,\\infty)$, $\\forall t_0\\in\\mathbb{R}$, $ \\forall (\\textbf{x}_0,\\textbf{y}_0-\\bm{\\varphi}_0(\\textbf{x}_0))\\in \\mathcal{B}_{\\textbf{x}}\\times \\mathcal{B}_{\\textbf{z}}$, and $\\forall t\\in[t_0,t_0+t_f]$, unique trajectories of the system (\\ref{eq:orig_sys}) exist and satisfy:\n\\begin{gather}\n    \\lVert \\textbf{x}(t)-\\bar{\\textbf{x}}(t) \\lVert < D \\\\\n     \\lVert \\textbf{y}(t)-\\bm{\\varphi}_0(\\textbf{x}(t))\\lVert < \\gamma \\lVert \\textbf{y}_0-\\bm{\\varphi}_0(\\textbf{x}_0)\\lVert \\,\\text{e}^{-\\omega(t-t_0)\\lambda } + D\n\\end{gather}\n\\end{prop}\n\\begin{proof}\nWe apply the time scaling $\\tau=\\omega (t-t_0)$, and we let $\\varepsilon=1\/\\sqrt{\\omega}$. In contrast to the standard singular perturbation analysis which starts with a coordinate shift for the singularly perturbed part of the system from $\\textbf{y}$ to $\\textbf{y}-\\bm{\\varphi}_0(\\textbf{x})$ (e.g. \\cite[Chapter 11]{khalil2002nonlinear}, \\cite[Section I]{durr2015singularly}), we augment the standard coordinate shift with a \\textit{near-identity} part:\n\\begin{equation}\\label{eq:nearid_shift}\n\\textbf{z}=\\textbf{y}-\\bm{\\varphi}_0(\\textbf{x})-\\varepsilon\\, \\bm{\\varphi}_1(\\textbf{x},\\tau)-\\varepsilon^2\\bm{\\varphi}_2(\\textbf{x},\\tau),\n\\end{equation}\nwhere the maps $\\bm{\\varphi}_i(\\textbf{x},\\tau)$ for $i\\in\\{1,2\\}$ are yet to be determined. This is coordinate shift is inspired by the standard near identity transform common in the higher order averaging literature \\cite[Section 2.8]{sanders2007averaging}. Observe that under this coordinate and time scale change, we have:\n\\begin{equation}\n\\begin{aligned}\\label{eq:shifted_orig_sys}\n    \\frac{d\\textbf{x}}{d\\tau}&=\\sum_{i=1}^2 \\varepsilon^i \\textbf{v}_i(\\textbf{x},\\textbf{z},\\tau) +\\varepsilon^3 \\textbf{v}_3(\\textbf{x},\\textbf{z},\\tau,\\varepsilon)\\\\\n    \\frac{d\\textbf{z}}{d\\tau}&= \\textbf{A}\\,\\textbf{z} + \\sum_{i=1}^2 \\varepsilon^i\\textbf{h}_i(\\textbf{x},\\textbf{z},\\tau) + \\varepsilon^3 \\textbf{h}_3(\\textbf{x},\\textbf{z},\\tau,\\varepsilon)\n\\end{aligned}\n\\end{equation}\nwhere the vector fields $\\textbf{v}_i$ and $\\textbf{h}_i$ for $i\\in\\{1,2\\}$ are given by:\n\\begin{align}\n    &\\begin{aligned}\n    \\textbf{v}_1(\\textbf{x},\\textbf{z},\\tau)&= \\textbf{f}_1(\\textbf{x},\\textbf{z}+\\bm{\\varphi}_0(\\textbf{x}),\\tau)\n    \\end{aligned}\\\\\n    &\\begin{aligned}\n    \\textbf{v}_2(\\textbf{x},\\textbf{z},\\tau)&= \\textbf{f}_2(\\textbf{x},\\textbf{z}+\\bm{\\varphi}_0(\\textbf{x}),\\tau)\\\\\n     &+\\textbf{C}(\\textbf{x},\\textbf{z}+\\bm{\\varphi}_0(\\textbf{x}),\\tau) \\bm{\\varphi}_1(\\textbf{x},\\tau)\n    \\end{aligned}\\\\\n    &\\begin{aligned}\n    \\textbf{h}_1(\\textbf{x},\\textbf{z},\\tau)&= \\textbf{A}\\,\\bm{\\varphi}_1(\\textbf{x},\\tau)-\\partial_\\tau\\bm{\\varphi}_1(\\textbf{x},\\tau) \\\\\n    &- \\partial_{\\textbf{x}} \\bm{\\varphi}_0(\\textbf{x}) \\textbf{f}_1(\\textbf{x},\\textbf{z}+\\bm{\\varphi}_0(\\textbf{x}),\\tau)\\\\\n    &+\\textbf{g}_1(\\textbf{x},\\textbf{z}+\\bm{\\varphi}_0(\\textbf{x}),\\tau)\n    \\end{aligned}\\\\\n    &\\begin{aligned}\n    \\textbf{h}_2(\\textbf{x},\\textbf{z},\\tau)&=\\textbf{A}\\, \\bm{\\varphi}_2(\\textbf{x},\\tau)-\\partial_\\tau\\bm{\\varphi}_2(\\textbf{x},\\tau)\\\\\n    &-\\partial_{\\textbf{x}} \\bm{\\varphi}_1(\\textbf{x},\\tau) \\textbf{f}_1(\\textbf{x},\\textbf{z}+\\bm{\\varphi}_0(\\textbf{x}),\\tau) \\\\\n    &- \\partial_{\\textbf{x}} \\bm{\\varphi}_0(\\textbf{x}) \\textbf{f}_2(\\textbf{x},\\textbf{z}+\\bm{\\varphi}_0(\\textbf{x}),\\tau)\\\\\n    &+\\partial_{\\textbf{w}} \\textbf{g}_1(\\textbf{x},\\textbf{w},\\tau)|_{\\textbf{w}=\\bm{\\varphi}_0(\\textbf{x})}\\bm{\\varphi}_1(\\textbf{x},\\tau) \\\\\n    &+\\textbf{g}_2(\\textbf{x},\\textbf{z}+\\bm{\\varphi}_0(\\textbf{x}),\\tau)\n    \\end{aligned}\n\\end{align}\nNow, we let $\\bm{\\varphi}_1(\\textbf{x},\\tau)$ and $\\bm{\\varphi}_2(\\textbf{x},\\tau)$ be the solutions of the linear non-homogeneous two point boundary value problems:\n\\begin{align}\\label{eq:BVPi_1}\n    \\partial_\\tau\\bm{\\varphi}_i(\\textbf{x},\\tau)&= \\textbf{A}\\, \\bm{\\varphi}_i(\\textbf{x},\\tau) + \\textbf{b}_i(\\textbf{x},\\tau)\\\\ \\label{eq:BVPi_2}\n    \\bm{\\varphi}_i(\\textbf{x},\\tau) &= \\bm{\\varphi}_i(\\textbf{x},\\tau+T)\n\\end{align}\nfor $i\\in\\{1,2\\}$, where:\n\\begin{align}\n     &\\textbf{b}_1(\\textbf{x},\\tau)= \\textbf{g}_1(\\textbf{x},\\bm{\\varphi}_0(\\textbf{x}),\\tau)-\\partial_{\\textbf{x}} \\bm{\\varphi}_0(\\textbf{x}) \\textbf{f}_1(\\textbf{x},\\bm{\\varphi}_0(\\textbf{x}),\\tau)\\\\\n     &\\begin{aligned}\n     \\textbf{b}_2(\\textbf{x},\\tau)&= \\textbf{g}_2(\\textbf{x},\\bm{\\varphi}_0(\\textbf{x}),\\tau)-\\partial_{\\textbf{x}} \\bm{\\varphi}_0(\\textbf{x}) \\textbf{f}_2(\\textbf{x},\\bm{\\varphi}_0(\\textbf{x}),\\tau)\\\\\n     &-\\partial_{\\textbf{x}} \\bm{\\varphi}_1(\\textbf{x},\\tau) \\textbf{f}_1(\\textbf{x},\\bm{\\varphi}_0(\\textbf{x}),\\tau) \\\\\n     &+\\partial_{\\textbf{w}} \\textbf{g}_1(\\textbf{x},\\textbf{w},\\tau)|_{\\textbf{w}=\\bm{\\varphi}_0(\\textbf{x})}\\bm{\\varphi}_1(\\textbf{x},\\tau)\n     \\end{aligned}\n\\end{align}\nThe following lemma is a simple consequence of \\thref{asmp:A} and standard linear systems theory:\n\\begin{lem}\\thlabel{lem:B}\nLet \\thref{asmp:A} be satisfied. Then, the non-homogeneous BVPs (\\ref{eq:BVPi_1})-(\\ref{eq:BVPi_2}) have unique solutions $\\bm{\\varphi}_i\\in\\mathcal{C}^{3-i}(\\mathbb{R}^{n};\\mathbb{R}^m)$ defined by:\n\\begin{align}\n    \\bm{\\varphi}_i(\\textbf{x},\\tau)&= \\left(\\text{Id}-\\text{e}^{T \\textbf{A}}\\right)^{-1}\\int_{0}^T \\text{e}^{ (T-s)\\textbf{A}}\\,\\textbf{b}_i(\\textbf{x},s+\\tau)ds\n\\end{align}\nwhere $\\text{Id}$ is the identity map on $\\mathbb{R}^m$. \n\\end{lem}\n\\begin{proof}\n    The result can be verified by direct substitution, and the regularity of the solutions follows from \\thref{asmp:A}.\n\\end{proof}\n\nWith this choice of the maps $\\bm{\\varphi}_i(\\textbf{x},\\tau)$ for $i\\in\\{1,2\\}$, observe that $\\textbf{v}_1(\\textbf{x},0,\\tau)=\\tilde{\\textbf{f}}_1(\\textbf{x},\\tau),\\,\\textbf{v}_2(\\textbf{x},0,\\tau)=\\tilde{\\textbf{f}}_2(\\textbf{x},\\tau)$, and that $\\textbf{h}_1(\\textbf{x},0,\\tau)=\\textbf{h}_2(\\textbf{x},0,\\tau)=0$, $\\forall \\textbf{x}\\in\\mathbb{R}^n,\\,\\forall \\tau\\in\\mathbb{R}$. That is, the origin $\\textbf{z}=0$ is an equilibrium point for the boundary layer model:\n\\begin{align}\n    \\label{eq:bl_model_z}\\frac{d\\textbf{z}}{d\\tau} &= \\textbf{A}\\,\\textbf{z} + \\sum_{i=1}^2 \\varepsilon^i\\textbf{h}_i(\\textbf{x},\\textbf{z},\\tau), & \\textbf{z}(0)&= \\textbf{z}_0\n\\end{align}\nMoreover, it can be shown that the vector fields $\\textbf{h}_i$ for $i\\in\\{1,2\\}$ are Lipschitz continuous and bounded on every compact subset $\\mathcal{K}\\subset\\mathbb{R}^n\\times\\mathbb{R}^m$, uniformly in $\\tau$, for some Lipschitz constants $L_{\\textbf{h}_i,\\mathcal{K}}>0$ and bounds $B_{\\textbf{h}_i,\\mathcal{K}}>0$, and that the remainder terms $\\textbf{h}_3$ and $\\textbf{v}_3$ are continuous and bounded on any compact subset $\\mathcal{K}\\subset\\mathbb{R}^n\\times\\mathbb{R}^m$ uniformly in $\\tau\\in\\mathbb{R}$ and $\\varepsilon\\in[0,\\varepsilon_0]$ for some $\\varepsilon_0>0$. Next, we have the following lemma:\n\\begin{lem}\\thlabel{lem:A}\nLet \\thref{asmp:A} be satisfied, and suppose that a compact subset $\\mathcal{S}\\subset\\mathbb{R}^n$ is globally uniformly asymptotically stable for the averaged reduced order system (\\ref{eq:reduced_avg_system}). Then, there exist constants $\\lambda >0 $ and $\\gamma > 0$ such that for every bounded subset $ \\mathcal{B}_{\\textbf{x}}\\times \\mathcal{B}_{\\textbf{z}}\\subset\\mathbb{R}^n\\times\\mathbb{R}^m$, $\\forall t_f\\in(0,\\infty)$, and $\\forall D\\in(0,\\infty)$, there exists $\\varepsilon^*\\in(0,\\varepsilon_0)$ such that $\\forall \\varepsilon\\in(0,\\varepsilon^*)$,  $ \\forall (\\textbf{x}_0,\\textbf{z}_0)\\in \\mathcal{B}_{\\textbf{x}}\\times \\mathcal{B}_{\\textbf{z}}$, and $\\forall \\tau\\in[0,t_f\/\\varepsilon^{2}]$, unique trajectories of the system (\\ref{eq:shifted_orig_sys}) exist and satisfy:\n\\begin{gather}\n    \\lVert \\textbf{x}(\\tau)-\\tilde{\\textbf{x}}(\\tau) \\lVert < D \\\\\n     \\lVert \\textbf{z}(\\tau)\\lVert < \\gamma \\lVert \\textbf{z}_0\\lVert \\,\\text{e}^{-\\lambda\\,\\tau} + D\n\\end{gather}\n\\end{lem}\n\\begin{proof}\n\\iffalse 0    \nThis lemma borrows ideas from \\cite[Lemma 1]{durr2015singularly} and \\cite[Lemma 2.8.2]{sanders2007averaging} but is rather long, so we refer the reader to the preprint of the journal version of the current manuscript for the full proof. Instead, we provide a sketch of the proof here. We emphasize that the lemma does not follow directly from the arguments in \\cite[Lemma 1]{durr2015singularly}.\n    \n    First, observe that the behavior of the singularly perturbed part of the system $\\textbf{z}$ is dominated by the linear vector field $\\textbf{A}\\,\\textbf{z}$ in the limit $\\varepsilon\\rightarrow 0$. Since the origin is an equilibrium point for the boundary layer model (\\ref{eq:bl_model_z}), only $O(\\varepsilon^3)$ terms affect the behavior of $\\textbf{z}$. Using an adapted version of ultimate boundedness \\cite[Theorem 4.18]{khalil2002nonlinear}, it can be shown that solutions to (\\ref{eq:shifted_orig_sys}) are such that $\\textbf{z}(t)$ satisfies:\n    \\begin{align}\n        \\lVert \\textbf{z}(\\tau)\\lVert < \\gamma\\,\\lVert \\textbf{z}_0\\lVert \\text{e}^{-\\lambda \\tau} + \\alpha\\,\\varepsilon^{\\frac{3}{2}}\n    \\end{align}\n    for some constants $\\gamma, \\,\\lambda, \\,\\alpha>0$ and small enough $\\varepsilon$. Next, one argues that the transient behavior of $\\textbf{z}(t)$ dies fast enough so as not to affect the behavior of $\\textbf{x}(t)$. This can be established by estimating the difference $\\lVert \\textbf{x}(\\tau) - \\tilde{\\textbf{x}}(\\tau)\\lVert$. Intuitively, $\\textbf{x}(\\tau)$ and $\\tilde{\\textbf{x}}(\\tau)$ move with an $O(\\varepsilon^2)$ velocity on average because the average of the $O(\\varepsilon)$ terms vanish. Hence, we expect the difference $\\lVert \\textbf{x}(\\tau) - \\tilde{\\textbf{x}}(\\tau)\\lVert$ to remain small on the time scale $O(1\/\\varepsilon^2)$, provided that $\\textbf{z}(\\tau)$ converges fast enough to a small enough neighborhood of the origin. Indeed, utilizing the fact that the time average of $\\textbf{f}_1$ is zero, one can obtain that for small enough $\\varepsilon$ the following estimate holds:\n    $$\\lVert \\textbf{x}(\\tau) - \\tilde{\\textbf{x}}(\\tau)\\lVert < \\delta(\\varepsilon) + M \\varepsilon^2\\int_{0}^{\\tau} \\lVert \\textbf{x}(s) - \\tilde{\\textbf{x}}(s)\\lVert ds$$\n    where $\\delta(\\varepsilon)$ is $o(1)$, i.e. $\\lim_{\\varepsilon\\rightarrow 0}\\delta(\\varepsilon) = 0$. An application of Gr\\\"onwall's inequality leads to:\n    \\begin{align}\n        \\lVert \\textbf{x}(\\tau) - \\tilde{\\textbf{x}}(\\tau)\\lVert<\\delta(\\varepsilon) \\text{e}^{ M\\varepsilon^2\\tau}\n    \\end{align}\n    Now it is clear that for any fixed $t_f>0$, and $\\forall\\tau\\in[0,t_f\/\\varepsilon^2]$, we have:\n    \\begin{align}\n        \\lVert \\textbf{x}(\\tau) - \\tilde{\\textbf{x}}(\\tau)\\lVert< o(1)\n    \\end{align}\n    In other words, the difference $\\lVert \\textbf{x}(\\tau) - \\tilde{\\textbf{x}}(\\tau)\\lVert$ can be made arbitrarily small $\\forall \\tau\\in[0,t_f\/\\varepsilon^2]$, which is what was to be proven. \n\\fi\nThe full proof of this Lemma is rather long. So we include it here for review purposes, but it will be replaced by a sketch in the final manuscript to conform with the page limit. The full proof will appear in a journal version of the current manuscript. The proof combines ideas from \\cite[Lemma 1]{durr2015singularly} and \\cite[Lemma 2.8.2]{sanders2007averaging}. \n\nFix an arbitrary bounded subset $\\mathcal{B}_{\\textbf{x}}\\times \\mathcal{B}_{\\textbf{z}}\\subset\\mathbb{R}^n\\times\\mathbb{R}^m$, an arbitrary $D\\in(0,\\infty)$, and an arbitrary $t_f\\in (0,\\infty)$. Due to \\thref{asmp:A}, we know that  $\\forall (\\textbf{x}_0,\\textbf{y}_0)\\in \\mathcal{B}_{\\textbf{x}}\\times \\mathcal{B}_{\\textbf{z}}$, and $\\forall \\varepsilon\\in(0,\\varepsilon_0]$, unique trajectories of the system (\\ref{eq:shifted_orig_sys}) exist.  Let $[0,\\tau_e)$ with $\\tau_e\\in(0,\\infty)$ be the maximal interval of existence and uniqueness of a given solution $(\\textbf{x}(\\tau),\\textbf{z}(\\tau))$, where the dependence of the solution on the initial condition is suppressed for brevity. By assumption, we know that the compact subset $\\mathcal{S}$ is globally uniformly asymptotically stable for the reduced order averaged system (\\ref{eq:reduced_avg_system}). Hence, according to \\cite{abdelgalil2022recursive,durr2013lie}, we know that the compact subset $\\mathcal{S}$ is semi-globally practically uniformly asymptotically stable for the reduced order system (\\ref{eq:reduced_system}). That is, we know that $\\exists \\varepsilon_1\\in (0,\\varepsilon_0)$ and a compact subset $\\mathcal{N}_{\\textbf{x}}\\subset \\mathbb{R}^n$ such that $\\forall \\textbf{x}_0\\in \\mathcal{B}_{\\textbf{x}}$, and $\\forall \\varepsilon\\in(0,\\varepsilon_1)$, solutions $\\tilde{\\textbf{x}}(\\tau)$ to system (\\ref{eq:reduced_system}) exist on the interval $[0,\\infty)$, and $\\tilde{\\textbf{x}}(\\tau)\\in \\mathcal{N}_{\\textbf{x}},\\,\\forall \\tau\\in [0,\\infty)$. Define an open tubular neighborhood $\\mathcal{O}_\\textbf{x}(\\tau)$ around $\\tilde{\\textbf{x}}(\\tau)$ by $\\mathcal{O}_\\textbf{x}(\\tau)=\\{\\textbf{x}\\in\\mathbb{R}^n\\,:\\,\\lVert \\textbf{x} - \\tilde{\\textbf{x}}(\\tau)\\lVert < D\\}$, and observe that the $\\textbf{x}$-component of the solution to (\\ref{eq:shifted_orig_sys}) is initially inside $\\mathcal{O}_\\textbf{x}(0)$, i.e. $\\textbf{x}(0)=\\textbf{x}_0\\in \\mathcal{O}_\\textbf{x}(0)$. Moreover, define the compact subset $\\mathcal{M}_\\textbf{x}=\\overline{\\{x\\in\\mathbb{R}^n:\\,\\inf_{\\textbf{x}'\\in \\mathcal{N}_\\textbf{x}}\\lVert \\textbf{x}-\\textbf{x}'\\lVert <D\\}}$, where the overline indicates the closure of a set. The continuity of the solutions $(\\textbf{x}(\\tau),\\textbf{z}(\\tau))$ implies that one of the following cases holds: C1) $\\exists \\tau_D\\in (0,\\tau_e)$ s.t. $\\textbf{x}(\\tau)\\in \\mathcal{O}_\\textbf{x}(\\tau),\\,\\forall \\tau\\in[0,\\tau_D)$, and $\\lVert \\textbf{x}(\\tau_D)-\\tilde{\\textbf{x}}(\\tau_D)\\lVert = D$, or C2) $\\textbf{x}(\\tau)\\in \\mathcal{O}_\\textbf{x}(\\tau),\\,\\forall \\tau\\in [0,\\tau_e)$. Suppose that case C1) holds and observe that $\\tilde{\\textbf{x}}(\\tau)\\in \\mathcal{M}_\\textbf{x},\\,\\forall \\tau\\in[0,\\tau_D]$. From \\thref{asmp:A}, we know that the matrix $\\textbf{A}$ is Hurwitz, which implies \\cite{khalil2002nonlinear} that there exists a function $ V\\in\\mathcal{C}^1(\\mathbb{R}^m;\\mathbb{R})$, and positive constants $\\alpha_i,\\,i\\in\\{1,2,3,4\\}$ such that:\n        \\begin{gather}\n            \\label{eq:Lyapunov_1}\\alpha_1\\lVert \\textbf{z} \\lVert^2 \\leq V(\\textbf{z}) \\leq \\alpha_2 \\lVert \\textbf{z} \\lVert ^2 \\\\\n            \\label{eq:Lyapunov_2}\\partial_{\\textbf{z}}V(\\textbf{z}) \\textbf{A}\\,\\textbf{z} \\leq -\\alpha_3 \\lVert \\textbf{z} \\lVert ^2, \\quad \\left\\lVert \\partial_{\\textbf{z}}V(\\textbf{z})  \\right\\lVert \\leq \\alpha_4 \\lVert \\textbf{z} \\lVert\n        \\end{gather}\n    \n    Let $c\\in (0,\\infty)$ be such that the compact subset $\\mathcal{N}_\\textbf{z}=\\{\\textbf{z}\\in\\mathbb{R}^m :\\, \\lVert \\textbf{z} \\lVert \\leq \\sqrt{c\/\\alpha_2} \\}$ contains the bounded set $\\mathcal{B}_{\\textbf{z}}$, and define the compact subset $\\mathcal{M}_\\textbf{z}=\\{z\\in\\mathbb{R}^m :\\, \\lVert z \\lVert \\leq \\sqrt{c\/\\alpha_1} \\}$. Furthermore, define the compact subset $\\mathcal{K}=\\mathcal{M}_\\textbf{x}\\times \\mathcal{M}_\\textbf{z}$, and compute the derivative of $V$ along the trajectories of (\\ref{eq:shifted_orig_sys}):\n    \\begin{align}\n        \\frac{dV}{d\\tau} &= \\partial_{\\textbf{z}}V(\\textbf{z}) \\textbf{A}\\,\\textbf{z} +\\sum_{i=1}^2 \\varepsilon^i\\partial_{\\textbf{z}}V(\\textbf{z})  \\textbf{h}_i(\\textbf{x},\\textbf{z},\\tau)+O(\\varepsilon^3)\\\\\n        &\\leq -\\left(\\alpha_3- \\alpha_4\\sum_{i=1}^2 \\varepsilon^i L_{\\textbf{h}_i,\\mathcal{K}}\\right) \\lVert \\textbf{z} \\lVert ^2 + B_{\\textbf{h}_3,\\mathcal{K}}\\,\\varepsilon^3\n    \\end{align}\n    where we used the fact that the $O(\\varepsilon^3)$ terms are uniformly bounded on the compact subset $\\mathcal{K}$, and the vector fields $\\textbf{h}_1$ and $\\textbf{h}_2$ have an equilibrium point at $\\textbf{z}=0$ and are Lipschitz continuous. Let $\\varepsilon_2=\\min\\{\\varepsilon_1,1,\\alpha_3\/(2\\alpha_4(L_{h_1,\\mathcal{K}}+L_{h^2,\\mathcal{K}}))\\}$, and observe that $\\forall \\varepsilon\\in(0,\\varepsilon_2)$, $\\forall (\\textbf{x},\\textbf{z})\\in \\mathcal{K}$, $\\forall \\tau\\in \\mathbb{R}$, we have:\n    \\begin{align}\n        \\frac{d V(\\textbf{z})}{d\\tau} \\leq -\\frac{\\alpha_3}{2}\\lVert \\textbf{z} \\lVert ^2 + B_{\\textbf{h}_3,\\mathcal{K}}\\, \\varepsilon^3\n    \\end{align}\n    Let $\\varepsilon_3=\\min\\{\\varepsilon_2,\\sqrt{\\alpha_3 c \/(4\\alpha_2 B_{\\textbf{h}_3,\\mathcal{K}})}\\}$, then observe that $\\forall\\varepsilon\\in(0,\\varepsilon_3),\\,\\forall \\tau\\in\\mathbb{R}$, and $\\forall (\\textbf{x},\\textbf{z})\\in \\mathcal{M}_\\textbf{x}\\times \\mathcal{M}_\\textbf{z}\\backslash \\mathcal{N}_\\textbf{z}$, we have that $\\dot{V}\\leq 0$, and so the solution $(\\textbf{x}(\\tau),\\textbf{z}(\\tau))$ stays inside $\\mathcal{M}_\\textbf{x}\\times \\mathcal{M}_\\textbf{z}$, $\\forall \\tau\\in[0,\\tau_D]$. In addition, similar to \\cite[Theorem 4.18]{khalil2002nonlinear}, we have that there exists constants $\\gamma,\\lambda,\\alpha >0$ such that the estimate $\\lVert \\textbf{z}(\\tau)\\lVert < \\gamma\\,\\lVert \\textbf{z}_0\\lVert \\text{e}^{-\\lambda \\tau} + \\alpha\\,\\varepsilon^{\\frac{3}{2}}$, holds on the time interval $\\tau\\in[0,\\tau_D]$, $\\forall (\\textbf{x}_0,\\textbf{z}_0)\\in\\mathcal{B}_\\textbf{x}\\times \\mathcal{B}_\\textbf{z}$, $\\forall \\varepsilon\\in(0,\\varepsilon_3)$. We emphasize that the constants $\\gamma, \\lambda, \\alpha$ depend on the constants $\\alpha_j,L_{\\textbf{h}_i,\\mathcal{K}},B_{\\textbf{h}_i,\\mathcal{K}}$, but do not depend on the choice of $\\varepsilon\\in(0,\\varepsilon_3)$. Similar arguments can be used to establish that in C2), we have that $(\\textbf{x}(\\tau),\\textbf{z}(\\tau))\\in \\mathcal{M}_\\textbf{x}\\times \\mathcal{M}_\\textbf{z},\\,\\forall \\tau\\in [0,\\tau_e)$, which implies that $[0,\\infty)\\subset [0,\\tau_e)$. Now observe that we may choose $\\varepsilon<(D\/\\alpha)^{\\frac23}$ in to ensure that the result of the lemma holds in case C2) (see also the proof of Lemma 1 in \\cite{durr2015singularly}). \n    \n    Next, we define an $\\varepsilon$-dependent time $\\tau_\\varepsilon$ by requiring that the following inequality is satisfied:\n    \\begin{align}\n        \\label{eq:t_epsilon} \\gamma\\,\\lVert \\textbf{z}_0\\lVert \\text{e}^{-\\lambda\\,\\tau} &< \\alpha\\, \\varepsilon^{\\frac32}, & \\forall \\textbf{z}_0&\\in \\mathcal{M}_\\textbf{z},\\,\\forall \\tau>\\tau_\\varepsilon\n    \\end{align}\n    and observe that this is always possible for $\\varepsilon >0$. In fact, it can be shown that $\\tau_\\varepsilon=$ $\\max\\{(3\/(2\\lambda)) $ $ \\log((\\gamma \\sqrt{c\/\\alpha_2})\/(\\alpha\\,\\varepsilon)),$ $0\\}$ satisfies the inequality (\\ref{eq:t_epsilon}).  Now, we show that $\\exists \\varepsilon_4\\in(0,\\varepsilon_3)$ such that $\\tau_\\varepsilon<\\tau_D,\\, \\forall \\varepsilon\\in (0,\\varepsilon_4)$. To obtain a contradiction, suppose that there exists a bounded subset $\\mathcal{B}_{\\textbf{x}}\\times \\mathcal{B}_{\\textbf{z}}\\subset\\mathbb{R}^n\\times \\mathbb{R}^m$, and a $D\\in(0,\\infty)$, such that $\\forall \\varepsilon_4\\in(0,\\varepsilon_3)$, $\\exists \\varepsilon\\in(0,\\varepsilon_4)$ such that $\\tau_\\varepsilon \\geq \\tau_D$. We estimate the difference $\\lVert \\textbf{x}(\\tau_D)-\\tilde{\\textbf{x}}(\\tau_D)\\lVert = \\lVert \\int_{0}^{\\tau_D}\\big(\\sum_{i=1}^2\\varepsilon^i (\\textbf{v}_i(\\textbf{x}(\\tau),\\textbf{z}(\\tau),\\tau))-\\tilde{\\textbf{f}}_i(\\tilde{\\textbf{x}}(\\tau),\\tau) + O(\\varepsilon^3)\\big) d\\tau\\lVert \\leq \\varepsilon \\int_{0}^{\\tau_D} B_{\\textbf{v}+\\textbf{f},\\mathcal{K}} d\\tau \\leq B_{\\textbf{v}+\\textbf{f},\\mathcal{K}} \\tau_D\\varepsilon \\leq B_{\\textbf{v}+\\textbf{f},\\mathcal{K}} \\tau_\\varepsilon\\,\\varepsilon$, where $B_{\\textbf{v}+\\textbf{f},\\mathcal{K}}$ is a uniform upper bound on the norm of the integrand inside the compact subset $\\mathcal{K}$ whose existence is guaranteed by \\thref{asmp:A}. Now, observe that $\\lim_{\\varepsilon\\rightarrow 0} \\tau_\\varepsilon\\,\\varepsilon = 0$, and so $\\forall D\\in(0,\\infty)$, $\\exists \\varepsilon_4\\in(0,\\varepsilon_3)$ such that $B_{\\textbf{v}+\\textbf{f},\\mathcal{K}} \\tau_\\varepsilon\\,\\varepsilon \\leq D\/2,\\,\\forall \\varepsilon\\in(0,\\varepsilon_4)$. Hence, we have that $\\forall \\varepsilon\\in(0,\\varepsilon_4)$, $\\lVert x(\\tau_D)-\\tilde{\\textbf{x}}(\\tau_D)\\lVert \\leq D\/2$ which contradicts the definition of $\\tau_D$. Accordingly, we have that for all bounded subsets $\\mathcal{B}_{\\textbf{x}}\\times \\mathcal{B}_{\\textbf{z}}\\subset\\mathbb{R}^n\\times \\mathbb{R}^m$, $\\forall D\\in(0,\\infty)$, $\\exists \\varepsilon_4\\in(0,\\varepsilon_3)$, such that $\\forall \\varepsilon\\in(0,\\varepsilon_4)$, $\\forall (\\textbf{x}_0,\\textbf{z}_0)\\in \\mathcal{B}_{\\textbf{x}}\\times \\mathcal{B}_{\\textbf{z}}$, we have that $\\tau_\\varepsilon < \\tau_D$.\n    \n    Next, we show that $\\exists\\varepsilon_5\\in(0,\\varepsilon_4)$ such that $t_f\/\\varepsilon^2< \\tau_D$ $\\forall \\varepsilon\\in(0,\\varepsilon_5)$. To obtain a contradiction, suppose that there exists a bounded subset $\\mathcal{B}_{\\textbf{x}}\\times \\mathcal{B}_{\\textbf{z}}\\subset\\mathbb{R}^n\\times \\mathbb{R}^m$, a $t_f\\in(0,\\infty)$, and a $D\\in(0,\\infty)$, such that $\\forall \\varepsilon_5\\in(0,\\varepsilon_4)$, $\\exists \\varepsilon\\in(0,\\varepsilon_5)$ such that $t_f\/\\varepsilon^2 \\geq \\tau_D$. Once again, we estimate the difference $\\lVert \\textbf{x}(\\tau)-\\tilde{\\textbf{x}}(\\tau)\\lVert$ on the interval $[0,\\tau_D]$. First, for $\\tau\\leq\\tau_\\varepsilon$, observe that $\\lVert \\textbf{x}(\\tau)-\\tilde{\\textbf{x}}(\\tau)\\lVert= \\lVert \\int_{0}^{\\tau}\\big(\\sum_{i=1}^2\\varepsilon^i (\\textbf{v}_i(\\textbf{x}(s),\\textbf{z}(s),s)-\\tilde{\\textbf{f}}_i(\\tilde{\\textbf{x}}(s),s)) + O(\\varepsilon^3)\\big) ds\\lVert\\leq B_{\\textbf{v}+\\textbf{f},\\mathcal{K}}\\tau_{\\varepsilon}\\varepsilon$ for some constant $B_{\\textbf{v}+\\textbf{f},\\mathcal{K}}$. Since $\\lim_{\\varepsilon\\rightarrow 0}\\tau_\\varepsilon\\varepsilon = 0$, we conclude that when $\\tau<\\tau\\varepsilon$, the difference $\\lVert \\textbf{x}(\\tau)-\\tilde{\\textbf{x}}(\\tau)\\lVert$ can be made arbitrarily small by choosing $\\varepsilon$ small enough. Second, for $\\tau>t_\\varepsilon$, we have that:\n    $\\lVert \\textbf{x}(\\tau)-\\tilde{\\textbf{x}}(\\tau)\\lVert= \\lVert \\int_{0}^{\\tau}\\big(\\sum_{i=1}^2\\varepsilon^i (\\textbf{v}_i(\\textbf{x}(s),\\textbf{z}(s),s)-\\tilde{\\textbf{f}}_i(\\tilde{\\textbf{x}}(s),s)) + O(\\varepsilon^3)\\big) ds\\lVert \\leq \\lVert \\int_{0}^{\\tau_\\varepsilon}\\big(\\sum_{i=1}^2\\varepsilon^i (\\textbf{v}_i(\\textbf{x}(s),\\textbf{z}(s),s)-\\tilde{\\textbf{f}}_i(\\tilde{\\textbf{x}}(s),s))\\big) ds\\lVert+\\lVert \\int_{\\tau_\\varepsilon}^{\\tau}\\big(\\sum_{i=1}^2\\varepsilon^i (\\textbf{v}_i(\\textbf{x}(s),\\textbf{z}(s),s)-\\tilde{\\textbf{f}}_i(\\tilde{\\textbf{x}}(s),s))\\big) ds\\lVert$ $+ B_{\\textbf{v}+\\textbf{f},\\mathcal{K}}\\tau_D\\varepsilon^3$, which leads to the estimate: \n    \\begin{gather}\\label{eq:I_estimate}\n    \\lVert \\textbf{x}(\\tau)-\\tilde{\\textbf{x}}(\\tau)\\lVert \\leq B_{\\textbf{v}+\\textbf{f},\\mathcal{K}}(\\tau_\\varepsilon+\\tau_D\\varepsilon^2)\\varepsilon+ \\lVert \\textbf{I}_1\\lVert\\\\\n    \\textbf{I}_1 = \\int_{\\tau_\\varepsilon}^{\\tau} \\sum_{i=1}^2\\varepsilon^i (\\textbf{v}_i(\\textbf{x}(s),\\textbf{z}(s),s))-\\tilde{\\textbf{f}}_i(\\tilde{\\textbf{x}}(s),s)) ds\n    \\end{gather}\n    on the interval $[0,\\tau_D]$. We proceed to estimate $\\lVert \\textbf{I}_1 \\lVert $ as follows:\n    \\begin{align}\\label{eq:I1_estimate}\n        \\lVert \\textbf{I}_1\\lVert &\\leq \\varepsilon (\\lVert \\textbf{I}_2\\lVert + \\lVert \\textbf{I}_3\\lVert ) + \\varepsilon^2 (\\lVert \\textbf{I}_4\\lVert +\\lVert \\textbf{I}_5\\lVert )\n    \\end{align}\n    where $\\textbf{I}_i$ for $i\\in\\{2,3,4,5\\}$ are given by:\n    \\begin{align}\n        \\textbf{I}_2&= \\int_{\\tau_\\varepsilon}^{\\tau}(\\textbf{v}_1(\\textbf{x}(s),\\textbf{z}(s),s))-\\textbf{v}_1(\\textbf{x}(s),0,s)) ds\\\\\n        \\textbf{I}_3&= \\int_{\\tau_\\varepsilon}^{\\tau} (\\textbf{v}_1(\\textbf{x}(s),0,s)-\\tilde{\\textbf{f}}_1(\\tilde{\\textbf{x}}(s),s)) ds\\\\\n        \\textbf{I}_4&= \\int_{\\tau_\\varepsilon}^{\\tau}(\\textbf{v}_2(\\textbf{x}(s),\\textbf{z}(s),s))-\\textbf{v}_2(\\textbf{x}(s),0,s)) ds\\\\\n        \\textbf{I}_5&= \\int_{\\tau_\\varepsilon}^{\\tau} (\\textbf{v}_2(\\textbf{x}(s),0,s)-\\tilde{\\textbf{f}}_2(\\tilde{\\textbf{x}}(s),s)) ds\n    \\end{align}\n    Observe that $\\textbf{v}_i(\\textbf{x},0,s)=\\tilde{\\textbf{f}}_i(\\textbf{x},s)$, and so we have that:\n    \\begin{align}\n        \\textbf{I}_3&= \\int_{\\tau_\\varepsilon}^{\\tau} (\\tilde{\\textbf{f}}_1(\\textbf{x}(s),s)-\\tilde{\\textbf{f}}_1(\\tilde{\\textbf{x}}(s),s)) ds \\\\\n        \\textbf{I}_5&= \\int_{\\tau_\\varepsilon}^{\\tau} (\\tilde{\\textbf{f}}_2(\\textbf{x}(s),s)-\\tilde{\\textbf{f}}_2(\\tilde{\\textbf{x}}(s),s)) ds\n    \\end{align}\n    We estimate each of the integrals above, starting by $\\textbf{I}_2$, $\\textbf{I}_4$ and $\\textbf{I}_5$, which can be estimated as:\n    \\begin{align}\\label{eq:I2_estimate}\n        \\lVert \\textbf{I}_2\\lVert &\\leq \\int_{\\tau_\\varepsilon}^{\\tau} L_{\\textbf{v}_1,\\mathcal{K}} \\lVert \\textbf{z}(s) \\lVert d\\tau \\\\\\label{eq:I4_estimate}\n        \\lVert \\textbf{I}_4\\lVert &\\leq \\int_{\\tau_\\varepsilon}^{\\tau} L_{\\textbf{v}_2,\\mathcal{K}} \\lVert \\textbf{z}(s) \\lVert ds \\\\\\label{eq:I5_estimate}\n        \\lVert \\textbf{I}_5\\lVert &\\leq \\int_{\\tau_\\varepsilon}^{\\tau} L_{\\textbf{f}_2,\\mathcal{K}} \\lVert \\textbf{x}(s)-\\tilde{\\textbf{x}}(s) \\lVert ds\n    \\end{align}\n    where $L_{\\textbf{v}_1,\\mathcal{K}},L_{\\textbf{v}_2,\\mathcal{K}},L_{\\textbf{f}_2,\\mathcal{K}}>0$ are Lipschitz constants. Next, we estimate $\\lVert \\textbf{I}_3\\lVert $. We proceed by dividing the interval $\\mathcal{I}=[\\tau_\\varepsilon,\\tau]$ into sub-intervals of length $T$ and a left over piece:\n    $$\\mathcal{I}=\\left(\\bigcup_{i=1}^{k(\\varepsilon)} [T_\n    {i-1},T_i]\\right) \\bigcup \\,[k(\\varepsilon)T,\\tau], $$ where $T_i=\\tau_\\varepsilon+i\\,T$, and $k(\\varepsilon)$ is the unique integer such that $k(\\varepsilon) T \\leq \\tau < k(\\varepsilon) T + T$. Then, we split $\\textbf{I}_3$ into a sum of sub-integrals:\n\\begin{align}\n      \\textbf{I}_3\n      &=\\sum_{i=1}^{k(\\varepsilon)} \\textbf{I}_{3,i}+ \\int_{k(\\varepsilon)T}^{\\tau}\\left(\\tilde{\\textbf{f}}_1(\\textbf{x}(s),s)-\\tilde{\\textbf{f}}_1(\\tilde{\\textbf{x}}(s),s) \\right) ds \\\\\n      \\textbf{I}_{3,i}&= \\int_{T_{i-1}}^{T_i}\\left(\\tilde{\\textbf{f}}_1(\\textbf{x}(s),s)-\\tilde{\\textbf{f}}_1(\\tilde{\\textbf{x}}(s),s) \\right)ds\n\\end{align}\nThe part of the integral on the leftover piece can be bounded independently from $\\varepsilon$ as follows:\n\\begin{align}\n    \\left\\lVert\\int_{k(\\varepsilon)T}^{\\tau}\\left(\\tilde{\\textbf{f}}_1(\\textbf{x}(s),s)-\\tilde{\\textbf{f}}_1(\\tilde{\\textbf{x}}(s),s) \\right) ds\\right\\lVert \\leq 2 B_{\\textbf{f}_1,\\mathcal{K}} T\n\\end{align}\nNext, we employ Hadamard's lemma to obtain:\n\\begin{equation}\n\\begin{aligned} \n      \\textbf{I}_{3,i}&=\\int_{T_{i-1}}^{T_i}\\textbf{F}_1(\\textbf{x}(s),\\tilde{\\textbf{x}}(s),s) (\\textbf{x}(s)-\\tilde{\\textbf{x}}(s)) ds\n\\end{aligned}\n\\end{equation}\nwhere the matrix valued map $\\textbf{F}_1$ is given by:\n\\begin{align}\n&\\textbf{F}_1(\\textbf{x},\\tilde{\\textbf{x}},s)  =\\int_{0}^{1}\\partial_{\\textbf{w}}\\tilde{\\textbf{f}}_1(\\textbf{w},s)|_{\\textbf{w}=\\tilde{\\textbf{x}}+\\lambda(\\textbf{x}-\\tilde{\\textbf{x}})} d\\lambda\n\\end{align}\nThrough adding and subtracting a term, we may write:\n\\begin{equation}\n\\begin{aligned} \n     \\textbf{I}_{3,i}=&\\int_{T_{i-1}}^{T_i} \\textbf{F}_1(\\textbf{x}(T_{i-1}),\\tilde{\\textbf{x}}(T_{i-1}),s)\\,(\\textbf{x}(s)-\\tilde{\\textbf{x}}(s)) ds \\\\\n     +&\\int_{T_{i-1}}^{T_i}\\Delta_i\\big[\\textbf{F}_1\\big](s)\\,(\\textbf{x}(s)-\\tilde{\\textbf{x}}(s)) ds\n\\end{aligned}\n\\end{equation}\nwhere the term $\\Delta_i[\\textbf{F}_1]$ is given by:\n\\begin{equation*}\n\\begin{aligned} \n    \\Delta_i\\big[\\textbf{F}_1\\big](s) &= \\textbf{F}_1(\\textbf{x}(s),\\tilde{\\textbf{x}}(s),s)-\\textbf{F}_1(\\textbf{x}(T_{i-1}),\\tilde{\\textbf{x}}(T_{i-1}),s)\n\\end{aligned}\n\\end{equation*}\nNext, since the matrix-valued map $\\textbf{F}_1$ is periodic with zero average over its third argument when the other arguments are fixed, we have that\n\\begin{align}\n    \\int_{T_{i-1}}^{T_i} \\textbf{F}_1(\\textbf{x}(T_{i-1}),\\tilde{\\textbf{x}}(T_{i-1}),s)\\,\\textbf{w}\\,ds = 0 \n\\end{align}\nfor any fixed $\\textbf{w}$. Thus, we may write:\n\\begin{equation}\n\\begin{aligned} \n    &\\textbf{I}_{3,i}=\\int_{T_{i-1}}^{T_i}\\Delta_i\\big[\\textbf{F}_1\\big]\\,(\\textbf{x}(s)-\\tilde{\\textbf{x}}(s)) ds \\\\\n     &+\\int_{T_{i-1}}^{T_i} \\textbf{F}_1(\\textbf{x}(T_{i-1}),\\tilde{\\textbf{x}}(T_{i-1}),s)\\Delta_i[\\textbf{x}-\\tilde{\\textbf{x}}] ds\n\\end{aligned}\n\\end{equation}\nwhere $\\Delta_i[\\textbf{x}-\\tilde{\\textbf{x}}] = (\\textbf{x}(s)-\\textbf{x}(T_{i-1}))-(\\tilde{\\textbf{x}}(s)-\\tilde{\\textbf{x}}(T_{i-1}))$. The fundamental theorem of calculus yields: \n\\begin{align}\n    &(\\textbf{x}(s)-\\textbf{x}(T_{i-1}))-(\\tilde{\\textbf{x}}(s)-\\tilde{\\textbf{x}}(T_{i-1}))\\\\\n    =&\\varepsilon \\int_{T_{i-1}}^{s}(\\textbf{v}_1(\\textbf{x}(\\nu),\\textbf{z}(\\nu),\\nu)-\\tilde{\\textbf{f}}_1(\\tilde{\\textbf{x}}(\\nu),\\nu))d\\nu + O(\\varepsilon^2) \\\\\n    =&\\varepsilon \\int_{T_{i-1}}^{s}(\\textbf{v}_1(\\textbf{x}(\\nu),\\textbf{z}(\\nu),\\nu)-\\textbf{v}_1(\\textbf{x}(\\nu),0,\\nu))d\\nu \\\\\n    +&\\varepsilon \\int_{T_{i-1}}^{s}(\\tilde{\\textbf{f}}_1(\\textbf{x}(\\nu),\\nu)-\\tilde{\\textbf{f}}_1(\\tilde{\\textbf{x}}(\\nu),\\nu))d\\nu+ O(\\varepsilon^2)\n\\end{align}\nThrough integration by parts, we obtain:\n\\begin{equation}\n\\begin{aligned} \n    &\\int_{T_{i-1}}^{T_i}\\textbf{F}_1(\\textbf{x}(T_{i-1}),\\tilde{\\textbf{x}}(T_{i-1}),s)\\Delta_i[\\textbf{x}-\\tilde{\\textbf{x}}] ds \\\\\n    =&\\textbf{I}_{\\textbf{F},i}(s)\\,\\Delta_i[\\textbf{x}-\\tilde{\\textbf{x}}]\\bigg|_{s=T_{i-1}}^{s=T_i}-\\varepsilon\\int_{T_{i-1}}^{T_i}\\textbf{I}_{\\textbf{F},i}(s)\\,   \\Delta[\\tilde{\\textbf{f}}_1] ds \\\\\n    -&\\varepsilon\\int_{T_{i-1}}^{T_i}\\textbf{I}_{\\textbf{F},i}(s)\\, \\Delta[\\textbf{v}_1]ds + O(\\varepsilon^2)\n\\end{aligned}\n\\end{equation}\nwhere we have that:\n\\begin{align}\n    \\Delta[\\textbf{v}_1] &= \\textbf{v}_1(\\textbf{x}(s),\\textbf{z}(s),s)-\\textbf{v}_1(\\textbf{x}(s),0,s) \\\\\n    \\Delta[\\tilde{\\textbf{f}}_1]&= \\tilde{\\textbf{f}}_1(\\textbf{x}(s),s)-\\tilde{\\textbf{f}}_1(\\tilde{\\textbf{x}}(s),s) \\\\\n    \\textbf{I}_{\\textbf{F},i}(s)&=\\int_{T_{i-1}}^s\\textbf{F}_1(\\textbf{x}(T_{i-1}),\\tilde{\\textbf{x}}(T_{i-1}),\\nu)d\\nu\\,\n\\end{align}\nThe boundary term coming out of the integration by parts vanishes because the right factor vanishes at $s=T_{i-1}$ and the left factor vanishes at $s=T_{i}$, leaving only the integral terms. Using Lipschitz continuity and boundedness on compact subsets, it is not hard to see that:\n\\begin{align}\n    &\\left\\lVert \\int_{T_{i-1}}^{T_i}\\textbf{I}_{\\textbf{F},i}(s)\\,\\Delta[\\textbf{v}_1]ds \\right\\lVert\\leq \\int_{T_{i-1}}^{T_i} M_{\\textbf{I}_\\textbf{F},\\textbf{v}_1,\\mathcal{K}}\\,\\lVert \\textbf{z}(s) \\lVert ds \\\\\n    &\\left\\lVert \\int_{T_{i-1}}^{T_i}\\textbf{I}_{\\textbf{F},i}(s)\\,\\Delta[\\tilde{\\textbf{f}}_1] ds\\right\\lVert \\leq \\int_{T_{i-1}}^{T_i} M_{\\textbf{I}_\\textbf{F},\\textbf{f}_1,\\mathcal{K}}\\,\\lVert \\Delta[\\textbf{x}] \\lVert ds\\\\\n    &\\left\\lVert\\int_{T_{i-1}}^{T_i}\\Delta_i\\big[\\textbf{F}_1\\big]\\,\\Delta[\\textbf{x}] ds\\right\\lVert \\leq \\varepsilon\\int_{T_{i-1}}^{T_i} L_{\\textbf{F}_1,\\mathcal{K}}\\,\\,\\lVert \\Delta[\\textbf{x}] \\lVert ds\n\\end{align}\nwhere $\\Delta[\\textbf{x}]=\\textbf{x}(s)-\\tilde{\\textbf{x}}(s)$. By utilizing the above estimates, the integral on the sub-intervals can be shown to satisfy the bound:\n\\begin{align}\n    \\left\\lVert\\textbf{I}_{3,i}\\right\\lVert \\leq L_{\\mathcal{K}} \\,\\varepsilon \\,\\int_{T_{i-1}}^{T_i} \\left(\\lVert \\Delta[\\textbf{x}] \\lVert + \\lVert \\textbf{z}(s) \\lVert\\right)ds\n\\end{align}\nfor some Lipschitz constant $L_{\\mathcal{K}}$ and consequently the integral term $\\textbf{I}_3$ satisfies the bound:\n\\begin{align}\\label{eq:I3_estimate}\n    \\lVert \\textbf{I}_3\\lVert &\\leq L_{\\mathcal{K}} \\,\\varepsilon \\,\\int_{\\tau_\\varepsilon}^{\\tau} \\left(\\lVert \\Delta[\\textbf{x}] \\lVert + \\lVert \\textbf{z}(s) \\lVert\\right)ds+ 2B_{\\textbf{f}_1,\\mathcal{K}}T\n\\end{align}\nCombining (\\ref{eq:I_estimate}), (\\ref{eq:I1_estimate}), (\\ref{eq:I2_estimate}), (\\ref{eq:I5_estimate}), (\\ref{eq:I4_estimate}), and  (\\ref{eq:I3_estimate}), in addition to the fact that $\\tau_\\varepsilon<\\tau_D,\\,\\forall \\varepsilon\\in(0,\\varepsilon_4)$, we can show that the following estimate holds:\n\\begin{equation}\n    \\begin{aligned}\n    &\\lVert \\textbf{x}(\\tau)-\\tilde{\\textbf{x}}(\\tau)\\lVert \\leq (M_{\\mathcal{K},1}+M_{\\mathcal{K},2} \\tau_\\varepsilon  + M_{\\mathcal{K},3}\\tau_D\\varepsilon^2)\\varepsilon \\\\\n    &+ M_{\\mathcal{K},4}\\varepsilon \\int_{\\tau_\\varepsilon}^{\\tau}\\lVert \\textbf{z}(s)\\lVert ds +M_{\\mathcal{K},5}\\varepsilon^2\\int_{\\tau_\\varepsilon}^{\\tau}\\lVert \\textbf{x}(s)-\\tilde{\\textbf{x}}(s)\\lVert ds\n    \\end{aligned}\n\\end{equation}\nUsing the fact that $\\lVert \\textbf{z}(\\tau)\\lVert<2 \\alpha\\,\\varepsilon^{\\frac32},\\,\\forall \\tau>\\tau_\\varepsilon$ by definition, we obtain that:\n\\begin{align}\n    \\int_{\\tau_\\varepsilon}^{\\tau}\\lVert \\textbf{z}(s)\\lVert ds \\leq \\alpha\\, \\tau\\,\\varepsilon^{\\frac32} \\leq \\alpha\\, \\tau_D\\,\\varepsilon^{\\frac32}\n\\end{align}\nNow, remember that in order to obtain a contradiction we assumed that $\\tau_D\\leq t_f\/\\varepsilon^2$, and so we will have:\n\\begin{align}\n\\lVert \\textbf{x}&(\\tau)-\\tilde{\\textbf{x}}(\\tau)\\lVert \\leq \\delta(\\varepsilon) +\\int_{0}^{\\tau} M_{\\mathcal{K},5}\\varepsilon^2 \\lVert \\textbf{x}(s)-\\tilde{\\textbf{x}}(s)\\lVert ds\n\\end{align}\nwhere the function $\\delta(\\varepsilon)$ is given by:\n\\begin{align}\n\\delta(\\varepsilon)&= M_{\\mathcal{K},1}\\varepsilon + M_{\\mathcal{K},2} \\tau_\\varepsilon\\varepsilon + M_{\\mathcal{K},3}t_f\\varepsilon + M_{\\mathcal{K},4}t_f \\varepsilon^{\\frac12}\n\\end{align}\nAn application of Gr\\\"onwall's inequality yields:\n\\begin{align}\n    \\lVert \\textbf{x}&(\\tau)-\\tilde{\\textbf{x}}(\\tau)\\lVert \\leq \\delta(\\varepsilon) \\text{e}^{M_{\\mathcal{K},5}\\varepsilon^2\\tau} \\leq \\delta(\\varepsilon) \\text{e}^{M_{\\mathcal{K},5}\\varepsilon^2\\tau_D}\n\\end{align}\non the interval $\\tau\\in[0,\\tau_D]$. Once again, recall that we assumed that $\\tau_D\\leq t_f\/\\varepsilon^2$, and so we have: \n\\begin{align}\n    \\lVert \\textbf{x}(\\tau)-\\tilde{\\textbf{x}}(\\tau)\\lVert \\leq \\delta(\\varepsilon) \\text{e}^{M_{\\mathcal{K},5}t_f}\n\\end{align}\nNow, observe that $\\lim_{\\varepsilon\\rightarrow 0} \\delta(\\varepsilon) = 0$, and so we are guaranteed the existence of an $\\varepsilon_5\\in(0,\\varepsilon_4)$ such that $\\forall\\varepsilon\\in(0,\\varepsilon_5)$ we will have:\n\\begin{align}\n     \\lVert \\textbf{x}&(\\tau_D)-\\tilde{\\textbf{x}}(\\tau_D)\\lVert \\leq D\/2\n\\end{align}\nwhich contradicts the definition of $\\tau_D$. Hence, the assumption that $\\tau_D\\leq t_f\/\\varepsilon^2$ is wrong, and we have that for all bounded subsets $\\mathcal{B}_{\\textbf{x}}\\times \\mathcal{B}_{\\textbf{z}}\\subset\\mathbb{R}^n\\times \\mathbb{R}^m$, $\\forall D\\in(0,\\infty)$, and $\\forall t_f\\in(0,\\infty)$, $\\exists \\varepsilon_5\\in(0,\\varepsilon_4)$, such that $\\forall \\varepsilon\\in(0,\\varepsilon_5)$, $\\forall (\\textbf{x}_0,\\textbf{z}_0)\\in \\mathcal{B}_{\\textbf{x}}\\times \\mathcal{B}_{\\textbf{z}}$, we have that $t_f\/\\varepsilon^2 < \\tau_D$, which shows that the lemma also holds in case C1).\n\\end{proof}\n\n\\thref{prop:A} follows from this lemma, after reversing the time scaling $\\tau=\\omega (t-t_0)$ and the near identity part of the coordinate shift (\\ref{eq:nearid_shift}), coupled with \\cite[Theorem 2.1]{abdelgalil2022recursive} and the fact that for $\\textbf{x}\\in\\mathcal{K}$ where $\\mathcal{K}$ is any compact subset, the maps $\\bm{\\varphi}_i$ for $i\\in\\{1,2\\}$ are uniformly bounded in time due to continuity and periodicity.\n\n\\end{proof}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn 2015 the Laser Interferometer Gravitational-Wave Observatory (LIGO) Scientific Collaboration and the Virgo Collaboration \\cite{Abbott:2016blz,TheLIGOScientific:2016agk} directly observed the first gravitational wave (GW) event GW150914 coming from the coalescence of binary black holes (BHs), \nand the discovery opened us a new window to understand the property of BHs and gravity in the nonlinear and strong field regimes.\nUntil now tens of GW events in the frequency range of tens to hundreds Hertz have been confirmed \\cite{LIGOScientific:2018mvr,LIGOScientific:2020ibl,LIGOScientific:2021usb,LIGOScientific:2021djp,LIGOScientific:2017vwq,LIGOScientific:2020aai,LIGOScientific:2021qlt}.\nHowever, due to the seismic noise and gravity gradient noise, the ground-based GW observatories can only measure transient GWs in the frequency range $10-10^3$ Hz, radiated by the coalescences of stellar-mass compact binaries.\nApart from transient GW sources, the future space-based GW detectors like the Laser Interferometer Space Antenna (LISA) \\cite{Danzmann:1997hm,Audley:2017drz}, TianQin \\cite{Luo:2015ght} and Taiji \\cite{Hu:2017mde} will help us uncover unprecedented information about GW sources and fundamental physics \\cite{LISA:2017pwj,TianQin:2020hid,Ruan:2018tsw,Amaro-Seoane:2022rxf,LISA:2022kgy}. \nOne of the most conspicuous sources for the future space-based GW detectors is the extreme mass-ratio inspirals (EMRIs) \\cite{Amaro-Seoane:2007osp,Babak:2017tow}.\nEMRIs, which consist of a stellar-mass compact object (secondary object) with mass $m_p\\sim1-100~M_{\\odot}$ such as BHs, neutron stars, white dwarfs, etc. orbiting around a supermassive black hole (SMBH) (primary object) with mass $M\\sim10^5-10^7~M_{\\odot}$, \nwith the mass ratio $m_p\/M$ in the range of $10^{-7}-10^{-4}$, \nradiate millihertz GWs expected to be observed by the future space-based GW detectors.\n\nFuture detections of EMRIs with space-based detectors can provide highly precise measurements on source parameters such as the BH masses, spins, etc.\nIn \\cite{Barack:2003fp}, the authors introduced a family of approximate waveforms for EMRIs to make parameter estimation with LISA.\nFor a typical source of $m_p=10~M_{\\odot}$ and $M=10^6~M_{\\odot}$ with a signal-to-noise ratio (SNR) of $30$, \nLISA can determine the masses of both primary and secondary objects to within a fractional error of $\\sim 10^{-4}$, measure the spin of the primary object to within $\\sim 10^{-4}$, and localize the source on the sky within $\\sim 10^{-3}$ steradians.\nThe improved augmented analytic kludge model \\cite{Chua:2017ujo} provides more accurate and efficient GW waveforms to improve the errors of parameters by one order of magnitude.\nThus, EMRIs can be used to precisely measure the slope of the black-hole mass function  \\cite{Gair:2010yu} or as standard sirens \\cite{Holz:2005df} to  constrain cosmological parameters and investigate the expansion history of the Universe \\cite{MacLeod:2007jd,Laghi:2021pqk,LISACosmologyWorkingGroup:2022jok}.\nThe observations of EMRIs can also help us explore gravitational physics.\nFor example, they can be used to figure out the spacetime structure around the central SMBH to high precision, \nallowing us to test if the spacetime geometry is described by general relativity or an alternative theory \nand analyze the environments such as dark matter surrounding the central SMBH \\cite{Amaro-Seoane:2007osp,eLISA:2013xep,Eda:2013gg,Eda:2014kra,Barausse:2014tra,Yue:2017iwc,Yue:2018vtk,Babak:2017tow,Berry:2019wgg,Hannuksela:2019vip,Destounis:2020kss,Burton:2020wnj,Torres:2020fye,Barausse:2020rsu,Cardoso:2019rou,Maselli:2020zgv,Maselli:2021men,Guo:2022euk,Zhang:2022rfr,Barsanti:2022ana,Cardoso:2021wlq,Dai:2021olt,Jiang:2021htl,Zhang:2022hbt,Gao:2022hho,Gao:2022hsn,Destounis:2022obl,Barsanti:2022vvl,Liang:2022gdk}.\nIn \\cite{Cardoso:2020iji,Liu:2020vsy,Liu:2020bag}, \nthe authors investigated the eccentricity and orbital evolution \n of BH binaries under the influence of accretion in addition to the scalar\/vector and gravitational radiations, and discussed the competition between radiative mechanisms and accretion effects on eccentricity evolution.\nHowever, these discussions were mainly based on the Newtonian orbit and dipole emission.\nA generic, fully-relativistic formalism to study EMRIs in spherically symmetric and non-vacuum BH spacetime was established in \\cite{Cardoso:2022whc}.\nConsidering the secondary object of mass $m_p$ orbiting the galactic BHs (GBHs) immersed in an astrophysical environment, \nlike an accretion disk or a dark matter halo, the authors found that the relative flux difference at $\\omega M=0.02$ between a vacuum and a GBH with the halo mass $M_{\\text{halo}}=0.1 M$ and the lengthscale $a_0=10^2 M_{\\text{halo}}$ and $10^3 M_{\\text{halo}}$ is $\\sim 10\\%$ and $1\\%$, respectively.\nThe results clearly indicate that EMRIs can constrain smaller-scale matter distributions around GBHs \\cite{Cardoso:2022whc}.\n\nAccording to the no-hair theorem, any BH can be described by three parameters: the mass, angular momentum, and electric charge.\nCurrent observations have not yet been able to confirm the  no-hair theorem or the existence of extra fields besides the gravitational fields in modified gravity theories.\nThe coupling between scalar fields and higher-order curvature invariants can invalidate the no-hair theorem so that BHs can carry scalar charge which depends on the mass of BH \\cite{Sotiriou:2013qea,Silva:2017uqg,Doneva:2017bvd,Antoniou:2017acq}.\nThe possible detection of scalar fields with EMRIs was discussed in \\cite{Maselli:2020zgv,Maselli:2021men,Barsanti:2022ana,Guo:2022euk,Zhang:2022rfr,Barsanti:2022vvl}.\nSMBHs are usually neutral because of long-time charge dissipation through the presence of the plasma around them or the spontaneous production of electron-positron pairs \\cite{Gibbons:1975kk,Eardley:1975kp,1982PhRvD..25.2509H,1982Prama..18..385J,Gong:2019aqa}.\nHowever, the existence of stable charged astrophysical stellar-mass compact objects such as BHs, neutron stars, white dwarfs, etc. in nature remains controversial.\nIn Refs. \\cite{Bekenstein:1971ej,de1995relativistic,deFelice:1999qp,Ivanov:2002jy,Majumdar:1947eu,zhang_influence_1982,Anninos:2001yb,Bonnor:1975gba}, \nit was shown that there can be a maximal huge amount of charge with the charge-to-mass ratio of the order one for highly compact stars, whose radius is on the verge of forming an event horizon.\nThe balance between the attractive gravitational force from the matter part and the repulsive force from the electrostatic part is unstable and charged compact stars will collapse to a charged BH due to a decrease in the electric field \\cite{Ray:2003gt}.\nAlso, BHs can be charged through the Wald mechanism by selectively accreting charge in a magnetic field \\cite{Wald:1974np}, or by accreting minicharged dark matter beyond the standard model \\cite{Cardoso:2016olt}.\nThe charge carried by compact objects can affect the parameter estimation of  the chirp mass and BH merger \\cite{Christiansen:2020pnv,Jai-akson:2017ldo}\nand the merger rate distribution of primordial BH binaries \\cite{Liu:2022wtq,Liu:2020cds}.\nThe electromagnetic self-force acting on a charged particle in an equatorial circular orbit of Kerr BH was calculated in \\cite{Torres:2020fye}. \nIt showed the dissipative self-force balances with the sum of the electromagnetic flux radiated to infinity and down the BH horizon,\nand prograde orbits can stimulate BH superradiance although the superradiance is not sufficient to support floating orbits even at the innermost stable circular orbit (ISCO) \\cite{Torres:2020fye}.\nThe observations of GW150914 can constrain the charge-to-mass ratio of charged BHs to be as high as 0.3 \\cite{Bozzola:2020mjx}.\n\nIn \\cite{Zhang:2022hbt}, the energy fluxes for both gravitational and electromagnetic waves induced by a charged particle orbiting around a Schwarzschild BH were studied.\nIt was demonstrated that the electric charge leaves a significant imprint on the phase of GWs and is observable with space-based GW detectors.\nIn this paper, based on the Teukolsky formalism for BH perturbations \\cite{Teukolsky:1973ha,Press:1973zz,Teukolsky:1974yv}, \nwe numerically calculate the energy fluxes for both tensor and vector perturbations induced by a charged particle moving in an equatorial circular orbit around a Kerr BH, the orbital evolution of EMRIs up to the ISCO, \nand investigate the capabilities to detect vector charge carried by the secondary compact object with space-based GW detectors such as LISA, TianQin, and Taiji. \nWe apply the methods of faithfulness and Fisher information matrix (FIM) to assess the capability of space-based GW detectors to detect the vector charge carried by the secondary compact object.\nThe paper is organized as follows.\nIn Sec. \\ref{sec2}, we introduce the model with vector charge and the Teukolsky perturbation formalism.\nIn Sec. \\ref{sec3}, we give the source terms as well as procedures for solving the inhomogeneous Teukolsky equations. \nThen we numerically calculate the energy fluxes for gravitational and vector fields using the Teukolsky and generalized Sasaki-Nakamura (SN) formalisms in the background of a Kerr BH.\nIn Sec. \\ref{sec4}, we give the numerical results of energy fluxes falling onto the horizon and radiated to infinity for gravitational and vector fields, \nthen we use the dephasing of GWs to constrain the charge.\nIn Sec. \\ref{sec5}, we calculate the faithfulness between GWs with and without vector charge and perform the FIM to estimate the errors of detecting vector charge with LISA, TianQin, and Taiji.\nSec. \\ref{sec5} is devoted to conclusions and discussions.\nIn this paper, we set $c=G=M=1$.\n\n\\section{Einstein-Maxwell field equations}\n\\label{sec2}\nThe simplest model including vector charges is Einstein-Maxwell theory, which is modeled via the massless vector field\n\\begin{equation}\nS=\\int d^4x\\frac{\\sqrt{-g}}{16\\pi}\\left[R-\\frac{1}{4}F^{\\mu\\nu}F_{\\mu\\nu}-A_{\\mu}J^{\\mu}\\right]-S_{\\text{matter}}(g_{\\mu\\nu},\\Phi),\n\\end{equation}\nwhere $R$ is the Ricci scalar, $A_\\mu$ is a massless vector field, $F_{\\mu\\nu}=\\nabla_{\\mu}A_{\\nu}-\\nabla_{\\nu}A_{\\mu}$ is the field strength and $\\Phi$ is the matter field, $J^\\mu$ is the electric current density.\nVarying the action with respect to the metric tensor and the vector field yields the Einstein-Maxwell field equations\n\\begin{equation}\nG^{\\mu\\nu}=8\\pi T^{\\mu\\nu}_p+8\\pi T^{\\mu\\nu}_e,\n\\end{equation}\n\\begin{equation}\n\\nabla_\\nu F^{\\mu\\nu}=4\\pi J^\\mu,\n\\end{equation}\nwhere $G_{\\mu\\nu}$ is the Einstein tensor, $T_p^{\\mu\\nu}$ and $T_e^{\\mu\\nu}$ are the particle's material stress-energy tensor and vector stress-energy tensor, respectively.\nSince the amplitude of the vector stress-energy $T_e^{\\mu\\nu}$ is quadratic in the vector field, the contribution to the background metric from the vector field is second order.\nFor an EMRI system $(m_p\\ll M)$ composing of a small compact object with mass $m_p$ and  charge-to-mass ratio $q$ orbiting around a Kerr BH with mass $M$ and spin $Ma$, we can ignore the contribution to the background metric from the vector field.\nThe perturbed Einstein and Maxwell equations for EMRIs are\n\\begin{equation}\nG^{\\mu\\nu}=8\\pi T^{\\mu\\nu}_p,\n\\end{equation}\n\\begin{equation}\n\\nabla_\\nu F^{\\mu\\nu}=4\\pi J^\\mu,\n\\end{equation}\nwhere\n\\begin{equation}\nT^{\\mu\\nu}_p(x)=m_p\\int d\\tau~u^{\\mu}u^\\nu\\frac{\\delta^{(4)}\\left[x-z(\\tau)\\right]}{\\sqrt{-g}},\n\\end{equation}\n\\begin{equation}\nJ^{\\mu}(x)=q m_p\\int d\\tau~ u^{\\mu}\\frac{\\delta^{(4)}\\left[x-z(\\tau)\\right]}{\\sqrt{-g}},\n\\end{equation}\nand $u^{\\mu}$ is the velocity of the particle.\nWe use the Newman-Penrose formalism \\cite{Newman:1966ub} to study perturbations around a Kerr BH induced by a charged particle with mass $m_p$ and charge $q$.\nIn Boyer-Lindquist coordinate,  the metric of Kerr BHs is\n\\begin{equation}\\label{SBH}\n\\begin{split}\nds^2=&(1-2r\/\\varSigma)dt^2+(4ar\\sin(\\theta)\/\\varSigma)dtd\\varphi-(\\varSigma\/\\Delta)dr^2-\\varSigma d\\theta^2\\\\\n&-\\sin^2{\\theta}(r^2+a^2+2a^2r\\sin^2{\\theta}\/\\varSigma)d\\varphi^2.\n\\end{split}\\end{equation}\nwhere  $\\varSigma=r^2+a^2\\cos^2{\\theta}$, and $\\Delta=r^2-2r+a^2$. When $a=0$, the metric reduces to the Schwarzschild metric.\nBased on the metric \\eqref{SBH}, we construct the null tetrad,\n\\begin{equation}\n\\begin{split}\n\\begin{split}\nl^\\mu&=[(r^2+a^2)\/\\Delta,1,0,a\/\\Delta],\\\\\nn^\\mu&=[r^2+a^2,-\\Delta,0,a]\/(2\\varSigma),\\\\\nm^\\mu&=[ia\\sin{\\theta},0,1,i\/\\sin{\\theta}]\/(2^{1\/2}(r+ia\\cos{\\theta})),\\\\\n\\bar m^\\mu&=[-ia\\sin{\\theta},0,1,-i\/\\sin{\\theta}]\/(2^{1\/2}(r-ia\\cos{\\theta})).\n\\end{split}\n\\end{split}\n\\end{equation}\nThe propagating vector field is described by the two complex quantities,\n\\begin{equation}\n\\phi_0=F_{\\mu\\nu}l^\\mu m^\\nu,\\qquad \\phi_2=F_{\\mu\\nu}\\bar{m}^\\mu n^\\nu.\n\\end{equation}\nThe propagating gravitational field is described by the two complex Newman-Penrose variables\n\\begin{equation}\n\\begin{split}\n\\psi_0&=-C_{\\alpha\\beta\\gamma\\delta}l^\\alpha m^\\beta l^\\gamma m^\\delta    ,\\\\\n\\psi_4&=-C_{\\alpha\\beta\\gamma\\delta}n^\\alpha \\bar{m}^\\beta n^\\gamma \\bar{m}^\\delta,\n\\end{split}\n\\end{equation}\nwhere $C_{\\alpha\\beta\\gamma\\delta}$ is the Weyl tensor.\nA single master equation for tensor ($s=-2$) and vector ($s=-1$) perturbations was derived as \\cite{Teukolsky:1973ha},\n\n\\begin{equation}\n\\label{TB}\n\\begin{split}\n&\\left[\\frac{(r^2+a^2)^2}{\\Delta}-a^2\\sin^2{\\theta}\\right]\\frac{\\partial^2\\psi}{\\partial t^2}+\\frac{4ar}{\\Delta}\\frac{\\partial^2\\psi}{\\partial t\\partial\\varphi}+\\left[\\frac{a^2}{\\Delta}-\\frac{1}{\\sin^2{\\theta}}\\right]\\frac{\\partial^2\\psi}{\\partial \\varphi^2}\\\\\n&\\qquad-\\Delta^{-s}\\frac{\\partial}{\\partial r}\\left(\\Delta^{s+1}\\frac{\\partial\\psi}{\\partial r}\\right)-\\frac{1}{\\sin\\theta}\\frac{\\partial}{\\partial \\theta}\\left(\\sin\\theta\\frac{\\partial\\psi}{\\partial \\theta}\\right)-2s\\left[\\frac{a(r-1)}{\\Delta}+\\frac{i\\cos\\theta}{\\sin^2{\\theta}}\\right]\\frac{\\partial\\psi}{\\partial \\varphi}\\\\\n&\\qquad\\qquad\\qquad\\qquad-2s\\left[\\frac{(r^2-a^2)}{\\Delta}-r-ia\\cos\\theta\\right]\\frac{\\partial\\psi}{\\partial t}+(s^2\\cot^2\\theta-s)\\psi=4\\pi\\varSigma T,\n\\end{split}\n\\end{equation}\nthe explicit field $\\psi$ and the corresponding source $T$ are given in  Table \\ref{source} \\cite{Teukolsky:1973ha}.\n \\begin{table}[h]\n  \\centering\n  \t\\begin{tabular}{|p{0.5cm}<{\\centering}|p{3.4cm}<{\\centering}|p{3.4cm}<{\\centering}|}\n\t\t\\hline\n$s$ & $\\psi$ &  $T$\\\\ \\hline\n-1   & $(r-i a\\cos\\theta)^{2}\\phi_2$ & $(r-i a\\cos\\theta)^{2}J_2$  \\\\ \\hline\n -2   & $(r-i a\\cos\\theta)^{4}\\psi_4$ &$2(r-i a\\cos\\theta)^{4}T_4$  \\\\ \\hline\n\t\\end{tabular}\n \\caption{The explicit expressions for the the field $\\psi$ and the corresponding source $T$.}\n    \\label{source}\n\\end{table}\nIn terms of the eigenfunctions ${_{s}}S_{lm}(\\theta)$ \\cite{Teukolsky:1973ha,Goldberg:1966uu}, the field $\\psi$ can be written as   %\n\\begin{equation}\n\\psi=\\int d\\omega \\sum_{l,m}R_{\\omega lm}(r)~{_{s}}S_{lm}(\\theta)e^{-i\\omega t+im\\varphi},\n\\end{equation}\nwhere the radial function $R_{\\omega lm}(r)$ satisfies the inhomogeneous Teukolsky equation\n\\begin{equation}\n\\label{Teukolsky}\n\\Delta^{-s}\\frac{d}{d r}\\left(\\Delta^{s+1}\\frac{d R_{\\omega lm}}{d r}\\right)-V_{T}(r)R_{\\omega lm}=T_{\\omega lm},\n\\end{equation}\nthe function\n\\begin{equation}\nV_{T}=-\\frac{K^2-2is(r-1)K}{\\Delta}-4is\\omega r+\\lambda_{lm\\omega},\n\\end{equation}\n $K=(r^2+a^2)\\omega-am$, $\\lambda_{lm\\omega}$ is the corresponding eigenvalue which can be computed by the BH Perturbation Toolkit \\cite{BHPToolkit}, and the source $T_{\\omega lm}(r)$ is\n\\begin{equation}\nT_{\\omega lm}(r)=\\frac{1}{2\\pi}\\int dt d\\Omega ~4\\pi \\Sigma T ~{_s}S_{lm}(\\theta)e^{i\\omega t-im\\varphi}.\n\\end{equation}\nFor the equatorial circular trajectory at $r_0$ under consideration,\nthe sources are\n\\begin{equation}\n\\begin{split}\nT^{\\mu\\nu}_p(x)&=\\frac{m_p}{r_0^2}\\frac{u^{\\mu}u^{\\nu}}{u^t}\\delta(r-r_0)\\delta(\\cos\\theta)\\delta(\\varphi-\\hat{\\omega} t),\\\\\nJ^{\\mu}(x)&=q\\frac{m_p}{r_0^2}\\frac{u^{\\mu}}{u^t}\\delta(r-r_0)\\delta(\\cos\\theta)\\delta(\\varphi-\\hat{\\omega} t),\n\\end{split}\n\\end{equation}\nwhere $\\hat{\\omega}$ is the orbital angular frequency.\nGeodesic motion in Kerr spacetime \nadmits three constants of motion: the specific energy $\\hat{E}$, the angular momentum $\\hat{L}$, and the Carter constant $\\hat{Q}$,\nand the geodesic equations are \n\\begin{eqnarray}\nm_p\\Sigma \\frac{d t}{d \\tau} &=&\\hat{E} \\frac{\\varpi^{4}}{\\Delta}+a \\hat{L}\\left(1-\\frac{\\varpi^{2}}{\\Delta}\\right)-a^{2} \\hat{E}\\sin^{2}\\theta, \\label{timequa}\\\\\nm_p\\Sigma \\frac{d r}{d \\tau} &=&\\pm \\sqrt{V_{r}\\left(r_{0}\\right)}, \\\\\nm_p\\Sigma \\frac{d \\theta}{d \\tau} &=&\\pm \\sqrt{V_{\\theta}\\left(\\theta\\right)}, \\\\\nm_p\\Sigma \\frac{d \\varphi}{d \\tau} &=&a \\hat{E}\\left(\\frac{\\varpi^{2}}{\\Delta}-1\\right)-\\frac{a^{2} \\hat{L}}{\\Delta}+ \\hat{L} \\csc ^{2} \\theta,\\label{anglequa}\n\\end{eqnarray}\nwhere $\\varpi\\equiv\\sqrt{r^2+a^2}$, the radial and polar potentials are\n\\begin{eqnarray}\n&V_{r}(r)& =\\left(\\hat{E} \\varpi^{2}-a \\hat{L}\\right)^{2}-\\Delta\\left(r^{2}+\\left(\\hat{L}-a \\hat{E}\\right)^{2}+\\hat{Q}\\right), \\\\\n&V_{\\theta}(\\theta)& = \\hat{Q}-\\hat{L}^{2} \\cot ^{2} \\theta-a^{2}\\left(1-\\hat{E}^{2}\\right) \\cos ^{2} \\theta.\n\\end{eqnarray}\n\nIn the adiabatic approximation, for a quasi-circular orbit on the equatorial plane,\nthe coordinates $r$ and $\\theta$ are considered as constants,\nthen Eqs. \\eqref{timequa}  and \\eqref{anglequa} are the remaining equations.\nThe conserved constants are \\cite{Detweiler:1978ge}\n\\begin{eqnarray}\\label{orbitE}\n\t\\hat{E}&=& m_p\\frac{r_0^{3 \/ 2}-2 r_{0}^{1 \/ 2} \\pm a }{r_{0}^{3 \/ 4}\\left(r_{0}^{3 \/ 2}-3  r_{0}^{1 \/ 2} \\pm 2 a \\right)^{1 \/ 2}},    \\\\\n\t\\hat{L}&=&m_p \\frac{\\pm (r_{0}^{2}\\mp 2a r_{0}^{1 \/ 2} + a^2)}{r_{0}^{3 \/ 4}\\left(r_{0}^{3 \/ 2}-3  r_{0}^{1 \/ 2} \\pm 2 a \\right)^{1 \/ 2}},\\\\\n\t\\hat{Q}&=&0.\n\\end{eqnarray}\nThe orbital angular frequency is\n\\begin{equation}\\label{orbitF}\n\\hat{\\omega} \\equiv \\frac{d\\varphi}{dt}=\\frac{\\pm 1}{r_{0}^{3 \/ 2} \\pm a},\n\\end{equation}\nwhere $\\pm$ corresponds to  co-rotating and counter-rotating, respectively.\nIn the following discussions, we use positive $a$ for co-rotating cases and negative $a$ for counter-rotating cases.\n\n\n\\section{Numerical calculation for the energy flux}\n\\label{sec3}\nThe homogeneous Teukolsky equation \\eqref{Teukolsky} admits two linearly independent solutions $R^{\\text{in}}_{\\omega lm}$ and $R^{\\text{up}}_{\\omega lm}$, with the following asymptotic values at the horizon $r_+$ and at infinity,\n\\begin{equation}\nR^{\\text{in}}_{\\omega lm}=\n\\begin{cases}\nB^{\\text{tran}}\\Delta^{-s}e^{-i\\kappa r^*},&\\quad (r\\to r_+)\\\\\nB^{\\text{out}}\\frac{e^{i\\omega r^*}}{r^{2s+1}}+B^{\\text{in}}\\frac{e^{-i\\omega r^*}}{r}, &\\quad (r\\to+\\infty)\n\\end{cases}\n\\end{equation}\n\\begin{equation}\nR^{\\text{up}}_{\\omega lm}=\n\\begin{cases}\nD^{\\text{out}}e^{i\\kappa r^*}+\\frac{D^{\\text{in}}}{\\Delta^{s}}e^{-i\\kappa r^*},&\\quad (r\\to r_+)\\\\\nD^{\\text{tran}}\\frac{e^{i\\omega r^*}}{r^{2s+1}},&\\quad (r\\to+\\infty)\\\\\n\\end{cases}\n\\end{equation}\nwhere $\\kappa=\\omega-m a\/(2r_+)$, $r_\\pm=1\\pm\\sqrt{1-a^2}$, and the tortoise radius of the Kerr metric\n\\begin{equation}\nr^*=r+\\frac{2r_+}{r_+-r_-}\\ln \\frac{r-r_+}{2}-\\frac{2r_-}{r_+-r_-}\\ln \\frac{r-r_-}{2}.\n\\end{equation}\nThe solutions $R^{\\text{in}}_{\\omega lm}$ and $R^{\\text{up}}_{\\omega lm}$  are purely outgoing at infinity and purely ingoing at the horizon.\nWith the help of these homogeneous solutions, the solution to Eq.~\\eqref{Teukolsky} is\n\n\\begin{equation}\n\\begin{split}\nR_{\\omega lm}(r)=\\frac{1}{W}\n\\left(R^{\\text{in}}_{\\omega lm}\\int_{r}^{+\\infty}\\Delta^{s}R^{\\text{up}}_{\\omega lm}T_{\\omega lm}dr+R^{\\text{up}}_{\\omega lm}\\int_{r_+}^{r}\\Delta^{s}R^{\\text{in}}_{\\omega lm}T_{\\omega lm}dr\\right).\n\\end{split}\n\\end{equation}\nwith the constant Wronskian given by\n\\begin{equation}\nW= \\Delta^{s+1} \\left(R^{\\text{in}}_{\\omega lm}\\frac{d R^{\\text{up}}_{\\omega lm}}{dr}-R^{\\text{up}}_{\\omega lm}\\frac{d R^{\\text{in}}_{\\omega lm}}{dr}\\right)=2i\\omega B^{\\text{in}} D^{\\text{tran}}.\n\\end{equation}\nThe solution is purely outgoing at infinity and purely ingoing at the horizon,\n\\begin{equation}\n\\begin{split}\nR_{\\omega lm}(r\\to r_+)=Z^{\\infty}_{\\omega lm}\\Delta^{-s}e^{-i\\kappa r^*},\\\\\nR_{\\omega lm}(r\\to \\infty)=Z^{H}_{\\omega lm}r^{-2s-1}e^{i\\omega r^*},\n\\end{split}\n\\end{equation}\nwith\n\\begin{equation}\n\\begin{split}\nZ^{\\infty}_{\\omega lm}&=\\frac{B^{\\text{tran}}}{W}\\int_{r_+}^{+\\infty}\\Delta^{s}R^{\\text{up}}_{\\omega lm}T_{\\omega lm}dr,\\\\\nZ^{H}_{\\omega lm}&=\\frac{D^{\\text{tran}}}{W}\\int_{r_+}^{+\\infty}\\Delta^{s}R^{\\text{in}}_{\\omega lm}T_{\\omega lm}dr.\n\\label{amplitudes}\n\\end{split}\n\\end{equation}\nFor a circular equatorial orbit with orbital angular frequency $\\hat{\\omega} $, we get\n\\begin{equation}\nZ^{H,\\infty}_{\\omega lm}=\\delta(\\omega-m \\hat{\\omega})\\mathcal{A}^{H,\\infty}_{\\omega lm}.\n\\end{equation}\nFor $s=-1$, the energy fluxes at infinity and the horizon read\n\\begin{equation}\n\\begin{split}\n\\dot{E}_q^{\\infty}=\\left(\\frac{d E}{dt}\\right)_{EM}^\\infty&=\\sum_{l=1}^{\\infty}\\sum_{m=1}^{l}\\frac{|\\mathcal{A}^{H}_{\\omega lm}|^2}{\\pi}, \\\\\n\\dot{E}_q^H=\\left(\\frac{d E}{dt}\\right)_{EM}^H&=\\sum_{l=1}^{\\infty}\\sum_{m=1}^{l}\\alpha^E_{lm}\\frac{|\\mathcal{A}^{\\infty}_{\\omega lm}|^2}{\\pi},\n\\end{split}\n\\end{equation}\nwhere the coefficient $\\alpha^E_{lm}$ is \\cite{Teukolsky:1974yv}\n\\begin{equation}\\label{energyformula}\n\\alpha^E_{lm}=\\frac{128\\omega\\kappa r_+^3(\\kappa^2+4\\epsilon^2)}{|B_E|^2}\n\\end{equation}\nwith $\\epsilon=\\sqrt{1-a^2}\/(4r_+)$ and\n\\begin{equation}\n|B_E|^2=\\lambda_{lm\\omega}^2+4ma\\omega-4a^2\\omega^2.\n\\end{equation}\n\nFor $s=-2$, the gravitational energy fluxes at infinity and the horizon are given by\n\\begin{equation}\n\\begin{split}\n\\dot{E}_{\\text{grav}}^{\\infty}=\\left(\\frac{d E}{dt}\\right)_{GW}^\\infty&=\\sum_{l=2}^{\\infty}\\sum_{m=1}^{l}\\frac{|\\mathcal{A}^{H}_{\\omega lm}|^2}{2\\pi\\omega^2}, \\\\\n\\dot{E}_{\\text{grav}}^H=\\left(\\frac{d E}{dt}\\right)_{GW}^H&=\\sum_{l=2}^{\\infty}\\sum_{m=1}^{l}\\alpha^G_{lm}\\frac{|\\mathcal{A}^{\\infty}_{\\omega lm}|^2}{2\\pi\\omega^2},\n\\end{split}\n\\end{equation}\nwhere the coefficient $\\alpha^G_{l m}$ is \\cite{Hughes:1999bq}\n\\begin{equation}\\label{energyformula2}\n\\alpha^G_{l m}=\\frac{256\\left(2 r_{+}\\right)^5 \\kappa\\left(\\kappa^2+4 \\epsilon^2\\right)\\left(\\kappa^2+16 \\epsilon^2\\right)\\omega^3}{\\left|B_G\\right|^2},\n\\end{equation}\nand\n\\begin{equation}\n\\begin{aligned}\n\\left|B_G\\right|^2 &=\\left[\\left(\\lambda_{l m \\omega}+2\\right)^2+4 a\\omega-4 a^2\\omega^2\\right]\\times\\left[\\lambda_{l m \\omega}^2+36 m a\\omega-36 a^2\\omega^2\\right] \\\\\n&+\\left(2 \\lambda_{l m \\omega}+3\\right)\\left[96 a^2\\omega^2-48 m a\\omega\\right]+144\\omega^2\\left(1-a^2\\right) .\n\\end{aligned}\n\\end{equation}\n\nTherefore, the total energy fluxes emitted from the EMRIs read\n\\begin{equation}\n\\dot{E}=    \\dot{E}_q+\\dot{E}_{\\text{grav}},\n\\end{equation}\nwhere\n\\begin{equation}\n    \\dot{E}_q=\\dot{E}_q^\\infty+\\dot{E}_q^H,\\ \\ \\dot{E}_{\\text{grav}}=\\dot{E}_{\\text{grav}}^\\infty+\\dot{E}_{\\text{grav}}^H.\n\\end{equation}\nThe detailed derivation of the above results is given in Appendix \\ref{gsne}.\nThe energy flux emitted by tensor fields can be computed with the BH Perturbation Toolkit \\cite{BHPToolkit}.\n\n\\section{Results}\\label{sec4}\nThe top  panel of Fig. \\ref{energyd} shows the normalized vector energy flux $m_p^{-2}M^2\\dot{E}_q$ for a charged particle with different charge values of $q$ on a circular orbit about a Kerr BH with the spin $a=0.9$, as a function of orbital radius.\nThe vector energy flux is proportional to the square of the vector charge $q^2$.\nThe vector energy flux increases as the charged particle inspirals into the central Kerr BH.\nThe ratio between the vector and gravitational energy flux is shown in the bottom panel of Fig. \\ref{energyd}.\nBoth the vector and gravitational fluxes are in the same order of $(m_p\/M)^2$,\nthe ratio of fluxes is independent of the mass ratio and increases as the orbital radius because the gravitational contribution falls off faster than the vector energy flux at a large orbital radius.\nFigure \\ref{energya} shows the normalized vector energy flux $m_p^{-2}M^2\\dot{E}_q$ and the ratio of energy fluxes $\\dot{E}_q\/\\dot{E}_{\\textrm{grav}}$ as a function of the orbital radius for different values of $a$.\nFor larger $a$, the ISCO is smaller, resulting in higher GW frequency at the coalescence.\nFor the same orbital radius, the vector energy flux is slightly larger for smaller $a$.\nHowever, the total energy flux increases with $a$ for one-year observations before the merger due to the smaller ISCO.\nFigure \\ref{energyHI} shows the ratio of energy flux falling onto the horizon to the energy flux radiated away to infinity, \nas a function of the orbital radius, for various spin  $a$, and for the vector and gravitational fields.\nIt is interesting to note that the sign of ratio becomes negative for Kerr BH with positive $a$ (co-rotating orbit) at  a small orbital radius.\nIn these cases, the vector and gravitational fields generate superradiance, leading to extraction of energy from the horizon.\nThe superradiance only happens when the coefficient $\\kappa$ in Eqs. \\eqref{energyformula} and \\eqref{energyformula2} becomes negative, which means that the orbital frequency slows down the rotation of the Kerr BH.\nOur results are consistent with those found in Ref. \\cite{Torres:2020fye}.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.95\\columnwidth]{energyd.pdf}\n    \\caption{The energy fluxes versus the orbital distance. \n    The top panel shows the vector flux normalized with the mass ratio from a charged particle orbiting around a Kerr BH with the spin of $a\/M=0.9$ for different values of the vector charge $q$.\n    The bottom panel shows the ratio between vector and gravitational energy fluxes for different values of the vector charge $q$.}\n    \\label{energyd}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.95\\columnwidth]{energya.pdf}\n    \\caption{Same as Fig. \\ref{energyd}, but for different values of the primary spin $a$.\n    The vector charge $q$ is set to 1.}\n    \\label{energya}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.95\\columnwidth]{energyHI.pdf}\n    \\caption{The ratio of the energy flux falling onto the horizon to the energy flux radiated to infinity, as a function of the orbital radius, for various spin $a$, and for the vector (the top panel) and gravitational cases (the bottom panel).}\n    \\label{energyHI}\n\\end{figure}\n\nThe extra energy leakage due to the vector field accelerates the coalescence of binaries.\nTherefore, we expect the vector charge to leave a significant imprint on the GW phase over one-year evolution before the merger for EMRIs.\nTo detect the vector charge carried by the small compact object in EMRIs, \nwe study the dephasing of GWs caused by the additional energy loss during inspirals.\nThe observation time is one year before the merger,\n\\begin{equation}\\label{sol-fre}\n\tT_{\\text{obs}}=\\int^{f_{\\text{max}}}_{f_{\\text{min}}}\\frac{1}{\\dot{f}}df=1\\ \\text{year},\n\\end{equation}\nwhere\n\\begin{equation}\n    f_{\\text{max}}=\\text{min}(f_{\\text{ISCO}},f_{\\text{up}}),~~~~~~f_{\\text{min}}=\\text{max}(f_{\\text{low}},f_{\\text{start}}),\n\\end{equation}\n$f=\\hat{\\omega}\/\\pi$ is the GW frequency, $f_{\\text{ISCO}}$ is the frequency at the ISCO \\cite{Jefremov:2015gza},\n$f_{\\text{start}}$ is the initial frequency at $t=0$,\nthe cutoff frequencies $f_{\\text{low}}=10^{-4}$ Hz and $f_{\\text{up}}=1$ Hz.\nThe orbit evolution is determined by\n \\begin{equation}\\label{orbittime}\n \\frac{d r}{dt}=-\\dot{E}\\left(\\frac{d \\hat{E}}{dr}\\right)^{-1},\\qquad \\frac{d \\varphi_{\\text{orb}}}{d t}=\\pi f,\n \\end{equation}\n where $\\dot{E}=\\dot{E}_q+\\dot{E}_{\\text{grav}}$.\nThe total number of GW cycles accumulated over one year before the merger is \\cite{Berti:2004bd}\n\\begin{equation}\\label{phase-end}\n\\mathcal{N}=\\int_{f_{\\text{min}}}^{f_{\\max }} \\frac{f}{\\dot{f}} d f.\n\\end{equation}\nConsidering EMRIs with the mass of the second compact object being fixed to be $m_p=10~M_{\\odot}$, \nwe calculate the dephasing $\\Delta\\mathcal{N}=\\mathcal{N}(q=0)-\\mathcal{N}(q)$ for different vector charge $q$, spin $a$ and mass $M$,\nand the results are shown in Fig. \\ref{phase}.\nFor one-year observations before the merger,\nthe charged particle starts further away from ISCO due to extra radiation of the vector field and the difference $\\Delta\\mathcal{N}$ is always positive.\nAs shown in Fig. \\ref{phase}, $\\Delta \\mathcal{N}$ increases monotonically with the spin $a$ and the charge-to-mass ratio $q$, and it strongly depends on the mass of the central BH such that lighter BHs have larger $\\Delta \\mathcal{N}$.\nThis means that the observations of EMRIs with a lighter and larger-spin Kerr BH can detect the vector charge easier.\nFor the same EMRI configuration in the Kerr background, the co-rotating orbit can detect the vector charge easier than the counter-rotating orbit.\nFollowing Refs. \\cite{Berti:2004bd,Maselli:2020zgv},\nwe take the threshold for a detectable dephasing that two signals are distinguishable by space-based GW detectors as $\\Delta \\mathcal{N}=1$.\nObservations of EMRIs over one year before the merger may be able to reveal the presence of a vector charge as small as $q\\sim 0.007$ for Kerr BHs with $a=0.9$ and $M=10^6~M_\\odot$, and $q\\sim 0.01$ for Schwarzschild BHs with $M=10^6~M_\\odot$.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.95\\columnwidth]{dephasing.pdf}\n    \\caption{The difference between the number of GW cycles accumulated by EMRIs with and without the vector charge in circular orbits. The left panel shows the dephasing as a function of the mass of the Kerr BH in the range $M\\in\\left[2\\times 10^5,2\\times 10^7\\right]M_\\odot$ for different spin values, the charge $q=0.01$.\n    The right panel shows the dephasing as a function of the charge-to-mass ratio $q$ for different $M$, the spin $a=0.9$. The red dashed line corresponds to the threshold above which two signals are distinguishable with space-based GW detectors. All observational time is one year before the merger.}\n    \\label{phase}\n\\end{figure}\n\n\\section{Parameter Estimation}\nTo make the analysis more accurate and account for the degeneracy among parameters, \nwe calculate the faithfulness between two GW waveforms and carry out parameter estimation with the FIM method.\n\\label{sec5}\n\\subsection{Signals}\nWe can obtain the inspiral trajectory from adiabatic evolution in Eq. \\eqref{orbittime}, then compute GWs in the quadrupole approximation.\nThe metric perturbation in the transverse-traceless (TT) gauge is\n\\begin{equation}\nh_{i j}^{\\mathrm{TT}}=\\frac{2}{d_L}\\left(P_{i l} P_{j m}-\\frac{1}{2} P_{i j} P_{l m}\\right) \\ddot{I}_{l m},\n\\end{equation}\nwhere $d_L$ is the luminosity distance of the source,\n$P_{ij}=\\delta_{ij}-n_i n_j$ is the projection operator acting onto GWs with the unit propagating direction $n_j$,\n$\\delta_{ij}$ is the Kronecker delta function,\nand $\\ddot{I}_{ij}$ is the second time derivative of the mass quadrupole moment.\nThe GW strain measured by the detector is\n\\begin{equation}\\label{signal}\nh(t)=h_{+}(t) F^{+}(t)+h_{\\times}(t) F^{\\times}(t),\n\\end{equation}\nwhere $h_+(t)=\\mathcal{A}\\cos\\left[2\\varphi_{\\rm orb}+2\\varphi_0\\right]\\left(1+\\cos^2\\iota\\right)$, $h_\\times(t)=-2\\mathcal{A}\\sin\\left[2\\varphi_{\\rm orb}+2\\varphi_0\\right]\\cos\\iota$,  $\\iota$ is the inclination angle between the binary orbital angular momentum and the line of sight,\nthe GW amplitude  $\\mathcal{A}=2m_{\\rm p}\\left[M\\hat{\\omega}(t)\\right]^{2\/3}\/d_L$ and $\\varphi_0$ is the initial phase.\nThe interferometer pattern functions $F^{+,\\times}(t)$ and $\\iota$ can be expressed in terms of four angles which specify the source orientation, $(\\theta_s,\\phi_s)$,\nand the orbital angular direction $(\\theta_1,\\phi_1)$.\nThe faithfulness between two signals is defined as\n\\begin{equation}\\label{eq:def_F}\n\\mathcal{F}_n[h_1,h_2]=\\max_{\\{t_c,\\phi_c\\}}\\frac{\\langle h_1\\vert\n\th_2\\rangle}{\\sqrt{\\langle h_1\\vert h_1\\rangle\\langle h_2\\vert h_2\\rangle}}\\ ,\n\\end{equation}\nwhere $(t_c,\\phi_c)$ are time and phase offsets \\cite{Lindblom:2008cm},\nthe noise-weighted inner product between two templates $h_1$ and $h_2$ is\n\\begin{equation}\\label{product}\n\\left\\langle h_{1} \\mid h_{2}\\right\\rangle=4 \\Re \\int_{f_{\\min }}^{f_{\\max }} \\frac{\\tilde{h}_{1}(f) \\tilde{h}_{2}^{*}(f)}{S_{n}(f)} df,\n\\end{equation}\n$\\tilde{h}_{1}(f)$ is the Fourier transform of the time-domain signal $h(t)$,\nits complex conjugate is $\\tilde{h}_{1}^{*}(f)$,\nand $S_n(f)$ is the noise spectral density  for space-based GW detectors.\nThe signal-to-noise ratio (SNR) can be obtained by calculating $\\rho=\\left\\langle h|h \\right\\rangle^{1\/2}$.\nThe sensitivity curves of LISA, TianQin, and Taiji are shown in Fig. \\ref{sensitivity}.\nAs pointed out in \\cite{Chatziioannou:2017tdw},\ntwo signals can be distinguished by LISA if $\\mathcal{F}_n\\leq0.988$.\nHere we choose the source masses $m_p=10~M_{\\odot}$, $M=10^6~M_{\\odot}$, \nthe source angles $\\theta_s=\\pi\/3,~\\phi_s=\\pi\/2$ and $\\theta_1=\\phi_1=\\pi\/4$, \nthe luminosity distance is scaled to ensure SNR $\\rho=30$, \nthe initial phase is set as $\\varphi_0=0$ and the initial orbital separation is adjusted to experience one-year adiabatic evolution before the plunge $r_{\\text{end}}=r_{\\text{ISCO}}+0.1~M$.\nIn Fig. \\ref{faithfulness}, we show\nthe faithfulness between GW signals with and without the vector charge for LISA as a function of the vector charge.\nThe results show that one-year observations of EMRIs with LISA may be able to reveal the presence of a vector charge as small as $q\\sim 0.002$ for Kerr BHs with $a=0.9$ and $M=10^6~M_\\odot$ (co-rotating orbit), \n$q\\sim 0.003$ for Schwarzschild BHs with $a=0$ and $M=10^6~M_\\odot$, and $q\\sim 0.004$ for Kerr BHs with $a=-0.9$ and $M=10^6~M_\\odot$ (counter-rotating orbit).\nLarger positive spin of the Kerr BH (co-rotating orbit) can help us detect the vector charge easier, which is consistent with the results obtained from the dephasing in the previous section.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.8\\columnwidth]{sensitivity.pdf}\n\t\\caption{The sensitivity curves for LISA, TianQin, and Taiji.\nThe horizontal solid lines represent the frequency band $f_{\\textrm{start}}$ to $f_{\\textrm{ISCO}}$ for EMRIs with $a=0.9$, and $M=2\\times10^5~M_{\\odot}$, $M=10^6~M_{\\odot}$ and $M=10^7~M_{\\odot}$ over one-year evolution before the merger.\n}\n\\label{sensitivity}\n\\end{figure}\n\n\\begin{figure}[thbp]\n\\center{\n\\includegraphics[scale=0.74]{Faithfulness.pdf}\n\\caption{Faithfulness between GW signals with and without the vector charge for LISA as a function of the charge $q$. The spin of the Kerr BH is $a=0$, $a=0.9$, and $a=-0.9$. The horizontal dashed line represents the detection limit with LISA, $\\mathcal{F}_n= 0.988$.\n}\\label{faithfulness}}\n\\end{figure}\n\n\\subsection{Fisher information matrix}\nThe signals \\eqref{signal} measured by the detector are determined by the following eleven parameters\n\\begin{equation}\n\\xi=(\\ln M, \\ln m_p, a, q, r_0, \\varphi_0, \\theta_s, \\phi_s, \\theta_1, \\phi_1, d_L).\n\\end{equation}\n In the large SNR limit,\nthe posterior probability distribution of the source parameters $\\xi$ can be approximated by a multivariate Gaussian distribution centered around the true values $\\hat{\\xi}$.\nAssuming flat or Gaussian priors on the source parameters $\\xi$,\ntheir covariances are given by the inverse of the FIM\n\\begin{equation}\n\\Gamma_{i j}=\\left\\langle\\left.\\frac{\\partial h}{\\partial \\xi_{i}}\\right| \\frac{\\partial h}{\\partial \\xi_{j}}\\right\\rangle_{\\xi=\\hat{\\xi}}.\n\\end{equation}\nThe statistical error on $\\xi$ and the correlation coefficients between the parameters are provided by the diagonal and non-diagonal parts of ${\\bf \\Sigma}={\\bf \\Gamma}^{-1}$, i.e.\n\\begin{equation}\n\\sigma_{i}=\\Sigma_{i i}^{1 \/ 2} \\quad, \\quad c_{\\xi_{i} \\xi_{j}}=\\Sigma_{i j} \/\\left(\\sigma_{\\xi_{i}} \\sigma_{\\xi_{j}}\\right).\n\\end{equation}\nBecause of the triangle configuration of the space-based GW detector, the total SNR is defined by $\\rho=\\sqrt{\\rho_1^2+\\rho_2^2}$, so the total covariance matrix of the binary parameters is obtained by inverting the sum of the Fisher matrices $\\sigma_{\\xi_i}^2=(\\Gamma_1+\\Gamma_2)^{-1}_{ii}$.\nHere we fix the source angles $\\theta_s=\\pi\/3,~\\phi_s=\\pi\/2$ and $\\theta_1=\\phi_1=\\pi\/4$, \nthe initial phase is set as $\\varphi_0=0$ and the initial orbital separation is adjusted to experience one-year adiabatic evolution before the plunge $r_{\\text{end}}=r_{\\text{ISCO}}+0.1~M$.\nThe luminosity distance $d_L$ is set to be $1$ Gpc.\nWe apply the FIM method for LISA, TianQin, and Taiji to estimate the errors of the vector charge.\n\nThe relative errors of the vector charge $q$ as a function of the vector charge with LISA, TianQin and Taiji are shown in Fig. \\ref{sigmaq}.\nFor one-year observations before the merger,\nthe charged particle starts further away from ISCO due to extra radiation of the vector field,\nso the $1\\sigma$ error for the charge decreases with the charge $q$.\nFor EMRIs with $M=2\\times 10^5~M_{\\odot}$ and $a=0.9$, as shown in the top panel,\nthe relative errors of the charge $q$ with TianQin are better than LISA and Taiji.\nFor $M=10^6~M_{\\odot}$ and $a=0.9$ as shown in the middle panel, \nthe relative errors of the charge $q$ with TianQin and LISA are almost the same.\nFor $M=10^7~M_{\\odot}$ and $a=0.9$ as shown in the bottom panel, \nthe relative errors of the charge $q$ with TianQin are worse than LISA and Taiji.\nIn all the cases, the relative errors of the charge $q$ with Taiji are better than LISA, for the reason that the sensitivity of Taiji is always better than LISA.\nFor $M=2\\times 10^5~M_{\\odot}$ and $a=0.9$, \nthe relative errors with TianQin are better than LISA and Taiji since the sensitivity of TianQin is better than LISA and Taiji in the high-frequency band, but worse than LISA and Taiji in the low-frequency band as shown in Fig. \\ref{sensitivity}.\nFor EMRIs with $m_p=10~M_{\\odot}$, $M=10^6~M_{\\odot}$ and $a=0.9$, \nthe vector charge can be constrained for LISA as small as $q\\sim0.021$, \nfor TianQin as small as $q\\sim0.028$ and for Taiji as small as $q\\sim0.016$.\nFor EMRIs with $m_p=10~M_{\\odot}$, $M=2\\times10^5~M_{\\odot}$ and $a=0.9$, \nthe vector charge can be constrained for LISA as small as $q\\sim0.0082$, \nfor TianQin as small as $q\\sim0.0049$ and for Taiji as small as $q\\sim0.0057$.\n\nFigure \\ref{sigmaa} shows the relative errors of the vector charge $q$ versus the spin $a$ with LISA, TianQin, and Taiji.\nIn general, the relative errors of the charge for the co-rotating orbit are better than those for the counter-rotating orbit.\nWe only consider the co-rotating orbit for simplicity.\nThe $1\\sigma$ error for the charge decreases with the spin $a$.\nComparing the relative errors for Kerr BHs with spin $a=0.9$ and spin $a=0$, \nwe find that the spin of Kerr BHs can decrease the charge uncertainty by about one or two orders of magnitude, depending on the mass of the Kerr BH.\nFor EMRIs with $M=10^6~M_\\odot$, $m_p=10~M_\\odot$, $q=0.05$, and different $a$,\nthe corner plots for source parameters with LISA are shown in Figs. \\ref{corner09}, \\ref{corner0} and \\ref{cornerm09}.\nFor comparison, we also show the corner plot for charged EMRIs in the Schwarzschild BH background in Fig. \\ref{corner0s}.\nFor $a=(0.9,0,-0.9)$, the corresponding errors of charge $\\sigma_q$ are $(0.0031,0.086,0.65)$, respectively.\nAs expected, $\\sigma_q$ is smaller for co-rotating orbits and bigger for counter-rotating orbits.\nFor EMRIs in the Kerr background, the co-rotating orbit can better detect the vector charge.\nIt is interesting to note that the charge $q$ are anti-correlated with the mass $M$  and the spin $a$ of Kerr BHs,\nand the correlations between $q$ and $M$, and $q$ and $m_p$ in the Kerr BH background\nare opposite to those in the Schwarzschild BH background.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.6\\columnwidth]{M2e5q.pdf}\\\\\n\\includegraphics[width=0.6\\columnwidth]{M1e6q.pdf}\\\\\n\\includegraphics[width=0.6\\columnwidth]{M1e7q.pdf}\\\\\n\\caption{\nThe $1\\sigma$ interval for the charge $q$ as a function of the charge $q$, inferred after one-year observations of EMRI with $a=0.9$ and different $M$ with LISA, Taiji, and TianQin. The horizontal dashed lines represent the $3\\sigma$ limit $33.3\\%$.\n}\n\\label{sigmaq}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.6\\columnwidth]{M2e5a.pdf}\\\\\n\\includegraphics[width=0.6\\columnwidth]{M1e6a.pdf}\\\\\n\\includegraphics[width=0.58\\columnwidth]{M1e7a.pdf}\\\\\n\\caption{\nThe $1\\sigma$ interval for the charge $q$ as a function of the spin $a$, inferred after one-year observations of EMRI with $q=0.05$ and different $M$ with LISA, Taiji, and TianQin.\n}\n\\label{sigmaa}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.9\\columnwidth]{LISA1e6A0.9.pdf}\n\\caption{Corner plot for the probability distribution of the source parameters $(\\ln M,\\ln m_p, a, q)$ with LISA, inferred after one-year observations of EMRIs with $q=0.05$ and $a=0.9$.\nVertical lines show the $1\\sigma$ interval for the source parameter.\nThe contours correspond to the $68\\%$, $95\\%$, and $99\\%$ probability confidence intervals.}\n\\label{corner09}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.9\\columnwidth]{LISA1e6A0.0.pdf}\n\\caption{Corner plot for the probability distribution of the source parameters $(\\ln M,\\ln m_p, a, q)$ with LISA, inferred after one-year observations of EMRIs with $q=0.05$ and $a=0$.\nVertical lines show the $1\\sigma$ interval for the source parameter.\nThe contours correspond to the $68\\%$, $95\\%$, and $99\\%$ probability confidence intervals.}\n\\label{corner0}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.9\\columnwidth]{LISA1e6A-0.9.pdf}\n\\caption{Corner plot for the probability distribution of the source parameters $(\\ln M,\\ln m_p, a, q)$ with LISA, inferred after one-year observations of EMRIs with $q=0.05$ and $a=-0.9$.\nVertical lines show the $1\\sigma$ interval for the source parameter.\nThe contours correspond to the $68\\%$, $95\\%$, and $99\\%$ probability confidence intervals.}\n\\label{cornerm09}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.9\\columnwidth]{LISA1e6A0.0s.pdf}\n\\caption{Corner plot for the probability distribution of the source parameters $(\\ln M,\\ln m_p, q)$ with LISA, inferred after one-year observations of EMRIs with $q=0.05$ in the Schwarzschild BH background.\nVertical lines show the $1\\sigma$ interval for the source parameter.\nThe contours correspond to the $68\\%$, $95\\%$, and $99\\%$ probability confidence intervals.}\n\\label{corner0s}\n\\end{figure}\n\n\\section{Conclusions}\n\\label{sec6}\nWe study the energy emissions and GWs from EMRIs consisting of a small charged compact object with mass $m_p$ and the charge to mass ratio $q$ inspiraling into a Kerr BH with spin $a$.\nWe derive the formula for solving the inhomogeneous Teukolsky equation with a vector field and calculate the power emission due to the vector field in the Kerr background.\nBy using the difference between the number of GW cycles $\\Delta\\mathcal{N}$ accumulated by EMRIs with and without the vector charge in circular orbits over one year before the merger, \nwe may reveal the presence of a vector charge as small as $q\\sim 0.007$ for Kerr BHs with $a=0.9$ and $M=10^6~M_\\odot$, and $q\\sim 0.01$ for Schwarzschild BHs with $M=10^6~M_\\odot$.\nThe dephasing increases monotonically with the charge-to-mass ratio $q$, and it strongly depends on the mass of the Kerr BH such that lighter BHs have larger dephasing.\n\nWe also apply the faithfulness between GW signals with and without the vector charge \nto discuss the detection of the vector charge $q$.\nWe find that positive larger spin of the Kerr BH can help us detect the vector charge easier.\nWe show that one-year observations of EMRIs with LISA may be able to reveal the presence of a vector charge as small as $q\\sim 0.002$ for Kerr BHs with $a=0.9$ and $M=10^6~M_\\odot$, \n$q\\sim 0.003$ for Schwarzschild BHs with $a=0$ and $M=10^6~M_\\odot$, and $q\\sim 0.004$ for Kerr BHs with $a=-0.9$ and $M=10^6~M_\\odot$.\n\nTo determine vector charge more accurately and account for the degeneracy among parameters, \nwe calculate the FIM to estimate the errors of the vector charge $q$.\nFor EMRIs with $M=2\\times10^5~M_{\\odot}$ and $a=0.9$, \nthe vector charge $q$ can be constrained as small as $q\\sim0.0049$ with TianQin, $q\\sim0.0057$ with Taiji, and $q\\sim 0.0082$ with LISA.\nFor EMRIs with $M=10^6~M_{\\odot}$ and $a=0.9$, \nthe vector charge can be constrained as small as $q\\sim 0.016$ with Taiji, $q\\sim 0.021$ with LISA, and $q\\sim 0.028$ with TianQin.\nSince the sensitivity of TianQin is better than LISA and Taiji in the high-frequency bands, \nthe ability to detect the vector charge for TianQin is better than LISA and Taiji when the mass $M$ of the Kerr BH with $a=0.9$ is lighter than $\\sim 2\\times 10^5~M_{\\odot}$.\nFor the mass $M$ of the Kerr BH with $a=0.9$ above $10^6~M_{\\odot}$, LISA and Taiji are more likely to detect smaller vector charges.\nThe relative errors of the charge $q$ with Taiji are always smaller than LISA  because the sensitivity of Taiji is always better than LISA.\nDue to the extra radiation of the vector field, \nthe charged particle starts further away from ISCO,\nso the $1\\sigma$ error for the charge decreases with the charge $q$.\nAs the spin $a$ of Kerr BHs increases, the ISCO becomes smaller, \nthe positive spin of Kerr BHs (co-rotating) can decrease the charge uncertainty by about one or two orders of magnitude, depending on the mass of the Kerr BH.\nFor EMRIs with $M=10^6~M_\\odot$, $m_p=10~M_\\odot$, $q=0.05$, and different $a=(0.9,0,-0.9)$, the corresponding errors of charge $\\sigma_q$ with LISA are $(0.0031,0.086,0.65)$, respectively,\nso co-rotating orbits can better detect the vector charge.\nIt is interesting to note that the charge $q$ are anti-correlated with the mass $M$  and the spin $a$ of Kerr BHs,\nand the correlations between $q$ and $M$, and $q$ and $m_p$ in the Kerr BH background\nare opposite to those in the Schwarzschild BH background.\nIn summary, the observations of EMRIs with a lighter and larger-spin Kerr BH can detect the vector charge easier.\n\n \n\n\\begin{acknowledgments}\nThis work makes use of the Black Hole Perturbation Toolkit package.\nThis research is supported in part by the National Key Research and Development Program of China under Grant No. 2020YFC2201504\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\tThe certification of quantum correlations is essential for the ever-accelerating development of quantum technologies.\n\tBeyond this practical demand, a fundamental understanding of quantum correlations, including their characterization and quantification, plays a key role when exploring the boundary between classical theories and the unique features of quantum physics.\n\tStill, the question remains which kinds of correlation are genuinely quantum.\n\tThat is, which of the many contenders---be it an established or recently proposed concept (e.g., quantum coherence and resource theory \\cite{SAP17,CG19}, entanglement \\cite{HHHH09}, discord \\cite{MBCPV12}, etc.)---does describe the concept of a nonclassical correlation best?\n\tIn this work, we show that the notion of nonclassicality in quantum optics \\cite{TG65,M86} can supersede contemporary forms of quantumness in its ability to unveil quantum correlations.\n\n\tIn the context of quantum optics, any observations which cannot be fully described in terms of Maxwell's wave theory of light are called nonclassical \\cite{TG65,M86}.\n\tThis well-established concept of nonclassicality is based on the impossibility of describing field correlations as done in classical electrodynamics, and it is commonly defined in terms of the Glauber-Sudarshan $P$ representation \\cite{G63,S63}.\n\tThe latter describes nonclassical light through phase-space distributions that are, in the case of quantum light, incompatible with a classical concept of a non-negative probability distribution.\n\n\tThe more recently developed concept of quantum coherence adapts some ideas of the notion of nonclassicality to quantify resources required for quantum information processing  \\cite{SAP17,CG19}.\n\tIn this framework, quantum superpositions equally serve as the origin of quantumness in a system.\n\tHowever, in most cases, the classical reference is defined through incoherent mixtures of orthonormal basis states, contrasting the notion of quantum-optical nonclassicality in terms of nonorthogonal eigenstates of the non-Hermitian annihilation operator.\n\n\tEntanglement, which can be embedded into the concept of coherence \\cite{SSDBA15,KSP16}, is by far the most frequently studied form of quantum correlation among the many contenders \\cite{HHHH09}.\n\tThis is due to its fundamental role as well as its many applications, e.g., in quantum metrology, cryptography, computing, and teleportation.\n\tThe phenomenon of entanglement was discovered in early seminal discussions about the implications of quantum physics \\cite{S35,EPR35}, long before the conception of the relatively young field of quantum information processing.\n\n\tMany other notions and measures of quantum correlations have been proposed too \\cite{MBCPV12}.\n\tFor instance, discord is a feature which includes correlations caused by entangled but also by nonentangled states \\cite{HV01,OZ01}, and it can be connected to quantum coherence as well \\cite{MYGVG16}.\n\tIn this context, it is worth mentioning that the label quantum for this sort of correlation is a topic of ongoing debates \\cite{GOS15}.\n\tNonetheless, it has been demonstrated that discord is maximally inequivalent to the notion of quantum-optical nonclassicality \\cite{FP12}.\n\tTo date, it remains an open problem to decide---not only in theory, but also experimentally---which of the candidates is best suited for characterizing quantum correlations.\n\n\tIn this Letter, we address this issue experimentally by realizing and analyzing a fully phase-randomized two-mode squeezed vacuum (TMSV) state \\cite{ASV13}.\n\tThis state of quantum light has the following properties:\n        It is nonentangled;\n        it has zero discord;\n        it does not exhibit quantum coherence in the photon-number basis;\n        its reduced single-mode states are classical;\n        and it has a non-negative, two-mode Wigner function.\n\tDespite these strong signatures of classicality, we demonstrate the presence of quantum correlations as defined through the notion of nonclassicality in quantum optics with a statistical significance next to certainty.\n\tBecause of the phase independence of the generated state, this nonclassical feature is robust against dephasing.\n\tFurthermore, the activation of entanglement using this kind of state is developed to show its resourcefulness for quantum information processing applications.\n\n\n\\section{Quantum correlations}\n\tFor analyzing quantum correlations, we consider a class of two-mode states that are phase insensitive \\cite{FP12,ASV13}.\n\tStill, intensity-intensity (likewise, photon-photon) correlations are present in such states.\n\tFor instance, this can be achieved by a full phase randomization of a TMSV state, resulting in\n\t\\begin{align}\n        \\label{eq:theo}\n\t\t\\hat\\rho=\\sum_{n\\in\\mathbb N} (1-p)p^n |n\\rangle\\langle n|\\otimes|n\\rangle\\langle n|,\n\t\\end{align}\n\twhere $p=\\tanh|\\xi|$ is a value between zero and one and $\\xi$ is the complex squeezing parameter.\n\tSuch states are, for example, a relevant resource for boson sampling tasks \\cite{SLR17}.\n\n\tIn terms of quantum correlations, it can be directly observed that the state in Eq. \\eqref{eq:theo} is an incoherent mixture of photon-number states, thus exhibiting no quantum coherence in the form of quantum superpositions of photon-number states;\n\tit is a classical mixture of tensor-product states, thus exhibiting no entanglement;\n\tand it has zero discord because $\\hat\\rho=\\sum_{n\\in\\mathbb N}\\hat\\rho_{A|n}\\otimes |n\\rangle\\langle n|$ holds true, where the photon-number states form the eigenbasis to $\\hat\\rho_{A|n}=(\\hat 1\\otimes \\langle n|)\\hat\\rho(\\hat 1\\otimes|n\\rangle)$ and $\\mathrm{tr}_A\\hat\\rho$ \\cite{D11}.\n\tFor those criteria of quantum correlations, it suffices in our scenario to consider the contribution of off-diagonal elements,\n\t\\begin{align}\n\t\t\\label{eq:CohMeasure}\n\t\t\\mathcal C(\\hat\\rho)\\stackrel{\\text{def.}}{=}\\sum_{\\substack{m,n,k,l\\in\\mathbb N: \\\\ m\\neq n, k\\neq l}}\n\t\t\\big|(\\langle m| \\otimes \\langle k|)\\hat\\rho(|n\\rangle\\otimes |l\\rangle)\\big|,\n\t\\end{align}\n\twhich quantifies the coherent contributions \\cite{BCP14} and is $\\mathcal C(\\hat\\rho)=0$ for the state in Eq. \\eqref{eq:theo}.\n\tIt is worth emphasizing that the nullity of coherence in the two-mode photon-number basis implies the nullity of discord which further implies no entanglement.\n\tIn addition, the incoherent mixture of photon-number states under study further implies a classical interpretation in a particle picture \\cite{SDNTBBS19}.\n\tFurthermore, the marginal states $\\mathrm{tr}_A\\hat\\rho$ and $\\mathrm{tr}_B\\hat\\rho$ are thermal states and, thus, classical, too.\n\tAlso, the two-mode state under study is a mixture of Gaussian TMSV states, implying a non-negative Wigner function.\n\n\tAt this point, one sees no indication of quantum correlations.\n\tYet, we have not considered the notion of nonclassicality in quantum optics so far.\n\tThis concept is defined through the Glauber-Sudarshan $P$ representation \\cite{G63,S63}\n\t\\begin{align}\n\t    \\label{eq:GS}\n\t\t\\hat\\rho=\\int d^2\\alpha \\int d^2\\beta\\, P(\\alpha,\\beta) |\\alpha\\rangle\\langle\\alpha|\\otimes |\\beta\\rangle\\langle\\beta |,\n\t\\end{align}\n\twhere $|\\alpha\\rangle$ and $|\\beta\\rangle$ denote classically coherent states of the harmonic oscillator \\cite{S26}.\n\tWhenever $P$ cannot be interpreted as a classical probability density, the state of light $\\hat\\rho$ refers to a nonclassical one \\cite{TG65,M86}.\n\tSince the $P$ distribution is in many cases highly singular \\cite{S16}, thus experimentally inaccessible, regularization and direct sampling procedures have been proposed and implemented to reconstruct a function $P_\\Omega$ which is always regular and non-negative for any classical states of light \\cite{KV10,KVHS11}.\n\tThis is achieved by a convolution of the Glauber-Sudarshan $P$ function with a suitable, non-Gaussian kernel $\\Omega$, resulting in $P_\\Omega$.\n\tOur previous theoretical studies suggest that the state in Eq. \\eqref{eq:theo} indeed demonstrates nonclassical correlations \\cite{ASV13}, i.e.,\n\t\\begin{align}\n\t    \\label{eq:Ncl}\n\t\tP_\\Omega(\\alpha,\\beta)\\stackrel{\\text{ncl.}}{<}0\n\t\\end{align}\n\tfor some complex phase-space amplitudes $\\alpha$ and $\\beta$.\n\n\\section{Experimental implementation}\n\tFigure \\ref{fig:setup} shows our experimental setup for the preparation and detection of the quantum state in Eq. \\eqref{eq:theo}.\n\tThe challenge here is that a full two-mode state tomography with long-term stability, phase control, and phase readout is paramount for our coherence analysis.\n\tSee Supplemental Material \\cite{SuppMat} for technical details.\n\t\n\\begin{figure}[b]\n\t\\includegraphics*[width=8cm]{setup.eps}\n\t\\caption{\n\t\tSetup outline.\n\t\tTwo single-mode squeezed states are prepared by two OPAs.\n\t\tCombining these beams on a $50{:}50$ beam splitter with a $\\pi\/2$ phase shift yields a TMSV state.\n\t\tIntroducing a phase randomization (indicated by the phase fluctuation $\\delta\\varphi$) in one of the output arms approximates the desired target state in Eq. \\eqref{eq:theo} \\cite{ASV13}.\n\t\tBoth final beams are probed by balanced homodyne detectors with controllable phases $\\varphi_{\\mathrm{LO},A}$ and $\\varphi_{\\mathrm{LO},B}$.\n\t}\\label{fig:setup}\n\\end{figure}\n\n\tTwo amplitude-squeezed fields at $1064\\,\\mathrm{nm}$ are produced by optical parametric amplifiers (OPAs).\n\tOne OPA consists of a type-I hemilithic, standing wave, nonlinear cavity with a $7\\%$ MgO:LiNbO\\textsubscript{3} crystal;\n\tthe other OPA uses a periodically poled potassium titanyl phosphate crystal instead.\n\tTo combine distinct sources of squeezed light in one setup, the seed and pump powers are adjusted such that the squeezed output fields of the two crystals are of equal intensity and squeezing.\n\tFor this purpose, the two pump powers of the second harmonic are chosen as $242$ and $50~\\mathrm{mW}$, respectively.\n\tMoreover, we have to minimize power fluctuations of the pump light since the optical parametric gain of each OPA responds differently \\cite{SuppMat}.\n\n\n\tThe two squeezed fields are superposed with a visibility of $96.5\\%$ and a relative phase of $\\pi\/2$ on a $50{:}50$ beam splitter.\n\tThe high visibility demonstrates a successful combination of the two sources that results in a TMSV state:\n\t\\begin{align}\n\t    |\\mathrm{TMSV}\\rangle=\\frac{1}{\\cosh|\\xi|}\\sum_{n=0}^\\infty \\left(\n\t        e^{i\\arg\\xi}\\tanh|\\xi| \\right)^n |n\\rangle\\otimes|n\\rangle.\n\t\\end{align}\n\n\tBoth output modes $A$ and $B$ of the state are probed by balanced homodyne detectors.\n\tWe observed $(96.7\\pm0.7)\\%$ visibility between the fields and their corresponding local oscillators.\n    For one OPA, we measured a single-mode squeezing variance of $-1.3~\\mathrm{dB}$ and $+3.7~\\mathrm{dB}$ antisqueezing.\n\tThis yields an initial squeezing of $-7.3~\\mathrm{dB}$ and an overall efficiency of $(63\\pm2)\\%$.\n\tThe latter figure was multiplied by two to compensate for the vacuum input, because blocking the second OPA effectively introduces additional $50\\%$ loss at the first (i.e., leftmost) beam splitter in Fig. \\ref{fig:setup}.\n\n    We use piezoelectric transducers to control the optical phases and realize a random phase shift $\\delta\\varphi$ in one of the arms.\n    To achieve a uniform dephasing over the full $ 2\\pi$ interval, white noise is applied with sufficiently high amplitude.\n    Because of the bandwidth limitations of the transducers, a uniformly distributed phase---as required to exactly obtain the state in Eq. \\eqref{eq:theo}--- can be approximated only via long measurement times, requiring to maintain a high stability of our setup.\n    Further technical details are provided in the Supplemental Material \\cite{SuppMat}.\n\n\\section{Results}\n\tFigure \\ref{fig:dment} depicts the first $625$ density-matrix elements of the reconstructed state, without (top panel) and with (bottom panel) phase randomization, which requires the full two-mode state tomography.\n\n\tThe initially generated state shows strong contributions of off-diagonal elements, relating to the dominance of quantum coherence [Eq. \\eqref{eq:CohMeasure}], $\\mathcal C(|\\mathrm{TMSV}\\rangle\\langle \\mathrm{TMSV}|)=1.789\\pm0.021$.\n\tThe implemented phase randomization then leads to a 45-fold suppression of the initial coherence, resulting in almost no subsisting coherence, $\\mathcal C(\\hat\\rho)=0.041\\pm0.005$.\n\t(Errors have been obtained through a Monte Carlo approach; see Supplemental Material \\cite{SuppMat} for details on data processing and a discussion of the residual amount of quantum coherence, caused by experimental imperfections.)\n\tThis loss of coherence implies that entanglement and discord do not contribute to the phase-randomized state that is almost completely characterized by its diagonal elements in the photon-number basis; see Eq. \\eqref{eq:theo} and the bottom plot in Fig. \\ref{fig:dment}.\n\n\\begin{figure}[t]\n\t\\includegraphics*[width=8cm]{dm-abs-rand0-noise27-63-nexp4-nc.eps}\n\t\\includegraphics*[width=8cm]{dm-abs-rand1-noise27-63-nexp4-nc.eps}\n\t\\caption{\n\t\tReconstructed density matrix elements of the TMSV state in photon-number bases, before (top) and after (bottom) phase randomization.\n\t\tEach entry provides the absolute values of the density matrix elements $|\\rho_{(k,m),(l,n)}|$, where $\\hat\\rho=\\sum_{k,l,m,n} \\rho_{(k,m),(l,n)} |k\\rangle\\langle l|\\otimes |m\\rangle\\langle n|$.\n\t\tPlease note the logarithmic scale.\n\t\tIn both plots, $k$ (bottom axis) and $l$ (right axis) denote photon numbers for $A$, and $m$ (left axis) and $n$ (top axis) indicate photon numbers for $B$.\n\t\tLarge off-diagonal contributions certify the presence of quantum coherence (top).\n\t\tStrongly diminished off-diagonal elements indicate the absence of quantum coherence (bottom).\n\t}\\label{fig:dment}\n\\end{figure}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics*[width=82mm]{regPw13bars.eps}\n\t\\caption{\n\t\tRegularized $P$ function sampled from experimental data for the phase-randomized TMSV state, including negativities.\n\t\n\t}\\label{fig:regP}\n\\end{figure}\n\n\tWe showed that quantum correlations in terms of quantum coherence are negligible for the produced state.\n\tThat is, the phase-averaged state is mostly consistent with a classical statistical mixture of orthonormal two-mode, tensor-product, photon-number states, thus also ruling out entanglement and discord as a source of quantum correlations as discussed earlier.\n    However, from our balanced homodyne detection data, we can further directly sample the regularized phase-space function $P_\\Omega$ \\cite{KVHS11}.\n    Here, the reconstruction is based on phase-insensitive pattern functions \\cite{SuppMat}.\n    Thus, no amount of residual coherence can contribute to nonclassicality in the form of Eq. \\eqref{eq:Ncl}.\n\n\tThe resulting distribution is shown in Fig. \\ref{fig:regP}.\n    We found a maximal statistical significance of more than 150 standard deviations for the negativity of the reconstructed quasiprobability distribution, $P_\\Omega(\\alpha=0,\\beta=1.5)=(-1.570\\pm0.010)\\times 10^{-3}$.\n\tBecause of these highly significant negativities, quantum correlations in the generated state are confirmed beyond the previously considered notions.\n\n\tAgain, we emphasize that quantum coherence, entanglement, and discord cannot contribute to the negativity as our approach is, by construction, insensitive to such phase-sensitive phenomena.\n\tFurthermore, our method is, to our knowledge, the only existing approach to uncover this form of quantumness.\n\tAs mentioned before, the Wigner function, for instance, is completely positive for our state.\n\tThus, we experimentally generated quantum correlations which are inaccessible via other means.\n\tMoreover, phase stability is not required for the kind of quantum effect.\n\tIn fact, we artificially introduced phase noise---which is often omnipresent in realistic quantum channels---to produce the sought-after state.\n\n\\section{Entanglement activation}\n\tBecause of the ever-growing importance for quantum information processing \\cite{HHHH09,NC00}, the question arises if entanglement can be activated, like for coherence and discord \\cite{SSDBA15,KSP16,MYGVG16}, from the nonclassically correlated state under study.\n\tAchieving such an activation renders this state a useful resource for quantum technologies.\n\n\tFor this purpose, let us recall that single-mode nonclassicality can be converted into entanglement via simple beam splitters \\cite{X02,KSBK02,VS14}.\n\tSimilarly, we consider combining each of our two modes separately on a 50:50 beam splitter with vacuum, where annihilation operators for the nonvacuum input map as $\\hat a\\mapsto(\\hat a+\\hat a')\/\\sqrt 2$ and $\\hat b\\mapsto(\\hat b+\\hat b')\/\\sqrt 2$;\n\tadditional modes obtained from the splitting are indicated by prime superscripts.\n\tSuch operations are free (i.e., classical) ones with respect to the reference $|\\alpha\\rangle\\otimes|\\alpha'\\rangle$ of the Glauber-Sudarshan representation [Eq. \\eqref{eq:GS}] since the beam-splitter output for mode $A$ remains in this family of states, $|(\\alpha+\\alpha')\/\\sqrt{2}\\rangle\\otimes|(\\alpha-\\alpha')\/\\sqrt 2\\rangle$, and likewise for $B$.\n\tFurthermore, applied to a photon-number input state $|n\\rangle\\otimes |0\\rangle$, the map yields the output state $|\\Psi_n\\rangle=2^{-n\/2}\\sum_{j=0}^n \\binom{n}{j}^{1\/2} (-1)^{n-j} |j\\rangle\\otimes|n-j\\rangle'$.\n\tTherefore, the state in Eq. \\eqref{eq:theo} results in the final four-mode state\n\t\\begin{align}\n\t\t\\label{eq:4mode}\n\t\t\\hat\\rho_{AA'BB'}=\\sum_{n\\in\\mathbb N} (1-p)p^n |\\Psi_n\\rangle\\langle\\Psi_n|\\otimes|\\Psi_n\\rangle\\langle\\Psi_n|.\n\t\\end{align}\n\n\tClearly, this state is still nonentangled when separating the joint subsystems $AA'$ and $BB'$ from each other.\n\tHowever, multimode entanglement is much richer since entanglement in various mode decompositions can be considered;\n\tsee, e.g., Refs. \\cite{GSVCRTF15,GSVCRTF16}.\n\tHere, we address the question whether there is entanglement between the primed and unprimed modes, i.e., with respect to the separation of $AB$ and $A'B'$.\n\n\tTo answer this question, we consider the partial transposition criterion \\cite{P96,HHH96}.\n\tIn one form \\cite{HHH96}, this criterion states that a state is entangled if the expectation value of a so-called entanglement witness $\\hat W=(|\\Phi\\rangle\\langle\\Phi|)^\\mathrm{PT'}$ is negative, where $\\mathrm{PT'}$ denotes the partial transposition of the primed modes.\n\tFor example, we can choose $|\\Phi\\rangle=|0\\rangle\\otimes|0\\rangle'\\otimes|1\\rangle\\otimes|1\\rangle'-|1\\rangle\\otimes|1\\rangle'\\otimes|0\\rangle\\otimes|0\\rangle'$, which results in\n\t\\begin{align}\n\t\t\\mathrm{tr}(\\hat W\\hat\\rho_{AA'BB'})=-\\frac{(1-p)p}{2}<0,\n\t\\end{align}\n\tfor all nontrivial parameters $0<p<1$ that define our states in Eq. \\eqref{eq:theo}.\n\tTherefore, the two-mode nonclassical correlations of the generated state can be successfully activated to produce four-mode entanglement, which is a useful mixed-state quantum resource, such as for teleportation \\cite{P94,G96,HH96,HHH01}, and which can be further enhanced via entanglement distillation \\cite{BBPSSW96,BBPS96,PSBZ01,KBSG01,HSDFFS08}.\n\n\\section{Conclusion}\n\tWe experimentally realized a quantum state with quantum correlations that are inaccessible by means of two-mode coherence, thus entanglement and discord, but can be intuitively visualized by negative quasiprobabilities.\n\tQuantum coherence---a recently explored resource for quantum information processing---exists in terms of superpositions of orthogonal photon-number states of the initial TMSV state.\n\tBut a phase averaging destroys this and related kinds of quantum correlations.\n\tThus, contemporary quantum-information-based concepts of correlation fail to certify the quantumness of the state under the challenging, but common scenario of dephasing.\n\tHowever, the Glauber-Sudarshan-based concept of nonclassiality of light---frequently considered to be a dated notion, or not considered at all--- is still capable of uncovering the quantum nature of the generated state.\n\n\tSince the Glauber-Sudarshan distribution often displays a highly singular behavior, we employ a technique which enables us to directly sample a regularized version of such a two-mode phase-space function, allowing us to certify nonclassical negativities with a statistical significance of more than 150 standard deviations.\n\tThus, the generated phase-independent state exhibits nonclassical correlations beyond entanglement and discord.\n\tFurthermore, we theoretically devised a method to activate multimode entanglement, utilizing only the produced state and simple beam splitter operations.\n\n\n\tIn conclusion, we realized a type of quantum correlation that can be accessed via phase-space approaches but not through more recent notions of quantumness.\n\tThis kind of correlation is intrinsically robust against dephasing and can be easily converted into multimode entanglement.\n\tTherefore, this finding offers useful quantum correlations that can be realized without experimentally costly phase stabilization and is resourceful for modern quantum information protocols.\n\n\\section*{Acknowledgements}\n\tE. A. acknowledges funding from the European Union's Horizon 2020 research and innovation program under the Marie Sk\\l{}odowska-Curie IF InDiQE (EU Project No. 845486).\n\tM. S. acknowledges financial support by the Deutsche Forschungsgemeinschaft through STA-543\/9-1.\n\tJ. S. thanks Torsten Meier and Tim Bartley for discussions.\n    This work was supported by the Deutsche Forschungsgemeinschaft through SFB 652, Projects No. B12 and No. B13.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\textit{Mesoscopic} electronic systems, those with physical dimensions comparable to the Fermi wavelength $\\lambda_F$, can display a variety of striking effects such as size quantisation, ballistic electron transport and quantum interference. Quantum interference effects such as Aharonov-Bohm oscillations, weak localisation and universal conductance fluctuations (UCFs) are all direct manifestations of phase coherent electron transport. Hence these are dependent on an additional lengthscale, namely the phase coherence length $l_\\phi$ over which the electron loses memory of its phase through inelastic processes. UCFs are aperiodic but reproducible features in the electrical conductivity $\\sigma$ as a function of an external parameter such as the magnetic field $B$ and are a consequence of the non-self-averaging nature of electron interference over lengthscales comparable to $l_\\phi$. The resulting fluctuations are not limited to $\\sigma$ but will tell on other transport parameters such as the thermopower as well. The thermopower or Seebeck coefficient $S$ of a system is defined as $V_{\\rm{th}}\/\\Delta T$ where $V_{\\rm{th}}$ is the voltage that develops between its end due to an imposed temperature difference $\\Delta T$. $S$ and $\\sigma$ are related by the Mott formula \\cite{1}: \n\n\\begin{equation}\n\\label{MottFormula}\nS_{\\rm{MOTT}} = \\frac{\\pi^2 k_B^2 T}{3e}\\frac{\\mbox{d}\\ln\\sigma}{\\mbox{d}E}\\vline_{E = \\mu}\n\\end{equation}\n\nHere $k_B$ is the Boltzmann constant, $T$ is the temperature, $e$ the electronic charge, $E$ the energy of the system and $\\mu$ the chemical potential. Indeed, 'universal thermopower fluctuations' have been experimentally demonstrated in quantum wires \\cite{2}, quantum dots \\cite{3} and, more recently, in other mesoscopic structures \\cite{4}. While mesoscopic fluctuations have been reported most often as a function of $B$, they are expected \\cite{4, 5, 6} and seen \\cite{7, 8, 9} even as a function of $\\mu$. The relative magnitude of the $S$-fluctuations can be much larger than those in $\\sigma$ by virtue of the derivate relation in equation \\ref{MottFormula}. Lesovik and Khmelnitskii (henceforth LK) show \\cite{10} that precisely for this reason, the mesoscopic contribution to the thermopower can be exceedingly large and either positive or negative. Consequently, they can potentially swamp the regular, 'bulk' contribution even causing sign reversals in $S$. The important difference between such fluctuations and thermopower sign-reversals resulting from Coulomb Blockade \\cite{11, 12, 13} is that the former occur in the high-conductivity regime $\\sigma >> h\/e^2$, while the latter become significant at $\\sigma \\sim h\/e^2$.\n\nRecently some of us reported $S$ measurements in low-density two-dimensional electron gases (2DEGs) of mesoscopic dimensions \\cite{14, 15} as a function of density $n_s$ and temperature $T$. Specifically, $S$ displayed large oscillations and even sign changes as the 2DEG density $n_s$ was varied whereas the electrical resistivity $\\rho \\equiv 1\/\\sigma$ was completely monotonic. However, these measurements focused on the low-conductivity regime of the samples (300$h\/e^2 > \\rho > 5h\/e^2$), where many-body effects suppressed the strong localisation (see also refs~\\cite{16} and \\cite{17}) and it is not clear whether weak localisation, and therefore, mesoscopic fluctuations are to be expected. In this work we measure $S$ in similar samples while concentrating specifically on the medium-$\\rho$ range around the onset of $S$-oscillations where, in principle, UCFs are expected. We perform a statistical analysis of these oscillations and examine whether they are consistent with the mesoscopic fluctuations described by LK.\n\nThe remainder of the article is structured as follows: Section 2 describes the experimental system and measurement setup, section 3 presents the experimental data and statistical analysis and Section 4 presents a discussion and concluding remarks.\n\n\\section{EXPERIMENTAL DETAILS}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=4in]{Figure1.jpg}\n\t\\caption{Schematic representation of the device and measurement scheme. The blue area represents the conducting mesa and ohmic contacts are represented in green. The yellow rectangles represent top-gates.}\n\t\\label{fig1}\n\\end{figure}\n\nOur experiments are all performed at $T$ = 0.3~K in GaAs-based 2DEGs with an as-grown mobility of 220~m$^2$\/Vs at $n_s$ = 2.1$\\times$10$^{15}$~m$^{-2}$. We use a wet etch to define a conducting mesa and deposit ohmic contacts to perform electrical and thermoelectric measurements. The mesoscopic device D (see figure 1) is defined by a metallic gate of lithographical dimensions L$\\times$W = $2 - 3~\\mu \\rm{m} \\times 8~\\mu \\rm{m}$. Figure \\ref{fig1} shows a schematic representation of the device. In addition to D, there are four gate-defined bar-gates (BGs) that serve as a pair of thermometers to measure the local electron temperature $T_{\\rm{e}}$on either side of D~\\cite{14}. To measure $S$ we heat one end of the device with an AC heating current $I_{\\rm{h}}$ = 4~$\\mu$A at frequency $f_{\\rm{h}}$ = 11~Hz. $V_{\\rm{th}}$ is measured using a lock-in amplifier at 2$f_{\\rm{h}}$ as shown in figure~\\ref{fig1}. The local electron temperatures on either side of the device $T_{\\rm{e1}}$ and $T_{\\rm{e2}}$ are measured using bar-gate pairs BG$_{\\rm{1a,b}}$ and BG$_{\\rm{2a,b}}$, respectively. Details of this measurement procedure can be found in Ref.~\\cite{14}. $S$ is then calculated as $V_{\\rm{th}}\/(T_{\\rm{e1}} - T_{\\rm{e2}})$. We measure the resistance $R$ using a standard four-probe lock-in technique with a constant current $I_{\\rm{ex}}$ = 1~nA at a frequency $f$ = 7~Hz. However, the results are unaffected upon increasing $I_{\\rm{ex}}$ by an order of magnitude to 10~nA. In all our measurements $V_{\\rm{th}}$ and $R$ are measured simultaneously. $\\Delta T$ is 20~mK in all the measurements reported here.\n\n\\section{Experimental Data}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=6in]{Figure2.jpg}\n\t\\caption{(a) $S$ as a function of $n_s$ (bottom axis) and $r_s$ (top axis). The inset shows the same data on a linear scale where the sign-reversals of $S$ are clearly visible. Figure 2b shows $R$ measured simultaneously with $S$ over the same $n_s$. Blue and red traces correspond to devices with L = 2~$\\mu$m and 3~$\\mu$m, respectively and the arrow marks the density below which thermopower oscillations are present.}\n\t\\label{fig2}\n\\end{figure}\n\nFigures \\ref{fig2}a and \\ref{fig2}b show $S$ and $R$ as functions of $n_s$ for two different samples labelled D1 and D2. The top horizontal axis in figure \\ref{fig2}a shows the interaction parameter $r_s \\equiv 1\/a_B\\sqrt{\\pi n_s}$, where $a_B$ is the effective Bohr radius in GaAs $\\approx$ 11~nm. D1 has L = 2~$\\mu$m and D2 has L = 3~$\\mu$m. Figure \\ref{fig2}a also shows $S_{\\rm{MOTT}}$ for $\\sigma = n_se^2\\tau\/m$\n\n\\begin{equation}\n\\label{DrudeMott}\nS_{\\rm{MOTT}} = \\frac{\\pi^2 k_B^2 T}{3e}\\frac{\\mbox{d}\\ln\\sigma}{\\mbox{d}E}\\vline_{E = \\mu} = \\frac{\\pi k_B^2 T m}{3e \\hbar^2} \\frac{1 + \\alpha}{n_s}\n\\end{equation}\n\nHere $\\tau$ is the inelastic scattering time and $\\alpha = (n_s\/\\tau)(\\mbox{d}\\tau\/\\mbox{d}n_s) \\approx$~1. It is seen that $S$, while agreeing closely with $S_{\\rm{MOTT}}$at high $n_s$, shows a sudden departure from Mott-like behaviour at $n_s \\approx$~2.5$\\times$10$^{14}$~m$^{-2}$ with strong oscillations and even sign reversals. The sign reversals are seen most clearly in the inset to figure \\ref{fig2}a which also serves to distinguish the smooth $S$ oscillations from the measurement noise ($\\sim$10~$\\mu$V\/K which corresponds to a voltage uncertainty $V_{err}$ of 200~nV), the latter being most prominent in the range $n_s >$ 2$\\times$10$^{14}$~m$^{-2}$. It is worth emphasising that the former are large oscillations and completely reproducible between $n_s$ sweeps (see figure \\ref{fig3}), while the latter correspond to measurement noise. In the same $n_s$-range, $R$ grows monotonically in both devices except for some broad features accentuated by the log scale which, importantly, bear no obvious correlation to the oscillations in $S$. Hence, this very unusual behaviour cannot simply be reconciled with $S$ oscillations in a disordered 2DEG that arise, say, due to Coulomb blockade resulting from the 2DEG fragmenting into charge puddles.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=6in]{Figure3.jpg}\n\t\\caption{(a) shows five separate $S$ traces vertically offset to each other. The inset shows the corresponding $R$ traces. All data are taken at $T$ = 300~mK. Figure 3(b) shows five $R(n_s)$ traces, again, with a relative vertical offset, at five different $T$ and concentrating on high-$n_s$ (boxed region in the inset to figure 3(a)). The inset shows the scale of fluctuations $\\delta_R$ as a function of $T$.}\n\t\\label{fig3}\n\\end{figure}\n\nIn figure~\\ref{fig3}a we plot five consecutive $S$ traces taken at $T$ = 0.3~K vertically offset relative to each other. In the inset we plot the simultaneously measured $R(n_s)$ corresponding to each $S(n_s)$ trace, again, with a relative vertical offset. These figures firmly establish both the reproducible nature of the $S$ oscillations as well as the smooth character of $R(n_s)$. In figure~\\ref{fig3}b we investigate whether the $R$ data at high-$n_s$ contain any signatures of UCFs. We see that in the five $R(n_s)$ traces at 0.3~K $< T <$ 1.5~K, vertically offset relative to each other by 100~$\\Omega$, there are no reproducible features but a fluctuation level that has no obvious $T$-dependence. This becomes clearer in the inset where we plot $\\delta R$, the magnitude of fluctuations in $R$ as a function of $T$. $\\delta R$ is defined as the standard deviation in $R$ after subtracting a smooth background. The dependence of $\\delta R$ on $T$ is not consistent with weak localisation and $\\delta R$ simply represents the experimental noise levels.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=6in]{Figure4.jpg}\n\t\\caption{The correlation function $C_{SS}(\\Delta n_s)$. The grey arrows indicate the minimum in each instance. The inset shows the same data without normalisation.}\n\t\\label{fig4}\n\\end{figure}\n\nWe now return to the $S$ oscillations and to analyse their statistical nature, we look at the autocorrelation of $S$. In figure~\\ref{fig4} we plot $C_{SS}(\\Delta n_s) \\equiv \\langle\\delta S(n_s) \\delta S(n_s + \\Delta n_s)\\rangle\/R^2(n_s)$ normalised to $C_{SS}(0)$ = 1. Here the angular brackets represent an average over $n_s$ and $\\delta S(\\Delta n_s)$ is defined as $S$ - $S_{\\rm{MOTT}}$.  The correlator is defined with $R^2(n_s)$ in the denominator rather than $S^2(n_s$) to avoid divergences at the zero-crossings of $S(n_s)$ and also to make a quantitative comparison with the LK theory (see next section). The inset shows the data without normalising and error bars as estimated from the standard deviation in the data. We find that $C_{SS}(\\Delta n_s)$ in both devices agree quantitatively and are consistent with an initial decay $C_{SS}(\\Delta n_s) \\sim \\Delta n_s^{-1\/2}$. Furthermore, there is a minimum and subsequent maximum that occur in the vicinity of $\\Delta n_s \\approx$ 2.5$\\times$10$^{13}$~m$^{-2}$. These features suggest a degree of periodicity in the $S$ oscillations which are further reflected in the fourier transforms (FTs) of the data shown in figure~\\ref{fig5}. The grey arrows mark the FT peaks which correspond to periodicities of $\\Delta n_s \\approx$~4.2$\\times$10$^{-14}$~m$^{-2}$ and 2.7$\\times$10$^{-14}$~m$^{-2}$ for D1 and D2, respectively. We note that figure~\\ref{fig4} shows several features at $\\Delta n_s$-values larger than 5$\\times$10$^{-13}$~m$^{-2}$ which can be correlated to peaks in the FT in figure~\\ref{fig5}a. However, these must be taken with caution due to the lack of sufficient statistics. In figure~\\ref{fig5}b we plot the FT of $S$ as a function of $r_s$. Interestingly we note that the periodicities correspond to $\\Delta r_s \\approx$ 0.4 \u2013- 0.5, reminiscent of conductance oscillations seen in disordered silicon inversion layers~\\cite{18}.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=6in]{Figure5.jpg}\n\t\\caption{Figures 5a and 5b show the fourier transform of the data in figure 2(a).}\n\t\\label{fig5}\n\\end{figure}\n\n\\section{Discussion}\n\nThe strong oscillations in $S$ without corresponding features in $R$ is extremely surprising and signifies a breakdown of the Mott formula (equation \\ref{MottFormula}). While there are instances where this is expected (see section 5), we first ascertain whether the mesoscopic nature of the 2DEG has a role to play in the observed experimental data. As mentioned earlier, mesoscopic fluctuations are necessarily present in all transport parameters of mesoscopic samples. However, oscillations in $S$ can be significantly amplified compared to those in $R$ due to the derivative relation between them. Conversely, therefore, one can expect situations where $S$ oscillations are visible but immeasurably small in $R$ and such situations would be wrongly perceived as a breakdown of the Mott relation. Motivated by this we now investigate whether the observed oscillations are consistent with the LK-type oscillations~\\cite{10} mentioned earlier.\n\nLK calculated the mesoscopic contribution to $S$ of a sample of small size as a function of $\\mu$ and the magnetic field $B$. Using the Mott formula (equation~\\ref{MottFormula}) they then showed that this contribution can be  positive or negative and even exceed the regular term in magnitude, potentially resulting in sign reversals of $S$. Furthermore, they constructed and evaluated the correlation function: $C_{SS}^{LK}(\\Delta \\mu, \\Delta B) \\equiv \\langle\\delta S(0,0) \\delta S(\\Delta \\mu, \\Delta B)\\rangle = (R^2\/\\hbar^2)F_S(\\Delta \\mu \\tau_F\/\\hbar, \\Delta B LW\/\\phi_0)$. Here $F_S$ is a scaling function, $\\tau_F$ is the time taken for an electron to diffuse across the sample of length $L$ and is given by $\\tau_F \\approx L^2\/D$, with $D$ being the electron diffusivity and $\\phi_0 = hc\/e$. At $\\Delta B$ = 0 and low $T$ this correlation functions is essentially identical to the correlation function evaluated in figure~\\ref{fig4}. The low-$T$ requirement is in order to approximate $\\mu$ by the Fermi energy $E_F$ and this is met in our experimental results since $k_BT\/E_F \\leq$ 0.25 in the $n_s$-range being considered. LK predict a universal form for the autocorrelation function which they numerically show to initially decay and then go through a minimum. This is in striking agreement with the result in figure~\\ref{fig4} in that the data from both devices has the same initial $\\Delta n_s^{-1\/2}$ decay and a clear minimum.\n\nAdditionally, LK consider the cross-correlation between $S$ and $R$: $C_{SR}^{LK}(\\Delta \\mu, \\Delta B) \\equiv \\langle\\delta S(0,0) \\delta R(\\Delta \\mu, \\Delta B)\\rangle$, where $\\delta R$ are fluctuations about the mean $R$. LK show $C_{SR}^{LK}(\\Delta \\mu, \\Delta B)$ to be 0 at $\\Delta \\mu$ = 0, i.e., oscillations in $S$ and $R$ at the same value of chemical potential are uncorrelated, but gradually grow as a function of $\\Delta \\mu$ and ultimately go through a maximum. While the statement that $S$ and $R$ oscillations at a given $\\mu$ are uncorrelated seems consistent with a visual inspection of the data in figure~\\ref{fig2}, a quantitative comparison reveals discrepancies with the LK picture. First, at the point where $S$ oscillations set in, there are absolutely no oscillations in $R$ (see figures~\\ref{fig2} and \\ref{fig3}), thereby trivially rendering $C_{SR}$ zero. In this region, $R \\approx$ 3~K$\\Omega$ and therefore the expected magnitude of $\\delta R = e^2R^2\/h \\approx$ 350~$\\Omega$. This is well within the limit of accuracy even if, considering the wide aspect ratio of the sample, we divide the expected $\\delta R$ by a factor $\\sqrt{W\/L} \\approx$~2. Given the above estimate for $\\delta R$, it is then reasonable to ask why UCFs are absent in the first place. While this is unclear at the moment, a possible explanation lies in the nature of the disorder being sampled by the device. The earlier experiments on mesoscopic 2DEGs~\\cite{16} that motivated the present studies were performed especially to circumvent the long-range disorder due to remote ionised dopants in the host GaAs wafer. The residual short-scale 'white' disorder would then restore the self-averaging nature of the electron paths, leading to the absence of UCFs. We hope to verify this argument by examining whether UCFs appear in larger mesoscopic samples.\n\nIt is worth mentioning here that preliminary results of $S$ and $R$ as a function of $B$ also indicate the absence of UCFs. Not only is this consistent with with the arguments in the previous paragraph, it is another discrepancy between the experimental data and the LK picture. The LK analysis predicts similar cross-correlation and autocorrelation functions as $B$ is tuned, but we observe no reproducible oscillations in $S$ or $R$ as functions of $B$, or at least no oscillations (other than the $B^{-1}$-periodic Shubnikov-de Haas oscillations) comparable in magnitude to those observed by tuning $n_s$. This data will be presented in a separate report.\n\n\\section{Conclusions}\n\nTo summarise, we find some aspects of the data to be consistent with oscillations due to the mesoscopic nature of the device, but several others that are at odds with it. Furthermore, the $S$ oscillations persist to much lower $n_s$ even up to $R \\approx$~100~$h\/e^2$~\\cite{14,15}, a regime in which there is no reason to expect LK-type oscillations. While the oscillations may indeed be related to those seen in Ref.~\\cite{18}, there have recently been several theoretical reports describing situations where the Mott relation (equation~\\ref{MottFormula}) breaks down. These include the vicinity of a quantum critical point~\\cite{19}, proximity to a Lifshitz transition~\\cite{20}, and, remarkably, even far away from a critical point~\\cite{21}.\n\nIn conclusion, we have observed intriguing oscillations in the thermopower $S$ of mesoscopic 2DEGs and analysed their statistical nature. We find that the experimental data from the 2 devices measured suggests several common aspects in the nature of oscillations despite the devices being of dissimilar sizes: 1. the decay of the autocorrelation function of $S$ is consistent with a $\\Delta n_s^{-1\/2}$ dependence; 2. the minima and maxima of the autocorrelation in the two devices occur at approximately similar locations; and 3. the oscillations have a degree of periodicity as revealed by the fourier transform. We have compared these results to the oscillations predicted by Lesovik and Khmelnitskii in Ref.~\\cite{10} and though there are suggestive similarities between the two, it seems unlikely that they are related.\n\n\\section{Acknowledgements}\n\nWe acknowledge funding from the Leverhulme Trust and EPSRC, UK. VN acknowledges a Fellowship from the Herchel Smith Fund, University of Cambridge. VN also acknowledges useful discussions with Sriram Shastry and Charles Smith.\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\tAperiodic sequences have been well-studied throughout the 20th century, with \\emph{substitution systems} being an important subclass in their study since the work of Thue \\cite{T:thue-morse}. These are sequences generated from an iterated replacement rule on a finite alphabet that assigns finite words to each letter. Substitutions are also called \\emph{morphisms} in the literature. There are several hundred entries containing the term \\emph{morphism} in the Online Encyclopedia of Integer Sequences \\cite{OEIS}. These sequences are of great interest to the field of combinatorics on words and have been extensively studied \\cite{F:book}.\n\t\n\tToday, aperiodic order is a topic of general interest to the study of tiling theory, dynamical systems, topology, Diophantine approximation, ergodic theory, computer graphics, mathematical physics and even virology \\cite{CH:codim-one-attractors, S:book, BS:beta-numerations, BG:book, L:comp-sci-book, BHZ:gap-labeling, T:virology-tilings}. Once it was discovered by Berger \\cite{B:undecidability} that sets of tiles exist which can only tile the plane aperiodically, a flurry of advances quickly followed, culminating in the celebrated discovery of the famous \\emph{Penrose tilings} \\cite{P:pentaplexity} and in the discovery of quasicrystals \\cite{S:nobel} for which Schectman was awarded the Nobel prize in Chemistry in 2011. However, quite apart from the higher-dimensional setting, there is still much that is not well-understood about the low-dimensional case of bi-infinite sequences. In many respects, substitution systems provide the simplest examples of aperiodic sequences.\n\n\tThe seminal work of Anderson and Putnam \\cite{AP} has led to a new algebraic topological approach to the study of aperiodic sequences and tilings associated with substitutions. It is this style of approach, and subsequent related methods for bi-infinite sequences, which we address here. In this setting, 1-dimensional tilings are combinatorially equivalent to bi-infinite sequences over an alphabet whereby letters from the alphabet are assigned to distinct tiles. Informally, the symbolic dynamicist takes the position that (in some instances) a sequence $s$ can be better understood by taking $s$ in concert with all other sequences which are \\emph{locally isomorphic} to $s$. One then provides a suitable metric or topological structure to this space of sequences. A simple dynamical system on such a space, given by the \\emph{shift map}, is then studied. Combinatorial properties of the original sequence $s$ translate to dynamical properties of this space of sequences. We can in turn translate these dynamical properties of the sequence space into topological properties of a related space---its so-called \\emph{tiling space}. Algebraic invariants of tiling spaces are hence important for combinatorially distinguishing aperiodic sequences.\n\t\n\tThere are still long-standing conjectures concerning the dynamics for symbolic substitutions which need to be settled; principal among them being the \\emph{Pisot conjecture} \\cite{ABBL:pisot} and related problems. It is hoped that the use of new programs will make testing conjectures in tiling theory and symbolic substitutional dynamics more efficient, as well as allowing for the confirmation of hand calculations and comparing different methods of calculation. In this direction, we present here supporting material for a program (\\emph{Grout}) developed to calculate some of these properties (especially methods of calculating cohomology). Analysis of large data sets which can be potentially generated by the Grout source code, and the recognition of underlying patterns in the data may also aid to further the theory.\n\t\n\tGrout has been designed with user experience in mind and includes many ease-of-use tools such as the ability to save and load examples, and convenient methods of sharing examples with other users via short strings that encode a substitution. There is also an option to export all of the data that has been calculated to a pre-formatted \\LaTeX{} file, including all the TikZ code for the considered complexes.  This is useful to those needing to typeset such diagrams by fully automating the generation of diagrams in TikZ.\n\t\n\n\t\nIn Section \\ref{SEC:background}, we introduce basic notions of subshifts (sequence spaces) and tiling spaces associated with substitutions on finite alphabets. In Section \\ref{SEC:main}, we introduce the relevant theory from symbolic dynamics. In this section we also include a new fully deterministic check for recognizability for primitive substitutions. The algorithm is simple enough to check by hand in a reasonable amount of time for small substitutions. Section \\ref{SEC:cohomology} will cover specifically those methods implemented to compute cohomology for tiling spaces. Throughout, we give instances of these methods being applied to well-known example substitutions. In Section \\ref{SEC:pisot} we give a brief overview of results relating to the \\emph{Pisot conjecture} and the \\emph{strong coincidence conjecture}. We focus our efforts on applying the functions developed for Grout to the study of Pisot substitutions. In particular, we perform a search for counterexamples to the strong coincidence conjecture.\n\nSection \\ref{SEC:pisot} is the only section in which original results appear (with the exception of the recognizability check). The preceding sections are meant as a survey of results in order to familiarize the reader with the types of calculations that Grout has been developed to compute.\n\n\t\\section{Background}\\label{SEC:background}\n\tMost of the definitions that follow can be found in any standard reference text on symbolic dynamics. For one such text, we refer the reader to the book of Fogg \\cite{F:book}.\n\t\t\\begin{definition}\n\t\t\tLet $\\mathcal{A}$ be a finite alphabet on $l$ letters (also called symbols). An alphabet $\\mathcal{A}$ may be a set of integers (e.g., $\\{0,1,2, \\ldots, (l-1)\\}$) or a set of letters (e.g., $\\{a,b,c\\}$). For all positive integers $n$, let $\\mathcal{A}^n$ be the $n$-fold cartesian product of the alphabet $\\mathcal{A}$ with itself. An \\emph{$n$-letter word} is an element of $\\mathcal{A}^n$. We formally identify an $n$-letter word, which is an $n$-tuple $(a_1, \\ldots, a_n)$ of letters from the alphabet $\\mathcal{A}$, with the finite string of letters $a_1 \\cdots a_n$. We let $\\mathcal{A}^+$ denote the union of the sets of $n$-letter words $\\bigcup_{n \\geq 1} \\mathcal{A}^n$. Further, if the empty word $\\epsilon$ is included, we let $\\mathcal{A}^\\ast$ denote the union $\\mathcal{A}^+ \\cup \\{\\epsilon\\}$. For $w \\in \\mathcal{A}^n$, we say that $n$ is the \\emph{length} of the word $w$ and write $|w|=n$. The empty word $\\epsilon$ has length $|\\epsilon| = 0$.\n\t\t\t\n\t\t\tLet $a_1, \\ldots, a_n, b_1, \\ldots, b_m \\in \\mathcal{A}$, $w \\in \\mathcal{A}^*$. \\emph{Concatenation} of finite words is a binary operation $\\bullet \\colon \\mathcal{A}^* \\times \\mathcal{A}^* \\to \\mathcal{A}^*$ defined by $a_1 \\cdots a_n \\bullet b_1 \\cdots b_m = a_1 \\cdots a_n b_1 \\cdots b_m$ and $\\epsilon \\bullet w = w \\bullet \\epsilon = w$. As concatenation is associative, we will omit this formal notation, and without loss of generality write $(uv)w = u(vw) = uvw$ for words $u,v,w \\in \\mathcal{A}^*$.\n\t\t\t\n\t\t\tA \\emph{substitution} $\\phi$ on $\\mathcal{A}$ is a function $\\phi \\colon \\mathcal{A} \\to \\mathcal{A}^+$ which assigns a non-empty word $\\phi(a) \\in \\mathcal{A}^+$ to each letter $a \\in \\mathcal{A}$. We extend $\\phi$ to a function $\\phi \\colon \\mathcal{A}^+ \\to \\mathcal{A}^+$ by concatenation; given a word $w = w_1 \\cdots w_n \\in \\mathcal{A}^n$, we set $\\phi(w) = \\phi(w_1) \\cdots \\phi(w_n)$. In this way, we can consider finite positive powers of the substitution $\\phi$ acting on $\\mathcal{A}^+$. We define $\\phi^1 = \\phi$ and $\\phi^{p+1} = \\phi \\circ \\phi^{p}$.\n\t\t\t\n\t\t\tAn infinite sequence $s$ over the alphabet $\\mathcal{A}$ is a function $s \\colon \\mathbb{N} \\to \\mathcal{A}$. We write $s$ sequentially as $s = s_0 s_1 s_2 \\cdots$.\n\t\t\tA bi-infinite sequence $s$ over the alphabet $\\mathcal{A}$ is a function $s \\colon \\mathbb{Z} \\to \\mathcal{A}$. We write $s$ sequentially as $s = \\cdots s_{-1} \\cdot s_0 s_1 \\cdots$ where a single dot $\\cdot$ is used to mark the position left of the \\emph{origin} $s(0) = s_0$.\n\t\t\t\n\t\t\tIf $s$ is an infinite sequence $s = s_0 s_1 s_2 \\cdots$, or a bi-infinite sequence $s = \\cdots s_{-1}\\cdot s_0 s_1 \\cdots$, then we define the shifted sequence $\\sigma(s)$ to be given by $\\sigma(s)_n = s_{n+1}$.\n\t\t\n\t\tWe say that a word $w$ is a \\emph{factor} of the word $w'$ if there exist words $u$ and $v$ (possibly empty) such that $u w v = w'$. Similarly, the word $w$ of length $|w| = n$ is a factor of the bi-infinite sequence $s$ if there exists an integer $k$, such that $w = s_{k+1} s_{k+2}\\cdots s_{k+n}$.\n\t\t\\end{definition}\n\n\t\tSome prefer to extend the definition of a substitution to maps whose codomain is $\\mathcal{A}^*$, where the empty word $\\epsilon$ is allowed to be the image of a letter under substitution. However, such substitutions are difficult to work with in our setting (and a further standing assumption to be introduced will negate this extended definition). So, for our purposes, we shall always require that the image of a letter under a substitution is non-empty.\n\n\t\t\\begin{definition}\n\t\t\tLet $\\phi \\colon \\mathcal{A} \\to \\mathcal{A}^+$ be a substitution. We say a word $w \\in \\mathcal{A}^\\ast$ is \\emph{admitted} by the substitution $\\phi$ if there exists a letter $i \\in \\mathcal{A}$ and a natural number $p \\geq 0$ such that $w$ is a factor of $\\phi^p(i)$. We let $\\mathcal{L}^n_\\phi \\subset \\mathcal{A}^n$ denote the set of all words of length $n$ admitted by $\\phi$. We form the \\emph{language} of $\\phi$ by taking the set of all admitted words $\\mathcal{L}_\\phi = \\bigcup_{n \\geq 0} \\mathcal{L}^n_\\phi$.\n\t\t\t\n\t\t\tWe say a bi-infinite sequence $s \\in \\mathcal{A}^\\mathbb{Z}$ is \\emph{admitted} by $\\phi$ if every factor of $s$ is admitted by $\\phi$. We let $X_\\phi$ denote the set of all bi-infinite sequences admitted by $\\phi$. The set $X_\\phi$ has a natural (metric) topology inherited from the product topology on $\\mathcal{A}^\\mathbb{Z}$ and a natural shift map $\\sigma \\colon X_\\phi \\to X_\\phi$ given by $\\sigma(s)_i= s_{i+1}$. We call the pair $(X_\\phi, \\sigma)$ the \\emph{subshift} associated with $\\phi$ and we will often abbreviate the pair to just $X_{\\phi}$ when the context is clear.\n\t\t\\end{definition}\n\t\t\n\t\tAn infinite sequence of words $u_0, u_1, \\ldots$ is \\emph{nested to the right} (or simply \\emph{nested} when context is clear) if, for each $n \\geq 0$, there exists a word $v_n$ such that $ u_{n+1} = u_n v_n$, where $v_n$ may be the empty word.\n\t\t\\begin{example}\\label{EX:fibonacci}\n\t\tLet $\\mathcal{A} = \\{0,1\\}$ and let $\\phi \\colon \\mathcal{A} \\to \\mathcal{A}^+$ be given by $\\phi \\colon 0 \\mapsto 01, 1 \\mapsto 0$. By iterating $\\phi$ on the initial seed letter $0$ we get a nested sequence of words:\n\t\t$$0 \\mapsto 01 \\mapsto 010 \\mapsto 01001 \\mapsto 01001010 \\mapsto 0100101001001 \\mapsto \\cdots, $$\n\t\twhose limit sequence is the \\emph{Fibonacci word} \\seqnum{A003849} \\cite{F:book}, (see also \\seqnum{A096270}):\n\t\t$$010010100100101001010010010100100101001010010010100 \\cdots.$$\t\t\n\t\tFor the purpose of studying the combinatorics and dynamics of such sequences, we do not distinguish this sequence from the version on the alphabet $\\{1,2\\}$ given by $1\\mapsto 12, 2\\mapsto 1$ \\seqnum{A003842} or on the alphabet $\\{a,b\\}$ given by $a\\mapsto ab, b\\mapsto a$, etc.\n\t\t\\end{example}\n\t\t\\begin{example}\\label{EX:thue-morse}\n\t\tLet $\\mathcal{A} = \\{0, 1\\}$ and let $\\phi \\colon \\mathcal{A} \\to \\mathcal{A}^+$ be given by $\\phi \\colon 0 \\mapsto 01, 1 \\mapsto 10$. In this case, the limit sequence given by repeated substitution on the seed letter $0$ is known as a \\emph{Thue-Morse sequence} \\seqnum{A010060} \\cite{P:thue-morse} (see also \\seqnum{A010059}, \\seqnum{A001285}):\n\t\t$$011010011001011010010110011010011001011001101001011 \\cdots.$$\n\t\t\\end{example}\n\t\tNote that, in general, a sequence of substituted iterates of a letter does not necessarily form a nested sequence of words extending to the right. However, if the substitution is suitably nice (if there exists an element $a$ of the alphabet, a natural number $n \\geq 1$ and a non-empty word $u$ such that $\\phi^n(a) = au$), then there exists a finite power of the substitution and some seed letter in the alphabet for which this property does hold. Infinite sequences of letters which are limits of such nested sequences of words whose lengths grow without bound are called \\emph{fixed points} of the substitution. Bi-infinite analogues also exist \\cite{SW:bi-infinite}, which we study here.\n\t\t\n\t\t\\begin{definition}\nLet $\\mathcal{A} = \\{0, \\ldots, (l-1)\\}$ be an alphabet with a substitution $\\phi \\colon \\mathcal{A} \\to \\mathcal{A}^+$. Then $\\phi$ has an associated \\emph{substitution matrix} $M_\\phi$ of dimension $l \\times l$ given by setting the entry $m_{ij}$ to be the number of times the letter $i$ appears in the substituted word $\\phi(j)$.\n\\end{definition}\n\\begin{example}\\label{EX:period-doubling}\nLet $\\mathcal{A} = \\{0,1\\}$ and let $\\phi \\colon \\mathcal{A} \\to \\mathcal{A}^+$ be given by $\\phi \\colon 0 \\mapsto 01, 1\\mapsto 00$, whose limit sequence with seed $0$ is the \\emph{period doubling sequence} \\seqnum{A096268} \\cite{D:period-doubling}, (see also \\seqnum{A035263}):\n$$01000101010001000100010101000101 \\cdots.$$\nThis substitution has associated substitution matrix\n\t\t$M_\\phi =\n\t\t\\begin{bmatrix}\n\t\t1 & 2 \\\\\n\t\t1 & 0\n\t\t\\end{bmatrix}$.\n\\end{example}\n\nThe first property that we check for a substitution matrix is primitivity. Other properties will be introduced in Section \\ref{SEC:matrix}.\n\n\\begin{definition}\nA substitution $\\phi \\colon \\mathcal{A} \\to \\mathcal{A}^+$ is called \\emph{primitive} if there exists a positive natural number $p$ such that the matrix $M^p_\\phi$ has strictly positive entries. Such a matrix $M_\\phi$ is also called \\emph{primitive}.  This condition is equivalent to having a positive natural number $p$ such that for all $i,j \\in \\mathcal{A}$ the letter $j$ appears in the word $\\phi^p(i)$.\n\\end{definition}\n\nIt is an easy consequence of the definition that any primitive substitution admits a letter $a \\in \\mathcal{A}$, a natural number $p \\geq 1$ and a word $u$ such that $\\phi^p(a) = au$. In particular, all primitive substitutions admit a power $p$, and a seed letter $a$, such that the sequence of words $a, \\phi^p(a), \\phi^{2p}(a), \\ldots$ is nested. Note that primitivity is only a sufficient condition for this nesting property to hold, and not necessary.\n\nChecking primitivity of a substitution is algorithmically simple: consider the sequence of matrices $M_\\phi, M_\\phi^2, M_\\phi^3, \\ldots$; as the number of zeros appearing as entries in the matrices of this sequence is non-increasing and will remain constant as soon as no difference occurs between steps, either the number of zeros in $M_\\phi^k$ and $M_\\phi^{k+1}$ are identical and positive for some $k$, in which case $\\phi$ is not primitive, or else there exists a power $p$ such that $M_\\phi^p$ has all positive entries, in which case $\\phi$ is primitive. Exactly one of these events occurs and in finite time.\n\nAll examples which have been presented so far are easily checked to be primitive.\n\\begin{example}\\label{EX:chacon}\nLet $\\mathcal{A} = \\{0,1\\}$ and let $\\phi \\colon \\mathcal{A} \\to \\mathcal{A}^+$ be given by $\\phi \\colon 0 \\mapsto 0010, 1 \\mapsto 1$, whose limit sequence with seed $0$ is the \\emph{non-primitive Chacon sequence} \\seqnum{A049320} \\cite{C:chacon-sub}:\n\n$$0010 0010 1 0010 0010 0010 1 0010 1 0010 0010 1 0010 \\cdots.$$\n\nAs the name suggests, this particular substitution that generates the non-primitive Chacon sequence fails to be primitive as its substitution matrix $M_\\phi = \\left[\\begin{smallmatrix}3 & 0 \\\\ 1 & 1\\end{smallmatrix}\\right]$ is lower triangular. Hence, $M^p_{\\phi}$ has a zero entry for all $p \\geq 1$.\n\nAlthough they are very closely related, the non-primitive Chacon sequence should not be confused with the \\emph{primitive Chacon sequence} \\seqnum{A049321} \\cite{F:chacon}:\n\n$$0012 0012 1 2012 0012 0012 1 2012 1 2012 0012 1 2012 \\cdots$$\n\ngiven as the limit sequence of the substitution $\\phi \\colon 0 \\mapsto 0012, 1 \\mapsto 12, 2 \\mapsto 012$ with seed $0$. This substitution \\emph{is} primitive.\n\\end{example}\n\nPrimitivity is a standing assumption that we ask all considered substitutions to satisfy. For many reasons, non-primitive substitutions are difficult to work with and Grout cannot be used to calculate properties of non-primitive substitutions. The study of non-primitive substitutions via their tiling spaces has only recently been seriously considered \\cite{MR:non-primitive}, and we are not aware of any results in the literature concerning the tiling spaces associated with substitutions with letters that have empty image.\n\t\t\n\n\n\\subsection{The subshift and tiling space}\n\t\t\tThe subshift associated with a substitution is one of the key objects with which we are concerned. We say $\\phi$ is a \\emph{periodic} substitution if $X_\\phi$ is finite, and $\\phi$ is non-periodic otherwise. Equivalently, $\\phi$ is periodic if $X_\\phi$ contains only periodic sequences, and $\\phi$ is non-periodic if $X_\\phi$ contains at least one non-periodic sequence (whereas a bi-infinite sequence $s$ is periodic if there exists a non-zero integer $n$ such that $\\sigma^n(s)=s$). We say $\\phi$ is \\emph{aperiodic} if $X_\\phi$ contains no periodic shift-invariant subspaces. Equivalently, $\\phi$ is aperiodic if $X_\\phi$ contains no periodic sequences. We say $\\sigma$ is a \\emph{minimal} action on $X_\\phi$ if the only closed shift-invariant subsets of $X_\\phi$ are the empty set $\\emptyset$ and the subshift itself $X_\\phi$. Equivalently, $\\sigma$ is minimal if the orbit of every point under $\\sigma$ is dense in $X_\\phi$. We say that $X_\\phi$ is \\emph{minimal} if the shift $\\sigma$ is a minimal action on $X_\\phi$.\n\t\t\t\n\t\t\tIf $\\phi$ is non-periodic and primitive, then $\\phi$ is automatically aperiodic (this follows from Proposition \\ref{PROP:minimal} below). If $\\phi$ is aperiodic, then $X_\\phi$ is a Cantor set and $\\sigma$ is a \\emph{minimal} action on $X_\\phi$.\n\t\t\t\n\t\t\tIf $X_\\phi$ is minimal in the sense given above, then one can replace this \\emph{internal} definition of a substitution subshift in terms of its language with an \\emph{external} definition in terms of the orbit closure of some fixed sequence of the substitution.\n\t\t\t\n\t\t\tLet $A$ be a subset of the metric space $X$. Let $\\overline{A}$ denote the closure of $A$. Recall that the closure of $A$ is the intersection of all closed subsets of $X$ which contain $A$. Equivalently, $$\\overline{A} = \\{ \\lim_{n \\to \\infty} (a_n)_{n \\geq 0} \\mid (a_n)_{n \\geq 0} \\in A^\\mathbb{N} \\}$$ is the collection of all the limit points of $A$.\n\t\t\t\\begin{proposition}\n\t\t\tLet $s \\in \\mathcal{A}^{\\mathbb{Z}}$ be a bi-infinite fixed point of the substitution $\\phi$ on $\\mathcal{A}$. \n\t\t\tIf $X_\\phi$ is minimal, then $X_\\phi = \\overline{\\{ \\sigma^n(s) \\mid n \\in \\mathbb{Z} \\}}$.\n\t\t\t\\end{proposition}\n\t\t\t\\begin{proof}\n\t\t\tLet $A = \\{ \\sigma^n(s) \\mid n \\in \\mathbb{Z} \\}$. As the fixed point $s$ is clearly admitted by $\\phi$, and $X_\\phi$ is closed under the action of the shift map $\\sigma$, we see that $A$ is a subset of $X_\\phi$. It is also clear that $A$ is non-empty and so $\\overline{A}$ is also non-empty. Suppose $x \\in \\overline{A}$. Then there exists a sequence $(a_n)_{n \\geq 0}$ of points $a_n \\in A$ whose limit is $x$. Note that the shift map is continuous on $X_\\phi$ and so $\\lim_{n \\to \\infty} \\sigma(a_n) = \\sigma (\\lim_{n \\to \\infty} a_n) = \\sigma(x)$. It follows that $\\sigma(x)$ is in the closure of $A$ because all the points $\\sigma(a_n)$ lie in $A$. This means that $\\overline{A}$ is a shift-invariant subset of $X_\\phi$. As $\\overline{A}$ is the intersection of closed sets, it is also closed, and so by shift-invariance and minimality we must have $\\overline{A} = X_\\phi$.\n\t\t\t\\end{proof}\n\t\t\tThe following result is well-known \\cite{F:book}.\n\t\t\t\\begin{proposition}\\label{PROP:minimal}\n\t\t\tIf $\\phi$ is primitive, then $X_\\phi$ is minimal.\n\t\t\t\\end{proposition}\n\t\t\n\t\t\\begin{definition}\n\t\t\tLet $\\phi$ be a substitution on the alphabet $\\mathcal{A}$ with associated subshift $X_\\phi$. The \\emph{tiling space} associated with $\\phi$ is the quotient space $$\\Omega_\\phi = (X_\\phi\\times [0,1]) \/{\\sim},$$ where $\\sim$ is the equivalence relation generated by the relation $(s,1)\\sim (\\sigma(s),0)$.\n\t\t\\end{definition}\n\t\tOne should imagine the point $(s,t)\\in\\Omega_\\phi$ as being a partition or tiling of the real line $\\mathbb{R}$ into labelled unit-length intervals (called tiles), where the labels are determined by the letters appearing in $s$, as shown in Figure \\ref{FIG:tiling}, and the origin of $\\mathbb{R}$ is shifted a distance $t$ to the right from the left of the tile labelled by the letter $s_0$. Two tilings $T$ and $T'$ in $\\Omega_\\phi$ are considered $\\epsilon$-\\emph{close} in this topology if, after a translate by a distance at most $\\epsilon$, the tiles around the origin in $T'-\\epsilon$ within a ball of radius $1\/\\epsilon$ lie exactly on top of the tiles around the origin in $T$ within a ball of the same radius and share the same labels.\n\n\\begin{figure}[h]\n\\centering\n\n\\begin{tikzpicture}[node distance=0.5cm]\n\\filldraw [fill=red, draw=black] (0.0,0.0) rectangle (0.5,0.25);\n\\filldraw [fill=blue, draw=black] (0.5,0.0) rectangle (1.0,0.25);\n\\filldraw [fill=blue, draw=black] (1.0,0.0) rectangle (1.5,0.25);\n\\filldraw [fill=red, draw=black] (1.5,0.0) rectangle (2.0,0.25);\n\n\\filldraw [fill=blue, draw=black] (2.0,0.0) rectangle (2.5,0.25);\n\\filldraw [fill=red, draw=black] (2.5,0.0) rectangle (3.0,0.25);\n\\filldraw [fill=red, draw=black] (3.0,0.0) rectangle (3.5,0.25);\n\\filldraw [fill=blue, draw=black] (3.5,0.0) rectangle (4.0,0.25);\n\n\\filldraw [fill=blue, draw=black] (4.0,0.0) rectangle (4.5,0.25);\n\\filldraw [fill=red, draw=black] (4.5,0.0) rectangle (5.0,0.25);\n\\filldraw [fill=red, draw=black] (5.0,0.0) rectangle (5.5,0.25);\n\\filldraw [fill=blue, draw=black] (5.5,0.0) rectangle (6.0,0.25);\n\n\\filldraw [fill=red, draw=black] (6.0,0.0) rectangle (6.5,0.25);\n\\filldraw [fill=blue, draw=black] (6.5,0.0) rectangle (7.0,0.25);\n\\filldraw [fill=blue, draw=black] (7.0,0.0) rectangle (7.5,0.25);\n\\filldraw [fill=red, draw=black] (7.5,0.0) rectangle (8.0,0.25);\n\n\\filldraw [fill=blue, draw=black] (8.0,0.0) rectangle (8.5,0.25);\n\\filldraw [fill=red, draw=black] (8.5,0.0) rectangle (9.0,0.25);\n\\filldraw [fill=red, draw=black] (9.0,0.0) rectangle (9.5,0.25);\n\\filldraw [fill=blue, draw=black] (9.5,0.0) rectangle (10.0,0.25);\n\n\\filldraw [fill=red, draw=black] (10.0,0.0) rectangle (10.5,0.25);\n\\filldraw [fill=blue, draw=black] (10.5,0.0) rectangle (11.0,0.25);\n\\filldraw [fill=blue, draw=black] (11.0,0.0) rectangle (11.5,0.25);\n\\filldraw [fill=red, draw=black] (11.5,0.0) rectangle (12.0,0.25);\n\n\\filldraw [fill=red, draw=black] (12.0,0.0) rectangle (12.5,0.25);\n\\filldraw [fill=blue, draw=black] (12.5,0.0) rectangle (13.0,0.25);\n\\filldraw [fill=blue, draw=black] (13.0,0.0) rectangle (13.5,0.25);\n\\filldraw [fill=red, draw=black] (13.5,0.0) rectangle (14.0,0.25);\n\n\\filldraw [fill=blue, draw=black] (14.0,0.0) rectangle (14.5,0.25);\n\\filldraw [fill=red, draw=black] (14.5,0.0) rectangle (15.0,0.25);\n\\filldraw [fill=red, draw=black] (15.0,0.0) rectangle (15.5,0.25);\n\\filldraw [fill=blue, draw=black] (15.5,0.0) rectangle (16.0,0.25);\n\n\n\\node  (0) at (0.25,0.5) {$0$}; \n\\node  (1) [right of=0] {$1$};\n\\node  (2) [right of=1] {$1$}; \n\\node  (3) [right of=2] {$0$}; \n\n\\node  (4) [right of=3] {$1$}; \n\\node  (5) [right of=4] {$0$}; \n\\node  (6) [right of=5] {$0$}; \n\\node  (7) [right of=6] {$1$}; \n\n\\node  (8) [right of=7] {$1$}; \n\\node  (9) [right of=8] {$0$}; \n\\node  (10) [right of=9] {$0$}; \n\\node  (11) [right of=10] {$1$}; \n\n\\node  (12) [right of=11] {$0$}; \n\\node  (13) [right of=12] {$1$}; \n\\node  (14) [right of=13] {$1$}; \n\\node  (15) [right of=14] {$0$}; \n\n\n\\node  (16) [right of=15] {$1$};\n\\node  (17) [right of=16] {$0$};\n\\node  (18) [right of=17] {$0$};\n\\node  (19) [right of=18] {$1$};\n\n\\node  (20) [right of=19] {$0$};\n\\node  (21) [right of=20] {$1$};\n\\node  (22) [right of=21] {$1$};\n\\node  (23) [right of=22] {$0$};\n\n\\node  (24) [right of=23] {$0$};\n\\node  (25) [right of=24] {$1$};\n\\node  (26) [right of=25] {$1$};\n\\node  (27) [right of=26] {$0$};\n\n\\node  (28) [right of=27] {$1$};\n\\node  (29) [right of=28] {$0$};\n\\node  (30) [right of=29] {$0$};\n\\node  (31) [right of=30] {$1$};\n\n\\end{tikzpicture}\n\n\\caption{A portion of the tiling associated with a Thue-Morse sequence.}\n\n\\label{FIG:tiling}\n\n\\end{figure}\n\t\t\n\t\tIf $\\phi$ is primitive and aperiodic, then $\\Omega_\\phi$ is a compact connected metric space which is nowhere locally connected. The tiling space $\\Omega_\\phi$ can be loosely seen as a twisted product of a Cantor set with the circle\\footnote{In topological language, $\\Omega_\\phi$ is a fiber bundle over the circle with Cantor set fibers.}. The natural translation $T\\mapsto T+t$ for $t\\in\\mathbb{R}$ equips $\\Omega_\\phi$ with a continuous $\\mathbb{R}$-action which is minimal (in the sense that all $\\mathbb{R}$-orbits are dense) as long as $\\phi$ is primitive. In this respect, tiling spaces are closely related to the more well-known spaces, the solenoids.\n\t\t\n\t\t\\begin{definition}\n\t\tFor substitutions $\\phi$ and $\\eta$, we say that a homeomorphism $f \\colon X_\\phi \\to X_\\eta$ is a \\emph{topological conjugacy} if $f \\circ \\sigma = \\sigma \\circ f$, where $\\sigma$ is the shift map on the respective subshifts. If such a topological conjugacy exists, we say that $\\phi$ and $\\eta$ are \\emph{conjugate} substitutions.\n\t\t\\end{definition}\n\t\t\\begin{example}\n\t\tAny letter-for-letter relabelling of a substitution describes a topological conjugacy. Suppose $\\phi$ is the Fibonacci substitution $\\phi \\colon 0 \\mapsto 01, 1 \\mapsto 0$ as in Example \\ref{EX:fibonacci}. Also consider the substitution $\\eta \\colon a \\mapsto b, b \\mapsto ba$. We can describe a rule $0 \\mapsto b, 1 \\mapsto a$ which extends globally to a topological conjugacy $X_\\phi \\to X_\\eta$.\n\t\t\\end{example}\n\t\t\\begin{figure}[H]\n\\centering\n\\includegraphics[scale=0.5]{fib-eps-converted-to}\n\\caption{The input of the substitution $a \\mapsto b, b \\mapsto ba$ into Grout.}\n\t\t\\end{figure}\n\t\tConjugacy is a strong form of equivalence for substitutions and it is an important task to be able to decide when two substitutions are conjugate. The next proposition hints that perhaps an easier object to study is the tiling space associated with a substitution.\n\t\t\\begin{proposition}\n\t\tIf $\\phi$ and $\\eta$ are conjugate, then $\\Omega_\\phi$ and $\\Omega_\\eta$ are homeomorphic.\n\t\t\\end{proposition}\n\t\tThe proof is an immediate consequence of the definition of the tiling space. The converse is false---as a simple counterexample, consider the periodic substitutions $a \\mapsto ab, b \\mapsto ab$ and $a \\mapsto aab, b \\mapsto aab$ whose tiling spaces are both homeomorphic to circles, but whose subshifts do not even have the same cardinality (2 for the former, 3 for the latter).\n\t\t\n\t\tWe will revisit the homeomorphism type of a tiling space and algebraic invariants of such spaces in the section on cohomology, Section \\ref{SEC:cohomology}.\n\n\\section{Combinatorics and geometry}\\label{SEC:main}\n\n\\subsection{Substitution matrices and their properties}\\label{SEC:matrix}\n\nAfter primitivity, the next properties that can be calculated from a substitution matrix are the natural tile frequencies and tile lengths of the substitution \\cite{S:frequency}.  This requires us to compute the eigenvalues of the matrix and make use of the Perron-Frobenius theorem \\cite{LM:introduction-to-symbolic}.\n\n\\begin{proposition}[Perron-Frobenius]\n\tLet $M$ be a primitive matrix.\n\t\\begin{enumerate}\n\t \\item[(i)] There is a positive real number $\\lambda_{PF}$, called the \\emph{Perron-Frobenius eigenvalue}, such that $\\lambda_{PF}$ is a simple eigenvalue of $M$ and any other eigenvalue $\\lambda$ is such that $|\\lambda|<\\lambda_{PF}$.\n\t \\item[(ii)] There exist left and right eigenvectors, called the \\emph{left and right Perron-Frobenius eigenvectors}, $\\mathbf{l}_{PF}$ and $\\mathbf{r}_{PF}$ associated with $\\lambda_{PF}$ whose entries are all positive and which are unique up to scaling.\n\t\\end{enumerate}\n\n\\end{proposition}\nGiven the above theorem, it is natural to ask what information is contained in the PF (Perron-Frobenius) eigenvalue and eigenvectors of $M_\\phi$ for a primitive substitution $\\phi$.\n\nBy assigning a length to the tiles labelled by each letter, we hope for such a length assignment to behave well with the given substitution. The left PF eigenvector offers a natural choice of length assignments. If we assign the length $(\\mathbf{l}_{PF})_i$ to the letter $i$, the $i$th component of the left PF eigenvector, then we can replace our combinatorial substitution by a geometric substitution---this is sometimes called an \\emph{inflation rule} \\cite{BG:book}. This geometric substitution expands the tile with label $i$ by a factor of $\\lambda_{PF}$ and then partitions this new interval into tiles of prescribed lengths and labels according to the symbolic substitution. In order to give a unique output, Grout normalizes the left PF eigenvector so that the smallest entry is $1$.\n\nThe information contained in the right PF eigenvector is also useful. The right PF eigenvector, once normalized so that the sum of the entries is $1$, gives the relative frequencies of each of the letters appearing in any particular bi-infinite sequence which is admitted by $\\phi$. That is, if $|w|$ is the length of the word $w$, $|w|_i$ is the number of times the letter $i$ appears in the word $w$, and letting $s_{[-k,k]} = s_{-k} \\cdots s_{-1} s_0 s_1 \\cdots s_k$, then $$\\lim_{k \\to \\infty} |s_{[-k,k]}|_i\/|s_{[-k,k]}| = (\\mathbf{r}_{PF})_i$$ for any $s \\in X_\\phi$.\n\\begin{example}\\label{EX:rudin-shapiro}\nLet $\\mathcal{A} = \\{0,1,2,3\\}$ and let $\\phi \\colon \\mathcal{A} \\to \\mathcal{A}^+$ be given by $\\phi \\colon 0 \\mapsto 01, 1 \\mapsto 02, 2 \\mapsto 31, 3 \\mapsto 32$ whose limit sequence with seed 0 is the fixed point of the \\emph{Rudin-Shapiro substitution} \\seqnum{A100260} \\cite{R:rudin-shapiro}, (see also \\seqnum{A073057}, \\seqnum{A020985}):\n$$0102 0131 0102 3202 0102 0131 3231 0102 \\cdots,$$\nwhich is related to the \\emph{paper-folding sequence} \\seqnum{A014577}, (see also \\seqnum{A014709}, \\seqnum{A014710}, \\seqnum{A106665}).\nThe substitution matrix for $\\phi$ has PF eigenvalue $\\lambda_{PF} = 2$ and PF eigenvectors $\\mathbf{l}_{PF} = (1,1,1,1)$ and $\\mathbf{r}_{PF} = (\\frac{1}{4},\\frac{1}{4},\\frac{1}{4},\\frac{1}{4})$. So, from the left PF eigenvector, we assign unit length to all four tiles. From the normalized right PF eigenvector, we can tell that each letter appears with equal frequency of $\\frac{1}{4}$.\n\\end{example}\nThe PF eigenvalues of a substitution matrix will play a role in Section \\ref{SEC:pisot} when we introduce Pisot substitutions. Eigenvalues and entries of eigenvectors are often irrational. Grout displays an approximation of these values, as shown in Figure \\ref{FIG:matrix}.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[scale=0.5]{matrix2-eps-converted-to}\n\\caption{The results of matrix calculations in Grout for the substitution $a \\mapsto b, b\\mapsto ba$.}\n\\label{FIG:matrix}\n\\end{figure}\n\n\n\n\n\n\\subsection{Enumerating \\texorpdfstring{$n$}{n}-letter words}\n\nNow that we have introduced the basic structure of the substitutions, and discussed the problem of primitivity and other matrix related calculations, we can discuss the first main combinatorial function in Grout.\n\n\\begin{definition}\nGiven a substitution $\\phi \\colon \\mathcal{A} \\to \\mathcal{A}^+$, we define the \\emph{complexity function} at $n$ to be the number of unique $n$-letter words admitted by $\\phi$.  We let $p_\\phi$ denote this function, and so $p_\\phi(n) = |\\mathcal{L}^n_\\phi|$. If $s$ is a sequence, we similarly let $p_s$ denote the complexity function of $s$, where $p_s(n)$ is the number of unique length-$n$ factors of $s$.\n\nIn the literature, one also encounter the terms \\emph{subword complexity} or \\emph{factor complexity}.\n\\end{definition}\nClearly, if $\\phi$ is primitive, then for all $s \\in X_\\phi$ we have $p_s(n) = p_\\phi(n)$.\n\\begin{example}\nThe period doubling sequence from Example \\ref{EX:period-doubling} is found to have a complexity function \\seqnum{A275202} whose first few terms are $$p_\\phi(n) \\colon 2, 3, 5, 6, 8, 10, 11, 12, 14, 16, 18, 20, 21, 22, 23, 24, 26, 28, 30, 32, 34, 36, 38, 40, 41, 42, \\ldots.$$\n\\end{example}\n\nComplexity functions are clearly non-decreasing, and for an alphabet $\\mathcal{A}$ on $l$ letters, we have $p_s(n) \\leq l^n$. The celebrated Morse-Hedlund theorem \\cite{MH:complexity} says that a bi-infinite sequence $s$ is periodic if and only if $p_s(n)$ is eventually constant if and only if there exists $n \\geq 1$ such that $p_s(n) \\leq n$. In particular, for aperiodic sequences one knows that $p_s(n) \\geq n+1$ for all natural numbers $n$. The non-periodic sequences which attain this minimal complexity are called the \\emph{Sturmian} sequences \\cite{F:book}. The complexity function of a sequence is a useful measure of disorder in a sequence. The asymptotic growth rate gives topological and dynamical information about the subshift and tiling space of a sequence \\cite{F:complexity} and, in particular, is a topological invariant \\cite{J:complexity}.\n\nOne is usually interested in either a deterministic formula for $p_\\phi$ or information about the asymptotic growth rate of $p_\\phi$ such as polynomial degree; Grout can be used to at least give circumstantial evidence for these, though has no means of calculating either (this appears to be a very difficult problem in general). A useful survey article by Ferenczi collates many of the key results on complexity functions \\cite{F:complexity2}.\n\nOf particular interest are the number of two-letter and three-letter words admitted by a substitution, as we use them later to compute cohomology.  For this reason, Grout not only enumerates the number of $n$-letter words, but prints out the two-letter and three-letter words as required.\n\n\n\n\nThe algorithm to find $n$-letter admitted words begins by generating a seed word\\footnote{By primitivity, we can begin with any letter as a seed, so the choice of $0$ here is arbitrary.} $w = \\phi^k(0)$ for some $k \\geq 1$ such that $|w|\\geq n$, and count all unique $n$-letter words appearing as factors of the seed.  We then iterate $\\phi$ on $w$ and add all new $n$-letter words appearing in $\\phi(w)$ to the result.  At each stage of the iteration, we count the size before and after adding the new words. If our list of admitted words does not change between iterates, we can stop as no new $n$-letter words are generated after a step that generates no new $n$-letter words.  It follows that the value $p_\\phi(n)$ is computable in finite time for any fixed $n \\geq 1$. Moreover, it is efficient to compute successive terms of the complexity function of a primitive substitution \\cite{GSS:subword-complexity}.\n\n\\begin{example}\nIt is well known that the complexity function for the Fibonacci substitution satisfies $p_\\phi(n)=n+1$ (so the Fibonacci word is Sturmian), and we can verify this for any value of $n$ by computing its complexity in Grout.\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[scale=0.5]{words-eps-converted-to}\n\\caption{The words results for the substitution $a \\mapsto b, b\\mapsto ba$ displayed in Grout.}\n\\end{figure}\n\\end{example}\n\n\\subsection{Barge-Diamond and Anderson-Putnam complexes}\\label{SEC:complexes}\nGrout has the ability to output two graphs as pdf files (provided that the user has PDFLaTeX installed).  The first of these is the \\emph{Barge-Diamond complex} \\cite{BD}, which is the key tool used in one of the methods of computing the \\v{C}ech{}  cohomology of the tiling space.\n\n\\begin{definition}\nLet $\\mathcal{A} = \\{0, \\ldots , (l-1)\\}$ be an alphabet with a substitution $\\phi \\colon \\mathcal{A} \\to \\mathcal{A}^+$. We construct the \\emph{Barge-Diamond complex} of $\\phi$ (\\emph{BD complex} for short) as follows:  we have two vertices for each $i \\in \\mathcal{A}$; an \\emph{in} node $v_i^+$ and an \\emph{out} node $v_i^-$.  We draw an edge from $v_i^+$ to $v_i^-$ for all $i$.  Then, for all two-letter words $ij \\in \\mathcal{L}^2_\\phi$ admitted by $\\phi$, we draw an edge from $v_i^-$ to $v_j^+$.\n\\end{definition}\n\n\\begin{example}\\label{EX:platinum}\nThe \\emph{platinum mean} \\cite[Ex.\\ 4.7, p.\\ 93]{BG:book} is the algebraic number $\\lambda_{pm} = 2 + \\sqrt{3}$, which is the PF eigenvalue of the matrix $\\left[\\begin{smallmatrix} 3 & 2 \\\\ 1 & 1 \\end{smallmatrix}\\right]$ (this matrix is obviously not unique). For this reason, we call the substitution $\\phi \\colon 0 \\mapsto 0001, 1 \\mapsto 001$ the \\emph{platinum mean substitution} with fixed point \\seqnum{A275855}:\n$$0 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 1 \\cdots.$$\nThe two-letter and three-letter admitted words for $\\phi$ are $\\mathcal{L}^2_\\phi = \\{00, 01, 10\\}$ and $\\mathcal{L}^3_\\phi = \\{000,001,010,100\\}$. The BD complex for $\\phi$ is given in Figure \\ref{FIG:BD}.\n\n\\begin{figure}[H]\n\\centering\n\n\\begin{tikzpicture}[->, node distance=2cm, auto] \n\\clip (-5,-3.4) rectangle (5,3.5);\n\\node [fill,circle,draw,inner sep = 0pt, outer sep = 0pt, minimum size=2mm] (ai) at (2.000000,0.000000) {}; \n\\node [fill,circle,draw,inner sep = 0pt, outer sep = 0pt, minimum size=2mm] (ao) at (0.000593,2.000000) {}; \n\\node [fill,circle,draw,inner sep = 0pt, outer sep = 0pt, minimum size=2mm] (bi) at (-2.000000,0.001185) {}; \n\\node [fill,circle,draw,inner sep = 0pt, outer sep = 0pt, minimum size=2mm] (bo) at (-0.001778,-1.999999) {}; \n\\draw (ai) edge[bend right=110, looseness=3, ->, ultra thick, color=red] node {$1$}(ao); \n\\draw (bi) edge[bend right=110, looseness=3, ->, ultra thick, color=blue] node {$0$}(bo); \n\\draw [-,ultra thick, bend right, draw=white, line width=6pt, looseness=0.7] (bo) to (ai); \n\\draw [->,ultra thick, bend right, looseness=0.7, swap] (bo) to node {$10$} (ai); \n\\draw [-,ultra thick, bend right, draw=white, line width=6pt, looseness=0.7] (ao) to (bi); \n\\draw [->,ultra thick, bend right, looseness=0.7, swap] (ao) to node {$01$} (bi); \n\\draw [-,ultra thick, bend right, draw=white, line width=6pt, looseness=0.7] (bo) to (bi); \n\\draw [->,ultra thick, bend left, looseness=0.7] (bo) to node {$00$} (bi); \n\n\\end{tikzpicture}\n\\caption{The Barge-Diamond complex for the platinum mean substitution.}\n\\label{FIG:BD}\n\\end{figure}\n\\end{example}\n\nThe other complex that we consider for a substitution is (a variant of) the \\emph{collared Anderson-Putnam complex} \\cite{AP}, which is again the key tool used in one of the methods of computing \\v{C}ech{}  cohomology of the tiling space. This particular definition is based on what G\\\"{a}hler and Maloney call the \\emph{Modified Anderson-Putnam complex} \\cite{GM:multi-one-d}.  The Anderson-Putnam complex is constructed in a similar fashion to the Barge-Diamond complex, but makes use of both the two-letter and three-letter words.\n\n\\begin{definition}\\label{DEF:ap-complex}\nLet $\\mathcal{A} = \\{0, \\ldots , (l-1)\\}$ be an alphabet with a substitution $\\phi \\colon \\mathcal{A} \\to \\mathcal{A}^+$. We construct the \\emph{Anderson-Putnam complex} of $\\phi$ (\\emph{AP complex} for short) as follows:  we have a vertex $v_{ij}$ for each two-letter word $ij \\in \\mathcal{L}^2_\\phi$ admitted by $\\phi$.  We draw an edge from $v_{ij}$ to $v_{jk}$ if and only if the three-letter word $ijk$ is admitted by $\\phi$.\n\\end{definition}\n\n\\begin{remark}\nReaders familiar with the family of Rauzy graphs \\cite{R:rauzy-graphs} associated with an infinite sequence over a finite alphabet will recognize a similarity between the Anderson-Putnam complex and the \\emph{second Rauzy graph} of an infinite sequence. Indeed, they are isomorphic as graphs. Anderson and Putnam originally introduced their complex in the context of tilings of $d$-dimensional Euclidean space, where the case $d=1$ corresponds to tilings of the line. Such one-dimensional tilings are combinatorially described by bi-infinite sequences over finite alphabets. In their definition, the edges of the AP complex also come equipped with data on the lengths of the corresponding tiles in the associated tiling. Although useful in other contexts, the cohomology groups that we later study via the AP complex are independent of this edge-length data.\n\\end{remark}\n\n\\begin{remark}\n\tOne should note that this definition of the AP complex is slightly different from the definition originally introduced by Anderson and Putnam. In particular, the original definition distinguishes between different occurrences of a two-letter word $ij$ if the occurrences of three-letter words containing $ij$ as a factor do not overlap on some admitted four-letter word. For example, if the language of a substitution includes the two-letter word $ab$, the three-letter words $xab, yab, abw, abz$, and the four-letter words $xabw, yabz$, but the words $xabz$ and $yabw$ do not belong to $\\mathcal{L}_\\phi$, then the original definition of the AP complex has two instances of vertices with the label $ab$, say $(ab)_1$ and $(ab)_2$. In our definition, these vertices are identified, so that $(ab)_1 \\sim (ab)_2 \\sim ab$. An example of such a substitution is given by $\\phi \\colon a \\mapsto bc, b \\mapsto baab, c \\mapsto caac$ where we label exactly one vertex with the label $aa$, but the original definition would require we include two distinct vertices labelled $(aa)_1$ and $(aa)_2$.\n\\end{remark}\nIn our discussion of cohomology calculated via AP complexes in Section \\ref{SEC:ap-cohomology}, we use the version of the AP complex given in Definition  \\ref{DEF:ap-complex} to describe the performed calculations. This is also the method implemented for Grout. It would have been possible to use the original definition, or one of the many variant AP complexes that have been defined in the literature. There are at least three such variants discussed by G\\\"{a}hler and Maloney \\cite{GM:multi-one-d}, of varying complexities and situations in which they can be used.\n\n\\begin{example}\nAs discussed in Example \\ref{EX:platinum}, the platinum mean substitution has two-letter and three-letter admitted words $\\mathcal{L}^2_\\phi = \\{00, 01, 10\\}$ and $\\mathcal{L}^3_\\phi = \\{000, 001, 010, 100\\}$. The AP complex for the platinum mean substitution is given in Figure \\ref{FIG:AP}.\n\n\\begin{figure}[H]\n\n\\centering\n\n\\begin{tikzpicture}[->, node distance=3cm, auto, text=black, line width=0.5mm] \n\\clip (-3,-4) rectangle (2.4,2);\n\\node [circle,inner sep = 0pt, outer sep = 2pt, minimum size=2mm] (abi) at (2.000000,0.000000) {$10$}; \n\\node [circle,inner sep = 0pt, outer sep = 2pt, minimum size=2mm] (bai) at (-0.999316,1.732446) {$01$}; \n\\node [circle,inner sep = 0pt, outer sep = 2pt, minimum size=2mm] (bbi) at (-1.001368,-1.731260) {$00$}; \n\\draw [-, bend left, draw=white, line width=2pt, looseness=0.7] (abi) to (bbi); \n\\draw [->, bend left, looseness=0.7] (abi) to (bbi); \n\\draw [-, bend left, draw=white, line width=2pt, looseness=0.7] (bai) to (abi); \n\\draw [->, bend left, looseness=0.7] (bai) to (abi); \n\\draw [-, bend left, draw=white, line width=2pt, looseness=0.7] (bbi) to (bai); \n\\draw [->, bend left, looseness=0.7] (bbi) to (bai);\n\\draw (bbi) to [out=300,in=180,looseness=20] (bbi);\n\n\\end{tikzpicture}\n\n\\caption{The Anderson-Putnam complex for the platinum mean substitution.}\n\\label{FIG:AP}\n\\end{figure}\n\\end{example}\n\n\\subsection{Recognizability}\\label{SEC:recog}\n\n\tThe results in this section on recognizability are not new, but are rarely cited in the literature. It follows that this section may be of interest to experts who may be unaware of such results, especially given the strength of Corollary \\ref{COR:recog}.\n\\begin{definition}\t\n\t\tLet $\\phi$ be a substitution on the alphabet $\\mathcal{A}$. We say $\\phi$ is \\emph{recognizable} if, for every bi-infinite sequence $s \\in X_\\phi$ admitted by $\\phi$, there is a unique way of decomposing $s$ into words of the form $\\phi(a)$ for $a \\in\\mathcal{A}$. That is, there exists a unique bi-infinite sequence $s' = \\cdots a_{-1} a_0 a_1 \\cdots$ and integer $n<|\\phi(a_0)|$ such that $s = \\sigma^n(\\phi(s'))$. In this sense, one can \\emph{recognize} which substituted letter each letter in $s$ has come from.\n\t\t\n\t\tEquivalently, we say $\\phi$ is recognizable if there exists a natural number $K \\geq 1$ such that for all admitted words $w \\in \\mathcal{L}_\\phi$ with $|w| > 2K$, there exist unique words $x, y$ of length $|x|, |y| \\leq K$ and a unique admitted word $u \\in \\mathcal{L}_\\phi$ such that $w = x \\phi(u) y$.\n\t\\end{definition}\nRecognizability is an important property of a tiling, as many of the tools used to study the topology of the associated tiling spaces rely on recognizability as a hypothesis, much like primitivity. Recognizability of a primitive substitution is equivalent to aperiodicity of the subshift $X_\\phi$ by a celebrated result of Moss\\'{e} \\cite{M:aperiodic} (later extended to higher dimensions by Solomyak \\cite{S:aperiodic}), and so any recognizability check for substitutions will also constitute an aperiodicity check for fixed points of primitive substitutions.  The algorithm designed to determine if a given substitution is recognizable relies on finding a fixed letter and return words to that fixed letter.\n\n\\begin{definition}\nGiven a substitution $\\phi$ on an alphabet $\\mathcal{A}$, the letter $a$ is said to be \\emph{fixed} (on the left) of \\emph{order} $k$ if there exists some integer $k$ such that $\\phi^k(a) = a u$ for some word $u$. Every substitution has at least one fixed letter and the value of $k$ for such a letter is bounded by the size of the alphabet.\n\\end{definition}\n\n\n\\begin{definition}\nGiven a fixed letter $a$, a \\emph{return word} to $a$ is a word $v$ such that $v = au $ for some (possibly empty) word $u \\in (\\mathcal{A}\\setminus\\{a\\})^\\ast$, and $va = aua$ is an admitted word of the substitution.\n\\end{definition}\n\nSince their introduction by Durand \\cite{D:derived-seqs}, return words have proved to be an extremely useful tool in studying substitutions. If $\\phi$ is primitive, then the set of return words to any letter is finite. We use these return words to determine whether a substitution is recognizable or not.\n\nThe following is a result of Harju and Linna \\cite{HL:periodicity}.\n\t\\begin{theorem}\n\t\tLet $\\phi$ be a primitive substitution on $\\mathcal{A}$ and let $a$ be a fixed letter. Let $\\mathcal{R}_{\\phi,a}$ be the set of all return words to $a$. So $\\mathcal{R}_{\\phi,a} = \\{v \\mid v = au,\\: aua \\in \\mathcal{L}_\\phi,\\: u \\in (\\mathcal{A} \\setminus \\{a\\})^\\ast\\}$. The substitution $\\phi$ is not recognizable if and only if, for all $v, v' \\in \\mathcal{R}_{\\phi,a}$, there exists a natural number $p \\geq 1$ such that $\\phi^p(vv') = \\phi^p(v'v)$.\n\t\\end{theorem}\n\n\tThe set of return words $\\mathcal{R}_{\\phi,a}$ is finite, but a priori one still needs to check arbitrarily high values of $p$ according to the above theorem. This, in fact, is not the case. The next result appears in the work of Culik \\cite{C:iterates-of-words} and also Ehrenfeucht and Rozenberg \\cite{ER:iterates-of-words}.\n\t\\begin{lemma}\n\t\tLet $\\phi$ be a substitution on $\\mathcal{A}$ and let $|\\mathcal{A}| = l$. For words $u, w \\in \\mathcal{A}^+$, there exists a natural number $p \\geq 1$ such that $\\phi^p(u) = \\phi^p(w)$ if and only if $\\phi^l(u) = \\phi^l(w)$.\n\t\\end{lemma}\n\tThat is, if some iterated substitution of $u$ and $w$ are ever equal, then their iterates must become equal at least by the $l$th iteration of the substitution, where $l$ is the size of the alphabet.\n\tTogether, these results give us the following corollary which provides us with a finite deterministic check for recognizability:\n\t\\begin{corollary}\\label{COR:recog}\n\tLet $\\phi$ be a primitive substitution on an alphabet with $l$ letters. Let $a \\in \\mathcal{A}$ be a fixed letter. The substitution $\\phi$ is recognizable if and only if, for all return words $v, v'$ to $a$ $$\\phi^l(vv') = \\phi^l(v'v) \\implies v = v'.$$\n\t\\end{corollary}\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[scale=0.5]{NEWrecog-eps-converted-to}\n\\caption{The recognizability results in Grout for the substitution $a \\mapsto b, b \\mapsto ba$.}\n\\end{figure}\n\\begin{example}\\label{EX:TM-return}\nLet $\\phi \\colon 0 \\mapsto 01, 1 \\mapsto 10$ be the Thue-Morse substitution. The letter $0$ is a fixed letter and the return words to $0$ are $\\mathcal{R}_{\\phi,0} = \\{0, 01, 011\\}$. We check all three distinct pairs:\n\n$$\\begin{array}{ll}\n\\phi^2(0\\:01) = 011001101001 \t\t\t\t& \\phi^2(01\\:0) = 011010010110\\\\\n\\phi^2(0\\:011) = 0110 011010011001 \t\t\t& \\phi^2(011\\:0) = 011010011001 0110\\\\\n\\phi^2(01\\:011) = 01101001 011010011001 \t& \\phi^2(011\\:01) =  011010011001 01101001.\n\\end{array}$$\nAs no row produces an equal pair of words, we conclude from Corollary \\ref{COR:recog} that the Thue-Morse substitution is recognizable.\n\\end{example}\n\n\\section{Cohomology of tiling spaces}\\label{SEC:cohomology}\nIn this section, we discuss methods for calculating the first \\v{C}ech{}  cohomology of the tiling space associated with a primitive recognizable substitution. In order to understand the topological structure of these tiling spaces, one would ordinarily need to be familiar with the notion of an inverse limit of topological spaces. Indeed, to prove that the following methods are correct, inverse limits are a necessary intermediary. Our approach, however, omits any discussion of inverse limits, and we instead refer the interested reader to the very approachable book by Sadun \\cite{S:book}. We replace such discussion instead with combinatorial and algebraic machinery.\n\nWe first introduce two important tools from the theory of abelian groups and homological algebra---\\emph{exact sequences} and \\emph{direct limits}. These notions and results can be found in any introductory text on homological algebra \\cite{CE:book}.\n\\begin{definition}\nLet $A_i$ be a bi-infinite sequence of abelian groups for $i \\in \\mathbb{Z}$ and let $F = \\{f_i \\colon A_i \\to A_{i+1}\\}$ be a sequence of group homomorphisms. As a diagram, we represent such a sequence of homomorphisms as follows:\n$$\\cdots \\to A_{i} \\stackrel{f_{i}}{\\to} A_{i+1} \\stackrel{f_{i+1}}{\\to} A_{i+2} \\to \\cdots.$$\nWe say that the sequence of homomorphisms $F$ is \\emph{exact}, and call the above diagram a \\emph{long exact sequence} (LES) if, for all $i \\in \\mathbb{Z}$, $\\ker f_{i+1} = \\operatorname{im} f_i$.\n\nIf all but a consecutive collection of three of the groups $A_i, A_{i+1}, A_{i+2}$ are trivial, we say that the diagram is a \\emph{short exact sequence} (SES). Generally, an SES will be written as a finite diagram capped by trivial groups:\n$$0 \\to A \\to B \\to C \\to 0.$$\n\\end{definition}\nThe following result is an easy exercise.\n\\begin{proposition}\nThe diagram $0 \\stackrel{f}{\\to} A \\stackrel{g}{\\to} B \\stackrel{h}{\\to} 0$ is exact if and only if $g$ is an isomorphism.\n\\end{proposition}\n\nThe next result appears in more general forms, but we give here only the version that we require. The proof is well-known \\cite{CE:book}.\n\nLet $0 \\to A \\stackrel{f}{\\to} B \\stackrel{g}{\\to} C \\to 0$ be a short exact sequence of abelian groups. If there exists a homomorphism $i \\colon C \\to B$ such that $g \\circ i = \\operatorname{Id}_C$, or there exists a homomorphism $j\\colon A \\to B$ such that $j \\circ f = \\operatorname{Id}_A$, then we say that the SES \\emph{splits}.\n\n\\begin{lemma}[Splitting lemma]\nIf $0 \\to A \\to B \\to C \\to 0$ is a split SES, then there exists an isomorphism $h \\colon B \\to A \\oplus C$.\n\\end{lemma}\nIt is useful to establish criteria under which a short exact sequence splits. One such criterion is that the third non-trivial group in the sequence is free abelian.\n\\begin{proposition}\nIf $0 \\to A \\to B \\stackrel{f}{\\to} \\mathbb{Z}^n \\to 0$ is an SES, then it is split. Further, $B \\cong A \\oplus \\mathbb{Z}^n$.\n\\end{proposition}\n\\begin{proof}\nAs the sequence is exact, the image of $f$ is the kernel of a homomorphism whose image is trivial, hence $\\operatorname{im}f = \\mathbb{Z}^n$. Let $\\{e_j \\mid 1,\n\\ldots, n \\}$ be a set of generating elements of $\\mathbb{Z}^n$ and let $b_1, \\ldots, b_n$ be such that $f(b_j) = e_j$. Define the map $i \\colon \\mathbb{Z}^n \\to B$ on generators by $i(e_j) = b_j$ and extend this to a homomorphism. We have $f (i (e_j)) = f(b_j) = e_j$ for all $j$ and so $f \\circ i = \\operatorname{Id}_{\\mathbb{Z}^n}$. Hence, the SES splits and, by the splitting lemma, we conclude that $B \\cong A \\oplus \\mathbb{Z}^n$.\n\\end{proof}\nThe groups which appear as cohomology groups of tiling spaces are usually presented as a so-called \\emph{direct limit}.\n\\begin{definition}\nLet $ A_0 \\stackrel{f_0}{\\to} A_1 \\stackrel{f_1}{\\to} A_2 \\to \\cdots$ be an infinite sequence of abelian groups $A_i$ with homomorphisms $f_i$ between them. We let the pair $(A_i,f_i)$ denote this \\emph{direct system}. For $j \\geq i$, write $f_{ij} = f_j \\circ \\cdots \\circ f_i \\colon A_i \\to A_{j+1}$ for the composition of a consecutive subsequence of these homomorphisms.\n\nWe define the \\emph{direct limit} of a direct system $(A_i,f_i)$ to be the disjoint union of the $A_i$ modulo the equivalence relation $a_i \\sim a_j$ for $a_i \\in A_i$, $a_j \\in A_j$ if there exists $N \\geq \\max\\{i,j\\}$ such that $f_{iN}(a_i) = f_{jN}(a_j)$. We write\n$$\\varinjlim(A_i,f_i) = \\bigsqcup A_i \\bigg\/{\\sim}.$$\nThere is an abelian group operation on $\\varinjlim(A_i,f_i)$ given by $[a_i]+[a_j] = [f_{iN}(a_i) + f_{jN}(a_j)]$ where $N = \\max\\{i,j\\}$ and square brackets $[-]$ denote an equivalence class with respect to $\\sim$.\n\\end{definition}\nThe equivalence relation $\\sim$ should be thought of as identifying elements in the disjoint union if they are `eventually equal' under applications of the homomorphisms $f_i$ as they move down the direct system of groups. When the context is clear, we may write a direct limit in terms of the homomorphisms appearing in the direct system $\\varinjlim (A_i,f_i) = \\varinjlim f_i$.\n\nA diagram of commuting homomorphisms $\\phi_i \\colon A_i \\to B_i$ between direct systems $(A_i,f_i)$ and $(B_i,g_i)$ induces a homomorphism $\\phi \\colon \\varinjlim f_i \\to \\varinjlim g_i$ by the rule $\\phi([a_i]) = [\\phi_i(a_i)]$.\n\\begin{example}\nIf $f_i$ is an isomorphism for all $i$, then $\\varinjlim (A_i,f_i) \\cong A_i$ for any of the $A_i$. In particular, an isomorphism $h \\colon \\varinjlim f_i \\to A_0$ is given by $h([a_k]) = f_{0k}^{-1}(a_k)$ for $a_k \\in A_k$.\n\\end{example}\n\\begin{example}\nIf the homomorphism $f_n$ is the zero homomorphism, then any element in the direct limit that appears before the $n$th position in the sequence is identified with zero by the equivalence relation $\\sim$. It follows that if $f_i$ is the zero homomorphism for infinitely many $i$, then $\\varinjlim f_i = 0$.\n\\end{example}\n\\begin{example}\nSuppose $A_i = \\mathbb{Z}$ for all $i$ and the homomorphisms are all given by the doubling map $\\times 2 \\colon n \\mapsto 2n$. The direct limit of this direct system is isomorphic to the set of \\emph{dyadic rationals} $\\mathbb{Z}[1\/2]$---the rational numbers whose denominator is a power of $2$ when written in reduced form. The isomorphism $h \\colon \\mathbb{Z}[1\/2] \\to \\varinjlim (\\mathbb{Z},\\times 2)$ is given by $h(n\/2^k) = [n_k]$ where $n_k$ is the copy of the integer $n$ that appears in the $k$th group $A_k$ in the direct system.\n\\end{example}\nIn general, it is difficult to reduce the direct limit of a sequence of matrices acting on free abelian groups to a form which is more familiar; for instance as a direct sum of free abelian parts and $n$-adic rational parts. In fact, it is often the case that this cannot be done \\cite{D:decomposable}. However, in certain cases we can give an explicit description in this form.\n\nTo an $(n \\times n)$-matrix $M$, we can associate a homomorphism $f \\colon \\mathbb{Z}^n \\to \\mathbb{Z}^n$ given by the action of the matrix $M$ on the left of column vectors. That is, $f(v) = Mv$. Without loss of generality, we may then write $\\varinjlim M = \\varinjlim f = \\varinjlim (\\mathbb{Z}^n,f_i)$ where $f_i = f$ for all $i$ in the case that every map in the direct system is given by the same action of the matrix $M$.\n\\begin{example}\\label{EX:TM-direct-limit}\nSuppose $A_i = \\mathbb{Z}^2$ for all $i$ and the homomorphisms $f_i$ are given by the action of the matrix $M = \\left[\\begin{smallmatrix}1 & 1 \\\\ 1 & 1\\end{smallmatrix}\\right]$ on the left of column vectors $\\left[\\begin{smallmatrix}n \\\\ m\\end{smallmatrix}\\right] \\in \\mathbb{Z}^2$. The direct limit $\\varinjlim f_i = \\varinjlim \\left[\\begin{smallmatrix}1 & 1 \\\\ 1 & 1\\end{smallmatrix}\\right]$ is isomorphic to the dyadic integers $\\mathbb{Z}[1\/2]$.\n\nWe note that each element $\\left[\\begin{smallmatrix}n \\\\ m\\end{smallmatrix}\\right] \\in \\mathbb{Z}^2$ in the group $A_k$ has a representative in $A_{k+1}$ which lies on the diagonal subgroup $D = \\{\\left[\\begin{smallmatrix}n \\\\ n\\end{smallmatrix}\\right] \\mid n \\in \\mathbb{Z}\\} < \\mathbb{Z}^2$. In particular, $\\left[\\begin{smallmatrix}1 & 1 \\\\ 1 & 1\\end{smallmatrix}\\right] \\left[\\begin{smallmatrix}n \\\\ m\\end{smallmatrix}\\right] = \\left[\\begin{smallmatrix}n+m \\\\ n+m \\end{smallmatrix}\\right]$, and so the direct limit is isomorphic to the direct limit of the induced action on this subgroup $\\varinjlim (D,f_i)$ where $f_i\\left[\\begin{smallmatrix}n \\\\ n\\end{smallmatrix}\\right] = \\left[\\begin{smallmatrix}2n\\\\ 2n\\end{smallmatrix}\\right]$. This direct limit is isomorphic to the direct limit of the doubling map on the integers. The isomorphism is given by mapping $\\left[\\begin{smallmatrix}n \\\\ n\\end{smallmatrix}\\right]$ in the $k$th copy of $D$ to $n$ in the $k$th copy of $\\mathbb{Z}$. This is illustrated in the following commutative diagram\n$$\n\t\t\\begin{tikzpicture}[node distance=2cm, auto]\n  \t\t\\node (00) {$\\mathbb{Z}$};\n  \t\t\\node (20) [above of=00] {$D$};\n  \t\t\\node (10) [above of=20] {$\\mathbb{Z}^2$};\n  \t\t\\node (01) [right of=00] {$\\mathbb{Z}$};\n  \t\t\\node (11) [right of=10] {$\\mathbb{Z}^2$};\n  \t\t\\node (21) [right of=20] {$D$};\n  \t\t\\node (02) [right of=01] {$\\mathbb{Z}$};\n  \t\t\\node (12) [right of=11] {$\\mathbb{Z}^2$};\n  \t\t\\node (22) [right of=21] {$D$};\n  \t\t\\node (03) [right of=02] {$\\cdots$};\n  \t\t\\node (13) [right of=12] {$\\cdots$};\n  \t\t\\node (23) [right of=22] {$\\cdots$};\n  \t\t\n  \t\t\\node (L2) [left of=20] {$\\varinjlim\\left(D,\\left[\\begin{smallmatrix}1 & 1 \\\\ 1 & 1\\end{smallmatrix}\\right]\\right):$};\n  \t\t\\node (L1) [left of=10] {$\\varinjlim\\left(\\mathbb{Z}^2,\\left[\\begin{smallmatrix}1 & 1 \\\\ 1 & 1\\end{smallmatrix}\\right]\\right):$};\n  \t\t\\node (L0) [left of=00] {$\\varinjlim\\left(\\mathbb{Z},\\times 2\\right):$};\n  \t\t\n  \t\t\n  \t\t\n  \t\t\\draw[->] (10) to node [swap] {} (20);\n  \t\t\\draw[->] (11) to node [swap] {} (21);\n  \t\t\\draw[->] (12) to node [swap] {} (22);\n  \t\t\\draw[->] (20) to node [swap] {} (00);\n  \t\t\\draw[->] (21) to node [swap] {} (01);\n  \t\t\\draw[->] (22) to node [swap] {} (02);\n  \t\t\\draw[->] (L2) to node [swap] {} (L0);\n  \t\t\\draw[->] (L1) to node [swap] {} (L2);\n  \t\t\\draw[->] (00) to node {$\\times 2$} (01);\n  \t\t\\draw[->] (01) to node {$\\times 2$} (02);\n  \t\t\\draw[->] (02) to node {$\\times 2$} (03);\n  \t\t\\draw[->] (10) to node {$\\left[\\begin{smallmatrix}1 & 1 \\\\ 1 & 1\\end{smallmatrix}\\right]$} (11);\n  \t\t\\draw[->] (11) to node {$\\left[\\begin{smallmatrix}1 & 1 \\\\ 1 & 1\\end{smallmatrix}\\right]$} (12);\n  \t\t\\draw[->] (12) to node {$\\left[\\begin{smallmatrix}1 & 1 \\\\ 1 & 1\\end{smallmatrix}\\right]$} (13);\n  \t\t\\draw[->] (20) to node {$\\left[\\begin{smallmatrix}1 & 1 \\\\ 1 & 1\\end{smallmatrix}\\right]$} (21);\n  \t\t\\draw[->] (21) to node {$\\left[\\begin{smallmatrix}1 & 1 \\\\ 1 & 1\\end{smallmatrix}\\right]$} (22);\n  \t\t\\draw[->] (22) to node {$\\left[\\begin{smallmatrix}1 & 1 \\\\ 1 & 1\\end{smallmatrix}\\right]$} (23); \n\t\t\\end{tikzpicture}\n$$\nwhere the top row of vertical homomorphisms are given by $\\left[\\begin{smallmatrix}n \\\\ m\\end{smallmatrix}\\right] \\mapsto \\left[\\begin{smallmatrix}n+m \\\\ n+m\\end{smallmatrix}\\right]$, and the bottom row of vertical homomorphisms are given by $\\left[\\begin{smallmatrix}n \\\\ n\\end{smallmatrix}\\right] \\mapsto n$. The induced homomorphisms on the direct limits are both isomorphisms by the reasoning above.\nHence, from the previous example, we know that $\\varinjlim(\\mathbb{Z}^2, \\left[\\begin{smallmatrix}1 & 1 \\\\ 1 & 1\\end{smallmatrix}\\right]) \\cong \\mathbb{Z}[1\/2]$; the dyadic rationals.\n\\end{example}\nEven if the direct limit is not easily parsable, we can at least easily extract the rank of a direct limit of a sequence of matrices when the sequence is constant. Here, the rank $\\operatorname{rk}A$ of an abelian group $A$ is the dimension of the $\\mathbb{Q}$-vector space $A\\otimes \\mathbb{Q}$, or equivalently, the cardinality of a maximal linearly independent subset.\n\nIf $M$ is an $(n\\times n)$-matrix, let $p$ be a power such that the rank of the matrix $M^p$ is the same as the rank of $M^{p+1}$ (after which point the ranks of successive powers remain constant). We call the rank $\\operatorname{rk} M^p$ the \\emph{eventual rank} of $M$ and let $\\operatorname{e-rk}M$ denote this value. The next result is an easy exercise in linear algebra and follows from the definition of the direct limit of a matrix.\n\\begin{proposition}\n$$\\operatorname{rk}\\varinjlim M = \\operatorname{e-rk}M.$$\n\\end{proposition}\nThere is an entire industry devoted to studying direct limits of integer-valued matrices, their invariants, and classifying such groups, possibly with an additional order structure. We refer the reader to the work of Boyle and Handelman \\cite{BH:shift-equivalence} and references therein.\n\nAssociated with any topological space $X$ is a collection of groups $\\check{H}^k(X)$ for all $k \\geq 0$ called the \\emph{\\v{C}ech{}  cohomology groups} of $X$. The \\v{C}ech{}  cohomology of a space is a topological invariant. That is, if $X \\cong Y$, then $\\check{H}^k(X) \\cong \\check{H}^k(Y)$ for all $k\\geq 0$. In particular, if $\\phi$ and $\\eta$ are conjugate substitutions, then $\\check{H}^k(\\Omega_\\phi) \\cong \\check{H}^k(\\Omega_\\eta)$ for all $k\\geq 0$. As we are only studying $1$-dimensional tiling spaces of primitive substitutions, it will always be the case for us that $\\check{H}^0(\\Omega_\\phi) = \\mathbb{Z}$ and $\\check{H}^k(\\Omega_\\phi) = 0$ for all $k \\geq 2$. Hence, the only interesting group to study is $\\check{H}^1(\\Omega_\\phi)$.\n\t\t\n\t\t\n\t\t\n\t\tThere are many standard texts providing an introduction to \\v{C}ech{}  cohomology \\cite{B:bott-tu}. \\v{C}ech{}  cohomology is an important topological invariant of tiling spaces and it is of general interest to be able to calculate and study these groups even beyond the task of classifying substitutions up to conjugacy \\cite{BGG:substitutions,BBJS:homological-pisot,CS:shape}. Grout implements three different methods for calculating the cohomology of tiling spaces associated with primitive aperiodic substitutions on finite alphabets:\n\t\t\\begin{enumerate}\n\t\t\t\\item The method of Barge-Diamond complexes \\cite{BD}.\n\t\t\t\\item The method of Anderson-Putnam complexes  \\cite{AP}.\n\t\t\t\\item The method of forming a conjugate \\emph{pre-left-proper} substitution \\cite{DHS:return-words}.\n\t\t\\end{enumerate}\n\t\tAll three outputs are algebraically equivalent---that is, they represent isomorphic groups---but it is not always obvious that this is the case given the presentations that each method produces. Compare Examples \\ref{EX:TM-BD}, \\ref{EX:TM-AP}, \\ref{EX:TM-proper}, where the Thue-Morse substitution is used as a reference example. These presentations take the form of direct limits of matrices (often of different sizes) and, in the case of the Barge-Diamond method, also direct sums of such groups and quotients by free abelian subgroups. This disparity between presentations of results for the equivalent methods was one of the major motivating factors for developing Grout.  These groups are extremely laborious to calculate by hand for large alphabets unless special criteria are met.\n\\subsection{Via Barge-Diamond}\n\t Let $\\phi$ be a primitive, recognizable substitution on the alphabet $\\mathcal{A}$. Let $\\Gamma_\\phi$ be the Barge-Diamond complex of $\\phi$ and let $S_\\phi$ be the subcomplex of $\\Gamma_\\phi$ formed by all those edges labelled with two-letter admitted words $ij$.\n\t\n\tLet $\\tilde{\\phi}\\colon S_\\phi \\to S_\\phi$ be a graph morphism defined in the following way on vertices: let $l(i)$ and $r(i)$ be the leftmost and rightmost letters of $\\phi(i)$ so $\\phi(i) = l(i) u r(i)$ (note that $u$ may be empty and $l(i)$ and $r(i)$ could overlap if $|\\phi(i)| = 1$). Define $\\tilde{\\phi}(v_i^+) = v_{l(i)}^+$ and $\\tilde{\\phi}(v_i^-) = v_{r(i)}^-$. Note that if $ij$ is admitted by $\\phi$, then $r(i) l(j)$ is also admitted by $\\phi$, and so $\\tilde{\\phi}$ is a well-defined graph morphism on $S_\\phi$. As $\\tilde{\\phi}(S_\\phi) \\subset S_\\phi$, we can define the eventual range $ER_\\phi = \\bigcap_{m \\geq 0} \\tilde{\\phi}^m(S_\\phi)$ (which stabilizes after finitely many substitutions).\n\t\n\tLet $\\tilde{H}^0(X) = \\mathbb{Z}^{k-1}$ where $k$ is the number of connected components of the space $X$---this group is called the \\emph{reduced cohomology} of $X$ in degree zero and always satisfies $\\check{H}^0(X) = \\tilde{H}^0(X) \\oplus \\mathbb{Z}$. For this method of computation we make use of the following result attributed to Barge and Diamond \\cite{BD}.\n\t\\begin{proposition}\n\t\tThere is an exact sequence\n\t\t$$0 \\to \\tilde{H}^0(ER_\\phi) \\stackrel{h_1}{\\to} \\varinjlim M_\\phi^T \\stackrel{h_2}{\\to} \\check{H}^1(\\Omega_\\phi) \\stackrel{h_3}{\\to} \\check{H}^1(ER_\\phi) \\to 0.$$\n\t\\end{proposition}\n\tThis exact sequence can be split into the two short exact sequences\n\t$$0 \\to \\tilde{H}^0(ER_\\phi) \\stackrel{h_1}{\\to} \\varinjlim M_\\phi^T \\to \\operatorname{im} h_2 \\to 0$$\n\tand\n\t$$0 \\to \\operatorname{im} h_2 \\to \\check{H}^1(\\Omega_\\phi) \\stackrel{h_3}{\\to} \\check{H}^1(ER_\\phi) \\to 0.$$\n\tUsing the first isomorphism theorem, we know that $\\operatorname{im}h_2$ is isomorphic to $\\varinjlim M_\\phi^T \/ \\ker h_2$. By exactness, $\\ker h_2 = \\operatorname{im} h_1$ and as $h_1$ is injective by exactness, the image of $h_1$ is isomorphic to its domain. Hence, $\\ker h_2 = \\operatorname{im} h_1 \\cong \\tilde{H}^0(ER_\\phi)$. It follows that $\\operatorname{im}h_2 \\cong \\varinjlim M_\\phi^T \/ \\tilde{H}^0(ER_\\phi)$.\n\t\n\tThe eventual range $ER_\\phi$ is a non-empty (possibly disconnected) graph, and so $\\tilde{H}^0(ER_\\phi)$ and $\\check{H}^1(ER_\\phi)$ are finitely generated free abelian groups. Using the splitting lemma on the second short exact sequence allows us to write $\\check{H}^1(\\Omega_\\phi)$ as a direct sum.\n\t\\begin{corollary}\\label{COR:BD} Let $\\phi$ be a substitution on an alphabet $\\mathcal{A}$ with $l$ letters. If the eventual range $ER_\\phi$ has $k$ connected components and $\\check{H}^1(ER_\\phi) \\cong \\mathbb{Z}^m$, then \n\t\t$$\\check{H}^1(\\Omega_\\phi)\\cong \\varinjlim (\\mathbb{Z}^{l},M_\\phi^T)\/ \\mathbb{Z}^{k-1} \\oplus \\mathbb{Z}^m.$$\n\t\\end{corollary}\n\t\n\tGrout displays the \\v{C}ech{}  cohomology in the above form when using the Barge-Diamond method.\n\t\n\tRecall that the Euler characteristic $\\chi$ of a graph $G$ is given by $\\chi = E - V$ where $E$ and $V$ are the number of edges and vertices of $G$ respectively. The Euler characteristic is also given as an alternating sum of ranks of cohomology groups; $\\chi = \\operatorname{rk}\\check{H}^1(G) - \\operatorname{rk}\\check{H}^0(G) = \\operatorname{rk}\\check{H}^1(G) - \\operatorname{rk}\\tilde{H}^0(G) - 1$. If $G$ has $k$ connected components, then $\\operatorname{rk}\\tilde{H}^0(G) = k-1$ and so $m = \\operatorname{rk}\\check{H}^1(G) = E - V + k$.\n\t\n\tWhen using this method via Corollary \\ref{COR:BD}, the only involved part of the calculation is finding the eventual range of the Barge-Diamond complex. The number of connected components of the eventual range then allows us to calculate its cohomology in degree zero. We can then calculate the cohomology in degree one by evaluating the above formula for $m$ derived from the Euler characteristic.\n\t\n\t\\begin{example}\\label{EX:TM-BD}\n\tLet $\\phi \\colon 0 \\mapsto 01, 1 \\mapsto 10$ be the Thue-Morse substitution. The Thue-Morse substitution is primitive and recognizable and has as its two-letter admitted words the set $\\mathcal{L}^2_\\phi = \\{00,01,10,11\\}$. The BD complex $\\Gamma_\\phi$ for $\\phi$ is given in Figure \\ref{FIG:BD-TM}. We calculate $\\tilde{\\phi}$ on the four vertices of $\\Gamma_\\phi$ to be \n\t$$\\tilde{\\phi} \\colon\n\t\\begin{cases}\n\tv_0^+  \\mapsto v_{l(0)}^+ & = v_{0}^+\\\\\n\tv_0^-  \\mapsto v_{r(0)}^- & = v_1^-\\\\\n\tv_1^+  \\mapsto v_{l(1)}^+ & = v_0^+\\\\\n\tv_1^-  \\mapsto v_{r(1)}^- & = v_1^-\n\t\\end{cases}$$\n\tand so on edges it acts by mapping $[00] \\mapsto [10], [01] \\mapsto [11], [10] \\mapsto [01], [11] \\mapsto [01]$ where we use square braces $[ij]$ to distinguish the two-letter word $ij$ from the edge in $\\Gamma_\\phi$ labeled by that word. As a map on the subcomplex $S_\\phi$, $\\tilde{\\phi}$ acts by permuting edges. In particular, $ER_\\phi = S_\\phi$. As $S_\\phi$ is topologically a circle, which has one connected component, we see that $k = 1$. As $S_\\phi$ has four edges and four vertices, we see that $E = V = 4$. This gives $m = E - V + k = 1$. We calculate that $\\tilde{H}^0(S_\\phi) = \\mathbb{Z}^{k-1} = \\mathbb{Z}^0 = 0$ and $\\check{H}^1(S_\\phi) = \\mathbb{Z}^m = \\mathbb{Z}$. By Corollary \\ref{COR:BD}, this tells us that $\\check{H}^1(\\Omega_\\phi) \\cong \\varinjlim \\left[\\begin{smallmatrix} 1 & 1 \\\\ 1 & 1 \\end{smallmatrix}\\right] \\oplus \\mathbb{Z}$. By Example \\ref{EX:TM-direct-limit}, $$\\check{H}^1(\\Omega_\\phi) \\cong \\mathbb{Z}[1\/2] \\oplus \\mathbb{Z}.$$\n\t\n\t\\begin{figure}[H]\n\\centering\n\n\\begin{tikzpicture}[->, node distance=2cm, auto] \n\\clip (-5,-3.4) rectangle (5,3.5);\n\\node [fill,circle,draw,inner sep = 0pt, outer sep = 0pt, minimum size=2mm] (ai) at (2.000000,0.000000) {}; \n\\node [fill,circle,draw,inner sep = 0pt, outer sep = 0pt, minimum size=2mm] (ao) at (0.000593,2.000000) {}; \n\\node [fill,circle,draw,inner sep = 0pt, outer sep = 0pt, minimum size=2mm] (bi) at (-2.000000,0.001185) {}; \n\\node [fill,circle,draw,inner sep = 0pt, outer sep = 0pt, minimum size=2mm] (bo) at (-0.001778,-1.999999) {}; \n\\draw (ai) edge[bend right=110, looseness=3, ->, ultra thick, color=red] node {$1$}(ao); \n\\draw (bi) edge[bend right=110, looseness=3, ->, ultra thick, color=blue] node {$0$}(bo); \n\\draw [-,ultra thick, bend right, draw=white, line width=6pt, looseness=0.7] (bo) to (ai); \n\\draw [->,ultra thick, bend right, looseness=0.7, swap] (bo) to node {$10$} (ai); \n\\draw [-,ultra thick, bend right, draw=white, line width=6pt, looseness=0.7] (ao) to (bi); \n\\draw [->,ultra thick, bend right, looseness=0.7, swap] (ao) to node {$01$} (bi); \n\\draw [-,ultra thick, bend right, draw=white, line width=6pt, looseness=0.7] (bo) to (bi); \n\\draw [->,ultra thick, bend left, looseness=0.7] (bo) to node {$00$} (bi);\n\\draw [-,ultra thick, bend right, draw=white, line width=6pt, looseness=0.7] (ao) to (ai); \n\\draw [->,ultra thick, bend left, looseness=0.7] (ao) to node {$11$} (ai); \n\n\\end{tikzpicture}\n\\caption{The Barge-Diamond complex for the Thue-Morse substitution.}\n\\label{FIG:BD-TM}\n\\end{figure}\n\n\t\\end{example}\n\t\n\\subsection{Via Anderson-Putnam}\\label{SEC:ap-cohomology}\n\tThe second technique for calculating cohomology is via the Anderson-Putnam complex. This method cannot as easily be reduced to the application of combinatorial tricks, such as the Euler characteristic formula and the splitting lemma, in order to circumvent complicated computational problems in homological algebra. We recommend the reader be familiar with simplicial and\/or cellular (co)homology $H^i$ when calculating cohomology using this method. In particular, the reader should be comfortable with calculating homomorphisms on cohomology groups induced by continuous maps. A standard text for an introduction to these concepts is Hatcher \\cite{H:book}.\n\n\tLet $\\phi$ be a primitive, recognizable substitution on the alphabet $\\mathcal{A}$. Let $K_\\phi$ be the Anderson-Putnam complex of $\\phi$. Anderson and Putnam have shown that the \\v{C}ech{}  cohomology of $\\Omega_\\phi$ is determined by the direct limit of an induced homomorphism acting on the cohomology of what is now called the Anderson-Putnam complex \\cite{AP}. Using the modified AP complex $K_\\phi$, G\\\"{a}hler and Maloney have shown that this complex, as defined in Section \\ref{SEC:complexes}, can be used in place of the originally defined AP complex \\cite{GM:multi-one-d}. We define the map acting on the AP complex $K_\\phi$ in the following way.\n\t\n\tSuppose we have an edge $e$ with label $ijk$. Suppose $\\phi(i) = i_1 i_2 \\cdots i_{|\\phi(i)|}$ and similarly for $\\phi(j)$ and $\\phi(k)$. Define a continuous map called the \\emph{collared substitution} $\\tilde{\\phi} \\colon K_\\phi \\to K_\\phi$ by mapping the edge $e$ to the ordered collection of edges with labels\n\t$$\n\t[i_{|\\phi(i)|} j_1 j_2]\n\t[j_1 j_2 j_3]\n\t\\cdots\n\t[j_{|\\phi(j)|-2} j_{|\\phi(j)|-1} j_{|\\phi(j)|}]\n\t[j_{|\\phi(j)|-1} j_{|\\phi(j)|} k_1]\n\t$$\n\tin an orientation preserving way (the parametrization of the map will not matter).\n\t\n\tLet $H^i$ be ordinary simplicial cohomology and, if $f \\colon X \\to Y$ is a continuous map between topological spaces, let $f^* \\colon H^i(Y) \\to H^i(X)$ be the induced homomorphism between cohomology groups.\n\t\\begin{example}\n\tLet $\\phi \\colon 0 \\mapsto 001, 1 \\mapsto 01$ be the proper version of the Fibonacci substitution (this will be made formal in Section \\ref{SEC:proper}). The word $010$ is admitted by $\\phi$ and so there is an edge labelled by $[010]$ in the AP complex $K_\\phi$. This edge is mapped by the collared substitution $\\tilde{\\phi}$ to the collection of edges $[101] [010]$. Similarly, the collared substitution acts on the full list of edges of $K_\\phi$ by the rule\n\t$$\n\t\\begin{array}{rcl}\n\t[001] & \\mapsto & [100] [001] [010]\\\\\n\t{[010]} & \\mapsto & [101] [010]\\\\\n\t{[100]} & \\mapsto & [100] [001] [010] \\\\\n\t{[101]} & \\mapsto & [100] [001] [010].\n\t\\end{array}\n\t$$\n\t\\end{example}\n\tThe map $\\tilde{\\phi}$ induces a homomorphism $\\tilde{\\phi}^\\ast \\colon H^1(K_\\phi) \\to H^1(K_\\phi)$ on cohomology. We use the following result \\cite{GM:multi-one-d} (Anderson and Putnam proved this result in the case that $K_\\phi$ is replaced with the original definition of the AP complex \\cite{AP}).\n\t\\begin{theorem}\\label{THM:AP}\n\tIf $\\phi$ is a primitive, recognizable substitution, then there is an isomorphism\n\t\t$$\\check{H}^1(\\Omega_\\phi) \\cong \\varinjlim (H^1(K_\\phi), \\tilde{\\phi}^*).$$\n\t\\end{theorem}\n\tLet $R = \\mbox{rk} H^1(K_\\phi)$, the rank of the cohomology of $K_\\phi$. Grout finds an explicit generating set of cocycles for the cohomology of $K_\\phi$ and then outputs the induced homomorphism $\\tilde{\\phi}^*$ as the associated $(R \\times R)$-matrix $M_{AP}$, which should be interpreted as acting on $\\mathbb{Z}^R$ with respect to this generating set. So $\\check{H}^1(\\Omega_\\phi) \\cong \\varinjlim M_{AP}$.  \n\t\n\tThe algorithm for this computation begins by constructing the boundary matrix for the AP complex.  To do this, take all of the admitted three-letter words $abc$ and use the convention that the boundary of such an edge is $bc-ab$. Construct the associated $m \\times n$ boundary matrix $B$, where $m$ is the number of two-letter words admitted by $\\phi$ and $n$ is the number of three-letter words admitted by $\\phi$. The entries of $B$ are given by\n\t$$(B)_{abc,xy} =\n\t\\begin{cases}\n\t1, & \\text{if $xy = bc$;}\\\\\n\t-1, & \\text{if $xy = ab$;}\\\\\n\t0, & \\text{otherwise.}\n\t\\end{cases}\n\t$$\n\tUsing standard methods from linear algebra, we find a maximal set of linearly independent $n$-dimensional integer vectors $g$ such that $Bg = 0$, searching over the set of all vectors of $0$s and $1$s. By construction, this set generates the kernel of the boundary homomorphism inside the simplicial $1$-chain group of $K_\\phi$. This gives us a generating set of cycle vectors for the first homology of $K_\\phi$.\n\t\n\tWe then apply the homomorphism induced by the collared substitution on the chain groups to each of these generating vectors, giving us a new set of image vectors.  Using Gaussian elimination, we find the coordinates of these image vectors in terms of the generating vectors.  This induced homomorphism on homology can be represented as a square matrix. The transpose of this matrix $M_{AP}$ then represents the induced homomorphism on cohomology. The matrix $M_{AP}$ is the output for the cohomology calculation via the Anderson-Putnam method.  It should be noted that this algorithm is not efficient in the case where the substitution admits many three-letter words, as the dimension $m$ of the $1$-chain complex is the dominant limiting factor when finding linearly independent generating cycles. The time complexity increases exponentially with respect to $m$.\n\t\\begin{example}\\label{EX:TM-AP}\n\tLet $\\phi \\colon 0 \\mapsto 01, 1 \\mapsto 10$ be the Thue-Morse substitution. The admitted two-letter words for $\\phi$ are given by $\\mathcal{L}^2_\\phi =\\{00, 01, 10, 11\\}$ and  the admitted three-letter words for $\\phi$ are given by $\\mathcal{L}^3_\\phi = \\{001, 010, 011, 100, 101, 110\\}$. The AP complex $K_\\phi$ for $\\phi$ is given in Figure \\ref{FIG:AP-TM}.\n\t\n\t\\begin{figure}[H]\n\\centering\n\\begin{tikzpicture}[->, node distance=3cm, auto, text=black, line width=0.5mm] \n\\clip (-2.5,-2.5) rectangle (2.5,2.5);\n\\node [circle,inner sep = 0pt, outer sep = 2pt, minimum size=2mm] (abi) at (2.000000,0.000000) {$01$}; \n\\node [circle,inner sep = 0pt, outer sep = 2pt, minimum size=2mm] (aai) at (0,2) {$00$};\n\\node [circle,inner sep = 0pt, outer sep = 2pt, minimum size=2mm] (bai) at (-2,0) {$10$}; \n\\node [circle,inner sep = 0pt, outer sep = 2pt, minimum size=2mm] (bbi) at (0,-2) {$11$}; \n\n\n\\draw [->, bend left, looseness=0.7] (abi) to (bbi); \n\n\\draw [->, bend left, looseness=0.7] (bai) to (abi); \n\n\\draw [->, bend left, looseness=0.7] (bbi) to (bai);\n\n\\draw [->, bend left, looseness=0.7] (abi) to (bai);\n\n\\draw [->, bend left, looseness=0.7] (bai) to (aai);\n\n\\draw [->, bend left, looseness=0.7] (aai) to (abi);\n\n\\end{tikzpicture}\n\\caption{The AP complex for the Thue-Morse substitution.}\\label{FIG:AP-TM}\n\\end{figure}\n\nThe graph $K_\\phi$ has Euler characteristic $\\chi = |\\mathcal{L}^3_\\phi| - |\\mathcal{L}^2_\\phi| = 2$. As $K_\\phi$ is connected, it follows that $\\operatorname{rk}H^0(K_\\phi) = 1$. We deduce that $R = \\operatorname{rk}H^1(K_\\phi) = \\chi + \\operatorname{rk}H^0(K_\\phi) = 3$.\n\nThe boundary matrix $B$ for $K_\\phi$ is\n$$B =\n\\left[\n\\begin{matrix}\n-1 & 0 & 0 & 1 & 0 & 0\\\\\n1 & -1 & -1 & 0 & 1 & 0\\\\\n0 & 1 & 0 & -1 & -1 & 1\\\\\n0 & 0 & 1 & 0 & 0 & -1\n\\end{matrix}\n\\right].$$\nWe find three linearly independent vectors $v_1, v_2, v_3$ such that $Bv_i = 0$ by inspection:\n$$\nv_1 = \\left[\\begin{matrix}0\\\\1\\\\0\\\\0\\\\1\\\\0\\end{matrix}\\right],\nv_2 = \\left[\\begin{matrix}1\\\\0\\\\1\\\\1\\\\0\\\\1\\end{matrix}\\right],\nv_3 = \\left[\\begin{matrix}0\\\\0\\\\1\\\\0\\\\1\\\\1\\end{matrix}\\right].\n$$\nThe action of substitution on the cycles represented by these vectors is $\\tilde{\\phi}^*(v_1) = v_2, \\tilde{\\phi}^*(v_2) = 2v_1 + v_2, \\tilde{\\phi}^*(v_3) = v_1 + v_2 $ and so the induced homomorphism on homology is represented by the matrix\n$$\\tilde{M}_{AP} =\n\\left[\n\\begin{matrix}\n0 & 2 & 1 \\\\\n1 & 1 & 1 \\\\\n0 & 0 & 0\n\\end{matrix}\n\\right].$$\nOn cohomology, the action is given by the transpose of $\\tilde{M}_{AP}$. That is, $M_{AP} = \\tilde{M}_{AP}^T$. By Theorem \\ref{THM:AP}, we calculate cohomology of the tiling space to be $$\\check{H}^1(\\Omega_\\phi) \\cong \\varinjlim\n\\left[\n\\begin{matrix}\n0 & 1 & 0 \\\\\n2 & 1 & 0 \\\\\n1 & 1 & 0\n\\end{matrix}\n\\right].$$\n\t\\end{example}\n\t\n\\subsection{Via properization}\\label{SEC:proper}\n\tFor this method of computation we make use of a technique involving the return words that were introduced in Section \\ref{SEC:recog}, as outlined in work of Durand, Host and Skau \\cite{DHS:return-words}, for replacing a primitive substitution with a conjugate \\emph{pre-left-proper} primitive substitution. One may then use the fact that if $\\phi$ is a recognizable pre-left-proper primitive substitution, then $\\check{H}^1(\\Omega_\\phi)\\cong \\varinjlim M_\\phi^T$ \\cite{S:book}.\n\n\tWe begin by defining what it means to be proper.\n\n\t\\begin{definition}\\leavevmode\n\t\t\\item A substitution is \\emph{left-proper} if there exists a letter $i \\in \\mathcal{A}$ such that the leftmost letter of $\\phi(j)$ is $i$ for all $j \\in \\mathcal{A}$. That is, $\\phi(j) = i w_j$ for some $w_j \\in \\mathcal{A}^\\ast$.\n\t\t\\item A substitution is \\emph{right-proper} if there exists a letter $i \\in \\mathcal{A}$ such that the rightmost letter of $\\phi(j)$ is $i$ for all $j \\in \\mathcal{A}$. That is, $\\phi(j) = w_j i$ for some $w_j \\in \\mathcal{A}^\\ast$.\n\t\t\\item A substitution is \\emph{fully-proper} (we will often abbreviate this to just \\emph{proper}) if it is both left-proper and right-proper.\n\t\t\\item A substitution is \\emph{pre-left-proper} if some power of the substitution is left-proper.\n\t\t\\item A substitution is \\emph{pre-right-proper} if some power of the substitution is right-proper.\n\t\t\\item A substitution is \\emph{pre-proper} if some power of the substitution is proper.\n\\end{definition}\n\n\tThe following algorithm produces what we call the \\emph{pre-left-properization} of a substitution, so-called because there exists a finite power of the new substitution which is left-proper. As usual, $k$ is the one from the definition of the fixed letter $a$. \n\n\tNote that if $v$ is a return word to the fixed letter $a$, then $va \\in \\mathcal{L}_\\phi$ and so $\\phi^k(va)$ must also be admitted by $\\phi$. We have $\\phi^k(va) = \\phi^k(v) \\phi^k(a)$, and both of $\\phi^k(v)$ and $\\phi^k(a)$ begin with the fixed letter $a$, hence $\\phi^k(v)$ is an exact concatenation of return words to $a$. So, if we apply $\\phi^k$ to a return word, then the result is a unique composition of return words.  We let $\\eta$ denote this newly constructed substitution rule on the new alphabet $\\mathcal{R}_{\\phi,a}$ of return words.\n\n\tThis process gives a new primitive substitution on a possibly larger alphabet \\cite{DHS:return-words}. By taking a power of this substitution, we arrive at a left-proper substitution. It is clear that such a power exists. Indeed, every return word $v \\in \\mathcal{R}_{\\phi,a}$ begins with the fixed letter $a$ and, by primitivity, $\\phi^n(a)$ contains at least two copies of the letter $a$ for large enough $n$, so $\\phi^n(a) = v_0 a u$ for some return word $v_0 \\in \\mathcal{R}_{\\phi,a}$ and some other word $u$. Then $\\eta^n(v)$ must begin with $v_0$. It follows that $\\eta^n$ is a left-proper substitution with leftmost letter $v_0$. \n\n\tWe may also form an associated proper substitution on $\\mathcal{R}_{\\phi,a}$ by composing $\\eta^n$ with its \\emph{right conjugate}. The right conjugate $\\phi^{(R)}$ of a left-proper substitution $\\phi$ is given by setting $\\phi^{(R)}(b) = w_ba$ where $a$ is the fixed letter such that $\\phi(b) = aw_b$ for all $b\\in\\mathcal{A}$. The right conjugate is a right-proper substitution, and the composition of a left-proper and right-proper substitution is both left-proper and right-proper, hence fully-proper.  It is easy to show that $X_{\\phi\\circ \\phi^{(R)}}$ and $X_\\phi$ are topologically conjugate subshifts \\cite{DL:properisation}. In fact, a word is admitted by $\\phi\\circ\\phi^{(R)}$ if and only if it is admitted by $\\phi$, so $X_{\\phi\\circ \\phi^{(R)}}$ and $X_\\phi$ are equal. Hence, $\\check{H}^1(\\Omega_{\\phi\\circ \\phi^{(R)}}) \\cong \\check{H}^1(\\Omega_\\phi)$.\n\t\\begin{example}\n\tThe Fibonacci substitution $\\phi \\colon 0 \\mapsto 01 , \\mapsto 0$ is left-proper. Its right conjugate is given by $\\phi^{(R)} \\colon 0 \\mapsto 10, 1 \\mapsto 0$ which is right-proper. Their composition is given by $\\phi \\circ \\phi^{(R)} \\colon 0 \\mapsto 001, 1 \\mapsto 01$ which is fully-proper.\n\t\\end{example}\n\t\n\tWe make use of the following which is a consequence of results appearing in the work of Durand, Host and Skau \\cite{DHS:return-words}.\n\t\\begin{proposition}\n\t\tLet $\\phi$ be a primitive substitution on $\\mathcal{A}$ and let $\\eta$ be the pre-left-properization of $\\phi$. The tiling space $\\Omega_\\eta$ is homeomorphic to $\\Omega_\\phi$.\n\t\\end{proposition}\n\tHence we get a corollary.\n\t\\begin{corollary}\\label{COR:pre-left}\n\t$$\\check{H}^1(\\Omega_\\eta) \\cong \\check{H}^1(\\Omega_\\phi).$$\n\t\\end{corollary}\n\tAs $\\eta^n$ is left-proper, $\\check{H}^1(\\Omega_{\\eta^n})\\cong \\varinjlim M_{\\eta^n}^T \\cong \\varinjlim (M_{\\eta}^n)^T \\cong \\varinjlim M_{\\eta}^T$. Hence $\\check{H}^1(\\Omega_\\phi)\\cong \\varinjlim M_\\eta^T$.\n\n\tGrout outputs the pre-left-properization $\\eta$, the left-properization $\\eta^n$, the full properization $\\eta^n \\circ (\\eta^n)^{(R)}$, and owing to the above, Grout also outputs the matrix $M_\\eta^T$ in the cohomology section.\n\n\\begin{figure}[H]\n\\centerline{\n\\includegraphics[scale=0.5]{NEWproper-eps-converted-to}\n}\n\\caption{The properization results for the substitution $a \\mapsto b, b\\mapsto ba$ in Grout.}\n\\end{figure}\n\n\\begin{remark}\nIf the substitution is already proper, the properization algorithm may return a different proper version of this substitution.  This may seem like it is a feature which has no use, but by iterating this process, we find that the sequence of substitutions is eventually periodic, first proved by Durand \\cite{D:derived-seqs}. This sequence of properizations is itself of interest.\n\\end{remark}\n\n\\begin{figure}[H]\n\\centerline{\n\\includegraphics[scale=0.5]{NEWcohom-eps-converted-to}\n}\n\\caption{The cohomology results for the substitution $a \\mapsto b, b\\mapsto ba$ in Grout.}\n\\end{figure}\n\\begin{example}\\label{EX:TM-proper}\nLet $\\phi \\colon 0 \\mapsto 01, 1\\mapsto 10$ be the Thue-Morse substitution. Recall from Example \\ref{EX:TM-return} that the Thue-Morse substitution has return words $\\mathcal{R}_{\\phi,0} = \\{0,01,011\\}$ which we relabel as $\\{[0] = i, [01] = j, [011] = k\\}$. The induced substitution on the new alphabet is then given by\n$$\\eta \\colon \\left \\{\\begin{array}{rclclcl}\ni & \\mapsto & [\\phi(0)] & = & [01] & = & j\\\\\nj & \\mapsto & [\\phi(01)] & = & [0110] & = & ki\\\\\nk & \\mapsto & [\\phi(011)] & = & [011010] & = & kji\n\\end{array}\\right.,$$\ngiving us the pre-left-properization. We then calculate cohomology via Corollary \\ref{COR:pre-left} to be $$\\check{H}^1(\\Omega_\\phi) \\cong \\varinjlim M_\\eta^T \\cong \\varinjlim \\left[\\begin{matrix}0&1&0\\\\1&0&1\\\\1&1&1\\end{matrix}\\right].$$\nWe invite the reader to check that the square of $\\eta$ is a left-proper substitution and that $\\eta^2 \\circ (\\eta^2)^{(R)}$ is a proper substitution.\n\\end{example}\n\n\n\\section{Pisot substitutions}\\label{SEC:pisot}\nIn this section, we direct our attention to a special class of substitutions known as \\emph{Pisot substitutions}. This class of substitutions has received considerable attention in the literature. For an overview of the various flavors of Pisot conjectures and a mostly up-to-date exposition of current knowledge, we refer the reader to the review article of Akiyama, Barge, Berth\\'{e} and Lee \\cite{ABBL:pisot}.\n\n\\begin{definition}\nLet $\\phi$ be a primitive aperiodic substitution on an alphabet $\\mathcal{A}$ with $l$ letters and let $M_\\phi$ be the associated substitution matrix with characteristic polynomial $q_{M_\\phi}$ and PF-eigenvector $\\lambda_{PF}$. We say that $\\phi$ is a \\emph{Pisot substitution} if for every root $\\lambda \\neq \\lambda_{PF}$ of the minimal polynomial of $\\lambda_{PF}$, we have $|\\lambda|<1$. If we further have that $p_{M_\\phi}$ is an irreducible polynomial over the rationals (equivalently over the integers) then we say that $\\phi$ is an \\emph{irreducible Pisot substitution}. Equivalently, $\\phi$ is irreducible Pisot if every eigenvalue $\\lambda \\neq \\lambda_{PF}$ of $M_\\phi$ satisfies $0 < |\\lambda| < 1$.\n\\end{definition}\nNote that irreducible Pisot substitutions on two or more letters are necessarily primitive and recognizable.\n\\begin{example}\nSome of the most well-known substitutions are irreducible Pisot:\n\\begin{itemize}\n\\item The Fibonacci substitution from Example \\ref{EX:fibonacci} has eigenvalues $\\lambda_{PF} = \\tau$ and $\\lambda_1 = \\tau - 1$ where $\\tau = (1+\\sqrt{5})\/2$ is the golden mean.\n\n\\item The \\emph{silver mean substitution} \\cite{BG:book} $\\phi \\colon 0 \\mapsto 001, 1\\mapsto 0$ with fixed point \\seqnum{A171588}, (see also \\seqnum{A049472}):\n$$001001000100100010010010001 \\cdots $$ has eigenvalues $\\lambda_{PF} = 1+\\sqrt{2}$ and $\\lambda_1 = 1-\\sqrt{2}$.\n\\item The \\emph{Tribonacci substitution} \\cite{R:rauzy-fractals} $\\phi \\colon 0 \\mapsto 01, 1 \\mapsto 02, 2 \\mapsto 0$ with fixed point \\seqnum{A092782}: $$01 02 01 0 01 02 01 01 02 01 0 01 02 01 02 \\cdots $$ has eigenvalues $\\lambda_{PF} \\simeq 1.83929, \\lambda_{1,2} \\simeq -0.419643 \\pm 0.606291 i$.\n\\item The \\emph{flipped Tribonacci substitution} \\cite{S:flipped-tribonacci} $\\phi \\colon 1 \\mapsto 12, 2 \\mapsto 31, 3 \\mapsto 1$ with fixed point \\seqnum{A100619}: $$123111212123112311231112123 \\cdots$$ has the same eigenvalues as the Tribonacci substitution as they have equal substitution matrices.\n\\end{itemize}\n\\end{example}\nThe Tribonacci substitution and the flipped Tribonacci substitution have very different dynamical properties, even though they appear extremely similar in their presentations and share eigenvalues \\cite{S:flipped-tribonacci}.\n\\begin{example}\nThe Thue-Morse substitution from Example \\ref{EX:thue-morse} is a Pisot substitution as it has PF-eigenvalue $\\lambda_{PF} = 2$ whose minimal polynomial is just $\\lambda - 2$, and is therefore trivially Pisot. However, this is not an irreducible Pisot substitution because $p_{M_\\phi} = \\lambda(\\lambda - 2)$ is reducible. Equivalently, because $0$ is also an eigenvalue of $M_\\phi$, the substitution cannot be irreducible Pisot.\n\\end{example}\n\\begin{example}\nThe substitution $\\phi \\colon 0 \\mapsto 001111, 1 \\mapsto 001$ is not Pisot, as the characteristic polynomial of $M_\\phi$ is $p_{M_\\phi} = \\lambda^2 - 3\\lambda - 6$, which is irreducible but with both roots of modulus greater than $1$.\n\\end{example}\n\n\nAn important tool for studying primitive aperiodic symbolic substitutions is the associated dynamical spectrum of both its subshift and tiling space. A good reference text for such notions is the book by Baake and Grimm \\cite{BG:book}. The notion of the dynamical spectrum for a compact measure dynamical system is beyond the scope of this work. Fortunately, there is a simple equivalent combinatorial definition of pure discrete spectrum for irreducible Pisot substitutions \\cite{BSW:pds}. We adopt this equivalent notion as our definition below in Theorem \\ref{THM:pure-discrete}.\n\n\\begin{remark}\nWe note that, due to results of Clark and Sadun \\cite{CS:shape}, the subshift of a Pisot substitution has pure discrete spectrum if and only if its corresponding tiling space (with possibly non-unital tile lengths) has pure discrete spectrum. With this in mind, we only mention the dynamical spectrum of the subshift, confident that everything also applies to the tiling space.\n\\end{remark}\n\nLet $\\psi(u) \\in \\mathbb{Z}^l$ be the abelianization of the word $u$. That is, the $i$th coordinate of the vector $\\psi(u)$ is the number of times the letter $i$ appears in the word $u$. We say that the pair of words $(u,v)$ are a \\emph{balanced pair} if $\\psi(u) = \\psi(v)$. If $u$ and $v$ cannot be factored as proper factors $u = u_1u_2$ and $v = v_1v_2$ such that $(u_1,v_1)$ and $(u_2,v_2)$ are balanced pairs, then we say that $(u,v)$ is an \\emph{irreducible balanced pair}. For every $i \\in \\mathcal{A}$, we call the balanced pair $(i,i)$ a \\emph{coincidence}. For pairs $(u_1,v_1)$ and $(u_2,v_2)$ we write $(u_1,v_1)(u_2,v_2) = (u_1u_2,v_1v_2)$ and call this \\emph{concatenation of pairs}. We say that $(u_1,v_1)$ and $(u_2,v_2)$ are \\emph{factors} of $(u_1u_2,v_1v_2)$. Clearly, every balanced pair can be factored uniquely into irreducible balanced pairs and if $(u,v)$ is a balanced pair, then $(\\phi(u),\\phi(v))$ is a balanced pair.\n\nFor a balanced pair $(u,v)$, let\n$$\n\\begin{array}{rcl}\nI(u,v) \t& = & \\{(u_i,v_i) \\mid \\exists n\\geq 0, \\mbox{ such that }(u_i,v_i) \\mbox{ appears as}\\\\\n\t\t&\t& \\mbox{an irreducible balanced pair factor of } (\\phi^n(u),\\phi^n(v))\\}.\n\\end{array}\n$$\n\n\\begin{definition}\nLet $\\phi$ be a primitive aperiodic substitution on an alphabet $\\mathcal{A}$ with $l$ letters. If $I(u,v)$ is finite, we say that the balanced pair algorithm for $\\phi$ \\emph{terminates} on $(u,v)$. If $I(u,v)$ contains a coincidence, then we say that the balanced pair algorithm for $\\phi$ \\emph{terminates with coincidence} on $(u,v)$.\n\\end{definition}\n\\begin{example}\\label{EX:Fib-coin}\nLet $\\phi \\colon 0 \\mapsto 01, 1 \\mapsto 0$ be the Fibonacci substitution. Consider the balanced pair $(01,10)$. We find that $\\left(\\phi(01),\\phi(10)\\right) = (010,001) = (0,0)(10,01)$ and the next iterate also contains the coincidence $(1,1)$. It is clear that no new irreducible balanced pairs are found as factors of higher iterates of the substitution. It follows that $I(01,10)$ is given by $$I(01,10) = \\{(0,0),(1,1),(01,10),(10,01)\\},$$ which is both finite and contains a coincidence. Hence, $\\phi$ terminates with coincidence on $(01,10)$.\n\\end{example}\nThe following theorem \\cite{BSW:pds} gives a simple criterion in terms of the balanced pair algorithm for determining if the subshift of an irreducible Pisot substitution has pure discrete spectrum. We take this as our definition for pure discrete spectrum.\n\\begin{theorem}\\label{THM:pure-discrete}\nIf $\\phi$ is an irreducible Pisot substitution, then for any distinct pair of letters $i,j \\in \\mathcal{A}$ the subshift $X_\\phi$ has pure discrete spectrum if and only if $\\phi$ terminates with coincidence on the balanced pair $(ij,ji)$.\n\\end{theorem}\n From Example \\ref{EX:Fib-coin}, the above theorem tells us that the Fibonacci substitution has pure discrete spectrum.\n\nThe following notion of strong coincidence is different from the balanced pair algorithm terminating with coincidence, though they are intimately related (and possibly equivalent) \\cite{BD:pisot-coincidence}.\n\\begin{definition}\nLet $\\phi$ be a primitive aperiodic substitution on an alphabet $\\mathcal{A}$ with $l$ letters. For a distinct pair of letters $i,j \\in \\mathcal{A}$, if there exists $n \\geq 1$ and a letter $k \\in \\mathcal{A}$ such that $\\phi^n(i) = ukv$, $\\phi^n(j) = \\tilde{u}k\\tilde{v}$ and $\\psi(u) = \\psi(\\tilde{u})$ then we say that $\\phi$ has a \\emph{coincidence} for the pair $i,j$. If $\\phi$ has a coincidence for every pair of letters $i,j \\in \\mathcal{A}$ then we say that $\\phi$ is \\emph{strongly coincident}.\n\\end{definition}\n\\begin{example}\nLet $\\phi \\colon 0 \\mapsto 10, 1 \\mapsto 0$ be the reversed Fibonacci substitution. Note that $\\phi^3(0) = (10)(0)(10)$ and $\\phi^3(1) = (01)(0)()$, where we have used brackets to partition the substituted words appropriately. So, $\\phi$ has a coincidence for the pair $0,1$ (with $\\psi(u) = \\psi(\\tilde{u}) = (1,1)$, $k=0$). As $\\mathcal{A} = \\{0,1\\}$, we have checked all distinct pairs of letters. It follows that the reversed Fibonacci substitution is strongly coincident.\n\\end{example}\nNote that, unlike in the above example, the ordinary Fibonacci substitution reaches a coincidence for the pair $0,1$ after just the first iteration. This is an artifact of the asymmetric definition for a substitution to have a coincidence.\n\nSolomyak has shown that if the subshift of a primitive substitution has pure discrete spectrum, then the substitution must be Pisot \\cite{S:pisot}. We are concerned with the converse of this statement in the case of irreducible Pisot substitutions and a close variant concerning strong coincidence\\footnote{Although the first of these conjectures is referred to just by the name \\emph{Pisot}, the conjecture is sometimes referred to as the \\emph{Pisot substitution conjecture} to distinguish it from conjectures in other areas adopting the same name.}.\n\\begin{conj}[Pisot conjecture]\nThe subshift of every irreducible Pisot substitution has pure discrete spectrum.\n\\end{conj}\n\n\\begin{conj}[Strong coincidence conjecture]\\label{scoin}\nEvery irreducible Pisot substitution is strongly coincident.\n\\end{conj}\n\nIt is not known, in general, if the subshift of every strongly coincident substitution has pure discrete spectrum. If this is the case, then the Pisot conjecture is equivalent to the strong coincidence conjecture.\n\nOther than in specific cases of single substitutions, there are relatively few families of substitutions for which we know that the Pisot conjecture holds:\nHollander and Solomyak have shown that for two-letter Pisot substitutions, the strong coincidence conjecture is equivalent to the Pisot conjecture \\cite{HS:pisot};\nBarge and Diamond had previously shown that the strong coincidence conjecture is true for two-letter substitutions and hence that the Pisot conjecture holds for all substitutions on two letters \\cite{BD:pisot-coincidence};\nBarge has shown that the Pisot conjecture holds for substitutions of $\\beta$-type \\cite{B:pisot-beta-subs}; Akiyama, G\\\"{a}hler and Lee have shown via an exhaustive search that the Pisot conjecture holds for all three-letter substitutions whose substitution matrices have trace at most 2 \\cite{AGL:pisot};\nBerth\\'{e}, Jolivet, and Siegel have shown that the Pisot conjecture holds for substitutions of Arnoux-Rauzy type \\cite{BJS:arnoux-rauzy};\nBarge has shown that the Pisot conjecture holds for all substitutions that are left-proper and right-bijective (or vice-versa) \\cite{B:proper-bijective};\nBarge, Bruin, Jones and Sadun have shown that a na\\\"{i}ve homological analogue of the Pisot conjecture admits counterexamples for all algebraic degrees, but that the coincidence rank conjecture (a variant of the homological Pisot conjecture relating the norm of the PF-eigenvalue and the \\emph{coincidence rank} of the substitution) holds for any substitution whose PF-eigenvalue has algebraic degree 1 \\cite{BBJS:homological-pisot}.\n\n\\subsection{Computational search for counterexamples}\nPerhaps the largest computer search for counterexamples to either conjecture in the literature appears in work of Akiyama, G\\\"{a}hler and Lee \\cite{AGL:pisot}, where all 446,683 irreducible Pisot substitutions on three letters whose substitution matrix has trace at most 2 were checked for counterexamples to the Pisot conjecture.  Instead, we choose to focus our search for counterexamples to the strong coincidence conjecture. We use the algorithms developed for Grout, as well as standard algorithms from linear algebra, to calculate eigenvalues and check for irreducibility over $\\mathbb{Z}$ \\cite{L:factorising}, in order to do a large scale search for counterexamples to the strong coincidence conjecture. For our search, we limit to substitutions over three or four letters, where irreducibility can be checked in a combinatorial fashion instead of full implementation of the irreducibility algorithms.\n\nWe would like to highlight a subtlety encountered while implementing the check that involved a deterministic check for when eigenvalues have modulus exactly 1 or are instead very close but strictly within or without the unit disc. Algorithms for finding eigenvalues are typically numeric and involve a converging sequence of approximate solutions to the true solution. In order for the process to halt in finite time, an acceptance threshold needs to be set. This is problematic when one of the conditions that needs to be checked is that an eigenvalue does not lie on the boundary of the unit disc. This problem was overcome by implementing an algorithm that was brought to our attention on Stack Exchange \\cite{R:stack-exchange}. Once it is determined that an eigenvalue does not lie on the boundary of the unit disc, the numerical process can be sure to satisfy a threshold around the approximate solutions which is either fully within or fully outside the unit disc, and so the algorithm halts with the correct evaluation of $|\\lambda| < 1$, $|\\lambda| = 1$, or $|\\lambda| > 1$.\n\nWe place an ordering on substitutions over the same alphabet (modulo the trivial equivalences given by reversing and letter permutations) via the lexicographical ordering of the words $(\\phi(1), \\phi(2) ,\\ldots, \\phi(l))$. For example, $(0,0,0) \\prec (0,0,1) \\prec (0,2,0) \\prec (00,0,0)$. We then methodically generate the substitutions in this list. Sample sizes were limited by allotted computation times; 24 hours for each run on a high performance computing cluster at the University of Leicester, UK.\n\\begin{res}[Three letters]\nThe first 321,425,442 three-letter substitutions were checked, of which 12,573,955 were irreducible Pisot.  All of these substitutions passed the strong coincidence check, and therefore no counterexample to the strong coincidence conjecture was found.  During these checks, the number of iterations of the substitution that it took for a strong coincidence to be found was recorded. This is summarized in Table \\ref{Tab:strong-coincidence}.\n\n\\begin{table}[H]\n\\centering\n\n\\begin{tabular}{c|c}\n\n\\textbf{Number of Iterations} & \\textbf{Number of Substitutions} \\\\ \\hline\n\n1                             & 1,921,144                        \\\\ \n2                             & 5,778,331                        \\\\ \n3                             & 3,777,324                        \\\\ \n4                             & 984,669                          \\\\\n5                             & 105,325                          \\\\\n6                             & 6,608                            \\\\\n7                             & 512                              \\\\\n8                             & 37                               \\\\\n9                             & 3                                \\\\\n10                            & 2                                \\\\\n11+                           & 0                                \\\\\n\\end{tabular}\n\\caption{Number of iterations to produce a strong coincidence.}\\label{Tab:strong-coincidence}\n\n\n\\end{table}\n\\end{res}\nThe substitutions that took 10 iterations to form their first coincidence for all pairs of letters were\n$$\n\\begin{array}{rlrlrl}\n0 \\mapsto & 2011   & 1 \\mapsto & 02 & 2 \\mapsto & 0, \\\\\n0 \\mapsto & 212101 & 1 \\mapsto & 0  & 2 \\mapsto & 1.\n\\end{array}\n$$\nThe result suggests a likelihood that there is no bound on the number of iterations necessary for a strongly coincident substitution to form its first coincidence for all pairs, although a proof of this statement is not known to us. Unfortunately, sampling the substitutions formed which produced a new `record' as the program ran did not provide any clear pattern that one might use to produce an infinite family of substitutions who each require a successively larger number of iterations.\n\\begin{res}[Four letters]\nThe first 200,399 four-letter substitutions were checked, of which 15,001 were irreducible Pisot. All of these substitutions passed the strong coincidence check, and therefore no counterexample to the strong coincidence conjecture was found.\n\\end{res}\n\n\\section{Supplementary resources}\nGrout is available to download for Windows and Mac OSX along with the supporting documentation at the following URL:\n\n\\vspace{3mm}\n\n\\centerline{\\url{https:\/\/github.com\/GroutGUI\/Grout}}\n\n\n\\section{Acknowledgments}\nThis research used the SPECTRE High Performance Computing Facility at the University of Leicester.\n\nThe authors would like to thank Fabien Durand for his helpful advice in piecing together a robust recognizability algorithm, and Etienne Pillin for helpful comments with regards to the coding.  We would also like to thank Alex Clark, Greg Maloney and Jamie Walton for helpful suggestions during the development of Grout and for testing early versions of the program, and Marcy Barge for suggestions and discussions relating to the section on Pisot substitutions.\n\n\n\\bibliographystyle{jis}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThis paper presents a Pareto analysis of time and fuel optimal trajectory planning for an autonomous ocean craft subject to direction-dependent ocean currents and flexible refueling constraints.\n\nThe study of unmanned surface vehicles (USV), specifically watercraft, spans several decades in the literature.  However, it was not until recently that USVs have become a viable option for diverse commercial, scientific, and military applications.  Relevant advances in controls and communications research are largely responsible for this development, enabling greater capabilities for autonomy and robust design.  Robust design of USVs is critical given the dynamic and vast ocean environments within which they operate.  For instance, in conventional military applications USVs are tasked with minesweeping missions, anti-submarine warfare, and surveying hazardous environments \\cite{Liu00}.\n\nModeling the dynamics of USVs is a challenging task because of the complex set of forces acting on the vehicle, and the significant geometric dependencies of these forces.  Model complexity varies drastically, with the most complex and comprehensive dynamical models giving direct consideration to ocean swells and motion in 6 degrees of freedom \\cite{Heins00}.  Other simple USV kinematics models utilize basic relations for surge, sway, and yaw movements \\cite{Wang00}.  However as always, the best modeling approach is subject to the demands and circumstances placed upon the vehicle.  There has been significant work to design path planning for vehicles with slow dynamics such as underwater gliders and other vehicles that predominantly rely on ocean current for motion \\cite{Yazdani00, Kim00}.  For these applications, it is critical that ocean current plays a strong role in the governing dynamics model of the vehicle.  For relevant estimation problems, ocean currents can also strongly impact both the quality of sensor measurements as well as potentially unmodeled disturbances \\cite{Horner00}.\n\nPast work has applied several tools from optimal control to address trajectory optimization problems which are immediately relevant to USVs. These optimization problems can take place over long-distance planning, or in over short distances to control events including docking maneuvers.  Due to the inherent non-linear nature of vehicle trajectory optimization, nonlinear programming (NLP) methods are frequently applied to solve these problems. One primary approach for time or energy efficient trajectory optimization has been implementing an initial trajectory based on global information and then updating this trajectory in real-time \\cite{Jones00}. Work by Mills et al. has used concepts from differential flatness and pseudospectral methods to solve an energy-aware trajectory optimization problem for unmanned aerial vehicle as an NLP \\cite{Mills00}. Similar research on unmanned underwater vehicles (UUVs) has explored the use of model predictive control (MPC). As demonstrated by Guerreiro et al., MPC is advantageous in the presence of disturbances such as ocean currents due to continuously updating the optimal path at each time step \\cite{Guerreiro00}.  MPC is, however, often employed on-line at substantially increased computational time and effort. To overcome the computational cost of methods such as MPC, some have employed quasi-optimal approaches \\cite{Yazdani00}. Obstacle avoidance also presents a unique challenge to trajectory optimization for USVs \\cite{Wang00, Svec00}. Relevant methods which address obstacle avoidance problems frequently utilize tracking control to avoid collisions during the routing process.  \n\nThe presence of ocean currents as direction dependent disturbances poses a significant effect on the optimal trajectory a USV may take.  Work by Hirsch et al.  has elegantly addressed this challenge \\cite{Dolinskaya00}. Ocean currents also can have a significant effect on the energy expenditure of a USV.  As a result, fuel-optimal strategies for USV trajectory optimization have been observed to exploit ocean currents to a similar degree.  Other research by Kim et al. has applied genetic optimization to explore the impact of environmental loads on the optimal trajectory taken by a USV \\cite{Kim01}.\n\n\nWhile time and fuel optimal USV trajectory optimization has been studied throughout the literature, an analysis of the tradeoff between the two objectives is relatively unexplored.  This is especially true in the case where refueling constraints must be considered.  This work seeks to address these open challenges by analyzing USV trajectory optimization with respect to fuel consumption and trip time when refueling constraints must be considered. First, we formulate a discrete-time optimal control problem conducive for a conventional dynamic programming solver.  Through Pareto optimization techniques, we explore the competing fuel and trip time tradeoff in detail.  Furthermore, we vary the initial fuel tank level of the USV to explore its impact on the refueling behavior of the vessel.  We also examine the effect of direction dependent ocean currents on the fuel consumption and overall routing of the USV.\n\nThe remainder of this paper is organized as follows.  Section II describes our formal problem statement, including the model we use to approximate the USV dynamics.  Section III details our reformulation of the problem in discrete time, which allows us to apply a dynamic programming solution approach.   Section IV illustrates our optimal trajectory results, including our Pareto analysis of the fuel and trip time tradeoff. Finally, our conclusion contextualizes our findings and their implications.\n\n\n\\section{Problem Formulation}\n\nThis sections describes our formulation of the USV trajectory optimization problem.  In the first subsection, we detail the dynamical model of the USV.  Then, we provide an explicit formulation of the optimal control problem statement.  Finally, we describe the ocean environment within which we optimize the vessel's trajectories subject to varying refueling requirements.\n\n\\subsection{USV Dynamical Model}\nTo solve the time and fuel optimal USV trajectory planning problem, we employ a relatively simple dynamical model of an ocean craft.  Namely, we assume that the vehicle is operating at steady-state, or that it operated on a timescale long enough such that we can neglect transient dynamics from propulsion and drag forces.  We do this so our representation of the dynamics can be addressed with a dynamic programming-based solver which possesses limitations regarding time and space complexity.  Our nonlinear 3rd-order dynamical model takes the following form:\n\\begin{multline} \\label{eqn::dyn1}\n    \\dot{x}(t) = \\begin{pmatrix}\n u_1(t)\\\\\nu_2(t)  \\\\\nu_3(t) - \\dot{m}_{f}(t)\n\\end{pmatrix} + \\begin{pmatrix}\n1\\\\1\\\\\n0\\\\\n\\end{pmatrix}*v_{cur}(x(t))\n\\end{multline}\nwhere $x(t) \\in {\\mathbb{R}}^3$ is the state vector at time $t$, $x_1(t)$ and $x_2(t)$ are the east-west and north-south positions of the vehicle, $x_3(t)$ is the current level of fuel in the vehicle's fuel tank, $u_1(t)$ and $u_2(t)$ are velocity inputs in the east-west and north-south directions, $u_3(t)$ is the input to the fuel tank level from refueling, $\\dot{m}_{fuel}(t)$ is the rate of fuel consumption, and $v_{cur}(x(t))$ is the effect of the ocean current on the trajectory of the craft, where ocean current direction and magnitude are functions of the current position $x(t)$. The refueling input takes the form:\n\\begin{equation}\\label{eqn::ncvx_con}\n    u_3(t) = \\left\\{ \\begin{array}{rcl}\nu_{3;max} & \\mbox{for} & {x},u \\in \\mathcal{P}\\\\ \n0 & \\mbox{for} & {x},u \\not\\in \\mathcal{P}\\\\\n\\end{array}\\right.\n\\end{equation}\nwhere $\\mathcal{P}$ is the set of states and inputs such that the vehicle is stationary at one of the discrete port locations.  We assume that rate of fuel consumption is directly proportional to power. Consider the simple relation $P = Fv$, where $P$ is power, $F$ is force, and $v$ is velocity.  If we assume drag forces take the form $F_{drag} = \\frac{1}{2} \\rho C_d A_f v^2$ and that $v = \\sqrt{(u_1(t))^2 + (u_2(t))^2}$, then $\\dot{m} \\propto{P}$ becomes:\n\\begin{equation}\n    \\begin{split}\n    \\dot{m}_f &= \\alpha F_{drag}v \\\\\n    &= \\alpha \\frac{1}{2} \\rho C_d A_f v^3 \\\\\n    &=\\frac{1}{2} \\alpha \\rho C_d A_f \\big{(}\\sqrt{(u_1(t))^2 + (u_2(t))^2}\\big{)}^3 \\\\\n    &=\\beta \\big{(}(u_1(t))^2 + (u_2(t))^2\\big{)}^{\\frac{3}{2}} \n\\end{split}\n\\end{equation}\nWe assume the ocean current does not affect the fuel consumption.  We base this assumption on the reasoning that the input velocity in our expression for $\\dot{m}_f$ is applied relative to the ocean current.\n\n\n\\subsection{Optimal Control Problem Statement}\nOur optimal control problem statement is as follows:\n\\begin{equation}\\label{eqn::ct1}\n    \\underset{u \\in {\\mathbb{R}}^3}\\min \\int_0^T\\big{[}(1-\\lambda)\\big{(}\\frac{\\dot{m}_{f}(t)}{\\dot{m}_{fuel;max}}\\big{)}+\\lambda t_{inc} \\big{]} dt \n\\end{equation}\n\\text{Subject to:}\n\\begin{align}\n\\dot{x}(t) &= \\begin{pmatrix}\n u_1(t)\\\\\nu_2(t)  \\\\\nu_3(t) - \\dot{m}_{f}(t)\n\\end{pmatrix} + \\begin{pmatrix}\n1\\\\1\\\\\n0\\\\\n\\end{pmatrix}*v_{cur}(x(t))\\\\\n\\dot{m}_{f}(t) &= \\beta \\big{(}(u_1(t))^2 + (u_2(t))^2\\big{)}^{\\frac{3}{2}} \\\\\nx_{min} &\\leq x(t) \\leq x_{max}\\\\\nu_{min} &\\leq u(t) \\leq u_{max}\\\\\nu_3(t) &= \\left\\{ \\begin{array}{rcl}\nu_{3;max} & \\mbox{for} & (x,u) \\in \\mathcal{P}\\\\ \n0 & \\mbox{for} & (x,u) \\not\\in \\mathcal{P}\\\\\n\\end{array}\\right.\\\\\nx(T) &\\in \\mathcal{X}_{terminal}\n\\end{align}\n\nwhere \n\\begin{equation}\\label{eqn::ctend}\n    t_{inc} = \\left\\{ \\begin{array}{rcl}\n1 & \\mbox{for} & {x} \\notin \\mathcal{X}_{terminal}\\\\ \n0 & \\mbox{for} & {x} \\in \\mathcal{X}_{terminal}\\\\\n\\end{array}\\right.,\n\\end{equation}\n$T$ is the final time, $\\lambda$ is the scalarization weight which controls the tradeoff between fuel consumption and trip time, and $\\mathcal{X}_{terminal}$ is the set of acceptable terminal states which only imposes conditions on the final x-y position of the vehicle.  We normalize the fuel consumption term in the cost function based on the fuel flow rate achieved when the velocity inputs are maximized.  An important constraint is that $x_{3;min} = 0$, meaning that the vehicle cannot run out of fuel at any point in its trip unless that point is at a refueling station or the final destination.  In Section III, we describe our discretization procedure which allows us to apply a standard dynamic programming algorithm to solve this optimal control problem.\n\n\n\n\\subsection{Ocean Environment}\nWe define the orientation of the time-invariant ocean currents based on arbitrary direction-dependent functions of both $x_1$ and $x_2$.  These nonconvex functions are given below:\n\n\\begin{equation}\n    v_{cur;e}(x_1,x_2) = \\gamma \\sin\\bigg(\\frac{2\\pi}{100}x_1\\bigg)-\\gamma \\sin\\bigg(\\frac{2\\pi}{100}x_2\\bigg)\n\\end{equation}\n\\begin{equation}\n    v_{cur;n}(x_2) = \\frac{\\gamma}{80}x_2 + \\frac{1}{9}\n\\end{equation}\nThe vector $v_{cur}(x(t)$ now takes the form:\n\\begin{equation}\n    v_{cur}(x(t)) = \\begin{pmatrix}\n v_{cur;e}(x_1(t),x_2(t))\\\\\nv_{cur;n}(x_2(t))  \\\\\n0\n\\end{pmatrix}\n\\end{equation}\nIn this case we specifically limit the mean current velocity to approximately $5 km\/hr$ with a maximum current velocity of approximately $9 km\/hr$, in order to match observed averages among typical ocean currents \\cite{NOAA00}. The overall state-space map of our problem, including the initial position and desired destination, is shown in Figure 1.  The vector field in Figure 1 describes the directions and relative magnitudes of the ocean currents at each position, and the refueling locations are marked with black squares.  We design this environment to create scenarios where the USV can exploit the ocean current speed and direction to either save fuel, or to reduce the overall travel time.\n\nTable 1 provides a description of each parameter in problem.  We tune the fuel consumption proportionality parameter $\\alpha$ such that our observations of fuel economy align with typical industry standards.\n\\begin{table}[h]\n\\caption{Model Parameters}\n\\label{table_example}\n\\begin{center}\n\\begin{tabular}{|c||c|c|}\n\\hline\nParameter & Description & Value\\\\\n\\hline\n$x_{1;min}$ & Minimum $x_1$ & 0 $[km]$\\\\\n\\hline\n$x_{1;max}$ & Maximum $x_1$ & 150 $[km]$\\\\\n\\hline\n$x_{2;min}$ & Minimum $x_2$ & 0 $[km]$\\\\\n\\hline\n$x_{2;max}$ & Maximum $x_2$ & 150 $[km]$\\\\\n\\hline\n$x_{3;min}$ & Minimum $x_3$ & 0 $[Gal]$\\\\\n\\hline\n$x_{3;max}$ & Maximum $x_3$ & 8 $[Gal]$\\\\\n\\hline\n$u_{1;min}$ & Minimum $u_1$ & -21.33 $[\\frac{km}{hr}]$\\\\\n\\hline\n$u_{1;max}$ & Maximum $u_1$ & 21.33 $[\\frac{km}{hr}]$\\\\\n\\hline\n$u_{2;min}$ & Minimum $u_2$ & -21.33 $[\\frac{km}{hr}]$\\\\\n\\hline\n$u_{2;max}$ & Maximum $u_2$ & 21.33 $[\\frac{km}{hr}]$\\\\\n\\hline\n$u_{3;min}$ & Minimum $u_3$ & 0 $[\\frac{Gal}{min}]$\\\\\n\\hline\n$u_{3;max}$ & Maximum $u_3$ & 0.533 $[\\frac{Gal}{min}]$\\\\\n\\hline\n$C_d$ & USV Drag Coefficient & 0.4 $[-]$\\\\\n\\hline\n$A_f$ & USV Frontal Area & 6 $[m^2]$\\\\\n\\hline\n$\\alpha$ & Power to Fuel Proportion & $1.2*10^{-7} [\\frac{s^2}{m^2}]$\\\\\n\\hline\n$\\gamma$ & Ocean Current Coefficient & 0.0556 [-]\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin{figure}\n      \\centering  \n      \\includegraphics[scale=0.57]{figures\/map2.pdf}\n      \\caption{Map of Ocean Space}\n      \\label{figurelabel}\n\\end{figure}\n\n\\section{Dynamic Programming Solver}\nThe nonlinearity of the dynamical system model as well as the presence of the refueling constraint (\\ref{eqn::ncvx_con}) make this a highly nonconvex mixed-integer program. Since our problem has a relatively low and tractable number of state and input variables, we elect to utilize a conventional dynamic programming approach to solve this problem across a finite time horizon.  To utilize a standard dynamic programming algorithm, we first discretize the continuous-time optimal control problem given by (\\ref{eqn::ct1}-\\ref{eqn::ctend}). Using a forward finite-difference method, the nonlinear system dynamics can be transformed into:\n\\begin{multline} \\label{eqn::dt1}\n    x(k+1) =  x(k) + \\begin{pmatrix}\n u_1(k) \\Delta t\\\\\nu_2(k) \\Delta t \\\\\nu_3(k) - \\dot{m}_{fuel}(k)\n\\end{pmatrix} \\\\+ \\begin{pmatrix}\n\\Delta t\\\\\n\\Delta  t\\\\\n0\\\\\n\\end{pmatrix}*v_{cur}(x(k))\n\\end{multline}\nwhere $k$ is the current time and $\\Delta t$ is the timestep length. The term $\\dot{m}_{fuel}(k) = \\dot{m}_f(t)* \\Delta t$. For a large enough timestep, we now assume the control input $u_3$ is sufficient to refuel the USV's fuel tank in a single timestep.\n\nUsing a dynamic programming based solution approach requires we divide the state space and input space into a series of uniform discretized meshes.  Table 2 provides the parameters which define this discretization.  These align with our maximum and minimum values for $x$ and $u$ listed in Table 1.\n\nNow, we can pose a discrete-time approximation of the continuous-time optimal control problem statement (\\ref{eqn::ct1}-\\ref{eqn::ctend}) as follows:\n\\begin{equation}\n    \\underset{u \\in {\\mathbb{R}}^3}\\min \\sum_{k=1}^N\\big{[}(1-\\lambda)\\big{(}\\frac{\\dot{m}_{fuel}(k)}{\\dot{m}_{fuel;max}}\\big{)}+\\lambda t_{inc} \\big{]}  \n\\end{equation}\n\\text{Subject to:}\n\\begin{align}\n&(\\ref{eqn::dt1})\\\\\n\\dot{m}_{fuel}(k) &= \\beta  \\big{(}(u_1(k))^2 + (u_2(k))^2\\big{)}^{\\frac{3}{2}} \\Delta t \\\\\nx_{min} &\\leq x(k) \\leq x_{max}\\\\\nu_{min} &\\leq u(k) \\leq u_{max}\\\\\nu_3(k) &= \\left\\{ \\begin{array}{rcl}\nu_{3;max}\\Delta t & \\mbox{for} & (x,u) \\in \\mathcal{P}\\\\ \n0 & \\mbox{for} & (x,u) \\not\\in \\mathcal{P}\\\\\n\\end{array}\\right.\\\\\nx(N) &\\in \\mathcal{X}_{terminal}\n\\end{align}\nwhere $N = \\frac{T}{\\Delta t}$ is the final timestep.  \n\n\\begin{table}[h]\n\\caption{Discretization Parameters}\n\\label{table_example}\n\\begin{center}  \n\\begin{tabular}{|c||c|}\n\\hline\nParameter & Value\\\\\n\\hline\n$\\Delta x_3$  &  0.1 $[Gal]$\\\\\n\\hline\n$\\Delta u_1$ & 10.67 $[\\frac{km}{hr}]$\\\\\n\\hline\n$\\Delta u_2$ & 10.67 $[\\frac{km}{hr}]$\\\\\n\\hline\n$\\Delta u_3$ & 8 $[\\frac{Gal}{\\Delta t}]$\\\\\n\\hline\n$\\Delta t$ & 15 $[min]$\\\\\n\\hline\n$N$ & 65 $[Steps]$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\nTo improve overall precision, we increase the relative coverage of our mesh by utilizing a Latin-hypercube sample to form the mesh of $x_1$ and $x_2$ points. Specifically, we generate a latin-hypercube sample ($n=7500$) within the bounds given by $x_{1;min}$, $x_{1;max}$, $x_{2;min}$, and $x_{2;max}$.  This overall discretization scheme necessitates implementing an interpolation rule when running the dynamic programming solver.  In this paper, we utilize a simple nearest-neighbor interpolation procedure.\n\n\n\n\n\n\n\\section{Results}\nThis section details our simulation results from our optimal trajectory planning.  The objective of this analysis is to explore the overall behavior of the USV when we vary several operating conditions of the optimal control problem.  These conditions include the initial fuel tank level and the scalarization parameter $\\lambda$ which governs the tradeoff between fuel economy and trip time.\n\n\\subsection{Optimal Path Planning}\nIn this subsection, we combine our discretized problem structure with a general dynamic programming algorithm to solve for fuel and time optimal USV trajectories. Figures 3 and 4 demonstrate a series of optimal trajectories as we vary the initial fuel tank level and the scalarization parameter $\\lambda$. \n\n\\begin{figure}\n      \\centering  \n      \\includegraphics[scale=0.55]{figures\/paretoF.pdf}\n      \\caption{Pareto Fronts}\n      \\label{figurelabel}\n\\end{figure}\n\\begin{figure*}\n      \\centering  \n      \\includegraphics[trim = 10mm 5mm 15mm 5mm, clip, width=\\textwidth]{figures\/traj_8_1.pdf}\n      \\caption{Comparison of optimal trajectories for initial fuel level of 8 gallons}\n      \\label{figurelabel}\n\\end{figure*}\n\\begin{figure*}\n      \\centering  \n      \\includegraphics[trim = 10mm 5mm 15mm 5mm, clip, width=\\textwidth]{figures\/traj_2_1.pdf}\n      \\caption{Comparison of optimal trajectories for initial fuel level of 2 gallons}\n      \\label{figurelabel}\n\\end{figure*}\n\nSeveral insights are clearly visible from these results.  First, in the case where $\\lambda=0$, the ocean current has much more visible effect on the trajectory the vehicle follows. The vehicle essentially allows the ocean current to do as much work as possible when fuel economy is the only consideration to the stage cost function. This is because the vehicle is defaulting to applying little to no input force whenever possible in order to save fuel.  This behavior can be seen in the leftmost portion of Figure 2, where the trajectory taken by the USV follows the ocean current vector field throughout a sizeable portion of its duration.  As $\\lambda$ increases, this drifting becomes less and less noticeable.  Specifically, as the value of $\\lambda$ increases the vehicle begins taking a more direct path to the desired destination.  This of course comes at the cost of additional fuel consumption, as the more direct trajectory necessitates neglecting and in some cases fighting directly against the direction of the ocean current.  As the value of $\\lambda \\rightarrow 1$, the vehicle will begin moving as fast as possible towards the destination as expected.  What is interesting is that this behavior requires the vehicle to in some cases refuel multiple times along its path.  In such cases, the vehicle defaults to roughly evenly spaced refueling ports which require the least detour from the most direct path.  \n\n\n\n\\subsection{Pareto Analysis}\nThis subsection details our comparative analysis between trip time and fuel consumption.  By varying the scalarization parameter $\\lambda$, we can adjust the degree to which our optimal controller weighs fuel economy compared to overall trip time.  The objective of this analysis is to explore the relationship between these two aspects of the USV trajectory, and to see if certain rules and trends can be exploited to yield more efficient behavior.  \n\nFor our Pareto analysis, we vary the weighting parameter in the objective function $\\lambda$ between zero and one in increments of 0.05.  Figure 2 shows the Pareto fronts obtained for four different initial fuel levels. We frequently find that some different values of $\\lambda$ will share the same resulting trajectory.\n\nThis series of Pareto fronts yields several meaningful insights.  First, it is almost always true that having a higher initial fuel level yields significant improvement to both overall trip time and overall fuel consumption simultaneously.  This could be due to several factors.  In the event the vehicle needs to refuel, it will likely have to travel outside the path leading directly towards the desired destination.  This is, of course, dependent on the specific locations of the refueling ports.  The refueling itself also takes time, momentarily stopping the vehicle from making progress. Another key insight we can take from these Pareto fronts is that regardless of the initial fuel level, optimizing with the bare minimum of consideration to fuel economy yields essentially the same overall trip time but at significantly reduced fuel consumption.  In each of the initial fuel level cases, changing $\\lambda$ from 1.0 to 0.95 had marginal change to trip time but decreased overall fuel consumption by 57.8\\%, 28.3\\%, 26.8\\%, and 21.1\\% (in increasing order of initial fuel level).\n\n\n\n\\section{Conclusion}\n\nThis paper presents a dynamic programming based analysis of multiobjective USV trajectory optimization.  Specifically, we explore the tradeoff between fuel consumption and overall trip time using Pareto optimization.  Our comparative analysis also examines how these competing objectives affect the trajectory taken by the vessel.  The principal novelty of our work comes from our implementation of refueling constraints, wherein the USV likely will need to stop to refuel on its way to its destination.  We augment this analysis with effects from direction-dependent ocean currents.\n\nOur paper yields several meaningful insights. First, we find it is almost universally better in terms of fuel consumption and trip time for the USV to start its trip with as much fuel in its tank as possible.  We also find that it is typically better to give some, even minimal, consideration to fuel consumption compared to only considering overall trip time.  When minimal consideration is given to fuel consumption, we find the USV takes trajectories with nearly identical trip times to the time-greedy case but at between 20-50\\% reduced fuel consumption.  Thus, this feature of a control algorithm can significantly extend the effective range of the USV. Furthermore, the relative effect of the ocean current on the shape of the optimal trajectory is considerable.  In general, the USV will avoid areas of high ocean current when the cost function heavily weights overall fuel economy.  This is true unless the direction of the ocean current aligns with the intended direction of the vessel.  In the same case, we frequently observe the USV being drawn seemingly off course by ocean currents.  However, the judicious use of control input allows the vessel to reach the destination with minimal overall fuel consumption.  \n\nAnother important note is that, in the event we wish to place the refueling ports in an optimal fashion, we expect higher average ocean currents will require a higher density of refueling ports. \n\n\n\n\\bibliographystyle{.\/IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro} The anomalous magnetic moment of the muon,\n$a_{\\mu}\\equiv\\left( g_{\\mu}-2\\right)\/2$ has recently been\ndetermined with a very high precision. A series of measurements with\npositive and negative muons by the E821 Collaboration at the\nBrookhaven National Laboratory resulted in the present average value\n\\cite{Bennett:2004pv}\n\\begin{eqnarray}\n\\label{g-2Res} a_{\\mu}^{\\rm exp}=(116\\,592\\,080 \\pm 60)\\times\n10^{-11}.\n\\label{exp}\n\\end{eqnarray}\nAt this level of a one half part per million precision, this quantity is\nsensitive to subtle effects predicted by the Standard Model (SM), including\nfive-loop quantum electrodynamics (QED), hadronic vacuum polarization and\nlight-by-light scattering, and electroweak interactions at two-loops.\n\nObviously, $g_\\mu-2$ may be affected also by interactions beyond\nthe SM.  For this reason, many researchers have analysed $g_\\mu-2$\nin various models of new physics and performed sophisticated\nstudies of the SM effects which are an irreducible background in\nthe search of unknown phenomena.  A summary of the SM prediction\ncan be found, for example, in the recent studies and reviews\n\\cite{Davier:2004gb,Nara,Melnikov:2003xd,Passera:2004bj}.\n\nAt present, the experimental result in Eq. (\\ref{exp}) exceeds the\nSM prediction by about 2.6 times the combined theoretical and\nexperimental uncertainty.  This tantalizing discrepancy may be due\nto an effect of a new interaction, perhaps supersymmetry. However,\nit is important to scrutinize the SM prediction before a conclusion\ncan be made. The present paper is devoted to a reevaluation of the\nlargest part of two-loop electroweak diagrams, namely those without\nclosed fermionic loops.\n\nElectroweak one-loop corrections to $g_\\mu-2$ were among the first\nquantum effects studied in the renormalizable electroweak theory, in\n1972  \\cite{fls72,Jackiw72,ACM72,Bars72,Bardeen72}.  They were found\nto increase $g_\\mu-2$ by\n\\begin{eqnarray}\n a_\\mu ^{EW}\n \\left(\n    {\\rm 1 loop}\n \\right)\n = \\frac{5}{3}\\frac{{G_\\mu m^2 }}{{8\\sqrt 2 \\pi ^2 }} \\times\n \\left[\n    {1 + \\frac{{\n    \\left(\n        {1 -4\\sin ^2 \\theta _w }\n    \\right)\n    ^2 }}{5} + O\n    \\left(\n        {\\frac{{m^2}}{{M^2 }}}\n    \\right)\n    }\n \\right]\n \\approx 195 \\times 10^{ - 11}\\,,\n\\label{oneLoop}\n\\end{eqnarray}\nwhere $G_{\\mu}=1.16639(1)\\times 10^{-5} GeV^{-2}$  and weak mixing\nangle $\\sin^2\\theta _w=1-M_W^2\/M_Z^2$.  The large mass parameter $M$\nrepresents the mass of a $W$, $Z$, or a Higgs boson.\n\nThis effect was too small to be measured in the then ongoing CERN\nexperiment.  The desire to observe it motivated in part the latest\nBrookhaven effort.\n\nTwo decades after the first electroweak result, Kukhto et al.\n\\cite{KKSS} found that an additional virtual photon may\nsignificantly modify the one-loop value in Eq. (\\ref{oneLoop}). They\nestimated that effect as a -22\\% reduction -- surprisingly large for\nhard virtual photons.  This and an analogous reduction of the rare\nmuon decay $\\mu\\to e\\gamma$ \\cite{Czarnecki:2001vf} are due to the\nlarge anomalous dimension of the dipole operators such as $\\bar \\mu\n\\sigma^{\\mu \\nu}\\mu F_{\\mu\\nu}$ and $\\bar e \\sigma^{\\mu \\nu}\\mu\nF_{\\mu\\nu}$.\n\nThat large effect found in a subset of two-loop electroweak\ncontributions was similar in size to the design precision of the\nE821 experiment.  It therefore appeared as warranted, even\nnecessary, to evaluate the complete two-loop result  -- a\ncalculation that had not been performed before in the electroweak\ntheory for any other observable.\n\nIn 1995 a complete set of 1678 electoweak two-loop diagrams for\n$g_\\mu-2$ was generated by a computer system \\cite{Kaneko:1995kr}.\nHowever, such large number of diagrams, many of which are divergent,\ncould not be numerically calculated, at least at that time.\nFortunately, it turned out that the majority of those diagrams are\nstrongly suppressed by extra factors of the\nmuon-to-intermediate-boson mass ratio and can be neglected.  Thus, a\ncomplete two-loop result was found \\cite{CKM96}.  The numerical\nvalue was found to be dominated by large logarithms arising due to\nphoton exchanges and was, somewhat accidentally, close to the value\nfound in its first studies \\cite{KKSS,KKS}.\n\nIn later work, the renormalization group equation was employed to\nestimate higher-order logarithmic effects, which however are not\nsizable \\cite{Czarnecki:2002nt,Degrassi:1998es}.\n\nThe result of \\cite{CKM96} was obtained in an approximation assuming\nthat the Higgs boson is sufficiently heavier than the $W$ and $Z$\nbosons such that $M_{W,Z}\/M_H$ can be used as an expansion\nparameter.  A recent study \\cite{Heinemeyer:2004yq} relaxed that\napproximation and found the value of two-loop contributions valid\nalso for a light Higgs (moreover, the two-loop supersymmetric\neffects were evaluated in that paper).\n\nIn this paper, we reanalyse the two-loop electroweak effects.  Our\ngoal is to check the previous results and study the Higgs mass\ndependence. We present our results in a semi-analytical form. That\nis, the dependence on the Higgs mass is presented analytically,\nwhile some parts of expressions that depend only on the well-known\nparticle masses are, for the sake of brevity, evaluated numerically.\nA method employed for obtaining analytical results is also described\nin some detail.\n\nIn this work we focus on diagrams with only bosonic loops (no closed\nfermion loops).  The fermionic subset of corrections was studied\nseparately \\cite{CKM95,Peris:1995bb,Knecht:2002hr}.  It was recently\nsubject of an interesting theoretical controversy which seems to be\nsettled now (see \\cite{Czarnecki:2002nt} for a thorough discussion\nand references).\n\n\n\nIn the next Section we briefly explain the asymptotic operation\nwhich is the main technical tool used to obtain the analytical\nresult for $g_\\mu-2$.  Section \\ref{SMcontribution} presents partial\nresults for various groups of contributing diagrams. First, we\ndivide all diagrams into five subsets of topologically equivalent\ndiagrams. Then we give analytical or semi-analytical result for each\ntopology. Finally we discuss the renormalization procedure and\nevaluate necessary counterterms, provide the final numerical result\nand compare it with the results of Ref.\n\\cite{CKM96,Heinemeyer:2004yq}.\n\n\n\\section{Techniques: Asymptotic operation and resulting integrals}\n\\label{techniques}\n\n\nThe method of asymptotic operation (see \\cite{Tkachev:1994gz} for a\nreview and references) has been an invaluable tool in numerous\nrecent studies of effects involving various energy scales. In the\npresent problem, the mass of the muon $m$ sets the soft scale, and\nthe masses of the $W$, $Z$, and Higgs bosons -- the hard scale:\n$M_{W,Z,H}\\gg m$.   As we explained in the Introduction, the\noriginal study \\cite{CKM96} assumed in addition $M_H \\gg M_{W,Z}$.\nHere, we will not assume any hierarchy between $M_{W,Z}$ and $M_H$\nat the price of certain complication of our results.\n\n\nWe would like to explain here the basic principles of asymptotic\noperation.  Instead of attempting a rigorous derivation or even a\nrigorous exposition, we will use a simple example to elucidate the\nmethod.\n\nIn this study, we are interested in two-loop Feynman graphs $G$,\nwhich have a soft-scale ($\\sim m$) external momentum and involve\ninternal lines with both soft-scale and  hard-scale ($\\sim M$)\nvirtual particles. The exact value of the two loop integrations is a\n(possibly very complicated) function of $m\/M$. However, for our\npurposes it is entirely sufficient to know only an expansion of that\nfunction up to $m^2\/M^2$.  The purpose of the asymptotic operation\nis to obtain the desired order of that expansion without having to\ncompute the whole function.\n\nFor our purposes, the action of the asymptotic operation on a\nFeynman graph $G$ may be described with the following formula,\n\\begin{eqnarray}\n\\label{GENas}\n {\\rm As} \\circ G =\n \\sum\\nolimits_{\n    {h }\n }\n \\left(\n    \n    {\\tau  \\circ h } }\n \\right)\n \\times\n \\left[\n    {G\\backslash\n    {h } }\n \\right].\n\\end{eqnarray}\nLet us first of all explain the notation.  $ {\\rm As} \\circ G$ is an\nexpansion of the exact value of $G$ in powers {\\em and logarithms}\nof $m\/M$.  $h$ are the subgraphs of $G$ in which all loop momenta\nare considered as hard.\n\nInstead of giving an exact definition of which $h$ are relevant,\nwe use as an example the diagram shown in Fig. \\ref{asympt}.\n\\begin{figure}[hbt]\n\\begin{tabular}{c@{\\hspace*{7mm}}c}\n    \\begin{tabular}{c}\n    \\\\\n        \\scalebox{0.9}\n        {\\includegraphics[width=.35\\textwidth]{1a.eps}}\n    \\\\\n        (a)\n    \\end{tabular}\n&\n    \\begin{tabular}{c@{\\hspace*{7mm}}c}\n        \\scalebox{0.6}\n        {\\includegraphics[width=.35\\textwidth]{1b.eps}}\n    &\n        \\scalebox{0.6}\n        {\\includegraphics[width=.35\\textwidth]{1c.eps}}\n    \\\\\n        (b)\n    &\n        (c)\n    \\\\\n    \\\\\n        \\scalebox{0.6}\n        {\\includegraphics[width=.35\\textwidth]{1d.eps}}\n    &\n        \\scalebox{0.6}\n        {\\includegraphics[width=.35\\textwidth]{1e.eps}}\n    \\\\\n        (d)\n    &\n        (e)\n    \\end{tabular}\n\\\\\n\\end{tabular}  \\\\\n\\caption{An example of applying the asymptotic operation to a graph $G$,\nshown in (a).  (b)-(e) show the full set of subgraphs, as described in the text.\nPropagators through which hard momenta flow are indicated with solid lines -- they constitute\na subgraph $h$.  Dashed lines contain only soft-scale momenta and masses -- they constitute $G\\backslash h$.}\n \\label{asympt}\n\\end{figure}\n In\nthat Figure, $G$ is shown in picture (a). Subgraphs $h$ are shown\nin (b)--(e), and range from a single heavy particle line to the\nwhole graph.   $\\tau$ is the Taylor expansion operator, expanding a\nsubgraph in all soft-scale parameters such as the light-particle\nmasses and external momenta of that subgraph (the latter include\nloop momenta of those loops that are not part of the subgraph).\nFinally, $ {G\\backslash {h } }$ denotes the graph $G$ with the\nsubgraph $h$ contracted into a point.  The physical intuition for\nthat concept is such that all masses and loop momenta in $\n{G\\backslash {h } }$ are soft; thus, from their point of view, the\nhard-scale subprocess  in $h$ occurs at a very short-distance\nscale and can be contracted to a point. In other words, the\nsubgraphs $h$ create effective vertices.\n\nWhat do we gain from this construction?  We completely separate soft\nand hard scales for the purposes of the integration.  Thus, the\nresulting integrals cannot contain both $m$ and $M$ -- they\nfactorize into a part that depends only on the soft scales and\nanother that depends on the hard ones.  No non-trivial functional\ndependence on $m\/M$ can arise and we obtain the expansion to a given\norder in $m\/M$ by taking a sufficient number of terms in the Taylor\nexpansion of the subgraphs.\n\nThe asymptotic operation greatly simplifies the types of integrals\nthat we need to evaluate, as shown in Fig. (\\ref{types}).\n\\begin{figure}[hbt]\n\\begin{tabular}{c@{\\hspace*{20mm}}c@{\\hspace*{20mm}}c}\n        {\\includegraphics[width=.35\\textwidth]{2a.eps}}\n&        \\scalebox{0.5}\n        {\\includegraphics[width=.35\\textwidth]{2b.eps}}\n &       \\scalebox{0.5}\n        {\\includegraphics[width=.45\\textwidth]{2c.eps}}\n        \\\\\n        (a) & (b)& (c)\n  \\end{tabular}  \\\\\n\\caption{Basic types of loop integrals needed for the evaluation of\ntwo-loop electroweak corrections to $g_\\mu-2$: (a) One-loop\nlight-mass on-shell integral multiplied by a one-loop  vacuum\nintegral with a large mass; (b) Two-loop vacuum integral (some\npropagators may be massless $(M=0)$); (c) Two-loop light-mass\non-shell integral.} \\label{types}\n\\end{figure}\n One-loop on-shell integrals and one-loop massive integrals,\npictured in Fig. (\\ref{types})(a), are trivial.\n\nTwo-loop integrals in Fig. (\\ref{types})(b), dependent on arbitrary\nmasses, were computed in Ref. \\cite{adt93,inlo15}. By means of\nintegration by parts identities  \\cite{Tkachov:1981wb} the initial\nintegral with arbitrary powers of propagators can be reduced to the\nintegral with all propagators in power one. For the case when all\nthree propagators are massive, we encounter only diagrams with two\ndifferent masses present. In this simplified case, the general\nresult for such integral, in the dimensional regularization, becomes\n\\begin{eqnarray}\n\\label{M3int}\n \\ {{I_{m m M} }} &\\equiv&\n \\frac{\\left(4\\pi\\right)^d}{\\Gamma^2(1+\\varepsilon)}\\int\\int\\frac{{\\rm d}^d p ~{\\rm d}^d\n q}{\\left(p^2+m^2\\right)\\left(q^2+m^2\\right)\\left(\\left(p+q\\right)^2+M^2\\right)}\n \\nonumber\\\\&= &\n \\frac{\\left(m^2\\right)^{1-2\\varepsilon}}{\\left(1-\\varepsilon\\right)\\left(1-2\\varepsilon\\right)}\n \\left[\n    -\\frac{1+2z}{\\varepsilon^2}+\\frac{4z \\ln 4z}{\\varepsilon}-2z \\ln^2 4z+2(1-z)\\Phi(z)\n \\right]\\,,\n\\end{eqnarray} where $d=4-2\\varepsilon$,\n $z=\\frac{M^2}{4m^2}$, and  the function\n$\\Phi$ is defined as\n\\begin{eqnarray}\n\\label{Phi1} \\Phi(z)= 4\n \\left(\n    \\frac{z}{1-z}\n \\right)\n ^{\\frac{1}{2}} {\\rm Cl_2}\n \\left[\n    2\\arcsin\n    \\left(\n        z^{\\frac{1}{2}}\n    \\right)\n \\right]\n ,\\hspace{10mm}\n {\\rm Cl_2}(\\theta)=-\\int_{0}^{\\theta}d\\theta \\ln|2\\sin(\\theta\/2)|\n \\,,\n \\end{eqnarray}\nfor $z<1$, and\n\\begin{eqnarray}\n\\label{Phi2} \\Phi(z)=\n \\left(\n    \\frac{z}{z-1}\n \\right)\n ^{\\frac{1}{2}}\n \\left\\{\n    -4{\\rm Li_2}(\\xi)+2 {\\rm ln}^2\\xi-{\\rm ln}^2 4z +\\frac{\\pi^2}{3}\n \\right\\}\n ,\\hspace{10mm}\n \\xi= \\frac{1-\n \\left(\n    \\frac{z-1}{z}\n \\right)\n ^\\frac{1}{2}}{2}\n    \\,,\n\\end{eqnarray} for $z>1$.\n\n For the case when one propagator is massless we introduce the function\n $I^{\\rm fin}$, which is  the finite\n part of the integral $I_{M_c M_a M_b } $ for the case of\n $M_c=0$; denoting $R=\\frac{M_b^2}{M_a^2}$ we have\n\\begin{eqnarray}\n\\label{M2int} \\ I^{\\rm fin}(R)=\n-{7\\over 2} (1+R)+3R\\ln R\n-(1-R){\\rm Li}_2(1-R) -{R\\over 2}\\ln^2 R.\n\\end{eqnarray}\nFinally, two-loop on-shell integrals shown in Fig. (\\ref{types})(c)\ncan be found, for example, in Ref. \\cite{bielefeld}.\n\nTo calculate the renormalization counter terms we shall also need\none-loop massive on-shell self-energy integrals shown in Fig.\n(\\ref{SE}).\n\\begin{figure}[!h]\n    \\centering\n    \\includegraphics[width=0.2\\textwidth]{3.eps}\n    \\caption{One-loop massive on-shell self energy integral}\n    \\label{SE}\n\\end{figure}\nThey are expressed as linear combinations of integrals of the type\n(we use $p^2=-M^2$):\n\\begin{eqnarray}\n\\label{SEint} \\ S^{a b} (M_1, M_2, M)= \\frac{\\Gamma\n(b-2+\\varepsilon)}{\\Gamma(b) \\Gamma(1+\\varepsilon)} \\int_0^1 {\\rm d}\nx \\frac{x^a}{\\left[(1-x)(M_1^2-xM^2)+x\nM_2^2\\right]^{(b-2+\\varepsilon)}}\\,.\n\\end{eqnarray}\nThese integrals can be represented as series in $\\varepsilon$ and\ncan be easily computed to the necessary order. We  introduce special\nfunctions $S_{\\rm fin}^{a b} (M_1, M_2, M)$ which represent the\nfinite parts of  (\\ref{SEint}). We shall need only the following\ncases,\n\\begin{eqnarray}\n\\label{S01} S_{\\rm fin}^{0 1}(M_1, M_2, M)&=& \\int_0^1{\\rm d}x\n\\left\\{ \\left[(1-x)\\left(M_1^2-xM^2\\right)+xM_2^2\\right] \\left[\\ln\n\\left((1-x)\\left(M_1^2-xM^2\\right)+xM_2^2\\right)-1\\right]\\right\\}\\,,\n\\nonumber \\\\\n\\label{S02} S_{\\rm fin}^{a 2}(M_1, M_2, M)&=&-\\int_0^1 {\\rm d}x\nx^{a}\\ln\\left[(1-x)\\left(M_1^2-xM^2\\right)+xM_2^2\\right]\n    \\,,\n\\end{eqnarray}\nwith $a$ taking one the values $0, 1$ or $2$.\n\n\n\n\\section{Bosonic electroweak two-loop contibutions in the SM}\n\\label{SMcontribution} In this section we apply the technique of\nasymptotic operation, as described in Sec.~\\ref{techniques}, to\nevaluate all two-loop bosonic contributions.  We divide the diagrams\ninto subsets of various topologies and mass assigments, and  present\ndetailed results for each subset.  We display the dependence on the\nHiggs mass in the analytical form. We also provide details of the\nrenormalization procedure.\n\nFollowing \\cite{CKM96}, we use the 't Hooft-Feynman non-linear gauge\n\\cite{Fujikawa:1973qs}. We choose it in such way that we\neliminate the vertex $\\gamma\nW^ \\pm G^ \\pm $. Since $g_\\mu-2$ involves an external photon in\nevery diagram, such choice greatly reduces the number of diagrams.\nThe set of diagrams we have to consider contains all two-loop\ndiagrams one can compose in $SM$ with the exclusion of pure QED\ndiagrams and diagrams with closed fermion loop; we drop all diagrams\nwith more than one scalar coupling to the muon line, since each such\ncoupling introduces an extra factor $\\frac{{m_\\mu }}{{M_W}}$. Taking\nadvantage of the mirror symmetry we reduce the number of diagrams\nthat must be individually calculated to 138; in addition, some\none-loop diagrams must be evaluated for the renormalization.\n\n\\subsection{Two-loop topologies and their evaluation}\n\nWe divide all two-loop diagrams into five topological types, as\nshown in Fig. (\\ref{TOPOS}).  Possible insertions of the external\nphoton vertex are indicated with a circle cross, and wavy lines\nstand for either scalar or vector bosons.  Each subset groups\ndiagrams with similar properties with respect to the asymptotic\noperation and thus can be calculated using the same algebraic code\n(we use FORM \\cite{form3} for most algebraic operations).\n \\normalsize\n \\linespread{1.0}\n\\begin{figure}[!h]\n    \\begin{tabular}{c}\n            \\begin{minipage}[c]{.45\\textwidth}\n                \\centering\n                \\begin{tabular}{|c|c|c|c|c|c|}\n                        \\hline\n                        ${\\rm Topology}$&$M_A$&$M_B$&$M_C$&$M_D$&$M_E$\n                        \\protect\\\\\n                        \\hline\n                        $1_A$&$M_Z$&$M_Z$&$m$&$m$&$m$\n                        \\protect\\\\\n                        \\hline\n                        $1_B$&$0$&$M_Z$&$m$&$m$&$m$\n                        \\protect\\\\\n                        \\hline\n                        $1_C$&$M_Z$&$M_W$&$m$&$0$&$0$\n                        \\protect\\\\\n                        \\hline\n                    \\end{tabular}\n            \\end{minipage}\n            \\hfill\n            \\begin{minipage}[c]{.5\\textwidth}\n                \\centering\n                \\scalebox{0.5}\n                {\\includegraphics{4a.eps}}\n            \\end{minipage}\n    \\\\\n    \\\\\n    \\\\\n        \\begin{minipage}[c]{.45\\textwidth}\n            \\begin{tabular}{|c|c|c|c|c|c|}\n                \\hline\n                ${\\rm Topology}$&$M_A$&$M_B$&$M_C$&$M_D$&$M_E$\n                \\protect\\\\\n                \\hline\n                $2_A$&$M_W$&$M_W$&$M_Z$&$0$&$m$\n                \\protect\\\\\n                \\hline\n                $2_B$&$M_W$&$M_W$&$M_H$&$0$&$m$\n                \\protect\\\\\n                \\hline\n                $2_C$&$M_Z$&$M_Z$&$M_H$&$0$&$0$\n                \\protect\\\\\n                \\hline\n                $2_D$&$M_W$&$M_Z$&$M_W$&$0$&$0$\n                \\protect\\\\\n                \\hline\n                $2_E$&$M_W$&$M_W$&$0$&$0$&$m$\n                \\protect\\\\\n                \\hline\n                $2_F$&$M_Z$&$M_H$&$M_Z$&$m$&$m$\n                \\protect\\\\\n                \\hline\n            \\end{tabular}\n        \\end{minipage}\n        \\hfill\n        \\begin{minipage}[c]{.5\\textwidth}\n            \\centering\n            \\scalebox{0.6}\n            {\\includegraphics{4b.eps}}\n        \\end{minipage}\n    \\\\\n    \\\\\n    \\\\\n        \\begin{minipage}[c]{.45\\textwidth}\n            \\begin{tabular}{|c|c|c|c|c|}\n                \\hline\n                ${\\rm Topology}$&$M_A$&$M_B$&$M_C$&$M_D$\n                \\protect\\\\\n                \\hline\n                $3_A$&$M_W$&$M_W$&$M_W$&$0$\n                \\protect\\\\\n                \\hline\n                $3_B$&$0$&$0$&$M_W$&$m$\n                \\protect\\\\\n                \\hline\n                $3_C$&$M_Z$&$M_Z\/0$&$M_W$&$m$\n                \\protect\\\\\n                \\hline\n                $3_D$&$M_W$&$M_W$&$M_Z$&$0$\n                \\protect\\\\\n                \\hline\n                $3_E$&$M_W$&$M_W$&$M_H$&$0$\n                \\protect\\\\\n                \\hline\n                $3_F$&$M_Z$&$M_Z$&$M_H$&$m$\n                \\protect\\\\\n                \\hline\n                $3_G$&$M_Z$&$M_Z$&$M_Z$&$m$\n                \\protect\\\\\n                \\hline\n            \\end    {tabular}\n        \\end{minipage}\n        \\hfill\n        \\begin{minipage}[c]{.5\\textwidth}\n            \\centering\n            \\scalebox{0.6}\n            {\\includegraphics{4c.eps}}\n        \\end{minipage}\n    \\\\\n    \\\\\n    \\\\\n        \\begin{minipage}[c]{.45\\textwidth}\n            \\begin{tabular}{|c|c|c|c|c|c|}\n                \\hline\n                ${\\rm Topology}$&$M_A$&$M_B$&$M_C$&$M_D$&$M_E$\n                \\protect\\\\\n                \\hline\n                $4_A$&$M_W$&$M_W$&$M_Z$&$M_W$&$0$\n                \\protect\\\\\n                \\hline\n                $4_B$&$M_W$&$M_W$&$M_H$&$M_W$&$0$\n                \\protect\\\\\n                \\hline\n                $4_C$&$M_Z$&$M_Z$&$M_H$&$M_Z$&$m$\n                \\protect\\\\\n                \\hline\n                $4_D$&$M_W$&$M_W$&$0$&$M_W$&$0$\n                \\protect\\\\\n                \\hline\n                $4_E$&$M_Z$&$M_Z$&$M_W$&$M_W$&$m$\n                \\protect\\\\\n                \\hline\n                $4_F$&$M_H$&$0$&$M_W$&$M_W$&$m$\n                \\protect\\\\\n                \\hline\n                $4_G$&$M_H$&$M_Z$&$M_W$&$M_W$&$m$\n                \\protect\\\\\n                \\hline\n                $4_J$&$0$&$0$&$M_W$&$M_W$&$m$\n                \\protect\\\\\n                \\hline\n            \\end{tabular}\n        \\end{minipage}\n        \\hfill\n        \\begin{minipage}[c]{.5\\textwidth}\n            \\centering\n            \\scalebox{0.6}\n            {\\includegraphics{4d.eps}}\n        \\end{minipage}\n    \\\\\n    \\\\\n    \\\\\n        \\begin{minipage}[c]{.45\\textwidth}\n            \\begin{tabular}{|c|c|c|c|c|c|}\n                \\hline\n                ${\\rm Topology}$&$M_A$&$M_B$&$M_C$&$M_D$&$M_E$\n                \\protect\\\\\n                \\hline\n                $5_A$&$M_Z$&$M_Z$&$m$&$m$&$m$\n                \\protect\\\\\n                \\hline\n                $5_B$&$M_W$&$M_W$&$0$&$m$&$0$\n                \\protect\\\\\n                \\hline\n                $5_C$&$0$&$M_W$&$m$&$0$&$m$\n                \\protect\\\\\n                \\hline\n                $5_D$&$M_Z$&$M_W$&$m$&$0$&$m$\n                \\protect\\\\\n                \\hline\n                $5_E$&$M_W$&$M_Z$&$0$&$0$&$0$\n                \\protect\\\\\n                \\hline\n                $5_F$&$0$&$M_Z$&$m$&$m$&$m$\n                \\protect\\\\\n                \\hline\n                $5_G$&$M_Z$&$0$&$m$&$m$&$m$\n                \\protect\\\\\n                \\hline\n            \\end{tabular}\n        \\end{minipage}\n        \\hfill\n        \\begin{minipage}[c]{.5\\textwidth}\n            \\centering\n            \\scalebox{0.6}\n            {\\includegraphics{4e.eps}}\n        \\end{minipage}\n\\end{tabular}\n\\caption{Diagram topologies and assignments of masses to their\nlines.} \\label{TOPOS}\n\\end{figure}\n\nApplying the general formula (\\ref{GENas}) for the asymptotic\noperation  and computing all resulting integrals, we obtain the\nfollowing results for the finite parts of various groups of\ndiagrams:\n \\normalsize\\linespread{1.0}\n\\begin{eqnarray}\n\\label{TsubHiggs}\n    \\ &&{\\rm {T1_{A,B,C}+T2_{A,D,E}+T3_{A,B,C,D,G}+T4_{A,D,E,J}+T5}}\n    \\nonumber \\\\\n    &=&\n    -\\frac{\\alpha ^2}{384 {\\rm c^2} {\\rm s^2} \\pi^2}  \\left\\{680 {\\rm c^4}-362 {\\rm c^2}-363+6\n     \\left(\n        \\left[  27-30 {\\rm c^2}     \\right]\n        \\ln m^2+    \\left[            54 {\\rm c^2}-56 {\\rm c^4}        \\right]\n        \\ln M_W^2\n     \\right.\n  \\right.\n\\nonumber\\\\&&\n \\left.\n    \\left.\n        +\n        \\left[\n            56 {\\rm c^4}-84 {\\rm c^2}+27\n        \\right]\n        \\ln M_Z^2\n     \\right)\n \\right\\}\n -\\alpha^2\\frac{m^2}{M_W^2}\n \\left(\n    4.6-0.197\\ln \\frac{m^2}{M_W^2}\n \\right)\n\\\\\n\\label{T2B} \\ {\\rm{T2_B}} &=& -\\frac{\\alpha ^2}{\n576\\pi^2}\\frac{m^2}{M_W^2}\\frac{1}{\\Delta^4\\left(\\Delta^2-1\\right)}\\frac{1}{{\\rm\ns^4}}\n  \\left\\{\n     6\\Delta^2\n     \\left(\n        \\left[\n            2\\Delta^6 - 6\\Delta^4+3\\Delta^2+1\n        \\right]\n        \\Phi\\left(\\frac{\\Delta^2}{4}\\right)\n     \\right.\n \\right.\n \\nonumber\\\\&&\n \\left.\n    \\left.\n        +\n        \\left[\n            4\\Delta^6 - 3\\Delta^2-4\n        \\right]\n        I^{\\rm fin}\\left(\\Delta^{-2}\\right)\n     \\right)\n    +\\pi^2\n    \\left[\n        3\\Delta^4+\\Delta^2-4\n    \\right]\n    + 84\\Delta^8+108\\Delta^6-111\\Delta^4-123\\Delta^2\n \\right.\n\\nonumber\\\\&&\n \\left.\n    -84+\n    \\ln \\Delta^2\n    \\left[\n        96\\Delta^6-78\\Delta^2-72\n    \\right]\n    -3\\ln^2 \\Delta^2\n    \\left[\n        4\\Delta^8-8\\Delta^6-3\\Delta^4+2\\Delta^2+8\n    \\right]\n \\right\\}\n \\\\\n\\label{T2C}\\ {\\rm{T2_C}} &=& -\\frac{\\alpha ^2}{\n1152\\pi^2}\\frac{m^2}{M_W^2}\\frac{1}{\\Lambda^4\\left(\\Lambda^2-1\\right)}\\frac{\\left(8{\\rm\nc^4}-12{\\rm c^2}+5\\right)}{{\\rm c^2}{\\rm s^4}}\n  \\left\\{\n     3\\Lambda^2\n     \\left(\n        \\left[\n            \\Lambda^6-3\\Lambda^4-6\\Lambda^2+8\n        \\right]\n        \\Phi\\left(\\frac{\\Lambda^2}{4}\\right)\n    \\right.\n  \\right.\n\\nonumber\\\\&&\n  \\left.\n    \\left.\n        +\n        2\\left[\n            \\Lambda^6- 3\\Lambda^2-4\n        \\right]\n        I^{\\rm fin}\\left(\\Lambda^{-2}\\right)\n     \\right)\n      +\\pi^2\n    \\left[\n        3\\Lambda^4+\\Lambda^2-4\n    \\right]\n     +3\n    \\left[\n        7\\Lambda^8+9\\Lambda^6-31\\Lambda^4-41\\Lambda^2-28\n    \\right]\n \\right.\n\\nonumber\\\\&&\n \\left.\n    +\\ln \\Lambda^2\n    \\left[\n        24\\Lambda^6+18\\Lambda^4-78\\Lambda^2-72\n    \\right]\n    -3\\ln^2 \\Lambda^2\n    \\left[\n        \\Lambda^8-2\\Lambda^6-3\\Lambda^4+2\\Lambda^2+8\n    \\right]\n \\right\\}\n\\\\\n\\label{T2F}\\ {\\rm{T2_F}} &=& \\frac{\\alpha^2}{192\\pi^2}\\frac{\nm^2}{M_W^2}\\frac{\\Lambda^2}{\\left(\\Lambda^2-1\\right)}\\frac{\\left({\\rm\nc^2}-{\\rm s^2}\\right)}{{\\rm {c^2 s^2}}}\n \\left\\{\n     3\\left[\n        \\Lambda^6-7\\Lambda^4+14\\Lambda^2-8\n     \\right]\n     \\Phi\\left(\\frac{\\Lambda^2}{4}\\right)\n  \\right.\n\\nonumber\\\\&&\n \\left.\n     +\n     12\\Lambda^2\\left[\n        \\Lambda^4-4\\Lambda^2+1\n     \\right]\n     I^{\\rm fin}\\left(\\Lambda^{-2}\\right)\n    +\\pi^2\\Lambda^2\n    \\left[\n        \\Lambda^4-5\\Lambda^2+4\n    \\right]\n    +6\n    \\left[\n        7\\Lambda^6-21\\Lambda^4-20\\Lambda^2+6\n    \\right]\n \\right.\n\\nonumber\\\\&&\n \\left.\n    +12\\ln \\Lambda^2\n    \\left[\n        3\\Lambda^4-11\\Lambda^2+2\n    \\right]\n    -3\\ln^2 \\Lambda^2\n    \\left[\n        \\Lambda^6-7\\Lambda^4+14\\Lambda^2-4\n    \\right]\n \\right\\}\n\\\\\n\\label{T3}\\ {\\rm{T3_{E,F}}} &=& -\\frac{\\alpha\n^2}{2304\\pi^2}\\frac{m^2}{M_W^2}\\Delta^2\\frac{1}{{\\rm s^4}}\n  \\left\\{\n    79-336{\\rm c^{2}}+224{\\rm c^{4}}\n    +72\\ln {m^2}\n    \\left[\n        1-3{\\rm c^{2}} +2{\\rm c^{4}}\n    \\right]\n \\right.\n\\nonumber\\\\&&\n \\left.\n    +6\\ln {M_H^2}\n    \\left[\n        7-12{\\rm c^{2}}+8{\\rm c^{4}}\n    \\right]\n    +30\\ln {M_W^2}-12\\ln {M_Z^2}\n    \\left[\n        5-12{\\rm c^{2}}+8{\\rm c^{4}}\n    \\right]\n \\right\\}\n\\\\\n\\label{T4B}\\ {\\rm{T4_B}} &=&\\frac{\\alpha ^2}{\n2304\\pi^2}\\frac{m^2}{M_W^2}\\frac{1}{\\left(\\Delta^2-1\\right)}\\frac{1}{{\\rm\ns^4}}\n  \\left\\{\n     -18\n     \\left[\n        \\Delta^8-6\\Delta^6+7\\Delta^4+4\\Delta^2-6\n     \\right]\n     \\Phi\\left(\\frac{\\Delta^2}{4}\\right)\n     - 36\\Delta^4\n     \\left[\n        \\Delta^4\n     \\right.\n  \\right.\n\\nonumber\\\\&&\n \\left.\n    \\left.\n        -3\\Delta^2-2\n     \\right]\n     I^{\\rm fin}\\left(\\Delta^{-2}\\right)\n    -126\\Delta^8+216\\Delta^6+635\\Delta^4+650\\Delta^2-367\n    -12\\Delta^2\\ln \\Delta^2\n    \\left[\n        12\\Delta^4\n    \\right.\n  \\right.\n\\nonumber\\\\&&\n \\left.\n    \\left.\n        -34\\Delta^2-17\n    \\right]\n    +18\\ln^2 \\Delta^2\n    \\left[\n        \\Delta^8-5\\Delta^6+4\\Delta^4+4\\Delta^2\n    \\right]\n    +60\\ln {M_W^2}\n    \\left[\n        \\Delta^4-4\\Delta^2+3\n    \\right]\n \\right\\}\n\\\\\n\\label{T4C}\\ {\\rm{T4_C}}&=& \\frac{\\alpha\n^2}{576\\pi^2}\\frac{m^2}{M_W^2}\\frac{1}{\\left(\\Lambda^2-1\\right)}\\frac{1}{{\\rm\ns^4}{\\rm c^{2}}} \\times\n  \\left\\{\n    \\left(\n        \\Lambda^2 - 1\n    \\right)\n    \\left(\n        38\\Lambda^2-125+\\rm{c^2}(2\\rm{c^2}-3)(25\\Lambda^2-91)\n    \\right)\n \\right.\n\\nonumber\\\\&&\n \\left.\n    +18\\ln {m^2}\n    \\left(\n        1-3{\\rm c^2}+2{\\rm c^4}\n    \\right)\n    \\left(\n        \\Lambda^4-4\\Lambda^2+3\n    \\right)\n    +3\\Lambda^2\\ln {M_H^2}\n    \\left(\n        \\Lambda^2-4\n    \\right)\n    \\left(\n        1-6{\\rm c^2}+4{\\rm c^4}\n    \\right)\n \\right.\n\\nonumber\\\\&&\n \\left.\n    -3\\ln {M_Z^2}\n    \\left[\n        12-20\\Lambda^2+5\\Lambda^4+\n        \\left(\n            \\rm{4c^4-6c^2}\n        \\right)\n        \\left(\n            3-8\\Lambda^2+2\\Lambda^4\n        \\right)\n    \\right]\n \\right\\}\n\\\\\n\\label{T4F}\\ {\\rm{T4_F}} &=& -\\frac{\\alpha^2}{\n16\\pi^2}\\frac{m^2}{M_H^2}\\frac{1}{\\Delta^2}\\frac{1}{{\\rm s^2}}\n  \\left\\{\n     \\left[\n        6-7\\Delta^2\n     \\right]\n     \\Phi\\left(\\frac{\\Delta^2}{4}\\right)+\n     \\Delta^2\\left(\n        6+\\Delta^2\n     \\right)\n     \\left(\n        \\ln \\Delta^2-2\n     \\right)\n \\right\\}\n\\\\\n\\label{T4G}\n \\ {\\rm T4_G}&=& \\frac{\\alpha^2}{\n128\\pi^2}\\frac{m^2}{M_H^2}\\frac{\\Lambda^2}{\\left(\\Lambda^2-1\\right)}\\frac{\\left(4{\\rm\nc^2}-3\\right)}{{\\rm s^4}}\n  \\left\\{\n     \\left[\n        14-3\\frac{\\Delta^2}{\\Lambda^2}+2\\frac{\\left(1-6{\\rm\n        c^2}\\right)}{\\Lambda^2}\n     \\right]\n     \\Phi\\left(\\frac{\\Delta^2}{4}\\right)\n \\right.\n\\nonumber\\\\&&\n \\left.\n     +\n     \\left(\n        2\\Lambda^2+12{\\rm\n        c^2}-\\Delta^2-18+\\frac{4\\Delta^2}{\\Lambda^2}\n     \\right)\n     \\Phi\\left(\\frac{1}{4{\\rm c^2}}\\right)\n     -\\ln \\Lambda^2\n     \\left(\n        2\\Delta^2+12-\\frac{\\Delta^4}{\\Lambda^2}-2\\frac{\\Delta^2}{\\Lambda^2}\n     \\right)\n \\right \\}\n \\,,\n\\end{eqnarray}\nIn the above expressions we have dropped the divergences (which\ncancel in the final sum with the counterterms).  We also express all\nmasses in units of $\\mu=1$ GeV.  The floating point coefficients in\n(\\ref{TsubHiggs}) were rounded to provide the precision of\n$10^{-11}$ of the final result.  They were obtained using\n$M_W = 80.423$ GeV and $M_Z = 91.1876$ GeV.  The functions $\\Phi(x)$ and $I^{\\rm\nfin}(x)$ are defined by means of (\\ref{Phi1}, \\ref{Phi2} and\n\\ref{M2int}) respectively.  We also use the notation $\\Delta \\equiv\n\\frac{M_H}{M_W}$,  $\\Lambda \\equiv \\frac{M_H}{M_Z}$,  ${\\rm c}\\equiv\n\\frac{M_W}{M_Z}$, and ${\\rm s} \\equiv \\sqrt{1-{\\rm c^2}}$.\n\nThe results for topologies $T1_A$, $ T1_B$, $ T1_C$, $ T2_A$, $\nT2_D$, $ T2_E$, $ T3_A$, $ T3_B$, $ T3_C$, $ T3_D$, $ T3_G$, $\nT4_A$, $ T4_D$, $ T4_E$, $ T4_J$, $ T5$, were collected into a\nsingle formula because the corresponding diagrams do not contain\nHiggs propagators and thus can be evaluated without any\nassumptions about $M_H$.\n\n\\subsection{Renormalization counterterms}\n\\label{renormalization}\n\nConceptually, the renormalization procedure for our calculation is\nidentical to the one we used in Ref. \\cite{CKM96}. The only\ntechnical difference is that in our calculation we computed one-loop\nmassive self-energy diagrams precisely, rather than in an expansion\nto the order of $ {\\sin^6 \\theta _W }$. In this\nsection we briefly review our renormalization procedure and list\nspecific expressions for the $W$ and $Z$ self-energies and for other\nrenormalization constants.\n\nThe counter terms for the two-loop $EW$ corrections   are generated\nby renormalizing vertices and propagators in one-loop diagrams. In\nthe non-linear 't Hooft-Feynman gauge there are just three such\ndiagrams: the Schwinger's QED diagram with a photon loop,  its\nanalog with the photon replaced by a $Z$ boson, and a diagram with\ntwo $W$ bosons.\n\nThe result for Schwinger's diagram in dimensional regularization\nis given by\n\\begin{eqnarray}\n\\label{Schwinger1} a^{\\rm Schw}=\\frac{\\alpha}{2\\pi}\n \\left[\n    1+\\varepsilon\n    \\left(\n        4-\\ln m^2\n    \\right)\n \\right]\\,.\n\\end{eqnarray}For the calculation of the counterterms we also need the\ncontribution of that diagram with one internal muon propagator\nsquared. It is\n\\begin{eqnarray}\n\\label{Schwinger2} a^{\\rm Schw}_2=\\frac{i \\alpha}{2\\pi m}\n \\left[\n    1+\\varepsilon\n    \\left(\n        1-\\ln m^2\n    \\right)\n \\right]\\,.\n\\end{eqnarray}Using  (\\ref{Schwinger1}) and  (\\ref{Schwinger2}),\nand bearing in mind that since for the calculation of the $EW$\ncorrections we droped the two-loop photonic corrections to the\nSchwinger diagram, and thus have to subtract the photonic\ncontributions from muon mass and wave function renormalization\nconstants as well, we find the counterterm to the $a_f$ at the\ntwo-loop level,\n\\begin{eqnarray}\n\\label{SchwingerCT} a^{\\rm Schw}_{\\rm CT}&=&a^{\\rm Schw}\n \\left[\n     \\frac{1}{2}\n     \\left(\n        \\delta Z^R_{\\mu}+\\delta Z^L_{\\mu}-2\\delta Z^{\\gamma}_{\\mu}\n     \\right)\n     +\\Sigma^{'AA}\n     \\left(\n        0\n     \\right)\n \\right]\n -2ia^{\\rm Schw}_2\n \\left(\n    \\delta m-\\delta m^{\\gamma}\n \\right)\n \\,.\n\\end{eqnarray}\n\nOne-loop contribution of the $Z$-loop diagram  is\n\\begin{eqnarray}\n\\label{Zloop1}\n &&a^Z  = \\frac{{m^2 }}{{M_Z^2 }}\\frac{\\alpha }{{4\\pi }}\n \\left(\n    {g_V^2 V + g_A^2 A}\n \\right)\n \\nonumber\\\\&&\n V \\equiv \\frac{1}{3} + \\varepsilon\n \\left(\n    { - \\ln m^2  + \\frac{2}{3}\\ln M_Z^2  - \\frac{{11}}{9}}\n \\right)\n \\nonumber\\\\&&\n A \\equiv  - \\frac{5}{3} + \\varepsilon\n \\left(\n    { \\ln m^2  + \\frac{2}{3}\\ln M_Z^2  - \\frac{{11}}{9}}\n \\right)\\,.\n\\end{eqnarray}\nThe corresponding diagram with one muon propagator squared gives\n\\begin{eqnarray}\n\\label{Zloop2}\n a_2^Z  = \\frac{{im^2 }}{{M_Z^2 }}\\frac{\\alpha }{{4\\pi }}\n \\left(\n    {g_V^2  - g_A^2 }\n \\right)\n \\left[\n    \\frac{1}{2} - \\varepsilon\n    \\left(\n        {\\frac{1}{2}\\ln m^2  + 1}\n    \\right)\n \\right]\n\\end{eqnarray}Combining expressions  (\\ref{Zloop1}) and\n (\\ref{Zloop2}) we obtain the counterterm generated by the $Z$-loop\ndiagram,\n \\normalsize\\linespread{1.0}\n\\begin{eqnarray}\n\\label{ZloopCT} a_{CT}^Z  &=&  - 2ia_2^Z \\delta m  +\n\\frac{{m^2}}{{M_Z^2 }}\\frac{\\alpha }{{4\\pi }} \\left[\n    Vg_V^2\n    \\left(\n        {\\frac{{\\delta Z_{\\mu}^R  + \\delta Z_{\\mu}^L }}{2} - \\frac{{\\delta\n        M_Z^2 }}{{M_Z^2 }} + 2\\frac{{\\delta g_V }}{{g_V }}}\n    \\right)\n    +Ag_A^2\n    \\left(\n        \\frac{{\\delta Z_{\\mu}^R  + \\delta Z_{\\mu}^L }}{2}\n    \\right.\n\\right.\n\\nonumber\\\\&&\n\\left.\n    \\left.\n        -\\frac{{\\delta M_Z^2 }}{{M_Z^2 }} + 2\\frac{{\\delta g_A }}{{g_A }}\n    \\right)\n    + g_V g_A \\frac{{V + A}}{2}\n    \\left(\n        {\\delta Z_{\\mu}^R  - \\delta Z_{\\mu}^L }\n    \\right)\n    + \\frac{2}{3}\\varepsilon\n    \\left(\n        {g_V^2  + g_A^2 }\n    \\right)\n    \\frac{{\\delta M_Z^2 }}{{M_Z^2 }}\n\\right]\n\\end{eqnarray}\n\nFinally, the one-loop contribution of the diagram with two $W$\nbosons  is\n\\begin{eqnarray}\n \\label{WW} a^W  = \\frac{{m^2 }}{{s^2 M_W^2}}\\frac{\\alpha }{{4\\pi }} \\left[\n    {\\frac{5}{6} + \\varepsilon\n    \\left(\n        { - \\frac{5}{6}\\ln M_W^2  + \\frac{{19}}{{36}}}\n    \\right)}\n\\right]\\,.\n\\end{eqnarray} Using this expression we obtain a\nrelatively simple counter term because there is no internal muon\nline and there are only left-handed couplings,\n\\begin{eqnarray}\n\\label{WWct} a_{CT}^W  = a^W\n \\left(\n    {2\\delta Z_e  - 2\\frac{{\\delta s }}{{s }} + \\delta Z_{\\mu}^L  -  \\frac{{\\delta M_W^2 }}{{M_W^2 }}}\n \\right)\n -\\frac{5}{6}\\frac{{m^2 }}{{s^2 M_W^2 }}\\frac{\\alpha}{{4\\pi }}\\varepsilon \\frac{{\\delta M_W^2 }}{{M_W^2\n }}\\,.\n\\end{eqnarray}\nThe constants appearing in the expressions\n (\\ref{Schwinger1})--(\\ref{WWct}) are defined as  follows:\n\\begin{eqnarray}\n\\label{gVgA} g_V  &=& \\frac{1}{{s c }} \\left(\n    { -\\frac{1}{2} + 2 s^2 }\n\\right) , \\qquad g_A  = \\frac{1}{{2 s c }}\\,, \\nonumber \\\\\n\\label{dgVdgAdsw} \\delta g_A & =&  - \\frac{1}{{2 s c }}\n \\left[\n    {\\delta Z_e  +\n    \\left(\n        {\\frac{{s^2}}{{c^2 }} - 1}\n    \\right)\n    \\frac{{\\delta s }}{{s}}}\n \\right],\n\\nonumber\\\\\n \\delta g_V & =&  - \\delta g_A  + 2 \\frac{{s}}{{c }}\n \\left(\n    {\\delta Z_e  + \\frac{1}{{c^2 }}\\frac{{\\delta s}}{{s }}}\n \\right),\n\\nonumber\\\\\n \\delta s &=&  - \\frac{{c^2}}{{2 s}}\n \\left(\n    {\\frac{{\\delta M_Z^2 }}{{M_Z^2 }}- \\frac{{\\delta M_W^2 }}{{M_W^2 }}}\n \\right)\n \\,,\n\\nonumber \\\\\n\\label{dZe} \\delta Z_e  &=& \\frac{{\\Sigma ^{'AA}\n \\left(\n    0\n \\right)\n }}{2},\\hspace{5mm}\\Sigma ^{'AA}\n \\left(\n    0\n \\right)\n = \\frac{\\alpha }{{4\\pi}}\n \\left(\n    { - \\frac{7}{\\varepsilon } + 7\\ln M_W^2  - \\frac{2}{3}}\n \\right)\\,,\n\\end{eqnarray}\nand $\\delta m , \\delta Z_{\\mu}^{L,R} $ are the muon mass and wave\nfunction renormalization defined as\n\\begin{eqnarray}\n\\label{dm} \\delta m &=&\\delta m^{\\gamma}+\\delta m^Z+\\delta\nm^W\\,,\\hspace{5mm} \\delta m^{\\gamma}= m\\frac{\\alpha}{4\\pi}\n \\left(\n    -4-\\frac{3}{\\varepsilon}+3\\ln m^2\n \\right)\\,,\n \\nonumber\\\\\n\\delta m^Z\n &=& \\frac{\\alpha }{{4\\pi}}\\frac{m}{s^2}\n \\left[\n    { -\\frac{1}{16}+ \\frac{1}{8\\varepsilon } -\\frac{\\ln M_Z^2}{8} - \\frac{1}{12}\\frac{m^2}{M^2_Z}}\n \\right]\n  \\nonumber\\\\&&\n +\\frac{\\alpha }{{4\\pi}}\\frac{m}{c^2}\n \\left[\n    -\\frac{21}{16}- \\frac{11}{8\\varepsilon } +\\frac{11 \\ln M_Z^2}{8} + \\frac{m^2}{M^2_Z}\n    \\left(\n        -\\frac{17}{12}+2\\ln\\frac{M_Z^2}{m^2}\n    \\right)\n \\right]\n \\nonumber\\\\&&\n +m\\frac{\\alpha }{{4\\pi}}\n \\left[\n    \\frac{5}{2}+ \\frac{3}{\\varepsilon } -3 \\ln M_Z^2+\n    \\frac{m^2}{M^2_Z}\n    \\left(\n        \\frac{8}{3}-4\\ln\\frac{M_Z^2}{m^2}\n    \\right)\n \\right]\\,,\n \\nonumber\\\\\n \\delta m^W\n &=& \\frac{\\alpha }{{4\\pi}}\\frac{m}{s^2}\n \\left[\n    { -\\frac{1}{8}+ \\frac{1}{4\\varepsilon } -\\frac{\\ln M_W^2}{4} + \\frac{1}{12}\\frac{m^2}{M^2_W}}\n \\right]\\,,\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\label{dZ} \\delta Z_{\\mu}^L &=& \\delta Z_{\\mu}^L ({\\gamma})+\\delta\nZ_{\\mu}^L(Z)+\\delta Z_{\\mu}^L(W)\\,,\n \\nonumber\\\\\n \\delta Z_{\\mu}^L(\\gamma)&=&\\delta Z_{\\mu}^R(\\gamma)=\\frac{\\alpha}{4\\pi}\n \\left(\n    -4-\\frac{3}{\\varepsilon}+3\\ln m^2\n \\right)\\,,\n \\nonumber\\\\\n \\delta Z_{\\mu}^L(Z)&=&\\frac{\\alpha }{{4\\pi}}\\frac{1}{16 s^2 c^2}\n \\left[\n    (2-4s^2)^2\n    \\left(\n        \\frac{1}{2}- \\frac{1}{\\varepsilon } +\\ln M_Z^2-\\frac{2 m^2}{3 M^2_Z}\n    \\right)\n \\right.\n \\nonumber\\\\ &&\\qquad\\qquad\\qquad\n \\left.\n    -\\frac{2 m^2}{3 M^2_Z}\n    \\left(\n        7-5(1-4 s^2)^2\n    \\right)\n \\right]\\,,\n\\nonumber\\\\\n \\delta Z_{\\mu}^L(W)&=&\\frac{\\alpha}{{4\\pi}}\\frac{1}{ s^2}\n \\left[\n    \\frac{1}{4}- \\frac{1}{2 \\varepsilon } +\\frac{\\ln M_W^2}{2}-\\frac{m^2}{3 M^2_W}\n \\right]\\,,\n\\nonumber \\\\\n\\delta Z_{\\mu}^R &=& \\delta Z_{\\mu}^R ({\\gamma})+\\delta\nZ_{\\mu}^R(Z)\\,,\n \\nonumber\\\\\n \\delta Z_{\\mu}^R(Z)&=&\\frac{\\alpha }{{4\\pi}}\\frac{1}{16 s^2 c^2}\n \\left[\n    16s^4\n    \\left(\n        \\frac{1}{2}- \\frac{1}{\\varepsilon } +\\ln M_Z^2-\\frac{2 m^2}{3 M^2_Z}\n    \\right)\n    -\\frac{2 m^2}{3 M^2_Z}\n    \\left(\n        7-5(1-4 s^2)^2\n    \\right)\n \\right]\\,,\n\\end{eqnarray}\nand $\\delta M^2_W$ and $\\delta M_Z^2$ are mass renormalization\nconstants.\n\nThe expressions  (\\ref{SchwingerCT}, \\ref{ZloopCT}, \\ref{WWct})\nprovide all counter terms necessary to renormalize two-loop\nbosonic corrections to $a_{\\mu}$. Taking the sum of the finite\nparts of these expressions we obtain for the counter terms,\n\\begin{eqnarray}\n\\label{CTfull} \\ {\\rm CT} &=&\n \\frac{\\alpha ^2}{384 {\\rm c^2} {\\rm s^2} \\pi^2}\n  \\left\\{\n    680 {\\rm c^4}-362 {\\rm c^2}-363\n  \\right.\n\\nonumber\\\\&&\n \\left.\n  +6\n     \\left(\n        \\left[\n            27-30 {\\rm c^2}\n        \\right]\n        \\ln m^2+\n        \\left[\n            54 {\\rm c^2}\n            -56 {\\rm c^4}\n        \\right]\n        \\ln M_W^2\n        +\n        \\left[\n            56 {\\rm c^4}-84 {\\rm c^2}+27\n        \\right]\n        \\ln M_Z^2\n     \\right)\n \\right\\}\n\\nonumber\\\\&&\n -\\alpha^2\\frac{m^2}{M_W^2}\n \\left(\n    0.74+0.058\\ln \\frac{m^2}{M_W^2}\n    -0.111\\Delta^2-0.0134\\Delta^2\\ln \\Delta^2\n \\right)\n\\nonumber\\\\&&\n -\\frac{\\alpha ^2}{384 {\\rm s^6} \\pi^2} \\frac{ m^2 }{M_W^2}\n \\left\\{\n    \\frac{S^{01}_{\\rm fin}(M_W, M_W, M_Z)}{M_W^2}\n    \\left(\n        {\\rm -96+475c^2-924c^4+828c^6-256c^8}\n    \\right)\n  \\right.\n\\nonumber\\\\&&\n \\left.\n    +\n    \\left(\n        \\rm{7-14c^2 + 4c^4}\n    \\right)\n    \\left(\n         \\frac{S^{01}_{\\rm fin}(M_W, M_H, M_W)}{M_W^2}\n        +\n        \\left(\n            \\rm{1+28c^2}\n        \\right)\n        \\frac{S^{01}_{\\rm fin}(M_Z, M_W, M_W)}{M_W^2}\n    \\right.\n \\right.\n\\nonumber\\\\&&\n \\left.\n    \\left.\n        +3S^{02}_{\\rm fin}(M_W, M_H, M_W)-\n        \\left(\n            \\Delta^2-2\n        \\right)\n        S^{12}_{\\rm fin}(M_W, M_H, M_W)-S^{22}_{\\rm fin}(M_W, M_H, M_W)\n   \\right.\n \\right.\n\\nonumber\\\\&&\n \\left.\n    \\left.\n            -\n            \\left(\n                \\rm{1+48c^2}\n            \\right)\n            S^{22}_{\\rm fin}(M_Z, M_W, M_W)\n    \\right)\n     +\n    \\left(\n        \\rm{5-16c^2+8c^4}\n    \\right)\n    \\times\n \\right.\n\\nonumber\\\\&&\n \\left.\n    \\left(\n        -\\rm{c^2} \\frac{S^{01}_{\\rm fin}(M_Z, M_H, M_Z)}{M_W^2}\n        +\n            \\left(\n                \\rm{1-4c^2 + 12c^4}\n            \\right)\n            \\left(\n                S^{22}_{\\rm fin}(M_W, M_W, M_Z)\n                -S^{12}_{\\rm fin}(M_W M_W M_Z)\n            \\right)\n    \\right.\n \\right.\n\\nonumber\\\\&&\n \\left.\n    \\left.\n            -3S^{02}_{\\rm fin}(M_Z, M_H, M_Z)\n            +\n            \\left(\n                \\Lambda^2-2\n            \\right)\n            S^{12}_{\\rm fin}(M_Z M_H M_Z)+S^{22}_{\\rm fin}(M_Z M_H M_Z)\n    \\right)\n \\right.\n\\nonumber\\\\&&\n \\left.\n    +\n    \\left(\n        \\rm{96-515c^2+1212c^4-1404c^6+512c^8}\n    \\right)\n    S^{02}_{\\rm fin}(M_W, M_W, M_Z)\n \\right.\n\\nonumber\\\\&&\n \\left.\n    +\\left(\n        \\rm{-133+82c^2+322c^4-169c^6+21c^8}\n    \\right)\n    S^{02}_{\\rm fin}(M_Z, M_W, M_W)\n \\right.\n\\nonumber\\\\&&\n \\left.\n    +\n    \\left(\n        \\rm{217-346c^2-42c^4+29c^6+7c^8}\n    \\right)\n    S^{12}_{\\rm fin}(M_Z, M_W, M_W)\n \\right\\}\\,,\n\\end{eqnarray}\nwhere, as in expression (\\ref{TsubHiggs}), the floating point\ncoefficients were rounded to provide the precision of $10^{-11}$ of\nthe result and do not depend on $M_W$ and $M_Z$ chosen within\nexperimentally accepted interval.  All special functions $S_{a,\nb}^{fin}$ are defined in Eq.~(\\ref{SEint}).\n\n\n\\subsection{Two-loop corrections to $g-2$}\n\\label{results}\n\nOur result (not yet in its final form -- see the next section)\nfor the two-loop bosonic\ncontribution to $g_\\mu-2$ is obtained by adding up\nall diagrams, Eqs.~(\\ref{TsubHiggs})--(\\ref{T4G}),\nand  counterterms (\\ref{CTfull}),\n\\begin{eqnarray}\n\\label{ResFull}\n \\ a_{\\mu}^{\\rm EW\\ bos}({\\rm\n two-loop})&=&T1_{A,B,C}+T2_{A,D,E}+T3_{A,B,C,D,G}+T4_{A,D,E,J}+T5+T2_{B}\n \\nonumber\\\\&&\n +T2_{C}+T2_{F}+T3_{E,F}+T4_{B}+T4_{C}+T4_{F}+T4_{G}+CT.\n\\label{eq:result}\n\\end{eqnarray}\n\nIts numerical value, normalized to the one-loop correction giving in Eq.~(\\ref{oneLoop}),\nis plotted in Fig.~(\\ref{comparison}) with a solid line,\nfor a range of the Higgs boson mass  50 GeV\n $\\le  M_H  \\le $ 700 GeV.  Also plotted, with a dashed line, is the approximate\nresult of Ref. \\cite{CKM96}.\n\\begin{figure}[!h]\n\\begin{tabular}{cc}\n&\n    \\centering\n    \\scalebox{0.75}\n    {\\includegraphics {5.eps}}\n\\end{tabular}\n    \\put(-220,-140){\\large $M_H$ [GeV]}\n    \\put(-310,80){\\large 114 GeV LEP lower limit on $M_H$}\n    \\put(-435, 10){\\large $[\\%]$}\n    \\caption{$\\frac{a_{\\mu}^{\\rm EW}(\\rm two-loop)}{a_{\\mu}^{\\rm EW}(\\rm one-loop)}$\n    as a function of $M_H$, expressed in percents. Dashed line represents the result\n    from Ref. \\cite{CKM96}, solid line is the result of this work.  The vertical dotted line\nshows the lower limit for the Higgs boson mass from direct searches, 114 GeV. }\n    \\label{comparison}\n\\end{figure}\nWe see that both results coincide well for the Higgs boson heavier than about 200 GeV.\nThe only difference in the large Higgs mass region is due to an additional approximation\nmade in Ref. \\cite{CKM96}, where the difference of the $W$ and $Z$ masses was treated as a\nsmall perturbation (first four terms in an expansion in $\\sin^2\\theta_W$ were retained there).\nAbove $M_H =250 $ GeV, the\nrelative difference between two results is less than $0.1\\%$,\nwell within the precision of the result in Ref.~\\cite{CKM96}.  However, the results differ\nstrongly in the low Higgs mass region.  The result of Ref.~\\cite{CKM96} is not valid in that region\nsince it was obtained under the assumption $M_{W,Z}\\ll M_H$.  We see that while the result\nof Ref.~\\cite{CKM96} seems to grow strongly for the light $M_H$, the actual bosonic correction (solid\nline) remains moderate.\n\n\n\\subsection{Reparametrization in terms of $G_{\\mu}$}\n\\label{reparametrization}\n\nEq.~(\\ref{eq:result}) gives a finite result for the two-loop correction,\nexpressed in terms   of the\nfine structure constant $\\alpha$, the $W$ mass, and the weak mixing angle.\nSome of the corrections computed\nin this way are universal for all weak processes and it is convenient\nto include them in the lower order result by expressing it  in\nterms of the Fermi constant $G_{\\mu}$. This amounts to the substitution, to\nbe made in all components of Eq.~(\\ref{eq:result}),\n\\begin{eqnarray}\n\\label{transform}\n    \\frac{e^2}{8~s^2~M^2_W}~\\to~\\frac{G_{\\mu}}{\\sqrt  2}(1-\\Delta r)\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n\\label{delta}\n    \\Delta r = 2\\delta Z_e-2\\frac{\\delta s}{s}-\\frac{\\delta\n    M_W^2}{M^2_W}+\\frac{\\Sigma^W(0)}{M_W^2}+\\frac{\\alpha}{4\\pi~s^2}\n    \\left(\n        6+\\frac{7-4s^2}{2s^2}\n    \\right)\n    \\ln c^2\\,.\n\\end{eqnarray}\nThis transformation decreases the central value of the ratio of\nthe two-loop to one-loop corrections by up to $7\\%$ (depending on\nthe Higgs mass) and, in addition, reduces its uncertainty, since\n$G_{\\mu}$ has been measured to much better precision than $M_W$\nand $s$.\n\n\\section{Conclusions}\n\nWe have presented a detailed evaluation of the two-loop bosonic corrections\nto $g_\\mu-2$.  Our result confirms the previous approximate evaluation\nin the limit of the heavy Higgs boson \\cite{CKM96}.  Our final number for this\ncorrection is\n\\begin{eqnarray}\n a_{\\mu}^{\\rm EW \\ bos}({\\rm two-loop})=(-22.2\\pm 1.6)\\times 10^{-11}\\,,\n\\label{final}\n\\end{eqnarray}\nwhere the central value was computed for the Higgs mass of 200\nGeV, and the error encompasses the interval 50 GeV\n $\\le  M_H  \\le $ 700 GeV. This is in agreement with the numerical results of an\nexact study in Ref.~\\cite{Heinemeyer:2004yq}.  In addition to the\nnumerical result we have provided here a number of semi-analytic\nintermediate results. They give insight into details of our\ncalculation, allow future checks, and simplify the evaluation of\nthe correction for any Higgs boson mass.\n\nThe uncertainty in Eq.~(\\ref{final}) is due to the unknown Higgs mass and is slightly reduced in\ncomparison with previous estimates based on the approximate evalulation.\nThe main variation of the correction occurs in the\nrelatively low Higgs mass range.  Thus, when more stringent limits on $M_H$ are\nobtained, that uncertainty will quickly decrease.  The main uncertainty of the\nelectroweak contributions will then be due to electroweak\nhadronic effects, as discussed in\n\\cite{Czarnecki:2002nt}.\n\n\n\n\n\\begin{acknowledgments}\nThis research was supported by the Natural Sciences and\nEngineering Research Council of Canada.  TG gratefully\nacknowledges support by the Golden Bell Jar Department of Physics\nGraduate Scholarship.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\n\\subsection{Phase retrieval problem}\n\\indent \n Recovering a signal from its intensity only measurements is called phase retrieval(PR) problem arised in a wild range of applications such as Fourier Ptychography Microscopy, diffraction imaging, X-ray crystallography and so on\\cite{Bunk2007Diffractive}\\cite{Miao1999Extending}\\cite{Zheng2013Wide}. PR problem can be an instance of solving a system of quadratic equations: \n\\begin{eqnarray}\n\\centering\ny_i=|\\mathbf{a}_i^*\\mathbf{x}|^2+\\mathbf{\\varepsilon}_i,~~i=1,...,m,\n\\end{eqnarray}\nwhere $\\mathbf{x}\\in\\mathbb{C}^n$ is the signal of interest, $\\{\\mathbf{a}_i\\}_{1\\leq i\\leq m}\\in\\mathbb{C}^n$ are the measurement vectors, $\\mathbf{y}=[y_1,y_2,...,y_m]^{T}\\in \\mathbb{R}^{m}$ is the observed measurement, vector $\\mathbf{\\varepsilon}=[\\varepsilon_1,\\varepsilon_2,...,\\varepsilon_n]^T$ is the noise.\\\\ \n\\indent\n(1.1) is a non-convex and NP-hard problem. Traditional methods usually fail to find the solutions. Besides, let $\\tilde{\\mathbf{x}}$ be the solution of (1.1), $\\tilde{\\mathbf{x}}e^{j\\theta}$ also satisfies (1.1) for any $\\theta\\in[0,2\\pi)$ where $j=\\sqrt{-1}$. So the uniqueness of the solution of (1.1) is often defined up to a global phase factor.\n\\subsection{Prior art}\nFor classical PR problem, $\\{\\mathbf{a}_i\\}_{1\\leq i\\leq m}$ are the Fourier measurement vectors. There were series of methods to solve (1.1). In 1970, error reduction methods such as Gerchberg-Saxton and Hybrid input and output method\\cite{Fienup1982Phase}\\cite{Gerchberg1971A} were proposed to tackle with PR by constantly projecting the evaluations between the transform domain and the spatial domain with some special constraints. These methods often get stuck into the local minimums. In addition, fundamental mathematical questions concerning their convergence remain unsolved. In fact, without any additional assumption about $\\mathbf{x}$, it is hard to recover $\\mathbf{x}$ from $\\{y_i\\}_{1\\leq i\\leq m}$. For Fourier measurement vectors, the trivial ambiguities of (1.1) include global phase shift, conjugate inversion and spatial shift. In fact, it has been proven that 1D Fourier phase retrieval problem has no unique solution even excluding those trivialities above. To relief those ill-condition characters, one way is to substitute Fourier measurements with other measurements owning the high redundancy property like Gaussian measurements\\cite{Cand2013PhaseLift}\\cite{Waldspurger2012Phase}\\cite{candes2015phase}, coded diffraction pattern\\cite{Cand2013Phase}, wavelet frame\\cite{Mallat2015Phase} and so on.\\\\\n\\indent\nSparsity priori has been considered in many problems with significant application meanings. Greedy sparse phase retrieval(Gespar) is a kind of heuristics to deal with phase retrieval problem\\cite{Shechtman2013GESPAR} based on but not restricted to Fourier measurements. It utilizes the damped Gaussian-Newton algorithm to search for the local minimum and then updates the support of signal by 2-opt method. In\\cite{Cand2013PhaseLift}\\cite{Waldspurger2012Phase}\\cite{Candes2011Phase}, SDP(Semi-Definite Programming) based algorithms were came up to deal with phase retrieval problem by lifting (1.1) into a higher dimension. This convex alternative can deal with phase retrieval problem but need high computational costs. Plugging the $l_1$ constraint into this convex objective function, Compressive Sensing Phase Retrieval via Lifting(CPRL) was proposed to tackle with the sparse phase retrival. On the other hand, to decrease the computational cost,  Alternating minimization(ALTMIN)\\cite{Netrapalli2013Phase} method alternativly updates signal and phase to search for the signal of interest. It can deal with sparse signal by adding hard-thresholding method in each iteration.\\\\\n\\indent\nAnother wildspread nonconvex method for phase retrival is the WF(Wirtinger flow)\\cite{candes2015phase} which directly utilizes the gradient descent method to search for the global minimum. Based on WF, there are several variants under different conditions. For Poisson likelihood function, truncated Wirtinger flow(TWF) was proposed\\cite{chen2015solving}. For amplitude based models, reshaped WF and TAF(Truncated Amplituded Flow) were considered in \\cite{zhang2016reshaped}\\cite{wang2016solving}. Those methods often need $m\/n>3$ for the exactly recovery. But, on the theoretical side, for a $k$ sparse signal $\\mathbf{x}\\in\\mathbb{R}^n$, $4k-1$ measurements are sufficient to guarantee uniqueness. The gap of the sampling complextiy is large. Thus, it is necessary to come up with an algorithm for sparse wirtinger flow phase retrival problem. In\\cite{wang2016solving1}\\cite{Cai2016Optimal}, thresholding WF and Sparse Phase Retrieval via Truncated Amplitude(SPARTA) were camp up utilizing the priority of sparsity. SPARTA can have a high recovery rate than thresholding WF with a faster converge rate. But SPARTA is sensitive to the priority $k$.\\\\\n\\indent\nBased on the thresholding WF and SPARTA, A sparse phase retrieval problem called SWF is proposed in this paper. We adopt the Gaussian maximum likelihood function as the objective which is a forth order smooth function with a benign geometrical property. Then we use a two-stage algorithm to find the global optimum. In the first stage, the support of the signal is estimated by a well justified rule in \\cite{Wang2016Sparse}, then we apply the truncated spectral method to evaluate the initialization which is restricted to the support evaluated above. In the second stage, the initialization is constantly refined by the hard thresholding based gradient descent method. The sample complexity and computational complexity of SWF can be seen in table 1. Theoretical results show that SWF can recover any $k$ sparse $n$ dimension signal $\\mathbf{x}$ through $\\mathcal{O}(k^2\\mathrm{log}n)$ measurements with the minimum nonzero entries's modulus on the order of $\\frac{1}{k}||\\mathbf{x}||$. Besides, SWF have a geometric convergence rate and need $\\mathcal{O}(k\\mathrm{log}\\frac{1}{\\varepsilon})$ iterations to get $\\varepsilon-$ accuracy. Though the computational complexity of SWF is $k$ times larger than SPARTA, numerical tests show that the SWF can have a better performance than state of the art methods. Specifically, SWF can have a 100\\% recovery rate when $m=0.7n$. Especially when the priori sparsity $k$ is unkown, SWF is also significantly superior to other algorithms. Moreover, SWF can be insensitive to the support misspecification and noise in some degree. For readers to be convenient to replicate the simulating tests, the codes of SWF are available at:\\url{https:\/\/github.com\/Ziyang1992\/Sparse-Wirtinger-flow.git}\\\\   \n\\begin{table}\n\t\t\\centering\n\t\\begin{tabular}{|c|c|c|}\n\t\t\\hline\n\t\t~~&Sample complexity&Computational complexity\\\\\n\t\t\\hline\n\t\tCPRL\\cite{Ohlsson2012CPRL}& $\\mathcal{O}(k^2\\mathrm{log}n)$&$\\mathcal{O}(n^3\/\\varepsilon^2)$\\\\\t\n\t\t\\hline\t\n\t\tALTMIN\\cite{Netrapalli2013Phase} &$\\mathcal{O}(k\\mathrm{log}n(k+\\mathrm{log}^3k+\\mathrm{log}\\frac{1}{\\varepsilon}\\mathrm{log}\\mathrm{log}\\frac{1}{\\varepsilon})$& $\\mathcal{O}(k^2\\mathrm{log}n(kn+\\mathrm{log}^2\\frac{1}{\\varepsilon}\\mathrm{log}\\mathrm{log}\\frac{1}{\\varepsilon})$\\\\\n\t\t\\hline\n  \t\tThresholding WF\\cite{Cai2016Optimal}&$\\mathcal{O}(k^2\\mathrm{log}n)$&$\\mathcal{O}(k^3n\\mathrm{log}n\\mathrm{log}\\frac{1}{\\varepsilon})$\\\\\n\t\t\\hline\n\tSPARTA\\cite{Wang2016Sparse}&$\\mathcal{O}(k^2\\mathrm{log}n)$&$\\mathcal{O}(k^2n\\mathrm{log}n\\mathrm{log}\\frac{1}{\\varepsilon})$\\\\\n\t\\hline\n\t SWF&$\\mathcal{O}(k^2\\mathrm{log}n)$&$\\mathcal{O}(k^3n\\mathrm{log}n\\mathrm{log}\\frac{1}{\\varepsilon})$\\\\\n\t\\hline\n\t\\end{tabular}\n\t\\caption{Comparison of State of art approach}\n\\end{table}  \n\\subsection{Contribution of this paper}\n\\indent\nOur contributions are in two folds. Firstly, we propose an algorithm called SWF to solve sparse phase retrieval problem. Though the amplitude based algorithm can be superior to the intensity based algorithm for the general signal when the ratio $m\/n$ is small\\cite{zhang2016reshaped}\\cite{wang2016solving}. In this paper, we find that when it comes to the sparse signal, the intensity based model \u2014\u2014SWF did have a better performance than the amplitude based one.\nThe second is our theoretical contribution. We prove that SWF has a linear convergence with measurements $m$ largely exceeding the sparsity $k$.\\\\  \n\\indent\nThe remainders of this paper are organized as follows. In section 2, we introduce the proposed SWF and establish its theoretical frames. In section 3, numerical tests compare SWF with state-of-the-art approaches. Section 4 is the conclusion. Technical details can be found in Appendix.\\\\\n\\indent \nIn this article, the bold uppercase and lowercase letters represent matrices and vectors. $(\\cdot)^{T}$ denotes the transpose. $|\\cdot|$ denotes the absolute value of a real number or the cardinality of a set. $||\\cdot||$ is the Euclidean norm of a vector. $||\\cdot||_0$ is the zero norm.\n\\section{Sparse Wirtinger Flow}\n\\subsection{Algorithm of SWF}\nSparse phase retrieval aims to find an evaluation $\\mathbf{z}$ approximating to $k$ sparse signal $\\mathbf{x}$ from (1.1),\n\\begin{eqnarray}\n\\nonumber&\\mathrm{Find}~~\\mathbf{z}&\\\\\n\\nonumber&s.t.~~ y_i=|\\mathbf{a}_i^T\\mathbf{z}|^2+\\mathbf{\\varepsilon}_i,~~i=1,...,m,&\\\\\n&||\\mathbf{z}||_0=k.&\n\\end{eqnarray} \nIn our paper, the sparstiy $k$ is assumed to be known as a priori for the theorectical simplicity. We also make simulation tests to show the performance of SWF when the priori sparsity $k$ is unknown. \\\\\nWe assume that $\\mathbf{a}_i\\sim\\mathcal{N}(\\mathbf{0},\\mathbf{I})$ and $\\varepsilon_i\\sim\\mathcal{N}(0,\\sigma)$, then the probability density function of $\\varepsilon_i$ is:\n\\begin{eqnarray}\n\\mathbb{P}(\\varepsilon_i)=\\frac{1}{\\sqrt{2\\pi}\\sigma}\\mathrm{exp}(-\\frac{(y_i-|\\mathbf{a}_i^T\\mathbf{z}|^2)^2}{2\\sigma^2}).\n\\end{eqnarray}\nAccording to (2.2), neglecting the effects of constants and assume the signal is real, we estimate the maximum likelihood function as\n\\begin{eqnarray}\n\\nonumber&\\mathop {\\mathrm{minimize}}\\limits_{\\mathbf{z}\\in\\mathbb{C}^n}~~\\mathit{f}(\\mathbf{z})=\\frac{1}{2m}\\sum_{i=1}^{m}\\big((\\mathbf{a}_i^T\\mathbf{z})^2-y_i\\big)^2,&\\\\\n&s.t.~~||\\mathbf{z}||_0=k.&\n\\end{eqnarray}\n(2.3) is a non-convex optimization problem which has many local minimums. As a result, it seems impossible to solve (2.3) with convex methods. Owning to the statistical property of Gaussian random vectors, (2.3) can have a benign geometrical structure. Without the sparisty constraint, a butch of algorithms were came up to search for the global optimums\\cite{candes2015phase}\\cite{gao2016gauss}\\cite{sun2016geometric}\\cite{li2016gradient}. Those method performs well when $m\/n$ is large enough. We can utilize the sparse condition to decrease the sample complexity. Thus for sparse signal, we come up with a more efficient algorithm called SWF. Firstly, we recover the support of $\\mathbf{x}$, then we apply the truncated spectral method to make a good initialization under the recovered supports. At last, we utilize a hard thresholding based gradient descent algorithm to search for the global optimum. Next, we will give the details of SWF without noise from these three parts.\\\\\n\\indent\nBefore explanation, we introduce several notations. The distance between the evaluation $\\mathbf{z}$ and real solution $\\mathbf{x}$ is defined as:\n\\begin{eqnarray*}\n\\mathrm{dist}(\\mathbf{z},\\mathbf{x})=\\mathop {\\mathrm{min}}\\limits_{\\phi\\in[0,2\\pi)}||\\mathbf{z}-\\mathbf{x}e^{j\\phi}||,\n\\end{eqnarray*}\nwhere $j=\\sqrt{-1}$.\\\\\n\\indent\nThen, for any vector $\\mathbf{z}$ and any support $S$, $\\mathbf{z}_S$ means vector $\\mathbf{z}$ deletes all the elements outside of support $S$. \n\\subsubsection{Support recovery}\n\\indent\nTo recover the support of $\\mathbf{x}$, we use the same method in \\cite{Wang2016Sparse}. Assuming $\\mathbf{x}\\in\\mathbb{R}^n$ is a $k$ sparse signal with support $S^*$, $|S^*|=k$. We define $D_{i,j}=(\\mathbf{a}_i^{T}\\mathbf{x})^2a_{i,j}$ where $a_{i,j}$ is the $j$th element of $\\mathbf{a}_i$. Note that $\\mathbf{a}_i\\sim\\mathcal{N}(\\mathbf{0},\\mathbf{I})$, by calculating the moment of Gaussian variables, we have $\\mathbb{E}(|a_{i,j}|^4)=3$, $\\mathbb{E}(|a_{i,j}|^2)=1$.\\\\\n\\indent\nSo,\n\\begin{eqnarray}\n\\mathbb{E}(D_{i,j})=\\mathbb{E}\\big((\\sum_{k=1}^{n}a_{i,k}x_k)^2a_{i,j}\\big)=||\\mathbf{x}||^2+2x_j^2.\n\\end{eqnarray}\nDenote $E_j=\\frac{1}{m}\\sum_{i=1}^{m}D_{i,j}$.\nFrom (2.4), the differences between $E_j$ are determined by $x_j$ if $m$ is large enough. If $x_j^2$ is larger, accordingly the $E_j$ is larger too. Then we can sort out the $k$ largest $E_j$ and record their indexes as the estimated support $S_0$. Lemma 1 guarantees the accuracy of this support recovery method.\\\\\n\\begin{lemma}\\cite{Wang2016Sparse}\n\tFor any $k$ sparse signal $x\\in\\mathbb{R}^n$ with support $S^*$ and minimum nonzero entries $x_{\\mathrm{min}}:=\\mathrm{min}_{j\\in S^*}|x_j|$ on the order of $(1\/\\sqrt{k})||\\mathbf{x}||_2$. If $\\mathbf{a}_i\\overset{i.i.d}\\sim\\mathcal{N}(\\mathbf{0},\\mathbf{I}_n)$, $i=1,...,m$. $S_0$ is equal to $S^*$ with a probability at least $1-6\/m$ provided $m\\geq C_0k^2\\mathrm{log}(mn)$ for some constant $C_0$.\n\\end{lemma}   \n\\indent\nLemma 1 shows that when $m$ is sufficiently large, $S_0$ approximates $S^*$ quite well. But from numerous tests, we find that SWF can still recover $\\mathbf{x}$ even when $S_0$ is quite different from $S^*$.\\\\\n\\subsubsection{Initialization evaluation}\nWe have estimated the support $S_0$ of $\\mathbf{x}$, now we constrain $\\mathbf{a}_i$ on $S_0$, i.e., deleting those elements which aren't in $S_0$. Under the guarantee of lemma 1, we will use the truncated spectral method to make an initialization. Specifically, we construct a matrix $\\mathbf{Y}$ as (2.5). When $m$ is sufficiently large, the eigenvector of $\\mathbf{Y}$ can be taken as an approximation of  $\\mathbf{x}$,\n\\begin{eqnarray}\n\t\\mathbf{Y}=\\frac{1}{m}\\sum_{i=1}^{m}y_i\\mathbf{a}_{i,S_0}\\mathbf{a}_{i,S_0}^T\\mathbf{1}_{\\{|y_i|\\leq\\alpha^2_y\\phi^2\\}},\n\\end{eqnarray}\nwhere $\\alpha_y$ is the truncation threshold. $\\phi^2=\\frac{1}{m}\\sum_{i=1}^{m}y_i$.\\\\\n\\indent\n Lemma 2 demonstrates the accuracy of the estimation made by the truncated spectral method.  \n\\begin{lemma}\n\tUnder the conditions of lemma 1, for any $\\delta>0$ and $\\mathbf{x}\\in\\mathbb{R}^n$, the solution $\\mathbf{z}^{0}_{S_0}\\in\\mathbb{R}^k$ returned by the truncated spectral method obeys:\n\t\\begin{eqnarray}\n\t\\mathrm{dist}(\\mathbf{z}^{0}_{S_0},\\mathbf{x}_{S_0})\\leq\\delta||\\mathbf{x}_{S_0}||.\n\t\\end{eqnarray}\n\twith probability not less than $1-\\mathrm{exp}(-C_1m)$, providing that $m>c_0k$ for some constant $C_1$ and $c_0>0$ which is determined by $\\delta$.\t\n\\end{lemma}\n\n\\noindent\n\\emph{Proof}:\\\\\n\\indent\nBased on the condition of lemma 1, we have $S_0=S^*$ with a probability at least $1-\\frac{6}{m}$ provided $m\\geq C_0k^2\\mathrm{log}(mn)$. Then (2.5) can be rewrited as\n\\begin{eqnarray}\n\\frac{1}{m}\\sum_{i=1}^{m}(\\mathbf{a}_i^{T}\\mathbf{x})^2\\mathbf{a}_{i,S_0}\\mathbf{a}_{i,S_0}^{T}=\\frac{1}{m}\\sum_{i=1}^{m}(\\mathbf{a}_{i,S^*}^{T}\\mathbf{x}_{S^*})^2\\mathbf{a}_{i,S_0}\\mathbf{a}_{i,S_0}^{T}=\\frac{1}{m}\\sum_{i=1}^{m}(\\mathbf{a}_{i,S_0}^{T}\\mathbf{x}_{S_0})^2\\mathbf{a}_{i,S_0}\\mathbf{a}_{i,S_0}^{T}.\n\\end{eqnarray}\nCombining with proposition 3 in \\cite{chen2015solving}, we can conclude lemma 2 is true.\\\\\n\\indent\nUtilizing power method in (2.5) to get the estimation $\\mathbf{z}^{0}_{S_0}\\in\\mathbb{R}^k$, we will scale  $||\\mathbf{z}^0_{S_0}||=\\phi$ and construct $\\mathbf{z}_0\\in\\mathbb{R}^n$ where elements in $S_0$ are equal to $\\mathbf{z}^{0}_{S_0}$, others are all zero. In the test, we generally set $\\alpha_y=3$ and run the power method with 100 iterations.\n\\subsubsection{Hard thresholding based gradient descent}\nWe utilize the hard-thresholding based gradient descent algorithm to search for the global optimum in each iteration with $\\mathbf{z}_0$ as an initialization.\\\\\nThe gradient of $f(\\mathbf{z})$ is calculated by the Wirtinger derivative.\n\\begin{eqnarray}\n\\nabla f(\\mathbf{z})=\\frac{2}{m}\\sum_{i=1}^{m}\\big((\\mathbf{a}_i^T\\mathbf{z})^2-y_i^2\\big)\\mathbf{a}_i\\mathbf{a}_i^T\\mathbf{z}.\n\\end{eqnarray} \nIn the $t$th iteration of gradient descent, we have:\n\\begin{eqnarray}\n\\tilde{\\mathbf{z}}^{t}=\\mathbf{z}^{t-1}-\\frac{\\mu_{t}}{\\phi^2}\\nabla f(\\mathbf{z}^{t-1}),\n\\end{eqnarray}\nwhere $\\mu_t$ is the step size.\nHere, we add a thresholding operator $\\mathcal{T}_k$ to $\\tilde{\\mathbf{z}}^{t}$.\n\\begin{eqnarray}\n\\mathcal{T}_k(\\tilde{\\mathbf{z}}^{t})=\\mathbf{z}^t,\n\\end{eqnarray}\nwhere $\\mathbf{z}^t$ keeps the $k$-largest absolute value of $\\tilde{\\mathbf{z}}^{t}$ and sets other elements to zero. $\\mathcal{T}_k$ projects $\\tilde{\\mathbf{z}}^{t}$ into the subspace $\\mathbb{V}_k=\\big\\{\\mathbf{v}\\in\\mathbb{R}^n\\big|||\\mathbf{v}||_0\\leq k\\big\\}$. This procedure can decrease the freedom dimension and constrain the searching domain. Numerous tests also show hard thresholding procedure is effective for sparse PR. Theorem 1 guarantees the convergence of this hard thresholding based gradient descent method. \n\\begin{theorem}\n\tBased on lemma 1 and lemma 2, with a proper stepsize $\\mu_t$, the $t+1$th estimation of $\\mathrm{SWF}$ $\\mathbf{z}^{t+1}$ satisfies:\n\t\\begin{eqnarray}\n\t||\\mathbf{z}^{t+1}-\\mathbf{x}||\\leq\\delta_1 (1-\\nu)^{t+1}||\\mathbf{x}||,\n\t\\end{eqnarray}\n\twith probability exceeding $1-c_1m^{-1}-c_2\\mathrm{exp}(-c_3k\/\\mathrm{log}m)$ provided $m\\geq C_0 k\\mathrm{log}k$, where $0<\\nu<1$, $C_0$, $c_1$, $c_2$, $\\delta_1$ are all constants. \n\\end{theorem}\nThe proofs of theorem 1 are in appendix. The stepsize $\\mu_t$ can be $0.1$ for all $t$, then $\\nu=0.19$. In our simulation tests, in order to make SWF have a good performance when $m$ isn't larege enough, we select a varying stepsize. The stepsize $\\mu_t=\\mathrm{min}\\Big(\\big(1-\\mathrm{exp}(\\frac{-t}{330})\\big)\/2,0.1\\Big)$ is utilized in the SWF. In the first few iterations, the stepsize is small to prevent iteration from stagnating into the local minimum easily, then we gradually increase the value of stepsize. Combining lemma 1, lemma 2 and theorem 1, we can get the exact recovery guarantee for SWF. The details of SWF can be seen clearly in algorithm 1.\\\\\n\\begin{algorithm}\n\t\\caption{\\textbf{Sparse Wirtinger Flow}($\\mathbf{SWF}$)}\n\t\\label{alg:Framwork} %\n\t\\renewcommand{\\algorithmicrequire}{\\textbf{Input:}}\n\t\\renewcommand\\algorithmicensure {\\textbf{Output:} }\n\n\n\t\\begin{algorithmic}   \n\t\t\\REQUIRE$\\{\\{y_i\\}_{1\\leq i\\leq m},\\{\\mathbf{a}_i\\}_{1\\leq i\\leq m},T,k,\\mu_t,\\alpha_y\\}$ \n\t\t\\STATE \n\t\t$\\{\\mathbf{a}_i\\}_{i=1}^m$: Gaussian vectors\\\\\n\t\t$y_i=|\\langle\\mathbf{a}_i,\\mathbf{x}\\rangle|^2$: measurements\\\\\n\t\t$\\varepsilon$: the accuracy required\\\\\n\t\t$k$: the sparsity of $\\mathbf{x}$\\\\\n\t\t$\\mu_t$: the step size\\\\\n\t\t$\\alpha_y$: truncation thresholds\\\\\n\t\t$T$: the maximum iteration times\\\\\n\t\t\\ENSURE$\\hat{\\mathbf{x}}$\n\t\t\\STATE $\\hat{\\mathbf{x}}$: the estimated signal\\\\\n\t\t\\vskip 4mm\n\t\t\\hrule\n\t\t\\vskip 2mm\n\t\t$\\mathbf{Support~recovery}$\n\t\t\\STATE\n\t\tset $S_0$ to be the set of $k$ largest indices of $\\{\\frac{1}{m}\\sum_{i=1}^{m}y_ia_{i,j}^2\\}_{1\\leq j\\leq n}$\\\\\n\t\t\\textbf{Initialization evaluation}\n\t\t\\STATE\n\t\t$\\phi^2=\\frac{1}{m}\\sum_{i=1}^{m}y_i$\\\\\n\t\tLet $\\mathbf{z}^0_{S_0}(||\\mathbf{z}^{0}_{S_0}||=\\phi)$ to be the eigenvector corresponding to the largest eigenvalue of\n\t\t\\begin{eqnarray*}\n\t\t\t\\mathbf{Y}=\\frac{1}{m}\\sum_{i=1}^{m}y_i\\mathbf{a}_{i,S_0}\\mathbf{a}_{i,S_0}^*\\mathbf{1}_{\\{|y_i|\\leq\\alpha^2_y\\phi^2\\}}\n\t\t\\end{eqnarray*}\n\t   set $\\mathbf{z}^0$ to be the vector where elements in $S_0$ are equal to $\\mathbf{z}^{0}_{S_0}$, others are all zero.\\\\\n\t  \\textbf{Hard thresholding based gradient descent}\\\\\n\t  $t=1$, $\\mathbf{z}^1=\\mathcal{T}_k(\\mathbf{z}^{0}-\\frac{\\mu_0}{\\phi^2}\\nabla\\mathit{f}(\\mathbf{z}^0))$\n\t\t\\WHILE{$||\\mathbf{z}^{t}-\\mathbf{z}^{t-1}||\\geq\\varepsilon$ and $t\\leq T$}\n\t\t\\STATE\n\t$\\mathbf{z}^{t+1}=\\mathcal{T}_k(\\mathbf{z}^{t}-\\frac{\\mu_t}{\\phi^2}\\nabla\\mathit{f}(\\mathbf{z}^t))$\\\\\n\t\\STATE $t=t+1$\n\t\t\\ENDWHILE\\\\\n\t\t$\\mathbf{\\hat{x}}=\\mathbf{z}^t$\n\t\\end{algorithmic}  \n\\end{algorithm}\n\\indent\nIn the next section, we will make several simulations to demonstrate the effectiveness of SWF with the comparison of state of the art methods.\\\\\n\\section{Numerical tests}\nNumerical results are given in this section which show the performance of SWF together with SPARTA\\cite{Wang2016Sparse}, ALTMIN\\cite{Netrapalli2013Phase}, TAF\\cite{wang2016solving} and Thresholding WF\\cite{Cai2016Optimal}. All the tests are carried out on the Lenovo desktop with a 3.60 GHz Intel Corel i7 processor and 4GB DDR3 memory. Here,  we are in favor of normalized mean square error ($\\mathbf{NMSE}$) which can be calculated as:\n\\begin{eqnarray*}\n\t\\centering\n\t\\mathbf{NMSE}=\\frac{||\\hat{\\mathbf{x}}-\\mathbf{x}||}{||\\mathbf{x}||},\n\\end{eqnarray*}\nwhere $\\hat{\\mathbf{x}}$ is the numerical estimation of $\\mathbf{x}$.\\\\\n\\indent\nIn all simulating tests, $\\mathbf{x}\\in\\mathbb{R}^{1000}$ is a real Gaussian random vector satisfying $\\mathcal{N}(\\mathbf{0},\\mathbf{I})$. $\\{\\mathbf{a}_i\\}_{1\\leq i\\leq m}$ are drawn from $\\mathcal{N}(\\mathbf{0},\\mathbf{I})$. The stepsize $\\mu_t=\\mathrm{min}\\Big(\\big(1-\\mathrm{exp}(\\frac{-t}{330})\\big)\/2,0.1\\Big)$ is utilized in the SWF. For all tests, if the $\\mathbf{NMSE}$ is below $10^{-5}$, we will regard it as a success. The successful times divided by testing times is the recovery rate. \\\\  \n\\noindent\n\\textbf{Test~1} \\\\\n\\indent\nIn the first test, we assume the sparsity $k$ is known. The signal $\\mathbf{x}$ is fixed with sparsity $k=10$. The ratio between $m$ and $n$ ranges from $0.1$ to $3$. At each ratio, we run 100 times tests. The recovery rate of different methods are shown in Figure 3.1.\\\\\n\\begin{figure}\t\n\t\\centering\n\t\\includegraphics[width=3in]{figure1.eps}\n\t\\caption{The comparion of different algorithms for fixed $k=10$ with different $m\/n$}\\label{fig:1a}\t\t\n\\end{figure}\n\\indent\nFrom Figure 3.1, we can see that SWF is a little superior to SPARTA. SWF and SPARTA can have a 100\\% recovery rate when $m\\geq0.7n$. But ALTMIN has 100\\% recovery rate only when $m\\geq1.1n$. TAF is one of the best algorithm for general phase retrieval which isn't designed for sparse signal. TAF~can have a 100\\% recovery rate when $m\\geq2.5n$. Thresholding WF can recover signal when $m\/n<1$, but it can't get a 100\\% recovery rate for all these ratios.\\\\\n\\noindent\n\\textbf{Test~2} \\\\\n\\indent\nAssume the sparsity $k$ is unkown. All test settings are the same with test 1. But the $k$ we known as a priori is taken as $\\sqrt{n}\\approx32$ according to the sample complexity in tabel 1. The results are shown in Figure 3.2.\\\\\n\\indent\nFrom Figure 3.2 we can find that SWF can be superior to SPARTA when the priori sparsity $k$ isn't known correctly. SWF can have a recovery rate about 90\\% at $m\/n=0.5$, but SPARTA nearly can't recover $\\mathbf{x}$ at the same ratio. ALTMIN can still have a 100\\% recovery rate when $m=1.1n$.\\\\\n\\begin{figure}\t\n\t\\centering\n\t\\includegraphics[width=3in]{figure2}\n\t\\caption{The comparion of different algorithm for fixed $k=10$ with different $m\/n$ ratio}\\label{fig:1a}\t\t\n\\end{figure}\nTable 2 shows the average iterations of each algorithm and corresponding time needed in test 1 and test 2. We select three different $m\/n$ ratios to make comparison. We refer SWF, SPARTA and ALTMIN in test 2 as SWF0, SPARTA0 and ALTMIN0. From table 2 we can find out that the SPARTA and SPARTA0 will need fewer iterations and less time to attain the required accuracy. Because they are all based on the truncated amplitude method which is one of the most efficient algorithms for general wirtinger flow phase retrieval. But it can't have a high recovery rate comparing to SWF. Besides the time and iterations SWF need cost is considerable.\\\\\n\\begin{table}[htp]\n\t\\centering\n\t\\caption{Computational complexity of different methods in test 1 and test 2}\n\t\\vspace{0.2cm}\n\t\\begin{tabular}{cccccccc}\n\t\t\\hline\n\t\t\\multicolumn{1}{c}{} & \\multicolumn{3}{c}{Iterations}& & \\multicolumn{3}{c}{Time}\\\\\n\t\t\\cline{2-4}\\cline{6-8}\n\t\t\\multicolumn{1}{c}{} & $m\/n=0.5$ & $m\/n=1$ & $m\/n=1.5$& & $m\/n=0.5$ & $m\/n=1$ & $m\/n=1.5$\\\\\n\t\t\\hline\n\t\t\\multirow{0}{*} SWF &79 &69 &66 & &0.360 &0.690 &1.070\\\\\n\t\tSPARTA&12&9&8& &0.007&0.012&0.0213\\\\\n\t\tALTMIN &-&-&4& &-&- &2.267\\\\\n\t\tThresholding WF &-&697&707& &-&6.953&11.784\\\\\n\t\tTAF &-&-&-& &-&-&-\\\\\n\t\tSWF0&123&90&80 & &0.575 &0.902 &1.289\\\\\n\t\tSPARTA0&-&24&16& &- &0.022 &0.034\\\\\n\t\tALTMIN0&-&-&6& &-&-&2.675\\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\\noindent\n\\textbf{Test~3} \\\\\n\\indent\n To find out the ability of those methods in resisting for the misspecific priori sparsity $k$, we fix $n=m$, and the sparsity of $\\mathbf{x}$ is 10. The priori sparisty $k$ ranges from $5$ to $100$. At each $k$, we also run 100 tests.\\\\\n \\indent\nFrom Figure 3.3, we can see that the misspecification ability of SWF are better than SPARTA. Especially for SWF, the recovery rate of it can be even 95\\% when the priori sparsity is $100$.\\\\ \n\\begin{figure}\t\n\t\t\\centering\n\t\t\\includegraphics[width=3in]{figure3}\n\t\\caption{The comparison between SWF and SPARTA on misspecific priori sparsity.}\n\\end{figure}\n\\indent\nWhy the SWF and SPARTA can have diffrent results with similar procedures? We think this can be attributed to the truncated procedure in SPARTA. Because SPARTA truncates some components of gradient to make the direction of truncated gradient heading to the global minimum. But this theory is based on the condition that $m$ is sufficiently large than $k$. When this condition can't be satisfied, the truncated procedure may neglect some positive information. Here is an example where $\\mathbf{x}=[1.0838,0,0]$ with sparsity $k=1$, $m=n=3$ which can be shown in Figure 3.4. The priority sparsity is $k=2$. Then we use the initialization estimated by SPARTA to recover $\\mathbf{x}$ by SWF and SPARTA. The black triangle is both algorithms' initialization $\\mathbf{z}_0=[0.8915, 0, -0.3987]$. Blue circles are the iterations made by SWF which converge to the point $\\hat{\\mathbf{x}}=[1.0838,0,-8.919\\times10^{-6}]$ after 70 iterations. $\\mathbf{NMSE}$ of SWF is below $10^{-5}$. But the iterations made by SPARTA which are shown by red circles are stagnated into the local minimum point $\\mathbf{z}^l=[0.8688, 0, -0.4806]$.\\\\\n\\begin{figure}\t\n\t\\centering\n\t\\includegraphics[width=3in]{figure4}\n\t\\caption{The itertations of SWF and SPARTA}\t\t\n\\end{figure}\n\n\\noindent\n\\textbf{Test~4} \\\\\n\\indent\nNext, we will research how the sparsity $k$ affects the recovery rate of those algorithms. Here, $m=1.5n$, the sparsity $k$ of $\\mathbf{x}$ varies from 10 to 100. There is no misspecification for priori sparsity. At each sparsity $k$, we also run 100 tests. The results are shown in Figure 3.5. We can find that SWF can be superior to othe algorithms and can have a mean recovery rate about $60 $\\% when sparsity $k=100$.\\\\ \n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=3in]{figure5}\n\t\\caption{Comparison of different algorithm for different sparsity $k$}\n\\end{figure}\n\n\\noindent\n\\textbf{Test~5} \\\\\n\\indent\nAt last, we will test the robustness of different algorithms. We assume the noise model is described as (2.1). The noise is Gaussian white noise and the SNR varies from $5$dB to $10$dB. We fix $k=10$ and $m=1.5n$. At each SNR, we run 100 tests and record the average \\textbf{NMSE}. The results are shown in Figure 3.6. We can see that SWF is more robust to the noise than SPARTA and ALTMIN.\\\\\n\\begin{figure}\t\n\t\t\\centering\n\t\t\\includegraphics[width=3in]{figure6}\n\t\t\\caption{Comparison of different algorithm to the Nosie}\t\t\n\\end{figure}\n\\indent\nAll in all, SWF has a high recovery rate than SPARTA and ALTMIN especially when the sparsity $k$ isn't known exactly. Besides it is also robust to noise. The iteration and time that SWF costs is also considerable .\\\\\n\\indent \n\n\\section{Conclusion}\nIn this paper, we propose a Wirtinger flow algorithm for sparse phase retrieval problem. It can have a high recovery rate when the sampling complexity is low with support recovery and hard-thresholding based gradient descent. This algorithm aims to the 1D signal and has a good performance when the signal is real. In the future work, we will be keen to broaden it and modify it to be appropriate for high dimensional signals. \n\\section{Acknowledgement}\nThis work was supported in part by National Natural Science foundation(China): 61571008.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nThe $\\Lambda(1405)$ resonance is getting renewed experimental attention \nand different reactions have been used to produce it. One of the recent reactions is the  $K^- p \\to \\pi^0 \\pi^0 \\Sigma^0$ measured at $p_{K^-}$ = 514 MeV\/c to 750 MeV\/c in \\cite{Prakhov:2004an}. COSY at Juelich has also produced it in the $p p \\to K^+ p \\pi^0 \\Sigma^0$ reaction \\cite{Zychor:2007gf}. The photoproduction of the resonance has been investigated at LEPS \\cite{Niiyama:2008rt} and CLAS\n\\cite{Moriya:2009mx} and plans are made to continue its study in DAFNE and J-PARC among other facilities. One of the motivations to search for it in different reactions was the theoretical observation in \\cite{Jido:2003cb} that\nthe use of chiral dynamics led to two nearby poles in the region of the \n$\\Lambda(1405)$ resonance, and that different reactions gave more weight to one or the other poles, such that the shape of the cross section would change from one reaction to another.  \n\nThe $\\Lambda(1405)$ has been a historical example of \na dynamically generated resonance in \nmeson-baryon coupled-channels dynamics with $S=-1$~\\cite{Dalitz:1967fp}.\nModern investigations based on chiral dynamics with a unitary framework \nalso reproduce well the observed spectrum of the $\\Lambda(1405)$ together \nwith cross sections of $K^{-}p$ to various \nchannels~\\cite{Kaiser:1995eg,Oset:1998it,Oller:2000fj,Oset:2001cn,GarciaRecio:2002td,Hyodo:2002pk,Hyodo:2003qa,Hyodo:2008xr}.\n\n The work of \\cite{Jido:2003cb} also motivated further theoretical studies introducing terms of higher order in the kernel of the $K^-p$ interaction with its coupled channels \\cite{Borasoy:2005ie,Oller:2006jw,Borasoy:2006sr}. The interesting thing observed in these works is that the two pole structure is always there independent of variations in the parameters allowed by the data.\n One of the states, located around 1420 MeV, couples dominantly to the $\\bar KN$\nchannel and is very stable, while the other one, sitting around 1390 MeV and with a width of 130 MeV or larger, \ncouples strongly to the $\\pi \\Sigma$ channel. This latter one is more model dependent but the mass is smaller than that of the other state and the width is considerably larger. Another interesting output of these works is that the results including the higher order kernel are compatible with those with the lowest order Lagrangian (the Weinberg Tomozawa term) within theoretical uncertainties \\cite{Borasoy:2006sr}. Depending on the reaction mechanism and which is the channel predominantly chosen by the reaction to create the $\\Lambda(1405)$, one obtains one shape or another from the superposition of the two resonances with different weights. Certainly, if one finds a reaction in which the $\\Lambda(1405)$ is basically created through the entrance channel $\\bar KN$, this will give more weight to the narrow, higher energy resonance, and one should see a spectrum peaking around 1420 MeV. The problem is that the $\\bar KN$ threshold is already above the resonance mass. An ideal method to produce the narrow resonance is the $K^- p \\to \\gamma \\Lambda(1405)$ reaction, since the mechanism where the photon is radiated from the initial $K^-$ is dominant. Then the emission of the photon reduces the $K^-$ energy, such as to be able to produce the  $\\Lambda(1405)$, and one ensures the creation of the resonance from $K^- p$. The theoretical study for this reaction was done in \\cite{Nacher:1999ni}, but the experiment is not yet done. Conversely, the study done in \\cite{Hyodo:2003jw} for the $\\pi^-p\\to K^0\\pi\\Sigma$ reaction showed that this was dominated by the $\\pi \\Sigma$ entrance channel and this justified the peak seen in the $\\pi \\Sigma$ invariant mass at around 1400 MeV in the experiment \\cite{Thomas:1973uh}. Other reactions like the  $p p \\to K^+ p \\pi^0 \\Sigma^0$ \\cite{Zychor:2007gf}, require a mixture of both resonances as seen in  \\cite{Geng:2007vm}.\n One of the strongest support for the idea of the two $\\Lambda(1405)$ states comes from the experiment of \\cite{Prakhov:2004an}, $K^- p \\to \\pi^0 \\pi^0 \\Sigma^0$, and the subsequent theoretical analysis of the reaction done in \\cite{magas}. In this case one of the $\\pi^0$ is ``radiated\" from the initial proton, the $K^- p$ system loses energy, and as a consequence the $\\Lambda(1405)$ production is initiated by the $K^- p$ system, exciting the higher mass state that couples mostly to the the $K^- p$ system and producing a peak in the $\\pi \\Sigma$ spectrum around 1420 MeV. \n \n Another great support for the idea of the two $\\Lambda(1405)$ states comes from the $K^{-} d \\to \\pi\\Sigma n$ reaction \\cite{Braun:1977wd} with kaons in flight, where a clear peak is seen in the $\\pi \\Sigma$ spectrum around 1420 MeV. This reaction was studied recently in \\cite{Jido:2009jf} and it was found that the rescattering mechanism of the $K^-$, in which the $K^-$ scatters on the neutron, loses some energy, propagates to the proton and produces the $\\Lambda(1405)$ on the proton target, dominates the reaction in this invariant mass region. Since  the production of the $\\Lambda(1405)$ is done from the $K^- p$ channel, it also excites predominantly the higher mass $\\Lambda(1405)$ state, which shows up in the \n$\\pi \\Sigma$ spectrum with a peak around 1420 MeV. \n\nThe detail of kaons in flight is important. Indeed, in \\cite{Jido:2009jf} it was shown that, together with the double scattering, where the $\\Lambda(1405)$ shows up cleanly, there is the contribution of the single scattering, the impulse approximation. This contribution peaks around a value of the $\\pi\\Sigma$ invariant mass, corresponding to the invariant mass of the kaon in flight and one nucleon at rest. If one has sufficient kaon kinetic energy this peak and the one of the \n $\\Lambda(1405)$ are quite separated. If one goes at threshold with kaons at rest, the single scattering peak is around $m_K +M_N$, does not have resonant structure and by interference with the double scattering distorts the shape of the  $\\Lambda(1405)$ contribution, and the reaction is not suited to investigate the properties of this resonance.  In \\cite{Esmaili:2009rf} a different opinion is expressed and claims are made that the impulse approximation can provide the shape of the resonance. We argue below that this is not the case. Indeed, one can get $K^- p $ invariant masses below threshold for kaons at rest, only through the Fermi motion of the deuteron. Yet, to reach invariant masses of the order of 1400 MeV, down from the threshold of 1432 MeV, one needs momentum components of the order of 300 MeV\/c. Obviously, our knowledge of the deuteron wave function for such large momentum components is extremely poor. In fact, standard wave functions of the deuteron have dropped by two orders of magnitude at 300 MeV\/c, with appreciable differences between different models. This is four orders of magnitude in the cross section, where the uncertainties are certainly not smaller than one order of magnitude. Even the concept of wave function, when  one has a transfer of energy of 32 MeV to a second nucleon, is ambiguous. Trying to reconstruct a shape of the resonance, by dividing an experimental cross section by the theoretical deuteron wave function squared has very large uncertainties and should not be used. The possible use of an ``experimental\" wave function for these high momentum components should be equally discouraged, since processes involving such momentum transfers have contributions from two body mechanisms difficult to quantify. \n \n   DAFNE produces intense kaon beams from $\\phi$ decay. Recently the $\\phi$ mesons can even be produced with a small momentum of 12 MeV\/c, by means of which the kaons have a range of 80 MeV\/c to 130 MeV\/c when they reach the target. We shall take advantage of this and investigate the production process for the most energetic kaons, looking at different kinematical conditions that maximize the clean production of the resonance. \n\n\n\n\n\n\n\n\n\\section{Formulation}\n\n\\label{sec:form}\n\nWe briefly sketch here the formalism of \\cite{Jido:2009jf} for this reaction. Further details can be seen in that paper.\n\n\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=6.5cm]{fig1.eps}}\n\\caption{Kinematics of the $K^{-} d \\to \\pi \\Sigma n$. \\label{fig1}}\n\\end{figure}\n\n\nWe consider $\\Lambda(1405)$ production induced by $K^{-}$ with a deuteron\ntarget, $K^{-} d \\rightarrow \\Lambda(1405) n$. \nThe $\\Lambda(1405)$ produced in this reaction decays into $\\pi \\Sigma$ with $I=0$\nas shown in fig.~\\ref{fig1}. The $\\Lambda(1405)$ is identified by the $\\pi \\Sigma$ invariant mass spectra of this reaction. \nFigure~\\ref{fig1} also gives the kinematical variables of\nthe initial and final particles. The kinematics of the three-body final state is completely\nfixed by five variables, the $\\pi\\Sigma$ invariant mass $M_{\\pi\\Sigma}$, the neutron \nsolid angle $\\Omega_{n}$ in the c.m.\\ frame and the pion solid angle $\\Omega_{\\pi}^{*}$\nin the rest frame of $\\pi$ and $\\Sigma$~\\cite{Amsler:2008zz}. \nThus the differential cross section of this reaction can be written as\n\\begin{equation}\nd \\sigma = \\frac{1}{ (2\\pi)^{5}} \n\\frac{M_{d}M_{\\Sigma} M_{n}}{4k_{\\rm c.m.}E_{\\rm c.m.}^{2}}\n   \\, |{\\cal T}|^{2}\n   |\\vec p_{\\pi}^{\\, *}|\\,\n   |\\vec p_{n}|\\, dM_{\\pi\\Sigma} d\\Omega_{\\pi}^{\\,*} d\\Omega_{n}\n   \\label{eq:difcross}\n\\end{equation}\nwhere $\\cal T$ is the $T$-matrix of this reaction, \n$E_{\\rm c.m.}$ is the center of mass energy, $k_{c.m.}$ is the kaon CM momentum\nand\n$\\vec p_{\\pi}^{\\, *} $ is the pion momentum in the rest frame of $\\pi$ and $\\Sigma$.\nThe pion momentum $|\\vec p_{\\pi}^{\\, *}|$ in the $\\pi \\Sigma$ rest frame \ncan be fixed by the invariant mass $M_{\\pi\\Sigma}$.\n\n\n\\begin{figure}\n\\begin{center}\n\\centerline{\\includegraphics[width=8.5cm]{fig2.eps}}\n\\caption{Diagrams for the calculation of the $K^{-}d \\to \\pi\\Sigma n$ reaction.\n$T_{1}$ and $T_{2}$ denote the scattering amplitudes for $\\bar KN \\to \\bar KN$\nand $\\bar K N \\to \\pi \\Sigma$, respectively.  \\label{fig2}}\n\\end{center}\n\\end{figure}\n\n\nThe $\\Lambda(1405)$ production is investigated by \nlimiting the kinematics with the invariant mass of the final $\\pi \\Sigma$ state\naround 1350 to 1450 MeV, in which the resonating $\\pi\\Sigma$ forms the \n$\\Lambda(1405)$\nand the resonance contribution may dominate the cross section.  \n\nIn this $\\Lambda(1405)$ dominance\napproximation, we have three diagrams for this reaction as shown in fig.~\\ref{fig2}.\nThe left diagram of fig.~\\ref{fig2} expresses  the $\\Lambda(1405)$ production \nin the impulse approximation. We refer to this diagram as direct production process.  The middle and right diagrams are for two-step\nprocesses with $\\bar K$ exchange. We refer to these diagrams \nas double scattering diagrams. \n\nIn fig.~\\ref{fig2}, $T_{1}$ and $T_{2}$ denote $s$-wave scattering amplitudes of \n$\\bar K N \\to \\bar KN$ and $\\bar K N \\to \\pi \\Sigma$, respectively. These amplitudes \nare calculated in a coupled-channel approach based on chiral dynamics. \n\n\n\nThe ${\\cal T}$  matrix appearing in Eq. (\\ref{eq:difcross}) is given by the sum of the contribution of the three diagrams of  fig.~\\ref{fig2}\n\n\\begin{equation}\n   {\\cal T} =  {\\cal T}_{1} + {\\cal T}_{2} + {\\cal T}_{3}\n\\end{equation}\nwhere the different amplitudes are given by \n\n\\begin{equation}\n  {\\cal T}_{1} = T_{K^{-}p \\rightarrow \\pi \\Sigma}(M_{\\pi\\Sigma}) \\, \\varphi(\\vec p_{n} - \\frac{\\vec p_{d}}{2}). \\label{eq:T1}\n\\end{equation}\n\n\\begin{eqnarray}\n  {\\cal T}_{2}& =&  T_{K^{-}p \\rightarrow \\pi \\Sigma}(M_{\\pi\\Sigma}) \n \\int \\frac{d^{3}q}{(2\\pi)^{3}} \\frac{\\tilde \\varphi (\\vec q+\\vec p_{n}-\\vec k - \\frac{\\vec p_{d}}{2})}{q^{2}-m_{K}^{2} + i\\epsilon}\n   \\nonumber \\\\ && \\times\n T_{K^{-}n \\rightarrow K^{-}n}(W_{1}) \\ . \\label{eq:T2}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n  {\\cal T}_{3}& =& - T_{\\bar K^{0}n \\rightarrow \\pi \\Sigma}(M_{\\pi\\Sigma}) \n  \\int \\frac{d^{3}q}{(2\\pi)^{3}} \\frac{\\tilde \\varphi (\\vec q+\\vec p_{n}-\\vec k - \\frac{\\vec p_{d}}{2})}{q^{2}-m_{K}^{2} + i\\epsilon}\n  \\nonumber \\\\ && \\times\n  T_{K^{-}p \\rightarrow \\bar K^{0}n}(W_{1})  \\ . \\label{eq:T3}\n\\end{eqnarray}\nwith $\\varphi(\\vec p_{n} - \\frac{\\vec p_{d}}{2})$ the deuteron wave function in momentum space and \n\\begin{eqnarray}\n   q^{0} &=& M_{N} + k^{0} - p_{n}^{0}\\ , \\\\\n   W_{1} &=& \\sqrt{(q^{0}+p_{n}^{0})^{2}-(\\vec q + \\vec p_{n})^{2}} \\ .\n\\end{eqnarray}\nFor $q^0$ we have assumed that the deuteron at rest has energy \n$2M_N -B$, and we have taken half of it for one nucleon, \nneglecting the small binding energy. The variable $W_1$ depends, \nhowever, on the running $\\vec{q}$ variable.\n\nIt should be noted, as it was also the case in \\cite{Jido:2009jf}, \nthat taking an average value of $\\vec q$,  we can fix $W_{1}$ by the \nexternal variables as\n\\begin{equation}\n   W_{1}  = \\sqrt{(M_{N}+k^{0})^{2} - \\vec k^{\\, 2}},\n   \\label{eq:W}\n\\end{equation}\nwhich allows us to take the $T_{\\bar KN \\to \\bar KN}$ amplitude \nout of the loop integral and \nproduces results practically equal to those obtained with the values of these variables depending on the loop variable. \nNote also that $T_{\\bar KN \\to \\pi\\Sigma}$, which is the amplitude producing \nthe resonance shape, has its argument fixed in all cases by the external  \n$\\pi \\Sigma$.\n\nThe $T$-matrix for the diagram 1 given in fig.~\\ref{fig2} has been calculated \nin the impulse approximation in which the incident $K^{-}$ and the proton in \nthe deuteron produce the $\\pi \\Sigma$ and the neutron behaves \nas a spectator of the reaction. As for the external particles, we have taken\n the wave functions of the incident kaon and the particles in the final state as plane waves.  \n\n\nThe double scattering requires a loop integral over the $\\vec q$ variable in \nEqs.~(\\ref{eq:T2}) and (\\ref{eq:T3}) with the deuteron wave function. \nAlthough the procedure is standard in multiple collision in nuclei, we find it appropriate to \nsketch the connection to a conventional loop evaluation with Feynman diagrams. \nThe loop integral would be\n\\begin{equation}\n    \\int \\frac{d^{3} q}{(2\\pi)^{3}} \\frac{1}{E_{\\rm tot} - k^{0} -H_{0NN}} t_{NN} \\phi_{NN}\n    \\frac{T_{\\bar KN \\to \\bar KN} T_{\\bar KN \\to \\pi\\Sigma}}{q^{2} - m_{K}^{2} + i \\epsilon} \\ ,\n\\end{equation}\nwhere $H_{0}$ is the free $NN$ Hamiltonian, $t_{NN}$ the appropriate $NN$ scattering\nmatrix and $\\phi_{NN}$ the free $NN$ wave function. By using the definition of \nthe scattering matrix \n\\begin{equation}\n   t_{NN} \\phi_{NN} = V_{NN} \\Psi_{NN}\n\\end{equation}\nwhere $V_{NN}$ is the $NN$ interaction potential and \n$\\Psi_{NN}$ is the actual $NN$ wave function, the integral becomes \n\\begin{equation}\n  \\int \\frac{d^{3} q}{(2\\pi)^{3}} \\frac{1}{E_{NN}  -H_{0NN}} V_{NN} \\Psi_{NN}\n    \\frac{T_{\\bar KN \\to \\bar KN} T_{\\bar KN \\to \\pi\\Sigma}}{q^{2} - m_{K}^{2} + i \\epsilon} \\ .\n\\end{equation}\nNow taking into account the Schr\\\"odinger equation \n\\begin{equation}\n  \\Psi_{NN} = \\frac{1}{E_{NN}-H_{0NN}} V_{NN} \\Psi_{NN}  \\ ,\n\\end{equation}\none rewrites the integral as \n\\begin{equation}\n    \\int \\frac{d^{3} q}{(2\\pi)^{3}} \\Psi_{NN} \\frac{T_{\\bar KN \\to \\bar KN} T_{\\bar KN \\to \\pi\\Sigma}}{q^{2} - m_{K}^{2} + i \\epsilon}\n\\end{equation}\nwhich is the integral that appears in Eqs.~(\\ref{eq:T2}) and (\\ref{eq:T3}).\n\n \n\\section{Results}\n\n In fig.~\\ref{fig3} we show $d \\sigma\/dM_{\\pi \\Sigma}$ for $\\pi^+ \\Sigma^-$, $\\pi^- \\Sigma^+$ and $\\pi^0 \\Sigma^0$ production for 130 MeV\/c $K^{-}$ momenta in the lab.~frame. What we see is a clear dominance of the single scattering.\n However, we observe a certain structure, more visible in the $\\pi^+ \\Sigma^-$ spectrum, with a peak \nat the invariant mass squared, $M_{\\pi\\Sigma}^{2}=(k+p_{N})^{2}$, with $p_{N}$ at rest in the deuteron rest frame, where the deuteron wave function has its largest weight. This peak has to do with the deuteron wave function and not with the amplitude $T_{K^{-}p\\to \\pi\\Sigma}$ of Eq.~(\\ref{eq:T1}). Then we see\nanother peak in  the region of 1420~MeV. \nThis second peak comes from the shape of the $T_{K^{-}p\\to \\pi\\Sigma}$ amplitude of Eq.~(\\ref{eq:T1}), which reflects the excitation of the higher energy $\\Lambda(1405)$ resonance appearing around 1420~MeV.\nWe also see that the mechanism of double scattering also peaks around 1420 MeV and there is interference with the single scattering amplitude which, nevertheless, leaves the peak at about the same position. Yet, since the shape of the double scattering is cleaner and does not have the double shoulder, it would be interesting to find some situation in which it becomes dominant. It is easy to find such kinematical set up, since in the impulse approximation the neutron is a spectator and goes backwards in the $K^- d$ CM frame. If we demand that the neutron goes forward we shall be reducing drastically the single scattering contribution, giving room to the double scattering one. The results of imposing angular cuts can be seen in fig.~\\ref{fig4}, \nin which $\\theta_{\\rm c.m.}^{n}$ is the angle \nbetween the outgoing neutron and the incident $K^{-}$ in the CM frame.  \nWe can see that for large angles of the neutron the impulse approximation is largely dominant, yet, as soon as we make the angles of the neutron smaller, the contribution of the single scattering becomes gradually smaller and at angles smaller than 30 degrees the double scattering has become dominant, with the total cross section showing a clean peak around 1420~MeV. To make it more precise, in fig.~\\ref{fig5} we show the quantity $d \\sigma\/dM_{\\pi \\Sigma}\\,d\\cos(\\theta_{\\rm c.m.}^{n})$ for different angles and we find that for angles below 32 degrees the double scattering contribution is dominant. Hence, for practical purposes, and in order to gain statistics, gathering all events with angles smaller than 32 degrees in the CM is ideal in order to show a clean contribution of the $\\Lambda(1405)$. The cross sections of 0.1 mb\/MeV, from Fig. \\ref{fig4}, are large enough to be observed in the DAFNE set up. A larger span of angles would also show a clear peak, with gain in statistics, since even at 60 degrees the peak is visible, as one can appreciate in figs.~\\ref{fig4} and~\\ref{fig5}.\n \n\n\\begin{figure}\n\\begin{center}\n\\centerline{\\includegraphics[width=7cm]{ChUM_PiSig_dia_130.eps}}\n\\caption{$\\pi\\Sigma$ invariant mass spectra of $K^{-}d \\to \\pi \\Sigma n$\nwith 130 MeV\/c of $K^{-}$ incident momentum. The three panels correspond to \n$\\pi\\Sigma$ spectra for different $\\pi\\Sigma$ charge combinations,\n$\\pi^{+}\\Sigma^{-}$ (upper panel), $\\pi^{-}\\Sigma^{+}$ (middle panel)\nand $\\pi^{0} \\Sigma^{0}$ (lower panel). In each panel, the solid line denotes \nthe total contributions of the three diagrams, while the dotted and \ndash-dotted lines show the calculations \nfrom the single and double\nscatterings, respectively. \n}\n\\label{fig3}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\centerline{\\includegraphics[width=8.5cm]{ChUM_Pi0Sig0_ang_int_130.eps}}\n\\caption{$\\pi\\Sigma$ invariant mass spectra of $K^{-}d \\to \\pi^{0}\\Sigma^{0}n$\nwith 130 MeV\/c of incident $K^{-}$ momentum imposing angular cuts \nfor the emitted neutron with respect to the incident $K^{-}$ in the CM frame, \n$90^{\\circ}<\\theta^{n}_{\\rm c.m.}<180^{\\circ}$ (up-left),\n$60^{\\circ}<\\theta^{n}_{\\rm c.m.}<90^{\\circ}$ (up-right),\n$30^{\\circ}<\\theta^{n}_{\\rm c.m.}<60^{\\circ}$ (down-left) and\n$0^{\\circ}<\\theta^{n}_{\\rm c.m.}<30^{\\circ}$ (down-right).\nIn each panel, the solid line denotes \nthe total contributions of the three diagrams, while the dotted and \ndash-dotted lines show the calculations \nfrom the single and double\nscatterings, respectively. \n}\n\\label{fig4}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\centerline{\\includegraphics[width=8.5cm]{ChUM_Pi0Sig0_ang_130.eps}}\n\\caption{Differential $\\pi\\Sigma$ invariant mass spectra of $K^{-}d \\to \\pi^{0}\\Sigma^{0}n$\nfor 130 MeV\/c of incident $K^{-}$ momentum fixing the angle \nbetween the emitted neutron and the incident $K^{-}$ in the CM frame.  \nIn each panel, the solid line denotes \nthe total contributions of the three diagrams, while the dotted and \ndash-dotted lines show the calculations \nfrom the single and double\nscatterings, respectively. \n}\n\\label{fig5}\n\\end{center}\n\\end{figure}\n\n \nBy producing the $\\phi$ with a finite small momentum in the new DAFNE set up, one obtains kaons with larger momentum than before, when the $\\phi$ was produced at rest. In what follows, we show the results for two other kaon momenta, within the present kaon momentum range, to emphasize the value of taking the kaons with larger momentum. \nIn fig.~\\ref{fig6}, we show the results equivalent to fig.~\\ref{fig5} for momenta of the kaons 100 MeV\/c and 80 MeV\/c.  At $k= 100$ MeV\/c we can see in fig.~\\ref{fig6} that the contribution of the single scattering dominates the cross section at neutron backward angles, and one finds two peaks, one coming from the resonance shape and the other one from the threshold effect. The two peaks are closer among themselves than in the case of  $k= 130$ MeV\/c. The final shape should then not be confused with the shape of the resonance. If one goes to neutron forward angles one can see that the single scattering is still as large as the double scattering one, and it shows its two peak structure, such that the shape of the global distribution is not the shape of the resonance.  This momentum is not good to extract the properties of the $\\Lambda(1405)$ resonance. The situation becomes worse for $k= 80$ MeV\/c, as one can see in fig.~\\ref{fig7}. Here, even at forward angles, the single scattering dominates the cross section and the shape that comes out does not reflect the shape of the resonance. Fortunately, the situation for  $k=$ 130 MeV\/c is satisfactory, with a clear dominance of the double scattering process and still a visible separation of the resonance and threshold peak contributions of the single scattering, such that the resulting peak around 1420 MeV clearly reflects the shape of the upper energy $\\Lambda(1405)$ state peaking at 1420 MeV. \n Hence, the small increase in the kaon momentum from the former situation, where the kaons came from $\\phi$ decay at rest,\nis most welcome and helps to get a cleaner signal of the  $\\Lambda(1405)$ resonance. \n\n One may wonder that, since double scattering becomes more important than single scattering in the kinematic conditions chosen, the triple scattering could also be important. However, as a general rule for inclusive processes, the single scattering is\n more important than the double and this one more important than the triple. We have observed this here, since the strength of the peak of single scattering is much bigger than that of double scattering. Yet, the\n single scattering has a very limited kinematics, a very restricted phase space,\n but this is not longer the case for double and triple scattering. Then the\n selection of forward neutron angles forces the single scattering out of its phase\n space, except for the finite momentum components of the deuteron, but does not\n put constraints on double scattering nor triple scattering, such that the\n triple scattering will remain small compared to the double.\n   Note that there is an important difference between the case of the break up process that we have here and the coherent scattering of $K^-$ with deuteron without deuteron breakup. In the latter case the two nucleons in the deuteron stick together and it is relatively easy for the kaons to rescatter. This is indeed the case as shown in \\cite{Toker:1981zh,Barrett:1999cw,kamalov}. But in the inclusive process, after the first scattering the nucleons move apart making more difficult the rescattering of the kaons with this pair of nucleons.\n\nWe also do not consider double scattering diagrams\nwith pion exchange in which the $\\pi$  and the $\\Sigma$ are emitted\nseparately in the \n$T_{1}$ and $T_{2}$ amplitudes given in fig.~\\ref{fig2},\nrespectively. It was discussed in \\cite{Jido:2009jf} that\nsuch diagrams would give a smooth background in the $\\pi \\Sigma$\ninvariant mass spectra.\n\n\n\n\\begin{figure}\n\\begin{center}\n\\centerline{\\includegraphics[width=8.5cm]{ChUM_Pi0Sig0_ang_100.eps}}\n\\caption{Differential $\\pi\\Sigma$ invariant mass spectra of $K^{-}d \\to \\pi^{0}\\Sigma^{0}n$\nfor 100 MeV\/c of incident $K^{-}$ momentum fixing the angle\nbetween the emitted neutron and the incident $K^{-}$ in the CM frame.  \nIn each panel, the solid line denotes \nthe total contributions of the three diagrams, while the dotted and \ndash-dotted lines show the calculations\nfrom the single and double\nscatterings, respectively.  \n}\n\\label{fig6}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\centerline{\\includegraphics[width=8.5cm]{ChUM_Pi0Sig0_ang_80.eps}}\n\\caption{Differential $\\pi\\Sigma$ invariant mass spectra of $K^{-}d \\to \\pi^{0}\\Sigma^{0}n$\nfor 80 MeV\/c of incident $K^{-}$ momentum fixing the angle between the emitted neutron and the incident $K^{-}$ in the CM frame.\nIn each panel, the solid line denotes \nthe total contributions of the three diagrams, while the dotted and \ndash-dotted lines show the calculations \nfrom the single and double\nscatterings, respectively. \n}\n\\label{fig7}\n\\end{center}\n\\end{figure}\n\n\\section{Summary}\n\\label{sec:summary}\n\nWe have studied the $K^{-}$ induced production of $\\Lambda(1405)$\nwith a deuteron target in the reaction $K^{-} d \\to \\pi \\Sigma n$, with the conditions of the DAFNE set up, where kaons are produced from the decay of slow moving $\\phi$ mesons. The process proved well suited to investigate the predicted $\\Lambda(1405)$ state around 1420 MeV, which couples mostly to $\\bar{K}N$, since in the $K^{-} d \\to \\pi \\Sigma n$ process,\nthe $\\Lambda(1405)$ resonance is produced by the $\\bar KN$ channel.\n\n We found that for the most energetic kaons coming from the decay of the moving $\\phi$ mesons, both the single scattering and the double scattering gave rise to a peak in the region of 1420 MeV for the spectrum of $\\pi \\Sigma $ invariant masses. Yet, it was also found that the contribution of single scattering can be easily distorted by a peak around threshold of the $K^-p$ channel. In order to obtain a cleaner signal, we evaluated the cross section at forward angles of the neutron and we found that the contribution of single scattering is drastically reduced and that of the double scattering takes over, leading to a cleaner signal of the $\\Lambda(1405)$ resonance. The general rule is that having kaons in flight, at moderate energies of the kaons, becomes advantageous to distinguish the  $\\Lambda(1405)$ signal, since this automatically reduces the contribution of the single scattering versus the double scattering one. This is not so because the contribution of the single scattering is smaller, but because it peaks at a different region of invariant masses. Yet, we could see that, given the energy constraints at DAFNE, it was still possible to drastically reduce the contribution of single scattering at the expense of looking at a kinematical region in the phase space, with neutrons forward, where the single scattering contribution is reduced but the double scattering, which has more flexibility on momentum sharing, is not so much affected.\n \n  In view of the results obtained here we can encourage to do this experiment at DAFNE where the fluxes of kaons are large enough to make the experiment feasible. Indeed, a first estimate \\cite{nevio} indicates that with the energy resolution for charged particles of 2-4 \\% of the kinetic energy and 4.2 MeV for neutrons for the energies of the experiment, measured with 20-35 \\% efficiency in the KLOE calorimeter (not necessarily needed if the $\\pi$ and $\\Sigma$ are detected), the shape of the distribution can be obtained with the needed resolution. Taking into account present fluxes and the calculated cross sections one would need about eight months of beam time \\cite{nevio}.\n\n\n\\section*{Acknowledgements}\nOne of us, E.~O.~wishes to acknowledge useful discussions with Nevio Grion and Alessandra Filippi. This work was partly supported by\nthe Grant-in-Aid for Scientific Research\nfrom MEXT and JSPS (Nos.\n   22105507,    \n   22740161,    \n   20540273        \n   and 22-3389),  \nthe collaboration agreement between the JSPS of Japan and the CSIC of Spain,  \nand the Grant-in-Aid for the Global COE Program \n``The Next Generation of Physics, Spun from Universality and Emergence\" \nfrom MEXT of Japan.\nThis work was also supported in part by DGICYT contract number\nFIS2006-03438. \nWe acknowledge the support of the European Community-Research\nInfrastructure Integrating Activity\n``Study of Strongly Interacting Matter\" \n(acronym HadronPhysics2, Grant Agreement n. 227431) under \nthe Seventh Framework Programme of EU.\nOne of us, T.~S.~acknowledges the support by the Grand-in-Aid for JSPS fellows. \nThis work was supported in part by DFG (SFB\/TR 16, ``Subnuclear Structure of Matter'').\nThis work was done under\nthe Yukawa International Program for Quark-hadron Sciences (YIPQS).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Re-investigating Intuitive\nInterpretability}\\label{re-investigating-intuitive-interpretability}\n\nPhilip Agre has argued that ``technology at present is covert\nphilosophy'' (Agre, 1997). While the scope of this claim is certainly\ndebatable, in the quest for interpretable machine learning models,\ncertain philosophical issues are evident, above all the fact that\ninterpretability itself is an intuitive notion. As Wolfgang Iser\nremarks: ``For a long time, interpretation was taken for an activity\nthat did not seem to require analysis of its own procedures. There was a\ntacit assumption that it came naturally, not least because human beings\nlive by constantly interpreting.'' (Iser, 2000)\n\nNevertheless, Kim and Doshi-Velez (2017) and many others have shown that\ninterpretability can be transformed into a more rigorous notion. While\nmost investigations into the interpretability of machine learning models\nthus focus on the further development of this rigorous notion of\ninterpretability, I suggest that a re-investigation of the intuitive\nnotion of interpretability can help to better understand the limits of\ninterpretability in general.\n\nWhen we talk about intuitive interpretability in the context of machine\nlearning, we assume a Cartesian concept of intuition that posits\nintuitive concepts as rational concepts, and thus intuition as an\nadequate measure of reality. The statement ``I know it when I see it'',\nwhich is often employed to illustrate this concept, indicates the\ndependency of such intuition on visualization. To intuitively understand\na machine learning model, we need to visualize it, make it accessible to\nthe senses. This process, however, is not as straightforward as it\nseems. I argue that specifically for machine learning models,\nvisualization -- and thus intuitive interpretation -- necessarily\nimplies two levels of pre-interpretation.\n\n\\section{Intuitive Interpretability Depends on Dimensionality\nReduction}\\label{intuitive-interpretability-depends-on-dimensionality-reduction}\n\nMachine learning models operate in high-dimensional vector spaces.\nHigh-dimensional vector spaces are geometrically counter-intuitive.\nWhile low-dimensional vector spaces can always be intuitively correlated\nwith our physical reality, with the existence of objects in space and\ntime, high-dimensional vector spaces have no intuitive equivalent in the\nreal world. Beyond this general inaccessibility, however,\nhigh-dimensional vector spaces also specifically impede interpretation,\nas distances between data points have a tendency to lose their\n\\emph{meaning} (Beyer et al., 1999) -- this is commonly known as the\n``curse of dimensionality''. Making a high-dimensional vector space\nintuitively interpretable thus requires its mathematical\npre-interpretation, its representation in human terms, i.e.~usually in\nno more than three dimensions.\n\nWhile there is certainly a quantifiable limit to the damage\ndimensionality reduction can inflict (Johnson and Lindenstrauss, 1984)\nit is nevertheless important to acknowledge the reason for the\ninevitability of this mathematical pre-interpretation. Internal states\nof machine learning models are non-concepts, concepts that have no\nintuitive equivalent in the real world and that can only be represented\n\\emph{in terms of what they are not}. This notion of the non-concept\nwill guide our further investigation of intuitive interpretability.\n\n\\section{Intuitive Interpretability Depends on\nRegularization}\\label{intuitive-interpretability-depends-on-regularization}\n\nArtificial neural networks trained on image data are notoriously opaque.\nParticularly for deep convolutional neural networks (Krizhevsky et al.,\n2012), it is very hard to infer from the training dataset and the final\nweights of the fully trained neural network how exactly the network\nmakes its decisions. Many different approaches to this problem have been\nsuggested, most prominently two types of feature visualization:\nactivation maximization (Zeiler and Fergus, 2014, Simonyan et al.\n(2014), Mahendran and Vedaldi (2015), Mahendran and Vedaldi (2016),\nNguyen, Dosovitskiy, et al. (2016), Nguyen, Yosinski, et al. (2016)) and\nsaliency maps, a technique also called attribution (Olah et al., 2017,\nSimonyan et al. (2014), Zeiler and Fergus (2014)). We will focus here on\nactivation maximization.\n\nNaive optimizations of an image to maximally activate a specific\n``neuron'', i.e.~the part of an artificial neural network that encodes a\nspecific feature, often result in noise and ``nonsensical high-frequency\npatterns'' (Olah et al., 2017) -- patterns that are without meaning and\nare thus, again, inaccessible to an intuitive interpretation. The\nregularization of this optimization is thus another pre-interpetation\nthat is necessary to establish intuitive interpretability. The goal of\nactivation maximizations is thus to generate ``natural'' pre-images\n(Mahendran and Vedaldi, 2015, Mahendran and Vedaldi (2016)) -- images\nthat are visual representations of intermediate stages in the neural\nnetwork, expressed \\emph{in terms of} a set of natural images. This\nregularization is achieved by introducing natural image priors into the\nobjective function.\n\n\\section{Non-Concepts and the Place of Semantic\nInformation}\\label{non-concepts-and-the-place-of-semantic-information}\n\nThe natural pre-images of activation maximization usually consist of an\narbitrary ``mix'' of different representations. This mix of\nrepresentations can either be a set of images that each show a different\naspect of the activation maximization, or a ``blend'' of different\nimages, i.e.~a single image that maximizes different aspects of the\nneuron. Most recently, such ``multifaceted'' representations have been\nimproved significantly (Nguyen, Dosovitskiy, et al., 2016, Nguyen,\nYosinski, et al. (2016)) through the automatic generation of natural\nimage priors with the help of an additional, generative adversarial\nneural network. Other approaches have used techniques from style\ntransfer to likewise increase the ``diversity'' (Olah et al., 2017) of\nthe visualization.\n\nHowever, as Olah et al. (2017) observe, many of the resulting images are\n``strange mixtures of ideas'' suggesting that single neurons are not\nnecessarily the right semantic units for understanding neural nets.\" In\nfact, as Szegedy et al. (2013) showed, looking for meaningful features\ndoes not necessarily lead to more meaningful visualizations than looking\nfor any features, i.e.~for arbitrary activation maximizations. This is\nalso the reason for the effectiveness of many adversarial strategies (Su\net al., 2017, Papernot et al. (2017), Kurakin et al. (2016), Goodfellow\net al. (2014)).\n\nIn other words, not only is the representation of non-concepts mediated\ntwice, by means of dimensionality reduction and regularization, it is\nalso questionable if non-concepts can be approximated at all in human\nterms. Szegedy et al. (2013) suggest that the entire space of\nactivations, rather than the individual units contain most of the the\nsemantic information.\\footnote{While we will not develop this idea\n  further within this limited context, it is worth noting that the\n  findings in Szegedy et al. (2013) in relation to the notion of\n  non-concepts mirror long-standing discussions in the humanities on\n  meaning and interpretation, with the most prominent concept being\n  Jacques Derrida's notion of diff\u00e9rance (Derrida, 1982).} As also\npointed out in Szegedy et al. (2013), a ``similar but even stronger\nconclusion'' was reached for word embedding models, and in fact the\nconcept of a distributed semantic structure becomes even more obvious\nwhen we look at the text and not the image domain.\n\nWord embedding models (Mikolov et al., 2013) employ shallow artificial\nneural networks to construct high-dimensional vector spaces that not\nonly reflect syntactic but also semantic properties of the source\ncorpus. Most prominently, word embedding models are able to solve\nanalogy queries, like ``what is to woman what king is to man''. This is\nachieved by not extending, but reducing the dimensionality of the vector\nspace in relation to the number of n-grams in the source corpus.\nAccordingly, no vector represents just a single n-gram. Instead, the\ntotality of vectors represents the totality of the semantic structure of\nthe source corpus. This distributed semantic structure, however, has\npeculiar consequences. The solution to an analogy query is given by the\nmodel not as a definite answer, but as a hierarchy of answers. Why?\nSimply because there are no ``intermediate'' words. If the best possible\nanalogy is a (new) data point right in between two (existing) data\npoints representing n-grams in the source corpus vocabulary, the best\npossible solution to the analogy query is neither of them, but it still\ncan only be described \\emph{in terms of} them. Even if the input\nvocabulary consisted of all words in the English language, the solution\nto the analogy query could still be a data point that is ``in between\neverything'' but has no equivalent in the real world -- a non-concept.\nEvery computational solution to an analogy task is thus, ironically,\nitself an analogy.\n\n\\section{Non-Concepts as a Critical Technical Practice: Revealing Human\nBias}\\label{non-concepts-as-a-critical-technical-practice-revealing-human-bias}\n\nA demonstration of this dilemma of non-concepts, and an example for a\ncritical technical practice based on it, is ``Image Synthesis from\nYahoo's open\\_nsfw''(Goh, 2016), a project by Gabriel Goh. Using the\ntechnique developed in (Nguyen, Dosovitskiy, et al., 2016) Goh produces\nimages that maximally activate certain neurons of a classifier network\ncalled ``open\\_nsfw'', which was created by Yahoo to distinguish\nworkplace-safe (``SFW'') from ``not-safe-for-work'' (``NSFW'') imagery:\na literal mathematical model of ``I know it when I see it''. By\ngenerating sets of images ranging from most to least pornographic, Goh\nproduces some interesting insights into Yahoo's specific interpretation\nof ``nsfw'', and the essence of the concept of pornography. Most\ninteresting, however, are the ``least pornographic'' images. What really\n\\emph{is} the ``opposite'' of pornography? Fully clothed people?\nNon-pornography, again, is a non-concept which has never been defined\nbut through its negation.\n\nExcept in this particular case it hasn't. Goh notes that the least\npornographic images ``all have a distinct pastoral quality -- depictions\nof hills, streams and generally pleasant scenery'' and concludes that,\nmost likely, this is the result of providing negative examples during\ntraining . Apparently, the non-concept of non-pornography was made into\na positive concept -- pastoral landscapes -- to improve the training of\nthe model. This, of course, becomes particularly problematic if, as it\nis the case with open\\_nsfw, a fully trained model is provided without\naccess to the training data. While the Github page for open\\_nsfw\nacknowledges that the ``definition of NSFW is subjective and\ncontextual'', what is at stake here is exactly the opposite: the fact\nthat ``SFW'' is subjective and contextual, and that regardless a very\nspecific notion of ``SFW'' was built into the model. More generally\nspeaking, the approximation of non-concepts with positive concepts\nnecesarily introduces a significant human -- aesthetic -- bias into the\nequation.\n\n\\section{Conclusion}\\label{conclusion}\n\nGoh's project serves to show the non-conceptual structure of machine\nlearning models and the problems this structure creates for intuitive\ninterpretability. While one possible way to address these problems (if\nintuitive interpretability is needed) is to avoid strategies that\npresent singular images as representations of internal model states\naltogether, and instead switch to multitudes of images, there is no\ngeneral solution to the problem of finding human-readable\nrepresentations of non-concepts. Research in interpretability thus has\nto take the non-conceptual structure of machine learning models into\naccount. To make the notion of interpretability more rigorous we have to\nfirst identify where it might still be impaired by intuitive\nconsiderations: we have to consider it precisely in terms of what it is\nnot.\n\n\\section*{References}\n\n\\small\n\nAgre, P.E., 1997. Computation and Human Experience. Cambridge University\nPress.\n\nBeyer, K., Goldstein, J., Ramakrishnan, R., Shaft, U., 1999. When is\n``nearest neighbor'' meaningful?, in: International Conference on\nDatabase Theory. Springer, pp. 217--235.\n\nDerrida, J., 1982. Diff\u00e9rance, in: Margins of Philosophy. University of\nChicago Press.\n\nGoh, G., 2016. Image synthesis from Yahoo's open\\_nsfw. Blog: Gabriel Goh.\n\nGoodfellow, I.J., Shlens, J., Szegedy, C., 2014. Explaining and\nharnessing adversarial examples. arXiv preprint arXiv:1412.6572.\n\nIser, W., 2000. The Range of Interpretation. Columbia University Press,\nNew York, NY.\n\nJohnson, W.B., Lindenstrauss, J., 1984. Extensions of Lipschitz mappings\ninto a Hilbert space. Contemporary Mathematics 26, 1.\n\nKim, B., Doshi-Velez, F., 2017. Towards a rigorous science of\ninterpretable machine learning. arXiv preprint arXiv:1702.08608.\n\nKrizhevsky, A., Sutskever, I., Hinton, G.E., 2012. Imagenet\nclassification with deep convolutional neural networks, in: Advances in\nNeural Information Processing Systems. pp. 1097--1105.\n\nKurakin, A., Goodfellow, I., Bengio, S., 2016. Adversarial examples in\nthe physical world. arXiv preprint arXiv:1607.02533.\n\nMahendran, A., Vedaldi, A., 2016. Visualizing deep convolutional neural\nnetworks using natural pre-images. International Journal of Computer\nVision 120, 233--255.\n\nMahendran, A., Vedaldi, A., 2015. Understanding deep image\nrepresentations by inverting them, in: Proceedings of the IEEE\nConference on Computer Vision and Pattern Recognition. pp. 5188--5196.\n\nMikolov, T., Sutskever, I., Chen, K., Corrado, G.S., Dean, J., 2013.\nDistributed representations of words and phrases and their\ncompositionality, in: Advances in Neural Information Processing Systems.\npp. 3111--3119.\n\nMordvintsev, A., Olah, C., Mike, T., 2015. Inceptionism: Going deeper\ninto neural networks. Google Research Blog.\n\nNguyen, A., Dosovitskiy, A., Yosinski, J., Brox, T., Clune, J., 2016.\nSynthesizing the preferred inputs for neurons in neural networks via\ndeep generator networks, in: Advances in Neural Information Processing\nSystems. pp. 3387--3395.\n\nNguyen, A., Yosinski, J., Clune, J., 2016. Multifaceted feature\nvisualization: Uncovering the different types of features learned by\neach neuron in deep neural networks. arXiv preprint arXiv:1602.03616.\n\nOlah, C., Mordvintsev, A., Schubert, L., 2017. Feature visualization.\nDistill.\n\nPapernot, N., McDaniel, P., Goodfellow, I., Jha, S., Celik, Z.B., Swami,\nA., 2017. Practical black-box attacks against machine learning, in:\nProceedings of the 2017 ACM Asia Conference on Computer and\nCommunications Security. ACM, pp. 506--519.\n\nSimonyan, K., Vedaldi, A., Zisserman, A., 2014. Deep inside\nconvolutional networks: Visualising image classification models and\nsaliency maps. arXiv preprint arXiv:1312.6034.\n\nSu, J., Vargas, D.V., Kouichi, S., 2017. One pixel attack for fooling\ndeep neural networks. arXiv preprint arXiv:1710.08864.\n\nSzegedy, C., Zaremba, W., Sutskever, I., Bruna, J., Erhan, D.,\nGoodfellow, I., Fergus, R., 2013. Intriguing properties of neural\nnetworks. arXiv preprint arXiv:1312.6199.\n\nZeiler, M.D., Fergus, R., 2014. Visualizing and understanding\nconvolutional networks, in: European Conference on Computer Vision.\nSpringer, pp. 818--833.\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{Introduction}\nRecent interest in the analytical properties of \nFeynman diagrams has been motivated by processes at the LHC. \nThe required precision demands the evaluation of a\nhuge number of diagrams having many scales to a high order, \nso that a new branch of mathematics emerges, which we may call \n{\\it the Mathematical Structure of Feynman Diagrams}\n~\\cite{hwa,pham}, which includes elements of algebraic geometry, algebraic \ntopology, the analytical theory of differential equations, multiple \nhypergeometric functions, \nelements of number theory, modular functions and elliptic curves, \nmultidimensional residues, and graph theory.     \nThis mathematical structure has been extensively developed, studied and \napplied. For a more detailed discussion of the oldest results and their relation \nto modern techniques, see\nRefs.\\ \\cite{golubeva}\\ and \\cite{Kalmykov:2008}). \nOne of these approaches is based on the treatment of Feynman \ndiagrams in terms of multiple hypergeometric \nfunctions~\\cite{kershaw}. For example, in the series of \npapers~\\cite{kreimer1,kreimer2,kreimer3}, the one-loop diagrams \nhave been associated with the $R$-function (a particular case \nof the $F_D$-function~\\cite{bateman,slater,srivastava}).\n\n\n\\subsection{Mellin-Barnes representation, \n\tasymptotic expansion, NDIM}\n\\label{Mellin-Barnes representation}\n\nA universal technique based on the Mellin-Barnes representation of Feynman \ndiagrams has been applied to one-loop diagrams in \nRef.\\ \\cite{boos-davydychev,davydychev:1991}\nand to two-loop propagator diagrams in \nRef.\\ \\cite{broadhurst,berends,bauberger,davydychev-grozin,weinzierl:2003}.\\footnote{Several \n\tprograms are available for the automatic generation of the  \n\tMellin-Barnes representation of Feynman \n\tdiagrams~\\cite{ambre,smirnov-smirnov,prausa}.}\nThe multiple Mellin-Barnes representation for a Feynman diagram \nin covariant gauge can be written in the form\n\\begin{eqnarray}\n\t\\Phi({\\bf A},\\vec{B};{\\bf C}, \\vec{D};\\vec{z}) \n\t& = & \n\t\\int_{-i \\infty}^{+i \\infty}\n\t\\phi(\\vec{t}) \n\td\\vec{t} \n\n\n\t= \n\t\\int_{-i \\infty}^{+i \\infty}\n\t\\prod_{a,b,c,r}\n\t\\frac{\\Gamma(\\sum_{i=1}^m A_{ai}t_i \\!+\\! B_a)}{\\Gamma(\\sum_{j=1}^r \n\tC_{bj}t_j \\!+\\! D_{b})}\n\tdt_c z_k^{\\sum_l \\alpha_{kl} t_l}\n\t\\;,\n\t\\nonumber \\\\ \n\t\\label{MB}\n\\end{eqnarray}\nwhere $z_k$ are ratios of Mandelstam variables and \n$A,B,C,D$ are matrices and vectors depending linearly on the \ndimension of space-time $n$ and powers of the propagators. \nClosing the contour of integration on the right\n(on the left), this integral can be presented  \naround zero values of $\\vec{z}$ in the form\n\\begin{eqnarray}\n\t&& \n\t\t\\Phi({\\bf A},\\vec{B};{\\bf C}, \\vec{D};\\vec{z}) \n\t= \n\t\\sum_{\\vec{\\alpha}} f_{\\vec{\\alpha}}\n\tH({\\bf A},\\vec{B};{\\bf C}, \\vec{D};\\vec{z}) \n\t\\vec{z}^{\\;\\vec{\\alpha}} \n\t\\;, \n\t\\label{representation}\n\\end{eqnarray}\t\nwhere the coefficients \n$\nf_{\\vec{\\alpha}}\n$\nare ratios of $\\Gamma$-functions and the functions $H$ are \nHorn-type hypergeometric functions~\\cite{horn} \n(see Section~\\ref{Horn-Functions} for details).\nThe analytic continuation of the hypergeometric functions \n$H(\\vec{z})$ into another region of the variables $\\vec{z}$\ncan be constructed via the integral representation \n(when available)~\\cite{wu,mano}, \n$H(\\vec{z}) \\to H(1-\\vec{z})$. \nHowever, for more complicated cases of Horn-type hypergeometric\nfunctions, this type of analytic continuation is still under \nconstruction~\\cite{friot1,friot2}.\n\nA major set of mathematical results (see, for example,~\\cite{bezrodnykh1,bezrodnykh2,bezrodnykh3}) is devoted to the \nconstruction of the analytic continuation of a series around ${z_j}=0$ \nto a series of the form\n$\\frac{z_A}{z_B}$: $H(\\vec{z}) \\to H(\\frac{z_A}{z_B})$, \nwhere the main physical application is the construction of an expansion about  \nLandau singularities $L(\\vec{z})$: $H(\\vec{z}) \\to \nH(L(\\vec{z}))$.\nFor example, the singular locus $L$ of the Appell function $F_4(z_1,z_2)$ \nis $L = \\{ (z_1,z_2) \\in \\mathbb{C}^2 |z_1 z_2 R(z_1,z_2)  =0 \\} \\cup L_\\infty$\nwhere $R(z_1,z_2) =(1\\!-\\!z_1\\!-\\!z_2)^2\\!-\\!4 z_1z_2$,\nand the physically interesting case of an expansion around the singularities \ncorresponds to an analytical continuation \n$F_4(z_1,z_2) \n \\to \n F_4\\left(\\frac{R(z_1,z_2)}{z_1}, \\frac{R(z_1,z_2)}{z_2} \\right)$.\n\nA similar problem, the construction of convergent series \nof multiple Mellin-Barnes integrals in different regions of parameters, \nhas been analyzed in detail for  \nthe case of two variables~\\cite{tsikh,passare,friot}. \nHowever, to our knowledge, there are no systematic analyses of the \nrelation between these series and the singularities of multiple \nMellin-Barnes integrals.  \n\nIt was understood long ago that there is a one-to-one correspondence \nbetween the construction of convergent series from Mellin-Barnes \nintegrals and the asymptotic expansions; see\nRef.\\ \\cite{MB:asymptotic} for example. The available software, \n{\\em e.g.} Ref.\\ \\cite{czakon}, allows the construction of\nthe analytical continuation of a Mellin-Barnes integral in \nthe limit when some of the variables $z$ goto to $0$, or $\\infty$. \nThese are quite useful in the evaluation of Feynman diagrams, but do not \nsolve our problem. The current status of the asymptotic expansions is discussed \nin Ref.\\ ~\\cite{asymptotic1,asymptotic2}. \n\nAnother technique for obtaining a hypergeometric representation \nis the so-called ``Negative Dimensional Integration Method'' (NDIM) \n~\\cite{ndim1,ndim2,ndim3,ndim4,ndim5,ndim6}. \nHowever, it is easy to show~\\cite{ndim} that all available results \nfollow directly from the Mellin-Barnes integrals~\\cite{boos-davydychev}.\n\nFor some Feynman diagrams, the hypergeometric representation follows from \na direct integration of the parametric representation, see \nRef.\\ \\cite{somogy,grozin-kotikov,\nbritto:2015,ablinger:2015,feng1,feng2,yang1,feng3,grozin}.\n\nWe also mention that the ``Symmetries of Feynman Integrals''\nmethod~\\cite{kol1,kol2,kol3} can also be used to obtain the \nhypergeometric representation for some types of diagrams. \n\n\\subsection{About GKZ and Feynman Diagrams}\n\\label{GKZ}\nThere are a number of different though entirely equivalent ways to describe \nhypergeometric functions:\n\\begin{itemize}\n\t\\item\n\tas a multiple series;\n\t\\item \n\tas a solution of a system of differential equations \n(hypergeometric D-module);\n\t\\item \n\tas an integral of the Euler type;\n\t\\item\n\tas a Mellin-Barnes integral. \n\\end{itemize}\nIn a series of papers, Gel'fand, Graev, Kapranov and Zelevinsky \n~\\cite{Gelfand1,Gelfand2,Gelfand3} \n(to mention only a few of their series of papers \ndevoted to the systematic development of this approach) \nhave developed a uniform  approach to the description of hypergeometric \nfunctions \\footnote{\n\tThe detailed discussion of $A$-functions and their properties \n\tis beyond our current consideration.  There are many interesting papers \n\ton that subject, including (to mention only a few) \n\tRefs.\\ \\cite{algorithm2,beukers2,review}.}.\nThe formal solution of the $A$-system is a\nso-called multiple $\\Gamma$-series having the following form:\n$$\n\\sum_{(l_1, \\cdots, l_N) \\in \\mathbf{L}}\n\\frac{ z_1^{l_1+\\gamma_1} \\cdots z_N^{l_N+\\gamma_M} }\n     {\\Gamma(l_1+\\gamma_1+1) \\cdots \\Gamma(l_N+\\gamma_N+1) \n}  \n\\;, \n$$\nwhere $\\Gamma$ is the Euler $\\Gamma$-function \nand the lattice $ \\mathbf{L}$  has rank $d$. \nWhen this formal series has a non-zero radius of convergence, \nit coincides  (up to a factor) \nwith a Horn-type hypergeometric series~\\cite{Gelfand3}\n(see Section~\\ref{Horn-Functions}). \nAny Horn-type hypergeometric function can be written in the form \nof a $\\Gamma$-series by applying the reflection formula \n$\\Gamma(a+n) = (-1)^n \\frac{\\Gamma(a) \\Gamma(1-a)}{\\Gamma(1-a-n)}$. \nMany examples of such a conversion -- \nall Horn-hypergeometric functions of two variables -- \nhave been considered in Ref.\\ \\cite{bod}. \n\nThe Mellin-Barnes representation was beyond Gelfand's consideration. \nIt was worked out later by Fritz Beukers~\\cite{beukers1};\nsee also the recent paper~\\cite{matsubara}. \nBeukers analyzed the Mellin-Barnes  integral\n$$\n\\int \n\\Pi_{i=1}^N \n\\Gamma(-\\gamma_i - \\vec{b}_i \\vec{s})\nv_i^{\\gamma_i+ \\vec{b}_i \\vec{s}} ds \\;, \n$$\nand pointed out that,\nunder the assumption that the Mellin-Barnes integral \nconverges absolutely, it satisfies the set\nof $A$-hypergeometric equations. \nThe domain of convergence \nfor the $A$-hypergeometric series and the associated Mellin-Barnes integrals\nhave been discussed recently in Ref.\\ \\cite{nilsson}. \n\nFollowing Beuker's results, we conclude that any \nFeynman diagram with a generic set of parameters (to guarantee convergence, we \nshould treat the powers of propagators as non-integer parameters) \ncould be treated as an $A$-function. \nHowever our analysis has shown that, typically, a real \nFeynman diagram corresponds to an $A$-function with reducible monodromy.  \n\nLet us explain our point of view. \nBy studying Feynman diagrams having a one-fold Mellin-Barnes \nrepresentation ~\\cite{bkk2009}, we have found that certain Feynman \ndiagrams \n($E^q_{1220}, B^2_{1220}, V^q_{1220}, J^q_{1220}$ in the notation of \nRef.\\ \\cite{bkk2009}) with powers of propagator equal to one\n(the so-called master-integrals) have the following hypergeometric \nstructure (we drop the normalization constant for simplicity): \n\\begin{eqnarray}\n\\Phi(n,\\vec{1};z)\n= \n{}_3F_2(a_1,a_2,a_3;b_1,b_2;z) \n+ \nz^\\sigma {}_4F_3(1,c_1,c_2,c_3;p_1,p_2,p_3;z) \\;,  \n\\label{example}\n\\end{eqnarray}\nwhere the dimension $n$ of space-time \\cite{dimreg} is not an integer\nand the difference between any two parameters of the hypergeometric \nfunction also are not integers. \nThe holonomic rank of the hypergeometric function \n${}_pF_{p-1}$ is equal to $p$, \nso that the Feynman diagram is a linear combination of two \nseries having different holonomic rank.  \nWhat could we say about the holonomic rank of a Feynman diagram $\\Phi$? \nTo answer that question, let us find the differential equation for \nthe Feynman diagram $\\Phi(n,\\vec{1};z)$ starting from  the\nrepresentation Eq.\\ (\\ref{example}).  This could be done by  the\nHolonomic Function Approach~\\cite{zeilberger} \nor with the help of a programs\ndeveloped by Frederic Chyzak \\cite{mgfun}\n(it is a MAPLE package)\nor by Christoph Koutschan\\footnote{\n\tSee Christoph's paper in the present volume.} \\cite{HolonomicFunctions} (it is MATHEMATICA package).\nWe used a private realization of this approach based on ideas from the \nGr\\\"obner basis technique. Finally, we obtained the result that  \nthe Feynman diagram $\\Phi$ satisfies the homogeneous differential equation \nof the hypergeometric type of order $4$\nwith a left-factorizable differential operator of order $1$:  \n\\begin{equation}\n(\\theta+A) \\left[\n(\\theta+B_1) (\\theta+B_2) (\\theta+B_3)\n+ z \\theta (\\theta+C_1) (\\theta+C_2) \n\\right] \\Phi(n,\\vec{1};z) = 0 \\;, \n\\label{MB:F}\n\\end{equation}\nwhere none of the $B_j$ and $C_a$ are integers\nand $\\theta = z \\frac{d}{dz}$. \n\nIt follows from this differential equation that\nthe holonomic rank of the Feynman diagram $\\Phi$ is equal to $4$, \nand factorization means that the space of solutions\nsplits into a direct sum of two spaces of dimension \none and three: $\\Phi_{\\mbox{dim}}  = 4 = 1 \\otimes 3$. \nAs it follows from~\\cite{beukers3}, the monodromy representation \nof Eq.\\ (\\ref{MB:F})\nis reducible and there is a one-dimensional invariant subspace.\nConsequently, \nthere are three non-trivial solutions (master-integrals) \nand the one-dimensional invariant subspace corresponds \nto an integral having a Puiseux-type solution \n(expressible in terms of $\\Gamma$-functions). \n\nWe pointed out in Ref.\\ \\cite{bkk2009} that \na Feynman diagram can be classified by the dimension \nof its irreducible representation.\nThis can be evaluated by the construction of differential equations \nor by using the dimension of the irreducible representation of the \nhypergeometric functions entering in the r.h.s.\\ of \nEq.\\ (\\ref{representation}). \nIndeed, in the example considered in Eq.\\ (\\ref{example}), the \ndimension of the irreducible representation \n${}_4F_3(1,\\vec{c};\\vec{p};z)$\nis equal to $3$ ~\\cite{beukers3}, \nso that \nthe dimension of the irreducible \nspace of the Feynman diagram $\\Phi$  is equal to $3$, \nsee Eq.(\\ref{MB:F}),\nand $\\Phi$ is expressible via \nthe sum of a series (see Eq.~\\ref{example}) with an irreducible representation of dimension $3$.\n \n  \nThe results of the analysis performed in Ref.\\ \\cite{bkk2009}, \nare summarized in the following proposition: \\\\[1em]\n{\\bf Proposition}:\\ {\\em A Feynman diagram can be \ntreated as a linear combination of Horn-type hypergeometric series  \nwhere each term has equal {\\bf irreducible} holonomic rank.}\\\\[1em]\n\nExamining this new ``quantum number,''\nirreducible holonomic rank, \nwe discover, and can rigorously prove, \nan extra relation between master-integrals~\\cite{KK:extra}.\nIn many other examples we found a complete agreement between the\nresults of differential reduction and \nand the results of a reduction based on the IBP \nrelations~\\cite{IBP1,IBP2}.\n\nThe Feynman diagram $J$\nconsidered in Ref.\\ \\cite{KK:extra}\nsatisfies the differential equation\n\\begin{eqnarray}\n\t&& \n\t\\left( \\theta \\!-\\! \\frac{n}{2} \\!+\\! I_1 \\right)\n\t\\left( \\theta \\!-\\! n \\!+\\! I_2 \\right)\n\t\\left[ \n\t\\theta \n\t\\left( \\theta \\!-\\! n \\!+\\! \\frac{1}{2} \\!+\\! I_3 \\right)\n\t+z\n\t\\left( \\theta \\!-\\! \\frac{3n}{2} \\!+\\! I_4 \\right)\n\t\\right] J = 0 \\;,\n\n\t\\label{J} \n\\end{eqnarray}\nwhere $I_1,I_2,I_3,I_4$ are integers, $n$ is the dimension of space-time \nand $\\theta = z \\frac{d}{dz}$.\nThe dimension of $J$ is $4$ and there are two one-dimensional \ninvariant subspaces, corresponding to two first-order differential \noperators:\n$\nJ_{\\mbox{dim}}  = 4 = 1 \\otimes 1 \\otimes 2 \\;.\n$\nIndeed, after integrating twice, we obtained\n$$\n\\left[ \n\\theta \n\\left( \\theta \\!-\\! n \\!+\\! \\frac{1}{2} \\!+\\! I_3 \\right)\n+z\n\\left( \\theta \\!-\\! \\frac{3n}{2} \\!+\\! I_4 \\right)\n\\right] J = C_1 z^{n\/2-I_1} + C_2 z^{n-I_2} \\;.\n$$\nSurprisingly, this simple relation has not been not reproduced \n(as of the end of $2016$)\nby any of the powerful programs for the reduction of Feynman diagrams\n(see the discussion in Chapter 6 of Ref.\\ \\cite{KK:sunset}).\n\tIn ~\\cite{SS}, it was shown that the extra relation \n\t~\\cite{KK:extra} could be  deduced from a diagram of more general \n\ttopology by exploring a new relation\n\tderived by taking of the derivative with respect to the mass \n\twith a subsequent reduction with the help of IBP relations.\n\tHowever, it was not shown that the derivative with respect to \n\tmass can be deduced from derivatives with respect to momenta, so \n\tthat the result of Ref.\\ \\cite{SS} could be considered as an alternative proof \n\tthat, in the massive case, there may exist an extra relation between \n\tdiagrams that does not follow from classical IBP relations.\n\n\nFinally, we have obtained a very simple result~\\cite{KK:MB}: \nEq.\\ (\\ref{MB:F}) follows directly from the\nMellin-Barnes representation for a Feynman diagram.\n(See Section~\\ref{Horn-Functions} and Eq.\\ (\\ref{MB:DE}) for details.)\nBased on this observation and on the results of our analysis performed \nin Ref.\\ \\cite{bkk2009}, and extending the idea of \nthe algorithm of Ref.\\ \\cite{algorithm1}, \nwe have constructed a simple and fast algorithm for the algebraic reduction  \nof any Feynman diagram having a one-fold Mellin-Barnes integral representation \nto a set of master-integrals without using the IBP relations. \nIn particular, our approach and our program cover some types of \nFeynman diagrams with arbitrary powers of propagators considered in \nRefs.\\ \\cite{mizera1,mizera2,mizera3}.\n\nIn a similar manner, one can consider the multiple Mellin-Barnes \nrepresentation of a Feynman diagram~\\cite{KK:sunset,hyperdire}. \nIn contrast to the one variable case, the factorization of the\npartial differential operator is much more complicated.\nThe dimension of the Pfaff system~\\footnote{Rigorously speaking, \n\tthis system of equations is correct when there is a\n\tcontour  in $\\mathbf{C}^n$ that is not changed \n\tunder translations by an arbitrary unit vector, see \n\tRef.\\ \\cite{Sadykov}, \n\tso that we treat the powers of propagator as parameters.} \nrelated to the multiple Mellin-Barnes \nintegral can be evaluated with the help of a prolongation procedure\n(see the discussion in Section~\\ref{Horn-Functions}). However, \nin this case, there may exist a Puiseux type solution even for a generic \nset of parameters (see for example Section~\\ref{FT:section}).\n\n\nExploring the idea~\\footnote{The monodromy group is  \n\tthe group of linear transformations of solutions \n\tof a system of hypergeometric differential equations under \n\trotations around its singular locus.\n\tIn the case when the monodromy is reducible, there is a \n\tfinite-dimensional subspace of \n\tholomorphic solutions of the hypergeometric system on which \n\tthe monodromy acts trivially.} \npresented in Ref.\\ \\cite{beukers3}, one possibility is to \nconstruct an explicit solution of the invariant subspace \n(see~\\cite{KK:sunset} and Section~\\ref{FT:section})\nand find the dimension of the irreducible representation.  \nOur results presented in ~\\cite{KK:sunset} were confirmed by another technique \nin Ref.~\\cite{bbkp}. \n\nLet us illustrate the notion of irreducible holonomic rank \n(or an irreducible representation) in an application to Feynman diagrams.\nAs follows from our analysis of sunset diagrams~\\cite{KK:sunset}, \nthe dimension of the\nirreducible representation of two-loop sunset with three different masses \nis equal to $4$. There is only one hypergeometric function of three \nvariables having holonomic rank $4$, the $F_D$ function. \nThen we expect that there is a linear combination of \nfour two-loop sunsets and the product of one-loop tadpoles that are\nexpressible in terms of a linear combination of the $F_D$ functions.  \n\nAnother approach to the construction of a GKZ representation \nof Feynman Diagrams\nwas done recently in the series of papers in Refs.\\ \\cite{GKZ1,GKZ2,GKZ3}.\nBased on the observation made in Ref.\\ \\cite{Nilsson-Passere}\nabout the direct relation between $A$-functions and \nMellin transforms of rational functions, and exploring the \nLee-Pomeransky representation~\\cite{Lee-Pomeransky}, \nthe authors studied a different aspect of the GKZ representation \nmainly considering the examples of massless or one-loop diagrams. \nTwo non-trivial examples \nhave been presented in Ref.~\\cite{GKZ2}: the two-loop sunset \nwith two different masses and one zero \nmass, which corresponds to a linear combination of two Appell functions \n$F_4$ (see Eq.\\ (3.11) in~\\cite{JK})~\\footnote{It is interesting to note, \n\tthat on-mass shell $z=1$, this diagram has two Puiseux type solutions\n\tthat do not have analytical continuations.}\nand a two-loop propagator with three different masses related \nto the functions $F_C$ of three variables~\\cite{berends}. \n\nA different idea on how to apply the GKZ technique to the analysis of Feynman Diagrams has been\npresented in \\cite{vanhove} and has received further development in\n\\cite{klemm1,klemm2}. \n\n\\subsection{One-Loop Feynman Diagrams}\n\\label{one-loop}\nLet us give special attention to one-loop Feynman diagrams. \nIn this case, two elegant approaches \nhave been developed~\\cite{DD,FJT} that allow us to obtain \ncompact hypergeometric representations for the master-integrals. \nThe authors of the first paper~\\cite{DD} explored \nthe internal symmetries of the Feynman parametric representation to get a one-fold integral \nrepresentation for one-loop Feynman diagrams (see also ~\\cite{bloch-kreimer,n-gon}). \nThe second approach~\\cite{FJT} \nis based on the solution of difference equations  \nwith respect to the dimension of space-time~\\cite{tarasov:d}\nfor the one-loop integrals.\nIn spite of different ideas on the analysis of Feynman diagrams,\nboth approaches, ~\\cite{DD} and ~\\cite{FJT},\nproduce the same results for one-loop propagator and vertex diagrams ~\\cite{one-loop:vertex1,one-loop:vertex2,one-loop:vertex3}. \nHowever, beyond these examples, the situation is less complete: it was \nshown in Ref.\\ \\cite{FJT} that the off-shell one-loop massive box is expressible in terms of a linear combination of \n$F_S$ Horn-type hypergeometric functions of three variables \n(see also discussions\nin Refs.\\ \\cite{bkm2013,riemann1,riemann2,phan}), \nor in terms of $F_N$ Horn-type hypergeometric functions of three variables ~\\cite{davydychev:box} \n(see Section ~\\ref{FN:section}).\n\nRecently, it was observed \n~\\cite{yangian1,yangian2,yangian3}\nthat massive conformal Feynman diagrams \nare invariant under a Yangian \nsymmetry that allows to get the hypergeometric representation \nfor the conformal Feynman diagrams. \n \n\\subsection{Construction of $\\ep$-expansion}\nFor physical applications, the construction of the analytical coefficients\nof the Laurent expansions of hypergeometric functions around particular values of parameters (integer, half-integer, rational) is necessary. \nSince the analytic continuations of hypergeometric functions \nis still an unsolved problem, the results are written \nin some region of variables in each order of the $\\ep$-expansion \nin terms of special functions like classical or multiple polylogarithms, \n~\\cite{lewin,harmonic,2dim,BBBL,mpl1,mpl2,mpl3},\nand then these functions are analytically continued to another region. For this reason, the analytical properties of special functions \nwere\n analyzed in detail~\\cite{mpl:properties1,mpl:properties2,mpl:properties3,\n\t  mpl:properties4,mpl:properties5,panzer2015,PolyLog}.\nAlso, tools for the numerical evaluation of the corresponding functions are important ingredients \n~\\cite{gr1,gr2,vollinga,logsine,maitre1,bonciani2011,maitre2,chaplin,\n\t li22,maple,harmonic8,handyG,duhr-tancredi,walden}.  \n\nEach of the hypergeometric function representations  (series, integral, Mellin-Barnes, differential equation) can be used for\nthe construction of the $\\ep$-expansion,\nand each of them has some technical advantages or disadvantages\nin comparison with the other ones.\nThe pioneering $\\ep$-expansion of the hypergeometric function \n${}_pF_{p-1}$ around $z=\\pm 1$ was done by David Broadhurst \n~\\cite{david1,david2}. \nThe expansion was based on the analysis of multiple series \nand it was interesting from a mathematical point of view~\\cite{david3}\nas well as for its application to quantum field theory~\\cite{david4}.\nThe integral representation was mainly developed by Andrei Davydychev \nand Bas Tausk~\\cite{dt1,dt2}, so that, finally, the all-order \n$\\ep$-expansion for the Gauss hypergeometric functions around \na rational parameter, a case\nthat covers an important class of diagrams, \nhas been constructed~\\cite{davydychev} in terms of \ngeneralized log-sine~\\cite{lewin} functions or in term of Nielsen polylogarithms \\cite{DK1,DK2}. \n\nThe integral representation was also the starting point for the \nconstruction of the\n$\\ep$-expansion of ${}_pF_{p-1}$ hypergeometric functions,~\\cite{huber-maitre1,huber-maitre2}, \nand also the $F_1$ \\cite{ndim4} and $F_D$ functions \naround integer values of parameters \n~\\cite{bogner-brown1,bogner-brown2,panzer2014,bogner-mpl}.\n\nPurely numerical approaches~\\cite{NumExp1,pentagon4,NumExp2}\ncan be applied for arbitrary values of the parameters. \nHowever, this technique typically does not produce a stable numerical\nresult in regions around singularities of the hypergeometric functions. \n\n\nA universal technique which does not \ndepend on the order of the differential equation \nis based on the algebra of multiple sums~\\cite{nested1,nested2,nested3}.\nFor the hypergeometric functions\\footnote{It was shown  \n\tin ~\\cite{smirnov,tausk} that multiple Mellin-Barnes integrals \n\trelated to Feynman diagrams could be evaluated analytically\/numerically \n\tat each order in $\\ep$ via multiple sums, without requiring a\n\tclosed expression in terms of Horn-type hypergeometric functions.}\nfor which the nested-sum algorithms~\\cite{nested1} are applicable, \nthe results \nof the $\\ep$-expansion are automatically obtained in terms of multiple \npolylogarithms.\n\nThe nested-sum algorithms~\\cite{nested1} have been implemented in a few\npackages~\\cite{nested1a,nested1b} and allow for the construction of the\n$\\ep$-expansion of hypergeometric functions ${}_pF_{p-1}$ and \nAppell functions $F_1$ and $F_2$ around integer values of parameters\\footnote{\n\tSee also Refs.\\ \\cite{series1,series2} for an alternative realization.}. \nHowever, the nested-sum approach fails for \nthe $\\ep$-expansion of hypergeometric functions \naround rational values of parameters and it is not applicable to \nsome specific classes of hypergeometric functions (for example, the $F_4$ \nfunction, see~\\cite{pentagon1}).\n\nIn the series of papers~\\cite{DK3,kalmykov2004}, the generating function \ntechnique~\\cite{generating1,generating2} has been developed \nfor the analytical evaluation of multiple sums. \nIndeed,  the series generated by the $\\ep$-expansion of hypergeometric \nfunctions has the form  \n$\n\\sum_k c(k) z^k\\;, \n$\nwhere the coefficients $c(k)$ include only products of the harmonic sums \n$\\Pi_{a,b} S_a(k-1) S_b(2k-1)$ and $S_a(k) = \\sum_{j=1}^k\n\\frac{1}{j^a}$. The harmonic sums satisfy the recurrence relations\n$$\nS_a(k) \n= S_a(k-1) \n+ \\frac{1}{k^a}\n\\quad, \\qquad\nS_a(2k+1) \n= S_a(2k-1) \n+ \\frac{1}{(2k+1)^a}\n+ \\frac{1}{(2k)^a}\\;,\n$$ \nso that the coefficients $c(k)$ satisfy \nthe first order difference equation\\footnote{In general, it could be \n\ta more generic recurrence, $\\sum_{1=0}^k p_{k+j}(k+j)c(k+j) = r(k)$.}: \n$$\nP(k+1) c(k+1) = Q(k) c(k) + R(k) \\;, \n$$ where $P$ and $Q$ are polynomial functions \nthat can be defined from the original series.\nThis equation could be converted into \na first order differential equation for the generating function\n$F(z) = \\sum_k c(k) z^k$, \n$$\n\\frac{1}{z} P \\left(z \\frac{d}{dz} \\right) F(z)\n- P(1)C(1) z = Q \\left(z \\frac{d}{dz} \\right) F(z)\n+ \\sum_{k=1} R(k) z^k \\;.\n$$\nOne of the \nremarkable properties of this technique, that the non-homogeneous part\nof the differential equation, the\nfunction $R(k)$,  has one-unit less depth in contrast to the \noriginal sums, so that, step-by-step, all sums could be evaluated analytically. \nBased on this technique, all series arising from \nthe $\\ep$-expansion of hypergeometric functions around \nhalf-integer values of parameters have been evaluated~\\cite{DK3} \nup to weight $4$.\nThe limits considered were mainly motivated by physical reasons (at $O$(NNLO) only \nfunctions of weight $4$ are generated) and, in this limit, only one new function~\\cite{harmonic} $H_{-1,0,0,1}(z)$ was necessary to introduce.\nThese results~\\cite{DK3} allow us to construct the $\\ep$-expansion of \nthe hypergeometric functions ${}_pF_{p-1}$\naround half-integer values of parameters, see~\\cite{JKV2003,MKL:Gauss}.\n\nOther results and theorems relevant for the evaluation of \nFeynman diagrams are related with the appearance of a factor \n$1\/\\sqrt{3}$ in the $\\ep$-expansion of some diagrams~\\cite{FK1999} \nexpressible in terms of hypergeometric functions  \n\\footnote{Recent results on the analytical evaluation of inverse binomial \n\tsums for particular values of the arguments  \n\thave been presented in~\\cite{binsum1,binsum2,binsum3}.}\nwere derived in Refs.\\ \\cite{DK2,KWY2007,KK2010}\\footnote{The appearance of \n\t$1\/\\sqrt{3}$ in RG functions in seven loops was quite \n\tintriguing~\\cite{schnetz1,panzer2015}.}.\n\nLet us consider typical problems arising in this program. We follow   \nour analysis presented in Ref.\\ \\cite{BKK2012}, see also the closely related \ndiscussion in Ref.\\ \\cite{abs2018}. \nFirst, the construction of the difference equation for the coefficient \nfunctions $c(z)$ is not an easy task~\\cite{schneider1,schneider2,schneider3}. \nIn the second step, the differential operator(s) coming from the difference \nequation\n$P \\left(z \\frac{d}{dz} \\right) -  z  Q \\left(z \\frac{d}{dz} \\right)$\nshould be factorized   into a product of differential operators of the first \norder, \n$$\nP \\left(z \\frac{d}{dz} \\right) -  z  Q \\left(z \\frac{d}{dz} \\right)\n= \n\\Pi_{k=1} \n\\left[ p_{k}(z) \\frac{d}{dz}  - q_{k}(z) \\right]\\;,\n$$\nwhere $ p_{k}(z)$ and $q_{k}(z)$ are rational functions.  Unfortunately, the \nfactorization of differential operators into irreducible factors is not \nunique~\\cite{landau}: \n$$\n\\left(\n\\frac{d^2}{dx^2} \\!-\\! \\frac{2}{x} \\frac{d}{dx} \\!+\\! \\frac{2}{x^2}\n\\right)\n= \n\\left(\n\\frac{d}{dx} \\!-\\! \\frac{1}{x}\n\\right)\n\\left(\n\\frac{d}{dx} \\!-\\! \\frac{1}{x}\n\\right)\n= \n\\left(\n\\frac{d}{dx} \\!-\\! \\frac{1}{x (1+ax)}\n\\right)\n\\left(\n\\frac{d}{dx} \\!-\\! \\frac{(1+2ax)}{x(1+ax)}\n\\right) \\;, \n$$\nwhere $a$ is a constant. \n\nHowever, the \nfollowing theorem is valid (see~\\cite{schwarz}):\nAny two decompositions of a linear differential operator $L^{(p)}$\ninto a product (composition) of irreducible linear differential \noperators \n$$\nL^{(p)} = \nL_1^{(a_1)}  \nL_2^{(a_2)}   \n\\cdots \nL_m^{(a_m)}  \n= \nP_1^{(r_1)}  \nP_2^{(r_2)}   \n\\cdots \nP_k^{(r_k)}\n$$\nhave equal numbers of components $m=k$ \nand the factors $L_j$ and $P_a$ \nhave the same order of differential operators:\n$L_a=P_j$ (up to commutation). In the application to the $\\ep$-expansion \nof hypergeometric functions this problem has been discussed in \nRef.\\ \\cite{yost2011}.\n\n\nAfter factorization, the iterated integral over \nrational functions (which is not uniquely defined, as seen in \nthe previous example) would be generated that in general is not \nexpressible in terms of hyperlogarithms. \nIndeed, the solution of the differential equation\n$$\n\t\\left[ R_1(z) \\frac{d}{dz}  \\!+\\! Q_1(z) \\right] \\left[ R_2(z) \\frac{d}{dz}  \\!+\\! Q_2(z) \\right] h(z) = F(z) \\;.\n\t\\label{de}\n$$\nhas the form\n$$\n\th(z) = \n\t\\int^{z} \\frac{dt_3}{R_2(t_3)} \\left[ \\exp^{-\\int_0^{t_3}\\frac{Q_2(t_4)}{R_2(t_4)} dt_4} \\right] \n\t\\int^{t_3} \\frac{dt_1}{R_1(t_1)} \\left[ \\exp^{-\\int_0^{t_1}\\frac{Q_1(t_2)}{R_1(t_2)} dt_2} \\right] F(t_1) \\;. \n$$\n    From this solution it follows~\\cite{KK2010B}\n\tthat the following conditions are enough to \n\tconvert the iterated integral into hyperlogarithms:\n\tthere are new variables $\\xi$ and $x$ so that  \n\t\\begin{eqnarray}\n\t\t&& \n\t\t\\int^z \\frac{Q_i(t)}{R_i(t)} dt  =  \\ln \\frac{M_i(\\xi)}{N_i(\\xi)} \n\t\t\\Rightarrow\n\t\t\\left. \\frac{dt}{R_i(t)} \\right|_{t = t(\\xi)} \n\t\t\\frac{N_i(\\xi)}{M_i(\\xi)}  =  dx \\frac{K_i(x)}{L_i(x)} \\;, \n\t\t\\nonumber \n\t\\end{eqnarray}\n\twhere $M_i,N_i,K_i,L_i$ are polynomial functions.\n\nThe last problem is related to the Abel-Ruffini theorem:  \nthe polynomial is factorizable into a product of its primitive roots, \nbut there are not solutions in radicals for \npolynomial equations of degree five or more. \nThe last problem got a very elegant solution by the introduction \nof cyclotomic polylogarithms~\\cite{nested3}, with the integration over \nirreducible cyclotomic polynomials $\\Phi_n(x)$.\nThe first two irreducible polynomials (see Eqs.\\ $(3.3)-(3.14)$ in~\\cite{nested3}) \nare  $\\Phi_7$ and $\\Phi_9$\n(the polynomial of order $6$). \nTwo other polynomials of order $4$, \n$\\Phi_5$ and $\\Phi_{10}$: \n$(x^4 \\pm x^3 + x^2 \\pm x +1)$, have \nnon-trivial primitive roots. But up to now, all these polynomials were not \ngenerated by Feynman diagrams. Surprisingly, \nby increasing the number of loops or number of scales, \nother mathematical structures are generated.~\\cite{bs1,bs2}.\nDetailed analyses of properties of the new functions have been presented in \nRefs.\\ \\cite{abs2013,abs2014} and automated by Jakob \nAblinger~\\cite{ablinger1,ablinger2,ablinger3}. \nThe problem of integration over algebraic functions \n(typically square roots of polynomials) \nwas solved by the introduction of a new type of functions,\n~\\cite{Bonciani}, intermediate between multiple and elliptic polylogarithms.\n\nThe series expansion is not very efficient for the construction \nof the $\\ep$-expansion, since the number of series increases with the \norder of the $\\ep$-expansion and increases the complexity of the individual \nsums.\nLet us recall that the Laurent expansion of a hypergeometric function \ncontains a linear combination of multiple sums. \nFrom this point of view, the construction of the analytical coefficients of\nthe $\\varepsilon$-expansion of a hypergeometric function\ncan be carried out independently\nof existing analytical results for each individual multiple sum. \nThe ``internal'' symmetry of a Horn-type hypergeometric function is \nuniquely defined by the corresponding system of differential equations. \nWhile exploring this idea, a new algorithm was presented in \nRefs.\\ \\cite{kwy2006,kwy2007}, based on factorization, looking for a \nlinear parametrization and direct iterative solution \nof the differential equation for a hypergeometric function.\nThis approach allows the construction of the analytical coefficients of the \n$\\varepsilon$-expansion of a hypergeometric \nfunction, as well as obtaining analytical expressions \nfor a large class of multiple series \nwithout referring to the algebra of nested sums.\n\nBased on this approach, the all-order $\\varepsilon$-expansion of the Gauss \nhypergeometric function around half-integer and rational values of parameters\nhas been constructed~\\cite{kwy2006,kk2008}, so that the first $20$ coefficients \naround half-integer values of parameters, the $12$ coefficients \nfor $q=4$ and $10$ coefficients for $q=6$\nhave been generated already in 2012~\\footnote{The results \n\thave been written in \n\tterms of hyperlogarithms of primitive $q$-roots of unity.}. \nAnother record is the generation of $24$ coefficients for the\nClausen hypergeometric function ${}_3F_2$ around integer values of \nparameters, relevant for the analysis~\\footnote{The results of \n\t~\\cite{david5} were relevant for the reduction of multiple zeta values\n\tto the minimal basis.} \nperformed in ~\\cite{boels}.\nTo our knowledge, at the present moment, this remains the fastest and most \nuniversal algorithm. \n\n\nMoreover, it was shown in Refs.\\ \\cite{kwy2006,kwy2007}, \nthat when the coefficients of the $\\varepsilon$-expansion of a hypergeometric \nfunction are expressible in terms of multiple polylogarithms, there is a set \nof parameters (not uniquely defined) such that, \nat each order of $\\varepsilon$, the coefficients of the $\\varepsilon$-expansion \ninclude multiple polylogarithms of a single uniform weight. \nA few years later, this property was established \nnot only for hypergeometric functions, but for Feynman Diagrams\n~\\cite{Henn:1}.\n\nA multivariable generalization~\\cite{BKK2012} \nof the algorithm of Refs.\\ \\cite{kwy2006,kwy2007} has been described.\nThe main difference with respect to the case of one variable is the \nconstruction of a system of differential equations of triangular form\n to avoid the appearance of elliptic functions.\nAs a demonstration of the validity of the algorithm, the first few coefficients \nof the $\\varepsilon$-expansion of the Appell hypergeometric functions \n$F_1,F_2,F_3$ and $F_D$ around integer values \nof parameters have been evaluated analytically~\\cite{bkm2013}. \n\nThe $\\varepsilon$-expansion of the hypergeometric \nfunctions $F_3$ and $F_D$ \nare not covered by the nested sums technique or its generalization. \nThe  differential equation\ntechnique can be applied to the construction of analytic coefficients of \nthe $\\varepsilon$-expansions of hypergeometric functions of several variables \n(which is equivalent to the multiple series of several variables)\naround any rational values of parameters via direct solution of the \nlinear systems of differential equations. \n\nThe differential equation approach~\\cite{kwy2006,kwy2007}\nallows us to analyze arbitrary sets of parameters simultaneously\nand to construct the solution in terms of iterated integrals, \nbut for any hypergeometric function\nthe Pfaff system of differential equations should be constructed.\nThat was the motivation for creation of the package(s) (the HYPERDIRE \nproject) ~\\cite{hyperdire} for the manipulation of the parameters of \nHorn-type hypergeometric functions of several variables. \nFor illustration, we describe in detail how it works in the application \nto the $F_3$ hypergeometric function in Section~\\ref{F3:general}.\n\nRecently, a new technique~\\cite{abreu1}\nfor the construction of the $\\ep$-expansion \nof Feynman diagrams~\\cite{abreu2}\nas well as for hypergeometric functions has been presented~\\cite{abreu3}.\nIt is based on the construction of a coaction~\\footnote{An interesting \n\tconstruction of the coaction for the Feynman graph has been presented \n\trecently in Ref.\\ \\cite{Kreimer2020}.} \nof certain hypergeometric functions. The structures of the $\\ep$-expansion of \nthe Appell hypergeometric functions $F_1,F_2,F_3$ and $F_4$ as well as $F_D$\n(for the last function $F_D$ see also the discussion in Ref.\\ \\cite{brown-dupont}) \naround integer values of parameters \nare in agreement with our analysis and partial results presented in\nRefs.\\ \\cite{BKK2012} and ~\\cite{bkm2013}. However, the \nstructure of the $\\ep$-expansion around rational values of parameters \nhas not been discussed in~\\cite{abreu3}, nor in ~\\cite{brown-dupont}.\n\n\\section{Horn-type hypergeometric functions}\n\\label{Horn-Functions}\n\\subsection{Definition and system of differential equations}\n\\label{Horn:defintion}\nThe study of solutions of linear partial \ndifferential equations (PDEs) of several variables in terms of multiple series,\n{\\em i.e.} a multi-variable generalization of the Gauss hypergeometric \nfunction~\\cite{Gauss}, began long ago~\\cite{Lauricella}.\n\nFollowing the Horn definition~\\cite{horn}, a multiple series is called a \n``Horn-type hypergeometric function,'' if, about the point $\\vec{z}=\\vec{0}$, \nthere is a series representation\n\\begin{equation}\nH(\\vec{z}) = \\sum_{\\vec{m}} C(\\vec{m}) \\vec{z}^{\\vec{m}},\n\\label{def}\n\\end{equation}\nwhere \n$\n\\vec{z}^{\\vec{m}} = z_1^{m_1} \\cdots z_r^{m_r}\n$\nfor any integer multi-index\n$\\vec{m} = (m_1, \\cdots, m_r)$, \nand the ratio of two coefficients can be represented as a ratio of\ntwo polynomials:\n\\begin{eqnarray}\n\\frac{C(\\vec{m}+\\vec{e}_j)}{C(\\vec{m})}  \n=  \\frac{P_j(\\vec{m})}{Q_j(\\vec{m})} \n\\;,\n\\label{horn}\n\\end{eqnarray}\nwhere\n$\n\\vec{e}_j\n$\ndenotes the unit vector with unity in its $j^{\\rm th}$ entry, \n$                   \n\\vec{e}_j = (0,\\cdots,0,1,0,\\cdots,0).  \n$\nThe coefficients $C(\\vec{m})$ of such a series can be expressed as \nproducts or ratios of Gamma-functions (up to some factors \nirrelevant for our consideration)~\\cite{Ore:Sato:1,Ore:Sato:2}:\n\\begin{eqnarray}                                              \n\tC(\\vec{m})=\n\\frac{\n\t\\prod\\limits_{j=1}^p\n\t\\Gamma\\left(\\sum_{a=1}^r \\mu_{ja}m_a+\\gamma_j \\right)}\n{\n\t\\prod\\limits_{k=1}^q\n\t\\Gamma\\left( \\sum_{b=1}^r \\nu_{kb}m_b+\\sigma_k \\right)\n} \\;,\n\\label{ore}\n\\end{eqnarray}\nwhere\n$                                                             \n\\mu_{ja}, \\nu_{kb}, \\sigma_j,\\gamma_j \\in \\mathbb{Z}\n$\nand $m_a$ are elements of ${\\vec{m}}$.\n\nThe Horn-type hypergeometric function, Eq.~(\\ref{horn}),  \nsatisfies the following system of differential equations:\n\\begin{equation}\n0 =\nL_j (\\vec{z})\nH(\\vec{z})\n=\n\\left[\nQ_j\\left(\n\\sum_{k=1}^r z_k\\frac{\\partial}{\\partial z_k}\n\\right)\n\\frac{1}{z_j}\n-\nP_j\\left(\n\\sum_{k=1}^r z_k\\frac{\\partial}{\\partial z_k}\n\\right)\n\\right]\nH(\\vec{z}) \\;,\n\\label{diff}\n\\end{equation}\nwhere $j=1, \\ldots, r$.\nIndeed, \n\\begin{eqnarray}\n&& \nQ_j\\left(\n\\sum_{k=1}^r z_k\\frac{\\partial}{\\partial z_k}\n\\right)\n\\frac{1}{z_j} \\sum_{\\vec{m}} C(\\vec{m}) \\vec{z}^{~\\vec{m}}\n=  \\sum_{\\vec{m}} Q_j(\\vec{m}) C(\\vec{m}\\!+\\!\\vec{e}_j) \\vec{z}^{~\\vec{m}}\n\\nonumber \\\\ && \n= \n\\sum_{\\vec{m}} P_j(\\vec{m}) C(\\vec{m}) \\vec{z}^{~\\vec{m}}\n=\nP_j\\left(\n\\sum_{k=1}^r z_k\\frac{\\partial}{\\partial z_k}\n\\right)\n\\sum_{\\vec{m}}  C(\\vec{m}) \\vec{z}^{~\\vec{m}} \\;.\n\\nonumber \n\\end{eqnarray}\nThe degrees of the polynomials $P_i$ and $Q_i$ \nare $p_i$ and $q_i$, respectively.\nThe largest of these, $r=\\max \\{p_i,q_j \\}$, is called the order of \nthe hypergeometric series. \nTo close the system of differential equations, the \n{\\it prolongation procedure} \nshould be applied: by\napplying the differential operator $\\partial_i$ to $L_j$\nwe can convert the system of linear PDEs with polynomial \ncoefficients into Pfaff form (for simplicity, we assume that system is closed): \n\n\\begin{eqnarray}\nL_j H(\\vec{z}) = 0 \n& \\Rightarrow & \n\\Biggl\\{\nd \\omega_i(\\vec{z}) = \\Omega_{ij}^k(\\vec{z}) \\omega_j(\\vec{z}) dz_k  \\;,\n\\quad \nd \\left[ d \\omega_i(\\vec{z}) \\right] = 0 \n\\Biggr\\} \\;.\n\\label{pfaff}\n\\end{eqnarray}\n\nInstead of a series representation, one can \nuse a Mellin-Barnes integral representation \n(see the  discussion in ~\\cite{KK:MB}). Indeed,  \nthe multiple Mellin-Barnes representation for a Feynman diagram \ncould be written in the form in Eq.\\ (\\ref{MB}).\nLet us define the polynomials $P_i$ and $Q_i$ as \n\\begin{equation}\n\\frac{P_i(\\vec{t})}{Q_i(\\vec{t})} = \\frac{\\phi(\\vec{t}+e_i)}{\\phi(\\vec{t})} \\;.\n\\end{equation}\nThe integral (\\ref{MB}) then satisfies the system of linear differential \nequations (\\ref{diff})\n\\begin{eqnarray}\n\\left. Q_i(\\vec{t}) \\right|_{t_j \\to \\theta_j }\\frac{1}{z_i} \\Phi({\\bf A},\n\t\\vec{B};{\\bf C}, \\vec{D};\\vec{z})  \n\t= \\left. P_i(\\vec{t}) \\right|_{t_j \\to \\theta_j } \\Phi({\\bf A},\n\t\\vec{B};{\\bf C}, \\vec{D};\\vec{z})  \\;,\n\\label{MB:DE}\n\\end{eqnarray}\nwhere $\\theta_i =  z_i\\frac{d}{dz_i}$.\nSystems of equations such as Eq. (\\ref{MB:DE}) are left ideals in \nthe Weyl algebra of linear differential operators with polynomial coefficients.\n\n\\subsection{Contiguous relations}\n\\label{Horn:contiguous}\nAny Horn-type hypergeometric function is a function of two types of variables, \n{\\it continuous} variables, $z_1,z_2, \\cdots, z_r$ and \n{\\it discrete} variables:\n$\\{ J_a \\}:= \\{\\gamma_k,\\sigma_r \\}$, \nwhere the latter can change by integer numbers \nand are often referred to as the {\\it parameters} of the hypergeometric \nfunction. \n\nFor any Horn-hypergeometric function, \nthere are linear differential operators \nchanging the value of the discrete variables by one unit. \nIndeed, let us consider a multiple series defined by \nEq.~(\\ref{def}).\n\nTwo hypergeometric functions $H$\nwith sets of parameters shifted by unity,\n$H(\\vec{\\gamma}+\\vec{e_c};\\vec{\\sigma};\\vec{z})$ and\n$H(\\vec{\\gamma};\\vec{\\sigma};\\vec{z})$,\nare related by a linear differential operator:\n\\begin{eqnarray}\nH(\\vec{\\gamma}+\\vec{e_c};\\vec{\\sigma};\\vec{z})\n& = &\n\\left ( \\sum_{a=1}^r \\mu_{ca} \nz_a \\frac{\\partial}{\\partial z_a}+\\gamma_c \\right)\nH(\\vec{\\gamma};\\vec{\\sigma};\\vec{z})\n\\;.\n\\label{do1}\n\\end{eqnarray}\nSimilar relations also exist for the lower parameters:\n\\begin{eqnarray}\nH(\\vec{\\gamma};\\vec{\\sigma}-\\vec{e}_c;\\vec{z})\n& = &\n\\left(\n\\sum_{b=1}^r\n\\nu_{cb} z_b \\frac{\\partial}{\\partial z_b} \\!+\\! \\sigma_c \\!-\\! 1\n\\right)\nH(\\vec{\\gamma};\\vec{\\sigma};\\vec{z})\n\\;.\n\\label{do2}\n\\end{eqnarray}\nLet us rewrite these relations in a symbolic form: \n\\begin{eqnarray}\nR_{K}(\\vec{z})\\frac{\\partial }{\\partial \\vec{z}_K} H(\\vec{J};\\vec{z}) \n= H(\\vec{J} \\pm e_K; \\vec{z}) \\;,\n\\label{direct}\n\\end{eqnarray}\nwhere $R_{K}(\\vec{z})$ are polynomial (rational) functions.\n\nIn Refs.\\ \\cite{algorithm1,algorithm2} it was shown that there is an \nalgorithmic construction of inverse linear differential operators: \n\\begin{eqnarray}\nB_{L,N}(\\vec{z})\\frac{\\partial^L }{\\partial \\vec{z}_N} \n\\left( R_{K}(\\vec{z})\\frac{\\partial }{\\partial \\vec{z}_K} \\right) \nH(\\vec{J};\\vec{z}) \n\\equiv \nB_{L,N}(\\vec{z})\\frac{\\partial^L }{\\partial \\vec{z}_N} H(\\vec{J} \\pm e_K; \\vec{z}) \n= \nH(\\vec{J};\\vec{z}) \n\\;.\n\\label{inverse}\n\\end{eqnarray}\nApplying the direct or inverse differential operators\nto the hypergeometric function\nthe values of the parameters can be changed by an arbitrary integer: \n\\begin{equation}\nS(\\vec{z}) H(\\vec{J}+\\vec{m}; \\vec{z})\n= \n\\sum_{j=0}^r S_j(\\vec{z}) \n\\frac{\\partial^j}{\\partial \\vec{z}} H(\\vec{J}; \\vec{z}) \\;,\n\\label{reduction}\n\\end{equation}\nwhere $\\vec{m}$ is a set of integers, $S$ and $S_j$ are polynomials \nand $r$ is the holonomic rank (the number of linearly independent solutions) \nof the system of differential equations Eq.~(\\ref{diff}).\nAt the end of the reduction, the differential operators acting on the \nfunction $H$ can be  replaced by a linear combination of the function \nevaluated with shifted parameters. \n\nWe note that special considerations are necessary when the system of \ndifferential operators,  Eq.~(\\ref{diff}), has a\nPuiseux-type solution (see Section \\ref{FT:section}). \nIn this case, the prolongation procedure gives \nrise to the Pfaffian form, but\nthis set of differential equations is not enough to construct \nthe inverse operators~\\cite{BK}, so that new \ndifferential equations should be introduced.\nIn the application to the Feynman diagrams, this problem \nis closely related with obtaining new relations between \nmaster integrals, see Ref.~\\cite{KK:extra} for details.\nIn the Section ~\\ref{FT:section} we present an example of the\nHorn-type hypergeometric equation of second order \nof three variables having a Puiseux-type  solution. \n\nAnother approach to the reduction of hypergeometric functions \nis based on the explicit algebraic solution of the contiguous relations, \nsee the discussion in Ref.\\ \\cite{Schlosser}. \nThis technique is applicable in many particular cases, including  \n${}_2F_1, {}_3F_2$, and the Appell functions $F_1,~F_2,~F_3,~F_4$ \n(see the references in Ref.\\ \\cite{acat}), and there is \na general expectation that it could be solved for any Horn-type hypergeometric function. However, to our knowledge, \nnobody has analyzed the algebraic reduction in the application to  \ngeneral hypergeometric functions having a Puiseux-type solution.  \n\nThe multiple Mellin-Barnes integral\n$\\Phi$ defined by Eq.~(\\ref{MB}) satisfies similar \ndifferential contiguous relations: \n\\begin{eqnarray}\n\\Phi({\\bf A}, \\vec{B} \\!+\\! e_a; {\\bf C}, \\vec{D};\\vec{z}) \n& = & \n\\left(\\sum_{i=1}^m A_{ai} \\theta_i \\!+\\! B_a \\right) \n\\Phi({\\bf A},\\vec{B}; {\\bf C}, \\vec{D};\\vec{z}) \\;,\n\\nonumber \\\\  \n\\quad \n\\Phi({\\bf A}, \\vec{B}; {\\bf C}, \\vec{D} \\!-\\! e_b;\\vec{z}) \n& = &  \n\\left(\\sum_{j=1}^r C_{bj} \\theta_j \\!+\\! D_{b} \\right)\n\\Phi({\\bf A},\\vec{B}; {\\bf C}, \\vec{D};\\vec{z}) \n\\;,\n\\label{step-up-down}\n\\end{eqnarray}\nso that  \nthe original diagram may be explicitly reduced to a set of basis functions\nwithout examining the IBP relations~\\cite{IBP1,IBP2}.\nA non-trivial example of this type of reduction beyond IBP relations \nhas been presented in Ref.\\ \\cite{KK:extra} \n(see also the discusion in Chapter 6 of Ref.\\ \\cite{KK:sunset}).\n\n\n\n\n\n\\section{Examples}\n\\label{Example}\n\\subsection{Holonomic rank \\& Puiseux-type solution} \nIn addition to the examples presented previously in our series of \npublications, we present here a few new examples.\n\n\\subsubsection{Evaluation of holonomic rank: the hypergeometric \n\tfunction $F_N$}\n\\label{FN:section}\nThe Lauricella-Saran hypergeometric function of three variables $F_N$ \n is defined about the point $z_1=z_2=z_3=0$ by\n\\begin{eqnarray}\n&& \nF_N(a_1,a_2,a_3;b_1,b_2; c_1,c_2; z_1,z_2,z_3)\n\\nonumber \\\\ && \n= \\sum_{m_1,m_2,m_3=0}^\\infty\n\\left[ \n\\Pi_{j=1}^3 (a_j)_{m_j} \\frac{z_j^{m_j}}{m_j!}\n\\right] \n\\frac{(b_1)_{m_1+m_3} (b_2)_{m_2}}      \n{(c_1)_{m_1} (c_2)_{m_2+m_3}} \n\\;. \n\\label{FN:series}\n\\end{eqnarray}\nThis function is related to one-loop box diagrams in an \narbitrary dimension considered by Andrei Davydychev~\\cite{davydychev:box}. \n\nFollowing the general algorithm~\\cite{bkm2013}, \nthe following result is easily derived: \n\\begin{theorem}\n\tFor generic values of the parameters, \n\tthe holonomic rank of the function $F_N$ \n\tis equal $8$. \n\\end{theorem}\nIn this way, for generic values of parameters, \nthe result of differential reduction, Eq.\\ (\\ref{reduction}), \nhave the following form: \n$$\nS(\\vec{z}) F_N(\\vec{J}+\\vec{m}; \\vec{z})\n= \n\\left[ S_0\n+ S_i \\sum_{j=1}^3 \\theta_j \n+ \\sum_{\\substack{i,j=1 \\\\ i<j}}^3  S_{i,j} \\theta_i \\theta_j \n+ S_{123} ~\\theta_1 \\theta_2 \\theta_3 \n\\right] F_N(\\vec{J}; \\vec{z}) \\;,\n$$\nwhere $\\theta_i =  z_i\\frac{d}{dz_i}$\nand inverse differential operators can be easily constructed\n(Note that  ``easy'' does not mean that these operators have a simple form, \nsee Ref.\\ \\cite{BK:FC}). \n\n\\begin{theorem}\nThe system of differential equations defined by the series \n(\\ref{FN:series})\nis reducible when the one of the following combinations of parameters \nis an integer: \n\t\\begin{eqnarray}\n\t&& \n\t\\left\\{ \n\ta_1, a_2, a_3, b_1, b_2,\n\t\\right\\}\n\t\\in \\mathbb{Z}  \\;, \n\t\\nonumber \\\\ &&  \n\t\\left\\{\n\ta_2\\!-\\!c_1\\!-\\!c_2\\!+\\!b_1, \\quad \n\ta_2\\!-\\!c_2\\!+\\!b_1, \\quad  \n\tb_2\\!-\\!c_1\\!-\\!c_2\\!+\\!b_1, \\quad  \n\tb_2\\!-\\!c_2\\!+\\!b_1 \\right\\}\n\t\\in \\mathbb{Z}  \n\t\\end{eqnarray}\n\t\\label{FN:monodromy}\n\\end{theorem}\nWhen one or more conditions of the Theorem \\ref{FN:monodromy}\nare valid, the number of independent differential equations \ndescribing the function $F_N$ reduces \nand some additional analysis is necessary (see for example \nRef. ~\\cite{KK:sunset}) to evaluate  \nthe value of irreducible holonomic rank.  \n\n\n\\subsubsection{Puiseux-type solution:  \nhypergeometric function $F_T$}\n\\label{FT:section}\nThe Lauricella-Saran hypergeometric function of three variables $F_T$  \nis defined about the point $z_1=z_2=z_3=0$ by\n\\begin{eqnarray}\n&& \nF_T(a_1;a_2;b_1,b_2; c; z_1,z_2,z_3)\n\\nonumber \\\\ && \n= \n\\sum_{m_1,m_2,m_3=0}^\\infty\n\\frac{(a_1)_{m_1} (a_2)_{m_2+m_3} (b_1)_{m_1+m_3} (b_2)_{m_2}}      \n{(c)_{m_1+m_2+m_3}} \n\\frac{z_1^{m_1}  z_2^{m_2}  z_3^{m_3}}{m_1! m_2! m_3!} \n\\;.\n\\nonumber \\\\ && \n\\label{FT:series}\n\\end{eqnarray}\nIn this case, the differential operators, Eq.~(\\ref{diff}), are the following: \n\\begin{subequations}\n\t\\begin{eqnarray}\n\tL_1 F_T: && \n\t\\quad \n\t\\theta_1 \\left(c \\!-\\! 1 \\!+\\! \\sum_{j=1}^3 \\theta_j \\right) F_T =  \n\tz_1 \\left(a_1 \\!+\\! \\theta_1 \\right) \\left(b_1 \\!+\\! \\theta_1 \\!+\\! \\theta_3 \\right) F_T \\;,\n\t\\\\  \n\tL_2 F_T: && \n\t\\quad \n\t\\theta_2 \\left(c-1 \\!+\\! \\sum_{j=1}^3 \\theta_j \\right) F_T =  \n\tz_2 \\left(a_2 \\!+\\! \\theta_2 \\!+\\! \\theta_3 \\right) \\left(b_2 \\!+\\! \\theta_2 \\right) F_T\n\t\\;, \n\t\\\\\n\tL_3 F_T: && \n\t\\quad \n\t\\theta_3 \\left(c-1 \\!+\\! \\sum_{j=1}^3 \\theta_j \\right) F_T =  \n\tz_3 \\left(a_2 \\!+\\! \\theta_2 \\!+\\! \\theta_3 \\right) \\left(b_1 \\!+\\! \\theta_1 \\!+\\! \\theta_3 \\right) F_T\n\t\\;, \n\t\\end{eqnarray}\n\t\\label{FT:diff}\n\\end{subequations}\nwhere \n$\nF_T = F_T(a_1;a_2;b_1,b_2; c; z_1,z_2,z_3) \\;.\n$\n\n\n\nLet us introduce the function \n\\begin{equation}\n\\Phi_T = \\frac{z_1^{1-c+a_2} z_2^{1-c+b_1}  }\n{ z_3^{1-c+b_1+a_2} } \\;.\n\\label{PhiT}\n\\end{equation}\nIt is easy to check that \n$$\nL_1 \\Phi_T = L_2 \\Phi_T = L_3 \\Phi_T = 0 \\;.\n$$\n\n\n\\begin{theorem}\n\tThe system of differential equations defined by Eq.~(\\ref{FT:diff}) \nhas a Puiseux-type  solution: \n\t\\begin{equation}\n\t\\Phi_T = \\frac{z_1^{1-c+a_2} z_2^{1-c+b_1}  }\n\t{ z_3^{1-c+b_1+a_2} } \\;.\n\t\\end{equation}\n\t\\end{theorem}\nIn particular, to construct the inverse contiguous relations \nfor the function $F_T$, one extra differential equation should be added. \n\nFor completeness, we also note the following result:\n\\begin{theorem}\n\tThe monodromy group of the system of differential equations \ndefined by Eq.~(\\ref{FT:diff}) \nis reducible when the one of the following combinations of parameters is \nan integer: \n$$\n\t\t\\left\\{ \n\t\ta_1, a_2, b_1, b_2, \\quad \n\t\tc\\!-\\!b_1\\!-\\!b_2, \\quad \n\t\tc\\!-\\!a_1\\!-\\!a_2\n\t\t\\right\\} \n\t\t\\in \\mathbb{Z}   \\;.\n$$\n\\end{theorem}\n\nA Puiseux-type solution for the hypergeometric differential equation\nof two variables was established by Erdelyi~\\cite{erdelyi:1950} still in the 50's and has been \nanalyzed in detail in ~\\cite{sadykov} in the framework of the GKZ approach. \n\n \n\\subsection{\nConstruction of the $\\ep$-expansion via differential equations: the Appell Function $F_3$:}\n\\label{F3:general}\nTo explain the technical details of our algorithm, \nlet us analyze and construct \nthe $\\ep$-expansion \nfor the Appell hypergeometric function $F_3$. The preliminary results \nhave been presented in ~\\cite{yost2011} and ~\\cite{BKK2012}. \n\n\\subsubsection{Notations}\nLet us consider the Appell hypergeometric function $F_3$ defined about\n $x=y=0$ as\n\n\\begin{eqnarray}\n\\omega_0 \\equiv F_3(a_1, a_2, b_1, b_2, c; x,y)\n=\n\\sum_{m=0}^\\infty\n\\sum_{n=0}^\\infty\n\\frac{(a_1)_{m} (a_2)_{n} (b_1)_m (b_2)_n}{(c)_{m+n} }\n\\frac{x^m}{m!}\n\\frac{y^n}{n!} \\;,\n\\label{definition:f3}\n\\end{eqnarray}\nIt is symmetric with respect to simultaneous exchange\n\\begin{eqnarray}\na_1 \\Leftrightarrow a_2 \\;, \\quad\nb_1 \\Leftrightarrow b_2 \\;, \\quad\nx  \\Leftrightarrow y  \\;.\n\\label{F3:symmetry}\n\\end{eqnarray}\nso that\n$\nF_3(a_1,a_2,b_1,b_2,c;x,y)\n=\nF_3(a_2,a_1,b_2,b_1,c;y,x) \\;.\n$\n\n\nThis function satisfies four differential equations\n(see Section 3.4 in ~\\cite{hyperdire}): \n\\begin{subequations}\n\t\\begin{eqnarray}\n\t(1\\!-\\!x) \\theta_{xx}  \\omega_0\n\t& = &  - \\theta_{xy} \\omega_0\n\t+\n\t\\left[\n\t(a_1 \\!+\\! b_1) x \\!-\\! (c \\!-\\! 1)\n\t\\right] \\theta_x \\omega_0\n\t\\!+\\! x a_1 b_1 \\omega_0 \\;,\n\t\\\\\n\n\n\n\n\n\n\n\n\t(xy\\!-\\!x\\!-\\!y) \\theta_{xxy} \\omega_0 & = &\n\t\\left[\n\t(1\\!-\\!y)(a_1 \\!+\\! b_1)x \\!-\\! y (a_2 \\!+\\! b_2 \\!+\\! 1 \\!-\\! c)\n\t\\right] \\theta_{xy} \\omega_0\n\t\\nonumber \\\\ &&\n\t+ (1\\!-\\!y) x a_1 b_1 \\theta_y \\omega_0\n\t\\!-\\! y a_2 b_2 \\theta_x \\omega_0 \\;,\n\n\n\n\n\n\n\n\n\t\\end{eqnarray}\n\t\\label{F3:system}\n\\end{subequations}\nand two other equations follow from  Eqs.~(\\ref{F3:system}) and the symmetry \nrelation (\\ref{F3:symmetry}). \nEqs.\\ (\\ref{F3:system}) can be written in projective space with homogeneous coordinates \non ${\\bf P}_2(C)$ with $x = X\/Z$ and $y = Y\/Z$.  \nWe want also to remark that the system of differential equations\nfor the Appell hypergeometric function \n$F_2(u,v)$ coincides with the system Eq.\\ (\\ref{F3:system})\nby a redefinition $u=1\/x,v=1\/y$. This result follows also from \nanalytic continuation of the Mellin-Barnes representation for \nAppell's functions $F_2$ and $F_3$.\n\n\\subsubsection{One-fold iterated solution}\nAssuming the most general form of the parameters\n$$\na_j =  \\frac{p_j}{q} \\!+\\! a_1 \\varepsilon \\;, \\quad \nb_j =  \\frac{r_j}{q} \\!+\\! b_j \\varepsilon \\;, \\quad  \nc =  1 - \\frac{p}{q} + c \\varepsilon \\; . \n$$\t\nwhere $p_i,r_i,p,q$ are integers, \nand writing the $\\varepsilon$-expansion for \nthe functions as \n\\begin{eqnarray}\n\\omega_j = \\sum_{k=0}^\\infty \\omega_j^{(k)} \\varepsilon^k \\;, \\quad j = 0,1,2,3,\n\\end{eqnarray}\nwe  obtain the full system of differential equations\n(under the conditions $p_j r_j = 0 \\;, j=1,2$):\n\\begin{subequations}\n\t\\begin{eqnarray}\n\t&& \n\t\\left[ (1\\!-\\!x) \\frac{d}{dx} \n\t\\!-\\! \\frac{s_1}{q} \n\t\\!-\\! \\frac{1}{x} \\frac{p}{q} \n\t\\right]\n\t\\omega_1^{(r)}\n\t\t= - \\frac{1}{x} \\omega_3^{(r)}\n\t\\!+\\!\n\t\\left[\n\t(a_1 \\!+\\! b_1) \\!-\\! \\frac{c}{x}\n\t\\right] \\omega_1^{(r-1)}\n\\nonumber \\\\ && \\hspace{5mm}\n\t+\n\t\\left[ \n\t\\frac{a_1 r_1\\!+\\!b_1 p_1}{q} \n\t\\right]  \\omega_0^{(r-1)}\n\t\\!+\\! a_1 b_1 \\omega_0^{(r-2)} \n\t\\;,\n\t\\label{f3:1}\n\t\\\\\n\t&& \n\t\\left[ (1\\!-\\!y) \\frac{d}{dy} - \\frac{s_2}{q} - \\frac{1}{y} \\frac{p}{q} \\right] \\omega_2^{(r)}\n\t=  \n\t- \\frac{1}{y} \\omega_3^{(r)}\n\t+\n\t\\left[\n\t(a_2 \\!+\\! b_2) \\!-\\! \\frac{c}{y}\n\t\\right] \\omega_2^{(r-1)}\n\t\\nonumber \\\\ && \\hspace{5mm}\n\t+ \n\t\\left[ \n\t\\frac{a_2 r_2+b_2 p_2}{q}  \n\t\\right] \\omega_0^{(r-1)}\n\t\\!+\\! a_2 b_2s \\omega_0^{(r-2)}\n\t\\;,\n\n\n\n\t\\\\ [1cm]\n\t&& \n\t\\left[\n\t(xy\\!-\\!x\\!-\\!y) \\frac{d}{dx} \n\t- (1-y) \\frac{s_1}{q}\n\t+ \\frac{y}{x} \\frac{(s_2 + p)}{q}\n\t\\right]\n\t\\omega_3^{(r)}  \n\t\\nonumber \\\\ && \\hspace{5mm}\n\t=\n\t\\left[\n\t(1\\!-\\!y)(a_1 \\!+\\! b_1) \\!-\\! \\frac{y}{x} (a_2 \\!+\\! b_2  \\!-\\! c)\n\t\\right] \\omega_3^{(r-1)}\n\t\\nonumber \\\\ && \\hspace{5mm}\n\t+ \n\t(1\\!-\\!y)\n\t\\left[ \n\t\\frac{a_1 r_1 \\!+\\! b_1 p_1}{q}  \\omega_2^{(r-1)} \n\t+ a_1 b_1  \\omega_2^{(r-2)}\n\t\\right]\n\t\\nonumber \\\\ && \\hspace{5mm}\n\n\n\n\t\\!-\\! \\frac{y}{x}  \n\t\\left[\n\t\\frac{a_2 r_2 \\!+\\! b_2 p_2}{q}  \\omega_1^{(r-1)} \n\t\\!+\\! a_2 b_2  \\omega_1^{(r-2)} \n\t\\right]\n\t\\;,\n\t\\nonumber \\\\\n\t\\label{f3:2}\n\t\\\\ [1cm]\n\t&& \n\t\\left[\n\t(xy\\!-\\!x\\!-\\!y) \\frac{d}{dy} \n\t- (1-x) \\frac{s_2}{q}\n\t+ \\frac{x}{y} \\frac{(s_1 + p)}{q}\n\t\\right]\n\t\\omega_3^{(r)} \n\t\\nonumber \\\\ &&\n\t= \n\t\\left[\n\t(1\\!-\\!x)(a_2 \\!+\\! b_2) \n\t\\!-\\! \n\t\\frac{x}{y} (a_1 \\!+\\! b_1  \\!-\\! c)\n\t\\right] \\omega_3^{(r-1)} \n\t\\nonumber \\\\ &&\n\t+ (1\\!-\\!x) \n\t\\left[ \n\t\\frac{a_2 r_2 + b_2 p_2}{q} \\omega_1^{(r-1)} \n\t+ a_2 b_2 \\omega_1^{(r-2)}  \n\t\\right]\n\t\\nonumber \\\\ && \\hspace{5mm}\n\t\\!-\\! \n\t\\frac{x}{y}\n\t\\left[\n\t\\frac{a_1 r_1 + b_1 p_1}{q} \\omega_2^{(r-1)} \n\t+ a_1 b_1 \\omega_2^{(r-2)} \n\t\\right]\n\t\\;,\n\t\\nonumber \\\\\n\t\\end{eqnarray}\n\t\\label{F3:system:expanded}\n\\end{subequations}\nwhere we have introduced new notations: \n\\begin{equation}\ns_1  = p_1 + r_1 \\;, \\quad \ns_2 = p_2 + r_2  \\;.\n\\label{s1s2}\n\\end{equation}\nWe want to mention that this system of differential equations does not have \nthe $\\ep$-form {\\em a-la} Henn's form in ~\\cite{Henn:1,Henn:2}. \nHowever, \nthe system of equations Eq.~(\\ref{F3:system:expanded})\ncan be straightforwardly solved iteratively. \nLet us redefine functions\n$\n\\omega_1, \\omega_2, \\omega_3, \n$\nas follows:\n\\begin{equation}\n\\omega_{1}^{(r)} = \\theta_{x} \\omega_0 =  h_1(x) \\phi_{1}^{(r)} \\;, \n\\quad \n\\omega_2^{(r)}  = \\theta_y \\omega_0 =  h_2(y) \\phi_2^{(r)} \\;, \n\\end{equation}\nwhere $h_{1,2}(x)$ are new functions, defined as\n\\begin{equation}\nh_1(x) =  \n\\sigma_1\n\\left[ \n\\frac{x^p}{(x-1)^{s_1+p}}\n\\right]^\\frac{1}{q} \n\\;, \n\\quad \nh_2(y)  =   \n\\sigma_2\n\\left[ \n\\frac{y^p}{(y-1)^{s_2+p}}\n\\right]^\\frac{1}{q} \n\\;, \n\\label{h}\n\\end{equation}\nand \n\\begin{eqnarray}\n\\omega_{3}^{(r)} & = & \\theta_{x} \\theta_y \\omega_0^{(r)} = H(x,y) \\phi_{3}^{(r)} \\;, \n\\label{H}\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\nH(x,y) & = &  \n\\sigma_3\n\\left[ \n\\frac{x^{s_2+p}y^{s_1+p}}\n{(xy-x-y)^{s_1+s_2+p}}\n\\right]^\\frac{1}{q} \n\\;, \n\\end{eqnarray}\nwhere $\\sigma_j, j=1,2,3$ are some normalization constants.\n\nSubstituting it to the original system, Eq.~(\\ref{F3:system:expanded}), we have \n(for completeness we show all four equations explicitly):\n\\begin{subequations}\n\\begin{eqnarray}\n\t&& \n\t(1\\!-\\!x) \\frac{d}{dx} \n\t\\phi_1^{(r)}\n\t= - \\frac{1}{x} \\frac{H(x,y)}{h_1(x)} \\phi_3^{(r)}\n\t\\\\ && \\hspace{5mm}\n\t+\n\t\\left[\n\t(a_1 \\!+\\! b_1) \\!-\\! \\frac{c}{x}\n\t\\right] \\phi_1^{(r-1)}\n\t+ \n\t\\left[ \n\t\\frac{a_1 r_1+b_1 p_1}{q} \n\t\\right]  \n\t\\frac{1}{h_1(x)} \\omega_0^{(r-1)}\n\t\\!+\\! \n\t\\frac{1}{h_1(x)} a_1 b_1 \\omega_0^{(r-2)} \n\t\\;,\n\t\\label{f3:3a}\n\t\\nonumber \\\\ [1cm]\n\t&& \n\t(1\\!-\\!y) \\frac{d}{dy} \n\t\\phi_2^{(r)}\n\t= - \\frac{1}{y} \\frac{H(x,y)}{h_2(y)} \\phi_3^{(r)}\n\t\\\\ && \\hspace{5mm}\n\t+\n\t\\left[\n\t(a_2 \\!+\\! b_2) \\!-\\! \\frac{c}{y}\n\t\\right] \\phi_2^{(r-1)}\n\t+ \n\t\\left[ \n\t\\frac{a_2 r_2+b_2 p_2}{q} \n\t\\right]  \n\t\\frac{1}{h_2(y)} \\omega_0^{(r-1)}\n\t\\!+\\! \n\t\\frac{1}{h_2(y)} a_2 b_2 \\omega_0^{(r-2)} \n\t\\;,\n\t\\nonumber \n\t\\label{f3:3b}\n\t\\end{eqnarray}\n\t\\label{F3:transformed1}\n\\end{subequations}\n\\begin{subequations}\n\t\\begin{eqnarray}\n\t&& \n\t(xy\\!-\\!x\\!-\\!y) \\frac{d}{dx} \n\t\\phi_3^{(r)}  \n\n\t=\n\t\\left[\n\t(1\\!-\\!y)(a_1 \\!+\\! b_1) \\!-\\! \\frac{y}{x} (a_2 \\!+\\! b_2  \\!-\\! c)\n\t\\right] \\phi_3^{(r-1)}\n\t\\nonumber \\\\ && \\hspace{5mm}\n\t+ \\frac{a_1 r_1 + b_1 p_1}{q} (1\\!-\\!y) \\frac{h_2(y)}{H(x,y)} \\phi_2^{(r-1)}  \n\t+ a_1 b_1 (1\\!-\\!y) \\frac{h_2(y)}{H(x,y)}  \\phi_2^{(r-2)}\n\t\\nonumber \\\\ && \\hspace{5mm}\n\n\n\n\t- \\frac{a_2 r_2 + b_2 p_2}{q}  \\frac{y}{x} \\frac{h_1(x)}{H(x,y)}  \\phi_1^{(r-1)} \n\t- a_2 b_2 \\frac{y}{x}  \\frac{h_1(x)}{H(x,y)}  \\phi_1^{(r-2)} \n\t\\;,\n\t\\label{f3:4a}\n\t\\\\ [1cm]\n\t&& \n\t(xy\\!-\\!x\\!-\\!y) \\frac{d}{dy} \n\t\\phi_3^{(r)}  \n\n\t=\n\t\\left[\n\t(1\\!-\\!x)(a_2 \\!+\\! b_2) \\!-\\! \\frac{x}{y} (a_1 \\!+\\! b_1  \\!-\\! c)\n\t\\right] \\phi_3^{(r-1)}\n\t\\nonumber \\\\ && \\hspace{5mm}\n\t+ \\frac{a_2 r_2 + b_2 p_2}{q} (1\\!-\\!x) \\frac{h_1(x)}{H(x,y)} \\phi_1^{(r-1)}  \n\t+ a_2 b_2 (1\\!-\\!x) \\frac{h_1(x)}{H(x,y)}  \\phi_1^{(r-2)}\n\t\\nonumber \\\\ && \\hspace{5mm}\n\n\n\n\t- \\frac{a_1 r_1 + b_1 p_1}{q}  \\frac{x}{y} \\frac{h_2(y)}{H(x,y)}  \\phi_2^{(r-1)} \n\t- a_1 b_1 \\frac{x}{y}  \\frac{h_2(y)}{H(x,y)}  \\phi_2^{(r-2)} \n\t\\;,\n\t\\label{f3:4b}\n\t\\end{eqnarray}\n\t\\label{F3:transformed2}\n\\end{subequations}\nThis is a system of linear differential equations \nwith algebraic coefficients. For completeness, \nit should be supplemented by the boundary condition. \nBoundary conditions are discussed below. \n\n\\noindent \n{\\bf Remark}:\\\\\nLet us recall that the surface of singularities $L(x,y)$ of the $F_3$ hypergeometric function is defined by the system of equations \nEq.\\ (\\ref{F3:system}) and has the form\n\\begin{equation}\nL: L_1 \\cup L_2 \\cup L_3 \\cup L_4 \\cup L_5\n   \\equiv x \\cup y \\cup (1-x) \\cup (1-y) \\cup (x+y-xy) \\;.\n\\label{L}\n\\end{equation}\nThe extra functions, Eqs.(\\ref{h}) and (\\ref{H}), can be \nunderstood as  the ratio of  elements $L_a$ of Eq.\\ (\\ref{L}), and $q$-root is related with \nangle of rotations of curves $L_j$ around zero.\n\nIndeed, a multiple polylogarithm can be understood as a\nsmooth map $U$  from one region of singularities, where the solution of differential equation exists, to another one:  \n$\\Li{A}{L_i} \\xrightarrow{U} \\Li{A}{L_j}$. Such map is nothing but an analytic continuation that mixed the singularities of the differential system Eq.\\ (\\ref{F3:system}). \n\n\\subsubsection{Boundary conditions}\n\\label{F3:boundary}\nThe boundary conditions for the system of equations, \nEqs.\\ (\\ref{F3:transformed1}),(\\ref{F3:transformed2}), are defined by the  series representation, so that  \n\\begin{eqnarray} \n\\left. \n\\omega_0(z_1,z_2) \\right|_{z_1=0} \n= {}_2F_1(a_2,b_2;c;z_2) \\;, \n\\quad \n\\left. \n\\omega_0(z_1,z_2) \\right|_{z_2=0} \n=  {}_2F_1(a_1,b_1;c;z_1) \\;. \n\\end{eqnarray}\nKeeping in mind that \n\\begin{eqnarray}\n\\omega_j(z_1,z_2) & = &  \nz_j \\frac{a_j b_j}{c} \n\\left. \nF_3(a_1, a_2, b_1, b_2, 1+c; z_1,z_2)\\right|_{\\substack{\n\ta_j \\to a_j+1 \\\\\n    b_j \\to b_j+1}} \\;, \\quad j = 1,2 \\;, \n\\nonumber \\\\ \n\\omega_3(z_1,z_2) \n& = & \nz_1 z_2 \\frac{a_1 a_2 b_1 b_2}{c(1+c)} \nF_3(1+a_1, 1+a_2, 1+b_1, 1+b_2; 2+c; z_1,z_2) \\;,\n\\nonumber \n\\end{eqnarray}\nwe have\n\\begin{eqnarray}\n\\left.\n\\omega_1(z_1,z_2) \n\\right|_{z_1=0}\n& = & 0 \\;, \n\\quad \n\\left. \n\\omega_1(z_1,z_2) \n\\right|_{z_2=0}\n=  z_1 \\frac{a_1 b_1}{c} {}_2F_1(1+a_1,1+b_1;1+c;z_1) \\;, \n\\nonumber \\\\ \n\\left.\n\\omega_2(z_1,z_2) \n\\right|_{z_2=0}\n& = & 0 \\;, \n\\quad \n\\left. \n\\omega_2(z_1,z_2) \n\\right|_{z_1=0}\n=  z_2 \\frac{a_2 b_2}{c} {}_2F_1(1+a_2,1+b_2;1+c;z_2) \\;, \n\\nonumber \\\\ \n\\left. \n\\omega_3(z_1,z_2) \n\\right|_{z_1=0}\n& = & \n\\left. \n\\omega_3(z_1,z_2) \n\\right|_{z_2=0}\n= 0 \\;. \n\\end{eqnarray}\nThe construction of the all-order $\\varepsilon$-expansion of Gauss hypergeometric functions around rational values of parameters in terms of \nmultiple polylogarithms \nhas been constructed in Refs.\\ \\cite{kwy2006},\\cite{kk2008}.\n\n\n\\subsubsection{The rational parametrization: \n\t            towards multiple polylogarithms}\nIt is well-known that one-fold iterated integrals over \nalgebraic functions are not, in general, expressible \nin terms of multiple polylogarithms \nbut demand the introduction \nof a new \nclass of functions~\\cite{non-polylog:1,non-polylog:2,non-polylog:3}\n\\footnote{Originally, \n\tsuch types of functions have been introduced in \nRef.\\ \\cite{Bonciani}.}.\n\nThe iterative solution of the system Eq.~(\\ref{F3:transformed1}),(\\ref{F3:transformed2}), \nhave the following form. \nIn the first two orders of $\\varepsilon$-expansion, we have\n(these results follow from the series representation and a \nspecial choice of parameters):\n\\begin{subequations}\n\t\\begin{eqnarray}\n\t&& \n\t\\omega_0^{(0)} = 1 \\;, \n\t\\quad \n\t\\phi_1^{(0)} = \\phi_2^{(0)} = \\phi_3^{(0)} = 0 \\;, \n\t\\\\ && \n\t\\omega_0^{(1)} =  \\phi_1^{(1)} = \\phi_2^{(1)} = \\phi_3^{(1)} = 0 \\;, \n\t\\end{eqnarray}\n\\end{subequations}\n\nThe second iteration produces: \n\\begin{subequations}\n\\begin{eqnarray}\n\\phi_3^{(2)}(x,y) & = & 0 \\;, \n\\\\  \n\\phi_1^{(2)}(x,y) & = & a_1 b_1 \\int_0^x \\frac{dt}{(1-t) h_1(t)}  \n \\equiv a_1 b_1 R_1(x) \\;, \n\\\\  \n\\phi_2^{(2)}(x,y) & = & a_2 b_2 \\int_0^y \\frac{dt}{(1-t) h_2(t)}  \n \\equiv  a_2 b_2 R_2(y)\n\\;, \n\\\\  \n\\omega_0^{(2)}(x,y) & = &  \n\\int_0^x \\frac{dt_1}{t_1} \\phi_1^{(2)} + {}_2F_1^{(2)}(x)\n+ \n\\int_0^y \\frac{dt_1}{t_1} \\phi_2^{(2)} + {}_2F_1^{(2)}(y)\n\\\\ && \n=  \n\\left[ a_1 b_1 \\int_0^x \\frac{dt_1}{d_1} R_1(t) + a_2 b_2 \\int_0^y \\frac{dt_1}{t_1} R_2(t) \\right]  \n+ {}_2F_1^{(2)}(x) + {}_2F_1^{(2)}(y)\n\\nonumber \n\\;, \n\\end{eqnarray}\n\\end{subequations}\nwhere ${}_2F_1^{(2)}(t)$ are the functions coming from boundary conditions.  \nThe finite part of the $F_3$ function is (in terms of $R$-functions):\n\n\t\\begin{eqnarray}\n\t\\phi_3^{(3)}(x,y) & = &  \n\t-\\left( \\frac{a_1 r_1 + b_1 p_1}{q}  \\right)  a_2 b_2 \n\th_2(y) R_2(y) \\int_0^x \\frac{dt}{H(t,y)\\left(t+\\frac{y}{1-y} \\right)} \n\t\\nonumber \\\\ && \n\t+ \\left( \\frac{a_2 r_2 + b_2 p_2}{q} \\right)  a_1 b_1 \n\t\\int_0^x dt \\frac{h_1(t) R_1(t)}{H(t,y)}\n\t\\left(\n\t\\frac{1}{t} - \\frac{1}{t+\\frac{y}{1-y}}\n\t\\right)\n\t\\nonumber \\\\ && \n\n\t- \\left( \\frac{a_2 r_2 + b_2 p_2}{q}  \\right) a_1 b_1 h_1(x) R_1(x)\n\t\\int_0^y dt \\frac{dt}{H(x,t)\\left(t+\\frac{x}{1-x} \\right)} \n\t\\nonumber \\\\ && \n\t+ \\left( \\frac{a_1 r_1 + b_1 p_1}{q} \\right) a_1 b_1 \n\t\\int_0^y dt \\frac{h_2(t) R_2(t)}{H(x,t)}\n\t\\left(\n\t\\frac{1}{t} - \\frac{1}{t+\\frac{x}{1-x}}\n\t\\right)\n\t\\;. \n\t\\end{eqnarray}\n\n\nUp to some factor, the function\n$R(t)$ coincides with the Gauss hypergeometric function with a rational set of parameters: \n$$\nR(z)\n\\equiv\n\\int_0^z \\frac{dt}{(1-t)} \\frac{1}{h(t)}\n\\sim\n{}_2F_1\n\\left(\n\\frac{r_j}{q}, \\frac{p_j}{q},\n1-\\frac{p}{q}; z\n\\right)\\;, \\quad j = 1,2,\n$$\nwhere $p_j r_j = 0$. \nThe $\\varepsilon$-expansion of the Gauss hypergeometric function\naround rational values of parameters\nhas been analyzed in detail in Ref.\\ \\cite{kk2008}.\nIt was shown that only the following cases are relevant: \n\\begin{itemize}\n\t\\item\n\tInteger set:\n\t$\t\n\tr_j = p_j = p = 0  \\;, \n\t$\n\t\\item\n\tZero-balance type: \n\t$r_j + p_j = -p \\;, $\n\t\\quad \n\t$r_j p_j = 0 \\;.$ \n\t\\item\n\tBinomial type: \n\t$p=p_j = 0 $\\;, \\quad  $r_j \\neq 0 $ \\;, \n\t(and symmetric one: \t\n\t$p=r_j = 0 $\\;, \\quad $p_j \\neq 0 $)\n\t\\item\n\tInverse binomial type:\n\t$r_j=p_j = 0 $\\;, \\quad $p \\neq 0 $\\;. \n\t\\item\n\tFull type: \n\t$r_j=p_j = -p \\;.$\n\\end{itemize}\nFor each particular set of parameters, \nthe rational parametrization of Eqs.~(\\ref{F3:transformed1}),(\\ref{F3:transformed2})\nshould be cross-checked. \nFor illustration,  let us consider \na few examples. \n\\subsubsection{The rational parametrization: set 1}\n\\label{F3:set1}\nLet us consider the hypergeometric function\n$$\nF_3\\left(a_1 \\varepsilon, \n         a_2  \\varepsilon, \n         b_1\\varepsilon, \n         b_2\\varepsilon, \n         1-\\frac{p}{q}+c\\varepsilon; x,y \\right) \\;, \n$$\nso that \n$$\nr_1 = p_1 = p_2 = s_2 = 0 \\;, \\quad p \\neq 0 \\;. \n$$\nFor this set of parameters, we have \n$\ns_1  =  0 \\;, s_2 = 0 \\;,  p \\neq 0 \\;, \n$\nand \n$$\nh(x) = \\left( \\frac{x}{x-1} \\right)^{\\frac{p}{q}} \\;, \n\\quad \nh(y)  =  \\left( \\frac{y}{y-1} \\right)^{\\frac{p}{q}} \\;, \n\\quad \nH(x,y)  =   \n\\left[ \n\\frac{xy}\n{((x-1)(y-1)-1)}\n\\right]^\\frac{p}{q} \\;.\n$$\nThere is a parametrization \nthat converts the  functions $h_1$ and $h_2$ \ninto rational functions $P_1$ and $P_2$:\n$$\n\\frac{x}{x-1} = P_1^q(\\xi_1,\\xi_2) \\;, \n\\quad \n\\frac{y}{y-1} = P_2^q(\\xi_1,\\xi_2) \\;, \n$$\nwhere the functions $P_1$ and $P_2$ have the form \n\\begin{equation}\nP_m(x,y) = \\Pi_{i,j,k,l} \\frac{(x-a_i)(y-b_j)}{(x-c_k) (y-d_l)}\\;, \\quad \nm=1,2 \\;, \n\\label{rational}\n\\end{equation}\nfor a set of algebraic numbers $\\{a_i,b_j,c_k,d_l\\}$. \nIn terms of new variables, the function $H$ has the form \n$$\nH(x,y) = \\frac{xy}{x+y-xy} = \\frac{1}{\\frac{1}{x} + \\frac{1}{y} - 1} = \n\\frac{1}{\\frac{1}{P_1^q} + \\frac{1}{P_2^q} + 1} = P_3^q \\;,\n$$\nwhere \n$P_3$ \nis again a rational function of two variables of the type (\\ref{rational}).\n\nAfter a redefinition \n$\n\\frac{1}{P_i^q}  = Q_i^q \\;, \\quad i =1,2,3\n$\nwe obtain the result  that a \nstatement about existence of a rational parametrization for the functions \n$h(x),~h(y)$ and $H(x,y)$ is equivalent to the existence of \nthree rational functions of two \nvariables satisfying the equation  \n\\begin{eqnarray}\nQ_1^q + Q_2^q + Q_3^q = 1\\;.\n\\label{rational-parametrization}\n\\end{eqnarray}\nTo our knowledge, for $q>2$, a solution exists only in terms \nof elliptic functions. However, it may happen that \nsuch a parametrization exists for $q=2$, but we are not able to find it.\n\nThis problem is closely related to  \nthe solution of a functional equation. \nFor example, for the equation \n$$\nf^n + g^n = 1\\;, \n$$\nthe solution can be characterized as follows: \n\\cite{gross,baker}:\n\\begin{itemize}\n\\item\nFor $n=2$, all solutions are the form \n$$\nf = \\frac{2 \\beta(z)}{ 1+\\beta^2(z)} \\;, \n\\quad \ng = \\frac{1-\\beta^2(z)}{1+\\beta^2(z)} \\;,\n$$\nwhere $\\beta(z)$ is an arbitrary function. \n\\item\nFor $n=3$, one solution is given by\n\\begin{equation}\nf = \\frac{1}{2 \\wp}\n\\left(\n1 + \\frac{1}{\\sqrt{3}} \\wp'\n\\right)\\;, \n\\quad \ng =  - \\frac{1}{2 \\wp}\n\\left(\n1 - \\frac{1}{\\sqrt{3}} \\wp'\n\\right)\\;, \n\\end{equation}\nwhere $\\wp$ is the Weierstrass $\\wp$-function satisfy \n$\n\\left( \\wp' \\right)^2 = 4 \\wp^3 - 1 \\;. \n$\nFor $n=3$, the original equation is of genus $1$, so that \nuniformization theorem assures the existence of an elliptic solution. \n\\end{itemize}\n\n\nOne of the most natural sets of variables for\nthe set of parameters under consideration is the following: \n$\nP_1 = \\xi_1 \\;, P_2 = \\xi_2 \\;, \n$\nso that \n\\begin{equation}\nH = \\frac{1}{\\xi_1^p \\xi_2^p} \n\\left(\\xi_1^q \\xi_2^q - \\xi_1^q - \\xi_2^q \\right)^\\frac{p}{q} \\;. \n\\label{surface}\n\\end{equation}\n\n\n\\subsubsection{The rational parametrization: set 2}\n\\label{F3:set2}\nIn a similar manner, let us analyze another set of parameters: \n$$\nF_3\\left(-\\frac{p}{q} + a_1 \\varepsilon, \n         a_2 \\varepsilon, \n         b_1 \\varepsilon, \n         -\\frac{p}{q}+b_2\\varepsilon, \n         1-\\frac{p}{q}+c\\varepsilon; x,y\n\\right) \\;.\n$$\nIn this case, \n$\ns_1 = s_2 = -p \\;,   \n$\nand the functions $h$ have the form\n$\nh(x) = x^\\frac{p}{q} \\;, \n\\quad \nh(y) = y^\\frac{p}{q} \\;. \n$\nApplying the same trick with the introduction of new \nfunctions $P_1$ and $P_2$, we would find that the existence of a rational parametrization corresponds in the present case to the \nvalidity of Eq.\\ (\\ref{rational-parametrization}). \nIn particular, by introducing new variables \n$\nx^\\frac{1}{q} = \\xi_1, y^\\frac{1}{q} = \\xi_2 \\;, \n$ \nwe obtain\n$\nH = \n\\left(\\xi_1^q \\xi_2^q - \\xi_1^q - \\xi_2^q \\right)^\\frac{p}{q} \\;. \n$\n\\subsubsection{The rational parametrization: set 3}\n\\label{F3:set3}\nLet us analyze the following set of parameters: \n$\np = 0 \\;,  s_1, s_2 \\neq 0 \\;, \n$\ncorresponding to \n$$\nF_3\\left( \\frac{p_1}{q} + a_1 \\varepsilon, \n          \\frac{p_2}{q} + b_1 \\varepsilon, \n                 b_2 \\varepsilon, \n                 1+c\\varepsilon; x,y\n\\right) \\;.\n$$\nIn this case, \n$$\nh_1(x) = \\left(1-x \\right)^\\frac{p_1}{q} \\;, \n\\quad \nh_2(y) = \\left(1-y \\right)^\\frac{p_2}{q} \\;, \n\\quad \nH(x,y) = \\left(\n\\frac{x^{s_2} y^{s_1}}{(xy-x-y)^{s_1+s_2}} \n\\right)^\\frac{1}{q} \\;. \n$$\nFor simplicity, we set $s_1=-s_2=s$, and put \n$1-x = P_1^q$ and $1-y= P_2^q$, so that \n$H = \\left(\\frac{1-P_2^q}{1-P_1^q} \\right)^{\\frac{s}{q}} \\equiv P_3$. In particular, for $P_1 = \\xi_1, P_2 = \\xi_2$, \n$H = \\left(\\frac{1-\\xi_2^q}{1-\\xi_1^q} \\right)^{\\frac{s}{q}}$. \n\n\\subsubsection{The rational parametrization: set 4}\n\\label{F3:set4}\nLet us analyze the following set of parameters: \n$\ns_1 = -p \\;, s_2 = 0 \\;, \n$\nso that hypergeometric function is \n$$\nF_3\\left(-\\frac{p_1}{q} + a_1 \\varepsilon, \n                          a_2 \\varepsilon, \nb_1 \\varepsilon, \nb_2 \\varepsilon, 1-\\frac{p}{q}+c\\varepsilon; x,y\n\\right) \\;.\n$$\nIn this case, \n\\begin{eqnarray}\nh_1(x) & = & x^\\frac{p}{q} \\;, \n\\quad \nh_2(y)  =  \\left(\\frac{y}{y-1} \\right)^\\frac{p}{q} \\;, \n\\quad \nH(x,y)  =   x^\\frac{p}{q} \\equiv h(x) \\;.\n\\end{eqnarray}\nLet us take a new set of variables $(x,y) \\to (\\xi_1,\\xi_2)$: \n\\begin{eqnarray}\n\\xi_1 & = & x^\\frac{1}{q} \\;, \n\\quad \n\\xi_2  =  \\left(\\frac{y}{y-1}\\right)^\\frac{1}{q} \\;, \n\\end{eqnarray}\nIn terms of a new variables we have: \n\\begin{eqnarray}\nH(x,y) & \\equiv & h_1(x) =  \\xi_1^p \\;, \n\\quad \nh_2(y)  =  \\xi_2^p \\;, \n\\quad \nx+y-xy = \n\\frac{\\xi_1^q-\\xi_2^q}{1-\\xi_2^q} \\;. \n\\end{eqnarray}\nThus, a rational parametrization exists. \n\n\\subsubsection{The rational parametrization: set 5}\n\\label{F3:set5}\nConsider the set of parameters defined by \n$\ns_2 = p = 0 \\;,  \ns_1 \\neq 0,\n$\nthat corresponds to the case\n$\nF_3\\left(-\\frac{p_1}{q} + a_1 \\varepsilon, \n                          a_2 \\varepsilon, \nb_1 \\varepsilon, \nb_2 \\varepsilon, \n1+c \\varepsilon; x,y\n\\right) \\;.\n$\nIn this case, \n\\begin{eqnarray}\nh_1(x) = \\left(1-x\\right)^{\\frac{p_1}{q}} \\;, \n\\quad \nh_2(y) = 1\\;, \n\\quad \nH(x,y) = \\left( \\frac{y}{xy-x-y} \\right)^\\frac{p_1}{q} \\;, \n\\end{eqnarray}\nLet us suggest that \n$$\n1-x = P^q \\;, \n\\quad \nH(x,y) = \\frac{y}{x+y-xy} = Q^q \\;,\n$$\nwhere $P$ and $Q$ are rational functions. \nThen, \n$\ny =  \\frac{(1-P^q)Q^q}{1-P^q Q^q} \\;.\n$\nAfter the redefinition,\n$\nP Q = R \\; \n$,  we get\n$ \ny =  \\frac{(1-P^q)}{(1-R^q)} \\frac{R^q}{P^q} \\;. \n$\nThe simplest version of $P$ and $R$ are polynomials: \n$\nP = \\xi_1 \\;, \n$\nand \n$ \nR = \\xi_2 \\;. \n$ \nIn this parametrization, \n$$\nh(x) = \\xi_1^{p_1} \\;, \n\\quad \nH(x,y) = \\left( \\frac{\\xi_2}{\\xi_1} \\right)^{p_1}\\;, \n\\quad \ny = \\frac{1-\\xi_1^q}{1-\\xi_2^q} \\left( \\frac{\\xi_2}{\\xi_1}\\right)^q  \\;. \n$$\nIn this case, the rational parametrization exists. \n\n\\subsubsection{Explicit construction of expansion: integer values of parameters}\nLet us consider the construction of the $\\ep$-expansion around \ninteger values of parameters. \nIf we put \n$$\n\\omega_0 = F_3(a_1 \\ep, b_1 \\ep, a_2 \\ep, b_2 \\ep, 1+c\\ep;x,y), \n$$\nthen the system of differential equations can be presented \nin the form \n\\begin{eqnarray}\n\\frac{\\partial}{\\partial x} \\omega_1 & = & \n\\left[ \n\\frac{1}{x-1}\n- \\frac{1}{x} \n\\right]\n\\omega_3 \n- \\left[ \\frac{c}{x} + \\frac{(a_1+b_1-c)}{x-1} \\right] \\ep \\omega_1 \n- a_1 b_1 \\frac{1}{x-1} \\ep^2 \\omega_0 \n\\;, \n\\\\ \n\\frac{\\partial}{\\partial y} \\omega_2 & = & \n\\left[ \n\\frac{1}{y-1}\n- \\frac{1}{y} \n\\right] \\omega_3 \n- \\left[ \\frac{c}{y} + \\frac{(a_2+b_2-c)}{y-1} \\right] \\ep \\omega_2 \n- a_2 b_2 \\frac{1}{y-1} \\ep^2 \\omega_0 \n\\;, \n\\\\\n\\frac{\\partial}{\\partial x} \\omega_3 \n& = &  \n\\left[ \n\\frac{(a_2\\!+\\!b_2\\!-\\!c) }{x}\n\\!-\\! \n\\frac{(a_1 \\!+\\! b_1 \\!+\\! a_2\\!+\\!b_2\\!-\\!c)}{x+\\frac{y}{1-y}} \n\\right] \\ep \\omega_3 \n\\nonumber \\\\ && \n- \n\\frac{a_1 b_1}{x+\\frac{y}{1-y}} \\ep^2 \\omega_2 \n\\!+\\! \n\\left[ \n\\frac{1}{x} \n- \n\\frac{1}{x+\\frac{y}{1-y}}\n\\right]\na_2 b_2 \\ep^2 \\omega_1 \n\\;, \n\\\\ \n\\frac{\\partial}{\\partial y} \\omega_3 \n& = &  \n\\left[ \n\\frac{(a_1\\!+\\!b_1\\!-\\!c) }{y}\n\\!-\\! \n\\frac{(a_1 \\!+\\! b_1 \\!+\\! a_2\\!+\\!b_2\\!-\\!c)}{y+\\frac{x}{1-x}} \n\\right] \\ep \\omega_3 \n\\nonumber \\\\ && \n- \n\\frac{a_2 b_2}{y+\\frac{x}{1-x}} \\ep^2 \\omega_1 \n\\!+\\! \n\\left[ \n\\frac{1}{y} \n- \n\\frac{1}{y+\\frac{x}{1-x}}\n\\right]\na_1 b_1 \\ep^2 \\omega_2 \n\\;. \n\\end{eqnarray}\nThis system can be straightforwardly integrated in terms of multiple polylogarithms, defined via a one-fold \niterated integral $G$, where \n\\begin{eqnarray}\nG(z;a_k,\\vec{a}) & = & \n\\int_0^{z} \\frac{dt}{t-a_k}\nG(t;\\vec{a}) \\;.\n\\label{G}\n\\end{eqnarray}\n \nIn addition, the $\\ep$-expansion of a Gauss hypergeometric function around integer values of parameters is needed. \nIt has the following form (see Eq.~(34) in ~\\cite{MKL:Gauss}):\n\\begin{eqnarray}\n&& \n{}_2F_1(a\\ep,b\\ep;1\\!+\\!c\\ep;z) \n= \n1  \\!+\\! \na b \\ep^2 \\Li{2}{z}\n\\nonumber \\\\ && \\hspace{5mm}\n+ \na b \\ep^3 \n\\left[(a \\!+\\! b \\!-\\! c) \\Snp{1,2}{z} \\!-\\! c \\Li{3}{z} \\right] \n+ O(\\ep^4)\n\\;.\n\\nonumber \\\\ \n\\end{eqnarray}\n\nThe first iteration gives rise to\n\\begin{eqnarray}\n\\omega_0^{(0)} & = & 1 \\;, \n\\quad \n\\omega_1^{(0)} = \\omega_2^{(0)} = \\omega_3^{(0)} = 0 \\;, \n\\quad \n\\omega_0^{(1)} = \\omega_1^{(1)}  =  \\omega_2^{(1)} = \\omega_3^{(1)} = 0 \\;. \n\\nonumber \n\\end{eqnarray}\n\nThe results of the second iteration are the following: \n\\begin{eqnarray}\n\\omega_3^{(2)} & = & 0 \\;, \n\\quad \n\\omega_1^{(2)}  =  -a_1 b_1 \\ln(1-x) \\;, \n\\quad \n\\omega_2^{(2)}  = - a_2 b_2 \\ln(1-y) \\;, \n\\nonumber \\\\ \n\\omega_0^{(2)} & = &  a_1 b_1 \\Li{2}{x} + a_2 b_2 \\Li{2}{y} \\;, \n\\end{eqnarray}\nwhere the classical polylogarithms $\\Li{n}{z}$ \nare defined as \n\\begin{equation}\n\\Li{1}{z} = - \\ln(1-z) \\; ,\n\\quad \n\\Li{n+1}{z} = \\int_0^z \\frac{dt}{t} \\Li{n}{t}, \\quad n\\ge 1.\n\\label{polylogarithm}\n\\end{equation}\n\nAfter the third iteration, we have\n\\begin{eqnarray}\n\\omega_{3}^{(3)} &  = & 0\n\\\\ \n\\omega_1^{(3)} & = &  \n\\frac{1}{2} a_1 b_1 (a_1+b_1-c) \\ln^2 (1-x) \n- a_1 b_1 c \\Li{2}{x} \n\\;, \n\\\\ \n\\omega_2^{(3)} & = &  \n\\frac{1}{2} a_2 b_2 (a_2+b_2-c) \\ln^2 (1-y) \n- a_2 b_2 c \\Li{2}{y} \n\\;, \n\\\\\n\\omega_0^{(3)} & = &  \n- a_1 b_1 c \\Li{3}{x} \n- a_2 b_2 c \\Li{3}{y} \n\\nonumber \\\\ && \n+ a_1 b_1 (a_1+b_1-c) \\Snp{1,2}{x}\n+ a_2 b_2 (a_2+b_2-c) \\Snp{1,2}{y}\n\\;,\n\\end{eqnarray}\nwhere $\\Snp{a,b}{z}$ are the Nielsen polylogarithms: \n\\begin{equation}\nz \\frac{d}{d z} \\Snp{a,b}{z}  =  \\Snp{a-1,b}{z} \\;, \n\\quad \n\\Snp{1,b}{z}  =  \\frac{(-1)^b}{b!} \\int_0^1 \\frac{dx}{x} \\ln^b(1-zx) \\;. \n\\label{nielsen}\n\\end{equation}\n\nThe result of the next iteration, $\\omega_3^{(4)}(x,y)$, \ncan be expressed in several equivalent forms: \n\\begin{eqnarray}\n\\frac{\\omega_3^{(4)}(x,y)}{a_1 a_2 b_1 b_2} \n& = &  \n\\ln(1-y) G_1\\left(x; -\\frac{y}{1-y} \\right)\n- G_2(x;1)\n+ G_{1,1}\\left(x; - \\frac{y}{1-y},1 \\right) \n\\nonumber \\;, \n\\\\ \n\\frac{\\omega_3^{(4)}(x,y)}{a_1 a_2 b_1 b_2 } \n& = & \n\\ln(1-x) G_1\\left(y; -\\frac{x}{1-x} \\right)\n- G_2(y;1)\n+ G_{1,1}\\left(y; - \\frac{x}{1-x},1 \\right)\n\\;. \n\\nonumber \n\\end{eqnarray}\nKeeping in mind that \n\\begin{eqnarray}\n&& \nG_{1,1}\\left(x; - \\frac{y}{1-y},1 \\right)\n= \n\\int_0^x \\frac{dt}{t+\\frac{y}{1-y}} \\ln (1-t) \n\\\\ && \n=  \n- \\ln (1-y) \\ln (x+y-xy) \n+ \\ln(1-y) \\ln y\n- \\Li{2}{x+y-xy} \n+ \\Li{2}{y} \n\\nonumber  \\;,\n\\end{eqnarray}\nthe result can be written in a very simple form,\n\\begin{eqnarray}\n\\frac{\\omega_3^{(4)}(x,y)}{a_1 a_2 b_1 b_2} \n= \\Li{2}{x} + \\Li{2}{y} - \\Li{2}{x+y-xy} \\;.  \n\\end{eqnarray}\n\nTaking into account that \n$\n\\frac{\\omega_3^{(4)}(x,y)}{a_1 a_2 b_1 b_2}\n=  \\frac{1}{2} xy F_3(1,1,1,1;3;x,y)\\;, \n$\nwe obtain the well-known result~\\cite{sanchis-lozano} \n$$\n\\frac{1}{2} xy F_3(1,1,1,1;3;x,y) = \\Li{2}{x} + \\Li{2}{y} - \\Li{2}{x+y-xy} \\;.\n$$\nThere is also another form ~\\cite{brychkov} for this integral,\n\\begin{eqnarray}\n\\frac{1}{2} xy F_3(1,1,1,1;3;x,y) &  = & \n\\Li{2}{\\frac{x}{x+y-xy}} \n- \n\\Li{2}{\\frac{x-xy}{x+y-xy}} \n\\nonumber \\\\ && \n- \\ln (1-y) \\ln \\left(\\frac{y}{x+y-xy} \\right) \\;.\n\\nonumber \n\\end{eqnarray}\nOne form can be converted to the other using the well-known \ndilogarithm identity \n$$\n\\Li{2}{\\frac{1}{z}}\n= \n-\\Li{2}{z} \n- \\frac{1}{2} \\ln^2 (-z)\n-\\zeta_2 \\;,\n$$\ntogether with the attendant functional relations~\\cite{lewin,Li2identities}. \n\nThe following expressions result from direct iterations in terms of $G$-functions: \n\\begin{eqnarray}\n&&\n\\frac{\\omega_1^{(4)}(x,y)}{a_1 b_1} = \na_2 b_2 \n\\Biggl\\{ \n\\ln(1-y) \n\\left[\nG_{1,1}\\left(x; 1, -\\frac{y}{1-y} \\right)\n- \nG_{2}\\left(x; -\\frac{y}{1-y} \\right)\n\\right]\n\\nonumber \\\\ && \n- G_{1,2}(x;1,1) + G_{3}(x;1)\n+ G_{1,1,1}\\left(x; 1, - \\frac{y}{1-y},1 \\right)  \n- G_{2,1}\\left(x; - \\frac{y}{1-y},1 \\right) \n\\Biggr\\} \n\\nonumber \\\\ && \n+ a_2 b_2 G_1(x;1) G_2(y;1) \n- c \\Delta_1 G_{2,1}(x;1,1)\n- \\Delta_1^2 G_{1,1,1}(x;1,1,1)\n- c^2 G_{3}(x;1)\n\\nonumber \\\\ && \n+ (a_1 b_1 \\!-\\! c \\Delta_1) G_{1,2}(x;1,1) \n\\;, \n\\end{eqnarray}\nwhere \n$$\n\\Delta_j  = a_j + b_j - c \\;.\n$$\nThe $G$-functions can be converted into classical or \nNielsen polylogarithms with the help of the following relations: \n\\begin{subequations}\n\\begin{eqnarray}\n&& \nG_{1,1}\n\\left( \nx; 1, - \\frac{y}{1-y}\n\\right)\n= \nG_1(x; 1)\nG_1\\left( \nx; - \\frac{y}{1-y}\n\\right)\n- \nG_{1,1}\\left( \nx; - \\frac{y}{1-y}, 1\n\\right) \\;,\n\\\\ && \nG_{1}\n\\left( \nx; 1\n\\right)\nG_{1,1}\n\\left( \nx; - \\frac{y}{1-y}, 1\n\\right)\n= \nG_{1,1,1}\n\\left( \nx; 1, - \\frac{y}{1-y}, 1\n\\right)\n\\\\ && \\hspace{40mm}\n+ \n2G_{1,1,1}\n\\left( \nx; - \\frac{y}{1-y}, 1, 1\n\\right) \n\\;, \n\\nonumber \\\\ && \nG_{1,1,1}\n\\left( \nx; - \\frac{y}{1-y}, 1, 1\n\\right) \n= \n\\Snp{1,2}{x+y-xy}- \\Snp{1,2}{y}\n\\\\ && \n+ \\frac{1}{2} \\ln^2 (1-y)\n\\left[ \n\\ln(x+y-xy) \n\\!-\\! \n\\ln y \n\\right]\n+ \\ln(1-y) \n\\left[ \n\\Li{2}{x+y-xy} \\!-\\! \\Li{2}{y} \n\\right]\n\\;, \n\\nonumber \\\\ &&\nG_{2,1}\\left(x; -\\frac{y}{1-y} ,1 \\right)\n+ \\ln(1-y)  G_{2}\\left(x; -\\frac{y}{1-y} \\right)\n= \nG_{1,2}\\left(1; 1-\\frac{1}{y}, \\frac{1}{x} \\right)\n+ G_{3}\\left(x; 1 \\right)\n\\nonumber \\\\  && \n= \\int_0^x \\frac{du}{u}\n\\left[ \n\\Li{2}{y} - \\Li{2}{u+y-uy} \n\\right] \\;.\n\\end{eqnarray}\n\\end{subequations}\nIn a similar manner, \n\\begin{eqnarray}\n&&\n\\frac{\\omega_2^{(4)}(x,y)}{a_2 b_2} = \na_1 b_1  \n\\Biggl\\{ \n\\ln(1-x) \n\\left[\nG_{1,1}\\left(y; 1, -\\frac{x}{1-x} \\right)\n- \nG_{2}\\left(y; -\\frac{x}{1-x} \\right)\n\\right]\n\\nonumber \\\\ && \n- G_{1,2}(y;1,1) + G_{3}(y;1)\n+ G_{1,1,1}\\left(y; 1, - \\frac{x}{1-x},1 \\right)  \n- G_{2,1}\\left(y; - \\frac{x}{1-x},1 \\right) \n\\Biggr\\} \n\\nonumber \\\\ && \n- c \\Delta_2 G_{2,1}(y;1,1)\n- \\Delta_2^2 G_{1,1,1}(y;1,1,1)\n- c^2 G_{3}(y;1)\n- c \\Delta_2 G_{1,2}(y;1,1)\n\\nonumber \\\\ && \n+ a_1 b_1 G_1(y;1) G_2(x;1) \n+ a_2 b_2 G_{1,2}(y;1,1) \n\\;, \n\\end{eqnarray}\nand the last term, expressible in terms of functions of order $3$, is $\\omega_3^{(5)}$. \n\n\n\n\\subsubsection{Construction of $\\ep$-expansion via \n\t           integral representation}\n\nAn alternative approach to construction of the \nhigher order $\\ep$-expansion of generalized hypergeometric \nfunctions is based on their \nintegral representation. \nWe collect here some representations\nfor the Appell hypergeometric function $F_3$\nextracted from Refs.\\ \\cite{slater} and ~\\cite{srivastava}.\n\nFor our purposes, the most useful expression is the following:\n(see Eq.\\ (16) in Section 9.4. of ~\\cite{srivastava}), \n~\\cite{sanchis-lozano,brychkov}:\n\\begin{eqnarray}\n&& \n\\frac{\\Gamma(c_1) \\Gamma(c_2)}{\\Gamma(c_1+c_2)}\nF_3(a_1,b_1,a_2,b_2,c_1+c_2;x,y)\n\\nonumber \\\\ && \n = \\int_0^1 du u^{c_1-1} (1-u)^{c_2-1}\n {}_2F_1(a_1,b_1;c_1;ux)\n {}_2F_1(a_2,b_2;c_2;(1-u)y) \\;.\n \\label{F3:1}\n\\end{eqnarray}\nIndeed, expanding one of the hypergeometric functions as a power series leads to \n$$\n\\int_0^1 u^{c_1-1} (1-u)^{j+c_2-1}\n{}_2F_1(a_1,b_1;c_1;ux)\n\\sum_{j=0}^\\infty\n\\frac{(a_2)_j (b_2)_j y^j}{(c_2)_j j!}\n \\;. \n$$\nThe order of summation and integration can be interchanged \nin the domain of convergence of the series, and after integration we obtain Eq.\\ (\\ref{F3:1}).\n\n\nThe two-fold integral representation~\\cite{slater},\n\\begin{eqnarray}\n&& \n\\frac{\\Gamma(b_1) \\Gamma(b_2)\\Gamma(c-b_1-b_2)}{\\Gamma(c)}\nF_3(a_1,a_2;b_1,b_2;c;x,y)\n= \n\\\\ && \n\\int \\int_{0 \\leq  u, 0 \\leq v, u+v \\leq 1} du dv \nu^{b_1-1} v^{b_2-1}\n(1-u-v)^{c-b_1-b_2-1}\n(1-ux)^{-a_1}\n(1-vy)^{-a_2} \\;,\n\\nonumber \n\\label{F3:slater} \n\\end{eqnarray}\ncan be reduced to the following integral: \n\\begin{eqnarray}\n&& \n\\frac{\\Gamma(b_1) \\Gamma(b_2)\\Gamma(c-b_1-b_2)}{\\Gamma(c)}\nF_3(a_1,a_2;b_1,b_2;c;x,y)\n\\label{F3:2}\n\\\\ && \n= \n\\int_0^1 du \\int_0^{1-u} dv \nu^{b_1-1} v^{b_2-1}\n(1-u-v)^{c-b_1-b_2-1}\n(1-ux)^{-a_1}\n(1-vy)^{-a_2}\n\\nonumber \\\\ && \n= \n\\frac{\\Gamma(b_2) \\Gamma(c-b_1-b_2)}{\\Gamma(c-b_1)}\n\\nonumber \\\\ && \\hspace{5mm}\n\\times \n\\int_0^1 du u^{b_1-1} (1-ux)^{-a_1} (1-u)^{c-b_1-1}\n~_2F_1\n\\left( \\begin{array}{c|}\na_2, b_2 \\\\\nc-b_1\n\\end{array}~ (1-u)y \\right) \\;.\n\\nonumber \n\\end{eqnarray}\n\nFor the particular values of parameters $(c_1=a_1,c_2=a_2)$, the \nintegral Eq.\\ (\\ref{F3:1}) can be reduced to the Appell function $F_1$: \n\\begin{eqnarray}\nF_3(a_1, a_2, b_1, b_2; a_1+a_2; x,y)\n& = & \n\\frac{1}{(1-y)^{b_2}}\nF_1\\left( a_1, b_1, b_2, a_1+a_2; x, -\\frac{y}{1-y} \\right) \\;.\n\\nonumber \\\\ \n& = &  \n\\frac{1}{(1-x)^{b_1}}\nF_1\\left( a_2, b_1, b_2, a_1+a_2; -\\frac{x}{1-x}, y  \\right) \\nonumber \\;.\n\\end{eqnarray}\n\nUsing the one-fold integral representation for the Appell function $F_1$, \nit is possible to prove the following relations:  \n\\begin{eqnarray}\n&& \nF_3(a,c-a,b,c-b,c,x,y)\n= \n(1-y)^{c+a-b}\n~_2F_1\n\\left( \\begin{array}{c|}\na, b \\\\\nc\n\\end{array}~ x+y-xy \\right) \\;.\n\\end{eqnarray}\n\nUsing Eqs.\\ (\\ref{F3:1}) and (\\ref{F3:2}), the one-fold \nintegral representation can be written for the coefficients of \nthe $\\ep$-expansion of the hypergeometric function $F_3$\nvia the $\\ep$-expansion of the  Gauss hypergeometric function, constructed in Refs.\\ \\cite{kwy2006}, ~\\cite{kk2008}.\n\nThe coefficients of the $\\ep$-expansion of the Gauss hypergeometric function can be expressed in terms \nof multiple polylogarithms of a $q$-root of unity with arguments \n$\\left(\\frac{z}{z-1}\\right)^\\frac{1}{q}$,\n$z^\\frac{1}{q}$ or $(1-z)^\\frac{1}{q}$ (see also ~\\cite{nested2}), so that the problem of finding a rational parametrization reduces to the problem of finding a rational parametrization of the integral kernel of Eqs.\\ (\\ref{F3:1}) and (\\ref{F3:2}) in terms of variables generated by the $\\ep$-expansion of the Gauss hypergeometric function. \n\nThe construction of the higher-order \n$\\ep$-expansion of the Gauss hypergeometric function \naround rational values of parameters ~\\cite{kwy2006,kk2008},\nplays an crucial role in construction of the higher-order \n$\\ep$-expansion of many (but not all)\nHorn-type hypergeometric functions.  \n\n\n\\subsubsection{Relationship to Feynman Diagrams}\nLet us consider the one-loop pentagon with vanishing external\nlegs.\n The higher-order $\\ep$-expansion for this diagram \nhas been constructed~\\cite{pentagon1} in terms of iterated one-fold integrals over algebraic functions. \nIn Ref.\\ \\cite{pentagon2}, the hypergeometric representation \nfor the one-loop \npentagon with vanishing external momenta has been constructed as a sum of  \nAppell hypergeometric functions $F_3$.\n\nIn ~\\cite{BKK2012}, where our differential equation approach is presented, it was pointed out that \nthe one-loop pentagon can be expressed in terms of multiple \npolylogarithms. Ref.\\ \\cite{pentagon4} verified the numerical agreement between the results of Refs.\\ \\cite{pentagon1} and \n\\cite{pentagon2}, and Ref.\\ \\cite{pentagon5} constructed \nthe iterative solution of the differential equation \\cite{kotikov}.\n\nLet us recall the results of Ref.\\ \\cite{pentagon2}.\nThe one-loop massless pentagon is expressible in terms of the Appell function $F_3$\nwith the following set of parameters: \n\\begin{eqnarray}\n\\Phi_5^{(d)} \\sim F_3\\left( 1,1,\\frac{7-d}{2},1,\\frac{10-d}{2};x,y \\right) \\;, \n\\label{pentagon1}\n\\end{eqnarray}\nwhere $d$ is dimension of space-time. \nAnother representation presented in \n\\cite{pentagon2} has the structure\n\\begin{eqnarray}\t\nH_5^{(d)}\n\\sim \nF_3\\left( \\frac{1}{2},1,1,\\frac{d-2}{2},\\frac{d+1}{2};\\frac{y}{x+y-xy}, \\frac{1}{x} \\right) \n\\;.\n\\label{pentagon2}\n\\end{eqnarray}\t\n\nLet us consider the case of $d=4-2\\ep$. \nThe first representation, Eq.\\ (\\ref{pentagon1}), is \n$$\n\\Phi_5^{(4-2\\ep)} \\sim F_3\\left( 1,1,\\frac{3}{2}-\\ep,1,4-\\ep;x,y \\right) \\;. \n$$\nThis case, \n$$\n\\{\np_1 = p_2 =r_2 = p = 0 \\}\\;, \n\\{r_1 = 1 \\;, q=2 \\}\n\\Longrightarrow\ns_1 \\neq 0; \\quad s_2 = 0 \\;, \\quad p = 0 \\;,\n$$\ncorresponds to our {\\bf set 5}, so that \nthe {\\bf $\\ep$-expansion is expressible in terms of multiple  polylogarithms}, defined by Eq.~(\\ref{G}). \n\nFor the other representation, Eq.~(\\ref{pentagon2}),   \n$$\t\nH_5^{(4-2\\ep)} \n\\sim\n\tF_3\\left( \\frac{1}{2},1,1,\\ep,\\frac{5}{2}-\\ep;\\frac{y}{x+y-xy}, \\frac{1}{x} \\right) \\;,  \n$$\nso that it is reducible to the following set of parameters: \n$$\n\\{p_2 = r_1 = r_2 = 0 \\}\\;, \n\\quad \n\\{p_1 = 1\\;, q=2, \\; p=1 \\}\\;\n\\Longrightarrow\n\ts_1 = 1; \\quad s_2 = 0  \\;, \\quad p = -1 \\;.\n$$\t\nThis is our {\\bf set 4}, so that the\n{\\bf $\\ep$-expansion is expressible in terms of multiple polylogarithms}, \ndefined by Eq.~(\\ref{G}).  \n\n\nThe $\\ep$-expansion of the one-loop pentagon\n about $d=3-2\\ep$ could be treated in a similar manner. In this case, \nthe first representation corresponds to {\\bf set 1} \n$$\n\t\\Phi^{(3-2\\ep)} \\sim  F_3\\left( 1,1,2+\\ep,1,\\frac{7}{2}+\\ep;x,y \\right) \n\\Longrightarrow\n\t\tp_1 = p_2 =r_1 = r_2 = 0, \\quad \n\t\t\tq=2 \\;, p=1 \n$$ \nand there is no rational parametrization, so that the result of the {\\bf $\\ep$-expansion is expressible in terms of a one-fold \niterated integral over algebraic functions}.\n\nThe other representation, Eq.\\ (\\ref{pentagon2}), also \ncannot be expressed in terms of multiple polylogarithms:\n$$\nH_5^{(3-2\\ep)} \t\\sim   \n\tF_3\\left( \\frac{1}{2},1,1,\\frac{1}{2} - \\ep,2-\\ep;\\frac{y}{x+y-xy}, \\frac{1}{x} \\right) \n\t\\Longrightarrow\n\t\\begin{array}{l}\n\tp_2 = r_1 = p = 0 \\;, \\\\\n\tq=2 \\; \\quad p_1 = r_2 = 1\\;, \\\\\n\ts_1 = 1; \\quad s_2 = 1 \\;.\n\t\\end{array}\n$$\nThis corresponds to {\\bf set 3}, and the \n{\\bf $\\ep$-expansion is expressible in terms of one-fold iterated integral over algebraic functions}.\n\nIn this way, the question\nof the all-order $\\ep$-expansion of a one-loop Feynman diagram in terms of multiple polylogarithms is reduced to the question of the existence of \na rational parametrization for the (ratio) of singularities.  \n\n{\\bf Remark}:\nThe dependence of the coefficients of the $\\ep$-expansion (multiple polylogarithms or elliptic function)\non the dimension of space-time is not new.\nIn particular, it is well known that the two-loop sunset \nin $3-2\\ep$ dimension is expressible in terms of polylogarithms~\\cite{rajantie,lee}\nand demands introduction of new functions in $4-2\\ep$ dimension \n\\cite{Laporta,tarasov:sunset,Bloch,Adams,Bloch:2016,Duhr,Bogner}.   \n\n\n\n\\section{Conclusion}\nThe deep relationship between Feynman diagrams and hypergeometric functions has been reviewed, and we\nhave tried to enumerate all approaches and recent results on that subject. Special attention was devoted to the discussion of \ndifferent algorithms for constructing the analytical \ncoefficients of the $\\ep$-expansion of multiple hypergeometric functions.\nWe have restricted ourselves to\nmultiple polylogarithms and functions related to \nintegration over rational functions \n(the next step after multiple polylogarithms). \nThe values of parameters related to elliptic polylogarithms \nwas beyond our consideration. \n\nWe have presented our technique for the construction of \ncoefficients of the higher order $\\ep$-expansion of multiple Horn-type hypergeometric functions, developed by the authors~\\footnote{Unfortunately, the further \n\tprolongation of this \n\tproject has not been supported by DFG, \n\tso that many interesting results remain unpublished.} \nduring the period 2006 -- 2013.\nOne of the main results of interest \nwas the observation~\\cite{DK1,DK2,DK3,kwy2006,kwy2007,kk2008,bkm2013} \nthat for each Horn-type hypergeometric function, a \nset of parameters can be found so that the coefficients of the \n$\\ep$-expansion include only functions of weight one\n(so-called ``pure functions,''  in a modern terminology). \nAs was understood in 2013 by Johannes Henn \n ~\\cite{Henn:1,Henn:2}, \nthis property is valid not only for \nhypergeometric functions but also for generic Feynman diagrams. \n\nOur approach is based on the systematic analysis of the\nsystem of hypergeometric differential equations (linear differential operators of hypergeometric type with polynomial coefficients) \nand does not demand the\nexistence of an integral representation, \nwhich is presently unknown for a large class\nof multiple Horn-type hypergeometric functions (they could be deduced, \nbut are not presently available in the mathematical literature). \n\nOur approach is based on the factorization of the system of \ndifferential equations into a product of differential operators,\ntogether with finding a rational parametrization and constructing\niterative solutions.  \nTo construct such a system, an auxiliary manipulation with \nparameters (shifting by integer values) is required,\nwhich can be done with the help of the HYPERDIRE set of \nprograms~\\cite{hyperdire}.\nThis technique\nis applicable not only to \nhypergeometric functions defined by series but also to multiple \nMellin-Barnes integrals~\\cite{KK:MB} -- one of the representations of Feynman diagrams in a covariant gauge.\nWe expect that the present technique is directly applicable (with some \ntechnical modifications) to the construction of the $\\ep$-expansion \nof hypergeometric functions \nbeyond multiple polylogarithms, specially, that the two-loop sunset \nhas a simple hypergeometric representation~\\cite{tarasov:sunset}.   \n\nThere are two of our considerations that have not \nbeen solved algorithmically:\n (a) the factorization of linear partial differential operators into irreducible factors\nis not unique, as has been ilustrated by Landau~\\cite{landau} \n(see also Ref.\\ \\cite{schwarz}):\n$\n(\\partial_x+1) (\\partial_x+1) (\\partial_x+x\\partial_y)\n= \n(\\partial^2_x+x \\partial_{xy}  \n+ \\partial_x+(2+x) \\partial_y) (\\partial_x+1) \\; ; \n$\n(b) The choice of parametrization is still an open problem, but there is essential progress in this direction~\\cite{root1,root2}.\n\nOur example has shown that such a parametrization is\ndefined by the locus of singularities of a system of differential \nequations, so that the problem of finding a rational parametrization \nis reduced to the parametrization of solutions of the\nDiophantine equation for the singular locus of a Feynman diagram and\/or hypergeometric function.\nIt is well known that in the case of a positive solution of this problem (which has no complete algorithmic solution), \nthe corresponding system of partial differential equations \nof a few variables takes the simplest structure.\nAt the same time, there is a relationship \nbetween the type of solution of the Diophantine equation \nfor the singular locus \nand the structure of the coefficients of the $\\varepsilon$-expansion: \na linear solution allows us to write the results of the\n$\\varepsilon$-expansion \nin terms of multiple polylogarithms. \nAn algebraic solution gives rise to functions different from multiple polylogarithms and elliptic functions, {\\em etc}. \nIt is natural to expect that, in the case of an elliptic solution of the Diophantine equation\nfor the singular locus, the results for the $\\varepsilon$-expansion are related to \nthe elliptic generalization of multiple polylogarithms.\n\nAnother quite interesting and still algorithmically open problem is the transformation of multiple Horn-type hypergeometric functions with reducible monodromy to \nhypergeometric functions with irreducible monodromy. \nIn the application to Feynman diagrams, such transformations correspond to functional relations, studied recently by Oleg Tarasov \n~\\cite{Tarasov:functional1,Tarasov:functional2,Tarasov:functional3}, \nand by Andrei Davydychev~\\cite{davydychev:box}. \n\n\nFurther analysis of the symmetries of the hypergeometric differential equations related to the Mellin-Barnes representation \nof Feynman diagrams (for simplicity we will call it the hypergeometric representation)  \nhas revealed their deep connection to the holonomic properties of Feynman diagrams. \nIn particular, a simple and fast algorithm was constructed for the \nreduction of any Feynman diagram \nhaving a one-fold Mellin-Barnes integral representation to a set of master integrals~\\cite{KK:MB}. \n\nThe importance of considering the\ndimension of the irreducible representation instead of generic \nholonomic rank has been pointed out in the application to Feynman \ndiagrams~\\cite{bkk2009}. In the framework of this approach, \nthe set of irreducible, non-trivial master-integrals \ncorresponds to the set of irreducible (with respect to analytical continuation of the variables, masses,\nand external momenta) solutions of the corresponding system of hypergeometric differential equations, \nwhereas diagrams expressible in terms of Gamma-functions correspond \nto Puiseux-type solutions (monomials with respect to Mandelstam variables) of the original system of hypergeometric equations. \n\n\n\\begin{acknowledgement}\n    MYK is very indebted to Johannes Bl\\\"umlein, Carsten Schneider \n\tand Peter Marquard for the\n\tinvitation  and for creating such a stimulating atmosphere\n\tin the workshop ``Anti-Differentiation and the\n\tCalculation of Feynman Amplitudes'' at DESY Zeuthen.\n\tBFLW thanks Carsten Schneider for kind the hospitality at RISC.\n    MYK thanks the Hamburg University for hospitality while \n\tworking on the manuscript. SAY acknowledges support from The Citadel Foundation.\n\\end{acknowledgement}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{The status quo}\n\nIn the system of peer review that is currently used in the sciences,\nan editor invites one or more referees to review an article submitted\nto a scientific journal. Based on the referees' recommendations, the\neditor will accept the article, demand modifications, or reject it.\nReferee reports are generally made available to the article's author\nin anonymized form only and are not otherwise published. (Some journals\nalso anonymize the article to be refereed, even though ascertaining\nthe true author of a submission is usually a simple matter of using\nan internet search engine.) \n\nThe system as described is completely opaque to outside observers.\nNeither the quality and timeliness of reviews, nor the standards of\na journal\\textquoteright{}s editors, nor the extent of modifications\nmade after initial review, nor the number of times an article has\nbeen rejected by other journals are publicly available. \n\nOther than professional integrity, referees have little legitimate\nincentive to produce timely, fair and high-quality reviews. Since\nthe reviews are not published, referees are not accountable for their\nwork and cannot use it to bolster a case for professional advancement\nor to improve their general standing in the academic community. Probably\nthe biggest (and most problematic) incentive for referees is the accumulation\nof editor goodwill, to be expended during future article submissions.\nIt is also conceivable that some referees reject articles whose authors\nthey dislike or whose approach or results interfere with their own\nresearch agenda. Finally, editors may circumvent the peer review process\naltogether in order to promote their own or their associates' work.\nSeveral case reports of dysfunction and breakdown of the peer review\nprocess in the mathematical and physical sciences have recently appeared\nin the literature. \\cite{Baez 2006,Schiermeier 2008,Trabino 2009}.\n\nAuthors, editors and referees are not paid for their work in this\npublication process. Nevertheless, publishers often charge exorbitant\namounts for the resulting product, journals which have typically ended\nup being hidden away in university libraries, inaccessible to the\npublic who funded the research in the first place. Independent workers\nas well as researchers in poor countries thus have often been cut\nout of the research loop entirely.\n\nThe need for a system of open electronic publishing of scientific\narticles has long been recognized (see e.g. \\cite{Odlyzko 1995}).\nSeveral electronic journals have now been created. Some of these charge\nreaders for access, others are free to read but charge authors for\npublication, and still others are free for all parties involved. Perhaps\nthe biggest success of the Open Access movement was a 2007 U.S. law\nrequiring all NIH-supported research to be submitted to an openly\naccessible archive one year after publication. \\cite{Weiss 2007} \n\nInternet-based alternatives to the prevalent peer review and publishing\nprocess have been discussed in \\cite{Harnard 2000} and \\cite{Nielsen 2008}.\nA trial in open peer review at the journal \\emph{Nature} in 2006 generated\nwidespread debate of the concept \\cite{Nature 2006-1}; the final\nreport concluded that, while the general concept was received enthusiastically,\nparticipation in and satisfaction with their particular model of open\ncommentary were disappointing. \\cite{Nature 2006-2}\n\n\n\\section{ArXiv.org}\n\nThe website \\noun{arXiv.org} (formerly \\noun{xxx.lanl.gov}) is an\nelectronic archive of freely accessible research preprints. \\cite{Ginsparg 1997}\nIt was started by physicist Paul Ginsparg in August 1991 and has since\nbecome an indispensable tool for researchers in physics, mathematics\nand, increasingly, computer science and quantitative biology. Authors\nsubmit their articles to the archive prior to peer review and official\npublication by a scientific journal; the preprints are posted on the\nwebsite in perpetuity after superficial moderator review. To participate,\nauthors need an affiliation with a recognized academic institution\nor an endorsement by an established author. Interested parties can\nsign up for regular e-mail announcements containing the abstracts\nof new preprints in their chosen fields.\n\nOnce a manuscript has been peer reviewed and accepted for publication,\nauthors should ideally post an updated version to the archive. Not\nall authors remember to do this, and some journals explicitly prohibit\nthe practice, claiming a copyright on the final result of peer review\n\\footnote{See for instance Elsevier's policy on electronic preprints at  \\url{http:\/\/www.elsevier.com\/wps\/find\/authorshome.authors\/preprints}\n(accessed 26 December 2008\n}\n\nConsequently, \\noun{arXiv.org} in its present incarnation and similar\npreprint archives in other fields do not serve as authoritative Open\nAccess repositories of peer reviewed research.\n\n\n\\section{A proposed solution}\n\nTo address the problems outlined in section 1, I propose the following\nextension to the \\noun{arXiv.org} preprint archive. A new class of\nusers is created, the {}``editors''. Each editor works for an electronic\njournal. Authors, after having uploaded a preprint to the archive,\nmay elect to submit their article for review and official publication\nin one of these electronic journals. An editor of that journal then\ndecides whether the article is appropriate for the journal in terms\nof scope and quality. If it is not, this decision is publicly attached\nto the article and the process ends; if it is, the editor invites\none or more referees to write public reviews, to be attached to the\narticle. The article author may subsequently post a public rebuttal\nto the reviews. Based on the referee reports and rebuttals, the editor\ndecides whether to accept, demand changes to, or reject the article.\nThe original article, reviews, rebuttal and publication decision are\npublished in perpetuity. If accepted, the author posts a final version\nof the article to arXiv.org; as a peer reviewed and officially published\narticle, it is visibly set apart from mere preprints and added to\nthe electronic journal's collection of published articles. Rejected\narticles may be submitted to another electronic journal.\n\nReviews should be signed with the referee's full name and affiliation.\nThis maximizes transparency and allows referees to receive academic\ncredit for their work. However, some reviewers might be reluctant\nto participate in such a system, for instance because they hesitate\nto openly reject the work of friends or influential researchers, or\nbecause they do not want to call attention to their ignorance of some\nof the issues discussed in the reviewed article. Thus it is probably\nnecessary to offer referees the option to publish their reviews under\na pseudonym. Over time, such a pseudonym might naturally develop a\nreputation as a solid reviewer, completely divorced from the writer's\nreal-world identity. Using a straightforward cryptographic scheme,\na referee could prove to selected others that he or she owns a certain\npseudonym; in this way even pseudonymous referees could receive academic\ncredit for their work at the time of tenure or promotion decisions.\n\nSome electronic journals may wish to develop a process for attaching\nnotes to published articles, for instance to point out prior work,\nmistakes or scientific misconduct discovered after publication. It\nwill also be desirable to attach a moderated discussion forum to each\narticle, as a natural gathering place of interested researchers. The\nquality of these forums would serve as a criterion to differentiate\nelectronic journals from each other. The pseudonyms used for refereeing\ncould also be used to sign forum contributions.\n\nOne may hope that the proposed system will engender several desirable\nconsequences. The act of refereeing will rise in prestige in accordance\nwith its importance for the scientific process. The quality of referee\nreports will improve. Outside evaluations and comparisons of the standards\nand practices of different electronic journals will become possible.\nThe process becomes completely transparent and its results are made\nfreely available. \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\n\n\nRadio astronomy aims to study radio emissions from the sky, in order to detect, identify new objects and observe known structures at higher resolution, in a specific electromagnetic spectrum \\cite{thompson2008interferometry}. This fundamental thematic shines a new light on our universe, revealing more about its nature and history. \nIn order  to carry out particularly sensitive observations in a large range of the spectrum, and to handle significant cosmological issues, largely distributed sensor arrays \nare currently being built or planned, such as the low frequency array (LOFAR) \\cite{van2013lofar} and the square kilometre array (SKA) \\cite{dewdney2009square}. They will notably be composed of a large number of relatively low-cost small antennas with wide field of view, resulting in a large collecting area and high resolution imaging. Nevertheless, to meet the theoretical optimal performances of such next generation radio interferometers, a plethora of signal processing challenges must be overcome, among them, calibration, data reduction and image synthesis \\cite{wijnholds2008fundamental,rau2009advances,deVos2009lofar, van2013signal}.\nThese aspects are intertwined and must be dealt with to take advantage of the new advanced radio interferometers. As an example, lack of calibration has dramatic effects in the image reconstruction by causing severe distortions. In this paper, we focus on calibration, which involves the estimation of all unknown perturbation effects and represents a cornerstone of the imaging step \\cite{mitchell2008real,wijnholds2014signal,wijnholds2010calibration}. \n\n\nArray calibration aspects have been tackled for a few decades in the array processing community leading to a variety of calibration algorithms \\cite{fuhrmann1994estimation,ng1996sensor,ng2009practical}. Such algorithms can be classified into two different approaches depending on the presence \\cite{lo1987eigenstructure,ng1992array,ng1995active}, or the absence \\cite{rockah1987array,weiss1989array,friedlander1991direction,wylie1994joint,qiong2003overview,flanagan2001array}, of one or more cooperative sources, named calibrator sources. In the radio astronomy context, calibration is commonly treated using the first approach as we have access to prior knowledge thanks to tables describing accuratly the position and flux of the brightest sources \\cite{baars1977absolute}.\n\n\n\n\nFollowing this methodology, the majority of proposed calibration schemes in radio interferometry are least squares-based approaches. The state-of-the-art consists in the so-called alternating least squares approach \\cite{boonstra2003gain, wijnholds2009multisource,wijnholds2010fish,salvini2014fast}, which leads to statistically efficient algorithm under a Gaussian model, since the least squares estimator is equivalent to the maximum likelihood (ML) estimator in this case. On the other hand, expectation maximization (EM) \\cite{dempster1977maximum, moon1996expectation,mclachlan2007algorithm} and EM-based algorithms, such as the space alternating generalized expectation maximization algorithm \\cite{fessler1994space}, have been proposed in order to enhance the convergence rate of the least squares-based calibration algorithms \\cite{yatawatta2009radio}. Nevertheless, the major drawback of these schemes is the Gaussianity assumption which is not realistic in the radio astronomy context. Specifically, the presence of outliers has multiple causes, among which i) the radio frequency interferers, which corrupt the observations and are not always perfectly filtered in practice \\cite{raza2002spatial,van2005performance}, ii) the presence of unknown weak sources in the background \\cite{yatawatta2014robust}, iii) the presence of some punctual events such as interference due to the Sun or due to strong sources in the sidelobes which can also randomly create outliers \\cite{boonstra2005radio}. To the best of our knowledge, the proposed scheme in \\cite{yatawatta2014robust}, represents the only alternative to the existing calibration algorithms based on a Gaussian noise model. \n\n\n\n\nIn \\cite{yatawatta2014robust}, theoretical and experimental analyses have been conducted in order to demonstrate that the effect of outliers in the radio astronomy context can indeed be modeled by a non-Gaussian heavy-tailed distributed noise process. Nevertheless, the algorithm presented in \\cite{yatawatta2014robust} has its own limits, since the noise is specifically modeled as a Student's t with independent identically distributed entries. To improve the robustness of the calibration, we propose, in this paper, a new scheme based on a broader class of distributions gathered under the so-called spherically invariant random noise modeling \\cite{jay2002detection,yao2003spherically}, which includes the Student's t distribution. A spherically invariant random vector (SIRV) is described as the product of a positive random variable, named texture, and the so-called speckle component which is  Gaussian, resulting in a two-scale compound Gaussian distribution \\cite{wang2006maximum}.\nThe flexibility of the SIRV modeling allows to consider non-Gaussian heavy-tailed distributed noise in the presence of outliers, but also to adaptively consider Gaussian noise in the extreme case when there are no outliers. Under the SIRV model, we estimate the unknown parameters iteratively based on a relaxed ML estimator, leading to closed-form expressions for the noise parameters while a block coordinate descent (BCD) algorithm \\cite{friedman2007pathwise,hong2015unified} is designed to obtain the estimates of parameters of interest efficiently and at a low cost. \n\n\n\nFinally, it is worth mentioning that the parametric model used in this paper to describe the perturbation effects is based on the so-called Jones matrices \\cite{hamaker1996understanding,smirnov2011revisiting}. Such formalism describes in a flexible way the conversion of the incident electric field into voltages. Indeed, along its propagation path, the signal is affected by various effects and transformations which correspond to matrix multiplications in the mathematical Jones framework. Multiple distortion effects caused by the environment and\/or the instruments can be easily incorporated into the model using an adequate parametrization of the Jones matrices. Such effects can represent, for example, the ionospheric phase delay resulting in angular shifts, the atmospheric distortions, the typical phase delay due to geometric pathlength difference, the voltage primary beam, the cross-leakage or also the electronic gains \\cite{noordam1996measurement,noordam2010meqtrees}. For the above reasons and due to its flexibility \\cite{thompson2008interferometry,yatawatta2009radio,noordam1996measurement,hamaker1996understanding,smirnov2011revisiting}, we adopt this parametric model. We make a distinction between the non-structured and the structured cases: in the first one, one total Jones matrix stands for all the effects along the full signal path while in the second case, we regard each physical effect separately thanks to individual Jones terms in a cumulative product. Thus, different corruptions are described by different kinds of Jones matrices.\n We emphasize that the proposed algorithm, entitled relaxed concentrated ML estimator, is a generic algorithm as it is based on a non-structured Jones matrices formulation as a first step. However, it can be adapted to various regimes \n describing distinct calibration scenarios in which an array can operate \\cite{lonsdale2004calibration}. In this paper, we consider the specific example of the direction dependent distortion regime with a compact set of antennas, which we refer to as the 3DC regime.\nThe array is therefore considered as a closely packed group of antennas but the array elements have a wide field of view. This is particularly well-adapted for calibration of compact arrays, typically a LOFAR station.\n \n The rest of the paper is organized as follows: in Section \\ref{data}, we present the data model  in the context of radio astronomy, first with non-structured Jones matrices and thereafter, we study an example of structured Jones matrices for the 3DC calibration regime. In Section \\ref{robust_estim}, we give an overview of the proposed robust ML estimator, based on spherically invariant random process (SIRP) noise modeling. An efficient estimation procedure of the distortions introduced on each signal propagation path is derived in Section \\ref{non_structure}. Then, the algorithm is adapted to the case of structured Jones matrices in Section \\ref{structure} for the 3DC calibration regime. Finally, we provide numerical simulations in Section \\ref{simus} to assess the robustness of the approach and draw  our conclusions in Section \\ref{ccl}.\n\n\n\n\n\n\nIn this paper, we use the following notation: symbols $\\left( \\cdot \\right) ^{T}$, $\\left( \\cdot \\right)^{\\ast }$, $\\left( \\cdot \\right) ^{H}$ denote, respectively, the transpose, the complex conjugate and the Hermitian transpose. The Kronecker product is represented by $\\otimes$, $\\mathrm{E}\\{\\cdot\\}$ denotes the expectation operator, $\\mathrm{bdiag}\\{\\cdot\\}$ is the block-diagonal operator, whereas $\\mathrm{diag}\\{\\cdot\\}$ converts a vector into a diagonal matrix. The trace and determinant operators are, respectively, referred by $\\mathrm{tr}\\left\\{ \\cdot \\right\\} $ and $ |\\cdot|$. The symbol $\\mathbf{I}_{B}$ represents the $B \\times B$ identity matrix, $\\mathrm{vec}(\\cdot)$ stacks the columns of a matrix on top of one another, $||\\cdot||_F$ is the Frobenius norm, while $||\\cdot||_2$ denotes the $l_2$ norm. Finally, $\\Re\\left\\{ \\cdot \\right\\} $ represents the real part \nand we note $j$ the complex number whose square equals $-1$.\n\n\n\n\\section{Data model}\n\\label{data}\n\n\\subsection{Case of non-structured Jones matrices}\n \nLet us consider $M$ antennas with known locations that receive $D$ signals emitted by calibrator sources. Each antenna is dual polarized and composed of two receptors, in order to provide sensitivity to the two orthogonal\npolarization directions $(x,y)$ of the incident electromagnetic plane wave. Consequently, the relation between the \\textit{i}-th source emission and the\nmeasured voltage at the \\textit{p}-th antenna is given by \\cite{hamaker1996understanding,KerThesis2012,smirnov2011revisiting}\n\\begin{equation}\n\\mathbf{v}_{i,p}(\\boldsymbol{\\theta}) = \\mathbf{J}_{i,p}(\\boldsymbol{\\theta}) \\mathbf{s}_{i}\n\\end{equation}\nwhere $\\mathbf{s}_{i} = [\n  s_{i_{x}},s_{i_{y}}]^{T}$ is the incoming signal, $ \\mathbf{v}_{i,p}(\\boldsymbol{\\theta}) = [\n  v_{i,p_{x}}(\\boldsymbol{\\theta}),\nv_{i,p_{y}}(\\boldsymbol{\\theta})]^{T}$ is the generated voltage with one output for each polarization direction and $\\mathbf{J}_{i,p}(\\boldsymbol{\\theta})$ denotes the so-called $2 \\times 2$ Jones matrix, parametrized by the unknown vector of interest $\\boldsymbol{\\theta}$. The Jones matrix models the array response and all the perturbations introduced along the path from the \\textit{i}-th source to the \\textit{p}-th sensor. Since each propagation path is particular,\nwe can associate a different Jones matrix with each source-antenna pair $(i,p)$, leading to a total number of $D M$ Jones matrices.\nIn this section, we  consider the non-structured case where no specific perturbation model is used to describe the physical mechanism behind each perturbation effect and the unknown elements correspond to the entries of all Jones matrices \\cite{yatawatta2009radio,nunhokeeghost2015} (a structured example is given for 3DC calibration regime, in Section \\ref{specific_regime}).\n\n\nFor each antenna pair, we compute the correlation of the output signals, resulting in the typical observations recorded by a radio interferometer.\nThe correlation between voltages is given, in the case of noise free measurements, for the $(p,q)$ antenna pair, by\n\\begin{align}\n\\label{ME}\n\\nonumber\n\\mathbf{V}_{pq}(\\boldsymbol{\\theta}) & =\\mathrm{E}\\left\\{\\left(\\sum_{i=1}^{D}\\mathbf{v}_{i,p}(\\boldsymbol{\\theta})\\right)\\left(\\sum_{i=1}^{D}\\mathbf{v}_{i,q}^{H}(\\boldsymbol{\\theta})\\right)\\right\\} \\\\ &=\n\\sum_{i=1}^{D}\\mathbf{J}_{i,p}(\\boldsymbol{\\theta})\\mathbf{C}_{i}\n\\mathbf{J}_{i,q}^{H}(\\boldsymbol{\\theta}) \\ \\\n\\text{for} \\ \\ p<q, \\ \\ p,q \\in \\{1, \\ldots, M\\},\n\\end{align}\nwhere the signals emitted by the sources are assumed uncorrelated and the $2 \\times 2$ matrix $\\mathbf{C}_{i}=\\mathrm{E}\\{\\mathbf{s}_{i}\\mathbf{s}_{i}^{H}\\}$\n is known from prior knowledge.\nLet us remark that autocorrelations are not considered as shown by the condition $p<q$ in (\\ref{ME})\n(this is a typical situation in the radio astronomy context where radio interferometric systems automatically flag the autocorrelations \\cite{van2013lofar}). \n\n\nUsing the property $\\mathrm{vec}(\\mathbf{A B C}) = (\\mathbf{C}^{T} \\otimes \\mathbf{A})\\mathrm{vec}(\\mathbf{B})$, we rewrite (\\ref{ME}) as a $4 \\times 1$ vector\n\\begin{equation}\n\\tilde{\\mathbf{v}}_{pq}(\\boldsymbol{\\theta})=\\mathrm{vec}\\Big(\\mathbf{V}_{pq}(\\boldsymbol{\\theta})\\Big)\n= \\sum_{i=1}^{D}\\mathbf{u}_{i,pq}(\\boldsymbol{\\theta})\n\\end{equation}\n in which $\\mathbf{u}_{i,pq}(\\boldsymbol{\\theta})=\\left(\\mathbf{J}^{\\ast}_{i,q}(\\boldsymbol{\\theta})\\otimes\n\\mathbf{J}_{i,p}(\\boldsymbol{\\theta})\\right)\\mathbf{c}_{i}$ with $\\mathbf{c}_{i}=\\mathrm{vec}(\\mathbf{C}_{i})$.\nWe stack all the noisy measurements within a full vector $\\mathbf{x}=\\left[\\mathbf{v}^{T}_{12},\n\\mathbf{v}^{T}_{13},\n\\ldots,\n\\mathbf{v}^{T}_{(M-1)M}\\right]^{T}$ $\\in\\mathbb{C}^{4 B \\times 1}$, where $B=\\frac{M(M-1)}{2}$ denotes the total number of antenna pairs and $\\mathbf{v}_{pq}=\\tilde{\\mathbf{v}}_{pq}(\\boldsymbol{\\theta})+\\mathbf{n}_{pq}$ with $\\mathbf{n}_{pq}$ the noise sample at a specific antenna pair. Specifically, $\\mathbf{x}$ reads\n\\begin{equation}\n\\label{express_x}\n\\mathbf{x}\n=\\sum_{i=1}^{D}\\mathbf{u}_{i}(\\boldsymbol{\\theta})+\\mathbf{n}\n\\end{equation}\nin which  $\\mathbf{u}_{i}(\\boldsymbol{\\theta})=\n  \\left[\\mathbf{u}^{T}_{i,12}(\\boldsymbol{\\theta}),\n\\mathbf{u}^{T}_{i,13}(\\boldsymbol{\\theta}),\n\\ldots,\n\\mathbf{u}^{T}_{i,(M-1)M}(\\boldsymbol{\\theta})\\right]^{T}$ and $\\mathbf{n}=\\left[\\mathbf{n}^{T}_{12},\n\\mathbf{n}^{T}_{13},\n\\ldots,\n\\mathbf{n}^{T}_{(M-1)M}\\right]^{T}$ is the full noise vector which accounts for Gaussian noise, but also the presence of outliers in our data. Therefore, the noise can no longer be considered Gaussian and a robust calibration method is required. \n To investigate non-Gaussian noise modeling and encompass a broad range of noise distributions, we propose to adopt the SIRP noise model \\cite{jay2002detection,yao2003spherically}. Specifically, the noise at each antenna pair is assumed to be generated as\n\\begin{equation}\n\\label{sirp}\n\\mathbf{n}_{pq} = \\sqrt{\\tau_{pq}} \\  \\mathbf{g}_{pq},\n\\end{equation}\nwhere the positive real random variable $\\tau_{pq}$ is referred to as texture, whereas the complex speckle component $\\mathbf{g}_{pq}$ follows a zero-mean Gaussian distribution\\footnotemark, i.e.,\n\\begin{equation}\n\\label{constraint}\n\\mathbf{g}_{pq} \\sim \\mathcal{CN}(\\mathbf{0},\\boldsymbol{\\Omega}).\n\\end{equation}\nIn order to remove scaling ambiguities, we impose $\\mathrm{tr}\\left\\{\\boldsymbol{\\Omega} \\right\\}=1$. Note that the choice of this constraint is arbitrary and does not affect the estimates of interest as argued in \\cite{zhang2016mimo}.\n\n\n\\footnotetext{Let us note that it is possible to consider a different covariance matrix $\\boldsymbol{\\Omega}_{pq}$ for each speckle component $\\mathbf{g}_{pq}$ in (\\ref{constraint}). In this case, the proposed algorithm requires a few modifications and the corresponding expressions are presented in Appendix A.\n}\n\nIn this section, we adopted the non-structured Jones matrices formulation which is relevant in the radio astronomical context \\cite{yatawatta2009radio,nunhokeeghost2015}. In this case, there is no need to specify the full propagation path, avoiding misspecification in the model. Besides, it is highly flexible and can be adapted to different scenarios \\cite{lonsdale2004calibration}. In the following, we present the direction dependent distortion regime with a compact set of antennas, named 3DC regime.\n\n\n\n\\subsection{Specific case of the 3DC calibration regime}\n\\label{specific_regime}\n\n  \\begin{figure}\n \\centering\n \\input{dessin}\n \\caption{3DC calibration regime, for which $V\\gg S$ and $S\\gg A$. All receiving elements in the station see the same ionosphere part but, due to their wide field of view, a multitude of sources are visible and perturbations are highly direction dependent.}\n \\label{regime3}\n\\end{figure}\n\nFor a specific propagation path, from the \\textit{i}-th source to the \\textit{p}-th antenna, the global Jones matrix $\\mathbf{J}_{i,p}$ accounts for multiple effects which can be described explicitly. Indeed, each global matrix can be decomposed into individual Jones terms which stand for specific physical effects \\cite{noordam1996measurement,smirnov2011revisiting}. This way, instead of estimating entries of all Jones matrices as done in the non-structured case, we will estimate physically meaningful parameters, thus reducing the total number of parameters to estimate. Introducing structured Jones matrices  can be done in the context of calibration scenarios\\cite{lonsdale2004calibration,wijnholds2010calibration}. In what follows, we target one specific commonly used calibration regime that we call 3DC calibration regime, described in Fig. \\ref{regime3}, which is well adapted for calibration of LOFAR on station level \\cite{wijnholds2010fish}, and also for stations of the future SKA radio interferometer \\cite{wijnholds2010calibration}. In this scenario, direction dependent distortions play a significant role since individual elements in the array have a wide field of view. Indeed, this implies different propagation conditions towards distinct sources in the field of view.\nHowever, the array being relatively compact, made of similar elements, some effects might be the same for all antennas.\nIn the following, we introduce a particular sequence of Jones matrices with specific parametrizations, in the context of 3DC calibration regime \\cite{yatawatta2012reduced}\n\\begin{equation}\n \\label{model_regime3}\n\\mathbf{J}_{i,p}(\\boldsymbol{\\theta}^{\\mathrm{3DC}}_{i,p})=\\mathbf{G}_p(\\mathbf{g}_p)\\mathbf{H}_{i,p}\\mathbf{Z}_{i,p}(\\boldsymbol{\\alpha}_i)\\mathbf{F}_{i}(\\vartheta_{i}) \n\\end{equation}\nfor $i \\in \\{1, \\hdots, D\\}$, $p \\in \\{1, \\hdots, M\\}$ and $\\boldsymbol{\\theta}^{\\mathrm{3DC}}_{i,p}=[\\vartheta_{i},\\mathbf{g}^T_p,\\boldsymbol{\\alpha}^T_i]^T$. We note $\\mathbf{H}_{i,p}$ the only assumed known matrix thanks to electromagnetic simulations in terms of antenna response and a priori knowledge given by calibrator source and antenna positions \\cite{yatawatta2012reduced,kazemi2015blind,noordam1996measurement,smirnov2011revisiting}, whereas the remaining matrices are explained in the following items.\n\n\n$\\bullet$ \\textit{Ionospheric effects} :\\\\\nPropagation through the ionosphere, the outer layer of the earth's atmosphere, introduces propagation delay on the signal which is affected by spatially variable fluctuations. \nIf these perturbation effects are not corrected for, the sources may appear shifted from their intrinsic positions \\cite{wijnholds2010calibration,van2007self}. In the case of a compact array, the ionospheric delay matrix is in fact a scalar direction-dependent phase given by\n  \\begin{equation}\n  \\label{form_Z}\n  \\mathbf{Z}_{i,p}(\\boldsymbol{\\alpha}_i)=\\exp\\Big\\{j\\varphi_{i,p}\\Big\\}\\mathbf{I}_2\n  \\end{equation}\n  in which $\\varphi_{i,p}=\\eta_i u_p+\\zeta_i v_p$ where $\\boldsymbol{\\alpha}_i=[\\eta_i,\\zeta_i]^T$ is the vector of unknown offsets resulting in a shift of the \\textit{i}-th source direction and $\\mathbf{r}_p=[u_p,v_p]^T$ is the vector of known antenna position in units of wavelength. \n  \nOn top of that, passing through the ionosphere is associated with a rotation of the polarisation plane of each signal source around the line of sight. We call it   the ionospheric Faraday rotation matrix $\\mathbf{F}_{i}(\\vartheta_{i})$ and write it as\n\\begin{equation}\n\\mathbf{F}_{i}(\\vartheta_{i}) = \\begin{bmatrix}\n    \\cos(\\vartheta_{i}) & -\\sin(\\vartheta_{i}) \\\\\n   \\sin(\\vartheta_{i}) & \\cos(\\vartheta_{i})\n\\end{bmatrix}\n\\end{equation}\nwhere $\\vartheta_i$ is the unknown Faraday rotation angle, assumed identical for all antennas, since the array has a limited spatial extent \\cite{noordam1996measurement}.\n  \n  $\\bullet$ \\textit{Instrumental effects} :\\\\\n  Individual antennas are described by electronic complex gains which appear in\n  $\\mathbf{G}_p(\\mathbf{g}_p)=\\mathrm{diag}\\{\\mathbf{g}_p\\}$ with $\\mathbf{g}_p$ the unknown electronic gain vector. \n  \n\n\n \n  Therefore, in this specific structured case, the physical model parameters in (\\ref{model_regime3}) are collected in the vector $\\boldsymbol{\\varepsilon}^{\\mathrm{3DC}}=\\mathbf{P}[\\boldsymbol{\\theta}^{\\mathrm{3DC}^T}_{1,1},\\boldsymbol{\\theta}^{\\mathrm{3DC}^T}_{1,2},\\hdots,\\boldsymbol{\\theta}^{\\mathrm{3DC}^T}_{D,M}]^T$ where $\\mathbf{P}$ is an appropriate rearrangement matrix such that $\\boldsymbol{\\varepsilon}^{\\mathrm{3DC}}=[\\vartheta_{1}, \\hdots,\\vartheta_{D}, \\mathbf{g}^T_1, \\hdots, \\mathbf{g}^T_M, \\boldsymbol{\\alpha}^T_1 , \\hdots, \\boldsymbol{\\alpha}^T_D]^T$.\n\n\n\n\n\n\\section{Robust calibration estimator}\n\\label{robust_estim}\n\nThis section is devoted to the design of a robust calibration estimator based on the model (\\ref{express_x}). As it can be seen from (\\ref{sirp}), one has to specify the probability density function (pdf) of each texture parameter $\\tau_{pq}$ in order to obtain the exact ML estimates. Nevertheless, in pratical scenarios, such prior knowledge is not available. Consequently, our idea is to make use of a relaxed version of the exact model, i.e., we assume deterministic but unknown texture realizations in the estimation process \\cite{conte2002recursive,pascal2008covariance}. This ensures more flexibility in our algorithm as the texture distribution is not precisely described and avoids any possible model misspecification, which is consistent with our motivation to design a broad robust estimator w.r.t.~the presence of outliers. On the other hand, we adopt here an iterative procedure in order to reduce the computational cost. In doing so, the proposed algorithm sequentially updates each block of unknown parameters while fixing the remaining parameters. This leads to a relaxed concentrated ML based calibration estimator for which the expression of the likelihood function, when independency is assumed between measurements,  is written as\n\\begin{align}\n\\label{likelihood}\n\\nonumber\n& f(\\mathbf{x}|\\boldsymbol{\\theta },\n\\boldsymbol{\\tau}, \\boldsymbol{\\Omega})= \\\\  &\n\\prod_{pq}\\frac{1}{| \\pi\n\\tau_{pq} \\boldsymbol{\\Omega} |} \\exp\n\\left\\{-\\frac{1}{\\tau_{pq}}\\mathbf{a}_{pq}^{H}(\\boldsymbol{\\theta})\n\\boldsymbol{\\Omega}^{-1}\\mathbf{a}_{pq}(\\boldsymbol{\\theta})\\right\\},\n\\end{align}\nwhere the vector composed of all texture realizations is $\\boldsymbol{\\tau}=[\\tau_{12},\\tau_{13},\\ldots,\\tau_{(M-1)M}]^{T}$ and $\\mathbf{a}_{pq}(\\boldsymbol{\\theta}) = \\mathbf{v}_{pq}- \\tilde{\\mathbf{v}}_{pq}(\\boldsymbol{\\theta})$.\nConsequently, the log-likelihood function reads\n\\begin{align}\n\\label{loglikelihood}\n\\nonumber\n & \\log f(\\mathbf{x}|\\boldsymbol{\\theta },\n\\boldsymbol{\\tau}, \\boldsymbol{\\Omega}) =\n-4B\\log\\pi \\\\  &\n-4\\sum_{pq}\\log\\tau_{pq}-\nB\\log |\\boldsymbol{\\Omega}|-\\sum_{pq}\\frac{1}{\\tau_{pq}}\\mathbf{a}_{pq}^{H}(\\boldsymbol{\\theta})\n\\boldsymbol{\\Omega}^{-1}\\mathbf{a}_{pq}(\\boldsymbol{\\theta}).\n\\end{align}\n\nIn the following, we present the sequential updates of each block of unknown parameters, namely, $\\boldsymbol{\\tau}$,  $\\boldsymbol{\\Omega}$ and $\\boldsymbol{\\theta}$, following the methodology as in \\cite{zhang2015maximum,ollier2016relaxed}.\n\n\\textbf{\\textit{1) Derivation of $\\hat{\\tau}_{pq}$}}:\nTaking the derivative of the log-likelihood function in (\\ref{loglikelihood}) w.r.t.~$\\tau_{pq}$\nleads to the following texture estimate\n\\begin{equation}\n\\label{tauExp} \\hat{\\tau}_{pq} = \\frac{1}{4} \\mathbf{a}_{pq}^{H}(\\boldsymbol{\\theta})\n\\boldsymbol{\\Omega}^{-1} \\mathbf{a}_{pq}(\\boldsymbol{\\theta}).\n\\end{equation}\n\n\\textbf{\\textit{2) Derivation of $\\hat{\\boldsymbol{\\Omega}}$}}:\nThe derivative of the log-likelihood function w.r.t.~the element\n$[\\boldsymbol{\\Omega}]_{k,l}$ of the speckle covariance matrix, using classical differential properties \\cite[p.~2741]{hjorungnes2007complex}, leads to\n\\begin{equation} \n\\label{OmegaDerive}\n-B\\mathrm{tr}\\left\\{\\hat{\\boldsymbol{\\Omega}}^{-1}\\mathbf{e}_{k}\\mathbf{e}_{l}^{T}\\right\\}+\n\\sum_{pq}\\frac{1}{\\tau_{pq}}\\mathbf{a}_{pq}^{H}(\\boldsymbol{\\theta})\n\\hat{\\boldsymbol{\\Omega}}^{-1}\\mathbf{e}_{k}\\mathbf{e}_{l}^{T}\\hat{\\boldsymbol{\\Omega}}^{-1}\\mathbf{a}_{pq}(\\boldsymbol{\\theta})=0\n\\end{equation}\nwhere the vector $\\mathbf{e}_{k}$ contains zeros except at the\n$k$-th position which is equal to unity. The permutation property of the trace operator enables to rewrite (\\ref{OmegaDerive}) as\n\\begin{equation}\n-B\\mathbf{e}_{l}^{T}\\hat{\\boldsymbol{\\Omega}}^{-1}\\mathbf{e}_{k}+\n\\sum_{pq}\\frac{1}{\\tau_{pq}}\\mathbf{e}_{l}^{T}\\hat{\\boldsymbol{\\Omega}}^{-1}\\mathbf{a}_{pq}(\\boldsymbol{\\theta})\\mathbf{a}_{pq}^{H}(\\boldsymbol{\\theta})\n\\hat{\\boldsymbol{\\Omega}}^{-1}\\mathbf{e}_{k}=0.\n\\end{equation}\nThis finally leads to the following estimate of the speckle covariance matrix \n\\begin{equation}\n\\label{OmegaEstimBefore}\n\\hat{\\boldsymbol{\\Omega}}=\\frac{1}{B}\\sum_{pq}\\frac{1}{\\tau_{pq}}\\mathbf{a}_{pq}(\\boldsymbol{\\theta})\\mathbf{a}_{pq}^{H}(\\boldsymbol{\\theta}).\n\\end{equation}\nInserting (\\ref{tauExp}) into (\\ref{OmegaEstimBefore}), we obtain\n\\begin{equation}\n\\label{OmegaEstim} \\hat{\\boldsymbol{\\Omega}}^{h+1} = \\frac{4}{B}\\sum_{pq}\n \\frac{\\mathbf{a}_{pq}(\\boldsymbol{\\theta})\n\\mathbf{a}_{pq}^{H}(\\boldsymbol{\\theta})}{\\mathbf{a}_{pq}^{H}(\\boldsymbol{\\theta})\\Big(\\hat{\\boldsymbol{\\Omega}}^{h}\\Big)^{-1}\\mathbf{a}_{pq}(\\boldsymbol{\\theta})}\n\\end{equation}\nwhere $h$ denotes the \\textit{h}-th iteration. Due to the introduced constraint, we normalize the estimate of $\\boldsymbol{\\Omega}$ by its trace, as follows\n\\begin{equation}\n\\label{trace}\n\\hat{\\boldsymbol{\\Omega}}^{h+1}=\\frac{\\hat{\\boldsymbol{\\Omega}}^{h+1}}{\\mathrm{tr}\\left\\{\\hat{\\boldsymbol{\\Omega}}^{h+1}\\right\\}}.\n\\end{equation}\n\n\\textbf{\\textit{3) Estimation of $\\hat{\\boldsymbol{\\theta}}$}}:\nFor given $\\boldsymbol{\\Omega}$ and $\\boldsymbol{\\tau}$, estimating $\\hat{\\boldsymbol{\\theta}}$ is equivalent to the following minimization problem\n\\begin{equation}\n\\label{optimtheta}\n\\hat{\\boldsymbol{\\theta }}=\\argmin_{\\boldsymbol{\\theta } }\n\\left\\{\\sum_{pq}\\frac{1}{\\tau_{pq}}\\mathbf{a}_{pq}^{H}(\\boldsymbol{\\theta})\n\\boldsymbol{\\Omega}^{-1}\\mathbf{a}_{pq}(\\boldsymbol{\\theta})\\right\\}.\n\\end{equation}\n In the following, we aim to reduce the computational cost of the minimization procedure in (\\ref{optimtheta}) by use of the EM algorithm. For generality, we first adopt the non-structured Jones matrix formulation, which can also be specified depending on the scenario, as shown in Section \\ref{structure}.\n\n\n\\section{Estimation of $\\hat{\\boldsymbol{\\theta}}$ for non-structured Jones matrices}\n\\label{non_structure}\n\nEstimating directly the entries of the Jones matrices avoids specifying any particular physical model, leading to a calibration algorithm which is less sensitive to model errors in comparison with algorithms based on the structured case. Due to the possible large size of $\\boldsymbol{\\theta}$, a multi-dimensional parameter search needs to be carried out to solve the optimization problem in (\\ref{optimtheta}) which requires significant computation time. To reduce this complexity, we make use of the EM algorithm.\n\nThe essence of this algorithm relies on a proper parameter vector partitioning as well as an adequate choice of the so-called complete data. \nAs mentioned above, the parameters of interest $\n\\boldsymbol{\\theta}$ represent the entries of all Jones matrices. Consequently, it is natural to consider the following partition\n\\begin{equation}\n\\label{express_theta}\n\\boldsymbol{\\theta} = [\\boldsymbol{\\theta}_{1}^{T}, \\ldots,\n\\boldsymbol{\\theta}_{D}^{T}]^{T} =\n[\\boldsymbol{\\theta}_{1,1}^{T}, \\ldots,\n\\boldsymbol{\\theta}_{1,M}^{T}, \\ldots,\n\\boldsymbol{\\theta}_{D,1}^{T},\n\\ldots,\\boldsymbol{\\theta}_{D,M}^{T} ]^{T},\n\\end{equation}\nfor which the vector $\\boldsymbol{\\theta}_{i,p}$ $\\in\\mathbb{R}^{8 \\times 1}$ is the parametrization of the path from the \\textit{i}-th calibrator source to the \\textit{p}-th sensor, i.e.,\n$\\mathbf{J}_{i,p}(\\boldsymbol{\\theta})=\\mathbf{J}_{i,p}(\\boldsymbol{\\theta}_{i,p})$.\n\n\\subsection{Use of the EM algorithm to solve (\\ref{optimtheta})}\nThe EM algorithm \\cite{dempster1977maximum,moon1996expectation, mclachlan2007algorithm} enables to compute the ML estimates and reduce the computational cost, via the iteration of two steps. The first one is the E-step which reduces, in our scenario, to the computation of the conditional expectation of the complete data given the observed data and the current estimate of parameters \\cite{feder1988parameter,yatawatta2009radio}. Afterwards, the log-likelihood function of this conditional distribution is maximized in the M-step. Therefore, this last step consists in an optimization process which can be performed numerically, for instance with the Levenberg-Marquardt (LM) algorithm \\cite{madsen1999methods,nocedal2006numerical,gavin2011levenberg}, or analytically if closed-form expressions are available. As we show in the following, the optimization step is carried out w.r.t.~to $\\boldsymbol{\\theta}_{i}$ $\\in\\mathbb{C}^{4 M \\times 1}$ instead of  $\\boldsymbol{\\theta}$ $\\in\\mathbb{C}^{4 D M \\times 1}$. Therefore, the global multiple source estimation problem is reduced to multiple single source sub-problems.\n\n\\textbf{\\textit{1) E-step}}:\nFor the \\textit{i}-th source, we introduce the so-called complete data vector $\\mathbf{w}_{i}$ such that \n\\begin{equation}\n\\mathbf{x}=\\sum_{i=1}^{D}\\mathbf{w}_{i}\n\\end{equation}\nwith $\\mathbf{w}_{i} =\n\\mathbf{u}_{i}(\\boldsymbol{\\theta}_{i})+\\mathbf{n}_{i}$ and $\\mathbf{n}=\\sum_{i=1}^{D}\\mathbf{n}_{i}$, in which $\\mathbf{n}_{i} \\sim \\mathcal{CN}(\\mathbf{0},\\beta_{i}\\boldsymbol{\\Psi})$. We have $\\sum_{i=1}^{D}\\beta_{i} =\n1$ and  $\\boldsymbol{\\Psi}$ is the covariance matrix of $\\mathbf{n}$.\nSince $\\mathbf{n}_{pq} \\sim \\mathcal{CN}(\\mathbf{0},\\tau_{pq} \\boldsymbol{\\Omega})$ and with the independence property, we obtain the following block-diagonal expression for $\\boldsymbol{\\Psi}$ \n\\begin{equation}\n\\label{formPsi}  \\boldsymbol{\\Psi}  =\\mathrm{bdiag}\\{\n    \\tau_{12} \\boldsymbol{\\Omega} ,\n    \\ldots, \\tau_{(M-1)M}\\boldsymbol{\\Omega}\\}.\n\\end{equation}\nLet us note $\\mathbf{w} =[\\mathbf{w}_{1}^{T}, \\ldots, \\mathbf{w}_{D}^{T}]^{T}$ the complete data vector, whose covariance matrix, denoted as $\\boldsymbol{\\Xi}$, has the following form\n\\begin{equation} \n\\boldsymbol{\\Xi}=\\mathrm{bdiag}\\{\n    \\beta_{1} \\boldsymbol{\\Psi}, \\ldots, \\beta_{D}\\boldsymbol{\\Psi}\\}.\n    \\end{equation}\n\n\nWith \\cite[p.~36]{anderson1958introduction}, and after some calculus, the expression of the conditional expectation is given by \n\\begin{align}\n\\label{wRealEstim}  \\nonumber \\hat{\\mathbf{w}}_{i}\n& = \\mathrm{E}\\{\\mathbf{w}_i|\\mathbf{x};\n\\boldsymbol{\\theta}, \\boldsymbol{\\tau}, \\boldsymbol{\\Omega}\\} \\\\ & =\\mathbf{u}_{i}(\\boldsymbol{\\theta}_{i}) +\n\\beta_{i}\\left(\\mathbf{x}-\\sum_{l=1}^{D}\\mathbf{u}_{l}(\\boldsymbol{\\theta}_{l})\\right).\n\\end{align}\n\n\\textbf{\\textit{2) M-step}}:\nThe goal of this step is to estimate $\\boldsymbol{\\theta}_{i}$. Once $\\hat{\\mathbf{w}}_{i}$ are computed for $ i \\in\n\\{1, \\ldots, D\\}$, the estimated complete data vector $\\hat{\\mathbf{w}}$ can be evaluated. The M-step is an optimization problem based on the following likelihood function where $\\mathbf{w}_{i}$ are independent \n\\begin{align}\n& \\nonumber  f(\\hat{\\mathbf{w}}|\\boldsymbol{\\theta },\\boldsymbol{\\tau},\\boldsymbol{\\Omega})  = \\\\ &\n\\nonumber\n\\frac{1}{|\\pi \\boldsymbol{\\Xi}|} \\exp\n\\left\\{-\\Big(\\hat{\\mathbf{w}}-\\mathbf{u}(\\boldsymbol{\\theta})\\Big)^{H}\\boldsymbol{\\Xi}^{-1}\\Big(\\hat{\\mathbf{w}}-\\mathbf{u}(\\boldsymbol{\\theta})\\Big)\\right\\}= \\\\ &\n\\prod_{i=1}^{D}\\frac{1}{|\\pi\n\\beta_{i}\\boldsymbol{\\Psi}|} \n\\exp\n\\left\\{-\\Big(\\hat{\\mathbf{w}}_{i}-\\mathbf{u}_{i}(\\boldsymbol{\\theta}_{i})\\Big)^{H}(\\beta_{i}\\boldsymbol{\\Psi})^{-1}\\Big(\\hat{\\mathbf{w}}_{i}-\\mathbf{u}_{i}(\\boldsymbol{\\theta}_{i})\\Big)\\right\\}.\n\\end{align}\nTo obtain an estimation of $\\boldsymbol{\\theta}_{i}$, we need to minimize the following cost function\n\\begin{equation}\n\\label{Mstep_i}\n\\phi_{i}(\\boldsymbol{\\theta}_{i}) = \\Big(\\hat{\\mathbf{w}}_{i}-\\mathbf{u}_{i}(\\boldsymbol{\\theta}_{i})\\Big)^{H}(\\beta_{i}\\boldsymbol{\\Psi})^{-1}\\Big(\\hat{\\mathbf{w}}_{i}-\\mathbf{u}_{i}(\\boldsymbol{\\theta}_{i})\\Big).\n\\end{equation}\n\n\nTo decrease even more the complexity cost of the proposed robust calibration scheme, we use the BCD algorithm \\cite{friedman2007pathwise,hong2015unified} in the M-step. Consequently, we obtain analytical solutions for each single source sub-problems in (\\ref{Mstep_i}), as shown below.\n\n\\subsection{Use of the BCD algorithm to minimize (\\ref{Mstep_i})}\n\n\nIn (\\ref{Mstep_i}), the optimization is performed w.r.t. $\\boldsymbol{\\theta}_{i}$. However, this unknown parameter vector can be partitioned according to the antennas, as expressed in (\\ref{express_theta}). In the following, we perform the optimization of the cost function w.r.t. each $\\boldsymbol{\\theta}_{i,p}$ (\\textit{p}-th antenna), given the current estimates of all the other $\\boldsymbol{\\theta}_{i,q}$ with $q\\neq p$. This leads to a closed-form expression of $\\hat{\\boldsymbol{\\theta}}_{i,p}$ as function of $\\boldsymbol{\\theta}_{i,q}$ for $q\\neq p$ and the optimization process is repeated for each component vector $\\boldsymbol{\\theta}_{i,p}$ for $p \\in \\{1,\\hdots,M\\}$.\n\n\nIf we want to minimize (\\ref{Mstep_i}) w.r.t. block-coordinate vector $\\boldsymbol{\\theta}_{i,p}$, we notice that only a subset of $\\mathbf{u}_i(\\boldsymbol{\\theta}_i )$ is actually dependent on $\\boldsymbol{\\theta}_{i,p}$, i.e., $\\{\\mathbf{u}_{i,pq}\\} \\ \\text{for} \\ \\ q > p, \\ \\ q \\in \\{1, \\ldots, M\\}$ and $\\{\\mathbf{u}_{i,qp}\\} \\ \\text{for} \\ \\ q<p, \\ \\ q \\in \\{1, \\ldots, M\\}$. Therefore, (\\ref{Mstep_i}) reads\n\\begin{align}\n\\label{Mstep_i_p}\n\\nonumber\n& \\phi_{i}(\\boldsymbol{\\theta}_{i,p})   = \\\\ & \\nonumber \\sum_{\\substack {q=1\\\\ q>p}}^{M} \\Big(\\mathbf{w}_{i,pq} - \\mathbf{u}_{i,pq}(\\boldsymbol{\\theta}_{i,p})\\Big)^H\n(\\beta_i \\tau_{pq} \\boldsymbol{\\Omega})^{-1} \n\\Big(\\mathbf{w}_{i,pq} - \\mathbf{u}_{i,pq}(\\boldsymbol{\\theta}_{i,p})\\Big)+ \\\\ &\n\\nonumber\n\\sum_{\\substack {q=1 \\\\ q<p}}^{M} \\Big(\\mathbf{w}_{i,qp} - \\mathbf{u}_{i,qp}(\\boldsymbol{\\theta}_{i,p})\\Big)^H\n(\\beta_i \\tau_{qp} \\boldsymbol{\\Omega})^{-1} \n\\Big(\\mathbf{w}_{i,qp} - \\mathbf{u}_{i,qp}(\\boldsymbol{\\theta}_{i,p})\\Big)+ \\\\ &\n\\mathrm{Constant}.\n\\end{align}\nBy $\\mathrm{Constant}$, we mean the expressions independent of $\\boldsymbol{\\theta}_{i,p}$.\nWe show in Appendix B that it is possible to write $\\mathbf{u}_{i,pq}$ directly as a function of $\\boldsymbol{\\theta}_{i,p}$, i.e., \n\\begin{equation}\n\\label{form_pq}\n\\mathbf{u}_{i,pq}(\\boldsymbol{\\theta}_{i,p})=\\boldsymbol{\\Sigma}_{i,q}\\boldsymbol{\\theta}_{i,p}.\n\\end{equation}\nIn the same way, we have \n\\begin{equation}\n\\label{form_qp}\n\\mathbf{u}_{i,qp}(\\boldsymbol{\\theta}_{i,p})=\\boldsymbol{\\Upsilon}_{i,q}\\boldsymbol{\\theta}^{\\ast}_{i,p}.\n\\end{equation}\nNotation and calculations being introduced in Appendix B, we only present here the results obtained, i.e., the expression of the estimated entries of the Jones matrix associated with the path from the \\textit{i}-th calibrator source to the \\textit{p}-th sensor which is given by\n\\begin{equation}\n\\label{theta_i_p_general}\n\\hat{\\boldsymbol{\\theta}}_{i,p} =\\begin{small}\n\\left\\lbrace\n\\begin{array}{lll}\n(\\boldsymbol{\\Sigma}^{H}_{i}\\mathbf{A}_{i,p}\\boldsymbol{\\Sigma}_{i}+\\boldsymbol{\\Upsilon}^{H}_{i}\\tilde{\\mathbf{A}}_{i,p}\\boldsymbol{\\Upsilon}_{i})^{-1} \\times \\\\ (\\boldsymbol{\\Sigma}^{H}_{i}\\mathbf{A}_{i,p}\\mathbf{w}_{i,p}+\n\\boldsymbol{\\Upsilon}^{H}_{i}\\tilde{\\mathbf{A}}_{i,p}\\tilde{\\mathbf{w}}_{i,p}) \\ \\ \\text{for} \\ \\ 1<p<M \\\\\n(\\boldsymbol{\\Sigma}^{H}_{i}\\mathbf{A}_{i,p}\\boldsymbol{\\Sigma}_{i})^{-1}\\boldsymbol{\\Sigma}^{H}_{i}\\mathbf{A}_{i,p}\\mathbf{w}_{i,p}\n \\ \\ \\ \\  \n  \\text{for} \\ \\  p=1\\\\\n (\\boldsymbol{\\Upsilon}^{H}_{i}\\tilde{\\mathbf{A}}_{i,p}\\boldsymbol{\\Upsilon}_{i})^{-1}\\boldsymbol{\\Upsilon}^{H}_{i}\\tilde{\\mathbf{A}}_{i,p}\\tilde{\\mathbf{w}}_{i,p}\n\\ \\ \\\n \\text{for} \\ \\ p=M\n\\end{array}\\right.\n \\end{small}\n\\end{equation}\n Therefore, $\\boldsymbol{\\theta}_{i,p}$ for $p\\in \\{1,\\hdots,M\\}$ are estimated in an iterative loop. With (\\ref{Mstep_i}) and (\\ref{theta_i_p_general}), it can be proven that the BCD algorithm leads to unique solutions and thus, convergence to at least a local maximizer, is ensured \\cite{bertsekas1999nonlinear}. If the M-step is performed exactly (i.e., the BCD gives the exact minimizer of (25) and consequently, the M step is exactly solved), convergence of the EM algorithm to a stationary point is ensured (to avoid the unusual case of convergence to a saddle point, a proper initialization is required) \\cite{mclachlan2007algorithm}, with a theoretical infinite number of iterations. Finally, in this case, convergence of the concentrated MLE is guaranteed for an infinite number of iterations since the value of the cost function at each step can either improve or maintain but cannot worsen \\cite{vorobyov2005maximum}. In practice, only a finite number of iterations is  considered in each loop, so we might not attain local convergence.\nHowever, we show in section  \\ref{simu_sirp} the relatively good numerical stability of the algorithm. \n \n \n The scheme of the proposed algorithm is described in Algorithm 1. \n\n \\begin{algorithm}\n\\SetAlgorithmName{Algorithm 1}{}{} \\caption{Relaxed concentrated ML based calibration algorithm}\n\\SetKwInOut{input}{input} \\SetKwInOut{output}{output}\n\\SetKwInOut{initialize}{initialize}\n\\input{$D$, $M$, $B$, $\\mathbf{C}_{i}$,  $\\beta_{i}$, $\\mathbf{x}$}\n\\output{$\\hat{\\boldsymbol{\\theta}}$}\n\\initialize{$\\hat{\\boldsymbol{\\Omega}}$ $\\leftarrow$ $\\boldsymbol{\\Omega}_{\\mathrm{init}}$, $\\hat{\\boldsymbol{\\tau}}$ $\\leftarrow$ $\\boldsymbol{\\tau}_{\\mathrm{init}}$, $\\hat{\\boldsymbol{\\theta}}\\leftarrow\\boldsymbol{\\theta}_{\\mathrm{init}}$}\n\\While\n{stop criterion unreached}\n{\n\\While \n{stop criterion unreached}\n{\n\\ShowLn E-step: $\\hat{\\mathbf{w}}_{i}$ obtained from (\\ref{wRealEstim}), $i \\in \\{1,\\hdots, D\\}$\\\\\n\\ShowLn M-step: $\\hat{\\boldsymbol{\\theta}}_{i}$ obtained as follows, $i\\in \\{1, \\hdots, D\\}$ \\\\\n\\SetAlgoNoLine\n\\Indp \\Indp \n\\While   \n{ stop criterion unreached}\n{ $\\hat{\\boldsymbol{\\theta}}_{i,p}$ obtained from (\\ref{theta_i_p_general}), $p \\in \\{1, \\hdots, M\\}$\n}\n}\n\\ShowLn Obtain $\\hat{\\boldsymbol{\\Omega}}$ from (\\ref{OmegaEstim}) and (\\ref{trace}),\\\\\n\\ShowLn Obtain $\\hat{\\boldsymbol{\\tau}}$ from (\\ref{tauExp})\\\\\n}\n\\end{algorithm}\n\n  \n  \\section{Structured Jones matrices}\n  \\label{structure}\n  \n  \n  \n  We recall that the output of Algorithm 1 is the estimate of each Jones matrix denoted by $\\hat{\\mathbf{J}}_{i,p}$ for $i\\in \\{1, \\hdots, D\\}$ and $p\\in \\{1, \\hdots, M\\}$. In the following, we consider the data model in (\\ref{model_regime3}) for the specific 3DC calibration regime, and intend to estimate the unknown parameter vector of interest $\\boldsymbol{\\varepsilon}^{\\mathrm{3DC}}$ in a sequential manner. To do so, we use an iterative estimation procedure by optimizing a cost function w.r.t.~one parameter while fixing the others.\n\n \n  \n  \n    \\textbf{\\textit{1) Estimation of $\\mathbf{g}_p$}}:\nThe diagonal elements of the gain matrix are given by     \n      \\begin{equation}\n  \\label{estim_gp}\n \\hat{\\mathbf{g}}_p= \\argmin_{\\mathbf{g}_p} \\kappa(\\mathbf{g}_p)\n  \\end{equation}\n  where $\\kappa(\\mathbf{g}_p)=\\sum_{i=1}^D  ||\\hat{\\mathbf{J}}_{i,p}-\\mathbf{G}_p(\\mathbf{g}_p)\\mathbf{H}_{i,p}\\mathbf{Z}_{i,p}\\mathbf{F}_{i}||^2_{F}$. We rewrite the cost function as\n  \\begin{equation}\n  \\kappa(\\mathbf{g}_p)=\\sum_{i=1}^D\\mathrm{Tr}\\Big\\{\\Big(\\hat{\\mathbf{J}}_{i,p}-\\mathbf{G}_p(\\mathbf{g}_p)\\mathbf{R}_{i,p}\\Big)\\Big(\\hat{\\mathbf{J}}_{i,p}-\\mathbf{G}_p(\\mathbf{g}_p)\\mathbf{R}_{i,p}\\Big)^H\\Big\\}\n  \\end{equation}\nin which $\\mathbf{R}_{i,p}=\\mathbf{H}_{i,p}\\mathbf{Z}_{i,p}\\mathbf{F}_{i}$.\n  The derivation of $ \\kappa(\\mathbf{g}_p)$ w.r.t. the \\textit{k}-th element  $[\\mathbf{g}_p]_k$ leads to\n    \\begin{equation}\n    \\label{derive_kappa}\n  \\frac{\\partial \\kappa(\\mathbf{g}_p)}{\\partial\n[\\mathbf{g}_p]_k}= \\sum_{i=1}^D\\mathrm{Tr}\\Big\\{-\\mathbf{e}_k\\mathbf{e}_k^T\\mathbf{R}_{i,p}\\hat{\\mathbf{J}}_{i,p}^H+\\mathbf{e}_k\\mathbf{e}_k^T\\mathbf{R}_{i,p}\\mathbf{R}_{i,p}^H\\mathbf{G}_p^H\\Big\\}.\n    \\end{equation}\n    Let us denote $\\mathbf{X}_{i,p}=\\mathbf{R}_{i,p}\\hat{\\mathbf{J}}_{i,p}^H$ and $\\mathbf{W}_{i,p}=\\mathbf{R}_{i,p}\\mathbf{R}_{i,p}^H$. From (\\ref{derive_kappa}), we deduce the equation satisfied by the gain matrix\n    \\begin{align}\n    \\nonumber\n\\sum_{i=1}^D[\\mathbf{X}_{i,p}]_{k,k} & =\\sum_{i=1}^D[\\mathbf{W}_{i,p}\\hat{\\mathbf{G}}_p^{H}]_{k,k} \\\\ & =\n\\sum_{i=1}^D[\\mathbf{W}_{i,p}]_{k,k}[\\hat{\\mathbf{G}}_p^{\\ast}]_{k,k}\n\\end{align}\n    for $k \\in \\{1,2\\}$, since $\\mathbf{G}_p$ is a diagonal matrix. Therefore, each complex gain element is estimated as \n  \\begin{equation}\n  \\label{estim_gp_final}\n[\\hat{\\mathbf{g}}_p]_k=\\Big(\\sum_{i=1}^D[\\mathbf{W}^{\\ast}_{i,p}]_{k,k}\\Big)^{-1} \\sum_{i=1}^D[\\mathbf{X}^{\\ast}_{i,p}]_{k,k}.\n\\end{equation}\n\n\n \\textbf{\\textit{2) Estimation of $\\boldsymbol{\\alpha}_i$}}:\n We first need to estimate $\\varphi_{i,p}$ (we recall that $\\varphi_{i,p}=\\eta_i u_p+\\zeta_i v_p$). This is done as follows\n       \\begin{equation}\n  \\label{estim_phi_i_p}\n \\hat{\\varphi}_{i,p}= \\argmin_{\\varphi_{i,p}} \\tilde{\\kappa}(\\varphi_{i,p})\n  \\end{equation}\n  where  $\\tilde{\\kappa}(\\varphi_{i,p})= ||\\hat{\\mathbf{J}}_{i,p}-\\mathbf{G}_p\\mathbf{H}_{i,p}\\mathbf{Z}_{i,p}(\\varphi_{i,p})\\mathbf{F}_{i}||^2_{F}$. Taking the derivative of $\\tilde{\\kappa}(\\varphi_{i,p})$ w.r.t. $\\varphi_{i,p}$ and setting the result to zero, we obtain\n \\begin{align}\n \\nonumber\n & \\mathrm{Tr}\\Big\\{j\\exp^{-j\\hat{\\varphi}_{i,p}}\\hat{\\mathbf{J}}_{i,p}\\mathbf{F}_{i}^H\\mathbf{H}_{i,p}^H\\mathbf{G}_p^H-j\\exp^{j\\hat{\\varphi}_{i,p}}\\mathbf{G}_p\\mathbf{H}_{i,p}\\mathbf{F}_{i}\\hat{\\mathbf{J}}_{i,p}^H\\Big\\} \\\\& =0\n \\end{align}\n which leads to \n  \\begin{equation}\n  \\exp\\Big\\{2j\\hat{\\varphi}_{i,p}\\Big\\}=\\frac{\\mathrm{Tr}\\Big\\{\\mathbf{M}_{i,p}\\Big\\}}{\\mathrm{Tr}\\Big\\{\\mathbf{M}_{i,p}^H\\Big\\}}\n \\end{equation}\n where $\\mathbf{M}_{i,p}=\\hat{\\mathbf{J}}_{i,p}\\mathbf{F}_{i}^H\\mathbf{H}_{i,p}^H\\mathbf{G}_p^H$. \n \n \nIn the case of a compact array, we can write for the \\textit{i}-th source\n   \\begin{equation}\n \\boldsymbol{\\varphi}_i^T=\\hat{\\boldsymbol{\\alpha}}_i^T\n\\boldsymbol{\\Lambda}\n  \\end{equation}\n where $\\boldsymbol{\\varphi}_i=[\\hat{\\varphi}_{i,1},\\hdots,\\hat{\\varphi}_{i,M}]^T$ and $\\boldsymbol{\\Lambda}= \\begin{bmatrix}\n   u_1, & \\hdots & ,u_M\\\\\nv_1, & \\hdots &  ,v_M\n  \\end{bmatrix}$.\n  Therefore, estimation of the directional shifts due to propagation in the ionosphere is given by\n  \\begin{equation}\n  \\label{estim_alpha_i}\n \\hat{\\boldsymbol{\\alpha}}_i^T=\n \\frac{\\boldsymbol{\\varphi}_i^T\\boldsymbol{\\Lambda}^H \\begin{bmatrix}\n  \\sum_{p=1}^M v_p^2 & -\\sum_{p=1}^M u_p v_p\\\\\n-\\sum_{p=1}^M v_p u_p &  \\sum_{p=1}^M u_p^2\n  \\end{bmatrix}}{\\sum_{p=1}^M u_p^2 \\sum_{p=1}^M v_p^2 - (\\sum_{p=1}^M u_p v_p)^2}.\n \\end{equation}\n \n   \\textbf{\\textit{3) Estimation of $\\vartheta_i$}}:\n  We consider the following minimization problem\n  \\begin{equation}\n  \\label{estim_thetai}\n \\hat{\\vartheta}_i= \\argmin_{\\vartheta_i} \\sum_{p=1}^M ||\\hat{\\mathbf{J}}_{i,p}-\\mathbf{G}_p\\mathbf{H}_{i,p}\\mathbf{Z}_{i,p}\\mathbf{F}_{i}(\\vartheta_{i})||^2_{F}.\n  \\end{equation}\n  We assume a large number of antennas $M$ while the number of calibrator sources $D$ is relatively reduced, such that observations outnumber unknown parameters. For each source, the 1D optimization in (\\ref{estim_thetai}) can be computed in a reasonable computational time through a classical data grid search or a Newton type algorithm.\n \n\n Finally, the proposed algorithm for the structured Jones matrices case regarding 3DC calibration regime is given in Algorithm 2.\n \\begin{algorithm}\n\\SetAlgorithmName{Algorithm 2}{}{} \\caption{Case of structured Jones matrices}\n\\SetKwInOut{input}{input} \\SetKwInOut{output}{output}\n\\SetKwInOut{initialize}{initialize}\n\\input{$D$, $M$, $B$, $\\mathbf{C}_{i}$,  $\\beta_{i}$, $\\mathbf{x}$, $\\hat{\\mathbf{J}}_{i,p}$ as output of Algorithm 1, $i \\in \\{1, \\hdots, D\\}$ and $p \\in \\{1, \\hdots, M\\}$} \n\\output{$\\hat{\\boldsymbol{\\varepsilon}}^{\\mathrm{3DC}}$}\n\\initialize{$\\hat{\\boldsymbol{\\varepsilon}}^{\\mathrm{3DC}} \\leftarrow \\boldsymbol{\\varepsilon}^{\\mathrm{3DC}}_{\\mathrm{init}}$}\n\\While\n{stop criterion unreached}\n{\\setcounter{AlgoLine}{0} \\ShowLn Obtain $\\hat{\\vartheta_i}$ from (\\ref{estim_thetai}), $i \\in \\{1, \\hdots, D\\}$ \\\\\n\\ShowLn Obtain $\\hat{\\mathbf{g}}_p$ from (\\ref{estim_gp_final}), $p \\in \\{1, \\hdots, M\\}$\\\\\n\\ShowLn Obtain $\\boldsymbol{\\alpha}_i=[\\eta_i,\\zeta_i]^T$ from (\\ref{estim_alpha_i}), $i \\in \\{1, \\hdots, D\\}$ \\\\\n}\n\\end{algorithm}\n\n\n\n\n  \n  \\section{Numerical simulations}\n  \\label{simus}\n  \n  In this part, we first aim to assess the statistical performance of Algorithm 1, when the noise model matches our noise assumption, i.e., a SIRP noise modeling. Afterwards, we intend to study our proposed algorithm, in a more realistic scenario where outliers are present. More specifically, the sky is composed of $D$ known bright calibrator sources but also $D'$ weak non-calibrator sources which are absorbed in the noise component and act as outliers. Under such assumption, we compare our scheme with the recently introduced robust calibration approach based on Student's t \\cite{yatawatta2014robust} and with the traditional Gaussian cases \\cite{yatawatta2009radio}. Finally, we apply Algorithm 2 to the introduced 3DC calibration regime where Jones matrices are structured.\n  \n  \\subsection{Numerical results under SIRP noise}\n  \\label{simu_sirp}\n  \nThe unknown parameters to estimate, $\\boldsymbol{\\theta}$ given in (\\ref{express_theta}), correspond to the real and imaginary parts of the entries of all Jones matrices. The additive noise in (\\ref{express_x}) is assumed to follow a SIRP as in (\\ref{sirp}) and the number of Monte Carlo runs is set to 100. \n\n \\begin{figure}[t!]\n \\centering\n\\subfigure[]{\\label{fig:SIRV-aa}\\includegraphics[width=0.45\\textwidth,height=0.35\\textwidth]{Fig1.eps}} \\hspace{0.5pt} \n\\subfigure[]{\\label{fig:SIRV-bb}\\includegraphics[width=0.45\\textwidth,height=0.35\\textwidth]{Fig2.eps}} \n\\caption{\\label{fig1:SIRV}(a) MSE of the real part of the first $32$ unknown parameters for a given SNR, (b) MSE vs. SNR for the real part of a given unknown parameter and the corresponding CRB, for $D=2$ bright signal sources and $M=8$ antennas, leading to $128$ real unknown parameters of interest to estimate and $224$ measurements. }\n\\end{figure}\n\n\n\\begin{figure}[t!]\n \\centering\n\\subfigure[]{\\label{fig2:SIRV-Conv}\\includegraphics[width=0.45\\textwidth,height=0.35\\textwidth]{Fig3.eps}} \\hspace{0.5pt} \n\\subfigure[]{\\label{fig2:SIRV-Conv_log}\\includegraphics[width=0.45\\textwidth,height=0.35\\textwidth]{Fig4.eps}} \n\\caption{\\label{fig2:SIRV-Conv}$\\epsilon_{\\Re\\left\\{\\boldsymbol{\\theta}\\right\\}}^h$ as function of the \\textit{h}-th iteration, for loop in line 2 (a), and  in line 1 (b) from Algorithm 1. }\n\\end{figure}\n\n\nIn order to evaluate the estimation performance of the relaxed concentrated ML based calibration algorithm, we fix the noise distribution, e.g., as a Student's t and make use of the Cram\\'{e}r-Rao bound (CRB) \\cite{stoica2005spectral}. To do so, each random texture component is supposed to follow an inverse gamma distribution\\cite{balleri2007maximum}.\nAs an example,\n\\begin{equation}\n\\tau_{pq} \\sim \\mathcal{IG}(\\nu\/2,\\nu\/2) ,\n\\end{equation}\nwith $\\nu$ degrees of freedom \\cite{ollila2012complex} and we choose for example $[\\boldsymbol{\\Omega}]_{k,l}=\\sigma^2 0.9^{|k-l|}\\exp^{j\\frac{\\pi}{2}(k-l)}$.\n\nThe covariance inequality principle states that, under quite general\/weak conditions, the variance satisfies\n  \\begin{equation}\n\\mathrm{MSE}([\\hat{\\boldsymbol{\\theta}}]_k)=\\mathrm{E}\\Big\\{\\Big([\\hat{\\boldsymbol{\\theta}}]_k-[\\boldsymbol{\\theta}]_k\\Big)^2\\Big\\}\\geqslant[\\mathrm{CRB}(\\boldsymbol{\\theta})]_{k,k}\n\\end{equation}\nwhere the CRB is given as the inverse of the Fisher information matrix (FIM) $\\mathbf{F}$. A Slepian-Bangs type formula of the FIM for SIRP observations is given in \\cite{besson2013fisher}  which can be adapted to our case and reads \n\\begin{equation}\n[\\mathbf{F}]_{k,l}= 2\\frac{\\nu+4}{\\nu+5}\\sum_{pq}\\Re\\left\\{\n\\frac{\\partial \\tilde{\\mathbf{v}}^H_{pq}(\\boldsymbol{\\theta})}{\\partial[\\boldsymbol{\\theta}]_k}\\boldsymbol{\\Omega}^{-1}\\frac{\\partial \\tilde{\\mathbf{v}}_{pq}(\\boldsymbol{\\theta})}{\\partial[\\boldsymbol{\\theta}]_l}\\right\\}.\n\\end{equation}\nNote that the noise parameters are decoupled from the parameters of interest. Consequently, only the part corresponding to the latter is kept in the FIM expression.\n\n\n  \n First, in Fig. \\ref{fig:SIRV-aa}, we plot the mean square error (MSE) of the real part of each unknown parameter, obtained with Algorithm 1, for a signal-to-noise ratio $\\mathrm{SNR}=15 \\mathrm{dB}$. We only plot the parameters relative to one given source, the behavior being the same for any source.  \n We also compare the MSE of one given parameter, as a function of the SNR, to its corresponding CRB  in Fig. \\ref{fig:SIRV-bb}. This enables to assess the statistical performance of Algorithm 1, and we notice that the MSE approaches the CRB. The small gap between the bound and the algorithm is explained by the relaxed nature hypothesis used in the design of Algorithm 1. Indeed, we have assumed unknown and deterministic texture parameters when we derived the estimates using Algorithm 1, but in the data model, these parameters are generated as random variables following inverse gamma distribution and the CRB was derived using the prior of the pdf of the texture.\n\n\n\n   \nSecond, we now aim to investigate numerically the convergence properties of our algorithm, which is composed of 3 loops. In each of these loops, $\\boldsymbol{\\theta}$ is updated at each iteration. We therefore consider the following quantity\n\\begin{equation}\n\\epsilon_{\\Re\\left\\{\\boldsymbol{\\theta}\\right\\}}^h=||\\Re\\left\\{\\boldsymbol{\\theta}^h -\\boldsymbol{\\theta}^{h-1}\\right\\}||^2_2\n\\end{equation}\nwhere $h$ refers to the \\textit{h}-th iteration.\nIn Fig. \\ref{fig2:SIRV-Conv} we present the convergence rate of loops described in Algorithm 1 at line 1 and 2 (the analysis of convergence of the third loop, given in line 5, has the same behavior as the loop in line 2 and thus, is not reported here).\nWe note that around 5 iterations are required in loop 2 to attain convergence while approximately 20 iterations are needed for the algorithm to be stable, in loop 1. Nevetheless, in simulations, we notice that only 3 to 4 iterations are sufficient to get close to the CRB.\n\n\n\n\n\\begin{figure}[t!]\n \\centering\n\\subfigure[]{\\label{fig:outlier_idem_temps-aa}\\includegraphics[width=0.45\\textwidth,height=0.35\\textwidth]{Fig5.eps}} \\hspace{0.5pt} \n\\subfigure[]{\\label{fig:outlier_idem_temps-bb}\\includegraphics[width=0.45\\textwidth,height=0.35\\textwidth]{Fig6.eps}} \n\\caption{\\label{fig1:outlier_idem_temps}(a) MSE of the real part of the $64$ unknown parameters for a given SNR, (b) MSE vs. SNR for the real part of a given unknown parameter, for $D=2$, $M=8$ and $D'=8$, leading to $128$ real parameters of interest to estimate and $224$ measurements. }\n\\end{figure}\n\n\n\\begin{figure}[t!]\n \\centering\n\\subfigure[]{\\label{fig:gain_param_AjoutNabil}\\includegraphics[width=0.45\\textwidth,height=0.35\\textwidth]{Fig7.eps}} \\hspace{0.5pt} \n\\subfigure[]{\\label{fig:ionos_param}\\includegraphics[width=0.45\\textwidth,height=0.35\\textwidth]{Fig8.eps}} \n\\caption{\\label{fig:all_structured}(a) MSE of the real part of the $16$ complex gains for a given SNR, (b) MSE of $\\zeta_1$ vs. SNR, for $D=2$, $M=8$, and $D'=4$, leading to $38$ real parameters of interest to estimate and $224$ measurements.}\n\\end{figure}\n\n\n\\subsection{Numerical results using a realistic model}\n\n\n\nLet us investigate the robustness of our proposed calibration procedure in a realistic situation, and compare it with the state-of-the-art. To do so, we consider $D$ calibrator sources, $D'$ weak outlier sources and Gaussian background noise in our data model.\nThe real parameters of interest to estimate still correspond to the real and imaginary entries of the Jones matrices associated to the calibrator sources paths. In the following, we compare Algorithm 1 with i) the calibration approach exposed in \\cite{yatawatta2014robust} which assumes a Student's t noise modeling using the so-called expectation-conditional maximization either algorithm \\cite{liu1995ml,li2006t} and ii) the traditional calibration scheme based on zero-mean white Gaussian noise modeling using a least squares approach \\cite{yatawatta2009radio}. The comparative results are plotted in Fig. \\ref{fig1:outlier_idem_temps}, for the same computation times. We notice better estimation performance with Algorithm 1, since we did not specify any particular noise distribution in our procedure and a SIRP includes many different types of distributions. Furthermore, no assumption was made about independent entries in the noise vector, thus ensuring more flexibility and robustness.\n\n\n\n\n\\subsection{Structured case}\n\nIn order to compare our proposed global algorithm (Algorithm 1 followed by Algorithm 2) with the approach based on Student's t  \\cite{yatawatta2014robust} and the Gaussian case \\cite{yatawatta2009radio} which were both introduced in the non-structured case, we apply Algorithm 2 on the output of these two latter algorithms.\n\n\n\nEach Jones matrix is generated according to (\\ref{model_regime3}) with $\\mathbf{g}=[\\mathbf{g}^T_1,\\hdots,\\mathbf{g}^T_M]^T$. In all exposed simulations, we consider similar computation times for the three presented patterns. We show the results only for the complex gains, cf. Fig. \\ref{fig:gain_param_AjoutNabil}, and the source offset $\\zeta_1$, cf. Fig \\ref{fig:ionos_param}, due to lack of space, the behavior being the same for the other parameters, i.e., $\\vartheta_1$, $\\vartheta_2$, $\\eta_2$, $\\eta_1$ and $\\zeta_2$. In the case of structured Jones matrices, adapted to the 3DC calibration regime, and in the presence of outliers, we still notice the better performances of Algorithm 1, compared to the state-of-the-art. This is expected since better estimation of the Jones entries leads to better estimation of the physical parameters describing the structured Jones matrices.\n\n\n\n\n\n \\section{Conclusion} \n \\label{ccl}\n \n In this paper, we proposed a robust calibration technique where perturbation effects are modeled thanks to Jones matrices. To deal with the presence of outliers in our data, the introduced ML estimation method is based on SIRP noise modeling, leading to a relaxed concentrated ML based calibration algorithm. Numerical simulations show that the proposed algorithm is more robust to the presence of outliers in comparison with the state-of-the-art, for both non-structured and structured Jones matrices, with a reasonable computational complexity.\n\n \n\n\\appendices\n\n\\section{}\n\nWe describe here the corresponding expressions of (\\ref{tauExp}) and (\\ref{OmegaEstim}) when we assume a different $\\boldsymbol{\\Omega}_{pq}$ for $p<q, \\ \\ p,q \\in \\{1, \\ldots, M\\}$. In this case, the log-likelihood function is written as\n\\begin{align}\n\\nonumber\n & \\log f(\\mathbf{x}|\\boldsymbol{\\theta },\n\\boldsymbol{\\tau}, \\boldsymbol{\\Omega}_{12},\\boldsymbol{\\Omega}_{13},\\hdots,\\boldsymbol{\\Omega}_{(M-1)M}) =\n-4B\\log\\pi \\\\  &\n-4\\sum_{pq}\\log\\tau_{pq}-\n\\sum_{pq}\\log |\\boldsymbol{\\Omega}_{pq}|-\\sum_{pq}\\frac{1}{\\tau_{pq}}\\mathbf{a}_{pq}^{H}(\\boldsymbol{\\theta})\n\\boldsymbol{\\Omega}_{pq}^{-1}\\mathbf{a}_{pq}(\\boldsymbol{\\theta}).\n\\end{align}\nFor each antenna pair, the texture estimate is given by\n\\begin{equation}\n \\hat{\\tau}_{pq} = \\frac{1}{4} \\mathbf{a}_{pq}^{H}(\\boldsymbol{\\theta})\n\\boldsymbol{\\Omega}_{pq}^{-1} \\mathbf{a}_{pq}(\\boldsymbol{\\theta})\n\\end{equation}\nwhile the speckle covariance estimate reads\n\\begin{equation}\n\\label{new_Omega}\n \\hat{\\boldsymbol{\\Omega}}_{pq}^{h+1} = 4\n \\frac{\\mathbf{a}_{pq}(\\boldsymbol{\\theta})\n\\mathbf{a}_{pq}^{H}(\\boldsymbol{\\theta})}{\\mathbf{a}_{pq}^{H}(\\boldsymbol{\\theta})\\Big(\\hat{\\boldsymbol{\\Omega}}_{pq}^{h}\\Big)^{-1}\\mathbf{a}_{pq}(\\boldsymbol{\\theta})}.\n\\end{equation}\nThe remainder of the algorithm is straightforwardly obtained using (\\ref{new_Omega}).\n\n\\section{}\nWe present here the steps to obtain (\\ref{theta_i_p_general}).\nFirstly, for sake of clarity, let us denote $\\mathbf{c}_{i}=[ c_{i_{1}},c_{i_{2}},c_{i_{3}},c_{i_{4}}]^T$ to refer to the four entries of the vectorization of source coherency matrix $\\mathbf{C}_{i}$. Likewise, for the \\textit{i}-th source, we write $\n\\mathbf{J}_{i,p}(\\boldsymbol{\\theta}_{i,p}) = \\begin{bmatrix}\n    p_{i_{1}} &  p_{i_{2}}\\\\\n    p_{i_{3}} & p_{i_{4}}\n  \\end{bmatrix}$ for the \\textit{p}-th antenna and $ \n\\mathbf{J}_{i,q}(\\boldsymbol{\\theta}_{i,q}) = \\begin{bmatrix}\n    q_{i_{1}} &  q_{i_{2}}\\\\\n    q_{i_{3}} & q_{i_{4}}\n  \\end{bmatrix}$ for the \\textit{q}-th antenna, i.e., $\\boldsymbol{\\theta}_{i,p}=[p_{i_{1}},p_{i_{2}},p_{i_{3}},p_{i_{4}}]^{T}$ and $\\boldsymbol{\\theta}_{i,q}=[q_{i_{1}},q_{i_{2}},q_{i_{3}},q_{i_{4}}]^{T}$. \nUsing these latter notation, we obtain (\\ref{form_pq})\nwhere \n\\begin{equation}\n\\boldsymbol{\\Sigma}_{i,q}= \\begin{bmatrix}\n   \\alpha_{i,q} & \\beta_{i,q} & 0 & 0\\\\\n0 & 0 & \\alpha_{i,q} & \\beta_{i,q}\\\\\n     \\gamma_{i,q} & \\rho_{i,q} & 0 & 0\\\\\n0 & 0 & \\gamma_{i,q} & \\rho_{i,q}\n  \\end{bmatrix}\n  \\end{equation}\n   in which $\\alpha_{i,q}=q^{\\ast}_{i_{1}}c_{i_{1}}+q^{\\ast}_{i_{2}}c_{i_{3}}$, $\\beta_{i,q}=q^{\\ast}_{i_{1}}c_{i_{2}}\n    + q^{\\ast}_{i_{2}}c_{i_{4}}$, $\\gamma_{i,q}=q^{\\ast}_{i_{3}}c_{i_{1}}+q^{\\ast}_{i_{4}}c_{i_{3}}$ and $\\rho_{i,q}=q^{\\ast}_{i_{3}}c_{i_{2}}\n    + q^{\\ast}_{i_{4}}c_{i_{4}}$.\\\\\nWe also obtain (\\ref{form_qp})\nwhere\n\\begin{equation}\n   \\boldsymbol{\\Upsilon}_{i,q}=\\begin{bmatrix}\n   \\lambda_{i,q} & \\mu_{i,q} & 0 & 0\\\\\n\\nu_{i,q} & \\xi_{i,q} & 0 & 0 \\\\\n      0 & 0 & \\lambda_{i,q} & \\mu_{i,q} \\\\\n0 & 0 & \\nu_{i,q} & \\xi_{i,q}\n  \\end{bmatrix}\n  \\end{equation}\n   in which $\\lambda_{i,q}=q_{i_{1}}c_{i_{1}}+q_{i_{2}}c_{i_{2}}$, $\\mu_{i,q}=q_{i_{1}}c_{i_{3}}\n    + q_{i_{2}}c_{i_{4}}$, $\\nu_{i,q}=q_{i_{3}}c_{i_{1}}+q_{i_{4}}c_{i_{2}}$ and $\\xi_{i,q}=q_{i_{3}}c_{i_{3}}\n    + q_{i_{4}}c_{i_{4}}$.\n\n\nFinally, the cost function in (\\ref{Mstep_i_p}) can be written as \n\\begin{align}\n\\label{compact_exp}\n& \\nonumber \\phi_{i}(\\boldsymbol{\\theta}_{i,p}) = \\Big(\\mathbf{w}_{i,p} - \\mathbf{u}_{i,p}(\\boldsymbol{\\theta}_{i,p})\\Big)^H\\mathbf{A}_{i,p}\\Big(\\mathbf{w}_{i,p} - \\mathbf{u}_{i,p}(\\boldsymbol{\\theta}_{i,p})\\Big) + \n\\\\ &\n\\Big(\\tilde{\\mathbf{w}}_{i,p} - \\tilde{\\mathbf{u}}_{i,p}(\\boldsymbol{\\theta}_{i,p})\\Big)^H\\tilde{\\mathbf{A}}_{i,p}\\Big(\\tilde{\\mathbf{w}}_{i,p} - \\tilde{\\mathbf{u}}_{i,p}(\\boldsymbol{\\theta}_{i,p})\\Big)+\\mathrm{Constant}\n\\end{align}\nwhere $\\mathbf{w}_{i,p}=[\\mathbf{w}_{i,p(p+1)}^T, \\hdots,\\mathbf{w}_{i,p M}^T]^T$, $\\mathbf{u}_{i,p}(\\boldsymbol{\\theta}_{i,p})=[  \\mathbf{u}_{i,p(p+1)}^T(\\boldsymbol{\\theta}_{i,p}),\\hdots,\\mathbf{u}_{i,p M}^T(\\boldsymbol{\\theta}_{i,p})]^T$ and $\\mathbf{A}_{i,p}=\\mathrm{bdiag}\\{\\beta_{i}\\tau_{p(p+1)} \\boldsymbol{\\Omega},\\hdots, \\beta_{i}\\tau_{pM}\\boldsymbol{\\Omega}\\}^{-1}$.\n\n Furthermore, we have $\\tilde{\\mathbf{w}}_{i,p}=[\n\\mathbf{w}^{\\ast^T}_{i,1p},\\hdots,\\mathbf{w}^{\\ast^T}_{i,(p-1)p}]^T$,  $\\tilde{\\mathbf{u}}_{i,p}(\\boldsymbol{\\theta}_{i,p})=[\n\\mathbf{u}^{\\ast^T}_{i,1p}(\\boldsymbol{\\theta}_{i,p}),\\hdots,\\mathbf{u}^{\\ast^T}_{i,(p-1)p}(\\boldsymbol{\\theta}_{i,p})]^T$ and $\\tilde{\\mathbf{A}}_{i,p}=\n\\mathrm{bdiag}\\{\\beta_{i}\\tau_{1p} \\boldsymbol{\\Omega}^{\\ast}, \\hdots, \\beta_{i}\\tau_{(p-1)p}\\boldsymbol{\\Omega}^{\\ast}\\}^{-1}$.\n\nWe make use of (\\ref{form_pq}) in what follows\n\\begin{equation}\n\\label{express_delta}\n\\mathbf{u}_{i,p}(\\boldsymbol{\\theta}_{i,p})=\\begin{bmatrix}\n   \\mathbf{u}_{i,p(p+1)}(\\boldsymbol{\\theta}_{i,p})\\\\\n\\vdots\\\\\n\\mathbf{u}_{i,p M}(\\boldsymbol{\\theta}_{i,p})\n  \\end{bmatrix} = \\begin{bmatrix}\n  \\boldsymbol{\\Sigma}_{i,p+1}\\boldsymbol{\\theta}_{i,p}\\\\\n\\vdots\\\\\n\\boldsymbol{\\Sigma}_{i,M}\\boldsymbol{\\theta}_{i,p}\n  \\end{bmatrix} = \\boldsymbol{\\Sigma}_{i} \\boldsymbol{\\theta}_{i,p}\n\\end{equation}\nwhere $\\boldsymbol{\\Sigma}_{i}=[ \\boldsymbol{\\Sigma}_{i,p+1}^T,\\cdots,\\boldsymbol{\\Sigma}_{i,M}^T]^T$.\nLikewise, we use (\\ref{form_qp}) in \n\\begin{equation}\n\\label{express_upsilon}\n\\tilde{\\mathbf{u}}_{i,p}(\\boldsymbol{\\theta}_{i,p})=\\begin{bmatrix}\n   \\mathbf{u}^{\\ast}_{i,1 p}(\\boldsymbol{\\theta}_{i,p})\\\\\n\\vdots\\\\\n\\mathbf{u}^{\\ast}_{i,(p-1) p}(\\boldsymbol{\\theta}_{i,p})\n  \\end{bmatrix} = \\begin{bmatrix}\n  \\boldsymbol{\\Upsilon}^{\\ast}_{i,1}\\boldsymbol{\\theta}_{i,p}\\\\\n\\vdots\\\\\n\\boldsymbol{\\Upsilon}^{\\ast}_{i,p-1}\\boldsymbol{\\theta}_{i,p}\n  \\end{bmatrix} = \\boldsymbol{\\Upsilon}_{i}\\boldsymbol{\\theta}_{i,p}\n\\end{equation}\nin which $\\boldsymbol{\\Upsilon}_{i}=[ \\boldsymbol{\\Upsilon}^{\\ast^T}_{i,1},\\cdots,\\boldsymbol{\\Upsilon}^{\\ast^T}_{i,p-1}]^T$.\nInserting (\\ref{express_delta}) and (\\ref{express_upsilon}) into (\\ref{compact_exp}) and taking the derivative w.r.t. $\\boldsymbol{\\theta}_{i,p}$ leads to the expressions in (\\ref{theta_i_p_general}), using the fact that $\\mathbf{A}_{i,p}$ and $\\tilde{\\mathbf{A}}_{i,p}$ are Hermitian.\n\n\n \n \\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Abstract}\n\nThis report will be a literature review on a result in algorithmic discrepancy theory. We will begin by providing a quick overview on discrepancy theory and some major results in the field, and then focus on an important result by Shachar Lovett and Raghu Meka in \\cite{Lovett}. We restate the main algorithm and ideas of the paper, and rewrite proofs for some of the major results in the paper.\n\n\\section{Introduction}\nThe discrepancy problem is as follows: given a finite family of finite sets of points, our goal is to color the underlying points (contained in the union of all the sets in the family) red and blue, such that each set has a roughly equal number of red and blue points. Formally, it is described as follows:\nGiven a universe $[n] = \\{1, ... ,n\\}$ and a collection of sets in the universe $S = \\{S_1,...,S_m \\subseteq [n]\\}$, We wish to find an assignment $\\chi: [n] \\rightarrow \\{-1, 1\\}$\nsuch that $\\mathsf{disc}(\\chi)$ is minimized, where $\\mathsf{disc}(\\chi)$ is defined as\n\\[ \\mathsf{disc}(\\chi) = \\max_{S_i \\in S} \\left|\\sum_{i \\in S_i} \\chi(i)\\right|. \\]\nPerhaps two of the most major results in Discrepancy Theory came in the 1980's, when two papers published proofs of the existence of assignments with surprisingly strong lower bounds. First, in 1981, J\\'ozsef Beck and Tibor Fiala showed that given an upper limit on the number of sets that each point is included in, we can find an assignment with discrepancy linear in that limit.\n\n\\begin{theorem}[\\cite{Beck-Fiala}]\nWe start with the assumption that for each $x \\in [n]$, it appears in at most $t$ sets. More formally, we have the constraint that $\\forall i \\in [n]$,\n$$|\\{j; i\\in S_j\\}| \\leq t$$\nThen, one can find an assignment $\\chi$ such that $\\mathsf{disc}(\\chi) \\leq 2t-1$.\n\\end{theorem}\n\nWe provide a proof of the Beck-Fiala theorem in the appendix in \\ref{Beck-Fiala}, using only arguments from linear algebra.\n\nThe other groundbreaking result in Discrepancy Theory is called Spencer's Six Standard Deviations and is given here:\n\\begin{theorem}[\\cite{Spencer}]\\label{Spencer}\nGiven any system of $n$ sets on a universe of $n$ points, there exists a coloring $\\chi$ such that $\\mathsf{disc}(\\chi) \\leq 6\\sqrt{n}$.\n\\end{theorem}\n\nBoth of these results remain cornerstones of Discrepancy Theory. Yet, despite their significance, they were both proven using nonconstructive methods, so we had no way to achieve them algorithmically. For some time, it was even conjectured that no algorithm could be provided. This question remained open until \\cite{Bansal}, where Nikhil Bansal provides a constructive randomized algorithm for discrepancy minimization based on an SDP relaxation. Later, Lovett and Meka propose a new constructive algorithm using only linear algebra \\cite{Lovett}. We will be focusing on this paper. \n\n\\section{Overview}\nThe paper provides a constructive algorithm for minimizing discrepancy, and uses it to prove that their bounds match the bounds given by the previously mentioned theorems. First, they demonstrate a result matching Spencer's Six Standard Deviations.\n\\begin{theorem}[\\cite{Lovett}]\\label{result}\nFor any system of $m$ sets on a universe of $n$ points, there exists a randomized algorithm that, in polynomial time and with at least $1\/2$ chance, computes a coloring $\\chi: [n] \\to \\{-1, 1\\}$ such that $\\mathsf{disc}(\\chi) < K\\sqrt{n\\log_2(m\/n)}$ for some universal constant $K$.\n\\end{theorem}\n\nWe note that for $m = n$, as in the case of Theorem \\ref{Spencer}, we reach the same asymptotic bound as Spencer provided. \n\\cite{Lovett} then provides a similar result for the \"Beck-Fiala case\" where the occurrence of each point is upper bounded\n\n\\begin{theorem}[\\cite{Lovett}]\nFor a system of $m$ sets on a universe of $n$ points where each point is contained in at most $t$ sets, there exists a randomized algorithm that, in polynomial time and with at least $1\/2$ chance, computes a coloring $\\chi: [n] \\to \\{-1, 1\\}$ such that $\\mathsf{disc}(\\chi) < K\\sqrt{t}\\log n$ for some universal constant $K$.\n\\end{theorem}\n\nIn this review we will focus on Theorem 3. The main idea behind the algorithm will be to first create a \\textit{partial coloring}. Given a set-system $(V,S)$, where $V=\\{1,..., n\\}$ and $|S| = m$, we assume that $m \\geq n$ (the other case can easily be reduced to this case by adding some empty sets to $S$). We call this partial coloring $\\chi : V \\rightarrow [-1,1]$ such that:\n\\begin{enumerate}\n    \\item For all $S_i \\in S, |\\chi(S_i)| = O(\\sqrt{n\\log(m\/n)})$\n    \\item $|\\{i : |\\chi(i)| = 1\\}| \\geq cn$ for a constant $c>0$\n    \n\\end{enumerate}\nThe idea is that if we are provided a good algorithm for finding a partial coloring, we can repeatedly apply this algorithm on the variables not yet \"colored\" by this partial coloring, while holding the colored ones constant. This will eventually converge to a full coloring and total discrepancy, as the number of points colored follows a geometric series with ratio $\\sqrt{1-c}$, and the discrepancy can be bounded by $O(\\sqrt{n\\log(m\/n)})$.\n\n\\section{Achieving a Partial Coloring}\n\nThe first and most important step is actually achieving a good partial coloring. We start with a convenient construction: We let $v_1,...,v_m \\in \\mathbb{R}^n$ be the indicator vectors for each of our subsets $S_1,...,S_m$ respectively. Then, the discrepancy of our collection $S$ can be very easily described as \n$$\\mathsf{disc}(S) = \\max_{i \\in [m]} |\\langle \\chi, v_i \\rangle|$$\n\\begin{theorem}[Main Partial Coloring Result]\\label{Coloring}\nLet $v_1,...,v_m \\in \\mathbb{R}^n$ be vectors, and $x_0 \\in [-1,1]^n$ be a \"starting point\". Further let $c_1,...,c_m \\geq 0$ be thresholds such that $\\sum_{j=1}^m\\exp(-c_j^2\/16) \\leq n\/16$. Then let $\\delta>0$ be an approximation parameter. Then there exists an efficient randomized algorithm which with probability $\\geq 0.1$ finds $x \\in [-1,1]^n$ such that\n\\begin{enumerate}\n    \\item discrepancy constraints: \\quad \\quad  $|\\langle x-x_0,v_j\\rangle| \\leq c_j \\|v_j\\|_2$ \n    \\item variable constraints: \\quad \\quad $|x_i| \\geq 1-\\delta$ for at least $n\/2$ indices $i \\in [n]$\n\\end{enumerate}\nMoreover, the algorithm runs in time $O((m+n)^3\\delta^{-2}\\log (nm\/\\delta))$\n\\end{theorem}\n\nThe reason for the constraint on the $c_j$'s will become apparent later, but for now we note that the smaller the $c_j$ are, the stronger the theorem is.  In other words, we want them to be small, but they can't be too small otherwise the theorem won't hold, hence the constraint. We also note that we can increase the probability of success by simply running the algorithm multiple times over.\n\n\\begin{center}\n    4.1. The Algorithm.\n\\end{center}\nWe begin with a general idea, before going into the details of the algorithm. We also assume without changing the problem that the $v_i$'s have all been normalized (we can simply adjust our $c_j$'s to account for this): $\\|v_i\\|_2 = 1, \\forall i$. Consider the following polytope, which describes the legal values that $x\\in \\mathbb{R}^n$ can take on:\n$$\\mathcal{P} = \\{x\\in \\mathbb{R}^n: |x_i|\\leq 1 \\forall i \\in [n], |\\langle x-x_0,v_j\\rangle| \\leq c_j\\}.$$\n\nThen the above theorem says we can find an $x\\in \\mathbb{R}^n$ such that at least $n\/2$ of the variable constraints are satisfied with (virtually) no slack, and it works with good probability as long as we have $\\sum \\exp(-c_j^2) << n$. The idea is to take very small, discrete, Gaussian steps (called Brownian motion) starting from $x_0$. We intuitively want to use these steps to find such an $x$ that is as far away from the origin ($x_0$) as possible, as this implies that more of the constraints are satisfied with no slack. \n\nWe are now ready to present the constructive algorithm that serves as a proof of Theorem \\ref{Coloring}. Let $\\gamma >0$ be a small step size such that $\\delta=O(\\gamma \\sqrt{\\log(nm\/\\gamma})$. The correctness of the algorithm will not be affected by the choice of $\\gamma$, only the runtime. Further let $T=K_1\/\\gamma^2$, where $K_1 = 16\/3$, and assume $\\delta < 0.1$. The algorithm then produces $X_0 = x_0, X_1,...,X_T \\in \\mathbb{R}^n$ according to the following algorithm\\\\\n\\begin{algorithm}[H]\n\\SetAlgoLined\n \n \\For{$t = 1,...,T$}{\n  $\\cdot$ Let $C_t^{var} \\gets C_t^{var}(X_{t-1}) = \\{i \\in [n]: |(X_{t-1})_i| \\geq 1-\\delta\\}$ be the set of variable constraints ``nearly hit\" on the previous iteration\\;\n  $\\cdot$ Let $C_t^{\\mathsf{disc}} \\gets C_t^{\\mathsf{disc}}(X_{t-1}) = \\{j \\in [m]: |\\langle X_{t-1}-x_0,v_j\\rangle| \\geq c_j-\\delta\\}$ be the set of discrepancy constraints \"nearly hit\" on the previous iteration\\;\n  $\\cdot$ Let $\\mathcal{V}_t \\gets \\mathcal{V}(X_{t-1}) = \\{u \\in \\mathbb{R}^n: u_i=0 \\forall i \\in C_t^{var}, \\quad \\langle u, v_j\\rangle = 0 \\forall j \\in C_t^{\\mathsf{disc}}\\}$ be the subspace orthogonal to the ``nearly hit\" variable and discrepancy constraints.\\;\n  $\\cdot$ Set $X_t \\gets X_{t-1} + \\gamma U_t$, where $U_t \\sim \\mathcal{N}(\\mathcal{V}_t)$\n }\n \\caption{The Brownian Motion process for Theorem \\ref{Coloring}}\n\\end{algorithm}\n\nWhen we say that $U \\sim \\mathcal{N}(\\mathcal{V}_t)$, we are referring to the standard multi-dimensional Gaussian distribution: $U = U_1v_1 + ... + U_dv_d$ where $\\{v_1,..,v_d\\}$ is an orthonormal basis for $\\mathcal{V}_t$ and $U_1, ...,U_d \\sim \\mathcal{N}(0,1)$ are all independent. \n\n\\newpage\n\n\\begin{center}\n    4.2. Analysis Outline.\n\\end{center}\n\nWe seek to prove the following:\n\\begin{lemma}\\label{coloring-analysis}\nWe have that Theorem \\ref{Coloring} holds for $X_T$ in the above algorithm, and that with probability at least $0.1$, $X_0,...,X_T \\in \\mathcal{P}$.\n\\end{lemma}\n\nWe begin with a useful claim regarding the behavior of the random walk.\n\n\\begin{claim}\\label{Ortho}\nFor all $t$ we have that $C_t^{var} \\subseteq C_{t+1}^{var}$ and similarly $C_t^{\\mathsf{disc}} \\subseteq C_{t+1}^{\\mathsf{disc}}$ for $t=0,...,T-1$. This further implies that $\\dim(\\mathcal{V}_t) \\geq \\dim(\\mathcal{V}_{t+1})$.\n\\end{claim}\n\\begin{proof}\nIntuitively, we are taking Gaussian steps orthogonal to the subspace $C_t$, so at each step we should never be able to remove any elements in $C_t^{var}$ or $C_t^{\\mathsf{disc}}$. Formally, let $i \\in C_t^{var}$. Then $U_t \\in \\mathcal{V}_t$ which implies $(U_t)_i = 0$. This implies that $(X_t)_i = (X_{t-1})_i$ and $i \\in C_{t+1}^{var}$ as desired. The argument is very similar for the discrepancy constraints.\n\\end{proof}\n\nNow, we can begin to look at the results of the algorithm. \n\nFirst, we can prove that with good probability, our Brownian motion will not leave the polytope. The ``nearly hit\" constraints serve this purpose; we select step size $\\gamma$ small enough that whenever a solution approaches a constraint, it is more likely to fall into the $\\delta$-band of the constraint than it is to break the polytope. Once it falls into this band, Claim \\ref{Ortho} implies that it will never break the polytope. This can be shown formally using Gaussian tailbounds.\n\nNext, we argue that the algorithm satisfies many variable constraints and few discrepancy constraints with high probability. Using our bound on the discrepancy coefficients as well as Gaussian tailbounds, we can demonstrate that the easily-satisfiable discrepancy constraints is small, and that it is unlikely that many other ones are met. With this in mind, at any time $t$ note that there are two scenarios for $C_t^{var}$: if it is large then we are done, and if it is small then our Brownian motion is less constrained so we expect to take steps of larger magnitude. Thus, we argue that by time $T$ it is likely that we ``nearly-hit\" many variable constraints.\n\nFinally, we look at the computational complexity of the algorithm, which is claimed to be $O((n+m)^3\\delta^{-2}\\log(nm\/\\delta))$. The paper does not provide a full justification of this runtime, but we believe it to be inaccurate. \n\nComputing $C_t^{var}$ and $C_t^{\\mathsf{disc}}$ given $X_{t-1}$ takes $O(nm)$ time, since computing $C_t^{\\mathsf{disc}}$ requires the computation of $m$ dot products in $\\mathbb{R}^n$. We can sample from $\\mathcal{N}(\\mathcal{V}_t)$ by constructing an orthonormal basis for $\\mathcal{V}_t$. We do this by constructing an orthogonal basis using our constraints, and using the completion theorem to find a basis of $\\mathcal{V}_t$. Finding a basis from $n+m$ constraint vectors requires Gaussian elimination, so it takes $O((n+m)^3)$ time. \n\nNow, we have to repeat this for $T$ rounds, so the runtime should be expressible as $O((n+m)^3 T)$. Note that $T = O(1\/\\gamma^2)$, so the runtime described by \\cite{Lovett} holds in the case where \n\\begin{align*}\n    \\frac{1}{\\gamma^2} &= O(\\delta^{-2}\\log(nm\/\\delta)) \\\\\n    \\frac{1}{\\gamma} &= O\\left(\\frac{1}{\\delta}\\sqrt{\\log(nm\/\\delta)}\\right) \\\\\n    \\delta &= O(\\gamma\\sqrt{\\log(nm\/\\delta)}).\n\\end{align*}\nHowever, in the paper, $\\gamma$ is selected under the condition $\\delta = O(\\gamma\\sqrt{nm\/\\gamma})$. Of course, this ends up being a small distinction for $nm \\gg \\delta$, but it is still worth noting.\n\nA full proof is provided in the appendix at section \\ref{full-proof}. \n\n\n\\section{The Discrepancy Minimizer}\nFor the purposes of brevity, we only provide a proof for Theorem \\ref{result}.\n\n\\begin{proof}[Proof (Theorem \\ref{result})] To find our full coloring, we will simply repeatedly use Theorem \\ref{Coloring}. For $m$ sets on a universe of size $n$, we'll select $\\delta = 1\/(8 \\log m)$ and $c_1, ..., c_m = 8\\sqrt{\\log(m\/n)}$, and denote by $v_i ... v_m$ the indicator vectors for the sets. We'll use the partial coloring algorithm starting with vector $\\vec{x}_0 = 0^n$ to find some vector $\\vec{x}_1$ where $|\\langle v_j, x_1 \\rangle| < \\sqrt{n}(8\\sqrt{\\log(m\/n)})$ for all $j$ and where more than half of the points have values within the ``nearly-hit\" bound. By Theorem \\ref{Coloring}, this has probability of at least $0.1$, which we can boost by repeating as needed. \n\nApplying this iteratively to the vectors that haven't yet been assigned a partial coloring, we find that within $t = O(\\log n)$ iterations every value in $x$ will be within $\\delta$ of an assignment. When this occurs, for any $j \\in [m]$, we note that $n_i < \\frac{n}{2^i}$, so we have \n\\begin{align*}\n    |\\langle v_j, x \\rangle| &< \\sum_{i=0}^t |\\langle v_j, x_t \\rangle| \\\\\n    &< \\sum \\sqrt{n_i}8\\sqrt{\\log(m\/n_i)} \\\\\n    &< 8\\sqrt{n} \\sum_{i=1}^{\\infty} \\sqrt{\\frac{i + \\log(m\/n)}{2^i}} \\\\\n    &< C\\sqrt{n\\log(m\/n)}\n\\end{align*}\nfor some constant $C$.\n\nWe then use this candidate solution and round it to an actual coloring. Knowing that each variable is within $\\delta$ of either $1$ or $-1$, we'll set each variable to the one it is closer to with probability $(1+|x_i|)\/2$, which means that $\\mathbb{E}[\\chi_i] = x_i$. Denoting $Y := \\chi - x$ we have that the discrepancy for any set $j$ follows  \n\\[ |\\langle \\chi, v_j \\rangle| \\leq |\\langle x, v_j \\rangle| + |\\langle Y, v_j \\rangle|\\]\ndue to triangle inequality. What's left, then, is to find an upper bound for $|\\langle Y, v_j \\rangle|$. Noting that $|Y_i| \\leq 2$, $\\mathbb{E}[Y_i]=0$, $\\sigma^2(Y_i) \\leq \\delta$ (which the paper claims but we are only able to show this is true for $2\\delta$), and $||v_j||_2 \\leq \\sqrt{n}$, the fact that $||v_j||_{\\infty} \\leq 1$ allows us to use a Chernoff bound and get \n\\begin{align*}\n    \\Pr[|\\langle Y, v_j \\rangle| > 2 \\sqrt{2\\log m} \\sqrt{n\\delta}] &\\leq 2\\exp(-2\\log m) \\\\\n    &\\leq 2\/m^2 \\\\\n    &\\leq 1\/2m\n\\end{align*}\nfor $m > 2$. Note that $\\delta = 1\/(8 \\log m)$, so $2 \\sqrt{2\\log m} \\sqrt{n\\delta} = \\sqrt{n}$, which means that across all $j$ we have $\\Pr[|\\langle Y, v_j \\rangle| > \\sqrt{n}] < 1\/2$. Therefore, with probability at least $1\/2$, $\\mathsf{disc}(\\chi) \\leq C\\sqrt{n\\log(m\/n)} + \\sqrt{n} < K\\sqrt{n\\log(m\/n)}$, as desired.\n\\end{proof}\n\n\\newpage\n\n\\section*{References}\n\\printbibliography[heading = none]\n\n\\newpage\n\n\\section{Appendix}\n\n\\subsection{A proof of the Beck-Fiala Theorem}\\label{Beck-Fiala}\nWe present a proof of the Beck-Fiala theorem using only arguments from linear algebra.\n\n\\begin{proof} \\cite{Chazelle}\nWe start by initializing all $\\chi(i) = 0, \\forall i \\in [n]$, and we call all of these variables \\textit{undecided}. We also call a set \\textit{stable} if it has less than or equal to $t$ undecided elements. We also note that due to the constraint, there must be less $n$ sets that contain strictly more than $t$ elements to start off with (all of which are undecided upon initialization). If we impose the constraints that all of the elements in each unstable set must be zero, we get a system of less than $n$ equations, and $n$ variables. This tells us that there is at least one nontrivial solution to the system of equations, that changes only undecided variables and maintains that the discrepancy of all unstable sets remains zero. We can normalize this solution until at least one of the undecided variables is $\\pm 1$. Then, this variable is decided, and we have a partial coloring. We now have at most $n-1$ undecided variables, and each undecided variable is in $(-1, 1)$. By the same argument from above, we have that the number of unstable sets is strictly less than the number of undecided variables, so we can repeat the procedure to find another nontrivial solution to our system of equations. We continue in the manner until all the sets are stable. Then we note that until a set is declared stable, its discrepancy is 0. Then, when it is declared stable, it has at most $t$ undecided variables, all of which are in $(-1,1)$. Then the process of deciding those variables changes the discrepancy of the set by strictly less than $2t$. And since the final discrepancy must be integral, we get the result.  \n\\end{proof}\n\n\n\\subsection{A Full Proof of Lemma \\ref{coloring-analysis}} \\label{full-proof}\nWe have already argued about the runtime of the algorithm. Here, we must show that the solution is unlikely to leave the polytope, that few discrepancy constraints are met, and that many variable constraints are met.\n\n\\begin{claim}\\label{Polytope}\nFor $\\gamma \\leq \\delta \/ \\sqrt{c\\log (mn\/\\gamma)}$ and $c$ a sufficiently large constant, with probability at least $1-1\/(mn)^{c-2}$ we have that $X_0,...,X_T \\in \\mathcal{P}$\n\\end{claim}\n\nTo prove the above claim, we will need to use a Gaussian tail bound:\n\\begin{claim} \\label{tailbound}\nFor any $\\lambda >0$, $P(|G| \\geq \\lambda ) \\leq 2\\exp(-\\lambda^2\/2)$, where $G \\sim \\mathcal{N}(0,1)$\n\\end{claim}\n\n\\begin{proof}\nWe have that \n$$P(|G|>\\lambda) = 2P(G > \\lambda) = 2 \\int_{\\lambda}^{\\infty} \\frac{1}{\\sqrt{2\\pi}}\\exp(-t^2\/2)dt \\leq 2 \\int_{\\lambda}^{\\infty} \\frac{t}{\\lambda}\\frac{1}{\\sqrt{2\\pi}}\\exp(-t^2\/2)dt  = \\frac{2\\exp(-\\lambda^2\/2)}{\\sqrt{2\\pi}\\lambda}$$\nFrom here, it is easy to see that for $\\lambda \\geq 1\/\\sqrt{2\\pi}$ we have that $\\frac{2\\exp(-\\lambda^2\/2)}{\\sqrt{2\\pi}\\lambda} \\leq 2\\exp(-\\lambda^2\/2) $ as desired. For the case when $\\lambda \\leq 1\/\\sqrt{2\\pi}$, it is easy to see that $2\\exp(-\\lambda^2\/2) > 1$ so the bound is trivial.\n\\end{proof}\n\n\\begin{proof}[Proof of Claim \\ref{Polytope}]\nClearly $X_0 = x_0 \\in \\mathcal{P}$. We further define $E_t := \\{X_t \\not\\in \\mathcal{P} | X_0,...,X_{t-1} \\in \\mathcal{P}\\}$ denote the event that $X_t$ is the first element of the sequence not in $\\mathcal{P}$. \\ We then have\n$$Pr(X_0,...,X_T \\in \\mathcal{P}) = 1-\\sum_{t=1}^T Pr(E_t).$$\n\nThe next step of the proof is clearly to calculate $Pr(E_t)$. In order for $E_t$ to happen, it must be the case that either a variable constraint or a discrepancy constraint was violated. Lets first look at the variable constraint case: say $(X_t)_i>1$. Since $X_{t-1} \\in \\mathcal{P}$, we must have that $(X_{t-1})_i \\leq 1$. Yet, if $(X_{t-1})_i \\geq 1-\\delta,$ then $i \\in C_t^{var}$ so $(X_{t-1})_i = (X_{t})_i$. Thus, for the constraint to be violated, we must have had that $(X_{t-1})_i < 1-\\delta $. Then, in order for $(X_t)_i$ to be greater than 1, and since $X_t = X_{t-1} + \\gamma U_t$, we must have that $|U(t)_i| \\geq \\delta\/\\gamma$.\n\nNow, let's look what must happen in order for $X_t$ to violate a discrepancy constraint. First we define $W := \\{e_1, ..., e_n, v_1,...,v_m\\}$. By our construction of $W$, we conclude that if $E_t$ holds then we must have that $|\\langle X_t - X_{t-1} ,w\\rangle| \\geq \\delta$ for some $w \\in W$. This is equivalent to saying that $|\\langle U_t, w \\rangle| \\geq \\delta\/\\gamma$ for that same $w$. We note here that, once again by construction of $W$, we have that if  $|U(t)_i| \\geq \\delta\/\\gamma$ holds, then we must have that $|\\langle U_t, w \\rangle| \\geq \\delta\/\\gamma$ holds for some $w$, in particular it holds if we pick $w=e_i$. However, the reverse does not hold. Therefore, it since the event of a variable constraint being violated is entirely contained in the event of a discrepancy constraint being violated, it suffices to bound $Pr[|\\langle U_t, w \\rangle| \\geq \\delta\/\\gamma]$. In order to bound this, we need the following claim:\n\n\\begin{claim} \\label{subspace}\nLet $V \\subseteq \\mathbb{R}^n$ be a subspace and let $G \\sim \\mathcal{N}(V)$. Then for all $u \\in \\mathbb{R}^n$, $\\langle G, u \\rangle \\sim \\mathcal{N}(0, \\sigma^2)$, where $\\sigma^2 \\leq \\|u\\|_2^2$\n\\end{claim}\n\n\\begin{proof}\nWe have that $G = G_1v_1 + ... + G_dv_d$, where $\\{v_1,...,v_d\\}$ is an arbitrary orthonormal basis for $V$ and $G_1, ..., G_d$ are all standard normals and independent. Then $\\langle G, u \\rangle = \\sum_{i=1}^d \\langle v_i, u \\rangle G_i$. This is a Gaussian RV which has mean zero, and variance $\\sum_{i=1}^d \\langle v_i, u \\rangle^2$. This equation is simply equal to $\\|\\mathsf{Proj}_V u\\|_2^2$, the norm squared of the projection of $u$ onto the span of $V$. Therefore, we have that $\\sum_{i=1}^d \\langle v_i, u \\rangle^2 \\leq \\|u\\|_2^2$, and we are done.\n\\end{proof}\nNow we can use the above claim, and we have that $\\langle U_t, w\\rangle$ is Gaussian with mean 0 and variance at most 1. Then, by claim \\ref{tailbound}, we have that:\n$$Pr[|\\langle U_t, w \\rangle| \\geq \\delta\/\\gamma] \\leq 2\\exp(-(\\delta\/\\gamma)^2\/2) $$\nNow, by our choices of variables, we have that $\\delta\/\\gamma = \\sqrt{C\\log (nm\/\\gamma})$ and $T = O(1\/\\gamma^2)$. Therefore, we have\n$$Pr[X_0 \\cup ... \\cup X_t \\not\\in \\mathcal{P}] = \\sum_{t-1}^T Pr[E_t]$$\nwhich, by a union bound, \n$$\\leq \\sum_{t-1}^T \\sum_{w \\in W} Pr[|\\langle U_t, w \\rangle| \\geq \\delta\/\\gamma] \\leq T (n+m)\\cdot 2\\exp(-(\\sqrt{C\\log (nm\/\\gamma}))^2\/2) = T(n+m)\\cdot 2\\left (\\frac{\\gamma}{nm}\\right )^C $$\n$$\\leq T(nm)\\frac{\\gamma^2}{(nm)^C} \\leq \\frac{1}{(nm)^{C-2}}$$\nFor large enough C, since we have that $\\gamma <1$ and $(nm) >1$.\n\\end{proof}\nWe are now well on our way to proving Lemma \\ref{coloring-analysis}. The intuition behind the remaining steps is as follows. We will use the constraint on our discrepancy thresholds $c_j, j \\in [m]$ to argue first that $\\mathbb{E}[|C_T^{var}|] \\ll n$. This will be useful because it will mean that $\\dim (\\mathcal{V}_{t-1})$ will be larger, which means that $\\mathbb{E}[\\|X_t\\|^2]$ should increase more appreciably compared to the previous timestep. At any given timestep ,either $|C_t^{var}|$ is large and we are done, or $|C_t^{var}|$ is small and once again $\\dim (\\mathcal{V}_{t-1})$ and we will be taking bigger steps. We note also that in order to prove the lemma, we really only need to show that $\\mathbb{E}[|C_t^{var}|] = \\Omega(n)$, since if we achieve this, then we can use this along with the fact that $|C_T^{var}|$ is upper bounded by $n$ to show that $Pr[|C_t^{var}| < n\/2] < 0.9$.\\\\\nWe first show that $\\mathbb{E}[|C_t^{\\mathsf{disc}}|]$ is small; that is, on average very few discrepancy constraints are ever nearly hit.\n\n\\begin{claim}\n$\\mathbb{E}[|C_t^{\\mathsf{disc}}|] < n\/4$\n\\end{claim}\n\\begin{proof}\nWe let $J := \\{j: c_j \\leq 10\\delta \\}$. In order to bound the size of $J$, we have that from our constraints\n$$n\/16 \\geq \\sum_{j \\in J} \\exp (-c_j^2\/16) \\geq |J| \\cdot \\exp(-100\\delta^2\/16) \\geq |J| \\cdot \\exp(-1\/16) > 9|J|\/10$$\nSince $\\delta <0.1$. So we have then that $|J| \\leq 1.2n\/16 < 2n\/16$. Now we consider the $j \\not \\in J$. If $j \\in C_T^{\\mathsf{disc}}$, then $|\\langle X_T-x_0, v_j\\rangle| \\geq c_j - \\delta \\geq 0.9c_j$. We want to bound the probability that his occur. Via our update formula, we have that $X_T = x_0 + \\gamma(U_1+...+U_T)$.  We then define $Y_i = \\langle U_i, v_j \\rangle$. We then have that for $j \\not\\in J$\n$$Pr[j \\in C_T^{\\mathsf{disc}}] = Pr[|Y_1 + ... + Y_T| \\geq 0.9c_j\/\\gamma]$$\nWe will also need the following Lemma:\n\\begin{lemma}[\\cite{Bansal}]\nLet $X_1,...X_T$ be random variables, and let $Y_1,...,Y_T$ be RVs where each $Y_i$ is a function of $X_i$. Suppose that for all $1 \\leq i \\leq T$, $Y_i|X_1, ..., X_{i-1}  $ is Gaussian with mean zero and variance at most one. Then for any $\\lambda >0$:\n$$Pr[|Y_1 + ...+ Y_T| \\geq \\lambda \\sqrt{T}] \\leq 2 \\exp(-\\lambda^2\/2)$$\n\\end{lemma}\nThe proof of the above lemma is a generalization of the proof for Claim \\ref{tailbound}, and is omitted. We note that $Y_i = \\langle U_i, v_j \\rangle \\sim \\mathcal{N}(0, \\sigma^2)$, where $\\sigma^2 \\leq \\|v_j\\|^2 = 1$ by our assumption at the beginning of the problem that we had normalized all the the $v_i$. We can apply the above lemma to our $Y_i$'s, since $Y_i$ is a function of $U_i$ and $Y_i|U_1,...,U_{i-1}$ is Gaussian with mean zero and variance at most one to get that \n$$Pr[j \\in C_T^{\\mathsf{disc}}] \\leq 2\\exp(-(0.9c_j)^2\/2\\gamma^2T$$\nwhich, since $T = K_1\/\\gamma^2$\n$$ = 2\\exp(-(0.9c_j)^2\/2K_1) < 2\\exp(-c_j^2\/2)$$\nWe therefore have that \n$$\\mathbb{E}[|C_T^{\\mathsf{disc}}|] \\leq |J| + \\sum_{j \\not\\in J} Pr[j \\in C_T^{\\mathsf{disc}}] < 2n\/16+2n\/16 = n\/4$$\nWhere above we have used conditional expectations, and assumed in the worst case that every element in $J$ is in $C_T^{\\mathsf{disc}}$ and we have also used the constraint $\\sum_{j=1}^m\\exp(-c_j^2\/16) \\leq n\/16$.\n\\end{proof}\n\n\\begin{claim} \\label{bound-x-t}\n$\\mathbb{E}[\\|X_T\\|_2^2] \\leq n$\n\\end{claim}\n\\begin{proof}\nWe start by noting that it suffices to show that $\\mathbb{E}[(X_T)_i^2] \\leq 1$ for all $i \\in [n]$, since $\\mathbb{E}[\\|X_T\\|_2^2] = \\sum_i \\mathbb{E}[(X_T)_i^2]$. We have that\n$$\\mathbb{E}[(X_T)_i^2] = \\mathbb{E}[(X_T)_i^2|i \\not \\in C_T^{var}]Pr[i \\not\\in C_T^{var}] + \\sum _{t=1}^T \\mathbb{E}[(X_T)_i^2|i \\in C_t^{var} \\setminus C_{t-1}^{var}]Pr[i \\in C_t^{var} \\setminus C_{t-1}^{var}]$$\nNow, we have that clearly $\\mathbb{E}[(X_T)_i^2|i \\not \\in C_T^{var}] \\leq 1$. For the rest of the terms, we have:\n$$\\mathbb{E}[(X_T)_i^2|i \\in C_t^{var} \\setminus C_{t-1}^{var}] = \\mathbb{E}[(X_t)_i^2|i \\in C_t^{var} \\setminus C_{t-1}^{var}]$$\n$$ = \\mathbb{E}[((X_{t-1})_i+\\gamma(U_t)_i)^2|i \\in C_t^{var} \\setminus C_{t-1}^{var}] \\leq \\mathbb{E}[(1-\\delta+\\gamma(U_t)_i)^2|i \\in C_t^{var} \\setminus C_{t-1}^{var}]$$\n$$ = (1-\\delta)^2 + 2(1-\\delta)\\gamma\\mathbb{E}[(U_t)_i] + \\gamma^2 \\mathbb{E}[(U_t)_i^2]$$\nHere, we note that $\\mathbb{E}[(U_t)_i^2] = 1$ and $\\mathbb{E}[(U_t)_i] = 0$, so we have\n$$ = (1-\\delta)^2 + \\gamma^2 \\leq 1-\\delta + \\gamma < 1$$\nby our construction of $\\gamma$.\n\\end{proof}\n\n\n\\begin{claim}\n$\\mathbb{E}[|C^{var}_T|] \\geq 0.56n$.\n\\end{claim}\n\\begin{proof}\nWe will use the high average norm of $X_t$ and low number of discrepancy constraints broken to demonstrate that the number of variable constraints broken is high with high probability. Note that \n\\begin{align*}\n    \\mathbb{E}[||X_t||^2_2] &= \\mathbb{E}[||X_{t-1} + \\gamma U_t||_2^2] \\\\\n    &= \\mathbb{E}[||X_{t-1}||^2_2] + 2\\mathbb{E}[||X_{t-1} \\cdot \\gamma U_t||_2] + \\mathbb{E}[||\\gamma U_t||_2^2] \\\\\n    &= \\mathbb{E}[||X_{t-1}||^2_2] + \\mathbb{E}[||\\gamma U_t||_2^2] \\\\\n    &= \\mathbb{E}[||X_{t-1}||^2_2] + \\gamma^2\\mathbb{E}[\\text{dim}(\\mathcal{V}_t)].\n\\end{align*}\nWe use the fact that $\\mathbb{E}[U_t]=0$, and we use Claim \\ref{subspace} as well. Then, by Claim \\ref{bound-x-t}, we have \n\\begin{align*}\n    n &\\geq \\mathbb{E}[||X_T||_2^2] \\\\\n    n &\\geq \\gamma^2 \\sum_{t=1}^T \\mathbb{E}[\\text{dim}(\\mathcal{V}_t)] \\\\\n    n &\\geq \\gamma^2 |T|\\mathbb{E}[\\text{dim}(\\mathcal{V}_T)] \\\\\n    n &\\geq K_1 \\mathbb{E}[n - |C_T^{var}| - |C_T^{\\mathsf{disc}}|] \\\\\n    n\/K_1 &\\geq \\mathbb{E}[n] - \\mathbb{E}[|C_T^{var}|] - \\mathbb{E}[|C_T^{\\mathsf{disc}}|] \\\\\n    \\mathbb{E}[|C_T^{var}|] &\\geq n (1 - 1\/K_1) - n\/4 \\\\\n    \\mathbb{E}[|C_T^{var}|] &\\geq n (1 - 3\/16 - 1\/4) \\\\\n    \\mathbb{E}[|C_T^{var}|] &\\geq 0.5625n.\n\\end{align*}\n\\end{proof}\n\nNow we can fully prove Lemma \\ref{coloring-analysis}. From Claim 14 and the fact that $|C_T^{var}| \\leq n$, in the worst case we have that with probability $0.88$, $|C_T^{var}| = n\/2$, and with probability $0.12$, $|C_T^{var}| = n$. This maximizes the number of instances where $|C_T^{var}| \\leq n\/2$, while still maintaining the fact that $\\mathbb{E}[|C_T^{var}|] \\geq 0.56$. Therefore we have that $Pr[|C_T^{var}| > n\/2] \\geq 0.12$. Combining this with claim 8 tells us that with probability at least $0.12 - 1\/poly(m,n)$ we achieve the partial coloring, and we are done.\n\n\\end{document}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nIsometric actions of Lie groups on Riemannian manifolds have been widely studied in the literature, as they constitute a powerful tool applied to deal with several interesting problems in both mathematics and physics, yielding important geometric and topological consequences. They appear in all branches of science where symmetries preserving length and angle measures play a role. In this paper we propose a generalization of the classical setting of isometric actions by studying a notion of isometric action of a categorified version of a Lie group \\cite{BD,BS} on a categorified version of a Riemannian manifold \\cite{dHF,dHF2,GGHR,PPT}. More precisely, we are interested in studying isometric actions of Lie $2$-groups on Riemannian groupoids. \n\nOn the one hand, as is explicitly mentioned by Baez and Lauda in \\cite{BD}, the notion of Lie 2-group goes back to Brown and Spencer in \\cite{BS} where it became clear that classical group theory is just the beginning of a larger subject that sometimes is called higher-dimensional group theory. In many contexts where we are tempted in using groups to tackle certain symmetries-involved problems, it turns out actually to be more natural to use a richer kind of structure where, in addition to group elements describing symmetries, we also have isomorphisms between these, thus describing symmetries between symmetries. On the other hand, Riemannian groupoids recently appeared into the picture as a differentiable model allowing to perform Riemannian geometry techniques over more general singular spaces than orbifolds or leaf spaces of singular foliations. Since the seminal works \\cite{dHF,dHF2}, the notion of Riemannian groupoid (stack) defined and studied therein has been applied to satisfactorily extend several theories where Riemannian manifolds have played an important role. For instance, some of the recent contributions in which Riemannian groupoid metrics have been used as a tool to describe topological and geometric feutures of Lie groupoids and their differentiable stacks are the following.\n\\begin{itemize}\n\\item Resolutions of proper Riemannian groupoids were introduced in \\cite{PTW} in order to obtain a desingularization of their underlying differentiable stacks via a successive blow-up construction.\n\\item A theory of stacky geodesics on Riemannian stacks was developed in \\cite{dHdM} allowing to establish a stacky version\nof the Hopf--Rinow Theorem.\n\\item The problem of understanding invariant linearization of proper Lie groupoids was addressed in \\cite{dHdM2} where the authors fixed and extended previous results in the literature as well as provided a sufficient criterion that uses compatible complete metrics and covers the case of proper group actions.\n\\item Recently, a notion of Morse Lie groupoid morphism was introduced in \\cite{OV}, allowing to extend the main results of classical Morse theory to the context of Lie groupoids. It is worth mentioning that the notion of isometric 2-action we will introduce in this paper was used in \\cite{OV} as one of the fundamental ingredients needed to construct an equivariant double Morse--Bott complex which computes the equivariant cohomology of a $2$-action as defined in \\cite{OBT}. \n\\end{itemize}\nThe paper is organized as follows. In Section \\ref{S:2} we briefly introduce the necessary terminology and notions about Riemannian Lie groupoids and Lie $2$-groups we will be dealing with along the paper. In Section \\ref{S:3} we define isometric Lie $2$-group actions on Riemannian groupoids and exhibit some of their immediate properties. We provide sufficient conditions to ensure that Riemannian groupoid metrics that are invariant by the action of a Lie $2$-group exist and use the nice ideas implemented to give a proof of the groupoid linearization theorem in \\cite{dHF} to describe an equivariant weakly groupoid linearization result. Based in the classical constructions we state versions of both the Slice Theorem and the Equivariant Tubular Neighborhood Theorem for proper and free Lie $2$-group actions. We also quickly explain a way to define a ``2-group orbit type topological stratification'' for the Lie groupoid we are working with, define orthogonal Lie $2$-groups and describe their infinitesimal counterpart, exhibit a few examples and provide some interesting applications using principal connection warpings and Cheeger deformations. Finally, in Section \\ref{S:4} we present a description of the Lie $2$-group of strong (weak) groupoid isometries of a Lie groupoid equipped with a $0$-metric and determine a model for its Lie $2$-algebra of strong (weak) Killing multiplicative vector fields. We apply some of the results of this section to the case in which we are equipped with an isometric Lie 2-group action. It is important to mention that our infinitesimal descriptions are motivated by those results developed in \\cite{OW} in order to describe the Lie $2$-algebra structure that the set of multiplicative vector fields on a Lie groupoid has. In particular, we show that the Lie $2$-algebra of weak Killing multiplicative vector fields is Morita invariant, thus yielding a good notion of geometric Killing vector field on a quotient Riemannian stack. This notion of geometric Killing vector field recovers the classical notions of Killing vector field on both a Riemannian manifold and a Riemannian orbifold as defined for instance in \\cite{BZ} as well as the notion of transverse Killing vector field on a regular Riemannian foliation as defined in \\cite[p. 84]{Mo}. We end the section by proving that the algebra of geometric Killing vector fields on a quotient Riemannian stack that is represented by a Riemannian foliation groupoid has finite dimension.\n\\medskip\n\n{\\bf Acknowledgments:} We would like to tank Mateus de Melo and Cristian Ortiz for several enlightened discussions, comments, and suggestions which allowed us to improve this work. Herrera was supported by the Brazilian Federal Foundation CAPES - Finance Code 001. Valencia was supported by Grant 2020\/07704-7 Sao Paulo Research Foundation - FAPESP.\n\n\\section{Preliminaries}\\label{S:2}\nIn this short section we briefly introduce the basics on Lie groupoids we shall be using throughout. For specific details the reader is recommended to visit for instance \\cite{BD,BS,dH,Ma,dHF}. \n\nA \\emph{Lie groupoid} $X_1\\rightrightarrows X_0$ consists of a manifold $X_0$ of objects and a manifold $X_1$ of arrows, two surjective submersions $s_X,t_X:X_1\\to X_0$ respectively indicating the source and the target of the arrows, and a smooth associative composition $m_X:X_2\\to X_1$ over the set of composable arrows $X_2=X_1\\times_{X_0} X_1$, admitting unit $u_X:X_0\\to X_1$ and inverse $i_X:X_1\\to X_1$, subject\nto the usual groupoid axioms. The collection of maps mentioned above are called \\emph{structural maps} of the Lie groupoid $X_1\\rightrightarrows X_0$. We shall drop up the sub-index notation for the structural maps only if there is no risk of confusion.\n\nSpecial instances of Lie groupoids are given by manifolds, Lie groups, Lie group actions, surjective submersions, foliations, pseudogroups, principal bundles, vector bundles, among others \\cite{dH, Ma}.\n\nLet $X_1\\rightrightarrows X_0$ be a Lie groupoid. For each $x\\in X_0$, its \\emph{isotropy group} $X_x:=s_X^{-1}(x)\\cap t_X^{-1}(x)$ is a Lie group and an embedded submanifold in $X_1$. There is an equivalence relation in $X_0$ defined by $x\\sim y$ if there exists $p\\in X_1$ with $s_X(p)=x$ and $t_X(p)=y$. The corresponding equivalence class of $x\\in X_0$ is denoted by $\\mathcal{O}_x\\subseteq X_0$ and called the \\emph{orbit} of $x$. The previous equivalence relation defines a quotient space $X_0\/X_1$ called the \\emph{orbit space} of $X_0\\rightrightarrows X_1$.  This space equipped with the quotient topology is in general a \\emph{singular space}, that is, it does not carry a differentiable structure making the quotient projection $X_0\\to X_0\/X_1$ a surjective submersion.\n\nA \\emph{Lie groupoid morphism} between two Lie groupoids $X_1\\rightrightarrows X_0$ and $X_1'\\rightrightarrows X_0'$ is a pair $\\phi:=(\\phi^1,\\phi^0)$ where $\\phi^1:X_1\\to X_1'$ and $\\phi^0:X_0\\to X_0'$ are smooth maps commuting with both source and target maps and preserving the composition maps.\n\nFinally, we recall that the \\emph{Lie algebroid} associated to the Lie groupoid $X_1\\rightrightarrows X_0$ is defined to be the vector bundle $A_X:=\\textnormal{ker}(ds_X)|_{X_0}\\subset TX_1$ with anchor map $\\rho:A_X\\to TX_0$ obtained by restricting $dt_X:TX_1\\to TX_0$ to $A_X$.\n\n\n{\\bf Riemannian groupoids:} The notion of Riemannian metric on a Lie groupoid that we will be working with along the paper was introduced in \\cite{dHF} (see also \\cite{GGHR,PPT}). Such a notion is compatible with the groupoid composition so that it plays an important role in several parts of our work. Because of our purposes we mainly focus in using both $1$-metrics and $0$-metrics which were initially defined in \\cite{GGHR} and \\cite{PPT}, respectively, but they were recovered later in \\cite{dHF} as particular cases of a more general notion. We start by recalling that a submersion $\\pi:(E,\\eta^E)\\to B$ with $(E,\\eta^E)$ a Riemannian manifold is said to be \\emph{Riemannian} if the fibers of it are equidistant (transverse condition). In this case the base $B$ gets an induced metric $\\eta^B:=\\pi_\\ast \\eta^E$ for which the linear map $d\\pi(e)=(\\textnormal{ker}(d\\pi(e)))^\\perp\\to T_{\\pi(e)}B$ is an isometry for all $e\\in E$. If $(\\eta^{E})^\\ast$ denotes the dual metric associated to $\\eta^{E}$ then the condition for a Riemannian submersion can be rephrased as follows. For all $e\\in E$ the map  $d\\pi(e)^\\ast: T_{\\pi(e)}^\\ast B \\to \\textnormal{ker}(d\\pi(e))^\\circ$ is an isometry, where $\\textnormal{ker}(d\\pi(e))^\\circ$ denotes the annihilator of the vectors tangent to the fiber.\n\nIt is well known that given a Lie groupoid $X_1\\rightrightarrows X_0$ every pair of composable arrows in $X_2$ may be identified with an element in the space of commutative triangles so that it admits an action of $S_3$ determined by permuting the vertices of such triangles. In these terms, a \\emph{Riemannian groupoid} is a pair $(X_1\\rightrightarrows X_0,\\eta)$ where $X_1\\rightrightarrows X_0$ is a Lie groupoid and $\\eta=\\eta^{(2)}$ is a Riemannian metric on $X_2$ that is invariant by the $S_3$-action and transverse to the composition map $m_X:X_2\\to X_1$. The metric $\\eta^{(2)}$ induces metrics $\\eta^{(1)}=((\\pi_2)_X)_\\ast \\eta^{(2)}=(m_X)_\\ast \\eta^{(2)}=((\\pi_1)_X)_\\ast \\eta^{(2)}$ on $X_1$ and $\\eta^{(0)}=(s_X)_\\ast \\eta^{(1)}=(t_X)_\\ast \\eta^{(1)}$ on $X_0$ such that $(\\pi_2)_X, m_X, (\\pi_1)_X:X_2\\to X_1$ and $s_X,t_X:X_1\\to X_0$ are Riemannian submersions and $i_X:X_1\\to X_1$ is an isometry. This is because the $S_3$-action permutes these face maps.\n\nThe metric $\\eta^{(j)}$, for $j=2,1,0$, is called a $j$-\\emph{metric}. It is important to mention that every proper groupoid can be endowed with a $2$-metric (more generally, an $n$-metric as defined below) and if a Lie groupoid admits a $2$-metric then it is weakly linearizable around any saturated submanifold. For more details visit \\cite{dHF}.\n\n\n{\\bf Lie 2-groups:} A \\emph{Lie 2-group} is a Lie groupoid in the category of Lie groups. In other words, a Lie 2-group is a Lie groupoid $G_1\\rightrightarrows G_0$ where both $G_1$ and $G_0$ are Lie groups and the structural maps of $G_1\\rightrightarrows G_0$ are Lie group homomorphisms \\cite{BD,BS}.  We will denote by $\\ast$ the composition of arrows in $G_1\\rightrightarrows G_0$ and by $\\cdot$ the product of arrows in $G_1$. For instance, the fact that the multiplication is a morphism of groups amounts to the identity:\n\\begin{equation}\\label{MultiProduct}\n\t(g_1\\ast g_2)\\cdot(g_1'\\ast g_2')=(g_1\\cdot g_1')\\ast(g_2\\cdot g_2'),\\quad \\forall (g_1,g_2),(g_1',g_2')\\in G_2,\n\\end{equation}\nwhere $G_2$ is the space of pairs of composable arrows of $G_1\\rightrightarrows G_0$.\n\nIt is well known that there are several alternative ways to think of Lie 2-groups. One of them is described in terms of crossed modules. Recall that a \\emph{crossed module of Lie groups} is a quadruple $(G,H,\\rho,\\alpha)$ consisting of a Lie group homomorphism $\\rho:H\\to G$ together with an action $\\alpha$ of $G$ on $H$, $(g,h)\\mapsto g h$, by Lie group automorphisms such that:\n\\[ \\rho(g h)=g \\rho(h) g^{-1},\\quad \\rho(h)h'=h h' h^{-1}, \\quad g\\in G,\\ h,h'\\in H. \\]\n\nTo such a crossed module one associates a Lie $2$-group with objects the Lie group $G_0:=G$, arrows the semi-direct product $G_1:=H\\rtimes G$ so that:\n\\[ (h_1,g_1)\\cdot (h_2,g_2):=(h_1 (g_1h_2), g_1 g_2), \\]\nand structure maps:\n\\[ s_G(h,g)=g,\\quad t_G(h,g)=\\rho(h)g, \\quad (h_1,\\rho(h_2)g_2)\\ast (h_2,g_2)=(h_1h_2,g_2). \\]\nConversely, any Lie 2-group $G_1 \\rightrightarrows G_0$ has an associated crossed module of Lie groups $(G,H,\\rho,\\alpha)$, where $G=G_0$, $H=\\textnormal{ker}(s_G)$, $\\rho:=t_G|_H:H\\to G$ and $G$ acts on $H$ by conjugation via the identity bisection: $g h:=1_g\\cdot h\\cdot 1_{g^{-1}}$. This defines an equivalence of categories between the category of Lie 2-groups and the category of crossed modules of Lie groups \\cite{BD}.\n\nA \\emph{Lie 2-algebra} is a Lie groupoid in the category of Lie algebras. More concretely, a Lie 2-algebra is a Lie groupoid $\\mathfrak{g}_1\\rightrightarrows \\mathfrak{g}_0$ where both $\\mathfrak{g}_1$ and $\\mathfrak{g}_0$ are Lie algebras and the structural maps of $\\mathfrak{g}_1\\rightrightarrows \\mathfrak{g}_0$ are Lie algebra homomorphisms. As expected, the Lie functor provides a one-to-one correspondence between Lie 2-groups and Lie 2-algebras.\n\nThe infinitesimal object associated to a crossed module of Lie groups is the so-called \\emph{crossed module of Lie algebras}. This is a quadruple $(\\mathfrak{g},\\mathfrak{h},\\partial,\\mathcal{L})$ where $\\mathfrak{g}$ and $\\mathfrak{h}$ are Lie algebras and $\\partial:\\mathfrak{h}\\to\\mathfrak{g}$ and $\\mathcal{L}:\\mathfrak{g}\\to\\textnormal{Der}(\\mathfrak{h})$ are Lie algebra homomorphisms verifying\n$$\\partial(\\mathcal{L}_xy)=[x,\\partial(y)]_\\mathfrak{g}\\quad\\textnormal{and}\\quad \\mathcal{L}_{\\partial(x)}y=[x,y]_\\mathfrak{h}.$$ \n\nA Lie $2$-algebra $\\mathfrak{g}_1\\rightrightarrows \\mathfrak{g}_0$ has an associated crossed module given by the data $\\mathfrak{h}=\\textnormal{ker}(s)$, $\\mathfrak{g}=\\mathfrak{g}_0$, $\\partial=t|_\\mathfrak{h}$ and $\\mathcal{L}_x=\\textnormal{ad}^{1}_{u(x)}$ for all $x\\in \\mathfrak{g}$. Conversely, a crossed module of Lie algebras $(\\mathfrak{g},\\mathfrak{h},\\partial,\\mathcal{L})$ has an associated Lie $2$-algebra which in turn is given by the data $\\mathfrak{g}_1=\\mathfrak{h}\\rtimes \\mathfrak{g}$ with Lie algebra structure provided by the semi-direct product with respect to $\\mathcal{L}$, $\\mathfrak{g}_1=\\mathfrak{g}$, and structural maps\n$$s(x,y)=y,\\quad t(x,y)=\\partial(x)+y,\\quad u(y)=(0,y),\\quad i(x,y)=(-x,y+\\partial(x)), $$\n$$m((x',y+\\partial(x)),(x,y))=(x+x',y).$$\n\nA \\emph{morphism} of crossed modules  $f:(\\mathfrak{g},\\mathfrak{h},\\partial,\\mathcal{L})\\to (\\mathfrak{g}',\\mathfrak{h}',\\partial',\\mathcal{L}')$ consists of two Lie algebra homomorphisms $f_0:\\mathfrak{g}\\to \\mathfrak{g}'$ and $f_1:\\mathfrak{h}\\to \\mathfrak{h}'$ such that $f_1\\circ \\partial =\\partial'\\circ f_0$ and $f_0(\\mathcal{L}_xy)=\\mathcal{L}'_{f_1(x)}(f_0(y))$. This induces a pair of Lie algebra homomorphisms $\\textnormal{ker}(\\partial)\\to \\textnormal{ker}(\\partial')$ and $\\textnormal{coker}(\\partial)\\to \\textnormal{coker}(\\partial')$. A morphism of crossed modules is a \\emph{quasi-isomorphism} if both of these Lie algebra morphisms are isomorphisms. The \\emph{derived category} of crossed modules of Lie algebras is defined to be the localization of the category of crossed modules of Lie algebras obtained by inverting all quasi-isomorphisms.\n\n\\section{Isometric Lie 2-group actions}\\label{S:3}\nIn the sequel we shall denote by $G_1\\rightrightarrows G_0$ a Lie 2-group and by $(X_1\\rightrightarrows X_0,\\eta)$ a Riemannian groupoid. A \\emph{left 2-action} of $G_1\\rightrightarrows G_0$ on $X_1\\rightrightarrows X_0$ is defined to be a Lie groupoid morphism $\\theta=(\\theta^1,\\theta^0):(G_1\\times X_1\\rightrightarrows G_0\\times X_0)\\to(X_1\\rightrightarrows X_0)$ such that both maps $\\theta^1$ and $\\theta^0$ are usual left Lie group actions. Here $G_1\\times X_1\\rightrightarrows G_0\\times X_0$ denotes the product Lie groupoid.  Note that given a left 2-action of $G_1\\rightrightarrows G_0$ on $X_1\\rightrightarrows X_0$ we immediately get that the structural maps of $X_1\\rightrightarrows X_0$ are equivariant with respect the structural maps of $G_1\\rightrightarrows G_0$. More precisely, if $p\\in X_1, x\\in X_0$ and $g\\in G_1, g_0\\in G_0$ then we obtain that\n\\begin{equation}\\label{Equivariant}\ns_X(gp)=s_G(g)s_X(p),\\quad t_X(gp)=t_G(g)t_X(p),\\quad u_X(g_0x)=u_G(g_0)u_X(x).\n\\end{equation}\n\n\nMoreover, we have that the action on arrows is \\emph{multiplicative}, meaning that for pairs of composable arrows $(p,q)\\in X_2$ and $(g,h)\\in G_2$ the following formula holds true\n\\begin{equation}\\label{MultAction}\n\t(p*q)(g*h)=(pg)*(qh).\n\\end{equation} \nHere we are denoting by $m_X(p,q)=p*q$ and $m_G(g,h)=g*h$. \n\\begin{definition}\nA $2$-action of $G_1\\rightrightarrows G_0$ on $(X_1\\rightrightarrows X_0,\\eta)$ is said to be \\emph{isometric} if $G_1$ acts by isometries on $(X_1,\\eta^{(1)})$.\n\\end{definition}\nAn immediate consequence of the previous definition is the following.\n\\begin{lemma}\\label{Rmk1}\nThe action $G_0$ on $(X_0,\\eta^{(0)})$ is also by isometries.\n\\end{lemma}\n\\begin{proof}\nLet $v,w\\in T_{x}X_0$ and $g_0\\in G_0$. It is clear that there are $\\tilde{v},\\tilde{w}\\in \\textnormal{ker}(d(s_X)_p)^{\\perp_{\\eta^{(1)}}}$ with $s_X(p)=x$ such that $d(s_X)_p(\\tilde{v})=v$ and $d(s_X)_p(\\tilde{w})=w$ and there is $g\\in G_1$ such that $s_G(g)=g_0$. Therefore,\n\\begin{eqnarray*}\n(\\theta^0_{g_0})_x^\\ast\\eta^{(0)}(v,w) & = & \\eta^{(0)}_{g_0x}(d(\\theta^0_{g_0})_x(v),d(\\theta^0_{g_0})_x(w))=\\eta^{(0)}_{g_0x}(d(\\theta^0_{g_0}\\circ s_X)_p(\\tilde{v}),d(\\theta^0_{g_0}\\circ s_X)_p(\\tilde{w}))\\\\\n\\star& = & \\eta^{(0)}_{g_0x}(d(s_X\\circ \\theta^1_{g})_p(\\tilde{v}),d(s_X\\circ \\theta^1_{g})_p(\\tilde{w}))=\\eta^{(1)}_{gp}(d(\\theta^1_{g})_p(\\tilde{v}),d(\\theta^1_{g})_p(\\tilde{w}))\\\\\n& = & \\eta^{(1)}_{p}(\\tilde{v},\\tilde{w})=\\eta^{(0)}_{x}(v,w).\n\\end{eqnarray*}\nIn the equality $\\star$ above we used the fact that $s_X$ is a Riemannian submersion and that the action $\\theta^1$ preserves the horizontal distribution $\\textnormal{ker}(ds_X)^{\\perp_{\\eta^{(1)}}}$. Note that this computation does not depend on the choice of $\\tilde{v},\\tilde{w},p$ and $g$. Furthermore, we may obtain the same conclusion by choosing $\\eta^{(0)}=(t_X)_\\ast \\eta^{(1)}$ instead of $\\eta^{(0)}=(s_X)_\\ast \\eta^{(1)}$. \n\\end{proof}\n\n\\begin{remark}\\label{Rmk2}\nIt is clear that $G_2$ is also a Lie group with the structure induced from the direct product $G_1\\times G_1$. With this in mind it is simple to see that the $2$-action $\\theta$ indices a canonical left action $\\theta^2$ of $G_2$ on $X_2$. We could have defined the notion of isometric $2$-action by requiring that $G_2$ acts on $(X_2,\\eta^{(2)})$ by isometries. If we do so then, just as we did in Lemma \\ref{Rmk1}, we can prove that the action of $G_1$ on $(X_1,\\eta^{(1)})$ is by isometries and, in turn, the action of $G_0$ on $(X_0,\\eta^{(0)})$ will be also by isometries. This is because $\\eta^{(1)}=(\\pi_{2,X})_\\ast \\eta^{(2)}=(m_X)_\\ast \\eta^{(2)}=(\\pi_{1,X})_\\ast \\eta^{(2)}$ and $\\pi_{2,X}, m_X, \\pi_{1,X}:X_2\\to X_1$ are Riemannian submersions. More generally, if $\\eta^{(n)}$ is an $n$-metric on $X_n$ in the sense of \\cite{dHF} and the induced left action $\\theta^n$ of $G_n$ on $(X_n,\\eta^{(n)})$ is by isometries then the action $\\theta^k$ of $G_k$ on $(X_k,\\eta^{(k)})$ will be by isometries for all $0\\leq k\\leq n-1$. Conversely, if we suppose that $G_0$ and $G_1$ act by isometries on $(X_0,\\eta^{(0)})$ and $(X_1,\\eta^{(1)})$, respectively, then  the induced action of $G_n$ on $(X_n,\\eta^{(n)})$ is by isometries for all $n\\geq 2$. Last assertion follows by using the formula\n$$\\eta^{(2)}=(\\pi_1)_X^\\ast \\eta^{(1)}+(\\pi_2)_X^\\ast \\eta^{(1)}-s_X^\\ast \\eta^{(0)}\\circ (d(\\pi_1)_X\\oplus d(\\pi_1)_X),$$\nwhich is derived from Remark 2.5 from \\cite{dHF} once the $n$-metric is chosen.\n\\end{remark}\nFor the purposes of this paper we only work with both $1$-metrics and $0$-metrics unless otherwise stated. Most of the notions we will introduce below shall have an analogous statement if the simplicial approach from Remark \\ref{Rmk2} is considered.\n\n\\begin{remark}\\label{Rmk3}\nThere is a different formulation for the notion of $2$-action in terms of crossed modules \\cite{hsz}. Namely, an action $\\beta=(\\beta^0, \\beta^1, \\beta^2)$ of a crossed module of Lie groups $(G,H,\\rho,\\alpha)$ on $X_1\\rightrightarrows X_0$ consists of three smooth actions $\\beta^0:G\\times X_0\\to X_0$, $\\beta^1:G\\times X_1\\to X_1$, and $\\beta^{2}:H\\times X_1\\to X_1$ satisfying the compatibility conditions (6.3.8)\u2013(6.3.12) from \\cite[Def. 6.3.4]{hsz}. Furthermore, from  \\cite[Lem. 6.3.13]{hsz} if $\\theta=(\\theta^1,\\theta^2)$ is a $2$-action of $G_1\\rightrightarrows G_0$ on $X_1\\rightrightarrows X_0$ then $\\beta^0=\\theta^0$, $\\beta^1=\\theta^0\\circ (u\\times \\textnormal{id}_{X_1})$, and $\\beta^2=\\theta^1|_H$ define an action of the associated crossed module on $X_1\\rightrightarrows X_0$. Conversely, given an action $\\beta=(\\beta^0, \\beta^1, \\beta^2)$ of a crossed module $(G,H,\\rho,\\alpha)$ on $X_1\\rightrightarrows X_0$ then $\\theta^0=\\beta^0$ and $\\theta^1_{(h,g)}=\\beta^2_{h}\\circ \\beta^1_g $ for all  $(h,g)\\in H\\rtimes G$, defines a $2$-action of the associated Lie $2$-group.\n\\end{remark}\nMotivated by what we just mentioned we set up the following definition.\n\\begin{definition}\nWe say that an action $\\beta=(\\beta^0, \\beta^1, \\beta^2)$ of a crossed module of Lie groups $(G,H,\\rho,\\alpha)$ on $(X_1\\rightrightarrows X_0,\\eta)$ is isometric if both $\\beta^1$ and $\\beta^2$ determine isometric actions on $(X_1,\\eta^{(1)})$.\n\\end{definition}\n Thus, the following result is clear:\n\\begin{lemma}\nThere exists a one-to-one correspondence between isometric actions of Lie $2$-groups and isometric actions of crossed modules of Lie groups.\n\\end{lemma}\nAn interesting consequence that comes up from our definition of isometric Lie $2$-group action is the following. The notion of Riemannian submersions in the Lie groupoid context was introduced in \\cite{dHF2}. So, the following result is expected:\n\\begin{proposition}\\label{PQuotient}\n\tIf $\\theta$ is a free and proper isometric $2$-action of $G_1\\rightrightarrows G_0$ on $(X_1\\rightrightarrows X_0,\\eta)$ then there is a structure of Riemannian groupoid $(X_1\/G_1\\rightrightarrows X_0\/G_0, \\overline{\\eta})$ so that the canonical projection $\\pi=(\\pi_1,\\pi_0):(X_1\\rightrightarrows X_0)\\to (X_1\/G_1\\rightrightarrows X_0\/G_0)$ becomes a Riemannian groupoid submersion.\n\\end{proposition}\n\\begin{proof}\n\tWe start by exhibiting the Lie groupoid structure on $X_1\/G_1\\rightrightarrows X_0\/G_0$. This fact was stated for instance in \\cite{GZ} but without proof so that we exhibit a proof of it by seek of completeness. It is well known that $X_j\/G_j$ admits a unique manifold structure so that $\\pi_j$ (for $j=0,1$) is a surjective submersion. We define source and target maps respectively as $\\overline{s}([p])=[s_X(p)]$ and $\\overline{t}([p])=[t_X(p)]$ for all $p\\in X_1$. As consequence of Identities \\eqref{Equivariant} these maps are well defined and, moreover, both of them are surjective submersions since $\\pi_j$, $s$, and $t$ are so. We have that $([p],[q])\\in (X_1\/G_1)_2$ if and only if $\\overline{s}([p])=\\overline{t}([q])$ which in turn holds true if and only if $s_X(p)=g_0t_X(q)$ for some $g_0\\in G_0$. Thus, we define $\\overline{m}([p],[q])=[m_X(p,gq)]$ for some $g\\in G_1$ such that $t_G(g)=g_0$. It is simple to check that $\\overline{m}$ does not depend on the choice of $g$ and it is well defined because of Property \\eqref{MultAction}. This is also clearly smooth and associative since $m_X$ is associative and Identity \\eqref{MultAction} is satisfied. The unit map and the inversion are respectively defined by $\\overline{u}([x])=[u_X(x)]$ and $\\overline{i}([p])=[i_X(p)]$ for all $x\\in X_0$ and $p\\in X_1$. It is also easy to verify that this maps are well defined, smooth, and they satisfy the required groupoid conditions.\n\t\n\tConsider, for $j=0,1$, the induced Riemannian metric $\\overline{\\eta}^{(j)}=(\\pi_j)_\\ast \\eta^{(j)}$ on $X_j\/G_j$ making of $\\pi_j$ a Riemannian submersion. Moreover, note that\n\t$$\\overline{i}_\\ast \\overline{\\eta}^{(1)}=(\\overline{i}\\circ \\pi_1)_\\ast \\eta^{(1)}=(\\pi_1\\circ i_X)_\\ast\\eta^{(1)}=(\\pi_1)_\\ast \\eta^{(1)}=\\overline{\\eta}^{(1)},$$\n\tsince $i$ is an isometry, and\n\t$$\\overline{s}_\\ast \\overline{\\eta}^{(1)}=(\\overline{s}\\circ \\pi_1)_\\ast \\eta^{(1)}=(\\pi_0\\circ s_X)_\\ast\\eta^{(1)}=(\\pi_0)_\\ast \\eta^{(0)}=(\\pi_0\\circ t_X)_\\ast\\eta^{(1)}=(\\overline{t}\\circ \\pi_1)_\\ast \\eta^{(1)}=\\overline{t}_\\ast \\overline{\\eta}^{(1)},$$\n\tso that $\\overline{s}_\\ast \\overline{\\eta}^{(1)}=\\overline{t}_\\ast \\overline{\\eta}^{(1)}=\\overline{\\eta}^{(0)}$. Therefore, $(X_1\/G_1\\rightrightarrows X_0\/G_0,\\overline{\\eta})$ with $\\overline{\\eta}=(\\overline{\\eta}^{(1)},\\overline{\\eta}^{(0)})$ is a Riemannian groupoid and $\\pi=(\\pi_1,\\pi_0)$ is a Riemannian submersion of groupoids, as desired.\n\\end{proof}\n\n\\begin{remark}\nLet us denote by $\\eta^\\ast$ the dual metric associated to a Riemannian metric $\\eta$. From \\cite{dHF} we know that if $\\pi:E\\to B$ is a submersion and $\\lbrace \\eta_1,\\cdots,\\eta_k\\rbrace$ is a collection of $\\pi$-transverse metrics then its tangent average $\\frac{1}{k}\\sum_{l=1}^k\\eta_l$ fails to be $\\pi$-transverse again in general. Nevertheless, its cotangent average $\\frac{1}{k}\\left( \\sum_{l=1}^k\\eta_l^\\ast\\right)^\\ast$ is always $\\pi$-transverse which sometimes makes more advantageous to take a cotangent space point of view in the study of Riemannian submersions.\n\\end{remark}\n\nWe will deal with the classical approach used to ensure the existence of invariant Riemannian metrics on $G$-manifolds to provide a similar construction in our context. Namely:\n\n\\begin{lemma}\\label{Existence1}\nIf $G$ is a compact Lie group seen as a Lie unit 2-group $G\\rightrightarrows G$ acting on $(X_1\\rightrightarrows X_0,\\eta)$, then there exists a $1$-metric $\\overline{\\eta}$ making of such an action isometric. In particular, if $G$ is compact and $X_1\\rightrightarrows X_0$ is proper then there always exists a $1$-metric making of such a $2$-action isometric.\n\\end{lemma}\n\\begin{proof}\nLet us consider the normalized Haar measure $\\mu$ on $G$. We define $\\overline{\\eta}=(\\overline{\\eta}^{(1)},\\overline{\\eta}^{(0)})$ by averaging its dual as follows:\n$$(\\overline{\\eta}^{(1)})^\\ast:=\\int_{G}(\\theta^1)^\\ast_g (\\eta^{(1)})^\\ast d\\mu(g).$$\nThe dual metric $(\\overline{\\eta}^{(0)})^\\ast$ has a similar defining formula but using $\\eta^{(0)}$ and $\\theta^0$ instead of $\\eta^{(1)}$ and $\\theta^1$. It is simple to check that $\\overline{\\eta}=(\\overline{\\eta}^{(1)},\\overline{\\eta}^{(0)})$ is the $1$-metric we are looking for. Last part of the assertion above follows from the fact that every proper Lie groupoid admits $n$-metrics \\cite{dHF}.\n\\end{proof}\nA similar result can be obtained only by requiring that $G\\rightrightarrows G$ acts properly on $(X_1\\rightrightarrows X_0,\\eta)$. Same proof as in Lemma \\ref{Existence1} goes through by applying \\cite[Lem. 4.2]{K}. Let us assume for a moment that $G_1$ is compact so that $G_0$ is also compact. If $\\mu_1$ is the normalized Haar measure on $G_1$ then, by uniqueness, the pushforward measure $s_{G\\ast}\\mu_1$ agrees with the normalized Haar measure $\\mu_0$ on $G_0$ since $s_G$ a surjective Lie group homomorphism and\n\\begin{equation}\\label{PushHaar}\n\\int_{G_0}f d(s_{G\\ast}\\mu_1)=\\int_{G_1} f\\circ s_G d\\mu_1,\n\\end{equation}\nfor each continuous function $f:G_0\\to \\mathbb{R}$. Analogously, $t_{G\\ast}\\mu_1=\\mu_0$ and $i_{G\\ast}\\mu_1=\\mu_1$. Note also that the fact that $i_{G\\ast}\\mu_1=\\mu_1$ and $s_G\\circ i_G=t_G$ immediately implies that $t_{G\\ast}\\mu_1=s_{G\\ast}\\mu_1$. Thus, we are in conditions to state:\n\n\\begin{theorem}\\label{Existence2}\nIf $G_1\\rightrightarrows G_0$ is a Lie $2$-group, with $G_1$ compact, acting on $(X_1\\rightrightarrows X_0,\\eta)$, then there exists a $1$-metric $\\overline{\\eta}$ making of such an $2$-action isometric.\n\\end{theorem}\n\\begin{proof}\nAs it was done in Lemma \\ref{Existence1}, by dual averaging  we define:\n$$(\\overline{\\eta}^{(1)})^\\ast:=\\int_{G_1}(\\theta^1)^\\ast_g(\\eta^{(1)})^\\ast d\\mu_1(g)\\quad\\textnormal{and}\\quad (\\overline{\\eta}^{(0)})^\\ast:=\\int_{G_0}(\\theta^0)^\\ast_{g_0}(\\eta^{(0)})^\\ast d\\mu_0(g_0).$$\nThe result will follow from the fact that $\\theta$ is a $2$-action, $\\eta$ is already a $1$-metric, and Identity \\eqref{PushHaar} holds true. Indeed, on the one hand, a straightforward computation using the fact that $i$ is an isometry of $(X_1,\\eta^{(1)})$ verifying both Identity \\eqref{Equivariant} and  $i_{G\\ast}\\mu_1=\\mu_1$ implies that $i_X$ is an isometry of $(X_1,\\overline{\\eta}^{(1)})$ as well. The fact that $s_X:X_1\\to X_0$ is a Riemannian submersion tells us that $ds_X(p)^\\ast: T_{s_X(p)}^\\ast X_0 \\to \\textnormal{ker}(ds_X(p))^\\circ$ is an isometry for all $p\\in X_1$ where $\\textnormal{ker}(ds_X(p))^\\circ$ denotes the annihilator of the vectors tangent to the fiber. Given $p\\in X_1$, covectors  $\\alpha,\\beta\\in T_{s_X(p)}^\\ast X_0$, and $g\\in G_1$ such that $s_G(g)=g_0$ we get the following chain of equalities:\n\n\n\n\n\n\n \n\\begin{eqnarray*}\n(\\overline{\\eta}^{(0)}_{s_X(p)})^\\ast (\\alpha,\\beta) &=& \\int_{G_0}(\\eta^{(0)}_{g_0s_X(p)})^\\ast ((\\theta^0_{g_0})^\\ast(\\alpha ),(\\theta^0_{g_0})^\\ast(\\beta ))d\\mu_0\\\\\n\\eqref{PushHaar}&=& \\int_{G_1}(\\eta^{(0)}_{s_G(g)s_X(p)})^\\ast ((\\theta^0_{s_G(g)})^\\ast(\\alpha ),(\\theta^0_{s_G(g)})^\\ast(\\beta ))d\\mu_1\\\\\n& = & \\int_{G_1}(\\eta^{(1)}_{gp})^\\ast (ds_X(gp)^\\ast((\\theta^0_{s_G(g)})^\\ast(\\alpha )),ds_X(gp)^\\ast((\\theta^0_{s_G(g)})^\\ast(\\beta )))d\\mu_1\\\\\n& = & \\int_{G_1}(\\eta^{(1)}_{gp})^\\ast ((\\theta^1_{g})^\\ast(ds_X(p)^\\ast(\\alpha )),(\\theta^1_{g})^\\ast(ds_X(p)^\\ast(\\beta )))d\\mu_1\\\\\n&= & (\\overline{\\eta}^{(1)}_{p})^\\ast (ds_X(p)^\\ast(\\alpha ),ds_X(p)^\\ast(\\beta )),\n\\end{eqnarray*}\nfrom which we conclude that $s_X$ is also Riemannian for the averaged metrics. With analogous computations we get a similar conclusion for $t_X:X_1\\to X_0$ so that the result follows as claimed.\n\\end{proof}\n\n\\begin{remark}\\label{Rmk4}\nIt is worth noticing that if we consider an $n$-metric on $X_n$ and take into account the simplicial approach described in Remark \\ref{Rmk2} then by applying similar averaging arguments as in the proof of Theorem \\ref{Existence2} it is possible to show the existence of another $n$-metric on $X_n$ that is invariant by the action of $G_n$ on $X_n$.  \n\\end{remark}\n\nThe \\emph{tangent groupoid} $TX_1\\rightrightarrows TX_0$ is obtained by applying the tangent functor at both manifolds of arrows and objects and all of its structural maps. If $S \\subset X_0$ is a saturated manifold (only an orbit for instance) then we can restrict the groupoid structure to $X_{S}=s_X^{-1}(S)=t_X^{-1}(S)$, thus obtaining a Lie subgroupoid $X_S\\rightrightarrows S$ of $X_1\\rightrightarrows X_0$. Furthermore, the Lie groupoid structure of $TX_1\\rightrightarrows TX_0$ can be restricted to define a new Lie groupoid $\\nu(X_S)\\rightrightarrows \\nu(S)$ having the property that all of its structural maps are fiberwise isomorphisms.\n\nThe notion of \\emph{linearization} around an orbit in the Lie groupoid setting is well known and it was a unsolvable problem for a long time when considering for instance only proper groupoids. We recommend the reader to visit \\cite{CS} with the aim of getting a quick introduction and contextualization about such a notion.\n\\begin{definition}\nSuppose that we have a Lie $2$-action of $G_1\\rightrightarrows G_0$ on $X_1\\rightrightarrows X_0$ and let us pick a $G_0$-invariant saturated submanifold $S\\subset X_0$. We say that $X_1\\rightrightarrows X_0$ is \\emph{equivariant weakly linearizable} at $S$ if there exist $G$-invariant Lie groupoid neighborhoods $\\widetilde{V}\\rightrightarrows V$ of $X_{S}\\rightrightarrows S$ in $\\nu(X_S)\\rightrightarrows \\nu(S)$ (seen as the zero section) and $\\widetilde{U}\\rightrightarrows U$ of $X_S\\rightrightarrows S$ in $X_1\\rightrightarrows X_0$, and a $G$-equivariant isomorphism of Lie groupoids $\\phi: (\\widetilde{V}\\rightrightarrows V)\\xrightarrow[]{\\cong} (\\widetilde{U}\\rightrightarrows U)$ which is the identity on $X_S\\rightrightarrows S$.\n\\end{definition}\nThe $G$-invariant property of the Lie groupoid neighborhood in $\\nu(X_S)\\rightrightarrows \\nu(S)$, used in the previous definition, makes sense because of the following facts. Let us assume in this case that we are given with a $2$-metric and that $G_2$ acts on $(X_2,\\eta^{(2)})$ by isometries (see Remark \\ref{Rmk2}). It is simple to check that every $2$-action $\\theta=(\\theta^1,\\theta^0)$ of $G_1\\rightrightarrows G_0$ on $X_1\\rightrightarrows X_0$ induces a $2$-action $T\\theta=(T\\theta^1,T\\theta^0)$ of $G_1\\rightrightarrows G_0$ on $TX_1 \\rightrightarrows TX_0$ by differentiating the actions $\\theta^1$ and $\\theta^0$. Let us pick a $G_0$-invariant saturated submanifold $S$ in $X_0$. This implies that $X_{S}$ is $G_1$-invariant and that $(X_{S})_2$ is $G_2$-invariant. If we respectively use $\\eta^{(2)}$, $\\eta^{(1)}$, and $\\eta^{(0)}$ to identify $\\nu(X_{S})_{2}\\cong \\nu((X_{S})_2)$ with $(T(X_{S})_2)^\\perp$,  $\\nu(X_{S})$ with $TX_S^\\perp$, and $\\nu(S)$ with $TS^\\perp$ then it follows that the $2$-action $T\\theta$ restrict to a well defined $2$-action $\\overline{T\\theta}$ of $G_1\\rightrightarrows G_0$ on $\\nu(X_{S})\\rightrightarrows \\nu(S)$ since $\\theta$ is an isometric $2$-action. Furthermore, the latter fact also implies that the exponential maps $\\textnormal{exp}^{(2)}$, $\\textnormal{exp}^{(1)}$, and $\\textnormal{exp}^{(0)}$ are equivariant local diffeomorphisms. As isometries preserve (horizontal) geodesics then by following the classical proofs of the equivariant tubular neighborhood theorem in \\cite[VI. Thm. 2.2]{B} and \\cite[Thm. 4.4]{IK,K} with the proof of weakly linearization theorem given in \\cite[Thm. 5.11]{dHF} we easily obtain:\n\n\\begin{proposition}[Equivariant weakly groupoid linearization]\\label{Lin1}\nLet $G_1\\rightrightarrows G_0$ be a Lie $2$-group, with $G_1$ compact, acting by isometries on $(X_1\\rightrightarrows X_0,\\eta)$. Then there exists an equivariant weakly linearization of $X_1\\rightrightarrows X_0$ around any $G_0$-invariant saturated submanifold $S$ in $X_0$.\n\\end{proposition}\nObserve that we may get a similar conclusion by requiring either that $G_1\\rightrightarrows G_0$ proper (or $s$-proper) and $G_0$ compact or else that $G_1\\rightrightarrows G_0$ acts properly.\n\n\n\\begin{definition}\\label{DefSlice}\nLet $\\theta$ be a $2$-action of $G_1\\rightrightarrows G_0$ on $X_1\\rightrightarrows X_0$ with $G_0$ acting freely. A \\emph{groupoid slice} at $x_0\\in X_0$ is defined to be a Lie subgroupoid $S_{1_{x_0}}\\rightrightarrows S_{x_0}$ of $X_1\\rightrightarrows X_0$ such that $S_{x_0}$ and $S_{1_{x_0}}$ are standard slices at $x_0$ and $1_{x_0}$, respectively.\n\\end{definition}\nThe following is a straightforward result.\n\\begin{lemma}\\label{IsoOrbit}\nTake any $x_0\\in X_0$. There are natural structures of:\n\\begin{itemize}\n\\item Lie 2-subgroup between the $G$-isotropy groups $\\textnormal{Iso}_{G_1}(1_{x_0})\\rightrightarrows \\textnormal{Iso}_{G_0}(x_0)$, and\n\\item Lie subgroupoid between the $G$-orbits $G_1\\cdot 1_{x_0}\\rightrightarrows G_0\\cdot x_0$.\n\\end{itemize}\n\\end{lemma}\nIt is important to mention that we asked $G_0$ to act freely in order to have a well defined groupoid composition on the Lie groupoids defined in the previous lemma. Let us state a version of the Slice Theorem in our context. Namely:\n\n\\begin{proposition}[Groupoid slice]\\label{SliceThm}\nIf $\\theta$ is a free and proper $2$-action of a Lie $2$-group $G_1\\rightrightarrows G_0$ on a proper groupoid $X_1\\rightrightarrows X_0$ then there exists a groupoid slice $S_{1_{x_0}}\\rightrightarrows S_{x_0}$ at each $x_0\\in X_0$.\n\n\\end{proposition}\n\\begin{proof}\nConsider the induced $2$-action of $\\textnormal{Iso}_{G_1}(1_{x_0})\\rightrightarrows \\textnormal{Iso}_{G_0}(x_0)$ on $X_1\\rightrightarrows X_0$. By the properness of $X_1\\rightrightarrows X_0$ and by applying Theorem \\ref{Existence2} together with Remark \\ref{Rmk4}, we may fix a $2$-metric on $X_2$ and use it to construct another $2$-metric on $X_2$ in such a way $\\textnormal{Iso}_{G_1}(1_{x_0})\\rightrightarrows \\textnormal{Iso}_{G_0}(x_0)$ acts isometrically on $X_1\\rightrightarrows X_0$. As in the classical case \\cite[Thm. 3.49]{AB}, we define $S_{x_0}$ by setting $S_{x_0}=\\exp^{(0)}_{x_0}(B_\\epsilon(0))$ where $B_\\epsilon(0)$ is an open ball of radius $\\epsilon>0$ around the origin in the normal space $\\nu_{x_0} (G_0 \\cdot x_0)$ to the $G_0$-orbit through $x_0$ (normal domain).\n\nAs $G_1\\cdot 1_{x_0}\\rightrightarrows G_0\\cdot x_0$ is a Lie subgroupoid of $X_1\\rightrightarrows X_0$ we have a well defined Lie subgroupoid $\\nu(G_1\\cdot 1_{x_0})\\rightrightarrows \\nu(G_0\\cdot x_0)$ of $TX_1\\rightrightarrows TX_0$ so that we may also consider the Lie groupoid $V_{B_\\epsilon(0)}\\rightrightarrows B_\\epsilon(0)$ where $V_{B_\\epsilon(0)}=\\overline{ds}_{1_{x_0}}^{-1}(B_\\epsilon(0))\\cap\\overline{dt}_{1_{x_0}}^{-1}(B_\\epsilon(0))$. By shrinking $B_\\epsilon(0)$ if necessary we may assume that $V_{B_\\epsilon(0)}$ is an open ball around the origin in the normal space $\\nu_{1_{x_0}} (G_1\\cdot 1_{x_0})$ to the $G_1$-orbit through $1_{x_0}$ on which $\\exp^{(1)}_{1_{x_0}}$ is well defined. Therefore, we now set $S_{1_{x_0}}= \\exp^{(1)}_{1_{x_0}}(V_{B_\\epsilon(0)})$. Hence, by arguing with similar arguments as those used to prove the multiplicative property of the exponential maps associated to a Riemannian $2$-metric in \\cite{dHF} together with the equivariant property these exponential maps have, we conclude that $S_{1_{x_0}}\\rightrightarrows S_{x_0}$ is the Lie subgroupoid of $X_1\\rightrightarrows X_0$ we are looking for.\n\\end{proof}\nThe previous result suggests that we may somehow prove a kind of $(G_1\\rightrightarrows G_0)$-equivariant ``Tubular Neighborhood Theorem'' for the $G$-orbit groupoid $G_1\\cdot 1_{x_0}\\rightrightarrows G_0\\cdot x_0$. To do so, we need to consider the following construction which has been defined in \\cite{HOV} (a particular case of such a construction can be found in \\cite[Lem. 9.1.2]{hsz}). \n\\begin{remark}\\label{AssociatedBundle}\nLet $\\pi: (P_1\\rightrightarrows P_0)\\to (X_1\\rightrightarrows X_0)$ be a  groupoid principal $2$-bundle with structural Lie $2$-group $G_1\\rightrightarrows G_0$. Assume that there exists a left $2$-action of $G_1\\rightrightarrows G_0$ over another Lie groupoid $F_1\\rightrightarrows F_0$. Given this data we can construct two associated fiber bundles $E_j:= P_j\\times_{G_j}F_j$ over $X_j$ for $j=0,1$. These are defined as the quotient spaces $(P_j\\times F_j)\/G_j$ with respect to the actions $g_j\\cdot (p_j,f_j)=(p_jg_j^{-1},g_jf_j)$, for all $g_j\\in G_j$, $p_j\\in P_j$ and $f_j\\in F_j$, together with projections $\\overline{\\pi}_j([p_j,f_j])=\\pi_j(p_j)$ onto $X_j$. From \\cite{HOV} we know that there exists a natural Lie groupoid structure $E_1\\rightrightarrows E_0$ for which the projection $\\overline{\\pi}: (E_1\\rightrightarrows E_0)\\to (X_1\\rightrightarrows X_0)$ becomes a Lie groupoid fibration. The source and target maps are the obvious ones, namely:\n$$s_E([p,f])=[s_P(p),s_F(f)]\\qquad\\textnormal{and}\\qquad t_E([p,f])=[t_P(p),t_F(f)],$$\nand the groupoid composition is defined as follows. If $s_E([p,f])=t_E([q,l])$ then there is $g_0\\in G_0$ such that $(s_P(p)g_0^{-1},g_0s_F(f))=(t_P(q),t_F(l))$. So, we set\n$$m_E([p,f],[q,l])=[(pg^{-1})\\ast q,(gf)\\ast l],$$\nfor some $g$ inside the $s_G$-fiber at $g_0$. The latter equality does not depend on the choice of $g\\in s_G^{-1}(g_0)$.\n\\end{remark}\n\nLet $\\theta$ be a free $2$-action of $G_1\\rightrightarrows G_0$ on $X_1\\rightrightarrows X_0$ and consider the classical $G$-invariant tubular neighborhoods $\\textnormal{Tub}(G_1\\cdot 1_{x_0})=\\theta^1(G_1,S_{1_{x_0}})$ and $\\textnormal{Tub}(G_0\\cdot x_0)=\\theta^0(G_0,S_{x_0})$ of $G_1\\cdot 1_{x_0}$ and $G_0\\cdot x_0$, respectively. It is clear that as consequence of Proposition \\ref{SliceThm} we have that $\\textnormal{Tub}(G_1\\cdot 1_{x_0})\\rightrightarrows \\textnormal{Tub}(G_0\\cdot x_0)$ is a Lie subgroupoid of $X_1\\rightrightarrows X_0$ and that the $2$-action $\\theta$ restricts well over it. Furthermore, $\\textnormal{Tub}(G_1\\cdot 1_{x_0})\\rightrightarrows \\textnormal{Tub}(G_0\\cdot x_0)$ determines an open Lie groupoid neighborhood of the $G$-orbit groupoid $G_1\\cdot 1_{x_0}\\rightrightarrows G_0\\cdot x_0$. \n\nFrom Lemma \\ref{IsoOrbit}, Proposition  \\ref{SliceThm}, the quotient construction described in Proposition \\ref{PQuotient}, and Remark \\ref{AssociatedBundle}, we easily deduce that:\n\\begin{lemma}\nThere exists a principal groupoid $2$-bundle $\\pi:(G_1\\rightrightarrows G_0)\\to (G_1\/\\textnormal{Iso}_{G_1}(1_{x_0})\\rightrightarrows G_0\/\\textnormal{Iso}_{G_0}(x_0))$ with structural Lie $2$-group $\\textnormal{Iso}_{G_1}(1_{x_0})\\rightrightarrows \\textnormal{Iso}_{G_0}(x_0)$, yielding an associated groupoid fibration $$(G_1\\times_{\\textnormal{Iso}_{G_1}(1_{x_0})}S_{1_{x_0}}\\rightrightarrows G_0\\times_{\\textnormal{Iso}_{G_0}(x_0)}S_{x_0})\\to (G_1\/\\textnormal{Iso}_{G_1}(1_{x_0})\\rightrightarrows G_0\/\\textnormal{Iso}_{G_0}(x_0)),$$ \nwith groupoid fiber $S_{1_{x_0}}\\rightrightarrows S_{x_0}$.\n\\end{lemma}\nWe are now in conditions to state our version of the Equivariant Tubular Neighborhood Theorem in this context. \n\\begin{theorem}[Equivariant groupoid tubular neighborhood]\\label{ETubular}\nSuppose that $\\theta$ is a free and proper $2$-action of a Lie $2$-group $G_1\\rightrightarrows G_0$ on a proper groupoid $X_1\\rightrightarrows X_0$. For every $x_0\\in X_0$ there exists a $(G_1\\rightrightarrows G_0)$-equivariant Lie groupoid isomorphism\n$$\\Psi: (G_1\\times_{\\textnormal{Iso}_{G_1}(1_{x_0})}S_{1_{x_0}}\\rightrightarrows G_0\\times_{\\textnormal{Iso}_{G_0}(x_0)}S_{x_0}) \\xrightarrow[]{\\cong} (\\textnormal{Tub}(G_1\\cdot 1_{x_0})\\rightrightarrows \\textnormal{Tub}(G_0\\cdot x_0)).$$\n\\end{theorem}\n\\begin{proof}\nThe left $2$-action of $G_1\\rightrightarrows G_0$ on $G_1\\times_{\\textnormal{Iso}_{G_1}(1_{x_0})}S_{1_{x_0}}\\rightrightarrows G_0\\times_{\\textnormal{Iso}_{G_0}(x_0)}S_{x_0}$ we will consider here is the one given by $\\overline{g_j}\\cdot [g_j,f_j]=[\\overline{g_j}g_j,f_j]$ for all $\\overline{g_j},g_j\\in G_j$ and $f_j\\in S_j$. Here $S_1=S_{1_{x_0}}$ and $S_0=S_{x_0}$.\nThe Lie groupoid isomorphism $\\Psi$ is defined as \n$$\\Psi^j([g_j,f_j])=\\theta^j(g_j,f_j).$$\nAs consequence of \\cite[Thm. 3.57]{AB} we only have to check that $\\Psi$ defines indeed a Lie groupoid morphism. By using the structural maps defined in Remark \\ref{AssociatedBundle} we have that\n$$(\\Psi^0\\circ s_E)([g,f])=\\Psi^0([s_G(g),s_X(f)])=s_G(g)s_X(f)=s_X(gf)=(s_X\\circ \\Psi^1)([g,f]).$$\nWe can similarly obtain that $\\Psi^0\\circ t_E=t_X\\circ \\Psi^1$. Moreover, by applying Formula \\eqref{MultAction} we get\n\\begin{eqnarray*}\n\\Psi^1([g,f]\\ast [h,l]) &=& \\Psi^1([(g\\overline{g}^{-1})\\ast h,(\\overline{g}f)\\ast l])=((g\\overline{g}^{-1})\\ast h)\\cdot((\\overline{g}f)\\ast l)\\\\\n&=& (gf)\\ast (hl)=\\Psi^1([g,f])\\ast \\Psi^1([h,l]).\n\\end{eqnarray*}\nHence, the result follows as desired.\n\\end{proof}\nWe finish this section by briefly commenting that it is possible to define a groupoid $G$-orbit type ``stratification'' for $X_1\\rightrightarrows X_0$. We say that $x_0$ and $y_0$ in $X_0$ have the same $G$-\\emph{orbit type} if there exists a Lie groupoid isomorphism $\\Phi$ between $G_1\\cdot 1_{x_0}\\rightrightarrows G_0\\cdot x_0$ and $G_1\\cdot 1_{y_0}\\rightrightarrows G_0\\cdot y_0$ such that $\\Phi^1$ is $G_1$-equivariant. This automatically implies that $\\Phi^0$ is $G_0$-equivariant. It is clear that this $G$-orbit type requirement defines an equivalent relation $\\sim$  on $X_0$ for which we denote by $M_{x_0}^\\sim$ the equivalent class at $x_0\\in X_0$ associated to such a relation. We claim that $M_{x_0}^\\sim$ is saturated in $X_0$, that is, $s_X^{-1}(M_{x_0}^\\sim)=t_X^{-1}(M_{x_0}^\\sim)$. If $p\\in s_X^{-1}(M_{x_0}^\\sim)$ then $s_X(p)\\sim x_0$ so that there is a $(G_1\\rightrightarrows G_0)$-equivariant isomorphism between $G_1\\cdot 1_{s_X(p)}\\rightrightarrows G_0\\cdot s_X(p)$ and $G_1\\cdot 1_{x_0}\\rightrightarrows G_0\\cdot x_0$. It is simple to check that $\\Phi_p$ defined by $\\Phi_p^1(g1_{t_X(p)})=g1_{s_X(p)}$ and $\\Phi_p^0(g_0t_X(p))=g_0s_X(p)$ defines another $(G_1\\rightrightarrows G_0)$-equivariant isomorphism between $G_1\\cdot 1_{t_X(p)}\\rightrightarrows G_0\\cdot t_X(p)$ and $G_1\\cdot 1_{s_X(p)}\\rightrightarrows G_0\\cdot s_X(p)$ so that by taking $\\Phi\\circ \\Phi_p$ we conclude that $t_X(p)\\sim x_0$. The other inclusion may be similarly verified. Thus, by setting $M_{1_{x_{0}}}^\\sim=s_X^{-1}(M_{x_0}^\\sim)=t_X^{-1}(M_{x_0}^\\sim)$ we obtain a collection of topological groupoids $\\lbrace M_{1_{x_{0}}}^\\sim \\rightrightarrows M_{x_0}^\\sim\\rbrace_{x_0\\in X_0}$ which somehow stratifies the Lie groupoid $X_1\\rightrightarrows X_0$. The previous observation suggests that by combining classical ideas from \\cite[s. 3.5]{AB} with some of the results obtained in this section it could be possible to show that each of the $M_{1_{x_{0}}}^\\sim \\rightrightarrows M_{x_0}^\\sim$ is a honest Lie subgroupoid. Nevertheless, the local $G$-orbit type notion from the classical case does not extended directly in our case. \\emph{We conjecture that our guess is true}.\n\\begin{comment}\n\\subsection{Linearization of equivariant groupoid fibrations}\nBy a \\emph{fibration} over $X_1\\rightrightarrows X_0$ we mean a Lie groupoid morphism $\\pi:(E_1\\rightrightarrows E_0)\\to (X_1\\rightrightarrows X_0)$, from another Lie groupoid $E_1\\rightrightarrows E_0$ such that:\n\\begin{enumerate}\n\t\\item the base map $\\pi_0:E_0\\to X_0$ is a surjective submersion, and \n\t\\item the map $\\tilde{\\pi}:E_1\\to X_1{}_s\\times_{\\pi_0} E_0$,  $e_1\\mapsto (\\pi_1(e_1),s_E(e_1))$, is a surjective submersion.\n\\end{enumerate}\nIt follows from (2) that the map $\\pi_1:E_1\\to X_1$ is also a surjective submersion. The \\emph{fiber} over a point $x_0\\in X_0$ to be the Lie subgroupoid:\n\\[ \\pi^{-1}(x_0):=(\\pi^{-1}_1(1_{x_0})\\rightrightarrows \\pi^{-1}_0(x_0))\\subset E.\\] \n\nSuppose that $G_1\\rightrightarrows G_0$ is a Lie $2$-group action on both $E_1\\rightrightarrows E_0$ and $X_1\\rightrightarrows X_0$. We say that the fibration $\\pi$ is $(G_1\\rightrightarrows G_0)$-equivariant if $\\pi_1$ is $G_1$-equivariant and $\\pi_0$ is $G_0$-equivariant.\n\nA \\emph{cleavage} of $\\pi$ is a smooth map $\\sigma:X_1 {}_{s_X}\\times_{\\pi_0}E_0\\to E_1$ which is section of the surjective submersion $\\tilde{\\pi}$. Note that we have an induced left action of the Lie group $G_1 {}_{s_G}\\times_{\\textnormal{id}}G_0$ (group structure induced by the direct product of groups) on $X_1 {}_{s_X}\\times_{\\pi_0}E_0$ given by $(g,g_0)(x,e_0)=(gx,g_0e_0)$. Therefore, it follows that $\\tilde{\\pi}(ge)=(g,s_G(g))\\tilde{\\pi}(e)$ for all $g\\in G_1$. With this in mind it is reasonable to say that a cleavage $\\sigma$ is equivariant if $\\sigma(gx,g_0e_0)=g\\sigma(x,e_0)$ for all $(g,g_0)\\in G_1 {}_{s_G}\\times_{\\textnormal{id}}G_0$. The cleavage is said to be \\emph{unital} if it preserves identities, meaning that \n$\\sigma(1_{\\pi_0(e_0)},e_0) = 1_{e_0}$, and is defined to be \\emph{flat} if it is closed under multiplication, namely $\\sigma(x\\ast y,e_0) =\n\\sigma(x,t_E(\\sigma(y,e_0)))\\ast \\sigma(y,e_0)$. A fibration admitting a flat unital cleavage is called a \\emph{split fibration}.\n\nWe want to adapt the results from \\cite{dHF2} about linearization of groupoid fibrations to the equivariant setting described above. Note that if $S$ is a $G_0$-invariant saturated submanifold in $X_0$ then $\\pi_0^{-1}(S)$ is so in $E_0$.\n\n\\begin{definition}\nWe say that $\\pi$ is \\emph{equivariant linearizable} around a $G_0$-invariant saturated submanifold $S\\subset X_0$ if there are equivariant weakly linearizations $\\widetilde{\\phi}$ of $E_1\\rightrightarrows E_0$ around $\\widetilde{S}=\\pi_0^{-1}(S)$ and $\\phi_X$ of $X_1\\rightrightarrows X_0$ around $S$ forming a commutative diagram\n$$\\xymatrix{\n\t\\nu(E_{\\widetilde{S}})\\supset \\widetilde{U} \\ar[r]^{\\widetilde{\\phi}} \\ar@{>}[d]_{\\overline{d\\pi}} & \\widetilde{V}\\subset E \\ar@{>}[d]^{\\pi}\\\\\n\t\\nu(X_S)\\supset U\\ar[r]_{\\phi}  & V\\subset X\n}.$$\n\\end{definition}\n\nFrom now on let us consider a Lie $2$-group $G_1\\rightrightarrows G_0$ with $G_1$ compact. Suppose that $\\pi$ is a Riemannian submersion, that is, both $E_1\\rightrightarrows E_0$ and $X_1\\rightrightarrows X_0$ can be equipped with $2$-metrics for which $\\pi^{(2)}$ becomes a Riemannian submersion. As consequence of Proposition \\ref{Existence2} we have that both $E_1\\rightrightarrows E_0$ and $X_1\\rightrightarrows X_0$ can be equipped with $(G_1\\rightrightarrows G_0)$-invariant $2$-metrics $\\widetilde{\\eta}$ and $\\eta$ in such a way $\\pi$ is still a Riemannian submersion.\n\\begin{lemma}\\label{Lin2}\nIf $\\pi$ is a Riemannian submersion then the equivariant exponential maps of $\\widetilde{\\eta}$ and $\\eta$ define an equivariant linearization of $\\pi$ around any $G_0$-invariant saturated submanifold $S$.\n\\end{lemma}\n\\begin{proof}\nAfter applying Proposition \\ref{Lin1} it follows that the proof of \\cite[Prop. 4.2.2]{dHF2} goes through in this context since $\\pi$ is formed by equivariant maps.\n\\end{proof}\nOn the one hand, let us consider the Lie groupoid determined by the homotopy fiber product $E'=X\\widetilde{\\times}_X E$ over $\\textnormal{id}_X$ and $\\pi$ which always exists since $\\pi$ is a groupoid fibration. The manifolds of objects and arrows of $E'$ are respectively given by\n$$E'_0=\\lbrace (x_0,k,e_0)\\in X_0\\times X_1\\times E_0:\\ \\pi_0(e_0)\\xleftarrow[]{\\it k}x_0\\rbrace,$$\nand \n$$E'_1=\\lbrace (x,k,e)\\in X_1\\times X_1\\times E_1:\\ s_X(p)=s_X(k),\\ \\pi_0(s_E(e))=t_X(k)\\rbrace.$$\nIts structural maps are defined in an obvious way by using those from $E$ and $X$. On the other hand, by using the Lie $2$-group $G_1\\rightrightarrows G_0$ we may obtain another Lie $2$-group determined by the homotopy fiber product $G'=G\\widetilde{\\times}_G G$ over the identity morphism $\\textnormal{id}_G$. As above, the manifolds of objects and arrows of $G'$ are respectively given by\n$$G'_0=\\lbrace (g_0,n,h_0)\\in G_0\\times G_1\\times G_0:\\ h_0\\xleftarrow[]{\\it n}g_0\\rbrace,$$\nand \n$$G'_1=\\lbrace (g,n,h)\\in G_1\\times G_1\\times G_1:\\ s_G(g)=s_G(n),\\ s_G(h)=t_G(n)\\rbrace.$$\nThe group structure in both $G'_0$ and $G'_1$ is the induced one from the direct product of groups so that the structural maps of $G'$ are clearly Lie group homomorphisms.\n\\begin{lemma}\\label{Lin3}\nThe $2$-actions of $G_1\\rightrightarrows G_0$ on both $E_1\\rightrightarrows E_0$ and $X_1\\rightrightarrows X_0$ induce a natural $2$-action of $G'_1\\rightrightarrows G'_0$ on $E'_1\\rightrightarrows E'_0$.\n\\end{lemma}\n\\begin{proof}\nIt is straightforward to check that the expressions\n$$(g_0,n,h_0)\\cdot (x_0,k,e_0)=(g_0x_0,nk,h_0e_0)\\quad\\textnormal{and}\\quad(g,n,h)\\cdot (x,k,e)=(gx,nk,he),$$\ndefine a $2$-action of $G'_1\\rightrightarrows G'_0$ on $E'_1\\rightrightarrows E'_0$ since $\\pi$ is $(G_1\\rightrightarrows G_0)$-equivariant. \n\\end{proof}\nThe morphism $\\pi$ has a canonical factorization \n$$\\xymatrix{\n\t& E' \\ar[dr]^{\\pi'}  & \\\\\n\tE \\ar@{>}[ur]^{\\iota}\\ar[r]_{\\pi} & & X\n}$$\nwhere the maps $\\iota$ and $\\pi'$ are respectively given, on objects and arrows, by $\\iota(e_0)=(\\pi_0(e_0), u_X(\\pi_0(e_0)),e_0)$, $\\iota(e)=(\\pi_1(e), u_X(\\pi_0(s_E(e)),e)$ and $\\pi'(x_0,k,e_0)=x_0$, $\\pi'(x,k,e)=x$.\n\nIf we apply the same factorization to $\\textnormal{id}_G$ we get a morphism of Lie $2$-groups $\\iota_G:G\\to G'$ defined as $\\iota_G(g_0)=(g_0,u_G(g_0),g_0)$ on objects and $\\iota_G(g)=(g,u_G(s_G(g)),g)$ on arrows. By using the $2$-action from Lemma \\ref{Lin3} we obtain that $\\iota$ is equivariant of type $\\iota_G$, meaning that $\\iota(g_0e_0)=\\iota_G(g_0)\\iota(e_0)$ and $\\iota(ge)=\\iota_G(g)\\iota(e)$ since $\\pi$ is $(G_1\\rightrightarrows G_0)$-equivariant. Analogously, $\\pi'((g_0,n,h_0)\\cdot (x_0,k,e_0))=\\textnormal{id}'_G(g_0,n,h_0)\\pi'(x_0,k,e_0)$ and $\\pi'((g,n,h)\\cdot (x,k,e))=\\textnormal{id}'_G(g,n,h)\\pi'(x,k,e)$.\n\n\nFrom \n\n\\textcolor{red}{I got stuck. I need to review the ideas I initially had}.\n\\end{comment}\n\n\n\n\n\\subsection{Orthogonal Lie 2-groups}\nLet $G_1\\rightrightarrows G_0$ be a Lie 2-group. Consider the pairs $L=(L^1,L^0)$ and $R=(R^1,R^0)$ where $L^j$ and $R^j$ for $j=0,1$ are respectively the actions of $G_j$ on itself determined by left and right multiplications. It is simple to check that as consequence of Identity \\eqref{MultiProduct} and the fact that the structural maps of $G_1\\rightrightarrows G_0$ are Lie group homomorphisms it follows that $L$ and $R$ determine left $2$-actions of $G_1\\rightrightarrows G_0$ on itself.\n\\begin{definition}\nA Lie $2$-group $G_1\\rightrightarrows G_0$ is said to be \\emph{orthogonal} if it may be equipped with a $1$-metric for which both $L$ and $R$ are isometric $2$-actions. Such a $1$-metric will be called \\emph{bi-invariant}.\n\\end{definition}\nWe will think of $1$-metrics on a Lie $2$-algebra $\\mathfrak{g}_1\\rightrightarrows \\mathfrak{g}_0$ as pairs of inner products $\\langle \\cdot,\\cdot\\rangle=(\\langle \\cdot,\\cdot\\rangle^{(1)},\\langle \\cdot,\\cdot\\rangle^{(0)})$ verifying the required conditions of $1$-metric. Let $G_1\\rightrightarrows G_0$ be a Lie 2-group with respective Lie 2-algebra $\\mathfrak{g}_1\\rightrightarrows \\mathfrak{g}_0$. We denote by $\\textnormal{Ad}=(\\textnormal{Ad}^1,\\textnormal{Ad}^0)$ the $2$-action of $G_1\\rightrightarrows G_0$ on $\\mathfrak{g}_1\\rightrightarrows \\mathfrak{g}_0$ determined by the adjoint actions $\\textnormal{Ad}^j$ of $G_j$ on $\\mathfrak{g}_j$ for $j=0,1$. This $2$-action will be called \\emph{adjoint $2$-action} of $G_1\\rightrightarrows G_0$. \n\n\\begin{proposition}\\label{Bi-invariant1}\nA Lie $2$-group $G_1\\rightrightarrows G_0$ is orthogonal if and only if there exists a $1$-metric $\\langle \\cdot,\\cdot\\rangle$ on its Lie algebra $\\mathfrak{g}_1\\rightrightarrows \\mathfrak{g}_0$ for which the adjoint $2$-action is by linear isometries.\n\\end{proposition}\n\\begin{proof}\nIt is clear that if $\\eta$ is a bi-invariant $1$-metric then $\\eta_e=(\\eta^1_{e_1},\\eta^0_{e_0})$, where $e_j$ is the identity element in $G_j$, defines a $1$-metric on $\\mathfrak{g}_1\\rightrightarrows \\mathfrak{g}_0$ for which the adjoint $2$-action is by linear isometries. Conversely, given such a $\\langle \\cdot,\\cdot\\rangle$, we define $\\eta$ on $G_1\\rightrightarrows G_0$ by setting\n$$\\eta^{(1)}_g(v,w)=\\langle d(L^1_{g^{-1}})_g(v),d(L^1_{g^{-1}})_g(w)\\rangle^{(1)}.$$\nThe metric $\\eta^{(0)}$ has a similar defining formula but using $L^0$ and $\\langle \\cdot,\\cdot\\rangle^{(0)}$ instead of $L^1$ and $\\langle \\cdot,\\cdot\\rangle^{(1)}$. It is clear that $\\eta^{(1)}$ is bi-invariant. Therefore, it remains to prove that $\\eta$ defines indeed a $1$-metric. Firstly, for $g\\in G_1$ and $v,w\\in T_g G_1$ we have\n\\begin{eqnarray*}\ni^\\ast_g\\eta^{(1)}(v,w) & = & \\langle d(L^1_{i(g)^{-1}})_{i(g)}(di_g(v)),d(L^1_{i(g)^{-1}})_{i(g)}(di_g(w))\\rangle^{(1)}\\\\\n&=&  \\langle d(L^1_{i(g^{-1})}\\circ i)_g(v),d(L^1_{i(g^{-1})}\\circ i)_g(w)\\rangle^{(1)}\\\\\n& = & \\langle d(i\\circ L^1_{g^{-1}})_g(v),d(i\\circ L^1_{g^{-1}})_g(w)\\rangle^{(1)}\\\\\n&=&\\langle di_{e_1}(d(L^1_{g^{-1}})_g(v)), di_{e_1}(d(L^1_{g^{-1}})_g(v))\\rangle^{(1)}=\\eta^{(1)}_g(v,w).\n\\end{eqnarray*}\nSecondly, let $g_0\\in G_0$ and $v,w\\in T_{g_0}G_0$. It is clear that there are $\\tilde{v},\\tilde{w}\\in \\textnormal{ker}(ds(g))^{\\perp_{\\eta^{(1)}}}$ with $s(g)=g_0$ such that $ds_g(\\tilde{v})=v$ and $ds_g(\\tilde{w})=w$. Thus\n\\begin{eqnarray*}\n(s_\\ast \\eta^{(1)})_{g_0}(v,w) & = & \\eta^{(1)}_g(\\tilde{v},\\tilde{w}) = \\langle d(L^1_{g^{-1}})_g(\\tilde{v}),d(L^1_{g^{-1}})_g(\\tilde{w})\\rangle^{(1)}\\\\\n& = & \\langle ds_{e_1}(d(L^1_{g^{-1}})_g(\\tilde{v})), ds_{e_1}(d(L^1_{g^{-1}})_g(\\tilde{w}))\\rangle^{(0)}\\\\\n&=& \\langle d(s\\circ L^1_{g^{-1}})_g(\\tilde{v}),d(i\\circ L^1_{g^{-1}})_g(\\tilde{w})\\rangle^{(0)}\\\\\n&=& \\langle d(L^0_{g_0^{-1}}\\circ s)_g(\\tilde{v}),d(L^0_{g_0^{-1}}\\circ s)_g(\\tilde{w})\\rangle^{(0)}\\\\\n&=& \\langle d(L^0_{g_0^{-1}})_{g_0}(v),d(L^0_{g_0^{-1}})_{g_0}(v)\\rangle^{(0)}=\\eta^{(0)}_{g_0}(v,w).\n\\end{eqnarray*}\nAnalogously, $t_\\ast \\eta^{(1)}=\\eta^{(0)}$. So, the result follows.\n\\end{proof}\nFrom now on we assume that the Lie groups we are working with are connected. It is well known that bi-invariant metrics on a Lie group are in one-to-one correspondence with inner products on its Lie algebra for which the adjoint representation determines infinitesimal isometries \\cite{Me,Mi}. A Lie $2$-algebra $\\mathfrak{g}_1\\rightrightarrows \\mathfrak{g}_0$ is said to be \\emph{orthogonal} if it admits a $1$-metric $\\langle \\cdot,\\cdot\\rangle=(\\langle \\cdot,\\cdot\\rangle^{(1)},\\langle \\cdot,\\cdot\\rangle^{(0)})$ for which the adjoint representation $\\textnormal{ad}^1:\\mathfrak{g}_1\\to \\textnormal{Der}(\\mathfrak{g}_1)$ acts by infinitesimal isometries on $(\\mathfrak{g}_1,\\langle \\cdot,\\cdot\\rangle^{(1)})$. Therefore, as consequence of Proposition \\ref{Bi-invariant1} and \\cite[Lem. 7.2]{Mi} we get:\n\\begin{corollary}\\label{MilnorCharacterization}\nA Lie $2$-group is orthogonal if and only if its Lie algebra $\\mathfrak{g}_1\\rightrightarrows \\mathfrak{g}_0$ is orthogonal.\n\\end{corollary}\nFrom Theorem \\ref{Existence2} we get that:\n\\begin{corollary}\nEvery proper Lie $2$-group with either $G_1$ or $G_0$ compact can be endowed with a bi-invariant $1$-metric.\n\\end{corollary}\n\n\\begin{remark}\nSuppose that we have a crossed module of Lie algebras $(\\mathfrak{g},\\mathfrak{h},\\partial,\\mathcal{L})$ which is $\\mathcal{L}$-orthogonal in the sense of \\cite{Ba,FMM}. That is, there exist inner products $\\langle \\cdot,\\cdot\\rangle_\\mathfrak{g}$ and $\\langle \\cdot,\\cdot\\rangle_\\mathfrak{h}$ such that $\\mathcal{L}$ acts by infinitesimal isometries on $(\\mathfrak{h},\\langle \\cdot,\\cdot\\rangle_\\mathfrak{h})$ and the adjoint representation of $\\mathfrak{g}$ acts by infinitesimal isometries on $(\\mathfrak{g},\\langle \\cdot,\\cdot\\rangle_\\mathfrak{g})$. Note that this directly implies that the adjoint representation of $\\mathfrak{h}$ acts by infinitesimal isometries on $(\\mathfrak{h},\\langle \\cdot,\\cdot\\rangle_\\mathfrak{h})$. Recall that the associated Lie $2$-algebra constructed with the crossed module data has $\\mathfrak{g}_1=\\mathfrak{h}\\rtimes \\mathfrak{g}$ with Lie algebra structure provided by the semi-direct product with respect to $\\mathcal{L}$. Therefore, a straightforward computation allows us to conclude that the adjoint representation of $\\mathfrak{g}_1$ acts by infinitesimal isometries with respect to $\\langle \\cdot,\\cdot\\rangle_\\mathfrak{h}+\\langle \\cdot,\\cdot\\rangle_\\mathfrak{g}$ if and only if $\\mathcal{L}=0$. As consequence, there is no a canonical correspondence between our notion of orthogonal Lie $2$-algebras and the notion of $\\mathcal{L}$-orthogonal crossed module of Lie algebras which is known in the literature.\n\\end{remark}\nGiven an orthogonal Lie $2$-algebra $\\mathfrak{g}_1\\rightrightarrows \\mathfrak{g}_0$ we may split $\\mathfrak{g}_1$ as a direct sum of ideals $\\mathfrak{g}_1=\\mathfrak{h}\\oplus \\mathfrak{h}^{\\perp_1}$ where $\\mathfrak{h}=\\textnormal{ker}(s)$. As the unit map $u$ is a canonical bisection we would expect that $u(x)\\in \\mathfrak{h}^{\\perp_1}$ for all $x\\in \\mathfrak{g}_0$ but, however, this is not true in general unless we assume ``non-canonical\" identifications. We say that an orthogonal Lie $2$-algebra is \\emph{trivial} if the latter condition holds true. This notion comes up by the following simple result.\n\n\\begin{lemma}\\label{0-orthogonal}\nIf $\\mathfrak{g}_1\\rightrightarrows \\mathfrak{g}_0$ is a trivial orthogonal Lie $2$-algebra then $\\textnormal{ad}^1_{u(x)}|_\\mathfrak{h}=0$ for all $x\\in \\mathfrak{g}_0$. In consequence, $\\mathfrak{h}$ is abelian and $\\textnormal{im}(t|_\\mathfrak{h})\\subseteq \\mathfrak{z}(\\mathfrak{g}_0)$.\n\\end{lemma}\n\\begin{proof}\nOn the one hand, note that for any $y\\in \\mathfrak{h}$ and $z\\in \\mathfrak{g}$ one gets\n$$\\langle \\textnormal{ad}^1_{u(x)}(y),z\\rangle^{(1)}=\\langle u(x),\\textnormal{ad}^1_{y}(z)\\rangle^{(1)}=0,$$\nsince $\\mathfrak{h}$ is an ideal. Thus, as $\\langle \\cdot,\\cdot\\rangle^{(1)}$ is nondegenerate we get that $\\textnormal{ad}^1_{u(x)}|_\\mathfrak{h}=0$ for all $x\\in \\mathfrak{g}_0$. On the other hand, for all $y,y'\\in \\mathfrak{h}$ and $x\\in \\mathfrak{g}$ it follows that $[y,y']=\\textnormal{ad}^1_{u(t(y))}(y')=0$ and $0=t([u(x),y])=[x,t(y)]$ since $t$ is a Lie algebra homomorphism.\n\\end{proof}\n\nMotivated by the previous result we set our notion of $0$-orthogonal crossed module. Namely:\n\\begin{definition}\nA crossed module of Lie algebras $(\\mathfrak{g},\\mathfrak{h},\\partial,0)$ is called 0-\\emph{orthogonal} if there exist inner products $\\langle \\cdot,\\cdot\\rangle_\\mathfrak{g}$ and $\\langle \\cdot,\\cdot\\rangle_\\mathfrak{h}$ such that the adjoint representation of $\\mathfrak{g}$ acts by infinitesimal isometries on $(\\mathfrak{g},\\langle \\cdot,\\cdot\\rangle_\\mathfrak{g})$ and the fibers of the canonical projection $\\pi:\\mathfrak{g}\\to \\mathfrak{g}\/\\textnormal{im}(\\partial)$ are equidistant.\n\\end{definition}\nNote that directly from the definition we get that $\\mathfrak{h}$ is abelian and $\\textnormal{im}(\\partial)\\subseteq \\mathfrak{z}(\\mathfrak{g})$. With this in mind we have:\n\\begin{proposition}\nThere exists a one-to-one correspondence between trivial orthogonal Lie $2$-algebras and $0$-orthogonal crossed modules of Lie algebras.\n\\end{proposition}\n\\begin{proof}\nIf $\\mathfrak{g}_1\\rightrightarrows \\mathfrak{g}_0$ is a trivial orthogonal Lie $2$-algebra then the result follows from Lemma \\ref{0-orthogonal} after setting $\\langle \\cdot,\\cdot\\rangle_\\mathfrak{g}=\\langle \\cdot,\\cdot\\rangle^{(0)}$ and $\\langle \\cdot,\\cdot\\rangle_\\mathfrak{h}=\\langle \\cdot,\\cdot\\rangle^{(1)}|_{\\mathfrak{h}\\times \\mathfrak{h}}$. Conversely, let us consider a $0$-orthogonal crossed module $(\\mathfrak{g},\\mathfrak{h},\\partial,0)$ and set $\\langle \\cdot,\\cdot\\rangle^{(1)}=\\langle \\cdot,\\cdot\\rangle_\\mathfrak{h}+\\langle \\cdot,\\cdot\\rangle_\\mathfrak{g}$ and $\\langle \\cdot,\\cdot\\rangle^{(0)}=\\langle \\cdot,\\cdot\\rangle_\\mathfrak{g}$. From the construction of the Lie $2$-algebra associated to $(\\mathfrak{g},\\mathfrak{h},\\partial,0)$ we easily see that $s$ is a linear Riemannian submersion. Now, on $\\mathfrak{g}_1=\\mathfrak{h}\\rtimes \\mathfrak{g}$ we get\n$$\\langle i(x,y),i(x',y')\\rangle^{(1)} = \\langle x,x'\\rangle_\\mathfrak{h}+\\langle y+\\partial(x),y'+\\partial(x')\\rangle_\\mathfrak{g}=\\langle x,x'\\rangle_\\mathfrak{h}+\\langle y,y'\\rangle_\\mathfrak{g}=\\langle (x,y),(x',y')\\rangle^{(1)},$$\nsince the fibers of the canonical projection $\\pi:\\mathfrak{g}\\to \\mathfrak{g}\/\\textnormal{im}(\\partial)$ are equidistant. As $t=s\\circ i$ it follows that $t$ is also a linear Riemannian submersion.\n\nThe Lie bracket on $\\mathfrak{g}_1$ is explicitly given as\n$$[(x,y),(x',y')]_0=(0,[y,y']_\\mathfrak{g}).$$\nHence, the adjoint representation of $\\mathfrak{g}_1$ acts by infinitesimal isometries with respect to $\\langle \\cdot,\\cdot\\rangle_\\mathfrak{h}+\\langle \\cdot,\\cdot\\rangle_\\mathfrak{g}$ since adjoint representation of $\\mathfrak{g}$ acts by infinitesimal isometries with respect to $\\langle \\cdot,\\cdot\\rangle_\\mathfrak{g}$.\n\\end{proof}\n\n\\begin{comment}\n\\textcolor{green}{Problem:} \\textcolor{blue}{If possible, interpret \\cite[Lem. 7.5]{Mi} with the aim of giving a complete characterization of orthogonal Lie $2$-groups}.\n\n\n\n\\textcolor{green}{Problem:} \\textcolor{blue}{Try to extend the construction called double orthogonal extension introduced in \\cite{MPh} by using the cohomology for Lie $2$-algebras of either \\cite{A} or \\cite{LSZ}}.\n\\end{comment}\n\n\n\\subsection{Some examples and applications}\nIt this short subsection we exhibit some toy examples and interesting applications in which isometric Lie $2$-actions naturally appear.\n\n\\begin{example}\nClassical isometric actions of Lie groups $G$ on Riemannian manifolds $M$ are recovered from isometric $2$-actions of unit Lie $2$-groups $G \\rightrightarrows G$ acting upon unit groupoids $M \\rightrightarrows M$.\n\\end{example}\n\n\\begin{example}\nLet $(X_1\\rightrightarrows X_0,\\eta)$ be a Riemannian groupoid and $(\\xi,v)$ be a complete multiplicative Killing vector field on $X_1\\rightrightarrows X_0$. Here we consider $\\xi$ a Killing vector field on $(X_1,\\eta^{(1)})$ and $v$ a Killing vector field on $(X_0,\\eta^{(0)})$ (as consequence of Lemma \\ref{Rmk1} it is actually enough to ask that $\\xi$ is Killing). From \\cite{MX} we know that the pair of flows defined by $(\\xi,v)$ determine global automorphisms on $X_1\\rightrightarrows X_0$ so that we get a well defined isometric $2$-action of $\\mathbb{R}\\rightrightarrows \\mathbb{R}$ on $(X_1\\rightrightarrows X_0,\\eta)$. In particular, the flow of the multiplicative vector field $((1_\\xi)_{X_1},\\xi_{X_0})$ formed by the fundamental vector fields of an isometric $2$-action determines another isometric $2$-action.\n\\end{example}\n\n\\begin{example}\\label{ExampleOrthogonal}\nLet $G$ be an orthogonal Lie group and $H\\leq G$ be a normal Lie subgroup. It is clear that $H$ acts on $G$ by left multiplication, leading to the action Lie groupoid $H\\times G\\rightrightarrows G$. Note that its space of arrows has a group structure, namely the semi-direct product by the\nconjugation action $C_g(h)=ghg^{-1}$ of $G$ on $H$ so that we get a well defined Lie $2$-group. More importantly, by applying the gauge trick construction behind Proposition 4.7 and Example 4.9 in \\cite{dHF} we conclude that it is possible to cook up a $1$-metric on $H\\rtimes G\\rightrightarrows G$ made out from the initial bi-invariant metric on $G$ in such a way it becomes an orthogonal Lie $2$-group.\n\\end{example}\n\n\n\n\n\\begin{example}\nLet $(M,\\mathcal{F})$ be a regular Riemannian foliation and consider a free and proper isometric foliated action $G\\times (M,\\mathcal{F})\\to (M,\\mathcal{F})$ of a Lie group $G$. From \\cite[Thm. 3.7]{GZ} it is known that the Lie $2$-group $G\\rtimes G\\rightrightarrows G$, as defined in Example \\ref{ExampleOrthogonal}, determines a canonical $2$-action on the holonomy groupoid $\\textnormal{Hol}(M,\\mathcal{F})\\rightrightarrows M$ which  extends the given action of $G$ on $M$. The Riemannian metric on $M$ completely determines a $0$-metric on $\\textnormal{Hol}(M,\\mathcal{F})\\rightrightarrows M$ which can be extended to a $1$-metric \\cite[Ex. 3.12]{dHF}. Thus, the fact that $G$ acts on $M$ isometrically implies that the extended $2$-action of $G\\rtimes G\\rightrightarrows G$ on $\\textnormal{Hol}(M,\\mathcal{F})\\rightrightarrows M$ is by isometries.\n\\end{example}\n\n\nIn the next example we will consider the notion of multiplicative $2$-connection $\\omega=(\\omega^1,\\omega^2)$ on a groupoid principal $2$-bundle $\\pi:(P_1\\rightrightarrows P_0)\\to (X_1\\rightrightarrows X_0)$ with structural Lie $2$-group $G_1\\rightrightarrows G_0$ as defined for instance in \\cite{HOV} (see also \\cite{CCK}). \n\\begin{example}[Principal groupoid warping]\nLet us prove that if $(X_1\\rightrightarrows X_0,\\eta)$ is a Riemannian groupoid and $G_1\\rightrightarrows G_0$ is orthogonal then there exists a $1$-metric $\\overline{\\eta}$ on $P_1\\rightrightarrows P_0$ for which the $2$-action of $G_1\\rightrightarrows G_0$ is isometric and such that $\\pi=(\\pi_1,\\pi_0):(P_1\\rightrightarrows P_0)\\to (X_1\\rightrightarrows X_0)$ is a Riemannian groupoid submersion. Consider the associated $1$-metric $\\langle \\cdot,\\cdot\\rangle$ on the Lie $2$-algebra $\\mathfrak{g}_1\\rightrightarrows \\mathfrak{g}_0$ of $G_1\\rightrightarrows G_0$ and define \n\t$$\\overline{\\eta}^{(1)}(v,w)=\\eta^{(1)}(d\\pi_1(v),d\\pi_1(w))+\\langle \\omega^1(v),\\omega^1(w)\\rangle^{(1)}.$$\n\tThe metric $\\overline{\\eta}^{(0)}$ is similarly defined by using instead $\\eta^{(0)}$, $\\pi_0$, $\\langle \\cdot,\\cdot\\rangle^{(0)}$, and $\\omega^0$. It is simple to check that this expression yields a well defined right $G_j$-invariant metric on $P_j$ for which $\\pi_j:P_j\\to X_j$ becomes a Riemannian submersion ($j=0,1$). This is because $\\pi_j$ is constant along the action orbits, $\\omega^j$ is of $\\textnormal{Ad}^j$-invariant type, and $\\textnormal{ker}(d\\pi_j)^{\\perp_{\\overline{\\eta}^{(j)}}}=\\textnormal{ker}(\\omega^j)$. Let us verify that $\\overline{\\eta}=(\\overline{\\eta}^{(1)},\\overline{\\eta}^{(0)})$ determines a $1$-metric on $P_1\\rightrightarrows P_0$. Note that, by abusing a little on the notation, we may rewrite $\\overline{\\eta}^{(j)}$ in a simpler way as $\\overline{\\eta}^{(j)}=(\\pi_j)^\\ast\\eta^{(j)}+(\\omega^j)^\\ast \\langle \\cdot,\\cdot\\rangle^{(j)}$. Recall that $\\pi$ and $\\omega:(TP_1 \\rightrightarrows TP_0)\\to (\\mathfrak{g}_1\\rightrightarrows \\mathfrak{g}_0)$ are Lie groupoid morphisms. So, on the one hand we get\n\t\\begin{eqnarray*}\n\t\t(i_P)^\\ast \\overline{\\eta}^{(1)} & = & (\\pi_1\\circ i_P)^\\ast\\eta^{(1)}+(\\omega^1\\circ i_P)^\\ast \\langle \\cdot,\\cdot\\rangle^{(1)}=(i_X\\circ \\pi_1)^\\ast\\eta^{(1)}+(d(i_G)_{e_1}\\circ \\omega^1)^\\ast \\langle \\cdot,\\cdot\\rangle^{(1)}\\\\\n\t\t&=& (\\pi_1)^\\ast\\eta^{(1)}+(\\omega^1)^\\ast \\langle \\cdot,\\cdot\\rangle^{(1)}=\\overline{\\eta}^{(1)}.\n\t\\end{eqnarray*}\n\tOn the other hand, if $v\\in \\textnormal{ker}(d(s_P)_{p})^{\\perp_{\\overline{\\eta}^{(1)}}}$ then it is simple to verify that the identities $\\pi_0\\circ s_P=s_X\\circ \\pi_1$ and $\\omega^0\\circ s_P=d(s_G)_{e_1}\\circ \\omega^1$ imply that $d(\\pi_1)_p(v)\\in \\textnormal{ker}(d(s_X)_{\\pi_1(p)})^{\\perp_{\\eta^{(1)}}}$ and $\\omega^1(v)\\in \\textnormal{ker}(d(s_G)_{e_1})^{\\perp_{\\langle \\cdot,\\cdot\\rangle^{(1)}}}$. Let us pick $v_1,v_2\\in \\textnormal{ker}(d(s_P)_{p})^{\\perp_{\\overline{\\eta}^{(1)}}}$. Thus,\n\t\\begin{eqnarray*}\n\t\t\\overline{\\eta}^{(0)}_{s_P(p)}(ds_P(p)(v_1),ds_P(p)(v_2)) & = & \\eta^{(0)}_{\\pi_0(s_P(p))}(d(\\pi_0)_{s_P(p)}(ds_P(p)(v_1)),d(\\pi_0)_{s_P(p)}(ds_P(p)(v_2)))\\\\\n\t\t& + & \\langle \\omega^0(ds_P(p)(v_1)),\\omega^0(ds_P(p)(v_2))\\rangle^{(0)}\\\\\n\t\t& = & \\eta^{(0)}_{s_X(\\pi_1(p))}(d(s_X)_{\\pi_1(p)}(d\\pi_1(p)(v_1)),d(s_X)_{\\pi_1(p)}(d\\pi_1(p)(v_1)))\\\\\n\t\t& + & \\langle d(s_G)_{e_1}(\\omega^1(v_1)),d(s_G)_{e_1}(\\omega^1(v_2))\\rangle^{(0)}\\\\\n\t\t& = & \\eta^{(1)}_{\\pi_1(p)}(d\\pi_1(p)(v_1),d\\pi_1(p)(v_1)) +  \\langle \\omega^1(v_1),\\omega^1(v_2)\\rangle^{(0)}\\\\\n\t\t& = & \\overline{\\eta}^{(1)}(v_1,v_2).\n\t\\end{eqnarray*}\n\tAnalogously, it follows that $(t_P)_\\ast \\overline{\\eta}^{(1)}=\\overline{\\eta}^{(0)}$. Hence, we have shown that $(P_1\\rightrightarrows P_0,\\overline{\\eta})$ is a Riemannian groupoid for which $G_1\\rightrightarrows G_0$ acts isometrically and $\\pi$ is a Riemannian submersion of groupoids, as claimed.\n\\end{example}\nNext example comes motivated by a beautiful construction known in the literature as \\emph{Cheeger deformation}. The classical construction can be found for instance in \\cite[s. 6.1]{AB}.\n\\begin{example}[Cheeger groupoid deformation]\n\tSuppose that $(G_1\\rightrightarrows G_0, Q)$ is an orthogonal Lie 2-group, with $G_1$ compact, acting isometrically on a Riemannian groupoid $(X_1\\rightrightarrows X_0,\\eta)$. On the Lie groupoid product $X_1\\times G_1\\rightrightarrows X_0\\times G_0$ we can consider the 1-metric $\\eta\\oplus \\frac{1}{\\tau}Q$.\n\t\n\tThere is a natural free $2$-action of $G_1\\rightrightarrows G_0$ on $X_1\\times G_1\\rightrightarrows X_0\\times G_0$ where the $G_j$-actions, $j=1,0$, on $X_j\\times G_j$ are given by\n\t\n\t\\begin{equation}\\label{CheegerD}\n\t\th_j\\cdot(p_j,g_j)=(h_jp_j,h_jg_j),\\qquad p_j\\in X_j,\\ h_j,g_j\\in G_j.\n\t\\end{equation} \n\t\n\tWe claim that the quotient groupoid $\\frac{X_1\\times G_1}{G_1}\\rightrightarrows\\frac{X_0\\times G_0}{G_0}$ determined by the previous actions is isomorphic to $X_1\\rightrightarrows X_0$. Let us consider the groupoid principal $2$-bundle $G_1\\rightrightarrows G_0$ over the point groupoid $\\lbrace e_1\\rbrace\\rightrightarrows\\lbrace e_0\\rbrace$ with structural Lie $2$-group $G_1\\rightrightarrows G_0$. By applying Remark \\ref{AssociatedBundle} we know that by using the $2$-action of $G_1\\rightrightarrows G_0$ on $X_1\\rightrightarrows X_0$ we can construct the associated Lie groupoid bundle $(X_1\\times_{G_1}G_1\\rightrightarrows X_0\\times_{G_0}G_0)\\to (\\lbrace e_1\\rbrace\\rightrightarrows\\lbrace e_0\\rbrace)$. It is important to notice that $X_1\\times_{G_1}G_1\\rightrightarrows X_0\\times_{G_0}G_0$ is precisely the quotient groupoid $\\frac{X_1\\times G_1}{G_1}\\rightrightarrows\\frac{X_0\\times G_0}{G_0}$. Therefore, as the new Lie groupoid bundle has groupoid fiber $X_1\\rightrightarrows X_0$ and base groupoid $\\lbrace e_1\\rbrace\\rightrightarrows\\lbrace e_0\\rbrace$ then we have the desired isomorphism. Under this identification the canonical groupoid projection $\\pi=(X_1\\times G_1\\rightrightarrows X_0\\times G_0)\\to (X_1\\rightrightarrows X_0)$ is formed by the maps $\\pi_j(p_j,g_j)=g_j^{-1}p_j$.\n\t\n\tObserve that the $2$-action \\eqref{CheegerD} is also isometric. Thus, as consequence of Proposition \\ref{PQuotient} there is a unique 1-metric $\\eta_\\tau$ on $X_1\\rightrightarrows X_0$ making of the projection $\\pi=(X_1\\times G_1\\rightrightarrows X_0\\times G_0,\\eta\\oplus \\frac{1}{\\tau}Q)\\to (X_1\\rightrightarrows X_0,\\eta_\\tau)$ a Riemannian groupoid submersion. \n\t\n\tIt is important to point out that by construction $\\eta_\\tau \\mapsto \\eta$ when $\\tau$ goes to $\\infty$ and the $2$-action of $G_1\\rightrightarrows G_0$ on $(X_1\\rightrightarrows X_0,\\eta_\\tau)$ is still isometric when $\\tau>0$. More importantly, the 1-parameter family of $1$-metrics $\\eta_\\tau$ on $X_1\\rightrightarrows X_0$ varies smoothly with $\\tau$ and extends smoothly to $\\tau = 0$ with $\\eta_0 = \\eta$. Hence, $\\eta_\\tau$ with $\\tau\\geq  0$ is a deformation of $\\eta$ by other $(G_1\\rightrightarrows G_0)$-invariant metrics on  $X_1\\rightrightarrows X_0$ which we call \\emph{Cheeger groupoid deformation} of $\\eta$. The reader is recommended to visit \\cite[s. 6,1]{AB} for specific details about the last assertions in the classical case. \n\\end{example}\n\n\\begin{remark}\nIt is worth mentioning that the notion of isometric 2-action we have introduced in this section was used in \\cite{OV} as one of the fundamental ingredients needed to construct an equivariant double Morse-Bott complex which computes the equivariant cohomology of a $2$-action as defined in \\cite{OBT}. \n\\end{remark}\n\n\n\\section{Isometries and multiplicative Killing vector fields}\\label{S:4}\nThe aim of this section is to bring a description of the group of isometries associated to a $0$-metric on a Lie groupoid. This will give rise to a notion of geometric Killing vector field on a quotient Riemannian stack. We start by describing what would be our attempt to set the \\emph{diffeomorphism group} of a differentiable stack. Some of the references we shall be following throughout are \\cite{Ma,OW} and \\cite[App. D]{An}. A \\emph{bisection} of a Lie groupoid $X_1\\rightrightarrows X_0$ is a smooth map $\\sigma:X_0\\to X_1$ such that $s_X\\circ \\sigma=\\textnormal{id}_{X_0}$ and $\\iota_{\\sigma}:X_0\\to X_0$ defined by $\\iota_{\\sigma}(x):=t_X(\\sigma(x))$ is a diffeomorphism. The set of all bisections of $X_1\\rightrightarrows X_0$ will be denoted by $\\textnormal{Bis}(X)$. This has the structure of an infinite-dimensional Lie group where the multiplication of two bisections $\\sigma$ and $\\sigma'$ is given by $\\sigma\\bullet \\sigma'(x):=\\sigma((t_X\\circ \\sigma')(x))*\\sigma'(x)$ for all $x\\in X_0$; see for instance \\cite{SW} and \\cite[s. 1.4]{Ma}. Let us denote by $\\textnormal{Aut}(X)$ the group of Lie groupoid automorphisms of $X_1\\rightrightarrows X_0$. Given a bisection $\\sigma \\in \\textnormal{Bis}(X)$ one has an inner automorphism $I_{\\sigma}:X_1\\to X_1$ defined by $$I_{\\sigma}(p):=\\sigma(t_X(p))*p* i_X(\\sigma(s_X(p))).$$\n\nClearly, $I_{\\sigma}$ covers the map $\\iota_{\\sigma}$. This inner automorphism allows us to define what we call the \\emph{crossed module of automorphisms of a Lie groupoid} $(\\textnormal{Aut}(X),\\textnormal{Bis}(X),I,\\alpha)$ where the map $\\alpha$ is defined as $\\alpha_{\\Phi}(\\sigma):=\\Phi\\circ \\sigma \\circ \\phi^{-1}$ for all $\\Phi \\in \\textnormal{Aut}(X)$ covering $\\phi:X_0\\to X_0$ and $\\sigma \\in \\textnormal{Bis}(X)$. Accordingly, we have a 2-group $\\textnormal{Bis}(X)\\ltimes \\textnormal{Aut}(X)\\rightrightarrows \\textnormal{Aut}(X)$ called the \\emph{2-group of Lie groupoid automorphisms}. \n\\begin{proposition}\n\tThe orbit space of the 2-group of Lie groupoid automorphisms equals the set of Lie groupoid automorphisms up to smooth natural equivalences. Namely:\n$$\\textnormal{Aut}(X)\/\\textnormal{Bis}(X)=\\left\\lbrace [\\Phi]\\,|\\, \\Phi\\in \\textnormal{Aut}(X),\\quad \\Psi\\sim \\Phi \\Leftrightarrow \\exists_{\\alpha}\\left(\\Psi\\stackrel{\\alpha}{\\Rightarrow}\\Phi\\right)\\right\\rbrace.$$\n\\end{proposition}\n\\begin{proof}\n\tLet us describe the orbit of an element in $\\Phi\\in \\textnormal{Aut}(X)$ covering $\\phi$. If we pick $\\sigma\\in \\textnormal{Bis}(X)$ and $\\Psi \\in \\textnormal{Aut}(X)$ covering $\\psi$ such that $I_{\\sigma}\\Phi=\\Psi$ then it follows that \n$$\\Psi(p)=I_{\\sigma}(\\Phi(p))=\\sigma(\\phi(t_X(p)))*\\Phi(p)*i_X(\\sigma(\\phi(s_X(p)))),$$\nfor some $p \\in X_1$ so that $\t\\Psi(p)*\\sigma(\\phi(s_X(p)))=\\sigma(\\phi(t_X(p)))*\\Phi(p)$. Therefore, by setting $\\alpha:=\\sigma\\circ \\phi$ we get a smooth natural transformation $\\Phi\\stackrel{\\alpha}{\\Rightarrow}\\Psi$. Conversely, note that if $\\Phi\\stackrel{\\alpha}{\\Rightarrow}\\Psi$  a smooth natural transformation then for some $p\\in X$ it holds $\\alpha(t_X(p))*\\Phi(p)=\\Psi(p)*\\alpha(s_X(p))$, thus obtaining that $t_X(\\Phi(p))=s_X(\\alpha(t_X(p)))$ and $s_X(\\Psi(p))=t_X(\\alpha(s_X(p)))$. On the one hand, by setting $\\sigma:=\\alpha\\circ \\phi^{-1}$ it follows that $s_X\\circ\\sigma=\\textnormal{id}_{X_0}$. On the other hand, observe that\n$$ \\psi(s_X(p))=t_X(\\alpha(s_X(p)))=t_X(\\sigma(\\phi( s_X)))(p)=\\iota_{\\sigma}(\\phi(s_X(p))).$$\nHence, we have obtained that $i_{\\sigma}=\\psi\\circ \\phi^{-1}$ so that it is a diffeomorphism. \n\\end{proof}\nLet $G_1\\rightrightarrows G_0$ be a Lie $2$-group acting on $X_1\\rightrightarrows X_0$ by the left.\n\n\\begin{lemma}\\label{2ActionDecomposition}\nThe normal subgroup $H=\\textnormal{ker}(s_{G})$ acts on $X_1\\rightrightarrows X_0$ by bisections and $G_0$ acts by Lie groupoid automorphisms. Moreover, the right multiplication map defined on $s_X$-fibers for each arrow is $H$-equivariant. \n\\end{lemma}\n\\begin{proof}\n\tConsider $h\\in H$ and define the map $\\sigma_{h}:X_0\\to X_1$ as $\\sigma_h(x):=h1_x.$ It is clear that $s_X(\\sigma_h(x))=x$ and $t(\\sigma_h(x))=\\rho(h)x$ so that $\\sigma_h$ is a well defined bisection.  Let us now take $g\\in G_0$ and define the map $\\Sigma_g:X_1\\to X_1$ as $\\Sigma_g(x):=1_{g}x$. Equation \\eqref{MultAction} implies that\n\t\\[1_{g}(x*y)=(1_g*1_g)(p*q)=(1_{g}p)*(1_{g}q),\\]\n\tfor all $g\\in G_0$ and $(p,q)\\in X_2$, thus obtaining that $\\Sigma_g$ is a Lie groupoid morphism which clearly satisfies $(\\Sigma_g)^{-1}=\\Sigma_{g^{-1}}$. Note that $s_{X}(hp)=s_{X}(p)$ for all $p\\in X_1$ and $h\\in H$ so that the left action of $H$ on $X_1$ preserves the $s_X$-fibers. Therefore, for each $y\\xleftarrow[]{\\it p}x$ the right action $R_{p}:s_X^{-1}(y)\\to s_X^{-1}(x)$ satisfies that\n\t\\[R_{p}(hq)=(hq)*(1_{e}p)=(h*1_e)(q*p)=hR_{p}(q).\\]\n\tIn consequence, $R_p$ is $H$-equivariant as claimed.\n\\end{proof}\nSame result can be obtained if we consider right 2-actions instead of left ones.\n\\begin{lemma}\\label{LemmaIso1}\nThere is a natural morphism of crossed modules of Lie groups $(\\sigma,\\Sigma):(G,H,\\rho,\\alpha) \\to (\\textnormal{Aut}(X),\\textnormal{Bis}(X),I,\\alpha)$ where $\\sigma_h$ and $\\Sigma_g$ are defined as in Lemma \\ref{2ActionDecomposition}.\n\\end{lemma}\n\\begin{proof}\nLet us check that $\\Sigma \\circ \\rho=I\\circ \\sigma$ and $\\sigma_{\\alpha_{g}(h)}=\\alpha_{\\Sigma_g}(\\sigma_h)$ for all $g \\in G$ and $h\\in H$. Firstly, for $ p\\in X_1$ and $h\\in H$ we obtain\n\\begin{eqnarray*}\n\t\tI_{\\sigma_h}(p)&=&\\sigma_h(t_X(p))*p*i(\\sigma_h(s_X(p)))=(h1_{t_X(p)})*p*i(h1_{s_X(p)})\\\\\n\t\t&=&(h*e)(1_{t_X(p)}*p)*i(h1_{s_X(p)})=hp*i_{G}(h)1_{s(p)}=1_{\\rho(h)}p=\\Sigma_{\\rho(h)}(p).\n\t\\end{eqnarray*}\n\tSecondly, for $x \\in X_0$, $g\\in G_0$ and $h\\in H$ we get\n$$\n\\alpha_{\\Sigma_g}(\\sigma_h)(x) =\\Sigma_g(h1_{(g^{-1}x)})=1_gh1_{g^{-1}}1_x=\\alpha_{g}(h)1_x=\\sigma_{\\alpha_g(h)}(x).\n$$\n\\end{proof}\nWe are now in conditions to define an \\emph{infinitesimal 2-action} associated to a right Lie 2-group action. Let $A_X$ be the Lie algebroid of $X_1\\rightrightarrows X_0$ and for each $\\xi \\in \\mathfrak{g}_0$ we denote its fundamental vector fields as $\\tilde{\\xi}$. We also denote by $\\mathfrak{X}_{m}(X)$ the set of multiplicative vector fields on $X_1\\rightrightarrows X_0$ and by $(\\mathfrak{X}_{m}(X),\\Gamma(A_{X}),\\delta,D)$ its associated crossed module. Here we have that $\\delta(\\alpha)=\\alpha^r-\\alpha^l$ and $D_{(\\xi,v)}\\alpha=[\\xi,\\alpha^r]|_{X_0}$ for all $\\alpha\\in \\Gamma(A_X)$ and $(\\xi,v)\\in \\mathfrak{X}_{m}(X)$; see \\cite{OW} for specific details.\n\n\\begin{theorem}\\label{InfinitesimalAction}\nSuppose that we have a right 2-action of $G_1\\rightrightarrows G_0$ on $X_1\\rightrightarrows X_0$. Then there is a canonical homomorphism of Lie 2-algebra $j=(j_{-1},j_0):(\\mathfrak{g},\\mathfrak{h},\\partial,\\mathcal{L})\\to (\\mathfrak{X}_{m}(X),\\Gamma(A_{X}),\\delta,D)$ defined by $j_{-1}(\\xi)=\\left.\\tilde{\\xi}\\right|_{X_0}$ and  $j_0(\\zeta)=(\\tilde{1_{\\zeta}},\\tilde{\\zeta})$ for all $\\xi \\in \\mathfrak{h}$ and $\\zeta \\in \\mathfrak{g}$.\n\\end{theorem}\n\\begin{proof}\nTo see that $j$ is a well defined morphism we have to check that for any $\\xi \\in \\mathfrak{h}$ its fundamental vector field $\\tilde{\\xi}$ belongs to $\\mathfrak{X}_{inv}^{s}(X)$ and that for any $\\zeta \\in \\mathfrak{g}$ it holds that $(\\tilde{1_{\\zeta}},\\tilde{\\zeta})$ is in $\\mathfrak{X}_{m}(X)$. The latter assertion is clear since if $\\exp{t\\zeta}\\in G_0$ then Lemma \\ref{2ActionDecomposition} implies that the pair of flows $(\\varphi_t^{\\tilde{1_\\zeta}},\\varphi_t^{\\tilde{\\zeta}})$ determines a Lie groupoid morphism. Now, if $\\xi \\in \\mathfrak{h}$ then again from Lemma \\ref{2ActionDecomposition} it follows that the flow of its fundamental vector field lies inside the $s$-fibers since $s_{X}(\\varphi_t^{\\tilde{\\xi}}(p))=s_{X}(p\\exp(t\\xi))=s_{X}(p)$. Therefore, $\\tilde{\\xi}\\in \\mathfrak{X}^s(X)$. Furthermore, for $y\\xleftarrow[]{\\it p}x$ and $q\\in s_X^{-1}(y)$ one has that\n$$\nR_{p}(\\varphi^{\\tilde{\\xi}}_t(q)) =(q\\exp(t\\xi))*p=(q*p)(\\exp{t\\xi}*1_e)=\\varphi_t^{\\tilde{\\xi}}(q*p)=\\varphi_t^{\\tilde{\\xi}}(R_p(q)),\n$$\nthus obtaining that $d(R_p)_q(\\tilde{\\xi}_q)=\\tilde{\\xi}_{q*p}$ so that $\\tilde{\\xi}\\in \\mathfrak{X}_{inv}^s(X)$ and $\\left.\\tilde{\\xi}\\right|_{X_0}\\in \\Gamma(A_X)$.\n\nLet us finally verify that for $\\xi\\in \\mathfrak{h}$ and $\\zeta \\in \\mathfrak{g}$ it satisfies that $\\delta(j_{-1}(\\xi))=j_{0}(\\partial \\xi)$ and $j_{-1}(\\mathcal{L}_{\\eta}\\xi)=D_{j_0(\\eta)}(j_{-1}(\\xi))$. On the one hand, by using the flow of vector field $\\delta(j_0(\\xi))=\\left.\\tilde{\\xi}\\right|_{X_0}^r-\\left.\\tilde{\\xi}\\right|_{X_0}^l$ and Equation \\eqref{MultAction} we get\n\\begin{eqnarray*}\n\\varphi_t^{\\delta(j_{-1}(\\xi))}(p)&=&\\varphi_{t}^{\\tilde{\\xi}}(1_{t_{X}(p)})*p*\\iota_X(\\varphi_t^{\\tilde{\\xi}}(1_{s_X(p)}))=1_{t_{X}(p)}\\exp(t\\xi)*p*i_X(1_{s_{X}(p)}\\exp(t\\xi))\\\\\n&=&(1_{t_X(p)}*p)\\exp(t\\xi)*1_{s_X(p)}i_G(\\exp(t\\xi))=p(\\exp(t\\xi)*i_G(\\exp(t\\xi)))\\\\\n&=&p(1_{t_G(\\exp(t\\xi))})=p1_{\\exp(t\\partial(\\xi))}=p\\exp(t1_{\\partial \\xi})=\\varphi_t^{{j_0(\\partial \\xi)}}(p).\n\t\\end{eqnarray*}\n\tHence, $\\delta(j_{-1}(\\xi))=j_{0}(\\partial \\xi)$. On the other hand, observe that\n\t\\[    D_{j_0(\\eta)}(j_{-1}(\\xi))=\\left.\\left[\\tilde{1_{\\eta}},\\tilde{\\xi}\\right]\\right|_{X_0}=\\left.\\tilde{\\left[1_{\\eta},\\xi\\right]}\\right|_{X_0}=\\left.\\tilde{\\mathcal{L}_{\\eta}(\\xi))}\\right|_{X_0}=j_{-1}(\\mathcal{L}_{\\eta}\\xi).\\]\n\\end{proof}\n\\subsection{Isometries of a 0-metric}\nRecall that a $0$-metric on a Lie groupoid $X_1\\rightrightarrows X_0$ is a Riemannian metric $\\eta$ on $X_0$ which is transversely invariant by the canonical left action of $X_1\\rightrightarrows X_0$ on $X_0$, compare \\cite{dHF,PPT}. Note that this is the same that requiting that $\\eta$ is transversely invariant by the action of the group of bisections $\\textnormal{Bis}(X)\\times X_0\\to X_0$ which is defined by $\\sigma\\cdot x:=\\iota_{\\sigma}(x)$. It is clear that this action preserves the orbits so that it induces a well defined action on the normal space of a orbit. In consequence, $\\eta$ is a 0-metric if and only if for all $\\sigma \\in \\textnormal{Bis}(X)$ the map $\\overline{d\\iota_{\\sigma}}:(\\nu_x(\\mathcal{O}),\\overline{\\eta})\\to (\\nu_{\\iota_{\\sigma}(x)}(\\mathcal{O}),\\overline{\\eta})$ is a linear isometry.\n\nLet $(X_1\\rightrightarrows X_0,\\eta)$ be a Lie groupoid equipped with a 0-metric $\\eta$ and consider the following sets \n$$\\textnormal{Bis}_{\\eta}(X)=\\left\\lbrace \\sigma \\in \\textnormal{Bis}(X)\\,|\\, \\iota_{\\sigma}^*\\eta=\\eta\\right\\rbrace\\quad\\textnormal{and}\\quad \\mathrm{Iso}(X,\\eta)=\\left \\lbrace (\\Phi,\\phi)\\in \\mathrm{Aut}(X)\\,|\\, \\phi^*\\eta=\\eta \\right\\rbrace.$$\n\n\\begin{proposition}\\label{IsoStrong1}\nThe quadruple $(\\mathrm{Iso}(X,\\eta),\\textnormal{Bis}_{\\eta}(X),I,\\alpha)$ determines a sub-crossed module structure  of $(\\textnormal{Aut}(X),\\textnormal{Bis}(X),I,\\alpha)$.\n\\end{proposition}\n\\begin{proof}\nIt is clear that $I(\\textnormal{Bis}_{\\eta}(X))\\subseteq \\mathrm{Iso}(X,\\eta)$. As $I_{\\alpha_{\\Phi}(\\sigma)}=\\Phi I_{\\sigma}\\Phi^{-1}$ for all $\\sigma \\in \\textnormal{Bis}_{\\eta}(X)$ and $\\Phi \\in \\textnormal{Iso}(X,\\eta)$ then when restricting to unities we have that $\\iota_{\\alpha_{\\Phi}(\\sigma)}=\\phi\\circ \\iota_{\\sigma}\\circ \\phi^{-1}$, thus obtaining an isometry.\n\\end{proof}\nThe Lie 2-group associated to the crossed module $(\\mathrm{Iso}(X,\\eta),\\textnormal{Bis}_{\\eta}(X),I,\\alpha)$ will be called \\emph{Lie 2-group of strong isometries} of $(X_1\\rightrightarrows X_0,\\eta)$. Let us now consider the sets \n$$\\Gamma_{\\eta}(A_X)=\\left\\lbrace \\alpha \\in \\Gamma(A_X)\\,|\\, \\rho(\\alpha) \\in \\mathfrak{o}(X_0,\\eta)\\right\\rbrace\\quad\\textnormal{and}\\quad \\mathfrak{o}_{m}(X)=\\left\\lbrace (\\xi,v)\\in  \\mathfrak{X}_{m}(X)\\,|\\, v \\in \\mathfrak{o}(X_0,\\eta)\\right\\rbrace$$\nwhere $\\mathfrak{o}(X_0,\\eta)$ denotes the Lie algebra of Killing vector fields of $(X_0,\\eta)$. In these terms we may describe the infinitesimal version of the Lie $2$-group of strong isometries as follows. \n\\begin{proposition}\\label{Strongkilling}\nThe quadruple $(\\mathfrak{o}_{m}(X),\\Gamma_{\\eta}(A_X),\\delta,D)$ defines a sub-crossed module structure of $(\\mathfrak{X}_{m}(X),\\Gamma(A_{X}),\\delta,D)$.\n\\end{proposition}\n\\begin{proof}\nFirstly, note that $\\mathfrak{o}_{m}(X)$ is a Lie subalgebra of $\\mathfrak{X}_{m}(X)$ since $\\mathfrak{o}(X_0,\\eta)$ is a Lie algebra. Secondly, if $\\alpha, \\beta \\in \\Gamma_{\\eta}(A_X)$ then $\\rho([\\alpha,\\beta])=[\\rho(\\alpha),\\rho(\\beta)]\\in \\mathfrak{o}(X_0,\\eta)$ so that $[\\alpha,\\beta]\\in \\Gamma_{\\eta}(A)$. If $(\\xi,v)\\in \\mathfrak{o}_{m}(X)$ and $\\alpha \\in \\Gamma_{\\eta}(A)$ then we have by definition that $D_{\\xi}(\\alpha)=\\left.[\\xi,\\alpha^r]\\right|_{X_0} \\in \\Gamma(A_X)$. However, the equivariance identity implies that $\\delta(D_{\\xi}(\\alpha))=[\\xi,\\delta(\\alpha)]$. Therefore, it holds that $\\left.\\delta(D_{\\xi}\\alpha)\\right|_{X_0}=\\left.[\\xi,\\delta(\\alpha)]\\right|_{X_0}$ which is the same thing that saying\n$\\rho(D_{\\xi}\\alpha)=[v,\\rho(\\alpha)] \\in \\mathfrak{o}(X_0,\\eta)$ since $v$ is also a Killing vector field.\n\\end{proof}\nThe Lie $2$-algebra associated to the crossed module $(\\mathfrak{o}_{m}(X),\\Gamma_{\\eta}(A_X),\\delta,D)$ will be called \\emph{Lie 2-algebra of strong multiplicative Killing vector fields} of $(X_1\\rightrightarrows X_0,\\eta)$.\n\\begin{remark}\nIt is worth noticing that if we have an isometric action of a Lie $2$-group $G_1\\rightrightarrows G_0$ on $(X_1\\rightrightarrows X_0,\\eta)$ then the assertions of Lemma \\ref{LemmaIso1} and Theorem \\ref{InfinitesimalAction} can be rewritten in terms of $(\\mathrm{Iso}(X,\\eta),\\textnormal{Bis}_{\\eta}(X),I,\\alpha)$ and $(\\mathfrak{o}_{m}(X),\\Gamma_{\\eta}(A_X),\\delta,D)$, respectively.\n\\end{remark}\nA diffeomorphism $\\phi:X_0\\to X_0$ is said to be a \\emph{transversal isometry} of $(X_0,\\eta)$ if $\\overline{d\\phi}:\\nu(\\mathcal{O}_x)\\to \\nu(\\mathcal{O}_{\\phi(x)})$ is a fiberwise isometry for every groupoid orbit $\\mathcal{O}_x$ in $X_0$. Consider the set\n$$\\textnormal{Iso}_{\\textnormal{w}}(X,\\eta)=\\left \\lbrace (\\Phi,\\phi)\\in \\mathrm{Aut}(X)\\,|\\, \\phi\\ \\textnormal{transversal isometry of}\\ (X_0,\\eta)\\right\\rbrace.$$\nNote that for every $\\sigma\\in \\textnormal{Bis}(X)$ it follows that $\\iota_\\sigma$ is a transversal isometry of $(X_0,\\eta)$. Thus, by arguing as in Proposition \\ref{IsoStrong1} we easily get that:\n\\begin{lemma}\nThe quadruple $(\\mathrm{Iso}_{\\textnormal{w}}(X,\\eta),\\textnormal{Bis}(X),I,\\alpha)$ determines a sub-crossed module structure  of $(\\textnormal{Aut}(X),\\textnormal{Bis}(X),I,\\alpha)$.\n\\end{lemma}\nThe Lie $2$-group determined by the crossed module $(\\mathrm{Iso}_{\\textnormal{w}}(X,\\eta),\\textnormal{Bis}(X),I,\\alpha)$ is called \\emph{Lie 2-group of weak isometries} of $(X_1\\rightrightarrows X_0,\\eta)$. Let us denote by $\\mathfrak{o}^{\\textnormal{w}}(X_0,\\eta)$ the set of vector fields on $X_0$ whose flow determines a (local) transversal isometry of $(X_0,\\eta)$ and consider the set $\\mathfrak{o}_{m}^{\\textnormal{w}}(X)=\\left\\lbrace (\\xi,v)\\in  \\mathfrak{X}_{m}(X)\\,|\\, v \\in \\mathfrak{o}^{\\textnormal{w}}(X_0,\\eta)\\right\\rbrace$. Observe that if $v_1,v_2\\in \\mathfrak{o}^{\\textnormal{w}}(X_0,\\eta)$ then for $t$ small enough we have that the commutator flow $\\varphi_{-\\sqrt{t}}^{v_2}\\varphi_{-\\sqrt{t}}^{v_1}\\varphi_{\\sqrt{t}}^{v_2}\\varphi_{\\sqrt{t}}^{v_1}$ is a transversal isometry so that $\\mathfrak{o}^{\\textnormal{w}}(X_0,\\eta)$ is a Lie subalgebra of $\\mathfrak{X}(X_0)$. From \\cite{SW} we know that $\\textnormal{Lie}(\\textnormal{Bis}(X))$ is identified with $\\Gamma(A_X)$. Therefore, by using similar arguments as those in Proposition \\ref{Strongkilling} we obtain a description of the Lie $2$-group of weak isometries of $(X_1\\rightrightarrows X_0,\\eta)$. Namely:\n\\begin{proposition}\nThe quadruple \n$(\\mathfrak{o}_{m}^{\\textnormal{w}}(X),\\Gamma(A_X),\\delta,D)$ defines a sub-crossed module structure of $(\\mathfrak{X}_{m}(X),\\Gamma(A_{X}),\\delta,D)$.\n\\end{proposition}\nAccordingly, the Lie $2$-algebra associated to the crossed module $(\\mathfrak{o}_{m}^{\\textnormal{w}}(X),\\Gamma(A_X),\\delta,D)$ will be called \\emph{Lie 2-algebra of weak multiplicative Killing vector fields} of $(X_1\\rightrightarrows X_0,\\eta)$.\n\\begin{theorem}\nLet $(X_1\\rightrightarrows X_0,\\eta)$ be a Riemannian groupoid with $\\eta$ a 1-metric and $X_1\\rightrightarrows X_0$ proper. Then there always exists a multiplicative vector field $(\\xi,v)$ with $v \\in \\mathfrak{o}^{\\textnormal{w}}(X_0,\\eta)$.\n\\end{theorem}\n\\begin{proof}\nLet us pick $v_1 \\in \\mathfrak{o}^{\\textnormal{w}}(X_0,\\eta)$. Since $s$ is a surjective submersion there is a vector field $\\xi_1$ on $X_1$ such that $\\xi_1$ is $s_X$-related with $v_1$. The fact that $\\overline{ds}:\\nu(G_{\\mathcal{O}})\\to \\nu(\\mathcal{O})$ is a fiberwise isometry ($s_X$ is a Riemannian submersion) implies that the flow of $\\xi_1$ is a local transversal isometry of $(X_1,\\eta)$ since $\\xi_1$ and $v_1$ are $s$-related.\n\nLet us now consider a proper Haar measure system $\\lbrace \\mu^x\\rbrace$ for $G\\rightrightarrows M$. By following results proved in \\cite{CS} we can construct a multiplicative vector field $\\xi:X_1\\to TX_1$ by taking the average with respect to $\\lbrace \\mu^x\\rbrace$:\n$$\\xi(p)=\\int_{a\\in t^{-1}(s(p))}dm_{(pa,a^{-1})}(\\xi_1(ap),di_a(\\xi_1(a))\\mu(a).$$\nThis vector field is $s_X$-related with the vector field $v$ on $X_0$ defined as\n$$v(x)=\\int_{a\\in t^{-1}(x)}dt_a(\\xi_1(a))\\mu(a).$$\nTherefore, our result will follow once we prove that the flow of $v$ is a local transversal isometry of $(X_0,\\eta)$ since $\\xi$ and $v$ are $s_X$-related. However, this follows from a straightforward computation after noting that the flow of $v$ is given by $\\varphi^v_r(x)=\\int_{a\\in t^{-1}(x)}(t\\circ \\varphi^{\\xi_1}_r)(a)\\mu(a)$ and $\\overline{dt}:\\nu(G_{\\mathcal{O}})\\to \\nu(\\mathcal{O})$ is a fiberwise isometry since $t_X$ is also a Riemannian submersion.\n\\end{proof}\n\n\\subsection{Morita invariance}\nIn this subsection we apply some of the result from \\cite{OW} to our context in order to define a notion of geometric Killing field on a quotient Riemannian stack.\n\nLet us start by introducing some necessary terminology.  A \\emph{Morita map} is a groupoid morphism $\\phi:(X_1\\rightrightarrows X_0) \\to (Y_1\\rightrightarrows Y_0)$ which is \\emph{fully faithful} and \\emph{essentially surjective}, in the sense that the source\/target maps define a fibred product of manifolds $X_1 \\cong (X_0 \\times X_0) \\times_{ (Y_0\\times Y_0)} Y_1$ and that the map $Y_1 \\times _{Y_0}X_0\\to X_0$ sending $(\\phi^0(x)\\to y)\\mapsto y$ is a surjective submersion \\cite{dH,MM}. An important fact shown in \\cite{dH} is that a Lie groupoid morphism is a Morita map if and only if it yields an isomorphism between \\emph{transversal data}. That is, the morphism must induce: a homeomorphism between the orbit spaces, a Lie group isomorphism $X_x\\cong Y_{\\phi^0(x)}$ between the isotropies and isomorphisms between the normal representations $X_x\\curvearrowright \\nu_x\\to Y_{\\phi^0(x)}\\curvearrowright \\nu'_{\\phi^0(x)}$. We think of a quotient stack as a Lie groupoid up to Morita equivalence in the sense that two Lie groupoids $X$ and $Y$ define the same stack if there is a third groupoid $Z$ and Morita maps $Z\\to X$ and $Z\\to Y$ \\cite{dH}. These two Morita maps may be assumed to be Morita fibrations (surjective submersion at the level of objects) \\cite{MM}. The quotient stack associated to the Lie groupoid $X_1\\rightrightarrows X_0$ will be denoted by $[X_0\/X_1]$. \n\nTwo Riemannian metrics $\\eta_1$ and $\\eta_2$ on $X_1\\rightrightarrows X_0$ are said to be \\emph{equivalent} if they induce the same inner product on the normal vector spaces over the groupoid orbits \\cite{dHF2}. More generally, we define a \\emph{Riemannian Morita map} (\\emph{fibration}) $\\phi: (Z_1 \\rightrightarrows Z_0) \\to (X_1\\rightrightarrows X_0)$ as a Morita map between Riemannian groupoids that induces isometries on the normal vector spaces to the groupoid orbits $\\nu_z(\\mathcal{O}^Z)\\to \\nu_{\\phi(z)}(\\mathcal{O}^X)$ (Riemannian submersion at the level of objects). By using this terminology we have that  $\\eta_1$ and $\\eta_2$ are equivalent if and only if the identity $\\textnormal{id}: (X_1 \\rightrightarrows X_0,\\eta_1) \\to (X_1\\rightrightarrows X_0,\\eta_2)$ is a Riemannian Morita map.\n\nFollowing \\cite{OW}, given a Morita fibration $\\phi: (Z_1 \\rightrightarrows Z_0) \\to (X_1\\rightrightarrows X_0)$ we denote the set of projectable sections \n$$\\Gamma(A_Z)^\\phi=\\lbrace \\alpha \\in \\Gamma(A_Z):\\textnormal{there exists}\\ \\alpha'\\in \\Gamma(A_X)\\ \\textnormal{such that}\\ \\phi\\alpha=\\alpha'\\phi\\rbrace.$$\nIf $\\alpha \\in \\Gamma(A_Z)$ then the surjectivity of $\\phi$ at the level of objects implies that there exists at most one section $\\alpha'\\in \\Gamma(A_X)$ such that  $\\phi\\alpha=\\alpha'\\phi$ so that it follows that there is a natural linear map $\\phi_\\ast : \\Gamma(A_Z)^\\phi\\to \\Gamma(A_X)$. We denote by $\\Gamma(A_Z)^\\phi \\hookrightarrow  \\Gamma(A_Z)$ the inclusion map. It is clear that we can similarly define the set of projectable multiplicative sections $\\mathfrak{X}_m(Z)^\\phi$, a natural map $\\phi_\\ast: \\mathfrak{X}_m(Z)^\\phi\\to \\mathfrak{X}_m(X)$ and an inclusion $\\mathfrak{X}_m(Z)^\\phi\\hookrightarrow \\mathfrak{X}_m(Z)$. From \\cite{OW} it follows that $(\\mathfrak{X}_m(Z)^\\phi,\\Gamma(A_{Z})^\\phi,\\delta,D)$ is a sub-crossed module of $(\\mathfrak{X}_m(Z),\\Gamma(A_{Z}),\\delta,D)$ and both maps $\\phi_\\ast: (\\mathfrak{X}_m(Z)^\\phi,\\Gamma(A_{Z})^\\phi,\\delta,D)\\to (\\mathfrak{X}_m(X),\\Gamma(A_{X}),\\delta,D)$ and $(\\mathfrak{X}_m(Z)^\\phi,\\Gamma(A_{Z})^\\phi,\\delta,D)\\hookrightarrow(\\mathfrak{X}_m(Z),\\Gamma(A_{Z}),\\delta,D)$  are morphisms of crossed-modules. More importantly,\n$$(\\mathfrak{X}_m(Z),\\Gamma(A_{Z}),\\delta,D) \\hookleftarrow (\\mathfrak{X}_m(Z)^\\phi,\\Gamma(A_{Z})^\\phi,\\delta,D) \\xrightarrow[]{\\it \\phi_\\ast} (\\mathfrak{X}_m(X),\\Gamma(A_{X}),\\delta,D),$$\nare quasi-isomorphisms of crossed modules. For specific details visit \\cite{OW}.\n\\begin{lemma}\\label{MoritaLemma1}\nLet $\\phi: (Z_1 \\rightrightarrows Z_0,\\eta^Z) \\to (X_1\\rightrightarrows X_0,\\eta^X)$ be a Morita Riemannian fibration. Then\n\\begin{itemize}\n\\item $(\\mathfrak{o}_m(Z)^\\phi,\\Gamma_{\\eta}(A_{Z})^\\phi,\\delta,D)$ is a sub-crossed module of $(\\mathfrak{o}_m(Z),\\Gamma_\\eta(A_{Z}),\\delta,D)$,\n\\item the inclusion $(\\mathfrak{o}_m(Z)^\\phi,\\Gamma_{\\eta}(A_{Z})^\\phi,\\delta,D)\\hookrightarrow(\\mathfrak{o}_m(Z),\\Gamma_{\\eta}(A_{Z}),\\delta,D)$ is a morphism of crossed modules, and\n\\item the projection $\\phi_\\ast: (\\mathfrak{o}_m(Z)^\\phi,\\Gamma_\\eta(A_{Z})^\\phi,\\delta,D)\\to (\\mathfrak{o}_m(X),\\Gamma_\\eta(A_{X}),\\delta,D)$ is a morphism of crossed modules.\n\\end{itemize}\nMoreover,\n$$(\\mathfrak{o}_m(Z),\\Gamma_\\eta(A_{Z}),\\delta,D) \\hookleftarrow (\\mathfrak{o}_m(Z)^\\phi,\\Gamma_\\eta(A_{Z})^\\phi,\\delta,D) \\xrightarrow[]{\\it \\phi_\\ast} (\\mathfrak{0}_m(X),\\Gamma_\\eta(A_{X}),\\delta,D),$$\nare quasi-isomorphisms of crossed modules. Same conclusion holds true for the weak counterpart.\n\\end{lemma}\n\\begin{proof}\nBy using similar arguments as those in Lemma \\ref{Rmk1} if follows that if $v$ is a (weak) Killing vector field on $(Z_0,\\eta^Z)$ then $\\phi_\\ast(v)$ is also a (weak) Killing vector field on $(X_0,\\eta^X)$. Therefore, the result follows by applying Proposition 7.4 and Theorem 7.3 from \\cite{OW} after restricting the structure.\n\\end{proof}\nThe following result is clear.\n\\begin{lemma}\\label{MoritaLemma2}\nIf $\\eta_1$ and $\\eta_2$ are equivalent Riemannian metrics on $X_1\\rightrightarrows X_0$ then the crossed modules $(\\mathfrak{o}_{m}^{\\textnormal{w}}(X,\\eta_1),\\Gamma(A_X),\\delta,D)$ and $(\\mathfrak{o}_{m}^{\\textnormal{w}}(X,\\eta_2),\\Gamma(A_X),\\delta,D)$ agree.\n\\end{lemma}\n\nSuppose that $X$ and $Y$ are Morita equivalent Lie groupoids so that there is a third Lie groupoid $Z$ with Morita fibrations $Z\\to X$ and $Z\\to Y$. From \\cite{dHF2} we know that if $\\eta^X$ is a Riemannian metric on $X$ then there exists a Riemannian metric $\\eta^Z$ on $Z$ that makes the fibration $Z\\to X$ Riemannian. We can slightly modify $\\eta^Z$ by a cotangent averaging procedure so that we get another Riemannian metric $\\tilde{\\eta}^Z$ on $Z$ which descends to $Y$ defining a Riemannian metric $\\eta^Y$ making of the fibration $Z\\to Y$ Riemannian. It turns out that these pullback and pushforward constructions are well-defined and mutually inverse modulo equivalence of metrics. This is because $\\eta^Z$ and $\\tilde{\\eta}^Z$ turn out to be equivalent. In this case we refer to $(X,\\eta^X)$ and $(Y,\\eta^Y)$ as being \\emph{Morita equivalent Riemannian groupoids}. It suggests a definition for Riemannian metrics over differentiable stacks. Namely, a stacky metric on the orbit stack $[X_0\/X_1]$ associated to a Lie groupoid $X_1\\rightrightarrows X_0$ is defined to be an equivalence class $[\\eta]$ of a Riemannian metric $\\eta$ on $X$. For further details the reader is recommended to visit \\cite{dHF2}. \n\n\\begin{theorem}\\label{MoritaTheorem}\nIf $(X_1 \\rightrightarrows X_0,\\eta^X)$ and $(Y_1 \\rightrightarrows Y_0,\\eta^Y)$ are Morita equivalent Riemannian groupoids then the crossed modules $(\\mathfrak{o}_{m}^{\\textnormal{w}}(X),\\Gamma(A_X),\\delta,D)$ and $(\\mathfrak{o}_{m}^{\\textnormal{w}}(Y),\\Gamma(A_Y),\\delta,D)$ are isomorphic in the derived category of crossed modules. In consequence, the following quotient spaces are isomorphic:\n$$\\mathfrak{o}_{m}^{\\textnormal{w}}(X)\/\\textnormal{im}(\\delta)\\cong \\mathfrak{o}_{m}^{\\textnormal{w}}(Y)\/\\textnormal{im}(\\delta).$$\n\\end{theorem}\n\\begin{proof}\nThis result is consequence of Lemmas \\ref{MoritaLemma1} and \\ref{MoritaLemma2} together with Theorem 7.4 and Corollary 7.2 from \\cite{OW}.\n\\end{proof}\nMotivated by the previous result and \\cite[Def. 8.1]{OW} we set up the following definition.\n\\begin{definition}\nLet $(X_1 \\rightrightarrows X_0,\\eta)$ be a Riemannian groupoid. A \\emph{geometric Killing vector field} on the quotient Riemannian stack $([X_0\/X_1],[\\eta])$ is defined to be \n$$\\mathfrak{o}([X_0\/X_1],[\\eta]):=\\mathfrak{o}_{m}^{\\textnormal{w}}(X)\/\\textnormal{im}(\\delta).$$\n\\end{definition}\nOn the one hand, if we consider proper \u00e9tale Riemannian groupoids then geometric Killing vector fields recover the classical notions of Killing vector fields on both Riemannian manifolds and Riemannian orbifolds as defined for instance in \\cite{BZ}. On the other hand, if we consider the Riemannian groupoid $\\textnormal{Hol}(M,\\mathcal{F})\\rightrightarrows M$ associated to a regular Riemannian foliation $(M,\\mathcal{F})$ then geometric Killing vector fields recover the notion of transverse Killing vector fields as defined in \\cite[p. 84]{Mo}.\n\nWe finish this section by proving a dimensional result for Riemannian foliation groupoids.  A \\emph{foliation groupoid} is a Lie groupoid $X_1 \\rightrightarrows X_0$ whose space of objects $X_0$ is Hausdorff and whose isotropy groups $X_x$ are discrete for all $x\\in X_0$. For instance, every \\'etale  Lie groupoid with Hausdorff objects manifold is a foliation groupoid. The converse is not true, however every foliation groupoid is Morita equivalent to an \\'etale groupoid. As shown in \\cite{C}, being a foliation groupoid is equivalent to the associated Lie algebroid anchor map $\\rho:A_X\\to TX_0$ being injective. As a consequence, the manifold $X_0$ comes with a regular foliation $\\mathcal{F}$ tangent to the leaves of $\\mathrm{im}(\\rho)\\subseteq TX_0$. Note that if $X_1 \\rightrightarrows X_0$ is source-connected then the leaves of $\\mathrm{im}(\\rho)\\subseteq TX_0$ coincide with the groupoid orbits. \n\n\\begin{proposition}\n\tIf $(X_1 \\rightrightarrows X_0,\\eta)$ is a Riemannian foliation groupoid then the algebra of geometric Killing vector fields on $([X_0\/X_1],[\\eta])$ has finite dimension.\n\\end{proposition}\n\\begin{proof}\nLet $\\iota: T\\hookrightarrow X_0$ be a complete transversal submanifold to the orbit foliation $\\mathcal{F}$ of $X_1 \\rightrightarrows X_0$  and consider its restricted groupoid $X_T\\rightrightarrows T$. As $X_T\\rightrightarrows T$ is \u00e9tale and Morita equivalent to $X_1 \\rightrightarrows X_0$ (see \\cite[136]{MM}), from Theorem \\ref{MoritaTheorem} it follows that \n$$\n\t\t\\mathfrak{o}([X_0\/X_1],[\\eta])=  \\mathfrak{o}_m^{\\textnormal{w}}(X_{T}, \\iota^*\\eta)\/\\textnormal{im}(\\delta)\\cong \\mathfrak{o}(T)^{X}. \n$$\nHere $\\mathfrak{o}(T)^{X}$ denotes the transversal Killing vector fields that are invariant by the normal action. Therefore, $\\mathrm{dim}\\mathfrak{o}([X_0\/X_1],[\\eta])\\leq \\frac{1}{2}\\mathrm{codim}(\\mathcal{F})(\\mathrm{codim}(\\mathcal{F})+1).$\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Abstract}\n{\\footnotesize\nSupernova 1604 is the last  Galactic supernova for which historical records exist.\nJohannes Kepler's name is attached to it, as he published\na detailed account of the observations made  by himself and European colleagues.\nSupernova 1604 was very likely a Type Ia supernova, which exploded\n 350~pc to 750~pc above the Galactic plane.\n Its supernova remnant, known as Kepler's supernova remnant, \n shows clear evidence for interaction with\n nitrogen-rich material in the north\/northwest part of the remnant,\n which, given the height above the Galactic plane, must find its\n origin in mass loss from the supernova progenitor system.\n The combination of a Type Ia supernova and the presence\n of circumstellar material makes Kepler's supernova remnant\n a unique object to study the origin of Type Ia supernovae.\n The evidence suggests that\n  the progenitor binary system  of supernova 1604\n consisted of a carbon-oxygen white dwarf and an evolved\n companion star, which most likely was  in the (post) asymptotic giant branch of its evolution.\n A problem with this scenario is that the companion star must have survived\n the explosion, but no trace of its existence has yet been found, despite a deep search.\n }\n \\tableofcontents\n\n \n \n \n \\section{Introduction}\n\\label{sec:intro}\n\nSupernova (SN) 1604 \\index{SN\\,1604 (SNR G4.5+6.8)} is the last of the historical supernovae. We know of at least\ntwo other supernovae with later explosion dates, those associated\nwith the supernova remnants Cassiopeia A     \\index{Cassiopeia A (Cas A, G111.7-2.1)}  \\index{SNR G111.7-2.1 (Cas A)}\nand G1.9+0.3 \\citep{green08}, \\index{SNR G1.9+0.3}\nbut no known historical records of the associated supernova events exist.\n\nThe discovery of SN\\,1604 occurred on October 9, 1604 (Gregorian calendar),\nearly in the evening when all interested in astronomy gazed the sky\nto watch the conjunction of  Mars, Jupiter and Saturn.\nThis special circumstance probably helped the early\ndiscovery of the supernova at around \n20 days before\nmaximum brightness \\citep{stephenson02}.\nThe occurrence of a new star in conjunction with three bright planets \nlead to fierce debates among contemporary astronomers,\nmost of whom still held partial medieval world views, with many elements\nof astrology. Of all these astronomers, it is in particular\nJohannes Kepler's \\index{Kepler, Johannes} name that is now attached to this supernova, \nbecause  he wrote a book ``De Stella Nova in Pede Serpentarii'' (\"On the new star in the foot of the Snake [Ophichius]\")\nabout the new star,\nin which he\npublished his own observations of the supernova and those by various European  colleagues,\nand discusses the importance of the new\nstar, including a possible link to the \\index{star of Bethlehem} star of Bethlehem.\nKepler only started observing SN 1604 relatively late, October 17, due to unfortunate  weather \ncircumstances in Prague, his city of residence.\nKepler and his contemporaries attached a great deal of metaphysical significance to the appearance of\nthe new star: {\\em ``The star's significance is a difficult matter to establish and we can be sure of only one thing: \nthat either the star signifies nothing at all for Mankind or it signifies something of such exalted importance that it is beyond the grasp and understanding of any man''}.\nKepler himself, in fact, was a bit more cautious about the metaphysical implications of the event than some of his contemporaries \\citep{granada05}.\n\nCurrently, the  remnant of SN 1604, labeled G4.5+6.8, \\index{SNR G4.5+6.8} but\nusually called \\index{Kepler's supernova remnant (SNR G4.5+6.8)} Kepler's supernova remnant (SNR), or even \"Kepler\" for short,\nis one of the best studied SNRs.  It appears that Kepler's SNR has indeed a wide ranging significance,\nbut not for metaphysical reasons, but  for understanding the nature of  \\index{Type Ia supernova} Type Ia supernovae.\nSN 1604 was almost certainly a  Type Ia supernova, but, peculiarly, the SNR itself is interacting\nwith a nitrogen-rich \\index{CSM (circumstellar medium)} \\index{circumstellar medium} circumstellar medium (CSM) at a distance of 2-3 pc from the explosion centre, \nsuggesting that the progenitor system had significant \\index{stellar mass loss} \\index{stellar wind} mass loss. \n\nAs described in more detail in Sections 3 and 6, two basic types of supernova Type Ia progenitor models exists  \\index{single degenerate Type Ia model} \\index{double degenerate Type Ia model} \n\\citep{maoz14}: 1) the single degenerate model,\naccording to  which a CO white dwarf \\index{white dwarf} accretes  from an (evolved) stellar companion until the pressure\nin the core leads to the thermonuclear explosion \\index{thermonuclear explosion} of the white dwarf, at the moment that the progenitor\nhas  a mass approaching the \\index{Chandrasekar mass} Chandrasekar mass (1.38 {M$_\\odot$}), and 2) the \ndouble degenerate model, according to which Type Ia explosions are triggered by the merging of two white dwarfs.\nThe time scale from birth of the binary system to the merging of the two white dwarfs is generally expected to  be a few $10^9$~yr \\citep{maoz14}, \nsince the two stars are brought closer to each other due  gravitational radiation.\nThe fact that Kepler's SNR is interacting with CSM is, therefore, more consistent\nwith the single degenerate scenario. However, as will be discussed, this scenario is also not fully consistent\nwith the properties of Kepler's SNR, in particular with the lack of a surviving donor star.\n\n\\begin{figure}\n\\includegraphics[trim=0 300 0 0,clip=true,width=\\textwidth]{Fig_1_kepler_mosaic.jpg}\n\\caption{\\label{fig:4images}\nKepler's SNR at different wavelengths: a) Radio VLA (4.85 GHz) map \\citep{delaney03}; b) 24$\\mu$m dust emission as\nobserved by the Spitzer MIPS instrument \\citep{blair07}; c) optical image (H$\\alpha$, [NII] and  [OIII]) obtained\nby the Hubble Space Telescope\/ACS instrument (credit: ESA\/NASA, The Hubble Heritage Team (STScI\/AURA));\nd) Chandra X-ray image in the iron-rich 0.7-1.0 keV band \\citep{reynolds07}.\nIntensity scaling of Fig. a, b, and d are square-root scalings, bringing out fainter details.\n}\n\\end{figure}\n\n\\section{The supernova remnant, its distance and its multiwavelength properties}\n\\label{sec:properties}\n\n\\subsection{Position, distance estimates and SN 1604 as a runaway system}\n\nKepler's SNR was discovered by Walter Baade  \\index{Walter Baade}  as \"a small patch of nebulosity close to the expected\nplace\" \\citep{baade43}. \\citet{minkowski59} revealed that the spectrum of the optical nebula displayed strong [NII]\nemission as compared to H$\\alpha$ emission. \nThe H$\\alpha$\/[NII] is concentrated toward the north\/northwestern region of the SNR and a bar-shape\nregion across the center, roughly oriented in the NW-SE direction (Fig.~\\ref{fig:4images}).\nThe overall extent of the SNR is more clear from radio and X-ray images of the SNR, revealing\na roughly spherical shell with two protrusions (\"Ears\") in the NW and SE \\citep[e.g.][]{delaney03,reynolds07}.\nThe SNR's center is located at $\\alpha_\\mathrm{J2000}=17{\\rm h}30{\\rm m}41{\\rm s} ,\\ \n\\delta_\\mathrm{J2000}=-21^\\circ 29^\\prime 32^{\\prime\\prime}$, in Galactic coordinates\n$l=4.52^\\circ, b=6.82^\\circ$. The angular radius of the SNR is approximately 1.8{$^\\prime$}\\ (not taking\ninto account the \"Ears\").\n\nThe distance to the SNR is uncertain, with estimates varying from $d=3.2$~kpc \\citep{danziger80} to  \n$d=12$~kpc \\citep[][]{vandenbergh73}.\nThe long distance was based on the maximum visual magnitude of the supernova event of\n$V_\\mathrm{max}\\approx -2.5$, as estimated by \\citet{baade43} from the historical records.\nAfter correcting for absorption \\citep[$A_\\mathrm{V}=2.17-3.47$,][]{vandenbergh77,danziger80}, the peak\nmagnitude can then be compared to the  absolute magnitude of Type Ia supernovae of $M_\\mathrm{V}=-19.0 \\pm 0.5$.\nA Hubble parameter was adopted of $H_0=100$~km\/s\/Mpc.\n\nMore recent distance estimates, not based on the historical peak brightness \\index{peak brightness} of the\nsupernova,  vary widely, ranging from $d=3.9\\pm 1.4$~kpc \\citep{sankrit05}\nbased on combining proper motion and Doppler broadening of H$\\alpha$ emission, \n up to, or even beyond, 6-7~kpc based on the angular size of\nthe SNR, combined with the energetics of the supernova \\citep[][see {Section}~\\ref{sec:dynamics} for details]{aharonian08,chiotellis12,patnaude12}.\nRadio absorption measurements\ntoward the SNR are consistent with a distance in the range 4.8-6.4~kpc \\citep{reynoso99}.\n\nThe Galactic latitude of Kepler's SNR ($b=6.8${$^{\\circ}$}) implies a large height above the Galactic plane\nof $z=594 d_5$~pc (with $d_5$ the distance in units of 5~kpc). The large separation from  active star\nforming regions suggests that SN 1604 was Type Ia supernova, in agreement with the reconstructed\nsupernova light-curve, as already pointed out by \\citet{baade43}. However, the large amounts of\nnitrogen present in the optical nebula can only have come from material processed by the CNO nucleosynthesis\ncycle, suggesting a  stellar wind origin for the optical nebula. This lead \\citet{bandiera87} to suggest  that\nthe progenitor was a massive star, which escaped the Galactic plane with a high velocity. The high\nproper motion of the progenitor is in agreement with observations of the kinematics of the optical nebula\n({Section}~\\ref{sec:dynamics}). Given the historical light curve, \na Type II origin of SN 1604 was excluded, but a Type Ib \\index{Type Ib supernova} supernova\nwas consistent with both the light curve and the idea of a massive runaway star as the progenitor of the supernova \\citet{bandiera87}.\n\n\n\n\n\\begin{figure}\n\\centerline{\n\\includegraphics[width=0.85\\textwidth]{Fig_2_kepler_chandra_oxygen_fel_si.png}\n}\n\\caption{\nChandra X-ray image, with the three image channels  (red, green, blue) \ncorresponding\nto oxygen emission (0.5-0.7 keV), Fe-L emission \\index{Fe-L complex} (0.7-1 keV) and Si-K \nemission (1.7-1.9 keV), respectively. It can be seen that the Si-K emission (blue)\npeaks more outward than the the Fe-L shell emission (green). Peaks in the oxygen emission\n(red\/yellowish) occur at the outskirts in the northeast and along the bar, suggesting\nthat the emission comes from shock-heated, circumstellar material.\n\\label{fig:Xray}}\n\\end{figure}\n\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=0.8\\textwidth]{Fig_3_kepler_chandra_spec_b.pdf}}\n\\caption{\nThe Chandra ACIS-S X-ray spectrum of Kepler's SNR, observed in 2006.\nThe \"bump\" between 0.7- 1.2 keV consists mainly of Fe-L emission \\index{Fe-L complex}, consisting\nof complex of lines from Fe XVII to Fe XXIV. The prominence of this Fe-L complex\nin the spectrum of Kepler's SNR is indicative of a Type Ia origin.\\label{fig:xspec}\n}\n\\end{figure}\n\n\n\n\\begin{figure}\n\\centerline{\n\\includegraphics[width=0.85\\textwidth]{Fig_4_kepler_24mu_sik_cont.png}\n}\n\\caption{\nMultiwavelength image of Kepler's SNR. The red channel represents the 24$~\\mu$m dust emission as\nobserved by the Spitzer MIPS instrument \\citep[][]{blair07}. The green channel is the Si-K band (1.7-2.0 keV), representing\nthe bulk of the thermal X-ray emission, associated with Si-rich ejecta. The Si-K emission comes  on average from a smaller radius than the dust emission,\nindicating that the dust originates from the swept up circumstellar medium, rather than the ejecta.\nThe blue channel (enhanced with contours) shows the X-ray continuum emission in\nthe 4-6 keV band. In the SE the emission is dominated by synchrotron emission from close to the shock front. \\index{X-ray synchrotron radiation}\nIn the NW it is probably a mixture of synchrotron emission and thermal bremsstrahlung.\nThe X-ray bands are taken from the 2006 Chandra observations of Kepler's SNR \\citep{reynolds07}.\n\\label{fig:morphology}}\n\\end{figure}\n\n\\index{X-ray imaging spectroscopy}\n\\subsection{X-ray imaging spectroscopy and SN 1604 as a Type Ia supernova }\n\\label{sec:xrays}\nThe idea of a Type Ib supernova origin for SN 1604 was discredited with the advent of X-ray imaging\nspectroscopy with CCDs. These type of detectors were first used by\n\\index{ASCA X-ray satellite} ASCA, and it provides now still the most often used instrumentation for X-ray observatories\nsuch as \\index{Chandra X-ray observatory} \\index{XMM-Newton X-ray space observatory} \n\\index{Suzaku X-ray satellite} \nChandra, XMM-Newton, and Suzaku. \nThe X-ray spectrum of Kepler's SNR (Fig.~\\ref{fig:xspec}) shows that the\ndominant line emission is caused by  \\index{Fe-L complex} Fe-L transitions \\citep{kinugasa99,cassam04,reynolds07}, which is very indicative of\nType Ia SNRs, since they produce 0.3-1.3 {M$_\\odot$}\\ of Fe and Fe-group elements.\nIn addition line emission from intermediate mass elements \\index{intermediate mass elements}\nlike silicon, sulphur, argon and calcium, are prominent.\nUnlike Kepler's SNR, young core collapse SNRs on the other hand, show dominant emission from oxygen, neon and magnesium.\nRecently, \\citet{katsuda15} confirmed the high nitrogen \\index{nitrogen} abundance in Kepler's SNR using high resolution X-ray spectroscopy with the XMM-Newton reflective\ngrating spectrometer.\n\n\nThe spatial distribution of the  X-ray line emission indicates that the ejecta are layered according to element mass, with the intermediate mass elements \nlocated, on average, at a larger radius than iron \\citep[][illustrated in Fig.~\\ref{fig:Xray}]{cassam04}. This distribution is similar to that of other Type Ia SNRs like Tycho's SNR \\citep[SN\\,1572,][]{hwang97} \\index{SNR G120.1+1.4} \\index{Tycho's supernova remnant (SNR G120.1+1.4)} \n\\index{SN\\,1572 (SNR G120.1+1.4)}\nand\nSNR  0519-69.0 \\citep{kosenko10}, but Fe-L emission is more prominent in Kepler's than in Tycho's SNR, whereas SNR 0519-69.0 and\nSN\\,1006  display more \\index{SNR\\,0519-69.0} \\index{SN\\,1006 (SNR G327.6+14.6)} \\index{SNR G327.6+14.6} \noxygen line emission from the outer layers\nThe relatively high iron abundance in Kepler's SNR,\nas compared to other Type Ia SNRs, suggests that SN 1604 was an iron\/nickel-rich supernova, similar to the peculiar Type Ia supernova SN1991T \\citep[][but see Section~\\ref{sec:lightcurve}]{patnaude12}.\n\n\nRecently it has become possible to detect line emission from the uneven elements manganese (Mn) and chromium (Cr).\nThe Mn yields for supernovae are a result of the presence of neutron-rich nuclei in the progenitors, which ultimately is linked to the nitrogen abundance of\nthe progenitor at the main sequence. Hence, it can be used to infer the metallicity of the progenitor. \\citet{park13} used the Mn\/Cr ratio to show\nthat SN 1604 must originate from a star with super-solar metallicity, ruling out a halo star progenitor\n\nThe continuum X-ray emission of Kepler's SNR, best distinguished in the 4-6 keV band (blue in Fig.~\\ref{fig:morphology}), shows narrow filaments tracing the outer region of\nthe SNR. \\index{X-ray synchrotron filaments} Most of this emission, especially the narrow filaments in the SE,\n are thought to be caused by X-ray synchrotron emission from\n10-100 TeV shock accelerated electrons \\citep[][for a review]{helder12b}. The widths of the filaments can be used to estimate the post-shock \nmagnetic fields, indicating for Kepler's SNR a magnetic field strength of $B\\approx 120$~$\\mu$G. Since the 10-100~TeV lose their energy very fast in such  a magnetic field ($\\sim 10-20$~yr), the X-ray synchrotron radiation is an excellent tracer of  the shock front. Some of the X-ray synchrotron filaments\nare situated along the edges of the central bar (Fig.~\\ref{fig:morphology}), suggesting that we observe the shock front  under a\n(nearly) edge-on viewing angle. Given the position of the bar in the center of the SNR, this  viewing angle  indicates  \na peculiar, perhaps halter-like, morphology of the SNR in three dimensions \\citep{burkey13}.\n\n\\subsection{The circumstellar medium as studied in the optical and infrared}\n\\label{sec:opt_ir}\nThe shocked interstellar medium in Kepler's SNR  is best studied in the optical and infrared. The optical emission, dominated by\nhydrogen line emission, comes from northwestern part of the SNR and from the central bar. \\index{radiative shocks} \\index{non-radiative shocks}\nThe optical emission can be divided in emission caused by either radiative or non-radiative shocks \\citep{dennefeld82,fesen87,blair91,sankrit08}. \nRadiative shock emission is characterised by a mix of hydrogen line emission (Balmer and Lyman lines) and\nforbidden line emission from NII, OIII, and SII. The H$\\alpha$ line emission is in those cases  double peaked, since NII has among\nothers line emission at $\\lambda=654.9$~nm, close to H$\\alpha$ ($\\lambda=656.4$~nm).\nForbidden line emission comes from regions heated by slow shocks ($\\lesssim 200$~{km\\,s$^{-1}$}), \nwhich heat plasma to temperatures below $\\sim 10^6$~K. For this temperature regime radiative losses by optical\/UV line emission causes rapid cooling. \nFor Kepler's SNR the overall shock speed is several thousand {km\\,s$^{-1}$}. \nSo the radiative emission must come from dense knots \\citep[$n \\gtrsim 10^3$~{cm$^{-3}$}][]{leibowitz83}, in which locally the shock wave\nhas decelerated considerably. The [NII]\/hydrogen line ratios measured for radiative shocks in Kepler's SNR \nsuggests an overabundance in nitrogen of a factor 2-3.5  \\citep{leibowitz83,blair91}.\n\nThe non-radiative emission is caused by neutral hydrogen atoms entering the shock, after which they may not be immediately ionised, but first\nget into an excited state, or undergo will charge exchange  with shock-heated protons behind the shock front. H$\\alpha$ line spectra of \nnon-radiative shocks typically consist of  narrow-line emission caused by direct excitation of the neutrals, and broad-line emission caused by\nhot protons that just picked up an electron through charge exchange\nThe narrow line width and line center can be used to measure the plasma speed\nand temperature of the un-shocked gas, whereas the broad-line width can be used to measure the temperature immediately ($\\lesssim 10^{15}\/n_\\mathrm{p}$~cm)\nbehind the shock (see {Section}~\\ref{sec:dynamics}). The H$\\alpha$ flux of the non-radiative filaments can be used to infer densities as well,\nwhich appears to be of the order of $\\sim 10$~{cm$^{-3}$} \\citep{blair91}. This is surprisingly large, given the height above the Galactic plane\nof $594 d_5$~pc. \nThe optical emission from Kepler's SNR, therefore indicates that the SNR is interacting\nwith  dense, CNO processed material, which, given the isolated nature of the SNR high above the Galactic plane, \ncan only be understood by assuming that the circumstellar material originates from \nthe progenitor system of SN\\,1604.\n\nIn the infrared Kepler's SNR has been observed in the wavelength range from 12-100$\\mu$m by all major\ninfrared observatories \\index{Infrared Astronomical Satellite (IRAS)} \\index{Spitzer Space Telescope} \\index{Herschel Space observatory} \\index{Infrared Space Observatory (ISO)}\n\\citep[IRAS, ISO, Spitzer, Herschel, see][]{braun87,douvion01,blair07,gomez12,williams12}.\nAdditional emission for wavelength larger than 100~$\\mu$m, even up to 850~$\\mu$m may be present \\citep{morgan03,gomez12}, but\nat long wavelengths background subtraction is difficult, and even at 100~$\\mu$m there is already a factor three discrepancy\nbetween IRAS and Herschel based flux measurements \\citep{gomez09}.\n\n\\index{dust emission}\nInfrared emission from SNRs is caused by warm dust, heated by collisions with hot electrons and ions. The dust temperature\n is established by the equilibrium between collisional heating and thermal emission \\citep[][]{draine81,dwek87}. \n\n Both heating\nand emission depend on the surface area of the dust grains, and for large dust grains the temperature is independent of the size\nof the dust particles. However, since the wavelength of the radiation and the dust particle size can be comparable, the dust emission\ncannot be adequately described by Stefan-Boltzmann's law. Moreover, the emission spectrum also depends on the composition of the dust.\n\nInfrared photometry of Kepler's SNR indicates dust temperatures of 80--120~K \\citep{braun87,douvion01,blair07,gomez12,williams12},\nwhereas more detailed spectra obtained with the ISO-SWS and Spitzer-IRS spectrometers  \\citep[respectively][]{douvion01,williams12} \nindicate that the dust particles themselves are  silicates (silicon oxides). Interestingly, the modelling of the spectra gives rise\nto different interpretations. \\citet{douvion01} conclude that the dust comes from regions with densities of $n_\\mathrm{e}=500-7500$~cm$^{-3}$\nwith plasma temperatures of $(0.4-6)\\times 10^5$~K. On the other hand, \\citet{williams12} estimates the  densities to be  $n_\\mathrm{e}\\approx 5$~cm$^{-3}$ in the South\nto $n_\\mathrm{e}\\approx 50$~cm$^{-3}$ in the North, but assumes that the plasma temperatures are consistent with the X-ray measured\ntemperatures ($T_\\mathrm{e}\\approx 10^7$~K, and  ion temperatures a factor ten higher). Both conditions may arise\nin Kepler's SNRs as the radiative shocks are going through very dense regions, whereas the non-radiative shocks are responsible\nfor the X-ray emitting plasma. Indeed, several of the regions analysed by \\citet{williams12} contain non-radiative shock emission. The infrared spectrum\nof the region in which radiative shock emission is dominant is characterised by bright forbidden line emission in the infrared (among others  [Si II], [Fe II]). \nIt is  important to note  that the location of the dust emission indicates that\nall the dust is associated with shocked CSM,\nand not with supernova ejecta. This appears to be a characteristic of Type Ia supernovae, which do not seem to produce dust in their ejecta \\citep{williams12}.\nThis is in contrast to a core collapse SNR like Cas A \\citep{lagage96} or SN 1987A, where dust clearly is formed from\nsupernova ejecta. \\index{Cassiopeia A (Cas A, G111.7-2.1)}  \\index{SNR G111.7-2.1 (Cas A)}\n\\index{SN 1987A}\n\\index{dust emission}\n\\index{dust production}\n\n\n\\section{The dynamics of Kepler's SNR}\n\\label{sec:dynamics}\n\n\\subsection{Velocity measurements}\nThe earliest measurements of  velocities in Kepler SNR's were  radial velocity measurements of the bright optical\nknots, indicating line emission that is blue shifted by about 230 {km\\,s$^{-1}$} \\citep{minkowski59}. Optical proper motion studies\n\\citep{vandenbergh77,bandiera91}\nalso indicated velocity offsets,  indicating that the bright optical knots have a bulk velocity of 212 $d_5${km\\,s$^{-1}$}\\ in northwestern direction,\naway from the Galactic plane,\nwhereas the expansion velocity, based on spherical expansion directed away from the centre of the SNR, is only $(74\\pm 29)d_5$~{km\\,s$^{-1}$}\n\\citep{bandiera91}. The slow expansion velocity, corresponding to an expansion time scale of 32000~yr, indicates that the velocity\nof the optical knots reflects mostly the velocity of these knots prior to interaction with the SNR blast wave. \n\nThe more diffuse non-radiative H$\\alpha$ emission \\citep{fesen87} provides information on the local shock wave velocities through\nthe width of the broad H$\\alpha$- component; and the narrow line component\nprovides information on the velocity of the unshocked, partial neutral gas,\n immediately ahead of the shock.\nThe latter informs us that the unshocked gas is blue shifted with about $180$~{km\\,s$^{-1}$} \\citep{blair91,sollerman03}.\n\nThe width of the broad line H$\\alpha$ component is caused by thermal Doppler broadening \\index{Doppler broadening} of shock-heated protons.\nThe reported H$\\alpha$ line FWHM of $\\sim 1800$~{km\\,s$^{-1}$}\\ translates into  shock velocities of $1500-2000$~{km\\,s$^{-1}$} \\citep{blair91}.\nPart of the uncertainty is caused by the uncertainty in electron-ion temperature ratio. \\cite{sankrit05} combined this shock velocity measurement\nwith a measure proper motion of diffuse filament in the NW of $1.45\\pm 0.03$ {$^{\\prime\\prime}$}\/16.33 yr to infer a distance to Kepler's SNR\nof 3.0-5.0~kpc. However, note that some of the Doppler broadening widths measured by  \\cite{blair91} indicate \nshock velocities larger than 2000~{km\\,s$^{-1}$}. Notably, the filament labeled \"NW diffuse emission\" has a width of $3409 \\pm 454$~{km\\,s$^{-1}$}, which corresponds\nto a shock velocity range of 2300-4000~{km\\,s$^{-1}$} \\citep{vanadelsberg08}, or even larger if cosmic-ray acceleration \\index{cosmic rays} absorbs part of the shock\nenergy-flux into the shock, which results in lower than expected post-shock temperatures \\citep[e.g.][]{vink10a}. \n\n\n\\begin{figure}\n\\centerline{\\includegraphics[trim=0 -30 0 0 0,width=0.4\\textwidth]{Fig_5_left_expansion3.pdf}\n{\\includegraphics[width=0.6\\textwidth]{Fig_5_right_expansion2.pdf}}\n}\n\\caption{\nLeft: Difference image between images in the 1-1.5 keV band in 2000 and 2006, as observed by Chandra.\nThe shadow-like features are caused  by proper motions.\nRight: X-ray expansion measurements based on these two Chandra images in various energy bands \\citep[adapted from figure in][]{vink08b}.\n\\label{fig:expansion}\n}\n\\end{figure}\n\n\n\nThe optical velocity measurements pertain to the densest regions of Kepler's SNR. The overall\nexpansion of Kepler's SNR has been measured in the radio \\citep[][with VLA]{dickel88} and X-rays \\citep[][see Fig.~\\ref{fig:expansion}]{vink08b,katsuda08} using proper\nmotion measurements. These measurements show that the northwestern part is expanding at a slower rate than the southwestern part.\nA useful way to characterise the expansion rate is in terms of the expansion parameter \n\\begin{equation}\nm \\equiv \\frac{R\/t}{V},\n\\end{equation}\nwith $R$ the radius, $t$ the age of the SNR and $V$ the velocity. For proper motions both $R$ and $V$ depend in a similar way on the distance,\nso one can replace $R$ and $V$ with angular radius and proper motion. \nOne can also calculate the \\index{expansion age} expansion age $\\tau_\\mathrm{exp}\\equiv R\/V$, in which case $m=t\/\\tau_\\mathrm{exp}$, \\index{expansion parameter}\nwith $t$ the true age\nof the SNR. For SNRs in the Sedov-Taylor phase of the evolution we\nexpect $R \\propto t^{2\/5}$ and hence $m=2\/5$, whereas in the ejecta dominated phase one expects $m=0.7-0.9$ \\citep{chevalier82}.\nIt turns out that for Kepler's SNR in the Northwest $m=0.3-0.35$, whereas in the Southwest the expansion is faster, with $m=0.6-0.7$.\nThe expansion parameter in the northwest is lower than $m=2\/5$, which indicates that the shock  must have encountered a density enhancement.\n\\citet{vink08b} estimates that the  total excess mass toward the northwestern region is about 1~{M$_\\odot$}.\n\nThe proper measured proper motions translate in shock velocities of\n\\begin{equation*}\nV_\\mathrm{sh}= m 9920(\\theta\/2.79) d_5~\\mathrm{km\\ s^{-1}}\n\\end{equation*} \nfor an age of 400~yr, with $\\theta=2.79${$^\\prime$}\\ the angular radius of Kepler's SNR.\nGiven the measured values of $m$ the shock velocities around Kepler's SNR are, therefore, in the range of 2900-7000 $d_5$~{km\\,s$^{-1}$}.\n\n\\subsection{Hydrodynamical simulations of Kepler's SNR}\n\nAs discussed above the density enhancements in the north-northwestern regions, as well as the evidence for high density,\nnitrogen-rich material suggests that the progenitor system of SN\\,1604 suffered significant mass loss. \n\\citet{borkowski94} simulated this situation in the context of a Type Ib origin of SN\\,1604 and \\citet{chiotellis12} adapted it\nfor the case of a Type Ia scenario, which I will briefly describe here.\n\nThe basic idea is that a dense wind has emanated  from the secondary star with  a mass-loss rate of $\\dot{M} $~{M$_\\odot$}\\,yr$^{-1}$,\nand a wind velocity $v_\\mathrm{w}$. The wind must have lasted long enough for the wind material to reach the current\nradius of Kepler's SNR ($2.6 d_5$~pc) implying that the wind  must have prolonged for\nat least $t_\\mathrm{w} \\gtrsim 2.5 \\times 10^5 d_5 (v_\\mathrm{w}\/10~\\mathrm{km\\,s}^{-1}$)~yr.\nThe total mass lost by the system  is given by \n\\begin{equation}\nM_\\mathrm{wind} \\gtrsim \\dot{M}t_\\mathrm{w}= \\dot{M} \\frac{R}{v_\\mathrm{w}}=\n2.5\\left( \n\\frac{\\dot{M}}{10^{-5}\\mathrm{M_\\odot\\,yr^{-1}}}\\right)\n\\left(\\frac{v_\\mathrm{w}}{10\\ {\\rm km\\,s}^{-1}}\\right)^{-1}\nd_5 \\ \\mathrm{M_\\odot}.\n\\end{equation}\nThis means that the mass loss rate cannot have been much higher than $10^{-5}$~{M$_\\odot$}\\,yr$^{-1}$, as the total\nmass loss of the donor could not exceed the mass of a viable donor star for a Type Ia progenitor,\nwhich must lie in the range of 3-6~{M$_\\odot$}.\n\nThe optical proper motions of the knots indicate a space of around 250-280~{km\\,s$^{-1}$}\\ (212 $d_5$~{km\\,s$^{-1}$} in NW direction and 180 km\/s\ntoward us). This likely corresponds to the space velocity of the progenitor system.  The combination of a spherical wind\nand a space velocity gives rise to a bow shock around the system in the direction of the motion of the system.\nThe density of the wind as a function of radius $r$ is given by\n\\begin{equation}\n\\rho_\\mathrm{w}(r)=\\frac{\\dot{M}}{4\\pi r^2 v_\\mathrm{w}} = 7.8\\times 10^{-25}\n\\left(\\frac{\\dot{M}}{10^{-5}\\,{\\rm M_\\odot}}\\right) \\left(\\frac{v_\\mathrm{w}}{10\\ {\\rm km\\,s}^{-1}}\\right)^{-1} \\left(\\frac{R}{2.6\\,{\\rm pc}}\\right)^{-2}\\, {\\rm g\\,cm}^{-3}.\n\\end{equation}\nThe densities in the Northwest appear to be  around 50~cm$^{-3}$ ({Section}~\\ref{sec:opt_ir}). The density in the wind\nmay be lower, as the gas encountered by the SNR shock may have been shocked by the wind termination shock,\nand later by the SNR shock. However, we see that the mass loss rate cannot have been substantially lower\n$10^{-5}${M$_\\odot$}\\,yr$^{-1}$, as otherwise the wind densities would fall below the densities of the CSM component in Kepler's SNR.\n\nThe termination radius of the wind is now given by assuming  equilibrium between the ram\npressure of the wind $P_\\mathrm{w}=\\rho_\\mathrm{w}v_\\mathrm{w}^2$ and the ram\npressure of the interstellar medium, $P_\\mathrm{ISM}=\\rho_\\mathrm{ISM}v_\\mathrm{prog}^2$, with\n$v_\\mathrm{prog}$ the velocity of the progenitor system. Hence,\n\\begin{equation}\nR_\\mathrm{ts}=  1.9\\ \n\\left(\\frac{\\dot{M}}{10^{-5}\\,{\\rm M_\\odot}}\\right)^{1\/2}\n\\left(\n\\frac{v_\\mathrm{w}}{10\\,\\mathrm{km\\,s^{-1}}}\n\\right)^{1\/2} \n\\left(\\frac{v_\\mathrm{prog}}{250\\ \\mathrm{km\\,s^{-1}}}\\right)^{-1}\\left(\\frac{n_\\mathrm{ISM}}{10^{-3}\\, \\mathrm{cm^{-3}}}\\right)^{-1\/2}~\\mathrm{pc},\n\\end{equation}\nwith $n_\\mathrm{ISM}=10^{-3}$  an estimate for the ISM at 500~pc above the Galactic plane.\nThis estimate for the termination shock radius is in reasonable agreement with the radius of Kepler's SNR.\n\nThe simulations of \\citet{chiotellis12} show that  both the bow-shock like shape of Kepler's SNR as well as the measured expansion parameters\ncan be reasonably well reproduced if SN\\,1604 indeed went off inside a stellar wind bubble with a systematic velocity of 250~{km\\,s$^{-1}$}. \nIt was assumed that the ejecta mass was 1.4~{M$_\\odot$}, typical for an exploding white dwarf. \nThe simulations showed that for a distance of 4~kpc the radius of the termination shock must be around 2~pc. However,\na SNR with an \\index{explosion energy} explosion energy of $10^{51}$~erg  going off inside such a smaller bubble, would have moved through the wind bubble\nand the shock would have penetrated into the ISM, which is in disagreement with the observations. \nOne can reproduce the characteristic of Kepler's SNR at this distance, but only if the explosion \nenergy is reduced to $2\\times 10^{50}$~erg. This is in disagreement with the typical  Type Ia explosion energies of $1.2 \\times 10^{51}$~erg\n\\citep{woosley07}. Moreover, the high Fe content of Kepler's SNR rather suggest that SN\\,1604 was a relatively energetic type Ia event\n\\citep{park13}. This problem does not occur if Kepler's SNR has a distance of $\\gtrsim 6$~kpc.\n\nA related hydrodynamical study by \\citet{patnaude12}\nconcentrates  on the impact of hydrodynamics on the resulting X-ray spectra. The wind bubble evolution itself is not modelled, but  an \n$1\/r^2$ density\ndistribution is assumed. They confirm that normal Type Ia explosion energies are not consistent with a distance of Kepler's SNR\nof 4~kpc, but that instead a distance of  7~kpc should be considered. \\citet{toledo14} follow more closely the scenario of \\citet{chiotellis12},\nbut the simulations are in 3D and the wind loss is assumed to be anisotropic. As a result more structure is found in the simulated X-ray maps,\nsome of which may explain for example the central bar in Kepler's SNR (e.g. their Fig.~10). In general, the anisotropy of the wind\ncan result in a peculiar structure. \\citet{burkey13} simulated the evolution of the SNR inside a wind strongly varying\nas a function of the polar axis. However, the simulation does not take into account the effects of the velocity of the system as a whole,\nand the expansion parameter of the plasma.\n\n\\section{The progenitor system of SN\\,1604}\n\n\\subsection{Elevated circumstellar nitrogen abundances, silicates  and a single degenerate scenario for SN\\,1604}\n\\citet{chiotellis12} argued that the mass loss properties inferred for the progenitor system of SN\\,1604 suggest that the progenitor\nsystem consisted of CO white dwarf accreting wind material from a Asymptotic Giant Branch (AGB) \\index{Asymptotic Giant Branch (AGB) star} star. \nIndeed, the high mass\nloss rates and the wind expansion time scale are in good agreement with what is generally inferred for AGB stars.\nGiven the relatively high total\nmass loss ($\\sim 2.5${M$_\\odot$}, plus matter that has accreted onto the white dwarf), \nit does mean that the donor star must have been a relatively massive star at the main sequence. \n\nIn addition, the nitrogen-rich abundance of the wind,  and the silicate dust  (Sect.~\\ref{sec:opt_ir}) also gives clues\nabout the donor star. The nitrogen-richness can be explained by \\index{hot bottom burning} hot bottom burning \\citep[e.g.][]{karakas10}, a process in which\nhydrogen is burned through CNO cycle at the base of the outer convective envelope. This brings nitrogen to the surface.\nHot bottom burning occurs in stars with initial masses larger than 4-5~{M$_\\odot$}.\nSilicate dust \\index{dust emission} \\index{sillicates} predominantly forms in winds in which the carbon\/oxygen ratio is smaller than 1. Otherwise the oxygen \nbinds to carbon, making CO, leaving no oxygen for the build up of silicates. In this light it is interesting that \\citet{mcsaveney07}\nfound two AGB stars in the Large Magellanic Clouds, which have elevated nitrogen abundance and a $C\/O<1$. The masses\nof these two stars are inferred to be 4~{M$_\\odot$}\\ and 6~{M$_\\odot$}.\n\nTaken all the evidence together we seem to have a rather consistent scenario for the progenitor system of SN\\,1604 and the evolution\nof the SNR: a single degenerate white dwarf, accreting wind material from a rather massive AGB star. The mass of the donor\nthen implies that the white dwarf was also rather massive at the main sequence (5-6~{M$_\\odot$}). {Kepler's SNR provides therefore the\nbest case that at least some Type Ia supernovae are caused by single degenerate white dwarf systems.}\n\n\\subsection{Problems with a single degenerate Type Ia scenario for SN\\,1604}\n\nThere two problems that need to be solved for the single degenerate scenario for SN\\,1604, or even for any Type Ia binary scenario:\n\n1) The donor star must be relatively massive (4-6{M$_\\odot$}), and therefore a luminous star. A recent search by \\citet{kerzendorf14} did not reveal\nany bright enough star near the centre of Kepler's SNR. The former donor star \\index{donor star} is likely to still be bright, although, its outer envelope\nmay have been removed by the supernova blast. Its space velocity should be close to that of the progenitor system.\nThe lack of a donor star argues instead for a double degenerate explosion for SN\\,1604. A similar conclusion was\n drawn for other Type Ia SNRs, SNR 0509-675 and SN\\,1006 \\citep{schaefer12,gonzalez12}, \\index{SNR G327.6+14.6}\nfor which also  no bright donor stars have been found. For Tycho's SNR a donor star has been identified \\citep{ruiz04}, but this identification\nis disputed \\citep{kerzendorf13}. \\index{SNR G120.1+1.4}\n\n2) The velocity of the progenitor system must have been $\\sim 250$~{km\\,s$^{-1}$}. This implies that the system must have left\nthe Galactic plane around 3~million year ago. \nSuch a time scale is too short for  any Type Ia scenario, which should have evolutionary time scales of $>50$ Myr \\citep[e.g.][]{claeys14}.\n One can of course speculate that Kepler's SNR\nis bound to the Galactic plane, and has been oscillating far up and below the Galactic plane for  considerable longer times.\nThis still creates, however, the problem of how to slingshot away a binary system with 250~{km\\,s$^{-1}$}. Runaway stars \\index{runaway stars}\nare usually created by\nclose interactions of triple systems, but in 90\\% of the cases single stars are being ejected,\nand the highest velocities are not expected to be obtained by the ejected binaries \\citep{leonhard90}. Moreover, in the AGB scenario\nfor SN\\,1604 the binary system must have been relatively wide \\citep{chiotellis12}. Nevertheless, observationally some high velocity binaries are known.\nA well known example is the Mira system, which has a space velocity of about 100~{km\\,s$^{-1}$}, and which consists also of an AGB star and\na probable white dwarf \\citep{martin07}.\n\nProblem 1) is a problem for only the single degenerate scenario, but problem 2) is a problem for any Type Ia  scenario involving\na binary system. \n\n\\subsection{Was SN\\,1604 a core-degenerate Type Ia explosion?}\n\\label{sec:core}\nA possible solution to problem 1 is so-called core degenerate scenario \\index{core degenerate scenario for Type Ia supernovae} \n\\citep{ilkov12,tsebrenko13} according to which a single degenerate system\nmay form a common envelope at the end of the evolution of the secondary star (an AGB star). The white dwarf\nand the core of the AGB star merge at the very end of the AGB star evolution, or shortly thereafter. The resulting white dwarf may\nbe super-Chandrasekhar, but gravitational collapse followed by an explosion can be delayed due to a rapid rotation of the massive white dwarf.\n Indeed, this scenario is somewhat\nintermediate between the single degenerate and the double degenerate scenario, as the supernova is clearly caused by a double\ndegenerate merger, but the time scales involved can be  similar to the single degenerate scenario. A consequence of this scenario\nis that the supernova mass does not have to be 1.38 {M$_\\odot$}, but could be more. Another possible consequence for SN\\,1604 may be that\nthere was high density material surrounding the progenitor, consisting of the envelope removed by the common envelope event. \nNo light echo has yet been detected of SN\\,1604 \\citep{rest08}, but if it will be detected and an optical spectrum can be obtained,\nperhaps we will learn more about the event itself and the presence of material in the immediate vicinity of the supernova.\n\n\\subsection{What can we learn from the historical light curve of SN\\,1604?}\n\\label{sec:lightcurve}\n\nUp to the 1990ies the hope was that the historical light curves \\index{supernova light curve} \\index{supernova light curve (SN 1604)} \nof SN\\,1572 and SN\\,1604 could shed  light on the maximum\nbrightness of Type Ia supernovae, and hence be used to constrain the Hubble parameter \\citep[e.g.][]{danziger80,schaefer96}. \nNowadays the Hubble parameter is known\nto sufficient detail, but the historical light curves remain of interest, as it may lead to sub-typing of the historical supernovae and\nprovide estimates on the distance of  SNRs ({Section}.~\\ref{sec:properties}). \\citet{baade43} was the first one to determine\nthe historical light curve of SN\\,1604, and this was later augmented with magnitude estimates based on Korean observations\nby \\citet{clark77}. These data have been compared to supernovae light curve by among others \\citet{schaefer96}, and most\nrecently \\citet{katsuda15}.\n\nThe historical light curve of SN\\,1604 is shown in Fig.~\\ref{fig:lc} with a typical magnitude error of 0.25-0.5. The historical light curve\nis compared to a number of recently observed Type Ia supernovae (Table~\\ref{tab:vmax} for an overview). \nThese lightcurves have been scaled to the light curve of SN\\,1604\nby taking into account the distance moduli ($\\mu$) and visual extinction parameters ($A_V$) of the supernovae, and assuming\na distance of 5~kpc to SN\\,1604 (somewhat in the middle of current estimates, Sect.~\\ref{sec:properties}) and $A_V=2.8$ for the historical supernova.\nThe latter is based on the estimate of \\citet{blair91} for the non-radiative shock \\index{non-radiative shocks} \nemission.\n\nAccording to Type supernova light curve models, the peak magnitude of Type Ia supernovae is determined by\nthe amount of radioactive $^{56}$Ni produced in the explosion (typically $\\sim 0.7${M$_\\odot$}), whereas the steepness\nof the decline after maximum is determined by the diffusion of heat generated by radioactivity to the photosphere of\nthe supernova; the more mass the slower the decline \\citep{arnett79,cappellaro97,dado15}. Hence,\nthe low peak magnitude and fast decline of SN\\,1991bg has been attributed to a sub-Chandrasekhar Type Ia explosion \\index{sub-Chandrasekhar Type Ia supernova}\n\\citep{turatto96,drout13}, whereas the very bright and slow declining SN\\,2009dc is thought to indicate \na super-Chandrasekhar \\index{super-Chandrasekhar Type Ia supernova} Type Ia \\citep{silverman11}. \nThe latter has a light curve that is even broader\nand brighter than SN\\,1991T, and its light curve has to be scaled by $\\Delta V=0.5$ in order to provide even an approximate\nfit to SN\\,1604.\n\n\nRecently \\citet{patnaude12} suggested that SN\\,1604 may have been similar to the bright Type Ia SN\\,1991T.\nFor a distance of 5~kpc the peak absolute peak magnitude of SN\\,1991T and SN\\,1604 match reasonably\nwell, but SN\\,1991T has a much brighter late time light curve, and a slower post maximum decline. \nIn order to make SN\\,1604 as luminous as\nSN\\,1991T at late times,  the distance of SN\\,1604 has to be $\\sim$7~kpc.\nSo the light curve of SN\\,1991T \\index{SN 1991T} does not seem to be an overall good template for SN\\,1604. \nThe situation is even worse for the probable super-Chandrasekhar Type Ia SN,2009dc, which, \neven when scaled down in brightness,\nshows a profile that is too broad to fit the historical light curve.\n\nNormal Type Ia supernova light curves (SN\\,1994D, SN\\,1996X, SN\\,2004eo) are in reasonable agreement with\nthe historical light curve of SN\\,1604. In particular, they fit reasonably well around the peak and they agree\nreasonably well at very late times (300--400 days), but they are brighter than SN\\,1604 for days 100--200.\n\nAt the faint extreme, there are peculiar  Type Ia supernovae such as SN\\,2009dc, whose light curve has to shifted up by\n$\\Delta V=1.5$ in Fig.~\\ref{fig:lc} in order to have the peak brightness coincide with the peak of SN\\,1604.\nSuch a shift would mean that SN\\,1604 would be at a distance of 2.5~kpc,\nwhich is lower than most independent distance estimates (Sect.~\\ref{sec:properties}).\nAs \\citet{katsuda15} showed, when properly scaled to match the peak magnitude of SN\\,1604, \nthe light curve of SN\\,2009dc is rather similar to the light curve of SN\\,1604,\nas both are fastly declining in brightness after peak brightness. \\index{SN 1991bg} \\index{SN 2005ek}\nA similarly fast decline Type Ia was SN\\,2005ek, whose light curve was used to estimate the ejecta mass of \n0.3-0.7~{M$_\\odot$}, and a kinetic energy of $(2-5)\\times 10^{50}$~erg. However, SN\\,2005ek was poor in $^{56}$Ni,\nat odds with what we know of SN\\,1604 (Sect.~\\ref{sec:xrays}). \nPerhaps one can still fit SN\\,1604 with a faint Type Ia model, if one allows for a small\n(sub-Chandrasekhar) progenitor mass, but with a large fraction of $^{56}$Ni.\nThis would make a fainter supernova, with a rapid decline, but it could\nstill allow for the high iron fraction in Kepler's SNR.\nThe reason this may work is that in X-rays the fraction of iron can be much better\ndetermined than the absolute mass of iron.\nA consequence is then that the distance of Kepler's SNR is  less than 5~kpc,\nas the supernova is fainter, but \nthis can be  accommodated\nwith hydrodynamical models since the supernova is then probably\nalso a sub-energetic Type Ia \\citep[$<10^{51}$~erg,][]{chiotellis12}.\n \nSo what can we conclude from the historical light curve, with some reserve as the magnitudes derived from the historical \nobservations have to be treated with some caution, and the extinction is uncertain?\nFirst of all, the light curves of normal Type Ia seem overall in better agreement with the light curve of SN\\,1604 than\nthe light curve of SN\\,1991T.\nSecondly, the light curve of SN\\,1604 may have declined rather rapidly after peak brightness. The evidence\nfor this is uncertain, but could indicate a sub-Chandrasekhar explosion. However,\nin order to account for the high iron fraction of Kepler's SNR, SN\\,1991bg and SN\\,2005ek are probably\nnot the right models. Moreover, a sub-Chandrasekhar model, would result in a shorter distance of Kepler's SNR.\n\nA possibility that needs to be explored better is\nwhether it is possible that the fast decline of SN\\,1604 could perhaps be the result of an overall faster expansion, and\/or\na distribution of nickel closer to the photosphere. This helps the heat generated by $^{56}$Ni to diffuse  faster to the photosphere,\nresulting in a more rapid post-maximum decline.\nIn any case, the fast decline is inconsistent with a super-Chandrasekhar explosion. This is of interest given the suggestion that\nSN\\,1604 could provide a evidence for the core-degenerate model  \\citep[Sect.~\\ref{sec:core}][]{tsebrenko13},\nwhich can result in super-Chandrasekhar Type Ia explosions.\n\n\\begin{table}\n  \n  \\caption{A comparison of SN\\,1604 to the Type Ia SNe shown in Fig.~\\ref{fig:lc}. \\label{tab:vmax}\n  }\n  \\scriptsize\n  \\begin{tabular}{l rcrcll}\\hline\\hline\\noalign{\\smallskip}\n    &\t$V_\\mathrm{max}$\t&\t$A_V$\t&\t$m-M$\t&\t$M_V$\t&\tRemark\t&\tReferences\t\\\\\n    \\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n    SN\\,1604:&\\\\\n    @ 3 kpc\t&\t-2.75\t&\t2.8\t&\t12.39\t&\t-17.94\t&\t\t\t\t&\t{\\scriptsize\t\\citet{blair91}\t}\t\\\\\n&\t-2.75\t&\t3.5\t&\t12.39\t&\t-18.64\t&\t\t\t\t&\t{\\scriptsize\t\\citet{danziger80}\t}\t\\\\\n @ 6 kpc\t&\t-2.75\t&\t2.8\t&\t13.89\t&\t-19.44\t&\t\t\t\t&\t{\\scriptsize\t\\citet{blair91}\t}\t\\\\\n \t&\t-2.75\t&\t3.5\t&\t13.89\t&\t-20.14\t&\t\t\t\t&\t{\\scriptsize\t\\citet{danziger80}\t}\t\\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\nSN\\,1991T\t&\\ \\ 11.50\t& \\ \\ 0.37\t& \\ \\ \t30.57\t&\\ \\\t-19.44 \\ \\ &\t{\\scriptsize\tBright Type Ia\t}\t&\t{\\scriptsize\t\\citet{sasdelli14}\t}\t\\\\\nSN\\,1991bg \\index{SN 1991b}\t&\t13.96\t& \\ \\ \t0.12\t&\t31.61\t&\t-17.77\t&{\\scriptsize\tSub-Chandrasekhar\t}\t&\t{\\scriptsize\t\\citet{turatto96,gomez04}\t}\t\\\\\nSN\\,1994D\t&\t11.90\t& \\ \\ \t0.08\t&\t30.82\t&\t-19.00\t&\t{\\scriptsize\tNormal Type Ia\t}\t&\t{\\scriptsize\t\\citet{cappellaro97,feldmeier07}\t}\t\\\\\nSN\\,1996X\t&\t13.21\t& \\ \\ \t0.22\t&\t31.15\t&\t-18.16\t&\t{\\scriptsize\tNormal Type Ia\t}\t&\t{\\scriptsize\t\\citet{salvo01}\t}\t\\\\\nSN\\,2004eo\t&\t15.35\t&\t0.34\t&\t34.12\t&\t-19.11\t&\t{\\scriptsize\tSub-Chandrasekhar?\t}\t&\t{\\scriptsize\t\\citet{pastorello07}\t}\t\\\\\nSN\\,2009dc\t&\t15.35\t& \\ \\ \t0.31\t&\t34.93\t&\t-19.89\t&\t{\\scriptsize\tSuper-Chandrasekhar\t}\t&\t{\\scriptsize\t\\citet{silverman11}\t}\t\\\\\n\\noalign{\\smallskip}\\hline\n  \\end{tabular}\n\\end{table}\n\n\n\\begin{figure}\n\\centerline{\n\\includegraphics[trim=50 50 50 50,clip=true,width=1.\\textwidth]{Fig_6_sn1604_lc.pdf}}\n\\caption{The light curve of SN\\,1604 based on the European observations as interpreted by \\citet[][black data points]{baade43} and\nthe Korean observations collected by \\citet[][red data points]{stephenson02}.\nFor comparison the light curves of several other Type Ia supernovae have been over-plotted.\nThese  curves have been scaled to that of SN\\,1604, assuming a distance for SN\\,1604 of 5~kpc and $A_V=2.8$.\nMore or less normal Type Ia supernovae light curves have been assigned bluish colours.\nSee Table~\\ref{tab:vmax} for details.\n\\label{fig:lc}\n}\n\\end{figure}\n\n\n\\section{Conclusions}\n\nThe remnant of SN\\,1604, Kepler's SNR, is one of the most remarkable SNRs, with its high bulk velocity of ~250~{km\\,s$^{-1}$}.\nSN\\,1604 is now generally regarded to  have been a Type Ia supernova, whose\nremnant is interacting with wind material from its progenitor system. One can even infer that the donor star must have been a 4-6 {M$_\\odot$}\\ star that\nhad evolved to the AGB phase. But this single degenerate Type Ia scenario for SN\\,1604 has only one problem: a surviving donor has not been detected.\nA core-degenerate supernova scenario, may offer a viable alternative theory, but it has its own problems.\nWhatever the explosion scenario, the high space velocity of the progenitor system remains mysterious, as it requires the ejection\nof a binary system out of the Galactic plane with $\\sim 250$~{km\\,s$^{-1}$}.\n\nIn Johannes Kepler's age SN\\,1604 was recognised as  a unique, mysterious event that was thought to have profound implications for mankind. \nWe may no longer think that SN\\,1604 is of prime importance for mankind, but SN\\,1604 is of prime\nimportance for understanding Type Ia supernovae.  \nSN\\,1604 has, even more than 400 yr after discovery, still profound implications for understanding cosmic phenomena, and\nit has still not shared all its mysteries with us.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\chapter{\\MakeUppercase{R-FFT: Functional Split at IFFT\/FFT in Unified LTE and Cable Access \n\t\tNetwork}}\n\n\\section{\\MakeUppercase{Introduction}} \n\\subsection{Motivation}\nToday the user Internet connectivity \nhas been dominated by the cellular wireless and \ncable\/DSL access network technologies \nFor a long time, wireless technologies, such as LTE \nhave enjoyed large growth opportunities due to the\nproliferation of wireless smart phones and hand-held devices \nin contrast to the cable technologies~\\cite{kuzlu2013assessment}.\nWith the recent development of DOCSIS~3.1 specifications for cable\ntechnologies, Multi-System Operators (MSOs)\nare able to offer the services which are equivalent \nto the advance wireless technologies supporting gigabit \nconnectivity to the users in both upstream and downstream~\\cite{Hamzeh2015}. \nThe next generation 5G technology~\\cite{agiwal2016next} \nfocuses on the unification of heterogeneous platforms \nat the access, backhaul and core networks especially through the \nsoftwarization and virtualization of the network infrastructures.\nTowards this end, integration of cable and wireless \ntechnologies could be the first step\nin the process of unifying the heterogeneous access platform.\n5G is also envisioned to support a wide range of applications \nranging from lower data rate Internet of Things (IoT) to\nultra-reliable health monitoring, and \nlarger data rates for 4K HD video streaming~\\cite{chen2015virtual}.\nChallenges in the 5G technologies~\\cite{chin2014emerging} include backhaul complexity\nresulting from densification of radio nodes, \ninterference with the neighboring cells \nand network deployment costs~\\cite{Chih-Lin2015}. \nPromising technologies, such as Software Defined \nOptical Networking (SDN)~\\cite{thyagaturu2016software} and\nNetwork Function Virtualization (NFV)~\\cite{blenk2016survey} \nare not only addressing the challenges posed by 5G but also \nfacilitating the faster integration of \nheterogeneous-access networks while supporting the larger date rate, \nlower latencies and high reliability.\nMoreover, current network infrastructures for both wireless and cable\ntechnologies can rely on common optical fiber technologies, such as \noptical digital Ethernet~\\cite{gomes2016new}. \nTherefore, MSOs are showing large interests in\nconverging to a common access platform through generic \nprotocols and standardization,  \nsuch as CPRI~\\cite{de2016overview} for integration of heterogeneous \ninfrastructures.\n\n\\subsection{Overview of Softwarization}\n\\subsubsection{Cloud-Radio Access Networks (CRAN)}\nSoftwarization of network functions can fundamentally \nreduce implementation complexity while increasing the \nflexibility~\\cite{galis2013softwarization}. Software implementation of \nthe RAN functions\nin a cloud environment is referred to as Cloud-RAN \nor CRAN~\\cite{makhanbet2016overview}.\nIn the CRAN, the functional implementation of the \nRadio Access Network (RAN) is split \nbetween Remote Radio Units (RRUs) which corresponds to\nantenna units with RF passband processing capabilities\nfor the physical transmission of the signal, \nand Base Band Units (BBUs) which \nimplements the complementary baseband signal processing.\nCommunication between CRAN and RRU is typically enabled by\noptical technologies, such as Passive Optical Networks (PON) and\nOptical Wavelength Division Multiplexing.\nTraditionally, the connection between RRU and BBU is established \nby a cable RF link. Cable RF contributes towards considerable signal degrade \nfor the larger distances between RRU and BBU.\nIn contrast to downlink signals, \nsignal degrade in the uplink direction has more negative consequences \ndue to the lower signal levels.\nThe signal degrade between BBU and RRU can be avoided by digitizing the \nRF signal and transmitting the digitized symbols over an optical fiber.\n\nAs noted earlier, important aspect of the RRUs in the context of\nCRAN is the reduced complexity in radio equipment hardware \nwhich results in lower CAPEX\/OPEX for\nthe large scale deployment of small cells.\nRRUs in a building environment are also\nreferred to as Distributed Antenna Systems (DAS)~\\cite{heath2013current}. \nDAS play an important role in realizing the large number of \nsmall cell deployments in meeting the coverage and capacity demands for\nindoor applications. \nCentralization of BBU can \nsupport multiple RRUs which can provide a common platform for \nthe centralized management of resources. \nBBUs are typically implemented on a generic computing \nhardware, where by a specific virtual machine (VM) \nimplements the base band processing operations \non a virtualized hardware~\\cite{dai2014uplink}.\n\n\\subsubsection{Distributed Converged Cable Access Platform (DCCAP) Architectures}\nConnectivity to the Cable Modems (CM) in the Hybrid Fiber Coax (HFC) network \nis delivered in part by optical and in part by cable segment.  \nThe Cable Modem Termination System (CMTS)\nconnects the CMs to Central Office (CO)\/headend\nthrough the protocol defined by Data Over Cable Service \nInterface Specifications (DOCSIS).\nIn the traditional HFC network deployments, analog signals from the\nCMTS are carried over an optical link to a remote analog fiber node where\nthe signal is converted to analog RF signal which can be \ntransmitted over the cable link.\nBut, an analog signal over an optical fiber gradually degrades\nas the optical propagation distance is increased \nlimiting the effective HFC operational distance (i.e., from the headend to CM). \nModular Headend Architecture version 2 (MHAv2) allows \nCMTS functions to be implemented in a modular fashion allowing \nsome of the modular entities to be implemented at the newly \ndesigned digital remote nodes. \nIn order to improve the signal quality over the cable segment,\nthe remote nodes are connected to headend through the digital \noptical link where the remote node can be deployed in close\nproximity to the users reducing the effective cable length.\nDistributed Converged Cable Access Platform (DCCAP) \narchitecture implements the modular CMTS functions at the\nremote nodes.\nRemote node is similar to a RRU in the CRAN whereby the \nRAN implementation at the cloud is similar to \nCMTS implementation at the headend.\nThe Remote-PHY (R-PHY) and Remote-MACPHY (R-MACPHY) technology \nof the DCCAP \nextends the modular CMTS and \nmodular headend architectures with a \ngoal to reduce the complexity and increasing \nserviceability of the cable \ninfrastructure. In particular, R-PHY implements\nthe DOCSIS PHY functions on a \nremote node, Remote-PHY Device (RPD), \nwhile centralizing the MAC and higher layer network functions at the headend. \nWhereas, the R-MAC implements the DOCSIS PHY and MAC \nfunctions on a remote node, Remote-MACPHY Device (PMD),\nwhile centralizing the IP forwarding\nand higher layer network\nfunctions at the headend. \nCentralized DOCSIS functions for RPD and RMD \ncan be implemented as a software on a \nvirtualized entity either at the headend or cloud.\n\nThe critical aspect of CRAN and DCCAP in the time domain I\/Q\ntransport are the lower latency and higher data rate requirements\non the optical fiber~\\cite{Wubben2014}.\nA constant high bit rate and low latency \nconnection must be maintained constantly between \nbaseband processing and radio node regardless of the user traffic.\nThe analog RF signals must be transmitted and received at all times even \nwhen there is no user activity. For e.g., the passband signal with the \ncell broadcast information, reference or pilot tones must be transmitted always. \nThus, I\/Q samples of the RF passband must always be transported at the \nconstant rate at all the times. \nMoreover, the requirements of an optical fiber\nincreases linearly with increase in the \nnumber of nodes. Therefore, numerous  \ntechniques such as~\\cite{nieman2013time, nanba2013new, guo2013lte, guo2012cpri} \nhave been proposed to enable the dynamic compression of RF I\/Q samples \nfor the effective transmissions over the optical fiber.\nHowever the compression techniques are lossy because of the quantization of \nRF signal reducing the sensitivity of the receiver in the upstream.\nNevertheless, the data rate requirements between cloud and radio unit\nleads to the dedicated deployments of\noptical fiber connections and static allocations of resources \nfor the CRAN and DCCAP. Functional split architectures~\\cite{Wubben2014,maeder2014towards} \ntransport the upper layer\ndata to the radio unit such that the constraints on\nthe optical connection are relaxed. However, the additional processing at the \nradio unit increases the complexity at the node as well as the flexibility\nof the infrastructure is reduced as compared to the traditional \nCRAN is due to the distributed implementation of split RAN functions. \n\n\\subsection{Contributions and Organization}\nIn this article, we primarily focus on the trade offs and benefits of \nthe functional-split in both LTE and cable networks. In particular, \nwe focus on the split-PHY architecture which implements the IFFT\/FFT at\nthe radio units of both LTE and cable networks. \nTowards this end, the main contributions of this article are as follows:\n\\begin{itemize}\n\t\\item[i)] Overview and trade off discussions on the functional split \n\tand split-PHY architectures for both LTE and cable networks. \n\tPresented in Sec.~\\ref{sec:backgroud}.\n\t\\item[ii)] A novel cross-split interaction \n\tmechanism for remote caching and prefetching of QAM symbols\n\tto reduce the data rate required for the \n\tI\/Q transmissions between cloud and \n\tradio unit, especially in the \\textit{downstream} direction \n\twhile implementing the IFFT\/FFT at the remote nodes. \n\tPresented in Sec.~\\ref{sec:split:inter}.\n\t\\item[iii)] A novel mechanism to commonly share the \n\tinfrastructure resources among LTE and cable networks \n\tThe discussion also includes the general timing analysis \n\tfor interleaving multiple IFFT\/FFT computations \n\tof LTE and cable on a single IFFT\/FFT module.\n\tPresented in Sec.~\\ref{sec:common:infra}.\n\t\\item[v)] In the end, we present the evaluation of \n\tFFT sharing mechanism and mean packet \n\tdelay performance evaluation of \n\tthe radio unit which implements IFFT\/ FFT for the \n\tsimultaneous LTE and cable operation in comparison\n\tto existing R-PHY technology.\n\tPresented in Sec.~\\ref{sec:perf:eval}.  \n\\end{itemize} \n\n\\subsection{Related work}\nChecko et al.~\\cite{checko2015cloud} has conducted the\nCRAN technology review which includes advantages, challenges, \nSDN\/NFV applications, as well as the compression techniques \nfor I\/Q transport over the fronthaul network. \nA more detailed discussions on the internals of the \nBBU and RRU units has been discussed by Wu et al. in~\\cite{wu2015cloud}.\nThey show that cooperative signal processing in the CRAN \nachieves better spectral efficiency and improves utilization \nas compared to traditional networks. \nCRAN also provides a platform to provide computing services \nto the users through the access network. \nA comprehensive survey on Mobile Cloud Computing (MCC) in the area\nof CRAN has been conducted by \nFernando et al.~\\cite{fernando2013mobile}. \nFundamentals of the functional and protocol split uplink as well as the \ndownlink and uplink compression techniques has been \npresented in~\\cite{peng2016recent}. The impact of overhead due to \npackatization on the data rate and latency \nof the fronthaul link while supporting \nvarious functional split in the CRAN\nhas been presented in~\\cite{chang2016impact}. \n\nMiyamoto et al.in their \nwireless performance evaluation~\\cite{miyamoto2016analysis}  \nclaim that their proposed split-PHY architecture reduces the \nfronthaul bandwidth by up to 97\\% compared to traditional\nCRAN with the penalty of 2~dB SNR. However, their proposed\nsplit-PHY architecture requires specialized \ntransceiver designs and optical transmission mechanisms\nwhich may increase the cost of CRAN deployments.\nIn a typical RRH deployment scenario, \nseveral RRHs experience different\nload due to the independent channel qualities of the UEs reducing\nthe overall load RRH. \nTherefore, especially when the higher functional splits are considered,\nall the RRHs connected to the CRAN would likely\nnot result in the peak load.\nUser specific traffic variation can provide the  \nmultiplexing opportunity and the impact of such multiplexing \nfor the RRH deployments has been studied by \nChang et al.~\\cite{chang2016impact}.\nFurther to split-PHY, Nishihara et al.~\\cite{nishihara2016study} \nhave evaluated the performance of \nMACPHY-split where both MAC and PHY are implemented at the RRH. \nTheir simulation results show\nthat MACPHY-split approach reduces the bandwidth from\n10~Gbps to 600~Mbps while supporting the multiplexing.\n\nCRAN involve large number of RRH deployments \nwhich can potentially\nbe the source of large power consumptions. \nEfforts to reduce the carbon foot print over the\ncellular networks supporting both CRAN as well as traditional\ndeployments has been presented in~\\cite{hasan2011green, oh2013dynamic,wu2015energy,de2014enabling}. \nWhereas the energy saving mechanism in the wireless\nnetworks has been presented in~\\cite{suarez2012overview}.\nStephen et al.~\\cite{stephen2016green} have designed an \ncache enabled OFDM resource allocation mechanism which saves\nthe transmissions over the fronthaul link. \nHowever, the content\nhas to be saved in parts marked with identifiers in the remote node\nIn contrast to all the previous CRAN studies, we have\nconsidered the caching of\nresource elements at the RRH on the split-PHY architecture \nto reduce the data rate over the fronthaul link potentially \nsaving the energy consumption when \nuser data is not present.\n\nComplementary to the wireless cellular networks\nHFC network provides the broadband access to the residential \nusers. Bisdikian et al.~\\cite{bisdikian1996cable}\nhas presented the overview of initial designs and protocol \nmechanisms for the cable networks. \nAs compared to the earlier designs, current and \nfuture broadband access networks extends the\ncapabilities of connectivity to Gigabit \nspeeds~\\cite{effenberger2016future}. \nEven further, present cable networks can be designed to support cellular \nnetworks services as described in Gambini et al.~\\cite{Gambini2012, gambini2010lte}. \nThe economic benefits of infrastructure sharing between residential wired and \ncellular wireless networks has been identified in~\\cite{pereira2012infrastructure}.\nAdditionally, economic benefits form the\nintegration of LTE and DOCSIS has also \nbeen discussed in~\\cite{gruber2014broadband}.\n\nArticles \\cite{liu2015bandwidth,zeng2016demonstration,liu2016cpri} have\ndemonstrated efficient transmissions of CPRI equivalent \n fronthaul data up  to 256 Gbps using the optical bandwidth of 10 GHz,\nespecially for supporting MIMO LTE applications. \nHowever, the proposed mechanism in the articles use specialized \noptical transmission with constant traffic over the optical link \nwithout allowing the possibility of multiplexing. In addition, \nwe propose the novel remote caching and prefetching of QAM symbols, \nwhich saves the fiber transmissions. Especially, when there are no \nusers, optical fiber connectivity can be effectively turned off to \nsave the power or share the resources with other nodes. \nIn comparison to most of the access networks, such as, \nmacro base stations, DAS, femto cells, and HFC,\nthe traditional macro base \nstation backhauled through a microwave link at low traffic\nresults in the most \nenergy economical option~\\cite{fiorani2016joint}.\nAs cable networks are one of the sources for growing\ncarbon footprint in the Internet connectivity,\nthere has been several studies proposed to address the issue\nsuch as~\\cite{zhu2011novel,Zhu2012,Lu2013}.\n\n\\section{\\MakeUppercase{Background on Functional Split in LTE and Cable Networks}} \n\\label{sec:backgroud}\n\\subsubsection{Downstream v\/s Upstream}\n\\paragraph{Upstream} In the upstream direction the RRU\nreceives the RF signal transmitted from the users. This analog\npassband signal is down converted to baseband and digitized \nfor the transmission to BBU for baseband processing. Unlike the \ncable link in the traditional cellar network and antenna infrastructures,\nthe CRAN connects the BBU and RRU with an digital optical fiber. \nThe cable link adds significant attenuation to the upstream signal especially\ndue to the low level of received signal from the devices. Whereas, the\ndigital fiber does not constitute towards the attenuation loss as it \ncarries the signal in the digital form. To note, an extreme care \nis needed at the RRUs for digitizing the uplink signal from the users \nas an additional loss cannot be afforded due to the low signal \nlevel of uplink RF signal at the RRU. For example, if the cable link \naccounts for $2$~dB of loss and noise floor is $-120$dB, then the received\nsignal at the RRU connected a BBU over cable link must be $\\leq-118$~dB \nfor the successful detection, whereas the received signal can be $\\leq \n-120$~dB if the RRU is connected to a BBU \nthrough the digital fronthaul link, thus increasing the dynamic range of the system by 2~dB. \nAlthough Single Carrier Orthogonal\nFrequency Multiplexing (SC-OFDM) \nuplink modulation format is used in the typical deployments, \nthe technology is advancing towards the uplink OFDM systems especially\nfor the MIMO applications~\\cite{studer2016quantized, pitarokoilis2016performance}.\nTherefore, in the scope of this article we focus on the symmetrical OFDM systems\nin both upstream and downstream directions. In addition to the increased dynamic \nrange, the processing of upstream \nfor the detection and extraction of information from the RF \nuplink signals can be centrally processed at the cloud on the generic hardware, \nsuch as the general purpose processors.\n\n\\paragraph{Downstream} In the downstream direction BBU sends the \ninformation to RRUs for the generation of passband signal to\ntransmit over the physical antennas. The RRUs can easily set the \ntransmit power level gain states for RF signals. As in the upstream direction,\nthere is no significant difference in terms of \nsignal generation or the dynamic range of the systems \nbetween cable and digital fronthaul links.\nSimilar to the centralized processing of upstream at the cloud, \nthe information is centrally processed on a generic hardware, such as the general \npurpose processors to generate the baseband downlink signals.\n\n\\subsection{Functional Split in LTE}\n\\begin{figure}[t!] \\centering\n\t\\includegraphics[width=5in]{fs\/fig_CRAN.pdf}\n\t\\caption{The Cloud-RAN (CRAN).} \n   \\label{fig_CRAN}\n\\end{figure}\nFigure~\\ref{fig_CRAN} shows the\ntraditional CRAN deployment in comparison conventional cellular deployments.\nA radio base station eNB of LTE protocol stack towards the UE \ncan be functionally split and implemented flexibly over radio node and cloud\nseprated by a fronthaul link which is typically an optical fiber connectivity\nThe traditional CRAN transports the baseband time domain I\/Q samples\nover the optical fiber to the RRUs. The number of RRUs that is\nsupported over a fiber deployment depends on the amount of traffic\nthat is required to support the operations of each RRU. Suppose if \n$C$ is the capacity of the fronthaul optical connectivity and $R_i$ \nis the data rate required by $i^th$ RRU,\nthen maximum number of RRUs $N$ that can be supported over that specific\nfronthaul link is $\\max_{N} \\{\\sum_{i = 1}^N R_{i} \\}$ such at \n$\\sum_{i = 1}^N R_{i} \\leq C$. In the present deployments of CRAN\nthe resources of the fronthaul link are statically allocated with dedicated\nconnectivity. Therefore in symmetrical and homogeneous deployment, \n$R_1 = R_2 \\ldots R_N = R$ resulting in the total number of RRUs \nthat can be supported over the fronthaul link as $N = C\/R$. The main\nbottleneck for the CRAN deployments is delay and capacity of the fronthaul link $C$.\nHeterogeneous fronthaul links where multiple RRUs requirements\nof delay and data rate changing over time independent of each other \nwould require an advance resource management mechanisms to avoid \nthe capacity wastages as compared to the \nworst case tolerance deployment practices.  \n\\begin{table}[t]\n\t\\caption{Typical LTE CRAN Parameters}\n    \\label{tab:CRANparams}\n\t\\centering\n\t\\begin{tabular}{|p{1cm}|p{5cm}|p{2cm}|} \n\t\t\\hline \n\t\t\\textbf{Param.} & \\textbf{Description} &  \\textbf{Value} \\\\ [.5ex]\n\t\t$N$ & Number of RRU per CRAN &  1 \\\\\n\t\t$B$ & LTE cell bandwidth &  20 MHz  \\\\\n\t\t$K$ & Bits per I\/Q & 10 bits   \\\\\n\t\t$W$ & Number of Tx\/Rx antennas & 2 \\\\\n\t\t$T_s$ &  OFDM symbol duration & 66.6 $\\mu$s \\\\\n\t\t$f_s$ &  Sampling frequency & 30.72 MHz \\\\\n\t\t$f_c$ &  Carrier frequency & 2 GHz \\\\\t\t\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\nTo understand the fronthaul requirements of an RRU, \nwe estimate the data rate $R$ required by the traditional CRAN where \nthe baseband I\/Q is transported from the cloud BBU to RRU \nfor the most common LTE deployment scenarios. \nTable~\\ref{tab:CRANparams} summarizes the important list of parameters\nused for evaluation in the context of LTE for the \nfronthaul optical link connecting RRU to CRAN. \nThe data rate comparisons of various functional \nsplit in the LTE protocol stack has \nbeen conducted in~\\cite{maeder2014towards, Wubben2014}. In contrast, \nwe take a closer look on the data rate requirements based on the implementation \nspecifics of the protocol stack. That is, we track the flow of information \nacross multiple protocol stack layers of the LTE and \nidentify the key characteristics that dictate the \nrequirements of fronthaul link. \n\nBased on the computationally intensive operation of the FFT The data flow \nbetween BBU and RRU can be categorically divided into \ntwo types i) time domain samples and ii) frequency domain samples.  \n\n\\subsubsection{Time Domain I\/Q Forwarding}\n\\label{sec:time:iq}\nThe time domain I\/Q samples\nrepresents the RF signal in the digital form either in the passband \nor baseband representation. Typically, the digital representation of the \npassband signal requires a very large data rate depending on the \nphysical transmission frequency band. Thus, passband time domain I\/Q forwarding\nis non economical. For example, in an LTE system, the passband signal\nis sampled at twice the carrier frequency $f_c$ with each sample requiring\n10 bits for digital representation, the passband \nI\/Q data rate $R_i^P$ required over the fronthaul link is\n\\begin{equation}\n\\begin{split}\nR^P_i = N \\times W  \\times 2f_c \\times 10 \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\\\\n= 1 \\times 2 \\times 2(2 \\times 10^9) \\times 10 = 80~\\text{Gbps}.\n\\end{split}\n\\end{equation}\n\nThe baseband signal for an OFDM symbol in the \ntime-domain consists time samples equal to the number of \nOFDM subcarriers because of the symmetric input and output samples \\\nfrom the IFFT\/FFT structure.\nThe cyclic prefix is added to the OFDM signal to avoid the \ninter symbol interference. In order to reduce the constraints on the \nRF signal generation at the RRU, the baseband signal is sampled\nat the frequency of 30.72 MHz with each sample requiring\n10 bits for digital representation and an oversampling factor of 2, \nthe baseband I\/Q data rate $R_i^B$ \nrequired over the fronthaul link is\n\\begin{equation}\n\\begin{split}\nR^B_i = N \\times W  \\times (2 \\times f_s) \\times (2 \\times K) \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\\\\n= 1 \\times 2 \\times (2\\times 30.72 \\times 10^6) \\times (2 \\times 10)\\\\\n = 2.46~\\text{Gbps}.\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\n \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\n\\end{split}\n\\end{equation}\n\nAlthough the data rate is significantly reduced as compared to the passband \nI\/Q forwarding, the baseband I\/Q data rate scales linearly with the\nnumber of antennas and bandwidth. Thus, for large number of \nantennas and larger aggregated bandwidth the data rate $R^B_i$ can be very large.\n  \n\\subsubsection{Frequency Domain I\/Q Forwarding}\n\\label{sec:freq:iq}\nIn the 20 MHz LTE system One OFDM symbol duration \nincluding the cyclic prefix is $71.3~\\mu$s which \ncorresponds to 2192 samples. The useful symbol duration in OFDM \nis $66.7~\\mu$s of 2048 samples while the cyclic prefix duration \nis $4.7~\\mu$s of 144 samples. \nThus, each of 2048 samples in the OFDM symbol (excluding the cyclic prefix) \ncorresponds to 2048 subcarriers ($B_{sub}$) when transformed by the FFT. \nHowever, only 1200 of these subcarriers are used for \nsignal transmission which corresponds to 100 resource blocks (RBs) \nof 12 subcarriers each the rest is zero-padded \nand serves as guard carriers. This \nleads to $(2048-1200)\/2048=0.41=41~\\%$ of unused guard carriers.\nEach subcarrier in the OFDM is modulated by a complex value mapped \nby a QAM alphabet. LTE QAM alphabet size is based on QAM bits \nsuch as, 64 QAM and 256 QAM. If only the subcarrier information which consists of\nthe complex values taken from the QAM alphabet i.e., only the frequency domain data\nis used to transport between BBU to RRU. \nThe resulting data rate $R^F_i$ is only dependent \non the number of subcarriers $B_{sub}$. A vector of \ncomplex valued QAM alphabet symbols of size $B_{sub}$ needs to be sent \nonce every OFDM symbol duration $T_s$ i.e., the data rate \n\\begin{equation}\n\\begin{split}\nR^F_i = N \\times W \\times B_{sub} \\times T_s^{-1} \\times (2 \\times K)\n\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\\\\n= 1 \\times 2 \\times 1200 \\times (66.7\\times 10^{-6})^{-1} \\times (2 \\times 10)\\\\\n= 720~\\text{Mbps}.\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\n\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\n\\end{split}\n\\end{equation}\nIn comparison to time domain baseband I\/Q data rate $R_i^B$, the $R_i^F$ is\napproximately reduced by $(R_i^B - R_i^F)\/R_i^B = (2460-720)\/2460 = 70~\\%$. \nFurther attempts to reduce the data rate $R_i$ \nrequire moving complex functions, such as MAC to the RRU.\nThe MAC HARQ (Hybrid ARQ) \nprocesses requires complex operations and \nlarge buffers to be implemented \nat the RRU increasing the complexity of the RRU.\nThus, with the savings of 70~\\%, keeping the simplicity at the RRU\nis worthwhile to explore the transport mechanism of \nfrequency domain I\/Q symbols over the fronthaul link. \n\n\\subsection{Functional Split in Cable Distributed Converged Cable Access Platform (DCCAP) Architectures}\n\\begin{figure}[t!] \\centering\n\t\\includegraphics[width=4.5in]{fs\/fig_remote_node.pdf}\n\t\\caption{Distributed Converged Cable Access Platform (DCCAP).} \n\t\\label{fig_remote_node}\n\\end{figure}\nTraditional CCAP architecture for the HFC network implements \nthe CMTS at the headend and \ntransports analog optical signal to an Optical Remote \nNode (ORN) over the optical fiber. \nORN then converts the optical\nanalog signal to electrical RF signal to transmit over the cable \nsegment. However, such architectures which rely on\nanalog optics suffer from attenuation of the analog signal in both \noptical fiber segment as well as cable segment of the HFC network because\nof the placement of the ORN. If the ORN is deployed far from the headend,\nthe attenuation in optical signal will be dominant, similarly\nif the ORN is deployed far from the CM (users) then the attenuation in \nthe RF signal of the cable will be dominant. Modular Headend\nArchitecture (MHA) overcomes the downside of CCAP architectures by\nenabling the functional splitting of CMTS allowing the modular \nimplementation of the CMTS functions. Implementation of \nmodular CMTS functions distributively across multiple nodes\nare referred to as Distributed Converged Cable \nAccess Platform (DCCAP) Architecture. As shown in the Fig.~\\ref{fig_remote_node}\nDCCAP architecture defines a new\nRemote Node (RN) which is connected to the headend through digital\nEthernet fiber. Digital connection between the RN and headend eliminates the \nattenuation of the analog optical signal allowing the RN to be \ndeployed more deeper into the users network \nreducing the cable segment length which in turn reduces \nthe analog RF attenuation improving the overall Signal to Noise Ratio (SNR)\nat the CM. The network which connects the RN to headend is referred \nto as Converged Interconnect Network (CIN). \nMHA version 2 (MHAv2)~\\cite{MHAv2} architecture\ndefines two DCCAP architectures Remote-PHY and Remote-MACPHY.  \n \n\\subsubsection{Remote-PHY (R-PHY)}\n\\paragraph{Overview}\nIn the R-PHY architecture, the DOCSIS PHY function  \nin the CMTS protocol stack is implemented at the \nRN referred to as Remote-PHY Device (RPD), while all the\nhigher layers in the CMTS protocol stack including \nthe MAC as well as the upstream scheduler are implemented at the headend.\nA virtual-MAC (vMAC) entity virtualizes the DOCSIS MAC on a \ngeneric hardware which can be flexibly implemented either at the headend \nor cloud\/remote data center. Since the upstream scheduler is implemented\nat the headend, the request from the CM is sent to the headend for processing\nthe grants to coordinate the upstream transmissions over the cable link. \nThe extensive performance evaluation of R-PHY and R-MACPHY have been conducted\nin~\\cite{ziyad2017,thyagaturu2017r}.\nTherefore, the CM request-to-grant delay depends on the distance of \nRN from the headend (i.e., the CIN distance)~\\cite{Chapman2014, Chapman2015}.\n\\paragraph{Advantages}\nR-PHY node is simple to implement and hence the CAPEX and OPEX of the\nRPD can be reduced. At the headend, more computational resources \ncan be dedicated to support the complex schedulers for \nmode advance coordination mechanism of CM transmissions over the \nbroadcast cable medium.\n\\paragraph{Disadvantages}\nAs the distance of RPD from headend increase, the CM request-to-grant\ndelay increases due to increases RTT between RPD and headend.\nApplications such as, virtual and augmented reality requires\nUltra Low Latency (ULL) over the cable link can be impacted\nby the larger CM request-to-grant delays.\n\n\\subsubsection{Remote-MACPHY (R-MACPHY)}\n\\paragraph{Overview}\nIn the R-MACPHY architecture, the DOCSIS PHY and MAC functions along with\nthe upstream scheduler in the CMTS protocol stack are implemented at the \nRN referred to as Remote-MAC Device (RMD), while all the\nhigher layers in the CMTS protocol stack are implemented at the headend.\nSince the upstream scheduler is implemented\nat the RMD, the requests from the CMs are processed for the \nthe grants at the RMD to coordinate the upstream transmissions \nover the cable link reducing the request-to-grant delay.\nTherefore, the CM request-to-grant delay is independent of the distance of \nRN from the headend.\n\\paragraph{Advantages}\nR-MACPHY produces significantly lower delay when compared to R-PHY especially\nthe CIN distance is very large. ULL applications can be readily supported as\nthe upstream scheduler is implemented very close the CM reducing the request-to-grant\ndelay. \n\\paragraph{Disadvantages}\nRMD device has to implement the scheduler on the remote node increasing the \nCAPEX and OPEX of the remote node. As remote node implements more \nfunctions of the CMTS protocol stack in a distributed fashion,\nthe flexibility of the network reduces as\ncompared to centralized architectures, such as R-PHY.\n\n\\subsection{Functional PHY-Split}\n\\begin{figure}[t!] \\centering\n\t\\includegraphics[width=5in]{fs\/fig_split_phy.pdf}\n\t\\caption{PHY-split architecture.} \n\t\\label{fig_split_phy}\n\\end{figure} \n\nThe maximization of the \"cloudification\" of CMTS DOCSIS and \nLTE functions results in the\nincreased flexibility and reduced CAPEX \nand OPEX for the Multiple System Operators (MSOs) and \ncellular operators. The cloud implementation of DOCSIS and LTE functions \ncan be easily achieved at the same physical location, remote data center or cloud.\nBoth LTE and DOCSIS 3.1 also share similar transceiver PHY characteristics for \nthe OFDM implementation that can be exploited to design \nthe simultaneous support for LTE and DOCSIS over the HFC network which is critical to\nthe 5G type of applications, such as health monitoring and security \\cite{veeranna2016hardware}. \nThe general overview of the physical layer for LTE and \nDOCSIS is shown in the Fig.~\\ref{fig_split_phy}. In the downstream direction,\nthe data from MAC layer is\nprocessed to PHY frames and mapped to OFDM resource locations which is then \nconverted to frequency domain QAM I\/Q symbols (see~\\ref{sec:freq:iq}) based on the \nmodulation and coding schemes. The\nQAM I\/Q symbols are then IFFT transformed to \nget the complex time domain samples. \nThis time domain samples (see~\\ref{sec:time:iq}) \nare then converted \nto analog RF signal for the transmission over the cable link. To note, the processing of I\/Q \ninformation before the IFFT\/FFT module is specific to DOCSIS and LTE \nprotocol whereas the processing of \nI\/Q after the IFFT\/FFT module is relatively same for both DOCSIS and LTE.\nThus, we can separate the functions based on the IFFT\/FFT module such \nthat IFFT\/FFT is implemented at the remote node to support the\nthe architectures and mechanisms to simultaneously \noperate LTE and DOCSIS over the HFC network. \nTo the best of our knowledge, there exists no prior research \nfor simultaneously supporting LTE and cable so as to efficiently \nutilize the optical fiber (fronthaul) \nresources of the already installed HFC plant. \n\n\n\\section{\\MakeUppercase{Proposed Cross-Functional Split Interaction}} \n\\label{sec:split:inter}\n\\begin{figure}[t!] \\centering\n\t\\includegraphics[width=4.5in]{fs\/fig_LTE_cable.pdf}\n\t\\caption{Deployment of the LTE eNB RRU at the Remote Node (RN).}\n\t\\label{fig_LTE_cable}\n\\end{figure} \nThe digital optical remote node (ORN) in the DCCAP architecture \nis deployed closer to the CM (users). Close proximity of the \nORN to the residential \nsubscribers would help in establishing the wireless LTE \nconnectivity by deploying\nan LTE eNB RRU at the ORN site. As show in Fig.~\\ref{fig_LTE_cable} \ninstalling an RRU at the ORN site \nis advantageous to the operators due to the close proximity of the users. \nWith the establishment of LTE connectivity by the MSOs users can be \nwirelessly connected to the MSO's core network \nfor the Internet connectivity increasing the service capabilities of the MSOs.\nIn addition, support of LTE eNB RRU at the ORN also reuses the \nexisting HFC infrastructure reducing the CAPEX and OPEX for the MSOs in \nproviding the additional services of LTE.\n\nTraditionally, the CRAN and DCCAP architectures split the \nfunctions linearly based on the protocol stack \n(i.e., MAC, PHY etc.), \nand implements the split parts at the ORN and cloud.\nAdditional to protocol stack split, our proposed mechanism \nenables cross-split interaction through signalling\nto facilitate the caching and prefetching (see Sec~\\ref{sec:caching}) \nof redundant information between the functional splits. More\nspecifically, cross-functional split interaction involves \nthe communication of control information between\nbaseband unit and radio unit.\n\\subsection{Proposed Shared Remote-FFT (R-FFT) Node}\n\\begin{figure}[t!] \\centering\n\t\\includegraphics[width=5in]{fs\/fig_rfft.pdf}\n\t\\caption{Remote FFT (R-FFT) implements the FFT module at the remote node.}\n\t\\label{fig_rfft}\n\\end{figure} \n\nIn the uplink direction, the proposed remote node R-FFT converts the \nincoming DOCSIS RF signal from the CM to an encapsulated data bits format \nthat can be transported over digital fiber link for additional processing and\nonward forwarding at the headend. In a similar way, for the downlink, \nRF signals are generated from the incoming formatted data bits and \nsent out on the RF link to the CMs. For LTE, an eNB can use a wide \nrange of licensed spectrum with a single largest carrier component \nof 20 MHz, however the bandwidth can be extended further by carrier \naggregation technique resulting in larger effective bandwidth. \nThe R-FFT node effectively converts the upstream LTE RF signal \nfrom the wireless users and transports the signal digitally to the \nBBU\/CRAN via the digital fiber link. In the downlink direction, \nthe R-FFT node converts the digital information to an LTE RF \nsignal that can be transmitted wirelessly to the users.\n\nA Remote-DAC\/ADC (R-DAC\/ADC) would require peak rate \ntransmissions \\cite{rdac}. Considering a system with a downlink \nbandwidth of 100 MHz and a 1024 QAM modulation format, \nwhich requires 12 bits per sample, results in a total bit rate over the fiber of (100 MHz) x (12) x (Frequency sampling rate). \nFor a frequency sampling rate of 2.5 times (slightly greater than the Nyquist rate),\n the total required bit rate is 3 Gbps. In contrast, \nan equivalent R-PHY system requires a data rate of (100 MHz) x (10 Bits\/Hz) x\n(90~\\%) =  900~Mbps; whereby, 100 MHz is for the resource allocation \nacross the entire bandwidth, 10 Bits\/Hz is for the 1024 QAM, and 90~\\% \nis for the effective channel with a 10~\\% guard band.\nThis simple example comparison indicates a threefold increase of the \ndata rate in the R-DAC\/ADC system compared to the R-PHY system. \nNote that the R-DAC\/ADC system requires a constant peak \ndata rate to transport the I\/Q samples, regardless of the \namount of user data present in the RF signal. \nIn contrast, similar to R-PHY\/MAC, in R-FFT system, \nthe data rate depends on the \namount of user data. In the above example data rate calculation, \nwe have considered the maximum allocation \nof the user data in the R-PHY system. \n\nOur motivation is to address the increased fiber data rate \nthrough a balanced split among the functions within the PHY node, \nwhile keeping the simplicity of the R-DAC\/ADC and the CRAN systems. \nThe existing R-DAC\/ADC node requires some digital circuitry, \nsuch as a CPU, for the DAC and ADC control. Therefore, we believe \nthat the FFT\/IFFT implementation is not a significant additional \nburden for the remote node~\\cite{jang2016study}. \nFigure~\\ref{fig_rfft} shows the implementation of vMAC \nfor both LTE and DOCSIS at the Headend and \na common FFT module at the remote node. \nThe advantages of proposed FFT implementation at the remote node include: \n\\begin{itemize}\n\t\\item[i)] flexible deployment support for LTE and DOCSIS\n\t\\item[ii)] requires lower data rate $(R_i^F)$ to transport frequency \n\tdomain I\/Q as compared to time-domain I\/Q $(R_i^B)$.\n\t\\item[iii)] data tones carrying no information are zero valued \n\tin the frequency I\/Q samples effectively resulting a \n\tlower date-rate over the fiber channel in both LTE and DOCSIS with the\n\tchance for statistical multiplexing, and\n\t\\item[iv)] possible caching of repetitive frequency QAM I\/Q samples, \n\tsuch as Reference Signals (RS) and pilot tones. \n\\end{itemize}\n\nTo emphasize the important data rate aspect, we note that in the \nproposed shared R-FFT, the data rate required over the fiber channel is \ndirectly proportional to the user traffic. We believe this is an \nimportant characteristic of the FFT functional-split whereby we \ncan achieve multiplexing gains by combining multiple R-FFT nodes or \nby the enabling the concurrent support of LTE as illustrated in \nFig.~\\ref{fig_LTE_cable}. With the similar characteristics of DOCSIS,\nthe LTE user traffic also translates to proportional data rate \nover the fiber channel. In addition, the proposed mechanism enables \nthe complex signal processing of the PHY layer to be implemented \nat the headend. Examples of the signal processing operations include channel\nestimation, equalization, and signal recovery, which can be implemented with\ngeneral-purpose hardware and software. In addition, the processing of \ndigital bits, such as the LDPC, which is necessary for the forward error\ncorrection, can also be implemented at the headend. Thus, the proposed approach\nreduces the cost of the remote nodes and increases the flexibility of changing \nthe operational technologies. The software implementations at the headend can be\neasily upgraded while retaining the R-FFT node hardware since the node hardware\nconsists only of common platform hardware, such as elementary DAC\/ADC and \nFFT\/IFFT components. Thus, the proposed approach eases the change\/upgrade of\ntechnologies. That is, the R-FFT node has minimal impact on technology \nadvancements because the blocks within a remote node are elementary \nor independent of most technology advancements. \n\n\\subsection{Proposed Remote Caching and Prefetching}\n\\label{sec:caching}\n\\begin{figure}[t!] \\centering\n\t\\includegraphics[width=5in]{fs\/fig_caching.pdf}\n\t\\caption{Caching at the remote node.}\t\n\t\\label{fig_caching}\n\\end{figure}\nIn order to further reduce the bandwidth in addition to functional split process, \nseveral techniques, such as I\/Q compression~\\cite{nanba2013new, nieman2013time,guo2012cpri, joung2013base} \ncan be employed. In contrast, we propose resource element (time and frequency slot) \nallocation based remote caching. If some part of the information is regularly and \nrepeatedly sent over the interface, a higher (orchestration, in case of SDN) \nlevel of the signaling process can coordinate caching mechanisms. For example, \nthere is no need to transmit the downlink I\/Q samples of the pilot tones as\nthey would remain constant in the DOCSIS. Figure~\\ref{fig_caching} illustrates the overview \nof repetitive QAM symbols in the LTE and DOCSIS. There is a process of \nup-sampling and zero padding before and after the FFT\/IFFT block, \ndepending on the FFT size and the effective subcarriers, which can \npotentially be controlled and cached through the headend (depending on \nthe implementation). Zero padded QAM symbols as well as certain reserved resource elements \ncan be excluded from the \ntransmissions while signaling the changes \nto the cache management at the R-FFT node.\nAs compared to downstream, upstream information must be entirely \ntransported to the headend to process all the signal component \nreceived by the R-FFT receiver. \n\\subsubsection{LTE Networks}\n\\paragraph{Reference Signal (RS) Tones Caching}\nRS tones are the pilot subcarriers which are embedded throughout the operational \nbandwidth of the wireless system for the channel estimation to equalize the \nimpairments of received wireless signal. Disturbances to the \nwireless signal are more prominent compared to propagation of signal \nin the wired channel and therefore\nthe RS tones are added in close proximity with \neach other to accurately estimate the channel based \non the channel characteristics, such as coherence-time and coherence-bandwidth. \nFor a single antenna, the RS tones are typically spaced 6 subcarriers apart \nin frequency such that 8 RS tones exists in a single subframe, i.e, 14 OFDM symbols\nand single Resource Block (RB), i.e., 12 subcarriers of LTE. Thus, the overhead\ndue to RS tones in the LTE resource grid is\n\\begin{equation}\n\\text{RS Overhead} = \\frac{8}{12 \\times 14 } = 4.7~\\%.\n\\label{eq:rs}\n\\end{equation}\nTherefore, approximately 4.7~\\% of I\/Q transmissions over the \ndigital fiber can be saved from the RS tones caching at the \nremote node regardless of the system bandwidth.  \n\n\\paragraph{PHY Broadcast Channel (PBCH) Caching}\nPHY Broadcast Channel (PBCH) carries the Master Information Block (MIB)\nwhich is broadcasted continuously by the\neNB regardless of the user connectivity.\nMIB includes the basic information about the LTE system \nsuch as the system bandwidth and control information specific to LTE channel.\nPBCH\/MIB always uses central\n6~RBs (i.e, 72~subcarriers) for the duration of \n4~OFDM symbols to broadcast the MIB data.\nPBCH space in the resource grid is inclusive of the \nRS tones used in the calculation of Eq.~\\ref{eq:rs} and therefore\nneeds to subtracting while calculating the overhead for MIB transmission.\nPBCH\/MIB occurs once every 40~ms and there exists 4~redundant versions of \nMIB which will be broadcasted with the offset of 10~ms. Thus, \nessentially PBCH\/MIB occurs once in every 10~ms (radio frame). The \nPBCH\/MIB overhead for the entire LTE system is\n\\begin{equation}\n\\text{PBCH Over.} = \\frac{6 \\times 12 \\times 4 - (8 \\times 6)}{1200 \\times 14 \\times 10} = 0.142~\\%.\n\\label{eq:mib}\n\\end{equation}\nAlthough 0.142~\\% is less, relative overhead of PBCH\/MIB data \nincreases significantly for lower bandwidth LTE system. Moreover, our evaluation of the\nPBCH overhead considers the full RBs i.e., 1200 subcarriers, 14 OFDM symbols and 10 subframes\nin the LTE. In a real system the allocation broadly varies user density i.e, \ngenerally only part of the RBs will be used increasing the overhead effects. \nFor example, for a 1.4~MHz system (used in an IoT type of applications) \nwith full RBs, the overhead jumps to \n\\begin{equation}\n\\text{PBCH Over.}_{1.4\\text{MHz}} = \\frac{6 \\times 12 \\times 4 - (8 \\times 6)}{72 \\times 14 \\times 10} = 2.3~\\%.\n\\label{eq:mib:14}\n\\end{equation}\nTherefore, up to 2.3~\\% I\/Q transmissions over the \ndigital fiber can be saved from the PBCH\/MIB caching at the remote node.\n\n\\paragraph{Synchronization Channel Caching}\nSynchronization Channel (SCH) consists of Primary Synchronization Sequence (PSS)\nand Secondary Synchronization Sequence (SSS) which is broadcasted continuously\nby the eNB regardless of the user connectivity. PSS and SSS are the special\nsequence which helps in the cell synchronization of wireless users by identifying\nthe physical cell ID and the frame boundaries of the LTE resource grid.\nPSS\/SSS occurs every 5~ms (twice per radio frame) and uses central 6~RBs over 2 \nOFDM symbols. Similar to Eq.~\\ref{eq:mib}~and~\\ref{eq:mib:14}, the overhead due to\nPSS\/SSS in 20~MHz and 1.4~MHz system is \n\\begin{equation}\n\\begin{split}\n\\text{SCH Over.} = \\frac{6 \\times 12 \\times 4}{1200 \\times 14 \\times 10} = 0.171~\\%. \\;\\;\\;\\:\\\\\n\\text{SCH Over.}_{1.4\\text{MHz}} \n= \\frac{6 \\times 12 \\times 4}{72 \\times 14 \\times 10} = 2.8~\\%. \\;\\;\\;\\;\\;\\;\\;\n\\end{split}\n\\label{eq:sch}\n\\end{equation}\n\n\\paragraph{System Information Block (SIB) Caching}\nIn a similar way, the caching mechanism can also be extended \nto the System Information Blocks (SIBs) broadcast messages \nwhich is through the PHY Downlink Shared Channel (PDSCH) of the LTE.\nThere are 13 different types of SIBs, from SIB1 to SIB13. \nSIB1 and SIB2 are mandatory broadcast messages while the transmission of other SIBs \ndepends on the relation between the \nserving and neighbor cell configurations. On a typical deployment, \nSIB3 to SIB9 are configured and can be combined in a single message block\nfor the resource block allocation. Typical configuration of the \nRB allocation type schedules the SIB over 8 RBs across\n14~OFDM symbols in time (1~subframe) are used for the transmission of SIB1 and SIB2, \nand with the effective periodicity (with redundancy version transmission) \nof 2 radio frames (20~ms). The overhead and caching gain from the \nof SIB1 and SIB2 transmissions while subtracting the corresponding \nRS tones overhead $8 \\times 8$, i.e., 8 tones per RB for 8 RBs is\n\\begin{equation}\n\\begin{split}\n\\text{SIB Over.} = \\frac{8 \\times 12 \\times 14 - (8 \\times 8)}{1200 \\times 14 \\times 20} = 0.381~\\%. \\;\\:\\\\\n\\text{SIB Over.}_{1.4\\text{MHz}} \n= \\frac{8 \\times 12 \\times 14 - (8 \\times 8)}{72 \\times 14 \\times 20} = 6.3~\\%. \\;\\;\\;\\;\\;\n\\end{split}\n\\label{eq:sib}\n\\end{equation}\nCaching of higher order SIBs i.e., from SIB3 to SIB 9 \ncan achieve further savings, however, the resource allocation \nand periodicity can widely vary to accurately estimate the overhead.\n\nThe stationary resource elements across the time domain, such as \nSIB information is allowed to change over a larger time scale in terms of \nhours and days, at such times the cached elements are refreshed or \nre-cached through cache management and signalling procedures, see Sec.~\\ref{sec:cache:mgt}.\n \n\\subsubsection{Cable Networks}\nIn the DOCSIS~3.1 downstream pilot subcarriers are modulated \nby the CMTS with a predefined modulation pattern which is known to all the\nCMs to allow the interoperability.\nTwo types of pilot patterns are defined in the DOCSIS~3.1 for OFDM time\nfrequency grid allocations, i) continuous, and ii) scattered. \nIn the continuous pilot pattern, the pilot tones with a predefined modulation \noccur at fixed frequencies in every symbol across time. \nWhereas, in the scattered pilot pattern, the pilot tones are sweeped to \nto occur at each frequency locations but at different symbols across time.\nThe scattered pilot pattern has a periodicity of 128 OFDM symbols along the time dimension such that the pattern repeat for the next cycle. \nScattered pilots assist in the channel estimation for the\nequalization process of demodulation. In a typical deployment~\\cite{cabledeploy}, \noperational bandwidth is 192~MHz corresponding to the FFT size of 8K\nwith the 25~kHz subcarrier spacing. The total number\nof subcarriers in a 192~MHz system is 7680 out of which \nthere are 80 guard band subcarriers, 88 continuous pilot subcarriers and \n60 scattered pilot subcarriers. Therefore, the overhead due to redundant \nsubcarriers in the DOCSIS system which can be cached at the remote node is\n\\begin{equation}\n\\text{Cable Over.} = \\frac{80 + 88 + 60}{7680} = 2.9~\\%.\n\\label{eq:cable}\n\\end{equation}  \n\n\\subsection{Memory Requirements for Caching}\nCaching of frequency domain I\/Q symbols in the OFDM \ncomes at the cost of caching memory implementation at the remote node.\nEach I\/Q symbol that needs to be cached is a complex number\nwith real and imaginary part. For the purpose of evaluation, we consider \neach part of the complex number represented by 10 bits resulting\nin 20 bits in total for each frequency domain QAM symbol. The caching of\nRS tones in the LTE results in 4.7~\\% of the savings in the \nfronthaul transmissions as show in the Eq.~\\ref{eq:rs}. In the duration of \neach OFDM symbol, 2 RS tones exists for every 12~subcarriers. For the \n20~MHz system, there would be 200 RS tones. Therefore,\ntotal memory required to cache the data\nof QAM symbols corresponding\nto the RS tones is \n\\begin{equation}\n\\centering\n\\text{RS Tones Mem.} = (2 \\times 100) \\times 2 \\times 10 = 4000~\\text{bits}.\n\\label{eq:rs:mem}\n\\end{equation}  \nSimilarly, caching of PBCH, SCH and SIB data requires \n\\begin{equation}\n\\begin{split}\n\\text{PBCH Mem.} = (4\\times12\\times6 - (8 \\times 6)) \\times 2 \\times 10 \\;\\;\\;\\;\\;\\;\\; \\\\\n= 4800~\\text{bits}, \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\n\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\: \\\\\n\\text{SCH Mem.} = (6 \\times 12 \\times 4) \\times 2 \\times 10 = 5760~\\text{bits},\\;\\;\\: \\\\\n\\text{SIB Mem.} = (8 \\times 12 \\times 14-(8 \\times 8)) \n\\times 2 \\times 10 \\;\\;\\;\\;\\; \\\\ =5760~\\text{bits}.\n\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\n\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\n\\end{split}\n\\label{eq:pbch:mem}\n\\end{equation}\nWhereas, for the DOCSIS, the the memory requirements for the\ncaching of continuous and scattered pilots is  \n\\begin{equation}\n\\centering\n\\text{Pilot Tones Mem.} = (80+88+60) \\times 2 \\times 10 = 4560~\\text{bits}.\n\\label{eq:pilot:mem}\n\\end{equation}\nThus, based on Eqs.~\\ref{eq:rs}-\\ref{eq:cable}, total  \nsavings of approx. 7~\\% to 18~\\% can be achieved in the\nfronthaul transmissions when the full resource allocation over the\nentire bandwidth is considered in both LTE and DOCSIS.\nFor lower allocations i.e, when there \nexists lesser user data, the caching process can achieve much \nlarger relative benefits. \nIn an extreme case, when the user data is not present, all the cell \nspecific broadcast data information can be cached at the \nremote node such that the fronthaul transmissions \ncan be seized reducing the power consumption.\nThe total memory for the caching required at the remote node based \non the Eqs.~\\ref{eq:pbch:mem}-\\ref{eq:pilot:mem} is approximately 19120~bits.\nThe implementation of cache at the remote node \nwhich is lesser than the size of 25~KB is relatively very simple \nand of no practical burden to the existing remote node. Therefore, \nwe believe the savings of more than 7~\\% with almost negligible implementation\nburden is a significant benefit in the CRAN functional PHY-split. \n \n\\subsubsection{Signalling and Cache Management}\n\\label{sec:cache:mgt}\nSignalling mechanism facilitates the cache management processes. \nCache management operation involves, \ni) transporting the caching information to the remote nodes,\nii) updating the cached information at the remote nodes with new \ninformation, and\niii) establishing the rules for prefetching of cached resource \nelements at the remote node.\nSignalling agents at the headend\/cloud\nand remote node coordinate with each other \nthrough a separate (i.e., non-I\/Q transport)\nlogical connection between headend\/cloud and remote node.\nSome of the information which has been cached may change over time but \nat a larger timescale compared to the transmission of \nI\/Q from headend to the cloud. The signalling overhead which arises \nfrom the cache management is negligible because of the timescale of\nsignalling operations. Headend\/cloud can notify the remote node \nwith the changes \nthrough signalling to keep the information up to date at the remote node.\nPrefetching of the cached content has to be precisely executed \nwith accurate placement of the subcarriers information in the particular \ntime and frequency locations to an IFFT\/FFT computational module. \n \n\\subsection{Transport Networks and Protocols}\n\\subsubsection{Transport Networks}\nAn R-FFT node can be connected to headend through variety of transport \nnetwork solutions over the optical fiber connectivity as well as dedicated \nmmWave links. The fundamental requirement is to support the CRAN and CCAP \ndemands while virtualizing the QAM functions at the headend. \nSome of the technologies that can be considered are:\n\\paragraph{Dark Fiber} A fiber network that is deployed in excess of \nthe existing requirement is referred to as a dark fiber network. \nTypically, dark fiber resources are abundant but with limited accessibility. \nIf a headend can be connected to an R-FFT node using dark fiber, a large data \nrate with low latency connections can be established, supporting safe \nand reliable transport of required I\/Q samples.\n\\paragraph{Optical Wavelength Division Multiplexing (WDM)} WDM establishes \nend to end connectivity with the dedicated wavelength aggregated to meet \nthe user demands. WDM with the potential to reach large data rates and low \nlatencies can easily support the connectivity of the R-FFT with the headend.\n\\paragraph{Passive Optical Network (PON)} A PON network typically provides \nhigh data rates in the Point to Multipoint Ethernet services for residential \nand Fiber to The Premises (FTTP) type of applications. If PON technology is \nused to establish the connectivity, careful latency limitations must be evaluated. \nSeveral interfaces, such as CPRI, exists to support the \nconnections over the PON transport network.\n\\paragraph{Wireless} A standard and dedicated microwave or millimeter-wave \nwireless links can support large data rate and low latency connection between \ntwo nodes. However, wireless links are limited by the range and \nsometimes require Line of Sight (LoS) operation.\n\n\\subsubsection{Protocols}\nProtocol is required to coordinate the transmissions of I\/Q data over the\ntransport network. A strict latency requirement for the CRAN and DCCAP \narchitectures limits the choice of generic protocols over \nEthernet. Some of the fronthaul protocol that exists for the \ntransport of information between headend\/cloud and radio node \nare:\n\n\\paragraph{Radio over Fiber (RoF)}\nRadio over fiber (RoF) transports the radio frequency signal \nover an optical fiber link by converting the electrically modulated signals\nto optical signal. RoF signals are not converted in frequency but superimposed\nover optical signals to achieve the benefits of optical transmissions,\nsuch as reduced sensitivity to noise and interference\nRoF signals are optically distributed to end nodes where the\noptical signals are directly converted to electrical signal with\nminimal processing reducing the cost of the remote node. \nThe downside of the RoF is the analog transmission of the optical \nsignal which attenuate over distance over the fiber as compared to \nthe digital data over the fiber.\n\n\\paragraph{Common Public Radio Interface (CPRI)}\nCommon Public Radio Interface (CPRI) \\cite{CPRI} defines the protocol\nto transport the digitized I\/Q data through the encapsulated CPRI frames. \nAs compared to RoF, CPRI provides more reliable end-to-end connection between\nheadend\/cloud and the remote node. Dedicated TDM channels can be established \nto supports multiple logical connections supporting different air interfaces. \nThe downsides of the CPRI are the strict timing and synchronization \nrequirement as well as support for only the fixed functional split to \ntransport time domain I\/Q samples. To overcome the fixed functional split\nin the eCPRI is being currently developed to support a new functional \nsplit with ten folds of decrease in the data rate requirement over the \nfronthaul link. In comparison to eCPRI and CPRI protocols, \nour proposal is an enhancement to \nthe base protocols where the repeated I\/Q can be cached when frequency \nI\/Q samples are transported over the fronthaul link.\n\n\\paragraph{Open Base Station Architecture Initiative (OBSAI)}\nOpen Base Station Architecture Initiative (OBSAI)~\\cite{obsai} is \nsimilar to CPRI in which the digitized time domain I\/Q samples are \ntransported over fronthaul interface.\nIn contrast to CPRI, OBSAI interface is an IP based connection.\nA digital interface which is based on an IP logical connection \ncan be implemented over any generic Ethernet link providing a flexible\nconnectivity between headend\/cloud to remote node.\n\n\\paragraph{External PHY Interfaces}\nDownstream External PHY Interface (DEPI)~\\cite{depi} and Upstream External PHY Interface\n(UEPI)~\\cite{uepi} enable the common transport mechanisms between \nR-PHY and CCAP core. DEPI\nand UEPI are based on the Layer 2 Tunneling Protocol version 3 (L2TPv3).\nThe L2TPv3 transparently transports the Layer 2 protocols \nover a Layer 3 network creating the psedudowires (logical connections).\nFor the R-MACPHY, since only the MAC payload is required to \nbe transported to the headend for processing of upper layer\ndata, strict requirements for the latency as in the case of R-PHY \ncan be relaxed, allowing\nany generic tunneling protocol to be used between headend and R-MACPHY.\n\n\\section{\\MakeUppercase{Proposed Common Platform for LTE and Cable Networks}}\n\\label{sec:common:infra}\n\\begin{figure*}[t!] \\centering\n\t\\includegraphics[width=\\textwidth]{fs\/fig_common_fft.pdf}\n\t\\caption{Reuse of infrastructure.}\n\t\\label{fig_common_fft}\n\\end{figure*}\nThe last hop connectivity to the user devices is typically \nenabled by wireless technologies. Although, any wireless interface, \nsuch as Bluetooth, ZigBee, or WiMax, can provide the wireless \nlast hop connectivity, WiFi is a very promising technology. \nWiFi has proven to be simple and efficient for managing large \nnumbers of connections as well as to serve large\ndata rates to connected nodes. Also, importantly, WiFi has the capability\nof delivering high data rates (up to Gbps) in the unlicensed bands\nover a reasonable coverage area (up to 100 m), and in indoor \nenvironments. In contrast, the LTE wireless interface, which is widely used \nfor commercial cellular communications in licensed bands, is capable of \ndelivering up to several hundreds of Mbps over larger geographical areas \n(up to several tens of miles). Several efforts are underway to integrate \nthese two wireless technologies so as to enable even higher data rates \non the order of several Gbps. LTE-U and LTE-LAA [6] are newly developed wireless \ninterfaces where a primary LTE link opportunistically utilizes the \nunlicensed WiFi bands. The next generation personal computers \nare expected to support LTE connectivity in conjunction with \nEthernet and WLAN. \nHowever, to meet the future demands \nto support concurrent connectivity, deployment and management of\nindependent LTE and WiFi technologies by the same operator \nincurs large expenses to the operators. We envision a solution\nwhere LTE base station can be established at the \nremote node in the DCCAP architecture.\nTo our knowledge, the\nimplementation of FFT at the remote node to simultaneously support\ncable and LTE infrastructure is novel to our proposal. \n\n\\subsection{Common IFFT\/FFT for LTE and DOCSIS}\nThe protocol of LTE and DOCSIS share the same physical properties as\nthey depend on OFDM for the physical layer modulation technique.\nImplementation of the OFDM is dependent on the design of \nFFT computations~\\cite{he1998designing}.\nThe property by which both LTE and DOCSIS require \nsame IFFT\/FFT operations to be \nperformed for each OFDM modulation and demodulation can be exploited\nto reuse the existing infrastructure provided such mechanism \ncan sustained on the computing hardware.\nThus, the main motivation of the FFT implementation at the remote node \nis to bring out  the common platform at the remote \nnodes while the transmission formats are \nflexibly realized at the headend for heterogeneous \nprotocols based on OFDM.\nFigure~\\ref{fig_common_fft} describe the internals of the\nR-FFT architecture simultaneously supporting cable and LTE. \nGenerally, in the downstream direction, an IFFT operation is performed \nonce every OFDM symbol duration. The LTE OFDM symbol duration is \napproximately 72 \u00b5s, and for DOCSIS, the OFDM symbol duration \nis 84.13 \u00b5s. However, the actual time to compute IFFT can span \nfrom few microseconds to several tens of microsecond. \nConsidering that there exists a large portion of idle \ntime durations in the IFFT module during the FFT computation, \nwe can interleave the I\/Q input in time such that same \nIFFT\/FFT module can be used for multiple technologies. \nBy reusing the IFFT\/FFT computing structures we can reduce \nthe complexity of the hardware, be more power efficient, \nand reduce the cost of R-FFT node. \n\n\\subsection{FFT Computations Interleaving Timing Discussion}\n\\begin{figure}[t!] \\centering\n\t\\includegraphics[width=5in]{fs\/fig_timing.pdf}\n\t\\caption{The IFFT\/FFT interleaving.}\n\t\\label{fig_timing}\n\\end{figure}\nIn this section we preset the timing schedule analysis \nfor the interleaving of IFFT\/FFT computations on \na single computing resource. Figure~\\ref{fig_timing} shows the\nbasic timing diagram to schedule the FFT computations on the\ncomputing resource. $T_D$ and $T_L$ are the OFDM symbol durations of \nDOCSIS and LTE respectively. Similarly, $\\tau_D$ and $\\tau_L$\nare the durations of FFT computations for DOCSIS and LTE.\nWhile $p$ and $q$ are the timing indexes for the frame of reference\nfor DOCSIS and LTE independent periodic events. \nWE also assume that $T_L$ and $T_D$ start at the time index p\nand q without any offset as show seen in the Fig.~\\ref{fig_timing}\n\nFor illustration, in this article we assume the OFDM\nsymbol duration of LTE is greater then the symbol duration of \nDOCSIS, i.e., $T_L > T_D$, Consequently, the duration\nof FFT computations depends on the FFT size where we assume \n$\\tau_D > \\tau_L$, as DOCSIS has the larger FFT size compared to LTE. \nAlso, it is most likely that heterogenous technologies operate with \ndifferent OFDM symbol times. Thus, \nwe define the difference in the OFDM symbol duration as \n$\\delta= T_L - T_D$. \n\n\\paragraph{Feasibility Discussion}\nSuppose if there is no offset between the cycles \nas shown in the start of timing schedule in Fig.~\\ref{fig_timing}\nand $k\/l$ denotes the fraction of OFDM symbol durations, \nwhere, $k$ and $l$ if exists are the minimum positive integers,\nsuch that $T_L\/T_D=k\/l$,\nfor every $k$ cycles of $T_D$ there would be $l$ cycles of $T_L$. \nThat is, when two periodic signal are overlapped, \nthe difference of periodicities $T_L$ and $T_D$, $\\delta$, \nat each cycle turns out to be integer multiple of the \nprevious cycle. \nFor example, in the Fig.~\\ref{fig_timing} at the\nfirst cycle we define $\\delta = T_L - T_D$, for the second cycle\nthe difference is $2T_L-2T_D = 2\\delta$ and similarly for the third\ncycle the difference is $3\\delta$. Since $T_L > T_D$ From \nthe fundamental principles we can evaluate that after  \n$k=\\lceil T_L\/\\delta \\rceil = \\lceil T_L\/(T_L-T_D) \\rceil $ \ncycles of $T_D$, the overlapping behavior repeats. Also,\nthe  For example,\nin the Fig.~\\ref{fig_timing}, for every $k=4$, which corresponds to\n4 cycles of $T_D$ and 3 cycles of $T_L$, the behavior repeats.\nTherefore, if the stability is ensured for $k$ cycles of $T_D$ or\n$l$ cycles of $T_L$, the system is stable and feasible.\n\nThus, considering the larger OFDM symbol duration $T_L$\nas the reference, where we have $l$ cycles of $T_L$ for the secondary\nperiodicity of combined $k$ and $l$ cycles. For the system \nto be stable, the total \nFFT computation durations from combined LTE and DOCSIS \ni.e., $l\\tau_L + k\\tau_D$ should not exceed \nthe duration of $l$ cycle of $T_L$ i.e.,\n\\begin{equation}\n\\centering\nl\\tau_L + k\\tau_D \\leq lT_L.\n\\label{eq:stability}\n\\end{equation}  \nWe know that $k$ cycles of $T_D$ is equal to the $l$ cycles of $T_L$,\ni.e., $kT_D = lT_L$ and therefore, $k=\\frac{lT_L}{T_D}$. Substituting $k$\nin the Eq.~\\ref{eq:stability} we get,\n\\begin{equation}\n\\centering\n\\begin{split}\n\\left(\\frac{lT_L}{T_D}\\right) \\tau_D+l\\tau_L\\leq lT_L \\\\\n\\frac{T_L \\tau_D}{T_D} + \\tau_L \\leq T_L\\\\\n\\tau_D T_L + \\tau_L T_D \\leq T_L T_D.\n\\label{eq:stability2}\n\\end{split}\n\\end{equation}\nIn addition to Eq.~\\ref{eq:stability2}, as shown in the Fig.~\\ref{fig_timing}\nwe need to ensure that\nFFT computation  durations are sufficiently small to support the \ninterleaving. That is, for interleaving $\\tau_{\\max(T_L, T_D)}$ on\nthe periodic cycles of $\\min(T_L, T_D)$, we need at least the space of  \n\\begin{equation}\n\\centering\n\\tau_{\\max(T_L, T_D)} \\leq 2\\min(T_L, T_D)-2(\\tau_{\\min(T_L, T_D)}).\n\\label{eq:stability3}\n\\end{equation} \n\n\\paragraph{Constraints}\nThe end of each OFDM symbol duration marks the\ntrigger point for the FFT computing scheduling request. \nOur fundamental analysis is based on\nthe fact that IFFT\/FFT computing takes much less time \ncompared to the actual OFDM symbol duration, especially\nas the computing hardwares are becoming more \nadvanced~\\cite{yeh2003high, arun2016design}.\nAs this factor is dependent on the hardware we impose \nfollowing constraints to design the interleaving scheduling\nprocedure of two heterogeneous technologies on the\nsame hardware, i) the FFT computing\ndurations should not exceed their OFDM symbol duration, and\nii) the FFT computations must be finished before the start of\nnext OFDM symbol, i.e.,\n\\begin{equation}\n\\centering\n\\begin{split}\n\\tau_L < T_L, \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\:\\\\\n\\tau_D < T_D, \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\:\\\\\np\\tau_L < (p+1)T_L,\\\\\nq\\tau_D < (q+1)T_D.\n\\end{split}\n\\label{eq:comp.constr}\n\\end{equation}\nThus, in addition to Eq.~\\ref{eq:stability2}, the constraints presented in\nthe Eq.~\\ref{eq:comp.constr} must be satisfied in the process interleaving\nof computations to have no adverse effects on the operational technologies.\n\n\\paragraph{Effect of Guard Time}\nGuard time separates two consecutive computations to compensate for the \nload and read time of the I\/Q to the FFT structure while supporting the \ncoexistence of two technologies on the same computation module.\nWe define $\\theta$ as the constant guard time which is applied \nto each scheduling set of the FFT computation. Based on Eq.~\\ref{eq:stability2},\nthe impact of guard time can be evaluated as\n\\begin{equation}\n\\centering\n(\\tau_D+\\theta)T_L + (\\tau_L+\\theta)T_D \\leq T_DT_L,\n\\label{eq:guardtime}\n\\end{equation}\nwhich can be written as,\n\\begin{equation}\n\\centering\n\\tau_DT_L + \\tau_LT_D \\leq T_DT_L - \\theta(T_L+T_D).\n\\label{eq:guardtime2}\n\\end{equation}\n\n\\begin{algorithm}[t!]\n\t\\caption{Caching and FFT Computation Procedure}\n\t\\label{algo:interleaving}\n\t\\SetKwInOut{Input}{input}\n\t\\SetKwInOut{Output}{output}\n\t\\nonl {\\bfseries{1. CRAN\/Headend}} \\newline\n\t\\nl (a) Identify cachable I\/Q samples. \\newline\n\t    (b) Create Caching rules. \\newline\n\t    (c) Signal the rules and data for caching.\t  \n\t\\newline\n\t\\nonl \\If{Cached I\/Q samples requires updating}{\n\t\t\\nonl  Signal remote node for cache renew or flush.\n\t\t\\nonl }\n \\nonl {\\bfseries{2. Remote Node}}   \\newline\n\\nonl\\ForEach{OFDM Symbol in $T_D$ and $T_L$}{\t\t\n  \\nonl          \\If{Caching is enabled}{\n  \\nonl\t\t\t\t  Read cache and I\/Q mapping\\; \n  \\nonl                    Insert I\/Q cache-read to recieved I\/Q\\;}     \n  \\nonl\t    \t\\If{FFT module is free}{\n  \\nonl\t\t\t\t  Schedule I\/Q for FFT based on Eqs.~\\ref{eq:stability}-\\ref{eq:comp.constr}\\;} \n  \\nonl          \\Else{\n  \\nonl\tSchedule at completion of current execution\\; \n  \\nonl }  \\nonl}\t   \t    \n\\end{algorithm}\n\n\\paragraph{Implementation}\n\\label{sec:tau}\nSharing of remote node infrastructure for multiple technologies can \nbe in both upstream and downstream, as the computations are \nperformed independent of each other \neven for wireless full-duplex communications. \nThe FFT computation duration \n$\\tau$ can be aggregate of multiple instances of OFDM symbol, for example, in\nthe case of carrier aggregation in LTE (or channel bonding in DOCSIS), \nthere would be an OFDM symbol \nfor each carrier component resulting in $\\tau_L=\\tau_1+\\tau_2\\dots\\tau_C$, \nwhere $C$ is the number of carrier components. Similarly, computations resulting \nfrom multiple LTE eNBs at a single node can be aggregated and abstracted to \na single $\\tau_L$ resulting in a lager $\\tau_L$ can be very \nclose the symbol duration $T_L$. Proposed approach can also be \neasily extended to more than two technologies which depend \non FFT computations by sharing the the remote node.\n \n\\section{\\MakeUppercase{Performance Evaluation}}\n\\label{sec:perf:eval}\nIn this section we present the performance evaluation of the R-FFT node where\nthe infrastructure of a remote node is \nshared for two independent LTE and DOCSIS technologies.\n\\begin{figure*}[t]\n\t\\begin{tabular}{cc}\n\t\n\t\t\\includegraphics[width=2.8in]{fs\/fig_tauD.pdf} \\vspace{-0.1cm}\n\t\t&\\includegraphics[width=2.8in]{fs\/fig_tauD_U.pdf} \\vspace{-0.1cm}\\\\\n\t\t\\vspace{+0cm}\n\t\t\\scriptsize (a) $T_L=71.4~\\mu\\textit{s, } \\tau_L=14.28~\\mu s\\text{, and } \\theta=0~\\mu\\textit{s}$\n\t\t&\\scriptsize (b) $T_L=71.4~\\mu\\textit{s, } \\tau_L=14.28~\\mu s\\text{, and } \\theta=0~\\mu\\textit{s}$\\\\\n\t\n\t\t\\includegraphics[width=2.8in]{fs\/fig_theta.pdf} \\vspace{-0.1cm}\n\t\t&\\includegraphics[width=2.8in]{fs\/fig_theta_U.pdf} \\vspace{-0.1cm}\\\\\n\t\t\\scriptsize (c) $T_L=71.4~\\mu\\textit{s, } \\tau_L=14.28~\\mu s\\text{, and } T_D=80~\\mu\\textit{s}$ \n\t\t&\\scriptsize (d) $T_L=71.4~\\mu\\textit{s, } \\tau_L=14.28~\\mu s\\text{, and } T_D=80~\\mu\\textit{s}$ \\\\\n\t\\end{tabular}\n\t\\caption{Simulation results for \n\t\tFFT interleaving procedure.} \n\t\\label{fig_tau_theta}\n\\end{figure*}\n\n\\subsection{FFT Module Sharing}\nIn the initial evaluation, we verify the proposed FFT interleaving procedure. \nwe have implemented the FFT interleaving mechanism \nas a discreet event simulation framework in OMNET++.\nA LTE and DOCSIS OFDM symbols are generated every \n$T_L$ and $T_D$ durations respectively. For LTE with normal cyclic prefix\nthe OFDM symbol duration $T_L$ is 71.4~$\\mu$s, while the DOCSIS \nsymbol duration can be either 40 or 80~$\\mu$s. \nFor each OFDM symbol arrival at the radio node, \nFFT computation is queued by the scheduler. Typically, the FFT size\nfor the LTE is 2K, whereas the FFT size for the DOCSIS can be 4K or 8K\nbased on the symbol duration (i.e., 40 or 80~$\\mu$s).\nComputation duration $\\tau$ for the FFT computation \ncan vary based on the number of carrier\ncomponents and number of radio nodes supported by the \nremote node as described\nin Sec.~\\ref{sec:tau}. \nInterleaving procedure should be aware of the computation duration for\neach technology to ensure the stable operation of infrastructure sharing. \nTo evaluate the system for wide range of FFT computation duration,\nwe define a computation duration factor\n\\begin{equation}\n\\centering\n\\beta=\\frac{\\tau_L}{\\tau_D}.  \n\\end{equation}\nWe choose the FFT computation duration $\\tau_L$ to be 20~\\% of \nthe $T_L$, which is $0.20 \\times 71.432~\\mu\\textit{s} = 14.2832~\\mu\\textit{s}$ \nand the computation factor $\\beta$ is varied \nfrom $0$ to $1$ with an effective sweep of \nthe FFT computation duration $\\tau_D$ (i.e., $\\min(T_L,T_D)$) \nfrom $0$ to $T_L$, which is effectively $0 \\leq \\tau_D \\leq T_L$. \nWe primarily evaluate the FFT sharing mechanism with \ntwo performance metrics, average wait time and utilization.\nAverage wait time is the average time required to\nschedule the FFT computation after the arrival of an OFDM symbol.\nWhereas, the utilization parameter is defined as the ratio of \ntotal computation time to total elapsed time.\n\nFigure.~\\ref{fig_tau_theta} presents the performance evaluation \nof proposed FFT sharing mechanism. \nThroughout the evaluation, the OFDM symbol duration \nof LTE is kept constant $T_L=71.4~\\mu$s.\nFigure.~\\ref{fig_tau_theta}(a) \nshows the average wait time of LTE and DOCSIS as a function of \ncomputation duration factor $\\beta$ for different values \nof DOCSIS OFDM symbol duration, $T_D=40\\mu$s and $T_D=80\\mu$s.\nWe can observe from the Fig.~\\ref{fig_tau_theta}(a), the \naverage wait time of LTE increases linearly as the computation factor \n$\\beta$ is increased which corresponds to increase\nin the FFT computation duration of DOCSIS $\\tau_D$. This shows that \nthe computation duration of one technology has the direct impact on \nthe average wait time of another as they compete with each other\nfor the computational resource. We also observe that the average\nwait time of both LTE and DOCSIS \nfor $T_D = 40~\\mu$s linearly increases up to $\\beta=0.45$ and \novershoots to a very large indicating the instability of the system.\nThis is because, for the FFT computation sharing mechanism to be \nstable, the Eq.~\\ref{eq:stability2} must be satisfied, i.e, \n$\\tau_D \\leq (T_L T_D - \\tau_L T_D)\/T_L$, which is, \n$\\tau_D \\leq (40 \\times 71.4 - 14.28 \\times 40)\/71.4 = 32~\\mu\\textit{s}$. \nWhen $\\beta=0.45$, the corresponding \n$\\tau_D = 0.45 \\times T_L = 0.45 \\times 71.4~\\mu\\textit{s} = 32.13~\\mu\\textit{s}$,\nsurpassing the stability limit based on Eq.~\\ref{eq:stability2}, which is $>32~\\mu\\textit{s}$. \nThis behavior can also be observed in Fig.~\\ref{fig_tau_theta}(b) \nwhich plots the utilization as a function of\ncomputation factor $\\beta$. While the utilization of FFT computation module \nincreases linear with $\\beta$, the system \napproaches nearly $100\\%$ utilization when $\\beta>0.45$ for $T_D=40~\\mu\\textit{s}$ \nreaching the stability limit of the system. \nSimilarly, when the DOCSIS OFDM symbol duration $T_D$ is\nchanged from $40~\\mu\\textit{s}$ to $80~\\mu\\textit{s}$, the \nbehavior of average wait time is similar to $T_D = 40~\\mu\\textit{s}$, but the\nsystem becomes unstable when $\\beta > 0.9$. When $\\beta$ is $0.9$, the value \nof $\\tau_D$ is $0.9 \\times T_L = 0.9 \\times 71.4~\\mu\\textit{s} = 64.26~\\mu\\textit{s}$,\nwhich is slightly  greater than stability limit of \n$\\tau_D \\leq (80 \\times 71.4 - 14.28 \\times 80)\/71.4$ or $\\tau_D \\leq 64~\\mu\\textit{s}$. \nConsequently, we can observe from Fig.~\\ref{fig_tau_theta}(b), the utilization\nof system approaches to $100\\%$ for $\\beta\\geq0.9$ for $T_D=80~\\mu\\textit{s}$.\n   \nFig.~\\ref{fig_tau_theta} (a) and (b) corresponds to the evaluation \nwhen guard time $\\theta=0~\\mu\\textit{s}$ for different values of\nDOCSIS OFDM symbol duration $T_D=40$ and $80~\\mu\\textit{s}$. In contrast, \nFig.~\\ref{fig_tau_theta} (c) and (d) evaluates the system for different \nguard times $\\theta=2$ and $10~\\mu\\textit{s}$ for $T_D=80~\\mu\\textit{s}$. \nFig.~\\ref{fig_tau_theta} (c) plots the average wait time as a function of\ncomputation factor $\\beta$, whereas, Fig.~\\ref{fig_tau_theta} (c) plots the\nsystem utilization as a function of $\\beta$. Fig.~\\ref{fig_tau_theta} (c) \nshows the same behavior as in Fig.~\\ref{fig_tau_theta} (a), where the average \nwait time for LTE increases linearly as the $\\beta$ is increased. \nThe scheduling of $\\tau_D$ experiences \na constant delay of $2.3~\\mu\\textit{s}$ until the stability limit as \nthe computation duration of LTE is retained constant in the simulation\n$\\tau_L=14.28~\\mu\\textit{s}$. \nIn contrast to Fig.~\\ref{fig_tau_theta} (a), \nsystem reaches stability limit for lower values of $\\beta$ in (c) \nand this behavior pronounced as the guard time is increased. \nMore specifically,\nwhen $\\theta$ is increased from 0 (see Fig.~\\ref{fig_tau_theta} (a) for $T_D = 80 ~\\mu\\textit{s}$) to $2~\\mu\\textit{s}$, the system becomes \nunstable for $\\beta=0.85$ as compared to $\\beta=0.9$. \nBased on Eq.~\\ref{eq:guardtime2}, for a system with guard time to be \nstable, $\\tau_DT_L + \\tau_LT_D \\leq T_DT_L - \\theta(T_L+T_D)$ or\n$\\tau_L \\leq (T_DT_L - \\theta(T_L+T_D)-\\tau_LT_D) \/ T_L$.\nWhen $\\beta=0.85$ which corresponds to $\\tau_D=60.69~\\mu\\textit{s}$, with $\\theta=2~\\mu\\textit{s}$, the stability\ncondition evaluates to be $tau_D \\leq 59.75$, therefore, the system is\nunstable for $\\beta=0.85$ indicated by the very \nlarge in the average wait time for $\\beta>0.85$.\nSimilarly, for $\\beta=0.6$ which corresponds to $\\tau_D=42.84~\\mu\\textit{s}$, with $\\theta=10~\\mu\\textit{s}$, the stability\ncondition evaluates to be $\\tau_D \\leq 42.79$, therefore, the system is\nunstable for $\\beta=0.6$.\nAdditionally, the effect of guard time has the direct impact on the\nutilization of the system. We see from Fig.~\\ref{fig_tau_theta} (d), the maximum\nachievable utilization of the system is only up to \n94~\\% when guard time $\\theta=2~\\mu\\textit{s}$ for $\\beta>0.8369$. \nAs value of $theta$ is increased\nto $10~\\mu\\textit{s}$, the maximum achievable utilization is\nreduced to 73.5\\% for $\\beta>0.59$. \nThe saturation in the utilization also\nindicates the instability of the system, where the average wait time\ngrows to a very large value for $\\beta>0.8369$ and $\\beta>0.59$,\nwhen guard time $\\theta=2$ and $2~\\mu\\textit{s}$ respectively. \n\n\\subsection{End-to-End Delay Evaluation}\n\\begin{figure*}[t]\n\t\\begin{tabular}{cc}\n\t\n\t\t\\includegraphics[width=2.9in]{fs\/fig_cable02.pdf} \\vspace{-0.1cm}\n\t\t&\\includegraphics[width=2.9in]{fs\/fig_LTE02.pdf} \\vspace{-0.1cm}\\\\\n\t\t\\vspace{+0cm}\n\t\t\\scriptsize (a) DOCSIS Delay (end-to-end), $\\rho_c=0.2$\n\t\t&\\scriptsize (b) LTE Delay (fronthaul), $\\rho_c=0.2$, $H=0.5$\\\\\n\t\n\t\t\\includegraphics[width=2.9in]{fs\/fig_cable06.pdf} \\vspace{-0.1cm}\n\t\t&\\includegraphics[width=2.9in]{fs\/fig_LTE06.pdf} \\vspace{-0.1cm}\\\\\n\t\t\\scriptsize (c) DOCSIS Delay (end-to-end), $\\rho_c=0.6$\n\t\t&\\scriptsize (d) LTE Delay (fronthaul), $\\rho_c=0.6$, $d=50$~km\\\\\n\t\\end{tabular}\n\t\\caption{Simulation results for RFFT and RPHY.}\n\t\\label{fig_rfft_perf}\n\\end{figure*}\n\\begin{figure}[t!]\n\t\\centering\n\t\\includegraphics[width=3.5in]{fs\/fig_tput06.pdf} \n\t\\vspace{-0.3cm}\n\t\\caption{Average DOCSIS (cable) and LTE throughput.}\n\t\\label{fig_rfft_tput}\n\\end{figure}\nSubsequently, in this evaluation, we study the delay characteristics of R-FFT\ndeployment in comparison to existing R-PHY deployments. In addition, \nwe also present the impact of cable traffic on the LTE fronthaul link when\nthe same cable infrastructure is shared between LTE and DOCSIS.\nWe developed a simulation framework in a discreet event \nsimulator OMNET++ to implement the DCCAP cable architecture of \nthe HFC network. A remote node (RPHY or RFFT node) is connected \nto the headend through an optical connectivity separated by a \ndistance $d$ and data rate of $R_o=10$~Gbps. \n200~Cable Modems (CMs) are connected to the remote node through\na analog broadcast cable connectivity. We assume the DOCSIS~3.1 protocol\nto coordinate the cable transmissions in the cable broadcast medium \nwith the bidirectional cable data rate $R_c=1$~Gbps.\nWe implemented a Double Phase Polling (DPP) protocol\\cite{cho2009dou}\nfor the scheduling of 200~CMs over the cable segment. In case of \nRPHY, DOCSIS PHY frames are digitized and transported over Upstream \nExternal PHY Interface (UEPI) where the REQ packets were prioritized. \nWhereas for the RFFT, the upstream cable \ndata is converted to frequency I\/Q and transported as a generic UDP packet.\nFFT size of $4K$ which corresponds to $T_D = 40~\\mu s$ and QAM size of $12$~bits,\nand the code rate of $9\/10$ are used in the \nprocess of converting data to frequency domain I\/Qs.\nEach complex number representing an I\/Q symbol\nis digitized with $20$~bits.\nWe assume the deployment of an LTE RRH at the site of \nremote node in conjunction to the operation of RFFT or RPHY \nnode sharing the optical connectivity \nof data rate $R_o$ to the headend where the BBU CRAN is implemented.\nWe note that our study is focused on deployment of LTE networks\nover the HFC cable infrastructures. Therefore, we implement and\nevaluate the performance of a cable network when an\nLTE fronthaul traffic is simultaneously injected to the shared optical link.\nWe have modeled a typical FIFO queue at the remote node to froward the \nLTE packets to CRAN\/BBU. \nWe model the self similar traffic generation \nat each CM with varying burst levels controlled \nby the hurst parameter $H$ with the average packet size of $472$~KB. \nThe hurst parameter $H=0.5$ corresponds to the Poisson traffic, and the\nburstiness increases as the value $H$ is increased.\n\nFigure~\\ref{fig_rfft_perf} presents the performance evaluation of the\nRFFT and RPHY network in presence of the LTE fronthaul traffic. \nIn Fig.~\\ref{fig_rfft_perf}(a) we show the mean packet delay of the DOCSIS\nas a function of LTE fronthaul traffic intensity $\\rho_L$ which corresponds \nto the LTE data rate $R_L = \\rho_L \\times R_o$ for different optical distance \n$d$ and traffic levels burstiness $H$, with fixed cable \ntraffic intensity of $\\rho_c = 0.2$ which corresponds to the \ntraffic of $\\rho_c \\times R_c = 0.2 \\times 1$~Gbps $=200$~Mbps. \nDelay is considered for an end-to-end connection of the cable \nconnectivity i.e., from the CM to headend.\nFrom Fig~\\ref{fig_rfft_perf}(a) we observe for the \nshorter optical distance which is typically deployed for an LTE\nfronthaul link has the minimal impact on the cable infrastructure for both\nRFFT and RPHY deployments. More specifically, \nfor $H=0.5$ in RFFT (and RPHY) for distance $10$~km and $50$~km \nwe observe the delay difference of roughly $0.2$~ms \n(delay of $3.73$~ms for $10$~km RFFT $H=0.5$ and \nthe delay of $3.93$~ms for $50$~km RFFT $H=0.5$)\nwhen the distance is changed from $10$~km to $50$~km. \nThis difference in the delay is from the propagation\ndelay in the optical fiber which is approximately \n$(d_2-d_1)\/c = 40~\\text{km}\/2\\times10^8~\\text{m\/s}=0.2$~ms.  \n\nWe also observe a minute difference in the\ndelay between RPHY and RFFT implementation at the remote node for \nthe Poisson traffic, but a larger difference for higher burstiness\ntraffic. For example, at $\\rho_L=0.5$ \nwe observe the mean packet\ndelay values of $3.75$~ms and $3.73$~ms for \nRFFT and RPHY respectively.\nThe difference in the delay of nearly $0.02$~ms \nis due to the average transmission delay of \nthe cable data which is inflated by the process of \nI\/Q conversion and digitization. For the \ncode rate of $9\/10$ and the QAM size of $12$~bits \nthe cable data would be inflated by \nfactor of $(10\/9) \\times (1\/12) \\times 2 \\times 10=1.85$. Thus,\nthe RFFT packets undergo an additional delay \nat the remote node for the \ncomplete transmission of data to headend. \nHowever, this effect is pronounced for $H=0.8$\nbecause of the traffic with the higher \nburstiness. \nThe behavior of the DOCSIS delay for RPHY and RFFT \nremains constant until $\\rho_L=0.96$ for $H=0.5$ and $\\rho_L=0.8$ \nfor $H=0.8$, as\nthe optical link is not congested. \nHowever for $\\rho_L>0.8$,\nthe effective throughput over the optical link \nreaches more than the capacity $R_o$ crossing \nthe stability limit of the system. For e.g., \nthe instantaneous cable traffic of RFFT on the \noptical link for $H=0.5$ and \n$\\rho_c=0.2$ would be approximately \n$200~$Mbps$\\times 1.85=370$~Mbps or $0.37$~Gbps.\nThus, when $\\rho_L>0.963$ the effective\ntraffic would be more than $R_o$ \nresulting in the very large value of delay due \nto instability of the system. \n\nAs seen in Fig.~\\ref{fig_rfft_perf}(b) which plots the \nmean packet delay for the LTE fronthaul traffic for RFFT and RPHY \nfor different optical distances of $10$ and $10$~km. LTE fronthaul delay\nthe directly impacted by deployed cable technology when the traffic from\nLTE and cable are aggregated at the remote node. \nRFFT induces more delay on the LTE fronthaul traffic compared to the RPHY, \nwhich is mainly resulting from the increase in the effective data rate\nover the optical link. Also, we can observe the \neffect of stability limit of the optical link which is \nin both Fig.~\\ref{fig_rfft_perf}(a) and (b) [see for e.g., \nthe overshoot of the delay \nwhen $\\rho_L=0.96$ for RFFT, $H=0.5$ and $d=50$~km \nat Fig.~\\ref{fig_rfft_perf}(a) and (b)]. \nTherefore, we can infer that RFFT and RPHY has the similar \ndelay characteristics for cable traffic as well as \nthe LTE fronthaul traffic.    \n\nTo further study effect of cable traffic load, in \nFig.~\\ref{fig_rfft_perf}(c) we evaluate the behavior of mean packet delay\nfor DOCSIS when cable load is $\\rho_c = 0.6$. Fig.~\\ref{fig_rfft_perf}(c) plots the\nmean packet delay as the function of LTE fronthaul traffic $\\rho_L$. \nSimilar to Fig.~\\ref{fig_rfft_perf}(a), we observe a narrow difference in the delay \nperformance between RFFT and RPHY for the Poisson traffic ($H=0.5$). But, for the \nself similar traffic with $H=0.8$, we observe the difference of more \nthan $10$~ms between RFFT and RPHY which is partly due to the \ntemporary cable link saturation resulting from the \ninstantaneous load reaching more than the cable link \ncapacity $R_i$ $\\rho_c$ and $20$~\\% which is reserved \nfor cable maintenance. This difference is even \nfurther pronounced by the \ncombined effect from the data inflation at the remote node. \nAs the LTE fronthaul traffic $\\rho_L$ approach $0.8$ and $0.94$ \nfor RFFT and RPHY respectively, the mean packet delay packet \nincreases to a very large value resulted by the\noptical link saturation. For $\\rho_c=0.6$, the effective rate over the \noptical link is $\\rho_L + (1.85 \\times \\rho_c)$, therefore, \nwhen $\\rho_c = 0.6$, the optical link is saturated for \n$(1 -1.85 \\times 0.6) = 0.89$ which results in the \nlarge value of delay at $\\rho_L = 0.8$ and $0.94$ for \nRFFT and RPHY with $H=0.5$. The throughput behavior \nis verified in Fig.~\\ref{fig_rfft_tput} which plots the \ncable and LTE throughput as a function of LTE traffic \nintensity $\\rho_L$. We can compare Fig.~\\ref{fig_rfft_perf}(c) and \nFig.~\\ref{fig_rfft_tput} to see that delay and the average\nthroughput reaching stability at $\\rho_L = 0.8$ and $0.94$ for \nRFFT and RPHY with $H=0.5$. On the other hand, when the distance is increased from $10$~km \nto $50$~km, we see the pronounced effect in \ndelay difference between the RPHY and RFFT \nin Fig.~\\ref{fig_rfft_perf}(c). \nThis effect is because of the \ncombined impact of propagation delay and transmission delay \ndue to data inflation of RFFT on the DOCSIS scheduler.\n\nTo understand the effect of \ntraffic burstiness along with the cable load\non the LTE fronthaul traffic,\nwe show the mean packet delay as a function of LTE fronthaul \ntraffic $\\rho_L$ in Fig.~\\ref{fig_rfft_perf}(d). \nWe can relate the results in \nFig.~\\ref{fig_rfft_perf}(c) to Fig.~\\ref{fig_rfft_perf}(d) \nas such the traffic with higher burstiness $H=0.8$ \nincreases delay of both cable and LTE fronthaul traffic. \nAdditionally, the large fronthaul delay values at  $\\rho_L = 0.8$ and $0.94$\nindicates the stability limits for the Poisson traffic ($H=0.5$) \nfor RPHY and RFFT. However,\nthe fronthaul delay is more impacted the higher burstiness ($H=0.5$) \nas compared to the Poission traffic ($H=0.8$) in the RFFT deployment.\nFor example, even when the fronthaul load is $\\rho_L=0.5$ \n(i.e, below the stability limit), \nthe fronthaul delay is almost increased\nby $4$ times from $0.27~$ to $2.04~$ms for $H=0.8$ in the RFFT deployments. \nTherefore, if the functional split of LTE requires a dedicated \nQoS requirements, then a resource allocation \nmechanism of optical link may be employed for the dedicated assignments. \n\n\\section{\\MakeUppercase{Conclusions}}\nWe have presented a mechanism for Time-Frequency resource elements (QAM) caching along with the \nFFT sharing for LTE and DOCSIS coexistence. We have comprehensively evaluated the FFT \ninterleaving procedure through the simulation evaluation.\nAdditionally we have also provided the \nperformance comparison between the RFFT and RPHY remote nodes while \ncoexisting with LTE networks.\nOur study  numerically evaluate and show that \nRFFT performance is very close to RPHY. As \nRFFT achieves more flexibility and more \nscalability RFFT can be preferred for deployment. Additionally we also \ndiscuss the impact of functional split cable traffic on \nthe LTE fronthaul traffic. \n\n\n\n\n\\chapter{\\MakeUppercase{Future Work Outline: The SDN-LayBack: An SDN-Based Layered Backhaul\n\tArchitecture for Dense Wireless Networks}}\n\n\\section{\\MakeUppercase{Introduction}}\n\n\\subsection{Need for a New Backhaul Architecture for Wireless Networks}\n\nA plethora of wireless devices running a wide range\nof applications connect to radio access networks (RANs), as illustrated in Figure~\\ref{Bottleneck}.\nRANs provide end device to radio node (base station) connectivity.\nThe radio nodes are connected to the core networks by technology-specific backhaul access\nnetworks, such as LTE or WiMax backhaul access networks.\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=1\\textwidth]{layback_commag\/Bottleneck2.pdf}\n \n    \\caption{Bottleneck in the wireless Internet access chain.}\n    \\label{Bottleneck}\n\\end{figure}\nPresent day hand-held wireless end devices have very high\nprocessing and memory capabilities, supporting various\napplications, including resource demanding ones such as live 4K video streaming.\nAdvanced RANs, such as LTE-Advanced,\nsupport up to several hundred Mb\/s in downstream\nby exploiting a range of physical wireless layer techniques, such as\nmulti-carrier aggregation, opportunistic utilization of unlicensed\nbands, and millimeter wave technologies.\nA dense infrastructure of small cells with appropriate interference\nmanagement is a critical technique for advanced RANs~\\cite{nak2013tre}.\nAt the same time, the core networks already employ high-capacity optical links\nproviding abundant transmission capacity.\n\nHowever, the backhaul access networks have emerged as a critical\nbottleneck in wireless Internet\naccess~\\cite{and2014ove,ghi2015rev,rat2010fem,wan2015bac}.\nToday's backhaul access networks typically require\nhighly-priced bulky network equipment with proprietary locked-in control\nsoftware~\\cite{ber2015sof}.\nThey also need to be highly reliable and have high\navailability.\nIt is therefore very difficult to\nadd new network functionalities, reconfigure operational states,\nor upgrade hardware as technology advances.\nThese aspects have combined to stifle progress in backhaul access\nnetworks, causing them to become a critical bottleneck that can stall\nthe progress of wireless networking and undermine the advances in the\ncomplementary RANs and core networks. It can therefore be concluded that there is a need for innovative approaches, such as a new architecture,\nto enable technological progress in terms of backhaul access networks. \n\n\n\\subsection{SDN-LayBack - an Advanced Wireless Network Backhaul Architecture}\nThis paper proposes \\emph{SDN-LayBack, a fundamentally novel backhaul architecture\nfor wireless backhaul access networks based on software defined networking\n(SDN)}~\\cite{Bradai,moh2015con}.\nIn contrast to other recently proposed backhaul architectures, such as\nCROWD \\cite{seb2015dyn},\niJOIN \\cite{Dongyao},\nU-WN \\cite{Shengli}, and Xhaul~\\cite{oli2015xha},\nour SDN-LayBack architecture\nconsistently decouples the wireless RAN technology\n(such as LTE or WiFi) from the backhaul access network.\nIn addition, our SDN-LayBack architecture flexibly accommodates\nhighly heterogeneous RANs,\nranging from sparse cell deployments in rural areas to\nextremely high density cells in crowded stadiums.\nFurther, to the best of our knowledge, SDN-LayBack is the first\narchitecture to simultaneously  provide a new clean-slate platform for\ncellular backhaul and support the coexistence with current cellular access backhaul technologies, such as LTE (4G\/3G),\nWCDMA (3G), and GSM (2G).\nPrevious clean-slate architecture proposals,\nsuch as the one proposed by Ameigeiras et al.~\\cite{Ameigeiras}, do \\textit{not}\nsupport existing cellular backhaul technologies, and\nhence have limited functionalities. SDN-LayBack is layered on top of the existing backhaul networks and flexibly supports a wide range of RANs through a network of SDN switches and a hierarchical SDN controller pool.\n\nThis paper also introduces \\emph{an innovative four-step intra-LayBack handover protocol} within a given\ngateway in the SDN-LayBack architecture. Evaluations indicate 60~\\%\nreductions of the signalling load in comparison with conventional LTE handover.\nFinally, the article discusses future research directions within the\nSDN-LayBack architecture framework, focusing on\ninterference mitigation, device-to-device (D2D) communication, and video streaming~\\cite{seeling2012video, seema2011towards, chikkerur2011objective,\nvan2009implications,van2008traffic, seeling2004network, kangasharju2002distributing, fitzek2001mpeg}.\n\n\\section{\\MakeUppercase{SDN-LayBack: a Novel SDN-Based Layered Backhaul Architecture}}  \\label{arch:sec}\n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=1\\textwidth]{layback_commag\/ProposedNewArchitecture.pdf}\n    \\caption{SDN-LayBack architecture.}\n    \\label{LA:fig}\n\\end{figure}\n\n\\subsection{Overview}\nAs already mentioned, the proposed SDN-LayBack is a novel SDN-based layered backhaul access network\narchitecture which can be layered on top of an existing backhaul access network (e.g., SAE-3GPP-LTE) in an evolutionary manner. Fig.~\\ref{LA:fig} illustrates SDN-LayBack's deployment on top of the classic LTE architecture.\n\nConventional backhaul access networks include technology-specific\nfunctional blocks for different RAN technologies.  For instance, in the LTE architecture, macro cells are\nserved by an LTE Serving-Gateway (S-GW), while pico\/femto cells can be\nserved by an Home-eNodeB-Gateway (HeNB-GW) (or S-GW), as illustrated in\nthe bottom left of Fig.~\\ref{LA:fig}.\nRAN and backhaul are thus coupled in conventional wireless networks, e.g., a\nWiMax radio node cannot be used within an LTE architecture. Only with\nan additional dedicated network entity, an evolved packet data\ngateway (ePDG), can a WiFi radio node be connected to LTE backhaul\nwith IPSec tunneling, as illustrated in the bottom middle of\nFig.~\\ref{LA:fig}.\n\nIn contrast, our proposed SDN-LayBack architecture decouples the RAN from\nthe backhaul, as illustrated in Fig.~\\ref{LA:fig}.\nThe figure also indicates SDN-LayBack's four architectural layers:\n\\emph{RAN Layer}, \\emph{SDN Backhaul Layer}, \\emph{Core Networks Layer},\n and \\emph{Evolutionary Internetworking Layer}.\n\nIn the SDN-LayBack architecture, heterogeneous RAN technologies, such as LTE\neNBs, pico\/femto LTE, WiFi, and WiMax, can be flexibly\ndeployed and accommodated in the \\emph{RAN Layer}. The \\emph{RAN Layer} interacts\nwith the \\emph{SDN-LayBack Backhaul Layer} through the network of flexible SDN (OpenFlow, OF)\nswitches and onwards through the programmable gateway pool to the Internet with the \\emph{Core\nNetworks Layer}.  The data plane of the SDN-LayBack backhaul network consists of\nthe network of SDN switches and the gateway pool. This data plane is\ncentrally controlled by the SDN controller pool and the applications (with corresponding databases). SDN-LayBack interfaces with the classic backhaul access network architecture through \\emph{the Evolutionary Internetworking Layer}.\n\n\\subsection{Outline of Layers in SDN-LayBack Architecture}\nThis section outlines the key components and functionalities of the\nproposed LayBack architecture and contrasts the LayBack architecture with previous designs. \n\n\\subsubsection{Radio Access Network (RAN) Layer} \n\n\\paragraph{Wireless End Devices}  \n\nMobile wireless end devices are heterogeneous and have a wide range of\nrequirements. Providing reasonable quality-oriented services to\nevery device on the network is a key challenge to the\nwireless network design.\nThe proposed SDN-LayBack architecture takes a unique approach to provide\nrequirement-specific network connectivity to every device that is\nconnected to the SDN-LayBack network.\nFuture devices will likely be\nhighly application-specific, such as a speed sensor on a race track\nor a health monitoring biosensor.\nThe SDN-LayBack architecture is designed to be adaptive\nto the different environments, such as an air port terminal, public park, or\nuniversity\/school. This enables SDN-LayBack to\nsupport a wide range of device requirements, such as\nreal-time D2D video streaming at a large sporting event or music concert.\n\n\\paragraph{Radio Nodes}  \nRadio nodes, such as the evolved NodeB (eNB) in LTE or an\naccess point (AP) in WiFi, provide RAN services to\nthe end devices. Aside from LTE and WiFi, there exists a wide range of\nwireless access technologies (and protocols), including\nWiMax, Zig-Bee, Bluetooth,\nand near field communication (NFC).\nThese wireless protocols have unique advantages and serve unique purposes.\nA fluidly flexible backhaul that homogeneously supports\nmultiple different types of radio nodes does not yet exist, but is\nhighly desirable.\nTherefore, our SDN-LayBack architecture is\nfundamentally designed to work with multiple RAN and communication technologies\nby isolating the RAN layer from the backhaul network.\n\nIn existing backhaul access networks, a given\nRAN technology requires a corresponding specific backhaul. For instance, the\nLTE radio access network can only operate with LTE\nbackhaul network entities. This restriction of\nspecific RAN technologies to specific backhaul\ntechnologies limits the usage of other RAN technologies on\nLTE backhaul networks and vice-versa.\nIn addition, femto cells are expected to operate\non multiple RAN technologies~\\cite{Bennis}.\nTo address this restrictive structure of present RAN-backhaul\ninter-networking, we propose our SDN-LayBack architecture to flexibly\nsupport multiple types of radio nodes, including software defined\nradios integrated with SDN enabled radio nodes.  \nIn an interesting approach, the reconfigurable antennas \n\\cite{saeed2012new, costantine2013new, saeed2015flexible, \n\tsaeed2016reflection, saeed2016radiation, costantine2014mimo, saeed2016inkjet} can be also \nbe further extended SDN to achieve more flexibility in the RAN design.\n\nMoreover, the system modeling for the RAN can be borrowed from the \ncontrol literature. Techniques to design the systems for multiple specifications have been extensively discussed in \n\\cite{puttannaiah_acc_2016,puttannaiah_acc_2015,puttannaiah_cdc_2015,puttannaiah_echols_gnc_2015,puttannaiah_2013_msthesis}. \nOptimization problems can be formulated that address multiple\nobjectives subject to constraints, by exploiting convex optimization ideas.\nProblems on a set of nonlinear hybrid systems, involving mixed-integer\nproblems, can be solved using convexification\n\\cite{puttannaiah_nandola_ecc_2013}. \nHigh fidelity numerical simulators and system identification techniques can\nbe used to efficiently solve optimization problems \\cite{puttannaiah_cartagena_acc_2016} using SDN.\n\n\\subsubsection{SDN Backhaul Layer}  \nThe SDN backhaul layer encompasses the SDN switches (with hypervisor), a\ncontroller pool, and a gateway pool, as well as the applications and corresponding databases.\nNext, the main aspects of the SDN switches, controller pool, applications, and gateways are briefly outlined in this section.\n\n\\paragraph{SDN (OpenFlow) Switches} \nSDN capable switches are today typically based on the OpenFlow (OF)\nprotocol\nand are therefore sometimes referred to as\nOpenFlow switches or OF switches.\nWe present SDN-LayBack in the context of OpenFlow switches, but emphasize that\nthe SDN-LayBack architecture applies to any type of SDN-capable switch.\nOF-switches are capable of a wide range of functions, such as forwarding a\npacket to any port, duplicating a packet on multiple ports, modifying the\ncontent inside a packet, or dropping the packet.\nThe switches can be connected in a mesh to allow disjoint flow paths\nfor load balancing.\n\n\\paragraph{Controller Pool} \nThe SDN-LayBack controller pool configures the entire OF-switch fabric\n(network of OF switches) by adding flow table entries on every switch.\nThe controller pool is also able to dynamically configure and program\nthe gateway pool functions, such as queuing policies, caching,\nbuffering, charging, as well as QoS \\cite{guck2016function, fitzek2002providing} and IP assignment for the end devices.\nThe SDN-LayBack controller pool coordinates with existing (legacy)\ncellular network entities, such as the mobility management entity (MME) and\nother 3GPP entities,\nto coexist with present architectures.\nConnections from the SDN-LayBack controller to the 3GPP entities can be\nprovided by extending the existing tunneling interfaces at the 3GPP entities.\nWe define extensions of the tunneling interfaces as a part of\nthe \\emph{Evolutionary Internetworking Layer} in Fig.~\\ref{LA:fig}.\nThus, the proposed SDN-LayBack architecture can enable communication\nbetween, the new\nflexible RAN architecture in the top left of Fig.~\\ref{LA:fig} as well as\nlegacy technology-specific RAN and backhaul networks, see bottom\nof Fig.~\\ref{LA:fig}. \n\n\\paragraph{Applications} \nApplications are programs executed by the controller for\ndelegating instructions to individual SDN switches. The controller\napplications realize all the network functions required for Internet\nconnectivity, such as authentication and\npolicy enforcement, through the switch flows.\nRadio nodes require RAN-specific interfaces for their operation\n(e.g., the X2 interface in LTE),\nWe realize these interfaces through controller applications on the network of\nSDN-switches. Therefore, for LTE in SDN-LayBack, eNBs are enabled with\ninter-radio node X2 interfaces along with other required interfaces.\nCellular networks have many network functions, such as the\nAutomatic Network Discovery and Selection Function\n(ANDSF)\nof 3GPP. As an example, to replace ANDSF\nin LayBack, a dedicated controller application will be responsible for\ndelegating network access policies to the devices.\nDatabases assist the SDN controller applications in their operations. \n\n\\paragraph{Gateways} \nGateway functions are programmed by the SDN controller\nto perform multiple network functionalities in order\nto simultaneously establish connectivity to heterogeneous RANs, such as\nLTE, Wi-Fi, and Wi-Max.\nFor example, the gateway in SDN-LayBack functions as\nboth S-GW and P-GW for an LTE radio node.\n\n\\subsection{SDN-LayBack Hierarchical Micro-Architectures}\n\\label{ma:sec}\nThe SDN-LayBack architecture can provide a wide range of\nheterogeneous services to localized regions through\nhierarchical SDN-LayBack micro-architectures.\nA micro-architecture hierarchy consists of a global (root) controller\nand multiple environment-specific local controllers located in the\nrespective environments, as illustrated in Fig.~\\ref{fig:microarchitectures}.\nThe local controllers can tailor applications for a specific\nenvironment, e.g., a park with a sparse user population that may be\nlocated adjacent to a stadium with a very high user density.\nAccordingly, micro-architectures allow to\nclassify applications specific to an environment\nas local applications and more common applications as global applications.\nFor example, WiFi offloading and adaptive video streaming\ncan be considered as local\napplications, whereas applications that serve a larger purpose, such as\ninterference management, can be considered as global applications.\n\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[width=1\\textwidth]{layback_commag\/MicroArchitecture2.pdf}\n\t\\caption{Environment-specific micro-architectures and controller\n\t\thierarchy.}\n\t\\label{fig:microarchitectures}\n\\end{figure}\n\n\\section{\\MakeUppercase{Intra-LayBack Handover: A Radically Simplified Handover Protocol}}\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=1\\textwidth]{layback_commag\/HOSigFlow.pdf}\n\\caption{Proposed simplified intra-LayBack handover protocol.}\n\t\\label{fig:handover}\n\\end{figure}\n\n\\subsection{Motivation: Signalling Load Due to Frequent Handovers}\nThe LTE protocol has generally higher signalling overhead\ncompared to legacy 3G protocols.\nA recent forecast from Oracle Corp.~\\cite{OracleSignalling}\nexpects a large increase of the global LTE signalling traffic, which is\npredicted to surpass the total global IP traffic growth by 2019.\nThe increase of signalling traffic can result in\nservice interruptions to customers and financial losses to cellular operators.\nTherefore, new architectures\nwith simplified signalling mechanisms are necessary to tackle the\ngrowth of signalling in the cellular networks.\n\nSignalling in the cellular backhaul is required for a wide range of\nactions, including initial attach and handovers.\nInitial attach procedures are completed only when a device requests\nconnectivity for the very first time. The initial attach procedure can result in\nlarge signalling overhead; however, it is executed only once.\nHandovers are initiated when a device measures and reports\npoor signal quality to the network.\nDepending on the configuration, the neighboring cell signal\nquality along with the quality of serving cell signal may be\nconsidered for handover decisions.\nHandovers require large\namounts of signalling in the backhaul and occur very frequently.\nEspecially in ultra dense networks (UDNs), where the cell\nrange is reduced to relatively small areas on the order of several\nmeters (e.g., 20~m in femto cells), handovers are very frequent.\nEven slow movements of devices\ncan cause device moves between multiple cells, resulting in handovers.\nHence, especially in UDNs, handovers are the source of serious bottlenecks in the backhaul entities.\n\n\\subsection{Proposed 4-Step Intra-LayBack Handover} \nWe propose \\emph{Intra Lay-Back Handover}, as the first mechanism\nto exploit the traffic duplication\nproperty of SDN to radically simplify the handover procedure in\ncellular backhaul.\nIn comparison to X2 based LTE handover \\textit{within} the same S-GW,\nour proposed method for handover within the same LayBack gateway\n(Intra-LayBack Handover) achieves:\n1) 100\\% reduction of the signalling load at the gateway,\ni.e., completely eliminates the handover signalling load on the LTE\ngateways,\n2) 69\\% reduction of signalling load compared to the LTE MME, and\n3) 60\\% reduction in the overall signalling cost.\n\nWe employ four steps in the handover protocol,\nas illustrated in Figure~\\ref{fig:handover}.\n\n\\emph{Step 1 Handover Preparation}\n\nWhen a measurement report satisfying\nthe handover conditions arrives at the serving eNB (SeNB), the device\nbearer context information is forwarded to the target eNB (TeNB) and a\nmessage requesting handover is sent from the SeNB to the SDN-LayBack controller.\n\n\\emph{Step 2 Enable Traffic Duplication}\n\nLayBack controllers enable the traffic\nduplication at the SDN switch through an OpenFlow\nmessage for all the downlink traffic related to the device waiting to\nbe handed over. The SDN switch continuously changes the headers (tunnel\nIDs) such that traffic related to the device reaches at TeNB. The bearer\ninformation pertaining to the device which has already been\nreceived by the TeNB from the SeNB is used to receive the duplicated traffic\nat the TeNB. Once the duplicated traffic related the device is\nreceived at the TeNB, a message is sent to the SeNB indicating the readiness\nfor the handover. The SeNB then sends a handover command to the device\n(i.e., RRC reconfiguration in LTE).\n\n\\emph{Step 3 Perform Wireless Handover}\n\nUpon receiving the handover command, the device\nbreaks the current wireless link (a hard handover)\nand establishes a new wireless connection to the TeNB (through reserved\nRACH preambles in LTE) \\cite{tyagi2014impact, tya2015con, gur2017hyb, vilgelm2017latmapa}. \nBy then, duplicated traffic arrived prior to wireless\nconnection reestablishment is waiting to be forwarded to the device in the\nnew connection. Downlink traffic is then forwarded to\nthe device as soon as the wireless connection is established.\n\n\\emph{Step 4 Stop Traffic Duplication}\n\nWhen the device traffic\nflow path through the new wireless connection\nis successful in both the uplink and downlink, then the\nTeNB sends a handover complete message to the LayBack controller.\nThe controller sends\nan OpenFlow message to the SDN switch to stop the duplication by\ndisabling the device traffic flow to the SeNB.\n\nOther handover types, such as inter-LayBack gateway\nhandover between different LayBack gateways, can be accomplished\nthrough extensions of the presented intra-LayBack handover.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=1\\textwidth]{layback_commag\/HOTable.pdf}\n\t\\caption{Handover cost and load comparison in LayBack and LTE backhaul}\n\t\\label{tab:handover}\n\\end{figure}\n\\subsection{Signalling Overhead Evaluation} \nTable~\\ref{tab:handover} characterizes the amount of required\nsignalling messages\nfor X2-based handovers in LTE and for intra-LayBack handovers\nwithin the same gateway.\nWe observe from Table~\\ref{tab:handover} that the overall signalling\ncost is reduced from 15 messages in LTE to 6 messages in the proposed\nintra-LayBack handover, a 60~\\% reduction of signalling overhead.\nWe note that we count an OpenFlow message sent from the controller to an SDN\nswitch as a signalling message in this evaluation.\nWe further observe from Table~\\ref{tab:handover} that the load\nof processing 9 handover signalling messages at the MME in LTE is reduced\nto processing 4 handover signalling messages at the controller in LayBack.\nMoreover, the P\/S-GWs in LTE need to process 4 signalling messages for\na handover, whereas the LayBack gateway is completely oblivious to the\nhandover mechanism. That is, the LayBack handover completely eliminates the\nhandover signalling load on the gateway.\n\nThe LTE handover changes the bearers (UE context) on the gateway,\nrequiring overall a high handover signalling load, whereby large\nsignalling loads have to be processed at the LTE MME and P\/S-GW.\nIn contrast, the LayBack handover reduces the overall signalling load\nand employs the SDN switches to change the packet flow path.\nEssentially, the LayBack handover moves the handover burden from\nthe  gateway to the network SDN switches.\nThe SDN switches can share the handover burden, avoiding bottlenecks.\n\n\\section{\\MakeUppercase{Summary and Future Directions}}  \n\\subsection{Summary of SDN-LayBack Architecture and Handover Protocol}\n\nWe have introduced SDN-LayBack,\na novel layered backhaul architecture for wireless networks.\nSDN-LayBack consistently decouples the radio access networks (RANs)\nfrom the backhaul, permitting the flexible backhauling\nof highly heterogeneous RANs.\nWithin the SDN-LayBack architectural framework, we have introduced\na first protocol mechanism, namely a 4-step handover protocol for\nhandovers within the scope of an SDN-LayBack gateway.\nThe proposed\nLayBack handover mechanism relieves the signalling bottleneck at the\nbackhaul gateways (e.g., LTE S-GW and P-GW)\nby (a) traffic duplication at SDN switches, and\n(b) bearer information forwarding from the currently serving eNB to the\ntarget eNB.\nMoreover, the LayBack handover reduces the signalling load on the\nbackhaul control entities (e.g., MME) by (a) reducing the required\nsignalling messages, and (b) moving some of the signalling messages\nfrom mobility control entities to the SDN controller.\nOur comparison of the handover signalling load indicates that\nLayBack achieves a 60~\\% reduction of the overall handover signalling load\ncompared to the conventional\nLTE handover. Moreover, LayBack completely avoids handover processing at\nthe gateway and distributes the handover processing load over a network\nof SDN switches, avoiding processing bottlenecks.\n\n\\subsection{Future Direction: Interference Management Protocol} \n\nThere are many exciting avenues for future protocol developments and\nevaluations within the SDN-LayBack architectural framework.\nSince interference is one of the main limitations for the deployment\nof small cells, an important future research direction is to\nexploit the capabilities of the SDN-LayBack architecture for\ninterference mitigation protocols.\nWe can exploit the central control (with a global holistic\nperspective) in SDN-LayBack for coordinating\nthe long-term wireless configurations, such as center\nfrequency, bandwidth, and time-sharing patterns,\nas well as protocol-specific configurations (e.g., random access channel\nparameters in LTE).\nWe emphasize that the goal of SDN-LayBack interference management\nwould \\textit{not} be to manage the physical\nresources, such as sub-frame power control in LTE, at the micro level.\nInstead, SDN-LayBack interference management assigns high-level configurations\nin a coordinated manner to the radio nodes so as to mitigate interference.\n\n\\subsection{Future Direction: D2D Communication Protocol}  \nDevice-to-Device (D2D) communication will likely play an important role in\nthe future of real-time content sharing and is another promising\ndirection for SDN-LayBack protocol development.\nThe LayBack architecture can be exploited for D2D communication so as\nto avoid flooding on the access network.\nMore specifically, the SDN controller can\nmaintain a subscriber (location) data base to coordinate the connection\nestablishment and routing between D2D communication partners.\n\n\\subsection{Future Direction: Video Streaming Protocol}  \nAnother direction is to develop and evaluate collaborative video streaming\nprotocols within the micro-architecture framework\n(Section~\\ref{ma:sec}) of SDN-LayBack.\nFor densely populated areas,\ne.g., airport terminals and stadiums, it is highly likely that\nmultiple RAN technologies, e.g., LTE and WiFi, are deployed,\nand multiple simultaneous (unicast) video streams are being\ntransmitted to (or from) devices within the coverage area of the\nconsidered micro-architecture, which is controlled by a local controller.\nA LayBack video streaming protocol can exploit the multiple ongoing video\nstreams and multiple RANs to collaboratively trade off the video traffic\ncharacteristics (e.g., traffic bursts from complex video scenes)\nwith RAN transmission conditions to achieve improved video experience\nacross an ensemble of video streams. In addition to existing techniques\nsuch \\cite{suresh2007superresolution, suresh2007robust, suresh2006robust, suresh2006discontinuity, aravind2012multispinning, suresh2014hog} can be applied to the SDN LayBack framework.\n\n\n\\chapter{\\MakeUppercase{Software Defined Optical Networks (SDONs): A Comprehensive Survey}}\n\n\\section{\\MakeUppercase{Introduction}} \\label{Sec1_Intro}\nAt least a decade ago \\cite{ComerNetMgmtBook} it was recognized that new\nnetwork abstraction layers for network control functions needed to be\ndeveloped to both simplify and automate network management. Software\nDefined Networking (SDN)~\\cite{HuHB14,jai2013b4,Vau11} is the design\nprinciple that emerged to structure the development of those new\nabstraction layers.\nFundamentally, SDN is defined by three architectural\nprinciples~\\cite{SDNarch11,KreRVR15}:\n$(i)$ the separation of control plane functions\nand data plane functions, $(ii)$ the logical centralization of control, and\n$(iii)$ programmability of network functions. The first two\narchitectural principles are related in that they combine to allow for network\ncontrol functions to have a wider perspective on the network. The idea is that\nnetworks can be made easier to manage (i.e., control and monitor) with a\nmove away from significantly distributed control. A tradeoff is then\nconsidered that balances ease of management arising from control\ncentralization and scalability issues that naturally arise from that\ncentralization.\n\nThe SDN abstraction layering consists of three generally accepted\nlayers~\\cite{SDNarch11} inspired by computing systems,\nfrom the bottom layer to the top layer:\n$(i)$ the \\textit{infrastructure} layer, $(ii)$ the \\textit{control} layer, and\n($iii$) the \\textit{application} layer,\nas illustrated in Fig.~\\ref{fig:sdnlayers}.\nThe interface between the\napplication layer and the control layer is referred to as the\nNorthBound Interface (NBI), while the interface between the control layer\nand the infrastructure layer is referred to as the SouthBound\nInterface (SBI). There are a variety of standards emerging for these\ninterfaces, e.g., the OpenFlow protocol~\\cite{LaKR14} for the SBI.\n\\begin{figure*}[t!]\t\\centering\n\t\\includegraphics[width=6in]{finman_sdon\/fig1.pdf}\n    \\caption{Illustration of Software Defined Networking (SDN) abstraction\n     layers.}\n\t\\label{fig:sdnlayers}\n\\end{figure*}\n\nThe \\textit{application} layer is modeled after software applications\nthat utilize computing resources to complete tasks.\nThe \\textit{control} layer is modeled after a computer's Operating System\n(OS) that manages computer resources (e.g., processors and memory),\nprovides an abstraction layer to simplify interfacing with the\ncomputer's devices, and provides a common set of services that all\napplications can leverage. Device drivers in a computer's OS hide the\ndetails of interfacing with many different devices from the\napplications by offering a simple and unified interface for various\ndevice types. In the SDN model both the unified SBI\nas well as the control layer functionality provide\nthe equivalent of a device driver for interfacing with devices in the\n\\textit{infrastructure} layer, e.g., packet switches.\n\nOptical networks play an important role in our modern information\ntechnology due to their high transmission capacities.\nAt the same time, the specific optical (photonic) transmission\nand switching characteristics, such as circuit, burst, and packet\nswitching on wavelength channels, pose challenges for controlling\noptical networks.\nThis article presents a comprehensive survey of Software\nDefined Optical Networks (SDONs).\nSDONs seek to leverage the flexibility of\nSDN control for supporting networking applications with an\nunderlying optical network infrastructure.\nThis survey comprehensively covers SDN related mechanisms that have\nbeen studied to date for optical networks.\n\n\\subsection{Related Work}\nThe general principles of SDN have been extensively covered in several\nsurveys, see for instance,~\\cite{aky2014roa,Chen2015a,cui2016big,far2015soft, FeRZ14, HuHB14, JaP13,\n  jar2014sur, JaZH14, kha2015jons, KreRVR15, LaKR14, li2014sof, Lopes2015,MiCK13,\n  NuMNO14,Racherla,Trois2016,van2014sca, wic2015sof, XiWF14}.\nSDN security has been surveyed in~\\cite{ahm2015sec,sco2015sur}, while\nmanagement of SDN networks has been surveyed in~\\cite{wic2015sof} and\nSDN-based satellite networking is considered in~\\cite{ber2015sof}.\n\nTo date, there have been relatively few overview and survey articles\non SDONs.\nZhang et al.~\\cite{zha2013surflexi} have presented a thorough survey\non flexible optical networking based on Orthogonal Frequency\nDivision Multiplexing (OFDM) in core (backbone) networks. The survey\nbriefly notes how OFDM-based elastic networking can facilitate\nnetwork virtualization and surveys a few studies on OFDM-based\nnetwork virtualization in core networks.\n\nBhaumik et al.~\\cite{BhZCS14} have presented an overview of\nSDN and network virtualization concepts and outlined principles\nfor extending SDN and network virtualization concepts to\nthe field of optical networking.\nTheir focus has been mainly on industry efforts, reviewing white papers\non SDN strategies from leading networking companies, such as\nCisco, Juniper, Hewlett-Packard, Alcatel-Lucent, and Huawei.\nA few selected academic research projects on general SDN optical\nnetworks, namely projects reported in the journal\narticles~\\cite{ChNS13,liu2013field} and a few related conference papers,\nhave also been reviewed by Bhaumik et al.~\\cite{BhZCS14}.\nIn contrast to Bhaumik et al.~\\cite{BhZCS14}, we provide a comprehensive\nup-to-date review of academic research on\nSDONs. Whereas Bhaumik et al.~\\cite{BhZCS14}\npresented a small sampling of SDON research organized by research projects,\nwe present a comprehensive SDON survey that is organized according\nto the SDN infrastructure, control, and application layer architecture.\n\nFor the SDON sub-domain of access networks,\nCvijetic~\\cite{Cvi14} has given an overview of access network challenges\nthat can be addressed with SDN.\nThese challenges include lack of support for on-demand modifications of\ntraffic transmission policies and rules and limitations to\nvendor-proprietary policies, rules, and software.\nCvijetic~\\cite{Cvi14} also offers a very brief overview of\nresearch progress for SDN-based optical access networks, mainly focusing on\nstudies on the physical (photonics) infrastructure layer.\nCvijetic~\\cite{CviSept14} has further expanded the overview of\nSDON challenges by considering the incorporation of 5G wireless\nsystems. Cvijetic~\\cite{CviSept14} has noted that\nSDN access networks are highly promising for low-latency and high-bandwidth\nback-hauling from 5G cell base stations and briefly surveyed\nthe requirements and areas of future research required for\nintegrating 5G with SDON access networks.\nA related overview of general software defined access networks\nbased on a variety of physical transmission media, including copper\nDigital Subscriber Line (DSL)~\\cite{dslbook}\nand Passive Optical Networks (PONs), has been\npresented by Kerpez et al.~\\cite{KeCGG14}.\n\nBitar~\\cite{Bi14} has surveyed use cases for SDN controlled broadband access,\nsuch as on-demand bandwidth boost, dynamic service\nre-provisioning, as well as value-add services and service protection.\nBitar~\\cite{Bi14} has discussed the commercial perspective of the\naccess networks that are enhanced with SDN to add\ncost-value to the network operation.\nAlmeida Amazonas et al.~\\cite{AmSS14} have surveyed the key issues\nof incorporating SDN in optical and wireless access networks.\nThey briefly outlined the obstacles posed by\n the different specific physical characteristics of\noptical and wireless access networks.\n\nAlthough our focus is on optical networks, for completeness we note that\nfor the field of wireless and mobile networks, SDN based networking mechanisms\nhave been surveyed\nin~\\cite{ArSR15,BeDSB14,haq2016wir,jag2014sof,SaCKA15,soo2015sof,yan2014sof}\nwhile network virtualization has been surveyed in~\\cite{LiY15} for\ngeneral wireless networks and in~\\cite{kha2015wir,lia2015wir} for\nwireless sensor networks.\nSDN and virtualization strategies for LTE wireless cellular networks\nhave been surveyed in~\\cite{ngu2015sdn}. SDN-based 5G wireless\nnetwork developments for mobile networks have been outlined\nin~\\cite{Huawei14,PeLZW14,TrGVS15,YaKO14}.\n\n\\subsection{Survey Organization}\nWe have mainly organize our survey according to the three-layer SDN architecture\nillustrated in Fig.~\\ref{fig:sdnlayers}.\nIn particular, we have organized the survey in a bottom-up manner,\nsurveying first SDON studies focused on the\ninfrastructure layer in Section~\\ref{sdninfra:sec}.\nSubsequently, we survey SDON studies focused on the control layer\nin Section~\\ref{sdnctl:sec}.\nThe virtualization of optical networks is commonly closely related to\nthe SDN control layer. Therefore, we survey SDON studies\nfocused on virtualization in Section~\\ref{virt:sec},\nright after the SDON control layer section.\nResuming the journey up the layers in Fig.~\\ref{fig:sdnlayers},\nwe survey SDON studies focused on the application layer in\nSection~\\ref{sdnapp:sec}.\nWe survey mechanisms for the overarching orchestration of the application\nlayer and lower layers, possibly across multiple network domains\n(see Fig.~\\ref{fig_control_orch}),\nin Section~\\ref{orch:sec}.\nFinally, we outline open challenges and future research directions in\nSection~\\ref{sec:open} and conclude the survey in Section~\\ref{sec:conclusion}.\n\n\n\\section{\\MakeUppercase{Background}}\n\\label{bg:sec}\nThis section first provides background on\nSoftware Defined Networking (SDN), followed by background on virtualization\nand optical networking.\nSDN, as defined by the Internet Engineering\nTask Force (IETF)~\\cite{rfc7426},\nis a networking paradigm enabling the programmability of networks.\nSDN abstracts and separates the data forwarding\nplane from the control plane, allowing faster\ntechnological development both in data and control planes.\nWe provide background on the SDN architecture, including its architectural\nlayers in Subsection~\\ref{sdnarch:sec}.\nThe network programmability provides the flexibility to dynamically initialize,\ncontrol, manipulate, and manage the end-to-end network behavior via open\ninterfaces, which are reviewed in Subsection~\\ref{sdnint:sec}.\nSubsequently, we provide background on network virtualization in\nSubsection~\\ref{bgnetvit:sec}\nand on optical networking in Subsection~\\ref{bg_access:sec}.\n\n\\subsection{Software Defined Networking (SDN) Architectural Layers}\n\\label{sdnarch:sec}\nSDN offers a simplified view of the underlying network infrastructure\nfor the network control and monitoring applications\nthrough the abstraction of each independent network layer.\nFig.~\\ref{fig:sdnlayers} illustrates the\nthree-layer SDN architecture model consisting of application,\ncontrol, and infrastructure layers as defined by the\nOpen Networking Foundation (ONF)~\\cite{SDNarch11}. The ONF is\nthe organization that is responsible for\nthe publication of specifications for the OpenFlow protocol.\nThe OpenFlow protocol~\\cite{HuHao2014,keo2008of,LaKR14}\nhas been the first protocol for the SouthBound Interface (SBI,\nalso referred to as Data-Controller Plane Interface (D-CPI))\nbetween the control and infrastructure layers.\nEach layer operates independently, allowing multiple solutions\nto coexist within each layer, e.g., the\ninfrastructure layer can be built from any\nprogrammable devices, which are commonly referred to as\nnetwork elements~\\cite{SDNarch11_521} or network devices~\\cite{rfc7426}\n(or sometimes as forwarding elements~\\cite{rfc3746}).\nWe will use the terminology network element throughout this survey.\nThe SouthBound Interface (SBI) and the NorthBound\nInterface (NBI, also referred to as Application-Controller Plane\nInterface (A-CPI)) are defined as the primary interfaces\ninterconnecting the SDN layers through abstractions.\nAn SDN network architecture can coexist with both\nconcurrent SDN architectures and non-SDN legacy network architectures.\nAdditional interfaces are defined namely the\nEastBound Interface (EBI) and the WestBound\nInterface (WBI)~\\cite{JaZH14} to interconnect the SDN architecture\nwith external network architectures\n(the EBI and WBI are also collectively\nreferred to as Intermediate-Controller Plane Interfaces (I-CPIs)).\nGenerally, EBIs establish communication links to\nlegacy network architectures (i.e., non-SDN networks); whereas,\nlinks to concurrent (side-by-side) SDN architectures are facilitated\nby the WBIs.\n\n\\subsubsection{Infrastructure Layer}\nThe infrastructure layer includes an environment for\n(payload) data traffic forwarding (data plane)\neither in virtual or actual hardware.\nThe data plane comprises a network of network elements,\nwhich expose their capabilities through the SBI\nto the control plane. In traditional networking, control mechanisms are\nembedded within an infrastructure,\ni.e., decision making capabilities are embedded within the\ninfrastructure to perform network actions, such as switching or routing.\nAdditionally, these forwarding actions in the traditional network elements\nare autonomously\nestablished based on self-evaluated topology information that is often\nobtained through proprietary vendor-specific algorithms.\nTherefore, the configuration setups of traditional network elements\nare generally not reconfigurable without a service disruption,\nlimiting the network flexibility.\nIn contrast, SDN decouples the autonomous control functions, such as\nforwarding algorithms and neighbor discovery of\nthe network nodes, and moves these control functions out of the infrastructure\nto a centrally controlled logical node, the controller.\nIn doing so, the network elements act only as dumb switches which\nact upon the instructions of the controller. This decoupling reduces the\nnetwork element complexity and improves reconfigurability.\n\nIn addition to decoupling the\ncontrol and data planes, packet modification capabilities at the line-rates\nof network elements have been significantly improved with SDN.\nP4~\\cite{bosshart2014p4} is a programmable\nprotocol-independent packet processor, that can arbitrarily\nmatch the fields within any formatted packet\nand is capable of applying any arbitrary actions (as programmed)\non the packet before forwarding. A similar forwarding mechanism,\nProtocol-oblivious Forwarding (PoF) has been\nproposed by Huawei Technologies~\\cite{Song2013}.\n\n\\subsubsection{Control Layer}\nThe control layer is responsible for programming (configuring)\nthe network elements (switches) via the SBIs.\nThe SDN controller is a logical entity that identifies the south bound\ninstructions to configure the network infrastructure\nbased on application layer requirements.\nTo efficiently manage the network, SDN controllers can\nrequest information from the SDN infrastructures,\nsuch as flow statistics, topology information, neighbor relations,\nand link status from the network elements (nodes).\nThe software entity that implements the SDN\ncontroller is often referred to as \\textit{Network Operating System (NOS)}.\nGenerally, a NOS can be implemented independently of\nSDN, i.e., without supporting SDN.\nOn the other hand, in addition to supporting SDN operations,\na NOS can provide advanced capabilities, such as virtualization,\napplication scheduling, and database management.\nThe Open Network Operating System (ONOS)~\\cite{onos}\nis an example of an SDN based NOS\nwith a distributed control architecture designed to operate over\nWide Area Networks (WANs).\nFurthermore, Cisco has recently developed the one Platform Kit\n(onePK)~\\cite{onepk}, which consists of a set of\nApplication Program Interfaces (APIs)\nthat allow the network applications to control Cisco network devices\nwithout a command line interface.\nThe onePK libraries act as an SBI for Cisco ONE controllers and\nare based on C and Java compilers.\n\n\\subsubsection{Application Layer}\nThe application layer comprises network applications and services\nthat utilize the control plane to realize\nnetwork functions over the physical or virtual infrastructure.\nExamples of network applications include\nnetwork topology discovery, provisioning, and fault restoration.\nThe SDN controller presents an abstracted view of the\nnetwork to the SDN applications to facilitate the realization of\napplication functionalities.\nThe applications can also include higher levels of network management,\nsuch as network data analytics, or specialized functions requiring processing\nin large data centers. For instance, the Central Office Re-architected as a\nData center (CORD)~\\cite{cord} is an SDN application based\non ONOS~\\cite{onos},\nthat implements the typical central office network functions, such as\noptical line termination, as well as BaseBand Unit (BBU) and\nData Over Cable Interface (DOCSIS)~\\cite{fel2001doc} processing as\nvirtualized software entities, i.e., as SDN applications.\n\n\\begin{figure}[t!]\n\t\\centering \t\\vspace{0cm}\n\t\\includegraphics[width=3.4in]{finman_sdon\/fig2.pdf}\n    \\caption{Overview of SDN orchestrator and SDN controllers.}\n\t\\label{fig_control_orch}\n\\end{figure}\n\\subsubsection{Orchestration Layer}  \\label{intro:orch:sec}\nAlthough the orchestration layer is commonly not considered one of the\nmain SDN architectural layers illustrated in Fig.~\\ref{fig:sdnlayers},\nas SDN systems become more complex, orchestration becomes increasingly\nimportant. We introduce therefore the orchestration layer as\nan important SDN architectural layer in this background section.\nTypically, an SDN orchestrator is the entity that coordinates\nsoftware modules within a single SDN controller,\na hierarchical structure of multiple SDN controllers, or a set of\nmultiple SDN controllers in a ``flat'' arrangement (i.e., without\na hierarchy) as illustrated in Fig.~\\ref{fig_control_orch}.\nAn SDN controller in contrast\ncan be viewed as a logically centralized single\ncontrol entity. This logically centralized single control entity appears as the\ndirectly controlling entity to the network elements.\nThe SDN controller is responsible for signaling the control actions or rules\nthat are typically predefined (e.g., through OpenFlow) to the network elements.\nIn contrast, the SDN orchestrator makes control\ndecisions that are generally not predefined.\nMore specifically, the SDN orchestrator could make an automated decision\nwith the help of SDN applications or seek a manual\nrecommendation from user inputs; therefore,\nresults are generally not predefined. These\norchestrator decisions (actions\/configurations)\nare then delegated via the SDN controllers\nand the SBIs to the network elements.\n\nIntuitively speaking, SDN orchestration can be viewed as a distinct\nabstracted (higher) layer for coordination and management that is\npositioned above the SDN control and application layers.  Therefore,\nwe generalize the term SDN orchestrator as an entity that realizes a\nwider, more general (more encompassing) network functionality as\ncompared to the SDN controllers. For instance, a cloud SDN\norchestrator can instantiate and tear down Virtual Machines (VMs)\naccording to the cloud\nworkload, i.e., make decisions that span across multiple network\ndomains and layers.  In contrast, SDN controllers realize more\nspecific network functions, such as routing and path computation.\n\n\\subsection{SDN Interfaces}  \\label{sdnint:sec}\n\\subsubsection{Northbound Interfaces (NBIs)}\nA logical interface that interconnects the SDN controller and a\nsoftware entity operating at the application layer is commonly\nreferred to as a NorthBound Interface (NBI), or as Application-Controller Plane\nInterface (A-CPI).\n\n\\paragraph{REST}\nREpresentational State Transfer (REST)~\\cite{REST15} is generally defined as\na software architectural style that supports flexibility, interoperability,\nand scalability.\nIn the context of the SDN NBI, REST is commonly defined as\nan API that meets the REST architectural style~\\cite{LiChou2016},\ni.e., is a so-called RESTful API:\n\\begin{itemize}\n\\item Client-Sever: Two software entities should follow the\n  client-server model.  In SDN, a controller can be a server and the\n  application can be the client.  This allows multiple\n  heterogeneous SDN applications to coexist and operate over a common\n  SDN controller.\n\\item Stateless: The client is responsible for managing all the states and\nthe server acts upon the client's request.\nIn SDN, the applications collect and maintain the states of the network, while\nthe controller follows the instructions from the applications.\n\\item Caching: The client has to support the temporary local storage\nof information such that interactions between the\nclient and server are reduced so as to improve performance and scalability.\n\\item Uniform\/Interface Contract:  An overarching technical\ninterface must be followed across all services using the REST API.\nFor example, the same data format, such as Java Script Object Notation (JSON)\nor eXtended Markup Language (XML), has to be followed for all\ninteractions sharing the common interface.\n\\item Layered System: In a multilayered architectural solution,\n  the interface should only be concerned with the next immediate node\n  and not beyond. Thus, allowing more layers to be inserted, modified,\n  or removed without affecting the rest of the system.\n\\end{itemize}\n\n\\subsubsection{Southbound Interfaces (SBIs)}\nA logical interface that interconnects the SDN controller and the network\nelement operating on the infrastructure layer (data plane) is commonly\nreferred to as a SouthBound Interface (SBI), or as the\nData-Controller Plane Interface (D-CPI).\nAlthough a higher level connection, such as a UDP or TCP connection, is\nsufficient for enabling the communication between two entities of the SDN\narchitecture, e.g., the controller and the network elements,\nspecific SBI protocols have been proposed.\nThese SBI protocols are typically not interoperable\nand thus are limited to work with SBI protocol-specific network elements\n(e.g., an OpenFlow switch does not work with the NETCONF protocol).\n\n\\paragraph{OpenFlow Protocol}\nThe interaction between an OpenFlow switching element\n(data plane) and an OpenFlow controller (control plane) is carried\nout through the OpenFlow protocol~\\cite{keo2008of,LaKR14}. This SBI (or D-CPI)\nis therefore also sometimes referred to as the OpenFlow control channel.\nSDN mainly operates through packet flows that are identified through\nmatches on prescribed packet fields that are specified in the\nOpenFlow protocol specification. For matched packets, SDN switches\nthen take prescribed actions, e.g., process the flow's packets in a\nparticular way, such as dropping the packet, duplicating it on a different\nport or modifying the header information.\n\n\\paragraph{Path Computation Element Protocol (PCEP)}\nThe PCEP enables communication between the Path Computation Client (PCC) of\nthe network elements and the Path Computation Element (PCE) residing within\nthe controller. The PCE centrally computes the paths based on constraints\nreceived from the network elements. Computed paths are then forwarded to the\nindividual network elements through the PCEP protocol~\\cite{rfc4655,rfc5440}.\n\n\\paragraph{Network Configuration Protocol (NETCONF) Protocol}\nThe NETCONF protocol~\\cite{rfc6241}\nprovides mechanisms to configure, modify, and delete configurations\non a network device.\nConfiguration of the data and protocol messages are encoded in the NETCONF\nprotocol using an eXtensible Markup Language (XML).\nRemote procedure calls are used to realize the NETCONF protocol\noperations. Therefore, only devices that are enabled\nwith required remote procedure calls allow the NETCONF protocol to remotely\nmodify device configurations.\n\n\\paragraph{Border Gateway Protocol Link State Distribution (BGP-LS) Protocol}\nThe central controller needs a topology information database,\nalso known as Traffic Engineering Database (TED), for\noptimized end-to-end path computation.\nThe controller has to request the information for building the TED,\nsuch as topology and bandwidth utilization, via the SBIs\nfrom the network elements.\nThis information can be gathered by a BGP extension,\nwhich is referred to as BGP-LS.\n\n\\subsection{Network Virtualization} \\label{bgnetvit:sec}\nAnalogously to the virtualization of computing\nresources~\\cite{gold1974sur,Douglis2013a},\nnetwork virtualization abstracts the underlying physical network\ninfrastructure so that one or multiple virtual networks\ncan operate on a given physical\nnetwork~\\cite{bel2012res,duan2012sur,fis2013vir,han2015net,leo2003vir,mij2015net,pen2015gue,RajJain2013,wan2013net}.\nVirtual networks can span over a\nsingle or multiple physical infrastructures (e.g., geographically\nseparated WAN segments).\nNetwork Virtualization (NV) can flexibly create independent virtual\nnetworks (slices) for distinct\nusers over a given physical infrastructure.  Each network slice\ncan be created with prescribed resource allocations.\nWhen no longer required, a\nslice can be deleted, freeing up the reserved physical resources.\n\nNetwork hypervisors~\\cite{Khan2012,she2009flo}\nare the network elements that abstract the\nphysical network infrastructure (including network elements,\ncommunication links, and control functions) into logically isolated\nvirtual network slices.\nIn particular, in the case of an underlying physical SDN network,\nan SDN hypervisor can create multiple isolated virtual SDN\nnetworks~\\cite{ble2015sur,dru2013sca}.\nThrough hypervisors, NV supports the implementation of a wide\nrange of network services belonging to the link and network protocol\nlayers (L2 and L3), such as switching and routing.\nAdditionally, virtualized infrastructures can also support\nhigher layer services, such as load-balancing of servers and firewalls.\nThe implementation of such higher layer services in a virtualized environment\nis commonly referred to as Network Function Virtualization\n(NFV)~\\cite{haw2014nfv,LiChen2015,lin2016dem,mat2015tow,ye2016joi}.\nNFV can be viewed as a special case of NV in which network\nfunctions, such as address translation and intrusion detection functions,\nare implemented in a virtualized environment. That is,\nthe virtualized functions are implemented in the form of software\nentities (modules) running on a data center (DC) or\nthe cloud~\\cite{Mijumbi2016}.\nIn contrast, the term NV emphasizes the virtualization of\nthe network resources, such as communication links and network nodes.\n\n\n\\subsection{Optical Networking Background}   \\label{bg_access:sec}\n\n\\subsubsection{Optical Switching Paradigms}\nOptical networks are networks that either maintain signals in the\noptical domain or at least utilize transmission channels that carry\nsignals in the optical domain. In optical networks that\nmaintain signals in the optical domain, switching can be\nperformed at the \\textit{circuit, packet, or burst} granularities.\n\n\\paragraph{Circuit Switching}\nOptical \\textit{circuit} switching can be performed in space,\nwaveband, wavelength, or time. The optical spectrum is divided into\nwavelengths either on a fixed wavelength grid or on a flexible\nwavelength grid. Spectrally adjacent wavelengths can be coalesced into\nwavebands. The fixed wavelength grid standard (ITU-T G.694.1)\nspecifies specific center frequencies that are either 12.5~GHz, 25~GHz,\n50~GHz, or 100~GHz apart. The flexible DWDM grid (flexi-grid) standard (ITU-T\nG.694.1)~\\cite{gon2015opt,jue2014sof,tom2014tut,zha2013surflexi}\nallows the center frequency to be any multiple of 6.25~GHz\naway from 193.1~THz and the spectral width to be any multiple of\n12.5~GHz. Elastic Optical Networks (EONs)~\\cite{cha2015rou,tal2014spe,yu2014spe}\nthat take advantage of the flexible\ngrid can make more efficient use of the optical spectrum but can cause\nspectral fragmentation, as lightpaths are set up and torn down, the spectral\nfragmentation counteracts the more efficient spectrum\nutilization~\\cite{Ger2012}.\n\n\\paragraph{Packet Switching}\nOptical \\textit{packet} switching performs packet-by-packet switching\nusing header fields in the optical domain as much as possible. An\nall-optical packet switch requires~\\cite{ram2009opt}:\n\\begin{itemize}\n  \\item Optical synchronization, demultiplexing, and multiplexing\n  \\item Optical packet forwarding table computation\n  \\item Optical packet forwarding table lookup\n  \\item Optical switch fabric\n  \\item Optical buffering\n\\end{itemize}\nOptical packet switches typically relegate some of these design\nelements to the electrical domain. Most commonly the packet forwarding\ntable computation and lookup is performed electrically. When there is\ncontention for a destination port, a packet needs to be buffered\noptically, this buffering can be accomplished with rather impractical\nfiber delay lines. Fiber delay lines are fiber optic cables whose lengths are\nconfigured to provide a certain time delay of the optical signal; e.g.,\n100 meters of fiber provides 500~ns of delay. An alternative to\nbuffering is to either drop the packet or to use deflection routing,\nwhereby a packet is routed to a different output that may or may not\nlead to the desired destination.\n\n\\paragraph{Burst Switching}\nOptical \\textit{burst} switching alleviates the requirements of\noptical packet forwarding table computation, forwarding table lookup,\nas well as buffering while accommodating bursty traffic that would\nlead to poor utilization of optical circuits. In essence, it permits\nthe rapid establishment of short-lived optical circuits to support the\ntransfer of one or more packets coalesced into a burst. A control\npacket is sent through the network that establishes the lightpath for\nthe burst and then the burst is transmitted on the short-lived circuit\nwith no packet lookup or buffering required along the path~\\cite{ram2009opt}.\nSince the circuit is only established for the length of the burst, network\nresources are not wasted during idle periods. To avoid any buffering\nof the burst in the optical network, the burst transmission can begin\nonce the lightpath establishment has been confirmed (tell-and-wait) or\na short time period after the control packet is sent\n(just-enough-time). \\textit{Note}: Sending the burst immediately after\nthe control packet (tell-and-go) would require some buffering of the\noptical burst at the switching nodes.\n\n\\subsubsection{Optical Network Structure}  \\label{optnetstruct:sec}\nOptical networks are typically structured into three main tiers, namely\naccess networks, metropolitan (metro) area networks,\nand backbone (core) networks~\\cite{sim2014opt}.\n\n\\paragraph{Access Networks}\nIn the area of optical access networks~\\cite{for2015nex}, so-called Passive\nOptical Networks (PONs), in particular, Ethernet PONs (EPONs) and\nGigabit PONs (GPONs)~\\cite{haj2006epo,sku2009com},\nhave been widely studied.\nA PON has typically an inverse tree structure with a central\nOptical Line Terminal (OLT) connecting multiple distributed Optical\nNetwork Units (ONUs; also referred to as Optical Network Terminals, ONTs)\nto metro networks.\nIn the downstream (OLT to ONUs) direction, the OLT broadcasts transmissions.\nHowever, in the upstream (ONUs to OLT) direction, the transmissions of the\ndistributed ONUs need to be coordinated to avoid collisions on the\nshared upstream wavelength channel.\nTypically, a cyclic polling based Medium Access Control (MAC) protocol,\ne.g., based on the MultiPoint Control Protocol (MPCP, IEEE 802.3ah),\nis employed.\nThe ONUs report their bandwidth demands to the OLT and the OLT then\nassigns upstream transmission windows according to a Dynamic\nBandwidth Allocation (DBA)\nalgorithm~\\cite{kan2012ban,mcg2010sho,mcg2012inv,zhe2009sur}.\nConventional PONs cover distances up to 20~km, while so-called\nLong-Reach (LR) PONs cover distances up to\naround 100~km~\\cite{mer2013off,nag2016n,son2010lon}.\n\nRecently, hybrid access networks that combine multiple transmission\nmedia, such as Fiber-Wireless (FiWi)\nnetworks~\\cite{aur2014fiw,gha2011fib,liu2016new,sar2015arc,tsa2011sur} and\nPON-DSL networks~\\cite{gur2014pon}, have been explored to take\nadvantage of the respective strengths of the different transmission\nmedia.\n\n\n\\paragraph{Networks Connected to Access Networks}\nOptical access networks provide Internet connectivity for a wide range\nof peripheral networks. Residential (home) wired or wireless\nlocal area networks~\\cite{che2014sur}\ntypically interconnect individual end devices (hosts) in a home or small\nbusiness and may connect directly with an optical access network.\nCellular wireless networks provide Internet access to a wide range of\nmobile devices~\\cite{cap2013dow,dam2011sur,sch2013pus}.\nSpecialized cellular backhaul\nnetworks~\\cite{LiPCYW14,LiZZW13,park2014fro,PeWLP15,raz2013bri,tip2011evo,YaLJS13} relay the traffic\nto\/from base stations of wireless cellular\nnetworks to either wireless access\nnetworks~\\cite{aky2005wir,alo2012sur,ben2012wir,kur2007sur,pat2011sur,vij2013dis} or optical access networks.\nMoreover, optical access networks are often employed to connect\nData Center (DC) networks to the Internet. DC networks interconnect highly\nspecialized server units\n  that process and store large data amounts with specialized\nnetworking technologies~\\cite{cai2013sur,kac2012sur,sam2016sof,yan2016sud,zha2013sur}.\nData centers are\ntypically employed to provide the so-called  ``cloud'' services for\n commercial and social media applications.\n\n\\paragraph{Metropolitan Area Networks}\nOptical Metropolitan (metro) Area Networks (MANs) interconnect the optical\naccess networks in a metropolitan area with each other and with\nwide-area (backbone, core) networks.\nMANs have typically a ring or star\ntopology~\\cite{bia2013cos,bia2015sho,cha2013tow,mai2003hyb,rot2013rou,sch2003wav}\nand commonly employ optical networking technologies.\n\n\\paragraph{Backbone Networks}\nOptical backbone (wide area) networks interconnect the individual MANs\non a national\nor international scale. Backbone networks have typically a mesh structure\nand employ very high speed optical transmission links.\n\\begin{figure*}[t!]\n\\footnotesize\n\\setlength{\\unitlength}{0.10in}\n\\centering\n\\begin{picture}(40,35)\n\\put(-4,33){\\textbf{SDN Controlled Photonic Communication Infrastructure Layer, Sec.~\\ref{sdninfra:sec}}}\n\\put(-7,30){\\line(1,0){50}}\n\\put(19,30){\\line(0,1){2}}\n\\put(-7,30){\\vector(0,-1){2}}\n\n\\put(-9,27){\\makebox(0,0)[lt]{\\shortstack[l]{\t\t\t\n\t\\textbf{Bandwidth Variable Transceivers (BVTs)},\\\\ Sec.~\\ref{transc:sec}\t\\\\\t\\\\ \t\t\n\tSingle-Flow BVTs~\\cite{aut2013eva,ElA12,gri2010fle}, Sec.~\\ref{SF-BVT:sec}\\\\\n\tMach-Zehnder Modulator Based~\\cite{Choi2013,Liu2013c}\\\\\n\tBVTs for PONs~\\cite{LaAVKS14,IiSSK12,yeh2010usi,yu2008cen,VaBPF13,bol2014dig,Bolea2015a}\\\\\n\tBVTs for DC Netw.~\\cite{Malacarne2014} \\\\ \\\\\n\tSliceable Multi-Flow BVTs~\\cite{jin2012mul}, \\\\ Sec.~\\ref{MF-BVT:sec}\t\\\\\t\t\t\t\n\tEncoder Based~\\cite{Sambo2014a,sam2015nex,sam2014sli} \\\\\t\t\t\t\t\n\tDSP Based~\\cite{Moreolo2016} \\\\\t\t\t\t\t\n\tSubcar. $+$ Mod. Pool Based~\\cite{Ou2016} \\\\\t\t\t\t\t\n\tHYDRA~\\cite{mat2015hyd} \n\n\n}}}\n\n\\put(18,30){\\vector(0,-1){26}}\n\n\\put(9,3){\\makebox(0,0)[lt]{\\shortstack[l]{\t\t\t\n\t\t\t\\textbf{Space Div. Multipl.} \\\\\n\t\t\t\\textbf{(SDM)-SDN}, \\\\ Sec.~\\ref{SDM_SDN:sec}\n\t\t\t\\cite{ama2013ful,ama2014sof,Galve2016}\n}}}\n\n\\put(25,30){\\vector(0,-1){2}}\n\\put(20,27){\\makebox(0,0)[lt]{\\shortstack[l]{\t\t\t\t\t\t\n\t\t\t\\textbf{Switching}, Sec.~\\ref{infra_sw:sec} \\\\ \\\\ \t\t\t\t\n\t\t\tSwitching Elements, Sec.~\\ref{sw_elem:sec}\t\\\\\n\t\t\tROADMs~\\cite{Co13,ama2013int,rof2013all,you2013eng,Way2013wav,Garrich2015}\t\\\\\t\t\t\n\t\t\tOpen Transport Switch (OTS)~\\cite{SaSPL13}\t\\\\\t\t\t\t\t\n\t\t\tLogical xBar~\\cite{PaSMK13}\t\\\\\t\t\t\t\t\n\t\t\tOptical White Box~\\cite{Nejabati2015}\t\\\\\t\t\t\t\t\n\t\t\tGPON Virt. Sw.~\\cite{Lee2016,gu2014sof,Amokrane2014,amo2015dyn,Yeh2015}\t\\\\\t\t\t\t\n\t\t\tFlexi Access Netw. Node~\\cite{FoG13,kon2015sdn} \\\\ \\\\\n\t\t\tSwitching Paradigm, Sec.~\\ref{ws_para:sec}\t\\\\\t\t\t\n\t\t\tConverged Pkt-Cir. Sw.~\\cite{DaPM09,DaPMSG10,VeBB13,AzNEJ11,ShJKG12,KaT12,CeLR13}\t\\\\\t\t\t\t\t\n\t\t\tR-LR-UFAN~\\cite{yin2013ult,ShYD14,Yin2015}\t\\\\\n\t\t\tFlexi-grid~\\cite{Cv13,CvTJSM14,OlSCH13,ZhZYY13}\t\n}}}\t\n\\put(43,30){\\vector(0,-1){26}}\n\\put(25, 3){\\makebox(0,0)[lt]{\\shortstack[l]{\t\t\t\t\t\n\t\t\t\\textbf{Opt. Perf. Monitoring}, Sec.~\\ref{opm:sec} \\\\ \\\\\n\t\t\tCognitive Netw. Infra.~\\cite{MiDJF13,Oliveira2015,Giglio2015} \\\\\n\t\t\tWavelength Selective Switch\/Amplifier \\\\ \\ \\ Control~\\cite{Moura2015,pao2015sup,Carvalho2015,Wang2015f}\n}}}\t\n\\end{picture}\n\\vspace{1cm}\n\\caption{Classification of physical infrastructure layer SDON studies.}\n\\label{infra_class:fig}\n\\end{figure*}\n\\section{\\MakeUppercase{SDN Controlled Photonic Communication Infrastructure Layer}}\n\\label{sdninfra:sec}\nThis section surveys mechanisms for controlling physical layer\naspects of the optical (photonic) communication infrastructure through SDN.\nEnabling the SDN control down to the photonic level operation of\noptical communications allows for flexible adaptation of the\nphotonic components supporting optical networking\nfunctionalities~\\cite{ChNS13,gri2013ext,jin2013vir,RoD08}.\nAs illustrated in Fig.~\\ref{infra_class:fig},\nthis section first surveys transmitters and receivers (collectively\nreferred to as transceivers or transponders) that permit SDN control of the\noptical signal transmission characteristics, such as modulation format.\nWe also survey SDN controlled space division\nmultiplexing (SDM), which provides an emerging avenue for highly\nefficient optical transmissions.\nThen, we survey SDN controlled optical switching, covering first\nswitching elements and then overall switching paradigms, such as converged\npacket and circuit switching.\nFinally, we survey cognitive photonic communication infrastructures that\nmonitor the optical signal quality. The optical signal quality\ninformation can be used to dynamically control the transceivers\nas well as the filters in switching elements.\n\n\\subsection{Transceivers}  \\label{transc:sec}\nSoftware defined optical transceivers are optical transmitters and\nreceivers that can be flexibly configured by SDN to transmit or\nreceive a wide range of optical signals~\\cite{Hillerkuss2016}.\nGenerally, software defined optical transceivers vary the modulation\nformat~\\cite{win2006adv} of the transmitted optical signal by\nadjusting the transmitter and receiver operation through Digital\nSignal Processing (DSP)\ntechniques~\\cite{cha2014adv,sch2010rea,yos2013dsp}. These\ntransceivers have evolved in recent years from Bandwidth Variable\nTransceivers (BVTs) generating a single signal flow to sliceable\nmulti-flow BVTs. Single-flow BVTs permit SDN control to adjust the\ntransmission bandwidth of the single generated signal flow. In\ncontrast, sliceable multi-flow BVTs allow for the independent SDN\ncontrol of multiple communication traffic flows generated by a\nsingle BVT.\n\n\\subsubsection{Single-Flow Bandwidth Variable Transceivers (BVTs)}\n\\label{SF-BVT:sec} Software defined optical transceivers have\ninitially been examined in the context of adjusting a single optical\nsignal flow for flexible WDM\nnetworking~\\cite{aut2013eva,ElA12,gri2010fle}. The goal has been to\nmake the photonic transmission characteristics of a given\ntransmitter fully programmable. We proceed to review a\nrepresentative single-flow BVT design for general optical mesh\nnetworks in detail and then summarize related single-flow BVTs for\nPONs and data center networks.\n\n\\paragraph{Mach-Zehnder Modulator Based Flexible Transmitter}\nChoi and Liu et al.~\\cite{Choi2013,Liu2013c} have demonstrated a\nflexible transmitter based on Mach-Zehnder Modulators\n(MZMs)~\\cite{bar2003wid} and a corresponding flexible receiver for\nSDN control in a general mesh network. The flexible transceiver\nemploys a single dual-drive MZM that is fed by two binary electric\nsignals as well as a parallel arrangement of two MZMs which are fed\nby two additional electrical signals. Through adjusting the direct\ncurrent bias voltages and amplitudes of drive signals the\ncombination of MZMs can vary the amplitude and phase of the\ngenerated optical signal~\\cite{cho2012ber}. Thus, modulation formats\nranging from Binary Phase Shift Keying (BPSK) to Quadrature Phase\nShift Keying (QPSK) as well as 8 and 16 quadrature amplitude\nmodulation~\\cite{win2006adv} can be generated. The amplitudes and\nbias voltages of the drive signals can be signaled through an SDN OpenFlow\ncontrol plane to achieve the different modulation formats. The\ncorresponding flexible receiver consists of a polarization filter\nthat feeds four parallel photodetectors, each followed by an\nAnalog-to-Digital Converter (ADC). The outputs of the four parallel\nADCs are then processed with DSP techniques to automatically\n(without SDN control) detect the modulation format. Experiments\nin~\\cite{Choi2013,Liu2013c} have evaluated the bit error rates and\ntransmission capacities of the different modulation formats and have\ndemonstrated the SDN control.\n\n\\paragraph{Single-Flow BVTs for PONs}\nFlexible optical networking with real-time bandwidth adjustments\nis also highly desirable for PON access and metro networks,\nalbeit the BVT technologies for access and metro networks should\nhave low cost and complexity~\\cite{LaAVKS14}.\nIiyama et al.~\\cite{IiSSK12} have developed a DSP based approach\nthat employs SDN to coordinate the downstream PON transmission of\nOn-Off Keying (OOK) modulation~\\cite{yeh2010usi} and\nQuadrature Amplitude Modulation (QAM)~\\cite{yu2008cen} signals.\nThe OOK-QAM-SDN scheme involves a\nnovel multiplexing method, wherein all the data\nare simultaneously sent from the OLT to the ONUs and the ONUs filter\nthe data they need.\nThe experimental setup in~\\cite{IiSSK12} also demonstrated\ndigital software ONUs that concurrently transmit data by exploiting\nthe coexistence of OOK and QAM.\nThe OOK-QAM-SDN evaluations demonstrated the control of the receiving\nsensitivity which is very useful for a wide range of transmission environments.\n\nIn a related study, Vacondio et al.~\\cite{VaBPF13} have examined\nSoftware-Defined Coherent Transponders (SDCT)\nfor TDMA PON access networks.\nThe proposed SDCT digitally processes the burst transmissions to\nachieve improved burst mode transmissions according to the distance of a\nuser from the OLT.\nThe performance results indicate that the proposed flexible\napproach more than doubles the average transmission capacity\nper user compared to  a static approach.\n\n\\begin{figure*}[t]\n    \\centering\n    \\vspace{0cm}\n    \\includegraphics[width=6in]{finman_sdon\/fig4.pdf}\n    \\vspace{0cm}\n    \\caption{Illustration of DSP reconfigurable ONU and OLT\n    designs~\\cite{Bolea2015a}.}\n    \\label{fig:bol2015}\n\\end{figure*}\nBolea et al.~\\cite{bol2014dig,Bolea2015a} have recently\ndeveloped low-complexity DSP reconfigurable ONU and OLT designs for\nSDN-controlled PON communication.\nThe proposed communication is based on carrierless amplitude and phase\nmodulation~\\cite{rod2011car} enhanced with optical Orthogonal\nfrequency Division Multiplexing (OFDM)~\\cite{bol2014dig}.\nThe different OFDM channels are manipulated through DSP filtering.\nAs illustrated in Fig.~\\ref{fig:bol2015}, the ONU consists of a DSP controller\nthat controls the filter coefficients of the shaping filter.\nThe filter output is then passed through a Digital-to-Analog Converter (DAC)\nand intensity modulator for electric-optical conversion.\nAt the OLT, a photo diode converts the optical signal to an electrical signal,\nwhich then passes through an Analog-to-Digital Converter (ADC).\nThe SDN controlled OLT DSP controller\nsets the filter coefficients in the matching filter to\ncorrespond to the filtering in the sending ONU.\nThe OLT DSP controller is also responsible for ensuring the\northogonality of all the ONU filters in the PON.\nThe performance evaluations in~\\cite{Bolea2015a} indicate that\nthe proposed DSP reconfigurable ONU and OLT system\nachieves ONU signal bitrates around 3.7~Gb\/s for eight ONUs transmitting\nupstream over a 25~km PON.\nThe performance evaluations also illustrate that long DSP filter lengths,\nwhich increase the filter complexity, improve performance.\n\n\\paragraph{Single-Flow BVTs for Data Center Networks}\nMalacarne et al.~\\cite{Malacarne2014} have developed a low-complexity\nand low-cost bandwidth adaptable transmitter for data center\nnetworking.\nThe transmitter can multiplex Amplitude Shift Keying (ASK),\nspecifically On-Off Keying (OOK), and Phase Shift Keying (PSK)\non the same optical carrier signal without any\nspecial synchronization or temporal alignment mechanism.\nIn particular, the transmitter design~\\cite{Malacarne2014}\nuses the OOK electronic signal to drive a Mach-Zehnder Modulator (MZM)\nthat is fed by the optical pulse modulated signal.\nSDN control can activate (or de-activate) the OOK signal stream, i.e.,\nadapt from transmitting only the PSK signal to transmitting\nboth the PSK and OOK signal and thus providing a higher transmission bit rate.\n\n\\subsubsection{Sliceable Multi-Flow Bandwidth Variable Transceivers}\n\\label{MF-BVT:sec} Whereas the single-flow transceivers surveyed in\nSection~\\ref{SF-BVT:sec} generate a single optical signal flow,\nparallelization efforts have resulted in multi-flow transceivers\n(transponders)~\\cite{jin2012mul}. Multi-flow transceivers can\ngenerate multiple parallel optical signal flows and thus form the\ninfrastructure basis for network virtualization.\n\n\\paragraph{Encoder Based Programmable Transponder}\nSambo et al.~\\cite{Sambo2014a,sam2015nex} have developed an\nSDN-programmable bandwidth-variable multi-flow transmitter and\ncorresponding SDN-programmable multi-flow bandwidth variable\nreceiver, referred to jointly as programmable bandwidth-variable\ntransponder. The transmitter mainly consists of a programmable\nencoder and multiple parallel Polarization-Multiplexing Quadrature\nPhase Shift Keying (PM-QPSK~\\cite{win2006adv}) laser transmitters,\nwhose signals are multiplexed by a coupler. The encoder is\nSDN-controlled to implement Low-Density Parity-Check (LDPC)\ncoding~\\cite{bon2011low} with different code rates. At the receiver,\nthe SDN control sets the local oscillators and LDPC decoder. The\ndeveloped transponder allows the setting of the number of\nsubcarriers, the subcarrier bitrate, and the LDPC coding rate\nthrough SDN. Related frequency conversion and defragmentation issues\nhave been examined in \\cite{Sambo2015}. In~\\cite{sam2014sli}, a\nlow-cost version of the SDN programmable transponder with a\nmultiwavelength source has been developed. The multiwavelength\nsource is based on a micro-ring resonator~\\cite{ras2009dem} that\ngenerates multiple signal carriers with only a single laser.\nAutomated configuration procedures for the comprehensive set of\ntransmission parameters, including modulation format, coding configuration,\nand carriers have been explored in~\\cite{cug2016tow}.\n\n\\paragraph{DSP Based Sliceable BVT}\nMoreolo et al.~\\cite{Moreolo2016} have developed an SDN controlled\nsliceable BVT based on adaptive Digital Signal Processing (DSP) of\nmultiple parallel signal subcarriers. Each subcarrier is fed by a\nDSP module that configures the modulation format, including the bit\nrate setting, and the power level of the carrier by adapting a gain\ncoefficient. The output of the DSP module is then passed through\ndigital to analog conversion that drives laser sources. The parallel\nflows can be combined with a wavelength selective switch; the\ncombined flow can be sliced into multiple distinct sub-flows for\ndistinct destinations. The functionality of the developed DSP based\nBVT has been verified for a metropolitan area network with links\nreaching up to 150~km.\n\n\\begin{figure}[t]\n    \\centering\n    \\vspace{0cm}\n    \\includegraphics[width=3.2in]{finman_sdon\/fig5.pdf}\n    \\vspace{0cm}\n    \\caption{Illustration of Subcarrier and Modulator Pool Based\n    Virtualizable Bandwidth Variable Transceiver (V-BVT)~\\cite{Ou2016}.}\n    \\label{fig:ou2016}\n\\end{figure}\n\\paragraph{Subcarrier and Modulator Pool Based Virtualizable BVT}\nOu et al.~\\cite{Ou2016,ou2015onl} have developed a Virtualizable BVT\n(V-BVT) based on a combination of an optical subcarriers pool\nwith an independent optical modulators pool, as illustrated in\nFig.~\\ref{fig:ou2016}.\nThe emphasis of the design is on implementing\nVirtual Optical Networks (VONs) at the transceiver level.\nThe optical subcarriers pool contains multiple optical carriers,\nwhereby channel spacing and central frequency (wavelength channel)\ncan be selected.\nThe optical modulators pool contains optical modulators that can\ngenerate a wide variety of modulation formats.\nThe SDN control interacts with a V-BVT Manager that implements a\nvirtualization algorithm. The virtualization algorithm\ngenerates a transceiver slice by combining a particular\nset of subcarriers (with specific number of subcarriers, channel\nspacing, and central frequencies) from the optical subcarriers pool\nwith a particular modulation (with specific number of modulators and\nmodulation formats) from the optical modulators pool.\nThe evaluations in~\\cite{Ou2016} have evaluated the proposed V-BVT in\na network testbed with path lengths up to 200~km with 20~GHz channel\nspacing and a variety of modulation formats, including\nBPSK as well as 16QAM and 32QAM.\n\n\\paragraph{S-BVT Based Hybrid Long-Reach Fiber Access Network (HYDRA)}\nHYDRA~\\cite{mat2015hyd} is a novel hybrid long-reach fiber access\nnetwork architecture based on sliceable BVTs. HYDRA supports\nlow-cost end-user ONUs through an Active Remote Node (ARN) that\ndirectly connects via a distribution fiber segment, a passive remote\nnode, and a trunk fiber segment to the core (backbone) network,\nbypassing the conventional metro network. The ARN is based on an SDN\ncontrolled S-BVT to optimize the modulation format. With the\nmodulation format optimization, the ARN can optimize the\ntransmission capacity for the given distance (via the distribution\nand trunk fiber segments) to the core network. The evaluations\nin~\\cite{mat2015hyd} demonstrate good bit error rate performance of\nrepresentative HYDRA scenarios with a 200~km trunk fiber segment and\ndistribution fiber lengths up to 100~km. In particular, distribution\nfiber lengths up to around 70~km can be supported without Forward\nError Correction (FEC), whereas distribution fiber lengths above\n70~km would require standard FEC. The consolidation of the access\nand metro network infrastructure~\\cite{wan2016mig} achieved through\nthe optimized S-BVT transmissions can significantly reduce the\nnetwork cost and power consumption.\n\n\\subsection{Space Division Multiplexing (SDM)-SDN}  \\label{SDM_SDN:sec}\nAmaya et al.~\\cite{ama2013ful,ama2014sof} have\ndemonstrated SDN control of\nSpace Division Multiplexing (SDM)~\\cite{ric2013spa} in optical networks.\nMore specifically, Amaya et al.~employ SDN to control the\nphysical layer so as to achieve a bandwidth-flexible and programmable\nSDM optical network. The SDN control can perform network slicing,\nresulting in sliceable superchannels.\nA superchannel consists of multiple spatial carriers to support\ndynamic bandwidth and QoS provisioning.\n\nGalve et al.~\\cite{Galve2016} have built on the flexible SDN controlled\nSDM communication principles to develop a reconfigurable Radio\nAccess Network (RAN). The RAN connects the BaseBand processing Units (BBUs)\nin a shared central office with the corresponding distributed\nRemote Radio Heads (RRHs) located at Base Stations (BSs).\nA multicore fiber operated with SDM~\\cite{ric2013spa} connects the\nRRHs to the BBUs in the central office.\nGalve et al.~introduce a radio over fiber operation mode where\nSDN controlled switching maps the subcarriers dynamically to\nspatial output ports.\nA complementary digitized radio over fiber operating mode maintains a\nBBU pool. Virtual BBUs are dynamically allocated to the cores of\nthe SDM operated multicore fiber.\n\n\n\\subsection{SDN-Controlled Switching} \\label{infra_sw:sec}\n\\subsubsection{Switching Elements}   \\label{sw_elem:sec}\n\\paragraph{ROADM} \\label{ROADMs:par}\nThe Reconfigurable Optical Add-Drop Multiplexer (ROADM) is an\nimportant photonic switching device for optical networks. Through\nwavelength selective optical switches, a ROADM can drop (or add) one\nor multiple wavelength channels carrying optical data signals from\n(to) a fiber without requiring the conversion of the optical signal\nto electric signals~\\cite{he2014sur}. The ROADM thus provides an\nelementary switching functionality in the optical wavelength domain.\nInitial ROADM based node architectures for cost-effectively\nsupporting flexible SDN networks have been presented\nin~\\cite{Co13}. Conventional ROADM networks have\ntypically statically configured wavelength channels that transport\ntraffic along a pre-configured route. Changes of wavelength channels\nor routes in the statically configured networks incur presently high\noperational costs due to required physical interventions and are\ntherefore typically avoided. New ROADM node designs allow changes of\nwavelength channels and routes through a management control plane.\nDue to these two flexibility dimensions (wavelength and route),\nthese new ROADM nodes are referred to as ``colorless'' and\n``directionless''. First designs for such colorless and\ndirectionless ROADM nodes have been outlined in~\\cite{Co13} and\nfurther elaborated in~\\cite{ama2013int,rof2013all}.\nIn addition to the colorless and directionless properties, the\ncontentionless property has emerged for ROADMs~\\cite{gri2010fle}.\nContentionless ROADM operation means that any port can be\nrouted on any wavelength (color) in any direction without causing\nresource contention.\nDesigns for such Colorless-Directionless-Contentionless (CDC)\nROADMs have been proposed in~\\cite{you2013eng,Way2013wav}.\nIn general, the ROADM designs consist of an express bank that\ninterconnects the input and output ports coming from\/leading to other ROADMs,\nand an add-drop bank that connects the express bank with the local\nreceivers for dropped wavelength channels or transmitters for added\nwavelength channels. The recent designs have focused on\nthe add-drop bank and explored different arrangements\nof wavelength selective switches and multicast switches to\nprovide add-drop bank functionality with the CDC\nproperty~\\cite{you2013eng,Way2013wav}.\n\nGarrich et al.~\\cite{Garrich2015} have recently designed\nand demonstrated a CDC ROADM with an add-drop bank based on\nan Optical Cross-Connect (OXC) backplane~\\cite{wan2012mul}.\nThe OXC backplane allows for highly flexible add\/drop configurations\nimplemented through SDN control.\nThe backplane based ROADM has been analytically\ncompared with prior designs based on\nwavelength selective and multicast switches and has been\nshown to achieve higher flexibility and lower losses.\nAn experimental evaluation has tested the backplane based ROADM for\na metropolitan area mesh network extending over 100~km\nwith an aggregate traffic load of  close to 9~Tb\/s.\n\n\\paragraph{Open Transport Switch (OTS)}\nThe Open Transport Switch (OTS) \\cite{SaSPL13} is an OpenFlow enabled\noptical virtual switch design. The OTS design abstracts the details\nof the underlying physical switching layer (which could be packet\nswitching or circuit switching) to a virtual switch element. The OTS\ndesign introduces three agent modules (discovery, control, and data\nplane) to interface with the physical switching hardware. These\nagent modules are controlled from an SDN controller through extended\nOpenFlow messages. Performance measurements for an example testbed\nnetwork setup indicate that the circuit path computation latencies\non the order of 2--3 s that can be reduced through faster processing\nin the controller.\n\n\\paragraph{Logical xBar}\nThe logical xBar~\\cite{PaSMK13} has been defined to represent a\nprogrammable switch. An elementary (small) xBar could consist of a\nsingle OpenFlow switch. Multiple small xBars can be recursively\nmerged to form a single large xBar with a single forwarding table.\nThe xBar concept envisions that xBars are the building blocks for\nforming large networks. Moreover, labels based on SDN and MPLS are\nenvisioned for managing the xBar data plane forwarding.\nThe xBar concepts have been further advanced in the Orion\nstudy~\\cite{fu2014ori} to achieve low computational complexity of\nthe SDN control plane.\n\n\\paragraph{Optical White Box}\nNejabati et al.~\\cite{Nejabati2015} have proposed an optical white\nbox switch design as a building block for a completely softwarized\noptical network. The optical white box design combines a\nprogrammable backplane with programmable switching node elements.\nMore specifically, the backplane consists of two slivers, namely an\noptical backplane sliver and an electronic backplane sliver. These\nslivers are set up to allow for flexible arbitrary connections\nbetween the switch node elements. The switch node elements include\nprogrammable interfaces that build on SDN-controlled BVTs (see\nSection~\\ref{transc:sec}), protocol agnostic switching, and DSP elements.\nThe protocol agnostic switching\nelement is envisioned to support both wavelength channel and time\nslot switching in the optical backplane as well as programmable\nswitching with a high-speed packet processor in the electronic\nbackplane. The DSP elements support both the network processing and\nthe signal processing for executing a wide range of network\nfunctions. A prototype of the optical white box has been built with\nonly a optical backplane sliver consisting of a $192 \\times 192$\noptical space switch. Experiments have indicated that the creation of\na virtual switching node with the OpenDayLight SDN controller takes\nroughly 400~ms.\n\n\\paragraph{GPON Virtual Switch}\nLee et al.~\\cite{Lee2016} have developed a GPON virtual switch\ndesign that makes the GPON fully programmable similar to a\nconventional OpenFlow switch. Preliminary steps towards the GPON\nvirtual switch design have been taken by Gu et al.~\\cite{gu2014sof}\nwho developed components for SDN control of a PON in a data center\nand Amokrane et al.~\\cite{Amokrane2014,amo2015dyn} who developed a\nmodule for mapping OpenFlow flow control requests into PON configuration\ncommands. Lee et al.~\\cite{Lee2016} have expanded on this groundwork\nto abstract the entire GPON into a virtual OpenFlow switch. More\nspecifically, Lee et al. have comprehensively designed a hardware\narchitecture and a software architecture to allow SDN control to\ninterface with the virtual GPON as if it were a standard OpenFlow switch.\nThe experimental performance evaluation of the designed GPON virtual\nswitch measured response times for flow entry modifications from an\nONU port (where a subscriber connects to the virtual GPON switch) to\nan SDN external port around 0.6~ms, which compares to 0.2~ms for a\ncorresponding flow entry modification in a conventional OFsoftswitch\nand 1.7~ms in a EdgeCore AS4600 switch. In a related study on SDN\ncontrolled switching in a PON, Yeh et al.~\\cite{Yeh2015} have\ndesigned an ONU with an optical switch that selects OFDM subchannels\nin a TWDM-PON. The switch in the ONU allows for flexible dynamic\nadaption of the downstream bandwidth through SDN.\nGu et al.~\\cite{gu2016eff} have examined the flexible SDN controlled\nre-arrangement of ONUs to OLTs so as to efficiently support PON\nservice with network coding~\\cite{bas2013net}.\n\n\\paragraph{Flexi Access Network Node}\nA flexi-node for an access network that flexibly aggregates\ntraffic flows from a wide range of networks,\nsuch as local area networks and base stations of wireless\nnetworks has been proposed in~\\cite{FoG13}.\nThe flexi-node design is motivated by the shortcomings of the\ncurrently deployed core\/metro\nnetwork architectures that attempt to consolidate the access and\nmetro networks.\nThis consolidation forces all traffic in the access network to\ntraverse the metro network, even if the traffic is destined to\ndestination nodes in the coverage area of an access network.\nIn contrast, the proposed flexi-node encompasses\nelectrical and optical forwarding capabilities that can be\ncontrolled through SDN. The flexi-node can thus serve as an\neffective aggregation node in access-metro networks.\nTraffic that is destined to other nodes in the coverage area of\nan access network can be sent directly to the access network.\n\nKondepu et al. have similarly presented an SDN based PON aggregation\nnode~\\cite{kon2015sdn}.\nIn their architecture, multiple ONUs\ncommunicate with the SDN controller within the aggregation node\nto request the scheduling of upstream transmission resources.\nONUs are then serviced by multiple Optical Service Units (OSUs)\nwhich exist\nwithin the aggregation node alongside with the SDN controller.\nOSUs are then configured by the controller based on Time and Wavelength\nDivision Multiplexed (TWDM) PON. The OSUs step between normal and sleep-mode\ndepending on the traffic loads, thus saving power.\n\n\\subsubsection{Switching Paradigms}  \\label{ws_para:sec}\n\\paragraph{Converged Packet-Circuit Switching}\nHybrid packet-circuit optical network infrastructures controlled by SDN\nhave been explored in a few studies.\nDas et al.~\\cite{DaPM09} have described how to unify the control and\nmanagement of circuit- and packet-switched networks using OpenFlow.\nSince packet- and circuit-switched networking are extensively employed\nin optical networks, examining their integration is an important research\ndirection.\nDas et al.~have given a high-level overview of a flow abstraction for\neach type of switched network and a common control paradigm.\nIn their follow-up work, Das et al.~\\cite{DaPMSG10}\nhave described how a packet and circuit switching\nnetwork can be implemented in the context of an OpenFlow-protocol based testbed. The testbed is a standard Ethernet network that could generally\nbe employed in any access network with Time Division Multiplexing (TDM).\nVeisllari et al.~\\cite{VeBB13} studied packet\/circuit hybrid optical\nlong-haul metro access networks.\nAlthough Veisllari et al.~indicated that SDN can be used for load balancing in\nthe proposed packet\/circuit network, no detailed study of\nsuch an SDN-based load balancing has been conducted in~\\cite{VeBB13}.\nRelated switching paradigms that integrate SDN with Generalized\nMultiple Protocol Label Switching (GMPLS)\nhave been examined in~\\cite{AzNEJ11,ShJKG12},\nwhile data center specific aspects have been surveyed in~\\cite{KaT12}.\n\nCerroni et al.~\\cite{CeLR13} have further developed the concept of\nunifying circuit- and packet-switching networks with OpenFlow,\nwhich was initiated by Das et al.~\\cite{DaPM09,DaPMSG10}.\nThe unification is accomplished with SDN on the network layer and\ncan be used in core networks. Specifically, Cerroni et\nal.~\\cite{CeLR13} have described an extension of the OpenFlow flow\nconcept to support hybrid networks. OpenFlow message format\nextensions to include matching rules and flow entries have also been\nprovided. The matching rules can represent different transport\nfunctions, such as a channel on which a packet is received in\noptical circuit-switched WDM networks, time slots in TDM networks,\nor transport class services (such as guaranteed circuit service or\nbest effort packet service). Cerroni et al.~\\cite{CeLR13} have\npresented a testbed setup and reported performance results for\nthroughput (in bit\/s and packets\/s) to demonstrate the feasibility\nof the proposed unified OpenFlow switching network.\n\n\\paragraph{R-LR-UFAN}  \\label{R_LR_UFAN:sec}\nThe Reconfigurable Long-Reach UltraFlow Access Network\n(R-LR-UFAN)~\\cite{yin2013ult,ShYD14} provides flexible dual-mode transport\nservice based on either the Internet Protocol (IP) or\nOptical Flow Switching (OFS).\nOFS~\\cite{chan2012opt} provides dedicated end-to-end network paths\nthrough purely optical switching, i.e., there is no electronic\nprocessing or buffering at intermediate network nodes. The R-LR-UFAN\narchitecture employs multiple feeder fibers to form subnets within\nthe network.\nUltraFlow coexists alongside the conventional PON OLT and ONUs. The\nR-LR-UFAN introduces new entities, namely the Optical Flow Network\nUnit (OFNU) and the SDN-controlled Optical Flow Line Terminal\n(OFLT). A Quasi-PAssive Reconfigurable (QPAR) node~\\cite{Yin2015} is\nintroduced between the OFNU and OFLT. The QPAR node can re-route\nintra PON traffic between OFNUs without having to pass through the\nOLFTs. The optically rerouted intra-PON channels can be used for\ncommunication between wireless base stations supporting inter cell\ndevice-to-device communication. The testbed evaluations indicate\nthat for an intra-PON traffic ratio of 0.3, the QPAR strategy\nachieves power savings up to 24\\%.\n\n\\paragraph{Flexi-grid}\nThe principle of flexi-grid (elastic) optical\nnetworking~\\cite{cha2015rou,gon2015opt,jue2014sof,she2016sur,tal2014spe,tom2014tut,yu2014spe,zha2013surflexi}\nhas been explored in several SDN infrastructure studies.\nGenerally, flexi-grid networking strives to enhance the efficiency\nof the optical transmissions by adapting physical (photonic)\ntransmission parameters, such as modulation format, symbol rate,\nnumber and spacing of subcarrier wavelength channels, as well as the\nratio of forward error correction to payload.\nFlexi-grid transmissions have become feasible with\nhigh-capacity flexible transceivers.\nFlexi-grid transmissions use narrower\nfrequency slots (e.g., 12.5~GHz) than classical Wavelength Division\nMultiplexing (WDM, with typically 50~GHz frequency slots for WDM)\nand can flexibly form optical transmission channels that span multiple\ncontiguous frequency slots.\n\nCvijetic~\\cite{Cv13} has proposed a hierarchical flexi-grid infrastructure\nfor multiservice broadband optical access utilizing\ncentralized software-reconfigurable resource management and digital signal\nprocessing. The proposed flexi-grid infrastructure incorporates mobile\nbackhaul, as well as SDN controlled transceivers~\\ref{transc:sec}.\nIn follow-up work, Cvijetic et al.~\\cite{CvTJSM14} have designed a\ndynamic flexi-grid optical access and aggregation network.\nThey employ SDN to control tunable lasers in the OLT for flexible\ndownstream transmissions.\nFlexi-grid wavelength selective switches are controlled through SDN\nto dynamically tune the passband for the upstream transmissions\narriving at the OLT.\nCvijetic et al.~\\cite{CvTJSM14} obtained good results for the upstream and\ndownstream bit error rate\nand were able to provide 150~Mb\/s per wireless network cell.\n\nOliveira et al.~\\cite{OlSCH13} have demonstrated a testbed for a\nReconfigurable Flexible Optical Network (RFON), which was one of the\nfirst physical layer SDN-based testbeds. The RFON testbed is\ncomprised of 4 ROADMs with flexi-grid Wavelength Selective Switching\n(WSS) modules, optical amplifiers, optical channel monitors and\nsupervisor boards. The controller daemon implements a node\nabstraction layer and provides configuration details for an overall\nview of the network. Also, virtualization of the GMPLS control plane\nwith topology discovery and Traffic Engineering (TE)-link\ninstantiation have been incorporated.\nInstead of using OpenFlow, the\nRFON testbed uses the controller language YANG~\\cite{Schonwalder2010}\nto obtain the topology information and collect monitoring data  for the\nlightpaths.\n\nZhao et al.~\\cite{ZhZYY13} have presented an architecture with\nOpenFlow-based optical interconnects for intra-data center networking\nand OpenFlow-based flexi-grid optical networks for inter-data center\nnetworking.\nZhao et al.~focus on the SDN benefits for inter-data center networking\nwith heterogeneous networks.\nThe proposed architecture includes a service controller, an IP controller,\nan and optical controller based on the Father Network Operating System\n(F-NOX)~\\cite{gud2008nox,zha2013uni}.\nThe performance evaluations in~\\cite{ZhZYY13} include results for\nblocking probability, release latency, and bandwidth spectrum characteristics.\n\n\\subsection{Optical Performance Monitoring}  \\label{opm:sec}\n\\label{sec:sdmonitoring}\n\\subsubsection{Cognitive Network Infrastructure}\nA Cognitive Heterogeneous Reconfigurable Dynamic Optical Network (CHRON)\narchitecture has been outlined in~\\cite{MiDJF13,cab2014cog,dur2016exp}.\nCHRON senses the current network\nconditions and adapts the network operation accordingly.\nThe three main components of CHRON are monitoring elements, software\nadaptable elements, and cognitive processes. The monitoring elements\nobserve two main types of optical transmission impairments, namely\nnon-catastrophic impairments and catastrophic impairments.\nNon-catastrophic impairments include the photonic impairments that\ndegrade the Optical Signal to Noise Ratio (OSNR), such as the\nvarious forms of dispersion, cross-talk, and non-linear propagation\neffects, but do not completely disrupt the communication. In\ncontrast, a catastrophic impairment, such as a fiber cut or\nmalfunctioning switch, can completely disrupt the communication.\nAdvances in optical performance monitoring allow for in-band OSNR\nmonitoring~\\cite{dah2011opt,sui2010osn,sch2011osn,sai2012ban} at\nmidpoints in the communication path, e.g., at optical amplifiers and\nROADMs.\n\nThe cognitive processes involve the collection of the monitoring\ninformation in the controller, executing control algorithms, and\ninstructing the software adaptable components to implement the\ncontrol decisions. SDN can provide the framework for implementing\nthese cognitive processes.\nTwo main types of software adaptable components have been considered\nso far~\\cite{Oliveira2015,Giglio2015}, namely control of transceivers and\ncontrol of wavelength selective switches\/amplifiers.\nFor transceiver control,\nthe cognitive control adjusts the transmission parameters.\nFor instance, transmission bit\nrates can be adjusted through varying the modulation format or the\nnumber of signal carriers in multicarrier communication\n(see Section~\\ref{transc:sec}).\n\n\\subsubsection{Wavelength Selective Switch\/Amplifier Control}\nIn general, ROADMs (see Section~\\ref{ROADMs:par}) employ wavelength\nselective switches based on filters to add or drop wavelength\nchannels for routing through an optical network. Detrimental\nnon-ideal filtering effects accumulate and impair the\nOSNR~\\cite{pao2015sup}. At the same time, Erbium Doped Fiber\nAmplifiers (EDFAs)~\\cite{zim2004amp} are widely deployed in optical\nnetworks to boost optical signal power that has been depleted\nthrough attenuation in fibers and ROADMs. However, depending on\ntheir operating points, EDFAs can introduce significant noise. Moura\net al.~\\cite{Moura2015,mou2016cog} have explored SDN based adaptation\nstrategies for EDFA operating points to increase the OSNR. In a\ncomplementary study, Paolucci et al.~\\cite{pao2015sup} have\nexploited SDN control to reduce the detrimental filtering effects.\nPaolucci group wavelength channels that jointly traverse a sequence\nof filters at successive switching nodes. Instead of passing these\nwavelength channels through individual (per-wavelength channel)\nfilters, the group of wavelength channels is jointly passed through\na superfilter that encompasses all grouped wavelength channels. This\njoint filtering significantly improves the OSNR.\n\nWhile the studies~\\cite{Moura2015,mou2016cog,pao2015sup} have focused on\neither the EDFA or the filters, Carvalho et al.~\\cite{Carvalho2015} and\nWang et al.~\\cite{Wang2015f} have jointly considered\nthe EDFA and filter control.\nMore specifically, the EDFA gain and the filter attenuation (and\nsignal equalization) profile were adapted to improve the OSNR.\nCarvalho et al.~\\cite{Carvalho2015} propose and evaluate a specific\njoint EDFA and filter optimization approach that exploits the global\nperspective of the SDN controller. The global optimization achieves\nONSR improvements close to 5~dB for a testbed consisting of four ROADMs\nwith 100~km fiber links.\nWang et al.~\\cite{Wang2015f} explore different combinations of\nEDFA gain control strategies and filter equalization strategies for a\nsimulated network with 14 nodes and 100~km fiber links.\nThey find mutual interactions between the EDFA gain control and the\nfilter equalization control as well as an additional wavelength assignment\nmodule.\nThey conclude that global SDN control is highly useful for synchronizing\nthe EDFA gain and filter equalization in conjunction with wavelength\nassignments so as to achieve improved OSNR.\n\n\\subsection{Infrastructure Layer: Summary and Discussion}\nThe research to date on the SDN controlled infrastructure layer has\nresulted in a variety of SDN controlled transceivers as well as a few designs\nof SDN controlled switching elements.\nMoreover, the SDN control of switching paradigms and optical\nperformance monitoring have been examined.\nThe SDN infrastructure studies have paid close attention to the physical\n(photonic) communication aspects. Principles of isolation of control plane and\ndata plane with the goals of simplifying network management and\nmaking the networks more flexible have been explored.\nThe completed SDN infrastructure layer studies have indicated\nthat the SDN control of the infrastructure layer can reduce costs, facilitate\nflexible reconfigurable resource management, increase utilizations,\nand lower latency.\nHowever, detailed comprehensive optimizations of the infrastructure components\nand paradigms\nthat minimize capital and operational expenditures are an important\narea for future research. Also, further refinements of the optical\ncomponents and switching paradigms are needed to ease the\ndeployment of SDONs and make the networks operating on\nthe SDON infrastructures more efficient.\nMoreover, the cost reduction of implementations, easy adoption by network\nproviders, flexible upgrades to adopt new technologies, and reduced\ncomplexity require thorough future research.\n\nMost SDON infrastructure studies have\nfocused on a particular network component or networking aspect, e.g.,\na transceiver or the hybrid\npacket-circuit switching paradigm, or a particular application context, e.g.,\ndata center networking. Future research should comprehensively\nexamine SDON infrastructure components and\nparadigms to optimize their interactions\nfor a wide set of networking scenarios and application contexts.\n\nThe SDON infrastructure studies to date\nhave primarily focused on the optical transmission medium.\nFuture research should explore complementary infrastructure\ncomponents and paradigms to\nsupport transmissions in hybrid fiber-wireless and other hybrid fiber-$X$\nnetworks, such as\nfiber-Digital Subscriber Line (DSL) or fiber-coax cable\nnetworks~\\cite{Fuentes2014,gur2014pon,luo2013act}.\nGenerally, the flexible SDN control can\nbe very advantageous for hybrid networks composed of heterogeneous\nnetwork segments. The OpenFlow protocol can facilitate the topology\nabstraction of the heterogeneous physical transmission media, which\nin turn facilitates control and optimization at the higher network\nprotocol layers.\n\n\\begin{figure*}[t!]\n\\footnotesize\n\\setlength{\\unitlength}{0.10in}\n\\centering\n\\begin{picture}(40,33)\n\\put(15,33){\\textbf{SDN Control Layer, Sec.~\\ref{sdnctl:sec}}}\n\\put(-7,30){\\line(1,0){49}}\n\\put(21,30){\\line(0,1){2}}\n\\put(-7,30){\\vector(0,-1){2}}\n\\put(-9,27){\\makebox(0,0)[lt]{\\shortstack[l]{\t\t\t\n\\textbf{Ctl. of Infra. Comp.}, \\\\ \\ Sec.~\\ref{ctlinfra:sec}  \\\\ \\\\\n\t\tTransceiver Ctl.\\\\~\\cite{YuZZG14,Chen2014,JiXia2014,liu2013field}    \\\\\n\t\tCircuit Sw. Ctl. \\\\ \\cite{ChNS13,DaPMSG10,DaPM09}, \\\\\n\t\t\t\\cite{DaPM09Jul,OFCircuitSwitch,ChNF13,Baik2014} \\\\\n\t\tPkt. $+$ Burst Sw. Ctl.~\\\\ \\cite{Cao2015,Harai2014,liu2013field}\n\t\t}}}\n\t\\put(5,30){\\vector(0,-1){20}}\n\t\\put(-2.5,9){\\makebox(0,0)[lt]{\\shortstack[l]{\t\t\t\n              \\textbf{Retro-fitting Devices},\\\\\n                 Sec.~\\ref{retfitctl:sec} \\\\\n              \\cite{ChNF13,ChNS13,Cao2015,liu2013field},\\\\\n\t\t\\cite{Alvizu2014,LiTMG11,LiuTMGW11,LiCTM12,LiZTV12,ClSLT14}\n\t}}}\n\t\\put(15,30){\\vector(0,-1){2}}\n\t\\put(12,27){\\makebox(0,0)[lt]{\\shortstack[l]{\t\t\t\n              \\textbf{Ctl. of Opt. Netw. Ops.}, \\\\\n                  Sec.~\\ref{ctlops:sec} \\\\ \\\\\n\t\tPON Ctl.~\\cite{KhRS14,LeK15,PaP13,par2014fut,YaZWZ14,KaCTW13} \\\\\n\t\tSpectrum Defrag.~\\\\\\cite{Ger2012,YuZZG14,Chen2014},\\\\\n\t\t\\cite{Munoz2014a,Zhu2015b,Meloni2016} \\\\ \\\\\n\t\tTandem Netw., Sec.~\\ref{ctltan:sec} \\\\\n\t\tMetro$+$Access~\\cite{wu2014glo,zha2015sof}\t\\\\\t\n\t\tAccess$+$Wireless~\\cite{boj2013adv,TaC14,Costa2015}\t\\\\\t\n\t\tAccess$+$Metro$+$Core~\\cite{SlR14} \\\\\t\t\n\t\tDC~\\cite{MaHSC14}\t   \\\\\t\n\t\tIoT~\\cite{wan2015nov}\n\t}}}\n\t\\put(30,30){\\vector(0,-1){22}}\t\t\n\t\\put(25,7){\\makebox(0,0)[lt]{\\shortstack[l]{\t\t\t\n\t      \\textbf{Hybrid SDN-GMPLS}, \\\\\n               Sec.~\\ref{hybsdnctl:sec}  \\\\\n\t\t\\cite{AzNEJ11,Alvizu2014,Munoz2014a}, \\\\\n\t\t\\cite{LiCTMMM12,Casellas2013b}\n\t}}}\n\t\\put(42,30){\\vector(0,-1){2}}\n\t\\put(35,27){\\makebox(0,0)[lt]{\\shortstack[l]{\t\t\t\n\\textbf{Ctl. Perform.}, Sec.~\\ref{cltperf:sec} \\\\ \\\\\n\t\tSDN vs. GMPLS~\\cite{ChNF13}, \\\\\n\t\t\\cite{LiTM12,ZhaoZYY13,CviAPJT13} \\\\\n\t\tFlow Setup Time~\\cite{VeSBR14,liu2013field} \\\\\n\t\tOut of Band Ctl.~\\cite{Sanchez2013} \\\\\n\t\tClust. Ctl~\\cite{Penna2014}\n\t}}}\t\t\n\t\\end{picture}\t\n\\caption{Classification of SDON control layer studies.}\n                  \\label{ctl_class:fig}\n\\end{figure*}\n\\section{\\MakeUppercase{SDN Control Layer}}\n\\label{sdnctl:sec}\nThis section surveys the SDON studies that are focused on applying the\nSDN principles at the SDN control layer to control the various optical network\nelements and operational aspects.\nThe main challenges of SDON control include extensions of the OpenFlow\nprotocol for specifically controlling the optical transmission\nand switching components surveyed in Section~\\ref{sdninfra:sec}\nand for controlling the optical\nspectrum as well as for controlling optical networks spanning\nmultiple optical network tiers (see Section~\\ref{optnetstruct:sec}).\nAs illustrated in Fig.~\\ref{ctl_class:fig},\nwe first survey SDN control mechanisms and frameworks for controlling\ninfrastructure layer components, namely transceivers as well as\noptical circuit, packet, and burst switches.\nMore specifically, we survey OpenFlow extensions for controlling the\noptical infrastructure components.\nWe then survey mechanisms for retro-fitting non-SDN optical network elements\nso that they can be controlled by OpenFlow.\nThe retro-fitting typically involves the insertion of an abstraction layer\ninto the network elements. The abstraction layer\nmakes the optical hardware controllable by OpenFlow.\nThe retro-fitting studies would also fit into Section~\\ref{sdninfra:sec} as\nthe abstraction layer is inserted into the network elements; however,\nthe abstraction mechanisms closely relate to the OpenFlow extensions for\noptical networking and we include the retro-fitting studies therefore in this\ncontrol layer section.\nWe then survey the various SDN control mechanisms for operational aspects of\noptical networks, including the control\nof tandem networks that include optical segments. Lastly, we survey\nSDON controller performance analysis studies.\n\n\\subsection{SDN Control of Optical Infrastructure Components}\n\\label{ctlinfra:sec}\n\n\\subsubsection{Controlling Optical Transceivers with OpenFlow}\nRecent generations of optical transceivers utilize digital signal\nprocessing techniques that allow many parameters of the transceiver to\nbe software controlled (see Sections~\\ref{SF-BVT:sec} and~\\ref{MF-BVT:sec}).\nThese parameters include modulation scheme, symbol rate, and wavelength.\nYu et al.~\\cite{YuZZG14} and Chen et al.~\\cite{Chen2014} proposed adding a ``modulation format'' field to the OpenFlow cross-connect table entries to support this\nprogrammable feature of some software defined optical transceivers.\n\nJi et al.~\\cite{JiXia2014} created a testbed that places super-channel optical transponders and optical\namplifiers under SDN control. An OpenFlow extension is proposed to control these devices. The modulation\ntechnique and FEC code for each optical subcarrier of the super-channel transponder\nand the optical amplifier power level can be controlled via OpenFlow. Ji et al. do not discuss this explicitly\nbut the transponder subcarriers can be treated as OpenFlow switch ports that can be configured through the\nOpenFlow protocol via port modification messages. It is unclear in \\cite{JiXia2014} how the amplifiers would be\ncontrolled via OpenFlow. However, doing so would allow the SDN controller to adaptively modify amplifiers to\ncompensate for channel impairments while minimizing energy consumption. Ji et al.~\\cite{JiXia2014}\nhave established a testbed demonstrating the placement of transponders and EDFA optical amplifiers under SDN control.\n\nLiu et al.~\\cite{liu2013field} propose configuring optical transponder operation via flow table entries with\nnew transponder specific fields (without providing details). They also propose capturing failure alarms from optical\ntransponders and sending them to the SDN controller via OpenFlow Packet-In messages. These messages are normally\nmeant to establish new flow connections. Alternatively, a new OpenFlow message type could be created for the purpose\nof capturing failure alarms~\\cite{liu2013field}. With failure alarm information, the SDN controller can implement\nprotection switching services.\n\n\\subsubsection{Controlling Optical Circuit Switches with OpenFlow}\nCircuit switching can be enabled by OpenFlow by adding new circuit switching flow table\nentries~\\cite{DaPM09Jul,DaPM09,DaPMSG10,Baik2014}. The OpenFlow circuit switching addendum~\\cite{OFCircuitSwitch}\ndiscusses the addition of cross-connect tables for this purpose.\nThese cross-connect tables are configured via OpenFlow messages inside the circuit switches. According to the addendum, a cross-connect table entry consists of the following\nfields to identify the input:\n\\begin{itemize}\n  \\item Input Port\n  \\item Input Wavelength\n  \\item Input Time Slot\n  \\item Virtual Concatenation Group\n\\end{itemize}\nand the following fields to identify the output:\n\\begin{itemize}\n  \\item Output Port\n  \\item Output Wavelength\n  \\item Output Time Slot\n  \\item Virtual Concatenation Group\n\\end{itemize}\nThese cross-connect tables cover circuit switching in space, fixed-grid wavelength, and time.\n\nChannegowda et al.~\\cite{ChNF13,ChNS13} extend the capabilities of the OpenFlow circuit switching addendum to\nsupport flexible wavelength grid optical switching. Specifically, the wavelength identifier specified in the\ncircuit switching addendum to OpenFlow is replaced with two fields: \\textit{center frequency}, and\n\\textit{slot width}. The \\textit{center frequency} is an integer specifying the multiple of 6.25~GHz the center\nfrequency is away from 193.1~Thz and the \\textit{slot width} is a positive integer specifying the spectral\nwidth in multiples of 12.5~GHz.\n\nAn SDN controlled optical network testbed at the University of Bristol has been established to\ndemonstrate the OpenFlow extensions for flexible grid DWDM~\\cite{ChNS13}. The testbed consists of\nboth fixed-grid and flexible-grid optical switching devices.\nSouth Korea Telekom has also\nbuilt an SDN controlled optical network testbed~\\cite{Shin2014}.\n\n\\subsubsection{Controlling Optical Packet and Burst Switches with OpenFlow}\nOpenFlow flow tables can be utilized in\noptical packet switches for expressing the forwarding table and its computation can be offloaded to an SDN\ncontroller. This offloading can simplify the design of highly complex optical packet switches~\\cite{Cao2015}.\n\nCao et al.~\\cite{Cao2015} extend the OpenFlow protocol to work with Optical Packet Switching (OPS) devices by creating:\n$(i)$ an abstraction layer that converts OpenFlow configuration messages to the native OPS configuration, $(ii)$ a process that\nconverts optical packets that do not match a flow table entry to the electrical domain for forwarding to the SDN\ncontroller, and $(iii)$ a wavelength identifier extension to the flow table entries. To compensate for either the\nlack of any optical buffering or limited optical buffering, an SDN controller, with its global view, can provide more\neffective means to resolve contention that would lead to packet loss in optical packet switches.\nSpecifically, Cao et al. suggest to select the path with the most available resources among multiple available paths between two nodes~\\cite{Cao2015}.\nPaths can be re-computed periodically or on-demand to account for changes\nin traffic conditions. Monitoring messages can be defined to keep the SDN controller updated of network traffic\nconditions.\n\nEngineers with Japan's National Institute of Information and Communications Technology~\\cite{Harai2014}\nhave created an optical circuit and packet switched demonstration system in which the packet portion is SDN\ncontrolled. The optical circuit switching is implemented with Wavelength Selective Switches (WSSs) and the\noptical packet switching is implemented with an Semiconductor Optical Amplifier (SOA) switch.\n\nOpenFlow flow tables can also be used to configure optical burst switching devices~\\cite{liu2013field}. When there is\nno flow table entry for a burst of packets, the optical burst switching device can send the Burst Header Packet (BHP) to\nthe SDN controller to process the addition of the new flow to the network~\\cite{liu2013field} rather than the first\npacket in the burst.\n\n\n\\subsection{Retro-fitting Devices to Support OpenFlow}  \\label{retfitctl:sec}\nAn abstraction layer can be used to turn non-SDN optical switching\ndevices into OpenFlow controllable switching\ndevices~\\cite{ChNF13,ChNS13,liu2013field,Alvizu2014,Cao2015}.\nAs illustrated in Fig.~\\ref{fig:retrofit}, the\nabstraction layer provides a conversion layer between OpenFlow\nconfiguration messages and the optical switching devices' native\nmanagement interface, e.g., the Simple Network Management Protocol\n(SNMP), the Transaction Language 1 (TL1) protocol, or a proprietary\n(vendor-specific) API. Additionally, a virtual OpenFlow switch with\nvirtual interfaces that correspond to physical switching ports on the\nnon-SDN switching device completes the abstraction\nlayer~\\cite{LiTMG11,LiuTMGW11,LiCTM12,LiZTV12,liu2013field}. When a\nflow entry is added between two virtual ports in the virtual OpenFlow\nswitch, the abstraction layer uses the switching devices' native\nmanagement interface to add the flow entry between the two\ncorresponding physical ports.\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=3in]{finman_sdon\/fig7.pdf}\n    \\caption{Traditional non-SDN network elements can be retro-fitted for control by an SDN controller using OpenFlow using\n    a hardware abstraction layer~\\cite{LiTMG11,LiuTMGW11,LiCTM12,LiZTV12,liu2013field}.}\n    \\label{fig:retrofit}\n\\end{figure}\n\nA non-SDN PON OLT can be supplemented with a two-port OpenFlow switch and a hardware abstraction layer that\nconverts OpenFlow forwarding rules to control messages understood by the non-SDN OLT~\\cite{ClSLT14}. Fig.~\\ref{fig:oltretrofit} illustrates this OLT retro-fit for SDN control via OpenFlow. In this way the\nPON has its switching functions controlled by OpenFlow.\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=3.5in]{finman_sdon\/fig8.pdf}\n    \\caption{Non-SDN OLTs can be retro-fitted for control by an SDN controller using OpenFlow~\\cite{ClSLT14}.}\n    \\label{fig:oltretrofit}\n\\end{figure}\n\n\\subsection{SDN Control of Optical Network Operation}  \\label{ctlops:sec}\n\n\\subsubsection{Controlling Passive Optical Networks with OpenFlow}\n\\label{ponctl:sec}\nAn SDN controlled PON can be created by upgrading OLTs to SDN-OLTs that can be\ncontrolled using a Southbound Interface, such as OpenFlow~\\cite{KhRS14,LeK15}. A centralized PON controller,\npotentially executing in a data center, controls one or more SDN-OLTs. The advantage of using SDN is the\nbroadened perspective of the PON controller as well as the potentially reduced cost of the SDN-OLT\ncompared to a non-SDN OLT.\n\nParol and Pawlowski~\\cite{PaP13,par2014fut} define OpenFlowPLUS to extend the OpenFlow SBI for GPON. OpenFlowPLUS extends SDN programmability to both OLT and ONU devices whereby each act\nas an OpenFlow switch through a programmable flow table. Non-switching functions (e.g., ONU registration,\ndynamic bandwidth allocation) are outside the scope of OpenFlowPLUS. OpenFlowPLUS extends OpenFlow\nby channeling OpenFlow messages through the GPON ONU Management and Control Interface (OMCI) control channel and adding PON specific action\ninstructions to flow table entries. The PON specific action instructions defined in OpenFlowPLUS are:\n\\begin{itemize}\n  \\item (new \\textit{gpon} action type): map matching packets to PON specific traffic identifiers, e.g.,\n  GPON Encapsulation Method (GEM) ports and GPON Traffic CONTainers (T-CONTs)\n  \\item (\\textit{output} action type): activate PON specific framing of matching packets\n\\end{itemize}\n\nMany of the OLT functions operate at timescales that are problematic for the controller due\nto the latency between the controller and OLTs. However, Khalili et al.~\\cite{KhRS14} identify ONU\nregistration policy and coarse timescale DBA policy as functions that operate at timescales that allow\neffective offloading to an SDN controller. Yan et al.~\\cite{YaZWZ14} further identify OLT and ONU power\ncontrol for energy savings as a function that can be effectively offloaded to an SDN controller.\n\nThere is also a movement to use PONs in edge networks to provide\nconnectivity inside a multitenant building or on a campus with multiple\nbuildings~\\cite{PaP13,par2014fut}. The use of PONs in this edge scenario\nrequires rapid re-provisioning from the OLT. A software controlled PON\ncan provide this needed rapid reprovisioning~\\cite{PaP13,par2014fut}.\n\nKanonakis et al. \\cite{KaCTW13} propose leveraging the\nbroad perspective that SDN can provide to perform dynamic bandwidth\nallocation across several Virtual PONs (VPONs). The VPONs are\nseparated on a physical PON by the wavelength bands that they\nutilize. Bandwidth allocation is performed at the granularity of OFDMA\nsubcarriers that compose the optical spectrum.\n\n\n\\subsubsection{SDN Control of Optical Spectrum Defragmentation}\nIn a departure from the fixed wavelength grid\n(ITU-T G.694.1), elastic optical networking allows flexible use of the optical spectrum. This flexibility can permit higher spectral efficiency by avoiding consuming an entire fixed-grid\nwavelength channel when unnecessary and avoiding unnecessary guard bands in certain circumstances \\cite{Ger2012}.\nHowever, this flexibility causes fragmentation of the optical spectrum as flexible grid lightpaths are established\nand terminated over time.\n\nSpectrum fragmentation leads to the circumstance in which there is enough spectral capacity to satisfy a demand but\nthat capacity is spread over several fragments rather than being consolidated in adjacent spectrum as required. If\nthe fragmentation is not counter-acted by a periodic defragmentation process than overall spectral utilization will\nsuffer. This resource fragmentation problem appears in computer systems in main memory and long term storage. In those\ncontexts the problem is typically solved by allowing the memory to be allocated using non-adjacent segments. Memory and\nstorage is partitioned into pages and blocks, respectively. The allocations of pages to a process or blocks to a file do\nnot need to be contiguous. With communication spectrum this would mean combining multiple small bandwidth channels\nthrough inverse multiplexing to create a larger channel~\\cite{Munoz2014a}.\n\nAn SDN controller can provide a broad network perspective to empower the periodic optical spectrum defragmentation\nprocess to be more effective~\\cite{Munoz2014a}. In general, optical spectrum defragmentation operations can reduce\nlightpath blocking probabilities from 3\\%~\\cite{YuZZG14} up to as much as 75\\%~\\cite{Chen2014,Zhu2015b}. Multicore\nfibers provide additional spectral resources through additional transmission cores to permit quasi-hitless\ndefragmentation~\\cite{Meloni2016}.\n\n\\subsubsection{SDN Control of Tandem Networks}  \\label{ctltan:sec}\n\n\\paragraph{Metro and Access}\nWu et al.~\\cite{wu2014glo,zha2015sof} propose leveraging the broad perspective that SDN can provide to\nimprove bandwidth allocation. Two cooperating stages of SDN controllers: $(i)$ access stage that controls each SDN\nOLT individually, and $(ii)$ metro stage that controls global bandwidth allocation strategy, can coordinate bandwidth\nallocation across several physical PONs~\\cite{wu2014glo,zha2015sof}. The bandwidth allocation is managed cooperatively\namong the two stages of SDN controllers to optimize the utilization of the access and metro network bandwidth. Simulation\nexperiments indicate a 40\\% increase in network bandwidth utilization as a result of the global coordination compared\nto operating the bandwidth allocation only within the individual PONs~\\cite{wu2014glo,zha2015sof}.\n\n\\paragraph{Access and Wireless}\nBojic et al. \\cite{boj2013adv} expand on the concept of SDN controlled OFDMA enabled VPONs~\\cite{KaCTW13} to\nprovide mobile backhaul service. The backhaul service can be provided for wireless small-cell sites (e.g., micro and femto cells)\nthat utilize millimeter wave frequencies. Each small-cell site contains an OFDMA-PON ONU that provides the backhaul service\nthrough the access network over a VPON. An SDN controller is utilized to assign bandwidth to each small-cell site through\nOFDMA subcarrier assignment in a VPON to the constituent ONU. The SDN controller leverages its broad view of the network\nto provide solutions to the joint bandwidth allocation and routing across several network segments. With this broad\nperspective of the network, the SDN controller can make globally rather than just locally optimal bandwidth allocation and\nrouting decisions. Efficient optimization algorithms, such as genetic algorithms, can be used to provide computationally\nefficient competitive solutions, mitigating computational complexity issues associated with optimization for large networks.\nAdditionally, network partitioning with an SDN controller for each partition can be used to mitigate unreasonable\ncomputational complexity that arises when scaling to large networks. Tanaka and Cvijetic~\\cite{TaC14} presented one\nsuch optimization formulation for maximizing throughput.\n\nCosta-Requena et al.~\\cite{Costa2015} described a proof-of-concept LTE testbed they have constructed\nwhereby the network consists of software defined base stations and various network functions\nexecuting on cloud resources. The testbed is described in broad qualitative terms, no technical\ndetails are provided. There was no mathematical or experimental analysis provided.\n\n\\paragraph{Access, Metro, and Core}\nSlyne and Ruffini~\\cite{SlR14} provide a use case for SDN switching control across network segments: use Layer 2 switching\nacross the access, metro, and core networks. Layer 2 (e.g., Ethernet) switching does not scale well due to a lack of hierarchy\nin its addresses. That lack of hierarchy does not allow for switching rules on aggregates of addresses thereby limiting the\nscaling of these networks. Slyne and Ruffini~\\cite{SlR14} propose using SDN to create hierarchical pseudo-MAC addresses that\npermit a small number of flow table entries to configure the switching of traffic using Layer 2 addresses across network\nsegments. The pseudo-MAC addresses encode information about the device location to permit simple switching rules. At the entry\nof the network, flow table entries are set up to translate from real (non-hierarchical) MAC addresses to hierarchical pseudo-MAC\naddresses. The reverse takes place at the exit point of the network.\n\n\\paragraph{DC Virtual Machine Migration}\nMandal et al.~\\cite{MaHSC14} provided a cloud computing use case for SDN bandwidth allocation across network segments:\nVirtual Machine (VM) migration between data centers. VM migrations require significant network bandwidth. Bandwidth\nallocation that utilizes the broad perspective that SDN can provide is critical for reasonable VM migration latencies\nwithout sacrificing network bandwidth utilization.\n\n\\paragraph{Internet of Things}\nWang et al.~\\cite{wan2015nov} examine another use case for SDN bandwidth allocation across network segments: the Internet\nof Things (IoT). Specifically, Wang et al. have developed a Dynamic Bandwidth Allocation (DBA) protocol that exploits\nSDN control for multicasting and suspending flows. This DBA protocol is studied in the context of a virtualized\nWDM optical access network that provides IoT services through the distributed ONUs to individual devices.\nThe SDN controller employs multicasting and flow suspension to efficiently prioritize the IoT service requests.\nMulticasting allows multiple requests to share resources in the central nodes that are responsible for processing a\nprescribed wavelength in the central office (OLT). Flow suspension allows high-priority requests (e.g., an emergency\ncall) to suspend ongoing low-priority traffic flows (e.g., routine meter readings). Performance results for a real-time\nSDN controller implementation indicate that the proposed bandwidth (resource) allocation with multicast and flow\nsuspension can improve several key performance metrics, such as request serving ratio, revenue, and delays\nby 30--50~\\%~\\cite{wan2015nov}.\n\n\n\\subsection{Hybrid SDN-GMPLS Control}  \\label{hybsdnctl:sec}\n\\subsubsection{Generalized MultiProtocol Label Switching (GMPLS)}\nPrior to SDN, MultiProtocol Label Switching (MPLS) offered a mechanism to separate the control and data planes\nthrough label switching. With MPLS, packets are forwarded in a connection-oriented manner through Label Switched\nPaths (LSPs) traversing Label Switching Routers (LSRs). An entity in the network establishes an LSP through a network\nof LSRs for a particular class of packets and then signals the label-based forwarding table entries to the LSRs. At\neach hop along an LSP, a packet is assigned a label that determines its forwarding rule at the next hop. At the next\nhop, that label determines that packet's output port and label for the next hop; the process repeats until the packet\nreaches the end of the LSP. Several signalling protocols for programming the label-based forwarding table entries\ninside LSRs have been defined, e.g., through the Resource Reservation Protocol (RSVP). Generalized MPLS (GMPLS) extends MPLS to offer circuit switching\ncapability. Although never commercially deployed~\\cite{liu2013field}, GMPLS and a centralized Path Computation Element\n(PCE)~\\cite{mun2014pce,oki2005dyn,pao2013sur,Casellas2013} have been considered for control of optical networks.\n\n\n\\subsubsection{Path Computation Element (PCE)}  \\label{PCE:sec}\nA PCE is a concept developed by the IETF (see RFC 4655) to refer to an entity that computes network paths given a topology\nand some criteria. The PCE concept breaks the path computation action from the forwarding action in switching devices.\nA PCE could be distributed in every switching element in a network domain or there could be a single centralized PCE\nfor an entire network domain. The network domain could be an area of an Autonomous System (AS), an AS, a conglomeration\nof several ASes, or just a group of switching devices relying on one PCE. Some of an SDN controller's functionality\nfalls under the classification of a centralized PCE. However, the PCE concept does not include the external\nconfiguration of forwarding tables. Thus, a centralized PCE device does not necessarily have a means to configure the\nswitching elements to provision a computed path.\n\nWhen the entity requesting path computation is not co-located with the PCE, a PCE Communication Protocol (PCEP) is used\nover TCP port 4189 to facilitate path computation requests and responses. The PCEP consists of the following message\ntypes:\n\\begin{itemize}\n  \\item Session establishment messages (open, keepalive, close)\n  \\item PCReq -- Path computation request\n  \\item PCRep -- Path computation reply\n  \\item PCNtf -- event notification\n  \\item PCErr -- signal a protocol error\n\\end{itemize}\n\nThe path computation request message must include the end points of the path and can optionally include the requested\nbandwidth, the metric to be optimized in the path computation, and a list of links to be included in the path. The Path\ncomputation reply includes the computed path expressed in the Explicit Route Object format (see RFC 3209) or an\nindication that there is no path. See RFC 5440 for more details on PCEP.\n\nA PCE has been proposed as a central entity to manage a GMPLS-enabled optical circuit switched network. Specifically,\nthe PCE maintains the network topology in a structure called the Traffic Engineering Database (TED). The traffic\nengineering modifier (see RFC 2702) signifies that the path computations are made to relieve congestion that is caused\nby the sub-optimal allocation of network resources. This modifier is used extensively in discussions of MPLS\/GMPLS\nbecause their use case is for traffic engineering; in acronym form the modifier is TE (e.g., TE LSP, RSVP-TE).\n\nIf the PCE is stateful with complete control over its network domain, it will also maintain an LSP database recording the provisioned GMPLS lightpaths. A lightpath request can be sent to the PCE, it will use the topology and LSP\ndatabase to find the optimal path and then configure the GMPLS-controlled optical circuit switching nodes using NETCONF\n(see RFC 6241) or proprietary command line interfaces (CLIs)~\\cite{Munoz2014a}. This stateful PCE with instantiation\ncapabilities (capabilities to provision lightpaths) operates similarly to an SDN controller. For that reason, GMPLS with a\ncentralized stateful PCE with instantiation capabilities can provide a baseline for performance analysis of an SDN\ncontroller as well as provide a mechanism to be blended with an SDN controller for hybrid control~\\cite{ChNF13,ChNS13,Alvizu2014}.\n\n\\subsubsection{Approaches to Hybrid SDN-GMPLS Control}\nHybrid GMPLS\/PCE and SDN control can be formed by allowing an SDN controller to leverage a centralized PCE to control a\nportion of the infrastructure using PCEP as the SBI~\\cite{AzNEJ11,Munoz2014a}; see illustration a) in Fig.~\\ref{fig:hybridctl}.\nThe SDN controller builds higher functionality above what the PCE provides and can possibly control a large network that\nutilizes several PCEs as well as OpenFlow controlled network elements.\n\nAlternatively, the SDN controller can leverage a PCE for its path computation abilities with the SDN controller handling\nthe configuration of the network elements to establish a path using an SBI protocol, such as\nOpenFlow~\\cite{LiCTMMM12,Casellas2013b,Alvizu2014};\nsee illustration b) in Fig.~\\ref{fig:hybridctl}.\n\\begin{figure}[t!]\n\t\\centering\n\t\\includegraphics[width=5in]{finman_sdon\/fig9.pdf}\n    \\caption{Hybrid GMPLS\/PCE and SDN network control~\\cite{LiCTMMM12,Casellas2013b,Alvizu2014}.}\n\t\\label{fig:hybridctl}\n\\end{figure}\n\n\\subsection{SDN Performance Analysis}  \\label{cltperf:sec}\n\\subsubsection{SDN vs. GMPLS}\nLiu et al.~\\cite{LiTM12} provided a qualitative comparison of GMPLS,\nGMPLS\/PCE, and SDN OpenFlow for control of wavelength switched optical\nnetworks. Liu et al.~noted that there is an evolution of centralized\ncontrol from GMPLS to GMPLS\/PCE to OpenFlow. Whereas GMPLS offers\ndistributed control, GMPLS\/PCE is commonly regarded as having centralized path\ncomputation but still distributed provisioning\/configuration; while\nOpenFlow centralizes all of the network control. In\nour discussion in Section \\ref{hybsdnctl:sec} we noted that a\nstateful PCE with instantiation capabilities centralizes all\nnetwork control and is therefore very similar to SDN. Liu et al.~have\nalso pointed out that GMPLS\/PCE is more technically mature compared to\nOpenFlow with IETF RFCs for GMPLS (see RFC 3471) and PCE (see RFC\n4655) that date back to 2003 and 2006, respectively. SDN has just\nrecently, in 2014, received standardization attention from the IETF\n(see RFC 7149).\n\nA comparison of GMPLS and OpenFlow has been conducted by Zhao et al.~\\cite{ZhaoZYY13} for large-scale optical networks.\nTwo testbeds were built, based on GMPLS and on Openflow, respectively. Performance metrics, such as\nblocking probability, wavelength utilization, and lightpath setup time were evaluated for a 1000 node topology. The\nresults indicated that GMPLS gives slightly lower blocking probability. However, OpenFlow gives higher wavelength\nutilization and shorter average lightpath setup time. Thus, the results suggest that OpenFlow is overall advantageous\ncompared to GMPLS in large-scale optical networks.\n\nCvijetic et al.~\\cite{CviAPJT13} conducted a numerical analysis to compare the computed shortest path\nlengths for non-SDN, partial-SDN, and full-SDN optical networks. A full-SDN network enables path lengths\nthat are approximately a third of those computed on a non-SDN network. These path lengths can also translate\ninto an energy consumption measure, with shortest paths resulting in reduced energy consumption. An SDN\ncontrolled network can result in smaller computed shortest paths that translates to smaller network latency\nand energy consumption~\\cite{CviAPJT13}.\n\nExperiments conducted on the testbed described in \\cite{ChNF13} show a 4~\\% reduction in lightpath blocking probability\nusing SDN OpenFlow compared to GMPLS for lightpath provisioning. The same experiments show that lightpath setup times\ncan be reduced to nearly half using SDN OpenFlow compared to GMPLS. Finally, the experiments show that an Open vSwitch\nbased controller can process about three times the number of flows per second as a NOX~\\cite{gud2008nox} based controller.\n\n\\subsubsection{SDN Controller Flow Setup}\nVeisllari et al.~\\cite{VeSBR14} evaluated the use of SDN to support both circuit and packet\nswitching in a metropolitan area ring network that interconnects access network segments\nwith a backbone network. This network is assumed to be controlled by a single SDN controller.\nThe objective of the study~\\cite{VeSBR14} was to determine the effect of packet service flow\nsize on the required SDN controller flow service time to meet stability conditions at the\ncontroller. Toward this end, Veisllari et al.~produced a mean arrival rate function of new packet and\ncircuit flows at that controller. This arrival rate function was visualized by varying the length\nof short-lived (``mice'') flows, the fraction of long-lived (``elephant'') flows, and the volume\nof traffic consumed by ``elephant'' flows. Veisllari et al.~discovered, through these visualizations,\nthat the length of ``mice'' flows is the dominating parameter in this model.\n\nVeisllari et al.~translated the arrival rate function analysis to an analysis of the ring MAN network\ndimensions that can be supported by a single SDN controller. The current state-of-the-art Beacon\ncontroller can handle a flow request every 571~ns. Assuming mice flows sizes of 20~kB and average circuit\nlifetimes of 1 second, as the fraction of packet traffic increases from 0.1 to 0.9, the network\ndimension supported by a single Beacon SDN controller decreases from 14 nodes with 92 wavelengths per\nnode to 5 nodes with 10 wavelengths per node.\n\nLiu et al.~\\cite{liu2013field} use a multinational (Japan, China, Spain) NOX:OpenFlow controlled\nfour-wavelength optical circuit and burst switched network to study path setup\/release times as well as path\nrestoration times. The optical transponders that can generate failure alarms were also under NOX:OpenFlow\ncontrol and these alarms were used to trigger protection switching. The single SDN controller was located\nin the Japanese portion of the network. The experiments found the path setup time to vary from 250--600 ms and the path release times to vary from 130--450 ms. Path restoration times varied\nfrom 250--500~ms. Liu et al.~noted that the major contributing factor to these times was the OpenFlow message delivery time~\\cite{liu2013field}.\n\n\\subsubsection{Out of Band Control}\nSanchez et al.~\\cite{Sanchez2013} have qualitatively compared four SDN controlled ring metropolitan network architectures. The architectures vary in whether the SDN control traffic is carried\nin-band with the data traffic or out-of-band separately from the data traffic. In a single wavelength ring\nnetwork, out-of-band control would require a separate physical network that would come at a high cost,\nbut provide reliability of the network control under failure of the ring network. In a multiwavelength\nring network, a separate wavelength can be allocated to carry the control traffic. Sanchez et al.~\\cite{Sanchez2013}\nfocused on a Tunable Transceiver Fixed Receiver (TTFR) WDM ring node architecture. In this architecture each\nnode receives data on a home wavelength channel and has the capability to transmit on any of the available\nwavelengths to reach any other node. The addition of the out-of-band control channel on a separate wavelength\nrequires each node to have an additional fixed receiver, thereby increasing cost. Sanchez et al. identified a clear tradeoff between cost and reliability when comparing the four architectures.\n\n\\subsubsection{Clustered SDN Control}\nPenna et al.~\\cite{Penna2014} described partitioning a wavelength-switched optical network into administrative\ndomains or clusters for control by a single SDN controller. The clustering should meet certain\nperformance criteria for the SDN controller. To permit lightpath establishment across clusters, an inter-cluster\nlightpath establishment protocol is established. Each SDN controller provides a lightpath establishment function\nbetween any two points in its associated cluster. Each SDN controller also keeps a global view of the network\ntopology. When an SDN controller receives a lightpath establishment request whose computed path traverses\nother clusters, the SDN controller requests lightpath establishment within those clusters via a WBI.\n\nThe formation of clusters can be performed such that for a specified number of clusters the average distance to\neach SDN controller is minimized~\\cite{Penna2014}. The lightpath establishment time decreases exponentially\nas the number of clusters increases.\n\n\\subsection{Control Layer: Summary and Discussion}\nA very large body of literature has explored how to expand the OpenFlow protocol to support various optical network\ntechnologies (e.g., optical circuit switching, optical packet switching, passive optical networks). A significant\nbody of literature has investigated methodologies for retro-fitting non-SDN network elements for OpenFlow control\nas well as integrating SDN\/OpenFlow with the GMPLS\/PCE control framework. A variety of SDN controller use cases have\nbeen identified that motivate the benefits of the centralized network control made possible with SDN (e.g., bandwidth\nallocation over large numbers of subscribers, controlling tandem networks).\n\nHowever, analyzing the performance of SDN controllers for optical network applications is still in a state of infancy.\nIt will be important to understand the connection between the implementation of the SDN controller (e.g., processor\ncore architecture, number of threads, operating system) and the network it can effectively control (e.g., network\ntraffic volume, network size) to meet certain performance objectives (e.g., maximum flow setup time). At present there\nare not enough relevant studies to gain an understanding of this connection. With this understanding network service\nproviders will be able to partition their networks into control domains in a manner that meets their performance\nobjectives.\n\n\n\\begin{figure*}[t!]\n\\footnotesize\n\\setlength{\\unitlength}{0.10in}\n\\centering\n\\begin{picture}(40,33)\n\\put(15,33){\\textbf{Virtualization, Sec.~\\ref{virt:sec}}}\n\\put(-8,30){\\line(1,0){42}}\n\\put(25,23){\\line(1,0){20}}\n\\put(27,26.2){\\textbf{Metro\/Core Netw.}, Sec.~\\ref{virt_core:sec}}\n\\put(34,25){\\line(0,-1){2}}\n\\put(21,30){\\line(0,1){2}}\n\\put(-8,30){\\vector(0,-1){2}}\n\\put(-10,27){\\makebox(0,0)[lt]{\\shortstack[l]{\t\t\t\n\\textbf{Access Networks}, Sec.\\ref{virt_access:sec}\t\\\\\t\\\\\nOFDMA PON~\\cite{wei2009pon,wei2009pro,wei2010ada,li2016pro,jin2009vir,zhou2015dem}\\\\\t\t\nVirtualized FiWi~\\cite{QiZSH14,QiGYZ13,ShGYZ13,he2013int,men2014eff,QiZHG14}\\\\\t\t\nWiMAX-VPON~\\cite{dha2010wim,DhPX10}\n}}}\n\n\\put(9,30){\\vector(0,-1){20}}\n\\put(3,10){\\makebox(0,0)[lt]{\\shortstack[l]{\t\t\t\n\\textbf{Data Ctr. Netw.}, \\\\\nSec.~\\ref{virt_dc:sec} \\\\ \\\\\nLIGHTNESS~\\cite{miao2015sdn,pen2015mul,pag2015opt,Saridis2016} \\\\\nCloudnets~\\cite{kan2015res,ahm2014enh,dov2015usi,xie2014dyn,vel2014tow,zha2015ren,tza2014con}\n}}}\t\t\t\t\n\\put(34,30){\\vector(0,-1){2}}\n\\put(25,23){\\vector(0,-1){2}}\n\\put(19,20){\\makebox(0,0)[lt]{\\shortstack[l]{\n\\textbf{Virt. Opt. Netw. Embedding}, Sec.~\\ref{virt_emb:sec} \\\\\nImpairment-Aware Emb.~\\cite{pen2011imp,PeNS13} \\\\\nWDM\/Flexi-grid \\\\ Emb.~\\cite{zha2013net,zha2013vir,gon2014vir,wan2015vir,pag2015opt} \\\\\nSurvivable Emb.~\\cite{hu2013sur,ye2015sur,xie2014sur,che2016cos,jia2015ava,son2011ban,pao2014mul,ass2016net,Hong2015} \\\\\nDynamic Emb.~\\cite{ye2014upg,zha2015dyn} \\\\\nEnergy-efficient Emb.~\\cite{non2015ene,che2016pow,she2014fol,wan2014hie}\n}}}\t\t\t\t\t\t\t\n\\put(45,23){\\vector(0,-1){15}}\t\t\n\\put(30,7){\\makebox(0,0)[lt]{\\shortstack[l]{\t\t\t\n\\textbf{VON Hypervisors}, Sec.~\\ref{virt_hv:sec} \\\\ \\\\\nT-NOS~\\cite{siq2015pro} \\\\\nOpenSlice~\\cite{LiuMCTMM13} \\\\\nOpt. FV~\\cite{Azod2012}\n}}}\n\\end{picture}\t\n\\caption{Classification of SDON virtualization studies.}\n\\label{virt_class:fig}\n\\end{figure*}\n\\section{\\MakeUppercase{Virtualization}}  \\label{virt:sec}\nThis section surveys control layer mechanisms for virtualizing\nSDONs.\nAs optical infrastructures have typically high costs, creating\nmultiple VONs over the optical network infrastructure is\nespecially important for access networks, where the costs need to\nbe amortized over relatively few users. Throughout, accounting\nfor the specific optical transmission and signal propagation characteristics\nis a key challenge for SDON virtualization.\nFollowing the classification structure illustrated in\nFig.~\\ref{virt_class:fig},\nwe initially survey virtualization mechanisms for access networks\nand data center networks, followed by virtualization mechanisms for\noptical core networks.\n\n\\begin{figure*}[t!]\n\t\\centering\n\t\\vspace{0cm}\n\t\\begin{tabular}{c}\n\t\t\\includegraphics[width=3.5in]{finman_sdon\/fig11a.pdf} \\\\\n\t\t\\footnotesize{(a) Overall virtualization structure} \\\\\n\t\t\\includegraphics[width=4in]{finman_sdon\/fig11b.pdf} \\\\\t\t\n\t\t\\footnotesize{(b) Virtualization of OLT}\n\t\\end{tabular}\n\t\\vspace{0cm}\n\t\\caption{Illustration of OFDMA based virtual access\n    network~\\cite{wei2009pon}.}\n\t\\vspace{0cm}\n\t\\label{fig_poniard}\n\\end{figure*}\n\n\\subsection{Access Networks}  \\label{virt_access:sec}\n\\subsubsection{OFDMA Based PON Access Network Virtualization}\nWei et al.~\\cite{wei2009pon,wei2009pro,wei2010ada} have developed\na link virtualization mechanism that can span from\noptical access to backbone networks\nbased on Orthogonal Frequency Division Multiplexing Access (OFDMA).\nSpecifically, for access networks, a Virtual PON (VPON) approach\nbased on multicarrier OFDMA over WDM has been proposed. Distinct\nnetwork slices (VPONs) utilize distinct OFDMA subcarriers, which\nprovide a level of isolation between the VPONs. Thus, different\nVPONs may operate with different MAC standards, e.g., as illustrated\nin Fig.~\\ref{fig_poniard}(a), VPON A may operate as an Ethernet PON\n(EPON) while VPON~B operates as a Gigabit PON (GPON). In addition,\nvirtual MAC queues and processors are isolated to store\n  and process the data from multiple VPONs, thus creating virtual\nMAC protocols, as illustrated in Fig.~\\ref{fig_poniard}(b).\nThe OFDMA transmissions and receptions are processed in a\nDSP module that is controlled by a central SDN control module.\nThe central SDN control module also controls the different\nvirtual MAC processes in Fig.~\\ref{fig_poniard}(b), which feed\/receive\ndata to\/from the DSP module.\nAdditional bandwidth partitioning between VPONs can be achieved through\nTime Division Multiple Access (TDMA).\nSimulation studies compared a static allocation of subcarriers to\nVPONs with a dynamic allocation based on traffic demands.\nThe dynamic allocation achieved significantly higher numbers of\nsupported VPONs on a given network infrastructure as well as lower\npacket delays than the static allocation.\nA similar strategy for flexibly employing different dynamic\nbandwidth allocation modules for different groups of ONU queues has been\nexamined in~\\cite{li2016pro}.\n\nSimilar OFDMA based slicing strategies for supporting\ncloud computing have been examined by Jinno et al.~\\cite{jin2009vir}.\nZhou et al.~\\cite{zhou2015dem} have explored a FlexPON with similar\nvirtualization capabilities. The FlexPON employs OFDM for adaptive\ntransmissions. The isolation of different VPONs is\nmainly achieved through separate MAC processing.\nThe resulting VPONs allow for flexible port\nassignments in ONUs and OLT, which have been demonstrated\nin a testbed~\\cite{zhou2015dem}.\n\n\\subsubsection{FiWi Access Network Virtualization}  \\label{virt_fiwi:sec}\n\\paragraph{Virtualized FiWi Network}\nDai et al.~\\cite{QiZSH14,QiGYZ13,ShGYZ13} have examined the\nvirtualization of FiWi networks~\\cite{Bock2014,Effenberger2015}\nto eliminate the differences between\nthe heterogeneous segments (fiber and wireless). The virtualization\nprovides a unified homogenous (virtual) view of the FiWi network.\nThe unified network view simplifies flow control and other\noperational algorithms for traffic transmissions over the\nheterogeneous network segments. In particular, a virtual resource\nmanager operates the heterogeneous segments. The resource manager\npermits multiple routes from a given source node to a given\ndestination node. Load balancing across the multiple paths has been\nexamined in~\\cite{he2013int,men2014eff}. Simulation results indicate\nthat the virtualized FiWi network with load balancing significantly\nreduces packet delays compared to a conventional FiWi network. An\nexperimental OpenFlow switch testbed of the virtualized FiWi\nnetwork has been presented in~\\cite{QiZHG14}. Testbed measurements\ndemonstrate the seamless networking across the heterogeneous fiber\nand wireless networks segments. Measurements for nodal throughput,\nlink bandwidth utilization, and packet delay indicate performance\nimprovements due to the virtualized FiWi networking approach.\nMoreover, the FiWi testbed performance is measured for a video\nservice scenario indicating that the virtualized FiWi networking approach\nimproves the Quality of Experience\n(QoE)~\\cite{che2014qos,seu2014sur} of the video streaming. A\nmathematical performance model of the virtualized FiWi network has\nbeen developed in~\\cite{QiZHG14}.\n\n\\paragraph{WiMAX-VPON}\nWiMAX-VPON~\\cite{dha2010wim,DhPX10} is a Layer-2 Virtual Private Network\n(VPN) design for FiWi access networks.\nWiMAX-VPON executes a common MAC protocol across the\nwireless and fiber network segments.\nA VPN based admission control mechanism in conjunction with a VPN\nbandwidth allocation ensures per-flow Quality of Service (QoS).\nResults from discrete event simulations demonstrate that the\nproposed WiMAX-VPON achieves favorable performance.\nAlso, Dhaini et al.~\\cite{dha2010wim,DhPX10}\ndemonstrate how the WiMAX-VPON design can be extended\nto different access network types with polling-based wireless and optical\nmedium access control.\n\n\\subsection{Data Centers}  \\label{virt_dc:sec}\n\n\\subsubsection{LIGHTNESS}\nLIGHTNESS~\\cite{miao2015sdn,pen2015mul,pag2015opt,Saridis2016} is a\nEuropean research project examining an optical Data\nCenter Network (DCN) capable of providing dynamic,\nprogrammable, and highly available DCN connectivity services.\nWhereas conventional DCNs have rigid control and management\nplatforms, LIGHTNESS strives to introduce flexible control and\nmanagement through SDN control.\nThe LIGHTNESSS architecture comprises server racks that are\ninterconnected through optical packet\nswitches, optical circuit switches, and hybrid Top-of-the-Rack\n(ToR) switches. The server racks and switches  are all\ncontrolled and managed by an SDN controller.\nLIGHTNESS control consists of an SDN controller above the optical\nphysical layer and OpenFlow agents that interact with the optical\nnetwork and server  elements. The SDN controller in cooperation with\nthe OpenFlow-agents provides a programmable data plane to the\nvirtualization modules.\nThe virtualization creates  multiple\nVirtual Data Centers (VDCs), each with its own virtual computing\nand memory resources, as well as virtual networking resources, based on\na given physical data center.\nThe virtualization is achieved through a VDC planner module and an\nNFV application that directly interact with\nthe SDN controller.\nThe VDC planner composes the VDC slices through mapping of the VDC requests\nto the physical SDN-controlled switches and server racks.\nThe VDC slices are monitored by the NFV application, which interfaces\nwith the VDC planner. Based on monitoring data, the NFV application\nand VDC planner may revise the VDC composition, e.g., transition from\noptical packet switches to optical circuit switches.\n\n\\subsubsection{Cloudnets}\nCloudnets~\\cite{azo2013sdn,ban2013mer,bari2013data,FerLR13,WoRSV11,WoRVS10}\nexploit network virtualization for\npooling resources among distributed data centers. Cloudnets support\nthe migration of virtual machines across networks to achieve\nresource pooling. Cloudnet designs can be supported through\noptical networks~\\cite{ShNS13}.\nKantarci and Mouftah~\\cite{kan2015res} have examined\ndesigns for a virtual cloud backbone network that interconnects\ndistributed backbone nodes, whereby each backbone node is\nassociated with one data center. A network resource manager\nperiodically executes a virtualization algorithm to accommodate\ntraffic demands through appropriate resource provisioning. Kantarci\nand Mouftah~\\cite{kan2015res} have developed and evaluated algorithms for\nthree provisioning objectives: minimize the outage probability of\nthe cloud, minimize the resource provisioning, and minimize a\ntradeoff between resource saving and cloud outage probability. The\nrange of performance characteristics for outage probability,\nresource consumption, and delays of the provisioning approaches have\nbeen evaluated through simulations. The outage probability of\noptical cloud networks has been reduced in~\\cite{ahm2014enh} through\noptimized service re-locations.\n\nSeveral complementary aspects of optical cloudnet networks have\nrecently been investigated.\nA multilayer network architecture with an SDN based\nnetwork management structure for cloud services has been\ndeveloped in~\\cite{dov2015usi}.\nA dynamic variation of the sharing of optical network resources\nfor intra- and inter-data center networking has been examined\nin~\\cite{xie2014dyn}.\nThe dynamic sharing does not statically assign optical network resources\nto virtual optical networks; instead, the network resources are\ndynamically assigned according to the time-varying traffic demands.\nAn SDN based optical transport mode for data center traffic has been\nexplored in~\\cite{vel2014tow}.\nVirtual machine migration mechanisms that take the characteristics\nof renewable energy into account have been examined in~\\cite{zha2015ren}\nwhile general energy efficiency mechanisms for optically networked\ncould computing resources have been examined in~\\cite{tza2014con}.\n\n\n\\subsection{Metro\/Core Networks}  \\label{virt_core:sec}\n\n\\subsubsection{Virtual Optical Network Embedding}  \\label{virt_emb:sec}\nVirtual optical network embedding seeks to map requests for virtual\noptical networks to a given physical optical network infrastructure (substrate).\nA virtual optical network consists of both a set of virtual\nnodes and a set of interconnecting links that need to\nbe mapped to the network substrate.\nThis mapping of virtual networks consisting of both\nnetwork nodes and links is fundamentally different from the\nextensively studied virtual topology design for optical wavelength\nrouted networks~\\cite{dut2000sur}, which only considered network links\n(and did not map nodes).\nVirtual network embedding of both nodes and link has already been\nextensively studied in general network graphs~\\cite{fis2013vir,rah2013svn}.\nHowever, virtual optical network embedding requires additional\nconstraints to account for the special optical transmission characteristics,\nsuch as the wavelength continuity constraint and the transmission reach\nconstraint.\nConsequently, several studies have begun to examine virtual network\nembedding algorithms specifically for optical networks.\n\n\\paragraph{Impairment-Aware Embedding}\nPeng et al.~\\cite{pen2011imp,PeNS13} have modeled the optical\ntransmission impairments to facilitate the embedding of isolated\nVONs in a given\nunderlying physical network infrastructure.\nSpecifically, they model the physical (photonic)\nlayer impairments of both single-line\nrate and mixed-line rates~\\cite{nag2010opt}.\nPeng et al.~\\cite{PeNS13} consider intra-VON impairments\nfrom Amplified Spontaneous Emission (ASE) and inter-VON impairments\nfrom non-linear impairments and four wave mixing.\nThese impairments are captured in a $Q$-factor~\\cite{azo2009sur,sar2009phy},\nwhich is considered in the mapping of\nvirtual links to the underlying physical link resources,\nsuch as wavelengths and wavebands.\n\n\\paragraph{Embedding on WDM and Flexi-grid Networks} \\label{wdm_flexi_emb:sec}\nZhang et al.~\\cite{zha2013net} have considered the embedding of\noverall virtual networks encompassing both virtual nodes and virtual links.\nZhang et al. have considered both conventional WDM networks as well as\nflexi-grid networks.\nFor each network type, they formulate the virtual node and virtual link\nmapping as a mixed integer linear program.\nConcluding that the mixed integer\nlinear program is NP-hard, heuristic solution approaches\nare developed. Specifically, the overall embedding (mapping) problem\nis divided into a node mapping problem and a link mapping problem.\nThe node mapping problem is heuristically solved through a greedy\nMinMapping strategy that maps the largest computing resource demand to the\nnode with the minimum remaining computing capacity (a complementary\nMaxMapping strategy that maps the largest demand to the\nnode with the maximum remaining capacity is also considered).\nAfter the node mapping, the link mapping problem is solved\nwith an extended grooming graph~\\cite{zhu2003nov}.\nComparisons for a small network indicate that\nthe MinMapping strategy approaches the optimal mixed integer linear program\nsolution quite closely;\nwhereas the MaxMapping strategy gives poor results.\nThe evaluations also indicate that the flexi-grid network requires only about\nhalf the spectrum compared to an equivalent WDM network for several\nevaluation scenarios.\n\nThe embedding of virtual optical networks in the context of\nelastic flexi-grid optical networking has been further examined in several\nstudies.\nFor a flexi-grid network based on OFDM~\\cite{zha2013surflexi},\nZhao et al.~\\cite{zha2013vir} have compared a greedy heuristic that maps\nrequests in decreasing order of the required resources with an arbitrary\nfirst-fit benchmark.\nGong et al.~\\cite{gon2014vir} have considered flexi-grid networks with\na similar overall strategy of node mapping followed by link mapping\nas Zhang et al.~\\cite{zha2013net}.\nBased on the local resource constraints at each node,\nGong et al.~have formed a layered auxiliary graph for the node mapping.\nThe link mapping is then solved with a shortest path routing approach.\nWang et al.~\\cite{wan2015vir} have examined an embedding approach\nbased on candidate mapping patterns that could provide the requested\nresources. The VON is then embedded according to a shortest path\nrouting.\nPages et al.~\\cite{pag2015opt} have considered embeddings that\nminimize the required optical transponders.\n\nche2016cos\n\n\\paragraph{Survivable Embedding}  \\label{surv_emb:sec}\nSurvivability of a virtual optical network, i.e., its continued\noperation in the face of physical node or link failures, is\nimportant for many applications that require dependable service. Hu\net al.~\\cite{hu2013sur} developed an embedding that can survive the\nfailure of a single physical node. Ye et al.~\\cite{ye2015sur} have\nexamined the embedding of virtual optical networks so as to survive\nthe failure of a single physical node or a physical link.\nSpecifically, Ye et al. ensure that each virtual\nnode request is mapped to a primary physical node as well as a\ndistinct backup physical node. Similarly, each virtual link is\nmapped to a primary physical route as well as a node-disjoint backup\nphysical route. Ye et al. mathematically formulate an optimization\nproblem for the survivable embedding and then propose a Parallel\nVirtual Infrastructure (VI) Mapping (PAR) algorithm. The PAR\nalgorithm finds distinct candidate physical nodes (with the highest\nremaining resources) for each virtual node request. The candidate\nphysical nodes are then jointly examined with pairs of\n shortest node-disjoint paths.\nThe evaluations in~\\cite{ye2015sur} indicate that the parallel\nPAR algorithm reduces the blocking probabilities of virtual network requests\nby 5--20~\\% compared to a sequential algorithm benchmark.\nA limitation of the survivable embedding~\\cite{ye2015sur} is that it\nprotects only from a single link or node failure.\nAs the optical infrastructure is expected to penetrate\ndeeper in the access network deployments (e.g., mobile backhaul),\nit will become necessary to consider multiple failure points.\nSimilar survivable network embedding algorithms that employ node-disjoint\nshortest paths in conjunction with specific cost metrics for\nnode mappings have been investigated by Xie et al.~\\cite{xie2014sur}\nand Chen et al.~\\cite{che2016cos}.\nJiang et al.~\\cite{jia2015ava} have examined a solution variant based\non maximum-weight maximum clique formation.\n\nThe studies~\\cite{son2011ban,pao2014mul,ass2016net} have examined\nso-called bandwidth squeezed restoration for virtual topologies.\nWith bandwidth squeezing, the back-up path\nbandwidths of the surviving virtual topologies are generally lower than\nthe bandwidths on the working paths.\n\nSurvivable virtual topology design in the context of multidomain optical\nnetworks has been studied by Hong et al.~\\cite{Hong2015}.\nHong et al.~focused on minimizing the total network link cost\nfor a given virtual traffic demand. A heuristic algorithm\nfor partition and contraction\nmechanisms based on cut set theory has been proposed for the\nmapping of virtual links onto multidomain optical networks.\nA hierarchical SDN control plane is split between local controllers\nthat to manage individual domains and a\nglobal controller for the overall management.\nThe partition and contraction\nmechanisms abstract inter- and intra-domain information as a\nmethod of contraction. Survivability conditions are ensured\nindividually for inter- and intra-domains such that survivability is\nmet for the entire network.\nThe evaluations in~\\cite{Hong2015} demonstrate successful virtual\nnetwork mapping at the scale required by commercial Internet\nservice providers and infrastructure providers.\n\n\n\\paragraph{Dynamic Embedding}\nThe embedding approaches surveyed so far have mainly focused on the\noffline embedding of a static set of virtual network requests.\nHowever, in the ongoing network operation the dynamic embedding of\nmodifications (upgrades) of existing virtual networks, or the\naddition of new virtual networks are important. Ye et\nal.~\\cite{ye2014upg} have examined a variety of strategies for\nupgrading existing virtual topologies. Ye et al. have considered\nboth scenarios without advance planning (knowledge) of virtual\nnetwork upgrades and scenarios that plan ahead for possible\n(anticipated) upgrades. For both scenarios, a divide-and-conquer\nstrategy and an integrate-and-cooperate strategy are\nexamined. The divide-and conquer strategy sequentially maps all the\nvirtual nodes\nand then the virtual links. In contrast, the integrate-and-cooperate\nstrategy jointly\nconsiders the virtual node and virtual link mappings. Without\nadvance planning, these strategies are applied sequentially, as the\nvirtual network requests arrive over time, whereas, with planning,\nthe initial and upgrade requests are jointly considered. Evaluation\nresults indicate that the integrate-and-cooperate strategy\nslightly increases a revenue\nmeasure and request acceptance ratio compared to the divide-and-conquer\nstrategy.\nThe results also indicate that planning has the potential to\nsubstantially increase the revenue and acceptance ratio. In a\nrelated study, Zhang et al.~\\cite{zha2015dyn} have examined\nembedding algorithms for virtual network requests that arrive\ndynamically to a multilayer network consisting of electrical and\noptical network substrates.\n\n\\paragraph{Energy-efficient Embedding}\nMotivated by the growing importance of green networking and information\ntechnology~\\cite{BiPCR12}, a few studies have begun to consider\nthe energy efficiency of the embedded virtual optical networks.\nNonde et al.~\\cite{non2015ene} have developed and evaluated\nmechanisms for embedding virtual cloud networks so as to minimize\nthe overall power consumption, i.e., the aggregate of the\npower consumption for communication and computing (in the data centers).\nNonde et al. have incorporated the power consumption of the\ncommunication components, such as transponders and optical switches,\nas well as the power consumption characteristics of data center servers\ninto a mathematical power minimization model.\nNonde et al. then develop a real-time heuristic for energy-optimized\nvirtual network embedding.\nThe heuristic strives to consolidate computing requests in the\nphysical nodes with the least residual computing capacity.\nThis consolidation strategy is motivated by the typical\npower consumption characteristic of a compute server that\nhas a significant idle power consumption and then grows linearly with\nincreasing computing load; thus a fully loaded server is more\nenergy-efficient than a lightly loaded server.\nThe bandwidth demands are then routed between the nodes according to\n a minimum hop algorithm.\nThe energy optimized embedding is compared with a cost optimized\n embedding that only seeks to minimize the number of utilized wavelength\n channels.\n The evaluation results in~\\cite{non2015ene} indicate\n that the energy optimized embedding significantly reduces the overall energy\n consumption for low to moderate loads on the physical infrastructure;\n for high loads, when all physical resources need to be utilized,\n there are no significant savings. Across the entire load range, the\n energy optimized embedding saves on average 20~\\% energy compared to the\n benchmark minimizing the wavelength channels.\n\nChen~\\cite{che2016pow} has examined a similar energy-efficient\nvirtual optical network embedding that considers primary and\nlink-disjoint backup paths, similar to the survivable embeddings in\nSection~\\ref{surv_emb:sec}. More specifically, virtual link requests\nare mapped in decreasing order of their bandwidth requirements to\nthe shortest physical transmission distance paths, i.e., the highest\nvirtual bandwidth demands are allocated to the shortest physical\npaths. Evaluations indicate that this link mapping approach roughly\nhalves the power consumption compared to a random node mapping\nbenchmark. Further studies focused on energy savings have examined\nvirtual link embeddings that maximize the usage of nodes with\nrenewable energy~\\cite{she2014fol} and the traffic\ngrooming~\\cite{wan2014hie} onto sliceable BVTs~\\cite{zha2015ene}.\n\n\\subsubsection{Hypervisors for VONs} \\label{virt_hv:sec}\nThe operation of VONs over a given\nunderlying physical (substrate) optical network requires an\nintermediate hypervisor. The hypervisor presents the physical\nnetwork as multiple isolated VONs to the corresponding VON\ncontrollers (with typically one VON controller per VON). In turn,\nthe hypervisor intercepts the control messages issued by a VON\ncontroller and controls the physical network to effect the control\nactions desired by the VON controller for the corresponding VON.\n\nTowards the development of an optical network hypervisor, Siquera et\nal.~\\cite{siq2015pro} have developed a SDN-based controller for an\noptical transport architecture. The controller implements a\nvirtualized GMPLS control plane with offloading to facilitate the\nimplementation of hypervisor functionalities, namely the creation\noptical virtual private networks, optical network slicing, and\noptical interface management. A major contribution of Siquera et\nal.~\\cite{siq2015pro} is a Transport Network Operating System\n(T-NOS), which abstracts the physical layer for the controller and\ncould be utilized for hypervisor functionalities.\n\nOpenSlice~\\cite{LiuMCTMM13} is a comprehensive\nOpenFlow-based hypervisor that creates VONs over underlying elastic\noptical networks~\\cite{cha2015rou,tal2014spe}. OpenSlice\ndynamically provisions end-to-end paths and offloads IP traffic by\nslicing the optical communications spectrum. The paths are set up\nthrough a handshake protocol that fills in cross-connection table\nentries. The control messages for slicing the optical communications\nspectrum, such as slot width and modulation format, are carried in\nextended OpenFlow protocol messages. OpenSlice relies on special\ndistributed network elements, namely bandwidth variable wavelength\ncross-connects~\\cite{jin2009spe} and multiflow optical\ntransponders~\\cite{jin2012mul} that have been extended for control\nthrough the extended OpenFlow messages. The OpenSlice evaluation\nincludes an experimental demonstration. The evaluation results\ninclude path provisioning latency comparisons with a GMPLS-based\ncontrol plane and indicate that OpenFlow outperforms GMPLS for paths\nwith more than three hops.\nOpenSlice extension and refinements to\nmultilayer and multidomain networks are surveyed in\nSection~\\ref{orch:sec}.\nAn alternate centralized Optical FlowVisor that does not require\nextensions to the distributed network elements has been investigated\nin~\\cite{Azod2012}.\n\n\\subsection{Virtualization: Summary and Discussion}\nThe virtualization studies on access\nnetworks~\\cite{wei2009pon,wei2009pro,wei2010ada,jin2009vir,zhou2015dem,QiZSH14,QiGYZ13,ShGYZ13,he2013int,men2014eff,QiZHG14,dha2010wim,DhPX10} have primarily\nfocused on exploiting and manipulating the specific properties of the\noptical physical layer (e.g., different OFDMA subcarriers)\nand MAC layer (e.g., polling based MAC protocol) of the optical\naccess networks for virtualization.\nIn addition, to virtualization studies on\npurely optical PON access networks, two sets of studies, namely\nsets~\\cite{QiZSH14,QiGYZ13,ShGYZ13,he2013int,men2014eff,QiZHG14}\nand WiMAX-VPON~\\cite{dha2010wim,DhPX10} have examined\nvirtualization for two forms of FiWi access networks.\nFuture research needs to consider virtualization of a wider set of\nFiWi network technologies, i.e., FiWi networks that consider\noptical access networks with a wider variety of wireless access\ntechnologies, such as different forms of cellular access or combinations\nof cellular with other forms of wireless access.\nAlso, virtualization of integrated access and metropolitan area\nnetworks~\\cite{ahm2012rpr,seg2007all,val2015exp,woe2013sdn} is\nan important future research direction.\n\nA set of studies has begun to explore optical networking support for\nSDN-enabled cloudnets that exploit virtualization to\ndynamically pool resources across distributed data centers.\nOne important direction for\nfuture work on cloudnets is to examine moving data center resources\ncloser to the users and the subsequent resource pooling across edge\nnetworks~\\cite{man2013clo}. Also, the exploration of the benefits of\nFiWi networks for decentralized\ncloudlets~\\cite{din2013sur,sat2009cas,ScSF12,ver2012clo} that\nsupport mobile wireless network services is an important future\nresearch direction~\\cite{mai2015inv}.\n\n\\begin{figure*}[t!]\n\t\\footnotesize\n\t\\setlength{\\unitlength}{0.10in}\n\t\\centering\n\t\\begin{picture}(40,33)\n\t\\put(15,33){\\textbf{Applications, Sec.~\\ref{sdnapp:sec}}}\n\t\\put(-6,30){\\line(1,0){51}}\n\t\\put(21,30){\\line(0,1){2}}\n\t\n\t\\put(-6,30){\\vector(0,-1){2}}\n\t\\put(-9,27){\\makebox(0,0)[lt]{\\shortstack[l]{\t\t\t\n\\textbf{QoS}, Sec.~\\ref{app_qos:sec} \\\\ \\\\\n\t\t\t\tLong-term QoS~\\cite{ZhZZ13,kho2016qua} \\\\\n\t\t\t\tShort-term QoS~\\cite{Li2014,PatelJiWang2013} \\\\\n\t\t\t\tVirt. Top. Reconfig.~~\\cite{WetteKarl2013} \\\\\n\t\t\t\tQoS Routing~\\cite{Tariq2015,SgPCVC13,Ilchmann2015,ChangLi2015} \\\\\n\t\t\t\tQoS Management~\\cite{RuBRK14,TeMAD14} \\\\\n\t\t\t\tVideo Appl.~\\cite{chi2015app,Chitimalla2015,Li2014video}\n\t\t\t}}}\n\t\t\t\n\t\t\\put(10,30){\\vector(0,-1){17}}\n\t\t\\put(5,12){\\makebox(0,0)[lt]{\\shortstack[l]{\t\t\n\\textbf{Access Control and} \\\\\n\\textbf{Security}, Sec.~\\ref{app_secur:sec}\\\\ \\\\  \t\n\t\t\t\tFlow-based Access \\\\ Ctl.~\\cite{MaGMT14,nay2009res}\t\\\\\t\t\t\n\t\t\t\tLightpath Hopping Sec.~\\cite{Li2016c}\t\\\\\t\t\t\n\t\t\t\tFlow Timeout~\\cite{ZhuFan2015}\n\t\t\t}}}\n\t\t\t\t\t\n\t\t\\put(31,30){\\vector(0,-1){2}}\n\t\t\\put(23,27){\\makebox(0,0)[lt]{\\shortstack[l]{\t\t\t\n\\textbf{Energy Eff.}, Sec.~\\ref{ene_eff:sec}\t\\\\ \\\\  \t\t\t\n\t\t\t\tAppl. Ctl.~\\cite{Ji2014,yan2013mul,yan2015per,yan2016per}\t\t\t\\\\\t\n\t\t\t\tRouting~\\cite{Tego2014,Wang2015,Yevsieieva2015,Yoon2015,val2015exp}\n\t\t\t}}}\n\t\n\t\t\\put(45,30){\\vector(0,-1){17}}\t\t\n\t\t\\put(33,12){\\makebox(0,0)[lt]{\\shortstack[l]{\t\t\t\n\\textbf{Failure Recov. $+$} \\\\\n\\textbf{Restoration}, Sec.~\\ref{app_fail_rec:sec} \\\\ \\\\\n\t\t\t\tNetw. Reprov.~\\cite{sav2015bac} \\\\\n\t\t\t\tRestoration~\\cite{Giorgetti2015} \\\\\n\t\tReconfig.~\\cite{Aguado2016,aib2016sof,SlKMPR14,Kim2015} \\\\\n\t\t\t\tHierarchical Surv.~\\cite{ZhangSong2014} \\\\\n\t\t\t\tRobust Power Grid~\\cite{Rastegarfar2016}\n\t\t\t}}}\n\t\t\t\t\t\t\t\t\t\n\\end{picture}\t\n\\caption{Classification of application layer SDON studies.}\n\\label{app_class:fig}\n\\end{figure*}\nA fairly extensive set of studies has examined virtual network embedding\nfor metro\/core networks.\nThe virtual network embedding studies have considered the\nspecific limitations and constraints of optical networks and have\nbegun to explore specialized embedding strategies that strive to\nmeet a specific optimization objective,\nsuch as survivability, dynamic adaptability,\nor energy efficiency.\nFuture research should seek to develop a comprehensive framework of\nembedding algorithms that can be tuned with weights to achieve\nprescribed degrees of the different optimization objectives.\n\nA relatively smaller set of studies has developed and refined hypervisors\nfor creating VONs over metro\/core optical networks.\nMuch of the SDON hypervisor research has centered on the OpenSlice\nhypervisor concept~\\cite{LiuMCTMM13}. While OpenSlice accounts\nfor the specific characteristics of the optical transmission medium,\nit is relatively complex as it requires a distributed implementation\nwith specialized optical networking components.\nFuture research should seek to achieve the hypervisor functionalities\nwith a wider set of common optical components so as to reduce cost\nand complexity.\nOverall, SDON hypervisor research should examine the performance-complexity\/cost\ntradeoffs of distributed versus centralized approaches.\nWithin this context of examining the spectrum of distributed to\ncentralized hypervisors,\nfuture hypervisor research should further refine and optimize the\nvirtualization mechanisms so as to achieve strict isolation between\nvirtual network slices, as well as low-complexity hypervisor\ndeployment, operation, and maintenance.\n\n\n\\section{\\MakeUppercase{SDN Application Layer}}  \\label{sdnapp:sec}\nIn the SDN paradigm, applications interact with the controllers to\nimplement network services. We organize the survey of the studies on\napplication layer aspects of SDONs according to the main application\ncategories of quality of service (QoS), access control and security,\nenergy efficiency, and failure recovery, as illustrated in\nFig.~\\ref{app_class:fig}.\n\n\\subsection{QoS}  \\label{app_qos:sec}\n\\subsubsection{Long-term QoS: Time-Aware SDN}\nData Center (DC) networks move data back and forth between\nDCs to balance the computing load and the data storage usage\n(for upload)~\\cite{DeCusatis2014}.\nThese data movements between DCs can span large geographical areas and help\nensure DC service QoS for the end users.\nLoad balancing algorithms can exploit the characteristics\nof the user requests.\nOne such request characteristic is the high degree of time-correlation over\nvarious time scales\nranging from several hours of a day (e.g., due to a sporting event)\nto several days in a year (e.g., due to a political event).\nZhao et al.~\\cite{ZhZZ13} have proposed a time-aware SDN application using\nOpenFlow extensions to dynamically balance the load across the\nDC resources so as to improve the QoS.\nSpecifically, a time correlated PCE algorithm based on flexi-grid optical\ntransport (see Section~\\ref{PCE:sec})\nhas been proposed. An SDN application monitors\nthe DC resources and applies network rules to preserve the QoS.\nEvaluations of the algorithm indicate improvements\nin terms of network blocking probability, global blocking probability, and\nspectrum consumption ratio.\nThis study did not consider short time scale traffic bursts,\nwhich can significantly affect the load conditions.\n\nWe believe that in order to avoid pitfalls in the operation of load balancing\nthrough PCE algorithms implemented with SDN, a wide range of\ntraffic conditions needs to be considered.\nThe considered traffic range should include short and long term traffic\nvariations, which should be traded off with various QoS aspects,\nsuch as type of application\nand delay constraints, as well as the resulting costs and control overheads.\nKhodakarami et al.~\\cite{kho2016qua} have taken steps in this direction by\nforming a traffic forecasting model for both long-term and short-term forecasts\nin a wide-area mesh network.\nOptical lightpaths are then configured based on the overall traffic forecast,\nwhile electronic switching capacities are allocated based on short-term\nforecasts.\n\n\\begin{figure}[t!]\n    \\centering\n    \\vspace{0cm}\n    \\includegraphics[width=4in]{finman_sdon\/fig13.pdf}\n    \\caption{Optical SDN-based QoS-aware burst switching\n      application~\\cite{PatelJiWang2013}.}\n    \\label{fig_app_qos}\n\\end{figure}\n\\subsubsection{Short Term QoS}\nUsers of a high-speed FTTH access network may request\nvery large bandwidths\ndue to simultaneously running applications that require high data rates.\nIn such a scenario,\napplications requiring very high data rates may affect each other.\nFor instance, a video conference running simultaneously with\nthe streaming of a sports video may result in\ncall drops in the video conference application and\nin stalls of the sports video.\nLi et al.~\\cite{Li2014} proposed an SDN based bandwidth provisioning\napplication in the broadband remote access server~\\cite{Dietz2015}\nnetwork. They defined and assigned the minimum bandwidth, which they named\n``sweet point'', required\nfor each application to experience good QoE.\nLi et al. showed that maintaining the ``sweet point'' bandwidth for each\napplication can significantly improve the QoE while other\napplications are being served\naccording to their bandwidth requirements.\n\nIn a similar study, Patel et al.~\\cite{PatelJiWang2013} proposed\na burst switching mechanism based on a software\ndefined optical network. Bursts typically originate\nat the edge nodes and the aggregation\npoints due to statistical multiplexing of high speed\noptical transmissions. To ensure QoS for multiple traffic classes,\nbursts at the edge nodes have to be managed by deciding their end-to-end\npath to meet their QoS requirements, such as minimum delay and data rate.\nIn non-SDN based mechanisms, complicated\ndistributed protocols, such as GMPLS~\\cite{mun2014pce,pao2013sur},\nare used to route the burst traffic.\nIn the proposed application,\nthe centralized unified control plane decides the routing\npath for the burst based on latency and QoS requirements.\nA simplified procedure involves $(i)$ burst evaluation at the edge node,\n$(ii)$ reporting burst information to the SDN controller, and\n$(iii)$ sending of configurations to the optical nodes by the controller\nto set up a lightpath as illustrated in Fig.~\\ref{fig_app_qos}.\nSimulations indicate an increase of performance in terms of\nthroughput, network blocking probability, and latency\nalong with improved QoS when compared to non-SDN GMPLS methods.\n\n\\subsubsection{Virtual Topology Reconfigurations}\nThe QoS experienced by traffic flows greatly depends on their\nroute through a network.\nWette et al.~\\cite{WetteKarl2013} have examined an application algorithm that\nreconfigures WDM network virtual topologies\n(see Section~\\ref{wdm_flexi_emb:sec})\naccording to the traffic levels.\nThe algorithm considers the localized traffic information and\noptical resource availability at the nodes.\nThe algorithm does not require synchronization,\nthus reducing the overhead while simplifying the network design.\nIn the proposed architecture, optical switches are connected to ROADMs.\nThe reconfiguration application manages\nand controls the optical switches through the SDN controller.\nA new WDM controller is introduced to configure the lightpaths taking\n wavelength conversion and lightpath switching at the ROADMs into consideration.\nThe SDN controller operates on the optical network which appears as a\nstatic network, while the WDM controller configures (and re-configures)\nthe ROADMs to create\nmultiple virtual optical networks according to the traffic levels.\nEvaluation results indicate improved utilization and throughput.\nThe results indicate that virtual topologies reconfigurations can\nsignificantly increase the flexibility of the network while achieving\nthe desired QoS.  However, the control overhead and the delay aspects\ndue to virtualization and separation of control and lightwave paths\nneeds to be carefully considered.\n\n\n\\subsubsection{End-to-End QoS Routing}\nInterconnections between DCs involve typically\nmultiple data paths. All the interfaces existing\nbetween DCs can be utilized by MultiPath TCP (MPTCP).\nEnsuring QoS in such an MPTCP setting\nwhile preserving throughput efficiency in a\nreconfigurable underlying burst switching optical network is a\nchallenging task. Tariq et al.~\\cite{Tariq2015} have\nproposed QoS-aware bandwidth reservation for\nMPTCP in an SDON.\nThe bandwidth reservation proceeds in two stages\n$(i)$ path selection for MPTCP, and\n$(ii)$ OBS wavelength reservation to assign the priorities for\nlatency-sensitive flows. Larger portions of a wavelength reservation are\nassigned to high priority flows, resulting in reduced\nburst blocking probability\nwhile achieving the higher MPTCP throughput.\nThe simulation results in~\\cite{Tariq2015} validate the two-stage algorithm\nfor QoS-aware  MPTCP over an SDON, indicating decreased dropping probabilities,\nand increased throughputs.\n\n\\begin{figure}[t!]\n    \\centering\n    \\vspace{0cm}\n    \\includegraphics[width=3.5in]{finman_sdon\/fig14.pdf}\n    \\vspace{0cm}\n    \\caption{Illustration of routing application.}  \n   \\vspace{0cm}\n    \\label{fig_app_i2rs}\n\\end{figure}\nInformation To the Routing System (I2RS)~\\cite{I2RSietf} is a high-level\narchitecture for communicating and interacting with routing systems,\nsuch as BGP routers.\nA routing system may consists of several complex\nfunctional entities, such as a Routing Information Base (RIB),\nan RIB manager, topology and policy databases,\nalong with routing and signalling units.\nThe I2RS provides a programmability platform that enables\naccess and modifications of the configurations of the routing system elements.\nThe I2RS can be extended with SDN principles to achieve global network\nmanagement and reconfiguration~\\cite{har2013sof}.\nSgambelluri et al.~\\cite{SgPCVC13} presented an SDN based routing application\nwithin the I2RS framework to integrate the control of the\naccess, metro, and core networks as illustrated in Fig.~\\ref{fig_app_i2rs}.\nThe SDN controller communicates with the Path Computation\nElements (PCEs) of the core network\nto create Label Switched Paths (LSPs)\nbased on the information received by the OLTs.\nExperimental demonstrations validated the routing optimization\nbased on the current traffic status and previous load as well as\nthe unified control interface for access, metro, and core networks.\n\nIlchmann et al.~\\cite{Ilchmann2015} developed an SDN application that communicates to an SDN controller via\nan HTTP-based REST API. Over time, lightpaths in an optical network can become inefficient for a number of\nreasons (e.g., optical spectrum fragmentation). For this reason, Ilchmann et al. developed an SDN application\nthat evaluates existing lightpaths in an optical network and offers an application user the option to\nre-optimize the lightpath routing to improve various performance metrics (e.g., path length). The application\nis user-interactive in that the user can see the number of proposed lightpath routing changes before they are made\nand can potentially select a subset of the proposed changes to minimize network down-time.\n\nAt the ingress and egress routers of optical networks (e.g., the\nedge routers between access and metro networks), buffers are highly\nnon-economical to implement, as they require large buffers sizes to\naccommodate the channel rates of 40~Mb\/s or more. To reduce the buffer\nrequirements at the edge routers, Chang et al.~\\cite{ChangLi2015}\nhave proposed a backpressure application referred to as Refill and\nSDN-based Random Early Detection (RS-RED).  RS-RED implements a\nrefill queue at the ingress device and a droptail\nqueue at the egress device, whereby both queues are centrally\nmanaged by the RS-RED algorithm running on the SDN\ncontroller. Simulation results showed that at the expense of small delay\nincreases, edge router buffer sizes can be significantly reduced.\n\n\\subsubsection{QoS Management}\nRukert et al.~\\cite{RuBRK14} proposed SDN based controlled home-gateway\nsupporting heterogeneous wired technologies, such as DSL, and\nwireless technologies, such as LTE and WiFi.\nSDN controllers managed by the ISPs optimize the traffic flows\nto each user while accommodating large numbers of users  and ensuring their\nminimum QoS.\nAdditionally, Tego et al.~\\cite{TeMAD14} demonstrated an experimental\nSDN based QoS management setup to optimize the energy utilization.\nGbE links are switched on and off based on the traffic levels.\nThe QoS management reroutes the traffic to avoid congestion and achieve\nefficient throughput. SDN applications conduct active QoS\nprobing to monitor the network QoS characteristics.\nEvaluations have indicated that the SDN based techniques\nachieve significantly higher throughput than non-SDN techniques~\\cite{TeMAD14}.\n\n\\subsubsection{Video Applications}\nThe application-aware SDN-enabled resource allocation\napplication has been introduced by Chitimalla et al.~\\cite{chi2015app}\nto improve the video QoE in a PON access network.\nThe resource allocation application\nuses application level feedback to schedule the optical\nresources. The video resolution is incrementally increased or\ndecreased based on the buffer utilization statistics that the client\nsends to the controller. The scheduler at the OLT schedules the\npackets based on weights calculated by the SDN controller, whereby\nthe video applications at the clients communicate with the\ncontroller to determine the weights. If the network is congested,\nthen the SDN controller communicates to the clients to reduce the\nvideo resolution so as to reduce the stalls and to improve the QoE.\n\n\\begin{figure}[t!]\n    \\centering\n    \\vspace{0cm}\n    \\includegraphics[width=5in]{finman_sdon\/fig15.pdf}\n    \\caption{SDN based video caching application in PON\n        for mobile users~\\cite{Li2014video}.}\n    \\label{fig_app_video} \n\\end{figure}\nCaching of video data close the users is generally beneficial for\nimproving the QoE of video services~\\cite{Ahlgren2012,Choi2012}.\nLi et al.~\\cite{Li2014video} have introduced\ncaching mechanisms for software-defined PONs.\nIn particular, Li et al.~have proposed\njoint provisioning of the bandwidth to service the video and\nthe cache management, as illustrated in Fig.~\\ref{fig_app_video}.\nBased on the request frequency for specific\nvideo content, the Base Station (BS) caches the content with the\nassistance of the SDN controller.\nThe proposed \\textit{push}-based mechanism delivers (pushes) the video\nto the BS caches when the PON is not congested.\nA specific PON transmission sub-band can be used to multicast video\ncontent that needs to be cached at multiple BSs.\nThe simulation evaluation in~\\cite{Li2014video} indicate\nthat up to 30\\% additional videos can be serviced while the service\nresponse delay is reduced to 50\\%.\n\n\n\\subsection{Access Control and Security} \\label{app_secur:sec}\n\\subsubsection{Flow-based Access Control}\nNetwork Access Control (NAC) is a networking application that\nregulates the access to network services~\\cite{cas2007eth,par2014fut}.\nA NAC based on traffic flows has been developed by Matias~\\cite{MaGMT14}.\nFlow-NAC exploits the forwarding rules of OpenFlow switches, which are set by a\ncentral SDN controller, to control the access of traffic flows to\nnetwork services.\nFlowNAC can implement the access control based on various flow identifiers,\nsuch as MAC addresses or IP source and destination addresses.\nPerformance evaluations measured\nthe connections times for flows on a testbed and found average connection\ntimes on the order of 100~ms for completing the flow access control.\n\nIn a related study, Nayak et al.~\\cite{nay2009res} developed the\nResonance flow based access control system for an enterprise\nnetwork. In the Resonance system, the network elements, such as the\nrouters themselves, dynamically enforce access control policies. The\naccess control policies are implemented through real-time alerts and\nflow based information that is exchanged with SDN principles. Nayak\net al. have demonstrated the Resonance system on a production\nnetwork at Georgia Tech. The Resonance design can be readily\nimplemented in SDON networks and can be readily extended to wide\narea networks. Consider for example multiple heterogeneous DCs of\nmultiple organizations that are connected by an optical backbone\nnetwork. The Resonance system can be extended to provide access\ncontrol mechanisms, such as authentication and authorization,\nthrough such a wide area SDON.\n\n\\subsubsection{Lightpath Hopping Security}\n\\begin{figure}[t!]\n    \\centering     \\vspace{0cm}\n    \\includegraphics[width=3.5in]{finman_sdon\/fig16.pdf} \n    \\caption{Overview of optical light path hopping mechanism~\\cite{Li2016c}.} \n    \\label{fig_app_hop}\n\\end{figure}\nThe broad network perspective of SDN controllers facilitates\nthe implementation of security functions that require this broad\nperspective~\\cite{ahm2015sec, sco2015sur, ShJJJ14}.\nHowever, SDN may also be vulnerable to a wide range of attacks and\nvulnerabilities, including unauthorized access, data leakage,\ndata modification, and misconfiguration.\nEavesdropping and jamming are security threats on the physical layer\nand are especially relevant for the optical layer of SDONs.\nIn order to prevent\neavesdropping and jamming in an optical lightpath, Li et al.~\\cite{Li2016c}\nhave proposed an SDN based fast lightpath hopping mechanism.\nAs illustrated in Fig.~\\ref{fig_app_hop}, the hopping mechanism operates\nover multiple lightpath channels. Conventional optical\nlightpath setup times range from several hundreds of milliseconds\nto several seconds and would result in a very low hopping frequency.\nTo avoid the\noptical setup times during each hopping period, an SDN based\nhigh precision time synchronization has been proposed.\nAs a result, a fast hopping\nmechanism can be implemented and executed in a coordinated manner. A hop frame\nis defined and guard periods are added in between hop frames.\nThe experimental evaluations indicate that a maximum hopping frequency\nof~1~MHz can be achieved with a BER of $1 \\times 10^{-3}$.\nHowever, shortcomings of such mechanisms are the secure exchange of hopping\nsequences between the transmitter and\nthe receiver. Although, centralized SDN control provides authenticated\nprovisioning of the hopping sequence, additional mechanisms to secure\nthe hopping sequence from\nbeing obtained through man-in-the-middle attacks should be investigated.\n\n\\subsubsection{Flow Timeout}\nSDN flow actions on the forwarding and switching elements have generally a\nvalidity period.\nUpon expiration of the validity period, i.e., the flow action timeout,\nthe forwarding or switching\nelement drops the flow action from the forwarding information base or the flow\ntable. The switching\nelement  CPU must be able to access the flow action information with very\nlow latency so as to perform switching actions at the line rate.\nTherefore, the flow actions are commonly stored in Ternary Content\nAddressable Memories (TCAMs)~\\cite{Pagiamtzis2006}, which are limited\nto storing on the order of thousands of distinct entries.\nIn SDONs, the optical network elements perform\nthe actions set by the SDN controller. These actions have to be stored in\na finite memory space.  Therefore, it is important to utilize the finite\nmemory space as efficiently as\npossible~\\cite{Bull2015, Liang2015, ngu2016rul, Xie2014d, Zhang2015timeout}.\nIn the dynamic timeout approach~\\cite{ZhuFan2015},\nthe SDN controller tracks the TCAM occupancy levels in the switches and adjusts\ntimeout durations accordingly.\nHowever, a shortcoming of such techniques\nis that the bookkeeping processes at the SDN controllers can become\ncumbersome for a large network.  Therefore, autonomous\ntimeout management techniques that are implemented at the hypervisors\ncan reduce the controller processing load and are\nan important future research direction.\n\n\\subsection{Energy Efficiency}  \\label{ene_eff:sec}\nThe separation of the control plane from the data plane\nand the global network perspective are unique advantages of SDN for\n improving the energy efficiency of networks,\nwhich is an important goal~\\cite{tuc2011gre,zha2010ene}.\n\n\\subsubsection{Power-saving Application Controller}\nJi et al.~\\cite{Ji2014} have proposed an all optical\nenergy-efficient network centered around an application\ncontroller~\\cite{yan2013mul,yan2015per} that monitors power consumption\ncharacteristics and enforces power savings policies.\nJi et al. first\nintroduce energy-efficient variations of Digital-to-Analog\nConverters (DACs) and wavelength selective ROADMs as components for\ntheir energy-efficient network. Second, Jie et al. introduce\nan energy-efficient switch architecture that consists of multiple\nparallel switching planes, whereby each plane consists of three\nstages with optical burst switching employed in the second (central)\nswitching stage.\nThird, Jie et al. detail a multilevel SDN based control architecture\nfor the network built from the introduced components and switch.\nThe control structure accommodates multiple networks domains,\nwhereby each network domain can involve multiple switching\ntechnologies, such as time-based and frequency-based optical switching.\nAll controllers for the various domains and technologies are placed\nunder the control of an application controller.\nDedicated power monitors that are distributed throughout the network\nupdate the SDN based application controller about the energy consumption\ncharacteristics of each network node.\nBased on the received energy consumption updates,\nthe application controller executes power-saving strategies.\nThe resulting control actions are signalled by the application\ncontroller to the various controllers for the different network domains\nand technologies.\nAn extension of this multi-level architecture to cloud-based\nradio access networks has been examined in~\\cite{yan2016per}.\n\n\\subsubsection{Energy-Saving Routing}\nTego et al.~\\cite{Tego2014} have proposed an energy-saving\napplication that switches off under-utilized GbE network links.\nSpecifically, Tego et al. proposed two methods: Fixed Upper Fixed\nLower (FUFL) and Dynamic Upper and Fixed Lower (DLFU). In FUFL, the\nIP routing and the connectivity of the logical topology are\n\\textit{fixed}. The utilization of physical GbE links (whereby\nmultiple parallel physical links form a logical link) is compared\nwith a threshold to determine whether to switch off or on individual\nphysical links (that support a given logical link). The traffic on a\nphysical link that is about to be switched off is rerouted on a\nparallel physical GbE link (within the same logical link). In\ncontrast, in the DLFU approach, the energy saving application\nmonitors the load levels on the virtual links. If the load level on\na given virtual link falls below a threshold value, then the virtual\nlink topology is reconfigured to eliminate the virtual link with the\nlow load. A general pitfall of such link switch-off techniques is\nthat energy savings may be achieved at the expense of deteriorating\nQoS. The QoS should therefore be closely monitored when switching\noff links and re-routing flows.\n\nA similar SDN based routing strategy that strives to save energy while\npreserving the QoS has been examined in the context of a GMPLS\noptical networks in~\\cite{Wang2015}.\nMultipath routing optimizing applications that strive to save\nenergy in an SDN based transport optical network have been presented\nin~\\cite{Yevsieieva2015}.\nA similar SDN based optimization approach\nfor reducing the energy consumption in\ndata centers has been examined by Yoon et al.~\\cite{Yoon2015}.\nYoon et al. formulated a mixed integer linear program that models the switches\nand hosts as queues. Essentially, the optimization decides on the\nswitches and hosts that could be turned off.\nAs the problem is NP-hard,\nannealing algorithms are examined.\nSimulations indicate that energy savings of more than 80\\% are possible for\nlow data center utilization rates, while the energy savings decrease to less\nthan 40\\% for high data center utilization rates.\nTraffic balancing in the metro optical access networks through the\nSDN based reconfiguration of optical subscriber units\nin a TWDM-PON systems for energy\nsavings has been additionally demonstrated in~\\cite{val2015exp}.\n\n\\subsection{Failure Recovery and Restoration}  \\label{app_fail_rec:sec}\n\\subsubsection{Network Reprovisioning}\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=3.5in]{finman_sdon\/fig17.pdf}\n    \\caption{Illustration of application layer\n        for disaster aware networking~\\cite{sav2015bac}.}\n    \\label{fig_app_reprov}\n\\end{figure}\nNetwork disruptions can occur due to various natural and\/or man-made\nfactors.\nNetwork resource reprovisioning is a process to\nchange the network configurations, e.g., the network topology and routes,\nto recover from failures.\nA Backup Reprovisioning with Path Protection (BRPP), based on SDN for\noptical networks has been presented by Savas et al.~\\cite{sav2015bac}.\nAn SDN application framework as illustrated in\nFig.~\\ref{fig_app_reprov} was designed to support the\nreprovisioning with services, such\nas provisioning the new connections, risk assessment, as well as service level\nand backup management.\nWhen new requests are received by the BRPP application\nframework, the statistics module evaluates the network state to find the\nprimary path and a link-disjoint backup path.  The computed backup\npaths are stored as logical links without being\nprovisioned on the physical network. The logical backup module manages\nand recalculates the logical links when a new backup path cannot be\naccommodated or to optimize the existing backup paths (e.g., minimize the\nbackup path distance). Savas et al.~introduce a degraded backup path mechanism\nthat reserves not the full, but a lower (degraded) transmission capacity\non the backup paths, so as to accommodate more requests.\nEmulations of the proposed mechanisms indicate improved\nnetwork utilization while effectively provisioning the backup paths\nfor restoring the network after network failures.\n\nAs a part of DARPA's core\noptical networks CORONET project, a non-SDN based Robust Optical Layer\nEnd-to-end X-connection (ROLEX)\nprotocol has been demonstrated and presented along with the lessons\nlearned~\\cite{VonLehmen2015}.\nROLEX is a distributed protocol for failure recovery\nwhich requires a considerable amount of signaling between nodes\nfor the distributed management.\nTherefore to avoid the pitfall of excessive signalling,\nit may be worthwhile to examine a ROLEX version with\ncentralized SDN control in future research\nto reduce the recovery time and\nsignaling overhead, as well as the costs of restored\npaths while ensuring the user QoS.\n\n\\subsubsection{Restoration Processing}\nDuring a restoration, the network control plane simultaneously triggers backup\nprovisioning of all disrupted paths.\nIn GMPLS restoration, along with signal flooding,\nthere can be contention of signal messages at the network nodes.\nContentions may arise due to spectrum conflicts of the lightpath,\nor node-configuration overrides, i.e., a new configuration request arrives\nwhile a preceding reconfiguration is under way.\nGiorgetti et al.~\\cite{Giorgetti2015} have proposed\ndynamic restoration in the elastic optical\nnetwork to avoid signaling contention in SDN (i.e., of OpenFlow messages).\nTwo SDN restoration mechanisms were presented:\n$(i)$ the independent restoration scheme (SDN-ind),\nand $(ii)$ the bundle restoration scheme (SDN-bund).\nIn SDN-ind, the controller triggers simultaneous independent\nflow modification (Flow-Mod) messages for each backup path\nto the switches involved in the reconfigurations.\nDuring contention, switches enqueue the multiple received Flow-Mod messages\nand process them sequentially. Although SDN-ind achieves reduced\nrecovery time as compared to non-SDN GMPLS, the waiting of messages in\nthe queue incurs a delay. In SDN-bund,\nthe backup path reconfigurations are bundled into a single message,\ni.e., a Bundle Flow-Mod message, and sent to each involved switch.\nEach switch then configures the flow modifications in\none reconfiguration, eliminating the delay incurred by the queuing of\nFlow-Mod messages. A similar OpenFlow enabled restoration\nin Elastic Optical Networks (EONs) has been studied in~\\cite{Liu2015d}.\n\n\\subsubsection{Reconfiguration}\n\\begin{figure}[t!]\n    \\centering\n    \\vspace{0cm}\n    \\includegraphics[width=3in]{finman_sdon\/fig18.pdf}\n    \\vspace{0cm}\n    \\caption{Illustration of Application-Based Network Operation (ABNO)\n         architecture~\\cite{Aguado2016}.}\n    \\label{fig_app_abno}\n\\end{figure}\nAguado et al.~\\cite{Aguado2016} have demonstrated\na failure recovery mechanism as part of the EU FP7 STRAUSS project with\ndynamic virtual reconfigurations using SDN.\nThey considered multidomain hypervisors and\ndomain-specific controllers to virtualize the multidomain networks.\nThe Application-Based Network Operations (ABNO) framework\nillustrated in Fig.~\\ref{fig_app_abno} enables network automation\nand programmability.\nABNO can compute end-to-end optical paths and delegate the\nconfigurations to lower layer domain SDN controllers.\nRequirements for fast recovery from network failures\nwould be in the order of tens of milliseconds, which is challenging to\nachieve in large scale networks. ABNO reduces the recovery\ntimes by pre-computing the backup connections after the first failure,\nwhile the Operation, Administration and Maintenance (OAM)\nmodule~\\cite{Paolucci2015}\ncommunicates with the ABNO controller to configure the new end-to-end\nconnections in response to a failure alarm. Failure alarms are triggered\nby the domain SDN controllers monitoring the traffic via the optical\npower meters when power is below $-20$~dBm.\nIn order to ensure survivability, an adaptive survivability scheme that\ntakes routing as well as spectrum assignment and modulation into\nconsideration has been explored in~\\cite{aib2016sof}.\n\nA similar design for end-to-end protection and failure\nrecovery has been demonstrated by Slyne et al.~\\cite{SlKMPR14} for a long-reach\n(LR) PON. LR-PON failures are highly\nlikely due to physical breaks in the long feeder fibers. Along with the\nhigh impact of connectivity break down or degraded service,\nphysical restoration time can be very long. Therefore, 1:1 protection for\nLR-PONs based on SDN has been proposed, where primary and secondary (backup)\nOLTs are used without traffic duplication.\nMore specifically, Slyne et al. have\ndevised and demonstrated an OpenFlow-Relay located at the switching unit.\nThe OpenFlow-Relay\ndetects and reports a failure along with fast updating of forwarding rules.\nExperimental demonstration show the backup OLT carrying protected traffic\nwithin $7.2$ ms after a failure event.\n\nAn experimental demonstration utilizing multiple paths in optical\ntransport networks for failure recovery has been discussed by\nKim et al.~\\cite{Kim2015}.\nKim et al. have used commercial grade IP WDM network equipment and\nimplemented multipath TCP\nin an SDN framework to emulate inter-DC communication.\nThey developed an SDN application, consisting of an\ncross-layer service manager module and a\ncross-layer multipath transport module to\nreconfigure the optical paths\nfor the recovery from connection impairments.\nTheir evaluations show increased bandwidth\nutilization and reduced cost while being resilient to network impairments\nas the cross-layer multipath transport module does not reserve the backup\npath on the transport network.\n\n\\subsubsection{Hierarchical Survivability}\nNetworks can be made survivable by introducing resource redundancy.\nHowever, the cost of the network increases with increased redundancy.\nZhang et al.~\\cite{ZhangSong2014} have demonstrated a highly survivable\nIP-Optical multilayered transport network.\nHierarchal controllers are placed for multilayer\nresource provisioning. Optical nodes are controlled by\nTransport Controllers (TCs), while higher\nlayers (IP) are controlled by unified controllers (UCs).\nThe UCs communicate with the TCs\nto optimize the routes based on cross-layer information. If a fiber causes a\nservice disruption, TCs may directly set up alternate routes or\nask the UCs for optimized routes. A pitfall of such\nhierarchical control techniques can be long restoration times. However,\nthe cross layer restorations\ncan recover from high degrees of failures, such as\nmultipoint and concurrent failures.\n\n\n\\subsubsection{Robust Power Grid}\nThe lack of a reliable communication infrastructure for power grid\nmanagement was one the many reasons for the widespread blackout in\nthe Northeastern U.S.A. in the year 2003, which affected the lives\nof 50 million people~\\cite{Parandehgheibi2014}. Since then building\na reliable communication infrastructure for the power grid has\nbecome an important priority. Rastegarfar et\nal.~\\cite{Rastegarfar2016} have proposed a communication\ninfrastructure that is focused on monitoring and can react to and recover from\nfailures so as to reliably support power grid applications.\nMore specifically, their\narchitecture was built on SDN based optical networking for\nimplementing robust power grid control applications. Control and\ninfrastructure in the SDN based power grid management exhibits an\ninterdependency i.e., the physical fiber relies on the control plane\nfor its operations and the logical control plane relies on the\nsame physical fiber for its signalling communications. Therefore,\nthey only focus on optical protection switching instead of IP layer\nprotection, for the resilience of the SDN control. Cascaded failure\nmechanisms were modeled and simulated for two geographical\ntopologies (U.S. and E.U.). In addition, the impacts of cascaded\nfailures were studied for two scenarios $(i)$ static optical layer\n(static OL), and $(ii)$ dynamic optical layer (dynamic OL). Results\nfor a static OL illustrated that the failure cascades are persistent\nand are closely dependent on the network topology. However, for a\ndynamic OL (i.e., with reconfiguration of the physical layer),\nfailure cascades were suppressed by an average of 73\\%.\n\n\\subsection{Application Layer: Summary and Discussion}\nThe SDON QoS application studies have mainly examined traffic and\nnetwork management mechanisms that are supported through the\nOpenFlow protocol and the central SDN controller.\nThe studied SDON QoS applications are structurally very similar in\nthat the traffic conditions or\nnetwork states (e.g., congestion levels) are probed\nor monitored by the central SDN controller.\nThe centralized knowledge of the traffic and network is then utilized\nto allocate or configure resources, such as DC resources in~\\cite{ZhZZ13},\napplication bandwidths in~\\cite{Li2014}, and\ntopology configurations or routes\nin~\\cite{WetteKarl2013,Tariq2015,SgPCVC13,ChangLi2015}.\nFuture research\n on SDON QoS needs to further optimize the interactions of the\ncontroller with the network applications and data plane\nto quickly and correctly react to changing user demands and\nnetwork conditions, so as to assure consistent QoS.\nThe specific characteristics and requirements\nof video streaming applications have\nbeen considered in the few studies on video\nQoS~\\cite{chi2015app,Chitimalla2015,Li2014video}.\nFuture SDON QoS research should consider a wider range of\nspecific prominent application traffic types with\nspecific characteristics and requirements, e.g., Voice over IP (VoIP)\ntraffic has relatively low bit rate requirements, but requires low\nend-to-end latency.\n\n\\begin{figure*}[t!]\n\t\\centering\n\t\\includegraphics[width=5in]{finman_sdon\/fig19.pdf}\n\t\\caption{Illustration of SDN orchestration of multilayer networking.}\n\t\\label{fig_control_multilay}\n\\end{figure*}\nVery few studies have considered security and access control for SDONs.\nThe thorough study of the broad topic area of security and privacy is\nan important future research direction in SDONs, as outlined in\nSection~\\ref{futworksec:sec}\nEnergy efficiency is similarly a highly important topic\nwithin the SDON research area that has received relatively little\nattention so far and presents overarching research challenges,\nsee Section~\\ref{futworkene:sec}.\n\nOne common theme of the SDON application layer studies focused\non failure recovery and restoration has been to exploit\nthe global perspective of the SDN control.\nThe global perspective has been exploited for\nfor improved planning of the recovery and\nrestoration~\\cite{sav2015bac,Aguado2016,ZhangSong2014} as well as\nfor improved coordination of the execution of the restoration\nprocesses~\\cite{Giorgetti2015,Liu2015d}.\nGenerally, the existing failure recovery and restoration studies\nhave focused on network (routing) domain that is owned by a particular\norganizational entity. Future research should seek to examine the\ntradeoffs when exploiting the\nglobal perspective of orchestration of multiple routing domains, i.e.,\nthe failure recovery and restoration techniques surveyed in this section\ncould be combined with the multidomain orchestration techniques\nsurveyed in Section~\\ref{orch:sec}.\nOne concrete example of multidomain orchestration could be to coordinate\nthe specific LR-PON access network protection and\nfailure recovery~\\cite{SlKMPR14}\nwith protection and recovery techniques for metropolitan and core\nnetwork domains, e.g.,~\\cite{sav2015bac,Aguado2016,Kim2015,ZhangSong2014},\nfor improved end-to-end protection and recovery.\n\n\\section{\\MakeUppercase{Orchestration}}  \\label{orch:sec}\nAs introduced in Section~\\ref{intro:orch:sec}, orchestration\naccomplishes higher layer abstract coordination of network services\nand operations. In the context of SDONs, orchestration has mainly\nbeen studied in support of multilayer networking.\nMultilayer networking in the context of SDN and network virtualization\ngenerally refers to networking across multiple network layers\nand their respective technologies, such as\nIP, MPLS, and WDM, in combination with networking across multiple\nrouting domains~\\cite{LeZGF11,leo2003vir,rui2011sur,tou2001net,vig2005mul}.\nThe concept of multilayer networking is generally an abstraction of\nproviding network services with multiple networking layers (technologies) and\nmultiple routing domains.\nThe different network layers and their technologies are sometimes classified into\nLayer~0 (e.g., fiber-switch capable), Layer~1 (e.g., lambda switching\ncapable), Layer~1.5 (e.g., TDM SONET\/SDH), Layer~2 (e.g., Ethernet),\nLayer~2.5 (e.g., packet switching capable using MPLS), and Layer~3\n(e.g., packet switching capable using IP routing)~\\cite{YoMKN14}.\nRouting domains are also commonly referred to as\nnetwork domains, routing areas, or levels~\\cite{LeZGF11}.\n\nThe recent multilayer\nnetworking review article~\\cite{LeZGF11} has introduced a range of capability\nplanes to represent the grouping of related functionalities for a\ngiven networking technology. The capability planes include the data\nplane for transmitting and switching data. The control plane and the\nmanagement plane directly interact with the data plane for\ncontrolling and provisioning data plane services as well as for\ntrouble shooting and monitoring the data plane. Furthermore, an\nauthentication and authorization plane, a service plane, and an\napplication plane have been introduced for providing network\nservices to users.\n\nMultilayer networking can involve vertical layering or\nhorizontal layering~\\cite{LeZGF11}, as illustrated in\nFig.~\\ref{fig_control_multilay}. In vertical layering, a given\nlayer, e.g., the routing layer, which may employ a\nparticular technology, e.g., the Internet Protocol (IP),\nuses another (underlying) layer, e.g., the Wavelength\nDivision Multiplexing (WDM) circuit switching layer, to provide\nservices to higher layers. In horizontal layering, services are\nprovided by ``stitching'' together a service path across multiple\nrouting domains.\n\nSDN provides a convenient control framework for these\nflexible multilayer networks~\\cite{LeZGF11}. Several research\nnetworks, such as ESnet, Internet2, GEANT, Science DMZ\n(Demilitarized Zone) have experimented with these multilayer\nnetworking concepts~\\cite{KiSTP13,RoMSS14}.\nIn particular, SDN based multilayer network\narchitectures, e.g.,~\\cite{woe2013sdn, iiz2016mul, mun2016nee},\nare formed by conjoining\nthe layered technology regions\n$(i)$ in vertical fashion i.e., multiple technology layers internetwork\nwithin a single domain, or\n$(ii)$ in horizontal layering fashion across multiple domains,\ni.e., technology layers internetwork across distinct domains.\nHorizontal multilayer networking can be viewed as a generalization of\nvertical multilayer networking in that the horizontal networking\nmay involve the same or different (or even multiple) layers in the\ndistinct domains. As illustrated in\nFig.~\\ref{fig_control_multilay}, the formed SDN based multilayer network\narchitecture is controlled by an SDN orchestrator.\n\\begin{figure*}[t!]\n\\footnotesize\n\\setlength{\\unitlength}{0.10in}\n\\centering\n\\begin{picture}(40,33)\n\\put(13,33){\\textbf{Orchestration, Sec.~\\ref{orch:sec}}}\n\\put(18,30){\\line(0,1){2}}\n\\put(0,30){\\line(1,0){34}}\n\\put(0,30){\\vector(0,-1){2}}\n\\put(-4,26.2){\\textbf{Multilayer Orch.}, Sec.~\\ref{mullayorch:sec} }\n\\put(0,25){\\line(0,-1){2}}\n\n\\put(25,23){\\line(1,0){20}}\n\\put(27,26.2){\\textbf{Multidomain Orch.}, Sec.~\\ref{muldomorch:sec}}\n\\put(34,25){\\line(0,-1){2}}\n\n\n\\put(-8,23){\\line(1,0){18}}\n\\put(-10,23){\\vector(0,-1){2}}\n\\put(10,23){\\vector(0,-1){11}}\n\n\\put(-10,20){\\makebox(0,0)[lt]{\\shortstack[l]{\t\t\t\n\\textbf{Frameworks}, Sec.~\\ref{mullayorchfr:sec} \\\\ \\\\\nHier. Multilay. Ctl.~\\cite{Felix2014,ger2013dem,sal2013inf,sun2014des,ShZBLT12}\\\\\nAppl. Centric Orch.~\\cite{Gerstel2015}\n}}}\n\n\\put(2,10){\\makebox(0,0)[lt]{\\shortstack[l]{\t\t\t\n\\textbf{Appl.-Specific Orch.}, \\\\ Sec.~\\ref{applorch:sec} \\\\ \\\\\nFailure Rec.~\\cite{Khaddam2015} \\\\\nRes. Util.~\\cite{Liu2015e} \\\\\nVirt. Opt. Netws.~\\cite{vil2015net}\n}}}\t\t\t\t\n\\put(34,30){\\vector(0,-1){2}}\n\\put(25,23){\\vector(0,-1){2}}\n\\put(20,20){\\makebox(0,0)[lt]{\\shortstack[l]{\n\\textbf{General Netw.}, Sec.~\\ref{muldomorchgen:sec} \\\\ \\\\\nOpt. Multidom. Multitechn.~\\cite{YoMKN14} \\\\\nHier. Multidom. Ctl.~\\cite{Jing2015} \\\\\nIDP~\\cite{Zhu2015} \\\\\nMultidom. Net. Hyperv.~\\cite{vil2015mul} \\\\\nABNO~\\cite{Munoz2015}\n}}}\t\t\t\t\t\t\t\n\\put(45,23){\\vector(0,-1){11}}\t\t\n\\put(30,10){\\makebox(0,0)[lt]{\\shortstack[l]{\t\t\t\n\\textbf{Data Center Orch.}, Sec.~\\ref{muldomorchdc:sec} \\\\ \\\\\nControl Arch.~\\cite{Liu2015b, may2016sdn} \\\\\nH-PCE~\\cite{cas2015sdn} \\\\\nVirt.-SDN Ctl.~\\cite{mun2015int,Vilalta2016}\n}}}\n\\end{picture}\t\n\\caption{Classification of SDON orchestration studies.}\n\\label{orch_class:fig}\n\\end{figure*}\nAs illustrated in Fig.~\\ref{orch_class:fig} we organize\nthe SDON orchestration studies according to their focus into studies\nthat primarily address the orchestration of vertical multilayer\n(multitechnology) networking,\ni.e., the vertical networking across multiple layers (that typically\nimplement different technologies) within a given domain,\nand into studies that primarily\naddress the orchestration of horizontal multilayer (multidomain)\nnetworking, i.e., the horizontal networking across multiple\nrouting domains (which may possibly involve different or multiple\nvertical layers in the different domains).\nWe subclassify the vertical multilayer studies into general (vertical)\nmultilayer networking frameworks and studies focused on supporting specific\napplications through vertical multilayer networking.\nWe subclassify the multidomain (horizontal multilayer) networking studies into\nstudies on general network domains and studies focused on internetworking\nwith Data Center (DC) network domains.\n\n\n\\subsection{Multilayer Orchestration}  \\label{mullayorch:sec}\n\n\\subsubsection{Multilayer Orchestration Frameworks} \\label{mullayorchfr:sec}\n\n\\paragraph{Hierarchical Multilayer Control}\nFelix et al.~\\cite{Felix2014} presented an hierarchical\nSDN control mechanism for packet optical networks.\nMultilayer optimization techniques\nare employed at the SDN orchestrator\nto integrate the optical transport technology\nwith packet services by provisioning end-to-end Ethernet services.\nTwo aspects are investigated, namely\n$(i)$ bandwidth optimization for the optical transport services,\nand $(ii)$ congestion control for packet network services\nin an integrated packet optical network.\nMore specifically, the SDN controller initially allocates the\nminimum available bandwidth required\nfor the services and then dynamically scales\nallocations based on the availability.\nOptical-Virtual Private Networks (O-VPNs) are created\nover the physical transport network. Services are then mapped\nto O-VPNs based on class of service requirements.\nWhen congestion is detected for a service,\nthe SDN controller switches the service to another O-VPN, thus\nbalancing the traffic to maintain the required class of service.\n\nSimilar steps towards the orchestration of multilayer networks\nhave been taken within the OFELIA\nproject~\\cite{ger2013dem,sal2013inf,sun2014des}. Specifically,\nShirazipour et al.~\\cite{ShZBLT12} have explored\nextensions to OpenFlow version 1.1 actions to enable\nmultitechnology transport layers, including Ethernet transport and\noptical transport. The explorations of the extensions include\njustifications of the use of SDN in circuit-based transport\nnetworks.\n\n\\paragraph{Application Centric Orchestration}\nGerstel et al.~\\cite{Gerstel2015} proposed an application centric network\nservice provisioning approach based on multilayer orchestration.\nThis approach enables the network applications\nto directly interact with the physical layer\nresource allocations to achieve the desired\nservice requirements.\nApplication requirements for\na network service may include\nmaximum end-to-end latency, connection setup and hold times,\nfailure protection, as well as security and encryption.\nIn traditional IP networking, packets from multiple applications\nrequiring heterogeneous services\nare simply aggregated and sent over a common transport link (IP services).\nAs a result, network applications are typically\nassigned to a single (common) transport service within an optical link.\nConsider a failure recovery process with multiple available paths.\nIP networking typically selects the single path\nwith the least end-to-end delay.\nHowever, some applications may tolerate higher\nlatencies and therefore, the traffic can be split\nover multiple restoration paths achieving better traffic management.\nThe orchestrator needs to interact with multiple\nnetwork controllers operating across multiple (vertical) layers\nsupported by north\/south bound interfaces\nto achieve the application centric control.\nDynamic additions of new IP links are demonstrated\nto accommodate the requirements of multiple application services\nwith multiple IP links when\nthe load on the existing IP link was increased.\n\n\\subsubsection{Application-specific Orchestration}  \\label{applorch:sec}\n\\paragraph{Failure Recovery}\nGenerally, network CapEx and OpEx increase as more protection against\nnetwork failures is added.\nKhaddam et al.~\\cite{Khaddam2015} propose\nan SDN based integration of multiple layers, such as WDM and IP, in a failure\nrecovery mechanism to improve the utilization\n(i.e., to eventually reduce CapEx and OpEx while maintaining\nhigh protection levels).\nAn observation study was conducted\nover a five year period to understand the impact of\nnetwork failures on the real deployment\nof backbone networks.\nResults showed $75$ distinct failures following a Pareto distribution,\nin which, $48\\%$ of the total deployed capacity was affected\nby the top (i.e., the highest impact) $20\\%$ of the failures.\nAnd, $10\\%$ of the total deployed\ncapacity was impacted by the top two failure instances.\nThese results emphasize the significance of backup\ncapacities in the optical links for restoration processes.\nHowever, attaining the optimal protection capacities while\nachieving a high utilization of the optical links is challenging.\nA failure recovery mechanism is proposed based\non a ``hybrid'' (i.e., combination of optical transport and IP)\nmultilayer optimization.\nThe hybrid mechanism improved the optical link utilization up to 50~\\%.\nSpecifically, 30~\\% increase of the transport\ncapacity utilization is achieved by dynamically reusing the remainder\ncapacities in the optical links, i.e.,\nthe capacity reserved for failure recoveries.\nThe multilayer optimization technique was validated on an experimental\ntestbed utilizing central path-computation (PCE)~\\cite{rfc5440}\nwithin the SDN framework.\nExperimental verification of failure\nrecovery mechanism resulted in recovery times\non the order of sub-seconds for MPLS restorations and\nseveral seconds for optical WSON restorations.\n\n\\paragraph{Resource Utilization}\nLiu et al.~\\cite{Liu2015e} proposed a method to improve\nresource utilization and to reduce transmission latencies\nthrough the processes of virtualization and service abstraction.\nA centralized SDN control implements the service abstraction\nlayer (to enable SDN orchestrations) in order to integrate the\nnetwork topology management (across both IP and WDM),\nand the spectrum resource allocation in a single control platform.\nThe SDN orchestrator also achieves dynamic and simultaneous\nconnection establishment across both IP and OTN layers\nreducing the transmission latencies. The control plane design is split\nbetween local (child) and root (parent) controllers.\nThe local controller realizes the label switched paths on the optical nodes\nwhile the root controller realizes the forwarding rules\nfor realizing the IP layer.\nExperimental evaluation of average transfer time measurements\nshowed IP layer\nlatencies on the order of several milliseconds, and\nseveral hundreds of milliseconds for the OTN latencies, validating\nthe feasibility of control plane unification for IP over\noptical transport networks.\n\n\\paragraph{Virtual Optical Networks (VONs)}\nVilalta et al.~\\cite{vil2015net} presented\ncontroller orchestration to integrate multiple transport network technologies,\nsuch as IP and GMPLS. The proposed architectural framework devises\nVONs to enable the virtualization\nof the physical resources within each domain.\nVONs are managed by lower level physical controllers (PCs), which\nare hierarchically managed by an SDN network orchestrator (NO). Network\nVirtualization Controllers (NVC) are introduced (on top of the NO) to\nabstract the virtualized multilayers across multiple domains.\nEnd-to-end provisioning of VONs is facilitated through\nhierarchical control interaction over three levels, the\ncustomer controller, the NO\\&NVCs, and the PCs.\nAn experimental evaluation demonstrated average VON\nprovisioning delays on the order of\nseveral seconds (5~s and 10~s), validating the flexibility of\ndynamic VON deployments over the optical transport networks.\nLonger provisioning delays may impact the network\napplication requirements, such as failure recovery processes,\ncongestion control, and traffic engineering.\nGeneral pitfalls of such hierarchical structures are\nincreased control plane complexity, risk of controller failures, and\nmaintenance of reliable communication links between control plane entities.\n\n\\subsection{Multidomain Orchestration}  \\label{muldomorch:sec}\nLarge scale network deployments typically involve multiple domains,\nwhich have often heterogeneous layer technologies.\nAchieve high utilization of the networking resources\nwhile provisioning end-to-end network paths and services across multiple\ndomains and their respective layers and respective technologies is\nhighly challenging~\\cite{Mayoral2015,Munoz2015b,Yu2015a}.\nMultidomain SDN orchestration studies have sought to exploit\nthe unified SDN control plane to aid the resource-efficient\nprovisioning across the multiple domains.\n\n\\subsubsection{General Multidomain Networks}  \\label{muldomorchgen:sec}\n\n\\paragraph{Optical Multitechnologies Across Multiple Domains}\nOptical nodes are becoming\nincreasingly reconfigurable (e.g., through variable BVTs and\nOFDM transceivers, see Section~\\ref{sdninfra:sec}),\nadding flexibility to the switching elements. When a single\nend-to-end service\nestablishment is considered, it is more likely that a service is supported by\ndifferent optical technologies that operate across multiple domains.\nYoshida et al.~\\cite{YoMKN14} have demonstrated SDN based orchestration with\nemphasis on the physical interconnects between multiple domains and multiple\ntechnology specific controllers so as to realize end-to-end services.\nOpenFlow capabilities have been extended for fixed-length variable capacity\noptical packet switching~\\cite{Losada2015}.\nThat is, when an optical switch matches the label on an\nincoming optical packet, if a rule\nexists in the switch (flow entry in the table) for a specific label,\na defined action is performed on the optical packet by the switch.\nOtherwise, the optical packet is dropped\nand the controller is notified. Interconnects between optical packet\nswitching networks and elastic optical networks are enabled\nthrough a novel OPS-EON interface card.\nThe OPS-EON interface is designed as an extension\nto a reconfigurable, programmable\nand flexi-grid EON supporting the OpenFlow protocol.\nThe testbed implementation of OPS-EON interface cards demonstrated\nthe orchestration of multiple domain controllers and the reconfigurability\nof FL-VC OPS across multidomain, multilayer, multitechnology scenarios.\n\n\\paragraph{Hierarchical Multidomain Control}\nJing et al.~\\cite{Jing2015} have also examined\nthe integration of multiple optical transport\ntechnologies from to multiple\nvendors across multiple domains, focusing on\nthe control mechanisms across multiple domains.\nJing et al.~proposed hierarchical SDN orchestration with parent and\ndomain controllers.\nDomain controllers abstract the physical layer by\nvirtualizing the network resources.\nA Parent Controller (PC) encompasses a Connection Controller (CC) and\na Routing Controller (RC) to process the abstracted virtual network.\nWhen a new connection setup request is\nreceived by the PC, the RC (within the PC) evaluates the end-to-end\nrouting mechanisms and forwards the\ninformation to the CC.\nThe CC breaks the end-to-end routing information into shorter\nlink segments belonging to a domain.\nSegmented routes are then sent to the respective domain\ncontrollers for link provisioning over the physical infrastructures.\nThe proposed mechanism was experimentally verified on a testbed\nbuilt with the commercial OTN equipment.\n\n\\paragraph{Inter-Domain Protocol}\n\\begin{figure}[t!]\n\t\\centering\n\t\\vspace{0cm}\n\t\\includegraphics[width=4.2in]{finman_sdon\/fig21.pdf}\n\t\\vspace{0cm}\n\t\\caption{Inter-domain lightpath provisioning mechanism.}\n\n\t\\label{fig_multilay_idp}\n\\end{figure}\nZhu et al.~\\cite{Zhu2015} followed a different approach for the\nSDN multidomain control mechanisms by considering the\nflat arrangement of controllers as shown in Fig.~\\ref{fig_multilay_idp}.\nEach domain is autonomously managed by\nan SDN controller specific to the domain.\nAn Inter-Domain Protocol (IDP) was devised to establish the communication\nbetween domain specific controllers\nto coordinate the lightpath setup across multiple domains.\nZhu et al.~also proposed a Routing and\nSpectrum Allocation (RSA) algorithm for the end-to-end provisioning\nof services in the SD-EONs.\nThe distributed RSA algorithm operates on the domain specific controllers\nusing the IDP protocol. The RSA considers both transparent lightpath\nconnections, i.e., all-optical lightpath,\nand translucent lightpath connections, i.e.,\noptical-electrical-optical connections.\nThe benefit of such techniques is privacy, since\nthe domain specific policies and topology information are not\nshared among other network entities.\nNeighbor discovery is  independently conducted by the domain specific controller\nor can initially be configured.\nA domain appears as an abstracted virtual node to all other domain specific\ncontrollers. Each controller then assigns the shortest path\nrouting within a domain between its border nodes.\nAn experimental setup validating the\nproposed mechanism was demonstrated across\ngeographically-distributed domains in the USA and China.\n\n\\paragraph{Multidomain Network Hypervisors}\n\\begin{figure}[t!]\n\t\\centering\n\t\\includegraphics[width=3in]{finman_sdon\/fig22.pdf}\n\t\\caption{Illustration of multilevel virtualization.} \n\t\\label{fig_multilay_mnh}\n\\end{figure}\nVilalta et al.~\\cite{vil2015mul} presented a mechanism for virtualizing\nmultitechnology optical, multitenant networks.\nThe Multidomain Network Hypervisor (MNH) creates customer specific\nvirtual network slices managed by the customer specific SDN controllers\n(residing at the customers' locations) as illustrated\nin Fig.~\\ref{fig_multilay_mnh}.\nPhysical resources are managed by their domain specific\nphysical SDN controllers.\nThe MNH operates over the network orchestrator and\nphysical SDN controllers for provisioning VONs\non the physical infrastructures.\nThe MNHs abstracts both $(i)$ multiple optical transport technologies,\nsuch as optical packet switching and Elastic Optical Networks (EONs),\nand $(ii)$ multiple control domains, such as GMPLS and OpenFlow.\nExperimental assessments on a testbed achieved VON provisioning\nwithin a few seconds (5~s), and\ncontrol overhead delay on the order of several tens of milliseconds.\nRelated virtualization mechanisms for multidomain optical SDN networks\nwith end-to-end provisioning have been\ninvestigated in~\\cite{SzAEK14,vil2016hie}.\n\n\\paragraph{Application-Based Network Operations}\n\\begin{figure}[t!]\n\t\\centering\n\t\\includegraphics[width=3.5in]{finman_sdon\/fig23.pdf}\n\t\\caption{The application-based network operations (ABNO)~\\cite{Munoz2015}}\n\t\\label{fig_multilay_abno}\n\\end{figure}\nMu{\\~{n}}oz et al.~\\cite{Munoz2015}, have presented an SDN\norchestration mechanism based on the\napplication-based network operations (ABNO) framework, which\nis being defined by the IETF~\\cite{rfc7491}.\nThe ABNO based SDN orchestrator integrates\nOpenFlow and GMPLS in transport networks.\nTwo SDN orchestration designs have been presented: $(i)$ with centralized\nphysical network topology aware path computation\n(illustrated in Fig.~\\ref{fig_multilay_abno}), and\n$(ii)$ with topology abstraction and distributed path computation.\nIn the centralized design, OpenFlow and GMPLS controllers (lower level control)\nexpose the physical\ntopology information to the ABNO-orchestrator (higher level control).\nThe PCE in the ABNO-orchestrator has the global view of the network\nand can compute end-to-end paths with complete knowledge of the network.\nComputed paths are then provisioned through\nthe lower level controllers. The pitfalls of such centralized designs\nare $(i)$ computationally intensive path computations,\n$(ii)$ continuous updates of topology and traffic information,\nand $(iii)$ sharing of confidential network information and policies\nwith other network elements.\nTo reduce the computational load at the orchestrator,\nthe second design implements distributed path computation\nat the lower level controllers (instead of path computation at the\ncentralized orchestrator). However, such distributed mechanisms may\nlead to suboptimal solutions due to the limited network knowledge.\n\n\\subsubsection{Multidomain Data Center Orchestration} \\label{muldomorchdc:sec}\n\\paragraph{Control Architectures}\nGeographically distributed DCs are typically\ninterconnected by links traversing multiple domains.\nThe traversed domains may be homogeneous i.e., have the same type of network\ntechnology, e.g., OpenFlow based ROADMs,\nor may be heterogeneous, i.e., have different types of network\ntechnologies, e.g.,\nOpenFlow based ROADMs and GMPLS based WSON. The SDN control structures\nfor a multidomain network can be broadly classified into the categories of\n$(i)$ single SDN orchestrator\/controller, $(ii)$ multiple mesh SDN\ncontrollers, and\n$(iii)$ multiple hierarchical SDN controllers~\\cite{Liu2015b, may2016sdn}.\nThe single SDN orchestrator\/controller has to support heterogeneous SBIs\nin order to operate with multiple heterogeneous domains, e.g.,\nthe Path Computation Element Protocol (PCEP) for GMPLS network domains\nand the OpenFlow protocol for OpenFlow supported ROADMs.\nAlso, domain specific details, such as topology, as well as\nnetwork statistics and configurations, have to be exposed to an external\nentity, namely the single SDN orchestrator\/controller,\nraising privacy concerns. Furthermore, a single\ncontroller may result in scalability issues.\nMesh SDN control connects the domain-specific controllers side-by-side\nby extending the east\/west bound interfaces.\nAlthough mesh SDN control addresses the scalability and privacy issues,\nthe distributed nature of the control mechanisms may lead to sub-optimal\nsolutions.\nWith hierarchical SDN control, a logically centralized controller\n(parent SDN controller) is placed above the domain-specific controllers\n(child SDN controllers), extending the north\/south bound interfaces.\nDomain-specific controllers virtualize\nthe underlying networks inside their domains, exposing only the abstracted\nview of the domains to the parent controller, which addresses the privacy\nconcerns. Centralized\npath computation at the parent controller can achieve optimal solutions.\nMultiple hierarchical levels can address the scalability issues.\nThese advantages of hierarchal SDN control are achieved at the expense of\nan increased number of network entities,\nresulting in the operational complexities.\n\n\\paragraph{Hierarchical PCE}\n\\begin{figure}[t!]\n\t\\centering\n\t\\includegraphics[width=3.5in]{finman_sdon\/fig24.pdf}\n  \\caption{Illustration of SDN orchestration based on Hierarchical\n  Path Computation Element (H-PCE)~\\cite{cas2015sdn}.}\n\t\\label{fig_multilay_HPCE}\n\\end{figure}\nCasellas et al.~\\cite{cas2015sdn} considered\nDC connectivities involving both intra-DC and inter-DC communications.\nIntra-DC communications enabled through OpenFlow networks\nare supported by an OpenFlow controller.\nThe inter-DC communications\nare enabled by optical transport\nnetworks involving more complex control,\nsuch as GMPLS, as illustrated in Fig.~\\ref{fig_multilay_HPCE}. To\nachieve the desired SDN benefits of flexibility and scalability,\na common centralized control platform spanning across heterogeneous\ncontrol domains is proposed.\nMore specifically, an Hierarchical PCE (H-PCE) aggregates PCE states\nfrom multiple domains.\nThe end-to-end path setup between DCs is orchestrated by a parent-PCE (pPCE)\nelement, while the paths are provisioned\nby the child-PCEs (cPCEs) on the physical resources, i.e.,\nthe OpenFlow and GMPLS domains.\nThe proposed mechanism utilizes existing protocol interfaces,\nsuch as BGP-LS and PCEP, which are extended with OpenFlow to support the H-PCE.\n\n\\paragraph{Virtual-SDN Control}\nMu{\\~n}oz et al.~\\cite{mun2015int,Vilalta2016} proposed a\nmechanism to virtualize the SDN control functions\nin a DC\/cloud by integrating SDN with Network Function Virtualization (NFV).\nIn the considered context, NFV refers to realizing network functions\nby software modules running on\ngeneric computing hardware inside a DC; these network functions were\nconventionally implemented on specialized hardware modules.\nThe orchestration of Virtual Network Functions (VNFs)\nis enabled by an integrated SDN and NFV management which\ndynamically instantiates virtual SDN controllers.\nThe virtual SDN controllers\ncontrol the Virtual Tenant Networks (VTNs), i.e., virtual\nmultidomain and multitechnology networks.\nMultiple VNFs running on a Virtual Machine (VM)\nin a DC are managed by a VNF manger.\nA virtual SDN controller is responsible for creating, managing, and tearing\ndown the VNF achieving the flexibility in the control plane management\nof the multilayer and the multidomain networks.\nAdditionally, as an extension to the proposed mechanism,\nthe virtualization of the control functions of the\nLTE Evolved Packet Core (EPC) has been discussed in~\\cite{mar2016int}.\n\n\\subsection{Orchestration: Summary and Discussion}\nRelatively few SDN orchestration studies to date have focused on\nvertical multilayer networking within a given domain. The few studies\nhave developed two general orchestration frameworks and have examined\na few orchestration strategies for some specific applications.\nMore specifically, one orchestration framework has\nfocused on optimal bandwidth allocation based mainly on\ncongestion~\\cite{Felix2014}, while the other\nframework has focused on exploiting application traffic tolerances for\ndelays for efficiently routing traffic~\\cite{Gerstel2015}.\nSDN orchestration of vertical multilayer optical networking is thus\nstill a relatively little explored area.\nFuture research can develop orchestration frameworks that accommodate\nthe specific optical communication technologies in the various layers and\nrigorously examine their performance-complexity tradeoffs.\nSimilarly, relatively few applications have been\nexamined to date in the application-specific orchestration\nstudies for vertical multilayer\nnetworking~\\cite{Khaddam2015,Liu2015e,vil2015net}.\nThe examination of the wide range of existing applications and\nany newly emerging network application in the context of\nSDN orchestrated vertical multilayer networking presents rich research\nopportunities.\nThe cross-layer perspective of the SDN orchestrator over a given\ndomain could, for instance, be exploited for strengthening security and\nprivacy mechanisms or for accommodating demanding real-time multimedia.\n\nRelatively more SDN orchestration studies to date have\nexamined multidomain networking than multilayer networking (within a single\ndomain). As the completed multidomain orchestration studies\nhave demonstrated, the SDN orchestration can help greatly in coordinating\ncomplex network management decisions across multiple distributed\nrouting domains.\nThe completed studies have illustrated the fundamental\ntradeoff between centralized decision making in a hierarchical\norchestration structure and distributed decision making in a flat\norchestration structure.\nIn particular, most studies have focused on hierarchical\nstructures~\\cite{Jing2015,cas2015sdn,mun2015int}, while only one study\nhas mainly focused on a flat orchestration structure~\\cite{Zhu2015}.\nIn the context of DC internetworking, the\nstudies~\\cite{Liu2015b, may2016sdn} have sought to bring out\nthe tradeoffs between these two structures by examining\na range of structures from centralized to distributed.\nWhile centralized orchestration can make decisions with\na wide knowledge horizon across the states in multiple domains,\ndistributed decision making preserves the privacy of network status\ninformation, reduces control traffic, and can make fast localized\ndecisions.\nFuture research needs to shed further light on these complex\ntradeoffs for a wide range of combinations of optical technologies employed\nin the various domains.\nThroughout, it will be critical to abstract and convey the key\ncharacteristics of optical physical layer components and switching nodes to\nthe overall orchestration protocols. Optimizing each abstraction\nstep as well as the overall orchestration and examining the various\nperformance tradeoffs are important future research directions.\n\n\\section{\\MakeUppercase{Open Challenges and Future SDON Research Directions}} \\label{sec:open}\nWe have outlined open challenges and future\nSoftware Defined Optical Network (SDON) research directions\nfor each sub-category of surveyed SDON studies in the Summary and\nDiscussion subsections in the preceding survey sections.  In this\nsection, we focus on the overall cross-cutting open challenges that\nspan across the preceding considered categories of SDON studies.\nThat is, we focus on open challenges and research directions\nthat span the vertical (inter-layer) and horizontal (inter-domain)\nSDON aspects. The vertical SDON aspects encompass the\nseamless integration of the various (vertical) layers of the\nSDON architecture; especially the optical layer, which is not considered\nin general SDN technology.\nThe horizontal SDON aspects include the integration of SDONs with\nexisting non-SDN optical networking elements, and the internetworking\nwith other domains, which may have  similar or different\nSDN architectures.\nA key challenge for SDON research is to enable the use of SDON concepts\nin operational real-time network infrastructures.\nImportantly, the SDON concepts need to demonstrate performance\ngains and cost reductions to be considered by network and service providers.\nTherefore, we cater some of the open challenges and\nfuture directions towards enabling and demonstrating the successful use\nof SDON in operational networks.\n\nThe SDON research and development effort to date\nhave resulted in insights for making the use of SDN in optical transport\nnetworks feasible and have demonstrated advantages of SDN based optical network\nmanagement.\nHowever, most network and service providers\ndepend on optical transport to integrate with multiple industries to\ncomplete the network infrastructure. Often, network and service providers\nstruggle to integrate hardware components and to provide accessible software\nmanagement to customers. For example, companies that develop hardware\noptical components do not always have a complete associated software stack\nfor the hardware components. Thus, network and service providers using the\nhardware optical components often have to maintain a software development team\nto integrate the various\nhardware components through software based management into their network,\nwhich is often a costly endeavor.\nThus, improving SDN technology so that it seamlessly integrates with\ncomponents of various industries\nand helps the integration of components from various industries\nis an essential underlying theme for future SDON research.\n\n\\subsection{Simplicity and Efficiency}   \\label{simpl_fut:sec}\nOptical network structures typically span heterogeneous devices ranging\nfrom the end user nodes and local area networks via ONUs and OLTs in the\naccess networks to edge routers and metro network nodes and on\nto backbone (core) network infrastructures.\nThese different devices often come from different vendors.\nThe heterogeneity of devices and their vendors often requires\nmanual configuration and maintenance of optical networks.\nMoreover, different communication technologies typically\nrequire the implementation of native functions that are specific\nto the communication technology characteristics, e.g., the transmission and\npropagation properties.\nBy centralizing the optical network control in an\nSDN controller, the SDN networking paradigm creates a unified view\nof the entire optical network.\nThe specific native functions for specific communication devices\ncan be migrated to the software layer and be implemented by a central node,\nrather than through manual node-by-node configurations.\nThe central node would typically be readily accessible and could reduce\nthe required physical accesses to distributed devices at their\non-site locations.\nThis centralization can simplify the network management and\nreduce operational expenditures.\nAn important challenge in this central management is the efficient\nSDN control of components from multiple vendors.\nDetailed vendor contract specifications of open-source middleware may be\nneeded to efficiently control components from different vendors.\n\nThe heterogeneity of devices may reduce the efficiency of\nnetwork infrastructures\ndue to the required multiple software and hardware modules for a\ncomplete networking solution. Future research should\ninvestigate efficient mechanisms for making complete networking solutions\navailable for specific use cases. For example, the use of\nSDON for an access network provider may require multiple SDN controllers\nco-located within the OLT to enable the control of the access network\ninfrastructure from one central location.  While the SDON studies reviewed in\nthis survey have led initial investigations of simple and dynamic\nnetwork management, future research needs to refine these management\nstrategies and optimize their operation across combinations of\nnetwork architecture structures and across various network\nprotocol layers.\nSimplicity is an essential part of this\nchallenge, since overly complicated solutions are generally not deployed\ndue to the risk of high expenditures.\n\n\\subsection{North Bound Interface} \\label{nbi:fut}\nThe NorthBound Interface (NBI) comprises the communication from the\ncontroller to the applications. This is an important area of\nfuture research as applications and their needs are generally\nthe driving force for deploying SDON infrastructures.\nAny application, such as video on demand, VoIP, file\ntransfer, or peer-to-peer networking, is applied from the NBI to the\nSDN controller which consequently conducts the necessary actions to implement\nthe service behaviors on the physical network infrastructure.\nApplications often require specific service behaviors that\nneed to be implemented on the overall network infrastructure.\nFor example,\napplications requiring high data rates and reliability, such as Netflix,\ndepend on data centers and the availability\nof data from servers with highly resilient failure protection mechanisms.\nThe associated management network needs to stack redundant devices as\nto safeguard against outages. Services are provided as\npolicies through the NBI to the SDN controller, which in turn generates flow\nrules for the switching devices. These flow rules can be\nprioritized based on the customer use cases.\nAn important challenge for future NBI research is to\nprovide a simple interface for a wide variety of\nservice deployments without vendor lock-in, as vendor lock-in\ngenerally drives costs up.\nAlso, new forms of communication to the controller, in\naddition to current techniques, such as\nREpresentational State Transfer (REST)~\\cite{REST15} and HTTP, should\nbe researched.\nMoreover, future research should develop an NBI framework that spans\nhorizontally across multiple controllers, so that service customers are not\nrestricted to using only a single controller.\n\nFuture research should examine control mechanisms that optimally\nexploit the central SDN control to provide simple and efficient\nmechanisms for automatic network management and dynamic service\ndeployment~\\cite{ZhYZ14}.  The NBI of SDONs is a\nchallenging facet of research and development because of the multitude\nof interfaces that need to be managed on the physical layer and\ntransport layer. Optical physical layer components and infrastructures\nrequire high capital and\noperational expenditures and their management is generally not associated with\nnetwork or service providers but rather with optical component\/infrastructure\nvendors. Future research should develop novel Application Program Interfaces\n(APIs) for optical layer components and infrastructures that facilitate\nSDN control and are amenable to efficient NBI communication.\nEssentially, the challenge of efficient NBI communication with the\nSDN controller should be\nconsidered when designing the APIs that interface with the\nphysical optical layer components and infrastructures.\n\nOne specific strategy for simplifying network management and operation\ncould be to explore the grouping of control policies of similar\nservice applications, e.g., applications with similar QoS requirements.\nThe grouping can reduce the number of control policies at the\nexpense of slightly coarser granularity of the service offerings.\nThe emerging Intent-Based Networking (IBN) paradigm, which\ndrafts intents for services and policies,\ncan provide a specific avenue for simplifying\ndynamic automatic configuration and virtualization~\\cite{BlDM08,CoBR13}.\nCurrently network applications are deployed based on how the network\nshould behave for a specific action. For example, for inter domain routing,\nthe Border Gateway Protocol (BGP) is used, and the network gateways\nare configured to communicate with the BGP protocol.\nThis complicates the provisioning of\nservices that typically require multiple protocols and limits the\nflexibility of service provisioning. With IBN, the application gives an intent,\nfor example, transferring video across multiple domains.\nThis intent is then\nassociated with automated dynamic configurations of the network elements\nto communicate data over the domains using appropriate protocols.\nThe grouping of service policies, such as intents, can facilitate\neasy and dynamic service provisioning.\nIntent groups can be described in a graph to simplify the compilation of\nservice policies and to resolve conflicts~\\cite{PrLT15}.\n\n\\subsection{Reliability, Security, and Privacy} \\label{futworksec:sec}\nThe SDN paradigm is based on a centrally managed network.\nFaulty behaviors, security infringements, or failures of the\ncontrol would likely result in extensive disruptions and\nperformance losses that are exacerbated by the centralized nature of the\nSDN control. Instances of extensive disruptions and losses due to\nSDN control failures or infringements would likely reduce the trust in\nSDN deployments. Therefore, it is very important to ensure reliable network\noperation~\\cite{rak2016inf}\nand to provision for security and privacy of the communication.\nHence, reliability, security, and privacy are prominent SDON research\nchallenges. Security in SDON techniques is a fairly open research\narea, with only few published findings.\nAs a few reviewed studies (see Section~\\ref{app_fail_rec:sec}) have explored,\nthe central SDN control can facilitate reliable network service through speeding\nup failure recovery.\nThe central SDN control can continuously scan the network and\nthe status messages from the network devices.\nOr, the SDN control can\nredirect the status messages to a monitoring service that analyzes\nthe data network.  Security breaches can be controlled by broadcasting\nmessages from the controller to all affected devices to block traffic\nin a specific direction. Future research should refine these\nreliability functions to optimize automated fault and performance\ndiagnostics and reconfigurations for quick failure recovery.\n\nNetwork failures can either occur within the physical layer\ninfrastructure, or as errors within the higher protocol\nlayers, e.g., in the classical data link (L2), network (L3), of\ntransport (L4) layers.\nIn the context of SDONs, physical\nlayer failures present important future research opportunities.\nPhysical layer devices need to be carefully monitored by sending\nfeedback from the devices to the controller.\nThe research and development on communication between\nthe SDN controller and the network devices has mainly focused on sending\nflow rules to the network devices while feedback communicated from the devices\nto the controller has received relatively little attention.\nFor example, there are three types of OpenFlow messages, namely Packet-In,\nPacket-Out, and Flow-Mod. The Packet-In messages are\nsent from the OpenFlow switches to the controller,\nthe Packet-Out message is sent from\nthe controller to the device, and the Flow-Mod message is used to\nmodify and monitor\nthe flow rules in the flow table. Future research should examine extensions of\nthe Packet-In message to send specific status updates in support of\nnetwork and device failure monitoring to the controller.\nThese status messages could be monitored\nby a dedicated failure monitoring service.\nThe status update messages could be broadly defined to\ncover a wide range of network management aspects, including\nsystem health monitoring and network failure protection.\n\nA related future research direction is to secure configuration and\noperation of SDONs through trusted encryption and key management\nsystems~\\cite{ahm2015sec}. Moreover, mechanisms to\nensure the privacy of the communication should be explored.\nThe security and privacy mechanisms should strive to exploit the\nnatural immunity of optical transmission segments to electro-magnetic\ninterferences.\n\nIn summary, security and privacy of SDON communication are largely open\nresearch areas.\nThe optical physical layer infrastructure has traditionally not\nbeen controlled remotely, which in general reduces the occurrences\nof security breaches.\nHowever, centralized SDN management and control increase the risk of\nsecurity breaches, requiring extensive research on SDON security, so\nas to reap the benefits of centralized SDN management and control in a\nsecure manner.\n\n\\subsection{Scalability} \\label{scalability:fut}\nOptical networks are expensive and used for high-bandwidth\nservices, such as long-distance network access and data center interconnections.\nOptical network infrastructures either\nspan long distances between multiple geographically distributed locations,\nor could be short-distance incremental additions (interconnects)\nof computing devices. Scalability in multiple dimensions\nis therefore an important aspect for future SDON research.\nFor example, a\nmyriad of tiny end devices need to be provided with network access in\nthe emerging Internet of Things (IoT) paradigm~\\cite{wan2015nov}.\nThe IoT requires access network architectures and\nprotocols to scale vertically (across protocol layers and technologies)\nand horizontally (across network domains).\nAt the same time, the ongoing growth of\nmultimedia services requires data centers to scale up optical network bandwidths\nto maintain the quality of experience of the multimedia services.\nBroadly speaking, scalability includes in the vertical\ndimension the support for multiple network devices and technologies.\nScalability in\nthe horizontal direction includes the communication between\na large number of different domains as well as support for existing non-SDON\ninfrastructures.\n\nA specific scalability challenge arising with SDN infrastructure is that the\nscalability of the control plane (OpenFlow protocol signalling)\ncommunication\nand the scalability of the data plane communication which transports\nthe data plane flows\nneed to be jointly considered. For example, the Openflow protocol~1.4\ncurrently supports 34 Flow-Mod messages~\\cite{OF2016}, which can\ncommunicate between the network devices and the controller. This\nnumber limits the functionality of the SBI communication. Recent studies\nhave explored a protocol-agnostic\napproach \\cite{HuLi2015, bosshart2014p4}, which is a data plane\nprotocol that extends the use of multiple protocols for\ncommunication between the control plane and data plane.\nThe protocol-agnostic approach resolves\nthe challenges faced by OpenFlow and, in general, any particular protocol.\nExploring this novel protocol-agnostic approach presents many new SDON\nresearch directions.\n\nScalability would also require SDN technology to overlay and scale\nover existing non-SDN infrastructures. Vendors provide support\nfor known non-SDN devices, but this area is still a challenge. There\nare no known protocols that could modify the flow tables of existing\npopularly described ``non-OpenFlow'' switches. In the case of optical\nnetworks, as SDN is still being incrementally deployed, the overlaying\nwith non-SDN infrastructure still requires significant attention.\nIdeally, the overlay mechanisms should ensure seamless integration and should\nscale with the growing deployment of SDN technologies while incurring only\nlow costs.\nOverall, scalability poses highly important future SDON research directions\nthat require economical solutions.\n\n\n\\subsection{Standardization}   \\label{std:sec}\nNetworking protocols have traditionally followed a uniform\nstandard system for all the\ncommunication across multiple domains. Standardization has helped vendors\nto provide products that work in and across different network infrastructures.\nIn order to ensure the compatible inter-operation of SDON components (both hardware and software) from a various vendors,\nkey aspects of the inter-operation protocols need to be standardized.\nTowards the standardization goal,\ncommunities, such as Open Networking Foundation (ONF), have created boards\nand committees to standardize protocols, such as OpenFlow.\nStandardization should ensure that SDON infrastructures can be flexibly\nconfigured and operated with components from various vendors.\nThe use of open-source\nsoftware can further facilitate the inter-operation.\nProprietary hardware and software components generally create vendor lock-in,\nwhich restricts the flexibility of network operation and reduces\nthe innovation of network and service providers.\n\nAs groundwork for standardization, it may be necessary to develop\nand optimize a common (or a small set) of SDON architectures\nand network protocol configurations that can serve as a basis\nfor standardization efforts.\nThe standardization process may involve a common platform that\nis built thorough the cooperation of multiple manufacturers.\nAnother thrust of standardization groundwork could be the development of\nopen-source software that supports SDON architectures.\nFor example, Openstack is a cloud based management framework that has been\nadopted and supported by multiple networking vendors. Such efforts should\nbe extended to SDONs in future work.\n\n\\subsection{Multilayer Networking} \\label{multilayer:fut}\nAs discussed in Section~\\ref{orch:sec}, multilayer networking\ninvolves vertical multilayer networking\nacross the vertical layers as well as horizontal multilayer (multidomain)\nnetworking across multiple domains.\nWe proceed to outline open challenges and\nfuture research directions for vertical multilayer networking in the context of\nSDON, which includes an optical physical layer, in this subsection.\nHorizontal multilayer (multidomain) networking is considered in\nSection~\\ref{multidomain:fut}.\n\nFor the vertical multilayer networking in a single domain,\nthe optical physical layer is the key distinguishing feature of\nSDONs compared to conventional SDN architectures for general IP networks.\nMost of the higher layers in SDONs have similar multilayer\nnetworking challenges as general IP networks.\nHowever, the optical physical layer\nrequires the provisioning of specific optical\ntransmission parameters, such as wavelengths and signal strengths.\nThese parameters are managed by optical devices, such as the OLT in PON\nnetworks. For SDON networks, so-called \\textit{optical orchestrators},\nwhich are commercially available, e.g., from ADVA Optical Networking,\nprovide a single interface to provision the optical layer parameters.\nWe illustrate this optical orchestrator layer in the context of an\nSDON multilayer network in the rightmost branch of Fig.~\\ref{fig_control_orch}.\nThe optical orchestrator resides\nabove the optical devices and below the SDN controller.\nThe optical orchestrator uses common SDN SBI interface protocols,\nsuch as OpenFlow,\nto communicate with the optical devices in the south-bound direction\nand with the controller in the north-bound direction.\n\nThe SDN controller in the control plane is responsible for the management of the\nSDN-enabled switches, potentially via an optical orchestrator.\nCommunicating over the SBI using different\nprotocols can be challenging for the controller.\nThis challenge can\nbe addressed by using south-bound renderers. South-bound renderers are\nAPIs that reside within the\ncontroller and provide a communication channel to any desired\nSBI protocol.  Most SDN controllers currently have an\nOpenFlow renderer to be able to communicate to Openflow network switches.\nBut there are also SNMP and NETCONF-based renderers, which\ncommunicate with traditional non-OpenFlow switches. This enables the\nexistence of hybrid networks with already existing switches.\nThe effective support of such hybrid networks, in conjunction with\nappropriate south-bound renderers and optical orchestrators, is an important\ndirection for future research.\n\n\\subsection{Multidomain Networks} \\label{multidomain:fut}\nA network domain usually belongs to a single organization that\nowns (i.e., financially supports and uses) the network domain.\nThe management of multidomain networking involves the\nimportant aspects of configuring the access control as well as the\nauthentication, authorization, and accounting. Efficient\nSDN control mechanisms for configuring these multidomain networking\naspects is an important direction for future research and development.\n\nMultidomain SDONs may also need novel routing\nalgorithm that enhance the capabilities of the currently used BGP\nprotocol. Multidomain research \\cite{phe2014dis} has now taken\ninterest in the Intent-Based Networking (NBI) paradigm for\nSDN control, where Intent-APIs can solve the\nproblems of spanning across multiple domains. For instance, the intent of an\napplication to transfer information across multiple domains is\ntranslated into service instances that access configurations between\ndomains that have been pre-configured based on contracts.\nCurrently, costly manual configurations between\ndomains are required for such applications.\nFuture research needs to develop concrete models for\nNBI based multidomain networking in SDONs.\n\n\\subsection{Fiber-Wireless (FiWi) Networking} \\label{wireless:fut}\nThe optical (fiber) and wireless network domains have\nmany differences.\nAt the physical layer, wireless networks are characterized\nby varying channel qualities, potentially high losses, and generally lower\ntransmission bit rates than optical fiber.\nWireless end nodes are typically mobile and may connect dynamically\nto wireless network domains.\nThe mobile wireless nodes are generally the end-nodes in a\nFiWi network and connect via intermediate optical nodes to\nthe Internet.\nDue to these different characteristics, the management of\nwireless networks with mobile end nodes is very different from the\nmanagement of optical network nodes.\nFor example, wireless access points should maintain their own\nrouting table to accommodate access to dynamically connected mobile\ndevices. Combining the control of both wireless and\noptical networks in a single SDN controller requires\nconcrete APIs that handle the respective control functions of\nwireless and optical networks.\nCurrently, service providers maintain separate physical management services\nwithout a unified logical control and management plane for\nFiWi networks.\nDeveloping integrated controls for FiWi networks\ncan be viewed as a special case of multilayer networking and integration.\n\nDeveloping specialized multilayer networking strategies for\nFiWi networks is an important future research directions as many aspects of\nwireless networks have dramatically advanced in recent\nyears. For instance, the cell structure of wireless cellular\nnetworks~\\cite{Ohlen2016} has advanced to femtocell\nnetworks~\\cite{cha2008fem} as well as heterogeneous and multitier\ncellular structures~\\cite{els2013sto,lou2011tow}. At the same time,\nmachine-to-machine communication~\\cite{has2013ran,lay2014ran} and\nenergy savings~\\cite{ana2015opt,has2011gre} have drawn research attention.\n\n\\subsection{QoS and Energy Efficiency}  \\label{futworkene:sec}\nDifferent types of applications have vastly different\ntraffic bit rate characteristics and QoS requirements.\nFor instance, streaming high-definition video requires high bit rates,\nbut can tolerate\nsome delays with appropriate playout buffering. On the other hand,\nVoIP (packet voice) or video conference applications have\ntypically low to moderate bit rates, but require low latencies.\nAchieving these application-dependent QoS levels in an energy-efficient\nmanner~\\cite{has2011gre,ShZL14,wan2015ene}\nis an important future research direction.\nA related future research direction is to\nexploit SDN control for QoS adaptations of real-time media and\nbroadcasting services.\nBroadcasting services involve typically data rates ranging from\n3--48~Gb\/s to deliver video at various resolutions to\nthe users within a reasonable time limit.\nIn addition to managing the QoS, the network\nhas to manage the multicast groups for efficient\nrouting of traffic to the users.\nRecent studies \\cite{Butler2015, Ellerton2015} discuss\nthe potential of SDN, NFV, and optical technologies\nto achieve the growing demands of broadcasters and media.\nMoreover, automated\nprovisioning strategies of QoS and the incorporation of quality of\nprotection and security with traditional QoS are important direction for\nfuture QoS research in SDONs.\n\n\\subsection{Performance Evaluation}\nComprehensive performance evaluation methodologies and metrics need to\nbe developed to assess the SDON designs addressing the\npreceding future research directions ranging from simplicity and\nefficiency (Section~\\ref{simpl_fut:sec}) to optical-wireless networks\n(Section~\\ref{wireless:fut}).\nThe performance evaluations need to encompass the data plane,\nthe control plane, as well as the overall data and control plane interactions\nwith the SDN interfaces and need to take virtualization\nand orchestration mechanisms into\nconsideration. In the case of the SDON infrastructure, the performance\nevaluations will need include the optical\nphysical layer~\\cite{Azodolmolky2014}.\nWhile there have been some efforts to develop evaluation frameworks for\ngeneral SDN switches~\\cite{OFTest,rot2014ope}, such evaluation frameworks\nneed to be adapted to the specific characteristics of SDON architectures.\nSimilarly, some evaluation frameworks for general SDN controllers have\nbeen explored~\\cite{jar2012fle,jar2014ofc}; these need to be extended\nto the specific SDON control mechanisms.\n\nGenerally, performance metrics obtained with SDN and virtualization\nmechanisms should be benchmarked against the corresponding\nconventional network without any SDN or virtualization components.\nThus, the performance tradeoffs and costs of the flexibility\ngained through SDN and virtualization mechanism can be quantified.\nThis quantified data would then need to be assessed and compared\nin the context of business needs. To identify some of the important aspects of\nperformance we analyze the sample architecture in Fig.~\\ref{fig_app_i2rs}.\nThe SDN controller in the SDON architecture in Fig.~\\ref{fig_app_i2rs}\nspans across multiple elements, such as ONUs, OLTs,\nrouters\/switches in the metro-section, as well as PCEs in the core section.\nA meaningful performance evaluation of such a network\nrequires comprehensive analysis of data plane performance aspects and\nrelated metrics,\nincluding noise spectral analysis, bandwidth and link rate monitoring,\nas well as evaluation of failure resilience.\nPerformance evaluation mechanisms need to be\ndeveloped to enable the SDON controller to obtain and analyze these\nperformance data. In addition, mechanisms for control layer\nperformance analysis are needed.\nThe control plane performance evaluation should, for instance\nassess the controller efficiency and performance characteristics,\nsuch as the OpenFlow message rates and the rates and delays of flow table\nmanagement actions.\n\n\\section{\\MakeUppercase{Conclusion}}  \\label{sec:conclusion}\nWe have presented a comprehensive survey of software defined\noptical networking (SDON) studies to date.\nWe have mainly organized our survey according to the SDN\ninfrastructure, control, and application layer structure. In addition,\nwe have dedicated sections to SDON virtualization and orchestration\nstudies.\nOur survey has found that SDON infrastructure studies\nhave examined optical (photonic) transmission and switching components\nthat are suitable for flexible SDN controlled operation.\nMoreover, flexible SDN controlled switching paradigms and optical\nperformance monitoring frameworks have been investigated.\n\nSDON control studies have developed and evaluated SDN control\nframeworks for the wide range of optical network transmission\napproaches and network structures. Virtualization allows for\nflexible operation of multiple Virtual Optical Networks (VONs) over a given\ninstalled physical optical network infrastructure.\nThe surveyed SDON virtualization studies have examined the provisioning\nof VONs for access networks, exploiting the\nspecific physical and Medium Access Control (MAC) layer characteristics of\naccess networks. The virtualization studies have also examined\nthe provisioning of VONs in metro and\nbackbone networks, examining algorithms for embedding the VON topologies\non the physical network topology under consideration of the\noptical transmission characteristics.\n\nSDON application layer studies have developed mechanisms for achieving\nQuality of Service (QoS), access control and security, as well as\nenergy efficiency and failure recovery.\nSDON orchestration studies have examined coordination mechanisms\nacross multiple layers (in the vertical dimension of the network protocol\nlayer stack) as well as across multiple network domains (that may belong\nto different organizations).\n\nWhile the SDON studies to date have established basic principles\nfor incorporating and exploiting SDN control in optical networks,\nthere remain many open research challenges. We have\noutlined open research challenges for each individual category of\nstudies as well as cross-cutting research challenges.\n\n\\chapter{\\MakeUppercase{SDN Based Smart Gateways (Sm-GWs) for Multi-Operator Small Cell\n\tNetwork Management}}\n\n\\section{\\MakeUppercase{Introduction}}\n\\subsection{Motivation: Small Cells}\nRecent wireless communications research has examined the\nbenefits of splitting the conventional cells in wireless cellular\ncommunications into small cells for supporting the growing\nwireless network traffic.\nSmall cells can coexist with\nneighboring small cells while sharing the same spectrum resources \\cite{khan2016cognitive}, and are thus\nan important potential strategy for accommodating wireless network traffic\ngrowth~\\cite{jaf2015sma}.\nSmall cells are also sometimes referred to as ``femto'' cells in\nthe context of the Third Generation Partnership Project (3GPP)\nLong Term Evolution (LTE) wireless standard; \nwe use the general terminology ``small'' cells throughout.\nHowever, small cells pose new challenges, including\ninterference coordination~\\cite{Cong2014},\nbackhaul complexity \\cite{Wei2013,Siddique2015},\nand increased network infrastructure cost~\\cite{Mugume2015}.\nIn this article~\\cite{thyagaturu2016sdn} we propose a solution to\nreduce the infrastructure cost and complexity of backhaul access\nnetworks supporting small cells.\n\nSmall cell networks are expected to be\nprivatively owned~\\cite{Haider2016}.\nTherefore it is important\nto enable usage flexibility and the freedom of investment in the new\nnetwork entities (e.g., gateways and servers) and the network infrastructures\n(e.g., switches and optical fiber)\nby the private owners of small cells.\nWhile a plethora of studies has examined advanced enhanced Node~B (eNB)\nresource\nmanagement, e.g.,~\\cite{chen2015virtual,liu2015qos,samdanis2016td},\nthe implications of small cell deployments for backhaul\ngateways have largely remained unexplored~\\cite{chuang2015resource}.\nGenerally, backhaul access networks that interconnect\n  small cell deployments with LTE gateways\n   can employ a wide variety of link layer (L2) technologies, including\nSONET\/SDH, native Ethernet, \nand Ethernet over MPLS~\\cite{Briggs2010,CiscoDeployment,men2013hie}.\nIn order to accommodate these heterogeneous L2 technologies, \ncellular LTE network interfaces, such as S1 and X2 interfaces,\nare purposefully made independent of the L2 technology between\nsmall cell deployments and gateways. Due to the independent nature \nof L2 technologies, a dedicated link with prescribed \nQoS, which can support the fundamental operations of cellular protocols,\nmust be established for each interface connection~\\cite{Ghebretensae2010}. \nStatistical multiplexing is then limited \nby the aggregate of the prescribed QoS \\cite{guck2016function, fitzek2002providing} requirements\nand only long-term re-configurations, e.g.,\nin response to deployment changes, can optimize the\nbackhaul transmissions~\\cite{Mathew2015}.\nPresent wireless network deployments based on the 3GPP LTE standard do \nnot provide feedback from the eNBs to a\ncentral decision entity, e.g., an SDN orchestrator, \nwhich could flexibly allocate network resources based on eNB traffic demands.\nThus, present wireless backhaul architectures are characterized by\n$(i)$ essentially static network\nresource allocations between eNBs and operator gateways,\ne.g., LTE Servicing\/Packet Gateways (S\/P-GWs), and\n$(ii)$ lack of coordination between the eNBs and the operator gateways\nin allocating these network resources,\nresulting in under-utilization of the backhaul transmission resources.\nAdditionally, exhaustion of available ports at the operator gateways can\nlimit the eNB deployment in practice.\n\nThe static resource allocations and lack of eNB-gateway cooperation\nare highly problematic since the aggregate uplink transmission bitrate of the\nsmall cells within a small geographic area, e.g., in a building,\nis typically much higher than the\nuplink transmission bitrate available from the cellular operators.\nThus, small cell deployments create a bottleneck between the eNBs and the\noperator gateways.  For instance, consider the deployment of\n100 small cells in a building, whereby each small cell supports\n1~Gbps uplink transmission bitrate.\nEither each small cell can be allocated only one hundredth of the\noperator bitrate for this building or the operator would need to\ninstall 100~Gbps uplink transmission bitrate for this\nsingle building, which would require cost-prohibitive operator gateway\ninstallations for an organization with several buildings in a small\ngeographical area.\nHowever, the uplink transmissions from the widespread data communication\napplications consist typically of short high-bitrate bursts, e.g.,\n100 Mbps bursts.\nIf typically no more than ten small cells burst simultaneously, then\nthe eNBs can dynamically share a 1~Gbps operator uplink\ntransmission bitrate.\nAn additional problem is that with the typically limited port counts on\noperator gateways, connections to many new small cells may require\nnew operator gateway installations.\nAn intermediate Sm-GW can aggregate the small cell connections\nand thus keep the required port count at operator gateways low.\n\n\\subsection{Overview of Network Management with SDN-based Sm-GW}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=4.5in]{finman_smgw\/fig1.pdf}\n    \\caption{The proposed smart gateway (Sm-GW).} \n    \\label{fig_enterprise}\n\\end{figure}\nWe present a new backhaul network framework for supporting small cell\ndeployments based on a new network entity, the Smart GateWay (Sm-GW).\nConsider an exemplary small cell deployment throughout multiple\nbuildings of a university.\nEach building has hundreds of small cells that are flexibly\nconnected to an Sm-GW, as illustrated in Fig.~\\ref{fig_enterprise}.\nMultiple Sm-GWs are then connected to core networks, i.e., the S-GWs\nand P-GWs, of multiple cellular operators via\nphysical links (e.g., optical or microwave links)\n\\cite{mcgarry2012investigation, mcgarry2008ethernet, aurzada2008delay, mcgarry2006wdm, mcgarry2004ethernet, reisslein6metropolitan, mercian2013offline, aurzada2014delay},\nas illustrated for a single Sm-GW in Fig.~\\ref{fig_SMGW_overall}.\nAn SDN orchestrator owned by the university\nmanages the cellular infrastructure of the entire university.\nThe SDN orchestrator coordinates the resource allocations from\nthe operators to the Sm-GWs.\n\nThe main original contributions of this article are:\n\\begin{enumerate}\n\\item A novel comprehensive Smart Gateway (Sm-GW) architecture\n  and protocol framework that accommodates a flexible number of eNBs\n  while reducing the requirements at the operator's core, e.g.,\n  at LTE S-GW and MME. The Sm-GW physically and logically aggregates\n the eNB connections so that a set of eNBs appears as a single virtual\n eNB to the operator gateways, see Section~\\ref{arch:sec}.\n    \\item A Sm-GW scheduling framework to flexibly share the limited\n     uplink transmission bitrate among all the small cell eNBs connected\n      to an Sm-GW,  see Section~\\ref{gwsch:sec}.\n    \\item An adaptive SDN-based multi-operator management framework that\n      dynamically shares the uplink transmission bitrates of multiple\n      operators among the Sm-GWs. An SDN orchestrator dynamically coordinates\n      the sharing among the Sm-GWs, the transport network connecting the\n      Sm-GWs to the operator gateways, and the operator gateways,\n  see Section~\\ref{MOM:sec}.\n\\end{enumerate}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{finman_smgw\/fig2.pdf}\n    \\caption{The Smart Gateway (Sm-GW) architecture.} \n    \\label{fig_SMGW_overall}\n\\end{figure}\n\n\\subsection{Related Work}\nRecently proposed SDN based backhaul architectures, such as\nCROWD~\\cite{AliAhmad,seb2015dyn}, iJOIN~\\cite{Dongyao},\nU-WN~\\cite{Shengli}, Xhaul~\\cite{oli2015xha},\nthe multi-tiered SDN based backhaul architecture~\\cite{elg2015thr},\nand similar\narchitectures~\\cite{cos2015sof,cos2015,dra2015dyn,hur2015sdn,\n  Jungnickel,liu2013cas},\nare revolutionary designs proposing new cellular infrastructure\n  installations.\nIn contrast, our proposed SDN-based Sm-GW enables the softwarization of\n\\textit{existing}\ncellular infrastructures consisting of eNBs and conventional operator gateways,\nsuch as the S\/P-GW in LTE core networks.\nThe proposed Sm-GW is inserted in the existing backhaul\ninfrastructure to inter-network and co-exist with the existing\nLTE network core entities, such as the S\/P-GW.\n\nSDN based backhaul architectures with centralized\ninterference coordination have been proposed\nin~\\cite{akh2016syn, gop2016gen, nie2015wir}.\nThe SDN controller in these architectures maintains a global database\nof spectrum resources \\cite{akhtar2016white} and dynamically assigns the resources to base stations so\nas to minimize the mutual interference among base stations in dense\ndeployments.\nWe note that wireless interference is a localized phenomenon.\nTherefore, a base station is most affected by its neighboring base stations\nin dense deployments.\nThe centralized interference coordination techniques\nin~\\cite{akh2016syn, gop2016gen, nie2015wir} are complementary\nto our proposed Sm-GW architecture in that they can be\nimplemented at the Sm-GW instead of the SDN controller\/orchestrator.\n\nSchedulers at the eNB allow multiple user equipment (UE) devices to\nshare the wireless resources at the eNB.  For example, the LTE standard medium\naccess control (MAC) protocol~\\cite{3gppLTEMAC} coordinates the\nscheduling of wireless resources between an eNB and multiple UEs.\nGenerally, most wireless resource scheduling studies to date have focused on the\nsharing of the wireless resources at a given single eNB.\nFor instance, quality of service\n(QoS) aware uplink scheduling and\nresource allocation at a given single small cell eNB in an LTE network have been\nexamined in~\\cite{Chaudhuri2015}.\nIn contrast, we propose a novel\nscheduling framework at the Sm-GW based on uplink transmission bitrate requests\nfrom \\textit{multiple eNBs}, i.e., we propose the sharing of\nthe backhaul network resources among multiple eNBs.\n\nA similar sharing of network resources among small cell base\nstations has been studied in~\\cite{lak2013h}.\nSpecifically, the H-infinity scheduler for limited capacity\nbackhaul links~\\cite{lak2013h} schedules the traffic\nin the downlink. The centralized H-infinity scheduler\nfocused on buffer size requirements at\nthe base stations in the small cell networks.\nIn contrast, we focus on the \\textit{uplink} traffic from the eNBs to the Sm-GW.\nTo the best of our knowledge, we propose the first\nnetwork protocol framework for the uplink transmissions\nfrom multiple eNBs to the operator gateways in the context of LTE small cells.\nWe note that our Sm-GW framework is complementary to several recently studied\nresource allocation mechanisms in cellular networks.\nFor instance, D2D resource allocation through traffic\noffloading to small cell networks has been studied in~\\cite{sem2015con}; this\nD2D approach can be readily supported by our proposed Sm-GW.\nCoordinated scheduling \\cite{mer2016ups, wei2014dyc, che2015sim, das2016gro} in the context of small cells\nwith dynamic cell muting to mitigate the interference has\nbeen discussed in~\\cite{wan2015dyn}. The cell muting technique can be\nfurther extended based on our approach of traffic scheduling to eNBs.\nA flexible wireless resource allocation mechanism\nbased on the SDN programmability of traffic flows\nfrom a single UE device to multiple\nbase stations in dense small cell networks\nhas been examined in~\\cite{gas2015pro}.\nThe offloading of UE traffic for efficient traffic management in\n  small cell networks has been examined in~\\cite{tal2015eff}.\nIn contrast, we propose an SDN-based multi-operator resource allocation\nmechanism that allocates limited backhaul link capacities to\nmultiple Sm-GWs (which in turn can flexibly allocate the capacities\nto multiple eNBs).\nThe UE to eNB communication approach from~\\cite{gas2015pro}\nand UE traffic offloading~\\cite{tal2015eff} are thus\ncomplementary to our eNB to Sm-GW and Sm-GW to S\/P GW\nnetwork management approaches.\n\n\\section{\\MakeUppercase{Background: Conventional LTE Small Cell Backhaul}}\nIn this section we describe the conventional architectural model for\nHome-eNodeB (HeNB) access  networks~\\cite{HeNB3GPP} and the network\nsharing mechanism in 3GPP LTE.\nHeNBs are the small cell base stations of the LTE standard.\n  We use the general terminology ``eNB'' to denote all types of\nsmall cell base stations.\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=4.5in]{finman_smgw\/fig3.pdf}\n    \\caption{HeNB architectural models in 3GPP LTE.}\n    \\label{fig_HeNB}\n\\end{figure}\n\n\\subsection{HeNB Architectural Models in 3GPP LTE}\nIn Figure \\ref{fig_HeNB}\nwe show the 3GPP HeNB architectural models:\n1) with dedicated HeNB-GateWay (HeNB-GW), 2) without HeNB-GW,\nand 3) with HeNB-GW for the control plane.\n\n\\subsubsection{With Dedicated HeNB-GW}\nWith a dedicated HeNB-GW, communication between\nthe HeNB and the HeNB-GW is secured by a mandatory security gateway\n(Se-GW) network function. The HeNB-GW aggregates the\ncontrol plane connections (S1-MME) and user plane connections (S1-U) of all\nHeNBs connected to the HeNB-GW to a single control and user plane\nconnection. The HeNB gateway appears as a single eNB to the outside\nentities, such as S-GW and MME.\nIn a similar way, the HeNB-GW appears as both an\nS-GW and an MME to the eNBs connected to the HeNB-GW.  The numbers of\nports required at the MME and S-GW are reduced through the\naggregation at the HeNB-GW.\nOur proposed Sm-GW architecture is similar to the\n  dedicated HeNB-GW architecture, in that the Sm-GW aggregates the eNBs\n  connections both physically and logically.\nIn addition, our Sm-GW flexibly allocates uplink transmission\nbitrates to small cell eNBs (see Section~\\ref{gwsch:sec})\nand allows for the adaptive allocation of\noperator uplink transmission bitrates to the Sm-GW by the SDN orchestrator\n(see Section~\\ref{MOM:sec}).\n\n\\subsubsection{Without HeNB-GW}\nDeployments of HeNBs without the HeNB-GWs increase the requirements on\nthe S-GW and MME to support large numbers of connections.  Large\ndeployments of small cells without gateway aggregation at the HeNBs\nwould greatly increase the total network infrastructure cost.\n\n\\subsubsection{With HeNB-GW for the Control Plane}\nHeNB control plane connections are terminated at the HeNB-GW and a\nsingle control plane connection is established from the\nHeNB gateway to the MME. Although\nthe number of connections required at the MME is reduced due to the\ncontrol plane aggregation at the HeNB-GW, data plane connections are still\nterminated directly at the S-GW, increasing requirements at the\nS-GW. The Se-GW typically secures the communication to and from the\nHeNB. In contrast, our proposed Sm-GW terminates all\nthe control and data connections from HeNBs.\n\n\\subsection{3GPP Network Sharing} \nNetwork sharing was introduced by 3GPP in Technical Specification\n  TS~23.951~\\cite{3GPPNetSharing}\n  with the main motivation to share expensive radio spectrum\n  resources among multiple operators.\n For instance, an operator without available spectrum \nin a particular geographic area can offer cellular\nservices in the area through sharing the spectrum of another operator. \nIn addition to spectrum sharing, 3GPP specifies\ncore network sharing among multiple operators\nthrough a gateway core network (GWCN) configuration~\\cite{3GPPNetSharing}.\nGWCN configurations are statically pre-configured at deployment\nfor fixed pre-planned core network sharing.\nThus, GWCN sharing can achieve only limited statistical\nmultiplexing gain as the sharing is based on the pre-configured \nQoS requirements of the eNB interface connections and not on the \nvarying eNB traffic demands. \nAlso, the GWCN configuration lacks a central entity for \noptimization of the resource allocations with global knowledge of the \neNB traffic demands.\nIn contrast, our Sm-GW framework includes a central SDN orchestrator for\noptimized allocations of backhaul transmission resources  \naccording to the varying eNB traffic demands (see Section~\\ref{MOM:sec}).\n \n\\section{\\MakeUppercase{Proposed Smart Gateway (Sm-GW)}} \\label{arch:sec}\nIn this section we introduce the proposed Smart Gateway (Sm-GW) network\narchitecture for existing LTE deployments.\nWe describe the fundamental protocol mechanisms and interfaces\nthat integrate the proposed Sm-GW into the conventional LTE\nprotocols.\n\n\\subsection{LTE Protocol Modifications}  \\label{prot:sec}\nFig.~\\ref{fig_protocol} illustrates the proposed protocol\nmechanisms between a set of $N_s$ eNBs and a given Sm-GW $s$.\n\\begin{figure*}[!t]\n    \\centering\n    \\includegraphics[width=6in]{finman_smgw\/fig4.pdf}\n    \\caption{Illustration of proposed protocol mechanisms at eNB,\nSm-GW, and S-GW.} \n    \\label{fig_protocol}\n\\end{figure*}\n\n\\subsubsection{eNB} \\label{prot:sec:eNB}\nAt the eNB, we newly introduce the eNB-to-Sm-GW\nreporting protocol, which\noperates on top of the GPRS tunneling protocol (GTP)~\\cite{GTP}\nand stream control transmission protocol (SCTP).\nThe reporting protocol $(i)$ evaluates the required uplink transmission\nbitrate, and $(ii)$ sends the bitrate request messages to the Sm-GW.\nThe reporting protocol formulates the operator specific uplink\ntransmission bitrate requests\nbased on the requests of the UEs that are connected\nvia the eNB to multiple operators $o,\\ o = 1, 2, \\ldots, O$.\n\nThe eNB wireless resource scheduler is responsible for the sharing\nof wireless resources between the eNB and the UEs.\nThe eNB wireless resource scheduler ensures\nthat only the resources available at the eNB are granted to the UEs.\nUEs periodically send buffer status reports\n(BSRs) to the eNB which they are connected to.\nTherefore, the eNB-to-Sm-GW reporting protocol can estimate the UE traffic\nrequirements by interacting with the wireless resource scheduler.\n\n\\subsubsection{Smart Gateway (Sm-GW)} \\label{prot:sec:Sm-GW}\nThe protocol stack at the Sm-GW is similar to the HeNB-GW protocol stack.\nHowever, in the Sm-GW, an additional eNB coordination protocol,\na scheduler for the dynamic resource\nallocation, and SDN capabilities are introduced.\n\nThe eNB coordination protocol collects request messages from eNBs.\nThe eNB uplink transmission grants are sized based on the\neNB requests and the available Sm-GW resources\naccording to the Sm-GW scheduling described in Section~\\ref{gwsch:sec}.\nThe eNB coordination protocol sends grant messages\nto all eNBs within a reasonable processing delay.\n\nS1 based handovers for the downlink transmissions are typically\nanchored at the S-GW. (For the uplink transmissions, an anchoring,\nor buffering of packets, at a network entity, e.g., eNBs or S-GW, is\nnot required.) We emphasize that the\nSm-GW will be transparent to all the downlink packets from the S-GW\nand hence not be limited by the network protocol scheduler.\nThis ensures that the S1 based handover mechanisms\nat the S-GW and eNBs continue to function normally.\n\n\\subsubsection{SDN Operations of Sm-GW}\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[width=3.5in]{finman_smgw\/fig5.pdf}\n\n        \\caption{Illustration of Sm-GW embedding in the SDN ecosystem.}\n\n\t\\label{fig_Sm-GW_SDN}\n\\end{figure}\n\n\\paragraph{SDN Infrastructure}\nThe Sm-GW SDN capabilities can be provided by an\nOpenFlow (OF) agent and\/or\na configuration manager at each Sm-GW, as illustrated\nin Fig.~\\ref{fig_Sm-GW_SDN}.\nOpenFlow is a popular protocol for the southbound\ninterface (SBI) and can be employed on the SBI\nbetween the Sm-GW SDN controller and Sm-GW.\nThe OpenFlow agent supports OpenFlow SDN functionalities at the Sm-GW,\nmaking the Sm-GW configurable through the OpenFlow protocol.\nThe Sm-GW configuration manager can be controlled by the Sm-GW SDN controller,\ne.g., through the NETCONF (or OpenFlow) SBI,\nto dynamically reconfigure the Sm-GW.\n\nThe Sm-GW SDN controller configures the Sm-GWs to enable the\ninternal LTE X2 tunnel interfaces among all connected small cell eNBs,\nas elaborated in Section~\\ref{X2int:sec}. Also, the Sm-GW SDN\ncontroller manages the external LTE X2 and S1 interfaces at the Sm-GW through\ntunnel establishments to the external LTE network core entities,\ni.e., MMEs and S\/P-GWs.\n\nWhereas the conventional LTE transport network between the eNBs and\nS\/P-GWs is configured with static MPLS\/IP paths~\\cite{Ghebretensae2010},\nthe flexible Sm-GW operation requires a flexible transport\nnetwork, that is controlled by a transport SDN controller, as illustrated in\nFig.~\\ref{fig_Sm-GW_SDN}.\nThe flexible transport network can, for instance, be implemented through a\nSoftware Defined Elastic Optical Network \n(SD-EON)~\\cite{thy2016sof,Yoshida2015}.\n\n\\paragraph{Sm-GW Virtualization}   \\label{smgwvirt:sec}\nThe SM-GW can support a variety of virtualization strategies, e.g., to\nprovide independent virtual networks for different operators.\nOne example virtualization strategy could let the\nSm-GWs abstract the connected eNBs. Sm-GWs could then be abstracted by\na hypervisor~\\cite{ble2015sur,mij2016ne,nak2015wif,rig2015pro}\nthat intercepts the SBI, as illustrated in Fig.~\\ref{fig_Sm-GW_SDN}.\nBased on the operator configurations that are sent via the SDN orchestrator\nto the Sm-GW SDN controller, resources at the Sm-GWs and the small cell eNBs\n(which are privately owned by an organization~\\cite{Haider2016}) can\nbe sliced to form operator-specific virtual networks of Sm-GWs and\neNBs. The configuration manger at each Sm-GW can allocate resources to each of\nthese virtual networks.\n\nFrom the operator perspective,\n  the Sm-GW virtualization \\cite{ble2016con} essentially allows multiple operators to share the\nphysical small cell infrastructure of Sm-GWs and eNBs.\nThus, simultaneous services can be enabled to large UE populations\nthat belong to multiple operators, i.e., that have contracts with multiple\noperators, while using the same physical small cell infrastructure.\nExisting conventional cellular deployment structures do not support\nthe infrastructure sharing among multiple operators.\n\n\\paragraph{SDN Orchestration}\nThe SDN orchestrator coordinates the network management across multiple\ndomains of Sm-GWs (whereby each Sm-GW domain is controlled by its own\nSm-GW SDN controller), transport networks, and core networks.\nThe SDN orchestrator implements the multi-operator management\nintroduced in Section~\\ref{MOM:sec} and configures the\nSm-GWs and transport networks based on\nglobal multi-operator network optimizations.\nFor example, the SDN orchestrator\ncommunicates with the path computation element (PCE) SDN application\non the transport SDN controller.\nThe PCE dynamically evaluates the label switched paths,\nsuch as MPLS\/IP paths, so as to flexibly enable and reconfigure\nthe transport network~\\cite{thy2016sof,Yoshida2015}.\n\n\\subsubsection{LTE X2 Interfaces of eNBs with Sm-GW}  \\label{X2int:sec}\nX2 interfaces enable critical functionalities in LTE small cells,\nsuch as X2-based handover as well as interference coordination and mitigation.\nTypically, each eNB connected to a given Sm-GW\npertaining to an operator shares the same\nMME; thus, each eNB needs an X2 interface to all other eNBs\nwithin the same MME coverage area, the so-called tracking area.\nHence, eNBs connected to an Sm-GW must be\ninterconnected with X2 interfaces.\n\n\\paragraph{To External Macro-eNBs}\nX2 traffic flows destined to eNBs located outside the scope of an\nSm-GW (typically to a macro-eNB) are not be limited by the scheduler\nat the Sm-GW. X2 packets flow out of Sm-GW into the backhaul (i.e., to an\nS-GW) as they originate at the eNBs. The Sm-GW appears as an\nexternal router (or gateway) to the X2 external interfaces.\n\n\\paragraph{To Internal Small-eNBs} \\label{prot:X2:conn}\nThe Sm-GW appears as a simple bridge or a router to the internal X2\ninterfaces, routing the internal X2 packets within.\nTherefore, the scheduler at the Sm-GW does not limit any X2 packets.\nFor small cell deployments, an eNB can have multiple neighboring eNBs\nin the tracking area;\nthese neighboring eNBs need to be interconnected with each other\nwith X2 connections.\nOn the order of $O(N(N-1))$ dedicated links would be required\n  to interconnect the X2 interfaces of $N$ eNBs in the tracking area in a full\n  mesh topology.\nIn contrast, a star\ntopology with the Sm-GW at the center requires only $O(N)$\nconnections to connect each eNB to the Sm-GW.\nIn summary, in our Sm-GW architecture, the Sm-GW manages the X2 interfaces\nof all the internal small cell eNBs, thus eliminating\nthe distributed management of X2 interfaces at each eNB.\n\n\\subsubsection{Authentication of Sm-GW with EPC Core}\nTypically HeNBs use IPSec tunneling for their\nsecurity and encryption, which creates overhead.\nIf Sm-GWs are authenticated, the HeNBs would no longer need IPsec tunneling.\nSpecifically, upon boot-up, the Sm-GW is authenticated with\nan LTE Evolved Packet Core (EPC)\nso as to eliminate the need for a network security-gateway\n(Se-GW) function or IPsec tunneling between the eNBs and the P-GWs.\nCritical cellular network functions, such as\nsecurity, authentication, and reliability,\nrequire additional effort to be enabled in WiFi networks.\nWiFi Passpoint~\\cite{Passpoint} (Hotspot 2.0)\naims at providing an experience similar to\ncellular connectivity in\nWiFi networks by providing the cellular authentication mechanisms.\nWith the authentication of Sm-GWs,\nthe simplicity of WiFi networks\ncan be achieved by the small cell cellular networks.\n\n\\subsection{Downlink vs.~Uplink Communication} \\label{updown:sec}\n\\subsubsection{Downlink Packets at the Sm-GW}\nTraffic flows in the conventional downlink path from\nan S\/P-GW to an eNB are typically sent at rates that do not\nexceed the wireless transmission rates from the eNB to the UE devices.\nThus, as long as the link rates from the S\/P-GW to the\ninserted Sm-GW and from the Sm-GW to the eNB are at least as high\nas the conventional S\/P-GW to eNB links, the Sm-GW can be\ntransparent to the downlink packets from the S\/P-GW.\n\n\\subsubsection{Uplink Packets at Sm-GW}\nIn contrast to the downlink data traffic, the uplink data\ntraffic from the eNBs to an Sm-GW needs to be\nregulated as the traffic flows from all the eNBs terminating at the\nSm-GW can overwhelm the outgoing link towards the operator S-GW.\nEnforcing QoS strategies and fairness among eNBs requires scheduling\nof the uplink packet traffic arriving from the eNBs at an Sm-GW.\nTherefore, our focus is on frameworks for the uplink transmission scheduling of\nthe communication $(i)$ from eNBs to an Sm-GW (Section~\\ref{gwsch:sec}),\nand $(ii)$ from Sm-GWs to S-GWs (Section~\\ref{MOM:sec}).\n\n\\section{\\MakeUppercase{Proposed Sm-GW Scheduling Framework}} \\label{gwsch:sec}\n\\subsection{Purpose}\nThe main purpose of the Sm-GW scheduling framework\nis to maximize the utilization of\nthe network resources, and to ensure fair uplink transmission service for\n all eNBs connected to an Sm-GW.\nWithout scheduling, highly\nloaded eNBs can impair the service for lightly loaded\neNBs connected to the same Sm-GW.\nWhen many eNBs are\nflexibly connected to an Sm-GW, traffic bursts from heavily loaded eNBs can\noverwhelm the queue of an Sm-GW, resulting in excessive packet\ndrops and high delays, even for lightly loaded eNBs.\nOn the other hand, with scheduling, a large number of\neNBs can be flexibly connected to the Sm-GW while ensuring\nprescribed QoS and fairness levels.\nEach eNB can possibly have a different service level agreement.\nThe Sm-GW allows for the flexible deployment of a wide variety of\nscheduling algorithms.\nWe outline two classes of Sm-GW scheduling algorithms, and illustrate\nan elementary algorithm for each class.\n\n\n\\begin{table}[t]\n\\caption{Summary of Notation of Sm-GW Network Management}\n     \t\\centering\n\\begin{tabular}{|p{0.4cm} p{7.2cm}|}\n\\hline\n\\multicolumn{2}{|c|}{\\textbf{\\rule{0pt}{1\\normalbaselineskip}\n    Sm-GW Sched. Framework (Sm-GW $\\leftrightarrow$ eNBs),\n           Sec.~\\ref{gwsch:sec}}} \\\\   [.5ex]\n$N_s$ &  Number of small cell eNBs at Sm-GW $s$ \\\\\n$G_{so}$ &  Available uplink transm. bitrate [bit\/s] from Sm-GW $s$ to operator $o$  \\\\\n$W$ &  Duration [s] of scheduling cycle   \\\\\n$\\Gamma_{so}$  &  $= G_{so} W \/ N_s$, Max. eNB uplink transm. data amount [bit] per cycle with equal sharing \\\\\n$\\rho_{son}$ & Data amount [bit] that eNB $n$ at Sm-GW $s$ wants to\n  transmit to operator $o$ in a cycle, i.e., request by eNB $n$ \\\\\n$\\gamma_{son}$ & Data amount [bit] that eNB $n$ at Sm-GW $s$ is allowed\nto transmit to operator $o$ in a cycle, i.e., grant by Sm-GW $s$ \\\\%[0.5ex]\n\\hline\n\\multicolumn{2}{|c|}{\\textbf{\\rule{0pt}{1\\normalbaselineskip}\n   SDN Based Multi-Operator Managm. Framework, Sec.~\\ref{MOM:sec}}} \\\\\n            \\multicolumn{2}{|c|}{\\textbf{\n    (Sm-GWs $\\leftrightarrow$ Operator Gateways)}} \\\\[.5ex]\n$o$  &  Index of operators, $o = 1, 2, \\ldots, O$\\\\\n$s$   & Index of Sm-GWs, $s = 1, 2, \\ldots, S$ \\\\\n$R_{so}$   &  Smoothed uplink transmission bitrate [bit\/s] request\n     from Sm-GW $s$ to operator $o$ \\\\\n$K_o$ &  Max. available uplink transm. bitrate through operator $o$\\\\ \n$G_{so}$ & Granted uplink transm. bitrate from Sm-GW $s$ to operator $o$\\\\\n$X_{so}$ & Actual uplink traffic bitrate from Sm-GW $s$ to operator $o$\\\\ [0.5ex]\n\t\t\\hline\n\t\t\\end{tabular}\n\t\t\\end{table}\n\n\\subsection{Configuration Adaptive Scheduling} \\label{gwsch:sec:equal}\nConfiguration adaptive scheduling adapts the scheduling,\ni.e., the allocation of uplink transmission bitrates, according to\nthe number of eNBs connected to a given Sm-GW.\nThe Sm-GW tracks the number of connected eNBs and sends\na configuration message to all eNBs in the event of a change in\nconnectivity at the Sm-GW, i.e., addition of new eNB or\ndisconnection of existing eNB.\nMore specifically, consider $N_s$ eNBs at a given Sm-GW $s$ that has been\nallocated the uplink transmission bitrate $G_{so}$ [bit\/s] toward a\ngiven operator $o$ (through the coordination techniques in\nSection~\\ref{MOM:sec}).\n\nAn elementary equal share scheduling shares the available uplink\ntransmission bitrate at the Sm-GW toward a given operator $o$\nequally among all eNBs connected to the Sm-GW.\nEach eNB $n,\\ n = 1, 2, \\ldots, N_s$, can then transmit at most\n$\\Gamma_{so} =  G_{so} W \/ N_s$ [Byte] of traffic during a cycle\nof duration $W$ [seconds].\nThe traffic amount limit $\\Gamma_{so}$ and cycle duration $W$\nare sent to the eNBs as a part of the initial configuration message.\nEach eNB schedules the uplink transmissions\nsuch that no more than $\\Gamma_{so}$ [Byte] of traffic are\nsend in a cycle of duration $W$ [seconds].\n\nThe simple equal share scheduler can flexibly accommodate\nlarge numbers $N_s$ of eNBs.\nHowever, the equal bandwidth assignments by the elementary\nequal share scheduler to\nthe eNBs under-utilize the network resources when some eNBs have very\nlittle traffic while other eNBs have high traffic loads.\n\n\\subsection{Traffic Adaptive Scheduling} \\label{gwsch:sec:excess}\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=3.5in]{finman_smgw\/fig6.pdf}\n\\caption{Illustration of traffic adaptive Smart Gateway (Sm-GW)\n    scheduling.} \n\\label{fig_ExcessTiming}\n\\end{figure}\nWith traffic adaptive scheduling, the Sm-GW collects uplink transmission\nrequests from the eNBs. The Sm-GW then adaptively allocates portions of\nthe uplink transmission bitrate $G_{so}$ to the individual eNBs\naccording to their requests.\nTraffic adaptive scheduling operates with a\nrequest-allocate-transmit cycle of duration $W$ [seconds]\nillustrated in Fig.~\\ref{fig_ExcessTiming}. At the start of the\ncycle, each eNB $n,\\ n = 1, 2, \\ldots, N_s$, sends an\nuplink transmission bitrate request to Sm-GW~$s$.\nWe let $\\rho_{son}$ denote the amount of traffic [in Byte] that\neNB $n$ wants to transmit to operator $o$ over the next cycle of duration $W$.\nOnce all requests have been received, i.e., following the\nprinciples of the offline scheduling\nframework~\\cite{zhe2009sur}, portions of $G_{so}$ can be\nallocated to the eNBs according to some scheduling policy.\n\nAn elementary excess share scheduling\npolicy~\\cite{bai2006fai}\nallocates the eNB grants as follows. Lightly loaded eNBs with\n$\\rho_{son} < \\Gamma_{so}$ are granted their full request, i.e.,\nreceive the grant size $\\gamma_{son} = \\rho_{son}$, while their\nunused (excess) portion of the equal share allocation is accumulated\nin an excess pool:\n\\begin{equation}\n\\xi = \\sum_{\\forall \\rho_{son} \\leq \\Gamma_{so}} \\Gamma_{so} - \\rho_{son}.\n\\label{ExcessAccum}\n\\end{equation}\nFollowing the principles of controlled equitable excess\nallocation~\\cite{bai2006fai}, highly loaded eNBs are allocated\nan equal share of the excess up to their request. That is,\nwith $|\\mathcal{H}|$ highly loaded eNBs, the grants are\n\\begin{equation}\n\\gamma_{son} = \\min \\left(\\rho_{son},\\; \\Gamma_{so} +\n       \\frac{\\xi}{|\\mathcal{H}|}\\right).\n\\label{HighGrant}\n\\end{equation}\n\n\\subsection{Scheduling Fairness} \\label{sec:fariness}\nWithin the context of our proposed Sm-GW scheduling framework, fairness is\nthe measure of network accessibility of all $N_s$ eNBs\nconnected to an Sm-GW $s$ based on individual eNB uplink throughput level\nrequirements.\nWe denote $T_{son}$ for the long-run average throughput level [bit\/s] of\nuplink traffic generated at eNB $n,\\ n = 1, 2, \\ldots, N_s$,\nat Sm-GW $s$ for operator $o$.\nThe throughput level $T_{son}$ can for instance be obtained through\nsmoothing of the requests $\\rho_{son}$ over successive cycles $w$.\nIn order to avoid clutter, we omit the subscripts $s$ and $o$ in the remainder\nof this fairness evaluation.\nWe define the following fair target throughput levels $\\Omega_n$ [bit\/s]:\nLightly loaded eNBs $l \\in \\mathcal{L}$\nwith throughput levels $T_l < \\Gamma\/W$,\nshould be able to transmit their full traffic load, i.e., $\\Omega_l = T_l$.\nNext, consider highly loaded eNBs $h \\in \\mathcal{H}$ with throughput\nlevels $T_h > \\Gamma\/W$.\nIf the total throughput requirement of all eNBs\n$\\sum_{l \\in \\mathcal{L}} T_{l} +\\sum_{h \\in \\mathcal{H}}T_h$\nis less than or equal to the\nuplink transmission bitrate $G$, then the highly loaded\neNBs should be able to transmit their full traffic load, i.e.,\n$\\Omega_h = T_h$.\nOn the other hand, if the total traffic load exceeds the uplink transmission\nbitrate, i.e., if $\\sum_{l \\in \\mathcal{L}} T_{l} +\\sum_{h \\in \\mathcal{H}}T_h > G$, then\nthe highly loaded eNBs should be able to transmit traffic up to\nan equitable share of the uplink\ntransmission bitrate not used by the lightly loaded eNBs.\nThus, overall:\n$\\Omega_h = \\min \\{T_h,\\ (G - \\sum\\limits_{l \\in \\mathcal{L}} T_{l} )  \/\n\t|\\mathcal{H}| \\}$. We define the normalized distance $\\mathcal{E}_n$\nof the actually achieved (observed) throughput $\\tau_n$ and the\ntarget throughput $\\Omega_n$, i.e.,\n$\\mathcal{E}_n = \\tau_n - \\Omega_n$.\n\nBased on the preceding target throughput definitions, we obtain\nthe normalized distance throughput fairness index~\\cite{JainIndex}\n\\begin{equation}\n\\mathcal{F}_T =\n\\frac{ \\sqrt{ \\sum_{n = 1}^N \\mathcal{E}^2_n}}\n{\\sqrt{\\sum_{n =1}^N\\Omega^2_n}},\n\\label{FairIndex}\n\\end{equation}\nwhereby $\\mathcal{F}_T$ close to zero indicates\nfair Sm-GW scheduling.\n\n\\subsection{Sm-GW Scheduling Overhead}\nIn configuration adaptive Sm-GW scheduling, a reconfiguration event,\ni.e., an eNB connect or disconnect event, triggers the re-evaluation\nof the grant size limit $\\Gamma_{so}$, see Section \\ref{gwsch:sec:equal}.\nThe new $\\Gamma_{so}$ value is sent to all eNBs.\nSince reconfiguration events occur typically only rarely, e.g., on\nthe time scale of minutes or hours, the overhead for\nconfiguration adaptive scheduling is negligible.\n\nTraffic adaptive Sm-GW scheduling requires each eNB $n$ to send a request\nevery cycle of duration $W$ seconds.\nUpon reception of the requests from all $N_{s}$ eNBs,\nthe Sm-GW evaluates and sends the grants to the respective eNBs,\nas illustrated in Fig.~\\ref{fig_ExcessTiming}(a).\nThe requests and grants can typically be sent simultaneously, i.e., in\nparallel, over the individual eNB-to-Sm-GW links.\nThus, within a cycle duration $W$,\nthe overhead amounts to the transmission delays of the request and\ngrant messages, the maximum round-trip propagation delay between eNBs and Sm-GW,\nand the schedule processing delay at the Sm-GW.\nFor typical parameter settings, such as\n70~Byte messages transmitted at 1~Gbps, up to 500~m eNB-to-Sm-GW\npropagation distance, $W=1$~ms cycle duration, and schedule processing\ndelay on the order of microseconds, the overhead is less than half a\npercent.\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\centering\n\t\\includegraphics[width=3.5in]{finman_smgw\/fig7a.pdf}\\\\\n\n\t\\footnotesize (a) Observed avg. throughput\n\tof lightly loaded eNBs  \\\\ \\footnotesize\n\t\\includegraphics[width=3.5in]{finman_smgw\/fig7b.pdf} \\\\\n\t(b) Observed avg. throughput of highly loaded eNBs\\\\\n\t\\label{fig_SmGW_sch2}\n\\end{figure*}\n\n\\begin{figure*}[t]\n      \\centering\n      \\centering\n      \\includegraphics[width=3.5in]{finman_smgw\/fig7c.pdf}  \\\\\n      \\footnotesize (c) Avg. delay of lightly loaded $T_L$ eNBs,\n      when $T_H = 80$ Mbps \\\\\n      \\includegraphics[width=3.5in]{finman_smgw\/fig7d.pdf} \\\\\n      \\footnotesize (d) Fairness Index $\\mathcal{F}_T$,\n      when $T_H$ = 200 Mbps.\n      \\caption{Simulation results for Sm-GW scheduling.}\n      \\label{fig_SmGW_sch}\n\\end{figure*}\n\n\\subsection{Evaluation of Sm-GW Scheduling}\n\\subsubsection{Simulation Setup}\nWe evaluate the performance of Sm-GW scheduling\nwith the discrete event simulator OMNET++.\nWe consider a given Sm-GW $s$ with an uplink\ntransmission bitrate to a given operator $o$ of\n$G_{so} = 1$~Gbps. We omit the subscripts $s$ and $o$ in the remainder\nof this evaluation section to avoid notational clutter.\nThe LTE access network typically requires the\npacket delay to be less than 50~ms~\\cite{3GPPQoS}.\nTherefore, we set the Sm-GW queue size to\n20~MBytes, which is equivalent to a maximum queuing delay of 20~ms\nover the $G = 1$~Gbps link. Without any specific scheduling,\nthe Sm-GW operates in first-come-first-served mode with taildrop.\n\nWe simulate the typical bursty eNB traffic\ngeneration pattern, with two eNB traffic rate states: low and heavy.\nThe sojourn time in\na given traffic rate state is randomly drawn from a\nuniform distribution over 1~ms to 4~ms. At the end of the sojourn time,\na switch to another state occurs with a probability of 70~\\%\nin the low traffic state\nand 30~\\% in the heavy traffic state. The traffic bitrate\nratio between the heavy and low traffic states is $4:1$.\nWithin a given traffic rate state, data packets are randomly\ngenerated according to independent Poisson processes.\n\nWe consider $|\\mathcal{L}| = 10$ lightly\nloaded eNBs and $|\\mathcal{H}| = 10$ highly loaded eNBs connected to\nthe considered Sm-GW.\nEach eNB, irrespective of whether it is lightly or highly loaded, generates\ntraffic according to the two traffic rate state (low and heavy) model.\nThe low and heavy traffic rates are set such that the long-run\naverage generated traffic rate corresponds to a prescribed required\nthroughput (load) level\n$T_L < G\/N = 50$~Mbps for a lightly loaded eNB and a prescribed required\nthroughput (load)\nlevel $T_H > G\/N$ for a highly loaded eNB.\nFor all simulations, the 95~\\% confidence intervals are less\nthan 5~\\% of the corresponding sample mean.\n\n\\subsubsection{Simulation Results}\n\\paragraph{Without Sm-GW Scheduling}\nIn Fig.~\\ref{fig_SmGW_sch}, we show representative evaluation results\ncomparing configuration adaptive equal-share Sm-GW scheduling\nand traffic adaptive excess-share Sm-GW scheduling with the\nconventional backhaul without Sm-GW scheduling.\nFigs.~\\ref{fig_SmGW_sch}(a) and (b) show the actual (achieved, observed)\nthroughput\n$\\tau$ of lightly loaded and highly loaded eNBs, respectively, as a function\nof the generated lightly loaded ($T_L$) and highly loaded ($T_H$)\nthroughput levels.\nWe observe from Figs.~\\ref{fig_SmGW_sch}(a) that without scheduling,\nthe lightly loaded eNB suffer reductions in the achieved throughput, that\nare especially pronounced (over 30~\\%) for the high $T_H = 200$~Mbps\nload of the highly loaded eNBs.\nAt the same time, we observe from Figure~\\ref{fig_SmGW_sch}(b)\nthat without scheduling, the highly loaded eNBs achieve more than\ntheir fair throughput share.\nFor instance, for the highly loaded eNB throughput requirement (load)\n$T_H = 140$~Mbps, and $T_L = 30$~Mbps, the observed throughout of the highly\nloaded eNBs is $\\tau_H = 76$~Mbps, which is significantly higher than\nthe fair share of $(G - |\\mathcal{L}| T_L) \/|\\mathcal{H}| = 70$~Mbps.\nThe unfairness arising without scheduling is further illustrated in\nFig.~\\ref{fig_SmGW_sch}(c), where we observe a sharp delay increase\nat $T_L = 20$~Mbps, when the total traffic load\n$|\\mathcal{L}| T_L + |\\mathcal{H}| T_H$ approaches the uplink transmission\nbitrate $G$.\nMoreover, from Fig.~\\ref{fig_SmGW_sch}(d), we observe an increasing\nfairness index $\\mathcal{F}_T$ as the lightly loaded eNBs generate more\ntraffic, i.e., as $T_L$ increases.\nThat is, as the lightly loaded eNBs try to transmit more traffic,\ntheir achieved throughput falls more and more below their\nfair share [see growing divergence between\nthe no scheduling curves and straight lines for\nscheduling in Fig.~\\ref{fig_SmGW_sch}(a)], leading to increasingly\nunfair treatment of the lightly loaded eNBs.\n\n\\paragraph{Equal-share Sm-GW Scheduling}\nWe observe from Fig.~\\ref{fig_SmGW_sch}(a) and (c) that\nlightly loaded eNBs benefit from equal-share scheduling in that they\nget the full share of their fair target throughput and experience\nlow delay.\nHowever, we observe from Fig.~\\ref{fig_SmGW_sch}(b) that\nhighly loaded eNBs achieve only a throughput of\n$G\/(|\\mathcal{L}| + |\\mathcal{H}|) = 50$~Mbps as\nequal-share Sm-GW scheduling assigns a configuration adaptive allocation of\nequal shares of the limited uplink transmission bitrate $G$ to all\neNBs irrespective of their traffic generation rates.\nCorrespondingly, we observe from Fig.~\\ref{fig_SmGW_sch}(d), a high\nfairness index $\\mathcal{F}_T$ for low traffic loads of the\nlightly loaded eNBs, as the highly loaded eNBs receive only\nunfairly small shares of the uplink transmission bitrate $G$.\n\n\\paragraph{Excess-share Sm-GW Scheduling}\nWe observe from Fig.~\\ref{fig_SmGW_sch}(a) and (b)\nthat with excess-share Sm-GW scheduling, both\nlightly loaded eNBs and highly loaded eNBs\nachieve their fair target throughput.\nWe further observe from Figs.~\\ref{fig_SmGW_sch}(c) and (d)\nthat excess-share Sm-GW scheduling gives also\nfavorable delay and fairness index performance.\n\n\\paragraph{Summary}\nWe conclude that scheduling\nof the Sm-GW uplink transmission bitrate $G$ is necessary to prevent backhaul\nbandwidth starvation of lightly loaded eNBs due to the overwhelming\ntraffic rates of highly loaded eNBs. On the other hand,\nsimple configuration adaptive\nallocation of equal uplink transmission bitrate\nshares to each eNB wastes bandwidth.\nFlexible traffic adaptive scheduling according to the\ntraffic loads of the eNBs, e.g., through excess-share\nscheduling, can ensure fairness while efficiently utilizing the\nuplink transmission bitrate.\n\n\\section{\\MakeUppercase{SDN Based Multi-Operator Management}}  \\label{MOM:sec}\n\\subsection{Overview}\nIn this section we introduce a novel SDN based network management\nframework for flexible sharing of the backhaul resources of\nmultiple operators.\nIn particular, the framework\nintroduced in Sections~\\ref{rap:sec}--\\ref{optex:sec}\nallows a set of Sm-GWs to flexibly share the\nuplink transmission bitrate of a given single operator.\nThe inter-operator sharing introduced in Section~\\ref{roam:sec}\nallows the Sm-GWs to flexibly share the uplink transmission bitrates\nof multiple operators.\nOur proposed multi-operator management framework accommodates\ndynamic changes of the traffic requirements of the small cells, such as\nchanges of the generated uplink traffic bitrates, as well as dynamic changes of\nthe operator characteristics, such as changes of the available\nuplink traffic bitrates.\nIn the proposed multi-operator management framework,\nan SDN orchestrator dynamically\nconfigures the Sm-GWs and the transport network connecting\nthe Sm-GWs to the operator gateways to flexibly adapt to changes in\nsmall cell traffic loads and the operator characteristics.\n\n\\subsection{Request and Allocation Procedures}  \\label{rap:sec}\nIn a small cell deployment environment, such as a large organization,\nmultiple Sm-GWs can serve multiple buildings.\nFor example, in a university setting,\na library can be equipped with an Sm-GW\nand the administration building can be equipped with another Sm-GW.\nThe throughput requirements and priorities\nof these buildings typically vary widely over time.\nFor instance, the administration building experiences a large\nvisitor influx during graduation and student admission periods, while\nmany students visit the library during exam week.\nMoreover, services from multiple operators may need to be\nshared among the buildings in a given organization,\ni.e., among multiple Sm-GWs.\nHence, there is a need for highly flexible traffic management\nwithin the large organization based on time-varying priorities\nand throughput requirements.\n\n\\begin{figure*}[t!]\n    \\centering\n    \\includegraphics[width=1\\textwidth]{finman_smgw\/fig8.pdf}  \n    \\caption{Illustration of SDN based multiple-operator management\nserving multiple Smart Gateways (Sm-GWs).}\n    \\label{fig_OPmanagement}\n\\end{figure*}\n\nSuppose, the Sm-GWs $s,\\ s = 1, 2, \\ldots, S$, and\noperators $o,\\ o = 1, 2, \\ldots, O$, are interconnected\nin a full mesh transport network, as illustrated in Fig.~\\ref{fig_OPmanagement}.\nAs described in Section~\\ref{gwsch:sec:excess}, with traffic adaptive\nSm-GW scheduling, each eNB $n$ sends its operator $o$ specific uplink\ntransmission bitrate request to Sm-GW $s$ in every cycle.\nThe requested uplink transmission data amounts $\\rho_{son}$\nwill typically vary over time scales that are long enough to\nreasonably permit adaptations of the Sm-GW configurations.\nFor instance, the requests will typically change\n on the time scales of several seconds or minutes, or possibly even\nlonger, such as the seasonal variations in the visitor volume to\nuniversity buildings.\nIn order to obtain the variational characteristics of the\neNB requirements, the operator specific requests at the Sm-GWs can be\naggregated over the eNBs and appropriately smoothed, i.e., averaged over time,\nto obtain an aggregate smoothed uplink transmission bitrate request\n$R_{so}$ [bit\/s] from Sm-GW $s$ to operator $o$.\n\nIdeally, the backhaul network should adapt to varying requirements at\nthe Sm-GWs to maximize the network utilization.\nWe exploit the centralized control property of SDN to\nadaptively configure the network for variable requirements.\nMore specifically, the SDN orchestrator in Fig.~\\ref{fig_OPmanagement}\noptimizes the allocations $G_{so}$ of operator $o$\nuplink transmission bitrate [bit\/s] to the individual Sm-GWs $s$.\nThe SDN orchestrator thus ensures that the grants to\nthe eNBs are globally maximized (subject to the operators' constraints\nand requirements).  When the optimized allocations $G_{so}$ are used at the\nSm-GW scheduler (Section~\\ref{gwsch:sec}),\nthe maximum allowed traffic flow is sent from the Sm-GWs\nto each operator core.\n\n\\subsection{Optimization Decision Variables and Constraints for Multi-Operator Management with Sm-GWs}\nIn this section,\n  we define a general optimization model for the multi-operator\nmanagement framework. Specifically, we define the constraints and\ndecision variables for optimizing the multi-operator management.\nThe defined decision variables and constraints are employed\nfor the operation of the SDN orchestrator,\nas detailed in Section~\\ref{sdnorchpro:sec}.\nThe SDN orchestrator can employ arbitrary objective functions and\nconstraint allocation strategies for the optimization, as illustrated\nfor an elementary example in Section~\\ref{optex:sec}.\n\n\\subsubsection{Constraints}\nRequests for the uplink transmission of $\\rho_{son}$ [bits] from\neNBs $n,\\ n = 1, 2, \\ldots, N_s$, arrive at Sm-GW $s,\\ s = 1, 2, \\ldots, S$,\nevery cycle of duration $W$ seconds, i.e., on the order of milliseconds,\nrequesting uplink transmission bitrates from operator $o,\\ o = 1, 2, \\ldots, O$.\nThe Sm-GW aggregates the requests over the eNBs $n$ and smoothes the\naggregated requests to obtain the smoothed aggregated requests $R_{so}$.\nDenoting $w$ for the cycle index, an elementary\nweighted sampling smoothing computes\n\\begin{equation}\n  R_{so}(w) = \\alpha\\; \\left( \\frac{1}{N_s}\n  \\sum_{n=1}^{N_s}  \\frac{\\rho_{son}(w)}{W}  \\right)\n   + (1-\\alpha)\\; R_{so}(w-1),\n\\label{filteredReports}\n\\end{equation}\nwhere $\\alpha$ denotes the weight for the most recent request sample.\nA wide variety of other smoothing mechanism can be employed and optimized\naccording to the specific deployment settings.\nThe smoothed requests $R_{so}$ are periodically (with a period typically\nmuch longer than the eNB reporting window) sent to the\nSDN orchestrator.\nIn particular, each Sm-GW $s$, sends a vector of smoothed requests\n$\\overrightarrow{R_{s}} = [R_{s1} \\; R_{s2} \\; \\cdots \\;R_{sO}]$\ncontaining the aggregated and smoothed requests for each operator $o$\nto the SDN orchestrator.\nThe SDN orchestrator combines the request vectors $\\overrightarrow{R_{s}}$ to\nform the request matrix\n\\begin{equation}  \\label{reportMatrix}\n\\mathbf{R} = [ R_{so} ],\\ s = 1, 2, \\ldots, S;\\ o = 1, 2, \\ldots, O.\n\\end{equation}\n\nEach operator $o,\\ o = 1, 2, \\ldots, O$, can enforce\na set of constraints $K_{oc},\\ c = 1, 2, \\ldots, C$,\nrepresented by a constraint vector\n $\\overrightarrow{K_{o}} = [K_{o1} \\; K_{o2} \\; \\cdots \\;K_{oC}]$\nthat is sent to the SDN orchestrator.\nEach constraint $c$ may be associated with a particular specification from\noperator $o$, e.g., for traffic shaping of the flows or for the aggregate\nmaximum bitrate.\nIn order to avoid clutter and not to obscure the main ideas of our\noverall multi-operator management framework, we consider in this study\na single constraint for each operator $o$.\nThat is, in place of the constraint vector $\\overrightarrow{K_{o}}$\nwe consider a single (scalar) constraint $K_o$.\nThe SDN orchestrator combines the scalar constraints from the\nvarious operators $o$ to form the constraint vector\n\\vspace{0cm}\n\\begin{equation}\n\\mathbf{K} = [K_{1} \\; K_{2} \\; \\cdots \\;K_{O}].\n\\label{constrantsVec}\n\\end{equation}\n\n\\subsubsection{Decision Variables}\nThe Sm-GW $s$ scheduler uses the operator $o$ specific grant size limits\n$\\Gamma_{so}$ to schedule\/assign uplink transmission grants to eNBs\n(see Sections~\\ref{gwsch:sec:equal} and~\\ref{gwsch:sec:excess}). By\ncontrolling the variable $\\Gamma_{so}$ specific to operator $o$ we\ncan control the flow of traffic outward from the Sm-GW, i.e.,\ntowards the respective operator $o$. The long-term average traffic\nflow rates $X_{so}$ [bit\/s] from the Sm-GW $s,\\ s = 1, 2, \\ldots, S$, to the\noperators $o,\\ o = 1, 2, \\ldots, O$, can be expressed as matrix\n\\begin{equation}   \\label{flowMatrix}\n\\mathbf{X} = [X_{so}],\\ s = 1, 2, \\ldots, S; o = 1, 2, \\ldots, O.\n\\end{equation}\n\nThe operator $o$ specific uplink transmission bitrates\n$G_{so}$ granted to the Sm-GWs are evaluated at the\nSDN orchestrator, based on the\nrequest matrix $\\mathbf{R}$ and the constraint vector $\\mathbf{K}$.\nThe orchestrator\nresponds to the request vector $\\overrightarrow{R_s}$ from each\nSm-GW $s$ with a grant vector $\\overrightarrow{G_s}$.\nAt the SDN orchestrator, the grant vectors $\\overrightarrow{G_{s}}$\ncan be combined to form the orchestrator grant matrix\n\\begin{equation}   \\label{GmaxMatrix}\n\\mathbf{G} = [G_{so}],\\ s = 1, 2, \\ldots, S; o = 1, 2, \\ldots, O.\n\\end{equation}\n$\\mathbf{G}$ is a positive (non-negative) matrix, since the\nmatrix elements $G_{so},\\ G_{so} \\geq 0$, correspond granted\nuplink transmission bitrates.\n\nOur objective is to maximize the traffic flow rates $X_{so}$ from the\nSm-GWs $s$ to the operators $o$ subject to the operator\nconstraints $\\mathbf{K}$.\nIn particular, the aggregated traffic\nsent from the Sm-GWs $s,\\ s = 1, 2, \\ldots, S$,\nto the operator $o$ core should satisfy the operator\nconstraint $K_o$, i.e.,\n\\begin{equation}\n\\sum_{s = 1}^S X_{so} \\leq K_o,\\ \\ \\ \\forall o,\\ o = 1, 2, \\ldots, O.\n\\label{Eq:OpCnstGen}\n\\end{equation}\n\nUsing the grant vector $\\overrightarrow{G_{s}}$ at Sm-GW $s$\nto assign, i.e., to schedule, uplink traffic grants to the eNBs\n(see Section~\\ref{gwsch:sec}) ensures that the\ntraffic flow rates $X_{so}$ from Sm-GW $s$ to operator $o$ are bounded\nby $G_{so}$, i.e., \\begin{equation}\nX_{so} \\leq G_{so},\\ \\ \\forall (s,o).\n\\label{eq:geqx}\n\\end{equation}\nThus, in order to ensure that the traffic flows\n$X_{so}$ satisfy the operator constraints $\\mathbf{K}$,\nthe grants $G_{so}$ must satisfy the operator constraints, i.e.,\n\\begin{equation}\n\\sum_{s = 1}^S G_{so} \\leq K_o, \\;\\;\\;\\;\\; \\forall o,\\ o = 1, 2, \\ldots, O.\n\\label{Eq:OpCnstGenGrant}\n\\end{equation}\nIn order to maximize the traffic flows $X_{so}$ to each operator $o$,\nthe SDN orchestrator needs to grant each Sm-GW $s$\nthe maximum permissible uplink transmission bitrate $G_{so}$.\n\n\n\\begin{algorithm}[t!]\n    \\caption{SDN Orchestrator procedure}\n    \\label{algo:SDN}\n    \\SetKwInOut{Input}{input}\n    \\SetKwInOut{Output}{output}\n    \\nonl {\\bfseries{1. Sm-GWs}} \\newline\n    \\nl (a) Evaluate aggregate smoothed requests $R_{so}$ from\n    eNB requests $\\rho_{son}$,  Eqn~(\\ref{filteredReports}). \\newline\n    (b) Periodically send request vector $\\overrightarrow{R_s}$\n     to SDN orchestrator\n\\newline\n    \\nonl \\If{Grant vector $\\overrightarrow{G_s}$ is received}{\n     \\nonl   Update SM-GW (to eNBs) grant size limits $\\Gamma_{so}$,\n    \\nonl }\n\n    \\nonl   {\\bfseries{2. Operators}} \\newline\n    (a) Send constraint $K_o$ to SDN orchestrator\n\n    \\nonl   {\\bfseries{3. SDN Orchestrator}}   \\newline\n    \\nonl     \\If{Request vector $\\overrightarrow{R_s}$ is received\n   \\textbf{OR} constraint $K_o$ is received}{\n     \\nonl   Re-optimize orchestrator (to Sm-GW) grants $\\mathbf{G}$ \\;\n      \\nonl  Send grant vector $\\overrightarrow{G_s}$ to Sm-GW $s$;\n    \\nonl }\n\n\\end{algorithm}\n\\subsection{SDN Orchestrator Operation}  \\label{sdnorchpro:sec}\nThe operational procedures for evaluating the\nSDN orchestrator grant matrix $\\mathbf{G}$ (\\ref{GmaxMatrix})\nare executed in parallel in the Sm-GWs, operators, and the SDN orchestrator,\nas summarized in Algorithm~\\ref{algo:SDN}.\nThe Sm-GWs aggregate and smooth the eNB requests and periodically\n  send the request vector $\\overrightarrow{R_s}$ to the SDN orchestrator.\nThe SDN orchestrator optimizes the grant matrix $\\mathbf{G}$\nupon the arrival of a new Sm-GW request vector $\\overrightarrow{R_s}$ or\n a change in an operator constraint $K_o$.\nThe orchestrator updates the Sm-GWs with the newly\nevaluated orchestrator grant vectors $\\overrightarrow{G_s}$,\nwhich update their grant size limits $\\Gamma_{so}$.\n\nOur SDN based multi-operator management framework\n  allows for a wide variety of\nresource (uplink transmission bitrate) allocations from the multiple operators\nto the Sm-GWs.\nIn order to illustrate the introduced framework, we consider\nnext an elementary specific optimization problem formulation with\na linear objective function and a proportional constraint allocation\nstrategy that allocates the uplink transmission bitrate constraints\nproportional to the requests.\nMore complex objective functions and allocation strategies,\ne.g., objective functions\nthat prioritize specific grants are an interesting direction for\nfuture research.\nWe note that this illustrative example does not exploit inter-operator\nsharing, which is examined in Section~\\ref{roam:sec}.\n\n\\subsection{Illustrative Optimization Example with Linear Objective\n  Function and Request-Proportional Constraint Allocations}  \\label{optex:sec}\nSince the grants $G_{so}$ are non-negative, an elementary objective\nfunction can linearly sum the grants $G_{so}$, i.e., as\n$\\sum_{s = 1}^S \\sum_{o = 1}^O G_{so}$.\nFor the constraint allocation,\nwe consider the aggregate over all Sm-GWs $s$ of the\naggregated smoothed requests $R_{so}$\nfor a specific operator $o$, i.e., we consider\nthe unit norm of the request vector\n$\\| \\overrightarrow{R_o} \\|_1 = \\sum_{s = 1}^S R_{so}$.\nIf $\\| \\overrightarrow{R_o} \\|_1$ is less than the operator constraint $K_o$,\nthen the corresponding grants $G_{so}$ are set to the requests,\ni.e., $G_{so} = R_{so}$.\nOn the other hand, if $\\| \\overrightarrow{R_o} \\|_1 > K_o$,\nthen we proportionally assign the operator $o$ backhaul bandwidth $K_o$, i.e.,\nwe assign the proportion $R_{so} \/ \\| \\overrightarrow{R_o} \\|_1$\nof the constraint $K_o$. Thus,\n\\begin{equation}\nG_{so} = \\min \\left( R_{so},\\\n    \\frac{R_{so}}{\\|\\overrightarrow{R_o}\\|_1} K_o\\right).\n\\label{Eq:propGrantThresh}\n\\end{equation}\nThe resulting elementary optimization problem can be summarized as:\n\\vspace{0cm}\n\\begin{equation}\n\\begin{split}\n\\text{Maximize} \\;\\;\\; \\sum_{s = 1}^S \\sum_{o = 1}^O G_{so}\\;\\;\\;\\;\n\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\\\\n\\text{Subject to:} \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\:\\:\\;\\;\\;\\;\\;\\;\\;\\;\n\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\:\\:\\;\\;\\;\\;\\\\\n\\forall s \\;\\in\\; \\{ 1, 2,\\ldots,S\\}\\;\\;\\; \\text{and}\n\\;\\;\\;\\forall o \\; \\in \\; \\{ 1, 2,\\ldots,O\\},\\\\\n-G_{so} \\leq 0,\\;\\;\\;\\;\\;\\;\\;\\:\\;\\;\\;\\;\\;\\;\\;\\;\n\\;\\;\\;\\:\\;\\;\\;\\;\\;\\;\\;\\;\\;\\\\\n\\;\\;\\; \\;G_{so} \\leq R_{so}, \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\n\\:\\:\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\\\\nG_{so} \\leq K_{o} \\frac{R_{so}}{\\| \\overrightarrow{R_o}\\|_1 }.\n\\;\\;\\;\\;\\;\\;\\;\\;\\;\\:\\;\\;\\;\\;\\;\\:\n\\end{split}\n\\label{optimization}  \\end{equation}\n\n\\subsection{Inter-Operator Sharing}\n\\label{roam:sec}\nWhen the aggregate backhaul bandwidth $\\|R_o\\|_1$ requested from\nan operator $o$ exceeds its constraint $K_o$, inter-operator sharing\ncan be employed to route the additional traffic through the\n network managed by another operator.\nOur proposed Sm-GW multi-operator management\nprovides a distinctive advantage in\nmaintaining active connections with multiple operators to\neasily route the excess traffic to a sharing operator.\nWe denote $o = m$ for the operator\nthat accepts the sharing traffic from an other operator $o = e$ whose\ntraffic constraint has been exceeded.\nIn this study, we focus on one operator accepting sharing traffic and\none operation with excess traffic.\nThe extension to sets of multiple operators accepting\nsharing traffic and multiple operators with excess traffic\nis left for future research. An operator in sharing $m$ should have\nlow incoming traffic as compared to the constraints $K_m$ in order to accept\nthe traffic from the operator in excess $e$. Therefore, for\nthe sharing ($o = m$) and excess ($o = e$) operators the requests\n$R_{so}$ need to satisfy,\n\\begin{equation}\n\\sum_{s = 1}^S R_{sm} < K_m,\\ \\ \\mbox{and}\\ \\\n\\sum_{s = 1}^S R_{se} > K_e.    \\label{Eq:op:class}\n\\end{equation}\nThe traffic rate from excess operator~$e$ that\ncan be carried by sharing operator $m$ depends on the\nunutilized slack uplink transmission bitrate of operator~$m$:\n\\begin{equation}\n\\begin{split}\n\\zeta = K_m - \\sum_{s = 1}^S R_{sm}.\n\\end{split}\n\\label{eq:op:ex}\n\\end{equation}\nIf $\\zeta > 0$, the last constraint in optimization problem\n(\\ref{optimization}) for the excess operator $e$ is replaced by the constraint\n\\begin{equation}\nG_{se} \\leq \\left(K_{e} + \\zeta\\right)\\frac{R_{se}}\n   {\\| \\overrightarrow{R_e}\\|_1 }\\ \\ \\ \\forall s.\n\\end{equation}\n\n\\begin{figure}[t!]  \\centering\n    \\includegraphics[width=3.2in]{finman_smgw\/fig9.pdf}  \n    \\caption{Traffic rates simulation.}\n    \\label{fig_opti1}  \\end{figure}\n\n\\begin{figure*}[t!]\n\t\\centering\n    \\includegraphics[width=0.9\\textwidth]{finman_smgw\/fig10.pdf}\n\t\\caption{Inter-operator sharing evaluation.}\n\t\\label{fig_roam2}\n\\end{figure*}\n\n\\subsection{Evaluation of Multi-Operator Management} \\label{OMres:sec}\nIn order to showcase the effectiveness of the SDN based\nmulti-operator management framework,\nwe conducted simulations for the elementary optimization with linear\nobjective function and proportional constraint sharing\n(see Section~\\ref{optex:sec}).\nWe consider $S=2$ Sm-GWs and $O=2$ operators.\nAs comparison benchmark, we consider a static equal allocation of\noperator uplink transmission bitrate $K_o$ to the $S$ Sm-GWs, i.e.,\neach Sm-GW $s,\\ s = 1, 2$, is allocated $K_o \/ S$ of the operator $o$\nuplink transmission bitrate.\n\n\\subsubsection{Without Inter-Operator Sharing}\nIn Fig.~\\ref{fig_opti1} we plot the Sm-GW $s$ to operator $o$\ntraffic flow rates $X_{so}$ resulting from the optimized\nSDN orchestrator grants $G_{so}$ as a function of\nthe uplink transmission bitrate requested by Sm-GWs $s = 1$ and $s = 2$\nfrom operator $o = 1$. Specifically, Sm-GW $s = 1$ requests bitrate\n$R_{11} = 2R$ and Sm-GW $s = 2$ requests bitrate $R_{21} = R$ from operator\n$o = 1$. The bitrate requests from operator $o = 2$ are fixed at 50~Mbps.\nEach operator $o,\\ o = 1, 2$,\nhas uplink transmission bitrate constraint $K_o = 100$~Mbps.\n\nWe observe from Fig.~\\ref{fig_opti1} that for requests for\n operator $o = 1$ bitrate up to $R = 25$~Mbps,\nthe traffic rates $X_{11}$ and $X_{21}$ are equal to the requests, irrespective\nof whether SDN orchestrated optimization is employed for not.\nIn contrast, as the requested bitrate increases above $R = 25$~Mbps,\ni.e., the bitrate $R_{11} = 2R$ requested by Sm-GW $s = 1$ from operator\n$o = 1$ increases above $K_1 \/ S = 50$~Mbps, the granted bitrate $G_{11}$\nwith SDN orchestration and the corresponding traffic flow $X_{11}$\ncontinue to increase. On the other hand, the granted bitrate $G_{11}$ and traffic\nflow $X_{11}$ without SDN orchestration stay limited at the static\nequal share $X_{11} = G_{11} = K_1 \/ S = 50$~Mbps.\n\nAs the requested bitrate $R$ increases above 33.3~Mbps, i.e., a total of\n$3R = 100$~Mbps requested from operator $o = 1$,\nwe observe from Fig.~\\ref{fig_opti1} that without orchestration,\nthe traffic flow $X_{21} $ from Sm-GW $s = 2$ to operator $o = 1$\ngrows to and then remains at the static equal share $K_1 \/ S = 50$~Mbps.\nThat is, the conventional static uplink transmission bitrate allocation\nresults in unfair disproportional backhaul service.\nIn contrast, our dynamic multi-operator management with\nSDN orchestrated optimization based on proportional sharing adaptively\nassigns the operator $o = 1$ bitrate to Sm-GWs $s=1$ and $s = 2$\nproportional to their requests.\n\n\\subsubsection{With Inter-Operator Sharing} \\label{sec:withroaming}\nIn Fig.~\\ref{fig_roam2}, we plot the Sm-GW $s$ to operator $o$\ntraffic flow rates $X_{so}$\nas a function of the uplink transmission bitrate $R_{11} = R$ requested by\nSm-GW $s = 1$ from operator $o = 1$ when inter-operator sharing is employed.\nSm-GW\n$s = 1$ requests bitrate $R_{12} = 100 - R$~Mbps from operator $o = 2$.\nAlso, Sm-GW $s = 2$ requests fixed bitrates\n$R_{21} = R_{22} = 20$~Mbps from each operator. Each operator $o$ has a fixed\nuplink transmission bitrate constraint of $K_o = 50$~Mbps. Note that operator\n$o = (m) = 1$ has slack uplink transmission bitrate\nwhen $R \\leq 30$~Mbps and can thus serve\nas roaming operator for the excess traffic to operator $e = 2$.\nAs $R$ increases and starts to exceed 70~Mbps, the roles are reversed, so\nthat operator $e = 1$ can send some of its excess traffic to roaming\noperator $m = 2$.\n\nFocusing initially on the case $R = 100$~Mbps, i.e., the right edge of\nFig.~\\ref{fig_roam2}, we observe that without SDN orchestrated optimization,\nSm-GW $s = 1$ can only transmit at its fixed static allocation of\n$X_{11} = K_1 \/ S = 25$~Mbps to operator $o = 1$, even though\nSm-GW $s = 1$ has a traffic load demanding $R_{11} = R = 100$~Mbps.\nAt the same time, Sm-GW $s = 2$ transmits at its requested rate\n$X_{21} = R_{21}  = 20~\\mbox{Mbps} < K_1\/S$. Thus, the operator $o = 1$\nuplink transmission bitrate $K_1$ is underutilized,\neven though Sm-GW $s = 1$ has\nmore traffic to send, but cannot due to the inflexible static\nuplink transmission bitrate allocations.\n\nWith SDN orchestrated optimization with proportional sharing,\nthe overloaded uplink transmission bitrate\n$K_1 = 50$~Mbps of operator $o = 1$ is shared between\nthe two Sm-GWs, allowing Sm-GW $s = 1$ to transmit\n$X_{11} = R_{11} \/ (R_{11} + R_{21}) = 41.7$~Mbps, while Sm-GW $s = 2$\ntransmits $X_{21} = R_{21} \/ (R_{11} + R_{21}) = 8.3$~Mbps.\nHowever, the uplink transmission bitrate\n$K_2$ of operator $o = 2$ is underutilized\nwith only Sm-GW $s = 2$ transmitting $X_{22} = 20$~Mbps.\n\nWith inter-operator sharing, the unutilized uplink transmission bitrate\n$\\zeta = K_2 - X_{22} = 30$~Mbps\nof operator $o = 2$, is used to carry excess traffic from\noperator $o = 1$.\nIn particular, the aggregate of the regular operator $o = 1$ uplink transmission\nbitrate $K_1$ and the uplink transmission bitrate available to operator $o = 1$ through\ntraffic sharing to operator $o = 2$ ($\\zeta = 30$~Mbps),\ni.e., $K_1 + \\zeta = 80$~Mbps is available to operator $o = 1$.\nWith proportional sharing, Sm-GW $s = 1$ can transmit\n$X_{11} = (K_1 + \\zeta) R_{11} \/ (R_{11} + R_{21}) = 66.7$~Mbps, while Sm-GW $s = 2$\ncan correspondingly transmit $X_{21} = 13.3$~Mbps, fully utilizing\nthe backhaul capacities of both operators.\n\nOverall, we observe from Fig.~\\ref{fig_roam2} that across the\nentire range of traffic loads $R$ from Sm-GW $s = 1$ for operator $o = 1$,\nour SDN based multi-operator orchestration with sharing is able to fully\nutilize the uplink transmission bitrates of both operators. Note in\nparticular, that depending on the traffic load, the roles\nof the two operators (excess or sharing) are dynamically adapted.\n\n\\section{\\MakeUppercase{Conclusions}}\nWe have introduced a new backhaul architecture by inserting a novel\nSmart Gateway (Sm-GW) between the wireless base stations (eNBs) and\nthe conventional operator gateways, e.g., LTE S\/P-GWs. The Sm-GW enables\nflexible support for large numbers of small cell base stations. In\nparticular, the Sm-GW adaptively schedules uplink\nbackhaul transmission grants to the individual eNBs on a\nfast (typically millisecond) timescale. In addition, an SDN\norchestrator adapts the allocation of the uplink transmission bitrate of the\nconventional gateways of multiple operators to the Sm-GWs on a slow\n(typically minutes or hours) time scale. Simulation results have\ndemonstrated that the scheduling of eNB grants by the Sm-GW can\ngreatly improve the backhaul service over conventional\nstatic backhaul uplink transmission bitrate allocations. Moreover, the\nSDN orchestrator substantially improves the utilization of the\nbackhaul bandwidth, especially when inter-operator sharing is permitted.\n\nThere are several important directions for future research on the\nSm-GW architecture and small cell backhaul in general.\nOne direction is to examine a variety of scheduling algorithms\nin the context of the Sm-GW.\nAnother direction\nis to examine different specific optimization objective functions\nwithin the general SDN orchestrator optimization introduced in this\narticle.\nMoreover, it is of interest to investigate\nQoS strategies for different traffic types, such as\ndata, voice, and video traffic \\cite{li2009ene, see2015wvs, pul2013tra,van2008tra, rei2002tra}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{MOSSCO concepts}\n\nThe modularity and coupling concepts proposed in this paper\nelaborate the design of a novel software system that emphasizes the needs of\nresearchers who want to  make maximum use of their existing knowledge in a  specific\nfield (e.g., geomorphology or marine ecology) but wish to conduct integrative research\nin a wider and flexible context. In strengthening modularity \\emph{sensu} independence\nof specific physical drivers, the new concept should, in addition to addressing the\nproblems listed above, support\n\\begin{inparaenum}[(1)]\n\\item liaisons  between traditionally separated modelling communities (e.g., coastal\nengineers, physical oceanographers and biologists),\n\\item inter-comparison  studies of, e.g., physical, geological, and biological modules, and\n\\item up-scaling studies where models developed at the  laboratory scale in a\nnon-dimensional context are applied to regional, global and Earth System scales.\n\\end{inparaenum}\n\nThe design of MOSSCO is application-oriented and driven by the demands for enabling and improving integrated\nregional coastal  modelling. It is targeted towards building coupled systems\nthat support decision making for local policies implementing the European Union Water\nFramework Directive (WFD) and Marine Strategic Planning Directive (MSPD).  From a design \npoint of view we envisioned a system that is foremost flexible and equitable.\n\n\\begin{description}\n\\item[Flexibility] means that the system itself is able to deal on the one hand \nwith a diverse small or large  constellation of coupled model components and on\nthe other hand with different orders of magnitude of spatial and temporal\nresolutions; it is able to deal equally well with zero-, one-, two- and three-dimensional\nrepresentations of the coastal system. Flexibility implies the capability to encapsulate\nalso existing legacy models to create one or more different ``ecosystems'' of models.\nThis feature should allow seamless replacement of individual model components, which\nis an important procedure in the continual development of integrated systems. Flexibly\nreplacing components finally creates a test-bed for model intercomparison studies.\n\n\\item[Equitability] means that all models in the coupled framework are treated as \nequally important, and that none is more important than any other.  This principle\ndissolves the primacy of the hydrodynamic or atmospheric models as the\ncentral hub in a coupled system.   Also, data components are as important as process \ncomponents or model output; any de facto difference in model importance should be\ngrounded on the research question, and not on technological legacy.  As complexity\ngrows by coupling\nmore and more models, this equitability also demands that experts in one particular\nmodel can rely on the functionality of other components in the system without having\nto be an expert in those models, as well. \n\\end{description}\n\nThe equitability design extends to participation:  contributions to the development\nof  components or the coupling framework itself is allowed and\nencouraged.  Anyone can use and modify the coupled framework or parts of both\nin a legal sense by open source licensing, and in an accessibility sense through\ntemplate codes and extensive documentation.\n\n\\subsection{Wrapping legacy models -- first steps in  \\texttt{PARSE}}\n\nAs MOSSCO is built around the ESMF hierarchy of components, any existing code\nthat can be wrapped in an ESMF component can be a component in MOSSCO, too.  \nThe ESMF user guide \\citep{ESMF2013usr} suggests a best \npractice method \\texttt{PARSE} to achieve this componentization of a legacy code.  \n\n\\begin{description}\n\\item[\\texttt{P}]repare the user code by splitting it into three phases that initialize, run and finalize a model;\n\\item[\\texttt{A}]dapt the model's data structures by wrapping them in ESMF infrastructure like states and fields;\n\\item[\\texttt{R}]egister the user's initialize, run, and finalize routines through ESMF;\n\\item[\\texttt{S}]chedule data exchange between components;\n\\item[\\texttt{E}]xecute a user application by calling it from an ESMF driver.\n\\end{description}\n\nThis \\texttt{PARSE} concept allows a smooth transition from a legacy model to an ESMF component. In this concept, the first \nthree steps have to be performed on the model side, and the latter two on the framework side and have been taken care \nof by the MOSSCO coupling layer. The \\emph{preparation} of the code is independent of the use of ESMF and provides \nthe basic couplability of the model; many existing models already implement this separation into initialize, run, and finalize phases, \neither structurally or more formally by implementing a Basic Model Interface \\citep[BMI,][]{Peckham2013}.  For the run \nphase, it is mandatory that this phase refers to a single model timestep and not to the entire run loop.\n\nThe \\emph{adaption} of a model's internal structures to ESMF consists of technically wrapping data into ESMF \ncommunication objects, and in providing sufficient metadata for communication.  Among these are grid definition and \ndecomposition, units and semantics of data, optimally following a metadata scheme like the widespread Climate and \nForecast \\citep[CF,][]{Eaton2011} or the more bottom-up Community Surface Dynamics Modeling System \n(CSDMS, following a scheme like  \nobject + [ operation ] + quantity, \\citealt{Peckham2014iemss}).  Both are currently being included in the emerging \nGeoscience Standard Names Ontology (GSN, \\href{http:\/\/geoscienceontology.org}{geoscienceontology.org}).  \n\nESMF provides the interfaces for models written in either the Fortran or C~programming languages; data \narrays are bundled together with related metadata in ESMF field objects.  All field objects from components are then \nbundled into exported and imported ESMF state objects to be passed between components.  As a third step, the\nESMF \\emph{registration} facility needs to be added to a user model; this step is achieved by using template code from \nany one of the examples or tutorials provided with ESMF.  The second and third step (\\emph{adapt} and \n\\emph{register}) are  typical tasks of what  \\citet[][]{Peckham2013} refers to as a component model interface (CMI);\nit  is very similar between models (and thus easily accessible from template code) and targets the interface of a \nspecific coupling framework.  \n\nMOSSCO contains CMIs for ESMF in all of its provided components (Fig.~\\ref{fig:bmicmi}).  The \ncurrent naming scheme  follows the CF convention for standard names except for quantities that are not defined by CF; \nthese names (often from biological processes) are modeled onto existing CF standard names as much as possible.  \nMOSSCO also allows the specification within other naming schemes and includes a name matching \nalgorithm to mediate between different schemes. For future development, adoption of the GSN ontology is \nforeseen.  \n\n\\subsection{Scheduling in a coupled system -- the ``S'' in  \\texttt{PARSE}}\n\nMOSSCO adds onto ESMF a scheduling system (corresponding to the fourth step in\n\\texttt{PARSE}) that  calls  the different phases of participating coupled models.  The\ncoupling time step duration of this new scheduler relies on the ESMF concept of alarms\nand a user specification of pairwise coupling intervals between models. The scheduler\nminimizes calls to participating models by flexibly adjusting time step duration to the\ngreatest common denominator of coupling intervals pertinent to each coupled model.\nUpon reading the user's coupling specification, \n\\begin{inparaenum}[(i)]\n\\item models are initialized in random order but with consideration of special initialization dependencies set by the user;\n\\item a list of alarm clocks is generated that considers all pairwise couplings a model is involved in;\n\\item special couplers associated with a pairwise coupling are executed;\n\\item the scheduler then tells each model to run until that model reaches its next alarm time;\n\\item  advancing of the scheduler to the minimum next alarm time repeats until the end of the simulation.\n\\end{inparaenum}\n\nThe MOSSCO scheduler allows for both sequential and for concurrent coupling of model components, or a hybrid \ncoupling mode.  In the concurrent mode, components run at the same time on different compute elements; in the \nsequential mode, components are executed one after another.  Recently,  \\citet{Balaji2016} demonstrated how a hybrid \ncoupling mode and fine granularity could be used to increase the performance of a system that consists of both highly \nscalable and less scalable components: In their system, they ran an ocean component concurrently with the radiation \ncode of the atmosphere sequentially to all other atmospheric process components.  \n\nFor both concurrent and sequential modes, the MOSSCO scheduler runs the connectors and mediator components\nthat exchange the data before the components are run, i.e. computations are performed on data with the same timestamp.  For sequential mode, the \ncoupling configuration allows also a scheme where consecutive components rely always on the most recently \ncalculated data from all other components (Fig.~\\ref{fig:scheduling}, see also Sect.~\\ref{sec:linkcopy}).\n\n\\subsection{Deployment of the coupled system -- the ``E'' in  \\texttt{PARSE}}\n\nMOSSCO provides a Python-based generator that dynamically creates an ESMF driver component\nin a star topology that then acts as the scheduler for the coupled system.  This generator reads a text-based\nspecification of pairwise couplings (including the coupling interval, dependencies and instantiation under\ndifferent names) and generates a Fortran source file that represents the scheduler component.  The\ngenerator takes care of compilation dependencies of the coupled models, and of coupling dependencies, such\nas grid inheritance; in addition to the basic init--run--finalize BMI scheme, it also honors multi-phase\ninitialization (as in the National Unified Operational  Prediction Capability, NUOPC, ESMF extension) and a restart phase.\n\nA MOSSCO command line utility provides a user-friendly interface to generating the scheduler,\n(re-)compiling all source codes into an executable and submitting the executable to a multi-processor system,\nincluding different high-performance computing (HPC) queueing implementations; this is the fifth step in \n\\texttt{PARSE}.  MOSSCO has been successfully deployed at several\nnational HPC centers, such as the Norddeutsche Verbund f\\\"ur Hoch- und H\\\"ochstleistungsrechnen (HLRN), \nthe  German Climate Computing Center (DKRZ), or the J\\\"ulich Supercomputing Centre  (JSC); equally,\nMOSSCO is currently functioning on a multitude of Linux and macOS laptops, desktops and multiprocessor\nworkstations using the same MOSSCO (bash-based) command line utility on all platforms.  \n\nThe MOSSCO coupling layer is coded in Fortran while most of the supporting structure is coded in\nPython and partially in Bash shell syntax.  The system requirements are a Fortran 2003 compliant\ncompiler, the CMake build system, the Git distributed version control system, Python with YAML\nsupport (version~2.6 or greater), a Network Common Data Form \\citep[NetCDF,][]{Rew1990}  library, and\nESMF (version~7 or greater).   For parallel applications, a Message Passing  library (e.g., OpenMPI) \nis  required. Many HPC centers have toolchains available that already meet all of\nthese requirements.  For an individual user installation, all requirements can be taken care of with\none of the package managers distributed with the operating system, except for the installation of\nESMF, which needs to be manually installed; MOSSCO provides a semiautomated tool for helping\nin this installation of ESMF. \n\n\\section{MOSSCO components and utilities}\n\nDriven by user needs, MOSSCO currently entails utilities for I\/O, an extensive model library, and\ncoupling functionalities (Fig.~\\ref{fig:functionalities} and Table~\\ref{tab:components}).   As a utility\nlayer on top of ESMF, MOSSCO also extends the  Application Programming Interface (API) of\nESMF by providing convenience methods to facilitate the handling of time, metadata (attributes),\nconfiguration, and to unify the provisioning and transfer of scientific data across the coupling\nframework.  The use of this utility layer is not mandatory;  any ESMF based component can be\ncoupled to the MOSSCO provided components without using this utility layer.\n\nOne of the major design principles of MOSSCO is  seamless scaling from zero-dimensional to three-dimensional spatial\nrepresentations, while maintaining the coupling configuration to the maximum extent possible.\nThis design principle builds on the dimensional-independency concept of FABM achieved by local description of processes\n(often referred to as a box model),  where the dimensionality is defined by the hydrodynamic model to which FABM is \ncoupled; MOSSCO generalizes this concept to enable especially the developers of new biological and chemical models to scale up from a\nbox-model (zero-dimensional) to a water-column (one-dimensional), sediment plate or a vertically resolved transect\n(two-dimensional), and a full atmosphere or ocean (three-dimensional) setup.\nAs a concrete example, upscaling of the novel Model for Adaptive Ecosystems in Coastal Seas \\citep[MAECS,][]\n{Wirtz2016,Kerimoglu2017} has been developed in this way going forth and back between the lab\nscale and the regional coastal ocean scale. \n\nAll utility functions and components, especially the generic I\/O facilities from MOSSCO, are able to\nhandle data of any spatial dimension.  MOSSCO communicates different dimensional information\nby grid inheritance: components that implement this are able to obtain the spatial information from another model in the coupled context.  \nUsually (but not necessarily) biological and chemical models inherit the spatial configuration from a hydrodynamic model; \nequally well, this information can be obtained from data in standardized grid description formats like Gridspec \n\\citep{Balaji2007} or the Spherical Coordinate Remapping and Interpolation Package  \\citep[SCRIP,][]{Jones1999mwr}.\n\n\\subsection{Model library: Basic Model Interfaces for scientific model components}\n\nThe model library (right branch in Fig.~\\ref{fig:functionalities}) includes new models (e.g., for filter\nfeeders and  surface waves) and wrappers to legacy models and frameworks such as FABM or\nGETM.  Some of these wrappers are  under development (e.g., Hamburg Shelf Ocean Model\n\\citep[HamSOM,][]{Harms1997} and a Lagrangian particle tracer model). Here, we briefly document\nthe model collection in particular with respect to their preparation and functioning within the new\ncoupling context. \n\n\\subsubsection{Pelagic ecosystem component}\\label{sec:fabm_pelagic}\n\nThe pelagic ecosystem component (\\texttt{fabm\\_pelagic\\_component}) collects (mostly biological) \nprocess models for aquatic systems. This component makes use of the Framework for Aquatic\nBiogeochemical Models  \\citep[][]{Bruggeman2014}.  FABM is a coupling layer to a multitude\nof biogeochemical models which provide the source-minus-sink terms for variables, their vertical\nlocal movement (e.g., due to sinking or active mobility), and diagnostic data. Each model variable \nis equipped with meta data, which is transferred by the ecosystem component into\nESMF field names and attributes. Similarly, the forcing required by the biogeochemical models\nis communicated within the framework and linked to FABM. The pelagic ecosystem component\nincludes a numerical integrator for the boundary fluxes and local state variable dynamics. \nAdvective and diffusive transport are not part of this component but are left to the hydrodynamical\nmodel through the \\texttt{transport\\_connector} (\\refsec{transport}). The close connection between\ntransport  and the pelagic ecosystem requires grid inheritance by the pelagic ecosystem\nmodel from the hydrodynamic model component.\n\nMany well known biogeochemical process models have been coded in the FABM standard by\nvarious institutes, such as the European Regional Seas Ecosystem Model\n\\citep[ERSEM,][]{Butenschoen2016}, ERGOM \\citep{Hinners2015}, PCLake \\citep{Hu2016} and\nthe Bottom RedOx Model \\citep[BROM,][]{Yakushev2017}.  All biogeochemical models complying\nwith the FABM standard can equally be used in MOSSCO, while retaining their functionality.\n\n\\subsubsection{Sediment\/soil component}\\label{sec:fabm_sediment}\n\nThe sediment component \\texttt{fabm\\_sediment\\_component}  hosts (mostly biogeochemical)\nprocess descriptions for aquatic soils. To allow efficient coupling to a pelagic ecosystem, the sediment \ncomponent inherits a horizontal grid or mesh from the coupled system and adds its own vertical coordinate, a number \nof layers of horizontally equal height for the upper soil (typical domain heights range from 10--50\\,\\unit{cm}). State \nvariables within the sediment are defined through the FABM framework within the 3D grid or mesh in the sediment. \nAs in the pelagic ecosystem component, the state variables, meta data, diagnostics and forcings are \ncommunicated via the ESMF framework to the coupled system. The sediment component is the numerical integrator for \nthe tracer dynamics within each sediment column in the horizontal grid or mesh, including diffusive transport, \ndriven by molecular diffusion of the nutrients or bioturbative mixing.\nAdditionally, a 3D variable porosity defines the fraction of pore water as part of the bulk \nsediment, while all state variables are measured per volume pore water in each cell.  FABM's infrastructure of state \nvariable properties is used to label the new Boolean property \\texttt{particulate} in FABM models to define whether a state \nvariable belongs to the solid phase within the domain.\nA typical model used in applications of the sediment component is the biogeochemical model of the \nOcean Margin Exchange Experiment OMEXDIA \\citep[][]{Soetaert1996}; a version of this model\nwith added phosphorous cycle is contained  in the FABM model library as OMEXDIA\\_P \\citep{Hofmeister2014iche}. \n\n\\subsubsection{1D Hydrodynamics: General Ocean Turbulence Model (GOTM))}\\label{sec:gotm}\n\nThe  General Ocean Turbulence Model \\citep[GOTM,][]{Burchard1999,Burchard2006jms} is\na one-dimensional  water column model for hydrodynamic and thermodynamic processes\nrelated to vertical mixing. MOSSCO provides a component for GOTM and a component\nhierarchy that considers a coupled GOTM with internally coupled FABM within one\ncomponent (\\texttt{gotm\\_fabm\\_component}), as many existing available model setups rely on the direct\ncoupling of FABM to GOTM.  This way, the modularization -- taking a coupled GOTM\/FABM apart and\nrecoupling it through the MOSSCO infrastructure, can be verified; the encapsulation of GOTM is implemented\nin the \\texttt{gotm\\_component}.\n\n\\subsubsection{3D Hydrodynamics: General Estuarine Transport Model (GETM)}\\label{sec:getm}\nMOSSCO provides an interface to the 3D coastal ocean model GETM \\citep{Burchard2002ec}.\nGETM solves the Navier-Stokes Equations under Boussinesq approximation, optionally\nincluding the nonhydrostatic pressure contribution \\citep{Klingbeil2013}. A direct interface to\nGOTM (see Section~\\ref{sec:gotm}) provides state-of-the-art turbulence closure in the vertical.\nGETM supports horizontally curvilinear and vertically adaptive meshes\n\\citep{Hofmeister2010,Graewe2015}; the different mesh topologies need to be made accessible\nas ESMF grid objects. Typically,  the GETM component exports its grid and subdomain decomposition to the\ncoupled system where the spatial and parallelization information is inherited by other \ncomponents. The interface to GETM is provided by the \n\\texttt{getm\\_component};  any model coupled to GETM via the transport component can\nhave its state variables conservatively transported by GETM (see Section~\\ref{sec:transport}).\n\n\\subsubsection{Model components for erosion, sedimentation, and their biological alteration}\\label{sec:erosed}\n\nThe erosion\/sedimentation routines of the Deltares Delft3D model \\citep[EROSED,][]\n{VanRijn2007} were encapsulated in a MOSSCO component.   EROSED uses a Partheniades--Krone equation \n\\citep{Partheniades1965} for calculating the net sediment flux of cohesive sediment at the water--sediment interface for \nmultiple SPM size classes.  MOSSCO's \\texttt{erosed\\_component} always uses the current version of \nEROSED maintained by Deltares, and with the help of subsidiary infrastructure isolates the EROSED code from the deeply intertwined \ndependencies of the original implementation.  The functionality of this erosion and sedimentation component is described in more \ndetail by \\citet{Nasermoaddeli2014ich}.\n\nFlow and sediment transport can be affected by the presence of benthic organisms in many ways. Protrusion of \nbenthic animals and macrophytes in the boundary layer changes the bed roughness and thus the bed shear stress and \nconsequently the sediment transport. The erodibility of sediment can be modified by the mucus produced by benthic \norganisms; the \nerodibility of the upper bed sediment can be altered by bioturbation generated by macrofauna \\citep{DeDeckere2001}.  \nIn the \\texttt{benthos\\_component}, these biological effects of microphytobenthos and of benthic macrofauna on \nsediment erodibility and critical bed shear stress are parameterized. The benthos effect model is described in detail by \n\\citeauthor{Nasermoaddeli2017} (submitted to Estuarine, Coastal, and Shelf Science).\n\n\\subsubsection{Filter feeding model}\\label{sec:filtration}\n\nThe generic filtration component describes the instantaneous filtration by filter feeders within the water column. The biological \nfiltration model follows \\citet{Bayne1993} and describes the filtration rate as a function of food supply; it  can be adapted \nto different species of filter feeders and was recently applied to describing the ecosystem effect of blue mussels on \noffshore wind farms as the \\texttt{filtration\\_component} of MOSSCO (\\citeauthor{Slavik2017}, to be submitted to Hydrobiologia).\nThe filtration model uses an arbitrary chemical species or compound, say phytoplankton carbon as the ``currency'' for \nprocessing.  The amount of ambient phytoplankton carbon concentration is sensed by the model organisms and it is \nfiltered along with the other nutrients (in stoichiometric proportion) out of the environment, creating a sink term for \nsubsequent numerical integration in the pelagic ecological model.  \n\n\\subsubsection{Wind waves}\\label{sec:waves}\n\nA simple wind wave model is part of the MOSSCO suite. Based on the parameterization by \\citealt{Breugem2007},\nsignificant wave height and peak wave period are estimated in terms of local water depth, wind speed and fetch length.\nThis wave data enable the inclusion\nof wave effects especially for idealized 1D water column studies, e.g. the consideration of erosion processes\ndue to wave-induced bottom stresses.\nCoupling to 3D ocean models and the calculation of additional wave-induced momentum forces there, following either the Radiation stress or\nVortex Force formulation \\citep{Moghimi2013a}, is possible as well.\nFor the inclusion of wave--wave or wave--current interaction in realistic 3D applications,\nthe coupling to a more advanced third generation wind wave model like SWAN, WaveWatch~III or Wave Atmospheric Model (WAM)\nwould be necessary. \n\n\\subsection{Input\/Output utilities} \nThe  I\/O utilities include generic coupling functionalities that deal with boundary conditions, provide a\nrestart facility, add surface, lateral and point source fluxes (lower left branch in Fig.~\\ref{fig:functionalities}).  \n\n\\subsubsection{Generic output}\\label{sec:output}\n\nThis utility component of MOSSCO provides a generic output facility \\texttt{netcdf\\_component} for any data that is \ncommunicated in the coupling framework. The component writes one- to three-dimensional time sliced data into a \nNetCDF \\citep[][]{Rew1990} file and adds metadata on the simulation to this output.  \nMultiple instances of this component can be used within a simulation, such that output of different variables, differently \nprocessed data, and output at various output time steps can be recorded. The output component is fully parallellized\nwith a grid decomposition inherited from one of the coupled science \ncomponents.\nIn order to reduce interprocess communication during runtime, each write process considers only the part of the data that resides within its compute domain. This comes \nat a cost to the user, who has to postprocess the output tiles to combine for later analysis;  a python script is provided \nwith MOSSCO that takes care of joining tiled files. \n \nThe generic output also adds meta data that is collected from the system and the user environment when the output is \nwritten to disk.   Diagnostics about the processing element and run time between output steps are recorded.   The \nstructure of the NetCDF output  follows the Climate and Forecast \\citep[CF,][]{Eaton2011} convention for physical \nvariables, geolocation, units, dimensions and methods modifying variables.  When (mostly biological) terms are not \navailable in the controlled vocabulary of CF,  names are built to resemble those contained in the standard.  \n\n\\subsubsection{Generic input}\\label{sec:input}\n\nThe generic  \\texttt{netcdf\\_input\\_component} of MOSSCO reads from NetCDF files and provides the file content \nwrapped in ESMF data structures (fields) to the coupling framework.  It is parallelized identical to the generic output \ncomponent, and inherits its decomposition from other components in the coupled system.  Data can be read for the \nentire domain or for all decomposed compute elements separately.  Upon reading of data, fields can be renamed and \nfiltered before they are passed on to the coupled system. \n\nThe input component is typically used to initialize other components, for restarting, to provide boundary conditions, and \nfor assimilating data into the coupled system.  The generic input facility supports interpolation of data in time upon \nreading the data, with nearest, most recent, and linear interpolation.   It also supports reading  climatological data and \ntranslates the climatological timestamp to a simulation present time stamp in the coupling framework.   \n\n\\subsection{MOSSCO connectors and mediators}\nInformation in the form of ESMF states that contain the output fields of every component are communicated to the\nESMF driver; requests for data by every component are also communicated to the ESMF driver component.\nMOSSCO connectors are separate components that link output and requested fields between pairwise coupled\ncomponents.   MOSSCO informally distinguishes between connector components that do not manipulate the field\ndata on transfer at all (or only slightly), and mediator components that extract and compute new data out of the input data.\n\n\\subsubsection{Link, copy and nudge connectors}\\label{sec:linkcopy}\n\nThe simplest and default connecting action between components is to share a reference (i.e. a link)\nto a single field that resides in memory and can be manipulated by each component; in contrast, the\n\\texttt{copy\\_connector} duplicates a field at coupling time.  The consideration of a link or copy connector is \nimportant for managing the data flow sequence in a coupled system: the copy mechanism ensures that two coupled  \ncomponents work on the same lagged state of data, whereas the link mechanism ensures that each\ncomponent works on the most recent data available.\n\nThe  \\texttt{nudge\\_connector} is used to consolidate output from two components by weighted averaging of the\nconnected fields.  It is typically used as a simple assimiliation tool to drive model states towards observed\nstates, or to impose boundary conditions.\n\nThese connectors can only be applied between components that run on the same grid (but maybe with a different subdomain decomposition).\nThe \\texttt{link\\_connector} can only be applied between components with an identical subdomain decomposition\nso that the components have access to the same memory.\nComponents on different grids require regridding, which is currently under development in MOSSCO.\n\n\\subsubsection{Transport connector}\\label{sec:transport}\n\nA model component qualifies as a transport component when it  offers to\ntransport  an arbitrary number of tracers in its numerical grid; this facility is\npresent, for example, in the current \\texttt{gotm\\_component} and\n\\texttt{getm\\_component}. The \\texttt{transport\\_connector} provides generic\ninfrastructure that communicates the tracer fields to be transported to the transporting\ncomponent based on the availability of both, the tracer concentrations as well as\ntheir rate of vertical movement independent of the water currents. This connector\nis usually called only once per coupled pair of components during the initialization phase.\n\n\\subsubsection{Mediators for soil--pelagic coupling}\\label{sec:bpcoupler}\n\nOne aspect of the generalized coupling infrastructure in MOSSCO is the use of\nconnecting components that mediate between technically or \nscientifically incompatible data field collections. The soil--pelagic coupling of biogeochemical\nmodel components with a variety of different state variables raises the need for these  mediators.\nThe use of mediators leaves\nthe level of data aggregation and dis-aggregation, and unit conversion to the \ncoupling routine, instead of requiring specific output from a model component depending\non its coupling partner component.\n\nFor soil--pelagic (or benthic--pelagic) coupling, the  \\texttt{soil\\_pelagic\\_connector}\nmediates the soil biogeochemistry output towards the pelagic ecosystem input and\nthe  \\texttt{pelagic\\_soil\\_connector} mediates the pelagic ecosystem output towards\nthe soil biogeochemistry input, e.g.:\n\\begin{inparaenum}[(i)]\n\\item dis-aggregation of dissolved inorganic nitrogen to dissolved ammonium and dissolved nitrate\n\\item filling missing pelagic state fields for phosphate using the Redfield-equivalent for dissolved inorganic nitrogen\n\\item calculation of the vertical flux of particulate organic matter (POM) from the water column into the sediment depending \non POM concentrations in the near-bottom water, its sinking velocity and a sedimentation efficiency depending on the \nnear-bottom turbulence. The effective vertical flux is communicated into the pelagic ecosystem component to budget the \nrespective loss, and is communicated to the soil biogeochemistry component to account for the respective new mass of \nPOM. The mediator also handles\n\\item dis-aggregation of a single oxygen concentration (allowing positive and negative values) into dissolved oxygen \nconcentration, if positive, and dissolved reduced substances, if negative.\n\\item aggregation of pelagic POM composition (variable nitrogen to carbon ratio) into fixed stoichiometry POM pools in \nthe soil biogeochemistry.\n\\end{inparaenum}\n\n\n\\section{Selected applications as feasibility tests and use cases}\\label{sec:apps}\n\nMOSSCO was designed for enhancing flexibility and equitability in environmental data and model coupling. These design goals \nhave been helpful in generating new integrated models for coastal research ranging from one-dimensional water-\ncolumn to three-dimensional, with applications at different marine stations, transects, and sea domains.  Below, we \ndescribe from a user perspective the added value and success of the design goals  obtained from using MOSSCO in \nselected applications; here, the focus is not on the scientific outcome of the application (these are described elsewhere \nby, e.g., \\citeauthor{Nasermoaddeli2017}, submitted, \\citeauthor{Slavik2017}, to be submitted, \\citealt{Wirtz2016}, \\citealt{Kerimoglu2017}). All setups\ndescribed in the use cases are available as open source (with limited forcing data due to space and bandwidth constraints).\n\n\\subsection{Helgoland station}\\label{sec:helgoland}\n\nThe seasonal dynamics of nutrients and turbidity emerges from\nthe interaction of physical, ecological and biogeochemical processes in the water column and the \nunderlying sea floor.  We resolve these dynamics in a coupled application for a \n1D~vertical water column for a station near the German offshore  island Helgoland.  Average water\ndepth around the island is 25\\,m; tidal currents are affected by\nthe M2 and S2~tides with a characteristic spring--neap cycle, with current velocity not \nexceeding 1 m\\,s$^{-1}$. \n\nThe Helgoland 1D application is realized by a coupled system consisting of GOTM hydrodynamics,\nthe pelagic FABM component with a nutrient--phytoplankton--zooplankton--detritus (NPZD) ecosystem model  \\citep{Burchard2005} and two SPM size classes,\ninteracting with the erosion and sedimentation module,  the sediment component with the OMEXDIA\\_P \nearly diagenesis submodel, and coupler components for soil--pelagic, pelagic--soil and \ntracer transport. This system and setup is described in more detail by \\citet{Hofmeister2014iche}.\n\nSimulations with this application show a prevailing seasonal cycle in the model states (Fig.~\\ref{fig:helgoland}). \nDissolved nutrients  (referred as dissolved inorganic nitrogen) are taken up by phytoplankton, which fills\nthe pool of particulate organic nitrogen during the spring bloom (Fig.~\\ref{fig:helgoland}d). The particulate organic matter sinks\n into the sediments, where it is remineralized along axis, sub-oxic and anoxic pathways;  denitrification, for example, shows a peak in late summer\n (Fig.~\\ref{fig:helgoland}b). At the end of a year, nutrient concentrations are high in the sediment and diffuse back into the\n water column up to winter values of 20--25 mmol\\,m$^{-3}$.  The seasonal variation of turbidity is a result of higher\n erosion in winter and reduced vertical transport in summer  (Fig.~\\ref{fig:helgoland}c).\n\n\\subsection{Idealized coastal 2D transect}\\label{sec:transect}\n\t\t\nThe coastal nitrogen cycle is resolved in an idealized coupled system\nfor a tidal shallow sea. This two-dimensional setup represents a\nvertically resolved cross-shore transect of 60\\,km length and 5--20\\,m\nwater depth and has  been used  by \\citet{Hofmeister2017}\nto simulate  sustained horizontal nutrient gradients by particulate matter\ntransport towards the coast.  Within the MOSSCO coupling framework,\nthe 2D~transect scenario additionally provides insights into horizontal variability of\nerosion\/sedimentation and benthic biogeochemistry. \nIts coupling configuration builds on the one used for the 1D~station\nHelgoland  (Sect.~\\ref{sec:helgoland}); the water-column hydrodynamic model\nGOTM, however, is replaced by the 3D~model GETM; a local wave component and\ndata components for open boundaries and restart has been added.\n\nFigure \\ref{fig:transect} shows exchange fluxes between the water column\nand the sediment for one year of simulation.  The simulation of turbidity, as a\nresult of pelagic SPM transport and resuspension by currents  and wave stress,\nprovides the light climate for the pelagic ecosystem. The flux of particulate\norganic carbon (POC) into the sediment reflects bloom \nevents in summer during calm weather conditions. Macrobenthic activity in the sea floor brings fresh organic matter into \nthe deeper suboxic layers of the sediment, where denitrification removes nitrogen from the pool of dissolved nutrients. \nThe coupled simulation reveals decoupled signals of benthic respiration, denitrification and nutrient reflux into the water \ncolumn, which is not resolved in monolithically coded regional ecosystem models of the North Sea \n\\citep{Lorkowski2012,Daewel2013}.\n\n\\subsection{Southern North Sea bivalve ecosystem applications}\n\nA Southern North Sea (SNS) domain was used in two  studies concerning the\neffects of bivalves on the pelagic ecosystem.  \\citeauthor{Slavik2017} (to be\nsubmitted) investigated  how the accumulation of epifauna on wind turbine\nstructures (Fig.~\\ref{fig:bivalves}d) impacts pelagic primary production and\necosystem functioning in the SNS at larger spatial scales. This study is the\nfirst of its kind that extrapolates ecosystem impacts of anthropogenic offshore\nwind farm structures from a local to a regional sea scale.  The authors use  a \nMOSSCO coupled system consisting of the hydrodynamic model GETM, the\necosystem model MAECS as described by \\citet{Kerimoglu2017}, the transport\nconnector, the filter feeder component, and several input and output components\n(Fig.~\\ref{fig:bivalves}e).   They assess the impact of anthropogenically\nenhanced filtration from  blue mussel (\\emph{Mytilus edulis}) settlement on\noffshore wind farms that are planned to meet the 40-fold increase in offshore\nwind electricity in the European Union until~2030; they find a small but \nnon-negligible large-scale effect in both phytoplankton stock and primary production, \nwhich possibly contributes to better water clarity (Fig.~\\ref{fig:bivalves}f).\n\nBiological activities of macrofauna on the sea floor mediate suspended sediment\ndynamics, at least locally. In the study  by \\citeauthor{Nasermoaddeli2017}\n(submitted), the large-scale biological contribution of benthic\nmacrofauna, represented by the key species \\emph{Fabulina fabula}\n(Fig.~\\ref{fig:bivalves}a), to suspension of sediment was investigated. Simulation\nresults for a typical winter month revealed that SPM is increased not only locally \nbut beyond the mussel inhabited zones. This effect is not limited to the near-bed\nwater layers but can be observed throughout the entire water column, especially\nduring storm events (Fig.~\\ref{fig:bivalves}c).  In this coupled application, the\nhydrodynamic model GETM, the pelagic ecosystem component with three SPM\nsize classes, the erosion--sedimentation and benthic mediation components,\nseveral input and one output components, and the transport connector were\nused (Fig.~\\ref{fig:bivalves}d).   \n\n\\section{Discussion and Outlook}\nIn merging existing frameworks that address distinct types of modularity and\nby developing a super-structure for making the multi-level coupling approach\napplicable in coastal research, the MOSSCO system largely  meets the design\ngoals \\emph{flexibility}  and \\emph{equitability}. In doing so, structural deficiencies of\nlegacy models and the need for practical compromises became very\napparent.   \n\nFor legacy reasons, \\emph{equitability} is the harder to achieve \ndesign goal.  Both the distribution of compute resources as well as the \nspatial grid definition can be in principle determined by any one of the participating\ncomponents; de facto, in marine or aquatic research, they are prescribed by\nthe hydrodynamic models \nthat have so far not been enabled to inherit a grid specification or a resource\ndistribution from a coupler or coupled system.  With the ongoing development and\ndiversification of hydrodynamic models, and no immediate benefit for the different physical\nmodels to outsource grid\/resource allocation, this situation is not likely to\nchange.  MOSSCO compromises here by its flexible grid inheritance scheme and\nwith the grid provisioning component that delivers this information to the \ncoupled system whenever a hydrodynamic component is not used.\n\nBeyond grid\/resource allocation, however, the \\emph{equitability} concept is successfully driving\nindependent developments of submodules.  We found that indeed experts\nin one particular model, e.g. the erosion module, could rely on the \nfunctionality of the other parts of the system without having to be an expert\nthemselves in all of the constituent models in the coupled application. \nThe limitations to this black-box approach became evident in the scientific\napplication and evaluation of the coupled model system, which was only\npossible when a collaboration with experts in these other model systems was\nsought.  By taking away the inaccessibility barrier and by enforcing clear separation\nof tasks the modular system stimulated a successful collaboration.  \nSustained granularity also helped to align with ongoing development\nin external packages. These can be integrated fast into the coupled system, which\ndoes not rely on specific versions of the externally provided software\nunless structural changes occur.\nLong-term supported interfaces on the external model side facilitate MOSSCO being\nup-to-date with e.g. the fast evolving GETM and FABM code bases.  \n\nWhen legacy codes were equipped with a framework-agnostic  interface\nwe encountered  four major difficulties:\n\\begin{enumerate}\n\\item For organizing the data flow between the components, MOSSCO\nuses standard names and units compatible with the infrastructure and \nlibrary of standard names and units provided in the pelagic component \nfor the FABM framework (mostly modelled on CF).  Other components, \nsuch as the BMIs of wrapped legacy models, do not provide such a \nstandard name in their own implementation, and in particular, often do\nnot adhere to a naming standard. We found ambiguity arising, e.g.,\nwith temperature to be represented as  \\texttt{temperature} vs.\\\n\\texttt{sea\\_water\\_temperature} vs.\\ \\texttt{temperature\\_in\\_water}.\nWhile this can be resolved\nbased on CF for temperature, most ecological and biogeochemical\nquantities currently lack a consistent naming scheme.  The forthcoming\nGSN ontology \\citep[building on CSDMS names,][]{Peckham2014iemss}\ncould adequately address this coupling challenge. \n\n\\item Deep subroutine hierarchies of existing models made it difficult\nto isolate desired functionality from the structural external overhead.\nIn one example, where a single functional module was taken out of the \ncontext of an existing third-party coupled system, the module depended on\nmany routines dispersed throughout that third-party system repository.\n\n\\item Components based on standalone models are developed and tested\nwith their own I\/O infrastructure and typically supply a BMI implementation\nonly for part of their state and input data fields. A new, coupled application or\ndata provisioning\/request within a coupled system can therefore easily require\na change in the model?s BMI. The implementation potential input and\noutput for all quantities, including replacement of the entire model-specific I\/O\nin the BMI is therefore desirable for new developments and refactoring.\n\n\\item Two-way coupling of mass and energy fluxes between components has\nto be integrated numerically in a conservative way, despite different time\ndiscretization schemes used within the different components. We relied on\nconservative integration of the transferred fluxes within pairwise coupled\ncomponents for thir respective coupling timestep, which is most flexible for\nasynchronous scheduling. The coupled system  itself, however, cannot ensure\nthe conservative integration of mass and energy fluxes between components. \nHere, the user needs to take care of a correct coupling configuration.\n\\end{enumerate}\n\nEfforts in making legacy models couplable, either for MOSSCO or \nsimilar frameworks, however, leads to additional benefits besides the \nimmediate applicability in an integrated context. Couplability strictly demands\nfor communication of sufficient metadata, which stimulates the quality and \nquantity of documentation and of scientific and technical reproducibility of\nlegacy models. Indeed, transparency has been greatly increased by wrapping legacy\nmodels in the MOSSO context.   All participating components  performed\nintrospection and leveraging of a collection of metadata at assembly time of the coupled\napplication and during output.  Transparency is\nexpected to be continuously increasing by new coupling demands and more\ngenerous metadata provisioning from wrapped science models.  MOSSCO \nis moving towards adopting the Common Information Model~(CIM) that is also \nrequired by Climate Model Intercomparison Project~(CMIP) participating\ncoupled models \\citep{Eyring2016}.  \n\nWith a current small development base of twelve contributors, the\nopenness concept of MOSSCO in terms of including contributions\nfrom outside the core developer team has not yet been tested; in the\ncategorization by \\citet{DeLaat2007} internal governance with \nsimple structure is sufficient at this size.  Formally, external contributions\ncan be included in MOSSCO by way of contributor license agreements.\nThe openness concept has been useful  in instigating\ndiscussions about the need for explicit (and preferably open)\nlicensing of related scientific software and data as demanded in current\nopen science strategies \\citep[e.g.,][]{Scheliga2016}.\n\nSo far, scalability in current MOSSCO applications is excellent: \nStrong scaling experiments with a coupled application using GETM,\nFABM with MAECS ($\\approx$20~additional transported 3D tracers), and FABM \nwith OMEXDIA\\_P on Jureca \\citep{Krause2016} show linear (perfect) speedup\nfrom 100 to 1000~cores, and a small leveling-off (to~85\\% of perfect scaling)\nat 3000~cores. We have not observed  loss of compute time  due to the\ninfrastructure and superstructure overhead of ESMF. \n\nProblems of multi-component systems \nneed to be solved in terms of acceptance by the research community. \nMulti-component systems are much harder to be implemented and maintained\nby individual groups, where researchers solve coastal ocean problems of a\nlarge range of complexity,  from purely hydrodynamic applications via coupled\nhydrodynamic--sediment dynamic applications to fully coupled systems. Many \nacademic problems  focus on specific mechanisms and thus do not require\nthe complete and fully coupled modular system, such that the application of the\nfull system might mean a large structural overhead and additional workload.\nMost researchers would agree on the potential necessity of following a holistic\napproach when tackling grand research questions in environmental science\nsuch as related to system responses to anthropogenic intervention. Yet, it is\nnot clear whether the up-scaling approach inherent to the addition of many\nmodular components can lead to meaningful results.\n\nAs evident form the test cases (\\refsec{apps}), MOSSCO also encourages\ncoupled applications that are far from a complete system level description.  \nWith few coupled components, the technical threshold to getting an application\nrunning on an arbitrary system is relatively low.  The user can reach a fast first success.\nMOSSCO provides a full documentation, step by step recipes, and a public bug tracker;\nit adopts abundant error reporting from ESMF and a fail fast design that\nstops a coupled applications as soon as a technical error is detected \\citep{Shore2004}.\nUsability is especially high due to an available master script that compiles,\ndeploys, and schedules a coupled application. To address a wide range of\nusers, the system is designed to run on a single processor or on a user's laptop\nequally well as on a high-performance computer using several\nthousand compute nodes.\n\nAn obvious advantage of modular coupling is the opportunity to bridge the gap\nbetween different scientific disciplines. It allows in principle to combine, e.g.,\nhydrodynamic models from oceanography with sediment transport models\nfrom coastal engineering. Thus different experts can work on their individual\nmodels but benefit from each others' progress. This seeming advantage, however,\nposes also a drawback for modular coupling approaches: An initial effort which is necessary\nfor individual models to meet the requirements of a modular modelling framework\nhas to be invested. This will only happen if there is either an urgent pressure\nto include specific model capabilities, which will be difficult to include\notherwise, or if convincing examples of possible benefits can be presented.\nIt cannot be expected that the coastal ocean modelling community\nwill agree about one coupler or one way of interfacing modules, such that it\nwill still require considerable implementation work to transfer a module from\none modular system to another. To solve this problem, coupling standards\nneed to become more general, but in turn this might even increase the\nstructural overhead in using these systems.\n\nOffsetting these concerns, the separate design of basic and component\ninterfaces (BMI\/CMI) ensures that the effort spent on wrapping an existing\nmodel, or on equipping a new model with a basic model interface is not tied\nto a particular coupling framework, or even a particular coupling framework\ntechnology.  A model that  follows BMI principles will be more easily\ninterfaced to other models no matter what coupler is used.  Wrapped legacy\nmodels from MOSSCO can thus be useful in non-ESMF contexts, as well; \nand models with an existing BMI can be integrated in MOSSCO more easily,\nin turn.\n\nOne demand for integrative modeling, which is likely best practised in open\nand flexible system approaches, arises from\ncurrent European Union legislation. The Water Framework Directive and the\nMarine Strategic Planning Directive require the description of marine\nenvironmental conditions and the development of action plans to achieve a\ngood environmental status. These objectives can initially be met by a monitoring\nprogram to determine present-day conditions but ultimately rely on numerical\nmodel studies to evaluate anthropogenic measures. This ecosystem based\napproach to management \\citep[e.g.,][]{Ruckelshaus2008} demands modelling\nsystems which are capable of taking into account hydrodynamics, biogeochemistry,\nsedimentology and their interactions to properly describe the\nenvironmental status. As further legal requirements can be expected for many\ncoastal seas worldwide, numerical\nmodelling systems applied for this task need to be flexible in terms of integrating\nadditional (e.g., site-specific) processes. In this ongoing process, the initial effort of creating a\nmodular system may be the only way forward that can take into account all relevant processes\nin the long run.\n\n\\subsection{Outlook}\n\nThe suite of components provided or encapsulated so far meets the demands\nthat were initially formulated by our users; they already allow for a wide range of\nnovel coupled applications to investigate the coastal sea.  To stimulate more\ncollaboration, however, and to bring forward a  general ``ecosystem'' of modular science\ncomponents, several legacy models could interface to MOSSCO components\nin the near future by building on complementary work at other institutions.  For example,\nthe Regional Earth\nSystem Model \\citep[RegESM,][]{Turuncoglu2013} provides ESMF interfaces for\nMITgcm, ROMS  and WAM, amongst others.\nConvergence of the development of MOSSCO and RegESM is feasible in the near term.\nAlso, the recently developed Icosahedral Non-Hydrostatic Atmospheric Model\n\\citep[ICON][]{Zaengli2015} is currently being equipped with an ESMF component model\ninterface.\n\nOnce ESMF interfaces have been developed for a legacy model, it is desirable that these\ndevelopments move out of the coupler system and are integrated into the development of\nthe legacy model itself.  This has been successfully  achieved with the ESMF interface\nfor the hydrodynamic model GETM, which is now distributed with the GETM code.\nMuch of the utility layer  developed in MOSSCO, or likewise in MAPL or\nin the ESMF extension of the WRF model, are expected to be propagated upstream into\nthe framework ESMF itself.   \n\nThe interoperability of current coupling standards will increase.   While currently there are\nthree flavors of ESMF: basic  ESMF as in MOSSCO, ESMF\/MAPL as in the GEOS-5 system,\nor ESMF\/NUOPC  as in the RegESM,  only a minor effort would be required to provide the basic\nESMF and ESMF\/MAPL implementations with a NUOPC cap and make them interoperable with\nthe entire ESMF ecosystem.    Even a coupling of ESMF based systems to OASIS\/MCT based\nsystems has been proposed; and investigation is ongoing on a \ncoupling of MOSSCO to the formal BMI for CSDMS.  \n\n\\conclusions\n\nWe problematized both the primacy of hydrodynamic models and the limited modularity in\ncoupled coastal modeling that can stand in the way of developing and applying novel and\ndiverse biogeochemical process descriptions. Such developmental potential is likely\nneeded to progress towards holistic regional coastal systems models.  We presented the\nnovel Modular System for Shelves and Coasts (MOSSCO), that is  built on coupling\nconcepts centered around equitability and flexibility to resolve the issue of obstructed modularity.\nThese concepts bring about also openness, usability, transparency and scalability. MOSSCO\nas an actual Fortran implementation of this concept includes the wrapped Framework\nfor Aquatic Biogeochemical Models (FABM) and a usability layer for the Earth System Modeling\nFramework (ESMF).\n\nMOSSCO's design principles emphasize basic couplability and rich meta information.  Basic\ncouplability requires that models communicate about flow control, compute resources, and\nabout exchanged data and metadata. We demonstrated that  the design principles flexibility\nand equitability enable the building of complex coupled models that adequately\nrepresent the complexity found in environmental modelling. In this first version, the MOSSCO\nsoftware wrapped several existing legacy models with basic model interfaces (BMI); \nwe added ESMF-specific component model interfaces (CMI) to these wrappers and other\nmodels and frameworks to build a suite of ESMF components that when coupled represent\na small part of a holistic coastal system.  These components deal with hydrodynamics,\nwaves, pelagic and sediment ecology and biogeochemistry, river loads, erosion,\nresuspension, biotic sediment modification and filter feeding. \n\nIn selected applications, each with a different research question, the applicability of the coupled\nsystem was successfully tested. MOSSCO  facilitates the development of new coupled\napplications for studying coastal processes that extend from the atmosphere through\nthe water column into the sea bed, and that range from laboratory studies to 3D\nsimulation studies of a regional sea. This system meets an infrastructural need that is\ndefined by experimenters and process modellers who develop biogeochemical, physical,\nsedimentological or ecological models at the lab scale first and who would like to\nseamlessly embed these models into the complex coupled three-dimensional coastal\nsystem. This upscaling procedure may ultimately support also the global Earth System community. \n\n\\codedataavailability{The MOSSCO software is licensed under the GNU General Public\nLicense~3.0, a copyleft open  source license that allows the  use, distribution and\nmodification of the software under the same terms.  All documentation for MOSSCO is\nlicensed under the Creative Commons Attribution Share-Alike~4.0 (CC-by-SA), a \ncopyleft open document license that allows use, distribution and modification of the\ndocumentation under the same terms. \n\nDevelopment code and documentation are currently  primarily hosted on Sourceforge\n(\\href{https:\/\/sf.net\/p\/mossco\/code}{https:\/\/sf.net\/p\/mossco\/code}) and mirrored on Github\n(\\href{https:\/\/github.com\/platipodium\/mossco-code}{https:\/\/github.com\/platipodium\/mossco-code}).\nThe release version 1.0.1 is permanently archived\non Zenodo and accessible  under the digital object identifier\n\\href{http:\/\/dx.doi.org\/10.5281\/zenodo.438922}{doi:10.5281\/zenodo.438922}.  \nAll wrapped legacy models are open source and freely available from the developing institutions;  \nfree registration is required  for accessing the Delft3D system at Deltares.\nSelected test cases are available from a separate Sourceforge repository\n\\href{https:\/\/sf.net\/p\/mossco\/setups}{https:\/\/sf.net\/p\/mossco\/setups}, where all of the data on\nwhich the presented use cases are based are freely available, with the \nexception of the meteorological forcing fields.  These are, for example, available at request online\nat \\href{http:\/\/www.coastdat.de}{http:\/\/www.coastdat.de}, from the coastDat model based\ndata base developed for the assessment of long-term changes by Helmholtz-Zentrum Geesthacht \\citep{Geyer2014}.\n}\\label{sec:availability}\n\n\\authorcontribution{C.L., R.H., K.K., H.N developed the MOSSCO components (CMI) and wrappers (BMI).  K.W, C.L., and K.K.\ndesigned the coupling philosophy, C.L. developed the user interface and the utility library. K.W., H.N., R.H., O.K. and C.L. \ncarried out and analysed simulations, based on contributions from all authors.  C.L., K.W. and R.H. wrote the manuscript with contributions from all other authors.}\n\n\\competinginterests{The authors declare that they have no conflict of interest.}\n\n\\begin{acknowledgements}\nMOSSCO is a project funded under the K\\\"ustenforschung Nordsee--Ostsee programme of the Forschung f\\\"ur \nNachhaltigkeit (FONA) agenda of the German Ministry of Education and Science (BMBF) under grant agreements \n03F0667A, 03F0667B, and 03FO668A.  This research contributes to the PACES~II programme of the Hermann von \nHelmholtz-Gemeinschaft Deutscher Forschungszentren.\nFurther financial support for K.K.\\ and H.B.\\ was provided by the Collaborative Research Centre TRR181\non Energy Transfers in Atmosphere and Ocean funded by the German Research Foundation (DFG).\nWe thank those MOSSCO developers that are not co-authors \nof this paper, amongst them Markus Kreus, Ulrich K\\\"orner and Niels Weiher, and acknowledge the support of \nWenyan Zhang in preparing the model setups.  This research is based on tremendous efforts by the open \nsource community, including but not limited to the developers of Delft3D, GETM, GOTM, FABM, ESMF, OpenMPI, \nPython, GCC and NetCDF who share their codes openly.\n\\end{acknowledgements}\n\n\\bibliographystyle{copernicus}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{SECT:intro}\n\nThe classical question in symbolic integration is whether the integral of\na given function can be written in \\lq\\lq closed form\\rq\\rq. In its most restricted form,\nthe question is whether for a given function~$f$ belonging to some domain $D$\nthere exists another function~$g$, also belonging to~$D$, such that $f=g'$. For\nexample, if $D$ is the field of rational functions, then for $f=1\/x^2$ we can\nfind $g=-1\/x$, while for $f=1\/x$ no suitable $g$ exists. When no $g$ exists\nin~$D$, there are several other questions we may ask. One possibility is to ask\nwhether there is some extension~$E$ of $D$ such that in $E$ there exists some\n$g$ with $g'=f$. For example, in the case of elementary functions, Liouville's\nprinciple restricts the possible extensions~$E$, and there are algorithms which\nconstruct such extensions whenever possible. Another possibility is\nto ask whether for some modification $\\tilde f\\in D$ of~$f$ there exists a $g\\in\nD$ such that $\\tilde f=g'$. Creative telescoping is a question of this\ntype. Here we are dealing with domains~$D$ containing functions in several\nvariables, say $x$ and~$t$, and the question is whether there is a linear\ndifferential operator~$P$, nonzero and free of~$x$, such that there exists a\n$g\\in D$ with $P\\cdot f=g'$, where $g'$ denotes the derivative of $g$ with\nrespect to~$x$. Typically, $g$~itself has the form $Q\\cdot f$ for some operator\n$Q$ (which may be zero and need not be free of~$x$). In this case, we call $P$\na telescoper for~$f$, and $Q$ a certificate for~$P$.\n\nCreative telescoping is the backbone of definite integration. Readers not\nfamiliar with this technique are referred to the literature~\\cite{PWZbook1996,Zeilberger1990c,Zeilberger1991,Zeilberger1990,Koepf1998}\nfor motivation, theory, algorithms, implementations, and applications. There are\nseveral ways to find telescopers for a given $f\\in D$. In recent years, an\napproach has become popular which has the feature that it can find a telescoper\nwithout also constructing the corresponding certificate. This is interesting\nbecause certificates tend to be much larger than telescopers, and in some\napplications only the telescoper is of interest. This approach was first\nformulated for rational functions $f\\in C(t,x)$ in~\\cite{BCCL2010} and later\ngeneralized to rational functions in several variables~\\cite{bostan13, lairez15}, to\nhyperexponential functions~\\cite{bostan13a} and, for the shift case, to hypergeometric\nterms~\\cite{chen15a} and binomial sums~\\cite{bostan15}. In the present paper, we will extend\nthe approach to algebraic functions.\n\nThe basic principle of the general approach is as follows. Assume that the\n$x$-constants $\\mathrm{Const}_x(D)=\\{\\,c\\in D:c'=0\\,\\}$ form a field and that $D$\nis a vector space over the field of $x$-constants. Assume further that there is\nsome $\\mathrm{Const}_x(D)$-linear map $[\\cdot]\\colon D\\to D$ such that for every\n$f\\in D$ there exists a $g\\in D$ with $f-[f]=g'$. Such a map is called a\n\\emph{reduction.} For example, in $D=C(t,x)$ Hermite reduction~\\cite{Hermite1872} produces for\nevery $f\\in D$ some $g\\in D$ such that $f-g'$ is either zero or a rational function\nwith a square-free denominator. In this case, we can take $[f]=f-g'$.\nIn order to find a telescoper, we can compute $[f]$, $[\\partial_t\\cdot f]$, $[\\partial_t^2\\cdot f]$, \\dots,\nuntil we find that they are linearly dependent over $\\mathrm{Const}_x(D)$.\nOnce we find a relation\n$p_0[f] + \\cdots + p_r[\\partial_t^r\\cdot f] = 0$,\nthen, by linearity,\n$[p_0 f + \\cdots + p_r \\partial_t^r\\cdot f] = 0$,\nand then, by definition of $[\\cdot]$, there exists a $g\\in D$ such that $(p_0+\\cdots + p_r\\partial_t^r)\\cdot f=g'$.\nIn other words, $P=p_0+\\cdots + p_r\\partial_t^r$ is a telescoper.\n\nThere are two ways to guarantee that this method terminates. The first requires that we already know for\nother reasons that a telescoper exists. The idea is then to show that the\nreduction $[\\cdot]$ has the property that when $f\\in D$ is such that there\nexists a $g\\in D$ with $g'=f$, then $[f]=0$. If this is the case and\n$P=p_0+\\cdots+p_r\\partial_t^r$ is a telescoper for~$f$, then $P\\cdot f$ is integrable\nin~$D$, so $[P\\cdot f]=0$, and by linearity $[f]$, \\dots, $[\\partial_t^r\\cdot f]$ are\nlinearly dependent over $\\mathrm{Const}_x(D)$. This means that the method won't\nmiss any telescoper. In particular, this argument has the nice feature that we\nare guaranteed to find a telescoper of smallest possible order~$r$. This\napproach was taken in~\\cite{chen15a}.\nThe second way consists in showing that $\\{\\,[f]:f\\in D\\,\\}$ is a finite-dimensional vector space over\n$\\mathrm{Const}_x(D)$. This approach was taken in~\\cite{BCCL2010,bostan13a}. It has the\nnice additional feature that every bound for the dimension of this vector space\ngives rise to a bound for the order of the telescoper. In particular, it implies\nthe existence of a telescoper.\n\nIn this paper, we show that Trager's Hermite reduction for algebraic\nfunctions directly gives rise to a reduction-based creative telescoping\nalgorithm via the first approach (Section~\\ref{SECT:CT-1}). We will combine Trager's Hermite reduction\nwith a second reduction, called polynomial reduction (Section~\\ref{sec:polynomial}), to obtain a reduction-based creative\ntelescoping algorithm for algebraic functions via the second approach (Section~\\ref{SECT:CT-2}).\nThis gives a new proof of a bound for the order of the telescopers, and in\nparticular an independent proof for their existence.\n\nA few years ago, Chen et al.~\\cite{chen12d} have already considered the problem of creative\ntelescoping for algebraic functions. They have pointed out that by canceling residues\nof the integrand, a given creative telescoping problem can be reduced to a creative\ntelescoping problem for a function with no residues, which may be much smaller than the\noriginal function. For this smaller function, however, they still need to construct a\ncertificate. The algorithms presented in the present paper are the first which can find\ntelescopers for algebraic functions without also constructing corresponding certificates.\nBy Theorem~6 of~\\cite{chen12d}, our results also translate into a certificate-free\ncreative telescoping algorithm for rational functions in three variables.\n\n\\section{Algebraic Functions}\n\nThroughout the paper, let $C$ be a field of characteristic zero, $K=C(t)$, and $\\bar K$ the algebraic closure of~$K$.\nWe consider algebraic functions over~$K$.  For some absolutely irreducible\npolynomial $m\\in K[x,y]$, we consider the field\n$A=K(x)[y]\/\\langle m\\rangle$. If $n=\\deg_ym$, then every element of $A$ can be written uniquely in the form\n$f=f_0+f_1y+\\cdots+f_{n-1}y^{n-1}$ for some $f_0,\\dots,f_{n-1}\\in K(x)$.\n\nThe element $y\\in A$ is a solution of the equation $m=0$,\nbecause in $A$ we have $m=0$ by construction. The polynomial $m$ also admits\n$n$ distinct solutions in the field\n\\[\n  \\bar K\\<x-a>:=\\bigcup_{r\\in\\set N\\setminus\\{0\\}} \\bar K(\\!(\\ (x-a)^{1\/r}\\ )\\!)\n\\]\nof formal Puiseux series around $a\\in\\bar K$. There are also $n$ distinct\nsolutions in the field\n\\[\n  \\bar K\\<x^{-1}>:=\\bigcup_{r\\in\\set N\\setminus\\{0\\}} \\bar K(\\!(x^{-1\/r})\\!)\n\\]\nof formal Puiseux series around~$\\infty$.\nSince $\\bar K\\<x^{-1}>$ and the $\\bar K\\<x-a>$ are fields, we can associate to every\n$f\\in A$ and every $a\\in\\bar K\\cup\\{\\infty\\}$ in a natural way $n$ distinct series\nobjects with fractional exponents, by plugging any of the $n$ distinct series solutions\nof $m$ into the representation $f=f_0+\\cdots+f_{n-1}y^{n-1}$.\nIn other words, for every $a\\in\\bar K\\cup\\{\\infty\\}$ there are $n$ distinct natural\nring homomorphisms from $A$ to $K\\<x-a>$ or $K\\<x^{-1}>$, respectively.\n\nIn the field $A$ as well as the fields $\\bar K\\<x-a>$ and $\\bar K\\<x^{-1}>$, we have\nnatural differentiations with respect to~$x$. For a series, differentiation is defined\ntermwise using the usual rules $\\bigl((x-a)^{\\nu+n}\\bigr)'=(\\nu+n)(x-a)^{\\nu+n-1}$ and\n$\\bigl((x^{-1})^{\\nu+n}\\bigr)'=-(\\nu+n)(x^{-1})^{\\nu+n+1}$. For the elements of~$A$, note\nfirst that $m(x,y)=0$ implies\n\\begin{equation}\\label{eq:yprime}\n  m(x,y)'=(\\frac d{dx}m)(x,y) + (\\frac d{dy}m)(x,y)y' = 0,\n\\end{equation}\nso $y'=-(\\frac d{dx}m)(x,y)\/(\\frac d{dy}m)(x,y)$. Regarding $m$ as element of $K(x)[y]$\nand observing that $0<\\deg_y\\frac d{dy}m<n$, we have $\\gcd(m,\\frac d{dy}m)=1$ in $K(x)[y]$,\nso $\\frac{d}{dy}m$ is invertible in $A=K(x)[y]\/\\langle m\\rangle$.\nNote that we have $x'=1$ and $c'=0$ for all $c\\in K=C(t)$, in particular also $t'=0$.\nThe derivative of an arbitrary\nelement $f\\in A$, say $f=p(x,y)$ for some $p\\in K(x)[y]$ of degree less than~$n$, is\n\\[f'=(\\frac{d}{dx}p)(x,y)+(\\frac d{dy}p)(x,y)y'.\\]\nThus we have an action of the algebra $K(x)[\\partial_x]$ of differential operators on~$A$.\n\nThe derivations on $A$ and on the series domains are compatible in the sense\nthat for every $f\\in A$, the series associated to $f'$ are precisely the\nderivatives of the series associated to~$f$.\n\nIn the context of creative telescoping, we will also need to differentiate with\nrespect to~$t$. The action of $K(x)[\\partial_x]$ on $A$ and on the series domains\nis extended to an action of $K(x)[\\partial_x,\\partial_t]$ on $A$ and on the series\ndomains. On $A$, the action of $\\partial_t$ is defined as the unique derivation with\n$\\partial_t\\cdot t=1$ and $\\partial_t\\cdot x=0$, analogously to the construction\nabove. For the series domains, $\\partial_t$ acts on the coefficients\n(which are elements of $\\bar K$) in the natural way, and does not affect~$x$.\nSince each particular element $c\\in\\bar K$ belongs to a finite algebraic extension of~$K$,\nthe result $\\partial_t\\cdot c$ is uniquely determined.\nThe actions of the larger operator algebra\n$K(x)[\\partial_x,\\partial_t]$ on $A$ and on the series domains are compatible to\neach other.\n\nIn this paper, the notation $f'$ will always refer to the derivative $\\partial_x\\cdot f$\nwith respect to~$x$, not with respect to~$t$.\n\n\\medskip\n\nTrager's Hermite reduction for algebraic functions rests on the notion of\nintegral bases. Let us recall the relevant definitions and properties.\nAlthough the elements of a Puiseux series ring $\\bar K\\<x-a>$ are formal\nobjects, the series notation suggests certain analogies with complex\nfunctions. Terms $(x-a)^\\alpha$ or $(\\tfrac1x)^\\alpha$ are\ncalled \\emph{integral} if $\\alpha\\geq0$. A series in\n$\\bar K\\<x-a>$ or $\\bar K\\<x^{-1}>$ is called integral if it only contains integral\nterms. A non-integral series is said to have a \\emph{pole} at the reference\npoint. Note that in this terminology also $1\/\\sqrt{x}$ has a pole\nat~$0$. Note also that the terminology only refers to $x$ but not to~$t$.\n\nIntegrality at $a\\in\\bar K$ is not preserved by differentiation,\nbut if $f$ is integral at~$a$, then so is $(x-a)f'$. Somewhat conversely,\nintegrality at infinity is preserved by differentiation, we even have the\nstronger property that when $f$ is integral at infinity, then not only $f'$ but also $xf'=(x^{-1})^{-1}f'$ is\nintegral at infinity.\n\nAn element $f\\in A=K(x)[y]\/\\langle m\\rangle$\nis called (locally) integral at $a\\in\\bar K\\cup\\{\\infty\\}$ if for every series\nassociated to $y$ the corresponding series for $f$ is integral.\nThe element $f$ is called (globally) integral if it is locally integral at every\n$a\\in\\bar K$ (``at all finite places'').\nThis is the case if and only if the minimal polynomial of $f$ in $K[x,y]$ is monic\nwith respect to~$y$.\nBecause of Chevalley's theorem~\\cite[page 9, Corollary 3]{Chevalley1951}, any\nnon-constant algebraic function has at least one pole. Equivalently, an element $f$ is\nintegral at all $a\\in\\bar K\\cup\\{\\infty\\}$ if and only if it is constant.\n\nFor an element $f\\in A$ to have a ``pole'' at $a\\in\\bar K\\cup\\{\\infty\\}$ means\nthat $f$ is not locally integral at~$a$; to have a ``double pole'' at $a$ means\nthat $(x-a)f$ (or $\\frac1xf$ if $a=\\infty$) is not integral; to have a ``double\nroot'' at $a$ means that $f\/(x-a)^2$ (or $f\/(\\frac1x)^2=x^2f$ if $a=\\infty$) is integral,\nand so on.\n\nThe set of all globally integral elements $f\\in A$ forms a $K[x]$-submodule of~$A$.\nA basis $\\{\\omega_1,\\dots,\\omega_n\\}$ of this module is called an \\emph{integral basis}\nfor~$A$. Such bases exist, and algorithms are known for computing them~\\cite{trager84,Rybowicz:1991:ACI:120694.120715,vanHoeij94}.\nFor a fixed $a\\in\\bar K$, let $\\bar K(x)_a$ be the ring of rational functions $p\/q$\nwith $q(a)\\neq0$, and write $\\bar K(x)_\\infty$ for the ring of all\nrational functions $p\/q$ with $\\deg_x(p)\\leq\\deg_x(q)$.\nThen the set of all $f\\in A$ which are locally integral at some\nfixed $a\\in\\bar K\\cup\\{\\infty\\}$ forms a $\\bar K(x)_a$-module. A basis of this module is\ncalled a \\emph{local integral basis} at $a$ for~$A$. Also local integral bases can\nbe computed.\n\nAn integral basis $\\{\\omega_1,\\dots,\\omega_n\\}$ is always also a $K(x)$-vector space\nbasis of~$A$. A key feature of integral bases is that they make poles explicit. Writing\nan element $f\\in A$ as a linear combination $f=\\sum_{i=1}^n f_i\\omega_i$ for some\n$f_i\\in K(x)$, we have that $f$ has a pole at $a\\in\\bar K$ if and only if at least one\nof the $f_i$ has a pole there.\n\n\\begin{lemma}\\label{lemma:1}\n  Let $\\{\\omega_1,\\dots,\\omega_n\\}$ be a local integral basis of $A$ at $a\\in\\bar K\\cup\\{\\infty\\}$.\n  Let $f\\in A$ and $f_1,\\dots,f_n\\in K(x)$ be such that $f=\\sum_{i=1}^nf_i\\omega_i$.\n  Then $f$ is integral at $a$ if and only if each $f_i\\omega_i$ is integral at~$a$.\n\\end{lemma}\n\\begin{proof}\n  The direction ``$\\Leftarrow$'' is obvious. To show ``$\\Rightarrow$'', suppose\n  that $f$ is integral at~$a$. Then there exist $w_1,\\dots,w_n\\in\\bar K(x)_a$ such that\n  $f=\\sum_{i=1}^nw_i\\omega_i$. Thus $\\sum_{i=1}^n(w_i-f_i)\\omega_i=0$, and then\n  $w_i=f_i$ for all $i$, because $\\omega_1,\\dots,\\omega_n$ is a vector space basis of~$A$.\n  As elements of $\\bar K(x)_a$, the $f_i$ are integral at~$a$, and hence also all the $f_i\\omega_i$\n  are integral at~$a$.\n\\end{proof}\n\nThe lemma says in particular that poles of the $f_i$ in a linear combination\n$\\sum_{i=1}^n f_i\\omega_i$ have no chance to cancel each other.\n\n\\begin{lemma}\\label{lemma:e}\n  Let $\\{\\omega_1,\\dots,\\omega_n\\}$ be an integral basis of~$A$.\n  Let $e\\in K[x]$ and\n  $M=((m_{i,j}))_{i,j=1}^n\\in K[x]^{n\\times n}$ be such that\n  \\[\n    e\\,\\omega_i'=\\sum_{j=1}^n m_{i,j}\\omega_j\n  \\]\n  for $i=1,\\dots,n$ and $\\gcd(e,m_{1,1},\\dots,m_{n,n})=1$.\n  Then $e$ is squarefree.\n\\end{lemma}\n\\begin{proof}\n  Let $a\\in\\bar K$ be a root of~$e$. We show that $a$ is not a multiple root.\n  Since $\\omega_i$ is integral, it is in particular locally integral at~$a$.\n  Therefore $(x-a)\\omega_i'$ is locally integral at~$a$.\n  Since $\\omega_1,\\dots,\\omega_n$ is an integral basis, it follows that\n  $(x-a)m_{i,j}\/e\\in\\bar K(x)_a$ for all~$i,j$.\n  Because of $\\gcd(e,m_{1,1},\\dots,m_{n,n})=1$, no factor $x-a$ of $e$\n  can be canceled by all the~$m_{i,j}$.\n  Therefore the factor $x-a$ can appear in $e$ only once.\n\\end{proof}\n\n\\begin{lemma} \\label{lemma:degM}\n  Let $\\{\\omega_1,\\dots,\\omega_n\\}$ be a local integral basis at infinity of~$A$.\n  Let $e\\in K[x]$ and $M=((m_{i,j}))_{i,j=1}^n\\in K[x]^{n\\times n}$\n  be defined as in Lemma~\\ref{lemma:e}. Then $\\deg_x(m_{i,j})<\\deg_x(e)$ for all $i,j$.\n\\end{lemma}\n\\begin{proof}\n  Since every $\\omega_i$ is locally integral at infinity, so is every $x\\,\\omega_i'$.\n  Since $\\omega_1,\\dots,\\omega_n$ is an integral basis at infinity, it follows that\n  $xm_{i,j}\/e\\in\\bar K(x)_\\infty$ for all~$i,j$. This means that $1+\\deg_x(m_{i,j})\\leq\\deg_x(e)$\n  for all~$i,j$, and therefore $\\deg_x(m_{i,j})<\\deg_x(e)$, as claimed.\n\\end{proof}\n\nA $K(x)$-vector space basis $\\{\\omega_1,\\dots,\\omega_n\\}$ of $A$ is\ncalled \\emph{normal} at $a\\in\\bar K\\cup\\{\\infty\\}$ if there exist $r_1,\\dots,r_n\\in\nK(x)$ such that $\\{r_1\\omega_1,\\dots,r_n\\omega_n\\}$ is a local integral basis\nat~$a$. Trager shows how to construct\nan integral basis which is normal at infinity from a given integral basis and\na given local integral basis at infinity~\\cite{trager84}.\n\nAlthough normality is a somewhat weaker condition on a basis than integrality,\nit also excludes the possibility that poles in the terms of a linear combination\nof basis elements can cancel:\n\n\\begin{lemma}\\label{lemma:3}\n  Let $\\{\\omega_1,\\dots,\\omega_n\\}$ be a basis of~$A$\n  which is normal at some $a\\in\\bar K\\cup\\{\\infty\\}$.\n  Let $f=\\sum_{i=1}^n f_i\\omega_i$ for some $f_1,\\dots,f_n\\in K(x)$.\n  Then $f$ has a pole at $a$ if and only if\n  there is some $i$ such that $f_i\\omega_i$ has a pole at~$a$.\n\\end{lemma}\n\\begin{proof}\n  Let $r_1,\\dots,r_n\\in K(x)$ be such that $r_1\\omega_1,\\dots,r_n\\omega_n$ is a\n  local integral basis at~$a$. By $f=\\sum_{i=1}^n\n  (f_ir_i^{-1})(r_i\\omega_i)$ and by Lemma~\\ref{lemma:1}, $f$~is integral at~$a$ iff all\n  $f_ir_i^{-1}r_i\\omega_i=f_i\\omega_i$ are integral at~$a$.\n\\end{proof}\n\n\\section{Hermite Reduction}\\label{sec:hermite}\n\nWe now recall the Hermite reduction for algebraic functions~\\cite{trager84,ACA1992,bronstein98}.\nLet $\\{\\omega_1,\\ldots,\\omega_n\\}$ be an integral basis for~$A$.\nFurther let $e, m_{i,j}\\in K[x]$ ($1\\leq i,j\\leq n$) be such that\n$e\\omega_i'=\\sum_{j=1}^n m_{i,j}\\omega_i$ and\n$\\gcd(e,m_{1,1},m_{1,2},\\ldots,m_{n,n})=1$.\nFor describing the Hermite reduction we fix an integrand $f\\in A$ and represent it in the\nintegral basis, i.e., $f=\\sum_{i=1}^n (f_i\/D)\\,\\omega_i$ with\n$D, f_1,\\ldots,f_n\\in K[x]$. The purpose is to find $g, h\\in A$ such that\n$f=g' + h$ and $h=\\sum_{i=1}^n(h_i\/D^\\ast)\\,\\omega_i$ with $h_1,\\ldots,h_n\\in K[x]$\nand $D^\\ast$ denoting the squarefree part of~$D$.\nAs differentiating the $\\omega_i$ can introduce\ndenominators, name\\-ly the factors of~$e$, it is convenient to consider those\ndenominators from the very beginning on, which means that we shall assume\n$e\\mid D$. Note that $\\gcd(D,f_1,\\ldots,f_n)$ can then be nontrivial.\nLet~$v\\in K[x]$ be a nontrivial squarefree factor of~$D$ of multiplicity~$\\mu>1$.\nThen~$D = uv^\\mu$ for some $u\\in K[x]$ with $\\gcd(u, v)=1$ and $\\gcd(v,v')=1$.\nOne step of the Hermite reduction is as follows:\n\\begin{equation}\\label{eq:hred}\n  \\sum_{i=1}^n \\frac{f_i}{uv^\\mu}\\omega_i =\n  \\biggl(\\sum_{i=1}^n\\frac{g_i}{v^{\\mu-1}}\\omega_i\\biggr)' +\n  \\sum_{i=1}^n \\frac{h_i}{uv^{\\mu-1}}\\omega_i,\n\\end{equation}\nwhere $g_i, h_i \\in K[x]$ and~$\\deg_x(g_i)< \\deg_x(v)$.\nThe existence of such~$g_i$'s and~$h_i$'s follows from the crucial fact that\nthe elements $s_i :=  uv^\\mu(v^{1-\\mu}\\omega_i)'$ with $i\\in \\{ 1, \\ldots, n\\}$\nform a local integral basis at each root of~$v$~\\cite[page 46]{trager84}.\nBy a repeated application of such reduction steps, one can decompose any $f\\in A$\nas $f=g' + h$ where the denominators of the coefficients of $h$ are squarefree\nand the coefficients of $g$ are proper rational functions (i.e., their numerators\nhave smaller degree than their denominators).\n\nIt was observed that Hermite reduction itself often takes less time than the construction\nof an integral basis. If Hermite reduction is applied to some other basis, for instance\nthe standard basis $\\{1,y,\\dots,y^{n-1}\\}$, it either succeeds or it runs into a division by zero.\nBronstein~\\cite{bronstein98a} noticed that when a division by zero occurs, then the basis can\nbe replaced by some other basis that is a little closer to an integral basis, just\nas much as is needed to avoid this particular division by zero. After finitely many\nsuch basis changes, the Hermite reduction will come to an end and produce a correct\noutput. This variant is known as lazy Hermite reduction.\n\n\\section{Telescoping via reductions: \\hskip0ptplus1fill\\break first approach} \\label{SECT:CT-1}\nRecall from the introduction that reduction-based creative telescoping requires\nsome $K$-linear map $[\\cdot]\\colon A\\to A$ with the property that\n$f-[f]$ is integrable in $A$ for every $f\\in A$. This is sufficient for the\ncorrectness of the method, but additional properties are needed in order to\nensure that the method terminates.\n\nAs also explained already in the introduction, one possibility consists in\nshowing that $[f]=0$ whenever $f$ is integrable. Trager showed that his\nHermite reduction has this property~\\cite[page 50, Theorem 1]{trager84}.\nFor the sake of completeness, we reproduce his proof here.\n\n\\begin{lemma}\\label{lemma:pole_at_inf}\nLet $W=\\{\\omega_1,\\dots,\\omega_n\\}$ be an integral basis for $A$ that is normal at\ninfinity. Let $g=\\sum_{i=1}^ng_i\\omega_i\\in A$ be such that all its\ncoefficients $g_i\\in K(x)$ are proper rational functions. If an integral\nelement $f\\in A$ has a pole at infinity, then also $f+g$ has a pole at\ninfinity.\n\\end{lemma}\n\\begin{proof}\nSince $f$ is assumed to be integral we can write it as\n$f=f_1\\omega_1+\\cdots+f_n\\omega_n$ with $f_i\\in K[x]$.\nIf $f$ has a pole at infinity, there is at least one index~$i$\nsuch that $f_i\\omega_i$ has a pole at infinity. There are two cases why this\ncan happen.\n\\renewcommand{\\labelenumi}{(\\alph{enumi})}\n\\begin{enumerate}\n\\item The polynomial~$f_i$ has positive degree.  This means that $f_i+g_i$ has a\n  pole at infinity, because the $g_i$ are proper rational functions.\n  Thus $(f_i+g_i)\\omega_i$ has a pole at infinity, because $\\omega_i$ has no poles\n  at finite places and therefore no root at infinity.\n\\item The integral basis element $\\omega_i$ is not constant and $f_i$ is not zero. Hence\n  $\\omega_i$ has a pole at infinity, and this also implies that $(f_i+g_i)\\omega_i$\n  has a pole at infinity, again employing the fact that $g_i$ is a proper rational function.\n\\end{enumerate}\nIn both cases, therefore, $f+g=\\sum_{i=1}^n(f_i+g_i)\\omega_i$ has a pole at\ninfinity by Lemma~\\ref{lemma:3}.\n\\end{proof}\n\n\\begin{theorem}\\label{thm:intiff0}\n  Suppose that $f\\in A$\n  has a double root at infinity (i.e., every series in $\\bar K\\<x^{-1}>$\n  associated to $f$ only contains monomials $(1\/x)^\\alpha$ with $\\alpha\\geq2$).\n  Let $W=\\{\\omega_1,\\dots,\\omega_n\\}$\n  be an integral basis for $A$ that is normal at infinity.\n  If $f=g'+h$ is the result of the Hermite reduction with respect to~$W$,\n  then $h=0$ if and only if $f$ is integrable in~$A$.\n\\end{theorem}\n\n\\begin{proof}\nThe direction ``$\\Rightarrow$'' is trivial. To show the implication\n``$\\Leftarrow$'' assume that $f$ is integrable in~$A$. From $f=g'+h$ it follows that\nthen also $h$ is integrable in~$A$; let $H\\in A$ be such that $H'=h$.  In order to show\nthat $h=0$, we show that $H$ is constant.  To this end, it suffices to show that\nit has neither finite poles nor a pole at infinity; the claim then follows from\nChevalley's theorem.\n\nIt is clear that $H$ has no finite poles because $h$ has at most simple poles\n(i.e., all series associated to $h$ have only exponents $\\geq-1$).\nThis follows from the facts that the $\\omega_i$ are integral and that\nthe coefficients of~$h$ have squarefree denominators.\n\nIf $H$ has a pole at infinity, then by Lemma~\\ref{lemma:pole_at_inf} also\n$g+H$ must have a pole at infinity, because Hermite reduction produces\n$g=\\sum_i g_i\\omega_i$ with proper rational functions~$g_i$.  On the other\nhand, since $f=g'+h=(g+H)'$ has at least a double root at infinity by\nassumption, $g+H$ must have at least a single root at infinity. This is\na contradiction.\n\\end{proof}\n\nNote that the condition in Theorem~\\ref{thm:intiff0} that $f$ has a double\nroot at infinity is not a restriction at all, as it can always be achieved by\na suitable change of variables. Let $a\\in C$ be a regular point; this means\nthat all series in $\\bar K\\<x-a>$ associated to $f$ are formal power series. By the substitution\n$x\\to a+1\/x$ the regular point~$a$ is moved to infinity. From\n\\[\n  \\int f(x) \\,\\mathrm{d}x = \\int f\\left(\\frac{1}{x}+a\\right)\\left(-\\frac{1}{x^2}\\right) \\mathrm{d}x\n\\]\nwe see that the new integrand has a double root at infinity.\n\nMoreover, since the action of $\\partial_t$ on series domains is defined coefficient-wise,\nit follows that when $f$ has at least a double root at infinity (with respect to~$x$),\nthen this is also true for $\\partial_t\\cdot f, \\partial_t^2\\cdot f, \\partial_t^3\\cdot f,\\dots$,\nand then also for every $K$-linear combination $p_0f+p_1\\partial_t\\cdot f+\\cdots+p_r\\partial_t^r\\cdot f$.\nThus Theorem~\\ref{thm:intiff0} implies that $p_0+p_1\\partial_t+\\cdots+p_r\\partial_t^r$ is a telescoper for $f$ if\n\\emph{and only if} $[p_0+p_1\\partial_t+\\cdots+p_r\\partial_t^r]=0$.\n\nWe already know for other reasons~\\cite{Zeilberger1990,chyzak00,chen12d} that\ntelescopers for algebraic functions exist, and therefore the re\\-duc\\-tion-based\ncreative telescoping procedure with Hermite reduction with respect to an\nintegral basis that is normal at infinity as reduction function succeeds when\napplied to an integrand $f\\in A$ that has a double root at infinity.\nIn particular, the method finds a telescoper of smallest possible order.\nAgain, if $f$ has no double root at infinity, we can produce one by a change of variables.\nNote that a change of variables $x\\to a+1\/x$ with $a\\in C$ has no effect on\nthe telescoper.\n\n\\begin{example}\\label{ex:ct}\nWe consider the algebraic function $f=y\/x^2$ where $y$ is a solution of the\nthird-degree polynomial equation $m(x,y) = y^3 + y + x + t = 0$. An integral\nbasis for $A=K(x)[y]\/\\langle m\\rangle$ that is normal at infinity is given by\n$\\omega_1=1$, $\\omega_2=y$, $\\omega_3=y^2$.  (This means that employing lazy\nHermite reduction avoids completely the computation of an integral basis in\nthis example.)\n\nBy solving Equation~\\eqref{eq:yprime} for $y'$ we obtain\n\\[\n  y' = \\frac{-6y^2 + 9(t+x)y - 4}{27x^2+54tx+27t^2+4}.\n\\]\nThen for the differentiation matrix~$\\frac1eM$, a simple calculation yields\n\\[\n  \\begin{pmatrix} \\omega_1' \\\\[1pt] \\omega_2' \\\\[1pt] \\omega_3' \\end{pmatrix} =\n  \\frac{1}{e} \\begin{pmatrix} 0 & 0 & 0 \\\\[1pt] -4 & 9 (t+x) & -6 \\\\[1pt] 12 (t+x) & 4 & 18 (t+x) \\\\ \\end{pmatrix}\n  \\begin{pmatrix} \\omega_1 \\\\[1pt] \\omega_2 \\\\[1pt] \\omega_3 \\end{pmatrix}\n\\]\nwith $e=27x^2+54xt+27t^2+4$. Thus we write $f=\\sum_{i=1}^3 (f_i\/D) \\omega_i$ with\n$f_1=f_3=0$, $f_2=e$, and $D=x^2e$.  After a single step the Hermite reduction\ndelivers the result\n\\[\n  f = \\biggl(\\, \\underbrace{\\vphantom{\\frac{1}{(x)}} -\\frac{y}{x}}_{\\textstyle=g_0} \\;\\biggr)' +\\>\n  \\underbrace{\\frac{-6y^2+9(x+t)y-4}{x(27x^2+54xt+27t^2+4)}}_{\\textstyle=h_0}.\n\\]\nAs the Hermite remainder~$h_0$ is nonzero, Theorem~\\ref{thm:intiff0} tells us that\n$f$ is not integrable in~$A$. Hence we continue by applying Hermite reduction to\n\\[\n  \\partial_t\\cdot f = \\frac{-6y^2+9(x+t)y-4}{x^2(27x^2+54xt+27t^2+4)}.\n\\]\nNote that we could as well take $\\partial_t\\cdot h_0$ instead of $\\partial_t\\cdot f$, which\nin general should result in a faster algorithm.\nAgain after a single reduction step, the decomposition $\\partial_t\\cdot f = g_1' + h_1$\nis obtained, where\n\\begin{align*}\n  g_1 &= \\frac{6y^2-9ty+4}{x(27t^2+4)} \\\\\n  h_1 &= \\frac{6\\bigl((9x+27t)y^2-(27xt+27t^2-2)y+6x+18t\\bigr)}{x(27t^2+4)(27x^2+54xt+27t^2+4)}.\n\\end{align*}\nSince $h_0$ and $h_1$ are linearly independent over $K=C(t)$, we continue with\n$\\partial_t^2\\cdot f$.\nThis time however, it is preferable to start the Hermite reduction\nwith $\\partial_t\\cdot h_1$, which is given by\n\\[\n  \\frac{1}{x(27t^2+4)^2(27x^2+54xt+27t^2+4)^2}.\n\\]\nSetting $v=27x^2+54xt+27t^2+4=e$ and doing one reduction step,\nthe Hermite remainder $h_2$ is found to be\n\\begin{multline*}\n \\bigl(6\\bigl((-729xt-1539t^2+96)y^2+(1215xt^2-144x+1215t^3-{}\\\\\n 306t)y-486xt-1026t^2+64\\bigr)\\bigr)\\mathrel{\\big\\slash}\\bigl(x(27t^2+4)^2e\\bigr).\n\\end{multline*}\nThe corresponding integrable part $g_2$ is not displayed here for space reasons.\n\nNow one can find a linear dependence between $h_0,h_1,h_2$ that gives rise to the telescoper\n$(27t^2+4)\\partial_t^2+81t\\partial_t+24$, which is indeed the minimal one for this example.\n\\end{example}\n\n\n\\section{Polynomial Reduction}\\label{sec:polynomial}\n\nRecall that instead of requesting that $[f]=0$ if and only if $f$ is integrable\n(first approach), we can also justify the termination of reduction-based\ncreative telescoping by showing that the $K$-vector space $\\{\\,[f]:f\\in A\\,\\}$\nhas finite dimension (second approach). If $[\\cdot]$ is just the Hermite\nreduction, we do not have this property. We therefore introduce below an\nadditional reduction, called \\emph{polynomial reduction,} which we apply after\nHermite reduction. We then show that the combined reduction (Hermite reduction\nfollowed by polynomial reduction) has the desired dimension property for the\nspace of remainders. As a result, we obtain a new bound on the order of the\ntelescoper, which is similar to those in~\\cite{chen12d,chen14a}.\n\nIn this approach, we use two integral bases. First we use a global integral basis (not\nnecessarily normal at infinity) in order to perform Hermite reduction. Then we write the\nremainder $h$ with respect to some local integral basis at infinity and perform the\npolynomial reduction on this representation.\n\nThroughout this section let $W=(\\omega_1,\\ldots,\\omega_n)^T\\in A^n$ be such\nthat $\\{\\omega_1, \\ldots, \\omega_n\\}$ is a global integral basis of~$A$, and\nlet $e\\in K[x]$ and $M=(m_{i,j})\\in K[x]^{n\\times n}$ be such that $eW'=MW$\nand $\\gcd(e, m_{1, 1}, m_{1, 2}, \\ldots, m_{n ,n})=1$. The Hermite reduction\ndescribed in Section~\\ref{sec:hermite} decomposes an input element $f\\in A$\ninto the form\n\\[\n  f = g' + h = g' + \\sum_{i=1}^n \\frac{h_i}{de} \\omega_i,\\qquad\n  g, h\\in A,\n\\]\nwith $h_i, d\\in K[x]$ such that $\\gcd(d, e)=\\gcd(h_i,de)=1$ and $d$ is squarefree.\n\\begin{lemma}\\label{LEM:d}\nIf $h$ is integrable in~$A$, then $d$ is in~$K$.\n\\end{lemma}\n\\begin{proof}\nSuppose that $h$ is integrable in~$A$, i.e., there exist $a, b_i\\in K[x]$\nsuch that $h = \\bigl(\\frac{1}{a}\\sum_{i=1}^n b_i \\omega_i\\bigr)'$. Then\n\\[\n  h = \\sum_{i=1}^n \\frac{h_i}{de}\\omega_i= \\sum_{i=1}^n \\biggl(\\Bigl(\\frac{b_i}{a}\\Bigr)' \\omega_i +\n  \\frac{b_i}{a e} \\sum_{j=1}^n  m_{i, j}\\omega_j\\biggr).\n\\]\nWe show that $a$ is constant. Otherwise, for any irreducible factor $p$ of~$a$, we would have that $h$ has a pole of\nmultiplicity greater than $1$ at the roots of~$p$. This contradicts\nthe fact that $d, e$ are squarefree. Thus, $d$ is a constant.\n\\end{proof}\n\nBy the extended Euclidean algorithm, we compute $u_i, v_i\\in K[x]$ such that\n$h_i = u_i d + v_i e$ and $\\deg_x(v_i) < \\deg_x(d)$. Then the Hermite remainder~$h$\ndecomposes as\n\\begin{equation}\\label{EQ:h}\n  \\sum_{i=1}^n \\frac{h_i}{de}\\omega_i =  \\sum_{i=1}^n \\frac{u_i}{e}\\omega_i + \\sum_{i=1}^n \\frac{v_i}{d}\\omega_i.\n\\end{equation}\n\nWe now introduce the \\emph{polynomial reduction} whose goal is to confine the $u_i$ to a finite-dimensional\nvector space over~$K$. Similar reductions have been introduced and used in creative telescoping\nfor hyperexponential functions~\\cite{bostan13a} and hypergeometric terms~\\cite{chen15a}.\nLet $V = (\\nu_1, \\ldots, \\nu_n)^T\\in A^n$ be such that its entries form a $K(x)$-basis of~$A$,\nand let $a\\in K[x]$ and $B = (b_{i, j})\\in K[x]^{n \\times n}$ be such that $aV'=BV$ and\n$\\gcd(a, b_{1, 1}, b_{1, 2}, \\ldots, b_{n ,n})=1$. Let $p = (p_1, \\ldots, p_n)\\in K[x]^n$. Then\n\\begin{equation} \\label{EQ:polyred}\n  (pV)' = \\sum_{i=1}^n (p_i \\nu_i)' = \\frac{ap' + pB}{a}\\, V.\n\\end{equation}\nThis motivates us to introduce the following definition.\n\\begin{defi}\nLet the map $\\phi_V\\colon K[x]^n \\rightarrow K[x]^n$\nbe defined by $\\phi_V(p) = ap' + pB$ for any $p\\in K[x]^n$.\nWe call $\\phi_V$ the \\emph{map for polynomial reduction} with respect to~$V$, and call\nthe subspace $\\im(\\phi_V) = \\{\\phi_V(p) \\mid p \\in K[x]^n\\}$\nthe \\emph{subspace for polynomial reduction} with respect to~$V$.\n\\end{defi}\n\nNote that, by construction and because of Lemma~\\ref{LEM:d}, $q\\in K[x]^n$ is in\n$\\im(\\phi_V)$ if and only if $\\frac{q}{a}V$ is integrable in~$A$.\n\nWe can always view an element of $K[x]^n$ (resp. $K[x]^{n\\times n}$) as a polynomial in~$x$\nwith coefficients in~$K^n$ (resp. $K^{n\\times n}$). In this sense we use the notation $\\lc(\\cdot)$\nfor the leading coefficient and $\\lt(\\cdot)$ for the leading term of a vector (resp. matrix).\nFor example, if $p\\in K[x]^n$ is of the form\n\\[\n  p = p^{(r)}x^r + \\dots + p^{(1)}x + p^{(0)},\\quad p^{(i)}\\in K^n,\n\\]\nthen $\\deg_x(p)=r$, $\\lc(p)=p^{(r)}$, and $\\lt(p)=p^{(r)}x^r$.\nLet $\\{e_1, \\ldots, e_n\\}$ be the standard basis of~$K^n$.\nThen the module $K[x]^n$ viewed as a $K$-vector space is generated by\n\\[\n  \\cS := \\bigl\\{e_ix^j \\mathrel{\\big|} 1\\leq i \\leq n,\\, j\\in \\bN\\bigr\\}.\n\\]\nWe define $K[x]_\\mu^n:=\\{p\\in K[x]^n \\mid \\deg_x(p) \\leq \\mu\\}$; as a $K$-vector\nspace it is generated by\n\\[\n  \\cS_\\mu := \\bigl\\{e_ix^j \\mathrel{\\big|} 1\\leq i \\leq n,\\, 0\\leq j\\leq \\mu\\bigr\\}.\n\\]\nAny element $p\\in K[x]_\\mu^n$ can be expressed in the\nbasis $\\cS_\\mu$ as a vector $\\vec{p}\\in K^{n(\\mu+1)}$ (in the following the\ndecoration~\\raisebox{-1pt}{$\\vec{}\\;$} always indicates such a typecast).\n\n\\begin{defi}\nLet $N_V$ be the $K$-subspace of $K[x]^n$ generated by\n\\[\n  \\bigl\\{t \\in \\cS \\mathrel{\\big|} t \\neq \\lt(p) \\ \\text{for all $p\\in \\im(\\phi_V)$}\\bigr\\}.\n\\]\nThen $K[x]^n = \\im(\\phi_V) \\oplus N_V$.\nWe call $N_V$ the \\emph{standard complement} of $\\im(\\phi_V)$.\nFor any $p\\in K[x]^n$, there exist $p_1\\in K[x]^n$ and~$p_2\\in N_V$ such that\n\\[\\frac{p}{a}V = (p_1V)' + \\frac{p_2}{a}V.\\]\nThis decomposition is called the \\emph{polynomial reduction} of~$p$\nwith respect to~$V$.\n\\end{defi}\n\n\\begin{prop}\\label{PROP:finite}\nLet $a\\in K[x]$ and $B\\in K[x]^{n \\times n}$ be such that $aV'=BV$, as before.\nIf $\\deg_x(B) \\leq \\deg_x(a)-1$, then $N_V$ is a finite-dimensional\n$K$-vector space.\n\\end{prop}\n\\begin{proof}\nIn addition to the proof of the assertion, we also explain how to determine\nthe dimension and a basis for $N_V$, for later use. For brevity, let\n$\\mu:=\\deg_x(a)-1$. We distinguish two cases.\n\n\\smallskip\n{\\it Case 1.}~\nAssume that $\\deg_x(B) < \\mu$. For any $p\\in K[x]^n$ of degree $s>0$, we have\n\\[\n  \\lt\\bigl(\\phi_V(p)\\bigr) = s\\lc(a)\\lc(p)x^{s+\\mu}.\n\\]\nThus all monomials $e_i x^j\\in \\cS$ with $1\\leq i\\leq n$ and $j\\geq \\mu+1$ are not in~$N_V$.\nLet $\\vec{B}_1, \\ldots, \\vec{B}_n$ be the columns of~$B$, expressed in the basis $\\cS_\\mu$.\nLet $C(B)$ be the $K$-subspace of $K[x]_\\mu^n$ generated by these column vectors.\nIf $q\\in \\im(\\phi_V)$, then $q = \\phi_V(p) = pB$ for some $p \\in K^n$, which implies that\n$\\vec{q}\\,$ is a linear combination of $\\vec{B}_i$'s. Then $K[x]_\\mu^n = C(B) \\oplus N_V$.\nSo $\\dim_K(N_V)= (\\mu+1)n - \\dim_K(C(B))$ and a basis of $N_V$ can be computed by\nlooking at the echelon form of the matrix $\\bigl(\\vec{B}_1, \\ldots, \\vec{B}_n\\bigr)$.\n\n\\smallskip\n{\\it Case 2.}~\nAssume that $\\deg_x(B) =\\mu$. For any $p\\in K[x]^n$ of degree $s$, we have\n\\[\n  \\lt\\bigl(\\phi_V(p)\\bigr) = \\lc(p)(s\\lc(a)I_n + \\lc(B))x^{s+\\mu}.\n\\]\nLet $\\ell$ be the largest nonnegative integer such that $-\\ell \\lc(a)$ is an\neigenvalue of $\\lc(B)\\in K^{n\\times n}$. Then for any $s>\\ell$,\nthe matrix $J_s = s\\lc(a)I_n + \\lc(B)$ is invertible. So any monomial $e_ix^j$ with $j> \\ell+\\mu$ is not in~$N_V$\nfor any $i=1, \\ldots, n$. Let $p = \\sum_{i=1}^n \\sum_{j=0}^{\\ell} p_{i, j} e_ix^j$.\nThen $\\phi_V(p)$ belongs to $K[x]_{\\ell+\\mu}^n$.\nIn the basis $\\cS_{\\ell+\\mu}$, we can\nexpress $\\phi_V(p)$ as a vector of length ${n(\\ell+\\mu+1)}$ with entries linear in the $p_{i, j}$'s.\nThis vector can be written in the form $M_{\\ell} \\vec{P}$,\nwhere $\\vec{P} = (p_{1, 0}, p_{2, 0}, \\ldots, p_{n, \\ell})^T$ and $M_{\\ell} \\in K^{n(\\ell+\\mu +1) \\times n(\\ell+1)}$.\nEvery $q\\in K[x]_{\\ell+\\mu}^n$ can be expressed as a vector $\\vec{q} \\in K^{n(\\ell + \\mu +1)}$.\nThen $q\\in\\im(\\phi_v)$ if and only if $\\vec{q}\\,$ is in the column space of~$M_{\\ell}$.\nTherefore,\n\\[K[x]_{\\ell+\\mu}^n = C({M_{\\ell}}) \\oplus N_V. \\]\nThis implies that $\\dim_K(N_V) = n(\\ell+\\mu+1) - \\rank({M_{\\ell}})$, and\na basis of $N_V$ can be computed by\nlooking at the echelon form of the matrix ${M_{\\ell}}$.\n\\end{proof}\n\nIn general, the condition $\\deg_x(B) \\leq \\deg_x(a)-1$ may not hold for an arbitrary basis~$V$ of~$A$.\nThe following lemma shows that we can perform a simple change of basis to make the condition hold.\n\n\\begin{lemma}\\label{LM:CB}\nLet~$W =\\{\\omega_1, \\ldots, \\omega_n\\}$ be an integral basis of~$A$ such that it is also normal at infinity. Then\nthere exist nonnegative integers~$\\tau_1, \\ldots, \\tau_n$ such that\n\\[ V := \\{\\nu_1, \\ldots, \\nu_n\\} \\quad \\text{with $\\nu_i = x^{-\\tau_i} \\omega_i$}\\]\nis a basis of~$A$ which is normal at $0$ and integral at all other places (including infinity).\n\\end{lemma}\n\\begin{proof}\nIt is clear that such a basis~$V$ will be normal at zero, because multiplying the generators by\nthe rational functions $x^{\\tau_i}$ brings it back to a global integral basis, which is in particular\na local integral basis at zero.\nIt is also clear that such a basis will be integral at every other point $a\\in\\bar K\\setminus\\{0\\}$, because the\nmultipliers $x^{-\\tau_i}$ are locally units at such~$a$.\nFinally, since the original basis is normal at infinity, there exist rational functions $u_1,\\dots,u_n$\nsuch that $\\{u_1\\omega_1,\\dots,u_n\\omega_n\\}$ is a local integral basis at infinity.\nSince $u_i$ can be written as $u_i=x^{-\\tau_i}\\tilde{u}_i$ with $\\tau_i\\in\\set Z$ and $\\tilde{u}_i$ being a unit\nin $\\bar{C}(x)_\\infty$, we see that also $V$ is a local integral basis at infinity.\nThe integers~$\\tau_i$ can only be nonnegative because the $\\omega_i$'s have no finite poles and therefore each\nof them is either constant or has a pole at infinity by Chevalley's theorem.\n\\end{proof}\n\n\n\nCombining the Hermite reduction and polynomial reduction, we get the following theorem.\n\\begin{theorem}\\label{THM:polyred}\nLet $W$ be an integral basis of~$A$ that is normal at infinity.\nLet $T := \\diag(x^{-\\tau_1}, \\ldots, x^{-\\tau_n}) \\in K(x)^{n\\times n}$\nbe such that $V = TW$ is integral at infinity.\nLet $e\\in K[x]$, $\\lambda \\in \\bN$, and $B, M \\in K[x]^{n \\times n} $ be such that\n$eW' = MW$ and $x^\\lambda eV' = BV$.\nThen any element $f\\in A$ can be decomposed into\n\\begin{equation}\\label{EQ:add}\nf = g' + \\frac{1}{d} PW + \\frac{1}{x^\\lambda e} QV,\n\\end{equation}\nwhere $g\\in A$, $d\\in K[x]$ is squarefree and $\\gcd(d, e)=1$, $P, Q\\in K[x]^n$ with $\\deg_x(P) < \\deg_x(d)$ and $Q\\in N_V$, which is\na finite-dimensional $K$-vector space. Moreover, $P, Q$ are zero if and only if $f$ is integrable in~$A$.\n\\end{theorem}\n\\begin{proof}\nAfter performing the Hermite reduction on $f$, we get\n\\[f = \\tilde{g}' + \\frac{1}{d} PW + \\frac{1}{e} UW,\\]\nwhere~$P = (v_1, \\ldots, v_n)\\in K[x]^n$ and $U = (u_1, \\ldots, u_n)\\in K[x]^n$\nwith~$u_i, v_i$ introduced in~\\eqref{EQ:h}. By Lemma~\\ref{LM:CB}, there exists\n$T := \\diag(x^{-\\tau_1}, \\ldots, x^{-\\tau_n}) \\in K(x)^{n\\times n}$\nsuch that $V = TW$ is normal at $0$ and integral at any other places (including infinity). Note that we can\nchoose~$T$ as the identity matrix if~$\\deg_x(M)\\leq \\deg_x(e)-1$.\nBy taking derivatives, we get\n\\[V' = \\left(T' + T\\frac{M}{e}\\right)T^{-1}V = \\frac{B}{a}V, \\]\nwhere $a=x^\\lambda e$ for some $\\lambda\\in \\bN$ and $B\\in K[x]^{n\\times n}$. Since $V$ is locally integral\nat infinity, $\\deg_x(B) \\leq \\deg_x(a)-1$ by Lemma~\\ref{lemma:degM}.\nBy expanding in terms of the new basis~$V$, we get\n\\[\\frac{1}{e} UW = \\frac{1}{a} \\tilde{U}V, \\]\nwhere $\\tilde{U} = x^\\lambda U T^{-1} \\in K[x]^n$. Next, we decompose $\\tilde{U}$ into\n$\\tilde{U} = \\phi_{V}(\\tilde{U}_1) + \\tilde{U}_2$ with $\\tilde{U}_1, \\tilde{U}_2\\in K[x]^n$ and\n$\\tilde{U}_2\\in N_V$. Then we get\n\\[\\frac{1}{e} UW = (\\tilde U_1 V)' + \\frac{1}{a} \\tilde U_2 V. \\]\nWe then get the decomposition~\\eqref{EQ:add} by setting\n$g = \\tilde g + \\tilde U_1 V$ and~$Q = U_2$.\n\nAssume that $f$ is integrable. Then Lemma~\\ref{LEM:d} implies that $d\\in K$.\nSince $\\deg_x(P) < \\deg_x(d)$, we have $P=0$. Then\n\\[\\frac{1}{x^\\lambda e} QV = \\sum_{i=1}^n (a_i \\nu_i)'\\]\nfor some $a_i\\in K[x]$. So $Q \\in \\im(\\phi_V)$.\nSince $\\im(\\phi_V) \\cap N_V = \\{0\\}$, it follows that $Q=0$.\n\\end{proof}\nThe decomposition in~\\eqref{EQ:add} is called an \\emph{additive decomposition} of~$f$ with respect to~$x$.\n\n\\section{Telescoping via reductions: \\hskip0ptplus1fill\\break second approach} \\label{SECT:CT-2}\n\nWe now discuss how to compute telescopers for algebraic functions via Hermite reduction and\npolynomial reduction.\n\nLet $W, V, e, \\lambda, M, B$ be as in Theorem~\\ref{THM:polyred}.\nTo construct a telescoper for~$f\\in A$,\nwe first consider the additive decompositions of the successive derivatives $\\partial_t^i\\cdot f$ for $i\\in \\bN$.\nAssume that\n\\[\\partial_t\\cdot W = \\frac{1}{\\tilde{e}} \\tilde{M}W \\quad \\text{and}\n\\quad \\partial_t\\cdot V = \\frac{1}{x^{\\tilde{\\lambda}} \\tilde{e}} \\tilde{B}V,\\]\nwhere $\\tilde e \\in K[x]$, $\\tilde M, \\tilde B\\in K[x]^{n\\times n}$, and $\\tilde \\lambda \\in \\bN$.\nSince $\\partial_t$ and~$\\partial_x$ commute, Proposition 7 in~\\cite{chen14a}\nimplies that $\\tilde e \\mid e$ and $x^{\\tilde{\\lambda}} \\tilde{e} \\mid x^\\lambda e$, as polynomials in $K[x]$.\nSo we can just take $\\tilde e = e$ and $\\tilde{\\lambda}  = \\lambda$ by multiplying $\\tilde M, \\tilde B$ by some factors\nof~$x^\\lambda e$. A direct calculation yields $\\partial_t\\cdot f = (\\partial_t\\cdot g)' + h$,\nwhere\n\\[h = \\left(\\partial_t\\cdot\\frac{P}{d}+\\frac{P\\tilde M}{de}\\right)W + \\left(\\partial_t\\cdot\\frac{Q}{x^\\lambda e}+ \\frac{Q\\tilde B}{x^{2\\lambda} e^2}\\right)V.\\]\nThis implies that the squarefree part of the denominator of $h$ divides $xde$. Applying Hermite reduction and polynomial reduction\nto~$h$ yields\n\\[ h = \\tilde g_1' + \\frac{1}{d} P_1W + \\frac{1}{x^\\lambda e} Q_1V,\\]\nwhere $P_1, Q_1\\in K[x]^n$ with $\\deg_x(P_1) < \\deg_x(d)$ and $Q_1\\in N_V$.\nRepeating this discussion, we get the following lemma.\n\\begin{lemma}\\label{LEM:idtf}\nFor any $i\\in \\bN$, the derivative $\\partial_t^i\\cdot f$ has an additive decomposition of the form\n\\[ \\partial_t^i\\cdot f = g_i' + \\frac{1}{d} P_iW + \\frac{1}{x^\\lambda e} Q_iV,\\]\nwhere $g_i\\in A$, $P_i, Q_i\\in K[x]^n$ with $\\deg_x(P_i) < \\deg_x(d)$ and $Q_i\\in N_V$.\n\\end{lemma}\nAs application of the above lemma, we can compute the minimal telescoper for $f$ by finding the first\nlinear dependence among the $(P_i, Q_i)$ over~$K$. We also obtain an upper bound for the order of telescopers.\n\\begin{corollary}\nEvery $f\\in A$ has a telescoper of order at most $n\\deg_x(d) + \\dim_K(N_V)$.\n\\end{corollary}\n\n\n\\begin{example}\nWe continue with Example~\\ref{ex:ct}, by applying the polynomial reduction\nto the Hermite remainders $h_0,h_1,h_2$. The matrix~$M$ computed before\nsatisfies the degree condition of Proposition~\\ref{PROP:finite}, so no\nchange of basis is needed. First we compute polynomials\n$u_i,v_i\\in K[x,y]$ such that for $i=0,1,2$ we have\n\\[\n  h_i = \\frac{u_i}{e} + \\frac{v_i}{d} = \\frac{u_i}{27x^2+54xt+27t^2+4} + \\frac{v_i}{x}.\n\\]\nBy noting that $\\deg_x(u_i)=1$ and $\\deg_x(e)=2$, we see that the\nmap for the polynomial reduction $\\phi(p) = ep' + pM$ can only be\napplied for $p\\in K^n$ so that it turns into $\\phi(p) = pM$.\nThis means that we reduce $xy^2$ using the third row of~$M$ and\n$xy$ using its second row. A straightforward calculation reveals\nthat $h_0$, $h_1$, and $h_2$ all reduce to~$0$. Hence we are left\nwith finding a $K$-linear combination among the $v_i$:\n\\begin{align*}\n v_0 &= \\frac{-6y^2+9yt-4}{27t^2+4},\\\\\n v_1 &= \\frac{6\\bigl(27y^2t-(27t^2-2)y+18t\\bigr)}{(27t^2+4)^2},\\\\\n v_2 &= \\frac{6\\bigl((96-1539t^2)y^2+(1215t^3-306t)y-1026t^2+64\\bigr)}{(27t^2+4)^3}.\n\\end{align*}\nAs expected, we obtain the same telescoper as in Example~\\ref{ex:ct}.\n\\end{example}\n\n\n\\section{The D-finite Case}\n\nWith algebraic functions being settled, it is natural to wonder about a possible\nreduction-based creative telescoping algorithm for D-finite functions. Recall\nthat in this setting we consider an operator $L\\in K(x)[\\partial_x]$ instead of a minimal\npolynomial $m\\in K[x,y]$ and instead of an algebraic field extension\n$K(x)[y]\/\\langle m\\rangle$ we consider the $K(x)[\\partial_x]$-left-module\n$A=K(x)[\\partial_x]\/\\langle L\\rangle$. Then the element $1\\in A$ is a solution of $L$\nbecause $L\\cdot 1=L=0$ in $A$ by construction. If $n=\\deg_{\\partial_x}L$, then\nthe general element of $A$ has the form\n$f=f_0+f_1\\partial_x+\\cdots+f_{n-1}\\partial_x^{n-1}$ for some\n$f_0,\\dots,f_{n-1}\\in K(x)$. Very much as in the algebraic case, there is a natural way\nto associate certain series objects to the elements of~$A$. Based on these\nseries objects, a notion of integrality was proposed last year~\\cite{kauers15b}, and an\nalgorithm for computing integral bases has been given for so-called Fuchsian\noperators~$L$.\n\nIt turns out that the Hermite reduction of Section~\\ref{sec:hermite} also works in this setting, if we say that\na term $(x-a)^\\alpha\\log(x)^\\beta$ in a generalized series solution is integral if and only if $\\alpha\\geq0$.\nNote that then $\\log(x)$ will then be considered integral at zero, despite the singularity of the complex\nfunction at this point. This has the somewhat counterintuitive consequence that $\\log(x)$ is integral at\nevery $a\\in\\bar K\\cup\\{\\infty\\}$ although it does not have a pole anywhere. For algebraic functions,\nthis is not possible by Chevalley's theorem, and this fact enters in an essential way in the proofs of\nSections~\\ref{SECT:CT-1} and~\\ref{sec:polynomial}. The lack of Chevalley's theorem is not an artefact of a (possibly\nwrong) treatment of logarithmic terms. Because of the Fuchs relation \\cite[p.~241]{schlesinger95} there exist operators\n$L\\in K(x)[\\partial_x]$ whose series solutions at any point $a\\in\\bar K\\cup\\{\\infty\\}$ have no logarithmic\nterms, only nonnegative exponents, and which are nevertheless not constant.\n\nFor the time being, the existence of such operators is a severe obstruction to a possible generalization of\nthe termination arguments for reduction-based creative telescoping from algebraic functions to Fuchsian D-finite\nfunctions. We hope to explore this topic further in the future.\n\n\\section*{Acknowledgements}\n\nWe would like to thank Ruyong Feng and Michael F.\\ Singer for helpful discussions.\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":" \nHow to Choose  \na Partner\n\nSusan Quilliam\n\nMACMILLAN\n_For Michael and Silvano. I could not wish for better friends._\n\n#### Contents\n\n1. Understanding\n\n2. Being Ready\n\n3. Looking Back\n\n4. Not Choosing\n\n5. Focussing\n\n6. Connecting\n\n7. Being in Love\n\n8. Knowing\n\nBibliography\n\nAcknowledgements\n\nPicture Acknowledgements\n\n#### 1. Understanding\n_When you make a choice, you change the future._\n\n(DEEPAK CHOPRA)\n\nChoosing a romantic partner is one of contemporary life's biggest adventures. Embark on the quest and we may meet fascinating people as well as some who make us crazy; we may rise to emotional heights as well as sinking into fury, fear and depression; we may lose direction completely before at last we find our way to love.\n\nThe real challenge is that we grow. Partner choice is a self-development journey, driving us to learn more about ourselves, about other people, about life and the way we want to live it. Take all that on board and we start to realize just how big an adventure choosing a partner is.\n\nWhat we may not realize is just how much bigger and more difficult that adventure is now more than ever before in history. For up to now, humankind has been sensible about partner choice. Of course lust and romance have had much to do with it \u2013 especially around affairs, liaisons or simple flings. But for serious lifetime pairing, people have historically leaned away from the romantic and towards the pragmatic. The rich have typically chosen a partner for honour, for fortune, for political expediency and to preserve the hereditary line. The less rich \u2013 with less to protect \u2013 have had more leeway to let hearts rule heads, but have still needed to guarantee financial security, secure practical support and bear children to provide for later years.\n\n_The Arnolfini Portrait_ by Jan van Eyck, a celebration of uber-traditional marriage with one main aim \u2013 an heir.\n\nEven in the glorious Age of Chivalry, when a knight's love for his lady was a key life aim, no one ever suggested that romance should lead to commitment; in the Court of King Arthur, Lancelot and Guinevere's real crime was not so much in adoring each other as in trying to make their adoration the basis for a 24\/7 relationship. As historian Stephanie Coontz points out in _Marriage, a History_ , while people have always enjoyed a love story, up until very recently 'our ancestors didn't live in one'.\n\nAn emotional revolution\n\nFast forward to the twentieth century and romance became an imperative. Claire Langhamer, in her book _The English in Love_ , explains that this emotional revolution had been simmering for a while, but was fully triggered into being by many and varied social changes \u2013 though who knows which of these was cause and which effect. The introduction of the contraceptive Pill making partnerships less focussed on procreation and more on emotional connection? Women becoming more educated, more highly paid and therefore more able to exit loveless marriages? The slaughter of two World Wars encouraging us to seize the day and prioritize short-term intensity over long-term commitment? The rise of social liberalization, mass education, global communication? The fall in religious belief, the rise of individual entitlement, the passing of divorce laws?\n\nWhatever the reasons, somewhere around the mid years of the last century, partnership became universally and inextricably linked with love. And that has tossed all the jigsaw-puzzle pieces into the air. For the first time, passion \u2013 sexual and emotional \u2013 has become the primary benchmark for relationship success. Think of the famous opening line of Jane Austen's _Pride and Prejudice_ : 'It is a truth universally acknowledged, that a single man in possession of a good fortune must be in want of a wife.' Then compare that with the 1949 pronouncement by Rev. Herbert Gray, chair of the National Marriage Guidance Council, that 'the only sufficient reason for marrying is that you . . . love somebody . . .' Where matrimonial ads in the mid-1900s \u2013 when my own mother was making her partnership decisions \u2013 marketed prospective mates on their cleanliness, honesty and weekly wage, our contemporary equivalent, online dating sites, now trumpet the glories of 'Chemistry', 'Encounters' and being 'Soulmates'.\n\nReality hits\n\nAll of which sounds enticing. But the reality's more problematic. For we now approach partner choice with bigger expectations, deeper confusion and heavier pressure than ever before. Blending love into the relationship mix may have promised fulfilment but it's created huge challenges.\n\nThe first issue is that we need to make more choices more often. Centre-staging love means we're likely to want to walk away from a relationship if the romance dies, while less insistence on marriage plus more liberal divorce laws means we can do so far more easily. The result is that we now have not just one window of partner choice at biological maturity \u2013 with an additional window if a spouse dies \u2013 but on average five windows through a lifetime. We may choose in our teens for first love, in our twenties for first commitment, in our thirties for parenting partnership, in our forties for post-divorce companionship, with a final choice for relationship support to take us through to death. All that plus any additional liaisons.\n\nNow set this increased need against decreased opportunity. We meet fewer partners because we are more globally mobile; as never before we move house, change jobs, relocate to new countries. We have less chance to create partnerships because we work long hours \u2013 then travel home, in different directions, to socially isolated conurbations. Plus, we're less resourced to find and choose a partner because we're less supported; we take the practical and emotional burden on our individual shoulders far more than when we only had to decide between marrying the boy\/girl next door or the one further up the street. The final outcome of this blend of more demand and less supply? Meeting a mate has rarely been so challenging.\n\nHappy endings?\n\nAnd rarely so important. As never before, loving coupledom is now regarded as the key task of the human lifetime, and even more vital because we live in a fractured and isolated society. Cue that wise verse in Genesis where God says 'it is not good for man to be alone'.\n\nWhich leads us to another problem. For now, religion has less and less place in loving relationships, just as it has in the human psyche, to the point where not only is God absent from partnership but partnership has become more significant than God. Philosopher Simon May, in his book _Love: A History_ , explains that where once we sought meaning in the divine, now that we can no longer find such meaning, we seek it elsewhere. Partnership is the source which is now expected to deliver all the hope and happiness that we originally expected to get from the deity.\n\nWhen you think about choosing a partner, do you envision a quest, a crusade, a battle, a treasure hunt . . . or a stroll in the park?\n\nNow, when we commit to someone, we're seeking a God-substitute \u2013 which means they have to be perfect. Then we have to become perfect God-substitutes for them, offering unconditional, everlasting and utterly selfless love. Coupledom has become not only a matter of practical support, continuing the line or personal fulfilment; it's now the route by which we gain sanctity and everlasting redemption.\n\nSurely that's impossible? Well, of course it is. Even in theory we can't reach the ideal expressed by philosopher Friedrich Schlegel: 'through love, humanity returns to its original state of divinity'. And in practice we have regular proof of how implausible that aim is. Proof in our own imperfect relationships, proof in the daily media coverage of failed celebrity partnerships, proof in the divorce figures that over recent years have reached 70 per cent in some European countries \u2013 and that doesn't include the endings of unmarried, therefore undocumented, relationships.\n\nIs it any wonder we panic about commitment? Traditional 'one time' selection limited our freedom, but once paired off we had the possibility of lifelong security, and a near-guarantee that if we stayed the course, society would call it a win. Now we fear that if we choose wrong, we will end up not only alone but condemned \u2013 even damned \u2013 for our failure to make love work.\n\nIf we ask older relatives what partner choice was like for their generation, and what contentment, as well as what constraints, they felt, we may be in for a surprise. Our ancestors may not have lived in a love story, but with lower expectations \u2013 both their own and their partner's \u2013 they may well have had more happy endings than we do.\n\nNew benefits\n\nAll that said, I'm hugely grateful to be living and loving today. For new order brings new benefits. Our partnerships are now our own, rather than those imposed on us by family or proscribed by community. More dating 'windows' throughout life means more go-rounds to discover which relationship decisions help us thrive. More arenas from which to choose mean more ways to find partnership outside traditional boundaries \u2013 across culture, belief, class and age range. More freedom to walk away if we pick wrongly means not being trapped for life in an unfulfilling half-death.\n\nAnd the current challenges are gradually finding solutions. Increased need and decreased opportunity are being met by a battery of ways to meet potential partners. My mother and her generation didn't imagine using dating agencies or matchmakers, but nowadays they often represent the elite level of the partner-choice range. My grandmother and her generation had never heard of the now ubiquitous speed dating \u2013 though I suspect she would have found it all great fun.\n\nThen there's new technology. In the past two decades the internet has extended choice from the few in one's 'village' to millions worldwide; has given us a plethora of extra ways to reach out to partners through websites, apps and social media; has transformed the courtship process \u2013 albeit with the downsides that any newly born innovation brings. The landscape of relationship decision-making has changed for ever; where in the early 1990s, 1 per cent of couples met through technology, now an estimated 33 per cent do and there are claims that by 2040 this number will reach an astonishing 70 per cent.\n\nThis enormous social shift is being supported by the development of new knowledge, new insights, new resources. My mother \u2013 who was a school teacher as well as an incredibly wise woman \u2013 often bemoaned the fact that 'how to love' was not on any classroom timetable; her wish is now reality, with the growing crop of relationship courses, workshops, coaching and counselling that has sprung up to meet the need. Love may never in history have been so challenging, but perhaps never before have we been so resourced to meet that challenge.\n\nStarting the adventure\n\nWhich leads us neatly to this book. I come to write it not only through my experiences as a teacher, coach and writer on relationship issues \u2013 as well, of course, as what I've learned through my own partnership decisions \u2013 but in particular through my association with the School of Life. Over the years I've worked with them, we've become more and more aware of a huge iceberg of concern around relationship choice, a concern that reaches across all genders, ages and nationalities.\n\n_How to Choose a Partner_ is a guide to finding the right partner for you \u2013 though be warned, it's not a map, not a tip-list, not an action manual. Instead, it is a series of reflections drawing on psychology, philosophy, culture and ordinary human experience. The book's wisdom is the wisdom not only of the many professionals who have considered the decisions that we make about love, but also of the class participants I have taught and the coaching clients I have worked with.\n\nThe aim is to inform, enhance and support your own thoughts, feelings and insights. Each chapter offers a different perspective on the issues, encouraging you to look not only at where you are now, but also at how your past has informed your present, how your criteria for a relationship can be clarified and refined, and how to explore whether you and a particular partner could be right for each other. In particular, the exercises and tasks that are scattered through the book invite you to consider the route you are taking on your journey and, if necessary, adjust it \u2013 to find, recognize and commit to a relationship in which you will thrive.\n\nHere is the first task, an initial question for you to consider. How do you fit into this contemporary relationship landscape? Where do you stand as regards the 'new deal' of partner choice? Do you see it as an exciting challenge or a hopeless task, a complex puzzle or a terrifying trial? You might want to complete the following sentence. 'When I think about choosing a partner, I feel\/realize\/wonder . . .' This simple exercise will tell you a great deal about your hopes, your fears, your attitudes, your feelings.\n\nAs you read on, a final optimistic thought. You are not alone. There are literally many millions of people out there who, like you, are looking for a deep connection. Like you, they have previously made the best decisions they could, given their circumstances and resources. Like you, they have sometimes suffered regrets and disappointment but are now once again wanting to love and to be loved. There are many options out there when you are ready.\n\nAnd, as the quotation at the head of this chapter suggests, by exploring these options you create a whole new set of possibilities for yourself. By taking on the adventure of choosing a partner, you have the opportunity of changing your future for ever. Starting now.\n\n#### 2. Being Ready\n_All things are ready if our mind be so._\n\n(WILLIAM SHAKESPEARE, _H ENRY V_, ACT IV, SCENE III)\n\nIt's very tempting to rush into love. It's very tempting to think we're ready to love because we want to \u2013 and there's nothing wrong with that wanting. But readiness to even look for a partner, let alone choose one, can be more complicated than it seems. Which is why this second chapter is something of an amber traffic light.\n\nThe first amber question to ask is this. Is now the right time to be seeking a committed relationship? There are many life situations, temporary transitions and extended periods where being single is essential. Perhaps our focus currently needs to flow inwards to ourselves because our energy needs to flow outwards \u2013 maybe to a demanding job, a sick parent (or child), a sudden life crisis. If so, though we may want the support of a relationship because we threaten to collapse without it, choosing a partner may actually be the last thing in the world we should be attempting. And not just because partnership's arguably the second hardest challenge of a lifetime \u2013 the first is parenting, if you're wondering \u2013 so it shouldn't be undertaken while vulnerable. But also because, vulnerable, we may choose a mate simply as a crutch; crisis over, life healed, that crutch may be superfluous. Unfair to both parties.\n\nThere are also many life phases when being single is enough \u2013 not because we are running on empty but because we are fulfilled. It can be a hugely enlightening exercise to list the people close to you then list the things they give to you, the things that enhance your life. Company, conversation, common history \u2013 or that simplest of support, a hug. Do this exercise and it may gradually dawn on you that most if not all of your needs are being met at the moment. If so, you may opt to put partner choice on hold \u2013 or choose a mate who fills the current gap even if they don't offer the traditional 24\/7 comprehensive companionship.\n\nStaying on amber\n\nThe next amber question is even more challenging. Is it ever the right time to be choosing a partner? It's said that the best thing in life is to be happily partnered and the next best thing is to be happily single \u2013 but for some people the hierarchy's reversed. Some of us are entirely whole without additions, flourish better without distractions, are simply happier alone.\n\nIf you suspect you're more contented when single, consider \u2013 and not just as a passing thought \u2013 whether partnership may not be what you are meant to do with your life. The ideal of singledom is highly valued in many spiritual traditions less because of puritanism than because it frees us to follow our real vocation. The composer Robert Schumann, when he achieved his initial musical success, is said to have compared it to his forthcoming marriage in these words: 'I doubt if being a bridegroom will be in the same class with these first joys of being a composer . . . I now . . . marry the wide world.' If you are seeking a relationship only because it is what 'everyone' does, try on for size the possibility that you are not everyone. You are special and the best way for you to thrive may be to 'marry the wide world'.\n\nPlanning\n\nFor the rest of us there are still the practicalities to consider. Do we actually have room in our lives right now to hold down a committed relationship? The reality is that online dating, for example, will likely take up an hour each night \u2013 the equivalent of a working day each week for anything up to a year.\n\nAnd once the search is over, life will be even more overextended. For love may be wonderful, but it demands time, space, energy and a willingness to accommodate. And while statistically our formal working hours at the start of the twenty-first century are apparently almost half those we endured at the start of the nineteenth, the additional claims on that time, space and energy \u2013 by family, friends, hobbies, housework, travelling, childcare, texts, email, Twitter, Facebook and the miscellaneous demands of living \u2013 are arguably double what they were a hundred years ago.\n\nThe extra sting in the tale is that the more successful we've been in life up to now, the more we've developed our career, expanded our social life and gained a rewarding lifestyle, the less room we have for partnership. The more established we are in our world, the less flexibility we have for allowing a partner into that world. To love, we may have to sacrifice at least some of the rest of our lives. (For moving stories from people who feel they've not sacrificed as they should have done, see the website www.notimeforlove.com, which describes itself as a project to acknowledge that 'in a reality where time is finite, prioritizing love, in any form, can be challenging'.)\n\nStop? Go? Wait? Hesitate? Panic and stay pinned to the pavement?\n\nTry this exercise. Map out your typical week \u2013 morning, afternoon, evening. Indicate which of those twenty-one time segments are currently full to the brim, then judge how many segments you would \u2013 or could \u2013 happily drain in order to make room for a relationship. Then ask what having a relationship might drain from the rest of your life and what the rest of your life might drain from a relationship. Is the bargain worth it? This is not, you understand, to dissuade you from the journey, but to make the advance planning more realistic.\n\nFacing fears\n\nWhich brings us to the question of emotional readiness. Here, the elephant in the room is typically the emotion of fear \u2013 particularly because choosing a partner also involves the intimidating challenge of being chosen.\n\nWhen I ask the men and women in my dating classes to think of one word to describe how they feel about partner choice, what is mentioned more than anything else is some variation on the fear concept; unease, wariness, anxiety, terror. As well as fear of making the wrong choice, we suffer fear of being rejected \u2013 'I'm too fat, too shy'; fear of being left on the shelf \u2013 'I'm too old, too boring'; and fear of being shamed by being left on the shelf, particularly when our peers all seem to have clambered down from said shelf and are now happily mated. Though on that last point, be aware that one never knows what goes on behind closed doors. Not to encourage _Schadenfreude_ , but many of the couples who seem most happy now will be the ones sobbing on your shoulder when they hit their first divorce in a few years' time.\n\nThere's also huge fear of admitting the fear. We don't want to confess \u2013 in this age of self-possession \u2013 that we're struggling, that we're not completely fine about the whole partner-choice business. _Bridget Jones's Diary_ made us laugh at Bridget's dating struggles not only because her errors redeemed ours but also because her fight for control over life and love made us more able to break the silence and admit our own struggle to ourselves and to others. And Bridget's final triumphant union with Mark Darcy, when it came, reassured us that there could be a good result even for those of us most petrified.\n\nSince Bridget, many books \u2013 both fiction and self-help \u2013 have urged a 'fake it till you make it' attitude to fear. But there is another way. More recently, and to my mind more usefully, we've started to embrace the suggestion that vulnerability may be better than bravado. Writer Bren\u00e8 Brown, in her book _Daring Greatly_ , brings the fear story up to date when she talks about the huge courage it takes to even consider entering a relationship, let alone doing so \u2013 but how the act of revealing the fear behind that courage is a key first step on the road to successful partnership. This is vulnerability. This is 'daring greatly'.\n\nMany of us already do just this. When my class participants confess their lack of self-confidence, I often look round the room awed by what these men and women are bringing to the field. Intelligence, personality, talent \u2013 but, above all, the courage and willingness, often despite earlier heartbreak, to dare greatly once again. Because of that, surely success is certain. As the Chinese proverb says, 'Pearls don't lie on the seashore. If you want one, you have to dive for it.' And we do.\n\nAvailable or not?\n\nBeing ready for love also depends not only on being emotionally open but also emotionally available. The crucial instance of this is that, however long it is since our last relationship, if we still grieve that story, if we secretly hope to open up its pages again, we can't move on to our life's next volume. Yes, it's common to believe that a new romantic connection will begin a new chapter. Scan the tabloid headlines and we find numerous triumphal reports of newly loved-up celebrities who underwent break-ups just a few weeks ago. Scan the online dating sites and we similarly discover the many profiles saying 'We separated last week so now I'm keen to find love again'.\n\nBut the assumption's unsound. Yes, humans are wired to bond \u2013 but when a bond breaks, humans are doubly wired to suffer and that makes us unfit to bond again for a while. Helen Fisher, Professor of Anthropology at Rutgers University, MRI-scanned fifteen newly single students and found their brains shimmering with a pain akin to that of going cold turkey from drug addiction. So it's not surprising that post-break-up is rarely the moment when we are emotionally available to connect with another. Problem is, that's precisely the moment in which we may be driven to connect in order to dull the pain \u2013 but then, once the injury is healed, find ourselves looking at our newly acquired partner and wondering what on earth we just did.\n\nSo how long does it take after losing a love to be truly and wisely available again? There are no schedules here. For recovery after a break-up, the litmus test is whether we can yet think about our former partner kindly \u2013 or if not, then at least not with hate but with love's true opposite, indifference. The delightful blog quantifiedbreakup.tumblr.com \u2013 by a blogger herself in 'relationship recovery' \u2013 lists several calculations for this, from 'at least two years' through 'half the length of the relationship' to 'one week for every month you were together'.\n\nWhen it comes to a variation on the theme of losing a partner \u2013 bereavement \u2013 the schedule may be more protracted; Jeanette Winterson's downbeat but arguably accurate appraisal in her book _Written on the Body_ , in which her heroine tries to come to terms with her lover's leukaemia, is 'To lose someone you love is to alter your life for ever.' Whatever the situation, the only definitive truth is that being ready for the next relationship will always take just as long as it takes.\n\nBeing grown-up\n\nThe final, tough-love question about readiness is this. Are we yet 'grown-up enough' to pick a partner? Mischievously, that very wording is a test; if we're willing to even consider the question, we're probably on the way to ready, for success in partner choice often lies in the ability to question ourselves with mature and undefended honesty.\n\nOn this point, we turn to philosopher and social psychologist Erich Fromm, whose master work, _The Art of Loving_ , is arguably the seminal statement of our contemporary view of what love can and should be. Fromm outlines in detail what he sees as the supremely 'grown-up' task of partnering another human being. In his view, mature partner choice needs the self-belief that we are worth loving, the self-insight to know what we need and the self-control to let go of our needs where necessary. Plus, the ability to teach a partner how to love us, the humility to learn from a partner how to love them and the insight to know that love doesn't just consist of partnership but constitutes, in Fromm's words, the 'only sane and satisfactory answer to the problem of human existence'. At which point, most of us will run screaming from the room, thinking that for this weight of responsibility we're not only unprepared right now but will always remain so.\n\nLet's not panic. If we can realize that the 'art' Fromm speaks of is not one we're born with but one which, like any art, we learn, then the challenge becomes more manageable. Love is surely not a single act but an ongoing course of lessons. Which means that at this point, as we choose a partner, all we really need to ask ourselves is whether we're ready to fill in the enrolment form.\n\nOf course, it isn't only we who need to be ready for love, but also the partner we choose. So it's more than a good idea to also do due diligence on a potential partner's readiness. We're not talking here about the relentless first-date quizzing that makes the person on the other side of the restaurant table reach for the bill after the first course, but a compassionate awareness of whether the person we are beginning to care for is truly free. Do they have time and room in their life for love? Are they over their past relationship or actually still yearning? Are they reaching out to us through genuine attraction or to fill a life gap?\n\nIf the answer to any of these questions is 'unclear', we may be wiser to step back and allow them the time and space they need. And allow ourselves to find a partner more ready to be chosen.\n\n#### 3. Looking Back\n_Study the past if you would define the future._\n\n(CONFUCIUS)\n\nWhen we actively search for a long-term partner, most of us tend to think ahead. We map out goals. We create aims. And the more serious a partnership we want, the further ahead we tend to think \u2013 not just to meeting a new date, but to moving in, to getting wed, to which gender our first child will be. There's wisdom in that. To choose well we have to gauge what the long-term will deliver.\n\nBut there's wisdom too \u2013 as Confucius says \u2013 in first putting attention back to the past. How has it made us who we are? What does that mean for who we choose? It's not only that our past partnerships have been preparation for this moment, giving us both ability and vulnerability around loving. It's also that every event in our past \u2013 from the moment we were born, let alone from the moment we began to date \u2013 has taught us messages about partnership and partnership choice. Whom we choose may be our decision alone, but why we choose will be influenced by a whole lifetime's cast of characters and scenes.\n\nYour influences\n\nIt's an interesting exercise to look back at our personal cast-list and our personal life-plot. Was it our mother, father or neither who taught us, by what they said or what they did, which kind of relationship we should aim for and which we should avoid? Was it our siblings, our friends, our teachers or our culture that told us we should choose a partner for their intelligence, their earning power, their beauty or their biddability? Was it being brought up in a conflict-ridden family or being a member of the debating society that has made us believe so completely that perfect partners either never argue or that, conversely, they regularly do? Is it our parents' ruby wedding anniversary or our own recent relationship break-up that leaves us convinced either that love lasts for ever or that it is utterly impossible? And what role do Jane Eyre, James Bond, _Fifty Shades of Grey_ or _The Selfish Gene_ have in all this?\n\nSo review. Who have been your biggest influencers on how best to choose a partner? What has most shaped your thoughts and feelings about the kind of person who will be the best mate for you?\n\n\u2022 Parents, siblings, extended family\n\n\u2022 Religious or cultural leaders, teachers\n\n\u2022 Your peer group and their partners\n\n\u2022 Your past loves, requited or unrequited\n\n\u2022 The media, be that news, books, films, the internet, television\n\n\u2022 Traumatic events in your own life or the lives of those close to you\n\n\u2022 Positive, affirming events in your own life or the lives of those close to you\n\nNow take this further. Consider what messages all of these have passed on to you. What definitions, expectations and presuppositions have you learned about the kind of relationship you should want, and so what kind of partner to pick? It may help to complete the following sentence: 'I learnt from ____ that the best relationship is ____ and so I should choose a partner who ____', and to complete that sentence at least ten times to get a range of differing messages. Then, see if there are any patterns, any surprises, any wake-up calls. Most crucially, how have these life lessons affected the partner choices you've made up to now?\n\nLove maps\n\nFor more insight let's turn to sexologist John Money, who calls these lessons 'love maps' \u2013 templates of how we see ideal partnership and how we see our ideal partner. Money suggests that we gain love maps instinctively and early \u2013 often between the ages of about five and eight \u2013 as if we're picking up an accent in our native language. So we don't question, perhaps hardly even notice, the internal image we're building of relationships, the specifications we're drawing up about the kind of partner we want \u2013 often down to race, height, build and manner. When we find someone fitting that map, we're compelled. Yes, we may know on a logical level that an alternative mate may be just as good for us; but somehow this person just feels right.\n\nNavigating one's way through the highways and byways of the heart can take concentration, courage and an accurate road map.\n\nVery often, the person feels right because they remind us of someone who felt right earlier in life or because we believe that with them we can reclaim the 'right' life events. One way of describing this would be 'transference'; we 'transfer' affection from someone who was important to us in the past to someone important to us now \u2013 or to someone we want to be important to us in the future. A tone of voice, a sideways glance, a certain strength, a certain gentleness \u2013 all of a sudden, typically without knowing why, we feel we're safe. No matter that we don't know this person and they don't know us; we feel inevitably drawn in.\n\nIt makes absolute sense to gravitate to the people most like the ones who have made us happy in life, and much of the time that's a wonderful strategy. But get expectations confused, and it can go wrong. We can end up assuming a prospective partner will deliver the same kind of wonderful experiences that our parents (or our best school friend, or our very first 'crush') gave us. But partners don't necessarily deliver, because they're not that person and this isn't the past \u2013 realizing this and accepting it is a key lesson of partner choice. Professor Sue Johnson tells how when she met her husband almost the first words he said to her were, 'I am not going to live up to your expectations'. In the face of such insight how could she do anything other than marry the man?\n\nThink back to when you were very young \u2013 say just starting school. Now identify three figures \u2013 perhaps parents, teachers, siblings \u2013 or events \u2013 successes, wins, triumphs \u2013 that at that period in your life made you feel deeply valid and loved. You'll not only consciously look for a partner who holds out the promise of replicating those feelings but also be unconsciously drawn to any potential partner who holds out that promise. Good idea, with just one caveat. You may also be drawn to any partner you _believe_ can deliver, even if they can't. So try before you buy.\n\nBad experiences\n\nThere's an added twist and it's this. Sometimes we get drawn to partners not because they could give us something wonderful but because they could give us something terrible \u2013 in the hope that they may also give us the chance to overcome that terror. So we may choose someone who reminds us how withdrawn our father was or how dominating our mother was, someone who creates demanding situations that remind us of exam failure in school or the time we got made redundant from work. We find all this painful but familiar, troubling but known; we aim, this time, to resolve, to cope, to survive.\n\nAnd it often comes good. Often, we get to turn things round, to cope with challenges in later life in a way we didn't in earlier life because we are now older and wiser. And the fact that we cope not only gives us victory here and now, but helps to resolve the previous sense of failure. The Israeli poet Yehuda Amichai wrote that 'people use each other as a healing for their pain' \u2013 but in this case we are actively choosing each other for the pain, in hopes of getting the healing as part of the package.\n\nAgain, think back to your early life. Which people or events made you feel unloved or taught you hard lessons about what it means to relate to others? Do these memories link to your adult partners and the experiences you have had with them? What does that tell you about your partner-search strategies?\n\nThere is also, of course, the impact of life events so traumatic that they create a deep vulnerability in us, so we end wary, grief-stricken, furious or in some way simply too wounded to make good partnership decisions. We may be confused about what love means, unable to recognize it when it happens, unresourced to take it when offered or give it when needed. The obvious \u2013 and publicized \u2013 traumas are bullying, abuse and violence, but other seemingly less serious occurrences can wound us too. If our world rocks on its axis \u2013 perhaps from a house move, a hospital stay, an absent parent \u2013 we can end up thinking love will let us down. If as a child the only way we got any attention was when we were punished for bad behaviour, we may end up prone to tantrums in our relationships. The shock has not only broken our spirit, but dented our capacity to make good partner decisions.\n\nMoving on\n\nAll the above works both ways. For our partners too the 'past is prologue', as Shakespeare wrote, and what happened in a partner's life before the scenes they write with us affects not only who they are but also who they are _when with u_ s. Partners steer their course by their own love maps, make their own transferences, have their own stream of negative or traumatic characters and episodes from their own lives. So they may choose us because our ways of being wonderful remind them of wonderful people and events in their past, or because our ways of being difficult remind them of difficult people and events in their past. They too may need to align their expectations, to work through the difference between their hallucination and the reality that is us \u2013 and to cope when we deliver the pain that they unconsciously wanted to resolve through being with us. If a partner seems to be reacting in a way that says more about their history than about the present reality of our relationship, it's worthwhile paying close attention.\n\nIt is not all bad news. Though it's tempting to believe that everything that comes to us from the past is detrimental, it's not so. Normal, kind, human love from those around us \u2013 whether given in childhood or in adulthood \u2013 not only provides us with solid ground for loving but goes a long way to redress any harm that comes our way. And while single intense experiences of betrayal can wound us, similar experiences of happiness, acceptance, success and security can emotionally vaccinate us against mistakes. For most of us, past bad experiences are not a car crash on the relationship road, but simply a bump on the way.\n\nPlus, we don't have to bring the past with us into the present. We can keep the bits that seem more helpful and most healthy; the others we can throw overboard. To which end, you may want to look back at all the lessons you've ever learned about love. Which messages are horribly outdated and need to be junked? Which are irrelevant to you now you're a grown-up? Which messages are so idealistic or perfectionist that no one has the slightest chance of matching up to them, and by continuing to try to match up, you are simply feeding your guilt monkey? Which were taught to you by people whose experience is not yours, whose word you no longer believe, or whose life you have no intention of living? Which lessons did you learn through events that were so painful that you need to cull them from your memory bank?\n\nWe can cull. With new awareness we can unbelieve our past beliefs, let them go and take on board a new and more helpful set. And if we can't do that alone, it's entirely possible \u2013 and utterly wise \u2013 to get help; there is a wealth of knowledge and guidance to help us overcome relationship issues. So if you suspect that some events or people have left you vulnerable to wrong choices or misguided decisions, let me encourage \u2013 even beg \u2013 you to see a professional. We can't change the life we've lived, but we can rethink it, understand it differently and so resolve the pain.\n\nConfucius said that the past needs to be studied in order for us to define the future. Once studied, though, we may want to complete the lesson and move on.\n\n#### 4. Not Choosing\n_I believe marriages would in general be as happy, and often more so, if they were all made by the Lord Chancellor, upon a due consideration of characters and circumstances, without the parties having any choice in the matter._\n\n(ATTRIBUTED TO SAMUEL JOHNSON, BOSWELL, _T HE LIFE OF SAMUEL JOHNSON_)\n\nIf you agree with Samuel Johnson, you're in a minority. Most people deeply believe that the best life outcomes are created by active and informed decisions. More, that if we ourselves don't make those decisions, we give up the chance of those best outcomes; we surrender control, cede responsibility and in so doing resign ourselves to compromise \u2013 as well as tacitly admitting that we're not up to the task of managing our own lives.\n\nBut let me argue the opposite. Maybe Johnson is right, and letting go of choice is a good idea. It's not just that love is one of the few areas in modern society where we may still cling to romantic notions of fate determining our future. But also, by handing over control to others, we might avoid repeating past mistakes or making new and future ones. When dealing with 'winged Cupid painted blind', as Shakespeare put it, taking our own blind prejudices out of the equation may be no bad thing.\n\nHeads or tails?\n\nOpt to step back from active decision-making, and the first possibility open to us is to toss a coin. When it comes to deciding a lifelong commitment, doing that literally may not seem wise, though devotees of the I Ching might disagree. A friend of mine did once use the system to decide whether to marry her boyfriend; the coins said 'no', which in hindsight proved to be excellent advice. And in Luke Rhinehart's _The Dice Man_ \u2013 as its title suggests, a novel about a man who makes all choices thus guided \u2013 several partner-decisions are made that way. The result is whole-scale erotic abandonment under the absolving aphorism 'Who am I to question the dice?'\n\nLet's consider next the American psychologist Barry Schwartz, whose work focusses on the reasoning behind choice. His argument is that randomness is a 'good enough' option. Once certain standards are met \u2013 we like each other well enough, we have similar life goals, we've checked issues such as substance abuse and criminal record \u2013 there are really no good reasons to discriminate between one suitor and the next. Chance, Schwartz suggests, might well be the 'most efficient' but also the 'fairest . . . most honest' option.\n\nInteresting ideas, though most of us are wary of making key life decisions with so little information. Which is why I don't advise either coin-tossing, dice-throwing or absolute randomness. But I do suggest that to circumvent our biases and introduce a little serendipity, daters should, online, approach one random person in every twenty profiles they access, and offline be open to the occasional blind date, opportunity dinner from a colleague or fortuitous encounter at the supermarket checkout. Randomness certainly can open a different, less blinkered and therefore sometimes better door.\n\nDestiny vs growth\n\nThe issue of fortuitous encounters brings us to an interesting double standard here. We may recoil in horror at the thought of haphazard chance determining our future life partner, but label that chance 'destiny' and we're entranced by the prospect. On the one hand we feel the need to be in control of our romantic choices; on the other hand the thought of losing control can be hugely seductive.\n\nEnter Professor 'Chip' Knee of the University of Houston, and his work on 'destiny love', his term for the conviction held by some couples that kismet rather than coincidence brought them together. Believe this, and we are likely to feel strongly and instantly attracted to a partner, the relationship is likely to be passionate and intense, and a magical certainty of success is likely to hover over the whole enterprise. As a perfect example, see that scene in the film _Sleepless in Seattle_ where Sam Baldwin describes meeting his first wife, speaks so movingly of taking her hand to help her out of a car, and \u2013 just with that initial touch \u2013 realizes 'we were supposed to be together . . . and I knew it.'\n\nSam Baldwin was lucky \u2013 thanks to a good scriptwriter creating the traditional rom-com happy ending \u2013 his ensuing marriage worked well. But the 'destiny' assumption may not survive the cold light of real-life daily commitment, because it comes with certain built-in structural defects. The problem is, if fate has generously provided us with a predestined partner, there may seem no need for effort to make the relationship work \u2013 and if effort is needed then that particular romance is clearly not as predestined as we thought it was and probably needs to end forthwith. Professor Knee's work suggests that those who believe in destiny love are likely to react badly when things go wrong, likely to exit relationships lightly, likely to move on quickly \u2013 to their next destiny.\n\nCompare and contrast couples convinced of what Knee calls 'growth love', who see partnerships as developing slowly over time, with any glitches along the way meaning nothing sinister, but simply indicating a need for more effort. Growth love couples tend to get involved more gradually, have lower expectations, but are more capable of the long haul. It is certainly worth remembering that while belief in fate-determined love \u2013 whether on our side or on a partner's \u2013 might add a breathless magic to romance, belief in growth-based love might mean the difference between a ruby wedding anniversary and being left at the altar.\n\nArranged\n\nLetting destiny decide partner choice may seem like giving away control, but at least destiny (or fate or God or Providence) is all-seeing, all-knowing and infallible. Letting other, all-too-human beings decide partner choice seems a far worse bet, for ordinary folk are all too capable of confusion, ignorance, error and their own personal prejudices. Which is why, while accepting support on partner choice from friends and family may be helpful, it's usually better to block one's ears and step away. 'You should' and 'you must', even when voiced by onlookers with the best of intentions, should not govern partner decisions.\n\nWhat does have a place, and a long history of some success \u2013 it's even mentioned in the Bible \u2013 is allowing others to 'arrange' possibilities which we can veto, or from which we make our own pick. Don't forget that until very recently, especially if a partnership was the way to cement political alliance or unite feuding families, it made perfect sense for elders rather than the happy couple to make the courtship decisions. In Britain, for example, it is only the last two generations of the monarchy that have had any kind of genuine personal freedom over marriage choice. (When certain of those love marriages failed, there was in some quarters a general feeling of 'Well, what did they expect?')\n\nIf done well, 'arranged' can triumph, delivering the objectivity of chance, the reassurance of destiny, the pragmatism of growth. Like chance, an externally determined partnership can avoid personal bias or the temptation to let lust dictate terms. Like destiny, it can remove the burden of choice, allowing us to relax rather than agonize about our own responsibility. Like growth-love, 'arranged' can mean we don't demand instant compatibility, or call foul if things don't prove perfect always and for ever.\n\nBut there does need to be a high level of competence here. Matchmakers \u2013 whether they be friends, family, the Jewish _Shadchanim_ or the upmarket urban introduction service \u2013 need to know us at least as well as we know ourselves, need to know our partners well enough to judge the fit, and need to have a deep understanding of how relationships play out. The problem is that this level of wisdom is unusual in today's society. Few people witness others profoundly and consistently enough to judge what is needed in a mate. Few relationships are sufficiently exposed to public gaze to deliver a real understanding of partnership dynamics. Which is why, though popular, modern matchmaking services can be infamous for high prices but low results. That said, in the right context and if the arranger is insightful and practised, 'arranged' can work well.\n\nLet us be grateful we don't live in 1917. Herbert Rawlinson and Alice Lake in the one-hundred-year-old black-and-white film _Come Through_.\n\nA quick note on the extreme of 'arranged', those serried ranks of mass marriages where hundreds of couples meet for the first time on their wedding day, their partner choice made entirely by religious leaders. For most of us, this is somewhere between incomprehensible and appalling; how on earth does it ever work? The answer is that spouses' contentment with the arrangement is constructed on their deep belief that they have the blessing of the deity; that their culture, community, family and religion will support them; and that love grows over time rather than instantly strikes. Despite individual horror stories, that's a pretty big resource to bring to the table.\n\nOnline choice\n\nDoes online dating take control out of our hands? At first glance, no; we seem to have a multitude of options. But look more closely and we see that what we get is pre-sorted. Most sites cherry-pick attractive users and profile them on the home page to catch incomers' attention. Many sites also highlight profiles of particularly popular users and present them as a separate and therefore more noticeable subcategory. We're getting a choice, but only after the site filters have done their secret work.\n\nAnd then there are the algorithms. It was way back in 1959 that a group of Stanford University maths students working on a final class project programmed their IBM 650 computer to pair up forty-nine men and forty-nine women according to their answers to a questionnaire; the result was one marriage and a deserved A-grade for the dissertation. More than half a century on, and that early study has led to a billion-dollar industry; almost every online dating site has a visible 'questionnaire' and they all have hidden algorithms to guide us firmly towards matches of our age, location and gender of preference. Choice is, if not taken out of our hands, at least slightly compromised.\n\nThere's also another problem: dating sites only do the top level of the job. Yes, most sites match those who, on tick-box criteria, are similar, which replicates the surface-level criteria that attracts us to a mate; some sites then aim for more, with personality questionnaires or hormone-based matching systems. But while these matching systems maintain that the criteria for compatibility are well-established and that it's therefore possible to predict relationship success, in fact 'neither of these assumptions is true'. (I quote here from a recent research review of studies of online love-search by Professor Eli Finkel of Northwestern University, Illinois.) By allowing sites to try to match us, we are not only limiting our own options, not only failing to take into account the deeper compatibility factor, but also entrusting our fate to completely unproven systems.\n\nTrusting chance\n\nI'm not suggesting that abandoning choice altogether is either a good or a bad idea. But it is a possibility. What you may find useful is to think about what degree of abandonment you've used in the past without even realizing it. Have you ever started chatting to someone near you at the theatre, during a party, on a train \u2013 then spent a delightful half-hour even though that didn't lead to romance? Probably. Have you ever met a partner by chance, circumstance or sheer happy coincidence? Again, probably. Have any of your intimate relationships been 100 per cent predetermined and predicted? Probably not. Start to gauge the extent to which you're comfortable with serendipity and you may feel more inclined to accept it as another tool in your decision-making kit.\n\nCoin-tossing, random chance and a belief in fate are all quite risky. But handing over some of the burden to others is certainly worth considering \u2013 so perhaps let online dating sites screen the undesirables or let matchmaking services do the legwork. Allowing ourselves to widen our initial love-search beyond our personal preference is also useful \u2013 so perhaps make a deliberate effort to contact (or start talking to, or agree a date with) someone who on the surface doesn't seem like a possible. Gathering emotional support is vital to get us through the process \u2013 so perhaps actively enrol trusted friends and turn to them for guidance, consolation and celebration.\n\nAbove all, when it comes to choice, be prepared to occasionally take a risk, accept uncertainty, let go of control. As ex US president Jimmy Carter once said, 'Go out on a limb. That's where the fruit is.'\n\nFrom time to time, they both wondered what the next forty years of marriage would bring.\n\n#### 5. Focussing\n_Focus is about saying No._\n\n(STEVE JOBS)\n\nSurely the more options we have, the more chance we have for love. Especially if we blame previous romantic disappointments on a lack of alternatives, then richness and variety of choice seem the obvious keys to success. Which is why the first question we may be asking about our love-search is how to find 'more' possibilities.\n\nThis is a great question if we're short on options, if our circle of single contacts has dwindled to nil; if we live in a town \u2013 or country \u2013 where we know no one; if we're not meeting any potential partners either at work or at play. It's an especially useful enquiry if we are in a majority gender for our life-stage: research suggests that men in their twenties have only half as many partner possibilities as women do, but that in the forties the balance begins to reverse. If we can't find a partner in these situations, the main problem is certainly the numbers. We're a seller in a buyers' market and the solution is to find more buyers.\n\nThe way forward is to steer clear of babbling brooks, avoid stagnant pools and find slow rivers. Stripped of the metaphors, what I mean is this: stay away from gatherings with no chance of 'get-to-know-you' conversations, or where meeting up again is unlikely. Don't get trapped in a social life where you see the same people over and over again. Instead, put your energy into groups which offer a steady and regular through-flow of different individuals, in situations where there's opportunity to mingle, meet, chat and bond. That's a slow river, and so long as it contains people with a similar background, outlook, values \u2013 it will deliver partner possibilities. Just as importantly, it will also deliver a fulfilled and fascinating existence as a base camp from which to embark on the partnership climb.\n\nFinding a slow river\n\nHow many slow rivers do you have in your life? If the answer is not many, and getting 'more' is your issue, pause now and brainstorm what your rivers could be; include as many ideas as possible, without censoring even the maddest thoughts. Start a new hobby? Go to a partnered dance class? Attend dating events? Organize your own dating events? Put a call out on Facebook? Put a call out on your local radio station \u2013 it worked for Sam Baldwin, aforementioned hero of _Sleepless in Seattle_. Take off on a three-country tour \u2013 it worked for Elizabeth Gilbert, author of _Eat, Pray, Love_.\n\nPlus \u2013 if it feels comfortable \u2013 sign up to a dating website. Whether we like it or not, online is without question the most accessible slow river currently available \u2013 one in which the whole agenda is to bond and the entire process ensures a regular through-flow. To enhance the expansion project, join a variety of sites \u2013 some paid, some free, some new, some established. Once on, be open-minded about possibilities, proactive rather than reactive and as flexible as possible about non-essential parameters. (Though not foolishly flexible. Tick the '200 miles away' location box on websites and you'll very soon face the reality of long-distance relationships and find yourself switching the location specification to 'same town'.)\n\nStagnant pools. It's worryingly easy to create a life so pleasant, established and secure that it delivers absolutely zero chance of meeting any suitable partner.\n\nMore or less?\n\nHaving noted the above strategies, and taken the first steps towards action, forget the advantages of 'more' and embrace the benefits of 'less'. And not only because quantity is less crucial than quality in partner choice, but also because the human brain treads a fine line between having a wide range of options and having too many for sanity.\n\nIn a famous experiment by Professor Sheena Iyengar of Columbia University, customers at a grocery store decided what jam to buy; of those who had a choice of twenty-four jams, only 3 per cent ended up purchasing, while of those who had a choice of six jams a full 30 per cent bought. Given we likely don't use the same criteria to pick a partner as to pick a preserve, let's also mention a less-famous but more relevant survey which gave parallel results. Prospective partners who looked at either four or twenty-four online profiles made more picks than those who looked at sixty-four. Significantly, the participants who were presented with only four options took time to consider everyone, while those who were presented with twenty-four or sixty-four options made only cursory decisions.\n\nThe point here is that too many options plunges us into what's called 'shopping mentality'. With a profusion of possibilities, we suffer mental overload. We get confused. Then we get anxious about getting confused. And then, to combat the increasing emotional paralysis resulting from that confusion, we try to simplify. Which in turn leads to our over-considering irrelevant criteria, rejecting without real consideration, and craving the 'next good thing' rather than focussing on the current one. (This last is part of the reason why the constant forward momentum of the left\/right swipe on some dating apps is so addictive.) Finally, when it comes to online dating in particular, even if we think we've found The One, we may still suffer buyer's remorse because the wealth of choice available suggests there's someone even better out there for us. If you've ever been enthusiastically emailed for days then suddenly dropped, you were probably the victim not of someone's rejection but of their shopping mentality.\n\nIf only choosing a partner were as simple as walking into a supermarket. On the other hand . . .\n\nWhat all this means is that any website which trumpets \u2013 as one of the global players did very recently \u2013 an astounding 197 million members might seem like a magnificent opportunity, but may well prove a disaster in disguise. Our real challenge, offline as well as on, is not how to expand possibilities but how to limit them, how to reach an equilibrium where we think clearly enough to make good judgement calls.\n\nElimination\n\nThe solution here is, not coincidentally, to take the route that nature intended. Intelligent elimination \u2013 the kind of shedding that Steve Jobs refers to in the quote at the head of this chapter \u2013 is the way evolution means us to mate. Walk into a room full of a hundred possibles \u2013 or log on to a website of a million \u2013 and we're doing just as our ancestors did thousands of years ago from the vantage point of their caves, unconsciously excluding those who don't meet our basic criteria (right gender, right age, right tribe). Elimination may sound cruel, but it's the way our instincts are meant to operate in the initial going \u2013 not saying 'yes' to one but 'no' to many, not making a single positive choice but first applying wholesale negative screening.\n\nIn a society more complex than the Neolithic, some screening's done before we even begin. Dive into any 'slow river' where we feel at home \u2013 a study group, a sports association, a dance course \u2013 and it's likely the event organizers have already aimed their publicity to eliminate people not in the tribe; we then informally cut the field by choosing to attend only certain events and then to mix only with certain people in the room.\n\nOnline, the system's parallel. From the start, some elimination's already done for us because we enter a world where those not looking for love have already self-excluded; many sites also helpfully (though covertly) screen out those with relationship-threatening issues such as drug abuse or long-term mental illness. We further narrow the range ourselves by signing up to sites that target our tribe even more specifically; there are now dedicated destinations for specific age groups, individual cities and almost any special interest you can name, along with some you can't even imagine.\n\nOnce signed up, the site tick-boxes create further focussing on surface compatibilities such as smoking (or not), drinking (or not), whether we follow sport or how passionately we love Chinese food. The algorithmic 'matching quizzes', if well done, then disqualify on deeper issues \u2013 though of course they can't predict face-to-face chemistry, and, as mentioned earlier, there's little correlation with long-term compatibility. Happily, the recent introduction of categories for 'serious', 'fling' or 'affair' now also achieves the elimination of those wanting a relationship of a kind we're not seeking, which is a huge relief for all those online daters single and looking for marriage who \u2013 up to a few years ago \u2013 kept bumping into those married and looking for a fling.\n\nThe funnel of love\n\nAll this pre-sorting is helpful. But the bottom line is that, however small the partner pool we fish in, there'll come a time when we need to do our own, more individual eliminations, partner by partner. My favourite metaphor here is one offered by the late Israeli psychologist Ayala Malach Pines. She imagined a kind of 'funnel of love' \u2013 her pun, not mine \u2013 into which we pour everyone we meet. But just as a funnel gets narrower as it deepens, and lets fewer and fewer elements through, so our criteria naturally get more and more focussed as we eliminate potential partners, until we eventually accept only the ones who really suit. (The criteria we use to create and operate our funnel form the substance of the following chapters of this book.)\n\nWhat often stops us from using the love funnel effectively is a kind of fear of focussing: a nervousness about elimination, a wariness of being choosy, a belief that we shouldn't \u2013 for which read 'aren't entitled to' \u2013 dismiss partners who mismatch, or partnership options which don't appeal. 'I need to spread the net wide or I won't find anyone'. . . 'If I set the bar too high no one will want me'. . . 'I can't say no to her\/him, they'll feel so rejected'. This panic is totally understandable if we've had \u2013 as we all have \u2013 past heartbreak. But it's nevertheless misguided; we do need to focus on what we want rather than going with the flow. I'm not a believer in 'The One', but unless we start saying no to those who aren't right for us, we won't get anywhere near those who are.\n\nIf you suspect that fear of focussing is holding you back, then try this. Imagine yourself sitting somewhere you feel most comfortable, and knowing that your 'right for you' partner is nearby. You're happy because he or she is there, you know that they care for you and you for them and you allow yourself to want them and to want the relationship you have with them. (If you're tempted into caveats or qualifications of the 'nobody loves me' or 'that'll never happen' kind, set these aside for the moment.) Now imagine your partner arrives. Imagine seeing them, hearing their voice as they speak to you, feeling their touch as they reach out for you. Allow yourself to experience their attention fully and know that you deserve that attention.\n\nThe point of this exercise is not to imagine what a future partner might actually be like. It's to have a mental experience of wanting that partner, feeling entitled to want them and being wanted in return. As explained at the very start of this book, partner choice is a quest, and as with all quests it's good to have a bit of feisty courage \u2013 courage to believe in yourself, courage to believe there are partners out there you can choose, courage to believe that there are partners out there who will choose you. Given that belief, you'll find it much easier to keep setting aside those you don't want and keep heading towards those you do.\n\nSpecifying\n\nSo how do we begin? How do we create our personal love funnel? The answer is to get more specific. Specificity clears the mind, orders the thinking, makes us feel in control, helps us understand what we're doing. We are right to believe that specifying will help us towards a result, even in an area as unpredictable as partner choice.\n\nThe historical wheel has come full circle here. When marriage decisions were made on measurable criteria \u2013 age, status, earnings, childbearing potential \u2013 specifying was the way to narrow the field. But when romance left its place in the wings and took centre stage, we sniffed at being too precise, because we wanted to let our emotions rule \u2013 if we loved each other, surely the details were irrelevant. Now things are coming back into balance. Much of the current coverage of partner choice presents its lessons in the form of specified lists \u2013 as seen on every online dating site, in many relationship-advice articles, and via the relationships section on Amazon. The current crop of urban tales featuring heroes\/heroines who found their princess\/prince through detailed specification inspire us not only because we too want to magically conjure up our own fairy-tale partners, but also because we recognize that specifying is necessary to the process.\n\nOf course we shouldn't over-specify. It's not only that, life being what it is, we can't have everything we want. It's also that many of the details of what we want will be irrelevant to our goal of a good relationship. It may matter hugely that a partner shares our love of animals, but if they do love animals \u2013 or if we fall for them regardless \u2013 it likely won't matter at all that they have blue eyes rather than brown. Proof of this is courtesy of another study by Professor Eli Finkel, in which he asked speed daters their partner criteria just before the event; said daters then completely ignored their own benchmarks when they started mingling only minutes later. However convinced we are that we need a partner who's extravert\/introvert, dark\/blonde, big\/small, if we find the right person, size won't matter.\n\nEven so, once our partner pool is big enough, specifying is the perfect starting point. It keeps us on logical track. It boundaries the challenge and keeps it doable. It engages our minds for a task which can otherwise all too easily become over-influenced by our hearts and other more lustful parts of our anatomy. Perhaps unexpectedly, specifying also opens the door to deeper realizations. Nobel Prize winner Daniel Kahneman, whose bestselling book _Thinking, Fast and Slow_ recently alerted us to the subtleties of decision-making \u2013 or in other words, choice \u2013 points out that specifying doesn't just call on logic but also on a more instinctive awareness. He believes that we 'improve . . . intuition by making a list then sleeping on it.' When we create a partner specification, we will often mysteriously find ourselves exploring much wider issues, not just about surface criteria but also about what we need from a partner on a deeper level, and how to instinctively recognize that when it arrives.\n\nYour wish-list\n\nSpecifying is the perfect starting point \u2013 and the right first step. Our assumption, offline as well as on, is that we should start with what we ourselves are 'selling' and only then consider what we want to 'buy' from a partner. But that's not only the wrong metaphor \u2013 partnership is not about trading but about relating \u2013 it's also the wrong way round. To attract someone, we first need to know what kind of someone to attract. We can't set the GPS effectively until we know the destination.\n\nIf you've already made a wish-list of the specifics you want in a partner, take this opportunity to revisit it. If you haven't already made a list, here's your opportunity. Write down all the elements which for you headline your ideal mate. Gender, age, appearance, cultural background, religious belief, lifestyle, career, earnings, leisure patterns, hobbies, interests, location \u2013 if you run out of categories, most online dating sites include an extensive tick-box list. Avoid vague, avoid abstract \u2013 the idea is to get a detailed, defined, quantifiable starter guide. Then prioritize in order of importance; as I've said, you can't have everything, so your top five at least need to be the things you couldn't live without.\n\nDeal-breakers\n\nNext, list the things you couldn't live _with_ , your deal-breakers. This part of the process aims to make sure we don't find ourselves involved with (worse, married to) someone utterly unsuitable; it's the bottom line which helps us less to find The One \u2013 for there are many Ones \u2013 as to make sure that The Totally Wrong One doesn't slip under our guard because they are tempting. Here are four classic deal-breakers, and that's about the number to aim for \u2013 more, and you're probably drawing the net too tight.\n\n\u2022 Different and uncomplimentary sexual leanings \u2013 such as a potential partner's being gay whereas you need straight, or vice versa\n\n\u2022 Mismatched relationship aims \u2013 such as their wanting marriage where you want casual, or vice versa\n\n\u2022 A conflict in deep values \u2013 such as their being highly religious whereas you are allergic to anything remotely spiritual\n\n\u2022 Incompatibility of interests \u2013 not a gap in enthusiasm, which can often be bridged by encouragement, but a serious mismatch, such as their being a passionate sailor whereas you are aquaphobic\n\nIt's tempting not to consider deal-breakers. But we should be clear about what we don't want, especially the elements we are uneasy about not wanting, as they're the ones we may most unwisely give ground on. If we truly couldn't live with someone who has a dangerous job, for example, it's fairer to everyone to be aware of that, rather than denying, compromising, then wobbling.\n\nA 'normal, happy day'\n\nWish-list made, deal-breakers noted, we next need to specify in a slightly different way. Rather than defining a partner, envision a partnership. What if you woke up one weekday morning five years in the future, having made the perfect choice. Imagine the rest of the day \u2013 not special, not peak experience, just normal, solid and satisfying. What might be your plan for this 'normal, happy day'? In what location would you see yourself? What would you do? How would you spend time \u2013 with your partner? By yourself? Who else might be there? What sort of lifestyle would you have? How would you feel? Above all, what would be especially rewarding about the relationship you'd created?\n\nThis exercise gives new perspective by drawing two crucial distinctions \u2013 between dream and actuality and between partner and partnership. Envisioning what we're after not as fantasy but as reality strips away many of the inessentials \u2013 things we may hanker after but are irrelevant to our happiness; we may find ourselves altering our wish-lists as a result. Plus the exercise broadens awareness out again, from the partner we might choose to the everyday life we would have with that partner. (This everyday life is, of course, the whole aim of choosing a partner in the first place: I did at one point wonder whether this book should have been called 'How to Choose a Partnership'.) Enhancing detail in this way leaves in place the core needs, but allows us to form a resonant and motivating picture of what we want our future to hold.\n\nWelcoming invitation\n\nHow to use these specifications? When doing distance dating, the wish-list and the deal-breakers inform the tick-boxes, while the 'normal, happy day' exercise will give you the material to write the free-text elements of profile and partner specification. There's an added bonus here. Research suggests that, online, it's more compelling \u2013 for which read attractive \u2013 to structure your profile not as a 'dating CV' (age, interests, holiday plans) but as a welcoming invitation to join you in a relationship; the 'normal, happy day' exercise provides all the essential raw material for such an invitation.\n\nIf you're motivated to put in the work, you could also use the material you've gathered to design and apply your own customized matching algorithm. This is precisely what was done by PhD student Chris McKinlay, who apparently had something of a eureka moment when he realized that, rather than relying entirely on personal charisma, he could use his talent as a mathematician to develop a personal formula for a perfect partner. He did so, found love and wrote the bestselling book to prove it. Details of _Optimal Cupid_ \u2013 and of _Data, a Love Story_ , from female counterpart Amy Webb \u2013 are in the bibliography.\n\nIf partner choice is face-to-face \u2013 speed dating, blind dating, meeting with an online suitor or simple right-place-right-time serendipity \u2013 the wish-list, deal-breakers and 'normal, happy day' specifications are vital touchstones. With these kept firmly in mind, we can begin to tally vision with reality, our desires with flesh and blood, and start asking ourselves the crucial questions: Does this person fit our theoretical specification and, if not, does that matter? We may well feel it doesn't, if in other ways there is a fit. Does this person contravene any of our deal-breakers, and if so, is that contravention writ in stone? It may well be resolvable with some negotiation. Finally, but most importantly, could this person help us create the daily life that we want for ever? If there's a real possibility they can, that's the best basis for continuing.\n\n#### 6. Connecting\n_Only connect._\n\n(E. M. FORSTER, _H OWARD'S END_)\n\nIt's surely clear that good relationship decisions are based on knowing far more about a partner than whether they simply tick the headline demands of our wish-list. To return to the 'love funnel' metaphor, the more we filter down our options to fewer possible partners, the more we need to aim for a broader and more extensive understanding of each one.\n\nFor there's an addendum to the earlier-mentioned jam experiment. Repeats of the research suggest that the 'shopping mentality' problem is not only down to too much choice but to too little information. Increase our knowledge about each jam (that is, each partner) and choice becomes not only easier but also more accurate, more emotionally intelligent, more successful. So, as we recognize which partners may be right for us, we need to view them from different perspectives, to witness them in different situations, to learn about them and allow them to learn about us \u2013 in short, to connect. The slow-river approach again, but this time trawling the deeper waters.\n\nLet's start with the simple fact of meeting. Once upon a time, we knew we were smitten because our eyes met across a crowded room (or at the village well, or as we were tilling the fields). The natural, biologically driven ritual of partner choice is founded on real-life contact.\n\nYes, some historical courtships were pursued at a distance. Europe in the Middle Ages was awash with ambassadors journeying from royal court to royal court while bearing portraits of beautiful princesses in the hope of arranging advantageous marriages. But without meeting, such long-distance strategies often went horribly wrong: King Henry VIII of England failed to even consummate his marriage to fourth wife Anne of Cleves because when she arrived for the wedding, he realized that 'she is nothing as fair as she hath been reported'.\n\nTo really connect with \u2013 and make a decision on \u2013 a partner, we need to see, to hear, to literally feel them. It was anthropologist David Givens who in 1978 mapped out for us the instinctive process of natural attraction. We glimpse a potential partner from afar, then engage by eye contact, then by talking, then touch, eventually getting close enough to smell, taste and, if the stars align, be sexual. At each stage we parallel our conscious appraisal by unconsciously rating the other's physical appearance, the way they move and speak, their hormonal invitations. At this early stage, closer and closer contact filters partners, not only because of what they say and do, but also because of the way they say it and do it. (Those who make the cut are likely not only to attract us but to be attracted to us; if face-to-face connection leads to a 'yes', it's likely to become a virtuous, circular reinforcing 'yes'.)\n\nTwenty-three days\n\nActually, we know all this. We intuitively trust face-to-face contact because we realize it gives us essential knowledge on which to base judgement. We are reluctant to develop a relationship before meeting someone, because we know that without face-to-face contact we can't make a full assessment. Which is why when it comes to the distance dating of new technology we may be fascinated but we're also wary; traditional partner choice follows a 'meet\u2013look\u2013talk\u2013touch' model, while new technology follows the new pattern of 'view\u2013read\u2013write\u2013talk\u2013meet\u2013touch', which leaves the really important bit to almost the very end.\n\nQuestion: How can that work? Answer: It often doesn't. We may find that, after crashing the internet with reciprocally enthusiastic emails for several weeks, when we finally get close to our online crush it's loathing at first sight. Moreover, studies suggest that disappointment when face-to-face rises in direct proportion to how long we've been in other sorts of contact before meeting. In other words, whether our expectations are met depends almost entirely on how long we've been stacking up those expectations. The key number here, according to a recent study at the University of South Florida, is twenty-three days of email; after that, a suitor would have to be a deity-come-to-earth in order to measure up face-to-face.\n\nIt's no coincidence that the site- and app-developers, having made their mark promoting a form of dating that was literally at a distance, are currently putting huge effort and budget into trying to reduce that distance and replicate the natural first step of crowded-room eye-catching. The current direction is not just for increased accessibility through profile pictures, audio and video, but for meet-ups, social evenings, dance classes and singles holidays, as well as apps that reveal which of those sitting in the same coffee bar are looking for love tonight. The litmus test will always be the 'close enough to smell' connection.\n\nDistance benefits\n\nThat said, we're wrong to be wholly wary of distance in a dating medium. For when it comes to thoughts and feelings, being apart can actually be more trustworthy and more revelatory. A study from Cornell University suggests we're more likely to tell the truth in an email than in a voice conversation, because we know our words are literally down in black and white and can be checked. We're also more likely to open up about significant thoughts and feelings online than face-to-face, because being less close to a prospective partner lowers the fear of rejection and leaves us more able to be authentic. The adrenalin rush that comes with the ping of 'you've got mail' stems not just from the potential of a new relationship but from the fact that the mail in question may contain deep emotional revelations \u2013 and the fact that we can respond in kind.\n\nSo here's how to drive distance dating to best effect. Benefit from the larger numbers to spot possible candidates, but rather than staying with 'winks' or 'likes', move to email as quickly as possible. Then, at that email level, without getting over-demanding, dive below surface chat; ask serious questions, give serious answers, open up and let partners do the same. The deeper revelations will help gauge the deeper compatibility; if it's there, move on to phoning and meeting \u2013 the gold standard of real life.\n\nProximity and chemistry\n\nFor contact makes everything tangible, not only physically but also emotionally. It adds in all the wonderful elements which circumvent our logical brain \u2013 there's a reason why Eros is shown with an arrow of love which pierces straight to the heart. It's almost a clich\u00e9, that total fixation of gaze and suspension of logical thought. Think Maria and Tony in _West Side Story_ , stepping across the dance floor to touch each other. Think Marius and Cosette in _Les Mis\u00e9rables_ , stopping in their tracks at the Luxembourg Gardens as their eyes meet for the first time.\n\nThis isn't, contrary to what we might think, only about desire. It's also that simple proximity matters hugely and biases hugely. Once there's a basic attraction, the more time we spend close to someone, the more they will seem appealing, lovable, simply better. And the longer, more frequently, more regularly we spend time with them, the deeper the impact. A 1930s study of 5,000 American couples showed that 45 per cent of them had got together when living within a few blocks of each other, which \u2013 while not as surprising in the 1930s as it would be nowadays \u2013 is still remarkable.\n\nA word here about 'chemistry', a factor so prized and persuasive that an entire online dating site is named after it. We've surely all experienced the sort of heartfelt mutual liking which leaves us feeling good about ourselves as well as about the other person. But it's difficult to pin down \u2013 and research hasn't shown \u2013 what exactly chemistry is, let alone what causes it. Basic biological urge? Complex psychological fit? And should we factor it into our choice strategy? Without chemistry most of us would hesitate to take things further with a partner, and many commentators feel its absence is a deal-breaker; the founder of eHarmony, Neil Clark Warren, states uncompromisingly that unless you feel the urge to kiss a prospective partner by the third date 'you're probably never going to feel it'. But other dating experts join with many traditional cultures in judging chemistry as no predictor of partnership success; perhaps the only thing that chemistry guarantees is chemistry. If so, then maybe instead of demanding it as a prerequisite for a relationship, we ought to be seeing it as a distracting delusion.\n\n_West Side Story_. Love at first sight with score by Leonard Bernstein, words by Stephen Sondheim and choreography by Jerome Robbins.\n\nDecision-making strategies\n\nWhich is why \u2013 unromantic though it may seem \u2013 it's good to add in some more measurable, more strategic, less instinctive criteria for judging. It's a wise exercise, as first meeting evolves into regular contact, to ask yourself what evidence you use as you form your opinion. (It may help to consider here how you judge not only partners, but anyone close \u2013 friends, colleagues, even family.)\n\nDo you reach conclusions about a person because of . . .\n\n. . . how they look?\n\n. . . what they say?\n\n. . . what they actually do in practice as opposed to what they say?\n\n. . . what their general reputation is?\n\n. . . what the important people in your life think about them?\n\n. . . how you feel emotionally when you are with them?\n\n. . . how you feel emotionally when they are not there but you are thinking about them?\n\n. . . your sexual contact, if any?\n\n. . . what their friends and family are like?\n\n. . . how they interact with their friends, family, colleagues?\n\n. . . how they interact with your friends, family, colleagues?\n\n. . . their previous life in general?\n\n. . . their relationship track-record in particular?\n\n. . . what your gut reaction tells you?\n\nTry dividing the elements into those you would always check, and those you'd never bother with because they seem unlinked, irrelevant, impossible to judge. Next, put the list in chronological order, according to those checks you pay attention to first, those you usually leave for later, and those you never attend to. Are there any surprises there?\n\nThen think back. How has this decision-making strategy of yours actually played out? If less than perfectly, how could you improve? Perhaps you could prioritize the elements that until now you've left until later. Perhaps you could place less reliance on some kinds of evidence and more on others. Perhaps you need to extend, to use a wider range of checks and balances so your decision-making's more effective.\n\nAn interesting thought about initial moments: one of my early psychology tutors suggested that the first experience we have of a partner provides a snapshot of our entire future relationship, sets the scene, writes the script for later plot development. So, if in their initial meeting Mary finds herself listening to Tom hammer on about his ex, she could find that Tom's past dominates their future relationship. If in their initial meeting Tom and Mary find themselves agreeing easily about where to go to eat, they're likely to have a cooperative time ahead.\n\nSo have there been any first meetings that, in hindsight, should have told you to turn and flee? Have there been any first meetings that, in hindsight, you should have taken as the starting point for something important? No regrets \u2013 the past is over \u2013 but maybe such meetings can be a touchstone as you go forward.\n\nThree elements\n\nIf face-to-face compatibility's confirmed and what we want is a simple liaison, we need demand very little more than passion and opportunity. If we are seeking something deeper, the criteria are more complex; below the intensity and the delight, there's a need for a deeper compatibility. As Joanne Woodward apparently said of her five-decade marriage to Paul Newman, 'Sexiness wears thin after a while and beauty fades. But to be married to a man who makes you laugh every day . . . that's a real treat.' So, can we get \u2013 can we give \u2013 that kind of treat, whether through laughter or otherwise, with our partner?\n\nIf we can, we're likely looking at connection across three elements: values, life goals and personality traits:\n\n\u2022 _Values:_ what makes our existence most worthwhile: safety, excitement, social recognition, happiness, self-respect, status.\n\n\u2022 _Life goals:_ the achievements we crave during a lifetime: career success, financial security, travel, adventure, marriage, children.\n\n\u2022 _Personality:_ a combination of character and temperament: honesty, mental acuity, kindness, generosity, bravery, commitment to hard work.\n\nThere's a numbers problem here. I've just enumerated a longish list of examples to cover all three elements, but I could have devoted an entire chapter (or indeed a whole book) to each. For, while a full tally of a partner's hobbies is typically achievable on the fingers of one hand, the possible subcategories of these three deeper levels might number more than the hairs on one's head. Far too many elements to keep count of, let alone keep in mind as a benchmark for partner choice. Better to explore these issues more broadly to reveal what's really important.\n\nThree questions\n\nThe next exercise lets you attempt this broader exploration. It has as its starting point a somewhat morbid topic, but it's useful to help you reflect. Imagine you are on your deathbed. You are looking back on a good life. You've experienced what you were meant to experience, done what you were meant to do, are ending as you are meant to end. Now ask yourself these questions:\n\nHe wondered how long it would take his partner to join him at the summit, given that they'd set off in different directions to begin with.\n\n1. What three values made your life most worthwhile? (Think benefits such as safety, happiness, etc.)\n\n2. Which three goals have you achieved in your life that you are most satisfied with and proud of? (Think aims such as career success, adventure, etc.)\n\n3. What three personality traits do you most want other people to praise you for when you have gone? (Think descriptors such as honesty, generosity, etc.)\n\nAnswer these three questions, get your nine answers, and you'll have a top-line list of what is deeply important in your life and thus what needs to be deeply important to your partner in his or her life. If one of your key values is status, you'll want a partner to rate that too. If your dream is to parent an entire football team of offspring, your ideal partner will be one who shares that goal \u2013 forgive the pun. If it matters to you to work hard, you'll need a partner willing to pull all-nighters alongside you, or at the least give you a genuinely appreciative hug when you finally come to bed at 6 a.m.\n\nProofs of love\n\nNine words are a great start, a good snapshot. Problem is, almost always we'll choose words that \u2013 while meaningful \u2013 are also abstract, indefinite, lacking detail. Example: almost everyone who does this exercise mentions the word 'love'. But what particular flavour of love is that \u2013 and what if it's not to a prospective partner's taste? What if our idea of 'love' consists of huge amounts of free time apart, and theirs is 24\/7 gazing into each other's eyes? What if we agree with St Paul that 'love is patient, love is kind', while our partner is more of the Woody Allen school of thought that says 'love is suffering'.\n\nFanfares please for counsellor Gary Chapman's bestselling book _The Five Love Languages_ , which makes just this point. He suggests we each have our own vocabulary of ways we feel cared for, as does our beloved \u2013 but discrepancy between our respective visions creates relationship booby traps. Chapman's list of five languages is as follows: words of affirmation, quality time, receiving gifts, acts of service, physical touch \u2013 though there are surely more one could creatively add to the list. And it's pretty clear where the traps might lie. If our partner's top love-language is gifts and ours is words of affirmation, then however many delightful presents they bring home, we'll still feel unappreciated if they don't say those three little words. If our top criterion is quality time spent together, while our partner defines love as acts of service, then however many romantic weekends we surprise them with, they won't be satisfied if we don't occasionally take the rubbish out.\n\nWhat are your proofs of love? Define them as concretely as you can. Then list just as concretely the 'proofs' of all your deathbed answers. What do you mean by 'safety' \u2013 is that financial, practical or emotional? What are you thinking of when you talk about 'career success' \u2013 promotion, work satisfaction or appreciation from your team? What do you have in mind when you talk of 'generosity' \u2013 is that giving of your time, your money, your energy? Think through these deeper meanings so that, when the time's right, you can explain them to a partner and teach them how to be good for you.\n\nSameness\n\nIt's clear that we need a partner to accept, appreciate and approve of these elements of us. But do we need them to copy, to match exactly, to be a 'bird of a feather' that 'flock[s] together'? Here we call on the insights of psychiatrist Hellmuth Kaiser, who in the 1940s, while watching identical twins ice-skate in perfect harmony, suddenly realized that the audience's rapt fascination was due more to the synchronicity of the siblings than to their technical skill. We human beings like sameness. It makes us feel secure; babies even a few minutes old gurgle with glee when adults mirror their movements. It makes us feel validated; imitation is flattering, so strengthens our good feeling about ourselves.\n\nThis good feeling's reciprocal, a virtuous circle; if we worship a partner because we're alike, they'll probably worship us back. Which is why most dating systems, from online to matchmaking, pair clients based on likeness. And certainly when it comes to values and life goals, this approach is quite correct. Pair with someone who's taking the same track in life and we'll be content; if we're on opposite tracks we'll either move apart at speed, or play tug-of-war. Love, as Antoine de Saint-Exup\u00e9ry famously wrote, consists 'not in two people looking at each other, but two people looking outwards in the same direction'.\n\nWith personality it's slightly more complicated, the research is more contradictory and it's possible opposites really do attract. Yes, same-personality partners may make good buddies. But there may not be enough spark for interest, let alone romance, not enough complementarity for day-to-day teamwork, let alone home-building, child-rearing and balancing all the spinning plates of a practical life together. As an example, in _Little Women_ , Louisa May Alcott's novel about girls growing up in Civil War America, Jo March makes a fast friend of Laurie, the boy next door. Their bond is close and strong, but when Laurie falls in love with Jo, she is sensible enough to realize that their similarly masculine attitudes and hot-tempered personalities mean they could never make a good match. Instead, she moves to New York and there meets academic Friedrich Bhaer, whose tender character complements rather than copies her own; under his guidance she fulfils her potential, while Laurie ends up equally happily married to more feminine, less tempestuous Amy March. The lesson Alcott offers us \u2013 and one which relationship-advice books have repeated down the ages \u2013 is that we should choose a mate with similar values and goals, but with a different personality.\n\nThat said, recent research from the University of Columbia suggests it's more complex than that. Apparently a better answer is to ask a further question: What do individuals want? Those of us who feel good around similarity will thrive with a partner as like us as possible. Those of us energized around difference will thrive with a dissimilar mate. It's not the dynamic itself \u2013 all variations can theoretically work \u2013 it's our comfort level that dictates success.\n\nWhat about the 'men are from Mars, women are from Venus' issue? We might wonder whether, if similarity is so very important, but X and Y chromosomes are so irreversibly unlike, all different-gender relationships are going to be fraught. Here's the reassurance: men and women are actually very similar. Some psychologists even suggest there are only two built-in gender differences \u2013 verbal skill and aggression \u2013 while anything else is down to nurture not nature and therefore adaptable. So if a relationship ends unhappily we shouldn't blame the gender difference, or point accusingly at 'men' or 'women'. And if a relationship seems well advised, we shouldn't worry that it's all doomed just because our beloved is of a particular gender. To put it another way: 'Men are from earth, women are from earth. Deal with it.' (The original of that quote is claimed by at least a dozen people, but it was far too apposite to omit from this book just because its author can't be identified!)\n\nDisrobing\n\nValues, life goals, personality. Given honest conversation, we can probably guestimate these three elements in the first few meetings with a potential partner and \u2013 as we should \u2013 bail out if there are reasons for doubt . . .\n\nBut real proof at this level takes a while, and it's as important to talk as it is to listen. Revealing what matters to us, doing the things that fascinate us, exploring the past and planning the future together will over time remove more and more individual 'veils', and, along with that disrobing, our judgement of each other will become more and more accurate. Studies suggest that goals will become obvious first, values next, with personality revealed last of all. To slightly hasten the process, you might try raising the 'deathbed questions' as a topic for mutual discussion; crassly posed, they may bring the conversation to a shuddering halt, but used at a stage where both partners are sufficiently at ease to go deeper, they're exceedingly good for highlighting compatibilities.\n\n'Commitment is an act, not a word.' \u2013 Jean-Paul Sartre\n\nEmbroidery\n\nSpeaking of veils, is it wise to 'embroider' them in the early going, perhaps omitting certain details and subtly altering others for effect? How should we present ourselves in order to attract a mate? Answer: don't 'present', simply be. Of course it's tempting to massage the headline figures \u2013 online, apparently women often lower their age and weight, while men increase their height and salary. And of course it's courteous not to be so 'open and honest' that we spend the first few dates revealing intimate details of our most recent relationship-breakdown or the exact figures of our current financial crisis. But if we're not at least factually truthful to begin with, and if we're not emotionally truthful very soon after, we're in grave danger of ending up with a partner who wants what we say we are, not what we actually are.\n\nWhich is why I'm no fan of dating systems such as _The Rules_ , which advise a 'play hard-to-get' approach to partner choice. I have to admit to feeling faintly nauseated by a book which, as early as line nineteen of its introduction, promises a way of acting around a guy so he becomes 'obsessed with you and wants to commit' \u2013 surely patronizing to both genders. But my objections are not only ideological but also pragmatic \u2013 any suitor, male or female, who is only enthralled by the thrill of the 'hard-to-get' chase will likely lose interest once the hunt is over. Although being authentic may feel scary, the more authentic we can be \u2013 about what we believe, what we want, who we are \u2013 the more chance we'll have of eventually meeting a suitable prince or princess, even if that also means that en route we drive off a lot of incompatible frogs.\n\nWhat if, conversely, we wonder if a partner is being inauthentic? This isn't so much about whether they are lying \u2013 if we suspect they are, we're almost certainly right. But what if nerves are holding them back, or lack of self-confidence is making them hide true thoughts and feelings. Usually only time, trust and the opportunity for in-depth conversation will tell. But \u2013 though it isn't under our control and shouldn't be magicked up to order \u2013 an external drama or crisis will often make it clear, snapping us out of early courtship timidity, giving us a reason to step out and show who we really are, for bad or for good.\n\nI am remembering one client whose turning point for leaving a boyfriend was when she was involved in an accident and he suggested she call a cab to take her to A & E as he was 'too tired' to drive her there. More hopefully for one's faith in human nature, I'm remembering another client who suffered a burglary; her partner's absolute support, practical action and complete understanding of how traumatized she was sent her rightly over the edge into commitment. It was sad about the burglary, but for the relationship, it was a result.\n\nSoothe then solve\n\nThe two stories above, highlighting two very different responses to the emotional needs of a partner, bring us neatly to another kind of connection. And if you're reading this book in the hope of a single 'top tip', here it comes. Without this element in a relationship, all the compatibility in the world won't keep us afloat; with it, all the alarms, excursions, dramas and crises won't sink us. Professor Sue Johnson, on whose work this section of the book is based, even hints that every other single factor involved in partner choice may be irrelevant. Get this right and you're sorted.\n\nI'm talking about emotional responsiveness \u2013 a partner's ability to pay loving attention to our emotional needs, and our ability to pay attention to theirs. Note the reciprocity. As well as needing to choose a partner who values our feelings, we need to choose a partner who motivates us to value theirs. However wonderful a suitor, if they don't inspire us to respond, they're the wrong choice.\n\nWe all have emotional needs. Abraham Maslow's famous hierarchy of needs \u2013 that triangle at whose base sit the human physical requirements for air, water, food, clothing, shelter and safety \u2013 has on its higher storeys the need for belonging, acceptance, security, respect, love. We generate much of this inside ourselves as we mature, but the meaning of John Donne's cry that 'no man is an island' is that none of us can do the job alone. Knowing our partner will respond when we need them, knowing we'll respond when they need us, is at the heart of the love bond.\n\nSo what is this responsiveness? Here's a definition which you may find useful:\n\n1. Being able to notice, pay attention to, reflect on, soothe and express our own emotions.\n\n2. Being able to notice, pay attention to, reflect on, soothe and respond to a partner's emotions.\n\n3. Being able to reflect on and discuss the interaction between 1 and 2.\n\n4. Being able to do all the above even when a partner can't, won't or doesn't want to.\n\nNotice what's missing: any mention of solutions. Solutions may be vital, but a partner's unlikely to even hear them, let alone do anything with them, until emotions are being honoured. So first soothe. Only then, solve.\n\nAt home by the fire\n\nLet's look at this another way, not as a list but a story. One of my favourite books is Thomas Hardy's _Far from the Madding Crowd_ \u2013 where headstrong Bathsheba Everdene turns down a proposal from good-hearted employee Gabriel Oak. Instead she marries dashing Sergeant Troy and, when he abandons her, accepts the courtship of reserved neighbour William Boldwood. Just to be clear, while there's tragedy, there are no absolute villains in this story. In their own way, everyone does their best.\n\nBut only one man \u2013 Oak \u2013 is worthy of Bathsheba. That's not just because he supports her practically through his work on the farm, but also because he is the only one who can stand firm emotionally. Sergeant Troy's passion seduces Bathsheba but his feelings are unreliable, while Boldwood is emotionally crippled. Only Oak is capable of empathic generosity; in Hardy's words, 'the [things] which affected Gabriel's personal well-being were not the most absorbing and important in his eyes'. If you've read the novel, you'll know the ending. If not, then take a clue from Gabriel's proposal to Bathsheba \u2013 which at first she rejects as uninspiring, but later learns to treasure: 'at home by the fire, whenever you look up there I shall be \u2013 and whenever I look up, there will be you.'\n\nEmotional engagement\n\nWhich makes it all sound very simple. But this kind of maturity is a big ask. It means, as Gabriel Oak knows, being there for a partner over time and under all circumstances. It means staying calm even when sad, angry or anxious. It means staying engaged even when our partner is sad, angry or anxious and we are sorely tempted to tell them to get a grip. It means staying rock-solid even when our partner is firing negative emotions, with us as the target \u2013 as they will do from time to time in even in the most loving relationship. Easy? Not at all.\n\nSo are we warranted in making emotional responsiveness a benchmark for partner choice? So long as we give back, surely it's entirely reasonable to expect a potential partner to deliver emotional support, entirely justifiable to walk away if they can't or won't. Science is on my side in this. A landmark fourteen-year study by Professor Ted Huston of the University of Texas at Austin suggests that where partners aren't emotionally concerned about each other during courtship, then even if they decide to wed they'll probably part in the end. If couples are consistently kind, warm, sympathetic and empathic from the start, they're hugely more likely to stay the course. We not only can, but should, make emotional responsiveness a relationship deal-breaker.\n\nAdam realized at once that this relationship was going to be emotionally demanding.\n\nYou may at this point be wondering about the widespread belief that such demands could never be made of those with the Y chromosome. So can men respond emotionally? The answer is absolutely yes. Any male wariness of emotion is not because men feel it less \u2013 physiologically they actually experience it more strongly, more painfully, hence the wariness. Men's main handicap is conditioning; little boys are told not to cry, big boys are told to 'be a man', but what they're rarely told is how to manage their emotions. But another study from the University of Texas suggests that when told empathy is attractive and that emotional responsiveness makes for good relationships, men are totally capable of stepping up. I know this first-hand. I have had as clients many men who fully take on board this particular relationship challenge, reach out to their partners and respond to their partners reaching out to them. These men are certainly brave \u2013 but they're not unique. There are lots of them out there.\n\nProofs of responsiveness\n\nEarlier in this chapter, we needed to get very specific about values, goals and personality \u2013 thinking through what our definitions were, deciding what 'proof' there might be that we can deliver what a partner might need and vice versa. In the same way, it's good to get specific about emotional responsiveness \u2013 what we mean by it, how we would know it if we saw it \u2013 or we again risk a relationship minefield where what one partner wants is not what the other can give.\n\nHow to unearth your definition? One good way is to think of the last time you actively felt emotionally supported; what did people say and do (or not say and do) to bring you back into balance? As contrast, when did you last feel unsupported; what went on that left you disappointed and discouraged? What were the differences between those times that made such a difference in how you felt?\n\nCan we tell if a potential partner is capable of responding? It's unwise to even try to judge this in the early going, when heavy-duty emotional interaction is rarely appropriate, let alone needed. That said, the following are excellent signs, their absence worrying: if a suitor listens with concentration when we are speaking, reveals emotional awareness when they are speaking, and \u2013 a sneaky one, this \u2013 shows empathy to those in the vicinity, even though these people are clearly not the object of courtship attention. One of my friends married a woman who on their first date behaved well to a waiter when he accidentally spilled soup in her lap. Another friend instantly dumped a beau who was rude to a nervous server. On both occasions my verdict was: good choice.\n\nSerious tests of this dimension will likely come when strong and uncomfortable emotions first bite \u2013 on occasions that might be as huge as a bereavement or as small as a hellish commute from work. A partner may be able to listen, comfort, sympathize and soothe \u2013 and we may accept that from them, or not. We may be able to listen, comfort, sympathize and soothe \u2013 and a partner may let us in, or not. We need to pay close attention, for it's here that both sides will show their true colours. And it's here we will find the answers to the two questions which Sue Johnson claims are most crucial to relationship success: 'Can I be there for you? Can you be there for me?'\n\nPaul and Linda\n\nWhat if the answer to the first question \u2013 can I be there for you? \u2013 is actually 'no'? If we don't feel inspired to deliver emotionally to a partner, we need to ask ourselves why and then listen carefully to whatever reasons our minds and hearts give back. 'Because I'm just not motivated enough with this person' is extremely useful information, even though it probably signals an ending.\n\nThe answer to the second question \u2013 can you be there for me? \u2013 may be unclear. We may sense a partner's willingness but the way they respond to our 'bids' for care may not be quite what we need \u2013 a loss of attention, a turn away, an unresponsive comment. Is this just cause for dismissal? If all other signs are good it's surely unfair to call a halt without giving a partner at least a chance to step up. So perhaps we ask, clearly and without criticism, for the response we need when we're emotional? Listening or cuddling. Asking questions or staying silent. Looking on the bright side or joining in the pessimism. Giving time alone or being willing to stay for a while. While we're on the subject, we might even turn the conversation to asking what response a partner might need from us when they are distressed. Just so we know.\n\nThis depth of conversation may not be possible at once. And those few preceding paragraphs may seem more like advice on how to relate to a partner than advice on how to choose one. But a huge element of successful decision-making is finding out whether each partner is both able and motivated to actually learn what the other person needs and what the relationship may demand. If either of us can't or won't learn then, harsh as it may sound, there is no future. If we're prepared to study, the future is shining bright.\n\nApparently when Paul McCartney and his first wife, Linda, married, they made a private vow: 'I will never put you down'. To my mind, this didn't only mean that they would never insult or disparage each other. It didn't only mean that they would never betray or abandon each other. It meant that they would never stop being aware of each other's feelings, never stop giving attention to each other's needs, never fail to open up to each other, never fail to reply. Emotional responsiveness at its best.\n\nFour tendencies\n\nWe come now to perhaps the most intricate of all issues by which we may assess relationship potential \u2013 how we relate, not just when one of us has an emotional need but when the connection between us threatens to fray.\n\nBecause all of us will inevitably, at some point in our relationship, feel insecure, even unloved. Where we differ is what we do with that. At the heart of understanding what we need in a partner is understanding just how we differ, and how that plays out.\n\nFor which insight, we thank psychologists John Bowlby and Mary Ainsworth and their work on child development (which, for easier consumption, I've here simplified). Bowlby began, in the mid years of the twentieth century, with a simple but innovative idea: that for humans, life is insecure. This starts early. We lose trust when we are propelled at birth into a world that is totally unlike the safe and comforting womb we just left. We lose more trust when, however devoted our carers, they're sometimes too distracted, busy or stressed to give us the attention we need. Yes, we survive, and largely we survive well. But underneath it all we're still mistrustful, fearing that we won't be loved and if we are that love will disappear \u2013 when such insecurity hits, we fall back on our own individual coping mechanisms, 'attachment' tendencies, as Bowlby called them.\n\nMary Ainsworth built on this theory with an in-depth study of seventy-six toddlers and their mothers, exploring more fully what these coping mechanisms might be. Mum and baby were shown into a room full of toys and with a research assistant on hand. Once toddler was happily playing, Mum left the room three times for three-minute intervals \u2013 during the last time the research assistant also left. Some little ones cried at first, then calmed, seeming 'secure' that Mum would come back. Some became 'anxious', and when Mum returned, clung on in case she went away again. Yet others protested by cutting off, ignoring Mum on her return, punishing her for her absence; Ainsworth called this 'avoidant' behaviour. Others got angry with Mum for leaving; I'm naming this tendency 'attacking'. Four toddler reactions, four emotional strategies, four tendencies to behave a particular way when feeling insecure, unloved and wary that love might never return.\n\nAs adults . . .\n\nA few decades on from Bowlby and Ainsworth's work and we see a growing realization that these tendencies don't just fade as a child grows up. As adults, we're certainly more secure because we're more in control of our world, but we also repeatedly learn \u2013 often through the route of heartbreak \u2013 that we can't always trust that world to deliver. And we still harbour those four attachment tendencies, sometimes majoring in one, mostly doing mix-and-match to differing degrees. When we are 'secure', trusting in ourselves and others, we handle love situations with calmness and confidence \u2013 even if in the end a relationship dies. When 'anxious' we are worried by love, unsure of our competence and needing reassurance. As 'avoidant', we fight shy of emotional engagement, pull away if commitment looms. 'Attacking', we feel an inner frustration, perhaps creating conflict in order to connect.\n\nIf any of this sounds familiar, you're right. We all have all these tendencies in different proportions. We may not manifest the more challenging ones except when we feel insecure or feel a relationship connection weakening \u2013 but they're there in all of us. For that, no blame, no shame; attachment tendencies are the human condition.\n\nWhich of course is why so many characters in literature and film display them (aside, of course, from the 'secure' tendency, which is so mature and wise that it has zero dramatic potential). The other tendencies, however, provide endless raw material for compulsive characters and angst-driven plot. For 'anxious', see Bridget Jones's romantic insecurity and her readiness to cling again and again to Daniel Cleaver even when he behaves badly (though thankfully, in the end, she sees the light and walks away). As an example of 'avoidant', we might turn to the butler Stevens who, in Kazuo Ishiguro's novel _The Remains of the Day_ , is so wary of the relationship that might develop if he showed his true feelings for housekeeper Miss Kenton that he fails to admit his love over a period of several decades. For 'attacking', we need look no further than Heathcliff and Catherine in Emily Bront\u00eb's novel _Wuthering Heights_ ; while they think of themselves as one \u2013 'Whatever souls are made of, his and mine are the same' \u2013 they are nevertheless endlessly intense with each other and Heathcliff in particular can often only express his love for Catherine through absolute rage.\n\nThese tendencies have had an undeserved bad press. As is clear from the last paragraph, particularly when they're presented in art rather than real life, they're often seen as being all downside, fault and weakness. 'Secure' is, rightly, viewed as the poster child for happy and mature relationships. But the other tendencies have their upsides too. Being anxious may mean we're motivated to ride out a partnership storm, stay loyal, stay committed. Being avoidant may mean a certain emotional independence, the ability to give partners space. An attacking tendency can mean being comfortable with dissension, or being able to summon huge intensity and passion.\n\nSo, to what extent do you stay calm and serene, trusting in life and love to give you what you need? How far do you stay loyal \u2013 even though you also sometimes cling? How far do you maintain your emotional independence \u2013 albeit you may get accused of not caring? How far do you engage energetically \u2013 granted that you occasionally pick fights? How do you respond when you feel love sliding away from you? What, in short, are you going to bring to the partnership table when you feel less than fully connected?\n\nPartner tendencies\n\nAnd what will a partner bring to the table? Discovering that is never going to be as simple as gauging height and weight, values and life goals, or a capacity for emotional responsiveness. Crucially, attachment tendencies may not kick in at the start of a relationship because at that point we likely feel perfectly loved and perfectly secure. So how can we make at least a top-level guess on a beloved's tendencies in time to make a reasoned decision?\n\nHere's a useful diagnostic. Sex. Sex is one of the arenas where a person is most themselves, and reveals that self most early and openly in a relationship. It is Sue Johnson who here again offers guidance. She suggests that sex which involves a healthy balance between physical pleasure and emotional bonding might be called 'synchrony sex' and reflects a secure tendency. Sex used largely for comfort and as a way to calm doubts or conflicts, which Johnson terms 'solace sex', suggests a tendency to be anxious. Sex that is self-focussed, performance-oriented, with little emotional openness, could be labelled 'sealed off' sex; it signals avoidant tendencies. I would add that energetic, forceful, 'fight it all out' passion is what one might call 'squabble sex' and might reflect an attacking tendency.\n\nAgain, these are tendencies, not types. All of us make love in all the above styles \u2013 with emotion, for reassurance, as self-gratification, to let out frustration \u2013 in fact my guess is that if you mixed all four together what you'd get would be sex labelled 'normal'. But a noticeable preponderance of one type or the other can act as a Rorschach test, revealing who a potential partner is and what a potential partnership with them might be like.\n\nTom and Jerry\n\nWhat do attachment tendencies suggest for partner choice? Similarity can work. When both of us are anxious or attacking we'll understand each other and thrive on our mutual need for security or strong engagement. When we're both avoidant \u2013 that is, if we ever manage to get together in the first place \u2013 we may feel constant gratitude for the reciprocal freedom.\n\nDissimilarity can play well too. If we compensate for each other, some anxious dependency may strengthen the bond; a small amount of avoidant independence may stabilize interactions; a little attacking engagement may mean issues get put on the table and dealt with in healthy short order. (A good exercise, at this point, is not only to remember your own past relationships but also look at the ones you see around you, as acted out by friends, family or colleagues. Playing 'spot the attachment tendency' is not only good practice for understanding partners-to-be but also good fun.)\n\nBut do be wary of extremes. If we meet a partner who pushes for commitment on the first date, one who's never had any relationships longer than a few weeks, or one who picks a fight within minutes of meeting, beware. We are facing respectively an off-the-scale anxious, off-the-scale avoidant and off-the-scale attacking personality. Unless we're totally convinced we can handle it, we shouldn't even try.\n\nThe more subtle, and often less easily spottable, combination from hell is when partners of different tendencies date but can't accommodate. Initial attraction may be strong \u2013 contrast makes for interest \u2013 and when we're safely in love there may be no trigger for attachment wobbles. But fast forward a little: inject any kind of stress or insecurity and the dynamic will make both sides crazy. Anxious plus avoidant means one of us clings, the other pulls away. Avoidant plus attacking means one of us runs, the other pushes to engage. Attacking plus anxious means one fights, the other fears. The result can be a Tom and Jerry cartoon-type chase, with A emotionally pursuing B round the room of the relationship. If you've ten years of commitment behind you by the time this sort of thing creeps in, it's absolutely worth taking time, energy and counselling to resolve it. But if it's happening a mere ten days or weeks into a new relationship, run for your life.\n\nThe ideal\n\nThe ideal here \u2013 and it is an ideal \u2013 is that mostly we are secure with each other and that at least one of us is secure most of the time. Yes, there'll be bouts of wobble \u2013 but in terms of choosing a partner, what to look for is someone largely honest and direct, who communicates clearly and is not into game-playing. Someone who is aware of emotion, not afraid of intimacy and is open to the possibility of commitment. In other words, human, but largely sorted.\n\nBut there's a problem. What I've just described can easily be experienced not as ideal but as boring. Remove the emotional roller-coaster tendencies of anxiety, avoidance and attack, and what's left can feel all too calm. It's easy to get confused and think that because someone isn't any kind of a problem, they are no kind of a partner. I have had clients who in the same breath as describing their relationship as 'the most secure I've had', express doubts that it is 'the real thing'; clients who, while reporting contentment, worry that what they're experiencing is too effortless to be true love.\n\nIn fact, secure is good, contented is something to be glad about, and the occasional tempest is exhilarating but constant storms are exhausting. If in the past we have experienced a feeling of secure contentment with a partner but didn't choose them, that was our loss. And if in the future we experience that kind of security in a relationship, then \u2013 assuming all other boxes are ticked \u2013 we should offer up heartfelt thanks and hang on for dear life.\n\n#### 7. Being in Love\n_Anthony Carthew: 'And [you are], I suppose, in love?'_\n\n_Lady Diana Spencer: 'Of course!'_\n\n_Charles, Prince of Wales: 'Whatever \"in love\" means.'_\n\n(ITN INTERVIEW ON THE ENGAGEMENT OF CHARLES AND DIANA, 1981)\n\nI am a huge believer not only in the possibility but also the wisdom of falling in love. It's a magical, sparkly feeling. The world seems bright and shiny, the future seems glorious, everything seems possible. Yes, being in love can sometimes be hard \u2013 as the classic French ballad has it, 'the pleasure of love lasts a moment, the pain of love lasts a lifetime.' But the excitement, the arousal and the adoration are surely irresistible. Everyone should be love-struck at least once in their life.\n\nTwo examples. In Charles Dickens's novel, the eponymous David Copperfield first meets Dora Spenlow when he visits her father's house. 'All was over . . . I was gone, headlong.' There follows a delightful description of the ecstasy that comes with having 'fulfilled my destiny . . . in an abyss of love.'\n\nAnd in case we think that fiction always exaggerates reality, let us also recall footballer David Beckham's equal wonderment at first seeing Victoria on a Spice Girls video: 'I thought \"she is . . . perfection\".' Their later face-to-face meeting, at a charity football match, happily led to David's infatuation being fully returned and an almost instant relationship, engagement and marriage.\n\nFixation\n\nBut what does being 'in love' have to do with effective relationship choice? In some ways, not a lot. In evolutionary terms, this rush of wonderful emotion was originally designed not to help us choose a compatible match, but to help us stand by a partner with whom we were making babies. Lust was there to get us rolling in the hay, being 'in love' was there to make us willing to push the pram alongside the one we'd originally sneaked off to the barn with. The 'in love' flurry of the hormones known as 'monoamines' exists to focus us on partnership through those early years when offspring need us to stay close, to the exclusion of all else. So much so that research at the University of Pisa has found key similarities between the monoamine levels of new partners and those who suffer obsessive\u2013compulsive disorder. When we fall for someone we become literally fixated on them \u2013 as David Copperfield describes it, 'a captive and a slave'.\n\nThere's a less evolutionary, more psychological and \u2013 in this age of family planning \u2013 less offspring-focussed set of reasons why 'in love' is so compelling. It's that when we fall in love, we're following an emotional dream of being the centre of the world. While very small, unless our childhood was damaged, those around us did their best to keep us absolutely safe, warm, cared for, loved. We leave that behind as we grow to adulthood, but we'll always be looking for it again, always be wanting to recreate the security and the validation that was ours in the early years. 'In love' holds out the promise that our beloved will make us the centre of their world, and for ever. No wonder it's an obsessive compulsion.\n\nWe build on that compulsion. When in love, we're likely to spend huge amounts of time together \u2013 proximity, you may remember, strengthens attraction. We stay close, connecting through sex but also through matched body language, emotional revelation and deep eye contact. The effectiveness of those last two elements was fascinatingly demonstrated by psychologist Arthur Aron, who got previously unintroduced subjects falling head over heels with each other simply by having them ask each other a series of thirty-six self-disclosure questions, then gazing into one another's eyes for a mere four minutes. This approach was later used in an experiment to raise the 'take-up factor' for speed daters \u2013 and indeed those daters who asked more personal questions and gave more revealing answers got more picks from the evening.\n\nThe wrong roller coaster?\n\nThe problem is that thereafter we may get stuck. For though I hate to be the bearer of bad news, none of that wonderful falling-in-love experience is guaranteed to produce long-term compatibility. Literature is riddled with stories of how initial adoration translates badly into ordinary everyday life: Anna Karenina and Count Vronsky; Madame Bovary and Rodolphe Boulanger; even Romeo and Juliet, if we believe Shakespeare's hints that had they actually set up house together in Mantua, the result would have been something of a car crash.\n\nIt's best to choose a roller-coaster ride that brings you back down to earth both safely and without nausea.\n\nGiving more cause for concern \u2013 for literature does dramatize for the sake of a good yarn \u2013 there's little scientific research showing a link between short-term attraction and long-term compatibility; Arthur Aron himself points out that while the closeness produced in his studies is similar to romantic passion, 'it seems unlikely that the procedure produces . . . commitment.' 'In love' might turn into a loss-leader for something deeper, but there's no absolute correlation between the two.\n\nThere is, however, correlation between 'in love' and anxiety. Here's another Arthur Aron experiment, this one involving eighty-five male subjects and a wobbly footbridge. Result: the men were significantly more likely to be charmed by a female researcher if they'd previously been scared out of their wits crossing the deep canyon at Capilano, Vancouver. (In case you're suspecting gender bias here, the experiment was replicated some years later with women as subjects and male researchers; it produced the same outcome.) What both groups experienced was a dynamic that's classic in the early stages of a relationship; our whole body is in a peak state of nervous tension, not only from wanting our beloved but also as a result of not knowing whether our beloved will want us. This arousal then provides the motivation to do exactly what we are genetically programmed to do, cling on to each other. Even the most delightful bonding involves some level of anxiety, and, in turn, anxiety often leads to bonding.\n\nYou can probably see where I'm going with this. We can easily confuse the strong emotion of the 'in love' kind with the strong emotion of more unhappy kinds. We may even find ourselves likely to bond in _any_ uncertain situation, with any uncertain partner \u2013 one who is emotionally unreliable ('I'll phone you' syndrome), one who is seriously uninterested ('s\/he's just not into you' syndrome), one who is terminally unavailable ('it's complicated' syndrome). In fact, when we keep choosing unsatisfying partners, that might be not despite the fact but _because_ of the fact that they are unsatisfying. If you recognize yourself in this scenario, don't self-blame; there's a very small distinction between an emotional roller coaster that makes us squeal with delight and one that makes us shriek with horror, and most of us at some time choose the wrong ride.\n\nThe earthquake of love\n\nThere's something else we need to remember about 'in love'. It's date-stamped. That original flurry of addicted monoamines will in the end give way to a lower-key flow of the 'cuddle hormone' oxytocin, designed to give us a more secure connection and get us not just pushing the pram but setting up house and home until the kids have grown and flown. There's a natural switch from high arousal to low-key stability, from excitement to steadiness, from absolute adoration to loving respect: as Louis de Berni\u00e8res writes in his novel _Captain Corelli's Mandolin_ , 'a temporary madness . . . an earthquake . . . then it subsides.'\n\nGiven this non-correlation between 'in love' and compatibility, it's therefore wise to avoid making choices from the earthquake zone, wise to wait until the earth has stopped moving before we pitch permanent camp. As de Berni\u00e8res suggests, at that point there is a decision to be made about whether we are so compatible, so committed that even without 'in love' we want to stay together. This stage beyond 'in love' may not feel as intensely exciting, but it's what makes a relationship stay the course. 'Love itself,' says de Berni\u00e8res, 'is what is left over, when being in love has burned away.' If we want a partner for the long term, we need to discover what is 'left over'.\n\nHere's an exercise to help in the discovery. Pick three friends you've known for a while and are emotionally at ease with. Now think of what makes them your friends; think of what you get from them that has maintained your connection over time. Fix in your mind that feeling of being comfortable with them, relaxed in their company, authentically yourself, simply content. And there's your point of reference, the touchstone to use when you fall in love. If you feel with a partner that same sense of comfort, relaxation, contentment \u2013 and if you sense that feeling could endure \u2013 then you are on safe ground.\n\nGood navigation\n\nWhich unfortunately doesn't mean that the ground will stay safe for ever. Yes, we're always told that when we love we will not only live 'happily' but will do so 'ever after'. We're assured that, apart from when it's absent, denied or ended, 'love is a many-splendoured thing', 'love lifts us up where we belong', 'all you need is love'. And so we tend to set our compass for partner choice by the pole star of joy . . .\n\nOften, that's good navigation. If we choose a partner who complements us, who completes our jigsaw and adds their skills and strengths to the ones we lack, we'll live a more effective but also a more contented life. In Jane Austen's _Pride and Prejudice_ , Elizabeth Bennet, having spent most of the novel thinking that Darcy is 'the last man on earth' she would marry, realizes at last \u2013 and possibly too late \u2013 just how complementary they are and how happy their relationship could be. 'It was a union that must have been to the advantage of both; by her ease and liveliness, his mind might have been softened, his manners improved; and from his judgement, information, and knowledge of the world, she must have received benefit of greater importance.' (If you've never read the book, be reassured: they get together in the end.)\n\nSimilarly, if we choose a partner who supports us, who by their attention and responsiveness heals our emotional wounds and makes us whole, we'll live a more fulfilled but also a more joyful life. Here is another Elizabeth, this time Elizabeth Barrett, writing to fellow poet Robert Browning after he has expressed his love for her: 'To receive such a proof of attachment from you not only overpowers every present evil, but seems to me a full and abundant amends for the merely personal sufferings of my whole life. [My tears] went away in the moisture of new, happy tears. Henceforward I am yours.' (If you don't know the history, they married; he then helped her to not only bear lifelong illness and a family estrangement, but also fulfil her promise as a writer.)\n\nPeople-growing\n\nBut despite these examples, love doesn't make us happy all the time and every time. Maybe an adorable partner has a less-than-lovable phase, an easy relationship starts to feel like hard work, a future that was glorious slowly becomes ordinary or even disagreeable. In Dickens's novel, David Copperfield finds it hard to come to terms with Dora's naivety and impracticality, while David Beckham's marriage has reportedly not been without its ups and downs. Faced with dissatisfaction, it's not surprising that we start to question our judgement. Does our unhappiness mean we chose the wrong partner? And what does it mean about how we made our decision and whether we should renege?\n\nHere's the thing. A relationship is what psychologist David Schnarch, in his book _Passionate Marriage_ , calls a 'people-growing process'. It invites us to flourish by learning to overcome not just the problems of everyday life, but the specific problems that our partner and our partnership present. It invites us to step up to the challenge of becoming more tolerant, more patient, more loving than we were before, in order to cope with the one we love. (If this seems unfair, be reassured. It cuts both ways; a partner has to step up in order to cope with us.)\n\nThis challenge of people-growing is never going to be a constantly happy one. It can't be. Poet David Whyte, in his work _The Three Marriages_ , offers the metaphor of buying two houses in order to join them together \u2013 but instead of being able to simply knock down the walls, we have to demolish both houses completely so we can rebuild, and 'from the razed foundations of our former individual [selves] make a new home'. Schnarch picks up on this metaphor but with an extra emotional challenge: that in order to do this we must become more loving not only to our partner, but to ourselves as well. We must be strong, self-contained, 'secure', comfortable in our own skin \u2013 for we need to feel 'at home' in ourselves in order to have 'a good place to invite a spouse to visit'.\n\nStanding in love\n\nBoth Whyte and Schnarch say the same \u2013 that this process is universal, that it is a good thing, that we need to welcome it as a way of maturing and developing, that long-term it will lead to happiness because it will help us thrive. Similarly, Erich Fromm, whom we met earlier in this book, says that falling in love is inevitably followed by a period of needing to 'stand' in love, if our relationship is to survive. Happiness will not necessarily be what marks progress; sometimes there will be pain.\n\nA common response to all this is to swallow hard as one tries to come to terms with the shock revelation that even the most successful 'love, sweet love' doesn't always feel so sweet. But there often follows a sigh of relief that relationship problems don't imply that the partners are wrong for each other or that they necessarily need to separate. For even if we make the most perfect partner-pick in the world, we will at some point meet challenges. We will always be required to master the essential human balancing act of trying to answer our own needs while meeting those of our beloved, of loving ourselves while loving another, and of growing through that process.\n\nSo the key aim in finding the right partner should not be to try to avoid that balancing act \u2013 it's inevitable \u2013 but to find someone for whose sake we'll attempt the act because we love them so much, someone who loves us so much that they'll make the attempt for our sake. The question for both sides becomes how to choose a partner who is so compelling that we're willing to demolish our own house in order to rebuild a more beautiful mutual home.\n\nDiamond-polishing\n\nThe good news is that we likely know how to choose correctly even when we're not doing so deliberately. For humans are drawn towards those who help them grow; the message from Whyte, Schnarch and Fromm is that partner choice is the way we actively, though often unconsciously, choose to mature. Schnarch even likens it to the meeting of two flawless but rough diamonds that rub away every part of themselves that doesn't fit in order to stay joined together. The end result is not only a loving relationship, but two sparkling jewels.\n\nSo, for example, we may be drawn to someone who needs something from us that we find it hugely difficult to deliver: attention even when they are angry, energy even when we are drained, emotional control in the face of difficult circumstances. Over time, because we care, we learn to step up and meet these challenges \u2013 and in this way develop a part of ourselves we would otherwise never have developed.\n\nOr, we may pair with someone who \u2013 once the rose-coloured glasses are off \u2013 manifests some vulnerability or fault that we dislike, or even deny, in ourselves. Because love means we are motivated to understand and accept that partner, we learn to accept and understand our own vulnerabilities \u2013 and so we thrive.\n\nMore unexpectedly, perhaps \u2013 for more explanation of this, reread Chapter 3 \u2013 we may find someone whose personality reflects that of someone from our early past with whom we've had a complicated relationship. That reflection may make our current relationship complicated \u2013 but given that we are now older and more mature, we may learn to relate to our partner in a way we never could to the original. We may grow up sufficiently to create a happy ending this time round.\n\nTake some time to think back to past relationships with partners (or friends, or family) where things were challenging but left you changed for the better, even in small ways. You've almost certainly, in those relationships, chosen to be with people who helped you with the diamond-polishing. So what did those relationships give you? What benefits have you gained? And what can you learn that will help you begin \u2013 and maintain \u2013 your future partnership?\n\nThe right order\n\nHow can we tell whether a partner will help us mature in these ways and whether we can help them in return? There's no formula, no guaranteed way of discovering if someone constitutes our growth opportunity \u2013 and as with attachment tendencies, real evidence may only be possible when the relationship's well beyond the point of first choosing.\n\nBut in the earlier going, certain signs are encouraging: feeling ourselves becoming more emotionally truthful, and seeing our partner becoming equally more authentic; feeling encouraged by them to reach our potential, and being able to encourage them to do the same; finding ourselves both learning and teaching, like Elizabeth Bennet; or finding ourselves emotionally healing and professionally achieving, like Elizabeth Barrett. In her novel _Beloved_ , the author Toni Morrison describes this kind of personal development as if the person in question were a jumble of jigsaw pieces, taken up by a partner, rearranged and returned 'in all the right order'. If we have chosen well, both partners may find their 'pieces' falling into place.\n\nThe sparkle\n\nThe lesson of this chapter is not that we should choose a partner with whom things are tough from the very beginning \u2013 if that's what's going on, it's not the right relationship. The lesson is that the best love is a three-part process, with the initial delights driving us on to ride out the medium-term challenges for the sake of long-term rewards. To complete the stories with which we began this chapter, David and Victoria Beckham are near to celebrating two decades together, while David Copperfield learned to accept his wife's failings and so allowed his marriage to gain stability, even if it was cut short by Dora's death.\n\nEnough, then, of any alarm bells. My original statement of belief in love still stands, and more. Because, firstly, whether or not a relationship lasts, being love-struck may be worth it just for the experience. And, secondly, if the relationship does last, love-struck is a wonderful place to start. The very fact we are so drawn to someone else \u2013 the lure of the obsession \u2013 is a huge motivator to stay loyal if things become challenging. The fact that we want so much from a partner means we're driven to deliver in return. The fact that 'falling in love' means we bond, connect, open up, reveal and respond to each other emotionally creates the foundation for 'standing in love' further down the line. If what's left after the earthquake subsides is a solid core of commitment, then we have the best of both worlds.\n\nSo let's embrace the romance, let's enjoy the passion. In short, bring on the sparkle.\n\n#### 8. Knowing\n_Seven years would be insufficient to make some people acquainted with each other, and seven days are more than enough for others._\n\n(JANE AUSTEN, _S ENSE AND SENSIBILITY_)\n\nThe question we're most likely to ask ourselves when we begin to date is 'How will I know?' What this really implies is two questions, 'how' and 'when': 'How and when will I know enough to choose this person, or to not choose them, or to decide that the moment of choice is past?' Let me say from the start that there's no answer here, nor are there guarantees that one ever will know. All we can do is explore the possibilities.\n\nInstant reaction\n\nPossibility number one is that the answer to 'how will I know?' is 'sheer gut reaction' and the answer to 'when will I know?' is 'instantly'. Eyes meet across a crowded room and the deal is sealed; as examples, see two separate if very different American presidents. In his memoirs, Bill Clinton reports that when his wife Hillary first spoke to him, he was so overwhelmed that he forgot his own name. Lyndon B. Johnson apparently asked 'Lady Bird', as his wife was known, for a date within minutes of meeting, and proposed at the end of that first date; they were still happily wed when he died almost forty years later.\n\nOf course, an instant and instinctive decision about partner suitability is often a vote against \u2013 the elimination principle at work. Which is why, if all other boxes are ticked, it's often wise to follow up even the most disastrous first meeting with a replay. That allows us to gather more, and more accurate, evidence \u2013 as well as reducing self-consciousness on both sides, allowing prospective partners to shine. This advice was passed on to me by a colleague who judged her blind date to be socially incompetent but gave him the benefit of a second hearing. She soon discovered he was simply overawed by her, a trait which soon transformed itself into socially competent adoration on both sides.\n\nThen there's the opposite danger. Over-speedy decisions can lead to being totally swept off one's feet by beauty and charm, only to realize that one's given house-room to a monster. That said, as I've confessed, I'm no enemy of a little carefully managed infatuation. And the danger's often resolved by the inevitable removal of rose-coloured glasses \u2013 inevitable because such instant passion is likely to implode at the first sniff of a problem; as Shakespeare put it, 'These violent delights have violent ends \/ . . . like fire and powder \/ Which as they kiss, consume.' But if instant enchantment makes us want to instantly commit in some comparatively irreversible way (mortgage, marriage, motherhood) then it's best to apply the necessary brakes.\n\nTaking time\n\nBill Clinton may have been so overwhelmed by Hillary that his brain stopped working, but the 'how' of knowing is normally more thoughtful. And it's usual to take longer than Lyndon Johnson's few minutes to decide that we have found our life partner; radioactivity pioneer Marie Curie, for instance, turned down husband Pierre's proposals a full seven times before she was convinced. Taking at least some time, space and consideration typically works best because it helps us to discover more about a partner, allows them to discover more about us. Where possible, dig deeper, search wider, allow both logic and emotion, head and heart to synchronize. As the Russian proverb runs, 'Trust, but verify'.\n\nWhich is why communities have traditionally built in delay: long engagements, no sex before marriage, betrothal for a year and a day before the wedding proper, all so we can verify the information and draw a considered conclusion. Our contemporary speedy courtship rituals \u2013 sleeping together without vows, moving in together without documentation \u2013 are only acceptable because we're now not bound together for life if love turns to hate. Ironically, the most modern method, online dating, has reintroduced some of that traditional delay, with users going through a sometimes lengthy and in-depth process of 'getting to know you' while they continue to 'get to know' several others online until they commit.\n\nBut it's also best not to be too slow and too considered, for there are dangers in hesitation; we may lose not only momentum but also faith. Especially in the key transition stages \u2013 having sex, becoming monogamous, moving in \u2013 if the moment passes, so too may the belief that it was a good idea. If you're worried about a timetable be reassured that, as the Jane Austen quotation at the beginning of this chapter shows, there is no ideal. That said, as very rough guidelines, two months of regular dating is long enough to know whether both sides want to declare themselves partners, two years long enough to know whether lifetime commitment is possible. Pass those breakpoints and it's justifiable to question the length of the journey.\n\nHow do you know?\n\nSo take stock. What is your journey to knowing? Do you rush into connection, physical or emotional, with spontaneous eagerness, enthusiasm and a marked inability either to doubt or to tolerate delayed gratification? Or do you progress so slowly and serenely in affairs of the heart that onlookers \u2013 or even the objects of your affection themselves \u2013 believe you're ambivalent? In other words, do you typically push for commitment because you're so quickly convinced that you've found perfection, or delay it because you're waiting for perfection to prove itself?\n\nThere is an argument here for acting against usual tendencies. Previous speedy commitment may have led to some lack of judgement, previous tardy commitment may have meant losing love. And, as the old saying goes, 'If you always do what you've always done, you'll get what you've always got.' So if we normally rush in, then, without playing 'treat 'em mean and keep 'em keen', we could maybe try holding back on that first kiss, that first sexual encounter, that first meeting with parents, until we know precisely what we're dealing with. Conversely, if we typically tend to hesitate, then, without playing 'wear your heart on your sleeve', we could maybe 'lean in' to suggesting that date, supplying that extra front-door key, declaring those intentions \u2013 even before it feels completely comfortable. Altering approach is, as always, a challenge. But perhaps if you do what you've never done, you might get what you've never before got.\n\nOne of the nicest, as well as one of the most effective, ways to make a partnership decision \u2013 which involves taking time to explore but doesn't drag things out \u2013 was suggested by my colleague Dr Charley Ferrer. Its approach is based not so much on information-gathering or timing as on attitude. For ninety days, Dr Ferrer advises, we should commit completely. We shouldn't hold back for fear of being taken for granted, shouldn't cling on for fear of being rejected. Instead, for a full three months, we should offer full engagement in giving and taking, full emotional responsiveness, full trust that the partnership will continue. In other words, we should behave as if we already 'know' and have already chosen. If after that time we have no hesitations, then we know enough to say a completely wholehearted 'yes'. If at the end of the ninety days we are still hesitating, we have gained more than enough knowledge to justify a 'no'.\n\nSaying no\n\nIn some cases that 'no' is obvious. Deceit. Infidelity. Drug dependency. Abuse. Violence. All these bad behaviours are perfectly good reasons to flee, even after making an initial choice.\n\nThere can also be good reason to flee when there's no bad behaviour at all. There are numerous partner combinations that don't augur well: deal-breakers in our preferences; differing values, goals, personalities; questions about one's enjoyment of a partner's company; a not-quite-good-enough sex life; an annoying level of conflict (even if it is generally bearable). As Ross from the TV comedy _Friends_ commented when asked why he and lesbian Carol weren't together: 'This is not a mix-and-match situation', and we shouldn't attempt to make round holes accommodate square pegs. (We equally, of course, shouldn't simply walk away in an attempt to force the pegs into the holes; break-up rarely works as a way to compel commitment in ourselves or others, and even if we reconcile, such abandonment will always strain the partnership, sometimes irreversibly.)\n\nSometimes too, the answer is neither to go nor to stay but to try the third alternative. Change. If the benefits of partnership are obvious, we could \u2013 kindly, supportively, specifically \u2013 ask for the adjustments we want. Our partner might then answer \u2013 willingly, readily, enthusiastically \u2013 that they're happy to oblige. Or vice versa. If that's the conversation, there's good reason to stay.\n\nThe bad news is that, without that conversation, there's every reason to go. Professor Ted Huston's fourteen-year couple study, mentioned earlier in this book, found that when women in the courting phase of a relationship predict future problems, their predictions usually come true. So if we decide to stay with someone in the hope they'll improve in time, we're not really committing to the person they are now but to a future ideal that's highly improbable. As Albert Einstein \u2013 who was something of a philosopher as well as a theoretical physicist \u2013 famously commented, 'Men marry women with the hope they will never change. Women marry men with the hope they will change. Invariably they are both disappointed.'\n\nIf having done due diligence we still feel a quiver of doubt, we could stay one more day to make absolutely sure. But if what we feel at the thought of staying is a flinch, a nausea, an exhaustion, that's our unconscious screaming 'go'. In which case, let me offer you an absolving 'get out of jail free' card. It is fine to turn someone down. In fact, it's actually best to turn them down if you have come to the conclusion that they aren't for you. Because by walking away you're not just freeing yourself to find someone you can love. You are also freeing your no-longer-potential partner to find someone who can love them. Don't feel guilty. If we know our heart is not in a relationship we do right by everyone if we leave.\n\nNot being chosen\n\nSometimes, though, the situation's more complex. Our heart is in, fully in. But our feelings aren't returned \u2013 or perhaps worse, not returned right now but with a hint that they might be at some future time. Then things become hugely more difficult \u2013 though this is also a timeless theme in our history and culture. The poet Dante's love for Beatrice. The Hunchback of Notre Dame's longing for Esmerelda. Mark in _Love Actually_ standing before his ideal (but already happily married) partner with a poster reading 'my wasted heart will always love you'. These tales touch us \u2013 and not only because we prefer a good ending to our love stories but also because they trigger a human terror that began when we were born: that we will not get everything we want in life.\n\n_Dante and Beatrice._ Sometimes, there's nothing more comfortable than focussing our choice on a prospect who, because we will never have them, will never disappoint us or be disappointed by us.\n\nHowever unfulfilled a relationship \u2013 Dante loved Beatrice for life, but only met her twice \u2013 not being chosen is a blow. If, or when, it happens to us, we shouldn't underestimate the impact; studies have equated this loss to an actual physical attack, a trauma, a bereavement. Then we need to cling to the fact that \u2013 given some weeping and wailing, but also given time, human support and a sense of perspective \u2013 we will likely move through the suffering.\n\nBut if we don't move through rejection, we can land in deep trouble. For sometimes the label 'non-available' spurs us to even more commitment; surely if we can be different, woo more successfully, offer more resources, deliver more favours or simply try harder, we will get what we want.\n\nHere the cautionary tale from literature is the Great Gatsby, F. Scott Fitzgerald's millionaire hero who invests everything he has in trying to rekindle the love of Daisy Buchanan's 'beautiful little fool'. He takes a luxurious mansion directly across the bay from her marital home, spends endless nights staring over at the green light at the end of her dock, throws hugely extravagant parties to tempt her to come to him.\n\nIt's clear from the start that there will be no happy ending here; Daisy's self-absorption makes it almost inevitable that, while she does restart her affair with Gatsby, she will in the end abandon him. But Gatsby himself creates his own tragedy by sacrificing all to win the love of someone who cannot truly love him, squandering his fortune in relentless decadence, abandoning his values and in the end losing his life to protect Daisy. The whole story is shot through with the pointlessness of clinging on in hope.\n\nIf our love story ever approaches that of Gatsby, the answer is not to twist our hearts and our souls out of shape as he did, but conversely to be even more authentic than usual. If after a while we are still not loved, then however much we mourn, it's been a lucky escape. In the same way as it's best to free up a partner we don't want, it is also best to be freed from a partner who doesn't want us. We deserve more than that.\n\nBeing chosen\n\nWhat if, instead, we are adored without adoring in return? If our immediate reaction to being picked is obvious repulsion, that's fairly straightforward; there likely follows a few embarrassing conversations, a little self-reproach and the occasional need to repel the boarder in question.\n\nMore difficult is when our reaction is not repulsion but temptation. For being the object of desire can be highly seductive. If we suffer the normal human tendency to self-negation, it may feel very good indeed when someone else puts us at the centre of their universe, giving us a control over them that's even more seductive because we aren't in any danger of ceding any control to them. (Plus, if we are feeling pessimistic because previous partner choices have misfired, such a person may be even more attractive because we believe they are our only remaining option.)\n\nGiven all this, being pursued can make us blind to danger signals. We may ignore huge signs of incompatibility, consign huge doubts to the file marked 'ignore'. And, particularly if our would-be partner stays the course, we may stay with them despite misgivings; there is something deeply romantic about Being Won Despite All Challenges. 'She wanted me even though she was married. . .'; 'I wasn't interested but he was so keen. . .'; 'I was convinced by her conviction that I was the one for her . . .' It was Wallis Simpson who \u2013 feeling bound to marry the abdicated Edward VIII long after she had tired of him \u2013 commented: 'You have no idea how hard it is to live out a great romance.'\n\nBalance, imbalance\n\nAgain, pause and take stock. Do you recognize in yourself a habit of staying in a relationship well beyond the point you should be leaving? Of leaving when you should be giving it another chance? Are you consistently drawn to those who don't pick you? To those who are married \u2013 whether to spouse or to job? To those who care, but only at certain times, only in certain situations and only when you offer attention, support or no-strings sex?\n\nOr do you find yourself in relationships with partners who do choose you but who you know, heart of hearts, will make you deeply unhappy? Do you find yourself persuaded by intensity (or by flattery, or by genuine if one-sided love), to the point of ignoring your own doubts and ending in uneasy partnership? In all these situations, it may be time to change.\n\nAll that said, don't panic if there's some imbalance in a relationship, particularly at the start. Every relationship suffers disequilibrium. Even for just a few moments, one person wonders while the other is certain; one person doubts while the other has faith. So long as the dynamic eventually settles into an equal match, that's fine. More, if what we experience at the start of a promising relationship is a certain deliberation on either side, this may well be good news. The resulting choice, if it comes, is likely to be more considered \u2013 and therefore more dependable.\n\nChoosing\n\nThe first chapter of this book painted partner choice as a journey \u2013 and the promise that description offers is that there is a conclusion to the trip. It's a promise that will almost certainly come true. We may have encountered detours or cul-de-sacs along the way, but the vast majority of us will reach our destination. We will let go of alternative possibilities and begin to focus more and more surely on one. We will come to a point where we believe that we can love, and that we can be loved in return.\n\nAnd there we are, decision made. We echo the moving words of poet Edwin Muir: 'yours, my love, is the right human face'. We know. We have committed. We are set for the happiest of endings.\n\nExcept, except . . . of course this isn't the end. Commitment is only the beginning, the first choice. There will be other decisions to come and it's wise to remember they are out there waiting for us.\n\nMoving in together. Getting engaged. Marrying. Having children. Raising those children and then staying together to the end of life. The thing about these later transitions is that by the time we make them, we're in a different place from before. As Somerset Maugham pointed out, we are not the same as we were even a year ago, and 'nor are those we love'.\n\nThrough the course of our relationship, we will likely several times come slap up against what can only be described as the Wall of Life \u2013 trials of illness, accident, job change, ageing, bereavement. Through these, and with the passing of time, we will learn more about our partner and they will learn more about us. And this may shift the goalposts. Soberingly, the second half of the Somerset Maugham quotation reminds us that it is only good fortune if 'we, changing, continue to love a changed person'.\n\nWhether or not we have such good fortune, much of what this book suggests is always relevant. In the context not of a hoped-for or recent connection but an existing and lengthy commitment, it is still useful to specify what we want from our partnership; to reassess the fit with our partner's values, goals and personality; to face fully whether we can still emotionally respond to each other. And sometimes that raises questions. If our partner promised us a rose garden but over the years has delivered only armfuls of brutally stinging nettles, it's understandable if we dump the greenery in the waste bin and head for the door.\n\nNettles aside, the most important factor in making future partnership decisions \u2013 just as for the initial choice to partner with that person \u2013 is not whether we are currently ecstatic but whether we are currently growing, whether we trust that we will grow in the future, and whether we believe our partner is growing too. If we are in a relationship that pushes us to mature, we may not be 100 per cent joyful, 24\/7 \u2013 ask any caterpillar in the process of becoming a butterfly. Yet if we are still evolving, that will make it worthwhile to keep loyal.\n\nLet me end with a tribute to continuing loyalty. In his novel _Love in the Time of Cholera_ , Gabriel Garc\u00eda M\u00e1rquez describes the experience of the multitude of ordinary couples who stay the course even though it proves hard. He pays tribute to the immense courage and willingness of those who have chosen again, beyond the first romantic decision, who have overcome the challenges of 'daily incomprehension', 'instantaneous hatred' and 'reciprocal nastiness' that characterize most relationships, and he celebrates that it is fully possible to come through and to triumph, to reach the point where partners 'love each other best'.\n\nThat's the hope. That if both of us keep evolving, keep learning, keep growing, then at some point in the future, we will be able to create a wonderful partnership, to relate to each other as never before, to love each other 'best'. And then we will know for absolute certain that we have made the right choice.\n\n#### Bibliography\n\n_Books_\n\nLouisa May Alcott, _Little Women_ , Vintage Children's Classics, 2012 edn\n\nJane Austen, _Pride and Prejudice_ , Wordsworth Classics, 1992 edn\n\n\u2014\u2014, _Sense and Sensibility_ , Wordsworth Classics, 1992 edn\n\nLouis de Berni\u00e8res, _Captain Corelli's Mandolin_ , Vintage, 1998\n\nEmily Bront\u00eb, _Wuthering Heights_ , Wordsworth Classics, 1992 edn\n\nBren\u00e9 Brown, _Daring Greatly_ , Avery Publishing Group, 2015\n\nGary Chapman, _The Five Love Languages_ , Moody Press, 2015\n\nStephanie Coontz, _Marriage, a History: How Love Conquered Marriage_ , Penguin USA, 2006\n\nHelen Fielding, _Bridget Jones's Diary_ , Picador, 2001\n\nF. Scott Fitzgerald, _The Great Gatsby_ , Wordsworth Classics, 1992 edn\n\nGustave Flaubert, _Madame Bovary_ , Wordsworth Classics, 1993 edn\n\nErich Fromm, _The Art of Loving_ , Thorsons, 2010\n\nElizabeth Gilbert, _Eat, Pray, Love_ , Bloomsbury, 2007\n\nDr Sue Johnson, _Hold Me Tight_ , Piatkus, 2011\n\n\u2014\u2014, _Love Sense_ , Little, Brown, 2013\n\nDaniel Kahneman, _Thinking Fast and Slow_ , Penguin, 2012\n\nClaire Langhamer, _The English in Love: The Intimate Story of an Emotional Revolution_ , OUP, 2013\n\nGabriel Garc\u00eda M\u00e1rquez, _Love in the Time of Cholera_ , Vintage, 2007\n\nSimon May, _Love: A History_ , Yale University Press, 2012\n\nChris McKinlay, _Optimal Cupid: Mastering the Hidden Logic of OkCupid_ , CreateSpace, 2014\n\nToni Morrison, _Beloved_ , Vintage Classics, 2007 edn\n\nDavid Schnarch, _Passionate Marriage: Keeping Love and Intimacy Alive in Committed Relationships_ , W. W. Norton, 2009\n\nWilliam Shakespeare, _Romeo and Juliet_ , Wordsworth Classics, 2000 edn\n\nLeo Tolstoy, _Anna Karenina_ , Wordsworth Classics, 1995 edn\n\nAmy Webb, _Data, a Love Story_ , Plume, 2014\n\nDavid Whyte, _The Three Marriages: Reimagining Work, Self and Relationship_ , Riverhead Books, 2010\n\nJeanette Winterson, _Written on the Body_ , Vintage, 1993\n\n_Websites and other resources_\n\nwww.notimeforlove.com\n\nEli J. Finkel et al., 'Online Dating: A Critical Analysis From the Perspective of Psychological Science', available at www.psychologicalscience.org\n\nwww.quantifiedbreakup.tumblr.com\n\nThe School of Life, 100 Questions: Love Edition (100 question cards with box \u2013 a toolkit for relationships)\n\nThe School of Life offers coaching and therapy for relationship enhancement: www.theschooloflife.com\/london\/shop\/therapy\/life-coaching\n\nIn Britain you can also find counsellors through the College of Sexual and Relationship Therapists: www.cosrt.org.uk, the British Association for Counselling and Psychotherapy: www.itsgoodtotalk.org.uk and Relate: www.relate.org.uk.\n\n#### Acknowledgements\n\nMy thanks go to everyone who has made this book possible, especially my agent Barbara Levy and all the staff at the School of Life and at Pan Macmillan, especially Morgwn Rimel, Cindy Chan, Robin Harvie, Zennor Compton, Laura Carr, Marcia Mihotich and Jonathan Baker, all of whom have been endlessly helpful. Above all, my thanks go to Caroline Brimmer, who originally invited me to help her develop the 'How to Choose a Partner' course at the School of Life, and whose talent, insight and support have been an inspiration; without Caroline, this book would not have happened.\n\n#### Picture Acknowledgements\n\nThe author and publisher would like to thank the following for permission to reproduce the images used in this book:\n\npagelink _The Arnolfini Portrait \u00a9_ Thaliastock \/ Mary Evans\n\npagelink _Sir Galahad \u2013 the Quest of the Holy Grail_ , 1870 (oil on canvas), Hughes, Arthur (1832\u20131915) \u00a9 Walker Art Gallery, National Museums Liverpool \/ Bridgeman Images\n\npagelink Traffic Sign With Heart Shape \u00a9 Richard Newstead \/ Getty Images\n\npagelink Human Heart Map \u00a9 Kate Hiscock \/ Getty Images\n\npagelink Come Through \u00a9 Everett Collection \/ REX Shutterstock\n\npagelink Watching the Roulette Wheel, Hollywood, California, 1930 (b\/w photo) \u00a9Underwood Archives \/ UIG \/ Bridgeman Images\n\npagelink Scene at Reelfoot Lake \u00a9 Andreas Feininger \/ Getty Images\n\npagelink Jam and Marmalade aisle, Woolworths store, 1956 (b\/w photo), English Photographer, (20th century) \u00a9 Private Collection \/ Bridgeman Images\n\npagelink _West Side Story_ Photo \u00a9 Michael Ochs Archive \/ Getty Images\n\npagelink A Man Hiking \u00a9 Lambert \/ Getty Images\n\npagelink Kissing Couple 1968 \u00a9Mary Evans Picture Library \/ Shirley Baker\n\npagelink _The Creation of Adam \u00a9_ Massimo Pizzotti \/ Getty Images\n\npagelink Corkscrew \u00a9 Keystone France \/ Getty Images\n\npagelink _Dante and Beatrice_ , 1883 (oil on canvas), Holiday, Henry (1839\u20131927) \u00a9 Walker Art Gallery, National Museums Liverpool \/ Bridgeman Images\n\n#### **TOOLS FOR THINKING**\n\nA RANGE OF THOUGHTFUL STATIONERY, GAMES & GIFTS FROM THE SCHOOL OF LIFE\n\nGood thinking requires good tools. To complement our classes, books and therapies, THE SCHOOL OF LIFE now offers a range of stationery, games and gifts that are both highly useful and stimulating for the eye and mind.\n\nTHESCHOOLOFLIFE.COM\n\nIf you enjoyed this book, we'd encourage you to check out other titles in the series:\n\nHOW TO MAKE A HOME | EDWARD HOLLIS\n\n---|---\n\nHOW TO LIVE IN THE CITY | HUGO MACDONALD\n\nHOW TO CHOOSE A PARTNER | SUSAN QUILLIAM\n\nHOW TO THINK LIKE AN ENTREPRENEUR | PHILIP DELVES BROUGHTON\n\nHOW TO BE A LEADER | MARTIN BJERGEGAARD & COSMINA POPA\n\nHOW TO BE BORED | EVA HOFFMAN\n\nAlso Available:\n\nHOW TO DEAL WITH ADVERSITY | CHRISTOPHER HAMILTON\n\n---|---\n\nHOW TO AGE | ANNE KARPF\n\nHOW TO DEVELOP EMOTIONAL HEALTH | OLIVER JAMES\n\nHOW TO CONNECT WITH NATURE | TRISTAN GOOLEY\n\nHOW TO THINK ABOUT EXERCISE | DAMON YOUNG\n\nHOW TO BE ALONE | SARA MAITLAND\n\nHOW TO FIND FULFILLING WORK | ROMAN KRZNARIC\n\nHOW TO STAY SANE | PHILIPPA PERRY\n\nHOW TO WORRY LESS ABOUT MONEY | JOHN ARMSTRONG\n\nHOW TO THINK MORE ABOUT SEX | ALAIN DE BOTTON\n\nHOW TO THRIVE IN THE DIGITAL AGE | TOM CHATFIELD\n\nHOW TO CHANGE THE WORLD | JOHN-PAUL FLINTOFF\n\nIf you'd like to explore more good ideas from everyday life, THE SCHOOL OF LIFE runs a regular programme of classes, workshops, and special events in London and other cities around the world.\n\nTHESCHOOLOFLIFE.COM\nSUSAN QUILLIAM is a relationship psychologist, coach, advice columnist and Faculty member of the School of Life. She is the author of twenty-two books published in thirty-three countries and twenty-four languages. She is a regular media expert on the subject of sex and relationships.\n\nTHE SCHOOL OF LIFE is dedicated to exploring life's big questions: _How do we find fulfilling work? Can we ever understand our past? Why are relationships so hard to master? If we could change the world, should we?_ Based in London, with campuses around the globe, The School of Life offers classes, therapies, books and other tools to help you create a more satisfying life. We don't have all the answers but we will direct you towards a variety of ideas from the humanities \u2013 from philosophy to literature, psychology to the visual arts \u2013 guaranteed to stimulate, provoke, nourish and console.\n\nwww.theschooloflife.com\n_By the same author:_\n\nThe New Joy of Sex (co-written with Dr Alex Comfort)\n\nStop Arguing, Start Talking\n\nThe Relate Guide to Staying Together\n\nFirst published 2016 by Macmillan\n\nThis electronic edition published 2016 by Macmillan  \nan imprint of Pan Macmillan  \n20 New Wharf Road, London N1 9RR  \nAssociated companies throughout the world  \nwww.panmacmillan.com\n\nISBN 978-1-4472-9330-9\n\nCopyright \u00a9 The School of Life 2016\n\nThe right of Susan Quilliam to be identified as the author of this work has been asserted by her in accordance with the Copyright, Designs and Patents Act 1988.\n\nThe picture acknowledgements here constitute an extension of this copyright page.\n\nPan Macmillan does not have any control over, or any responsibility for, any author or third party websites referred to in or on this book.\n\nYou may not copy, store, distribute, transmit, reproduce or otherwise make available this publication (or any part of it) in any form, or by any means (electronic, digital, optical, mechanical, photocopying, recording or otherwise), without the prior written permission of the publisher. Any person who does any unauthorized act in relation to this publication may be liable to criminal prosecution and civil claims for damages.\n\nA CIP catalogue record for this book is available from the British Library.\n\nVisit **www.panmacmillan.com** to read more about all our books and to buy them. You will also find features, author interviews and news of any author events, and you can sign up for e-newsletters so that you're always first to hear about our new releases.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"QUESTION***\n\n\nE-text prepared by the Online Distributed Proofreading Team\n(http:\/\/www.pgdp.net) from page images generously made available by\nInternet Archive\/Canadian Libraries\n(http:\/\/www.archive.org\/details\/toronto)\n\n\n\nNote: Images of the original pages are available through\n      Internet Archive\/Canadian Libraries. See\n      http:\/\/www.archive.org\/details\/shelleyandthema00todhuoft\n\n\n\n\n\nSHELLEY AND MARRIAGE.\n\nOf this Book Twenty-Five Copies only have been printed.\n\n\nSHELLEY AND THE MARRIAGE QUESTION.\n\nby\n\nJOHN TODHUNTER, M.D.,\n\nAuthor of _Notes on \"The Triumph of Life,\" A Study of Shelley, etc._\n\n\n\n\n\n\n\nLondon:\nPrinted for Private Circulation Only.\n1889.\n\n\n\n\nSHELLEY AND THE MARRIAGE QUESTION.\n\n\nNow that marriage, like most other time-honoured institutions, has come\nto stand, a thing accused, at the bar of public opinion, it may be\ninteresting to see what Shelley has to say about it. The marriage\nproblem is a complex one, involving many questions not very easy to\nanswer offhand or even after much consideration. What is marriage? Of\ndivine or human institution? For what ends was it instituted? How far\ndoes it attain these ends? And a dozen others involved in these.\n\nThe very idea of marriage implies some kind of bond imposed by society\nupon the sexual relations of its members, male and female; some kind of\nrestriction upon the absolute promiscuity and absolute instability of\nthese relations--such restriction taking the form of a contract between\nindividuals, endorsed by society, and enforced with more or less\nstringency by public opinion. Its object at first was probably simply to\nensure to each male member of the tribe the quiet enjoyment of his wife\nor wives, and the free exploitation of the children she or they\nproduced. The patriarchal tyranny was established, and through the\nsanction of primitive religion and law became a divine institution.\nThen, as civilization progressed, the wife and children became less and\nless the mere slaves, more and more the respected subjects, of the\npatriarch. The paternal instinct (like the maternal) became developed,\nand family affection came into existence. At present the whirligig of\ntime is bringing its revenges. The patriarchal tyranny begins to\ntotter; parents are often more the slaves than the masters of their\nchildren. And even wives begin to rebel against wifedom, and threaten to\nrevolutionize marriage in their own interest. Woman, like everybody\nelse, is beginning to strike for higher wages. There are more than the\nfirst mutterings of that revolution in the Golden City of Divine\ninstitutions prophesied of by Shelley in _Laon and Cythna_. There are a\ngood many Cythnas ready to rush about on their black Tartarian hobbies,\nof whom Mrs. Mona Caird is the one who has recently made most noise.\n\nThere is a little design of Blake's in _The Gates of Paradise_, which\nrepresents a man standing on the earth who leans a ladder against the\nmoon and prepares to mount; the motto underneath being: \"I want! I\nwant!\" This is a type of our own age. Never was such an age of\ndiscontent, never such a Babel of voices crying: \"I want! I want!\" We\nhave become very conscious of our pain, and are not ashamed to cry out\nand proclaim it on the house-tops in these hysterical times--simply\nbecause the ancient sanctions and anodynes have lost their sanctity and\ncomfort for us. The very \"priests in black gowns\" who used to \"walk\ntheir rounds and bind with briers our joys and desires,\" have been\nthemselves corrupted with a longing for a little present happiness, and\nthat Old Woman in the shoe, Mrs. Grundy herself, instead of whipping us\nall round and putting us to bed in the old summary fashion, when we\nventure to complain that the shoe pinches here and there, has herself\nbecome lachrymose. We cry out because, having neither the old\nrepressions nor the old opiates to restrain us, there is no valid reason\nwhy we should hold our tongues. By crying loud enough and long enough we\nmay get some help. We may even find some good-natured person to stop\ncrying himself and help us; and then for very shame we may go and do\nlikewise. In this lies the age's hope. It is really in its best aspect\nan unselfish age, an age in which sympathy and justice are vital forces,\nin which the miseries of others are felt as our own. There are thousands\nnow who feel themselves \"as nerves o'er which do creep the else unfelt\noppressions of the earth.\" We are not wise enough yet to conceive and\norganize those vital adjustments between conflicting wants, interests,\nand principles, which shall be of deeper efficiency than mere\nsuperficial compromises; but this wisdom will come in due time, if we do\nnot rush into anarchy through that licentious impatience which is the\ncurse of revolutionary periods.\n\nNow, of all the bitter cries ringing in the air at the present time,\nabout the bitterest and most persistent is that not merely of women, but\nof woman with a capital W. It is the most appalling note of change that\ncan pierce the ear of self-satisfied Conservatism. The patient Griselda\nhas begun to protest against the tyranny of her lord and master. Love's\nmartyr has at last begun to think that her martyrdom must have its\nlimits. It is as if the Lamb, whose function we thought was to be dumb\nbefore its shearers and even sacrificers, had found a voice of\nprotestation. It is a portent. And even men are constrained to listen to\nthe cry; for it sounds like the birth-cry of regenerated Love. Not now\n\"Love self-slain in some sweet shameful way,\" but Love the winged angel\nwho shall finally cast out Lust, the adversary. But many things must\ncome to pass before this triumph of love can be brought about; and in\nmany respects the horoscope looks unpropitious enough. The first effect\nof the birth, or coming to the surface of a higher ideal, gradually\nevolved by the progress of society, is apparently to make confusion\nworse confounded. Not peace but a sword is the first gift of the Prince\nof Peace. Liberty comes masked like Tyranny, and cries \"Fraternity or\ndeath!\" Love goes wantonly about with the Maenads of licentiousness at\nhis heels. But the divine Logos, incarnate as the Son of man, always\ncomes not to destroy but to fulfil.\n\nJust now that highly moral being, Man in the masculine gender, is much\nshocked at the strangely immoral conduct of his feminine counterpart. In\nthe first place, she has dared to look at the realities of things with\nher own eyes, not through the rose-coloured spectacles with which he has\nbeen at pains to provide her; and not only that, but to peep behind the\nsacred veil which man has modestly cast over many ugly things. Secondly,\nshe has begun to talk openly about these ugly things, and to call them\nby non-euphemistic, ugly names, in a manner quite unprecedented.\nThirdly, she has dared to attempt her own solution of things insoluble,\nher own achievement of things impossible. And fourthly, she has dared to\nformulate a demand for liberty, equality, fraternity on her own\naccount--a demand which every day comes more and more within the sphere\nof practical politics. Here are pure women making common cause with\nprostitutes, married women crying out against the holy institution of\nmatrimony, mothers rebelling against the tyranny of the beatific\nbaby--nay, absolutely on strike against child-bearing, or at least\ndemanding limited liability as regards that important function. Finally,\nhere is Woman, whether as virgin, wife, or widow, demanding independence\nas to property and a fair share of the world's goods in return for a\nfair share of the general work of the world outside of her special\nwomanly functions. \"D----n it, sir, I say that women are unsexing\nthemselves--unsexing themselves, by Jove!\" as Major Pendennis might\nexclaim. And the worst of it is that there are so many men, traitors to\ntheir sex, who are casting in their lot with women in this terrible\nWomen's Rights movement--\"unsexing themselves,\" too, no doubt--so that\nwe shall all soon become either a-sexual or hermaphrodite beings! And\nhere let us leave for a moment the more or less limited and prosaic\nCythnas of the day, the terrible women who ride about upon Tartarian\nhobby-horses in novels and magazine articles, who spout on platforms and\npractise medicine and other dreadful trades--the scientific Mrs.\nSomervilles, and medical Mrs. Garrett Andersons, and pious Mrs.\nJosephine Butlers, and impious Mrs. Mona Cairds, and get back to Shelley\nhimself, the poet of this shocking social aberration.\n\nShelley, as Mr. Cordy Jeafferson has taken great pains to demonstrate,\nwas an exceedingly immoral young man. He outraged the conventional\nmorality of his day by his actions as well as in his writings in the\nmost shameless manner; but this shamelessness was due to his intense\nconviction that he thus outraged _conventional_ in the interests of\n_ideal_ morality. His life and writings are so full of the paradoxical\ncharacter which I have ascribed to the social agitation of the present\nday, and some of his utterances are so prophetic of it, that we may\nfairly regard him as its precursor.\n\nShelley, as we know, started rather as an anarchist than as a mere\nreformer. His ideas were cataclysmal rather than evolutional. But he was\nan optimistic not a pessimistic anarchist, and he endeavoured to destroy\nin order to rebuild with all possible expedition. The kingdom of heaven\nwas, for him, at the very doors, ready to take shape as soon as man\nwilled it; and man _would_ will it as soon as the mind-forged fetters of\nhis mind were loosed. Accordingly he endeavoured to loose them. He\ndethroned God that the Spirit of Nature might be enthroned; and then he\nproceeded to abolish marriage that free love might regenerate mankind.\nHe believed in regeneration by incantation--a few words murmured in\nmen's ears would make them as obedient to the ideas those sacred words\nrepresented as spirits to the spells of a magician. Abolish marriage\n(and what could be easier?), and love, being set free, prostitution\nwould cease. We may pass by such puerilities of inexperienced idealism,\nto be found by the score in _Queen Mab_, and pass on to Shelley's more\nmature utterances, always remembering that he died, as the _Triumph of\nLife_ shows, in the very process of maturation. His whole history is\nthat of an idealist, who first seeks his ideal in the actual, and not\nfinding it endeavours to bring the actual into harmony with his ideal.\nHis imagination hacks at the rude block of the world with the divine\nfury of a Pygmalion; thinking at first that he has but to remove the\ndull superfluous husks of custom to find the living idea in the centre;\nbut gradually perceiving it was but created an inanimate image, which\ncan only come to life by the invocation of Venus Urania. All the\nweaknesses, faults, and follies of his life and his writings, as well as\nthat \"power in weakness veiled\" which he felt himself to be, come from\nthis. He is driven to reform society by attacking the conventional\nmorality of marriage, because he is first a transcendental lover; just\nas Mr. William Morris is driven into socialism, because he is first a\nvery practical decorative artist. To speak irreverently, both men want\nelbow-room for their fads. But Shelley's fad is of even more importance\nto us than Morris's. It is better to have a beautiful love, than to have\na beautiful house to put him in. Shelley is, above all things, the poet\nof modern love. Dante's love, fantastic and supersensuous, was not\nmodern love. We do not want angels, either in heaven or in the house, to\ncondescend to our depravity and lead us upward. We do not want the\ndivine school-mistress to bring us to something not ourselves which may\nor may not make for righteousness, but the divine mistress, passionate\nas well as pure, to bring us to our best selves, and live with us in\nperfect union. Shakespeare showed us glimpses of this love defeated by\ncircumstances in _Romeo and Juliet_, triumphant over circumstances in\nPosthumus and Imogen; but Shelley has had a fuller vision of it. Since\nShakespeare's time both manhood and womanhood, and especially womanhood,\nhave by pressure of circumstances become more self-conscious, and the\nconditions of their union through love more complex.\n\nAnd what is this modern ideal of love, of which Shelley is the exponent?\nWhat is this strange affection, love, whether ancient or modern? It is\nthat most paradoxical of passions, that compound of selfishness and\nself-renunciation, that forlorn desire which strives to reconcile all\nthings, and found an eternal home on the shifting sands of time, of\nwhich we all know something. Blake has expressed this paradoxical\ncharacter of love once for all in his little poem \"The Clod and the\nPebble.\"\n\n  \"Love seeketh not itself to please,\n    Nor for itself hath any care,\n  But for another gives its ease,\n    And builds a heaven in hell's despair.\n\n  Love seeketh only self to please,\n    To bind another to its delight,\n  Joys in another's loss of ease,\n    And builds a hell in heaven's despite.\"\n\nWe may call these the masculine and feminine elements in love; though of\ncourse both exist in all love, whether of man to woman or woman to man.\nBoth sexes give more than they receive, and receive more than they\ngive. In all love, from the first step beyond mere physical appetite, to\nthe most transcendental Platonism, there are these two antagonistic\nelements. If the merely self-indulgent element prevails, we tend in\nthe direction of lust, one of the most cruel diseases that plague\nhumanity, which Milton rightly places \"hard by hate.\" If the merely\nself-renouncing, we tend in the direction of monastic chastity, which\nthough not so distinctly an evil thing, may become cruel and inhuman,\nand a bar to human progress. Asceticism is not, like lust, a disease,\nphysical and spiritual, but it may lead to disease, spiritual if not\nphysical. There is an asceticism, the Greek [Greek: aschesis], a\ntraining of the lower faculties to act in subordination to the higher,\nwhich is the strait gate by which we enter upon the arduous ascent\ntoward noble passion and noble action. There is another asceticism which\nif not truly Christian, came in the wake of Christianity, which, denying\nthe rights of the body, was less a training than a mortification. Both\nunrestrained sensuality and monastic chastity, in their injustice to the\nbody outrage the sexual principle, the former by regarding it as a toy\nto be polluted by base pleasure, the latter by regarding it as a thing\nunclean in itself to be cast out and killed, or at best tolerated and\ncleansed by the Church's holy water. To the present day the average\nman's, or at least the average Englishman's great temptation is to sin\nagainst love, through dull unimaginative lust, the average\nEnglishwoman's through dull unimaginative chastity. Men live too much in\nthe sensuous, and women in the supersensuous, to meet fairly. Love, the\nreconciler, himself is too weak fully to reconcile them and to bring\nthem together in that perfect ecstasy, body to body, spirit to spirit,\nsoul to soul, that \"unreserve of mingled being,\" which Shelley, giving a\nvoice to the desire of all ages, but especially to modern desire, sighed\nfor. To understand Shelley's protest against marriage, we must\nunderstand his ideal of love--the unconstrained rush together of two\npersonalities of opposite sexes, in whom the body is but the vehicle of\nthe spirit. This love is not born merely of the flickering fire of the\nsenses. It is a divine flame, kindled alike in body, soul, and spirit,\nand fusing them into unity. Of course, if this love is to be the great\nend of life, marriage is somewhat of an impertinence. While the divine\nfire burns, what need of artificial ties to keep the two lovers\ntogether? If it goes out why should they be kept together? To which the\nprosaic moralist replies: \"Your ideal of love is very beautiful, no\ndoubt. Get as much as you can of this divine flame into your Hymen's\ntorch; and after all, every young couple start with some such high-flown\nnotions in their heads; but I must have some guarantee that your wife\nand children are not left as burdens upon the parish, when you begin to\nfeel the pinch of real life, and the glamour of your imagination fades\nfrom your 'divine mistress.' Marriage was not ordained to be the\nparadise of ideal love, but for the sober discipline of the affections\nof men and women, and above all for the production and rearing up of\ngood citizens of the commonwealth. To judge by your own writings, Mr.\nShelley, you seem to have been running after a will-o'-the-wisp all your\nlife in this ideal love. And if _you_ did not catch it, is it likely\nthat Tom, Dick, and Harry will? In any case the pursuit of it seems just\nas likely to make inconstant lovers as that sensuality you affect to\nlook down upon. You always had the word 'for ever' on your tongue; but\nhow long did your for evers last? No, no, my dear sir, the good of\nsociety demands fidelity to incurred responsibilities, and we find by\npractical experience that both men and women, but especially men, are\ninclined to shirk the responsibilities which indulgence of the sexual\npassion brings in its train. Hence the marriage contract. It does not\nconcern itself primarily with either love or lovers, but it helps to\nkeep husbands and wives together, and women and children maintained\ndecently without coming upon the rates. And, mind you, it does not by\nany means leave love out in the cold. It may not rise to your\ntranscendental ecstasy; but it is love all the same, good honest\ndomestic affection, when your young couples get well broken to harness.\nDid you not say yourself that one might as well go to a gin-shop for a\nleg of mutton as to you for anything human? Well, give me the wholesome\nleg of mutton--none of your gin for me. Egad, sir, when I see some\nhonest couple going to church of a Sunday morning, with half-a-dozen\npretty children about them, I call that a poem--ay, and a better poem,\nMr. Shelley, than all the fantastic Epipsychidions you ever put upon\npaper. Hang it all, sir, let a man make love to his own wife, and stick\nto her when he has got her. I'm a plain man, sir, but I hope a moral\nman, and them's my sentiments.\" To all which, let Shelley reply as best\nhe may. The fact is that he has given no satisfactory reply, simply\nbecause it was only just before his death that he realised the\ncomplexity of the problem of life. He did, however, see clearly that the\nbringing of men and women into more complete harmony, by raising the\nideal of love, was the most important step towards that renewal of the\nworld, that living of the most perfect life attainable by man, for which\nhe sighed and after which he strove; and he saw clearly that our\nsolution of the marriage problem was imperfect, not merely in practice,\nbut to some extent in theory. As regards the subjection of women, he\nseems to have considered this wholly an artificial product of religious\ndogma, and not, as it is, the natural result of an imperfect\ncivilization. Man protects woman because, on the whole, she adds to his\ncomfort. Protection implies subjection, and subjection to a tyrant is\nslavery; and man, if not altogether a tyrant in these later times, has\nalways the temptation to become one, and the tyrannical traditions of\nbygone times have a strong tendency to persist. Laws and even customs\nlag far behind the highest public opinion of the day.\n\nNow, men being in possession of the capital of the world, the material\nmeans of life, women stand to them in the position of what the\nsocialists call wage-slaves. They must do what their employers require\nof them on pain of starvation, and there is no true freedom of contract.\nAnd so far men have almost without exception required of them\nconcubinage or menial service, or a mixture of both. English marriage,\nwhile recognizing the existing fact of the subjection of women, has done\nsomething to raise their status, chiefly by making the bond between the\ncontracting parties theoretically, and to a great extent practically,\none of love and mutual service. It has indeed been much more than\nShelley seems to have realized, the _nidus_ of a love pure and\nwholesome, if not very passionate. Theoretically strictly monogamic, it\nhas been so practically to a very respectable extent. It has put a\nperceptible curb upon the strong polygamous instinct of men, and it has\nfostered the monogamous habit in women enormously. English women are for\nthe most part faithful wives. Even transitory prostitution does not kill\nthe monogamous propensity in them. They settle down into marriage, or\nlive faithfully with one man, if they get the chance.\n\nStill, Englishwomen are not satisfied with marriage as it exists. Let us\nhear Mrs. Mona Caird on the subject. She is much more prosaic than\nShelley; she looks at the subject, chiefly from the standpoint of\npractical comfort. She sees that from this standpoint, from various\nreasons, which may be summed up in the phrase \"incompatibility of\ntemper,\" marriage does not induce even that amount of mutual toleration,\nnot to say happiness, without which it is impossible for man and wife to\nlive decently together. She therefore asks, What good purpose is served\nby keeping two people together who are evidently unfit to live together?\nWhy indeed? if, as Mrs. Caird says, \"The matter is one in which any\ninterposition, whether of law or society, is an impertinence.\" But,\nunfortunately, law and society are the most impertinent things in the\nworld, always binding with briers our joys and desires, and poking their\nugly noses into our private affairs in the interests of the British\nratepayer. We shall never be happy until we have got rid of them--if\neven then, and it is quite impossible to get rid of them for some time\nto come. Now the British ratepayer cares nothing about women and\nchildren, except in so far as there is a danger of their coming upon the\nrates. And he is a little scared about giving greater liberty of\ndivorce, \"saving for the cause of adultery,\" as he piously ejaculates.\nHe does not like stray women and children going about the world. But\nafter all, adultery is only a particular, perhaps even a minor, case of\nincompatibility. Marriage was made for man, and not man for marriage,\nand although marriage may work well in nine cases out of ten, the tenth\ncase must be considered, and relief given if possible. The individual is\nright to demand relief, and the mode of giving relief is a question for\nthe legislator. Greater facility of divorce must come, and will come,\nnow that both men and women demand it.\n\nMrs. Caird's demand for greater laxity of the marriage bond _ab initio_,\nthe nature of the contract being left to the contracting parties, like a\nmarriage settlement, is quite outside the sphere of practical politics,\nas she is herself quite aware. If men were but educated up to the\nShelleyan ideal, then we might try all sorts of delightful experiments\nin marriage, and gradually arrive at absolute freedom of contract, which\nwould _not_ mean that absolutely unsentimental hygienic promiscuity\nwhich is the ideal of the highly advanced physiologist. But men are not\nyet harmonious creatures, like Wordsworth's cloud, which \"moveth\naltogether if it move at all.\" They are torn by their lusts which war in\ntheir members. Hence these bonds. Lust, lust, lust: this is the most\nconcentrated form of selfishness--the undying worm at the root of the\nTree of Life. This is the tyrant that women have at last begun to\nrecognize as their deadly adversary and to fight against. Shelley, a\nbetter physician than Goethe, laid his finger on this plague-spot, and\ntold the age plainly: \"Thou ailest here.\" But he did not see that\ninstead of saying, \"Abolish marriage and prostitution will cease,\" he\nought to have said, \"Abolish prostitution and marriage will\ncease\"--marriage without love being only a particular form of\nprostitution. He did not see that the abolition of marriage would no\nmore get rid of lust than the abolition of private property would get\nrid of selfishness. We have already, in monogamic marriage, struggled\npainfully upward to the level of the higher animals; let us not imperil\nthis progress rashly.\n\nThe Cythnas of the present day have felt their burthens more directly\nthan Shelley did. Hence their demand for economic independence, that\nthey may not be forced into marriage or prostitution by the various\ndegrees of starvation. Their demand is a just one, and must be satisfied\nsomehow, even if we have to put a bonus upon womanhood and pay women,\nnot merely fair wages for their work of all kinds, but a tribute to them\nas women, as potential mothers, which shall fairly handicap the sexes\nin the struggle for existence, and put men more on their good behaviour.\n\nShelley, the mystic, who looked for a miraculous change in nature\ncoincident with a miraculous change in man, seems to have seen, almost\nas little as the average socialist of the present day, who believes in\nthe spiritual efficacy of a purely material revolution, that the ideals\nand interests of the two sexes are widely apart, more so now than ever\nbefore probably. He, like the socialist, in his impatience to arrive at\na practical solution of the life-problem, did not take the trouble to\nunderstand the true bearing of the doctrine of Malthus. He did not see\nthat whether Malthus's figures be right or wrong, it is a fact that the\npopulation of any given district (be it an English barony, or the world\nitself) tends to increase up to the limits of its food-supply, taking\nthe word _food_ in its very widest sense to signify all the means of\nwell-being; and that this tendency is a fundamental element in all\nsocial problems, just as friction is in all mechanical problems. He did\nnot see that, other things being the same, a higher standard of comfort,\nwhile, finally tending to diminish the rate of increase of population,\nfirst increases its pressure. He did not contemplate that strike against\nchild-bearing on the part of women, which is induced, not merely by the\ndesire for personal comfort, but is largely due to the vague influence\nof those new ideals of which he was himself the prophet. He, like the\nsocialist, thought that we might go on increasing and multiplying _ad\nlibitum_, till we reached the ultimate limit of standing-room on the\nearth, and of miraculous chemical food out of the air, and began, as\nastral bodies, to emigrate to Mars. Women know better than this; and\nfeel the pinch of population, when what they just now consider their\nhigher life is hampered by children. The woman who has one child more\nthan she wants is an over-populated woman; and the advanced woman of the\npresent day, having her own higher culture, and the culture of humanity,\non the brain, possibly with a high ideal of the duties of maternity, and\nfrequently a sickly and weary creature, morbid in body and mind, is very\neasily over-populated. Hence much social discomfort. Shelley does not\nseem to have contemplated this, nor seen that the good-natured\nacceptance of the feminine ideal by man might lead him, like poor St.\nPeter in his old age, \"whither he would not.\" How all this is going to\nend I confess I don't know. I trust in more delicate adjustments, a\nhigher and more wholesome life all round; but the ascent of man is\nalways a painful process. Meanwhile it is quite time for this bald,\ndisjointed chat of mine to come to an end.\n\n\n\n\n  _London:\n  Printed by Richard Clay & Sons, Limited, Bread Street Hill.\n  September, 1889._\n\n\n\n\n      *      *      *      *      *      *\n\n\n\n\nTranscriber's note:\n\nText in italics is enclosed by underscores (_italics_).\n\nThe original text includes Greek characters. For this text version these\nletters have been replaced with transliterations.\n\nThe misprint \"tempation\" has been corrected to \"temptation\" (page 15).\n\n\n\n***","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nMore Praise for the _Best Food Writing_ Series\n\n\"This is a book worth devouring.\"\u2014 _Sacramento Bee_\n\n\"The cream of the crop of food writing compilations.\" _\u2014Milwaukee Journal Sentinel_\n\n\"An exceptional collection worth revisiting, this will be a surefire hit with epicureans and cooks.\"\u2014 _Publishers Weekly,_ starred review\n\n\"If you're looking to find new authors and voices about food, there's an abundance to chew on here.\"\u2014 _Tampa Tribune_\n\n\"Fascinating to read now, this book will also be interesting to pick up a year from now, or ten years from now.\"\u2014Popmatters.com\n\n\"Some of these stories can make you burn with a need to taste what they're writing about.\"\u2014 _Los Angeles Times_\n\n\"The book captures the gastronomic zeitgeist in a broad range of essays.\"\u2014 _San Jose Mercury News_\n\n\"The next best thing to eating there is.\"\u2014 _New York Metro_\n\n\"Stories for connoisseurs, celebrations of the specialized, the odd, or simply the excellent.\"\u2014 _Entertainment Weekly_\n\n\"Spans the globe and palate.\"\u2014 _Houston Chronicle_\n\n\"The perfect gift for the literate food lover.\" _\u2014Pittsburgh Post-Gazette_\n\n\"With this typically delectable and eclectic collection of culinary prose, editor Holly Hughes proves her point made in the intro that the death of 68-year-old _Gourmet_ magazine a year ago didn't lead to the demise of quality food journalism . . . There's a mess of vital, provocative, funny and tender stuff . . . in these pages.\" _\u2014USA Today_\n\nALSO EDITED BY HOLLY HUGHES\n\n_Best Food Writing 2011_\n\n_Best Food Writing 2010_\n\n_Best Food Writing 2009_\n\n_Best Food Writing 2008_\n\n_Best Food Writing 2007_\n\n_Best Food Writing 2006_\n\n_Best Food Writing 2005_\n\n_Best Food Writing 2004_\n\n_Best Food Writing 2003_\n\n_Best Food Writing 2002_\n\n_Best Food Writing 2001_\n\n_Best Food Writing 2000_\n\nALSO BY HOLLY HUGHES\n\n_Frommer's 500 Places for Food and Wine Lovers_\n\n_Frommer's 500 Places to See Before They Disappear_\n\n_Frommer's 500 Places to Take the Kids  \nBefore They Grow Up_\n\n_Edited by_\n\nHOLLY HUGHES\n\nA Member of the  \nPerseus Books Group\nMany of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks. Where those designations appear in this book and Da Capo Press was aware of a trademark claim, the designations have been printed in initial capital letters.\n\nCopyright \u00a9 2012 by Holly Hughes\n\nPages 380\u2013384 constitute an extension of the copyright page.\n\nAll rights reserved. No 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 written permission of the publisher. For information, address Da Capo Press, 44 Farnsworth Street, Third Floor, Boston, MA 02210.\n\nSet in 11 point Bembo by the Perseus Books Group\n\nCataloging-in-Publication data for this book is available from the Library of Congress.\n\nFirst Da Capo Press edition 2012\n\nISBN 978-0-7382-1619-5\n\nPublished by Da Capo Press\n\nA Member of the Perseus Books Group\n\nwww.dacapopress.com\n\nDa Capo Press books are available at special discounts for bulk purchases in the United States by corporations, institutions, and other organizations. For more information, please contact the Special Markets Department at the Perseus Books Group, 2300 Chestnut Street, Suite 200, Philadelphia, PA 19103, or call (800) 255-1514, or e-mail special.markets@perseusbooks.com.\n\n10 9 8 7 6 5 4 3 2 1\n\n#### CONTENTS\n\nIntroduction\n\nBy Holly Hughes\n\nFOOD FIGHTS\n\n**On Killing,** From HunterAnglerGardenerCook.com\n\nBy Hank Shaw\n\n**The Gumbo Chronicles,** From _Outside_\n\nBy Rowan Jacobsen\n\n**Serving Up Sustainability,** From _Edible Boston_\n\nBy Erin Byers Murray\n\n[**Kids Battle the Lure of Junk Food,**  \nFrom _Pacific Northwest Magazine_](chapter04.html#ch4)\n\nBy Maureen O'Hagan\n\n**Pastoral Romance,** From _Lapham's Quarterly_\n\nBy Brent Cunningham\n\nFARM TO TABLE\n\n**Sweet Spot,** From _Alimentum_\n\nBy Paul Graham\n\n[**Snowville Creamery Has a Modest Goal: Save the World,**  \nFrom _Edible Columbus_](chapter07.html#ch7)\n\nBy Eric LeMay\n\n**Matters of Taste,** From _Tomatoland_\n\nBy Barry Estabrook\n\n**Olives and Lives,** From _Extra Virginity_\n\nBy Tom Mueller\n\n**This Little Piggy Went to Market,** From _Memoir Journal_\n\nBy Laura R. Zandstra\n\nHOME COOKING\n\n**How to Live Well,** From _An Everlasting Meal_\n\nBy Tamar Adler\n\n**Still Life with Mayonnaise,** From _At the Kitchen Table_\n\nBy Greg Atkinson\n\n**The Fried Chicken Evangelist,** From Leite's Culinaria\n\nBy Lorraine Eaton\n\n**Lasagna Bolognese,** From SmittenKitchen.com\n\nBy Deb Perelman\n\n**The Forager at Rest,** From _Bon Appetit_\n\nBy Christine Muhlke\n\n**Better Cooking Through Technology,** From _Technology Review_\n\nBy Corby Kummer\n\nFOODWAYS\n\n**The Pastrami Dilemma,** From Chow.com\n\nBy John Birdsall\n\n**Passover Goes Gourmet,** From _Sunset_\n\nBy Rachel Levin\n\n**The 2011 Dyke March Wiener Taste Test,** From _The Stranger_\n\nBy Bethany Jean Clement\n\n**The Missing Link,** From _The Times-Picayune_\n\nBy Brett Anderson\n\n**Foraging and Fishing Through the Big Bend,** From _Desert Terroir_\n\nBy Gary Paul Nabhan\n\n**Italian America,** From _Saveur_\n\nBy John Mariani\n\n**What Makes Sushi Great?,** From GiltTaste.com\n\nBy Francis Lam\n\n**Food for Thought,** From the _New York Times_\n\nBy Jeff Gordinier\n\nDUDE FOOD\n\n[**Learning to Barbecue Helped Make Me a Man,**  \nFrom _Food & Wine_](chapter25.html#ch25)\n\nBy Joel Stein\n\n**Memphis in May: Pork-a-Looza,** From _Garden & Gun_\n\nBy Wright Thompson\n\n**Truffle in Paradise,** From _Gastronomica_\n\nBy John Gutekanst\n\n**A Slice of Family History,** From _Food & Wine_\n\nBy Daniel Duane\n\n**Barbecue Road Trip: The Smoke Road,** From _Garden & Gun_\n\nBy John T. Edge\n\nTHE FAMILY TABLE\n\n**The Food-Critic Father,** From _The Washingtonian_\n\nBy Todd Kliman\n\n[**The Legacy That Wasn't: Wonton Soup,**  \nFrom _A Spoonful of Promises_](chapter31.html#ch31)\n\nBy T. Susan Chang\n\n**Curious Cookies,** From _Edible Vancouver_\n\nBy Eagranie Yuh\n\n**Chicken Brick,** From _Fire & Knives_\n\nBy Henrietta Clancy\n\n**Angry Breakfast Eggs,** From Poor Man's Feast.com\n\nBy Elissa Altman\n\n**Sweet Southern Dream,** From _Saveur_\n\nBy Ben Mims\n\nSOMEONE'S IN THE KITCHEN\n\n**The King of Pop-Up,** From GQ\n\nBy Brett Martin\n\n**Hot Plate,** From _Minnesota Monthly_\n\nBy Rachel Hutton\n\n**Austria's Culinary Ambassador,** From _Edible Manhattan_\n\nBy St. John Frizell\n\n**Remembering Savoy,** From _Edible Manhattan_\n\nBy Rachel Wharton\n\n**Appetite for Perfection** , From _Los Angeles Magazine_\n\nBy Ed Leibowitz\n\n**Supper Clubs in Denver,** From the _Denver Post_\n\nBy John Broening\n\n**Why Chefs Sell Out,** From Chow.com\n\nBy Richie Nakano\n\n**A Chef's Painful Road to Rehab,** From the _Chicago Tribune_\n\nBy Kevin Pang\n\n**Bitter Start to a Life of Sweets,** From _Sacramento Bee_\n\nBy Chris Macias\n\nPERSONAL TASTES\n\n**Kitchen Confessional: Burnin' Down Da House** , From Leites Culinaria\n\nBy David Leite\n\n**Do I Dare to Eat a Peach?,** From _Texas Monthly_\n\nBy John Spong\n\n[**A Proposal for Feeding the Fat and Anxious,**  \nFrom _Gastronomica_](chapter47.html#ch47)\n\nBy Josh Ozersky\n\n**The Bone Gatherer,** From _Saveur_\n\nBy Mei Chin\n\n**They Don't Have Tacos in the Suck,** From _Houston Press_\n\nBy Katharine Shilcutt\n\n**I Won't Have the Stomach for This,** From the _New York Times_\n\nBy Anna Stoessinger\n\nRecipe Index\n\nPermissions Acknowledgments\n\nAbout the Editor\n\n#### INTRODUCTION\n\n#### By Holly Hughes\n\nI'm easily mesmerized when it comes to food shopping\u2014inhaling the yeasty scent of the bakery, gently plucking ripe items from the produce bins, shivering in the frosty air of the freezer section. But lately it seems that all I do in the grocery aisle is pore over package labels. Yes, I'm following Michael Pollan's sage advice in _Food Rules_ (no food products with more than five ingredients, no ingredients you can't pronounce, nothing your great-grandmother wouldn't have recognized\u2014you know the drill). But I have two other compelling reasons to vet the foods I feed my family.\n\nOur kitchen now must be totally nut-free, after my college-age son went into anaphylactic shock from a dinner of Thai shrimp and cashews. A game-changer? Absolutely. So now I scrutinize the fine print on every package of food that enters our home. Even when the ingredient list doesn't include nuts, there's the dreaded caveat: _manufactured in a facility that also processes tree nuts._ Maybe it's a slim chance of cross-contamination, but I can't take that risk\u2014put the Le Petit Ecolier cookies back on the shelf and choose Choco Leibnitz instead.\n\nWe'd just gotten used to that New Normal when my younger daughter threw another wrench in the works: She's decided to go vegan three days a week, filling our refrigerator with tubs of tempeh, seitan, and Tofurkey. Usually I'm wary of any dietary regimen that's so exclusionary, but I'm going along with this one, because A) this too shall pass, and B) until it passes, she's been inspired to try all sorts of healthy foods she wouldn't touch before. It's actually expanding her culinary horizons instead of narrowing them, and I'm all in favor of that. But now I have to apply a second filter when I read food labels. And lo and behold, what's the main source of protein in many vegan products? You guessed it, nuts. So my kitchen has become a bit of a battleground.\n\nSound familiar? When it comes to food, our entire society seems to be a battleground these days. Americans were once known as a nation of slapdash, thoughtless eaters; now it almost seems we think about nothing else. On the one hand, we obsess over food as entertainment, fetishizing \"decadent\" desserts and all-you-can-eat buffets and trophy high-end dining. On the other, we relentlessly worry about nutrition, health, and the environmental impact of what we eat. People feel so invested in their dietary choices that the age-old concept of sharing a common meal\u2014breaking bread together, even if it's gluten-free\u2014gets short shrift. It's almost impossible to throw a dinner party these days without negotiating a minefield of various guests' food demands.\n\nEnter food writers, who every year become more and more indispensable as guides to this shifting gastronomic landscape. Since editing the first edition of _Best Food Writing_ in the year 2000, I've witnessed an explosion in the number of magazines, websites, newsletters, and TV shows devoted to food; the shelves of my local culinary bookshop (New York City's esteemed Kitchen Arts & Letters) are crammed with expensively produced cookbooks, best-selling culinary memoirs, and scholarly works on all aspects of food. It's been a thrilling metamorphosis to observe firsthand.\n\nIn those years, I've watched food stories move from the \"women's pages\" of newspapers onto the front pages and op-ed pages. Check out the issue-oriented pieces in the opening section, Food Fights\u2014stories like Rowan Jacobsen's venture into post-oil-spill Louisiana fishing waters (page 7), Brent Cunningham's questioning of farm-to-table as a cure-all for America's food supply (page 36), and Hank Shaw's defense of hunting (page 2). And here's another sign of the times: Many of this year's writers earned their stars as bloggers (Elissa Altman of PoorMansFeast, page 246; Smitten Kitchen's Deb Perelman, page 103; Katharine Shilcutt of EatingOurWords and SheEats, page 364), a source of food writing that was barely on anyone's radar thirteen years ago.\n\nIn a food-obsessed culture, trend-spotting is always risky. Nevertheless, when you spend months combing through bookstores and magazines and websites, as I do, every year certain themes pop out. I think of 2012 as the Year of the Three F's: fermentation, foraging, and fennel. What inspired the fermentation craze is anybody's guess, but it's a fair bet that foraging rose to the forefront thanks to Rene Redzepi's Copenhagen restaurant Noma (see Christine Muhlke's profile on page 111). The fennel? It may just be me, but ever since reading Tamar Adler's _How to Live Well_ (page 82), I've noticed roast fennel and shaved raw fennel on menus everywhere.\n\n2012 also produced a bumper crop of pieces concerned with cooking as a Guy Thing. Hence our whole new section on Dude Food, congregating a tailgate party's worth of male food writers, no less than two of whom\u2014Joel Stein (page 180) and Daniel Duane (page 204)\u2014have recently written entire books on the subject of Manliness.\n\nAlways looking for new frontiers, Americans have recently developed an avid curiosity about diverse food cultures. Our Foodways section examines a wide spectrum of those, from sushi (Francis Lam, page 169) to Passover seders (Rachel Levin, page 133) to red-sauce Italian-American (John Mariani, page 162). And because chefs are the new rock stars, Someone's in the Kitchen profiles all sorts of chefs\u2014from Kevin Pang's cautionary tale of a chef on the skids (page 320) to Chris Macias' inspiring profile of one redeemed (page 327).\n\nSome pieces in this year's _Best Food Writing_ feature the very newest developments, such as pop-up restaurants (Richie Nakano, page 317, and Brett Martin, page 258) and the high-tech wonders of _Modernist Cuisine_ (Corby Kummer's review, page 117). But the Old always has a place alongside the New: witness Paul Graham's lyrical essay on syrup making (page 48), Jeff Gordinier's look at eating as an act of meditation (page 173), or Mei Chin's musings on broth and bones (page 359).\n\nSpeaking of balancing the Old with the New, this year's book features a number of writers who have graced these pages often: Deans of the food writing world such as Southern food champion John T. Edge (page 208), political watchdog Barry Estabrook (page 62), locavore crusader Gary Paul Nabhan (page 149), the always witty David Leite (page 336), philosopher of home cooking Greg Atkinson (page 92), and the man whose food writing helped save New Orleans, Brett Anderson (page 140). But the robust state of food writing ensures that there are always new personalities bursting onto the scene\u2014writers such as Vancouver chocolatier Eagranie Yuh (page 237), San Francisco's John Birdsall (page 128), New England's T. Susan Chang (page 230), Londoner Henrietta Clancy (page 241), and transplanted Southerner and sweetaholic Ben Mims (page 250).\n\nSo while fretting over package labels has become a necessity for me, I gratefully turn to food writing to remind me that food should also be a pleasure. Instead of limiting their options, I've found ways to help my kids expand their food choices\u2014teaching my son to make a stir-fry that's better than risky takeout, surprising my daughter with a savory lentil salad full of diced raw veggies. Here's hoping that this year's _Best Food Writing_ will help you too navigate the gastronomic landscape with zest and an open mind. The food scene doesn't have to be a battleground, after all\u2014there's room enough for all of us at the table.\n\n## Food Fights\n\n### ON KILLING\n\n### By Hank Shaw\n\n### From _HunterAnglerGardenerCook.com_\n\n### Foraging and fishing have gained hipster cachet recently, but hunters still are often shunned by politically correct foodies. Hank Shaw\u2014former line cook and commercial fisherman, and author of _Hunt, Gather, Cook\u2014_ demands equal respect for those food lovers who slay their meat.\n\nI have been dealing a lot of death lately. I've hunted five of the past eight days and have killed birds on each trip. My larder is filling, and Holly and I are eating well. Lots of duck, some pheasant and even a little of the venison I have left over from the 2010 season. That is the good side of all this, the side of hunting that most people can embrace. I hunt for a lot of reasons, but for me the endgame is always the table.\n\nIt is the journey to that table that can sometimes give people pause. What I do to put meat in my freezer is alien to most, anathema to some. In the past seven years, I can count on one hand the times I've had to buy meat for the home. This fact alone makes me an outlier, an anomaly. And that I am unashamed\u2014proud, really\u2014of this seems to cause a lot of folks I meet to look at me funny: I am a killer in their midst.\n\nNot too long ago, I was at a book signing event for _Hunt, Gather, Cook_ when a young woman approached me. She was very excited about foraging, and she had loved that section of my book. Then her face darkened. She told me she'd also read my section on hunting. \"How can you enjoy killing so much? I just don't understand it. You seem like such a nice person, too.\" It took a few minutes for me to explain myself to her, and I am grateful that she listened. She left, I think, with a different opinion.\n\nA few weeks later, I was at the University of Oregon talking about wild food to some students. When I mentioned hunting, I could feel the temperature in the room drop. It occurred to me that no one there was a hunter, nor were they close to any hunters. I called for a show of hands. One guy raised his. I asked him briefly about his hunting experience, and it was obvious that it had been traumatic for the poor kid. I let the topic slide and moved on to mushrooms.\n\nWhen I was in Cambridge, Massachusetts, I spoke with more than 100 diners during my book dinner at Craigie on Main. Only four were hunters, although a few more wanted to start. Over the course of the night, I fielded weird question after weird question from diner after diner. _Have I ever shot someone? Did I actually eat what I shot? Wasn't I afraid of diseases?_ It was a stark show of ignorance. Not stupidity, mind you, just an utter lack of knowledge of what hunting is all about.\n\nTo be sure, these encounters were in college towns among a certain set of people. I had some book events, notably those in Montana, Pittsburgh and Austin, where most everyone who attended either hunted or was at least familiar with it. And in most places I could be assured of a healthy smattering of fellow hook-and-bullet types or farmers, who are equally familiar with the death of animals.\n\nBut the fact remains: Most people reading this have never killed anything larger than an insect, and among those who have it's usually been a fish, or an accident\u2014like running over someone's dog. Most people have no idea what it's like to take the life of another creature, let alone why someone would actively seek to do so. Let me try to explain to you the way I did to my young foraging friend on book tour. Let me tell you what it means to kill, at least for me.\n\nTo deal death is to experience your world exploding. It is an avalanche of emotion and thought and action.\n\nArmed with a shotgun, it is often done without thought, on instinct alone. A flushing grouse gives you no more than a few seconds to pull the trigger before it disappears into the alders. A rabbit can leap back into the brambles in even less time. Unless you are perfect in that split second, the animal wins. And being human, we are far from perfect. Even with ducks, where you often have plenty of time to prepare for the shot, their speed and agility are more than adequate defenses. We hunters fail more than we succeed.\n\nThis is why we will often whoop it up when we finally bring a bird down: We are not being callous, rejoicing in the animal's death. It is a hard-wired reaction to succeeding at something you have been working for days, months, even years to achieve. In some corner of your brain, it means you will eat today. This reaction can look repulsive from the outside.\n\nShould you arm yourself with a rifle, you then must wrestle your conscious mind. Buck fever is real. A huge set of antlers will hypnotize the best of us, man and woman alike. Even if the animal lacks antlers, as mine often do, you have to contend with The Twin Voices: On one shoulder sits a voice shouting, _Shoot! Shoot! You might not get another chance!_ On the other shoulder sits another voice, grave and calm: _Be careful. You must not put that bullet in a place where the animal will suffer. Better to pass a shot than wound an animal._ A wise hunter does not kill lightly.\n\nIn that moment when the game shows itself and you ready yourself to shoot, all that matters is that you do your job correctly. And that job is to kill cleanly and quickly. The animal deserves it; we would want no less were the tables turned. And make no mistake: A great many hunters, myself included, do this mental table-turning with some frequency. Seeing animals die so often makes us think of our own death, and I can assure you most of us would rather die with a well-placed shot than wither in a hospital.\n\nWe also know all too well that we are fallible creatures. When we fail to kill cleanly, when we wound the animals we seek, it is our duty to end their suffering ourselves. If there is a moment in this whole process that breaks my heart, it is this one. Everything wants to live, and will try anything it can to escape you. We see ourselves in this struggle, feel tremendous empathy for the struggling bird, the fleeing deer. It is a soul-searing moment where part of you marvels at the animal's drive to live\u2014 _to escape!_ \u2014at the same time the rest of you is consumed with capturing it as fast as possible so you can end this miserable business. This internal conflict is, to me, what being human is all about. A coyote or a hawk has no remorse. We do.\n\nI am not ashamed to tell you that I have shed a tear more than once when I've had to deliver the coup de grace to a duck. I'm not sure what it is about ducks, but they affect me more than other animals. I always apologize to it, knowing full well that this is a weak gesture designed mostly to help me feel better. But it does help me feel better. At least a little. So I keep doing it.\n\nAs the moment of killing fades, death rides home with you in the back of the truck. Once home, you must transmogrify the animal you killed into meat. The transformation is a mystical one, and every time I \"dress\" game\u2014such a pleasant euphemism, that\u2014I marvel at how fast my mind toggles from hunter to butcher to cook.\n\nIt is a necessary process, and one that is vital to why I have chosen this life, why I am a hunter.\n\nI look down at my keyboard and see death under my fingernails. I smell the fat and gore and meat of dead ducks upon me; it's been a good week of hunting. And because I **eat everything on a duck but the quack,** I have become intimate with the insides of waterfowl. Over the years, I've gutted and taken apart so many animals that I know the roadmap blindfolded. And that road leads to meals long remembered. I reach into a deer's guts without thought: I want those kidneys, and that liver. I turn my arm upwards and wrap my fingers around its stopped heart, slick and firm. It will become **heart cutlets,** or **jaeger schnitzel.**\n\nOnce plucked and gutted, I can take apart a duck in 90 seconds. Maybe less. My fingers intuitively know which way and how hard to pluck each feather from a pheasant's carcass. I know just where to put my boning knife, sharp as lightning, to slice the tendons that hold a hog's tongue into its head. I use the same knife to caress its hind legs, separating the natural muscle groups apart along each seam. Some will become roasts, others salami. Animal becomes food. The pop of a goose's thigh bone disjointing from its body no longer sickens me; all it means is that I need to slip my knife under that bone and around the coveted \"oyster,\" the best bite on any bird.\n\nWasting meat is the sin I cannot forgive. When I kill an animal, its death is on my hands, and those animals to whom I've had to deliver the coup de grace are especially close to me. There is a bond between us that requires that I do my part to ensure they did not die for nothing. This is why I spend so much time creating recipes for every part of the animal. Nature wastes nothing, and neither should I. It pains me to know that some hunters do not share this feeling, that they care only for backstraps or breasts\u2014and while I know that coyotes and buzzards will eat what we do not, I do not hunt to feed those creatures.\n\nYou might ask me that with all this, why bother eating meat at all? Why deal with all the moral and emotional implications? In the face of such constant death, is it not better to be a vegetarian?\n\nFor me, no. It is a cold fact that no matter what your dietary choice, animals die so you can eat. Just because you choose not to eat the flesh of animals does not mean that their homes did not fall to the plow to become acres of vegetables and soybeans, wheat and corn. Habitat, more than anything, determines the health of a species. The passenger pigeon may have been snuffed out by hunting, but it was the massive destruction of virgin forest\u2014forest cleared to grow crops\u2014that brought the pigeon to the brink. I have nothing against vegetarians, and the vast majority I've met understand what I do and respect it. But to those few who do not, I say this: We all have blood on our hands, only I can see mine.\n\nIt all boils down to intimacy. Hunting has created an uncommon closeness between the animals I pursue, the meat I eat, and my own sense of self. There is a terrible seriousness to it all that underlies the thrill of the chase, the camaraderie of being with my fellow hunters and the deep sense of calm I feel when alone in the wild. I welcome this weight: It fuels my desire to make something magical with the mortal remains of the game I manage to bring home. It is a feeling every hunter who's ever stared into the freezer at that special strip of backstrap, or hard-won bird or beast understands.\n\nMeat should be special. It has been for most of human existence. And no modern human understands this more than a hunter. I am at peace with killing my own meat because for me, every duck breast, every boar tongue, every deer heart is a story, not of conquest, but of communion.\n\n### THE GUMBO CHRONICLES\n\n### By Rowan Jacobsen\n\n### From _Outside_\n\n### How successful was BP's much-touted cleanup of the Gulf of Mexico after 2010's Deepwater Horizon oil spill? Food writer\/environmentalist Rowan Jacobsen (author of American _Terroir_ ) sailed out with Louisiana crabbers, oystermen, and shrimpers to fish for the real truth.\n\nIn the predawn steam of a Louisiana night, I stood in a yard surrounded by catfish heads. The headlights of dualies towing fiberglass crabbing boats swept into the yard and illuminated wooden pallets stacked six feet high, holding tens of thousands of fish heads: eyes, whiskers, stringy stuff coming out the back. Men in baseball caps stepped out of the trucks, loaded pallets into their boats, and pulled away.\n\nIt was 4:30 A.M., and I was in the yard of Vincent Comardelle, 67, who supplies bait to the crabbers in Larose, a small Cajun town abutting Bayou Lafourche (pronounced la _-foosh_ ), a 109-mile, shellfish-heavy waterway that peels off from the Mississippi River above New Orleans and winds through the marshes of southern Louisiana to the Gulf of Mexico. I was waiting to meet Ryan Comardelle, Vincent's son and one of the top crabbers in the area, who had promised to take me out on the marshes. Finally, a white pickup pulled up. Ryan stepped out and peered at me. He was wearing a tight red T-shirt over a massive chest and Popeye biceps. Buzz cut, goatee, merciless blue eyes.\n\n\"You bring any food?\" he asked. I hadn't. He shot me an unimpressed look. \"We gonna be out there all day,\" he said. \"I don't like to share my food.\"\n\nVincent shambled into the house\u2014he has a limp from an old boating accident that ended with an outboard propeller buried in his back\u2014and returned with a package of peanut butter crackers. We launched Ryan's boat from a nearby dock and motored through a series of shallow, brackish marshes\u2014crab heaven\u2014until we reached Little Lake, which is no longer little, thanks to erosion and land subsidence. Southern Louisiana is sinking, and every year the salty Gulf of Mexico covers more of it, killing the grasses and trees that hold the land together. This is a problem for native shellfish, which rely on a delicate, finely tuned balance of fresh inland water and salty tidal flow.\n\nA full moon hung in the west, silhouetting the rocket-ship spires of gas rigs and the bones of dead oak trees, killed by encroaching water. Ryan, who has crabbed for most of his 40 years, steered while his friend Reggie, an athletic guy sporting a Bud Light cap and a bewildered expression, handled the traps. Reggie was a little off, which Ryan kept pointing out.\n\n\"He ain't exactly stupid,\" Ryan said. \"He just got no sense.\"\n\n\"Oh yeah?\" Reggie said.\n\n\"Yeah.\"\n\nReggie tried to think of a comeback, failed, and went back to the traps. We could just make out ghostly foam floats bobbing on the water. Attached to the side of Ryan's boat was a metal rake, its teeth combing the surface and snatching the line dangling beneath each passing float. When the rake hooked a float, a wire trap came up, dragging near the back of the boat. Reggie opened each trap, shook the captured crabs into a plastic crate, and removed the rotting catfish heads, tossing them overboard. Then he added two new heads and dropped the trap back into the water. For eight hours this is what we did. The boat never stopped. Reggie never stopped. The crabs piled up: olive shells, turquoise legs, orange claws reaching for Reggie when he got close. Every time they pinched him, he'd yelp, \"Oh mercy!\"\n\nMost traps held just two or three crabs and the occasional flounder. \"I used to run 400 cages,\" Ryan said. \"Now I'm running 700 and making less money.\"\n\n\"Bad season?\" I asked.\n\nRyan glared. \"Today might be my worst day ever,\" he said. \"Normally I fill 20 pans. Last time I filled 13. I'm hoping to catch half that today. That don't even cover my expenses.\"\n\nRyan was going to give up for the year if things didn't change in a week or two, and he wasn't alone. When I was in Bayou Lafourche, in September 2011, many crabbers didn't even bother to go out. Apparently, they hadn't got the memo that everything in the Gulf was fine. A year and a half after BP's April 2010 Deepwater Horizon oil spill, virtually the entire fishery was open for business, and federal and state officials were happily trumpeting the health of the Gulf's marine life. Ryan had nothing but contempt for those officials.\n\n\"Dey tellin' everybody everything's OK,\" he said, in the region's ubiquitous Cajun accent, which features a lot of _dis_ and _dat._ \"And it's not. The crabs are not getting fat. A lot of dem are dying right when dey shed. The biologists say everything's normal. Well, shit. We out here on the water almost every day of our lives. We know what changes from one day to the next. Where the little crabs? Before BP hit, they'd be all over this boat. Where dey at? We screwed.\"\n\nRyan was much happier talking about arm wrestling. He was ranked number one in Louisiana for his weight division. \"No guy around here can put my arm down. You want to see why?\"\n\n\"Sure,\" I said.\n\nRyan grabbed the next float from the rake and curled it tightly against his chest as the boat churned along. The cage dragged in the water, half-submerged and spraying a wake. \"That's how I won my biggest match!\" Ryan said, veins popping. \"Other guys use weights. I just come out here and work. Wanna try?\"\n\nI did. The cage jerked me to the back of the boat, and mercifully, the float ripped out of my hand before I was dragged overboard. Ryan smiled and went back to crabbing. We struggled to fill six crates. I counted time in catfish heads. Around noon, I broke out the crackers.\n\nIn the immediate aftermath of the Deepwater Horizon spill, many observers\u2014myself included\u2014anticipated an unthinkable disaster: the death of the Gulf Coast fishery and the final ruination of the Gulf's beleaguered estuary system. Nearly 206 million gallons of oil leaked from the Macondo well. More than 88,000 square miles of fishing grounds\u201437 percent of all federal Gulf of Mexico waters\u2014were closed, along with most affected Louisiana waters, including Bayou Lafourche. Oil filled the estuaries and worked its way into the bayou marshes. Destruction appeared imminent.\n\nAnd then the oil just seemed to fade like disappearing ink. This happened, we were told, thanks to naturally occurring hydrocarbon-eating bacteria, aided by beneficial ocean currents at the site of the leaking well. By October 2010, most Louisiana state fisheries were reopened, including Bayou Lafourche. Since then thousands of seafood samples have been tested for the carcinogens found in hydrocarbons and deemed safe. On April 12, 2011, with the one-year anniversary of the spill approaching, Eric Schwaab, assistant administrator for fisheries at the National Oceanic and Atmospheric Administration (NOAA), proudly declared that \"not one piece of tainted seafood has entered the market.\" Meanwhile, BP has mitigated the damage to its image with a $20 billion fund to pay the claims of affected businesses, plus a slick TV ad campaign to lure tourists to the region. In January, _The New York Times_ ran an editorial lauding the recovery effort as a shining example for future industrial cleanups. These days the consensus from state and local governments, BP, and pundits alike is that everything is pretty much fine. We dodged a bullet, they tell us.\n\nAll of which seemed incredible to me. I reported from the Gulf during the spill, and I watched the Vessels of Opportunity (local boats hired by BP to lay boom and corral oil) and saw how ineffective they were. I worried about what polycyclic aromatic hydrocarbons\u2014the mutagens and carcinogens found in crude oil\u2014might do to the generation of shrimp and crab larvae that were getting soaked in gunk. I wondered about the effects of the 1.8 million gallons of chemical dispersants that BP sprayed to prevent the oil from accumulating on the ocean's surface. And I thought about the scientists who said it would take years before we could comprehend the effect of the spill, much less make grand diagnoses about the Gulf's health.\n\nThese were guys I quoted in my 2010 book on the spill, _Shadows on the Gulf,_ among them Florida State University oceanographer Ian MacDonald, who said, \"I expect the hydrocarbon imprint of the BP discharge will be detectable in the marine environment for the rest of my life.\" And University of South Florida oceanographer John Paul, who said, \"The impact on commercially important larvae that are bathed in this stuff is hard to say. We might see grouper with tumors three years from now.\"\n\nI wanted to believe in the recovery\u2014who doesn't like miracles?\u2014but the spin was hard to cut through. So I decided to do some personal recon to see if the facts on the ground jibed with the reports being fed to the public. In my mind, there was no better way to assess the Gulf's health than by traveling around and sourcing one of my favorite meals\u2014seafood gumbo\u2014in the heart of bayou country.\n\nI've written a lot about food traditions, and I've always had a thing for gumbo, a delicious stew that came about after a bunch of poor people from elsewhere were stuck in a marginal environment, handed a cooking pot, and told to fend for themselves. The poor people were Acadians, French Catholic farmers who got booted out of Nova Scotia by the British in 1755, arrived in Louisiana, looked around at the soupy, unpromising environment, and thought, How do you make dinner out of this? Not only did they succeed, but they designed a culinary miracle, a spontaneous dish born of the holy trinity of crab, shrimp, and oyster that's as American as anything we've got.\n\nIt's no exaggeration to say that Bayou Lafourche has traditionally been the finest place in the world to find ingredients for gumbo. Route 1, which hugs Lafourche, is often called \"the longest Main Street in the world\" because of the way one linear town eases into the next. There are few significant side streets; the only solid land is the two parallel strips of high ground built up on either side of the bayou by millennia of flood sediment. As soon as you get away from that, you sink into the marshes. Those marshes are some of the most fabled crab, shrimp, and oyster grounds on the planet. And they were directly in the crosshairs of the spill. The way I figured it, if I could still make an all-Lafourche gumbo, then America's seafood soul was intact. So I bought a plane ticket and called my buddy Jim.\n\nJim Gossen is the CEO of Louisiana Foods Global Seafood Service, a major Gulf Coast supplier. We first met a few years ago, at the International Boston Seafood Show, and bonded over our enthusiasm for all things oyster. Gossen, who is 64, grew up in bayou country and has been buying from its fishermen since the 1970s. Not incidentally, he has a beach house on Grand Isle, a long barrier island that sits where Bayou Lafourche meets the Gulf of Mexico. When I told him about my plan to toast the non-death of the Gulf with gumbo, he offered to guide me to local fishermen and host the celebratory cook-off. \"I just need to cancel a week's worth of appointments,\" he said.\n\nGossen, who has slicked-back, thinning hair and a gourmand's earthmover-size spare tire, picked me up from the airport in his white SUV on a warm September afternoon, air conditioning blasting. We rolled down Route 1, past moored shrimp boats and ramshackle stores advertising bait and boudin, a Cajun sausage. Our destination was the house of Wilbert Collins, a legendary septuagenarian oysterman. \"There's something about oysters,\" Gossen said. \"I always loved them. My dad loved them. My grandfather loved them. We'd buy sacks in the wintertime and sit in the backyard and shuck.\"\n\nLouisiana oysters tend to grow faster than other kinds, thanks to the warm waters of the lower Mississippi River and an extraordinary abundance of phytoplankton. Collins's specimens were particularly fat. I'd eaten them with Gossen before. They were too large to fit in any human mouth, which stymied me until I watched Gossen take the edge of one in his lips and slurp it like a massive linguine.\n\nAt one time, 16 family-run oyster businesses plied the waters of Lafourche Parish. Gossen used to send a tractor-trailer down Bayou Lafourche and load the entire thing with oysters. But Lafourche's marshes have been under assault for years, long before oil started fouling its beaches. Those marshes, along with the rest of the lower Mississippi, constitute the fastest-disappearing landmass on earth. There are two reasons for this: the leveeing of the Mississippi River by the Army Corps of Engineers, and the digging of canals through the marshes by the oil-and-gas industry in search of new reserves. The levees prevented natural flooding, which over time had deposited enough sediment in the marshes to compensate for natural subsidence. The new canals allowed pure saltwater to intrude inland and kill the marsh grass that held the mud in place. The result: sinking, disintegrating land as the Gulf of Mexico chewed its way farther north every year.\n\nOysters do best in brackish water. Fresh water kills them outright, and while they can grow just fine in salt water, the ocean brings with it numerous critters that relish young oysters, especially snails and drumfish. As the salt water has encroached, many traditional oyster reefs have succumbed to predation and hurricanes. By the early nineties, the reefs near Bayou Lafourche had declined so much that Gossen stopped sending the truck. By the time Deepwater Horizon blew, only the Collins Oyster Company was still in operation. When we arrived, we discovered that it, too, had gone under.\n\n\"Collins Oyster Co. Out of Business After 90 Yrs.,\" read the sign in the yard, \"Because of BP's Oil & Governor Jindals Fresh Water.\" In May 2010, as the oil crept closer to the Louisiana wetlands, Governor Bobby Jindal and local officials tried to push it back by fully opening seven of Louisiana's floodgates on the Mississippi for the first time in history. River water poured through the brackish marshes, turning them fresh. After being threatened for years by overly salty water, most of the reefs in the Barataria and Lake Pontchartrain basins\u2014some of the state's richest oyster grounds\u2014were decimated.\n\nGossen and I stared at Collins's boat, dry-docked beside the bayou. \"Whenever I drove down to my beach house,\" Gossen said, \"I used to stop at Wilbert's and say, 'Please pick me out some good ones! A couple of sacks\u2014make sure they're real nice.'\" He shook his head. \"I don't want to see this die.\"\n\nLouisiana harvested some seven million pounds of oysters in 2010, down from 15 million the year before. Crabs aren't faring much better. The state's crabbers pulled in 53 million pounds of crustaceans in 2009 and 31 million in 2010, when some areas were closed because of the spill. With the fishery reopened in 2011, everyone expected a rebound, but that didn't happen. In the normally plentiful month of May, crab landings for the region that includes Bayou Lafourche were 460,068 pounds, down from 953,503 in May 2010 and 1,471,987 in May 2009. There were so many reports of crabs dying with mysterious infections that the state launched an investigation. Normally, when landings plummet, prices rise, but not when consumers suspect your product is contaminated. Prices had plunged from $1.42 per pound in 2009 to between 35 and 55 cents at the time of my visit.\n\nGossen and I continued down to Grand Isle. As we drove, he told me about another program he'd started at Louisiana Foods, to buy and promote forgotten fish like drum, almaco jack, and wild catfish. \"The goal is to cultivate new fishermen,\" Gossen said. \"As I get older, I've seen less and less fishermen. We losin' 'em, one by one. If there's no way for a young guy to get into it and make a living, he's going to go into gas or oil.\"\n\nLuckily, we had a Plan B: farmed oysters. Last year, Gossen teamed up with an oysterman named Jules Melancon and a professor from Louisiana State University, John \"Soup\" Supan, to build a nursery on Grand Isle. They wanted to grow oysters in wire cages, as is done in most of the world. This had never been attempted in Louisiana, thanks to the abundance of wild reefs. But after watching saltwater-loving predators deplete Bayou Lafourche for years, Gossen decided that farming was Louisiana's future.\n\nIn June 2011, Soup seeded the waters of Barataria Bay, which borders Grand Isle, with 100 million oyster larvae and supplied 250,000 baby oysters the size of match heads to the Gossen and Melancon's project. The idea was to grow the oysters on land, in barrels of water, until they were large enough to be transferred to the bay, where they would live in submerged cages that would protect them from predators. Melancon, a fourth-generation oysterman with underwater leases in Barataria Bay, provided the real estate, the boat, and the labor. Gossen provided the startup cash. The oysters weren't very big yet, according to Gossen, but I just wanted enough to fill out our gumbo.\n\nWhen we pulled up to the docks at Grand Isle, Melancon, 54, was scrubbing algae from plastic barrels filled with seawater and oysters the size of fingernails. He has huge shoulders and hands that seem to be made of petrified wood, the result of spending his teenage summers hauling 100-pound sacks of oysters from his grandfather's 400 acres of reefs, which were located in a marsh a couple of miles from where we stood. Now those reefs were entirely under saltwater, thanks to erosion, and surrounded by predators. The only option was cages.\n\nMelancon had no oysters for us. He took us out to his leases in his 23-foot flat-bottomed boat to show us why. For an hour we motored gently through the calm, flat water, passing flocks of ibises and pelicans, until we moored the boat and Melancon winched up a cage. It surfaced dripping black slime. The oysters were coated in shiny goop. Melancon plunged his tongs into the water and pulled up a jiggling mound of black pudding.\n\n\"What the hell is that?\" I asked.\n\nMelancon scooped some up with his hand. It held like shaving cream. \"I think it's oil,\" he said. \"Look how it stains your hand. Dat's carbon.\" Grand Isle is the sandiest spot in Louisiana. Its reefs are normally a pleasing jumble of shell and grit. But when Melancon checked this one on August 26, two weeks before I arrived, it was buried under a foot of black gunk. \"Dis land was clean,\" he said. \"Just shells and sand. Not all dis fuckin' bullshit.\"\n\nUpon discovering the gunk, Melancon called the mayor of Grand Isle. The next day, the Coast Guard and a state fisheries employee turned up to collect samples, which they delivered to the Gulf Coast Claims Facility\u2014the organization run by \"claims czar\" Kenneth Feinberg that adjudicates damage payments for BP. At the time, Feinberg had distributed about $5 billion of BP's $20 billion fund to 200,000 victims of the spill, who were granted payments of twice their documented 2010 losses in exchange for waiving the right to future compensation. (The exception was oystermen, who were granted up to four times their 2010 losses.) But an estimated 100,000 claimants have refused to settle, instead filing a monstrous class action. Now BP is arguing that future payments should be curtailed, since the Gulf's tourism and fishing industries have rebounded.\n\nSince the samples were submitted, Melancon had heard nothing. Now we scooped more samples into ziplock bags to send to an independent researcher at the University of Georgia.\n\nAccording to some scientists, the presence of oil near shore is a distinct possibility, though tracing it to Deepwater Horizon could be tough. \"It would not necessarily be surprising that there is some residual oil in these wetlands,\" said Don Boesch, professor of marine science at the University of Maryland Center for Environmental Sciences, who served on the National Commission on the BP Deepwater Horizon Oil Spill in 2010. \"However, that area is one of intense development by oil-and-gas companies. After a year and a half, the natural process would change the chemical composition of the oil from Deepwater Horizon. It becomes more difficult over time to identify oil with a specific well.\"\n\nSuch statements are of little comfort to Melancon. He dropped the mess over the side and motored back in. We drove the samples up Route 1, looking for a place to FedEx them. As we rode, Melancon told me how, after the spill, he had worked for BP skippering a Vessel of Opportunity, hauling boom on his boat. He was paid $3,300 a day. By my calculations, he earned half a million dollars, an amount that takes some of the sting out of losing your life's work. Many other \"spillionaires\" on Grand Isle saw similar paydays. The island is crawling with antique roadsters. But a lot of the spillionaires, like fishermen everywhere, have only a passing familiarity with the Internal Revenue Service. They had no idea a tax bill was coming and saved no money for it.\n\nMelancon, on the other hand, bought nothing. As a well-known boat operator with a 500-ton hauling license and no oysters to harvest, he was in high demand. As we drove, he took a call from a guy who offered him a job in Texas working as a barge captain for a chemical company. One hundred and thirty grand per year, plus benefits.\n\n\"Can't do it right now,\" Melancon said. \"I'm tied up with Jim on dis seed business.\"\n\nHe hung up, and then his phone rang again. It was a state official with the test results. \"Uh-huh,\" he said. \"But it's carbon. Smear it on your finger. You can tell. It's contamination.\" He hung up. \"Dey say it ain't oil,\" he told us. \"Dey say it's sludge. What's dat mean?\"\n\n\"Could mean a lot of things,\" I said. \"Maybe it's stuff that washed down the Mississippi.\"\n\n\"Never did before.\" Melancon Googled \"sludge\" on his phone. \"Semisolid material left from industrial wastewater,\" he read aloud.\n\n\"Maybe somebody opened up their holding tank over your reef,\" Gossen offered.\n\n\"I'm worried about my gumbo,\" I said.\n\n\"There's a place up here called Bob's Seafood,\" Melancon said. \"Maybe he's got oysters.\" He kept poking at his phone, eyebrows knit. \"Sludge. Sounds like contamination to me.\"\n\nBob's shop looked pretty barren: the stainless steel oyster-shucking station was gleaming and clean, and a nearby walk-in fridge was just about empty. \"Got any oysters?\" Melancon asked.\n\n\"I got one half-gallon tub from Chauvin,\" Bob said. Chauvin had dodged both the oil and the fresh water. \"Why?\"\n\n\"We gonna make gumbo,\" Melancon said.\n\nBob marveled at Melancon. \"Oysterman wants to buy oysters?\"\n\nIt was looking like our gumbo might be skimpy on crabs and oysters, so we decided to go heavy on the shrimp. Grand Isle, after all, is America's unofficial shrimp capital, complete with its own shrimp king. Actually, _don_ might be a better word.\n\nDean Blanchard, 53, is the only guy I know who's done time in prison for illegally transporting red snapper across state lines. He once drove to New York City's Fulton Fish Market in a Corvette and threatened a deadbeat seafood buyer with an Uzi. He fits in well on Grand Isle, a place that's proud of its pirate heritage, which dates to 1810, when the buccaneer Jean Lafitte set up camp nearby, smuggling slaves and rum into New Orleans by poling them up Bayou Lafourche.\n\nBlanchard is the top shrimp broker in the country, single-handedly responsible for 7 percent of the shrimp bought and sold in the U.S. At least he was. Since the oil spill, Grand Isle shrimp have been few and far between. Chances are you haven't noticed. While the Gulf Coast is still America's top shellfish producer, 90 percent of the shrimp you see in stores is imported from Asian farms. Louisiana shrimpers can't compete; three-quarters of them have disappeared in the past 25 years, a result primarily of competition from the East. When the last Gulf fisherman hangs up his hip waders, Red Lobster will not be fazed. Blanchard is not only the biggest shrimp buyer in Grand Isle; he's the last.\n\nWhen Gossen and I pulled up to Blanchard's house\u2014a newly renovated, _Sopranos-esque_ affair on an island of beach camps, built with internal steel beams to withstand hurricanes\u2014there were four men, one on a forklift, polishing his massive motor home, which he had just driven to and from the Saints-Packers game in Green Bay, Wisconsin. Next to it was a Hummer with decorative Saints plates. Dean was inside watching football in his recliner, wearing shorts and a sleeveless T-shirt.\n\n\"How are your shrimp?\" I asked.\n\n\"I wouldn't eat dat shit,\" he said, rubbing his spiky gray hair.\n\n\"Really?\"\n\n\"Dat water's poison. Every day, dey haulin' dead porpoises in front of my place. Wildlife says dey was dying before the oil spill. I say, Yeah? I musta been sleepin'. Cause I ain't never seen y'all haul one in front of my place before. Now I see y'all haul a hundred of these motherfuckers!\"\n\nIt should be noted that Dean is one of the claimants in the class action angling for a massive settlement with BP. He therefore may not be the most objective source. But his story more or less checks out. NOAA reported some 80 abnormal dolphin deaths between January and April 2010. Since the spill, more than 450 dead dolphins have washed up on Gulf Coast beaches. Many were newborns or stillborns, leading some biologists to hypothesize that ingested oil had contributed to a wave of miscarriages. NOAA has declared the situation an \"unusual mortality event.\"\n\nIf none of the seafood tested by NOAA showed oil contamination, how could the Gulf's marine life be so affected? A recent paper by LSU biologist Andrew Whitehead provides a clue. Whitehead examined Gulf killifish\u2014minnows that live in the marshes and are an important food source for many species\u2014before, during, and after the oil hit. He found that even tiny amounts of oil caused genetic abnormalities and tissue damage in the fish, enough to impair their reproductive abilities. And you wouldn't have known this simply by testing them for contamination.\n\n\"Though the fish may be safe to eat,\" Whitehead said, summarizing the report, \"that doesn't mean they are capable of reproducing normally.\" This problem may extend to other marine life. And many fishermen blame low yields on BP's dispersants, though the scientific jury is still out.\n\n\"The dispersant is biodegradable,\" said Ralph Portier, professor of environmental science and oceanography at LSU. \"The oil is biodegradable. So we're not worried about their presence over a long period. The real issue is whether the mixture of dispersant and oil made it to the marsh and had a catastrophic effect on key organisms. There are literally hundreds of scientists working on these problems, but right now there are too many variables and not enough data.\"\n\nPerched on his dock at the very edge of the Gulf, Blanchard has a unique vantage point to speculate on such things. And what he wanted to tell me was that the white shrimp season was a bust. \"White shrimp are born on the beach,\" he said. \"Dey ain't got a chance to go nowhere. Dey layin' in polluted fuckin' water. Dey dead! Unless dey like Jesus and can raise from the dead, dey ain't comin' back. Usually, I buy about 250,000 pounds opening day. This year I didn't buy one. First time in my life.\"\n\nAgain, Blanchard's not lying. According to Clint Guidry, president of the Louisiana Shrimp Association, the white shrimp haul was down 80 percent across Louisiana in 2011. On November 30, in the face of overwhelming evidence of the poor season, Feinberg, the claims czar, announced that commercial crab and shrimp harvesters would be entitled to four times their documented 2010 losses, instead of two times, in exchange for waiving their right to sue. This was good news for the 1,000 or so shrimpers and crabbers still holding out but an affront to the 4,000 who had already accepted a settlement and were told that there would be no retroactive payments.\n\nI asked Blanchard whom he blamed for the disaster. \"The French media came and asked me whose fault it was. I told 'em Napoleon. He should've killed all the British at Waterloo. The German media came. I told 'em Hitler. He should've bombed 'em with the fuckin' Luftwaffe. The American media came. I told 'em it was George Washington's fuckin' fault. We been fightin' the British on this island since 1673.\"\n\nWe decided not to buy our shrimp from Blanchard. Instead, we bought it back up the bayou, in areas less affected by oil, from an old guy on a boat called the _Tootsie._ I paid for five pounds, and he shoveled about 20 into our cooler. As he handed it over, he said, \"Hey, lemme show you dis shit we found on the west side of the bayou.\" From the cabin of his boat he produced a tar ball and a big chunk of hard, black crude. \"Never seen nothin' like this in my life,\" he said.\n\nOn the way back to Grand Isle, we tuned in the local talk-radio station. A caller had hijacked the show and was on a full-blown BP rant. \"I blame Napoleon!\" he yelled.\n\nMaking gumbo is a social affair. That's the point. People gather, work side by side, drink beer, celebrate the bounty of the sea. Against all odds, that's what we did. Word spread around Grand Isle: gumbo at Jim's house. Melancon showed up with his wife, Melanie. Vincent and Ryan Comardelle didn't make it, but a guy who ran the port commission did, having heard the news. So did \"Soup\" Supan from LSU. He stared in mute horror at the photos we'd taken of Melancon's reef. \"Jules,\" he said, \"get those cages out of there.\"\n\nGossen boiled the shrimp to make a stock. I peeled off the tails. Gossen made the roux\u2014flour and oil simmered and stirred until it turns mahogany brown. Then he threw in onions, celery, and bell peppers. _Gumbo_ is an African word for okra, a mucilaginous vegetable that came over with the slave trade. Around New Orleans, okra is the standard thickener for gumbo. But down in bayou country, where there were no plantations, the gumbo has always been thin, heavy on the seafood.\n\nMelanie pulled a box of live crabs from the fridge. The chill had slowed them considerably. \"We down here at the end of the world,\" said Melanie, who has lived on Grand Isle all of her 50 years. \"Nobody knows what goes on here.\" Then she ripped a live crab in half. I'd never seen a woman do that before. Every now and then, a crab would warm up enough to sink a claw into her, eliciting a giggle. \"Oooh, he ticklin' me!\" she'd say. \"Stop dat, now.\" Then, with a twist, she'd pop his top shell off, pull out his lungs, snap off his mouth, wrap a meaty hand around each half of his body, and snap it in two. That put an end to the tickling.\n\nThe crabs went into the pot, guts and all. Once the crabs and veggies were cooked, we threw in the shrimp and oysters and turned off the heat. I'd never made a gumbo with live crabs and shrimp right off the boat. It was sweet as all get-out. I dipped my spoon in and fished up some shrimp and oysters. A crab claw protruded from my bowl.\n\n\"Nice job, Jim,\" I said, raising my beer.\n\nHis eyes got big. \"I just want to see it come back!\" he said.\n\nMe, too. But I wonder. Complex systems do complicated things. The night after Hurricane Katrina passed through southern Louisiana, everyone breathed a sigh of relief. We dodged a bullet, they said. And then the river came pouring through the levees. Since the spill cleanup ended, the government, the media, and the oil industry have been only too happy to announce that we got lucky. Wait and see, though. You can still hustle up a meal from the marshes of Bayou Lafourche. But a way of life is ending along the lengths of the longest Main Street in the world.\n\n### SERVING UP SUSTAINABILITY\n\n### By Erin Byers Murray\n\n### From _Edible Boston_\n\n### Cold New England waters offer very different seafood harvests, as no one knows better than Erin Byers Murray, author of _Shucked: Life on a New England Oyster Farm._ So how do Boston chefs and restaurateurs define sensitive issues of seafood \"sustainability\"?\n\n\"Is this sustainable?\" Restaurant chefs are faced with the question regularly, especially when it comes to seafood. But as most of them are learning, there is no easy answer.\n\nDepending on what's on the plate (salmon, sea bass, tuna or lobster), the answer might be yes\u2014with the justification that it's caught or farmed in a way that maintains healthy fish stocks; that it keeps the ocean floor intact; and that harvesting it results in limited bycatch. All three, if true, would be valid answers, at least on the surface. But look deeper into any single species and the chef might be falling into murky territory.\n\nTake Atlantic bluefin tuna, for example. Right now, this local species is one to \"avoid\" according to the Monterey Bay Aquarium's Seafood Watch list. This list uses what the aquarium calls \"science-based, peer-reviewed and ecosystem-based criteria\" and many restaurant chefs rely on it, or others like it, to determine what to put on their menus. Bluefin has been on the list for several years, and in November 2010 the Center for Biological Diversity, a group that campaigns to protect endangered species, initiated a \"bluefin boycott,\" asking chefs to pledge to avoid serving the species and diners to pledge to avoid restaurants that serve it.\n\nBut if a chef were to talk to a small day boat fisherman working out of Plymouth or New Bedford last summer, he might reconsider the boycott. Fishermen working in and around those waters found the tuna to be considerably abundant, so much so that guys were hitting their season quota in just three or four days.\n\nBuyers were heeding the watch lists and avoiding bluefin altogether so the fishermen found that the wholesale demand had dried up. Yet the tuna fishery in Massachusetts is closely regulated so while landing a tuna might not have been considered morally correct, it was (and is) still very much legal. In light of the boycott and a dying market, would a fisherman throw the tuna back? Why, when landing a large one might bring him a large profit\u2014a profit that would help his business, feed his family and put money into the local economy?\n\nLooking at it that way, the chef's choice is easy: he's buying tuna from a local fisherman who caught the product legally and within regulations and in turn, he's showcasing a quality fish on his menu, which he likely got for a great deal because the market has pushed prices down. He realizes that he can turn a nice profit on the plate with that bluefin tuna\u2014and what's more, despite the boycott, people are going to order and eat it. So what if he's going against a watch list and an industry ban: Can the chef afford not to put that bluefin on the menu?\n\nThe answer is probably no. Restaurants all over Boston, no matter the size, shape or concept of the restaurant, will tell you that they have, in fact, put bluefin and other species like it on their menu\u2014and that customers continue to order it.\n\nIt is, undoubtedly, a complex web of issues that chefs struggle with every day. And because there are as many types of restaurants as there are fish in the sea, each chef will handle the situation differently. In the end, each chef needs to create his or her own definition of sustainability. Is it about the restaurant's bottom line? Is it about the fisherman's livelihood and buying local? Or, is it maintaining the population of a species?\n\nIt's about all of the above, and more.\n\nThe good news is that in Boston, many chefs weigh their buying decisions carefully. They study watch lists, they discuss options with their distributor, they ask questions of local fishermen. Sometimes, they make compromises. Other times, they draw a hard line and refuse to cross it. Either way, putting sustainable seafood options on their menu is a job that chefs have started taking very, very seriously.\n\n##### **Defining Priorities**\n\nChef Michael Leviton, who owns Lumiere in Newton as well as the newly opened Area Four in Cambridge, is well known for focusing on sustainable practices across his menu. (This spring, he was named the director of the board of overseers at Chefs Collaborative, a nonprofit dedicated to sustainability education.) When it comes to seafood, he defines sustainability through three different domains: environmental (a species that is caught or farmed using low-impact methods but also doesn't have to travel thousands of miles); economic (micro in terms of \"can he afford that fish?\" and \"can he put it on the plate for an amount his customers will pay?\"; macro in terms of the economics of our state and regional fishing industry); and social diversity (maintaining a 400-year-old industry). He also adheres to a strictly local policy by only using seafood from waters between the Chesapeake Bay and northern Canada and prides himself on knowing the name of every boat his fish comes off of, which he gets directly from purveyors.\n\n\"I drew the line at a certain point,\" Leviton says about his seafood choices. He long ago cut salmon from his menu altogether due to the environmental impact of shipping fish from thousands of miles away. \"So as long as there are other alternatives that are equally good I don't think it's an issue,\" he says. \"The first rule of running a sustainable restaurant is to keep your doors open. I am no good to the sustainable agenda if I don't have my pulpit,\" he adds.\n\nLeviton's practices might seem extreme but they resonate with other like-minded chefs. Steve Johnson, chef-owner of Rendezvous in Central Square, maintains a similar philosophy: Sourcing locally actually helps narrow his seafood buying decisions.\n\n\"It's made it easier for me to develop a very focused approach, which is that almost all of the species we serve are north Atlantic,\" he says. His priorities start with healthy fish stocks, so migratory species that are carefully regulated such as striped bass and blue fish, are served in season while a species like squid, caught off of Point Judith, Rhode Island, are on the menu year-round.\n\nBut Johnson admits that certain factors have to fall to the bottom of the hierarchy. Catch methods, for example, are less of a concern for him. While he's aware that squid and scallops caught with nets may have an impact on the ocean floor, he justifies purchasing them, saying, \"I have to believe that someone is working to improve the technologies so that we can have fewer issues environmentally.\"\n\nIn general, small restaurants that can update their menus frequently seem to have an easier time with local options. Fish Market, a tiny sushi restaurant in Allston, forgoes exotic species from Japan in order to utilize local species. Chef and co-owner Kin Chan says about 70% to 80% of his menu is locally sourced. \"There are plenty of options here: mackerel, sea bass, fluke, sea urchin,\" he says. \"I'd rather get it here than something that's been on a ship from Japan for two weeks. It's very important to me.\"\n\nOn the other end of the spectrum, larger seafood restaurants like Summer Shack have as many as five locations and serve several hundred diners a day.\n\nSummer Shack owner Jasper White says their seafood orders are tremendous. But White, who advocates strongly for the Boston harbor advocacy group Save the Harbor\/Save the Bay still puts sustainability at the top of his priority list\u2014and in his eyes, sustainability does not come down to watch lists. \"I question everyone. I question the integrity of some of those groups. I don't think they have much concern for the humans involved,\" he says. After opening his first Summer Shack in Cambridge, he became so frustrated with the regular seafood supply chain that he launched his own wholesale company. Run by former commercial fisherman Max Harvey, Summer Shack's wholesale arm supplies all of White's restaurants along with a few other select accounts. Because of Harvey's fishing background, he chooses to get information directly from the men and women who are out on the water every day. From his vantage point, he says he's seeing good change: \"There's been a lot of work done to rebuild certain stocks.\"\n\n##### **Gathering Information**\n\nDetermining where to look for trustworthy information takes practice, most chefs say. While some use watch lists as a guide, others dig deeper. Leviton speaks to his distributors\u2014Kim Marden of Captain Marden's Seafood in Wellesley and Ingrid Bengis, a seafood purveyor out of Stonington, Maine\u2014almost daily to learn what's coming off the boats. Others go directly to the fishermen themselves.\n\n\"I get as much of my information as I can from the people who are harvesting, growing and catching the fish,\" says chef Jeremy Sewall, a partner in Kenmore Square's Island Creek Oyster Bar who not only works closely with oyster farmer Skip Bennett of Island Creek Oysters but also with his own cousin, Mark Sewall, a lobsterman up in Maine. Those direct relationships have given him connections to other commercial fishermen who are helping him grasp what should or should not go onto his menu. So not only is he getting information directly from the source, he's also asking questions of his purveyors and others in the industry in order to get a well-rounded look at what is happening on the water. In some cases, he finds that he's the one educating his purveyors\u2014a role that more and more chefs find themselves in these days.\n\nAll of that information gathering gives Sewall the unique advantage of being able to read the market daily. \"You're seeing different species become more popular out of necessity rather than creativity,\" he says. he brings up the examples of fluke and sea trout, which are both delicious fish that have been scattered on menus for a long time but have recently become more popular now that some mainstream species (Atlantic halibut, cod) are showing up on watch lists. \"One fish is taking the pressure off another, which means fishermen have had to go to alternative species to fill the void of things they just can't offer like they used to,\" says Sewall.\n\nIn Gloucester, restaurant owner Mark McDonough is located close enough to the docks so that he can literally have conversations with fishermen every day. And he does. his restaurant Latitude 43 buys directly from the smaller, family-run day boats that come in and out of Gloucester harbor because, he says, \"we believe that if we buy local, we're safe.\"\n\nChris Parsons, the chef-owner of Parsons Table in Winchester chooses to take it even one step further: he discusses many of his seafood buying decisions with two marine biologist friends from Rhode Island. \"They opened my eyes to the larger story,\" Parsons says, explaining that they've given him in-depth background information on what goes into salmon farming. That knowledge has helped him shape a plan for buying from farms that use low-impact methods. \"I'm not an expert,\" he admits. \"I could read some articles online and base my decisions on a little bit of research but that won't give me the full picture. Working with these guys really helps me cut to the chase.\"\n\n##### **Factoring Cost**\n\nMaking responsible choices is just part of the equation. Most chefs will admit that ultimately what they serve comes down to price. \"There will always be critics that say farm-raised salmon is bad. And yes, I think wild king salmon is the top of the line in terms of a well-managed species but it is ungodly expensive,\" says Jeremy Sewall. \"It's hard to put a piece of fish on the menu at $44 per plate. Your guests aren't going to feel like they can come in every day and order it.\" Instead, he says, the farmed salmon he uses, farmed salmon from the Faroe Islands, \"has a minimal impact on the environment and they're pretty responsibly raised.\" And he only charges $26 for it.\n\nFor other chefs and restaurants, dollars and cents equal out to more than just the final tab. Karen Masterson, a co-owner of Nourish restaurant in Lexington, considers the cost to the fishermen themselves. This spring, she started working with Cape Ann Fresh Catch (CAFC), a community-supported fishery, on a pilot program that incorporates restaurants into the community-supported model; Cape Ann now delivers weekly shipments of fish directly from Gloucester day boats to her restaurant kitchen. (CAFC is looking to launch the program at other restaurants, including Dog Caf\u00e9 in Gloucester, this summer.) The setup is risky in that Masterson never knows what she might get (whole yellow-tail flounder one week, cod filets the next). But once she receives the shipment, her kitchen staff comes up with specials and that's what she serves through the weekend.\n\nWhile it is undoubtedly more expensive than working with a traditional wholesaler, she's chosen this route because CAFC provides a direct conduit to the fishermen. So instead of spending money with a third party week after week, she's paying slightly more to receive fish directly from the boat\u2014and her dollars are going right back to the fishermen. \"It's a truer cost of our food. From my perspective, that's a more just purchase,\" she says.\n\n##### **The Hook**\n\nEven while chefs struggle with making sustainable decisions, they do it knowing their efforts are part of a larger movement. For Steve Johnson, that means keeping his dollars in the local economy. \"People in our own community, in Gloucester, are suffering. In this small way, I know I can keep my money here,\" he says.\n\nMichael Leviton sees a larger impact. \"Part of doing what I do is the hope that there is this eventual trickle-down in sustainability,\" says Leviton. \"That it just becomes the way we do things so that everyone can afford it.\"\n\n### KIDS BATTLE THE LURE OF JUNK FOOD\n\n### By Maureen O'Hagan\n\n### From _Pacific Northwest Magazine_\n\n### Battling America's childhood obesity \"epidemic\" takes more than preachifying, as _Seattle Times_ reporter Maureen O'Hagan examines in this award-winning three-part feature story. This first installment profiles a pair of obese teens and the daily minefield of temptations they navigate.\n\nNathan Stoltzfus has a problem.\n\nIt starts first thing in the morning, when he catches five or 10 minutes of TV just before heading out the door to Northshore Junior High. \"There's these commercials for Cookie Crisp cereal or Pop-Tarts,\" he says. \"They show some really happy kid eating them.\"\n\nOver at school, he makes his way to homeroom as the smell of French toast drifts from the cafeteria.\n\nAt lunchtime, he sees a sign over the ice cream and cookies. It says, \"Treat yourself today.\" He tries to focus on his veggie-filled sandwich. But invariably, one of his buddies will jump up for another run through the lunch line. \"Who wants to come with me to get some cookies?\" the kid'll ask.\n\nTo Nathan, this innocent question is a trap. As the heaviest kid at the table, he tells himself, keep your eyes on the veggies. But he also wants to fit in. \"It's very tempting,\" Nathan says. \"You feel like you almost have to do it in order to be friends.\"\n\nNow, Nathan's just 14, but he's no slouch. He's articulate, creative, has a good group of friends and seems to take time to think about what he's doing. He's dedicated, too, singing for years with the Northwest Boychoir.\n\nHe's also been overweight for most of his life. To him, it feels like a curse.\n\nAnd the pressure never subsides. There's the school's annual pie day to celebrate \"pi\"; there's the group Slurpee runs after school; the torment of his skinny brother devouring Oreos; the weekend trips to the mall. \"A couple weeks ago, there was like 16 of us. One of the guys said, 'Oh my gosh you have to try this ice cream,'\" he recalls.\n\nWhat's a kid to do?\n\nRight now, leaders here are trying hard to help. In the past decade, local agencies have been awarded at least $53 million in grants and other funding to combat obesity\u2014most of it in the past year. They've enlisted hundreds of partners in this effort, and they're using some of the most newfangled approaches.\n\nWithin the next year, for example, they'll have spent at least $1.8 million getting healthy produce into corner stores. In the next two years, another $276,000 will be spent on building school gardens. Money will go to rejigger P.E. curriculums, train cafeteria workers and try to get kids to walk to school.\n\nStill, all this effort will miss one of the biggest battlegrounds of all: what's going on inside Nathan's head.\n\n\"Treat yourself today,\" the sign in the cafeteria commands. Just say no, Nathan tries to tell himself. Let's call this the Temptation Complication.\n\nIt's a problem for Nathan. It's a problem for tens of thousands of overweight kids in Washington. And it's a problem for all of us.\n\nIt'll take a lot more than school gardens to dig our way out of this one.\n\nPeople who work on childhood obesity often talk about how different the world was a generation ago. When she was a kid, University of Washington researcher Donna Johnson told me, soda was a special treat. Now, sugar-sweetened beverages are just a few quarters away, in the school vending machine. In surveys of Washington adolescents, about 40 percent said they drank at least one soda yesterday.\n\nWhen he was a kid, former FDA Commissioner David Kessler says, there weren't coffee shops on every corner selling super-duper, fat-and-sugar, grande frappa-yummies. There wasn't the cacophony of chips and cookies at every gas station.\n\nWhen I was a kid, I recall, we ate pretty much what our parents ate. Now, vast product lines are designed just for youngsters: the Go-Gurts and Lunchables and drink pouches. Experts say more new food products are introduced each year for kids than for adults. And guess what: Studies show kid food has more sugar than the adult version.\n\nAnd don't even get me started on the \"fruit leathers\" and \"fruit snacks\" with nary a drop of juice. \"When I was a kid,\" snickers Margo Wootan, director of nutrition policy at the Center for Science in the Public Interest, \"they were called jelly beans.\"\n\nThis special kid food is sold on special TV stations created just for kids by special advertising execs who study youth culture as if they were researching a doctoral thesis.\n\nMeanwhile, kids are noshing practically nonstop. \"Your product doesn't have to be for a meal anymore,\" says Laurie Demeritt, president of the Bellevue market-research firm the Hartman Group. \"Now there are 10 different 'eating occasions' throughout the day.\"\n\nWhat we have here is a supply problem. Food is everywhere. That means temptation is everywhere. If Nathan makes healthy choices half the time, it's probably not enough.\n\nBut the Temptation Complication has another layer, one that's older than Go-Gurt and Nickelodeon and Mountain Dew. It has to do with biology.\n\nPatsy Treece says her daughter, Hannah, has a \"face that radiates kindness.\" She's right. Her eyes are a beautiful brown; her smile shines. At age 13,Treece says, Hannah's the kind of kid who'll stop to help if someone gets hurt on the playground.\n\nShe also has an appetite that won't quit.\n\nAt dinner, she'll ask for seconds, even thirds. \"I'm really, really hungry,\" she explains.\n\nWell, maybe not hungry exactly.\n\n\"It's just that if I see something good,\" she sighs, \"I automatically pick it up and eat it.\" Like a lot of us, she gets pleasure from food. But afterward, she also feels pain.\n\nHannah's twin brother is slim and athletic, but her mom is also overweight. Treece has tried different diets with Hannah. She's tried sports. Delay. Portion control, using 100-calorie snack packs. \"She'll have a couple of them,\" Treece says.\n\nDon't even mention Flamin' Hot Cheetos. \"She would kill somebody to get a package of those,\" Treece says. \"It's almost like a compulsion.\" No kidding. Remember the old ad campaign, \"Betcha can't eat just one\"? That was Lay's Potato Chips. The same company makes Hannah's Cheetos.\n\nSome people simply _can't_ stop. There's science to prove it, says former FDA Commissioner Kessler.\n\nHere's a guy who fought Big Tobacco. He's a doctor; he knows what's healthful. Yet until recently, he could come undone over a chocolate-chip cookie. \"I have suits in every size,\" he says ruefully.\n\nOur desire to eat doesn't originate in the stomach, really. We're wired to crave salt, sugar and fat, Kessler says. You see it in laboratory rats, too. They'll brave the possibility of electric shocks to keep eating their junk food. Even _bacteria_ swim to sugar.\n\nThere's more.\n\n\"We used to think people were lazy or it was a question of willpower,\" Kessler explains. \"I can now show you the (brain) scans. The vast majority of people who have a hard time controlling their eating have excessive activation of the brain's emotional core.\"\n\nWhen you eat things like cookies or Cheetos you get an immediate reward. You feel good. Your brain actually changes when you eat that stuff, Kessler explains. The neurocircuitry gets rewired. Stumble across a \"food cue\"\u2014maybe an ad for cereal, or the smell of French toast\u2014and suddenly, your brain lights up. Your thoughts slip into those newly laid tracks and can't get off, like the way your skis follow along in a cross-country trail. Your brain, Kessler says, gets \"hijacked.\" And the new pathway is reinforced further. Scientists say it's not exactly that we're addicted to food, but it sure is an awful lot like that.\n\nProblem is, these food cues are everywhere. \"We're living in a food carnival,\" Kessler says.\n\nSo we're not just fighting temptation like Nathan. We're battling our very biology, that automatic response that makes Kessler crumble at the thought of cookies and Hannah unshakable in the face of orangey-red salt.\n\nLet's call this the Cheeto Compulsion. Betcha can't eat just one.\n\nHilary Bromberg, strategy director at the Seattle brand\/communications firm Egg, says they sometimes put clients through a little exercise: _Talk about your food history._ \"They enter almost a trance state,\" she says. \"They say, my grandmother made me this, or my mother made me this. There's this visceral attachment.\"\n\nEating, in other words, isn't some sort of clinical calculation of calories. Most people aren't thinking, _Gee, I better have carrots instead of cake because I didn't get my five servings of vegetables today._ \"Food is a source of sensual pleasure,\" Bromberg explains. \"The emotions around food are profound.\" She's saying this as a marketing maven. She's also saying this as someone who studied cognitive neuroscience at Harvard and MIT.\n\nNathan's mother, Susan Stoltzfus, knows this, too. \"Food has been that comfort or that source of consolation or that sense of belonging,\" she says. It's immediate, too\u2014unlike losing weight, which requires forgoing that sense of pleasure over and over and over. \"How do you live for that delayed gratification?\" she wonders.\n\nNathan and Hannah might not explain it the same way. But they understand. Food is pleasure. Food is family, culture and tradition. Food is love.\n\nLet's call this the Comfort Connection. And I'm willing to bet it's within arm's reach right now.\n\nMarlene Schwartz, a clinical psychologist at Yale University, suggested a little experiment. Ask a roomful of people, _Who thinks junk-food marketing works?_ All the hands will go up.\n\nAsk, _does it work on you?_ The hands will vanish.\n\n\"I think people decide, once it's in their pantry, it's no longer junk food,\" she says. Talk about getting inside your head. These marketing guys can put an idea in there and you don't even know it.\n\nAnd it's growing even more sophisticated. Food producers have long hired experts to survey thousands of consumers at once. But in the past decade or so, the behind-the-scenes work has become even more stunning in its scope and highly particular in its findings. Some of the topics food marketers have studied recently: mother\/daughter baking rituals; parents' stress level on car trips; the habits of kids at sporting events; how preschoolers are using their parents' smartphones; the traits of parents who are strict versus parents who are more permissive\u2014and what it all means for their kids' eating habits. There's an entirely new genre of marketing and advertising firms, in fact, that focus on youth culture.\n\nMeanwhile, in the past decade or two, says Demeritt of the Bellevue market-research firm, food-marketing companies also started going directly into people's homes. Shopping with them. Asking detailed personal questions. It's called ethnographic research, and it's being done in many cases by social anthropologists. Hordes of psychologists have been enlisted, as well, to better get inside consumers' heads.\n\nThe result is that marketers know exactly how families eat, what they eat and why. They know what makes them keep coming back, even when they know it's probably less nutritious.\n\nTake Go-Gurt, for instance. It's one of the most successful kid foods ever. Moms are doling it out for breakfast, yet it's got more sugar per ounce than Coke. Yoplait sells $129 million worth of this stuff a year. Unnatural-hued goo in a squeezable tube!\n\nHow did Go-Gurt come to be? Really good research. In phone interviews, typical consumers would say breakfast was a sit-down, family affair. But when General Mills hired an anthropologist to spend time with ordinary families, she discovered they were actually eating on the go. A niche was identified: Families could use something portable. And squeezable yogurt was born.\n\nIt appeals to a kid's taste buds for sure. But the makers of Go-Gurt aren't selling that, per se. They're selling fun, coolness. Remember Kessler and the way food activates the emotional core? Bingo.\n\nThe opportunities to hit that emotional core are greater than ever. Advertisers can reach kids on their own cable channels. They can reach them on the Internet\u2014for a lot less money. At the same time, they'll learn even more about them. Marketers know the search terms consumers enter, the information they put on social-networking sites, the pages they view and countless other metrics. Companies are prohibited from collecting personal data on users less than 13 without parental permission, but a _Wall Street Journal_ investigation found even youngsters' online activities were being tracked, and in some cases offered for sale to advertisers. \"Youth in 2010 will be the first generation in the post-digital economy the retailer will know by name,\" one marketing report said last year.\n\nCheck it out. There are Go-Gurt pages on Facebook. Go-Gurt videos on YouTube. There's even a Go-Gurt game. Betcha can't quit at one.\n\nLast December, Demeritt's firm conducted a nationally representative survey about weight issues. Forty-two percent of people said childhood obesity is a big problem. But among the same people, only 3 percent agreed it was a problem in _their_ family. \"It shows you why people don't really do anything about weight,\" she says.\n\nOther studies have found that most people aren't driven by a desire to be healthy. Instead, they judge their own weight in relation to their peers. If your peers are heavy, there's less motivation to reduce. As it turns out, we have a lot of heavy peers.\n\nAnd look what we're up against: the Temptation Complication, the Cheeto Compulsion, the Comfort Connection.\n\nThen there's the sheer firepower of food producers. \"I remember a honcho at the CDC looked at me and said, 'They're way smarter than we are, and they have more money,'\" the UW's Johnson recalls. She spends a lot of time nowadays thinking about the food environment, how it's easier to get Cheetos than it is to get an apple. But even if apples were everywhere, I ask her, would Nathan choose them? Would Hannah forgo her Cheetos? That's a tough one, she says.\n\n\"Most of us,\" she says sadly, \"are going to choose fat and salt and sugar over foods that don't have those things in them.\" It's biology. It's culture.\n\nThen she thinks, what if apples were made to seem more appealing? \"It's not like Madison Avenue is inherently evil, right?\" she muses. \"If we could harness that . . . oh, man . . . Think of the potential of what they could sell.\"\n\n### PASTORAL ROMANCE\n\n### By Brent Cunningham\n\n### From _Lapham's Quarterly_\n\n### Is back-to-the-land agrarianism the answer to American food reform, or just another salvo in the culture wars? Political journalist Brent Cunningham, who is deputy editor of the _Columbia Journalism Review,_ offers some healthy skepticism, based on grassroots research in a struggling Rust Belt city.\n\nBetty Jo Patton spent her childhood on a 240-acre farm in Mason County, West Virginia, in the 1930s. Her family raised what it ate, from tomatoes to turkeys, pears to pigs. They picked, plucked, slaughtered, butchered, cured, canned, preserved, and rendered. They drew water from a well, cooked on a wood stove, and the bathroom was an outhouse.\n\nPhoebe Patton Randolph, Betty Jo's thirty-two-year-old granddaughter, has a dream of returning to the farm, which has been in the family since 1863 and is an hour's drive from her home in the suburbs of Huntington, a city of nearly fifty thousand people along the Ohio River. Phoebe is an architect and a mother of one (soon to be two) boys, who is deeply involved in efforts to revitalize Huntington, a moribund Rust Belt community unsure of what can replace the defunct factories that drove its economy for a hundred years. She grew up with stories of life on the farm as she watched the empty farmhouse sag into disrepair.\n\nRecently, over lunch in Betty Jo's cozy house in a quiet Huntington neighborhood, I listened to them talk about the farm, and I eventually asked Betty Jo what she thought of her granddaughter's notion of returning to the land. Betty Jo smiled, but was blunt: \"Leave it. There's nothing romantic about it.\"\n\nLeave it? But isn't Green Acres the place to be? Listening to the conversation about food reform that has unspooled in this country over the last decade, it's hard to avoid the idea that in terms of food production and consumption, we once had it right\u2014before industrialization and then globalization sullied our Eden. Nostalgia glistens on that conversation like dew on an heirloom tomato: the belief that in a not-so-distant past, families routinely sat down to happy meals whipped up from scratch by mom or grandma. That in the 1950s, housewives had to be tricked by Madison Avenue marketers into abandoning beloved family recipes in favor of new Betty Crocker cake mixes. That the family farm was at the center of an ennobling way of life.\n\nEvidence of the nostalgia abounds. There is an endless series of books by urban food revolutionaries who flee the professional world for the simple pleasures of rural life, if only for a year or so: _Growing a Farmer: How I Learned to Live Off the Land; Coop: A Family, a Farm, and the Pursuit of One Good Egg; The Bucolic Plague: How Two Manhattanites Became Gentlemen Farmers: An Unconventional Memoir._ A new crop sprouts each year. There's Michael Pollan's admonition, in his best-selling book _Food Rules,_ to not \"eat anything your great-grandmother wouldn't recognize as food.\" And then there are countless articles about the young and educated putting off grad school to become organic farmers. A March 5 piece in the _New York Times_ is typical. Under the headline, \"In New Food Culture, a Young Generation of Farmers Emerges,\" it delivers a predictable blend: twenty-somethings who quit engineering jobs to farm in Corvallis, Oregon\u2014microbrews, Subarus, multiple piercings, indie rock, yoga. This back-to-the-landism is of a piece with the nineteenth-century, do-it-yourself fever that has swept certain neighborhoods of Brooklyn, complete with handlebar mustaches, jodhpur boots, classic cocktails, soda shops, and restaurants with wagon wheels on the walls.\n\nThe surest sign that this nostalgia has reached a critical mass, though, is that food companies have begun to board the retro bus. PepsiCo now has throwback cans for Pepsi (the red-white-and-blue one Cindy Crawford famously guzzled in the 1990s) and Mountain Dew (featuring a cartoon hillbilly from the 1960s) in which they've replaced \"bad\" high-fructose corn syrup with \"good\" cane sugar. Frito-Lay is resurrecting a Doritos chip from the 1980s (taco-flavored, a sombrero on the package).When nostalgia is co-opted by corporate America and sold back to us, as it invariably is, the backlash can't be far behind. Consider this the opening salvo.\n\nIt's unlikely that most serious food reformers think America can or should dismantle our industrial food system and return to an agrarian way of life. But the idea that \"Food used to be better\" so pervades the rhetoric about what ails our modern food system that it is hard not to conclude that rolling back the clock would provide at least some of the answers. The trouble is, it wouldn't. And even if it would, the prospect of a return to Green Acres just isn't very appealing to a lot of people who know what life there is really like.\n\nI came to Huntington last November with my wife, the food writer Jane Black, to research a book about the effort to build a healthier food culture there. This is where celebrity chef Jamie Oliver last year debuted his reality television show, _Jamie Oliver's Food Revolution,_ after the Huntington metro area was labeled the nation's most unhealthy community by a 2008 Centers for Disease Control study. It is a place that has suffered the familiar litany of postindustrial woes: a decimated manufacturing base, a shrinking population, a drug problem. It is also precisely the kind of place where the food-reform movement must take hold if it is to deliver on its promise of large-scale and enduring change.\n\nHow would the messages and assumptions that have powered the movement in the elite enclaves where it took root over the last decade\u2014like Brooklyn, where we live, Berkeley, Washington, DC, etc.\u2014play in communities like Huntington? Places where most people don't consider Applebee's and Wal-Mart to be the enemy. Where the familiar and the consistent are valued over the new and the exotic, especially when it comes to what's for dinner. Where a significant portion of the population lives in poverty or perilously close to it.\n\nJane and I suspected that the environmental, social justice, it-just-tastes-better case for eating seasonally and sustainably that our foodie friends consider self-evident would be met with skepticism\u2014or shrugs\u2014by people who have more pressing concerns than the plight of tomato pickers in Florida or the fact that cows are meant to eat grass, not corn. Nostalgia, though, did not immediately register with us as part of the movement's message problem. Perhaps because we live in the same world as the people who write those My-Year-Doing-X books, foodie nostalgia only seemed an innocuous, if annoying, bit of yuppie indulgence.\n\nBut in Huntington we kept meeting people like Betty Jo. Alma Keeney, for instance, who also grew up on a farm, is baffled by her daughter-in-law Shelly's decision to launch a goat-cheese business. Shelly runs the fledgling Yellow Goat Farm with her friend, Dominique Wong, and together they tend their Nubian and Alpine dairy goats on a small plot in Proctorville, Ohio, just across the river from Huntington. The eighty-seven-year-old Alma, Shelly told us, prefers individually wrapped American slices of cheese, not \"farm food,\" which brings back memories of hard times. Jane and I started thinking about the uncritical, even simplistic way that our agricultural past\u2014and our kitchen-table past\u2014are referenced in American society generally, and in the conversation about food reform specifically.\n\nThe farmer is among the most enduring figures in the American pantheon. \"Those who labor in the earth are the chosen people of God,\" wrote Thomas Jefferson in _Notes on the State of Virginia,_ his classic work on the promise of the American experiment. The agrarian ideal\u2014a belief that the family farm is the soul of the nation, a pure embodiment of our democracy\u2014is a recurring theme in the national narrative. In 1782, J. Hector St. John de Cr\u00e8vecoeur, in his _Letters from an American Farmer,_ celebrated the notion of independence and self-sufficiency that is central to the story: \"Where is that station which can confer a more substantial system of felicity than that of an American farmer, possessing freedom of action, freedom of thoughts, ruled by a mode of government which requires but little from us?\"\n\nThe exalted status of the farmer has influenced political strategy and policy decisions throughout our history: in New Deal legislation that sought to place the family farm, which struggled mightily during the Depression, on par with other industries primarily through price supports; in an amendment to the Selective Service Act of 1940, which granted deferments to young men who were \"necessary to and regularly engaged in an agricultural occupation\"; in the creation of the U.S. food-assistance program in 1954, which pitted the stalwart American farmer against the menace of Soviet collectivized agriculture. And it surely informs the nostalgia that shrouds today's food-reform movement. One can essentially trace a through line from Thomas Jefferson's romantic image of the farmer to a recent defense of rural America in the _Washington Post_ by Tom Vilsack, the U.S. Secretary of Agriculture: \"There's a value system there. Service is important for rural folks. Country is important, patriotism is important.\"\n\nToday most of us are so removed from the agricultural life, and so ignorant about its realities that this wholesome and nostalgic lens is the only one we know. Research by the FrameWorks Institute, a think tank employed by nonprofits to strategically reframe public conversation about social issues, found that for Americans, \"Rural Utopia\" is the dominant image of life beyond the cities and suburbs: a countryside \"filled with poor but noble, tough and hardworking people living healthier and fundamentally better lives than the rest of us.\" This despite the fact that the reality in rural America today is one of decline: unemployment, rising divorce rates, a scramble to get out. According to the Bureau of Labor Statistics, farming is the nation's fourth-most-dangerous job.\n\nStill, nostalgia has been a useful tool for the food-reform movement. It has provided a blueprint for how to think about and act on the daunting environmental, moral, and health problems associated with our industrial food-system for people who have the resources\u2014financial, social, and educational\u2014that allow them to participate in the movement if they so choose, and that predispose them to be sympathetic to the cause in the first place. Whether they started raising chickens in their backyards or simply became better informed about how their food is produced, this idea that we've lost our way has helped make food _important,_ and in ways that go beyond simple sustenance.\n\nMost of these food revolutionaries won't become actual farmers, and most of those who do\u2014including those microbrew-swilling kids in Corvallis\u2014won't make a career of it. But the movement has, I suspect, permanently changed their attitudes toward food, and this alone is already forcing modest systemic change. Since 1994 the number of farmers' markets in the U.S. has risen from 1,700 to more than 6,000. And between 2000 and 2009, organic-food sales grew from $6 billion to nearly $25 billion\u2014still less than 4 percent of total U.S. food sales, but it's a start. Twenty years from now, most of these young \"farmers\" will have rejoined the professional ranks. Like their middle-class forefathers who tuned in, turned on, and dropped out in the 1960s, the appeal of financial security and a climate-controlled office will, in most cases, win out. That said, they probably won't be regulars at McDonald's, and they'll instill these values in their own kids.\n\nNevertheless, a \"bourgeois nostalgia\" pervades the food-reform movement, as Amy Trubek, an anthropologist at the University of Vermont who studies food and culture, points out. This is a perception of our food history that is the luxury of people who have little or no experience with farming, or more generally with manual labor. A perception that appeals to those who have never had to cook from scratch, let alone milk cows, kill chickens, and bake bread, just to get food on the table every day. A perception of people for whom it makes perfect sense to redefine their _leisure_ time to include things like making _guanciale_ or Meyer-lemon marmalade. As such, it may not resonate with great swaths of the public who don't fit this demographic profile, and it is a perception that ignores some crucial truths about our food history.\n\nThe reality of America's food past is far more complicated, and troubling, than is suggested by the romantic image at the heart of our foodie nostalgia. In _Revolution at the Table_ and its sequel, _Paradox of Plenty,_ the historian Harvey Levenstein provides a more sober, and ultimately more useful, accounting of that past. Levenstein shows how, starting in the late nineteenth century and continuing through the twentieth, food preparation steadily migrated outside the home. The reason is simple: if you have no choice but to plan and prepare multiple meals every day, cooking not only isn't cool, it's tedious and damned hard work.\n\nJane and I experienced this firsthand in West Virginia. We both are skilled and enthusiastic cooks, and as part of the reporting for the book, we wanted to see how well, and local, we could eat, and for how much money, preparing three meals a day. But we also understood that we were the kind of people for whom cooking is a hobby. Outside our door in Brooklyn, there is a cornucopia of options for the nights when we are busy or not in the mood to cook. In Huntington, though, most of those options are missing. Three months in we began to notice, with dismay, that as soon as one meal was finished, we had to start thinking about the next. Four months in, the joy of cooking was replaced by a growing irritation, a longing to amble down the block for _banh mi_ or a bowl of ramen. By mid-March, Jane wrote in her journal, \"Officially sick of cooking.\"\n\nBetween 1880 and 1930, the fruits of industrialization\u2014canning, bottling, the growth of food manufacturers and restaurants\u2014enabled the outsourcing of food preparation that Levenstein describes. Improved transportation\u2014first the railroad and then the automobile\u2014and food-preservation processes\u2014refrigerated rail cars, for instance\u2014brought an end to seasonal and regional restrictions on what we ate. Soon, people in Kentucky had the same food choices as those in New York or California.\n\nThe standardization of the American diet, so bemoaned by people like me, is what many\u2014maybe even most\u2014people want at mealtime. It is reassuring to have what everyone else has. The desire to have the same Big Mac in Syracuse as in San Diego is a big part of why fast-food outlets became America's default dining-out option, and why suggesting that as a nation we return to a more seasonal and regional way of eating will be a tough sell.\n\nThe family farm itself was not immune to these developments. By the 1920s and '30s, the gap between city and farm diets had begun to collapse, as processed foods became high-status items in rural areas. Poor Appalachian farmers began to prefer canned hams to country hams; farm women who could afford store-bought canned vegetables and other processed food embraced this new convenience without a second thought that they were abandoning a purer, nobler way of life.\n\nThere's a reason that less than 2 percent of people in this country are engaged in farming today, and it isn't simply that they've been driven off the land by Cargill and ADM. Just like Betty Jo Patton, many of them wanted things to be easier. This revolution at the table\u2014the one that produced the food culture that today's revolutionaries are trying to counter\u2014was considered a tremendous leap forward. It was modern. It gave people time for things other than keeping the family fed.\n\nThere is an even more fundamental concern about our nostalgia: America's food system has always depended on the exploitation of someone, whether it was indentured servants, slaves, tenant farmers, braceros, and other guest workers, or, now, immigrants. In his ode to the American farmer, Cr\u00e9vecoeur made it clear that he had a little help on his farm. \"My Negroes are tolerably faithful and healthy,\" he wrote. This is an aspect of our agricultural heritage that rarely gets mentioned in the mainstream conversation about food-system reform, and it raises thorny questions about who actually grows, harvests, processes, and prepares the food in a capitalist society. We have no history of a food system that does not depend on oppression of some sort, and it seems unlikely that we will be able to create a future system that avoids this fate. The leaders of the food revolution have, in recent years, begun to speak out on the matter of farmworker rights. But few acknowledge\u2014at least in the public debate\u2014that if a central goal of the movement is a more equitable food system, then the notion that we once had it right is deeply problematic.\n\nExploitation is as true in the kitchen as in the field. Women have always borne the burden of transforming the raw to the cooked in the American home. Interestingly, it was a confluence of these two inconvenient truths about our food past\u2014its reliance on women and exploited labor\u2014that helped set the stage for our national embrace of fast food.\n\nDuring the Gilded Age following the end of the Civil War, and continuing into the early twentieth century, America's rapidly expanding ranks of wealthy industrialists used extravagant dinner parties, featuring French haute cuisine, as a way to showcase their status. Hosts and hostesses sought to outdo one another: chefs were imported from France; eight courses were standard, as were menu cards, elaborate centerpieces, and a labor-intensive style of service known as _\u00c3 la Russe,_ which involved a butler carving and arranging the food on plates at a sideboard, which were then delivered to guests by servants. (The traditional style had been to fill the table with platters and bowls and let the guests serve themselves.)\n\nThe fetish for French cuisine, and all the attendant showmanship, quickly trickled down, and the nation's middle-class, which also was expanding, sought pecuniary emulation of this conspicuous consumption. Competitive dinner parties became a fixture of middle-class social life. And it wasn't just at dinner; there were also multicourse luncheons and high teas to pull together. The problem, though, was that middle-class households couldn't afford the number and quality of servants necessary for this kind of entertaining. This \"servant problem,\" as Levenstein calls it, became something of an obsession for American housewives, who saw it as the main obstacle to fulfilling society's expectations of them.\n\nTheir plight led to various time-saving experiments, including cooperative kitchens\u2014in which meals for multiple families were prepared for pickup in a central location\u2014and the first home-meal delivery services. The former failed because they were regarded as a violation of the \"ideal of American family life,\" a critique that had more than a whiff of antisocialist sentiment. The latter, it turns out, was simply an idea ahead of its time. These delivery services conformed to what was then considered the standard for a \"proper meal\": three courses and a menu that changed daily. As such, they were too expensive to be sustainable. The inability to solve the middle-class servant problem led, eventually, to a new conception in American society of what constituted a proper meal: simpler, cheaper, and of course, faster. We know how that story turned out.\n\nBy misrepresenting\u2014or misunderstanding\u2014our food history, we make a realistic conversation about what to change and how to change it more difficult than it already is. America will not revert to a nation of family farms. Convenience will always be important. Seasonal and regional limitations on what we eat can only go so far. If Americans want to cook like their grandmothers, fine, but the fact is our grandmothers, by and large, made only a handful of meals, they made them over and over again, and they used plenty of shortcuts, courtesy of the industrial age. My grandmother's cornbread, which still remains the gold standard for cornbread in my family twenty years after her death, began with a Martha White mix.\n\nNostalgia is part of a larger message problem that food revolutionaries face as they attempt to broaden the appeal of their cause. For example, when Wal-Mart announced earlier this year that it would, over the next five years, reduce the amount of sodium by 25 percent and added sugars by 10 percent in its house brands, and pressure other food manufacturers whose products it carries to follow suit, the overwhelming response from within the food-reform community was, \"That's not good enough.\"\n\nIn Huntington, and in communities across the country, Wal-Mart is where a lot of people get their food. They like the way the food there tastes. If that food has less sugar and salt\u2014incrementally less so that they will still like the way it tastes\u2014that is an important, and realistic, step toward a healthier food culture. Wal-Mart has many bad policies, but it's shortsighted to write off every initiative just because it comes from Wal-Mart. New ideas about food need to conform to people's social and economic aspirations, and those aspirations are going to be different in 2011 than they were in 1900, and they will be different, too, in Huntington, West Virginia, than in Brooklyn, New York. Achieving fundamental and lasting change in our food system will require the efforts of those yuppie farmers in Oregon who can afford to step outside the mainstream food culture and, as they say, vote with their forks. It will also require the more hard-won, incremental reforms at the big food processors and sellers, like Wal-Mart, that feed the great mass of people who either can't or won't vote with their forks.\n\nSomewhere in the middle of these two efforts, hopefully, we can eventually arrive at a food system that makes sense for the twenty-first century. But the process of figuring out what that will look like needs to begin with a full and honest accounting of where we've been, and what's possible given where we are.\n\n## Farm to Table\n\n### SWEET SPOT\n\n### By Paul Graham\n\n### From _Alimentum_\n\n### When novelist and English professor Paul Graham drove a tap into the one maple in his yard, he bottled more than syrup\u2014he captured in words the poetry of coaxing golden elixir from a tree. His debut short story collection _Crazy Season_ was released in summer 2012.\n\n##### **1. Reveille**\n\nThe other day, I turned a corner while walking my dog in town and suddenly, finally, knew the _bon mot_ for the taste of the late-winter air in northern New York. It tasted _tarnished._ I'd been worrying the language for weeks, and then there the word appeared, dull on my tongue, a little bitter. The light, the wind, the dampness: these all tasted faintly oxidized, like a dented old bugle my grandfather had played with when he was a child and one year presented to my brother. The bugle slept in a closet until someone impulsively decided to hold the bell to the other's face while he lay soundly sleeping, tongue the brassy mouthpiece, tasting metal and neglect, and explode one flatulent, croaking note. In March I seem to hear that sound constantly\u2014spring's pathetic reveille\u2014as the days lengthen but bring no other relief. The snowy silence no longer feels refreshing, as it did early in December, when the flakes fell big and clung to the pine-boughs. Forget my stores of books and firewood, forget my full freezer, my loving wife and good dog. By now the sky has reabsorbed the sand and rock salt grit, the frozen dogshit and trash in the jagged plow-furrows, seemingly with the intention of flinging them back in my face.\n\nI'm waiting for 40\u201320. Forties during the day, twenties at night. Not a big deal most places, but here, it's better news than the arrival of the first robin. Robins can lie to you like any other courier. You also can't eat them, or at least _I_ haven't.\n\nWhen I finally see those numbers on the nightly news around the first week of March, I look at my wife and ask, Is this guy to be trusted? How many forecasts has he blown recently?\n\nThe urge to cheat, to set the taps early, is strong. It's not bad for the tree, only your pride; you'll be the one sad fool in the county with galvanized buckets hanging from his tree while the snow swirls like sand.\n\nI choose a weekend day to tap, so I can boil the first catch of sap that very evening. Even if the yield is only a few tablespoons, like holy water, the first run is powerful stuff. The syrup speaks of hope, and of promises kept, in the way of all seasonal foods: apples, tomatoes, berries, venison. Early in the morning I walk outside with the Makita. I bought it years ago for home improvements, knowing even then I'm not that type. I use it once a year, on average, to drill \u00be\" holes in a sugar maple. I can hear my furniture-maker grandfather smacking his forehead with his palm somewhere on the other side of the grave.\n\nWe have only the one tree, a haphazardly trimmed 60-footer that turns orange in October. I'd never have thought to tap it if our old neighbor, a poet who worked in an Adirondack sugarbush with several hundred trees, hadn't tapped _his_ one tree. One, it turns out, can be more than enough and not nearly enough at the same time. I've tried to picture the yield from 1500 taps and can only get as far as a fleet of tanker trucks and an acre of fire.\n\nThe holes go waist-high on the western side, which receives more sun and delivers more sap. I sink two taps with a rubber mallet. I reserve a third in case one dries up. Immediately the juice starts running, dribbles to the metal lip, and trembles in the sunlight before falling into the bucket with a flat _plunk._ It seems impossible that I'll empty these buckets twenty, thirty times, but I will.\n\nThe sap, when I lift a drop to my lips, tastes of the purest, cleanest water.\n\n##### **2. First Haul**\n\nThe first day is always a good run, the tree primed like a well-oiled pump, and when I walk outside at four o'clock, I can't believe my eyes: the buckets are half full. If I don't get started on a boil now, I'll be up all night. I pour the two buckets into one and pour that into a heavy stockpot on the strongest burner. Fire it up. Walk away.\n\nIn thirty minutes, the sap will boil. A layer of scum needs to be skimmed off the top.\n\nIn three hours, the sap reduces by half. The kitchen smells honeyed. The brew in the pot is the color of light beer.\n\nIn four hours, it needs to be transferred to a wider, shallower pot, filtered again, and re-boiled.\n\nIn five hours, the kitchen smells of vanilla. There may be almonds, honey, butter, or hazelnuts, too, depending upon the year. There's surprisingly little \"maple\" flavor. The poet-friend accurately compared maple syrup made from one tree to single-malt Scotch.\n\nThen the syrup begins to foam in the pot with a tidal fizzing. I must watch it carefully now, observe how it coats a spoon. There are instruments, hydrometers and thermometers, but I don't do precision. This explains why, although I'm a good cook, I'm a miserable baker\u2014I can't be bothered, even though I know better.\n\nHow close to disaster do I want to ride? The longer the boil, the more intense the flavors, but if I push too far, the syrup will harden when it cools. I'll have to chip the crystals out of the jar with a fork and suck on the shards.\n\nIt's late. The dog chases footballs in his sleep, his nails clicking on the floor. My wife's knitting needles tap. I'm tired, so I kill the burner and pour the syrup into a clean ramekin. There's no need to sterilize a jar. This stuff won't make it to morning.\n\n##### **_Recipe for First-Boil Syrup_**\n\nIngredients:\n\nAs much maple syrup as you just made\n\nA spoon\n\nA ramekin or small bowl\n\nA partner, if you're the generous type, in which case you'll need two spoons and ramekins\n\nIn the steamed-over windows and soft light, pour the limpid syrup into the ramekin and eat it still warm. Close your eyes. Marvel at the power of nature, wonder at the Indians who were first bored or desperate or clever enough to boil down tree-water for a whole day _just to see what happened._ Who the hell _does_ that? You're in their debt. Taste honey, vanilla, floral notes. Feel the back of your throat burn with the sweetness of winter's end.\n\nGo ahead, make yourself sick. Pancakes? Overrated and too much trouble.\n\nIf you have an embarrassment of riches, pour the warm syrup over vanilla ice cream. But just how much syrup would that _be,_ though?\n\nAnd look\u2014it's already gone.\n\n##### **3. Doubters**\n\nEven up here, in the heart of the country, they exist. Like this one: sixty-something, windbreaker, her pant cuffs stained by puddle-splash. Dogless. I've never seen this snow troll before, and I wonder what thawed snowbank she's crawled out of. She looks at me kind of walleyed as I'm at the end of my driveway collecting sap one week into the season.\n\n\"You're not boiling that _inside your house,_ are you?\"\n\n\"Why?\"\n\n\"It'll take your wallpaper right off. It'll turn your walls yellow.\"\n\nWe've been home-making maple syrup for three years now, and we have had zero wall-issues. But we don't have wallpaper. After stripping nearly all that came with our house, we have unflattering things to say about people who do have wallpaper. Charlotte Perkins-Gilman nailed it. I once called the stuff The Official Decorating Choice of Hell. After fifty-some boils, my kitchen walls are still white. True, the exhaust fan over the stove burned up in the middle of last season, but that could have been anything\u2014a faulty part, spiders in the wiring, or, most likely my aggressiveness in general as a cook. I love heat. _Sear_ is one of my favorite verbs, _smoke point_ one of my favorite temperatures. I revel in culinary accelerants: brandy, amaretto, oil, chili paste. There is often fire\u2014intentionally. The hiss of deglazing makes me hoot. I've been meaning to replace that exhaust hood, but electrical wiring is not my forte, and what are windows for? It's 44 degrees already, after all.\n\nI pity this woman and all of her kind. She appears unaware that nothing great ever gets made the easy way.\n\n##### **4. Regarding the Holes in the Tree**\n\nThey cannot be plugged up. Not with old wine corks, not with DAP, not with cement, rubber or real. Once I've drilled them, the sap won't stop flowing for a month. I won't stop boiling until the yield is so low that it's no longer worth the trouble. Unless I'm using all of those hours while the stove steams to build a time machine that will enable me to go back two weeks _and not pick up that drill,_ I'm in it for the duration.\n\nI know this in theory, but on days when I have other things to do, because I'm running a typical household, not a sugarbush, it will seem as if that maple tree is cranking out liquid just to spite me. On a good day the buckets fill several times, and I pour them into a 5 gallon water cooler barrel. One barrel is about eight hours' boiling with all the burners running, meaning that night dinner will be Thai takeout, or Mexican, or cheese and fruit, but never out at a restaurant, because only a fool would walk away from a kitchen with that much fire pouring out of it. I'd like to expedite the process, but that seems begging for trouble.\n\nJust yesterday I heard about someone on the other side of town who burned his garage down while trying to do a boil in\u2014of all things\u2014a turkey fryer. He had too much sap and he needed to raise the temperature _fast,_ apparently.\n\nThese are my people. Sort of.\n\n##### **5. A Steep and Slippery Slope**\n\nMeaning both downward and upward. Downward in the sense that after only _one_ season, let alone several, you can never go back. Log Cabin, Aunt Jemima, Mrs. Butterworth, anything that comes in a plastic jar or, Godhelpus, a tiny rectangular pack from Kraft with a peel-away top\u2014it's criminal. I'll pay extra for a vial of the real stuff if I have to, though in northern New York we have syrup like the Gulf has shrimp, so it often comes for free. A year we lived in western Pennsylvania stands out as one of the darkest in my life for many reasons, not the least of which is the crap they expected you to dump on your pancakes.\n\nAnd upward, in the sense that someone always seems to have a better setup. Like a friend who taps his maples _and_ his black walnut tree. One year I finally got my hands on a jar of his black walnut syrup, trading it, I'm sad to say, for some amateurish bottles of homebrewed beer I'd made. Not even the guilt of sticking it to him could keep my brain from lighting up in neon\u2014black walnut syrup is earthy, sweet, nutty-green, with the texture of molasses. I took a walk through our tiny village lot after pouring the syrup on oatcakes as he'd recommended, hoping to find that I'd mistaken a walnut tree for a scrub pine. Thwarted, I surveyed the treeline for anything else that could be tapped. _What about that tree over there? Or that one? Arborvitae sap? Yellow birch? Why not?_ People have gotten themselves killed this way, I'm certain.\n\n##### **6. Potential**\n\nMarch is the time of the year for maple festivals, where you can watch what the big boys do and sample all sorts of maple products. We've never been. We have our own maple festival nonstop for several weeks, and we seem to invent a new product every day. Of course there are the usual suspects: pancakes, waffles, French toast, regular toast with a small beaded drizzle in the butter; but we also pour the syrup over ice cream, into ice cream, and can make sorbets and gelatos; use it in place of sugar in breads, cakes, and cookies; in homemade barbeque sauce, salad dressing, and mayonnaise; as a background to sauteed Thai chiles, pad Thai, and other stir fries; with fruit, yogurt, Kefir; alone, out of the jar, like a shot of whiskey on a rough day; shot _with_ whiskey, like a sweet boiler-maker, on a rougher day; as a glaze for chicken, salmon, pork, and shrimp; over oatmeal, with cinnamon and nutmeg; in a pan sauce, like Madeira; in place of honey with strong cheeses. The limits are your own apprehensions and the amount you're willing to risk wasting should the flavors fall into dissonance. The only way I _haven't_ had it is as the Canadians do, dumped into a pile of clean snow on a cutting board outdoors and stirred up with a stick, then eaten, communally, with a spoon.\n\n##### **7. Gifts**\n\nWhether any recipient of DIY maple syrup knows it or not, the value of the gift is always high, might be called equal in dollars to the yield in ounces squared (at least), where yield is the product of a completely idiosyncratic and therefore unpredictable product of temperature, time, and sugar level. I don't know a damn thing about higher mathematics, but it seems chaos theory might be useful here.\n\nWhat I do know is that most maple producers say 40 gallons of sap yields an average of 1 gallon of Grade-A syrup. Our numbers may be higher early on, lower a little later. We'll put up 1 gallon or so a year, spread across an oddball collection of jam, jelly, sauce, mason, and other jars. Some jars, the best, we'll mark with smiley faces, a special cache.\n\nI give jars away reluctantly, and only after attempting to verify that Aunt Jemima is nowhere on the recipient's premises, the apparition of her broad-shouldered, placating form in a kitchen cabinet being, like wallpaper, deeply revealing in a number of ways.\n\n##### **8. Anxiety and Its Cure**\n\nYou begin anticipating the temperature and then move to obsessing _over_ the temperature. The end can come quickly, and not always predictably. One week the temperatures hold steady in the 40s during the day, maybe toeing 50 as if considering the idea, and then one afternoon the wind shifts and the mercury shoots into the 60s. You can recover from only one 60 degree day, but not three or four. It doesn't matter if the temperatures drop again; the syrup will be thinner, the taste not as complex, suitable for baking, perhaps, but barely. In a good year there are two or three weeks in the forties and you can see the warm-up coming. I hate to curse the spring, but I'm never ready to let go unless the season has been so good that the tree has bullied me to exhaustion. I always want just one more jar socked away for August, when the blueberries are as deep as the January snows, and for January mornings, when the snows themselves are deep and we self-medicate with elaborate breakfasts of French toast and local bacon.\n\nBut the season _does_ end, of course. It must. The arrival of the bees in April tells you all you need to know. One day they just appear around the buckets, a little drunk on the sunlight, circling wildly, too drowsy to sting. The rest of the run is theirs. If you keep collecting and boiling, you're only the fox from Aesop. I grab the pliers and pull out the taps, dropping them into the bucket with a bittersweet, metallic clank. The sap begins weeping down the trunk of the tree, staining the sidewalk.\n\nInside, I count the final take, squirreling away jars in various cabinets and corners of the kitchen and pantry so we can forget and rediscover them one night in November. The next season seems a long way off, but now, standing in the sunlight, it's easy to idealize the winter that must come first. Spring will atone for ending the boils with wild leeks and mushrooms in May. Not too long after, there will be garlic and strawberries. Then tomatoes, and watermelons, and winter squash. You can play your favorites, and we do, but the truth is that it's never too far to the next good thing.\n\n### SNOWVILLE CREAMERY HAS A MODEST GOAL: SAVE THE WORLD\n\n### By Eric LeMay\n\n### From _Edible Columbus_\n\n### Poet, essayist, writing teacher, webmaster Eric LeMay\u2014author of _Immortal Milk: Adventures in Cheese\u2014_ pursues his longtime obsession with dairy products from his home in Athens, Ohio.\n\nPouring milk from Snowville Creamery feels blissful. When you pick up the carton, you're greeted by a lovely dairymaid who seems to embody the countryside, with its green pastures and rustic fences. She wears a white fluffy bonnet and wholesome dress. At her bosom, she cradles a pitcher, as though she were Mother Earth pouring out the milk of human kindness. Behind her, the sun rises, encircling her with its hopeful glow.\n\nThis is milk made mythic, and it's one of the reasons I wanted to meet Warren Taylor, owner of Snowville Creamery. The other is the milk itself, which is thick and frothy and delicious. The first time I tried it, I drank a quarter of a gallon, glass after glass. I thought it was a milkshake.\n\nHow, I wondered, did a small diary nooked in the hills of Southeast Ohio produce milk this good? Who was this man behind the maid?\n\nNow, having spent some time with Warren, I see that maid differently: She's wearing a red bandana around her forehead and raising a revolutionary fist in the air. She's marching down Independence Avenue, passing out leaflets on the dangers of genetically modified foods and hydraulic fracturing. And she's smiling, because she's looking forward to a fight. She's going to take on huge corporations who want to strip our food of its nutrients and flavor. She supports our local businesses serving our local communities. She wants our kids drinking healthy milk from healthy cows raised on a healthy soil.\n\n\"Love your food,\" she cries. \"Love each other.\"\n\nShe wants us to join a revolution.\n\n\"This isn't about _this_ ,\" says Warren.\n\nWe're outside, standing on the slope of a hill, looking up at what's essentially a pole barn. It's the milk plant that Warren designed from the ground up, and he's telling me how it takes advantage of gravity: Every time Snowville processes and packages milk, they have to flush the plant's metal pipes with hot water and cleaner. This happens multiple times and generates waste. Thanks to the hill, however, this waste funnels down into a huge tank on the plant's lower level. But then what? You've still got to deal with hundreds of gallons of cleaner-filled wastewater. Turns out that Warren uses cleaners that are different from those used in most dairies. His contain nitrogen and potassium hydroxide.\n\n\"And you know what those are, don't you?\" he asks.\n\nI don't. I don't know 1% of what Warren knows about dairy. Luckily, Warren gets a kick out of teaching it. He has a catchy \"Isn't this so cool?\" vibe about him, even when he's describing chemical cleaners.\n\n\"Fertilizer!\" he booms, nodding toward a nearby field where cows graze. In the distance, an unassuming sprinkler spritzes the grass.\n\nGradually, I get it: That's the wastewater. \"Instead of contaminating the water supply,\" says Warren, already moving on, \"we're nourishing the soil.\"\n\nI'm hustling after him, trying to keep up, as I fit this latest information into my growing picture of the plant. So far I've learned about\n\n\u2022The genetic makeup of the 260 cows that provide Snowville with their milk. They're Jersey, Guernsey, Brown Swiss, Milking Shorthorn, Friesian and Holstein, all intermixed by years of cross-breeding. They're heartier and give healthier milk than the huge Holsteins that are confined on most dairy farms.\n\n\u2022The grazing practices used to feed the herd, which rotate through many different pastures rather than remaining confined to one, so that the cows can rebuild the topsoil instead of depleting it.\n\n\u2022The minimal amount of processing Snowville does to its milk, which leaves the milk's nutrients and flavor intact rather than pasteurizing and homogenizing them out of existence.\n\n\u2022The eco-friendly and taste-preserving cartons in which Snowville packages its milk. And the delivery schedule\u2014from cow to store in less than 48 hours\u2014that keeps its milk so fresh.\n\nAnd that's just what I've managed to scribble down, because Warren is now detailing the spatial layout of the plant. Its core, where the raw milk goes from storage tanks to finished cartons, takes up only 600 square feet, which saves space, energy and money.\n\n\"So what _is_ this about?\" I finally get to ask. I can't imagine putting so much care into creating a milk plant that isn't about milk.\n\n\"This,\" says Warren, \"is about building a canoe. This is about where you go with it and what you do with it.\"\n\nA canoe seems a bizarre metaphor for a milk plant, but I press on. \"What do you want to do with it?\"\n\nWarren doesn't hesitate: \"Save the world.\"\n\nSave the world? With milk? The idea sounds absurd, the sort of thing proclaimed by a zealot or a madman. That absurdity isn't lost on Warren.\n\n\"I'm hoping to be the Che Guevara of dairy,\" he says.\n\nAnd sure enough, on the flipside of his business card, hiding behind that milkmaid, is a smiling Che. There's also a quote: \"The true revolutionary is guided by great feelings of love.\" Che isn't Warren's only guiding spirit; you can add John the Baptist and Neal Cassady.\n\n\"And what about that milkmaid?\" I ask.\n\n\"That's not me,\" says Warren, throwing a friendly, quit-messing-with-me elbow into my chest. \"I'm the Hunter S. Thompson of dairy.\"\n\nWarren grins when he says stuff like this\u2014it's not every dairyman who compares himself to a Marxist revolutionary, a religious prophet, a psychedelic muse and an antiauthoritarian wild man\u2014but his eyes have an edgy gleam. He's not joking. He's got the messianic, near-manic glow of a visionary, because he has a vision: This small milk plant in Pomeroy, Ohio, is going to spark an international movement of social justice.\n\nBut how? That's a fair question. And when you ask it, you see that Warren also possesses the keen intelligence of an engineer. He can make his case coolly and methodically, even when it comes to saving the world:\n\nFirst, take a food as essential to human existence as milk. Then build a dairy plant that provides this food for people in a safe, natural and healthy state. Run that plant in an environmentally sustainable manner. Then integrate that food and that plant into the local community and economy, so that it provides both good jobs and good nourishment for the people it serves. Then make that plant a model, make its design freely available to anyone who wants it, and mentor those who want to emulate it, so that milk plants like Snowville pop up around the country and around the world.\n\nAnd don't stop with dairy. Let the principles that guide Snowville\u2014sustainability, community, harmony and love\u2014catch on and spread. Let them inform how we grow all of our food, how we run all of our businesses. And show people that this is really possible, right now, by starting with one small milk plant in Pomeroy, Ohio. Save the world.\n\nIf that sounds like unattainable vision, so be it. Warren pursues it tirelessly. He regularly works 100-hour weeks at the plant, and when he's not dealing with broken delivery trucks or carton fillers, he's speaking at universities, community centers and city council meetings about the need for sustainability and the dangers of hydraulic fracturing. He's traveled to Italy to represent Ohio as part of the worldwide Slow Food Movement. He's testified at USDA hearings on behalf of small dairies against regulations that unfairly favor mass producers. He's started initiatives for feeding livestock with grain that isn't genetically modified and worked for more honest and clear labeling of dairy products. He's helping to create a food distribution center in Columbus, so he and other producers can pool their resources. In essence, he's working to develop an infrastructure for delivering local food to local people.\n\nAnd yet, amidst all this, he can't resist riffing on the best way to eat an ice cream sandwich, or the mindset of a martial artist (\"You've got to expect to get hit\"), or how to domesticate a wolf.\n\n\"I wouldn't want to control Warren,\" says Victoria Taylor, Warren's wife and partner. Victoria is the plant's co-owner and general manager and she's equally savvy about Snowville's larger vision. She'll quietly turn from giving a group of employees shipping instructions to telling you about the omega acids in milk or the effect of herd grazing on the North American landscape. She struck me as the still point of Snowville's spinning world. After 25 years of marriage, she's come to a conclusion: \"You adapt to Warren.\"\n\nSure enough, once you hear about Warren's family history, you understand that he couldn't be anyone or anyway else. Warren's brother is the president of Daisy Brand, the largest sour cream producer in the world, and his father, a renowned dairy taster, worked in the dairy industry for 35 years. Warren got his dairy tech degree from Ohio State University in 1974 and three years later he was working for Safeway, the largest milk processor in the country. He spent three decades as a dairy engineer and consultant for plants that processed up to 300,000 gallons of milk a day. (Snowville, by contrast, processes 60,000 gallons a month.) He knows every facet of the industry.\n\n\"Cut me and I bleed white,\" he'll say, and you get the sense that dairy and destiny fuse in Warren, that his family made him into who he is.\n\nThat's true, but not how you'd think. For me, what most revealed the man behind the milk wasn't when I heard about Warren's father the dairyman, but Warren's father the Navy pilot. During the Second World War, he led the Medical Evacuation Squadron in the Pacific. Warren told me about one of his father's rescue missions to evacuate sailors who had survived the sinking of the USS _Indianapolis._ The ship was torpedoed by a Japanese submarine after delivering the atomic bombs to an air base on Tinian island. Of the ship's 1,196 crewmen, only 316 survived. On the way to pick them up, Warren's father and the pilots of the six other planes who were flying alongside him learned they were heading into a typhoon. Four of the six turned back, but not his father. And he made it.\n\n\"I realize now, my father didn't expect to live,\" says Warren, his voice going quiet, \"but he didn't turn back.\" He pauses and takes a deep breath. Then, like a bull readying for the charge, he digs at the ground with his foot. \"That's what I told the USDA the last time I was in Washington: I'm not turning back.\"\n\n### MATTERS OF TASTE\n\n### By Barry Estabrook\n\n### From _Tomatoland_\n\n### In _Tomatoland,_ Barry Estabrook exposes how agribusinesses compromise the environment, exploit migrant workers, and obliterate the taste of Florida tomatoes. Estabrook's investigative food journalism has appeared in _Gourmet, The Atlantic, Gastronomica,_ and on his blog politicsoftheplate.com.\n\nIn early 2010, I enjoyed a supermarket experience that I'd never had before. I bought a pretty, stridently red winter tomato that actually tasted like something. It was by no means a great tomato, harboring only hints of the flavor wattage of a vine-ripened August tomato, but it was nonetheless unmistakably a tomato, in taste as well as appearance. As is de rigueur with so-called premium produce nowadays, my purchase, which weighed three-quarters of a pound and cost $3.47, versus 80 cents a pound for its nearby commodity cousins, bore a little sticker with the trademarked name Tasti-Lee. A few days later, I returned to the same store hoping to replenish my tomato larder, but the Tasti-Lees had all been sold. And I was left to ask, what made this tomato so different? How come I had never heard of it? Why don't all supermarket tomatoes taste like it did? And where could I get more?\n\nWith those questions on my mind, I drove to the University of Florida's Gulf Coast Research and Education Center, a curvilinear structure faced with pinkish bricks that rises like a space station from the endless fields of strawberries and tomatoes a half hour's drive southeast of Tampa. There, in an office crammed with all manner of tomato kitsch\u2014coffee mugs, antique labels for packing boxes, framed vintage magazine advertisements for Campbell's Tomato Soup, tomato piggy banks, tomato salt and pepper shakers, and teetering stacks of\u2014who knew\u2014 _The Tomato_ magazine\u2014I met John Warner Scott, a professor of horticultural sciences. Scott, who is known far and wide in the tomato business as \"Jay,\" is one of the most prolific breeders of new tomatoes in the state. Over his three-decade career at the university, he has developed more than thirty varieties, although he doesn't keep track. \"I haven't gone back and counted in a while,\" he told me. Unlike seedsmen who rely on molecular biology, DNA sequencing, and in the case of some crops, genetic modification, Scott is the last of a dying line of old-fashioned plant breeders. His tools are the same ones the great nineteenth-century tomato grower Alexander Livingston used to develop the Paragon: a keen eye, a disciplined palate, and superhuman reserves of patience. Each year, Scott grows several hundred different varieties of tomatoes, called \"parent lines,\" in test plots surrounding the Gulf Coast center. His goal is to find plants with complementary traits\u2014one may have disease resistance but low yields, another high yields but weak immunity\u2014and crossbreed them hoping that some of the offspring will carry the best traits of both parents. Toiling in the hot sun, Scott pulled a floppy sun hat over his close-cropped graying hair and plodded through the rows, notebook in hand, carefully examining each plant and ticking off a mental checklist: How many fruits has it set? Are they big? Do they have cracks? Are their bottoms smooth and rounded, or do they still have scar tissue where the blossoms fell off? What's the color like? \"With some of them, you can just look at the plant and just throw it out,\" he said.\n\nIf a plant passed visual muster, Scott took out his pocket knife and, still standing in the field, lopped off a slice and tasted it. \"Plant breeding is a matter of seeing what's good,\" he said. \"But you can't make any decisions based on one season. You have to grow a variety a lot of times in a lot of environments to see if it's really good.\"\n\nTasti-Lee is not perfect. Its fruits are smaller than commercial growers like. But it is as close as Scott has ever come to finding Tomatoland's Holy Grail\u2014a fruit thick-skinned enough to shrug off the insults of modern agribusiness, but still tender at heart and tasting like a tomato should. . . .\n\nDeveloping a better tomato can take years, and even then, there is no guarantee that it will be picked up by professional growers and have a shot at commercial acceptance. Florida's multimillion-dollar tomato industry is littered with once promising but now forgotten varieties. But in early 2010, after more than a decade of painstaking growing, breeding, and crossbreeding, Tasti-Lee left the rarified confines of academic test plots and rigorously monitored consumer-tasting panels to try to make its way in the competitive hurly-burly of the produce section. If Tasti-Lee lives up to its early promise, Scott will achieve a plant breeder's version of immortality. The rest of us finally will be able to head to the local supermarket any day of the year and bring home a half-decent-tasting tomato.\n\nBut it won't be an inexpensive tomato. Scott developed the Tasti-Lee to provide farmers in his state with a crop that can be planted outside to compete with hydroponic, greenhouse-grown tomatoes, the latest competitive threat to the Florida fresh tomato industry. Beginning from almost nothing in the early 1990s, greenhouse tomatoes expanded from a tiny niche-market novelty mostly imported from Europe to a mainstream produce item. They are now in every supermarket and account for about 10 percent of fresh tomato sales. Although Florida's field-grown slicing tomatoes remain as popular as ever in the food-service industry, sales have declined sharply in supermarkets. With Tasti-Lee, Scott hopes to give growers a baseball-size tomato that packs the same flavor as the popular ping pong ball\u2013size salad tomatoes produced in greenhouses and often sold in clusters on the vine. \"It seems to me that it would be a win-win situation,\" said Scott. \"Consumers tend to be spoiled. They go into the grocery store and they expect to see fresh tomatoes any time of year, even if they grumble about the quality. I want people to buy Tasti-Lees because they like them, not just because they are the only tomatoes there.\"\n\nLike many plant varieties, Tasti-Lee owes its existence to a combination of serendipity and the time-sharpened instincts of a great plant breeder. In Florida, the summer of 1998 was a terrible season for anyone trying to grow a tasty tomato. For some unknown reason\u2014too wet, too cloudy, too hot\u2014Scott's tomato field tests failed to produce fruits with any sweetness. Even tried-and-true varieties that had been sweet during previous years tasted dull. But one morning after tasting fifty varieties, each more bland than the other, Scott spotted a nice-looking tomato called Florida 7907. He picked a fruit, cut off a wedge, and popped it into his mouth. \"Aha!\" he said.\n\nIt was sweet, but Florida had one big flaw that made the variety a nonstarter for commercial production: It was too spherical. Florida growers like their fruits to have defined shoulders and slightly flattened bottoms. And that's only one item on a list of must-haves. Because producers are paid strictly by the pound, plants first and foremost must produce high yields of large, uniform fruit. They have to be able to resist diseases and tolerate extremes of heat and cold. And their tomatoes need to have a long shelf life. Taste enters the equation, if it enters at all, only after all those conditions are met. \"Sometimes I wonder why we even bother with flavor,\" said Scott. \"There is no easy way to breed for taste. It's not like there's one genetic marker that tomatoes must have to taste good,\" he said.\n\nThe structure of a tomato also makes breeding for both taste and toughness a difficult balancing act. The grocery part of a tomato, called locular jelly, has most of the all-important acidity. The pericarp tissue, the walls of a tomato, give it strength and some sweetness, but no acidity. The harder a tomato is, the more bland it is likely to taste. Even if you have a perfect balance of sugars and acids, there are still many obstacles in getting decent-tasting tomatoes from field to consumers' kitchens. Most Florida tomatoes are picked at the so-called mature green stage. Under ideal circumstances, a mature green tomato, reddened by being exposed to ethylene gas, will ripen and develop a measure of taste\u2014not great taste, but something. The problem is that short of cutting one open, there is no definite way to tell a mature green tomato from one that is simply green. Inevitably, some immature tomatoes get picked and they will never develop flavor, although the ethylene will give them the appearance of ripeness. Finally, even if all else goes according to plan, a tomato can lose its taste if exposed to cold temperatures at any time between harvest and being eaten, after which point it can never recover it. Crop specialists even have a scientific term for this process: \"chilling injury.\" Whether it happens in a truck, warehouse, produce section, or home refrigerator, a tomato that is held at temperatures lower than 50 degrees soon becomes a tasteless tomato. For reasons unknown, chilling reduces the fragrant volatile chemicals that are all-important in giving the fruit its distinctive flavor. Unfortunately, keeping tomatoes cool extends their shelf life, too, so the temptation to refrigerate dogs tomatoes every step of their journey to the table. Years of efforts by a plant breeder can be destroyed by a few days in a refrigerator.\n\nScott was also developing a line of what he calls \"ultrafirm\" tomatoes during the same season he happened on the sweet-flavored 7907. Among those he was developing was a tomato called Florida 8059. It was hard and had the right shape. Sensing a match made in heaven, Scott crossbred the sweet but too-spherical 7907 with the firmer 8059, and in the fall of 2002 the first of what was then referred to as Florida 8153 ripened. Scott thought the new hybrid carried the best traits of both parents. At trials conducted by the university, consumers on test panels agreed. Time after time, 8153 beat out other tomatoes. Subsequent chemical analyses showed that the fruit had a desirable balance of sugars, acids, and volatiles. It also had a surprise bonus: Both of its parents possessed what plant breeders call the \"crimson\" gene, which was originally revealed when the pioneering tomato geneticist Charlie Rick crossed a wild _L. chilense_ (a relative of the domestic tomato) with a commonly grown variety. The crimson gene gives 8153 a striking fire-engine red color and an extraordinarily high level of lycopene, a sought-after antioxidant. \"It sounds like magic, doesn't it?\" said Scott. \"It really is, in a way.\"\n\nFlorida 8153 had everything going for it, except for a catchy, appetizing name. Scott christened and trademarked his new baby Tasti-Lee, Lee being the first name of his mother-in-law, a tomato lover who had encouraged and supported his research through the years. \"You hear lots of stories about bad mothers-in-law, I had a great mother-in-law,\" Scott said, a flash of emotion overcoming his usual deadpan. \"She had tasted what was then still just called Florida 8153. She really liked it and encouraged me. Sadly, she fell terminally ill. I went to visit her in the hospital. She was in a coma at that point, but I took in a tomato anyway and showed it to her and told her that I was going to name it after her. I like to think she heard me.\"\n\nFour seed companies lined up to bid for rights from the university to produce and distribute Tasti-Lee seeds. The winner was Bejo Seeds, Inc. A large, family-owned, Dutch firm with offices around the world, Bejo's specialties are cabbage, carrots, and other cool-weather crops. \"We felt that marketing would be a key to Tasti-Lee's success,\" said Scott. \"It seemed like Bejo would be hungry to get into the tomato market and that they would push Tasti-Lee pretty hard.\"\n\nThe job of giving Tasti-Lee that push fell to Greg Styers, Bejo's sales and product development manager for the southeastern United States, who has been known to board airplanes lugging twenty-five-pound boxes of tomatoes as carry-on baggage. \"We had a vision to start with a grassroots movement,\" said Styers. \"We were going to start with roadside growers and chefs. People who were interested in good flavor and good quality. Then we were going to work our way up.\" It didn't turn out as planned. Styers, who was looking for a grower who shared his vision that Tasti-Lee was \"born to be a premium tomato,\" approached Whitworth Farms, which grows vegetables on seven hundred acres near Boca Raton, making it a small player in the Florida tomato business. \"Whitworth was big enough to deal with some large retailers, but small enough that they were willing to take a chance on Tasti-Lee. It was a perfect fit for us,\" said Styers.\n\nOne of Whitworth's customers was Whole Foods Market. Glenn Whitworth, who owns the farm along with his sister and two brothers, approached one of the company's produce buyers. Weeks went by before the buyer would even schedule a meeting with Styers and Whitworth. When they did finally get some time, Styers stopped by a Whole Foods store beforehand and bought one of every tomato on display and added a Tasti-Lee to the mix. On the basis of that impromptu conference room taste test, the buyer agreed to test-market Tasti-Lee. In February 2010,Tasti-Lees began appearing in sixteen Whole Foods stores in Florida. By late March, reorders were coming in faster than Whitworth could grow Tasti-Lees. Later that spring, Whole Foods stores as far north as Washington, DC, began to carry Tasti-Lees, and by the end of the year, other retailers and even a few restaurant chains were expressing interest. \"I think the stars really lined up for Jay when he developed this variety. It truly is remarkable,\" Styers said.\n\nScott, who drawls his carefully chosen words with little inflection and almost no emotion, didn't go that far. \"I stand behind it,\" he said. \"For a full-size tomato, it's better in my opinion than what's out there. Hopefully, it goes.\" If it doesn't, Scott has plenty to keep him busy. He's currently developing heat-tolerant tomatoes, tomatoes with resistance to the virulent leaf-curl virus, and tomatoes that can be grown on the ground and theoretically harvested by machine. And he hasn't given up on flavor. \"In some work we've done, there is this fruity-floral note that adds pique to the sweetness,\" he said. \"We've crossed a big, crimson tomato with that trait into one of Tasti-Lee's parents. The result might have even better flavor.\"\n\n### OLIVES AND LIVES\n\n### By Tom Mueller\n\n### From _Extra Virginity_\n\n### Living in Liguria in an ancient stone farmhouse surrounded by olive groves, writer Tom Mueller found ready inspiration for this groundbreaking book about the international olive oil trade\u2014its frauds, its failures, and the artisanal producers striving to restore honor to the label \"extra virgin.\"\n\nFor the De Carlo family olives mean home, not only because their family tree has intertwined with their groves and with oil-making for the last four centuries, but literally as well; their house is perched atop their mill like the keep of a castle. The impression of defensiveness at casa De Carlo is accentuated by the imposing security wall which rings the property, as well as the surveillance cameras which film everyone who approaches the main gate and project them on screens inside the house. Local producers are periodically held up by armed oil bandits, who drive tanker trucks with high-pressure pumps to siphon oil out of storage silos. \"After a certain hour we don't open the gates,\" Saverio said as we returned to the house for lunch, after touring the De Carlo groves. Francesco and Marina were waiting for us beside a large olive-wood fire, Francesco resting his head in his sister's lap as they ate pickled _cima di Mola_ olives from a small porcelain bowl and tossed the pits onto the coals. (Olive pits are an excellent fuel, and oil-makers often sell their olive pomace\u2014the solid residue from oil extraction, consisting mostly of crushed pits\u2014to electric companies and other industries, to be burned in furnaces.)\n\nGrazia knelt on the hearth beside them, and set several pounds of suckling lamb on a grill over the coals to cook. While the meat sizzled and popped and we watched the little eruptions of flame from each drip of fat, she told a true story of poisoning, blindness, and death, and said she half wished it would happen again.\n\nIn March 1986, she said, hospitals in northwest Italy began to admit dozens of people suffering from acute nausea, lack of coordination, fainting spells, and blurred vision. Twenty-six died, and twenty more went blind. Investigators eventually discovered that each victim had recently drunk a local white wine; several producers, they found, had been raising the alcohol levels of their wines by cutting them with methanol, a highly toxic substance also called wood alcohol. The scandal, and the resulting government crackdown, devastated the Italian wine industry. Consumption plummeted, and hundreds of producers, most of them honest, went bankrupt. Ultimately, however, the crisis radically improved Italian wine-making and forced a generalized shift from quantity to quality.\n\n\"Before the methanol scandal, people around here didn't make wine like this,\" Grazia said, pouring glasses of Rivera Il Falcone 2004, a garnet-red wine made from a local grape varietal, _nero di Troia,_ by a producer at Castel del Monte, a nearby medieval castle. \"And even if they had, nobody would have bought it. Most people just bought their wine in big jugs without labels. You'd see them on tables in restaurants, where they'd been sitting open for days. Most people wouldn't dream of buying a bottle of wine with a label on it.\"\n\nAfter the methanol crisis, consumers grew more particular, and the producers who survived the market consolidation learned to use techniques and technology pioneers by French enologists. \"After the scandal, producers started creating brand names they were proud of and wanted to defend. Only after methanol did people start thinking about what they were buying and drinking, and become willing to pay for the good stuff. And only after methanol did the government get really serious about checking quality, and making sure that the bottle contained just what the label said.\" During the 1990s, dozens of premier Italian wines emerged, and wine became a major export product (wine recently topped $1 billion in annual sales in Italy).\n\nGrazia brought the wine to her lips, then stopped and put it down without tasting it. \"In olive oil, we're where the wine-makers were before methanol,\" she said. \"We're stuck in the dark ages.\" She shook her head disconsolately. \"It would be awful to see my children's livelihood damaged, even destroyed. And I'd certainly never want to see anyone hurt. But sometimes I wish there could be a methanol scandal in olive oil, which would obliterate this corrupt industry completely, and rebuild it in a healthy way. It's been Babylon around here for far too long.\"\n\nOur lunch began with a succession of seasonal vegetables, mostly from the De Carlos' own garden: _lampascioni,_ a small wild hyacinth bulb marinated in oil and vinegar; meaty, densely flavored cherry tomatoes; _puntarelle,_ the tender tips of a local chicory; and flat little artichokes as big around as a pound coin or a quarter, lightly fried. _\"Pugliesi_ eat an incredible amount of vegetables\u2014we're like goats,\" said Francesco, a rangy twenty-four-year-old with a crew cut and large, dark, serious eyes that watch you unblinkingly, though their intensity is softened by a faint, unsarcastic smile that never leaves his lips. He holds a degree in food quality and a diploma in olive oil tasting from the University of Naples and recently launched a De Carlo line of vegetables in extra virgin olive oil: mushrooms, artichokes, peppers, and other produce grown on their lands, as well as green table olives of the _picholine_ and _cima di Mola_ cultivars. \"I introduced them to broaden our product offering so that our facilities would remain active throughout the year,\" he explained. \"But given the sorry state of oil prices nowadays, they're a much higher-margin business and help us stay profitable.\" His modern financial jargon was so different from his father's homespun way of talking about the oil business that I instinctively asked if he and Saverio worked well together.\n\nFrancesco didn't miss a beat. \"No. We argue every day,\" he said. \"Every single day!\" And when the laughter, a shade nervous, subsided, he added, with a quick, testing look at Saverio: \"Disagreeing, sharing different opinions, deciding together the best way forward\u2014that's the best way to collaborate, no?\"\n\nFor all his university training, Francesco clearly shares his father's visceral enthusiasm for olive oil. His earliest memories also concern the family mill\u2014such as the time, as a three-year-old, when he fell asleep in a little nest between sacks of olives, and slept through the increasingly despairing cries of his family as they searched for him amid the whirring blades and grinding wheels.\n\n\"If you'd come a couple of weeks earlier, or later, you'd be eating a completely different meal,\" Francesco said, looping a green ribbon of Arcamone oil over a big bowl of a half-dozen different wild-looking greens, most of which I'd never seen before, and whose names he knew only in the local dialect, not in Italian: _cuolacidd, spunz\u00e1l, sev\u00f3n, cicuredd._ \"We _pugliesi_ are demanding about these things,\" he continued as he stirred the glistening leaves. \"We try to eat only vegetables and fruits that are in season. Many Italians are the same. They prefer fresh things from local gardens to the brown, tired-looking produce in supermarkets, even when local crops cost more. So why don't they buy their oil the same way? Olives are a seasonal fruit, and olive oil is a fresh-pressed fruit juice\u2014it's best shortly after it's made, and goes downhill from there. Why on earth do people buy expensive vegetables like these, and dress them with the cheapest oil they can find?\"\n\nDe Carlo oil flowed for the rest of the meal, gushing over the _burrata,_ a rich curdy cousin of mozzarella, and pooling in the little cups of the _orecchiette_ pasta with _boragine,_ a wild herb. At first I thought the De Carlos were showing off for me, but I soon saw that they used olive oil this way every day, choosing from the four different oils on the table the one that best fit each dish they were dressing. Saverio sloshed so much Tenuta Torre di Mossa, the family's pepperiest oil, over his grilled lamb that the others giggled and pointed. He bobbed his head and smiled happily, the first smile I'd seen from him. \"I've spent my whole life making oil, but I can never eat enough of it. What other job gives you this?\"\n\nHe handed me the oil, and I poured some over my lamb. As if it had catalyzed subtle chemical reactions in the meat, I tasted dense new flavors which I hadn't noticed in my previous, unoiled bites: the rosemary and _santoregia_ Grazia had used to season it, the browned fat, the light charring from the olive-wood grill\u2014each flavor had a new depth and intensity. The meat even felt different, more supple and juicy. This oil wasn't just a condiment, but had entered into the dish.\n\nWhen I observed this, Francesco snorted. \"Try telling that to a chef!\" He explained that he'd recently given an oil-tasting course in Naples to twenty head chefs of prominent restaurants, most of whom had shown the most abject ignorance about olive oil. \"Each of these guys ran a top-flight restaurant, right? Some had Michelin stars. They had highly developed palates for wine and for foods of all kinds. But every last one of them was using a refined olive oil or a cut-rate extra virgin in their kitchens, and even on their tables. They'd been using bad oils so long that they didn't even know what a good oil tasted like.\"\n\nGrazia, who had been silent for some time, spoke with sudden force. \"Then we've got to teach them. The road we've got to follow is _la cultura_ ; educating people about good oil is the only way out of this crisis. Because once someone tries a real extra virgin\u2014an adult or a child, anybody with taste buds\u2014they'll never go back to the fake kind. It's distinctive, complex, the freshest thing you've ever eaten. It makes you realize how rotten the other stuff is, literally _rotten._ But there has to be a first time. Somehow we have to get those first drops of real extra virgin oil into their mouths, to break them free from the habituation to bad oil, and from the brainwashing of advertising. There has to be some good oil left in the world for people to taste.\"\n\nShe stood and went into the kitchen to get dessert, leaving a sudden silence in the room. Everyone seemed to be thinking about what she'd said, and what she'd omitted: that if the economics of oil-making don't change soon, no one will be left to make real extra virgin oil. Not even the De Carlos.\n\n### THIS LITTLE PIGGY WENT TO MARKET\n\n### By Laura R. Zandstra\n\n### From _Memoir Journal_\n\n### Before she became a creative nonfiction writer and oral historian, teenaged Laura Zandstra helped her Indiana farming family sell produce at a weekly farmer's market in Chicago. With impressionistic strokes, she evokes the market day experience as seen from behind the scales and cash register.\n\nChicago at four a.m. smells like two hundred years of dirt and one hundred years of oil and gasoline spilled on pavement and then congealed in the dew before daylight. About that time on Saturday mornings, our 1979 Mercedes truck pulled into the high school parking lot at 29th and King Drive, thirty miles from our Indiana fields, and eight of us would tumble out into the darkness, throw open the rear roll-up door, and set to work unloading our wares.\n\nThe cabbages were in fifty pound boxes that, try as I might, I could not heave off the back of the truck by myself. The beans were in square wooden crates, the green onions in fancy waxed cartons with pictures on them. We bought the green onions from our neighbors, the DeJong brothers. They sold us radishes, too, because we didn't quite have the space or the dirt or the patience to grow them ourselves, as prone to pests and in need of constant looking after as radishes are. The tomatoes, however, were ours, and kept in cardboard that cushioned their soft red flesh. The spinach was ours, too, packed into bushel baskets with lids held in place by wooden slats secured under metal handles.\n\nThe beets, the carrots, the turnips, the corn: load by load, we paced the lot from the rear of the truck to unmarked spots on the pavement, arranging these boxes and bushels on the blacktop where their contents would be displayed once the sun was up and the community emerged. While we worked, a handful of people would stray past, illuminated beneath the streetlights the last ones to sleep and the first ones to wake. They'd wish us good morning, or stop to tilt their heads and watch our efforts.\n\nBrother Al was one of this handful. He always appeared long before darkness had melted from the sky, with hair that looked like an old Brillo pad and gray coveralls that smelled like sweat and dust.\n\n\"You got any butta' beans, Baby?\" he'd ask me.\n\nI'd shrug. \"I don't know yet.\"\n\n\"You don't know yet?\" he'd tease with a grin and a gentle elbow jab.\n\n\"It's too early,\" I'd tell him with a scowl.\n\n\"Too early for what? How old a' you?\"\n\n\"Eleven.\"\n\n\"You just a baby.\"\n\n\"The truck isn't all the way unloaded yet,\" I'd say, mad. \"So it's too early for me to know if there are any lima beans today.\" Then defiantly under my breath: \"I'm not a baby.\"\n\nOur vegetable stand was the most primitive at the South Side farmer's market. My father and his brother had purchased canopies from a man with a fancy welder who lived a few towns over. The man had cut one-inch metal pipe into long pieces that served as beams and short pieces that served as rafters. These made sturdy rectangular frames once fitted into welded steel corners that looked like hollow-limbed spiders. We'd bungee gray tarps over two such frames and then erect them, three medium-length pipe legs on each long side of a rectangle.\n\nBeneath the tarps we balanced old doors on tall, conical bushel baskets that arrived from southern states packed tight with pole beans and purple hull peas. The doors had come from deep in the junk stocks of our farm where they'd congregated for years, gathered two and three at a time from dumps and construction sites and neighbors who were remodeling. There is room on a farm for all manner of almost-garbage\u2014car carcasses and plywood scraps and retired traffic signs\u2014and always the sense that one day, hoarded odds and ends will fall precisely into place in the puzzling mill of farm projects.\n\nThe doors and baskets had found just such fruition, and once they were lined up in long skinny rows, we piled them with produce picked from our fields in the days before. Yellow squash next to zucchini, leaf lettuce next to green peppers, pickles next to tall stalks of dill. The melons arrived on their own flatbed truck and were too heavy to arrange on our rickety doors, so we'd line up from truck to table front, tossing Millionaires and Sangrias, Saticoys, muskmelons, and honeydew one man to the next, arranging them on the pavement until handsome foothills of fruit girded the walkway beneath our gray tarps.\n\nTo the left and around the corner from us in the line-up of vendors was Lyon's Orchards, whose apple cider is still the best I've ever had. Tom Lyon\u2014a dashing football player of a farmer's son with wavy black hair slicked back from his forehead\u2014enjoyed harassing me as much as Brother Al did. He'd throw rotten fruit at me from time to time just to watch me get mad, which I always did in spite of myself.\n\nDon the flower guy was to our right. His sandy blonde hair stuck out in wings beneath his green seed company cap and hung in a thick mustache under his nose. He chain-smoked from open to close, a cigarette dangling between his lips while he handed out change or put bouquet stems in plastic bags so as not to drip mud and water onto paying customers. He gave me leftover bunches of baby's breath at the end of the day.\n\n\"Here comes trouble,\" he'd always say when he saw me approaching.\n\nAt the far end of the parking lot, across a wide expanse of gravel and cracked pavement, were two vegetable growers\u2014our competition. They tried to outsell us by offering cheap produce bought in bulk from California, or so the farmer rumor went. I squinted my eyes in their general direction whenever I had a moment to spare and assumed that they squinted back.\n\nIt was my job to scout out the nature of these competitors to make sure we were on level playing fields. After setup, I'd nose around their fancy, store bought display tables and bright blue tents, memorizing the price per pound of everything I could, and then present the information, along with note of any dubious products, like red peppers in June, to Uncle Butch, my boss. He'd sigh and heave a bit, then say, \"Good work, Laura. We couldn't do this without you,\" and send me off to sell melons.\n\nMelon prices were always set on dollars and quarters, so one could conduct sales and dig change from the green money-apron pockets without the need for plastic bags or proper scales or speed in simple math.\n\n\"Muskmelons are two for a dollar,\" I'd tell those who stopped to sniff at the alligator skin of the fruit.\n\n\"Mush melons?\" people would snarl in disgust.\n\n\"No, muskmelons. They're just like cantaloupe.\"\n\n\"Got any half-price melons?\"\n\n\"Not yet, Brother Al. Come back later and I'll have some set aside for you.\"\n\n\"Come back later?\" he'd chuckle. \"You just a baby, but I'll see if I can't come back after lunch, get me some melons and butta' beans. What about okra? You got any okra today?\"\n\n\"I think I saw some halfway down the table,\" I'd say and point timidly, knowing that if I was wrong, further harassment would ensue.\n\nBrother Al would snort with a smile and pat my shoulder before ambling off into the crowds.\n\nBy the time I turned twelve, I had graduated from melons up to the greens table. I was shocked to learn of its existence. Unbeknownst to me, somewhere hidden deep in its flat acres, our farm produced endless bushel baskets of turnip greens, curly mustard greens, slick mustard greens, collards, and kale. Unlike the rest of the vegetables at our stand, these items had never made an appearance at our dinner table. I'd never even heard of them, and yet they were significant enough to require a sizable display and separate staffing. Fortunately, my lack of knowledge mattered little. The greens went like hot-cakes no matter who was selling them.\n\n\"Now,\" my uncle instructed, joining me briefly on my first day behind the greens table where two bushels of each variety were pushed up against each other in haphazard showcase, \"when one basket is almost empty, you dump the last few handfuls into the other bushel of the same kind and then put a new one up on the table. Got it? I'll come and check on you in a while.\"\n\nWithin moments of him leaving my side, women crowded in to claim five pounds or five bushels, stuffing handfuls of leaves into plastic bags that we bought at discount because they'd been misprinted. \"Tank you!\" smeared in red, four times across white bags engorged with produce. \"Fred's Discont Grocer,\" \"Mike's Meaatts Welcomes You!\"\n\nThe misprints didn't matter either. The women jabbered together and rolled deep laughter over their purchases as they planned for good food and family. Then, while still scrutinizing the spread as if they might have missed a perfect handful of leaves in a bushel basket not yet considered, they'd absent-mindedly hand me their bags to be weighed.\n\n\"Baby, how much I got there?\"\n\nI'd place the bag on top of the scale and then drag my finger right across the price table, which calculated from fifteen cents a pound all the way up to a dollar twenty-five.\n\n\"Four pounds at fifty cents a pound . . .\" I'd figure to myself. \"Two dollars.\"\n\n\"What d'you say, Baby?\"\n\nWe had to pay for our scales to be reckoned and licensed. A man in a nice button-up shirt would come with a briefcase full of weights on the first day of market each summer and examine the front and back panel of the display windows as he gingerly placed two pounds, five pounds, ten pounds in the pan on top. Without fail, he'd renew our machines with an official sticker slapped onto the front. He'd smile, shake hands with us and take our money, gone within moments of appearing, yet the event always caused a nervous stir in my Uncle Butch, which infected me even though I knew that we would never deliberately cheat anyone. My family was raised in the tradition of John Calvin and the Heidelberg Catechism; vigorous moral standards were part and parcel to the ins and outs of days. Short-changing people on their vegetable purchases was almost as bad as rejecting the church, and this religious intentionalism was only enhanced by an existence that hinged on the wiles of the earth.\n\nThe Lord giveth and the Lord taketh away, I knew. There were floods and droughts and there were perfect warm springs and fields made rich from the old swamps of Lake Michigan. There were years with too many locusts, with lightning storms and early frosts that burnt whole fields of tomatoes, but there was always enough food to keep our family far from hunger, so we were in a position to do the giving and not the taking away.\n\nThe scales, though, were prized and pricey possessions to be handled with extreme care. Drop them once and they were ruined, rejected by the man in the button-up shirt, thrown irrevocably out of whack, the glass cracked, the mechanics jarred so that no more than half a pound could be correctly ascertained. Sometimes Uncle Butch would shove a screwdriver into the metal guts of an ailing machine, but the prognosis for such a procedure was generally grim. I was therefore not to pick up the scales when they were loaded on and off the truck, my arms too flimsy to be trusted with the weight and investment of the things.\n\nI was allowed only to take bags from customers, place them on top of the scale, figure the weight, and then state the price. \"Four pounds. Two dollars.\"\n\n\"Four pounds? Gimme 'notha bag. I need more than that. Got the pastor's family comin' over for dinner tomorrow after church! Phew! Pastor loves those curly mustids. You ever try those?\"\n\n\"No.\"\n\n\"Well, why not? Here, try one. Take a bite.\"\n\nI shrugged slightly at the challenge, the picture of nonchalance, then reached into a bushel basket, grabbed hold of a leaf and wiped it off on my money-apron.\n\n\"Little dirt won't hurt you none,\" the woman said, watching with a mischievous grin on her big, brown lips.\n\nI nodded and bit down, grinding the green between my molars for a moment with disinterest, but when the taste spread over my tongue I didn't try to hide my shock and disgust. The woman broke into giggles and those at her elbows looked up to see what was going on.\n\n\"Curly mustid,\" she said by way of explanation as she nodded at me, and then everyone was chuckling.\n\n\"Got a little bite, don't it?\" another one said with a knowing flop of her wrist.\n\n\"Yeah,\" I said, examining the curly mustard as if to gain a better understanding.\n\nI never imagined that a frilly leaf colored such a happy shade of green could have so much heat and pepper coursing through its veins. This was almost as mean as feeding jalape\u00f1os to my younger cousins, but the ladies loved me for my innocence. I was the scrawny farm girl with glasses and freckles and a limp brownish ponytail straggling over ratty work clothes, selling greens, easily coerced into taking big bites of fire leaf. They came back every weekend for years, knowing me as I finished junior high and started high school, knowing me as the girl who sold the mustids and took the food stamps.\n\n\"How you doin', Baby?\" they'd say every week. \"Mm-mm-mm, look at these greens! I need six pounds. You got any butta' beans today?\"\n\nThe markets stretched out across the summer, starting in mid-June when the asparagus and rhubarb and peas were just in, and ending with the last days of October. As the months passed, more and more doors were balanced on bushel baskets and covered with the burgeoning harvest until the frosts set in and the fields went brown. By then, days were so cold that I would wear four pairs of pants to work, making myself sick with all that elastic squeezing at my middle, and five layers of warmth on top so my movements were slow and padded. I despised the taste of coffee, acidic and bitter, but I started drinking it to warm my insides and keep my fingers from freezing on those mornings at the greens table, when I scooped up the leaves myself so the ladies didn't have to get their hands out of their gloves and make them wet and dirty and cold.\n\n## Home Cooking\n\n### HOW TO LIVE WELL\n\n### By Tamar Adler\n\n### From _An Everlasting Meal_\n\n### Tamar Adler's quirky culinary education has included working under such talented writer-chefs as Gabrielle Hamilton, Dan Barber, and Alice Waters. In _How to Live Well,_ Adler shares recipes, kitchen tips, and a compelling philosophy of cooking as an approach to life.\n\n### _Si stava meglio quando si stava peggio._  \nWe were better off when things were worse.  \n\u2014FIFTEENTH-CENTURY TUSCAN SAYING\n\nBeans have always been associated if not with poverty, with the sweating classes. Fava beans, whose slightly bitter flavor is so refreshing it's common to see them being peeled and eaten raw, were called, in ancient Rome, _faba,_ a play on words with _faber,_ the Latin word for \"worker.\" The Roman physician Galen said of beans: \"Legumes are those grains of Demeter that are not used to make bread.\" He then chose them over less wholesome wheat loaves as the staple of Roman gladiators' diets.\n\nMost of us regard beans with suspicion, as we do stale bread and cooking in water. Prejudices are always best dispatched, but not always unfounded. When food is boiled badly, it's fair to turn away from it, and if stale bread isn't cooked with, or toasted, but served dry and harsh, it's awful.\n\nBeyond the indelible stain the poor little things will never shake, the distaste we feel for beans is not unfounded either. Our beans are rarely as good as they can be. They're usually so bad, in fact, that basing an opinion of their merit on prior experience is very much like deciding you don't like Bach after having heard the Goldberg Variations played on kazoo.\n\nI suggest you set your doubt aside, fill a pot with cold water and two cups of dried beans, put it on your counter, and leave it there overnight. You will be on your way toward making beans that taste like that that have fed laborers and fighters for centuries.\n\nYou will also have plowed effortlessly through the hurdle of \"soaking beans,\" a hurdle whose existence and gnarliness is a pure invention of food writers' proclivities for making cooking seem difficult.\n\nThe way to keep bean soaking from getting in the way of your cooking beans is to detach the process from today's hunger and expectations and pour dried beans into a pot and cover them with cold water whenever you think of it. Their needing to stay where they are until being cooked tomorrow won't be a problem, and you'll have soaked your beans.\n\nA lot of bean recipes advise soaking in the refrigerator: beans are vegetables, and warmed too gently in water may think they're being asked to sprout. I soak mine wherever there's room in the kitchen, and they keep their vegetal ambitions well in check for a day.\n\nOnce the sun has set and risen, drain the beans through a colander and cover them by two inches with fresh, cold water. What gets flushed out of the beans on their overnight wallow is what inspires musicality in eaters. Feed their soaking water to your plants, who will digest it more quietly, if you like.\n\nIf you didn't put two cups of beans in a pot of cold water last night, get on the bandwagon today by putting them in a pot, covering them with five inches of water, bringing it to a boil, turning off the heat, and leaving them sitting in hot water, covered, for an hour. Then drain them and cover them with new water. This has the same effect as overnight soaking and is a good alternative.\n\nThe cooks who make the best beans are the ones who hold simplicity in high esteem. Romans and Tuscans value spare eating and living. Both of their legislative histories are peppered with sumptuary laws limiting the length and content of meals, passed whenever their citizens' affection for simple living got flabby.\n\nTuscans, though, make the best beans. They are known in Italy as _mangiafagioli,_ or \"bean eaters.\" Tuscans believe that frugality is next to godliness and give the humblest ingredients their finest treatment. Tuscan cooks are extravagant with good olive oil, pressed from dark trees, and with vegetable scraps and Parmesan rinds, which, along with salt and more of that fine oil, make transcendent pots of beans.\n\nThose odds and ends are as crucial to pots of beans as fresh water. Your pot will benefit from a piece of carrot, whatever is left of a stalk of celery, half an onion or its skin, a clove of garlic, fibrous leek tops. If you must decide what to save for your chicken pot and what for stock and what for beans, save your fennel scraps with pots of beans in mind. I make notes to myself after meals, and there are enough torn pieces of paper attesting that \"The best bean broth has fennel in it!\" for it to have become axiomatic.\n\nYour pot also wants parsley stems, whole sprigs of thyme, and a bay leaf. It can all be tied into a neat bundle in cheesecloth or with kitchen twine, or it can be left bobbing around, as everything in my bean pot always is.\n\nBeans need salt. There is a myth that adding salt to beans keeps them crunchy and unlovable. Not cooking beans for long enough keeps them crunchy, and undersalting them is a leading culprit in their being unlovable. They also need an immoderate, Tuscan amount of olive oil. This is different from adding oil to a boiling pot of water for pasta. Pasta doesn't cook in its water long enough to benefit from the oil, and you use only a small amount of pasta's cooking water to help sauce and noodle get acquainted.\n\nThe liquid in a bean pot becomes broth as beans cook in it just as the water in which you boil a piece of meat does. No ounce of the water that goes into a bean pot should be discarded. Tuscan food is based as much on the broth made by the beans on which Tuscans lavish their affection as on the beans themselves. Harold McGee, who writes about the chemistry of food simply, writes that beans make their own sauce. He is right. Their sauce must be well made and it must be kept.\n\nCooking beans is like boiling a chicken or boiling an egg: only their water boils, and only for a brief second. The rest of their cooking is slow and steady. Light the burner under your beans, and as soon as the pot has come to a boil, turn the heat down to just below a simmer. Gray scum will rise to the top of the pot and gather around the edges. Skim it off and discard it.\n\nThe best instruction I've read for how long to cook beans comes from a collection of recipes called _The Best in American Cooking,_ by Clementine Paddleford. The book instructs to simmer \"until beans have gorged themselves with fat and water and swelled like the fat boy in his prime.\" The description is so perfectly illustrative I don't think anyone should write another word on the subject. I don't know who the fat boy is, but I feel I understand his prime perfectly, and it is what I want for my bean.\n\nAs they cook, beans should look like they're bathing. Their tops should stay under the surface of the liquid, or they will get cracked and leathery, and they shouldn't ever be in so much water that they're swimming. Taste their broth as they cook to make sure it is well seasoned. It should taste not like the pleasant seawater of the pasta pot, but like a sauce or soup.\n\nThe second good piece of advice from the same book is in one of its recipes for black beans: \"Soak beans overnight; drain. Put in pot, cover with water. Add onion, celery, carrot, parsley, salt, and pepper. Simmer until bean skins burst when blown upon, about three hours.\" This is the only recipe I've ever read that takes the doneness of beans as seriously as it should be taken: a cooked bean is so tender that the mere flutter of your breath should disturb its skin right off.\n\nBeans are done when they are velvety to their absolute middles. You should feel, as soon as you taste one, as though you want to eat another. The whole pot is only ready when five beans meet that description. If one doesn't, let the beans keep cooking. (My \"five bean\" method is good, but ever since reading Mrs. Paddleford's book, I feel like a brute when I practice it, and am quite intent, moving forward, on whistling the skin off a bean.)\n\nCool and store your beans in their broth. The exchange of goodness between bean and broth will continue as long as the two are left together, and the broth helps the beans stay tender through chilling, freezing, and warming up again.\n\nThose are instructions for cooking all beans. The only exception to these instructions is lentils. On the timescale of beans, lentils are instant. They do not need to be soaked and take only a half hour to cook. It is smart to keep cans of cooked beans around, but there is no reason to buy precooked lentils. Cooking them from dried does not take any more planning than putting a pot of water on the stove, lighting a burner, and opening a plastic bag.\n\nOther than how good they are when they're cooked well, and how many good meals you can get out of them, beans are economical because they're a cheap habit. I keep an assortment of different beans stored separately in little glass jars. I have jars of little green French flageolets, marsh-brown cranberry beans, inky black beans, turtle beans, speckled Jacob's cattle beans, and plain burnt umber kidneys.\n\nPots of beans have an admirable, long-term perspective on eating. It's the same to them whether you eat them tonight or in three days. Beans get better over a few days' sitting, gorged and swelled, like happy fat boy. Any longer and you should freeze them, but they'll thaw ungrudgingly when you want them back.\n\nA bean pot has a lot of meals in it, and you've already done much of the cooking you need to for many. A bowl _of pasta e fagioli_ is a pot of boiling water away. Bring a few cups of beans and broth to a simmer in a deep pan or pot along with the rind from a piece of Parmesan. Smash the beans with a spoon as they warm. Cook a short pasta like ditalini or orecchiette until it is still quite firm. When the pasta is nearly done, remove the cheese rind from the beans and scoop the pasta into the bean pot to finish cooking. Serve drizzled with olive oil, and top with freshly cracked black pepper and freshly grated Parmesan.\n\nSimple and delicious beans and rice also only requires that you boil a pot of water and add rice. Warm your beans in their broth until they're very hot, make rice, and ladle the beans on top. Or, if it's spring, cook halved little white turnips with their long greens still attached, or English or snap peas in butter and bean broth or water, and cut little wedges of artichokes and cook them in olive oil and butter. When everything is tender, combine it in one big pan, add beans, a lot of broth, and a big handful of whatever soft herbs you have\u2014chervil, chives, mint, fennel fronds, celery leaves\u2014and ladle the bright, springy stew over rice. If you don't want to make rice, add a little extra butter and herbs to the vegetables and beans and serve it over toast.\n\nA deeply comforting supper for one or two is beans and egg. Warm cooked beans in a little pan. Add saut\u00e9ed kale, or roasted squash, or a little bit of roasted tomato, or add nothing at all. Crack an egg or two onto the beans, cover the pan, and cook. If you have stale bread, put a toasted piece, rubbed with garlic, in each bowl. Spoon the beans and egg over the toast, salt each egg, grind it with fresh black pepper, drizzle the beans and egg copiously with olive oil, grate them thickly with Parmesan, and dine like a Roman plebeian, or a Tuscan pauper, prince, or pope.\n\nCassoulet is a bean dish from southern France, where austerity is not considered next to godliness. If you can tell such things from what people eat, for Toulousians, pork, goose, and duck, all slow-cooked in fat, occupy that station.\n\nTraditional cassoulet contains all three, plus copious quantities of fat, pork skin, and a great quantity of beans. If they were lingering, any vestigial associations between bean meals and deprivation should be erased by the very existence of cassoulet. To make an authentic one, follow any of a million good recipes. They're involved but worth the trouble.\n\nOr make a simpler, utterly satisfying version by cooking a mixture of finely chopped onion, carrot, and celery, called mirepoix, in olive oil, browning a small, garlicky fresh sausage per person, spooning beans and mirepoix into a baking dish big enough to fit them happily, and nestling the sausages among the beans. Add bean broth to come up just halfway and put it in a 300-degree oven.\n\nIt takes about one hour for the sausages to cook through at low heat, which gives them time to get tender and for the beans to sip up some of their juices. Take the dish out when it's bubbly and the sausages are cooked, scatter the top heavily with toasted breadcrumbs, then put it under the broiler for a few minutes, until the top is crisp and brown.\n\nThere are similar dishes, made of mostly beans with some meat, in every bean-eating cuisine in the world. They range from franks and beans to black-eyed peas and ham, from chili to the majestic Brazilian _feijoada. The_ principle of all is the same and the principle is good.\n\nIf there's already meat on the table, or you can go without, skip the sausages and ladle beans an inch or two deep in a small ceramic roasting pan and turn them into a rustic, herby French bean gratin. Cook mirepoix as above and mix it into the beans. Bake the gratin in the oven until it begins to bubble. Mix a big handful of any combination of chopped parsley, rosemary, and safe into toasted breadcrumbs, top the gratin thickly, and let it cook until the top is quite brown.\n\nThe world of bean soups is populous. Its population is for the most part exemplary. If you'd like to make the most straightforward one, put more broth than beans in a pot and heat it up. For the second most straightforward one, pur\u00e9e the mixture with a little olive oil and a squeeze of lemon.\n\nMinestrone is much more than a bean soup; it is the complete expression of the bean's generosity, its raison d'\u00eatre. Minestrone underlines all sensible cooking practices. Like the other great Tuscan soup, _ribollita,_ minestrone is a beacon. If you have the ingredients to make either one of those soups, two of which are beans and their broth, it means you're cooking steadily, buying good ingredients, and saving the parts of them you don't cook immediately to cook later.\n\nMinestrone is a precisely seasonal soup: it should reflect the season inside and outside your kitchen at all times. The beans you have cooked will always be at its center, but the rest will change throughout the year. In the winter, it will be chock-full of beans and pasta and thick enough to stand a spoon in. In spring, you will leave out the dark greens and include English peas and new onions; in summer, include the first slim green beans and basil, and little zucchini and ripe tomatoes, cut into cubes.\n\nMinestrone is the perfect food. I advise eating it for as many meals as you can bear or that number plus one.\n\n##### **_Minestrone_**\n\n1 cup diced onion, carrot, celery, leek, fennel\n\n3 cloves garlic, sliced\n\n\u00bd cup olive oil\n\na small pinch of chile flakes\n\nthe end of a piece of cured meat or hard salami, diced\n\n1 cup any combination parsley, rhyme, marjoram, basil leaves\n\n2 to 3 cups roughly chopped any combination kale, collard greens, Swiss chard, spinach, mustard greens, dandelion greens, broccoli raab, escarole, cabbage (cooked or raw), any stems and leaves, ribs, and cores, cooked or raw\n\n\u00bd cup whole tomatoes, well chopped, or drained canned tomatoes\n\noptional: \u00bd to 1 cup chopped root vegetables (if they are there and need to be cooked, or cooked and need to be eaten)\n\n6 cups cooked beans\n\na Parmesan rind\n\n8 cups any combination bean broth, stock, and liquid from cans of tomatoes\n\n1 cup small pasta such as orecchiette, little tubes, or small penne\n\npesto, olive tapenade, fresh ricotta, or parsley for garnish\n\nCook the onion, carrot, celery, leek, fennel, and garlic in the olive oil until tender in a big pot. Add the chile flakes and any cured meat. Stir to combine. Add the herbs, greens, tomatoes, root vegetables, beans, and cheese rind, crushing the tomatoes against the side of the pot. Add liquid to cover. Simmer for 45 to 60 minutes, until everything has agreed to become minestrone. Just before you eat the soup, cook the pasta in a pot of salted, boiling water, only enough for the soup you're planning to eat that week, and add it to the week's soup. If you freeze minestrone, cook new pasta whenever you eat the minestrone you've frozen.\n\nGarnish with pesto or olive tapenade, or a big dollop of fresh ricotta, or simply parsley.\n\nI once lived with a Tuscan in a house in San Francisco. I would cook a pot of beans weekly, and our bean meals followed a regular pattern. The cooked beans would sit in their broth for a half hour, contenting themselves with their last swallows of olive oil and herbs. When my Tuscan decided their time was up, he would stand ceremoniously, clear his throat, slice bread, open wine, and put olive oil on the table.\n\nThen we would eat just beans and bread, and we would drink wine. I would do it all happily, he intently, glowing with genetically imprinted joy at his great fortune to be sitting there, eating beans, beans, beans.\n\nThere are a good number of bean-loving Americans who agree that cooked beans need no further fussing, and eat beans, on their own, as whole meals, as Tuscans do. In New Mexico, big pots of beans are cooked studded with pork and served for dinner. They're called _borrachos_ and eaten plain and hot with an accompanying stack of warm corn tortillas and beer. In Texas, the same beans, cooked the same way, are _called frijoles_ and are eaten plain and hot with plenty of corn tortillas and beer. In the South, you can still get bowls of black-eyed peas or crowder peas accompanied by chopped scallions and watermelon pickles and pepper vinegar to eat with sliced white bread and beer.\n\nTuscans may treat dried beans with reverence, but it is a fresh bean they worship. When you know the taste of a fresh bean, you taste in dried ones the invisible mark all true loves bear: a memory of what it was we first fell in love with. Fall in love with a fresh bean, and you will stay in love with a dried one.\n\nFresh beans are in season in the summer, and come in as many shapes and colors as you can imagine. To shell fresh beans, practice a technique a friend calls \"the twist and tickle\": twist a bean's ends, one in each direction, and then, once its seam opens, tickle its beans into a bowl. This won't work for fava beans. Their pods' insides are sticky, and if you tickle them, they tickle back.\n\nI usually use up my bean broth in minestrone or a bowl of pasta, or warm it up and make some odd, delicious thing of stale bread and whatever else is around, and probably cook an egg on it in the end. But I've been served plain bean broth twice, and been inspired to serve it myself several times.\n\nThe first time it was served to me was at a convent in Oaxaca, Mexico. The broth was the first course of a meal so pure and simple, the air seemed to thin as we ate. The soup was smooth and golden and tasted of grass. After it there were five tiny, glossy meatballs, on a pool of serene, dark amber tomato sauce. It was a simple meal, and it was calming.\n\nThe second time was in a weathered dining room in Turin, Italy. The broth was ladled out of a ceramic jug in which beans had cooked in the fireplace. The beans themselves were served separately with torn kerchiefs of fresh pasta, but the beanless soup was hot and each spoonful told the story of the beans' slow bubbling amid herbs and garlic.\n\nIf you decide to serve bean broth, I have only the advice of a guest to whom I served it once. He thanked me for the meal after saying good evening, and suggested that the next time I might serve it hot. If you serve bean broth as a soup, do remember what I forgot and was too proud to rectify during dinner. If you are ladling it from anywhere other than an earthenware jug in a fireplace, the broth will have cooled as it sat and needs to be ladled into its own pot and heated up again before being served.\n\nThe writer Waverley Root did a thorough survey of Italian food, top to bottom, in his book _The Food of Italy._ He was deeply enamored of the noble Tuscan and insisted that the Tuscan obsession with frugality was nothing but \"finesse.\"\n\nI cannot associate the word _finesse_ with bean cookery. It doesn't take finesse, but dried beans, good olive oil, a big pot, and time to do it well, and it takes only common sense to appreciate.\n\nBut there is great dignity in allowing oneself to keep clear about what is good, and it is what I think of when I hear the term \"good taste.\"Whether things were ever simpler than they are now, or better if they were, we can't know. We do know that people have always found ways to eat and live well, whether on boiling water or bread or beans, and that some of our best eating hasn't been our most foreign or expensive or elaborate, but quite plain and quite familiar. And knowing that is probably the best way to cook, and certainly the best way to live.\n\n### STILL LIFE WITH MAYONNAISE\n\n### By Greg Atkinson\n\n### From _At the Kitchen Table_\n\n### Though chef-writer Greg Atkinson earned his stars in high-end restaurant kitchens, both in Europe and in the Seattle area, his meditative essays on food and cooking are less about dazzling technique than about the quiet, honest rhythms of home cooking.\n\n### \"When she picked up her lunch the bag felt very light. She reached inside and there was only crumpled paper. They had taken her tomato sandwich.\"\n\n\u2014LOUISE FITZHUGH, FROM _Harriet the Spy_\n\nSince I am both a chef and a writer, I am sometimes compelled to contemplate what cooking and writing have in common. What draws me to both pursuits is the simple joy I find in making something, and I have often said that baking a cake or writing a story satisfies the same impulse. I believe that this creative impulse is a basic human need. We all like to make things. And since I am not particularly good with power tools, I don't make houses.\n\nBut among creative outlets, cooking and writing are unique in that both endeavors produce something that ultimately becomes a part of whoever partakes in them. If I cook a meal and someone eats it, and if everything proceeds as it should, then something in that food will become a part of that person. If I read something and internalize that dialogue, then the words on the page will be incorporated into my own thoughts. Ideas expressed on the page will be reformulated in my mind into thoughts of my own.\n\nIf I write a recipe and you make it, then we are sharing both the words and the dish that results from them. Of course, you'll change the recipe. Of course, you'll hear the words differently in your head than I would in mine, but a connection is made nevertheless, and that connection is what writing recipes is all about.\n\nWhen she was compiling the recipes that would eventually become _Mastering the Art of French Cooking,_ Julia Child was living in France with her husband, Paul, who worked for the Office of Strategic Services (OSS), the agency that would eventually become the CIA. So it's not surprising that she maintained strict security about her recipes.\n\n\"Perhaps it was my old OSS training kicking in, or just my natural protectiveness,\" wrote Julia in her memoir, _My Life in France._ \"But,\" she wrote in a letter to her sister, \"the form we think is new, and certainly some of our explanations, such as that on our beloved mayonnaise, are personal discoveries.\" And so she sandwiched each recipe between pink sheets of paper on which she wrote \"Confidential . . . to be kept under lock and key and never mentioned.\"\n\nSince I learned to make mayonnaise at a very early age, I never thought of the technique as particularly secret. My mother and her seven siblings all learned it from their mother, and they in turn taught it to any members of my generation willing to learn. In our family, dishes like potato salad and Waldorf salad just had to be made with homemade mayonnaise. But some people feel just as strongly about certain brands of store-bought mayonnaise.\n\nThe novelist Tom Robbins is quite devoted to Best Foods\u2013brand mayonnaise. \"A lot of people in my hometown are loyal to Duke's,\" he says, \"but I like Best Foods, which is the same thing as Hellmann's in the South.\" Robbins hails from Blowing Rock, North Carolina, and Duke's is made in Greenville, South Carolina. Hellmann's, which originated in New York City, was purchased by California-based Best Foods in 1932, and the two brands utilize the same formula and market it in similar packaging on their respective coasts.\n\nIn his 2003 novel, _Villa Incognito,_ Robbins waxes poetic about mayonnaise. \"Yellow as summer sunlight,\" he writes, \"soft as young thighs, smooth as a Baptist preacher's rant, falsely innocent as a magician's handkerchief, mayonnaise will cloak a lettuce leaf, some shreds of cabbage, a few hunks of cold potato in the simplest splendor, restyling their dull character, making them lively and attractive again.\" The rave prompted many of his fans to start sending Robbins samples of their favorite brands of mayonnaise.\n\n\"There are some surprisingly good Mexican brands,\" says Robbins, \"and the Japanese make extraordinary mayonnaise. I think I have about twenty-three brands in my refrigerator right now.\"\n\nWhen Tom's wife, Alexa, invited my wife, Betsy, and me up to their place in La Conner for a private mayonnaise tasting, we hit the road with a few jars and bottles of our favorite brands. I also had, secreted away in a canvas shopping bag, a wire whisk, a deep mixing bowl, a fresh egg, a bottle of organic canola oil, some white balsamic vinegar, and a bottle of good Dijon mustard. It occurred to me that Tom and Alexa might like to learn how to make their own mayonnaise, and I wanted to see how the homemade stuff stood up in a taste test with the commercial brands, especially Robbins' beloved Best Foods.\n\nBut our evaluation of mayonnaises at Chez Robbins involved more than just the condiment itself. Mayonnaise is just one of the essential components of Robbins' favorite food, the tomato sandwich, a culinary delight he celebrated in his book of essays, _Wild Ducks Flying Backward._ In addition to various musings and critiques, the collection of stories and poems includes a piece called \"Till Lunch Do Us Part,\" in which Robbins answers the age-old question, \"What would you have for your last meal?\" with an eloquent treatise on the tomato sandwich and its essential components, soft white bread and Best Foods mayonnaise. \"But the mayonnaise would have to be the right mayo,\" Robbins reminds us. \"The bread would have to be the correct bread. I don't want to leave the world on an inferior tomato sandwich.\"\n\nSo along with the various jars, tubes, and squeeze bottles of mayonnaise set out for our consideration, Alexa had acquired a soft and wonderful commercial white bread and several perfectly ripe red tomatoes. Then I pulled out my bag of tricks and went to work. But when I set about making a batch of homemade mayonnaise so that we could compare it to the store-bought stuff, Robbins did not appear to be interested. In fact, he seemed to deliberately avoid getting too close.\n\n\"I was occasionally watching you out of the corner of my eye,\" said Robbins later, \"because I did find it interesting. But I didn't want to see exactly how it's made because I kind of like the idea of it being a mystery. I have been eating mayo for sixty years, and until ten years ago, I didn't even know what the ingredients are. I preferred to think of it as some kind of substance dug out of an underground cave in the French Alps.\n\n\"Socrates said, 'the unexamined life is not worth living,' but Oedipus Rex and I are not so sure. I like the mystery. I think Oedipus might have had a long and happy marriage with his mother if he hadn't found out the truth.\n\n\"I had a 1969 Mercury Montego, and in two hundred thousand miles, the head was never off the engine. I attribute that to the fact that I never once looked under the hood. I thought there was a ball of mystic light that kept the motor running.\n\n\"Beneath that silliness is a propensity for mystery. Every great work of art, whether it's a painting or a film, has an element of mystery. Mayonnaise is not a work of art, but it is the food of the gods; it is ambrosia.\n\n\"I have been quoted as saying that I don't know how to write a novel,\" he said, \"and that was construed as a confession of incompetence. But that's not what I was saying. I'm saying I don't have a formula; I don't have a recipe for a novel.\" Rather, for Robbins, the creative process is something of a mystery. \"I used to cook quite a bit, too,\" he said. \"But I didn't use recipes. When I cooked, I cooked from vibration.\"\n\nI like the idea of this well enough, and even though I write recipes for a living, I almost always cook without them, feeling my way from one step to the next. First this happens, then that happens. While the onions soften, I'm cutting the celery, and on a back burner, the rice is simmering away. But eventually, my left brain kicks in and I start to codify things because I want to share them. How much olive oil did I swirl into the pan? Was that a medium onion or a large one? Was it chopped or sliced? I like the geometric proof-like formula of a recipe, and I feel that if the precision of writing it down doesn't get in the way of the thing, it can be like an incantation, a magic formula for transforming a bunch of ingredients into something completely unlike its component parts. Mayonnaise is, after all, nothing like eggs and oil.\n\nMaking a recipe is not unlike making a sandwich. There is a formula, and when it is followed, real transformation occurs. That is magical.\n\n##### **_Homemade Mayonnaise_**\n\n_Homemade mayonnaise is not only easy to make, it's also an exercise in practical magic. The end result is definitely greater than the sum of its parts. Many recipes, including some of mine, call for a food processor. But for a small batch of the stuff, especially someone's first batch, hand whisking is better. It helps to have a second pair of hands; one person handles the whisk and the bowl while the other person slowly dribbles in the oil. If white balsamic vinegar is not available, use white wine vinegar with a teaspoon of sugar._\n\n_Makes about 2 cups_\n\n1 egg\n\n1 tablespoon white balsamic vinegar\n\n1 tablespoon Dijon mustard\n\n\u00be teaspoon fine sea salt\n\n1\u00bd cups canola oil, preferably organic\n\nWhisk the egg in a medium mixing bowl with the vinegar, mustard, and salt until the mixture is very smooth, almost fluffy. Whisk for at least 1 full minute before adding any oil in order to set a good foundation.\n\nWhisk in a few drops of oil. Then, whisking all the while, build to a very slow but steady stream until all the oil is incorporated. As the sauce comes together to make a stable emulsion, the last of the oil can be added somewhat more steadily than the first few tentative dribbles.\n\n### THE FRIED CHICKEN EVANGELIST\n\n### By Lorraine Eaton\n\n### From Leite's Culinaria\n\n### Covering the local food scene for southeast Virginia's _The Virginian-Pilot_ newspaper, columnist-blogger Lorraine Eaton delights in the characters and culture of Southern cooking\u2014like Mississippi-born fried-chicken master Sydney Meers.\n\nI was raised in the South in a home that never knew a grease-splattered stove.\n\nMy parents were displaced New Yorkers. So despite my growing up in a region where everything is fried\u2014tomatoes, bologna, okra, pies, and, at the State Fair of Virginia, even Pepsi\u2014I never learned the fine art of frying\u2014especially how to make fried chicken.\n\nOver the years, this bothered me some, sort of like a loose pot handle that was in need of fixing yet easily ignored. Now, though, my job title is Staff Epicure for Virginia's largest newspaper. Readers look to me for advice on how to dry-age steaks, open oysters, roast goose. What would they think if they knew of my fear of frying?\n\nWhen I reached the half-century mark\u2014an occasion for celebration as much as assessment\u2014I figured the time had come to correct this culinary shortcoming. I set my sights on how to make Southern fried chicken and sought professional help from Sydney Meers, a Mississippi native who owns Stove, a quaint and quirky restaurant in Portsmouth, the Virginia town where I was raised. Every neo-Southern meal at this 32-seater comes with a side of Syd, who intermittently cooks behind the tiled half-wall of his open kitchen and repairs to the dining room to sip whiskey, neat, alongside dinner guests.\n\nSyd grew up in Senatobia, Mississippi. He spent many a day hanging out in the restaurant kitchen run by his Grandma Winnie Lee Johnson. There were also many mornings spent tending her half-acre garden, pulling weeds, and tamping down leaves between rows, until she'd say, \"Let's go make some lunch.\" Then they'd cook. She shared all that she knew\u2014all manner of pies, pork, and fried chicken\u2014and encouraged Syd to do the same. \"It's part of your heritage,\" she'd say. \"You have to pass it on.\"\n\nThat notion is now a part of him. Syd is an evangelical cook, spreading the gospel of Southern fare by way of his menu and occasional cooking classes. \"To say something is a secret recipe is bullshit,\" he once told me.\n\nSo when I called, Syd graciously agreed to bring me into the fold of folks who fry\u2014with Grandmother Winnie Lee's recipe, no less.\n\nTwo days later, Syd hands me an apron. We're standing in front of an eight-burner gas stove in his speck of a kitchen, a honey-colored chopping block before us and pots and pans dangling above us. Metal shelving units that are stacked to the ceiling practically spill their ingredients and equipment, they're so crammed. Classical music plays from a radio that's hidden somewhere, maybe on one of those shelves.\n\nSyd inspects the organic, free-range bird that I proffer for my lesson in fried chicken.\n\n\"Now, you've done real good, Low-raine,\" Syd says in a voice seasoned with the South. \"This is beautiful.\"\n\nOrganic is fine and dandy, he explains, but that alone doesn't deem the bird worthy. It has to also be free-range and raised without hormones, just like the ones that ran around his grandmother's backyard when he was young, eating bugs and grasses, before Winnie Lee wrung their necks.\n\nSyd makes a cut between the leg and wing with a flick of his knife. He notes that the drumette is slender, not bulging. Proof of the absence of hormones. I may not know how to fry, but I do know how to cut up a chicken. So when Syd flattens the breasts of the bird against the cutting board with his hand, I'm surprised to see that instead of slicing out the backbone, he leaves it attached to one breast.\n\n\"Sometimes grandmother would fry it up separate,\" he says. \"Now most people just cut it out, but that's the choicest piece. The bone holds so much flavor.\" Huh? The things you learn.\n\nSyd has already prepared the brine, a critical element, he says, that imparts flavor and extracts bacteria and toxins. It sits in a stockpot beside us, a simple solution of salty water with a sprinkling of crushed bay leaves and fresh rosemary sprigs from his garden.\n\nI submerge the raw meat in the brine. Syd sets his spattered red kitchen timer for 30 minutes. We slide into his tiny eight-seat bar which he calls The Cougar Lounge. Staring down at us from the wall are mounted game\u2014buffalo, wildebeest, and antelope\u2014that came to him by way of a friend of a friend with an IRS problem of which Syd will say no more. He opens a split of champagne and pours us each a glass.\n\nSyd opened his first restaurant, The Calico Cat, in Senatobia. After leaving Mississippi, he joined the Air Force and attended culinary school on the GI bill. He's owned several popular restaurants in the Tidewater region along Virginia's southern coast. His \"Cowboy Syd's Sextuple Truffle Tart with Bittersweet Chocolate Whiskey Cream and Sensuously Sensational Chocolate Espresso Ice Cream\" is included in the James Beard award-winning cookbook \"Death by Chocolate.\"\n\nHeady stuff. But sitting in the bar, waiting for the bird to brine, Syd says it all goes back to his grandmother.\n\nAs a child, he sat atop sacks of flour in her restaurant watching the line cooks while his mother waited tables. He takes a picture from the wall and hands it to me. It's a faded snapshot, circa 1955, of the diner-ish place. The sign out front says \"Johnson Cafe\u2014Bar B Q Steaks.\" A second sign, emblazoned with \"air-conditioned,\" hangs over the door, a beacon to the hot and weary. A man-sized ice cream cone stands at the entrance.\n\n\"Winnie Lee,\" says Syd, looking at the picture. \"What a girl she was.\"\n\nThe timer dings.\n\nBack in the kitchen, Syd removes a hotel pan from a shelf and dumps three cups of flour and one cup of cornmeal into it. This is the start of his grandmother's dredge. Syd uses his hands to mix it up, cautioning me to rely on all-purpose flour because cake flour will absorb too much moisture and turn to clumps. His grandmother sifted her flour, but because most come pre-sifted these days there's no need to do so unless it's humid. It's always humid down South.\n\n\"We like cornmeal,\" he adds, \"because of the crunch factor and because it helps hold the flour on the skin.\" He says the slightly coarse texture of stone-ground works best.\n\n\"Here's the way you've got to do the salt,\" Syd says. He sprinkles four pinches into flour and cornmeal and offers me a taste. The salt registers in specks on my tongue.\n\n\"Not enough?\" I venture a guess.\n\nSyd agrees. He adds two more pinches and mixes it in. I take another taste. A more uniform salt sensation spreads across my palate.\n\n\"Just right,\" Syd says.\n\nThe pepper that Syd sprinkles into the pan was gleaned from the bottom of the pepper grinders in the dining room. Shrewd restaurateurs don't waste a thing. I place a smidgen of the dredge on my tongue. There's just a whisper of heat. I'm surprised that Syd says the seasoning is correct given that it's much less pronounced than the salt. I voice my concern.\n\n\"Girl, that's because pepper holds up better during cooking than the salt,\" he says. \"The salt will dissolve.\"\n\nOh.\n\nIt's time. I remove the chicken from the brine, flicking off any herbs that stick to the skin. I catch the faint aroma of rosemary and bay as I roll a drumstick in the dredge.\n\n\"No, stop doin' that,\" Syd says, taking over. He covers the meat with handfuls of flour, flipping and flouring and flipping and flouring the drumstick until it's completely coated. \"Handle the flour, not the bird.\"\n\nI can't recall what kind of oil I used during my past frying foibles, but I know it wasn't what Syd pulls out of the fridge: a clear plastic tub containing lard that's slightly off-white and as smooth as buttercream icing. At the bottom of the tub is a glistening brown gel\u00e9e\u2014drippings from braised pork bellies, a staple on Syd's dinner menu.\n\nIn this kitchen, the contents of that tub are considered gold, all but guaranteeing chicken in the same league as Winnie Lee's. Settle for nothing less, Syd says. If you don't have lard, fry up about five pounds of bacon and use the drippings. If all you have is Crisco, don't fry chicken.\n\nAnother nonnegotiable item: Syd's decades-old cast-iron skillet. Black as a cat, the pan has been wiped clean but never washed, same as Winnie Lee's. She figured the difference was her skillet, which held all the seasonings of every food she ever cooked in it. No secret. Just the soul of the pan.\n\nSyd ignites a gas burner and adjusts the flame to medium-high. The lard melts into a pool about 1\/2 inch deep. When it begins to bubble, the pitch becomes more a gurgle than a sizzle.\n\n\"They now have fancy thermometers, but this is the way you fry chicken,\" Syd says, noting the sound. \"You've got to listen.\"\n\nThe throaty gurgle signals that the lard is hot enough to fry. Using tongs, we quickly nestle the chicken in the skillet, skin side up, careful to handle only the ends of the bones so we don't mar the coating or inadvertently squeeze out any moisture. The gurgling continues. Soon the meat is rimmed with a frenzy of fine bubbles, which Syd attributes to moisture from the bird. Only when these tiny globules create a goldenrod outline around each piece is it time to turn the chicken.\n\nWe take turns carefully turning the fried chicken with tongs. Grease spits and snipes at us, but Syd nixes my plea to use a spatter screen, saying it would trap moisture in the pan and turn the chicken soggy. When the pieces turn a shimmering golden brown, we tong them from the pan and set them on paper towels to rest for a spell. It's by sight, not sound, that you know when fried chicken is done.\n\nA few long minutes later we sit in the empty bar, white linen napkins in hand, the gorgeous platter of browned chicken before us.\n\nI bite into a thigh, piercing the perfectly crisped skin. It's juicy but not greasy. I detect a hint of herb from the brine. Syd selects a breast\u2014the one with the backbone attached\u2014and proclaims it just like Winnie Lee's. After we finish, he packs up the remaining thigh and breast for me and offers one last morsel of wisdom.\n\n\"You can do this, girl,\" he says. \"The more you fail, the better you get. That's the fun of cooking.\"\n\nI ask if he ever failed at channeling his grandmother's cooking. He raises an eyebrow. \"Everything I've tried, I got it right the first time right on the spot,\" he says, without hesitation. \"I think it might be a gene.\"\n\n### LASAGNA BOLOGNESE\n\n### By Deb Perelman\n\n### From Smitten Kitchen.com\n\n### The runaway success of Deb Perelman's chatty blog Smitten Kitchen\u2014featuring detailed photos of her cooking adventures in a tiny New York City apartment\u2014won her every food blogger's dream: a book contract. _The Smitten Kitchen Cookbook_ is due out in October _2012._\n\nThis, this is my culinary Mount Everest. This twenty-layer striation of noodles, ragu, b\u00e9chamel and cheese, repeated four times and then some, took me more than five years to conquer. To be honest, six years ago I didn't know what it was. Sure, I had heard of lasagna but I wasn't terribly fond of it because I don't much care for the texture of ricotta once it has baked. (Ricotta, I'd argue, is best rich, fresh and cold on toast.) But I was galloping through a post on an Italian food blog and I stumbled upon a parenthesised side-thought that stopped me dead in my tracks. It said something along the lines of \"I don't know whose idea it was to put ricotta in lasagna but . . . shudder.\" And I thought, but wait! _What's supposed to go in lasagna?_ But there was no answer, so I set out to find one.\n\nLasagna alla Bolognese is an epic dish. Oh sure, it looks like an ordinary broiled mass of cheese, pasta and meaty tomato sauce but it's so much more. To make it as I dreamed from that day forward I wanted to, everything gets a lot of love and time. The ragu is cooked for hours. The b\u00e9chamel (ahem, _besciamella_ ), although the simplest of the five \"Mother Sauces,\" is still a set of ingredients that must be cooked separately, and in a prescribed order. The pasta doesn't have to be fresh, but I figured if I was going to do this, I was going to really, really do this, and I wanted fresh, delicious sheets of pasta to support the other cast members I'd so lovingly craft. And the cheese? There's just one, Parmesan, and it doesn't overwhelm.\n\nSo why did it take the better part of six years to conquer? First, I had to find the ragu of my dreams. I realize that most people have a bolognese they like\u2014maybe it has milk or a mix of meats, not just beef (mine doesn't), maybe it goes easy on the wine (mine doesn't), maybe it can't be cooked for less than six hours (mine can) and maybe it just has a slip of tomatoes inside (mine doesn't). I sometimes think that there are as many interpretations of bolognese as there are people who make it; it's totally cool to use your favorite. But if you're still bolognese-hunting, oh, I do love Anne Burrell's above all else. You could forgo the pasta, the white sauce and the cheese and enjoy it straight from a bowl. But today, we won't.\n\nEven once I found my ragu nirvana, it took a couple rounds to get the lasagna right. The first time I made the noodles, I rolled them too thin and put them on towels, where they proceeded to stick. Miserably. The dish was intended for a 2 p.m. lunch in New Jersey; at 3, Alex was running to a bodega in Manhattan to buy a box of dried pasta. On the plus side, we're still talking to each other. On the minus, we had \"lunch\" at almost 8 that night. I had a few other mishaps; recipes I found seemed out of balance or evasive in directions. One b\u00e9chamel was too thin. And I kept ending up with too much ragu, too little white sauce, too many noodles, not enough directions, too little time. It was not until this week that I finally got the recipe exactly as I'd always dreamed of it, with I hope a level of detail that will make it replicable for anyone at home. Even if you, like me, got to the final inning and realized you were out of cheese, requiring a run to the bodega to pick up _I don't want to even talk about it_ variety of so-called Parmesan. Yes, even for people like me.\n\nNow here's the part where I know you're not going to believe me, but I implore you to consider it: This lasagna, it feels light, almost ethereal, or as close as a decidedly hearty dish can. Maybe it's the absence of ricotta and mozzarella, or the thinness of the homemade noodles but something about it feels utterly decadent, mind-bogglingly delicious, completely warming but not . . . gutting. It needn't immediately lead to a nap on the sofa. It's a miracle. A miracle in twenty parts. Let's get started.\n\n##### **_Lasagna Bolognese_**\n\nRagu adapted from Anne Burrell, everything else from trial and error.\n\nServes 12 (in hearty portions) to 15 (in generous 3-inch squares). You will have double the bolognese sauce that you need because I cannot in good conscience let you spend several hours simmering a sauce that will only yield 4-ish cups of sauce. Trust me, you'll want extra.\n\nThis is a beast of a dish, and worth every second you put into it. I recommend making the meat sauce a day or longer before you need it; then, do everything else on the second day. My advice is to give yourself way more time than you could possibly need on the second day, so that you can make the dish from a place of leisure and love, and not one that is frenzied and not particularly fun. You'll be glad you did. This is a perfect project for a lazy winter weekend, something two people could then eat dreamily all week.\n\nA note on authenticity: This is the kind of dish that gets, ahem, passionate cooks out in droves. I've been told that you cannot call it bolognese if you simmer it for less than __ hours or that it can\/can't have tomato\/milk\/wine\/only beef in it. Others will pfft over the lack of color on the crust (I had a word with my dinky oven about it). I absolutely love this about cooking\u2014the way we care so deeply about the way our food is made, and how much I'm lucky enough to learn about the different ways people approach the same dish. But, my other favorite part about cooking is that it's just you in the kitchen and you can make your food the way you alone like it. Feel free to tweak this to your taste by replacing portions of the beef with other meats, using less tomato paste or wine if desired or replacing some wine with milk.\n\n**Bolognese sauce**\n\n1 medium onion, coarsely chopped (1-inch pieces are fine)\n\n1 large or 2 slim carrots, coarsely chopped\n\n2 ribs celery, coarsely chopped\n\n3 cloves garlic, coarsely chopped\n\n2 to 3 tablespoons olive oil\n\nKosher salt\n\nFreshly ground black pepper\n\n2 pounds ground chuck, brisket or round or combination\n\n1 \u00bc cups tomato paste (from 2 6-ounce cans)\n\n2 cups red wine, preferably hearty but really, anything you like to drink\n\nWater as needed\n\n2 bay leaves\n\nA few sprigs thyme, tied in a bundle\n\n**Pasta**\n\n1 \u00bd cups all-purpose flour\n\n2 large eggs\n\n\u00bd teaspoon table salt\n\n1 to 2 tablespoons water, if needed\n\n**B\u00e9chamel sauce**\n\n\u00bd cup (8 tablespoons) unsalted butter\n\n\u00bd cup all-purpose flour\n\n4 cups whole milk\n\n1 teaspoon table salt\n\n1 clove minced garlic\n\nFreshly grated nutmeg, to taste\n\nFreshly ground black pepper\n\n_To assemble_\n\n1 \u2154 cups grated Parmesan cheese\n\n_Day 1: Make the bolognese sauce:_ In a food processor, pulse onion, carrots, celery, and garlic until finely chopped. Heat a moderatesized Dutch oven (4 to 5 quarts) over medium-high heat. Once hot, coat the bottom of the pan with two to three tablespoons of oil. Once it is hot, add the chopped vegetables and season them generously with salt and pepper. Cook the vegetables until they are evenly brown, stirring frequently, about 15 minutes.\n\nI'm going to insert my favorite Burrell-ism here: Brown food tastes good! Don't skimp on the cooking times as this creates the big flavors that will carry right through to your plated lasagna. And now I'm going to insert my own-ism: Don't worry about sticking bits of food or uneven pieces or anything. It's all going to work out in the end.\n\nAdd the ground beef and season again with salt and pepper. Brown the beef well and again, don't rush this step. Cook for another 15 minutes. Add the tomato paste and cook for 3 to 4 minutes. Add the red wine, using it to scrape up any stuck bits in the pan. Cook the wine until it has reduced by half, about 5 more minutes. Add water to the pan until the water is about 1 inch above the meat. Toss in the bay leaves and the bundle of thyme and stir to combine everything, bringing it to a low simmer.\n\n_Here's how the next 3 to 4 hours will go:_ You'll keep a pitcher of water near the stove. You'll stir the sauce from time to time. As the water in the sauce cooks off, you'll want to add more but you don't want to add more than 1 to 2 cups at a time or you'll have boiled meat sauce (bleh) rather than something thick and robust with flavor. Taste it from time to time and add more seasoning if needed. Simmer for 3 to 4 hours.\n\nYou'll have about 8 to 8\u00bd cups of sauce but will only need 4 for the lasagna. Discard the thyme and bay leaves and put half in the fridge for lasagna assembly tomorrow and the other half in the freezer for up to a couple months. Ours was still as good as day one after 6 weeks.\n\n_Day 2: Make your pasta:_ Combine all of the pasta ingredients in a food processor. Run the machine until the mixture begins to form a ball. You're looking for a dough that is firm but not sticky. If needed, add water a drop at a time until it comes together. Place ball of dough on a lightly floured surface and invert a bowl over it. Let it rest for an hour. (You'll have about 10 ounces or a little less than   pound of fresh pasta dough.)\n\nGet your work area ready; I like to line a large tray with waxed paper. Dust the waxed paper with flour. Keep more waxed paper and flour nearby.\n\nWorking with a quarter of the dough at a time, run it through your pasta roller on the widest setting (usually \"0\"), then repeat this process with the roller set increasingly smaller (1, 2, 3) until the pasta is very thin. My Atlas machine goes to 9, but I almost always stop at 8 because this setting makes for thin, delicate pasta that's not so fragile that I'm pulling my hair out with frustration trying to move it around.\n\nIf you find your dough sticking, lightly flour it. If it gets too big to handle, cut it in half. If the piece gets too wide for the machine or becomes annoyingly irregularly shaped, I re-\"fold\" the dough by folding the sides of the dough into the middle, like an envelope, and press it flat. Then, run the piece back through the machine with the open sides up and down on the widest setting again (0) working your way thinner. This allows the machine to \"press\" any trapped air out.\n\nLay your pasta on the floured waxed paper in a single layer, trying to keep the pieces from touching. Flour the tops of them and place another sheet of floured wax paper on top. Repeat this process with the remaining dough and as many layers of pasta as you need.\n\n_Next, cook your pasta:_ Cut your pasta lengths into square-ish shapes. The fun thing about making fresh pasta for lasagna is that the shape doesn't much matter; you're going to tile together whatever you have and nobody will care if it took 9 or 16 bits to patch the layer together. Bring a large pot of water to boil. Have ready a skimmer, a large bowl of ice water and a large tray or platter that you've drizzled or spritzed with oil. Boil several squares of noodle at a time for 1 to 2 minutes each (1 minute if you, indeed, went to the thinnest setting on your machine; 2 if you, like me, stopped one shy of thinnest). Scoop them out with your skimmer, swish them in the ice water and lay them out (still wet is fine) on the oiled platter. Repeat with remaining pasta. It's okay to have your noodles touch; they shouldn't stick together in the short period of time until you begin assembling but if you're nervous, you can drizzle or spritz each layer very lightly with more oil.\n\n_Make your b\u00e9chamel:_ Melt your butter in the bottom of a medium-to-large saucepan over medium heat. Once melted, add your flour and stir it into the butter until smooth. Cook the mixture together for a minute, stirring constantly. Pour in a small drizzle of your milk*, whisking constantly into the butter-flour mixture until smooth. Continue to drizzle a very small amount at a time, whisking constantly. Once you've added a little over half of your milk, you'll find that you have more of a thick sauce or batter, and you can start adding the milk in larger splashes, being sure to keep mixing. Once all of the milk is added, add the salt, garlic, nutmeg (if using) and few grinds of black pepper, and bring the mixture to a lower simmer and cook it, stirring frequently, for 10 minutes. Taste and adjust seasonings if needed.\n\n_At last, you may assemble your dish:_ Preheat oven to 400 degrees. In a 9\u00d713-inch or equivalent rectangular baking dish, spread a generous \u00bc cup of the b\u00e9chamel. I mostly use this to keep the noodles from sticking. Add your first layer of cooked noodles, patching and slightly overlapping them however is needed to form a single layer. Ladle 1 cup bolognese sauce over the noodles, spreading it evenly. Drizzle \u00bd cup b\u00e9chamel over the bolognese; don't worry about getting it perfectly smooth or even. Sprinkle the layer with \u2153 cup parmesan cheese. Repeat this process\u2014pasta + 1 cup bolognese + \u00bd cup b\u00e9chamel + \u2153 cup parmesan\u2014three more times, then add one more layer of pasta. You'll use 5 layers of pasta total.\n\nThere are two ways to finish the dish. You can simply sprinkle the top layer of pasta with your remaining parmesan before baking. This makes the crunchiest lid. I like a semi-crunchy lid and first spread 1\/4 cup b\u00e9chamel over the top layer of pasta before sprinkling it with the remaining cheese. It still gets crunchy\u2014and has corners that are worth fighting over\u2014but never unpleasantly so.\n\nBake your lasagna for 30 to 45 minutes, until bubbly all over and browned on top. You should do absolutely nothing but put your feet up and drink a glass of wine while you do; you've earned it. When it comes out of the oven, I like to let it rest for 10 minutes before serving it.\n\n_Do ahead:_ Lasagna can be prepared right up until the baking point a day in advance, and kept wrapped in plastic in the fridge. Theoretically, you could also freeze it at this point but I haven't tried this. I'll update this to say \"go for it\" if many people respond in the comments that they've done so successfully. Lasagna will also reheat well for up to three days, possibly longer but in my apartment, we've never had the chance to find out.\n\n* Yes, cold is fine. I divert from the proper b\u00e9chamel method here as I've found that as long as you add your milk slowly, you do not need to heat it separately first. Hooray for fewer steps and pots!\n\n### THE FORAGER AT REST\n\n### By Christine Muhlke\n\n### From _Bon Appetit_\n\n### Two of the world's hottest chefs\u2014Noma's Rene Redzepi and Momofuku's David Chang\u2014meet for Sunday lunch, and who's lucky enough to cover it? _Bon Appetit_ executive editor Christine Muhlke, who co-authored _On the Line_ with another star chef, Le Bernardin's Eric Ripert.\n\nIt's Sunday morning, and Rene Redzepi is gliding around the kitchen in his socks while obscure Aboriginal music drifts through his bright, cavernous, just-moved-into Copenhagen apartment. His wife, Nadine, smiles and hands the couple's 13-week-old daughter to her mother so she can make everyone a second cappuccino. New York chef David Chang, whose photo hangs in the kitchen next to family portraits, is in town and has called to say he'll be by in an hour. It's the type of scene\u2014kind, calm, lovely people in a spare yet creatively furnished space\u2014that makes you wonder why we don't all live in Scandinavia.\n\nAs the chef at Copenhagen's esteemed Noma, Redzepi has inspired cooks around the world to find their ingredients as close to home as possible. So what does that mean when the man behind what many consider the world's best restaurant\u2014a place where fried reindeer moss and coastal flowers have appeared on the menu\u2014is actually cooking at home?\n\nToday's menu is, of course, seasonal and local. But it's also quick. Nothing takes more than 90 minutes to cook. Reindeer-horn-handled knife at the ready, Redzepi takes a parsley plant off the windowsill and cuts off a handful, leaving the stems. \"There's so much flavor in the stems!\" says the man who has made foraging the ultimate in locavorism. \"In the winter we saute them\u2014incredible.\" Next, he slices small biodynamic fennel bulbs so that the open V's resemble the slender mussels that were gathered in a fjord: \"I'm imagining that if a mussel slides out of its shell, maybe it will slide in here.\" So this is how his restaurant dishes come to mimic nature\u2014a gnarled breadstick mistaken for a twig, a snail replacing a nasturtium stamen.\n\nRedzepi's talent for reconnecting natural flavors runs through the rest of the dish: The mussels will be steamed open in gooseberry juice, pressed from the sour berries that grow wild in Scandinavia. \"Berries are so underestimated in savory cooking,\" he says. \"They're so versatile, and they're not all sweet\u2014you just associate them with pies.\" White wine balances the acidity. \"People cook savory dishes with wine, and that's fruit juice, so what's so strange about this?\"\n\nMussel prep finished, he starts work on the appetizer, a toast inspired by the _bo ssam_ (pork shoulder served with a dozen raw oysters) served at Chang's Momofuku Ssam Bar in New York's East Village. Wild chanterelles are sauteed over high heat on the induction cooktop (below which is a clever ventilated IKEA drawer that keeps butter and other ingredients cool and close at hand; cue Scandi lifestyle envy). The mushrooms get a few tablespoons of cream, some minced shallots, parsley, and roughly chopped oysters. After several brisk shakes of the pan, the rich mixture is spooned onto sourdough toast that has practically been caramelized in butter. Translucent sheets of paper-thin Speck fat are laid atop a layer of sliced raw chanterelles just as Chang arrives.\n\nThe Momofuku chef loves the expansive, earth-forest-sea flavors of the toast, even though it's not yet breakfast time in New York, where he was just 36 hours prior. The toasts are even better with the spicy Noma beer brewed from birch sap by Skovlyst. Then the mussels are cooked for five minutes and served in mismatched bowls made by Noma's ceramist. \"Samples,\" explains Nadine.\n\nFor the entree, there's a pot roast. But this one is celery root, not beef. At Noma, Redzepi has been treating vegetables like meat: braising them, basting them, flavoring them with lots of herbs and butter (preferably that made from sweet, rich goat's milk). He'll even throw in a handful of coffee beans from the excellent local roaster Coffee Collective to see what happens. Earthy celery root takes extremely well to the treatment, browned and tender with herbal undertones. A quick sauce of warm buttermilk and olive oil\u2014\"basically a vinaigrette\"\u2014adds a complex tanginess. Garnished with black olives heated through in the herb butter, it's deeply, shockingly satisfying.\n\n\"This is a very good example of how we eat at home,\" Redzepi says: \"It's vegetable based, there's a little twist to it, and it's very, very simple.\"\n\nSeated at the sprawling dining table is an easy crowd. There's Alessandro Porcelli, himself an important tastemaker in the international food scene. He's a co-founder of the Cook It Raw chefs' adventure series, as well as closely involved in the inaugural MAD Foodcamp symposium, which is what has brought Chang to town. Noma sous-chef Trevor Moran, an Irishman with the most fantastic hair, is tucking into his day off. Nadine's mother holds a very sleepy infant, while toddler Arwen works on her princess drawing. And then there is the serene young Nadine, a reservationist at Noma, where the 12 tables are booked three months ahead. If people knew how beautiful she is, an Ingres portrait come to life with a smile as quietly sure, they wouldn't be upset when she tells them that the dining room is full. Possibly forever.\n\nMaybe Nadine could send them some of her walnut cake to alleviate the sting. It's made with over a pound of nuts and a half pound of butter. \"This is one of the best desserts I've had,\" Moran says a little sheepishly as he glances at his boss. But Redzepi isn't slighted; the British food critic Jay Rayner is coming over for dinner tomorrow, and Redzepi asks Nadine to please make another one. Everyone nods imploringly.\n\nAfter coffee, Redzepi buckles Arwen into the front carriage of his dad bike and everyone walks alongside them to the King's Gardens. \"This is where the king used to grow his vegetables. You can still get food here!\" he says. While locals laze in the late-afternoon sun or play petanque, he heads straight for the trees. In the middle of the city\u2014just blocks from his home\u2014is a mulberry tree jeweled with untouched fruit. A few dish ideas are batted around by the chefs as they stare up into the leaves. \"I know where we can find walnuts, too,\" Redzepi says and bikes ahead. Soon he and Chang are cracking open the green orbs and picking at the jellied flesh, running it between their fingers. \"Trevor,\" he says decisively, \"it's time for walnuts.\" Noma might be closed Sundays, but a true forager knows no bounds.\n\n##### **_Pot-Roasted Celery Root with Olives and Buttermilk_**\n\n_In this surprising main course, Redzepi pot-roasts whole celery roots. Be sure to use small celery roots; larger ones will not cook evenly. The chef also cooks small heads of cauliflower in this way._\n\n_6 servings_\n\n**Ingredients**\n\n3 tablespoons plus 2 teaspoons extra-virgin olive oil plus more for drizzling\n\n6 small celery roots (celeriac; each about 4 oz.), unpeeled, trimmed with some stem still attached\n\n\u00bd cup (1 stick) unsalted butter, cut into 1\" cubes\n\nKosher salt\n\n12 sprigs thyme\n\n6 sprigs rosemary\n\n6 sprigs sage\n\n1 tablespoon coffee beans (optional)\n\n1 cup buttermilk\n\n\u00bd lemon\n\n\u00bc cup oil-cured black olives, pitted, quartered lengthwise\n\n**Preparation**\n\nHeat 3 Tbsp. oil in a large heavy pot over medium heat. Add celery roots and cook, turning frequently, until golden, 8\u201310 minutes. Add butter and a large pinch of salt. When butter begins to foam, reduce heat to medium-low. Add herb sprigs, coffee beans, if using, and 1 1\/2 cups water. Cover and gently cook, adding more water by tablespoonfuls if pan is dry, until celery roots are very tender, about 1 hour.\n\nRemove herbs and coffee beans (if using) from pot and discard. Baste celery roots with buttery juices in pot. Remove from heat. Transfer celery roots to a cutting board, slice in half through stems, and place 2 halves on each plate.\n\nMeanwhile, very gently warm buttermilk in a small saucepan over low heat (it will break if warmed too much). Squeeze in a few drops of lemon juice and add remaining 2 tsp. oil and a pinch of salt to make a loose sauce. Spoon sauce around celery roots on plates. Garnish with olives, drizzle with oil, and squeeze a few more drops of lemon juice over each.\n\n##### **_Walnut Cake_**\n\n_Nadine Levy Redzepi created this incredibly rich, moist cake. \"Fat with fat\u2014what could be better?\" asks her husband. Serve it for dessert or with coffee or tea for breakfast._\n\n_18\u201324 servings_\n\n**Ingredients**\n\n1 cup (2 sticks) unsalted butter, room temperature, plus more for pan\n\n6 tablespoons raw sugar, divided\n\n7 cups walnut halves\n\n\u00be cup all-purpose flour\n\n1 \u00bd cups almond flour or almond meal\n\n\u00be cup granulated sugar\n\n6 large eggs\n\n\u00be cup heavy cream\n\n\u00bd cup plain whole-milk yogurt\n\n1 teaspoon kosher salt\n\n1 vanilla bean, split lengthwise\n\nWhipped cream\n\n**Ingredient Info**\n\nAlmond flour is available at some supermarkets and at natural foods stores and specialty markets.\n\n**Preparation**\n\nPreheat oven to 350\u00b0. Butter a 13x9x2\" metal or glass baking dish; sprinkle bottom evenly with 3 Tbsp. raw sugar. Set aside.\n\nPulse walnuts in a food processor until coarsely chopped. Set 2 cups aside. Add all-purpose flour to processor and pulse until walnuts are very finely ground, 1\u20132 minutes. Add almond flour; pulse to blend. Set aside.\n\nUsing an electric mixer, beat 1 cup butter and granulated sugar in a large bowl until light and fluffy, 2\u20133 minutes. Add eggs, cream, yogurt, and salt. Scrape in seeds from vanilla bean (reserve bean for another use). Beat until well combined, 1\u20132 minutes. Add ground-walnut mixture and beat just to blend. Gently fold in chopped walnuts, being careful not to over-mix. Pour batter into prepared dish; smooth top. Sprinkle with remaining 3 Tbsp. raw sugar.\n\nBake until cooked through and a tester inserted into center comes out clean, 50\u201355 minutes. Let cool in pan on a wire rack. Serve with whipped cream. **DO AHEAD:** _Can be made 3 days ahead. Cover and chill. Cake is best served cold._\n\n### BETTER COOKING THROUGH TECHNOLOGY\n\n### By Corby Kummer\n\n### From _Technology Review_\n\n###\n\n### Poised at the apex where gourmet cooking meets high-tech technique, Nathan Myhrvold dazzled the food world with his 2011 multi-volume _Modernist Cuisine._ Corby Kummer, longtime senior editor for _The Atlantic,_ reviewed and interviewed Myhrvold for a tech-savvy audience.\n\nTo see _Modernist Cuisine_ is to covet it. Which is why, one day in May, the team that spent six years creating the oversized, over-everything five-volume work came from Bellevue, Washington, to New York City to demonstrate the wondrous object. And it is why a group of chefs, writers, and TV personalities (so stellar that one guest remarked, \"The only other event that could bring these people together is a funeral\") gathered one morning at Jean-Georges, the flagship restaurant of Jean-Georges Vongerichten, at the invitation of Tim and Nina Zagat. They were there to meet Nathan Myhrvold, the mastermind and financier of a book so expensive to create that he refuses to say how much he spent (other than to say it was more than $1 million but less than $10 million). They wanted to try the pastrami cooked sous-vide for 72 hours, the \"tater tots\" dunked in liquid nitrogen before being fried, the fruit juices spun in a centrifuge, the mushroom omelet striped with powdered-mushroom batter so that it looked like a piece of upholstery, with a perfectly spherical, magically just-cooked egg yolk right in the middle. But they really wanted to see the book.\n\nAnd it _is_ a wondrous object. _Modernist Cuisine's_ five volumes comprise 1,522 recipes and 1,150,000 words of text on 2,438 pages, almost every one of them illustrated with color photography and charts, with dozens of gee-whiz, never-before-seen photographs of beautiful free-form color swirls that could be textile designs but turn out to be life-threatening pathogens; or sculptural objects that could be outdoor art installations but are mussels suspended in clear gelatin; or stunning anatomies of a painstakingly shelled lobster or flayed monkfish or whole chicken; or spectacular cross-section cutaways of pieces of equipment you never thought would or should be sawed in half, like ovens, woks full of hot oil, and kettle grills with white-hot smoldering coals. It weighs 40 pounds, four of them just ink. When Wayt Gibbs, the book's editor in chief, met me later that week in Cambridge, Massachusetts, at Toscanini's, an ice-cream parlor and intellectual salon heavy with MIT students and faculty, he painstakingly unwrapped the gigantic carton he had lugged on a portable dolly from Bellevue to New York and then to Boston. The caf\u00e9-goers grew silent and stared at the huge white volumes in their clear Lucite case, one of them later wrote me, as if they were the monolith in _2001._\n\nThe long-awaited publication of _Modernist Cuisine,_ in March, was the most significant event in the food world since . . . well, there might not be a precedent. The 6,000 copies that Myhrvold printed privately\u2014against more conservative advice from what he describes as \"cooler heads\" in book publishing\u2014immediately sold out at the introductory price of $465. \"We sold 9,000 of those 6,000 copies,\" Myhrvold says with satisfaction. He quickly ordered 25,000 more copies to be printed.\n\nTo research the book, Myhrvold built a 4,000-square-foot laboratory\u2013kitchen\u2013photo studio in an 18,000-square-foot former motorcycle showroom in Bellevue, where an ever-expanding team of cooks experimented with machinery usually restricted to doctor's offices, hospitals, and commercial food processing, using powders and essences and chemicals similarly typical of the food industry.\n\nSuch experimentation had been going on for years, of course, most famously starting in the early 1990s at Ferran Adri\u00e0's El Bulli in Catalonia, Spain, and in the mid-1990s at Heston Blumenthal's Fat Duck in Berkshire, England\u2014the two main inspirations for Myhrvold and his team, whose lead members trained with Blumenthal. In this country, it was hard for ambitious young chefs to visit a similar nucleus of cooking research unless they could get into Grant Achatz's Alinea, in Chicago ( _see \"The Alchemist,\" January\/February 200_ 7). Or unless they were among the favored few to be invited to one of the 30-course tasting dinners in the Bellevue lab, which were reserved mostly for cooks and industry leaders who had lent expertise or machines to the team, and for reporters like me who wanted a look at just what went into the three years of decision making, recipe testing, writing, and editing that preceded publication.\n\nMyhrvold himself is an object of intense interest. The former chief technology officer of Microsoft and the founder of the patent investment company Intellectual Ventures, he is a genius and billionaire who still indulges his boyish enthusiasms, which include photography and dinosaurs but revolve mostly around cooking. The interest began at age nine, when he nearly set his mother's kitchen on fire in a plan to flamb\u00e9 everything for a Thanksgiving dinner. Myhrvold is a charming, even twinkling, spokesman for his mad invention processes\u2014quite unlike, say, Ferran Adri\u00e0, whose trademark is a messianic intensity and utter imperviousness to anyone not as focused as he on the windstorm of creativity ever blowing round his brain. Myhrvold's voice is strong enough to come through in many sections of the book, though they're unsigned: a droll account in the third volume of traveling to Greenland and eating rotten shark, which \"doesn't taste like chicken,\" is probably his. So is much of an excellent chapter in the fourth volume on coffee, in which he goes in search of what baristas call the \"God shot\" and in the process learns and shares a terrific amount of information on roasting, grinding, foaming milk, and pulling espresso shots.\n\n##### **Justifiable Cost**\n\nI managed to get my hands on the copy Gibbs unpacked at Toscanini's. Opening any volume brings you right to cutting-edge tech techniques that produce food unlike anything anyone has ever tasted outside El Bulli or the Fat Duck or Alinea.\n\nBut the high-tech toys and futuristic food are not why I think you should put yourself on the wait list and spend the $478 the book now costs (unless you want to spend the $800 and up being asked for \"used\" copies). Although some of the futuristic food is fabulous, as I learned from the 30 courses I got to try at the lab, the reason to pony up the going rate is that _Modernist Cuisine_ is an incomparable introduction to many of the basic techniques of food and cooking. Within its five volumes (six, actually, including a spiral-bound book of recipes for the professional kitchen) are several long chapters that are as comprehensive and readable and valuable as any books I've seen on subjects essential for anyone interested in food.\n\nThe first volume alone contains a long, definitive introduction to food pathogens and food safety, a subject cooks ignore at their peril. The other volumes give basic information on science, ingredients, and techniques common to all cooking, not only \"modernist\" cuisine. And, of course, the book is a guide to the avant-garde\u2014one far more comprehensive and usable than anything else yet written. As for the food, there are those 1,522 recipes, and if you can lay out a fairly substantial sum and clear enough counter space to start trying them\u2014well, more on that later.\n\nI didn't read all 1,150,000 words\u2014no one other than Gibbs has claimed to, he told me when I spent a few days in the Bellevue kitchen. (He also admits to having tried only a few recipes, because he's not a cook.) But I'll claim a good 750,000. Watching the media appearances of the buoyant and unfailingly enthusiastic Myhrvold; visiting the book's website for cool videos of machines and pots being sawed in half or a kernel of popcorn dancing across a black screen until it explodes and soars up and off like a rocket; even eating 30 courses in Bellevue\u2014none of it prepares you for the experience of reading _Modernist Cuisine._\n\nEverything about the book has been designed to keep a reader going, with bits of information in the margins and pages-long interruptions for techniques, cooking charts, and \"parametric\" and \"example\" recipes. Some sidebars go on for a few pages; they're printed in white type against black, as are most of the charts, so that the flow of text, though unusually complex, becomes intuitive. As in a magazine or textbook, captions provide complementary information and pr\u00e9cis that make you feel you've got the gist of the main text.\n\nThe similarity to magazines and textbooks is not accidental. Myhrvold and Chris Young, a scientist and former Fat Duck chef who is listed as an author along with Maxime Bilet, another Fat Duck alumnus and the head chef of the Bellevue kitchen, were influenced by the illustrated Time-Life series of cookbooks from the late 1970s\u2014books, like this one, assembled by veterans of the magazine and book worlds who knew how to unite text and photographs for maximal informative value. Gibbs, who's been a writer and editor at _Scientific American_ and has extensive experience creating illustrated features, served as producer for a total of 44 writers, photographers, designers, researchers, and editors whose combined efforts bring life and interest to every page.\n\nMyhrvold has been derided for producing a book, that most old-fashioned of objects, rather than a $5 app. His stock reply: a \"really good\" electronic version, with interactive features to scale recipes, animations for key techniques, and video clips, would be a project \"bigger than the one to do this book.\" A book is still the best way to publish so much information, and the spiral-bound supplement, unlike an iPad, is waterproof. And with a few exceptions (including much of the fourth volume, which is devoted chiefly to thickeners, gels, emulsions, and foams\u2014the trickiest in the new chef's bag of tricks), my interest never flagged.\n\n##### **Cool Tools**\n\nThe book I'd make required reading for any cooking student is Volume 2, on techniques and equipment, which gives as good a description as I've seen of basic processes like baking and frying. This is also the volume that lists the toys in the toy box. Number one on the list is a sous-vide water bath\u2014a tool that is ubiquitous in the recipes, particularly for meat. The bulk of the volume, unsurprisingly given the project's origins as an eGullet chat forum Myhrvold opened about sous-vide, is devoted to this cookery technique, which Myhrvold and many other cooks value for its precise control and predictability. Thanks to the enthusiasm of chefs like Achatz, Thomas Keller (who wrote a book on the method), and Philip Preston, of PolyScience, a manufacturer of controlled-temperature equipment who worked closely with Achatz and Myhrvold, the water bath has gone from science-lab \"immersion circulator\" to almost-affordable kitchen tool.\n\nI don't have the patience for sous-vide, and I find that it produces too soft a texture in meat and fish. The piece of equipment I'd like to buy is a combination dry-and steam-heat \"combi\" oven, which so far hasn't found a manufacturer like Preston willing to work on one for the Williams-Sonoma crowd. There are tiny ones for $2,000, but they hold almost nothing; models not much bigger than a big microwave easily cost $12,000, and the authors say you need a couple of those. Still, the fact that the ovens thaw, steam, poach, and roast makes them as appealing to me as the microwave\u2014which, hearteningly, the authors endorse for cooking vegetables, frying tender herbs, and turning vegetable juices into \"perfect powders.\"\n\nThe authors do list cool tools that are within the reach of many home cooks, like digital scales and thermometers; the carbonator, for foam; a Toddy cold-brewing coffee kit, for deriving extracts of many flavorings besides coffee; and my favorite all-purpose tool, a pressure cooker, something I use nearly every night. The reason that stocks made in pressure cookers are perfectly clear, they point out, is that the water inside never boils, and the motion of boiling is what emulsifies oil and creates scum in normal stocks. They give everyday tips for ways to use the device, including making risotto (a longtime guilty secret of time-pressed Italian cooks, who will reveal it only after receiving compliments on how good their risotto is) and adding calcium chloride to the water for beans to let them soften without splitting their skins.\n\nThe tool many professional chefs may decide they need is a centrifuge, which costs $10,000 to $30,000 and can take up as much room as a washing machine. Myhrvold's team used a centrifuge to clarify juices from citrus and from sous-vide bananas, which became translucent and serum-like. Thomas Keller, chef of the French Laundry in Yountville, California, decided he needed a centrifuge after he saw the Bellevue lab use one to separate the fatty solids from peas to make \"pea butter,\" spreadably thick and perfectly smooth.\n\nAnd some chefs might sign on to the tank of liquid nitrogen that the team says is second in utility only to the sous-vide water bath. Dunking a food in liquid nitro before you fry it\u2014\"cryofrying,\" the lab calls it\u2014makes the outside of, say, cubed pork or sous-vide chicken or \"tater tots\" hot and crisp while the interior turns out just warm and not overcooked. It also makes soft foods manageable to slice thin or to grind. Cubes of beef can be put into meat grinders, drawn out in parallel, extruded strands, and carefully rolled into plastic-wrapped cylinders. After being submerged in liquid nitro, they're cut into patties and deep-fried for hamburgers that, as Jean-Georges Vongerichten reported with wonder, are juicy and crumbly without being fatty or tough. But the grownup boy magicians on the _Modernist_ team use the cooling agent, they admit, \"for just about every food,\" because it's \"just plain fun\"\u2014for instance, to \"cryoshatter\" olive oil for a garnish.\n\n##### **Possibilities**\n\nVolume 3, on animals and plants, is both an anatomy class and a guide to how proteins and fibrous plants react to heat. The section on meat includes wonderful color diagrams of muscle fibers and collagen that do more than anything else I've ever seen to explain the structure of meat and make it clear why different cuts cook differently. The section on vegetables is a good bit shorter, perhaps reflecting the cooks' degree of interest. (They seem to have been intensely interested in the heat scale of peppers, though!)\n\nVolume 4, on ingredients and preparations, spends the most time on thickeners, this being \"truly the best age ever in which to thicken a liquid.\" I was glad to learn about viscosity and fluid gels, and to discover that alginate, a hydrocolloid extracted from brown seaweed, is the key to the \"spherification\" that Adri\u00e0 has made almost as popular and widespread as foams. But for anyone who doesn't plan to buy Ultra-Sperse 3 or Ultra-Tex, or N-Zorbit M or even xanthan gum\u2014all of which are turning up in modernist-inspired kitchens, and all of which appear in dozens and dozens of recipes in the book\u2014this will make the least absorbing reading.\n\nI'd argue that the plated-dish recipes in Volume 5 will date the book faster than any other part. They're included to demonstrate the possibilities of all the techniques and ingredients we've learned about in other volumes, and to conclude the argument started in Volume 1 that all history builds to their inevitability. So classic recipes are updated and adapted, using a panoply of time-consuming steps few home cooks would attempt. Blanquette de veau, the classic veal stew, is liquefied to a warm cream called \"veal nog\" that requires a rotor-stator homogenizer and a centrifuge. Boeuf en gel\u00e9e, the gel hot rather than cold, demands a homemade oxtail stock, xanthan gum, and low-acyl gellan.\n\nSome cooks might attempt all this\u2014perhaps ambitious professionals who haven't been able to apprentice in any of the new-wave kitchens, perhaps semi-obsessed hobbyists. But I'm not one of them, and the 30 courses at Bellevue, every one of which I tried and took notes on, didn't make me a convert. Some of the flavors and textures were revelatory: a clear, strong \"beef tea\" that came from a sous-vide bag; cocoa pasta, something impossible without \"vital gluten\" (cocoa powder has no gluten of its own), with pur\u00e9ed, cured sea urchin cooked sous-vide. But much of it still seems mere trickery: freeze-dried corn kernels and powders of brown butter and lime and ash in a version of the Mexican street food corn elote, the powdered fat unpleasantly greasy on the tongue; a cream of mushroom and bacon soup infused with dark miso and gelled into a too-intense foam; smoked butter made in a rotor-stator homogenizer that overwhelmed a delicate piece of fresh-caught, unfortunately brined albino salmon (cooked sous-vide, of course).\n\nBut these are matters of personal taste, and the night I visited Bellevue I was fascinated every moment. As I did at the Jean-Georges breakfast, I came away convinced that these techniques and ingredients will be essential for cooks of the future. It's too early to know how they'll be adapted, and which will be most frequently used, but my feeling when sampling the 30 courses was that as prices for homogenizers and centrifuges come down, thickening agents become easy to find, and even liquid nitrogen becomes commonplace in professional and then home kitchens, we'll make our own ketchup and many other staples, and come to cook dishes as basic as fried chicken and hamburgers in completely different ways.\n\nEvery big-name chef, however rooted in classic techniques, is already interested. A couple of hours after the Zagat breakfast, Vongerichten persuaded me to stay for lunch, to eat the tasting menu he was giving the chef Daniel Boulud as a birthday gift. He didn't send us 30 courses, but the number approached 20, and the flavors were a kind of musical composition that varied in volume and intensity but never in virtuosity. Almost none of them used any of the new techniques described in _Modernist Cuisine_ ; almost all of them strove to find innovative but nonrevolutionary ways to extract the maximum flavor and fragrance from the herbs, fish, and meat Vongerichten had in the kitchen.\n\nBut Vongerichten told me he's getting ready to be a not-quiteearly adopter, even though \"I tried meat glue, and I just don't understand\u2014why do I need meat glue?\" As we discussed the book after lunch\u2014he had paged through it in the morning like a child with his first train set\u2014he had a look in his eyes that was both wistful and determined. \"I've got to make burgers that crumbly,\" he said. Another tank of liquid nitro sold.\n\n## Foodways\n\n### THE PASTRAMI DILEMMA\n\n### By John Birdsall\n\n### From Chow.com\n\n### Chow senior editor John Birdsall\u2014a former chef who segued into food writing for the Bay Area's SF Weekly and _East Bay Express\u2014_ traces the fortunes of this Jewish deli staple, finally getting its long-overdue artisanal revival. Reuben sandwiches will never taste the same.\n\nThe joke goes: A gorilla walks into Sol's delicatessen and orders a pastrami sandwich.\n\n\"That'll be $15,\" says Sol, handing the gorilla a sandwich. \"I gotta say, I never expected to see a gorilla in my deli.\"\n\n\"At $15 for a pastrami sandwich,\" snaps the gorilla, \"you never will again.\"\n\nOnce, pastrami was cheap. It came from Jews on New York's Lower East Side in the 19th century, when beef was abundant. They adapted it from Romanian _pastram  ,_ originally made from salted and pressed geese and ducks. And since then, it's been viewed as a well-loved but lowbrow American original, like the hot dog or the hoagie. You could say it's the soul of the American deli.\n\nBut even while chefs in other parts of the restaurant industry are making their own charcuterie, pastrami has, until recently, been stuck in the deli case of the 1950s: industrialized and mass-produced.\n\nYou know all those discussions about who has the best, Stage or Katz's, Langer's or Canter's? Forget about it. Three factories make nearly all of that famous deli pastrami and distribute it around the country.\n\nThe owners of these three pastrami factories, kind of like dons in a benign pastrami mafia, each control their own turf. In Brooklyn, for example, there's Eddie Weinberg, who supplied the old 2nd Ave Deli. In Detroit, it's Sy Ginsberg, who does _Zingerman's._ Lou Sandoval of Burbank, California, supplies _Langer's._\n\nBut finally, there's a pastrami revival going on: Artisans all over the country are starting to make pastrami the old-fashioned way. From Portland to Brooklyn to San Francisco, young chefs are opening traditional Jewish delis and bringing back the nearly extinct tradition of hand-cured, hand-smoked pastrami. But it turns out the old way is a lot harder than you'd think.\n\n##### **Hot Beef Injection**\n\nPicture your favorite Jewish deli: You probably have visions of a guy in a white apron slicing spicy, steaming brisket into a gorgeous tower. The truth is, that guy didn't make that brisket. Almost all pastrami in this country is made more like a Honda on an assembly line than like a craft project.\n\nFirst, the meat goes into an injection machine, where it's shot up with a salty solution. Then it gets a quick brine, is coated in spices, and gets smoke flavor\u2014either briefly, over actual smoldering wood chips, or more likely on a grill like at a backyard barbecue, where the \"smoke\" comes from fatty juices flaring up against hot metal. Or even in a chamber of aerosolized liquid smoke. The whole process is over in a matter of hours, after which the meat is vacuum-sealed in plastic so it can sit in a cooler for weeks before it's dumped in some deli's steamer to cook for a few hours.\n\nAnd probably not surprisingly, the factory dons are vague about where they get the meat for their clients' famous pastrami. We can guess it's not grass-fed.\n\nThe truth about pastrami isn't exactly a secret. David Sax's book _Save the Deli,_ published in 2009, revealed the state of industrialization of high-end deli meats. And it's not that pastrami's an unpalatable product. A pastrami sandwich from one of America's storied delis can be great. (LA food critic Jonathan Gold, for instance, thinks _Langer's_ makes the best pastrami sandwich in the country.)\n\nBut handmade everything is the new ethos among a certain breed of chef. And those opening up nouveau Jewish delis think they can do better. As it turns out, though, making pastrami is a lot more challenging than making rugelach and whitefish salad.\n\n##### **Navel Gazing**\n\nProblem number one: To make old-timey pastrami, the way the original preindustrial New Yorkers did it, you have to use the navel: a fatty belly cut akin to bacon on a hog. It's got thick streaks of fat that ensure the meat stays moist.\n\nAt Saul's, a New York\u2013style Jewish deli in Berkeley, California, co-owner Peter Levitt learned it's nearly impossible to use navel without going broke. Levitt, a former Chez Panisse chef, makes some essential deli products in-house: sodas, pickles, kreplach. But when it came to pastrami, the traditional ingredients were out of reach. The navel used to be a cheap cut of meat. Nowadays it's expensive, thanks to demand from Asia. (In China and South Korea, the navel is prized.) Adding to his problems was Levitt's desire to find _sustainably raised_ navel\u2014after all, Michael Pollan is a regular at Saul's.\n\nFor a few weeks, while he was trying to locate the right navel, Levitt took pastrami off the menu. A Jewish deli without pastrami? You can imagine how well that went over. For nearly two weeks, servers wore buttons on their aprons reading \"Pastrami Under Construction.\" Some customers were pissed, and Levitt continues to get calls from people asking, with trepidation, if pastrami is still \"not on the menu.\"\n\nWhen Levitt was finally able to source sustainable navel, it was cost-prohibitive because the meat is so fatty that the usable portion from one slab of navel is very small. There's a reason why most pastrami these days is made from brisket, with a tendency toward dryness. And now Saul's is too.\n\n##### **Mystery Meat**\n\nProblem number two: If nobody's making \"real\" pastrami anymore, how are you supposed to know what it's supposed to taste like? Sort of a funny question, but that was a real conundrum for Leo Beckerman and Evan Bloom. In late 2010 when the two former college buddies started Wise Sons, a pop-up Jewish deli in San Francisco, they knew they wanted to make their own pastrami, but they had no models to follow. Nobody they knew was making honest-to-God nonfactory pastrami, so they had no one to learn from. They had to turn to the Canadians for help.\n\nMile End Deli in Brooklyn, opened by Canadians Noah Bernamoff and his wife, Rae Cohen, in 2009, makes its own Montreal smoked beef. (Canada never lost its tradition of delis smoking and curing their own beef the way America did.) Mile End's product was as close to traditional pastrami as Beckerman and Bloom could find. The most noticeable thing about it was the intense smokiness that most American pastrami has lost.\n\nUsing the Bernamoffs as their guide, Beckerman and Bloom began to experiment. After trial and error, they developed an authentic-tasting recipe. First the meat (they experimented with navel but soon abandoned it for brisket) gets \"wet cured\" in brine and pickling spices for a week. Then it sits another couple of days on a rack, coated in powdered spices, until it dries out and develops a crust (the technical term is _pellicle_ ). Then it smokes for six to ten hours over hickory chunks.\n\nAt the end, they get meat that's both smokier and more beefy-tasting than factory pastrami, that's chewier, and that has an almost creamy texture from fat marbling. After nearly two weeks of work, it better be good.\n\n##### **Is It Worth It?**\n\nOf course, not every old-fashioned process works in the modern world. Ever tried making your own salt by evaporating seawater, or not washing your hair like the Native Americans did? At Portland, Oregon, deli Kenny & Zuke's there's been some backsliding on the road to authenticity. One of the earliest pastrami revivalists, co-owner Ken Gordon started making his own while still a chef in a Portland bistro, six years ago. He used the navel. But at Kenny & Zuke's, he found the original recipe to be off-putting for many Portlanders.\n\n\"It was too fatty,\" he says. \"My grandparents came over from Russia and Warsaw and ate serious chunks of fat and died in their 60s. These days customers want it lean.\"\n\nSo the deli switched to what people are used to: brisket.\n\nNow Kenny & Zuke's pastrami has a cult following. (One reviewer described their sandwich as capable of bringing a woman to orgasm.)\n\nAnd yet a week's worth of curing is a long time. Imagine you're burning through, say, 2,000 pounds of pastrami a week. How are you going to keep up? And where are you going to find the walk-in space? Such were the problems for Kenny & Zuke's. Add to that the fact that Gordon has designs on Whole Foods\u2014he hopes eventually to get FDA approval to package his pastrami to sell to retailers.\n\nAnd so, there have been some tweaks to Kenny & Zuke's traditional-ness. To shave a few days off the curing process, the meat no longer goes through the long brining. Instead, it's injected, just like the factories do it.\n\nBerkeley's Levitt has started injecting, too. He _was_ a pastrami purist, but these days he says he doesn't think the old-timey wet brine makes any difference to the taste of the meat.\n\nAnd a week after opening their brick-and-mortar delicatessen, Wise Sons' Evan Bloom is already thinking about how he can speed up the curing process.\n\nThe revivalists' pastrami still tastes great, though. It still sells like crazy. And it's still handmade. After all, Gordon's and Levitt's employees are the ones holding the syringe. For now.\n\nSomebody tell the gorilla that he can come back for a pastrami sandwich.\n\n### PASSOVER GOES GOURMET\n\n### By Rachel Levin\n\n### From _Sunset_\n\n### Once you've conquered pastrami making, why not throw a hipster Seder for the Bay Area? Freelance travel and food writer Rachel Levin laid to rest ghosts of Passovers past at this pop-up event in a San Francisco warehouse.\n\nWhy was this night different from all other nights? For starters, there was a bar. And not a bottle of sticky-sweet Manischewitz behind it. People at this Passover Seder were drinking. _Good_ wine. _Before_ the first of the traditional four glasses was poured.\n\nSecond, people were dressed in jeans. My mother never let me wear even my very best Jordache to any Jewish holiday. Skirts only, and tights that would sag around my ankles. Now, three decades later, I swapped a pair of faded cords for a stylish purple number and heels. I hadn't felt this overdressed since I wore a bathing suit to the Big Sur hot springs.\n\nThird, this wasn't my grandparents' house in a manicured Boston suburb, but a mod cafe in a former warehouse in San Francisco. My aunts and crazy cousins were clear across the country. There were no conversations-cum-arguments about what route everyone took to get there. Or kids' tables topped with Dixie cups of Welch's white grape juice.\n\nAbove all, apart from my Caribbean-born gentile friend George\u2014whom I'd dragged here while my Jewish husband was, uh, at a Black Crowes concert\u2014this was a Seder of strangers. All different backgrounds. Fifty folks here _voluntarily,_ not because their parents forced them.\n\nThe big draw? The food. Cooked, not by Grandma Hannah, but by Leo Beckerman and Evan Bloom, whose Wise Sons Jewish Delicatessen pop-up here had an instant cult following. Lines snaked down sidewalks for their hand-sliced artisanal spin on pastrami. The demand for good deli\u2014in a city long lacking it\u2014grew so strong that the duo recently opened a real-deal restaurant. Last April, their first-ever public Passover Seder sold out within minutes by word of mouth.\n\nImagine, the promise of gefilte fish _that_ good.\n\n##### **Strangers become friends**\n\nCandles were lit. Communal tables were set. Sparely. No lacy white tablecloths or Blue Danube china. Playing silently on a screen was the '50s classic film _The Ten Commandments._ I mean, Charlton Heston's low-tech parting of the Red Sea is the kind of Seder entertainment I could've used as a kid.\n\nI loved my grandpa Orrin, I really did. He was a kind, lanky doctor in a knit tie and corduroy blazer. But his Seders were by-the-book snooze.\n\nHere was fresh-faced 28-year-old Leo! With waist-length dreadlocks pulled back in a ponytail, he had a cool, confident command over the room that would no doubt make his own grandfather proud. After the blessing over the wine, servers presented plates of matzo. It was blistered, cracker-thin, imperfectly shaped. And _not_ from a box, but made by Bay Area local Blake Joffe of Beauty's Bagel Shop\u2014with more than just the requisite flour and water. If all it takes is a little sea salt and olive oil to enhance matzo's typically dry-mouth taste, then I vote for a minor overhaul of tradition.\n\nStill, this was a legit Seder. Everyone had a photocopy of a Haggadah, the book of prayers, songs, and biblical tales that recount the Israelites' exodus from Egypt and freedom from slavery.\n\nIt's a good story. But as a kid, taking turns around the table reading The. Entire. Freaking. Thing meant we didn't eat for _hours._ I'd steal sprigs of parsley from the tabletop (long after we'd dipped it in the ritual salt water)\u2014and sit, starving and bored as hell. Grandma's dense-as-rocks matzo balls and gray, leather-tough brisket weren't any prize. But by the time dinner was actually served, I would've eaten the jar of Heinz Chili Sauce she'd \"seasoned\" it with.\n\n\"Tonight, we're going to move through the Passover story pretty quickly,\" announced Leo. \"We've got eating to do!\"Amen to that.\n\nAnd so it began: the explanation of the Seder plate, the Four Questions (typically, the youngest person at the table is charged with tone-deaf singing this integral part of the evening, but on this night, the lone tween was too shy; instead we were treated to a woman who actually had a beautiful voice), and the Ten Plagues, detailing Old World woes. By the festive song _\"Dayenu\"_ (\"Enough\"), we'd lost count of glasses of wine and were all one big, happy family\u2014singing, clapping, exchanging smiles. George turned to me and exclaimed: \"I love this! I'm with my people!\"\n\nBefore we knew it, dinner was served, family-style: pickled heirloom carrots and Bull's Blood beets. \"Mock liver\"\u2014a mash of organic peas and Blue Lake beans. The prettiest, most perfectly pungent, handgrated fluorescent-fuchsia horseradish I've ever had. (Note to Wise Sons: Jar that stuff!) The soup was a clean, flavorful broth buoying matzo balls as God intended them to be: feather-light and fluffy. The gefilte fish was a custom-grind of carp and whitefish in a fennel-thyme fumet\u2014a far cry from the congealed stuff in jars you see every season at the supermarket. And the brisket: not gray! Not tough! Just fork-tender shreds of peppery sweet meat.\n\nThree hours later, when we were down to the last sips of madeira, matched with a creamy, rich chocolate pot de cr\u00e8me (single-handedly bringing Passover desserts back from the dead), there was laughter; career-advice-giving; gossip about embarrassing wedding toasts and bad breakups involving people we didn't know. No barking between relatives or help-clear-the-table mandates from Mom. But hugs good-bye. And sincere cries of \"Next Year\u2014with Wise Sons!\"\n\n### THE 2011 DYKE MARCH WIENER TASTE TEST\n\n### By Bethany Jean Clement\n\n### From _The Stranger_\n\n### The Fourth of July: The iconic American holiday, made for parades, fireworks, and cooking hot dogs on the grill. Seattle's wacky spin on this tradition is perfect fodder for _The Stranger_ managing editor Bethany Jean Clement's subversive, droll restaurant review style.\n\nAnd so we come to Independence Day, when Americans turn to thoughts of wieners.\n\nFact (according to a press release received by _The Stranger_ last week about something called \"Grillebration,\" which sounds like an emergency room procedure): Our countrypeople will enjoy seven billion hot dogs between Memorial Day and Labor Day. On July 4th, we will engulf more than 150 million of them. One hundred and fifty million hot dogs: We must all do our part. So as a public service in advance of your patriotic barbecue, we present the 2011 Dyke March Wiener Taste Test.\n\nFor the purposes of the test, three representative weenies were selected. (We shall set aside the veggie dogs as a dangerous fringe element.) First, the low: the bottom-shelf supermarket dog. When I was a child, we were fed Bar-S\u2014containing, among other things, unspecified tidbits of pork and beef and mechanically separated chicken, as well as corn syrup (2 percent or less of the latter, the label hastens to reassure).While among a certain set nowadays, serving kids Bar-S would get you reported to Child Protective Services, it claims to be the number-one-selling hot dog in America (with the dubious corollary claim \"Only the best is branded Bar-S\"). Interpreting the runes on my supermarket receipt, it appears that last Saturday, a package of Bar-S Classic Franks was on special for just one dollar. That's 12 cents per dog. TWELVE CENTS. Is this a great country or what? (Or what.)\n\nThen there is what might arguably be called the middle way: Hebrew National, the kosher frank that brags about being all beef (unspecified \"premium cuts,\" riiiight), with no artificial flavors or fillers (animated buns on the website wave protest-style signs about this). It's worth noting, however, that Hebrew National\u2014just like Bar-S\u2014contains 2-percent-or-less of four kinds of sodiums, including our friend nitrate. Furthermore, while Hebrew National touts its \"humble beginnings in New York City's turn-of-the-century immigrant neighborhoods,\" it is now a subsidiary of food giant ConAgra, the _Wikipedia page of which_ \u2014with allegations of environmental irresponsibility, labor issues, health violations, salmonella and E. coli\u2013related product recalls, and delicious, delicious more\u2014truly merits all Americans' attention. The price of Hebrew National beef franks at QFC last Saturday: just about 43 cents each.\n\nLast but not least, and just in time for the taste test, Seattle has a brand-new high-class dog: Rain Shadow Meats' house-made wieners. When I phoned the all-local-meats Capitol Hill butcher to inquire what they sold that was most like a hot dog, they reported that after extensive research and development, they'd just debuted their own\u2014just like a regular hot dog, but made with reduced-guilt-and-ick-factor Carlton Farms pork, Painted Hills beef, \"a little bit of ham,\" and a proprietary spice blend, all inside a lamb casing. Rain Shadow proprietor and great-name-haver Russ Flint said that while he only eats a hot dog once in a blue moon (which is really how often you should be eating a hot dog, America), the people deserve a high-quality option. Rain Shadow's wieners are peachy-colored and pornographically large in both length and girth; they're about a quarter-pound each. Uncooked, they are redolent of traditional baloney. At $6.99 a pound, each weenie will run you about $1.65.\n\nThe obvious time to conduct the taste test, as a celebration of all things U.S. of A., was during a friend's annual barbecue along the route of the Seattle Dyke March last Saturday. A gas grill was fired up on the sidewalk; the three kinds of weenies were cooked until nice and hot, with good grill-markage (except the Bar-S, which due to an unforeseen grill hot spot obtained an all-over char to which no one objected). The dogs were ensconced in cheap, squishy buns (the only proper hot-dog conveyance, no matter what Macrina Bakery may offer). The smaller dogs were cut into halves and the Rain Shadow behemoths cut into thirds, as no one in their right mind wants to eat three entire hot dogs; ketchup and yellow mustard were made available, though if utilized, had to be applied to all three samples for test consistency. The dogs were fed, unidentified, to study participants (though without actual blindfolding, as that seemed too complicated). Now: the results.\n\n**Bar-S Classic Frank:** Most test subjects were administered this weenie first, and the general consensus was that, lacking any basis for comparison, it was completely adequate if \"not exciting\" (Ben K., attorney). One subject pronounced it \"an all-American, delicious hot dog\" (sound engineer and education coordinator Jeffery). Small-business-owner Greg theorized that, due to its highly processed look and squishy consistency, the Bar-S sample might be a veggie dog; upon sampling, however, he said, \"It tastes like it's real meat.\" His hypothesis briefly spread, leading Toshi (artist) to say that the dog \"looked fake . . . I don't feel like it's meat.\" Zac (another artist) reported simply, \"It tastes like a hot dog, straight-up.\" Greg, with the gimlet eye of a capitalist, offered the only real condemnation: \"If these were your hot dogs, I wouldn't invest in your business.\"\n\n**Hebrew National Kosher Frank:** Several test subjects whose parents were less skinflinty than mine identified this as the taste of childhood. Ben K. got a bit misty-eyed, saying, \"It's reminiscent of the classic hot dogs of my youth.\" It was judged less squishy than Bar-S, as well as saltier; \"Salty-delicious,\" said Sara (arts administrator), while another subject (me, not blind, but whatever) felt it was oversalted (and I like salt). Upon visual inspection, Greg pronounced this dog to have the appearance of real meat; after tasting, it was judged to have better flavor, \"a little smoky\u2014good texture\u2014I'm impressed.\" Jeffery cannily identified it as a kosher dog. \"It's real good,\" he said succinctly.\n\n**Rain Shadow Meats' Quality Dog:** Well, what do you know? This was hands down the wiener-winner. Jeffery reported (again, cannily) that it had \"lots of actual flavor\u2014less like a hot dog, more like a sausage. I feel like the pig was killed within 50 miles of here.\" Conjecturing about the methodology of the test, Zac said, \"I feel like we're going up in quality\u2014maybe it's psychosomatic, but I feel like we went up in meatiness.\" Ben B. (floor refinisher) admired both the crunch and the spice. And yet the gains in texture and taste were not at the expense of archetypal hot dog flavor; as Sara put it, \"Mmmmm\u2014hot dog.\" Greg liked the looks of both the size and the color, and after one bite said, \"This is it. This is the shit. This is in my refrigerator. It tastes like people just cared more, whether it's quality ingredients, craftsmanship\u2014it has more dimension. Someone cared.\" Informed about the added expense, he opined that if the hot dog was still only a couple bucks, it was entirely worth it\u2014you do get what you pay for. Ben K. concurred, lauding the Rain Shadow dog's \"multiple flavor notes.\" He then summed up the taste test thusly: \"Three wieners in my mouth\u2014what a perfect Pride weekend.\"\n\nNot long after, the Dyke March went by with beating drums, aglow with the fresh legalization of gay marriage in New York and the late-evening sun. March participants wore rainbows and trilby hats and gold lam\u00e9 shirts and no shirts at all, flashing peace signs and waving. The taste-testers screamed in support until all were hoarse. Is this a great country or what? (Great country!)\n\n### THE MISSING LINK\n\n### By Brett Anderson\n\n### From _The Times-Picayune_\n\n### After eleven years of covering New Orleans' food scene\u2014most memorably through the tragedies of Hurricane Katrina and the BP oil spill\u2014Brett Anderson recently took a year's leave from the _Times-Picayune_ (in the throes of a controversial downsizing) to accept a Nieman Fellowship at Harvard.\n\nDonald Link barely gave the chickens a chance to stop sizzling before he put his hands around them, subjecting each to a tactile examination that looked like nothing so much as a quarterback blindly feeling his way to a football's seam.\n\nOne of the chickens looked like wild game, its flesh darkened by injected Cajun spices vivified by the flames in the wood-fired oven behind it. The other, which had been brined overnight, wore the more typical mottled gold-brown armor of roasted farm-raised fowl. Both stood upright in cast-iron pans, impaled by beer cans. Link appraised them while sucking the grease off his fingers. \"Turns out there are a lot of ways to cook a chicken,\" he said.\n\nThat statement of the obvious prompted laughter in the peanut gallery behind him, at the edge of the open kitchen inside Cochon in Lafayette.\n\nThe opening of the restaurant, a spin-off of the original Cochon in New Orleans, was still three weeks away, but on this hot August night, Link and his team had crossed the threshold where obsessive planning gives way to undressed rehearsals. Ryan Prewitt, until recently chef de cuisine at Herbsaint, Link's flagship New Orleans restaurant, explained, \"We talked about (cooking chicken) for like three hours last night.\"\n\n\"It got pretty heated,\" chuckled Stephen Stryjewski, who is, along with Link, chef and co-owner of both Cochon locations.\n\nCochon Lafayette has more than just a name in common with its New Orleans counterpart. The most important similarity is a concept that encapsulates Link's vision of what, to use his words, \"Cajun food has become.\" Not since Paul Prudhomme opened K-Paul's Louisiana Kitchen more than three decades ago has New Orleans seen a new restaurant elicit such a phenomenal response from such an array of diners. That both happen to hang their hats on the food of Acadiana is a topic ripe for academic inquiry.\n\nEvidence of Cochon's success goes beyond the crowds that regularly congest the restaurant's corner of New Orleans' Warehouse District.\n\nSince Cochon opened in 2006, both Link and Stryjewski have won prestigious chef awards from the James Beard Foundation. In 2008, _The New York Times_ ranked Cochon the third best new American restaurant outside New York. Link also won a Beard for \"Real Cajun,\" his provocatively titled cookbook that delves deep into the pot that inspired Cochon's creation. \"Gossip Girl\" star Blake Lively was so besotted with the restaurant's sweet potato sauce at a recent visit that she tried to persuade staff to circumvent FDA regulations by sending her some inside a disemboweled teddy bear. (The response she received from Cochon, according to _Glamour_ magazine: \"We are not the drug cartel.\")\n\nStill, the new Cochon is a re-creation of the old one, not a straight replication. (An item not central to the New Orleans Cochon repertoire: roast chicken.) The fine distinction begins to explain why Link's journey back to his native Cajun country\u2014Lafayette is its putative capital\u2014has been filled with trepidation as well as joy.\n\nThe chef's family roots run deep in the region: He was raised in Lake Charles, on Cajun country's southwestern edge. But as much as Link identifies with the cooking of his\u2014and perhaps as importantly, his family members'\u2014youth, there is no erasing that he became a big shot in a city whose relationship to Cajun country has dysfunctional dimensions. As Billy Link, a Crowley soybean, rice and crawfish farmer, put it, \"New Orleans is New Orleans, and there's a line between New Orleans and here. They don't mix well, in a way.\"\n\nBilly Link, who is either Donald's third or fourth cousin (it depends on whom\u2014and when\u2014you ask), was leaning against the poured-concrete counter separating the restaurant's kitchen from one of two main dining rooms. He'd arrived with his wife, Becky, and their two young sons to feast on the dishes Cochon's chefs were fine-tuning while test-driving the new kitchen equipment.\n\nThe banquet included the two roast chickens, along with one that had been cooked in an outside smoker built by Dwane Link, another cousin; two darkly crusted pork shoulders; a whole ribeye roast cut into bite-size strips; a pan of shrimp in a butter sauce spiked with the Brazilian peppers that Donald Link grows in his Lakeview backyard; and smothered rabbit provided by yet another cousin, served with rice that Billy Link is supplying the Lafayette restaurant.\n\n\"That's the old Cajun style right here!\" Billy Link proclaimed, delighted by the sight of the rabbit, which he called, in an exaggerated French-Cajun accent, _lapin._ \"If they cook it like this, they'll be all right.\"\n\nBilly Link has known his cousin only as a successful chef, having first met Donald at a family reunion six or seven years ago, and relishes his role as an unofficial critic of Cochon's food. He's playfully dismissive of meat smokers as an influence of the Zaunbrechers, the Cajun family on Donald Link's mother's side. (\"The Link side? Non-smokers.\") He wore a Cochon T-shirt but actually prefers Herbsaint, where the food reflects European traditions as much as Louisiana ones.\n\nThe preference could be a simple matter of taste. It also could have something to do with the games Cochon plays on native Cajuns' memory and sense of pride. The phenomenon might be summed up by the review Billy Link said a group of his friends gave the New Orleans restaurant after visiting: \"We can cook better than that.\"\n\nDonald Link responds to his family's ribbings the way he responds to irritants both mild and severe: with a crooked smile that causes him to resemble a cat that just made a snack of a pet canary. While he's no stranger to cameras or laudatory press, by the standards of a moment where chefs can become television stars without ever running a restaurant, Link counts as a throwback to the days when chefs let the food speak for itself. He insists, \"I am not trying to be a celebrity chef.\"\n\nThe strategy has served Link well, and not just economically. Anthony Bourdain, the acerbic chef, television host and best-selling author, has said of Link, \"there's no one in the business with more credibility.\"\n\nThe challenge in Lafayette is that credibility earned for cooking Cajun food in New Orleans isn't exactly a recognized currency. In fact, it could be a liability.\n\n\"A lot of (Cajuns) know of him who haven't tried his food, but they know of him because he's up here,\" Billy Link said of his cousin, raising his hand up high to illustrate the chef's exalted status. \"And they're waiting for him.\"\n\nOne of the many ironies attending Cochon Lafayette is that its owners don't regard area Cajun restaurants to be their primary competition. Ask Link or Stryjewski what inspired them to open in Lafayette, as opposed to, say, Houston or Covington, both of which were considered, and they will invariably talk with amazement about the crowds at Pamplona Tapas Bar or the slick Japanese restaurant Tsunami in Lafayette's old downtown.\n\n\"You need to check that place out on a Friday night,\" Stryjewski said ofTsunami. The night-clubby restaurant is not the sort one would expect to impress Stryjewski, a tattooed, bearishly boyish man who in plain clothes often appears to have just stepped off a skateboard.\n\nBut Tsunami, like Pamplona, captured the chefs' attention because its crowd, particularly on weekends, exposes an indigenous population of young adults whose interests clearly go well beyond\u2014and possibly don't even include\u2014boudin and zydeco.\n\nThe opportunity Cochon's chefs see in Lafayette has as much to do with business as it does aesthetics, and it is similar to the one they road to fame and profit in New Orleans. While the level of attention Prudhomme had brought to Cajun food in the 1980s altered the way Americans eat, the Opelousas-born chef's fame was so widely felt\u2014and his food so widely misinterpreted\u2014that it sparked a debate over authenticity of the cuisine that has yet to quiet.\n\n\"Stir the Pot: The History of Cajun Cuisine,\" a definitive book on the subject, debunks the myth that Cajun food was developed in a vacuum; authors Marcelle Bienvenu, Carl A. Brasseaux and Ryan A. Brasseaux call it a \"cross-cultural borrowing of the diverse ethnic and racial groups that have co-existed in the Bayou Country since the late eighteenth century.\"\n\nCochon entered into this historical fray by exposing one thing Prudhomme's revolution did not spawn: modern Cajun restaurants that uphold the highest standards of quality and service. The very fact that Link's team drew a bead on Lafayette suggests that this has been the case not just in New Orleans but in Cajun country itself, an implication that steers the age-old debate over authenticity into uncharted, potentially turbulent waters.\n\nLink and Stryjewski are intense students of Cajun cuisine and its evolution. And the chefs' idea of \"real\" Cajun food around Lafayette tends to be found in the same places Cajun food purists look for it: on home cooks' stoves or in decidedly blue-collar restaurants like T-Coon's and Laura's II, both order-at-the-counter, rice-and-gravy plate lunch places that have almost nothing in common, at least atmospherically, with Cochon.\n\nBut in the crowds found at the more modern non-Cajun restaurants, Link sees \"an indication of the desire of this city. If you want to feel metropolitan, if you want to go for a glass of wine and a decent meal, where do you go? It's usually fried food and beer and cocktails, and if there is wine, it's not good wine. I think there's a lot more sophistication going on in these small towns that's not being reflected in the restaurants.\"\n\nDemographic evidence supports Link's hunch. The 2010 Census data puts the median household income in Lafayette Parish at $47,901.That compares to a $35,505 median household income in Orleans Parish, according to data provided by the Greater New Orleans Community Data Center.\n\nBut is it possible tonier Cajun-style restaurants are few and far between in Cajun country because Cajun diners don't trust restaurants with wine lists to properly represent a folk-art form born of subsistence living many people still remember?\n\nPat Mould was the chef at Charlie G's when the restaurant opened in Lafayette in 1985. He remembers raising eyebrows with the restaurant's contemporary Louisiana cooking and sleek interior, which was designed by a prominent Chicago architect.\n\n\"Because we had this perception of being a citified restaurant, we got a ton of (criticism),\" Mould said. \"People were suspicious. We have this jaded perspective that no one's going to cook it better than mama.\"\n\nThat Charlie G's remains one of the region's relatively few high-end restaurants interpreting Louisiana cuisine also points to persistent assumptions about the corrupting effects of elevated social status. In an oral history conducted for the Southern Foodways Alliance by the New Orleans historian (and Lafayette native) Rien Fertel, T-Coon's owner David Billeaud said, \"I'm not a chef. I'm a cook. Cooks work hard.\"\n\nIn New Orleans, Cochon proves daily that questions surrounding its food's authenticity are matters of semantics and style, not substance. The food's rusticity provides cloud cover to a technical proficiency that is the mark of professionals who regard the chef title as an honor earned through labor. The results\u2014the fried rabbit livers riding pepper jellied toasts, the hogs-head cheese shaved over fresh peas, the skillet-cooked rabbit and dumplings based on a Link relative's recipe for squirrel\u2014are almost always prettier than anything a Cajun grandmother has ever served.\n\nLink and Stryjewski, after all, are not Cajun grandmothers. They're chefs whose skill-sets and sensibilities were formed in restaurants as far away as northern California. But because the ingredients and recipes ground Cochon's food in Cajun country's bayous, prairies and marsh, Cochon cuts through the social baggage\u2014name another Beard-winning restaurant offering iceberg lettuce salad, unironically\u2014that has weighed on Cajun cuisine since its commercialization.\n\nDiners can reasonably argue that they've never had anything like Cochon's braised pork cheeks in Abbeville, or that it is heresy to charge $8 for an oyster-meat pie, flaky as Cochon's is. But it would be impossible to conclude after eating either dish at Cochon, perhaps with a bottle of Burgundy wine alongside a free cone of fried pigs ears, that the chefs regard their source of inspiration as a backwater.\n\nStill, theories that hold true in New Orleans are being tested all over again at Cochon Lafayette. And Link knows it.\n\n\"As a chef your (work is) always on the table. You're always up for discussion, \" Link said. The difference in Lafayette, he explained, is \"everybody is a food critic who could make or break you. It's a whole other level.\" You can't \"PR your way out of\" a bad night's performance when there are no waves of tourists coming in behind the diners you may have disappointed on an off night.\n\n\"I'm feeling way more pressure performance-wise,\" Link added, comparing Lafayette to New Orleans, \"because it's deeper.\"\n\nThe chef was sitting on the deck outside Cochon after dinner service on Sept. 15, the restaurant's opening night. The task of convincing Lafayette diners that the restaurant is adequately respectful of Cajun cuisine's hardscrabble roots is further complicated by its location in the city's River Ranch development, a model of mixed-use New Urbanism that architecturally looks more suburban than urban.\n\nT-Coon's is only three miles away, but Cochon Lafayette's closest and fiercest business competition may be the Bonefish Grill. Earlier in the night, the River Ranch outpost of the Florida-based national chain hosted overflow crowds, mostly locals angling for a good view of the LSU football game. (Bonefish is also where Link's father, who lives in Lake Charles, drove to celebrate his most recent birthday.)\n\nCochon Lafayette won't conjure visions of the rural idylls on full-color display in Link's cookbook, but it is beautiful, particularly at night, when its lights cast a soft glow on the Vermilion River running just below the herb-and citrus-tree-lined deck and terrace. At 6,000 square feet, the restaurant seats around 250, twice as many diners as its sister location in New Orleans. It also evokes the original, with its pigmented concrete floors, blonde wood accents and open kitchen.\n\nBeth Hebert ate her lemon-and-garlic-scented oven-baked shrimp at the restaurant's expansive bar, which overlooks the river and suggests what a fishing camp might look like if renovated by an architect specializing in urban lofts. Hebert was in Lafayette visiting relatives from her home in Los Angeles. She declared Cochon's food \"not unrecognizable\" from what she knew growing up. Still, she said, \"I couldn't take my parents here. They're old. They'd be confused.\" Hebert's friends in L.A.? \"They'd absolutely love this place.\"\n\nIf Cochon Lafayette succeeds, bridging this generational divide could be its greatest accomplishment. Cognizant of this challenge, the owners larded the opening-night staff with seasoned Link Restaurant Group operatives. Among them were Stryjewski and Prewitt, who was recently promoted to oversee all of the company's restaurants' kitchens, including Cochon Butcher, the sandwich cafe and Cajun-style butcher shop in New Orleans.\n\n\"One of the reasons we're doing this is to give our people new opportunities, provide them a career and a life,\" Link said. \"If we didn't have the talent to open this place, we wouldn't be.\"\n\nTogether with Kyle Waters III, Cochon Lafayette's chef de cuisine, the restaurant's team built a menu around Cochon classics while, as Link put it, \"trying not to appear like we're competing with anyone's grandma.\"\n\nThe smothered chicken nods to a regional mainstay, only Cochon's is smoked and gilded with pickled onions and mustard seeds. One older man told Link he enjoyed the grilled skirt steak \"but didn't know what to do with\" its side of collard green slaw. (Major difference between the slaw and traditional smothered collards, which are also on the menu: length of cooking.)\n\nThe fried redfish collar was certainly familiar to any Cajun fisher who has refused to waste any morsel of a day's catch\u2014never mind that other diners may recognize the dish as a staple of Japanese cooking, too.\n\nCochon Lafayette's boudin is smoked, its casing charred, the space in the cavity where it's split filled with a pinch of sliced pickled chile peppers. The dish represents a reversal of Link's original vow to stay away from boudin in Lafayette for fear of becoming ensnared in the contentious regional argument over whose is best. So what if his doesn't resemble anything anyone has ever eaten in the cab of a pick-up?\n\n\"There are a lot of cultures involved in Cajun food,\" Link said. \"No one can really lay claim to it. Why does it have to be one thing? Why can't it be different? Why can't it be in an entirely different context?\"\n\nOn opening night, Cochon Lafayette was already addressing these and other questions. The restaurant's parking lot was full.\n\n### FORAGING AND FISHING THROUGH THE BIG BEND\n\n### By Gary Paul Nabhan\n\n### From _Desert Terroir: Exploring the Unique Flavors and Sundry Places of the Borderlands_\n\n### Long before it became a foodie mantra, \"eating local\" was a guiding principle of Gary Paul Nabhan's pioneering work as a farmer, food writer, and conservation biologist. In _Desert Terroir,_ he gives us a deeply personal look at how his life in the American Southwest forged that connection to the land.\n\nThere was a particular moment in my life when I knew I must live where I could fully taste the place in which I lived every week for the rest of my life, if not every day. I call it my Seek-No-Further moment, named for an heirloom apple that sometimes grows in old streamside orchards of the Desert Southwest. This revelatory moment did not occur exactly where I live today, but along the Big Bend of the Rio Grande, where barren rocks and towering yuccas dwarf any gardens, fields, orchards, or pastures.\n\nIt was along the Big Bend that I learned the pleasure of making a meal of foodstuffs gleaned only from the surrounding landscape. I gained the urge to make such a meal by meeting some poor Mexican families who had to eat off the land in order to survive. My time among them humbled me, for it suggested that I should never again ignore or waste the harvestable foods within reach of where I stood. To do otherwise would certainly dishonor the borderline families that so generously shared with me the little foods that they had, but it would also dishonor the plant and animal foods that emerged from some of the driest reaches of this earth.\n\nMy revelatory moment of homing took place some twenty-one years before I write this, but it has taken me all this time to fully acknowledge how that epiphany has guided my life since then. Before Big Bend, I had moved along listlessly, without much of a place or taste of my own. I meandered through life like a disengaged visitor, strolling through the garden of earthly delights merely reading the signs in front of each plant rather than reaching out to pluck the ripest among them.\n\nAnd then, as I broke out of the chaos of cactus and brambles on that day of epiphany, I could at last see, as well as savor, the fruits of the tree of life growing right before me.\n\nIronically, my Garden of Eden was devoid of apples (no Seek-No-Furthers nor any other heirloom varieties). Instead it tasted of cornmeal-crusted catfish, wild oregano, chile pequin, and a few poisonous nightshade berries tossed into some goat's milk to make it curdle. If you can wait a little longer for me to cast the rest of this story, I think you'll understand how a catfish, some goat cheese, chiles, and oregano came to lead me home.\n\nOddly, I can't remember much about my life at that time. I had been living rather unhappily in the sprawling metropolis called Phoenix, working at a botanical garden filled with desert plants that seemed to be assaulted on all sides by asphalt, concrete, plywood, wallboard, and two-by-fours. Fortunately, someone suggested to me that I should study the edible flora of the Big Bend, and that I might be able to get a deal serving as a guest guide with a rafting outfit that regularly ran through the canyons there. I had never spent much time in the wilds of Texas nor in the _monte_ of Coahuila, but portions of those areas had recently been conserved as biosphere reserves. I knew little about biosphere reserves except that they allowed the local, more sustainable uses of edible and medicinal plants to continue within their boundaries. I needed to get out of the city, and Big Bend was the ticket.\n\nThe Big Bend, as you may know, is a raggedy dogleg of the Rio Grande. That river\u2014hardly more than sixty yards across and four feet deep through much of the Big Bend\u2014briefly cuts through a series of canyons with walls four hundred feet tall. The population density of that region of the U.S.\/Mexico border rivals that of the bleakest reaches of North Dakota. There are more cacti than people along the Big Bend, and more emptiness than busy-ness.\n\nPerceived by most folks as an empty, desolate, and desultory place, Big Bend has also been neglected by wild food enthusiasts, but is held in great esteem by desert rats. I myself had not thought much about nor thought much of the Tex-Mex borderlands until I was granted the chance to float and eat my way through Big Bend on a raft trip with Far-Flung Adventures in the autumn of 1988. When the proprietors of Far-Flung sent me a postcard of their six-month-old baby smiling, sitting naked in a pile of jalapeno peppers, I became intrigued.\n\nThere was a lot more than jalapenos, however, among the groceries when I and the other rafters arrived at the \"put-in\" site known as Solis, where we loaded the rafts with waterproof river bags, grills, canteens, and coolers with a half ton of canned food and beer. I began to wonder whether the familiar foods we'd brought along might serve as a disincentive for discerning the unique flavors of the Big Bend, but I readily accepted a cold one when someone popped open a beer and offered it to me.\n\nMexico's Sierra San Vicente was in sight as the trip began, and as we glanced downstream, we saw the glistening river disappear into a three-mile \"tunnel\" called San Vicente Canyon. Its limestone walls rose above us in subtle shades of grays, buffs, and creams; where the sun shined upon them, they were so barren and bright that we could have gone stone-blind by simply staring at them for more than a minute's time.\n\n\"This ain't Eden,\" I grumbled to myself, wondering how in the hell I was going to lead an \"edible plant walk\" later in the day when everything in sight looked as though it might bite at _me_ before I could take a bite out of _it._\n\nIf there was anything edible along that first stretch of San Vicente, it remained hidden from me. However, Marcos Paredes, our head guide, reminded me that we were visiting during the season of hunger weather, when food was hard to come by. That's why Mexican cowboys brought the cowhides they cured and _candelilla_ wax they refined across the river to get cash so that they could buy canned groceries and beer. Yes, there were some desert foods out there, but they would be slow to reveal themselves. Instead, we turned our immediate attention to all the other plants that poked their thorns and spines and stickers out from the otherwise naked canyon walls.\n\nAfter our first riverside lunch of cold cuts, chips, and beer in the shade of an old mesquite tree, we put the rafts in and began to float the Rio Grande in earnest. Like the Rio Colorado, the River Nile, and most other desert rivers I have floated, the Rio Grande has its banks choked by a dense thicket of shrubby tamarisks and a canebrake of bamboo-like grass that the Mexicans call _carrizo._ As the sinuous river curved and looped back on itself like a coiled snake, grassy curls of _carrizo_ offered a minimalist's hint of greenery in swirls and corkscrews and commas on the water's edge. Beyond this narrow little lip of lushness, the limestone slopes of San Vicente Canyon appeared barren, that is, unless you got down on your hands and knees beneath the creosote bushes and took a good hard look at the ground. There, hunkered down amid the dull-colored stones, were beautiful \"rainbows,\" highly prized by cactus collectors, along with a dozen other species of cryptic cacti unique to the Chihuahuan Desert. In some places, as many as thirty-five kinds of spiny gray cacti were hiding out on each acre, some of them bearing delicious strawberry-like fruit. To recognize the cacti and to eat their succulent fruit, I first had to learn how \"to get over the color green\" (as Wallace Stegner once urged us to do) so as to take note of _their world._\n\nDown on the river, the most obvious objects other than our rafts were two-liter Coke bottles bobbing up and down, especially in the eddies just downstream from where the river bent.\n\n\"Holy _mole!\"_ one of my raft companions exclaimed, pointing and frowning. \"The Mexican folks around here must toss all their Coke bottles into the river with no respect for the river itself.\"\n\n\"Well, I'm not saying they're pretty, but at least they're functional,\" explained Sammy, a boatman from the Far-Flung crew. Sammy was skinny, leathery, gray-haired, and goofy, and wore as few clothes as possible, rain or shine. He rowed as he spoke:\n\n\"A lot of those bottles . . . I guess they've been scavenged from our camps or rescued from the river. The Mexican folks retrofit 'em, turn 'em into floats for jug-lines to catch catfish.\"\n\n\"What do you mean . . . they're not all free-floating trash?\"\n\nSammy shook his head. \"No, not at all. Especially during the drought, people need to eat from the river cuz, well, they can't eat a helluva lot from the land. So they gather up four or five of those two-liter Coke bottles, put a little gravel in them, sometimes paint 'em white, caulk them closed, screw the caps back on, and they've got some pretty cheap floats for their jug-lines. Then they tie some braided twine tight around the bottlenecks. They might extend the line, oh, maybe some twenty feet or so, and start adding hooks, leaders, or sinkers, or anchor them with bricks. Or tie them to a lead line tied to a tree off the bank. Now, when a catfish hits, the bottle dips, the gravel rolls to one side, and weight shifts to the end with the cap. They come by and see the jug has shifted position, check it, and find a big of catfish on the hook.\"\n\n_\"Muy suave._ Pretty slick. Wish we could do it,\" I said.\n\n\"Well, hell, just keep your eye out for one of them jugs that has got loose.\" Sammy put one oar in his lap and pointed over toward an eddy, where the drift lines tend to get waylaid. \"We like to pull in the drifters, anyway. Maybe it'll already have a fish on it, maybe it won't, but you can float it behind the raft for a while. As long as you don't let it drift loose again, and check it every half hour or so, I doubt anyone will bother you.\"\n\nLate that afternoon, after we had set up camp in San Vicente Canyon, not far above the town of Ojo Caliente, I found a ghost line drifting along with a jug behind it. I pulled it in, refurbished it a little, and set it out. Within fifteen minutes of placing it in the water, I saw my Coke bottle float by, bobbing. I grabbed some gloves out of my pack, and a bucket from the cooking area, and went over to pull up my first fish of the trip.\n\nIt was most likely a Chihuahua catfish, close kin to the widespread channel cat, but a species more common on the middle and lower Rio Grande. Its tail has a shallower fork than those of both channel and headwater catfish, and the Chihuahua seldom reaches a full foot in length. Differences aside, they are all good eating, as we learned later that evening after gutting and grilling four or five more that we had caught (by rod as well as by jug).\n\nWhile the Far-Flung crew reheated a pre-prepared pan of enchiladas, poured out a can of beans into a skillet to refry, and mashed some avocados into guacamole with the bottom of a beer bottle, a couple of us sat in the sand next to the grill and pan-fried our catfish. We passed around nuggets of catfish as finger food just after dark, and most everyone agreed that fresh catfish was much tastier than canned beans. The catfish was soon forgotten, however, as more beer appeared out of the coolers\u2014Dos Equis, Tres Esquis\u2014all the X's you could count or drink. A little beat-up guitar was passed around the group, and we tried our best to remember the words of songs by Gary P. Nunn, Doug Sahm, Steve Fromholz, Butch Hancock, and other Texas songwriters.\n\nWith the jagged lines of the canyon rim above us, I put my sleeping bag on a tarp ten feet from the river and looked up at the sky to watch for the moonrise from my sleeping bag. As usual, I faded off to sleep before the moon arrived.\n\nIn the twilight time the Mexicans call _la madrugada,_ I awakened to roosters crowing not all that far away. The stars were still visible in the slot of sky framed by the canyon, but I could see from a rose-colored patch of the Sierra del Carmen in the distance that the day was coming on. I pulled on my jeans, slipped into my huaraches, put on my straw cowboy hat, grabbed my plant collection bag, and began to walk toward the sound of the roosters, where I presumed I would find the little spring-side settlement of Ojo Caliente.\n\nThe hot springs had been tapped to irrigate a few small fields, and they were in the last throes of being harvested. There were melons and squashes, corn and beans, broom sorghum, sugar cane sorghum, and pasture grass. It was hard to tell who had harvested more of the crops, the Mexicanos or the grasshoppers. Grasshoppers were everywhere, hopping around as if they had bit parts in a motion picture about biblical plagues. I wondered if these _grillos_ were ever grilled and eaten when other foods were scarce. (Later in life, I carried _grillos asados_ from Mexico along with me as a trail food whenever I ran rivers.)\n\nJust as the dawn's light spilled into the canyon, I met a young Coahuiltecan woman who was going to fetch some drinking water from the springs. She carried two beat-up plastic water jugs. She had three small children in tow, and another one or two on the way, judging from her bulging shape. She was willing to talk for a while in exchange for me carrying the water back to her house. Her name was Maria de Lourdes.\n\nWhen we were going back to her place, she stopped to rest in the shade of a big mesquite, so I opened the bag of plants I had been collecting and dumped them in the sand before her and the children.\n\n\"Would you mind telling me if any of these are eaten here?\" I queried her.\n\n_\"Pues,\"_ Maria said haltingly, as she sorted through the plants freshly picked from the river's edge. \"Well, you should really talk to an older, more knowledgeable woman like the _yerbera_ downstream in Boquillas del Carmen.\"\n\n\"The _yerbera_?\" I asked.\n\n\"You know, well, she's the _curandera._ Her name is Isabela. Everyone knows her in Boquillas. Even the gringos come to see her.\"\n\n\"Well, I'll try to find her when I get to Boquillas, but perhaps there are plants here too that you yourself use as food . . .\"\n\n\"Well, yes, there are some, but most of these plants you have picked this morning are medicines, not foods. I can tell you a little about them, but Isabela knows them better.\"\n\nMaria de Lourdes paused for a second to rummage through the pile, and her children crowded in around her.\n\n\"Look, you have the _gobernadora_ [creosote bush]. Very good for stomachache and for kidneys. You have _calabacilla loca_ [buffalo gourd]. That's not good to eat. It's good for bleaching out our bowls and towels and pots and pans. And there, you have the _hierba de la golondrina. We_ give it to the babies to stop their diarrhea.\"\n\n\"Sounds like you have got a lot of plant medicines around here that can help you deal with stomach problems,\" I noted.\n\n\"What would you expect?\" Maria said flatly, running her fingers back through her braided hair. Her mouth closed tightly before she began again. _\"Asi es la vida_. That's how it is where you have bad well water. Here, you got river water contaminated with _mierda_ from the cows. The meat and the vegetables spoil quickly in the heat if you don't have any refrigeration. On top of that, the drought all the time, too. It can keep you hungry, too.\"\n\n\"Well, what do you try to grow over here?\" I asked.\n\n\"The _nopal_ and the _tuna_ \u2014what you call the prickly pear cactus\u2014that's the most reliable thing we got. Even in drought, we can rely on it. Not so the beans and the corn and the squash and the melons over there in the field. Even if we irrigate them, maybe we get something two, maybe three out of five years. If the heat doesn't get them then it's the grasshoppers. Or else the cattle break in. I'll show you what we grow, what we keep inside the fence around our house. . . . I can always grow some onions there. Sometimes the plants of chiles, tomatillos, and cilantro survive. You know, I just throw the dishwater on them. That's all they get.\"\n\nMaria de Lourdes got up to go, motioning the kids and me to follow. I grabbed the water jugs she had filled at the springs.\n\n\"What else can you eat when the corn and beans fail?\"\n\n\"You know, you have some of them here in your pile [of plants], but it takes work to pick enough of them to eat. Like this one here, the _garambullo_ [hackberry]. Or that other red berry, _agrito_ [wolfberry]. Sometimes, we'll even make a drink from the mesquite bean pods. Not too often anymore. Like anybody else, we'll eat the wild greens if they come up in the fields. _Qaelites. Verdolagas._ But some years, they don't even sprout\u2014too dry here.\"\n\nWhen we approached her little adobe house\u2014one of only five left in the village\u2014I was ready to drop the jugs of water from my shoulders. I looked at her, so young, pregnant for the fourth time, and wondered how she did it each day. Of course I had guessed that it wasn't easy to fill the larder from this stretch of desert, but I had never realized that a shallow well, some edible wild greens, and berries might be the only buffers against thirst and starvation.\n\n\"So how do you make it? Do the men here gain much cash from the livestock? I mean, do you gain any income from your own work\u2014cooking, crafts, sewing?\"\n\n\"With the children this age? I can't do much more than keep them fed and clean. The men proudly say they are just poor _vaqueros._ You know, they don't really keep enough _vacas_ to gain much cash. If a cow gets hurt, we have to butcher it right away, no matter what the size. Butcher and jerk and dry the meat for later. My husband, he cuts _candelilla_ for wax, cuts _lechuguilla_ for fiber, cuts the _sotoi_ for the bootleggers, picks oregano or _chile pequins._ It's whatever people will buy. Anything so we can keep the kids fed.\"\n\n\"You have wild oregano and wild chiles here?\"\n\nShe pointed to the backyard as I set the water jugs down for her. \"We have a plant or two in our _huerta_ behind the kitchen. But you'll see them down by the river, up on canyon walls, if you know how to look.\"\n\n_If you know how to look_ . . . as I had to learn to look for the ground-hugging cacti and their fruit on the canyon slopes. If you're hungry enough, you _have_ to learn how and where to look for them. You would have to learn how to pick them with as much speed as you could possibly muster. You would have to learn how to dry them so that they wouldn't spoil before the buyer arrives. You would have to learn how to do all of this as efficiently as possible because your family needs food and the drink and the medicines, as well as the cash for the things you can't grow or forage. This desert is not for wimps. It exacts its price. I looked at Maria's hands: a few scars, a few burn marks, chipped fingernails . . .\n\n\"So how much might they pay you or your husband for the harvest of chile pequins?\"\n\n\"Two dollars a pound,\" Maria whispered, as if trying to keep her neighbors from knowing what a deal she received from the buyer who came from the other side of the border.\n\n_Two dollars a pound?_ The same chiles retailed in stores from West Texas across New Mexico and sometimes into Arizona for _sixty dollars a pound_! The harvester\u2014the poorest player in the entire value chain, since the chiles changed hands five or six times before reaching a kitchen table\u2014received only one-thirtieth of the final market value. With the entire economic deck stacked against them, no wonder Maria de Lourdes and her husband could hardly keep their children fed and clothed.\n\n\"Do you have any _chile pequins_ I can buy from you?\"\n\nMaria motioned for me to stay on the porch, and she went inside. She brought me back a glass Nescafe jar filled with small bright-red wild chiles, some of them round, but some of them beaked. I couldn't easily guess their weight, so I just handed her a hundred-peso bill.\n\n\"No, that's too much, and I don't have any change.\" She sighed, looking embarrassed. \"You just take them as a gift. After all, you carried my water.\"\n\nNow I was the one who was embarrassed, if not humbled. \"Keep it all. Get something for the children. . . . I can eat chiles, but they need something else to eat that isn't hot.\"\n\n\"I know,\" she said, relenting. \"I've had to stop eating so many chiles now that I'm pregnant since it makes the baby inside me kick. . . . Here, if you're going to give me money, why don't you take all of our chiles?\"\n\nI heard Marcos or Sammy whistling for everyone to come back into camp, get some coffee and breakfast, and pack up. _\"Es la hora para embarcar_ ,\" I explained, and she nodded.\n\nAs I picked up my plants and turned to go, Maria de Lourdes put a small gunnysack of _chile pequins_ in my hands. She then tucked the hundred-peso bill into her blouse and shooed me out the door. I bowed to her, departing from her ramshackle home with enough wild chiles to last me for a year.\n\nThe next village we visited, Boquillas del Carmen, was many times larger than Ojo Caliente, claiming some fifteen houses down below a mesa, another nineteen houses up on top, and five more intermingled among a few tourist stores. In Boquillas, they will try to sell you anything that can move, that has moved since the time of the dinosaurs, or that will appear to move if you ingest enough peyote to send it into flight. When I was there, I could get bootlegged sotol for five dollars a quart, and false peyote (the star cactus) for the same price. If I had persisted, I might have been able to find some real peyote for far more than that per button. I could also purchase petrified wood, ponchos, ammonites, crystals, leather purses, sombreros, leather vests, baskets, miniature _huapango_ guitars, gallon jugs of cheap tequila, Christmas cacti, cheeses, tortillas, and hot sauces galore.\n\nWhile my rafting companions decided whether or not to try a shot of sotol or to purchase various and sundry curios, I walked around the village until I noticed an elderly woman making something in big bowls on her porch. Thinking that she might be Isabela the Curandera, I approached her and asked if she was an herbalist.\n\n\"No, I'm not her. But is it that you are interested in plants? I am using a little poisonous berry to make cheese right now.\"\n\n\"A berry? May I see what kind of berry?\" I asked.\n\n\"Sure, come on up here onto the porch. I am using the berries of trompillo . . . making a batch _of queso asadero._ See, the trompillo berries are in that bowl,\" she said as she wiped her hands on an apron that covered her flowered cotton dress. Her gray hair was largely covered by a scarf of the same flower pattern.\n\nI stood behind her and peeked into the bowl. There were five little golden berries of trompillo, a deadly nightshade that I guessed to be _Solanum eleaginifolium,_ which is sometimes called buffalo berry in English. It is a thorny, poisonous weed that colonizes overgrazed or recently flooded areas.\n\n\"I crush two fresh berries and three old ones in a half of a cup of tepid water. See? Let them soak until the water is yellow. Like this. See? I strain out the seeds and skins of the berries,\" she explained. \"Then I use it to congeal the curds out of the whey.\"\n\n\"Is the cheese made from cow milk or goat milk?\"\n\n\"Our cows aren't here no more. They're being impounded on the other side. So this batch is from our goats. Bi-national goats,\" she said with a smile. \"They browse on this side of the river, then they sneak over to that side too. They are too fast and crafty for those _Rinches de Tejas_ to catch them and put them in jail. What do you call them? Texas rangers? Park rangers?\" She laughed.\n\n\"Watch. I put the half cup of trompillo water in with three gallons of goat's milk. Here, I stir it for ten seconds. I'm using one-day-old _leche de agria_ [sour milk] for half of the mix, and fresh _leche dulce_ [sweet milk] for the other half. I can't tell you how long before the milk will have begun to curdle. If I leave this batch in the sun, it will turn to curds in three hours! Later in this afternoon, that's when I'll separate the curds from the whey. That's when I'll concentrate the whey. Make into a cream cheese we call _riquezon.\"_\n\n\"But why do they call what they do with the curds _queso asadero_?\" I asked, having wondered why the cheese is said to be \"grilled.\" \"Do they say that the berries cook the cheese?\"\n\n\"No, it's because we actually take the curds and melt them. We do that by kind of roasting or grilling them\u2014haven't you ever seen it?\u2014by heating them on a comal griddle over a wood fire. As they begin to melt on the griddle, we pick them up. Shape them into something like a fat tortilla, you know, a gordita. We turn them over on the griddle and leave them 'til the texture is even. Then we pick them up. We stack them, you've seen them that way, between wax paper in a pile. That's how we make the true _queso asadero.\"_\n\nMilk from bi-national goats. Berries from a deadly nightshade. Cheese tortillas that are grilled as if they were hamburger patties. I had to buy them and try them for dinner.\n\n\"How much does it cost for two dozen of them?\" I asked. She named her price and I complied. She went into her house and brought me out two neat stacks of _queso asadero,_ wrapped them both in a paper sheet, then enveloped the piles within the paper sheet in a pale cotton cloth. She pulled the ends of the cloth over the cheese, then braided them together and tied the braids into a knot that also served as a carrying device.\n\n\"Will I get sick if you put too many trompillo berries into the milk?\" I asked as I counted out my money to pay her.\n\n\"No,\" she said with a straight face, \"you'll be dead! Why do you gringos worry so much about getting sick and dying? By the grace of God, it's to happen to all of us sooner or later. Just enjoy the few days, the ones you have left with us here in this desert,\" she said matter-of-factly. And then she went inside and closed her door. That was the last I saw of her. . . .\n\nOf course, I could not undo the absurdity of floating down a river at a hundred dollars a day through a place where the few remaining residents would make only three dollars an hour for most of their lives. My catching a couple of fish did not make the loads of store-bought groceries in the eight coolers suddenly disappear. I had been floating down the River of Inequity, and all of its blatant juxtapositions were still immediately before me.\n\nBut that catfish, _queso asadero,_ chile, and oregano gave me a way to be nourished by both sides of the river, and by the life of the river itself. I would no longer privilege one side over the other, nor let myself be fed by something utterly remote from where I actually stood.\n\nThe next day, as we floated out of Big Bend National Park and arrived at our \"take-out\" location, one kind of journey came to a close, but another one had started. My imagination and my palate had been sown with the seeds of a Seek-No-Further sensibility that would soon germinate in the desert soil of the Southwest borderlands, where wild chiles and oreganos still grew whether we noticed them or not. But I had noticed them, and I decided to put my own roots into the same soil.\n\n### ITALIAN AMERICA\n\n### By John Mariani\n\n### From _Saveur_\n\n### John Mariani's eclectic tastes in food and wine have for years informed his reviews in _Esquire, Bloomberg News,_ and his weekly Virtual Gourmet newsletter. In his latest book, _How Italian Food Conquered the World,_ he comes home to the cuisine he grew up on: New York City Italian-American.\n\nAt my family's home in the Bronx, we ate slices of fresh, milky mozzarella with seeded bread from the Italian bakery down the block, macaroni shells stuffed with Polly-O ricotta, lasagne with little meatballs between the layers, baked rigatoni, eggplant parmigiana, chicken cacciatore, beef braciola. We drank chianti that came in a straw-covered flask and espresso from a drip pot, with a sliver of lemon peel. This was the 1950s and '60s, and though Mom was always cooking for our family and friends, Dad knew his way around the kitchen, too. He took to the stove on weekends, concentrating on a single dish: lobster fra diavolo, because my mother hated handling live lobsters.\n\nWe thought we were eating authentic Italian food, because the dishes were the same ones all the other Italian families we knew cooked and ate. But in reality, our cuisine was an American invention: an amalgam of hearty, rustic dishes brought here, primarily by southern Italian immigrants (my grandparents came from Abruzzo and Campania), then adapted and embellished upon in American kitchens. By the time I started writing about food in the mid 1970s, this homegrown cuisine had fallen out of favor as northern Italian\u2014inspired dishes, deemed (sometimes erroneously) lighter and more authentic, became all the rage. I can't say that I didn't welcome the new trend of delicate fresh egg pasta, or celebrate the fact that grilled branzino had replaced shrimp scampi on so many Italian menus. But I will never deny my love for a supersize plate of spaghetti with homemade meatballs, or an eggplant parmesan hero, with its ample breading and sauce and molten mozzarella. There's a beauty and succor to Italian-American food, and it's for a good reason that so many chefs have been returning to those classics recently, preparing them with a newfound zeal and sense of respect.\n\nThe story of the rise and fall and rise again of Italian-American food is a fascinating one; it's an American story, its plot interwoven with the entrepreneurial drive, embrace of pop culture, proliferation of convenience foods, and creativity of home cooks that has informed our country's culinary spirit. It began authentically enough, with Italian immigrants who were skilled at making the very most from the very least. The _abbondanza_ for which Italian-American cooking is known stems from the fact that these immigrant cooks, most of whom came from dire poverty, took pride in being able to feed family and friends sumptuously on the kinds of foods they couldn't afford back home. Ingredients like mozzarella and ricotta were no longer used as accents, or as meals in themselves: They were added to dishes with abandon. My father's lobster fra diavolo, which was likely inspired by tomato-based seafood stews made with small spiny or inexpensive rock lobster in Italy, was another example: When it was popularized in the 1950s in Italian-American restaurants, it became a lavish dish\u2014far bigger in size and flavor than its predecessors\u2014of fat New England lobsters cooked in a fiery tomato sauce. Another ingredient considered an extravagance in southern Italian cooking, veal, could be found on early Italian-American menus in myriad forms: _alla parmigiana_ (breaded and covered with sauce and cheese), _alla marsala_ (doused with fortified wine), and as massive one-pound veal chops, often stuffed with mozzarella and prosciutto, then smothered with tomato sauce.\n\nFoods from the homeland became springboards for invention in the States. Take pizza, which evolved from its simple Neapolitan roots into styles unlike anything found in Italy (see _Any Way You Slice It_ ), with more cheese, more sauce, and more toppings. Or, tomato sauce, for that matter: When my wife and I first traveled across Italy, on our honeymoon in 1977, we saw neither marinara (that quickly cooked sauce of just tomatoes, garlic, and oil) or the long-simmered \"Sunday sauce,\" stocked with all kinds of meats, which most families I knew while growing up served on Sunday. Of course, tomato sauce exists in Italy; the irony is that tomatoes were brought to Italy from the Americas in the 16th century and considered poisonous by all but southerners, who found them a delicious addition to their meager diet and discovered that they flourished in their sunny clime. Which explains why when more than 4 million Italians, a vast majority of them from the south, immigrated to America between 1890 and 1910, they brought tomatoes, and tomato sauce, with them. Every cook had a version, and it became the food on which immigrant mothers staked their eminence within their neighborhoods. Growing up, we would no more insult a friend's sauce than we would his mother or grandmother. The sauce was sacred.\n\nSoon enough, red sauce became emblematic of Italian food in the United States, embraced by Americans from every ethnic group and marketed by savvy restaurateurs as part and parcel of the cuisine's abundance. My family dined out at least once a week, usually at a place called Amerigo's in the Bronx, which began as a pizza stand in the 1930s and evolved into a restaurant of extraordinary breadth, with a menu that ranged from antipasti to zabaglione, and a dining room decked out with an illuminated waterfall and a mural of the nearby Throgs Neck Bridge. It was at restaurants like Amerigo's that we feasted on the kind of fancy dishes that Mom didn't make on weeknights: I always had the gnocchi with tomato sauce and my brother, the manicotti. My father would order a massive New York strip steak, introduced to the city's steakhouses by Italian-American butchers, and my mother would have filet of sole \"Livornese,\" a dish with clams and mussels, white wine, and a moderate amount of garlic. Portions were huge, including the cheesecake and cannoli for dessert. A waiter came to the table to whip zabaglione in a big copper pot.\n\nThe epitome of this style of dining was Mamma Leone's on 48th Street in Manhattan. That multistoried spectacle of Italian kitsch, with nude statuary and blocks of mozzarella and provolone cheese on every table, opened in 1906 and was operated by the same family until it was sold to a restaurant group in 1959, eventually closing in 1994. Had Verdi lived to eat there, he would have written an opera about it, and Enrico Caruso and Luisa Tetrazzini\u2014both of whom were immortalized in pasta dishes that bore their names\u2014would have sung the leads.\n\nAs much as Americans adored places like Mamma Leone's, Italian-American food was often referred to as grub for \"greasers\" and \"garlic eaters.\" It wasn't until the arrival of first-rate Italian ingredients\u2014many of which had been kept out of the U.S. by trade laws\u2014in the 1970s and '80s that Italian-American cooks were able to reproduce the regional flavors that travelers to Italy complained they could never find in the States. This included prosciutto di Parma, extra-virgin olive oil from different locales, parmigiano-reggiano, arborio rice, funghi porcini, balsamic vinegar, and outstanding Italian wines from producers like Angelo Gaja and Giovanni di Piero Antinori.\n\nBy that time, many Italian-American restaurants had become tired and tiresome, and some restaurateurs tried to refine the clich\u00e9s\u2014and justify higher prices\u2014by turning to northern Italy for inspiration. In New York there were Romeo Salta (opened in 1953), Nanni (1968), and Il Nido (1979); in Santa Monica, California, Valentino's (1972).They all downplayed the red sauce factor, substituting butter and cream sauces and adding\u2014at $20 a plate\u2014risotto and, of all things, polenta, a dish that had been the thrice-a-day staple of poor northern Italians who could afford to eat little else. This food was welcomed as authentic regional Italian: Lasagne with meatballs and meat sauce was dismissed in favor of _lasagne alla Bolognese,_ with _besciamella_ and spinach pasta. Italian-American cheesecake and cannoli were replaced by tiramis\u00f9 and _panna cotta._ The old chianti bottles in straw _fiaschi_ baskets were abandoned in favor of expensive barolos, barbarescos, and \"super-Tuscans.\" Murals of Mount Vesuvio were painted over in favor of blown-up photos of Sophia Loren and Marcello Mastroianni.\n\nRed-checkered tablecloths disappeared; now the tables were set with Frette linens.\n\nThe zeitgeist looked north for another reason: Italian fashion and design (centered in northern cities like Milan and Florence) was all the rage in the 1980s. The chicest new restaurants in the U.S. proclaimed they were Tuscan-style trattorias or grills (even if they didn't serve Tuscan food). Among the first to promote their Tuscan origins were Da Silvano, opened in 1975, and Il Cantinori (1983). Both, still operating in New York's Greenwich Village, became darlings of magazine editors, art gallery owners, and other members of the cultural elite. Before long, their menus were copied across the country, and extra-virgin olive oil became the new red sauce.\n\nThese northern Italian\u2013inspired places adopted the tenets of the Mediterranean Diet, named for a book written in 1994 by cookbook author (and _Saveur_ contributor) Nancy Harmon Jenkins. The basic argument was that what _real_ Italians ate\u2014a diet abundant in vegetables, seafood, grains, and olive oil\u2014was far more healthful than the meaty, rich, fried, cheese-laden, red sauce\u2013drowned food of Italian-Americans. Quick saut\u00e9ed greens and _farro_ were in; chicken parmesan and meatballs were out.\n\nBut who doesn't love meatballs? As influential as the Mediterranean Diet has been, the Italian food you are most likely to encounter in London, Berlin, Moscow, or Mumbai will be far closer to the old \"red sauce\" archetype than to regional Italian menus featuring Alba's white truffles, Sardinia's cheeses, or Venice's cuttlefish ink risotto. Even the most trailblazing purveyors of modern Italian cuisine in America, while proudly serving regional specialties, still champion the good old-fashioned Italian-American classics, even if they change the names. The addictive, confectioners' sugar-dusted fried doughballs known in the Italian-American canon as _zeppole_ often show up as similar _bombolini,_ and what used to be called plain macaroni is now broken down into specific subclassifications, whether it's rigatoni rigate, garganelli, or casarecci. At Osteria Morini in New York City, Michael White\u2014born in Wisconsin and trained at San Domenico, the Michelin 3-star outside of Bologna\u2014serves mostly Emilia-Romagna-style food, but he also offers meatballs in tomato sauce and pasta with white clams.\n\nMoreover, traditional Italian-American restaurants are opening in higher profile, trendier spots. The 10-table, infamously hard-to-get-into Rao's in New York City opened in Vegas in 2006, serving classics like meatballs and veal chops. And, Rubirosa, owned by the same family that runs the 51-year-old Staten Island pizzeria Joe and Pat's, recently opened in downtown Manhattan, serving the kind of food I grew up on, in a dining room painstakingly designed to evoke the classic midcentury neighborhood red-sauce joint. The chef, Albert Di Meglio, and owner Angelo Pappalardo, both worked at the Manhattan restaurant Circo in the 1990s, owned by Sirio Maccioni, the man responsible for inventing pasta primavera and making it one of the most popular dishes of the 1980s at his celebrated Le Cirque. Rubirosa is returning to the Italian-American classics by serving the likes of saut\u00e9ed broccoli rabe and stuffed clams.\n\nPerhaps the place offering the most creative take on Italian-American cooking is the widely praised\u2014\"aggressively Italian-American,\" as Sam Sifton, former restaurant critic for _The New York Times_ put it\u2014Torrisi Italian Specialties. In a storefront on Mulberry Street, Little Italy's main drag, Mario Carbone and Rich Torrisi typically serve more than 300 Italian-American-style hero sandwiches at lunch, and at the neighborhood's annual San Gennaro festival, they set up a booth hawking mozzarella sticks. But where they really capture Italian-American food's melting pot qualities and spirit of innovation is at dinner. In a dining room decorated with a poster of Billy Joel, fifty bucks gets you five courses, which may include their inspired take on garlic bread, slathered with tomato-garlic butter; bowls of still-warm, made-to-order mozzarella; _gemelli_ from Severino, a 40-year-old pasta company in New Jersey, in a hearty duck rag\u00f9; maybe tilefish with pickled green tomato relish or duck breast with broccoli rabe and mulberry mustard. The meal ends with a paper cup of Italian ice and a plate of cookies. That was last night; tonight it will all be different.\n\nWhat's also different is that Carbone and Torrisi, who often incorporate Asian influences from neighboring Chinatown into their menus, use only American products. Nothing's imported, not the prosciutto, not the tomatoes, not the spaghetti, not the bread crumbs\u2014because, what's wrong with Progresso? The message is clear: It doesn't have to be straight from Italia to be special. And if it was good enough for Italian home cooks, then it's good enough for us.\n\nWhenever I eat at these new school, or cheffy, Italian-American restaurants, I never expect the food to taste just like my mother's. These restaurants are a testament to the fact that Italian-American food is its own living, breathing cuisine; that can evolve just like any other. What I love most about where Italian-American cooking is now is that there's an equal respect for the tried and the true, as well as the changing tastes of the day: even at home, the dishes I prepare tend to be lighter, and maybe a bit brighter with fresh herbs, than they used to be. But they still embody all that is genuine, and generously wholesome, about Italian-American food. And they're served with gusto and with just one intention: to make me and my family and friends very, very happy.\n\n### WHAT MAKES SUSHI GREAT?\n\n### By Francis Lam\n\n### From GiltTaste.com\n\n### Francis Lam's incisive food writing for _Gourmet,_ Salon, and currently Gilt Taste deconstructs the cultural value of everything from cherries Jubilee to Chinatown roast duck. You may also know Lam as a commentator for the Food Channel's _Food (ography)_ and a judge on _Top Chef Masters._\n\nA friend of mine once met a delegation of revered Japanese chefs. There was a wizened gentleman among them who was clearly the leader. He spoke little, but the other star chefs deferred to him, paid him obvious respect. My friend finally asked, quietly, \"So, what does the old guy do?\" The response: \"He has mastered rice.\"\n\nTo be honest, I don't know what that means. I mean, I know the difference between a pot of rice that I like eating and a pot that's gluey, but there aren't a whole lot of points between the two. And yet here is a man whose claim to fame among master chefs is that he makes _rice_ better than the rest of them, and to accept that is to accept that there is a level of cooking that most of us will never comprehend. At some point, cooking is not a matter of skill; it's a matter of _understanding,_ of learning to see the differences between one perfectly good pot of rice and another, of the minute details in something that, for most anyone else, is pure pearly blandness. Truly great cooking is, in this way, first an act of learning to see, and then a striving to do. This is why, among chefs, the truism is that simple food is hard.\n\nSushi, of course, is the ultimate in simple food: mostly just rice and a piece of raw fish, it would seem that anyone with a knife and one functioning hand can make it. But take an impossible eye for detail and apply it to fish\u2014Where did it come from? How long should you age it before serving for best flavor? How long should you massage it to make it tender, but still have texture? Where should you cut a piece from, and at what angle, to highlight the flavors of different parts of the muscle? Since temperature affects aroma, how warm should you let the fish get in your hand before serving it? How hard do you press the fish into the rice to form a bite that has integrity, but is not dense?\u2014and you begin to see where a simple food is not so simple. You don't have to buy into all the minutiae a sushi master trades in to know that the pleasures of great sushi span from the animal to the emotional and the intellectual, which is a great trick for anything to pull off, let alone a piece of raw fish on rice.\n\nWhat animates a sushi master? What drives someone to be so focused, to be a god of small things?\n\nJiro Ono, 85 years old and counting, is a revered sushi chef who runs a restaurant inside a Tokyo subway station, and _Jiro Dreams of Sushi_ is easily the best, most beautiful movie about sushi you will see this year, or, let's face it, probably any other. The film is part documentary bio-pic, part food-blogger's wet dream. (OMG, did you _see_ the super-macro shot of that tuna??!? NOM NOM. Etc.) It doesn't take us into the world of technique: Jiro has mastered rice, too\u2014his rice dealer claims that he doesn't bother to sell his best stuff to anyone else because they wouldn't know what to do with it\u2014but while he describes how he does it, the film never shows us the whys and what-fors of his method. (Though, as Silvia Killingsworth reports for the _New Yorker,_ the French-American star chef Eric Ripert describes Jiro's rice as \"tasting like a cloud.\")\n\nInstead, the movie focuses on the life of a man who is utterly devoted to his craft. Jiro doesn't have a secret to why his sushi is more astonishing than anyone else's. What he says, over and over, is that great sushi\u2014and, by extension, greatness itself\u2014is the result of hard work, of dedication, of a commitment to excellence that, in the end, trumps everything else in life.\n\nHis search for perfection is eternal. At 85, he hasn't stopped working; he says he hates holidays because they are too long to spend away from the restaurant. Chefs, in particular, who have seen the film don't hesitate to call it \"inspiring.\" To watch the gorgeously shot scenes of him forming pieces of sushi, jewel-like and dripping with soy sauce and life, is to wish that you might one day make so much beauty. (Indeed, a film critic friend said that her reaction to seeing this was not hunger, but to want to go home and make jewelry.)\n\nStill, there is another side to this mastery, to this inspiring devotion. Jiro has two sons, and it's hard to tell exactly what their relationship to each other is, or to their esteemed father. The master admits to not being at home when they were young, telling a story of how one day he slept in, and his children complained to their mother that there was a strange man in the house. The younger one seemed, at first, to be the favorite, because the father helped him open his own restaurant. The older son, Yoshikazu, is still an apprentice to the father . . . at 50. But Jiro tells the camera, with a laugh, that when he helped his younger son open his restaurant, he told him, \"Now, you can never come home again.\" As he recounts his own life, leaving his home to begin his career at nine, it's not clear that he was kidding with his kid.\n\nWith an inflection of either humble pride or resignation, Yoshikazu says that in Japan, it's the oldest son's role to take over for the father. He works dutifully; he has taken over the selection and buying of fish since Jiro had a heart attack 15 years before. He, not the acclaimed master, was the one who served the inspectors who granted Jiro three Michelin stars, the highest recognition in the restaurant world. And yet, Jiro's restlessness keeps his son forever in his shadow, unwilling to let him stand for himself.\n\n\"You must fall in love with your work,\" Jiro says. He refers to himself as a _shokunin,_ literally an \"artisan,\" but more accurately someone who commits the entirety of himself to his work. It's a term with gravity; you won't find _shokunin_ bread in the grocery store. One of his young apprentices wells up when he tells the camera of how he finally earned the term from his master. It was after he'd worked for Jiro for 10 years. He's signed up for a life of dignity and honor and hard work. He's signed up for the life of Jiro's sons, men who may or may not have their own sons to mentor and pass their restaurants down to. He's signed up for a life given\u2014or lost?\u2014to the making of beautiful things.\n\n### FOOD FOR THOUGHT\n\n### By Jeff Gordinier\n\n### From the _New York Times_\n\n### Bringing a cultural critic's sensibility to his food writing, Jeff Gordinier writes often for the _New York Times_ Dining section. He has also published essays and interviews in _Esquire, Details, GQ, Elle, Spin,_ and _Outside,_ and authored X Saves _the World,_ a manifesto for the slacker generation.\n\nTry this: place a forkful of food in your mouth. It doesn't matter what the food is, but make it something you love\u2014let's say it's that first nibble from three hot, fragrant, perfectly cooked ravioli.\n\nNow comes the hard part. Put the fork down. This could be a lot more challenging than you imagine, because that first bite was very good and another immediately beckons. You're hungry.\n\nToday's experiment in eating, however, involves becoming aware of that reflexive urge to plow through your meal like Cookie Monster on a shortbread bender. Resist it. Leave the fork on the table. Chew slowly. Stop talking. Tune in to the texture of the pasta, the flavor of the cheese, the bright color of the sauce in the bowl, the aroma of the rising steam.\n\nContinue this way throughout the course of a meal, and you'll experience the third-eye-opening pleasures and frustrations of a practice known as mindful eating.\n\nThe concept has roots in Buddhist teachings. Just as there are forms of meditation that involve sitting, breathing, standing and walking, many Buddhist teachers encourage their students to meditate with food, expanding consciousness by paying close attention to the sensation and purpose of each morsel. In one common exercise, a student is given three raisins, or a tangerine, to spend 10 or 20 minutes gazing at, musing on, holding and patiently masticating.\n\nLately, though, such experiments of the mouth and mind have begun to seep into a secular arena, from the Harvard School of Public Health to the California campus of Google. In the eyes of some experts, what seems like the simplest of acts\u2014eating slowly and genuinely relishing each bite\u2014could be the remedy for a fast-paced Paula Deen Nation in which an endless parade of new diets never seems to slow a stampede toward obesity.\n\nMindful eating is not a diet, or about giving up anything at all. It's about experiencing food more intensely\u2014especially the pleasure of it. You can eat a cheeseburger mindfully, if you wish. You might enjoy it a lot more. Or you might decide, halfway through, that your body has had enough. Or that it really needs some salad.\n\n\"This is anti-diet,\" said Dr. Jan Chozen Bays, a pediatrician and meditation teacher in Oregon and the author of \"Mindful Eating: A Guide to Rediscovering a Healthy and Joyful Relationship with Food.\" \"I think the fundamental problem is that we go unconscious when we eat.\"\n\nThe last few years have brought a spate of books, blogs and videos about hyper-conscious eating. A Harvard nutritionist, Dr. Lilian Cheung, has devoted herself to studying its benefits, and is passionately encouraging corporations and health care providers to try it.\n\nAt the Food and Brand Lab at Cornell University, Prof. Brian Wansink, the author of \"Mindless Eating: Why We Eat More Than We Think,\" has conducted scores of experiments on the psychological factors that lead to our bottomless bingeing. A mindful lunch hour recently became part of the schedule at Google, and self-help gurus like Oprah Winfrey and Kathy Freston have become cheerleaders for the practice.\n\nWith the annual chow-downs of Thanksgiving, Christmas and Super Bowl Sunday behind us, and Lent coming, it's worth pondering whether mindful eating is something that the mainstream ought to be, well, more mindful of. Could a discipline pioneered by Buddhist monks and nuns help teach us how to get healthy, relieve stress and shed many of the neuroses that we've come to associate with food?\n\nDr. Cheung is convinced that it can. Last week, she met with team members at Harvard Pilgrim Health Care and asked them to spend quality time with a chocolate-covered almond.\n\n\"The rhythm of life is becoming faster and faster, so we really don't have the same awareness and the same ability to check into ourselves,\" said Dr. Cheung, who, with the Vietnamese Buddhist monk Thich Nhat Hanh, co-wrote \"Savor: Mindful Eating, Mindful Life.\" \"That's why mindful eating is becoming more important. We need to be coming back to ourselves and saying: 'Does my body need this? Why am I eating this? Is it just because I'm so sad and stressed out?' \"\n\nThe topic has even found its way into culinary circles that tend to be more focused on Rabelaisian excess than monastic restraint. In January, Dr. Michael Finkelstein, a holistic physician who oversees SunRaven, a holistic-living center in Bedford, N.Y., gave a talk about mindful gardening and eating at the smorgasbord-friendly headquarters of the James Beard Foundation in New York City.\n\n\"The question isn't what are the foods to eat, in my mind,\" he said in an interview. \"Most people have a general sense of what the healthy foods are, but they're not eating them. What's on your mind when you're eating: that's mindful eating to me.\"\n\nA good place to try it is the Blue Cliff Monastery, in Pine Bush, N.Y., a Hudson Valley hamlet. At the serene refuge about 75 miles northwest of Manhattan, curious lay people can join Buddhist brothers and sisters for a free \"day of mindfulness\" twice a week.\n\nAt a gathering in January, visitors watched a videotaped lecture by Thich Nhat Hanh (pronounced tik-nyot-HAHN), who founded this and other monasteries around the world; they strolled methodically around the grounds as part of a walking meditation, then filed into a dining room for lunch.\n\nNo one spoke, in keeping with a key principle of mindful eating. The point is simply to eat, as opposed to eating and talking, eating and watching TV, or eating and watching TV and gossiping on the phone while Tweeting and updating one's Facebook status.\n\nA long buffet table of food awaited, all of it vegan and mindfully prepared by two monks in the kitchen. There was plenty of rice, herbed chickpeas, a soup made with cubes of taro, a stew of fried tofu in tomato sauce.\n\nIn silence, people piled their plates with food, added a squirt or two of condiments (eating mindfully doesn't mean forsaking the hot sauce) and sat down together with eyes closed during a Buddhist prayer for gratitude and moderation.\n\nWhat followed was captivating and mysterious. Surrounded by a murmur of clinking forks, spoons and chopsticks, the Blue Cliff congregation, or sangha, spent the lunch hour contemplating the enjoyment of spice, crunch, saltiness, warmth, tenderness and like-minded company.\n\nSome were thinking, too, about the origins of the food: the thousands of farmers, truck drivers and laborers whose work had brought it here.\n\nAs their jaws moved slowly, their faces took on expressions of deep focus. Every now and then came a pause within the pause: A chime would sound, and, according to the monastery's custom, all would stop moving and chewing in order to breathe and explore an even deeper level of sensory awareness.\n\nIt looked peaceful, but inside some of those heads, a struggle was afoot.\n\n\"It's much more challenging than we would imagine,\" said Carolyn Cronin, 64, who lives near the monastery and regularly attends the mindfulness days. \"People are used to eating so fast. This is a practice of stopping, and we don't realize how much we're not stopping.\"\n\nFor many people, eating fast means eating more. Mindful eating is meant to nudge us beyond what we're craving so that we wake up to why we're craving it and what factors might be stoking the habit of belly-stuffing.\n\n\"As we practice this regularly, we become aware that we don't need to eat as much,\" said Phap Khoi, 43, a robed monk who has been stationed at Blue Cliff since it opened in 2007. \"Whereas when people just gulp down food, they can eat a lot and not feel full.\"\n\nIt's this byproduct of mindful eating\u2014its potential as a psychological barrier to overeating\u2014that has generated excitement among nutritionists like Dr. Cheung.\n\n\"Thich Nhat Hanh often talks about our craving being like a crying baby who is trying to draw our attention,\" she said. \"When the baby cries, the mother cradles the baby to try to calm the baby right away. By acknowledging and embracing our cravings through a few breaths, we can stop our autopilot of reaching out to the pint of ice cream or the bag of chips.\"\n\nThe average American doesn't have the luxury of ruminating on the intense tang of sriracha sauce at a monastery. \"Most of us are not going to be Buddhist monks,\" said Dr. Finkelstein, the holistic physician. \"What I've learned is that it has to work at home.\"\n\nTo that end, he and others suggest that people start with a few baby steps. \"Don't be too hard on yourself,\" Dr. Cheung said. \"You're not supposed to be able to switch on your mindfulness button and be able to do it 100 percent. It's a practice you keep working toward.\"\n\nDr. Bays, the pediatrician, has recommendations that can sound like a return to the simple rhythms of Mayberry, if not \"Little House on the Prairie.\" If it's impossible to eat mindfully every day, consider planning one special repast a week. Click off the TV. Sit at the table with loved ones.\n\n\"How about the first five minutes we eat, we just eat in silence and really enjoy our food?\" she said. \"It happens step by step.\"\n\nSometimes, even she is too busy to contemplate a chickpea. So there are days when Dr. Bays will take three mindful sips of tea, \"and then, O.K., I've got to go do my work,\" she said. \"Anybody can do that. Anywhere.\"\n\nEven scarfing down a burrito in the car offers an opportunity for insight. \"Mindful eating includes mindless eating,\" she said. \"'I am aware that I am eating and driving.'\"\n\nFew places in America are as frantically abuzz with activity as the Google headquarters in Mountain View, Calif., but when Thich Nhat Hanh dropped by for a day of mindfulness in September, hundreds of employees showed up.\n\nPart of the event was devoted to eating thoughtfully in silence, and the practice was so well received that an hourlong wordless vegan lunch is now a monthly observance on the Google campus.\n\n\"Interestingly enough, a lot of the participants are the engineers, which pleases us very much,\" said Olivia Wu, an executive chef at the company. \"I think it quiets the mind. I think there is a real sense of feeling restored so that they can go back to the crazy pace that they came from.\"\n\nIt's not often, after all, that those workhorse technicians get to stop and smell the pesto. \"Somebody will say, 'I ate so much less,' \"Ms. Wu said. \"And someone else will say, 'You know, I never noticed how spicy arugula tastes.'\"\n\nAnd that could be the ingredient that helps mindful eating gain traction in mainstream American culture: flavor.\n\n\"So many people now have found themselves in an adversarial relationship with food, which is very tragic,\" Dr. Bays said. \"Eating should be a pleasurable activity.\"\n\n## Dude Food\n\n### LEARNING TO BARREGUE HELPED MAKE ME A MAN\n\n### By Joel Stein\n\n### From _Food & Wine_\n\n### In _Man-Made: A Stupid Quest for Masculinity,_ humor columnist Joel Stein\u2014an often controversial contributor to _Time_ magazine, the _Los Angeles Times,_ and many other magazines\u2014adroitly deconstructs American manhood. Here he tackles the most manly of all arts: Outdoor barbecuing.\n\nLike most of what I do, my cooking is not manly. I fuss, apologize and occasionally\u2014embarrassingly\u2014look at recipes. But for the past two years, I've been trying to boost my masculinity, working on a book for which, finally, at age 39, I learn how to be a man. I've gone one round with Ultimate Fighter Randy Couture, done three days of basic training at Fort Knox, earned a badge on a Boy Scouts camping trip. But I still prepare bouillabaisse like a little girl, crying over onions and talking in French.\n\nThe manliest form of cooking that doesn't involve mounting the animal's head on a wall is barbecue; if I were really going to learn to be a man, I'd have to master it. So I went to Houston, a town that specializes in the manliest meat, beef. My plan was to eat as much barbecue as was physically possible, so I could understand life-changing 'cue. Once I knew what I was aiming for, I'd get lessons from a barbecue expert and then test my new knowledge at a barbecue contest.\n\nI go straight from the airport to Catalan Food & Wine restaurant, where, for the moment, Chris Shepherd is chef. This fall he'll open his own restaurant, Underbelly, which will feature a full-scale butcher room (he's so manly, he calls butchering pigs his \"happy place\"). Shepherd is a heap of a man, incredibly friendly and passionate; the kind of guy who can get away with calling both men and women \"baby.\" Last year, he helped create \"Where the Chefs Eat\" tours of Houston. And now he's going to take me on the barbecue tour, his favorite.\n\nWe head to Gatlin's Barbecue & Catering, where the pit boss's mother, Mary Gatlin, tells customers to \"have a blessed day.\" Sitting outside at picnic tables, we eat soft, rich brisket from the pointy side of the cut, known in most places as \"fatty meat.\" Because so much of the tip is exposed to the smoke, it has the most \"bark\"\u2014the blackened, tarry bits where smoke, fat and meat mix together. To Shepherd, this is the asparagus tip of barbecue. He prizes the top, boneless cap of the sparerib for the same reason.\n\nWe eat our way around the city, making a detour to the legendary BBQ Pits by Klose, where Shepherd bought his $1,800 home pit. David Klose speaks quickly, delivering entirely unprovoked self-proclamations such as, \"I'm a real quick draw.\" He is the most Texan person I've ever met, in his Wrangler jeans and brown camouflage cap that says \"Deer Predator.\" Klose has made barbecue pits out of a phone booth, a mailbox and a police car. The more steel in the pit, the steadier the heat will be, and that, he insists, is the trick to good barbecue. This seems to justify why men are happy to spend thousands of dollars to cook $1.75-per-pound meat.\n\nOur final stop is Pierson & Company Bar-B-Que, where we get a mixed trinity of brisket, ribs and sausage links that are smokier and sweeter than some we've tried elsewhere and nearly as satisfying. Texans believe that, just as you wouldn't cover Kobe beef in b\u00e9arnaise, you don't smear barbecue sauce on brisket that's been cooking for 12 hours in an oven you'd need to put a down payment on. You want to taste meat, salt and spice. This is such a basic idea that I cannot believe there isn't a Doritos flavor named Meat, Salt and Spice.\n\nThe other lesson I learn is that barbecue is about timing: The brisket at Gatlin's wouldn't be nearly as delicious a few hours past its prime. Barbecue exists in time as much as it does in space. It's why so many great, small places run out of meat by 11 a.m. Why they don't get a bigger pit and buy more meat is something that leads me to many more questions about the South than I can possibly answer.\n\nMy eating tour complete, I am ready to actually try cooking. I wake up early the next day to meet curmudgeonly Robb Walsh, a recently retired restaurant critic and author of the _Legends of Texas Barbecue Cookbook._ He recently opened a Tex-Mex restaurant, El Real Tex-Mex Caf\u00e9, with _Food & Wine_ Best New Chef 2009 Bryan Caswell right across the street from the space that will become Underbelly. And for the past year, he's been running a barbecue course out of his home. He's going to give me the simplified version. The one for Northerners.\n\nThere are three grills in Walsh's backyard. He's got a pricey offset smoker\u2014in which you light a wood fire in a small cylinder called the firebox, which is below and off to the side of a big grill with an exhaust pipe on top. He's got a \"Mexican hibachi,\" an oil drum cut in half\u2014you build a fire on one side and put the meat on the other. And he's got a gas grill, just like I do. Walsh makes it clear that I am never, ever to refer to anything I do on the gas grill as \"barbecuing.\" Barbecuing is cooking with smoke. Grilling is what you do to your children's hamburgers.\n\nHe teaches me to put some charcoal in a starter chimney, which looks like a big metal travel mug, and light it with a burning newspaper. We drop the charcoal in the offset smoker's firebox, then add oak logs along the sides, not in a pile\u2014exactly the opposite of the way the Boy Scouts taught me\u2014so the fire smokes instead of roars. By controlling the amount of fuel and oxygen inside the smoker, we aim to keep the heat at around 275 degrees\u2014a nice, slow, long cook to break down the tough, fatty cuts of meat that work best in barbecue.\n\nOnce the temperature hits 275 degrees, we place some pork ribs, brisket rubbed with just salt and pepper and a chicken marinated in Kraft Italian dressing on the grate. Walsh explains that bar-becue people are \"weird,\" spending wildly on gadgets but using inexpensive supermarket rubs and marinades. What he means is that barbecue people are men.\n\nUnlike the brisket my Jewish family eats at Passover, this one is half fat. We set it on the grate fat side down for two hours, then flip it fat side up so it self-bastes for the next eight hours. Because we don't have to be there while all this cooking goes on, we head out on a long drive to pick up supplies and a breakfast taco.\n\nD. W. Vasbinder's, where we stop to buy logs, looks like a junkyard, with metal sculptures of cowboys, cow skulls and a sign with wood prices that explains the honor system of payment. We pick up some oak, which Walsh says is the purist's choice, since it imparts a neutral smoky flavor. (Mesquite can be tangy and resiny, whereas pecan gives a sweet, sooty flavor that can blacken your food.) As Walsh stops a welder to ask for a piece of metal mesh for the bottom of his hibachi grill, he says, \"Places like this are why you can make great barbecue in Texas. It's like car culture in L.A.\" I pretend I am manly enough to be part of car culture in L.A.\n\nWhen we get back, we check on the meat. The chicken and ribs are done, but the brisket has stalled out at an internal temperature of 165 degrees, about 20 degrees shy of what we need. We wait another hour, during which we work hard opening and drinking local Saint Arnold beers. But the brisket is still at 165. It's a classic barbecue phenomenon\u2014the meat gets to a point where the internal temperature just won't budge. We have a few choices: We can keep waiting and hope the problem fixes itself. We can wrap the brisket in foil, which will raise its temperature but might make it soggy. Or we can bring it inside and put it in the oven at 275 degrees, which could dry it out. I refuse to bring it inside, because inside is for womenfolk. Walsh decides to wrap the meat in foil for just 1 hour and leave it in the smoker. Which works. We lop the giant fat cap off the parts we're going to eat first, leaving the rest on to keep the meat moist.\n\nLike most men, I feel that one day of training makes me qualified to compete, so I am headed to the World's Championship Bar-B-Que Contest, where more than 400 competitors fill the parking lot of Reliant Stadium with tents bearing giant corporate logos. I join team Drillin' & Grillin', which won for the best ribs last year. The seven-man team is run by Ernest Ramirez, a pipe fitter who has built his own enormous pit. Ramirez has a good oak fire going, but he shows me a bag of pellets made out of hickory, cherry, hard maple and apple that he sprinkles on the flames to sweeten the smoke. He hands me a syringe so I can inject chickens with marinade to keep them moist; for the same reason, we place a pan of water and chicken broth on the bottom rack. But we're mostly focused on the spareribs, which are cooking with a simple rub. There is a lot of hanging out and chatting, but we've got to pay attention: If we screw up by letting the fire get too hot, too cool or too smoky, there's no chance to recover. Because starting over takes 14 hours.\n\nRamirez believes ribs need a glaze to balance the sugar and saltiness, so when they're just about finished, he pats some brown sugar on top and puts them back in the pit. Then, 30 minutes later, he brushes on a glaze of brown sugar, melted butter and pepper jelly, puts the ribs in a Styrofoam container and takes them to the judges' tent.\n\nI help slice the brisket, stealing several pieces of bark for myself. At 5 p.m., a committee member tells Ramirez that our team didn't place in the top 10 to compete for the finals. Ramirez thanks her, smiles politely and goes right back to being the caterer for his corporate sponsor's rodeo tailgate, slicing a huge brisket against the grain to keep it tender. This is what I've learned about being a man. We've come to call it repression, but it's really self-control. It's a respect not just for the contest and the people you compete with, but for yourself.\n\nI'm going to start barbecuing at home. Not on an $1,800 pit that takes up half my yard, but on a little Weber with the wood off to the side. Mostly because nothing really tastes like smoked meat. But also because, unlike women, men need an activity as an excuse to talk. And we're not going to talk about salad.\n\n### MEMPHIS IN MAY: PORK-A-LOOZA\n\n### By Wright Thompson\n\n### From _Garden & Gun_\n\n### ESPN senior writer Wright Thompson writes mainly about sports, though as a native of Clarksdale, Mississippi, he's occasionally lured into food writing by his love of Southern food. And what could be more Southern than barbecue?\n\nBefore a man with one leg got women to take their shirts off while he poured liquor into their mouths via an ice luge; before the wild-eyed guy who provides the pigs to the French Laundry walked around the party slipping packages in people's hands, which at least two of us thought were drugs but turned out to be bacon; before Donald Link's boudin for lunch and John Currence's andouille for happy hour and Sean Brock's soft-shell crab for dinner; before ten-year-old Jess Edge asked his daddy, the Southern food guru John T. Edge, \"What's a Jell-O shot?\"; before we waited to hear if we'd made the finals of the Memphis in May World Championship Barbecue Cooking Contest; before the revolutionary act of creating an all-star team that included four James Beard Award\u2013winning chefs, three old-school Southern pit masters, and one boozehound writer; before any of that, there was a pretentious but earnest idea: Could we rescue a barbecue contest, and maybe even barbecue itself, from a crushing sameness?\n\nThe line was drawn. On one side, us: a cooking team called the Fatback Collective, organized by barbecue industrialist Nick Pihakis, who founded the Jim 'N Nick's Bar-B-Q chain of restaurants. On the other, Memphis and its world-famous barbecue contest, entering its thirty-fourth year and drawing an estimated 100,000 fans. Now, Memphis in May is many things: a place for Parrotheads to gather between nautical-pun tours, a grown-up frat party with a hundred thousand pledges, a place where friends commune over a smoking pig, and, maybe most important, a driver of where our barbecue culture will go. Here's what it isn't: a reflection of where our barbecue has been. We wanted to turn back the clock.\n\nMaybe that's silly. Maybe that's an idea fueled by ten cases of whiskey and two thousand Jell-O shots, but from the belly of the beast, surrounded by what many of us consider to be a threat to authentic barbecue\u2014lean pigs, tricked out with injections, cooked not as a reflection of a family or place, not as a connection to our past, but, rather, gamed to the strange tastes of the Memphis judges\u2014we all realized the mission of the Fatback Collective: redemption.\n\nRedemption and about seventeen thousand calories a day, most of them liquid.\n\n##### **Ready, Set, Strip**\n\nIt all starts with a screaming buzz saw and the smell of burning pig bone.\n\nA hog is splayed out and gutted, with a half dozen hands reaching inside its carcass. Pat Martin, a Nashville pit master, revs the blade and digs into the backbone. Pig shrapnel flies around the tent.\n\nWhen the job is done, the truly heartbreaking part begins, trimming out pounds of glorious, expensive, and carefully cultivated fat. Our team is consulting with a former grand champion, someone who is gracefully helping us understand the intractable customs of Memphis in May. He points at the thick layers of white.\n\nThe chefs look at each other, then at the pig. Reluctantly, they start stripping. Every so often, they'll make eye contact with one another and shake their heads. Someone mutters. Brock stands to the left of the pig, and Link on the right, each cutting back ribs to expose more shoulder meat. A pile of fat forms on the table.\n\n\"Too much,\" says Ryan Prewitt, the chef de cuisine at Herbsaint in New Orleans.\n\nStephen Stryjewski, the chef at Cochon who won his Beard award just four days ago, asks the expert once more if he's sure.\n\n\"They don't want marbling?\" he asks.\n\n\"You don't find that in other hogs,\" the former grand champion says. \"Technically, they don't want to see that.\"\n\n\"That's so 180 degrees to what I do every day,\" Stryjewski says.\n\nLink, the mind behind Cochon and Herbsaint, watches in silence. This is what he's thinking: Arrrgh! There are all these Beard winners with their hands in the hog, but something is being lost in translation. A chef's job is to cook food that stays true to the essence of the ingredients. The job of a Memphis in May contestant is to deliver what the judges want, and, more important, to stay away from things they don't. Here, as best as I can tell, are some of the things the judges don't like:\n\n1. Fat\n\n2. Spice\n\n3. Pork that tastes like pork, as opposed to pork that taste like it got pistol-whipped by MSG and sugar\n\n4. Puppies\n\n##### **Gaming the System**\n\nThe more I learn, the more I realize that winning this thing has less to do with great barbecue and more to do with anticipating the judges. They like sweet. They don't like spice. They like tenderloin. They don't like belly. On and on. So competitors study past winners, then go Mr. Wizard on the pigs. They fill the cavity with bricks of cold butter. They pack iced pillowcases around their tenderloins to stop the cooking. Some pigs are souped up with culinary nitrous oxide: liquid fat\u2013laden injections.\n\nThis isn't happening in a vacuum. For many, these traveling cooking teams are the face of Southern barbecue. Not the guys, like the pit masters on our team, who cook pigs in the same pit three hundred days a year. The most wonderful thing about barbecue has always been its regional differences. Each pig told a story. Rodney Scott, pit master at Scott's Bar-B-Que in Hemingway, South Carolina, uses wood he chops himself. Pat Martin was born in Mississippi but worked as a bond trader in Charlotte before realizing his calling, and his Beach Road 12 sauce, with the Carolina tang and a touch of the Memphis sweet, is a reflection of his own journey to the pit. Barbecue changes from town to town, an entire style morphing at the Tennessee River, or at the Piedmont, or when you sweep down onto Highway 61 from Memphis to Clarksdale, Mississippi. Memphis in May, the most important barbecue event in the world, rewards homogeneity. If you live in the South, maybe you've noticed how hard it's becoming to find a good, simple barbecue sandwich. Traditional barbecue is fading as competition barbecue is rewarding smoke and mirrors.\n\nSo while we do cut away some of the fat, more than the chefs would have liked, there is still plenty left on the hog. I've never seen a pig like this. It's marbled, laced with thin lines of fat. It looks like a rib eye steak. There are fewer than a thousand Mangalitsas in North America, and this is the first time, to the best of everyone's knowledge, that one has ever been barbecued. (We're putting two on the coals.) It could put a new face on barbecue or, more accurately, give barbecue its old face back. The guys shake on a rub, the extent of the doctoring, and look down at the pit-ready pig.\n\n\"I think we should do a shot of bourbon,\" says Drew Robinson from Jim 'N Nick's.\n\nThere are murmurs of agreement. Hell, yeah. Breakfast.\n\n\"I'll get the Pappy Van Winkle,\" says Brock, chef\/owner of Husk in Charleston, South Carolina.\n\nPappy is poured into those flimsy cone-shaped water cups. I hold mine over the pig and knock it back.\n\n\"Bourbon and pigs,\" Link says.\n\n##### **The Wee Hours**\n\nBourbon and pigs. That fairly sums up my next twenty hours. The pit doors shut and smoke rolls out. Nothing to do but wait. The chefs and pit masters hold little summits, conversations that food nerds would freak out over. There is laughter. There is boudin and soft-shell crabs and oysters and crawfish. There are trays of Jell-O shots, and big cups of bourbon, and people dance until the speakers overheat. There is some drama at the pig; the fire gets too hot, but Rodney Scott finesses the coals, brings the temperature down. There's no pit problem he can't fix. You've seen Pulp Fiction? He's the Mr. Wolfe of pork.\n\nI fall asleep in a chair by the pit and later move to a couch. I wake up at 4:30 a.m. to find Scott and Brock still awake. Brock and I tag out, and I settle in next to Scott. He's bulletproof. I'm not. The Pappy is gone. There's a dead tall boy of Pabst and a cashed bottle of Patr\u00f3n on a table, along with the empty shell casings of Jell-O shots.\n\n\"We look rough,\" says Sarah Johnson from Jim 'N Nick's.\n\n\"I feel rough,\" Brock says.\n\nScott puts R&B on the speakers. Al Green brings us back to life. We've been goofballs for the past two days, but when the end comes, it's all business.\n\n\"Do you feel good about this?\" Nick Pihakis asks.\n\nBrock nods.\n\nIt's time to create the Box. We've talked about the Box endlessly, lending it the importance of an advanced policy initiative, which seems slightly ridiculous, given that the Box is a Styrofoam container of cooked pig. But the Box is for the most important part of judging, the blind taste test, so there are four Beard-winning chefs on the Jim 'N Nick's mobile smoking rig. Everyone is calm, quiet, with a few jokes and short, precise comments. The Box has to go at noon. Link has a knife in his hand. Some of the Beach Road 12 sauce is in a jug; abstract ideas and theories are great, but these guys didn't become who they are by trying to lose. Some of my snark evaporates as I realize every team is doing this exact same thing.\n\n\"It's 11:50, guys,\" John Currence, from City Grocery in Oxford, Mississippi, says quietly.\n\nWhat follows is a damn impressive ten minutes. My boys are stone cold. Nobody ever raises his voice or appears to rush, and I realize that before these guys were famous, they spent their lives in hot kitchens, cranking out dinner night after night. All thirty-four other whole hog teams are equally concerned with sticking the landing.\n\nThe Box is off, and three in-person judges are coming through. They are given the Gospel According to Fatback. Pat Martin does most of the talking. They hear about the Mangalitsas, about the pit masters and the chefs, about Martin's dream of a win at Memphis in May changing the arc of the pig industry, replacing the flavorless factory hogs with ones more like our grandparents ate. His son, he testifies, might one day enjoy the results of our work today. Finally, the last judge leaves, and Scott walks back to the pit. Sam Jones, whose joint, the Skylight Inn in Ayden, North Carolina, grew out of a family barbecue tradition that dates back two hundred years, is standing there.\n\n\"We are so full of it,\" Scott says.\n\n\"Full of what?\" Jones asks.\n\n\"Truth,\" Scott says.\n\n##### **Fatback Glory**\n\nPihakis is worthy of love for many reasons, for his generosity, for his terrible dancing, and for his creation of the Fatback Collective. But the moment I really want to spoon him is when he arranges for $500 worth of Gus's World Famous fried chicken to be delivered to the tent just after the judges leave. People make love to that chicken, coming up for air with faces and fingers covered in grease\u2014\"That's my last meal,\" Brock gushes, orbiting the tables like some sort of bird of prey\u2014and I eat, very quickly, four chicken thighs and two pieces of Wonder bread. So, we're pigging out, and someone is telling Drew Robinson that bad news comes via a messenger on foot, but if you make it to the finals, a judge will arrive in a golf cart. At that exact moment, a golf cart pulls up.\n\n\"You mean like that?\" Brock asks.\n\nThe Fatback Collective is in the final three.\n\nThe last group of judges arrive, four this time, and the show is smooth. They eat the pig and head out to make their decision. The confidence is palpable. People are flocking to the pit to pull out meat. Word has spread that the team of ringers has the greatest barbecue ever cooked. There are judges coming by just to eat. One tells me: \"You've got it in the bag.\" A feeling arises.\n\n\"If we win this thing,\" Pihakis says, \"people are gonna look at pork differently.\"\n\nWe go stand by the stage. Someone there has a pig's head on a stake, with a cigarette in its mouth, a picture of excess that defines much of Memphis in May.\n\n\"This is the old model,\" Edge says.\n\nHowever, we are not entirely innocent. We consulted a past champion. We did a taste test on various sauces. We tried to game the judges, too, while preserving our integrity. We came here wanting to change this competition, but now, in the last moments, we want to win it. I haven't even cooked anything\u2014my role is to be a hyper-partisan observer\u2014and even I want to win. I've rolled my eyes for the past two days at the other teams, and now I realize my ire is misplaced. Everybody else isn't trying to educate the judges\u2014they are trying to kick ass. And, as a look at the tense Fatback Collective confirms, so are we. Finally, the results are announced. The winner is Yazoo's Delta Q, the same team that won last year.\n\nFatback Collective is third.\n\nWe shuffle to the stage, trying to smile. The crowd cheers, and all our hopes of stopping the homogeneous 'cue train suddenly feel like sour grapes. We are lots of things, but we are not sore losers. Martin takes the microphone and tells the other cooks that we all respect the hell out of what they do. Those are his exact words, and after, teammates slip up to him to tell him he nailed it. We go back to the tent, and there is talk of how great it is to finish third in our first year, and hugs, and drinks, but the DJ cuts to the heart of the feelings in the tent. We are serenaded with \"Auld Lang Syne.\"\n\n##### **Truth in Barbecue**\n\nAfter Sean Brock grew up \"dirt, dirt, dirt poor,\" with a dream that seemed impossible from the forgotten corner of Virginia he called home; after he won his Beard medal at age thirty-two and hid with his cell phone in a Lincoln Center bathroom, weeping, calling to tell his mama, I did it; after Donald Link did for boudin what Arnold Palmer did for golf; after Sam Jones tended his pit the same way his ancestors did two hundred years ago; after Rodney Scott showed up to man the fires at midnight the night he graduated from high school; after John Currence learned to cook on a tugboat the morning after he graduated; after we traveled to Memphis to try to change the way people think about barbecue; after we succeeded, and also failed; after all that, I can't shake an image that I'll cherish long after my cardiologist buys a new ski boat with the money he'll make off the weekend: Pat Martin, our loud, opinionated mouth of the South, sitting in the corner, waiting to find out if we'd won.\n\nHe's quiet now, with his little boy on his lap. They have the same haircut. Martin gives his son a kiss and rubs his forehead. He holds him tight. Something becomes clear in this moment. The real barbecue we love, that we pretentiously and earnestly came to save, might be under siege, but it isn't dead. It lives in anyone who believes in doing things the way their grandfathers did, who believes that what we eat tells a story about who we are. It lives in anyone who cares enough to sit all night with a hog. It lives in the fading notes of \"Auld Lang Syne\" and in the sparks popping off the burn barrel past midnight. It lives in the way a father holds his boy when the cooking is done.\n\n### TRUFFLE IN PARADISE\n\n### By John Gutekanst\n\n### From _Gastronomica_\n\n### The proprietor of Avalanche Pizza Bakers in Athens, Ohio, John Gutekanst has worked as everything from a pot scrubber to a sommelier, but his great passion is pizza. His creative pies have won numerous awards, including several in Italy, pizza's homeland. He blogs about pizza on PizzaGoon.com.\n\nOur car climbs the mountainous roads above the northern Italian spa town of Salsomaggiore. It is late March, and we have a job to do. I've convinced my three teammates\u2014Tony, Bruno, and Justin\u2014to accompany me on my quest for _tartufi bianchetti,_ Italian spring truffles. The _bianchetti_ aren't the hardest truffle to find in all of Italy, or the most fragrant, but they are the best at this time of year because they're fresh. By now, the white truffles have lost their intensity, and the blacks, all of their flavor. I need the _bianchetto_ to make an award-winning entry for the 2008 World Pizza Championship, and these elegant truffles, paired with sweet Maine diver scallops, are perfect partners. The judges will be looking for taste, innovation, and execution. If I can find the truffle guy, I'll nail the championship.\n\nThe town of Salsomaggiore Terme, located in the Parma region of Emilia-Romagna, dates from the time the Etruscans first found mineral springs nestled in this snug valley. Today, people come for massages and bathe in the brown stink of the healing egg water, thanks to Benito Mussolini, the fortieth prime minister of Italy, who built a palatial building around the springs, called the Berzieri Spa. The waters are famed for healing.\n\nBut I am a pizza man with just one purpose: to win first place, USA, at this year's competition. Normally, I confine myself to my small pizzeria in Athens, Ohio, and suffer no illusions as to my business's place in a small college town: customers are locals and college kids; business is brisk until 4:00 A.M.; volume is high and profit is small. My customers love my products, and I'm happy but not complacent. I know the next three days won't bring me any fame, but just maybe they'll bring some validation that even a small guy can win sometimes.\n\nThe championships are held annually in a huge stadium just outside Salsomaggiore. Blooming flowers and squawking pheasants on the steep, vine-covered slope outside the building belie the intense human conflict within. This year I'm entering four pizzas: scallop and truffle pizza in the Pizza Classico category; spinach, feta, chicken, and sun-dried tomato pizza in the Pan Style category; fresh mozzarella with basil and San Marzano tomatoes in the Neapolitan category; and in Non Gluten, a new category, an onion-linguine and sweet potato orecchiette-topped pizza with dough made from rice and garbanzo-bean flour. All culinary scores are figured from 800 as perfect, with each contestant rated by cooking prowess, sanitation, presentation, taste, and knowledge of the pizza. From prior year competitions, I have also learned the unwritten rules: look Italian, cook the Italian way, make your presentation short, have confidence and a pinch of arrogance. My show of detached condescension must make the judges think I am the most knowledgeable pizza maker ever, and, as a gift, I am going to turn them on to the best pizza they've ever had. Once I get my truffles, that is.\n\nWe see the worn sign in front, Ristorante Al Tartufo, flickering across the white gravel parking lot, and we turn, circling to the back of the building. Tony Gemignani, founder and president of our American team, leans across the front seat to get a better look at the building, dark bangs falling across his forehead. He has just driven Justin Wadstein, Bruno di Fabio, and me on this four-hour expedition. He is probably weary from the winding roads and the jetlag, but Tony looks calm, as usual, and intense.\n\n\"We're here, I think.\" A quick smile accompanies his classically handsome Italian looks and puts us all at ease. Tony is an international legend in the pizza industry. With his brother Frank, he owns Pisano's Pizzeria in Castro Valley, California. He has won more acrobatic and culinary awards for pizza than any other person in this country and probably the world. Tony and Bruno recently opened Tony's Pizza Napoletana, a successful pizza school and pizzeria in San Francisco. Tony has already driven us around to purchase radicchio, zucchini flowers, and broccoli rabe. These truffle guys are our last stop.\n\n\"So, John,\" Bruno says to me. \"This is the place with your amazing truffle guy, huh?\" Bruno beams. He can't help himself. There on the gravel he looks like an actor about to go on stage. Both he and Tony live for strange new situations; after hearing of my last visit to the truffle guy in 2006, they had to see for themselves.\n\nJust like last time, the quest began with creepiness, a James Bond moment by the hotel front desk when we asked for directions to the truffle guy's restaurant. Two men in pressed blue suits asked why we wanted these truffles and how we knew of the truffle guy. It seemed more like an impending drug deal than a stinky fungus inquiry. If we could get fresh truffle at the local stores, life would be simpler, but that's not how it works in Italy.\n\nTruffle hunters gather the _bianchetto,_ also known as the Tuscan truffle, from January to March, from Tuscany north to this part of Emiglia-Romagna. Its lobes, unlike the round, sandpaper-like skins of the black truffle, look like bulbous bumps on a small animal brain thrown in the dirt. It's tough to nail down the exact aroma and taste of the spring truffle: cheesy, woody, earthy, and garlicky are the usual adjectives. While not as aromatic as the white or black truffle, the _bianchetto_ is particularly suited to pizza\u2014at low, warm temperatures it gives off a mildly pungent aroma. Introducing strong flavors to a pizza can be disastrous. A great _pizzaiolo_ (pizza dude) has to make sure that the competing flavors of bread, sauce, cheese, protein, and vegetable are compatible and not overpowered by another element.\n\nNow I'm just moments away from a stash of _bianchetti._ I can almost smell them. But Justin turns and looks off into the dark green hills beyond the parking lot. \"I'm not gonna get whacked for a truffle,\" he says. \"I'm staying in the car.\" Justin is no coward. He's a world-class pizza-dough-tossing acrobat who once tossed three rounds of pizza dough high into the air while jumping up and down on a soccer ball. When Justin won the World Pizza Championship of 2007, the crowd jumped out of their seats, screaming in adulation and surprise.\n\nBruno leans in toward Justin, who's still sitting in the car. \"Okay, Justin, this is the deal. If you hear some loud pops, then see me hauling ass to the van with a big bag of truffles and John's and Tony's brains splattered on my face, start the car, and we'll be gone.\" Bruno laughs his usual high, nasal guffaw and kicks the loose gravel. He then punches my arm and nods a \"let's go\" toward the back of the restaurant. We crunch our way through the parking lot. The sunlight is waning now. Clouds moving in make the mottled and muted blue sky meld with the valley below.\n\nThe restaurant's door is a huge slab of beaten, scratched wood that to my mind looks like a door to a gladiator ring. Tony, in the lead, stops to face me. \"Hey, John, how do you know that this guy's your buddy? This area was a Mussolini stronghold in World War II, and there's still lots of ill will toward Americans. I've also heard of some strange deaths attributed to the truffle black market. It's a big business here, taken very seriously. Why don't you go in first and I'll back you up?\"\n\nI hesitate. Perhaps I overstated how familiar I was with the truffle guy, having met him only once, and that was two years ago. What if Justin is right and the truffle mafia decides we're interlopers?\n\nBruno chimes in with a chuckle. \"Yeah, dude, we're right behind you. Anything happens, we'll be the first to run and get the cops or the ambulance.\"\n\nI stop at the threshold and sniff the air. \"Smell that?\" I ask.\n\n\"Porcini, cepes. Unmistakable. Food of the gods,\" Tony mumbles. \"You sure this is the place you visited last time?\" My hands grip the near-mythical brass doorknob. I smile nervously and throw open the door.\n\nI have indeed been here before. I'm transported back two years ago, to a late March night in the marble lobby of the Grand Hotel et de Milan in the heart of Salsomaggiore Terme. Feeling a little like Jack Nicholson at the start of _The Shining,_ I had blurted out, \"I need to find some truffles. Do you know where I can buy some?\"\n\n\"Who wants to know?\" a slender, elegant man in his mid-fifties asked. His perfectly pressed blue suit with white lapels was more suited to a movie than a hotel. He looked beyond me with such curiosity that I turned and looked myself. No one there.\n\n\"Follow me, please,\" he said as he clicked his way across the marble, circling behind me and out the door. The lights of downtown and the disco across the street twinkled as we walked through a black-green garden filled with huge marble statues. The fence around it reminded me of a cemetery.\n\n\"My name is Giorgio.\" He tapped a cigarette on his pack. \"I can get you _tartufi_ ,\" he continued, his voice dropping to a whisper. \"It is . . . how do you say? Not simple.\"\n\n\"Complicated,\" I said.\n\n\"Yes.\" Giorgio lit his cigarette and took a deep puff. He pulled out his card, wrote an address on it, and said, \"Take a taxi, give him this address. When you get there, go in the back door. Ask for the owner of the restaurant and give him this card.\" I looked down at the cursive Italian writing. The only word I could distinguish was the last one, _tartufo._\n\nAs I walked away, Giorgio cautioned, \"Be careful, Giovanni . . . John. And keep your hands out of your pockets.\"\n\nThirty minutes later my cab reached the mountain restaurant. I walked across a parking lot under the yellow glare of its sign and opened a big wooden door\u2014the same one I had just opened with Tony and Bruno behind me. The room was not well lit. Pale green walls showed splotches where the plaster had chipped off. Shelves all around held jars of unidentifiable brown things floating in liquid.\n\nA dozen or so locals were bantering with their friends. Four old men at an old wooden tabletop played cards to my right. Middle-aged waitresses lounged and cackled at a small bar on the other side. I stepped into the middle of the room.\n\n\"Hi,\" I blurted out to no one in particular. A drop of sweat trickled down the side of my face.\n\nEverything stopped. The abruptness was so intense that it could only have been better choreographed if someone had taken a record player and scratched the needle hard: _RRRRRRRRRIIIIIIIIIP._ I looked around, my hands and brow sweaty. Even my brain seemed to sweat. All the faces were hard and expressionless. It was one of the longest silences I've experienced in my life. I looked down. \"Oh shit,\" I thought. There were my hands, in the pockets of my coat. The oldies at the card table had their hands under the table now. I removed my hands slowly.\n\n_\"Buona sera.\" A_ man appeared from behind a patterned curtain, arms outstretched in the clothesline position. He was over six feet tall, with graying, balding hair and a bushy unibrow held up in a \"what do you want\" look. He cracked his neck with a snap and cocked it to one side, expressing, in timeless guy language, \"Gimme what you got, punk.\"\n\n\"I came to get some truffles,\" I squeaked, cursing myself for sounding like a little girl.\n\n_\"Americano?\"_ he asked.\n\n\"Yes, er, _si,\"_ I replied. \"I need them for my pizza at the World Pizza Competition.\"\n\n\"Who send you?\" the tall man demanded.\n\n\"Giorgio, at the hotel.\" I pulled out his card, trying to keep my hand steady.\n\nA smile flashed across the guy's face. It was like someone had put the needle back on the record. All was good, all was right, just a stupid Americano.\n\nThe tall man shook my hand. From the back room he brought out a large, plastic container, covered with a cloth. He looked around again, and then summoned me with his hand. He grabbed the cloth and stared at me as though he was about to show me my future or the meaning of life. Then he lifted the lid.\n\nThe truffle fumes were overpowering. I smelled black pepper and garlic with a hint of gunpowder, cashews, and ozone. The truffle guy nodded his chin for me to take a closer whiff.\n\n\"Heaven,\" I muttered appreciatively.\n\n_\"Tartufi bianchetti,\"_ the truffle guy said.\n\nI paid forty dollars for three large, marble-sized truffles and a big bag of dried porcini, and I made a new friend. We completed our deal with a toast of _limoncello,_ and I dashed to the waiting taxi. Considering that the porcini go for fifty dollars a pound in the United States, I calculated I was holding over two hundred dollars' worth of great mushrooms. Not to mention the truffles.\n\nI used the truffles to create a pizza with American bison, _fontina di Aosta,_ arugula, and a b\u00e9chamel sauce. I scored in the upper echelon but did not win.\n\nBuying truffles this time around is bound to be different. I have Bruno to translate and Tony to back me up. But as I open the heavy door, I'm hit with a major d\u00e9j\u00e0 vu: four old men at the same table turn to stare at me. \"No freakin' way. Not the same guys in the same chairs,\" I say to no one and everyone. I stand there with my mouth open. Tony and Bruno push me into the pale green room with the jars. Nothing has changed. We get the same silent treatment, the same record player scratch, only this time I'm armed. My weapon is Bruno di Fabio.\n\nBruno sidles up to a waitress at the bar. _\"Buona sera.\"_ I watch her face closely as he continues, because Bruno, besides being a consummate pizza aficionado and fluent in Italian, never hesitates to have fun at my expense. For all I know, he could be saying, \"My two Hungarian friends over there are lepers. They want to know if the dishwasher found a finger in a soup bowl last night.\"\n\nBruno turns. He must have noticed me boring holes into the back of his head. \"What! I just asked her where the truffle guy was.\"\n\n\"Those guys staring at us look harmless,\" Tony says, nodding toward the old guys. \"Why do they all have their hands under the table?\"\n\n\"They're probably holding old rusty shotguns under the table ready to blow us to smithereens,\" I whisper. \"Didn't you ever see _Godfather II?\"_\n\nI'm interrupted by a loud _\"EEEEEEEEEEAAAAAYYYYYY\"_ that sounds more like a shotgun blast than a greeting. Tony and I jump. It's my truffle guy, appearing from behind the curtain. _\"Buona sera,_ my pizza friend!\" The truffle man hugs me with his lanky arms. His gray unibrow now seems welcoming. I beam. Now my buddies know who's the real pizza guy, I think. Truffle guy mutters something to the old men and they relax, but they still keep their hands under the table.\n\n\"You want _tartufi_? Porcini?\" he asks as he walks to the front door, where he pokes his head out and scopes out the parking lot. Bruno takes over communications and explains what I want, accurately for once.\n\n\"Wait,\" the truffle guy says, disappearing behind the worn tapestry.\n\n\"I wonder what these guys would do if I grabbed this jar of porcinis and ran to the door?\" Bruno asks playfully. Tony and I giggle.\n\n_\"Tartufi bianchetti,\"_ the truffle guy announces. He puts a two-by-one-foot plastic box on the bar, and we all crowd around. I'm surprised to see the waitress crowding too. The grand presentation begins with a _bbblllllurrrrp_ from the vacuum-sealed plastic container. Musky, pungent truffle fumes waft out. I imagine shucking an almost rotten ear of corn, popping a can of Planters nut mix, then sticking my nose next to diesel truck exhaust for half a second. I smack my tongue against my upper palate and breathe out slowly. I experience disgust and fascination, repelled and attracted in perfect, intense equilibrium.\n\nWith the pressure of the contest tomorrow, all I want is to get my truffles, nod in ignorance, drink my _limoncello_ and haul ass out of there. I hold up three large truffle marbles and ask how much.\n\nThe truffle guy holds up fifteen fingers (not all at once), indicating fifteen Euro.\n\n\"Less than thirty bucks for all that?\" Bruno exclaims.\n\n\"Great deal,\" Tony said. As a bonus, the truffle guy gives me half a bag of dried porcini that he has handpicked.\n\nThe next day, as we enter the stadium to register at the Pizza Championship, we see our competition\u2014the world. The world sees us, too. To enter with such a pizza legend as Tony Gemignani is like being a bodyguard for the president. Heads turn to stare at him. I am proud to be on his team.\n\nAll countries are represented: Iran, Bangladesh, Ireland, Scotland, Brazil. The French show up and try to bribe the judges with some of their kick-ass Champagne. Each contender wants the title of \"Best Pizza in the World.\" Now in its ninth year, the World Pizza Championship makes _Iron Chef_ look like a yoga class at a retirement center.\n\nThe best pizza dough I've ever tasted is from Bruno's _biga._ He's such a fanatic that he'd probably stand at the mixing bowl all night telling bedtime stories to his _biga_ if he knew it would make his pizza crust better. _Biga_ is pre-fermented dough, Italian style\u2014basically, a batch of dough left at room temperature for twenty-four hours or more and added to a new batch. This approach concentrates the flavor, adds complexity, and leads to better digestibility and taste.\n\nBruno's goal at this year's competition is to become the \"World's Fastest Pizza Maker.\" Five dough balls rest in a pile of fine-grained Italian flour on his table. He has to stretch them faster than all his competitors. Bruno walks three feet to the back wall, facing it as if in greeting, then pulls up his checkered cook's pants and squats, sumo style. His nose is just inches from the surface; his face is gnarled in knotted confidence as he beats his chest with closed, flour-covered fists. Puffs of flour shoot up the wall with every beat. Back at the table, he pushes each dough ball with both hands, stretching the gluten strands outward as fast and economically as possible. Bruno makes the time of 48.93 seconds, beating his close Italian friend Salvatore Salviani's 52.68-second score. When he hears the news, Bruno's primordial yell of \"Yeah!\" is probably audible in Bulgaria. He struts to the edge of the stage, gazes out at the crowd, and crows, \"That's gold, baaaaaaby!\" as only a New Yorker can.\n\nMy other USA teammate, Justin, progresses to the finals of the acrobatic championship with a great score. He places third in the world in acrobatic dough tossing.\n\nAs for me, my _biga_ is ready for the fire, just twelve hours old, and incorporated into my seven-ounce dough ball that rests on the marble pizza table near the oven. Three Italian contestants burned the bottoms of their pizzas just before me. This is not promising: they are all better wood-fired pizza men than I am. The optimal temperature for baking a traditional Neapolitan pizza, 905 degrees Fahrenheit, is set by the Verace Pizza Napoletana, an organization founded to protect the way true Neapolitan pizzas are made. In this traditional method, the pizza can bake for only up to ninety seconds. But the actual temperature depends on whether the oven judges stay with an existing 900-degree temperature or whether they throw another couple of oak logs on the fire. The latter is obviously the case, as bottom bricks are burning every pie. The oven temp must be well over 1,000 degrees.\n\nThe oven judge watches as I hand stretch the _tipo 00_ flour, a more refined and higher-gluten flour perfect for making high-temperature, wood-fired pizzas. I jab the tips of my fingers into the soft dough and create my eleven-inch disk, spreading Italian goat cheese from the Piedmont _(caprino di Rimella)_ over it. Next, I shred fresh _asiago pressa_ (un-aged, unlike the cheddar-like _asiago d'Allevo_ more common in the United States) over the pizzas, followed by small squares of _mozzarella di bufala._ I add thinly sliced braised baby fennel, fresh thyme, Parmigiano-Reggiano, and my par-cooked Maine diver scallops cut in half horizontally. I sprinkle a fine chiffonade of roasted red peppers over all.\n\nOne of the pitfalls for many contestants is not making certain that the ashes are brushed out from the previous pizza or from fallout of the burning wood. Ashes ruin the bottom of a true Neapolitan pizza. Any good Italian oven judge can see the ashes, because they stick to the dough and turn the crust gray. Some oven judges see a chump coming to the final bake and \"accidentally\" forget to brush the oven bottom. Any contestant must remember that this guy docks points for just about anything, scratching away at his little scorecard at the tiniest screw-up. Too much flour on your dough? Scratch. Put sauce on the wrong way? Scratch. Forgot to pull the dough onto the pizza peel? Scratch. Neglect to clean your area? Scratch. Hail from Ohio? Scratch. I nod to the judge and grab the oven broom. He takes it from me and sweeps the oven himself.\n\nI use the dome method to cook my pizza. I lift the pizza off the too-hot bottom bricks with the long pizza peel and hold it aloft into the domed upper roof of the pizza oven. This technique ensures an even bake and would have saved previous contestants from their burnt bottoms. Ninety seconds, and I withdraw my pie from the oven and place it on a plate, then pull out my truffle slicer and glide my _tartufo bianchetto_ across the razor-sharp blade, depositing small, fragrant slices onto my pizza. The heat of the pizza enhances the scent. Finally, I garnish my pie with a fresh arugula and sunflower-sprout salad, and then a spritz of lemon. A great pizza, if I say so myself.\n\nBut it's Tony who wins \"Best Pizza in the USA\" in the Pizza Classica category, with a whopping 713 points out of 800, for his pie of white asparagus wrapped with prosciutto, Parmigiano-Reggiano, San Marzano tomato sauce, fresh _buffalo_ mozzarella, and fresh basil.\n\nStill, I end up scoring higher than any other American for all categories, having successfully completed every culinary competition at the World Pizza Championship. My pan pizza with feta, chicken, sun-dried tomato, and spinach scores a respectable 607. I place first in the USA with a 550 score for my Neapolitan pizza. Even my last place in Non-Gluten garners 438 points. The 600 for my truffle and Maine diver scallop pizza puts me in First Place USA with a total of 2,195 points.\n\nI see my American team of Bruno, Tony, and Justin only once a year, but we always bond, despite the prickly circumstances and stress that break us down to exactly who we are inside. Our shiny varnish and outward personas vanish with the jetlag, lack of space, surly hotel-kitchen chefs, hostile competitors, and the mere fact of being at the toughest pizza competition in the world. A smart guy once said, \"In prosperity, our friends know us; in adversity, we know our friends.\"\n\nAs our last night in Italy winds down and Bruno's pathetic karaoke version of \"Baby, We Can Talk All Night\" wafts across the disco, I think about our _tartufo_ adventure. Was there really danger or impending doom at the hands of the truffle mafia? The intrigue made the trip more exciting. Besides, I've saved the biggest truffle for myself. It's stinking up my hotel room, along with a big bag of dried porcini mushrooms. After I smuggle them home, I can tell the story of the truffle guy to my family while shaving my white beauty over freshly made risotto. Then, as we taste the memory of the World Pizza Championship, I can exclaim, \"That's gold, baaaaabby!\"\n\n### A SLICE OF FAMILY HISTORY\n\n### By Daniel Duane\n\n### From _Food & Wine_\n\n### Hobbies have a way of turning into book subjects for San Francisco\u2013based novelist\/lifestyle writer Daniel Duane\u2014surfing, rock-climbing, and now food and wine. His latest book: 2012's _How to Cook Like a Man: A Memoir of Cookbook Obsession._\n\nSeveral months ago, I received an unexpected UPS delivery: a cardboard box from my wife Liz's Aunt Mary in Omaha, Nebraska. Inside, buried among the packing peanuts, I found an old black leather knife roll\u2014a huge one\u2014heavily worn and without a note. But I knew instantly what it was: the knives of my wife's grandfather, Mary's father. And I wondered if it might not carry a message from the past, if only I could figure out what.\n\nI never met my wife's grandfather. He died relatively young, back when she was in second grade. But when I first met Liz, a picture of the man chopping vegetables hung in her kitchen, as well as in the kitchens of all her food-besotted cousins. \"Papa\"\u2014his name was Bernard Schimmel\u2014loomed like Moses in that family; over time, I grew more and more curious about him. A professional chef who had trained at the world's oldest and finest hotel school, in Lausanne, Switzerland, Bernie seems to have had an outsize, gregarious personality, a natural party host who loved his Scotch, his lemon drops and his three daughters, one of whom was Liz's mother, Judy.\n\nBernie was also the head chef and dining consultant for the hotels that his own father had built along the rail lines south of Chicago in the mid-20th century. At one of those hotels, Omaha's Blackstone, family lore has it that Bernie invented the Reuben sandwich for one of his father's regular poker buddies, Reuben Kulakofsky. (A competing origin story credits an earlier \"Reuben Special,\" from a New York deli, but that sandwich had neither sauerkraut nor pastrami and was not grilled.)\n\nI'm something of a knife geek, so my first hope, as I opened Bernie's knife roll, was that I would find treasures\u2014maybe a collection of midcentury carbon-steel Sabatiers, from Bernie's time in Switzerland. Instead, I found run-of-the-mill American stuff: stainless steel slicers for bread and big roasts; a lightweight boning knife for lamb and poultry; a long fish-filleting knife; and a bunch of accessories, like a melon baller, a butter curler, an orange peeler, an oyster-shucking knife, old salad tongs, a carving fork and a plain wooden spoon. What I found, in other words, was the functional tool kit, heavily used but well maintained, of a working Midwestern hotel chef from the 1950s.\n\nTools are for using, even heirloom tools, but I felt a little stumped: If I dumped all of Bernie's things into my own kitchen drawers, the collection would get lost among my many implements, fast losing their identity. If I left them all in the knife roll, they would stay there forever, unused and under-appreciated.\n\nAll this mattered to me because I never cooked before I met Liz. In fact, I never thought much about food. But after we got engaged, Liz's parents started taking us along on their competitive restaurant-going, hitting every great San Francisco spot: Zuni Caf\u00e9, Gary Danko, Spruce, Charles Nob Hill, Aqua. In their company and on their dime, I ate my first foie gras _torchon,_ my first fresh truffles.\n\nI knew I would never be able to afford regular Michelin-starred fabulosity on my own, so I taught myself to cook. By the time Liz and I had two daughters, cooking had become my primary pastime. I blew the first few meals I made for Liz's parents\u2014some through poor menu selection, like serving a main course of pig's liver _caillettes_ (sausage patties made from liver and spices), wrapped in lacy caul fat and baked. Other meals flopped through my sheer ineptitude, like the time I opened the oven, with dinner already an hour late, to find my chickens raw and cold. But I fought my way toward triumph, culminating in a nine-course menu from _The French Laundry Cookbook,_ prepared with help from my then-seven-year-old daughter Hannah, a cheerful and brilliant assistant.\n\nAfter that, Judy began to say that \"Dad\" would have loved knowing me, that he would've loved my fascination with the craft of cooking. I felt flattered by this, and I suspected that I would have loved knowing Bernie, too\u2014not just the chance to learn from him, but also the chance to eat, drink and party with a man whose appetites apparently matched my own. The gift of the knives, I thought, might reflect nothing more complicated than that. They were just knives\u2014less fancy and less materially thrilling, to be honest, than all the handmade Japanese blades I'd accumulated on my own. The talismanic power of all this stuff, if it had any, came from its integrity as a collection, and the link that it provided to Bernie himself.\n\nFor this reason, I felt grateful when Judy asked me one day to bring the knife roll to a family reunion. She was going to serve Reuben sandwiches at a picnic dinner, and she thought it would be fun to let her cousins cut those sandwiches with Bernie's big slicers. So I did, handing them over before the meal and receiving them back again afterwards. On the way home, in the car, liking how the day had felt, I thought perhaps this was the answer: As Guardian of the Knives, I would store the collection in my home as a complete entity, but I would bring it out for key family functions.\n\nMy daughter Hannah, wise beyond her years, found this ridiculous: \"Daddy!\" she said, exasperated. \"Stuff is supposed to be used. You're not supposed to just hide it somewhere!\"\n\nI did not own a 12-inch chef's knife, so I figured I would add that one first to my regular rotation, if only for slicing big piles of leafy greens into chiffonades. Pulling that hefty knife from the roll, I noticed a few fine, bright scratches near its blade edge, as if it had been sharpened only days earlier. This caught my eye because the knife had not been touched at the luncheon. So I scraped the blade softly sideways across my forearm, a standard sharpness test: Arm hairs flicked free, effortlessly, one after another. The blade was an absolute razor, so finely honed it wouldn't have stayed that way through more than a few days' cooking.\n\nI phoned Judy first, assuming she had paid to have the knives sharpened after her father passed away. Judy had no idea what I was talking about. So I called Mary, who laughed out loud. \"Dad must have left them like that,\" she said, chuckling. \"I haven't touched those knives since he died.\"\n\nAnd that's how it happened\u2014that's how I felt Bernie reaching through time, making contact with me. Every chef drags a fingertip sideways across the blade edges of his knives, testing their sharpness; Bernie's own fingers, therefore, had felt precisely the edge I was feeling.\n\nHannah, by this point, was poring over the contents of the knife roll, marveling at all these little tools from her great-grandfather, the legendary cook. \"And what's this thing?\" she asked me.\n\nShe held a battered old steel tasting spoon. It was stamped, on its face, with the names of the Schimmel hotels in curiously tiny lettering: Omaha's Blackstone, of course; but also the Cornhusker in Lincoln, Nebraska; and the Lassen in Wichita, Kansas; and the Custer in Galesburg, Illinois. Not one of these hotels remained in business, and yet this spoon had served in Bernie's most personal, intuitive act: tasting. Holding it in my hands, thinking of all the times I'd brought a sauce to my own lips, I felt a link to Bernie, and to the simple love of food he had passed along to his daughters. And, through them, to me.\n\n### [BARBECUE ROAD TRIP:  \nTHE SMOKE ROAD](contents.html#ch29)\n\n### By John T. Edge\n\n### From _Garden & Gun_\n\n### Social historian, food writer, and tireless promoter of Southern culture, John T. Edge is director of the Southern Foodways Alliance and a columnist for _The New York Times, Garden & Gun,_ and _The Oxford American._ His most recent book, _The Truck Food Cookbook,_ was released in May 2012.\n\nJess shook his head, tapped his nose, and mouthed the words No smoke. But I insisted. How often do we get to Covington, Tennessee? I thought, as we pondered our third barbecue pit stop of the day. We might as well give their sandwich a shot.\n\nMy boy remained dubious. He stepped to the counter and asked the gentleman in charge how they cook their pigs. \"With charcoal,\" the fellow said. \"Gets it tender, and the sauce does the rest.\" Jess looked up at me with all the world-weariness a ten-year-old with a shock of blond hair and a snaggletoothed grin can muster. He didn't have to say a word.\n\nAs we drove away, bound for the last stop on our semiannual road trip through rural western Tennessee\u2014a region I've come to think of as a Land of the Lost for hickory-cooked pork barbecue believers, where roadside purveyors still fuel masonry pits with hickory and oak logs and undiscovered treasures always seem to lurk around the bend\u2014I caught his eye in the rearview mirror. Jess smiled, shook his head, and tapped his nose again.\n\nJess knows barbecue. He ate his first solid food, a shoulder sandwich with slaw and sweet red sauce, in the parking lot of Spruce's Bar-B-Q in Griffin, Georgia. Through the years, on family trips to Alabama, where my wife grew up, and Georgia, where I was born and raised, he's learned to case a joint. In addition to working his nose, he knows how to survey the woodpile, ferret out freezer case menu conceits, and spot steam table pork at twenty paces.\n\nA couple of years back, Jess and I began to take just-the-guys jaunts north, from our home in Oxford, Mississippi, through the small Tennessee towns that sprawl between Jackson and Memphis. Jess discovered how to scan for armadas of martin houses, made from whitewashed gourds. He wondered, rightly, what really goes on inside tanning salons, twirling academies, and taxidermy studios. And he learned to eat, with discernment, intelligence, and gusto.\n\nOn one of those trips, I tried to get Jess to read the text of a historical marker, erected in tribute to some long-forgotten graybeard statesman. He feigned interest, as any dutiful son would. Five minutes later, he recovered his gumption. \"What if I could play laser tag and you could get a really good barbecue sandwich in one place?\" he asked. \"That's what we really want, right?\"\n\nOver time, Jess and I have learned each other's palates. I'm a thin vinegar sauce guy. Jess likes molasses and ketchup-thickened concoctions. I prefer whole hog, pulled into long strands. Jess likes shoulder meat, hacked into shards and piled high on a white-bread bun.\n\nMore important, we have developed a shared love of western Tennessee barbecue and the people who cook it. As he grows from a goofball boy, who sings in the shower and hugs me good night, to a querulous teenager, who questions my authority and demands the keys to the family sedan, I hope we're also honing an enduring friendship that won't depend solely on blood and genes.\n\nOn our most recent expedition, Jess and I hit two stellar spots before that misstep in Covington. (We also did a drive-by of the Mindfield, a skyscraping folk art environment on the edge of downtown Brownsville, and made a quick tour of the Alex Haley Museum, an homage to the author of _Roots_ tucked in a modest neighborhood in nearby Henning.)\n\nHelen's Bar-B-Q in Brownsville came first. Set in a metal hutch, on a hard curve north of the courthouse square, the six-seat caf\u00e9 sits next to a retailer of discount jewelry and hair extensions. Across the street is a derelict tourist-court-style motel. Inside, Helen Turner, one of the few female pit masters in the South, stokes the fire and works the sandwich board. She's a dervish. Behind the caf\u00e9, in a screened pit room, she burns hickory and oak down into coals between two sheets of corrugated tin. With a long-handled shovel, she slides those coals into a concrete block berth. After they spend ten to twelve hours in a swirl of smoke, she pulls pork shoulders from that pit, chops them with a cleaver, piles the meat high on buns, and douses all with a red sauce that straddles the line between ketchup and vinegar, between sweet and hot.\n\nJess ate his sandwich in four greedy bites. He moaned as he ate. Literally. But he also kept glancing up at Ms. Turner, who was wearing a blue floral-print hairnet and pink hospital scrubs. As she loped about the pit room, Jess watched. As he chewed through smoke-blackened hunks of pork flesh, his eyes tracked her movements. And his mind engaged.\n\nWhen we drove away, Jess was cradling a pickle jar of Helen's sauce and plotting the many ways he could employ it when we got back home. His eyes were wide. And so was his smile. He got it, I told myself. I didn't have to make some fatherly point about how women can do anything men can do. About how women work as hard as men. Ms. Turner had made those points for me. And she had dished a stupendously great sandwich, too.\n\nSam's Bar-B-Q in Humboldt was next. When Jess spied the woodpile, just south of downtown, he yelled for me to brake. A jumble of hickory trunks and oak stumps\u2014delivered by friends and neighbors as they cleaned up from last spring's tornadoes\u2014that wood served as a kind of free-form art installation and a de facto advertisement that announced HONEST BARBECUE COOKED OVER WOOD HERE.\n\nThe building is made of white block, patined with soot. The pit, set in an adjacent building, is an iron-lung-shaped double-decker, with wood burning down to coals on the bottom and shoulders smoking on racks up top. It's a feat of country-boy engineering, designed by the founder, Sam Donald, and now manned by his son-in-law John Ivory. The efficacy of that design gets proved every time customers heft a sandwich to their maw.\n\nAnd that's just what Jess and I did. We hefted two beautiful sandwiches and a slice of chocolate chess pie, still hot from the oven. And we listened as locals came streaming in. \"Now, you're the baby girl, aren't you?\" Mr. Ivory asked a woman who ordered a barbecue bologna sandwich. \"Give me a sandwich with fatty meat and slaw,\" said a fireman with soot on his hands.\n\nWe could have stayed all afternoon, rubbing our bellies and listening to the dulcet tones of happy eaters. But we motored on, bound for Covington. Two stops don't make a barbecue road trip, I told Jess. Three is the minimum. Everybody knows that.\n\nBy the time we left Covington, we didn't need another sandwich. We had eaten well and often. But we pushed on. That's what you do on a barbecue road trip. You push on, no matter how full you might be, goaded by the promise that the next experience might prove to be the best.\n\nAs rain began to splatter the windshield and night began to fall, we pulled into Millington, Tennessee. Our goal was Woodstock Store N' Deli, a cinderblock rectangle where Anthony Bledsoe serves a green-hickory-smoked shoulder sandwich, piled perilously high and capped with slaw. He calls it the Sleeper.\n\nThe source of the moniker became clear when Jess and I stepped back into the gloam and spread paper towels on the hood of our station wagon. Our sandwich was an overstuffed behemoth. Fatter than fat. Stacked like a napoleon of meat, slaw, and sauce. Wretched and lovely excess. Enough to put any eater to sleep.\n\nAs we picked at stray bits of meat that had fallen from the sandwich and worked up the courage to take yet another bite, a woman approached us. She was stooped. And slight. She carried a ragged umbrella. Her hand was out. She said her belly was empty. She looked directly at Jess. I reached into my pocket for some cash. Jess folded the sandwich back into its tinfoil pouch and asked if she would accept it. She did. And then she was gone.\n\nFive miles from home, I asked Jess what he learned on our barbecue road trip. \"Respect what you have while you have it,\" he said. That insight applies to issues of hunger and poverty. And to the barbecue traditions of the South. Not to mention father-son relations.\n\nThough our barbecue buzz had faded by the time we rolled back into Oxford, my road-trip-fueled appreciation for Jess had grown, as had my conviction that we needed to plot another expedition. Before his hunger fades. Before he grows up.\n\n## _The Family Table_\n\n### THE FOOD-CRITIC FATHER\n\n### By Todd Kliman\n\n### From _The Washingtonian_\n\n### Every new parent knows how a new baby upends family life. But what if you're a restaurant reviewer, obligated to dine out several nights a week? Todd Kliman, award-winning dining critic for the monthly _Washingtonian_ magazine, chronicles his heroic efforts to accommodate the new addition.\n\n\"Welcome to my world,\" my friend Laura said when I called to tell her my wife and I were having a baby.\n\n\"You mean the world of early-morning feedings and interrupted sleep?\"\n\n\"I mean,\" she said, \"the world of chicken tenders. You're in for a rude awakening, my food-critic friend.\"\n\nThere are things every parent-to-be fears. The grand, existential worries: Let him be breathing, I said to myself at the moment of delivery. Let there not be a cord wrapped around his neck. Let him be normal. Then there are the little, quotidian matters, the myriad concerns of getting along with what is, let's face it, a new housemate. A wailing, demanding housemate. A housemate who makes the slob you once roomed with\u2014the one who raided the fridge and lounged on the couch in his underwear\u2014look not half bad by comparison.\n\nIn the months preceding our son's arrival, I kept hearing that life was about to change, that being a parent would alter my reality in ways big and small. But it wasn't until Laura framed the terms of my soon-to-be life that the existential abstraction\u2014change\u2014became quotidian and concrete.\n\nI thought I had prepared myself. Long before our son was ever a notion in our heads, my wife and I sat down one night and figured it all out\u2014discussing, with the certainty of people for whom nothing is at stake, how we'd raise our Hypothetical Kid. We'd never buy a minivan, we wouldn't build our existence around our Hypothetical Kid's world. Hypothetical Kid would enter ours.\n\nBut what was this world I'd created? As a food critic, I ate dinner out every night\u2014often following a long lunch. Sometimes I went out to two dinners in a single night, and the wine sometimes meant that those nights stretched into the morning.\n\nI understood when I accepted this job that I was to think of myself as a kind of public functionary\u2014a Designated Eater. I endured the caloric overload and the punishments to the body so readers could spend their time and money more sensibly. Not that I ever complained. This was the other thing I accepted\u2014griping was bad form when you were eating out nightly on someone else's dime.\n\nIf it wasn't all Champagne and truffles, the fact remained that every meal was a restaurant meal. I lived a fantasy life, and I nurtured one, too. As much as I existed to feed my readers tips, I existed to feed their fantasy of the bon vivant. Some bon vivant I was about to be, I thought, spooning strained carrots and mashed peas to a newborn.\n\n\"You have to sacrifice certain things now,\" an obstetrician said, preparing us for our new life.\n\nBut there were sacrifices I wasn't sure I was prepared to make.\n\nIn the beginning, things were a relative breeze. In their first few months, most babies sleep 18 hours a day or more. The car seat was an ungainly purse we carted everywhere, leaving it on a chair beside us or on the floor next to our table. Jesse slept in his protective pod or stared blankly and drooled. Half the time we brought him with us to restaurants, I didn't even notice he was there until it was time to leave.\n\n\"Wait,\" my wife said. \"It gets harder.\"\n\nWe hit more than 100 eateries in those first three or four months\u2014white-tablecloth spots, dives, and everything in between. By our son's first birthday, he had been to more than 250\u2014a fact I shared with the friends and family who gathered at our house to eat cake and watch Jesse open presents.\n\n\"You're kidding,\" someone said.\n\n\"Nope.\"\n\n\"Two hundred fifty,\" she said. \"You're keeping a total?\"\n\n\"Actually, 272,\" I said. \"But who's counting?\"\n\nShe shook her head. \"You're such a guy.\"\n\nIn a sense, we entered a new phase when Jesse graduated from the car seat, where he mostly slept, to a high chair, where he mostly didn't. But it was the night I took my one-year-old son out for barbecue for the first time that we truly entered a new stage of parenthood.\n\nBarbecue has always held a certain primacy for me among all the foods I cook and crave, and I wanted it to be special the first time my son dug into a pile of smoky 'cue.\n\nIt was a long drive. My wife had expressed doubts about chancing a place this far from home in rush hour, and now as we hit a patch of traffic on the Beltway, she worried we'd be getting home too late. Jesse wailed and wailed. I told her we could bail at the next exit. \"Stick to the plan,\" she said, slipping Jesse animal crackers.\n\nI suspected she encouraged me to keep going only so she'd have the ideal incriminating illustration to use against me the next time I suggested we venture out too late for a restaurant.\n\nTraffic eased up. We managed to be only 40 minutes late. Fortunately, the food\u2014ribs, pulled pork, beans, cornbread, coleslaw\u2014came quickly. Jesse, despite the in-car appetizer, was ravenous. He attacked his plate, shoving bits of spare rib into his mouth as if satisfying some deep need. I'd never seen him go at any food like this. But I'd never turned him loose on barbecue. He's got it in his genes, I thought, marveling at his sauce-smeared face. I took pictures.\n\nWe drove home in one of those warm, expansive moods when your earlier fears have been exposed as premature and overstated and life feels loose and easy. We laughed and celebrated his initiation into the culture of barbecue and the world of adult eating.\n\nAnd then at 3 in the morning, my son, snugly sleeping between Mommy and Daddy, spewed the contents of his stomach all over himself, our bed, and us. Even after we changed the sheets, the smell of wood smoke and stomach acid was unmistakable.\n\nYay, solid foods!\n\nMy wife laid down certain rules after that: In a high chair by 6 at the latest. If traffic is bad and we have to bail, so be it.\n\nDinner became something to prepare for twice. There were the preparations I needed to make for him and the preparations I needed to make for me, to ensure that my stealth mission\u2014my anonymous visit to a restaurant as a critic\u2014came off without a hitch.\n\nAfternoons usually went like this: Make a reservation under an assumed name\u2014my nom de mange, I like to call it. E-mail my wife to be ready to leave the house at the appointed time. Choose all of the dishes we'd order. E-mail our guests the \"rules\" for the night: what can and can't be talked about at the table, when and how to pass plates so that I'm able to sample each dish, how to quiz the waitstaff about the Syrah and the veal without lapsing into an SS-style interrogation.\n\nAt the same time: Load his bag with necessities\u2014change of clothes, diapers, sippy cup. Choose an assortment of books along with crayons and paper. Fill and pack Tupperware containers with Cheez-Its, Cheerios, and animal crackers while relishing the irony of sneaking cheap processed food into a restaurant that charges $16 for a glass of wine.\n\nSometimes I prepared in vain. One night as we were nearing the restaurant, I glanced into the rearview mirror. There he was, a limp pile of limbs. \"What should we do now?\" I asked.\n\nThe look on my wife's face said: Tell me that's a rhetorical question.\n\nWe'd already taken him out three nights in a row. If it were just any restaurant, I said, I'd understand. But this was Important Restaurant X. I'd been waiting five weeks to get in. I proposed we give it 15 minutes, that maybe he'd wake up then and we could have our meal. I drove around the block three times.\n\nHe did not wake up.\n\nFive more minutes, I said.\n\n\"No,\" she said, peeved that I'd thought to push through.\n\nMy son, if he'd known I had put my needs as a critic above his, would probably have been peeved, too.\n\nThe hostess professed understanding when I called to cancel five minutes after we were supposed to be there, but her blithe, sing-songy tone oozed the opposite. It was all peeve. As for me, I had left peeved and was approaching pissed. Dinner at Important Restaurant X turned out to be takeout pizza from Chain Restaurant Y. It was cold by the time we got it home.\n\nIt was rare that we missed a meal. But there were many I wished we had. One night when Jesse was almost two, we were at an Italian restaurant in Alexandria. The waitress was excellent, and we put in orders for drinks and appetizers. All was well until my son dropped his fork.\n\nHe was inconsolable.\n\n\"It's okay,\" I said, \"they'll get you a new one.\"\n\nYou would have thought a stranger had snatched his Elmo doll. Tears streamed down his pinched red face.\n\nI handed him my fork.\n\nNo. He wanted his fork.\n\nHow did he know this wasn't his fork? He just knew.\n\nThe stares were coming from every corner of the restaurant as he wailed. It dawned on me: I've become the diner I used to despise. I once shot dirty looks at tables with kids who wouldn't pipe down. I never thought twice about asking to be moved to another table, away from the offending baby. Now I always found myself scanning the room for tables with small children, curious to know how their dinner was going. Reading online reviews, with their frequent, raging criticism of parents who brought young children to restaurants, made me cringe. \"Who do these arrogant, entitled yuppie assholes think they are?\" went a typical review on Yelp. \"What makes them think their oh-so-precious princes and princesses are welcome at every goddamn restaurant in the city?\"\n\nMy wife, at the limit of her frustration, gathered Jesse up and took him outside. Ten minutes later, he returned . . . and remembered why he'd been so upset.\n\n\"My fork!\" he cried.\n\nHe and my wife spent the remainder of the meal outside while I ate parts of two appetizers, three entr\u00e9es, and two salads.\n\nIn American culture, there are restaurants for adults and there are restaurants for kids, and the two are not expected to mix. It's different at Latin and Asian restaurants, where no such divide seems to exist.\n\nThe first time I took Jesse to a roadside joint in Bladensburg's Little Mexico\u2014he was probably six months old\u2014the waitress fussed over him for a minute or two before extending her arms like a forgotten aunt. The expression on my wife's face was priceless: diplomacy vying with primal anxiety. You want me to hand over my baby? But she relented. The waitress scooped him up, cuddled him, cooed over him\u2014and whisked him out of the dining room.\n\nNow my wife's alarm was palpable.\n\nWe found our waitress in the kitchen, parading Jesse around as the cooks entertained him with their shiny tools.\n\nEven better was the restaurant in Annandale's Koreatown where the co-owner descended on our table as if she were his bub-bub and hadn't seen him in weeks. She took him to a nearby table where her staff was chopping vegetables. For the next 20 minutes, she bounced him on her knee and dangled scallions in front of him as if they were car keys, entreating us to seize the chance to eat in quiet, just the two of us.\n\nThese were among the most extraordinary restaurant experiences I've ever had. Not only was our son showered with affection, but we were made to feel like part of an extended family. We were being embraced, too.\n\nBy contrast, whenever I walked into an American restaurant with him in those first two years, there was seldom a smile, much less an embrace. My son was a problem to be solved. \"You can't bring that in here,\" the general manager of a popular restaurant commanded us one afternoon.\n\nWe think he was referring to the stroller.\n\nAround the time Jesse turned two, my wife made a decision. Her decisions are never the result of minor epiphanies. They never involve switching cell-phone providers or choosing a new doctor. When she says, \"I've made a decision,\" I brace myself.\n\n\"It's too hard taking us along,\" she said. \"And I don't enjoy it if he doesn't enjoy it, and who can tell if he's going to? There are just so many things that can happen.\"\n\nThe precipitating event had occurred a few nights earlier at a restaurant in DC's Palisades I was reevaluating. She was feeding Jesse a hunk of bread and butter, and he began to choke and then cry. Every head turned. My wife quieted him after several moments, enabling us to hear all the disparaging things everyone around us was saying\u2014condemnations of our parenting style, of our effrontery in taking him out to eat at a place like this. (Never mind that the restaurant had a high chair!) We spent the late-summer night taking turns with him out on the sidewalk, one of us holding down the table, the other walking him back and forth on the pavement, all three of us bitten by mosquitoes. At least somebody ate.\n\n\"You've got a job to do, and you're trying to make everybody happy and making nobody happy,\" my wife said. \"So just forget it.\"\n\n\"What about one of the noisy places?\" I suggested.\n\nI'd come to love the noisy places, those casual midlevel restaurants with exposed ceilings and cement floors that had been popping up in DC. Young people loved them, finding in their cool industrial aesthetic a validation of their decision to live in the expensive but bustling big city; older people hated them, despairing of ever having a normal conversation. As a food critic, I saw the merits of both arguments. But as a new parent, I was a fierce partisan. I loved the protective cover they offered. My son could screech like an electrocuted cat, as he did one night at a clattering bistro, and no one would glance up from the scallop crudo. If only every new restaurant that opened had decibel levels akin to that of a construction site, I sometimes thought.\n\n\"No,\" my wife said, \"not even the noisy places.\" The only exception she'd make was for a Latin or Asian restaurant.\n\n\"How about we revisit this again in a few months?\" I said.\n\nShe didn't agree; she didn't disagree. \"Look, we're learning,\" she said. \"He's going through phases, but we're going through them, too. This isn't forever. And that doesn't mean you can't go out with him, just the two of you. It'll be good for you to have that time together. Daddy-and-Jesse time.\"\n\nThings were harder without my wife there\u2014like being-denied-the-use-of-my-limbs harder. I hadn't realized just how smooth and skilled she was until I was forced to handle it all myself. How did she do it? How did she attend to all his needs\u2014cutting his food into small bites, reading him a story, swabbing him down whenever his face became one big schmutz\u2014while also serving as a sounding board for my impressions and ideas?\n\nIt took all of half a meal for me to realize that being a daddy and being a food critic were incompatible, like pairing foie gras with ice cream. I couldn't make mental notes about the red wine that was served too warm, the sauce that hadn't been properly reduced, the waitress who neglected to fill the water glasses . . . and also read him a story, color with him, and cut his food into 47 pieces.\n\nRestaurant visit number 466: I ducked into the bathroom to make notes. I figured the sanctuary of the stall would reduce the static in my head. I began pecking out letters on my iPhone: \"Scallops = slight translucence in center, dark crusting on top. Microgreens, dabs of corn sauce. Gazpacho blended smooth, a thickness there\u2014bread or bread crumbs?\" But I still had my son with me.\n\n\"Daddy, whatcha doin'?\"\n\n\"Making notes,\" I said.\n\nI heard someone enter the bathroom. \"Making notes,\" Jesse sing-songed as the door to the next stall slammed shut. \"Daddy's making notes.\"\n\nMy son the mole.\n\nMy stall-mate could have been a manager, a waiter. I imagined walking back to my table to the stares of the staff as the news made the rounds: critic in the house.\n\nAmong the first laws of being a food critic is not to take things personally. Our recourse is our column. A bad night is writing material. But my dinners out with Jesse several nights a week were an exercise in double consciousness. I was both the discerning critic and the anxious father. The critic in me focused on the server's knowledge of the menu and of food generally, his ability to keep the flow of the meal smooth and steady. The father in me was fixated on the waiter's regard for the little person at the table. Servers were either with you, I was learning, or they were against you. There was rarely an in-between.\n\nOne night a waiter set down a giant steak knife in front of my son. Servers routinely gave him stemware. A few brought him apple juice or milk in a wine goblet.\n\nI'll never forget restaurant visit number 684 and the cast-iron skillet of mushrooms and polenta. \"Hot\u2014careful,\" the waiter warned. He walked away, leaving the scalding handle much too near my son's face. Jesse's little fingers drifted toward it. I grabbed his forearm and squeezed hard. He began to cry at the pain I'd caused\u2014a pain to forestall a nastier, more lasting one.\n\nHoping to avert disaster, I created it: I knocked over a glass, spilling water all over. The waiter materialized with napkins and stood watching as I blotted while the spill, moving as inexorably as floodwaters, dripped onto Jesse's lap. The manager swung by, and the men stood watching. Doing nothing. Like men do.\n\nThe male staff at restaurants were, almost without exception, awful. That's not to say all the waitresses were superior at their jobs, just that the women were more inclined to understand that having a toddler at the table meant taking care of things other than drink requests and asking if everything was seasoned properly. More inclined to improvise entertainments to distract Jesse\u2014spoons and straws to play with, pen and paper for drawing, even a cookie fetched from the petit four plate. Some people prefer a female doctor\u2014believing, rightly or wrongly, that a woman will be more sensitive to their needs. I had never understood that position until now. I didn't ask to be taken care of by a waitress and not a waiter when my son was with me, but the father in me pleaded silently to be given a waitress.\n\nIt was hard with him. It was hard without him. The nights that neither he nor she came with me felt strange, as if I had one life and they had another. I sometimes felt like a traveling salesman.\n\nThe texts my wife liked to send me while I was out at dinner were meant to impart a sense of closeness but only reinforced this feeling of being alone on the road. Restaurant-going had been a huge part of our dating life. Our first dinner date had been at Red Sea, the now-defunct Ethiopian place on 18th Street in DC's Adams Morgan. Nothing fancy. I'd wanted to see whether she was my kind of woman\u2014a real eater, up for an adventure\u2014or a princess in need of pampering.\n\n\"How's dinner?\" she would text me. If the food was good, I downplayed it. If it was mediocre, I made her think I had spared her from something awful. I declined to give dish descriptions, rendering my meal in single-word summations: chicken, polenta, sorbet.\n\n\"I made Jesse some mashed yams with cinnamon, which he seemed to love,\" she texted one night. \"I had the rest of the chicken from Friday plus heated up some ramen.\" Ramen! The last resort of the penny-pinching college student. I felt a stab in my stomach.\n\nI often brought home leftovers, which my wife dug into the next day at lunch, or sometimes in the middle of the night. Whether veal cheeks over polenta from a French restaurant or pollo a la brasa from a carryout rotisserie, it didn't matter. All distinctions were erased by refrigeration. Everything tasted like fast food. \"I guess you had to be there,\" she'd say.\n\nStill, I persisted: \"Let me bring you something.\"\n\n\"Whatever you want. Enjoy yourself. I love you.\"\n\n\"Tell him Daddy loves him and misses him. And I love you, too.\"\n\nThinking I might persuade my wife to relax the rules, I told her one night about a new non-Asian, non-Latin place that advertised itself as kid-friendly. I had already done my homework, calling ahead to ask the manager about the menu (hot dogs and chicken tenders) and the provided-for distractions (a coloring book and crayons). What a parent is paying for in patronizing a kid-friendly restaurant is immunity from the dirty looks of other diners. Nobody who comes to a place with coloring books and crayons ought to complain about a kid acting up at the next table\u2014or so my wife contended. \"If they have a high chair,\" she said, \"they've made the decision that kids are welcome.\"\n\nNot that she trusted restaurants to understand that that's what they'd decided. My wife no longer took restaurants at their word when it came to children. \"What makes this place kid-friendly?\" she wanted to know, pouncing on the phrase like a prosecutor.\n\n\"Besides the menu, I don't know. A high chair? Coloring books?\"\n\n\"Just because they have a high chair,\" she said, \"doesn't mean they're going to be competent.\"\n\nMy God, I thought. It's happened: My wife has become a mother.\n\nI knew the moment she pushed him out into the world that her life had changed and that ours, together, had changed along with it. But for some reason it wasn't emotionally real to me until now. This fierce protective instinct\u2014where had it been hiding all these years? She'd always been willing to give things a try, always relished the opportunity to turn a bad situation into something good. I was under no illusion that she believed it was her job in life to make me happy, but if there was anyone, before, whom she sought to comfort or appease or please, it was me.\n\nWe never did visit that \"kid-friendly\" place.\n\nThe line between discernment and snobbery can seem razor-thin in the world of food. I've never wanted to be the kind of critic concerned chiefly with handing out demerits and judging dishes according to some Olympian notion of correctness. I've never wanted readers to feel that I equate a discriminating palate with a sense of my own superiority.\n\nIn my reviews, I've always strived to remember I'm writing for all sorts of readers and that my goal in critiquing a restaurant is to craft an interesting piece that diners who might never get to the restaurant would want to read. It hasn't always been easy, because to talk with any degree of specificity about food is to lapse necessarily into a kind of inside baseball. And to engage professionally in any form of artistic criticism is to choose a life in which you're constantly working out your theories and ideas\u2014intellectualizing experiences that most people never bother to analyze.\n\nI felt a version of this tension playing out in me as Jesse racked up the restaurant visits\u2014not as many as in his first year but still two, three, even four times a week.\n\n\"Your son's going to be such a foodie!\" people would say when they learned what sorts of foods he considered normal.\n\nI would cringe.\n\nSome years ago for this magazine, I edited a story about prepubescent food snobs\u2014kids who sneered at fish sticks and pined for bacalao and who knew to order off the secret menu. My Hypothetical Kid would never be like those kids, I remember vowing. He'd eat widely and learn to love food and gain exposure to all the cultures and cuisines of the globe. He wasn't going to be one of those children whose sophistication comes across as a party trick. I wasn't going to let him become an adult in kiddie clothing.\n\nIt was, of course, premature to tell what sort of kid he would become\u2014he was three.\n\nA parents' group met in our neighborhood once a month, with mothers, fathers, and children coming together for a couple of hours at a playground or someone's home. Inevitably, talk of sleeping routines gave way to talk of food. Emma was a very good eater and had recently discovered broccoli, which of course she adored, while Sebastian not only ate hummus but actually liked to eat the chickpeas themselves, and Gabriella, who had only recently weaned herself from the bottle, had developed an inexplicable taste for pickles. Can you imagine? I preferred to downplay Jesse's catholicism in these sorts of conversations. I was leery of characterizing him as a precocious toddler foodie.\n\nThere was another reason I kept quiet: I feared exposing myself as the food-world equivalent of the arrogant Little League coach. I liked to joke to my wife that, eating out as often as he did and being exposed to such an astonishing array of cuisines, Jesse had already surpassed his peers who were blissfully munching away on McNuggets, ignorant of what they were doing to their bodies and (of greater concern to me) their taste buds.\n\nBut all humor conceals a darker truth.\n\nI was proud.\n\nIn his three years on this earth, Jesse had pretty much devoured it. Thai, Indian, Salvadoran, Vietnamese, Afghan, Cuban, Greek, Turkish, Ethiopian, French, Brazilian, Portuguese, Japanese, Chinese, Armenian, Peruvian, Bolivian, Kazakhstani, Bosnian\u2014there was nothing he hadn't tried. The list of foods he loved and asked for was long: edamame and mango lassi and chicken tikka and char siu bao and grilled stuffed grape leaves and kanom jeeb and chicken curry puffs.\n\nHe made the other kids from the parents' group look like pikers. He made me look like a piker. I had been writing professionally about food for a decade but had never tasted many of these things until I was in my mid-twenties.\n\nWas there a correlation between trying so many kinds of foods at such a tender age and becoming a richer, more broad-minded person? I liked to think so. Jesse wasn't being dragged out to dinner night after night at Daddy's whim, I told my wife; he was getting an education. And what an education it was. You couldn't buy an education like this.\n\nAs I made this point one night in the car after restaurant visit number 872\u2014the third night in a row I had kept him out two hours past bedtime as I worked feverishly to finish my work for a dining guide\u2014Jesse, giddy with tiredness, was belting out songs in the back as if he'd just discovered Ethel Merman on YouTube.\n\n\"Daddy? Sing?\"\n\n\"You like to think you're the guide, in control,\" a friend with two kids in their twenties told me. \"But you're not. All you can do is direct them a little. You're not the bike. You're more like the training wheels. And what you have to remember is eventually the training wheels come off.\"\n\nWe were on our way to restaurant visit 924 when my son pointed out the pair of golden arches looming ahead. This is what's commonly referred to as a teachable moment, a chance to put forward a philosophy of food and, in this way, install a belief system that will last a lifetime. \"McDonald's,\" I said, \"is yucky.\"\n\n\"It's not yucky,\" my son replied.\n\n\"Ick,\" I said. \"Ugh.\" I shuddered.\n\nJesse became agitated, thrashing against the constraints of his car seat. He was crying. \"It's not yucky,\" he said and then, gathering himself, uttered the words with an almost sinister deliberation, driving the dagger in more deeply: \"McDonald's is delicious.\"\n\nIt required every ounce of concentration on my part not to hit the car in front of me.\n\nIt required every ounce of concentration on my wife's part not to burst out laughing.\n\n\"How'd this happen?\" I demanded. \"Did someone take him to a McDonald's? One of the babysitters?\"\n\nSilence. A guilty silence.\n\n\"I mean,\" my wife said cautiously, \"he's had fries a couple of times.\"\n\n\"You've taken him for fries?\"\n\n\"A couple times.\"\n\n\"A couple\u2014\"\n\n\"Okay, a few. Maybe a burger once or twice.\" She winced, cutely.\n\n\"How could you?\"\n\nNumber 1,000 loomed, a figure I'd been pointing to for nearly a year. But it seemed ridiculous now, all that number-keeping. All that striving.\n\n\"He's three, you know,\" my wife said.\n\n\"I know.\"\n\n\"Do you?\"\n\n\"What are you saying?\"\n\n\"He just enjoys being with you.\"\n\nMy son had had a rough week: two doctors' visits, an upset stomach, and\u2014either because he'd just turned three or because the gods had decided to complicate our life even more\u2014a growth spurt that made him cranky. I hadn't been around much, having been cramming in final visits to several restaurants before an upcoming family trip. That Saturday, I decided, would be Daddy-and-Jesse day.\n\n\"Aww,\" my wife said. \"He really could use some Daddy time. What's the plan?\"\n\nI told her.\n\n\"You mean,\" she said, \"you plan to take him to work with you.\"\n\n\"No. To dim sum.\"\n\n\"You're sure about that.\"\n\n\"He loves dim sum.\"\n\n\"I meant you're sure this isn't work? This isn't a place you're reviewing, or checking up on? What's the restaurant?\"\n\nI told her.\n\n\"I've never heard of it.\"\n\n\"It's new.\"\n\n\"Uh-huh,\" she said. That tone again. \"Just try to have a good time, okay?\"\n\nIt was noon when we arrived for restaurant visit 1,027. The place was a scene of happy chaos. Extended families huddled around big circular tables with lazy Susans, servers dashed through the room like rush-hour commuters in pursuit of a departing subway train, the hawkers with their metal carts circled in search of a sale, lifting lids and sending up little clouds of steam.\n\nOur table was a scene of happy chaos, too. Steamed pork buns and roast pork buns. Shrimp dumplings and shrimp balls. Rice-noodle crepes stuffed with shrimp and stuffed with ground pork. Tiny custard pies and pineapple buns for dessert. I don't think I had ever seen Jesse eat so much at any one meal. For a feeder descended from a long line of feeders, there's nothing quite as gratifying as seeing someone you love eat with gusto.\n\nSomewhere between the shrimp balls and the shrimp noodle crepes, my wife texted me: \"How's it going?\"\n\n\"He's loving it,\" I wrote back. \"He looks relaxed and happy, and he's eating like someone who hasn't seen food in a week.\"\n\n\"I'm glad he's getting his fill. He loves being with his daddy.\"\n\nA triumph.\n\nBack home, I couldn't help myself. Before I had unstrapped the diaper bag from my shoulder, before I had helped him out of his shoes, before I had even stepped ten feet into the house, I began exulting. I was dangerously skirting the line of gloating. There were, I said, all sorts of ways to have a good time and all sorts of venues. It didn't always have to be a zoo, a park, a playground. He could come into our world, I said; we didn't have to go into his.\n\n\"I'm glad it went so well,\" my wife said, walking into the living room only to be crushed by my son's running, lunging hug.\n\n\"Amazingly well,\" I corrected.\n\n\"Jesse, did you have a good time with Daddy?\"\n\nFor the next ten minutes, as my wife helped him into a new set of clothes and then all three of us went into the playroom, I listened to my son tell the story of the day. It came in bursts, a little at a time, and what he said wasn't as interesting as what he didn't say.\n\nThe afternoon, according to Jesse, amounted to the following: Sitting next to Daddy. Eating with Daddy. Blowing bubbles in the water glass with Daddy. Laughing at the waiter with the funny glasses with Daddy. Making the bunched-up straw wrapper into a snake with Daddy.\n\nHe didn't mention a single dish. He didn't mention any food at all.\n\nMy son is up to something like 1,300 visits by now.\n\nI don't know how many exactly.\n\nMy wife and I just had another child, a son, and life is busier than ever.\n\nIn any case, I've stopped counting.\n\n### [THE LEGACY THAT WASN'T:  \nWONTON SOUP](contents.html#ch31)\n\n### By T. Susan Chang\n\n### From _A Spoonful of Promises_\n\n### Cookbook reviewer, blogger, and essayist T. Susan Chang looks through a prism of food memories to meditate on her many families\u2014the emigrated Chinese clan she grew up with, the young singles tribe of her 20s, and the nuclear family she and her husband have begun.\n\nThe secret of the wontons, I think, was always in the smell. It was only ground pork, with maybe some shrimp, and it was seasoned with all the usual suspects\u2014dry sherry, soy sauce, scallions, a little ginger, a little sesame oil. But when you finally had it just right, you'd know it instantly, because it didn't smell like raw meat any more. It smelled different\u2014fresh, sweet, and good.\n\nMy mother learned to make wontons from her mother. It's possible her mother learned it from hers. It's possible that there, beneath the misted camel-hump caves of Guizhou, generations of women related to me made wontons just like my mom's. I really can't say. What I do know is that nobody has to learn to eat a wonton. That just sort of takes care of itself.\n\nThe wonton wrappers came from the Chinese grocery store, and since this was the '70s, that meant an excursion to Chinatown, in the city. My mom would park our ugly-yet-iconic beige Plymouth Volar\u00e9 station wagon and stroll from store to store, returning only to check in on me and feed the meter. I would lie in back, trying to make sense of _David Copperfield_ while slowly being imprinted with the vinyl pattern of the seat. It never struck me as odd that I could better comprehend the idiom spoken in a nineteenth-century London factory than the sidewalk Cantonese being shouted just outside the window. My sister, incidentally, had named the station wagon Ouagoudougou, after the capital of then\u2013Upper Volta (Burkina Faso now), and that seemed pretty normal too.\n\nMy mom made the filling at home. How, I couldn't tell you. I was probably busy reading the essay on Indo-European roots in the back of the dictionary, another favorite pastime. In any case, as I lolled bookishly under the bed, the smell of wontons would travel down the hallway to my room, followed with momlike efficacy by an exhortation to come to the kitchen, where the bowl of filling waited.\n\nThen, as in Chinese families everywhere, everyone would sit around the table and wrap wontons. Even my dad sometimes helped. We'd talk about violin and piano practice, about the books my dad's company was publishing, about who was doing well in school. It was sort of like a shareholders' meeting, but with better food.\n\nWhat we didn't talk about so much was the past that brought us here. Or rather, that brought our grandparents here\u2014in a hurry, and for good.\n\nBetween 1938 and 1949, the Chang family and the Pu family, like so many other families, were on the move. My dad's dad was a financier, antiques dealer, millionaire, governor, mobster (probably), and, for less than a year, the Chinese premier. He even had a mobster nickname\u2014the \"Reclining Cicada.\" My mom's dad came to study English literature in Michigan, where he came by his own nickname: \"Mr. Pu from Kalamazoo.\" My dad's family brought precious antiques, and a brilliant, perilous political past. My mom's family came with hardly anything but its resilience, and its considerable wits.\n\nI wish I could say that the wonton soup recipe was a tangible inheritance, that it arrived on the plane in New York in the suitcases of my fleeing family (for the Reclining Cicada, it wasn't so much flight as \"strategic relocation,\" I like to think). Maybe in my grandfather's steamer trunk, which was stamped \"CKC\" in gold.\n\nBut no, there was no physical recipe to puzzle over, divine, and interpret, and in any case, the wonton soup didn't come from my dad's family at all. It came from my mom's, and somehow or other, it got here safely, as did my mom's mom, my Po Po. She made wonton soup for her children, and she made it for an increasing array of grandchildren\u2014two, five, eight, ten, finally thirteen in all. Then she started in on her great-grandchildren. Their hair might have been a little lighter colored, the eyes a little more Caucasian, but for fifty years the scene remained pretty much the same: a little head bent over a steaming bowl, focused on a small regiment of wontons staring back up from the spoon with wrinkly, Shar-Pei-like faces. An indelicate slurping provided the soundtrack. Unlike my grandfather's antiques, Po Po's wonton soup was typically gone within ten minutes of being served. But it was not hard to interpret.\n\nIt's the nature of inheritance to stick close in some places and fall free in others. As our family propagated and throve in its new home, bits of our cultural heritage dropped away. Gradually, we lost our panic, our notoriety, our accent. Our names commingled with a United Nations of new names: Yudkovsky, Fasquelle, Eramian, Barnum, McLean, te Velde. Two generations after Ellis Island, our family's China stories are evolving into myths\u2014the last stop of a memory before it vanishes for good.\n\nFor me, the greatest loss was language. My parents spoke different dialects, and when they spoke Chinese, it was a secret language, for keeping things from the kids. By the time I tried to learn it at school, I couldn't pick it up easily. What I learned, I was too shy to practice, and therefore quickly lost.\n\nSome of my cousins did better, becoming fluent in Chinese. To learn a language, you have to be willing to make mistakes\u2014big, loud ones sometimes. Being a shy perfectionist with a taste for Dickens is a recipe for learning a lot of one language; it's not a recipe for learning a second or a third. So Po Po and I sat at the table smiling at each other and eating wonton soup in more or less silence, divided by language. I could taste the plain fact of my grandmother's affection, which didn't require translation into English. But I could no more ask the questions that would help me learn how to make it\u2014however much I might wish to\u2014than I could question the affection itself. Even a simple question like \"Should you use shrimp in the wontons?\" was out of the question, especially the \"should\" part.\n\nLearning to make wontons didn't seem to matter much when I was studying ancient Greek at eighteen, or learning the saxophone at twenty-two, or salsa dancing at twenty-six. But when my own children were born, I began to lecture myself: Surely, I thought, your family didn't come all this way only for the wonton soup to be forgotten in two generations. And so, setting aside the problems of language for the time being, I began.\n\nI started with a Joyce Chen recipe (written in English), dutifully measuring a teaspoon of this, a teaspoon of that. But then, after a while, I ditched the teaspoons and just sniffed and splashed my way back to what I remembered.\n\nThe smell of wonton filling evolves as you make it. First it just smells porky, with a high overtone of shrimp if you used it. When you add the rice wine, the smell staggers a bit and opens up. Then you add the soy, and it takes on salty, definite edges. You add in a bit of stinging scallion and a bit of bracing ginger, and the smell sits up at attention. You throw in a pinch of sugar, to mollify it a bit. You put in a bit of oil to spread the word, and a little chicken broth to loosen up the attitude. And then you have that final smell that I most remember: fresh, sweet, and good.\n\nLike life, it's a series of sequential approximations. If you trust the famous parable from Herodotus\u2014and why wouldn't you? the old liar\u2014you never know if you've had a good life until it's over. So it is with wontons, but thankfully, the wait's a little shorter. It's only an hour or two of sniffing and folding before you arrive at a steaming pot of certainty.\n\nIt's strange to produce something so concrete from so imprecise a sense. You can't record a smell on a hard drive; you can't upload it from a server. The exactness of the written word means nothing when you don't speak the language. But smell, however vague, is indelible. It's never turned off, and it's not erasable.\n\nWhen it comes to culture, the lines of transmission are never continuous. Yet, from nothing more than a smell you can patch together broken bits and pieces of memory and common sense and find, to your shock, years later, that you have something your children recognize as their own, as if it were always whole and perfect. Like it or not, you're not just a descendant\u2014you're a forebear too.\n\nToday, Po Po is ninety-seven, still alert and smiling, though very deaf and certainly too frail to cook. Having had my fill of Dickens\u2014at least for the moment\u2014I am finally learning Chinese the Information Age way: with Rosetta Stone, while walking a treadmill. I've still not going to ask about the shrimp, though. I would say something incomprehensible, and Po Po would laugh and shake her head at me, the mutest of her granddaughters. Knowing she loves me that much is enough. I know it for a wordless fact, just as I know that the wontons I eat from this point forward will all be my own.\n\n**_Best Guess Wonton Soup_**\n\n_I try to double the wonton part of the recipe when I feel up to it. They're so good, and you can freeze them in little quart-size ziplock bags for a later lunch. I freeze them raw, but it's also possible to blanch them\u2014cook them in boiling water for just a couple of minutes, till half-done\u2014drain them, and then freeze them that way. If you do this, you also have the messy but outrageously good option of deep-frying them, which everyone should try at least once._\n\n_Serves 4_\n\n**For the filling:**\n\n1 pound ground pork, or pork butt\/shoulder (if you have a grinder)\n\n\u00bd pound shrimp, peeled and deveined\n\n2 tablespoons Shaoxing cooking wine or dry sherry\n\n1 tablespoon soy sauce\n\n1 teaspoon minced fresh ginger\n\n1 teaspoon minced fresh garlic\n\n2 fat or 4 skinny scallions, whites only (reserve the greens for later use), minced\n\n1 \u00bd teaspoons cornstarch\n\nIf you have a meat grinder or a meat grinder attachment for your mixer, run the pork and shrimp through it, using a disk with medium-sized holes. If you don't, you are probably working with ground pork and whole peeled and deveined shrimp, and you'll need to mince the shrimp as finely as you can; it's easier if you freeze them for 10 minutes first. Cut the shrimp into 1\/8-to 1\/4-inch slices, then turn them crosswise and slice the other way. When you have minced all the shrimp in this manner, hash them a bit finer by holding the knife tip down with one hand and seesawing the blade through the shrimp mass against the cutting board. Precision isn't as important as thoroughness here.\n\n1 tablespoon chicken broth or homemade chicken stock\n\n1 \u00bd teaspoons vegetable oil, canola oil, or corn oil\n\n\u00bd teaspoon sesame oil\n\nPinch of sugar\n\n1 package wonton wrappers (about 45\u201350)\n\n**For the soup:**\n\n\u00bd head Chinese (napa) cabbage\n\n6 cups chicken broth or homemade stock\n\nGreens reserved from 4 scallions (see above)\n\n1-inch piece fresh ginger, unpeeled and crushed\n\n1. Place the pork and shrimp in a large mixing bowl and add the rest of the filling ingredients. Toss gently but thoroughly with a fork or chopsticks; the mixture should remain loose and smell fresh, briny, and gingery.\n\n2. To fill the wontons: Set the filling and a small bowl of water next to your work area. Take out a wonton wrapper and set 1 teaspoon (more if you dare, and as you get better at it) of filling in the center. Dip your finger in the water and trace along the top edge and halfway down the sides of the wrapper. Fold the wrapper in half so that the dry edges meet the wet edges. You should have a rectangle with a big lump of filling in the middle. Moisten one corner of the folded edge. Draw the other corner of the folded edge over it\u2014not face to face as if you were closing a book, but front to back\u2014and press them together firmly. Set your finished wonton aside and repeat until you've exhausted the filling.\n\n3. Chop the cabbage roughly into pieces about 1 inch square.\n\n4. For the soup: Bring the broth, scallion greens, and crushed ginger to a simmer. Add the wontons. If the broth doesn't quite cover them, add a bit of water. Salt to taste. Add the cabbage, and any extra wrappers if you have them, atop the broth and return to a simmer. As it simmers, tuck the thickest bits of cabbage into the broth with a wooden spoon to help cook them through. The wontons should be cooked through, with the wrapper puckering around the filling, and the cabbage tender within about 8 or 9 minutes at a steady simmer (if you had more broth, you would see the wonton float to the surface, but I prefer a less brothy soup). Serve immediately, with Special Sauce.\n\n**Special Sauce**\n\n1 tablespoon finely minced ginger\n\n1 clove garlic, finely minced\n\n1 tablespoon finely minced scallions\n\n\u00bd teaspoon white sugar\n\n\u00bd teaspoon black vinegar (Chinkiang vinegar)\n\nA few drops white vinegar (rice vinegar)\n\n1 teaspoon dark sesame oil\n\n\u2153\u2013\u00bd cup dark soy sauce (e.g., Kikkoman)\n\n1. In a small bowl combine the ginger, garlic, and scallions. Add the sugar, vinegars, and sesame oil. Crush the mixture lightly with a fork or chopsticks to help release the flavors. Let it rest for at least 5\u201310 minutes.\n\n2. Add the soy sauce gradually to taste. Less soy makes for a thick, pungent sauce; more for a milder dressing.\n\n### CURIOUS COOKIES\n\n### By Eagranie Yuh\n\n### From _Edible Vancouver_\n\n### Vancouver-based Eagranie Yuh teaches classes about chocolate\u2014a logical focus for a pastry chef with a master's degree in chemistry. As a copywriter, blogger, and freelancer, she is a frequent contributor to _Northwest Palate_ and _Edible Vancouver_ and edits IACP's newsletter _Words._\n\nHere's how most chocolate-chip cookie recipes go: in a mixing bowl, cream sugar and butter until light and fluffy. Dribble in eggs and vanilla. In a separate bowl, combine flour, baking powder, baking soda, and salt. Fold the dry ingredients into the wet ingredients, then sprinkle in chocolate chips and stir just to combine. Drop identical dollops of cookie dough onto a baking sheet, leaving gaps between neighbours. Bake into perfectly golden-brown cookies, evenly studded with chocolate chips.\n\nHere's my mom's method for making cookies: In a saucepan, melt margarine and sugar. Crack in an egg, and stir furiously to minimize scrambling. Dump in a mixture of whole-wheat flour, baking powder, baking soda, and salt. Toss in a few handfuls of miniature carob chips, then mix roughly until the batter comes together. Slap the batter onto an ungreased cookie sheet, glossing over instructions like \"place the cookies two inches apart\" and bake one enormous frankencookie that covers the entire pan. Snap off the blackened edges, and divide it into rough squares.\n\nI was the only kid at school with square cookies, but it didn't bother me. I didn't know the difference between creaming and melting the fat, that butter and margarine have different properties, or that whole-wheat flour can make baked goods dense and heavy. And I certainly didn't know that carob is a legume masquerading as chocolate.\n\nFor every recipe that begins, \"cream the butter and sugar,\" there's someone who looks confused. It simply means to mix the butter and sugar vigorously until light and fluffy, but it might as well be secret code, because there's no way for you to figure that out. By creaming, you incorporate air, thereby creating a lighter, fluffier cookie.\n\nWhen my mom melted, rather than creamed, the margarine and sugar, she set things up for flat, dense cookies. Even worse, the melted margarine resulted in a warm, fluid batter that gave the cookies no choice but to smoosh into each other as they baked into a crisp carob-chip pancake.\n\nHydrogenated vegetable fats\u2014like margarine and shortening\u2014lead to crisper, crumblier cookies with little flavour. When it comes to chocolate chip, I prefer butter. Aside from being tasty, it results in a cookie that spreads slightly when it bakes, leading to crispy edges and soft centres. So why did she use margarine?\n\nIn the '80s there was a backlash against saturated fats. Consumption of red meat plummeted, prompting pork to advertise itself as \"the other white meat.\" Sales of skinless, boneless chicken breast shot through the roof. And well-meaning people, like my mom, eschewed butter in favour of margarine. And not just any margarine\u2014hard margarine, sold in bricks, full of trans-fatty acids that have since been linked to heart disease. Whoops.\n\nWhile she was substituting margarine for butter, my mom also swapped out all-purpose flour for whole wheat. Whole-wheat flour, as the name suggests, is milled from the entire wheat grain and so contains the bran, germ, and endosperm. All that added stuff can throw off the balance of dry and wet ingredients in a recipe, so simply substituting whole-wheat for all-purpose flour can be risky. Dry, crumbly cookies weren't the only result; I also ate my share of hockey puck muffins.\n\nFinally, let's be clear: carob is not chocolate. It's a legume. In some countries, carob is a legitimate food, used as a sweetener and in hot drinks. When I was growing up, carob was faux chocolate, synonymous with health-conscious hippies. It came from a bulk food store that smelled like stale spices, and it tasted like dirt\u2014with a hint of cumin and coriander, common carob neighbours in the bulk food section.\n\nTo this day, my mom hates to cook and baking mystifies her. Still, my childhood memories are punctuated with a steady stream of square chocolate-chip cookies. When I was small, my mom sat me on the kitchen counter while she made them. From my perch, I stared intently at the kitchen floor, lost in the kaleidoscope of orange, brown, and avocado blotches on the linoleum.\n\nOne day, I was old enough to unwrap the pre-portioned bricks of margarine from their waxed paper. And when I could see above the counter, we made square cookies together. I remember scraping a spoon across the bottom of the saucepan, the crunch of the sugar, the smell of the margarine. Those cookies may have broken all the rules, but to me, they were perfect.\n\n**_Sneaky Whole-Wheat Chocolate Chip Cookies_**\n\n_These handsome cookies are crispy on the outside, chewy on the inside, and full of melty chocolate goodness. You'd never guess that they contain whole-wheat flour, which lends a pleasant chewiness and nuttiness without being heavy or dense._\n\n_Patience is a virtue, especially where these cookies are involved. First, to avoid scrambled eggs, make sure your butter-sugar mixture has cooled to room temperature before you add the eggs. Second, chill the cookie batter for at least two hours before baking, to give the butter enough time to re-solidify. And if you are saintly enough to wait for the dough to rest overnight, the cookies are even better._\n\n_Makes 24 cookies._\n\n\u00bd cup (114g) butter\n\n\u00be cup (150g) brown sugar\n\n\u00bc cup (55g) granulated sugar\n\n\u00be cup (95g) all-purpose flour\n\n\u00bd cup (80g) whole-wheat flour\n\n\u00be teaspoon (4g or 3mL) baking powder\n\n\u00bd teaspoon (2g or 2mL) baking soda\n\n\u00bd teaspoon (2g or 2mL) salt\n\n1 egg\n\n\u00be teaspoon (3g or 3mL) vanilla\n\n3 oz (81g) dark chocolate (70\u201380% cocoa solids), coarsely chopped\n\nIn a medium saucepan, heat butter, brown sugar, and sugar over low-medium heat. Stir occasionally until the butter melts. Remove from the heat and set aside for 15 minutes or until it cools to room temperature.\n\nIn a medium bowl, combine all-purpose flour, whole-wheat flour, baking powder, baking soda, salt, and chocolate. Set aside.\n\nWhen the butter-sugar mixture is room temperature, add the egg and vanilla to the saucepan. (CAUTION: If the butter-sugar mixture is still warm, you will cook your eggs.) Using a spatula, stir to incorporate. Add the flour mixture and stir just until there are no flecks of flour remaining. Transfer the batter to a sheet of plastic wrap, wrap tightly, and chill in the refrigerator for at least 2 hours until firm, and preferably overnight.\n\nPreheat the oven to 350\u00b0F. Lightly grease two cookie sheets, or line them with parchment.\n\nBreak off teaspoon-sized balls of dough and roll them into balls, allowing at least 10cm between nearest neighbours. Flatten each ball to a thickness of 1.5 cm. Bake for 12\u201314 minutes until golden brown on top. Let the cookies cool for one minute on the cookie sheet, then transfer to a cooling rack.\n\nNote: Refrigerated cookie dough will keep for three days. Alternately, you can bake and cool all the cookies, then freeze the extras.\n\n### CHICKEN BRICK\n\n### By Henrietta Clancy\n\n### From _Fire & Knives_\n\n### South Londoner Henrietta Clancy writes on food, drink, and travel for various publications such as _Imbibe, Square Meal, Fresh Escapes_ and DrinkBritain.com. A sometime beekeeper and hard cider fanatic, she trained at the Ballymaloe Cookery School in Ireland.\n\nBefore Dad married Mum, he lived in what he now refers to as his \"bachelor pad\" in Ealing; a place that I've painted a vague picture of from the small snippets I've gathered about his premarital existence. In the living room there was a moleskin sofa and a dark wooden bowl of walnuts with a nutcracker; in the bedroom he'd done a bit of late 70s man cave decoration and fashioned himself a waist-high mustard-brown border. This last piece of info is a fact that's often volunteered by my mother\u2014I gather she was quite impressed at the time, what with being 20 and still living in the parental home in Enfield, amid fake flowers and floral carpets.\n\nIn the kitchen, among some obligatory kitchen paraphernalia of the era\u2014most of it orange, brown or cream\u2014there was a chicken brick, not hidden behind the doors of a Formica cabinet, but proudly sitting on a Formica surface. This chicken brick got so much use that storing it in a cupboard out of arm's length would have been plain silly; however, that's not why it was on the surface.\n\nNo, it lived perpetually beside the oven because it was always cooling down having perfectly cooked one tender chicken\u2014presumably a hormone-munching monster of a thing, because being kind to edible livelihood hadn't been invented yet\u2014and awaiting the raw carcass of the next one. That's how much use it got: it literally never stopped. He fed moist chicken to the masses, and they were all impressed. The more he used it the better it, and consequently he, became. As that layer of chicken fat coating the brown clay interior of the brick developed and mollified, so he, by doing nothing more than staying faithful to his brick, became a better cook. Armed with the brick, there was nothing my father couldn't do.\n\nThat was until my mother moved in, and everything changed. She was a graduate of Leith's School of Food and Wine, and a maker of directors' lunches. Naturally she brought her tools with her. I can see her now, whirling around the kitchen brandishing a piping bag\u2014she had at least seven different nozzles\u2014whipping up all manner of decorative toppings for p\u00e2t\u00e9s and pavlovas. Anyone could use a brick, you see\u2014she had qualifications. Prue was lapping up the ways of her contemporaries and showing my mother the ways of the International Kitchen; there was duck _\u00e0 l'orange, b\u0153uf bourguignon,_ steak _au poivre._ When they ate chicken it was in goujons, or as Kiev; perhaps the occasional _coq au vin._\n\nNo doubt there were some roast chickens in there too, but by that point, Dad had handed over the reins, relinquished his kitchen power. Roast chicken was cooked the way mum wanted it\u2014on a tray, basted with lard, cavity filled with mushroom stuffing, sitting on the shelf above some courgettes, also stuffed. The chicken was separate from the veg, not inappropriately nuzzling them, and most importantly it was exposed to the crispifying roof of the oven.\n\nThe brick lay dormant for many years, in the back of a never-used utility room cupboard, sharing space with Tupperware lids who'd lost their better halves and things that looked like they might belong to a piece of kitchen equipment, but in reality didn't. Not that I ever saw it, nor seemingly ever did my father, but one post-millennial day its absence was noticed.\n\nI was but a fresh-faced fresher at the time, not long flown the nest, when I missed a phone call from my father, and beheld a voicemail that contained a roaring accusation: \"You have stolen my chicken brick!\"\n\nI listened to it several times. Breathless, outraged, desperate and confused, he was indeed talking about a chicken brick.\n\nI had done no such thing, of course; the brick was, to me, a mythical piece of kitchenware that was only referenced once every three years when, on the rare occasion that the subject of my father's cooking skills came up, it was put forward as evidence. I had never clapped eyes on the thing.\n\nI'm the eldest of five children and the pattern repeated itself: child left home, child received out-of-the-blue chicken brick accusation, Mum located brick in never-used cupboard, cupboard door was closed again.\n\nThe situation became farcical, but it did give me time to consider dad's perplexing relationship with the chicken brick. My mother's prowess in the kitchen had rendered the brick comatose, yes, but where did the brick burglary accusations stem from? I began to brew theories. Was this a flirting tool\u2014something he could fall back on if he was left alone? Was it sitting in our cupboard to keep Mum on her toes, silently reminding her that it had seduced many a fine lady before she came along, and if she misbehaved it was quite capable of doing it again? Or was it simpler than that: the brick represented independence for him, so maybe he assumed that in order to properly grasp ours, we'd need a tool?\n\nOr was it more of a practical thing: maybe on a subconscious level he assumed that, without our mummy around to fill our belly, we would need a brick to take care of us. He viewed it as the provider, the caretaker, the other.\n\nI theorized, but perhaps I overtheorized. Essentially he could have bought us all brand new bricks as \"off to uni\" presents. No, it was this specific brick to which he had a personal attachment.\n\nI paid an unannounced midweek visit to my parents' house not too long ago. I had hopes of being fed, but there was no car in the drive when I arrived and only a single kitchen light on. As I put my key in the door I half expected an alarm to go off, but no, I could hear the hubbub of sports noise coming from the TV room and the sound\u2014comforting and infuriating in equal measure\u2014of a happy dog's tail slapping kitchen cabinets.\n\nIn the kitchen, dinner had been started, just\u2014there was a whole chicken still hugged by plastic wrap, an unnecessarily large and perpetually blunt Sabatier, and a mess of onion and carrot\u2014but the cook was missing. And there it was, the brick, sat on the central unit. I don't think I'd ever actually seen it before, and it looked smaller, less able than the legacy that rested upon its shoulders. Almost like a sleeping woodlouse. I stopped suddenly: no car, no mum, a forlorn brick. . . .\n\nOh right, I thought, she's left him, and he's coping with it just how he always knew he would; he's following his well-made plan and he's dealing with it. Just as I began to seriously envision the painful and uncomfortable chicken dinner I was going to have to chew my way through, staring at a man who'd sadly and swiftly replaced the woman he'd loved for 36 years with a piece of insect-like terracotta, my apron-clad mother trotted in\u2014she never walks\u2014and amidst her salutations resumed chopping.\n\n\"So you're actually using the chicken brick?\" I said, unable to mask my intrigue.\n\nShe looked confused.\n\n\"The brick,\" I said.\n\nNothing.\n\n\"The chicken brick!\" I wailed, pointing.\n\n\"This!\" I slapped my hand on its lid.\n\nShe registered it, and promptly reeled with the sheer absurdity of the idea. \"NO! No, no, no, no, no. . . . No, I'm doing this new recipe from my, you know that book you bought me with all of the nice, different recipes in it, I've used it loads . . . the one with the caramelized garlic and goa. . . .\"\n\n_\"Ottolenghi.\"_\n\n\"That's it. It's a roast chicken dish from there. It's with honey, saffron and hazelnuts. . . . I just really fancied it. So, anyway, let me tell you, I was teaching that lovely little boy today. . . .\"\n\n\"Hold on, back to the brick\u2014why is it out?\"\n\n\"Erm. . . . Daddy was looking for it so I just left it out for him. What was I saying before?\"\n\nOf course, how silly of me. My youngest brother had recently left for university and the coming-of-age accusation had been hot on his heels.\n\nTelevised match over, dad sailed in and registered me _en route_ to a wine bottle, poured three glasses and handed them out, then started searching for a music video on YouTube (a fairly recent and incredibly odd phase my parents have found themselves in, eating dinner to a backdrop of pop ballads), settled for Adele and spotted the brick.\n\n\"Ah, great, you found it.\"\n\nThe time had come to ask him the question that had been perplexing me and my siblings for over a decade.\n\n\"Dad, why is it that you always think we're after your chicken brick?\"\n\nHe smiled, wiggled his eyebrows up and down on his face for a while, picked up the brick and did a little Irish jig or sorts (cue some silly laughter from mum) and just before placing it back in the cupboard\u2014presumably forevermore, the nest is now empty\u2014he said, \"It doesn't matter. It's mine.\"\n\n### ANGRY BREAKFAST EGGS\n\n### By Elissa Altman\n\n### From PoorMansFeast. com\n\n### Award-winning blogger Elissa Altman does a lot of things well: She's a humorist, political commentator, cookbook editor, and food columnist whose work has appeared everywhere from the Huffington Post to GiltTaste to _Saveur._ Her book _Poor Man's Feast: A Love_ Story will appear in 2013.\n\nShe has never slept, for as long as I can remember. First, there was the hair, which, when I was very small, was very tall; these were the days of teasing, and to keep her updo in place, she climbed into bed every night next to my father with three feet of toilet paper wrapped around her head, a six inch tail of Charmin hanging off the pillow, blowing in the air-conditioned breeze like a Coppertone banner dragged behind a beach plane. She lay there stiffly all night, immobile and exhausted, and sat up the next morning, her hair perfect.\n\nEventually, it was just plain pique that kept her awake\u2014the constant working of herself into a lather over imaginary transgressions, while my father and I and the world around her, ever the transgressors, slept soundly. When the black and white numbers on her bedside clock flipped over to 6:30 a.m. and the alarm went off, she swung her legs off the side of the bed and stood up, already furious and seething.\n\nAnd then she made eggs.\n\nA lot of eggs.\n\nAt first, when things were still good and happy, they were soft boiled, and sat in the broad end of our porcelain egg cups, their tips sliced away so that my father and I\u2014perched side by side at the breakfast counter half an hour before he dropped me off at the school bus stop on his way to the subway\u2014could dunk untoasted fingers of Pepperidge Farm Diet White into the runny yolk. As my parents' marriage wore on and she grew angrier, the eggs were medium boiled, their firm yolks like thick golden velvet, with spots of remaining tenderness just barely discernible.\n\nWhen I turned fourteen, my mother began hard boiling our eggs; she'd put them in a small pot filled with a shallow inch or two of water, set them on the stove, crank up the flame, and walk away. Eventually, they'd explode, their snow white glair erupting like Vesuvius through the fissures of her discontent. I'd refuse to eat them at that point, and when she came back into the kitchen, she'd grab the black plastic handle of the pot and dump its contents\u2014the water had long since evaporated\u2014directly into the trash.\n\nMy parents divorced the following year.\n\nMy mother still doesn't sleep, and she still cooks eggs every single morning, even with cholesterol that hovers near the 400s if she's forgotten her to take her Lipitor. She's been through a passel of saucepans\u2014the brown and white Dansk pan that followed her into the city after her divorce, and that she burned until its white enameled interior melted away into a noxious cloud; two Revere-Ware pans that we brought to her apartment from our basement stash\u2014they'd belonged to Susan's mother who had them for fifty years. My mother burned them until their insides turned black as coal. Now she uses a tiny butter warmer, big enough to hold exactly one jumbo egg.\n\nEggs are my mother's mood barometer: when she's happy, she'll deftly separate yolk from albumen, throw out the former, dump the whites into the one tiny stick-proof pan she owns, and while they bubble and spread, she'll lay a piece of Diet White bread right down in the middle of it, and top it off with a dollop of honey. This, she says, is her version of French toast, and she loves it. If Susan and I are staying there and she's feeling glad, she'll insist on scrambling some whites for us because, she says, they're low fat and good diet food, and together we'll sit at her dining room table, having breakfast, while the traffic rumbles down West End Avenue twenty-one stories below. Not overcooked and not runny, the eggs bear no evidence of seasoning; it's just them and us, a piece of bread, and my mother's favorite morning cup of hot water. If we're staying there and she's furious, she'll boil the eggs until a sulfuric haze wafts out into the living room; we'll leave while the pan is still rattling over the flame.\n\n\"I _had to throw them OUT_ ,\" she'll tell me later.\n\nThe correlation between cooking and scorn is a fraught, famous one; food created by angry people seems, somehow, to be bitter, and so attuned to their off flavors and textures am I because of my mother's eggs that once, when a conversation with a well-known cookbook author took a sudden and surprising turn south, I had to get rid of her book, because every one of the dishes I cooked from it after our argument tasted of her rage; no matter what I did, none of the recipes worked anymore. Food cooked in anger becomes collateral damage; meat is carbonized, pasta becomes starchy mush, vegetables go limp and sad, and it's not like you can\u2014or even want to\u2014revive them, to coddle or comfort them, or to save them for another meal. You simply can't do it. If the optimum way to cook and live and run a kitchen is, as Tamar Adler says, with economy and grace\u2014use everything, every shard and peeling and drop of fat with care, kindness, and thoughtfulness\u2014scornful cooking results in the opposite: profligate waste and clumsy distraction.\n\nIt was six in the morning last Sunday; I lay in bed, listening to the ticking of the ignition on my Viking's pilot light. There was the sound of running water, the clank of a pan on a burner. When my mother came to visit us last weekend and awoke in the throes of pre-dawn Bad Mood, she rifled through our refrigerator, pulled out four eggs, set them in shallow water, turned the burner on high, and cooked them until they burst with fury.\n\n\"I couldn't sleep,\" she barked from the guest bed where she'd laid back down after preparing the breakfast she decided I needed to eat, \"so I made you eggs. THIS is what you should be eating for breakfast\u2014not the _heel of a baguette_ and a piece of cheese.\"\n\nShe had been watching me _that_ closely the previous morning; to my mother, a piece of bread\u2014no matter how small\u2014spells o-b-e-s-i-t-y. She was in a rage.\n\n\"But I don't have any eggs,\" I answered, suddenly remembering the half-crate of six local duck eggs that were hovering in the back of the fridge, waiting for a recipe test.\n\n\"They're in the SINK\u2014\" she shouted from the guest room.\n\nI walked into the kitchen and there they were, in a now-dry All-Clad saucepan, the shells cracked and broken, their whites extruding like Elizabethan collars. Susan broke one into a cup to see if the yolk was hard-cooked, and somehow salvageable; it was raw and cold. The eggs had been sitting out at room temperature for over two hours.\n\nMy mother marched into the kitchen behind me and watched Susan put on the tea kettle; I stepped on the pedal of the trashcan and tossed each duck egg out, one by one, like small grenades.\n\n### SWEET SOUTHERN DREAMS\n\n### By Ben Mims\n\n### From _Saveur_\n\n### It's been a long journey from Koskiusko, Mississippi, to his current home in New York City for associate food editor Ben Mims, along a sweets-paved road passing through the French Culinary Institute, the Saveur test kitchen, and an Ice Cream Takedown victory in Brooklyn.\n\nPrior to October 7, 2010, my mother and I were the best of friends. A consummate Southern lady, Judy Mims is a fantastic cook, gossiper, and mom\u2014and in her relationship with me she had always drawn on all those talents. But on that October day, I flew from New York City to my childhood home in Kosciusko, Mississippi, to come out, at 25 years old, as a gay man to my parents. As anyone who grew up in the Bible Belt can imagine, the outcome was heartbreaking. My mother and I used to talk at least weekly; now months go by without a call. I miss her. And I can't help feeling like I've lost touch with not only my mother, but also my lifeline to the world I grew up in. Thank goodness I still have the cakes.\n\nLayer cakes originated in the South, and with their over-the-top grandeur and unapologetic sweetness, they're inextricably linked to the culture I grew up in. The drama, excitement, and praise\u2014they all speak to the South.\n\nMy childhood in rural Mississippi was filled with fantastic bakers: my mother, of course; her sister, Barbara Jane; my paternal grandmother, Carol; and Mom's friends, those church ladies decked out in hats who produced a never-ending procession of astounding cakes. My grandmother's neighbor Louise Hodges made a cake three yellow layers tall, draped in warm caramel fudge icing, which exuded a fragrance of vanilla and browned butter that could knock you over. Carol, who bought those cakes from Mrs. Hodges, served one to our family virtually every Sunday after church. We would sometimes have two slices each, and when I'd tease my grandmother, asking her who made the cake, she would primp her curly blonde hair, give me a wink, and reply, \"Why, who do you think?\"\n\nMy mother, for her part, turned out mammoth sour cream Bundt cakes, domed lemon and cream cheese pound cakes, and a ludicrously rich cheesecake that was my staple birthday cake. I spent my childhood at her elbow, watching her pour glaze down the grooved sides of a Bundt cake, mirroring her smile as she passed me a beater with batter barely clinging to it. Unlike Mrs. Hodges, Mom was never big on making layer cakes. Maybe she didn't have the patience to stack and frost all those layers, though she liked them just fine as an effortless treat baked by someone else. The only layer cake in her repertoire was red velvet, for just as most Southern women have a subscription to _Southern Living_ magazine and at least one gilded holiday wreath in their attic, most also have a red velvet cake up their sleeve. The deep crimson cake against the luminous white frosting is pure Southern drama. It's Shirley MacLaine in _Steel Magnolias_ hacking into the blood-colored tail of an armadillo-shaped groom's cake. It's my mother's ceramic-white skin contrasted by her lips, always burnished with brick-red lipstick.\n\nMy mother stacked her red velvet only two layers deep and almost always made it with cream cheese frosting. One Christmas during my teen years, though, she got adventurous with a whipped cream frosting consisting of cooked flour, sugar, and milk beaten into butter. When executed correctly, a frosting like this holds up like a dream and provides just the right balance\u2014not too sweet or rich\u2014for the slightly acidic, chocolate-flavored red velvet layers. She labored over that cake all day, and we carried it in our car two hours away to my grandfather's house in Holly Springs. After dinner, once the coffee was perking, the cake dome was lifted, and my mom sliced into the scarlet layers and snow-white frosting. Everyone took bites, and then spit them out. The frosting was as chalky and tacky as wallpaper paste; my mom was nearly in tears.\n\nShe never tried her hand at it again, but that whipped cream icing had a profound effect on me. Motivated by my mother's failure, I made it my mission to learn how to make the cake she had envisioned. Schooled at her apron strings, I was already an avid baker, and nailing that recipe helped direct my life's path. In my current job, I'm able to hone my skills every day to produce the platonic ideals of the cakes my mother raised me to love.\n\nSome of my best recipes were passed down from my mother's own mother, Jane Newson, who died the year before my birth. The very morning following the red velvet cake disaster, my mother sat with Barbara Jane and me on my grandfather's living room floor and sorted through hundreds of her mother's recipe cards. By all accounts, Jane was a fantastic maker of layer cakes: prune and fig in a cinnamon meringue; Lane cake filled with boozy nut and raisin custard; walnut spice laden with cinnamon, allspice, and cloves. It was the coconut cake recipe, though, that the sisters agreed was the one to save if ever the house caught on fire. The first time I tried the recipe, it exceeded all my expectations. The cake was filled with freshly grated coconut, the sweet water seeping into the yellow layers surrounded by fluffy Italian meringue. Left for a day to \"mature\" in the refrigerator, every inch of it was suffused with rich coconut flavor.\n\nStill, of all the Southern layer cakes I have known, the one that sticks with me the most these days is lemon. It goes back to the summer before my senior year of college, when I moved to Vicksburg, home of the Miss Mississippi Pageant, to work as a reporter for _The Vicksburg Post._ Toward the end of my stay, just before the beauty queen pomp began, my mom came to visit, and we took a walk along the riverfront. When we ducked into a little caf\u00e9 for coffee, we noticed a case full of beautiful layer cakes. We ordered a pastel-yellow slice of lemon cake to share. It was a stunner: four layers of citrusy butter cake drenched in lemon syrup and enrobed in a lemon buttercream frosting. We sat and chatted, and every bite of cake tingled our cheeks with delicious tartness. Now, whenever I get nostalgic for the South, I break out my cake pans, butter, and sugar, and whip up a lemon cake like the one we shared; it buoys my hope for a future in which my mother and I are as close as we once were. The result\u2014bittersweet and beautiful\u2014reminds me of that afternoon, four years before our lives changed, when we sat together in that caf\u00e9 without a care in the world and just talked about cake.\n\n**_Coconut Cake_**\n\n_Serves 10\u201312_\n\n**For the Cake:**\n\n16 tbsp. unsalted butter, softened, plus more for pans\n\n2 \u00bd cups cake flour, plus more for pans, sifted\n\n1 tsp. baking soda\n\n1 tsp. kosher salt\n\n1 cup buttermilk\n\n1 tbsp. vanilla extract\n\n2 cups sugar\n\n5 eggs\n\n**For the Frosting:**\n\n4 egg whites\n\n\u00bd tsp. cream of tartar\n\n2 \u00bc cups sugar\n\n\u00bc cup light corn syrup\n\n1 tsp. kosher salt\n\n2 tsp. vanilla extract\n\n\u00be cup fresh coconut water\n\n3 cups freshly grated coconut\n\n**Instructions**\n\n1. Make the cake: Heat oven to 350\u00b0. Butter and flour two 9\" cake pans, and set aside. Whisk together flour, baking soda, and salt in a bowl; set aside. Whisk together buttermilk and vanilla in a bowl; set aside. In the bowl of a stand mixer fitted with a paddle, cream butter and sugar on medium-high speed until pale and fluffy, about 3 minutes. Add eggs one at a time, beating well after each addition. On low speed, alternately add dry ingredients in 3 batches and wet ingredients in 2 batches. Increase speed to high, and beat until batter is smooth, about 5 seconds. Divide batter between prepared pans, and smooth top with a rubber spatula; drop pans lightly on a counter to expel large air bubbles. Bake cakes until a toothpick inserted in middle comes out clean, about 35 minutes. Let cakes cool for 20 minutes in pans; invert onto wire racks, and let cool. Using a serrated knife, halve each cake horizontally, producing four layers; set aside.\n\n2. Make the frosting: Place egg whites and cream of tartar in the bowl of a stand mixer fitted with a whisk, and beat on medium-high speed until soft peaks form; turn mixer off. Bring sugar, syrup, salt, and 1\/2 cup tap water to a boil in a 2-qt. saucepan over high heat, stirring to dissolve sugar; attach a candy thermometer to side of pan, and cook, without stirring, until thermometer reads 250\u00b0, 4\u20135 minutes. Turn mixer to medium speed, and very slowly drizzle hot syrup into beating egg whites. Add vanilla, and increase speed to high; beat until meringue forms stiff peaks and is slightly warm to the touch, about 3 minutes.\n\n3. To assemble, place one layer on a cake stand, drizzle with 3 tbsp. coconut water, spread with 1 1\/2 cups frosting, and sprinkle with 1\/2 cup grated coconut; top with another cake, drizzle with 3 tbsp. coconut water, spread with 1 1\/2 cups frosting, and sprinkle with 1\/2 cup coconut. Place another cake over frosting, drizzle with 3 tbsp. coconut water, spread with 11\/2 cups frosting, and sprinkle with 12 cup coconut; top with remaining cake and drizzle with remaining coconut water. Cover top and sides with remaining frosting, and cover outside of cake with remaining coconut, pressing it lightly to adhere; chill cake to firm frosting. Serve chilled or at room temperature.\n\n**_Lemon Layer Cake_**\n\n_Serves 10\u201312_\n\n**For the Cake and Syrup:**\n\n16 tbsp. unsalted butter, softened, plus more for pans\n\n2 \u00bd cups cake flour, plus more for pans, sifted\n\n2 \u00bd tsp. baking powder\n\n1 tsp. kosher salt\n\n\u00bd cup milk\n\n1 tsp. vanilla extract\n\n1 \u00be cups sugar\n\n1 tbsp. lemon zest\n\n4 eggs\n\n\u2153 cup fresh lemon juice\n\n**For the Frosting:**\n\n1 \u00bd cups sugar\n\n\u00bc cup cornstarch\n\n\u00bc cup lemon zest\n\n1 tsp. kosher salt\n\n10 egg yolks\n\n1 cup fresh lemon juice\n\n1 \u00bd cups unsalted butter, softened\n\n1 tsp. vanilla extract\n\n**Instructions**\n\n1. Make the cake: Heat oven to 350\u00b0. Butter and flour two 9\" cake pans, and set aside. Whisk together flour, baking powder, and salt in a bowl; set aside. Whisk together milk and vanilla in a bowl; set aside. In the bowl of a stand mixer fitted with a paddle, cream butter, 1 1\/2 cups sugar, and zest on medium-high speed until pale and fluffy, about 3 minutes. Add eggs, one at a time, beating well after each addition. On low speed, alternately add dry ingredients in 3 batches and wet ingredients in 2 batches. Increase speed to high and beat until batter is smooth, about 5 seconds. Divide batter between prepared pans, and smooth top with a rubber spatula; drop pans lightly on a counter to expel any large air bubbles. Bake cakes until a toothpick inserted in middle comes out clean, about 30 minutes. Let cakes cool for 20 minutes in pans; invert onto wire racks, and let cool. Using a serrated knife, halve each cake horizontally to produce four layers; set aside. Bring remaining sugar and juice to a boil in a small saucepan over high heat. Remove from heat, and set syrup aside.\n\n2. Make the frosting: Whisk together sugar, cornstarch, zest, and salt in a 4-qt. saucepan. Add yolks, and whisk until smooth; stir in juice. Stirring often, bring to a boil over medium heat; cook, stirring constantly, until very thick, about 3 minutes. Remove from heat, let cool, and transfer to a bowl; chill the lemon curd. In the bowl of a stand mixer fitted with a paddle, beat butter and \u00bc of the curd on medium-high speed until fluffy and smooth, about 1 minute. Add half the remaining curd, beating until smooth, and then add remaining curd and vanilla. Increase speed to high; beat frosting until pale and fluffy, about 3 minutes.\n\n3. To assemble, place one cake layer on a cake stand, drizzle with 2 tbsp. syrup, and spread with \u00be cup frosting; top with another cake, drizzle with 2 tbsp. syrup, and spread with \u00be cup frosting. Place another cake over the frosting, drizzle with 2 tbsp. syrup, and spread with \u00be cup frosting; top with remaining cake, and drizzle with remaining syrup. Cover top and sides with remaining frosting; chill cake to firm frosting. Serve at room temperature.\n\n## Someone's in the Kitchen\n\n### THE KING OF POP-UP\n\n### By Brett Martin\n\n### _From GQ_\n\n### Brett Martin's profiles, food and travel pieces, and essays appear regularly in _GQ,_ as well as _Vanity Fair, The New Yorker, Bon App\u00e9tit, Food & Wine,_ and public radio's _This American Life._ Connoisseur of the edgy and trendy, here he parses a modern dining scene phenomenon: The pop-up restaurant.\n\nThere is, naturally, a truck. Just as surely as there are the tattoos, the facial hair, the coif, the Twitter feed, the reality-TV show, the silent _pop pop pop_ of the bloggers' cell-phone cameras. He who has these\u2014all the signs and signifiers that have replaced whisk and toque as emblems of modern chefdom\u2014is 40-year-old Ludovic Lefebvre, a classically trained, preternaturally handsome Burgundian who's become Los Angeles's most talked-about cook. What he doesn't have is a restaurant. Instead, Lefebvre and his wife, partner, and brand manager, Krissy, run a series of wildly popular pop-ups called LudoBites, which occupy, hermitcrab-like, the off-hour shells of other eateries\u2014first in obscure corners of L.A. and, more recently, across the United States for the Sundance Channel's _Ludo Bites America,_ debuting July 18. Lefebvre is a kind of walking, talking, preening manifestation of all the blessings and damnations, transcendence and silliness, that mark this moment in American dining.\n\nIn any other context, Gram & Papa's, a soup-and-sandwich spot in L.A.'s Garment District, would hardly suggest culinary adventure. Nevertheless, that's where the Lefebvres staged the fifth iteration of LudoBites\u2014LudoBites 5.0\u2014for six weeks last summer. By day, this area downtown was crowded with shoppers at rows of fabric stores and zipper distributors; after dark, it was all but deserted\u2014save, in those weeks, for questing foodies dubiously checking their GPS units.\n\nThe production of high-end food in low-end and otherwise improbable settings is, of course, part of the pop-up phenomenon. But that doesn't mean a chef raised in the great kitchens of Paris won't grumble. On this morning, the Lefebvres arrived with Ludo in full temperamental-artist mode. The immediate object of his pique: the state of the walk-in fridge.\n\n\"It's like fucking Baghdad in there,\" he muttered to Mike Ilic, Gram & Papa's owner.\n\nBehind the register, Ilic raised his eyebrows. He had invited the Lefebvres to assume weeknight squatter rights in exchange for a fee, a cut of the profits, and the publicity; it was obviously a sufficient trade-off to make him tolerant of his tenant's moods.\n\n\"They cleaned up Baghdad, Ludo,\" he said. \"It's gonna have to be like Afghanistan.\"\n\n\"Every morning, the same routine,\" said Krissy, rolling her eyes. She was sitting at a nearby table, looking over that night's reservation list.\n\nUpstairs, in a tiny storage area, it wasn't difficult to tell which shelves belonged to Gram & Papa's and which to LudoBites: On one side, ketchup and La Choy Chinese noodles; on the other, kombu seaweed, industrial-grade gelatin, star anise. Nearby was a pile of Lefebvre's niftier gadgets: an immersion circulator, which looks like the mating of a heating coil and a medical device, and a Gastrovac, which first cooks ingredients in a vacuum and then reimpregnates them with flavor the moment the seal is broken.\n\nIn the kitchen, Lefebvre's skeleton staff was assembling. One sous-chef was cutting perfect rectangles of pork belly, uniform and creamy as frosted sheet cake. Lefebvre runs his kitchen in a manner in keeping with his French education. That is, he yells. And the staff is expected to yell back. \"I want the kitchen clean before I start!\" he hollered now. \"Use your fucking heads!\"\n\n\"Yes, Chef!\" came the chorus.\n\nLefebvre's eye fell on a young cook named Joon Sung.\n\n\"Joon!\"\n\n\"Yes, Chef!\"\n\n\"Do we have dashi made?\"\n\n\"No, Chef!\"\n\nLefebvre made a face that suggested it might be easier simply to end it all via Gastrovac right there. The dashi broth, infused with kombu, was for a dish he had woken up intent on adding to that night's menu. It would act as a poaching liquid for oysters that would then be served with a froth of butter infused with the briny taste of their own smashed shells. All this he patiently explained to Sung.\n\n\"You understand, Joon?\"\n\n\"Yes, Chef!\" Sung thought it over for a moment.\n\n\"Chef!\"\n\n\"What, Joon?\"\n\n\"The oysters. They'll be poached _\u00e0 la minute_?\"\u2014meaning \"to order.\"\n\n\"Of course!\" said Lefebvre. \"Or else it's no fun!\"\n\nHe came out front, where Krissy was still at work, and wiped his forehead with a kitchen towel.\n\n\"This is my last LudoBites,\" said Ludo.\n\n\"Nice try,\" said Krissy.\n\nThey are a pretty couple, in an easily alliterative way: he Gallically goateed, she classically Californian (or blonde and buxom, if you prefer).With his pierced ears and arms covered in ink, Ludo could be an instructional diagram for the Metrosexual Pirate look that has dominated kitchens for the past decade. Both Lefebvres are camera-ready, and even before the Sundance show, both did time on reality TV: Ludo as a contestant, and designated villain, on _Top Chef Masters,_ Krissy on a season of _The Apprentice._ (She subsequently posed for a _Playboy_ cover.) They met, in an oft-told story, when Krissy, at the time an intellectual-property attorney, was dining at L'Orangerie, the late old-guard French restaurant that was Ludo's first stop in Los Angeles; she thought the amuse-bouche was a flirtatious gift just for her. In addition to running the front of the house, she anticipates his moods, minds his malapropisms (\"I cook through instant.\" \"Instinct, honey\"), and acts as adoring PR agent. That the adoration is obviously genuine and reciprocated isn't inconsistent with a parallel impression she gives: that of a business-savvy cat with a particularly telegenic bird in its teeth.\n\nNot that star looks are necessarily good for a man's culinary cred\u2014just as a _Playboy_ spread doesn't result in instant respect for one's litigation skills. This is especially true in Los Angeles, where the local eatocracy has an uneasy relationship with the rules of celebrity that dominate the rest of town. Woe to the big-name chef who arrives from somewhere else and is perceived to worship at the altar of Scene over that of Food. Take San Francisco's Michael Mina, whose L.A. outpost, XIV, a barn on the Sunset Strip, might as well be Olive Garden for all its reputation among local foodies. Likewise, Rick Bayless's Red O, which has been pummeled with unequaled ferocity since opening last year. Bayless's crime, in part, was daring to offer \"authentic\" Mexican food to a town that thinks it pretty well knows its Mexican. But one got the feeling that the bigger sin was the fedora-wearing valets.\n\nAngelenos are, of course, susceptible to the status-seeking inanity that infects eaters everywhere. But the logic of the moment demands that their culinary heroes\u2014Jon Shook and Vinny Dotolo of Animal and Son of a Gun; Sang Yoon of Father's Office and Lukshon\u2014be immune to the usual course of Hollywood power, that they be in some sense of the People.\n\nLefebvre walks this line closely. A crucial element of LudoBites is its reservations system\u2014a characteristically savvy confluence of idealism and show business. There is no phone number, just a Web site that opens at a random hour posted on Twitter. When LudoBites 5.0 was announced, 3,000 hopeful diners crashed the page within six minutes.\n\nThere have been missteps on the path to fame and credibility. The most egregious is Lefebvre's cookbook, _Crave: The Feast of the Five Senses_ , published by Judith Regan in 2005. On the cover, Ludo stares intensely into the camera, holding forth a split pomegranate and a spoon. He looks like nothing so much as Chris Gaines, Garth Brooks's short-lived Goth alter ego. Most infamously, there is the Fish Photo, in which a bare-chested Lefebvre, dressed in tight jeans, stands shin-deep in the surf, a large, glittering striped bass in each hand.\n\nAmong other things, the image presented logical problems: Were we to believe he'd _caught_ the fish with his bare hands? If so: Both simultaneously? Or was he so used to catching fish in this manner that it was hardly worth heading in with just one? If not: Had he brought the fish with him? Or had he found them floating there already dead?\u2014a notion that detracts substantially from the overall sexiness of the picture.\n\nMore to the point: It made him look like a world-class douche.\n\nBoth Lefebvres now express appropriate embarrassment at the Fish Photo. Krissy recalls that it could have been worse; Regan, she says, had already demanded three new photo shoots, each \"sexier\" than the last.\n\n\"She wanted him to be rolling around in the sand with the fish,\" Krissy says.\n\nThese are the kinds of semiotic negotiations that have become de rigueur in the era of the celebrity chef. It helps that Ludo's path has taken him to both extremes. He arrived in L.A. in 1996 as starstruck as any would-be actor getting off the bus. \"I had never tasted sushi! I discovered jalape\u00f1os! Green tea! It was crazy,\" he says. Paradoxically, he had a more difficult time exploring such wonders at conservative L'Orangerie than he would have back in Paris, where he had worked under acclaimed and innovative chefs including Alain Passard and Pierre Gagnaire. \"I was a very typical French chef. No tattoos. Short hair. Perfect for the army,\" he says.\n\nThat changed when Joe Pytka, the mercurial commercial director and impresario, decided to fire the beloved chef at his restaurant Bastide and installed Lefebvre in his place. Up went the sleeves, revealing a chiaroscuro of Hawaiian girls, dragons, and Sanskrit he had secretly accumulated, and out came the liquid nitrogen, gelatins, and other accoutrements of molecular gastronomy. Some critics were smitten by dishes like chicken crusted with popcorn and foie gras pi\u00f1a colada, others less so\u2014 _L.A. Times_ restaurant critic S. Irene Virbila wrote, \"I feel as if I've been mugged,\" and busted Bastide down from four stars to one. (She now says that two stars may have been more appropriate, but stands by the rest of the review: \"He was trying interesting things, but it just wasn't very good.\")\n\n\"She was so mean,\" Ludo says, clearly still angry. \"I decided I won't cook for critics anymore.\"\n\nThat decision was one half of the epiphany that led to LudoBites; the other\u2014not to cook for investors, either\u2014came soon after. That's when Lefebvre was lured to Las Vegas to open a 300-seat restaurant at the Palazzo casino. Named Lavo, it came with a \"Mediterranean\" menu (which, oddly, included a Reuben-stuffed knish \"slider\") already set in stone; Lefebvre wasn't even allowed to add nightly specials.\n\n'\"Money,\"' he says sadly, when asked what the investors could possibly have said to lure him into such a creative disaster. \"They said, 'Money.' I used to go home at night and cry.\"\n\nSo what's a tortured-artist chef to do? Why, throw off the yoke of the restaurant altogether! LudoBites 1.0 and 2.0 took place in 2007 and 2009 at BreadBar, a bakery on the border of Beverly Hills, and were immediate sensations. Iteration 3.0 was staged at a cavernous Culver City gallery-caf\u00e9 called Royal\/T. Krissy set up a professional light box at one end of the room, the better for the stream of bloggers to photograph each dish. \"If they were taking pictures anyway, why not make them as beautiful as possible?\" she says.\n\nNext came the LudoTruck, a thirty-foot-long beauty wrapped in lurid red, decorated with roosters and called the Big Red Coq. In keeping with the lowbrow obsession of the moment, it serves fried chicken\u2014albeit fried chicken whose recipe begins, \"Day One.\" (There are three in all.) When the LudoTruck debuted at the L.A. Street Food Fest, a three-hour line developed, surprising even the Lefebvres. \"There's nothing three-hours-and-$5 good,\" Krissy says.\n\nSince, the couple set up shop for six weeks in an Italian restaurant in Sherman Oaks, and LudoBites 7.0 is expected to happen this summer. The Lefebvres spent the intervening time filming _Ludo Bites America_ , for which Ludo took on barbecue in North Carolina, chilies in Santa Fe, and, quite literally, buffalo outside Denver: Krissy tweeted a photo of the chef sinking his teeth into a freshly killed bison's heart.\n\nThe pop-up life has its drawbacks. Lefebvre never gets to work in a kitchen he's designed for his own needs. He has trouble keeping quality staff at either the front or the back of the house. He can't build lasting relationships with butchers, fishmongers, and the like.\n\nIn return? No permit issues. No dishwasher and refrigerator maintenance. No electric bills, oil-disposal problems, breakage costs, laundry bills, venting-regulation compliance\u2014all the bullshit that comes along with running a permanent restaurant. And of course, there's the freedom to cook whatever his heart desires, to treat each night as a piece of harrowing theater, a magnificent fire that only he can extinguish.\n\n\"Joon!\"\n\n\"Yes, Chef!\"\n\n\"The dashi!\"\n\n\"Done, Chef!\"\n\nWith service approaching, the kitchen was in full swing. An intern scooped whole steaming octopi from a bubbling pot. Another was monitoring the circulator; the eggs inside would be used in one of the evening's less obviously pyrotechnic but most sublime offerings\u2014a dish of feathery-smooth potato mousseline capping, shepherd's-pie style, the barely poached egg and a warm bed of chorizo. As with most of Lefebvre's food these days, it made use of technology and imagination without seeming to do so for its own sake. The stuntiest item on the LudoBites 5.0 menu\u2014a cheese \"cupcake\" frosted with chicken-liver mousse\u2014was also the least successful.\n\n\"It's a miracle what we do every day, working in these conditions. You see that?\" he said, pointing to the beat-up four burner stove. \"That's magical.\"\n\nIt _is_ magical. It could also, one worries, become a dodge\u2014a perpetual deflection of the question of what someone with this much talent might do with the kitchen and staff of his dreams. Krissy says as much, remembering the thinking that went into LudoBites 1.0: \"Just using 'Bites' diminished expectations. It could never be a failure, because it had a beginning and an end.\"\n\nJonathan Gold, the Pulitzer-winning dean of L.A. food writing and one of Lefebvre's earliest and most consistent champions, likens LudoBites to a series of first records, with all the excitement, energy, and rough edges first records entail. Eventually, though, you want your favorite band to risk it all with a double-length theme album.\n\nPut another way, the question is this: Is it possible to be a great chef without a great restaurant? The kind of place, as Animal's Jon Shook memorably puts it, where \"you go back after it's been open ten years and it's still fuck-your-mouth good\"? The question matters, because at the very heart of celebrity chefdom there lies a problem that may prove irreconcilably hostile to the future of the restaurant: the boredom of chefs.\n\nRemember that the stars of the dining world used to be in the front of the house. They were the great maitre d's, wizards of the seating plan, social impresarios who got their names above the door. That made sense, since every night in the dining room was like a freewheeling improvisational concert, complete with an ever shifting lineup and the ever present potential for disaster. The kitchen was more of a stellar rhythm section: steady, reliable, above all consistent.\n\nNowadays, chefs have completed the radical shift from anonymous laborers to celebrated artists. For some, that's meant leaving the kitchen altogether, traveling the world, expanding their brands. But for those who can't give up the particular adrenaline rush of being behind the stove, there's no getting around the fact that the essence of the job\u2014turning out the same dish over and over again, night after night\u2014is deeply, profoundly dull. And that means we may soon be living in a pop-up world.\n\nAt Gram & Papa's, it was almost curtain time. French rap played over the sound system as Krissy's team of young waitresses set tables. Taking his position in the open kitchen, Lefebvre carefully laid out his tools: a silver quenelle spoon, a tiny grater with bamboo brush, a Sharpie. At the head of this little shrine, he lit a Mexican prayer candle.\n\nThe doors opened, and the first seating poured in. At one table, each of five diners held cameras or camera phones. One customer presented Lefebvre with a bundle of locally picked Spanish garlic. \"Tomorrow we have garlic soup,\" the chef announced. \"With escargots.\"\n\nOrders came in. Dishes flowed out: a cheese \"cupcake\" frosted with chicken liver and foie gras; a feathery-smooth potato mousseline capping, shepherd's-pie style, a barely poached egg and a warm bed of chorizo; a version of a classic French _fris\u00e9e aux lardons_ reimagined as a tower of greens teetering in a bowl of smooth, rich goat-cheese soup. (When I tried the dish at dinner with Gold, I nervously approached it with knife and fork. \"It's hard to cut soup,\" the critic deadpanned.) All but the oyster dish, which nobody had yet plated or tasted\u2014leaving no assurance that Lefebvre's morning inspiration would even work.\n\nOne by one, the tickets for oysters piled up. Lefebvre hovered over the simmering dashi, watching as a test run poached. Krissy looked on in a state of bemused panic. Finally, Lefebvre lifted two of the oysters from the water and pried them open, revealing the pair of perfectly swollen iridescent bubbles within. He swiveled toward the counter where Joon Sung was supposed to be assembling the dishes.\n\n\"Joon!\"\n\n\"Yes, Chef!\"\n\n\"Why haven't you fucking prepped for the oysters?\"\n\nSomething in Sung snapped: \"Because I don't know what the fuck I'm supposed to do, Chef!\"\n\nThe kitchen came to a halt. The chef spun slowly in his clogs. Sung waited, blinking. Then Lefebvre grinned, clapped his cook on the shoulder, and screamed right back:\n\n\"Well, neither do I!\"\n\n### HOT PLATE\n\n### By Rachel Hutton\n\n### From _Minnesota Monthly_\n\n### One index of a regional restaurant culture's vitality is the number of talented food writers it supports. Among the leading lights of the Twin Cities' lively dining scene is Rachel Hutton, formerly of _CityPages_ and newly named food editor of _Minnesota Monthly._\n\n**7:00 a.m.**\n\nIt's going to be another big night at The Bachelor Farmer and Marvel Bar.\n\nThe cooler is stocked. The reservation book is full. And the overnight cleaning crew just eradicated the last trace of yesterday's service. On this chilly morning, 50 North Second Avenue looks much as it must have a century ago, when the brick-and-timber warehouse was new to Minneapolis's North Loop. The streets are deserted and the sky is dark except for the florescent lights streaming from the building's kitchen\u2014a beacon for the hottest thing going in the neighborhood.\n\nThe former Northwestern Hide and Fur Building wasn't attracting much attention until Eric Dayton, son of Minnesota Governor Mark Dayton, purchased it in 2008. Its most recent occupant was an industrial wire-cutting business called Marvel Rack, so architect James Dayton (yes, he's a relative) faced no small task in converting the space into a modern restaurant, bar, and retail shop, while preserving its historic character.\n\nWhen Eric and his brother, Andrew, conceived of their contemporary Scandinavian restaurant and subterranean speakeasy, they imagined an intimate neighborhood hangout. But as soon as the kitchen served its first plate of Swedish meatballs, patrons flocked from near and far, packing the place night after night. How were a couple of hospitality greenhorns able to pull off such a feat?\n\nThe Bachelor Farmer's baker and pastry chef are the first to arrive, as always. Their first stop: the storage room. The restaurant makes almost everything from scratch using premium ingredients\u2014even growing some produce on the building's roof, which is buried in snow. The pastry chef bypasses a tub labeled \"Top Secret Pickle Project\" and half a hog carcass to collect dozens of Larry Schultz eggs, pounds of Hope butter, and gallons of Organic Valley cream. She has less than six hours to create a dessert that will earn a spot on tonight's menu.\n\nBack up in the kitchen, the two feed bread starter, melt chocolate, heat milk for fresh cheese, and roll dough into long sheets of flatbread. It's too early for conversation, so they work in near silence, save for the hum of the ovens, vents, and refrigeration units.\n\n**8:58 a.m.**\n\nChef Paul Berglund enters looking rather like a lanky graduate student with his narrow glasses and day's worth of beard growth. Rather than rely on Eric's experience with line cooking at Good-fellows restaurant back in high school, the Dayton brothers called upon Paul, an alumnus of Oliveto, a highly respected restaurant in Oakland, California, to lead The Bachelor Farmer's kitchen. Paul starts his day with a clipboard in hand, reading through the to-do list: ducks, lentil base, pat\u00e9; render pork fat. His first task is to fillet last night's shipment of rockfish\u2014beautiful specimens, with red-and-white skins, clear eyes, and resilient flesh.\n\n**9:35 a.m.**\n\nEric and Andrew arrive. Although they're both already clutching Dunn Brothers cups, Andrew starts the coffee maker. Their preppy outfits look as if they might have come from the same closet. The brothers' similar taste in clothes has become something of a running joke. If they accidentally dress alike, Andrew says, \"We do rock-paper-scissors to see who has to go home and change.\"\n\nAs if the brothers didn't have enough to do between overseeing the bar and restaurant, they followed in their forefathers' retail footsteps and opened a men's shop in the building's storefront. It's called Askov Finlayson, a name that should sound familiar to anyone who's driven I-35 between the Twin Cities and Duluth and paid attention to the exit signs.\n\nThe restaurant and the store often share clients: fans of farm-to-table cuisine seem to appreciate the classic clothing and accessories. One night, the Daytons sold two handcrafted leather iPad cases when a Bachelor Farmer guest showed the one she'd just purchased to her dining companion, who promptly went down and did the same.\n\nWhile they wait for customers, Eric and Andrew meet with the retail manager to plan their itinerary for the menswear shows in New York and time an order of British umbrellas. The shop is minimally furnished with leather chairs and an area rug pulled from Andrew's emptying apartment. Eric, too, has sacrificed home d\u00e9cor for the business, having lent the restaurant several of his photographs by local-boy-made-good Alec Soth. \"My apartment looks like it's been robbed,\" Eric explains. \"There are bare hooks where art used to be.\" The brothers spend far more time at work than home these days, so at least here they can enjoy their possessions.\n\nAskov Finlayson's merchandise tends toward timeless pieces: dress shirts, slacks, Danish wool sweaters, sailcloth duffel bags. The common denominator, Andrew explains, is quality, partial justification for the $125 price tag on the store's most casual item: sweatpants with their brand name, \"Warriors of Radness,\" spelled out down the leg in rainbow-colored cursive script.\n\n**11:14 a.m.**\n\nIn the office upstairs, the general manager sits at his computer and scans the evening's reservation list, which already shows 150 diners on the books. He's going to need every last table in the dining room, including the corner booth that went out of commission last night due to an icy draft, so he heads downstairs armed with a caulk gun.\n\nAt Andrew's computer, the brothers update the Marvel Bar's Facebook page. Today's post describes a new drink called the Strong-water, which the Marvel Bar's head bartender, Pip Hanson, recently invented. Bourbon, cognac, thyme liqueur, and lemon zest are mixed together and then highly diluted. It's a surprisingly pleasurable technique: the water smooths out any harshness and highlights the spirits' subtleties. Pip developed a reputation for precise, innovative bartending at his previous position at Caf\u00e9 Maude, where the brothers recruited him after he poured the best Manhattans they'd ever sipped.\n\nNa\u00efvet\u00e9 more than confidence may have spurred the Daytons to tackle their ambitious project, but what the brothers lack in entrepreneurial experience, they seem to make up for in diligent management. (Presumably they have picked up a few tips from their father's overseeing of the entire state of Minnesota.)\n\nThe brothers plan to add brunch service to the restaurant as well as private events, but they have prioritized fine-tuning their current operation versus expanding\u2014don't expect to see The Bachelor Farmer franchises out in Bloomington and Maple Grove. Building wealth would seem less of a motivator to the Daytons than creating a restaurant experience that proud locals show off to out-of-towners. Considering the Dayton family's legacy in retail and political leadership, thinking small would be uncharacteristic. And who better to understand the collective consciousness of Minnesotans than the Daytons? They practically invented our aspirations of what it means to be one.\n\n**1:02 p.m.**\n\nThe kitchen staff gathers in the dining room for their daily meeting. Paul instructs one of his cooks on how to prepare an appetizer made with small fish called sand dabs. \"You're going to need capers, white wine, olive oil, and mullet roe,\" he says in confident rapid-fire, like Brad Pitt trading baseball players in _Moneyball._\n\n\"You need a shaver for that,\" he adds, talking the cook through the steps. \"Microplane. Same one you're using for the Bibb.\"\n\n\"Large flake?\" the cook asks, furiously scribbling in a notebook.\n\n\"I don't want a microplane-microplane,\" Paul clarifies, in language only cooks can understand. \"But a microplane is preferable to really large flakes.\"\n\n**1:43 p.m.**\n\nPaul and his sous chef have just finished testing a new shrimp appetizer and roast-chicken entr\u00e9e. It's now the moment of truth for the pastry chef's dessert: a flourless chocolate cake with salted-caramel semifreddo, lingonberry p\u00e2tes de fruit jellies, and a dollop of cream. Paul takes a bite and says he loves the flavor combination, but he's concerned about the p\u00e2tes de fruit sticking to the plate. The two discuss a few options before deciding to simply plop the gummies into the cream.\n\nAfter they've finished, Paul sets forth on one of the kitchen's lowliest tasks: dicing an array of onions with mechanical precision. The former naval officer isn't exactly _Food Network_ material\u2014he's too calm, too disciplined.\n\n**2:35 p.m.**\n\nA deliveryman wheels a 300-pound block of sculptor's ice into the Marvel Bar. When even the world's purest ice cubes can't give the Marvel bartenders the drink-cooling control they desire, they chip their own ice from large blocks. Two bartenders help the deliveryman heft the thing onto the counter and immediately start attacking it. One scores the block into eighths with what looks like a large putty knife, while the other splits it with a hammer and chisel. The first bartender breaks the block down further by aggressively stabbing it with a metal pick, like he's a human sewing-machine needle. The smaller blocks are neatly stacked in the freezer within a matter of minutes. \"It reminds me of a cave man taking down a mammoth,\" Pip remarks.\n\n**3:59 p.m.**\n\nAfter a staff meal\u2014a substantial spread that includes the roast-chicken tester, two types of potatoes, popovers, salad, and a hearty split-pea soup\u2014the front-of-the-house staff gathers upstairs for their pre-shift meeting, which takes place in one of the restaurant's private dining rooms. The walls are covered with a funky collage of crocheted afghans, which lends the otherwise modern space a warm familiarity.\n\nFirst on the agenda: the sous chef explains the new dishes. After subjecting the servers to a beer-list pop quiz, the general manager issues a warning: \"Do not cap the stack.\" More than one blank look results. \"When stacking similar dishes for the dishwashing crew,\" he explains, \"you should never add a different dish to the top of the pile\"\u2014not only is it inefficient, but the whole thing could topple.\n\nAndrew previews some exclusive new Alec Soth merchandise for the store, including coloring books from Soth's publishing company that may be the first in the genre to feature Bronko Nagurski and the Coen brothers. The general manager then pours the staff samples of a hard-to-find new wine he's acquired. \"I buy wine like they buy Soth,\" he says of the Daytons. \"Take it all so nobody else can get it.\"\n\n**4:45 p.m.**\n\nSeated in the Norsten Bar, next to the dining room, the assistant general manager undergoes her daily ritual of reviewing the comment books that the servers drop off with the check. She initials each remark, incorporating the feedback in her mental databank. \"Coming from a bachelor farmer, we love this place.\" \"Uffdah! Very good.\" \"It's not Manhattan, but it's surprisingly hip, delicious, and cool.\" Minnesotans always seem to find the need to make coastal comparisons, don't they?\n\nThe books contain poems, a fake mustache, a lipstick kiss, and, inexplicably, what appears to be someone's senior-class photograph. One of the servers has received multiple marriage proposals from anonymous guests. Books that have recently received \"edgier\" messages\u2014\"I f\u2014-ing love you guys,\" for example\u2014circulate in the bar. Only a few book pages with \"inappropriate\" sketches (a group of the Dayton brothers' friends are prime suspects) had to be removed.\n\n**5:20 p.m.**\n\nGuests with 5:30 p.m. reservations are already peeking through the windows and the assistant general manager doesn't want them to wait in the cold. \"We're opening the doors,\" she announces to the kitchen. A cook makes shots of espresso for the entire line. \"Double? Or triple?\" she asks.\n\n**5:24 p.m.**\n\nA server cuts butter pats for the restaurant's complimentary appetizer of flatbread and radishes and ferries the first plates to the dining room. Restocking the butter pats is among the hundreds of details that front-of-the-house staff has to track. Dozens more tips are posted on a list called \"Service 101\" near the servers' order terminal. No. 1 is a no-brainer: \"Acknowledge all guests with eye contact and a smile immediately.\" But others are more obscure, like \"12. Level the art.\" Several deal with appropriate guest interaction: \"18. Do not react to the amount of the tip; 30. Do not ask a question while a guest's mouth is full; 31. Do not enter a guest's conversation unless clearly invited.\"\n\n**6:00 p.m.**\n\nThe sous chef stands at the pass, a counter that divides the kitchen from the dining room, lining up tickets and calling out orders to the various cooks. She's essentially playing air-traffic controller so each table's orders will be ready at the same time. \"Fire Camembert, no shallots,\" she hollers. Two old tickets are taped to the wall with messages scrawled on top, \"10-top in 23 minutes,\" and, \"9-top in 20 minutes\": house records in putting together big orders, the kitchen equivalent of scalps.\n\nPaul runs his kitchen like the Daytons run their business: hire people you trust and let them do their thing. He lets his staff handle tonight's dinner service as he scores the fat on dozens of duck breasts and makes other preparations for the upcoming days. From the back of the kitchen, he can watch the orders flow and anticipate any problems\u2014demonstrating strong \"field awareness,\" as they say in team sports.\n\n**7:04 p.m.**\n\nMuch like The Bachelor Farmer's dining room, the accompanying Norsten Bar has a mellow vibe despite being nearly full. No matter the guest's dress\u2014lumberjack plaid and sequins inhabit adjacent barstools\u2014they seem to appreciate the food and beverage. A man approaches the bartender and gruffly asks her what's in one of the drinks. She shows him the bottles: aquavit, gin, orange bitters, and Cocchi Americano, an aperitif wine. \"Is that your favorite cocktail?\" she asks. \"It's delicious,\" he says. \"In fact, I should probably tip you.\" He pulls a stack of folded bills from his pocket and tosses $5 on the counter.\n\n**8:32 p.m.**\n\nThe kitchen is running full tilt. Eggshells pile up. A stove burner flares. The cooks fill the pass with plates and servers rush to distribute them. Meanwhile, in the relative serenity of the basement cooler, Paul deconstructs the pig carcass with a hacksaw.\n\nIn the dining room, guests appear rapt in conversation, enjoying the evening and blissfully unaware of all these activities. The lights are low, the music is soft, and wine glasses cover nearly every table. Diners may never make a conscious note of all the tiny details that made their meal great versus simply good\u2014the house-butchered pork, the hand-chipped ice, the plating of the p\u00e2te de fruit, the minute difference between a \"microplane-microplane\" and a \"large flake.\" When they next recall this meal, all they will remember is that they ate well and enjoyed themselves.\n\nIn fact, The Bachelor Farmer's secret to success may be that its food isn't necessarily the star: the cooking is novel enough to discuss, but not at the expense of other topics. It's the restaurant's vibe\u2014its contemporary spin on nostalgic comforts and Volvo-like sense of understated luxury\u2014that accounts for its enduring appeal. Sure, a few guests may have been hoping to rub shoulders with politicians and scions, but mostly they are drawn in by the way the place reflects their own sophisticated populist sensibilities. And besides, who doesn't love the idea of hopping from shop to restaurant to bar without the hassle of re-parking the car?\n\n**10:49 p.m.**\n\nEric and Andrew finally sit down to eat dinner in the Norsten Bar. Paul comes in to say goodbye, sporting a new Band-aid on his finger\u2014he pricked himself with the giant syringe he was using to pump brine into cuts of pork. Trying to brine himself, it seems.\n\n**11:26 p.m.**\n\nEric stops in the kitchen and says goodnight. Andrew will leave a few minutes later and both will be back tomorrow morning by 10 a.m. to open Askov Finlayson. The cooks clean up as the servers start to cash out, counting their tips and dispensing a few appreciative bills to the hostesses and food runner.\n\n**12:05 a.m. Sunday**\n\nA guest is upset that his party can't get into the packed Marvel Bar and a couple staffers gracefully smooth his ruffled feathers. Restaurant diners are given priority access, but walk-ups must take their place in the queue\u2014no VIPs, no favorites. Supposedly, even Governor Dayton doesn't get to cut the line, but that scenario has yet to arise.\n\n**1:52 a.m.**\n\nThe Marvel Bar's last patrons head out the door and someone flips on the lights. A bartender pours a round of beers and passes out a loaf of homemade banana bread. (The only food the bar serves is Cheetos, and often the bar staff doesn't stop to eat during their shift.) After nearly 12 hours spent thinking about booze, two devoted bartenders discuss the merits of 110-proof gin as they clean up.\n\n**2:56 a.m.**\n\nThe last of the crew bundles up\u2014one bartender pulls on an Askov Finlayson sweater\u2014and spills out into the cold, dark night. The kitchen emits the same florescent light it did 20 hours earlier, a reminder that in roughly four hours, the cycle will start all over again.\n\n### AUSTRIA'S CULINARY AMBASSADOR\n\n### By St. John Frizell\n\n### From _Edible Manhattan_\n\n### St. John Frizell has an insider's perspective on the restaurant biz, as the proprietor of Brooklyn's popular bar-caf\u00e9 Fort Defiance. Meanwhile, he plies his trade as a food-and-drink writer in such publications as _Bon App\u00e9tit, Saveur, Islands, The Oxford American, Edible Brooklyn_ and _Edible Manhattan._\n\nHigh above the Austrian village of Ratsch, on a hilltop overlooking a sun-soaked valley planted with rows of grapevines, I'm all set to tuck into the biggest plate of fried chicken that I've ever seen. Walnut trees shade the deck of Weinlokal Maitz, a restaurant on the Southern Styrian wine route, where I sit with Kurt Gutenbrunner, the Austrian-born chef-owner of five restaurants in Manhattan, and Michael Gross, the charming young scion of a local wine family. The chicken, served with lemon wedges and a salad made with purple-specked k\u00e4ferbohnen\u2014\" beetle beans,\" what we know as \"scarlet runners\"\u2014is a local specialty called backhenderl, and a dream come true. Just under a crust that has the airy crispness of a perfect Wiener schnitzel, the meat bursts with gamy juice that makes me want to find the chicken farmer and shake his hand. And I could, without much trouble; at this point in my travels in Austria, I know the country is a locavore's fantasy\u2014and that the chicken I'm about to eat was clucking in someone's yard the day before yesterday. (The English menu reinforces my belief in plain language: \"Needless to say, we know all of our distributors personally.\")\n\nThe wine we're drinking is local, too\u2014on the hillside across the valley is the Gross winery and vineyards, where Michael lives and makes wine with his brother, Johannes, and their father, Alois. From here we can see the terraced slopes where they grow gr\u00fcner veltliner. Down in the valley, we can make out the roof of the buschenschank that Michael's grandparents still run. The buschenschank is a glorious Austrian tradition\u2014a little tavern where winemakers, with special governmental permission, serve their own vintages and a small selection of hot and cold dishes to tourists and locals alike. \"In the fall, we live to go to the buschenschanks,\" Michael says. \"People come for the leaves\u2014every tree you can see in this valley turns a different shade of green, gold and red. And they come for the sturm.\"That's the newest wine of the year's harvest, the grape juice that's just started to ferment. The men at the table smile and let out a collective groan, as if they've all gotten into plenty of trouble drinking sturm, and can't wait to do it again. \"I think it continues to ferment in your stomach,\" Kurt says.\n\nJust then, the server brings Michael a plate of schafkase im mantel\u2014sheep's milk cheese wrapped in speck. What, no backhenderl? \"It's delicious, but no. I had it for lunch yesterday,\" Michael says, sheepishly. \"And the day before.\"\n\nSince he opened Wallse, his Michelin-starred homage to Viennese cuisine, in the West Village in 2000, Kurt Gutenbrunner has been turning Manhattanites on to schnitzel, spaetzle and gruner veltliner. He's the primary advocate and most recognized representative of Austrian food culture in America, and in Austria, a country with a population about the same size as New York City's, he's something of a local hero. For six days, Kurt and I scoured the country in search of new ingredients, producers and dishes that will inspire his menus in the year to come at Wallse and his other restaurants: Blaue Gans, the Austrian bistro in TriBeCa; Caf\u00e9 Sabarsky, the Viennese caf\u00e9 at the Neue Gallerie; Caf\u00e9 Kristall, at the Swarovski Building; and his wine bar, the Upholstery Shop. To kick off the week, we meet at Plachutta, one of Vienna's most respected traditional restaurants, for a crash course in Austrian Cuisine 101.\n\nLooking at the menu, Kurt explains how Austria has always been a crossroads, a place in the middle. Once the seat of power of the Austro-Hungarian Empire, Vienna was also the easternmost city in Western Europe during the Cold War. Centuries of trade, diplomacy and conquest have left their mark on the city's cuisine. \"Gulasch? From the Hungarians,\" explains Kurt. \"Schnitzel, that's essentially a Milanese, from Italy. A lot of these pastries are Czech. And the coffee in our famous caf\u00e9s? From the Turks.\"\n\nBut since the end of the Empire, the major influence on Austrian cuisine has come from within. In the early 20th century, Austrian thinker Rudolf Steiner provided the philosophical underpinnings for what would become the organic farming movement.\n\nNaturally, Austria has become one of the world's leaders in organic farming; it was one of the first countries to set official organic guidelines, and its government continues to subsidize ecological farming practices. As a result, close to 20 percent of its farms are organic, more that in any other European country (except tiny Liechtenstein).\n\nFrom the hippest Viennese nightspot to the most rustic small-town gasthaus, we see these two forces working in harmony, and they define contemporary Austrian cuisine\u2014traditional dishes, like Wiener schnitzel and krautsalat, are thoughtfully prepared so that the flavors of the fresh, local meat and produce shine through. Kurt is tickled by the recent rise of Austrian cuisine in New York's dining scene, where Midtown's Seasonal now has a Michelin star, and dozens of beer gardens have sprouted up over the past 10 years.\n\n\"When I put tafelspitz on the menu in New York, people said, 'You're serving us boiled beef?' It wasn't very cool,\" Kurt says. Plachutta is famous for this quintessential Austrian dish, which is essentially a simple boiled dinner; the name refers to the cut of beef, which comes from the round. Other cuts\u2014kavalierspitz, tafelst\u00fcck and so forth\u2014are commonly used, but \"tafelspitz in particular is accompanied by a myriad of legends, and no other dish has a comparable historic significance,\" according to the Plachutta cookbook. The service is warm and formal; the food, simple and hearty as corned beef and cabbage. Kurt serves kavalierspitz at Blaue Gans, his casual Austrian bistro in TriBeCa. Does he lighten it up a bit for his downtown Manhattan crowd? \"No, of course not,\" he says, between forkfuls of beef. \"Does the Viennese Philharmonic play Mozart any differently when they come to New York?\"\n\nAfter getting through almost a pound of beef, I put my fork down, while Kurt continues to eat. He is a man of voracious appetites and seemingly boundless energy\u2014it's no wonder at all that his little hometown of Wallsee, a little village in Upper Austria, couldn't hold him.\n\nIn 1988, after a two-year stint at Munich's then-Michelin-Three-Star restaurant Tantris, he was introduced to Hermann Reiner, chef of Windows on the World in New York, who hired his 26-year-old fellow Austrian as sous chef. Gutenbrunner fell in love with New York, but found the experience of cooking at the top of the World Trade Center disconcerting. \"I was used to buying produce off a truck in back of the restaurant; now, we were buying produce 110 floors below. It was like working on a submarine,\" Kurt says.\n\nIn the meantime, an American chef with ties to Austria and impeccable French training was earning rave reviews in TriBeCa; before opening his eponymous restaurant, David Bouley had worked at the then four-star restaurant Vienna 79 with chef Peter Grunauer. In 1990, Gutenbrunner found his first real home in the States in the kitchen at Bouley, and started a working relationship with chef David Bouley that would last, off and on, for 10 years. The restaurant Bouley has become a proving ground for American chefs\u2014Eric Ripert, Dan Barber and C\u00e9sar Ramirez all passed through in the 1990s, to name a few. Gutenbrunner was suitably challenged and invigorated by the other chefs in the kitchen. \"We were a great team. No one could beat us. It was a shitload of work, and there was a lot of pressure, but when people push each other like that, a lot of extreme things can happen,\" Kurt says, before turning wistful. \"I think if things happened on schedule, David and I would still be working together.\"\n\n\"What are you doing, Johnny?\" Kurt asks, gesturing at my half-full plate as he scoops the last bites of kavalierspitz from his own. At our first meeting, Kurt started calling me Johnny, and after a couple of days, I stopped correcting him. I told Kurt I was stuffed; I probably should have passed on the chanterelle omelet that we had as an appetizer. \"What's the matter? You only ate half a cow,\" he says. \"Let's get dessert.\"\n\nWhen we meet early the next morning, we have dessert again. \"I love cake,\" Kurt tells me as he quickly demolishes a slice of Landtmann torte, with its decadent layers of walnut cream and marzipan. Vienna's caf\u00e9s are justly famous for their extravagant cakes\u2014like sachertorte, linzertorte and apfelstrudel, to name a few\u2014and during our week in Austria, Kurt never misses a dessert. His trim silhouette can only be explained by his inexhaustible energy and outright speed; I often have to jog to keep up with him as he navigates the twisting streets of Vienna like he's being chased. He only stops to eat, as he is now, at Caf\u00e9 Landtmann, a fabled Viennese caf\u00e9 a stone's throw from Vienna's University, the City Hall and the monumental Burgtheater.\n\nAn hour later, we stumble onto a frozen yogurt shop called \"Kurt\" on a cobblestone street; Kurt warmly greets the owners (\"You're Kurt? I'm Kurt!\"), drops business cards on all of the shop's patrons and orders a blueberry-acai yogurt, in the \"Classic Kurt\" size. \"Once, I went to Caf\u00e9 Sabarsky for a business meeting, and I ate four apricot cakes before it was over,\" he says. Kurt opened Caf\u00e9 Sabarsky in the Neue Gallerie, Manhattan's museum of Austrian and German art and design, in 2001. Kurt had met the museum's founder, Ronald Lauder, when they both ordered Thonet chairs from the same Austrian company. After visiting Vienna, the likeness of Caf\u00e9 Sabarsky to its Viennese cousins is almost eerie\u2014from the waiters' uniforms to the Thonet hat racks to the menu, with its Staud's jams and sachertortes and grosser brauner (coffee with steamed milk), it's all the same.\n\nGutenbrunner comes to Austria several times a year, in part to see old friends and family, and in part to source new ingredients for his New York kitchens. This is what has brought us to Heimschuh, a sleepy town in the southern state of Styria, near the Slovenian border. It's home to the Hartlieb mill, where some of the world's best pumpkinseed oil is pressed. \"This was once oil of the poor,\" says Thomas Hartlieb, whose great-grandfather opened the Hartlieb mill in 1896, when they used river power to mill lumber as well as grain. \"People thought it was low quality, because of its dark color.\" In the bottle, pumpkinseed oil is a dark, almost purplish green; when Hartlieb holds a bottle high and pours it so that the afternoon sun shines through the oil, it's a vivid crimson. Now, it is to Austria what EVOO is to Italy; in a week, I think I had it at every meal, dressing greens, tomatoes, cheese and fish with its distinctive nutty flavor.\n\nHartlieb keeps a collection of antique pumpkinseed presses in a makeshift museum on the mill's second floor, but on the ground floor, high-tech presses and grinders do the work today. Local farmers grow special Styrian \"oil pumpkins,\" whose seeds grow without hulls. This squash's harvest is the inversion of its American cousins: they keep the seeds and discard the flesh. At Hartlieb, those prized seeds are ground and roasted, which causes the proteins in the seed puree to separate from the oil. That mixture is pressed, the oil collected, and what's left behind\u2014a protein-rich puck of pressed pumpkinseeds\u2014becomes livestock feed on nearby farms.\n\nOver a seidel of Puntigamer, the go-to lager in Styria, at the caf\u00e9 across the street, Gutenbrunner and Hartlieb talk about pumpkinseed oil's appearance on the gourmet food scene; until about 20 years ago, it wasn't even common in upper Austria, though it's been used in Styria for centuries. \"It wasn't that we didn't want it; you Styrian guys just didn't want to give it to us,\" says Kurt. It's an ingredient Kurt has always showcased to great effect. Ruth Reichl, in typical for-mature-audiences-only prose, reviewed Gutenbrunner's food when he was chef at the Monkey Bar, a clubbish Midtown restaurant then owned by steakhouse czar Peter Glazier, in the New York Times in 1998: \"Just take a spoonful of his butternut squash soup. Hold it in your mouth, rejoicing in the deep richness of the pumpkinseed oil on top. . . . It is irresistible.\"\n\nEight years earlier, when his first son was born, Kurt had left Bouley and moved to Germany\u2014he thought Europe would be a better place to raise a family\u2014but returned to New York to work with his old boss whenever time allowed. In 1996, he moved to New York for good and hatched plans with Bouley to open the Austrian restaurant that would become Danube. But plans stalled, Kurt became impatient and steakhouse czar Glazier made Kurt an offer he couldn't refuse\u2014his first executive chef gig, at Monkey Bar. \"He gave me everything and beyond. I didn't want to do it at first, but it's hard to see everyone else moving ahead when you're standing still,\" Kurt says. After two years there, he met the investors who would help him fund Wallse\u2014parents of a kid on his son's soccer team\u2014and a restaurant empire was born.\n\nFrom the roof of the Gegenbauer vinegar factory, we can see the broken roof tiles of the surrounding apartment blocks in this unglamorous neighborhood of Vienna's 10th district. We've come to check in on an old friend of Kurt's, Erwin Gegenbauer, whose grandfather Ignaz started making sauerkraut and pickled vegetables in this building in 1929. In the 1990s, Erwin sold off most of the company's assets to focus on his true aspiration: to make the world's best fruit vinegars. Gegenbauer makes vinegars from every kind of foodstuff imaginable: apples and grapes, but also honey, figs, cucumbers and asparagus. These are not the vinegars on your typical grocery shelf, flavored with raspberries or other fruit. \"Those vinegars are made by adding fruit flavors to wine vinegar. That is chemistry, and I don't do that,\" Gegenbauer says, in near-perfect English. \"My raspberry vinegar is all raspberries, no other ingredients. You could say it's more simple this way. But the simplest products can be the most complicated.\"\n\nGegenbauer starts his process by working with local farmers, selecting fruit with the careful attention of a winemaker. \"How many leaves per branch is optimal? When do we harvest? How do we press the fruit to get the juice? These are the questions we ask,\" he says. The fruit juice is allowed to ferment, creating a wine; Gegenbauer then introduces specific strains of bacteria\u2014he keeps several hundred on hand\u2014which will, over several weeks, convert the alcohol in the wine to acid, creating vinegar. Some vinegars are then aged in oak wine barrels, either in his cellar or on his roof, exposed to the elements. I ask why there's no tradition of making this kind of vinegar in Austria, or anywhere else. \"There's a popular perception that vinegar must be cheap, that vinegar is wine that's gone bad,\" Gegenbauer says. \"That's changing.\"\n\nGegenbauer brings out his newest project to show Kurt, an oil made from pressed raspberry seeds\u2014a by-product of his vinegar process. \"I love working with Kurt,\" Gegenbauer says. \"I give him vinegar, we taste it and discuss. Sometimes I work with him in the kitchen, and together we create a new dish. I'm the craftsman; he's the creative, pushing me to experiment with new flavors.\"\n\nHe places a few drops on the back of Kurt's hand and mine; Kurt licks it off and stares at Gegenbauer as he rolls it around in his mouth. The flavor is subtle and woody at first, then slowly blossoms into something like raspberry jam on toast. \"The berry flavor comes late,\" Kurt says.\n\n\"But it stays a long time,\" Gegenbauer says. \"That's amazing! Can I take this with me?\" Kurt says. Our hands will smell, pleasantly, like raspberry bushes for the rest of the day.\n\nA photographer from an Austrian society magazine arrives; she asks the two men to pose between the batteries of casks on the roof. Gutenbrunner is totally relaxed in front of a camera, posing for pictures like an aging rock star, to which he sometimes compares himself (\"You know what they say about British rock and rollers? They don't fucking die! Keith Richards. Robert Plant. I'm like that. You can't kill me.\"). His body totally relaxed, Kurt looks directly into the camera, eyelids heavy, his lips slightly curled in the suggestion of a smile.\n\n\"Let's go see Claus, Johnny,\" Kurt says. The open fields of Burgenland, planted with chest-high cornstalks and sunflowers, whip past the windows of our Mini Countryman, as Kurt drops gears to pass another ambling truck. \"You'll like this guy. He's a little crazy.\"\n\nClaus Preisinger's winery is a strikingly modern poured-concrete bunker filled with strikingly low-tech equipment\u2014just a bunch of stainless steel tanks and wooden barrels. There's not even a pump in sight\u2014when Preisinger needs to move his wine from tank to barrel, he uses a length of tubing to siphon it, concerned that mechanical pumping will disturb his wines. In the corner, there's a stack of crates of mineral water (\"Good for breakfast,\" Preisinger says.) and Budweiser Budvar (\"Very important after a day in the vineyards: cold beer.\") and, off to the side, the winery's most advanced piece of machinery: a 500-liter teapot, where he makes chamomile tea to feed his vines during times of stress. He hasn't used pesticides or herbicides in years and cites Rudolf Steiner as a direct influence.\n\nKurt includes a wide swath of Austrian winemakers on the lists at his New York restaurants, from the most traditional old houses to the newest, most cutting-edge vintners, like Preisinger. With the tousled good looks of an emo rocker, Preisinger is, at 31, the youngest member of Pannobile, a group of nine local winemakers who have banded together to form their own appellation\u2014like a French AOC, which controls what grapes can be used in what wines, but without the government. Each year, the winemakers in the group submit their wines to a tasting panel of their peers; to be considered a Pannobile wine, all nine winemakers must unanimously approve.\n\nHe pours his 2008 Pannobile, a blend of zweigelt, blaufr\u00e4nkisch and a little bit of St. Laurent, a rare, highly aromatic grape. \"It's tricky to grow,\" says Preisinger, \"but sometimes the trickiest grapes make the best wines.\" It's lovely stuff, light-bodied and subtle, with flavors of black currant and earth. As we drink, watching the sun set over Lake Neuseidl into a bank of clouds, a burly farmer with mud-spattered boots walks in\u2014it's Paul Achs, another Pannobile winemaker, carrying a bottle of his 2000 blaufr\u00e4nkisch, and the glasses are filled again. The two winemakers are eager to take him out to dinner, to a restaurant called Blaue Gans\u2014the same as Kurt's TriBeCa bistro. But Kurt begs off, and we get back on the road. \"I know these guys,\" says Kurt. \"We go out with them to dinner and it'll be sunrise before we get to our hotel.\"\n\nErich Stekovics, with his round red cheeks, deep-set eyes and red polo shirt covering a round belly, fits his nickname: Kaiser der Paradeiser (the emperor of tomatoes). In the fertile plains of Burgenland, he raises more than 800 varieties of tomatoes every year; in his stores, he keeps the seeds for 2,000 more. \"I think he named one of his daughters Tomato,\" Kurt whispers, as we follow Stekovics into his greenhouses.\n\nAs we stalk quickly through rows of six-foot plants heavy with fruit, he pulls tomatoes off the vines for us to taste. With his knife, he splits a small, dark tomato and shows us the purplish flesh. \"Black cherry,\" he says, before popping half in his mouth and motioning for me to do the same. The flesh is lush and flavorful; the juice is thick and heady, like a swig of Achs's blaufr\u00e4nkisch. The tomatoes follow in quick succession, and Stekovics rattles off the names; he can identify hundreds of varieties by sight. There's de Barao, yellow and plum-shaped; Russian pear, sweet and smooth, as fragrant as a ripe peach; vibrant red Schlessian raspberry\u2014each more flavorful and succulent than the one before.\n\nHis secret? He doesn't water his plants. Ever. Bred for hardiness, they're forced to sink their roots ever deeper into the earth, giving them greater access to resources and, he believes, producing the world's most flavorful tomatoes. His methods buck conventional wisdom, which dictates that tomatoes need plenty of water, and baffle experts; a research team from the University of Innsbruck took three days to excavate the root-ball of one of his plants. Home gardeners from all over Europe visit to see Stekovics's plants and hear his gospel\u2014but their hearts are not strong enough to follow him. \"They're afraid,\" he says, obviously a little hurt. \"They go home and water their plants.\"\n\nAs Kurt tries to convince Stekovics to visit the U.S. (\"I want to introduce you to some of my farmers,\" he says), Stekovics brings us to one of his favorite plants: the Firework tomato, a Russian variety that's 450 years old, its carmine flesh streaked with the yellow and gold flecks that give it its name. As he cuts into it, red juice drips down his hand; the flesh is dark red to its core. It's just delicious, fruity and aromatic, like summer savory and wildflowers, the Platonic ideal of a tomato. \"You want a restaurant?\" Stekovics shouts, flourishing the knife in his seed-stained hand. \"Bring your table in here. I'll make you a feast you won't forget. Seven courses of tomatoes.\" He leans in close, and fixes his hound-dog eyes on mine. \"If you write about this, no one will believe you.\"\n\nWithin minutes of arriving in the town of Axberg in northern Austria, Hans Reisetbauer, perhaps Austria's most respected distiller, is making us coffee, and after hours on the road, we need it. We started the day just past dawn in the farmers market of Graz, where Kurt loaded my arms with sunflowers, tiny raspberries, dry pork sausage studded with pumpkinseeds, a huge bag of ripe apricots, and some slices of poppy-seed cake as he flitted from stall to stall, chatting with the vendors. \"It's good to talk to these old ladies,\" he told me, as he handed me a squash. \"They know best. This pumpkin? She told me you can cook the whole thing, leave the skin on. Makes good soup.\"\n\nIt's the start of Reisetbauer's busy season\u2014he makes his living turning Austria's best fruit into award-winning eaux-de-vie, or schnapps, and as we saw in the market this morning, the first fruit of the summer was already ripe. He distilled his first batch of raspberries the week before our visit; through the porthole in one of his copper stills, I see his first batch of apricots cooking away, bubbling like a pot of jam.\n\nReisetbauer grows all of his own apples and pears on his estate, just outside the distillery; other fruit, like these apricots, he sources from farmers who grow fruit especially for him. \"I have to find farmers who are as crazy as I am,\" Reisetbauer tells me. He doesn't look crazy, he looks like fun\u2014like a retired actor, with an easy smile, graying hair pushed back, and shoulders as big as a bear's. \"Most customers want to see the perfect color of an apricot. Me, I want the perfect taste. I need the best fruit to make the best schnapps.\"\n\n\"It's the same in the kitchen,\" Kurt says. \"You have to work with the farmers to get what you want. See, Johnny? It's always the same.\"\n\n\"Once we have the perfect fruit, it's up to us not to make any mistakes,\" Reisetbauer says.\n\nSchnapps are not always pretty spirits\u2014some are just fiery moonshine, roughly distilled by farmers and drunk by same to fend off cold, fatigue and boredom (and forget the cinnamon-and watermelon-flavored liqueurs called schnapps in the U.S.\u2014they're completely unknown in Austria). But schnapps can be magical; clear, high-proof spirits, enjoyed after dinner, that somehow evoke through taste and smell the sensation of biting into a ripe fruit, at the peak of its season, right off the vine\u2014or better.\n\nReisetbauer's pear schnapps is better than any French poire Williams I'd ever had\u2014as the flavors spread across my tongue and waft up through my sinuses, I have the illusion of tasting a pear with the backs of my eyes. His elsbeere (\"serviceberry\") schnapps is the most expensive in the world; he needs more than 35 kilos of the rare fruit to make one liter of spirit. Last year, the only three bottles exported to the U.S. went to Wallse. It tastes like blueberry marzipan, in between layers of God's own wedding cake.\n\nAfter an amble through his orchards, it's time for lunch; we head to his kitchen in time to see his entire staff\u2014farmhands, still operators and marketers alike\u2014sitting down to eat with his young children. His wife plates slabs of the pork neck she's been roasting all morning with hunks of browned cabbage, herb dumplings and a good ladleful of dark brown jus. Everything is perfectly cooked, and the gem\u00fctlichkeit, that sense of warm hospitality in which all Austrians take pride, has never been stronger. We all help ourselves to glasses of apple juice from the orchards outside.\n\n\"Here's where I fished for semling, that fish you had the other day,\" Kurt says, standing on the grassy banks of the Danube in Wallsee, his hometown, a quiet village of about 3,000. \"Up there, by those reeds, I caught eel and catfish.\" While fishing as a child, Kurt would watch the river cruise ships pass by, some on their way to Vienna. He didn't know much about them, but he knew they had cooks, and they seemed as good a way as any out of Wallsee. In the 1970s, aspiring Austrian chefs didn't aim too high; if he paid attention and worked hard in culinary school, he might even make it to the kitchen of a hotel in Switzerland. Before Kurt, Wallsee's only claim to fame was a castle that was once home to Archduchess Marie Valerie, daughter of Emperor Franz Joseph. Now it can claim a famous American chef, too. This fall, Kurt's cookbook _Neue Cuisine_ will be published by Rizzoli, and this fall he'll return to Austria to receive the Decoration of Honor for Services to the Republic of Austria, a prestigious award given by the Austrian president to citizens who promote Austrian culture abroad.\n\nHis parents still live in town; so do his brother and sister. When we stop by his old family homestead, his mother, as hospitable and charming as her son, brings out a homemade plum cake and some coffee. I grill her for embarrassing stories from her son's youth, but she comes up short. \"When he was about 10, he made me a cake for Mother's Day. After dinner, his brother said, 'Kurt, where's the cake?' but Kurt just shushed him. It turns out he had hidden the cake under his bed.\"\n\n\"It didn't come out right! I had high-quality standards, even then,\" Kurt says.\n\nOur stay isn't long; we have dinner reservations in Vienna that night. Before we leave, Kurt carries in bags bursting with the bounty of Austria's fields, orchards, cellars and shops, and starts to unload them onto the kitchen table, despite his mother's protestations: the market sunflowers; sausage studded with pumpkin seeds; bottles of Gegenbauer's vinegar and Hartlieb's pumpkinseed oil; fresh apple juice from Reisetbauer's orchard; tomatoes from Stekovics's miraculous vines; and from the farmers market in Graz, a small mountain of apricots. \"We love it when Kurt comes to visit,\" his mother says. She beams with pride for her son, the chef, who has made good in New York and is the reason Americans have heard of Wallsee. Then she looks distractedly back at the kitchen table. \"But now I have to do something with all of these apricots.\"\n\n### REMEMBERING SAVOY\n\n### By Rachel Wharton\n\n### From _Edible Manhattan_\n\n### North Carolina native Rachel Wharton has become immersed in New York City food culture since earning a master's in food studies from New York University. A former food reporter for the _New York Daily News,_ she is now deputy editor _of Edible Manhattan_ and _Edible Brooklyn_ and a contributor to GiltTaste.\n\nWhen Peter Hoffman announced that on June 18 he would shutter Savoy\u2014the SoHo restaurant he has run for 21 years at the cobblestoned corner of Prince and Crosby\u2014Manhattan learned it would lose more than just a fine dining landmark, a longstanding icon of its culinary landscape. Hoffman, you may have heard, promises to open a more casual, up-to-date place in the space by September. But when Savoy's golden glow goes dark, with it goes the urban version of our very own hearth, our collective spiritual home\u2014the kind of place where people gather around the fire not just to eat, but to commune with kin and take the long view of life.\n\nThe hearth, after all, is a literal one: Until the end of next month, at least, the restaurant boasts two fireplaces, both used for cooking. The restaurant's many regulars will long mourn the loss of Savoy's cassoulet suppers simmering away inside the fire each fall in embertopped cast-iron Dutch ovens, and the shad feasts celebrating the return of that fish to the Hudson River each spring. For this year's shad dinner, Hoffman and his executive chef, Ryan Tate, nailed the boned and bacon-larded fish to planks inserted right into the flames the way the Colonists did. In true Savoy style, they served those fillets with smoked shad fritters and a charred spring onion aioli; a briny-sweet bite of house-pickled fish tucked between a buttery crisp of bread and a layer of green garlicky omelet; and a wedge of its creamy roe with brown butter cream in a lemony sorrel broth.\n\nIf you'd been paying attention at Union Square Greenmarket just a few days before, you'd have seen Hoffman tuck the sorrel into the back basket of his trademark giant tricycle, emblazoned with a sticker reading \"The revolution will not be motorized.\" That line is exactly the kind of idea those family-style, special-occasion dinners are meant to highlight. Savoy has long been Manhattan's place not just to share in one season's harvest and plan the next, but to critically evaluate what we eat, how we live and the American state of home economics, in the Wendell Berry sense of those words. (Berry, for decades regarded as the back-to-the-land poet laureate, has personally spoken during one such Savoy dinner, as have Michael Pollan, Alice Waters, Mark Kurlansky and just about every other living luminary in the farm-to-table world.)\n\nSavoy has always been a Greenmarket-showcasing pioneer, starting in 1990 when Hoffman and his wife, Susan Rosenfeld, opened its doors in what had been a shiny luncheonette. (And before that, a barber shop: The old painted pole still stands guard in the downstairs dining room.) Savoy has since steadied city souls with its \u00fcber-sustainable sustenance: A crusty wedge of real bread redolent with rustic local grains and slathered with cultured Vermont butter; a snarl of saber-toothed dandelion greens slicked with anchovy dressing and crowned with a slow-poached egg laid by a hen Hoffman may have even met.\n\nTrue, if farm-to-table awards were given out by how much product a chef buys from local farmers, Hoffman wouldn't win. Even combined, the volume at his two restaurants (Back Forty, his more casual spot, opened on Avenue B in 2007) couldn't come close to a place like Gramercy Tavern, which could serve a hundred hungry locavores at lunch alone. He wasn't Manhattan's first cook to obsess over Union Square, nor would he score first-place for flat-out farm-driven deliciousness, either. Which is understandable: Many of the best city chefs regard ingredients with integrity as a win-win, but their main goal is taste; for Hoffman, one gets the feeling it's the other way around.\n\nBut Savoy never sought to be a Fine Dining Experience, as Blue Hill or Del Posto or Gramercy Tavern all are. It's the kind of place to order a pint of local suds and a few fat slices of housemade mortadella at one of the city's loveliest little bars, or where your Eileen Fisher\u2013wearing aunt can come after her environmental book club for a wedge of silky, oil-poached wild-caught striped bass, perhaps paired with house-pickled ramps and spring's first spinach with walnut-mustard dressing. And Back Forty is decidedly down home, an Alphabet City outpost serving outstanding burgers, berry crisps and beer milkshakes\u2014dishes that Dan Barber, who long ago supplanted Hoffman as the locavore spokesmodel, would never offer on his extraordinary\u2014and extraordinarily refined\u2014menus.\n\nBut while Hoffman's fare may be more rustic than rarified, it's often wonderful\u2014and if you're hungry for a heaping helping of meaning, he serves up multiple kinds of satisfaction. That's because for him, farm-to-table is not a cooking style or a purchasing preference: It's a belief system that's just one piece of his progressive worldview. Ask him what he ate for lunch, and he'll likely connect the answer to healthcare reform or congestion pricing or this morning's op-ed about fracking in our foodshed.\n\nThis is a man who would never willingly miss a single day at Union Square Greenmarket, and not just for the grub. Other chefs may drop in for a frenetic Saturday spree before hailing a cab (or simply send their sous-chefs), but Hoffman's approach is to savor the experience. He'll park that giant trike and lean against box trucks and benches for hours, discussing pests, peapods and parenting, what the Japanese nuclear crisis means for Obama's energy agenda and whether it's possible to taste the difference between maple sap collected in metal buckets and sap that ran through plastic tubing. (Hoffman claims he can.)\n\nThere are hellos to the grad student working Flying Pigs' stand (ever the mensch's mensch, he asks after her dating life); a chat with the president of Abrams publishing company (\"your bar was the best thing about working at Scholastic,\" he says wistfully of his time in that SoHo office); and a stop to score ricotta at Dancing Ewe, a sheep dairy whose owners, like so many farmers here, credit Hoffman with their success. Hours later, the chef has still not bought the sorrel he originally came for.\n\n\"That's part of the whole point,\" says Hoffman of all these exchanges, \"there's a conversation that doesn't happen at Whole Foods.\" Here you don't just read a sign that tells you how much sea scallops cost; you spend 20 minutes talking with the fisher about the social-political-ecological meaning behind their place in history.\n\nThese are all ideas he's been pondering since adolescence. As a 16-year-old in Tenafly, New Jersey, in the early 1970s, he knew he wanted to do something \"intellectually stimulating but still physical,\" he says: \"I didn't want a desk job.\" Seeking work connected to the natural world, he considered forestry and biology, but was set on a culinary path by an unlikely friendship with a retired commercial fisherman and a shot of a chef clutching vegetables to his chest on the cover of _Time._\n\nThe fisherman was Chris Letts, now a Hudson River Foundation frontman in his 70s who long ago introduced the teenage Hoffman to his lifelong love of the sea and to the concept of foraging via Euell Gibbons' _Stalking the Wild Asparagus._\n\nThe chef was Paul Bocuse, the Frenchman famous for his farm-forward food. \"I think the title was _Cuisine du March\u00e9,\"_ recalls Hoffman of the _Time_ cover. \"I was like, 'that's it.' I don't think I even read the article.\" (Skimming is not standard practice for Hoffman, who's celebrated Proust with a four-course meal and counts _New Yorker_ writer Adam Gopnik as a friend.)\n\nHoffman decided to fast-track high school and took an entry-level job at a resort restaurant in Stowe, Vermont. The continental cuisine didn't thrill him but the kitchen energy did. \"I loved the theater of it, the community, the performance and the climax of Saturday night at 8:30.\"\n\nWhen the weather warmed, Letts landed him a fishing gig back home, plucking writhing shad from nets off 138th Street, when the waters were still alive with the herring cousin heading up the Hudson River to spawn. (Hoffman chronicled the experience in this magazine in 2009.) But before he could go back to cooking, there was college to attend, as per parental demands. So Hoffman enrolled at UC Santa Cruz, where he counted gray seals for a professor and learned about the perils of the green revolution before dropping out to cook at a local restaurant. He again loved the line but, ready to come home, ignored a friend's advice that he should go work at this place in Berkeley called Chez Panisse.\n\nInstead, back in Manhattan, he cooked at La Colombe d'Or, one of the first French restaurants in New York to serve rustic Proven\u00e7al cooking rather than haute cuisine. That sensibility appealed to Hoffman\u2014simpler food that wasn't \"Frenchie La French,\" as he puts it. So did John McPhee's 1979 _New Yorker_ piece Brigade de Cuisine, about a European-trained chef in a tiny town in Pennsylvania raising his own trout and cooking what grew nearby.\n\nWhen the opportunity to master that kind of cooking arose, Hoffman jumped at the chance. He had wanted to study in Paris with chef Madeline Kamman after reading her book _The Making of a Cook_ but her classes were full. \"I got a phone call on the pay phone at [La Colombe d'Or],\" he recounts: '\"There's been a cancellation,' a voice informed him. 'If you can be in France in two weeks, you can be in Madeleine's class.'\" He made it, and spent the last of his college money on three months cooking with Kamman, whose core concepts have formed the framework of his cooking since: that regional foods were rooted in geography and social history, and that the true way to cook was using, he recalls, \"what was in the moment and what was in the market.\"\n\nThose seem like no-brainers now that everyone has turnip tattoos, but at the time, the approach was just taking root stateside, coalescing as something called New American cuisine. Back home, Hoffman immersed himself in it, reveling in the food of trailblazing restaurants like An American Place, originally opened on the Upper East Side by chef Larry Forgione\u2014\"I ate my first morel in his restaurant,\" Hoffman dreamily recalls\u2014and the Quilted Giraffe, the quirky French farmhouse-y spot in Turtle Bay where Hoffman cooked alongside Ray Bradley, who would go on to be one of the Greenmarket's most beloved farmers. He moved to the kitchens of Hubert's on 22nd, a townhouse turned New American bistro where he met Susan Rosenfeld, who would eventually become his wife.\n\nThey knew they wanted to open a restaurant, but first he would complete two more stints abroad\u2014one in Provence with Richard Olney, the Iowa-born author of cookbooks on French country fare, the second at a restaurant in Japan. (\"I saw an aesthetic that would bring the natural world to the plate,\" says Hoffman of the latter trip, counterbalancing the French penchant for what he calls \"manipulation of the natural world: dots and cubes and brunoise.\")\n\nHoffman married Rosenfeld in 1988, and they set out in search of a space to open the restaurant they'd been imagining together. When they stumbled on that shuttered SoHo luncheonette, Hoffman looked up the lease in the city's records. It was coming up for grabs, and thus Savoy was born in a fledgling artists' outpost in the wasteland east of Broadway, a lifetime before direct trade macchiatos proliferated and shoppers' stilettos provided the sidewalk soundtrack. Susan worked the simple, 40-seat dining room, behind which guests could glimpse Peter in the tiny kitchen. He made connections with local farmers and fishers and foragers, cooking their harvests into dishes like marinated halibut with cucumber salad and braised duck with Concord grape puree. In 1995 they annexed the upstairs apartment, got a liquor license, and put in a bar and a working fireplace. The Times noticed, and critic Ruth Reichl awarded the restaurant two stars.\n\nHoffman was still Savoy's sole chef, and his handiwork would go on to become signatures of a scene: He has an \"instinctive understanding of vegetables,\" Reichl wrote, a way with salads (\"you will instantly be seduced\"), plus a penchant for breaking rules: \"Who would expect that rosy slices of grilled peppered tuna in a vinaigrette based on the classic Catalan romesco sauce (ground nuts, peppers and tomatoes), would be served with good old American onion rings? Fabulous onion rings, I might add, made of sweet red onions.\" Reichl nailed the feel of the place itself: \"The small restaurant is so casual, so comfortable and so unpretentious,\" she wrote, \"that it is hard to believe it is in Manhattan.\"\n\nMany restaurateurs have taken the same approach, and several have done it better, but Hoffman's serious study of farm and food issues remains a rarity. Which makes it easy to see why, when Frank Bruni gave Savoy its second glowing two-star _Times_ review in 2009, he called the restaurant New York's Chez Panisse, an East Coast answer to Alice Waters's Berkeley-based birthplace of the good food revolution.\n\nBoth Waters and Hoffman drew inspiration from youthful tours of France and Italy, but both ultimately formed consistent allegiance to Americana\u2014from half-wild cress foraged on a Hudson Valley riverbank to wild salmon caught by Native Americans off the Alaskan coast. (\"It's a delicious fish, and it's supporting indigenous fishing communities,\" says Hoffman, for whom the latter is as important a trait as the former.) Both kitchens have served as springboards, their line-cook lineage traced like one big family tree across the country's locavore landscape. Hoffman's culinary offspring all but created what _New York_ magazine dubbed \"New Brooklyn Cuisine:\" Minds behind RoseWater, Diner, the Grocery and Franny's were all sparked at his stoves.\n\nBut moreover, like Waters, Hoffman has long been happy to leave the stoves to hired hands like Ryan Tate at Savoy and Shanna Pacifico at Back Forty, freeing him up to save the world. While others pursue television spots and lucrative advertising sponsorships\u2014Hoffman has turned down both\u2014this guy is after another kind of action, the kind whose payoff is a different kind of change.\n\nHe's always been much more than a chef, even back when he was still the one cooking. The man spent 10 years as the only chef on the Greenmarket's advisory board, helping guide and grow the city's market system to the marvel and model it is today. He is a founding member of Chefs Collaborative, the national network of professional cooks working to teach their colleagues where, how and why to find sustainably sourced products. And when he's not considering the concept and a name for his new restaurant\u2014it will balance his philosophy with a price point and feel that fits Soho's touristy clientele, he says\u2014he's hard at work on a marketdriven memoir (with recipes) that we can't wait to read.\n\nThose projects are only possible when you delegate the day-to-day: Having traded his whites for brown cords and button-downs, he still shows up at both restaurants nearly every day with new product and new ideas, but insists on changes only when something's really wrong. (Missteps, like a basil cocktail from a new bartender in February, are pretty rare.) \"I'm not the chef anymore,\" Hoffman freely admits, \"but I'm still the culinary director. I hold out a set of guidelines by which the restaurants operate.\"\n\nThose go beyond sourcing to stuff like caring about your staff: \"One good thing about Peter,\" says Tate, \"is he wants his employees to have a life.\" For the most part, he lets Tate and Pacifico do what they want if they follow his heart. \"It doesn't have to be his food,\" says Tate, \"it just has to be his philosophy.\"\n\nThis summer he'll focus that philosophy at Back Forty, which he named after the hidden quarter of land on the Midwestern parcels doled out by the Homestead Act of 1862. The government gave away 120-acre blocks, Hoffman explains, sketching rectangles on a scrap of paper, and the quarter farthest from the road was most likely to be left wild and wooded. It was also the place, he adds with a smile, \"where you would go to make out.\"\n\nThat's maybe true of his own Back Forty, too, thanks to its sweet backyard, communal tables and that laid-back menu\u2014\"great ingredients in casual delicious ways,\" as Hoffman puts it\u2014that will likely inspire what happens at the space where Savoy now stands.\n\nBack Forty's menu is largely driven by the butchery skills Pacifico mastered in order to break down whole carcasses from Fleisher's Grass-Fed Meats upstate. While her vegetables shine\u2014roasted roots and tangles of the sweetest spring greens\u2014Back Forty is a carnivore's delight: fried pork jowl nuggets with pepper jelly; fat housemade sausages that change nearly every week; a terrine with stoneground mustard; grilled flatbread made with lard and topped with a mouthful of pig's trotters, bacon, melted onions and thyme. Yet Back Forty is best known for Pacifico's contender for best-in-city patty, transformed from trimmings of her two weekly deliveries of a half steer, served on a buttered sesame bun with spicy ketchup and a housemade pickle. (Tomatoes, too, says Pacifico, but only in season).\n\nDespite the handiwork of his chefs, Hoffman is still master of ceremonies, especially during those idea-centric fireside feasts held for 18 years at Savoy. At those family-style nights of dinner and discussion, all manner of tastemaker\u2014from Betty Fussell to Joel Salatin to Stephen Jay Gould\u2014have held forth in front of the upstairs fireplace and, between each course, discussed the fate of local fisheries, the flavors of the Riviera, the follies of the Farm Bill or the forestry skills of Umbrian truffle hunters. Way back in 1995, Michael Pollan spoke about the history of apples in America\u2014a decade before his Omnivore's Dilemma would hit nearly every nightstand in the nation.\n\nAs much about philosophy as they are about food, those communal hearth-side chats are what we'll miss most about Savoy, those nights when the place transcended typecast and become nothing less than a salon. Savoy's final shad dinner this March featured a lecture on the species (and a recipe for the pickled fish) by none other than Chris Letts, who took Hoffman fishing for shad back in 1974 and put him on the path to Prince Street. When he and Hoffman took the floor that night it became a pulpit, as they re-inspired eaters that when it comes to voting with your fork, hell yes you can, and hell yes you should.\n\nStill, while Hoffman is often called the ur-locavore, he doesn't identify himself as such: \"It's a simplistic look at an extremely complex topic,\" he says of the term. Citrus, olives, chocolate and countless other foods from afar share pride of place on both his menus, just as long as his sustainable sensibilities\u2014that those foods come from independent groves or eco-enlightened fishers\u2014stay true. Moreover, Hoffman claims the question of what to call his culinary philosophy isn't what matters. \"You don't think Jackson Pollock called himself an experimentalist, do you? He was too busy painting. I could talk about it,\" he says of his life's work, \"but mostly I was just doing it.\"\n\nAs it happens, for once we disagree. It's not so much Hoffman's cuisine that draws us back time after time, or Savoy's menu that we will miss most, but his fireside wisdom, and his longstanding role in illuminating our place in the world. \"Savoy has been my life for the past 21 years,\" he says wistfully, \"that place, that style and all the rest . . . but now I am getting excited about what it will become.\" If that includes conversation by the hearth, then we are, too.\n\n### APPETITE FOR PERFECTION\n\n### By Ed Leibowitz\n\n### From _Los Angeles Magazine_\n\n### Writer-at-large Ed Leibowitz has profiled everyone from actor Charlton Heston to controversial defense attorney Gloria Allred for _Los Angeles Magazine._ He has also written about culture and politics for _The Atlantic, Smithsonian, Men's Journal, Money, BusinessWeek,_ and _the New York Times._\n\nRobyn Sewitz is almost done flipping through the latest _Bon App\u00e9tit_ this Sunday afternoon when she makes a discovery she just has to share with her son. Brushing back a lock of her auburn hair, she calls to him across her spacious kitchen. \"Jon,\" she says, \"here's boar hunting for beginners!\"\n\n\"Oh yeah?\" he says, halting his knife's progress. Several weeks shy of his 16th birthday, Jon is tall and lean, his full cheeks not yet ready for their first serious shave, but after all those months he's spent studying cookbooks, mastering culinary modernism, and apprenticing in professional kitchens, he's a more capable cook than either of his parents will ever be. Jon's a sophomore at the Oakwood School, a private progressive academy in North Hollywood. His mother is a psychotherapist, while his father, David, designs furniture and window treatments for clients of enormous net worth.\n\nLike many Oakwood parents, the Sewitzes regard their child's artistic ambitions not as some passing teenage fancy but as a creative flowering that could lead to great things. When Jon thought he wanted to be a musician, they signed him up for lessons and invested in a Bellafina double bass. He plays in the school orchestra and jazz band, but much of the time now his instrument sits on its side in the deserted music parlor of their Encino home, like a dejected mastiff.\n\nJon's food epiphany came during a family trip to Spain a couple of years ago. Robyn wasn't trying to make a chef out of him when she booked dinner reservations at El Celler de Can Roca, which earned three stars in the 2011 Michelin Guide. She wanted to expose him to something that would pique his interest more than the art museums were. Jon's palate still carries the sensations of that meal: oysters served in the bottom half of a wine bottle with carbonated _cava_ sauce; eggplant souffl\u00e9 wrapped in white sardines; a lineup of mussels, one bathed in bergamot foam, another in nectarine jelly and caramelized rose petal, and another in \"distilled earth jelly,\" a clear sauce derived from a dab of mud that was boiled for hours at low temperatures in an evaporator. \"It tasted like dirt you'd try when you were a kid,\" he says, \"when it didn't taste bad.\" Dessert was even more spectacular. \"It had apple in it, cinnamon, and vanilla cr\u00e8me,\" he says. \"They brought out a DKNY perfume bottle and had us taste the dessert as we smelled the perfume, and the dessert tasted exactly like the perfume's aroma.\" In nine courses Jon was transformed. \"I would always go back to that experience in my mind,\" he says, \"how food could be so amazing you could remember it forever.\"\n\nAs Jon continues with his knife at the marble-topped kitchen island, he's joined by 17-year-old Sam Yehros and 18-year-old Macklin Casnoff, both seniors at Oakwood. They dice and pulverize, clarify and puree, strain and scour. For almost two years the friends have been collaborating on original multicourse dinners, charging only for ingredients, that they prepare for special occasions\u2014a brother pushing off to college or a mom celebrating a birthday along with a tableful of relatives and friends.\n\nIf there were any single event that brought these three together as a cooking collective, it would be the Oakwood winter immersion program of 2009. While other students headed out with teachers to roam China or photograph Death Valley, Macklin found himself with 15 classmates in a cabin in Utah contemplating Euclid's Golden Ratio. They cross-country skied and discussed whether the ancient Greek concept of beauty and proportion could apply to their world. Macklin's world had been sharks, then skateboards, but lately he had rekindled an old passion for food. Back home, when he wasn't watching _Iron Chef,_ Macklin and his friend Henry Kwapis would explore how to apply the Golden Ratio to fine cuisine. They tried cold-calling some of the best chefs in L.A. to see if they could come in and ask them about their craft. Jos\u00e9 Andr\u00e9s, meeting them at his Beverly Hills restaurant the Bazaar, was so impressed by the philosophical sweep of their questions that he answered in a 45-minute stream of consciousness, punctuating his thoughts with a liquid olive and molecular _caprese_ in a pipette. \"Look at the light above your head,\" he commanded, loud and jolly. \"Now try to eat it. This is how I look at food.\"\n\nWhen Jon heard about what Macklin and Henry had been doing, he asked to tag along for their talks with Michael Cimarusti at Providence and David Myers at his now-defunct dining room, Sona. Before long Jon was an unpaid apprentice in Sona's kitchen, and Macklin was learning by Cimarusti's side. As the two boys began cooking meals for friends and family, Macklin started experimenting in the kitchen with Sam, his Hancock Park neighbor, who had his own apprenticeship with Neal Fraser of Grace. Within a few months the two operations merged.\n\n\"See, look how cool,\" says Jon's mother, holding up the magazine article, which has shots of the hunter and roasted pork, but no dead pigs. \"This is a vacation you should take. You can get in touch with your inner Michael Pollan.\"\n\n\"What?\" says Jon, distracted.\n\n\"You haven't read _The Omnivore's Dilemma_?\" asks Sam, referring to Pollan's book, a James Beard award winner that ends with the author trying his luck at hunting and gathering.\n\n\"I did,\" Jon says.\n\n\"You didn't read all of it,\" Macklin says with a smile.\n\n\"Yeah, I did.\"\n\n\"You admitted to me you hadn't read all of it.\"\n\n\"Well,\" Jon says, \"I read a _lot_ of it.\"\n\n\"Jon is dying to kill an animal, which I don't approve of,\" Robyn tells me. \"I don't like the idea of holding a gun and killing anything, to be honest. It goes against my whole belief system, but it's better than suffering on one of those terrible farms.\"\n\nOften Sam, Macklin, and Jon seem more like they're members of a teen rock group than three exceedingly capable cooks. One moment they'll be chomping blueberry Airheads or playing basketball on the Xbox or downloading Toto's \"Africa\" from iTunes just to goof on it. The next they'll be transfixed by a YouTube video of Chicago chef Grant Achatz demonstrating his solid-sauce technique or discussing the radical foraging philosophy of Ren\u00e9 Redzepi, the chef at Noma restaurant in Copenhagen. Like a band, the boys have given themselves a name: Samacon (Sam, Macklin, and Jon squished together). For their gigs they even bring in backup players\u2014Henry, who's 17, and another Oakwood friend, Brendan Garrett, who's 16, to help out as sous-chefs.\n\nUnlike teen rockers, though, the Samacon chefs have taken up an art form thoroughly rooted in the adult world and have mastered it with an idealism and fellowship that usually disappears with age. They aren't cocky about their work or egomaniacal\u2014they are constantly challenging each other but never competing for supremacy. I could tell they knew they were good, but they had no way to measure _how_ good. The friends and relatives at their dinners were cheering for them regardless of what came out on the plate, and the chefs they've worked for hadn't tried the boys' creations.\n\nSo one afternoon over burgers I asked them if they'd like to put together one of their nine-course meals for the chefs they admire most. \"It would be nerve-racking,\" Jon told me, \"probably the most nerve-racking thing we've ever done. But it would be amazing if we could cook for great chefs.\"\n\nAfter wresting a blank check from my editor to pay for ingredients, I asked the boys for their ultimate guest list and began making calls. Five agreed to attend: Nancy Silverton of La Brea Bakery and Osteria Mozza; Vinny Dotolo and Jon Shook of Animal and the recently launched Son of a Gun; Fraser, of Grace; and Ludo Lefebvre, creator of LudoBites pop-up restaurants, an occasional judge on _Top Chef,_ and starting next month, the star of his own reality show on the Sundance Channel. Now all the boys needed to do was pull off the best meal of their young lives.\n\n**A La Carte**\n\nA week and a day before their big dinner, the Samacon chefs gather at Jon's kitchen table to hammer out their menu. They agree on the squid kimchi _amuse-bouche_ ; the scallop with radish and black sesame; the hay-roasted, kale-wrapped pork medallions; and a whiskey-tangerine-nutmeg palate cleanser. Consulting the list in Macklin's disintegrating Moleskine notebook, they even agree on the raw oyster with bittersweet chocolate\u2014admittedly a risk. They only have to settle on one more entr\u00e9e. To Macklin, what their meal lacks is a strong narrative, and he has a solution: \"Chefs, I think, are less concerned about being served the type of food they might be doing in their own restaurants,\" he says. \"I feel like a chef's favorite thing to eat is, like, a roasted chicken or a fatty piece of pork.\"\n\n\"That would be _so_ good,\" says Sam, his brown eyes almost moistening behind his horn-rimmed glasses. It's as if he's just turned 70, not 17, and has bitten into Proust's madeleine. \"My mom makes roast chicken, and then you add quartered sweet potatoes and then potatoes and carrots, and the fat from the chicken soaks them as you're roasting.\"\n\n\"We roast a chicken and bring it to the table,\" Macklin says, \"and they pull it apart. It's not like we're serving people who want everything to be done for them. Chefs love to get involved.\"\n\nJon is aghast. \"We'd have to carve it for them,\" he says with the crumpled brow of somebody whose universe is near collapse.\n\n\"No, we don't,\" Macklin says. \"I see one of the chefs cutting it. It can be a communal thing, and we can even be out there talking because that's sort of where there's a little bit of a break in the meal.\"\n\n\"Yeah,\" says Jon, \"but they're not coming to this dinner for that. They're expecting teen chefs who've worked at some of the best restaurants. I know it's cool, but there's no restaurant that has beautiful, sophisticated plating, and suddenly it goes family style and back to beautiful plating. The menu _is_ all about a story, and it has to flow. If it doesn't flow, the diners are not happy, and they just leave confused.\"\n\nLike many Macklin concepts\u2014the seemingly impossible wheat grass puree he executed for their last dinner, the rosemary soda company that hasn't gotten off the ground\u2014his homey chicken interlude could lead to counterintuitive triumph or the abyss. Jon brings to the enterprise his expertise in modernism and molecular gastronomy, and Sam, his more traditional approach. But Samacon wouldn't be what it is without their combined talent for managing Macklin.\n\n\"I believe,\" Macklin says, \"that doing a dish that's family style and the most simple, perfect thing in the world would show our reverence for what food is. What food means. The fact that it brings people together.\"\n\nEventually Macklin loses his ally. \"If we're trying to make the whole meal communal,\" Sam says, \"then we have to change the entire menu.\" Instead they opt for _loup de mer,_ which will evoke the simplicity of country French cuisine, though Macklin has trouble letting go. \"I don't want to argue about roast chicken anymore,\" he says. \"Because I have very strong feelings about roast chicken. OK?\"\n\n**Locally Grown**\n\nHis generation of L.A. teenagers has taken to food like no other before it. They've grown up in a messy, polyglot city where often only the shared experience of eating Vietnamese _pho_ and Salvadoran _pupusas_ and Ethiopian lamb cakes can seem capable of holding everything together. They've spent countless hours watching the culinary blood sport of _Top Chef_ and _Iron Chef_ and (for those whose parents can swing it) learned their way around the menus of high-end restaurants. They'll opine authoritatively on Yelp about tripe soup just as quickly as they'll wince when Grandpa asks the waiter what the hell a _pappardelle_ is.\n\nBut Sam, Macklin, and Jon are a category unto themselves. Of the five guests at their chefs' dinner, only Ludo Lefebvre hasn't met at least one of them before. Shook and Dotolo got to know Jon, Macklin, and Henry the way many other L.A. chefs did: a series of phone calls and e-mails, unanswered in this case, followed by a teenage siege on their restaurant. \"We kind of blew it off at first,\" says Shook. \"Then we saw a couple of bicycles parked outside. 'Like, it's so cool that somebody stopped in on a bike ride here,' we thought, and the manager was like, 'Oh, it's those kids.'\" Shook and Dotolo invited them into the kitchen to observe and help out. By closing time there was a steady rain. \"And I was like, 'Put the bikes in the back of the Jeep, and I'll drive you home,'\" Shook recalls.\n\nJon and Brendan, the Samacon sous-chef, met Nancy Silverton through Oakwood. They're close friends with her son, who also attends the school. In fact, they're all spending time together with Silverton at her house in Umbria this summer as soon as Jon and Brendan finish the two-week apprenticeship she's arranged with Dario Cecchini, Italy's most famous butcher. \"Jonathan Gold is going to be in town working on a _Saveur_ article they're doing about Umbria,\" she says of the food critic. \"So it will be fun for those guys to eat with Jonathan.\"\n\nMacklin may meet them there, too. The Silverton influence reaches far back into his past, when he carpooled with her son to kindergarten. At the time Silverton and Mark Peel were married, running Campanile and La Brea Bakery together. Picking up Macklin in the morning, they were already well into their workday. \"They would come from the fish market or the vegetable market,\" he says. \"I was always interested in their double life.\" When Macklin was 13, he bumped into Peel at school. \"I was like, 'Hey, I've been thinking about it,'\" he remembers, \"'and I'd love to come in and work.'\"\n\n\"He spent one day peeling garbanzo beans,\" says Macklin's father, Philip Casnoff, an actor who had his breakthrough playing Patrick Swayze's nemesis in the 1985 miniseries _North and South. \"I_ never saw him look so exhausted in his life. His second day he cut his finger badly, and after the third day he was gone.\"\n\n\"I was too young to appreciate it,\" Macklin tells me, \"and I didn't force myself to stay there long enough to really get it. I thought, 'This isn't for me\u2014it's too much.' I really loved it, though.\"\n\nHe's had an easier time working on Sundays with Cimarusti at Providence. \"Coming out of cooking school, three or four years into a career, people can be making great progress, but they may also have developed bad habits,\" says Cimarusti, who will be in France the night of the chefs' dinner. \"They believe what they've learned so far is gospel. But Macklin came here to be programmed, and whether it's the sous-chefs teaching him or myself, we've molded him to be exactly what we wanted. Also, he realizes quality\u2014what are good ingredients, what are great ingredients, and what are bad ingredients\u2014and that's something that doesn't come naturally to most people.\"\n\nNeal Fraser says much the same thing about Sam. \"I've had people who've come in from culinary school who were very game, then they'd get their Gucci loafers dirty,\" he says. \"With Sam, he'd burn or cut himself, and all you would see was him in the corner taping himself up.\" From that first conversation, when Sam cold-called Fraser, they hit it off. \"I rode my bike down to Grace,\" he says. \"Neal had told me to ask for Jason, the sous-chef. I got there at about two o'clock and started chopping carrots.\" He began as a prep cook, then graduated to garnishing salads before Fraser let him work the fish station and the grill.\n\nSam's success in landing a job at Grace shocked his father. \"When he said, 'I'm going to call these restaurants and ask for an internship,' and mentioned some highbrow ones in Los Angeles, I just rolled my eyes,\" says Ilan Yehros, who waited tables and worked in the kitchens at several restaurants in Toronto. \"There's no way in hell that somebody would let a 15-year-old in with all the knives and equipment around. Kitchens are busy places. No one has the patience.\"\n\nIlan still liked to cook after he became a banker. He would wake up at four in the morning and pad into the kitchen to check his beef stock for some bravura French meal weeks in the making. Back then Sam and his siblings would occasionally serve their parents breakfast in bed and cook simple dinners. Without cable TV, Sam didn't have the Food Network for inspiration, so he turned to his father's copy of _Larousse Gastronomique,_ the classic-French culinary encyclopedia.\n\nDuring the Yehroses' vacation in Provence last summer, Sam and his dad took a seven-and-a-half-hour excursion to chef Edouard Loubet's two-star Michelin restaurant at the hotel La Bastide de Capelongue. \"I wanted Sam to be enlightened about food,\" says Ilan, \"to understand it as a chef clearly at the top of his profession understands it.\" They got there early, and Loubet gave them a tour of the kitchen. Sam worked his way through 14 courses on a hilltop patio, awestruck. \"The first course was a really light vinegary soup and escargots,\" he tells me. \"It was amazing, with this beautiful bouquet of tiny flowers on the edge of the bowl.\" He loved the snails and the frogs' legs but not the veal kidneys. \"I could only eat one and a half of them,\" he tells me. \"It's an acquired taste.\"\n\n**Garnishing**\n\nOne more day until dinner is served. The Lakers are fighting hard against the Celtics and their own lethargy, which isn't good for saut\u00e9ing vegetables. \"Henry,\" Jon yells. \"You haven't been watching your oil, and now it's burnt.\" Henry springs from his seat in the den and sprints to the Thermidor range in his skinny jeans and distressed leather shoes. The damage turns out to be minimal. They all watch part of the game during lunchtime, when Brendan, gentle faced and built like a linebacker, refuels everybody with helpings of his home-cooked stew.\n\nEarlier this morning the chefs went shopping at the Hollywood Farmers' Market. They knew right where to find the vendors with the plumpest Kumamoto oysters, the most recently harvested pea tendrils, and were gone within an hour. From there they visited McCall's Meat and Fish Shop in Silver Lake, whose owner had cooked in several highly regarded restaurants before becoming a butcher.\n\n\"I swear,\" Macklin told me earlier, \"McCall's is like the best butcher shop in the world, probably.\" At which point Sam offered a little perspective. \"They are not the best butcher shop in the world. There are so many instances where Macklin says, 'This is the _best_ thing I've ever had.'\"\n\n\"I was just excited,\" Macklin explained.\n\n\"You just haven't had a lot of life experience,\" Sam said.\n\nAll week, with lesser ingredients, the chefs had rehearsed almost every dish. Today they work the kitchen with practiced efficiency and no trace of nerves. They don't use measuring spoons or timers, and aside from Henry's incident with the oil, they don't lose any ingredients to mishaps or crossed signals. Jon, Samacon's youngest and most focused chef, assigns Henry and Brendan tasks that send them scurrying across the bamboo floor. Sam tosses two scorched eggplants into Jon's Vitamix 5200 and grinds them to a sweet, smoky paste. He tries a teaspoonful, letting it settle on his palate. \"Needs sherry,\" he says.\n\n\"Don't put too much sherry in it because of the red pepper,\" Jon warns.\n\nMacklin drops a slotted spoon into a tall, steaming pot and captures a sweetbread. He gives it a jab with his finger and then throws it back in.\n\n\"They're being poached,\" Henry tells me, his blond bangs wilting in the steam. Then turning to Macklin, \"Did I use the right terminology?\"\n\n\"Poached in lime and lemon juice and chicken stock,\" says Macklin. He's dressed in a vintage short-sleeved shirt with thick red and white stripes\u2014standard issue for selling saltwater taffy on the Atlantic City boardwalk in 1952. \"This is how they do sweetbreads at Animal,\" he explains. \"They said that when you're pre-blanching sweetbreads, you have to cook them to the consistency of. . .\" he trails off.\n\nOf what? I ask him.\n\n\"Silicone breast implants.\"\n\n\"Macklin, Jesus,\" Jon says.\n\n\"I had visions of lots of things other than that, when he was so circumspect about talking about it,\" says Jon's mother, Robyn, at her usual post in the kitchen. The boys have slept at her house for the past couple of nights. Twice they've tried to watch the Disney movie _Ratatouille_ on Jon's flat screen, and both times they've fallen asleep before Remy the Rat becomes the greatest chef in all of France.\n\n**Table for Five**\n\n7 p.m. The table has been set in the Sewitz dining room, with its polished limestone floor, antique Chinese doors, and high-backed seats designed by Jon's dad. To psych themselves up, Sam, Macklin, and Jon are sporting white headbands, the kind favored by Ginsuwielding chefs at those rock-and-roll sushi palaces of the '90s. The look clashes with the Zen-like tranquillity they've maintained as they've prepped the dinner. They might be more anxious if they were able to read the thoughts of the guests now trickling in for the meal.\n\nNancy Silverton is the first to arrive, in chunky eyeglasses and a dark coat, her hair a mass of curls gathered tightly on top of her head. She plants a big kiss on Robyn's cheek and gives the boys a maternal hug. All the while she's puzzling over how she's going to give the boys honest feedback about a meal she doesn't expect to be spectacular.\n\nA few minutes later come Vinny Dotolo and Jon Shook, hirsute and stocky in respectable dark sweaters. Dotolo has been worrying about the food as well. On the ride over he'd asked Shook what they should do if the meal bombs. \"I think it's going to be pretty good,\" Shook assured him. \"They didn't do that bad when they were in the kitchen at Animal.\"\n\nThen Ludo Lefebvre strides in, the rolled-up sleeves of his plaid shirt revealing a riot of tattoos, his hair thick and stylishly cut. As the boys awkwardly shake his hand, he's not sure what to make of them. He just knows that he wouldn't have felt comfortable cooking for a bunch of chefs when he was their age. In fact, Neal Fraser is the only chef who comes to the dinner table with high expectations.\n\nThe guests don't have much time to settle into their seats before Sam, Jon, and Henry swoop down in dive-bomber formation with the amuse-bouche. Brendan follows, filling the glasses halfway. Sly and the Family Stone play on hidden speakers. \"So this is sort of an amuse,\" Macklin says with an adrenaline grin. \"It's squid marinated in kimchi, and there's a little bit of kimchi at the bottom. There is lime zest and parsley, and then we have unfiltered sake. I don't know much about pairing, but we went to a wine place, and they told us that this would go well with it.\"\n\nThe chefs wait until the boys are safely back in the kitchen before anyone gives the dish a try. Silverton scoops up the squid, finishes it in a bite, and scans the table for reactions. _\"All right,\"_ she says. \"I'm impressed.\"\n\n\"As an amuse,\" Lefebvre says, \"it's good.\"\n\nThe mood dampens with the arrival of the second course, which Jon introduces as a Kumamoto oyster with buttermilk sauce, a disk of dark chocolate, radish, and fennel fronds on top. The guests spend a few moments contemplating the glistening bulk on their plates with the gallowslike expression of that Dr. Seuss character compelled to try a mouthful of green eggs and ham. \"The first time I cooked an oyster for my family,\" Lefebvre says, \"I was 14 years old. I didn't prepare the oysters very well. I kept them out all day.\"\n\n\"And everyone got sick?\" Silverton says, laughing. Lefebvre gives her a sheepish nod.\n\n\"Are you trying to warn me?\" Fraser asks Lefebvre, looking up from his dish. \"Is that what you're telling me?\"\n\nWhen they finally put it in their mouths, most seem astonished to find that the combination isn't awful. Fraser is almost enthusiastic. \"The fattiness of the chocolate is completely different from the fattiness of the oysters, you know what I mean?\" he says. \"Usually you just swallow oysters. Because of the texture of the chocolate, though, you can almost chew them together.\"\n\n\"You know what I appreciate?\" says Silverton. \"That the chocolate is so thin.\"\n\n\"I don't taste the chocolate,\" says Lefebvre. \"I don't know about the dish, but I like the risk. I love risk in cooking. And it's not bad.\"\n\nThe third course\u2014cubed scallop, radish, and black sesame\u2014is a hit. \"It's cool that they have that edge, to think of that combination at this age,\" Lefebvre says. By the time she's tried the sweetbreads with burnt eggplant puree, almond marshmallow, and two-toned parsley bread crumbs, Silverton is verging on exuberance. \"These bread crumbs are beautiful,\" she says. \"And their technique from start to finish is spot-on.\"\n\n\"It's a solid dish,\" Lefebvre says. \"I have some cooks working for me for a long time\u2014I don't know if they could do a dish like this. And the sweetbread is cooked perfectly. It's very pink inside and crispy. Perfect! To see this technique at this age . . .\"\n\n\"You know what I think we need? We need a photo of our plates,\" Silverton says. She waves over the photographer who's shooting the meal for this story. Turning to me, she says, \"We're not being polite.\"\n\n\"I am from France,\" Lefebvre says. \"I'm _not_ polite.\"\n\nIn the kitchen there's no clattering, no commotion, no voices raised in panic or frustration, no rushing feet or bodies bumping into each other. Macklin rests his knuckles against the loup de mer on the cooling rack. \"The fish is done,\" he says. \"The skin could be a bit crispier.\" He gently lowers each fillet back into the frying pan. The seasonal vegetables in their butter glaze are bright as protoplasm beneath the lights. Sam, Jon, Brendan, and Henry plate them in a tight circle. The loup de mer is placed atop, slightly cantilevered, skin side up.\n\nIt's fallen to Jon again to introduce the dish. \"On the bottom is our attempt to have a traditional French sauce, and it's a sauce _bercy_ ,\" he says, pronouncing it \"ber- _say_.\"\n\n\"It's what?\" Lefebvre asks.\n\n\"Ber- _say_ ,\" Jon says.\n\n\"Ah, ber-see,\" Lefebvre says. Then, like a schoolteacher: \"Remember it the next time when you speak French.\"\n\n_\"Oui, oui_ ,\" says Jon, backing out of the room.\n\n\"So why do you think he said 'attempted' to do a sauce?\" Silverton asks after the boys have gone.\n\n\"'Cause Ludo's sitting here,\" Fraser says.\n\n\"It's cool that they do a classic like this,\" says Lefebvre. \"It's well balanced. You have molecular dishes and then a little classic. And what I'm impressed with is really their technique, and the fish is cooked perfectly. The skin is crispy.\"\n\n\"The vegetables are cooked really nice,\" says Dotolo.\n\n\"I like this even better than the sweetbreads,\" says Shook. \"I wish I could get a dish this caliber at most restaurants in the city.\"\n\n\"This city?\" says Silverton. \" _Any_ city.\"\n\nThe dishes the boys had cooked during their practice sessions looked beautiful, and what I'd tried tasted pretty amazing. But just as Silverton came to this meal worrying about the euphemisms she'd have to come up with, I was concerned about having to hear praise that would betray itself as little more than a pat on the head. Nonetheless this is getting ridiculous\u2014five of the city's leading chefs cleaning their plates, raving about the food, and as best as I can tell, meaning every word. Seeking shelter from the hosannas, I return to the kitchen. Sam asks how they like the meal. After I repeat some choice quotes, the boys are hugging each other and high-fiving.\n\n\"We should all stick together,\" Sam says. \"I don't really want to graduate anymore.\"\n\n\"I just want to cook for chefs,\" Jon says.\n\n\"I just don't want to go to school tomorrow,\" Macklin says. \"Let's just open a restaurant.\"\n\n\"Yeah,\" Sam says. \"Can we open a restaurant now?\"\n\n\"Hey,\" says Henry, \"we're getting way ahead of ourselves.\"\n\nTonight will be the culmination of their work together. Jon, though he'll have two more years to decide, is thinking of skipping higher education, since it would be too big a detour from his goal of becoming a professional chef. This fall Macklin is bound for Bard College, in the Catskills, and Sam might take a year off to travel before beginning at Reed College, in Oregon. They may still end up opening restaurants, or they'll be pulled away by fresh possibilities. Jon will be getting his driver's license soon, able to transport himself to a new restaurant job\u2014wherever it may be\u2014on weekends. But it seems certain that none of the chefs they've worked under or might work under, and none of the other friends they've made or will make, are likely to have nearly the same impact on their creative growth as they've had on each other.\n\nThe next dish\u2014hay-roasted pork medallions with coleslaw, minimalist barbecue sauce, and mahogany-tinged roasted potatoes on a cat-eye-shaped plate\u2014seems to flow right out of them. So too the whiskey-tangerine-nutmeg palate cleanser and the hazelnut streusel with coffee soil, hay ice cream, and caramel broth. In the dining room Fraser admires the spare Scandinavian beauty of the final course: pear sorbet, sponge cake, apricot puree, lemongrass, lemon rind, and chamomile gel. \"This,\" he says, \"is the essence of winter. It's kind of stoic. Like if Ingmar Bergman made a dessert, this is what it would taste like, right?\"\n\n\"I'm glad that there is one smart person at this table,\" says Shook.\n\nTentatively the teen chefs peek in, then pull up a few chairs.\n\n\"You're the calmest cooks I've ever seen,\" Silverton says. \"We think you did . . .\"\n\n\"An amazing job,\" says Shook.\n\n\"To cook a fish perfectly,\" says Lefebvre, \"or do a sweetbread perfectly, or cook pork perfectly\u2014that's not easy.\"\n\n\"Everyone at this table was so blown away by the technique that you guys put into the dishes,\" says Dotolo, \"and the thoughtfulness of them all\u2014even though we all taste the food differently. If you stay in this business, it will constantly happen where some people taste something and they might love it, and some people might hate it, especially if you guys are taking risks, which you're obviously doing, which is cool. Definitely don't stop doing that.\"\n\n\"We all said that we could have been eating at anyone's restaurant as a contemporary,\" says Silverton. \"Anywhere, easily . . .\"\n\n\"Now we're going to do one of the most embarrassing things,\" says Shook. \"The applause.\"\n\nSlumped in their chairs or over the table, the Samacon chefs absorb it all, exhilarated but beyond exhaustion, their faces still flushed after the clapping dies down. \"So,\" says Macklin, grinning at the dinner party. \"Now are you guys going to cook for us?\"\n\n### SUPPER CLUBS IN DENVER\n\n### By John Broening\n\n### From the _Denver Post_\n\n### Having worked in various big-name kitchens in New York, San Francisco, and Paris, John Broening is now the executive chef of three Denver restaurants, Duo, Olivea, and Spuntino. An ex-English major and son of a foreign correspondent, he keeps his writing skills honed with a weekly column for the _Denver Post._\n\n\"If Paul Prudhomme were dead, he'd be rolling in his grave,\" says Chris MacGillivray as he seasons his gumbo with a few decidedly nontraditional ingredients: sherry vinegar, agave syrup, and butter.\n\nThe gumbo is a sauce for what MacGillivray calls an \"Asian-Cajun dish\"\u2014Andouille sausage and shrimp-filled shumai dumplings garnished with a cross-section of fried okra.\n\nIt is the third course of Noble Swine Supper Club's August dinner, hosted by the owners of Crema, a scruffy luncheonette\/coffee shop in the heart of Denver's warehouse district north of Coors field.\n\nThe Noble Swine Supper Club, a loose collective of line cooks, chefs, managers, servers and sommeliers, has been around since 2010. Liz Batkin, a front-of-the-house manager who calls herself the Club's \"cat herder,\" describes Noble Swine as a \"floating dinner party.\" Batkin prefers to limit the events to about 30 people and mix half regulars with half newcomers.\n\nNoble Swine events feature place cards with assigned seating (\"we aim to inspire unexpected community\" the website declares). Democratic, informal, inexpensive to mount, spontaneous, and often wildly varying in quality, supper clubs are the perfect culinary vehicle for the Internet age\u2014they are the culinary equivalent of blogs. Unlike pop-up restaurants, which usually offer an a la carte menu and can run as long as several months, a supper club offers one fixed menu and a single seating. But both formats give otherwise unheralded cooks the opportunity to flex and shine.\n\nBatkin's husband, Andrew Van Stee, a slender, bearded wood-oven cook at Potager who helped found Noble Swine, enjoys the freedom to experiment and the freedom from having to run Noble Swine as a for-profit business.\n\nTraditional restaurant kitchens are usually the expression of the vision and personality of one person and value consistency and obedience to that vision. In a supper club, mistakes, experimentation, involved group discussion, and last-minute, improvisations are not frowned upon but actively encouraged. The menus are, more or less, conceived and executed collectively.\n\nA group e-mail about the event goes out about two weeks beforehand, listing the date and the site. Previous sites have included warehouses, empty apartments, and backyards. The menu is unknown to the guests until they sit down, but most of the regulars like that just fine. \"You're going out on a limb, but it's a controlled limb,\" says Carl Nixon, a regular who describes himself as an Internet abuse analyst.\n\nAn hour before a recent 7 o'clock dinner, the dining room at Crema is empty of people and furniture. But by 6:30, two folding tables and a few dozen mismatched chairs have been set up, a few rumpled tablecloths have been smoothed out and decorated with clumps of dried lavender, flowering dill and eucalyptus branches. Red-and-white wine glasses are set on the table, flanked by Mason jars for water.\n\nIt's been a hot day, and by the time the guests start to trickle in, the dining room thermostat reads a sweltering 83 degrees. Andrew Burch, the Supper Club's sommelier, has toweled off his glistening shaved head and changed his sweat-soaked T-shirt for a marginally dressier plaid button-down.\n\nNoble Swine does \"concept\" dinners: a Mexican dinner, a vegetarian dinner, an all-tomato dinner; and the Breakfast for Dinner menu, which featured a now-notorious take on chicken and waffles made with a huge slab of guinea hen. Tonight's dinner is a market menu, eight courses plus a cocktail, five wines, and coffee for $50.The menu features late-summer fruits and vegetables from the farmers market.\n\nThe vibe in the kitchen is chatty and casual. The food simmers away on makeshift equipment, camping stoves balanced on narrow counters, a three-tiered plastic steamer from Bed Bath & Beyond that looks like something you'd find at a yard sale. Stubby bottles of beer appear on workspaces after the second course goes out. A call for fresh herbs on a chicken liver dish demands a quick foray through a tenant's apartment to reach herb boxes behind the restaurant.\n\nMost of the cooks wear street clothes and sneakers. The best-groomed Swiner is the dishwasher, a barista at Crema who works in two-tone wingtips and sports a complicated hairstyle. (\"We pay him in high fives and beer,\" Van Stee says.)\n\nThe first course is a chilled soup made with Rocky Ford melons, garnished with slivers of yellow Peach tomatoes and shavings of pinkish Coppa salami from Il Mondo Vecchio. The dish pulses with color in the bowl, and on the palate, it tingles with farm-freshness. The soup brings out, surprisingly, the melonlike notes in the tomato. (MacGillivray admits that the combination of melons and tomatoes came from California chef David Kinch.)\n\nIf I've had a better restaurant dish in Denver, I can't remember what it was.\n\nThe chef in me wonders if a few dishes could have been improved by the intercession of a single, authoritative guiding hand. But the batting average is high. Crispy, pungent buttermilk-battered chicken livers with a deeply flavored jam made of heirloom peppers are followed by a moist grilled quail with pickled tomatoes and a sauce made from charred peaches.\n\nIn the long wait between the livers and the quail, BlackBerrys, cellphones, and digital cameras are taken out at the dinner tables. A few of the guests wander over to Crema's coffee nook, which doubles as the plating area, and chat easily with the cooks.\n\nDessert courses appear. A simple, vibrant dish of Red Heart plums marinated in Meyer lemon juice and sweet basil. A rectangle of zucchini cake garnished with pecan praline and brown butter ice cream. The zucchini cake is super-moist and more deeply flavored than most quick breads: Van Stee has replaced the traditional neutral oil in the recipe with cold-pressed hazelnut oil.\n\nFor the Noble Swine Supper Club, this is success\u2014cachet in a small corner of the food world and an opportunity to do the food they want without shortcuts. And enough money to buy equipment and beer and, occasionally, to pay themselves.\n\n### WHY CHEFS SELL OUT\n\n### By Richie Nakano\n\n### From Chow. com\n\n### After several years as a line cook at San Francisco dining hotspot NOPA\u2014a high-octane existence chronicled in podcasts, tweets, and his entertaining blog LineCook415\u2014Richie Nakano launched his own trendsetting noodle shop, Hapa Ramen, a pop-up based at the Ferry Plaza Farmer's Market.\n\nI used to think I was a real cook, legit: burnt arms, late nights, and an unusually high tolerance for whiskey. I believed I was part of something, a generational stance on cooking that was a strange mix of punk ethos and military discipline\u2014a savage precision. It felt like the food I cooked was real food. Honest food.\n\nOn any given night you could find me spouting off over a shift drink about all the food world's bullshit. I would rant about cooking shows I hated, restaurants that were corny, and above all which chefs were sellouts. Guys who pandered to critics made me crazy. And whenever I found myself at another glossy event plating bites of yellowtail crudo, I would scowl and mumble disdain for all the elite foodies I was serving.\n\nIn my days as a line cook there was no degree of compromise I found acceptable. I had come up reading _White Heat_ by Marco Pierre White. Anthony Bourdain's _Kitchen Confidential_ informed and usually validated my behavior. Rick Bayless was the crazy guy talking Mexican food in his bathrobe on PBS.\n\n**CHEFS, INC.**\n\nThen came _Top Chef_ and the rise of foodie blogger-ism, Bayless endorsing a shitty chicken sandwich for Burger King, White bending over to do a reality show, and suddenly it was all Bourdain, all the time. Tom Colicchio was hawking Diet Coke, and chefs who'd won James Beard Awards were taking jobs at Chipotle. Just a few weeks ago I saw Thomas Keller (the brand, not the man) in an American Express commercial.\n\nWhat does it mean to sell out when you're a chef? Consulting for a chain restaurant on the side? Sellout. Being slathered in makeup for a TV appearance? Sellout. Pebble Beach\/Aspen\/South Beach food festivals for the elites? YOU ARE A SELLOUT. And it got worse. Guys I knew who became personal chefs, caterers\u2014they were all compromised. I was righteously purist even with my own kind.\n\nThat is, until I sold out.\n\n**The Fame Racket**\n\nKitchen lifers are notoriously underpaid. It's part of the game, what you signed up for. In the beginning, this is fine: You get hooked up at friends' restaurants, live in a rough neighborhood that compensates with good banh mi. You scrape by. Then you get older. And when a child enters your life, all of a sudden things look very different.\n\nSo when some ad agency for a housewares manufacturer contacted me to appear in a new \"edgy\" magazine campaign, I listened. They said: Here's the equivalent of a few months of line-cook wages; all you have to do is fly to New York and have your picture taken for the ads. That's how I found myself with an endorsement contract in hand, getting caked in makeup for the photos, and plotting out monthly cooking demos I'm legally obligated to do for the next year.\n\nBack in San Francisco, I ran into a chef I know at a market one day and told him what I'd done. His judgment was quick: \"Sellout!\" But other chefs\u2014guys who are also dads\u2014they understood. Corporate chefs get to spend evenings home with their kids, make a better life for their families. Shit, with that money I was going to be able to pay off my Diapers. com bill and put some away for the restaurant I'm planning to open. Selling out is always a calculation, a weighing of benefit against cost.\n\nI have limits. I won't do any endorsement for food corporations, especially for industrial or fast foods. I refuse to appear at those mega-exclusive events in Aspen and Pebble Beach\u2014I have no idea who those festivals are for, or why they exist.\n\nBut I do know that selling out comes with new responsibilities. Will I totally lose any credibility I've built to this point? It's one thing for an ad executive to look at my tattoos and think I'm edgy; it's another thing for my food to live up to the hype. The result: I'm under new pressure to be a better chef. The truth is, I took the cash for my kid, but money's never free. Sometimes the sellout is the realest guy in the room.\n\n### A CHEF'S PAINFUL ROAD TO REHAB\n\n### By Kevin Pang\n\n### From _Chicago Tribune_\n\n### As the _Tribune's_ feature writer and Cheap Eats columnist, the prolific Kevin Pang works a dual beat: food and popular culture. Though his approach is usually snappy and humorous, this cautionary tale of a chef's self-destructive downward spiral reveals the darker aspects of life on the line.\n\nHere the man falls, all gruff and bravado, falling like a rocket that exploded midflight. One week he's poised to be the wunderkind chef of a big-time downtown restaurant; the next, he's slipping away from the success he'd hoped to achieve, falling from the grace he's yet to taste.\n\nBrandon Baltzley slides into the middle seat of a taxi. Sitting by the door might give him second thoughts about what he's about to do. The cab swings out the driveway of the Gold Coast high-rise where his girlfriend lives and onto southbound State Street.\n\nIt is 8:35 a.m. His belly feels scorched from the bottle of Jim Beam and the nearly entire deep-dish pizza consumed the night before, when the thought of the cab ride and where he was heading filled him with such anxiety that he stayed up all night. Arriving at rehab exhausted and hung over isn't the best idea, but Baltzley has been through this before and is keenly aware of what he is about to face.\n\n\"I definitely don't want to go,\" Baltzley says to his girlfriend, Emily Belden, 24. He exhales loudly, clears his throat. \"These 30 days are going to be f\u2014\u2014\u2014rough.\"\n\nA week earlier, Baltzley, 26, was the head chef at Tribute, an ambitious, 170-seat restaurant set to open in the Essex Inn in the South Loop. He spent months developing his menu, crafting a document to tell the world: This is who I am. Instead, on this morning in late May, he will check himself into a drug rehabilitation program on the Southwest Side.\n\nThe night before, he paid $100 he owed his dealer. He gave his apartment keys to a friend with instructions on locating his cocaine paraphernalia. Throw it all away, he told him.\n\nIn his duffel bag are clothes, two cooking books, toiletries and paper for letters he promised to write Belden every day.\n\nThe pressures of the kitchen drive an untold number of chefs into substance abuse. \"Aside from officers and firefighters that put their lives on the line, there's no other profession that puts demands on an individual and sets (its workers up) so well to fall into substance abuse and failed marriages,\" said chef Phillip Foss of the forthcoming El restaurant. \"And the vast majority of substance abusers just let it slide.\" But Baltzley sought treatment voluntarily, and in the process let go of a high-profile position many cooks would kill for.\n\nThe cab will arrive at the rehab center in eight miles, 26 minutes.\n\nIn the mid-1990s, years before he frittered away what he called his dream job, 9-year-old Brandon needed a stepping stool to reach the kitchen counter. He'd cut corn from the cob, mom would slice cabbage.\n\nIt was the two of them\u2014always the two of them in life\u2014working in The Whistlestop, the cafe Amber Baltzley owned in Jacksonville, Fla., where Baltzley made his first bowl of white corn turkey chowder. His mother sought ways to spend time with her only child and cooking held his attention like no other.\n\nHe was less devoted to his studies, and he dropped out two weeks into high school. Amber Baltzley knew forcing her son to return was a waste of energy, so she cut him a deal: If you don't go to school, you work full time. By 16, Baltzley was cooking at one of the toniest restaurants in Jacksonville, Stella's Piano Cafe.\n\nBaltzley found parallels with cooking and his other love, playing drums. Both provided immediate and intense tactile gratification. For two years he toured with the metal band Kylesa, and he submitted to all the rock star vices\u2014booze, girls, weed, his first line of cocaine. Amber Baltzley first realized her son was dabbling while watching a show. He looked as if he'd been awake for a week, she recalled. After the last song, Baltzley collapsed onto his drums.\n\nTired of traveling and eager to cook again, he landed at an Italian restaurant in Savannah, Ga., at 21, but the drug habit lurked. He said he stayed clean in Georgia but lapsed when he moved to Washington, D.C.\n\nMoving to New York only made things worse. Getting cocaine was as easy as pizza, Baltzley said\u2014you called and they'd deliver in 30 minutes. He was making good money at restaurants like Allen & Delancy and Bouley Upstairs. But the jobs were more pit stops: six months here, nine months there. On days off, he'd disappear from the world, snorting cocaine alone in his apartment, always fearing the crash that followed the high. In a single four-day binge, he recalled going through $2,000 of product.\n\nBaltzley was, however, lucid enough to check himself into a rehab facility early last year. He called it the worst 30 days of his life. The withdrawals were hell. His three roommates were legally obligated to be there, he said, and offered no support. And for a chef\u2014the indignity of hospital food! Square slabs of fish served on compartmentalized trays, well, that just put it over the top.\n\nHe emerged a shaken man, with a result that would not hold.\n\nThree miles, 14 minutes in, the cab turns onto the Eisenhower Expressway on-ramp. He stares down at nothing in particular; Belden is looking out the window. Their arms are hooked at the elbow.\n\nAs Baltzley describes what he's feeling, the conversation turns to vaccinations, and the feeling of knowing a needle's coming. How the nurse removes the plastic cap and you catch a glimpse of metal, and every muscle tenses up, and you know the needle's coming, and your heart's pounding, and you're rubbing the tops of your knees for distraction, _and still you know the needle's coming . . ._\n\n\"That's exactly what it's like,\" Baltzley says.\n\nHe sent his resume to Alinea on a whim. Not a chance, Baltzley thought. So when he was hired in September, Baltzley was over the moon. He moved to Chicago, and his trajectory was headed in one direction: to the skies.\n\nBut two weeks after he started at Alinea, his mom's Jacksonville home was hit by bullets. She lived in a part of town plagued by gang violence. Baltzley flew down to comfort his mother, knowing fully well that if he went home, his old friends would come around, and the urge . . .\n\nWhen he returned to Chicago a week later, Baltzley was in bad shape. He'd taken up drinking again. He wrote chef Grant Achatz, and, in one of his most humiliating moments, apologized and said he wasn't in any state to continue.\n\nA month after Alinea, Baltzley took over the head job at Mado in Bucktown. He didn't realize the mess he was in for: He was told he had 48 hours to open, so he scrambled and assembled a kitchen crew mostly of culinary school students. When squabbles with the owner over finances began to boil over, he and his staff walked out right before Thanksgiving.\n\nAround this time, restaurateur Simon Lamb was looking to open a restaurant at the Essex Inn on Michigan Avenue. (Lamb oversaw daily operations at Gioco and Redlight.) Lamb called Baltzley and suggested he apply, and, from 100-plus applicants, Baltzley made it to the round of a dozen finalists. Then came the cooking tryout.\n\n\"Brandon blew everybody away,\" Lamb said. \"He cooked food where I was later dreaming about the dish.\" Baltzley wowed him with a modern take on Carolina barbecue: pork belly confit with Brussels sprouts coleslaw, sweet potato panna cotta and mustard barbecue sauce. \"He was driven, entertaining, nontraditional, very relaxed and very funny. I thought, I gotta give this guy a shot.\"\n\nBaltzley landed the highest-profile job of his career\u2014a tattooed dropout heading a restaurant on Chicago's most famous avenue. But his newly steady paycheck also enabled his drug habit. Gaps of time appeared in the otherwise constant chatter on his Twitter feed; phone calls went unanswered.\n\nIt is difficult to quantify substance abuse among chefs, or, when it comes to drinking, at least, even to define it. Television would have you believe that cooking is a glamorous industry when a reality of 14-hour days and $350 a week is closer to the norm.\n\nBefore Rick Gresh became executive chef at David Burke's Primehouse, he witnessed friends strung out during service. At one restaurant, Gresh recalled asking a line cook working next to him, \"Dude, are you alive?\"The cook collapsed moments later.\n\nAnecdotes suggest a combination of factors that make work in the kitchen conducive to substance abuse after hours: a high-pressure environment, the type of people the job attracts and a social hour that begins after midnight.\n\n\"We're in a business where you can get anything you want, any time of day, any day of the week,\" Gresh said. \"It's just how it is.\"\n\nSeven miles, 23 minutes in. The cab turns off the Eisenhower and onto Independence Boulevard.\n\n\"Cooking gives me a gratification that nothing else gives me,\" Baltzley says. \"It's the fact that I'm so f\u2014\u2014-up and I do all these horrible things in my life, but when I can cook for someone they don't think about those things. Cooking is my mask.\"\n\nSilence inside the cab.\n\n\"I'm secretly wishing they'd turn me away . . .\"\n\nNervous laughter.\n\nBaltzley is asked if he's done his last line of cocaine. Seven seconds tick by.\n\n\"Yeah. I think I'm done.\"\n\nWhy the long pause?\n\n\"It's been in my life for seven years.\"\n\nOn the right, a three-story brick building appears.\n\n\"Oh God, we're here,\"he says. _\"Oh my God.\"_\n\nEmily Belden doesn't know why she fell for him. She was the product of a stable, drug-free household in the suburbs and dated clean-cut men with steady careers. An hour into their first meeting, three months ago, Baltzley gave his full disclosure\u2014\"I'm self-destructive. I'm an addict. I pretty much would ruin everything.\"\u2014and yet it developed into romance.\n\nBelden first saw him use on their second date. They spent a perfect Sunday together, but that night Baltzley said he needed to get high, and, being naive about cocaine's effects, she didn't protest. In hindsight, she said, it was a mistake.\n\nThe first sign of trouble came in late April. Baltzley, between apartments and staying at the Essex Inn, left work one night and started doing lines in his hotel room. It was a few days until anyone reached him, and by then, Baltzley was in cocaine's full grip.\n\nHis kitchen staff found him shaking, sweating, vomiting, suffering from chest pains. Belden walked into his room and found empty wine bottles, cigarette butts and drug bags strewn about. She was massaging out fist-size bumps of lactic acid built up on Baltzley's back.\n\nLamb entered and saw shame in Baltzley's eyes. They read, \"I let you down.\" Lamb was more concerned than angry\u2014by that time the two had become friends first and colleagues second. Still, Lamb had a lot of investors' money riding on Tribute. He made Baltzley sign an agreement saying he'd find help.\n\nBut it happened again in mid-May. He called Belden: \"I'm doing drugs tonight. You don't want to be around me.\"\n\nThe next five days were a blur for him. Baltzley simply vanished. Lamb said he left 20 messages for him and called around to all the hospitals. Thursday, Friday, Saturday, Sunday, Monday\u2014nothing.\n\nTuesday morning, May 17, Lamb got a text from Baltzley. It read: \"We have to talk.\"\n\nThe cab meter stops at $23. Baltzley arrives at the Gateway Foundation Alcohol & Drug Treatment center at 9:01 a.m. One week has passed since he and Tribute agreed to end their relationship. He and Lamb remain amicable. (On May 18, Tribute announced Lawrence Letrero, who worked under Baltzley, as its new chef.)\n\nBaltzley says he doesn't know which is more frightening: the next 30 days, or the first day he re-enters the world. He doesn't know if people will ever take him seriously again. But he says he's sorry for the people he has let down.\n\nSaid Belden: \"Seeing him do drugs has changed my opinion of what a drug addict is. Brandon does this alone on a Sunday night. Yes, he decides to take the first hit . . . but once he does that, he loses all control over it. I feel like he's trapped in a body that's convincing him that he needs it.\"\n\nA patient stares out the window, as if sizing him up. Then, he waves a friendly wave. Baltzley lights up one last cigarette.\n\n\"I don't want to be here for the next month, but I have to if I want to . . . if I want to be hirable. If I want the opportunity again for someone to put their trust in me.\"\n\nHe takes one final puff and extinguishes the cigarette. He rubs the back of Belden's neck with his hand. She drapes her arm across his back. Brandon Baltzley walks inside.\n\n### BITTER START TO A LIFE OF SWEETS\n\n### By Chris Macias\n\n### From _Sacramento Bee_\n\n### Then there's the other side of the coin: A chef whose life was turned around by the opportunity to perform in a high-end kitchen. _Sacramento Bee_ food reporter and wine columnist Chris Macias profiles an unlikely pastry chef and his against-all-odds success.\n\nEdward Martinez gently places a chocolate orb in the center of a white bowl. To the touch, this confection is hard and impeccably smooth, like a small eight ball.\n\nHis right hand, emblazoned with a skull tattoo, holds a small ladle of warmed chocolate-infused milk. He drizzles the liquid over the orb, accompanied by chocolate streusel and toasted hazelnuts. It soon breaks open, revealing a sumptuous filling of hazelnut and milk chocolate pudding, mixed with more crispy bits of chocolate and hazelnut.\n\n\"I like making pretty food,\" Martinez said. \"The first thing you do is eat with your eyes. You want it to be beautiful. If the flavors work, it brings that whole dish together.\"\n\nMartinez serves as executive pastry chef of Hawks in Granite Bay, which specializes in seasonal ingredients and is among the region's finest restaurants. Even in a chocolate-stained apron, Martinez doesn't look like a guy you'd want to mess with. He stands over 6 feet tall with a shaved head and a black widow spider on the back of his neck. His body is an evolving canvas of tattoos, some of which hark back to a past that he's since left behind: membership in one of California's most notorious street gangs.\n\nLearning to make pastries may have saved Martinez's life, or at least spared him a stretch in the state penitentiary. In 2005, facing three felony charges, Martinez promised to enroll in a pastry-making program, leading to a reduced sentence\u2014and perhaps a last chance at an honest life.\n\nMartinez's Facebook photos show a collage of the sweet and a bitter taste of his past. There's a shot of his moelloux of white chocolate, compressed mandarins, pistachio macaron and mandarin sorbet; an \"I heart foie gras\" T-shirt sported by his baby son; and the casket of one of Martinez's homeboys from his Fresno gang days being lowered into the earth.\n\n\"I never expected to get this far,\" said Martinez, who recently turned 27. \"I expected . . . (to be) in jail, or dead.\"\n\nNow, Martinez surrounds himself with sugars, ripe seasonal fruits and delicate desserts. He's devouring \"Modernist Cuisine,\" the six-volume book of cutting-edge cooking techniques. His repertoire at Hawks includes nitrogen-frozen chocolate mousse with gianduja cr\u00e9meux and hazelnut pudding.\n\n\"He's the best working pastry chef I've seen,\" said Pajo Bruich, midtown's Lounge ON20 executive chef, known for his complex cooking techniques. \"Hands down, nobody in the Sacramento market is doing the creative elements he's doing.\"\n\n**The Rise of Baby Gangster**\n\nBaby Gangster was always ready to fight.\n\nThat's what the Bulldogs gang members called Martinez, after he was \"jumped into\" the gang at age 13.\n\n\"I was at the homeboy's house, in the backyard,\" Martinez recalled, between sips of coffee at a midtown Sacramento cafe. \"I'm telling them, 'I want to be in. This is what I want. I want to be a Bulldog.' And they said, 'OK, let's do it.'They beat me up for about 30 seconds. It's weird. You're beating up your friend so they can hang out with you. I got \"FRESNO\" tattooed across my chest about six months after that.\"\n\nThe Bulldogs have few friends, except for those also inked with the dog paws and \"BD\" tattoos. Bulldogs are recognized as a violent California gang, based primarily in Fresno. Law enforcement estimates the gang has more than 6,000 members. The Bulldogs, who take the name and logo from the mascot at California State University, Fresno, have no allies and no leadership structure. Crips, Bloods, Norte\u00f1o and Sure\u00f1o gangs are all sworn Bulldogs enemies.\n\nBoth of Martinez's older brothers were Bulldogs; so were other close family members. One cousin was nicknamed \"Big Gangster,\" while an older brother was \"Lil Gangster.\" Baby Gangster Martinez was \"Baby G\" for short\u2014and had it tattooed into his left forearm.\n\nHe said his turf was on the east side of Fresno, where he claimed \"Mariposa Street Gangsters\"\u2014or \"MSG\" for short. He'd moved there from San Jose at the age of 9, about two years after his mother, Theodora, died in a car accident. He said he still thinks of her baking in the kitchen, surrounded by the smells of sugar and frosting.\n\nHis father, Joe Martinez, said his son didn't cope well after her death. The elder Martinez, who earned an economics degree from Stanford University, had hoped his four children would get educations, but his wife's death fractured the family spirit.\n\n\"With Edward, he kept a lot inside and started getting into trouble at school,\" said Joe Martinez. \"Prior to that, he was doing excellent in school.\"\n\nBaby Gangster developed a taste for stealing. He was charged and later convicted in 2004 with grand theft for stealing $2,000 worth of DVD players and other merchandise from a Blockbuster Video.\n\nIn April 2005, while at a Fresno fast food restaurant, Baby Gangster thought someone looked at his girlfriend the wrong way. He attacked, punched the victim and fled. According to documents in Fresno Superior Court, the victim identified his attacker as a gang member because of his tattoos.\n\nThe victim and two witnesses picked Edward Martinez out of a photo lineup. Martinez was already on parole for the second-degree burglary at Blockbuster. Baby Gangster went on the run for more than three weeks.\n\nHe knew he couldn't hide forever.\n\n\"I finally got tired of running and went to my dad's house,\" said Martinez. \"I knew they were going to get me there. When they came to the door, there were cops everywhere. I was going to jail.\"\n\n**Sweet Salvation**\n\nMartinez's dad had heard all the talk before about changing for good. So had judges. Martinez was 20 and had served stints in county jail.\n\nNow he faced felony charges of assault and battery, both with gang enhancements, and street terrorism. Facing eight years in state prison, he said he wanted to enroll in a local baking program.\n\n\"I pretty much begged,\" Martinez said. \"I knew if I was going to prison that I would do the whole eight years. It would be me gang-banging harder than ever before, trying to fight with everybody.\"\n\nMartinez pleaded no contest to misdemeanor battery. The other charges were dropped, but he'd have to honor his parole and enroll in the pastry program as promised.\n\nBy this time, Martinez had spent nine months in county jail. He was ready to bid his Baby Gangster persona goodbye.\n\n\"I needed to prove I wasn't a (screw) up,\" said Martinez. \"I just had to prove to my dad that this is what I was going to do. Baking and pastries was something I could have fun doing. I remembered being happy doing that.\"\n\nIn the Fresno suburb of Clovis, Martinez enrolled at the Institute of Technology's baking and pastry specialist program. He felt self-conscious at first, still sporting a nearly bald head with a \"BD\" tattoo he had inked in county jail.\n\n\"When he came into my class he wasn't very talkative, but when he did talk he had a lot of questions,\" said Thomas Mendoza, a culinary mentor and instructor there. \"He was very inquisitive on things that were new, and when he got a basic technique down, he wanted to take it above and beyond, and make it his own.\"\n\nHis older brothers were still in the gang. In many cases, leaving a gang means \"blood in, blood out\"\u2014you can only leave with your life.\n\n\"There were times when I had a friend come over and tell me some stuff happened in the 'hood and we needed to go handle it,\" said Martinez. \"I'm like, 'I can't.' I'd never told anyone that. He kind of gave me a look like, 'Are you serious?' I said, 'I just can't.'\"\n\nMartinez dug into his textbooks and other reading, including \"The French Laundry Cookbook.\" He learned a new vocabulary: cr\u00e8me anglaise, mignardises, cr\u00e8me de farine and velout\u00e9 of bittersweet chocolate.\n\n\"I was loving it,\" said Martinez. \"When I was making breads for the first time, they would look exactly like the stuff in the book. I started showing my dad, 'Look what I made!'\"\n\nFor his final exam, after the nine-month program, Martinez presented a complex tuile cookie cone with garnishes and the point side down in the center of the plate. His attendance had been perfect, and Martinez made the dean's list.\n\n\"He had all the awards you could receive,\" said Mendoza. \"He was one of the leaders in the class. He's one of those students that just gets it.\"\n\nMartinez applied for a job at Slate's, one of Fresno's finest restaurants. The interview was the first time he'd ever stepped into a fancy restaurant.\n\n\"I took him on because no one would probably hire the kid,\" said Roy Harland, former executive chef of Slate's. \"A lot of the ultraconservative Fresno clientele would not be comfortable with a former Bulldog gang member walking through the dining room. I immediately knew this guy has talent and could create.\"\n\n**His New Persona**\n\nLike the chocolate orb, Martinez's \"Baby Gangster\" persona has melted away. When he's not working at Hawks, Martinez raises four children in Antelope with his wife of seven years, Michelle.\n\nMartinez moved his family to the Sacramento area two years ago from Napa, where he worked at the Michelin-starred Bistro Jeanty. He's had other job opportunities, but lost some after potential employers checked his background. Either way, Martinez says, Hawks and Sacramento are happy homes for him.\n\n\"My kids are happy and they're going to a good school,\" said Martinez. \"It's about my wife and my kids now. That's why I do everything I do. If it wasn't for them, I wouldn't be working 14 to 15 hours a day.\"\n\nPast the restaurant's houndstooth chairs and blue walls, Martinez towers over the kitchen's pastry station. He keeps a tank of liquid nitrogen close to whip up new pastries. One looks like something from a mad scientist's laboratory: frozen coconut mousse with coconut sorbet and cilantro oil drizzle.\n\nThe infusion of liquid nitrogen to the coconut mousse adds a theatrical kind of fog as the dessert freezes, with the final product looking like delicate cauliflower. The dish's coconut flavors are perfectly pronounced, with Martinez's cilantro oil adding a pleasing herbal accompaniment.\n\n\"I like to do modern stuff, but with classic techniques,\" said Martinez. \"I can put some liquid nitrogen into a mousse, but I can also knock out some perfect crepes for you. People will say, 'Oh, this is so beautiful, can we speak to the pastry chef? Where is she?' They'll look at me like, 'You made these plates?'\"\n\nHe visits other restaurants for inspiration. In February, he and his wife traveled to New York City, dining at wd-50, Eleven Madison Park and Per Se. Over 14 courses at Per Se, dressed in a suit with his collar barely concealing his neck tattoos, he thought of how far he'd come.\n\n\"When I walked in, they were all, 'Hi, how are you, Mr. Martinez?'\" said Martinez. \"I'm pretty sure I'm the only one from my neighborhood that's ever going to do anything like this.\"\n\nSometimes he feels the shadow of his past. Before St. Patrick's Day this year, Martinez and others in the Hawks crew shaved their heads for charity. Everyone could see the \"BD\" tattoo. Two days later, he covered it with a giant skull tattoo.\n\nHis goal is to ink over all of his gang tattoos.\n\n\"I don't want to be somewhere, like at the beach with my wife, and all of a sudden someone's like, 'What's that FRESNO for?'\" said Martinez. \"I don't give off the same vibe that I used to.\"\n\nStill, in the rush of a packed night at Hawks, and especially if someone botches one of Martinez's desserts, he can snap. The difference now, Martinez said, is that he'll apologize.\n\n\"To some degree, I think he still has some issues there,\" said Joe Martinez. \"But, he's managed to control it quite a bit. . . . He's done pretty good, and I'm very proud.\"\n\nMartinez said that both his older brothers have also left the Bulldogs gang. A younger brother, Matt, lives in Sacramento now and works as a line cook at Lounge ON20.\n\nMartinez dreams of opening his own dessert bar, hoping to be known around the country for his pastries.\n\n\"I'm Edward now,\" said Martinez. \"I'm not a gangster. He's gone. He's no longer there. I don't look back.\"\n\n## Personal Tastes\n\n### [KITCHEN CONFESSIONAL:  \nBURNIN' DOWN DA HOUSE](contents.html#ch45)\n\n### By David Leite\n\n### From Leite's Culinaria\n\n### This award-winning website's title says it all: It centers on food and cooking, and its founder and editor-in-chief is David Leite, author of _The New Portuguese Table_ (among many other publishing credits). Leite's hallmark self-deprecating humor sparkles through an all-too familiar scenario of kitchen disaster.\n\nNow that the turkey leftovers are gone, the tryptophan torpor has receded, and we've physically and emotionally pushed away from the Thanksgiving table, I need to get something off my chest. A kitchen confessional, if you will: On the Holiest of Holy Days for culinistas all over the country, I failed miserably at the stove. Twice.\n\nIt was far and away the worst hatchet job I've ever committed\u2014and it was at baking, my bailiwick. In the 20-something years that I've been cooking Thanksgiving dinner, yes, I've forgotten to take the giblets packet out of the bird; yes, I've both under- and overcooked the turkey; and, yes, I've neglected to heat the stuffing to the ideal (read: salmonella-free) temperature. But I've never, ever failed to whip up gasp-inducing desserts. But I can't take full responsibility for my fumble: I mostly blame Twitter and Instagram, because if it weren't for me snapping pictures of my marvelosity in the kitchen for public consumption, I would've had a relaxing holiday, and the members of the Roxbury volunteer fire department would've been able to finish their meal undisturbed.\n\nLet me backtrack. Please.\n\nThe Tuesday night before Thanksgiving I was planning to make my pumpkin cake with maple-cream cheese frosting and Melissa Clark's spiced maple pecan pie for dessert. The One is a pumpkin freak and demands the cake every year. The pie was a concession, a peace offering to those poor friends of ours who've been politely eating the same dessert for nearly a decade. I thought they might need a change.\n\nKnowing that some of my blogging brethren, among them Ree Drummond, Shauna James Ahern, David Lebovitz, Gail Dosik, Sara Kate Gillingham-Ryan, are quite adept at snapping cell phone pics of their kitchen hijinks and tweeting them while cooking, I decided I could, too. So with iPhone in hand, and iPad in its kitchen condom, I began clicking away. But instead of waiting until the cake was safely in the oven to upload the shots and check Twitter for the inevitable onslaught of kudos from you all, I decided to reply to every single response while baking.\n\nBasking in your immediate adulation and unconditional love with one hand while meticulously dividing, weighing, and smoothing the batter with the other, I noticed something odd. As in the batter spreading as thick as spackle. I had to work it into the edges of the pan, where the sides meet the bottom. _No big deal,_ I thought. _I've made this a million times, and it always comes out perfectly. Must be the dry weather._ With that, I slid all three pans into the oven and returned to my 4G iNeedConstantLoveMachine.\n\nForty minutes later, I pulled the cake layers from the oven to discover they hadn't risen much. _No big deal,_ I told myself again. _I'm using three nine-inch pans instead of the usual two eight-inchers._ They're bound to be a little thinner.\n\nI tipped the cakes out of the pans, and instead of steaming circles of spicy pumpkin loveliness, I was affronted by what can only be described as mutants. Each layer was riddled with worm holes. Entire sections were curdled and dry, with huge gaps in them. _No big deal, that's why God made frosting._ It was while reaching for my iPhone, to see who else liked my photos on Instagram, that I spotted them sitting on the counter, mocking me: a chorus line of three cans of unopened solid-packed pumpkin. I'D FORGOTTEN TO ADD PUMPKIN TO THE PUMPKIN CAKE.\n\nFor a brief, dark moment, I contemplated passing off this castrato of a cake as the real thing. Chances are my guests wouldn't know, and, most important, neither would you. I imagined millions of you sitting at your computers or holding your cellphones while watching \"Body of Proof\" just waiting for the final shot of my towering creation. Guilt, my constant sniggering companion, won out. I dumped the damn thing into a plastic trash bag like so many dead bodies on TV.\n\nThe next morning, refreshed but hours behind, I turned out what The One later called the best pumpkin cake ever. I tweeted its headshot, of course.\n\nThe cake redo slapped me all the way into the middle of Wednesday afternoon. If I worked quickly and efficiently, I could knock out the spiced maple pecan pie and prep my three side dishes: Virginia Willis's bourbon sweet potatoes, roasted carrots with an agresto sauce (a to-die-for mix of chopped nuts, lemon juice, vinegar, wine, parsley, and spices), and homemade green-bean salad.\n\nMelissa's recipe calls for maple syrup and demerara sugar to be simmered until reduced by about a third. Being in a hurry, I calculated I could save almost 20 minutes if I let it _boil_ down\u2014and who the hell has demerara sugar in the middle of rural Connecticut? So I used granulated sugar instead. It was then that I walked out of the kitchen into the family room to get a recipe. I'm talking all of 60 feet, people. I was flipping through a cookbook when what sounded liked a nuclear-disaster siren went off.\n\nI ran to the kitchen and from the pot billowed the blackest, foulest-smelling smoke I ever had the misfortune to encounter. Now, I'm good in emergencies. The One and I were like hopped-up Eagle Scouts on 9\/11, filling bathtubs and sinks with water; withdrawing huge sums of cash from all of our accounts; and shopping for food, flashlights, batteries, and the current issue of _People_ magazine. But on this day, as I ping-ponged between four fire alarms and three French doors, shooing out the smoke with my apron and a spatula ( _spatula_?), what's the one thing I forgot to do? Turn off the stove. So as soon as I got the air raid under control, it started again. And again. And again. Finally, I tossed the pan in the sink then thought better of it and flung it out into the yard.\n\nWith the bleating now over, the phone rang. _Holy go to war, the alarm company_. I smoothed my sooty apron and cleared my throat. \"Hello?\" I said, as if I were the top earner at a phone sex company.\n\n\"Sir, we have a report of an alarm trigger at this residence. Who am I speaking with?\"\n\n\"David Leite.\" My voice was all warm caramel and Cognac.\n\n\"Who else is on this account?\"\n\n\"_______________,\" I replied, using The One's real name.\n\n\"What's the passcode, sir?\" _Passcode? **What** passcode?_\n\nAnd as if reading a roll call, I listed every single password I could remember. (Note: None of these are real. What do you think? I'm crazy?) \"Ginger, Gilligan, Miss Piggy, Marcia Brady, Julia Child, Tom and Jerry, Mr. Spock.\"\n\n\"Sir . . . \"\n\n\"Murphy Brown . . . \"\n\n\"Sir!\"\n\n\"I DON'T KNOW THE FREAKING PASSCODE, ALL RIGHT? BUT IT'S ME, DAVID LE\u2013\"\n\nDial tone. He'd hung up on me. Then the most sickening sound pierced the air: the wail of the town's fire alarm. \"Noooooooooooo!\" _The One is going to kill me._ I could see the headlines in the _Litchfield County Times_ : \"Lauded Food Writer Almost Burns Down the House.\" Frantic, I called 411 and asked for the Roxbury Fire Department.\n\n\"Sir,\" said the operator, \"you don't need to call the fire department. You just need to dial 911.\"\n\n\"No, I don't need to report a fire\u2013\"\n\n\"Then why are you calling the fire department?\"\n\n\"Because . . . \"\n\n\"Sir, I'm required to connect you to 911\u2013\"\n\nI pressed \"End Call\" and dropped my iPhone on the couch as if I were letting go of a putrid piece of pork. Lying there, it chimed an alert: \"Instagram: Talon245 liked your photo.\" _Oh, how sweet of him._ I instinctively reached out to see what he'd written. \"No!,\" I shouted, shaking my head trying to gain perspective.\n\nAfter a few minutes, The One and our friend Caroline, who was spending the holiday with us, came home. He looked around the kitchen and out into the backyard at the tar-colored pot, slack jawed. \"Don't ask,\" I said before he could say anything. \"Please, don't ask.\" As we stared at each other the whine of another siren grew louder.\n\n\"Don't tell me . . .,\" he said pointing over his shoulder to the sound, realizing it had my name on it. I nodded my head. \"Oh, David\" was all he could get out before flashing red lights splashed across the family room walls. I rose to go to the door. \"Sit,\" he said. \"SIT!\" I obeyed.\n\n\"Think this will end up in the newspaper's police blotter?\" I asked Caroline, looking for some sympathy.\n\nEver immune to subtle interpersonal cues, she said flatly, \"Probably.\"\n\nI ran through the kitchen cutting off The One before he got to the door and opened it. A man in a flannel jacket and a bruised fire helmet poked his head in. \"Um, is there a fire here?\" he asked, unsure he got the right address.\n\nSuddenly self-conscious about what I looked like\u2014after all I was in my Warner Bros. pajamas and a sooty apron\u2014I smoothed my hair.\n\n\"Hi, officer,\" I said, smiling. Behind him was a fire truck and several men putting on gear. \"Um, is it _officer_ ,\" I continued trying to sound nonchalant, \"or _fire marshall?\"_\n\n\"John. It's John.\"\n\n\"John,\" I replied, emphasizing his name, \"this is rather embarrassing, but I kind of messed up my Thanksgiving dessert. Just a bunch of smoke and drama, but no fire.\" He looked at The One who was behind me for some kind of assurance. The One nodded.\n\n\"I hope I didn't pull you all away from anything important.\"\n\n\"Well, some of the guys were just having an early Thanksgiving at the firehouse.\" It's amazing how small a 295-pound man can feel.\n\n\"Stay away from the stove, will ya?\" he said as he jumped back on the truck. \"And happy Thanksgiving.\"\n\n\"You, too.\" I waved off my own personal fire brigade parade.\n\nExhausted, I curled up on the couch and fell asleep for the rest of the afternoon. I awoke after dark, shivering. The windows were still open; the kitchen still smelled acrid. I avoided The One's gaze as I quietly made my fallback chocolate pecan pie. When I pulled it from the oven, it was a picture of baking mastery. Forgetting myself, I held it out for him. \"Look!\" He just nodded. Realizing that the coolness in the room wasn't coming from just the windows, I slid the pie on a rack, and then I couldn't help myself.\n\nI took a picture and posted it.\n\n### DO I DARE TO EAT A PEACH?\n\n### By John Spong\n\n### From _Texas Monthly_\n\n### SM senior editor and Texas native son John Spong covers everything from dance halls and outlaw country to Texas Longhorns football and _Friday Night Lights._ Mulling over his childhood as a card-carrying Picky Eater, Spong ponders how his own children might escape a similar fate.\n\nI've never eaten a pickle, at least not on purpose. It's not a claim I make with pride, though it comes up somewhat often, especially in the summer months. Backyard-beer-and-burger-flip season. For much of my life, such occasions were actually harrowing affairs, hardly conducive to the relaxation for which they were purposed. The stress typically kicked in at the end of hour one, just as the congregants moved to the fixings table. The sun might shine and the birds might sing. A pi\u00f1ata might even hang in the yard. But the spread would stretch out like a minefield. Plates stacked with onions, tomatoes, and lettuce, items that, to my mind, had no more business on a burger than peanut butter. Bowls filled with potato salad and coleslaw, two concoctions whose very names I preferred not to let pass my lips. For dessert, the dreaded watermelon. My only solace would come when the chef called, \"Who wants cheese on their burger?\" at which point, if I was lucky, I'd spot a five-year-old wearing my same look of disgust. A compatriot. We'd get our burgers first\u2014less time was spent in their construction\u2014then go eat at the swing set. \"You know,\" I'd explain, \"I've never eaten a pickle, at least not on purpose.\"\n\nOn one such occasion a friend's son got curious. \"Does that mean you've had one on accident?\" he asked.\n\n\"Actually, your father once snuck four pickle slices and some mustard on a hamburger he fixed for me. It was at a cookout shortly after we got out of college, an engagement party for him and your mother.\"\n\n\"What did you do?\"\n\n\"I took one bite and spit it all over the table. I think your grandmother was pretty grossed out.\"\n\nHe looked up at me skeptically, causing me to worry for a moment that he might be pro-pickle. But as he turned to examine the burger on the paper plate in his lap, I knew it didn't matter. I could make him understand by likening the pickle to the beet. Or to broccoli. For that is the essence of the picky eater's dilemma: Whatever that foodstuff is that he finds most objectionable, nothing will be as terrifying as the thought of having it in his mouth.\n\nI say that with intimate authority. I grew up the worst eater I'd ever heard of, the kid that my friends' parents always sent home at suppertime, a sufferer of bizarre food phobias that were absolutely nonnegotiable. I'd refuse to eat cheese, except on pizza, and then only with pepperoni. Mac and cheese and grilled cheese sandwiches were out. By a similar logic, french fries were in but mashed potatoes were out. Condiments were unthinkable, and so too soup, fruit, and any vegetable that wasn't corn. Those few foods I did eat could never be allowed to touch on the plate; \"casserole\" was the dirtiest word I could think of. I would eat a peanut butter sandwich but had no use for jelly and would refuse to take a bite within an inch of the crust. Chicken was fine, turkey was not, and fish was just weird. Essentially, all I ate willingly was plain-and-dry hot dogs and burgers, breakfast cereal with \"sugar\" in bold letters on the box, and anything with Chef Boyardee's picture on the label. Or, rather, almost anything. I didn't fully trust the shape of his ravioli; something told me cheese might be lurking within.\n\nSuch proclivities came at a cost. In elementary school, I was regularly disciplined for not eating enough of my lunch, sequestered to the \"baby table,\" where talking was forbidden and cafeteria monitors would loom overhead, pushing me to eat. When summer came, my parents would no doubt have loved to ship me off to camp but didn't out of a legitimate fear that I'd starve. That was fine by me. I was similarly terrified that some camp counselor would force me to drink iced tea.\n\nAt home, my parents did what they could but never had much heart for the battle. According to my dad, the opening skirmish was over a sweet potato, when I was two. Though I remember nothing of the encounter, my guess is\u2014given that my parents were children of the Depression and were neither adventuresome eaters nor particularly adept in the kitchen\u2014that the sweet potato had been boiled, probably for longer than it needed to be. I looked at it and told him that I didn't eat those. He responded that this was the first sweet potato I'd seen. At his strong insistence I took a bite, then airmailed it onto his chin.\n\nMeals became a combination of accommodation and subterfuge. My mom served dinner on steel cafeteria trays purchased at an Army surplus store. That allowed her to segregate my food. She'd sprinkle Jell-O mix on banana slices to make them seem closer to candy. She'd even turn a blind eye\u2014occasionally\u2014when I'd slide objectionable items to my two younger brothers, neither of whom suffered from finickiness. One of them actually ate crayons and cigarettes.\n\nMy palate did broaden as I got older, though none of these victories were won at my parents' table. And so ingrained were the food phobias that I can clearly remember each time I branched out. I first tried ketchup as a tenth grader, at the old Holiday House on Austin's Ben White Boulevard, in an effort to look sophisticated in front of two much cooler upperclassmen. I was a University of Texas sophomore standing on the corner of Speedway and what is now Dean Keeton when I became an acknowledged fan of caramelized onions. A friend argued that they were the primary attraction in the $1.50 fajitas we'd just bought from a campus vendor, then opened one up to prove it. I was shocked. At that point I'd been enjoying them unwittingly for more than a year.\n\nAnd then there were tomatoes. I'd long heard that garden-fresh tomatoes were nothing like the canned ones I'd picked out of my mom's spaghetti. I could even recite the lyrics to Guy Clark's celebratory hymn \"Homegrown Tomatoes.\" But I'd never been willing to try one until an afternoon twelve years ago at the home of the writer Jan Reid. The occasion was a reunion of sorts. Four months earlier some friends and I had been with Jan in Mexico City. Our cab had been hijacked by two pistoleros, and Jan had fought back, ending up with a gunshot wound in his belly and a bullet near his spine. While rehabbing in Houston, he had asked me to water his cherished tomato plants. When he finally got home, the Gang of Four, as he called us, met at his house for dinner.\n\nAs we sat down, he announced he was serving BLTs, casually mentioning how good it had felt to have been able to pick the tomatoes that afternoon. He thanked me for keeping them alive while he'd been in the hospital. It didn't seem an appropriate time to say, \"I don't eat those.\" They tasted as great as food served by someone who's saved your life should. And the affinity held up; the next time I encountered a homegrown tomato I bit into it as if it were an apple.\n\nBy then I was 33 years old. And though nowadays I'll eat just about anything\u2014and have never really wondered what my life would have been like if only I'd met tomatoes sooner\u2014a new concern has arisen. At 44, I've finally gotten married, and my wife and I are talking about starting a family. We've seen enough friends have children to know that wearing regurgitated yams will be part of the bargain. But we'd like to find a way to make that stop sometime before the kids go to college. Since my genes will get the credit for any picky eaters produced, the burden of learning why they happen and how best to deal with them has fallen to me. So I started doing some research.\n\nImagine a caveman is eyeballing a hamburger. His reaction will be as instinctual as going to the bathroom or looking for love. The sight and smell will alert his brain that proteins and calories are available. With the first bite, chemical reactions between the burger's ingredients and taste receptors in his tongue will send messages through his nervous system, primarily the chorda tympani nerve, which stretches around his eardrum to the stem of his brain. If there's a tomato on it, or maybe some ketchup, he'll get a sweet taste, which upon arrival upstairs will trigger a small dopamine release. His body will read that as good news. The same will happen with the salty fat in the meat and cheese. But if by chance there's some arugula onboard, a bitter taste will register, signifier of potential poison. He'll likely spit that out and pick it off the rest of the burger. As he continues, chewing and swallowing each bite, a second, internal smelling process will take place every time he exhales. This information will be more detailed than that from the tongue, which can read only the five basic tastes: salty, sweet, sour, bitter, and the newly discovered, ever-nebulous umami. The news will combine in the brain and be read as distinct flavors. He'll go about the rest of his day with a good supply of energy and remember that meal as a fine thing.\n\nNow picture the caveman eating at Austin's Counter Cafe, rightfully considered home to the city's best burger. Sitting next to him and regarding an identical lunch is a member of that class of Austinite that considers itself the town's most evolved: the trendy hipster. (Though they share the same bedhead and beard, the hipster will be identifiable by the pair of Ray-Bans folded next to his plate.) His relationship with the burger will be much more complicated. Assuming his parents were middle- to upper-class, he's at least one generation removed from foods of necessity, so he's known only the luxury of choice. If he grew up in the seventies or eighties, his earliest exposure to vegetables was probably via Del Monte and Green Giant, black-magic alchemists who, through canning and freezing, confused an entire nation on the meaning of \"garden fresh.\" If he suffered from chronic ear infections as a kid, his chorda tympani may have been damaged and his sense of taste permanently altered. Or he may even be a supertaster, one of that quarter of the populace whose tongues can have twice as many taste receptors as the average eater's. In that case, every taste will be magnified, particularly the bitter ones. Given all the variables, if the hipster chooses to leave everything off his meat patty but the bun, there'd be plenty of potential reasons why.\n\n\"When we talk about picky eating, we are talking about pleasure and people who don't get the same hit from eating that others do,\" instructs Linda Bartoshuk, the director of human research at the University of Florida's Center for Smell and Taste. She was one of the first experts I called, a legend in the tight circle of neuroscientists, psychologists, and nutritionists who study the way people eat. She's researched taste for 45 years, and among her discoveries is the supertasting phenomenon. \"There are major categories of things that affect how much pleasure we take from food. One is sensory, and that's where the supertasters fit in. We don't all taste things the same way. That's hardwired. The other is experience, the pathologies you have encountered. That is all learned.\"\n\nThose lessons come early. When Bartoshuk explained the fundamental nature of conditioned food preferences and aversions, she pointed to baby rats, who sniff their mother's breath to learn what is safe to eat. In finicky humans, the primary pathology is gastrointestinal problems. If a person of any age throws up shortly after eating, he'll automatically develop an aversion to whatever he just ate, regardless of any causal connection between it and getting sick. \"When I see a picky kid, the first thing I try to find out is his medical history. If the parents say he threw up a lot when he was young, I've got a pretty good idea why he finds many foods disgusting. It's a brain mechanism he can't help.\"\n\nThe neuroscientists I consulted stressed the same kinds of physical problems as Bartoshuk. Psychiatrists and psychologists, on the other hand, steered the conversation to the behavioral side of the equation. They said that many kids between the ages of two and four will experience some measure of pickiness. It's as natural as learning to say no. Timid children may have an ingrained distrust of things that are new. Tactilely sensitive kids, like the ones who need the tags cut out of their T-shirts, may have trouble with food textures. Others may live in the neon food world of a supertaster. In these instances, the key is the parents' reactions. If the parent forces the kid to eat food he doesn't like, meals will turn into power plays. With a strong-willed child, that's the kind of problem that can stretch well into adolescence. (The chefs I talked to, by the way, piled on the parents even harder. The problem, they said, is that most moms and dads can't cook.)\n\nAs the experts ticked off the things that typically go wrong, they sounded as if they had had access to my childhood scrapbooks. My first extended hospital stay came shortly before I turned three, during a frightful bout with epiglottitis. Because of a virus, my throat was closing shut, producing the kind of prolonged, painful eating trauma that the shrinks and neuroscientists said could lead a kid to reject a whole host of foods. But the sole connection my parents ever made to that event and my diet was of a different sort: They cited it as an example of how obstinate I could be. The hospital stay had been cut short because I wouldn't eat the food. My folks got tired of bringing me Spaghetti-O's.\n\nAs my teen years approached, every meal became a battle of wills. My parents would tell me to eat, I would refuse, and they'd wait me out. My brothers would finish dinner and be excused to their rooms before I could sneak them my green beans. The family dog, a supremely overfed basset hound named Bobo who was my greatest ally in such matters, would be shooed to the garage. While Mom cleaned the kitchen, I'd remain at the table. Eventually she'd sit and watch me, sometimes for as long as an hour. She never turned cruel. One doctor I talked to described parents who tell their children, \"If you don't want it for dinner, you'll have it for breakfast,\" then put the plate in the fridge to serve it again in the morning. That sounds like torture, and that didn't happen. Instead, I'd ultimately give in, choke down my two green beans, and wash off my plate.\n\nBut those wars were fought just once a week. My dad worked days and my mom worked nights; Thursdays were the only time we assembled for what we called \"sit-down family meals.\" Only years later did I recognize another dynamic at work. My folks split at the start of my senior year at UT, after 29 years of marriage. Suddenly it dawned on me that they'd never exactly been crazy about each other. That explained their work schedules and the tension around mealtime and the fact that my dad moved into my room when I left for college. It also provided a new name for the suppers he had cooked solo: Dysfunctional-Family Recipes. We ate a lot of fried bologna sandwiches and pancakes made with Bisquick and water when there was no milk in the house. A favorite among us three boys was something my dad called \"barbecued hot dog casserole,\" which consisted of butterflied foot-long wieners spread out in a glass dish, bathed in a full jar of hickory sauce, and baked. I'd always thought that eating a condiment and a casserole represented growth.\n\nOn weekends we'd occasionally hit the McDonald's drive-through as a full unit. I was, of course, unwilling to eat any of the already prepared items that give fast food its name. The burgers under the heat lamp sported mustard, pickles, and onions, and I wouldn't touch one, even with everything scraped off. Instead I'd insist on one specially made.\n\nThe cashier at the window would direct us to a corner of the parking lot, where we would sit in the station wagon and wait. My mom didn't believe in air-conditioning, and my dad didn't believe in bickering, so the interludes were quiet and uncomfortable. He might fiddle with the radio; she might comment that the car needed washing. My brothers and I would turn around to stare out the back window at the McDonald's front door.\n\nEventually an employee would emerge and bring out our order, then wait by the car while I inspected my burger. If so much as a hint of yellow mustard showed up on the outside of the wrapper, I'd send it back.\n\nChef Andrew Zimmern is the co-creator and star of a program on the Travel Channel called _Bizarre Foods with Andrew Zimmern._ For six seasons, he's played the part of the cheerfully daring food tourist, landing each week in a new spot on the globe to sample local staples, always something that would shock any eater back at his Minneapolis home. He's become a devotee, for instance, of spoiled foods. \"Whether it's fermented skate wing in Japan, or h\u00e1karl [fermented shark] in Iceland, or stinkhead in Alaska, fermented and rancid foods are eaten all over the world,\" he told me. He's had bat meat on three continents. \"Fruit bats are actually really clean. You can even eat all the innards because they have a very small diet in a very-small-ranging area.\" He once stood with members of the Masai tribe in a corral inside the Ngorongoro Crater, in Tanzania, drinking cow's blood directly from the source. \"That was a big jump,\" he admitted.\n\nThe segments are essentially snuff films for picky eaters, the kind of TV that would have once given me nightmares. \"It's been amazing to watch my gag reflex get less responsive,\" he said over the phone after a weekend exploring Montreal's finest seal meat dishes. \"I was certainly more trepidatious about food when I started. But when you taste something that at first scares you, that you don't understand or just don't want to eat\u2014maybe you've had a bad version of it before\u2014if it's good you learn to stop practicing contempt before investigation.\"\n\nZimmern the world traveler blames limited diets on cultural forces. \"I've been running a kind of experiment with my son, who's six. I've tried to get to him before the cultural guardians can. He had a book called _Yummy Yucky_ , and it associated worms with yucky. So he won't eat worms, which is very interesting to me. Because he loves crickets and june bugs and all of the other funky little things that are edible in our garden in the summertime. Sometimes we just sit and eat them off the ground.\"\n\nZimmern makes meals in his household sound uncomfortably close to meals on his show. But assuming he's not telling his son that he can't go inside until he finishes his bugs, his experiment isn't far from the fix suggested by every expert I consulted on getting past picky eating: Kids learn to enjoy food from parents who model\u2014not demand\u2014healthy eating habits. There's no way to predict how a child will react to a food; identical twins can have completely different diets. But as soon as a parent tells a kid that his preference is something that needs correcting, the discussion stops being about food. Nutritionists say most children need to be exposed to an objectionable food twelve to fifteen times to develop a taste for it. Psychiatrists insist that every plate at the table should have the same foods on it, but in portions that reflect what each person wants. Chefs suggest giving the kid authorship of his meals. Let him pick an item or two, then encourage him to help cook. If possible, plant a garden together. But above all, don't create a problem where none exists. The key is to provide regular, stress-free family meals.\n\nIt's worth noting, however, that none of the experts said those family meals had to be with people you were actually related to. Shortly after my parents divorced, I started law school at UT, and a new economic reality set in. With nothing but a student loan to fund my first year, I had to make adjustments in every aspect of living, and particularly in eating. Most meals came from boxes of frozen chicken breasts that my mom bought at Sam's Club. But each Sunday night I had dinner at the table of Marisol Vidal-Ribas Brown.\n\nAn elegant, aging daughter of upper crust Catalonia, Mrs. Brown had moved to Washington, D.C., in the mid-sixties and gone to work for the CIA, where she fell madly in love with a spy named Glenn. They married almost instantly, then raised two sons at various Pan-American locales where he was stationed before settling down in Austin in 1979. But Mr. Brown died in 1990, while their older son, Carlos, a college roommate of mine, was off at medical school. I started stopping in Sunday evenings to tutor Glenn the younger and get a free meal.\n\nHer dinners were different from anything I'd ever known. The dining room walls were covered with black and white photos of her parents and twelve siblings from before she left Barcelona, the women in gowns and the men in morning coats, some with a hand tucked inside their lapel. They each had Mrs. Brown's same long, somber Spanish face and seemed to be watching to make sure she held to her Old World upbringing. She did. There was always a crisp white tablecloth and polished chargers, along with the rest of Mr. Brown's family silver. We said grace. We drank wine, but never to excess. And we never stacked plates when clearing the dishes.\n\nShe would hold court at the table's head, fingering her pearls and Mr. Brown's wedding band, which she wore on a long chain around her neck. Her stories were incredible, often summoned by the meal she was serving. If she had been lazy that day and only managed to fix chicken, it might be good, but never as good as the chicken roasted by her governess, Tata, on a beach in Genoa when her family fled Barcelona during the Spanish Civil War. If she made Spanish rice, she'd point out that it wasn't Spanish at all but a variant of something she'd first tasted in Honduras, when Mr. Brown was keeping an eye on the Cubans.\n\nThe great lesson from her wasn't just to try food but to experience it. Well-mannered as she was, she wasn't above dropping her fork at a satisfying bite and grunting loudly, \"Oh, wow!\" And though she took an immigrant's pride in her American citizenship, she never let go of an ounce of her Spanishness. \"In Spain,\" she explained, in an accent that grew thicker as she got older, \"food is as big a part of who we are as Picasso or Gaud\u00ed.\" Gradually, because Carlos had been a picky eater too\u2014he and I didn't fully bond until he introduced me to the magic of late-night ketchup-only Whoppers at UT\u2014she started bringing out her native dishes. Paella made with saffron sent by one of her sisters. White almond gazpacho with frozen green grapes sunken in and topped with a dollop of aioli. She cleaned out the fridge like her mother had, by making what she called a \"tortilla apartment building\": four egg omelets, each with a different \"roommate,\" like potatoes, mushrooms, spinach, and shallots. She'd stack them one on top of the other, cover them with a simple red sauce, then cut slices, like a cake.\n\nI ate there once a week for the next ten years, continuing after Glenn left for law school, with a regular group of his friends that she called her \"stray dogs.\" As I started to experience adult life's little victories and defeats, she coaxed me through career changes and romantic entanglements, and our relationship became about more than meals. But food was how we expressed it. Before she moved away in 2001 to join Carlos's family in Los Angeles, we determined that our last outing together should be a trip to Central Market, where she would teach me how to \"buy Spanish.\"\n\nOn the day after Christmas 2006, I joined her and Glenn in Barcelona for a week of meeting her family and seeing her country. All I remember are the meals. Each day a three-hour afternoon feast was scheduled at someone's apartment, every one a rerun of our Sunday nights. But we decided to skip out on the final day's invite. Mrs. Brown wanted to take me to a famous restaurant near the harbor called Les Set Portes, which is Catalan\u2014not Spanish\u2014for the Seven Doors. \"I know it is touristy now,\" she said, \"but this is where my family came when I was a little girl.\"\n\nIn a huge formal dining hall with two attendants at our table, Glenn made the boring order, a simple seafood paella. Mrs. Brown had monkfish roasted in romesco sauce, a traditional Catalan accompaniment that looked like a creamy tomato sauce but was actually made from almonds, pine nuts, olive oil, and roasted sweet peppers. But I ordered best. I had a fideu\u00e0 negra, a paella variant with tiny Catalan pastas that looked like minced straw. They were soaked in squid ink and cooked with mussels, oysters, shrimp, and small, whole squid. The fish had clearly been caught that morning, and the taste was as rich as cake icing. It was the single greatest meal I've ever eaten.\n\nShortly before I left on that trip, my mother asked me to bring her back a gift. By that time, the nature of our food fights had changed, if not the outcome. When picking a place to eat, she would suggest something Southern fried and I'd push for sushi, just to get under her skin. She'd get as irritated at that torture as she once did the tantrums. But on this occasion she had a surprise: She asked me to return with a Spanish cookbook.\n\nWe flipped through it together when I got home, and I showed her some of the dishes I'd eaten. Most of them struck her as far too exotic. But then she saw a recipe for a lightly battered, pan-fried tilapia. We agreed that would be a meal we should prepare together.\n\nSweetly, she made no mention of pickles. To this day I've never tried one. Maybe I'll wait and do that with my own kids.\n\n### A PROPOSAL FOR FEEDING THE FAT AND ANXIOUS\n\n### By Josh Ozersky\n\n### From _Gastronomica_\n\n### Columnist ( _Time, Esquire_ ), founder of the Grub Street food blog, and star of his own web-based OzerskyTV, Josh Ozersky is the author of _Hamburger: A History_ and _Colonel Sanders and the American_ Dream. A man of prodigious appetite, Ozersky can be as entertaining as he is often controversial.\n\nI want to design a restaurant for fat people. You may be thinking, \"Wait, aren't all restaurants designed for fat people?\" They're not, not really. For one thing, almost all good restaurants are designed by slim androgynies wearing Buddy Holly glasses. Their friends are slim, and the people who eat there are slim. The servers are slim, and frequently slim and winsome. The cooks themselves, who in happier times were the very images of portly mirth, are now sinewy whippets, the cords of their young muscles visible beneath full-sleeve tattoos. Even the very chairs and tables themselves are designed for thin people. In short, what is needed is a full, radical rethinking of restaurants from the point of view of fat people\u2014something like what Temple Grandin did for beef cows, but further up the food chain.\n\nI believe that I am the man for that job. The fat have a fellowship, a shared knowledge that regular people can never grasp. Once Michael White, a chef of no small bulk himself, looked at my shoes and said, \"I see you have 'fat man laces.'\" \"Fat man laces?\" I asked. \"What's that?\"\n\n\"They're tied on one side of your shoe. Fat guys tie their shoes with their legs crossed because they hate to bend down.\" The inescapable truth of this hit me; a glimpsed image in a passing mirror, and I knew he was right. I did have fat man laces! Sherlock Holmes, cadaverous though he was, might have figured out such a thing\u2014but he would never have known the indignity of pushing and peering over his own stomach, of it getting in the way of his feet and knees like the gross, distorted imposition a fat man's stomach really is. Nor would the genius of Baker Street ever deduce the general unsteadiness that threatens a fat man on those rare occasions when he bends down, trying to hold bodies in balance when every law of physics seeks to topple him unhappily to the floor.\n\nI know about this because I've been varying degrees of fat for most of my life. I'm at a low ebb at the moment, and this has allowed me to reflect on how to make a restaurant for fat people. We're only really happy at restaurants, you know\u2014and then only for a few minutes. (We prefer to be at home, eating over the sink or munching away in front of a monitor of some sort; but the more gregarious of our race, when we do go out, generally head to restaurants\u2014or would, if they were better designed.)\n\nFor example:\n\n\u2022Fat people have bad backs and poor posture. They don't like to sit in narrow, hard-backed chairs. What they really want most is a padded La-Z-Boy or some such contraption. But since eating requires an upright posture, a well-padded chair, with rests for bulky broad arms and plenty of lumbar support down low, is a must. Fat people live in a world with twice the gravity of Earth, after all. That's why they wheeze and waddle the way they do. We're not saying it's OK. It's not.\n\n\u2022The restaurant should be cold, too cold for thin people. This will have the doubly beneficial effect of driving thin people out, because, really, who wants to look at thin people? And of course fat people, their swollen, unhealthy bodies working hard just to pointlessly stay alive, are fiery furnaces deep within, churning and chewing away beneath troubled brows. We require constant refrigeration just to keep going. Thermostats should be set for a frigid 55 degrees, with all fans set on high for maximum cooling.\n\n\u2022The servers need to be fat themselves, but fat in a non-threatening way. The last thing a fat person wants to see in his or her server is a sweltering, shameful wretch, wincing under the stigma of her body image. On the contrary! What's needed is a cheerful vulgarian, a postmenopausal mother figure with a ready smile and a sprightly line of patter. The sort of lady who might say, \"Do you want some more coffee, Hon?\" were you somewhere more downscale. I say, \"The last thing a fat person wants to see\"\u2014but of course, there is something a fat person wants to see less: a thin, long-limbed, carefree twenty-something, glorying in her sexual prime and regarding the customers as so many hideous zoo animals, waiting to be fed. (They don't need to actually have this attitude; just being young and attractive is enough.)\n\n\u2022Speaking of sexuality, there shouldn't be any. Dining here is a solitary and celibate experience, in which both sexes are protected from even a hint of having to socialize. For this reason, every table is a solo one, kidney-shaped\u2014its full space angled to the diner's sad eyes and ravenous maw, its shape allowing advanced convex gut docking, as well as maximal hand and arm reach in every direction.\n\n\u2022Certain design elements are mandated for the Fat Restaurant. Obviously, there will be no mirrors, frosted or otherwise, anywhere, and the lighting will consist of a dim and melancholy twilight interrupted only by chiaroscuro spotlights not on the table, indeed, but rather on the food itself. Always the food itself\u2014only the food.\n\nSpeaking of which, the menu will consist of, but not be limited to, the following:\n\n\u2022Large joints of meat\u2014most notably the shoulder, leg, buttock or round, saddle, baron, and ham, suitably burnished with a luminous glaze, to dazzle the weak, beady eyes of gourmands, and bring a temporary sparkle to them. The imposing size of large cuts dignifies the act of eating, and the fact of having one entirely to yourself gives a temporary sense of value and worth to the customer. Uneaten portions can be used for subsequent courses, their fat rendered for hash browns, the exposed pink flesh seared off in saucepans with olive oil or brown butter, the more difficult pieces ground up for hash, the bones split and seared for marrow (to then be used as a dressing on the hash).\n\n\u2022High-piled platters of fried foods, including chicken, cutlets, country-fried steak with cream gravy, crusty onion rings breaded with panko, matzo, fine flour, coarse flour, and\/or pork cracklings; non-vegetable tempura items; semi-boned chicken wings; untrimmed shoulder pork chops in joyously shatterable beer batter; cod filets; shoestring French fries; slow-braised short ribs; tender melting lamb breast; long-simmered veal stew chunks; and other softened meats, pressed and refrigerated and bound with their own collagen, and then plunged into cauldrons of the appropriate boiling animal fat; and tater tots, lots of them, dressed with fried garlic and Maldon salt. (These should also be available as bar snacks, petit fours, and bathroom mints.)\n\n\u2022Grilled cheese prepared on the airiest conceivable bread, thin and diaphanous to the point of abstraction, orgiastically slathered with oleomargarine, and containing nourishing viscous, mild and rich slices of bright-orange American cheese, such as gluttons remember from the faint mists of their childhood, when a future entombed in necrotic, immobilizing tallow still lay unimagined.\n\n\u2022Heavy stews, civets, porridges, congee, risotto, lush plovs, and pilafs, and other starchy, melting media for fat and flavor. Each bite is prized by the portly for their soporific effect and the brief periods of torpid slumber that result.\n\n\u2022All variations of the hamburger, including meat loaf sandwiches, sliders, meatball subs, leftover Salisbury steak served on slices of untoasted potato bread, massive steakhouse burgers with carbonized char marks and red bleeding interiors redolent of zinc and Roquefort; flattened coffee-shop discs, served on large toasted white buns, on each half of which a slice of tangerine-colored cheese (see above) has been melted; maid-rites; Jucy Lucys; sloppy joes; steak tartare on toast points; and Manwiches.\n\n\u2022High-piled layer cakes with copious amounts of lemon frosting, chocolate fudge, or coconut, depending on age and region of the diner, with or without heavy dollops of fresh whipped cream, and who's kidding who, it's obviously going to be \"with.\" (And let's have another dollop over here, too, thank you.)\n\nOnce the diner has finished eating, no further presence on his or her part is required: no labored shifting in the chair to accommodate a burdened body, no polite excuses to go out and take an \"air bath,\" no endless wait for the server to return. Fat diners, at the moment of having eaten a big meal, are at the absolute nadir of their day-to-day existence: their best and only pleasure and goal has been sated, leaving in its wake an aching, angry nausea and a self-hatred almost as deep as the pleasure they've just taken at table. They no longer wish to sit alone with their thoughts. They don't want to face the far side of the meal now working its way through the bilious labyrinth of their innards. All they really want is to get out, and quick. So an EZ Pass-type device, perhaps implanted in an earlobe or neck fold, will debit their bank account as they pass through the gate of the Fat Restaurant back out into the world, until they are ready to return.\n\n### BONE GATHERER\n\n### By Mei Chin\n\n### From _Saveur_\n\n### Essayist and fiction writer Mei Chin often stretches the bounds of food writing, mixing in elements of memoir and magical realism. Besides award-winning _Saveur_ articles, her writing has appeared in _Gourmet, Vogue,_ the _New York Times,_ and on her website bastethebook.com.\n\nI've always loved meat on the bone\u2014spicy, messy chicken wings; pan-fried pork chops; the beef ribs my mom used to bake, coated in bread crumbs and mustard butter\u2014but I never really thought about bones until a recent trip to South America forced me to take them seriously. I had signed on as a camp cook for a birding expedition to a remote part of Central Suriname. We were helicoptered onto a patch of bare rock several thousand feet above sea level, a place where, we were told, no human had been before. We were surrounded by jungle filled not only with birds, but venomous snakes. I was assured by my companions that any resident jaguars would mistake me for a small mammal and, hence, lunch. Torrential rains flooded camp every night, and our waterlogged satellite phone died, leaving us with no contact to the outside world.\n\nIt was a feral life. I hacked through bamboo with a machete, washed my hair in a stream. I cooked with peanut butter, rice, and from time to time, the roasted carcasses of the birds that we had collected. I had learned the weird but beautiful art of preparing specimens, a painstaking process in which you separate the bird's skin from its flesh while leaving much of the skeleton intact. We had set up bird-checking nets a bit higher on the mountain, and I would check these while my mates were out exploring. The rocks above camp were slippery with moss and rain. I have been clumsy since I was a child, but in the mountains, I got very good at falling\u2014indeed, I became kind of addicted to it.\n\nIt wasn't until a couple of weeks after my return to the States, when I took a spectacular spill on some hotel stairs, that my falls on the mountain came back to me with a vengeance. I found myself in excruciating pain, with a broken hip, and doctors were telling me that my left femur\u2014the leg bone between the pelvis and knee\u2014was so messed up, it would have to be replaced. I'd be off my feet for months. According to the older Chinese women in my life, I was supposed to eat a lot of bones, in keeping with the traditional Far Eastern belief that you should eat whatever body part is ailing\u2014owl eyes for myopia, pig lungs for emphysema. Condemned as I was to crutches and virtual house arrest, the old beliefs started to make sense.\n\nBesides, despite my own fragility, bones are powerful things. In ancient China, they were used to make prophecies. In the Gabriel Garc\u00eda M\u00e1rquez novel _One Hundred Years of Solitude,_ a girl carries her parents' bones around in a sack, where they clank and groan until a spot is located for their burial. In my favorite fairy tale, the bone of a murdered man is carved into a flute, which plays a song that reveals the killer. There is a restlessness in bones, a personality that endures long after the owner has passed on.\n\nI began my convalescence by making a stock from beef necks and veal knuckles the color of old lace. When I saw them\u2014beautiful, haunting\u2014I was reminded of the animals from which they came. When you look at a steak you don't necessarily think of a steer, but the neck bones, shaped like giant jacks, conjured the massiveness of the animal and how it moved.\n\nNothing demonstrates the elemental magic of bones more aptly than a stock. Any Chinese child with the flu will know the taste of pork bone and ginger stock, hot, heady, and healing. Korean babies are weaned on _sullongtang,_ the milk-white soup made from beef bones simmered for anywhere from 12 hours to days on end. The French chef Auguste Escoffier claimed that a great kitchen is founded on great stock; serious cooks approach their stocks with shamanic intensity. It's a matter of extraction. Protein, sugar, and fat break down during cooking and are released from the meat and bones into the water in which they steep. And while the meat contributes to flavor, the bones, loaded with collagen, impart body and a velvety mouth-feel.\n\nHome from the hospital, staring at the stove in my apartment's small kitchen, revisiting old volumes on my shelves\u2014the fairy-tale collections and cookbooks and photo albums\u2014I began to entertain a romantic notion of the perfect broth, based on the memory of a _brodo_ I had when I was 11 years old on a chilly March evening in Venice. Limpid, sweet, and nuanced, it was as fortifying as wine or tea\u2014a rich yet balanced infusion of meat, bone, and aromatics. Broth has always been part of my cooking repertoire, but I've frequently allowed mine to boil because I could not be bothered to watch the pot. If there's one thing all the cookbooks I now pored over agreed on, it's that should your broth ever so much as begin to boil, you should throw it away. During boiling, particles of fat and protein are agitated and become suspended in the liquid; a boiled broth is murky and greasy.\n\nIn pajamas and slightly stoned from daytime television and Percocet, I had plenty of time on my hands\u2014time enough, finally, to heed the experts. I set my stove to its lowest heat and prepared to wait a very long time: By all accounts, the water\u2014and beef bones and turkey wings, carrots, onion, garlic, celery, and bay leaf\u2014would take more than an hour just to come up to temperature. I left the pot on the stove overnight and all the next day. A _brodo_ should barely simmer; several seconds should pass between bubbles. At a very low and constant heat, unwanted impurities released from the meat and bones will coagulate and rise to the top or cling to the sides of the pot, and they can be easily skimmed off.\n\nWhen at last I strained the broth, the result was pure alchemy: a clear, golden liquid with a perfume much greater than the sum of its parts\u2014there were notes of caramel and nutmeg, butter and clove. It was one of the most thrilling moments I've experienced as a cook. How often do we manage to duplicate perfectly a romantic notion? I garnished my first bowl with curls of Parmesan and sipped it slowly, inhaling the sweet steam.\n\nOf course, these days it's trendy to be into bones, not only wings and ribs, but chicken necks and ham hocks and shanks. Much as many chefs can now be found flaunting their affinity for bones, they're still a fantastic bargain: At my local butcher, marrow bones go for $2.99 a pound. This is true of all sorts of bones and bony cuts. Sometimes, if a customer orders a noisette\u2014the meaty eye of the rack of lamb\u2014my butcher will even give me the bony remainder for free.\n\nYears ago, the same butcher had taught me how to french a rack of lamb, a technique that involves scraping some of the meat away with a long, thin knife to lay bare a fringe of elegantly curved bones. Now, laid up and armed with a boning knife, I found the taxidermy skills I'd acquired in Suriname useful. I started frenching everything in sight, and was alarmingly good at it. I turned chicken wings into chicken lollipops and frenched itty-bitty rabbit racks. I found out that the technique also worked wonderfully with shank\u2014the length of bone and meat just below the knee\u2014by far my favorite part of any animal.\n\nThe marrow was silk on my tongue, and yet the white bone on the plate retained an echo of the visceral and the wild.\n\nLamb shanks braised low and slow, until the meat is tender and the bones release their marrow to enrich the braising liquid, are always marvelous served with something starchy to soak up the sauce\u2014polenta, mashed potatoes, risotto\u2014but I like them best when they're set, gigantic and resplendent, on a bed of white bean pur\u00e9e. Frenching the shanks makes the presentation that much more spectacular, a hunk of meat beckoning at the end of a length of parchment-colored bone. When I tried it, I browned the shanks thoroughly before putting them in the oven, and I made sure to turn the meat every half hour or so for an evenly caramelized exterior. Cooking a shank in this way is virtually foolproof due to its high ratio of bone to meat; because the bone absorbs heat, the meat immediately surrounding it cooks slowly and is the most succulent. And let us again not forget the collagen that attaches the meat to the bone. Over the course of cooking it turns to gelatin\u2014a special treat to enjoy once you've dispensed with the meat.\n\nThen there's marrow. When the creamy, voluptuous stuff is scooped from the bone's hollow, it can be stirred into a sauce to add lushness. It is the best part of an osso buco\u2014that's Italian for \"bone with a hole\"\u2014and once you've stripped the meat from the long-braised veal shank and devoured it, inside that bone you'll find a final treat, a secret store, best coaxed out with a long, slender spoon.\n\nTo eat marrow\u2014the tissue that produces new blood\u2014is to indulge in an act that treads the boundary between the rude and the refined. There was, in fact, a time not too long ago when my supermarket was selling marrow only as a dog treat. But once I felt well enough to put on a dress and hail a cab, marrow was the first thing I sought. Together with my new hip\u2014a man-made bone fashioned from enameled metal\u2014I headed to a Manhattan restaurant called Ai Fiori.\n\nIf eating marrow is typically a messy, primal, hands-on affair, at Ai Fiori, chef Michael White has resolved the issue by halving the bone lengthwise. For the dish he calls _Mare e Monte_ (\"sea and mountain\" in Italian, a play on surf and turf), White lines the halved bone with celery root pur\u00e9e, nestles in overlapping disks of steamed scallop and black truffle, lays out a layer of marrow on top, and then broils the whole thing. It was silk on my tongue, marrow I could eat with a knife and fork, a subtle balance of flavors and textures\u2014and yet the white bone on the plate retained an echo of the visceral and the wild. It conjured what lurked in the shadows on that mountaintop where I fell so many times, and it evoked my mending body ensconced in that gleaming haute dining room, my crutch still at my side.\n\nWith bones, in other words, the possibilities for reincarnation are endless. A joint becomes a stock, which then becomes the base for _pot au feu,_ or another rich, meaty stew. I've even taken to roasting cuts from animals on racks made from their bones, a roast beef on a bed of marrow bones. It's culinary id. At some point, though, my fridge started to look like a boneyard, my hair smelled like veal, and I began to long for another life, one away from the stove and skipping on both legs. Still, I am grateful for the chance that being hobbled for a while presented me: to linger in the kitchen while things cooked slowly; then to grip the bones in my fist, use my teeth to strip the meat, and quietly relish the savagery\u2014and the delicacy\u2014of it all.\n\n### THEY DON'T HAVE TACOS IN THE SUCK\n\n### By Katharine Shilcutt\n\n### From _Houston Press_\n\n### In June 2010, blogger and web editor Katharine Shilcutt inherited Robb Walsh's mantle as food critic for the alternative weekly _Houston Press._ Since then, Shilcutt rarely has time to update her original blog SheEats, but she makes up for that on the Press's blog Eating our Words.\n\n\"Can I have the hot dog, please?\" I asked the woman inside the bright green Tacos D.F. truck on Long Point at Witte.\n\n\"You're ordering a hot dog?\" teased my friend Ryan with a chuckle. He'd already placed his order for a _pastor_ taco and a can of Coke at the window. \"I thought we were doing a taco truck crawl.\"\n\n\"I'm getting a taco, too!\" I grinned sheepishly, before placing an additional order for a taco _de cabeza._\n\n\"Is that what I think it is?\" asked Ryan as he eyed the cabeza. Shreds of fine beef from a cow's head a la _barbacoa_ filled the double corn tortilla that the woman handed through the window, topped with a handful of raw white onions and cilantro leaves. Despite his initial misgivings over its provenance, he ate his half of the taco with relish\u2014pronouncing it \"great\" when he was finished\u2014and I remembered why I'd missed him so much.\n\nRyan was my best friend in college, where we fancied ourselves a couple of misfits at a highly conservative university that made both of us itchy and desperate with discomfort. We met on the first day of school our freshman year, both of us shunted into an off-campus apartment complex because the dorms were overflowing in the late '90s and, somehow, releasing 17-year-olds into the wild seemed like a good idea at the time.\n\nRyan couldn't cook. I had roommates that I hated. We bonded over shared meals in his apartment and nights spent commiserating with each other about the limited kinds of politics and religious dogma that teenagers understand while the other kids rushed sororities or went to Bible study. Until last week, I hadn't seen him in ten years.\n\nAfter a few minutes, the woman in the Tacos D.F. truck handed over my hot dog. It was a small frank inside a small bun, but the whole thing was topped with a confetti blast of ketchup, mustard, mayonnaise, diced tomatoes, raw onions and pickled jalape\u00f1os that packed remarkably little heat. While I mused over the surprisingly sweet peppers, Ryan finished his other taco with gusto.\n\n\"Where else can you get real meat that someone bought themselves for $1?\" he mused rhetorically. \"Where else can you get real meat that someone bought and then cooked right in front of you and handed to you for $1? You can't do that at Taco Bell.\"\n\nWhen he'd emailed me that he was coming to Houston for the day, I assumed that it was to visit some of his Texas family he'd left behind after joining the Air Force one day out of the blue during college.\n\nRyan had been attracted to EOD\u2014explosives ordnance disposal\u2014upon enlisting and quickly advanced to Tech Sergeant as he discovered within himself a serious and previously unknown talent for defusing bombs. He'd done four tours since 2002, in both Iraq and Afghanistan. He married a lovely German girl in between and finally settled in Florida when he wasn't in some far-flung region with a terp at his side, sweeping for mines in vast deserts.\n\nBut the trip to Houston was for the tacos.\n\n\"What is the planned criteria\/theme of our hunt?\" Ryan wrote me in an email a few weeks before he came to town. \"Keep an eye out for goat tacos.\"\n\nWe didn't find any goat tacos on Thursday afternoon, but it didn't matter. It was as though a decade hadn't passed, and we fell into the same easy rhythms of bullshitting and storytelling that we always had.\n\n\"So, you're really a food critic?\" he asked as we finished our cans of Coke outside Tacos D.F.\n\n\"Yeah,\" I responded, with a little elaboration after some prompting on his part. Yes, it's my full-time job. Yes, I really do get paid to eat. No, I'm not anonymous. \"But I want to hear about _your_ job.\"\n\n\"Most of my friends are dead,\" Ryan responded immediately, point-blank. \"My boss was killed last year.\" His faced darkened briefly. I didn't know what to say and stammered softly until he started talking again.\n\n\"Let's go hit the next one.\"\n\nWe climbed back into my SUV and drove until he pointed another taco truck out, his eyes scanning both sides of the road in a practiced motion. \"Tacos Arcelia. Let's try them next.\"\n\nTacos Arcelia has two things going for it: The first thing is the 99-cent tacos that it advertises in bold black letters on the side of its second thing, a school bus that's been painted bright red and silver. There was already a line forming around noon on Thursday, so Ryan and I figured it was a sure bet.\n\nWe ordered a taco each\u2014 _lengua_ for me, _chicharrones_ for him\u2014and stepped back to await our orders. Even working in the larger-than-average confines of a school bus, the crew was moving at a slow clip.\n\n\"You know that part in _Black Hawk Down_ where an RPG gets lodged in a guy's chest but it doesn't go off?\" asked Ryan idly while we waited.\n\n\"Uh, yeah. Although I hadn't thought of that movie\u2014or that scene in years.\" I didn't ask why he asked me, wary of the answer. He told me anyway.\n\n\"That really happens.\"\n\nI thought back to the time when Ryan and I were making dinner at his apartment one night, both 18 years old, and I'd stupidly thrown a handful of frozen okra into a deep pan of hot grease to fry, not knowing any better. I started a minor grease fire which we quickly put out, but my face and hands were pockmarked with grease burns that took a few years to fade. The burns hurt terribly and I avoided frying anything at all for at least another five years, scared to death by such a minor injury.\n\n\"Have you considered moving out of EOD?\" I asked finally. \"I know there are other areas of the Air Force you could go into,\" I added with a little laugh, hoping he wouldn't be offended by the suggestion that he leave an area which poses clear and constant danger to his life every single day that he's on duty.\n\n\"No way,\" Ryan replied. \"If I stick it out another 10 years, I can retire on a full pension. Retired at 42. Can you imagine?\"\n\nI chuckled. \"No, I definitely can't.\" Just then, our orders came up.\n\nThe corn tortillas were listless and anemic-looking, a pale color that was closer in hue to flour tortillas. My pile of diced lengua was equally pallid, and a bite of the tongue confirmed that it tasted as bland as it looked.\n\nRyan's taco, on the other hand, was filled with more vibrant-looking pieces of chicharron. The fatty skin was puffy and thick with a spicy red sauce that made me mourn the terrible lengua even more. Ryan was clearly proud of his choice, too, grinning as he finished the rest of the tortilla off.\n\nThe grin never left his face as he told me about his EOD training, about the dozens of minute tactical decisions and assessments that have to be made before even approaching a bomb or a mine or an IED. He also told me about how he rarely wears the 90-pound bombsuit meant to protect him from the 132 explosive devices he's defused in his decade with the Air Force.\n\n\"Everyone knows the bombsuit and everyone associates it with EOD,\" he said. \"It's like everyone knows a firefighter's jacket and helmet. But it also weighs 90 pounds. So we make the choice: Carry around 90 pounds worth of equipment all day long, or be better and faster without it.\"\n\n\"And if I can be better and faster,\" he finished, \"that means a bunch of 18-year-old kids can go home safely from the war. I'd rather sacrifice one of me than a bunch of them. At least, that's the way I look at it.\"\n\nWe stood in silence for a few seconds after that. I contemplated the ways in which one makes a decision like this every day, and the ways in which so many of my own memories of Ryan are tied to us being 18-year-old kids ourselves. I balled up our trash to throw it away, then we walked quietly back to my SUV to continue the tour.\n\n\"What will you do when you retire then?\" I asked as we headed out. Would he and his wife enjoy their home in Florida, the new boat he just bought? He was briefly contemplative before answering.\n\n\"I want to do something quiet,\" he said. \"You know, like become a firefighter.\"\n\nWe both laughed, although I knew he was quite serious. And suddenly the conversation had turned back again to our old favorite subject.\n\n\"I ate at this crazy Puerto Rican buffet up in Dallas recently,\" Ryan began as we drove on.\n\n\"Let's get a palate cleanser,\" I told Ryan as we pulled into the parking lot of the New Flea Market on Long Point at Pech. On the weekends, you can't find a space to park in the asphalt lot. But today, on an overcast Thursday afternoon, it was empty except for a few trucks parked haphazardly around Refresqueria Rio Verde.\n\n\"What are we getting here?\" asked Ryan as we climbed out. \"Do they have tacos?\"\n\n\"Sure, they have tacos,\" I said. \"But I thought we'd get something different in between. Do you like _elotes_?\"\n\nElote, as I explained to Ryan, is basically corn on the cob. But instead of serving it with butter and salt, as us white folks tend to do, elote is served with _crema,_ chile powder, lime juice and a host of other condiments that only seem foreign until you taste them all mixed together. Elote in a cup, the shaved kernels topped with a thick dollop of cream and a rough shake of chile powder, is mystifyingly comforting even if you've never had it before.\n\nI ordered a cup for myself and a giant glass of _tamarindo_ for us to split, while Ryan went whole hog and got an elote-on-the-cob. \"I have corn on a stick!\" he called out to me like a little kid. And between swigs of the sweet, apple-like tamarind juice, Ryan bluntly asked: \"So, what happened? You were married for, like, a second.\"\n\nRyan himself has been married for six years. As so often happens with Air Force men, he met a pretty German girl while stationed at the Rammstein Air Base in southwestern Germany on the edge of the hilly, green _Pf\u00e4lzerwald_ forest. They were married in a castle. She is beautiful, with expressive blue eyes and a kind face.\n\nI gave Ryan a brief rundown of my own fumbling attempt at marriage, the millions of tiny ways in which my ex-husband and I both failed at the institution every single day until we were both relieved to finally call it quits a year and a half later. Ryan listened with a playful smirk on his face as I explained how I fell into the trap of being pursued by a good-looking athletic-type\u2014the weak spot of too many nerdy wallflowers the world over, men and women alike\u2014and refuted at least one point.\n\n\"You're not really a nerd,\" he laughed. \"You're more of a pop culture dork. You're a female Chuck Klosterman.\"\n\n\"I don't know that being a female Chuck Klosterman is such a great thing!\" I replied. The smirk was still on his face. He was waiting for his turn; I could tell. I promptly shut my mouth and let him have it.\n\n\"Well, the missus and I,\" he began grandly, \"have been together since day one.\" He told me the story of their brief courtship and the mutually agreed-upon eventuality that they were destined to be together, so why spend useless years dating? It was sweeping and romantic and beautiful and everything you could want for your best friend, or for anyone with a good heart who deserves to meet another good-hearted soul in this world.\n\n\"When we got married,\" he told me, \"I asked her: 'How much of what happens over there do you want me to tell you?'\" It suddenly occurred to me that I wouldn't know the answer to that question were I married to a military man myself. But Ryan's wife knew the answer immediately: She wanted to know everything.\n\nThose shared experiences became a bond between them, and Ryan grew even closer to her over time than he imagined possible. One day he told her: \"You're my reason.\"\n\n\"My reason?\" she wanted to know.\n\n\"You're the reason I want to come home after every deployment,\" he told her. \"When I'm sent away on a six-month deployment, I just picture myself walking home to you. It's what gets me through. I picture myself walking over a huge mountain for six months until I see you again.\"\n\nWe sat and grinned goofily at the mountain Ryan had traced in the air with his hands. His corn-on-a-stick was gone, my cup was empty.\n\n\"My palate is cleansed,\" he announced happily. It was time to move on.\n\nRyan and I had been driving for a while, for many blocks since our \"palate cleanser\" of _elotes_ at Refresqueria Rio Verde. I knew he was wondering why I passed other taco trucks and failed to pull up to them, but I had a plan.\n\nIn my mind, I knew this stop would be our last taco truck of the day. I had to pick my cousin up from the airport soon, and Ryan had to get back on the road.\n\nI pulled into our final destination: El Ultimo, a brightly decorated taco truck near Long Point and Wirt. Its parking lot was already busy, a line had already formed outside that was composed entirely of blue collar workers off for lunch, equal parts white, black and Hispanic. I've made no secret of the fact that El Ultimo is my favorite taco truck in town, and I have followed it over the years as it moves a few blocks up and down Long Point.\n\n\"On the weekends,\" I told an impressed Ryan, \"it has a waitress who takes your order, since the line gets so long.\"\n\n\"So this is your favorite, huh?\" he said, eying the simple menu and wondering what exactly made this spot so special.\n\n\"Yes. You'll see.\"\n\nThe wait at El Ultimo was the longest of the afternoon, and Ryan and I had run out of polite conversation. He told me about the few phrases he's learned working in Afghanistan, about how Pashto and Dari only sound alike on the surface. Once you get to know them, he said, you can immediately spot the differences when you hear them spoken, intermingled, on the streets.\n\nHe tried to teach me a few phrases in Pashto. \"Move it, asshole!\" was one of them. I couldn't pick it up. I was too busy laughing absurdly, thinking of Ryan in a wholly foreign country, yelling out practiced Pashto phrases like these to his terp in what must now seem like a completely normal occurrence to him.\n\nWe commiserated about how rusty our Spanish had gotten over the years, useful these days only for ordering food at taco trucks. Ryan was even more out of practice, blaming it on the sad dearth of taco trucks back home in Florida. \"There are only, like, two where I live,\" he grumbled. And he told me about how his German wife was startled one morning by the realization that she had started to dream in English.\n\nWhen our tacos came out, Ryan finally saw what I did in El Ultimo: The tacos here are on soft, homemade flour tortillas\u2014not corn, interestingly\u2014and come with more than just the standard handful of cilantro and onions. Green slices of avocado and white crumbles of _queso fresco_ fill the tacos, too, along with our chosen meats: fatty shreds of barbacoa and annatto-hued _pastor._\n\nRyan gulped his taco down, pausing only briefly to admire the avocado and cheese on top. Then he ordered three more.\n\n\"This is your favorite, huh?\" he asked again.\n\nI nodded once more, pleased.\n\n\"Well, it's my favorite too.\" He smiled. And then: \"I don't think I can eat any more tacos.\"\n\n\"Neither can I,\" I laughed back. We went back to the car with his extra tacos, and were nearly ready to go when Ryan said: \"Katie, aren't you forgetting something?\"\n\nOnly a few people call me Katie anymore; I thrilled to the sound of hearing the adolescent version of my name, as if time hadn't passed at all and I was still Katie, still 17-years-old and big-eyed and strong.\n\nI was forgetting something: I hadn't taken his photo in front of the truck, as I'd done with all the others. I dug my camera back out of my purse as Ryan posed, hand out and thumb up, grinning. I snapped one final picture, and we packed it in.\n\nThe car ride back to Ryan's truck seemed to last almost as long as the crawl itself had. I listened hard to every last one of Ryan's words, even the awful ones and the cruel ones that involved horror stories of friends killed in battle. I listened as he told me about watching as weeping fathers carried their children into makeshift hospitals, limbs absent and blood reeling out of charred wounds. I listened as he told me about fathers who strapped bombs to their children's thin chests and sent them out to fight the battles their cowardly parents could not. I listened as he told me about watching a young girl's leg blown off by a crudely designed IED that he had not seen, was not able to defuse.\n\nI listened as he told me about being blown up twice himself. He stared forward the entire time as he spoke, and I noticed for the first time what looked to be shrapnel wounds on his head. The curved wounds were barely noticeable except where the hair had stubbornly refused to grow back. I didn't ask about them. My chest burned as he spoke about being lucky enough to survive both times he was attacked. He'd never shot anyone, he assured me. But he'd shot _at_ them.\n\nI thought back to something he'd said earlier that day: \"I'm happy to get bad guys and help people,\" he'd put it, simply and succinctly. This terrifying job makes him happy. This job that could wipe him out of existence in one trembling second makes him happy. Instead of happy thoughts, my mind was filled with horrific visions of Ryan dying in battle. I was ashamed of myself for thinking such a thing.\n\nBefore I could get a word out, we were back at his truck. Ryan was unstrapping his seat belt. Here was 10 years, gone in an afternoon.\n\n\"You know, I looked for you,\" he said, suddenly and without warning. Ryan had entered the Air Force on a whim between our junior and senior years of college, finally disillusioned enough with college after three years to make the leap. And although we kept in touch for a while, we never saw each other again. We finally lost contact entirely after his first deployment.\n\n\"I looked for you everywhere. I looked on Facebook and even MySpace, back in the day, and Googled you and then one day I found you. It was totally by accident. I was reading an article, and you had written it.\" He had sent me a message on Facebook later that day. I had been thrilled to hear from him, with no idea of how long he'd searched for me.\n\nI had no idea what to say. Finally, all I could get out was: \"I'm glad you did.\"\n\nAnd then, because the hour was drawing so near: \"I've got to get going.\" And I tried to follow it up with a casual, \"What else are you doing to do in Houston today?\"\n\n\"Nothing,\" Ryan said. \"I'm driving back today. I really just came to see you.\"\n\nSo it wasn't just the tacos. And again, all I could manage was a short: \"I'm glad you did.\" A smile. Unblinking eyes, because if I blinked, the tears would spill over and I'd be done for. I couldn't see Ryan's eyes at all; he never removed his sunglasses all day.\n\nAnd just like that, his door was open. A brief hug and promises of a future visit\u2014this time with his wife\u2014and Ryan was gone. Here was 10 years and two hours, gone. I drove out of the parking lot, unable to look back, and drank the last of the apple-sweet tamarind juice until it was gone and I was home once again.\n\n### I WON'T HAVE THE STOMACH FOR THIS\n\n### By Anna Stoessinger\n\n### From the _New York Times_\n\n### Native New Yorker, advertising writer, and self-professed gastronome Anna Stoessinger inspired floods of blog posts with this moving op-ed piece in the Sunday _Times._ In it, she answers a question few of us will ever have to ask ourselves.\n\nI am a ravenous, ungraceful eater. I have been compared to a dog and a wolf, and have not infrequently been reminded to chew. I am always the first to finish what's on my plate, and ever since I was a child at my mother's table, have perfected the art of stealthily helping myself to seconds before anyone else has even touched fork to frog leg. My husband and I have been known to spend our rent money on the tasting menu at Jean Georges, our savings on caviar or wagyu tartare. We plan our vacations around food\u2014the province of China known for its chicken feet, the village in Turkey that grows the sweetest figs, the town in northwest France with the very best raclette.\n\nSo it was a jarring experience when, a few months ago, at 36 years old, I learned I had stomach cancer.\n\nI had only mild symptoms at first: a slight pain below the breastbone when I swallowed, discomfort that felt like nerves or indigestion. Two doctors told me it was nothing. \"Take some Prilosec,\" they said, which made sense. We had just returned from a trip to Italy. In Florence, we had eaten mounds of roast duck, crostini and rich fish stews; maybe I just had heartburn. But the feeling lingered, and the hypochondriac in me went to the gastroenterologist.\n\nIt was a tumor. We got the call early on Friday morning. My husband and I were still in bed, and it took more than a moment to register. At my age, I am not supposed to have stomach cancer. In the United States, it's a disease that most commonly afflicts older, Asian men, and I am none of these. I have also parted with all my vices, save the occasional sugar binge. But after years of worrying that I might have cancer, years of, \"Can you look at this? Is this a lump? What's this right here? No, here,\" I actually did.\n\nI had only one thought about the possibility of death: the fear that I would have to part from my husband a half-century too soon. We had just married in October. We had just moved into a cottage in Connecticut. We had just discovered the simple pleasures of a happy routine. A calendar on the fridge. Roast chicken with leeks for dinner. Losing our life together was what death meant to me, and that, I think, is love.\n\nThankfully, my doctors assured me that death was a remote possibility. But I wasn't getting off easily; there were things to lose. First, with three rounds of intense chemotherapy, I lost my appetite. But that was only temporary. Then my surgeon told me that I needed a total gastrectomy\u2014I would have part of my esophagus and all of my stomach permanently removed.\n\nWith nothing but a small intestine left to digest food, my gastronomic future would hold only small, frequent meals, consumed slowly and deliberately, without my characteristic gusto. Without abandon. Without\u2014there would be a lot of without.\n\n\"You can live without a stomach,\" my doctor told me. I have often thought about what I could live without, if I had to: a savings account, an extra bedroom, the new Prada suede platform pump in burgundy. But a stomach never entered my mind. And food? It was so much more. As a little girl, sharing food with my mother was a solace, a joy, and a way of communicating. Sharing it with my husband has been as intimate as anything I've experienced. We fell in love one taste at a time: roadside cheeseburgers, bonito with ginger sauce, hazelnut gelato. After the first bite had lingered on our tongues, we'd say to each other: Wait for it. And then: Did you get that? The smoke? The spice? The texture? We always did.\n\nAnd so, with just 10 days left with my trusted stomach, we set out to capture all that food meant\u2014all the memories it conjured, all the happiness it brought. We were determined to eat as much and as well as possible. We made lists. What categories of food needed attention? Which meals did we want to recreate? We went from lowbrow to high, and everywhere in between. Peanut butter and jelly doughnuts, ginger ice cream, sashimi, grilled porterhouse, wild blueberries. We came up with a plan. Travel options were limited (health, timing), but we would go from Connecticut to Maine to New Brunswick, and finish in New York City three days before my surgery.\n\nOn the road, we ate candy in the car like kids. Then, at the White Barn Inn near Kennebunkport, Me., we ate a foie gras and fig torchon, which was velvety, buttery and dusted with pistachios; we ate butter-poached smoked lobster, the summery steam wafting up from the meat; and we tasted scallops with passion fruit coulis, thinly sliced disks of silky pleasure in a sweet, tangy sauce.\n\nMy mother made scallops like nobody else. Perfectly seared and turned in butter. Simple and divine. And she served them at her hugely popular, often impromptu, dinner parties. Watching her cook was what I imagined it was like to watch Jackson Pollock paint. She hurled salt and spices. Spun sugar like a sculptor. Emptied a bottle of rosemary onto a leg of lamb, massaged it with butter into the meat, and turned out a masterpiece. I surged with pride when the first guests arrived and remarked on the wonderful smells sailing out of the kitchen, to whose creation I alone had been witness.\n\nMy father was something of a tyrant, and every year my mother and I went to southern France to escape him. We were like war buddies on leave there, and we ate like queens. We drank tea out of giant bowls and picked lavender and stayed at wonderful old inns with names like L'Hermitage. There were cheese courses and pastries and the most delicious filet of sole I've ever encountered. There was also a deep and unwavering friendship between my mother and me, the tastes and smells of the food we shared overpowering even our worst memories of my father.\n\nThose summers came back to me at our next stop: the Kingsbrae Arms in St. Andrews, New Brunswick, which had an exquisite dining room, gardens full of lavender and a chef who studied in the south of France. There we sat down to a wild boar terrine and Guinness vegetable soup with rosemary whipped cream. It was sublime and hinted of beef, celery, sweet carrot and earth. Finally, there was a warm apple and cinnamon tarte tartin\u2014not too sweet, not too tart and not quite large enough. I ate mine and half of my husband's as well, and yearned for more.\n\nIt had been a long time since I had experienced such satisfying fullness. There was comfort and exuberance, a familiar feeling like a long embrace, a coming in from the cold\u2014that I fear I will not know again. I know I will mourn my loss. Because for me, food\u2014and eating it with abandon\u2014is about shared experience. It's about love and memory and the capacity to conquer even the worst hours with something warm and wonderful.\n\nBut let me be clear: I am unspeakably lucky. Had my diagnosis come even three or four months later, my prognosis would have been much, much darker. I had the surgery two weeks ago, and thankfully everything went smoothly. Once I've recovered a bit more, I will be able to eat again. In the future, my meals will be little intermissions throughout the day. Overtures, not full symphonies. They will be small, but I will try to make them grand. Even if it's just a spoonful of pudding. And I would give up all of my organs for the possibility of many more years with my beloved husband.\n\nWe had our last good meal together\u2014our last of the old meals\u2014in Manhattan, at Le Bernardin. It's the best place in the city for a final meal with a stomach, the best place in the city, arguably, for any meal. When I called the hostess for a last-minute table, I was told that the only seating they had was at 10:45. I pulled out the big guns: \"I have stomach cancer, and this is literally my last meal with a stomach.\"\n\n\"Well,\" she said, irritated, \"I suppose we can seat you at 5:30.\"\n\nWhat a town. And what a magnificent meal it was.\nRECIPE INDEX\n\nFirst-Boil Syrup, (from \"Sweet Spot\"),\n\nMinestrone (from \"How to Live Well\"), 88\u201389\n\nHomemade Mayonnaise (from \"Still Life with Mayonnaise\"),\n\nLasagna Bolognese (from \"Lasagne Bolognese\"), 105-110\n\nPot-Roasted Celery Root with Olives and Buttermilk (from \"The Forager at Rest\"), 114-115.\n\nWalnut Cake (from \"The Forager at Rest\"), 115-116\n\nBest Guess Wonton Soup (from \"The Legacy that Wasn't: Wonton Soup\"), 234-236\n\nSneaky Whole-Wheat Chocolate Chip Cookies (from \"Curious Cookies\"), 239-240\n\nCoconut Cake (from \"Sweet Southern Dream\"), 253-254\n\nLemon Layer Cake (from \"Sweet Southern Dream\"), 255-256\nPERMISSIONS ACKNOWLEDGMENTS\n\nGrateful acknowledgement is made to all those who gave permissions for written material to appear in this book. Every effort has been made to trace and contact copyright holders. If an error or omission is brought to our notice, we will be pleased to remedy the situation in subsequent editions of this book. For further information, please contact the publisher.\n\nShaw, Hank. \"On Killing.\" Copyright \u00a9 2011 by Hank Shaw. Used by permission of Hunter Angler Gardener Cook. Originally appeared on www.honest-food.net, December 7, 2011.\n\nJacobsen, Rowan. \"The Gumbo Chronicles.\" Copyright \u00a9 2012 by Rowan Jacobsen. Used by permission of the author. Originally appeared in _Outside,_ April 2012.\n\nMurray, Erin Byers. \"Serving Up Sustainability.\" Copyright \u00a9 2012 by Erin Byers Murray. Used by permission of the publisher. Originally appeared in _Edible Boston,_ 2012.\n\nO'Hagan, Maureen. \"Kids Battle the Lure of Junk Food.\" Copyright \u00a9 2011 by _The Seattle Times._ Used by permission of the publisher. Originally appeared in _Pacific Northwest,_ June 2011.\n\nCunningham, Brent. \"Pastoral Romance.\" Copyright \u00a9 2011 by Brent Cunningham. Used by permission of Brent Cunningham. Originally appeared in _Lapham's Quarterly,_ June 2011.\n\nGraham, Paul. \"Sweet Spot.\" Copyright \u00a9 2012 by Paul Graham. Used by permission of the author. Originally appeared in _Alimentum,_ Winter 2012.\n\nLeMay, Eric. \"Snowville Creamery Has a Modest Goal: Save the World.\" Copyright \u00a9 2012 by Eric LeMay. Used by permission of the publisher. Originally appeared in _Edible Columbus,_ Winter 2011.\n\nEstabrook, Barry. \"Matters of Taste.\" From _Tomatoland: How Modern Industrial Agriculture Destroyed Our Most Alluring Fruit._ Copyright \u00a9 2011 by Barry Estabrook. Used by permission of Andrews McMeel Publishing, pp: ix-xii, 139\u2013144.\n\nMueller, Tom. \"Olives and Lives.\" From _Extra Virginity: The Sublime and Scandalous World of Olive Oil._ Copyright \u00a9 2012 by Tom Mueller. Used by permission of W.W. Norton & Company, Inc.\n\nZandstra, Laura R. \"This Little Piggy Went to Market.\" Copyright \u00a9 2012 by Laura Zandstra. Used by permission of the publisher. Originally appeared in _Memoir Journal,_ Spring 2012.\n\nAdler, Tamar. \"How to Live Well.\" From _An Everlasting Meal: Cooking with Economy and Grace._ Copyright \u00a9 2011 by Tamar Adler. Reprinted with the permission of Scribner, a Division of Simon & Schuster, Inc. All rights reserved.\n\nAktinson, Greg. \"Still Life with Mayonnaise.\" From _At the Kitchen Table: The Craft of Cooking at Home._ Copyright \u00a9 2011 by Greg Atkinson. Used by permission of Sasquatch Books.\n\nEaton, Lorraine. \"The Fried Chicken Evangelist.\" Copyright \u00a9 2011 by Leite's Culinaria. Used by permission of Leite's Culinaria. Originally appeared in Leite's Culinaria, June 26, 2011.\n\nPerelman, Deb. \"Lasagna Bolognese.\" Copyright \u00a9 2012 by Deb Perelman. Used by permission of the author. Originally appeared on www.SmittenKitchen.com, February 12, 2012.\n\nMuhlke, Christine. \"The Forager at Rest.\" Copyright \u00a9 2012 by Cond\u00e9 Nast. All rights reserved. Originally appeared in _Bon Appetit,_ March 2012. Reprinted by permission.\n\nKummer, Corby. \"Better Cooking Through Technology.\" Copyright \u00a9 2011 by Corby Kummer. Used by permission of Wrights Media. Originally appeared in Technology Review, July\/August 2011.\n\nBirdsall, John. \"The Pastrami Dilemma.\" Copyright \u00a9 2012 by CBS Intereactive Inc. Used by permission of CBS Interactive Inc. Originally appeared on CHOW. com, February 2012.\n\nLevin, Rachel. \"Passover Goes Gourmet.\" Copyright \u00a9 2011 by Rachel Levin. Courtesy of _Sunset Magazine._ Sunset is a registered trademark of Sunset Publishing Corporation. Used with permission.\n\nClement, Bethany Jean. \"The 2011 Dyke March Wiener Taste Test.\" Copyright \u00a9 2011 by Bethany Jean Clement. Used by permission of the author. Originally appeared in _The Stranger,_ 2011.\n\nAnderson, Brett. \"The Missing Link.\" Copyright \u00a9 2012 by Brett Anderson. Used by permission of the author. Originally appeared in _The Times-Picayune,_ 2012.\n\nNabhan, Gary Paul. \"Foraging and Fishing Through the Big Bend.\" From _Desert Terroir: Exploring the Unique Flavors and Sundry Places of the Boderlands_ by Gary Paul Nabhan, illustrations by Paul Mirocha. Copyright \u00a9 2012 by Gary Paul Nabhan. Reprinted with the permission of the University of Texas Press.\n\nMariani, John. \"Italian America.\" Copyright \u00a9 2011 by John Mariani. Used by permission of the publisher. Originally appeared in _Saveur,_ December 2011.\n\nLam, Francis. \"What Makes Sushi Great.\" Copyright \u00a9 2012 by Francis Lam. Used by permission of the publisher. Originally appeared in _Gilt Taste,_ March 12, 2012.\n\nGordinier, Jeff. \"Food For Thought.\" Copyright \u00a9 2012 by the _New York Times._ All rights reserved. Used by permission and protected by the Copyright Laws of the United States. The printing, copying, redistribution, or retransmission of this Content without express written permission is prohibited. Originally appeared in the _New York Times,_ February 8, 2012.\n\nStein, Joel. \"Learning to Barbecue Helped Make Me A Man.\" Copyright \u00a9 2011 by Joel Stein. Used by permission of the author. Originally appeared in _Food & Wine,_ June 2011.\n\nThompson, Wright. \"Memphis in May: Pork-a-Looza.\" Copyright \u00a9 2011 by Wright Thompson. Used by permission of the publisher. Originally appeared in _Garden & Gun,_ October\/November 2011.\n\nJohn Gutekanst, \"Truffle in Paradise.\" Copyright \u00a9 2012 by the Regents of the University of California. Used by the permission of the University of California Press. Originally appeared in _Gastronomica: The Journal of Food and Culture_ vol. 12, no. 1, (Spring 2012), pp. 66\u201371.\n\nDuane, Daniel. \"A Slice of Family History.\" Copyright \u00a9 2011 by Daniel Duane. Used by permission of the author. Originally appeared in _Food & Wine,_ December 2011.\n\nEdge, John T. \"Barbecue Road Trip: The Smoke Road.\" Copyright \u00a9 2012 by John T. Edge. Used by permission of the author. Originally appeared in _Garden & Gun,_ June\/July 2012.\n\nKliman, Todd. \"The Food-Critic Father.\" Copyright \u00a9 2012 by Todd Kliman. Used by permission of the publisher. Originally appeared in _Washingtonian,_ February 2012\n\nChang, T. Susan. \"The Legacy That Wasn't: Wonton Soup.\" _A Spoonful of Promises: Stories & Recipes from a Well-Tempered Table._ Copyright \u00a9 2012 by T. Susan Chang. Lyons Press: pp. 33\u201340.\n\nYuh, Eagranie. \"Curious Cookies.\" Copyright \u00a9 2012 by Eagranie Yuh. Used by permission of the author. Originally appeared in _Edible Vancouver,_ Spring 2012.\n\nClancy, Henrietta. \"Chicken Brick.\" Copyright \u00a9 2012 by Henrietta Clancy. Used by permission of the publisher. Originally appeared in _Fire & Knives,_ Issue #9.\n\nAltman, Elissa. \"Angry Breakfast Eggs.\" Copyright \u00a9 2012 by Elissa Altman. Used by permission of the author. Originally appeared on www.poormansfeast.com, April 25, 2012.\n\nMims, Ben. \"Sweet Southern Dream.\" Copyright \u00a9 2012 by Ben Mims. Used by permission of the publisher. Originally appeared in _Saveur,_ March 2012.\n\nMartin, Brett. \"The King of Pop-Up.\" Copyright \u00a9 2011 by Cond\u00e9 Nast. All rights reserved. Reprinted by permission. Originally appeared in GQ, June 2011.\n\nHutton, Rachel. \"Hot Plate.\" Copyright \u00a9 2012 by Rachel Hutton. Used by permission of the publisher. Originally appeared in _Minnesota Monthly,_ March 2012.\n\nFrizell, St. John. \"Austria's Culinary Ambassador.\" Copyright \u00a9 2011 by St. John Frizell. Used by permission of the author. Originally appeared in _Edible Manhattan,_ September\u2013October 2011.\n\nWharton, Rachel. \"Remembering Savoy.\" Copyright \u00a9 2012 by Rachel Wharton. Used by permission of the author. Originally appeared in _Edible Manhattan,_ May\u2013June 2012.\n\nLeibowitz, Ed. \"Appetite for Perfection.\" Copyright \u00a9 2011 by Ed Leibowitz. Used by permission of the author. Originally appeared in Los Angeles Magazine, June 1, 2011.\n\nBroening, John. \"Supper Clubs in Denver: Informal, Spontaneous and Inexpensive.\" Copyright \u00a9 2011 by John Broening. Used by permission of John Broening. Originally appeared in the _Denver Post,_ September 14, 2011.\n\nNakano, Richie. \"Why Chefs Sell Out.\" Copyright \u00a9 2012 by CBS Interactive Inc. Used by permission of the CBS Interactive Inc. Originally appeared on CHOW. com, March 2012.\n\nPang, Kevin. \"A Chef's Painful Road to Rehab.\" Copyright \u00a9 2011 by _Chicago Tribune._ All rights reserved. Used by permission and protected by the Copyright Laws of the United States. The printing, copying, redistribution, or retransmission of this Content without express written permission is prohibited. Originally appeared in _Chicago Tribune,_ June 6, 2011.\n\nMacias, Chris. \"Bitter Start to a Life of Sweets.\" Copyright \u00a9 2012 by Chris Macias. Used by permission of the publisher. Originally appeared in _Sacramento Bee,_ 2012.\n\nLeite, David. \"Kitchen Confessional: Burnin' Down Da House.\" Copyright \u00a9 2011 by Leite's Culinaria. Used by permission of Leite's Culinaria. Originally appeared in Leite's Culinaria, December 7, 2011.\n\nSpong, John. \"Do I Dare to Eat a Peach?\" Copyright \u00a9 2011 by _Texas Monthly._ Used by permission of the publisher. Originally appeared in _Texas Monthly,_ July 2011.\n\nJosh Ozersky, \"A Proposal for Feeding the Fat and Anxious.\" Copyright \u00a9 2011 by the Regents of the University of California. Used by the permission of the University of California Press. Originally appeared in _Gastronomica: The Journal of Food and Culture_ vol.11, no. 4 (Winter 2011), pp. 91\u201392.\n\nChin, Mei. \"Bone Gatherer.\" Copyright \u00a9 2012 by Mei Chin. Used by permission of the publisher. Originally appeared in _Saveur,_ March 2012.\n\nShilcutt, Katharine. \"They Don't Have Tacos in the Suck.\" Copyright \u00a9 2012 by Katharine Shilcutt. Used by permission of the publisher. Originally appeared in _Houston Press,_ 2012.\n\nStoessiner, Anna. \"I Won't Have the Stomach for This.\" Copyright \u00a9 2011 by The _New York Times._ All rights reserved. Used by permission and protected by the Copyright Laws of the United States. The printing, copying, redistribution, or retransmission of this Content without express written permission is prohibited. Originally appeared in the _New York_ Times, August 14, 2011.\nABOUT THE EDITOR\n\nHolly Hughes is a writer, the former executive editor of Fodor's Travel Publications, and author of _Frommer's 500 Places for Food and Wine Lovers._ She has edited the Best Food Writing series since its inception in 2000. Visit her website at hollyahughes.net.\n\n**Submissions for Best Food Writing 2013**\n\nSubmissions and nominations for _Best Food Writing 2013_ should be forwarded no later than May 15, 2013, to Holly Hughes at _Best Food Writing 2013,_ c\/o Da Capo Press, 44 Farnsworth Street, Boston MA 02210, or emailed to best.food@perseusbooks.com. We regret that, due to volume, we cannot acknowledge receipt of all submissions.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n**CONTENTS**\n\n_TITLE PAGE_\n\n_DEDICATION_\n\n_EPIGRAPH_\n\n_ACKNOWLEDGEMENTS_\n\n_PROLOGUE_\n\nNESTLING\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\nFLEDGLING\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\nFLIGHT\n\nCHAPTER 28\n\nCHAPTER 29\n\nCHAPTER 30\n\nCHAPTER 31\n\nCHAPTER 32\n\nCHAPTER 33\n\nCHAPTER 34\n\nCHAPTER 35\n\nCHAPTER 36\n\nCHAPTER 37\n\nCHAPTER 38\n\nCHAPTER 39\n\nCHAPTER 40\n\nCHAPTER 41\n\nCHAPTER 42\n\nCHAPTER 43\n\nCHAPTER 44\n\nCHAPTER 45\n\n_ABOUT THE AUTHOR_\n\n_COPYRIGHT_\n_To Jane Gunther for always asking\u2014did you get any writing done?  \nThere wouldn't be a book without you_\n_Why should you love him whom the world hates so? Because he loves me more than all the world._\n\n_\u2014Edward II,_ Act I, sc. iv Christopher Marlowe\n**\u00b7 ACKNOWLEDGMENTS \u00b7**\n\n**N** OTHING IS CREATED in a vacuum. So, thank you to my family, who never once said, You want to do what? To Odyssey Workshop and Jeanne Cavelos for showing me that I not only could write day in day out, but love doing so. To Larry Taylor for endless patience and invaluable critiquing when I needed it most. Thanks also to Caitlin Blasdell and Fleetwood Robbins for their enthusiasm, support, and excellent efforts on my behalf. All of it is deeply appreciated.\n**\u00b7 PROLOGUE \u00b7**\n\n**_The Stirring of Wings_**\n\n**I** N THE SOUTHERNMOST TIP of the island kingdom of Antyre, a carriage set a rapid pace through the city streets of Murne. The horse-and-four racketed down the broken cobblestone street, shuddering and jolting on the uneven surface. Midmorning sunlight lanced off the blue-lacquered carriage, lighting it like a jewel in a tarnished crown. On either side, narrow houses listed and shed fragments of their fa\u00e7ades, littering the streets below with rubble. Once this had been a prosperous merchant neighborhood, the most common thoroughfare between the palace and the sea\u2014before a girl's prayer had been heard by Black-Winged Ani, that treacherous god of love and vengeance.\n\nOnce the road had been filled with horse carts and carriages, a throbbing artery pulsing with bustle. Now, corpselike, it rotted; the ground crumbled beneath the ruined buildings and cobbled streets, homes for those desperate enough to live in the merchants' Relicts.\n\nInside the coach, an elaborately dressed dark man clenched his hands into fists. Kritos stared at the far wall of the coach, ignoring the Relicts outside, his thoughts occupied entirely with his grievances. He cursed himself for an overobedient fool, and he cursed Last and the man's filthy sense of humor.\n\nKritos levered himself out of the quilted seat, reaching for his stout malacca cane, which had been flung, haphazard, across the coach. Recovering it, he pounded the roof of the carriage. \"Slow down, you damned fool. You'll have the wheels off and I'll have your hide.\"\n\n_Slow down,_ he thought. He was in no hurry to find the boy who meant an end to his comfortable position as Last's heir. He raised the cane, hesitated. He could order the carriage around now. Tell Last that there was no boy, that the bastard son had died, unwanted, in Antyre's Relicts.\n\nHe lowered the cane, indecision and fear warring in his belly. John Coachman would know, damn his eyes, and the boy might surface on his own, seeking his birthright. If the boy lived, Last would find him, no matter what Kritos told him. Last was too well aware of the creditors hounding Kritos to trust his word alone.\n\nHis hands knotted around the cane as he pondered more drastic measures. The boy was only a boy, and he was a strong man and armed. Hefting the cane, he swung it in short, brutal strokes, slamming it against the cushioned seat with a satisfying impact. If the boy were dead\u2014\n\nBut his hands clenched again in a paroxysm of stifled frustration. Last did not trust him, and Kritos knew well enough that the coachman had undoubtedly been warned to prevent such a thing.\n\nInevitability slumped his broad shoulders, made him discard his plans with a breath that sounded like a sob. Last was a treacherous bastard, he thought, wholly without irony.\n\nAhead of the coach, the cobblestones had been worn to a bare edge around what once was a central square, its pits filled in with scavenged rubble, smoothed over with dirt. In it, the Relicts children circled one lad with a stick, watching with an intentness that betrayed that this was no game the children played.\n\n\"So you hit 'im onna head, then inna cods, inna belly, and onna head again, 'til they don't never get up no more. Then you cut their purse, innit right?\"\n\n\"Right, Roach,\" they chorused. But Roach sought his approval from the two youths standing apart, in a closed circle of their own. He stood, shifting from dusty foot to dusty foot, waiting.\n\nWhat did they look like, these leaders of children? Children themselves, though not for very much longer. In a year or so, they'd be hunting up a new trade: highwayman, whore, gaolbird, or hangman's bait if they were particularly unlucky. Currently, they appeared much like the others. Rag-clad, barefoot, begrimed, and feral. One was, perhaps, blond under the dust and dirt. The other was as dark as night, with eyes of the same hue, startling and dramatic in a pale, pinched face. No amount of filth could darken hair like that any further.\n\nThe might-be-blond had vivid blue eyes and a placid face. He was bigger by a head than his snarl-maned companion. This was Janus, the unknowing object of Kritos's search.\n\nJanus ignored Roach's demand for approval, studying instead his friend's dusty, extended fists. Janus reached out, ready to choose, ready to tap one fist over the other, and paused. Raising his gaze to the smirking silent mouth, to the mocking eyes, Janus touched the brunette's lips, winning the game.\n\nThe brunette stuck out a pink tongue, and on it a ring quivered\u2014tarnished, but undeniably gold. The children's eyes lingered on it enviously. A man's ring, it was a little too big for his forefinger, but Janus slid it on, clenching his hand around it. He smiled.\n\nOnly then did he turn his attention to the waiting Roach. \"You'll never learn, Roach.\" His voice was roughened with the onset of manhood, and the accent was as different from Roach's as night was from day, elegant, cleanly spoken, not the gutter patois of the other.\n\n\"So what'd I forget?\" Roach asked.\n\nThe brunette, freed from silence, burst into speech. \"The same thing you forgot last time and the time before. The shoes. Always remove their shoes. Good shoes sell.\" Accented like Janus's, an aping of the aristocracy, her voice held a bite to each word that made it utterly distinctive.\n\n\"Sorry, Miranda.\"\n\n\"Regret will not put food in your belly when the stolen coin runs out. Then it's back to stealing and whoring again.\"\n\nThe pack tightened, shifting from predators to an assortment of frightened children.\n\nRoach stepped out of the circle. \"I'm no whore.\"\n\n\"Not yet,\" Miranda said, unrelenting.\n\n\"Not never,\" he swore.\n\n\"It's his belly, let him learn,\" Janus said. He fingered the ring and stared at the other children, the fallen houses, the gaping sinkholes, like a young lord surveying the ruin of his empire, holding court one final time.\n\nCarriage wheels clattered, sparking unusual echoes on this forgotten street, and sprayed jagged shards of stone in their wake. At the carriage's approach, the children scattered, only re-forming the pack when it came to a sliding halt. The horses reared in their traces before settling down to heave giant, wet-flanked breaths.\n\nThe children perched in empty windows, on piles of broken cobbles. Their thin fingers clenched sticks and loose stone, waiting in silence.\n\nThe liveried coachman put one hand on his pistol, one on his whip, daring them to act. Freed of his touch on their reins, the horses danced and bridled until the coachman snapped his whip above their heads.\n\nKritos stepped from the carriage, sleek in his embroidered finery. Feral eyes grew warily speculative, and Kritos paused, boots still on the carriage rung. He pulled a linen handkerchief from his sleeve, damped with scent, and pressed it to his nose and mouth for a moment.\n\n\"Which of you miserable bastards is Janus, son of Celia?\"\n\nAt this sign of specific interest, Roach and the pack spooked and fled like a burst of hunted doves. Miranda and Janus stayed. Their faces might have been made of the same stone as the houses\u2014blank and still, hiding secrets.\n\nKritos took the last step down. His mouth twisted as his champagne-shined boots slid in the greasy dirt. \"Is one of you Janus?\"\n\nMiranda spoke first. \"How much to show you where he is?\"\n\nKritos didn't listen to her past his first start of surprise at her accent. His eyes fixed on Janus, on the echoes of Last in Janus's straight jaw and nose, in the gas-flame blue eyes. So this was the prize he was sent to secure. His replacement. The murderous temptation rose again, but Kritos, the strong man, felt distinctly leery of tackling this pair.\n\nWhile he contemplated, he watched Janus watching Miranda. Signals flowed like water between them, a code learned in faint lifts of eyebrows, of tightened lips and tilted chins, of twitched fingers. No, he had no intention of giving his back to these hellions.\n\n\"Pay you,\" he said, \"when I can thrash the information from you?\"\n\n\"Like to see you try,\" Miranda said.\n\n\"Like to see you catch us,\" Janus said.\n\nTheir lovely aristocratic voices made the words a taunt. Their bodies tensed like strung wire, poised for flight and the shelter of the labyrinthine alleyways. He would have to lull them, let them presume he abided by their rules, all the better to take them by surprise.\n\nKritos said, \"How much?\"\n\n\"Not so poor as copper. Not so dear as sols,\" she said.\n\n\"Lunas,\" Janus said.\n\n\"Silver? For a gutter-born bastard?\" Outrage thickened Kritos's voice and drew Miranda's lips into a cold approximation of a smile.\n\n\"Pay what is fair for us. Not for you,\" she said.\n\n\"One luna, then. When you lead me to him.\" Kritos pressed the scented linen to his mouth.\n\n\"Done,\" Janus said.\n\nKritos paused, wary. The haggling ended too soon; Kritos had expected them to demand the coin up front, expected them to lose him in the Relict's nooks and alleys. He knew they had no more intention of fulfilling the bargain than he did; for them to agree meant they were not content with the thought of a single luna. They wanted it all.\n\n\"We'll lead you.\" She eeled off, followed by Janus. Neither looked back, the challenge implicit as they entered the narrow, dark alley torn between two houses.\n\nKritos set his shoulders, judging their span against the gap. The coachman leaned off his perch. \"Sir, be careful.\"\n\n\"Do you think me feared of urchins?\" Kritos blustered. \"Hold the cattle for my return.\"\n\nKritos was not afraid, not now when the moment was imminent. Undoubtedly, they planned to trap him, but they had let him take their backs, left themselves at risk.\n\nCaught in his thoughts, squinting in the murky light of the alleyway, Kritos saw the girl brush by the wall, a shadow in the dark; a spill of stones tumbled free, driving at his head. _You'll need more than that, girl,_ he thought, leaping aside. The ground gave beneath his feet.\n\nThe cascade of stone hadn't been the ambush; the loosely covered pit had. Kritos hit the bottom hard, rolled to his feet, found himself chest-deep in the ground, with damp stone beneath. It was probably the only reason the pit didn't go deep enough to break a man's neck.\n\nHe raised his cane just as the girl swung her stick at him, striking the soft tissue beneath his chin. Kritos gagged with pain; his teeth snapped against each other. Slapping her legs with the cane, he made her dance away, but she reversed her grip on the stick, making up the distance, and jabbed the end of it at his face. Kritos jerked and took the blow meant for his temple in his eye instead. Pain flared, and fear. Howling in shock, he flailed with his cane, but couldn't avoid the blow from behind that forced him to his knees.\n\n\"His purse, Miranda, hurry,\" Janus said.\n\n\"He's not down yet,\" Miranda protested.\n\n\"Near enough to make no difference, and the coachman might come after him. With the pistol.\"\n\nMiranda pounced, fingers working on Kritos's purse strings. Janus squeezed beside her to help. Wheezing for breath, tears streaking his face, Kritos fumbled for a weapon and found rough stone at his hand, one of the little avalanche that betrayed him. He swung, felt satisfaction push pain away when the rock made contact.\n\nMiranda screamed, a banshee yelp of rage and shock, and her hands flew to cover the bloody gash along her jaw.\n\n\"Bastard!\" Janus brought his stick down again, knocking the rock from Kritos's hand, and reaching for the purse himself.\n\nKritos lunged upward, got a grip on Janus's white throat, and held on. After three stifled breaths, Janus's face started mottling with lack of air. Miranda scrabbled for her dropped stick, and Kritos leaned back against the supporting wall and kicked her in the stomach. She flew backward and stunned herself against the stone.\n\nJanus clawed Kritos's hands, then groped upward, gouging at the man's damaged eye. His thumb slipped along the same rut the girl's stick had made. Kritos screamed and slammed Janus against the earthen and stone wall, dazing him. Kritos kept the boy's slackening form before him, using him as a shield, and struggled to his feet, sobbing with pain and anger. His strangle-hold on the boy's throat loosened, and Janus showed signs of recovering.\n\nMiranda crept forward, barely visible against the pit's floor, her hands fisted. She reared back on her haunches; pebbles and dirt stung Kritos's face, and he kicked out again and caught her in the thigh. She tottered and fell at his feet. He kicked her in the face, and she made a small, sighing sound, eyes fluttering shut.\n\nKritos tightened his grip on Janus's neck, squeezing. The boy choked and spluttered, his nails leaving tracks on Kritos's wrists. Behind him, the girl muttered and swayed to her hands and knees. Kritos swore. If he took the time to kill the boy, she'd be on him again with the tenacity of a weasel.\n\nKritos pulled Janus after him, dragging him out of the pit, keeping his watering good eye on the dark shape in the dark pit. He backed up, dragging Janus into the uncertain sunlight of the Relicts. The coachman pulled his pistol and swore.\n\n\"Put that away,\" Kritos said. \"Get ready to go.\" Janus had a neck like a young ox, and Kritos's hands were slick with his own blood. Janus muttered and twisted, made every step a battle, digging at Kritos's hands with his own, fumbling at his face, slumping and dragging his weight to slow him. The ring, so newly acquired, slipped and fell into the dirt without a sound.\n\nKritos resolved his struggle by slamming Janus's head into the edge of the carriage doorframe and dropping the sprawling, limp form onto the floor of the coach. Janus fell half-in, half-out, his bare feet trailing in the gritty dust.\n\nThe coachman warned, \"Last never said nothing about half-killing nobody.\"\n\n\"Last never had to deal with that hellcat,\" Kritos said, panting for breath, shuddering all over with belated nerves and denied fear. His hands shook as he bent and forced Janus's slack limbs into the carriage. His face throbbed as the maltreated eye complained and continued its slow weeping of bloody tears.\n\nHe clambered into the carriage, uncertain in his balance, and paused, his back to the open door, to kick Janus once, then twice in the ribs, listening for the crack of bone. It was all the vengeance he could allow himself, and it was out of sight of the coachman.\n\nThe coachman's cry came nearly too late. Kritos spun, but too slowly, his vision too impaired to see more than a quick dark blur.\n\nThe stick caught him in the side, punched through the skin but no further. It was, after all, only a piece of well-worn wood. \"Let him go. Pervert. Thief.\" Even with her voice shrill and panicked, her accent remained unchanged, and for the first time Kritos wondered who the girl was.\n\nFrom his high perch, the coachman angled his pistol, trying to get a clear shot. He pulled the trigger; the gun exploded, the ball expelled forward, burning powder and shards of metal flying back to scald the coachman's hand. Kritos flung Miranda from him at the same moment, and the ball bypassed Miranda to add yet more stone chips to the earth.\n\nAs she lunged at him again, the coachman ripped the air with his whip and wrapped the lash around her chest, spinning her around, leaving her in the dirt, gasping, sniveling, nose running blood.\n\nThe sight of her sprawled in the rubble did more to restore Kritos's courage than the coachman coiling his whip for another blow.\n\n\"You must want your silver,\" Kritos said.\n\nHe flung his coin purse into her face; the leather pouch burst at the seams and scattered his own trap\u2014the silver lunas that were only painted wood. \"What he's worth. Coachman, drive on.\"\n\nThe carriage turned slowly in the open square and headed back the way it had come, faster and faster, as the coachman gave the spooked horses their heads.\n\nLEFT ALONE, Miranda gathered her stick and the tarnished ring with shaking hands. Her head spun and her jaw burned where grit had been ground into the open wound. The whip marks crossing her torso through the thin, ripped shirt seeped. And all of that pain was nothing to her. All she could think of was Kritos, carrying Janus away like a hunter's trophy.\n\nShe limped over to the edge of the street and fell, her blood dampening the dirt. She waited for the strength in her legs to come back, and she turned the tarnished ring over and over and over in her hands, peering in to see the words within. _Only each other at the last._ Heart pounding, breath seizing in her chest, refusing to cry, she clenched the ring in fisted hands and curled herself around it.\n\nWHEN HER BREATH RETURNED, she rose and stumbled home\u2014a room with two women, one old bed, a press of mismatched clothing, and some few bits of furniture salvaged out of some refuse bin closer to the civilized parts of the city and hauled back to the Relicts.\n\nNearest the door, a pale, enervated woman swayed in her seat, her hand loosely caging a bottle of Petal, a potent mixture of Laudable syrup and cheap spirits that turned any grief to a distant dream. She stared into the air as if she could see her past unfolding behind her, her days as the pampered daughter of a lord. Before Janus. Before the stigma of bearing a child, unwed, sent her to the Relicts to find a new life as occasional whore and mother, though she was dismal at both. Her customers only ever visited once, and that for the novelty of fucking an aristocrat. Her sole act of generosity had been to teach the children to read and write, though she leavened that by raining stinging slaps whenever their accents faltered into Relicts cant.\n\nThe other woman, Ella, sat on a rough footstool near the fire, coaxing it into life with callused hands. She had been at her peak of attractiveness when she birthed Miranda, fourteen long years ago, and her looks had faded to nonexistence. Her hair, coarse, gray, and unconfined, stood out in rough snarls. But unlike Celia, she could at least manage a gap-toothed smile and a bit of routine coquetry for her progressively more infrequent customers.\n\nMiranda banged the door shut and came in, leaning on her stick. When Miranda moved into the dim glow of the fire, Ella's mouth twisted in dismay. \"Not your face. What have you done, child?\" Ella cried.\n\nMiranda's rage simmered. Both women were too worn to be successful whores. But Miranda was meant to be a courtesan, meant to pass her earnings on to the older women.\n\nElla rushed to the clothespress and pulled out a wooden box. In a different world, it might have held a lady's less valuable jewels. Here it held the tricks of a whore's trade: powders and cosmetics, abortifacients, and a scattering of remedies. She dragged Miranda before the footstool, tutting and fretting.\n\nMiranda yanked her arm free. \"Celia, they took him.\"\n\nThe pale woman roused herself to a reluctant semblance of life, setting the bottle on the earthen floor. \"And who took him from you?\"\n\n\"A town buck in a blue and gold carriage with a crest.\" Miranda kept her voice calm, but tears welled with the effort. She was only waiting for the vital information\u2014a name. Without an identity, there was no one to fight.\n\n\"Last, it seems,\" Celia said, picking up her bottle and letting the thick fluid trickle into her open mouth.\n\n\"Last?\" Miranda said. The name meant nothing, a word in a ring. It should have sounded in her ears like a clarion. But her enemy's name meant nothing. Ella took the opportunity to pull her close, dabbing the long rip along Miranda's jaw with powdered alum before packing it with saved cobweb.\n\n\"How do I get him back?\" Miranda asked, breaking in that instant years of tradition. She and Janus had learned long ago that their mothers had no answers to the slightest of dilemmas.\n\n\"Hush,\" her mother said. \"Don't you talk while I'm cleaning this out.\"\n\n\"Get him back?\" Celia repeated. \"Why? He only caused us pain and trouble. Let he and Last spite each other and spare us.\"\n\n\"'Tis a pity about the scarring there'll be,\" Ella said, finishing her ministrations and turning Miranda's face in her hands, judging the results. \"Still\u2014such eyes, such a mouth...what a price you'll command and never mind the scar. And with Janus gone, well, you'll get over that stubbornness of yours and do as I say.\"\n\nMiranda screamed, an animal cry of wordless rejection and rage. She shoved her mother with enough force to send the older woman to the floor. Celia's hooded eyes widened even in her drug-induced haze. Miranda dropped her hand to the rough-edged knife her mother used to shave bits of wood for the fire, knotted her fist around the handle. How dare they simply dismiss him as if he were of no more importance than a customer? She would make them regret their callousness. She stood, hand shaking, body tensed, waiting for release; then, unclenching her fingers, she let the knife fall.\n\n_Why kill them?_ she thought. At this moment, they were dead to her, and she knew that without Janus, without her, the two women would fade away. The rage faded as fast as it came, leaving contempt. She despised their sickening passivity, their languid acceptance of their decaying lives. She would not accept the same.\n\nMiranda paused in the doorway, taking a long last look. She was leaving now. She would never see these women, this room again. She would not rue it.\n\nNight had folded over the Relicts, and the only lights were the small fires set by the desperate, conflagrations of wood and burning cloth. Miranda felt her way down the street with her stick, tapping before her like the blind beggars who worked the boundary between moneyed and poor.\n\nA name was all she had, and it was not enough, but she headed for the border of the Relicts as if she were welcome on the other side. In the worst section, while climbing over fallen masonry, she put her foot through a toppled door, spiderwebbed with rot, and tumbled headlong. She plummeted down the heap, scraping her hands, her shins, and landing with a painful crash against a stone slab. She fought tears, tried to rise, but stumbled and fell to one knee. Her breath labored and the wounds of the day, so long suppressed, made themselves felt with belly-wrenching intensity. Clawing at the stone slab, her fingers caught on rough frescoes, and she drew herself up, found it was a wall. Leaning close, she made out wings and knew what the building was, though she'd never seen it before.\n\nThe room was even more damaged than most Relict buildings; the walls had all fallen inward, making a precarious lean-to of stone, but then, this was a temple to Black-Winged Ani, She who had spread ruin through the Relicts. Carved wings created a slanted tunnel, which Miranda crept through. Her destination was the altar. But not to pray for a dead god's aid. Even in this extremity, Miranda was not stirred to begging.\n\nThe altar was the sturdiest place in the temple, sheltered deep within the wreckage. Miranda crawled beneath it, let her body relax. In the darkness, the grinning face of Black-Winged Ani stared down at her from every angle. Miranda tucked herself into a ball, wrapping herself around the stick, missing Janus's warmth with an ache greater than the throbbing wounds. She fell asleep to the whispering mutter of prayers, trapped like memories in the stone, fell asleep to Ani's looming scrutiny.\n\nOutside, a cold wind rose.\n\n**\u00b7 1 \u00b7**\n\n_Maledicte lived and Maledicte died And only at his birth did anybody cry. How many people did he kill? One, two, three..._\n\n\u2014Children's skipping song\n\n**B** ARON VORNATTI WAS AN OLD MAN, hunched in his chair, staring at the wonders of his extensive library with a jaded and bleary eye. A sable pelt poured over his wasted legs. Absently, he ruffled the furs while he flipped pages of the book of pornographic woodcuts on his lap. A hedonist and a sensualist, he was much withered by time and pain; on a cold winter's night, he fondled old memories as he once did flesh. But all his precious memories, of women's softly rounded shoulders and mounded breasts, the sweet juncture at their thighs, of young men's ripe buttocks, greedy mouths, and strong square hands, all these could not distract him as they used to do.\n\nHis back flared and spasmed. His glassy eyes flew to the old grandfather clock by the door. \"Gilly,\" he roared. \"Time!\"\n\nGrinding his teeth, Vornatti sagged forward in the chair to ease the strain. The book fell to the floor, splayed on opened pages. He wanted a distraction, something beyond the torment of his bones and illusory remembrances of the flesh. Once, he had found engrossment in the bloody game of court intrigue, but even that had palled with his mastery of it.\n\nIn the distant recesses of the library, beyond the firelight, beyond the lamps, glass broke with a sound like cracking ice. Slow, crunching footsteps echoed.\n\nA chill serpent of air wound around Vornatti's ankles, hissing with blown snow.\n\n\"My lord?\" Gilly said from the doorway. The large silver tray of drug and drink was dwarfed in his hands. His voice put a temporary stop to the footsteps.\n\n\"We have an intruder,\" Vornatti said, straightening with a wince.\n\n\"Who's there?\" Gilly said, as the footsteps resumed their slow progress, now thudding against the bare wooden floor. He squinted against the glare of the built-up fire, set the tray down on the thick carpeting beside Vornatti.\n\nThe footsteps gained the carpet and disappeared in the muffling softness. Gilly lifted a book pole, holding it across his chest, the hook facing the shadows.\n\n\"Put it down, damn it, Gilly. Put it down, and give me my Elysia. Let the bastard wait.\"\n\nGilly hesitated, but finally set the book pole back against Vornatti's chair.\n\nHe bent, turning his back to the shadows, and cradled Vornatti's withered arm against his own. He drew the Elysia into the glass syringe in a cloudy swirl that held something of its origin in it, the elixir left in Naga's serpent-scaled wake. Letting it settle long enough so that the contents stopped their eddying, Gilly pushed the needle into the old man's ropy veins. Vornatti closed his eyes as Gilly worked the plunger, hissed against the bite of it in his blood.\n\nWhen Gilly looked up, they were no longer alone in the circle of firelight. The intruder shared it with them. It was only a boy, shivering in his thin shirt, blue at the lips. He had shadowed eyes, made darker by a cropped tumble of black curls that seemed to spread shadow out behind him. A thin scar sliced along the left side of his jaw, and he held his right hand behind his back.\n\nVornatti's eyes opened and he smiled as if a bit of his past had come back in salacious detail.\n\n\"What young toothsome have we here?\" he murmured, lazy on a release of pain and burgeoning euphoria. \"Gilly, only look what the gods have brought me.\"\n\n\"Be silent, old man,\" the boy said, drawing his arm from behind him. In his hand was a sword.\n\nAnd such a sword. It was black-bladed, black-hilted. The pommel was a burnished mirrorstone, and the edges were so sharp as to seem blurred in human sight. The cross-hilt was made of stilled, dark wings with wickedly edged feathers more reminiscent of daggers than of flight. Like some remnant from the god-touched times, the sword radiated presence beyond its workmanship.\n\n\"Are you Last?\" The boy raised the sword, its glitter matched by the wildness in his eyes.\n\nVornatti wheezed into laughter, slapping Gilly's arm, startling him. \"Last.\"\n\nThe boy's face grew red temper lines around the jaw and nose. The scar flared to whiteness. \"Don't laugh at me.\" He pushed the blade forward; Vornatti parried with the book pole, still laughing despite the slice the sword had carved in the wood.\n\nGilly stepped between Vornatti and the blade, and Vornatti stilled his laughter. \"There is more than one noble house in Graston. No, boy, I am not Last. Look here.\" He thumped the end of the pole on the carpet. \"Gilly, get your big feet out of the way and fetch us all drinks.\"\n\nHe stabbed at the floor again. \"See that, boy?\"\n\nThe carpet was burgundy and blue, and in the center a fantastical creature writhed in golden embroidery.\n\n\"I am Vornatti, _Baron_ Vornatti, since you are not likely to know that. An Itarusine subject, now living in Antyre; Aris's brother-by-law, and accountant. The winged serpent is my crest; Last's crest is a twisted hourglass, and his motto is 'only a Last at the last.' Smug bastard.\"\n\n\"Who's Aris?\" the boy said.\n\n\"Our king,\" Gilly said, carrying the tray to the table.\n\n\"Oh. Him.\" The boy studied the elaborate embroidery on the carpet, the slippers Vornatti wore, the crest imprinted on the spine of the long-forgotten book.\n\n\"Are you listening, boy?\"\n\nThe boy didn't answer, but his dark eyes flickered to Vornatti's face, studying the sagging, spotted flesh, the dark rheumy eyes. He let the sword point lower to threaten the floor. \"Where do I find Last?\"\n\n\"What sends you seeking him, blade in hand? Answer me that, first.\"\n\nThe boy frowned, but visibly needed the answer too badly to play coy. \"Last took Janus from me, and left me for dead. I will return the favor, reclaim Janus and leave Last dead. Though I will be less careless and see his heart stop before I go.\"\n\nSo much emotion was invested in the reply that it seemed a cantrip or incantation, and as rote as a nightly prayer. It silenced Vornatti, and the only sound was Gilly and the chinking of crystal as he poured the requested drinks.\n\n\"I doubt that was Last himself,\" Vornatti said. \"He so rarely soils his hands, and is far more dedicated to a task once undertaken. If it were Last, make no doubts, you would be dead, boy. But who is Janus to warrant such attention?\" Vornatti took the goblet Gilly offered him, sipped at the steaming negus. He gestured to Gilly, and Gilly turned the second goblet, the glass meant for him more usually, toward the boy.\n\nDistrust furrowed the boy's features, and he finally said, \"The earl of Last's bastard son.\"\n\n\"Last has no living children,\" Vornatti said.\n\nGilly pressed the cup into the boy's left hand, spoke to Vornatti. \"There were rumors. You remember. Celia Rosamunde, the admiral's daughter.\"\n\n\"Oh. Her.\" Vornatti mocked the boy's earlier words. \"That weak-willed, wanton wench.\" He laughed at his own wordplay, and to Gilly's surprise, even the boy flickered a bitter smile.\n\n\"I thought she died,\" Vornatti said.\n\n\"She was disowned and abandoned when her condition became evident. I heard she made her way to the Relicts and died there.\"\n\n\"Not yet,\" the boy said. Across the room, the boy's stomach growled audibly. He raised the goblet, and swallowed three great gulps of the sweetened, spiced wine, throat working. Making a face, he dropped the goblet to the carpet.\n\nVornatti shifted his slippers away from the spreading stain. \"If you didn't want it, all you had to do was give it to Gilly,\" Vornatti said, without heat, his mind occupied. \"He reclaimed his bastard son? I had heard that Last's newest wife died of childbed fever and the babe of milksickness.\"\n\n\"This talk means nothing,\" the boy said. \"Tell me where to find Last.\" He dragged the sword point up to menace Vornatti once more.\n\n\"Let's have less of that,\" Vornatti said. \"It grows wearisome. I will tell you what you want to know, for all the good it will do you. Last's estate is yet ten miles from here. Will you walk it in ice and snow?\"\n\nThe boy's face sagged into momentary despair, and then the determined mask slid back into place. He started for the door.\n\nVornatti caught his arm with a sudden movement, surprising in a man so seemingly infirm. \"Stay the night,\" Vornatti said. He stroked the boy's scarred cheek.\n\nThe boy jerked free, no longer listening, caught up in his own thoughts, bent on following some inner drive that denied obstruction.\n\n\"Gilly,\" Vornatti commanded.\n\nReluctantly, Gilly roused himself to comply, though had he a choice, he'd be relieved to see the boy's back. Still, obedience was ingrained, and he stepped before the boy, keeping a wary eye on the boy's sword hand. \"Come on then. Humor the old bastard and stay.\"\n\nThe boy halted, staring at Gilly. \"Get out of my way.\" Shadows danced in the depths of his eyes, and Gilly stepped out of easy harm's range. Still, he balked the boy at the door.\n\nGilly was good at anticipating his Lord's requests. From the moment that Vornatti smiled at the intruder, Gilly knew he desired the boy. Other intruders had been summarily and unpleasantly dealt with, the pistol fished out from beneath Vornatti's lap furs, not teased to flushing. With that in the forefront of his thoughts, Gilly had drugged the boy's negus, the Laudable's sticky sweetness masked by the honeyed spice of the heated wine. Gilly was surprised the boy still stood. A single mouthful should have been enough to incapacitate one skinny youth.\n\nStill more surprising was the force of the boy's presence. Gilly found it harder and harder to keep himself blocking the boy's egress. Only Vornatti's expression of cupidity and interest kept Gilly still. Vornatti's mood could shift like the tide; he'd grow bored soon enough with the filthy, bad-tempered lad, but until that moment occurred, Gilly had best obey or suffer Vornatti's own bad temper.\n\n\"I'll pay you,\" Gilly said, inspired. \"Enough to rent a hack to Lastrest in the morning.\"\n\nThe boy put out a hand, palm upward, waiting.\n\n\"In the morning,\" Gilly said.\n\nHis hand clutched the black hilt. \"I should trust you after you've drugged the wine?\"\n\nGilly saw it now, the slackening mouth, the loosening fingers. The signs he had expected long minutes ago\u2014but the boy was fighting the effects of the drug with the considerable force of his will. Gilly wondered how much longer the boy could stay standing.\n\n\"I'm tired, Gilly. Show our guest to a room and be done with it.\" Vornatti leaned forward, creaked out of his chair, and reached for his fallen book. \"The painted room, mind you.\"\n\nGilly nodded. He bowed to the young savage as if he were truly a guest, and said, \"Follow me.\"\n\nHis back was tense with his concern that the boy was not following, and tense with the cat-feeling that the boy _was_ following, albeit on footsteps too silent to hear. Gilly turned and found the boy paused in the long stone-paved hall.\n\nThe boy stared into the great, clouded mirror that hung on the gilded and flocked walls, a spot of uncertain shadow in the midst of rich colors and elaborate hangings. Touching the rippled glass, the boy leaned close, fingered his reflection.\n\n\"You are comely enough,\" Gilly said, wondering if the boy had ever seen his face in a looking glass before. Despite the jarring notes of sword and accent, Gilly knew the boy was no more of the aristocracy than he himself was. \"But it's no assignation we head to, only rest.\"\n\n\"There is nothing before me but a rendezvous,\" the boy said, his thickening tongue slurring his words. The boy pressed his face against the cool glass, closing his eyes.\n\nGilly took the boy's arm, and the boy leaned against him, looked up at him with enormously pupiled eyes. \"Have you seen Janus?\"\n\n\"I have not. Is he fair like Last?\"\n\n\"Fair,\" he agreed on a sigh. He tugged Gilly's blond tail of hair; his dark-fringed eyes closed, then flickered open in sudden awareness. \"Bastard. I'd better get my lunas in the morning.\" He shoved Gilly away.\n\nGilly led the boy away from the mirror, and looked back over his shoulder, expecting to see the boy's reflection lingering behind, as stubborn as the boy himself.\n\nAt the painted room, Gilly unlocked the door, went inside, and lit the gas lamps. The boy stared at the furnishings in a near stupor.\n\nThe room was shadowed. The gaslight illumined only a circle the size of a man's outstretched arms, and the chamber was easily thrice that, if not more. The bed itself was a small room, walled of swagged draperies, embroidered with gilded serpents. Even to Gilly's eyes, they seemed to undulate in the wavering light; how must they appear to the drugged boy? Thick carpeting underfoot stifled sound, turning each movement into a secretive whisper. Heavy, dense curtains draped the distant walls, though Gilly knew there were no windows behind them, only the murals that gave the room its name. One drape, drawn back, revealed nothing but the image of rushing water, full of movement without progress. This room was a well-appointed prison.\n\nThe boy shivered as if he had sensed Gilly's thought, but headed for the swaddled bed as if for a long-sought rest. Gilly watched the boy clamber up the bed steps and lie down. Only then did the boy release the sword from his grip. It sprawled over the counterpane beside him like a living thing.\n\nGilly closed the door, drew out the key, and locked it, sealing away the boy and his sword.\n\nThe click of the bolt sliding home sounded as final as a headman's ax in the silent hall. Gilly winced, expecting the boy to rouse, sputtering curses and making futile strikes against the heft of the oaken door, but the moment passed in peace. Tucking the key into his vest pocket, Gilly returned to the library.\n\nVornatti waited, slumped down in his chair, too worn tonight to make the walk to his chambers without aid.\n\n\"The boy?\" Vornatti said, without raising his rough-whiskered chin from his chest.\n\n\"Caged. Asleep,\" Gilly said.\n\n\"Good. I'm not inclined to conduct business at this hour.\" Vornatti pressed his hands into the arms of the chair, trying to raise himself. Though his face grayed with effort and his hands whitened, his body stayed motionless. Gilly forestalled further effort, slipping the rug from Vornatti's lap, setting the pistol aside, and, his hands beneath Vornatti's shoulders, hefted the diminished weight of what had once been a big man. Vornatti fell against Gilly's side, muttering. \"Too much Elysia,\" he said.\n\n\"Too much winter,\" Gilly said. \"I measure your dose most carefully.\" He lifted the old man into his arms like a child, chary of his grip on brittle limbs, and carried him down the hall to his quarters.\n\nDespite the prevailing fashion for the master's quarters to be housed higher than the common riffraff of ground dwellers, Vornatti's rooms were on the main floor, dictated by his long illness, and guarded from intruders by Gilly. Gilly's footsteps grew muffled as he set foot to the thickly piled carpets of Vornatti's room.\n\nA barbarian's bedroom, Gilly remembered thinking, when he first came to stand wide-eyed and tentative on the sill. An oddity in the kingdom of Antyre\u2014Vornatti's bedroom had been furnished in Itarusine fashion, a carved, curved bed with high sloping sides, covered not only with Antyrrian linens, wools, and velvets, but the heavy pelts of winter snowbears, imported from Itarus at exorbitant cost.\n\nAfraid of falling in his later years, of exacerbating the swelling in his back, Vornatti kept adding carpets, until the floor had risen high enough to make Gilly step up into the room. The fireplace along the exterior wall glowed darkly, muted by ash and spent coal. Gilly frowned at it, at the ghost of his breath in the air. He set Vornatti down on the bed, and helped him remove his dressing gown, draping it over the bedstead. He knelt and removed the slippers from his feet. Despite the homely wool stockings, Vornatti's feet were white and cold to the touch. Gilly chafed them until Vornatti drew away. \"Enough, Gilly, leave me my skin.\"\n\n\"You're the one who complains that cold feet keep you from your rest,\" Gilly said, and Vornatti settled his feet back into Gilly's hands.\n\nVornatti tangled his fingers in Gilly's hair, pulled the tie loose, and combed his locks with gnarled motions. \"My Gilly,\" Vornatti said, and Gilly, still kneeling, assessed the tone. Amorous? He hoped not, thought not.\n\nHe straightened, took himself away from Vornatti's caress, and drew back the linens to the scent of hot earth. The heated bricks were still warm to the touch when he tugged them out, and helped Vornatti into their place.\n\nVornatti stroked the skin at Gilly's throat, the soft cotton of his shirt over his chest, but the caresses were cursory. Again he stepped away from Vornatti's touch where normally he would have allowed it, or even prolonged it, making such small contact the opening move in their barter of desire and favor. He didn't need anything tonight but information, and Vornatti had always been free with that.\n\n\"Are you going to send for the magistrate? Turn the boy over to him and his workhouse?\" Gilly's belly clenched with anxiety. He was worried by Vornatti's interest, by his own conviction that the boy's presence could alter the fragile balance of his life with the baron.\n\n\"That boy's too delicate for the workhouse,\" Vornatti said, wincing as Gilly folded back the bedding and his bones took the weight of the furs, linens, and quilts.\n\n\"You think him delicate?\" Gilly asked, remembering the boy's grip on the sword, the willful refusal to give in to the drugged wine. The boy and his sword raised the fine hairs on his neck. Vornatti would not find this boy as pliable as the city whores he used to collect.\n\nVornatti laughed, a harsh, quick bray. \"Perhaps not. But still, I intend to keep him.\"\n\n\"He's not a dog,\" Gilly said. \"You can't keep people.\"\n\n\"Kept you, didn't I?\" Vornatti said. \"Found you, liked you, took you home, and here you stay.\"\n\n\"You bought me,\" Gilly said, watching the red coals of the fire dim further, the room grow more chill.\n\n\"And what's keeping me from buying him? Who knows what he'd do for food, for warmth?\" Vornatti stroked the fur coverlet in a speculative fashion.\n\n\"What would you do with him?\" Gilly asked.\n\n\"You saw his face and form and need ask? But don't fuss yourself; he won't replace you,\" Vornatti said, splitting his lips into a malevolent grin. \"I've trained you to a nicety.\"\n\n\"Another pet then?\" Gilly said, flushing.\n\nVornatti sighed and closed his eyes. \"You know my sentiments regarding Last. Do you think I would turn that boy away? When his desires so mirror my own?\"\n\nGilly paused in his restless tidying to sit on the bed, shaken. \"You plot treason....\" Gilly was hushed by Vornatti's fingers tightening on his thigh, by the malevolence in his dark eyes, unwinking.\n\n\"Plotting? I do no such thing. If I aid a starveling boy, that's charity. If I give him information, that's education. All I am doing is putting the piece in play.\"\n**\u00b7 2 \u00b7**\n\n**F** ULL DAYLIGHT SEEPED beneath the heavy, pooled draperies in Gilly's room, waking him from a familiar dream turned nightmare. He clawed free from the winding cloths his linens had become, shuddering at their clammy touch, reminded irresistibly of the catafalques. As was not uncommon, Gilly had dreamed deep and dark, and found himself alone among the tombs of the five dead gods. Familiar, the sight of the white-shrouded coffins, the vast room around them smelling of ancient dust and decaying opulence. The carvings beneath the restless shrouds peeked through as the linens shifted with his approach, revealing the aspects and symbols he understood only while dreaming, shaping the great names: Baxit, Ani, Naga, Espit, Haith. The dead gods.\n\nFamiliar and yet\u2014Some tremor lingered in the silent mausoleum, as if something had run through, leaving stirred air in its wake, as if something stood in the shadows of the tombs, holding an indrawn breath, preparing to speak. A single sudden thought took life in his skin, as chilling as ice water: What if the sound had been the gods Themselves, the echoes of Their final words?\n\nThe threat had been enough to wake Gilly into gasping awareness. To dream of the dead gods was a chancy thing at best; to hear Their words was a burden few mortals could bear. Gilly's mother had once lamented the timing of Gilly's birth, assuring him that such dreams should have seen him an intercessor. But the gods were gone, and Gilly's path had led him not to the contemplative life, but to a life as Vornatti's companion.\n\nA groan shivered into the day-lit silence; the sobbing of tortured wood, and Gilly's breath seized while he tried the sound against the one he hadn't heard. The echo offorced wood, a coffin shifting.... Sense reasserted itself. His room was next to the prisoned boy's. That was all he heard, the boy hunting a way out of his cage. Ignoring the bed steps, he dropped to the floor and hastened to pull back the curtains.\n\nThe old woman who acted as their housekeeper had been about her rounds. The fire was newly laid, but unlit; the morning tray of tea and toast was chill to the touch. Gilly sighed. The old woman had her way of making her disapproval felt. Baron Vornatti kept city hours\u2014late nights and later mornings. Chrisanthe might work for Vornatti's coin, but she would not compromise. The tray had been deposited with the dawn, many hours ago. Gilly hadn't seen dawn light since the time he attended court with Vornatti and they returned home with the paling grayness. The last time he woke with the dawn was before Vornatti, back on the farm, when he had dreamed in torn sheets, shared with the two brothers closest in age.\n\nGilly drank down the tea in a gulp; cold and overbrewed, it did nothing to soothe his nerves. Not for the first time, he considered dismissing Chrisanthe. He would do so in a heartbeat, were it not that Vornatti's reputation precluded a household of competent servants. The country servants were afraid of Vornatti; reputations were all servants had, and scandals lingered. A lad or girl who worked for Vornatti would find no other position after. The thought made Gilly shiver. Vornatti was an old man, after all, and Gilly still young. What would happen to him when Vornatti gave in to mortality?\n\nBut scandals, though they discommoded their household, also kept it a power in the king's court. One of Gilly's chores, when Vornatti could bear to part from him, was to ferret out the latest gossip, faint whispers of things not yet known. Vornatti used such information to keep himself a player in the game of politics. Bartered information and his considerable fortune kept Vornatti a welcomed presence in the Antyrrian court, rather than a shunned creature, the hated Itarusine warden.\n\nA longer, more shuddering moan drew Gilly's thoughts back to what he had been avoiding. The boy.\n\nVornatti would still be sleeping, unconscious with Elysia and drink; it fell on Gilly to care for this strange guest. Gilly poured another cup of tea, lifted the tray. Hunger had been the boy's downfall last night. He would still be hungry.\n\nInside the room, the activity had stopped. Gilly set the tray down and listened. Silence. Gilly fished out the key, slipped it into the lock. The latch drew back with a protest. Gilly pushed the door but it refused to budge.\n\nShoving, Gilly pitted his not-inconsiderable strength against it. The door yielded a scant inch, and yielded it with the shuddering groan of shifting furniture. Gilly peered through the crack into the dimness beyond, jerking back as the blade slid toward him with a slow shine.\n\n\"You've barricaded the door,\" Gilly said.\n\n\"A prison can also repel,\" the boy said, his voice fierce and quiet.\n\nGilly sighed with enough force that his shoulders rose and fell. \"And here I've brought you tea and toast.\" Exasperation laced his words. Overnight, he had forgotten the boy's bloody-mindedness, his irritability. Had he really expected to find the boy meekly waiting, grateful to be let out, and willing to help entertain Vornatti over a long winter? Gilly chastised himself because he had. Had thought only of the boy's hunger and obvious poverty, and not of his pride.\n\n\"You can't stay in there forever,\" Gilly said. He slid down and sat on the exquisitely tiled floor of the hall, tracing the leafy patterns, green and gold, with a forefinger, listening for a reply. Patience, he thought, would be the only way. As patient as he must be during Vornatti's worst megrims. But cautious, too; he took care to sit beyond the reach of the sword.\n\nHe heard a faint sound, the wild-animal complaint of a starving body, and said, \"You'll get hungrier if you don't come out. And you're too thin to miss meals.\"\n\n\"I can live at least a week on the mice you let infest this place. Crack their heads, drink their blood, chew their flesh. Gnaw their tails, their feet when there's no meat left elsewhere. I'd last at least a week, maybe a fortnight without any aid from you.\"\n\nGilly fought a reluctant smile, a little charmed by the boy's manifest obstinance. \"You'll waste that blade's edge on mice? Don't do that. Come on out,\" Gilly said. \"Have tea, have toast; we'll see what old Chrisanthe has left for us in the chafing dishes. There is no need to barricade yourself away.\"\n\n\"I don't know who you are or what you want,\" the boy said.\n\n\"I'm just a servant,\" Gilly said. The usual burn touched his throat at the admittance. \"I don't want anything, except for you to come out. And you have not been forthcoming yourself. Have you no name, no history?\"\n\n\"None to share with a servant,\" the boy said.\n\nGilly flinched, flicked on the raw, and peered through the crack in the door again. The boy sat, wedged in between the wardrobe and bed steps that barricaded the door. He had added strength by ripping down the drapes and tying them around the blockade. He sat now in a dim cavern of cloth and wood, just barely big enough to fit him, and shivered with cold and hunger in the thin bar of light that streamed in over Gilly's shoulder.\n\nSilently, Gilly passed a piece of toast through the crack, felt the boy take it, and waited for the complaint about the coldness of it. But either the boy's bravado was lacking, or his knowledge didn't go so far as to encompass toast and heat. He ate it in three silent bites.\n\n\"You owe me silver.\"\n\n\"Do I,\" Gilly said. \"Will you trust a servant's word then?\"\n\n\"You promised,\" the boy said.\n\n\"I did,\" Gilly said. \"Lunas enough to get you to Lastrest, but they won't help you.\"\n\n\"Why?\"\n\n\"It's winter, and Last will be abroad, in the Itarusine court. It is his habit. Supposedly, he loves the ice and snows, but it's an open secret that he visits only to keep abreast of the Itarusine king's schemes. Last will have emptied his house of all but a few servants, taken the rest with him. He will not be there.\" Gilly smiled, getting a little pleasure in the boy's body curling tighter on itself; the lad wasn't the only one who could wound with words. \"You've chosen the wrong season for your hunt.\"\n\n\"Lies.\" A faint breath of air.\n\n\"I just hate to see lunas go to waste. When you could use them to buy food.\"\n\n\"Stop talking about food,\" the boy said, his voice cracking.\n\nRepenting, Gilly passed him the rest of the toast. \"I will take you to Lastrest if you must go, but you will see that I am telling the truth.\"\n\nThere was nothing but silence from the boy's side. Pity flared in him suddenly, seeing the boy's unwelcome choices: starvation or servitude. Was it any wonder he was uncivil?\n\n\"Stay with us,\" Gilly said. \"The old bastard's not so bad to work for. Feeds me well, and I sleep on warmed sheets.\"\n\n\"Not a whore,\" the boy whispered, going quieter, his words fading into a bare susurration. A hissed name? Janus.\n\n\"I'll call for Vornatti's coach. It will take you to Lastrest, faster than the stagecoach, faster than a hack, and you can see that I am only telling you the truth. Then it will bring you back here.\"\n\n\"No,\" the boy said.\n\n\"Where else will you go?\" Gilly said. \"Even if I gave you silver, it wouldn't last you the winter.\"\n\n\"A ship,\" the boy said.\n\n\"Go abroad? After Last?\" Gilly said. \"I don't have enough lunas for that. And even if you did get there, it's colder there, snow all year round, and they'd kill you for your sword. Listen, boy. Listen. Vornatti would see Last ruined. Vornatti sees his vengeance in you.\"\n\nFurniture stuttered into reluctant movement, and after a time, the boy eeled out the widened crack. His face was dusty and blanched beneath, his hands shaking with the effort of moving the furniture again. Gilly swallowed. Another thing he had forgotten\u2014the uncanny effect of those dark eyes against the pallor of his skin. Vornatti would devour him entire. The sword, unsheathed at his side, twitched as if it had read Gilly's thought. The boy leaned against the wall, raised his head, and said, \"You owe me lunas, still.\"\n\n\"How do you figure that?\" Gilly said, grateful for the distraction. \"I am loaning you Vornatti's coach.\"\n\n\"Your promise was enough lunas for a hack, without reliance on whether I took one or not.\"\n\nGilly paused a moment, surprised to find that even after what must be a severe setback to his hopes, the boy could haggle like a merchant and reason like a solicitor. Then he laughed, pleased at such an agile wit; he fished into his pocket and passed three lunas to the grimy hand.\n\nThe boy smiled, not the faint flicker from last night, but a slow thing that bloomed into a mockery of pleasure. \"How enlightening. To know what my price is. Some silver, the offer of food, and nebulous mention of vengeance. I just hope the food is good.\"\n\nGilly caught his breath at the self-loathing in the boy's voice, sought for words, and finally said, \"There's usually venison steaks, fresh-made bread, milk, and eggs.\"\n\nHe reached for the boy's elbow to escort him to the breakfast room, but the boy jerked away. Gilly gestured him down the hallway.\n\nThe boy's eyes widened in the breakfast room as Gilly handed him a loaded plate. \"Here. If you want more, look on the sideboard. I'll be back presently.\"\n\nIt was a gamble leaving the boy alone, but Gilly thought that the food, in such quantity, would occupy the boy long enough. After all, on that first morning in Vornatti's house, Gilly himself hadn't fled. And Vornatti offered the boy more than he had ever offered Gilly\u2014his vengeance.\n\nBaron Vornatti was only now rousing to querulous waking. He gave Gilly no time to speak, but began issuing complaints and commands. \"My hands are numb, Gilly. Rub them, get me warmed water, and some damned hot tea\u2014that raddled bitch left the trays too early again. I want my gold dressing gown today, and it's near time for...\" His voice trailed away and the dissatisfaction dragging his tone sharpened to anticipation. \"The boy, Gilly. The boy.\"\n\n\"The boy is eating, and will be doing so, I judge, for some time. He is starving.\" Gilly yanked the bell cord, and when Chrisanthe appeared, demanded hot tea and warmed water for the washbasin. Vornatti's fire was burning low, and Gilly stoked it to life again.\n\nGilly poured the tea, washed the old bastard's hands with the rest of the warm water, watched the knotted fingers uncurl. He helped him into the quilted robe, found slippers for his feet, and added yet another log to the fire until it blazed. When he would have aided Vornatti to his bath, the old man shrugged him off. \"I want to see the boy. See if he's worth the trouble I mean to expend on him.\"\n\nIn the breakfast room, the boy had finished his sampling of the chafing dishes, and paced the length of the room, pausing to stare out the windows at the blustering wind and snow. He spun at their entrance, poised to flee.\n\n\"Ah, come here, boy,\" Vornatti said, from the depths of the wheeled chair. \"Come and let me look at you.\"\n\nCatlike, the boy approached. He took his time, pausing to look at a portrait blurred with age, the ice-silvered windows, the pattern on the parquet, the engraved dishes still heating on the sideboard, making it clear that he was obeying only his own whim and that if he ended up before Vornatti, well, that was only happenstance. But he stood before Vornatti docilely enough, the sword hanging loose in his hand, waiting for the man's eyes to stop tracing his features.\n\n\"How old are you, boy?\"\n\n\"Near fifteen.\"\n\n\"Small for your age. And slight. Do you know how to use that weapon of yours?\" Vornatti said.\n\n\"Well enough to strike from behind,\" the boy said.\n\nVornatti laughed, delighted by this boy's audacity, by his blunt scheming. At least, Gilly thought, delighted by audacity in such a pleasant form.\n\n\"Don't laugh at me,\" the boy ground out.\n\n\"You've a lot of Itarusine blood in you, right enough,\" Vornatti said. \"More than just your looks\u2014it's in your very manner, that touchy pride. I don't suppose your heritage is anything more than the usual Relicts tangle of sailor and whore?\"\n\n\"If my mother thought my father Quality instead of a conscripted sailor, she'd have battened on him like a lamprey. One useless woman feeding off a useless kind. Aristocrats.\" He spat on the floor.\n\n\"Such an insolent tongue. Once, I would have challenged you for such in-civility, and you would have trembled. I was held a master of the blade in my youth,\" Vornatti said.\n\nIn the first act of courtesy Gilly had seen, the boy did not laugh, though Gilly saw the retort quivering on those curling lips. But perhaps it was not courtesy, only belated calculation. Gilly had invoked the specter of Vornatti's aid against Last, offered a coach to Last's estate. Would the boy jeopardize that for a retort better swallowed?\n\n\"What is your name?\"\n\nThe boy ignored the question, paced back to the windows, looked over his shoulder, a black shadow against the daylight. \"Will the coach travel in this snow?\"\n\n\"The open coach, in this weather?\" Vornatti asked. \"No. Gilly, go with him, take the carriage. Take hot bricks, and wine, and furs to keep warm. It's near a day's journey in this muck.\" He clutched his robe tighter about him as if he could feel the bite of the wind, the snow flung from the horses' hooves.\n\n\"I can't leave you,\" Gilly said, though the chance of a day free from Vornatti's demands tempted him. Even if it were spent nursemaiding an entirely too irritable boy.\n\n\"Give me my Elysia and I'll doze the day with wine and books,\" Vornatti said.\n\nGilly kept his eyes lowered as he nodded, but a quick series of thoughts had crossed his mind. The boy wanted coin and a trip to Lastrest. Gilly could provide both as payment to see the problematic boy gone, without Vornatti there to object.\n\n\"Gilly, I expect you to bring our lad back safe and sound,\" Vornatti said, and Gilly nodded again, mutiny fading. Vornatti always knew.\n\nHe called for the carriage to be brought around, the coachman rousted from his warm rooms in the stable. Vornatti insisted that they take every precaution against the cold, and Gilly set the housekeeper to heating bricks in her oven, to filling flasks with spiced tea and a basket with food. Gilly watched the boy's face, hoping to see some hint of appreciation that all this was done for him. But the boy's face stayed blank, as sublimely arrogant as the Quality he claimed to despise.\n\nAt Vornatti's instruction, Gilly brought Vornatti's best greatcoat out from the wardrobe, a supple thing of leather and fur. Throughout all this, the boy stared out the window, his breath clouding the glass, his hand knotted around the sword.\n\n\"May I see that?\" Gilly asked, his hand outstretched.\n\nThe boy twitched, his reverie shattered, the blade swinging up in startled reaction. Gilly caught it in his gloved hand, and the thick glove parted beneath the blade. A thin stripe of blood crossed his palm. \"Well, it's sharp enough,\" Gilly said, \"But you cannot continue carrying it like that. You need a sheath, or better still, leave it behind.\"\n\n\"And if Last is there, then what will I do? Pelt him with stones and snowballs?\"\n\n\"Last will not be there,\" Gilly said.\n\n\"So you say. I will keep the sword by my side.\" But he accepted the sheath Gilly found, even suffered him to strap the belt around his narrow hips. The boy shrugged into the heavy coat Vornatti offered, allowed the man to fumble the fastenings shut.\n\nVornatti stood on careful legs, his fingers at the top button, soft fur touching his hands, the boy's dark hair mixed with sable. He stooped, caught the boy's tense mouth with his own, and fell back into his chair. \"Travel well.\"\n\nVornatti had calculated well. The boy's face flushed red, but the sword was safely sheathed and buckled beneath the greatcoat, essentially unreachable. The boy scrubbed at his mouth with a gloved hand, stormed out of the room. Gilly followed, and despite the inherent unfairness of Vornatti, rich and powerful, controlling the boy, found himself suppressing a laugh at Vornatti's incorrigible nature, and at the would-be murderer's indignation. Perhaps they deserved each other.\n**\u00b7 3 \u00b7**\n\n**G** ILLY REGAINED HIS CUSTOMARY PLACIDITY by the time he joined the boy inside the carriage. The boy sulked in the corner, face nearly buried in the fur collar, eyes shuttered by dark lashes.\n\n\"Drive on,\" Gilly called. \"The sooner there, the sooner back.\"\n\nThe coach lurched to motion. A strange, silent ride it would be, too, Gilly thought, traveling through the snowfall with a shadow for a companion. When the hiss of snow and silence numbed Gilly's ears, he determined to coax the boy out of his megrims. \"Are you warm enough?\" Gilly asked.\n\nThe boy turned his head to meet Gilly's eyes but made no answer, his face scornful. The boy, wrapped in fur, a hot brick at his feet, full-bellied, was probably warm for the first winter of his life.\n\nBasic civilities having failed, Gilly tried bluntness. \"Tell me about this Janus of yours.\"\n\nThe boy looked out at the falling snow.\n\n\"You intend to rescue him, so there must be some fondness there. Is Janus your protector? Your friend? Your brother? Lover?\"\n\n\"Stop saying his name like you have a right to it,\" the boy said, a frantic edge to his voice. His chest heaved, visible even beneath the bulky coat, and Gilly sighed.\n\n\"You are a nervy creature. I only thought to pass the time by getting better acquainted. Since I don't know you, I asked questions, but you may ask them of me, if you prefer.\"\n\nDrifting snow and the muffled rhythm of hoofbeats were the only response he got. Gilly shrugged, sank down into the seat, and composed himself for a nap.\n\n\"Are you his whore?\" The question jerked him out of his doze, made heat scald his face and ears.\n\n\"All servants are such,\" Gilly said, taking shelter in philosophy. \"We do as we're told for money, whether we want to or not. The most we can do is choose our masters.\"\n\n\"I'm my own master,\" the boy said. \"And semantics aside, you're a whore.\"\n\nGilly flinched at the weary contempt in the boy's voice, reminded of the only time he went home. His father had spoken to him in just that tone. Un-fair, Gilly had thought then. They'd sold him to Vornatti; what had they expected? \"I\u2014Vornatti\u2014I obey his whims and accept his advances. Why shouldn't I? He took me from a farm that couldn't feed me, and gave me a library, good food, a room to myself, and free license when in the city.\" By the end of his speech, Gilly had nearly convinced himself once more of the wisdom of his bargain. \"He will grant you similar gifts.\"\n\n\"If I please him,\" the boy muttered. \"Make myself a toy for his whims and desires. Make myself a thing rather than a person. A possession easily replaced when he tires of it. How long have you been with him?\" When Gilly choked on his answer, the boy smiled before staring out the window; his breath frosted on the cloudy glass.\n\nQuiet minutes passed with the only sound being that of the wet impact of snow spattering the carriage like sea spume.\n\n\"How uncivil you are,\" Gilly said. \"Vornatti's offered to aid you\u2014\"\n\n\"Catch me believing anything that an aristocrat says,\" the boy bit out. \"They all cleave together.\"\n\n\"And you know so much of their ways,\" Gilly said, gently mocking. \"Do you think you're the only one to hate Last? I promise you, Vornatti's distaste for the man runs to the bone.\"\n\nAt the boy's skeptical expression, Gilly said, \"It's true. They met more than thirty years ago in the Itarusine court. It's a dangerous place, rife with assassin princes and poisonous noblewomen. A frozen land of coldhearted people who pride themselves on their courage, their aggression, and their willingness to do anything to see their desires met. It makes our court seem milkwater in comparison.\n\n\"For Last, only a fourth son even if of royal blood, a season in the Itarusine court was his opportunity to marry well, to gain a fortune he would not inherit on his own, perhaps make an alliance between the two courts. But Last proved too stiff-necked, too conservative in his views to thrive there, and when he met Vornatti\u2014well, I believe they arranged a duel before they finished making their first bows to each other.\"\n\nThe boy gazed out the clouded glass again, seemingly uninterested, but his fingers sketched brittle shapes in the fog his breath left. Tiny crosses that could be daggers, could be swords. Gilly said, \"By the time the duel became fact, Last had absorbed enough of Itarusine ways that he paid Vornatti's whore to render him insensible. When Vornatti missed the duel, Last declared him a craven. It wasn't done out of fear; Last is an admirable swordsman. Rather, it was done out of spite. It ruined Vornatti, far more effectively than even a lost duel could. It took until Xipos for him to regain his reputation.\"\n\n\"One thwarted duel and you think Vornatti can hate Last as I do?\"\n\n\"There's more, there's always more. And far too much to explain now.\" It wasn't time so much that stilled Gilly's tongue as consideration. Vornatti had few weaknesses, but the reminder of his sister was one of them. Aurora Vornatti had been the old bastard's heart, the only person he loved purely. When Aris Ixion had chosen to wed her, Last had spread slander wide and far trying to dissuade him. Whispers of wantonness, of inbreeding, even of flesh turned poisonous. Aris had earned Vornatti's friendship by denying the rumors. But when the long-awaited heir proved damaged, when Aurora died of his birth, the slander rose again and followed her to the grave.\n\nSometimes, it seemed to Gilly that Antyre itself was trailing after her into the grave. Xipos the first blow, and Aurora a deadly second thrust. Aris seemed unable to recover from either.\n\nGilly frowned. If the boy hadn't known Aris, their king\u2014\"You understand about Xipos?\"\n\nAfter a blank, black look, during which Gilly recalled the utter self-absorption of his audience and the lack of education, he explained the Xipos War. As much as that prolonged and bloody decade could be explained. The cause was simple enough: Itarus attempted to seize Xipos and its winter ports from Antyre's grip. But the battling grew so protracted and vicious that even the gods grew sick of it, the cause of their vanishment.\n\nThe poisonous offshoot of battle, assassination, claimed the king and crown prince; the second son died on the front; and the third son, the quiet scholar, Aris Ixion, inherited a kingdom at war. Aris, who valued life more than land, acted decisively. He surrendered, no matter that the terms of concession were ruinous: tithes, taxes, forced exports to Itarus that were sold back to Antyre at a hefty profit. If it weren't for Antyre's colonies in the Explorations, Itarus would have conquered Antyre one coin at a time.\n\n\"Grigor, the Itarusine king, sent Vornatti to be Antyre's auditor and warden, which places him ideally to aid you.\" The boy raised his head from where he had been resting it against the cushions. \"I thought that might wake you,\" Gilly said, smiling.\n\n\"I don't need his aid,\" the boy said.\n\n\"You'd be frozen back at the first hedgerow without it,\" Gilly countered.\n\nThe carriage came to a halt, rocking gently on its wheels in the wind and snow. The coachman leaned his snow-crowned head in, pulled the ice-crusted muffler from his mouth. His breath plumed in the still air. \"Gates are closed to Lastrest, Gilly.\"\n\n\"Open them and drive on.\"\n\nA rush of cold air greeted Gilly's words. The boy had clambered out and was floundering through the deeper snow beside the road. Gilly framed himself in the door and called out, \"It's still a mile or more. Too far to walk, boy.\"\n\nThe boy stepped between the bars of the gate while the coachman shoved at them, trying to free the gate from the clutch of the snowdrifts. Gilly cursed and went to help. On the other side, the boy tripped over the long skirt of the coat, and sprawled, frosting the sable fur, then got up again, heading for the trackless white of the drive.\n\nOnce the gates were ajar, Gilly's longer legs allowed him to catch the boy. He grabbed the boy's shoulder and shoved him back toward the coach. \"Get in. You'll freeze and Vornatti will be angry with me.\"\n\nThe boy's face was white, blanched by strong emotion, and he shivered under Gilly's rough grasp. \"There will be no one there. Don't fuss yourself so.\" The boy sat with deceptive patience as the coach furrowed its way through the snow.\n\nAt the manor, the coachman pointed out the shuttered windows, the door knocker taken off the latch. \"No one's here, Gilly.\"\n\n\"We'll ask the servants, to be sure. At the least, they might offer us some hospitality from the cold,\" Gilly said. He pounded on the door. Snow crusted his shoulders, and his gloved fingers had gone numb before an old man answered.\n\n\"Last has gone abroad to join his son. He will not return until the spring.\" The old man spoke it all in one breath, as inanimate as a puppet, and as disinterested. He started to shut the door, but Gilly leaned against it.\n\n\"Join his son?\"\n\n\"Yes, the boy has been educated abroad these sixteen years, and the earl wishes to see how his lessoning has gone before he introduces him to the court.\" More puppetry speech, but irritation surfaced in the old man's eyes as the wind stung his exposed face.\n\n\"His _bastard_ son, Janus?\" Gilly said.\n\nThe butler's mouth primmed. \"It is my understanding that Janus is the son of a prior, unreported marriage.\"\n\n\"Of course,\" Gilly agreed. \"My mistake.\" _An unrecorded marriage, and a new heir for the earl of Last, a new member of the royal family?_ The immensity of the news left him stunned. _A bastard\u2014heir to the earldom, to the throne?_ Gilly turned to see what the boy was making of all this, and found him gone. The door slammed at his back. \"Where is he?\" Gilly asked the coachman.\n\n\"Went round 'longside, like he knew what he was doing.\"\n\n\"You didn't stop him?\"\n\nThe coachman shrugged.\n\nFaintly, Gilly heard the chime of glass breaking. Exasperated, he hurried after the boy, stumbling in the deep footprints of his path.\n\nThe coat had been discarded, a dark blotch in the snow beside an ivy-covered wall. The ivy was brown and sere, withered by the cold, but up above a window flashed, and something weightier than ice fell to the drift beside him. Gilly stooped, picked up the shard of glass, and swore. He backtracked, pounded on the door until his hands stung with impact, but this time no one answered.\n\n\"Drive down to the gate, blanket the horses, and wait for us. I'd rather them not grow suspicious of our continued presence.\"\n\nHe returned to the wall of ivy and stared upward. Gilly estimated that he had at least eighty pounds over the boy, but the ivy showed no breakages from the boy's ascent. Maybe it would hold his weight, he thought. Or maybe he would plummet to the snow beneath and break a leg or his neck.\n\nGilly took off his coat, dropped it over the boy's, firmed his gloves around his fingers, and began to climb. Ivy leaves crumbled beneath his hands; the vines stayed firm, until, within an armspan of the window, they started to peel away from the mortar. Gilly lunged upward, hooked his hands over the sill, and pulled himself inward, landing on the dusty floor of an unused bedchamber. He wrinkled his nose, repressed a sneeze and a sneer that Last couldn't get good servants either, and set off tracking the boy's damp footprints.\n\nHe found the boy standing in the shadowed alcove of a long hallway. \"What are you\u2014\"\n\nThe boy put an icy hand over Gilly's mouth, drew him into the alcove. \"There's someone coming.\" His whisper warmed Gilly's ear.\n\nThey watched the maid carrying the bundles of whites pass them and head down distant, uncarpeted stairs.\n\n\"What are you doing?\" Gilly repeated.\n\n\"Learning my enemy.\"\n\nGilly sighed. \"I can tell you about him. And in the comfort of Vornatti's library. Or at his dining table if you're hungry again.\"\n\nThe boy wandered into the hallway, looked down the stairs into a dim long room, near bare of furnishings. \"What is that?\"\n\nGilly peered over the boy's shoulder. \"Portrait gallery. Pictures of his ancestors.\"\n\n\"He knows what they all look like?\" The boy was impressed by that, by the simple fact that the earl of Last knew who fathered him, who fathered his father, and so on, and more, could trace images of himself in their features. \"I want to see.\" He descended the stairs and Gilly hastened after him.\n\nThe nearest panel was blank. The boy turned to Gilly, his face demanding explanation. Gilly, keeping a nervous eye and ear out, said, \"That's for the next earl, the current earl's son. The portraits go by birth order.\"\n\n\"So this is Last?\" The boy walked on to the next portrait, heading farther into the house, farther from the window and escape.\n\nGilly caught him up. \"Yes. This is Michel Ixion, the fourteenth earl of Last.\"\n\nThe boy's eyes narrowed. \"Is it like him?\"\n\n\"Enough,\" Gilly said. The boy touched the painting, put his palm flat against the canvas, then drew his hand into a claw as if he might start tearing at it, as if he could slake his bloodlust on a picture. Gilly tugged the boy away, memories of rural superstition making his skin crawl, thinking of pins stuck in dolls and left on altars for godly intercession, never mind that the gods could not answer.\n\nThe boy's wrist trembled against Gilly's hand, the fine bones taut under Gilly's fingers, but he didn't resist. His eyes fell on the blank panel again and his breath caught. \"This is for Janus?\"\n\n\"It's unlikely. He is a bastard, no matter the story they intend to put out. It's far more likely that Last means to use Janus as a bargaining chip in his next marriage\u2014a tangible, albeit scandalous, counter to the rumors of tainted blood.\"\n\nEven as Gilly said it, he wondered, _Why legitimize the bastard at all?_ It wasn't like Last, a stalwart traditionalist, to fly in the face of custom, to not only recognize his bastard son, but to legitimize him. But maybe someone had commanded him.... Gilly collected rumors for Vornatti, and at the heart of them was the king, that melancholy scholar who'd been saddled with the burden of the throne\u2014a burden he could set down only for an heir. Last was no solution\u2014only a year's difference lay between the two\u2014and Kritos, though younger, was a wastrel, a gambler, and a fool. Perhaps Aris, trapped on his throne, dreamed of Janus.\n\nBut the three counselors would be hard to convince. Like Last, the duke of Love was a traditionalist, and would condemn Janus for the irregularity of his birth. But perhaps he could be bought; he had a marriageable daughter. DeGuerre was a believer in blood and a military man; he might accept the bloodline, and ignore the lack of marriage papers. After all, Celia was an admiral's daughter. And Westfall, despite his trappings of egalitarianism, had a young man's awe of the royal blood, even watered down. _A bastard king?_\n\nThe boy murmured at his side, waking him from his political reverie. \"This could be his?\"\n\n\"It is impossible to allow him the earldom and not put him in line for the throne,\" Gilly said. \"So it's unlikely, but I suppose, if Last died suddenly\u2014\"\n\n\"He will.\" The boy touched the sword hilt, smiled a little, then stared at the empty spot again. \"The throne. This house?\" Incredulity laced the boy's voice at the idea of the house more than of the throne. Gilly understood that. The throne was so far distant from his experience it might as well be a dream, but the possibility that Janus could live in a house like this\u2014\n\n\"The king has but one child, and that one born simple. He whiles his time away in padded nurseries, playing with dolls. There are few members left in the house of Last. The king, the earl, and their nephew, Kritos.\"\n\n\"Kritos,\" the boy said, a bare whisper.\n\nFootsteps echoed down the hallway; Gilly snatched at the boy to drag him back up the stairs, but the boy eluded him, passed through another door. \"What's this room?\"\n\n\"Last's study,\" Gilly said. He closed the door behind them, turning the key in the lock.\n\nThe boy skimmed around the room, pocketing trinkets: an enameled snuff box, a gold-handled letter opener, a quill pen and an ink bottle, a silver paperweight, and a crystal carving of Baxit, the cat-headed god of indolence and reason. He unearthed a gilt-edged porcelain dish of old toffees and, after a quick sniff, put two in his mouth, closed his eyes, and chewed. Then he tilted the rest into his shirt. As a visible afterthought, he dropped the delicate dish into his sleeve as well. Gilly bit back a laugh. \"We must go. The horses\u2014\"\n\nThe boy investigated the books on the shelves, touching brightly colored leather bindings and tracing his finger over the gilded titles. Sitting at the desk, the boy used the letter opener to pry open the locked drawer. Sheaves of paper curled out.\n\n\"Let me see,\" Gilly said, reading. \"Creditors, debts, and bills from Kritos. Such a wastrel. Won't please Last, that's for certain.\" A smile quirked his lips. \"Perhaps Kritos will drive Last to apoplexy and spare you the trouble.\"\n\nThe boy's eyes sharpened, went black with rage, and Gilly felt the smile vanish from his lips as if it never existed.\n\n\"I dream of killing him,\" the boy said, \"his blood painting my blade, his cry in my ears as I touch his heart....\" His fist tightened around the hilt, fingers whitening as if he meant to withdraw it.\n\n\"Because of Janus,\" Gilly said, stepping back out of reach of the boy's sword, edgy again. He had almost forgotten this boy's vendetta in a strange enjoyment of this leisurely housebreaking.\n\n\"Janus,\" the boy echoed. Something softer warmed the bleak fury in his eyes, and his grip lessened. His face grew still and troubled; Gilly wondered what the boy was thinking on\u2014his bloody plans for Last, or the butler's unwelcome confirmation that his prey had slipped his grasp. In this quiet state, the boy was malleable, allowing Gilly to usher him out of the study.\n\nAfter a brief consideration of the state of the ivy, Gilly dragged the boy down another flight of stairs to find a ground-floor window. Gilly dropped out into the deep snow, and then held up his hands. \"Come on.\"\n\n\"I don't want your help,\" the boy said, flinging himself into the snow and frost.\n\n\"At least put on your coat,\" Gilly said, reclaiming them from the snow-bank, and flinging the boy's at him.\n\nThe boy snarled and Gilly walked on, leaving the boy to flounder his way through the drifts, hampered by the heavy coat. Gilly reached the coach long before the boy, climbed into it, and sat, sipping hot, whiskey-laced tea from the flask. The boy staggered up, white from head to toe with blown snow, and shuddering with chill. His eyelashes were frosted and his face showed signs of suspicious dampness. Gilly wondered if the boy had been crying as he fought his way down the drive.\n\nThe boy clung to the edge of the coach, panting, shaking, soaked through. Again, Gilly offered a hand. The boy flinched, put his hands over his face, and then let out a sigh. He reached out and Gilly tugged him into the coach, rapped on the roof to let the coachman know to start.\n\n\"What'd you do? Swim your way through the drifts?\" Gilly asked, peeling the sodden coat from the boy's arms and back. \"I've heard that in Itarus there are sports like that, where bored lordlings drag themselves behind sleighs, but I don't think even they manage to get so much snow packed into their skin.\" He pulled a woolen blanket from their basket, draped it around the boy, then passed him the flask. \"Drink this. It'll warm you.\"\n\nThe boy's teeth chattered on the edge of the flask, but a faint tinge of color seeped into his cheeks after the first few gulps. He looked up at Gilly with a cringing wariness. \"Can I get to Itarus?\"\n\n\"If you sell everything you stole from Last's house and that sword, you might have enough. But then what? You'd be alone, hunting Last, hunting Janus in a country where you didn't speak the language. In a country of poisoners and duelists who'd make mincemeat of you before you ever reached your goal.\"\n\nThe boy turned his face, drew in a breath and held it. In his lap, his hands clawed at each other. \"If I stay\u2014\"\n\n\"If you stay, you'll wait out the year in comfort, in warmth, fed well, with a man who can explain the ways of the court and the aristocracy\u2014who might even aid you.\" Gilly took the flask away from the boy, drank another draft, more for the whiskey in it than for the warmth. He felt like a procurer.\n\nThe boy wrapped the blanket tighter and tucked his head into it, like a bird ducking its head beneath its wing.\n\nThis time, Gilly found no need to break the silence of the ride and the ever-darkening sky.\n\nChrisanthe greeted them at the door with a sour \"He's been waiting for you. Having fits, thinking you was tipped into a ditch.\"\n\nGilly went straight into the library, shrugging off the lashings of snow that adhered to his sleeves and hair. The boy's footsteps followed, but Gilly didn't look back. The boy had to make his own decision.\n\nHe knelt down. \"How was your day, old bastard? Did Chrisanthe give you your Elysia?\"\n\n\"Near broke the needle off in my arm, stupid cow. But you've come back and you've brought the boy with you.\" He patted Gilly's cheek. \"We'll feast. Stir up Cook.\"\n\nGilly returned, the cook's grumbling ringing in his ears, to hear Vornatti speaking. \"Well, boy, did you find we were telling you true?\"\n\nThe boy didn't reply in words; instead, he opened his pockets and showed Vornatti the small valuables he had pilfered, laying them out like offerings. Vornatti reached forward and picked up the little crystal figure. \"Baxit. Not surprising. Some men are too stubborn to give up on the past.\"\n\n\"You can have it,\" the boy said. \"The other things too. For my board.\"\n\nGilly raised a brow, waited for Vornatti's response, to see if he would let the boy buy himself a position as guest.\n\n\"I don't take renters. And I have trinkets enough,\" Vornatti said. \"Keep these trifles to decorate your room.\"\n\nThe boy sucked in his breath, moved to the fireplace, stared out at the snow. Finally, moving as stiffly as a wounded man, he walked back toward Vornatti, hand on sword hilt. Gilly tensed, but the boy only scooped up the fallen toffees, the little dish, and took a step back.\n\n\"You're staying then?\" Vornatti said, reaching out to fold his fingers in the boy's hair, caressing the dark, snow-damp locks.\n\nThe boy remained still with a small but visible effort; his eyes flickered again to the fireplace, to the fur coat shedding its icy rills of melted snow, to Chrisanthe grumbling in under the weight of a laden tray, and said, \"Until spring.\"\n**\u00b7 4 \u00b7**\n\n**I** WON'T DO IT,\" the boy said, taking a step back, away from Gilly, closer to the door and escape.\n\nGilly, sweating with effort, emptied another iron kettle full of near-boiling water into Vornatti's bath. Pushing the steam-damp hair from his face, Gilly assessed the boy, standing as rigid as a nervy horse, looking at the sloping marble tub with every evidence of horror.\n\n\"You will,\" Gilly said. \"You stink. And you likely have lice. Two things Vornatti doesn't much care for. You're lucky he's been as patient as he has. He had me scrubbed the very first moment he brought me home, and I was far cleaner than you.\"\n\n\"You die if you wash in winter,\" the boy said, taking another step back at the next gush of water added.\n\n\"That's ridiculous,\" Gilly said. \"My mother scrubbed us all once a week, no matter the season. Vornatti insists on a bath daily, and you've seen his advanced age.\"\n\n\"I don't want a bath,\" the boy said, withdrawing like a repulsed cat. \"If my stink keeps him away, so much the better.\"\n\nGilly grunted as he hefted another of the water-heavy kettles. \"You'll have one. The only choice you have in the matter is whether you want to be held down and scrubbed\u2014\" He ignored the boy snarling and drawing the sword, and continued, \"or whether I leave you here with the soap and your dignity. Think, boy\u2014at best you'd avoid the waters tonight\u2014but you'd find yourself drugged again, and bathed all unwitting. The baron may be an old man, but his sense of smell is keen.\"\n\nLetting the last kettle fall with a clang, Gilly wiped his hands on his breeches, then opened an armoire. \"There are dressing gowns here. Put one on while you dry off and you'll catch no chill. We'll find you clothes later.\" The boy's eyes were still wild, and Gilly sighed, let the vexation in his tone ease. He supposed, to a boy like this, brought up city-poor, immersion in water might be frightening.\n\n\"I'll leave you the key to the door. You know, boy, some people, myself included, enjoy a bath after a cold day. The water is very pleasant\u2014as long as you don't let it grow chill.\" He laid the key beside the bath and left the boy, sword still drawn, staring at the steaming water.\n\nTHE CLOSING DOOR woke the boy from his stillness. Reaching forward, he closed the key in his fist, then turned to the door. He turned the key in the lock, tested the latch, then set the sword down on a wide bench. _Vornatti must sit there,_ he thought, _before his bath, drawing off his clothes. No,_ he thought, his mouth twisting, _Vornatti sits there and_ Gilly _draws off his clothes._\n\nThe boy touched the steaming water with cautious fingers, setting off small ripples. He put his fingers to his mouth, then sat in Vornatti's chair. Toeing off his boots, he hesitated, looking at the locked door once again. He leaned his head against the door, listening; the dense wood gave back only silence.\n\nGingerly, the boy rose and unfastened the rough strip of canvas that made his belt. His breeches sagged past his knees and he stepped out of them. His stained linen shirt cloaked him from neck to thigh, and after another wary moment, he pulled that off as well with the air of a conjurer.\n\nAnd with a conjurer's touch, the moment changed. One moment a grubby, skinny stripling boy stood before the bath\u2014the next, a young woman, unbinding another strip of dirty canvas from across her budding chest. Her side was mottled dark with old blood, spilled from a wound that had healed long since. She touched the flaking residue, touched the pale pink weal of the whip mark, frowning.\n\nTaking up the soap, she held her breath, then stepped into the bath. After her first shuddering moment, when the heat of the water and the chill of the air warred over her, she calmed, sank down into the water.\n\nNerving herself, she took a breath and ducked her head; she came up to a sudden draft in the room. Spinning, water slopping over the edge, she clawed at the rim of the bath. Vornatti laughed, closed the door behind him. \"Looking for this?\" he said, taking cautious steps forward, holding her blade. His eyes glittered.\n\n\"Get out,\" she said, clutching the soap with shaking fingers.\n\n\"You have so much to learn,\" he said, voice full of amusement. \"You've learned two things now already. One\u2014a door that is locked can be unlocked. Guard your secrets accordingly. Two\u2014keep your sword by your side. A blade's no good, no matter how sharp, if it's out of reach.\"\n\nHe settled stiffly onto the bench, and leaned forward, his eyes lingering on her skin. \"A girl, then.\" He smiled. \"It's been too long since I've had a girl.\"\n\nShe threw the soap at him; he raised her blade and bisected the soap, then winced. \"Elysia only takes the pain away, my girl, it doesn't restore youth. This sword is a young man's weapon.\"\n\n\"It's mine,\" she said, surging out of the water, snatching it from his hand.\n\n\"Forgive me,\" Vornatti said. \"But tell me then\u2014was it chance or choice that made us take you for a lad?\"\n\nShe pulled on a dressing gown, sank into a sulky heap near the fire. \"I am not a fool. A girl with a sword is asking only for someone to take it from her.\"\n\n\"You intend to face Last as a boy.\"\n\n\"Would he face a girl? I think he would not. I think he'd call forth his coachman to beat me down again, and ride on.\" As she spoke, she tapped the tip of the blade on the hearth, chipping bits of brick loose like old blood.\n\nVornatti leaned forward and laid his hand on her shoulder. \"What's your name, girl? After all, we're to be intimates. I'd like to know what to call you, what _Janus_ called you\u2014\" He broke off, the sword pressing up against the crepey skin of his neck.\n\n\"What he called me is of no matter. That girl is dead. And you don't need my name, don't need it to call me to heel. After all, I'm rarely to be out of your reach. Unless\u2014\" The sword shifted a tiny, meaningful increment.\n\n\"Will you kill me? Then what? Flee my home back into the snows, as desperate as you came?\"\n\n\"I'll rob this place blind,\" she said, rising, the blade steady, depositing a line of brick dust against his pale skin.\n\n\"And what about Gilly? All I need do is call out\u2014then your secret would be shared with one more.\"\n\n\"He'd probably thank me for killing you,\" she said.\n\n\"Would he? If he had to go back to the farm where he came from, bury his wit in the soil? Till the fields alone, next to the graves of his family, dead of the plague? He has no one, no one but myself, and nothing but what I provide. I own him as surely as I own my horses, which would suffer if set free.\" Vornatti pushed the blade aside, touched her face, her neck. \"It's not so much I want from you. A name.\"\n\nShe shivered as his fingers spidered into the vee of the dressing gown, cupped her breast, touched the curling scar beside it. \"I will not give it.\"\n\n\"You're too thin,\" he said, withdrawing. \"Get Gilly to feed you more. If I wanted to stroke drawn skin and bones, I'd find my pleasure in myself and spare myself troublesome chits and lads.\"\n\nHe slumped back on the bench; she took advantage of the space to move away.\n\n\"So tell me, girl,\" he said, voice growing weaker. \"Shall I have Gilly find you breeches or a skirt?\"\n\n\"Breeches,\" she said.\n\nVornatti dozed, jerked awake. \"Well, I own I'm glad not to share your secret with Gilly. He's a devil with the maids. Thinks I don't know he spends his allowance on willing barmaids in Graston village.\" He coughed, breathed heavily for a moment, studied the heap of her fallen clothing.\n\n\"You bind your breasts? The scant handfuls that they are? Well, I can help you there. One of my\u2014friends was an actress who specialized in male roles. Her corset should fit you and be more secure than any length of linen. Come now, girl, aren't you going to thank me? It's not every man who'd help a girl find vengeance....\"\n\nHe patted his cheek, his mouth. Clutching the robe closer about her, she leaned forward, touched his cheek, his lips with her own. Vornatti smiled.\n\n\"Let me tell you one thing more, if boy you'll be: To play the part, you must believe the part\u2014forget who you were. Rumor and gossip are everywhere in this country, even when it involves insignificant little chits like yourself.\"\n\n\"That was the mistake Kritos made, thinking me insignificant enough to leave me alive,\" she said, _he_ said, the suppressed savagery in his voice enough to stifle Vornatti's smile.\n\nGILLY, WAITING OUTSIDE the baron's quarters, went in at the sound of the slamming door. Vornatti staggered over to his wheeled chair, panting. Gilly took the handles and drew him over to the bed. Vornatti laughed. \"Such a lovely surprise under the filth, Gilly, you've no idea....\"\n\nThe door to the bath slammed open; the boy stalked out, clad in Gilly's old breeches and shirt, long ago outgrown. He shot Vornatti a look composed of equal parts anger and wariness, but the black look Gilly earned was all rage. Gilly stepped back under the weight of it.\n\n\"I'm tired, Gilly,\" Vornatti said, holding up his arms. \"I won't want dinner.\"\n\nGilly put him to bed and went after the boy. He hadn't gone far; Gilly stepped out, and found himself skipping back against the door, the sword at his chest.\n\n\"There were two keys,\" the boy said. \"You let me believe\u2014\"\n\n\"Enough,\" Gilly said, too tired to be wary. He ducked the sword and, cat-quick, seized the boy's thin wrists in his hands. The boy kicked his shins, and Gilly, remembering squabbles with his hot-tempered little brother, twisted the boy's wrists sharply, making him drop the sword. When the boy still fought, twisting and biting, Gilly lifted him by his wrists, dangled him in the air. \"Enough,\" he repeated. It had always worked on his brother, on fighting dogs, and feral cats. It worked now. The boy sagged in his grip, wiggling only a little.\n\nGilly released him. The boy fell to the floor. Gilly winced as the boy turned a wary face up to him.\n\n\"Sorry,\" Gilly said. He picked up the sword; the curling hilt scraped his knuckles, and the whole thing seemed to whisper against his palm. \"Here,\" he said, \"take it.\" He thrust it out at the boy, regretted touching it at all. It hadn't felt quite like steel, felt born, not forged, and malign by nature. Perhaps it _had_ been god-created, but such artifacts were few and jealously hoarded. Gilly fisted his hand, ridding himself of the sensation it left behind. With growing concern, he watched the boy sheathe it: Where had the boy found such a blade? He knew better than to expect an answer were he to ask.\n\n\"Are you hungry? There's dinner waiting. It's venison again. It's mostly venison all winter. You'll be sick of it come spring.\"\n\nThe boy stood. \"I'm not hungry,\" he said.\n\n\"You're skin and bones,\" Gilly said, wondering if he was always doomed to argue with the boy.\n\n\"So he said. But why I should gain flesh simply to please his lecherous\u2014\" The boy's jaw snapped shut; his eyes blazed.\n\nGilly took the boy's elbow in his hand, walked him toward the library, wanting to be, if not friends, at least amicable, if the boy's temper could allow such a thing. To that end, Gilly said, \"I know something that might make you feel better.\" He drew the boy past the books to the frosted doors. They stepped outside into the winter night; their breaths fled from them like ghosts.\n\nGilly bent, pulled up a handful of broken marble, snow-dusted. \"It's the old facing from the house. I like to get my anger out this way.\" He hefted the fragments, tested the weight, and pivoted, hurling the missiles at the orchard. The rocks hit the nearest tree, scattering icicles.\n\nGilly collected another handful, thinking of Vornatti giving him precise instructions regarding keys and bathing rooms. He sent another tree-load of ice to the ground, letting the sound drown out his guilt.\n\nHe handed the boy the next stone, cold and damp with snow. \"Imagine you're throwing your anger, your frustration out.\" Another game he'd played with his brother, who could only be distracted from his tempers, rarely soothed.\n\nThe boy closed his eyes, his jaw clenched, the scar flared red, and then he threw. The stone sailed forward, effortlessly hitting the tree. Icicles cascaded, but before Gilly could hand him another stone, the next tree shed its icy teeth. Then the next and the next, until the entire orchard was crashing and shattering with one thrown stone. Gilly caught a shuddering breath at the glittering wreckage the boy had made.\n\nVORNATTI RUSTLED PAPER in his lap, unfolding the envelope. Gilly watched, intrigued. Usually the old bastard tossed Gilly his post, trusting Gilly to file away the gossip, the bills, Aris's reports on profits sent to Itarus, and to act on the few business letters he received. But this letter lacked the creamy color of Antyrrian vellum, was tinted slightly blue, nearly translucent. The thick lines of script shone through the paper.\n\n\"What think you of this?\" Vornatti said, passing the letter to Gilly.\n\nAcross the library, the boy looked up from his contemplation of Vornatti's book.\n\n\"Read it aloud, Gilly, since it concerns our young friend.\"\n\nThe boy shut the book, not bothering to mark his page. And why would he, Gilly thought, when he could only be looking at the pictures, and not the text?\n\n\"Gilly,\" Vornatti warned.\n\n\"Sir,\" Gilly said, began. \"It's from Itarus,\" he said, surprised. \"How did you\u2014\"\n\n\"I have my ways, Gilly. You'd do well to remember that.\" Vornatti closed his eyes. \"Read.\"\n\n\"It's a copy of a letter from Kritos to Last,\" Gilly said. The boy stiffened, silent. Gilly angled the letter to get the most of the firelight on the crossed words.\n\n_\"Michel, cousin, while I acknowledge that you have come to Itarus as I requested, I did not intend for you to dally within the foreign court, and leave me with your ill-begotten, ill-tempered bastard son. He is unmanageable. A feral dog would have more gratitude. A rabid animal would have shown less rage. We've had to lock him in the turret, to keep him from escape attempts. If we were not on Ice Island, he would have succeeded. I can not even enter his room without his attack._\n\n_\"It's all very well to suggest threatening him, to derive obedience through fear, but he's not so blind as that. He knows you want him alive, and what else have I to threaten him with? He cares not for hunger nor cold nor beatings, though at least those serve to weaken his outbursts._\n\n_\"No, cousin, if you have any hopes of firing him off among the Itarusine court, and then among our own, you will have to take a hand. As it is, I have the severest doubts he can ever learn our ways. I wash my hands of his education. I will be his jailer only. If you would have me do otherwise, you must take a hand yourself._\n\n_\"Kritos.\"_\n\nThe boy stood, his hands shaking. \"Kritos.\" The loathing in his voice darkened the atmosphere of the room, bringing winter darkness to the fire-lit circle.\n\n\"How did you get this?\" Gilly asked, turning the letter over in his hands, looking for some hint of the sender.\n\n\"A matter of enmity,\" Vornatti said. \"Last hates me. As does my heir, Dantalion. As such, they are acquaintances, at least during the days of the Winter Court. It's a small matter to pay one of Dantalion's servants to copy any interesting letters.\"\n\n\"Is there anything else?\" the boy asked. \"Did Last go to aid Kritos?\"\n\nGilly watched the tremor move from the boy's hands through his spine and disappear, leaving him as still as a crouching cat.\n\n\"So greedy,\" Vornatti said. \"Here I've worked one prodigious collection of information, from my chair, mind you, and you only ask for more. Will you thank me?\"\n\n\"You derive too much pleasure from your intrigues to need my thanks,\" the boy said. \"Tell me.\"\n\n\"I did receive word that Last has retired to Ice Island,\" Vornatti said.\n\nThe boy's face shuttered, locking away emotion, but Gilly had seen a quick wash of perplexity cross his face, as if he didn't know whether to take Last's involvement as a good thing or bad.\n\n\"Come then, thank me,\" Vornatti said. \"Or are you as unmannered as Last's whelp?\"\n\n\"I'm worse,\" the boy said.\n\nVornatti laughed. \"How do you figure that, boy? You've been brought to heel, domesticated by food and a little frost. The only independence you have left is your stubborn refusal to grant us your name.\"\n\nThe boy looked to the barren trees in the orchard outside. This winter, they had not gathered icicles for more than a night without rocks being hurled at them. The black rage in the boy's eyes sank back; he dutifully crawled into Vornatti's lap and kissed him. Vornatti stroked the black curls, but the moment Vornatti's lips left his to draw breath, the boy was across the room, never mind that he left strands of his hair in Vornatti's clutching fingers.\n\n\"Gilly,\" Vornatti said, smiling. \"I'm tired.\"\n\nObediently, Gilly rose, folded the letter in neat quarters, and set it on the desk.\n\nWhen he returned an hour later, flushed and straightening his clothes, he found the boy still in the library. Gilly hastily tucked his shirt back into his breeches, embarrassed anew under the boy's dark eyes.\n\nSeeking distraction, he discovered it in the boy sprawled beside the fireplace, in the book spread open before him. Gilly remembered the frustration he had felt once, touching the incomprehensible secrets of letters and words.\n\n\"I'll teach you to read if you like. And write,\" Gilly offered.\n\nThe boy propped himself on his elbows. \"Do you think I come to look at the pictures?\" He passed the book to Gilly.\n\nThe book was not one of Vornatti's pornographic woodcuts. It was instead Sofia Grigorian's text-dense treatise on exotic poisons used in the Itarusine court.\n\n\"Are you suggesting you can read?\"\n\n\"I am telling you I can. And write.\" The boy's lips curled in a smirk that Gilly was beginning to recognize. It betokened the boy's worst tempers. The news from Itarus was not to his liking, Gilly thought. Despite everything, the boy had hoped for Janus's return this year.\n\n\"So you see how little I need you,\" the boy continued. \"I can read my own damn letters. And I don't need Vornatti's lecherous aid, either.\"\n\nGilly's own temper quickened as the boy's words woke the caresses Vornatti had pressed to his skin.\n\nHe yanked the boy to his feet. Gilly handed him a quill and the Itarusine envelope. \"Prove it. Write something for me then.\"\n\n\"Anything,\" the boy said, defiant.\n\n\"Anything?\" Gilly grinned. \"Promise?\"\n\nThe boy hesitated in the face of that smile. But then he raised his chin. The smirk deepened. \"Anything.\"\n\n\"Your name.\"\n\nThe boy's face froze and he whispered, \"Bastard. And you'll run off to tell him, won't you?\"\n\n\"If you are incapable...\" Gilly said, goading him.\n\nThe boy dipped the quill into the inkwell, shook the excess off, and bent over the paper with a faint awkwardness that spoke of inexperience. But the scrolling ink spread over the silky parchment smoothly and quickly, stirring Gilly's breath while he read the letters as they formed.\n\nThe boy stepped back, bowed, tossed the quill onto the desk with a spattering of inky drops, and left the room, all so smoothly done that he was gone before Gilly's eyes rose from the paper and the single word that the boy claimed as his name.\n\n_Maledicte._\n\nGILLY WOKE TO THE ROUGH sound of Vornatti's labored breathing in his ear and, from farther down the hall, the distant protest of moving furniture. Gilly wondered drowsily if something new had distressed the boy and he had built barricades in his room last night, or if he was thieving furniture from the other rooms. A settee had already disappeared into the boy's quarters, and once, Gilly had found the boy preparing to move an enameled table down the wide, slippery stairs. Gilly had carried it down himself, but the boy, as suspicious as a mother cat, had maneuvered it inside without Gilly's help. The boy\u2014 _Maledicte,_ Gilly thought, jerking awake all at once, unnerved again. The name rang in his ears like the voices of mad intercessors and witches, ill-omened.\n\nVornatti's gnarled hand sought Gilly's thigh. \"Who would have thought,\" Vornatti rasped, \"the boy would find such tame pursuits to amuse him through the cold season.\"\n\nGilly smiled, but when Vornatti's hands stroked higher, he pulled away, freed himself from the smothering weight of eiderdown and fur. \"I'll start the fire,\" he said.\n\n\"Linger yet,\" Vornatti commanded. \"It's rare enough I wake with you in my bed these days. It makes me wonder what sent you fleeing into my arms last night.\"\n\nGilly shrugged, fed the spills into the redly burning coals, grew a little flamelet, and fed the first log in.\n\n\"That's not an answer, Gilly,\" Vornatti said, mood souring along with his voice. He gasped, and Gilly knew the old man's pains had caught up with him once again.\n\nGilly stirred a spoonful of Laudable into the leftovers of last night's wine. \"Drink this.\"\n\nVornatti gulped it. \"Tell me why, Gilly. Do you want something out of the ordinary way?\"\n\n\"I'm not a whore,\" Gilly said, stoppering the lid so hard the seal cracked in his hands.\n\n\"Well, not _just_ a whore,\" Vornatti said, mocking. \"There are endless supplies of reasonably intelligent young men. There are endless supplies of reasonably willing young men. But there are few who are both. And gentle\u2014\" Vornatti touched the rough stubble on Gilly's cheeks, his tone losing its petulance. \"What was it that frightened you? The boy?\"\n\n\"I suppose,\" Gilly said. \"I didn't want to be alone in the dark, with only the boy in my head for company.\"\n\n\"But such fascinating company,\" Vornatti said, gloating.\n\nGilly knelt beside the bed, found Vornatti's slippers, and slid them onto his feet. Head still lowered, he said. \"Sir, have you never thought that this might be a dangerous thing? This boy\u2014sometimes he seems merely a youth with a temper; at other times, he seems uncanny, his rage unnatural, that sword with raven wings like Black-Winged Ani....\"\n\n\"Black Ani,\" Vornatti said. In his voice, Gilly heard old remembrances, and wondered what it had been like, to live under the eyes of the gods.\n\n\"The sword, the hunger for vengeance. His will. His determination. Even his name. Ani could\u2014\"\n\n\"The gods are dead, Gilly. Any man who fought at Xipos in the endgame knows that. Xipos proved it; men made offerings grim and great, and men died, churned into mud and blood, screaming for Haith's mercy and hearing _nothing._ That _sword_ is nothing\u2014stolen from some incautious aristocrat, nothing more. The boy has a magpie heart, we've seen that.\" Vornatti tugged his dressing gown closer across his shoulders; it sagged where his flesh had once filled it, revealing the great, pitted scars over his spine and hip, the source of his pains and problems, the place where a warhorse had danced across his back with rough iron shoes.\n\n\"But\u2014\" Gilly started, remembering the feel of the sword beneath his hand and shivering.\n\n\"The gods are gone,\" Vornatti said. \"Baxit Himself gave us that gift. Though some swear it was His curse. To live at our own behest. To answer our own prayers.\"\n\nGilly nodded, obediently.\n\n\"My superstitious Gilly, I am an old man,\" Vornatti said. \"I grew up in the god times. And I saw one god-possessed.... If this boy were Ani's, he would have slaughtered us both rather than falter in his forward steps. There is an old book of such histories in the library, should you doubt me. I think you merely mazed with nightmares. Haven't I heard you call out in your sleep while you dream of dead things?\"\n\nGilly nodded, this time with more belief. Maledicte was likely nothing but a clever actor, skilled in evoking dread. It would serve him well, should he ever come to grips with Last, Gilly thought. He refused to think on the sword and the feeling it left in his skin.\n\n\"But you learned his name?\" Vornatti said. \"Tell me.\"\n\n\"Maledicte,\" Gilly said.\n\nVornatti threw back his head and laughed.\n\nA SHADOW CROSSED GILLY'S LINE of sight as he crouched beside the shelves, pulling out the books rarely read. His hand closed on the spine of one old enough to have grown foxed and spotted, the leather cracking. _The Book of Vengeances._\n\nVornatti, Gilly thought, had never succumbed to the worst affliction of old age, that of a faulty memory. The book opened in Gilly's hands to an illustrated page black with ink and a raven's eyes, to a man battling, though knives pierced his flesh. The shadow moved over him again, and he twitched, closing the book reflexively.\n\nThe boy stood behind him, eyes calculating. \"You don't guard your back very well.\"\n\n\"I'm only a servant,\" Gilly said. \"I don't need to.\"\n\n\"I suppose that's true. And you don't have to fight for your food, your clothes, or your hair as Relicts children do.\"\n\n\"Is that\u2014\"\n\n\"Where I come from? Of course. You've known that all along,\" Maledicte said. \"Or did you think Ani birthed me from an egg?\" His lips curled in amusement.\n\nGilly sighed in embarrassment. \"Vornatti told you.\"\n\n\"Vornatti found it funny; you, fearing me.\" Maledicte's face darkened. \"I could take you, though.\"\n\nGilly said, \"I shook you once, and I can do it again.\" He kept his tone matter-of-fact, and the boy slid away from the confrontation.\n\n\"Your hair's all over cobwebs,\" he said. \"No one would buy it in that state, not even for pillow stuffing.\" The boy set the sword down, reached out, and tugged the ribbon from Gilly's hair with agile fingers. \"Turn around.\"\n\nHesitantly, Gilly did. Maledicte moved behind him, unfastened his hair, and stroked cool fingers through to Gilly's nape. Gilly tensed; the boy's gentler moods all too often presaged a sting so delicate that only later did it smart and bleed. \"What do you want?\"\n\nMaledicte backed away, spread his arms wide, and said, \"Tell me what you see.\"\n\nGingerly, uncertain of Maledicte's mood, Gilly said, \"A boy pretty enough to attract attention, disconcerting enough to repel, and very young.\"\n\nMaledicte's brows snapped down. \"Not dangerous? And why so young? I am near a man's age.\"\n\n\"You are not wearing the sword,\" Gilly said. \"And your slightness, coupled with your light voice, will always strike men as youthful.\"\n\nMaledicte sank onto the library stool, ran his hands through his hair, first pushing the curls back, then raking them forward to leave only his dark eyes visible. \"I would not want Last to laugh at me,\" Maledicte whispered. \"Tell me how to be feared, Gilly. You who know so much.\"\n\nGilly sat beside Maledicte, flattered that the boy sought his advice, a sign that perhaps he might think Gilly something more than just Vornatti's pet. \"You have time.\"\n\n\"The trees are budding, the songbirds sing, and daylight grows. It is almost spring. Last will return, though not Janus....\" Maledicte's breath flowed out and with it, seemingly his strength. He slumped against the bookshelf, restless hands knotting Gilly's ribbon, shaping the cloth as he could not shape the future. \"If I kill Last, what will it avail me with Janus still caged in Itarus? Last sent him there, and only Last will bring him back.\"\n\n\"Kritos would kill him if he could,\" Gilly agreed, \"rather than spend one copper on him. But perhaps Vornatti could be relied upon.\"\n\n\"Vornatti, least of all.\" Maledicte rose to pace the room. \"Vornatti has his own agenda for me, his own desires. I am but a reflection of what he wants me to be. He shelters me, indulges me, and whispers sweet vengeful fancies in my ears, but they are his fancies, his vengeance. Not mine.\n\n\"My schemes are...far from complete. And I know of no way to make them more so. I will not have Last mock me, a boy with a blade. I cannot face him at all; my skills are too uneven, and there is no satisfaction in being cut down by Last's men.\n\n\"So you see my dilemma, Gilly?\" The boy held his hands out in calculated supplication. \"You have all the answers for Vornatti; have you none for me?\"\n\nGilly's eyes fell before Maledicte's steady ones, startled at the boy's candor and need. The boy waited silently for Gilly's advice, seemingly patient, though Gilly could see the boy's hands twisting into fists.\n\nGilly hastened into speech. \"Wait. Learn to use the sword. Last is a swordsman of some skill. And you\u2014\"\n\n\"Vornatti has tried to instruct me, but he is too old, his directions meaningless.\"\n\n\"Vornatti will hire a master if you ask,\" Gilly said, fighting a sudden sense of disloyalty. It was more than advice Maledicte needed; he needed an ally. \"In the right way.\" He grabbed Maledicte's hands, pulled him near. \"Listen, and let me tell you how to manage him.\"\n\nHe bent his blond head to Maledicte's dark one and began to speak, voicing things he had never consciously plotted, small details of pleasure and drugs and wine, and when to ask, and how. The lecture made him cringe, made him realize he worked Vornatti like any paid companion. If Maledicte said anything scathing now, he'd never be able to speak so again, but the boy stayed attentive and blessedly silent. When Maledicte left, Gilly turned, scalded in his own skin, to distract himself with _The Book of Vengeances._ But it was gone in the boy's light-fingered wake.\n\nVORNATTI SAT IN HIS CHAIR in the library. The fire burned low, and a sure sign of encroaching spring was that Vornatti did not demand that it be built up at once. Gilly dusted the books; their butler had quit after entering Maledicte's room without knocking and finding his cravat ruined by a black blade and his own blood.\n\nMaledicte sat on the floor beside Vornatti's chair, reading aloud from the pages spread over his lap, pausing every other page to sip from the goblet beside him.\n\nVornatti's right hand rested in the boy's dark hair, fingers lost in the tumbled curls, moving in lazy increments.\n\nGilly looked back to his dusting and smiled a little sourly. So tranquil, so falsely domestic\u2014Maledicte had been heeding his advice.\n\nMaledicte paused in his reading. It was an account of a young girl's first visit to the court, supposedly true, but from the detailed and violent debaucheries awaiting her on the successive pages, hopefully false. Vornatti chose it this evening, saying, \"This is why one always wants a protector in the court, Maledicte. Of course, you have the sword, and you're no girl, so you'll find this tale more amusing than cautionary.\"\n\n\"Is the court really so decadent?\" Maledicte asked.\n\n\"Quite so. It's a hard time, this, though the nobles refuse to acknowledge it. And with the gods gone, there is little men fear. But for all of that, the court is beautiful: there is no time so lovely as twilight, after all.\n\n\"The courtiers meet in gilded ballrooms; they compete with each other to be the most beautiful, the most noticed.\" Vornatti smiled down at the boy and continued.\n\n\"Everything is gilt or silver; you'd want to thieve it all. The rooms, the furniture, the clothing\u2014their clothing is something to see. Every shade under the sun and moon, every stone under the earth and sea is there, burnished and made perfect.\n\n\"The courtiers' tongues seem gilded as well, their manners stiffened by the same embroidery that clings to their gowns and jackets. The men carry swords they may not draw in the ballrooms, to show fierceness many of them lack. And they whisper, like dry leaves in autumn, all the things they dare not say aloud.\"\n\nVornatti looked down and said, \"A savage, unrestrained tongue like yours, my boy, would scandalize them, raise their whispering into sounds of the surf rushing at your back.\"\n\nMaledicte smiled. \"They do love a good scandal. You've told me so yourself.\" He brushed his fingers over Vornatti's mouth, allowed the old man to press a kiss to his palm.\n\n\"Good night,\" Maledicte said, withdrawing his hand, his smiles. He left the room; Vornatti stared hungrily after his slim, retreating back.\n\nGilly joined Vornatti, levered himself down to the floor, and picked up Maledicte's near-empty goblet. He swirled the wine within, ruby against crystal, heart's blood on ice.\n\n\"He means you to take him to court. To wait for Janus's return there,\" Gilly said.\n\n\"I believe he does.\"\n\n\"Is he mad? It is one thing to aid him in his quest, to loose him like a falcon in the field, another altogether to set him in the king's court.\" Gilly chose his goad as carefully as Maledicte had chosen his.\n\n\"Why shouldn't I provide entree? It'll give me something new to draw his affections. And why not? It's been too long since I was there in the midst of it all.\"\n\n\"He's not even an aristocrat. He's...common.\" The word choked Gilly even as he forced it over his tongue.\n\nVornatti chuckled. \"There's nothing common about our boy, and well you know it, Gilly.\" He contemplated aloud, \"Still, he will need training in swordplay, dance, dress, and manners, of course, though you have rubbed off the worst excesses already, and some new clothes. His accent is already acceptable.\" He was as pleased with this idea as if he had thought of it himself.\n\n\"Best of all,\" Vornatti said, touching his lips as if he could still feel Maledicte's caress. \"Best of all, none of this can be accomplished overnight. Even should he prove an excellent student, it'll be next spring at the earliest. For some small outlay of lessons and wardrobe, I'll have him twice as long. And you doubted he could be tamed so easily.\"\n\nGilly swallowed the lees from Maledicte's cup, tasting their bitterness. Vornatti grinned, malevolent glee touching his eyes, livening his old face. \"Can't you see it? Their faces as he enters the court\u2014elegant, wicked, and entirely too beautiful.\"\n\n\"Perhaps,\" Gilly said. Beyond the opened door, Maledicte lingered in the hall, listening. Maledicte touched two fingers to his mouth, and inclined them toward Gilly.\n\n\"He'll have to have some rank, some right to be there among them,\" Gilly said. \"And his antecedents do not bear scrutiny, no matter his appearance.\" Another soft guide.\n\n\"He'll be my ward, of course. Last thought Aurora base-born when she was not, thought me foisting an impostor on the court when I was not. I wonder what he'll make of Maledicte.\"\n**\u00b7 5 \u00b7**\n\n_Of all the myriad choices open to a clever poisoner, perhaps none is more versatile than the commonly scorned stonethroat. An aspyhxiant and paralytic, it is most often employed to rid one's home of rats, and oneself of enemies._\n\n_However, there are more subtle uses...._\n\n_\u2014A Lady's Treatise,_ attributed to Sofia Grigorian\n\n**M** ALEDICTE STOOD, MUTE AND REBELLIOUS, while Vornatti raged at him. \"You are impossible,\" Vornatti shouted. \"Bad enough you went through four dance instructors in two years, wounding two of them seriously, and scarring another, so that I ended up paying _Gilly_ to teach you. Now, you attempt the same on your newest swordmaster, which shows both disrespect and a serious lack of judgment. You may have learned more than your last master could teach, but Thorn has much still to show you. Do you know what it cost me to keep him here?\"\n\n\"He called me little _girl,_ mocking me,\" Maledicte said, voice shrill with outrage.\n\n\"And why shouldn't he?\" Vornatti snapped. \"I was a fool to think this could work. You've not learned anything, not dared anything. I think your vaunted vengeance is nothing more than an excuse to allow yourself to linger here, fed and pampered, indulged and petted.\"\n\n\"I will kill Last,\" Maledicte said.\n\n\"Master Thorn sent you sprawling. Last would skewer you without a moment's thought.\"\n\nMaledicte paced the floor, breath coming fast, regretting the sword left in his room, and hating the fact that he feared Vornatti would be rid of him.\n\n\"You've not learned anything so far as I can tell\u2014though I've paid dearly for the lessons.\"\n\n\"You've taken it out on my hide,\" Maledicte spat. \"Your hands on me\u2014\"\n\n\"You rate your charms too highly,\" Vornatti said. \"I could find the same in any brothel, and sweeter-tempered.\"\n\nMaledicte fisted his hands, strangled by emotion again, fear and rage warring in him.\n\n\"Show me you've mastered one lesson,\" Vornatti said. \"Make me believe your disguise can hold. A dance, a duel, or even the delicate uses of poison that I've taught you. Show me you've learned anything at all, or I'll send you back to the Relicts.\"\n\nMaledicte slammed the door behind himself and fled to the gardens, rage scorching his belly; he couldn't turn his ire on Vornatti, not without retribution he was unwilling to court, but the swordmaster\u2014 _little girl\u2014_ Rage reddened his gaze and he sought out the poison chest Vornatti had given him.\n\nNow he sat, coolheaded and cold-palmed in the grotto at the far edge of Vornatti's estate, watching the tenth cat lick up fish paste, oblivious of its nine dead predecessors. His temper had chilled, leaving the path clear. Master Thorn was no fool; he would not take food or drink from Maledicte's hand. Vornatti's proof would have to be found elsewhere. _Little girl._ Corsets and clothing could only take him so far.\n\nMaledicte watched the cat, hands chilled, white-knuckled around the crystal vial; he would sate the snake-eye glitter in Vornatti's face, would prove his worth. Hadn't risked anything? Maledicte would risk everything....\n\nThe cat staggered, mouth working in silent, pained outrage, and finally slunk, spitting, beneath the bench.\n\nMaledicte raised the vial, crystal warmed by his death grip on it, and tapped the last dose onto his tongue. The clay taste of cold graves filled his mouth, and he nearly gagged before swallowing. His breath fled; his throat seized; tears scalded his eyes as he choked. The sound, the pain reminded him of the Relicts battle, gasping for air, and Janus gone....\n\nHe clutched the pain tight, fought for breath. Miranda had lived through that battle, that pain: Maledicte would live through this one. Spots danced before his eyes, his need for breath frenzied now. \"Janus,\" he moaned, a bare thread of sound. \"Janus.\"\n\nGILLY TAPPED the door once more, listening to the sound of an empty room. Wherever Maledicte had hidden himself away, it wasn't in the painted room, that combination prison and shelter. Gilly turned the knob, the brass cold under his nervous fingers, and went in.\n\nThis was the first time in two years that he had been inside, the first time since the room had become Maledicte's, and he half expected it to be filled with remnants of the boy's bloody dreams of vengeance. Too many nights, woken from his own recurring nightmares, Gilly had walked the hall and heard the boy muttering behind the closed door. He always moved on quickly, imagining Maledicte within, wild-eyed and raving, a madman with a feathered sword. Some mornings it was a shock to see that the boy was not the savage of his imagination, but a youngster quick to tease, and equally quick to help Gilly defy Vornatti's more objectionable whims.\n\nStill, Gilly found himself thinking more of Maledicte as the would-be killer, as he encroached into his room. All the drapes had been drawn, baring contradictory murals of snowfall and spring, of velvet night and golden days. Gilly found the effect oddly unsettling, flinched at the unveiled image of a lurking wolf, eyes gleaming through a snowscape. Not for the first time, Gilly thought that Vornatti had uncomfortable tastes. No matter the luxury, the predator lurked beneath.\n\nLooking away from the walls, Gilly tallied furniture, thinking, _Oh, so that's what's become of the divan, the chinoiserie table, the best candlesticks. A magpie heart._ Gilly smiled, but lost his amusement as he looked closer at the low table near the high, four-postered bed. The sword lay there, mute testimony that Maledicte had returned after his lesson's abrupt cessation. Beside it, a lady's embroidery box rested, an elaborate thing of interlocking wood and small, carved flowers. Old, Gilly thought, and odd; he doubted Maledicte soothed his nerves sewing primroses onto linen.\n\nCloser, Gilly saw that it was a puzzle box. There had been a rage for the elaborate toys some years back. Gilly had always liked them, the reward of a sealed box blossoming beneath his fingers. He sat down on the edge of the bed, picked the box up; the lid gaped, left open, as if Maledicte, having agreed to wait, having bided not one but three winters, had no patience left to spare for small things.\n\nInside, where there should have been skeins of silk on ivory bobbins, there was a dazzling gloss of crystal vials, miniature works of glass-blowing art, each sealed. One space yawned, empty. Gilly pulled a vial free, turned it up to the light, peering into the smoky glass. Something that looked like coarse salt, grayed with ash. It sparked memories of rat poison and traps his father had set on the farm. _Arsenixa._\n\nGilly's fingers shook and he set the vial back. Vornatti had taught Gilly to read and to write, but Maledicte already knew both. And Vornatti liked to give lessons. Looking at the demure little chest again, Gilly's stomach roiled.\n\nSlumping onto the bed in distress, Gilly found something thicker than linens pressed against his palm. He pulled the coverlet back, then the ticking itself. The leather-bound book had lost some of its faded gilding: _of Vengeances,_ Gilly read. The long-missing book, squirreled away. Gilly collected the book and fled the room.\n\n\"THERE YOU ARE,\" GILLY SAID. The early-evening light blued the air, and made of Maledicte a hunched darkness crouched in the stony outcropping. Around his feet lay rigid shadows with stiffened tails and legs, opened mouths, and black tongues. The smell of must and murdered cats lingered in the enclosure.\n\nMaledicte rose unhurriedly, cradling a cat in his arms. In the pale, filtered sunlight, its color seemed the dusty gray of old cobwebs; its soulless amber eyes winked and gleamed.\n\nIts mouth gaped and its tongue curled back, its ears flattened, but no sullen complaint reached Gilly's ears. Maledicte touched his lips to its head, set it down on the stone and earthen floor. His hands slid into a sunbeam, showed forearms red with bloody gouges. For all that, Maledicte looked smug as the cat slunk from sight.\n\nGilly stepped closer to Maledicte; his boot struck glass and sent it scattering over the floor to splinter against a wall.\n\n\"So clumsy,\" Maledicte said. His eyes burned with wicked amusement when Gilly's head whipped back to him. The boy's light voice was changed, made furred, raspy, as if he had traded with the cat. Remembering the cat's silence, Gilly amended himself. Stolen.\n\nThe half-seen memory of curving crystal spurting away from him flickered back into his mind, and he said, \"Poison?,\" thinking of the empty space in the embroidery box.\n\nMaledicte said, \"Stonethroat.\"\n\n\"You tested dosages on the cook's cats? To see what would change but not kill?\"\n\n\"I would have used hounds, being more man-sized, but Vornatti doesn't keep kennels.\"\n\n\"Why do this?\"\n\nMaledicte coughed, hand flying to his neck. He dropped his hands to his side. \"I will not be mocked, not by Thorn, nor by Vornatti.\"\n\n\"You poisoned yourself to lower your voice?\" Gilly said. \"You couldn't wait for nature?\"\n\n\"I've done nothing but wait,\" Maledicte said, his face flushing. \"While Vornatti snarls and paws at me and time passes. A third winter approaches and Janus is as far from me as he has ever been.\"\n\nGilly folded himself onto the grotto bench, shivering at the clamminess of damp stone seeping through his breeches. \"Maybe his lessons go no more smoothly than yours.\"\n\nMaledicte shrugged, eyes still worried.\n\n\"Do you fear he will forget you?\" Gilly asked.\n\nMaledicte turned his face up, startled and horrified. Gilly shuddered. Had the boy never thought that time passed for Janus also?\n\n\"If he has forgotten...\" Maledicte said, his ruined voice as devastated as his eyes.\n\nGilly winced away from the raw pain, and Maledicte levered himself onto the bench with a cough and a sigh. Gilly smelled blood, sweat, and a pungency to both that reminded him of the poisonous trial Maledicte had inflicted upon himself.\n\nMaledicte turned a curved fragment of glass about in his fingers, stilled them, and looked at the glass. \"Am I forgettable?\"\n\n\"No,\" Gilly whispered. Maddening. Mercurial. Charming. Never forgettable.\n\nMaledicte coughed again, a series of quick outward breaths like a man puffing to liven a fire.\n\n\"Are you well?\" Gilly asked. His fingers trembled as he took Maledicte's damp wrist in his grip. Maledicte's pulse hammered steadily, more so than Gilly's. Gilly was all too aware of the boy's ashy pallor, the warm stickiness of blood on his hands.\n\n\"Well enough,\" Maledicte said. He freed his wrist, slid down to lean his head back on the bench. He kicked a dead cat from under his boot with a moue of disgust.\n\n\"Except the months speed by and I am no more forward.\"\n\nDaring, Gilly stroked the damp, dark hair. Maledicte sighed, rolled his head, settled it in Gilly's lap. Gilly froze, as startled as if a wild creature had unaccountably failed to bite. He twitched his fingers into life again, slid them over Maledicte's nape.\n\n\"Maledicte,\" Gilly whispered, the word an invocation. \"Dark words, dark paths, a heart laden with secrets, and no one to rely on.\"\n\n\"There's always you,\" Maledicte said, so softly, so muffled by damaged throat, by the sweep of Gilly's sleeves curtaining his face, that Gilly felt that his trust was no more than a distant rumor, fragile and easily disproved. His fingers worked loose a tangle from the dark head.\n\nIt was with some reluctance that Gilly roughed his voice to speech, pointing out Maledicte's slow-bleeding arms, the bodies that needed to be disposed of before they lost a cook, and the lateness of the hour for sitting in a damp grotto.\n\nGilly would not allow Maledicte to help with the dead cats, concerned that some taint of death would sift free and unbalance Maledicte's fragile control over the stonethroat. So Maledicte watched, his arms bound in wide strips torn from Gilly's shirt, his eyes as flat and opaque as the stones Gilly cleaned. Still, Gilly thought he heard the soft falling weight of blood on earth. Maybe it was only ghostly steps from the dead creatures Gilly shoveled into a sack, or maybe\u2014maybe it was the faint ticking of an unseen clock, counting down the moments until Maledicte must act.\n**\u00b7 6 \u00b7**\n\n_Maledicte snooped and Maledicte pried._\n\n_No one escaped from Maledicte's spy._\n\n_How many secrets did he find?_\n\n_One...two...three..._\n\n**S** ORNATTI PUSHED BACK HIS DINNER PLATE, the roast hen only pulled apart, not eaten, and turned on Maledicte. \"Your appetite seems well enough.\"\n\n\"Shouldn't it be?\" Maledicte said, licking his fingers, his voice the mutter of a feral cat.\n\nVornatti slapped him. \"Manners!\" Maledicte surged out of his seat as if he meant to return the blow.\n\nGilly said, \"Mal,\" quietly in a warning. It seemed to him that in the cold months since the self-inflicted poisoning, Maledicte's temper had grown apace. Or perhaps it was only the uncanny rasp, a menace bred purely by sound. Gilly thought that Last would have little desire to laugh now, if he encountered the boy and his sword.\n\n\"To think I thought you could act as a courtier,\" Vornatti continued. \"Dogs lick themselves. People do not.\"\n\n\"But I am your dog, am I not?\" Maledicte said, visibly warring with his own temper. \"You've trained me to heel.\"\n\nBefore Vornatti could retort, Gilly knelt beside Vornatti's chair. \"Tell me, my lord, what have we done to displease you?\" There was something, some balance that had changed; ever since Maledicte had poisoned himself, Vornatti had veered between pride in Maledicte's lessons and rage at the smallest infraction. Gilly thought perhaps Vornatti had also been spooked by Maledicte's success with the stonethroat. He sighed. _Fanciful._ Vornatti's moods were, as ever, dictated by pain, Elysia, and events, not by superstition.\n\nVornatti's face, drawn into tight lines, eased at Gilly's conciliatory tone. He stroked the line of his jaw, his neck. Maledicte leaned up against the wall; the silent weight of his watchful presence heated Gilly's skin with embarrassment.\n\n\"I've heard from Aris,\" Vornatti said finally. \"I have his permission to present my ward to the court without the usual petty testing of manner and dress. Aris,\" Vornatti said, \"grants a favor for a favor.\"\n\nAt Maledicte's questioning gaze, Gilly said, \"On occasion, Vornatti...pads Aris's financial reports to Itarus and to Antyre's benefit.\"\n\n\"Still, it seems Aris's approval matters little,\" Vornatti said. \"I also received the broadsheets today. Look you at them, and tell me what you see.\" He fished the folded sheets out from his chair, passed them to Gilly. Gilly spread them over the tablecloth, smoothing the crumpled lines. His breath caught. The infamous artist Poole had turned his attention to the court.\n\nThe caricature claimed most of the front page, a myopically drawn king, a book in one hand, a trailing leash in the other. The hounds, named for members of the court and for political entities like Itarus and the antimachinists, ran freely around him, fighting, fornicating, and fouling the palace. It was titled \"The Learned King.\"\n\n\"Is Poole mad?\" Gilly asked. \"Was he arrested?\"\n\n\"No,\" Vornatti said. \"Aris couldn't be bothered. The disrespect of the court and papers is ingrained.\"\n\nGilly sucked in his breath, but it was Maledicte who said it for him, paying more attention than Gilly had thought. \"So the king's welcome does not insure my acceptance? Without such acceptance, I have no way to reach Last.\"\n\n\"Not only your vengeance is at stake, but my position. If they spurn my ward\u2014I have been shunned before, and I will not suffer it again. Gilly! We need to be assured of our acceptance. You will go to town and find such assurance. I want one of the counselors\u2014Lovesy, DeGuerre, or Westfall\u2014to greet us with open arms. Do what you must. Dig up what you must.\"\n\n\"Leave here?\"\n\n\"Maledicte can care for me, can he not?\" Vornatti traded a long look with Maledicte; after a time, Maledicte dropped his eyes and nodded. Vornatti smiled and gestured to his side. Maledicte came silently over, and nestled down, leaning against Vornatti's thigh. \"Good dog,\" Vornatti said. \"You can growl all you want, but you know better than to bite.\"\n\nGilly, watching the redness rise and fall in Maledicte's cheek, wondered if that was entirely true.\n\n\"Gilly, start with the betting books,\" Vornatti said. \"There's always scandal there, if you know how to look.\"\n\nHe nodded understanding and obedience, and left the room, glancing back once to see Vornatti leaning over Maledicte, biting at the marble curve of his neck.\n\nIN MURNE, Gilly found his duty more tedious than taxing, the obstacles many but responsive to his handling. While his target was apparent from the first study of the betting books at the Horned Bull, that rough tavern where the most disreputable bets were laid, the evidence proved more difficult to gather. Still, several weeks later, Gilly held proof of a scandal in his hand, not regarding a counselor himself, but a counselor's close kin\u2014more than potent enough for their needs. Vornatti, notified by letter, had agreed, and sent Gilly a bonus, as well as further instructions.\n\nA bonus Gilly felt well earned. Gilly left the meeting with their chosen victim feeling that only luck and good planning had kept him alive. The Marquis DeGuerre was a very angry young man; Gilly was glad to immerse himself in the less dangerous details of preparing Vornatti's town house after his long absence, taking care in the meantime to stay safely away from DeGuerre's reach. With such determination and little distraction, he was able to send word to Vornatti that the house was readied ahead of schedule.\n\n\"Welcome to Murne, my lord,\" Gilly said, greeting Vornatti at the door of the Dove Street residence. Vornatti, white with the strain of two days' travel, nevertheless walked across the threshhold, leaning on a stout cane. Maledicte pushed the wheeled chair behind and came in, dressed in city finery\u2014delicate lawn shirt, leather breeches, a satin vest, all the opalescent black of a raven's wing. Gilly helped Vornatti into his chair, and turned back, drawn like a magnet to the elegance of the boy.\n\n\"You look the part now,\" Gilly said.\n\nMaledicte dropped into a bow, smiled up at him. \"Gilly.\" His voice held distinct pleasure, a purr beneath the rasp, and Gilly hoped it wasn't merely for his success. He hoped that the month apart hadn't undone their tenuous friendship, that a month alone with Vornatti hadn't raised Maledicte's temper to a razor edge.\n\n\"You're rather elegant yourself,\" Maledicte said, tugging Gilly's blond queue, eyeing his embroidered livery. \"But what's all this?\" He gestured to the collection of flowers and wrapped packages.\n\nGilly cast a cautious look around at the other staff, waiting along the wall for Vornatti's acknowledgment, and spoke quietly. \"Once DeGuerre folded, other courtiers remembered Vornatti's ways and sent tribute, an urging to look elsewhere.\"\n\nMaledicte laughed. \"Afraid of you and your watchful eyes. And they've not even met me yet.\"\n\n\"Gilly,\" Vornatti said, interrupting their chatter. \"I'm tired. Show me to my room, and supervise the unpacking. I don't trust the maids; they look a shifty lot,\" Vornatti said, then grinned. \"Especially that saucy one.\" His cane swung out, jabbed at her ankle-high skirts.\n\n\"What's your name?\" he asked.\n\n\"Livia, sir,\" she said, dropping a curtsy. Vornatti's cane stirred her skirts, revealing her calf. He patted her wrist. \"You weren't hired by the housekeeper, I'm sure. Not with those legs. I'll wager Gilly hired you personally. He has a weakness for red-haired maids.\"\n\nLivia nodded, dimpling.\n\nVornatti dismissed them all, letting his eyes linger on Livia's retreating skirts, and when Gilly moved to carry the baron's personal luggage, Vornatti put his cane in his path, slapping Gilly's shins with that always-deceptive turn of speed the old man had. Like his crest, the serpent, which struck at speed and without mercy, Gilly thought, wincing.\n\n\"Gilly,\" Vornatti said. \"Your dalliances stop the moment you fail to please me.\"\n\n\"Yes,\" Gilly said, flushing.\n\n\"You've had free run for a month now. Don't forget who pays\u2014\"\n\n\"Leave off, old bastard,\" Maledicte said, stepping between the two, touching Gilly's hand in passing. \"You don't need to crack the whip.\"\n\n\"And you, youngling, need to recall that we're in town now, and the rules are different. Gilly, help me to my room.\"\n\nObedient and silent, Gilly did, guiding Vornatti's chair down the hallway to a room that had once been the second parlor, before Gilly's frantic redecoration. The last time Vornatti had lived in town, he could manage the wide, polished stairs to the upper floors.\n\nMaledicte, as ward to Vornatti, had been given the room Vornatti would no longer occupy, the master chamber on the second floor. Thinking of Maledicte's quick defense, Gilly was pleased that he'd taken the time and some of his bonus to fill a bedside dish with toffees of the kind Maledicte had pilfered from Last.\n\nGilly's room was also on the floor for family and guests, showing his position for what it was, neither fish nor fowl. The maids served him, Cook chivied him like a mischievous lordling, the butler grudgingly conferred with him. Yet his room overlooked the mews and the trash bins; the furnishings were pieces not good enough for the baron or his guests.\n\nBut late in the night, Gilly didn't dwell on arbitrary inequalities, though his eyes lingered on the scarred dresser opposite his bed. Instead, he listened to the peaceful silence of the sleeping house, wishing he could rejoin it. Sweat glistened at the neck of his nightshirt, damped his back and arms, catching the light of his bedside candle. A handful of spent matches attested to his failed attempts to light the candle with trembling hands. He had not dreamed this past month, and yet, the very day Maledicte set foot under the Dove Street roof, the nightmare returned.\n\nHe shook himself like a wet dog, shrugging off tendrils of nervousness and fear as if they were droplets of water. Settling himself into the sheets again, he reached to snuff his hard-earned flame. Then, instead, he rolled his back to the light, pulling the linens over his shoulders.\n\nThe nightmare returned as if he had never managed to wake from it. The catafalques again, and one tomb split asunder, the crushing, underground darkness, lightless save for the sullen bloody glow around Her. She perched, talons dug into a dead man's chest, Her beaked face stabbing into the soft, opened belly. Clotted gore blurred Her features. All Gilly could see were Her starving eyes and a few strands of pale hair gleaming in the offal smearing Her face. Gilly took hesitant steps, wanting to name Her victim, but She had been there already and the eyes were gone, the face ruined.\n\nThe unlight that showed him Black-Winged Ani coiled, shifted, and revealed Maledicte, down on one knee, leaning on the sword. He looked at Gilly with eyes as hungry as Ani's and said, \"Is this my vengeance completed? This is not how I expected it to be.\"\n\nAni rose up behind him, Her wings shutting out even that bloody light, a taloned foot reaching for Maledicte's shoulder. Maledicte's hand ghosted up, a pale spider in the darkness, and rested atop Her clawed foot; in protest or acceptance, Gilly couldn't tell. She bent Her face to Maledicte's, Her beak hovering closer and closer to Maledicte's eyes.\n\n\"Mine\" was all She said, but Her voice was as merciless as floodwaters.\n\nGilly woke for the second time with a racing heart and nausea stirring his belly. His ears rang with the aftermath of Her voice. The candle flame danced with the rushing wind of his breath and Gilly reached for it. As he moved, he saw someone standing in the doorway, a shadow blooming against the small flame.\n\nThe maid Livia? Gilly thought, hoping to lose his fear in the game of pleasure. But she wouldn't have come to his room, risking her position, not now that Vornatti had taken residence.\n\n\"Awake, finally?\" The rasp identified the speaker beyond any doubt. Maledicte strolled over to the bed, stopping a few feet from the edge.\n\nStill fully dressed in his fanciful black, the candlelight coiling around him, he woke some of the dream dread in Gilly. But instead of despair, Maledicte's current expression hovered toward offense.\n\n\"What were you dreaming, Gilly?\" he asked. \"You tossed and turned so, and you\u2014\" His voice, brittle, broke and faded.\n\n\"It was a nightmare. A most unpleasant one.\"\n\n\"You said my name in it. I thought you'd seen me come in, but you were sleeping. What were you dreaming?\" It was more than interest. It was an angry demand.\n\nGilly pushed his sweaty hair back, feeling worn beyond his measure, and not in any mood to decipher the why of Maledicte's anger.\n\n\"You have no right to dream of me,\" Maledicte said.\n\nGilly sighed. \"Dreaming is a magic beyond reason, Mal. I am sorry that your presence in my dream offended you. It doesn't mean anything.\" Except that Ani, supposedly dead and gone, lived well in his dreams.\n\n\"Are you trying to burn the house down?\" Maledicte said, changing the subject. \"Or have you taken to tippling the old bastard's opiates?\"\n\n\"Did you come in to cut up at me?\" Gilly asked. \"If so, please go away. I'm tired. While you explored your new territory, I worked.\"\n\nAs changeable as mercury, Maledicte said, \"My poor Gilly. I'll let you sleep, but I do recommend that you put out the candle. I can think of more pleasant deaths than burning in your bed.\"\n\n\"What do you need?\" Gilly said. \"I am quite awake and intend to stay so.\" To prove his point, he propped himself up into a sitting position against the mounded pillows.\n\nMaledicte drifted to the side of the bed, settled himself. The feather mattress shifted under his weight. \"I want to know what magic you worked to gain our entrance. Whose secrets were so interesting...?\"\n\nGilly mistrusted the dark humor in Maledicte's eyes and tried to delay. \"I met a boy who knew your Janus.\"\n\nMaledicte's face shuttered into blankness. \"Did you.\"\n\n\"A boy called Roach? Was working in the Bull and robbing their customers, given the chance.\"\n\n\"Roach,\" Maledicte said, recognition evident in the wariness of his voice. \"What had he to say?\"\n\nGilly shrugged. \"Our paths crossed only briefly.\" Roach, skulking in the alley, had tried to rob Gilly of his hard-won letter and the scandalous information within. Gilly, pleased with his success, had merely shaken Roach silly, and told him he was a fool to try to steal something he couldn't even read. \"He said Janus taught him to read. Said that Janus killed his girl.\"\n\n\"Roach is a fool,\" Maledicte said. \"Best forget him and his words, Gilly.\" Maledicte rose, and lit another candle from the first. But instead of warming the room, the second flame only added more shadows to lurk in the corners.\n\nGilly tore his apprehensive gaze from them, focused on Maledicte's acid voice. \"Didn't you have better things to do than listen to a Relicts rat?\"\n\n\"I had what I needed by then,\" Gilly said. He took the candle back from Maledicte and settled it firmly beside him once more, wondering what kept the dream still so close. His own fear? Or Maledicte's presence?\n\n\"Which was\u2014?\" Maledicte drawled. \"So far I've not learned what secret was so powerful.\"\n\n\"An indiscreet letter from the Marquis DeGuerre to his sister,\" Gilly said. \"Not a counselor himself, but a counselor's nephew. It sufficed.\"\n\n\"You played housebreaker?\" Maledicte asked. \"I cannot imagine you doing so. You're rather too big.\"\n\n\"I hired Livia away from DeGuerre,\" Gilly admitted. \"She pilfered the letter as she left his employ.\"\n\n\"Very clever,\" Maledicte said, his tone mocking. \"But Gilly, don't you think Vornatti's household is rather full of riffraff by now? The old profligate himself, his pet blackmailer, a thieving maidservant\u2014\"\n\n\"And a stripling killer,\" Gilly said. He had meant the words to be a gentle tease, but with Ani's presence lingering in his mind, the words came out like a taunt.\n\nMaledicte frowned, temper risen. \"We'll see whether Last finds me as amusing as you do. Go back to your dreams, Gilly.\" He snuffed the candle with quick, angry fingers, and Gilly caught his arm.\n\n\"Mal, don't,\" he said.\n\n\"Frighted of the dark?\" Maledicte said, freeing himself, stumbling over a book beside the bed. \"No wonder, if this is the nonsense you read.\" He tossed _The Book of Vengeances_ onto Gilly's lap and started for the door.\n\n\"Don't be so touchy,\" Gilly said. \"I meant no offense.\" But after three years of feeding and training, the boy hadn't grown much in truth.\n\nMaledicte slumped into a chair, put his feet up on Gilly's bed. \"Last cannot come too soon to suit me. This waiting palls.\" His eyes grew as dark as the shadows, and Gilly's mouth dried, imagining Ani reaching out to claim Maledicte from the gloom encircling him. Even once Maledicte left, Gilly watched the flame and found tenuous solace in its light until it burned down with sunrise.\n**\u00b7 7 \u00b7**\n\n**M** ALEDICTE STOOD, limned in the sulfurous candlelight that was de rigueur for formal occasions, flattering to the aged rou\u00e9s and dames, adding glamour to insipid youths. The nobles' ballroom was a half-moon bordered with elaborate gardens on the curving side, shuttered with gilded doors along the straight edge. For special nights, the king threw open the doors, folding them in on themselves, making a full moon of the ballroom. But the nobles' ballroom was there for their delectation; it was full from spring to fall and, if boredom weighed too heavily, through the winter as well.\n\nIn the antechamber behind Maledicte, Vornatti penned his signature in the guest book. Maledicte had already signed, and Gilly added his under the line of servant-attendant.\n\nVornatti snapped the book shut, to the irritation of those trying to read the name of the new attendee over his shoulder. \"Haven't the discretion to wait 'til we're in?\"\n\nMaledicte heard all this faintly, watching the ballroom. One or two dancers paused in their steps, their eyes slewing to the doorway. A cluster of young gentlemen began an endless night of betting and gambling. Jewels flashed like captured sunlight fed back to the sky. The inlaid marble flooring was patterned like broken seashells, and the drift of dresses and seafoam lace made the room sway like ocean waves. Blue-gray drapes fluttered and whispered at alcoves, at exits, at every furtive movement.\n\nMaledicte stood in the door, neither in nor out. If he stepped forward, the game became inevitable, even if the result remained uncertain. If he stepped back, he forfeited the prize he so dearly craved. He was wrong, he thought; the floor didn't resemble the sea, but shadowy wings rising in a twilight sky. At his side, the sword hilt brushed his shaking fingers; a shudder rippled across his nape, traveled down his rigid spine in a convulsive bout of nerve storms, and was done. He stepped forward.\n\nMaledicte wore dove gray tonight, a demure color, and yet all eyes moved toward him. He bowed to those who followed their gazes and came to meet him. One nobleman, fox-haired, broad-shouldered, and lean, glanced up, his eyes registering hatred. But he moved across the floor, made a clipped bow to their party, and said, \"Baron Vornatti.\"\n\n\"Ah, Marquis, I don't think you know my ward, Maledicte.\"\n\nThe marquis nodded. \"Maledicte. I bid you\u2014welcome.\"\n\n\"So formal,\" Maledicte said. He heard Gilly suck breath in beside him, and stifled a smile. \"When it seems we know you so well. Tell me, sir, how is your sweet sister?\"\n\nDeGuerre's face stiffened; he turned on his heel, walking away. Vornatti laughed.\n\nGilly rushed into speech. \"Be careful, Mal, he is a very angry man. And angry men are hard to hold by secrets. I do not know how long it will be before he strikes at you. If I'd known you would speak so, I would never have told you the contents of that letter.\"\n\n\"Don't fret, Gilly,\" Maledicte said. \"I only said it to amuse.\"\n\n\"Amuse?\" Gilly said. \"Amuse who?\"\n\n\"Myself,\" Maledicte said. \"Don't lecture. Tell me instead who these people are.\"\n\nVornatti smiled again, and said, \"Yes, Gilly. Show me how well you remember your faces.\"\n\n\"The gentleman in the corner is Dominick Isley, Lord Echo, Mal, and perhaps even you've heard of him. He heads Echo's Particulars, his private band of thief takers, bill collectors, and gallowsmen.\"\n\n\"I know them,\" Maledicte said, thinking of frantic scrambles in the Relicts, dodging the sound of Echo's bells.\n\n\"Our scrutiny has drawn his attention,\" Gilly warned.\n\nEcho strolled over, nodded curtly to Vornatti. \"Still alive, old reprobate?\"\n\n\"Solely to spite you, I've found myself an heir,\" Vornatti said.\n\nEcho surveyed Maledicte, nostrils flaring as if he scented the Relicts lingering on Maledicte's skin. \"Your kin?\"\n\nMaledicte said, \"In temperament, perhaps.\"\n\n\"Your parentage?\" Echo demanded. \"If not kin to Vornatti, then who? Who was your sire?\"\n\nMaledicte raised his brows at Echo's interrogation but drawled, \"A bit of a scandal there.\" Vornatti set his hand on Maledicte's arm, hand closing tightly, gloved fingers pinching the long nerve. \"And one my kind guardian would prefer I not discuss.\"\n\nMaledicte sighed; Vornatti wanted to cloak him in rumor and speculation, and had set Gilly to plant lies, slandering various dead noblemen. Maledicte wanted to spit the truth in their face, show them that a Relict rat was human, regardless of blood. He knew, though, that even had he done so, brought Ella cringing and fawning forward, the nobles would deny it, and turn their belief back to more-palatable rumor.\n\nEcho blinked, unused to being denied. \"Your mother, then?\"\n\nMaledicte tugged his arm free from Vornatti's painful grip, and leaned closer to Echo. \"I'll give you a hint. They called her Lady Night, and she collected men's tithes with a smile, a moan, and a curse,\" Maledicte said.\n\nNearby, a woman in a bronze-green gown laughed, her eyes meeting Maledicte's with wicked amusement. \"Such a scandal.\" She mimicked his earlier words.\n\nMaledicte smiled at her before turning to Echo once more. \"You seem perplexed. I will let you ponder her identity on your own.\"\n\nEcho flushed at the dismissal in Maledicte's voice, cast him a fulminating glance, and left.\n\n\"Be careful, Mal,\" Gilly said. \"He's more clever than he appears.\"\n\n\"He'd have to be,\" Maledicte said, frowning. \"That's twice tonight you've told me to be careful, Gilly. I think you don't want me to enjoy myself at all.\"\n\n\"Aris likes him,\" Vornatti said. \"That's reason enough to be cautious. Aris would have him head of the Kingsguard would Echo only agree. But Echo enjoys his thief-taking ways too well to change his prey from rats to aristocrats.\"\n\n\"Unless they're _poor_ aristocrats,\" Gilly said. \"He has jailed several of those, and so pretends to evenhandedness.\"\n\n\"Is that so,\" Maledicte said, watching Echo make his way across the room, inclining his head to some, and fetching up near a young man sipping moodily from his goblet. \"Who stands beside him?\"\n\n\"Can't you guess?\" Gilly said.\n\n\"Limp cravat, mud on his boots, hair disordered and yet\u2014people smile at him, bow to him. Lord Westfall.\"\n\n\"The same,\" Gilly said. \"The third of Aris's advisers and the only one here tonight. A financier of Echo's Particulars, and Aris's gesture toward the future. Westfall is machine-mad, his mind occupied with gears and levers that will miraculously insure Antyre's prosperity\u2014if he can keep the antimachinists from destroying his factories as he builds them. So far, they're winning.\"\n\n\"Why isn't Last one of Aris's counselors?\" Maledicte asked.\n\nVornatti answered, leaning back in his chair to take the wine Gilly offered him. \"He's a traditionalist past the point of sense. Had he ascended the throne, Itarus and Antyre would war openly yet. It's the gods, though, that did him in. His refusal to admit Their absence. Aris is a modern king, uninterested in the ways of dead gods.\"\n\n\"And of course, there _are_ whispers that Last attempted to wrest the throne from Aris in the first moments of his ascension. A hard thing to forgive, even for a kindly king.\" The new voice belonged to the noble lady in the bronze-green gown that burnished her auburn hair to a flame. \"You are new come to the court,\" she accused.\n\n\"I am,\" Maledicte said.\n\n\"Tell me then, is it gaucherie or insouciance that allows you to stare so scornfully?\"\n\n\"Insouciance, of course,\" Maledicte said, with the first pure enjoyment he'd had. \"What cavalier ever admits to gaucherie? But, Lady, a question in return. I thought it not the thing for a lady to approach a strange courtier. Is it lack of manners or audacity that drives you?\"\n\nThe lady laughed, a delicate trill of sound. \"Neither in my case, though what lady would ever declare herself mannerless? I am quite well acquainted with your guardian.\" She curtsied toward Vornatti, her skirts pooling outward in elegant sweeps. \"Will you introduce us then, sir? Or have you been so long from the court that you forget our friendship?\"\n\nVornatti kissed her pale hand a breath too long, and said, \"Only a madman or fool would allow himself to forget the charming Lady Mirabile. May I make my ward known to you?\"\n\n\"I think not,\" she said, her lips curling with amusement. \"His tongue is perhaps too rough, unless\u2014\" She paused to flash a dimpled cheek at Maledicte. \"Unless you've had him schooled in dancing.\"\n\n\"Expensively schooled,\" Vornatti said, glaring briefly at Maledicte.\n\n\"Then,\" Mirabile said, \"you may present me. And you may take my hand for the after-dinner dance set.\" She curtsied again, and departed.\n\n\"Gilly, I've been upstaged,\" Maledicte whispered, half smiling, half offended. \"Who is she?\"\n\n\"A woman, wicked and wild-natured enough to seduce a onetime intercessor. Darian Chancel's widow and murderess. As well as Vornatti's onetime paramour.\"\n\n\"Watch your tongue, Gilly,\" Vornatti cautioned. \"What courtiers prattle about could see a servant whipped.\"\n\n\"Echo jailed her,\" Gilly lowered his voice. \"But evidence was hard to come by. There was a matter of another man who she accused of the crime. It cost her everything to buy her freedom\u2014her estate, her fortune, her reputation. Now she leeches off friends and hunts a husband again.\"\n\nVornatti said, \"You see what wonders civility affords, Maledicte? Mirabile's dearest friend is Brierly Westfall, and so she lives on Westfall's estate, where Echo visits daily. I hear they often sit to tea together.\" He laughed. \"And I wager Echo is more uncomfortable than she. Mirabile is a most dangerous woman.\"\n\nGILLY RETURNED TO THE BALLROOM after ferrying a weary Vornatti to the Dove Street house, wondering if Maledicte's training had held without Vornatti to insure discipline, without Gilly to gesture disapproval. Peering around the room, he ignored the thump and rattle of a dowager tapping her cane on the tile. \"Servant,\" she said. \"Servant!\"\n\n\"Lady,\" Gilly turned, bowing hastily, recognizing the temperamental and inquisitive baroness they called Lady Secret for her inability to keep one.\n\n\"You're Vornatti's, are you not?\" The diamonds piled in her falsely dark hair winked in emphasis as she nodded in answer to her own question. \"His ward is Itarusine, is he not? Has the look of one, all dark eyes and bones. Like Vornatti in his youth. Like the queen.\"\n\nLike a lowly Itarusine sailor, Gilly thought, marveling again at the strange magic of flesh that created Maledicte. A creature fey and beautiful from blood as common as seawater. \"He is Antyrrian,\" Gilly said. \"He makes his bows to Aris, not Grigor.\"\n\nShe snorted, irritated at being corrected. Gilly bowed and escaped. Mulling what it meant for Maledicte's success that the baroness had taken an interest, it took him more time than it should have to notice the change in the air. A silent current ran the room, carried on whisper and scandal, as dangerous as a snake at twilight. Gilly tracked the source through widened eyes and bent heads, a trail of murmuring. At its center, of course, was Maledicte. Hastily, Gilly headed toward his charge, overcome with irritation. Not even a solitary hour; Maledicte was dangerously hard work.\n\nEven now, Maledicte leaned close enough to the Lady Mirabile to warm her marble flesh with his breath as he spoke. Her eyes widened and gleamed; her teeth flashed in a small, practiced smile.\n\nGilly, close enough to overhear Maledicte's words, took Maledicte's sleeve in white-gloved hands and tugged like a demanding child. Maledicte paused, his mouth hovering by the spider-shaped patch on Mirabile's cheek.\n\n\"Is something wrong, Gilly?\" Maledicte asked.\n\nLady Mirabile twined her arm around Maledicte's, forcing Gilly to move his hands or touch her skin. He let go his grip and met the wicked, jaded eyes of Lady Mirabile and, worse, the astonished gray gaze of Brierly Westfall. A servant interrupting his master? Mirabile laughed musically at his discomfiture.\n\nAcross the ballroom, DeGuerre heard Mirabile's triumphant voice, turned to look, and, espying Maledicte, turned his back.\n\nThe blackness of that brief glance restored Gilly's courage. \"Outside, please?\"\n\nEither the \"please\" mollified Maledicte or Mirabile had begun to bore him; Maledicte took Gilly's gloved hand in his own. \"Ladies. Forgive me.\" He sketched a bow; his hair, worn loose, fell curling over his shoulders. Mirabile's fingers twitched as if she would like to tangle those dark locks in her bloodless hands. She tossed her head in mechanical, charming disappointment, but Gilly knew her irritation and chagrin were real.\n\nGilly ushered Maledicte through the ballroom, toward the gilded antechamber, where heaps of discarded floral tributes perfumed the air and dusted the floor with bright, fallen petals.\n\n\"Have I been errant, Gilly? Mirabile seemed to admire my audacious tongue. Of course, I think she expects me to spend some time later with her, where my audacious tongue would be only for her enjoyment.\"\n\nGilly yanked Maledicte into the blue-curtained alcove that served as the cloakroom. Maledicte stumbled against Gilly and snarled, \"Surely my sins, whatever they are, do not merit manhandling.\"\n\n\"Shh,\" Gilly said. \"Be silent.\"\n\nMaledicte's lips thinned, his eyes blackened, but the rage never surfaced. \"You're angry at me.\"\n\nGilly shoved Maledicte toward a seat near the curtain.\n\n\"Temper, temper,\" Maledicte teased. His mouth opened; his tongue touched his teeth. He scented trouble and was pleased at the prospect. He folded himself down on a woman's fur-lined cloak, smoothing the pelt against his face. \"Tell me why you're angry?\"\n\n\"DeGuerre.\" Gilly tore the cloak from Mal's hands, hurled it over another hook.\n\n\"Everything is going quite well,\" Maledicte said.\n\n\"Well?\" Gilly said. \"Blackmail requires one person to hold a secret over another, and to keep that secret in exchange for goods, money, or services.\"\n\n\"I am passing familiar with the concept.\" Maledicte reached past the curtains to capture a goblet from a startled waiter, heading out toward the balconies and a rendezvous. He sipped, curled his lips in appreciation.\n\n\"Maledicte,\" Gilly said. A single word containing three years of exasperation. He raked his hands through his hair, catching his fingers in the queue and leaving it in tufty disarray. \"You told Lady Mirabile and Brierly Westfall, the biggest gossips in the court, about Lilia DeGuerre.\"\n\n\"I hinted, Gilly.\"\n\n\"Every whisper you spread loosens our grip on DeGuerre. We need DeGuerre.\"\n\n\"No, we don't.\" Maledicte stood, pushed the goblet into Gilly's startled hands, the wine sloshing. \"Sit.\"\n\nGilly did, his anger draining away, fear leaving him weak. DeGuerre was a powerful man with a bad temper. A counselor's nephew. A man not to be taunted. Maledicte smiled, slow and smug, and Gilly's concern grew. \"If this is not some vicious whimsy, what is it?\"\n\n\"DeGuerre gained us admittance to the court. Lady Westfall will keep us here to please her friend Mirabile.\"\n\n\"That frees you to mock a dangerous man, to spill his secrets to his peers? Mirabile is undoubtedly at DeGuerre's side even now, whispering your oh-so-audacious words into his ears, saying of course there's no truth in the matter, is there, but she thought he should know what is being said.\"\n\n\"If she's not, I've been most misled as to her character.\"\n\nGilly drank the excellent wine in one long draft. Maledicte moved behind him, unfastened the drooping laces in Gilly's hair. He smoothed the wheaten strands, refastened the ties. \"There. All better. But really, Gilly, you should have more care for your appearance.\"\n\n\"DeGuerre will try to disprove your words with steel.\"\n\nMaledicte touched the frown lines etching themselves between Gilly's brows as if he could remove Gilly's fears by the smoothing of his flesh. \"I welcome it,\" Maledicte said. \"I have been trained to fight, Gilly, but have not yet dueled. How can I face Last without knowing how my skills, belatedly learned, stack up against one trained from birth?\"\n\n\"It's a foolish risk,\" Gilly said.\n\n\"But mine to take,\" Maledicte said, touching the hilt of his sword with contemplative fingers. \"Come, Gilly, I haven't much time if I'm to goad DeGuerre to a duel. Aris is coming, and I've heard he frowns on such activities.\"\n\n\"I told you that, if you'll remember,\" Gilly said.\n\n\"Ah, that might explain why I believe it.\" Maledicte slid past the heavy drapes. \"Wait a moment or two before returning to the ballroom. To be circumspect.\"\n\nGilly sighed. \"I am your servant, Maledicte. I am beneath notice.\" He set the goblet on the floor of the cloakroom, followed Maledicte.\n\nIn the ballroom's wide doorway, DeGuerre stood, an arm outstretched as if he would bar Maledicte's reentry; his other hand hovered near the sword hilt on his hip.\n\nMaledicte stood before him, smaller, slighter, and smiling. \"Leaving the field of battle early?\" He slipped past the man's locked arm like a shadow, and looked back over his shoulder. \"My regards to Lilia. You have no idea how fondly I think of her.\"\n\nDeGuerre spun, snatched Maledicte's sleeve; the seam gave in a slow syncopation of popping thread. \"You're nothing. You're nothing at all. A common little catamite.\"\n\nMaledicte stepped closer to the angry, bull-like figure. \"Do you suppose that's what Lilia says\u2014writhing, moaning, under her husband's thrusting? He was nothing to me, my darling, nothing at all....\" Maledicte forced his ruined voice into a parody of a woman's high tones. The rasp lent an air of gasping breathiness to the words, the sound of a woman in the throes of ecstasy or torment.\n\nDeGuerre's face blanched. His eyes shone.\n\nMaledicte said, \"How it must gnaw at you. Loving her as you do, knowing she's lying with another, unable to protest. Knowing that there is one person for you, loving them through all hardships, and then, the sudden shock when it's all ripped away, like an unexpected gut wound that stinks and festers. Love rules you, Leonides DeGuerre, and torments you. We are not so different after all.\"\n\nGilly let out his breath, relaxing, but the sympathetic tone did what no taunt had. DeGuerre clenched his fist.\n\nIt was no openhanded slap that he landed, a gentleman's response to an affront, but a boxer's blow. Maledicte stumbled, head snapping to the side. The stiff cicatrix along his jaw cracked; a thin, red-beaded line welled up, touching the high edges of his lace collar. The violence rippled outward, quieting the court. Only the musicians continued, sawing out tunes for people no longer dancing.\n\n\"Not mannerly,\" Maledicte said. \"And worse, it leaves your intentions in doubt. Are you inviting me to duel? I warn you, I have a bad temper in the mornings.\"\n\n\"Duel over filthy lies that no one believes,\" DeGuerre said, raising his voice for the court's listening ears. \"I think not.\"\n\n\"You've killed to stifle those whispers before. Are you afraid of me?\" Maledicte smiled.\n\nGilly bit his lip; if Maledicte's words stung too harshly, DeGuerre wouldn't wait until dawn, but would strike now, heedless of the court's traditions.\n\n\"One doesn't duel with vermin,\" DeGuerre said. \"Or acknowledge their lies.\"\n\n\"You could sue me for slander,\" Maledicte said. \"If you could prove my words false.\"\n\nDeGuerre struck him again, backhanding him from the other direction.\n\nMaledicte raised his head. Blood rouged his mouth. \"If you intend to beat me to death, don't expect me to abstain from steel. Otherwise, declare the duel.\"\n\n\"Never with you. Relict rat.\" DeGuerre's breath came as fast and as hard as if he had been running; his hands shook. He took a stilling breath, then said. \"Everyone knows commoners lack the moral sense to understand honor.\"\n\nBefore him, Maledicte seemed composed and faintly amused. He lowered his voice, luring DeGuerre closer. \"But at least we don't fuck our siblings. It takes a nobleman to think of that.\" DeGuerre's face reddened in patches over his cheekbones, as if Maledicte's vulgarity had been an actual blow.\n\nDeGuerre drew his sword in one flash of economic motion. The metallic rasp of the drawn blade spread, and the whispers rose. Blade drawn in the court. Lady Westfall, the highest-ranking hostess present, stepped forward, but said nothing; Mirabile's nails dug into her friend's hand, her eyes avid.\n\nThe two men circled each other like angry cats, DeGuerre's grip steady under the long weight of his blade, Maledicte's arms held out to his side, flaunting his still-empty hands. \"Are you wronged or am I, DeGuerre? Do you claim affront? Or do I?\"\n\n\"Draw your blade,\" DeGuerre said, \"and stop your mouth.\"\n\n\"I will not draw until you admit I am your equal,\" Maledicte said.\n\n_Damn fool,_ Gilly thought. Anxiety rose in him like pain. _Draw your sword and have done with it. How will you take your vengeance from the grave?_\n\n\"You cower behind words. Draw your blade,\" DeGuerre repeated. He thrust his sword forward.\n\nMaledicte leaped back, as quick and precise as an insect. The second thrust he ducked under, his curls ruffled by the blade's passage. Gilly could hear his panting, and the sound of whickering horses from outside.\n\nDeGuerre's third thrust met steel. Maledicte rose from his half crouch, the black blade held before him.\n\nThis was the first time anyone within the court had seen the blade, the reality of it below the elaborate hilt. The sword might as well have flamed for all the horrified attention it claimed. It woke hungry shadows in the room, and changed Maledicte from merely another asp-tongued courtier to something much more, something dangerous.\n**\u00b7 8 \u00b7**\n\n**N** OW THAT I HAVE YOUR REGARD,\" Maledicte said, \"shall we agree to continue this at a more civilized time?\" \"I would rather see if you've earned that fancy blade,\" DeGuerre said. He stepped forward, silver flashing from his blade, from the argent lace on Maledicte's quick-moving sleeves. The bell-ring of steel against steel tolled once, twice, growing louder, more resonant. Maledicte evaded another slash with boneless grace, dancing six steps back, out of range.\n\n\"Tradition demands we fight in the dawn.\"\n\n\"I'll trade tradition to see you die,\" DeGuerre said, rushing forward. Maledicte pivoted, regained his distance.\n\n\"You aren't good enough,\" Maledicte said, smiling.\n\nThey closed again, the shuff of their boots over the polished tiles a whisper beneath the chiming rasp of metal.\n\nSmall flickers of triumph darted over Maledicte's pale features, small moments where a touch could have been made. Instead, Maledicte bypassed openings; he prolonged the duel, playing with DeGuerre, testing himself.\n\n\"A natural gift,\" Master Thorn had said grudgingly as he left Vornatti's employ, bandages wound the length of his arm, a white swath around his neck. \"His timing, his footwork, his extension, and his balance\u2014\" He touched his throat and said, \"deadly.\" Watching now, Gilly shivered. Didn't a gift imply a giver?\n\nA new sound entered the room\u2014hoarse panting, the clicking of nails on the marble tiles. Gilly blanched; the duel had gone on too long, and whether DeGuerre improved or not, Maledicte had lost. Only one man brought his hound to the ballroom.\n\nHe raised his head. The king stood in the wide doorway, his hand resting on the brindle mastiff 's withers, face layered with weariness and surprise. Beside him, the Kingsguard, clad in lapis and gold, hastily spread out, encircling him, pistols drawn. A sandy-bearded man pushed past them, cheeks flushing. \"Who dares this?\"\n\n\"Isn't that my question, brother?\" Aris asked, releasing the hound. The mastiff pushed through the two front guards, and Aris followed him through the space.\n\nGilly caught a wheeling glimpse of the room, the interest in jaded eyes, the ashen dismay on Lady Westfall's face, the two men wagering at the most distant point of the room, the musicians' silence.\n\nMaledicte, his back to the door, sucked in a breath as if Gilly's alarm had been transmitted, wordlessly, to him. He cast down his sword, though it seemed to writhe in his hand, and fell to his knees, lowering his head before the king's approach. Scarlet with rage, DeGuerre finished his extension, and his blade sliced the edge of Maledicte's shoulder. Maledicte hissed; his jaw clenched. Gilly's hands tore at his own sleeve, watching the wound's red tide rise.\n\n\"Sire,\" Maledicte said.\n\nDeGuerre dropped his sword. Blood spattered the pale marble. \"Sire.\" He knelt, as stiffly as an old man.\n\n\"To bare blades in the king's presence is treason,\" Last spoke, his eyes lingering on Maledicte, the stranger in the court's midst.\n\n\"Who drew first blade?\" Aris asked, through lips compressed and pale.\n\n\"I did, sire,\" DeGuerre confessed, at the same moment Lady Westfall, pinched by Mirabile, said, \"DeGuerre.\"\n\n\"Dueling in the ballroom is forbidden. As well you know, DeGuerre.\" Once more, the earl of Last spoke before Aris could. The earl's disapproval was marked in the downward sweeps of his brows, adding more rigid lines to his austere features.\n\n\"What matters where they duel, in the park or the courts? Blood shed is blood lost, be it on marble or dirt,\" Aris said, raking the assembled nobles with scorn in his face and voice. \"But to stab a man as he kneels in fealty, DeGuerre...\"\n\n\"How came you to do such a thing,\" Last said, \"you with the best of our blood in your lines? Your uncle a king's counselor? Were you mad?\" Last cast a wary glance at Maledicte's slender form. \"Or witched?\"\n\n\"Michel, search for your demons elsewhere. The court is mine, the offense mine. And the sentence _mine,_ \" Aris said. His brows drew down, so like his brother's, and Gilly felt a spurt of hope. If Last pushed, Aris would pull.\n\n\"Leonides DeGuerre, of late I have heard distressing things about you. I think perhaps you would be better off for several years abroad, away from the...temptations and miseries you find so readily here. Seek the Explorations or Kyrda and make your fortune elsewhere. You may rise and go.\"\n\n\"As for the lad\u2014\" The king swept his eyes over the dark, bent head, the slim form. \"Get the lad a physician and send him back to the schoolroom,\" he said, turning on his heel and tapping his thigh for the hound to follow.\n\nGilly, freed from his obeisance, darted to Maledicte's side, touched him with gentle hands, the red stain darkening the pale sleeve.\n\n\"Aris, the lad is guilty of more than\u2014\" Last trailed off as, beneath Aris's tensing hand, the mastiff growled.\n\n\" _My_ court, brother,\" Aris said, and then, with a spurt of open irritation, \"Oh, do get up, lad.\" He reached for Maledicte's shoulder, and paused as Maledicte raised his eyes to meet Aris's.\n\nThe king startled at Maledicte's blackly lashed eyes, at his curling hair, at his mouth, at the pale skin. He came closer, pushing the hound out of his way.\n\n\"You\u2014you are Vornatti's ward?\" Aris said, voice low.\n\n\"I am,\" Maledicte said.\n\nThe king's face grew shuttered. The silence in the room strained in Gilly's ears. Maledicte swayed, jerked himself to rigidity again. A few new drops of blood spattered on the marble.\n\n\"You seem guilty of nothing but impetuosity,\" Aris said, the words a bare breath, his eyes locked on Maledicte's dark ones. He cleared his throat, spoke again. \"You may return to this court as you will.\"\n\nMaledicte bent his head, hiding his strained face. \"You are kind.\"\n\nThe king's eyes never left the bowed, dark head. \"Get to a physician. Lad.\"\n\n\"Some lessons in manners would be more to the point, Aris,\" Last said. \"Or are we to have yet another foreign decadent making hash of our tradition? Is the court not tired of such?\"\n\n\"The court, Michel,\" Aris said wearily, \"is entranced at the wonderful entertainment we've had this night.\"\n\nLast stiffened, a marionette instead of a man. He bent at the waist, and left.\n\nThe king watched him go, and then said, \"And yet he is correct. Will you swear, lad, that you will never draw a blade again in my presence?\"\n\n\"I so swear, my liege.\" When the king proffered his hand, Maledicte kissed the crested ring.\n\nThe king turned and left, his guards following and half the court. The remaining nobles clustered in little knots of bright fabric, to discuss and whisper. Gilly heard a scrap of conversation as he drew Maledicte to his feet, moved toward the door.\n\n\"Last was right. DeGuerre must have been witched to so lose his senses.\"\n\nGilly turned, seeking the dangerous speaker. Witchcraft was not a word to be spread lightly, not when he'd seen Aris's face go so still and empty for that one moment, as if the page of his thoughts had been erased and rewritten. Not when witchcraft was the only force left that the nobles feared.\n\n\"Hardly a sign of bewitchment,\" Mirabile's elegant voice said. \"To be so goaded by an agile tongue and mind.\"\n\n\"But that sword, Mirabile, and if not witchcraft then what else can it be?\" Lady Secret asked, her voice pitched to carry while at the same time still pretending to a whisper. \"Your Chancel was a theologian, surely you must\u2014\"\n\n\"Oh la, Secret, what makes you think I ever had any interest in his prosing on about dead things?\" Mirabile laughed, but her eyes on Maledicte held speculation and a faint hint of surprise. Gilly dragged Maledicte toward the door, away from that too-intent gaze.\n\nMaledicte clutched Gilly's arm, halting their steps. \"The sword, Gilly. Let me go. I cannot leave it.\"\n\nIts blade had sunk into the marble floor nearly a finger's length; Maledicte yanked it free with an impatient grace that made the court widen the clearing around him.\n\nOutside, in the cool, damp air, their boots crunched unevenly over the oyster-shell drive as Gilly supported Maledicte. Gilly called out as they reached the entrance to the stables; their hired coachman rose from the grouping of his fellows and their game of dice. He lit the lanterns inside the coach and held the door for them.\n\nGilly ripped off his servant's cravat, only loosely starched, and pressed it over Maledicte's bloody sleeve. \"Hold this,\" he said.\n\nMaledicte's white fingers pressed the cloth tight. \"I know what to do. It's not much of a wound.\"\n\n\"Enough to make you stagger and faint,\" Gilly said. He fumbled for his handkerchief to pad the wound.\n\n\"Last came in with Aris. Within reach of my sword,\" Maledicte whispered through white lips. \"And I did nothing.\"\n\nGilly, securing the rough bandage of cravat and handkerchief with his hair ribbon, paused. \"Yes,\" Gilly said, taken aback, realizing that Maledicte's sudden weakness came from that rather than blood loss\u2014that pain was less to him than vengeance. Gilly's hands trembled; he stilled them.\n\n\"What if I've missed my moment?\" Maledicte said, his color fading further.\n\n\"Don't be ridiculous,\" Gilly said roughly, frowning as he saw a bloody line forming over Maledicte's ribs also. \"The summer-solstice ball comes in a month. And the moment will repeat itself: you can still run Last through in a crowd full of witnesses and get shot by the Kingsguard for your pains. That is what would have happened tonight, had you not held your\u2014\" Gilly jerked his hand from Maledicte's shirt buttons when Maledicte slapped at him.\n\n\"Leave it,\" Maledicte snapped. \"I've told you it's not much; it only stings and burns.\"\n\nAs the coach lurched to motion, Maledicte leaned back into the seat cushions with a wince, and no thought for his blood painting the embroidered fabric.\n\n\"You are fortunate,\" Gilly said.\n\n\"DeGuerre could never have touched me had I not thrown down my blade. I should have finished him first,\" Maledicte muttered.\n\n\"Not the duel,\" Gilly said. \"Fortunate that Aris forgave you. He could have banished you as easily as he did DeGuerre.\"\n\n\"He forgave me for Vornatti's sake,\" Maledicte said. \"Their paths are linked after all, kin by marriage, bound by money.\" He shifted in the seat. A small sound that might have been a groan stifled itself behind clenched teeth.\n\n\"We'll be home in moments.\" Gilly put his head out the carriage window and demanded the coachman's flask, passed it to Maledicte.\n\nMaledicte tilted the flask; his anticipatory wince gave way to startlement and a smile. \"He has raided Vornatti's good spirits, Gilly.\"\n\nGilly paid little attention beyond noting color seeping back into Maledicte's lips, thinking instead about the expression frozen on the king's face\u2014the interest heating the cool eyes. \"For Vornatti's sake only? I think there is more to it.\"\n\n\"What does it matter? He forgave me and that's enough. Aris is of no interest to me, save that he seems to hold Last in dislike. Still, not so much as I do....\" His lips compressed, his hand clenched on the hilt of the naked blade resting beside him. \"I should have struck instead of knelt.\"\n\nShivering a little at the hunger in Maledicte's voice, wondering if that hunger would be so intense if Maledicte were not touching the blade, his vengeful instrument, Gilly said, \"I thought you intended to wait on Janus?\"\n\n\"Janus,\" Maledicte said, his hand unknotting. \"I must see what they've made of him before I know how best to act. If he loves me not\u2014\" Maledicte's voice caught. \"If he loves me not, I will kill him, and Last for taking him from me.\n\n\"If he loves me, I will still murder Last for taking him from me. Either way it means blood.\" Maledicte sat forward, hunched himself over his knees, a restless savor in his eyes, clasping the sword hilt again.\n\nGilly's skin crawled, the hair raising along his arms. Surreptitiously, he made a little **X** of his forefingers, invoking the old country charm against the god-touched. He had not had time to peruse _The Book of Vengeances_ as he would have liked, but the little he had translated, slowly changing the Itarusine words for Antyrrian, had chilled him.\n\nHistory claimed that a compact, irreversible, could be entered into, binding Ani and Her devotee to a single task, but that the compact became active only after the first kill. Were Gilly's fears real, his dreams more than dreams, were Ani not so dead as She once was, those implications would distress him beyond measure\u2014that Maledicte, who hunted Last, would build strength in shed blood.\n\nGilly's mouth dried; he snagged the flask back and sipped. He was a fool, dreading the impossible. The gods were gone, and Maledicte\u2014Maledicte was merely a man. The dreams were not evidence, the sword's presence inconclusive, his dueling skills purchased\u2014if Maledicte survived poison, it owed only to caution; if he never sickened, it owed only to luck. All of it was sea-fire proof, prone to disappearing when Gilly sought answers, leaving him only with a knot in his belly and the taste of copper in his mouth where he had bitten his tongue.\n\nThe pleasure in Maledicte's eyes while dueling came back to him, that gloating joy as he danced around DeGuerre's strikes. Even while Gilly dwelled on nightmare possibilities, Maledicte smiled, stroking the feathered hilt, eyes black and clouded.\n**\u00b7 9 \u00b7**\n\n**I** T WAS A FINE DAY, just past the cusp of noon, and Gilly chose to walk to the baker's, escaping the cluster of noblewomen who had come to gossip again with the most scandalous courtier in several seasons. The first day the crush had arrived, Maledicte had found it amusing to parade himself. But a week into the season, Maledicte's temper sharpened as his interest waned; today he'd abandoned his guests to Vornatti's care.\n\nGilly, summoned to the formal parlor, found the room more crowded than before; the very walls, the spinet, even the small stage, seemed overlaid with women. Mirabile had seized the position of hostess and sat entirely too close to Vornatti, distributing tea and spite with an equal hand. When Gilly presented himself, careful not to tread on any trailing hems, Vornatti had dispatched him to roust Maledicte from wherever he had hidden himself.\n\nA lucky word with the cook had sent Gilly out-of-doors, hunting tea cakes as well as Maledicte. He stretched under the sunlight and found himself smiling. Of late, he'd been too much in stuffy, overcrowded rooms. The brisk wind and the faint smell of the sea were balm to his senses. He passed the Dove Square speakers, the men standing on rough-made pulpits, preaching egalitarianism, sedition, economics. In the midst of this, one man, dressed in country best, dared to speak for the absent gods, and was booed to silence.\n\nGilly dropped a few coppers into the intercessor's cast-off coat. The intercessor stroked the symbol of Baxit, the god of indolence and reason, above Gilly's head. _Let it stop the dreams,_ Gilly thought, nodding his head in thanks before moving on toward the shops.\n\nHis business done with the baker, Gilly paused, unwilling to go back. He had not found Maledicte and had no desire to be scolded by Vornatti or thrown into polite confines with women he could not touch. Sifting ideas and excuses, he kept walking, heading toward the quay. He would bring back fresh fish and crab for Vornatti, and some of the succulent oysters that Maledicte and Gilly shared an unfashionable taste for. He took the winding way, through the alleys behind the shops, the way the cart horses went.\n\nThe alleys were intermittently crowded; Gilly stepped aside for a cart bearing sacks of sugar and fine spices from the Explorations. The stamp on the bags, a blue moon, told Gilly that this was the best of the imports, destined not to be sold to Itarus and Dainand, but to be made into elaborate sweets for the Antyrrian court. The redolence of cinnamon and raw cocoa lingered.\n\nThe soughing of the waves, the spluttering suck of water around the pier and ship hulls, announced the docks before he saw them. The last twist of the alley dropped and provided a cobblestoned view that ran, illusory, into the gray waters.\n\nGilly went down, intending to watch the ships and sailors at dock, perhaps find his friend Reg's ship at berth, when a flutter of delicate cloth caught his eye. A sprig of the nobility stood on the quay, his pale shirt gold-shot in the sunlight, his hair black and wind-tossed. Gilly stopped. Not just any young noble, but his.\n\nMaledicte stared at the water, the ships coming in, the new ships being built, his body rocking slightly with the movement of the sea as if he were imagining himself on it.\n\n\"Planning on catching a ship abroad?\" Gilly asked.\n\nMaledicte shifted his gaze to a ship with a red and gold-spotted prow and a figurehead like a dolphin. \"Why should I make all the effort? Besides, I have Vornatti's assurance he'll return. I can trust that, can't I?\"\n\nThe bitterness in his voice slowed Gilly's approach. \"Of course,\" Gilly said. \"What's got you so cross so early?\" Gilly sat down on the pier, dangled his legs over the eddying water. This close to shore, the waves carried refuse: draggled gull feathers, floating fish, silver bellies up, and ropes of seaweed torn loose from their beds by rough anchors.\n\n\"Mirabile,\" Maledicte said. \"She shadows my every move, clinging to my arm, matching my clothes\u2014it's uncanny.\"\n\nGilly laughed. \"Mal, we broker in information, sift through servants' tales for our benefit, why not Mirabile? She must pay someone in the house.\"\n\n\"You?\" Maledicte said. Gilly looked up into sun dazzle and Maledicte's shadowed face.\n\n\"Livia, likely,\" Gilly said. \"She likes coin. It's harmless enough.\"\n\n\"Well, tell Livia to stop, or to feed Mirabile lies, that I'm wearing rose when I'm wearing blue. She's too vain to cling if she clashes.\" Maledicte kicked a small strip of tar-daubed wood into the water.\n\n\"If she learned you had misled her deliberately, she'd be offended, and she's not one to take offense lightly,\" Gilly said. \"She's courting you, Mal.\"\n\nMaledicte snarled. \"Why me? No, her reason doesn't matter. Stop her.\"\n\n\"All right,\" Gilly said, and Maledicte ceased his fidgeting.\n\n\"Just like that?\"\n\nGilly grinned; for once he had surprised Maledicte. \"I know what to say. It's only a matter of feeding the information to her.\"\n\nMaledicte let out a long sigh, his shoulders loosening. \"No one taught me how to repel the nobles. All my lessons were to fit in. It was easier before I learned proper etiquette.\"\n\nGilly stifled a laugh. \"How would you have rid yourself of her before you became such a pattern card of propriety?\"\n\nMaledicte shrugged. \"With a stick.\"\n\nGilly let the laugh free. But when the first wash of amusement had faded, he knew it was the truth. He'd seen the boy Roach and his rude weaponry, knew the damage a savage hand and a stick could inflict. And Maledicte wasn't just any Relict rat; a glossy dark feather washed by, and Gilly's good humor died with resurgence of his fears, waking something in its stead.\n\nGilly didn't understand it\u2014why that one drifting feather should spur him to the point that he had avoided for a week, for far longer, were he honest with himself. A dream of Ani. A boy with a feather-hilt sword and a thirst for vengeance. The words rose in his throat, the question, the need for an answer. Knowledge had to be preferable to this gnawing uncertainty. But Maledicte's moods were tricky, and Gilly swallowed the first simple question for a more cautious approach, attempting to creep up on truth. He cleared his throat of nervousness.\n\n\"You came from the Relicts,\" Gilly said. \"Have you ever heard the story of how they came to be?\"\n\n\"Of course I have. The noble girl, spurned by her merchant lover, prayed to Ani. Ani answered _her_ prayer by destroying the merchant, the shops, the streets, everything he ever loved.\" Maledicte sank down to sit beside Gilly, sheltering in the lee of Gilly's broader body. \"Just proves the power of the nobles, even over the gods.\"\n\nGilly shrugged, kept it casual with an effort. \"I know a different version. Should I tell you?\"\n\nMaledicte aped Gilly's shrug. \"If you must.\"\n\n\"The noble girl's name was Liana, the merchant was no merchant at all, being even more common than that. A delivery man named Edward. She loved him beyond all reason. She gave him everything\u2014her body, her heart, the jewelry she stole from her home. And when she had nothing more to give, he left her for another. The rage and pain she felt were too much to bear and she cried out to Ani to avenge her hurt, to deny that he could love someone else, that the other woman could exist.\"\n\n\"So the Relicts\u2014\" Maledicte said.\n\n\" _No._ Nothing happened. Ani didn't act Herself, but She crept into Liana's dreams, bartering love for vengeance, rage for power. Liana drowned her rival in a swan pond. Edward found his new love there, soaked in black feathers though all the swans were gray, and knew Liana had asked Ani for intervention.\"\n\nGilly fisted his hand; even in _tales_ the proof of Ani's touch seemed ephemeral at best, until it was far too late for any doubt. The rook feather eddied below them, riding the waves, and Gilly looked from it to Maledicte, wondering if Ani might be listening behind those black eyes.\n\n\"Go on then,\" Maledicte said.\n\n\"Once the deed was done, Liana found nothing left in her heart but grief. Ani muttered, reminding her that she had cried out against her lover also and their compact must be completed. But there was no hatred left for Ani to fuel; though Liana sought Edward, it was only to beg forgiveness.\n\n\"It was then that Ani roared to life, Her wingbeats leveling the Relicts. Liana and Edward disappeared beneath the rubble, entombed together in the city. Vengeance, once begun, cannot be stopped. Black-Winged Ani has no pity in Her, and Her wings are carrion wings.\"\n\nMaledicte watched the wavelets foam and fade against the pilings, wordless. Gilly tilted his chin up, peered into the dark eyes, his fingers trembling as he finally braved the question. \"Where did you get the sword, Maledicte? Why do I dream of Ani when you're near? Did you call out to Her? Did She answer?\" His fingers tightened as Maledicte's silence continued.\n\nMaledicte's eyes stayed enigmatic behind barring lashes. He tugged his chin free from Gilly's fingers, dropped a piece of shell into the water, watching the ripples overtake the waves and fade before answering. \"I never called Her.\" He flicked a quick glance at Gilly, cooling Gilly's burgeoning relief, and continued, voice low. \"Yet, while I was dreaming, She woke in me, whispered such things\u2014When I slept, it was summer. I woke to winter, the feel of feathers in my skull and skin, and a black sword at my side.\"\n\nIn that moment, Gilly knew he had expected Maledicte to laugh at him, to shelve his doubts behind a wall of scorn for his gullible nature. Gilly had expected to laugh at himself, and to compliment Maledicte's talents for acting. This impossible admission woke shock in his belly, set his blood to racing, and rendered him mute. The gods not gone. Maledicte bound to Black-Winged Ani. He shuddered, wanting to surge to his feet and flee, heedless that he might offend Mal\u2014or Ani!\u2014mortally. His breath seized in his chest.\n\nThe thing that balked him, cooled him, kept him from panic, however, was a memory of another quiet moment, and how lovely it had felt to hold Maledicte's trust, to be the one who could tease truth out from semblance.\n\nMaledicte shivered as if he would unsay his words, remove Her looming presence. Gilly dropped a wary arm over his shoulders, seeking something to take the chill from between them, to chase the nearly solid mass of fear from his belly.\n\n\"That ship, you see it?\" He pointed to a massive square-rigged ship entering the cove, setting up a frenzy of motion on a far pier. Its figurehead shone molten in the sunlight, a curled cat with a fish's flukes.\n\n\"It looks like gold,\" Maledicte said, ignoring the quiver in Gilly's voice, focusing on the ship with an avidity that suggested that he also hunted an escape from his confession.\n\n\"It is.\" Gilly found a shaky smile at Maledicte's astonishment. As always the ships soothed him as nothing else could. The fear unclenched; his voice evened out. \"That's the _Virga._ She sails to the Explorations and comes back with treasures\u2014spices, wood for our shipbuilders, and strange pets, birds, and small scampering monkeys that look nearly human. Someday I'll be on that ship. Headed for the new world, where people build ascending temples of dirt and stone to speak to their sky gods, see what they have to teach me. Though, according to most accounts, they're only savages.\"\n\n\"Why do we call them savages if they have temples and religion? That's more than we have,\" Maledicte said, his hair whipping in the sea breeze, his booted feet swinging off the pier, his frozen stillness broken.\n\nGilly laughed, drew closer to Maledicte, inclined his head in the studied manner of a professional gossip. He raised his brows, and exclaimed in falsely arch tones, \"Oh, my dear, haven't you _heard,_ don't you _know\u2014_?\"\n\nMaledicte's ease faded. He cast a slantwise glance at Gilly, testing for unexpected mockery.\n\nLaying his hand on Maledicte's arm, careful of the bandages beneath the full sleeve, Gilly dropped his voice to a penetrating whisper. \"My dear, they wear _feathers_ where we wear _leathers._ \"\n\nMaledicte's eyes widened and he laughed, a stuttering, raw thing in his ruined throat.\n\nGilly grinned, pleased with the result of his teasing. He had wanted Maledicte to laugh earlier; he shook back the shiver that wanted free, concentrating instead on their innocuous conversation.\n\n\"For breeches? Are the feathers ticklish?\"\n\n\"I suppose there's a hide backing. But they wear feathers all over. On their feet, in their hair, all shades of red, gold, blue, and green. They have birds down there bigger than our owls, and more brightly patterned than our pheasants.\"\n\n\"What else, Gilly?\"\n\nMaledicte seemed honestly interested, completely at ease, and Gilly wondered if perhaps he had only been teasing. But\u2014 _I woke, a black sword in my hand,_ Maledicte said, his voice drowned in memory. Gilly shuddered. He fell into the security of speech, nearly babbling. \"They find gold on the ground, in the waters, and they make soft, hand-malleable jewelry from it. Wide necklaces, armbands, earrings, rings. They even press gold between their teeth so that their smiles are as bright as their feathered clothing. They have dark eyes, like yours, but their skin is the color of strong tea, and they draw pictures on it with clays and dyes. They drink chocolate with every meal under warm blue skies.\" Gilly spoke mostly for himself, remembering the tales his sailor friend, Reg, told. Maledicte, rapt, watched the _Virga,_ resting his chin on his drawn-up knees, setting Gilly to wondering where Maledicte's interests lay: The gold? The tropical warmth? The images of strange cities and stranger men?\n\nMaledicte shivered and said, \"I'm hungry.\" The complaint was blessedly familiar, and Gilly relaxed into it.\n\n\"Cook said you missed breakfast, and I know you missed tea. Let's go get you an ice.\" Gilly stood, offered his hand.\n\nMaledicte took it, shook the dust from the pier off, and said. \"I have no money. Vornatti was angry this morning and wouldn't make me my allowance.\"\n\n\"A true aristocrat doesn't even think about money. He assumes all shops offer credit and are pleased to do so.\"\n\nMaledicte merely nodded, his face pale, his lips drawn.\n\n\"Your side? Your arm?\" Gilly asked, stopping in his tracks. Since Maledicte's injuries, Gilly had feared infection. The boy had not let him see to the wounds, instead had trusted his skin to Vornatti's suturing.\n\n\"Sore,\" Maledicte admitted.\n\n\"You'll be lucky if the wounds don't fester. Vornatti is no physician.\"\n\n\"It heals, regardless,\" Maledicte said.\n\nYes, Gilly thought, the knowledge assailing him again, a slap of frigid seawater, scouring and impossible to digest. Wasn't healing one of Ani's gifts? Something good turned to malign purpose; it was hard to stop a man immune to violence. But Maledicte had bled enough\u2014\n\n\"Stop staring,\" Maledicte said. \"You promised me food.\"\n\nGilly found them a table in the public rooms of the Glorious, the ice shop popular among the maidservants and merchants, secretaries, sailors, and laborers. It had once been a temple to Naga, the serpentine god of health and avarice, and the rooms still boasted elaborate murals of undulating waves and scale; the columns were Naga rising from the sea depths, fanged mouths gaping and holding coats.\n\nIn the midst of this they sat, eating tart lime ices and sugar pastries, drinking bitter coffee with sweet sludge at the bottom. Maledicte's lips reddened with the cold kiss of the confection, his cheeks flushed by the steaming drink.\n\n\"Vornatti must have grown bored with his company,\" Gilly said, looking at the carriage drawing up to a discreet storefront, marked only by three silver balls on a cord.\n\nFollowing Gilly's gaze, Maledicte turned his head. They watched Mirabile step out of the carriage, her dress loosely cloaked for anonymity, carrying a parcel. She disappeared into the dark recesses of the shop.\n\n\"A pawnshop?\" Maledicte said, shifting to shelter behind Gilly, out of sight.\n\nGilly said, \"She's popping her valuables. Or more likely Westfall's. I doubt she has anything left of value. But if she wed someone wealthy...\"\n\nMaledicte pushed his plate away. \"She can't think of anything else?\"\n\n\"There's nothing else for her to do,\" Gilly said. \"She's an aristocrat, not trained to do anything. Or allowed to. Women in this society are ruined so easily.\"\n\n\"You sound sorry for her.\"\n\n\"No,\" Gilly said. \"She had a rich husband and killed him. You might keep that in mind when you speak with her.\"\n\n\"I don't have to,\" Maledicte said, recovering his appetite, stealing the rest of Gilly's pastry. \"My tasks require swords. This one doesn't. This task is yours.\"\n**\u00b7 10 \u00b7**\n\n_Throughout time, men have been driven by outraged pride or loss to commit terrible acts of vengeance, demonstrating how dangerous, how vile man can be when he chooses to turn intellect to malign purpose. But nothing man can do is so dreadful as one aided by Black-Winged Ani, the god of love and vengeance. Under Her aegis, a single man's vengeance can consume not only families, but cities._\n\n\u2014Darian Chancel. \"On Theology\"\n\n**M** IRABILE GLIMMERED in shades of orange and flame that brought warmth to her icy perfection, and echoed the fire and gilt of Maledicte's coat. At his entrance into the ballroom, she joined him as neatly as if they had planned it. Unusually, there was a faint tint of color in her cheeks; Maledicte, having heard the gossip, didn't wonder at it.\n\nA trill of laughter touched his ears. \"No, really my dear, Westfall had to pay for his own silver-backed brushes. Can you imagine\u2014\" Lady Secret and her listeners fell silent as Maledicte and Mirabile passed, stifling their smiles.\n\nOutside, Maledicte had heard much of the same, that Adam Westfall tired of his unwanted guest, and pressured his wife to be rid of her. Maledicte only wished the man would do it soon and spare him yet another series of encounters with her delicately acid tongue.\n\n\"Tell me, Maledicte, how fares Vornatti, that you missed the Lovesys' ball? I thought you had meant to attend.\"\n\n\"That's the difficulty with bribing servant girls,\" Maledicte said.. \"They cost you coin and are unreliable.\"\n\nMirabile laughed. \"You do say such terrible things.\" She leaned closer, offering a tantalizing glimpse of perfumed, powdered skin. \"But you haven't excused your absence, and to spurn a counselor's ball requires an apology at least.\"\n\n\"I, too, noted your absence. You sent no word.\" The pleasant voice dropped Mirabile into a curtsy, Maledicte into a bow, as Aris joined them.\n\n\"Sire,\" Maledicte said. \"Vornatti was ailing and I felt my place was beside him.\" It was as close as he would come to the truth, that after Gilly's and his disobedience that day by the sea, they had been punished. Gilly had been sent to sleep in the stables for a week, and Maledicte\u2014Vornatti had kept him so close he might have been wearing a leash.\n\nMirabile murmured, \"Yes, I believe I've seen what Vornatti considers your place.\" Maledicte felt a sudden crest of hatred for her tongue, for the fact that Mirabile, a favored visitor, had witnessed Vornatti's dominance with laughing eyes.\n\n\"You're pale tonight, lad. Do not let the idleness of town life cheat you of your health. You should dance more,\" Aris said, frown easing. \"Put color in those fair cheeks.\"\n\n\"As I have been urging him to do,\" Mirabile said, tapping Maledicte's shoulder with her fan. \"But will he dance? No, he will not.\" She held out her hand with expectant grace as the musicians began opening measures to a country dance, as if all the days of watching Maledicte obey Vornatti's whims made her wishes inviolate also.\n\nMaledicte stepped out of reach without thinking. Mirabile's perfect features etched a quick frown and smoothed again. \"You see, sire?\"\n\n\"Ill-done of you, lad,\" Aris said. \"We noblemen must never disappoint a charming and beautiful lady.\"\n\nMirabile claimed Maledicte's hand with a possessiveness that made his skin itch. \"Yet you do not dance,\" Maledicte said, irritation bleeding into his voice. Belatedly, he tried to mask it with flattery, as he would for Vornatti. \"And to be bold, my king, you are far more a maiden's dream than I.\"\n\nAris laughed, flushing a little. He reached for Mirabile's hand. \"We will assay the floor together, Mirabile, and teach this pup some manners.\"\n\nMirabile curtsied again, topazes winking in her ruddy hair. \"You honor me, sire.\"\n\n\"Come lad, find a partner,\" Aris said, smiling. He held up a gloved hand; the musicians paused.\n\nIn the silence, Maledicte's eyes slewed around. For a bare moment, they lit on Gilly, near hidden in the shadows of the balcony, before falling on a tiny, porcelain doll of a debutante whose chaperone had her head bent away in gossip. Maledicte took quick strides to her side. \"Lady?\"\n\nThe musicians surged into the involved patterns of the Labyrinthine. Maledicte and his partner moved neatly, with careful grace and the physical wariness of two people unacquainted. When Maledicte raised his eyes from the girl's downturned face, he found Aris's intelligent blue eyes fixed on him, and Maledicte stumbled.\n\nMaledicte dropped into the final bow, brushed his lips over his partner's hand. She faded away, rejoining her frowning chaperone.\n\nAris bent over Mirabile's hand, and Maledicte took the moment to escape toward the balcony's evening shadows. Gilly saw him coming, raised the flask from his coat pocket.\n\nA hand on his arm halted Maledicte. He spun and swallowed his bile. \"Sire.\"\n\n\"Maledicte, come with me.\" The king released his hold on Maledicte's silk-covered arm, walked on, sure that Maledicte would follow. The scalloped balconies and quiet alcoves were popular enough that Aris had to search several doorways until he found a vacant one.\n\nAris sank down onto one of the carved marble benches that ran the perimeter of the balcony. On either side, tree roses shielded them from view. Below, the gardens smelled of damp moss and night flowers opening. Maledicte stood before him, hesitant and worried. He knew his temper was foul tonight, knew also that it led him into incautious behavior.\n\n\"You are new to my court and with Vornatti as your only guide, perhaps less informed than you should be of the social niceties.\"\n\n\"I apologize for my reluctance to dance, sire. I will make amends and dance every set left this evening. If you will it,\" Maledicte said, despite his aversion for such things. He did not care to stand so close to the other men, to hold women his height, fearing that it would only point out his slightness, risking his mask. Was that all\u2014such a small thing to incur a king's displeasure. Maledicte bit his lips, closed his eyes, wishing again that he could simply reclaim Janus without all this mummery.\n\n\"You may do as you please, but Mal\u2014\" Aris's voice shifted as he assumed the intimacy of a friend. \"Mal, a word of caution. While it is understood that certain young men find the company of other men preferable to the ladies, I would not have the lines of the dance ruined by such a pair. It requires discretion. Can you be discreet?\"\n\n\"Do you find me so gauche as to expect such from me?\" His tone was more insulted than concerned, but he was irritated out of reason that the king's interest extended so far into his life.\n\n\"I find you\u2014\" Aris hesitated, pulling a rose from the tree beside him and breathing in its scent. Its petals were near blown and browning at the edge; they shed at the touch of his breath. \"I find you hard to predict. A creature of impulse in a rigid court, and I would not see my brother set against you. He has more power than I would like....\"\n\nMaledicte paused, listening to the silence of what Aris had not mentioned\u2014the effort it took Aris to tread the path between pleasing Vornatti, who held the purse strings of Antyre, and satisfying his ambitious brother.\n\nBefore Maledicte could speak, strains of music drifted outward, and Aris smiled. \"The Labyrinthine again.\" He rose.\n\n\"Shall I dance it?\" Maledicte asked.\n\n\"Not until you rectify your steps,\" Aris said.\n\nMaledicte flushed, annoyed that Aris had seen him stumble.\n\n\"It's not so hard,\" Aris said, \"But it takes some thought for one not brought up on it.\" He held out his hand.\n\nMIRABILE, THWARTED IN HER PURSUIT by Aris's easy theft of her partner, stalked toward a balcony. She faltered when she saw it occupied, but then, with a sudden smile, came forward. \"Such a moody creature, your master,\" Mirabile said, joining Gilly in the shadowed alcove.\n\n\"Lady?\" Gilly said, his eyes on Maledicte vanishing after the king.\n\n\"Don't look so foolish,\" Mirabile said. \"Sweet Livia tells me you're the man behind the scenes. Gilly, is it? Tell me about Maledicte.\"\n\n\"What do you want to know that Livia can't tell you?\" Gilly said absently. Maledicte had seemed ordinary enough tonight, or as ordinary as he could be. Ani's presence seemed more dreamlike now than in his nightmares.\n\n\"I want to know what all women want to know. How much he dotes on me.\"\n\nGilly's attention sharpened. \"Shouldn't you ask instead about his prospects? Or do you seek a marriage purely for love?\"\n\n\"Purely for love?\" Mirabile said, eyes flattening with wariness. \"Maledicte has more to offer than love alone.\"\n\n\"Not money,\" Gilly said, leaning against a pillar. \"Vornatti grants him an allowance, that's all.\" Her pleasant expression changed to one of slit-eyed anger.\n\n\"You lie. Livia says Maledicte has coin of his own.\"\n\n\"Livia,\" Gilly said, with a rush of anger, \"is a _servant._ She thinks ten sols is a fortune. Instead of the overlace on your dress.\"\n\nMirabile twitched, hands clawing at her long skirts as if she'd recoup the money spent on them. \"But his future prospects...Vornatti will settle a yearly allowance on him, surely.\"\n\n\"He prefers to keep Maledicte under his own roof. Should Maledicte wed, Vornatti will cease funding him, he's that possessive.\" Gilly's tone soured, the very bitterness in it adding weight to his words. But a week spent ostracized from the house, spent worrying about his own position, left him a pessimist. Alone in the stable with only the dreams for company\u2014dreams of Ani's laughter, waking to find the horses kicking and thrashing as if they, too, felt Her.\n\n\"You know nothing,\" Mirabile said, nearly spitting in her anger. \"You're just a servant.\"\n\n\"The one behind the scenes,\" Gilly retorted.\n\n\"Between the bedsheets. You're nothing but Vornatti's toy.\"\n\nGilly flinched. \"Nonetheless, what I tell you is true. Though Vornatti may be wealthy, he has no obligation to Maledicte. Indeed, he cut back his allowance a week ago.\"\n\nMirabile's face blanched, her green eyes closing. Her hands still twisted around each other. \"Good night, Mirabile,\" Gilly said.\n\nShe slapped him, curving her nails inward. He jerked back, saved himself from the worst of their sting, though his cheek burned.\n\n\" _Lady_ Mirabile. If I have nothing else, I have that, and you are only a servant.\"\n\nPushing past him, she hesitated in the ballroom doors, then, raising her head, returned to the court.\n\n\"LIKE SO,\" ARIS SAID, as the tune came around, his hand clasping Maledicte's. Maledicte took a breath, made the delicate feint inward, the retreat, then the elaborate pivot and bow, all the time aware of the king's hand on his. He tripped, and Aris, patiently, said, \"Again.\"\n\n\"You're quite the teacher,\" Maledicte said.\n\nAris smiled. \"I've always been thorough in my studies.\"\n\n\"And I, apparently, shirked mine,\" Maledicte said.\n\n\"You waltz splendidly,\" the king said. \"Your teacher should be commended.\"\n\nMaledicte hid a smile, remembering spinning Gilly in the waltz until he pled dizziness and shortness of breath.\n\nIn the ballroom, the measure came round again, and Aris held out his hand. \"Once more?\"\n\n\"Aris!\" Last said, looming behind Aris, body blocking the glow of the ballroom.\n\n\"My brother, the hound,\" Aris muttered.\n\nMaledicte stepped away from Aris, and Last's mouth, outlined by his pale beard, turned downward.\n\n\"Remember discretion,\" Aris said, stepping aside to let Maledicte return to the ballroom.\n\nMaledicte touched his sword hilt, stroked the feathers, considering Last's presence, anger eating through his veins. The earl had taken Janus, had sent Kritos to recover him as if he were nothing more than a strayed possession\u2014Wings fluttered in his chest, a heartbeat of rage and pain. To strike _now_ and be done with it...\n\n\"Maledicte?\" Aris's brows drew downward as Maledicte stood, his hand locked on the sword hilt.\n\nMaledicte's hand flew from the sword. He was badly startled. How long had he gaped at Last like a rabid dog? He sketched a hasty bow, gave it an elaborate fillip to make Aris smile, and fled into the ballroom. Ani shrieked within him. A second time, to be so close, and not to strike...\"Not yet,\" he said, speaking to that heat in his blood.\n\nComing onto the floor, he saw Gilly peering through a doorway and detoured again. This time he reached the safety of Gilly's side. \"I need a drink.\"\n\nGilly paused in his search for his pocket flask. \"What did the king want?\"\n\nMaledicte shrugged, slung himself down onto a bench, laid his legs along the length of it, precluding Gilly's joining him. \"To teach me to dance. To lecture me on my behavior. Between him and Mirabile, my card is full.\" Maledicte sulked, studying the toes of his polished boots.\n\n\"I've put a stop to Mirabile,\" Gilly said. \"And apparently she picked my pockets while I did so. No flask, Mal, I'm sorry.\"\n\n\"Gilly!\" Maledicte said. \"What matters a flask when you've removed the huntress from my trail? Dare I ask how?\"\n\n\"Simple truth made you unsuitable,\" Gilly said, still touching his coat pockets with a faint frown on his face, as if trying to recall the exact moment the flask disappeared.\n\n\"Truth?\" Maledicte said, coldness shifting in his belly like a snake. \"What truth is that?\"\n\n\"That Vornatti's fortune is not your own.\"\n\n\"To think a lack of a fortune could ever be beneficial.\" His grim amusement faltered. He stepped closer, touched Gilly's cheek. \"What's this?\"\n\nGilly touched the scratch at the cheekbone. \"She wasn't best pleased with what I had to say.\"\n\n\"Should I repay her for that?\" Maledicte asked, his tamped-down rage resurfacing, redirecting. \"Spill the fact that I won't have her? If the duns are after her\u2014\"\n\n\"No,\" Gilly said. \"Leave her alone. She's rat-vicious, best not cornered.\"\n\nMaledicte sighed. \"Take me home. I'm bored with the wonders of the court, sick of the people in it; I can't kill the ones I want to, so why stay?\" He tugged Gilly's hair, touched the red mark once more, wiping away a quick smear of blood from the thin scratch, and headed back into the ballroom and the main doors.\n\nTHE CROWD PARTED FOR THEM QUICKLY, and Gilly, unable to push his way through for fear of damaging noble flesh and feelings, watched Maledicte slip away.\n\n\"Going home to your master like a faithful dog?\" Mirabile said, appearing next to him. Gilly said nothing. On the balcony, unseen, he had been able to speak his mind. Here, a wrong word could see him whipped.\n\nShe circled him, radiating anger and danger, like a predatory beast. She stroked the length of his spine, and whispered, \"Do you wag your tail for anyone? Or just Vornatti?\"\n\nGilly bit his lip and tasted blood.\n\n\"He listens to me. I could spin him such tales\u2014he'd have you cast out....\"\n\n\"Mirabile,\" Maledicte said, returning, his face white, his eyes hot. \"Watch yourself. Gossip is a knife, and it's at your throat. Would you like me to push it closer still?\"\n\nGilly shuddered at the quiet rage in Maledicte's voice, at the surprise in Mirabile's eyes, the reassessment of Gilly's status in the Vornatti household.\n\n\"Come, Gilly,\" Maledicte said, seizing Gilly's hand and tugging him along, heedless of the nobles in their path.\n\n\"The main door's back that way,\" Gilly said when he could speak. \"Mal, you shouldn't have defended me.\"\n\n\"We'll go through the gardens and avoid any more display of noble manners. My temper is as sharp as my blade and eager to be loosed.\" He dropped from the rail of the curving balcony to the earth four feet below. Gilly followed, landing soundlessly in the soft moss.\n\nNear the entrance to the garden maze that lay between them and the road, Maledicte put his hand out to halt Gilly. Gilly stepped back until they were both in the deep shadow of statuary and hedge, looking up at a dark balcony on the king's side of the ballroom. Two men stood in the shadows, and at their feet a great dog raised its head, sniffing the night air.\n\n\"...eager to meet this boy of yours, Michel, no matter the irregularities of his birth. Bring him at once when he arrives. We need more young men in the court, men not spoiled as we are, with old secrets and schemes, soured by battles fought decades ago.\"\n\nLast said, \"Youth is no great thing, Aris. It masks threat and schemes as well as any old face.\"\n\nThe king said, \"You mean Vornatti's ward. Maledicte.\"\n\n\"I do.\"\n\n\"He's but a young man with his own pleasures to seek, his own wants.\"\n\n\"He carries hate and hunger with him. His eyes burn with it.\" Last swung his cane, clipped roses from the hedge before him.\n\n\"Mmm,\" the king murmured. \"I see no such thing in his eyes. Yours though\u2014\"\n\n\"You're a fool, Aris. Shall I tell you what I hear, whispered in the air of the court? One word, blown like leaves: 'witch.' They know him for what he is, an accursed creature.\"\n\n\"You sound like a country intercessor, seeing the old gods in every shadow. But you forget, as the gods are dead, so are your witches. Without the gods' power to scavenge, a witch is nothing but caged spite. Maledicte seems a pleasant boy, albeit one with an unfortunate mentor.\"\n\nLast snarled. \"Fool, twice over. To have loved that Vornatti woman who brought neither healthy child nor power, and to defend his creature, now. Black-Winged Ani has touched him, made him Her lover\u2014\"\n\nBeneath them, Gilly shuddered and Maledicte moved closer, a gesture of support, or perhaps for shelter.\n\n\"Michel, superstition is the mark of a fool,\" the king said; Last drew his lips closed over set teeth and jaw. He stalked back through the doorway, setting the hound to growling after him. Aris brushed back his hair, displacing his circlet, and resettled it. \"Eavesdropping is a standard of the court. I see you're practicing noble manners.\" He looked into the shadows, pinning Maledicte and Gilly with his amused gaze. His eyes flickered downward. \"But remember discretion.\"\n\nMaledicte's hand, resting on Gilly's hip, recoiled. Gilly dropped into a hasty bow.\n\nThe king grinned like a young man and sauntered into the darkness, the hound rumbling to its feet with a sigh.\n\n\"Come on,\" Maledicte said, and they plunged into the green moonstruck darkness of ivy-covered stone and thick hedges. Tiny white flowers coiled around animal statues like a spattering of stars.\n\nGilly pointed to a small carving within a mortared niche. \"As long as we follow the mouse we should come to the center and then the exit.\"\n\n\"Do you know every secret, Gilly?\" Maledicte asked, taking that first turning, disappearing into shadows, leaving only his voice behind. The breeze painted each shaking leaf with moonlight and dappled the pathway so that it seemed silvered with frost.\n\nGilly trailed after him, on smooth grassy paths designed to be strolled in the night. Moonflowers spotlighted a lover's bench; a stone mouse leaped on its side and directed their next turn.\n\nThe trail opened into a garden, ringed with pathways like the spokes of a wheel, its shape an echo of the city itself. Night-blooming jasmine laced the air with heady fumes. Maledicte spun in the center of it, staring at the starry sky. \"Can you see Her, Gilly?\"\n\nGilly saw nothing, but heard the rasp and rush of feathers. Hoarse calls and rattles came from all around them, and the leafy walls of the maze gave birth to dozens of rooks. The air filled with the drumbeat sound of their wings, and the stars above flickered. When he could speak, his heartbeat slowing back to a normal pace, Gilly said, \"We must have disturbed their nesting.\"\n\n\"You don't think She's watching me?\" Maledicte said.\n\n\"They're just rooks, startled to find intruders in the maze.\" The maze, which had seemed peaceful and secluded, now closed about him like a net. Was this how it would be? His eyes opened now? Not for the first time since the pier, Gilly wished for blindness, for his question to have died unspoken.\n\n\"This way, Gilly.\" Maledicte moved on, and Gilly followed. But somewhere in the maze of trails and turns, shaken by the soaring rooks and led astray by moonlight, they lost the correct path and found themselves in a cul-de-sac of whispering leaves. Frowning, Gilly headed back for the last turn, but Maledicte's stillness halted him.\n\nPaused, his feet no longer stirring the grass, Gilly heard it: the clipped, echoing sounds of hooves drawing closer on cobbled streets. He parted the ivy, revealing the training wires beneath, and peered through. \"It's a carriage,\" he said. \"A rented hack.\" Around them, the rooks settled like blight on the trees that lined the drive.\n\nMaledicte's face was bleached of color in the night. He drew his sword, slashed the ivy. It parted like paper, and Maledicte stepped out of the maze, though he kept to the shadows. A man stepped down from the hack, dark and well dressed, far too well dressed to require the services of a rental driver. A large man with glossy dark hair, and an elaborate malacca cane. Maledicte tensed like a dog on point. Faint chords of familiarity woke in Gilly, but it wasn't until the noble turned his head, exposing an eye filmed and scarred, that Gilly knew him.\n\nKRITOS MET THEIR GAZES and his face grew disquieted, as if there were some danger to seeing a tall blond and a slight brunette coming out of the shadows. He was slow to put his back to them, slow to climb the steps toward the court, though perhaps some of his deliberate pace was due to the heavy aroma of spirits that lingered around him like a cloud.\n\nMaledicte stood like marble, only his rocketing heart betraying him, sending a bloody flush to his cheek. The rooks' wings, shifting, whispered partite beats: Kritos. Hissing the name. Tolling the name. Maledicte had not thought he would feel anything but hate for this man, but a wild wash of joy made his mouth quiver. And why not? Kritos's return heralded Janus's. Janus was within reach at long last\u2014Maledicte clenched his hands, took steady breaths, and watched Kritos belabor the great doors. This drunkard had been a threat to him once?\n\nThe doors slammed back, the footmen scrambling to shield the delicate inlays from contact with the stone. Kritos stepped forward as if the doors had flung themselves wide for him, and was immediately distracted. \"Last, there you are,\" Kritos said. \"What do you mean, denying me your house? I am your blood.\"\n\n\"Where is Janus?\" the earl asked.\n\n\"At Lastrest. Recuperating from the sea voyage. Answer me. Why was I denied entrance to your house?\" Kritos's voice rasped with desperation.\n\n\"I am weary of your debts. I warned you that I would stand for no more of it. Yet, not returned a full day, and I hear you've lost your coach and its team to the tables. I will not have you as an anchor on my purse any longer. I have let the moneylenders know this.\" Last brushed past Kritos and was stopped by Kritos's grip on his arm.\n\n\"Aris won't stand for it. I'm blood kin,\" Kritos said.\n\n\"Oddly enough,\" Last said, \"this is the only action Aris and I have agreed on in years. But he thinks responsibility will make a better man of you, while I...doubt it.\n\n\"So were I you, nephew, I would not waste my limited time arguing. You had best find yourself an heiress to take on your debts.\" The earl shook off Kritos's clutching fingers, cast an inimical glance at Gilly and Maledicte, and lowered his voice.\n\nMaledicte closed his eyes, the better to hear words that were balm to his senses. _Let Kritos suffer,_ he thought. _Let him face the streets, the rats, the poverty._\n\nLast's voice, clipped with anger, slid into his ears, jolting him with one name. \"...Janus is more clever at card playing than you are at card sharping. I do have some family feeling. Consider those debts cleared.\" Ignoring Kritos's choleric flush, the earl proceeded down the stairs, signaling for his own coachman to pull up behind Kritos's waiting hack.\n\nBehind him, Kritos lashed out with his cane, the heavy wood meant to crush Last's fair head. Maledicte froze, imagining his chance at vengeance gone, but Last pivoted as smoothly as a serpent; the cane cracked into his gloved palm and he yanked it from Kritos's hands. Overbalanced and overwrought, Kritos tumbled down the wide stairs, landing in the oyster-shell gravel to the detriment of his skin and clothes. He moaned as he staggered to his feet.\n\n\"Take my nephew someplace to sober up,\" Last said, speaking to the coachman. The man shook his head, mute, unwilling, until Last tossed him a luna.\n\nLast looked on Kritos's limping form dispassionately. \"As always, you make poor gambles, Kritos. Attack the man who can and will disinherit you without further qualm? And to do so before Vornatti's catamite and his spy. How fast the word will spread of your straits, and you with no one to blame but yourself.\"\n\nMaledicte sketched the briefest of bows when Kritos turned a furious face toward them and said, \"Such delicious gossip it is, too. The family loyalty of the House of Last.\" He shivered in small spurts along his spine, the only outlet he afforded his rage. So close, and yet, Janus still eluded his grasp. Killing Last now when he could be caught before he reached Janus was\u2014unthinkable. He could not move; the urge to kill and the need to wait warred in him, keeping him frozen as Last paced forward.\n\n\"What is your game, boy?\" Last looked down his narrow nose; he was close enough that Maledicte had to look up to see those pale, icy eyes.\n\n\"Is this a game? I never thought it one. As for my conduct, I do as I see fit.\" His voice, raspy, covered the shiver in it.\n\n\"Were you not Vornatti's ward, you'd find out how little that arrogance would avail you,\" Last said. \"But tell me, boy, what do you want with my brother?\"\n\nMaledicte forced a smile, despite the ache in his guts that pointed out how close Last was, how sharp the black blade was, how quickly the deed could be done. A pastel froth of dresses spilled down the stairs as the youngest debutantes and their chaperones came outside, looking for their coaches. They stopped; one lady giggled uncertainly, sensing the charged atmosphere, the muttering rooks flanking the drive.\n\n\"I want nothing from Aris that he has not already given me, but what I want from you\u2014\" Maledicte said.\n\n\"I have done nothing to you, and yet I could swear to your enmity.\"\n\n\"Have you something on your conscience? Some wrong done? Shall I remind you? You gave me a gift once before you learned its value and took it back,\" Maledicte said, breath catching in anger. Careless, he thought. If Last understood, raised his mind from petty offenses to himself, from the confusion over Aris's support, would he not whisk Janus away once again?\n\nHe tempered the ragged edge from his voice and said, \"Your coach is waiting, your grace, and I have nothing more to say to you tonight.\" The cool dismissal whitened the skin around Last's nose even as it flushed his thick neck.\n\nLast reached out as if he would shake or strangle Maledicte, but his hands dropped to his side at a gasp from the throng of women. \"I will see you gone from the court,\" he said, as he stepped up into his coach. \"Revealed for what you are.\"\n\n\"What am I?\" Maledicte whispered as Last closed the coach's door. \"I wish you would tell me.\"\n\nLast gave the coachman a signal and the horses drew him away. Maledicte stood trembling, until Gilly took his hand and led him home.\n**\u00b7 11 \u00b7**\n\n**S** ORNATTI'S DOVE STREET RESIDENCE was lit against the silky fogs creeping inward from the sea, making itself a welcoming beacon in the twilight. Inside, that echo of hospitality continued as maids laid silver out, as the cook prepared her courses, as Vornatti waited for his dinner guests.\n\nIn his bedchamber, Maledicte paced, irritated that Vornatti staged this party now, when he wanted to flee the city and hunt Janus down at Lastrest. But Vornatti had insisted on the party; more, he had disabled the coach and disallowed Maledicte any coin. Once, Maledicte thought, such obstacles would have only slowed him, not stopped him. But now, he had grown soft\u2014or practical\u2014and knew there was no point attempting the forty-mile journey afoot. Not when Janus was destined to attend the solstice ball; not when morning might see Vornatti more agreeable.\n\nStill, his temper was bad, and at the sight of a carriage come unfashionably early, he slammed his window shut, cracking the glass.\n\nSince his bloodless confrontation of Last, Ani had gnawed at him, muttering and seething, until his entire body ached with fluttering wings and razor beaks. His mind, like feathers in an eddy, kept coming round and round, always returning to blood-drenched dreams. His hand cramped on the sword hilt, seized yet again to no purpose. Ani, restless, threatened to withdraw Her support, the compact annulled by his dilatoriness. _Coward,_ Ani jeered, Her message in his clenching fist on the hilt: _Last must die._\n\n\"He will,\" Maledicte muttered. \"I swore it. I swore.\"\n\nBehind him, the door, left ajar for Gilly, whispered open. Maledicte tensed his shoulders, pulled the drapes across the blank-eyed glass. \"Not yet,\" he said.\n\n\"Are you practicing your lines?\" The voice wasn't Gilly's low voice, husked with the indelible country accent, but a woman's, delicately arch. \"I thought your wit more ready than that.\"\n\nRage muted Maledicte's response. He turned. Mirabile lingered in the doorway, and as he met her eyes, she took a step inward, her hands trailing across the jambs, emphasizing her invasion of his room. Like the night sky, she swept inward, all in dark satin and fog-gray trim.\n\n\"Apparently, I was mistaken,\" she said. \"A flaw in your nature. You should correct it. Those who practice their thoughts are often caught flat-footed.\"\n\n\"Get out\u2014\" Maledicte whispered, then with effort removed his hand from the sword hilt again, and assayed a reasonable fa\u00e7ade of courtly speech. \"I have a care for my reputation, even if you do not. A man's bedchamber is no place for a lady.\"\n\n\"What _has_ the baron been teaching you?\" Mirabile said, her lips curving. \"Shall I show you otherwise? Prove that a lady indeed has a place in a man's bedchamber?\" She glided toward him, her dark skirts creeping ahead of her.\n\nMaledicte stepped back, bumped the wall. She laughed and settled herself on his bed, rested her cheek against the canopy post, stroked its length. \"As nervous as a virgin. How unflattering,\" she said. \"And feared for your reputation? Let me teach you this, Maledicte, that scandalous creatures such as myself, such as you\"\u2014she nodded toward him with a regal incline of her head\u2014\"need not fear the strictures of propriety. The peerage expects misbehavior from us. We are free in ways they will never be, granted license by their hunger for scandalous gossip.\"\n\n\"They don't need our actions to feed their gossip. I believe they make it out of whole cloth.\"\n\n\"Strong words from a man who owes much of his place to the collection and manipulation of rumor,\" Mirabile said, laughing. \"If you claim such disinterest, I will stifle a whisper I meant to share\u2014a gift of sorts, to mend this awkward dislike you bear for me.\" She rose, crossed the room, her presence as warm beside him as an animal's.\n\nMaledicte leaned against the window, wishing he were back in the ballroom with a low balcony and velvet grass behind him instead of a steep drop to a thorny garden. \"Dislike?\" he said. \"Is that what you term my feelings to be?\"\n\n\"Hush,\" she said, putting her hand to his mouth. Perfume rose from her skin, the dizzying, cloying attar of imported jasmine. She tipped her mouth toward his. He turned his head, and she, determined, followed, brought her lips to his. Again, he reached for the sword, but Mirabile caught his hand and brought it to her bodice, the scented intersection of flesh and satin.\n\nSnarling, Maledicte shoved her away, drew the sword in a long hiss. Evading its blade, Mirabile fell over her long skirts. Her eyes darkened; her face stiffened in insult. But her voice stayed sweet. \"So very gauche. If you were not so lovely, I wonder if I'd bother with you. Still, you might at least pretend to civility. Were I to report your behavior to Vornatti\u2014\"\n\n\"If you report this, Vornatti would see me cast out, which would gain you neither husband nor the wealth you crave.\" Maledicte sheathed the sword again, fighting the urge to see blood wet its blade. Any blood.\n\nMirabile rose, brushing at a creased panel on her skirt. \"But would gain me the satisfaction of seeing you so discommoded. Still, you're quite correct, that's not the result I wish.\" She shook out her skirts, then settled herself at his dressing table, sorting the discarded jumble of stickpins and fobs, leafing through _The Book of Vengeances,_ and smiling at the illustrations. \"So I'll return to my first purpose. Shall I tell you the gossip?\"\n\n\"I'll hear it from a dozen mouths by morning,\" Maledicte said, watching her hands, thinking of Westfall's stolen belongings, of Gilly's missing flask, but, unwilling to get closer, allowed her to set the book aside for his embroidery box.\n\n\"How you wrong me,\" she said. \"My gossip is never the ordinary. I am as clever as your baron when it comes to ferreting secrets.\" Her agile fingers sought the catch.\n\n\"Tell me if you must,\" Maledicte said.\n\nHer fingers stilled as she smiled at him. \"Last dislikes you very much; you return the sentiment,\" she said.\n\n\"Old news,\" Maledicte interrupted.\n\n\"But Kritos dislikes him beyond that. You'll not suffer Last for long. Kritos means to kill him before Last has a chance to disinherit him in favor of Janus. It is his only way to salvage his debts. So he raises coin, even now, to hire an assassin.\"\n\nOutrage scoured him, all unexpected. His vengeance\u2014stolen? And by the man who had done him such wrong before?\n\nMirabile's fingers found the catch; the lid opened. Her painted mouth made a delicate \"oh\" of surprise, and then her lips curled. \"Not what I expected,\" she said. \"But perhaps you and I can understand each other. Shall I make myself clear?\"\n\n\"I wish you would,\" Maledicte said, the rage fading slowly as he wrestled for control. He watched her select the vial of arsenixa, admiring it in the lamplight as if it were a gemstone.\n\n\"I need funds desperately,\" she said, \"thus a husband. But my reputation is such that the moneyed men, who can afford to be selective, will not be caught. And time grows short.\n\n\"My lord Westfall is a most impatient host, and Brierly has had the poor judgment to allow herself to become with child. Her vanity is so great that she will not be seen while she is increasing, and so we must rusticate with her. And I will not stomach a country clod for a husband.\" She ran her nails along the crystalline vials until they hummed.\n\nMaledicte reminded her, \"My fortunes are as negligible as yours.\"\n\n\"So your dog said, barking most convincingly. I see only one option left me,\" she said.\n\nMaledicte stiffened. \"Surely\u2014\"\n\n\"Hmm\u2014\" she said, interrupting him, tugging out a soft roll of cloth-bound powders. \"You are not so indifferent to the female sex as you pretend. Not and keep Harlot's Friend in such ready supply. I own I am glad to see it here.\"\n\n\"Potions be damned,\" Maledicte said. \"Make your point and be gone, Lady.\"\n\nHer hands clenched, then eased as she mastered her temper. \"Shall I be blunt? Even Vornatti must grow bored in his bed. He has his rough servant and he has your elegance\u2014don't try to deny it,\" she said. \"I know what they call you in the court when your back is to them\u2014Vornatti's catamite. And I must believe them, since you so nobly defended your fellow, Gilly\u2014you're both whores, one simply better dressed than the other.\"\n\n\"I am not a whore,\" he said, reflexively. His anger leaped but fell back beneath caution. She wanted something of him, more than the obvious. Her entire visit felt a sortie, its purpose cloaked in layers. She wanted his alliance in her schemes; he believed that. But his acceptance or denial was not the answer to quench the burning question in her eyes. Maledicte wanted badly to know what she truly sought; he would deny her everything he could.\n\n\"Whore or not, it doesn't signify,\" she said, \"except that lovely as you both are\u2014surely he's surfeited with male flesh. He's willing to pay and pay well for his desires. My price isn't so dear, a simple ceremony and ring.\"\n\n\"You think he'd wed you? He can find other women without wedding them,\" Maledicte said.\n\n\"He will wed me,\" she said, the fury so long missing from her face flaring to life. \"Be careful, Maledicte. I would like to have you at my side, in my bed, a partner in this, but I will not tolerate your disrespect.\"\n\n\"Respect you? When my contempt grows apace?\" Maledicte said.\n\nMirabile's hand curled inward, her forearm tensed; that warning was enough for Maledicte. As her hand rose, fingers clawed, he freed the sword, using the blade like a shield.\n\nHer sharp nails sang along its length; her hand closed over his on the hilt. She shuddered as the steel feathers bit into her skin, raising blood on her white hands. The rage faded from her eyes, her face; her jaw slackened, lips quivered. \"What\u2014\" she gasped. \"I hear...whispers, a question\u2014\" For a moment, her face went slack, as her own driving question was answered with another.\n\nMaledicte jerked the sword away, her fingers welling blood. She brought her hand to her mouth, licked the wounds, her eyes never leaving his. \"It's true. Chancel, blight him, was right. All true,\" she said breathlessly. \"You _are_ Ani's creature\u2014the gods are no more gone than the sun at sundown.... Tell me how you did it,\" she said, her voice feverish, the white flesh of her breasts flushing. \"Tell me! Is it true? All true? Are you immune to poison, to injury? Such power you must have, perhaps enough to share?\" She leaned toward him, all fervor and white teeth bared in wild rejoicing.\n\n\"You're mad,\" he said. \"And long past your welcome.\" He pushed past her and fled, hating to leave her loose in his chamber, but equally loath to stay with Ani stirring to an interest beyond Maledicte's vengeance. Coldness grew in his belly, coiling, twisting; what if Ani, grown sick of his slow scheming, left him?\n\nWhat of Kritos? Though Maledicte had little respect for the man's ability, what if Kritos succeeded? What would Ani do then?\n\nPolite laughter rose from the floor below and he checked in his stairward movement, thinking the laughter directed at him, at his allowing Mirabile to cozen the truth from him that he had never meant to reveal. He growled, retraced his steps, and slipped down the servants's tairs, stroking his fingers for balance along the ill-lit walls.\n\nGILLY LOOKED UP from the household books. This, too, was his task, the endless daily budgeting and balancing of accounts. Of late, he found the minutiae soothing. So he sat now, lamp wicking down on the desk, one long leg curled back beneath the chair rungs. When Maledicte drifted into the study, a vision in dark silks, he found his first response not pleasure but dismay.\n\n\"Shouldn't you be with your dinner guests?\" he asked.\n\n\"Absolutely not,\" Maledicte said, sulking into the deep, plush chair that Vornatti usually claimed. \"What did you say to her, Gilly?\"\n\n\"Her?\" Gilly echoed.\n\n\"Mirabile,\" Maledicte said, kicking at the carved legs of the chair in bad temper.\n\n\"She's not still hunting you?\" Gilly said, surprised.\n\n\"No. Yes,\" Maledicte said. \"She's decided to wed Vornatti and make me her paramour.\"\n\nGilly scoffed, unconcerned with the black look Maledicte turned on him. \"And you deciphered this how?\"\n\n\"No effort at all,\" Maledicte said, rising smoothly, stalking the room. \"She told me so herself in my chambers.\" He growled wordlessly and for the first time, Gilly saw the knife edge beneath the familiar petulance and temper.\n\nGilly said, \"She was in your room?\"\n\n\"I left her there,\" Maledicte said.\n\n\"Mal\u2014\" Gilly trailed off, thinking of Mirabile in Maledicte's room, snooping, finding more than she could have imagined. \"Go roust her out. You cannot withstand her scrutiny. Her husband believed in the gods' survival\u2014\"\n\n\"Gilly, don't be foolish,\" Maledicte said, \"The damage is most thoroughly done. She is now quite introduced to Ani.\" He settled down on the desk, careful of the ink bottle and the fallen pen, as if Gilly's outburst soothed his own.\n\n\"She knows?\" Gilly said.\n\n\"She cornered me until she provoked the response she wanted. And while I despise her, I must admire her boldness. It's quite inspired me.\"\n\n\"To what end?\" Gilly asked, closing the ledger over his fingertip. He misliked the angry precision of Maledicte's words, the fey light in his dark eyes.\n\n\"I thought to dedicate my night to gambling,\" he said, taking the ledger from Gilly and flipping through it.\n\n\"Your entire existence seems one gamble after another. You have a dinner party to attend and the old bastard to soothe\u2014what do you seek at the gambling tables?\" Gilly took the ledger back, set it in the drawer, and shut it.\n\n\"Kritos.\"\n**\u00b7 12 \u00b7**\n\n**K** RITOS?\" GILLY REPEATED. \"YOU ARE a gambler, then. Vornatti will be furious if you slight him again.\"\n\n\"I am well aware that Vornatti will not support me in this. Of late, he counsels temperance. So it must be done quickly. You'll have to raid the house box for me, Gilly.\"\n\n\"I thought you meant to kill Kritos. Not game with him,\" Gilly said.\n\n\"I want to beat him first.\" A brief, ugly light flared and smoldered in Maledicte's eyes; Gilly looked away, fed more of the wick into the lamp, chasing the clustering shadows from the desk, fearing more than simple temper in Maledicte's eyes.\n\n\"Do you even know how to play?\" Gilly asked.\n\n\"Well enough to win,\" Maledicte said.\n\nGilly's breath snagged in his chest, not only at the resurgence of the smothering rage in Maledicte's eyes, but at his own realization that this was the moment he had been dreading, the moment when Ani's compact would be sealed with blood, no matter that it came cloaked in ridiculous questions of card games. Gilly sought something to dissuade Maledicte. \"You believe you can win at will?\"\n\n\"Of course. It's called card-sharping, Gilly; have you never heard of it? I assure you, it's all the rage in some circles.\" Maledicte smiled, sharp-toothed, as if he or Ani understood Gilly's attempts at delay.\n\n\"Why am I not surprised?\" Gilly sat back in the chair, tilted the seat for a less-shadowed view. \"Are you good enough, Mal? This will be different than rooking sotted sailors at the pier, or wherever you gained your experience. Kritos belongs to the worst hells in the city. Where getting caught sharping will see far more than a reputation ruined.\"\n\n\"Don't fret, Gilly,\" Maledicte said. \"I do excellently well. Shall I prove it to you?\"\n\n\"Go to Vornatti's dinner, Mal.\"\n\n\"No more delay,\" Maledicte said.\n\nGilly clenched his hands. \"Mal\u2014\"\n\n\"It will be done, with or without your aid,\" Maledicte said. \"Kritos has decided to deprive us of our prey. To kill Last for his own selfish desires. Did you expect me to grow bored of my enemies? My hatred? I am not Vornatti, content to nurture a grudge for years with no satiation in sight. Janus has returned. I could go to Lastrest this moment and slaughter Last; we could live like royalty until the Kingsguard or Echo's Particulars managed to bring us down. Be thankful I'm content with Kritos tonight.\"\n\n\"I'll call for a hack,\" Gilly said, his mouth dry.\n\nMaledicte bowed and left, claiming he needed to change into something more suitable, with all the airy insouciance of a man going out for a night's pleasure instead of a murder. Left alone, Gilly laid his head down on the desk, tangled his fingers in his hair, thinking. Short of drugging Maledicte, he saw no way to avert this. And Maledicte, though he trusted Gilly, would be unlikely to take food or drink from Gilly's hand tonight. The solution was as distant as ever by the time the hack arrived, and Gilly found himself following Maledicte into it in a brooding silence of his own.\n\nTHEY PAID THE HACK TO take them as far as it would, the driver loath to travel Jove Street at night.\n\n\"I do believe you, you know,\" Gilly said, watching Maledicte palm another card with a demonstrative flourish, and wishing he'd never expressed his doubts. Maledicte had made him play and lose hand after hand in the swaying carriage, until his head pounded. Gilly focused and caught the cheat this time, tapped Maledicte's wrist. \"Be careful, Mal. Sleight of hand relies on vision driven by expectation. Some men see more clearly than you think. The Fiery Hell is no place to be caught cheating.\"\n\n\"You'll be at my back,\" Maledicte said, smiling in a way that did nothing to soothe Gilly's nerves.\n\nThe hack rattled to a stop, the horses blowing out steam in the foggy dark. The last of the twilight was making way for the darkness seeping down from the sky. On other streets, the young boys employed as lamplighters would be running from post to post with their long tapers, making small, defiant blazes against the invading night.\n\nBut on Jove Street, light was unwelcomed. Only a small torch affixed to the jamb gave enough of a glow to allow a visitor to decipher the address: Fire 3 Jove Street\u2014known as the Fiery Hell to its regulars.\n\nThere were other gambling houses, more genteel in their fa\u00e7ades, more tempting to the dallying aristocracy, where women and men, lords and ladies went to be gently daring, frivolous, and above all, seen. But the Fiery Hell took its games seriously, unleavened with court manners or excess speech. No waiters circled, offering sweetened wines or spirits. No music played, and no whores waited for a lucky gambler. There were only the tables and the players, including impoverished aristocrats so desperate to recoup fortunes they would risk a knife in the ribs. The Fiery Hell was Kritos's best chance.\n\nGilly hefted the brass knocker in his hand, dropped it down on the face-plate. The sharp rapping left muffled cracks echoing down the cooling, darkening street.\n\nThe door opened; Gilly looked up, surprised at having to do so, but the man filled the doorframe as effectively as the oak door itself. Gilly nudged Maledicte. \"Stormy Jack,\" he whispered. \"The boxer.\"\n\nMaledicte looked at Gilly blankly and shrugged. Gilly sighed. \"Just don't irritate him.\"\n\nPausing in the foyer, Maledicte and Gilly studied the tables, while Jack latched the door behind them. The Fiery Hell had been stripped of any pretension of being a domicile. In every available space the tables dominated the house, each one surrounded by close-packed men, some muttering, some gray and silent.\n\nMaledicte headed toward the darkest corner as if he could sense his quarry. Gilly went after him, seeing not the crowds, but the pistols in coat pockets, the knives and swords hung on chairs or slung onto tables as prizes. All around the walls lurked employees as big as the doorman.\n\nMaledicte intended to cheat here? Gilly had known it was a foolish idea; now he found it a suicidal one.\n\nIn the corner of the room, at a small table, Kritos sat alone, his back to the wall and its peeling flocking. Maledicte slung himself into the chair opposite.\n\n\"What do you want?\" Kritos asked, his good eye bleared with drink and bitterness. Maledicte touched the table before him, as if feeling the games that had gone before.\n\n\"To gamble with you, of course,\" he said.\n\nGilly pressed in close behind Maledicte, listening, watching that slim, rigid spine.\n\n\"When Vornatti keeps you as short of coin as Last keeps me? I know who you are and I'll not waste my time,\" Kritos said, starting to rise. Maledicte shoved the table across the broken parquet and pushed Kritos back into his seat.\n\n\"You listen to too much gossip. That's the mark of a fool,\" Maledicte said.\n\n\"Watch your tongue,\" Kritos said, but wearily, as if he lacked the energy to take offense. Gilly, watching his face, thought that Kritos intended to parlay his entrance fees into a night's lodging.\n\n\"Surely you can come up with some small stake, or you wouldn't be here,\" Maledicte said, leaning across the table. \"I'm not as particular as these gentlemen. I'll wager for things other than coin.\"\n\nKritos laughed, hoarse and unamused. \"Why should I? What have I to win?\"\n\nMaledicte tipped a handful of coins out on the table. They gleamed with the moon's cool frost in the dark room. \"Not so poor as coppers,\" Maledicte said. \"Not so dangerous as sols. Lunas.\"\n\nKritos licked his lips. \"I've no coin.\"\n\n\"When has that ever balked an aristocrat?\" Maledicte asked.\n\nKritos hesitated a minute longer, looking at Maledicte. Gilly saw the moment when Kritos made his decision, saw the quick flash of scorn on Kritos's face as he relegated Maledicte to fool and fop. \"If you're that eager to play...\" He waved over one of the Hell's attendants, one with a curved scar through his mouth. \"A house stake for me, Smiles.\"\n\n\"Your hide,\" Smiles said, and dropped a roll of mixed coins to the table. \"House takes thirty percent of your winnings, plus the stake back.\" Kritos waved him off, though his lips tightened at the reminder.\n\nTwo hours later, Kritos was richer a tidy pile of coins, as well as the random miscellany Maledicte had allowed to represent coins when Kritos upped the stakes beyond his current coin: a palm-sized miniature meant for Aris, a sailor's compass, the jadestone buttons from his vest, and the Itarusine-made lace from his cuffs. Two of the room's guards, drawn by the scent of unusual play, prowled around the table.\n\nGilly thought Maledicte had let it run too long, had allowed Kritos to win too often and risked losing him from his net. As if in unity with his thoughts, Kritos made to rise.\n\nMaledicte sighed. \"One more?\" He eyed the table with an inexplicable avarice until Gilly realized that his attention focused on the miniature, and who the subject must be. The newest member of the House of Last, Janus Ixion.\n\n\"I think not,\" Kritos said, scooping the pile toward himself.\n\nWordlessly, Maledicte tipped the last contents of his purse out. Sols, this time, and Kritos halted where he stood.\n\n\"Very well,\" Kritos said. \"But I'll want a new deck. Just to be sure, you understand.\"\n\n\"Of course,\" Maledicte said.\n\nHANDS LATER, Maledicte had lost sols to gain lunas and was frowning over his cards. Kritos smiled. \"My game again, I think,\" he said, laying out the elements. \"Land, sea, sky, fire, all mine,\" he said. \"Kings all.\"\n\nGilly peered into Maledicte's hands and stifled a wince. While he had a representation of all the suits, Maledicte lacked high cards enough to beat Kritos, but it was close. He had three queens to trump three kings, but the last suit, that of air, had only a scattering of seven seabirds as his high card. Not enough.\n\nMaledicte shifted in his seat, and the guards leaned closer, watching his hands; Kritos's eye narrowed. Maledicte smiled at them all, and set the cards down. \"Your fortune has waned. My hand.\" He laid down the fire cards and queen, the earth cards and queen, sea and sea queen. He set out the air cards one by one: butterflies and clouds and starry nights. The suspicion in Kritos's eye deepened. The seven-gulls card lingered in Maledicte's hand last, and Gilly shivered, wondering if Maledicte was going to try to cheat now, under all these eyes, but Maledicte merely smiled and set it down.\n\nThe queen of air, Black-Winged Ani, Her dark feathers filling the card. Kritos lunged to his feet. \"Impossible.\"\n\nBehind him, a pasteboard fluttered to the floor, unseen by the others\u2014a dark card. The queen of air, palmed to prevent Maledicte's win. But even as it fell, Gilly watched the color leach from it, the dark pinions traded for the sun-speckled backs of seven gulls.\n\nMaledicte shoved the table again, harder, knocking Kritos back against the wall, winding him. \"Are you calling me a cheat? I'm sure these gentlemen will tell you they saw no such thing.\"\n\nRising unhurriedly, Maledicte swept the coins and trinkets into his purse, and handed it to Gilly. The miniature went into his own pocket. \"Good night.\"\n\n\"Another game,\" Kritos demanded.\n\n\"No,\" Maledicte said, heading for the door. Gilly stumbled after, still waiting for the protest to go up. His back burned with the heat of it. Looking back over his shoulder, he saw Kritos watching them, his face furious, even as the house guards closed in to reclaim their stake.\n\nHe hurried Maledicte outside, and started casting for a hack, looking down the dark streets, listening for approaching hoofbeats, but heard only Maledicte's stuttering laughter.\n\n\"I thought you meant to kill him,\" Gilly said, giving up the futile hunt.\n\n\"I do,\" Maledicte said, \"and I will, but Gilly, wasn't it wonderful? Such a foolish thing to please me so, and yet the games went so well.\"\n\n\"With Ani's aid,\" Gilly said. \"Or dare you say that the transformation of the queen was a cheat learned in the Relicts?\"\n\n\"Transformation?\" Maledicte said. At the pure incomprehension in his tone, Gilly's stomach clenched.\n\n\"Never mind,\" he said. \"Let's go home.\"\n\n\"Did you see his face, Gilly? Wasn't it perfect?\"\n\n\"Yes,\" Gilly said, casting cautious glances back over his shoulder at the near-empty street, at the disreputable building they had just left. He _had_ seen Kritos's expression; fury and barely cloaked desperation. It made him leery of the distance they must travel afoot to find a hire carriage. This late, no carriage would appear on Jove Street, in the shadow world of narrow, winding alleys and blinded gas lamps, no matter how he wished it. No god listened to his will. \"We need to be gone, before he manages to convince the Fiery Hell he can best regain their money by hunting you.\"\n\nMaledicte merely shrugged, his steps still slow.\n\nGilly turned his head again. Was that a man following in their wake, sidling along the dark walls? Was that the tap of booted heels trying for silence, or the drip of water condensing against stone walls and sliding to the street? Gilly was certain of one thing only: On this street, at this hour, no one they met would be a friend. He picked up his pace, hooking his arm through Maledicte's. \"Let's not dally. We're some ways yet from Sybarite Street, and there'll be no carriages closer than that.\"\n\n\"There's an alley through the houses,\" Maledicte said. \"It'll save us the entire walk up this street and then the walk back down to the carriage stand on Sybarite. Somewhere. Ah, here it is.\"\n\nGilly pictured the streets of the city, spread out like a spiderweb, the long sweep of the seven main streets, and the nameless curling crossings that linked them all. Maledicte was right. But only the change of sound, the lack of reverberation, told Gilly that the shadow Maledicte faced was more than a recessed door. The darkness swallowed his words and fed nothing back.\n\n\"This should bring us out near Clara's, and there are always hacks there, waiting for the men to stagger away. If you're nervous of the night air, the dark...\"\n\nMaledicte turned; the quick flash of teeth told Gilly Maledicte was mocking his fears. \"Not the night, the people in it.\" Gilly peered into the alley, saw the wall curve away into velvety darkness, and yet, as he studied it, a lambent glow traveled back, carrying the flushed color and acrid scent of the additives used in Syb Street gas lamps. \"Let me go first,\" he said, and stepped into the alley.\n\n\"Why didn't I think of that?\" Maledicte asked. \"It makes so much sense to let the unarmed man go first into danger. Are you sure you're as clever as you think?\" The rasp of his voice was leavened by a breathy amusement. They headed into the alley, their boots sliding on fog-slick cobbles and unseen effluvia.\n\n\"You're going to lose that tongue of yours yet,\" Gilly warned. The light ahead grew marginally brighter, coated the dark walls with blushed contours.\n\nMaledicte opened his mouth, and Gilly said, \"I think you want us to be attacked. Just hush now, I can't hear if we're being followed if you keep talking.\"\n\n\"Someone _is_ following you,\" Maledicte said, in a carrying mutter. \"Me. How you expect to hear anything but my boots stumbling around in this muck...\"\n\nGilly turned in laughing exasperation, ready to put Maledicte before him, just to shut his mouth, and the startled inhalation of breath was all the warning he could manage to give.\n\nIT WAS ALL MALEDICTE HAD been waiting for. He stepped aside, pivoted, drew the sword. There was barely room for its length in the narrow alley; the tip scraped, unseen, against the opposite wall, and Kritos's cane impacted on the sheltering steel of the blade rather than on Maledicte's vulnerable nape.\n\n\"Not very good with that thing,\" Maledicte said. \"You couldn't hit old man Last; what makes you think you can touch me?\"\n\nThe cane slid along the flat of the blade. Beyond words, Kritos continued to grind his cane against the unseen blade. In the shadows, the blade was only an extension of darkness; frustration knotted Kritos's jaw, bared his teeth as he fought what must have seemed like invisible forces bent on his failure.\n\nMaledicte's wrist trembled with effort, but he succeeded in turning his hand, and with it, the angle of the blade. The wooden cane gave against the sword edge, splitting, and Kritos stumbled forward, roaring.\n\nMaledicte skipped back, nearly falling over Gilly as he attempted to get between the combatants. \"Don't be a fool,\" Maledicte spared the breath to say.\n\nConcentrating on the weight at the end of his wrist, Maledicte thrust at Kritos, backing him away, cursing because he, too, was hampered by this dim light. Kritos avoided a lunge with a desperate effort. Maledicte, overextended, felt the rush of warm air on his throat, saw Kritos stretching one hand out for his neck. It ghosted toward him, pale and indistinct, then clear, the fingers clenching as if Maledicte's neck were already within their grip.\n\n\"Mal, watch out,\" Gilly cried.\n\n\"Hush, hush,\" Maledicte said, recovering his balance, raising the sword and batting Kritos's outreaching arm into the wall. Quickly, Maledicte threw his weight onto the blade, the wall, and the arm pinned between.\n\nA solid slap of flash against flesh exploded by his head. Gilly had reached over Maledicte's shoulder and caught Kritos's other fist in his own.\n\n\"I can do this without your help,\" Maledicte gritted out, though a small part of him sang at the comfort of Gilly's help, reminding him of days long gone. Thinking of them, Maledicte stepped forward and brought his knee up with brutal force into the man's belly; when Kritos curled forward, Maledicte slammed his left fist into his throat.\n\nGagging, Kritos folded to his hands and knees, blood splashing where the blade had ripped at his escaping arm. He scrabbled back and Maledicte thrust the blade at the pale target of his good eye. Kritos cried out, hurling himself backward, overbalancing, landing on his back, still struggling for air. Maledicte grinned, raised the sword, heard Ani moaning Her pleasure inside him, and drove it downward. Beside him, Gilly flinched.\n\nBut Maledicte only used the sword to pin the man to the cobbles and the earth below, through one ornate and now-ruined sleeve. \"I suppose I should let you up, fight you like a gentleman. But then, the first gentleman I ever met didn't fight fair either.\"\n\n\"Mal,\" Gilly said, reaching outward.\n\nMaledicte evaded his grip, dropping to put both his knees into Kritos's belly. He fumbled in his pocket, pulled out a small purse.\n\nKritos's struggle increased. He shoved at Maledicte with his free arm, ripping at Maledicte's hair, reddening his jawline. \"You\u2014\" Kritos said, on a stifled breath.\n\n\"None of that,\" Maledicte said, dodging another blow, clamping his hand over Kritos's mouth lest anything secret slip out. But he had wanted Kritos to know him, and that the man did, at the last, when it was too late, made both him and Ani purr with delight. The loose hand tore at his face again, scrabbling. \"Gilly, get that arm, would you?\"\n\nThe feeble blows stopped. Maledicte upended the purse of wooden coins over Kritos's heaving chest. Faint glimmers of silvering still lingered on them. \"I saved them for you,\" Maledicte said.\n\n\"No\u2014\" Kritos said, true fear reaching his face, past the outrage, the bluster, and the desperation. He tried to surge upward, and Maledicte ground his weight downward, stabilizing himself by gripping the sword's hilt.\n\nDone with words, Maledicte yanked at the sword hilt, not bothering to pull the tip of the blade from its earthen sheath, but rather slanting the entire blade sideways like the closing blade of a scissors. Kritos yelled but the blade silenced him in a rush of wheezy air and blood. Maledicte put his thinly gloved hand down on the blade edge and pushed, in a step that should have seen him lose fingers, but only made the sword edge sink deeper until it ground against bone.\n\nKritos convulsed; his hands, freed by Gilly's repulsed recoil, clutched at Maledicte's forearms. Maledicte rode the spasms until they ceased, until Kritos's grip fell slack. Blood ran down the sword, pooling in the feathered hilt. Maledicte pulled his gloves off, ran his finger down the same path, chasing the blood. The dim alleyway robbed some of his satisfaction; in the darkness, the blood looked black rather than crimson, but the tang of it in the air...He stood, feeling as unsteady on his feet as a newborn creature.\n\nGilly bent and picked his gloves up when Maledicte turned away from the corpse. \"Careless,\" he said. His voice cracked.\n\nMaledicte recovered his sword and wiped it off with the muddy remains of Kritos's cloak, humming tunelessly. He broke off to laugh, giddy with satisfaction. \"That will teach him to interfere in my life. Or would, if he weren't past any lessoning now.\"\n\nGilly's face whitened until he seemed a ghost in the dark alley; he bent over the cobbles, retching. Maledicte rose hastily, took him down the alley away from the body. \"What is it, Gilly? What's the matter?\"\n\n\"I've never killed anyone before,\" Gilly said, swallowing audibly.\n\n\"You haven't yet. Don't get greedy. Kritos was all mine, and I could have done it without you,\" Maledicte said.\n\n\"You'd have had your brains dashed out on the alley wall without me,\" Gilly said.\n\n\"No,\" Maledicte said, \"But your aid was timely all the same.\" He reached up to touch Gilly's face. Gilly flinched, and Maledicte let his hand drop, scowling. \"Wallow in your conscience if you will. But remember, I told you when we set out that I meant to see him dead tonight. I am a man of my word and will after all\u2014\"\n\nGILLY CLOSED HIS EYES and ears to Maledicte's contented ramblings; he felt as if he had strayed into one of his nightmares. Behind his closed eyes, the alley felt overfull of presences and scents: the acid smell of his own sickness, the smell of turned earth and blood that lingered, and something else, something out of place in this dark world of cobbles and stone\u2014the scent of musty feathers. Opening his eyes, he saw Maledicte had returned to the body, admiring his work, saw the doubled shadow clinging to him in the low, fevered light, rough-edged like wings.\n\nGilly rubbed his face, the bridge of his nose between his eyes, and pulled his hand away, repelled. His fingers were damp and sticky, splashed with Kritos's blood. Ani's compact sealed in blood. He leaned against the wall and retched in dry miserable heaves.\n\nMaledicte came back and took Gilly's sleeve. Gilly shuddered, imagining something new within Maledicte, something hungry. But Maledicte's voice, raspy and cool, was his own. \"There's a rain barrel up ahead. Come on.\"\n\nThe shock of the cold water cleared Gilly's senses, and he washed his hands with steady fingers. Maledicte splashed happily about in it, rinsing his hands, the blade.\n\nGilly found a smile himself, a brittle thing, but Maledicte, delighted, was a charming companion, and Kritos\u2014Gilly managed to lose the iron weight that had settled in his chest, at least until the hack they found disgorged them into Dove Street Square.\n\nGilly felt the eyes on him as soon as he stepped out of the carriage. Looking around, he saw no one at first, then made out the dusty gray coat of the homeless intercessor, huddled up against the central fountain. The man nodded his head in greeting, but when Maledicte stepped out onto the cobbles, smiling, the intercessor's eyes widened and he sketched a symbol in the air.\n\nWith sudden dismay, Gilly recognized it, the inverted blessing of Baxit, the _avert_ against the god-possessed. As Maledicte passed the fountain, the intercessor disappeared into the deeper shadows with an alacrity that spoke of fear.\n**\u00b7 13 \u00b7**\n\n**G** ILLY WOKE WITH VORNATTI'S BELL ringing in his ears, and the nagging sense that there had been earlier bells still, not only Vornatti's summoning bell, but deep, resonating tones. Perhaps it was only guilt; Kritos's body could not have been found soon enough for the palace bells to bemoan the loss of one of their own. The sunlight, though, streamed dark gold across the floor, the color of midmorning. Swearing, Gilly staggered to his feet, cursing himself for staying up past the time of their return, brooding over the murder, the intercessor's fear, and dulling those thoughts with drinking. A glance at the clock in the hall warned him it was nearer noon than morning, and again he was late with Vornatti's shot.\n\nVornatti hunched, still abed, sleep-bleared and roaring at Livia. She held the loaded syringe in trembling hands.\n\n\"Give it here,\" Gilly said, taking it from her. She passed it over with a gasp of relief, and Gilly bent to find an unbruised vein in Vornatti's arm. His guilt, malleable, shifted from Kritos to a miserable wondering of how dreadful it must be for Vornatti to wake to pain while Gilly slept, babying an aching head.\n\nVornatti's teeth gritted as the Elysia slid into his vein, and then, slowly, his body started to relax.\n\n\"Late again, Gilly\u2014\" Vornatti growled, seizing Gilly's collar as if to shake him.\n\nLivia, busying herself near the door, returned with a cup of tea, hot and steaming. \"Here, sir.\"\n\nVornatti wrapped his knotted hands around the teacup, distracted by Livia's skirts, inches shorter than propriety suggested, and damped besides, Gilly thought. Livia shot a quick conspiratorial smile at him as he backed away.\n\n\"Did you hear the news, sir? Such goings-on as the milkman brought us this morning. Kritos found stripped and gutted in the alleys near the wharves.\"\n\nVornatti set his tea down, untouched. \"Last night?\"\n\n\"Yes, sir.\"\n\nIn the distant recesses of the house, the butler could be heard opening the door, and a clear, cool voice rising upward. \"Livia,\" Vornatti said. \"Go downstairs and bring the Lady Mirabile to my chamber.\"\n\n\"Here, sir?\" she repeated.\n\n\"Give me time to get him dressed,\" Gilly added.\n\nHe bent over Vornatti, undid the sash of the man's robe. \"Do you want the\u2014\"\n\nVornatti slapped him. \"I should give your job to Livia. Do you forget who pays you?\"\n\n\"No,\" Gilly said, kneeling beside the bed, head throbbing.\n\nVornatti put his hand into Gilly's hair, turned his face upward to meet his. \"Worse for drink, Gilly? My drink? While I held my dinner party, my faithful servant and my ward were where? Where was Maledicte?\"\n\n\"Lady Mirabile is impatient,\" Gilly said, slipping Vornatti's robe off. \"You wouldn't want her to find you still in bedclothes.\"\n\n\"Did he do it, Gilly?\" Vornatti said, voice low, maddened. \"Did he kill Kritos?\"\n\nGilly found a loose vest and coat that could cover Vornatti's nightshirt and settled him in his wheeled chair, hoping that the Elysia would erase Vornatti's temper, his inquisitiveness. His hands trembled as he continued dressing Vornatti, imagining what Vornatti might do to punish them.\n\nLivia scratched once on the door and ushered Mirabile in; Vornatti caught Gilly's hand and muttered, \"I'll have words for you, later.\"\n\nGilly bowed his head to Mirabile, avoiding the eager, wild light in her eyes, and fled.\n\nHer voice carried into the hallway. \"\u2014dead and by such vile means as one can scarcely comprehend\u2014\"\n\nGilly leaned against the wall, felt the silk wallpaper smooth and slick under his hot cheek, remembered the blood on his hands, and Kritos's death throes. But he also remembered Kritos coming after them, in the dark and from behind.\n\nLivia put a hand on his arm, startling him with her presence. \"He's in a right mood today, isn't he? And his party went so well last night, I'd've thought he'd be sleeping all the day.\"\n\nGilly shrugged, still irritated at the gossip she had chosen to bring. \"He's contrary. Have you seen Maledicte?\"\n\nLivia made a face at the abruptness of his tone, but answered readily enough. \"He's in the parlor with Lord Echo. Is it true Maledicte gamed with Kritos last night?\"\n\n\"Go back to your duties,\" Gilly said. He wouldn't be responsible for any more information slipping from Livia's tongue to Mirabile's ear. Livia sighed, but disappeared obediently.\n\nGilly hurried toward the parlor. Echo was not a foolish man, and Maledicte's temperament led too often to incautious words for him to feel sanguine about the two of them closeted away together. A sudden wash of guilt stopped him as he reached for the door, and he slumped against the wall instead, weakened by a pounding heart, and a face that he knew would betray him instantly.\n\nThrough the door, he heard Echo speaking. \"...realized you were acquainted enough with Kritos to play cards with him.\"\n\n\"How acquainted must one be to take another's coin?\" Maledicte said, in full court archness.\n\n\"True enough,\" Echo said, \"Still, to gamble with Kritos showed a distinct lack of caution. The man was a well-known scoundrel.\"\n\n\"You came to lecture me? Or did you have a higher purpose?\" Maledicte said.\n\nAfter a silent second where Gilly imagined Echo biting back his temper, he heard him say, \"Did you see anyone who might have wished Kritos ill?\"\n\n\"Have you been to the Fiery Hell?\" Maledicte asked. \"Even _I_ found it a veritable snake pit.\"\n\n\"You have nothing more useful to say?\"\n\n\"I so rarely do,\" Maledicte agreed, and Echo's quick-moving footsteps were the only warning Gilly had to back away from being an obvious eavesdropper. Still, the quickness of it insured that the guilt he felt written on his skin was overlaid with startlement. Echo's own temper did the rest for him; he left without further word.\n\nGilly tapped at the door and went in. Maledicte looked up from his seat, his boots propped on the hearthstone, and smiled. He set something small down on the chair arm beside him.\n\n\"Did you hear the bells this morning?\" Maledicte asked. \"A lovely way to wake.\"\n\n\"I missed them,\" Gilly said. \"Mal\u2014\"\n\n\"Vornatti?\" Maledicte asked.\n\n\"He knows now, whether he heard the bells or not. Livia brought the news, and Mirabile has brought the gossip. He's furious.\" Gilly paced, unable to settle, and picked up the miniature from beside Maledicte, holding it up toward the lamplight. \"He wants to see you immediately.\"\n\n\"And interrupt Mirabile's visit?\" Maledicte said, rising and claiming the portrait, tucking it away in his vest. \"Or worse, give her the pleasure of seeing me dressed down before her? No, Gilly. Vornatti can wait.\"\n\n\"They found Kritos in the Relicts,\" Gilly said. He lowered himself into Maledicte's vacated seat.\n\n\"Not surprising,\" Maledicte said. He picked up the blade and began stalking shadows with impatient stabs and thrusts. \"You don't think Kritos walked the streets unnoticed, do you? The scavengers would have started to work with their little knives, making sure they found all he had to hide. After the scavengers had at him, they would have discarded him as far from their territory as they could, unwilling to find Echo on their doorsteps. Probably the Relicts. The rats would have finished the job then, taking his laces and boots and hair, the pieces the scavengers left. If he'd near rotted before he was found, perhaps one of your rough sailor friends would have seen him chopped for chum\u2014\"\n\n\"Shut up,\" Gilly said, the nausea swirling in his belly, churning, mingling with his dread of Vornatti's punishment.\n\nMaledicte paused, blade inscribing an uplift in the air, a stroke like a wingbeat. \"That is where I was born, Gilly.\"\n\nThey were sitting in near silence when footsteps paused in the hall outside. Maledicte rose to drop the latch on the door, but it opened before he could do so. Mirabile came in, smiling.\n\n\"Maledicte\u2014\" she said, holding her hands out. Gilly, making himself unobtrusive, watched Maledicte's muscles shift, as if he couldn't decide between flight or fight.\n\n\"Lady,\" he said, his voice neutral.\n\n\"So formal?\" she asked, her tone arch. \"When we are to be family soon?\"\n\nMaledicte stiffened like an affronted cat.\n\nMirabile trilled laughter, and held out her wrist again, showing the bright band of gemstones circling it. Emeralds, Gilly thought numbly. The old bastard's given her emeralds to match her eyes.\n\n\"From your so-generous guardian,\" Mirabile said. She leaned closer to let Maledicte admire the bracelet. \"I expect the rest of the set when I return from the country. The necklace, the earrings, the ring...\"\n\n\"You're deluded,\" Maledicte said, \"a woman who cannot discern the difference between a bridal gift and a whore's trinket.\"\n\nMirabile's green eyes darkened, but before she could speak, Maledicte continued, his voice as delicately acid as it had been the night he faced DeGuerre. \"Or are you telling me he offered for your hand? That Vornatti propelled himself from his chair and to his knees before you? I think not.\"\n\nHer smile held with effort. \"You had best watch yourself, Maledicte. He is most displeased with you, and I am as elegant and as lovely as you.\"\n\n\"But so much older,\" Maledicte murmured, and Gilly winced.\n\n\"I see you're in no civil mood,\" she said. \"Perhaps your guardian's lecture will teach you to mind your manners.\" She dropped a tiny curtsy toward Maledicte and turned, flirting her skirts as she left.\n\nLooking after her, Gilly missed Maledicte's first words, but the sullen rage in them reclaimed his attention. \"Did you buy that bracelet for Vornatti and not tell me?\"\n\n\"No,\" Gilly said, hastily. \"No, Mal.\" At the black anger in Maledicte's eyes, Gilly stifled his next words\u2014that likely Livia had done it, and pocketed herself a few coppers as well.\n\n\"Think you that he favors her enough for marriage?\" Despite the rage cording his throat, tightening the silken lines of his cravat, Maledicte's voice was quiet.\n\n\"You said it yourself\u2014it's a whore's gift, a trinket he can use to keep her dancing to his tune. And a small price to pay for raising your doubts, I'd lay wager. He's angry, Maledicte. Not an idiot.\"\n\n\"If he wanted to wed Mirabile, he'd have to be, wouldn't he?\" Maledicte said, the flush leaving his cheeks, the tension slackening from his hands. \"A woman who's murdered one husband already and who plots to cuckold her second before the marriage lines are even written.\"\n\n\"Still,\" Gilly said, \"Best go to him now, and take your punishment. Soothe him if you can.\"\n\n\"Soothe him?\" Maledicte said. \"He gave her jewelry.\"\n\n\"He's so angry, Mal, please. Last time, I ended in the stables, while you only had to serve him as I do. He's angrier this time, and I always pay for it....\" He trailed off, unsure where the bitterness in his voice had come from.\n\nMaledicte's eyes widened and then he said, \"I'll go to him at once.\"\n\nVORNATTI'S CHAMBER SEEMED SUBDUED, AS if the death bells had shocked it to stillness. Maledicte looked at the room with new eyes, eyes that were looking into the uncertain future. Vornatti was a scarecrow of a man, hunched in a wing chair, drawn up before an unlit fire, dozing, but Maledicte knew better than to presume helplessness. Maledicte looked at the bed, plush heaps of featherdown and velvet and linen, followed the line of the posts up to the ceiling and its obscene fresco of fornicating cupids.\n\nHe found himself amused at Vornatti's unflagging concupiscence, at the determination that filled the man's hours. The smile was fragile, though. Though Maledicte flouted Vornatti's strictures, gave in to his own whims, still he dreaded displeasing Vornatti too greatly, wary of the man's vindictiveness and temper.\n\nVornatti woke, coughed, then said, \"Boy, come here.\"\n\nMaledicte turned. Surely he had done this already, the slender youth, the sword, an old man's lures. He dropped the sword on the cluttered bed chest, beside the potions and formulas, the shaving soap and scent. A bottle tipped with a crystalline ring, but kept its stopper. Maledicte crouched beside Vornatti. \"What do you want, Vornatti?\"\n\nThe baron's dark eyes fixed on Maledicte's upturned face. \"You killed Kritos.\"\n\n\"I did,\" Maledicte said. A denial would only feed Vornatti's anger. _Soothe him,_ Gilly had pled.\n\nTo that end, Maledicte stood, slipping off his coat, his boots, the stiff, brocaded vest, undressing piece by piece.\n\nThe baron's eyes softened a little, anger tempered by a more familiar appreciation.\n\nMaledicte settled gently in Vornatti's lap, resting his head against his shoulder, as falsely obedient as a young wife, allowing Vornatti's hand to slip along his thigh. \"Echo seems unsuspecting.\"\n\n\"Kritos was a fool and a bad gambler; such men come to bad ends routinely,\" Vornatti said, absently. His knotted fingers slid upward, traced circles over Maledicte's hipbones.\n\n\"See there, no harm done. Don't begrudge me Kritos's death. In turn, I'll\u2014\"\n\nVornatti put a gnarled hand to Maledicte's lips. \"A bargain, my boy.\"\n\n\"We struck one already. In the library of your country home.\"\n\n\"I am not doddering, Maledicte. I remember our agreement.\" Vornatti laid his hand on Maledicte's head, holding it to his shoulder. \"I was wrong.\"\n\nMaledicte waited, heart pounding, wondering whether Vornatti intended to be rid of him. Despite Gilly's words, Vornatti didn't seem angry, and that raised nervous hackles on Maledicte's neck. The old man plotted.\n\n\"You can only lose in this quest of yours. You will always be the outsider, always an object of suspicion, and they will turn on you without hesitation. Better I had let you stab Last in the back than teach you to think of honor and nobility. Antyre is not Itarus to admire the cunning of those trained like assassin princes.\"\n\nMaledicte nipped the fingers so near his mouth, and Vornatti pulled his hand away. \"Is that what you have taught me? And here I called your lessons vice.\"\n\nVornatti chuckled. \"Ah lad, your wicked, disrespectful tongue.\" The humor faded from his face, draining to melancholy. \"I am an old man, Maledicte, and I find myself prey to an old man's most insidious and foolish disease. I would keep you, your wicked mouth, your liquid eyes, your tempers and tempests, only for myself. Keep you mine alone.\"\n\nDisgust flared in him, twisting his lips. Stay here? Limit his touches to Vornatti's withered flesh when Janus awaited? Maledicte started to rise, patience gone, blood drumming in his veins; Vornatti tangled a fist in Maledicte's hair, sent him to his knees by his chair. \"Listen to me, boy. Bide your time. Last will keep until you are better established, or until I am gone and cannot watch you fall. My name grants you some safety but not enough for a direct attack. Such can only end in death or prison. Stay with me. Continue as we have been, and I'll make it worth your while. You know you can trust me to keep my word. Haven't I kept your other secrets safe, my _girl_? Stay and I'll reward you. Make you my ward in truth. My heir.\"\n\n\"Do you think I can be bought?\" Maledicte asked, fisting his hands in his lap. The sword slid from the chest with a protesting scrape and fell to the rugs below.\n\n\"Haven't I bought you once already? Now, all I'm buying is your time. You're a young...man. Last, curse him, is a healthy man. And I, Maledicte, am an old man. My blood fails beneath my skin, but even dying men have favors to bestow. Wait and you'll have money enough to escape from Antyre when they turn on you, teeth bared and bloody.\"\n\nMaledicte said, \"You swore you hated Last.\"\n\n\"And I am content to know you will destroy him. I have no need to see it. Perhaps my vengeance should have been taken when my blood first burned. However it occurs, that fire is cold now and I'd exchange chill for warmth. Yours would be preferable. If you persist on your impatient course, I will make do with Mirabile, and set you back to the streets.\"\n\n\"You speak to me of vengeance fading, yet you would have me balk and delay? I have not that luxury. You had not the spur to act that I have.\" Maledicte escaped from Vornatti's anchoring hand, seized up his shirt, and shrugged it on.\n\n\"Janus,\" Vornatti growled.\n\n\"Ani,\" Maledicte countered to Vornatti's scoffing laugh. He had woken, newly aware of Her, Her contentment that he had sealed their compact, satisfaction that more blood would be forthcoming. Vornatti's prattling of delay made Her shift like a snake coiling to strike.\n\nVornatti leaned forward in his chair, hands clutching the padded leather armrests. \"Forget the boy; he has surely forgotten you. What can he offer you? He's only a bastard nephew to a dreaming king. He'll never be earl, never inherit. Last will see to that, count on it. And he has never attempted to find you. Your desire is one-sided, boy. Stay your hand.\"\n\nMaledicte took refuge in the inane persiflage of Vornatti's favorite literature. \"Why sir, this is all so sudden.\" The acid snarl to his voice removed all humor from the words.\n\nVornatti's eyes squinched, peering at him. Maledicte stepped back farther into the shadows, out of Vornatti's sight, given over to an uncontrollable shaking. He trembled like a spooked horse, from head to toe, while he thought. It was too soon to dispense with Vornatti's patronage\u2014and inciting his wrath so near the solstice ball\u2014\n\n\"Is that a refusal?\" Vornatti asked. \"Casting you back to the streets not enough? I could expose you first, girl. Or cast Gilly out alongside you. He's begun to bore me anyhow, and I know others who seek his services, though they would not treat him as kindly.\"\n\nMaledicte's shivering ceased as quickly as it had come, his composure restored. \"Won't you even grant me the time to think on it? The heroines in your novels always have time to think on it.\"\n\n\"You're no heroine,\" Vornatti said.\n\n\"And you're no gentleman.\"\n\nVornatti laughed. \"Stay or go. Yes or no, Maledicte.\"\n\n\"Damn you,\" Maledicte said, plunged away from Vornatti as if he would flee, but then returned. \"Yes, damn you.\"\n\n\"As greedy and as fickle as I thought. After all, Janus is nothing but a boy you no longer know.\" Vornatti leaned forward, took Maledicte's hands. \"Thank me, boy. You've learned something it has taken me years to learn: We all outgrow our pasts. Now kiss me and cry friends.\"\n\nMaledicte kissed his dry cheek, amazed that the choked rage within him wasn't enough to scald Vornatti's skin. \"Shall I send for Gilly, let him dress you for court?\"\n\nVornatti said, \"Call Gilly by all means. We'll let him know you're staying. But first\u2014\" He drew Maledicte to his lap again, slipped the shirt away.\n\nMALEDICTE SEETHED QUIETLY while Vornatti tugged at the bell rope, his face carefully controlled while Gilly heard the news. He raised his eyes to see the expression on Gilly's face: pure, unadulterated alarm. But then, Gilly believed in Ani's presence, and Vornatti, fool several times over, did not. There was no future but vengeance for him.\n\nMaledicte ascended the stairs, turning the gas lamps down as he went, leaving a smothering trail of darkness behind him, hoping to balk Gilly in pursuit. Though it had been his choice to stay, the easy temper in his blood also blamed Gilly.\n\nBut Gilly, with longer legs, caught him at the first landing, seized his shoulder. \"What are you planning?\"\n\n\"Don't,\" Maledicte said, Ani already an angry presence in his blood. To be manhandled was more than he could stand. Heedless, Gilly shook him. \"Tell me why you agreed to put off your vengeance.\"\n\nMaledicte put his hand on Gilly's chest, shoved him away without effort, watched Gilly fly back and hit the far wall. Maledicte's hands shook, a resurgence of the eager trembling that had beset him in Vornatti's room, spreading over his entire body.\n\n\"Maledicte?\" Gilly said, rising, caution on his face.\n\nMaledicte slid down the wall, crouched in the shadows, ashamed of himself. \"Who else would I be?\"\n\nThe name hung in the air between them. Gilly hesitated, then dropped to his knees beside Maledicte. \"Lean on me. I'll help you upstairs. I'll bring you up some milk, warmed and scented with vanilla and almonds.\"\n\n\"As if Ani can be cured like a cold, or Vornatti's caresses made sweet,\" Maledicte said, acid in his voice, then in a different tone. \"Thank you, Gilly, my gentle Gilly....\" He lapsed into silence as they made their way up the dark stairs, Gilly looking back over their shoulders, as if he expected to see Ani sweeping after them.\n**\u00b7 14 \u00b7**\n\n**O** N THE EVENING OF THE SOLSTICE BALL, Gilly and Maledicte found themselves part of a line of coaches, wending their way through the city streets to the palace at so slow a pace that noblemen sauntered from coach to coach, visiting, chatting, flirting, admiring costumes. From his perch on the driver's bench, Gilly watched it all, and couldn't help but contrast the general giddiness with his passenger's stillness. Costumed forlornly as the Heartsore Chevalier, that tragic figure of legend, Maledicte drew the coach curtains whenever nobles drew near, sulking into silence.\n\nGilly hated the costume, the sleek white wrappings of vest and coat and pants, hated the crimson touches at wrist and neck; most of all he hated the expression in Maledicte's eyes, as if this moment might be too much to bear.\n\nGilly's nerves were strung tight enough as it was; hadn't they left Vornatti home, lost in a drugged sleep when he had meant to attend? Maledicte's doing, of course, and done so swiftly that Gilly had not understood until he tried to rouse Vornatti. His remonstrance had died when Maledicte turned on him, raging. \"Do you think I could have borne it? Hunting Janus with Vornatti draped over my skin? Touching me as if he had my welcome?\"\n\nAt the ballroom, Gilly tossed the reins to a waiting stableboy, and opened the carriage door. Maledicte stepped out like a ghost, one hand on the sword.\n\nMaledicte started up the stairs with Gilly behind, and paused at the great Book of Names. On the last page, recently scribed, his goal was marked.\n\nJanus Ixion, Lord Last: the name was scrawled with such black finality that Gilly was not surprised to see the next names rough and surrounded by splatters. Janus had destroyed the nib.\n\nMaledicte touched the ink with his gloved fingers. The ink sank in, still wet, black staining into the red silk gloves. Maledicte wiped his hand against his mouth, shoved past Gilly, and escaped into the night air, past the cloying sweetness of heaped violets and jasmine, lilies and heliotrope, and the slow-burning haze of beeswax candles. Gilly found Maledicte outside, pacing beside the ivy maze.\n\n\"Mal\u2014\"\n\n\"It stinks in there, like rot. Do you think anyone has ever told Aris that crushed flowers smell like a grave?\"\n\nGilly held out a gentling hand.\n\nMaledicte turned, retched into the leaves. Gilly stepped back when Maledicte looked up. His eyes were wild. His hands shook like those of a man with fever, and his voice trembled. \"Comes the moment when everything changes. This idyll dies, and it has been an idyll, hasn't it? Even with Vornatti's tempers and demands and threats? I'm feared to see it end.\" He stretched up and pressed his lips against the drawn corner of Gilly's mouth.\n\nGilly could smell Maledicte's skin, scenting faintly of lilac, could feel the smoothness of Maledicte's cheek against the stubble rising on his own. \"Feared of Last? Of Janus? I won't believe either with the course you've set.\"\n\n\"Not them. Last is a dead man, and Janus is neither alive nor dead 'til he speaks. I fear myself, Gilly, the brush of feathers in my mind. If Janus spurns me\u2014Her feathers urge me to darker hungers, and Her wings smell of death and bloody iron.\" He hid his face in Gilly's neck, but shied away when Gilly reached up a comforting hand.\n\n\"If Janus spurns me, or remembers me not, there will be nothing left of me. Only Ani's puppet. But still, there is no going back.\"\n\n\"Would you go back, if you could?\" Gilly asked, voice rough.\n\n\"No,\" Maledicte answered immediately, without the need for thought, his eyes black and very cold. \"Why are we standing in the dark, Gilly, when the ball awaits? I've killed one Last already. Let's see how many this night holds.\"\n\nGilly followed him inside, where the nobles' ballroom had been doubled with the drawing back of the barriers between the king's ballroom and theirs. While the nobles' ballroom was painted in washes of blue and dust, Aris's ballroom was all rose and gold, and so the dancers whirled from twilight to sunrise. Maledicte walked on, unaware, his eyes flicking from one bare face to the next. The Bright Solstice required costumes but not masks; those were saved for winter's Dark Solstice, where one wore masks to shield identity from the hungry dead.\n\nGilly paced beside him, looking for a man he'd never seen, but felt sure he would recognize. As the moments passed, and Maledicte's expression grew fixed, Gilly whispered, \"Follow the gossip, the bent heads. A newcomer leaves such in his tracks. It will lead you to him.\"\n\nMaledicte granted Gilly a shaky smile, then stiffened like a hound on scent. Gilly followed his gaze.\n\nThe young man entering from the balconies could only be Janus Ixion; he was the butter stamp of Last, pale-eyed, gilt-haired, tall, and broad-shouldered. The brief impression Gilly had gleaned from the miniature had been of a vapid nobleman, but he had assumed it due to an artist overeager to please Last.\n\nBut the reality was no better; Gilly felt disappointment turn his stomach. This was the face that had driven Maledicte so far? This was Janus, this elegantly draped figure in blue velveteen and gold? His face was as empty as that of any longtime court rou\u00e9. Where Maledicte still carried a rat's wariness, Janus seemed pampered from birth, the perfect son of an aging aristocrat, with an expression as devoid of intelligence as it was of interest. Here, in the glittering heart of Antyre, Janus conveyed only boredom.\n\nGilly felt a shiver in the air, turned. Maledicte was no longer at his side, but had disappeared while Gilly gaped. He caught a glimpse of him, moving along the perimeter of the room, following in Janus's idle path like his shadow.\n\nGilly hissed, watching the game begin. Cat catch mouse, with both men playing. Janus was aware of his shadow, acted the complicit mouse, limited himself to small turns of his neck and head, trying not to catch sight of his pursuer. If he spotted the shadow too soon, too obviously, the game would end, and interest and amusement sparked in those incandescent blue eyes, livening the mask of his face.\n\nThe court grew progressively more silent, watching in avid delight.\n\n\"Must we continue with this roundaboutation?\" Janus spoke aloud, his voice laced with amusement, though he had yet to acknowledge his pursuer with even a glance. In his voice, Gilly heard the same careless arrogance that drove Maledicte's speech, but layered in tones like sculpted velvet. \"I'm but new come to this court, and fail to see how I have erred in your opinion. If I have offended your sister, mother, lover, I apologize. If it's other than that, let it wait. We have affrighted the musicians to silence.\"\n\n\"They sounded like cats strangling anyway, and I should know,\" Maledicte said. His raspy voice was shocking after Janus's polished one.\n\nJanus, startled, turned to confront his shadow, and the amused smoothness of his face shifted. Even Gilly, standing so near, could not name the sentiment, the emotion fleeing too quickly to identify, like a ripple over deep water and gone.\n\nJanus took a step toward Maledicte; the courtiers, the maidens caught between slipped away, and the whispering court found their eyes drawn not to either man but to the emptiness between them, the nexus of space that slowly closed.\n\nMaledicte took another step. His face was as pale as his shirt.\n\n\"Have we danced enough?\" Janus said. \"So come, then, declare yourself and have at me.\" His lips stayed parted after his words; his face tightened as Maledicte took the space between them and made it an illusion, not the impenetrable barrier it seemed.\n\n\"Janus Ixion\u2014\" Maledicte said, at the heart of the circle. His voice caressed the syllables, and again that flicker of emotion swept Janus's face.\n\n\"Lord Last,\" Janus said, dropping into a bow, his golden hair sliding, gleaming over sky-blue shoulders.\n\nThe sweeping arc of the black blade stopped his descent. He tilted his head up, pale throat like marble. \"Not in the mood, hmmm?\" Janus stood straight, spread his arms. \"Have at me then; I will not fight you.\"\n\nMaledicte wavered, visibly unable to move forward or back. Janus's arms closed; he caught Maledicte's wrist, his other hand caught the shoulder of Maledicte's embroidered coat and drew him closer. Then he released Maledicte's sword hand, all as smoothly done as if it were only the steps of a dance and not a potential duel.\n\n\"Will you strike me?\" Janus asked. His voice, which so far had been pitched for the horrified, fascinated, scandalized audience, dropped to a husk. There was the faintest sound of despairing entreaty in his words, as if Maledicte's enmity was too heavy a weight to shoulder.\n\nThe black blade shivered in the light, a shadow chased by candle flames, moving. Falling. It clattered to the marble floor and Janus smiled. He slid his hand over Maledicte's shoulder, into the dark hair, and bending close, put his kiss first on Maledicte's mouth, then on the silk-covered throat. Maledicte threw his head back, in a movement as voluptuous as any woman's.\n\nJanus murmured something too low for the riveted crowd. Gilly strained, but even his clever ears missed the word. A name? A prayer?\n\nWhat Janus said, of course, in his crushed-velvet voice, was _Miranda._\n\nThe silence faltered as whispers broke over the court like the tide. Janus stepped apart from Maledicte, dropped into a bow again, elegant and courtly. Maledicte returned it after a moment, and where Janus's bow was all Antyrrian languor, Maledicte's carried the stiff perfection of Vornatti's teaching. Maledicte spoke a few quiet words, drowned in the hiss of the court, and turned away.\n\nThe courtiers flooded inward, erasing the stage Janus and Maledicte had created with their presence; scandalmongers sailed from one side of the room to another, tongues preparing to wag. Trying to follow who might have the most dangerous words to spill, Gilly lost sight of Maledicte in the mass. A faint whisper in his ear, a quick scent of lilac, and Maledicte slipped by him and disappeared with an eerie grace. Gilly turned, trying to track him, and instead caught sight of Mirabile standing, frozen, her face a mere mask. Shock, Gilly thought, and worse\u2014 _betrayal._ Janus fit nowhere in her plans for Maledicte.\n\nStill near the epicenter of the storm, Janus accepted a glass from Westfall's hand, smiled his thanks, and headed toward the balcony doors.\n\nHis cue, Gilly knew. Maledicte's command ghosted through his mind. _Show Janus to the carriage. We're stealing him away._\n\nExcept Gilly could think of nothing he would like less than to take Janus through the romantic tangles of the king's maze where he had walked with Maledicte. He told himself it was relief that Maledicte was not launched on his erasure of self, his bloody vendetta without care for his own life.\n\nA quick movement, checked, drew his attention to the dais, to Aris staying the Kingsguard in their search for Maledicte, and wearing a fine, high flush on his cheekbones. Anger, Gilly feared. The king's eyes shifted to meet his. Gilly dropped his gaze immediately, caught staring at the king like a country fool. But more disturbing to his composure was the unwilling recognition of their shared emotion, the bite of unreasoning jealousy.\n\nThe voices of the court were roaring now, the musicians fighting to be heard, belaboring their instruments to make up for their earlier silences. Gilly, making his way out, collected comments like tiles from a mosaic. \"Sword in the court. Again. And yet Aris does nothing\u2014\"\n\n\"Last will not be amused that his son set us such a scene.\"\n\n\"It's enchantment, I tell you.\" Mirabile's face was livening finally to well-controlled rage, taking Maledicte's actions as an affront to her own charms. \"First he claims Aris's approval and now his nephew's. But _how,_ is the question. I have seen things he would not like me to tell, an altar, books of spells...\"\n\n\"A devotee of the dead gods?\" her hearer, Micah Chalefont, sneered. \"Only fools believe in them.\"\n\n\"Fools deny the evidence of their own senses. I am not so witless,\" Mirabile said, snapping her fan closed. The certainty in her voice stifled Chalefont and made Gilly hasten his steps.\n\nHe found Janus waiting silently in the dimness; when Gilly approached, those blank blue eyes showed only disinterest. \"This way,\" Gilly said, gesturing into the maze, and Janus's eyes burned with eagerness.\n\nMALEDICTE PULLED HIMSELF from Janus's embrace, settled on the opposite side of the carriage, leaned forward and knotted Janus's hands in his own. He laughed soundlessly, his constricted throat unable to voice the emotion. \"I dreaded this moment, feared I'd never see you again. That you wouldn't know me if I did...What a fool to forget how we fit together.\"\n\n\"I'd know you anywhere, Miranda\u2014\"\n\n\"No,\" Maledicte frowned, a fleeting thing, displaced by joy. \"Miranda's dead. Murdered in the Relicts like the rat she was.\"\n\n\"That's as Roach told me. That you were dead, and at my hand. Absurd. As if I could ever harm you....\"\n\nMaledicte was glad of the dimness of the carriage, the swaying that shadowed their faces and granted them a sweet intimacy, all too aware of the flush on his cheeks, the rush of pleasure at Janus's careless words. \"You went back?\"\n\n\"Soon as the _Kiss_ docked. Soon as I could convince Kritos that the voyage had made me ill. The only familiar face I found was Roach.\" Janus laughed and kissed Maledicte with a greedy mouth. \"And here we are.\"\n\nTheir noses bumped, their foreheads jarred each other's when the carriage bumped over rough stones. Janus, distracted, peered out the window at Dove Street passing by with its line of tall, trim houses and sculpted lawns. \"Where are we going?\"\n\n\"My town house.\" Maledicte slid on the seat, pushing his back into the plush warmth of the cushions, his fingers slipping from Janus's grip, then leaning forward again, unwilling to let go.\n\n\"This isn't just some raid on the nobles' court then?\"\n\n\"I told you. Miranda's dead. Despite the masquerade, I am much as you see me. I am Maledicte, ward to Baron Vornatti, a courtier and not a lady.\" Maledicte gave a little half bow, constrained by the seat. \"I warn you now. I am a known entity in the court, and you've undoubtedly blacked your reputation tonight.\"\n\n\"You always were the clever one, Mir\u2014Maledicte, was it? Quite a mouthful, love. But tell me, do you know that Kritos is dead now? Do you know that?\" Janus said, settling back. \"Struck down by an unseen hand, left for rat food.\" A smile played about his lips, the same secret communication in his eyes that Maledicte had missed so sorely.\n\n\"He took you from me. Should I have let that go unrevenged?\" Maledicte said. Ani grumbled beneath the joy in his blood, and he said, \"But let's not talk of vengeance now.\" He put himself into Janus's lap and kissed him again and again.\n**\u00b7 15 \u00b7**\n\n**M** ALEDICTE SECURED HIS BEDCHAMBER DOOR while Janus's teeth teased his nape. Janus raised his head, looked around at the sumptuous room, at the fireplace, still faintly red with burning coals, the wide chair beside it, the tall windows with their heavy crimson drapes, the plush, high bed. \"Your baron treats you well.\" The question lurked in his tone.\n\n\"For a price,\" Maledicte admitted, then skirted the pointed truth for a few lesser ones. \"The keeper of my secrets, and my own personal blackmailer. If I displease him, he threatens me with exposure in the court.\"\n\n\"Would that be such a dreadful thing?\"\n\n\"I would be ruined. A woman in Vornatti's house, unchaperoned? A woman with a sword? Not even Vornatti's novels tell such audacious tales. Besides,\" Maledicte said, \"I'm rather attached to my persona and the freedom it brings.\" It was all the explanation he could give on the matter, and he shivered, wondering what Janus thought, to find Miranda in a circumstance she had always sworn she would never be in.\n\nJanus kissed the silk cravat on Maledicte's throat once more, then loosed the knot with careful fingers. \"They truly believe you a man? With your sleek skin and smooth throat\u2014\" He traced the bared lines of chin and neck and collarbone. \"I expected a scar to match your voice. It is not an affectation.\"\n\n\"No,\" Maledicte said, \"stonethroat.\"\n\nJanus kissed the pale skin at the base of Maledicte's throat, touched his tongue to the hollow there, a tiny flicker of warmth that sparked likewise in his veins. \"Such a gamble.\"\n\n\"Risks are necessary when one seeks a prize of inimitable value,\" Maledicte said, opening Janus's shirt, finishing the job begun in the carriage, easing the stiff, formal vest off, then the silken lawn. In the low-lit room, Janus's skin gleamed like gold, and heated Maledicte's blood like fine brandy. Maledicte shoved him toward the wing chair, and once Janus was seated, pulled off his boots, glad beyond measure that Vornatti seemed to have faded from Janus's thoughts. Maledicte looked again at Janus, at the lazy, hungry expression in those familiar eyes, and gave himself to the moment\u2014more, to their future, for the first time feeling he had more than a tentative grasp on it. He found himself smiling again.\n\nJanus tugged him into his lap, unbuttoned Maledicte's shirt, slipped the silk away, and paused, amused. \"A corset?\"\n\n\"Padded,\" Maledicte said, touching his sides, his belly, his back. \"Throughout here, to bulk up my waist, flatten my breasts. This masquerade would be less successful otherwise. Unhook me.\" He presented the elaborate back to Janus's waiting hands.\n\n\"How do you manage to lace this by yourself?\" Janus laughed. \"Every lady I've met needed a maid or a man.\"\n\nMaledicte moaned as the hooks parted with faint pops and the corset fell free. \"Practice. Necessity. And it needn't be as tight as the ladies' corsets. It needs only to disguise,\" Maledicte said into Janus's hair, nipping at his ear-lobes.\n\nJanus slid his hands around the exposed narrow waist, the curve of spine and hip, the small, soft breasts. Maledicte sighed, and leaned into his hands, arching back as Janus kissed each tender tip. Things did change, he thought a little deliriously. Janus had changed, grown taller, broader, stronger, harder. Maledicte basked against him, the scent of his skin rising over the court colognes, and smiled hungrily.\n\nHe stroked his hands down Janus's ribs, the sleek, muscle-padded heat of them, trailing light fingertips down Janus's belly. Janus's breath grew a little more rapid, and Maledicte chose to slide away from a more intimate touch, even as Janus shifted his hips toward his hands. A delightful thought occurred: This time, they had all the time in the world\u2014no stolen moment made fragile by Ella's importunate callers, by Celia's drunken rages, by Roach's jealous dogging of their heels. This time was theirs alone, and he intended to savor every moment, to relearn the feel of Janus's skin against his own.\n\n\"What's this?\" Janus asked, stopping his caresses to touch the red lines on Maledicte's left arm and side.\n\n\"Sword strike from the Marquis DeGuerre.\" Maledicte studied it again, briefly bewildered by a history not shared.\n\n\"And this one?\" Janus traced the long, serpentine scar that wrapped her left hipbone, veered around her back, and licked the base of her right breast.\n\n\"Whip,\" Maledicte said. \"Kritos, in the Relicts.\"\n\n\"Bastard,\" Janus muttered, bending to kiss the upward curl of the weal, a wash of breath and heat that made Maledicte gasp, draw him closer.\n\n\"Just another dead aristocrat now,\" Maledicte said, clutching Janus's shoulders as his lips left the scar and moved down the pale, soft skin of Maledicte's belly.\n\nJanus untied the laces in Maledicte's hair, setting it tumbling free, framing Maledicte's face and shoulders in whispery tendrils that made him shudder with sensation. \"None so blind...\" Janus murmured. \"You look like no man I've ever seen.\"\n\nMaledicte preened. \"Gilly calls it vision driven by expectation.\"\n\n\"Gilly knows?\" Janus asked. His lips paused in their brushing over Maledicte's skin, tightened in a frown.\n\n\"No,\" Maledicte said, flushing at the very thought. \"The fewer to know this secret, the fewer to tell it.\"\n\n\"Are you so sure he is unaware?\" Janus asked. \"If he has mentioned\u2014\"\n\n\"He was referring to cheating at cards, and simple sleight of hand. He has not thought to apply his rule to me, I assure you.\" Maledicte ran trembling fingers through Janus's hair, admiring the sparks of sunlight captured in the golden strands.\n\n\"He's aided you, acted your partner,\" Janus said. He slipped away from Maledicte, paced across the room to poke at the coals, sending sparks upward in swirls of angry heat.\n\n\"My partner on my path to retrieving you,\" Maledicte said. \"You must be my partner now as well.\"\n\n\"Like secrets, partners are often best kept to two,\" Janus said. \"Vornatti knows\u2014\"\n\n\"Janus,\" Maledicte said, impatient with the subject, only aware of the flickering firelight over Janus's skin, and the answering heat in his own. \"Come here and free me of these boots and breeches, unless you fancy taking me as if we were gentlemen in the stables, fumbling and baring only what we must.\"\n\nJanus grinned, mood sweetened. \"Another time, perhaps.\" He knelt, and tugged Maledicte's boots off; he ran his hands up the thin leather of Maledicte's breeches, began sliding them down, lingering to kiss the inside of his bared thighs. \"Very nice legs...for a gentleman of the court.\" The words tickled against his skin, made him shiver, made him writhe.\n\n\"I admit the court finds me a rather girlish young man in appearance.\" But his good spirits chilled. First Vornatti, now this. He felt that every moment exposed pitfalls he hadn't imagined, every moment revealing a threat to this fragile joy. He touched Janus's mouth, halting further banter, and stepped out of the entangling leather. \"But Janus, they do believe me a man, and while it is understood, so says Aris, that some men have appetites only for their own, it is not a fashionable thing. And I will not give up this role. I am Maledicte now, and so I think myself male, all evidence to the contrary aside.\" Maledicte gestured, encompassing bared flesh. \"Your reputation may suffer if you are seen in my company overmuch.\"\n\nThe last words were pained; only now did Maledicte realize the trap he had laid for himself. To become female again was unthinkable, and yet his guise could cost him Janus.\n\nHe turned, studied himself in the mirror, distracted from worrying in the shock of self-exploration. It had been so long since he had taken the risk of loitering unclothed, or even thought of himself as Miranda; though he had all her desires, her dreams, he spoke truly to Janus when he declared her dead. Maledicte could not put himself back in her position, could not re-make time, unable to remember how it felt to not carry this secret.\n\n\"I suppose I should be grateful I don't resemble Ella,\" Maledicte said, \"or this rebirth would have been impossible. My lines are more male than female.\"\n\nJanus laughed, snaked an arm over her shoulders; she shivered in relief and want at the sight of his form alongside her in the glass. \"You are blind yourself. You are barely taller than most women of the court. Your voice is your most believable attribute, but this\u2014\" He cupped one breast, then the other, stroked his fingers over her nipples, making them stiffen. \"This is purely female.\"\n\nMaledicte's heart raced; he leaned back against Janus's chest, playing now, directing his attention. \"My hips are not broad enough.\"\n\nJanus slid his hand downward, splaying his fingers down Maledicte's belly, lingering, teasing, his voice furred by desire. \"The women of the court wear corsets, boning, bustle, and padding to make them shapes different from their own.\"\n\nHis fingers slipped into the warm cleft of her thighs, moving in gentle patterns, growing warm, her skin growing slick against his touch, and Maledicte trembled. \"Do you still think yourself male?\" Janus whispered. \"Do you fear I will leave you at the say-so of the court? What care I for their approval when I have you in my arms again?\"\n\nAgain Maledicte flushed, the pale skin staining pink over cheeks and throat. Maledicte turned in Janus's arms, kissed him as if to devour the taste of him. Maledicte guided Janus's steps until he tumbled backward onto the bed, a golden expanse over rich crimson. Maledicte crept up Janus's body, touching, kissing, tasting, with the same greedy, gloating hunger a starving man mustered for a sudden feast.\n\nJanus arched his body into a bow, let Maledicte slide his breeches off and to the floor. Maledicte nestled warmly between his legs and allowed himself a leisurely reacquaintance with Janus's body. Measuring tongue tip by tongue tip how Janus had grown, how he continued to do so, until Janus gasped and strained against Maledicte's teasing kisses. Janus drew him up, and they tangled, each trying to map the other in touches and kisses and the shiver of skin against skin. Janus licked the shell of Maledicte's ear, stirred gentle fingers through her heat. In return, Maledicte lapped at Janus's throat, tasted their mingled salt, and chased the taste up to his jaw. Janus obliged him, tilting his head back, and then erupted into choking laughter. Maledicte raised his head.\n\n\"There are some quite perverse cupids watching,\" Janus said.\n\n\"Vornatti's obscenely fond of them. He had them commissioned for every private room of the house,\" Maledicte said, flinching even as Vornatti's name slipped his lips.\n\n\"Vornatti,\" Janus breathed, catching Maledicte's hands, stilling the caresses. \"What are we to do with him? I gather he will not be pleased to share you. Come to think on it, neither am I.\" Though the words were indifferent, the tone was not.\n\nAgain, Maledicte hovered on the brink of explanation. Again, he slid away\u2014what could he say? That to regain Janus, Miranda would have done far worse than forswear oft-stated avowals and barter her body? Surely Janus knew that already. So instead of an explanation of how this came to be, he found himself murmuring an explanation of why it would continue, sweetening the sting of it with a meandering touch that teased nipples, traced ribs, delved into his navel, and wrapped warmly around his shaft.\n\n\"I have no name of my own, no funds that he has not granted. To leave him would be to leave with nothing save what we could carry.\"\n\nJanus laid his hands over Maledicte's, slowing the pleasure so he could find words. \"I have no funds either, save for what I won from Kritos. And we can't risk your exposure, so you say\u2014still, something must be done. You killed Kritos....\" Janus lay back, rested his head on his hands, silently urging Maledicte's caresses to resume.\n\n\"A gambler with a multitude of foes,\" Maledicte said, frowning. He traced swords and feathers across Janus's skin.\n\n\"One old man should prove little challenge. Especially one who grants you such access to his person.\" Janus's jaw tightened; Maledicte licked; the tension in Janus's face traded anger for pleasure.\n\n\"True enough,\" Maledicte said slowly, letting his thoughts turn dark, his movements still, sifting memories of Vornatti's demands, his threats, comparing rewards and chastisements, against the lure of money to hand. He shuddered a little and climbed up to nestle into Janus's strength, soaking in his surety, the pleasure of scheming with him once again.\n\n\"Is there any reason to wait?\" Janus asked. \"If not\u2014\"\n\nMaledicte kissed him again, stopping his words. \"He's promised me an inheritance. I'm minded to collect it.\"\n\n\"Then I'll not stop you. Not when you look so fierce. So mercenary,\" Janus said, toying with the black curls that lay over Maledicte's shoulder. He raised the lock to his mouth, kissed it. \"So bewitching. I've missed you....\" He rolled them both over, pinned Maledicte between his arms, under his thighs. \"But inherit soonest, Mal.\"\n\nMaledicte drew Janus's head down to kiss him. \"Anything for you.\" He gasped, parted his thighs, and lost his hold on his courtier's mask. This physical definition was so much more real than the shadowy grasp of personality and will.\n\nWith Janus sliding in, possessing her, there was nothing left but Miranda, clawing Janus still closer. \"Janus\u2014\" she breathed, her voice caught by the damage in her throat, muted.\n\n\"Shh,\" Janus said, \"my love, my courtier, my dark cavalier...\"\n\nMaledicte's fingers tensed and dug into Janus's back, scraping the sleek indentation of the spine between muscle, the gas lights streaming and filtering through the pale gold mesh of Janus's hair, the cherubs watching, coaxing, laughing. Maledicte closed her lashes against the incandescent blue flame of Janus's eyes, lost in this blissful heat of touch and friction, of scent and sound. Janus's panting was in her ear, and for a brief moment it sounded like the rasp of feathery wings, and Maledicte's eyes flew open, trading quick startlement for rushing pleasure in the wash of blue and gold and velvet voice that was Janus in ecstasy.\n\nJanus's moan gave way to a breathless laugh, his blond hair drifting like spiderwebs. \"And they think you a man....\"\n\nMaledicte ran speechless fingers up Janus's chest, tugged him down to lie beside her, and slowly reassembled the guise she lived within.\n\nJanus continued, \"You always were good at misdirection, though. Remember when you bullied all the rats into pretending to be street players at the market? While we muddled along, shouting our lines, jeered at by everyone within earshot, you and Roach stole a feast for us all.\"\n\nMaledicte turned his face in the pillow; his lips quirked. \"Small potatoes only. I have a larger scheme now.\"\n\n\"How can that be? Am I not your end-all and be-all? Are you not completed now that I am here?\" Janus said, looking at Maledicte with apparent sincerity. \"Are we not bound together from childhood to death?\"\n\nMaledicte broke into a rasping laugh. \"False sentimentality from you? I'd say you were disguised, and yet I know you are sober.\"\n\n\"Drunk on you,\" Janus said. The archness that underlaid his last set of pretty words was missing, though his expression never changed.\n\nMaledicte touched Janus's pale lips, slid from the bed in a sleek line of white flesh. He pulled the drapes back from one wall.\n\nThe curtain pull revealed not the outside world, glazed through silvery glass, but a small alcove with a tall, narrow table. After Mirabile's visit, Maledicte had stored the poisons chest out of easy sight. Maledicte dug through the bottles, and retrieved a small bit of bright gold.\n\nMaledicte approached, hands held out, fisted before him, to play their long-gone guessing game. At the last second, as Janus reached forward, he opened his right hand to reveal the ring. He could not bear to have Janus misguess.\n\n\"I saved it; Kritos let it fall. It's not much now, but it's still for you.\" Maledicte looked at it once more, remembering it through Relict eyes, the treasure that had fallen into his hands. Gilly said such rings were common during the war\u2014bits of jewelry melted down, reshaped, and engraved with some false trumpery, extolling the glories of battle and the hearts left at home. Knowing now the gold was of questionable quality, and the sentiment looted from a dead man's hand, Maledicte still found it apt.\n\nJanus took the ring, rolled it in his hands, chasing the chill of the metal away, measuring its width against his forefinger. Once, it had been too big. Now it was almost too small; it required some effort to slide it down to sit above his signet ring. He took it off again, tilted it so he could see the inscription. \"I remember this. _Only each other at the last._ \" He turned the ring around in his hands. \"It was warm where you had kept it in your mouth and when I put it on, it felt like a kiss. My signet reads: _Only a Last at the Last._ \" His mouth twisted, and he tugged his crested signet off, trading it to his left hand. \"I like this motto better,\" he said, sliding the plainer ring on in its place.\n\nHe reached for Maledicte, drew him back into the nest of linens, stroked the dark head resting on his chest. \"It's been so long, I don't want to loose my grip for fear you'll slip away. All this because I wanted to rob Kritos before he was properly out, steal his gold for you. And still, I haven't any gold to give you, nor jewelry.\" He grinned lazily. \"Probably for the best. I would never have thought of stickpins, watches, and cuff links.\"\n\n\"Can you get jewelry?\" Maledicte asked.\n\nJanus laughed again. \"Some, I suppose. Greedy?\"\n\nMaledicte leaned close, listening to Janus's heartbeat, a steady sound, a companion from long ago, listening to the quiet sound of Ani's wings answering. \"It's only that I mean to kill him, you understand. And if Vornatti's promises are lies, we'll be penniless.\"\n\n\"Last,\" Janus said, his voice flattening.\n\n\"The jewels I own will be needed for our flight. If we're careful, we could stay in Antyre,\" Maledicte said, voicing plans he had barely allowed himself to think through, too afraid to plan beyond this reunion. \"We could live in the country, away from this rat-hunted city. But if the murder goes badly, we'll have to flee Antyre completely and that takes funds.\"\n\nJanus's supple mouth frowned, his pale eyes narrowing.\n\n\"What is it?\" Maledicte said, frightened. For one moment everything had been as planned, but in the wolf-pale eyes of his lover, something forced changes. \"Never tell me you love him,\" Maledicte said. \"I've sworn to kill him. I must kill him.\"\n\n\"You need not scruple otherwise on my account,\" Janus said, pushing himself up against the pillows, propping his chin on his knee. \"I bear him no fondness, but his title, his land\u2014\" Janus's tone dropped to an intimate whisper. \"It's a tricky business being a nobleman's bastard, especially if one has ambition. To allow me access to the courts, Last pretended the past had happened otherwise, that Celia and he had wed, that I am legitimate. No one believes him, of course. Whoever heard of an earl not searching for an infant heir stolen away? But Aris supports me, and if I can gain the support of the counselors, then\u2014\"\n\n\"Then what?\" Maledicte interrupted. \"How will this see Last dead?\"\n\n\"It won't, you'll have to do that,\" Janus said, \"but it will give me his title if done at the right moment. You crave Vornatti's money. I want the title.\"\n\nMaledicte said, \"It's only a word\u2014\"\n\nJanus shook his head. \"A title is power. Listen, Mir\u2014Mal, listen. This position of ours, of _yours,_ is precarious.\"\n\n\"I'll kill Last; we can rob his coffers and flee.\"\n\n\"And then what?\" Janus said. \"Money runs out. You used to teach that to those rats of yours. What happens then? Stealing? Starving? Whoring? Hasn't Vornatti been enough for you? I suppose we could take up a profession, but what are we suited for?\"\n\nMaledicte trembled; Janus stroked her belly, soothing.\n\n\"You used to cry,\" Janus said, sliding back down to lie beside Maledicte. \"In the cold, when the hunger was so bad. I'd bleed myself just so you could have something warm in your mouth. I swore every year it would be better and it never was. I never want to feel so desperate again.\"\n\nTurning, Maledicte kissed Janus's throat and whispered, \"I must kill him.\" As if in counterpoint, the sword, resting against the bed, fell with a hiss of scraped velvet.\n\n\"With your pretty little blade?\" Janus asked. \"Last is a brute but a damn good swordsman. And I mistrust the steel in painted blades.\" Janus reached out a long arm, picked up the sheathed sword, and drew it. \"Too often the paint dulls the edge.\"\n\nJanus fell silent, studying the blade, his fingers caged in the feather hilt. He raised a hand, curled it around the thin edges of the blade, and flinched. Blood beaded up along his fingertips and thumb and dripped to the sheets. \"Where did you get this?\"\n\nWhen Maledicte hesitated, afraid to wake Ani from Her cautious slumber by invoking Her name, Janus shrugged. \"Vornatti? Generous of him.\" He attempted to sheathe the sword; the feather hilt clung and bloodied his knuckles. He dropped the blade.\n\n\"Ani gave it to me,\" Maledicte said, the first time he had acknowledged Her gift aloud, a small act of worship. But far better to wake Her attention than to let Janus think Vornatti's touch reached so far into his life. Another frisson licked his nerves\u2014would Janus share this, the specter of a vengeful god?\n\n\"Black-Winged Ani is a myth meant to frighten superstitious bastards like my father. The dead gods returned? They never existed at all.\"\n\nMaledicte clambered over Janus, recovering sheath and sword and mating them with a practiced motion, albeit with a tinge of temper. \"Ani exists. I swore I would kill Last. I swore it to Her.\"\n\n\"All I ask for is delay,\" Janus said. \"Time to insure myself the earldom. After that it doesn't matter what happens\u2014we'd be as safe as we never were before.\"\n\nMaledicte trembled again, not in half-remembered dread of the Relicts, but at Ani listening to Janus's casual blasphemy, at the thought of staying his hand when She yearned for the kill.\n\n\"So you'll wait?\" Janus said, lying back, licking the tiny cuts on his fingers closed.\n\n\"Ani willing,\" Maledicte whispered, too low for Janus to hear.\n\n\"I think I'd make a splendid earl,\" Janus said, smiling. He snuffed out the last low-burning lamp and dropped them into darkness.\n**\u00b7 16 \u00b7**\n\n**I** T WAS EARLY AFTERNOON BEFORE Janus and Maledicte bestirred themselves. Maledicte, hunting Gilly, found him reading in the parlor. He leaned over Gilly's shoulder, eliciting a start, a flush, and a guilty twitch. Tweaking the book from Gilly's unresisting hand, Maledicte sighed. \" _The Book of Vengeances_ again? You spend more time thinking on Ani than I do.\" He flipped the book into the ashy fireplace, avoiding the snatch Gilly made, and shifted to stand before the hearth.\n\n\"Come, we're going to Whitspur Street. Janus wants to explore the city. We'll stop at Rosany's Booksellers and you can buy something less inclined to bring nightmares.\"\n\n\"Mal\u2014\" Gilly started, but Maledicte, hearing footsteps in the hall, turned, his skin warming as Janus approached. Janus met his eyes and smiled, leaning in and kissing his temple. Maledicte wove his fingers in Janus's hands, content, Gilly's dark dreams and Ani's rage insignificant.\n\n\"Ready, Mal?\"\n\n\"We both are,\" Maledicte said, releasing Janus to pull Gilly to his feet.\n\n\"Without luncheon? What kind of host would I be if I allowed my unexpected guest to leave hungry?\" Vornatti rasped, drawing three heads to where he rested against the doorjamb. Maledicte stepped away from Janus, unnerved by the intensity of Vornatti's gaze, by the simple fact that, though Vornatti's morning dose of Elysia would have worn off, the man confronted them on his feet. He should have been a pitiable sight, all grayed age and aches; instead, he radiated the wary strength of a veteran soldier.\n\nMaledicte's thoughts raced. He had assured Janus that Vornatti would be still abed, had played up the baron's poor flesh and feebleness to soothe Janus's jealousy, and more, to keep Janus from slipping out at dawn. He had believed it himself; this moment found him flat-footed. Janus was more sanguine than he. A bare flicker of distaste crossed his lips before he smiled at Vornatti. \"Too gracious of you, sir. I hope my presence hasn't troubled you overmuch.\"\n\n\"Visitors are always a pleasure. Trouble only comes from allowing them to stay past their welcome,\" Vornatti said. He limped heavily into the room, and said, \"I do warn you it is only bachelor fare. I have no hostess, though this is a lack I mean to remedy.\"\n\nMaledicte said, \"If it's Mirabile you mean, she leaves today with the Westfalls to the countryside. You'll have to be quick, old man, or chase after her like a hound on a scent.\" He tried for insouciance though his lips were cold with dread and his body crackling with nervous energy. Vornatti had the power to throw him to the streets, to throw Gilly out; did he have the power to finish the blow and see Janus sent away also, when it was Aris who wanted him in Murne? But Aris preferred lives to land once; he might put the kingdom's fortunes above his own this time, if Vornatti made it too costly to do otherwise....\n\n\"A message will suffice to bring her to my side,\" Vornatti said.\n\n\"Messengers are often unreliable,\" Maledicte said.\n\nVornatti grimaced at him, then glared at Gilly, who lowered his gaze in wordless agreement. Janus studied the bookshelves with polite courtesy.\n\n\"Perhaps you could spare me the trouble,\" Vornatti said, \"of hunting a reliable messenger, and play hostess yourself. I could find you a dress\u2014\" He crooked an arm; Maledicte saw no alternative other than an immediate unmasking, so with a quick look at Gilly, he took Vornatti's arm in his own.\n\nVornatti leaned on him, wrapping a possessive arm about his waist, and pressed his lips to Maledicte's cheek.\n\nIn silence, Maledicte led Vornatti into the dining room, all too aware of Janus's watchful eyes on his back, on Vornatti's stroking fingers. Maledicte settled Vornatti into his seat and attempted to slip free of his grasp. Vornatti only shifted his grip, tugging. Face scalding, Maledicte sat before him, pressed tightly against Vornatti's chest and wandering hands. Janus sank into the seat opposite and Maledicte shivered at the placidity in Janus's face, wondering what the mask hid. Rage at Vornatti's manhandling? Or, worse, kindling disgust at Maledicte's obedience?\n\nVornatti bent him back, hand in his hair, and tasted the hollow beneath Maledicte's ear, overlaying the bruise Janus's kiss had made. Maledicte jerked free, rocking the chair, and winding Vornatti. \"Our bargain,\" Vornatti warned.\n\n\"Still holds,\" Maledicte said, biting back rage, trading it for calculation. \"But surely your generosity will allow me one day with my old friend.... Like a bride-to-be bidding her old life farewell.\"\n\nVornatti chuffed with disgust, but let Maledicte claim an empty seat, out of his reach. Janus drank tea as if their conversation were only the usual pleasantries. Once served, the three dined in silence, Vornatti pushing his food around the plate, his eyes never leaving Maledicte; Janus eating steadily and with appetite. Maledicte removed bones from the fish without eating anything, finding solace in the steady ruination of flesh before him.\n\n\"'Tis a pity I had no way of knowing I would be at your table,\" Janus said. \"Your cousin Dantalion asked if I could relay a message, but having heard that you were rarely in Murne, I denied him. I do apologize.\"\n\n\"Dantalion has nothing to say of interest to me,\" Vornatti snapped. \"His only interest lies in knowing how near I am to dying, and how close he is to his presumed inheritance.\" Vornatti grinned malevolently at Maledicte. \"But that is my business, and none of his.\"\n\nMaledicte smiled vaguely in response, all the while feeling the sharp bite of anxiety in his belly as he twisted his long-held scheme into new configurations. To kill Last, to do so in a way that enabled Janus to inherit, to do so from a position of power\u2014it all rested on Vornatti's whim. One wrong word, and Vornatti might change his mind on the importance of kin, might recognize the truth Maledicte felt naked in his eyes: He would never relinquish Janus. Time was short. With Janus's kiss so recent on his skin, his warmth still lingering between his thighs, Maledicte could not imagine accepting a caress from Vornatti, never mind feigning welcome.\n\nJanus mopped his roll over his plate in the Itarusine fashion, chasing savory juices from the smoked fish. \"I suppose I should have expected that. After all, my father and Dantalion are rather cronies. And I've heard of the enmity between my father and yourself. He swears you a craven, running from a duel.\"\n\nVornatti slapped the table beside his plate, making his glass jerk and teeter. \"He dares\u2014\"\n\n\"But then,\" Janus added hastily, \"the nobles of two courts can attest to Last's easily offended nature. I am quite prepared to find you unobjectionable.\"\n\n\"How very kind of you,\" Vornatti said. \"Mal, isn't it wonderful\u2014such condescension from Last's bastard?\" Maledicte felt the blood rush to his face, his tongue leap to defend Janus; instead he fisted his hands beneath the sheltering cloth. Vornatti had to believe that Maledicte was his. The scheme aborning in Maledicte's mind demanded such.\n\n\"Don't,\" Vornatti said abruptly, turning his gaze from Maledicte to Janus, \"labor under the impression that you're fooling me with your agreeable manners, Ixion. I, too, was taught in the Itarusine court. I, too, know how to smile and spit poison. But having lived so long, I've found I much prefer bluntness. So I tell you\u2014you are not as clever as you think, and this is the last time an Ixion will run tame beneath my roof.\"\n\nJanus dabbed at his mouth and rose. \"So manners yield to candor and temper; I'm vanquished, sir. I will quit your house and never bother you more. Mal? I'll see you on the promenade.\" Without waiting for a response, he kissed Maledicte leisurely and left.\n\nMaledicte licked his lips, savoring the taste left behind like a promise, like absolution. He opened his eyes to find Vornatti glaring. \"That smacked of later, not farewell. Did you dismiss him or no?\"\n\n\"A day only,\" Maledicte said, facing Vornatti and turning to his meal with more appetite. \"You were correct, after all. Janus is not as I remember him.\" That was even true, Maledicte thought complacently, at least in the details. The Janus he had known was an impulsive, temperamental boy; this Janus\u2014Maledicte's lips curved against his will\u2014this Janus was subtle, and infinitely more dangerous. \"I will see him out,\" Maledicte said. \"Given your obvious disdain, you'll want me to make sure he hasn't lifted any of the silver....\" He escaped before Vornatti could laugh or protest.\n\nJanus caught him up as Maledicte reached the hall, leaned him against the wall and kissed him. Maledicte twined his arms around Janus's neck, keeping a wary eye on the closed door of the dining room.\n\n\"You must kill him,\" Janus whispered. \"It's intolerable.\" He took Maledicte's wrists in his hand and caged them against the wall above Maledicte's head. Closing his eyes, Maledicte shivered, gave himself over to Janus's confident touch. Let Vornatti come out; if he complained, Maledicte would spit him on the sword without another thought. Janus nipped at his throat, and murmured against his pulse, \"Do it soon.\"\n\nLINGERING IN THE KITCHEN, Gilly heard the front door open and close, and wondered who had left, and who had won. Whether Janus had gone with Maledicte by his side, or whether Maledicte was closeted with Vornatti, spitting useless anger. Gilly bit his lip; Vornatti was willfully blind if he thought Maledicte would tolerate his ownership much longer. No matter Vornatti's influence and strength of will, Maledicte was every inch the savage creature Vornatti liked to call him. And Vornatti was old now.\n\nGilly remembered the first time he'd met Vornatti\u2014the tall, elegant man complimenting Gilly's parents on their fine crop. Even then Gilly had been aware of undercurrents. While his father preened at Vornatti's praise for his fields, Gilly had seen the dark eyes assessing them all, and knew the crop Vornatti meant was himself and his brothers.\n\nThe bell rang fiercely in the kitchen, jangling on the board, barely stilling before it rang again.\n\nMaledicte the victor, Gilly thought, and Vornatti left alone and angry. He shuddered. While Janus and Maledicte had been here, Vornatti's attention and outrage had centered on them. Now the man's violent whims would turn to him.\n\n\"Best go to him before he has the bell from the board,\" Cook said, turning from her assessment of the pantry, looking at him with pity. \"He'll only get worse.\"\n\n\"I know,\" Gilly said, knowing he'd be kneeling before him, choking in the close scent of age and Elysia, all in the name of soothing the man's outraged pride. For a moment, he envied Maledicte and his bloody approach to life, the certainty that Vornatti was only a temporary affliction.\n\n\"Why do you put up with his ways? My boys wouldn't stand for it. You should find a new place, though I'd miss you sorely, Gilly lad.\"\n\n\"No one will have me, knowing the uses I've been put to. At least, no one who won't expect the same,\" Gilly said.\n\nCook turned back to her inventory, her silence only confirmation of his fear. She made a note or two, and finally said, \"Kettle's on. Take some tea afore you go.\"\n\nThe bell rang again, and Gilly shook a handful of tea leaves into a mug of steaming water before leaving.\n\nBut knowing Vornatti, knowing his moods, Gilly detoured first to the library, searching for something to distract him. He gulped the tea while skimming the shelves for something Vornatti hadn't read recently, or for new purchases not read at all. Grimacing at the acrid cling of tea leaves on his tongue, Gilly dribbled them back into the cup.\n\nLike the nobles he dined with, Gilly rarely had his tea unstrained, and the damp leaves woke lingering superstitions. He swirled the dregs around once, twice, then once again. Mindful of the varnish on the shelves, he found a sheet of blotting paper, and with an almost forgotten motion, upended the mug. Raising it, he stared at the blurred heaps of leaves, trying to read the pattern. But there was no symbol he recognized in the L-shaped spread, no chair, no hourglass, no raised hand.\n\nSuperstitious foolery, he chastised himself. What had he expected? He picked up the book he had laid aside; when he looked back, his breath caught\u2014not a symbol, but the thing itself. The leaves made a perfect gallows tree.\n\nWHITSPUR STREET WAS A FRANTIC cluster of millinery shops and tailors, divertissements, and gossip. Janus studied the broadsheets pinned above the boy hawking them, the images of courtiers at play, and incendiary articles urging Aris to shun the most recent trade delegates from Dainand. \"Don't buy that,\" Maledicte said. \"It's only gossip, and days-old gossip at that.\"\n\nJanus tossed the boy a copper anyway, and folded the sheet under his arm. \"I'm more interested in the news. Westfall mentioned a potential treaty with Kyrda, one that might offset some of the damage done by Aris's Xipos surrender. The broadsheets run several pages\u2014surely there must be some substance to it.\"\n\nMaledicte laughed. \"You'll be disappointed.\" He took Janus's elbow in his hand, and they strolled the raised walkway along the shop entries, while carriages clattered by on the cobbles below. From the distant green paths of Jackal Park, faint shouting came across the still air, the chanting of angry citizens protesting Aris's new ban on Itarusine imports. Janus listened to them for a minute and sighed. \"Shortsighted.\" Whether he meant Aris or the protesting men, Maledicte didn't know or care, simply pleased to have Janus at his side.\n\nJanus's clothes, still the fine wear of the evening before, spoke quiet scandal and drew several glances from passing nobles. Maledicte teased, \"A good thing you didn't go in costume.\"\n\nJanus leaned close as if to leave a kiss, but whispered instead, \"I did. I went as Last's dutiful, obedient son.\" His words warmed more than Maledicte's nape, set Black-Winged Ani to heated delight.\n\nLord Edgebrooke and his wife stepped from the walkway and threaded the crowded street rather than be forced to acknowledge them. For the open scandal of it, or for something less tangible? Maledicte shrugged. Let Gilly worry about that; he would filter the rumors and feed back all Maledicte needed to know.\n\nMaledicte paused in Rosany's doorway, at the display in the windows, looking at books. \"I should select something for poor Gilly, left to Vornatti's mercies.\"\n\n\"He's a servant. You needn't reward him for doing his duty,\" Janus said.\n\nMaledicte laid a hand on his arm. \"Gilly's my friend.\"\n\nJanus sighed, the temper fading from his eyes. \"I apologize. It's only that he's had your companionship while I've been deprived of it. I find myself envious of all the moments I've missed, of all the moments he had with you.\"\n\nMaledicte's lips curved. \"Pretty words. You've been trained well in courtly ways.\" His smile faltered. \"I suppose you had occasion to practice such things with the Itarusine ladies.\"\n\n\"As if I could ever care for vapid noblewomen who think of nothing but gossip.\"\n\nAs they dallied, a shadow fell across them, an approaching nobleman who chose not to step from the path. Maledicte looked up and his face stiffened to feral stillness. \"Last.\"\n\nJanus smiled, lips curling to malicious amusement. \"This should prove entertaining,\" he said, voice low in Maledicte's ear. \"But do restrain yourself, hmm?\"\n\nAny response Maledicte would have made was stymied by Last's nearness. \"Father,\" Janus said, tipping his head. \"Have you met Maledicte?\"\n\n\"To my chagrin,\" Last said, his face darkening above his high collar. \"Is this the kind of companion you seek? A scandalous courtier?\"\n\n\"I am not the only scandalous one, surely,\" Maledicte said. \"Or is my presence so overwhelming that the court can think of nothing but me?\" Maledicte felt a vicious triumph when Last's color intensified. Janus might force him to postpone the kill, but he would not give up baiting the man for anything. Janus's hand closed on his nape in warning, and Maledicte realized that Aris approached.\n\n\"Surely, Michel, you will not add to scandal by enacting a scene on Whitspur Street. After all, what happens in the court can only be reported secondhand in the scandal sheets, implausible hearsay. But lose your temper here, and there are a dozen witnesses who work for the broadsheets. Do try to leash your temper. For once.\" Aris joined them, two of the Kingsguard idling at his back and the brindled hound pacing beside him.\n\nLast turned. \"Aris?\"\n\n\"Am I unrecognizable without my crown, brother?\"\n\n\"You will acknowledge this creature on the streets? In front of the same audience you warn me of?\"\n\n\"I will,\" Aris said, turning his faded eyes on Maledicte. \"Though, my impetuous courtier, I remind you that I urged discretion; instead you create the season's greatest scandal,\" Aris said. \"How come you to know my nephew?\"\n\n\"Last's spurning of Celia Rosamunde sent Janus to me,\" Maledicte said, with a little bow in the earl's direction. \"I thank him for it. As for scandal, sire\u2014though I am loath to say it, your court thrives on scandal and spite. Mirabile, and others like her, sell tales to the scandal sheets purely so they can see their gossip in print and picture.\"\n\nLast spluttered, and Janus laid a warning hand on Maledicte's sleeve. Maledicte shook him off, aware of Aris's gaze on them both.\n\n\"Scandal and spite, perhaps. But it also thrives on decorum and rules. My rules, Maledicte. Do you realize they wait to see me banish you? You have put me in a difficult position. To flout my own rules or to displease your guardian when I need his goodwill\u2014\"\n\n\"Banish me?\" Maledicte echoed, his heart skipping for the first time since their conversation began. Banished. Away from Janus? He clutched Janus's sleeve.\n\nAt Aris's side, Last smiled, savoring the moment. It sparked such bloodlust in Maledicte that he felt his eyes must be reddened with it. The sword could have Last before anyone could pull him back. The world narrowed to red simplicity.\n\n\"Did you not swear me an oath that you would never draw your blade in court again? After I stayed your punishment once before?\" Aris's words came from a distance.\n\nMaledicte looked away from Last, his thoughts calming, turning. He rested his hand on the hilt. \"I swore...I would never draw it in your presence, sire.\"\n\n\"You would sidestep my strictures so carefully? Laws are more than the words composing them, Maledicte.\"\n\n\"I did not think at all, acted the impetuous youth you called me,\" Maledicte said, trying to shape words fast enough to soothe Aris. At his back, Janus's steady breathing brushed his nape, and the sound, the sensation staved off his growing alarm.\n\n\"That is no excuse, nor even an acknowledgment of wrongdoing,\" Last said.\n\n\"Father,\" Janus said, his respectful courtesy never faltering. \"It is King Aris's offense, his decision. But should you be allowed a say, so should I.\"\n\nLast purpled again at the unexpected insolence.\n\nAris's face relaxed at his brother's discomposure; the tightness left his voice. \"What would you say in Maledicte's defense?\"\n\n\"Only that deeds are misunderstood all too often. Only that if I took no offense, saw no wrong, and had the blade at my throat, perhaps there was no wrong meant.\"\n\nMaledicte would have smiled were he not afraid it would be misinterpreted. But he was pleased and surprised; not at Janus's defense\u2014he expected nothing less\u2014but at the sweep and subtlety of the words. Janus had changed, had learned the discretion Maledicte forgot when his temper was raised.\n\n\"You are presumptuous, Janus. You set your wrongs above the king's,\" Last said. \"Perhaps you are not as ready for the court as I assumed, and require another year's training.\" Maledicte went cold. He could not allow Last to take Janus from him again.\n\n\"Leave him be, Michel,\" Aris said. \"I, for one, am pleased to find someone so loyal to a companion. Tell me, Janus, would you be resentful if I removed Maledicte from the court?\"\n\n\"Saddened, say instead. You are my king, my uncle, my kin, and as such, incapable of wrongdoing.\"\n\nMaledicte wanted badly to applaud. Janus had mastered what he could not\u2014the art of cynical humor without the edge that offended.\n\n\"Have his words won me a reprieve?\" Maledicte asked, unable to keep silent longer.\n\nLast started to speak, but Aris overrode him: \"If you swear, without reservation, without duplicitous intent, that your blade stays sheathed within my court. And the gods alone know what the papers will have to say.\" Aris lowered his voice, stepped closer. \"But Mal, remember discretion.\"\n\n\"I swear,\" Maledicte said, bending his head. He felt the king's hand hover above it, barely touching his dark curls. He heard Last's chuff of disgust, watched him stalk off without further word, and raised his eyes to Aris's. The pale eyes flickered from his to Janus, standing so close, and a quick frown crossed his mouth. He reached out and pulled Maledicte a pace away.\n\n\"I accept your oath for a second time, and yet I await an apology,\" Aris said.\n\nMaledicte felt his temper stirring and stifled it. Too much was at risk and yet...he could not be other than he was, and his words came out edged. \"Shall I kneel before you, here and now, begging you to show mercy, sire? Speak the word and I will prostrate myself before you. My future is in your hands.\"\n\nAris tilted Maledicte's face up to his own. \"Michel would have your tongue removed for speaking so\u2014\" He released Maledicte's chin. \"But I am not my brother; I do not seek insult in every speech. I will forgive you, but as penalty, I will steal your companion from you. Janus and I have had little chance yet to speak.\"\n\nAris gestured ahead of him. \"Nephew.\"\n\nUsing the king's body to shield them from most of the watchers, Janus pressed his lips to Maledicte's palm before following his uncle. Maledicte shivered, chilled by Janus's absence, his mood plummeting. He wanted to run after them, refuse to let Janus from his sight. \"Restraint,\" Janus had whispered, his breath warm in the shell of his ear. \"Discretion,\" Aris had demanded. Maledicte watched Janus board the king's carriage and did nothing. When the carriage was gone, he thought of home, the delight gone from the day with Janus. But at the thought of Dove Street, of Vornatti, his plan shifted. He had other tasks to complete before he could return home.\n**\u00b7 17 \u00b7**\n\n**W** ILL YOU TAKE WINE?\" ARIS asked, waving away the young page who brought a sheaf of papers toward him, and closing Janus and himself into his sitting room. \"Please,\" Janus said, accepting the crystal goblet with graceful hands. Aris studied the lad, liking what he saw\u2014the unmistakable stamp of family blood, the confident manner. He found it hard to believe that the young man had spent most of his life in the worst slum Antyre had.\n\n\"Did Celia teach you the ways of the court?\" Aris asked, sitting down in a dark, velvet-upholstered chair. The mastiff settled on his boots with a groan, and Aris groaned back. \"Off my feet, dammit, Bane.\"\n\n\"Celia?\" Janus said, lowering his goblet and swirling the claret; a ruby whirlpool formed and faded. \"She taught me to speak properly, when she remembered my existence,\" Janus said. \"Her world is bounded by her supply of the old Laudable.\"\n\n\"I am sorry to hear it. Perhaps that explains her absence. I rather expected her to return once your name appeared in the broadsheets, claiming her bloodright, or sols for her part in keeping your past quiet,\" Aris said. \"You have not seen her since your return?\"\n\nBlue eyes met Aris's, and he was aware that he had startled Janus; yet Aris himself found surprise in Janus's reaction. Had the boy no family feeling at all?\n\n\"I went back,\" Janus said, finally. \"They were gone.\"\n\n\"They?\"\n\n\"Her compatriot,\" Janus said. \"Another whore. I presume they found a patron, or died of rat fever.\"\n\nJanus's words were empty of emotion, and Aris was bothered by this; surely it was natural for a boy to mourn his mother. But then, whores and addicts did not make for comfortable family.\n\n\"How do you and Michel get on?\" The door swung open as another mastiff sought out its master, pushing the door with its heavy head. Bane raised his lip, rumbled, and the newcomer settled down near the fireplace.\n\nJanus smiled. \"I like his dogs immensely.\"\n\n\"Only his dogs?\" Aris laughed. \"My poor brother. But have you no other response to his care?\"\n\nStanding, Janus paced the room. \"I do not love him. His reclamation of me was too clumsy for that. Yet I am grateful to him for this new life. Should I tell you how I feel for my uncle?\" He sat on the floor beside the fireplace and stroked the fine stripes in the bitch's fur. \"Or will you tell me something instead\u2014what you would say that requires such privacy? While I am honored to bear you company, I sense a motive beyond socializing.\"\n\n\"Of course you're right. But can you not think of anything you've done that might require discreet discussion?\"\n\nJanus bent his head over the hound, rubbed her soft ears. Her tail thumped against the granite hearth. \"Maledicte,\" he said.\n\n\"Maledicte,\" Aris agreed. \"Your father would have me command you cease relations with the lad. Make it a matter for law, not family.\"\n\n\"Will you?\" Janus asked, as if it were only a matter of small importance.\n\nAris didn't answer right away; he watched his placid nephew and found himself wondering what Maledicte admired in him, what fire, what source of desire. Aris thought Janus a pleasant addition to their dwindling family, but milk-watered for his taste, and, he would think, for Maledicte's. A tiny thread of suspicion arose; perhaps Janus played a part. Earnestness and honesty were not common traits for men trained in either royal court, nor, he would think, for the Relicts. But then, he recalled Janus's arguments with Last and sighed. A man playing the part of utter amiability would work not to offend anyone.\n\n\"Uncle?\" Janus said.\n\n\"Do you love Maledicte?\" Aris asked.\n\n\"Beyond all reason,\" Janus said.\n\n\"Rumor declares you bewitched.\"\n\n\"Only by the oldest magic, that of lover and loved.\"\n\nAris could not help but smile at the romantic simplicity of the declaration. He remembered arguing with Michel in the great ivy bower of Lastrest, arguing his reasons for marrying Aurora Vornatti, an Itarusine noble, and moreover, kin to a man his brother despised. \"With Kritos gone, you are Last's heir,\" Aris pointed out.\n\n\"I am not heir as yet,\" Janus said. \"Father feels my progress incomplete.\"\n\n\"You could aid him,\" Aris said.\n\n\"By giving up Mal?\" Janus said. \"Please do not ask that of me.\"\n\n\"I will not,\" Aris said, surprising himself. \"You will not be the first nobleman to keep company with a courtier. But you must convince Michel that you mean to honor our line. Do you understand?\" Aris leaned forward, resting his hands on the great dog's back as if it were a lectern, he the tutor and Janus the student.\n\n\"I must marry,\" Janus said. \"Produce an heir. A healthy one.\"\n\n\"Marry _well._ A girl of impeccable lineage to offset the irregularity of yours,\" Aris said. \"I have heard of your exploits in Itarus. If there is any accuracy to them, you should have no difficulty with a wife.\"\n\n\"With the bedding, you mean,\" Janus said. His mouth, so long sober, slid into a grin. \"No difficulties. But choosing a wife\u2014\"\n\n\"I could name one for you,\" Aris said.\n\nDiffidently, Janus said, \"Grant me some time to choose my own?\"\n\nAris set his goblet down. \"If you select a bride by the close of the year and present her for approval, you may find your own. If you do not, I will choose for you, and you may be thankful it will be my task and not Michel's.\" The dogs, at Aris's subtle signal, rose and stretched, their tongues lolling.\n\nJanus stood and waited for dismissal.\n\n\"Have you\u2014\" Aris paused. \"Have you met your cousin yet?\"\n\n\"I have not,\" Janus said.\n\n\"Come then, he likes visitors.\"\n\nAris opened the door, releasing the dogs, and the page, slumped against the wall, hastily stood. \"Sire, Captain Jasper says will you\u2014\"\n\n\"Not at the moment, Marcus.\"\n\nJanus said, \"If there's something that demands your attention, Uncle...I understand there are some accords to be made.\"\n\nAris strode down the hall, talking over his shoulder, his words clipped. \"The Dainanders seek to renew our trade agreement, but like Itarus, they want it all to their benefit and none to ours. They think to take advantage of my ban on Itarusine imports. But they discount the Explorations, which are beginning to bear fruit\u2014the nobles may exclaim all they like over exotic fripperies and sideshow spectacles of savages, but the last six ships from the Explorations brought us corn, rice, and wheat, far dearer to my heart. So, I see no need to cede to Dainand's unreasonable demands. Let them wait and rethink their avarice.\"\n\n\"And the Kyrdic delegation?\" Janus said.\n\nBefore him, Aris stopped, and looked back. \"You seem to be well informed.\"\n\n\"I am Last's son,\" Janus said. \"Am I not supposed to take an interest in Antyre's affairs?\"\n\nAris smiled. \"I'm pleased you are. But the Kyrdics may wait also\u2014to be blunt, I am not so sure that they are not a stalking horse for Itarus, with Grigor grown weary of our failure to be annexed. I see no other reason for Kyrda's interest in our shipbuilding.\"\n\nThe dogs loped up the stairs ahead of them, rushing down the long hallway. Aris smiled at the sight, his good humor restoring itself in fondness for the brutes. \"They're Adi's hounds really, and grudge the hours I keep them beside me.\"\n\nThe nursery guard had let the dogs in and Aris could hear their wagging tails thumping the carpet. The guard opened the door, bowing.\n\n\"Papa,\" the boy said, rolling on the thick carpet with the hounds. He rose to his knees, saw Janus, and went silent.\n\nAris tried to see his son from different eyes, and yet the tragedy was still there\u2014the good-looking lad of twelve, who could not be made to think, learn, or even clean himself as a two-year-old might. Thin-boned for his age, he lacked the gawkiness that preceded adolescence. For Adiran, there would be no adulthood, only this fairy-child existence in a boundless present.\n\nThe boy darted to Aris, clung to his side, and stared at Janus. Janus bowed. \"Your highness?\"\n\n\"Blue.\" The prince advanced, hand outstretched. At the last moment, Janus caught his hand gently.\n\n\"Yes, my eyes are as blue as yours.\" Janus's nostrils flared slightly, as if he could smell it as an animal could, the wrongness in his boy.\n\n\"Adiran, this is your cousin, Janus,\" Aris said. \"Will you greet him?\"\n\nTen years of training, ten years of repetition, ten years of concentrated effort on the part of Aris, and Adiran responded to the cue with a clumsy bow.\n\nThen he whirled and claimed his boiled sweet from Aris's pockets. Aris pulled his son into his lap. The boy tucked his head under Aris's chin and worked on the candy, taking it out to look at it, putting it back in. \"He has a sweet nature,\" Aris said, \"which makes it easier for us and for him. But sometimes I wonder if he's not aware, imprisoned within his own mind....\"\n\nJanus raised his hands, dropped them; his words died away and Aris liked him the better for it.\n\n\"My poor son will be king, at least in name,\" Aris said. \"Itarus will devour him entire.\" He shuttered his heart against the pain of that. On the fireplace mantel, the icon of Espit, god of creation and despair, mocked him from Her tangled web, Her laughing mouth at war with Her veiled, teary eyes. Aris, who had removed all other traces of the gods from his quarters, had let this one remain, perhaps simply because it was the loveliest version of Weeping Espit he had seen. It had something of his wife in it, in the way tears caught on Her smile.\n\nJanus lowered his eyes, then said tentatively, \"Uncle, you are not an old man. Will you not marry again?\"\n\nAris rocked the child, hearing in Janus's words the echo of Michel. \"I will not risk prisoning another child in a broken mind. You must take that risk for me, Janus. Do not deny me that.\" He stood, shifted Adiran's weight to more even distribution along his hip and side. The boy tangled his thin arms around his father like spiderwebbing, fragile and yet binding. \"Can you find your way out, Janus, or shall I have Marcus guide you?\"\n\n\"Please,\" Janus said.\n\nThe door opened again and Jasper, the head of his Kingsguard, entered with a cursory bow. \"Sire, the antimachinists have burned Westfall's newest engine, and he expects the Kingsguard to act. We need your command\u2014\" Aris sighed at the frustration pinking Jasper's fair face, and sighed again at Marcus peering around Jasper's solid form, papers still clasped close. \"Of course,\" he said, letting Adiran down to play.\n\nDismissed, Janus bowed, and followed Marcus out, retracing their steps down quiet corridors, stone overlaid with wood and plaster, and was let out into the courtyard, illuminated with hung lamps and candles. From his vantage point in the nursery over the yard, from behind barred windows, Aris watched him go.\n\nIT WAS LATE EVENING BEFORE Maledicte returned, and Gilly, hearing his footsteps in the hall, crept away from a dozing Vornatti. He found the hall deserted, Maledicte's coat abandoned over the stair railing. He finally ran Maledicte to earth in the formal parlor when he heard the sound of quiet laughter.\n\nTo his nameless relief, he found Maledicte alone, kneeling before the stage; he had expected Maledicte to return in Janus's company, braving Vornatti's wrath.\n\nOn the little stage, a toy puppet theater rested. Without looking back, Maledicte said, \"See what Janus has sent me?\" Maledicte dragged the tiny crow-god across the false world, laughing. Within Ani's beak dangled the threads of a smaller puppet, jerking as Ani twitched, strings within strings within strings. Gilly raised his eyes, saw the strings extending beyond the theater, saw them stretching beyond Maledicte and himself and Vornatti, stretching to encompass the far reaches of the city.\n\n\"Only a fool plays puppets with gods,\" Gilly snapped. The day had been one well-devised torment after another; Vornatti still kept his Itarusine inventiveness as well as his temper.\n\nMaledicte only said, \"Then many people are fools. These theaters are apparently quite popular.\"\n\n\"They were meant to tell the gods' tale,\" Gilly said, remembering his mother telling the story.\n\n\"Tell it to me,\" Maledicte said.\n\n\"No,\" Gilly said. \"You know it already\u2014our demands and dreams drove the gods first to quarreling, then to fighting, and finally, on Baxit's command, to Their own oblivion. The only way They could escape us.\"\n\n\"If you think Ani dead, then I wonder that you fear Her at all,\" Maledicte said, taking the figure from Gilly's hand. \"Still, I suppose it stands to reason. Baxit seems much like Aris, trying to guide those who, while crying for help, disregard his words.\"\n\nGilly hesitated, a frisson touching his spine. \"Baxit? You've encountered\u2014\"\n\n\"Do you think Ani shares?\" Maledicte said. \"Don't be a fool. I was merely speculating.\"\n\nIn the distance, they could hear the bell shrilling as Vornatti, woken, yanked the bell rope.\n\n\"Come soothe him,\" Gilly said, taking Maledicte's arm. \"And don't mention Janus.\"\n\nMaledicte laughed. \"Don't fret so, Gilly. I brought him gifts. Won't he be pleased to know I thought of him?\"\n\nGilly hesitated, alerted by something off in Maledicte's tone. He seemed entirely too blithe, a child with a gleeful secret. \"Mal?\" Gilly asked.\n\n\"I brought something for you too,\" Maledicte said, selecting a small parcel layered in translucent cloth.\n\n\"For me?\"\n\nMaledicte passed him the flat package. \"I bought it off your sailor friend, Reg. He swore you would like it, as if I had any doubts.\" He took the remaining packages and went toward Vornatti's room.\n\nGilly peeled back the gilt-edged organza until the object came clear. It was an etched piece of whale ivory, the lines filled in with ink and gold leaf, detailing an elaborate scene. A feather-clad man climbed stairs toward the clouds, a streak of golden sunlight leading the way.\n\nGilly smiled, touched out of all proportion. He placed the engraving in his room, amid his small collection of treasures: an elaborate puzzle box Vornatti gave him that held his meager savings, his four books, edges fuzzed with repeated readings, a curved piece of sea glass cradling a twisted seashell, gold on the outside, luscious pink within. He touched the whale ivory at the pinnacle of sun and sky, the gilt warming beneath his fingers.\n\nThen, recalling Maledicte's strange cheer, he hastened to Vornatti's chamber. Vornatti was still echoing variants of Gilly's own unvoiced question. \"But where have you been?\"\n\nMaledicte shrugged. \"I've told you and told you.\" He dropped a nosegay of lilies and evening primroses on the bedside table. \"I even brought you flowers, since you spent all day closeted inside, and missed the gardens in bloom. Though, I admit, they're nothing like as lovely as the bracelet you gave Mirabile.\"\n\nVornatti said, \"So she showed you my gift.\"\n\n\"Yes, and a foolish thing it was,\" Maledicte said. \"Like any beggar, she'll come back for more.\"\n\n\"What makes you think I'll disappoint her?\" Vornatti asked, leaning back against his pillows, smirking. \"Perhaps there will be a wedding upon her return to the city. Didn't I say as much earlier, or were your senses too taken up with Last's whelp?\"\n\nMaledicte's eyes darkened, then he shrugged. \"You're far too wily to be caught by the kind who'd see you cuckolded on your wedding night. Besides, we have our bargain, and as I abstain from Janus, so you must abstain from Mirabile.\" He seized Vornatti's grayed head in his hands, kissed his forehead and lips. \"Don't be irascible, you'll spoil my good temper.\"\n\n\"You are done with Janus?\" Vornatti asked, skeptically. \"Your grand passion burnt out in a day?\"\n\n\"I solemnly swear,\" Maledicte said, \"to any god you care to claim, that you will never see me dealing with an Ixion again. Not Janus, not Last. Do you know, Janus has no interest in gossip at all? It's all news and trade agreements, and the plight of ex-soldiers. He has as many dreary opinions as Westfall. He asked my opinion of Aris's ban on Itarusine imports. I ask you\u2014\"\n\nVornatti smiled and Maledicte brought up the remaining packages. The first one, lumpen under lashings of gauze, revealed a statuette in the best brothel art tradition, which Maledicte danced along Vornatti's bedsheets. \"I thought of you at once when I saw it.\"\n\nGilly choked on a gasp. The little monkey leered up at Vornatti, its hands locked in a lewd self-caress.\n\n\"Impudence,\" Vornatti said, but there was laughter lurking beneath. \"And paid for with my name, no doubt.\"\n\n\"I had to resort to it; you've kept me short of coin of late. If you'd given me jewelry, I could pawn it, as Mirabile undoubtedly has done with your bracelet,\" Maledicte said. \"But here\u2014\" He brought up two silver-wrapped boxes, one small, one large, both ribbon-bedecked. \"Chocolates from the Explorations.\"\n\nHe laid the big box on Vornatti's lap, untied the ribbon, and parted the tissue. Maledicte chose a chocolate for himself, popped it into his mouth with delight. \"They're wonderful\u2014try them, Vornatti. You too, Gilly.\" Maledicte tossed the small box to Gilly, who fielded it with quick hands. \"Go on. Have it before dinner, be indulgent with us.\"\n\nHe held another to the old man's mouth. \"A peace offering, my lord?\" Vornatti's eyes met Maledicte's over the confection before accepting it; it collapsed under his tongue, and Maledicte let Vornatti lick the chocolate from his fingers without protest.\n\nMaledicte sprawled across the velvet coverlet, his lacy sleeves foaming over the candy box. He crossed his booted feet, and fished for another chocolate. Vornatti smacked his hand. \"Mine. But I'll share.\" Maledicte accepted a sweet from Vornatti's shaking hand, trapped it neatly with tongue and teeth, and sucked the sweet filling out from the darker coating. Vornatti watched him eat, and took another chocolate himself.\n\nGilly turned the package over in his hands; the label was DELIGHT'S, the confectionery shop where Aris bought the prince's candy. He held a piece in his hand and the smell rose temptingly and yet\u2014\n\n\"Don't you want it?\" Maledicte asked, lolling his head onto Vornatti's shoulder, letting Vornatti kiss the lingering traces of confection from his mouth.\n\nGilly took a bite. Sweetness spilled over his tongue, rich, smooth, cloying. He swallowed hard, as suddenly sickened as if he had found a worm in an apple. The gallows image lingered in his mind.\n\n\"Not to your taste, Gilly?\" Maledicte said.\n\n\"Never mind about him,\" Vornatti said. \"He's just ungrateful.\" At the warning note in the baron's voice, Maledicte curled closer to Vornatti.\n\n\"I'm grateful,\" he said, passing him another chocolate.\n\n\"Pretty little liar,\" Vornatti said, but the old man's thin-skinned cheeks flushed with pleasure. Gilly finished his chocolate in one bite, seeking distraction from Vornatti stroking the juncture of Maledicte's thigh and hip.\n\n\"Don't force yourself, Gilly,\" Maledicte said. \"I sent oysters for our suppers. Will you join us, Vornatti?\"\n\n\"I think not. I'll stay abed, eat chocolate and be an indulged old fool,\" Vornatti said, brought to a rare good humor by Maledicte's obedience. \"But stay and let me feed you chocolates until the dinner bell sounds.\"\n\n\"Like a Kyrdic harem, only much less sandy,\" Maledicte said. \"Should we invite Gilly to join us?\"\n\n\"No,\" the two men said as one, and Maledicte laughed, even while Vornatti curled greedy hands around his shoulders.\n\nGilly's stomach churned at Maledicte sprawled so in Vornatti's bed, and yet he was afraid to go. Maledicte's giddy, uncharacteristic behavior struck him as dangerous.\n\nThe gong rang; Gilly jumped, the small foil box tumbling from his lap to the carpet. Maledicte disentangled himself from Vornatti's hands, lips rouged with chocolate liqueur, face flushed with something that might have been pleasure. Or well-masked rage.\n\n\"Come on, Gilly. Let's leave Vornatti to his desserts.\" Maledicte tugged Gilly from the room.\n\nIN THE DINING ROOM, Gilly picked at his meal, eyeing the stuffed oysters with repugnance.\n\n\"Aren't you hungry?\" Maledicte asked.\n\nHis face seemed luminescent in the candlelight, and dark wings spread out from his shoulders. Gilly rubbed his eyes, his aching head, and the vision was gone. But the room felt wavering and fluid, as if the walls were only curtains about to be drawn. He shook his head, the taste of chocolate strong in his mouth.\n\n\"What did you give me, Maledicte? What was in the chocolate?\" Gilly asked, his voice rising.\n\n\"Shadowplay,\" Maledicte said, setting his fork down. \"It's not harmful. It's only a sedative, though some claim it has visionary qualities.\"\n\n\"Why?\"\n\n\"Because you'd recognize the taste of Laudable.\" Maledicte's voice was faintly surprised. The answer was not to the question Gilly meant\u2014why drug him at all\u2014but the room shivered; the overhead beams grew sputtering halos as if they were the masts of a ship beset by seafires. Gilly cupped his hands over his eyes.\n\n\"Are you seeing things? Tell me what you see.\"\n\nGilly peered through the swirling brightness that leaked from the candles, watching feathers sprout from Maledicte's skin like spring blossoms.\n\nMaledicte crossed the space between them and sat on the table. Gilly looked up into his face. \"Death in your eyes,\" he whispered.\n\n\"But not for you, never for you.\" The touch on his cheek was light and Maledicte was gone. Gilly rose; the floor fell away from him and he tumbled down, sliding across the room, rolling against the closed door. Scratching traveled through the wood, the scrabbling of a large bird. Gilly knelt, clawing at the knob. The door opened, and darkness rushed in on great black wings.\n\nShe crouched on the table, mouth agape, Her breathing like the gasps of a dying man. With each exhalation, the room darkened until the only light was one guttering candle, the flame streaming high and thin. The table became an altar, the disrupted meal Her offerings. Gilly crawled backward, trying to escape Her notice. She leaned over the edge of the table, Her pale feet dangling like gibbet corpses, Her wings upraised. \"He is Mine. He will worship Me. He will love Me. Nothing you do will keep him from My kiss at the end.\" Her voice, a god's voice, seared his mind.\n\nGilly cried out and woke, head on the table, neck stiff, numb hands dangling off the sides of his chair. He touched the plate nearest him and found it cool. The candles had burned to half their length, spreading wax into the spillways below. The clock hands had jumped; hours had passed. He shoved back the chair with a shudder of protesting effort, ears still ringing. Staggering, he made his way from the dining room and down the hall.\n\nVornatti's door was closed. Gilly touched the blank, dark wood, and hesitated. He opened the door to darkness and cringed, but this darkness was only that of a room dimly lit. In the center of the room, Vornatti's bed was shrouded by the drawn bed curtains and the deafening silence within them. Gilly stumbled forward in the low light, reaching for the cloth. His fingers clenched velvet, but he could not bring himself to fling the panels back and accept their revelation. A clinking of glass on glass made him twitch galvanically.\n\nMaledicte rose from the shadows of the wheeled chair, a goblet in his hand. The wine had darkened his lips to the color of old blood.\n\n\"Did you kill him?\" Gilly said, his voice a rasp to match Maledicte's. \"Tell me the truth, Mal....\" Gilly slumped against the wall, shivering. \"Is he dead?\"\n\nMaledicte poured a small snifter of brandy and handed the glass to Gilly. \"Your hands are shaking.\"\n\nGilly thought of murder done, and murderers caught red-handed, of cornered rats and poisons, but raised the glass to his lips and gulped the liquid without hesitation. The brandy warmed his tongue, his throat, his belly.\n\n\"Yes,\" Maledicte said, taking his seat again, setting his feet up on the bed, boots parting the hangings.\n\n\"How?\" Gilly asked. \"Poison? Or like _Kritos_?\" His voice cracked, imagining the sheets sodden with blood.\n\n\"Peacefully.\" Maledicte drained his glass, poured another.\n\nGilly set his snifter down and yanked the drapes back, still expecting Vornatti to wake into furious complaint. He lifted the feather-heavy pillow from Vornatti's face, the fabric as malleable as liquid and as drowningly lethal. Vornatti's mouth was open, his eyes shut, his gnarled hands limp with a relaxation life and drugs had not granted. Gilly rubbed his wet face. \"I should have warned you.\"\n\n\"When did he ever listen, Gilly?\" Maledicte said.\n\n\"Why?\"\n\n\"You know why.\" Maledicte leaned his head on the back of Gilly's shoulder, took his hands in his. Panic spiked him beyond brandy's ability to soothe. This murder might have freed Maledicte, but it cast Gilly into unemployment.\n\n\"You couldn't wait?\" Gilly asked.\n\n\"To have Janus at hand and beyond my touch? Impossible. And time is fickle, Gilly, as was Vornatti. He would have grown bored with the victory handed him, might even have thrown me over for Mirabile, despite his promises. I thought him near death, but every month he seemed to improve. I could not chance it.\"\n\n\"You've gambled on other things,\" Gilly said.\n\n\"Not this,\" Maledicte said. \"Gilly, I am in your hands. A murderer. What will you do?\"\n\nGilly turned, freeing his hands from Maledicte's cool touch. He tilted Maledicte's face to meet his, to see what the dark eyes held: fear, hope, pain, the lashes spiked with dampness.\n\n\"Do you regret this?\"\n\nMaledicte met Gilly's eyes. \"No.\"\n\n\"What would you have me do?\" Gilly whispered.\n\n\"Nothing.\" Maledicte's voice was tight. \"His death should pass without scrutiny; his habits were known to be precarious\u2014drinking mixed with Elysia. But do find me his will. I need to be sure he didn't append some recent codicil. I wouldn't put it past Mirabile to finagle one out of him.\"\n\nGilly nodded, feeling oddly numb, as if the Shadowplay lingered yet and this was only another dream. He knelt and prised up the floorboard near the hearth, revealing the strongbox beneath.\n\nGilly lifted the parchment out, smoothing the creamy vellum from its rolled shape. He weighted one end with the strongbox, the other with his hand.\n\nMaledicte knelt beside him, so close they were nearly bumping heads. Above them, the pillow Gilly had taken from Vornatti's face shifted and Gilly jumped to his feet, heart pounding.\n\nMaledicte flattened the curling edge that Gilly's abrupt movement had allowed, and skimmed the elaborate language, sorting and reading. Gilly, turning back, thought again that Maledicte had the instinct of a solicitor.\n\nMaledicte smiled for the first time since Gilly had entered the room. \"Seems he was not so much a fool as that,\" Maledicte said. \"Mirabile is nowhere mentioned. Nor are his Itarusine relatives. I suppose he held that grudge right and true enough.\"\n\nGilly slid the document away from Maledicte, sought out his own name, fearing, hoping. It was with the other servants'. Though his bequest was by far the largest, it still knotted his belly with resentment and fear. Tired of him, Gilly thought, his dismissal imminent. The sum allotted was a year's salary, no more. Enough to buy himself a berth to the Explorations, but not on the swift _Virga._ Enough to take him slowly away, and set him down in the Explorations, penniless, with no funds to return. Alone and friendless, he thought, as chilled as if a dash of blown snow had touched him; it would leave him without Maledicte.\n\n\"Don't worry, Gilly. You'll stay with me,\" Maledicte said, reading over Gilly's shoulder. \"I told you. I'll take care of you.\"\n\n\"As a servant,\" Gilly said.\n\n\"As a friend.\" The quiet word resonated in this room, this city of purchase and patronage, manipulation and deception. The silence gave weight to the word, and Gilly realized with sudden disbelief that this was the measure of the city's moral decay\u2014that his closest ally and dearest friend was a murderer, with more bodies yet to reap.\n\n**\u00b7 18 \u00b7**\n\n**G** ILLY AND MALEDICTE HAD BEEN returned from the funeral for only minutes; in silence both of them fled to the dining room and the warming comfort of the liquor on the sideboard. Maledicte poured two glasses, and raised a toast. \"Done and be-damned,\" he said. Gilly swallowed his whiskey without a response, too shaken by Mirabile's conduct at the funeral. They had not expected her to attend at all, considered her safely rusticating in the country. But Brierly Westfall's sudden miscarriage had kept the Westfalls in the city, and Mirabile with them.\n\nStill, funerals were not the ceremonies they once were. With the gods gone, there was no one to impress with their piety but themselves, and Vornatti's funeral was sparsely attended: Mirabile, Echo, two representatives from the palace, indistinguishable from each other, and Vornatti's solicitor, Bellington. There was no ritual, nothing but two cemetery workers covering the hole in the earth, shadowed by a chapel now used for harvest storage, and overlooked by the great stone god chairs, overgrown by weeds.\n\nIt was when the grave was nearly full that Mirabile had whispered, \"You murdered him. To keep him from me. I would have shared everything with you\u2014now we'll see what rumors I can spread, see how fast your welcome disappears.\"\n\n\"Do so, and I will comment on the timeliness of Brierly's miscarriage, your access to Harlot's Friend, and your hatred of the countryside. Whileeveryone knows of your murderous past, they know nothing of mine,\" Maledicte said.\n\n\"Bastard,\" she hissed, trembling with frustrated rage, then as suddenly as a shadow chased by sunlight, her face cleared. \"I'll strike a bargain with you. We'll keep each other's secrets, each other's counsel, and each other's company\u2014\"\n\n\"No. I've made one dangerous bargain already and it's consumed any desire to make another.\"\n\nShe growled under her breath, a distinct animal sound, and Maledicte cast a wary glance at her. She dimpled and said, her voice sweet again, \"Mal, remember me, and this, the moment you've spurned me. I told you once before\u2014I am as clever and as determined as you. I have been playing too gently, but that's done now. I have a mind to level the field. I know what you fear\u2014\"\n\nMaledicte seized her shoulders, grip bruising, suddenly washed with rage at her nebulous threats, but instead of the fright he hoped to see, she laughed, honestly amused. \"Such a savage,\" she said. \"Is unthinking force always your solution when there are subtler resources to draw on?\"\n\nHeads were beginning to turn, and Maledicte felt trapped, unwilling to back away, conceding her this round, and equally unwilling to keep Echo's scrutiny on him. Gilly put his hands on Maledicte's arms, and Maledicte relaxed, given a reason to release her. Mirabile leaned forward, closing the distance between them once more, even as Maledicte attempted to back away and was blocked by Gilly.\n\nMirabile kissed his mouth, her lips cold on his, and he shivered. She left the gravesite, the only sign of her anger the fisted hands at her sides. Maledicte turned back, aware of Gilly muttering quietly to their coachman, and the man slipping away. Then it was done, and Maledicte and Gilly had come home, Gilly taking up the reins of the coach.\n\n\"You sent the coachman after her?\" Maledicte said.\n\n\"She seemed too confident. I want to know where she goes,\" Gilly said.\n\n\"Yes,\" Maledicte agreed. Unsettled, he took refuge in peevishness granted by the front door opening, heralded by its usual creak and Livia's voice as she played butler. \"Who's that now? All these cards and flowers, all this fuss for one old man\u2014\"\n\n\"It's Bellington,\" Gilly said, looking into the hall. \"With Echo at his side.\" He set down his glass; it clattered on the tray. \"With Vornatti's death so sudden, with Last's dislike of you, with Mirabile spreading her venom, Echo will be looking for something actionable.\"\n\n\"I don't fear Echo,\" Maledicte said.\n\n\"You should. He has followers beyond that rabble of Particulars. Powerful men like Westfall and Last. Even Aris listens to him.\"\n\n\"So what do you counsel?\" Maledicte said.\n\n\"Attend to Bellington's reading of the will without any asides or insults. Be silent as best you can and pretend to grieve. Please. Or Echo'll have you in jail.\" He ushered Maledicte into the hallway. From the library, Gilly heard the stilted tones of Echo conversing with Bellington.\n\n\"I could remove his threatening presence for good,\" Maledicte said. \"A doctored drink\u2014some of my stock is quite tasteless. It would be a small matter to\u2014\"\n\n\"No,\" Gilly yelped. \"Are you mad?\" He dropped his voice to the barest whisper. \"And to discuss such a thing so close to Echo.\"\n\n\"He's nothing but a man. Not some avenging creature of a dark god,\" Maledicte said, a faint smile curling his mouth.\n\nBy the time Gilly had his panicked urge to laugh under control, Maledicte was greeting the two men, Echo first as was due his rank. \"Lord Echo, what brings you here? I find it hard to imagine that you intend to pay your respects to Vornatti since you had none for him while he was alive.\"\n\nEcho's dark eyes narrowed. \"I find it odd that Vornatti took you in, and suspicious that he died so abruptly.\"\n\n\"The ways of the heart are not easily understood,\" Maledicte said. \"Neither why he cared for me, nor why his heart stopped. But if it gives you pleasure, you may join me for the reading of the will.\"\n\nBellington started into speech, portly form rocking back onto his heels. \"If it's your will that Lord Echo be privy to the contents, then I withdraw my objection.\"\n\nMaledicte settled himself as Bellington took the will from his worn leather valise. Bellington coughed, face reddening. \"You are familiar with the late baron's will?\"\n\nA tap on the door interrupted Maledicte's response, and drew a snarl from Echo. \"Your servants don't know their place.\" He yanked the door open, startling Livia.\n\nBehind her, Janus stood, elegant in the color the court called Last blue. Echo mimicked Livia's startlement and stepped back. \"You visit a house of mourning?\" Maledicte's smile bloomed, and Bellington coughed again.\n\n\"Aris sent me,\" Janus said, with a half bow in Echo's direction, \"to carry his condolences.\" He held out a letter sealed in gold-edged blue. Echo moved to take it, and Maledicte forestalled him.\n\n\"First you pry into the will, now my correspondence? How deadly dull your life must be, Echo, to find mine so fascinating.\"\n\nHe claimed the missive from Janus, and Janus bent and brushed his lips over Maledicte's fingers. \"He waits on a response, my dark cavalier.\"\n\nBellington stood, \"Perhaps I should return\u2014\"\n\n\"Sit,\" Maledicte said, \"read away. Let us hear my guardian's last thoughts.\"\n\n\"In broadest outlines, the entailed properties in Itarus and his title go to his next of kin, Dantalion Vornatti; his Antyrrian country estate, being a residence for life, reverts to the Crown; the Dove Street residence and his considerable fortune fall to you, Maledicte.\"\n\nEcho grew more intent. As if sensing their master's mood, the Particulars in the garden straightened.\n\n\"Perhaps we should take another look at his cadaver,\" Echo said. \"To leave a fortune to a stranger and slight his own blood\u2014\"\n\n\"If it pleases you. Only make sure you tamp down the grave dirt well after, or you'll find him up yet again, and burgled,\" Maledicte said, even while Janus stiffened minutely. Gilly's throat felt thick, and he concentrated on looking merely miserable, rather than guilty.\n\n\"By the gods, your tongue is foul\u2014\"\n\n\"Tell Aris,\" Maledicte said, his voice overriding Echo's. \"Tell him I am only too glad to accept his condolences, and to accede to his request. With pleasure.\" The opened letter whispered stiffly in the close room, the vellum brushing Maledicte's sleeves.\n\n\"Perhaps Echo can deliver your reply,\" Janus said, \"if he truly intends to petition Aris for an exhumation.\"\n\nEcho stormed for the door, and Maledicte said, \"Gilly, it seems Lord Echo has had a surfeit of our company. Show him out, and Bellington as well, please.\"\n\n\"Sir,\" Bellington said, hesitating. \"We should go over the details. Besides the usual estate matters, there's the Antyrrian audit books to be dealt with. They need to be sent abroad to wait for the next auditor.\"\n\n\"Another day will suit, surely,\" Maledicte said.\n\nBellington nodded. \"It may take some time for Itarus to name a replacement for Vornatti's post. I understand the court abroad is most competitive. Though I believe Dantalion Vornatti is in the running, if only for familiarity with the baron's script.\"\n\nGilly herded the men to the door. Bellington stepped out with the step of a man relieved of an onerous duty.\n\nJanus nodded to Echo and said, \"Shall I walk you to your coach, my lord?\"\n\n\"No,\" Echo said, letting his gaze linger on the letter in Maledicte's hands.\n\nGilly shut the door and slumped against it, slid down to rest on the cool tiles of the foyer, exhausted.\n\nA shadow came between him and the light and he looked up. Maledicte hesitated on the stair. \"Are you well, Gilly?\"\n\n\"Well enough for having Echo in the house three days after a murder.\"\n\n\"Watch yourself,\" Janus snapped. \"Echo will be snooping for days. Only Aris's interest sent him off so soon.\"\n\n\"And sooner or later, someone will tell him of my chest of poisons and potions. It will be a sore disappointment to him that Vornatti died of nothing so exotic,\" Maledicte said.\n\nHe drifted up the stairs after Janus, paused again. \"Gilly, I'll need your help.\"\n\nGilly nodded, wondering what Maledicte wanted now.\n\n\"Vornatti's Antyrrian ledgers need to be copied. The private ones that detail all the funds diverted back to Aris. You needn't worry about copying Vornatti's hand, just the information,\" Maledicte said, continuing up the stairs, untying his black cravat, slinging his black coat over the banister.\n\n\"It may take days,\" Gilly said. The private ledgers filled nearly an entire shelf.\n\n\"Then it takes days,\" Maledicte said. \"Aris wishes to see those ledgers gone, and as he barters so nicely for them, I cannot help but think they may have other uses in the future. And don't fret, Gilly, I'll stay out from underfoot while you work.\"\n\n\"You're in mourning, Mal. Your activities must be curtailed\u2014\"\n\nHis only answer was the flutter of tossed paper as Maledicte continued on his path up the stairs.\n\nGilly smoothed the paper out, the thick foolscap, the weight of the seal against his palm letting him know that this was Aris's letter in truth, and not some convenient forgery.\n\n_Maledicte,_\n\n_Let me express my condolences for your loss, and relay an unusual request. As you have no doubt been aware, a financial agreement existed between myself and your guardian, wherein he arranged certain figures to the benefit of us both. As I cannot rely on the next auditor to be so amenable, please bring me those ledgers. In recompense, I am prepared to grant you dispensation from whatever scandal-broth your impetuous heart contrives to create._\n\n_Aris_\n\nThe request, couched so openly, in a letter passed hand to hand, left Gilly breathless. The king was perhaps the fool the newspapers called him to think Maledicte's behavior might always be so easily condoned. Maledicte's own nature stirred trouble, but with Ani's wings urging him on, Maledicte was capable of anything. And Aris thought to dismiss all trouble with a smile and a gracious word. It went beyond foolishness and into madness.\n\nARIS ENTERED THE THRONE ROOM from the king's entrance at the rear of the dais. By the main door, two of the Kingsguard stood to attention, dressed in the armor the palace etiquette required, enamel over steel, Last blue over silver. The elaborate gate that closed off the anteroom was drawn back, showing Aris his petitioners\u2014two Dainanders in their customary gray cloaks, and on the receiving end of their horrified gazes, Maledicte. Though he was dressed in gray as they were, it was entirely evident that he held none of their abstemious views: His grays were silk and satin, his hair curled and glistened; they wore wool and linen, and kept their hair cropped close.\n\nAris hid a smile as he nodded greetings. With Maledicte a visible reminder of Antyre's decadent court, the straitlaced Dainanders' eagerness to return home might outweigh their avarice. Aris waved Maledicte forward, and once he had entered, the guards shut the anteroom gate on the waiting emissaries.\n\nMaledicte dropped into a bow as he approached Aris; it felt amused to Aris, as if Maledicte mocked the role of court and courtier, the positions of king and servant, when they were only two men.\n\nAris settled himself on his throne, conscious of his own smile. \"You came. My request done?\" His gaze flicked to the single book held in a gloved hand.\n\n\"Delivered to your quarters, I believe. Your Captain Jasper seemed most eager to relieve me of the burden.\" Maledicte offered up the ledger he held.\n\nAris opened it, and Maledicte said, \"Careful, the ink may still be damp.\"\n\nWariness replaced the pleasure Aris had felt. He tapped the open book with an agitated finger, smudged the ink, and said, \"So I see.\" He waited for an explanation.\n\n\"Vornatti's hand was crabbed and nigh unreadable, quite gave me the headache looking at them. I thought to spare you that pain.\"\n\n\"The originals?\" Aris asked.\n\n\"Quite safe,\" Maledicte said. \"I am careful of my possessions.\" The dark gaze that met his was confident, and under its power, Aris bit back his first instinct to shout for the guards.\n\nAris tore his eyes from those dark ones, and said, \"My brother thinks you dangerous.\"\n\n\"What courtier is not?\" Maledicte said. \"I have a sword to let men's blood, and a wit to make them wish their wounds fatal. But I am hardly the only such in your court.\"\n\n\"I could wish otherwise,\" Aris said, impetuous himself. \"It makes me feel less a shepherd of my people and more a serpent charmer.\"\n\nMaledicte smiled, and uninvited, dropped to sit on the stair, looking up at Aris. \"I am charmed,\" he said.\n\n\"And yet you deny me what I asked for.\"\n\nMaledicte sighed, traced the horned image of Haith worked into the side of the throne. \"Vornatti had the training of me, sire, and so I am constitutionally unable to part with anything of such potential value.\"\n\n\"Your intentions?\" Aris's anger roiled, turning belly-up, and exposing that thread of betrayal and hurt that lay beneath it. This young courtier had drawn his liking from the first. Aris had imagined the sentiment returned.\n\n\"I have none.\"\n\n\"Then give them to me,\" Aris commanded.\n\n\"I will not,\" Maledicte said.\n\nAris let his breath out, stung and chilled at once. \"Little fool. I am your king. Should I so command, you must obey. To do otherwise is treason.\"\n\n\"I will not,\" Maledicte repeated.\n\nAris put his hand beneath the mulish jaw and tilted the dark eyes to meet his own. \"I could have the guards take your stubborn head from your shoulders.\" Aris slid his other hand around Maledicte's jawline, into the silken curls. \"I do not think your stiff neck will prove obstacle enough for steel.\"\n\n\"Can you truly fault me for my caution? My craving for security in a world where I've known so little?\" Maledicte met Aris's eyes without fear or apology. Aris contemplated the face turned up to his, his fingers moving idly through Maledicte's hair, feeling the fineness of his neck and skin, the sleek curve of the skull, the rough touch of the boy's scarred cheek against his fingers. Maledicte smelled of lilacs, and Aris found himself staring into the wide, dark eyes as if they were the only thing in the room.\n\nAris was suddenly, uncomfortably aware that he was breathing faster, that his touch had turned to a caress, that Maledicte had woken long-dormant desire he thought gone with his dead queen. He started to withdraw his hands, and Maledicte caught one in his own gloved hand, caged the palm over his mouth.\n\n\"Please, Aris.\" Maledicte's breath warmed Aris's palm, and Aris was wholly mindful of the soft brush of lips against his skin. The rasping voice was as intimate as any Aris had ever heard; that agile tongue swept out, licked ink from Aris's fingertips.\n\nAris took his hand away, hid the warmth of it against his side. Maledicte's lips curled.\n\n\"Leave us. We dismiss you from our presence.\" Aris took his refuge in unusual formality of speech, uncomfortably stirred.\n\nMaledicte rose and bowed. He backed down the dais stairs, then bowed again and turned toward the door and the guards, the slim line of his back a gray smudge in the white room. Only after he had left was Aris able to recall the ledgers denied him. A brief spurt of bewildered irritation rushed his veins: Maledicte thought he required the ledgers to protect himself from Aris? When he had just proven to his satisfaction that words and touch alone would suffice?\n**\u00b7 19 \u00b7**\n\n_Women, by their nature, are more susceptible to sudden nerve-storms, to crying out to the gods for succor. However, they are also fickle creatures, no more able to hold to one course than a ship without its sail, and their petty outrages often die stillborn on their lips, forgotten as the next emotion crests. Though there are histories of women summoning the aid of gods, few are substantiated, and indeed, I have never met the woman determined enough to deserve a god's attention._\n\n\u2014Darian Chancel, \"On Theology\"\n\n**G** ILLY, TELL ME ABOUT THE debutantes this year. The king has it in mind for me to wed.\"\n\nJanus was in smallclothes only, his hair sleek along the thick column of his neck. Gilly held the shaving basin and mirror steady for him, studying Janus surreptitiously. Janus's skin was marked with scars, wounds neglected and healed without care. The young noblemen of the court had hides of gilt and marble; Janus recalled to mind the older generation, the ones who had fought and bled, the ones who had their history etched into their skin. Gilly wondered if Maledicte's skin carried that same violent history.\n\nWatching, Maledicte lounged amid rumpled sheets, fully dressed, admiring Janus.\n\nJanus nicked himself and flinched at the spot of bright blood. \"You keep this thing too damn sharp,\" he said.\n\n\"What purpose has a dull blade?\" Maledicte said. \"But I'll act your valet.\" He sat and smoothed the bedsheets, reaching for the razor.\n\n\"You're too fond of sharp edges to play barber,\" Janus said.\n\nMaledicte subsided back into the massed pillows, propping himself on his elbows. \"And so I find out how much you trust me.\"\n\nJanus pounced, rolling Maledicte into his arms for a lazy kiss.\n\n\"You're bleeding on me,\" Maledicte said, rubbing the smear near his mouth. Janus bent, licked the smudge away.\n\n\"There are several acceptable candidates, depending on your intentions,\" Gilly said, unwilling to watch more of this play.\n\n\"I want my birthright,\" Janus said, voice muffled by the soft skin of Maledicte's throat. \"The title and all it entails.\"\n\nMaledicte wrestled free of Janus, sat behind him, tilted Janus's head back. \"Consider while I shave him.\" Maledicte collected the basin and razor over Janus's objection.\n\n\"Amarantha Lovesy,\" Gilly said, after several moments of reflection.\n\n\"The duke of Love's daughter? She's no debutante, too old and serious-minded, mad about books and horses. Father told me of her,\" Janus said, nearly getting a mouthful of soap.\n\n\"And above a bastard's touch,\" Gilly said. \"Except she's blotted her book, got caught with her skirts up with a stablehand. But she's got breeding, and she has a reasonable dowry. Besides being a counselor's daughter.\"\n\n\"Aris expects an heir. Is she barren?\" Janus said.\n\nGilly shrugged. \"Nothing's been said. But she's not so old as all that.\"\n\n\"Is she pretty, Gilly?\" Janus asked. \"I've not laid eyes on her yet.\"\n\nMaledicte's fingers tensed about the razor.\n\nGilly recalled his first court attendance with Vornatti and his first sight of Amarantha. A girl near his own age and as far removed from his farming sisters as to be a separate race. Lady Perfection, the court called her. When her skirts had foamed over Gilly's boots in passing, he had nearly melted with desire.\n\n\" _Very_ pretty,\" Maledicte said, before Gilly could.\n\n\"Beautiful,\" Gilly amended.\n\nMaledicte snarled; his hands clenched about the razor's handle, and Janus put a hand up to keep the blade away.\n\n\"I suppose a beautiful wife is more palatable than an ugly one. Why hasn't she wed, Gilly?\" Janus asked, still caging Maledicte's hand and the razor.\n\n\"Her father's holding out for a title. Even now. Rumor has it he's been thinking of an Itarusine duke, and so regain some of his money that's fled abroad with the Itarusine tithe. But your title might prove more tempting with its hint of the throne for future generations.\"\n\n\"Still, she's old,\" Janus said.\n\n\"If you don't want her, Gilly can bed her. He's already admitted to finding her desirable.\" Maledicte's voice was edged.\n\n\"I won't share with Gilly,\" Janus said.\n\nHis light eyes met Gilly's and Gilly deciphered the warning as clearly as if it had been spoken aloud. \"I might find myself saddled with a farmboy's brat for heir.\"\n\nMaledicte slapped him, a blow that reddened his newly shaved skin and filled the room with its impact. Janus winced and seized the razor again, tugging it from Maledicte's hand.\n\n\"Oh, then bed your beautiful wife yourself.\" Maledicte shoved Janus forward and surged off the bed. Soapy water splashed Janus's lap.\n\nJanus ignored his tantrum and said to Gilly, \"Amarantha it is, then.\" Wedding the Lovesy chit turns a necessary marriage into more than a way to please Aris. Love's support would be invaluable in my quest for the title.\"\n\n\"She has her pride. She might not accept,\" Gilly said, watching Maledicte's face. He touched Maledicte's wrist, intending to soothe, but Maledicte twitched away, all nerves and temper.\n\n\"Does she truly have a choice?\" Janus asked. \"With Aris urging me to wed, and her father eager to see her settled before she finds another stablehand to liven her days? Tell me, is she still considered respectable? I've no desire to be burdened with a socially unacceptable wife.\"\n\n\"She is Love's daughter,\" Gilly said. \"Her sins can be forgotten with a ring.\"\n\n\"Enough,\" Maledicte said, voice cracking.\n\nJanus's face softened; he held out his arms. \"Come here, my dark cavalier.\" Maledicte dropped to his knees before Janus and put his head in his lap.\n\nJanus stroked his dark hair, then said, \"What else can I do, Mal? If not Amarantha, who then? Will there be anyone you accept?\"\n\n\"I hate them all,\" Maledicte whispered, voice edged.\n\n\"So will I, I promise,\" Janus said, brushing aside Maledicte's hair to kiss the back of his nape.\n\nMaledicte turned his mouth away when Janus sought it, but said, \"Gilly, arrange for Janus to meet Amarantha. Find out where she goes, what she does, what she likes. Treat her as any other enemy we mean to vanquish.\"\n\nGILLY RETURNED TO THE HOUSE in the afternoon after a series of meetings with his fellow servant-spies and informants. Though most of them had revolved around Amarantha, Gilly had taken the time to meet with a maid in the Westfalls' employ, to discover if Mirabile had yet returned. The answer had been no, but Gilly, watching the girl's eyes slip away from his, wondered if she were truthful. The last they had heard of Mirabile's doings had been the coachman's report. After Vornatti's funeral, he had followed her to the edge of the Relicts and balked while she delved farther in, her skirts vanishing into the winding rubble. The Relicts and their denizens should have been the end of her, but Gilly had uncomfortable doubts. Mirabile was a dangerous woman.\n\nThe castle of rooks on the roofline muttered growling agreement and Gilly shivered, slipping inside the house without looking upward, afraid of catching their black gaze. Inside, he found Livia waiting to intercept him. \"Watch yourself, Gilly. Once Lord Last left\u2014well, he's in a temper right enough. He threw the kettle at me. Broke the mirror on the landing, too.\"\n\nGilly looked up the dark stairs. \"Why don't you go out?\"\n\n\"Don't have to ask me twice,\" she said. Livia held out her hand, and Gilly, conscious that her pay wasn't due for another week, passed her a handful of coppers. \"I'll give you a luna also, if you take the other two maids with you.\"\n\n\"Those dull mice?\" she complained, but nodded.\n\nGilly took the stairs two at a time and paused. Livia hadn't said Maledicte had used the bronze serpent in the hall to break the mirror. The glass was still caged by the frame; only a few silvery fragments from the heart of the shattered mirror dotted the carpet. He stepped over them, tapped on Maledicte's door, and opened it on a curse.\n\nMaledicte sat on the bed, sawing cravats apart with the shaving razor. The floor around him was littered with scraps of mangled linens and thrown objects: a boot on the hearth, the water basin beside the wall, a long splash before the upended kettle near the door. \"Whose throat are you wishing cut?\" Gilly asked.\n\n\"Mine,\" Maledicte said, his face blotched with tears and temper. Grief wrapped the sulky lines of his mouth. Gilly's heart turned over.\n\n\"His marriage,\" Gilly said.\n\nMaledicte rubbed his swollen eyes. The razor moved perilously close to his skin and Gilly took it away, folding it closed.\n\nMaledicte's eyes darkened. \"I wasn't done.\"\n\n\"Why don't I pour you a drink instead?\" Gilly looked over at the liquor tray, littered with broken crystal. \"Or better yet, let's finish the job you've started, and frighten out Cook tonight. We'll raid her kitchen, find out what spirits she's been snaffling from your cellar.\"\n\n\"I'm not a child to be humored,\" Maledicte warned.\n\n\"Why shouldn't I humor you?\" Gilly said. \"You look like your heart is breaking.\"\n\nMaledicte let Gilly tug him to his feet. \"When he marries\u2014\"\n\n\"Shh,\" Gilly said. \"It's remarkable how much improved things seem after a drink.\"\n\n\"An attitude like that, it's a marvel you're not a sot,\" Maledicte said, but allowed Gilly to lead him down the stairs, past the grand, empty, and shadowed rooms into the warm recesses of the kitchen. The cook looked up from her chopping, startled.\n\n\"We won't need you tonight,\" Gilly said.\n\nThe woman eyed Gilly, flickered her eyes over Maledicte's face, and pulled off her apron. \"There's bread in the oven. Be a good boy, Gilly, and don't let it burn or you'll have no toast tomorrow.\"\n\nWhen she had bustled away, Maledicte said, \"She likes you.\"\n\n\"She's a motherly sort. If you weren't so off-putting, she'd never leave you be. She thinks you need feeding up.\"\n\n\"Is that motherly?\" Maledicte said. \"My mother wasn't like that.\" He settled down at the scarred wooden table, the unusual surroundings distracting him from his tantrum, and poked at the chopped almonds with the knife tip.\n\nGilly set a battered tin saucepan on the stove, checked the fire, and poured milk into the heating pan. \"No?\"\n\n\"She was just another Relicts whore. Like me.\" The tremble in the rasping voice sounded more like a rattlesnake warning than tears, but Gilly had memorized the nuances of his voice, and spun.\n\n\"Shh, shh, don't do that,\" Gilly said, daring to brush his lips over Maledicte's forehead as if he were no more than an unhappy sibling. \"You're not that; you're an aristocrat.\"\n\n\"When he marries\u2014all I become is his whore. Yet I chose this path. Vornatti didn't matter. I used him as he used me. But once I kill Last\u2014what will I be if Janus is married? Exactly what my mother intended. A rich man's pet.\"\n\nGilly poured out the milk, added brandy with a liberal hand, and set it before him. \"Never seen a pet with so many claws and teeth,\" Gilly said lightly. \"Drink, and I'll tell you tales of the court.\"\n\nMaledicte brought the cup to his lips, swallowed. \"I don't know why I listen to your sentimental stories.\"\n\n\"Because you know I'll put up with your tempers and moods in return,\" Gilly said. \"But if you're sick of love, I'll tell you about the sinking of the _Redoubtable_ and the _Deviltry._ \"\n\n\"Is there blood?\" Maledicte said.\n\n\"It was war,\" Gilly said, \"There's always blood. This was during the first days of Xipos, when the gods were still with us. The _Redoubtable_ was captained by Bellane, and the _Deviltry_ by one of the Itarusine princes. Their cannons were loaded with iron, and their chests were packed with gold, the better to coax greedy Naga to their aid. They battled and bribed and bled, throwing sols overboard as often as they fired their cannons, and finally Scaled Naga, god of health and avarice, thrashing below in an agony of greed, raised Himself out of the sea and took it all. Ships, men, cannonballs, and two king's ransoms of gold. Bloody enough?\"\n\n\"Mmm,\" Maledicte agreed on a hum of pleasure. \"No one tried to reclaim the gold?\"\n\n\"What the gods have touched is changed forever. Better left safely away from men's hands.\"\n\nGilly rose and fielded the hot bread from the oven, dropping it onto the cooling racks. He took one loaf to the table, found fresh-churned butter in the larder, and settled back at the table. He ripped a piece free, handed the warm bread to Maledicte. \"I wager you've not eaten today, but wallowed in your temper.\"\n\n\"Don't lecture me,\" Maledicte said, but he reached out and slathered the butter on his bread.\n\n\"Eat, and I'll tell you another story. An older story of a knight and his squire and their petition to Espit to grant them a child of their own.\"\n\nMaledicte rolled his eyes. \"And back to love. Gilly, you're a romantic.\"\n\n\"It's an incurable disease,\" Gilly said, judging Maledicte's mood. His eyes were shadowed, drawn with weeping, but the sulkiness had left his mouth; even now his lips curled faintly.\n\nMaledicte finished his cup of milk, walked over to the stove, and poured himself another. He sat down and ate chopped almonds and warmed bread, waiting. \"Love stories are too often dull\u2014\"\n\n\"Should I take a leaf out of one of Vornatti's pornographic stories, give you ribaldry instead of romance?\" Gilly teased.\n\n\"Whatever you want, Gilly, I am only your audience.\"\n\n\"There was a knight\u2014\" Gilly smiled as he told the story, not for the subject matter, but for Maledicte's reluctant attention, like a child coaxed into interest against his will. It was an old tale, and sad. The men's petition to Espit, the god of creation and despair, had been answered. A mare in the stables swelled with a human child. But during her birthing convulsions, the mare kicked the squire in the throat, and the sound of their daughter's first cry was mingled with the squire's death rattle.\n\nMaledicte's eyes were shadowed again when he finished, his mouth down-drawn; Gilly took a rueful breath and retold it as farce, where the men petitioned Espit, the horse was a stallion; the two men ended pregnant, and the horse...well satisfied. Maledicte's moodiness gave way to laughter.\n\n\"I never guessed you knew tales like that,\" Maledicte said when his breath returned.\n\n\"I lived with the old bastard for eight years,\" Gilly said, \"and before that I lived on a farm. It's only wonderful I don't talk like that all the time.\"\n\nMaledicte stretched his arms across the table, his hands open. \"Thank you, Gilly.\"\n\nWith Maledicte's bad temper assuaged for the moment, Gilly's thoughts turned to the wreckage upstairs. \"Let's tidy up so that you'll have someplace to sleep without worrying about glass shards in your sheets.\" He tugged Maledicte to his feet, and herded him up the stairs, ignoring Maledicte's complaints and mocking claims of being aristocracy.\n\nMaledicte held a handful of shredded lace, and Gilly had the linens stripped and piled neatly, still glittering with thrown porcelain, when Janus returned. Janus opened the door and paused.\n\nMaledicte dropped his bundle, kicked it beneath the bedsteps.\n\nJanus righted the hearthside chair and sank into it. \"Temper again?\"\n\n\"Better out than in, as Celia used to say.\"\n\n\"Celia used the axiom to excuse her drug fits,\" Janus said. He reached down, picked up the boot resting on the hearth, stroked the long scrape down its side.\n\n\"Are you angry?\" Maledicte asked, crouching before Janus.\n\n\"They're your things,\" Janus said.\n\nGilly picked up the kettle; its spout was cracked and he added it to the wastebin.\n\nAfter the effort Gilly had taken to soothe Maledicte, he was not inclined to let Janus rile him again so he busied himself around the room.\n\n\"What are you thinking about to make you so quiet?\" Maledicte said, settling into Janus's lap.\n\n\"About boots. This one is ruined.\" He dropped it from his hands, wrapped his arms around Maledicte's waist. \"At least, we consider it ruined. Now.\"\n\nMaledicte touched the supple, scarred leather with slow fingertips, tracing the damage. \"I haven't thought about that in years. We could have eaten off a pair of boots like this for a week.\"\n\n\"You ate boots?\" Gilly asked.\n\n\"No, fool,\" Janus said. \"We sold them for coppers, maybe lunas, if Mal did the haggling. Ragmen painted the flaws over, sold them at four times what they paid us.\"\n\n\"Can't eat boots, Gilly. They don't digest, and if you use them to flavor water, it only tastes like feet,\" Maledicte said. \"If you could get the water at all. I was always thirsty in the Relicts.\"\n\nGilly sat down on the bedsteps.\n\n\"Had to put a pebble beneath your tongue to stave off thirst,\" Janus said.\n\n\"Rise in the dawn to wipe the dew from the walls. But so close to the sea, even dew tastes of salt,\" Maledicte said. \"I haven't woken at dawn now in years.\"\n\n\"I did, at first, no matter that I was in a gilded cage. I woke with the sun, but there was always a pitcher of fresh water by my bed, and later, the maids came to bring me tea.\" Janus sighed into Maledicte's neck. \"It seems so hard to recall being hungry.\"\n\n\"I remember hunger,\" Maledicte said. His mouth drew down as if he felt that bite in his belly now, bread and milk and nuts notwithstanding.\n\n\"You were always hungrier than I was,\" Janus said. \"It's amazing you haven't gone to fat with the feasts you can have now.\" He raised his hand, circled Maledicte's wrist, spoke in a voice near dreaming. \"It was so hard. And no one cared if we starved.\"\n\n\"Not our mothers,\" Maledicte said. \"We're well rid of them.\"\n\n\"They ate what they would out of our hoard, and if there was nothing left, well then, wasn't it past time for us to go get more? Never mind that we had to steal or beg for it.\"\n\n\"You make me hungry now,\" Maledicte complained.\n\n\"I can't help you with past want, but if you don't mind a simple dinner, I can make that,\" Gilly said.\n\n\"Thank you, Gilly,\" Janus said.\n\nGilly startled at the lack of condescension in Janus's voice. Gilly nodded and went out the door, wondering what Janus was thinking. While Maledicte was all temper and secrets, Janus's apparent openness was still harder to read.\n\nMALEDICTE UNDRESSED IN THE NEAR darkness of his bedchamber, the lamps both turned low, watching his shadow flicker and shrink. If he listened carefully, he could hear Gilly and Janus discussing Amarantha below in the unusual silence of a house emptied of servants. Maledicte chose not to make the effort, and let their words fade into a pleasant murmur like the crackling of a low-burning fire.\n\nCarefully, he concealed his padded corset in the back of the wardrobe, trading it for a crisp white nightshirt. He caught sight of his reflection, ghostly in the mirror, and lingered, touching the snowy folds of cloth, the blunt cut of his unbound hair, and wondered, in a melancholy moment, if Amarantha hunted sleep attired in silks and lace.\n\nBut he wore silks aplenty during the day, and in the colors he chose. He went where he pleased; he carried a sword. The thought of the sword reassured him; the familiar lean length of it beckoned.\n\nUnsheathing it, he sparred with shadows until the sulky set of his mouth shifted into a fierce grin, until the dark hair on his nape grew damp with the effort. Two final, quick slashes sliced the wicks from the oil lamps.\n\nHe woke to sumptuous darkness interrupted by wavering golden light, a flame in the room. His hand opened and closed, found the surety of the hilt in his palm. \"Janus?\"\n\n\"Who else?\"\n\n\"I thought you were for home tonight,\" Maledicte said, opening the bed curtains to allow himself the sleepy pleasure of watching Janus undress by lamplight, all planes and angles, alternately shadowed and limned in flame. The fine hairs on his arms and legs gleamed.\n\n\"When I could be here?\" He slid into bed, all warm limbs and skin, and Maledicte sighed into the feel of him.\n\n\"And you've brought your wardrobe with you,\" Maledicte said, catching sight of a valise by the door. He smiled and pushed Janus back into the nested pillows, arranging him for his own comfort before resting his head in the juncture of Janus's neck and shoulder.\n\nJanus raised up enough to tug at the bed curtain, sealing them into a cocoon, then lay back again. \"I've scandalized the court once by wearing the same clothes when I should not. I won't do so again. Damn.\"\n\n\"Hmm?\" Maledicte said, half drowsing.\n\n\"I left the light burning.\"\n\n\"It will burn itself out,\" Maledicte said. \"We're rich. We can waste lamp oil.\" He yawned, rubbed his cheek over Janus's chest and finally chose the spot over his heartbeat.\n\n\"It's not the oil, nor the light that bothers me,\" Janus said, tightening his arm around Maledicte's shoulders. \"It's those cupids. Watching.\"\n\nMaledicte's slackening mouth quirked into a smile; he let out a few puffs of silent laughter that stirred Janus's hair on the pillow. \"I suppose we could hire someone to paint them over, but I loathe the smell of paint, and I hate the fuss and bother.\"\n\n\"You love fuss and bother,\" Janus said, tenting his elbow over his eyes. \"As long as you're inflicting it.\" His voice slowed, relaxed; his body slowly un-tensed, stretching out to fill the space. \"Rats take it!\"\n\nMaledicte jerked back to wakefulness. \"What now?\"\n\n\"Your damn sword bit me. Why in hell have you given it its own pillow?\" Janus sat up, dislodging Maledicte. A bleeding scratch etched the width of his biceps, a line of darkness against the paler skin, as if the night had left its own mark. \"Look at that.\"\n\n\"I wanted company,\" Maledicte said.\n\n\"You have mine,\" Janus snapped, pushing the sword out of the bed with all the distaste of a man removing vermin.\n\nMaledicte's mouth tightened as the sword hilt rasped along the edge of the bed before falling. \"Don't dump it there. You'll wake in the morning and tread on it, and that will be my fault too. Get up and put it away.\"\n\n\"It's your sword,\" Janus said, dabbing at the scratch with the lace edge of the pillowcase.\n\n\"You left the lamp burning.\" Maledicte drew the blankets more firmly about his neck, burrowing after warmth. After a moment the sheets rustled and the mattress shifted as Janus ceded. Maledicte rolled over, stared at the ceiling, his mouth curling. \"While you're up, will you\u2014\"\n\n\"Will I what?\" Janus interrupted. \"Make you tea? Bring you a biscuit?\"\n\n\"Since you mention it, I am hungry.\"\n\n\"You should have stayed to dinner then,\" Janus said. \"Gilly makes an acceptable cook.\"\n\nMaledicte smiled. \"Thank you, Janus, for being kinder to him.\"\n\n\"He has his place,\" Janus said. \"As long as he realizes it's not in your bed, I have no quarrel with him.\" His face, exaggerated by faint light, stayed grim, belying his words. \"Where do you keep this blade?\"\n\n\"As with all my favorite possessions, I keep it near to hand,\" Maledicte said. \"Set it beside the trunk. There are biscuits in the trunk also.\"\n\nJanus paused, his hand on the lamp, then sighed. \"Your sword by your bed, the sweets within reach\u2014I am surprised you do not have Gilly sleeping outside your door. After all, he is also one of your favorite possessions.\" He fished the tin out and tossed two biscuits toward Maledicte's outstretched hands. \"Will those suffice, or should I stay my hand on the lamp?\"\n\n\"Put it out,\" Maledicte said, nibbling on the first biscuit, cupping his palm to catch the tender crumbs, keeping them from the sheets. Belated recognition of Janus's words filtered through his mind. \"Gilly is no possession. You cannot own a friend.\"\n\n\"You own him as surely as you owned Roach,\" Janus said, moving through the darkness. He finagled his way beneath the sheets, drew the curtain shut. \"I do not understand it,\" he said, tugging Maledicte into his arms. \"I make friends easily. You offend people with every outborne breath, and yet you end with worshippers. Roach, Gilly, even Aris.\"\n\n\"And yourself?\" Maledicte asked, wiping his fingers on the coverlet.\n\n\"No,\" Janus said, catching Maledicte's hands, and kissing the crumbs away. \"I know you too well. I can only be your lover.\"\n\n\"Only,\" Maledicte said. \"Isn't that everything? Let them follow me as they will. I will follow you.\"\n**\u00b7 20 \u00b7**\n\n**A** T TEN O'CLOCK IN THE MORNING, the elaborate lawns and paths of Jackal Park swarmed with aristocracy exercising their mounts and strolling along the rows of honored vendors permitted to hawk their wares in this playground. Maledicte clutched the reins and tightened his legs about his steed, trying not to collide with anyone, tensing as they passed the barricade that kept the antimachinist protestors from encroaching. If they shouted or threw stones, as they were wont to do\u2014\n\n\"He'll have you off if you don't relax,\" Janus said, frowning. \"Vornatti taught you dancing, dueling, and etiquette, but not horsemanship?\"\n\n\"Vornatti tried,\" Maledicte said.\n\nJanus sighed. He slowed his horse, reached out, and drew Maledicte's hands back on the reins. \"Don't clench.\"\n\n\"They can't like being ridden,\" Maledicte said, but forced his hands to loosen. Beneath him, the horse stopped feeling like a pile of agitated muscle.\n\n\"Better,\" Janus said.\n\n\"Still, I don't see why we had to ride,\" Maledicte said. \"And at such an hour.\"\n\n\"This is the hour to be seen, you know that,\" Janus said, shaking his hair free from his collar.\n\n\"I'd rather not be seen falling off a horse, thank you,\" Maledicte said acidly, but followed Janus along the hedge. The hedges, carved into hounds and hares, alternately pursued and fled as they passed. Ahead, the trail broadened to incorporate the promenading aristocracy, the small, decorative carriages, and more riders.\n\nBeside him, Janus drove his horse into a sudden, flashy canter. At the end of the path, he slowed to a showy halt. Maledicte kept his horse to its nervous walk, glaring at the amused glances he garnered. A thin, dapper man in an Itarusine frock coat laughed aloud, teeth flashing within his neat ring of mustache and beard, and Maledicte spurred his horse forward, drawing up beside Janus. \"Was there a purpose to that display?\"\n\n\"Mating dance,\" Janus said, smiling. \"The air is sweet, and courting is everywhere.\"\n\nMaledicte's lips softened until he followed Janus's gaze and found it lingering on two well-attended women promenading along a side path. Their dresses were the height of fashion, and their eyes were raised, discreetly watching Janus.\n\nThe older woman was well into her fifties, Maledicte knew, but as un-lined as powder and potions could make her. The younger woman's beauty needed no such aid. \"And Amarantha Lovesy is easily impressed by horse-men,\" Maledicte finished.\n\n\"So they say,\" Janus said. \"Will you excuse me? I do not think my chances so good that you should come with me.\"\n\n\"Then tell me why I accompanied you at all?\" Maledicte said. \"Why I must rise and ride with you, when I hate horses and despise mornings?\" His horse crow-hopped beneath him, and Janus caught its bridle.\n\n\"I thought you'd prefer witnessing my wooing of Amarantha to imagining it.\"\n\n\"No,\" Maledicte said. He kicked his horse and sawed on the reins, trying to turn its head. Janus cantered away. Maledicte's horse, as restless as he at being deserted, made an attempt to follow. Maledicte yanked the reins; the horse danced beneath him, and Maledicte slacked his grip, clutching its mane.\n\n\"Too much beast for you? Perhaps you should join me for a stroll instead.\" Mirabile dimpled at him and tucked her gloved hand over the curve of boot and stirrup. Lacking his sword, Maledicte's hands knotted around the riding crop; he was startled to find her alive and well after her disappearance into the Relicts. Maledicte had not dwelled much on her threats, but to run across her of a sudden\u2014he found himself remembering the animal fury in her eyes. To see her so poised now when he knew her enmity made him cat-nervous.\n\n\"No steed for you, Lady, or are the rumors true\u2014have you sold your riding habits for pin money?\" He was rewarded by the tightening of her rosebud lips.\n\n\"You'd best keep your grip on the reins, or you'll be at my feet before you know it.\"\n\n\"Take your hand away.\"\n\nShe stepped back, gloved hands spread wide, laughing. He was at a loss for her shifting moods, at her returning to his side time and time again, and the damn horse kept tugging at his hands. Maledicte could not keep himself from glancing over his shoulder, hoping for aid. But, now leading his steed, Janus was deep in conversation with the duchess of Love.\n\n\"He does the pretty very well,\" Mirabile said, leaning her weight against the horse's velvet side. \"I hear he was quite well versed as a lover of women\u2014do you suppose he's reverting to type?\"\n\n\"Perhaps he already has, and the ladies were the anomaly,\" Maledicte said, lured into speech. \"Some men lose all sense of self abroad, or so I'm told.\"\n\n\"So confident in his affections? I hope your loyalty is not misplaced. But let us not quarrel today. Instead, come and have tea with me.\"\n\n\"What have we to say to each other?\" Maledicte said, his jaw tight.\n\n\"At the very least, tea would grant you an excuse to dismount. Come now, Mal. Is that horse really preferable to my company?\" She leaned forward and blew into its flaring nostrils. The entire animal seized under him, going as rigid as a corpse, then it reared, hooves striking at the sky. Maledicte wrestled it down, panting, then dismounted with more haste than grace.\n\n\"I never did like having discourse on an unequal footing,\" she said, smiling.\n\nMaledicte wound the reins in his hands, reconsidering. There was rage in her voice, barely contained. But he was unwilling to back down, or worse, attempt to remount beneath her gaze. He hoped to see Janus returning, escaping the vapid confines of polite first conversation between suitor and sought, but Janus lounged against a tree, one boot propped on a mounting block, smiling down at Amarantha. Even from a distance, Maledicte could see him working to hold Amarantha's interest. The duchess was his, but Amarantha looked away, plucking fitfully at her gloves.\n\nMirabile insinuated her hand into the crook of his arm. \"Come and have tea,\" she said.\n\n\"It's early yet\u2014\" Maledicte said, looking at the angry shadows in her red-brown eyes. Something shifted and moved behind them, something sleek and dark, drowning his objections, as if her words were law.\n\n\"There, no protests,\" Mirabile said. \"Such kindred spirits as we should be allies.\"\n\nHe walked with her, her hand around his elbow and caged by his free hand. Walking as if they were lovers. The horse? he wondered briefly, dragging his gaze away to look back. Had he loosed it in the park?\n\n\"Here we are,\" she said, settling herself onto a marble bench in the shade of a beech tree. A small table, its top a maze of inlaid tiles, had been laid out with a teapot, two cups of wafer-thin china, and a covered tray. Maledicte sat beside her, took the cup she handed him.\n\nShadows fell from the tree above, one crow and then another, followed by a slew of rooks, all come to scavenge for scraps. The two sets of birds squabbled and jabbered while Mirabile laughed and threw them tea cakes. Maledicte watched their glossy wings, the slick emptiness of their dark eyes. What could drive a noblewoman to the Relicts? He very much feared he had the answer.\n\n\"Your tea's growing cold,\" Mirabile said.\n\nThe same slick darkness rested in her eyes, Maledicte realized, the blank gaze of a predatory creature. The cup hovered at his lips, smelling of sweet jasmine and warmth, and reflecting the crow-blackness of his own gaze. He set it down with nervous fingers. \"I am not thirsty,\" he said, standing.\n\nShe rose with him as if they were linked. She collected his cup and swallowed several mouthfuls. \"There. In case your fearful heart cried poison, I have drunk from it as well.\" She folded his fingers around the cup again.\n\nHer eyes on his, the hush of the leaves in the faint breeze, and the squabbling crows at his feet all conspired together, making him feel he had stumbled into a dream. But he looked at the shadows in her gaze and forced a smile. \"No.\" He set down the cup; without looking back, he walked away, ignoring the quiver in his spine that urged him to run before her mask fell again and showed him more than he could bear to know.\n\nGILLY WAS READING in the parlor when the front door shut with enough force to rattle a sour note out of the spinet. \"How was the park?\" Gilly asked, as the carved door opened.\n\n\"Vile,\" Maledicte said, settling down on a delicately curved love seat. \"Janus went haring after Amarantha; Mirabile leeched onto me and tried to feed me dismal tea.\"\n\nGilly folded the pages of his new book closed with casual fingers, hoping to distract Maledicte from it. A moment's reflection showed him that Maledicte was unlikely to notice anything. \"Mirabile? Are you well? You look...scared.\"\n\n\"I am no coward,\" Maledicte said, the words quick and hot, ragged in his throat. \"At least, I never was before. But something was wrong. Mirabile's...changed. She had shadows in her eyes, Gilly.\"\n\n\"Shadows,\" Gilly parroted, heart sinking.\n\n\"I know, such melodrama,\" Maledicte said. \"But I swear to you\u2014No, I will think no more on it.\"\n\nGilly shivered, thinking of other eyes, all too often shadowed. \"Mal, did you drink her tea?\"\n\n\"No,\" Maledicte said, turning in his seat and gouging at the upholstery buttons. \"I know there's no rule against declining tea, so you needn't frown at me like that.\"\n\n\"It might have been poisoned,\" Gilly said. \"She hates you enough for that.\"\n\nMaledicte paused in his destruction of the chair. \"I sincerely hope it was. When I chose not to drink, she swallowed it. Perhaps she's ended herself?\"\n\n\"Or found herself,\" Gilly said. \"Shadows and poison. Mal\u2014you said that she had changed. Could she have sought out Ani's aid as you did?\"\n\n\"I never sought Her,\" Maledicte snapped. \"As for Mirabile seeking Ani\u2014\" His hands clenched on the chair, his voice tightening as he rose. \"It's those damned books you read. You see Her hand everywhere, when the simple fact is that I fled from Mirabile like a frightened child, afraid she'd pour poison down my throat.\"\n\nGilly seized Maledicte by the shoulders, stilled his restless pacing. Something moved over Maledicte's eyes, like the reflections of dark feathers, and Maledicte slumped.\n\n\"Let go of me.\"\n\n\"Ani supposedly protects Her own from poison,\" Gilly said. \"Even had you drunk\u2014\"\n\n\"You say that\u2014with stonethroat's effects branded in my voice? You have read far too many tales, Gilly.\"\n\n\"But that was before you sealed Her compact. Before you killed Kritos.\"\n\nMaledicte said, \"The only gift Ani brings is the only curse She brings, that of resolve and obsession. No more nonsense.\"\n\n\"And the sword?\" Gilly said, watching Maledicte retrieve it from the divan where Janus had forced him to leave it before exiting the house. \"She gave it to you. What might She have given Mirabile?\"\n\n\"Gilly!\" Maledicte said. \"Are you trying to make me fear Mirabile more or less?\"\n\nGilly sat, the book beneath him rustling as he did so, and Maledicte's attention shifted like a cat's. \"What's that? Another tract on the dead gods?\"\n\n\"It is,\" Gilly admitted, pulling it out and laying it on the floor between them. \"Written by Mirabile's husband, as it occurs.\"\n\n\"I should have it burned,\" Maledicte said, looking at the gaudy cover with an expression composed equally of wariness and contempt.\n\n\"You gave me the money that bought the book. I suppose it's yours. Everything is, even me.\"\n\n\"No.\" Maledicte turned, the shadows fading from his face. \"The money I gave you was only your share.\"\n\n\"An accomplice to murder,\" Gilly muttered.\n\nMaledicte touched Gilly's cheek and said, \"Don't fret, sweet Gilly. Or if you must, fret yourself to find something to entertain me until Janus returns.\"\n\nGilly's spirits lifted at the familiar petulance. Or so he told himself, dismissing the touch and casual endearment. Flushing, he cast about for diversion. \"Want to learn to play the spinet?\"\n\n\"No,\" Maledicte said. \"Do you know how to play?\"\n\n\"Vornatti had me take lessons when he thought it might be pleasant to have private entertainment on command. Before he decided his private entertainments didn't involve music.\"\n\n\"Then play for me. It can't be worse than the amateur talent they have at the courts.\"\n\nGilly sat at the spinet, but shifted on the seat, ill at ease. \"Stop staring at my back. It's too much to ask of me, to play and to perform at the same time.\"\n\nMaledicte rose and joined Gilly on the bench. \"What if I sit here? Then you cannot mistake me for a critical audience.\"\n\nGilly set his hands on the keys and ran out a scale. The notes vibrated in the air, going flat as the untuned strings sounded. \"Vornatti said my hands on the keys were too big. He was right.\"\n\n\"Excuses,\" Maledicte said. \"I have found one thing you cannot do perfectly and you're ruining it by making reasonable excuses. Just play, Gilly.\"\n\nGilly turned his head to object and got lost in the sweep of dark hair sliding over Maledicte's cheek and throat. He took his hands from the keys, brushed Maledicte's hair away from his face.\n\n\"Are you going to play that instrument, Gilly?\" Janus said from the doorway. \"Or are you playing at fashionable music master instead?\" At the palpable edge in Janus's voice, Gilly stood, leaving Maledicte possessor of the bench.\n\n\"I hear you made contact with Amarantha Lovesy,\" Gilly said. Behind him, Maledicte picked out notes at random.\n\nJanus heaved a sigh, came into the room fully, and slung himself into a chair. \"What a harridan. Despite her mother's enthusiasm, she made it clear the only reason she would even consider me was that she coveted Lastrest. All that beauty cannot mask her greed.\"\n\n\"Choose someone else,\" Maledicte said, head still bent over the keys, adding trembling dissonances to the air.\n\n\"What other wife could grant me a counselor's support so neatly? Lilia DeGuerre is wed and bred already, and Westfall has no child. No, I'll wed the bitch, and leave her in the country house she admires so much.\" Janus levered himself out of the chair, paced between Gilly and Maledicte.\n\n\"I thought we were to live at Lastrest,\" Maledicte said, eyes fixed on the spinet keys.\n\n\"It only needs to be for a little while. So many of the Last countesses have died of childbearing, we can create one more tragedy without much suspicion.\" Janus dropped a kiss on Maledicte's bent head, and pulled Maledicte from the bench. He lifted him onto the low stage. \"But as for now\u2014her parents will push her to accept my suit, we'll put my father in the ground, and you'll be consorting with an earl before you know it.\"\n\nMaledicte smiled. \"You hate her.\"\n\n\"Utterly. Set your heart at rest.\" Janus bent Maledicte over his arm to kiss his throat. \"Gilly, give us a waltz.\"\n\n\"Please do,\" Maledicte added.\n\nGilly thumped out a waltz, ignoring his mistakes and the pitch of the untuned spinet.\n\nJanus and Maledicte tussled for a moment, hands shifting and regripping, until Janus laughed and said, \"Stop trying to lead, Mal.\" He raised his voice, carrying the tune himself, humming, a warm, intimate sound in the room. Maledicte leaned into Janus's arms.\n\nWhen the waltz ended, Maledicte said, \"Play something else, Gilly. Something that doesn't want an in-tune instrument.\"\n\n\"You don't ask for much,\" Gilly said. But he searched his memory for a folk jig of single notes at a time.\n\nJanus shifted his grip, and Maledicte laughed, and then they were swinging each other like children, clasping each other's wrists, pulling and spinning until there was no dance, only the dizziness and laughter, Maledicte's voice disappearing under stonethroat's leash. Then Janus stumbled over the sword and swore. \"Damn thing. Enough, Gilly.\"\n\nJanus limped over and settled himself on the edge of the stage. \"Why do you carry that in the house?\"\n\n\"I like to,\" Maledicte said.\n\n\"Savage tastes, my dark cavalier,\" Janus said, rubbing his shin.\n\nMaledicte rejoined Gilly and touched the keys with curious fingers. \"We really should have it tuned.\" He tugged Gilly's hair. \"Maybe even hire an instructor.\"\n\nGilly laughed. \"Is that your subtle way of telling me I'm an abysmal player? I'm not used to such from you.\"\n\n\"Say better than some, worse than many,\" Maledicte said. \"A thing of no moment since you are perfection itself in all other fields of endeavor.\" He tucked his legs up beneath him and sat on the floor.\n\n\"A compliment and a sting at once. I applaud you,\" Gilly said.\n\n\"Gilly, it's not too soon to invite the Lovesys to Lastrest, is it?\" Janus said, interrupting their banter. \"I intend to ask them to Lastrest tomorrow.\"\n\n\"Sudden, but acceptable,\" Gilly said. \"They will be aware of Aris's command to wed.\"\n\n\"How long will you be gone?\" Maledicte said, laughter wiped away as quickly as blown sand.\n\n\"The standard visit is a fortnight,\" Gilly said. \"Add time for travel, laden with luggage and the stops noble ladies insist upon? Three weeks.\"\n\nMaledicte said nothing, still curled up like a boy on the floor. Janus went to his knee. \"Mal?\"\n\n\"I will go with you,\" he said, his voice a bare whisper, as if he recognized the impossibility even as he said it. Gilly heard Maledicte's breath coming faster, realized that somehow this step had caught him unawares. Maledicte would have to release Janus from his side, and Gilly, looking at Maledicte's stricken face, wondered if Maledicte would allow it.\n\n\"It won't be straightaway. Not only must I inform Lastrest's staff, and Father, but the spoiled chit probably will require a week to pack,\" Janus said, kissing the dark hair, tilting the pale face and kissing the tight lips. \"But I cannot bring you while I court her. For a title so close to king, the duke and duchess seem willing to overlook you, but that is far easier if you're not nearby.\"\n\nGilly caught Janus looking not at Maledicte, but at him, and with an expression very close to hatred. \"Gilly will be here. He can tell you stories, play the spinet badly for you, make you laugh. I'll be back as soon as I can. Rats take it, love, how long can I stand to be apart from you? You may have me running back within a week.\"\n\nMaledicte's shivering passed to Janus. Gilly saw their past in their trembling bodies, the pain that Maledicte felt when Janus was stolen from his side. Gilly was dwarfed by it, his own uncomfortable urges made irrelevant. He could not see himself anything but an unwelcomed interloper, and it was left to Janus to soothe Maledicte while Gilly sat, trapped at the bench.\n**\u00b7 21 \u00b7**\n\n**G** ILLY WALKED INTO THE COOLING EVENING, seeking to clear his head, and found his steps taking a familiar path into the city. At Sybarite Street, he turned toward the brothel with the ship drawn above the door. As it catered to sailors, Gilly found more than simple carnal amusement there; his fantasies of the Explorations were fed. But tonight he bypassed the salon, with its laughing, drinking sailors, and headed upstairs. He tapped on a closed door. It was the night he usually reserved, but he hadn't let her know he was coming. Just as he decided she was with someone, the door opened and Lizette stood there, rubbing her red hair out of her face and yawning. \"Gilly, I thought you weren't coming.\" She kissed his mouth. He leaned into her warmth, her encircling arms.\n\n\"But you look so sad tonight,\" she said, drawing him into the room. \"That love of yours giving you trouble?\"\n\n\"Not mine at all,\" Gilly said. \"Never was. There's someone else.\" He kissed her neck and stroked her shoulders beneath her silken robe.\n\nShe took herself out of his reach and lit the candles by the bed while he set lunas down on the dresser. \"Well, she's a right fool then. You're sweet and gentle and generous.\"\n\nPaid compliments though they were, Gilly relaxed under them. \"My employer already gets those things from me, without needing my love.\"\n\nLizette drew back. \"Your master's that courtier, ain't it? Why would you want someone like that?\"\n\n\"I didn't know you followed the court,\" Gilly said, settling himself onto the smooth sheets. He paid extra for clean ones on his nights, and she'd been sure enough of his custom that they were freshly laundered, smelling of nothing more than the iron and a faint trace of her perfume.\n\n\"Not the courts, Gilly. Just you. I saw your man once. At a distance and all. Thinks he's a king, don't he, the way he walks. But pretty enough to be a girl.\"\n\n\"Watch yourself,\" Gilly said. \"He's fast with a sword and doesn't like being called a girl. No matter his tastes.\"\n\n\"Mmm, well maybe he'd make a bad girl at that. Too scrawny. Not like me.\" She guided his hand to her voluptuous breasts. He bent his head to greet them with a kiss.\n\n\"Lizette,\" he murmured.\n\n\"That's it, Gilly-boy. Don't waste your thoughts on the likes of him.\"\n\nHe stopped her mouth with his and she tickled his ribs until he laughed. She rolled him over, teased him with her trailing hair until he growled and tangled his hands in her locks, pulling her to him, merging his body with hers, thinking yes, Lizette was right. This was simple. This was easy. This was welcoming and warm, and the only shadows in the room were those from flickering candle flames, not unseen gods. But he kept his eyes open, to make sure he didn't trade the vision of her warmth and curves for the cool, austere, and oft-imagined lines of Maledicte.\n\nStill, once they'd finished, Gilly left her side after only a cursory attempt at sleep, haunted by the premonitory instinct that warned him he would only dream of Black-Winged Ani. Better awake than that. He slipped out into the night, wending his way home through the back streets, and came across Echo's Particulars rousting a drunken man from his stupor at the base of a fountain.\n\nAfter a passing glance, Gilly paused and went back, exchanging coins for the drunken man's freedom. Briskly, he walked the man back and forth until he moaned, \"At least in a cell, I could have obtained rest.\"\n\n\"I'm only trying to help,\" Gilly said.\n\n\"Help? Buy me a drink,\" the intercessor said, staggering away toward the nearest pub. Gilly hesitated before following; he had heard the old man speak, had seen his eyes when he recognized Ani in Maledicte; this might be his only chance for answers beyond his books and pamphlets.\n\nGilly sat down in the seat opposite him, wincing at the smells of old stew and drunken leftovers. The intercessor sighed. \"The servant in Dove Street, correct?\"\n\n\"Yes,\" Gilly said, \"Your name, Intercessor?\"\n\n\"Not that anymore,\" the man said. \"I've given up shouting the truth to a city of the deaf and forgetful. I'll join them in their willful oblivion.\"\n\n\"I need your advice,\" Gilly said, gesturing to a barmaid when it looked as if the intercessor would walk away. The barmaid brought two ales to the table and the intercessor settled back.\n\n\"No one listens,\" the intercessor said, raising his drink, draining half of it.\n\n\"I listen. It's my job,\" Gilly said.\n\n\"It was mine once,\" the intercessor said. \"I spoke for the gods. Filtering their words until my ears bled. And now\u2014I'm forgotten along with them. You\u2014do you understand how it was? To be like a child, forced to hear his parents come to blows? Months of strife and no surcease\u2014only the visions of the gods battling each other. Intercessors died in their sleep. Others avoided sleep and went mad. Madder. Then came blessed silence\u2014a silence that rendered our lives without meaning, except to bear the blame of Baxit's final message. On Xipos, we were stoned. In Itarus, Grigor rounded up the intercessors, sick, mad, despairing, and plunged them into a frozen sea. So now, when the gods stir again, there's none to hear....\"\n\n\"I hear,\" Gilly said, his voice ragged, remembering Ani claiming Maledicte in his dreams.\n\nThe intercessor paused, set the drink down, and looked at Gilly. \"You do. But then, you should have been one of us. I can see it in your face. Do you dream of tombs, boy? Where the occupants lie sleeping, but restless?\"\n\nGilly took a gulp of the sour ale, and said, \"I do.\"\n\nThe intercessor pushed his empty tankard aside, and when Gilly would have gestured the barmaid back, the man shook his head, looking weary. \"What would you know?\"\n\n\"They're not dead, are they?\" Gilly asked.\n\n\"They're gods and immortal,\" the intercessor said. \"They've only withdrawn. Was that your question, because I believe you knew the answer already.\"\n\n\"I need to know how to break a compact between Black-Winged Ani and Her follower.\"\n\nThe intercessor gaped, then said, \"She's a god, and Her compact is more binding than anything mortal man can understand. There is no escape save in fulfillment. Oh, perhaps you could distract Her with charms asking for Baxit's aid, Her opposing force, but even if He bestirred himself from His indolence, it would only buy you moments. It would not undo Her will.\"\n\n\"Nothing can be done?\" Gilly asked, aware of the desperate edge to his voice. He hadn't understood how much he had hoped for another possibility until the intercessor scoffed at it.\n\n\"If you truly have a care for your friend, you will help him accomplish Her bloody goal and complete the Compact.\"\n\n\"He'll die,\" Gilly said. \"I can't allow\u2014\"\n\n\"Die?\" the intercessor said. \"You've been reading _Vengeances_ by the sensation-monger Grayle. Nothing but corrupt scholarship there.\"\n\n\"He'll survive then?\" Gilly said, the relief enormous.\n\n\"Quicker he acts, the likelier it is. But Ani's compact takes a toll. She grants gifts. Grayle will have told you that in his own hysterical fashion. But She also takes. I once visited a woman who is kept walled in a country asylum. Years ago, she climbed a turret in her wedding gown, carrying a dagger in one hand. She should have fallen. She didn't. She killed her husband, waiting for his lover. But she was left with the mind of a child. Ani is a creature of instinct and emotion, violence and passion, not intellect.\n\n\"In Elisande's case, it was a kindness, I think. Her mindlessness. Others have taken their own lives after, unable to bear the remorse, the grief. Ani feeds on their triumph and leaves them nothing. Aid your boy or not, care for him after, but do not expect him to remain the same.\"\n\n\"There must be some way to fight,\" Gilly cried.\n\n\"There is none,\" the intercessor snapped. \"Do not treat the gods as if they are human.\" He snagged Gilly's ale and gulped it down, then when Gilly continued to sit in mute misery, said, \"Boy\u2014let me warn you of one thing further. I have seen several compacts play out, and never have I seen Her shadow so strongly as I did in your master. Grayle, for all his melodrama, is right in one thing. A certain type of follower, strong-willed, fierce-natured, clever, might be enough to let Her manifest, creating Her Avatar, a mingling of god and man. A creature who could destroy the city.\n\n\"Most of Ani's children hunger and kill, the compact finished before danger ever arises. But the longer they delay, the more Ani invests of Herself, and the greater the gifts: Immunity to poison. Immunity to hurt. Witchcraft. Finally, transformation of the flesh. If your master is as strong as I fear, you'll not only lose him, but likely your own life, and the lives of all those around you.\"\n\nGilly fled the man's bleak eyes and sought the tranquil dark waters of the nighttime pier, contemplating flight. But his panic paled in the memory of dark eyes and a mouth coaxed to sulky laughter. When dawn crept over the sea in streaks of gray and yellow, he turned his steps back toward Dove Street and Maledicte.\n\nThe house was silent when he returned, creeping through the kitchen door. The sun just risen, even Cook was barely awake. Setting the tea to steep, she jumped at the sight of him, and the teapot clattered from the hearth.\n\n\"Sorry,\" Gilly said, recovering it before it spilled.\n\n\"So you should be, sneaking up on a body like that,\" she said. \"Where've you been, Gilly lad?\"\n\n\"Out,\" Gilly said. Without his asking, she poured him tea. Gilly took the kitchen mug in his hand. He drank it in one scalding gulp, then tipped the cup over.\n\nIt was not what he wanted to see, but in the wake of his conversation with the intercessor, he was not surprised to find that again the leaves gave him no shape but the gallows. He shivered in the warming kitchen. There would be nothing but the gallows tree until Maledicte's vengeance was done. He knew that now. Another death approached? _Let it be Last,_ he thought, _and the end of it._\n\n\"You're too levelheaded a lad to believe in such things,\" she said, wiping the leaves up with a dishrag.\n\n\"Thank you for the tea,\" he said, escaping her scolding.\n\nA light limned the edge of the library door, spilled a faint dusting of gold over the dark, carpeted hall, and Gilly paused before pushing the door open.\n\n\"Maledicte?\"\n\n\"Come look at this, will you?\" Maledicte said, bent over the desk. \"Vornatti's solicitor, Bellington, brought it over last night after you'd gone. He wanted coin for it. I'm not used to paying for information, Gilly. I gave him what he asked without haggling. If you're going to be gone all night, you need to teach me such things.\" The look Maledicte sent him was faintly accusatory.\n\n\"How much did you give him?\" Gilly asked.\n\n\"Ten lunas,\" Maledicte said.\n\nGilly winced. \"Oh, he'll be back then. I hope the information was worth it.\"\n\nMaledicte took his hand from the scroll of paper and it coiled again. \"It's from Vornatti's spies abroad. The same ones who told us of Janus's return. Read it, Gilly.\"\n\nGilly did so, fighting through the tight script. \"Vornatti's cousin, Dantalion, has an agent in Antyre?\"\n\n\"A solicitor, supposedly. Janus and I are torn on what it means that, though we know he's in the kingdom, he's made no attempt to challenge the will that disinherited his client. No one's heard of him at all.\"\n\nGilly sat down, nerves singing.\n\n\"Gilly, what do you call a solicitor who shirks the law in favor of secrecy and prying?\" Maledicte said, eyes dark. \"I have a word in mind.\"\n\n\"Assassin,\" Gilly breathed. \"Mal, you must be careful. An Itarusine lord is a dangerous foe on his own, and Dantalion is a crony of Last, and so will know more about us than perhaps is safe. Last could tell him your haunts and your favorite pasttimes.\"\n\nMaledicte said, \"Should I mew myself up behind these walls? I do not want to be a prisoner, kept away from all my hard-won freedoms.\" He was weary; Gilly saw it in the droop of his mouth, the pallor of his skin.\n\n\"We will hire agents of our own to find this man. Once found, he'll be no threat. Assassins thrive on secrecy, and Dantalion will have to find another route to recoup his losses,\" Gilly said, taking the letter. \"I'll send runners out, and you\u2014go back to bed.\"\n\n\"I could crawl into his arms and stay there forever, were it not that our enemies would find us,\" Maledicte said. \"I know the path I've taken, and yet it galls me that I have nothing but enemies at my back. Even I weary of the fight.\"\n\n\"You may have enemies to spare, but you have allies as well. Janus, myself, even the king.\" Gilly tugged Mal to his feet. \"It will be well, how can it not be? Are you not the scourge of the court, the terror of Last?\" He cupped Maledicte's face in his hands, and allowed himself to kiss Maledicte's forehead. \"Go to bed.\"\n\nMaledicte rubbed his cheek into Gilly's palm like a contented cat, then stiffened and freed himself. \"I trust you are right, that finding the man is as good as killing him. Because I intend to do so. I have no time for Dantalion's nonsense. I have an earl to kill, and an earl to create. Fortunately, though I may tire, Ani does not, and Her blade is sharp,\" Maledicte said, with a sudden surge of strength, an angry glitter in the black eyes.\n\nGilly, chilled again, watched Maledicte leave the room, and thought of the gallows tree and blood spilled beneath.\n**\u00b7 22 \u00b7**\n\n_Stillheart, mostly used to facilitate battlefield surgeries, is a chancy powder at best. The correct dosage is notoriously difficult to quantify, and many a surgeon has finished his task only to find that the corpselike stillness of his patient is nothing less than death in truth._\n\n_\u2014A Lady's Treatise,_ attributed to Sofia Grigorian\n\n**A** LITTLE PAST THE FASHIONABLE hour of the evening, Maledicte and Janus were still at home, sitting in moody silence in the library.\n\nFrom the dining room, they heard the maids laying the table for their last meal together. Without a word, Maledicte rose from his seat and straddled Janus's lap. Janus tugged Maledicte closer, resting his chin in the dark, loose curls.\n\n\"Sir,\" the newest butler said, entering. \"There is a solicitor to see you.\"\n\n\"Gilly said Bellington would be back. But so quickly? I must have over-paid him dramatically.\"\n\nThe butler coughed. \"It is not Bellington, sir, but another gentleman and, from the cut of his coat, foreign. Shall I show him in?\"\n\n\"Why not?\" Maledicte stood, pushing away from Janus's chest. \"Perhaps he's heard I sought him and chose to save me the effort.\" Maledicte collected the sword, strapping it to his hip, ready for use. \"But show him into the dining room. I see no reason to hold back supper.\"\n\nJanus said, \"I mislike him coming to us. It seems counter to his mission.\"\n\n\"Never question fortune,\" Maledicte said. \"But I agree, he must be a most confident gentleman indeed.\"\n\nThe man stood in the dining room, the lean, well-dressed man Maledicte had seen in the park. Laughing at him. Maledicte's mood darkened.\n\n\"Maledicte, and Lord Last, is it not? How gracious of you to receive me.\"\n\n\"How obliging of you to come to us,\" Maledicte said, faintly startled to see that the man's wary eyes fixed not on Maledicte and his sword, but on Janus.\n\n\"I've had a task to do first,\" he said, \"but with that accomplished, I'm aware of how little Dantalion is paying me.\" Unasked, he took a seat at the table. Deliberately, he laid a pistol before him, like an unexpected part of their place setting. \"It is primed,\" he said. \"And quicker to hand than that blade of yours. Still, keep it sheathed and all will be well.\"\n\n\"You want money from me?\" Maledicte said. \"Like some distant relative come a-begging?\"\n\n\"You have no relatives,\" the solicitor said. \"Unless you count a Relict whore. Dantalion chose not to challenge the will outright, not with Ixion, and some say Aris himself, sotted on you. So I sought levers and found such a lovely one, I can barely believe it, even now.\"\n\nMaledicte sat down, leaned forward. \"Tell me.\"\n\nThe solicitor said, \"Not until Ixion takes his seat. I know his reputation and I don't want him at my throat. I doubt you'd grant me my pension if I had to shoot him.\"\n\nMaledicte's breath sailed out in a rush. When he took it back, there was nothing but rage filling him, pure, cold, and smelling of feathers. \"So far you've told me nothing worth a copper.\" He poured himself a glass of wine, settled himself, hitching his hip to allow for the sword's presence.\n\n\"I sought information on this black-haired boy from the Relicts and found nothing at all. As if he never existed. But I did hear stories about Janus. I found a quick talker in a Relict rat called Roach\u2014you know him? A useful boy, hungry for coin, and short on moral qualms. He agreed to act my courier for certain letters if I fail to return to him tonight.\" The solicitor reached out for a goblet. \"Will you quench my thirst, _Sir_ Maledicte?\"\n\nMaledicte poured the glass. The solicitor's eyes never left his hands. \"Do I have your attention?\"\n\nJanus said, \"Roach is a liar.\"\n\n\"Only an inadvertent one. He told me Miranda was dead. I thought nothing of it, until I saw you in the park, Maledicte.\"\n\nMaledicte shuddered, shaken to the core by hearing that name unexpectedly voiced.\n\nThe solicitor laughed. \"I couldn't believe it\u2014there you were, in a crowd of blind men. No one takes the time to look past expectation anymore. No one but me. So tell me, girl, what do I do now? Tell Dantalion that the will is invalid, or tell him nothing, and let you take care of me?\"\n\n\"I will kill you,\" Maledicte said, voice raw with outrage, nearly shaking. He sought control. Murder in an aristocratic house took concentrated effort and planning.\n\n\"You're nothing but a girl, and I won't let your man get close enough.\"\n\nGilly tapped and entered. \"Mal, Cook's waiting to serve.\" He paused, eyeing the stranger at the table. \"Mal?\"\n\n\"This is Dantalion's solicitor, come to blackmail us. I see no reason he cannot stay, so long as he understands that talking business during dinner is a killing offense.\" Maledicte's thoughts raced, calculating risk against risk, exposure versus letting the man disappear back into the shadows.\n\n\"Your house, _sir,_ \" the solicitor said. \"But I'm afraid I've taken someone's place; the table is set for three.\"\n\n\"I'll eat in the kitchen,\" Gilly said.\n\n\"I think not,\" the solicitor said. \"I will not trust any of you near my food's preparation. Come, sit beside me and share my plate.\"\n\nMaledicte lowered his head to disguise his sudden chagrin. He wanted the man dead, wanted it so badly he was willing to take any risk, with Ani spurring him on. He met Janus's eyes, read the communication there, that they could kill the solicitor before he could do more than wound one of them, but the servants were one wall away, and secrets were hard to keep. Best for the man to die silent and quick. If Dantalion's man could not trust them with his food, neither could they trust him to leave this house and keep to his word.\n\nMaledicte dropped his hand to his sword, and watched the solicitor's eyes follow the movement and narrow. Death wouldn't be by blade; the man was warier than his confidence painted him.\n\n\"What is for supper, Gilly?\" Maledicte said, buying time and thinking there was always poison, an Itarusine's love, but all Maledicte's potions were upstairs.\n\n\"Oysters from market.\"\n\nMaledicte leaned back as he was served, watching the solicitor, trying to judge whether his wariness was only the usual thing, weapon-focused, or something more troublesome. He picked up an oyster shell, feeling the sharp, rippled edge.\n\n\"Gilly, is it?\" the solicitor said. \"I'll let you have the first bite.\"\n\n\"Too late,\" Maledicte said, dropping the empty shell to his plate. \"They're good, Gilly.\"\n\n\"So you say,\" Janus said, poking at his with open revulsion. \"If you loved me, Mal, you'd never serve them, no matter Aris's strictures on imported foods. I've seen the harbors here. The water's vile.\"\n\n\"The price of tablestuffs is ridiculous; Itarus gets the lion's share and leaves us to squabble over the rest, driving the price beyond reason. The nobles should learn to eat rats, as we did. Cure their impecunious ways.\"\n\n\"Must you bring the past into every conversation, Mal?\" Janus scowled. \"And we _never_ ate rats. Filthy animals.\"\n\nThe solicitor grinned. \"Hard to convince everyone you're Quality when they picture you in rags and rubble.\"\n\nJanus surged from his seat, hand tightening around his dull table knife. Gilly flinched. The solicitor's eyes swung in Janus's direction, even at that minimal threat; his hand found the pistol beside his plate.\n\n\"Sit down, Janus. Don't let him distress you. After all, your breeding, illicit though it was, is surely more genteel than his,\" Maledicte said.\n\nJanus nodded at Maledicte. \"Ever my voice of reason.\"\n\nMaledicte smiled back. The temper tantrum had done what he needed, shown the solicitor's predilections toward conventional weaponry. After all, he had focused on Janus and the dull knife, rather than Maledicte with a better edge to hand.\n\n\"Gilly, I'm done. And you've had to share. Would you like my last oyster?\" Maledicte rose; the solicitor watched him walk, watched Janus, and rested his hand on the pistol.\n\n\"Open your mouth, Gilly,\" Maledicte said, sitting on the arm of Gilly's chair. \"I'll feed you.\" Distantly, he trusted Gilly would play along, but Ani spread Her wings and the feeling of imminent bloodshed was so pleasurable, he smiled.\n\nGilly raised his head, parted his lips, and Maledicte tipped the oyster down. \"Good?\"\n\n\"Yes,\" Gilly said, swallowing, lips pale.\n\nMaledicte kissed his forehead, and as he did so, the solicitor said, \"Not content with one man? You are a wanton creature\u2014\"\n\nThe quick, slicing kiss of the ragged oyster shell across his neck shocked him into silence, into groping for the trigger. The wound was not deep; the breadth of Gilly's body and the chair robbed Maledicte of a killing blow. The solicitor shoved his seat back, a hand covering the bloody line. Maledicte swung again before the solicitor could bring the pistol to bear, and the second blow was messier, deeper, and quite fatal.\n\nMaledicte slammed the shell into the solicitor's gasping mouth, silencing any attempt at an outcry, and then shoved him over the back of his chair.\n\n\"Gilly, get the doors,\" Maledicte said.\n\nGilly started out of his shock and darted to the doors, locking them. Janus seized the pistol from the solicitor's thrashing hands. \"Mind the trigger,\" Maledicte warned. They stepped back; the man convulsed in silence, boots kicking at the overturned chair until Janus righted it.\n\nGilly's hands were at his mouth, shaking. He picked up Maledicte's wineglass, and drained its contents. Still, his hands trembled.\n\nMaledicte joined him, reached out for his hands; Gilly drew back, his eyes on Maledicte's bloody fingers. \"Go then. Flee if you must. But it had to be done.\" Maledicte toed the corpse; the solicitor made a spasming gasp. Maledicte wrinkled his nose in distaste. Gilly paled and fled.\n\nJanus settled back into his seat and began picking through his cooling dinner. \"I must do something about Roach. I should have done it when he told me I'd killed you. But I'll do it before I go. The solicitor was right about that. Roach does love his coin.\"\n\n\"Roach was my friend.\" Maledicte picked up a cloth, wet it in the finger bowl, and started washing his hands.\n\n\"Roach's tongue is a danger. Those letters are a danger.\" Janus's voice soothed and coaxed.\n\n\"Roach's probably lost them already,\" Maledicte said. \"Or drunk away the coin meant to frank them. But if you worry so, go find him and bring him back here.\"\n\n\"Then there'd be two who knew your identity and soon after, multitudes, when Roach slips your secret to Gilly, to the maids, to the gossips on the street.\"\n\n\"We could warn him not to talk.\"\n\nBut Maledicte knew that Roach's discretion was not to be relied upon. Even as a child, he let information slip at the worst moments. Might as well expect Ella to learn modesty. \"We could send him away?\"\n\n\"Where?\" Janus said. \"He can barely manage the skills of a Relict rat.\"\n\nMaledicte closed his eyes, his heart pounding. Janus could have been killed. Or Gilly, if the pistol had misfired. \"We haven't time to hunt him down. Not now. We have to dispose of the solicitor tonight.\"\n\n\"The timing is unfortunate,\" Janus agreed. \"But inescapable. Intercepting the letters before they're sent, taking them from Roach, is feasible. Intercepting them after they're sent is not. The solicitor gave us no word on the recipients. Dantalion, Aris, Echo, Last? Who knows how many? But Roach is easy; you could lure Roach out.\"\n\n\"Go back to the Relicts and chance being recognized? How many men do you want to see dead tonight? I've killed one man already.\" And night in the Relicts meant Ella would be trolling. What if she saw him, recognized him\u2014would she see Vornatti lingering on his skin, know what he had become? \"If you must go after Roach, you'll do it on your own.\"\n\nThe doors opened and Gilly returned, pale-faced but steady. \"I sent the butler on an errand.\" His mouth twisted. \"I blamed your eccentric desire for some _absente_ and never mind about the hour. We'll have to get the body out the door\u2014\" Gilly blanched again, and Maledicte went around the table to see what had disturbed him.\n\nJanus peered over the table and swore.\n\n\"What a mess. Indoor bloodlettings are so unforgiving without the earth to soak up the fluids.\" Janus joined them, and looked down at the corpse. \"Still, dead is dead. That's the crux of it. Gilly, get the floor cleaned before the blood sets. Mal, get his feet\u2014no, wait.\" Janus cleared the table, then yanked the tablecloth free and laid it out on the floor. \"Now.\"\n\nJanus and Maledicte levered the body onto the cloth; Janus went through the solicitor's pocket with quick, agile fingers, sorting coin and rubbish, before allowing Maledicte to wind the cloth tight. He handed the mixed pile of currency to Gilly, who looked at it with horror before dumping it onto the table.\n\n\"Where to?\" Maledicte said. \"The docks and the sea beyond?\"\n\n\"No,\" Gilly said, his voice ragged. \"Too many eyes.\"\n\n\"Loath as I am to agree with Gilly, he's right,\" Janus said. \"The docks rarely sleep, or the Relicts. Wrap him tight and I will take him courting with me.\"\n\nGilly asked, \"Will you really do that?\"\n\n\"What else is there to do? Bury him out back? His ghost will not haunt me, I assure you.\" Janus bent, checked for seepage.\n\n\"Gilly, go confine the maids to their quarters so we can fetch water from the laundry,\" Maledicte said.\n\nWith the servants out of the way, the cleaning went faster; Gilly brought water and soap and removed the blood from the carpet, while Maledicte and Janus practiced winding cloths. When they had finished, they headed to the attic to locate a suitable trunk.\n\nAlone in the room, Gilly was aware of his hands shaking again, pale red to the wrists. He kept tasting the sweet firmness of the oyster in his throat, followed by the sound of the solicitor's blood-soaked gasp. Gilly swore off blackmail on the spot.\n\nJanus's casual appropriation of the solicitor's purse and pistol lingered with him, the absentminded way he wiped the smeared gore on his fingers on the man's shirt. He contrasted it to Maledicte's hungry stillness and sudden violence and wondered, not for the first time, which of them was more dangerous.\n\nThe slam of the trunk hitting the exposed floorboards jarred him from his thoughts.\n\n\"There,\" Janus said, hefting the body up across his shoulders with deceptive ease. \"Not a leak to be sprung.\" He forced the body into the trunk and snapped the lid closed. Bending, he picked up the trunk itself and carried it out toward the carriage house.\n\nGilly scrubbed at the carpet; the wet cloth, pink-tinged now, shredded.\n\n\"Gilly, it's enough,\" Maledicte said, kneeling beside him.\n\n\"I just don't want to be able to see the stain.\"\n\n\"A new rug is in order,\" Maledicte said. \"I find I no longer like the looks of it myself. But it could have been Janus's blood spilled there, or yours, or mine. The fact that he underestimated me made him no less of a danger. I acted as I had to. Forgive me for involving you?\"\n\nWadding up the cloth in his hands, Gilly nodded.\n\n\"Will you do one thing more for me tonight?\"\n\n\"Yes,\" Gilly said, hearing Lizette in his mind\u2014 _why love him?_ \u2014and he could not answer it now either, sick to his stomach and frightened, yet he met Maledicte's dark eyes, and the tallies began: freedom, friendship, money, and the strange workings that created desire where there should be none.\n\n\"Three deaths to my hand now, and I am no closer to killing the earl of Last than I was three years ago,\" Maledicte said, quietly. \"But I tell you, Gilly, no matter the outcome of this marriage, I will have Last dead within the cold season. But I must have my chance at him; Janus feels there is another danger. Will you go with him tonight? Aid him as you can?\"\n\nMaledicte's eyes were glossy; Gilly could not tell if they were wet with unshed tears, or with anticipation. But, bloody rags in hand, he swore again to aid Maledicte.\n\nGILLY LOOKED OVER HIS SHOULDER, eliciting an exasperated sigh from Janus. \"What are you expecting to see? The solicitor climbing free from the trunk? You'll have us off the road.\"\n\n\"Why did we bring it along?\" Gilly said.\n\n\"Do you think Mal wanted it underfoot?\" Janus said. \"We'll start with the docks. I wager Roach has his roots there. When I saw him, he was cleaning salt from his clothes. He's a skinny thing and dark, taller than Mal, but not nearly so pretty.\"\n\n\"I met a boy with that name some months ago,\" Gilly said. \"A would-be thief working in a tavern called the Horned Bull.\"\n\nJanus's hands tensed so tightly around the edge of the seat that they seemed carved of alabaster. \"You spoke to him?\" His voice, as quietly knotted as his hands, made Gilly nervous.\n\n\"Not much,\" Gilly said, trying to make that unaccountable anger disappear, feeling out his words in increments of Janus's stiffening or loosening hands. \"He offered to housebreak at DeGuerre's for a fee. I turned him down.\"\n\n\"Nothing else?\"\n\n\"He mentioned you. Said you taught him to read. He doesn't like you. Said you'd killed his girl. How are you going to get him to come to you?\" Gilly said, clopping his tongue at the horses as they shied from a drunk staggering down the cobblestone streets.\n\nJanus sighed. \"He's greedy and lazy and undoubtedly in need of coin. I expect he's whoring somewhere, passing time and waiting for the solicitor to return.\"\n\n\"You think him a whore?\" Gilly could not reconcile the feral, defiant boy in the street outside the tavern with a pliant whore. \"He seemed too thin for that.\"\n\n\"They're not all like your pretty one,\" Janus said, \"put in gilded rooms where they eat sweets and wait for their men. The Relict whores are so different you might not even recognize them as human. They're not.\" The bitter edge to Janus's voice kept Gilly silent.\n\n\"Mal fell ill once when we were children; I thought he would die. He couldn't stop coughing and shivered so violently that I could barely hold him. Ella dosed him with enough Laudable to damn near drown him because his moaning and shaking was scaring away her customers, who couldn't tell Relict fever from plague. I spent the night with him in my arms, curled beneath the bed, wondering if he was going to wake, and Ella spent the night fucking above us. One sailor after another. That was before she realized Mal was going to be beautiful. Then she cared. But I cared first, and Mal is mine.\"\n\nThe whole speech was a near rasp, so choked with rage that Gilly felt it was Maledicte telling him this slice of nightmare. He drove on wordlessly; he barely knew how to soothe Maledicte; Janus was a mystery still.\n\nBeside him, Janus's ragged breathing steadied, but he didn't speak again until they were at the Horned Bull. \"Go see if Roach still works there.\"\n\nGilly clambered off the bench and went inside. He nodded at the taverner and slipped into the kitchens.\n\n\"What do you want?\" A heavyset woman looked up from the hearth where she was stirring a fish stew so old and salty Gilly could smell it across the room.\n\n\"Looking for Roach,\" Gilly said.\n\n\"He don't work here anymore. Not that he ever did more than rob our customers when they got too castaway to notice. Lift your purse, did he?\"\n\n\"Something like that,\" Gilly said.\n\n\"Try down by the cheap brothels. He's like to be trying his hand at robbing drunken sailors. Sooner or later, he'll try the wrong man, get hisself killed. If Echo's Particulars don't catch him first. Stupid rat.\"\n\nGilly nodded his thanks, and continued out the back door, circling around the building, checking the shadows. A black-haired young man in cheap clothing stepped out of an expensive carriage, drawing Gilly's attention. The boy tossed a luna from palm to palm, and blew a kiss after the retreating coach. \"Mal\u2014\" Gilly breathed, but even as he did so, he realized his error. The boy vamped at him, all painted lips and eyes, and headed into the tavern.\n\n\"Another Itarusine sailor's get, I'd imagine,\" Janus said, his voice velvet and sudden in Gilly's ear. \"But a startling resemblance, nonetheless.\"\n\nGilly jumped. \"Yes,\" he said.\n\nJanus flashed a quick, malicious smile. \"Though if I were you, I'd not tell Mal you mistook a rented boy for him.\"\n\n\"I'm not a fool,\" Gilly said.\n\nJanus raised a brow. \"What did they say, within?\"\n\n\"That he had turned to robbing brothel customers.\" He climbed up onto the bench. Janus joined him and took up the reins.\n\n\"I bet he's not even stripping them of their boots,\" Janus said, urging the horses into a bone-jarring trot across the cobbles. \"Let's finish this and go home. I shudder to think what Maledicte has done to the house this time.\"\n\n\"He has no regrets over the solicitor's death, and he's past the worst part of accepting your marriage. What is there to upset him?\"\n\n\"With Maledicte, sometimes I think it's the shifting of the wind.\"\n\nGilly turned his head as they came onto Sybarite Street, smiled at the sight of the familiar door painted with the sailing ship.\n\n\"Thinking of your girl?\" Janus said. \"What's her name?\"\n\n\"It doesn't matter,\" Gilly said, leery of Janus's interest.\n\nJanus drove, avoiding the whores advertising in the street, avoiding the clusters of young men daring each other to bravery in the fields of love, the hired whisperers who haunted the street, murmuring of places where one could go to buy smuggled Itarusine whiskey or other illicit imports, and as the street grew darker and less well-kept, avoiding the rubble they could barely see. \"Gilly, get down. Look for him. If you find him, don't waste time chatting, but bring him back. I'd do it, but I think you would not like being left with the trunk.\"\n\n\"No,\" Gilly agreed, dropping from the side of the carriage, and wandering into the dark alone, hunting for a half-remembered pickpocket. Abstraction lent an air of drunkenness he hadn't intended, and the first he knew of Roach's presence was the skinny wrist reaching for his purse, even as the stick missed Gilly's head entirely. \"Stop that,\" Gilly said, grabbing the stick, the thin-boned wrist with practiced quickness.\n\n\"I remember you. What do you want?\" Roach rubbed his wrist ruefully. \"Caught me just as quick this time.\"\n\n\"Information. I'll pay,\" Gilly said. He closed his fist, opened it, and a silver luna caught the light, a strayed bit of starshine.\n\n\"All right then,\" Roach said, snatching the coin. \"What do you want to know?\"\n\n\"Not me, a friend. He's a poet, wants to write verses about the Relicts, wants it to be romantic.\" Gilly was surprised he even bothered to lie. The hunger in Roach's eyes, the lack of caution, spoke volumes. Roach didn't care for anything but the sight of silver.\n\n\"Verses about the Relicts? Is he touched?\" Roach frowned. \"Going to write about rats and boots and fever, is he?\"\n\n\"No, he's another one who wants to write about Black-Winged Ani bringing down the Relicts for love,\" Gilly said, thinking he had been too long exposed to the court, finding it all too easy to envision his imaginary poet. He took Roach's arm in his hand.\n\n\"How much is he gonna pay?\" Roach said.\n\n\"He's a poet without a patron. Not much. Maybe two lunas.\" More and even Roach might find suspicion, but Gilly had given him coin before, easy earnings, and this looked the same.\n\n\"Can't take too long. I'm meeting someone,\" Roach said, even as he followed in Gilly's wake.\n\n\"Ain't that a fancy rig?\" Roach said as they approached the carriage. \"And your man ain't got more coins than that?\"\n\n\"Paper's expensive, as are quills,\" Gilly said.\n\n\"I bet they are,\" Roach said, turning over new thoughts. \"Where's your fancy man?\"\n\n\"He was here,\" Gilly said.\n\nThe reins were weighted with a cobble; the carriage door hung off the latch. Gilly wondered what had happened, a soundless struggle? Or had Janus simply grown bored with waiting?\n\nRoach opened the carriage door and peered inside. \"Ain't that fine.\"\n\nGilly picked up the reins and looked up to see Janus come out of a shadow, the knife aimed for Roach's nape. Gilly gasped, but Roach was already crumpling forward. Janus bent in the same quick economy of motion and shoved the body into the carriage, closing the door.\n\n\"Don't gape, get on the bench, and let's go.\" Janus swung himself up, setting the carriage to rocking, and held down his hand. \"Gilly! Let's not attract more notice than the carriage will have already. Get up, or I leave you here.\"\n\nGilly saw the temper flaring in Janus's pinched nostrils, in the swelling blueness of his eyes, but he could not respond. He remembered Roach's desperate insouciance, the thinness of his arm beneath his hand, and his awe of the carriage, all snuffed out in one moment.\n\nJanus snapped the reins and turned the carriage. \"Last chance, Gilly. There is still some night left and I don't mean to spend it haranguing you when I could spend it in Mal's arms. No? Fine.\" The carriage wheels spattered loose bits of sand and gravel over Gilly's boots. He watched it go, and only then started the long walk home.\n\nIt was nearer dawn than midnight when he opened the doors and crept through dark halls. As he approached the stairs, a light flared and smoked, setting shadows to dancing in the narrow stairwell. \"Do you realize you always use the servants' stairs when you're upset?\"\n\nMaledicte was still dressed as he had been for dinner; dark splotches remained where the blood stained his cuffs.\n\n\"Did you know he was going to kill him when you sent me?\" Gilly asked. His voice shook. He took a seat on the riser below Maledicte when the tremble in his voice reached his legs.\n\n\"Yes,\" Maledicte said, looking away.\n\n\"Why?\" Gilly cried. \"He was just a boy. The solicitor I understand, Kritos, even Vornatti, but this boy...\"\n\n\"He knew things Janus thought best left unsaid, a gossip as deadly as Mirabile,\" Maledicte said, setting the lamp on the stairs above his head, haloing them both.\n\nMaledicte's voice was all Gilly heard. He refused to look at him, not wanting to be distracted by the dark eyes, by the lush mouth, by his own desire. \"You killed the solicitor. Who else would think to talk to a rat? Why not pay him to go away? Or send him to the sea as aristocrats have done as long as there have been sailing vessels and inconvenient people\u2014\" Gilly's voice broke.\n\nThe lamp flared and popped as impurities in the oil burned, and in the silence after those small explosions, Maledicte said, \"There was more to it, Gilly. I was scared, and Janus was so sure. But he never liked Roach. Not ever.\" Maledicte let his breath out in a shivering gust, as if he had caught Gilly's quaking.\n\n\"Don't ever send me out again to lure someone to their death,\" Gilly said. \"I cannot do that. I will aid you in any other capacity I can. But please, Mal\u2014don't make me kill for you.\"\n\n\"Hush.\" Maledicte drew Gilly's head into his lap, stroked back the fair hair, streaked damp with night fogs. \"I promise. Never again. Not ever. I'm so sorry, Gilly. So sorry.\"\n\nHe smoothed the pale hair, spreading it over his lap like a gilt mesh, until Gilly's trembling ceased, and the lamp was beginning to gutter, unaware of Janus standing at the top of the staircase.\n**\u00b7 23 \u00b7**\n\n**A** RIS PAUSED IN THE FOYER of the Westfall city estate and watched Westfall's face, never as restrained as the other nobility, reveal his surprise as Aris nodded and moved on, dog and guards in tow. Aris smiled; he was surprised himself. Westfall's periodic afternoon parties were really no such thing. If the attendees found amusements, it was through avoiding Westfall, who used the occasions to argue his interest in the equality of the classes, his remarkable engines, and the future world to come.\n\n\"Sire,\" Lord Echo said, stepping alongside him. \"I didn't think to see you here.\"\n\n\"I'm sure that will be a sentiment I hear frequently this afternoon. And I'm equally sure that Westfall's cause will draw new interest. I do like to keep my counselors content,\" Aris said, and while true, it was far from the entirety of the truth: He had attended for one reason only.\n\n\"Thus you attend an affair you have little interest in,\" Echo said, with the ease of long acquaintance.\n\n\"It's the industrial aspect that flummoxes me, Dominick,\" Aris said. \"How Adam thinks that machines that increase the rate of production will aid us when the problem remains the same\u2014that our profits are not our own. I see little reason to benefit Itarus.\"\n\n\"Eventually the terms of surrender will end, our concessions fulfilled,\" Echo said.\n\n\"And in the meantime, Westfall's engines will replace the working poor.\" Aris shook his head. \"What both you and Westfall forget is that a country is not built on machines, but its people. All of them.\"\n\n\"The poor are\u2014\"\n\nAris put a hand up, halting Echo. \"If you'll excuse me, I came here to appease Adam, not listen to his ideas. And I think you spend too much time among the sordid types of our city. It makes you bitter, unable to see that good exists on all levels. My offer still stands. Join my Kingsguard and I'll see you at its head.\"\n\n\"Your Kingsguard, sire, is not a thinking thing, but an army. I prefer to solve problems.\"\n\n\"As you will, then,\" Aris said. When Echo would have followed, Aris shook his head, and Echo dropped back. Aris moved on, unimpeded, as the small crowd bowed before him. So many familiar faces turned toward him, and yet the one he wished to see eluded him. He frowned at the sight of Mirabile whispering into Brierly's ear, wondering what poison the woman cared to spread now, and if her presence meant Maledicte's absence. Aris tapped his fingers impatiently against his thigh, and Bane trotted forward to pant heavily by his side.\n\nAris slipped into the cool hallways, still seeking. He had expected to see more of Maledicte, now that the young man held the ledgers over his head. Until Janus had left Murne with Amarantha Lovesy, Aris had expected Maledicte to use the ledgers to derail Aris's requirement that Janus wed. Perhaps Maledicte had been in earnest after all, when he claimed he had no intentions of using the ledgers against Aris. Aris wanted badly to believe that, and not simply for the safety of Antyre.\n\nHe finally found Maledicte studying a portrait of a fair-haired woman in Westfall's family gallery. Maledicte frowned and spoke quietly to his servant.\n\n\"Lady Rosamunde, Celia Rosamunde's mother, and Janus's grandmother. I believe she was a distant cousin of the Westfalls.\" The servant's words came clear as Aris approached.\n\nMaledicte seemed uncharacteristically at a loss for words, lips parting. He touched the painting. \"Celia looks much like her.\"\n\n\"The whole point of aristocratic breeding,\" the servant said. \"Put a stamp of heredity on the children's faces, even if it requires inbreeding and creates idiots.\"\n\nThe guard coughed, and the servant turned, paling. He dropped to his knee. Maledicte's smile faded, his eyes growing wary. He put a hand on his servant's shoulder.\n\n\"You may rise,\" Aris said. \"Your sentiments are not new, I assure you. If I spent all my time punishing them, I'd have no time left to rule.\"\n\nThe servant nodded and stood, uncertain.\n\n\"You may leave us,\" Aris said. The servant glanced at Maledicte for permission, caught himself, and bowed again before leaving.\n\nMaledicte shook his head. \"He'll be fretted for weeks about that. It's my fault entirely. I've allowed him to be as free with his thoughts as he likes.\"\n\n\"It's not his thoughts that will see him in trouble,\" Aris said, \"but his tongue.\"\n\n\"Still\u2014it's pleasant to have an honest opinion,\" Maledicte said. \"Rare and pleasant.\" He turned back to the wall of portraits.\n\n\"Yes,\" Aris said, nearly a whisper, startled again at the warmth he felt near this lad. He cleared his throat. \"I have to admit to some surprise at finding you here. Adam's afternoon entertainments are known to be deadly bores.\"\n\n\"Then why do you attend?\" Maledicte asked, frowning at another painting of a corpulent gentleman in furs.\n\n\"I asked for the attendee list,\" Aris said. \"Once I saw that you meant to attend\u2014\" Pleased, he watched a fragile blush touch Maledicte's cheek.\n\n\"Oh, and what have I to say that lures you to brave such dullness as egalitarianism and economics?\" Maledicte asked, his tone a little stiff.\n\nAris caught Maledicte's quick glance at the guard and changed the subject. \"I thought you might tell me how goes Janus's courting? Michel knows nothing. And Amarantha\u2014\"\n\n\"You ask me that?\" Maledicte said, his voice dropping to an offended hush. \"Do you think Janus pauses in his wooing to send me accounts of it?\"\n\n\"I don't know what to think,\" Aris said, his own temper sparking. \"Janus has sent no word. To be honest, I had not expected such a political choice of wife from him. I thought him unconcerned with court ways and yet...Amarantha is a difficult woman at best.\"\n\n\"But very beautiful,\" Maledicte said, his tone still distant. \"And perhaps he thought the choice would be pleasing to you. A counselor's daughter.\"\n\n\"You hold his wooing her against me,\" Aris said. The weight of his distress startled him. But what had he expected?\n\n\"We all do what we must,\" Maledicte said, and to Aris, the quiet weariness in his voice sounded like forgiveness.\n\n\"Still, I hate to think I played a part in making you unhappy. Were it in my power\u2014I'd grant your wishes.\" Daring, he stroked the soft curls clustered at Maledicte's nape.\n\n\"I don't believe you would,\" Maledicte said, lips curling into a slow smile.\n\n\"No?\" Aris said. \"Won't you tell me what drives you? What brought you to my court?\"\n\nMaledicte looked up through dark lashes, his eyes merciless. \"Janus.\"\n\n\"Just that,\" Aris sighed. He took a step back. \"Is it true what the gossips say? That you knew him before? That you were\u2014\"\n\n\"A Relict rat?\" Maledicte finished. \"I fear my secrets are no secrets at all. But I'd rather not discuss that, if you'll indulge this wish. The past is past, Aris.\"\n\n\"No,\" Aris said. \"It's never past. Not when everything reminds you of what you've done or lost.\" He cupped Maledicte's chin, brushed his thumb over the soft lips, and watched the dark eyes shade yet darker. \"Was I wrong?\" he whispered. \"To surrender Xipos to Itarus, to barter our future away for our present?\"\n\nMaledicte laughed. \"You ask me? The war was before my time, sire.\"\n\n_That young?_ Aris thought, finding it a strange relief that here was one person who could not force him to relive the pains of his past. He drew closer.\n\nMaledicte's eyes flickered over Aris's shoulder.\n\n\"He's only my guard,\" Aris said. \"He doesn't matter.\"\n\nMaledicte slipped from his grasp. \"Aris\u2014how can you say that here? In Westfall's home, where all men are equal?\" The mockery seemed evenly apportioned between Westfall's follies and Aris's own.\n\nA sudden shout tore the quiet, and the mutter of voices in the next rooms rose.\n\nThe guard swore; the mastiff lunged to attention, pushing between Aris and Maledicte, growling. His hackles bristled.\n\n\"Find out,\" Aris said, resting a hand on Bane's withers. \"I'm safe enough.\"\n\nAnother mocking smile bloomed in response. \"Do you think your counselors, your _brother_ would agree to that? Alone with me\u2014and safe?\"\n\n\"Why must it matter so much what they think?\" Aris said, answering the tone and not the words. He felt suffocated.\n\n\"Because, sire,\" Maledicte said, \"you belong to them, not they to you.\"\n\nAris knew the truth of that by the weight it left on his heart.\n\n\"Do you still think my company safe?\" Maledicte asked, reaching out. He laid his hand over the king's. Bane's growling ceased.\n\nAris couldn't tell if it were his hand trembling or the hound beneath it. \"Safe enough,\" Aris whispered. He leaned forward to taste Maledicte's lips, to see if they were as sweet as they looked, or as bitter as the words he spoke.\n\n\"Nothing of importance, sire,\" the guard said, returning. Maledicte stepped away. \"Lady Mirabile's hem was torn by a servant. Done deliberately, she swore. Your brother is taking the whip to him now.\"\n\n\"That temper of his,\" Aris said, shaking his head. \"Find out whose servant I will need to replace.\"\n\n\"It's yours, Maledicte.\" The guard acknowledged Maledicte's presence for the first time.\n\nRage washed Maledicte's face, transforming it so utterly that Aris froze and Bane keened uncertainly. Courtier or not, Maledicte ran from the room like an arrow in flight. \"Go!\" Aris said to the guard, but to protect Maledicte from Michel or the reverse, he didn't know. Hand wrapped around Bane's collar, he followed more discreetly.\n\nThe green lines of the garden maze were marred with the violence. Westfall dithered, plucking at Last's sleeve, while Last worked the whip back into his palm. Beside him, Mirabile watched, a tiny smile on her lips.\n\nThe servant knelt, half-fallen, pressed back against the hedge, his shirt torn and the skin beneath bloody. He put his hand to the wound, heedless of the whip being drawn back again, and the look of such utter shock on his face told Aris that whatever flaws Maledicte had, beating his servants was not one of them.\n\nMaledicte interposed his slender shape between Aris's sight and the servant. A faint growl of pleasure rose from Last's throat as he set the whip flying again. The whistle of it sang in the air. \"No, Michel, I forbid it,\" Aris shouted, but the stroke had been sent.\n\nTurning as if possessed of a snake's quickness, Maledicte moved to meet the lash head-on. Aris flinched, but when the snap-crack of contact sounded, the scene had changed. Maledicte held the lash's tip in his hand, firm against Last's tugging. The air darkened and drew close.\n\nLast put his weight into the effort to free the whip, his face purpling with rage and embarrassment. Maledicte stood unbudging, his face remote. His other hand dropped to the hilt of his sword and drew a fist length of steel.\n\nBane growled, low and uncertain, crowding against Aris's hip, nudging Aris's numb fingers.\n\nMaledicte's gaze, as black as city smoke, fell on him and Aris looked away, unable to meet the empty rage in it. Then Maledicte's dark lashes fell and rose and he was simply another courtier in a temper. He released his grip on the hilt; the sheath swallowed the blade.\n\n\"If this is a taste of your vaunted equality, Westfall,\" Maledicte said, \"and the people you choose to build futures with, I don't fancy your chances.\"\n\nWestfall flushed.\n\n\"You insult the king,\" Last said.\n\n\"Do I?\" Maledicte asked, tugging his servant to his feet and supporting him. \"I thought I insulted you.\"\n\nLast snarled, his hand clenched on the whip handle again. \"I will insure that Janus never sees you again.\"\n\n\"Janus cannot leave me alone. He's mad for the touch of my hands, my mouth; he begs for me at night. When you are dead, I will lie in your bed while you lie in the ground, one more unmourned ancestor, and I will be free to do as I see fit,\" he said, his voice so laced with venom that Aris half expected to see Last finally subject to apoplexy.\n\n\"May I take my leave, sire?\" Maledicte asked but turned away without waiting for an answer.\n\n\"Mal,\" Aris said, his voice rough. \"Are you\u2014hurt? Your hand is bloody.\"\n\n\"It's not mine,\" Maledicte said, closing his fingers over the clotting gore. He turned his head and said something softly to the servant, and then they made their way out of the garden. Mirabile dropped a curtsy as Maledicte neared her, and he widened his path to give her a wide berth. Aris watched and worried. Maledicte had faced Last without hesitation or the merest sign of fear, yet skirted Mirabile. Her expression was no longer that pleased half smile, content that a man was whipped for her whim, but something darker, and far more calculating. She swept back into the house, her skirts trailing behind her, undamaged despite her claims otherwise. Aris frowned.\n\n\"It's witchcraft,\" Last spat, drawing his attention. \"Did you not hear anything he said?\"\n\n\"I heard a young man in a temper, showing remarkable loyalty to a servant,\" Aris said. He turned his unease on his brother, and his tone was cold and unwelcoming.\n\n\"You are witched. Do you think I failed to notice you seeking him out? Ask yourself what draws you so?\" Last said again, and as if he meant it more than angry words. \"Ask yourself what kind of man can take the lash's touch unscathed?\"\n\n\"A proud one who refuses to acknowledge hurt,\" Aris said. \"What man brings a whip to beat a clumsy servant and turns it on a peer of the realm? I think you've been too much in the city, Michel. Some rest might suit you. Go home to Lastrest.\"\n\nLast grated out, \"As you command, sire.\" He turned toward the stables, and said, more temperately, \"He is dangerous, Aris. I hope you never have cause to regret the license you grant him.\"\n\n\"Michel,\" Aris said, his temper fading, but Last walked away. Aris sighed, and sat on a garden bench, taking care to choose one that did not overlook the hedge where blood still spattered the leaves and lawn.\n\nMICHEL IXION, EARL OF LAST, lay in wait. He had arrived at Lastrest to find that Janus and Lady Amarantha were riding the grounds, but his temper demanded instant expression. So instead of busying himself with his correspondence, his bills, his petitioners, he sat in his reading room, the double doors wide to the hall and the front door, the leaded windows opened over rosebushes and the smoothly clipped lawn, waiting for the first sound of their return.\n\nA day and a half 's hard travel and distance from Maledicte's insolent mouth had not eased his temper; firecracker spurts of rage still flared beneath his outward composure.\n\nHe heard the hoofbeats first, drumming toward the stable, hooves spattering the fine gravel of the drive, and he frowned. They should have more care for the tender hooves of his livestock. Even as he thought that, he heard a woman's voice raised and the horses slowed.\n\nSome minutes later, the front doors opened and two sets of footsteps rang against the marble floor of the foyer. He rose and went into the hall. Janus and Amarantha dallied there, shedding gloves. Amarantha's cheeks were flushed, and Last thought the heat under her skin due to temper and not exertion, judging so from the rigidity of Janus's smiling countenance, from the slap of her leather gloves striking the hall table.\n\n\"Janus, I must speak with you at once,\" he said, in lieu of greeting.\n\n\"If you must,\" Janus said.\n\n\"At once,\" Last repeated, irritated anew at Janus's laconic acknowledgment.\n\nAmarantha said, in a remote tone, \"Courtesy is owed to one's elders.\"\n\nJanus's face went brittle; then he recovered his smile. \"Father, have you met Lady Amarantha?\" Janus said, drawing her forward. \"I believe you well acquainted with her parents.\"\n\n\"It's been some time since I've last had the pleasure,\" the earl said, frustrated in his desire to speak his mind at once. He bowed with careful formality.\n\nLady Amarantha shrugged out of her riding coat, ignored Janus's waiting hand, and dropped it over her gloves. She curtsied and held her hand toward Last. \"I remember you, of course. Father speaks highly of you. I am most grateful for your hospitality, my lord. This is a lovely and well-ordered estate.\"\n\nLast took her hand in his and raised her curled fingers to his mouth, thinking that Lovesy's daughter had grown into an uncommonly pretty woman. \"I hope Janus has been making you welcome.\"\n\n\"He has done his best,\" Amarantha said. \"If you will excuse me, I will change.\" She nodded again in the earl's direction before ascending the broad stairs.\n\n\"She seems a bit short with you,\" Last said, bemusement doing what time had not, cooling his temper. \"What have you done?\"\n\n\"Nothing pleasing, according to her,\" Janus said, looking after the retreating tail of her riding habit. \"However, I have vowed reform.\"\n\n\"Yes,\" Last said, his brows folding downward again. \"Come with me.\" He led the way into the study, closing the doors behind them.\n\n\"I am at your disposal,\" Janus said, sketching a bow, darkening the earl's frown.\n\n\"Perhaps she finds your manner flippant. It is remarkable that Lovesy will even consider a match. Were I you, I would not jeopardize my chances with mannerisms borrowed from a scandalous source.\" Last closed the windows, and turned the key of the gas lamp up to push away the resultant dimness.\n\n\"Maledicte, you mean,\" Janus said, sitting opposite Last's study desk.\n\n\"I do,\" Last said, his temper flaring like a wick touched by flame. He took his chair, aware of protesting stiffness in his hips and other joints. Once, the journey from the city to Lastrest took him a day, galloping on horseback. Now, a carriage trip left him stiff and sore.\n\n\"This marriage of yours is well thought,\" he said, testing Janus's mood, since the calm face revealed nothing. \"Despite the story we have put about, you are nothing but a bastard.\"\n\n\"And Amarantha is no longer virgin. Which is more scorned by the court? Bastard or slut?\" Boredom flashed over his face; he stifled a yawn.\n\n\"You will not speak of a lady in that fashion.\" The earl continued, \"I am not as unquestioning as Aris. To truly convince me of your sincerity to the line of Last, you must end your unsuitable alliance with that creature and eradicate the unfortunate influence he has had on your tongue and manners. Or I will take steps of my own.\"\n\nJanus smiled. \"Every boy sows his wild oats to the distress of his parents. Actresses, whores, dancers, peasants...scandalous courtiers. I daresay Mal's attraction will pall; for now, I find too much enjoyment in his company to be rid of him so precipitously. But I will endeavor to convince Lady Amarantha and yourself of my sincerity toward the line of Last.\"\n\n\"Your creature thinks to set up house here,\" the earl said, his tone darkening at the memory.\n\nJanus sighed. \"Such a wild mouth on him. He says things merely for effect and to watch people rage. You must have pleased him mightily.\"\n\n\"Until I took the whip to him,\" Last said.\n\nThe distant amusement in Janus's face vanished, then reappeared so quickly that if Last had not been studying Janus closely, he would have missed it. The realization chilled him. His son was not only a stranger, but an able liar. Last was forced to recognize that he could not believe anything Janus said. That phantom blaze in the pale eyes, that hot rage, convinced the earl of one thing: Maledicte, wretched creature that he was, spoke truth. More truth than he had from this stranger sitting across from him. Dread seeped into his bones, quenching his anger with caution.\n\n\"Go away,\" Last said, rising from his seat. To make too much of the conversation would be to reveal his shock, so he resorted to a simpler anger. \"Repair your tongue before you speak with Amarantha again. We will discuss your influences when you are in more of a humor to take this seriously.\"\n\n\"Sir.\" Janus bowed and left the room.\n\nLast sat alone in his study, thinking. Perhaps Maledicte, with his open enmity, was the lesser danger. Resolve tightened his mouth, shook the last trepidation from his mind. Janus would not find his way to the title so easy. Last would insure that.\n\nA folded letter caught his attention; discarded a week ago, now it beckoned him. Dantalion bemoaned that his agent had disappeared and asked counsel. Last had further advice for him now.\n\nLooking up from his work with quill and parchment, he saw Lady Amarantha descend the stairs in a wash of buttery yellow that set off her pale beauty like a diamond set in gold. Rising, Last went to the window, broke off the first scarlet blossom to hand, and met her in the hall.\n**\u00b7 24 \u00b7**\n\n**W** HAT SAYS JANUS?\" GILLY ASKED, twisting in his seat, testing the sting and pull of healing scabs. He glanced over the emptied plates and soiled silverware to Maledicte, lounging in his chair, boots on the table, letter in hand.\n\n\"Stop squirming, you'll have them open again. And as you were such a baby about their bandaging, I'd prefer not to do it again,\" Maledicte said without looking up. His booted feet knocked over his wineglass as he shifted; his brows furrowed as he squinted at the letter, a little beyond the range of candlelight.\n\n\"Janus says that all is well, save that Last and Amarantha are trying to drown him in examples of proper etiquette. He asks if I am well, after Last's assault with the whip\u2014hmm, some misunderstanding there, I see, though perhaps Last has rewritten his memories.\"\n\n\"I don't wonder at it,\" Gilly said. \"There seems to be a good deal of that in the court.\" Maledicte shrugged, leaving Gilly to his own dark memories, that he had not gone within ten feet of Mirabile, fearing her nails, her temper, and yet, they all believed her when she spoke against him. He fisted his hands, wishing it were only that her words had the weight of aristocracy behind them, and fearing it was more.\n\nMaledicte continued reading. \"He writes that Amarantha is a difficult woman but that Last seems inclined to make the match work, spending time with Amarantha when Janus cannot.\" Maledicte reached for his glass, chuffed in disgust when he found it tipped and his fingers wet. He rose and went to the sideboard to pour himself another glassful.\n\n\"I don't like that,\" Gilly said.\n\n\"No,\" Maledicte agreed, concern pinching his features. \"Are you going out tonight?\"\n\n\"I meant to. I'll stay if you want me.\" Gilly reconsidered his visit to Lizette's. He felt as if he were back in the country, watching his family race to protect their crops from a looming storm.\n\n\"And deprive you of your fun? No, if I do that, you'll make do with Livia, and I'm running low on Harlot's Friend. I swear you are a worse rake than Vornatti.\"\n\n\"Quantity makes up for quality,\" Gilly said, startled as the words came out of his mouth, unsure of his own meaning. Belated awareness dawned red and hot on his face and neck. He pulled his glass back to him and drank, avoiding Maledicte's eyes, trying not to let Maledicte see his hunger. Quality, he thought, letting his eyes rest on Maledicte's pale hands.\n\nMaledicte laughed. \"And they think my tongue vicious. If that is the case, by all means, go to your girl.\"\n\n\"I can stay,\" Gilly said.\n\n\"Don't be foolish,\" Maledicte said. \"Go on, then. She'll be disappointed, otherwise.\"\n\n\"Only at the lack of coin,\" Gilly said, conscious that he wronged Lizette, but torn between her softness and the odd coziness of being alone with Maledicte.\n\n\"You sell yourself short,\" Maledicte said. \"Those shoulders, that sweet face.\"\n\nHis gaze raked Gilly from head to toe in a speculative, appreciative fashion and a second flush of embarrassment rushed Gilly's skin.\n\nGilly remembered Maledicte bandaging the whip's stripe, bent so close that his hair had brushed Gilly's bared chest. It had been that touch that set him squirming. He stood, half-naked, with Maledicte's light fingers on him, and yet Gilly had never so much as seen the white lines of Maledicte's shoulders or the sinews of his back or thigh.\n\nGilly had shifted from foot to foot until Maledicte laughed. \"If you're that agile, you can't be much hurt,\" he had said, \"But really, Gilly, did no one ever teach you to step away from the whip?\" He stroked the last plaster into place; it tugged the skin taut over his ribs. Gilly winced, desire and embarrassment forgotten.\n\n\"This hurts out of all reason,\" he said.\n\n\"It's the hatred behind it that festers,\" Maledicte said. \"Be glad you were not born a sailor on one of your beloved ships, or you would have felt the whip long ago.\"\n\n\"Not I,\" Gilly said, \"I would have been a perfect sailor. They would have made me first mate by now.\"\n\nMaledicte laughed. \"With your hair bleached white, and your skin burned brown, running around in torn breeches\u2014\" He smiled at Gilly with a considering light in his eyes and Gilly felt warmth bloom deep within his body. Maledicte stepped closer, stroked Gilly's cheek, and Gilly's ardor cooled. He moved away from the touch, from Maledicte's hand.\n\nShadows chased themselves across Maledicte's eyes.\n\n\"Your hand,\" Gilly said. \"Where you caught the whip\u2014your injury's gone without a mark. I still bleed.\"\n\nGilly remembered that now, as he dithered between leaving and staying, the storm-cloud feel of fate in the air, and the lingering presence of Ani in Maledicte's eyes.\n\n\"What, a compliment renders you dumb?\" Maledicte said.\n\nGilly flickered a smile, and nodded. \"I will go out.\"\n\nHe shrugged on his coat against the evening's coolness and opened the front door, startling a young messenger, hand upraised to knock. Gilly closed his eyes in acknowledgment. The storm, he thought, had come.\n\n\"Message?\" the boy squeaked, still nervous.\n\n\"Yes,\" Gilly said, \"I'll take it.\" He handed the boy a coin, and went back in, holding the sheet sealed with blue wax as if the weight of the world hung from it.\n\n\"Not gone?\" Maledicte said.\n\n\"Janus sent this,\" Gilly said. He dropped it before Maledicte, on the white tablecloth marred with spilled wine.\n\nMaledicte broke the seal. He let the note fall to the table, his face white, his fingers clenching on open air as if he would do battle if he only had the blade at his side.\n\nGilly recognized Janus's black scrawl at once. There was only one line on the note.\n\n_Father wed Amarantha Lovesy last night._\n\n\"I should have killed him the moment he first stepped within my reach and damned the consequences,\" Maledicte said, surging to his feet.\n\n\"Only a setback,\" Gilly said, trying to keep the bloody light in Maledicte's eyes from flooding outward.\n\nMaledicte hurled first his plate against the door, then Gilly's; at the shattering china, a maid poked her head in and as quickly withdrew it. \"Setback? Last has wed. Had he ever intended Janus to follow him, he would never have married. It is more than a counter, Gilly, it's Last's declaration that he will stymie us any fashion he can. I've waited too long, too lulled by your talk of consequences and Janus's mercenary considerations and caution.\"\n\n\"And how will you kill Last? His murder will be treason, Mal. Janus might well inherit by murder, it's happened before, but you...you would be forced to flee. Alone. Or do you think Janus would give up a title for\u2014\"\n\n\"Get out, Gilly,\" Maledicte said, voice thinned to a thread. The sword bloomed in his hand and the black light rose in his eyes.\n\nGilly edged away from the blade that tracked him, from the insane anger in dark eyes, until he reached the door.\n\nIn the center of the room, Maledicte sparred with shadows that crawled out from the walls to meet him. Maledicte shivered as if in a fit, the shadows ripping at the touch of the blade, spilling slow darkness. Kritos's dark shape, last seen lunging toward them in an alley, rose up behind Maledicte, and Gilly gasped. But Maledicte turned like a somnambulist and struck him down once more. Remembered triumph lit his face and he slowed his swordplay, tension in his eyes slackening, his lips parting. He shuddered all over, and he pressed the sword into the floor, skewering Kritos's corpse. He slumped. The shadows hovered.\n\nGilly returned in a rush, bearing Maledicte away from the blade, grabbing the slender shoulders hard enough that he knew Maledicte would bear bruises tomorrow, if Ani allowed it. Maledicte emitted a wordless shriek of rage, a vibration in the ravaged throat, and then Gilly was holding a madman.\n\nGilly yowled as Maledicte's teeth bloodied his knuckles, as Maledicte's elbows found his sore ribs. But he held on, like trying to cage a raptor, pinning it without damaging the fragile bones, all the time aware of the beak and the talons and the snapping edges of wings pulled taut.\n\n\"You cannot give in to this,\" Gilly said, his mouth near choked with Maledicte's hair. \"Now, more than ever, you need reason and patience, not blind rage. You must be cunning, must be clever, must be careful.\" His breath was rapid, his words mere pants of air and sound, and he wondered if they still contained meaning. He pinned Maledicte's arms behind his back, unnerved at the bowstring tension in them. If he let go, what would happen?\n\n\"Ani loves blood for blood alone. Vengeance begets death and nothing more. You tell me She does not rule you\u2014make me believe it. Or you'll end as others have, mad or dead.\" In a desperate hope, he whispered one of the little paeans to Baxit, god of indolence and reason. Either aspect would aid Gilly now.\n\nMaledicte went limp in his hands, slumping forward like a puppet freed from its strings. Gilly, taking the sudden weight in his arms, found his pulse hammering with more anxiety. Was this a feint? If he let go, would he find Maledicte coming back at him and with the sword in his hand? Beneath him, Maledicte grumbled.\n\n\"Mal?\"\n\n\"Get off me,\" Maledicte said. \"What are you doing? Trying to wrench my arms out of true over a few broken plates?\"\n\nGilly slackened his vise-grip on Maledicte's wrists, feeling like a bullyboy, aware of how Maledicte's slender bones ground beneath his hands.\n\nGilly released him and Maledicte turned, his face filled with simple irritation. \"Rats take it, Gilly. If you're not even going to let me throw a tantrum in my own house, there'll be no fun left at all in being wealthy.\"\n\n\"A tantrum,\" Gilly said. \"Is that what you call it?\"\n\n\"Two broken plates. And you knock me down,\" Maledicte said. \"And my sword? Were you trying to break it, wedging it in the floor like that? You've bloodied your knuckles on it and serve you right. I need it to kill Last.\"\n\nMaledicte stepped over and put his hand on the hilt, tugging it free. Gilly flinched, but Maledicte only complained, \"I've lost the sheath again.\"\n\n\"It's probably still abovestairs in your room,\" Gilly said, his lips numb with shock. Did Maledicte remember nothing of the moments between broken crockery and Gilly's hold?\n\n\"I don't carry the blade unsheathed. I am not the callow boy I once was.\" Maledicte sat at the table, laid the blade out before him, ran his fingers down the steel, sheeting blood from the blade as if the shadows had been flesh enough to bleed.\n\n\"Gilly?\" Maledicte said. \"Did you hurt yourself badly on it?\" A faint tremor touched his fingers.\n\nGilly touched his own hands, scraped and bloodied, hands that had never touched the blade at all. \"You came down without the blade,\" Gilly said. \"It came down later.\"\n\n\"Ridiculous,\" Maledicte said, his tone so uninflected that Gilly had no way of telling if Maledicte believed him or not. \"It's only as word. But think how useful it would be to have one that heeled like a dog.\" Maledicte's lips lifted in a movement too faint to be smile or snarl, though it held something of both, and more, the death's-head rictus of a gallows corpse. Gilly, picking up the largest pieces of shattered china, flinched at his expression, and watched his blood roll over the porcelain. For the first time in a long, long while, Gilly found himself frightened of Maledicte and the violence that eddied around him like storm winds, merciless to friend or foe alike.\n\nJANUS ARRIVED JUST AFTER TWILIGHT. Gilly opened the door to him silently, still listening to the quiet voice within himself warning that Ani was gaining Her ascendancy with each passing day.\n\n\"Where is Maledicte?\"\n\n\"In his rooms.\"\n\nJanus hesitated on the stair. \"Angry?\"\n\n\"Possessed,\" Gilly said. The ugly word hung between them.\n\n\"Rats eat your nonsense.\" Janus turned. \"Maledicte is no one's creature; there is no god of love and vengeance anymore.\"\n\n\"Maledicte made his vow to Her. He's bound to Her as surely as he's bound to you.\"\n\nJanus's face flushed, quick temper rising in his cheeks, then fading. \"You know so little of him. Maledicte has an odd and morbid sense of humor, prone to elaborate charades.\"\n\n\"I know him well enough to declare that humor, like perspective, is something Maledicte lacks,\" Gilly said.\n\n\"Perspective?\" Janus said, eyes paling further yet, until Gilly was minded of distant lightning.\n\n\"He is determined to see you earl, even at the cost of his own life, while you sit back and allow him the risk\u2014\"\n\nGilly didn't see Janus move as much as felt the rush. Then his back slammed against the flocked wallpaper with enough force to set little motes of color floating free. His head and ears rang with the impact. Like an insect pinned, he struggled until Janus's gloved fingers closed on his throat. Janus pinched the pulse on either side of Gilly's neck with delicate inquiry, his face placid, his eyes mad.\n\n\"I could choke your life away and Maledicte would forgive me. He would forgive me anything. Remember that before you speak so. You know nothing of me if you think I would endanger Maledicte. Do you not listen? I've told you time and time again. He's mine.\" Janus released Gilly, ghosted past him and up the stairs.\n\nGilly shuddered, left to himself in the empty hall. Again the _Virga_ crept into his mind, the siren song of sea and sail.\n\nMALEDICTE HEARD THE DOOR open and close, unheralded, and knew, without looking away from the window, that Janus had entered. Only he made himself so free in Maledicte's home. But numbed with rage, Maledicte waited for Janus to come to him. Janus's hands rested on his shoulders, turning him, drawing him close. \"Janus,\" Maledicte whispered.\n\nJanus buried his face in Maledicte's neck. \"I am sorry,\" he said, his voice roughed with exhaustion. \"Sorry I did not win her hand. Sorry I made mincemeat of my duty. Though by the gods, I am not sorry to see her tongue hitched to another.\"\n\n\"Will you loose me on him now?\" Maledicte said, raising his eyes to Janus's. \"He plots against us, Janus. He'll have you prisoner, soon enough, confined to Lastrest or the town house, watched and spied upon.\"\n\n\"He already does that,\" Janus said. \"Haven't I crept away from their wedding reception? But kill him? Not without a plan.\"\n\nMaledicte stepped out of the warm circle of his arms. \"My blade, his heart. What more plan do I need? Set me free to act. Didn't I rid us of Kritos?\"\n\n\"Kritos was a rogue and a gambler. That he died as he did was not worthy of comment or even much concern. To beard Last, who rarely leaves the courts, or his estates, is a far thornier problem.\"\n\n\"I want him dead. Need him dead. Blood flows the same regardless of one's surroundings. If it goes wrong, if suspicion brands us guilty, we can quit the earldom entirely. We'll gut his estate and live like kings in the Explorations.\"\n\n\"Flee like rats? Die like rats, when the money runs out?\" Janus said, shaking Maledicte. \"I'll not revisit that life again.\" His hands stilled on Maledicte's shoulders, caressed where they had bruised. \"You remember hardship. Even were you minded to risk it again, it would be worse than you think, knowing what we do now. The taste of fresh bread in the morning. The warmth of fireplaces in the winter.\" Janus kissed Maledicte's mouth when he started to speak again, sealing the words away with his tongue.\n\nJanus walked him back toward the bed, and when their lips parted, continued, \"The softness of silk on our skin, the luxury of sheets and velvet coverlets. It's more than luxury, Mal. It's safety.\"\n\nMaledicte fell back into the feather mattress, his shirt falling open, the corset sliding down to his waist. Janus pinned Maledicte's hands above his head, buried them in the soft drape of down pillows. \"I will endeavor to put Last in the path of your blade, winkle him out of his secure areas, but you must wait for it. I don't intend to lose you to a verdict of treason.\"\n\nMaledicte looked up at the pale eyes, kissed the fine scattering of gold down at the back of Janus's neck. \"Make it soon?\"\n\n\"Very soon,\" Janus whispered. \"My dark and bloodthirsty cavalier.\"\n\n\"Time is our enemy,\" Maledicte said. \"Amarantha must not\u2014\"\n\n\"Shh,\" Janus said, \"It's been near a fortnight without you.\"\n\nMaledicte gave himself over to the familiar, marvelous touch; sighing, moaning, biting where such response was called for, and all the time he thought of Last's machinations working against them. Again he found himself balancing the simple act of vengeance and flight weighed opposite this elaborate charade of parry and counterthrust, of politics and power. The rushing pleasure that lit Janus's face only dimly touched Maledicte, lost in his thoughts of blood and patience.\n\nJanus disentangled himself from the loose knot of their legs, stretched, and said, a little crossly, \"Even whores feign their pleasures.\"\n\nMaledicte stroked the furrow on Janus's brow. \"Whores do not have such schedules and schemes as I do. Amarantha is a threat to us. But to kill _her_ would be to only delay the problem. Until you deliver Last to my blade, she might quicken and deprive you of your birthright. We must keep her barren until I have Last's heart spitted.\" He slipped from Janus's grasp, the comforting softness of the bedsheets, and opened his chest of poisons. Dragging his finger along the vials until they sang, he found the first glimmer of pleasure. It wasn't death, but it was a small vengeance even if it moved through Amarantha instead of Last.\n\nHe pulled out a handful of waxed paper twists, each pale green at their heart, and returned to the bed, spilling the papers out between them. \"Harlot's Friend,\" he said. \"You recognize it?\"\n\n\"Yes,\" Janus said. \"Our mothers used such to prevent pregnancy. But Mal, how am I to get her to drink it daily?\"\n\n\"Daily prevents. Monthly\u2014she'll miscarry if she's gravid. Rougher on her, but far easier for you. All you need do is slip it into her wine. Though I warn you now, Janus, I will not tolerate a wait of months.\"\n\nJanus tucked the twists of paper into his pockets, sat up and began straightening his clothes, fastening buttons and retying laces.\n\n\"Janus?\"\n\n\"I'd best be back before my absence is noted. Now that Father feels he has the upper hand, he has laid new strictures on me.\"\n\n\"Were he dead, he could set no such rules.\"\n\n\"Mal\u2014\" Janus warned.\n\nMaledicte looked away. \"If you're bent on leaving, there are fresh linens in my armoire. You'd best have one. I've clawed your cravat past discretion.\"\n\nJanus searched out the snowy lengths of cloths, sorted through them while Maledicte watched with an amused eye. \"The one on the end is starched silk, if that's what your pampered skin demands.\"\n\n\"Tie it for me,\" Janus said.\n\nMaledicte let the sheets unwind and reached out. \"Ever the aristocrat\u2014lift your chin more, hmm? I wear my linens to disguise. You will not leave yours off even for a covert jaunt across town.\"\n\n\"Dress is the easiest aspect of rank to mimic and a visible sign of breeding,\" Janus said, rising to check his appearance. He ran his fingers through his hair, smoothing the sleek locks. \"Westfall may gad about town like a rustic in shirtsleeves and open collar...he has ten impeccable generations behind him. But if I appear with a veneer of dust marring the shine on my boots, it is because I am only a bastard, aping my betters.\" Janus curled his mouth in a smile unlike his usual pleasant one. \"But a title will change that. Or at least take the whispers from my hearing.\"\n\nMaledicte moved Janus's sweep of hair aside, kissed the nape of the neck, silk stiff under his lips. \"When will you be back?\"\n\n\"Two nights from now. Father and Amarantha attend another wedding reception,\" Janus said. \"I will creep out of my window like some lovesick suitor, and come courting.\"\n\n\"Promise?\" Maledicte asked.\n\nJanus turned, cupped Maledicte's face in his hands, pressed their foreheads together. \"I swear.\"\n\nLOOKING OUT over the crowded room, the Duke of Love turned to Last and said, \"Has your son forgiven you? I saw him only briefly and thought him a little grim.\"\n\n\"It matters not,\" Last said, sipping from his glass. \"His spirit has been too independent for my tastes. This may chasten him. He is too much Celia's son, willful and selfish.\"\n\n\"Some of that may be due to venal influences,\" Love said.\n\n\"Maledicte,\" Last growled. \"That damned...I'd see him gone, only Aris is unaccountably fond of him.\"\n\n\"Yes,\" Love said. \"Have you seen this week's paper? And Poole's scurrilous caricature? I've set Echo on the artist, but unless Aris objects, nothing will come of it.\"\n\n\"I missed it,\" Last said. \"But it can hardly be worse than Poole's previous images, my brother the king of slavering hounds.\"\n\n\"It's vile,\" Love said, drew Last aside, and into the quiet recesses of his study. He unfolded the broadsheet.\n\nLast's mouth parted and his cheeks flushed. The image was drawn with careful simplicity, not the usual cluttered style of the artist, but something designed to be quickly memorable. Maledicte, all dark hair and lush mouth, lounged on the throne, his sword naked in one hand, and in the other\u2014a long leash fastened about Aris's neck. Aris himself had been drawn so that his face was nearly witless, blindly turning away from the corpse labeled Vornatti at his feet, looking up at Maledicte with adoration. Last's hand crumpled the page. \"Tell Echo to jail Poole. We'll see if he has influence enough to free himself. And then we'll know who aids him in spreading such slander.\"\n\n\"Mirabile, no doubt, started the rumor; her whispers have increased tenfold in scope. And people believe where before they only listened,\" Love said. \"But that still leaves us with the ultimate source of your troubles and mine. Maledicte.\"\n\nLast sighed. \"I thought that problem solved once. Dantalion, Vornatti's displaced heir, sent an assassin, but either the man failed or was bought. I believe Dantalion plans to handle it himself once his position as Antyre's new auditor is official.\"\n\n\"A foreign assassin on our shores? Dantalion hunting our courtiers?\" Love said, scowling. \"Itarus is too free with their manners and too blatant in their hatreds. You should have mentioned such to me or Captain Jasper at once. Had he succeeded with Maledicte, he might have turned his attentions to the throne. Itarus would reward an assassin well for Aris. Still\u2014\" He raised his head and met Last's eyes. \"The idea is sound. Perhaps we can accomplish the task on our own. I have a manservant who might be of assistance. A man I've used before to rid myself of a troublesome stableboy. Why not a troublesome courtier?\"\n\nLast smiled. \"Do this thing, and I'll renegotiate Amarantha's bride portion.\"\n\nLove said, \"Let me grant you this as a wedding gift. All you need do is insure Janus's absence from Maledicte's side. Despite his posturing, I think the boy only a stripling. On his own, he should prove no threat.\"\n**\u00b7 25 \u00b7**\n\n**L** ATE TE NIGHT, the town house fell into silence as the maids retired, the cook settled to sleep, and the empty butler's chambers cooled. In his bedroom, Maledicte alone sat awake. In the hall below, the great clock tolled out the quarter hour, and Maledicte set his book aside. With every chimed note, Janus's promised arrival grew less likely. With every chime a new fear took hold.\n\nPerhaps Last had drugged Janus and sent him abroad, a prisoner until Amarantha's worthiness could be proved. Or maybe Last had chosen to gamble the future entirely on Amarantha and had drowned Janus like an unwanted cat. The old wounds on Maledicte's side and face burned, a scorching reminder that Last had defeated Miranda before, had robbed her of Janus before.\n\n_Before Me_ \u2014Ani's whisper throbbed in his blood, turning fears to fury, but without any comfort. Janus had no such protection. Ani's wings sheltered Maledicte only. Maledicte picked up the sword, dancing his way across the room with it, stabbing and slicing as if his fears could be defeated by bladework. Were Gilly home and dreaming, Maledicte would creep in to find solace in the placid face. He would wake him to hear him grumble and find his fears pushed back by the familiar voice. But Gilly was out, visiting the damned brothel girl.\n\nWith a flourish of the blade, Maledicte whirled and took his sudden temper out on the heavy mahogany door. Red wood peeled back like parting flesh and Maledicte sighed.\n\nHe discarded the sword, a little ashamed of his tantrums, ashamed of his doubts and fears. Janus would come.\n\nBehind him, the air shifted, the dark scent of the night fogs creeping in. Maledicte turned to see the windows parting and a man surging through. He leaped for the sword, his fingers falling short as the assassin lunged forward, and tumbled them both across the room. Maledicte fell to his knees beneath the brute's weight, and, snarling, turned to claw at his face.\n\nThe garrote caught him by surprise, still warm where the assassin had held it close. It fell over his head, and, sword forgotten, Maledicte strove instead to deter the closing noose. His hands changed direction, forcing panic away for planning. Instead of clawing at the assassin, he clawed at the wire, managing to slip a hand up between the tightening loop and his neck.\n\nMaledicte wheezed for breath, forcing his hand farther through the loop, skinning the flesh on his forearm, and nearing the elbow. If he could only get the wire past his elbow, he could slip out of it, its chokehold vanished with the inclusion of shoulder and rib. But the wire tightened, making it hard to muster the energy needed. Doggedly, breathlessly, Maledicte wormed his arm up another inch; the assassin rolled them both forward and put his knees in Maledicte's back.\n\nMaledicte could see the sword now, only a yard away, and thought if Gilly were right, if the sword could come to him at will, now would be the time above all. Maledicte's last breath faded and fled, scorching his lungs with its haste. Where was Ani's aid now? Or was this assailant unworthy of Her notice, being no part of their bargain.... Maledicte's blood drummed in his ears; he clawed backward, one-handed, trying to find leverage.\n\nThe door flew open and the assassin's grip slackened in surprise.\n\nMaledicte sucked in air, put his shoulder up, and squirmed through the garrote loop, heading for the sword. His hand was on it when he heard the crash. Turning on unsteady feet, he saw Gilly holding the downstairs poker and standing above the assassin. One look at the huddled form and Maledicte dropped the blade. \"You saved me the effort. Do I thank you?\" he croaked.\n\nWhite to the lips, Gilly held his death grip on the dripping iron of the poker as if frozen in horror and Maledicte repented his flippant words.\n\nHe took Gilly by the shirtsleeve and steered him away from the corpse, opening his fingers and letting the poker fall.\n\n\"I saw him, a shadow going through the window. I didn't think I'd make it.\" Gilly's voice trembled and faded like an amateur singer at her debut.\n\n\"You did,\" Maledicte said, pressing Gilly back into the softness of the chair, tucking a blanket around his shoulders. \"You're shaking.\" His own composure was returning, the familiar rush of breath and anger, both so essential to him.\n\n\"I just\u2014Did I kill him?\"\n\n\"He's not like to rise after you knocked that piece of skull loose. Tell me, did you ever play at stick as a child?\" Maledicte poured a glass of brandy for Gilly, curled his fingers around the heavy crystal.\n\nGilly raised the glass to his lips, managed a sip, and then let it settle in his lap again. \"Are you hurt?\"\n\n\"Only knocked about.\" Maledicte sat down on the bed and rubbed his throat with tentative fingers. His other hand drummed and twitched with residual nervousness and he started to his feet again.\n\nGilly passed him the brandy glass. Maledicte tossed it back, coughing as the liquor hit strained flesh.\n\n\"He's the duke of Love's man,\" Gilly said, shifting in his seat so that his head rested against the sheltering side. \"I recognized him as I struck. I forget his name, but not his face.\"\n\n\"The duke? What wrong have I done him?\" Maledicte felt as bewildered as a child, the clarity of battle gone.\n\n\"The king's favor, perhaps,\" Gilly said. \"The scandal of it. Or Last may have borrowed him.\"\n\n\"Last,\" Maledicte said, rather more pleased than not to gain another reason to hate the man. He knelt beside the body and began rifling the assassin's clothing. An inner pocket yielded a note on expensive parchment. He unfolded it, trying to keep the blood sliding down his arm from obscuring the words.\n\n\"What is that?\" Gilly said, roused from his shock by the sound of paper. He took it from Maledicte's hands. \"It's instructions, but no names, not even yours.\"\n\n\"Well then, if he can't be useful, let's get him out of here. I swear he's beginning to stink.\" Maledicte picked up the poker. \"Do you have a lucifer on you?\"\n\n\"You're not going to burn him?\"\n\nMaledicte found a reluctant smile at Gilly's evident horror. \"Only the blood and hair on the poker. He's far too fresh for that kind of thing. But I am open to suggestions, Gilly. The night will not last forever, and I am tired. You know everything; tell me what to do with him.\" Maledicte's tone was soft, conciliatory.\n\n\"I never had to know how to dispose of corpses until I took service with you,\" Gilly said. He stared down at the body, at the ruined head, and flexed his fingers. \"We bluff.\"\n\nTHEY STAGGERED QUIETLY down the dim, carpeted hall, past the sweeping main staircase, to the narrow attic stairs, the corpse hanging heavy between them. Gilly's hands, wet with sweat, slipped on the man's booted ankles, and he made a hasty catch to prevent them from banging on the stair riser. Maledicte, breathing harshly, signaled a rest, propping the corpse's torso up along the wall. Gilly looked away from the staring eyes, hating that this was his idea, and so disallowed him complaint\u2014his idea to turn the assassin into an unlucky thief who had fallen to his death.\n\nMaledicte, kicking speculatively at the cooling body, had suggested his bedroom window as the \"thief 's\" point of attempted entry, but Gilly pointed out the plush lawn and rose beds below. A man might break his neck, but not open his brain box in such a fall to such a surface.\n\nSo they carried his weight toward the high attic window, where the decaying trellis made plausible both an attempt at entry and the successive fall to the stone path below.\n\nMaledicte's other suggestion\u2014that they claim self-defense, and allow Echo the pleasure of removing the body\u2014Gilly had rejected out of hand, citing the inadvisability of allowing Echo within Maledicte's rooms at all, given the man's desire to prove Maledicte culpable of something, anything. Gilly had held his breath until Maledicte agreed, but knew from the look in the sharp, dark eyes that his real motive had not gone unnoticed: If it were a thief and an accident, then Gilly would not have to dwell on the thing he had done....\n\nGilly looked up to find the dull shine of the corpse's eyes fixed on him. Something slick and wet slid down the dead face, reminding him of a melting waxwork. Gilly fought bile. No wonder the aristocrats dueled; they could kill someone and be away, even as the body fell. They never had to tidy up after. Maledicte's cold fingers touched his wrist, startling him.\n\n\"Stop gawping. You've been blooded now and no distress on your part will undo it. Take his ankles and be done with your vapors,\" Maledicte said. \"But let me go up the stairs first. If we take him head down, we'll have to clean the stairs after. The night is long enough without that.\"\n\nDeath\u2014aristocrats turned it into sport, Gilly thought savagely. He took a step too fast and Maledicte stumbled.\n\nToo near the top and the maid's rooms to apologize, Gilly merely shrugged. Maledicte tugged again and started up, his hair and eyes slipping into shadows, then the shadows flowing forward and enveloping them entire.\n\nAt the top of the stairs, they hesitated, looking for thin lines of light that might betoken a maid still awake. But the darkness was near absolute; only the faint gleam of night sky sketched out the floor beams as they crept by the maids' rooms and up the last half stairs to the attic proper.\n\nMaledicte set the body down with a sigh, stretched his hands up to the roof, easing his back and shoulders. \"Why are the dead so damn heavy?\"\n\n\"Awkward, not heavy,\" Gilly said. \"You or I could have carried him alone, were it not for the stairs and our own squeamishness. Cradled like a lover, there'd be little difficulty.\" Though looking again at Maledicte's slight form, he doubted his words.\n\n\"He's stiffening,\" Maledicte said, prodding the corpse, then locking his grip around the assassin's chest so that his hands met and knotted into each other, white-knuckled. \"Let's finish this.\"\n\nGilly shifted his grip from dead ankles to thighs, trying to take more of the weight, seeing the tension in Maledicte's hands echoed in his neck and shoulders.\n\n\"I am going to make Love regret this,\" Maledicte panted as they maneuvered the body up to the narrow windowsill.\n\nThe corpse stuck for a moment, then it dropped with a rustle of ivy and the sudden burst of black wings. Maledicte and Gilly jumped back as a handful of rooks came in the window, their sleep disturbed, their wings beating like Gilly's startled heartbeat.\n\n\"Did you know they were nesting there?\" Gilly asked, when the last rook had found its way back into the sky.\n\n\"It's convenient. Gives him a lovely reason to have been startled and fall to his death. You've done well by me tonight, Gilly. My lucky piece, faithful friend.\" Maledicte brushed Gilly's cheek with his lips.\n\nGilly found himself wondering what all his distress had been for. An assassin who would have killed Maledicte?\n\nDown below, the blurred black shape of the assassin, broken on the pale stones, shifted and seethed. Gilly flinched, and looked at Maledicte, startled at the apparent movement and sick at the thought that perhaps they had dropped a living man. But Maledicte merely smiled and Gilly realized the moving blackness was not the man, but the man covered in feeding rooks.\n**\u00b7 26 \u00b7**\n\n**O** NCE AGAIN, MALEDICTE FOUND HIMSELF at the heart of scandal; the young courtier nearly burgled by the duke of Love's own valet. As if that were not enough to keep tongues wagging, the confrontation between Maledicte and the duke, which started with mutual recriminations and threats, ended in Love's apoplexy and death. And all, rumor had it, without Maledicte even needing to draw his sword, or raise his voice. There were bets laid, trying to guess what it was Maledicte had said. Some rumors said it wasn't words, but the rudeness that had kept Maledicte from allowing Love and Echo entrance, kept them standing in the cold, morning drizzle. Brierly Westfall, Mirabile's sweet-voiced mouthpiece, said she had heard that Maledicte cursed Love's family. Others claimed he mocked Love's pride and so sent him into apoplectic rage.\n\nWhen the bets grew high enough midweek, Maledicte entered the game himself, swearing he would tell them what he had said, if the victor would split the spoils. He visited the Horned Bull, over Gilly's objections of discretion and propriety, and went through the betting book himself, finally awarding a young poet the purse. \"It wasn't so much,\" Maledicte said, smiling at the wildest fancies of witchcraft and exotic poisons. \"All I did was bar my door and tell Love it was in fear that, as his man was a sneak thief, why should the master be any better? Love had a weak heart. His own temper undid him.\"\n\nPoole, the caricaturist, still ensconced in Stones, made the scene his next work when he heard of it: Maledicte, cloaked in shadow, ringed by sly faces, with coins spilled before them all. He captioned it \"Betting on Death,\" and behind Maledicte's image, the horned brow of Haith, god of death and victory, loomed. Gilly fussed when he saw it, but Maledicte only grinned.\n\n\"You must be more discreet, stay home and be\u2014\"\n\n\"Besieged by visitors curious to see where first Love's man and then himself died? No, Gilly, that's begun to pall.\"\n\nBut in the midst of all the notes Maledicte received, the invitations to scandalous gatherings, the reawakening interest in the dark cavalier who had been silent for a season, one omission stood out. From Janus, there had been nothing, and Gilly knew it was that which had driven Maledicte out and about, courting his attention.\n\nWhen Janus came, it was days later and late at night. He came cloaked in secrets and wearing all the signs of a late appointment to a lover's rendezvous, dark linens, newly applied scent, and barbered cheeks.\n\nGilly let him in and Janus swept into the parlor as if he knew that since the attack Maledicte had not slept well within his bedchamber. Gilly, following Janus, resented his glib assurance of welcome, even a week late; he was appeased at Janus's sudden pause at the sight of Maledicte stretched out on the chaise longue beside the spinet. All in dove gray, Maledicte embodied both demureness and danger.\n\n\"Was it an assassin? The duke's man?\" Janus said, dropping his cloak over a chair, ignoring the slow slide of velvet-trimmed wool as it puddled over the edge and fell to the floor. Gilly picked it up, draped it again.\n\n\"The duke's man. Your father's hand. Sit, if you're not going to drink; Gilly's teaching me to cheat at pennywhist.\"\n\nJanus bent to kiss Maledicte and Maledicte turned his head like an affronted maid, refusing to be turned up sweet.\n\n\"That's a servant's game,\" Janus said, seemingly willing to humor Maledicte's mood. He settled himself on the chaise, bumping his hip against Maledicte's.\n\n\"Victory is victory, no matter the stakes.\" Maledicte stood, stretched in another series of deliberate movements.\n\nWatching, always watching, Gilly was reminded of smoldering fires in Maledicte's stillness, of hot lampblack, of the mountainous volcanoes abroad, rumbling.\n\n\"The body,\" Janus said. \"Why the charade?\" He gripped Maledicte's hand, tugged him back to the seat, settled him between his thighs.\n\n\"I thought a dead burglar would be less commented on than a dead assassin,\" Gilly answered when Maledicte did not.\n\n\"I wish you had waited,\" Janus said.\n\n\"Wait, always wait. How you love that song. Should I have just propped him in a closet?\" The edge rose in Maledicte's voice suddenly. \"Or send a card, begging Last's pardon, but could Janus come aid me\u2014I seem to have a corpse in my bedchamber? I was sick of the sight of him, and Gilly was just sick. And you were not here.\"\n\n\"Father changed his plans and kept me at his side, late into the night, going over land management for land I may never own while I acted the obedient son. But I'm done with that,\" Janus said, nestling his face into Maledicte's hair. His eyes grew blue and bright. \"Is your sword sharp?\"\n\n\"No,\" Gilly said, \"No.\"\n\n\"Don't tease,\" Maledicte said, after a frozen moment.\n\n\"I am in deadly earnest,\" Janus said. \"He tried to kill you. He cannot be allowed to try it again. I have flushed our quarry, Mal, goaded them into movement. It is time to hunt.\"\n\n\"You're hiding something,\" Gilly accused, meeting the gas-flame blue eyes with his own. He wouldn't flinch. Not anymore.\n\nJanus slapped his gloves back and forth in his palms. \"Only that I gambled on our plan, our odds of success, when I am not entirely sanguine of the result. I am not a gambler by nature. But after Father's attempt on Maledicte's life, waiting seemed even more of a fool's game.\"\n\n\"What did you do?\" Gilly asked.\n\n\"I poured Amarantha a drink, taking care that she saw\u2014\"\n\n\"What does it matter?\" Maledicte interrupted. \"Just tell me where\u2014\"\n\n\"The docks,\" Janus said. \"To the _Winter's Kiss_ and thence to Itarus, where they can ally with another who wishes you dead.\"\n\n\"Dantalion,\" Gilly said in Maledicte's place.\n\nMaledicte seemed uncaring, eyes half closed, a smile curling his mouth. A man lost in a delightful dream of murder. He opened his eyes, pure triumph simmering in their depths. \"You doubt those odds, Janus? I'll slaughter Last ere he ever sets foot on the _Kiss._ \"\n\n\"You cannot run into the night waving your sword like a barbarian,\" Gilly said, standing before the door into the hall and the city outside as if to bar their way.\n\n\"Don't fret, Gilly. It does you no good and cannot change what we do,\" Maledicte said.\n\n\"We could hire footpads to kill Last for us. Take a page from his book,\" Janus said.\n\n\"No,\" Maledicte said, rising again, as slim as his sword. \"What joy is there in blood spilled by proxy?\"\n\n\"But sword wounds look like nothing else. If you made it appear accidental, an attempted robbery or some such\u2014\"\n\n\"Burn it,\" Maledicte said, settling his sword around his narrow hips. \"I have been patient. I will be so no longer.\"\n\n\"I set a boy to watch the house,\" Janus said. \"He'll send word when Last leaves. You'll wait that long.\" Half command, half entreaty, it worked well enough that Maledicte stopped in his forward path toward the door.\n\nMaledicte paced the confines of the hallway, much like the caged animals of the city gardens, waiting, refusing to be drawn into further speech, though Gilly tried. It was nearly dawn before the message came, a one-word note. _Now._ Maledicte was out the door before the paper had fluttered to the tiles.\n\nThe rooks fled the roof, creeling and calling; in the stables, the horses shrieked as if their stalls were afire. Maledicte turned and walked into the retreating night, following the sting of salt in the air, and dragging shadows after him like the sweep of dark wings.\n\nTHE EARL OF LAST SWUNG down out of the carriage and gestured his coachman onward to the silent quay and the waiting Itarusine ship, distinctive with its ice-breaking prow. The tiger dropped from his post behind the carriage, not the usual skinny lad to run messages but a man Gilly's size. While Last looked after the coachman and crew now unloading his valises, the tiger kept watch, peering into shadows, a hand on his knife. In a dark niche of broken wall and alley, Maledicte's eyes narrowed. Did Last think a single, extra guard would be enough? Even one so large\u2014Maledicte had long ago stopped fearing men who outweighed him. They all bled the same. Insult lanced through him; that Last, who supposedly feared for his life, would rely on only one guard and his coachman for protection.\n\nThe dank, morning fogs hugging the streets and pier reached pale tendrils into Maledicte's lair and wrapped a cloak of obscurity around Last's form. Maledicte welcomed the warming trickle of hatred creeping through his heart and brain, Ani's shared rage eclipsing all else. The foggy fingers touching him slowly tinted to the ink of starless night. He stepped out of his lair, onto the pier, eyes focused on the tall blur that was Last, and the shadows in the alley followed him out, spreading soundlessly, blanketing the wharf in untimely dark.\n\nThe tiger, for all his jumping at shadows, was woefully unprepared when Maledicte emerged, blade already drawn and moving. The size, the strength, all for show, Maledicte thought dimly, as he moved past the man falling to the street, blood pouring from his ruined throat. Last would have done better to bring a hound, which at least could have been counted on to bark.\n\n\"Last,\" he said, when he was a swordstrike away.\n\nThe earl spun, one hand clenched the haft of his cane, the other the handle, while he quickly sought sign of the guard. Finding no one, he twisted the cane. It clicked and showed a faint gleam of metal. No surprise showed on his features, only resignation and rage. \"He set you on? So be it. You will not dispose of me so easily as you think.\"\n\nMaledicte lunged without response, thrusting the blade forward. It made an eager hiss all its own, and the earl parried, using his cane to deflect the blow. The wood split, revealing the sword beneath.\n\nThe earl shoved, using the steel core as a lever, and forced Maledicte's retreat, giving him a moment to strip the sword of its ruined sheath. Maledicte danced forward, and Last evaded the next blow by pivoting on one heel. Last called for help, but the words were swallowed by the black fogs. \"Witchcraft,\" he spat. \"You're nothing without it. You couldn't take me without the god's aid.\"\n\n\"All She does is hold the world at bay,\" Maledicte said, lunging. His sword bit into the bulky coat. Last winced, then clamped the fabric close, trying to use it to disarm Maledicte.\n\nMaledicte hung on to his blade, slipped back, and Last ripped the hampering greatcoat off with a quick agility that made Maledicte snarl.\n\nLast snapped the weighted edge into Maledicte's face. Maledicte ducked away with no space to spare and ripped the coat from Last's hand, enraged that the man thought to use it as a shield.\n\nAt this long-delayed moment, it became apparent that the earl of Last was a swordsman of some skill, and, of course, was possessed of the urge to stay alive. But Maledicte, shadowed and dark, moving as if he had no more limitations than the fog itself, was merely possessed. Gradually, strike after strike, he began to win.\n\nThe darkness was as complete as if the deepest sea had reached up and drowned the city. In its black embrace, Maledicte's world narrowed to the ring of steel on steel and their panting breaths. There wasn't even room for rage or triumph; with Last dancing so well with him, all Maledicte could feel was the enjoyment of physical exertion, and the breath-stealing anticipation of blood. The dark sword, ghostly in the fogs, swooped forward and took a bite; Maledicte moaned for Last, the pleasure deep in his belly surfacing. Last cursed and stumbled back, bleeding along his upper arm.\n\nThe fog carried the muffled prayers of frightened sailors. Nearby, the coachman called out but Last had Maledicte's sword pressing him, and no breath to answer.\n\nLast gained ground, took them from the treacherous slick wood of the quay to the cobbled edge of pier and street, but the effort left him bleeding from arm and thigh.\n\nHis face, set in lines of desperate concentration, broke and filled with a wild triumph as the coachman blundered into their duelists' circle. Maledicte danced back and turned, the blade flying from his hand to strike home in the coachman's chest. The primed pistol dropped from the coachman's hand, falling onto the cobbles with a crimson roar, its shot spent. Maledicte's pleasure turned to anger. The coachman thought to interfere; who might next?\n\nLast lunged forward, his blade aimed for the sweat-wet cravat at Maledicte's neck and Maledicte slipped away in time to save his neck, if not his waistcoat and shirt. The fabric ripped, the dark brocade, the pale linen, the dense cotton, all rent with one swift stroke, baring Maledicte's moon-white flesh and ending with one drop of blood blooming and fading at his collarbone.\n\nMaledicte growled at the touch of the fog on his skin. He was a body's length from the coachman's corpse and his sword, and his temper was turning foul.\n\nThis went on too long; every minute brought Last closer to reprieve. The pistol shot would have been heard, and though the fog played havoc with sound, eventually they would be found. Even now, Maledicte heard a woman shrieking, a high, shrill sound that set his teeth on edge. Last hesitated, his face confounded; Maledicte lunged in a long, low dive for his distant sword, even as Last shook the astonishment from his face, and returned to the fight.\n\nLast's blade missed his tumbling form by a hair's breadth. More silk ripped; the riband holding his hair slithered down his neck and pointed out how close the blade had come, but the coachman's body and Maledicte's sword were within reach. Maledicte's hand closed on the sharp-feathered hilt of his sword and it slipped from flesh and bone as if a human body were just another hilt.\n\nMaledicte shook the hair from his face and backed Last toward the pier. The duel was done. The set blankness on Last's face told Maledicte such, the fear unmasked in blue eyes. Maledicte darted out with his blade, a bird's quick stoop for prey, narrowed his vision to Last's throat. The man parried at the last, but Maledicte's blade shrieked over steel and sliced through Last's ear.\n\n\"Who are you?\" Last gasped.\n\n\"Is that what ails you?\" Maledicte laughed, trying to imagine what Last saw. His hair falling loose, shadowing his eyes, the feral snarl of teeth and tongue, the pale skin and delicate breasts bared to the night air.... Were Last less of an aristocrat, he would fear less, but in his experience, women were nothing but playthings and pawns. That one could wield a sword and more, fight him to the death, would be beyond his experience.\n\n\"Who are you?\" he cried again.\n\n\"Black-Winged Ani.\" The name forced itself free of Maledicte's throat without his willing it.\n\nLast staggered and Maledicte punched the sword into his chest, twisted the blade for the pleasure of hearing the man cry out as he fell. Blood welled, bubbling through Last's sweat-soaked linens, spurting against the fabric as blood and air fought to escape. Left alone, Last would bleed to death. Maledicte knelt beside him, touching the wound. \"Is it your heart I've hit, or your lungs? You took my heart once.\"\n\n\"I will not beg,\" Last said, a rough whisper, blood frothing his mouth.\n\n\"Nor did I, and it made little difference.\" Maledicte brought his hand to his mouth and nose, smelling the hot tang of blood. He licked his fingers; blood warmed his raw throat, meeting his thirst, but not slaking it.\n\nDizzy with rage, he stood. It was not enough. Maledicte swayed, wondering whose hunger he felt, whose bloodlust. Last was near dead and in a moment Maledicte would finish the deed, his revenge accomplished in blood and shadows. But he felt nowhere near sated, and inside his belly, wings fluttered.\n\nA bare gasp behind him reminded him of Last's physical presence and he turned, wanting to see the light fade in his eyes, the life leave. But Last was not alone. Another man had found his way to the heart of the fogs.\n\n\"Filth,\" Last gasped, \"come to see your leman kill me? I should have killed you at birth.\"\n\nJanus bent and slit his throat ear to ear.\n\n\"He was mine!\" Maledicte said, rage erasing his voice so that all he could do was whisper.\n\n\"You were too slow. I was worried about you. And rightly, I see, mooning about, half-naked,\" Janus said. He tugged the gaping sides of Maledicte's shirt together. Maledicte's fingers curled into a fist and, feather-blind, he struck. Janus reeled, licking his split lip, eyes darkening. \"What matters who killed him? You made him bleed, you made him fear you.\"\n\n\"He was mine,\" Maledicte said, \"Mine.\" He tightened his hand on the sword hilt; it seemed to surge in his hand. He lunged and buried the sword in Last's body once more. Janus spun, stepping out of range. Maledicte wrenched the sword through Last's guts, hoping for some final groan, some final hurt he could wring out.\n\n\"Enough, Mal,\" Janus said.\n\nMaledicte, not listening, pulled the sword from Last's chest, then sheathed it when he realized that all he wanted was to spill blood and that the only person within reach was Janus. He stared out at the sea, dark waters surging and receding, at the ships slowly becoming visible as the fog thinned and faded. The shining figurehead of the _Winter's Kiss,_ a young man seemingly carved of ice, glimmered as the fogs parted. Gilly would like that one, he thought, and with that the rage faded. The earl was dead. His vengeance done, Maledicte stared down into the blank eyes, the gaping, slack mouth. Ani's touch subsided into sulky confusion.\n\nMaledicte watched the body, waiting for some sign, his eyes never leaving the corpse, though Janus spoke his name twice, then finally in exasperation\u2014\"Miranda!\"\n\nShe turned, searching for words to explain her dismay. The earl of Last was dead and her victory was as meaningless as when she first heard his name. Last\u2014it had meant nothing to her then. Dead, he meant nothing to her now.\n\n\"Over your tantrum?\" Janus asked. \"Grab his ankles and let's sink him into the water; we'll weight his coat with cobbles and delay the discovery that much longer.\"\n\nMaledicte mutely did as Janus said. He watched the body drop by slow degrees deeper into the water, fading. \"What are you doing here?\"\n\n\"Aiding you,\" Janus said. \"Amarantha?\"\n\n\"She's here?\" Maledicte said.\n\n\"Aboard the _Kiss,_ I suspect,\" Janus said, lips thinning. \"Well, there's no help for that. Unless we want to fight the entire crew.\"\n\n_We could,_ Ani whispered. _Turn the waters red._\n\nMaledicte shook his head, and at Janus's imperious wave, went over to help him with disposing of Last's men.\n\nThey were still stooped, the sagging weight of the burly tiger held between them, when two sailors came out of the night, drunk and staggering. They blinked and one of them frowned. \"Milord Last, is that you?\" Janus pulled his cloak over his pale hair, and traded a glance with Maledicte, one that needed no special communion to be understood. The sailors had to disappear. Maledicte reclaimed his sword and flew at them, and they, panicking, headed not for the safety of the ship, but back into the dark streets. Janus yanked his cravat up over his nose and cheeks and followed as silent and supple as a cat. Maledicte still gained on the two sailors, moving as light and quick as a shattering of glass.\n\nHe darted through a shadowed alley and tore the shadows after him. Looking behind him, one sailor tried to sign an avert charm and stumbled. Maledicte pounced, jamming his head against the rocks. He freed his sword from the hilt, and Janus, flashing by, said, \"Not the sword.\"\n\nMaledicte snarled but balked, understanding. The sailor struggled to rise, and Maledicte picked up a rock. He brought it crashing down on the man's skull, heard the wet crack, and remembered Miranda's lectures. He bent over the corpse and plucked the few lunas left from the bag.\n\nJanus returned, out of breath, but dragging the second sailor behind him. He dropped the body near the other. \"Did you get his purse?\" he asked.\n\nMaledicte nodded.\n\n\"Then let's leave this. There's no great mystery to drunken sailors being set upon for coin,\" Janus said. \"And I need to be home, setting arnica to my mouth. A bruise might be hard to explain, should Last surface with the next tide.\" His tone invited apology, but Maledicte denied him that.\n\nIn truth, Maledicte wouldn't be able to apologize even were he so inclined. This final death stole his voice; he fought a surprising urge for tears. The sailor was no part of his revenge. Just a man visiting the city whores and finding his pleasure where he could.\n\nJanus glanced at him and said coldly, \"If Ella had had her way, you'd be bedding men like this.\" He tugged Maledicte closer, and wound a narrow strip of cording around his waist, cinching the fabric closed. \"Try not to run into Gilly, hmm?\"\n\nMaledicte touched the dead man's belt and nodded. He strode off into the darkness, heading for home, clinging to the shadows of the Relicts like a ghost, daring someone to confront him, to recognize him. Invoked by Janus's words, he almost expected to see Ella staggering out of an alley, shaking down her skirts. Maledicte wondered what would happen then. Would Ella even see Miranda in Maledicte's guise? Most like, she'd not even look, but scuttle away, recognizing danger, if not her child. Maledicte wondered briefly if Ella had grieved when Miranda left, if she had meant anything to her at all, beyond merchandise.\n\nSudden thoughts stilled his feet, left him numb. Miranda was nothing, a rat, but Janus\u2014Celia knew Last had taken him\u2014why had she not followed? Surely the gossip reached this far\u2014surely she knew that Janus had come into coin.... Her absence meant she was either dead or so far lost in her Laudable dreams she might as well be dead. And what befell Celia undoubtedly befell Ella. Maledicte shed a shaky breath and went on, dreading his past rising up to meet him. But alleys passed in these uncanny fogs, peopled only in his imagination; Maledicte saw no one.\n\nHe arrived home and crept in through the kitchen. Cook drowsed in her chair, and fresh dough rose on the counter. The dark, sour scent of the yeast made his stomach clench. He fled the kitchen, halting when he found Gilly dozing on the main stair, a guttering candle spilling wax over the riser. Maledicte backed away, and crept up the servants' stairs.\n\nIt was done, he thought as he stripped the bloody shirt from his flesh. Even now, Last plagued him, the cloth sticking to his skin, making its removal close to pain.\n\nDone, and Janus to be earl. Maledicte winced. If they escaped conviction. He ripped off the ruined corset and dropped it in the pile of stiffening cloth.\n\nNaked, he fumbled to the hearth, searching for lucifers. He thrust the fabric into the fireplace. Kneeling, he blew at the struggling flame and succeeded only in scattering the fine ash left behind from last season's fire. It stung his eyes and brought him to tears again.\n\nWhy this weakness? Why this grief? Maledicte could not understand it. No such thing had plagued him before.\n\nDaylight showed through the curtained glass of his rooms, and, despairing, he sloshed the brandy over the slow-singeing clothes. Then and only then did the room fill with the stink of burning blood and embroidery. Some of the rigidity left Maledicte's shoulders. He scrubbed his face with cold washwater, trying to wipe the ashes away. The cloth came away speckled with red as well as black, and he squeezed it out until the basin turned pink. The face, the figure that looked back at him from the glass was that of a madwoman and he refused to let his eyes rest on it. He was Maledicte, the dark cavalier, Last's scourge. Ani's servant. _Still_ Her servant. And with Last dead, whose blood could release him?\n\nDaylight reminded him that the maid would come to bring fresh water and morning tea. The clothes had best be burnt by then, burnt to ashes. He stirred them with the poker, set glowing sparks free to sting his bare arms and hands. These faint pains woke him from his stupor, and he dressed with his usual elaborate care. Fine leather breeches laced along the thighs and belly for the fashionable fit. Another corset, secreted in the space that once held Vornatti's will, bound his chest and thickened his narrow waist. He layered on the shirt, finest lawn, and the brocade vest, tied his cravat in a reasonable facsimile of the popular Leaffall, and called it done just as Gilly's familiar tap sounded on the door. \"Mal?\" Anxiety laced his voice.\n\nAt his invitation Gilly came in and coughed. \"What are you burning?\" he asked, flinging the windows open without waiting for permission. He poked at the black remains of the fire, and said, \"Oh. Did you do it then? Kill him?\" His voice grew tentative and pained. \"Were you seen? What happens next?\"\n\nMaledicte hefted the brandy bottle, but the liquor was gone, fed to the fire. \"It's all ashes,\" he said. \"Nothing but ashes. Can you tell me why this should be so unsatisfying?\" He took the poker from Gilly and stirred the fire, breaking the charred fabric into black dust. \"I plotted this death for years. Gloated over it, imagined it, fed my rage on it. And it's ashes in my mouth.\"\n\n\"Ani goads you, spurs you on to mad rages, and when you've done as She wished, She recedes, taking it all for Herself. And you're left with nothing but the aftermath of blood. Now that your vengeance is done, She'll have left you completely.\" Gilly let out his breath and studied the carpet beneath his boots, scuffing at soot that had spilled out with the withdrawn fire-iron. \"Most of Her previous children have gone mad with Her loss.\" His voice shook.\n\nMaledicte said, \"Gilly. Look at me and tell me She's left me. That Ani's mark is gone. You who see it so clearly. My compact completed\u2014that I am no longer Her stalking horse.\"\n\nGilly sucked in his breath. He touched the shadows under Maledicte's dark eyes, and then looked into the shadows in the eyes themselves. \"No. She rides you still.\"\n\n\"The earl is dead, but She lingers. Is this all my future holds? Blood and fighting? I wanted Janus back and I have that. I wanted riches enough to never starve. I have that. If this night's bloody work remains a mystery, Janus will be earl, and there will be nothing left to need.\" He wrenched his chin from Gilly's grip, and collapsed into the chair beside the fire.\n\n\"Janus will always want more. He's near as hungry as Ani Herself,\" Gilly said, bitterness in his tone. \"Why is Ani still with you? What happened, Mal?\"\n\n\"It's all ashes,\" Maledicte repeated. He slumped, his face in his hands, tears in his throat.\n\n\"I know something that is all ashes, without question,\" Gilly said, changing his mood with an audible effort. \"What did you do, clean the hearth with your hair?\"\n\nDistracted, Maledicte touched his hair, and frowned at the dusting it left on his fingers. Gilly collected the ivory-backed brush from the dresser and said, \"Lean forward.\"\n\nMaledicte bent his head, letting his hair dangle over the hearth, and Gilly brushed the ashes out, brushed until Maledicte's hair gleamed like a rook's wing.\n\n\"What color ribbon do you want?\" Gilly said. \"Your usual black, or something more dramatic?\"\n\n\"Black will do,\" Maledicte said. \"I am expecting Echo at any moment. Even if Last obliges by staying disposed of, his disappearance will cause comment. And that means Echo again. I never thought how apt his name was before\u2014he keeps returning, my words distorted on his lips.\"\n\nGilly gathered the dark strands into a queue, tied it off at Maledicte's nape, making a neat knot and letting the ribbons dangle. \"What will you say?\"\n\n\"It hardly matters,\" Maledicte said. \"I am not the only one with animus against Last, and until his body surfaces, Echo will be hard pressed to charge me with a crime. He'll assume I killed Last. But he'll have to prove it\u2014and he'll find no witness to the deed.\"\n\nGilly frowned, thinking beyond his immediate worries for Maledicte's sanity, for the compact still unfinished. \"I'll send out runners and spies, those who can differentiate whispered fact from wishful rumor. We'll know what Echo plans before he does. And Amarantha?\"\n\n\"On the _Kiss,_ we think,\" Maledicte said.\n\n\"I'll send a letter in her wake, then, asking Vornatti's spies in Itarus to keep their ears open. If she's abroad, she'll seek Dantalion as Last would have done.\" Gilly sucked in a breath, beginning to hope that, Ani's lingering presence notwithstanding, Last's murder had been accomplished without jeopardizing Maledicte's neck.\n**\u00b7 27 \u00b7**\n\n**G** ILLY PACED OUTSIDE MALEDICTE'S ROOM, waiting. It was the dark night of the year, and superstition held that the unshriven dead returned on the Dark Solstice. Once there had been solemn ceremonies to placate the dead, but in the bored hands of the aristocracy, the Dark Solstice had become another excuse to play.\n\nDid the dead come back and to the court, Gilly thought, they would return to their graves, ashamed for the antics of the attics-to-let and the debauched, for the gamblers who took advantage of their costumes to cheat with abandon. Though none was truly masked in the select crowd, caught in the nets of well-recognized foible and mannerism, they could pretend and act accordingly. Despite his creeping unease at the idea of Vornatti's ghostly step sounding in the hall, Gilly wished Maledicte had chosen not to attend, not when Mirabile had paved their path with rumors centering on Last's disappearance and Maledicte's possible role in it.\n\nThe Dark Masque, though, had put a spark into Maledicte's eyes, replacing the sullen temper and brooding fits Maledicte had been prone to since Last's death. Gilly had expected to see only release in Maledicte's eyes after the earl's death; everything the boy had craved was granted. Janus, his lover; the earl of Last, dead. Gilly knew where his own unease lay; Ani's continuing presence as obvious now as it had ever been, darkening Maledicte's nights, sending him into muttering rages and long bouts of sword practice against enemies only he could see.\n\nPerhaps, it was simply that the changes had proved fickle and fleeting. Janus was Maledicte's, true, but Janus was also at Aris's beck and call, and Aris still hunted him a wife. The earl was dead, but earl was a title and it could be bestowed else\u2014\n\nThe floorboards creaked along the stairwell; Gilly twitched but didn't take his eyes from the closed door. For one moment he even wished Janus here, waiting in this dim hall. On the darkest night, Gilly would prefer to spend it in Lizette's arms, letting the flame of her hair warm him like sunlight, letting the ampleness of her charms drive dead men from his mind. But Janus attended the masque as part of the king's entourage, and not Maledicte's escort. Gilly believed that it wasn't the masque itself that had brought the smile to Maledicte's face, but the simple fact that he would see Janus this evening, after a long month where Maledicte had seen Janus only rarely.\n\nThe stairs creaked again and Gilly turned his head in time to see a gray figure slip away. For a bone-shuddering moment, he believed in the dead as wholeheartedly as a child, but it was only Livia, stealing down the main stairs to avoid Cook's watchful eye.\n\n\"What do you think?\" Maledicte asked, ghosting from his room.\n\nGilly shivered, caught by the dark eyes behind the raven mask. It was as if he saw Maledicte for the first time, and he was surprised all over again at how lovely Maledicte was. The lush mouth beneath the jutting black beak curled in amusement, changing the shape of the face. For one dizzying moment, Gilly saw a woman in a mask, behind the mask.\n\n\"Speechless?\" Maledicte asked, the harsh rasp of his voice breaking the illusion. He stepped closer and the rook feathers sewn into his coat glimmered and shifted green, gold, and returned to black. The scent of dusty feathers washed over Gilly, the sweet pungency of lilac. He imagined touching the lush mouth, and the sudden violence of his desire made him quake.\n\nLizette teased him often about his love for Maledicte, and he allowed it, acknowledging his fascination, and acknowledging that Maledicte was not his. A bittersweet pang. But tonight, for the first time, he wondered if he could change that, wondered why his heart and body had run so counter to his tastes. He took Maledicte's slim shoulders in his hands, the feathers rustling against his palms, and drew him closer.\n\nMaledicte looked up, inquiring and impatient. \"Say something. I thought you would like the feathers. And what a time I had collecting them.\" Gilly imagined kissing the sulkiness from Maledicte's mouth, but the beaked mask was more than proof against such incursions.\n\n\"Do you think it wise to parade your allegiance with Black-Winged Ani?\" he asked instead.\n\n\"It's not the fashion to believe in gods, remember?\" Maledicte said, pushing away from Gilly. He paused in his path down the hall, and said, \"Come along, Gilly. We're running late, and we must get you a mask. Should Vornatti come hunting for someone to tend his needs, I will not have him find you.\"\n\nONCE AGAIN THE TWO HALF-MOON courts were opened to each other, Aris's sunrise half and the nobles' secretive twilight. But the whirl of time and sky was fractured, made into mazes with hanging mirrors, swaying gently in the press of bodies, reflecting dizzying views of gilded traceries on the walls and the movement of the revelers, clad in fantastic concoctions limited only by their pockets or sense\u2014lace and leather, masks and fur, and gems.\n\nSmall groups clustered near these illusory walls, where once they could have expected the gods' eyes to peer out at them, overseeing the Dark Solstice and its intersection of the living and the dead. Now the mirrors only served to double the attendees' numbers and the only ghosts were their own cloudy reflections. Masked royal servants, clad in gray velvet, moved like wraiths through the crowd, a reminder of the deaths the aristocrats evaded.\n\nIn the heart of the room, a raised dais stood, draped from above with a pale gray cloth that shimmered like rain in moonlight. The gilded chairs on it were empty, the backs of the chairs draped in black\u2014the only symbol of mourning Aris allowed himself for a brother not proved dead.\n\nMaledicte's face was stern behind the mask; his mouth, starred by a small velvet feather, grew tighter as he scanned the room, dismissing the crowd one costumed aristocrat at a time. In turn, Gilly noticed how few faces, even shielded, braved Maledicte's gaze.\n\n\"Janus will be on the dais with Aris,\" Gilly said. \"Do you know his mask?\"\n\n\"Do you think a mask can hide him from me?\"\n\nThe royal dais became a hive of motion, of servants moving back and forth. Through their smoky shapes, behind the silvery drapes, the diminished family of Last could be seen. Aris, in festival white, nonetheless wore black armbands, and his mask was shaped like Sorrow.\n\nAdiran sat at his father's right, in the queen's chair, which dwarfed his fragile body. On either side, the mastiffs, tongues lolling, watched the crowds, gulping foodstuffs that Adiran tossed them. A servant, clumsy with nervousness, stumbled over his own feet, and Hela chastised him with a deep, sudden bark. Adi giggled, tugged at her ears, holding his mask on with his other hand. He wore only a half-mask, white satin, trimmed with blue, and kept pushing it up his forehead, ruffling his hair.\n\nBehind him, hands on the high back of the throne, was a dog-masked man with blue eyes. Adi looked up at him and barked, echoing Hela with a curious, imitative precision. Janus smiled and straightened his cousin's mask once again. He raised his head and met Maledicte's gaze, smiling.\n\n\"Will you dance?\" a woman asked in Maledicte's ear, her hands on his shoulders, her breath on his nape.\n\nDespite the dulcet words, there was a edge to the voice that Maledicte recognized. He knew her, mask or no mask.\n\n\"What reason will you give me, lady?\" he asked, taking in her costume, the twin to his own in spirit if not in shade. Where Maledicte's feathers were un-seasonably black, Mirabile's were unnaturally white, though more in tune with fashion. Her beaked mask had rubies crusted around the eyeholes, the only spark of color about her. Even her hair had been powdered to whiteness.\n\n\"Think how well we will look,\" she coaxed.\n\n\"Is that more important than enmity and spite?\" Despite his glibness, her presence made him wary. He had not forgotten his atypical docility alongside her in Jackal Park, the way he had bent to the certainty in her eyes. Time had not diminished that strength.\n\n\"Infinitely,\" she said, holding out her hands. He had been wrong about the rubies being her only color. Her ungloved hands showed red nails, painted like any harlot's.\n\nMaledicte drew back, turned, and found his path thwarted by a swaying mirror. Reflected in it, Janus bowed over a young woman's hand, drew her into a dance with a smile while Aris looked on approvingly. Mirabile ghosted up, a shimmer of red-eyed white in the mirror, and Maledicte found his hands entwined with hers as the dance began.\n\nThey danced in silence, Maledicte fighting the drowning sensation of being nothing more than her shadow, of having as little mind as a shadow\u2014He forced through the numbness finally, his voice rough, \"What do you want?\"\n\n\"Found your tongue, I see,\" Mirabile said. Her lips curled in approval. \"I was sure you would.\"\n\n\"Found my tongue, my will, and my senses,\" Maledicte said. He forced his steps to a halt, and freed his hands. \"I am done with you\u2014\"\n\n\"Don't be ridiculous,\" Mirabile said, and fit herself back into his arms as if his rejection had only been a flirt within the dance pattern. \"You ask me what I want? Nothing so dire as to make you frown so. I only wish to aid you. And in doing so, aid myself. Our common goal\u2014\" Her eyes darkened as his grip tightened, grinding the small bones of her hands together, but showed no other sign of pain.\n\nMaledicte said, \"What have we in common? _Nothing._ \"\n\n\"Nearly everything,\" she said.\n\nMask to mask, Maledicte faltered a step, seeing Ani mirrored in her, and made no reply. She smiled sweetly, savagely at him, and said, swaying close, her voice a whisper, \"You think your task is done, your compact fulfilled? When Last's death is not on your hands, but on those of your impetuous lover, who stole your kill?\"\n\nFor one moment, Maledicte knew sympathy for Gilly, who preached caution with an ever-increasing avidity. For Mirabile to speak so in the king's presence, among witnesses, for her even to _know\u2014_ Maledicte shivered, suddenly unsure of who he held in his arms, Mirabile, the vicious-tongued harridan, or Ani Herself. The pale feathers on her costume brushed his, whispering.\n\nMirabile leaned in, warmed his cheek with her breath, and said, \"Intercessors dream of the gods through an imperfect window, but I am one of Ani's chosen, and I...I dream of you. I saw you in the blurred shadows of sleep, you and Ixion, killing him.\"\n\n\"How dull your wits have become,\" Maledicte said. \"To think to entertain me with past events that I experienced for myself.\" His mouth dried with unease. Once he had chastised Gilly for dreaming of him; to have Mirabile doing the same was far less bearable. Were they not the cynosure of the room, he would claw her eyes out that she not see him, tear her tongue out that she not speak of him.\n\n\"Now, now,\" she said. \"Ixion favored you when he struck, whether you know it or not. To keep Ani close, clutched in your heart, can only be a boon. She grants such gifts\u2014\" Mirabile's eyes fluttered, opened again, russet eyes red-tinted as if they had taken on some of the rubies' bloody splendor. \"Everything you ask, She grants, and all you need do is allow Her in as deep as She will go. You've asked so very little of Her, caught up in your petty obsession with Ixion. My advice is simple. Forgo this business of love and settle into power.\"\n\n\"Such things you say,\" Maledicte said. \"I believe you are mad.\" Over the wheeling of the dance, the flashing mirrors, the rustling of feathers meeting feathers, he saw Gilly watching, concern etched in his furrowed brow.\n\n\"Why play the fool?\" she said. \"We are kin, the children of the carrion crow. Be my complement, my comrade, and we will do as we will. Isn't it seductive, my dark cavalier, to see the knowledge in their eyes\u2014that their lives are in your hands or mine?\"\n\n\"Your hands are too dainty for such work,\" Maledicte said. \"You're only a spoiled aristocrat and delusional with despair. A gift of dreaming? Dreams are useless compared to a blade.\"\n\nShe tensed her fingers on his hands, digging her nails into the skin until blood welled.\n\n\"Weakling,\" she said. \"Gifted and you do nothing with it. Take power for yourself\u2014it will not satisfy you otherwise. Will you watch your lover rise, and stay weak as a woman, at his mercy?\"\n\nMaledicte stared at the distortion in her fine features as she shivered with rage. It touched ice to his bones, the realization that she was his mirror, or worse, his future. She had given herself wholly to Ani, and she was as terrible as a specter.\n\n\"Ani drives you mad,\" Maledicte said. \"I will not join you on that road. Your vaunted alliance would be only to lull me into complacency so you could stab me in the back.\"\n\n\"Knife work is your m\u00e9tier. I prefer subtler arts. But you doubt me, doubt my skills?\" She smiled, her eyes going distant. \"The ice,\" she said, \"breaks under the ship's prow. Shall I see what can be stirred to the surface? Bring your sins to light? I shall prove my skills to you, tonight,\" she said, curtseying and disappearing behind the nearest partition.\n\nSweat broke out along Maledicte's back and brow. Cold settled in his stomach. His hands shook, and he jammed them against his sides, striding purposefully for the doors and outside air. Gilly shadowed him, and Maledicte turned. \"Back off, Gilly. I'm in a killing mood.\"\n\n\"You're bleeding,\" Gilly said, flinching as Maledicte snarled.\n\nMaledicte forced his temper down, the pain radiating up his torn hands, and said plainly, \"I believe the bitch poisoned me. _Tested_ me.\" He moved onward to the balcony, shivering, hunching his shoulders against that spreading internal chill.\n\n\"Mal\u2014\" Gilly said, voice tight.\n\n\"I'm thirsty,\" Maledicte said. \"Bring me a drink.\" His hand swung again and again to his sword hilt; droplets of blood flecked the marble floor. Maledicte leaned over the edge of the rail, panting a little, then recovered his poise. \"Go on, Gilly. I'll be here. I've got my mask, after all, to protect me from death.\"\n\nOn the balcony, Maledicte watched the small wounds puff and swell with a near-indifferent eye, though he shook with chill and his feathers grew spiky with his sweat. Numbness swept over him, stiffening his legs, arms, and face, as if the chill in his veins had turned to ice. His breath labored. Then a convulsive shudder shook him and the small wounds spat back blood and something darker, something that trickled like a spill of greenish syrup. It pooled on the stone at his feet, and when it was done, he licked the scratches closed.\n\n\"Mal?\" Janus came out of the light, into the shadows on the balcony, and Maledicte rose from his crouch.\n\n\"Here.\"\n\n\"And hale? Gilly spun me a tale of poison,\" Janus said, setting the glass on the balcony railing. It chattered and clinked against the stone. The velvet quality to his voice was ruffled. \"I should have known he lied.\"\n\n\"Gilly so rarely lies,\" Maledicte said, picking up the glass and draining it. His raw throat eased.\n\nJanus sucked his breath in, grabbed Maledicte to him, touching his damp skin and peering into his eyes. \"You seem well enough.\"\n\n\"Yes,\" Maledicte said, curling into Janus's arms. Their hearts beat steadily against each other's: Janus's slowing as his fright eased, and Maledicte's speeding up from the dirge it had been dragged into by Mirabile's poison.\n\nJanus sighed into Maledicte's hair, bent to kiss him, and finding the mask's black beak in his way, pushed it off.\n\nMaledicte moved into Janus, pressing against him as if they could become one creature. Give up love for power? Mirabile was madder than he thought.\n\nToo soon, Janus broke the embrace. \"I must return.\"\n\n\"I have not touched you in weeks. A single kiss is all you spare me? Was this only an interlude in your quest for social acceptance?\" Maledicte's fear made him bitter. The poison came too close, woke fears long dormant, reminding him that the world could separate them at any time, and still, Janus chose to play games. \"Should I thank you for the time you've condescended to grant me?\"\n\n\"Mal,\" Janus said. \"I'm only trying to keep tabs on what is being said regarding Last, assuring Aris that you are blameless in Last's disappearance. And with Echo replacing Love as Counselor, I have much to defend against. But I'll come home tonight. I've missed you.\" He tugged Maledicte's hands to his chest, pressed them to his heart. Maledicte brought Janus's mouth to his again. The kiss lingered, but once more Janus broke it, this time walking away into the court.\n\nMaledicte stared after him for a hungry moment, wishing he could make Janus see that they needed none of this. Let them retire to Lastrest, out of Mirabile's threats and entreaties, out of sight of Aris and his attempts to continue the line. But he had sworn to aid Janus, and Janus wanted the court, the title\u2014Aris would likely reward a marriage with the title, once Last's death was accepted, and though that was what they wished, the idea of it woke some slow-burning worry in Maledicte's chest that he couldn't understand.\n\nSpying Janus's earlier dance partner, the tiny girl dressed as a wood nymph in green and gold, with a veil instead of a mask, Maledicte deserted the balcony abruptly.\n\nHe made his bow before the wood nymph's chaperone and took the nymph's hands. A mask was no aid to the little doll girl; her lack of inches made her obvious.\n\nThey danced in stifling silence for a full four measures before Maledicte said, \"If you hate me so much that you can't be bothered to be civil, I wonder why you agreed to dance at all. Surely no one would fault you for turning me away.\"\n\nShe looked up at him and flushed, the redness visible through the fine pale linen of her veil. Maledicte waited until the color had faded, then provoked it again. \"Can't you answer me? Or were you never trained to talk?\"\n\n\"Mother's desperate,\" she said in a breathless rush. \"I have six younger sisters waiting for me to wed.\"\n\n\"Does she think to attach me?\" he asked, incredulous. \"I might as well be the plague for a chit like you.\" Compared to these noblewomen, Ella was an amateur schemer when it came to profiting from her daughter.\n\nShe lowered her head and mumbled some words that, though unintelligible, made her flush again.\n\nImpatient, he tilted her head up, his bare fingers beneath the delicate veil. Her heart raced beneath his fingertips. \"Be brave, girl. You're masked. Perhaps I don't even know who you are\u2014as improbable as it seems.\"\n\nThe girl either took heart or umbrage, it was hard to tell, but the result was the same. She raised her head, a grim determination settling on her blurred face, and said. \"She wants me to wed Lord Last.\"\n\nMaledicte's temper turned in his belly. \"Brazen or desperate indeed. To use Janus's lover to meet him. Does she want me to tell you what pleases him? What makes him sweat and cry out? What his skin feels like under my lips? What he says to me while we're abed?\"\n\nThe girl's breathing quickened in shock. Such plain speaking was hard enough to hear for a maiden; for Maledicte to speak with such venom undid her composure completely. Her shoulders shook, and the veil over her eyes grew damp and dark with tears.\n\n\"Stop that,\" Maledicte said. \"I will not have you start another scandal with me at the heart of it.\"\n\nShe stopped struggling, and he loosened his painful tourniquet on her arm. They took another round of the dance without any speech; the spots of moisture on the veil shrank and dried, leaving only quivering lips and shaking hands to convey the shock she still felt.\n\n\"I think your mother must be mad,\" he said, though mad brought to mind not a matchmaking aristocrat, but Mirabile with her red eyes and bloody nails.\n\n\"Why\u2014\" The girl paused, then continued, her voice gaining strength, \"Why are you so cruel? I've never said a thing to hurt you. But you'll say anything to hurt me. You have every reason to be kind. You're handsome, and rich, and no one tells you what to do.\"\n\n\"You've never met my Gilly, if you think no one dictates to me,\" Maledicte said.\n\nThe dance ended and he bowed, but stayed at her side. She blanched as he drew her toward one curtained partition, a seat at the nexus of three mirrors. \"Your virtue is safe,\" Maledicte said. \"I merely want a word with you.\"\n\nShe nodded, biting her lips so hard that he thought the veil might darken with blood. He tugged it away from her face entirely, watched her eyes widen like morning-glories at sunrise.\n\nHe sat down in one of the quilted chairs, still holding her arm so that she had to bend with him. He drew her closer, put his lips by her ear. \"Your mother may want Janus for you. Aris may want the same. But if you take Janus from me\u2014\" His breath hissed out at the very thought. \"If you take him, I'll kill you. Best say no, should he come courting.\"\n\nShe whimpered and he said, \"We are understood?\"\n\nBacking away, she stumbled and nearly fell, stepping on her skirts in her haste to be out of his reach. Heads turned as she floundered across the floor, toward the shelter of her abigail and the cluster of debutantes drinking toasts to the dawn, safely arrived.\n\nMirabile passed through the debutantes with a word here, a touch there, and a smile for Maledicte cast over her shoulder as she bypassed the nymph sobbing in her chaperone's arms.\n\n\"That was not well done of you,\" Gilly said. \"Besides being a cousin to Westfall, Psyke Bellane is a gentle, inoffensive girl.\"\n\n\"Gentle, yes. Inoffensive?\" Maledicte said, \"No. But killing her would be like crushing a sparrow, so easy as to arouse more pity than satisfaction. By warning her off, I've done both of us a kindness.\"\n\n\"Only you could argue that way,\" Gilly said.\n\nMaledicte pulled his mask off, dropped it to the marble floor. \"I've had enough. I'm going home.\"\n\nGilly's response was drowned in the sudden tolling of deep bells. This close to the palace, the sound was as powerful as the tide and as inexorable. Maledicte turned to catch Janus's eyes and met Aris's instead, and saw the quick shattering in them. Maledicte flinched.\n\n\"Last,\" Gilly whispered. \"The unshriven dead.\"\n\n\"Hush, Gilly,\" Maledicte said, taking his hand. \"Hush.\"\n\nOn the dais, Adiran, startled by the clamor, clapped his hands over his ears and wailed, his voice rising over the low pitch of the bells like a descant. Aris pulled him into his lap, soothing him. Jasper and Echo moved toward Aris at a trot.\n\nThe bells faded into silence though the mirrors still shivered with their echoes. In the sudden hush, a startled shriek rang out as a debutante fainted in her escort's grip. Gilly stepped closer to Maledicte, and Maledicte tightened his grip on Gilly's trembling fingers. \"Shh, Gilly.\"\n\nThe girl's abigail fanned her face, and her escort chafed her wrists, ever more frantically. He looked up with wide eyes. \"She's not breathing.\"\n\nBefore his words stopped sounding, a second girl fell, an heiress of some repute. By the time her people converged around her, the first debutante was dead.\n\nOn the dais, Aris stood and started as if he could see Death walking the floor. Jasper gestured madly and the Kingsguard enclosed the king, surrounding him. Hela barked, long, deep, and hoarse, the sound reminiscent of the death bells, and Janus closed her muzzle with his hand. Aris nodded his shaky thanks and they fled the court. As the king's doors sealed tight behind him, the chaos spread unchecked, as a third, then fourth girl collapsed.\n\nIn the doorway, shadowed by the rising sun, Mirabile smiled at Maledicte, and dropped the tiniest of curtsies, a performer acknowledging praise.\n\n**\u00b7 28 \u00b7**\n\n**A** RIS LOOKED DOWN AT THE wreck of a body resting on the marble slab, lying between Haith's sculpted hands, sheltering in the grasp of the god. Though pains had been taken with the corpse, the worst of the torn and waterlogged flesh hidden beneath the blue cloak, still the face was barely human. Only the chill of the winter sea had kept flesh and bone together, and Aris, remembering his first horrified look at the sea dreck his brother had become, knew that beneath the softening cloak were sections of bare bone.\n\n\"Sire,\" a kingsguard said, \"the courtier Maledicte has arrived. Where will you receive him?\"\n\n\"Bring him here,\" Aris said.\n\nThe guard's sandy brows rose nearly to his hairline, but he merely nodded. Aris turned his attention back to his brother's body, barely hearing the man leave.\n\n\"So it came to this,\" he said. \"Nearly alone, our family winnowed by time...You should have been the older, Michel,\" Aris said, feeling as if a weight had settled over his neck and shoulders, sinking toward his heart. \"You would have made a better king than I, I think, shortsighted and reactionary though you were. Antyre loves me not, and more, respects me not. You would have forced respect on them. Or fear. And I would not be left with this\u2014\" Footsteps echoed in the hall, the shuffle of feet on stone.\n\nAris raised his head. Few enough people came down the winding, dark hallways toward the chapel that he knew who it must be.\n\nMaledicte dropped a bow. \"You sent for me?\" Behind him, a guard lurked.\n\n\"Yes,\" Aris said, his throat rough. Maledicte's eyes were heavy-lidded, his hair loosely and hastily tied back. Small jet feathers sieved from them, and Aris remembered the two dancers, one black, one white, whirling around the ballroom, heads bent close together. His hands fisted.\n\n\"Come and see what has befallen my brother,\" Aris said, stepping away from the bier.\n\nMaledicte came forward, wavering like one of the shadows in the dimly lit room. He leaned over the ruined head, and stepped back, his pale face expressionless.\n\n\"Does it satisfy you to see him dead?\" Aris said. \"I know you hated him, though never the why of it.\"\n\n\"Yes,\" Maledicte said.\n\n\"Give me your hand,\" Aris said.\n\nMaledicte proffered his right hand. Aris seized it and tugged him back to the edge of the bier. The sickly sweet odor of putrefaction washed over them, driving Maledicte's perfume back. His hand in Aris's struggled. Maledicte turned his face away, leaving Aris to speak to the wing of dark hair sheltering him.\n\n\"Touch him,\" Aris said, voice ragged. The guard leaned forward to watch, witness. Maledicte resisted, and Aris yanked his arm forward, stretching it toward the body. Tears started in his eyes.\n\nMaledicte turned, caged Aris's prisoning hands with his free hand, stopping him. \"A learned man so wild with grief,\" Maledicte whispered, his voice meant only for Aris's ears. Maledicte's dark lashes lifted; the black eyes met Aris's and Aris shivered. \"Which superstition are we chasing, sire? Is Last supposed to bleed at my touch?\" He freed his hand from Aris's grip, as gently and precisely as a pickpocket liberating coins.\n\n\"I doubt you can claim squeamishness.\"\n\n\"Why don't you ask me?\" Maledicte said.\n\n\"You must do it,\" Aris said, looking away from the sweet mouth turned down in distaste, hardening his ears to the intimacy of that raspy voice, so close, sounding so caring. False or not, it made Aris tremble.\n\nMaledicte stepped forward and touched his fingers to Last's forehead, then touched the sodden fabric over Last's heart. Then he spread his hand to show Aris the unmarked flesh. \"I did not kill your brother.\"\n\nAris no longer knew what to believe, his mind as cold and as numb as Last's corpse. As cold as the fallen debutantes, awaiting their spring burials. He only knew that death had come to his country on Maledicte's heels, that any member of his court must be an able liar, well versed in apparent sincerity, and that Maledicte had an unsavory reputation as a swordsman and blackmailer. The one Aris could testify to, thinking of the damning Antyrrian audit ledgers kept hidden by this boy's pale hands, weapons more worrying to Aris than the blade.\n\nAs for the sword\u2014those delicate hands were smooth, barely callused, and Aris knew that reputations were often based on gossip. Neither Aris, nor any of his guards, had ever seen Maledicte dueling. Even Echo, ready to condemn, had qualms imagining Last taken by Maledicte's sword. Last himself granted no aid; though he had spoken out against Maledicte in life, his body, caught up in the _Fleur_ 's anchor and dragged along the keel before breaking free, was too mangled to make any mute accusations.\n\nAris covered his eyes as if he could blot out the images, blot out the panic surging in his blood.\n\nAris shivered as Maledicte put his hand on his arm, unasked. He heard the guards shifting uneasily, but said nothing, instead allowing Maledicte to tow him away from the bier. He opened his eyes to see what expression he could catch on Maledicte's face, as if he could sneak up on verity when it eluded words, but learned nothing new, save that compassion sat uneasily on Maledicte's clever face.\n\n\"I thought you meant to question him, sire,\" Echo said, arriving in the doorway on an upswing of anger. Aris put another body width between himself and Maledicte.\n\nMaledicte said, \"Ask at will. I will give you no more difficulty.\"\n\n\"The debutantes,\" Aris said, his words overriding Echo's attempt to take control of the room. \"Had you anything to do with their deaths?\"\n\n\"No,\" Maledicte said, startled and frowning. In the background, Echo scoffed, and it was to him that Maledicte addressed the rest. \"I am no killer of feckless girls.\"\n\n\"Whispers speak of a evil pact between yourself and Mirabile,\" Echo said, coming closer.\n\n\"Are you serving as counselor of gossip?\" Maledicte asked, his customary acid eating into his tone.\n\n\"You danced with her,\" Echo said, \"your heads bent close as if you shared secrets and schemes\u2014\"\n\n\"Never with her,\" Maledicte said. \"She's quite crazed. Send for her, Echo. I doubt she'll deny her wrongdoing.\"\n\n\"I sent Jasper to collect her,\" Aris said, noting the lack of concern in Maledicte's gaze. \"Echo, it's early. Let us continue this later, after we've spoken to Mirabile...\" He drew Maledicte away from Echo, closer to himself, and lowered his voice. \"I will see you out of this, but you will repay me with your discretion and silence this winter. I want no gossip to reach my ears of your doings.\"\n\n\"Such a thing you ask of me,\" Maledicte said. \"Surely, it is not within my power to still idle tongues\u2014\"\n\n\"Enough,\" Aris said, in no mood for banter, not with his brother's corpse so near, not with the young man suspected of killing him. \"The court is closed. Until spring comes and wipes away death with life, there will be no balls, no celebrations, no masques. The nobles will rusticate in their country homes, or in town estates should they feel inclined. You will do likewise. Do so and I will give you Janus.\"\n\n\"He's not yours to bestow,\" Maledicte said.\n\n\"But his absence is mine to command. I could see him mewed at Lastrest, trapped in mourning clothes and customs. Or I could keep him at my side exclusively\u2014\" Aris trailed off. In the dimly lit room, cold with death and pain, Maledicte's burgeoning anger felt huge, a looming, dark presence. Echo moved closer, hand dropping to his sword hilt.\n\n\"Little fool,\" Aris said, seizing Maledicte's shoulders and shaking him with all his pent-up frustrations. \"Do not force me to heed the whispers.\"\n\nMaledicte suffered the shaking meekly, his hair falling from its loose queue and hiding his face. His harsh voice came like a whisper of scale on stone. \"I will be discreet, sire, to the best of my ability, which is not inconsiderable, you'll agree?\"\n\n\"Aris,\" Echo objected. \"Better to hold him until we see what Mirabile has to say\u2014\"\n\n\"Are you so eager to keep me from my luncheon and my books?\" Maledicte asked. Aris felt like one of his dogs gone to point. The specter of the ledgers had been in the room since Maledicte's entrance. He had expected Maledicte to invoke their power sooner, and more bluntly. This reminder, so gently spoken, could be a threat, or merely a reminder that Maledicte could, indeed, be discreet.\n\n\"Go then,\" Aris said. His hand, still resting on Maledicte's shoulder, lifted and twined the dark hair around his fingers, turning his face up for study. It wasn't innocence that greeted him; Aris would have distrusted such an expression on Maledicte's face, but there was no triumph either. Its lack softened Aris's offense. \"Go on then. Off with you. Let me hear nothing of you but praise, and come back in the spring.\"\n\nMaledicte bowed, his hair slipping through Aris's grasp.\n\n\"Sire\u2014\" Echo objected, but never had time to finish. Jasper returned, white-faced, two guards walking behind him, their hands on their pistols, their gazes nervous, as if they had seen devils.\n\n\"Jasper?\" Aris asked, his voice unsteady. \"What's happened?\"\n\n\"Westfall's dead...his house afire\u2014\"\n\n\"The antimachinists? They dared to\u2014\"\n\nJasper wiped a hand over his mouth and his sweating face as he interrupted his king. \"Not them. Mirabile's run mad, sire. She's killed them both, poisoned Brierly and murdered Westfall. She took his eyes and heart.\" He shuddered. \"We saw her, gown bloodstained from hem to hip, as if she'd been wading through blood. But before we could lay hands on her, she was gone, like a shadow disappearing under the noon sun.\n\n\"Gone,\" Aris repeated, dumbly.\n\n\"It's witchcraft, sire, I swear. No matter that the gods are gone...she's found some way to touch power, and no one will be safe until she's stopped.\"\n\nAris sank onto the bench, looking up at the painted gods, and for the first time in thirty years offered a whisper of a prayer to Baxit, praying that reason would return to his kingdom.\n**\u00b7 29 \u00b7**\n\n**T** HE FIRST SIGNS OF SPRING inside the city limits were the groans and creaking of the ice breaking up near the docks, moaning like live things in torment. Maledicte had slept badly this winter, and would have slept a good deal worse, saving Janus's presence and the gossip mill turning from him to Mirabile. As the Dark Solstice deaths faded in urgent memory, his rumored part in it fell beneath the waves of Mirabile's continued depredations.\n\nThe slaughter of the Westfalls paled beside the subsequent deaths of the four kingsguards who had run her to ground. All four men were found rent and eyeless, and on that violent topic, tongues wagged. Mirabile, some cried, was a phantom, returned to plague the living. A witch, cried others, and one who meant to curse the aristocracy. Others, more cautious, whispered of returned gods and Ani's touch, whispered so quietly that only Gilly, sifting information, heard that rumor.\n\nOne further tidbit kept bored tongues busy. The whisper that Aris had chosen Janus to replace Westfall as counselor. The rumors claimed first that this was merely Aris's way of leashing his scandalous nephew and keeping the last of a line close. More acidly voiced rumors said that Aris always liked one of his counselors to be in touch with the common folk, and what was more common than a bastard?\n\nStill the season passed, with Janus often at the palace, acting as Aris's aide. It was Janus who greeted foreign merchants at the dock, haggling for Aris, and spurning the bulk of the Itarusine cargo. And it was Janus, or so it was murmured, who met with Captain Tarrant, that pardoned war pirate, to strike a surreptitious bargain, smuggling those same spurned Itarusine goods into the country, thus relieving the exorbitant prices on staples, and silencing some of the protesting poor. But if Janus spent his days at Aris's beck and call, his nights were Maledicte's exclusively.\n\nEven with Janus's near-constant presence, Gilly knew that Maledicte was more often haunted than not, nights given over to nightmares, and saw, with increasing regularity, the shadows drifting in Maledicte's eyes as Ani, stymied, made Herself felt in a hundred small black tantrums and nightmares. From the brittle tension that rose between Maledicte and Janus, from the near-resentful looks Maledicte cast Janus on occasion, Gilly thought he understood what had happened. As of yet, he had not found a tactful way to ask for confirmation.\n\nTonight, Gilly came in with the groaning of the ice, feeling as grave as if it were he doing the moaning. He had, in his hand, the instrument that would shatter their fragile peace.\n\n\"Do you know you have frost in your hair?\" Maledicte said, lounging in the hall with a glass of wine in his hand. \"And you look chilled through.\" Maledicte set his glass down on the empty receiving salver. As Aris had requested, Maledicte had refused to attend any of the makeshift festivities, though in truth, few had requested his presence. Maledicte dusted the frost from Gilly's coat and sleeves. Parchment crackled like breaking ice, and Maledicte tugged the paper from Gilly's hand.\n\n\"What's this? A note from Lizette? Does your ladybird know how to write?\" Maledicte teased. Gilly reached for the letter, but Maledicte evaded him.\n\nTaking up his glass again, Maledicte propped himself against the wall, and began to read the gathered gossip and speculation Gilly paid Bellington for.\n\nThe glass splintered in his hand. \"When did you know about this?\" Maledicte demanded.\n\n\"Just this afternoon,\" Gilly said. \"I got word from the solicitor and went down to the docks to talk to the captain of the _Kiss._ He confirmed it, said she was showing signs even on the journey out.\"\n\nMaledicte let out a strangled sound\u2014whimper, snarl, or both together, combined of rage and despair. \"We should have thrown her into the sea with her damned husband. But who would have calculated the odds to be so against us? Five years it took for Last to seed his previous wife, and several slips after that. But Amarantha\u2014wife for a bare sennight\u2014\" His breath sobbed in his chest, unequal to his rage.\n\n\"What's the matter now, the soup served cold?\" Janus asked, coming into the hall. As he looked from Gilly to Maledicte, the bored humor drained from his face.\n\n\"The countess of Last, Amarantha Ixion, is near to term with your father's child, and she returns to lay claim to the estate and title.\" Maledicte belatedly noticed the broken glass in his hand. He opened his clenched fist, let the shards patter down like rain from his unmarred skin.\n\nJanus blanched. \"A blow to be sure.\" He raised his hand to his forehead, rubbed the narrow spot between his eyes. \"Is it Father's child for sure? Not some bastard thing she's using to gain control of the estate?\"\n\n\"If it is not your father's babe, it is so close that we will never prove otherwise,\" Gilly said.\n\n\"What do we do?\" Maledicte said. \"If the child is born, if it is a boy, our plans are thrown over, Janus.\"\n\nJanus stroked Maledicte's dark hair. \"You'll just have to kill her before she gives birth. But be careful, Mal. Aris seems most...interested in your activities. Echo counsels mistrust, while I scoff, and yet we only attain stalemate. Best we heed Aris's obsession, and be discreet in her death.\"\n\nWHEN ARIS SENT A RUNNER to Janus, informing him that he would be sent to meet Amarantha's ship with the royal carriage and a slew of guards, Maledicte said, \"I don't suppose you could drown her by accident.\" He said it with no particular energy, lounging on the chaise, slowly moving to fill the area that Janus had vacated.\n\n\"No,\" Janus said, though his lips quirked.\n\nMaledicte marked the smile with one of his own. \"At least we know that Aris holds you innocent of your father's fate, to send you to fetch Amarantha.\"\n\n\"He sends me with an armed escort. That argues no particular trust,\" Janus said, pulling his coat on, settling the shoulders, and checking the lines in the mirror.\n\n\"Well, you did murder Last,\" Maledicte said.\n\nWatching from the chair beside the parlor door, Gilly raised his head sharply. Janus cast him a baleful look, and spoke to Maledicte. \"We are not discussing this again. You killed him, I merely sped him on his way.\"\n\n\"All your own way,\" Maledicte said, still lazily. \"The fun of patricide and treason and none of the blame.\"\n\nJanus stooped, pulled Maledicte up, and shook him, once. \"Enough. What do you want me to do to apologize?\"\n\nMaledicte smiled at him. \"I can't think of anything.\"\n\n\"I can,\" Gilly said, drawing two sets of eyes to him. \"Wouldn't it be appropriate to have a welcoming celebration? Urge Aris to hold one.\"\n\n\"She wouldn't attend,\" Maledicte said. \"The letters our spies sent said she had become quite mad with suspicion. To expect her to attend a ball, where others have died\u2014\"\n\n\"Take it up with the king. If Aris commands it, she will attend,\" Gilly said, his mouth dry as he argued for murder. But if Janus meant to see Amarantha dead, Gilly would do what he could to ensure Maledicte's survival. Without a plan, Maledicte would be far too prone to give in to Ani's careless bloodlust. And unlike Last, a pregnant Countess would rarely be alone.\n\nGilly let out a shaky breath. Perhaps this was a second chance to free Maledicte from Ani's touch. There was no earl of Last, but Amarantha was the titular head of the line\u2014perhaps her death would be enough; perhaps whatever had been done the first time to invalidate Last's death, Maledicte could undo. It came to him, suddenly, that he was hoping for the death of a woman with child, and his whole body rang with the shock of it.\n\nJanus paced the room. \"If Aris agreed, it would be a well-guarded thing, Mal. I doubt you could you kill her there.\"\n\n\"Mirabile did well enough,\" Maledicte said. \"You gave me aid when none was wanted, Janus. Give me aid now when I ask for it. Our enemies grow like the hydra. One dead, two created. Let's destroy this head before we have to kill an infant as well.\" Maledicte's voice shifted.\n\n\"Be as honest with yourself as you were with me when you accused me of patricide. Were Amarantha not gravid, you would not need to raise your sword. Infanticide is your goal, Mal. Can you stomach it?\"\n\n\"I have no choice,\" Maledicte said, \"if you would be earl.\"\n\nAfter Janus left, Maledicte sank onto the chaise and covered his eyes. Gilly went to his side, hesitant, then reaching out, took one hand. Maledicte's fingers curled around Gilly's.\n\n\"He killed Last?\" Gilly asked, shying away from future murders in favor of past ones. But the confirmation of his fear laced his heart with dismay. If Janus had done so, was it any wonder that Ani lingered, foul-tempered and growing? \"That could not have satisfied Ani.\"\n\n\"In all your books,\" Maledicte said pensively, \"all your pamphlets and gossip, have you ever heard that Ani can be satisfied? Mirabile seemed to think otherwise.\"\n\n\"Tell me what happened,\" Gilly asked. \"How you meant to kill Last, and instead had Janus kill Last and some sailors.\"\n\nMaledicte raised his eyes, ringed with sudden weariness. \"If you know that, you know it all. We should have taken Amarantha then. If I had, I would not be facing this now.\"\n\n\"You could wait,\" Gilly said. \"Perhaps the child will be female, or, like Adiran, will be born flawed, unable to inherit. Or it may die of its own accord, as Last's most recent son did. Murdering Last is one thing, this is another.\"\n\n\"Enough, Gilly, I am done with talking about it. If Amarantha attends the party, she dies.\" Maledicte burst from the chair, yanking his hand from Gilly's, and stormed toward the door. He paused at the last moment and turned back, his voice ragged and wild. \"I cannot have this, Gilly. I cannot take on your conscience. I need to be free to draw blood at will, be it man, woman, or babe. As there are deaths behind me, that is all that is before me as well. Do not weaken me.\"\n**\u00b7 30 \u00b7**\n\n_...he withdrew his knife and stabbed her thrice, seeking her heart, but she merely mocked his prowess with the blade, for Ani's unnatural children scoff at injury and fear no man. She clawed out his eyes with sharpened nails and when dawn came, she was found still feasting on his heart..._\n\n\u2014Grayle's _Book of Vengeances,_ \"The Savage of Issey\"\n\n**S** NOW SPOTTED THE FIRST BRAVE leaves of the spring crocus. Maledicte looked up at the leaden sky, and at the faint sparks of spiraling white drifting down to edge the palace grounds. \"Spring?\"\n\nGilly, attending him, said, \"Snow's not unheard of this early in the season. But it is damaging to silk. Best go in now.\"\n\nMaledicte smiled at him. \"Oh yes, because spotted silk is a terrible sin.\" The fey cheerfulness to his manner made Gilly's stomach ache. He had seen this before. It was as if Ani, knowing Maledicte's plans, was curled up in sulky approval, sated before the act as She never was after.\n\n\"I think you just don't want to be alone in the dark with a murderer,\" Maledicte said, tugging Gilly's tied-back tail of hair.\n\n\"Mal, hush,\" Gilly said, looking around. No one was within earshot, but his heart pounded all the same. Halfway to the king's court, in the winter-riven lines of the garden, with the stables at their back\u2014Gilly couldn't imagine what Maledicte was playing at.\n\n\"Admit it, Gilly. You fear me.\"\n\n\"Fear _for_ you,\" Gilly said, taking Maledicte's arm and pulling him deeper into the skeleton of the garden. He pressed Maledicte back into the prickly embrace of a hedge, its leaves only faint smudges of starting greenery, and said, \"What ails you?\"\n\nMaledicte closed his eyes, letting the snow lay ephemeral patterns on his skin. Gilly touched Maledicte's cheek. Had it not been for the quick, cold nip of snow melt, the dampness on his palm could have been tears. \"Mal?\"\n\n\"I am,\" Maledicte said, opening dark eyes. \"I am afraid to be alone in the dark by myself.\"\n\nTongue caught, Gilly could say nothing.\n\n\"I do things I never expected I could. And that's cause enough to fear, but more, I do not feel alone in my own mind, in my own skin. She's there, wanting out. It's getting crowded, Gilly. The person I was, the person I am, and the crow. We're all jostling for ascendancy, and I don't know who's going to win.\"\n\nGilly opened his mouth; Maledicte put his gloved hand over it. \"Listen, Gilly. If Ani wins, leave me. Don't stay. I would never hurt you, but She would devour you entire. Promise me.\"\n\nGilly shook his head, and Maledicte frowned. Footsteps crunched on the seashell paths as a coachman walked steadily into the dark, undoubtedly heading for the wall the stable staff used as a privy. \"Come on,\" Maledicte said, ducking under Gilly's caging arms.\n\nTurning in the direction of the court, Gilly found himself walking alone. Maledicte had gone back the way they had come, heading into the stables.\n\nGilly caught up, trying to move soundlessly, grimacing with the effort. Maledicte smiled at him when he arrived. \"Watch for anyone coming?\" He went down the silent rows of detached coaches. Twenty feet away, stableboys fed horses, rubbed them down, and cleaned the stalls.\n\nMaledicte ghosted along the rows of coaches until he reached the glossy blue of Last's coach, gone drowned and greenish in the flickering lamplight. Maledicte climbed onto the coachman's bench and insinuated his hand into the juncture of carriage and seat, recovering a worn flask. He joined Gilly again in the sheltering darkness of an unused stall. \"Hold this,\" he said. He stripped off his jacket and felt inside the seams. Gilly watched, mouth falling open. As Maledicte pulled out two tiny crystal vials, Gilly said in a furious whisper, \"You brought poison to court? After what happened to those girls? You are mad.\"\n\n\"We're not in court,\" Maledicte said, \"and these vials never will be.\" He levered out the wax stopper and trickled a thin syrup into the coachman's flask. Closing the flask's lid, he sloshed it gently.\n\n\"You're going to poison the coachman?\" Gilly said.\n\n\"Would you prefer me to stalk into the court and strike Amarantha dead by blade, or pour her a drink and have her fall at my feet? This way is more chancy, but far more likely to pass as accidental.\"\n\nMaledicte sloshed the flask a moment more, then opened it and sniffed. \"Perfect.\"\n\n\"And if he drinks it all now? While waiting for his masters?\"\n\n\"I'm counting on it,\" Maledicte said. \"Janus is supposed to goad Amarantha into flight. Failing that, my presence alone should do it.\" He looked over Gilly's shoulder and scowled. Two stableboys had skived off their chores and crept into the coach aisle, and were playing dice in the carriages' shadows.\n\n\"Rats take it,\" Maledicte muttered.\n\n\"We've time,\" Gilly said.\n\n\"That we do not,\" Maledicte said. \"Dantalion is too careful. He will not allow his coachman to linger in the yard with the others, playing cards. He'll want him here. Guarding the carriage against saboteurs.\"\n\n\"Give it to me, then,\" Gilly said. \"I look enough like a coachman. I'll return it; those boys won't remember me at all.\"\n\nMaledicte relinquished his hold on the flask and Gilly sauntered out into the lamplit alley between coaches. The two boys paused in their game, bodies wary, ready to bolt should Gilly show any signs of noticing them.\n\nGilly realized halfway to Last's coach that his was more than a little errand that Maledicte could not do, that what he was doing would result in at least one man's death, maybe more. But the fear that if he balked Maledicte would choose a more dangerous path kept him from freezing in his tracks. \"Don't ask me to kill for you,\" he had said once. Now it seemed he volunteered.\n\nFeeling as if he ascended the gallows, Gilly climbed to the bench. He had just reached to return the flask when he heard the cry. \"Hoy! What're you doing?\"\n\nHe turned, aware of the two stableboys scattering\u2014directed at them or not, the words were too close to the ones they dreaded\u2014and found Dantalion's coachman staring up at him.\n\n\"Get off of there\u2014hey, that's mine,\" he said, his indignation darkening to suspicion and anger. He held out his hand for the flask and Gilly, seeing no alternative, put it in his hand.\n\n\"What were you doing with that?\"\n\nGilly, reaching for a plausible explanation, was forestalled by Maledicte. \"I asked him to find me a drink,\" Maledicte said.\n\n\"There's fancy guff inside. What d'you need mine for?\" The coachman scowled at the slim aristocratic shape.\n\n\"The last time drink was taken within those walls, people fell dead. Call me overcautious,\" Maledicte said, leaning against the stall.\n\n\"Mirabile killed fillies,\" the coachman said, but after another sneering look, he continued, \"though you've got more than a touch of the mare about you, don't you?\"\n\nMaledicte's cheeks flushed, and he dropped his hand to where his sword hilt would have been, had he not left it in the hay when he removed his jacket.\n\nAssessing that motion, the coachman paused. \"You're that one, aren't you? That cursed cavalier my master natters on about. Maybe you'd better have my flask after all. Take a drink of it, just in case.\" He tossed it to Maledicte.\n\n\"Too gracious,\" Maledicte said, tilting the tarnished metal to his lips.\n\nGilly's heart was in his mouth, choking back protest, as he watched for the trick, the movement that betrayed that Maledicte was not really swallowing mouthfuls of his own poison. A trickle of adulterated whiskey ran from the edge of his mouth, and it was too much for the coachman.\n\n\"Here! Leave me some. Gi' me that.\" He snatched it from Maledicte's hand. He shook the flask, and swore. \"Drank near half of it, damn you.\"\n\nMaledicte wiped his mouth with a lazy hand. \"That stuff 's rot; you really should get your employer to give you better.\"\n\nThe coachman spat on the floor, and Maledicte moved the tip of his polished boot away from the glistening, wet spot with a moue of distaste. \"And people find my manners lamentable? Gilly, bring my coat with you.\" He stalked off without waiting for reply.\n\n\"I don't envy you your master, boy,\" the coachman said.\n\nGilly jammed his shaking hands into his coat pockets. \"He pays well.\" His words were near as hoarse as Maledicte's, tight with dread. Dread that Maledicte drank his own brew. Dread that the coachman would drink and die and make Gilly a murderer.\n\nGilly cast a frantic glance into the gardens, but Maledicte had disappeared from sight. His stomach clenched to the point of pain, imagining Maledicte fallen, convulsing. Would Ani protect him? With Last's death denied Her?\n\nHe snatched up Maledicte's coat, hearing a faint rip as the embroidery snagged on the baled hay, and hurried toward the stable doors. Gilly looked back once to see the coachman take a great pull from his flask, making Gilly a murderer.\n\nAfter a few panicky moments, he found Maledicte back in the quiet shadows of the thorny hedge again. His eyes glittered like black water. \"Did he drink?\" Maledicte asked. Coatless, he seemed smaller, more fragile than he was.\n\n\"Did you?\" Gilly asked, his voice trembling, a bare whisper.\n\n\"You saw me,\" Maledicte said.\n\nGilly tugged at him. \"Let's go home. We'll find you the antidote. Or maybe you won't really need one. Ani protects Her own, right? But we can't risk it.\"\n\nMaledicte slipped from his grip. \"Gilly, I've already taken it. Two vials, remember? You were afraid I was dying? I don't trust Ani that far. And I'm not stupid enough to die in an attempt on Amarantha's life. Not when it's a chancy death at best. I think I'm offend\u2014\"\n\nGilly seized him close, held him, heart beating against his own. This close, Maledicte surprised him by not being awkwardly tall or broad; he fit as snugly in his arms as Livia did. Maledicte sagged against him, giving Gilly license to let his hands rove down across the narrow back and slender hips, pulling him closer still. Maledicte looked up at him, and Gilly bent; at the last, Maledicte avoided his mouth. Gilly's kiss ended on the slightly slick length of the scar on his jaw. He tasted the flesh there, a tongue tip at a time, and Maledicte made a faint sound in his arms, of appreciation or protest, Gilly wasn't sure.\n\nGilly's clever fingers transmitted a piece of information to him. \"You wear a corset?\"\n\n\"I eat too much,\" Maledicte muttered, and while he didn't take himself from Gilly's arms, nonetheless Gilly was aware of some wary withdrawal.\n\nGilly touched the line of his jaw, guided Maledicte's mouth toward his own, but even as he did, desire faltered to curiosity. The lean bones of Maledicte's arms and legs argued against such a need.\n\nMaledicte's sigh against his skin stifled curiosity, and Gilly pressed his suit, aware of his own hunger made evident in the fit of his breeches, his thighs against Maledicte's.\n\n\"Let me go,\" Maledicte said. \"Enough, Gilly.\" The whisper was faint enough that Gilly could ignore it if he chose. But while the tremble in Maledicte's back, the kneading of his hands on Gilly's chest urged him on, Gilly was all too aware that Maledicte's desire did not equal his own, that if they were to step apart, there would be no telltale swelling to mar the smooth fit of Maledicte's breeches.\n\n\"Gilly,\" Maledicte's voice was more urgent. \"Let me go, or I'll make a eunuch of you.\"\n\nStartled, Gilly released him. Maledicte staggered away, fell to his knees, and vomited in the hedges. Snow hares rustled and darted away from his sudden descent into their domain. Gilly crouched beside him and Maledicte gasped that he was well. Gilly drew Maledicte's hair from his face while he was sick.\n\nMaledicte rose and took steps back to the main path, sat down on a stone bench, covered with a thin drifting of blown snow. He wet his hands with it, the snow melting at his touch, and wiped his face. \"Sometimes the antidote is worse than the poison.\"\n\nGilly sat down heavily by his side, his heart feeling overtaxed. \"I thought you were dying.\"\n\n\"We've had this conversation,\" Maledicte said. \"And it led us\u2014\" He put snow in his mouth like a child. It reddened his lips.\n\n\"Led us where?\" Gilly asked. It hurt to do so, to probe at the disconnection between them, but he was no more capable of not asking than he was of walking away.\n\n\"Astray,\" Maledicte said. \"Decidedly astray.\" He leaned his elbows on his knees, traced images in the frost at their feet. Raven wings, eyes, a sword. \"I am his entirely, remember. What I do, I do for him.\" His mouth twisted, as if he found the fact not as much of a boon as it once was.\n\n\"And he'll be looking for you,\" Gilly said, standing and holding out an arm.\n\nMaledicte hesitated, then took Gilly's arm. As they walked toward the yellow glow of candlelight and warmth, the drifting voices that held an edge of fear, Maledicte said, \"Besides, Gilly, I'm no partner for you. You need a nice girl, one who'll give you babies, not ask you to kill them.\"\n\nGilly let out his breath. \"Lizette's a whore, and no fonder of me than she is of my money, and our little Livia's a spy. Yet I care for them both. So what's the addition of one murderer to my affections?\"\n\n\"Livia\u2014a spy?\" Maledicte said, his eyes hooded by speculation.\n\nGilly bit his lip, but words once said were impossible to cage again. \"She has far more coin than she should and she creeps out nights. And none of our trinkets or teaspoons are missing. Unless she's thieving other houses, it's information she's selling.\"\n\n\"A spy,\" Maledicte said, dismay in his voice. \"And we have such secrets to sell.\" He drifted up the lawn, boots leaving dark tracks in the rime, and paused. \"Perhaps we can turn it to our advantage. Do nothing directly until we know who's buying.\"\n\n\"Dantalion,\" Gilly suggested.\n\n\"Or Mad Mirabile, or even Aris, as unpalatable as that thought is. We'd best find out.\"\n\nGilly nodded, a little shamed that he had needed telling.\n\nMaledicte looked toward the lit rooms, spills of light raying out like slow lightning, flickering in the wakes of skirts and coats, and his mouth tipped into a deeper frown. This near, they could hear the forced gaiety, the musicians sawing out newly written tunes, lest anyone be reminded of the Dark Night deaths. \"Gilly, go prepare the coach. I will enter only long enough to spook Amarantha, if Janus has not already done so. Tonight, I prefer my nest.\"\n\nMALEDICTE WIPED HIS MOUTH one last time; the bitter taste of bile, tannic fluids, and belladonna lingered. He climbed the wide steps from the garden and gained access to the balconies, unwilling to enter under the watchful, fearful eyes of the other attendees. Seen through the open doors of the ballroom, Janus danced attendance on Psyke Bellane, his eyes alight with an amusement she didn't share.\n\nWhite around her rosebud lips, the china doll curtsied and attempted to take her leave. Janus stopped her with another question, a hand on her silkdraped arm, smiling down at her. Her chaperone watched with a smile. When Janus lifted his hand from her sleeve, she flew like a dove.\n\n\"What kept you?\" Janus said, turning as if he had sensed Maledicte's approach. \"I've had to entertain myself with sweet, scared Psyke. What did you say to her?\"\n\n\"Nothing she took seriously enough,\" Maledicte said, watching the slight girl slip through the crowds. He fought an absurd sense of betrayal, as if he had expected Psyke to forswear Janus simply because the time spent listening to Maledicte's threats had spared her from Mirabile's touch.\n\nHe swayed on his feet a moment, off balance by the imagined weight of his hatred. Janus smiled at him and led him back out to the seclusion of topiary and stone.\n\n\"You look so fierce,\" Janus said, kissing Maledicte's throat and cheek. Maledicte turned his head to avoid his lips, thinking of toxins on his, feeling as if he'd stumbled into an odd, repetitive dream world. Turn from Gilly's warmth, turn from Janus's arms; face the cold wind and the dark alone. He dreaded Ani rising up through his throat, stabbing out from his mouth, the long beak terrible and gore-smeared, Her wings scratching through his chest, pushing out past his lungs and ribs, dragging him up into the night sky, a soaring, bloody puppet. Distantly, Maledicte realized some of the belladonna must have lingered beyond the antidote, sparking hallucinations.\n\n\"What kept you?\" Janus asked.\n\n\"Gilly kissed me in the snow gardens,\" Maledicte said. \"He killed the coachman and kissed me....\" He shook his head, shaking the moment from his mind. \"I fear his ethics have been severely compromised by our association.\"\n\nJanus shook him. \"Mal, are you mad?\"\n\n\"Yes,\" Maledicte said. \"I think I must be.\" Lips moving against the roughness of the brocade, Maledicte imagined the threads snaking out to drag him into Janus's skin.\n\n\"To let Gilly kiss you, I must agree,\" Janus snapped, pushing Maledicte away. He paced a quick circle and then came back, blue eyes smoldering. \"Do you want him, desire him? Is that it? Why you would risk all for a tumble? In the king's garden? Tell me, Mal\u2014do you love him?\"\n\nJanus's hushed words sounded wounded, stripped of strength, but Maledicte, dark-dreaming with the belladonna's aid, saw what the low tones disguised\u2014the red cloud settling around his bright form, splintering out from the steady flame of rage behind his pale eyes.\n\n\"You are all my desires,\" Maledicte said, twining his arms about Janus's neck. \"And so I told him. That I am yours and yours alone. Though you are not exclusive to me...you will marry.\"\n\n\"And why shouldn't you?\" Janus twitched within Maledicte's arms at the familiar deep voice. Maledicte felt near panic himself\u2014how long had Aris been listening? Behind the king, Psyke stood in the doorway. Maledicte fought a growl\u2014had she led the king to Janus?\n\n\"Sire,\" Maledicte said, sinking into a bow.\n\nBeside him, Janus nodded. \"Uncle.\"\n\n\"I would speak with Maledicte,\" Aris said. \"Janus, I owe Psyke this dance. Please take my place.\"\n\nJanus bowed and left, Psyke on his arm once more; Maledicte stood as still as a wild creature unexpectedly cornered. His heart pounded. Inside him, Ani stretched Her wings, whispering. His fingers itched for the sword hilt, for its cold security, with a desire not his own.\n\n\"Be easy,\" Aris said, settling himself onto a bench with visible weariness. \"You are in the fortunate position of the king requesting a favor.\" Aris patted the bench.\n\nMaledicte slipped over to him, sat on the very edge of the stone, among the carved vinework. \"You want me to marry?\"\n\n\"All men should marry, if only to see themselves through others' eyes. Wedding Aurora changed the way I saw my kingdom and myself. Under her tutelage, I saw my court as it was\u2014decadent, violent, concerned more with matters of style than of state. We have chosen to mirror Itarus, but we have chosen to reflect only their surface. Their courtiers vie and kill, but they further the kingdom as a whole, whereas my court\u2014cares only for entertainment.\" He sighed, breath loud in the night; frosty clouds carried his words away, scented with the sting of wine.\n\n\"Aurora was a queen to be revered. She knew we had to change, and I did so. I have attempted to make the court do so as well, and failed on that count.\"\n\n\"Such foolish heads can be led,\" Maledicte said. \"Make them bend to your will.\"\n\nAris laughed. \"I am merely a king and not a god.\" His laughter faded into bitterness. \"And while some kings can ape gods, make their will and their people's one and the same, I am all too aware that I am only a man. But I understand Baxit, who looked on His own court and despaired, who forced the gods into oblivion. Were it not for Adiran, for other innocents who would suffer, I could do the same. Stop the struggle and let us fall\u2014\"\n\nMaledicte shivered. Aris's face, so like Janus's in structure, might have been uncarved marble for all that Maledicte could learn of his mood in it.\n\n\"Were it not for love\u2014\" Aris said, his gloved hand touching Maledicte's chin, his voice thinned by exhaustion. Then his mouth claimed Maledicte's, his lips not tentative like his words, but so fierce that Maledicte felt Ani dwarfed beneath the sensation. It wasn't simple desire that Maledicte felt in Aris's hunger, but bleakness, a desperate attempt to quicken the blood.\n\nAris's tongue touched his, wine-rich, and Maledicte shoved him away, panic racing in his veins. It was done without grace or subtlety, but all he could think was what if the belladonna lacing his mouth was enough to kill a king? He fell off the bench, awkward in fright and dismay.\n\nMaledicte knelt on the cold, damp stone, silent and waiting, his agile tongue gone dry along with his bravado, tracing the tangling lines of the stony vines.\n\n\"It is bewitchment,\" Aris said, \"that feeds this fascination. But I think one of my own making.\"\n\nHis voice, empty of any emotion but despair, wicked some of the dread from Maledicte's spine. \"Aris,\" Maledicte breathed.\n\n\"Shh,\" Aris said, laying his hand over Maledicte's mouth, then taking it away as if desire would spill over again. \"I will be done with this nonsense, and ask my favor of you. The Lady Amarantha fears you. Fears your eyes on her belly, beyond all reason. I would have you avoid my court until her child is born.\"\n\n\"Exile?\" Maledicte said, striving to put some flippancy in his voice, striving to restore himself. He was Maledicte, the unflappable, dark cavalier. Why should a weary king and a despairing kiss have had the power to over-set him so? \"On a woman's whims\u2014you are a gentle man indeed.\"\n\n\"It is my brother's child,\" Aris said. \"Antyre's future.\"\n\n\"I will do as you ask,\" Maledicte said, rising to his feet and stepping toward the balustrade. \"But you know, Aris, you needn't have asked my compliance. You could have demanded it.\" Without waiting for leave, Maledicte dropped down the few feet to the garden and fled back to the stables, to Gilly, who was fiddling with harness straps and buckles.\n\n\"Done so soon?\" Gilly asked, without looking up from his hands.\n\n\"Yes,\" Maledicte said. He leaned against a mossy wall and closed his eyes, stopped fighting the belladonna; it took him into the dark clouds above, a raven's-eye view of the city wheeling and spinning beneath him. He wouldn't want to be in any coach driven by a man hallucinating the way he was. \"Poor Aris,\" Maledicte murmured, thinking of the king with distant regret. \"If only I could trust it to be a girl\u2014\"\n\n\"Are we waiting for Janus?\" Gilly called back, busy with the harness.\n\n\"No. He'll have to chase Amarantha away. I've been banned,\" Maledicte said, sliding down the wall, pressing his back against it until the stone's dampness sank through the layers of silk and linen, touching his skin with the intimacy he had denied Gilly and Aris. Gilly spoke but Maledicte heard only the comforting sound of his voice, watching as coaches came and went in the spaces of his blinking.\n\n\"Come on now,\" Gilly murmured in his ear, pulling him to his feet. \"You've dozed enough to miss Amarantha on the move. Best we leave before her coachman spies us lurking.... Mal, you're shaking,\" he said, his calm slipping away.\n\nMaledicte's thoughts tangled in his mind, strangling the words of reassurance in his throat.\n\n\"The antidote _is_ working?\" Gilly said.\n\nAgain, Maledicte's response died stillborn. That the belladonna was more potent than he had thought, that the antidote was less effective than he had been led to believe, that Ani sulked and shirked Her aid.\n\nClutching the hilt of his sword, Maledicte staggered to the coach. Gilly caught him, his words lost in the rushing murmur of Maledicte's blood. Gilly bundled him into the coach, tucked him round with heavy warmth, and shut the door.\n\n\"Ani,\" Maledicte whispered. Inside his heart, his belly, his bones, the whisper of wings stirred and rustled, sounding their susurrant reassurance. Maledicte sprawled across the seat, wrapped in the rough leather of Gilly's greatcoat. Rocked by the movement of the coach, he slipped into waking dreams.\n\nAni pressed out through his ribs, sending out long feathers to row through the air. Rising, She soared above Maledicte's coach, the cold winds parting beneath Her strokes. She rose above the wide streets of the palace surrounds, the smooth cobbles glistening like scales beneath Her. Circling above the palace, She watched the coaches moving like bright beetles, finally spotting Her goal\u2014the glossy blue coach, its color robbed by darkness, trundling slowly along the cobbled road out of the city.\n\nHow afraid Amarantha must be, She gloated, to brave the overnight journey to Lastrest with a weary coachman. Flanking the coach, four kingsguards on gray horses and Dantalion on a blood-colored bay insured her safe passage.\n\nAs She neared, the coachman yanked on the reins, frightening the team into arrhythmic canters. His face blanched. The kingsguard wheeled their mounts and wheeled them again, confusion and concern written on their faces.\n\n\"She's there!\" the coachman screamed, his voice spiraling into the sky like a prayer. She reveled in it, dropping closer. The kingsguards gaped at the road, at the sky, at the trees alongside; Dantalion kept his eyes where it mattered\u2014the coach. He drew his steed nearer the door, preparing to dismount and climb aboard. But the coachman snapped the reins, lashed out with the whip, and set the horses to a panicked gallop, leaving Dantalion still reaching for the frame.\n\nCaught flat-footed, the kingsguards milled for a moment, a tangle of reins and stirrups and pistoning hooves, then they streamed after the swaying coach. Dantalion was a length ahead and gaining when She opened Her wings to their fullest extent, spreading the stench of carrion fields, the sweet rot of the grave. The horses reared and frothed. Two kingsguards were thrown, rolling hastily to avoid being trampled by their maddened horses.\n\nDantalion savagely held his horse to his will, but he lost ground, and the coach hurtled away, Amarantha's screams trailing in its wake. The coachman still peered over his shoulder, panicked, trusting the horses to stay on the road. Their hooves pounded out the cadence of a frantic heart.\n\nHer feathers sliced the air, driving Her over and beyond the coach. The coachman's head swiveled, his mouth slackened. She wheeled, soared, and came back at the coach. The coachman's face, seen head-on, was that of a ghost, gibbering and hollow-eyed.\n\nHe sawed on the reins and the stressed leather snapped. Kicking their heels, heads flat out and flecked with foam, the horses bolted. The coach tipped to the left, putting one edge in the dirt, skidding, rolling, broken wheels crashing through the enamel and gilt, and coming to a shuddering halt. Lying in the road, the coachman whimpered, \"Ani.\" She devoured his prayer, his worship.\n\nDantalion gained the scene, his mouth taut with rage. He dismounted his chastened horse, tied it to a piece of the wreckage, and started sorting through the remains of the coach. Lifting the door, he found Amarantha, her eyes staring at the sky, her belly huge. Dantalion knelt....\n\n\"Mal?\"\n\nThe voice distracted Her, and the scene faltered. Strong hands confined Her, dragging Her away, Her feathers dwindling, Her sight gone. She protested.\n\n\"Easy, Mal,\" Gilly murmured in his ear. \"Or you'll have us tumbling down the stairs.\"\n\nBlinking, Maledicte pieced the details together. That steady rush and thump was not the downbeat of wings, but Gilly's chest beneath his cheek, the swaying sense of flight nothing but Gilly's slow ascent up the staircase, cradling Maledicte in his arms. The shattering of wood and wheel was the damage done by Maledicte's trailing scabbard against the delicate ornaments in the railing. \"Put me down.\"\n\n\"Two more steps,\" Gilly said, tightening his grip.\n\nMaledicte tensed, uncomfortable with such proximity to Gilly, too aware of secrets, Janus's potential arrival, and his own weakness that urged him to slide his arms around Gilly's neck.\n\nAt the top of the stairs, Gilly set him down, patiently making sure he had his balance before stepping back. Throughout it all, his eyes never met Maledicte's. \"Are you well now? Your shaking has stopped.\"\n\n\"Yes,\" Maledicte said.\n\n\"I thought you were immune to poisons.\"\n\n\"I'm not dead, am I?\" Maledicte croaked; his throat felt stiff, as if it wanted to voice words not his own, to finish Ani's triumphant cry.\n\nGilly nodded, eyes sluing toward the stairs and the front hall.\n\n\"Thank you, Gilly,\" Maledicte said, touching his cheek.\n\nBeneath his fingers, Gilly flinched. \"I'm going out,\" he said.\n\n\"Are you well?\" Maledicte asked.\n\n\"No,\" Gilly said. \"I killed a man tonight. You nearly poisoned yourself, and all the way back, I listened to Ani ranting in your voice. All I want is to be someplace far from death. I know Lizette won't ask me to kill anyone.\"\n\n\"Gilly,\" Maledicte said, \"don't\u2014\"\n\n\"Don't what? Don't feel guilt? Don't dream of them? The coachman, Amarantha, the babe? My head is already full of Vornatti, Kritos, that assassin, Love's man, and poor Roach.\"\n\n\"I need you,\" Maledicte said. \"You agreed it had to be done. I didn't ask you to kill him.\"\n\nGilly sighed. \"I know. But tonight, I didn't kill for you, in your defense. Tonight, I killed to make Janus's path easier. And I can't think of a single reason I should let myself be used by him, the way he's using you.\"\n\nMaledicte shoved him, despair replaced with something stronger, hotter, more palatable. Gilly stumbled backward, missed the top step, and fell. He caught the railing with one quick hand before he fell more than a few risers. He righted himself, looked up at Maledicte.\n\nBreathing quickly, Maledicte waited, aching for the fight. For something he could win. Once, he would have been able to use words to sway Gilly, but he found nothing to say now, all churned under Ani's wings.\n\n\"And you worried that Ani would hurt me,\" Gilly said. \"That was all you.\"\n\n\"Gilly,\" Maledicte said, voice a thread of sound, forcing words through the rage that choked him.\n\n\"Think about what you want and need of me. I will not kill for Janus. If that's what you want, you'll have to find a new ally.\"\n\n\"No,\" Maledicte said. He stretched a hand out, but Gilly had already turned and finished going down the stairs. The door shut with a bang.\n\nHis hands fisted. Gilly just didn't understand. He would apologize, explain that so close to their goal, he was unsettled, make him the promise he'd made before: that Gilly wouldn't have to kill for him. This time, he'd make sure it was kept. If Gilly returned. If the blood on his hands hadn't been too much for his honest nature.\n\nBlond hair gleamed in the light and Maledicte's breath caught. \"Gilly?\"\n\n\"No,\" Janus said. \"What are you doing on the stairs? Come down, let's wait out the night and see death in with the morning.\"\n\nMaledicte stretched his hand out, and Janus tugged him to his feet, kissed his temple, driving away his moodiness, his anger and fear at Gilly. \"What did Aris want of you?\"\n\n\"Nothing,\" Maledicte said, then laughed. \"He asked me to stay out of the court while Amarantha was attending.\"\n\nJanus smiled. \"You promised, of course.\"\n\n\"Knowing what I know, how could I not?\" Maledicte leaned against Janus, and they went down the stairs, through the quiet house, hand in hand.\n**\u00b7 31 \u00b7**\n\n**P** INK HAD JUST CREPT INTO the sky when the great bells of the palace began to toll. Maledicte, dozing against Janus's shoulder, sat upright, anticipation chasing the last sleep from his face. Janus turned his head, smiling. \"A sweet sound of funeral bells in the air. You've done it. Amarantha's dead.\"\n\nMaledicte didn't respond, too caught up in the deep, slow voice of the bells. When they came to a stop, like a faltering heartbeat finding rest at last, Maledicte let out his pent breath in a languorous sigh. \"It's done. Finally done.\" A bubble of lightness started in his belly, a seed of relief.\n\nJanus kissed his forehead, his mouth. \"Thank you, my cavalier, my dark swordsman. Now you may rest your sword.\"\n\nThe relief in Maledicte's belly refused to grow. Even as he murmured agreement, he wondered if Janus could sense his forebodings. Would Ani leave him now?\n\nWhen he closed his eyes and listened to the dark recesses of Miranda's body, he believed Ani had taken root like a child not easily ridded by potions and poison. \"What will I do?\" he said aloud.\n\n\"Anything you like,\" Janus said. \"We've won, Mal.\" At the hushed velvet quality in his voice, a tone saved for long moments between the sheets, whispers in the dark, Maledicte let the last of his tightwire energy drain away.\n\nWere Gilly in the room, he might see past the disguise now, see beyond his expectations. In Maledicte's softening limbs and giddy smile, in the way he folded himself into Janus's arms...all these had more in common with Miranda than any courtier. But Gilly was still gone from the town house, though no longer closeted in Lizette's sheltering embrace. Instead, he roamed the early-morning streets, seeking information the bells could not give him\u2014was the coachman alive?\n\nWhat he heard, in whispers from servant to servant, from merchant to customer, and finally from the broadsheet criers, sent him home, running through the narrow streets.\n\nMALEDICTE RESTED HIS HEAD in Janus's lap, let his eyes drift closed. Janus trailed his fingers through Maledicte's hair, planning aloud. \"I'll need to attend Aris. There may be questions. Amarantha made no secret of her fears\u2014\"\n\nThe sound made them both stiffen, made Maledicte raise his head, eyes flaring dark and wild. \"What is that?\" The bright carillon continued, ringing off stone and rebounding, filling the air. Tumbling off the chaise, Maledicte put his hands over his ears. Within him, Ani twisted, churned, waking to malevolence.\n\nGilly burst into the room, and Maledicte looked up, near blind with nameless anxiety. \"What is that sound, Gilly? What is it?\"\n\nGilly panted for breath, his chest shuddering, too distraught to mince words. \"A child has been born to the royal family,\" he said, staring at Maledicte's face, as white as milk or marble. \"Dantalion cut him from her belly. The bells mean they expect him to live.\"\n\nMaledicte screamed, the sound soaring up over the bells, ripping free of the confines of his maimed throat, beyond human range. Outside, the rooks burst into panicked flight, wheeling and setting dark flickers behind the window glass. Janus released him, face blank in alarm and chagrin.\n\n\"Gilly, are you sure?\" Janus said, but Gilly had no time for Janus, no time for anything but the swelling blackness in the slim form before him. Gilly stroked countercharms in the air with all the fervor of a country intercessor, but the empty wildness in Maledicte's eyes remained unchanged.\n\n\"The earl is dead...Long live the earl...I will not allow it.\" The voice was barely recognizable as human; it raised hackles along Gilly's nape, the rattle and rasp of it like old bones, like his dream of Ani brought to life.\n\n\"Maledicte,\" he breathed. \"Please.\"\n\nSword drawn, Maledicte moved toward the door, inexorably dragging the shadows after him. \"Stop him,\" Gilly said.\n\nJanus reached out with alarming casualness and seized Maledicte's arm, his face annoyed. \"Mal, enough with the melodrama. We need to\u2014\" He sucked in his breath and lunged back as the sword sliced toward his belly. Gilly leaped forward, taking advantage of Maledicte's half-turned body, taking that slim form in his rush and bearing it to the floor. Maledicte shrieked again, thwarted blood in his voice; the rooks crashed through the windows, shredding themselves on the glass, pelting them with bone and feather and blood.\n\n\"The sword,\" Gilly gasped, trying to keep Maledicte down, when it felt as if Maledicte was as muscular and as agile as a serpent. If the countercharms were worthless, removing the sword from Maledicte's grip might be the only chance left. Sliding over Maledicte's back, he pushed Maledicte's arm out, spreading the sword hand farther away from himself.\n\nJanus, assessing, shook himself and then stamped on Maledicte's out-spread hand. Despite his desperation, Gilly winced when the bones cracked. In an elegant motion, as well suited to a dance as to a duel, Janus swept the sword across the floor with a booted foot.\n\n\"Elysia, in the butler's pantry,\" Gilly panted.\n\nMaledicte, heedless or insensible of the pain, heaved himself to his hands and knees, reaching for the sword. Gilly exhaled, made himself heavy, thought of immovable boulders, of nets. Janus's footsteps moved swiftly away, and Gilly thought _hurry, hurry._ He could not hold him much longer; with every pulse of his heart, Maledicte gained on the sword.\n\nGilly yanked Maledicte's leading arm up and back, spilling him from his inexorable crawl. Then Maledicte slipped sideways, rolled, got his knees between his body and Gilly's, and kicked. The blow was all out of proportion painful; Maledicte shook free of Gilly's spasming fingers, and only Janus's quick grasp saved the sword from making its way back to Maledicte. Janus backed away, the sword held awkwardly in his grip, bloodying his fingers, the Elysia bottle in the other hand, the syringe slipping through the cage of his hand. Gilly made a gasping effort and caught it, rolling clear of the space between them.\n\n\"That's mine,\" Maledicte growled; as if Ani tired of the pretense, of the games, the sword twisted in Janus's hand and clattered across the floor, skidding up against Maledicte's boot. He scooped it up with his foot, kicking it into the air, and caught it with his sword hand.\n\nThe bones reknitted, the tendons flexed, and the sword shifted to a better grip. Janus dropped the Elysia bottle, staring at Maledicte's burgeoning shadow, at the drift of bloody feathers saturating the air. Maledicte stepped forward and broke the bottle underfoot.\n\nJanus met Gilly's eyes and for once, his poise was stripped from him. \"Keep him here,\" Gilly said.\n\nJanus stepped between Maledicte and the door. Faintly, a frown crossed the blank mask of Maledicte's face. Gilly wished it concern, but was far more afraid that the emotion was outrage.\n\n\"Hold him!\" he called, then ran for the stairs and Maledicte's rooms. Slamming the door back, heedless of damage, he started searching for the poison chest. Below him, steel crashed against steel, and Gilly wondered, his heart in his throat, how much time Janus could grant him. More, how little time would pass before Janus realized that he was preventing Maledicte from his goal, a goal that Janus wholeheartedly craved.\n\nThe chest in his hands, Gilly pawed through the contents carelessly. All the little crystal vials seemed maddeningly identical to his frantic eyes. But beneath them, a bottle, bigger than the others, caught his attention\u2014what had Maledicte planned for that? Shaking the question off, he snatched it and bolted for the parlor.\n\nJanus, backed against the door, panted, holding Maledicte at bay with the parlor poker; Janus's sword, notched and scarred, lay trembling across the room. Feathers littered the air as the maddened rooks spilled unceasingly into the room.\n\nGilly gritted his teeth and pulled off his shirt. He soaked the fabric with the bottle's contents. Janus lunged and ducked and parried, the poker thrust punching Maledicte's sternum. When Maledicte staggered, Gilly flung the cloth over Maledicte's head, pressing the fabric close to his face, his bared teeth.\n\nThe sword stroked back and Gilly leaned into Maledicte's body, trying to hide in the shelter of his back. In his arms, Maledicte contorted and fought. Gilly, holding his breath, had time for the single despairing thought that this was not going to succeed, that Maledicte would step free and slash his way to the palace.\n\nJanus took advantage of Maledicte's cloth blindness to strike another blow, breaking the delicate elbow joint and sending the sword spinning away. In Gilly's arms, Maledicte collapsed all at once.\n\nGilly fell with him, sprawled on the floor, nerves singing, shaking as with an ague. The living rooks fled. Janus kicked a few of their bodies out of their way with a fastidious foot and knelt beside Gilly and Maledicte. He lifted the cloth and wrinkled his nose.\n\n\"Ether,\" Gilly said, but Janus wasn't listening. He touched Maledicte's slack face, the hand that had been broken, the elbow that even now mended itself.\n\nFinally he looked up and met Gilly's eyes. \"What the devil was that?\" His voice was a near whisper, as if he feared Maledicte would wake. \"I hurt him. I broke his hand, I broke his ribs, his elbow, and nothing mattered.\"\n\n\"She's insane, and infinitely more powerful than we are.\" Gilly dragged the cloth back over Maledicte's face. \"Fortunately, Maledicte is not, being mere bone and blood like the rest of us...no matter how powerful She is.\"\n\n\"She?\" Janus said, his sword in his hand, though when it had been recovered, Gilly couldn't say. Janus's expression was blank.\n\n\"The danger's past, I believe. You can put that away,\" Gilly said. \"She, my lord Last, is Black-Winged Ani.\" He shifted his weight, dragged a dead rook out from beneath his knee, and settled back again. \"And She grants Her followers certain abilities. Freedom from poison, from injury, and all She asks is their bodies. The longer the vengeance takes, the stronger She grows. She has no cares beside the shedding of blood.\"\n\n\"But he fell to the ether,\" Janus said. \"None of your nonsense, Gilly....\"\n\n\"I think immunity from poison is a mistranslation,\" Gilly said. \"It affects him, but not for long.\"\n\n\"She can heal wounds? All wounds?\" Janus said, touching Maledicte's shrouded form again.\n\n\"Some say so,\" Gilly said. In his arms, Maledicte stirred, despite the ether-soaked cloth over his nose and mouth. Gilly put his hand back to the bottle and soaked the cloth again. Maledicte subsided.\n\n\"Be careful, Gilly,\" Janus snapped. \"He's not very big. You'll kill\u2014\" The wolf paleness of his eyes flickered, the shock of belief hitting home. \"Will he wake maddened?\"\n\n\"I don't know. I don't think so. I think it takes effort for Her to manifest Herself. I think it had to do with the belladonna Mal drank last night.\"\n\n\"Belladonna,\" Janus said, his voice low. \"How much of it?\"\n\n\"Enough to kill,\" Gilly said. \"All for you.\"\n\nJanus made a small, choked sound, his face whitening. He gathered Maledicte into his arms and put his face into Maledicte's neck, rocking them both.\n\nStiffly, Gilly stood, and surveyed the wreckage. Another mess too difficult to explain to the few servants they had remaining. He picked up one dead bird by its wing, dropped it out the shattered windows.\n\nBehind him, Janus whispered, \"I'm so sorry I hurt you.\"\n\n\"Didn't hurt.\" The ghost of a whisper turned Gilly about. Maledicte's eyelids flickered. \"Like pain in a dream. Not real.\"\n\nJanus grunted with effort but brought both himself and Maledicte off the floor in a single movement, Maledicte cradled in his arms. \"Real or not, you need to rest.\"\n\nMaledicte slipped from Janus's arms, and picked up the sword, flexing his hand around it, parrying with a few still floating feathers. \"See, not hurt.\"\n\nJanus and Gilly tensed, and Maledicte smiled at them both as acidly as he had ever smiled at his enemies. \"I would have had it done, had you two not balked me.\"\n\n\"You would have died,\" Gilly said. Despite the guilt this relationship had sparked, it was nothing compared to the pain of imagining the loss of it.\n\n\"Would I? With Her touch on me?\" Maledicte shrugged as if it were a matter of no import, and sheathed the sword. \"Perhaps, but not until the babe was dead. My release from Ani is contingent upon my vengeance.\"\n\n\"We want more than simple vengeance, remember,\" Janus said. \"We want the court, the title, the safety.\"\n\n\" _I_ remember,\" Maledicte said. He sat down at the spinet, flicked a wing from the stained keys, and pressed a few notes, oddly muted. He reached inward, tugged another bird free from the strings, and dropped it to the floor. \"Ani doesn't care.\"\n**\u00b7 32 \u00b7**\n\n**A** RIS STOOD BESIDE THE EMPTY CRADLE, rocking it with a trembling hand. In the chair next to it, the wet nurse, holding the infant against her, shot nervous glances at the king and the attendant guards.\n\n\"Adi was never so small,\" he said aloud. The woman opened her mouth and closed it again. Against her breast, the baby suckled. In the corner of the nursery, Adi played with Hela, uninterested in the new baby, or in the newly partitioned section of his domain.\n\n\"Speak, if you would,\" Aris said.\n\n\"Your son was a full-term child, and this one, this little one\u2014it is a miracle he survived at all.\" She rocked the child; his lips rolled back, showing pink gums and a milky tongue.\n\n\"He must be whole.\" His voice broke on the last word, as he wondered if his brother's child would be another Adiran. Sound of body, lacking mind. Right now, he could not decide which would be better. He needed an heir, a child of sound mind and blameless parentage, and yet\u2014if the babe were deficient, there was no danger to him. And Aris, who had seen the wreckage of the coach, the twisted wood and wheel, the coachman's broken body, Amarantha's gutted flesh, knew there was an undeniable danger.\n\nFleeing, he had sought the sanctuary of the nursery, away from the fearsome images evoked by the wreckage, away from the horror of Jasper's reports. Dantalion had come in with the child just after dawn, the infant still bloody in his arms, but alive. Jasper had brought Amarantha's body in and laid her respectfully down, his fair face flushed and distraught. \"He didn't wait, sire. Not for her to live or die. He just cut her open and took the babe. Left her body like refuse on the road.\"\n\nAt first, Aris had been nearly afraid to look on the child, afraid that Jasper's words would have left a taint of atrocity on the boy, but the child was an infant pure and sweet. Briefly, Aris let himself remember that too-short moment when Aurora had held Adiran up to him, smiling. Before they knew she was dying. Before Adi's flaws became apparent.\n\nGingerly, he touched the infant's soft skull, cupped it, warm and pulsing, in the cradle of his palm. He owed Dantalion's decisiveness for this moment, but still he could not trust the man. Last night, only last night, he had heard Dantalion extolling the virtues of culling the Itarusine children, insuring that only the fittest lived. The ones found wanting were plunged into the icy seas. Aris had thought of his sweet Adiran and had fled the court room.\n\n\"Sire,\" Jasper said, entering and dropping his voice to lower tones immediately on seeing the sleeping child. Echo followed him in and averted his gaze from the blushing wet nurse.\n\n\"Have you found the cause of the accident?\" Aris asked, but lost parts of their answers, studying the delicate veins in the child's eyelids.\n\n\"Coachman spooked...lost control of the reins, though why\u2014\" Jasper said.\n\n\"We've questioned the stablehands...\" Echo said. \"And found silver embroidery thread in an unused stall.\"\n\n\"Which means little,\" Jasper said. \"Could be off livery, could be signs of a noble girl's dalliance. Silver's popular this year\u2014\"\n\n\"Maledicte made it so,\" Echo said.\n\nAris took the child from the nurse with a careful hand. The infant curled his fingers around Aris's forefinger, and he smiled. \"A good grip. And I believe I saw a glimmer of awareness in his face.\"\n\n\"He's a right one,\" the nurse said. \"Small but perfect.\"\n\n\"He is,\" Aris said, veering between joy and worry. An heir. A release from his burden. But such a court to leave a beautiful child\u2014\n\nBehind him, Echo and Jasper's voices rose, growing harsh and brittle as they argued with each other.\n\n\"I tell you, you cannot blame your favorite in this matter,\" Jasper said. \"You cannot blame a death on a man who did not attend.\"\n\n\"And you\u2014I suppose you see the hand of witchcraft in this?\" Echo spat. \"Your Mad Mirabile creeping onto the palace grounds, still in her ballgown, poisoning the coachman with her spells? The same spells you blame for allowing her to escape your net?\"\n\n\"Perhaps if you could be convinced to share your resources with my men\u2014\" Jasper said. \"We are at home in the palace, and in the main thoroughfares of the city. The alleys and Relicts are your Particulars' job, and yet they've not found her either. But that has nothing to do with this. I think you must admit\u2014\"\n\n\"To you, nothing,\" Echo said.\n\n\"We need look elsewhere. Amarantha feared\u2014\" Jasper flinched under Aris's sudden gaze.\n\n\"Feared who?\" Aris asked.\n\n\"More than one man,\" Jasper said, refusing to meet the king's eyes.\n\n\"Your bastard nephew,\" Echo said, unafraid of Aris. \"Your newest counselor. She feared Janus but I still believe the blame lies where it is most obvious. Maledicte.\"\n\n\"Echo, you seem slow to learn. Maledicte did not attend the ball,\" Jasper said.\n\nAris laid the infant down, his throat suddenly numb and cold. He forced the words from his throat, but frozen, they did not carry above their argument. \"He was there.\" Aris rocked the cradle, all the while remembering the rough silk feel of Maledicte's lips against his. \"I saw him coming from the gardens and stable.\"\n\nJasper and Echo fell silent, staring at him.\n\n\"He was there, his eyes wild, his manner\u2014\" Aris trailed off, knowing himself for a fool at last. He sank down onto the padded bench that Adi often napped on, and covered his eyes. He had looked the other way when Kritos died, when Vornatti died, neither man long for the world, a debt-ridden gambler and an old rou\u00e9.\n\nEven when Last fell, Aris had sought other foes, had looked into the dark eyes and thought, shamefully, that perhaps Last was not entirely innocent of his own death. After all, the enmity had been mutual and undeniable. Maledicte might have only defended himself. Aris had even presumed that with Last's death, Maledicte had no one to hate, and so was defanged.\n\nBut hate, Aris knew, was addictive; why had he never considered that? Instead, he had accepted gentle blackmail from the lad with mute passivity, trusting Maledicte to need no more than diversion from scandal. He had even, Baxit forgive him, found it almost a game of wits between them.\n\n\"Echo,\" he said, his voice rough. \"You blame him. Tell me what motive he held.\" He had been a fool perhaps, but one capable of learning.\n\n\"Janus,\" Echo said. \"What he does, he does for him.\" He held up a hand at Aris's protest. \"I don't know that Janus understands what kind of man he's allied himself with.\"\n\n\"Janus,\" Aris said, \"is one of my counselors and my nephew. Your peer. He has never given me reason to distrust him. Perhaps, like me, he is only too trusting.\"\n\n\"I'll send the guard for Maledicte\u2014\"\n\n\"No,\" Aris said. The refusal came instinctively. There were the ledgers to worry about. They would have to be recovered before any steps could be taken. But Echo waited impatiently on an explanation. \"He thinks too quickly for that. I would deprive him of time to prepare glib assurances. Jasper and I will go to him and see if I can surprise truth from his lips.\"\n\nEcho said, \"If it were as easy as that, I'd have had truth from him long ago. We should simply arrest him, and let him prove his innocence.\"\n\n\"I'd rather you prove his guilt before I see a member of the court imprisoned,\" Aris said. Echo glowered, and Aris forced himself to the intricate steps of manipulation so necessary to his court. \"Find me incontrovertible proof of a crime committed, Echo, and you may have him. Talk to Dantalion, who knows more of poisons than I would like, being an Itarusine. Ask him if he knows of a potion to send a coachman mad. I will ask Maledicte the same.\"\n\nHe leaned over the cradle once more, breathing deeply, as if the child's innocence would not only grant him clarity and strength, but wisdom.\n\nARIS RETURNED from the Dove Street town house, having gone to confront Maledicte over Echo's objections. The visit had been fruitless and unsettling, Maledicte not at home and the house and hall so spattered with rook feathers and blood that it seemed a nigh-impossible task for the manservant left to clean them away. Shaken, Aris retreated without attempting a hunt for the ledgers. At the palace, he found one member of that eccentric household in the nursery, leaning over the sleeping infant, alone. Nearby, Adiran sang quietly and stacked blocks. \"What do you call him?\" Janus said, without looking up. \"My brother....\"\n\nHalf-formed fear melted at the interest in Janus's blue eyes. Aris came forward, and joined his small family.\n\n\"Auron,\" he said. \"After my wife.\"\n\nJanus touched the sleeping baby's soft mouth with a finger that dwarfed it. Aris watched as Janus rocked the cradle, setting it to sea-tide swaying until Auron opened cloudy blue eyes.\n\n\"Auron Ixion,\" Janus said, \"Welcome, your grace.\"\n\nAris let his breath out in a steady hiss.\n\n\"It's true, isn't it?\" Janus said. \"Little brother is the earl. I am only a bastard.\"\n\n\"Janus, you will always have a place at court, always be cared for,\" Aris said. \"My counselor.\"\n\n\"I know that, Uncle,\" Janus said. \"I wonder if Auron will feel the same....\" He smiled and said, \"If he's as sweet-natured as Adi, as kind as you, I will never want for anything.\"\n\nAris said, \"I thought to name you guardian to Auron and one of his regents, should I not live to see his ascension.\"\n\nHe saw sudden startlement in the pale eyes. \"I, guardian?\"\n\n\"Who better than family?\" Aris said.\n\n\"I think most of your court would say who is not better...but if it is your will, I am honored,\" Janus said, sinking to his knee before Aris.\n\n\"Two caveats,\" Aris said. \"Two conditions fulfilled before I name you guardian.\"\n\n\"Sire?\" Janus said, face growing still.\n\n\"Do you know where Maledicte is?\" Aris asked.\n\n\"At this moment?\" Janus asked. \"No.\"\n\nAris studied Janus, seeking honesty, seeking rebelliousness. Instead, he saw only resignation and a glimmer of anticipated pain. \"He will be questioned in Amarantha's death.\"\n\n\"He had no reason to injure her,\" Janus said.\n\n\"None at all?\" Aris said. \"Not even for you?\"\n\nJanus's eyes widened and he laughed. \"Once Amarantha wed Last instead of me, Maledicte ceased to notice her at all. As for more...elaborate motives...\" He paused, as if feeling his way. \"I am not unaware that had Auron died, it might have been seen as beneficial to myself. But this sequence of events? No, if you want inducements to violence, consider Dantalion himself. He saved the babe. He put Antyre in Itarus's debt. He will collect.\"\n\n\"You know what he will ask?\" Aris said, gesturing for Janus to rise. He grew dizzy looking into the pale depths of Janus's eyes, so like water shifting beneath thin ice.\n\n\"The Itarusine court is much occupied with honor. Dantalion's disinheritance must rankle like salt in an open wound. But if you owe him a boon, and if Maledicte is suspect and out of favor\u2014\"\n\n\"You think Dantalion would kill Amarantha merely to regain his inheritance?\" Aris leaned against the wall, jostling the cradle and setting Auron to fussing. Absently, he smoothed the blanket, stroked the soft skin until the drowsy complaint stopped.\n\n\"Men have done worse for less. Still, I think it more likely the accident was only that, and Dantalion capitalizes on it, cleverly shaping events to his benefit,\" Janus said.\n\n\"Maledicte hated Michel and he died. Amarantha believed he sought her death as well and she is dead. Elaborations aside, how do you know he did not kill them both?\"\n\n\"I was with Maledicte when Father died. I swear he did not strike the blow. As for Amarantha\u2014did Maledicte ride after the carriage like a highwayman, shatter their wheels all unseen and send them to their deaths? Or are you imagining poison in a court so fearful that no one partakes of refreshments since the Dark Solstice?\"\n\n\"If I had those answers,\" Aris said. \"There would be no need for this conversation. But Maledicte\u2014\"\n\nAuron made a sleepy sound of contentment, and Janus smiled down at him. \"He's perfect, isn't he?\"\n\nAris hewed doggedly to the topic. \"Janus. My first condition\u2014Maledicte holds some books for me that I would require returned.\"\n\nJanus nodded understanding. \"The Antyrrian audits. I believe I can put my hands on them. Your second stipulation, sire?\"\n\n\"Should you become guardian, you will not be allowed the freedom you have now. You will not live in Maledicte's house, but the palace only. You will never bring him here, and indeed I prefer you not see him at all. His situation becomes too\u2014irregular.\"\n\n\"A perfect model of a courtier,\" Janus murmured.\n\n\"With an empty house full of blood and feathers,\" Aris said.\n\nJanus flinched, his composure faltering. For once, Aris could read a thought in his nephew's face\u2014astonishment and dismay that Aris had visited Maledicte's house. \"An accident,\" he said. \"Startled birds, nothing of moment\u2014\"\n\n\"Janus,\" Aris said. \"It has been borne in on me that I've been a fool. I would like to see you escape that same realization. Of late, I have done nothing for Antyre but watch it, dreamlike, crumble. I've woken now; I have Auron and his future to protect. Do not think me blind.\"\n\nJanus worried his lower lip like a schoolboy, and Aris felt some of his anger mellow. \"Whatever hold Maledicte has on you, whatever has happened, with your knowledge or not, it stops now.\n\n\"If you must keep him, you may. But you will do so as men keep mistresses, discreetly, and never at the expense of your own responsibilities. I will find you a wife, and you will wed her. And should you misstep\u2014\"\n\n\"What, then,\" Janus said. \"Banishment?\"\n\n\"Yes,\" Aris said. \"But not for you. Your slightest misdeed will see Maledicte sent beyond your reach.\"\n**\u00b7 33 \u00b7**\n\n**A** LONE AT A TABLE IN THE BACK, Maledicte surveyed the tavern with an incredulous eye. The Seadog had been the height of imagined luxury to him and to Janus not so long ago. They used to peer in at the smoky rooms, at the jewel tones of firelight on Naga's scales fetched up from the deeps, on sea buoys hanging from the crossbeams like necklaces of great sea creatures. The men themselves all seemed made of coin, scattering moon spills of silver over the teak bar with careless fingers. Miranda, who rarely held more than one luna in her palm, and that clutched tight, had leaned against Janus's shoulder and marveled at such wealth.\n\nNow the Seadog was evident for what it was, a run-down shanty near the edge of the pier, patronized only for its proximity to the salt-weary sailors and by nobles recovering from their visits at the cheapest brothels, the drug dens, and gaming tables so rigged that two nights running saw them in different surroundings. Even now, Maledicte watched two young nobles ruefully and dazedly taking stock of themselves, cataloguing damage done to their purses or persons.\n\nFools, he thought. They might think themselves beggared, but their clothes, their boots, even their perfumed hair meant money down here, and yet, they stood in the heart of the lamplight.\n\nHe sank farther back into his corner, into the dark shadows of his dimly turned lamp. It cast a dying glow over his table, made the bottle burn with hidden lights. Maledicte poured the contents into his glass, straining it through sugared mesh.\n\n\"You never got that here,\" Janus said, settling into the seat opposite him.\n\nMaledicte paused in his pouring and smiled, taking up a broken chunk of brown sugar, and grating it over the whole. \"No, _absente_ is definitely beyond the Dog's cellar.\" He took a sip, but the warmth that moved through him had less to do with the liquor than Janus's arrival. \"You found me.\"\n\n\"Always,\" Janus said. \"Though I admit to a few false starts before I remembered the Seadog. Vile place, really.\" Janus studied the smoky room with a contemptuous glance.\n\n\"We thought it so fine,\" Maledicte said. He closed his eyes, trying to overlay one image on another.\n\n\"We were fools,\" Janus said, taking the bottle from Maledicte's grasp. \"And _absente_ is nearly as vile. It drives men mad.\"\n\n\"I'm already mad,\" Maledicte said, a whisper in the lamplight. \"Haven't you realized? Gilly does. And as for the _absente_ \u2014I haven't offered you any, so you needn't sneer at it.\"\n\nJanus edged his seat around the table, drawing closer to Maledicte. \"Mal, I need you to listen to me.\"\n\n\"I always do,\" Maledicte said, taking another languid sip. Janus removed the glass from his hands, set it on the floor beside him.\n\n\"Aris went to Dove Street and found the house as we left it, bloodied,\" he said. \"Aris suspects that you caused Amarantha's death, more so than he ever did for Last. I don't understand why, but you must be prepared. What will you say if he charges you in her death?\" Before Maledicte could answer, Janus leaned in. \"I did my best, shifting his eye to Dantalion, but can you keep it there?\"\n\nMaledicte stroked the line of Janus's jaw, admiring the gold-stubbled sheen in the lamplight. \"I suppose. Best just to stay clear as long as I can.\" He frowned, reaching for his glass on the floor. Right now, its bitterness leavened with sweet suited his mood.\n\nJanus said, more to himself than to Maledicte, \"Who knew he would grow so suspicious so swiftly?\"\n\n\"We should have planned for it,\" Maledicte said. \"Even the most docile of men might choke on the death we served him. Still, had I known\u2014I would have let him kiss me longer, instead of fleeing like a virtuous maid.\"\n\n\"You let Aris\u2014\" Janus said, rage flickering in his eyes. \"You risked the intimacy, the scrutiny? He's no stripling to be blinded to the difference between man and woman. In his arms\u2014\"\n\n\"He discovered nothing, near drunk on his own unhappiness,\" Maledicte said, allowing Janus to tug Maledicte into sharing his seat. \"And I? I was thinking of you dancing attendance on Psyke. He took me by surprise and meant little of it. Unreasoning jealousy will be your downfall,\" Maledicte said, shifting his weight until he sat on Janus's lap. \"So possessive of what's yours...\"\n\nJanus cut his words off with a kiss, careless of spectators. Maledicte laughed against his lips and the triumph in it bled through. \"But you're mine too, aren't you? It's still just us against the world.\"\n\n\"Of course,\" Janus said, drawing his mouth back to his. \"Of course.\"\n\nMaledicte turned his mouth up for Janus's kiss, closed his eyes at the familiar warmth of Janus's tongue touching his own.\n\n\"Such a tender moment.\" The voice cut into their intimacy, and Maledicte felt Janus stiffen with recognition. A dark man, heavyset and tall, his words accented in the Itarusine manner. Maledicte stood and smiled. \"You must be Dantalion, Vornatti's despised kin.\"\n\n\"And you, Maledicte,\" Dantalion said, eyes widening slightly. \"You, the dark cavalier of Aris's court? A puling youth with a sword too good for him?\"\n\nMaledicte's hand dropped to the sword hilt, even though he saw Echo appear behind Dantalion. \"Janus, next time, try to avoid leading the bores to us?\" Despite his easy words, he felt the surging wingbeat in his blood, the rage that they dared interrupt an all-too-rare moment of pleasure. He knew he should feel caution; Dantalion and Echo were an alliance that meant him nothing but ill.\n\n\"My uncle's catamite,\" Dantalion said. \"I'd heard such things about you. I nearly believed them, but you're barely worth my sword at all.\"\n\nThe other patrons, sailors and scattered nobles, watched with avid eyes in dulled faces.\n\n\"Mal,\" Janus said, laying a restraining hand on his forearm. \"You mustn't.\"\n\n\"But I _want_ to,\" Maledicte said. His eyes never left Dantalion's. He felt the dreamy tone lace his voice, the languor of the _absente_ reaching his head. All he wanted at this moment was the sound of steel and sweep of flashing metal. He savored the unwinding of Ani's coiled hatred, its warmth seeping through his bones.\n\n\"Waiting for an excuse?\" Dantalion asked, then struck. Janus tried to deflect the blow, but Dantalion was too close and too fast. Maledicte caught Dantalion's fist in his own. Dantalion's eyes grew thoughtful at the speed of it.\n\n\"That will do,\" Maledicte said, releasing the man's fist. \"Shall we duel?\"\n\n\"No,\" Janus said, rising, interposing himself between the two men.\n\n\"I never thought you a coward, Ixion,\" Dantalion said. \"And I had planned on spitting you next.\"\n\nMaledicte drew his sword in quick economy of motion, evading Janus's grasp. \"You have to take me first,\" he said.\n\n\"Mal,\" Janus said, voice low. \"I don't like this. It reeks of calculation. Echo is trying to\u2014\"\n\nMaledicte grinned at him. \"I don't care.\" He wondered what was in his face that made Janus blanch so, but lost that in the delight welling in him. To be free of another cur that nipped at his heels\u2014the lure was too great to deny.\n\n\"Outside, think you?\" he asked Dantalion.\n\n\"The cobbles are slick with dew,\" Dantalion said, shaking his head. \"And the alleys too full of those who would interrupt our sport for the coin we carry. No, we'll stick to tradition and the dueling grounds at the park.\"\n\nJanus grasped Maledicte's hand once more, held him back when he would have followed. \"It's a trap, Mal.\" As if to underscore his words, Maledicte noticed Echo slipping away, his task obviously accomplished.\n\n\"It doesn't matter,\" Maledicte said, touching Janus's cheek. \"Don't you see that? If not now, then later. Dantalion means to see us all dead.\"\n\n\"But now, when you're near drunk on _absente\u2014_ \"\n\n\"Drunk on blood,\" Maledicte whispered into Janus's ear, laughed at the expression on his face, the whisper of feathers in his mind. He pulled free of his grip and followed Dantalion into the gray-dawning sky.\n\nDAWN BATHED THE MEN in the park in pearly light and cast their shadows like spider-legged creatures over the lawn. Maledicte stood before Dantalion, sword unsheathed, dangling lazily from his hand.\n\nDantalion leaned close and spoke at length, his words lost in distance, but Maledicte's face darkened. All that, Aris saw as the carriage drew close, and more, Janus pacing madly behind the fighters. Janus drew forward and Maledicte pushed him back, his eyes never leaving Dantalion's blade.\n\nEcho was right, Aris thought. Maledicte did intend to duel Dantalion, flaunting his disobedience. But as they approached, Aris found himself dwelling not on the illegality of it, but the monstrously uneven odds. Dantalion was head and shoulders above Maledicte, and muscled with it.\n\nBeside him, Echo sank back into the seat, smiling. Dantalion's blade was of the Itarusine style, heavy and curved, the length of a man's arm. A savage weapon. And Maledicte's dark, slim blade seemed more a child's toy in comparison, though wickedly sharp.\n\nDantalion stepped forward, and the duel began.\n\n\"Should we intervene, sire?\" Jasper said. \"Before it goes further?\"\n\n\"We can always cry halt,\" Echo coaxed. \"I am interested in seeing Maledicte's ability.\"\n\nAris nodded, his eyes on the two figures. Maledicte made a sound like a laugh; it carried over the still air, and raised the hackles on Aris's neck. It wasn't a sound he associated with men, but the sound of feeding birds on a battlefield. Dantalion took a hasty step back as Maledicte darted in, as agile as a crow in flight.\n\nDantalion struck back, his great curved sword carving the air, but missed Maledicte entirely. His blade bit only Maledicte's black-fluttering sleeves.\n\nMaledicte danced forward, his sword skidding along the width of Dantalion's blade, raising sparks of outraged metal, and ending with a little flourish at the end that nearly touched Dantalion's shoulder. Dantalion jerked back, pivoted his weight, and came on again, thrusting, parrying, slicing at Maledicte's delicate form.\n\nDantalion dripped sweat, growing ponderous under his own weight. Maledicte, facing him, seemed as intangible as a shadow. His blade wavered, judged, and then they were moving again; Maledicte thrust forward, extending his arm, his blade, and kept himself out of range of the returned slash. His blade took the force of Dantalion's parry, unyielding. Aris let out his breath, watched the narrow focus of concentration on Maledicte's face, the dark glee in his eyes, and knew Dantalion was going to lose. That Maledicte had enough skill to have killed Last and anyone else he chose.\n\n\"End it,\" Aris said.\n\nMaledicte swept forward, danced under Dantalion's swing, not bothering to parry, and put his blade across Dantalion's throat. Dantalion jerked back at the last second, and the blade left only a wet red line behind.\n\n\"In the king's name,\" the guard called, spurring his steed forward. \"Halt!\"\n\nMaledicte fell back, out of the reach of Dantalion's sword, his teeth bared. \"Your salvation,\" he spat, spreading his arms like wings, inviting Dantalion's stroke. \"Too much a coward to finish this?\"\n\nDantalion lunged, and Maledicte flowed backward until Dantalion was overextended. The sword slipping past him, Maledicte reached out with his free hand and caught Dantalion's arm against his body, prisoning it, the sword useless. With agile fingers, Maledicte pinched the nerve inside Dantalion's elbow; Dantalion cried out and dropped his blade.\n\nAris breathed a sigh of relief to see the battle so neatly won, and then Maledicte reached forward with his sword hand and inscribed the same line over Dantalion's heaving throat.\n\nThis time, the blood was not content to spill over a thin trench, but spouted instead. Maledicte released Dantalion's arm and grabbed his hair, yanking his head farther back, widening that bloody smile. Blood sprayed Maledicte's face, his shirt, his sword, and ran in steaming droplets onto the fresh spring grass.\n\nAris froze; the duel had been won. Had Aris any remaining doubts about Maledicte as a killer, they were gone now. Watching the blood fountain under Maledicte's manipulation, he found it impossible to see any trace of the impetuous young cavalier he had so often declared him to be.\n\nWhen the blood slowed, Maledicte shoved Dantalion's body away. He wiped his blade on the grass, tore handfuls of the grass up to wash the mask of blood from his skin. When the guard put his sword at his back and commanded him to be still, Maledicte settled the sword on the ground. He stood, bloody hands dangling; the sword quivered on the grass.\n\n\"Your Majesty,\" he said.\n\n\"I cried halt,\" Aris said.\n\n\"He would have killed me had it been his chance,\" Maledicte said, his voice uninflected, a quiet rasp. \"He has tried to do so by proxy, and would have done so again. Should I have turned my back on a dangerous enemy?\"\n\n\"If you had proof, you could have petitioned me,\" Aris said.\n\n\"Proof is all too often hard to find,\" Maledicte said, his words a bitter echo of Aris's own.\n\nJanus said, \"Dantalion forced the fight.\" He put his arms around Maledicte, heedless of blood, heedless of the guardsman's expression.\n\nAris sat silent, and Janus said, \"Uncle,\" a quiet plea, his eyes eloquent.\n\n\"I asked for your discretion,\" Aris said. \"This is very far from discreet. A member of the foreign courts, murdered\u2014and it was murder I saw, Maledicte, you had him disarmed, and yet you struck....\" Aris rubbed his hands over his face.\n\nEcho said, \"I'll have him taken to Stones immediately.\"\n\n\"No,\" Janus said, his voice as abrupt, as commanding as Aris's was not. \"You set Dantalion on; you are as guilty of his death as Maledicte. Please, Aris, let me take Maledicte home. Let him be prisoned there until you decide his fate.\"\n\nAris could not think around the blood in his mind, this surfeit of death. But Echo had presumed\u2014that was as clear to him as Maledicte's act of murder. And the ledgers were not yet in his hands. To act now would risk them.\n\n\"No, I see no need for the cells. Not yet. Take him home, Janus. Remember our bargain. I am more desirous than ever of you fulfilling your end.\" Nerving himself, Aris stepped down from the coach, and joined the two men. He reached out and touched Maledicte's blood-spattered face. Blank black eyes looked back at him, as enigmatic as a starless night. \"Maledicte, tell me why.\"\n\n\"Death to us all,\" Maledicte whispered. \"My blood, then Janus's, then my poor Gilly's. That's what Dantalion promised.\" The gaze sharpened, lifted, held Aris's eyes with their intensity. \"He boasted of killing her, of granting you an heir. How long do you think you would have survived? Or the babe? After all, Adiran is the son of Aurora Vornatti, and if Janus died, if you died, then Dantalion was Adiran's nearest kin, an Itarusine nobleman, hungry for power and blood.\"\n\n\"You speak treason,\" Echo spat.\n\n\"Do I?\" Maledicte said, voice fading to a bare thread of sound. \"I thought I only repeated it.\"\n\nAris frowned, and nodded to his guards. \"My guards will see you home.\"\n**\u00b7 34 \u00b7**\n\n**G** ILLY MET THEM at the door to the town house, blanching at the blood spray on Maledicte's clothing. \"What happened?\" Janus slapped him hard, twice, and would have done so a third time but for the sudden hiss of the black blade unsheathing. He lowered his hand. Maledicte lowered the sword.\n\n\"Don't take your temper out on him,\" Maledicte said, slipping past them and disappearing into the library. Their footsteps followed him in like distant drums, their words like the ocean, quiet, senseless, repetitive. He slung himself into a chair, and watched them argue, wondering if he would need to intervene again. He supposed he should be angry, but all he felt was the wet seep of blood through the linen shirt, and the numbed weariness in his soul.\n\nHe interrupted their quarreling to say, \"Gilly, run me a bath, please. I'm all over blood. And when you're done with that\u2014I have another errand for you. I want you to take our funds from the banks and hide them where Aris and his guards can't reach. I've heard enough of Itarusine ways to know they'll attempt to take recompense for Dantalion's death in a monetary way.\"\n\nJanus opened his mouth to object and Maledicte rounded on him savagely. \"You allowed Aris to send guards to follow me home. You cannot object to the steps I choose to take now.\"\n\nMALEDICTE LEANED FORWARD in the warmth of the bathwater, setting off small, tidal sloshes. Tinged red and perceptibly cooling, still the water was soothing, as soothing as Janus's slow detangling of Maledicte's blood-matted hair. Mindlessly, he trailed his fingers through the water, and grimaced at the bloody foam sheeting out of his hair.\n\n\"You were a fool to kill Dantalion,\" Janus said quietly, as if he kept close rein on his temper. He paused in his ministrations, trading fingers for a wide-toothed comb, ivory overlaid with silver. \"Twice a fool to kill him in such a manner\u2014to show Aris the blood on your blade. Aris could see you hanged.\"\n\n\"No fear of that,\" Maledicte said. \"Ani is more potent than kings. I'll slip the gallows knot yet.\"\n\nMaledicte reached for the sponge and rinsed his arm, removing a streak of blood that had so far evaded the water. Tension curled in his belly at the sight.\n\n\"You are mad,\" Janus said. \"I accept Black-Winged Ani as your patron\u2014I have seen the evidence myself\u2014but that's no reason to overthrow sense. I need you to be clever, and all you think of is blood.\"\n\n\"Your doing,\" Maledicte said. \"I would have been content with Last's death, had you not robbed me of it. Instead, I must kill again and again to keep the ground we've gained. Is it any wonder it's become habit?\" Ani's offense mingled with his own and his words grew edged; beneath the water, his hands fisted. \"But I'd give it all up in a heartbeat. For you. But you\u2014you made a bargain with Aris,\" Maledicte said, letting the sponge drift. \"You promised him\u2014What exactly did you promise him?\"\n\n\"Discretion, as he said,\" Janus said, but the shift of his eyes told Maledicte more than he wanted to know.\n\n\"How much discretion?\" he asked. \"To not live here at my side? To not walk beside me in the public streets? To not acknowledge me at all? Did you promise to set me aside?\"\n\nJanus stood and claimed a towel. \"The water's grown cold. You're all over gooseflesh.\"\n\n\"It's not the water's chill,\" Maledicte said, but rose and allowed Janus to wrap him in warmth. \"Tell me how things have changed with Amarantha's death. With Dantalion's. The longer you balk, the more I fear what you've done.\"\n\nJanus sat down on the edge of the tub, looked up with clear blue eyes. \"In exchange for staying third counselor and guardian to Auron, I am to live in the palace. I am not to see you more often than a man sees a mistress. I am not to take you about in public. I am to wed and\u2014\"\n\nMaledicte dropped the towel, wrapped himself in a dressing gown, tying the sash with angry jerks. \"You are to play Aris's lapdog. You ruled the Relicts, and yet you'll take such orders from the king, just to play second fiddle to a true-born infant?\"\n\nJanus said, lowering his face, his eyes blazing, \"To be kept at arm's distance when I am this close to power, to fortune, to everything we've ever wanted, galls me. But I must act the part. And act it well and for some time. Were little Auron to die so soon in my guardianship, it would ruin us far more than his survival. Aris\u2014\" He frowned. \"Aris doesn't trust me. So I will live in the palace, act the dutiful guardian, fond older brother. But I will visit you\u2014\"\n\n\"Like a whore set up in a bijou,\" Maledicte said. \"How Ella would laugh\u2014you, who she blamed for my recalcitrance to sell my body; you making me a whore when she could not. Except you don't have to pay for me or my rent. She wouldn't like that at all.... Please, tell me you'll set me up in a fashionable part of town, my dear. I would so hate to cross paths with my mother.\"\n\n\"Ella's long cold,\" Janus said quietly, hands dabbling in the bloodied water.\n\nMaledicte's breath caught, startled out of his rage for the moment. He had always known that Ella would not have survived to the fullness of old age, but had never thought of her as dead. Just gone. But Janus's voice held more than that simple understanding.\n\n\"You went back,\" Maledicte said, remembering. \"You told me you found R-Roach\u2014\" He couldn't help the tiny stagger in his voice, but he went on. \"Did\u2014did you find them too?\"\n\n\"Yes,\" Janus said. \"It wasn't hard. Once I realized they could tell me nothing of you\u2014\"\n\n\"You killed them,\" Maledicte said, eyes on his blood-tinged bath. His heart gave a sudden, unexpected lurch; his ears rang.\n\n\"The moment Last reclaimed me, Celia's life was forfeit. I could not have her dogging my heels, begging for coin, for her precious rank, her Laudable-addled mind making hash of my plans. And Ella, who had the poor taste to scream when I killed Celia\u2014do not tell me you will cry for her; not that selfish woman who gave you nothing beyond your bare existence.\" Janus's blue eyes simmered with heat, the expression recalling kicks and curses, and days spent scrounging for scraps while the two of them plotted and seethed.\n\nUntrue, Maledicte thought numbly. Ella had given Miranda one precious gift, even unknowingly, when she had opened her door to an aristocratic castoff and her infant son. A gift so valued that some of its virtue rubbed off on the giver: Maledicte's heart, tangled in loathing and contempt, still held room for that spark of gratitude. He could not have killed Ella. But Janus\u2014\n\nMaledicte shuddered all over, temper surging back. \"How is it that you raise your blade to whomever you choose, and I\u2014I only garner censure and suspicion? You walk the courts and I might be banished, impoverished, or hanged.\"\n\n\"I will never let that happen,\" Janus said.\n\nMaledicte threw the robe off, began dressing with shaking hands. \"And how will you prevent it? Time may be on your side, allowing you to worm your way into acceptance, but it's my enemy. Banished, impoverished, or hanged; none of them appeals to me overmuch. Janus, let's give this up. We did in the Relicts, remember, when our plans went wrong or grew too dangerous.\" At the sight of Janus shaking his head, Maledicte tried entreaty though it stuck in his throat like dusty feathers. \"Janus, you're all I have. All I value.\"\n\n\"Be rid of Gilly, and I'll believe you. You trust him with everything. Sometimes I think you trust him more than me. Giving him control over your accounts, really Mal...\"\n\n\"This isn't about him,\" Maledicte said. \"It's about Auron, about claiming your birthright, about making our future. We've failed at claiming your title. I see no way to succeed.\"\n\n\"When you gained Ani's favor, you lost your mind,\" Janus said. \"There's always a way.\"\n\n\"So you say, and yet the babe lives, his future safeguarded by one who should be the wolf. You've been playing dog too long. Aris holds your leash, and you fawn at his feet, saving your teeth for those weaker than yourself, helpless women...\" Maledicte heard his voice, hoarse and ranting, taunting Janus as if he were one of his enemies, willing his words to bruise and sting, and wondered what was becoming of him. In the rippling water, he thought he saw the reflection of dark wings.\n\nJanus slapped him, knocking him back against the edge of the tub. Maledicte growled low in his throat and launched himself at Janus, knocking over the vanity, sending them sprawling. The shaving mirror broke, scattering their reflections across the floor, their tangled bodies distorted in its shards. Fighting for control, Janus twisted Maledicte's arm behind his back.\n\nPain erupted in his shoulder and Maledicte grabbed a handful of mirror glass, heedless of the stinging slice against his palm, and swung. For the second time that day, blood stained his hand. His own and Janus's. The pressure on his arm eased and Maledicte turned, remorse sickening him.\n\nJanus mopped his shoulder with his sleeve, blood dripping onto his forearm. \"That's two battles in one day,\" Janus said. \"A record even for your temper.\"\n\nMaledicte curled his fingers into his palms, aware of the blood sliding between them. He fumbled for words, tried to control his angry breathing. \"Let me see,\" Maledicte said, his temper vanishing under concern and shame. \"I'll get bandages.\"\n\n\"Don't bother,\" Janus said, blue eyes smoldering. He twisted Maledicte's hand over and studied the gash in the palm, watched the welling blood slow and stop with a calculating eye. \"You don't need it, and I'll get mine cleaned elsewhere. With the mood you're in, I'd get _arsenixa_ on the bandage.\"\n\n\"No,\" Maledicte said. \"Janus, I'd never hurt\u2014\"\n\nJanus smiled, a thin, tight thing. \"Maybe not, but I can do it myself.\"\n\nMaledicte slumped back against the floor, watching as Janus raided his room for bandages, alum powder, and aloe.\n\n\"You're going back to the palace,\" Maledicte said.\n\n\"I have to go play lapdog to Aris,\" Janus said, his tone savage as he bandaged the wound. \"You need to trust me, Miranda. Once you trusted me implicitly. Do so again, and all will be well. You forget that there is no separation between us. No plans for me, and plans for you. There's only us, though you seem to fear otherwise. Your job done, you must let me work, now. You must be patient. You used to know how.\"\n\nMaledicte trailed into the bedroom, sat down by the fire. \"I used to be someone else.\"\n\nJanus poured him a tumbler of whiskey and folded his fingers around it. \"You're becoming maudlin.\" He sat beside Maledicte, the temper still in his eyes, though less bright, less hot. \"We must lull Aris. You need to give me time to undo the damage you did by killing Dantalion.\"\n\nMaledicte nodded, drinking deeply. \"But you won't stay.\" He forced his voice level, when he wanted to beg. It felt like years ago, lying in the street rubble, watching Ella's gift, her Janus, taken from her; now Janus took himself away, and she was more powerless than before.\n\nJanus shook his head. \"I must prove myself to Aris.\"\n\nAfter Janus left, Maledicte tilted the rest of the whiskey past his lips, and then poured himself another. He stared into the empty fireplace until the darkness settled into the room, filling it with the sound of wings. The house creaked, Livia going on tiptoe out the door. In the kitchen, Cook sluiced down the tables. Eventually silence fell, and still, Gilly had not returned. Maledicte swallowed the last of a third whiskey and fled the house.\n**\u00b7 35 \u00b7**\n\n**G** ILLY AND LIZETTE TUMBLED AND teased each other over the sheets, but Gilly's pleasure was leavened by guilt. He was all too aware that he should have returned to the town house, that Janus was not to be relied upon, and that Maledicte's temper was uncertain at best. But he dreaded the fraught rooms and the lingering presence of Black-Winged Ani. So when Lizette hailed him on the street, he gave in to temptation, and followed her home as shamelessly as an alley cat. Tonight, Lizette's simple charms were panacea for what ailed him. He was bent between her breasts, making her laugh, making her gasp, when she stiffened in his arms.\n\n\"Get out, you. Wrong room.\" She held Gilly's head to her chest, but the shiver up his spine had already warned him. There was no surprise in him when he heard the raspy voice responding.\n\n\"No,\" Maledicte said. \"This is the right room. But there are too many people in it.\" His words still held their bite, but their customary precision had been traded for a drinker's slur. Gilly turned, drawing the sheets to his chest like a maiden under a lecher's stare, though Maledicte's gaze lingered on Lizette and not him.\n\n\"Well, she's clean enough, I suppose, if a little overripe. I expected the worst, it took me so long to get directions to this brothel. And she's redheaded. Predictable, Gilly.\"\n\n\"Mal\u2014what are you doing here?\"\n\nMaledicte ignored him. \"Go away,\" he told Lizette.\n\n\"He's paid for me. You leave.\" Her eyes were slightly protuberant; from the study of her throat, Maledicte considered squeezing her neck, making them pop.\n\nGilly intervened. \"It's all right, Lizette. The money's yours. Just leave us.\" His heart pounded in his chest, but for what cause? Fear for Lizette's safety, or anticipation at being alone with Maledicte?\n\n\"You still want the room? Cause Ma's going to want to know.\" Lizette drew her dressing gown on, and Gilly, seeing it through Maledicte's critical eyes, was aware of the clash of violet silk against her ruddy skin and hair.\n\n\"Go away,\" Maledicte repeated. He stumbled forward, ripped the sword from his belt, dropped it on the floor, sat down, and pulled off his boots. With a last moue, Lizette left, banging the door behind her. Maledicte shot the bolt, locking them in, and leaned against the door.\n\nThe silence grew, and Gilly, tiring of waiting, said, \"I thought you house-bound.\"\n\n\"The guards only watched me go,\" Maledicte said. \"Perhaps they followed and are downstairs, sampling the house delights on Aris's coin.\"\n\n\"Janus?\" Gilly asked.\n\n\"The palace. But I came here to be away from all that.\" He crawled onto the bed, swaying, his balance shot, and lay down beside Gilly. He studied the draped shape of Gilly's body. \"I came to let you distract me.\"\n\nDrunk, Gilly thought, and wondered how much whiskey it took to override Ani's effect on poison. But even as he wondered, Maledicte's eyes lost the glassy quality they had held a moment ago, growing sharper, darker.\n\nGilly forced a laugh. Distract him. Maledicte had driven him to distraction and now all he could think about was the nearness of Maledicte's body to his. He could not think of a story now to save his life. \"How am I to do that?\"\n\nMaledicte rolled over in a swift move that belied his drunkenness, pinned Gilly, and kissed him, like the fierce first kiss of a child. Gasping, Gilly clutched Maledicte's neck, brought his mouth back down, and deepened the kiss. He tasted whiskey on Maledicte's tongue, like the sting within a liqueur-laced chocolate.\n\nMaledicte slid the sheet down, followed the retreating fabric with his mouth and tongue and teeth, no child's kiss this. Gilly groaned at the blaze of hot breath on bare skin, his hands clutching the thick layers of brocaded jacket and vest, embroidered shirt. He tangled his hands in Maledicte's hair, drew him up for another kiss, tasting the sweet, hungry mouth, as pliant as any woman's, but coupled with the sinew and strength of the body pressed to his.\n\nMaledicte shuddered against him, and Gilly bit at his neck, tasting starched silk. He lipped Maledicte's delicate fingers, ran his hands over the sleek shoulders knotted with muscle, and groaned again, nearly undone by the combination of female and male, of softness and steel. Maledicte laughed, a soft cat-rasp of pleasure, and rolled over, caging Gilly's hands beneath his flesh. Gilly knelt up, crawled over him, pressing his face into the black curls at his nape, licking, biting. Some blood scent still lingered in the dark hair, and Gilly moaned. Maledicte rocked back beneath him, rough embroidery scratching teasingly against Gilly's flushed skin. He gripped Maledicte's hips, pressed closer, and was rewarded with a breathy gasp that might have been his name.\n\nDrowning in the scent of his black hair, fumbling blindly, Gilly unwound Maledicte's cravat, kissed his bared nape, craned around, and tried for the divot at the heart of his collarbone. Maledicte turned, sought Gilly's mouth again, agile hands slipping down Gilly's flanks, pulling him closer. Gilly reached for the vest buttons between them, that maddening barrier of silk and wool, and Maledicte pulled away in a convulsive movement. \"Ugh. I can taste her on you.\" He rolled away from Gilly's questing hands, put his back to Gilly.\n\nGilly had no way of knowing that this was untrue, that the only thing Maledicte tasted was the warm salt of Gilly's flesh, the flavor of a beating heart. But belated caution skulked into Maledicte's mind\u2014if Gilly uncovered Miranda's secret, what would Janus do?\n\nAll Gilly knew, felt, was his pounding heart, his aching body. Breathing faster, he tried to tug Maledicte back into his arms, but Maledicte snarled, \"Get off me.\"\n\n\"Mal\u2014\" Gilly said, his word a plea, a breath, a groan.\n\nMaledicte huddled himself up, a slim line condensing itself. \"Leave me alone. Let me sleep.\" Maledicte's voice was rougher than usual, thicker. Gilly wondered if he would see tears or desire as the cause if he forced Maledicte to face him. Belated recognition of the words filtered into his mind.\n\n\"Are you staying?\"\n\n\"Why not? It's late, I'm drunk, and the bed seems free of vermin. My apologies if I ruined your sport. But more fool you if you paid her before you took your pleasures.\"\n\nGilly stared at the ceiling, counting the crystal stars pasted on it. Lizette's client list. Each star a patron. In the center of the ceiling was his own favored patronage, marked out in spirals and dots, a constellation made of desire. Lizette swore the constellation was one seen in the Explorations, taught to her by a sailor, but Gilly had no proof of that. Not yet. Perhaps not ever.\n\nBeside him, Maledicte's breath steadied and quieted.\n\n\"Mal?\" Gilly said.\n\nWhen he was met with silence, he pulled the quilt off the floor where he and Lizette had dropped it, and draped it over Maledicte. He bent to tuck it around Maledicte's shoulders and hesitated, finding Maledicte watching him with steady black eyes. \"Here,\" Gilly said, awkwardly finishing the motion, aware of the rough silk of Maledicte's hair pinned between his fingers and the coverlet.\n\n\"I'm not cold.\"\n\n\"Colder than marble,\" Gilly said. He stroked the smooth cheek, feeling the dip and sway of the flesh between cheekbone and scarred jaw. \"You should go back to Dove Street.\"\n\n\"I don't want to.\"\n\n\"Petulant as a child,\" Gilly said. \"But if you won't, you won't. It would serve you right if I brought Lizette back in.\"\n\n\"Don't,\" Maledicte said. \"Please.\"\n\n\"I won't. Best move over some, you're going to fall off the edge.\"\n\nGilly slid beneath the sheets, unwound them as best he could, allowing for Maledicte's weight atop them. Gingerly, he stroked the softness of Maledicte's hair fanning out over the pillows. When Maledicte didn't object, he tucked himself around that slim form, and felt the tension rise and fall in Maledicte's bones.\n\nSnuggled together in such surroundings, Gilly found himself wondering if this was how it had been for Maledicte and Janus. Away from the town house, this moment felt fragile, endangered by any opening door, by the rumble of voices down the hall, by the shouts of laughter and anger that rose from the streets. No wonder they clung to each other, he thought, they grew up with no haven but each other. Still, he thought, the situation was no longer the same.\n\n\"I've been thinking about the Explorations again,\" Gilly said, testing the waters.\n\nIn his arms, Maledicte made a sound of protest more felt than heard. It gave Gilly courage.\n\n\"I want you to come with me. The _Virga_ sails in five months, just before the fall. We could ride out beneath its tall sails, out through the harbor, into the deep waters where it's so blue you can't tell sea from sky. We could watch sea beasts at play; the great whales spouting and diving, and stranger creatures still, so strange that no sailor ever mentions them unless you've seen them already for fear of being mocked. We'd land in a new world. No Relicts, no court, just the land and the sky and the stars. The sailors say it's a different sky entirely down there, that it never goes black, just to darker and darker shades of blue\u2014\"\n\n\"And do what?\" Maledicte whispered. \"Live like paupers? Or fish for a living, at the mercy of storm and sea?\"\n\n\"I'd be a chocolate farmer,\" Gilly said, inventing on the spot. \"Feed you sweets for breakfast, until you grew fat or sick from them.\"\n\nMaledicte laughed, his warm breath brushing Gilly's forearm, raising the hairs on it and on his neck. \"Dreamer.\"\n\n\"But not a fool,\" Gilly said. \"Before I go, I'd buy up small luxuries here, to take with me. Sell them to the Antyrrian \u00e9migr\u00e9s over there, desperate for a taste of home, and use the money for things exotic to Antyre and send them back for sale. Feathers for gowns and hats, pelts for pelisses, illustrations. Maybe I'd even write a book for the libraries.\" His words came stiff and slow, awkward as his stories never were, fearful of being mocked. It was the first time he'd spoken aloud of his dreams. And even then he balked at spilling it all, that he would include Maledicte as more than a whim. That he couldn't imagine life without him.\n\n\"A trader overseas,\" Maledicte said.\n\n\"I know, a dreamer,\" Gilly said.\n\nMaledicte rose up, turned, and kissed Gilly's forehead, stroked his cheek. He slid back into Gilly's loose embrace, and only once his face was hidden again did Gilly hear him speak. \"What about Janus?\"\n\nIf Gilly's words of salt and sea and sky carried sunlight and tropical flavors, the mention of Janus brought the first taste of winter into the room.\n\nHeart pounding, realizing that Maledicte was tempted, Gilly said, \"Can't you forget him?\" His words were a whisper. Vornatti died for voicing a similar thought, but Gilly gambled that Maledicte was far fonder of him.\n\n\"I could easier forget my own name,\" Maledicte said. \"He needs me.\"\n\n\"I need you,\" Gilly said, sudden hunger darkening his voice to a growl, rolling Maledicte to face him.\n\nMaledicte scowled. \"You want me. Which is not the same thing at all.\"\n\nGilly started to protest and Maledicte put a hand over his mouth. \"Listen to me, Gilly. When you'd had me, what then? You'd be as weary of me as you were of the old bastard...you'd be longing for your sweet-fleshed, sweet-tempered maids and not a dark-natured creature like myself.\"\n\nGilly kissed the fingers overlying his mouth in soundless retort. Maledicte withdrew his hand. \"No, Gilly. Leave me be. I'm tired beyond belief....\" Hetried to unravel himself from the blanket but Gilly stopped him.\n\n\"Shh, just sleep. I'll guard your sleep. Even from myself.\" Gilly forced a lightness into his voice he didn't feel, was rewarded with Maledicte relaxing into his embrace. He lay with Maledicte a swaddled bundle in his arms, and tried to sort out the truth of it. Was Maledicte right, would he repent of this unseasonal desire? Gilly couldn't imagine doing so, but when he slept his dreams were full of Black-Winged Ani cradled in his arms, covered in blood.\n\nMALEDICTE WOKE, rubbed grit from his dry eyes, and took a startled breath at finding himself in the brothel. The corset pinched his ribs and he gasped. Untangling himself from the blankets, he staggered to the wash-stand, stared at his reflection in the still water. Gilly. Maledicte turned, breathing shallowly, breathing with small hitches of pain. In the emptiness of the room, he shucked out of the crumpled coat, the vest, and reached into his shirt to loosen the first laces on his corset. The relief was as sweet as the memory of his restraint. Janus would have killed him. Like Roach. Like Ella. Salt stung his eyes.\n\nHe splashed water on his face. The water was tepid but clear. Gilly must have asked them to bring another basin when he was done washing and shaving. Without Gilly's presence, the room seemed too full of the whore's trade, draggled lace, fine fabrics worn thin with use, the narrow bed and sagging mattress, the cloying odor of rose-scented powder and sweat in the air. If Maledicte had been less fortunate, less determined, without Janus to aid him, Ella might have sold Miranda to a place like this.\n\nNausea churned in his belly. Never to this, he thought. He would spill blood on the roads first, turn highwayman and waylay rich men's coaches. The thought calmed him; the sword on the bed soothed him with its bird's eye glitter.\n\nThe door opened behind him. In the mirror, water blurring his vision, he saw Lizette enter. \"What do you want?\"\n\nLizette grinned. \"Gilly said to make you comfortable. I came to offer you a razor.\"\n\nAware of the dampness at his throat and the loosened laces around his chest, Maledicte took up his vest, buttoning it with his back to her. \"I hardly think to be here long enough to require one.\"\n\n\"I would be amazed if you did,\" she said. She closed the door, leaned against it much as he had last night. He remembered that. The stability of the rough wood when his heart was pounding with possibilities.\n\n\"Have you something to say?\" Maledicte settled his coat as best he could, adjusting the shoulders. His head throbbed, imagining her laughing over how he had routed her from her bed to lie with his servant.\n\n\"Poor Gilly,\" she said. \"His head's in a swivet about you, desiring you, loving you.\" The scorn in her eyes took away any sweetness left in her face. \"And he don't know the first thing about you, does he, _my lady_?\"\n\nMaledicte's breath stopped. All his worries about Gilly knowing, about the court finding out, about Aris looking at him without preconceptions, just _once,_ and it was this whore who guessed. \"I've killed one man for suggesting I was effeminate. What makes you\u2014\"\n\n\"Whores know things, we've got eyes for artifice, don't we? Appearances is our trade, more'n anything else. How to look better, smaller, fuller\u2014each of us has played the man at least once, going out with a fellow where we wouldn't be wanted or escorting ourselves places where women don't walk alone. You're just better than most. Without that sword though, you're nothing but a tall, skinny\u2014\"\n\n\"Shut your mouth,\" Maledicte said. He grabbed the sword, yanked it free, then fought to resheathe it. He didn't need to murder her. Whores were easily bought.\n\n\"Or you'll shut it for me?\" She vamped at him, flashing her skirts, wrinkling her nose, fluttering her eyelashes.\n\nMaledicte grabbed her neck and slammed her into the door.\n\nShe coughed, then laughed. \"Rough play do it for you? You'll be disappointed with Gilly then. He's a sweetheart, through and through, my Gilly is\u2014\"\n\n\"Shut up,\" Maledicte said, pinching his fingers inward like the claws of a crab, collecting her attention along with the air in her windpipe.\n\nWary now, she opened her mouth to cry out. As quickly as she did, he barred her mouth with his fingers. \"Listen to me, Lizette. Should you unmask me, I will make you suffer. I know Itarusine potions to make your blood surge and foam within your skin, seeking egress. You'll bleed and keep bleeding from your eyes, your mouth, your overused sex...and you'll suffer pain you can't imagine. You'll die slowly while your blood swells like the surf and your skin splits to make way for it. And when you're dead, even the crows won't touch your flesh.\" He released her. \"And no one will even care. Or investigate. You'll be just another dead whore.\"\n\nShe slid down the door, soiled violet silk and blotchy face. \"I wasn't going to say nothing. Whores don't say nothing.\"\n\n\"Not if they're wise.\" Maledicte stepped back. She wiped her teary eyes and nose with the edge of her gown, looking up at him. He put a hand to her shoulder and she flinched. \"I see you understand me.\"\n\nHe left the door sagging open, left Lizette huddled on the floor, and went home.\n\nGILLY, LOUNGING IN THE LIBRARY, looked up at the bang of the door. He folded the broadsheet, set it down at his feet. \"Did you sleep well?\" He doubted it, the way Maledicte clung to the shadows of the room, pulling curtains.\n\n\"Did you send Lizette in to me?\" Maledicte asked.\n\nGilly winced at the ugly edge in Maledicte's voice. \"No. But the greedy little cat probably liked the rich looks of you. Did she wake you?\"\n\n\"The screeching of the matron did, rousting some sailors who out-stayed their coin. Don't I give you enough to establish a bijou in a peaceful neighborhood?\"\n\n\"Woke up temperamental, that's obvious,\" Gilly said. Aware that Maledicte hadn't yet met his eyes, Gilly wondered what ailed him, embarrassment or anger. \"Come here.\" When Maledicte hesitated, he repeated himself. \"Come here.\"\n\nMaledicte stood before him, stiff and spiky like a child uncertain of chastisement.\n\n\"Woke with a head, I've no doubts,\" Gilly said. \"Poor Mal. You were very drunk last night. I am amazed Ani allowed it.\"\n\nMaledicte knelt before Gilly. \"You're not angry?\"\n\n\"No,\" Gilly said. Why should he be angry? He knew something he hadn't known before, that Maledicte desired him. That knowledge made him lazy and content. He stroked Maledicte's neck, his shoulders, his dark hair. Maledicte laid his head in Gilly's lap, sighing.\n\n\"I'm sorry you woke unpleasantly,\" Gilly said, separating strands of dark hair and twining them again. \"You looked so peaceful when I left.\"\n\n\"That was your mistake,\" Janus said. Gilly flinched in his seat, felt Maledicte carry the movement through. \"When Mal is peaceful, it's always deceptive. Usually means he's going to kill someone.\"\n\n\"I have to wash,\" Maledicte said. \"Those sheets probably had fleas.\" He pushed away from Gilly's loose embrace.\n\nJanus snagged his arm, studied him with a sapphirine gaze. \"Aris restricted you to these four walls. You went out?\"\n\n\"You disobey him at will. Why shouldn't I?\" Maledicte said, twitching his arm free.\n\n\"I am not on sufferance,\" Janus said.\n\n\"Aren't you?\" Maledicte said.\n\nJanus's face darkened, and Maledicte sighed. \"My temper is foul today, Janus, so go cautiously.\"\n\n\"I remind you,\" Janus said. \"Aris has guards watching the house, watching you. Remember that, should you feel the need to draw your blade.\"\n\nMaledicte shivered; his hands clenched, but he made no further response to the tightening noose of suspicion he found himself in. Instead, he drew Janus's head down and kissed him fiercely, after a quick, burdened glance at Gilly.\n\n\"I'm tired. You need to time your visits better. Until then, Gilly will take care of me.\" Maledicte slipped out of Janus's arms and went upstairs.\n\nJanus smiled thinly at Gilly, and prowled the room, unspeaking. Gilly rose to go and Janus forestalled him. \"Something you want to confess to me, Gilly? You're jumpy today. As if you had a guilty conscience.\"\n\n\"Is yours any more pristine?\" Gilly countered.\n\n\"Do I need to tell you again to stay away from Maledicte? He's more than you can handle.\"\n\n\"I handled things well enough last night,\" Gilly said.\n\nThe sudden blankness in Janus's face gave him enough warning to duck the blow. But then Janus seized his shoulders in a grip that trembled with rage; Gilly felt bruises starting.\n\n\"You dared,\" Janus said.\n\n\"Why shouldn't I?\" Gilly said. \"If you can change your appetite from women for Mal's sake, why can I not do the same? He was willing enough.\"\n\nThe tension in Janus's arms eased enough for Gilly to free himself. What he had said that defused the worst of Janus's temper, Gilly didn't know, and he found himself regretting it. This fight had been a long time coming, and he welcomed it as much as he feared it.\n\n\"Get out,\" Janus said.\n\n\"This is my house,\" Gilly said. \"You go.\" He grinned. His heart raced with exhilaration and fear. He found Maledicte's evil genius poking him, as if the night spent together had left him with more than frustrated desire. \"Of course, you could stay. Could hit me again, threaten to kill me. Again. But I know why you balk...you are not so sure that Mal would forgive you\u2014\"\n\nJanus struck and Gilly blocked, catching the fist in his own grip, twisting it. \"No more idle threats, Janus. I've gotten your measure. Maledicte's love protects me from you. And he does love me, whether he wants to or not.\"\n\n\"Then be honest with me,\" Janus said. \"You mean to steal him from me.\"\n\nGilly said, \"Your love will send him to blood and death.\"\n\n\"You don't care about the blood, about the court. All you care about is having him for yourself. Don't dress your motives in fine words. You want him.\"\n\n\"I do,\" Gilly said.\n\n\"I'll kill you first,\" Janus said.\n\n\"And that brings us around to where we started this quarrel,\" Gilly said. \"Like two dogs fighting over a bitch in season.\"\n\n\"That's a flattering thought. Be sure to share it with Mal. He'll gut you for me,\" Janus said.\n\n\"He promised he'd never hurt me. I believe him. He may be many things, but he's not a liar.\"\n\n\"He'd forgive me anything,\" Janus said.\n\n\"Are you willing to test it?\" Gilly said. He stepped back, raised his arms wide, inviting Janus. \"Not that you'd have it all your own way. I may not be a swordsman, but I outreach you.\"\n\nJanus snarled. \"You forget your place. Maledicte may call you friend, but you are a servant born, and a servant until death.\"\n\n\"And you haven't forgotten yours?\" Gilly asked. \"You're so far out of your place that you're dangling from a rope marked treason.\"\n\nJanus hissed, his hand clenching around his sword, but as soon as his knuckles whitened, they relaxed, the rage cresting and disappearing as if it had never been. He turned a placid face to Gilly, leaving him off-balance. Where did the rage go? Where would Janus vent it? Not on Maledicte, surely; it was no longer safe to do so.\n\nGilly put his back to Janus and walked out, though his skin crawled. If Janus could be rid of him by accident or manipulation, Maledicte's protection would be useless. And Janus's anger, though better controlled than Maledicte's, always erupted in the end.\n\nHe listened for Janus's footsteps in the hall, in case Janus chose to continue their quarrel. But instead, he heard them going up the stairs, chasing after Maledicte.\n**\u00b7 36 \u00b7**\n\n_For a most enlightening murder, in times when subtlety is not as prized as spectacle, one can do no better than to seek out tincture Precatorius, imported from the Explorations. A single death by its means is always enough to open the eyes of the most recalcitrant subject._\n\n\u2014 _A Lady's Treatise,_ attributed to Sofia Grigorian\n\n**T** HE MESSENGER ARRIVED EARLY in the morning, rousting Gilly from his bed after a night full of stealthy leavings, first Livia creeping out yet again, then Janus seeking the palace. Sleepily, Gilly paid the boy and flipped open the note, curious to see which of their spies had something to report, or if perhaps the coachman hadn't lost Livia this time\u2014the girl was clever and careful. But the terse lines didn't involve Maledicte and his schemes at all. The note, straggling words written in a hand unused to a pen, read simply: _Lizette very sick. Need help._ Gilly crumpled the quarter sheet of cheap paper in his fist, releasing the scent of the brothel and desperation.\n\nGilly was torn between agitation and irritation. Lizette had been furious with Maledicte's intrusion and threats, had failed to meet with him last night, sent Ma Desire herself down to make her sentiments known. He half suspected this emergency mere stratagem, showing whether he valued her.\n\nWhen he entered, he smelled the hot tang of blood over all the other odors, and knew the need was real. The madam met him at the door, her skirts splotched with blood.\n\n\"It's too late,\" she said. \"She's gone.\"\n\n\"Gone,\" Gilly repeated, and went where the madam beckoned.\n\nLizette's boudoir was drenched. The blood, mostly stiff and browning, still had a few spots of freshness to it. One rivulet dripped slowly from the bed.\n\nHer back arched; her eyes were open but obscured by blood, her hands locked on the sheets in her last spasm. Gilly gagged. _Lizette._\n\n\"What happened?\"\n\n\"Poison,\" Ma Desire said.\n\n\"Poison,\" Gilly repeated, his ears numbing.\n\n\"She got a box of chocolates, last night, after she sent you away. She ate them up, didn't she, most of them at once. And a note\u2014didn't I read it for her.\" She spoke to the room at large, though Gilly was the only one listening. Over by the hearth, another whore, her hands gloved, shoved bloody sheets into the fireplace. A second girl scrubbed at the spots on Lizette's finest dress, attempting to salvage it.\n\n\"Said as how you were sorry. That she should forgive you. Real gentlemanly, it was.\"\n\n\"I didn't send it,\" Gilly said. But he might as well have, he thought. Somehow this blame fell at his door.\n\n\"Figured that out when she started to bleed. She knew then who done it.\"\n\n\"Who?\" Gilly said.\n\nThe madam turned her head, studied the room with a speaking silence. Gilly's breath shuddered out of him. Throat tight, he reached into his pocket for coins.\n\n\"'Tain't for me,\" Ma Desire said, as she tucked them into her bodice. \"For her. Someone's got to pay for the burying.\"\n\n\"Who did this?\" Beneath the grief, a flicker of anger grew. He was not Janus, was not Maledicte, to find forgiveness of anything.\n\n\"She raved about your other lover, your highborn lady, said the crows were at her, tearing her insides. She felt their beaks. Said your lover had warned her. Said she'd bleed. Said Black-Winged Ani was killing her sure. Said she stole the crow's man and doomed herself. That true?\"\n\n\"What?\" Gilly said, his mind quaking away from the ruined woman on the bed, the fevered words attributed to her. Maledicte? Maledicte found killing offenses entirely too easily, Gilly thought, sick at heart. And penned at home with watchful guards, unable to bring the sword to bear, a box of poisoned sweets would be all too easy to arrange.\n\n\"That you're one of Black Ani's creatures.\"\n\n\"No,\" Gilly said. \"My master\u2014my friend\u2014\" His voice broke. He sat on the bed, touched Lizette's distorted face, cold, waxy, and faintly sticky.\n\n\"You want to protect yourself. There's charms and such,\" she said. \"You're a good boy. Don't get caught up in the crow's feathers.\" She reached out, touched his hair, the hardness fading from her face. \"You was good to my girl.\"\n\nTHE KINGSGUARDS POSTED near the house eyed him curiously as he pounded up the entry stairs, slamming the door open, but did no more than watch.\n\nInside, Maledicte, muted in gray wool with a scarlet shirt peeking out, sat to luncheon, his head bent over a book. Gilly paused, the anger in him churning, and he bypassed the dining room for the main stairs. He pushed open Maledicte's doors, ransacked drawers and wardrobe until he found the wooden case that held the poisons and dumped them across the bed, greenish-gray powder spilling out of white twists, dusting his hands. He rummaged through the small vials, looking for the one that could make flesh melt to blood.\n\n\"You've only been out for a few hours,\" Maledicte said from the doorway. \"Surely no one's offended you so badly in that time that you would turn to poison. But if they have, let me know, and I'll take care of it for you.\"\n\nGilly's hand closed around the vial. A scant few purple drops clung to the curved bottom. \"You didn't have to kill her. All you had to do was ask me to give her up. I would have done anything for you. I have done everything for you.\"\n\n\"Gilly?\"\n\nGilly threw the vial at him. Maledicte caught it easily, looked at it with wary eyes. \"Precatorius syrup. She bled to death, as you threatened.\"\n\n\"Lizette.\" Just her name drew Maledicte's supple mouth into a scowl.\n\nTo Gilly, it felt like confession. \"She bled out and your bottle is near empty. Why kill her like that? Why make her suffer? Why kill her at all?\" Gilly's eyes blurred with tears.\n\n\"Is she worth all this fuss? She was just a whore.\" Maledicte's face twisted. \"A creature without value.\"\n\nGilly's fingers clenched; he raised his fist, and dropped it. Maledicte hadn't flinched.\n\n\"I liked her. She was uncomplicated and mouthy. What did she do to you? What did she say? Did she laugh at you? Give me a reason\u2014\" He raised his hand to Maledicte's cheek. \"Please.\" He needed something, anything to stop the rage and pain churning inside him. He waited in frozen silence for Maledicte's response, waited to be freed to anger or bittersweet relief.\n\n\"I didn't kill the bitch,\" Maledicte said, slapping Gilly's hand away. \"Are you my hanging judge? Go away, Gilly.\"\n\n\"Chocolate and poison. A sweet with a sting. A note she couldn't read but had my name on it. It apologized for our interrupted sport. You expect me to believe you didn't do this?\"\n\n\"Burn your soul, I\u2014\" His voice refused to rise, the rasp giving way to forced silence. Thwarted, Maledicte bared his teeth and shoved past Gilly like a departing evil spirit. The parlor door downstairs slammed with a sound of cracking glass, leaving Gilly cut off from his answers. Small crashes shattered silence like distant cannonfire as Maledicte took his temper out on frangibles.\n\nGilly's own rage simmered and roiled. He fled the house, past the lurking guards, and into the city. He was nearly into the merchants's treets before the fog of temper and pain cleared way for a single thought. Maledicte had never denied his wrongdoings before. Still, it was Maledicte's bottle that had been emptied....\n\nGilly moaned, resting his sweating face against his hands. He forgot that he had enemies himself, one of whom resided under the same roof, privy to Lizette's existence, to her location, to Gilly's thrice-weekly visits. Grim, Gilly traced his way back in the twilight. He would ask once again, and this time, he would listen.\n\nThe door was not locked against him as he half expected it to be. The parlor was awash in wreckage, as if it was the spill point for the tides' refuse. The mirrored door was broken; winking glass met Gilly's gaze from every angle; the spinet stool lay beneath the lintel, one leg snapped.\n\nGilly's boot crunched in the soft pile of the rugs. A curled bit of porcelain stuck out from beneath his boot. He picked it up\u2014a small porcelain arm. The silk thread and dangling stick were all that told him he held the remains of one of a series of puppeteer figures. He raised his eyes to their shelf. Not one remained, and though he found more identifiable pieces, a dog's head with a high, ruffled collar, a serpent's rattle, a minuscule puppet's puppet with its arms snapped off, he found no whole survivor. Some of them had been broken so fiercely it seemed as if Maledicte had attempted to grind them underfoot.\n\nGilly set down his handful of parts on the curtained altar with a speculative expression. Janus had gifted Maledicte with these puppets. Maledicte held them dear. Or had. Likened himself to the puppeteer of the gods, but perhaps he felt more a puppet today.\n\nGilly ascended the stairs to the first level, turning the gas lamps to glowing life as he went.\n\n\"Maledicte?\" Gilly called. The house was as hushed as if Gilly was the only breathing thing within its walls, and his heart beat faster. \"Maledicte?\" In the hallway, Gilly hesitated, then chose to climb the dark attic steps. Faint glimmers of porcelain dust traced a footstep six steps before him and he took the rest of the stairs with more surety.\n\nCool evening air swirled down, whistled under the attic door. Gilly pushed it open. The attic window gaped with jagged glass. Maledicte sat before it, perched on a pile of trunks. The heavy sweep of his scarlet shirt, the sleeves uncorded, unrestrained by a jacket, draped like bloody wings. His knees were drawn up, wrists crossed over their peaks.\n\n\"I'm sorry, Gilly.\" Maledicte turned his head to look back over his shoulder; the heavy hair whispered and shifted. His face gleamed in the faint starshine and reflected gaslight straining through the city fogs.\n\nA scrabble and the brushed, whiskery sound of feathers kept Gilly from instant speech as the rooks hopped his foot and lifted off, one wheeling out the window, the other perching on the pile of discarded clothing. It dipped its beak, exposed a rent in an embroidered jacket, and flew out the window, trailing golden strands.\n\n\"I didn't know you cared so much about her,\" Maledicte said. His voice was muffled; he laid his head into the space between his arms. \"I hated her.\"\n\nGilly sat on a low trunk, peering up at the huddled shape. \"You didn't even know her. I barely knew her beyond her profession. How could you know her enough to hate her?\"\n\n\"She came in while I was washing, teasing me. I told her to go away. She wouldn't, just leaned against the wall, her dress falling off her breasts, flaunting herself. I tried to scare her away and she laughed at me. I only did it to make her stop.\"\n\n\"So instead of stabbing her there, you came home and sent her poisoned chocolates, leaving her time to spread the story. No, you came home. You spoke to Janus. And Janus killed her.\"\n\n\"No,\" Maledicte said. \"Gilly, you have all the evidence. What more do you want? I am a murderer after all, several times over.\"\n\nGilly let out his breath; it left blueness in the chill air. \"Tell me you killed her. Tell me you sent death to her, wrapped in pink paper.\"\n\nMaledicte stared down at Gilly. \"I killed Lizette.\"\n\nThe shadows made patchwork of his face, created dark holes where his eyes should shine, and his voice was as calm as ever. Yet Gilly felt his pulse jump, his breath catch as he recognized the lie. Maledicte, the competent killer, was a bad liar, more used to half truth and misdirection.\n\n\"I'm sorry, Gilly,\" Maledicte said. He levered himself to his feet, standing before the open window. He stretched forward, slipped a clenched fist out into the night sky, rainwater washing over his fist, and then he opened it. Small, and glittering malevolently, the carved puppet of Ani plunged to the street below. Maledicte swayed in its wake and Gilly put a steadying hand on his ankle.\n\n\"Sorry for something you didn't do.\" Gilly plucked Maledicte from the chests, and let him go, listening to him stumble down the stairs. Gilly looked out the window, down to where the statue had disappeared.\n\n\"You'd forgive him anything,\" Gilly said. \"Even making me believe you killed her. Trying to set us at odds.\" Bile twisted in his belly at the sheer callousness of it, at turning Lizette's life into a pawn move. In the attic's soothing darkness, Gilly, like Maledicte before him, crouched and wept. If he had any doubts before\u2014he had none now. Janus was a killer, and like his father, like Dantalion, preferred to smile and kill at a distance.\n\nGilly bowed his head. The fault, after all, was his. He had goaded Janus, knowing that the man would retaliate. But to imply that Maledicte was to blame\u2014Outrage settled the anger in his belly to a steady flame.\n\nHis fisted hands touched the stiff, dark patches of Lizette's blood, transferred when he sat beside her corpse, and he turned to seek his bath. But through the broken window, he heard the coach draw up outside, the horses' hooves loud on the cobbles. _Janus,_ he thought, and decided to put off his cleaning in favor of confrontation.\n\nWhen Gilly reached the entry hall, he found Maledicte, blank-faced and white, facing Lord Echo and a brace of Particulars.\n\n\"I have a warrant sworn out for your arrest,\" Echo said, smiling grimly, his hand on his pistol.\n\n\"In what matter?\" Maledicte asked. \"You've sought to blame me for so\u2014\"\n\n\"One incontrovertible death. The murder of Dantalion Vornatti.\"\n**\u00b7 37 \u00b7**\n\n**H** ARDLY MURDER,\" MALEDICTE SAID. \"A duel.\" \"His blood, your sword. You cannot deny that.\" \"Why would I deny it? When I enjoyed myself so much?\" Maledicte said, though in truth he had almost forgotten it. Dantalion's death swallowed under Ani's blood tide; to find himself accountable for it now\u2014His hands shook but his voice remained light. He tightened the small muscles in his hands, and they too obeyed his will, stilling. The sword hilt shifted against his palm, though he wasn't aware of seeking it out. The feathers coaxed and whispered against his skin. Gilly's clothes, scented with Lizette's passing, kept blood in Maledicte's mind.\n\nThere were only three men after all. He could have his sword through the protruding belly of the nearest Particular without much effort, spill the blood out and dance toward the young Particular to Echo's left. Already his dewy skin paled at the audacity of arresting a nobleman. If Maledicte gutted the first man, the youngster would bolt. He'd lay sols on the matter. Only Echo promised a fight.\n\nMaledicte sucked air through his teeth, felt it cool the furnace of his blood. Yes or no. Fight or fly\u2014he'd have to chase the stripling if he fled. He'd had enough of witnesses. But three were manageable.\n\n\"Maledicte,\" Echo said. \"Lay down your sword.\"\n\n\"If I choose otherwise?\" Maledicte said, still listening to the clock of his blood, ticking away.\n\nEcho pulled his pistol, cocked it, and leveled it at Gilly's chest. \"Do I need to use your friend as a surety for your behavior? Should I see one gesture that speaks of weapon, poison, or even enchantment, I will kill him.\"\n\nRage reddened Maledicte's vision; his heartbeat, reacting, deafened him. _Kill them all,_ Ani whispered. _Bathe in their blood. Feast on their eyes. Even Gilly. He mistrusted you after all, accused you of something you never did._ The sword rose in the sheath; his fingers coiled down, touched the cool metal of the blade itself.\n\nThe violent simplicity of the idea held Maledicte hostage. _Kill them all._\n\nMaledicte felt the movement in the air and he spun on instinct, the sword free of the sheath, registering the shock in Echo's face even as he did so. It was Gilly moving, only Gilly, and instead of slicing skin and bone, Maledicte twisted the blade, letting the flat of the sword strike Gilly's broad wrist. It welted the skin and left a fine line of blood where the edge had nipped. But the hand was whole, the wrist entire, the fingers closing on his shoulder. Maledicte panted, watching that slow beading of blood on Gilly's fair skin.\n\n\"Maledicte,\" Gilly said. \"Be still.\" He stepped past him, blocked Maledicte's view of Echo. \"Does the king know you're here?\"\n\n\"The king can call me off\u2014he has that privilege. But he need not set me on. There were many witnesses.\" Echo smiled. \"Maledicte is mine.\"\n\nBehind Gilly's sheltering back, Maledicte started shaking again, this time beyond his ability to control. Stonegate Prison. How could he accomplish Ani's goal then? How could he even survive, locked in the dark, with constant company\u2014how could he be Maledicte?\n\nThe thin thought crossed his mind, a ghost of reason. Echo had not shot Gilly, though Maledicte had drawn his sword; Echo bluffed.\n\nMaledicte bolted for the stairs, for the scent of the sky, and Echo shouted. Behind him, Maledicte heard the report of a gun, but no outcry from Gilly. If he hadn't been so desperate, he could have wept with relief. But trying to think around the flapping blackness of Ani's rioting emotions left him little but the frantic intellect of a cornered rat.\n\nEcho's hard hands grabbed his shoulders and Maledicte kicked back in a Relict rat's dirty blow. But Echo, though he faltered, was wise enough to have anticipated such a trick. Maledicte twisted to bring the blade to bear, freeing himself from Echo's clawing grasp, and found himself borne back into the wall by Gilly.\n\nMaledicte fought Gilly's grip, breath sobbing in his throat. Ani whispered, _Kill him and be gone._ Maledicte gasped his refusal, even as Gilly pulled him closer, clutched him to his body, pressing him between his solid warmth and the unyielding wall.\n\n\"Mal,\" Gilly said. \"Maledicte, please. If you flee now, there will be no Janus, no future, only blood and death.\" Gilly's breath warmed his cheek; his fingers traced soothing patterns on his wrists and back.\n\nOver Gilly's shoulder Maledicte saw the two Particulars nervously watching, saw Echo's eyes narrow, and Maledicte hissed at him. Echo took an involuntary step back and Maledicte laughed.\n\n\"Hush,\" Gilly said. \"Hush, this is what we'll do. Where one man can be paid to do his duty, another can be paid to ease your way. Go with Echo. It'll be only temporary.\"\n\nMaledicte burrowed into Gilly's warmth, listened to the heartbeat pounding beneath his ear, not as calm as the words Gilly spoke. Beneath the patterns Gilly traced, Maledicte felt Ani retreat, muttering, leaving Maledicte drained but capable of thought.\n\n\"You won't leave me there?\" Maledicte said.\n\n\"No,\" Gilly whispered, stroking Maledicte's hair, heedless of Echo's furious gaze. \"I promise,\" Gilly said, \"I will always come for you.\"\n\n\"Tell him.\"\n\n\"Yes,\" Gilly said.\n\n\"He'll get me free,\" Maledicte said.\n\nGilly nodded. Maledicte reversed his grip on the sword; Echo raised his pistol again, but lowered it as Maledicte handed the sword to Gilly hilt-first. \"Take care of this. I'll need it again.\" Then he stepped past Gilly's sheltering arms, and into the rough grasp of the Particulars.\n\nAN HOUR LATER, shoved into a filthy communal cell, Maledicte reminded himself of the satisfaction that had filled him when he had taken Dantalion's throat, reminding himself that the bloodlust had been worth the price he paid now. The remembered smell of blood kept away the stink of unwashed bodies, of rank straw, of fouled water and illness, soothed the panicky flutter of his heart.\n\nThe cell fell silent at his entrance. In his fine clothing, his perfumed hair, he was a world away from their existence. Usually the nobles met with Damastes, the jailer, handed over their valuables for a private cell, for fresh water, for a mouthful of bread not gone blue. But Echo had brushed by the jailer, ignoring the man's covetous looks at Maledicte's finery, and forced Maledicte into the common cell. The rattle and thump of the heavy door woke those who had learned to be wary, and made others flinch in their sleep. Maledicte's heart leaped again at the long rattle of chains being drawn through iron rings, the wooden bar sealing the cell door shut behind him. Caging him. His mouth dried.\n\nTwo women, huddled in the corner, averted their eyes, pulling ragged shawls up to cover their faces. Beside them, a man rose to his feet, bare arms showing the dark ink of a conscripted soldier, a survivor of Xipos Island, and undoubtedly an enemy of the aristocracy that had used him and discarded him. Wary, Maledicte watched him stand. \"You're even taller than my Gilly,\" he said aloud. The torchlight wavered through the grill on the door, casting ruddy shadows into the room.\n\n\"And you're dressed for pleasure, not prison,\" the man said, his voice equal to Maledicte's rasp. \"Those shiny buttons, that stickpin\u2014hope you won't mind sharing.\"\n\n\"I do,\" Maledicte said. His hand itched for his sword, but when he was Miranda he'd taken on grown men, unarmed, save for a stick. Though even Miranda, half-mad with starvation, might have balked at this fellow.\n\nThe man lumbered at him, meaty hands outstretched, and Maledicte laughed. Snatching up a handful of stiff straw, he lunged to meet him, stuck his makeshift weapon into the man's eyesockets, and twisted. The man screamed, his voice gone high and hoarse. \"You're too slow and fat,\" Maledicte said.\n\nHe pivoted, aware of others slowly joining the fray, eager for revenge on a noble, for the temptation of riches enough to pay off their petty crimes or debts.\n\nIn the back of his mind, Miranda began to panic\u2014she knew what happened to girls who got overwhelmed, torn down\u2014but Ani raised Her wings and Maledicte let his will slip away, gave himself entirely over to Her hungers.\n\nHe reached up to the thrashing giant, climbed his shirt, and bit through the skin at his neck. The man fell, whimpering, covering his bloody neck. Ani spat the tiny scraps of flesh out and they were lost on the floor.\n\nThe other men hesitated a moment, and Ani grinned a bloody smile at them. In the corner, the women gibbered, whether in support or terror he couldn't tell. Ani sucked in a breath, took in the foul vapors of the room, of the death lingering in corners, and spat it all back out. Black foam flecked the floor where his spit landed, splashed on their faces.\n\n\"Rot you,\" Maledicte said. \"Rot you all.\"\n\nThey backed off, hands touching their faces, wiping the spittle away as if it burned them.\n\nMaledicte's throat itched as if his saliva had been caustic. He reached for a water pail, skimmed the top of it, and drank the clearer water below. Where his lips touched the dipper, the metal blackened, Ani moving through him in waves of heat.\n\n\"What's all this?\" the jailer said from the doorway, keys jangling self-importantly. He checked on seeing the big man whimpering in the middle of the floor. Maledicte looked at Damastes blankly for a moment, trying to recover the courtly ways that Ani had eclipsed.\n\n\"He wanted to share the things I'm saving for you,\" Maledicte said, touching his jeweled cuffs, his gemstone-buttoned vest, the fine weave of his coat.\n\n\"You shouldn't be in here. Not with the likes of them. You're Quality,\" Damastes said. \"Quality\"\u2014he drew the word out again, raising his head to stare Maledicte down. His eyes were the color of dirty slate, and oddly opaque, his hair a faded brown, as if he took his coloring from the stone and earth around them.\n\n\"I've always thought so.\" Maledicte said. \"Shall we adjourn to your office? Maybe have some wine sent in. That water is foul.\" His flippancy felt strained.\n\nThe jailer nodded, his eyes assessing. \"Yes, let's talk about your situation.\" He bowed with as much mockery as Maledicte had ever managed, and gestured him out of the common cell.\n\nAs Maledicte passed through the doorway, guards fell in step beside him from the places on either side of the door, letting Maledicte see that as greedy as Damastes was, he was also wary.\n\nEcho and his damned interference again, Maledicte thought. The jailer was unlikely to treat his other noble patrons with such caution. Too often now, Echo had created obstacles for him. Maledicte, walking down the narrow hallway, ignored the stone walls, the damp, spending his thoughts on sweeter dreams of killing Echo. His fingers curled, seeking the hilt of his sword, and for a moment, the familiar memories of it were so strong, so real that he felt the weight of the blade waiting, smelled the steel tang of it in the dank air.\n\nHis hand snatched at empty air; he faltered in his steps as the sense of steel faded to nothing, like smoke in his grasp. \"Keep moving,\" a guard said, reaching out to prod Maledicte into motion. Maledicte evaded the careless hand and started up the uneven stairs he had been pushed down barely an hour before.\n\nThe jailer's office and quarters were only cells with their walls knocked out, leaving cut masonry edges visible. Narrow windows allowed an unbarred view over the approaching street, but were too thin to permit egress. Around the room, heaped on elegant furniture, jumbled piles of aristocratic castoffs gave the impression of a disorganized pawnshop. Small jewels spilled over the edge of a mahogany dresser, gleaming like water, pouring into the half-open drawers. A riot of chairs made the room a maze of gilded legs and scrollwork, of tapestry and velvet and leather. At the heart of the room, a clerk's desk, all pigeonholes and paperwork, rested. A fireplace peeked out from behind a stack of dust-felted books.\n\nIdly, Maledicte bent and picked up a pocket watch from a pile of others. Lapis sails, a nacre ship, enameled on washed gold. He swung it from his hands, the chain slipping through his fingers with the heft of a living serpent.\n\n\"Sit,\" Damastes said.\n\nTucking the watch and chain up his cuff with the same economy of motion that he had used while card-sharping, Maledicte felt more at ease. If the jailer and his guards missed his small theft, they were not so observant as he feared. He turned his attention to choosing a chair, looking at gilded legs, carved frogs, or lions rampant on leather.\n\n\"This is not a shop for your perusal,\" the jailer said, brows drawing down over hooded eyes.\n\n\"No,\" Maledicte said. \"A shop would be better organized, and considerably cleaner.\" He hauled a lady's chair up, all delicate legs and filigree, took care to sprawl over it, overflowing it.\n\nDamastes sat in a velvet chair opposite, put his filthy boots up on a carved ivory footstool that creaked under their weight. Maledicte flickered his eyes downward, studied the worn soles of the jailer's boots.\n\n\"All this plunder and you need new boots,\" Maledicte said. \"Is it false economy that hinders you, or do you just not know a decent shop?\" Over the man's shoulder, he watched the night sky split by darting bats and the sleek flow of rooks.\n\nHe was minded to draw this bickering out as long as he could, lulled by the sight of the sky. Only underground for minutes and already he felt buried alive. It was Ani within him who loathed the dirt, he knew; the underground dark had always been Miranda's friend, her kingdom found beneath the beds, beneath the rubble, beneath the storm-cloud overhangs of stone eaves.\n\nDamastes grinned at him, brown teeth in a turned-down smile. \"Say what you want. I've been abused by aristocrats before\u2014but remember, you're here to beg for my favors.\"\n\n\"Is that what drives you?\" Maledicte said. \"All this stolen wealth and it means only humbled aristocrats to you? You're a fool. You could buy yourself a title abroad and live like a lord. If it's begging you want, I have nothing to offer you.\"\n\n\"Make him kneel,\" Damastes said, his strange, slaty eyes hardening.\n\nThey reached for him; Maledicte evaded their hands, stepping behind a wing chair, making them stumble over the heaped greatcoats he pushed from its seat.\n\n\"No need,\" Maledicte said. \"I'm tired of my clothes being manhandled. First my Gilly, who should know better, and Echo, then that oaf in the cells. I see no reason to add two more pairs of damp handprints to my coat. You want me to kneel?\"\n\nMaledicte searched out a clean spot on the floor, finding one just as the guards reached for him again. He dropped, letting them grope the air. He grinned at Damastes. \"Here I am. Kneeling before you...but very far from begging, I assure you.\"\n\nThe jailer surged out of his chair, a thin hand knotting into a fist, and paused, his shoulders rising and falling with a laden breath. \"I could break you,\" he said, his voice striving to match Maledicte's insouciance.\n\n\"My bones perhaps,\" Maledicte said. \"But what then? Will you gamble that I am to be incarcerated forever? Or will you strike me, and see me freed tomorrow, full of rancor? My lover does not care to see me abused.\"\n\n\"Your lover\u2014the king's nephew,\" the jailer said.\n\n\"No secret there, an old scandal in the court.\"\n\n\"You're as much a bauble as any of these jewels,\" Damastes said. \"A favored possession. Close to royalty. You've been bedded on crested sheets.\"\n\n\"Sometimes in crested carriages,\" Maledicte agreed, all silken tones, like steel withdrawing.\n\n\"A collectible and rare. They say even Aris has touched you\u2014\" The jailer's voice dropped to a whisper; he darted a quick glance at his guards.\n\n\"That would be indiscriminate of me, surely, to bed both nephew and uncle,\" Maledicte said, relaxing into the familiar thrust and parry of spite and gossip. Damastes was simply another fool to be manipulated.\n\n\"To add you to my collection, to have something that was theirs...I could\u2014\" The jailer paused, an ugly, triumphant light in his eyes. He touched Maledicte's throat, drew closer, a hand on his own breeches.\n\nMaledicte smiled. \"My teeth are as sharp as my wit.\"\n\nDamastes took his hand away. Maledicte shrugged, a loose liquid thing, as if he had been only chatting with friends. \"Are we not to barter at all? Or have you brought me here only to enact the worst examples of boring pornographies?\"\n\nHe made no attempt to lower his voice and Damastes snapped, \"Shut up, or I'll gag you.\"\n\n\"Back to the cell, then?\" Maledicte said. \"You'll never get your trophies that way.\"\n\n\"What have you got?\" the jailer said grudgingly, sinking back into his seat.\n\n\"No furniture, I'm afraid, I haven't been here long enough to have furnishings brought, nor do I intend to be. But then, this room is rather bewildered with furniture. All I have is the usual bric-a-brac of a gentleman's life.\"\n\nHe turned out his purse. \"Two sols, how lucky for you\u2014enough to get your boots resoled. After all, gold is no trophy, gold spends. A stickpin, ruby, jet, and silver.\" He dropped it onto the desk. \"Had I known I was to be arrested, I would have worn one I liked less. Jet buttons on my waistcoat. Cuff links, ruby again.\" They landed beside the stickpin, rolled, and fell to the floor with faint thumps.\n\n\"In my pockets, well, Gilly says it's the mark of a gentleman to have nothing marring the line of my coat, but luckily for you, I am not so much a gentleman as all that. A luna and a snuffbox\"\u2014he frowned\u2014\"that I stole from Dantalion's corpse. I'd be careful with it. Knowing the man's reputation, I'd expect it to be full of something that would do you no good at all to inhale.\" He tugged at his coat sleeves, and withdrew another handful of small objects. \"Broken porcelain, nothing to interest you there, I'm afraid. That looks to be it. What do you think? Enough for a solitary cell aboveground? That bottle of wine we discussed?\"\n\nDamastes jerked his head at the guards. One left and returned with an opened bottle, passed it to Maledicte. Maledicte sniffed, and made a face, acting the spoiled lord. \"Adequate, I suppose.\" He drank deeply, taking the dryness from his throat, the scratching sensation that the dirt was trying to crawl into his mouth. He craved the night air, fouled with fog as it was, yearned to go over and put his face to the windows.\n\n\"All right then,\" the jailer said. \"Bargaining's concluded. Guards, take him back to the common cell.\"\n\nMaledicte snarled, caught flat-footed long enough for the first guard to take hold of his arm. The second guard caught the bottle square in the jaw, and fell backward, teeth broken and bleeding.\n\nDamastes swung himself over the desk, and helped pin Maledicte, knees digging into Maledicte's back. He said, \"You're right. Sols do spend. And Echo gave me plenty of them to keep you caged with the other rats.\" He wrenched Maledicte's head up by his hair. \"If you want out of that cell, you'll have to beg.\"\n\nMaledicte struggled, clawing and kicking, until Damastes called for more guards to secure him. Even with the shock still ringing through his body that he had misread Damastes so, Maledicte growled, \"You'll be dead before I ever come begging to you.\" The jailer's hand swung around, crashing across Maledicte's face and ear. When the ringing stopped, Maledicte ran his tongue over his bloody lip, and spat the blood back at him.\n\nThey dragged him down the stairs and threw him into the cells. He crawled away from the door into a dark corner, his head swimming, his body aching, and in his chest, Ani and Miranda vying for panic. Miranda felt the corset loosening as a result of the rough handling, her bladder already protesting the water and wine, and wondered how long she could hold out, how she could repair the corset strings without attracting notice.\n\nA shadow crossed her. She raised her head and hissed; the men, allies of the earlier oaf, backed away. But she knew they'd watch and wait for their chance.\n\nAni flapped wings through him, setting his heart to racing, his blood pumping; he wanted to fly, but there was no escape from the surrounding earth and stone. He whimpered but swallowed the sound, and refused to make another.\n\nGilly would tell Janus. Janus would get him out. They wouldn't leave him here. Gilly hadn't believed him when he confessed to killing Lizette. He would come, tell him tales to soothe him, make him laugh. Maledicte sank back against the stone, felt a small impact in his forearm, and reached trembling fingers up his sleeve. The pocket watch spun on the end of the chain, catching the faint torchlight from the hall, making a small sun and sea in the dimness of the prison cell. He refused to acknowledge the pressing walls and earth, choosing to dwell on images of the sea and sky and Gilly's low voice telling him improbable stories.\n\nAni, displaced by Miranda's panic, by Maledicte's careful control, spread outward, seeking egress.\n\nAcross the room, one of the predatory men began to beat his head against the stone to the rhythm of Maledicte's imaginary oarsmen. The sleeping prisoners whimpered without waking. By the time the needs of his body sent him into knotted coils, no one was left to notice. The oaf staved his head in with a sudden last blow. Around his fallen body, his two allies stood and began to beat out the same fatal rhythm. One woman screamed, her face welting up with black bruises that burst when she touched them. People scattered away from her, shrieking, some of them already blistering.\n\nMaledicte dragged over a chamber pot and used it without worry, still imagining the blueness of the sea, and gulls reeling overhead, but sounding like rooks.\n**\u00b7 38 \u00b7**\n\n**A** T FIRST LIGHT, Gilly sought the palace, slipping through the maze to the dark side of the king's ballroom, skirting it until he saw the house servants at their morning chores. He followed a maidservant burdened by wet linens to the rear entrance of the residential side of the palace. Following her in, he was halted not by an upper servant, which he had expected, but by an armed guard.\n\n\"You're not employed here. What's your business?\" the guard asked.\n\n\"Message for Janus Ixion, Lord Last,\" Gilly said.\n\n\"You can leave a message at the front gate,\" the guard said, then scowled. \"Wait, I know you. Your master's Maledicte. I saw you going in and out of his home.\"\n\nGilly nodded when his startled hesitation made any other answer a lie. But he was dismayed at his own incompetence; he hadn't recognized the guard, though he had passed him more than once. Such notice used to be his task. He hoped the guard was less aware of Maledicte's current status, of his arrest\u2014or that, even if aware, had no reason to deny a message.\n\nThe guard said, \"Ixion's in quarters next to the nursery. You know where that is?\"\n\n\"Yes,\" Gilly lied, gambling that it was better to be familiar with the palace. He strode away, unwilling to give the guard a chance to decide that Gilly should wait, and wait, and wait for Janus. Not while Maledicte was prisoned.\n\n\"You,\" the guard said. Gilly turned. \"You take the servants' stairs.\" He pointed to the small doors Gilly had passed.\n\nGilly bent his head, and went into the labyrinthine world of the palace servants. Dark, ill-lit, and narrow, the stairs rose at a leg-burning angle, then suddenly veered. Heat flushed Gilly's skin, and he thought he must be behind a fireplace. He found himself dallying on the stairs, trying to map the castle in his mind. He acknowledged that he didn't want to see Janus at all. He shouldn't have had to, except that when Gilly had gone to retrieve the Antyrrian audit ledgers, intending to use them to buy Maledicte's freedom, he found them gone.\n\nHidden as they had been in the recesses of Maledicte's bedchamber, Gilly had no doubts that Janus had used them for his own purpose. Without the ledgers for leverage, Gilly had tried bribing the jailer directly, but the man refused his coin. So Gilly was left to beg aid from Lizette's murderer, from the man who had taken Maledicte's security for his own.\n\nCoupled with that loathing, fear crawled along his spine. If Janus had faulted Gilly for Maledicte's behavior before, what would he think now, when he learned of Maledicte's arrest?\n\n\"Be wary,\" Gilly whispered to himself. \"Be careful.\"\n\nHe found the nursery door by the simple expedient of the two guards flanking it. These guards wore mail as well as leathers, pistols as well as swords. Gilly shuddered. Aris knew the babe was still threatened. For a brief moment Gilly found himself heart-glad of Maledicte's imprisonment, the murderous plan stymied. But he had come to release Maledicte....\n\n\"Janus Ixion.\" Despite himself, he couldn't help the growl that came out. He had expected them to allow him to pass farther down the hallway, toward Janus's quarters, but the guards opened the nursery door with little more than a glance.\n\nGilly's unruly emotions, fear, loathing, and worry, gave way to a far simpler one. Wonder. So this was how royal children lived. The long room was appointed as richly as any room he had ever seen, adorned with tapestries and carpets, ornate furniture and shelves full of books tooled in gold.\n\nThe carpets piled thickly enough across the floor in a careless riot of scarlet, lapis, and gold that even the most clumsy child could find no injury in falling. At one end of the room, opened windows overlooked the gardens below, their panes barred with iron. But even the iron had been wrapped with batting to protect the children from hurt. A firescreen locked to the stone fireplace attested to more precautionary measures. At the opposite end of the room, wide doors, paned with mirrors, stood closed.\n\nNear them, reflected in bits and pieces, the heart of all these small worries, Adiran played with painted blocks, stacking them with an air of weary boredom. The mastiff beside him whuffed at Gilly, halting his approach.\n\nGilly had heard about Adiran, of course, had shared that knowledge with Maledicte long ago. Gossip about the king's son could fill every ear in the kingdom were all the rumors spoken at once, but he had never seen him so close. Disbelief shaded his thoughts. Adiran seemed hale and entire; then the boy looked up at him with such exquisite vacuity that Gilly's breath lodged in his throat.\n\nAdiran stood, and approached like an uncertain pup, cautiously pleased. Behind him, the mirrored doors flashed, scattering reflections as they opened.\n\n\"He thinks you're the servant who brings him his morning sweet,\" Janus said, standing framed within the mirrored doors.\n\n\"Oh,\" Gilly said, as Adiran reached out and tugged at his pockets, then held up an empty hand. Gilly obediently searched his pockets, finding coin, but no candy. His fingers closed on something smooth and cool, and he brought it out to look at it. The porcelain puppeteer, least damaged by Maledicte's temper, barely chipped by Her fall from the attic. He handed Black-Winged Ani to Adiran, who cupped Her wings in his hands and laughed. He returned to his building, placing the puppet atop the blocks.\n\nGilly watched the boy, horribly aware of Janus's eyes on him, of his own simmering anger in this peaceful place.\n\n\"So you came to give little Adi a toy Aris will surely dislike\u2014or is there another reason?\"\n\n\"Maledicte's been arrested. He was taken to Stones last night. Echo has seen to it that money alone will not free him; the guards turned my offering away without a moment's thought. It wants an influence that seems to have gone missing.\" Gilly turned to see the result of his blunt words.\n\n\"I'll see Echo gutted and spread on the docks for the gulls,\" Janus said, a whisper of rage. \"And you\u2014where were you that you allowed this to happen?\"\n\nGilly, unused to lying, found a lie on his lips now, a lie to serve two purposes, to shield him from Janus's wrath and a small, barbed retaliation for Janus's actions. \"With Lizette. Seeing what could be done to ease her passing.\"\n\n\"You should never have left him,\" Janus said.\n\n\"It's you he wants. Not me. And you're here.\" His voice cracked, bitter with the taste of it, and Janus curled his mouth into a smile. Gilly drove it away with his next words. \"Tell me, Janus, did you buy this position with the ledgers? Trade Maledicte's security for your own power?\"\n\n\"I had little choice. Maledicte's impatience has seen him ruined. What good would it do us to have me fall alongside?\"\n\n\"Ani rides him too fiercely for patience or reason. Your doing also, I believe.\" Goading Janus wasn't wise, Gilly thought, but he seemed unable to stop, and worse, unable to provoke the reaction he wanted: guilt.\n\n\"Maledicte and Ani are not the same creature. What he lacks in patience he should make up in trust. But he doesn't understand....\" Janus turned, looked back in at Auron's small, huddled form. \"Guardian to the earl is not so different from being the earl. By the time Auron is grown enough to take the title, well, boys of that age are notoriously careless. Carriage racing, dueling, drinking in the bad parts of town...It'll be as much a wonder if he survives his first year as a young man as it was that he survived the carriage wreck.\"\n\nGilly's breath knotted at the pale serenity in Janus's eyes, at the pleasant tone to his words. Surely there should be some outward taint, some hint of the viciousness beneath, but even knowing Janus as he did, knowing what lay beneath the mask, Gilly's first impression of Janus still lingered, a bored, amiable young aristocrat.\n\nThough aghast at Auron's coolly planned fate, Gilly refused to let it distract him from his current purpose. \"What about Maledicte? Will you free him or is he simply a casualty of your schemes?\"\n\n\"Don't be insulting.\" Janus said, frowning.\n\n\"Then you'd best go soon,\" Gilly said. \"Best sweep down on them like an avenging godling and remove Mal before they stop to think that king's nephew or no\u2014your influence is fragile, your breeding suspect, and your pockets to let.\"\n\nJanus's hands clenched, but his voice remained pleasant. \"You're aping your betters, Gilly. Trying to sound like him. You haven't the tongue for it.\"\n\nAny rejoinder Gilly would have made was stifled by the guards opening the door, not the crack they opened it for Gilly, but flinging it wide, stepping back.\n\nAris came into the room, dressed casually, breeches and linens under a dressing gown. Adiran cried, \"Papa,\" and flung himself on the king.\n\nGilly dropped to his knees, shivering, and when he looked up, Aris was watching him, startlement in his eyes, as if Gilly's master had been much on his mind.\n\nThe weariness and drawn lines of the king's face made Gilly's guilty heart turn over. How many of those lines had Gilly helped put there?\n\n\"You've brought the news, then,\" Aris said.\n\n\"Sire,\" Gilly said in agreement.\n\n\"Then you've done your duty and we need not keep you,\" Aris said.\n\nGilly bowed out and headed for the servants' stairs, shaken, and desperately worried.\n\nIn the darkness of the servants' stairwell, Gilly hesitated, seeing again the other doors. They would open into other rooms. Once, Gilly earned his pocket money by gathering secrets, by being unobtrusive and silent, by sneaking and prying himself, instead of paying others to do so. He should never have stopped his snooping, he thought bitterly now. He should have had word of the warrant signed for Maledicte's arrest\u2014but either his spies had failed or the message had been intercepted, his fault either way.\n\nGilly tried to judge which door would allow him to eavesdrop without being caught. He eased open the next door, the one that should be Janus's and therefore empty. The room was cool and dim; he listened for movement and heard nothing. He leaned up against the interior wall, near the hearth, and the white-clad maid he had taken for a curtain in the dull light made a quick squeak of surprise. Gilly put his hand over her mouth. \"Shh, I'm just here to listen. Like you.\"\n\n\"It's the only way,\" she said, keeping her voice low. \"That one has a temper on him, if you don't watch out\u2014best to know his moods well ahead.\"\n\n\"The king?\" Gilly said, though he knew the answer.\n\n\"He never notices us at all\u2014it's the bastard you've got to watch.\"\n\nGilly leaned closer to the wall, drowning her words in the rumble of voices filtered through plaster and brick.\n\n\"\u2014in charge, and his holdings seized,\" Aris said.\n\n\"With your approval? Echo is sure of himself, but not so confident as all that,\" Janus said.\n\n\"With my approval,\" Aris said, and Janus hissed out his breath. \"Maledicte cannot escape punishment, Janus.\"\n\n\"I'll secure his release.\"\n\n\"It will be impossible to do so without my permission.\"\n\nBehind the wall, Gilly gritted his teeth, trying to think of who he could bribe or blackmail and failing that, how to free Maledicte without permission at all.\n\n\"...prison,\" Janus said, in quiet tones. \"Is there no alternative?\"\n\n\"Want to watch?\" the maidservant said.\n\nGilly nodded again. She tugged at a brick, gingerly sliding it out.\n\n\"They'll see the hole,\" Gilly said, his hand on hers, halting her progress.\n\nShe put her hand to his lips, rough with brick dust, and shook her head. She pulled the brick out completely, cradled it in her apron pocket.\n\nGilly, picturing the infant's room from the narrow slice he'd seen over Janus's shoulder, recalled the extensive firescreen that spread beyond the confines of the hearth, encompassing much of the wall.\n\nHe peered through, saw the two men standing beside the cradle, their words clearer now.\n\n\"Janus, don't take on so,\" Aris said. \"I never meant his imprisonment to last. A few days locked alone in a cell, and Maledicte will be more amenable to his fate.\"\n\nSeen through the woven mesh of the firescreen, Janus's face was as still as marble, his eyes as blank as Adiran's blue-sky ones. The expression stirred familiar notes in Gilly's head.\n\n\"Enlighten me as to your plans?\" Janus asked.\n\n\"He will live,\" Aris said. \"Though for all practical purposes, he is dead to you.\n\n\"There's a village called Ennisere on the north coast. It's a cold place, and desolate, but I have comfortable holdings there. I will send Maledicte there with a competence to live on, servants to care for his whims and to watch over him. It will be a prison, Janus, but one far more pleasant than Stones. And you will never see him again.\" He raised a hand as if Janus had started to interrupt, but Gilly could see that Janus's face was as frozen as lakewater in winter. \"He should have been hanged, Janus.\"\n\nJanus let his breath out. \"May I at least take him from the cells, tell him what you've done for him?\"\n\n\"I would prefer...\"\n\n\"We fought, Uncle, the last time we were together. Will you deny me the chance to apologize to him? To leave us both with a sweeter memory?\" Janus's tendons were white in his neck, white in his hands as he dared to interrupt the king.\n\nAris turned Janus's face up to his, searched the open eyes for signs of rebellion, and then nodded. \"A brief meeting only. And my guards will go with you, should his temper hold sway.\"\n\n\"Thank you, Uncle,\" Janus said, and Gilly was horrified by the clear blueness in Janus's eyes. He had placed the memory, placed that empty exaltation in Maledicte's eyes in the moments when he turned to murder.\n\n\"Do not thank me,\" Aris said. \"I should have taken Michel's advice in this instance. Maledicte is not fit for civilized society. Take a wife, Janus, and if you crave your male flesh, take a lover, but one less disposed to mayhem and more disposed to discretion.\" His mouth firmed, then relaxed. \"Janus, it is for the best. Such a companion is not fit for a counselor.\"\n\n\"As you say,\" Janus said. His eyes reminded Gilly of heat lightning, and he wondered that the king couldn't see or feel the danger. All that rage, ruthlessly tamped down, until Janus found something, someone to release it on.\n\nGilly slotted the brick back into place, keeping his hands from trembling by the greater fear that if his hands shook, made the brick chatter, Janus would see him.\n\n\"Stay out of his way,\" Gilly urged the maid.\n\n\"No fear,\" she said. \"I've seen that look before. Do you know your way back to the street?\"\n\n\"I came by carriage,\" Gilly said.\n\n\"Aren't you the one?\" she said. \"Your master treats you well then. It's a pity he's not likely to need another maid.\"\n\nGilly kissed her cheek and she giggled at him before ushering him down the stairs. Each step down, Gilly thought giddily, was one step closer to Maledicte's side. Aris meant to send servants of his own, but surely Gilly would be allowed to attend Maledicte's needs in Ennisere. Still, his happiness was bittersweet; Gilly knew Maledicte would rail at the prison, no matter how fine the cage, and Ani would drive him mad, and as for himself\u2014his distant dream of the Explorations died in his chest.\n\nHe rested against the dark walls, trying to sort the mixture of relief and glee, of pain and dismay, of fear and doubt into some more palatable sensation, and failed. Long minutes later, he let himself out of the servants' stairwell and headed for the stables.\n\nAt the carriage, the door was open. Gilly hesitated at the unexpected sight, and while he did so, Janus stepped out. \"You took your sweet time,\" he said. \"Get in.\"\n\n\"I'll walk,\" Gilly said, mistrustful of Janus's smile.\n\n\"Don't you want to help Maledicte?\" Janus said. Again the storm flickers washed his eyes.\n\n\"I don't see the guards Aris spoke of, your escort to the prison,\" Gilly said.\n\n\"Eavesdropper,\" Janus said, without heat. \"But you dally. I thought you'd be chafing at the bit, ready to seek banishment with him all to yourself.\"\n\nGilly stifled all reply, mistrusting that hot light still luminous in Janus's gaze.\n\n\"Do come on. I have errands aplenty. Before we release Mal, lock him into the Kingsguard's care, I want to go to the town house to take what we can salvage. The king's competency is likely to be adequate, but Maledicte is most particular.\"\n\n\"He'll want the sword,\" Gilly said, thinking of it left waiting for Maledicte's return.\n\n\"Didn't Echo take it from him?\" Janus said.\n\n\"Maledicte gave it to me for safekeeping,\" Gilly said.\n\nJanus fingered his own sword thoughtfully. \"Well, I'm not one to take his toys from him. We'll collect it, and load the carriage with his possessions while we're at it. Or do you want to explain to him how he comes to be a hundred miles from the nearest tailor and without his favorite vest?\" Janus climbed back into the carriage, leaned against the seat, and said. \"Go up and drive, Gilly.\"\n\nGilly, relieved not to be closed in the carriage with Janus, did as he was told. The town house stable, when they arrived, was emptied of horses. The door to the house was marked with Echo's seal, but Gilly ignored it. In the entry hall, Maledicte's sword rested on the marble table where calling cards usually littered the surface, as if it too waited a response.\n\nJanus picked up the sheathed sword, swearing as the feathered hilt bit through his thin gloves. His voice echoed in the house, striking no response from the shadows. When Gilly looked into the kitchen below, Cook's belongings had gone. He wondered who the little maids would work for now, wondered if Livia had smelled this coming, as cunning as a rat, and had found herself a new place.\n\nJanus came into the kitchen after him, his boots ringing on the stone floor. \"Tell me something, Gilly. How could Maledicte give you the blade if you weren't here?\" The sudden storm feel of the room caught Gilly by surprise. Janus thrust Ani's sheathed sword at him hard enough to break ribs; Gilly flung himself backward, tripping over the raised brick hearth.\n\n\"You allowed his capture and blame me for it,\" Janus said. \"He'd be free now, fought through them all, but for you\u2014how did you manage it? Did you drug him again?\"\n\nGilly said, \"They would have killed him, Janus. This way he lives.\"\n\n\"Killed him, when he heals, when poison flees his blood? I have my own plans set in motion, and you had to interfere. You and he go north, banished together? Mal said you were intelligent\u2014did you plan this? Itarusines make long plans, and Vornatti had the training of you\u2014\" Janus dropped the sword, unsheathed his own. Gilly grabbed the abandoned rolling pin, and took the strike on its marble surface. The sword skidded, shrieking, and Gilly pushed back, throwing the pin at Janus.\n\nWatching the blade, he missed the bare-handed blow that hit his neck and shoulder, stiffening them into instant pain, and hurling him off balance. In desperation, Gilly threw himself forward, landing a blow of his own that split his knuckles and Janus's lip. Rage swept him and he forgot he was facing a man with a sword, bent on extracting at least a small measure of Lizette's pain from Janus.\n\nJanus's head rocked with the blow; he spat blood at him, and said, \"Fool. You'll break your hand before you hurt me that way.\"\n\nGilly punched out again, and Janus used his momentum against him, letting the rush take them both to the ground among a smashing of the cook's old chair. Rolling to land atop, Janus put his knees in Gilly's belly, bearing down, his hands sliding around Gilly's neck, the sword dropped and forgotten.\n\nAlready breathless from the exigencies of the fight, Gilly began to gasp in earnest. He pried Janus's hands away, doing his best to break the thumbs, and Janus let go. Gilly sucked in air, tried to push Janus's weight off of him, and barely avoided the elbow aimed at his face.\n\nJanus's hands wound through Gilly's hair, and pounded his head into the floor, then the edge of the hearth. The room reeled, spinning into a moment's dull blackness; his vision cleared to Janus risen above him. Janus kicked him in the jaw, setting off another bout of spinning dizziness. Gilly knew he had to rise\u2014another blow caught his shoulder as he tried to roll to hands and knees, tried to reach either sword.\n\nThe next kick cracked ribs and dropped him to his belly. There was blood in his mouth and dripping into his eyes; Gilly crawled up again, the kitchen spinning and dipping as if it were a galley in a shipwreck and not one safely at shore.\n\nThe explosion of pain in his side staggered him. He kept his balance, but barely, trying to pull himself up the table legs, wondering when Janus was going to remember the swords. The next blow, to the side of his knee, sent him writhing to the floor. Vision tunneling and clearing, pain a tide washing over him, he could barely make out Janus standing above him, incandescent as flame, grinning.\n\nGilly knew he was dead; it gave him breath enough to say, \"Maledicte\u2014\"\n\n\"I'm going to give you to the sea. He'll think you abandoned him, took his coin and fled to the Explorations. He'll hate you for it,\" Janus said.\n\nGilly struggled to his feet, rested his hands on his thighs, his right leg buckling, and said, \"Rot you, he'll know otherwise.\"\n\nJanus drew back a moment, and Gilly stumbled toward the kitchen door and outside, hoping for a witness, even for the Kingsguard or Echo. A faint rasp of metal set him to fumbling for the handle, the iron slick in his palms, when Janus stepped behind him, as patient and as mad as an outcast wolf. He raised his sword. Gilly closed his eyes, whispered, \"Mal.\"\n**\u00b7 39 \u00b7**\n\n**A** S THE DOOR TO THE communal cell opened, Maledicte looked up from his seat on the corpses of those who'd died overnight. Damastes slammed the door shut again and Ani, who'd seen the dark welts on his face, laughed through Maledicte's throat, flecking his lips with blood. Above him, in other cells, people screamed and wept as Ani's glee rose through the darkness and touched their dreams.\n\nThere were rough sounds of argument in the hall and then the door opened again.\n\n\"What a mess you've made,\" Janus said, holding the keys in a casual hand. \"Damastes is cowering in his quarters, muttering about rat fever and devils; the kingsguard refused to come inside at all, and here you sit, laughing.\" Though insouciant, his voice held a hint of tremor.\n\n\"Janus.\" The name worked some of its old magic, driving some of the madness back; he fled his throne of corpses, belatedly repulsed.\n\n\"I brought your sword,\" Janus said. \"Thought you might like to come out and use it.\"\n\nMaledicte joined Janus at the door, each slow step returning him to himself. He took the sword in his hand, grimaced at the blood on his skin, and said, \"I hope Gilly has a bath run. And despite his wishes, I am never wearing gray again, it's far too funereal.\" He forced the words out, trying to collect the courtier's mask about him, but finding that it didn't fit as well as it had.\n\n\"Are you unhurt?\" Janus asked.\n\n\"I am,\" Maledicte said. \"Some of these others cannot say the same. And I want my belongings back.\"\n\nJanus drew him into the hall, folded him into his arms. \"That jailer\u2014Damastes, he didn't find out?\"\n\n\"No,\" Maledicte said. He shivered in Janus's arms. \"Let's go. I want to see the sky.\" Tears streaked his face, ran through the dirt and blood; he only noticed them when they trickled into his mouth, bitter with dust.\n\n\"Of course,\" Janus said, kissing Maledicte's mouth, delaying their exit.\n\nMaledicte leaned against Janus, smelling the clean heat of the sun on his skin, tasting the sweetness of his tongue against his own. It pushed more of Ani's madness away, increased his shaking. \"Janus, call for a physician.\"\n\n\"I thought you unhurt?\" Janus held Maledicte at arm's length.\n\nMaledicte shrugged under that piercing gaze. \"For them. I don't know what I let loose in there.\"\n\n\"It's an outbreak of rat fever. Common enough in prisons. You're no witch, Mal, god-driven or not. And as for them? They're nothing,\" Janus said. \"They would have rotted here regardless. Come now, Mal, shelve such unreasonable concerns and dry your tears\u2014or do you want Damastes to see them?\"\n\nMaledicte let his breath out in relief as Janus reminded him of an enemy to face, shunted the poisonous guilt back, let Ani dissolve it with the clean heat of Her hatred. \"He put us underground.\" His fingers tightened around the sword hilt; his mouth drew into a hungry grin.\n\n\"Mmm,\" Janus said. \"Why don't _I_ go talk to Damastes, get your things back? Let you wait in the carriage.\"\n\n\"The Kingsguard,\" Maledicte said, the words filtering through slowly, as if he was still half lost in nightmares. \"Why are they here?\"\n\n\"Did you truly think there would be no penalty?\" Janus said. \"The town house is sealed against you. They are here to escort you to a hotel, and to make sure you don't leave it. We're just trading one cell for another.\" Bitterness seeped through his voice.\n\n\"I cannot live in a hotel forever,\" Maledicte said. \"What has Aris planned?\"\n\nJanus urged him up the stairs without answering, and Maledicte, sheathed in dirt and stone, was willing to allow evasion, eager to make the sky his own again.\n\n\"Look there,\" Janus said, laughing. \"Damastes is not such a fool as all that.\" The piled belongings near the door sparkled in the low light. Maledicte swept them up into his hands, then passed them to Janus, preferring to keep his blade ready.\n\nMaledicte stepped out into afternoon sunlight and winced. The Kingsguard standing beside Last's carriage stood to attention, then drew back as they saw the naked blade in Maledicte's hand.\n\n\"Where's Gilly?\" Maledicte asked. \"I thought sure he'd be here.\"\n\nJanus helped him into the carriage, and Maledicte picked up the sheath lurking on the seat cushions. He buckled it on and sighed.\n\nJanus gave the coachman the signal to go, and settled beside Maledicte. A kingsguard passed alongside the window, and Maledicte put his hand on the hilt of the sword.\n\n\"Take these back,\" Janus said, distracting him from contemplations of flight and murder.\n\nThe scatter of small stones and coins made him release the sword so he could catch them before they tumbled from his lap. Moodily, Maledicte sorted buttons from cuff links, stickpin from coins. The pocket watch fell into his fingers again and he pulled it out, setting it to spinning in the sunlight. \"You never answered me. Where is Gilly?\"\n\nJanus's silence went on a moment too long, long enough for Maledicte's interest to turn to concern. \"Janus, tell me.\"\n\n\"I haven't seen him,\" Janus said, tapping the watch in Maledicte's hands, making it swing. \"When Aris told me of your arrest, I went to the town house to collect your belongings. The house was empty. No one had stayed behind\u2014all the rooms were stripped. Your accounts too, undoubtedly. I told you your trust was misplaced.\"\n\n\"Gilly,\" Maledicte whispered, clutching that sudden hurt to his heart. Ani, Her attention diverted from the sky by his pain, turned the hurt around, studied it, and let it drop. There was nothing to be mined; Maledicte had already replaced the pain with wariness and hope. Gilly would return.\n\n\"He's fled,\" Janus said. \"Count on it. Gone to the sea as he threatened to do so often.\"\n\nJanus studied the guards maneuvering outside, his expression hidden. Maledicte turned Janus to face him, stared into the guileless blue eyes, and felt his heart constrict. Roach, Celia, and Ella\u2014all had been helpless before Janus. \"Did you...did you kill him?\"\n\n\"Burn it, Mal,\" Janus said, irritation drawing his brows down, his lips thinning. \"I begged Aris to free you, swore promises I hate to keep, and all you can ask is if I've killed your servant? I did not. Likely he's decided that our ways are too rough for his tender heart.\"\n\n\"Lizette died,\" Maledicte said. \"Seemingly at my hand. Why did you do it?\"\n\nJanus cast another glance outside the carriage, at the kingsguard nearest, and leaned forward. \"You know why. To punish Gilly, since you won't let me lay a hand on him. You should be glad of my restraint. And why you sought the brothel in the first place\u2014\"\n\n\"You murdered Ella, and kept it from me...I despise secrets from you,\" Maledicte said, waiting. When Janus only shrugged irritably, Maledicte asked, \"What is it that Aris has planned? You seem remarkably loath to mention it.\"\n\n\"Ennisere,\" Janus said. \"You're to live out your time there, on an estate staffed by guards.\"\n\nMaledicte thought of maps and distance, but his knowledge was sketchy. Vornatti had taught him about the city and its fashionable retreats. Janus had told him about Itarus, and Gilly had sweetened his dreams with descriptions of the Explorations. Ennisere meant nothing, a foggy blur on an unfinished map of the world. \"What of you?\"\n\n\"I stay at Aris's side, and work to further our plans.\"\n\n\" _Your_ plans,\" Maledicte said. \"My plan was always simple, god-guided. Kill the earl of Last, and reclaim you. And I have yet to do the first. That child survives\u2014\"\n\nJanus said, \"Listen Mal, listen to me. I have my schemes. You're correct. Maledicte is ruined. So let Aris send you to Ennisere, bide your patience only a little. I know of a black-haired boy with pale skin, a poor mirror of you. We'll kill him, leave his body at Ennisere, and you can become Miranda again, and return to my side.\"\n\n\"You're a fool,\" Maledicte said. \"Miranda with a ruined voice, a distinctive scar, and no antecedents? You may play puppets with the king but he is not so mindless as all that.\" He could not keep the threads of his argument together, losing them in the pale calculation in Janus's eyes, the clatter of hooves outside the carriage, the line of blood marring Janus's mouth. \"You're bruised.\"\n\nJanus touched his mouth. \"You struck me, don't you recall?\"\n\nThe blood was fresher than that, Maledicte thought, but that too was sucked away in the skirl of feathers within him. Above the coach, the rooks swarmed, darkening the sky prematurely with their wings. \"Where's Gilly?\" he asked again.\n\nThe coach drew up to the hotel; the horses milled uncertainly as the kingsmen conferred. Finally, two guards dismounted, flanked Maledicte as he and Janus went up the front stair. At the desk, the owner made a surreptitious charm against evil, and Maledicte smiled at him, showing all his teeth.\n\n\"The second floor,\" the guard said. \"Go ahead of us.\"\n\nMaledicte walked into the rooms without protest. The quarters were roomy enough, a bedchamber, a valet's chamber, sitting room, and bath. He peered out the window, drawing back the curtain. \"No balcony. No trellis.\"\n\n\"It's a prison, Mal,\" Janus said, taking a seat on the bed, and waving the guards out irritably. They shut the door, but Maledicte could hear the faint jingle of their mail as they leaned against the wall.\n\n\"So it is,\" Maledicte said, dropping the curtain. \"When am I transported north?\"\n\n\"Tomorrow,\" Janus began and Maledicte growled.\n\n\"So soon?\" He paced the room, boot heels muffled against the carpet, the blade swinging freely. \"For how long?\"\n\n\"Until Aris\u2014\"\n\n\"What? Until Aris dies\u2014\" Maledicte's voice rasped in the quiet room, and Janus pressed his hand close over his mouth.\n\n\"Hush,\" he said. \"The guards are just outside.\"\n\n\"You're taking it all from me,\" Maledicte said. \"I wanted the earl dead and you denied me, and I wanted you. Now you're walking away, because the king asks it of you. Why can't we just flee? Kill the guards and run for it?\"\n\n\"A Last doesn't run, he conquers,\" Janus said.\n\nMaledicte let his breath out in a hiss. \"You choose playing for power over me.\"\n\n\"Not over\u2014\" Janus said. \"With. I want both. You must be patient. Let me plan since your sense seems to have been buried with Amarantha. Trust me. I'll win through. See us both rich and powerful.\"\n\n\"It's all gone wrong in my head. It's all beaks and wings and blood.... Where's Gilly? He can make it better,\" Maledicte said, slumping back onto the feather mattress.\n\nJanus kissed his forehead. \"You're overtired, overwrought. You should never have had to go to Stones.\"\n\n\"Not underground,\" Maledicte said. \"Wings want sky.\"\n\n\"Not anywhere within those walls. But your discomfort will be repaid. I promise that.\"\n\nMaledicte nodded, the words washing over him like the empty chatter of songbirds, soothing but meaningless. He let Janus undress him like a child, stood docilely in the hip bath while Janus sponged the filth of Stones from his body. He tangled his hands in Janus's pale hair, kissed his mouth, and let his mind drift away entirely. Janus laid him over the bed, kissing, stroking, soothing, and Maledicte clutched him close. When they were done and dressed, Janus gone, Maledicte sat by the window, staring at the sky.\n\nGilly kept creeping into his mind, the earnest eyes, the worried half frown that had become his common expression; his image was displaced only by Janus, and the slow ache that grew inside Maledicte. He couldn't keep them in his mind at the same moment; when he tried, all he saw was blood.\n\nAcross the room, the sword muttered and whispered until he cradled it in his lap. \"I will, I promised you. In exchange for the sword. I'll spill his blood yet.\" Outside, the rooks settled atop the hotel, their chatter quieting.\n**\u00b7 40 \u00b7**\n\n**R** OCKING WATER, AND THE STINK of salt brine and tarred ropes, woke Gilly. He opened his eyes to a room made of shadows and lapping water, fractured and shivering with the pulsing of his aching head.\n\nAlive. Why? Gilly wondered. There'd been murder enough in his face and strength enough in his hands.\n\nGilly tried to raise himself on limbs that were too numb to support him, and fell forward, splashing face-first into dark water. Panic woke him from his stupor. He scrambled back on unwieldy legs, sucked in air, and reassessed. His hands were knotted in a nest of twine and hemp, his ankles likewise. He was in the bilge of a ship. Gilly let out his breath in horrified understanding. A conscripted sailor.\n\nWas death not enough for Janus; was it suffering he wanted? In the dark hold, dizzy and sick, surrounded by dank water and the strange oil scent of piled metal, Gilly found himself thinking with a clarity that surprised him. He'd been sold for a luna or two to line Janus's pockets, and more, the ability to tell Maledicte that he hadn't killed Gilly should Maledicte ask. Gone to sea finally, Gilly thought, and shuddered. Janus had piled the irony even higher; the strange metal shapes could only be bound for the Explorations, to build one of Westfall's engines there.\n\nHe started picking the ropes apart with his teeth, the tar and sodden hemp making him gag. They were still near shore; the slapping of the waves against pilings and other nearby hulls told him that. He had friends on nearly every pier, sailors, harbor clerks, dockworkers, who might aid him. If only he could get free....\n\nThe shadows in the bilge massed and roiled as if they were water, stirred by an unseen tide. In the distance, Gilly heard a crow's call carried on the shrieks of gulls. The shadows seemed to vibrate to its resonance; the pain in Gilly's head crested and blurred his vision.\n\nHe chewed diligently at the knots linking his hands until they gave, but didn't fuss himself with the tight, salt-sodden loops left about his wrists. Though they chafed and burned, they could wait. He bent to work on the cords around his ankles and a rook flew out of the shadows on silent wings.\n\nIt landed on a jut of scrap metal, its talons making no sound as they contracted. Its eyes were matte black, as empty as a doll's, lacking the shine of a living creature's, and Gilly swallowed. It opened its beak, fluffed its wings, and bloomed bigger, a crow now, birthed of shadows.\n\n\"Mal\u2014\" Gilly whispered. The bird fluttered to the edge of the bilge, to the narrow ladder that rose to the deck and freedom. It fluffed its wings, again, and waited, rasping its beak against the splintered wood.\n\nGilly bent back to the ropes at his ankles, though keeping his head down increased the spinning languor of his body. He wanted nothing so much as to lie down. Instead, he dragged himself to the other side of the bilge and the aid implicit in the metal scraps. The right tool would be quicker than teeth surely, and far more efficient than fingers numbed by swollen wrists.\n\nThe ropes parted, surrendered strand by tarred strand, shredding with maddening slowness. When Gilly looked to share his triumph with the crow, he was alone. A flicker of movement pulled his attention upward.\n\n\"Mal\u2014\" he breathed again. The bird-shade, caught in midtransformation, flopped, wings unwieldy, folding inward, stretching itself tall and thin. A familiar human shape darted up the last rungs of the ladder, pausing for a bare moment to look back before flowing out onto the deck.\n\nGilly dragged himself to the ladder, up the first rung, sweat collecting on his abused body, chilling him like a layer of hoarfrost. His senses reeled and swam, nearly deserting him entirely. He felt as if he wandered in a dream.\n\n\"Mal,\" he whispered. Was it Last's window he was climbing to, hunting the nameless boy, ivy brittle under his gloved fingers, and snowmelt refreezing in his eyelashes, making him blink cold tears? Or was it underground, going further back, before everything he knew, following a staggering boy, newly hatched from Ani's wings, the sword naked and gleaming in his hands as he climbed into the Relicts.\n\n\"Mal,\" Gilly repeated, pulling himself up another rung, chasing that delicate phantom. Time stopped, sped up, shadows and light shifting across Gilly's vision left him standing at the king's palace, looking up at the high tower, at the slim shape, as black as the blade, standing at bay. Gilly reached up to climb to his aid, and his hand struck empty air. The salt smell of the sea woke him from his dreaming. \"Mal....\"\n\nNo vision this, but a fate he wanted to escape. The foredeck bristled with sailors, drinking away their last hours ashore, telling each other stories, and repairing the fishing nets that would keep them fed on the long journey. The gangplank lay stretched to the pier before them, for easy access to the Relicts' bars and whores.\n\nOnce Gilly would have considered the crew good company. Now, he could only think of them as enemies, and all he could hope was that they had drunk enough to be careless. But so castaway as to watch him escape before their very eyes? He doubted it.\n\nGilly clung to the top of the ladder, leaned his head against the salt-scoured planks, watched the sun burning down into the sea, setting shadows roaming over the deck. He had lost all sense of time. Were it not for the spider constellation sparking to life in the sky, he could believe he'd slept for years, lost in the bilge.\n\nThe shadow, _his_ shadow, divorced itself from its brethren on the deck, and flowed toward him. Not so human-shaped now, it bled outward like watered ink, growing fuzzed around the edges. It wafted toward him, swallowed him in an embrace chilly and dank, and a voice breathed into his skin, like no voice he'd ever heard before. _Hurry._\n\nStaggering like a drunkard, Gilly gave his fate to the shadow and wandered toward the sailors, toward the gangplank with its lure of safety beyond. Though it made his flesh crawl, and his heart pound as hard as his battered head, he made his way past the sailors and to the gangplank. They made no sign that they had noticed anything out of the ordinary way, not even when the worn plank sagged and moaned beneath his weight.\n\nThe water below him churned in odd eddies, dark and flecked with luminous foam, splashing upward toward him. He fixed his faltering vision on the pier, and at the end of it, waiting by a coach, a pale face in shadow.\n\nMaledicte, Gilly thought on a crest of relief, come to take him home, and showing a rare subtlety for once, coaxing him from beneath the eyes of the sailors, rather than forcing Gilly's freedom at swordspoint.\n\nThe water beneath him surged, a sudden high tide rising as he descended, and it slapped salt water over his feet, his ankles, and burned the shadow away. Naga's touch inimical to Ani's uncommonly delicate working.\n\nGilly urged himself onward, finally reaching the salt-weathered planks of the pier. He stumbled, pushed himself to his feet, concentrated on walking normally. With the cloaking shadow gone, he thought the illusion might have gone with it. The dark sky might hide his identity, but he was still too close to the ship to be anything but their prisoner escaping.\n\nThe shout went up, and Gilly staggered into the closest thing to a run he could approximate, a listing, limping thing that set his head and ribs to throbbing, the world shuddering like an opera curtain, whisking back and forth.\n\n\"Gilly,\" a low raspy voice called, \"hurry.\" Reaching the end of the pier, he found cool, smooth fingers on his arm; the pursuing captain drew to a halt.\n\n\"Lady,\" he said, wary.\n\nLady? Gilly craned his head to look but was defeated by the dizziness. The rasping voice took on a clear sweetness that Gilly had heard before. \"Why ever are you hunting my servant? Has he been brawling with the crew?\"\n\n\"He's mine. Four lunas he cost me.\"\n\n\"Forced labor is illegal,\" she said. \"Such a shame, too.\" Gilly tried to tug free; her nails slid into his skin, waking new pains, and Gilly subsided.\n\n\"Purchasing a man's services is not.\" But the captain's voice already faltered. Gilly, his eyes drifting, found himself staring at a sweep of tattered silk, stained dark around the hem. A ruined ballgown.\n\n\"When those services are already promised\u2014\"\n\nGilly moaned and she halted herself with a wild laugh. \"And here I am going on as if I need to win by words. He's mine, Captain. Do not argue further. I am most unpleasant when offended.\" As verbose as Maledicte, he thought, teeth chattering. But far more inimical to him.\n\n\"But still, you lost coin, and I know how dearly money can be needed. I'll repay you.\" She threw coins at the captain. While he scrabbled for them, keeping them from rolling through the cracks between the planks, she said in a tone like exposed steel, \"Any further complaints?\"\n\n\"No, my lady,\" the captain said, still kneeling, shivering. He knew who she was now, Gilly thought. Even the sailors had heard the tales of Mad Mirabile.\n\nMirabile laughed, the sound not as pleasant as it once was, like a bell cracked and off tune. She walked Gilly toward her carriage like a marionette. Wordless, he sprawled on its floor, dripping salt water and blood. She tangled her fingers in his wet hair, dragging her nails across his scalp, setting the long gash from contact with the hearth to bleeding again. \"So Ixion finally removed you\u2014or was it Maledicte who cast you into the sea?\"\n\nGilly winced, but did not reply, concentrating on regaining his equilibrium with the sway of the moving carriage.\n\n\"No answer, and I've gone to the expense and effort of saving you. I suppose that means you're not grateful either.\"\n\n\"Let me go,\" Gilly said, sick at heart. He'd followed her lure as blindly as a hound on scent, thinking only of Maledicte.\n\n\"You'll serve me now,\" she said.\n\n\"No,\" Gilly said.\n\n\"You will,\" she said. \"In one fashion or another. I've waited for my vengeance too long. Ani's beak has grown sharp, and I would share that pain with others.\"\n\nHe jerked away, reaching for the door handle, hoping to tip himself out onto the cobbles. The handle writhed in his hand, supple and scaled like a serpent, coiling around to strike him, and he let go in sheer horror. She laughed and he turned toward her, spoke the words of Baxit's countercharm. She winced, then slapped him across the face, sending him to the floor again. She slid closer, put her hand beneath his chin, forced his head up. \"Such a waste,\" she said. \"A comely young man doomed because of one man's refusal to share himself with me. I'd feel sorry for you. If I could.\"\n\n\"Maledicte will kill you,\" Gilly breathed.\n\nShe leaned closer, confiding. \"Black-Winged Ani granted your master a sword. She saw to it that I would never need one. Confused, my sweet Gilly? Shall I spell it out for you? She granted me power....\" In her eyes, red fires flared and sank back to a simmer.\n\nGilly turned his head away from the madness in her gaze, and she dragged it back, effortlessly. \"Look at me, Gilly. Am I not more beautiful than your master? More beautiful than those foolish debutantes?\" She rolled her fingers together, opened her palm, and blew dust into his face. Coughing, he tried not to breathe but the stupor in his head settled into his bones.\n\n\"You do love me, don't you, Gilly?\" She touched her lips to his; he shivered all over and felt the heat scorch from her mouth to his groin. \"Tell me you love me.\"\n\n\"Love you,\" Gilly said, the words dragged from his throat.\n\n\"You'll love me until the day you die....\"\n\n\"Yes,\" Gilly said, his heart pounding under the twin stresses of fear and lust.\n\n\"More than you love him,\" she said.\n\nGilly closed his eyes. Maledicte. The image, dark hair, dark eyes, soft mouth against his own, did nothing to cool his body or his fear. Her nails tightened on his face, and he said, \"Yes.\"\n\n\"I'll let him know you said so, when I gift him with your body. Let him see what it's like to lose someone through the caprice of another.\"\n\n\"\u2014loves Janus more...\" Gilly said, as her mouth descended on his.\n\nShe drew back. \"I'll have him later. But Maledicte must come first.\"\n\nThe carriage drew to a halt, tumbling him into her skirts. \"Clumsy thing,\" she said. \"I'll expect better of you.\" She pushed him from the carriage; he got his feet under him just in time, and stood there, swaying. They were deep in the heart of Sybarite Street, past the brothels, beyond even the insalubrious dens that specialized in drug dreams and poison selling. This section of Sybarite bordered on the Relicts, the buildings more fallen than run-down. Still, if he fled, he could get to Ma Desire's, maybe to safety. If he could move.\n\nMirabile took his hand in her cold one, tugged him into movement like a puppet. Mirabile's coachman slipped off the driver's bench in a flurry of skirts and cloak, a familiar tail of red hair and brown eyes: Livia. Betrayed rage gave Gilly momentary strength, and he pulled away.\n\nMirabile snarled, \"Stop.\" His limbs locked up at her word. Livia drew her hood up about her face, and edged past him, shifting piled-up boards to reveal a low, dark opening. The ruined building looked as if no one but rats could fit within, yet with the opening revealed, Gilly saw clear rooms inside.\n\n\"Well,\" Mirabile said, guiding him in, \"Welcome to my parlor.\" His shocked gaze recognized the place, even as he started to shiver. The walls were covered with Her image; Mirabile dwelled in the ruins of Ani's temple, slept in the lee of Her wings. Livia lit lamps around the room, each one revealing another depiction of Ani. Some of them smelled new, smelled as if Mirabile had painted their rough shape with blood. On the altar itself, a dark shape muttered and croaked at their return.\n\nWhile he stood numb and helpless, she drew off the ruins of his shirt, his breeches, and smiled. \"Don't look so frightened, lambling. I'm not going to kill you right away.\"\n\nMirabile circled Gilly, her expression as proprietary as Vornatti's had ever been, and far crueler. Gilly felt fourteen again, remembered the dread washing over him with the soapy water, the dull light in Vornatti's eyes growing brighter with each limb washed clean. But the dread then had been fear of the adult world pressing in on him; he had trusted Vornatti not to hurt him. Gilly had no such illusion with Mirabile, not with her nail marks bleeding sluggishly on his cold flesh, or the hunger he saw in her face.\n\nLivia, after another averted glance, busied herself lighting the rest of the gas lamps, as silent as she never had been in the town house. The small flames caught and flared under her shaking hands, illuminating wet streaks on her face. Throughout her task, she twisted her head to avoid meeting Gilly's eyes.\n\n\"When you're done, Livia, you may go. Unless I am mistaken, you have no desire to watch me at my play.\"\n\nLivia shook her head, so mute that Gilly imagined atrocities\u2014that Mirabile had torn out Livia's tongue, or bespelled her to a future as a slave.\n\n\"Come back for your coins in the morning,\" Mirabile said. \"I'll need you to do the washing up, after. But don't return too early. I intend to be about this business until the late hours.\"\n\nLivia flinched; her eyes met Gilly's for a brief, scalded moment, then blurred and ran with tears. She left with a deliberate pace, as if she wanted to run, but controlled herself.\n\nNot enough fear, Gilly thought, and Mirabile would kill her. Too much fear and the result would be the same. Like a predator, Mirabile would hunt the fleeing creature out of instinct. Weakness spilled through him, and he slumped, unable to fall while her potions and will held him upright.\n\n\"She'll go for coin tonight. Go for Maledicte,\" Gilly said. \"Greedy little girl.\" Each word was an effort to push out through his stiff tongue and lips.\n\n\"I do hope so,\" Mirabile said. \"I doubted her for a moment there\u2014thought I might have to send a messenger less trustworthy, or one that Maledicte might kill on sight, and then where would I be? Without my audience.\"\n\n\"Maledicte's in Stones,\" Gilly said, finding a sudden perverse pleasure in the fact that had troubled him so greatly earlier.\n\n\"Was in Stones,\" she said. \"You used to be better at keeping abreast of the gossip, Gilly. The rooks have all moved again. They follow him, you know.\" She drew her hands along his flanks, trailed inward; his muscles jumped and flinched at her touch and she smiled. \"They nested at Stones while he was there, and now his little birds have flown to the Grand Hotel. They darkened the sky like a whirlwind. All that power at his will, and he refuses to reach out and grasp it. He could control them, their eyes, their secrets, would he only admit complete fealty to Ani.\n\n\"But no matter,\" she said, \"That he fails to reach his power is only of assistance to me. But think of it, Gilly, what a sight it would make, Maledicte in the ballroom with the rooks wheeling about him, calling and excreting over all the nobles.\" She grinned; were it not for the mad eyes, Gilly could have enjoyed the mischief in her face.\n\n\"A pity it will never happen,\" she said, laughing, and wrapped her hands firmly around his genitals.\n\nHe tried to force her hands off him, but she squeezed and his breath went short with unwilled pleasure. Her nails sliced into the tender flesh and he cried out, the pain lancing over his body, then settling back into steady throbbing.\n\n\"Gilly,\" she said. \"Take up my skirts.\"\n\nChary of her grip, her touch, he knelt, breathing more easily as her hand slid away to allow his descent. He folded her draggled skirts up about her waist. Under the finery, where the noblewomen wore their slips and petticoats, their lawn chemises, where even the poorest maids wore pantaloons, she was bare. Just above her sex, above the flame of hair, feathers had burst from her skin, small and black. At first he thought she had decorated herself as an honor to Ani, but when she urged his hands to her skin, he knew it was the inverse, that Ani was decorating her.\n\n\"My bodice,\" she said. He reached behind her; she knelt before him, pressed her hips to his as if she was nothing but an eager lover. Her bodice fell loose in his hands; she shrugged it from her shoulders, baring more white flesh, patterned with tiny black feathers so small they seemed like scales. He gasped, his hands flying away. She grabbed them, pressed them to her breasts, sank herself onto him. He groaned.\n\n\"You love me, Gilly.\"\n\n\"Yes,\" he said, her words in his mouth. His own words drowned as she rocked herself over him.\n\nIn his head, he began whispering prayers though the only god present was Ani. Her teeth bit into the welts left by the barnacles on the pier, raised blood again; her nails dug into the deep bruises left by Janus's fists, scribed the edges of his raw wrists. She tongued the wound on his head, lapping the blood until it stopped, then biting until it bled again. His prayers dissolved into one internal plea. Maledicte.\n\nMALEDICTE PACED THE ROOM, agitated without cause. He had the sky now, through the high windows, and yet...the sword throbbed in his hands, seeking blood.\n\nA tap on the door sent him spinning around, sword bared.\n\n\"Sir, I've brought your dinner.\" The girl's voice, though tight with tension, was familiar.\n\nMaledicte drew open the door; the guards stepped back, out of reach of his sword, too cautious to let him use the maid as a distraction. One guard spoke. \"Are you certain you want to go in with him, miss?\"\n\n\"He's my master,\" she said. \"I brought the food from his own table, what's left of it, and he'll be hungry.\"\n\nThe other guard shrugged. \"It's your neck.\"\n\n\"May I go in?\" Livia asked. \"You've already looked me over, peered in the bowls. You know I have nothing to aid him.\" She shifted the heavy tray on her hip, and the guard nodded her in, latched the door behind her.\n\nMaledicte watched her red hair slide over the shoulders of her damp cloak like a scarf. He raised the sword and brought it winging to her nape, halting it at the very last.\n\nShe gave a stifled shriek, too frightened to move. Then the long braided tail of her hair slithered to the floor, cut. He picked it up. \"Get undressed,\" he said. \"And don't think of crying out. I've no need to hurt you but I must find Gilly.\"\n\nHer skin paled white as marble; her mouth worked, soundless. Maledicte read the word on her lips. \"Gilly?\"\n\nBehind his cold rage, the hunger, something as warm as baked bread rose, soothing his temper, then settled back into rage. Anything Livia knew, with her eyes like a dead woman's, was not going to please him.\n\nLivia licked dry lips.\n\n\"If you don't find your voice, I'll hunt it with my sword,\" Maledicte said.\n\n\"You have to help him. Mirabile will kill him.\"\n\nMaledicte put his hands around her neck, found a vicious satisfaction in making her flinch, and undid the knotted strings of her cloak. \"Get undressed,\" he said again.\n\n\"They'll never believe it\u2014\" she said, fumbling her bodice off, her skirts.\n\n\"They don't have to for more than a moment. It's vision driven by expectation, but never mind all that,\" he said, tugging her skirt up over his breeches, watching her blink astonishment when the buttons closed around him. \"Your bodice,\" he said. \"Your cloak.\"\n\n\"If you're taking my cloak, I don't see why you need\u2014\"\n\n\"Because a cloak over breeches looks like a cloak over breeches, and a skirt is an entirely different thing,\" he said. \"As glad as I am you found your voice, now I want you to be silent again.\"\n\nShe stood stripped to her chemise, shivering in the chill room.\n\n\"My dressing gown,\" Maledicte said, motioning to the bed and the heavy drape of quilted fabric lying across the bottom. \"Put it on.\"\n\nPulling it off the bed, she pulled it on, her hands shaking as she tightened the tie around her waist. He bit back the rage swelling in him, and, unwilling to risk the guard's overhearing, said with hushed impatience, \"For gods' sake, don't show off your narrow waist. Have you no sense at all? Tie it around your hips. Turn around. Stand in the window.\"\n\nShe did, visibly reluctant to turn her back to him. Maledicte snarled. Her hair, rough cut by his sword, stood out like flame in the dark glass. He looked at the fireplace, long cold, long cleaned, and turned to the oil lamps instead.\n\n\"Gilly,\" she said again. \"You have to go to him.\"\n\n\"I am endeavoring to do so. Or would you have me call a challenge to the guards in the hall, forcing me to dally in bloodlust until dawn? Gilly would be long dead by the time I fought my way clear.\" He pinched out the wick, pulled off the glass, ran his hands over the residue inside; his fingers came away streaked black. He scrubbed his hands into her hair, pushing her against the window. She grabbed the frame with shaking hands, clung to it as if she feared he would push her through it and onto the cobbles below.\n\nHe blew out another lamp, dimming the room, rubbed the lampblack into her hair again, and stepped back to look at her. \"Unconvincing. Stand up straight,\" he said. \"Like you're so frightened your spine is an icicle.\" She stiffened, her hands on the window frame whitening.\n\n\"Better,\" he said. \"A few lessons in comportment and you might be able to pass as a lady. Or a lord.\" One more thing was needed, one last piece to anchor belief, even fleetingly. The fireplace would aid him after all. He drew out the poker from its rack with a rasp that made her shudder. \"Take this,\" he said.\n\nShe clutched it.\n\n\"Like a sword, Livia, like a sword.\"\n\nHe threw the food into the fireplace, tucked the loose braid of her severed hair into the neck of the cloak, and drew the hood around his face, leaving only the flare of redness hanging out, the rustle of lace and skirt.\n\nMaledicte took the sword up by its blade, held it below the hilt, angled it so it lay under his forearm and extended only a foot past his fist. Picking up the dinner tray, he laid it over the visible blade, and then tapped on the door.\n\nThe guard opened it warily, gaze slipping over the tray, the hair, the cloak, and lit on the figure shadowed beyond. Maledicte stepped up to him, and slid the sword through his throat.\n\nThese guards were not the simple Particulars who had come with Echo to arrest him; the other guard had stayed out of easy reach, and even now turned to shout for aid. Maledicte threw the tray, caught him in the throat, and while he was reeling from that, brought the sword up and made him as mute as his thrashing friend.\n\n\"Livia,\" Maledicte said. \"Come.\"\n\n\"I thought I was to stay,\" she said, but hustled toward him anyway. \"I thought you were going to sneak out, and leave me behind so they wouldn't notice.\"\n\n\"They'll notice the bodies in the hall. And even had I time to dispose of them, I do not have time to scrub the carpet clean. When you came in, where were the main force of the guards? And where were the balconies? I foresee a climb in our future.\"\n\n\"The front,\" she said. \"Both at the front of the hotel.\"\n\n\"They would be together, of course. Still, no help for it. Let's find a front-facing room,\" Maledicte said. Despite the fear for Gilly, Maledicte almost enjoyed having a goal at hand with the promise of bloodshed at the end of it. For this moment, Ani and he moved in rare concert.\n\nHe darted down the corridor, pulling at the skirts and cloak, trying to keep them from tangling his legs in a hindering embrace. Behind him, Livia trailed, and he reached his hand back and tugged her alongside him. \"Hurry, Livia.\"\n\nThey turned a corner, startling a drunkard returning to his room. \"Are you lovely girls come to warm my bed?\"\n\n\"Of course we have,\" Maledicte whispered. \"A noble with a room with a view. You gladden my heart.\" He pushed past the man as he fumbled to close the door, threw open the glass-paned windows.\n\n\"Perfect,\" he said. He tucked the sword into Livia's skirt, and looked down. \"Livia, look, climbing roses, how lovely.\" He swung his leg over, settled his boots onto the thickest branch. \"Livia.\"\n\nShe dodged away from the drunk, pushed him back outside the room, and slammed the door. Maledicte began his descent, wincing at the sharp needle kiss of the thorns.\n\nLivia's face peered down at him. \"Oh, I can't.\"\n\nMaledicte called up in a hoarse whisper. \"You'll be hanged for helping me, if you don't come down\u2014\"\n\nAshen, she clambered over the balcony rail, tearing the dressing gown on its wrought-iron finials, and reached her toes out for a foothold. She let out a little shriek as her weight settled.\n\n\"Hush,\" Maledicte said. Above, he could hear the drunkard coming to at least a fraction of his senses, pounding on the door. The second story was going to be full of people soon and there were two dead guards waiting to be found. Maledicte looked down; the ground, dark with distance, seemed to recede. A droplet, warmer than rain, dripped onto his cheek, rolled toward his mouth. Salt and iron. Blood. He licked it up, looked up. Livia's soft slippers were wet with blood.\n\nMaledicte settled his hand on a wickedly large thorn, watched the blood well up and stop when he removed it, the pain vanishing. \"After all,\" he murmured, \"we can't hold a sword with damaged hands.\"\n\nHe dropped the last few feet, skidded on a rounded cobble, and fell hard, wrenching his ankle. \"Ani,\" he said. \"We can't fight with a bad limb.\" The soreness retreated, the swelling receded, and he stood.\n\n\"Drop, Livia,\" he said and she was either so exhausted or so frightened that her body obeyed without hesitation. He steadied her as she rocked on sore feet, muffled her cries in the cloak. \"Shh.\"\n\nLights flared on the second story, bobbed from window to window; faintly he could hear a woman screeching.\n\n\"Rot them all,\" Maledicte said. \"Does no one sleep anight anymore?\"\n\nHe dragged Livia forward. \"Tell me where he is.\" She was too slow to keep up with him, too fragile to fight.\n\n\"Her temple. Sir, it's my fault, all my fault,\" she moaned. \"I told her\u2014told her you loved Gilly. If he dies\u2014I never meant\u2014\"\n\nBehind them, the hotel doors were flung wide, disgorging the Kingsguard. Shouts rang through the night, including the one Maledicte had been dreading. \"There they are! By the wall!\"\n\nMaledicte reached out, intending to shove Livia toward the shadows and dubious safety, but Ani had other thoughts. His hands pulled her before him, into the torchlight; his rumpled dressing gown, the short sooty hair\u2014the guards fired at once. The shot sent her reeling backward, falling into the cobbles. Maledicte turned and ran, hands clenching at his side, shivering, refusing to feel guilt, not while Gilly needed him.\n\n\"Her temple,\" he muttered, thinking of the elaborate and twisted length of Sybarite Street between him and the ruined Relict temple, the only temple to Ani he knew. He had no doubt at all that Mirabile had made her quarters there where he had begun his own quest.\n\nMaledicte slowed his steps as he reached Sybarite Street and the evening's crowds. He drew the cloak tighter, keeping a wary eye out. The guards would know their mistake soon enough.\n\nLights bloomed in the windows of pleasure houses, and slow, drugged laughter spilled like syrup on a cold day. Maledicte moved on, hand clenching around the sword, seeking the darker shadows, tracking his way back to a place he had never thought to revisit. But as if he had mapped the route, he guided himself as steadily, as surely as if he were going home.\n**\u00b7 41 \u00b7**\n\n_And he gave his soul into Ani's keeping and became Her Avatar, winged and blood-mantled, a sorcerous nightmare in human flesh, who carved his way through the battlefields and laughed. And his words became ravens, and where he walked, men died of plague...._\n\n\u2014Grayle's _Book of Vengeances_\n\n**M** IRABILE ROSE FROM GILLY'S BODY, shaking her skirts down, stretching her arms above her head. Along her breasts, the feathers shivered. Gilly rolled onto his side, huddled on an earthen floor warmed with his heat and blood.\n\n\"Poor Gilly, but the sea captain might have used you as roughly. I spared you that at least.\" Mirabile knelt, turned his chin up to peer into his face.\n\nGilly tried not to meet her eyes.\n\n\"Such blueness,\" she said. \"So clear.\" She dropped her voice to a whisper. \"Do you see the gods with those clear eyes?\"\n\n\"Only Ani,\" Gilly said, wrenching his head away.\n\n\"Well, we can't have that. You spying on Her at all hours, watching Her, judging Her; it's the raven for you.\" Mirabile stood, walked away from him.\n\nGilly took a careful breath and rolled onto his hands and knees. Her spell, whatever it had been, cantrip or poison, had left him. His body was his again. Slow, weak as an infant, and hurting, but his.\n\nAcross the room, Mirabile tugged the weight of rotting fabric from the covered bulk on the altar. The raven in the cage beneath woke to raucous complaint.\n\n\"Easy, love, easy. Haven't I got the bluest eyes ever seen this side of the sea? And they're all for you.\" She freed the massive bird from the cage. It turned its head up to look at her with one glossy black eye and she dropped a kiss on its head. It clattered its beak and croaked at her.\n\nGilly rolled to his belly, wrapped his arms around his head. \"Please,\" he whispered, remembering Westfall and the others, found eyeless, their faces shredded until all that identified them were their clothes.\n\n\"None of that now,\" she said, toeing him in the side, trying to turn him from playing turtle, but he clenched himself farther into the floor, knotted his eyes closed.\n\n\"Gilly, I could bespell you again. But wouldn't you rather die your own man?\" she asked, pulling his hair until his scalp stung. When that didn't work, she slipped her hand beneath him, slashed her nails over his bare skin. He twitched; she levered him over and dropped the bird onto his chest.\n\nGilly recoiled as the stink of its blood-matted feathers washed over him. He tried to shove it away with leaden hands. Its talons scrabbled for purchase on his chest; its flapping wings slapped his ears, left them ringing, and its hungry beak stabbed at his defensive hands. Mirabile reached around the bird's wings and pinned Gilly's hands to his sides. He closed his eyes, waited, numb and sick.\n\n\"Go on then,\" she said. \"Peck, bite, maim.\"\n\nThe feathers rustled as the bird settled again, wafting stale blood and feather molt into his face. Gilly shuddered and opened his eyes, stared at the foreshortened beak.\n\nMirabile slapped the ground. \"Do it now, bird. I command you.\"\n\nBehind her, movement. Gilly's heart gave a great leap in his chest. \"Livia, please....\"\n\nMirabile shifted her weight, silencing him. \"Back already? Did you see him? Is he coming?\"\n\nLivia nodded, her disheveled braid slipping over her shoulder, her heavy cloak shielding her face from the sight.\n\n\"Your time is near, my pet,\" Mirabile told the raven. \"It's his eyes or yours.\" The bird jerked forward at her pinch; Gilly felt the beak rip the skin of his face, tracing a careful, delicate line along his jaw.\n\nMirabile hissed, following the gash with her finger. \"Maled\u2014\" The raven went wild in her grip, wings pelting them both, before it turned and clawed its way across Mirabile's face and hair. She leaped up, and slapped it down to the floor, crushed its body with her heels.\n\nGilly could only stare at Livia, at the braid sliding loose and limp to the floor, at the pale shine of the face beneath the hood and the black eyes. He knew those eyes, the rage within them, but not like this, not wrapped in woman's flesh. Maledicte, he thought, dizzy and worn past sense, made an almost convincing woman\u2014if it weren't for the spare lines of his shoulder and chest, the strength in the white hands, even now pulling the sword free.\n\n\"You came,\" Mirabile said, brushing her disordered hair back in long-ingrained habit, playing the coquette. On her face, the weals left by the dead bird sealed flawlessly.\n\n\"With such an invitation, how could I not?\" Maledicte said.\n\n\"But, my intemperate guest, you have come before the hostess is quite ready,\" Mirabile said. \"Still, I can find something to occupy you while I finish with your pet.\"\n\nMaledicte threw himself forward, the sword ripping through the cloak, the skirt, without effort; he dove toward Mirabile, sword extended, aiming at the dark blossom of feathers on her breast.\n\nMirabile raised her hands, cried a single word, and flew out of reach of the sword like smoke blown across the room. Maledicte's eyes widened, then grew thoughtful. The shadows in the room caressed Mirabile's ragged skirt, and she drew their darkness up her body.\n\n\"You're weak, Maledicte. You carry a remnant of conscience, and you'll never know vengeance or true power until it dies. Until you take Ani in fully....\"\n\nMaledicte shrugged out of the cloak remnants. \"Why all this advice\u2014you want me? Fight me.\"\n\n\"I have no desire to kill you,\" Mirabile said. \"I want to rule you. And in turn, you and I, ruling this kingdom, the sky dark with Her wings\u2014\" She stirred the shadows alongside her, her breath rasping.\n\n\"Mal,\" Gilly said, his warning caught in his throat, drowned by his heartbeat. The shadows shaped themselves under her command, taking familiar form. Something skinny and tall, someone holding a stick in a clenched fist.\n\nMaledicte's eyes narrowed. \"What is this?\"\n\n\"You want a fight, I'll set you fighting yourself first.\"\n\nThe shadow snapped into flesh, corpse-pale, with wild snarled hair, a ragged stick held like a sword. Gilly knew this image: a scarecrow boy standing in a pile of broken glass and snow in an old man's library, a feral child crawling from beneath an altar in his dreams. Maledicte's own childhood.\n\nMirabile swayed on her feet, panting, the feathers on her skin ruffling with exhaustion. Maledicte stepped forward again, the sword gleaming; the shadow child bared his teeth and charged at the sword, stick raised and fearless.\n\nTrying to gain his feet, Gilly stumbled, found himself clutching Mirabile's tattered skirts for support. She staggered beneath his weight and he yanked harder, wanting her to fall, wanting his hands around her neck. He brought her crashing down, and struggled to crawl up her body enough to put his hands to use.\n\nShe scrabbled at the floor and swung a piece of stinking darkness at his face; the raven's body, he realized, even as its beak scraped his neck and shoulders. Mirabile was laughing, high and wild, as she flung the bird at him again. He deflected it, and she kicked him in the stomach, sending him rolling back, giving him a surging view of the room. Of Maledicte diving at his shadow self, the sword barely missing the boy's head.\n\n\"No, Mal, no,\" Gilly said. \"No.\" If Maledicte killed that child self, what would he become, freed from the child's innocence? Ani would swallow him entire.\n\nMirabile grabbed his hair, reminding him of his own battle, and slapped his face. \"I'd wanted him for my audience, but I suppose it's going to be the other way around. You can watch if you like. A man with a sword against a boy with a stick. And when it's done...he'll kill you himself for all those whispered prayers, all those little warnings you dared to voice. Against _me._ \" Blood touched her lips for a moment as Black-Winged Ani surfaced and faded, leaving Gilly shaking in the presence of the god.\n\nThe sword flashed down, impacted on the boy's stick. Gilly winced, but the stick held firm, the stick and the sword grating against each other. The shadow boy kicked Maledicte on the shin, and Maledicte broke their clinch with a curse. Following the small advantage, the shadow boy slashed at Maledicte's head with the stick; when Maledicte reached up to block it, the boy darted the stick toward his stomach instead, and Maledicte jumped aside.\n\nGilly moaned, sick with dread. There could be no good end to this. For Maledicte to destroy his own conscience, or to lose Maledicte to a shadow of himself...Mirabile laughed, pressing herself against Gilly's side, excoriating his tender skin with her feathers. The boy wiped his face; his shoulders heaved with effort, but the stick, held out before him, never wavered. Maledicte's sword hand shook; the blade tip magnified that tremor into a palsy.\n\nThe boy danced forward and Gilly saw the confidence in his face, realized the boy's mistake: Maledicte was shamming. The boy had all of Maledicte's ferocity, the bloodthirsty desire to rush for the throat, but no idea of swordsmanship or strategy.\n\n\"No,\" he cried out. \"No.\"\n\nBut the boy was extended, the stick thrust out too far, and Maledicte knocked it aside. The stick, loosed from the boy's hand, disappeared into shadow, unmaking itself before it touched the floor. The boy gritted his teeth, eyes wild with panic and rage. Maledicte's blade, unhindered now, slipped through the boy's flesh without a sound, and passed through his throat. Maledicte's hand and hilt protruded from the boy's nape, dripping shadow plasm.\n\nHorrified, Gilly could only stare, the boy dead, trying to imagine Maledicte without even the smallest leavening of conscience or kindness.\n\nMirabile stood, hands outstretched, smiling. \"My compatriot\u2014\"\n\nMaledicte carried his forward momentum on, stepped through the dissolute shadow that had been himself, and took her head from her body, the blood spray splashing Gilly's skin.\n\nMirabile fell, her blood spreading outward in a tide. Gilly crabbed away from it, scuttling on weak limbs to avoid its touch. Maledicte stepped into it, unconcerned, and pierced her heart, then ripped her body open, spilling her guts out onto the floor. \"Need I do more?\" he asked, voice a wisp. \"Will Ani heal that?\"\n\n\"I don't know,\" Gilly said. The numbness of his limbs had spread to his lips; he felt as chilled as a corpse. His muscles, so long tensed in struggle, deserted him. He collapsed to the floor, sprawled out, face-to-face with Mirabile's head. He could see Mirabile's unwinking eyes staring back at him, and he retched drily.\n\n\"I quite agree,\" Maledicte said. He picked up the head and threw it into the raven's cage, covering it with the altar cloth.\n\nGilly curled up, shaking, tears scalding his cheeks. \"Gilly,\" Maledicte said, kneeling beside him. Gilly felt the soft warmth of Livia's ragged cloak en-folding him, felt the floorboards shift as Maledicte sat beside him. \"Gilly, are you\u2014\" The rasping voice cracked, resumed. \"Will you be all right?\"\n\nGilly had no words at all, nothing but the tears that streamed from him, as if anxious to wash away the spilled blood. He folded into Maledicte's lap, pressing his face against Maledicte's thighs, sobbing.\n\n\"My poor Gilly,\" Maledicte said, voice so soft that Gilly had to strain to hear it, stifling his tears. \"Vornatti should have cast me back to the snows, never disturbed the pattern to your days.\" Maledicte's fingers traced soothing lines on his back, bringing slow warmth to his frozen skin. So gentle. Gilly winced. But the shadow boy\u2014\n\n\"You killed\u2014\" Gilly said.\n\n\"Am I supposed to regret it? Woman or not, Mirabile was a monster.\"\n\n\"But the boy,\" Gilly said. \"Your own shadow.\" He forced himself to look into those dark eyes that he feared to see soulless now.\n\nThe black eyes were dark-ringed with fatigue and worry, but they were calmer than Gilly could ever remember.\n\n\"Yes,\" Maledicte said. \"I should have done it long ago.\"\n\n\"But your innocence\u2014\"\n\nMaledicte laughed, as silent as a cat, his shoulders shaking, near hysteria. \"Did you think that Relict rat was innocent? That creature who knew no kindness, only hunger, fear, and rage\u2014whose only virtue was a love so mad that Ani could find purchase in my soul? I am not that thing anymore. How could I be, with you teaching me kindness? For all that I've corrupted you, my sweet Gilly, you've bettered me. I would not have made the same choice I did then, were I offered it now.\"\n\nGilly's breath let out on a gasp, his chest pounded. \"Mal\u2014\" Maledicte gathered him close, kissed his ear, his temple, drew back when his lips touched the wound on his head.\n\n\"She used you so hard,\" Maledicte said. \"How did you fall into her hands...did Livia entrap you?\"\n\n\"No,\" Gilly said, flushing with embarrassment and revulsion at the resurgent aches of his body. For a moment, he had forgotten pain in hope.\n\nMaledicte awkwardly rocked him in his arms, Gilly overflowing his narrow lap. \"Shh, I'll take you someplace safe.\"\n\n\"Janus,\" Gilly said.\n\nMaledicte stilled, waiting.\n\n\"He found out I let you go to Stones, instead of fighting them off; he beat me, and sold me to the sea.\"\n\n\"And Mirabile?\" Maledicte's voice was cool, as disinterested as if these names were those of strangers.\n\n\"She was on the docks when I escaped. I think she had been watching us all,\" Gilly said.\n\nMaledicte's lips thinned; absently he stroked Gilly's shoulders. \"We'd best get you someplace safe, then. I've escaped the Kingsguard. Echo will be hunting me. And Janus\u2014best not chance his temper again. Will the madam at Lizette's take you in, do you think? It's near enough.\"\n\n\"If we pay her,\" Gilly said, dismayed at Maledicte's cool abstraction, his willingness to ignore Janus's murder attempt.\n\n\"Mirabile will have coin somewhere,\" Maledicte said. \"It only remains to find it.\" Gently Maledicte set Gilly from his lap, and searched the room. He riffled through a stack of papers, pausing as he found the note from the palace spy warning of Maledicte's imminent arrest. \"Mirabile's been stealing our correspondence, Gilly.\" Gilly made no answer, and Maledicte bit his lip, turned back to searching with more urgency. He found the coin purse beneath the altar stones, and tucked it into his shirtsleeve. In another makeshift safe, he found her poisons and rummaged through, muttering to himself. Then he knelt back beside Gilly. \"Drink this\u2014it should fire your blood. I cannot carry you.\"\n\nGilly couldn't focus on the vial, only on the red-washed skin of Maledicte's hands. He flinched. \"Drink it,\" Maledicte said.\n\nThe liquid, bitter as gall, flamed down his throat, spreading heat to cold limbs. His heart drummed for a frantic, caged moment, then settled.\n\n\"Better?\" Maledicte asked, sliding his arm beneath Gilly's shoulders.\n\n\"Yes,\" Gilly said, kissing the frowning face bent so near his own.\n\nMaledicte pulled away. \"Not now, Gilly.\" He tugged, and Gilly raised himself into Maledicte's bracing arms. The temple wheeled around him; the trickle of blood on his cheek shifted direction, dripping over his collarbone and chest. \"What was that?\" Gilly asked.\n\n\"Sailor's Dream, I think,\" Maledicte said.\n\n\"You think,\" Gilly said, licking his lips.\n\n\"Gilly, don't fuss at me,\" Maledicte said, \"I'm bone-tired, and doing what I can.\" His arms trembled around Gilly's chest, and Gilly forced himself to his own support, aware again of Maledicte's slightness.\n\n\"As slight as a girl,\" Gilly said, patting the embroidery on Maledicte's cuffs.\n\n\"Yes,\" Maledicte said. \"I am. And you're as big as an ox, and about as easy to steer.\"\n\nGilly nodded, forced himself to concentrate not on the wonderful warmth seeping through his mind and body, but on the blood-damp footprints they left on the temple floor.\n\nMALEDICTE SWORE Gilly had stopped again, and weariness was ripping through his bones, weighing him so that he felt he might sink into the earth at any moment. \"Come on, Gilly,\" Maledicte said, pulling.\n\nGilly still balked, and, overbalanced, Maledicte fell up against him, warm skin exposed by Livia's inadequate cloak. Gilly's hands wandered again, and Maledicte, trying to secure the cloak, didn't step away. Damn Mirabile, he thought with a snarl. If she'd wanted him so badly, all she had to do was give him a dose of Dream, and he would have been hers. But she chose to hurt him, instead.\n\nHe shook with a rage that was all his, without the taint of Ani at all. Where was Ani? Maledicte wondered. She'd been still and quiet since the shadow boy's death, since Mirabile's death, when he had expected Her to rise screaming from his belly, expected to have to fight Her. But She was silent; it made him nervous, like a sailor beneath a storm-clouded sky.\n\nGilly's hands were under his shirt now, Maledicte noticed with a sudden warmth of his own, tracing lazy circles on his back, slipping lower.\n\nBlood touched his face, and Maledicte looked up, watched another thin rivulet sneak past the slow crust forming on Gilly's head wound. Rising to his toes, he licked at it, hoping some of Ani's healing might be found in his spit. The copper taste woke him to the urgency of moving on, of not being found near Mirabile's body, of not being found at all.\n\nGilly fumbled at Maledicte's breeches and frowned in drugged puzzlement. \"Livia?\" Gilly whispered.\n\n\"No,\" Maledicte said, stung. He shoved Gilly into movement again, promising himself a confessional with Gilly as soon as he was sober. He had nothing to hide anymore; the unexpected freedom of it washed over his skin. Maledicte was a dead man, on the run, ruined. And he, who had changed his identity once, was free to do so again. The sword twanged against the doorjamb as they passed it, sparking a muttered response from Ani lurking within.\n\nNot free yet, Maledicte thought, sobered. He had his vengeance to complete, though he knew it was an empty gesture now. Last was long dead and buried, his enemies gone\u2014Kritos, Last, Amarantha, Dantalion, all fed to the crow-bitch, leaving him never sated. If he killed the child, would Ani leave him then, their compact finished with the infant earl's death? Maledicte felt Her whispering inside his blood, murmurs of agreement and coaxing. Just one more and then he'd be free.\n\nFree, Maledicte thought, to do what? Change his name, leave the city, leave Janus? The breath fell out of him; his heart throbbed. Janus\u2014back-to-back, fighting the world, only each other at the last. Maledicte, swaying under Gilly's weight as they shambled up the street like two drunkards, felt clearheaded for the first time in months.\n\nVengeance was a cold thing, and his prize...he had bartered his soul for Janus, and his prize was not all he'd expected. Golden Janus, his lover, his most trusted friend, had sent Gilly to the sea....\n\nRosy light washed over them, flushed their skin with a false health as they passed beneath the brothel windows. Maledicte dragged a protesting Gilly down the alley and hammered on the back door. When it opened, he drew his sword and levered himself and Gilly past the girls.\n\n\"A room,\" he said. \"Now.\"\n\nA butterfly flutter of silk told him one girl had run for the madam and likely for whatever protector she hired for the brothel. Maledicte intended to have Gilly ensconced before their arrival. He went toward the stairs, and Gilly, moving by muscle memory, stumbled up them, and chose a room without hesitation. Empty, thankfully, Maledicte thought. He could bribe the whores, but a customer might be another matter. And how long had it been since his escape? Had they posted guards on the streets yet?\n\nHe pressed Gilly back into the sheets, wrapped him in blankets, and sat down on the edge of the bed.\n\n\"Will you forgive him this, too? If he has killed me?\" Gilly whispered.\n\n\"You're not dying, Gilly.\" Spoken, the fear was real in the room. Maledicte took Gilly's hand in his own, sought out the steady throb of the pulse, the warmth of his fingers, and repeated, \"You're not dying. You're hurt. But you'll recover. Street urchins get beat worse than this by their parents in the Relicts, and look how well they grow.\"\n\n\"Stop crying then,\" Gilly said. \"If I'm not dying.\"\n\nMaledicte put his hand to his face; it was wet and stinging with tears that he hadn't noticed. He sniffed them back, letting them add to the pressure within him.\n\n\"I don't forgive him,\" Gilly said. \"Even if I'm not dying.\" He closed his eyes, blooming bruises and exhaustion spreading shadows under them.\n\n\"Nor do I,\" Maledicte whispered. \"Not this. Not you.\" The door opened again, and the madam stood there without the protector Maledicte had expected. A sudden dismay rose in him that he wouldn't have to fight anyone. He took a slow breath, forcing Ani back again.\n\n\"The girls said it was Gilly,\" she spat, seeing Maledicte seated on the bed.\n\n\"It is,\" Maledicte said, leaning back so she could see Gilly's sprawled form.\n\n\"And you've done this to him?\" she asked. \"Like my Lizette?\"\n\n\"No, to both,\" Maledicte said. \"I need a safe place for him to stay. To heal.\" He felt as if he were talking at a distance, the room seen down a telescope.\n\n\"He can stay. You try to, and I'll summon the guards.\"\n\n\"I wasn't going to,\" Maledicte said. Gilly clutched his hand, and Maledicte returned the pressure absently. \"He's hurt, though. And on Dream. He'll need stitches for his head, and balm for the rest.\" Maledicte dropped the pouch of coins to the floor; the madam scooped it up and disappeared through the door.\n\n\"You can't go,\" Gilly said. \"Where are you going?\"\n\n\"Away,\" Maledicte said. \"It seems that killing Mirabile was not wise, despite my satisfaction. I had not realized how much of Ani's concentration was focused on Mirabile. And now, there's only me to see to Her whims.\" Bile seared the back of his throat; he coughed it back. Gilly pushed himself up to his elbows, eyes going wide, even in his drugged state.\n\n\"Mal, I can hear Her in you\u2014\"\n\n\"Yes,\" Maledicte said. \"Ani's coming back. I can't stay near you. It's not safe.\" He stroked Gilly's arm, felt his other hand seize the hilt of the sword.\n\n\"Mal,\" Gilly said. \"Let's leave Antyre. Let's go to the Explorations, please.\"\n\nYes, Maledicte thought, yes. Away from the kingdom, away from Ani's tyranny and Her never-ending vengeance. But he was gagged by the taste of Her feathers and Her searing hatred.\n\nGilly touched his throat, traced the god- _avert_ over his flesh with trembling fingers, cooling his heated skin. The obstruction in his throat lifted.\n\n\"Yes,\" Maledicte said. \"Yes, I'm done with this. With this fruitless vengeance, with ashes in my heart, with\u2014\" The heat scalded him, raced up his spine, his throat, scorched back into his belly, his arms; the sword jerked and quivered, demanding that Maledicte remove Gilly's offensive, charm-using hands.\n\nMaledicte screamed under the weight of Ani's will, Her thundering voice demanding his loyalty, his promise completed, obliterating everything else in his mind. The shadows moved inward, blinding him.\n\n_Not Gilly,_ he thought. _Let me finish my compact instead. You are the god of love as well as vengeance; let me leave Gilly alive. Please._ Maledicte remembered Gilly saying Ani destroyed the Relicts when Her follower denied her. Maledicte shivered, trying to keep the image of the brothel slipping into the earth at bay. _Anything,_ he pled. _Anyone. I'll bring them to you._ Ani bent her head to his first prayer and Her wings fluttered in triumph.\n\nMaledicte stood, hand on sword hilt, and brushed by Ma Desire, who stood trembling in the doorway, hands full of clean bandages. Her shocked face was the last thing he saw before Ani took complete control.\n\nGILLY STRUGGLED WITH THE CLOAK, with the sheets, hearing again that stifled raw shriek, trying to get his stubborn legs sorted out so he could rise and go after Maledicte. Ma Desire hurried over and pressed him back. \"No, you let that one go to the hell he's headed for. You don't go with him.\"\n\n\"But\u2014\" Gilly said.\n\nShe covered his mouth with her hand. \"There's nothing you can do. He's wing-bent.\"\n\n\"No\u2014\" he said, and she tipped a glass against his open mouth, made him sputter even as he recognized the taste of the Laudable. Behind the window, in the night sky, he could see darker clouds flowing lowly across the sky, full of the feather-rasp of flying rooks. A dark cloud in the night, moving through the heart of the city, following their master toward the palace.\n**\u00b7 42 \u00b7**\n\n**M** ALEDICTE WOKE TO HIMSELF on the grounds of the palace, staring up at the brick wall of the residential house, with only a dreamlike idea of how he'd arrived. The sky was dark, he knew that, and the rooks were everywhere. Had he flown? He raised his arms, peeled back his sleeves, looking for feathers, but saw only smooth skin, unmarred.\n\nThe light in the window beckoned him; the bars on the frame told him it was the nursery. Maledicte sheathed the sword and reached upward. He pressed his fingers into the brick mortar and it gave, creating a fingerhold. Raising himself one handhold at a time, he climbed, the birds swooping around him, carrying shrouding darkness on their wings, delaying morning.\n\nOn the streets below he could see lamps guttering and being relit against the thick darkness. Being lit against him, loose in the night, and within those glows, the gleaming gilt of the Kingsguard, huddled close.\n\nA shadow moved across the window, and Maledicte clung to the wall with a predator's caution and patience. He shook his head, trying to clear this dreamlike sensation from his skin, trying to feel something other than Ani's fevered determination. Mortar and brick crumbled under his fingers, spat one hand into the air, left him hanging by the other. Below him, duller uniforms mingled with gold and blue: the Particulars with their pistols close to hand. Maledicte shifted his weight gingerly, trying to ease the cramp threatening to destroy his precarious grip on the wall.\n\nTrust Me. He heard Ani's whisper, not in his ears, or his mind, but in the tides of his blood. It strengthened him like a tonic, and he clawed another foot upward.\n\nSweating, gasping, Maledicte gave himself over to the simplicity of Ani's will, of climbing the wall. He crept up to the lighted square of the nursery window, braced one foot on the sill, and peered inward. A kingsguard leaned up against the glass, spreading his bulk between the light and Maledicte.\n\nHanging motionless, Maledicte watched, wondering why the guard never turned to look outside, then understood. This man was there to watch the inside of the room, secure in knowing that there were guards posted at every entrance, and that the window was barred and, moreover, three stories above ground.\n\nMaledicte, clinging with one hand to the stone sill above the window and braced by his feet, his ribs pressed against the sharp corner of the sill, reached for his sword. He slid the blade through the age-bubbled glass as smoothly and as cleanly as if it had been through paper, pressed it home before the guard could turn at the tiny chime of breaking glass. The sword bit deep into the guard's heart, and he stiffened against the pane. Maledicte withdrew the sword, leaving a ring of blood on the glass as the wet sword returned.\n\nThe guard slumped, and Maledicte, clinging to the bars of the window, waited a moment, to see if anyone within the room would object\u2014wet nurse, child, or another guard. But the minutes passed in silence, with only Ani's urging to be heard.\n\nHe measured his shoulders against the bars, measured his head; the bars were too narrowly spaced for him. Designed to keep small children within, they also kept larger predators out. But as he ran his hands against their iron length, waiting for Ani to act, he realized they didn't go all the way to the top; their pointed finials stopped before the window did, leaving a gap. Too high for a child to climb to, but for him on the outside\u2014it was the only way in, unless he expected Ani to peel back the bars one by one.\n\nThe gap wasn't much, a space of eight inches high, and only as wide as the window. Maledicte raised himself up, slid his legs past the iron prongs and slowly, gingerly, worked his way through, the sharp tips pressing against his rib cage, ripping a line through his shirt and spilling a tuft of padding from his corset. Pressed between the bars and the glass, he used the sword hilt to work up the latch.\n\nDropping into the room, he landed on the dead guard, and rolled away, came up with the sword extended. He bent and pulled the guard to his feet, using the man's belt to fasten him to the bars. A brief look in the night-dim nursery would see the man dozing against the window. But he hadn't played puppetmaster with the body unseen, he realized, as he heard the steady breath catch in surprise.\n\nAcross the dimly lit playroom, Adiran stared at him. Nested in blankets, surrounded by his blocks, he hadn't slept in his bed. Ani moved Maledicte's feet toward the boy. At the end of the playroom, closed in Adiran's room, a dog barked sharply.\n\nAdiran smiled up at him, fumbled in the blankets beside him, and held up a hand. Ani smiled and accepted the token, the little porcelain puppet with black wings. She touched Adiran's head, and said, \"Sleep, wingless one.\" As he had in Stones, Maledicte felt something transfer through him, not the same toxic wave of sickness, but something small and sharp, a crystalline seed. Beneath his hand, Adiran's eyes fluttered. Sighing, Adiran folded back into his blankets. The dog scratched madly at the door, and Ani hissed. It whimpered and fell silent.\n\n\"Hela?\" The main door started to open, and Ani fought a brief battle with Maledicte over the necessary movement. Blood or stealth? Maledicte won by a bare margin, and ducked behind the carved toy chest, sheltering in its shadowed bulk.\n\nThe guard looked in and about, saw Adiran sleeping, and shut the door again, oblivious of the scent of blood that filled Maledicte's senses.\n\nOn silent feet, Maledicte ghosted toward the other end of the room and the other bedroom door. He opened it, the sword slipping free, but the wet nurse snored in her chair, her gas light guttering.\n\nAni prodded her with the tip of the sword, drawing blood, but no flinch, no waking; surprise and thwarted bloodlust drove Her back again. Maledicte touched her cup; sniffed the dregs of tea. Drugged. He smiled, a lean, cold thing that had more of Maledicte in it than Ani. Janus had been here. Janus was working with him. But where was he? Maledicte turned to search and Ani showed him the crib instead.\n\n_The earl of Last. My enemy. The last death._ Maledicte bit his lip at the idea. The freedom from Ani, their goal met, but the idea warred with a simpler image\u2014Gilly's face, flushed with distress over murdering an infant. Ani snarled within him, reminded him that She had let Gilly live. \"I promised,\" Maledicte said, took a step forward; the cradle linens seemed crimson with blood, the copper tang of it rich in the air. He curled cold fingers around the hilt. One more, he thought, and inched forward.\n\nHe reached into the cradle and touched warmth and wet, and pulled his hand back. The blood on his hands was not the child's murder played out of time, not Maledicte's imagination, or Ani's vengeful illusion. The blood on his hands was real. And the infant lay in a sleep from which it would never awaken.\n\nMaledicte made a noise in his throat of utter protest, a double-throated thing, his choked cry of pity and revulsion, and Ani's harsh gasp of thwarted rage. A shadow detached itself from the wall, took his wrist. \"Shh, Mal, not yet....\"\n\n\"Janus,\" Maledicte breathed, the room shivering around him like something in a dream, like it might fly apart and show itself to be mere delusion.\n\nJanus touched his mouth; the odor of blood washed over Maledicte with the touch. It soaked Janus's cuff. Maledicte backed away, leaned against the closed doors. \"You killed\u2014\"\n\n\"Saved you the grief,\" Janus said, his voice low. \"I saw how it distressed you. The idea of killing Auron. But it had to be done. When I heard you'd fled the hotel, I knew you'd be coming here. I thought you'd be quicker, though.\"\n\n\"I had to retrieve Gilly,\" Maledicte said, numbly. He waited to feel something, but Ani's rage, though white-hot, only dimly touched him. He wished it would wash over him, comfort him, take this cold horror from his belly, that this man with bloody hands and cold eyes was his lover, his companion for years, his beloved.\n\n\"Gilly, again,\" Janus said, scowling. \"Timing is important, Mal. My plan\u2014\"\n\n\"To be earl, I know. You hated this child,\" Maledicte said.\n\n\"It was only a child, unworthy of hate. I never hated Auron. And I don't care about being earl any longer.\"\n\n\"What?\" Maledicte whispered, startled out of his dream world, back to the solidity of this room, this moment, his breath fast in his chest, the ruined, wet texture of the baby's skin still warm against his hand. He shuddered all over, wanting out, wanting to run, but the guards were outside the doors and he couldn't imagine climbing down the way he knew he'd come.\n\n\"Janus,\" he breathed, seeking understanding and freedom from this room that had become a trap.\n\n\"I intend to be king,\" Janus said, the words cool and measured in the quiet of the room. \"My blood's good enough, and why not, there've been bastards on the throne before. I can rally some support already. DeGuerre, some of Westfall's friends. But I had to make a choice. Kill Aris, that vacillating, sentimental fool, or the babe? Assassination of a king's always a chancy thing. But kill Auron, and who's left for Aris to turn to but me, when the only blood left is mine or Adiran's?\"\n\nMaledicte leaned against the wall, chilled at the pale fire in Janus's eyes. Ani surged in him, screaming so harshly that nothing of Her words was distinguishable, only the shrieking desire to kill. At this moment, Maledicte didn't know who she hungered for. The earl of Last was his promise\u2014and the babe was dead.\n\n\"Ani still rides you,\" Janus said, stepping back and away, calculation in his face. \"Does She know, does She understand\u2014I am the earl of Last, now?\"\n\nMaledicte moaned, the sword leaping in his hands, darting toward Janus. Maledicte fought it, but Janus stepped closer, let the blade bite into his arm. Janus savaged his lip, but did not cry out.\n\nJanus danced back, hand clutching his wound. Blood rose and welled between his fingers, flowing down his sleeve and mingling with Auron's spilled blood. \"I knew you would understand. Perfect.\" His eyes widened suddenly and he rolled beneath the cradle to avoid the next swing, rose on the other side. \"Once is enough, Mal. You must control Ani. Use Her abilities for our ends.\"\n\nMaledicte gasped for breath, shuddering with exhaustion and dread that Janus thought to play puppets with the god. The sword burned in his grasp, the feathered hilt sinking into his skin. Janus's eyes narrowed, gas-flame blue, as the sword moved toward him like a needle on a compass. Maledicte lunged again and found Janus using the same gambit Maledicte had used in all his duels, stepping too close for the sword to be brought to bear. Janus grabbed Maledicte's wrist, holding it out like a pinned wing.\n\n\"Shh,\" he whispered into Maledicte's ear. \"You cannot kill me. I am both your Love and your Vengeance now. The thing you wanted and the thing you hated. Ani cannot kill me without breaking your compact. We've caged Her perfectly within you. After all, Her skills are far too valuable to lose.\"\n\nMaledicte dropped the sword, trembling all over, wordless. Kaleidoscope images burst behind his lids, of Last dying, of Auron's blood, of Mirabile's feather-studded skin. He whimpered, sobbing for air and reason. Janus's blood perfumed the air, the wound near his face. \"Miranda, trust me. I know what I'm doing,\" Janus said. \"Now pick up your sword. You'll need it.\"\n\nJanus stepped back, and Maledicte, blank-minded, did as his lover bid. As his fingers touched the hilt, there was a sudden shatter of glass from the other room as the crack made in the window by the sword raced side to side. Rooks blew through it, and the mastiff broke into frantic barking.\n\nJanus fell back against the cradle, bloody wound clutched in a hand, smiling. \"Go.\"\n\nMaledicte fled the pale, ecstatic light in Janus's eyes, the sword shivering at his side, ran through the swirling cloud of rooks, leaping over the crumpled guard. He made a leap for the barred window, but his hand, still slick with the infant's blood, slipped down its length without catching.\n\nThe guards burst into the room at Hela's barking, and Maledicte reacted, slicing into them, severing the first guard's arm from his body and driving the sword through the chest of the next one. Panting, he put his foot on the corpse, levered his sword free; it stuck on a rib, and he yanked harder. Dimly, he saw Adiran, awakened, standing beside him, blue eyes wide and worried.\n\nBefore the next guard could reach him, Maledicte freed the sword, and grabbed up Adiran. The guard faltered. Adiran clung to his neck and began to cry. Behind him, he heard Janus stumbling into the room and checking also, as if only waking from an assault.\n\nMaledicte swallowed hard, the child's wailing in his ear. He moved toward the door, and first one guard, then the next, stepped out of his path.\n\nAdiran pushed feebly in his grip, his wailing breaking into uncertain hic-coughs. Maledicte clutched him closer, his mind twisting ideas together, trying to think of escape and only imagining a noose. The guards would follow him to the ends of the earth as long as he held Adiran. He could set him down and flee\u2014he had nearly the length of the room on them\u2014could kick the door closed, delay them that second longer. But, to set Adiran down now\u2014the guards would surge after him like hounds, leaving Aris's beloved son, Aris's heart, behind in the nursery. Alone with Janus.\n\nMaledicte turned and raced the long hallway, found guards pelting up the main stairs nearly on him. Jasper headed them, his eyes fever-bright with anger. Seeing Adiran clutched so close, he waved the rest to a halt. They paused, piling into each other, but despite Maledicte's fervent wishes, stayed upright. The second mastiff, pushing through them, had no hesitation at all. Despite Jasper's snatch, Bane came roaring through, savaging his restraining hands.\n\nMaledicte dropped Adiran and bolted. The child, startled again and terrified at the rage in the air, began wailing. Bane gained his side, and, frantic, began slicing the air with his teeth, keeping everyone away from his charge, and obstructing the hall. Adiran clung to Bane and howled. The guards were stymied. For the moment.\n\nBut the floor shivered beneath Maledicte's feet with the arrival of more guards. He shuddered. The palace was worse than a beehive struck unthinkingly.\n\nWithin him, Ani whispered, let Me make it better, let Me make them all suffer. Give yourself to Me.\n\nNo, Maledicte thought, pushing away from a wall, taking the corner too fast, his boots skidding on the polished wood. He saw another stairwell and raced for it. Janus had a plan; Maledicte had to trust him. There was no alternative. It was only their old game, made more risky. Miranda had done the running before, dashed away with stolen goods, or the weapons to be hidden. She had always been able to outrun the blame, and Janus\u2014had always been able to deny it.\n\nThis was more of the same, all part of the plan. _Janus's plan,_ Maledicte thought, savagely. _Not mine._ His breath tore in his chest, his heart hammered; he grabbed the railing of the stairs, saw more guards coming up them, just two, roughly woken and still addled with sleep. He shrieked and dove forward. The first man took the blade in the face and collapsed instantly, blood bubbling through the wreck of his nose. Maledicte tumbled down the stairs on top of the other, using the man's body to cushion his own bones against the risers' edges.\n\nPanting, Maledicte slit the man's throat when he started, clumsily, to fight back at the base of the stairs.\n\nIf he could only get outside the palace, the night itself would hide him; the clouds of rooks would shelter him, as safe as any babe\u2014In the disused dining room, Maledicte leaned against the wall and retched, wiped his bloody blade clean on the shrouded table.\n\nTrust in Me, Ani whispered, coaxing, gentle, as compelling as Her first words to Miranda had been. Huddled beneath the altar, the salt burning her eyes, her skin, her scraped flesh, and Ani asking, What wrong has been done to you, little one? Tell Me what you want....\n\nNow Her words were gentle again, the strident, bloody harridan only a nightmare image in his heart. Why trust Janus? Everything you've done, everything you've been, you've done for him. And is he the man you thought him to be? Hasn't he lied to you? Can you trust him? There's only Me to protect you, now.\n\nMaledicte sucked in his breath, quieting its wheeze, ignoring Ani as best he could. They had lost him, albeit briefly. Best to make the most of it. Curtains draped the far wall, and Maledicte, hoping for windows, yanked them back. Painted gardens, sunlit, even in the dark of night. Maledicte laughed wildly; he hated this court, the overwhelming falsity of it all, where not even the architecture could be relied upon to be honest.\n\nFootsteps sounded outside the doors. He ran for the servants' entrance, yanked the door open at the expense of its hinges, and dashed into the dark corridor beyond, the door listing in the jamb, a clear pointer to his direction.\n\nDarkness and shadows and enclosed walls struck both Maledicte and Ani nerveless\u2014the specter of Stones again. If he were caught\u2014to spend the last moments of his life in a cell\u2014Maledicte ran blindly down the hall, toward a faint spark of light growing in the distance. A maid with a lantern crept out of a room to see what was happening. She opened her mouth to shriek, but Maledicte pounced, snatched the lantern, and pushed her into the center of the hall. Gasping for breath, shocked, she sprawled across the smoothed floorboards, watching as he retreated. Maledicte grinned. Let her lie there in a stupor; let the damn guards trip over her, and buy him a few precious moments.\n\nHe wanted more stairs, more windows, some hint of where he was. Why had Janus never given him a map of the palace when he had known it must come to this?\n\nMaledicte shivered, though his skin was hot with sweat, and the lantern's heat burned his left hand. He had no answer for himself. He was bent on escape, and thinking was for later.\n\nHis feet pounded along the hall; the servants' passageway, though narrow enough to prevent the guards from surrounding him, was stripped of carpeting, and his steps echoed like pistol shots. They could track him by that alone, and he had no idea which of the doors held more stairs, winding their ways, mazelike, through the palace. There had been stairs in the dining room, but he had fled mindlessly past them, and the pursuing guards, their cries audible now, made doubling back impossible. The dining room would have needed to be connected to the kitchens, and the kitchens always opened out to the world. Maledicte pushed open the next door, slid through it, and shut the door again.\n\nA woman repairing sheets looked up at him, the needle held in her mouth, the thread dangling. It dropped and Maledicte lunged at her. \"Not a word.\" He blew out the lantern, slid himself under the sheltering drape of the sheet she was sewing, pressed the sword tip up against her belly. \"Not a word,\" he said again, his voice rough with fear. Had he been in court, he would have done his best to disguise that weakness, but here his desperation could only insure her obedience.\n\nThe door swung open and guards spilled in like a piled mass of hunting dogs.\n\n\"What do you want?\" she said, her voice shrill, going shriller as Maledicte leaned his weight on the blade. A thin line slid down the blade, as thin as her linen thread, but dark, and forming a slow droplet at the end. Maledicte caught the drop on his fingertips, lest somehow the guards hear that small act of violence over their searching. They yanked open all the connecting doors, threw the loose piles of sheets around the room, until the seamstress cowered, bending her face near to her waist. Maledicte could see her features, distorted by fear, through her pale linens.\n\nThe guards left, slamming the door again, and Maledicte slid away from her. \"Please,\" she said. \"Please.\"\n\nMaledicte knew killing her would buy him time, prevent her from shrieking that he'd turned rabbit and bolted back the way he'd come, but her blood was already streaking his blade; the sight of it made his stomach churn. A fine time to lose the taste for it, he thought bitterly, but Ani only laughed.\n\nIf you won't come to Me, why should I help you? She asked.\n\nMaledicte put his hand over the seamstress's mouth, put the blade to her throat; the woman paled, her tongue licked out nervously to touch dry lips.\n\nMaledicte pulled away, the blade no more bloodied than before, and ran. He had reached the dining room again when he heard the muffled violence of her screaming.\n\nFool, Ani said within him. Betrayed fool. Lose yourself in Me and I will aid you. He clattered down the stairs, burst into the kitchen, and found it overfull of guards, watching the exits.\n\nMaledicte turned and fled back upward, aware of the upstairs contingent approaching. \"Help me,\" he whispered.\n\nYes, Ani said, Go always upward, and the rooks will aid you. He kicked the stair doors shut in the guard's face as the first man reached it, and he kept going up, past the landing to the servants' quarters, past the point where the stairs were kept in good condition, and became friable, bowed with time. He stumbled, but kept going, secure in the knowledge that these stairs were blind. There were no doorways to open up at his side, disgorging guards or Particulars. No maidservants to trip over, just a straight shot to the sky.\n\nAn explosion snapped through the air in the hall; the plaster near his face puffed into dust, and Maledicte spat. Pistols.\n\nHe turned and cursed them for cowards. The Particular drew another pistol and fired again, then screamed as the pistol exploded in his hand. Ani's doing, or pure luck. It didn't seem to matter. The stairs came to an abrupt end, spilling him out into a jumbled attic.\n\nUpward. In the shadowed ceiling, the door to the rooftop was hinted at by a darker patch, a square with a telltale latch. He climbed the pile of aristocratic refuse and forced the latch back, even as the guards swarmed in and spread out, creating a net of flesh and swords.\n\nMaledicte levered himself up and through, and found himself on the flat roof of the palace, the night air cool and crisp in his face, and the sky alive with wings. Within him, Ani spread Her wings, stroking his fears back.\n\nHe laughed, stood over the trapdoor, and took the head of the first guardsman to climb through, pushing the body back down onto his colleagues. Maledicte kicked the head through as an afterthought and dropped the trapdoor closed.\n\nThere was nothing there to hold it closed; the latch was on the other side, but the very fact that only one man could come through at a time acted like a weight on the guards below. Maledicte left the trapdoor, ran to the edge of the roof, and looked down. Dizzyingly far, the ground seemed as unattainable as the sky as a means of escape. He leaned over the edge, testing the wall for scalability. In this part of the castle, it was old stone, not soft mortar and jutting brick. More, Ani showed no inclination to grant him preternatural skills again, and only a fool tried to descend a sheer stone wall.\n\nBeneath his feet, the muttering panic of the guards went quiet and orderly; one voice cracked out above them all. Echo, taking charge. At least there was that at the end.\n\nHe watched the trapdoor lift, disgorging Echo, who rose like a stage demon, flung aloft by his guards, pistol in one hand, sword in the other, and a length of chain mesh guarding his throat.\n\nMaledicte danced toward him as Echo leveled the pistol, eyes narrowing. The puff of smoke, the ricochet of sound struck Maledicte a moment after the lead did. He stumbled, but the ball had only penetrated his leg; Ani chased it out, healing its intrusive heat, absorbing the hurt. Maledicte reached out with his sword and took the pistol from Echo's hand, flung it off the roof.\n\n\"I'm glad you came,\" Maledicte said. \"This wouldn't have been the same without you.\"\n\n\"I'll see you dead,\" Echo said. Behind him, the guards started to join them, and Maledicte pivoted, kicked the first one in the throat, and sent him backward. Echo's blade whistled in the sky, coming for his chest, and the air was suddenly full of rooks. Echo flailed his sword, trying to clear them from his face, the stabbing beaks, the snatching claws, and Maledicte screamed, \"He's mine.\"\n\nThe space between them cleared, the rooks pulling away into the sky like a windspout, flowing upward and then falling back toward them, circling them. \"All your tricks won't help you, now. Aris will see you hanged,\" Echo said, closing.\n\nMaledicte took the blow on his blade, skidded under the man's weight, and stepped aside at the last, forcing Echo off his blade. Maledicte thrust, aiming for Echo's exposed side, but the man pivoted and parried.\n\nMaledicte stepped back, trying to keep an eye on both Echo and the door to the attic. Echo was not any of the fools that Maledicte had dueled previously, buoyed by tradition and stupidity; should Maledicte be struck from behind by a guard, Echo would finish him from the front without hesitation.\n\nMaledicte jumped the low thrust Echo aimed in an attempt to hamstring him, and swept the blade outward, pushing him back. The trapdoor started to rise, and Maledicte leaped on it, his sudden weight forcing the guards back. But only for a moment. They shoved upward; Maledicte felt the wood shift beneath his feet, saw Echo's blade coming for him, and tumbled forward, going head over heels away from the strike.\n\nHe knelt, heart pounding, blood singing in his ears, listening to Echo's approach. Not an honorable fool, Maledicte thought, not loath to strike down a fallen man, but a fool nonetheless. Maledicte dropped from his knees to his thigh, rolling and turning. Echo leaned inward just as Maledicte pushed the blade up into his chest. He worked the blade through, then worked it free, letting Echo fall as the guards gained the roof.\n\nSuddenly leaderless, they hesitated. Echo bled out before them; the rooks swirled, filling the air with their cries and feathers, and Maledicte levered himself to his feet, panting through bared teeth. They spread out loosely, but none approached.\n\nMaledicte leaned back against the parapet, looking over them all. What now, he wondered. Fly, Ani urged him. Fly.\n\n\"I cannot,\" he said, not caring that he spoke aloud. The guards flinched and one, braver than the rest, stepped forward. Maledicte shifted his grip on his sword, and said, \"Don't do that. Your colleagues aren't going to support you and I'll kill you. We'll wait.\"\n\nThe trapdoor rose again, and Maledicte found a smile at the pale gilt hair, at the blue eyes. \"Janus,\" he said. Now the game could continue\u2014Janus's plan unfurl.\n\n\"Sir,\" the guard said. \"Be careful. He killed Lord Echo.\"\n\n\"And so you are all waiting for someone else to stop him. How brave of you all,\" Janus said, his tone caustic. \"Had you all rushed him, this would be done with by now.\"\n\nMaledicte felt his heart jerk and flutter, as if wings were beating in it. It was an act, he knew it was, but it felt\u2014\n\nHe's betrayed you, Ani crowed. You know he has. Thrown you over for ambition and a golden throne. Give your future to me. I'll bring their city down on them all.\n\nJanus would not turn on him\u2014Maledicte clung to that certainty as he had clung to the wall earlier. Janus loved him more than the world itself.\n\nJanus took a sword from the nearest guard and paced forward, his face set, as white as the marble busts that lined the king's hall.\n\nIt was not the sword, or the implacability of his face that seeded doubt into Maledicte's heart, but the clammy remembrance that Janus had waited for Maledicte to arrive before striking the child. That he had plotted to use the gods, and one so mad as that might dare anything. More still, the simple fact that Janus had had his wound dressed and his shirt exchanged for a fresh one. While Maledicte ran, Janus had been dressing for this moment.\n\n\"Ani,\" Maledicte called out to the sky, his voice a croaking plea, his pain choking him. \"Ani.\"\n\nThe sensation burst over his skin like a thousand needles stabbing; his hands shook, and the sword fell with a clatter on the stone. He could smell it, the rough scent of new feathers springing out, cloaking his skin, letting Ani free. Beneath his feet, the palace twitched, like an animal waking, like a horse shaking off a bothersome fly, like wings unfurling.\n\nThe guards rocked on their feet, their eyes wild. Two of the Particulars fled for the stairwell and disappeared into it.\n\n\"No,\" Janus said, stepping forward, his sword hand moving.\n\n\"Sir, the king wants him for a trial,\" a guard called.\n\n\"And risk the palace?\" Janus countered. \"Risk the ceilings crushing his son?\" He struck like a snake.\n\nMaledicte rolled away from the blow. He backed away from Janus, wishing Ani would hurry with this transformation, would hurry with his obliteration.\n\nA faint thought crossed his mind, shocking him to stillness\u2014Gilly, waiting for him. Gilly. Ani shrieked within his skin, the rooks flew into the ranks of guards, blinding one man, and causing chaos.\n\n\"Mal\u2014\" Janus said. \"Mal\u2014\" A warning in his voice, or entreaty. Maledicte didn't know which it was, only knew that his lover came for him with a sword, that Ani raged within him, and that the small quiet space left in him was weeping for Gilly. Then there was nothing in his mind but the black arrival of Ani, holding him immobile as She sought control of his body, making it over as She wished.\n\nThe moonlight reflected off the blade as Janus pulled his arm back and thrust forward. Despair let him break Ani's grip; Maledicte's hand flashed out, pressing the blade aside as it moved, blood spattering from his palm as he pushed, trying to shift the sword, trying to shift it away from his tender flesh. He moved it a bare half inch. Not enough.\n\nThe sword sank into him as if it had always belonged there, the slick heat of it intimate within his chest, nestled inside.\n\nJanus's eyes were wild; his mouth slack, as if he hadn't believed he could do it at all. The blade slid free despite Maledicte clutching at it, holding it to him. Blood sprayed over Janus's clean shirt, hit his face and mouth. Janus flinched, and closed his eyes. Maledicte, still standing, felt his body going numb and distant. He watched Janus wipe the blade clean with his hand, sheeting his blood from the steel to the stones of the roof. The guards' faces were stupid with shock, as if they hadn't believed Maledicte could be killed any more than Maledicte had.\n\n\"Ani,\" Maledicte breathed. Ani poured Herself into the wound and found it mortal.\n\nShe screamed, Her burgeoning power pushed back and redirected, fighting to stay ahead of Maledicte's death. The stones of the roof birthed ravens, and the guards stumbled, waving their swords blindly in a blizzard of black feathers.\n\nThe rooks skied away, shrieking, as the ravens rose. But as Ani's hold on the world faded, the ravens slowly became stone once more, shattering at the touch of a sword. Maledicte watched it all, sinking back against the edge of the roof, sliding down to lie on the surface.\n\nJanus knelt, pressing his hand to the wound. \"Mal\u2014\"\n\n\"I wouldn't have\u2014\" Maledicte whispered. Blood washed up and brushed his lips. \"You didn't have\u2014Why?\" His breath gave out, and he saw tears standing out in Janus's eyes. They made him angry, but his blood was too thin for the old fire to catch, too thin and spreading out over his shirt, Janus's hands, the roof tiles.\n\n\"Shh,\" Janus said, bending forward, pressing him close. It woke pain in the wound and Maledicte moaned in the dark shelter of Janus's shoulder. \"Drink this,\" Janus said, a small vial in his hand. \"Please. It'll ease the pain.\"\n\nMaledicte let the liquid trickle into his mouth, felt Janus rubbing his throat to make him swallow, and the pain receded into numbness. Ease it, he thought muzzily, when they hang him. A lover's last gift\u2014the gift of oblivion in the face of a slower death. He closed his eyes and welcomed it.\n**\u00b7 43 \u00b7**\n\n**D** O YOU SEE THE BIRDS?\" The whisper woke Gilly from his Laudable stupor. \"Did you hear the bells?\" another whore asked. Gilly could hear it now, through senses dulled and fogged, the deep tolling of the castle bell. He choked on an indrawn breath. He forced himself to his feet; something heavy and cool slithered down his legs. He caught it absently and staggered to the window. The women squeaked as he fell into the frame beside them.\n\nThe rooks were wild, flying without pattern, without sense, over the city, flickers of a dark night lingering into the dawn. _The rooks follow him,_ Mirabile had whispered. But now, they flew without purpose, without destination, and Gilly felt their panic and loss sinking into his bones. He knew, without words, without telling, that there was no one for them to follow now.\n\nHis hand clenched around the object that had been on the bed with him. He opened his palm, saw a watch engraved with sailing ships, and his breath left him, overwhelmed by despair.\n\nJANUS STOLE THROUGH THE HALLWAYS of the palace; those who saw him backed away. He couldn't blame them. He knew his temper, knew what it was like at the best of times, and this was far from being that. Those unafraid of his rages waited to see what his position would be, now that his lover had murdered the infant earl.\n\nPanic gripped his throat again. What if he had been wrong? What if he'd miscalculated? The threat of Aris deciding that Janus was to blame for Auron's death was nothing compared to this. Even had he calculated right, the window of safety was so narrow\u2014they meant to hang his body from the turrets in a few hours.\n\nThe chapel was silent and dark, empty except for Maledicte. Aris had refused to let Auron lie in the same room as his murderer, had kept Auron by his side. _But such only aids me,_ Janus thought, _keeps Aris stupid with grief, and Mal\u2014_\n\nHe hesitated near the marble bier. So white that even the dried blood on his mouth seemed scarlet instead of brown. No one had cleaned him, given him the courtesies granted the dead. No one would. But in denying the rites, they aided Maledicte one last time by keeping his secret. More, their neglect kept her life...Janus forced a smile at the irony, forced himself to believe that he had been right, that the books thieved from Gilly's possession had been accurate.\n\nJanus nerved himself to rest his hand on Maledicte's chest. It was cool to the touch, as still as marble. But when he pressed his fingers against the wound, they came away touched with fresh blood.\n\nJanus dropped to his knees. Thank you, Ani\u2014thank you. As possessive as the books stated, Ani would not relinquish Her hold while the compact remained undone. The relief unmanned him as the fear had not. It took him long moments to regain his composure.\n\nAni's strength would keep Maledicte from death, while the poison from the Itarusine court would mimic the symptoms of it. Janus had planned it to a nicety, all variables controlled, and still, it had gone wrong. Maledicte had moved. Had endured a far more lethal blow than the one Janus had intended, had let the blade bite into the heart itself.\n\nShaking, he wrapped Maledicte around with the shrouding cloth, lifted him into his arms, and headed into the hall.\n\nHe left the main halls for the servants' corridors, hurrying along, careless of noise. This deep into the old palace, the corridors were thick with dust and cobwebs. But once he reached more modern segments, he hesitated. Only one last stretch lingered between him and the stables and his waiting carriage. But a single servant now could see his plans ruined. Or a guard resting in the stables. He closed his eyes, trusting to chance and a castle steeped in mourning.\n\n\"Just a little longer, Mal,\" he said. \"Then it'll be all over. We'll have won and we'll be together.\" He forced his mouth shut; a whisper where none should be might bring a servant to investigate or overhear. And there would be mayhem enough when the discovery of Maledicte's disappearance was made. Better they think him mad with grief, determined to preserve dignity for his lover, than to even suspect that Maledicte might live yet.\n\nThe passage stayed silent, and Janus brought Maledicte out into the midmorning sunlight. Janus flinched at the brightness after the dark corridors, but Maledicte's face stayed fixed. Janus shivered.\n\nHe laid him on the seat of the carriage, and called up to the driver, \"To Lastrest, and stop for nothing.\"\n\nThe paid driver looked back at the bundle in Janus's arms and shuddered, but snapped the reins, spurring the team into a trot.\n\nJanus folded Maledicte's fingers about his own, but they refused to stay there, slipped away from him, limp and chill. Janus pushed panic away. Maledicte could still die. The wound was so deep, and if Ani's touch faded...If She forsook their bargain, Maledicte might wake only to bleed to death in his arms.\n\nJanus compared the risks of rough travel and the virtues of speed, and yelled up at the coachman to spring them. The horses surged into a gallop. In the jolting coach, Janus held Maledicte to him more tightly, shielding his body from the worst of the rattling, and thought, _Now, if only that damn boy has done what I asked._\n\nHis stomach clenched and roiled. When the rain started, he relaxed a little. The rain could only help. He wondered where the pursuit was now, whether Maledicte had been missed yet, and if so, if the guards had gone to Aris first, disturbing his solitude, or if they'd simply taken off after him. With Echo dead and Jasper dog-mauled, Janus assumed the Kingsguard would dither for some space of time before intruding on Aris.\n\nThe coach clattered through the gates of Lastrest, and Janus sprang out of it, carrying Maledicte into the house.\n\nHis sudden arrival startled the servants into action; Janus ignored them, headed for his bedroom. He laid Maledicte on the bed, then shut the door firmly behind him, latching it. \"Mal?\" he said.\n\nBut Maledicte was still white, still unresponsive, and cool to the touch. Janus winced; the serum should have worn off by now, and his control veered into panic again. What would he do if Maledicte were gone? If he had killed\u2014\n\n\"Sir?\" the boy said, the hidden door sliding open. \"You've been gone a long time.\"\n\n\"Did anyone see you?\" Janus asked, staring at the young man. Slim as a sword blade, dark-haired, and as pale as powder could make his skin\u2014he made Janus's heart clench.\n\n\"No, you said not to come out. And I've been so bored.\" The boy turned his lips down in a sullen pout and Janus laughed.\n\n\"Sorry, Mal.\"\n\n\"My name isn't\u2014\"\n\n\"It doesn't really matter,\" Janus said, standing. \"Does it?\"\n\n\"What's that?\" the young man asked, as curious as a cat, and as fickle with his attention as the whore he was.\n\n\"You,\" Janus said, unwrapping Maledicte with careful fingers. \"Come and take a look.\"\n\n\"It's a wax doll,\" the boy said, coming closer.\n\n\"No,\" Janus said, laughing as Maledicte's lips tightened, his eyelashes flickered at the light shining on his face.\n\n\"I don't like it,\" the boy said, backing away. \"I want to go back to the brothel.\"\n\nJanus fought the urge to just grab the sword. Chasing the boy around the room would do no good, but he burned to have the deed done, the time spent sealing Maledicte's wounds. \"Without being paid?\" Janus said, letting disdain slip into his voice. \"I haven't had you yet. But you've had my bed to lie in, the food and drink I gave you.\"\n\nThe young man came closer, licking his lips. \"You won't\u2014\"\n\n\"Won't what?\" Janus asked, leaning back against the bedsheets, slipping his hand onto the hilt of his sword, hidden by the shroud.\n\n\"Won't make me touch that\u2014\" The boy jerked his head toward Maledicte, still more corpse than living flesh. Janus found a hot ember of his temper left, and said, \"If you touched him, I'd have to kill you. I don't share him.\"\n\n\"Well, that's all right, then, 'cause I ain't going to touch it,\" the boy said, slipping around to Janus's side. Janus drew the boy close, kissed the soft mouth, and pressed the sword home. The boy jerked and gasped, blood spilling up into Janus's mouth, hot and salty.\n\n\"I'm sorry, Mal, I'm so sorry,\" he whispered, licking the blood away. \"I never wanted to hurt you.\" A footstep in the hall woke him to the reality. Not the rooftop again, not Maledicte in his arms, but a paid boy. Janus let him drop, wiped his mouth, and bent his attention to Maledicte. He picked Maledicte up again, passed through the door the boy had used, entered the secret room. Aris would know about it, of course, having grown up at Lastrest, but he might not think to look in it. Not with a body to be found elsewhere.\n\nJanus laid Maledicte down, shoved the debris of the boy's meal away, and picked up the medical supplies he'd laid in. He stitched the wound closed, reaching in past the skin, beneath the small breast, to sew up muscle and tendon, working with shaking hands. What if it didn't work\u2014He poured whiskey over the stitching, and Maledicte arced his back, the tendons in his throat standing out.\n\n\"Shh,\" Janus said, kissing his forehead, wrapping a layer of cotton gauze around the damaged hands where Maledicte had tried to stop the sword. It made him uneasy that these smaller wounds hadn't healed yet. \"I'll be back. Just rest.\"\n\nHe shut the door behind him, and wrapped the whore in the shroud, carried it downstairs past the gaping servants and into the gardens, wet with rain.\n\nWhen he returned to the house, mud-covered and shaking with exertion, wound reopened and bloody, Aris was waiting for him.\n\n\"Where is his body?\" Aris said; his face was lined, wet with tears and rage. \"Where is it?\"\n\nJanus knelt. \"Find him yourself. Sire.\" Aris moved forward and struck him across the face, the blow knocking Janus back across the floor. Exhausted and in no mood to feign obedience, Janus still kept control enough not to strike back, to continue his schemes. Besides, the guards even now clustered around Aris. Instead, Janus let tears spring to his eyes, and whispered, \"I loved him too well to see his body displayed, to let them bet on when the crows would take his eyes, and which bones would fall away first, to see the rabble fight over his dropped finger bones.\"\n\n\"You've buried him on the grounds? Here, on the family property of the child he's slain? He cannot stay here. And he will be hung high. As all traitors are.\" Aris paced the room, peered out through the rain-streaked glass, gestured to the guards. \"Start searching. Look for turned earth.\"\n\nThey bowed and went out. Janus sank down to a crouch again. \"Uncle, please.\" He made himself think of Maledicte outside, in the earth, trying not to let any trace of his triumph show.\n\n\"Did you know?\" Aris said, tugging Janus's face up to meet his. A very different king, this, Janus thought. No longer passive and beaten, but charged with grief and rage. \"Did you know what your lover intended?\"\n\n\"No,\" Janus said. \"No.\"\n\n\"The only thing sparing your neck is that he came up the wall, like some damned demon. And he didn't know how to escape; had you aided him, I would have expected him to sneak in through the doors and flee with more ease.\"\n\nJanus wondered if another denial would be more or less convincing than the first. He kept silent, waiting.\n\n\"I trusted Maledicte too long, played the fool. You lived in his pocket and yet claim I should trust you\u2014I wish I could believe you,\" Aris said. \"Wish I could trust my own flesh and blood, but death seems overinterested in smoothing your path.\" Aris leaned against the table, his face older than Janus had ever seen it.\n\n\"I could have you imprisoned or executed, but there is no other left of our blood. If the line of Last is to continue, it must be through you.\" Aris's hands knotted and unknotted uselessly against his coat. \"As for banishment, Itarus would be only too glad to take you in, to use you against Antyre. How would Adiran fare then? I have not the stomach to fight another war over my throne.\"\n\nAris held out a hand, face grim. Janus cautiously took it. Aris clenched his hand tight, drew Janus close. \"This is my sin\u2014and my guilt. That I would prefer a conspirator on my throne to the bloodshed that would follow a war or my death without a viable heir. So you have won, Janus. To a degree.\n\n\"You will continue as a member of my court. My third counselor. But the moment you approach Adiran, I will have the dogs at your throat. Without hesitation. Your life is now linked to his. If he contracts fever, should he suffer hurt of any kind, you will pay for it. Do you understand me, nephew?\" Aris's eyes were the cold blue of winter skies, and Janus found himself looking away first for once.\n\n\"I would never hurt Adiran,\" he said, finding his voice. There was no need. Adiran's presence could only aid him. If he were regent for the simple young man Adi would be, it would be no different from being king in name. But all those thoughts passed in a driving need to return to Maledicte, to make sure the stitches were holding as the poison wore off, to ease his pain.\n\n\"Janus, I am sick of your meaningless words. Go upstairs; you are confined to your quarters until we have recovered his body.\"\n\nJanus fought a surge of angry temper, reminding himself again that it was the boy, the bait, that they hunted. Doors slammed upstairs, and he focused his eyes on the floor, dropping into a bow, though alarm shot through him. He had expected the guards to confine their seeking to the grounds of Lastrest, ignoring the house itself. Casting a final glance at Aris's ravaged face, at the guards in the drive passing out spades, he left the room as if reluctant, even while his blood whispered, _hurry hurry._\n\nUpstairs, he found the door to his rooms open, his armoire opened, guards looking through it. He parted his lips to object and his blood froze in his veins. The opened armoire door had blocked the wall from his sight, and the hidden room's door gaped wide.\n\n\"There's blood on the bed and the floor,\" the guard said.\n\n\"I was injured earlier. Defending Auron.\" His voice was without conscious control; the entirety of his being vibrated with the need to shut the door to the hidden room, though he knew it was too late.\n\nThe guards shrugged and left, locking him in as they did, and Janus stumbled into the hidden room, finding only shadows and darkness within its narrow confines, the candles gone out and cold.\n\nJanus fumbled his numb, blind way toward the dusty chaise where he had laid Maledicte. Surely he had not recovered, was not playing a second, lethal game of cat and mouse even now.... Janus's breath caught; he bit back a sob, imagining the guards finding so much more than they expected, not a corpse but a revenant, weak but alive. Easy prey. He dropped to his knees, reaching out in entreaty, and then his hands touched clammy, sweating flesh; a whisper of a moan reached him, and when he drew his hands back, blood marked his fingers.\n\nAni's doing, Janus realized; this empty room not empty at all. The same weaving of shadow that had snared Last now spared Her vessel. He burst into laughter, tinged with hysteria, and startled himself silent. But such power She held, and Miranda held. What he couldn't do with it at his side.... He and Maledicte would rule this country as surely as they had ruled the Relicts. If Maledicte lived.\n\nJanus pressed his body against the chaise, clutching those cool, twitching fingers in his own.\n\nTHE FLAVOR OF DUST and blood filled her mouth, and a faint tang of oiled steel, as if it had risen through her veins from the wound. Pain radiated out, central, devastating, lethal. Miranda opened her eyes with bleary effort. Dim shadows draped her, and steep walls surrounded her. Too steep to be those of a coffin, and too far away. But Maledicte was dead, she knew that; Janus had done that, chasing even Ani back down into the depths of her body, cowering.\n\nRaising her hand took all her will, and she let it drop on her chest, tangling in the stiff linen stitching at the heart of that pain. The shadows pressed in, and her eyes closed; the room promenaded around her, spinning her in elaborations as fanciful as the dance steps Aris had taught her in the gardens. When she could open her eyes again, the walls had returned to the static stone that they were.\n\nA pallid gleam swam toward her, a square of darkness pivoting to birth a white-shirted figure with gilded hair.\n\n\"Don't touch that. It's so close to your heart. I thought I'd lost you.\"\n\n\"Janus\u2014\" Miranda breathed his name out on a delicate exhalation. More than the faintest motion of her chest woke wet heat and lung-locking agony.\n\n\"Shh.\" He knelt beside her, took her bandaged hand in his own. It brought a new smell to her senses, the dark scent of turned, wet earth and leaves; it overwhelmed even the blood scent of her skin and the stale exertion of his. Black earth and loam, cold soil from deep beneath the surface where the sun could never reach. Miranda knew the scent, had sniffed it at Vornatti's interment. Janus had been digging graves.\n\n\"I had it all planned, Mal. Why did you move? I almost killed you.\"\n\nMaledicte breathed. Tried to. Forced his mouth into a rictus movement, put his tongue to his teeth, shaped words without breath. \"Ani never stops fighting. I couldn't stop fighting. Even you. Thought you meant to kill me.\" He let the darkness roll over him, muffling his senses, his fingers numb on his chest, his legs as inert as lead.\n\n\"I'd never hurt you.\" Janus's hand, so hot against his cold flesh, seared him to wakefulness once more.\n\n\"Hurts now....\"\n\n\"I thought She'd heal you faster than this. That it would be a matter of minutes. Not this.\" Janus dropped his head; his hair trickled muddy water onto the pillow, splashing Maledicte's face, startling him with little bursts of sensation. When Janus raised his head again, Maledicte could see wet tracks coursing through the begrimed skin. Janus fumbled a bottle out of his shirt. \"I brought you Laudable. Do you think you can swallow?\"\n\nMaledicte said, \"Elysia\u2014\"\n\n\"I haven't any, and with Aris here, I can't send for it. I'm sorry.\"\n\nPain swept over him again, stabbing outward, throbbing, setting sweat to slicking his side. \"Send Gilly,\" he said. The threat of continuing torment scared him as nothing else had, save being buried in Stones.\n\n\"Shh.\" Janus helped Maledicte raise his head, lifting gently at his nape. A bare inch from the pillow, the movement contracted tiny muscles along his neck, his rib cage, and Maledicte lost the room to an inner blackness.\n\n\"Mal?\"\n\nParting his lips, Maledicte tried to focus. Janus trickled the Laudable in; Maledicte choked, and rolled his head, spilling most of it back out onto the pillow. Rather that than a paroxysm of coughing as it burned down his lungs. Maledicte would rather drown on the syrup than deal with that anticipated agony. A bare taste sank down his throat, scorching it, and spreading a blaze in his chest.\n\nMaledicte breathed. The room throbbed dark and darker. Janus's hands stayed steady at Maledicte's nape. When he could see Janus's face again, Janus spoke. \"Another mouthful?\"\n\nMaledicte tilted his chin a little higher, waited. \"Slower,\" he breathed, felt the glass lip rest against his tongue, the liquid spreading thinly over his tongue, coating it. Maledicte swallowed, a deeper mouthful this time, managed another, before letting the rest trickle over his cheeks and chin.\n\nThe pinpoint heat of candlelight in Miranda's eyes brought her back to awareness. Numbness spread over her body like a shroud, and Janus, bent over her, seemed only a dream. He touched the bandaging on her chest and pain flared anew. He peeled back the bandaging, washed it with spirits and salve. Miranda shuddered, and wondered if time had passed or not. If this were her death, and it would repeat forever, Janus, the wounds, the pain, the words. But the dirt was gone from Janus's face, the hair dried in awkward waves.\n\n\"You had to move. I had it all planned, a simple stroke, a clean miss of everything vital, but you had to move. I thought you loved me more than that, Mal. Knew me better than that. You thought I meant you dead....\n\n\"I don't know if you'll heal. I don't know if I can save you....\" Janus's voice cracked and faded away.\n\n\"Love you?\" Maledicte said, his breath coming a little easier now. \"I do love you, loved you for so long, my Janus. My king. But I don't know that I trust you.\"\n\nJanus's hands paused in his ministrations. \"You can always trust me.\" The quick heat in his eyes seemed brighter than the candle flames. He smoothed the bandages back down, sealing the wound closed.\n\nThe counter to that was on Maledicte's tongue, a single name, Gilly, but he swallowed it instead, let the word nourish him.\n\n\"Here,\" Janus said, bringing the glass back up to his mouth. \"Rest.\"\n\nA faint sound reached Maledicte's ears, and Janus pressed the glass to his chest. \"Not a sound.\" He disappeared back through the door, sealing Maledicte into the shadows again. Gingerly, Maledicte tipped the bottle to his mouth, swallowed several deep drafts, before letting the bottle roll away over the sheets, soaking them.\n\nSHE WOKE AGAIN TO HEAT and fever, stretching walls and stone. Trying to piece together even the simplest things\u2014who was she now? Miranda was dead, had died in the Relicts, and Maledicte had met death on the palace tower. Janus had dug his grave. The wet earth smell lingered in his nose; the sheets were muddied with it where they swept the floor. Blood spattered the floor, his blood, and tear tracks on Janus's face. Maledicte must be dead; his body lay numb and mute, cold as clay. But he could think.... A ghost, then. Some pale half-life, not one thing or another, as dusty, as empty as this room. But the fever that burned in him made him feel alive. Death wouldn't hurt so much, surely?\n\n\"And yet, no proof either way.\" Rambling, muttering aloud, the room sent his raspy voice back at him like the skitterings of rats. And yet there were other voices in the air, whispers traveling through the walls, secrets overheard by stone. He got out of the bed, forcing numb legs to react, falling, swooning, stumbling until he leaned against that whispering spot in the wall. Words filtered through, meaningless to the ghost-creature, yet he stored them in his memory just the same.\n\n\"Janus, you must tell us where you've buried him. I will not have Auron's murderer lie here. We will find him. Why delay the inevitable?\"\n\nOther voices shouted outside, their voices spiraling away in clear air, creeping in through the shielded, narrow slit near the eaves. Maledicte raised his arms, gingerly.... Were he a ghost he would fly to that spot, peer down at the scramble of living soldiers, watch them unearth his bones from the raw earth, the spades slicing the soil as his sword had sliced him from the court.\n\n\"We've found it, sir.\" A new voice, closer by, respectful.\n\nAris sighed. \"At last. Bundle it up. Take it to the palace for display.\"\n\nMy body, Maledicte thought, the Laudable's effects fading with the weary pain in Aris's voice, the quiet defeat in Janus's. \"You will not allow him to lie here.\"\n\n\"He will not lie anywhere in Antyre,\" Aris said. \"When the birds are done with him, his bones will go to the sea.\"\n\nMaledicte slid down the door, unable to stand upright, but the shock that ran through him on hitting the floor stabilized reality for him, even as it washed him with waves of breath-stealing discomfort. If he were alive\u2014\n\nThe door opened, spilling him at Janus's feet. Janus swore, lifted him into his arms. \"Are you mad? That door could have opened at any time with you leaning on it. What a sight for Aris that would have been.\" He carried him over to the chaise, and with a grimace at the stained sheets, set him down beside it.\n\n\"Who was it?\" Maledicte asked.\n\nJanus stripped the sheet from the furniture, laid out another with the awkwardness of a man who rarely had to do such things. \"Just some boy. He looked enough alike, and four days in the dirt will have helped it along.\"\n\n\"Some innocent who died because he looked like me,\" Maledicte said.\n\n\"Don't\u2014\" Janus said. \"Gilly was the worst influence on you. Giving you a veneer of morality and conscience. The boy was nothing compared to you. The moment I saw him, I knew his fate would be to spare you yours.\" Janus lifted him onto the cushioned seat; Maledicte bit his lip with the pain, and then relaxed into the softness.\n\nJanus smiled at him, \"You're doing so much better. For a while there, I thought I'd lost you. That Ani had left you and you were vulnerable.\"\n\n\"Ani,\" Maledicte said. \"No, She's still within.\" But so small, so hidden; Maledicte had to search for the spark of her presence, that black well of anger buried under the weariness and ache of his bones. \"Have you won, then?\" he asked, drowsing. \"Has Aris forgiven you for loving me? Does he believe you blameless?\"\n\nJanus sank down beside him, stroking Maledicte's matted hair. \"I am not punished, but neither am I trusted.\"\n\nLacking the energy to move his head away from the stroking fingers, Maledicte tried to push him off with words. \"You cannot seem to hold trust for long, can you?\" The heat in his voice woke answering pain in his chest.\n\nJanus paused in his caresses, then continued. \"I'm sorry you doubted me. I'll teach you, and Aris, to trust me again.\"\n\n\"Doesn't matter if I do,\" Maledicte said. \"I am a dead man after all.\"\n\n\"Shh,\" Janus said, bent close and pressed his mouth to Maledicte's. \"All will be well. We'll be where we've always wanted to be. You'll see.\"\n**\u00b7 44 \u00b7**\n\n**M** ALEDICTE WATCHED the blank walls moodily, pacing with his eyes since his body could not. His chest burned, but with the heat of healing wounds instead of outraged flesh. He stared at the window slit, back at the door. It had been silent for hours, or days; he was still not sure how much time had passed, lost in Laudable dreams and delirium.\n\nThe blood-heavy scent and the dark tang of deep grave dirt had gone from the room. The linens that draped him carried only the aromas of starch and the iron. A confectioner's assortment lay untouched beside him, one of Janus's attempts to nourish him. Maledicte opened it, but the chocolates only raised memories of drugging Gilly and Lizette's untidy death.\n\nA tray beside the bed was cold, despite the covering cloths, the teapot stained with tannin from the oversteeped leaves. The wine bottle was half full, the Laudable bottle near empty. A hunk of bread, still fresh enough to be tempting, lay beside the pot. He took a bite, though the effort to chew made his body ache. He dropped it back to the plate, and it rolled off, tumbling down a cliff of piled novels.\n\nThe whole room maddened him. It was like something out of one of Vornatti's mindless tales, the invalid girl beset by suitors' gifts and doomed to a tragic end.\n\nHe shifted gingerly to his side and when the pain, his most faithful attendant, stayed with him but pressed no closer, he foundered to his feet, breath whistling in his throat.\n\nHe stumbled the length of the room and rested against the door, seeking the catch. He forced it open with a whimper of exertion, letting himself out into Janus's empty chambers. The room was dark, the curtains drawn over the windows, and the door out of the room, when he tested it, resisted opening. Bending, Maledicte saw the key in the hole and sighed. A caged bird still. But where would he go\u2014when Maledicte had been so well-known?\n\nHe made his creeping way to the window, and pulled back the curtains a small inch. Sunlight sloping over the grounds gave him his first solid time\u2014it was early evening, with twilight closing in. A fitting time for a ghost to walk, he thought. Below, he heard carriage wheels on the oyster-shell drive, approaching without haste.\n\nIn the gardens, limned by the setting sun, servants' children, dressed neatly in patched hand-me-downs, whispered to each other and then scattered as the carriage swept by. Maledicte watched them race to their respective places, envying them their small freedoms; one boy paused and looked up at him, eyes going wide.\n\nMaledicte dropped the drape and stepped back, heart pounding. But what could the child have seen? Only a shadowy figure in a darkened room.\n\nThe key turned behind him, and Maledicte darted for the bed curtains, vision swirling with the sudden effort. He sucked his breath in, fought to stay silent when his body ached. When Janus stepped in, Maledicte released it in a rush.\n\n\"Mal\u2014\" Janus said, in a startled whisper. \"Someone might see you.\"\n\n\"In a locked room?\" Maledicte asked, sinking down onto the bed, holding the ache in his chest.\n\nJanus came to him, leaned over, and the chain of roses around his neck slipped free. Maledicte reached out and broke the string, sending petals and leaves over the sheets. \"A betrothal charm?\"\n\n\"I wed Psyke Bellane in three days, by special license and the king's decree,\" Janus said, biting back a grin. \"I presume Aris means to use her as a spy. Poor child, and how like Aris to mistake intelligence for competence.\"\n\n\"You're so clever,\" Maledicte said, lying back against the mounded pillows, slanting his forearm over his eyes. \"I always thought I was the clever one. And yet, you've gotten everything you've wanted. And I\u2014I am a ghost.\"\n\n\"Too much Laudable for you, love,\" Janus said. \"You're quite alive.\"\n\nAnger, as always, restored his strength and breath. \"But mewed up like a corpse. I am not your mistress, your lover, your courtier; I am your secret, kept behind stone walls, hidden from the servants and living on your stealthy leavings.\n\n\"Now you're to wed; you'll be off on your wedding tour, and tell me, my love, who will feed me? I am utterly dependent on your goodwill. Will you trust a man with your secret\u2014and such a dangerous, treasonous secret I am\u2014or will I creep like a rat through your home, stealing a loaf of bread here, a sausage there, and hearing the servants quarrel and split blame for their loss?\"\n\n\"Mal, enough.\" Janus kissed his mouth, sealed the complaints with his lips.\n\nMaledicte kissed back with teeth and protest, and tore his mouth away when he could, wishing he could stand and storm out.\n\n\"Haven't I made that room more a haven than a prison? The finest linens, the finest furniture, the newest books and treats. It won't be forever. Only until they forget.\"\n\nMaledicte drew himself up, long hair spilling over white flesh, livid scars on cheek and chest, acid in his voice. \"Am I so easily forgotten?\"\n\nJanus stroked the dark hair back from Maledicte's face, and Maledicte shook his head away, refusing Janus's touch.\n\n\"No glib answer for me?\" Maledicte said. \"I believe your son, should you have one, will still hear my name in whispers. I will be dead in truth long before they forget me. You've seen to that. I am the monster of ballads.\"\n\n\"There will be no wedding tour. Even could I stand to be gone from your side, Aris wants me under his eye. I will bring her here.\"\n\n\"Bring her here...\" Maledicte echoed. He stood and headed back toward the hidden chamber. \"Won't that be pleasant for you, dividing your time between your wife and your dead lover.\"\n\nHe worked the catch just as Janus approached, eyes wary, and had the satisfaction of closing the door in his face. Were Maledicte feeling stronger, he would push the table before the door and let Janus explain the noise away as best he could. As it was, he lay down on the mattress, and stared up at the bare ceiling. His hand shifted against the sheets, questing for a nearly forgotten comfort, and he sat in one quick movement, pain dismissed. Where was it?\n\nWithin him, Ani stirred for the first time since the night at the palace, hungry.\n\nHe lay back, closed his eyes, and remembered the way the hilt fit his hand, the roughness of the metal feathers caging his fingers, the easy weight of it balanced in his palm, along his arm. Maledicte closed his hand on the cold touch and opened his eyes, rolling to look at the sword. Dark with dirt and blotched with damp, it was as much a revenant as he. Maledicte wiped it off on the linens, leaving rusty trails of old blood and earth, and blew along its length. Where his breath touched, the mottled damp faded to matte black. The raven's eyes gleamed, and he curled himself around it, remembering how it had been. Miranda, under the altar, curled around her pain, and Ani's voice cresting in her mind.\n\nMALEDICTE HEARD UNEASY LAUGHTER in the air, distant but startling in the silent house, and drew himself away from his meal. The elaborate food and spun-sugar decorations had told him what day it was, even without Janus's unusual absence. Now he rose and went to the door, levered the catch open, just enough to release him from his cage. Sword in hand, he stalked through Janus's empty room, and braved the hall.\n\nThe door immediately opposite drew him. He had heard the servants moving furniture and gossiping, knew whose room it would be. Maledicte opened the door a bare fraction, giving him a narrow view of the chamber beyond. Janus teased Psyke, soothing her, leading her toward the bed. Psyke's cheeks flushed with wine and nervousness, and Janus paused to kiss the palms of her hands, making her laugh.\n\nMaledicte clenched the hilt of the sword as he watched Janus bed Psyke, enjoying himself, enjoying debauching her, teaching her things she hadn't imagined, until Psyke gasped and laughed and cried out. She kissed Janus's neck, and his eyes, bored, roamed the room, widening as he saw Maledicte. He pressed Psyke's face closer to his shoulder, blocking her view.\n\nMaledicte smiled and reached out with the sword. Not to touch bare flesh, but to push the gilded frame of the marriage portrait from the wall. It crashed to the floor; Maledicte had sealed himself back in the room before Psyke's first shrieks rang out, hiding his own laughter.\n\nWITHIN DAYS, Maledicte found himself glad of the marriage, glad to have something to occupy his time and attention. While it palled, the idea that the terror of the court was reduced to \"haunting\" a timid girl, it was more satisfying than watching her coo over Janus. Satisfying to wake her in the night by dropping a pillow over her face, touching her with a chilled hand, making her hate Lastrest, making her hate the husband who would not take her back to the city.\n\n\"You're driving her mad, Mal.\" Janus slammed the door back, burst in heedless of noise.\n\n\"Careful,\" Maledicte said. \"Do you really want your wife to hear you yelling at a ghost?\"\n\n\"Why are you doing this? I'm supposed to keep her content. Instead, she's jumping at shadows. Worse, Mal, she's beginning to ask questions. Clever ones. Little Psyke is not as foolish as she appears.\"\n\n\"Why don't you just kill her?\" Maledicte said, daring Janus to obey. The flame in Janus's eyes sparked one in his own chest, making his breathing rapid, making the wound sting and pull.\n\n\"She is a favorite of the court. She is Aris's pet. I cannot do such a thing.\"\n\n\"Will not. A harmless babe gave you no trouble. Or is it just that you fear you will have no one to blame for her death?\" Maledicte grinned.\n\n\"Mal! Stop.\"\n\nMaledicte raised the sword, tested his endurance by taking three quick fencer's steps. The pain tugged his lips back from his teeth; he let the sword tip drop.\n\n\"Where did you get that?\" Janus said, stepping away.\n\n\"It's mine,\" Maledicte said. \"You left it in the dirt.\"\n\n\"It's dangerous, Mal.\"\n\n\"Of course it is. It's a sword. You kill people with it; enemies, babies...wives.\" Maledicte set it down with a sigh. \"Unfortunately, I'm not up to strength yet. You'll have to do it yourself.\"\n\nJanus flinched, and Maledicte said, \"Too late to be squeamish, now. Is it fear that binds you? Or is the obstacle something else\u2014do you care for her?\"\n\n\"You're jealous....\"\n\n\"No,\" Maledicte denied, quick and hot, then, \"Yes. Of her position, her freedom. And she wastes it, blindly listening to you. She's a dog, not a woman. Obedient but mindless. Should I whisper in her ear at night, tell her what trusting you can lead to?\"\n\n\"I've given you everything. We have money and power, now. The safety and luxury we've always craved. What more can I do?\" Janus's temper sank, left him looking strained and miserable.\n\nMaledicte's breath caught, remembering the boy companion of the Relicts, his lover, his beloved. They had come so far, with nothing but each other.... But sentiment was not enough. He steeled himself and said, \"I have neither money nor power. I am relegated to being your prisoner. And you say you love me. If you love me\u2014\" Maledicte paused, rawness slipping into his throat, making his words more obvious than he meant. \"Find Gilly for me. Let him serve me. Be my eyes and ears while you are gone....\" Maledicte felt the tears start behind his eyes, blinked them back furiously; Gilly's absence worried at him like the wound, catching him at unexpected moments with pain.\n\n\"Gilly's dead,\" Janus said. The simple words seemed louder in the room than his previous shouting. Maledicte sucked in his breath.\n\n\"Think sense, Mal, do you really think Aris would have let him live? Your confidant, your eyes and ears.... His skull's up there next to yours.\"\n\nMaledicte shuttered his face, unwilling to let Janus see how much that hurt, how much he hated him in that moment, for telling him, for taking that last dream. He closed his eyes against tears. \"Go away.\"\n\n\"Mal, it's just you and me again. Only each other at the last, remember?\" Janus said, reaching out and pulling Maledicte into his arms.\n\nWeak with shock, Maledicte leaned up against Janus, rested his head on his shoulder, let him wrap his arms more securely about him. \"But it's not just us,\" he said in a whisper. \"It's you and the court and your wife and your king. I'm nothing now.\"\n\n\"You're everything,\" Janus said. \"Mal, this melancholy is only lingering effects of the Laudable. You've been ill, you've been hurt\u2014I confess, right now I see few paths to your freedom, but I found a way to dispose of Auron; I'll find a way to restore you to court. I depend on you. You'll stand at my side\u2014\"\n\n\"In your shadow\u2014\"\n\n\"\u2014ruling Antyre yet, my dark cavalier.\"\n\nMiranda let Janus press her back against the sheets, touching her gently. He peeled up her gown, kissed the slow-healing scar over her chest. \"We'll have you back on your feet, a force in the world again, your blade at my command...even if we have to kill everyone who ever laid eyes on you as Maledicte.\" He kissed her mouth, and she closed her eyes, listening to his promises, his body moving against hers. She ran her fingers through his silken hair, trying to convince herself that Janus's love, Janus's schemes would be enough. Only each other at the last, she thought, biting her lips. It was everything she had fought for.\n**\u00b7 45 \u00b7**\n\n**G** ILLY PASSED BY THE PALACE, his collar turned up high around his face. Despite himself he looked up, stared at the sad remnants tattered by wind and rain. He had been there when they were hung; Ma Desire hadn't been able to dose him with Laudable enough to keep him away.\n\nSwaying on his feet, heart numb, Gilly had still found a faint surprise in him as the body was exposed. So ordinary. Just an assembly of ruined flesh and bones, the personality gone with the breath. And yet\u2014some suspicion so small he couldn't name it had surfaced and sunk without ever coming to light. All he knew was that the body displayed was the body of his master, fed on by the rooks and ravens that had once followed him. Outrage had welled in him at that, the anger that Ani allowed Her feathered disciples to feast on one of Her own.\n\nHe had picked up stones, intending to knock them away, but Ma Desire had tugged him back to the brothel, kept him there for days, drunk and despairing. Kept him there through the outcry of Mirabile's body being found, kept him there through the state funeral for the infant earl. Gilly couldn't imagine attending anyway, couldn't stomach the idea of seeing Janus standing at the king's side.\n\n\"He was a wrong one,\" Ma Desire said. \"You're better off without him.\"\n\nGilly had nodded in polite obedience, but inside, he rebelled. The infant\u2014he couldn't imagine Maledicte killing the infant, no matter how hard Ani had ridden him. Especially after his triumph over the shadow boy, his baser self. That quiet belief, bitter in the face of the charges laid to Maledicte's account, slowly woke Gilly from his stupor. Antyre held nothing for him now, and the _Virga_ shipped out soon. He meant to be on it.\n\nIt had taken some time, collecting money from the banks, careful in case the Kingsguard sought him. Wasted care, Gilly realized after the first few transactions. He was only a servant, after all, in a city full of servants, invisible.\n\nThe wind shifted, bumping the ragged body against the tower with a faint rattling as of wind chimes. Some children shrieked and laughed beside him, jumped to their feet, letting their ropes reach hand to hand.\n\n_\"Maledicte lived and Maledicte died_\n\n_Only at his birth did anybody cry_\n\n_How many people did he kill?\"_\n\nThe young girl tripped on the third skip and they all switched places, started again.\n\n_\"Maledicte snooped and Maledicte pried_\n\n_Not a soul escaped the notice of his spies_\n\n_How many secrets did he buy?\"_\n\nThe boy kept the rope moving so long that Gilly, after the first flinch, tuned them out.\n\n_\"Maledicte fought with a blade so black_\n\n_Can't be beat with Ani at his back_\n\n_How many duels did he win?\"_\n\nAnother girl took over the chanting, her voice as sweet as Mirabile's, as sweet as poison. Gilly watched her skip, her curls flying.\n\n_\"Maledicte fled and sought the sky,_\n\n_Ended bent and broken, hung on high_\n\n_How many times can Maledicte die?_\n\n_One!\"_\n\nShe stopped immediately and burst out laughing.\n\nGilly felt the tears start in his eyes, realized that a guard was looking at the playing children, at his distressed face, and he tugged his coat closer and turned away.\n\nOne. Even with Ani's aid, he was only mortal. Gilly made his way down the main street, past Vornatti's town house, still closed and dark, waiting the pleasure of its Itarusine owners. He let himself in, forcing the lock on the kitchen door, and drifted through the empty house, thinking, _Here I told him stories, and made him supper, and here I played the spinet and watched them dance. Here he made me try on all of Vornatti's clothes, for fit. And here, I killed a man for him._\n\nBut the house stayed cold and shadowed, refused to be peopled with his ghosts of memories. He left the door open, walked down to the docks, sat on the quay where he and Maledicte had sat once watching the _Virga_ come in. Then he rose, and went to the shipyards to buy his ticket to the Explorations.\n\nHe left the harbormaster with his ticket and a tight throat, fighting the urge to return and request a second, pretending just a little longer that Maledicte might be coming with him.\n\nA blue-lacquered carriage passed him by, and he turned to watch it go even as he stepped into a shadowed alcove. Janus, in town? Since the murder, Janus spent all his time at Lastrest, so much so that rumor whispered he'd been banished there. Gilly followed the carriage at a distance, watched Janus hand Psyke down and follow her into the DeGuerres' estate house, smiling.\n\n\"Is he visiting?\" Gilly asked the driver, stopping to pet the horse's nose.\n\n\"He and his wife. Stopping for a fortnight. Though he'll hare off home soon enough. Doesn't like to leave Lastrest, he don't. But she's like all wives, wants the city life, the shops, the culture.\"\n\n\"I see,\" Gilly said, handing the man a luna. A fortnight. The _Virga_ didn't leave for three days. Time enough to go to Lastrest. One final pilgrimage. The place where Maledicte had found rest, even briefly.\n\nWHEN HE REINED THE HORSE to a halt, he saw children skipping in the courtyard, as they had outside the palace. The children of farmers and house servants took the opportunity of Janus's departure to play over the grounds unhindered.\n\nGilly caught only a fragment of their skipping song, enough to know it was the same one making the rounds of the city. A housemaid in a starched apron came out and slapped the eldest boy. \"You know how Lord Last feels about that one,\" she said.\n\n\"He's not here, is he?\" the boy said.\n\n\"You'll forget and sing it when he's back and then we'll be out of a job. Mind your tongue.\" She marched back to the house.\n\nGilly swung down from the horse; the boy rushed up to hold it. \"The master's not here.\"\n\n\"That's all right,\" Gilly said. \"I haven't come to see him.\"\n\n\"You're one of the Kingsguard, aren't you?\" the boy asked. \"They come every time he leaves, and snoop around. Don't know what you're looking for, do you?\"\n\n\"No,\" Gilly said. \"Not really.\" He walked up to the front door, lifted the latch.\n\n\"Hey, mister,\" the boy said. Gilly looked back.\n\n\"Watch out for that ghost. It's a mean one. Ripped up the master's room something awful....\"\n\n\"Just watch my horse,\" Gilly said. He went inside; with Janus and Psyke gone, the main hall was dark and silent, all the liveliness of the house behind the scenes in the servants' quarters. Gilly drifted up the stairs; the long portrait hall had a new picture now. Janus and his wife, the frame chipped at one edge. Gilly turned his face away, wandered into the study where Maledicte had first shown his talents for petty burglary.\n\nFor spite, Gilly palmed a letter opener, and then, reconsidering, put it back. If the servants decided he wasn't a kingsguard, he didn't want to be found a thief, either. He went upstairs along the central hall, and opened a door, one of two master rooms. This was a lady's chamber, Psyke's: fussy, ornate, and boring. He closed the door, opened the opposite door.\n\n\"Ripped it up something awful.\" The relish in the boy's voice came back. Gilly stared. The window glass was cracked, the drapes sagging down in long tatters. The decanters on the dresser had been unstoppered and tipped over, allowed to spill into opened drawers, over clothes and furbelows. A chair, its tufted back sliced open, oozed stuffing. Gilly turned at a whisper of sound, almost expecting to see Maledicte, sitting shamefaced in the midst of his wreckage, tidying after his tantrum.\n\nBut it was only the shuff of pale feathers blowing across the floor, tangling in the bed curtains like fish in nets. Gilly pulled back the bed curtains and nearly choked.\n\nThe sword. Ani's sword. Gilly found his hands shaking. The sword was driven through the mattress, feathers bleeding out of the ticking, snowing the room.\n\nJanus kept the sword, Gilly thought, aghast. He touched the hilt, and twitched as if a spark had touched his skin. The sword was warm to the touch, and fluttered against his palm. Whispering.\n\nGilly could almost hear the words. Something of pain, something of loss, of love torn away\u2014if he listened harder, maybe he would hear what had driven Maledicte. If he listened long enough, maybe Ani would hear his pain as well. The cage of the hilt warmed, shifted, making room for his fingers and palm.\n\nGilly smiled, imagining taking the blade to Janus, destroying the smug, golden monster with a single stroke and erasing the raw burn in his chest that cried protest that Janus survived when Maledicte did not. Slow images trickled behind his closed eyelids, a visual parade of encouragement\u2014Janus, on his knees before him, that gas-flame blue of his eyes dulling, as Gilly pulled the sword back. In a sudden flash, the image forced itself wider, not Ani's doing, but Gilly's own, showing him more\u2014the carcasses strewn behind him, casualties of his quest.\n\nHe jerked his hand away, scraping his knuckles and bloodying his hand. He wouldn't share his hate with Ani, or his pain, wouldn't let it twist and distort his mind the way it had Maledicte's.\n\nHe licked the blood from his hand, turning around and around in the room, tracking Maledicte's presence. It seemed so strong in the room, a scent, a warmth lingering, here in this place where his sword stood, here where he never was in life. It drove Gilly to melancholy wonderings of Janus haunted as the boy said. Janus deserved it, but Gilly wished\u2014\n\nHis horse whickered outside and he fled the room. The boy looked up at his approach. \"Did you see the ghost?\"\n\n\"No,\" Gilly said. \"I saw nothing.\" He hesitated. The boy holding the horse looked at the other children, a little distance away, and lowered his voice.\n\n\"I bet I know what you really want to see.\"\n\n\"What is that?\" Gilly said.\n\n\"Where they dug up the murderer. I'll show you for two lunas.\"\n\nGilly closed his eyes. Did he want to see that? The pain in his heart said finish it, see it all, and go. But touching the sword woke an anger in him, raising the numb curtain he had been living behind. He wanted to leave a mark here, something to hurt and sting, a last message to Janus.\n\n\"One luna,\" Gilly said, reminded of haggling with Maledicte. \"And another verse to your skipping song.\"\n\nThe children stepped closer. \"You know the song?\"\n\n\"All the verses by heart,\" Gilly said. \"Including the real end one.\"\n\n\"What is it?\" the boy said. \"If it's good, I'll show you the grave.\"\n\nThe girls lowered the rope, started it spinning. A small boy jumped into it, and Gilly spoke his lines.\n\n_\"Maledicte loved and so Maledicte died._\n\n_He never saw the truth behind his lover's lies._\n\n_How many lies can blue eyes hide?_\n\n_One, two, three\u2014\"_\n\nThe children picked it up, chanting it, skipping its measures into their memories. It would pass on, Gilly knew, from one voice to the next. Even the palace wouldn't be exempt. Faintly, Gilly smiled.\n\nThe boy holding the horse's reins watched the skipping with a thoughtful gaze. \"But what does it mean?\"\n\n\"It means there are two sides to every story. Even this one,\" Gilly said. \"Show me the grave?\"\n\nThe boy tied the horse to the gatepost, beckoned Gilly into the shadows. The site was not too far from the curve of the drive, within a stand of trees, and within sight of Janus's bedchamber window. Gilly drew his lips down. Even dead, Janus didn't want Maledicte out of his sight. A stone marker lay there, left in the dirt. The boy went as far as the shadow of the trees, then balked, obviously afraid to go closer. Gilly went the rest of the way alone.\n\nGilly touched the stone. It was plain basalt with only a name carved into it, and some wearing of the stone that looked like feathering. \"I wish you were coming with me, instead of your memory,\" he said, voice rough. \"I always hoped you would.\" It wasn't enough; like an altar, the gravestone expected offerings.\n\nFumbling in his pockets, Gilly found the receipt for his berth on the _Virga,_ and a stub of a pencil. He hesitated, then put the words to paper that he had never been given a chance to say. He folded the parchment and put it at the base of the marker, weighing it with a rock.\n\nThe horse, restless, pawed at the ground. Behind him, the children sang the song again, starting at the beginning. Gilly looked over and up; he caught a faint movement in the window of the bedchamber. If he pretended, it could have been a dark-eyed, pale face, but the thought of Maledicte linked to Janus even in death was bitter, and he looked away. He unfastened the horse's reins, and swung into the saddle, left the children practicing their song with careful volume, wary of adult chastisement.\n\nGILLY RETURNED THE HORSE TO the tavernkeeper, and asked when the next coach to the city was coming through. When the answer was dawn, he chose not to take a room, not wanting to be alone. He sat in the main tavern room until the hour grew late, and then settled himself on the bench outside, sleeping among the others too poor to take a room, or too thrifty.\n\nWhen he woke to the sound of the team's hooves, his coat was wet with morning dew, and stiff with chill. He stretched, watching his fellow passengers loom out of the morning dimness. They gathered slowly, drawn from inside the inn and from the nearby fields by the music made in the jingling harnesses and stamping hooves. The coachman and his assistants unharnessed the first team, changing horses for the trip to the city. The youngest assistant crept inside. Gilly saw him coaxing bread from the innkeeper and returned his attention to the coach, looking toward the future, and not the past.\n\nA young family waited, one whining child shifting from foot to foot until his mother picked him up and held him. A young man, either tutor or clerk, stood stiff and self-conscious in a shiny new coat. Two young men waited, dressed like the nobility, and muttering about the cost of the coach. Gilly surmised a country romp, maybe a gaming hell, and their pocket money all but gone. They'd learn better, or not; Gilly couldn't find it in him to care.\n\nA girl in a much-turned dress of bottle green, clutching a case to her side, waited near the door of the tavern, visibly nervous; her chaperone, an old woman, glared at Gilly and he looked away. The last to take shape was a maid in a heavy dark dress, coming slowly down the tree-lined path. The shadows clutched at her, only resentfully relinquishing her to the thin morning sunlight.\n\nHer thick wool dress was too big for her; she seemed bent under it. Her face was obscured by an ugly bonnet, years out of fashion, and her dark hair spilled dully from behind it. She held herself with the fragile rigidity of sickness or hurt. Consumption, Gilly thought, or simple hunger, maybe a beating\u2014He wondered what her story was, whether she was waiting for someone to arrive, or going somewhere, and if she would be welcome when she got there. He welcomed the curiosity, indulged it, knowing it heralded healing, to feel interest beyond the scope of pain, love, and vengeance.\n\nGilly was the last to get on the coach, still studying his enigma, wishing he could see her face, know if she were young or old, worn or serene. All his conclusions could be dismissed with the face. He left the little mystery with reluctance and climbed the steps into the coach. When he looked back, he saw her approach, a passenger after all. From his height he saw only the top of her bonnet. She paused at the step, and held up a thin, sinewy hand.\n\nA faint whisper, like a voice from a dream. \"Help me up, Gilly.\" She tilted her face to his. So thin, so pale, the scar white as bone, her eyes dark as the grave.\n\n_Is this what Janus felt,_ Gilly wondered, _that first night at the solstice ball?_ This heart-racing immobility? This disbelief, this impossible joy?\n\nWordless, he stretched out his hand, jolted when her hand in his was real, and not some product of his grieving heart. She climbed up beside him, her weight mostly in his hands, managing her skirts with visible impatience. She folded herself into the space beside Gilly with a gasp and a wince. He found his voice, breathless but audible. \"Mal\u2014\"\n\nHer raised hand cut him off. The horses started, and, flinching, she rocked with their movement. He slipped off his coat and tucked it behind her back, trying to cushion her body. Run through, they'd said. Through the heart.\n\n\"What's your name?\" Gilly asked, the question of years ago, new again.\n\n\"Can't you guess?\" The rasping voice was an anomaly in the maid's throat. Her lips quirked downward.\n\nGilly shook his head. He couldn't put two rumors together at this moment. She only smiled at him, and he found he had an answer after all, a collection of pieces belatedly put together. \"Miranda,\" he said.\n\n\"Miranda,\" she agreed. She wove her fingers into his, sat upright, with only that tenuous connection between them.\n\nGilly could only stare, wondering at his blindness all these years. It was as if a painting had been blurred and out of focus, and he had finally found his spectacles. After a cautious glance at the other passengers, she said, \"I own I am surprised to find you live and hale. I had heard otherwise.\"\n\n\"Your surprise is nothing to mine, I assure you,\" Gilly said, smiling like an idiot. It hadn't been a ghost after all, peering down at him, but flesh and blood. \"But how?\"\n\n\"It wasn't complete,\" she said. \"Ani would not loose me, not while my vengeance lay undone. And Janus made it impossible.\"\n\n\"The babe?\" Gilly said, his voice a whisper. The coach seemed full of prying ears, and his happiness felt fragile.\n\n\"Not by my hand,\" she said.\n\nGilly brought her fingers to his mouth, kissed them, studied the veins standing out in such relief. \"You're so thin,\" he said.\n\n\"I need looking after,\" Miranda said. \"I admit as much. Maybe I can find someone to do so.\" She curved her mouth in a way that made him want to laugh. So falsely demure.\n\n\"Where are you journeying, Miranda?\" Gilly asked, teasing in return. \"Only the city? Or farther still\u2014\" The question, asked in amusement, ended in uncertainty. He was afraid to hope, though sense told him otherwise.\n\n\"I thought to seek the Explorations,\" she said. \"I want to see some savages. Did you know they wear feathers instead of leather?\"\n\n\"I told you that,\" he said.\n\n\"Ah, that's why I believe it then, but Gilly, I hope you have money. The pawnbrokers took ruthless advantage of me; I wasn't in any fit mood for haggling. All I've got are the clothes on my back, and I'd rather not have them.\"\n\n\"Anything you want. Just tell me, will it be breeches or skirts?\"\n\n\"Skirts until we reach the Explorations. I cannot chance recognition. After that,\" she said, smiling up at Gilly, \"breeches, I think. I had forgotten how unwieldy these things can be.\"\n\nGilly's smile grew wider as another thought touched his giddy mind. \"Ship captains are notoriously choosy about female passengers. Shall I claim you as sister\u2014?\"\n\nMiranda smiled back. \"I think not. Perhaps you'd best reassure the captain. Prevail on him to marry us. What's more common than that? A wedded pair seeking a new life in a new land.\"\n\n\"I will speak to the captain as soon as we step aboard,\" Gilly vowed, laughing. Then he sobered. A question burned in his mind, one he was loath to ask, loath even to mention. But it had to be asked. \"What about Janus?\"\n\n\"Janus\u2014\" She turned her head as if she could see her past behind her along the road. \"He was taken from me and I fought to reclaim him. I would have died for him without hesitation, were it not that he was not as I remembered. Were it not for my vow. Ani knows, Ani told me, but I didn't want to listen. Where there's love, there must be vengeance. But I'm done with vengeance now.\"\n\n\"You sated Ani?\" Gilly asked.\n\n\"Ani cannot be sated. I will bear Her company for the rest of my life. Janus, clever creature that he is, saw to that. I cannot wreak vengeance on the one I vowed to love. On the one I vowed to hate, when they are one and the same. Still,\" Miranda said, sighing, \"I can leave him, and find some small measure of satisfaction there. But be warned, you take a perilous creature to wife....\"\n\n\"I'd have no other kind,\" Gilly said. He kissed her closed eyes, tasting salt in the soft tangles of her lashes, and dreamed of the sea waves ahead of them.\n**_About the Author_**\n\nLANE ROBINS was born in Miami, Florida, the daughter of two scientists, and grew up as the first human member of their menagerie. When it came time for a career, it was a hard choice between veterinarian and writer. It turned out to be far more fun to write about blood than to work with it. She received her B.A. in Creative Writing from Beloit College, and currently lives in Lawrence, Kansas.\n_Maledicte_ is a work of fiction. Names, characters, places, and incidents are the products of the author's imagination or are used fictitiously. Any resemblance to actual events, locales, or persons, living or dead, is entirely coincidental.\n\nA Del Rey Books Trade Paperback Original\n\nCopyright \u00a9 2007 by Lane Robins\n\nAll rights reserved.\n\nPublished in the United States by Del Rey Books, an imprint of The Random House Publishing Group, a division of Random House, Inc., New York.\n\nDEL REY is a registered trademark and the Del Rey colophon is a trademark of Random House, Inc.\n\nLibrary of Congress Cataloging-in-Publication Data\n\nRobins, Lane.\n\nMaledicte \/ Lane Robins.\n\np. cm.\n\n\"A Del Rey Books trade paperback original\"\u2014T.p. verso.\n\nI. Title.\n\nPS3618.03177M35 2007\n\n813'+6\u2014dc22 2006035571\n\nwww.delreybooks.com\n\neISBN: 978-0-345-50049-6\n\nv3.0\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n#  Reconstruction: A Very Short Introduction\nVERY SHORT INTRODUCTIONS are for anyone wanting a stimulating and accessible way into a new subject. They are written by experts, and have been translated into more than 45 different languages.\n\nThe series began in 1995, and now covers a wide variety of topics in every discipline. The VSI library currently contains over 550 volumes\u2014a Very Short Introduction to everything from Psychology and Philosophy of Science to American History and Relativity\u2014and continues to grow in every subject area.\n\nVery Short Introductions available now:\n\nABOLITIONISM Richard S. Newman\n\nTHE ABRAHAMIC RELIGIONS Charles L. Cohen\n\nACCOUNTING Christopher Nobes\n\nADAM SMITH Christopher J. Berry\n\nADOLESCENCE Peter K. Smith\n\nADVERTISING Winston Fletcher\n\nAESTHETICS Bence Nanay\n\nAFRICAN AMERICAN RELIGION Eddie S. Glaude Jr\n\nAFRICAN HISTORY John Parker and Richard Rathbone\n\nAFRICAN POLITICS Ian Taylor\n\nAFRICAN RELIGIONS Jacob K. Olupona\n\nAGEING Nancy A. Pachana\n\nAGNOSTICISM Robin Le Poidevin\n\nAGRICULTURE Paul Brassley and Richard Soffe\n\nALEXANDER THE GREAT Hugh Bowden\n\nALGEBRA Peter M. Higgins\n\nAMERICAN CULTURAL HISTORY Eric Avila\n\nAMERICAN FOREIGN RELATIONS Andrew Preston\n\nAMERICAN HISTORY Paul S. Boyer\n\nAMERICAN IMMIGRATION David A. Gerber\n\nAMERICAN LEGAL HISTORY G. Edward White\n\nAMERICAN NAVAL HISTORY Craig L. Symonds\n\nAMERICAN POLITICAL HISTORY Donald Critchlow\n\nAMERICAN POLITICAL PARTIES AND ELECTIONS L. Sandy Maisel\n\nAMERICAN POLITICS Richard M. Valelly\n\nTHE AMERICAN PRESIDENCY Charles O. Jones\n\nTHE AMERICAN REVOLUTION Robert J. Allison\n\nAMERICAN SLAVERY Heather Andrea Williams\n\nTHE AMERICAN WEST Stephen Aron\n\nAMERICAN WOMEN'S HISTORY Susan Ware\n\nANAESTHESIA Aidan O'Donnell\n\nANALYTIC PHILOSOPHY Michael Beaney\n\nANARCHISM Colin Ward\n\nANCIENT ASSYRIA Karen Radner\n\nANCIENT EGYPT Ian Shaw\n\nANCIENT EGYPTIAN ART AND ARCHITECTURE Christina Riggs\n\nANCIENT GREECE Paul Cartledge\n\nTHE ANCIENT NEAR EAST Amanda H. Podany\n\nANCIENT PHILOSOPHY Julia Annas\n\nANCIENT WARFARE Harry Sidebottom\n\nANGELS David Albert Jones\n\nANGLICANISM Mark Chapman\n\nTHE ANGLO-SAXON AGE John Blair\n\nANIMAL BEHAVIOUR Tristram D. Wyatt\n\nTHE ANIMAL KINGDOM Peter Holland\n\nANIMAL RIGHTS David DeGrazia\n\nTHE ANTARCTIC Klaus Dodds\n\nANTHROPOCENE Erle C. Ellis\n\nANTISEMITISM Steven Beller\n\nANXIETY Daniel Freeman and Jason Freeman\n\nTHE APOCRYPHAL GOSPELS Paul Foster\n\nAPPLIED MATHEMATICS Alain Goriely\n\nARCHAEOLOGY Paul Bahn\n\nARCHITECTURE Andrew Ballantyne\n\nARISTOCRACY William Doyle\n\nARISTOTLE Jonathan Barnes\n\nART HISTORY Dana Arnold\n\nART THEORY Cynthia Freeland\n\nARTIFICIAL INTELLIGENCE Margaret A. Boden\n\nASIAN AMERICAN HISTORY Madeline Y. Hsu\n\nASTROBIOLOGY David C. Catling\n\nASTROPHYSICS James Binney\n\nATHEISM Julian Baggini\n\nTHE ATMOSPHERE Paul I. Palmer\n\nAUGUSTINE Henry Chadwick\n\nAUSTRALIA Kenneth Morgan\n\nAUTISM Uta Frith\n\nAUTOBIOGRAPHY Laura Marcus\n\nTHE AVANT GARDE David Cottington\n\nTHE AZTECS Dav\u00edd Carrasco\n\nBABYLONIA Trevor Bryce\n\nBACTERIA Sebastian G. B. Amyes\n\nBANKING John Goddard and John O. S. Wilson\n\nBARTHES Jonathan Culler\n\nTHE BEATS David Sterritt\n\nBEAUTY Roger Scruton\n\nBEHAVIOURAL ECONOMICS Michelle Baddeley\n\nBESTSELLERS John Sutherland\n\nTHE BIBLE John Riches\n\nBIBLICAL ARCHAEOLOGY Eric H. Cline\n\nBIG DATA Dawn E. Holmes\n\nBIOGRAPHY Hermione Lee\n\nBIOMETRICS Michael Fairhurst\n\nBLACK HOLES Katherine Blundell\n\nBLOOD Chris Cooper\n\nTHE BLUES Elijah Wald\n\nTHE BODY Chris Shilling\n\nTHE BOOK OF COMMON PRAYER Brian Cummings\n\nTHE BOOK OF MORMON Terryl Givens\n\nBORDERS Alexander C. Diener and Joshua Hagen\n\nTHE BRAIN Michael O'Shea\n\nBRANDING Robert Jones\n\nTHE BRICS Andrew F. Cooper\n\nTHE BRITISH CONSTITUTION Martin Loughlin\n\nTHE BRITISH EMPIRE Ashley Jackson\n\nBRITISH POLITICS Anthony Wright\n\nBUDDHA Michael Carrithers\n\nBUDDHISM Damien Keown\n\nBUDDHIST ETHICS Damien Keown\n\nBYZANTIUM Peter Sarris\n\nC. S. LEWIS James Como\n\nCALVINISM Jon Balserak\n\nCANCER Nicholas James\n\nCAPITALISM James Fulcher\n\nCATHOLICISM Gerald O'Collins\n\nCAUSATION Stephen Mumford and Rani Lill Anjum\n\nTHE CELL Terence Allen and Graham Cowling\n\nTHE CELTS Barry Cunliffe\n\nCHAOS Leonard Smith\n\nCHARLES DICKENS Jenny Hartley\n\nCHEMISTRY Peter Atkins\n\nCHILD PSYCHOLOGY Usha Goswami\n\nCHILDREN'S LITERATURE Kimberley Reynolds\n\nCHINESE LITERATURE Sabina Knight\n\nCHOICE THEORY Michael Allingham\n\nCHRISTIAN ART Beth Williamson\n\nCHRISTIAN ETHICS D. Stephen Long\n\nCHRISTIANITY Linda Woodhead\n\nCIRCADIAN RHYTHMS Russell Foster and Leon Kreitzman\n\nCITIZENSHIP Richard Bellamy\n\nCIVIL ENGINEERING David Muir Wood\n\nCLASSICAL LITERATURE William Allan\n\nCLASSICAL MYTHOLOGY Helen Morales\n\nCLASSICS Mary Beard and John Henderson\n\nCLAUSEWITZ Michael Howard\n\nCLIMATE Mark Maslin\n\nCLIMATE CHANGE Mark Maslin\n\nCLINICAL PSYCHOLOGY Susan Llewelyn and Katie Aafjes-van Doorn\n\nCOGNITIVE NEUROSCIENCE Richard Passingham\n\nTHE COLD WAR Robert McMahon\n\nCOLONIAL AMERICA Alan Taylor\n\nCOLONIAL LATIN AMERICAN LITERATURE Rolena Adorno\n\nCOMBINATORICS Robin Wilson\n\nCOMEDY Matthew Bevis\n\nCOMMUNISM Leslie Holmes\n\nCOMPARATIVE LITERATURE Ben Hutchinson\n\nCOMPLEXITY John H. Holland\n\nTHE COMPUTER Darrel Ince\n\nCOMPUTER SCIENCE Subrata Dasgupta\n\nCONCENTRATION CAMPS Dan Stone\n\nCONFUCIANISM Daniel K. Gardner\n\nTHE CONQUISTADORS Matthew Restall and Felipe Fern\u00e1ndez-Armesto\n\nCONSCIENCE Paul Strohm\n\nCONSCIOUSNESS Susan Blackmore\n\nCONTEMPORARY ART Julian Stallabrass\n\nCONTEMPORARY FICTION Robert Eaglestone\n\nCONTINENTAL PHILOSOPHY Simon Critchley\n\nCOPERNICUS Owen Gingerich\n\nCORAL REEFS Charles Sheppard\n\nCORPORATE SOCIAL RESPONSIBILITY Jeremy Moon\n\nCORRUPTION Leslie Holmes\n\nCOSMOLOGY Peter Coles\n\nCOUNTRY MUSIC Richard Carlin\n\nCRIME FICTION Richard Bradford\n\nCRIMINAL JUSTICE Julian V. Roberts\n\nCRIMINOLOGY Tim Newburn\n\nCRITICAL THEORY Stephen Eric Bronner\n\nTHE CRUSADES Christopher Tyerman\n\nCRYPTOGRAPHY Fred Piper and Sean Murphy\n\nCRYSTALLOGRAPHY A. M. Glazer\n\nTHE CULTURAL REVOLUTION Richard Curt Kraus\n\nDADA AND SURREALISM David Hopkins\n\nDANTE Peter Hainsworth and David Robey\n\nDARWIN Jonathan Howard\n\nTHE DEAD SEA SCROLLS Timothy H. Lim\n\nDECADENCE David Weir\n\nDECOLONIZATION Dane Kennedy\n\nDEMOCRACY Bernard Crick\n\nDEMOGRAPHY Sarah Harper\n\nDEPRESSION Jan Scott and Mary Jane Tacchi\n\nDERRIDA Simon Glendinning\n\nDESCARTES Tom Sorell\n\nDESERTS Nick Middleton\n\nDESIGN John Heskett\n\nDEVELOPMENT Ian Goldin\n\nDEVELOPMENTAL BIOLOGY Lewis Wolpert\n\nTHE DEVIL Darren Oldridge\n\nDIASPORA Kevin Kenny\n\nDICTIONARIES Lynda Mugglestone\n\nDINOSAURS David Norman\n\nDIPLOMACY Joseph M. Siracusa\n\nDOCUMENTARY FILM Patricia Aufderheide\n\nDREAMING J. Allan Hobson\n\nDRUGS Les Iversen\n\nDRUIDS Barry Cunliffe\n\nDYNASTY Jeroen Duindam\n\nDYSLEXIA Margaret J. Snowling\n\nEARLY MUSIC Thomas Forrest Kelly\n\nTHE EARTH Martin Redfern\n\nEARTH SYSTEM SCIENCE Tim Lenton\n\nECONOMICS Partha Dasgupta\n\nEDUCATION Gary Thomas\n\nEGYPTIAN MYTH Geraldine Pinch\n\nEIGHTEENTH\u2011CENTURY BRITAIN Paul Langford\n\nTHE ELEMENTS Philip Ball\n\nEMOTION Dylan Evans\n\nEMPIRE Stephen Howe\n\nENERGY SYSTEMS Nick Jenkins\n\nENGELS Terrell Carver\n\nENGINEERING David Blockley\n\nTHE ENGLISH LANGUAGE Simon Horobin\n\nENGLISH LITERATURE Jonathan Bate\n\nTHE ENLIGHTENMENT John Robertson\n\nENTREPRENEURSHIP Paul Westhead and Mike Wright\n\nENVIRONMENTAL ECONOMICS Stephen Smith\n\nENVIRONMENTAL ETHICS Robin Attfield\n\nENVIRONMENTAL LAW Elizabeth Fisher\n\nENVIRONMENTAL POLITICS Andrew Dobson\n\nEPICUREANISM Catherine Wilson\n\nEPIDEMIOLOGY Rodolfo Saracci\n\nETHICS Simon Blackburn\n\nETHNOMUSICOLOGY Timothy Rice\n\nTHE ETRUSCANS Christopher Smith\n\nEUGENICS Philippa Levine\n\nTHE EUROPEAN UNION Simon Usherwood and John Pinder\n\nEUROPEAN UNION LAW Anthony Arnull\n\nEVOLUTION Brian and Deborah Charlesworth\n\nEXISTENTIALISM Thomas Flynn\n\nEXPLORATION Stewart A. Weaver\n\nEXTINCTION Paul B. Wignall\n\nTHE EYE Michael Land\n\nFAIRY TALE Marina Warner\n\nFAMILY LAW Jonathan Herring\n\nFASCISM Kevin Passmore\n\nFASHION Rebecca Arnold\n\nFEDERALISM Mark J. Rozell and Clyde Wilcox\n\nFEMINISM Margaret Walters\n\nFILM Michael Wood\n\nFILM MUSIC Kathryn Kalinak\n\nFILM NOIR James Naremore\n\nTHE FIRST WORLD WAR Michael Howard\n\nFOLK MUSIC Mark Slobin\n\nFOOD John Krebs\n\nFORENSIC PSYCHOLOGY David Canter\n\nFORENSIC SCIENCE Jim Fraser\n\nFORESTS Jaboury Ghazoul\n\nFOSSILS Keith Thomson\n\nFOUCAULT Gary Gutting\n\nTHE FOUNDING FATHERS R. B. Bernstein\n\nFRACTALS Kenneth Falconer\n\nFREE SPEECH Nigel Warburton\n\nFREE WILL Thomas Pink\n\nFREEMASONRY Andreas \u00d6nnerfors\n\nFRENCH LITERATURE John D. Lyons\n\nTHE FRENCH REVOLUTION William Doyle\n\nFREUD Anthony Storr\n\nFUNDAMENTALISM Malise Ruthven\n\nFUNGI Nicholas P. Money\n\nTHE FUTURE Jennifer M. Gidley\n\nGALAXIES John Gribbin\n\nGALILEO Stillman Drake\n\nGAME THEORY Ken Binmore\n\nGANDHI Bhikhu Parekh\n\nGARDEN HISTORY Gordon Campbell\n\nGENES Jonathan Slack\n\nGENIUS Andrew Robinson\n\nGENOMICS John Archibald\n\nGEOFFREY CHAUCER David Wallace\n\nGEOGRAPHY John Matthews and David Herbert\n\nGEOLOGY Jan Zalasiewicz\n\nGEOPHYSICS William Lowrie\n\nGEOPOLITICS Klaus Dodds\n\nGERMAN LITERATURE Nicholas Boyle\n\nGERMAN PHILOSOPHY Andrew Bowie\n\nGLACIATION David J. A. Evans\n\nGLOBAL CATASTROPHES Bill McGuire\n\nGLOBAL ECONOMIC HISTORY Robert C. Allen\n\nGLOBALIZATION Manfred Steger\n\nGOD John Bowker\n\nGOETHE Ritchie Robertson\n\nTHE GOTHIC Nick Groom\n\nGOVERNANCE Mark Bevir\n\nGRAVITY Timothy Clifton\n\nTHE GREAT DEPRESSION AND THE NEW DEAL Eric Rauchway\n\nHABERMAS James Gordon Finlayson\n\nTHE HABSBURG EMPIRE Martyn Rady\n\nHAPPINESS Daniel M. Haybron\n\nTHE HARLEM RENAISSANCE Cheryl A. Wall\n\nTHE HEBREW BIBLE AS LITERATURE Tod Linafelt\n\nHEGEL Peter Singer\n\nHEIDEGGER Michael Inwood\n\nTHE HELLENISTIC AGE Peter Thonemann\n\nHEREDITY John Waller\n\nHERMENEUTICS Jens Zimmermann\n\nHERODOTUS Jennifer T. Roberts\n\nHIEROGLYPHS Penelope Wilson\n\nHINDUISM Kim Knott\n\nHISTORY John H. Arnold\n\nTHE HISTORY OF ASTRONOMY Michael Hoskin\n\nTHE HISTORY OF CHEMISTRY William H. Brock\n\nTHE HISTORY OF CHILDHOOD James Marten\n\nTHE HISTORY OF CINEMA Geoffrey Nowell-Smith\n\nTHE HISTORY OF LIFE Michael Benton\n\nTHE HISTORY OF MATHEMATICS Jacqueline Stedall\n\nTHE HISTORY OF MEDICINE William Bynum\n\nTHE HISTORY OF PHYSICS J. L. Heilbron\n\nTHE HISTORY OF TIME Leofranc Holford\u2011Strevens\n\nHIV AND AIDS Alan Whiteside\n\nHOBBES Richard Tuck\n\nHOLLYWOOD Peter Decherney\n\nTHE HOLY ROMAN EMPIRE Joachim Whaley\n\nHOME Michael Allen Fox\n\nHOMER Barbara Graziosi\n\nHORMONES Martin Luck\n\nHUMAN ANATOMY Leslie Klenerman\n\nHUMAN EVOLUTION Bernard Wood\n\nHUMAN RIGHTS Andrew Clapham\n\nHUMANISM Stephen Law\n\nHUME A. J. Ayer\n\nHUMOUR No\u00ebl Carroll\n\nTHE ICE AGE Jamie Woodward\n\nIDENTITY Florian Coulmas\n\nIDEOLOGY Michael Freeden\n\nTHE IMMUNE SYSTEM Paul Klenerman\n\nINDIAN CINEMA Ashish Rajadhyaksha\n\nINDIAN PHILOSOPHY Sue Hamilton\n\nTHE INDUSTRIAL REVOLUTION Robert C. Allen\n\nINFECTIOUS DISEASE Marta L. Wayne and Benjamin M. Bolker\n\nINFINITY Ian Stewart\n\nINFORMATION Luciano Floridi\n\nINNOVATION Mark Dodgson and David Gann\n\nINTELLECTUAL PROPERTY Siva Vaidhyanathan\n\nINTELLIGENCE Ian J. Deary\n\nINTERNATIONAL LAW Vaughan Lowe\n\nINTERNATIONAL MIGRATION Khalid Koser\n\nINTERNATIONAL RELATIONS Paul Wilkinson\n\nINTERNATIONAL SECURITY Christopher S. Browning\n\nIRAN Ali M. Ansari\n\nISLAM Malise Ruthven\n\nISLAMIC HISTORY Adam Silverstein\n\nISOTOPES Rob Ellam\n\nITALIAN LITERATURE Peter Hainsworth and David Robey\n\nJESUS Richard Bauckham\n\nJEWISH HISTORY David N. Myers\n\nJOURNALISM Ian Hargreaves\n\nJUDAISM Norman Solomon\n\nJUNG Anthony Stevens\n\nKABBALAH Joseph Dan\n\nKAFKA Ritchie Robertson\n\nKANT Roger Scruton\n\nKEYNES Robert Skidelsky\n\nKIERKEGAARD Patrick Gardiner\n\nKNOWLEDGE Jennifer Nagel\n\nTHE KORAN Michael Cook\n\nKOREA Michael J. Seth\n\nLAKES Warwick F. Vincent\n\nLANDSCAPE ARCHITECTURE Ian H. Thompson\n\nLANDSCAPES AND GEOMORPHOLOGY Andrew Goudie and Heather Viles\n\nLANGUAGES Stephen R. Anderson\n\nLATE ANTIQUITY Gillian Clark\n\nLAW Raymond Wacks\n\nTHE LAWS OF THERMODYNAMICS Peter Atkins\n\nLEADERSHIP Keith Grint\n\nLEARNING Mark Haselgrove\n\nLEIBNIZ Maria Rosa Antognazza\n\nLEO TOLSTOY Liza Knapp\n\nLIBERALISM Michael Freeden\n\nLIGHT Ian Walmsley\n\nLINCOLN Allen C. Guelzo\n\nLINGUISTICS Peter Matthews\n\nLITERARY THEORY Jonathan Culler\n\nLOCKE John Dunn\n\nLOGIC Graham Priest\n\nLOVE Ronald de Sousa\n\nMACHIAVELLI Quentin Skinner\n\nMADNESS Andrew Scull\n\nMAGIC Owen Davies\n\nMAGNA CARTA Nicholas Vincent\n\nMAGNETISM Stephen Blundell\n\nMALTHUS Donald Winch\n\nMAMMALS T. S. Kemp\n\nMANAGEMENT John Hendry\n\nMAO Delia Davin\n\nMARINE BIOLOGY Philip V. Mladenov\n\nTHE MARQUIS DE SADE John Phillips\n\nMARTIN LUTHER Scott H. Hendrix\n\nMARTYRDOM Jolyon Mitchell\n\nMARX Peter Singer\n\nMATERIALS Christopher Hall\n\nMATHEMATICAL FINANCE Mark H. A. Davis\n\nMATHEMATICS Timothy Gowers\n\nMATTER Geoff Cottrell\n\nTHE MEANING OF LIFE Terry Eagleton\n\nMEASUREMENT David Hand\n\nMEDICAL ETHICS Michael Dunn and Tony Hope\n\nMEDICAL LAW Charles Foster\n\nMEDIEVAL BRITAIN John Gillingham and Ralph A. Griffiths\n\nMEDIEVAL LITERATURE Elaine Treharne\n\nMEDIEVAL PHILOSOPHY John Marenbon\n\nMEMORY Jonathan K. Foster\n\nMETAPHYSICS Stephen Mumford\n\nMETHODISM William J. Abraham\n\nTHE MEXICAN REVOLUTION Alan Knight\n\nMICHAEL FARADAY Frank A. J. L. James\n\nMICROBIOLOGY Nicholas P. Money\n\nMICROECONOMICS Avinash Dixit\n\nMICROSCOPY Terence Allen\n\nTHE MIDDLE AGES Miri Rubin\n\nMILITARY JUSTICE Eugene R. Fidell\n\nMILITARY STRATEGY Antulio J. Echevarria II\n\nMINERALS David Vaughan\n\nMIRACLES Yujin Nagasawa\n\nMODERN ARCHITECTURE Adam Sharr\n\nMODERN ART David Cottington\n\nMODERN CHINA Rana Mitter\n\nMODERN DRAMA Kirsten E. Shepherd-Barr\n\nMODERN FRANCE Vanessa R. Schwartz\n\nMODERN INDIA Craig Jeffrey\n\nMODERN IRELAND Senia Pa\u0161eta\n\nMODERN ITALY Anna Cento Bull\n\nMODERN JAPAN Christopher Goto-Jones\n\nMODERN LATIN AMERICAN LITERATURE Roberto Gonz\u00e1lez Echevarr\u00eda\n\nMODERN WAR Richard English\n\nMODERNISM Christopher Butler\n\nMOLECULAR BIOLOGY Aysha Divan and Janice A. Royds\n\nMOLECULES Philip Ball\n\nMONASTICISM Stephen J. Davis\n\nTHE MONGOLS Morris Rossabi\n\nMOONS David A. Rothery\n\nMORMONISM Richard Lyman Bushman\n\nMOUNTAINS Martin F. Price\n\nMUHAMMAD Jonathan A. C. Brown\n\nMULTICULTURALISM Ali Rattansi\n\nMULTILINGUALISM John C. Maher\n\nMUSIC Nicholas Cook\n\nMYTH Robert A. Segal\n\nNAPOLEON David Bell\n\nTHE NAPOLEONIC WARS Mike Rapport\n\nNATIONALISM Steven Grosby\n\nNATIVE AMERICAN LITERATURE Sean Teuton\n\nNAVIGATION Jim Bennett\n\nNAZI GERMANY Jane Caplan\n\nNELSON MANDELA Elleke Boehmer\n\nNEOLIBERALISM Manfred Steger and Ravi Roy\n\nNETWORKS Guido Caldarelli and Michele Catanzaro\n\nTHE NEW TESTAMENT Luke Timothy Johnson\n\nTHE NEW TESTAMENT AS LITERATURE Kyle Keefer\n\nNEWTON Robert Iliffe\n\nNIELS BOHR J. L. Heilbron\n\nNIETZSCHE Michael Tanner\n\nNINETEENTH\u2011CENTURY BRITAIN Christopher Harvie and H. C. G. Matthew\n\nTHE NORMAN CONQUEST George Garnett\n\nNORTH AMERICAN INDIANS Theda Perdue and Michael D. Green\n\nNORTHERN IRELAND Marc Mulholland\n\nNOTHING Frank Close\n\nNUCLEAR PHYSICS Frank Close\n\nNUCLEAR POWER Maxwell Irvine\n\nNUCLEAR WEAPONS Joseph M. Siracusa\n\nNUMBERS Peter M. Higgins\n\nNUTRITION David A. Bender\n\nOBJECTIVITY Stephen Gaukroger\n\nOCEANS Dorrik Stow\n\nTHE OLD TESTAMENT Michael D. Coogan\n\nTHE ORCHESTRA D. Kern Holoman\n\nORGANIC CHEMISTRY Graham Patrick\n\nORGANIZATIONS Mary Jo Hatch\n\nORGANIZED CRIME Georgios A. Antonopoulos and Georgios Papanicolaou\n\nORTHODOX CHRISTIANITY A. Edward Siecienski\n\nPAGANISM Owen Davies\n\nPAIN Rob Boddice\n\nTHE PALESTINIAN-ISRAELI CONFLICT Martin Bunton\n\nPANDEMICS Christian W. McMillen\n\nPARTICLE PHYSICS Frank Close\n\nPAUL E. P. Sanders\n\nPEACE Oliver P. Richmond\n\nPENTECOSTALISM William K. Kay\n\nPERCEPTION Brian Rogers\n\nTHE PERIODIC TABLE Eric R. Scerri\n\nPHILOSOPHY Edward Craig\n\nPHILOSOPHY IN THE ISLAMIC WORLD Peter Adamson\n\nPHILOSOPHY OF BIOLOGY Samir Okasha\n\nPHILOSOPHY OF LAW Raymond Wacks\n\nPHILOSOPHY OF SCIENCE Samir Okasha\n\nPHILOSOPHY OF RELIGION Tim Bayne\n\nPHOTOGRAPHY Steve Edwards\n\nPHYSICAL CHEMISTRY Peter Atkins\n\nPHYSICS Sidney Perkowitz\n\nPILGRIMAGE Ian Reader\n\nPLAGUE Paul Slack\n\nPLANETS David A. Rothery\n\nPLANTS Timothy Walker\n\nPLATE TECTONICS Peter Molnar\n\nPLATO Julia Annas\n\nPOETRY Bernard O'Donoghue\n\nPOLITICAL PHILOSOPHY David Miller\n\nPOLITICS Kenneth Minogue\n\nPOPULISM Cas Mudde and Crist\u00f3bal Rovira Kaltwasser\n\nPOSTCOLONIALISM Robert Young\n\nPOSTMODERNISM Christopher Butler\n\nPOSTSTRUCTURALISM Catherine Belsey\n\nPOVERTY Philip N. Jefferson\n\nPREHISTORY Chris Gosden\n\nPRESOCRATIC PHILOSOPHY Catherine Osborne\n\nPRIVACY Raymond Wacks\n\nPROBABILITY John Haigh\n\nPROGRESSIVISM Walter Nugent\n\nPROHIBITION W. J. Rorabaugh\n\nPROJECTS Andrew Davies\n\nPROTESTANTISM Mark A. Noll\n\nPSYCHIATRY Tom Burns\n\nPSYCHOANALYSIS Daniel Pick\n\nPSYCHOLOGY Gillian Butler and Freda McManus\n\nPSYCHOLOGY OF MUSIC Elizabeth Hellmuth Margulis\n\nPSYCHOPATHY Essi Viding\n\nPSYCHOTHERAPY Tom Burns and Eva Burns-Lundgren\n\nPUBLIC ADMINISTRATION Stella Z. Theodoulou and Ravi K. Roy\n\nPUBLIC HEALTH Virginia Berridge\n\nPURITANISM Francis J. Bremer\n\nTHE QUAKERS Pink Dandelion\n\nQUANTUM THEORY John Polkinghorne\n\nRACISM Ali Rattansi\n\nRADIOACTIVITY Claudio Tuniz\n\nRASTAFARI Ennis B. Edmonds\n\nREADING Belinda Jack\n\nTHE REAGAN REVOLUTION Gil Troy\n\nREALITY Jan Westerhoff\n\nRECONSTRUCTION Allen C. Guelzo\n\nTHE REFORMATION Peter Marshall\n\nRELATIVITY Russell Stannard\n\nRELIGION IN AMERICA Timothy Beal\n\nTHE RENAISSANCE Jerry Brotton\n\nRENAISSANCE ART Geraldine A. Johnson\n\nREPTILES T.S. Kemp\n\nREVOLUTIONS Jack A. Goldstone\n\nRHETORIC Richard Toye\n\nRISK Baruch Fischhoff and John Kadvany\n\nRITUAL Barry Stephenson\n\nRIVERS Nick Middleton\n\nROBOTICS Alan Winfield\n\nROCKS Jan Zalasiewicz\n\nROMAN BRITAIN Peter Salway\n\nTHE ROMAN EMPIRE Christopher Kelly\n\nTHE ROMAN REPUBLIC David M. Gwynn\n\nROMANTICISM Michael Ferber\n\nROUSSEAU Robert Wokler\n\nRUSSELL A. C. Grayling\n\nRUSSIAN HISTORY Geoffrey Hosking\n\nRUSSIAN LITERATURE Catriona Kelly\n\nTHE RUSSIAN REVOLUTION S. A. Smith\n\nTHE SAINTS Simon Yarrow\n\nSAVANNAS Peter A. Furley\n\nSCEPTICISM Duncan Pritchard\n\nSCHIZOPHRENIA Chris Frith and Eve Johnstone\n\nSCHOPENHAUER Christopher Janaway\n\nSCIENCE AND RELIGION Thomas Dixon\n\nSCIENCE FICTION David Seed\n\nTHE SCIENTIFIC REVOLUTION Lawrence M. Principe\n\nSCOTLAND Rab Houston\n\nSECULARISM Andrew Copson\n\nSEXUAL SELECTION Marlene Zuk and Leigh W. Simmons\n\nSEXUALITY V\u00e9ronique Mottier\n\nSHAKESPEARE'S COMEDIES Bart van Es\n\nSHAKESPEARE'S SONNETS AND POEMS Jonathan F. S. Post\n\nSHAKESPEARE'S TRAGEDIES Stanley Wells\n\nSIKHISM Eleanor Nesbitt\n\nTHE SILK ROAD James A. Millward\n\nSLANG Jonathon Green\n\nSLEEP Steven W. Lockley and Russell G. Foster\n\nSOCIAL AND CULTURAL ANTHROPOLOGY John Monaghan and Peter Just\n\nSOCIAL PSYCHOLOGY Richard J. Crisp\n\nSOCIAL WORK Sally Holland and Jonathan Scourfield\n\nSOCIALISM Michael Newman\n\nSOCIOLINGUISTICS John Edwards\n\nSOCIOLOGY Steve Bruce\n\nSOCRATES C. C. W. Taylor\n\nSOUND Mike Goldsmith\n\nSOUTHEAST ASIA James R. Rush\n\nTHE SOVIET UNION Stephen Lovell\n\nTHE SPANISH CIVIL WAR Helen Graham\n\nSPANISH LITERATURE Jo Labanyi\n\nSPINOZA Roger Scruton\n\nSPIRITUALITY Philip Sheldrake\n\nSPORT Mike Cronin\n\nSTARS Andrew King\n\nSTATISTICS David J. Hand\n\nSTEM CELLS Jonathan Slack\n\nSTOICISM Brad Inwood\n\nSTRUCTURAL ENGINEERING David Blockley\n\nSTUART BRITAIN John Morrill\n\nSUPERCONDUCTIVITY Stephen Blundell\n\nSUPERSTITION Stuart Vyse\n\nSYMMETRY Ian Stewart\n\nSYNAESTHESIA Julia Simner\n\nSYNTHETIC BIOLOGY Jamie A. Davies\n\nTAXATION Stephen Smith\n\nTEETH Peter S. Ungar\n\nTELESCOPES Geoff Cottrell\n\nTERRORISM Charles Townshend\n\nTHEATRE Marvin Carlson\n\nTHEOLOGY David F. Ford\n\nTHINKING AND REASONING Jonathan St B. T. Evans\n\nTHOMAS AQUINAS Fergus Kerr\n\nTHOUGHT Tim Bayne\n\nTIBETAN BUDDHISM Matthew T. Kapstein\n\nTIDES David George Bowers and Emyr Martyn Roberts\n\nTOCQUEVILLE Harvey C. Mansfield\n\nTOPOLOGY Richard Earl\n\nTRAGEDY Adrian Poole\n\nTRANSLATION Matthew Reynolds\n\nTHE TREATY OF VERSAILLES Michael S. Neiberg\n\nTRIGONOMETRY Glen Van Brummelen\n\nTHE TROJAN WAR Eric H. Cline\n\nTRUST Katherine Hawley\n\nTHE TUDORS John Guy\n\nTWENTIETH\u2011CENTURY BRITAIN Kenneth O. Morgan\n\nTYPOGRAPHY Paul Luna\n\nTHE UNITED NATIONS Jussi M. Hanhim\u00e4ki\n\nUNIVERSITIES AND COLLEGES David Palfreyman and Paul Temple\n\nTHE U.S. CONGRESS Donald A. Ritchie\n\nTHE U.S. CONSTITUTION David J. Bodenhamer\n\nTHE U.S. SUPREME COURT Linda Greenhouse\n\nUTILITARIANISM Katarzyna de Lazari-Radek and Peter Singer\n\nUTOPIANISM Lyman Tower Sargent\n\nVETERINARY SCIENCE James Yeates\n\nTHE VIKINGS Julian D. Richards\n\nVIRUSES Dorothy H. Crawford\n\nVOLTAIRE Nicholas Cronk\n\nWAR AND TECHNOLOGY Alex Roland\n\nWATER John Finney\n\nWAVES Mike Goldsmith\n\nWEATHER Storm Dunlop\n\nTHE WELFARE STATE David Garland\n\nWILLIAM SHAKESPEARE Stanley Wells\n\nWITCHCRAFT Malcolm Gaskill\n\nWITTGENSTEIN A. C. Grayling\n\nWORK Stephen Fineman\n\nWORLD MUSIC Philip Bohlman\n\nTHE WORLD TRADE ORGANIZATION Amrita Narlikar\n\nWORLD WAR II Gerhard L. Weinberg\n\nWRITING AND SCRIPT Andrew Robinson\n\nZIONISM Michael Stanislawski\n\nAvailable soon:\n\nSMELL Matthew Cobb\n\nTHE SUN Philip Judge\n\nAERIAL WARFARE Frank Ledwidge\n\nRENEWABLE ENERGY Nick Jelley\n\nEVOLUTIONARY PSYCHOLOGY Maryanne Fisher\n\nFor more information visit our web site\n\nwww.oup.com\/vsi\/\nAllen C. Guelzo\n\n#  Reconstruction\n\n#  A Very Short Introduction\n\nOxford University Press is a department of the University of Oxford. It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and certain other countries.\n\nPublished in the United States of America by Oxford University Press\n\n198 Madison Avenue, New York, NY 10016, United States of America.\n\n\u00a9 Oxford University Press 2018, 2020\n\nPublished in hardcover as _Reconstruction: A Concise History_ in 2018\n\nAll rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by license, or under terms agreed with the appropriate reproduction rights organization. Inquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above.\n\nYou must not circulate this work in any other form and you must impose this same condition on any acquirer.\n\nLibrary of Congress Control Number: 2019948412\n\nISBN 978\u20130\u201319\u2013045479\u20131\n\nebook ISBN 978\u20130\u201319\u2013045481\u20134\n\n1 3 5 7 9 8 6 4 2\n\nPrinted in Great Britain by Ashford Colour Press Ltd., Gosport, Hants.\n\n#  Contents\n\nAcknowledgments\n\nList of illustrations\n\nIntroduction\n\n 1 Vengeance: April\u2013December 1865\n\n 2 Alienation: December 1865\u2013March 1867\n\n 3 Arrogance: March 1867\u2013May 1868\n\n 4 Resistance: May 1868\u2013March 1869\n\n 5 Distraction: March 1869\u2013May 1872\n\n 6 Law: 1866\u20131876\n\n 7 Dissension: September 1872\u2013April 1877\n\nEpilogue\n\nTimeline\n\nReferences\n\nFurther reading\n\nIndex\n\n#  Acknowledgments\n\nSeveral talented and thoughtful historians and commentators took away time from their own schedules and their own more important tasks to read and review the manuscript of _Reconstruction_ ; I regret to admit that I have no better means of recognizing their generosity and perspicacity than these simple acknowledgments. James Oakes (City University of New York) was the first on whom I tried out the idea of Reconstruction as a bourgeois revolution, and his unfeigned interest fed the spark until it glowed. Louis Masur (Rutgers University) combed through each chapter, relentlessly sniffing out every gaffe and indiscretion, as did Jonathan White (Christopher Newport University). Michael Lind (New America Foundation), Steven Woodworth (Texas Christian University), Matthew Norman (University of Cincinnati), and Brian M. Jordan (Sam Houston State University) also volunteered themselves to read parts here and there. Any errors that might unfortunately appear in the interpretations and writing here must be blamed on me; however, any praise must be shared equally with all of those named above.\n\n#  List of illustrations\n\n[ 1 Andrew Johnson   \n_Library of Congress, LC-DIG-cwpbh-00112_](010_chapter.xhtml#actrade-9780190454791-chapter-2-figureGroup-1)\n\n[ 2 Passage of the Civil Rights Bill, 1866   \n_Architect of the Capitol_](011_chapter.xhtml#actrade-9780190454791-chapter-3-figureGroup-2)\n\n[ 3 Riots in Memphis, May 2, 1866   \n_Harper's Weekly_ , May 26, 1866](011_chapter.xhtml#actrade-9780190454791-chapter-3-figureGroup-3)\n\n[ 4 Impeachment trial of Andrew Johnson, March 4, 1868   \n_Frank Leslie's Illustrated Newspaper_ , March 28, 1868](012_chapter.xhtml#actrade-9780190454791-chapter-4-figureGroup-4)\n\n[ 5 Jury of whites and blacks   \n_Frank Leslie's Illustrated Magazine_ , November 30, 1867](013_chapter.xhtml#actrade-9780190454791-chapter-5-figureGroup-5)\n\n[ 6 Democratic party campaign card for the 1868 presidential election   \n_Author's collection_](013_chapter.xhtml#actrade-9780190454791-chapter-5-figureGroup-6)\n\n[ 7 Ku Klux Klan party in North Carolina, August 1871   \n_Courtesy of the Abraham Lincoln Foundation of the Union League of Philadelphia, Union League of Philadelphia Archives, UL.VI.4.7_](013_chapter.xhtml#actrade-9780190454791-chapter-5-figureGroup-7)\n\n[ 8 Murder of Philadelphia black political organizer Octavius Catto   \n_Washington Evening Star_ , October 12, 1871](014_chapter.xhtml#actrade-9780190454791-chapter-6-figureGroup-8)\n\n[ 9 Free blacks trying to vote in the 1868 elections, Taylor County, Georgia   \nCharles Stearns, _The Black Man of the South and the Rebels_ (1872)](016_chapter.xhtml#actrade-9780190454791-chapter-8-figureGroup-9)\n\n[ 10 Memorial to Massachusetts senator Charles Sumner   \n_Frank Leslie's Illustrated Newspaper_ , March 28, 1874](016_chapter.xhtml#actrade-9780190454791-chapter-8-figureGroup-10)\n\n#  Introduction\n\nThe era in US history known as Reconstruction forms a sort of coda to the traumatic years of the American Civil War, which lasted from 1861 to 1865. It embraces the twelve years of active effort to rebuild and reconstitute the American Union after the attempt by the Confederate States of America to secede from it, and in some sense (because it had no official starting or ending date) it spluttered on well into the 1890s.\n\nBut Reconstruction can also reasonably be characterized as the ugly duckling of American history. The twelve years that are the conventional designation of the Reconstruction period, from 1865 to 1877, teem with associations and developments that seem regrettable, if not simply baleful. They left a long legacy of bitterness, especially among Southerners who believed that they had fought an honorable war and were handed a dishonorable peace, as well as Southerners who refused to accept defeat and manufactured the myth of a glorious \"Lost Cause\" to justify themselves and their continuing belief in the rightness of the Confederate project. Reconstruction also coincided with an eruption of notorious levels of graft, corruption, and fraud in American civil governments\u2014not least in the ones erected by federal force in the former rebel states. But Reconstruction is probably best known, and least liked, as the greatest missed opportunity Americans ever had to erase the treacherous impact of slavery and race in a reconstructed and unified nation. There is, in other words, something in Reconstruction for nearly every American to regret.\n\nThe term \"reconstruction\" actually surfaced even before the Civil War began in 1861, although in its first form it was a way of describing how the Constitution would have to be amended in order to accommodate the demands of the Southern states and head off their secession. \"Sooner or later,\" predicted New York Democratic financier August Belmont, there must be \"a national convention for the reconstruction of one government over all the States.\" The term resurfaced in 1862, this time to describe the pacification policies that the federal government might deploy once the Union armies had suppressed the Confederate rebellion. Articulating these policies turned out to be no easy matter. Abraham Lincoln, the president whose inauguration had triggered Southern secession in the first place, was never at ease using the word \"reconstruction\"\u2014he qualified it with add-ons like \"what is called reconstruction\" or \"a plan of reconstruction (as the phrase goes)\"\u2014and preferred to speak of the \"re-inauguration of the national authority\" or the need to \"re-inaugurate loyal state governments.\"\n\nBut use it Lincoln did, however grudgingly, and he built all of his assumptions about the shape of Reconstruction on one single presupposition: the constitutional impossibility of secession. The Constitution granted no right to the states to secede; ergo, the Southern states had never really left the Union, and the so-called Confederate States were really only insurrectionary regimes. \"I hold, that in contemplation of universal law, and of the Constitution, the Union of these States is perpetual,\" Lincoln said in his first inaugural address. On that basis, he sanctioned the creation of a Unionist Virginia government-in-exile and installed temporary \"military governors\" in areas of Tennessee, North Carolina, and Arkansas, who proposed to carry on the usual functions of state government as though secession (and the secessionists) had never existed.\n\nBut whatever the legal and constitutional arguments, the Confederacy did not look at all like a mere insurrection, like the Whiskey Rebellion or the Dorr Rebellion. It comprised eleven contiguous states, with a population of more than nine million (Belgium, by comparison, had a population of less than five million, Scandinavia eighteen million), and it created a new government larger than most European nation-states, along with armies to defend it. The \"leaders of the South have made an army, and they have made what is more, they have made a nation,\" declared William Ewart Gladstone, the United Kingdom's chancellor of the exchequer.\n\nIronically, Gladstone's view was echoed by several of the most fervent members of Lincoln's own party, the Republicans. They had become known as \"radicals\" from the outset of the war, and though the term described only about half of the Senate's Republicans, and slightly more than that in the House of Representatives, they were unapologetic in asserting that the Confederate states had lost all title to statehood by seceding.\n\nWhat had really happened when the Southern states seceded, Massachusetts senator Charles Sumner countered, was \"State suicide or State forfeiture or State abdication.\" Whatever the name, secession had converted the old Southern states into something less than what they had been. What was more, it was the prerogative of Congress, not the president, to define that status and create the policies which control any ideas of reconstructing the Union. Thaddeus Stevens, Sumner's counterpart in the House of Representatives, wanted to go even further: \"We propose to confiscate all the estates of every rebel belligerent whose estate was worth $10,000, or whose land exceeded two hundred acres in quantity.\" As for the Confederate leadership\u2014and \"how many captive enemies it would be proper to execute, as an example to nations\"\u2014Stevens would bleakly \"leave others to judge.\"\n\nLincoln, however, wanted no part of such a draconian reconstruction. For one thing, it was tantamount to conceding that the Confederates had been correct all along in claiming a right to withdraw from the Union. Lincoln laid out his first detailed strategy for \"Amnesty and Reconstruction\" in December 1863. He proposed to grant full pardons \"with restoration of all rights of property, except as to slaves\" to any inhabitants of Southern states reoccupied by federal military forces upon taking an oath of future loyalty (and keeping \"said oath inviolate\"). There were several categories of exceptions to this opportunity: \"civil or diplomatic agents\" of the Confederacy, military and naval officers, and those who had mistreated Union prisoners of war. But the purpose behind this apparent generosity was really political, because this strategy would grant Lincoln the authority to \"re-establish a State government\" as soon as the oath-takers numbered just 10 percent of the 1860 voting population in each state\u2014while also ensuring that the \"reinaugurated\" state governments abolished slavery.\n\nMany Republicans applauded Lincoln's plan as \"glorious\" and saw it as an enticement to Southerners not only to end the rebellion but to abandon slavery as well. \"The President,\" wrote Ohio Republican James A. Garfield, \"has struck a great blow for the country and himself.\" But so broad an offer infuriated Stevens and Sumner, who interpreted this approach to mean that traitors were being invited back into the Union with full privileges and with only one significant punishment, and by a president who meant to sideline Congress in overseeing the process. Opposition to Lincoln's plan within Congress (and within Lincoln's own party) was spearheaded by Ohio Senator Benjamin Wade and Maryland Congressman Henry Winter Davis, who designed a Reconstruction plan of their own: first, the required number of oath-takers was increased from 10 to 50 percent; then, a civilian provisional governor would be appointed, and the eligible voters would elect a state convention to write a new state constitution that would ban slavery and forbid rebel officers from serving in the legislature or as governor. Only then could these states resume their proper place in the Union and send representatives and senators to Congress. \"Until majorities can be found loyal and trustworthy for state government,\" declared Wade, the South \"must be governed by a stronger hand\" than either Lincoln or its own repentant Unionists.\n\nLincoln scoffed at the Wade-Davis plan, and at the criticisms of his own plan, as tantamount to conceding that \"states whenever they please may of their own motion dissolve their connection with the Union,\" the very constitutional issue that began the war. Moreover, Reconstruction was an executive branch responsibility, just as managing the war had been; Congress had nothing to do with it, any more than it had the authority to trespass on his military authority as commander in chief. (Lincoln said nothing about the role of the federal judiciary, which had been mostly silent during the war, but the judicial branch would soon seek to join the debate, too). So, although Lincoln protested that he was not \"inflexibly committed to any single plan of restoration,\" he pocket-vetoed the Wade-Davis plan on July 8, 1864. Congressional Republicans made an effort to mollify Lincoln later that year when the House of Representatives' Committee on the Rebellious States (chaired by Henry Winter Davis) produced a Reconstruction bill that left open a shared role for the president and Congress in the process of Reconstruction. But in February 1865, opposition from uncooperative Democrats and unappeased Radical Republicans tabled it.\n\nNot that Lincoln's own schemes for Reconstructing the Confederacy had shone with any success. When Union forces overran eastern North Carolina in the spring of 1862, Lincoln appointed Edward Stanly, a North Carolina Unionist, as \"military governor\" of North Carolina, and urged him to arrange the election of a Unionist member of Congress in the occupied zone. But Stanly could recruit only 864 voters for the election, and the House of Representatives refused to seat Stanly's candidate. Stanly was also at odds with Lincoln about the Emancipation Proclamation, and on January 15, 1863, Stanly resigned. Lincoln did not appoint a successor.\n\nThis inauspicious beginning was followed by another failure. Union forces recaptured more than half of Arkansas in 1862, and on July 19 of that year, Lincoln appointed John S. Phelps as military governor. A state constitutional convention met in Little Rock in January 1864 and elected Isaac Murphy as provisional governor. But the two senators and three congressmen they elected were also refused seats in Congress. The same pattern, with still more embarrassing developments, repeated itself in Louisiana. A military governor, George Shepley, was appointed by Lincoln for Louisiana after the US Navy captured New Orleans in April, 1862, and Lincoln urged him to organize congressional elections without waiting for a new state constitution.\n\nThe two congressman elected under Shepley's oversight on December 3, 1862, managed to persuade Congress to seat them. But after a Unionist state convention wrote a free-state constitution for Louisiana in 1864, the two senators elected by the Louisiana legislature were blocked from their seats by the Senate Judiciary Committee \"till by some joint action of both Houses there shall be some recognition of an existing State government acting in harmony with the government of the United States.\" One of Lincoln's military governors, Andrew J. Hamilton, never even set foot in his home state of Texas. Only Andrew Johnson, the lone US senator from a seceding state to remain loyal to the Union, enjoyed any kind of success after Lincoln appointed him military governor of Tennessee, and even then, Johnson did not convene a new state constitutional convention until January 1865. This did not look much like the reconstruction of anything.\n\nA second fundamental problem was the future status of the freed slaves. Not a single one of the haphazard experiments in Reconstruction conducted before the end of the war had specified what the future status of the freedpeople would be. Were they now supposed to sign contracts and be paid for their labor? Who would guarantee that the contracts would not turn out to be simply a newer version of bondage? Should provision be made out of the public purse to educate them? Should they be considered citizens, and entitled to all the \"privileges and immunities\" guaranteed to citizens by Article 4, section 2, of the Constitution? And what were those \"privileges and immunities\" anyway? Office-holding, jury service, bearing witness in court, voting, election to office?\n\nOne thing was politically certain: on the day slavery was abolished, the Constitutional rules on representation in the House of Representatives would undergo a complete change. The so-called three-fifths rule in the Constitution (which had prevented slaveholding states from counting more than three-fifths of their enslaved population as part of their total population in order to determine the number of representatives each slave state could send to Congress) would disappear, and going forward, every freed slave would be counted as a full person\u2014yet without any right to vote. It was not beyond the realm of possibility that the defeated Southern states might send back to a postwar Congress, not only the same number of representatives they had been able to send before the eruption of war, but an additional thirteen representatives (by Thaddeus Stevens's reckoning) beyond what they had once elected. This unlooked-for increase in Southern representatives, if it was elected only by white Southern votes, would likely be composed entirely of old Southern Democrats who would find common cause with their old Northern Democratic allies. Together, they could put a swift end to the Republicans' wartime control of Congress and enable Congress to pass legislation repealing the Republicans' wartime domestic achievements (such as protective tariffs, government assistance to the railroads, the Homestead Act, and the national banking system), while burdening the US Treasury with the Confederacy's wartime debts. The idea of enfranchising the freed slaves would disappear entirely as a political possibility.\n\nFor that reason, Charles Sumner made a final desperate attempt to assert that \"there can be no power under the Constitution to disenfranchise loyal people... especially when it may hand over the loyal majority to the government of the disloyal minority.\" Lincoln signaled that he understood this in his last speech, on April 11, 1865. He simultaneously chided Congress for refusing to admit the Louisiana senators, and urged the Louisiana Unionists to reconsider granting at least limited black voting rights by \"extending the vote to the very intelligent, and... those who serve our cause as soldiers.\" They, at least, would be reliable safeguards for \"the jewel of liberty.\" Beyond that, however, beckoned a future he described in Richmond on April 4 as one in which free black Americans \"shall have all the rights which God has given to every other free citizen of this republic.\" As for the Confederates, he instinctively erred on the side of \"malice toward none.\" He urged his generals to \"let them have their horses to plow with, and, if you like, their guns to shoot crows with.\" He \"wanted no one punished; treat them liberally all round,\" which he believed was the only way to get \"those people to return to their allegiance to the Union and submit to the laws.\" Whatever else might have been contained in that strategy disappeared with Lincoln's assassination three days after his last speech.\n\nOne of the most-asked questions in American history must surely be, what would have happened in the Reconstruction era if Lincoln had lived? Lincoln was such a private person, and so tight-lipped a politician, that it is impossible to project what further plans he would have developed. And, if he had obeyed the unspoken two-term rule for occupying the presidency, he would only have been in office until March 1869, which is not a substantial period in which to effect something as momentous as Reconstruction. Nor were there any generally received models to guide Lincoln in such a process\u2014or, indeed, anyone else throughout history who had faced a similar dilemma. The Roman civil wars divided on postwar policy, with Sulla crushing his surviving rivals with an iron fist, but with Pompey and Caesar opting for clemency and reconciliation. Henri IV brought an end to the strife of the sixteenth-century French Wars of Religion because he could advertise himself as an outsider, willing to conciliate all parties and eager to bring peace and prosperity to all. But across the English Channel, the restored monarchy of Charles II abandoned any notion of reconciliation, and hanged, drew, and quartered even the corpse of Oliver Cromwell, not to mention his living supporters. Closer in time to Reconstruction, the Taiping Rebellion in China was suppressed in 1864 in an orgy of massacres. Had Reconstruction been planned according to the Sulla, Stuart, or Taiping scripts, then proscription, trials, and executions might have continued for another generation.\n\nBut instead, American Reconstruction wears the garb of improvisation, uncertainty, and experiment\u2014which historians have difficulty containing within narratives that thrive on direction, purpose, and determinism. The first non-participant historians of Reconstruction, James Ford Rhodes and William Archibald Dunning of Columbia University, bore down harshly on Reconstruction's missteps, largely because both were Democrats politically and had little sympathy for a Republican program. Although Rhodes and Dunning professed a kind of objective relativism, refusing to offer judgments on the faults or virtues of Reconstruction, the faults they found were usually with Republicans and the virtues Democratic. As a Progressive, Dunning (and those who followed in his train: Ulrich B. Phillips, J. G. Hamilton, Walter L. Fleming, Charles Ramsdell, and Merton Coulter) was suspicious of the follies and inefficiencies of democracy when spread too broadly. In his mind, Reconstruction brought not democracy to the South but mob rule and to Washington, nothing but vindictiveness and plunder.\n\nCriticism of the Dunning School made its first appearance in the 1930s, beginning with the attacks launched at the Dunningites by William Edward Burghardt Du Bois in _Black Reconstruction_ (1935) and James S. Allen in _Reconstruction: The Battle for Democracy_ (1937). Reconstruction might not have been a proud achievement, but, Du Bois objected, Reconstruction led directly \"to democratic government and the labor movement today.\" Allen agreed: \"The destruction of the slave power was the basis for real national unity and the further development of capitalism, which would produce conditions most favorable for the growth of the labor movement.\"\n\nUnhappily, neither Du Bois nor Allen possessed a broad platform from which to rally a countermovement, partly because of Du Bois's imperious self-isolation and Allen's identification with the Communist Party. It would not be until the 1960s, after the emergence of the civil rights movement as a \"second Reconstruction,\" that the idols of the Dunning School began to fall. John Hope Franklin's _Reconstruction after the Civil War_ (1961) and Kenneth Stampp's _The Era of Reconstruction, 1865\u201377_ (1965) started the trend, to be followed by John and LaWanda Cox, George Bentley, Richard Current, Allen W. Trelease, Herman Belz, and finally by Eric Foner's massive _Reconstruction: America's Unfinished Revolution, 1863\u20131877_ (1988).\n\nNoble as their intentions were, the anti-Dunningites had their foibles, too. Du Bois and Allen were both writing from self-consciously Marxist frameworks that forbade any other understanding of Reconstruction but through class and revolution, with race sometimes deployed as a surrogate for class. \"The emancipation of man is the emancipation of labor,\" Du Bois wrote, \"and the emancipation of labor is the freeing of that basic majority of workers who are yellow, brown and black.\" Thus, Reconstruction should have been the moment when working-class blacks and whites together had an opportunity to create a new American economic and political order, only to have it yanked away by a nervous white Northern bourgeoisie who preferred making peace with the defeated Confederates to licensing a genuinely radical, biracial workers' movement.\n\nReconstruction was, in other words, seen as a typical bourgeois-democratic revolution, not unlike the initial phase of the French Revolution in 1789 or the liberal democratic revolutions across Europe in 1848. Like those uprisings, it pitted a capitalist, middle-class bourgeoisie over a slaveholding aristocracy, with the former striking up alliances with peasants on the land and workers in the tenements to overthrow the rule of the planter elite. This newly empowered bourgeoisie derived their authority, first, as the Southern Unionist allies of Union military victory, and second as the owners of capital and the possessors of professional and commercial income (rather than birthright or status). Alas, bourgeois revolutions frighten their own architects, who quickly come to see that in encouraging peasants and workers, they have created a Frankenstein monster that has no more respect for the bourgeoisie than it had for the aristocrats.\n\nAt that moment, \"the bourgeoisie,\" wrote Lenin, \"strives to put an end to the bourgeois revolution halfway from its destination, when freedom has been only half won, by a deal with the old authorities and the landlords.\" They strive \"to reach a tacit pact with the old-landed aristocracy in order to preserve their power.\" But the genie cannot be stuffed back into the bottle; it is only stunned, and in time it will reawaken with renewed strength as the guide and leader of the socialist revolution, and finish off industrial capitalism, just as the bourgeoisie finished off the aristocrats. Du Bois in particular bears the impress of this notion of Reconstruction as a bourgeois revolution, for in Du Bois's telling, Reconstruction's \"vision of democracy across racial lines\" was undone by a \"counterrevolution of property.\"\n\nThe principal difficulty with such a conventional Marxist narrative is that neither the Civil War nor Reconstruction fit neatly into it. Both the Civil War and Reconstruction belong to a chapter in American history in which the United States was still an overwhelmingly agricultural economy, and the contest that was waged between 1861 and 1865 was largely an argument (in economic terms) between the free-labor family farm and the slave-labor cotton plantation.\n\nNor is there any evidence that the victorious Republicans who attempted to build a bourgeois South among the ruins of the old plantation order ever panicked at the prospect of empowering blacks or poor whites, or betrayed them by establishing a self-protecting alliance with the dethroned aristocrats. And the freedpeople hardly experienced a taste of Marxist alienation; they instead experienced bourgeois frustration at their exclusion from material accumulation and democratic and judicial process, and that was how they articulated it. If Reconstruction was indeed a bourgeois revolution, it was a _pure_ bourgeois revolution\u2014a self-contained revolutionary event outside the boundaries of Marxist theory. And if it failed, it was not because it sold out, but because it was overthrown by the resurgent political power of a bloodied but unbowed aristocracy.\n\nIt was also easy, in the midst of so many shortcomings and failures in Reconstruction, for the anti-Dunningites to overlook four important ways in which Reconstruction actually succeeded:\n\n\u2022 Reconstruction restored a federal Union, for which the North had been fighting from the start, and corrected the centrifugal forces of the American federal Union that had brought on the war in the first place.\n\n\u2022 Reconstruction followed the route of generosity\u2014it created no conquered provinces, no mass executions for treason. As Walt Whitman wrote, almost in self-congratulation, Reconstruction \"has been paralleled nowhere in the world\u2014in any other country on the globe the whole batch of the Confederate leaders would have had their heads cut off.\" Ironically, most of the violence that pockmarked Reconstruction was inflicted on the victors, not the vanquished.\n\n\u2022 The freedpeople made only modest economic gains in moving out of the shadow of slavery into freedom and self-ownership. But there were still beachheads for black Southerners all across the South in terms of property ownership and embourgeoisment, which would form the soil out of which the civil rights movement would flourish eighty years later.\n\n\u2022 In the same fashion, Reconstruction established, beyond a doubt, the legal equality of all Americans under the banner of citizenship. Much of that equality was compromised by racial prejudice, vigilante violence, and the twisting of law. But it was not extinguished, and the Reconstruction-era amendments to the Constitution (the Thirteenth, Fourteenth, and Fifteenth) have together formed the last on which injustice, racial prejudice, and inequality have repeatedly been hammered down.\n\nNot everything that should have been gained was gained in Reconstruction; but not everything was lost, either.\n\nHistorical writing on Reconstruction has expanded exponentially since the 1960s, pushing the boundaries of Reconstruction historiography into new subfields of time, labor, geography, gender, family, and economics. The American West has increasingly become part of the Reconstruction schema, starting with the military subjugation of the Plains Indian tribes, but increasingly reaching to include the challenge posed to an American Protestant culture by the Mormon colony of Utah and the racial triangle formed by Cherokee slavery and segregation. There is now a greater sense of the continuity of Reconstruction backward to the war years (as in, for example, connecting the activities of Civil War guerrillas with the postwar insurgency of the Ku Klux Klan) and forward to the civil rights movement of the 1960s. Studies of the experience of veterans, both Northern and Southern, in Reconstruction have only just begun to come to the fore, as have also inquiries into how, with a certain postmodern twist, Reconstruction influenced the memory of the Civil War and, in addition, generated its own historical memory. Even the conventional chronology of Reconstruction has been reworked, so that in some cases the starting point has been pushed back well into the Civil War years, and in others substantially far forward, to the beginning of Jim Crow segregation in the 1890s. For the purposes here, it will be easiest if Reconstruction is thought of as four discrete movements:\n\n\u2022 Early Reconstruction, from the first of Lincoln's experiments in 1862 until the announcement of Andrew Johnson's appointment of provisional governors for the Southern states in 1865;\n\n\u2022 Presidential (or executive) Reconstruction, covering the short-lived Johnson governments, from mid-1865 to the passage of the Congressional Reconstruction Acts in 1867, which attempted to curtail the liberties of the freed slaves and return ex-Confederates to Washington as members of Congress;\n\n\u2022 Congressional Reconstruction, which begins with the Reconstruction Acts and concludes with the readmission, under the terms of those statutes, of the last of the one-time Confederate states to representation in Congress in 1870, during which time the Fourteenth and Fifteenth Amendments to the Constitution empowered the emergence of a black political leadership class; and\n\n\u2022 Overthrow of Reconstruction, from 1870, when the first white Democratic regimes were elected to \"redeem\" the southern states from Republican control, until 1877, when the last Southern Republican governments, in South Carolina and Louisiana, were extinguished.\n\nOne may also speak of an \"Aftermath\" of Reconstruction, from 1877 until 1896, to include the increasingly oppressive nature of the \"Redeemer\" regimes, the capture of both houses of Congress and the executive branch by Democrats in 1893, and the disastrous decision in the case of _Plessy v. Ferguson_ to give federal countenance to racial segregation. Some of these categories are porous: Virginia, for instance, was readmitted under the Reconstruction Acts in 1870, but had already returned the Democratic party to effective power four months before, only to witness the rise of a \"Readjuster\" movement in the 1880s which fused black Republicans and moderate Democrats; and Tennessee was never included in the Reconstruction Acts and was readmitted to Congress in 1866. But they will serve as a general timeline.\n\nAwareness of the increasing varieties of Reconstruction interpretation will not prevent _Reconstruction: A Concise History_ from committing some offenses, mostly in the interest of remaining short. Extensive explorations of gender, family, veterans, philosophy, literature\u2014all of them, unhappily, are beyond the scope of a series whose volumes, to be fair, are intended to each offer but a brief introduction to the topic at hand. This small offering will attempt no more than to fashion a basic scaffolding for understanding Reconstruction, leaving the vaster structures of elaboration and interpretation to improvised _ad libitum_.\n\n# Chapter 1\n\n#  Vengeance: April\u2013December 1865\n\n\"Today the city is wild with grief and anger over the report that President Lincoln was assassinated last night by a Southerner, John Wilkes Booth,\" wrote a newly discharged Union lieutenant in Lowell, Massachusetts, on April 15, 1865. \"All flags are at half-mast, public and private buildings draped in black, and business is suspended.... It is all a man's life is worth to show any feeling but sorrow or anger.\" The scene in Lowell was repeated across the North that day. In Philadelphia, \"old men bowed their heads in sorrow and wept like children.\" On every block of Chestnut Street, between Third and Thirteenth, \"the whole street, as far as the eye could stretch, was wailing the loss of the Chief Magistrate of the nation.\" And not the least in mourning was Washington, DC, where only a few hours were allowed to elapse in the capital before Lincoln's vice president, Andrew Johnson, was sworn in as the seventeenth president by Chief Justice Salmon P. Chase.\n\nJohnson had not been an obvious choice as Lincoln's vice president. When Lincoln was elected president in 1860, his vice president was Hannibal Hamlin, a Maine abolitionist who had left the Democratic Party for the Republicans. But Hamlin had not been a consequential figure, and in 1864 the Republican National Convention substituted Johnson, a lifelong Tennessee Democrat, on Lincoln's reelection ticket as a way of demonstrating the Republicans' bipartisan dedication to re-union. Although Johnson never openly embraced the Republicans, he had earned warm applause in the North for his lonely refusal, at the beginning of the war, to abandon his seat in the US Senate and follow Tennessee into secession. He also had the advantage of having been reasonably successful as Lincoln's military governor of Tennessee from 1862 to 1865, and he had commended himself to the most radical members of Lincoln's Republican Party for promising black Tennesseans in October 1864 that he would be \"your Moses, and lead you through the Red Sea of war and bondage to a fairer future of liberty and peace.\"\n\nAt the same time, Johnson was also a loner, an outsider in Washington politics who concealed his insecurities behind a wall of quiet. Lincoln was taken aback when Johnson suggested that political business in Tennessee should excuse him from attending his own inauguration, and he nearly sank himself into political oblivion when he did show up for the inauguration, visibly intoxicated, and proceeded to deliver a rambling monologue that sent eyes rolling helplessly upward. \"This Johnson is a queer man,\" Lincoln remarked, and in April, when Lincoln visited Richmond, he ignored a suggestion that he and Johnson meet in the ruined ashes of the Confederate capital.\n\nAny uncertainties Johnson generated, though, were banished in the wake of Lincoln's assassination by Johnson's promises to deal harshly with the guilty Confederacy. Indeed, it was not only Lincoln's death that motivated Johnson; John Langdon Sibley, the librarian of Harvard College, was deeply depressed to see the swelling numbers of \"infirm men & cripples in the streets.... The terrible battles which have been fought have been turning out more & more, & it is painful to see the large number... who have lost an arm or a leg or are otherwise maimed & move on crutches.\" When the members of Congress's Joint Committee on the Conduct of the War called on Johnson the day after Lincoln's death, he assured them that he would deal harshly with the rebels who had been responsible for the war and the assassination. \"Robbery is a crime; rape is a crime; murder is a crime; _treason_ is a crime, and _crime_ must be punished.\" The perpetrators of those crimes included many more than just John Wilkes Booth (who was still at large). Johnson continued, \"Treason must be made infamous and traitors must be impoverished.\" Radical Republicans were delighted: the Confederacy, wrote Michigan senator Jacob Howard, \"should be made to bear the brand of dishonor & enduring shame.\"\n\nHow this sentiment was to be translated into policy was another question, however. Reconstruction of the Union would require dealing with a thorny hedge of legal, constitutional, and political questions. Were the eleven states that seceded to form the Confederacy in 1861 still states in any meaningful sense of the term? Radical Republicans certainly thought not, and now that Lincoln was no more, the way seemed clear for Congress to step forward and reshape the South in the same way the Constitution authorized Congress to oversee the federal territories in the West. And what was the standing of the Confederacy's ordinary legal actions? Was the federal government obligated to assume the debts that the Confederate states had contracted during the war? Above all, what was to become of the 3.9 million emancipated African Americans whose legal status as chattel had been extinguished by the war, by Lincoln's Emancipation Proclamation, and by the Thirteenth Amendment? \"Shall they be men; be clothed with the rights and duties of freemen,\" asked the Union League of Philadelphia, \"or shall they be returned to a worse slavery than that from which we have freed them?... Shall our old notions about race and color shut our eyes to the manifest march of the times or shall we accept and solve the problem with truth and reason?\"\n\nFor the moment, Andrew Johnson had to concentrate his attention on ending the war. The rebel army surrendered by Robert E. Lee, and the paroled by General Ulysses S. Grant, at Appomattox Court House on April 9, 1865, was the most important, but not the only, Confederate army still in existence. Negotiations were already under way for the surrender of the next-largest Confederate force, in North Carolina. Meanwhile, the assassin Booth dodged federal pursuers until he was cornered in a tobacco barn in Virginia on April 26 and shot to death, and Confederate president Jefferson Davis and his cabinet would remain beyond federal reach until Union cavalry captured them near Irwinville, Georgia, on May 10. Two other Confederate commands in Alabama and Texas would not surrender until May 4 and May 26.\n\n1. In 1865, Andrew Johnson was described by the newspaper editor Alexander K. McClure as \"about five feet ten in height, rather stoutly and symmetrically built,... rather a cold grey eye that looks as if in its calmest glances there slumbers behind it quite enough to quicken it; a finely chiseled Roman face, usually sad in expression, at time relieved by a genial smile, and in manner and dress serenely plain and unaffected.\"\n\nUnlike the rebel parolees at Appomattox, few of these other defeated Confederates had much hope for mercy, especially from Andrew Johnson. The new president had already authorized the trial of John Wilkes Booth's collaborators in a military tribunal rather than in a civil court, and the only question seemed to be how many of them would be hanged. When Union general William T. Sherman appeared to have granted excessively lenient terms to the Confederate army in North Carolina that included \"recognition\" of the wartime Confederate state governments and the \"practical\" cancellation of \"the confiscation laws, and relieved rebels of every degree... from all pains and penalties for their crimes,\" Sherman was swiftly rebuked and ordered to renegotiate the surrender. It was \"clear and settled\" in Johnson's mind \"that no civil authority should be recognized which has its source in rebel election or appointment.\" This gave Johnson all the appearance of an avenging angel, and even hesitant Northerners suspected that \"the presidential chair is occupied by a man who has pledged himself to make the blood of the educated and influential rebels run in streams.\"\n\nBut despite his newfound radical reputation, Johnson's bark was very different from his bite. Lincoln and the leadership of the Republican radicals had been Whigs (before 1856, when the Whig Party collapsed over the slavery issue); Johnson was, as one English observer shrewdly perceived, \"a Democrat... of the [Andrew] Jackson type\" who believed \"that freedom in the United States ought to tend toward social equality\"\u2014although the equality Jacksonians had in mind did not extend across the divide of race. He \"had a latent hostility towards money-power, and the aggregation of property, as essentially aristocratic.\" But he had owned slaves himself and (as Indiana congressman George W. Julian discovered) \"was, at heart, as decided a hater of the negro and of everything savoring of abolitionism, as the rebels from whom he had separated.\" His real animus was directed at the plantation nabobs, and he extended his grudge against the planter class to Northern financiers. \"The aristocracy based on negro property disappears at the Southern end of the line,\" he snarled, \"but only to reappear in an oligarchy of bonds and national securities in the States which suppressed the rebellion.\" Although he did not mind promising to be the freedmen's Moses, he did not expect the Promised Land to make them the social or political equals of whites. \"I am for a _white_ Mans [ _sic_ ] Government in America,\" he told a Tennessee supporter.\n\nGradually, Johnson began rolling back the wartime measures Lincoln had improvised from 1861 onward. At first, these rollbacks were uncontroversial: on April 29, he lifted the sanctions on trade with the former Confederate states east of the Mississippi; on May 9, he formally recognized the Virginia government-in-exile (which had been established in 1862 under Francis Pierpont) as the new Virginia state government; and on May 22, he formally dissolved the US Navy's blockade of Confederate ports. He gave particular cheer to Radical Republican hearts on May 15, when he appointed the \"Christian soldier,\" the unapologetically evangelical Major General Oliver Otis Howard, to head the newly created Bureau of Refugees, Freedmen and Abandoned Lands (which became known simply as the Freedmen's Bureau) and begin redistributing to the freedpeople \"all the abandoned lands, and those in the Government's possession under the Confiscation Act, and all those acquired by Treasury tax sales or otherwise.\"\n\nThe cheer began to dissipate, however, on May 29, when Johnson issued a proclamation, built on Lincoln's December 1863 amnesty proclamation, granting \"to all persons who have, directly or indirectly, participated in the existing rebellion....amnesty and pardon, with restoration of all rights and property, except as to slaves and except in cases where legal proceedings under the laws of the United States providing for the confiscation of property... have been instituted.\" There followed a long list of exclusions\u2014\"civil or diplomatic officers\" of the Confederacy, \"military or naval officers... above the rank of colonel in the army or lieutenant in the navy,\" and any Southerners \"who have voluntarily participated in said rebellion... whose taxable property is over $20,000.\" But even the most notorious could still make \"special application\" to Johnson for executive clemency.\n\nThe intent of the proclamation puzzled congressional Republicans. \"It was supposed that President Johnson would err, if at all, in imposing too harsh terms\" on the rebels, recalled Ohio US senator John Sherman. After all, Johnson himself repeatedly claimed that he wanted \"the prompt initiation of legal proceedings against the leaders of the civil war.\" But as it turned out, what he really wanted was for them to come to him on bended knee and experience some of the humiliation he had lived with as a \"plebian.\" After that, he was more than satisfied to trade the garb of Moses for that of Pharoah. When John Eaton, who had been the general commissioner of freedmen in Tennessee, pressed Johnson about \"establishing schools and organizing new industries\" for the freedmen, Johnson's habitual curtain of silence descended. \"He was quite obviously bored, and all that might have been said on the subject had no more inclination to stay by him than has water to stay on a duck's back.\"\n\nPuzzlement turned to anger with the release of a second presidential proclamation, this time authorizing the appointment of a civilian \"provisional governor\" for North Carolina. The person chosen for this new position was William W. Holden, who had voted in favor of secession in 1861 but then transformed himself into a \"peace\" candidate for governor of North Carolina in 1864. Holden was charged with \"convening a convention... for the purpose of altering or amending\" the state constitution and \"to restore said State to its constitutional relations to the Federal government.\" In short order, through June and July, Johnson proceeded to do likewise for Mississippi (where he appointed as governor William L. Sharkey, \"a man of mind and culture,\" a judge and former Whig, but \"offensive... on the subject of slavery\"), Georgia (where he installed James Johnson, a Unionist lawyer and personal friend from Johnson's days in Congress, in the governorship), Texas (where Andrew Hamilton, who had fumbled Lincoln's appointment as a military governor of Texas, now became provisional civilian governor), Alabama (Lewis Parsons, a New York\u2013born lawyer who had briefly held a Confederate army commission), South Carolina (Benjamin F. Perry, a Unionist newspaper editor who had also agreed to serve as a Confederate judge), and finally Florida (here the nominee was William Marvin, a former federal district judge who had actually spent most of the war practicing law in New York City).\n\nIt took some convincing to believe that these nominees were the best men to reestablish the vacant courts and the wrecked machinery of day-to-day governance, to summon state conventions, and to encourage the conventions to erase all traces of slavery from the state statute books. \"For Gods [ _sic_ ] Sake appoint a Sober man,\" erupted a fellow Tennessee Unionist, \"instead of A.J. Hamilton; better known as drunken Jack Hamilton.\" What was more appalling was the realization that Johnson's governors promised to do little more than return the South, and the freed slaves, to a status only marginally different from what had prevailed before the war. These \"quondam rebels,\" complained a Louisianan who had suffered real \"rebel persecution,\" may \"talk like union men and have ears like union men but they don't smell much like union men.\"\n\nWhat incensed Republicans in particular was that Johnson set up civilian governments without so much as a by-your-leave to Congress, which had closed the last session of the Thirty-Eighth Congress on March 3, 1865 (the Senate stayed in special session until March 14) and would not reconvene as the new Thirty-Ninth Congress until December. It was one thing to create military governors, since their authority came strictly from Johnson's constitutional position as commander in chief of the army and navy and would expire the moment Johnson declared the war emergency over. But appointing _civilian_ governors was a gesture ominously similar to Congress's authority to appoint civilian governors for the territories, and it implied that Johnson was reaching for much more power than Lincoln had ever dreamt or that Congress was willing to relinquish. \"How the executive can remoddle [sic] the _States in the union_ is past my comprehension,\" roared Thaddeus Stevens. He could see \"how he could govern them through military governors until they are recognized,\" but civilian governments were \"a question for the Legislative power exclusively.\" But worse still were the omissions in Johnson's proclamations: not a word about the Confederate and state war debts, nor any eligibility restrictions on ex-Confederates who had received amnesty or who would receive it after applying to Johnson, nor the slightest indication that the state constitutional conventions were obliged to consider the civil rights\u2014and especially voting rights\u2014of the freedmen.\n\nThe \u00e9migr\u00e9 Republican Carl Schurz, who had spent most of the war as a Union general, chided Johnson for proclamations that have \"been generally interpreted as a declaration of policy on your part adverse to the introduction of negro suffrage.... The old pro-slavery and disloyal element, I mean the oath-taking rebels,\" he predicted, would seize control. To Charles Sumner, it seemed pure political apostasy. \"Andrew Johnson is the impersonation of the tyrannical slave power. In him it lives again.\" But Johnson waved the protests away. He had never stopped considering himself a Southerner and \"cherished the kindest feeling towards the people of the Southern States.\" He told a South Carolina delegation that Reconstruction was really a \"question of restoration,\" and he expected that the procedures described in his proclamations would so accelerate that process that the entire work of Reconstruction might be finished by the time Congress reconvened.\n\nTo recalcitrant Southerners, the proclamations were like a second wind. In the weeks after Appomattox, Hannah Rawlins (of Orange County, Virginia) confessed to feeling \"as if it could not be reality, but... some hideous nightmare\" that the Confederacy was no more. Stunned into passivity by the surrender, Rawlins remembered that \"for three days after we learned of the fate of our devoted army, I don't think there were a dozen sentences spoken in the family where I was. A stranger would have thought there was a corpse in the house.\" But the passivity was only temporary and not the same thing as acquiescence. The war had pushed many Southerners beyond the point of reconciliation to their conquerors. \"A lady near by,\" recorded Confederate War Department clerk J. B. Jones, \"while surveying her dilapidated shoes, and the tattered sleeping gowns of her children, burst forth... 'I pray that I may live to see the United States involved in a war with some foreign power, which will make refugees of her people, and lay her cities in ashes!' \" One Virginia woman believed that \"the feeling here against the North is intense, tho' smothered.... Mothers will teach their young children to abhor the slayers of their fathers and brothers, they will teach it to them from their earliest infancy. Had I sons, this is the religion that I would inculcate from the time they could lisp.\" The Johnson proclamations revived Southern hopes for pulling some form of victory back from the abyss of defeat and seemed to the Cincinnati journalist Whitelaw Reid \"to have called into active utterance all the hostility to Northerners.\" Strategies of resistance now began to take substance, and Confederate veteran Reuban Wilson hoped that \"with the aid of the democratic party (which is bound to be very strong) of the north we will be able to check the republican party in their wild scheme.\"\n\nMeanwhile, Johnson unwittingly fed the tidal return of Southern defiance. Petitions for pardons came in by the hundreds, and were given the presidential signature by the hundreds, from General George Pickett (of \"Pickett's Charge\" fame at Gettysburg) and Alexander H. Stephens (the Confederate vice president) to James A. Cheatham, a Confederate postmaster and railroad station agent. Ultimately, Johnson would issue 13,500 pardons by the fall of 1867 and had to appoint a pardon clerk, Matthew Pleasants, just to deal with the correspondence.\n\nOnce Johnson's provisional governors had arranged for the election of their state conventions, Republican confidence turned to disbelieving fury, as offices and legislatures filled up with generously pardoned Confederates, who only a few months before had been striving to overturn the government they now expected to rejoin. William Holden called for the election of North Carolina's state convention on September 21, but when it assembled on October 2 in Raleigh, it was top-heavy with ex-Confederate officers: Dennis D. Ferebee, the colonel of the Fifty-Ninth North Carolina; Samuel H. Walkup, lieutenant colonel of the Forty-Eighth North Carolina; Giles Mebane, a member of the 1861 secession convention and speaker of the North Carolina state senate. Almost their first concerns were \"the State's assuming the debt contracted during the rebellion\" (as though North Carolina's Confederate expenses were as legal as any other debt) and granting immunity from prosecution to any \"person who have aided in the passage or execution of any law of the State of North-Carolina or of the Confederate States.\" This pattern was repeated in Mississippi's convention in August, over the protest of \"the colored citizens of Mississippi\" who feared \"that the Legislature will pass such proscriptive laws as will drive the freedmen from the State, or practically re-enslave them.\"\n\nNothing, however, offered more calculated defiance than the representatives and senators whom the newly revived Southern states proposed to send to the opening of the Thirty-Ninth Congress. The new Georgia legislature defied reason by dispatching as Georgia's two senators the former Confederate vice president, Alexander Stephens, and former Confederate senator Herschel V. Johnson; South Carolinians elected to the House of Representatives a Confederate general, Samuel McGowan, as did Alabama, which elected Cullen A. Battle. All told, Johnson's self-reconstructed states chose for senators and representatives (in addition to Stephens) six Confederate cabinet officers, four Confederate generals, and fifty-eight members of the Confederate Congress.\n\nThroughout all of these deliberations, not a single one of the rehabilitated legislatures took any step in favor of civil rights for the freedpeople. To the contrary, the new state legislature created by the Alabama state convention ratified the Thirteenth Amendment with the express reservation that \"it does not confer upon Congress the power to legislate upon the political status of freedmen in this State.\" Even worse, freshly elected state legislatures in six states created a hedge of \"black codes,\" designed to bind the freedmen into, for all practical purposes, peonage. Beginning in the Mississippi legislature on November 22, the \"black codes\" defined as \"vagrants\" or \"paupers\" any freedperson who appeared unemployed, and allowed local officials to bid them out (as they had once been at slave auctions) as laborers for up to a year. Other provisions of the codes forbade black-white intermarriage; curtailed free speech (including \"insulting gestures\"); and most ominous of all, banned freepeople from owning firearms, ammunition, and even knives.\n\nThis was not what Johnson had been expecting. He warned the provisional governors to mind their states' behavior, given that Congress would have the ultimate word on their future, and he even pressed his governor in Mississippi, William Sharkey, to \"extend the elective franchise to all persons of color who can read the Constitution of the United States in English and write their names, and to all persons of color who own real estate valued at not less than two hundred and fifty dollars, and pay taxes thereon.\" In November, he tried to make it clear to Alexander Stephens that showing up for the next session of Congress as though he was only suffering from four years' worth of amnesia would be \"exceedingly impolitic.\" Above all, Johnson did not withdraw federal troops from the South. Although the Union armies began demobilizing at rapid rates, there were still ninety thousand soldiers in uniform at the end of the year, spread over three hundred posts across the former Confederacy, and exercising a dual jurisdiction with Johnson's self-reconstructed governments.\n\nThese cautions garnered Johnson no sympathy whatsoever from the Republicans, whose disenchantment with the president rose hand in hand with their incredulity at the actions of new Southern state governments. \"We almost believed him honest,\" charged the abolitionist and feminist journalist Jane Grey Swisshelm, but \"we now fall back upon first impressions\":\n\nThe first time we saw him was at the great Union meeting in the Capitol, in April 1863. His speech then was like all his succeeding speeches, all about Andrew Johnson, and him crucified, and every idea three times expressed. We greatly regretted his nomination, but friends persuaded us, almost, that we were mistaken, and we made our second effort at playing conservative by coddling poor Andy Johnson by way of aiding thousands of honest men in their vain efforts to troll him along the path of rectitude, as sheep are led by walking before with a pan of salt.\n\nParty conventions in Massachusetts and Ohio that summer saw barely restrained outbreaks of protest against the Johnson governments. By July, Charles Sumner was suspected by Navy Secretary Gideon Welles of \"organizing and drilling... to make war upon the Administration policy and the Administration itself.\" Ohio's Benjamin Wade worried about the practical effect of Johnson's decisions on the Republican Party itself. \"To admit the States on Mr. Johnson's plan,\" Wade confided to Sumner, would compromise the party's control of Congress and amount to \"nothing less than political suicide.\"\n\nRemembering the unhappy result of the Wade-Davis reconstruction plan, Ben Wade was not eager to engage in another political war with a president. Johnson, after all, had control of the vast web of executive patronage, and Wade feared that \"too many men... wanted their brothers and friends appointed to office, and knew that Andy was bidding high for votes.\" But when Wade called on Johnson at the White House, he found the president's \"ante-room full of rebels and Copperheads,\" and Wade soon concluded that Johnson intended to \"subordinate... Congress and the Judiciary\" to the \"Executive.\"\n\nMatters came to a head two days before the opening of Congress, as newly arrived congressional Republicans in Washington caucused in the Capitol. Thaddeus Stevens, who was convinced that \"we must put the rebel states under territorial governments at once,\" called for the creation of a joint House-Senate committee of fifteen members, modeled on the wartime Joint Committee on the Conduct of the War, to determine whether the Johnson-appointed state governments were sufficiently sanitized of rebellion to merit the readmission of their representatives to Congress. The joint committee's report would decide the standing, not only of Johnson's rehabilitated states, but also Tennessee (which had reinstituted a civilian government in February 1865, and then elected eight congressmen in July), Louisiana, and Francis Pierpont's Virginia. Until such a report was submitted, \"no member shall be received into either House from any of the so-called Confederate States.\" But lurking at the back of Radical minds was a bigger question: would the political confrontation that was brewing tip the country back into a fresh civil war? That would very much depend on what happened on the day the new Congress assembled.\n\n# Chapter 2\n\n#  Alienation: December 1865\u2013March 1867\n\nThe Radical Republicans arrived in Washington for the opening of Congress on December 4, 1865, with two major weapons at their disposal. One was the party caucus, especially in the Senate, where, Minnesota senator Morton Wilkinson reported, \"Republican Senators used to meet almost every day... so as to leave no chance for hesitation, or division.\" The other was the lopsided majorities Republicans had won on Lincoln's coattails in both the House and Senate\u201437 out of 48 seats in the Senate, 132 out of 191 in the House\u2014both of them sufficient to override presidential vetoes. The caucus moved to protect those majorities by the simple device of instructing the clerk of the House of Representatives, Edward McPherson, to \"omit the names of the Representatives elect from all of the States heretofore declared in insurrection\" from the initial roll call of the House \"and close the doors of Congress to this rebel invasion.\" As McPherson called the truncated roll, Thaddeus Stevens was ready with the caucus's resolution to create a joint committee \"for the purpose of considering the condition of the so-called Confederate States of America.\" It was adopted by an outsize 129 to 35 vote.\n\nJohnson was not unaware of what Stevens and the caucus had been plotting. Nevertheless, he was still obliged to act as the head of the coalition that had dominated the wartime Congresses, and he hoped to defuse Radical wrath through the annual presidential message to Congress the next day. No one, he argued, should think that the solution to Reconstruction was to manage it from the Capitol. If \"it is not one of the rights of any State government to renounce its own place in the Union,\" then Congress does not have the right to demote them to territorial status or exclude their representatives from the US government. He also struggled to shut down the possibility of granting voting rights to the freed slaves. The federal government had no more constitutional power to require the enfranchising of Southern freedpeople than it had to enfranchise free blacks \"in the Northern, Middle, and Western States\"\u2014and he knew very well that three states (Connecticut, Wisconsin, and Minnesota) had just voted such measures down, and that Congress knew it, too.\n\nThis was not an unreasonable appeal, and Johnson sought to add to it some of his favorite distractions\u2014\"monopolies, perpetuities, and class legislation,\" and a Jacksonian \"return to the standard of gold and silver\"\u2014to make it appear even more reasonable. He was wasting his words. A Republican juggernaut was already in motion that would produce, over the next seven months, a flurry of legislation, speeches, and reports designed to dissolve the self-reconstructed governments, extend voting rights to the freedmen by national authority, and reach past Johnson's hands to seize the reins of Reconstruction for Congress.\n\nThe first gestures were the easiest. The formation of a Joint Committee on Reconstruction was rolled through the House on the first day of the session. On December 5, 1865, William D. Kelley (a wartime Democrat-turned-Republican who was known as \"Pig-Iron Kelley\" for his efforts to shield Pennsylvania coal behind import tariffs) introduced a bill to legalize black voting rights in the District of Columbia\u2014one jurisdiction where there was no question about Congress's direct authority to legislate. When the roll was called on it five and a half weeks later, the District voting bill sailed through, 116 to 54, and, as the _Chicago Tribune_ reported, was \"greeted in the galleries and on the floor with loud demonstrations of applause, which the Speaker [of the House, Schuyler Colfax] was for some time unable to check.\"\n\nThen, on December 19, Lyman Trumbull introduced a bill in the Senate to extend the life of the Freedmen's Bureau beyond its original one-year authorization, giving it new support from the War Department and new responsibilities as the \"guardian\" of the freedpeople; it would arbitrate disputes; oversee the establishment of schools; \"set apart for the use of freedmen... unoccupied public lands in Florida, Mississippi, Alabama, Louisiana, and Arkansas\"; and subdivide land seized under the wartime Confiscation Acts into \"parcels not exceeding forty acres\" for \"the loyal refugees and freedmen.\" It, too, easily passed in the Senate on January 25, and in the House, on February 6, by a luxuriously comfortable margin of 136 to 33.\n\nA fourth proposal dealt directly with the nationalization of civil rights. Lyman Trumbull once more stepped forward as a legislative ghostwriter, introducing \"An Act to protect all Persons in the United States in their Civil Rights, and furnish the Means of their Vindication\" on January 5, 1866. The bill opened by affirming that\n\nall persons born in the United States and not subject to any foreign power, excluding Indians not taxed, are hereby declared to be citizens of the United States; and such citizens, of every race and color, without regard to any previous condition of slavery or involuntary servitude... shall have the same right, in every State and Territory in the United States, to make and enforce contracts, to sue, be parties, and give evidence, inherit, purchase, lease, sell, hold, and convey real and personal property, and to full and equal benefit of all laws and proceedings for the security of person and property.\n\nPart of this bill faced backward, to the US Supreme Court's decision in the Dred Scott case of 1857. _Dred Scott v. Sanford_ denied that blacks, slave or free, could ever qualify as citizens, or enjoy any of the Constitution's other \"privileges and immunities.\" Curiously, the Constitution had never defined citizenship, either at the federal or state level. But this oversight only allowed Chief Justice Roger B. Taney, the author of the _Dred Scott_ decision, to insert his own definition, on the basis of _jus sanguinis_ \u2014that is, citizenship by specific birthright. The birthright Taney had in mind was whiteness, and he used it to deny citizenship to anyone of \"African descent.\" But in 1866, Taney was dead and gone, and Trumbull's civil rights bill aimed to bury Dred Scott with him by defining US citizenship by the rule of _jus soli_ \u2014or, by birth on US soil.\n\nNeither the new joint committee nor Trumbull's civil rights bill was designed to make Andrew Johnson happy, and Navy Secretary Welles heard him express \"himself... with sharpness\" in a cabinet meeting about \"the manner in which things had been got up by the Radicals before the session.\" But on the other hand, none of these bills exactly offered him a direct challenge. Trumbull had gone out of his way to mollify Johnson on the civil rights bill. Private interviews with Johnson had given Trumbull \"just expectations\" that the civil rights bill would satisfy the president, especially because Trumbull had cautiously avoided any reference to the act of voting itself as a \"protected\" right. When the Senate approved the bill on February 2, by a 33 to 12 vote, Trumbull had every reason to assume \"that the President's aims, like his own, were in the direction of peace and concord.\"\n\nIt was not the first time a politician had mistaken Andrew Johnson's moody silences for assent. The critics of Johnson's self-reconstruction regimes had lit a slow-burning fire inside of him, and the activities of the Freedmen's Bureau only fanned it. Disturbed by the \"small, endless, mean little injustice of every day\" for the freedpeople, Otis Howard created a system of bureau courts to handle magistrate-level cases\u2014and thereby take them out of the hands of Southern civil courts. In July, Howard issued a circular to his assistant commissioners in the bureau in July, authorizing them \"to select... such confiscated and abandoned property as they deemed necessary\" and designate it \"for the immediate use for the life and comfort of refugees and freedmen.\" Johnson saw that this would collide with his amnesty proclamation, and abruptly ordered Howard \"to return all abandoned lands to owners who were pardoned by the President, and provided no indemnity whatever for the occupants, refugees, or freedmen.\"\n\nJohnson expected Congress to take the hints he had so helpfully sprinkled through his annual message in December and curtail Howard's activities. When it did not, he followed those hints on December 18 with a second message, laying out a little more impatiently how his measures were the best strategy to persuade the restored states \"to confer upon the freedmen the privileges which are essential of their comfort, protection, and security.\" He finally applied the torch on February 19, by issuing a veto of the Freedmen's Bureau Bill and following that a month later with a veto of Trumbull's civil rights bill. And when the District of Columbia voting rights bill finally emerged from the Senate later in the year, Johnson vetoed that, too.\n\nThe veto of the Freedmen's Bureau Bill touched off an uproar in Congress. Johnson, complained the _Chicago Tribune_ , had yielded to \"the whisperings of the tempting copperhead serpent, and... treacherously stabbed to the heart this measure of benevolence.\" Nevertheless, some Republicans in the Senate now wavered, rather than force an outright split with the president. Painfully, an attempt to override Johnson's veto fell two votes short of the necessary two-thirds majority.\n\n2. A modern mural in the Capitol rotunda portrays the passage of the Civil Rights Bill in 1866. In the foreground, Henry Highland Garnet, who had been born a slave in Maryland, speaks with newspaper editor Horace Greeley, who supported African American suffrage. The vote was greeted with applause and cheering that lasted several minutes.\n\nThese divisions emboldened Johnson: on April 2, he issued yet another presidential order, declaring that \"there now exists no organized armed resistance... to the authority of the United States.\" The wartime powers which had justified \"military occupation, martial law, military tribunals, and the suspension of the privilege of the writ of habeas corpus\" evaporated at a click of the telegraph key. The new Southern governors would no longer have to cope with suspicious department commandants, and the Freedmen's Bureau would lose any enforcement power over the freedmen's land claims and working conditions.\n\nCongress usually avoided butting heads with a president of its own party; John Tyler had done likewise when he, like Johnson, inherited the presidency in 1841 from the deceased William Henry Harrison. But on this occasion, Johnson was listening to voices that prophesied that \"another civil war is pending and that the Radical leaders design and are preparing for it.\" Johnson imagined he could parry this blow by crossing the party aisle and uniting whatever Republicans could be persuaded or compelled to follow him with the old Northern Democrats to create a new political coalition\u2014a \"third, or Tyler party\"\u2014with himself at its head.\n\nThis was not an impossible strategy. Democrats had managed to win more than three hundred thousand more popular votes than Lincoln in 1860, and then 45 percent of the presidential vote in 1864; they were still a force to be reckoned with\u2014and clearly a force hostile to anything resembling black civil rights. \"We want no Negro equality,\" Wisconsin Democrats exclaimed, as \"it would degrade and brutify our race, giving Negro Husbands and Negro progeny to our fair daughters and sisters.\" A determined push-back by the president and a new bipartisan coalition to unseat the Radicals in the fall of 1866 elections might give Johnson the triumph he wanted, and the leadership of a new, postwar political world. At the prompting of Secretary of State Seward, on June 25 the _New York Times_ published a call for \"a National Convention, for the purpose of adopting... a platform of principles upon which the Northern and Southern States could take common political action.\"\n\nThe timing of the Johnson convention could not have been worse. By the time it met in Philadelphia on August 14, the fifteen members of the Joint Committee on Reconstruction, with the lofty William Pitt Fessenden in the chair, had finished recording six grueling months of witness examination and testimony from 144 individuals (ranging from Robert E. Lee to Clara Barton), and the published report they submitted on June 18 teemed with eyebrow-raising accounts of how the self-reconstructed states had done everything within their power to return the freedmen to little better than slavery. \"Congress,\" the report soberly concluded, \"cannot be expected to recognize as valid the election of representatives from disorganized communities\" or \"admitting such communities to a participation in the government.\"\n\nAs if on cue, white Southerners confirmed every bleak suspicion in the report in an eruption of race riots in Southern cities. For three days at the beginning of May, working-class whites in Memphis launched bitter and bloody attacks, first on mustered-out black soldiers whom they feared would become rivals for their jobs, and then at African Americans generally in South Memphis, which had drawn up to twenty thousand black fugitives during the war. It did not help Johnson's reputation that the federal commandant at Memphis, Major General George Stoneman, had been warned not to intervene \"until orders are rec'd from Washington\"\u2014orders that had not been forthcoming. By the time Stoneman finally moved in to disarm the rioters, forty-eight black Memphians were dead and \"every negro church and schoolhouse in the city\" had been burned to the ground.\n\nMore rioting followed. In Norfolk, a freedmen's parade degenerated into a shoot-out that left two whites and two blacks dead. In June, Charleston was \"disturbed and humanity shocked by serious and bloody riots, between white and colored citizens and troops.\" In July, rioting convulsed New Orleans. Since Louisiana already had a civilian state government in place at the time of Lincoln's death, under the governorship of James Madison Wells, Johnson allowed Wells to hold self-reconstruction elections in November 1865. But Louisiana had a larger share of Radical sympathizers than any of the other rebel states, and in June, they proposed to call the original 1864 state constitutional convention back to life in a meeting at the Mechanics Institute in New Orleans on July 30. Instead, a crowd of white police and suspiciously well-armed civilians tried to prevent the convention from taking place in the Institute, and a general melee broke out. By mid-afternoon, when federal troops at last arrived to restore order, 38 people had been killed and 146 wounded, most of them black Louisianans who been ruthlessly shot down by their attackers. It was, said federal Major General Philip Sheridan (who commanded the military district in New Orleans) \"no riot; it was an absolute massacre by the police.\"\n\nJohnson instantly accepted any explanations that blamed the violence on \"the Radical members of Congress,\" but the country thought otherwise. Nothing was more \"alarming in this sad affair,\" wrote E. L. Godkin of _The Nation_ , than \"the coolness with which he refrained from expressing one word of honest indignation at the slaughter.\" When the pro-Johnson Philadelphia convention met in August, the organizers did Johnson no service by inviting the two most notorious anti-war Democrats, Fernando Wood and Clement Vallandigham.\n\n3. On the morning of May 2, 1866, the city recorder urged the white citizens of Memphis to arm themselves and \"kill every Negro and drive the last one from the city.\" That night, _Harper's Weekly_ reported, \"the Negroes were hunted down by police, firemen and other white citizens, shot, assaulted, robbed, and in many instances their houses searched under the pretense of hunting for concealed arms, plundered, and then set on fire, during which no resistance so far as we can learn was offered by the Negroes.\"\n\nBut no one was more capable of doing harm to Johnson than the man himself. Each member of Johnson's cabinet was invited to the Philadelphia convention, and Johnson made acceptance of the invitation a condition of remaining in office. Three of them\u2014Attorney General James Speed, Postmaster General William Dennison, and Interior Secretary James Harlan\u2014refused, and then resigned. Then, a week after the close of the convention, Johnson set off on what was billed as a political pilgrimage to Chicago to dedicate a monument at the tomb of Stephen Douglas. The notion of honoring the man who had twice opposed the martyred Lincoln was as close as Johnson could get to announcing his defection from the Republican Party. Johnson proceeded to turn each stop on his \"Swing Round the Circle\" into a stump speech against Radical candidates, thus racking up further cumulative political errors.\n\nPeople who remembered Johnson's inaugural performance had good reason to fear that this was a mistake\u2014and indeed it was. In Cleveland, Johnson lashed out at the Radicals as \"this gang of office-holders, these blood-suckers and cormorants\"; in St. Louis, he dismissed the New Orleans massacre as \"substantially planned\" by the Radicals and called for \"the emancipation of the white men as well as the colored ones.\" Soon enough, his shotgun style attracted hecklers, and Johnson could not resist doing what they wanted him most to do, which was to descend into an insult match. When someone in the audience called out \"Traitor!\" Johnson shouted back, \"I wish I could see you,\" continuing,\n\nI will bet now, if there could be a light reflected on your face, that cowardice and treachery could be seen in it. Show yourself. Come out here where we can see you. If ever you shoot a man, you will stand in the dark and pull your trigger... Those men\u2014such a one as insulted me here tonight\u2014you may say, has ceased to be a man, and in ceasing to be a man shrunk into the denomination of a reptile, and having so shrunken, as an honest man, I tread on him.\n\nThe cost of this behavior surprised Johnson: \"the respect of the whole country and a hundred thousand votes.\" It was even more costly for his allies. Harvard students booed one of Johnson's apologists in Massachusetts, the venerable Robert C. Winthrop, at their commencement ceremony. As the congressional election tallies began to arrive in September, it was clear that Johnson-affiliated candidates, at both the state and federal levels, had lost New England; the elections in Pennsylvania and Ohio in October were even worse. When the political bloodletting had finished, the Republicans in the Fortieth Congress would have five more seats in the Senate and six more in the House\u2014and no reason in the upcoming final session of the Thirty-Ninth Congress to offer Johnson more than the frostiest welcome.\n\n\"If the President does not take the present terms,\" Iowa senator James W. Grimes gloomily informed Gideon Welles, \"harder ones will be proposed.\" But Andrew Johnson was no more inclined to yield in defeat than he had been in victory a year before. He kept vetoing legislation of various sorts\u2014bills covering everything from permission for a mining company to purchase public lands, to Colorado statehood\u2014and Congress merrily proceeded to override them. Congress did likewise with Trumbull's civil rights bill two weeks after Johnson's veto. A new bill for the Freedmen's Bureau, guaranteeing to the freedmen \"any of the civil rights or immunities belonging to white persons,\" was vetoed by Johnson on July, 16, 1866; this time around, the Senate and the House overrode the veto within twenty-four hours.\n\nRelentlessly, Johnson kept banging his head against the Radicals' brick wall. In the annual message he sent to the second session of the Thirty-Ninth Congress in December, he criticized the failure \"to admit to seats loyal Senators and Representatives from... States whose inhabitants... had engaged in the rebellion.\" The rebel states \"are nothing less than States of the Union,\" he declared, and by refusing them recognition, Congress was standing in the way of \"the revival of fraternal relations, the complete obliteration of our past differences, and the reinauguration of all the pursuits of peace.\"\n\nHow Johnson could expect anyone to agree after the Memphis and New Orleans shoot-outs defied comprehension. But he continued to delude himself with the advice of cronies who whispered that Johnson's troubles were only the product of a conspiracy \"to thwart the President's great scheme of adjusting the Union.\" Ulysses Grant, now the general of the army, began to worry that Johnson would brand Congress as \"illegal, unconstitutional and revolutionary.\" And there were vast rumors that Johnson was toying with the idea of preventing the Fortieth Congress from convening, after the pattern of the coup which brought Louis Napoleon to power in France in 1851, by employing the state militia of Maryland and putting \"Maryland and the District of Columbia under martial law.\"\n\nNothing came of this plan, however, because Johnson was weaker than he realized. Beginning in February 1867, Congress swept control of Reconstruction entirely out of executive hands with a series of four Reconstruction bills. The first dissolved the self-reconstructed governments and imposed an overlay of five military occupation districts on the one-time Confederate states (except for Tennessee). In each district, commandants would exercise plenary military and civilian rule \"to protect all persons in their rights of person and property, to suppress insurrection, disorder, and violence, and to punish... all disturbers of the public peace.\"\n\nThe second Reconstruction Act on March 23 made the commandants' principal responsibility the creation, for the first time in the history of American elections, of a registry of eligible voters whose qualifications\u2014namely, that they had \"never been a member of any State legislature, nor held any executive or judicial office in any State, and afterward engaged in insurrection or rebellion against the United States, or given aid or comfort to the enemies thereof\"\u2014ensured that no ex-Confederates could participate. Only those eligible voters who met these requirements would then elect delegates to state conventions, and write state constitutions which would entitle them \"to representation, and senators and representatives\"\u2014but only if Congress agreed. The third and fourth Reconstruction Acts further defined how the commandants could unseat uncooperative civil officials, laid out schedules for the registration process, and established residency requirements.\n\nJohnson, of course, vetoed each bill, raging that they placed Americans \"under the absolute domination of military rulers.\" Congress overrode these vetoes, as well. Presidential-style Reconstruction was dead.\n\n# Chapter 3\n\n#  Arrogance: March 1867\u2013 May 1868\n\nFor the survivors of the defeated Confederacy, the first issue was simple survival. The war had wrecked the Southern economy beyond most implications of the word \"wrecked.\" \"Financial ruin seems to be universal,\" wailed the South Carolina planter, Henry William Ravenel. \"There was not a man in the place who was thirty years of age who had not fought in some capacity,\" wrote an up-country South Carolinian, \"and there was not a woman who had not gone hungry for weeks and badly clad for years.\"\n\nThe losses were so great that the numbers cease to register. In what Walt Whitman called \"the desolated, ruined South... nearly the whole generation of young men between seventeen and thirty\" had been \"destroyed or maimed; the rich impoverished; the plantations covered with weeds; the slaves unloosed and become the masters; and the name of Southerner blackened with every shame.\" Emancipation alone had wiped between $1.6 and $2.7 billion of capital investment off the books. Per capita income in the South collapsed by more than 40 percent, and real estate fell in value, ranging from an 18 percent loss in Tennessee to a whopping 70 percent devaluation in Louisiana. A third of Southern livestock and half of the South's farm machinery had disappeared, and with them, the agriculture that depended on them, as land under cultivation shrank by 34 percent in South Carolina and 30 percent in Louisiana. Banking capital, much of it invested in Confederate securities, suffered losses of 28 percent, sending interest rates on what remained for borrowing through the few intact roofs. Total direct and indirect costs\u2014including the value of enslaved labor\u2014were probably close to $13.6 billion\u2014which does not even begin to convey the sense of demoralization that accompanied the losses. \"The whole country is alive with robbers,\" shivered one Tennessee woman. \"Every night we hear of a new robbery and sometimes murder.\"\n\nNo reconstruction would succeed which left the former Confederacy an impoverished shell, and victorious Northerners thought they knew exactly the sort of new economy that needed to be substituted for the old one. The great Republican goal of abolishing slavery was not entirely a crusade to right a racial injustice; abolishing slavery was not, in fact, much of a racial question at all, but rather an economic one. \"I see National glory in the future such as the past has never seen,\" rejoiced Benjamin Brown French, and not just because slavery is \"forever abolished,\" but because the South would soon be \"thriving under Free labor & Free rule! No more Cotton lords, but plenty of Cotton Commons, and all the land pouring out its productions & becoming immensely rich!\" The Union \"represents the principles of free labor,\" declared a New York pamphlet, and only when \"the victory of the Northern society of free labor over the landed monopoly of the Southern aristocracy\" was complete would the war be over.\n\nThe United States is truly the land\u2014the very paradise of labor.... Even capital, which in Europe controls labor, here becomes subordinate to and serves labor. It is the free and intelligent labor of the country that creates the Administration or the Government.... Whatever may be said against the doings and intrigues of the politicians, yet it is the highest honor of the system of government of the United States that Free labor\u2014that is to say, the equal right of all men to the pursuit of happiness\u2014has been recognized as the first natural and inalienable right.\n\nIn the most basic sense, \"free labor\" was simply a shorthand term for liberal economic democracy, of the same sort advocated by the \"Manchester School\" in Great Britain and by liberals in France, Italy, and Prussia in the mid-nineteenth century. Among free labor's fundamental tenets were the encouragement of small-scale manufacturing and industry, especially through government-sponsored \"internal improvements\" (in the form of canals, highways, and railroads); economic mobility, with constant movement up the ladder of classes; and the practice of a constellation of bourgeois virtues\u2014such as thrift, prudence, industry, religious faith, temperance, rationality, and nationalism\u2014which would thus dignify those whom the _New York American_ described as \"the enterprising mechanic, who raises himself by his ingenious labors from the dust and turmoil of his workshop, to an abode of ease and elegance\" and \"the industrious tradesman, whose patient frugality enables him at last to accumulate enough to forego the duties of the counter and indulge a well-earned leisure.\"\n\nIn the eyes of free-labor Republicans, the mistake of the South had been to allow the thousand-bale planters to turn the Enlightenment clock back, toward what was essentially a replica of medieval serfdom. \"Who knows,\" the _New-York Tribune_ had once asked, \"but we may see revived there the feudal tenures\u2014maiden-right, wardship, baronial robberies, the seizure of white children for the market, military service, and the horrible hardships of villenage which men have fondly deemed forever abolished.\" In the South, the ruling class of \"monarchists and aristocrats\" had shunned government-sponsored improvements, cultivated a style based on braggadocio, and held poor whites and black slaves in the grip of a permanent and oppressive hierarchy. \"There labor has been degraded,\" the _Chicago Tribune_ lamented, \"the laborer left untaught... thus converting half the Union into a charnel house of despotism, without a free religion, free speech, free press or free schools.\"\n\nThe war, however, had swept this \"despotism\" away, and the ash heaps that dotted the Southern countryside were blessings in disguise, since they could now be cleared for Northerners to introduce into the South a New England-style high type of culture\u2014what Republican periodicals like the _Continental Monthly_ described as \"the cultivated valley, the peaceful village, the church, the school-house, and thronging cities.\" The South \"under the old system\" was \"adverse to manufacturing and commercial enterprises.\" But now, the South's \"worn-out plantations will become thriving farms, its mines and inexhaustible water-powers will call into play the incessant demand and supply of vigorous industry and active capital.\"\n\nReconstruction would thus have twin aims: a \"reconstruction such as will protect loyal men, black and white, in their persons,\" but, added Frederick Douglass, one which will also \"cause Northern industry, Northern capital, and Northern civilization to flow into the South, and make a man from New England as much at home in Carolina as elsewhere in the Republic.\" In John Greenleaf Whittier's vision, Reconstruction would\n\nThe cruel lie of caste refute,\n\nOld forms remould, and substitute\n\nFor Slavery's lash the freeman's will,\n\nFor blind routine, wise-handed skill;\n\nA school-house plant on every hill,\n\nStretching in radiate nerve-lines thence\n\nThe quick wires of intelligence;\n\nTill North and South together brought\n\nShall own the same electric thought,\n\nIn peace a common flag salute,\n\nAnd, side by side in labor's free\n\nAnd unresentful rivalry,\n\nHarvest the fields wherein they fought.\n\nThis was a vision captured by Thomas Nast's famous cartoon, \"Emancipation,\" whose ten vignettes hopefully pictured the full integration of the freedpeople into a middle-class future. Here was a bourgeois revolution\u2014not in the Marxist sense of being a necessary footstool to the \"real\" proletarian revolution, but an end in itself, as the triumph of rights and liberal democracy, an Enlightenment counterrevolution against what the Northern middle classes feared was the real wave of the future: namely, the renascence of oligarchy and monarchy.\n\nThe principal obstacle to realizing this dream was the refusal of the defeated Southern planter class to admit that it had been defeated. Whatever else had been lost, title to Southern lands remained in the hands of the same landowners who had owned them before the war. In western Alabama's \"Black Belt,\" 236 landowners possessed at least $10,000 in real estate in 1860 (with the median landholding amounting to 1,600 acres); by 1870, 101 of those landowners were still in possession\u2014which was about the same rate of persistence over time that had prevailed before the war. They, too, had lived by an ideology, but one composed of Romantic medievalism, Jeffersonian agrarianism, white supremacy, and a general suspicion of bourgeois ambitions. \"The typical Southerner,\" warned a contributor to the _Atlantic Monthly_ , \"possessed a... cast of character which was founded mainly on family, distinction, social culture, exemption from toil, and command over the lives and fortunes of his underlings.\" Free labor meant nothing to the planter, since \"it is the highest gentility in him not to work,\" and one Northern reporter believed that \"nine-tenths of the people must be taught that labor is not debasing.\"\n\nIf Johnson was fool enough to issue them a free pass, Frederick Douglass warned, Southerners would simply default to the prewar order of things, even after the ratification of the Thirteenth Amendment at the end of 1865, and especially if they were allowed to remain in possession of the lands they had owned before the war. \"Slavery is so strong that it could exist, not only without law, but even against law. Custom, manners, morals, religion, are all on its side everywhere in the South.\" Everything was governed by a pervasive and \"rigid spirit of caste.\" Black codes simply exchanged the legal reality of slavery for what Georgia planter William Hodgson called a \"state of serfage or ascription to the soil.\" In Georgia, \"colored people are not allowed to travel without passes from their former owners, while others still are afraid to acknowledge themselves free. In many places their meetings are broken up, and schools strictly forbidden... while in others murder and other horrid outrages are the ordinary suffering of our people.... Several who worked for a share of the crop were driven off, as soon as it was gathered.\"\n\nWorst of all, the great landowners seized on the Johnson amnesty to reclaim the lands the army had confiscated during the war and turned over to the freedmen. In 1865, William Tecumseh Sherman (through Special Field Order No. 15) set aside a thirty-mile-wide belt of four hundred thousand acres of prime rice- and cotton-growing land along the Georgia and South Carolina coast for forty thousand freedmen; in North Carolina, the military department commander ordered chaplain Horace James to \"take possession of all unoccupied lands\" on Roanoke Island \"and lay them out and assign them... to the families of colored soldiers... and other blacks in the employ of the Government\"; around Fortress Monroe in Virginia, contrabands simply set up for themselves on land abandoned by Confederate owners. \"The colored man is not content when given simple emancipation,\" lectured John Mercer Langston, \"he demands... to acquire, hold, and transmit property.\"\n\nThe end of the war, however, brought back the refugee landowners, and when they attempted to enforce legal title and evict the freedmen, angry confrontations ensued. George Benjamin West came home from the Confederate army \"without a cent of money,\" found \"30 axemen\" cutting wood on his father's land \"for Jerry Lee, a slave of my sister's,\" and chased them away. Elizabeth Allston's mother revisited her South Carolina plantation to demand the keys to the barns from the freedpeople who had occupied them. She had almost persuaded \"the head man\" to surrender the keys when a \"young man who had stood near, with a threatening expression sprang forward and shouted, 'Ef yu gie up de key, blood'll flow'... and a deafening clamor followed.\" The \"head man\" paused, then \"returned the keys to the depths of his pocket.\" On Edisto Island, in the Sherman Reserve, a group of landowners who tried to evict black squatters were told, \"You had better go back to Charleston and go to work there, and if you can do nothing else, you can pick oysters and earn your living as the loyal people have done\u2014by the sweat of their brows.\" On St. Catharine's Island, black farmers organized by a former hotel waiter named Tunis Campbell armed themselves to prevent the island's owner from regaining possession \"and would not allow any white person to land.\" In the lowlands of southern North Carolina, a mixed-race farmer named Henry Berry Lowry and a small band of fighters struck back at white landowners, raiding farms and disappearing into the swamps of the Pee Dee and Lumber Rivers.\n\nIt did little good. Samuel Thomas, who had commanded the Sixty-Fourth US Colored Troops during the war and in 1865 was named a Freedmen's Bureau commissioner in Vicksburg, Mississippi, warned that Southern whites \"still have the ingrained feeling that the black people at large belong to the whites at large,\" and \"to take property away from a negro they do not consider robbery.\" The Roanoke Island freedmen lost control of their colony in 1866, and most of the freedmen moved away. Tunis Campbell was expelled to the Georgia mainland, and within ten years had been imprisoned on trumped-up charges. The Sherman Reservation shrank under lawsuits for reclaimed titles, until it amounted to no more than one-fifth of the original set-aside.\n\nIt did not help, either, that the freedpeople, at this early stage, had no press to advertise their needs or the injustices inflicted on them. Nor was there a large fund of politically experienced black leadership ready to take up the freedpeople's banner. In South Carolina, Robert Smalls made headlines during the war by boldly piloting the steamboat _Planter_ right past the noses of Charleston Confederates and into the hands of the US Navy; in Florida, Josiah Walls had served in the Thirty-Fifth United States Colored Troops; John Mercer Langston was born in Virginia, educated at Oberlin College, and actually won election as one of the first African American public officials in Ohio in 1855. But in many other places, black political leadership was condemned to emerge slowly, and in the teeth of white hostility as well as intra-racial jealousies and suspicion. \"In such a contest,\" sighed Mississippi's first black congressman, John Roy Lynch, \"they were neither organized nor armed.... They and their white allies were entirely at the mercy of their political adversaries.\"\n\nIf they could not defend their land, the freedmen found other ways to assert themselves. The one most often employed was simply to leave, especially for the towns and cities. Before the end of April, five thousand freed blacks were crowding into Montgomery, Alabama, and when Union soldiers \"advised them to stay with their old masters,\" they only replied, \"No, da was going to be free.\" A Boston correspondent near Macon, Georgia, discovered an old freedman living in a small hut, and when he found that the man had been born in Tennessee, the correspondent was curious to know why the man had come so far for so little. \"I likes to be a free man,\" the old man replied, \"whar Ise can go an' cum, an' nobody says not'ing.\"\n\nThose who did stay behind nevertheless wanted nothing more to do with \"the servile arts.\" Confederate general and would-be congressman Cullen Battle thought that James Alston, \"when he belonged to me... was quite an orderly negro.\" But since then, Battle was amazed to find Alston had become \"exceedingly turbulent and sometimes... over-bearing... and insolent in his manner toward white people.\" The Georgian, Frances Butler Leigh (the daughter of the celebrated English actress Fanny Kemble), was furious to see that \"their whole manner was changed: they took to calling their former owners by their last name without any title before it... and tried speaking to me with their hats on, or not touching them to me when they passed me.\" Black women were particularly irritating to Southern whites, since \"the women do not like to work\u2014it is not ladylike.\"\n\nThe most remarkable improvisation of the freedpeople was the school. Northern abolitionists had been dispatching teachers as \"missionaries\" to runaways and contrabands since the first such contrabands had shown up at Fortress Monroe in 1861. By the end of the war, the American Missionary Association, the American Freedman's Union Commission, and the Methodist Freedman's Aid Society had become launching pads for teachers who endeavored to teach basic literacy and numeracy skills to children and adults alike. \"The pupils in Northern seminaries might often take a lesson from the pains with which these colored students... put into immediate use the lessons of the class-room,\" wrote Helen Ludlow, a teacher in the schools surrounding Fortress Monroe. In New Bern, North Carolina, \"nearly a thousand joyfully accepted the privilege\" of Freedmen's Bureau schools. Hostile whites \"chased and stoned\" teachers and pupils, but by 1866, there were 965 bureau schools across the South.\n\nBut nothing weakened the desire to own land. What the freedman wanted most, said Martin Delany, the black abolitionist, \"is a home\u2014one that he can call his own, and possess in fee simple.\" And the only agency likely to override the disaster of Johnson's amnesty and the truculence of the self-reconstructed state governments would be Congress.\n\nThis was exactly what Congress now proposed to do with the four Reconstruction bills it passed between February 1867 and March 1868. Andrew Johnson vetoed three of them; Congress overrode the vetoes and Johnson allowed the remaining bill to become law by refusing to respond, and the process of dissolving the self-reconstructed governments began. The Arkansas legislature adjourned on March 23, 1867, just as the Second Reconstruction Act was passed, and never reconvened; in Mississippi, the newly appointed commandant of what was now the Fourth Military District, Major General Edward O. C. Ord, allowed the state legislature to remain in session, but nothing could be done by it without his approval. With President Johnson's blessing, the Pierpont legislature in Virginia kept on sitting, too, but any further elections were suspended by the new commandant, Major General John Schofield, on April 2, and state offices were to be filled by military appointment.\n\nThe commandants, however, were merely a means, rather than an end. Under the terms of the Reconstruction Acts, their principal task was to create an entirely new electorate in the South that carefully excluded recusant Confederates and their sympathizers among the white population, and certified the enfranchisement of the freedmen through the creation of vast registries of eligible voters. Major General John Pope began his registration program only a week after taking command in Georgia, dividing the state into forty-seven election districts and creating three-man registration boards (each was required to include two whites and one African American) to publish lists and issue certificates of registration. Pope's boards certified 188,000 voters in Georgia (as opposed to the 106,000 who voted in the 1860 presidential election), almost equally split between black and white; in Alabama (which also belonged to Pope's Third Military District), 165,000 voters were certified, split unevenly between 104,000 black voters and 61,000 white voters. General Ord completed registration in Mississippi by the beginning of September, certifying 106,000 voters in the state, 60,000 of them black. In South Carolina, 121,000 voters were recorded, with previously voiceless black South Carolinians now accounting for 79,000 and with black majorities in twenty-one of thirty-one election districts. Taken together, the registration process would identify 1.3 million voters in the five military districts\u2014and to the horror of white Southerners, 700,000 were black, with black majorities in five states.\n\nRegistration, in turn, was the signal for organization. In Georgia, the Freedmen's Bureau encouraged the creation of the Georgia Equal Rights Association, with its own weekly newspaper, the _Loyal Georgian_ , and a host of \"associations... in every town and neighborhood for the relief of [the] poor and suffering, and to see after the education of our children.\" In North Carolina, a statewide convention for a State Equal Rights league was called in October 1866. Union (or Loyal) Leagues, modeled after the wartime urban political organizations which had supported the Union war effort, appeared in Alabama and Mississippi \"and gave the negroes their first notions of parliamentary law and debating,\" as well as a taste of fighting back by creating Unionist para-military militias \"to murder or drive out... every man who sympathized with the Confederate cause.\"\n\nThe leagues then formed the foundation for organizing state Republican parties. Republican (or \"Union\") political conventions had met in Louisiana in 1865 and in Virginia in 1866; now, they sprouted biracial conventions in South Carolina a week after the passage of the first Reconstruction bill, in Houston in April, in Atlanta in May, and in Jackson, Mississippi, and Raleigh, North Carolina, in September. By October 1867, Republicans were congratulating themselves that \"there are forty thousand white men enrolled in the Leagues, and these, with the blacks, will give us the state in perpetuity if we are only wise.\"\n\nUnfortunately, they were too eager to believe. The principal defect of the Reconstruction Acts was that the five military districts were military, and thus presided over by commandants who had to be appointed by the president, given his position as commander in chief. Just how ready Andrew Johnson was to use this appointment power to his own ends became evident from the commandants he appointed: Ord, in the Fourth District, had once commanded black troops in the Army of the Potomac, but privately despised the Union Leagues and discouraged the freedmen's political gathering; John Schofield, in the First District (comprising Virginia), was convinced of \"the absolute unfitness of the negroes, as a class\" for \"an equal voice with ourselves in Government\"; Pope, who had threatened to close down all-white juries in Georgia, was abruptly removed from command in December 1867, and replaced by Major General George G. Meade, the hero of Gettysburg who was nevertheless utterly unsympathetic to the cause of black voting rights.\n\nPhilip Sheridan, who was installed at the head of the Fifth District (Louisiana and Texas), originally owed his appointment more to the concern of his chief and mentor, Lieutenant General Ulysses Grant, for the possibility of war in Mexico to depose the French puppet emperor, Maximilian. Once in charge of the Fifth District, Sheridan acted swiftly to remove Louisianan officials whom he held responsible for the Mechanics Institute riot, starting with the mayor and police chief of New Orleans, twenty-two members of the city council, and finally the attorney general and the governor of the state. \"Game was scarce down that way then,\" Sheridan sarcastically remarked, \"and some gentlemen amused themselves by shooting negroes.... I stopped this sport and they loved me not.\" Johnson, likewise, quickly tired of Sheridan, and in August 1867 moved to replace him with Major General Winfield Scott Hancock, one of the most unapologetic Democrats in the Union army.\n\nCongressional Radicals had long suspected that Johnson would use his military authority to undermine the Reconstruction Acts, and Johnson did nothing to discourage the rumors by ordering the creation of a new military district to include the District of Columbia. \"If I should have trouble with Congress,\" Johnson had once asked Ulysses Grant, \"whose side would you support?\"\u2014a question loaded enough to confirm every suspicion the Radicals had about Johnson's plans for a coup. On the same day that Congress passed the first Reconstruction bill, it also overrode a presidential veto of a Tenure of Office Act, forbidding the president to dismiss civil servants \"until a successor shall have been in like manner appointed and duly qualified... with the advice and consent of the Senate,\" and then adopted an army appropriations bill that compelled the president to issue all orders to the army through the commanding general.\n\nSheridan was not a civil servant, but he did enjoy at least the limited protection of the army appropriations bill. So Johnson had to order Grant to perform the dismissal of Sheridan, which Grant was very reluctant to do. Only after arguing with Johnson and warning the president that Sheridan was \"universally, and deservedly, beloved by the people who sustained this government through its trials,\" did he finally obey.\n\nSheridan was only one target. At the same time, Johnson charged straight at the Tenure of Office Act by \"suspending\" his increasingly uncooperative secretary of war, Edwin M. Stanton, and replacing him with Grant as interim secretary. Grant, who by now wanted nothing to do with anything that looked like cooperation with Johnson, tried to decline, and when the Fortieth Congress assembled on December 2, it quickly became clear that they would disallow Stanton's suspension. On January 14, 1868, with ill-concealed relief, Grant surrendered the keys to the War Department and Stanton returned to his office. Infuriated, Johnson ordered the adjutant general of the army, Lorenzo Thomas, to evict Stanton. The temperamental secretary instead barricaded himself in his office, defying repeated summonses from Thomas to surrender. In the end, though, it was Johnson who surrendered. On February 21, John Covode of Pennsylvania rose in the House of Representatives to offer a resolution, that \"Andrew Johnson, President of the United States, be impeached of high crimes and misdemeanors.\"\n\n4. On March 4, 1868, Chief Justice Salmon P. Chase presided as the impeachment trial of Andrew Johnson opened in the Senate chamber. \"A large crowd filled the galleries and corridors, making the trial a public spectacle.\"\n\nCongress had used its impeachment powers (under Article 2, section 4, of the Constitution) on only five occasions; only twice had the House of Representatives ever considered impeaching a sitting president, and those discussions had been, in an eerie parallel, about John Tyler in 1843 and (by motion of the same John Covode) James Buchanan in 1860. But Radicals in (and beyond) Congress had been talking about impeaching Andrew Johnson for two years. \"The president should long ago have been impeached,\" declared _The Nation_ in 1866, and Ohio representative James Ashley made repeated motions for impeachment in January and March 1867. But the Radicals were not able to persuade the rest of the Republican caucus to trigger a constitutional crisis until Johnson, by sending Thomas to replace Stanton, clearly defied the Tenure of Office Act.\n\n\"The country was buzzing with rumors of iniquities which Johnson was meditating and would surely attempt if he were not disarmed,\" remembered Carl Schurz. \"There was a widespread feeling among well-meaning and sober people that the country was really in some sort of peril.\" Grudgingly, the reluctant yielded. After three days of \"excited eloquence\" and a final damning speech by Thaddeus Stevens, while \"a driving storm of hail and sleet\" raged outside, the House impeached Andrew Johnson on a straight party-line vote of 126 to 47. But Stevens and the House managers of the ensuing trial in the Senate were unable to elevate Johnson's disregard of the Tenure of Office Act to the level of \"high crimes and misdemeanors.\" By April, it was clear that the Senate would fail to convict, and on May 16, the Senate was unable to muster the necessary two-thirds majority to convict Johnson of the primary charge. Ten days later, the remaining charges also fell short.\n\nAndrew Johnson had survived, but barely, and his presidency, even with ten months remaining in his term, was from that point on as good as over.\n\n# Chapter 4\n\n#  Resistance: May 1868\u2013 March 1869\n\nThe year 1868 should have been the high moment of achievement for Reconstruction. The four Reconstruction Acts reversed the disastrous self-reconstruction initiatives of 1865, and Andrew Johnson had been resoundingly rebuked, if not actually convicted, by an impeachment trial. The Southern states, politically purged of disloyal whites and abundant with freedmen primed to assume a major role, now proceeded to organize a new round of state conventions (known with unintended humor as the \"black-and-tan\" conventions), which would write a new round of state constitutions, guaranteeing a new order of free labor and liberal democracy in the old Confederacy.\n\nIn Alabama, Republicans easily dominated the state convention called under General Pope's oversight. Ninety-six of the 100 delegates were Republicans, and 18 were black. The convention's president, Elisha W. Peck, was born in New York, moved to Alabama in the 1820s, and was forced to flee the state for Illinois during the war. Mississippi's new convention met in January 1868, with 97 delegates, all but 18 of them safely Republican, and chaired by Beroth Eggleston, a transplanted Ohioan and former Union general. North Carolina's convention also met in January, and also with a lopsided Republican majority, 107 to 13, 15 of whom were black.\n\nThe fall of 1868 would also mean another presidential and congressional election season. It was clear that Andrew Johnson had no political future with the Republicans, but few Democrats were eager to welcome him back to the party's ranks, much less nominate him for a second term. A few Tennessee admirers put his name up at the Democratic national convention in July anyway, but after twenty-two ballots the Democrats nominated former New York governor Horatio Seymour, and Johnson ended the convention with only four votes. \"I have experienced ingratitude so often,\" he complained, still in self-crucifixion mode, \"that any result will not surprise me.\"\n\nLike John Tyler before him, Johnson had become a pariah in the eyes of both parties. Meanwhile, the Republicans had the strongest possible presidential candidate waiting in the wings: the war-winning general Ulysses Grant, who had been edging closer and closer to the Radicals as Andrew Johnson grew nearer and nearer to impeachment. When the Republican National Convention went to balloting for a nominee, there was no question that \"there is not one loyal heart,\" declared the New York state delegation, \"that does not beat in unison with the sentiment that calls upon us to select that great chieftain.\" Grant was unanimously nominated by all 650 delegates, \"with swinging hats and waving handkerchiefs\" and a spontaneous chorus of \"The Battle Cry of Freedom,\" and went on in November to win the presidency by 300,000 votes and a 214 to 80 victory in the electoral college.\n\nEven before the election, congressional reconstruction was bringing Southern states back into the Union, this time firmly under Republican control. Between June 22 and July 15, Congress readmitted seven of the ten rebel states\u2014Florida, Georgia, Alabama, Arkansas, Louisiana, and North and South Carolina\u2014under new constitutions. South Carolina's constitution enfranchised any \"resident of this State\" who was also a \"male citizen of the United States, of the age of twenty-one years and upwards... without distinction of race, color or former condition,\" and forbade the state legislature from passing \"any law that will deprive any of the citizens of this State of the right of suffrage, except for treason, murder, robbery or dueling, whereof the persons shall have been duly tried and convicted.\" And there was serious discussion of appealing to the federal government for a loan of $1 million to be used in buying land for the freedmen. \"There is but one way to make a man love his country,\" argued Franklin J. Moses, an ex-Confederate who had transformed himself into a Radical Republican, \"Give them lands; give them houses.\"\n\nThe most startling aspect of the new state governments was the role played by African Americans. Of the eighty-four Republicans in the lower house of the Georgia legislature, twenty-nine were black. In Arkansas, eight were black, and the average age was thirty-seven; five were biracial, three were ministers, three were farmers, and one was a postmaster. Florida elected fifty-three members to its lower house, thirty-seven of them Republican, and of those Republicans, seventeen were African American. In North Carolina, sixteen African Americans were elected to the state House of Representatives and three to the state Senate. As a group, they impressed a _New York Times_ correspondent as possessing \"by long odds the largest share of mental calibre.\" By contrast, \"there is scarcely a Southern white man\" sitting in the state offices \"who has character enough to keep him out of the Penitentiary.\"\n\nOnly a little less startling were the numbers of Northerners who heeded the call for a culturally transformed South and reinvented themselves as missionaries of free labor. John Townsend Trowbridge, a journalist who journeyed through the South in 1865, warned that converting the white population to free labor principles would be an uphill battle. He found it \"impossible for the people of Mississippi\u2014and the same may be said of the Southern people generally\u2014to understand the first principle of the free-labor system.\" For generations, Southern whites had been taught to believe that labor was \"beneath the dignity of a gentleman.\" But Reconstruction, wrote Albion Tourg\u00e9e, now offered a means of refashioning the entire labor system of the South\u2014provided the South was \"desouthernized and thoroughly nationalized\" by importing a new white population to replace the old planter class and act as missionaries of the free labor cause.\n\n**Reconstruction by State**\n\nStates reconstructed under the Reconstruction Acts | Readmitted to the Union  \n---|---  \nArkansas  |  June 22, 1868   \nFlorida  |  June 25, 1868   \nLouisiana  |  June 25, 1868   \nNorth Carolina  |  July 4, 1868   \nSouth Carolina  |  July 9, 1868   \nAlabama  |  July 14, 1868   \nVirginia  |  January 26, 1870   \nMississippi  |  February 23, 1870   \nTexas  |  March 30, 1870   \nGeorgia  |  July 15, 1870\n\nTourg\u00e9e was an example of how eager Northerners were to help this process along. Born in Ohio and educated in New York, Tourg\u00e9e had served in an Ohio regiment, endured the sufferings of Richmond's Libby Prison as a prisoner of war, and settled in Greensboro, North Carolina, at the end of the war in order to find relief in a warmer climate for a wound that had damaged his spine. He opened a law office and became president of a small wood-handle business, the Snow Turning Company, whose success left him \"perfectly thunderstruck at the profits,\" as well as the good wages paid to its largely black workforce. John Hay, who had been Lincoln's private secretary, was another example. Hay had been sent in 1864 to register Southerners willing to take the oath of allegiance, and came away sufficiently intrigued by Florida (\"It is the only thing that smells of the Original Eden on the Continent\") that he bought land to grow oranges near St. Augustine. Albert T. Morgan, who had been a student at Oberlin College at the war's beginning and served in the Second Wisconsin at Gettysburg, had a vision of \"a tide of thrifty emigrants and others with capital settling southward,\" and within twenty-five years making \"the two million people of the Mississippi lowlands twenty millions, and in a century a hundred millions.\" Even Harriet Beecher Stowe, the author of _Uncle Tom's Cabin_ , bought orange groves near Jacksonville, moved South, and created a free-labor colony around the village of Mandarin.\n\n5. Civil rights legislation under congressional reconstruction admitted blacks to jury service in southern states alongside whites for the first time. The _New York Times_ observed: \"The sensation is peculiar... to see a Court in session, where former slaves sit side by side with their old owners in the jury, where white men are tried by a mixed jury, where colored lawyers plead, and where white and colored officers maintain order. But this is done at every Court, and justice is not overwhelmed.\"\n\nDisgruntled Southern whites stigmatized the free-labor apostles as carpetbaggers, a nickname that sprang into usage in late 1867, after the first of the \"black and tan\" conventions met in Alabama, to describe a Northerner \"who turns up here and there and everywhere,\" like a cheap valise made from carpet remnants in-hand, \"ready to run for office or to do any other job that will pay expenses.\" Nevertheless, this Northern free-labor vanguard supplied twenty-eight of the fifty-one white delegates to South Carolina's constitutional convention in 1868 and another thirty-eight to the one hundred delegates at Alabama's constitutional convention. The new governors of Georgia (Rufus Bullock), Louisiana (Henry Clay Warmoth), and South Carolina (Robert K. Scott) were Northerners; of the two hundred office holders appointed by Governor Harrison Reed in Florida, fifty were Northern-born, and Reed himself had only arrived in the state in 1865. \"We are, in fact and from absolute necessity, transplanting the whole South with the higher Civilization of the North,\" rejoiced Frederick Douglass. \"The New England schoolhouse is bound to take the place of the Southern whipping post.\"\n\nIn hindsight, it is clear that it all happened too fast. Little more than a year elapsed between the passage of the Reconstruction Acts and the readmission of the first seven Southern states, and during that time, voter registration had to be carried out, the old Confederate power structure dismantled or disabled, state constitutional conventions called and constitutions written, and elections held for both state and federal legislatures\u2014not to mention a presidential election as well. In the absence of any clear blueprints for rebuilding an entire political order, much was going to be overlooked, and the overlooking would provide uncertain foundations for further rebuilding. Worse, in the welter of political reordering, little was done to address the problem of land ownership, the restructuring of commerce, or the reorganization of public resources.\n\nNot that the new state governments would get much direction from the national executive. Andrew Johnson continued to stick stubbornly to the path of noncooperation, vetoing bills until his last day in office, issuing a blanket amnesty to former Confederates on Christmas Day 1868, and declining to participate in his successor's inauguration. At least there was little more harm Johnson could do by that point, apart from writing a stinging farewell address that described the Reconstruction Acts as a \"catalogue of crimes\" against the Constitution. His Christmas \"universal amnesty and pardon\" would only relieve the last ex-Confederates (including Jefferson Davis, still under indictment for treason but free on bail since 1867) of federal criminal penalties; the political exclusions written into the new state constitutions would remain in place.\n\nMuch greater difficulties were generated by fumbles in the actual implementation of the new Reconstruction regimes. For one thing, the exclusions in the Reconstruction Acts were intended to remove large numbers of disgruntled and disenfranchised ex-Confederates from political participation, but the registration process proved easy to game and most of the military governors indulgent. Only 14 state officials from self-reconstruction out of 342 were removed in Arkansas, and only 12 in North Carolina. In 1860, 62,000 votes were cast in the presidential election, all of them, of course, by white voters; in 1868, the total number of registered whites had shrunk to only 57,000, alongside 47,000 new black voters. White Louisianans had tallied 50,000 votes in 1860; the number of registered white voters in 1867 sank only to about 45,000, beside 84,000 black Louisianans.\n\nBut those ex-Confederates sidelined by the registration process could still find ways to bring public attention to their political positions. In Georgia, unbowed Democrats met to denounce \"the effort to establish the supremacy of the negro race in the South, and to place the destinies of those states in the hands of adventurers and irresponsible persons,\" and nominated the ex-Confederate General John B. Gordon (who had taken the Johnson oath in 1865) for governor. The Republican nominee, Rufus Bullock, defeated Gordon, 83,000 votes to 76,000. But it was worth noticing that in 1860, a total of 116,000 votes had been cast by white Georgians, which meant that Bullock had been elected largely on the strength of black voter turnout, and that 48 percent of the reconstructed Georgia electorate had chosen his opponent. Not that white Republicans always proved steadfast, either: no sooner had Georgia Republicans taken control of the Georgia legislature than, with the connivance of the military commandant, George Meade, they turned away three black state senators and twenty-nine black representatives on the flimsy assertion that the state constitution made no provision for black legislators; in Arkansas, the state constitutional convention enfranchised black voters but then debated criminalizing interracial marriage.\n\n6. The New York Democratic party distributed this campaign card to white voters during the 1868 presidential election. It advised \"White Men of New York\" that \"the Republicans, in Convention, declared in favor of Negro Suffrage,\" that \"they want Negro children to be educated in the same schools with White Children,\" and \"have exempted from taxation the Bonds of the rich [as] class legislation.\"\n\nA more direct problem was created by the determination of Southern whites to impose a kind of counter-disfranchisement on blacks through intimidation. In a hierarchical society, honor had always been a volatile component of politics, and Southerners displayed a cultural weakness for a rush to the sword whenever honor was challenged. Whipping, rape, starvation, and murder had long been the chief means of controlling nearly four million slaves, whether it was the routine application of physical punishment by owners or overseers, or the slave patrols organized from slaveholders and non-slaveholders alike to stop and investigate blacks found, for any reason, off the plantation. And, the war itself had immersed numbers of Southerners in guerrilla warfare, partly against Union forces but just as often against each other, and brought new recruits to a long-time strategy for settling regional differences across the South.\n\nThe practices of patrolling for slaves and guerrilla warfare now passed easily over into terrorism. \"If persecution and confiscation are to follow,\" Confederate clerk J. B. Jones heard one Confederate official prophesy in 1865, \"instead of organized armies we shall have bands of assassins everywhere in the field, and the stiletto and the torch will take the place of the sword and the musket\u2014and there can be no solid reconstruction.\" With over half a million paroled Confederate veterans at large\u2014mostly unemployed and smoldering with resentments\u2014large pools of clandestine white resistance easily sprang to life: the White League and the Knights of the White Camellia in Louisiana, the Society of Pale Faces in Tennessee, the Regulators in Florida, and the Constitutional Guards in Virginia. They made no effort to conceal their intention to recapture control of their states by white Democrats. The mission of the White League, as \"the white man's party,\" was to \"rescue Louisiana from the polluting embraces of... a hybrid pack of lecherous pimps... brought forth in pollution, nursed by filthy horbies, and dropped in Louisiana to show to the world to what depth of corruption, disgrace and infamy human nature can stoop.\"\n\nFresh riots broke out in Georgia, where five cases of Henry repeating rifles were used by local Democrats to attack a Republican rally in Camilla, killing nine, and in Louisiana, accompanied by over a thousand murders between November 1867 and Election Day in 1868. In St. Landry Parish, in the town of Opelousas, between two hundred and three hundred blacks were killed, and black prisoners in the parish jail were marched out and executed. \"The negroes all over the Parish have been disarmed, and have gone to work briskly,\" rejoiced a Democratic newspaper. \"Their loyal league clubs and been broken up\" and \"St. Landry is quiet for the first time since the War.\"\n\nThe tactics of these white resistance groups could include economic bullying as well as political violence. \"Thousands of negroes had liens on their crops released, land rented them at nothing, supplies promised for next year, or money paid them outright in consideration of their turning democrats, or of staying away from the polls,\" reported the _Atlantic Monthly._ Those who could not be bought could be ruined, and at election times, \"the torch began its terrible work all over the country.\" Likewise, white Northerners who employed freedmen\u2014and who were thus drawing black Southerners away from labor peonage under the old masters\u2014could find their properties and improvements reduced to cinders in the night, or their workers waylaid and beaten to force them back to the cotton fields.\n\nThe most infamous of these white jacobins was the Ku Klux Klan, whose \"night visits and whippings and murders was the legitimate offspring of the patrol.\" Originally organized in Tennessee in 1866 as a carnivalesque club with a Greek-style name similar to a fraternity (\u03ba\u03cd\u03ba\u03bb\u03bf\u03c2 simply means \"circle\"), the club's hijinks concealed a profound racial hatred of blacks and a determination to overthrow Reconstruction. The Klan quickly became, by 1867, a night-riding posse, complete with graveyard costumes, bizarre ranks and titles, and a mission (as Nashville editor S. C. Mercer described it in 1868) to \"overawe union men, both black and white\" and \"put the negro in a semi-serf condition.\"\n\nDespite their grotesque rituals and preposterous outfits (in Alabama, Charles E. Robert saw them wearing \"a red gown, trimmed with white braid; pants of the same; high hat, with vail over the face; holes for the eye and mouth\"), the Klan spread by both extension and imitation. Nathan Bedford Forrest, the rough-hewn Confederate general who had presided over the massacre of black soldiers at Fort Pillow in 1864, and who became the unofficial head of the Klan in 1867, estimated that the Klan could count forty thousand members in Tennessee and more than half a million in the rest of the South. In \"small squads of masked men\" or in battalions of \"two or three hundred masked and mounted men,\" they broke into houses; shot, whipped, and raped the inhabitants; and warned any freedmen who \"thought we were all free; that we could vote,\" that \"we will stop all of that.\" In Tennessee, between three thousand and four thousand black refugees escaping Klan violence streamed into Nashville in 1868; on June 13, hooded Klansmen armed with pistols and ropes even hijacked a passenger train in Columbia, Tennessee, searching the cars for a Republican congressman.\n\nIt was hard to ascertain whom the Klan hated most: the freedmen, the carpetbaggers, or home-grown Southern unionists (who received their own title at the same time: \"scalawags\"). In most states, the scalawags were drawn from the tiny prewar Southern bourgeoisie\u2014lawyers, journalists, physicians\u2014and more often than not, they had received Northern educations. In Louisiana, out of 172 prominent unionists who turned Republican, 65 percent had been born in the North or other countries. Even though they had long resided in the South, to their unreconciled neighbors they were nothing but turncoats who had gone \"over to the republicans for the sake of office or plunder.\" But James L. Orr, who had once been Speaker of the US House and a member of the Confederate Senate, retorted that the South's past was nothing to be proud of, and that what he wanted for his home state of South Carolina was \"the material prosperity of New England. I would have her acres teem with life and vigor and industry and intelligence, as do those of Massachusetts.\"\n\n7. This Ku Klux Klan party was captured in eastern North Carolina in August, 1871, by Deputy US Marshal Joseph G. Hester as they were attempting to lynch \"a strict and earnest\" Republican, John Campbell. Hester had the group photographed in Raleigh so that \"the people at large may know... what a Ku Klux tribunal looks like.\"\n\nAt least scalawags like Orr could claim some prior political experience; it was, in the end, inexperience which proved a deadlier poison in Reconstruction's cup than murderous white violence. Of the twelve carpetbaggers elected to Congress from Alabama, Louisiana, Mississippi, Texas, and Virginia, only one had ever held political office; all but two had been army officers who decided to stay in the South at the end of the war as merchants or landowners. Of the carpetbagger governors\u2014Powell Clayton (Arkansas), Adelbert Ames (Mississippi), Rufus Bullock (Georgia), Harrison Reed (Florida), Henry C. Warmoth (Louisiana), Robert Scott (South Carolina), and Gilbert Walker (Virginia)\u2014not a single one had previously been active in politics. This created a deep well of political amateurism, the mistakes of which would be held up as proof across the country of the political evil of the entire Reconstruction project and used to obscure the despotic behavior of Southern resisters. In Georgia, Rufus Bullock quarreled with John Emory Bryant, the president of the Georgia Equal Rights Association; the Mississippi scalawag governor, James Lusk Alcorn, tried to win over his one-time rebel friends, which brought him into direct conflict with the Maine-born Adelbert Ames. Louisiana's Republicans split over control of office appointments in the federal customs house in New Orleans, leading to the impeachment of Governor Warmoth, and the attempted assassination of Warmoth's successor, William Pitt Kellogg.\n\nInexperience also created manifold opportunities for factionalism, swapping political favors, and most dangerous of all, a willingness to seek alliances with ex-Confederates on the \"the enemy of my enemy is my friend\" principle. Corruption, too, was a problem: Florida's Republicans were \"solid citizens,\" recalled black journalist T. Thomas Fortune, whose father sat in the legislature; but they also had \"wide-open eyes to the main chance, which was not always reached by the straight and narrow, and much crooked business went on which helped to bring the Reconstruction government into disrepute.\"\n\nDominating all these political weaknesses was the political situation of the new president, Ulysses S. Grant. No one's reputation stood higher with Northerners after Lincoln's death than Grant's; \"The election of Grant as President of the United States\" was seen by John Langdon Sibley as the occasion for \"great rejoicings by the Republican party, in as much as it indicates a purpose to put an end to the unrighteous alliance of the Democratic party of the North with the rebel spirit at the South, which still lingers with the hope that this combination will again put the southerners in the political ascendance.\" Yet, even though Grant won the presidency with 58 percent of the popular vote, he lost New York by ten thousand votes, and New York City by sixty thousand votes. Five states had given him victory margins of less than 5 percent, and Democrats had gained twenty-two seats in the House of Representatives. Even where Republicans managed to hold seats in Congress, the margins of victory were unnervingly thin. Nor could it save some of Radical Republicanism's most favored sons. In Ohio, James Ashley (who had led the charge to impeach Andrew Johnson) was beaten with only 48 percent in his district, while Ashley's coadjutor in Radical Republicanism, John Bingham, survived by only 416 votes. Four governorships fell into Democratic hands; in Maryland, every state legislative district and all five US representatives went Democratic, while in New Jersey, three of the five congressional districts elected Democrats, and Democrats held the state legislature.\n\nThe Democratic Party was, in fact, already entering a period of recovery that turned out to be remarkable in its speed and extent, and both Northern and Southern Democrats would soon enough find common cause in shaking off the hated rule of Republicans. In another year, this would convince former Confederates to stop hammering their fists against Republican walls, and look instead to promoting fissures in Republican unity, and wooing the weak-kneed and disenchanted to some new banner.\n\n# Chapter 5\n\n#  Distraction: March 1869\u2013 May 1872\n\nUlysses Grant's 1868 presidential campaign slogan, \"Let us have peace,\" was a study in vagueness\u2014and not without reason, since Grant believed that \"in times like the present it is impossible... to lay down a policy to be adhered to, right or wrong, through an Administration of four years.\" This was intended as a statement of flexibility, but it could also be read as an attempt to duck hard decisions. His inaugural address on March 4, 1869, was only 1,100 words long; nearly half of it was devoted to his goal of liquidating the national debt: \"Principal and interest, as well as the return to a specie basis as soon as it can be accomplished without material detriment to the debtor class or to the country at large.\" He spent more time on promising \"the proper treatment of the original occupants of this land\u2014the Indians\" than he did on \"the question of suffrage\" in the South.\n\nThe day after his inauguration, he sent his list of cabinet appointees to the Senate for confirmation, and the national gasp of dismay was almost audible. \"The announcement of the Cabinet has caused a general commotion,\" declared the _Chicago Tribune_ , and no wonder: his old political sponsor, Illinois congressman Elihu Washburne, was given leadership over the State Department, despite his reputation as \"coarse, comparatively illiterate, a demagogue without statesmanship or enlarged views,\" and two political unknowns\u2014Alexander Stewart, a New York multimillionaire, and Adolph Borie\u2014were given the nod for treasury and the navy. At least, Grant nominated for attorney general the well-respected Massachusetts jurist E. Rockwood Hoar. But Borie was so little known that one editor burst out in frustration, \"The Hon. Adolph E. Bovie [sic], Secretary of the Navy, is\u2014is\u2014well, who in hell is Bovie, anyway?\" Stewart withdrew his own nomination on March 9; Borie was gone by June; and Elihu Washburne, after eleven days as secretary of state, resigned and was bundled out of the country as American minister to France.\n\n\"No one doubted that Grant's intention had been one of reform; that his aim had been to place his administration above politics,\" Henry Adams recalled. The real problem was with Grant himself. \"Grant appeared as an intermittent energy, immensely powerful when awake.\" But \"for stretches of time his mind seemed torpid,\" and at those times he appeared \"simple-minded beyond the experience of Wall Street or State Street.\" Grant did not initiate policies so much as react to crises. Never having held political office before, he displayed a fatal naivet\u00e9 about people and government and allowed his inner circle of advisors and secretaries to direct affairs until embarrassment over their corruption or their folly roused him to action. There was never any doubt that he was well intentioned, especially on the subject of the freedpeople's rights\u2014he declared the protection of black voting rights to be \"a measure of grander importance than any other one act of the kind from the foundation of our free government to the present day,\" and he was the first president to employ black diplomats (to Haiti and Liberia). Yet he could also be inert in protecting those rights until they were actually assaulted and utterly blind to the incompetence and dissimulation of the officials he appointed to public trusts. Grant was, in the unforgiving eyes of Gideon Welles, \"a pitiable object, wholly unfit for his position, the duties of which he no more comprehends now, than when he was acting as porter in his brother's store in Galena, or carting wood in St. Louis.\"\n\nOn the day Grant was sworn in as president, Virginia, Mississippi, Texas, and Georgia were still under military rule, per the first of the 1867 Reconstruction Acts. Nor did it look like they would be making any significant steps toward restoration any time soon. In Mississippi, an election under self-reconstruction had installed former Confederate general Benjamin Humphreys as governor. The military governor under congressional Reconstruction, Edward Ord, kept Humphreys in place until a new constitutional convention could produce a fresh document, have it ratified by the registered voters, and hold elections. But already, anti-Republican resistance was stiffening. A convention of Mississippi Democrats meeting in Jackson organized themselves as the Democratic White Man's Party. They denounced the \"nefarious design of the Republican party to place the white men of the Southern states under governmental control of... the African negro,\" and called \"upon the people of Mississippi to vindicate alike the superiority of their race\"\u2014which they did, by rejecting the new state constitution, 63,000 to 56,000.\n\nMeanwhile, the Ku Klux Klan tracked down Northern-born schoolmasters (Allen P. Huggins was whipped and warned to leave Monroe County in ten days), invaded courthouses (and killed the judge, E. L. Bramlette, in Meridian), and kidnapped uncooperative blacks (in Sumter County). The situation came so close to chaos that in the summer of 1868, Humphreys was unceremoniously evicted from the governor's mansion by military order and replaced by Adelbert Ames, a Maine abolitionist and the son-in-law of Benjamin Butler, who had infuriated Southerners with his strict occupation policies as military governor of New Orleans during the war, and encouraged fugitive slaves by welcoming them as \"contraband of war.\" Ames went at his work with a vengeance, dismissing sixty sheriffs, seventy-two judges, fifty mayors, and even local constables. The state constitution was resubmitted, and James Lusk Alcorn, who had sat in the 1861 secession convention but was now (as W. E. B. Du Bois described him) \"the most advanced reconstructionist,\" was duly elected governor.\n\nBut instead of these confrontations signaling caution to him, Grant proposed to move ahead as swiftly as possible to restore the remaining Southern states. Although Reconstruction \"has not met with the success in all particulars that might have been desired,\" as he conceded in his first annual message to Congress on December 6, 1869, \"on the whole they have been more successful than could have been reasonably anticipated\"\u2014a judgment that might have surprised a number of people on the ground in the South. Nevertheless, Virginia's legislature, at least, had met all the technical expectations of Reconstruction, and thus was entitled once more to send representatives and senators to Congress. Texas and Mississippi were at that moment in the middle of holding elections, and Grant hoped that \"the acts of the legislatures of these States, when they meet, will be such as to receive your approval, and thus close the work of reconstruction.\"\n\nGrant might have had more difficulty in persuading Congress to readmit the last hold-out states had it not been for the changes Congress itself had undergone. The old Radical guard was still represented in the Senate by Zachariah Chandler, Jacob Howard, Lyman Trumbull, Charles Sumner, and Henry Wilson. But Benjamin Wade was gone\u2014the Ohio legislature had been captured by Democrats in the 1867 state elections, which doomed his chances for reelection\u2014as was old Jacob Collamer, who died in 1866, and William Pitt Fessenden, who died in the fall of 1869. In the House of Representatives, the single most Radical voice of them all was stilled with the death of Thaddeus Stevens on August 11, 1868. As one by one the veterans of the Radical struggle fell silent, the expectations for a radical Reconstruction shrank as well, and Grant himself became the new radical baseline. The _New York Herald_ 's reporter noticed that, during the reading of Grant's message aloud by the clerk of the House of Representatives, \"some one on the floor said, 'Well, that's radical enough.' \" And indeed it was, under these new circumstances. On January 24, 1870, the House passed an enabling bill to readmit Virginia by a lopsided 136 to 58 vote; Grant signed it two days later. Mississippi followed a month later, Texas a month after that, and Georgia, at last, on July 15, 1870.\n\nThis should have marked the conclusion of Reconstruction, and officially, it did\u2014except that the Republican governments in the Southern statehouses were fragile and unstable and led by political nonentities willing to do anything to retain their offices. In Georgia, Rufus Bullock forgot the promises he had made to black Georgians and courted alliances with Georgia Democrats; they, in turn, cheerfully used Bullock to obtain majorities in the Georgia legislature in 1870 and then moved to impeach him. Franklin J. Moses, the Speaker of the South Carolina House and later governor, was described by Josephus Woodruff, the clerk of the state senate, as \"a blackmailer\" with \"no conscience\"; Francis Cardozo, South Carolina's secretary of state and the first African American to occupy a state office in Reconstruction, was a man whom Woodruff said \"nobody ever knew to keep his word.\" They were more incompetent than corrupt; and the corrupt among them were not more corrupt than the slaveholding regimes that had preceded them. But incompetence and corruption were not what the hour called for, and the political corner-cutting that seemed routine under prewar state governments was now denounced as unholy fraud by disenfranchised whites and Democrats.\n\nThe Republican South was also economically shaky. During the five years since the end of the war, the crusade for a free labor economy in the former Confederacy had barely taken its first steps. The South owned only 12 percent of the nation's mills and factories, and employed only 7 percent of its population as laborers in those establishments. Cotton agriculture remained, as it had been before the war, the producer of the republic's single most valuable export commodity (some 32 percent of all exports as late as 1889). And no wonder: while commodity prices for wheat, corn, and coal had operated (except during the war years) within fairly narrow ranges, cotton was selling above all its prewar highs; in Georgia, Alabama, Mississippi and Arkansas, cotton acreage and production expanded, employing a black labor force indistinguishable from that under slavery. Great Britain still bought 58 percent of the cotton it imported for textile manufacturing from the United States, and that would continue to rise through 1876.\n\nStruggling to jump-start a New Englandized economy, Southern Republicans spent unprecedented amounts of money on internal improvements, including railroads, turnpikes, and public education. Alabama and Texas doubled their railroad mileage between 1866 and 1872; Arkansas, which counted only thirty-eight miles of railroad at the end of the war, had 258 by 1872. The number of elementary and secondary schools more than doubled in Mississippi, North Carolina, South Carolina, and Florida between 1870 and 1879; in Georgia, the number quadrupled.\n\nBut the railroads were not, as it turned out, a magic bullet. Construction costs for railroads fluctuated between $30,000 and $50,000 per mile, and new rail lines built on speculation folded as fast as they were finished. In the Natchez district, six new railroad companies were established in the early 1870s; all of them failed. Georgia's state-owned railroad, the Western & Atlantic, plunged the state into $750,000 worth of debt in two years. And the Ku Klux Klan, seeing the railroads as the agents of Northern free labor, made railroad repair shops and new construction sites the targets of murderous raids\u2014not the least because the railroads hired freedmen. In North Carolina's Alamance County, the Klan accounted for twelve murders and fourteen arsons on railroad property in one twelve-month period.\n\nNor were the schools entirely successful. Literacy would be critical to the economic and political awakening of the freedpeople, and their Republican allies, in the form of the Freedmen's Bureau and the American Missionary Association (AMA) moved south even before the end of the war to open schools. But the resistance they met was intimidating. When the AMA's William L. Coan left Chelsea, Massachusetts, in 1865 to open a freedpeoples' school in Lexington, Virginia, he was assaulted at a train station at Gordonsville. Coan persevered, and opened an AMA school in Lexington for three hundred freedpeople of all ages. But he was constantly harassed, his landlady was forced to leave Lexington, and the school's meetings were disrupted by rowdy cadets from the nearby Virginia Military Institute. Once Southern states were legally reconstructed, the costs of education were shifted to the new Republican legislatures, but they were frequently unprepared to accept the fiscal burden the schools represented. In Southern states where no public schooling systems had existed before the war, the construction and staffing costs were staggering. In Mississippi alone, 432 schoolhouses were built in just the 1871\u201372 fiscal year; the teaching staffs alone numbered 4,800, and the overall cost for funding the Mississippi system was just over a million dollars. Nor were the schools always welcomed by African Americans who could ill afford to pay fees and teachers. \"You preachers and teachers are hard on us,\" one old man complained to Charles W. Chestnutt, a teacher (and the grandson of a slaveowner who could have easily \"passed\" as white and later became a major American novelist). \"You want us to pay you thirty or forty dollars a month for sitting in the shade, and that is as much as we can make in 2 or 3 months.\"\n\nThese investments left Republican governments in Southern states mired in debt\u2014Louisiana, for example, held $2.2 million dollars in unfunded debt, tax receipts of only $4.3 million, and a budget deficit of $60,000. South Carolina was in even worse shape: it had $5.3 million in unfunded indebtedness and only $1.6 million in revenue. Only Georgia and Texas managed balanced budgets. The debt, in turn, added to the new tax levies of Reconstruction's state Republican governments and gave further fuel to resentment among Southern whites.\n\nAnd as if these were not obstacles enough in the path of a new culture of free labor, there was always the intractable problem of race. Former slaveholders, thanks in large measure to the Johnson amnesties and the failure to break up or confiscate Confederate property, were thus free to use cotton profits to maintain a version of the plantation system and force the freedpeople into peonage; this, in turn, gave white Democrats the power to control black voting; and control of voting would spell the end of Republican governments and free-labor economics, and, as a Vermont journalist put it, \"keep the negro in his condition of ignorance, that they may retain him as nearly as possible in his old state of slavery.\" Major General John Pope, who briefly supervised the Third Military District (comprising George, Alabama, and Florida) warned Grant that it would not take much before \"the Union men & Freedmen\" were reduced to being \"the slaves of the old negro rebel aristocracy.\"\n\n\"The relation of master and slave no longer exists here,\" wrote one Mississippi valley planter, \"but out of it has evolved that of patron and retainer.\" Nor were Republicans at all successful in appealing to small-scale Southern white yeomen, since the tax burden for schools, railroads, and infrastructure which, before the war, had been carried by taxes on slave ownership, now fell onto the shoulders of the small farmers and drove them into supporting the cotton elite. \"Here and there through all the cotton states,\" marveled _Harper's New Monthly Magazine_ , \"are reappearing the planter princes of old time, still lords of acres though not of slaves.\" Slavery might have been dead, but it was only being replaced by hutted serfdom. \"General,\" read a petition from the freedpeople of Edisto Island to Otis Howard, \"we want Homesteads.... We can only do one of three things Step Into the public road or the sea or remain on them working as In former time and subject to their will as then.\" This might be emancipation, but it \"is not the condition of really freemen.\"\n\nThe Southern Republican regimes might have had a better chance if the federal government could have suppressed the most direct threat to their survival, which was the violence, political intimidation, and terror inspired by the Ku Klux Klan and its fellow travelers among the White Leagues, Red Shirts, and Pale Faces. \"In my state since emancipation,\" complained Jefferson Long, Georgia's first black congressman, \"there have been over five hundred loyal men shot down by the disloyal men there, and not one of those who took part in committing those outrages has ever been brought to justice.\" Even if \"we take the men who commit these outrages before judges and juries,\" added Long, \"we find that they are in the hands of the very Ku Klux themselves who protect them.\" Although the Klan had been active in Tennessee, Arkansas, Alabama (where Edmund Pettus was its \"dragon\"), and Mississippi, it generated particularly violent collisions in North Carolina and South Carolina in 1870. South Carolina's Republican governor, Robert Scott, wrote to Grant in horrifying detail of \"colored men and women... dragged from their homes at the dead hour of the night and most cruelly and brutally scourged for the sole reason that they dared to exercise their own opinions upon political subjects.\" Scott called out the mostly black state militia to ensure peaceful voting in the October 1870 state elections. But a day after the election, a pitched battle broke out in Laurens County between the militia and 2,500 armed white Klansmen, followed by still more fighting in Spartanburg, Union, and York counties.\n\nThe prospect of what the _New York Herald_ called \"the lawless and malignant elements of the late rebellion\" on the rampage stirred President Grant into action. Grant first called the newly elected Forty-Second Congress into emergency session on March 4, 1871, demanding that the Speaker of the House, James G. Blaine, focus Congress's attention on \"the single subject of providing means for the protection of life and property\" in South Carolina. Grant then issued a cease-and-desist proclamation on March 24; when that seemed to make no impression, he declared nine South Carolina counties to be in a state of insurrection, and suspended the writ of habeas corpus. Congress, likewise, passed a series of three Enforcement Acts that guaranteed federal protection for those seeking \"to become qualified to vote without distinction of race, color, or previous condition of servitude,\" authorized the deployment of federal troops, and permitted individuals suffering harms to recover damages in federal court.\n\nThe federal government also had the advantage of being served by a tenacious attorney general, Amos Akerman (a former Confederate who had succeeded Judge Hoar in June 1870 and vigorously pursued the Klan for crimes in Louisiana and North Carolina), and a veteran solicitor, David T. Corbin, as federal district attorney. Akerman and Corbin wasted no time at all in obtaining 785 indictments and going to trial in the federal district court in Columbia, South Carolina. \"The courts are going on here,\" Akerman jubilantly announced, and \"There is trepidation everywhere.\" Or almost everywhere: the pursuit of the Klan induced a certain _schadenfreude_ among black Republicans. \"There is not one in fifty of them now but is uneasy and trembling at the sight of an officer or a blue-coat,\" chuckled Elias Hill, a Baptist preacher who had been beaten by the Klan, \"and they are staying out in the woods by day, and some by night, like we used to.\"\n\nStill, even in the face of federal authority, the Klan found ways to resist. Corbin was able to obtain only five convictions at the first round of trials in Columbia; he managed twenty-eight more at the next session of the district court in Charleston in April 1872, but only nine more in the third round back in Columbia the following November.\n\nThe volume of prosecutions strained Akerman's budget so severely that he had to turn down a US marshal's request for $200 to hire a detective. Worse still, Grant fired Akerman in December 1871 (Secretary of State Hamilton Fish had complained that Akerman had the Klan \"on the brain\"), and although his replacement, George H. Williams, continued to press for convictions in Klan cases across the South, the costs of the anti-Klan initiative were mounting. In 1872, Williams instructed Corbin to enter a _nolle prosequi_ (\"do not prosecute\") in his remaining cases \"for the sake of the public good.\" Nevertheless, the convictions Akerman and Corbin did manage to obtain under the Enforcement Acts allowed Grant to declare victory over the Klan insurgency\u2014and turn his attention to other insurgencies that required reconstruction.\n\n8. Large numbers of free blacks gathered in Taylor County, Georgia, to vote in the 1868 elections, only to be turned away \"on account that the voters (colored) had not paid their poll tax.... The whole crowd inside of the election room, the canvassers, challengers, or what their name may be, were composed of real Southern democrats, and their action a mere farce.\"\n\nBy 1870, Arizona, New Mexico, Utah, Idaho, Montana, and Wyoming all had territorial governments, and Kansas, Nevada, and Nebraska had been admitted to the Union as states. The wartime Homestead Acts unloaded publicly owned land there at fire-sale prices, often without much regard for the arability of lands where average rainfall fell undependably short of what was needed to sustain commercial agriculture. Nevertheless, settlers began moving into the new lands, and railroads laid track to open conduits to markets, beginning with the completion of the transcontinental railroad at Promontory Point, Utah, in 1869. Almost all of the financing of the railroads came through federal grants which were the virtual equivalent of the Homestead Acts\u2014over 131 million acres west of the Mississippi. The chief hindrance to this expansion was the presence of the Plains Indians, who had been crowded westward by successive Indian removal campaigns between the Revolution and the Civil War. President Grant's impulses toward the Plains tribes were well-intentioned but paternalistic. \"The Indians require as much protection from the whites as the white does from the Indians,\" Grant sharply observed in 1865. \"My own experience has been that little trouble would have been had from them but for the encroachments & influence of bad whites.\" But at the same time, Grant's notion of protection included \"placing all the Indians on large reservations\" where \"they will live in houses, have schoolhouses and churches, and will be pursuing peaceful and self sustaining avocations\"\u2014which, in a word, meant that they too would become part of the new world of free labor. While still general-in-chief, he invited Ely S. Parker, his military secretary and a Tonawanda Seneca sachem, to create a national inspection board to oversee and enforce treaties with the tribes. In 1869, Grant persuaded Congress to create a Board of Indian Commissioners \"eminent for their intelligence and philanthropy,\" and in the summer of 1870, Grant took the unprecedented step of hosting a meeting in Washington with Red Cloud, the chief of the Oglala Sioux, and Spotted Tail, chief of the Brul\u00e9 Sioux.\n\nStill, little of what the commissioners recommended in their first report in 1869 would have struck the Plains Indians as especially philanthropic and the so-called Five Civilized Tribes of the Indian Territory resented the suggestion that they were a racial problem similar to Southern blacks. (The Cherokee, in particular, complained that they were \"vastly superior, in every respect, to any portion of the Negro race\" and urged whites \"to draw the necessary distinction between Indians and negroes.\") Even Ely Parker dismissed the notion of signing treaties with the Plains tribes because \"the Indian tribes of the United States are not sovereign nations capable of making treaties, as none of them have an organized government.\" Moreover, whatever effectiveness the commissioners hoped to exercise was undermined by squabbling within the board, accusations of corruption (which led to Parker's resignation and a congressional investigation in 1871), bickering over jurisdiction between the Interior Department and the War Department, and an ongoing rumble of tribal war involving the Utes, Comanches, the Chiricahua Apache, and the Sioux (which in 1874 erupted into full-scale warfare). Only the comparatively small numbers of the Plains tribes\u2014fewer than four thousand Comanche, fewer than two thousand Pawnee, and fourteen thousand in the seven Lakota Sioux tribes\u2014made it possible for the US Army to suppress Indian resistance in the West in ways it failed to do so in the white South, and even then, armed resistance did not subside entirely until after 1890.\n\nThe Grant administration had no more success in dealing with the Mormon enclave in Utah. Mormonism was one of the more peculiar offshoots of American religious enthusiasm in the nineteenth century, and one of the more successful once it established its own quasi-independent colony around the Great Salt Lake in 1847. It was governed as a Mormon theocracy, with its prophet, Brigham Young, acting as virtual dictator. Like slaveholders, free-labor Republicans raged against Young as an economic despot who \"by espousing the interests of one class against those of the other, instead of attempting to reconcile them... introduced a new element of discord into his already inharmonious kingdom.\" But Mormonism offended bourgeois sensibilities even more with its espousal of polygamy, which the 1860 Republican platform had denounced as a gendered form of slavery.\n\nEven while the Civil War was still in progress, Radical Republicans in Congress demanded that Utah be de-Mormonized \"by force\" and passed a statute in 1862 to ban polygamy \"in a territory or other place over which the United States shall have exclusive jurisdiction.\" But again, budget restraints on federal prosecutors in Utah prevented aggressive action against the Mormon leadership. \"Brigham Young is rich,\" complained federal district attorney George Bates, \"but the US authorities are penniless.\" Utah remained a defiantly \"Theo-Democracy\" where \"the petrified truth\" is that the territory \"is an absolute monarchy and Brigham Young is king.\" Meanwhile, calls for \"the moral power of bayonets\" to be used in subduing Utah were met with the same chill that was growing over the use of federal military intervention elsewhere. In October 1871, after a two-day fire devastated large portions of Chicago and generated \"murder and plunder on all sides,\" federal troops were sent in to restore order without informing the governor of Illinois, John M. Palmer. The governor at once accused the Grant administration of reaching for \"powers they did not possess, and that while free government endures cannot be conferred upon them.\" This delighted Democrats, who jubilantly insisted that what was good for the goose was good for the gander. \"A northern State now tries her hand against usurpation,\" rejoiced one Southern newspaper, \"We shall see whether circumstances alter cases.\" After all, \"Governor Palmer is asserting only what we are asserting every day.\"\n\nGrant's last insurgency was the only one he could be said to have triumphed over, and that was within his own party. Dissatisfied Radical Republicans who saw nothing in Grant but slow-footedness, and who suspected that Grant was using the Klan prosecutions and his penchant for appointing loyal nonentities to major federal positions to create a soft dictatorship, rallied around the call of Missouri's US senator Carl Schurz to dump Grant and nominate a \"liberal Republican\" replacement. Liberal Republicanism was largely an elite rebellion within the party ranks at the Radicals, whose racial egalitarianism they did not share and whom they blamed for the political chaos in the South. But even the Southern States Convention of Colored Men, which brought together the South's most prominent black leaders in Charleston, South Carolina, in October 1871, almost balked at P. B. S. Pinchback's motion to endorse Grant's reelection. Dissatisfied Liberals arranged to hold a national convention in Cincinnati on May 1, 1872, and assembled some of the bluest blood of Northern Republicanism: Charles Francis Adams, Lyman Trumbull, William Grosvenor, Morton Wilkinson, and Oliver Perry Morton. For many of them, the still-smoldering embers of the Chicago fire illustrated the error of Reconstruction's military ways. \"The war of the rebellion is ended,\" said Ohio Judge Stanley Matthews in one of the convention's opening addresses, and \"as the war has ended so ought military rule and military principles.\"\n\nThe problem in Cincinnati was that too many of the Liberal Republicans each imagined that they were best suited to receive the presidential nomination. The consensus heading into the Liberal Republican convention had been that Charles Francis Adams, the son and grandson of two presidents and the American minister to Great Britain during the war, ought to head the Liberal Republican ticket, with Lyman Trumbull as his running mate. But the platform committee was grievously divided, and Adams had no floor manager to round up delegate votes. On the sixth ballot, the convention nominated instead Horace Greeley, the editor of the _New-York Tribune_ and one of the flightiest political minds in the country.\n\nThe nomination of Greeley\u2014\"the worst political fiasco ever enacted in American politics,\" snarled one Chicago newspaper\u2014took the air out of the Liberal Republican movement even before the convention adjourned. When Schurz went to dinner afterward, a one-time Illinois political ally of Lincoln, Gustave Koerner, remembered that Schurz first sat down at the parlor piano and played Chopin's funeral march. \"The discontent with Grant was far-reaching,\" recalled George Frisbie Hoar (the brother of Grant's first attorney general), \"but the nomination of Greeley was ludicrous and preposterous.\" In July, the Democratic national convention made an even more startling decision by endorsing Greeley, who was now running, in the scathing estimate of William Lloyd Garrison, as \"a stool-pigeon for the Democracy to capture the Presidency.\" But there was never any serious hope that Greeley could prevail. It was, as _Harper's Weekly_ described it, an act of \"political suicide\" by \"a political assassin.\" Grant was unanimously renominated by the Republican national convention in June, and handily won reelection with 56 percent of the popular vote; Horace Greeley died three weeks after the election and won only three posthumous votes in the Electoral College. Republicans even gained back seats in the House of Representatives, increasing their majority to 203 seats.\n\nFor Grant, it was \"my vindication\"\u2014but it would not last long.\n\n# Chapter 6\n\n#  Law: 1866\u20131876\n\nSo much of Reconstruction is understood as a struggle over race, politics, and the nature of state sovereignty within a federal system that not enough attention is paid to the fact that it was also a constitutional struggle between the branches of the federal government. On the occasions when intra-branch conflict in Reconstruction actually does put in an appearance, it is usually in terms of the executive's clash with the legislature, whether in the figure of Lincoln versus Congress or Andrew Johnson versus Congress. The judiciary appears only in terms of isolated cases and their impact on the freedpeople's rights. But this incomplete analysis misses what is, in fact, a major part of the Reconstruction story.\n\nThe US Constitution apportioned various federal responsibilities among the three federal branches\u2014executive (the presidency), legislative (Congress), and judicial (the federal courts)\u2014but it did not do so evenly or with the same amount of detail. The Constitution gave the federal courts, beginning with the US Supreme Court, only the sketchiest of descriptions, and then, with a fine disregard for the separation of powers, left much of the construction of a federal court system in the hands of Congress. John Jay, sitting as the first chief justice of the United States from 1790 to 1795, wrote opinions on only four cases and \"left the Bench perfectly convinced that under a system so defective it would not obtain the energy, weight and dignity which are essential... nor acquire the public confidence and respect which, as the last resort of the justice of the nation, it should possess.\"\n\nJohn Marshall, the fourth chief justice, pushed back strenuously against both state overreach and congressional encroachment on the judicial branch until his death in 1835; his successor, Roger Brooke Taney, sat as chief justice for another twenty-eight years, and in the opinion Taney wrote in _Dred Scott v. Sanford_ (1857), he overruled congressional authority to restrict slavery, even in the Western territories. If the executive and legislative branches could not solve the slavery question, then it was Taney's resolution that the judiciary would.\n\nThe case of _Dred Scott v. Sanford_ aggravated the national temper over slavery, rather than soothing it. But when the Civil War broke out, Taney did not hesitate to give the courts primary oversight of how the war should be waged by insisting, in Ex parte _Merryman_ (1861), that Lincoln had no power to deal with the rebellion by suspending the writ of habeas corpus. But _Merryman_ was only Taney's opinion as a circuit judge, not a full Supreme Court decision, and Abraham Lincoln simply ignored it. More than that, Lincoln invoked the presumption of a set of \"war powers\" in the executive branch that justified an emancipation proclamation and a general suspension of habeas corpus. This rendered the Supreme Court nearly mute; if anything, its wartime decisions in the _Metropolitan Bank v. Van Dyck_ (1863), Ex parte _Vallandigham_ (1864), and _Prize Cases_ (1863) tamely endorsed the administration's policies on paper money, military tribunals, and the US Navy's blockade of the Southern coast, respectively.\n\nNevertheless, Lincoln remained anxious that once the war ended, the rationale for emancipation which he had built around the \"war powers\" would evaporate, and a revived judiciary might attempt to dismantle what the executive and legislative branches had accomplished. \"The emancipation proclamation,\" Lincoln wrote in 1863, \"I think... is valid in law, and will be so held by the courts.\" But he could not be sure, especially with Taney as chief justice. So it came as a great relief to Lincoln when, first, Taney died in October 1864 and he was able to appoint the unquestionably pro-emancipation Salmon P. Chase in Taney's place, and second, Congress followed his promptings and passed a thirteenth amendment to the Constitution, banning slavery entirely and putting emancipation beyond recall by the federal courts.\n\nStill, even though Lincoln was confident that Chase's appointment would render emancipation secure from the federal judiciary, Chase's tenure as Lincoln's secretary of the treasury had been marred by his unceasing schemes for self-promotion, and Lincoln feared \"that if I make him chief justice, he will simply become more restless and uneasy and neglect the place in his strife and intrigue to make himself president.\" Lincoln was not wrong to suspect Chase's ambitions, for Chase almost at once began a campaign to rebuild the Supreme Court's damaged reputation and reassert the prestige it had enjoyed under Marshall and Taney. The Reconstruction federal courts, under Chase (and then Morrison Waite) as chief justice, would expand federal judicial oversight at the expense of state courts, reassert the centrality of habeas corpus (to the denigration of wartime military tribunals), enjoy an expanded control of the award of money damages through the creation of the Court of Claims, and give federal courts increased jurisdiction over bankruptcy cases (and through that, increased powers over the national economy). At the same time, the federal courts would also become, in the name of restraining overreach by the executive and legislative branches, a potent agent in the undermining of Reconstruction.\n\nThe federal court system at the end of the Civil War was not large. Nine justices sat on the US Supreme Court, but beneath them were only fifty-four federal district judges in as many districts. Appeals from the district courts were heard in ten federal circuit courts (a jurisdiction that combined several districts, where the local district judge and a Supreme Court justice for that circuit presided); in turn, appeals from the circuit courts went to the Supreme Court. The membership of the high court itself had undergone some serious reorganization. Alabamian John Archibald Campbell resigned his seat as an associate justice at the outbreak of the war, and was replaced by Lincoln's old friend and judge on the Illinois 8th Judicial Circuit, David Davis. Justice John McLean, who had dissented from Dred Scott, died on April 4, 1861, and was replaced by Noah Swayne of Ohio; Samuel Freeman Miller was confirmed by acclamation in Congress to replace a seat left vacant for two years by the death of Peter V. Daniel; and Stephen J. Field was appointed to a new seat on the court created by Congress in 1863. But the most significant change was at the very top, as Salmon Chase assumed the role of chief justice on December 15, 1864.\n\nOriginally a Democrat, Chase had risen in Ohio politics as a fervent opponent of slavery, to the point where he had been tagged as \"the attorney-general for fugitive slaves.\" He moved to the Republican Party in 1855, and during the war he had agitated ceaselessly for a Radical Republican agenda in the Lincoln administration. Everything in his record suggested this would be the same path he would follow as chief justice. \"It will hereafter be counted equally a crime and a folly if the colored loyalists of the rebel States are left to the control of the restored rebels,\" Chase warned Lincoln four days before Lincoln's death; merely \"to confine the right of suffrage to the whites... will enable them to make all sorts of invidious and unjust discriminations; nor will they be slow to do so.\"\n\nBut Chase described himself as a Radical only \"in principle, and [I]\u200c have never disclaimed the name; but I have tried to be a conservative in working.\" This was not because Chase was indifferent to black rights but rather because federal jurists lived with a distinction between civil and natural rights, which saw the federal government as responsible for defending specific constitutional privileges and immunities, but not the wider scope of natural rights. \"Human Rights do not depend on the equality of Men or Races, but are wholly independent of them,\" declared the _New-York Tribune_ in 1862, which is to say that they exist independently of statutes and codes and even constitutions. The US Constitution was therefore a hybrid document, in that it vested the protection of some natural rights in the federal government, but left the bulk of them\u2014marriage, personal liberties\u2014with the states, and it would be up to the states to recognize the freedpeople as a new class having claim to natural-rights protections. As Lyman Trumbull explained in the debates over the Freedmen's Bureau, \"It is the policy of the Government that the rights of the colored men are to be protected by the States if they will, but by the Federal Government if they will not.\" But federal intervention was a secondary, compensatory step, not a primary one.\n\nThe design... is not... to consolidate all power in the Federal Government, or to interfere in the domestic regulations of any of the States, except so far as to carry out a constitutional provision which is the supreme law of the land....Let the people of the rebellious States now be as zealous and as active in the passage of laws and the inauguration of measures to elevate, develop, and improve the negro, as they have hitherto been to enslave and degrade him... and we shall all be moving on in harmony together.\n\n\"So long as the states\" did not attempt an explicit abolition of those rights, \"the national government had no more power in the areas of traditional state jurisdiction than it had before the war.\" Not until states deliberately acted to deprive (or neglect protection for) those rights did the national government have the constitutional power to intervene.\n\nThus, the court would display a perfect willingness throughout Reconstruction to protect the freedpeople's property rights and physical safety when the states refused, but hesitate at enforcing equal access to streetcars, schools, and public spaces unless the states had clearly abdicated that responsibility, and not just as examples of criminal malice. Chase would also draw similar lines of jurisdiction between the branches of the federal government. He refused to preside in the traditional seat of the chief justice in the Virginia circuit courts because \"it was neither right nor proper that the chief justice or any justice of the Supreme Court of the United States\u2014the highest tribunal of the nation, and the head of one of the coordinate departments of the Government\u2014should hold a court subject to the control or supervision of the executive Department.\" Even a justice as sympathetic to the Radicals as Samuel Freeman Miller felt it necessary to insist that \"there can be no liberty where the power of judging is not kept separate from the executive and legislative powers.\"\n\nThe first test of postwar judicial assertiveness arrived quickly. In the fall of 1864, the commandant of the Military District of Indiana, Alvan Hovey, arrested a former candidate for the Indiana governor's chair, Lambdin Milligan, who had been part of a hare-brained, pro-Confederate plot to raise an insurrection in the Midwest. Milligan was duly arraigned before a military tribunal, found guilty of treason on December 18, 1864, and sentenced to be hanged. Milligan seems to have been persuaded that Lincoln (who delayed signing Milligan's death warrant) would issue a pardon, so it was not until after Lincoln's murder\u2014and Andrew Johnson's mutterings about treason being made \"odious\"\u2014that Milligan became nervous and petitioned in the US Circuit Court for the District of Indiana for a writ of habeas corpus, so that he could be retried in a civilian (and presumably less bloodthirsty) court.\n\nMilligan's appeal for a writ was referred to the US Supreme Court, which heard arguments in March 1866 and announced a five-to-four decision in April. Written by David Davis (with a _caveat_ from Chase), the decision discharged Milligan and declared military tribunals unconstitutional in areas where the civil courts were open and operating. \"The Constitution of the United States is a law for rulers and people, equally in war and in peace,\" Davis wrote in ex parte _Milligan_ , \"and covers with the shield of its protection all classes of men, at all times, and under all circumstances.\"\n\nRadical Republicans treated ex parte _Milligan_ as a stab in the back, binding the ability of the army, and especially the Freedmen's Bureau, to intervene when Johnson's self-reconstructed regimes passed newer and more oppressive versions of the \"black codes.\" A group of Radical congressmen wrote a series of articles for the _Washington Daily Chronicle,_ protesting that \"This opinion of the Supreme Court\" shows a \"total want of sympathy with the spirit in which the war for the Union was prosecuted,\" and offered a path by which Reconstruction could \"be reversed by the very theory of construction which led to rebellion.\" Coming hard on the heels of Andrew Johnson's humiliation in the 1866 congressional elections, it now appeared that \"there will be a new conflict,\" this time between Congress and the judiciary.\n\nSuch a showdown seemed all too likely when, in the same month the Milligan decision was released, the Chase Court moved to strike down \"iron-clad\" loyalty oaths (\"iron-clad,\" meaning oaths which averred that the oath-taker would not only be loyal to the United States, but always had been). In January 1865, Missouri (one of the four slave states that remained loyal to the Union) adopted a new ordinance that outlawed slavery, but also added an oath to be taken by anyone acting in any public capacity, including clergy \"of any religious persuasion, sect or denomination,\" that they had \"always been truly and loyally on the side of the United States against all enemies thereof, foreign and domestic.\"\n\nThe Roman Catholic archbishop of St. Louis treated this ordinance as an unconstitutional trespass on religious freedom, and one of his priests, John A. Cummings, proceeded to say Mass and preach on September 3, 1865, without taking the oath. Cummings was indicted, tried, and imprisoned. He appealed to the Missouri Supreme Court, then to the US Supreme Court, and in December 1866 the court branded the oath as a bill of attainder and an ex post facto law (because it imposed punishments for acts that \"previous to the adoption of this Constitution... had not been declared punishable or illegal\" and that had not been criminal at the time they were committed), violating the rule of presumption of innocence. This brought still more violent eruptions in Congress. \"That supreme tribunal of justice has no power in the premises,\" declared Ohio congressman John Bingham. Reconstruction policies \"are political not judicial questions, and can be decided only by the political department... and from that decision there is no appeal.\"\n\nBut Bingham needed more than fury to save Reconstruction from the hands of the courts, especially given that the Milligan and Cummings cases offered the possibility that the Supreme Court might send Lyman Trumbull's 1866 Civil Rights Act to the same dustbin. Bingham's solution, like Lincoln's on emancipation, was to cut the Gordian knot with a constitutional amendment that would place civil equality beyond the power of the courts to alter.\n\nBingham had proposed such a fourteenth amendment in December 1865, at the very opening of the Thirty-Ninth Congress. But it was not until after Andrew Johnson had vetoed Trumbull's Civil Rights Act (which Congress overrode) that Bingham's amendment was re-introduced, now expressly declaring that states were obligated to protect the \"privileges or immunities of citizens of the United States.\" The Fourteenth Amendment was approved by the Senate on June 6, 1866, and by the House of Representatives the following week. But it took more than two years for the amendment to achieve ratification, partly because Andrew Johnson encouraged resistance to ratification at every point, and partly because the amendment itself was so awkwardly constructed. Of the amendment's five sections, only one dealt with the \"privileges or immunities of citizens of the United States\"; the others were an omnibus that forbade assumption of the Confederate debt, imposed a further ban on Confederate office holders, and threatened to proportion representation in Congress on the basis of actual voters (an indirect way to prevent Southern states from swelling their delegations in the House of Representatives on the basis of black citizens who were denied voting rights).\n\nThe weak link in the Fourteenth Amendment, however, was the failure of the amendment to specify just what was included in the \"privileges or immunities\" that the federal government now had authority to enforce. The court's 1823 decision in _Corfield v. Coryell_ had narrowly defined the \"privileges and immunities of citizens\" (Art. 4, sec. 2) as simply matters of comity that states extended to citizens of other states; in 1833, John Marshall himself had declared (in _Barron v. Baltimore_ ) that the \"privileges and immunities\" clause was designed to restrict the federal government, not the states. So it remained possible that, even after the ratification of the Fourteenth Amendment, black voting rights could be kept at bay, since voting rights belonged to a category of rights over which the states, not the federal government, had primary jurisdiction.\n\nThe Fortieth Congress, assembling for its lame-duck session in December 1868, took up a resolution from the House Judiciary Committee, calling for a fifteenth constitutional amendment specifically to ban \"discrimination... in the exercise of the elective franchise, or in the right to hold office in any State, on account of race, color, nativity, property, education or creed.\" This amendment was bitterly opposed by Border State Southerners who had never been subjected to Reconstruction, and by Northern Democrats who resented the extension of voting rights to African Americans in their own states. The lame-duck Republican majority settled for half a loaf, adopting a stripped-down version of the amendment that simply provided that \"the right of citizens of the United States to vote shall not be denied or abridged by the United States or by any State on account of race, color, or previous condition of servitude.\"\n\nThe Fourteenth and Fifteenth Amendments put the executive and legislative branches' shoulders to the wheel of voting rights, but they did nothing to prevent the Chase Court from digging potholes in their path, or chipping away at the severities of the Reconstruction statutes. In February 1867, Texas state authorities under the Reconstruction Acts brought suit against three brokers to recover US bonds that had been conveyed to Texas before the war as part of the financial settlement of Texas's affairs once it was annexed to the United States. In January 1862, the Confederate state of Texas authorized the sale of $634,000 worth of the remaining bonds to purchase stores and ordnance. But once Johnson's self-reconstructed Texas regime was in place, the state set out to recover the bonds from the brokers, George White and John Chiles, who had purchased them. If, as Lincoln had always claimed, the Confederacy was a legal nonentity, then the state regimes under the Confederacy could have no better claim to legal existence, and all transactions conducted under their aegis were as invalid as Confederate government debts.\n\nBut when _Texas v. White_ landed in the lap of the Chase Court in 1867, the conclusion drawn by Chase was a good deal more complicated. Yes, Texas's secession was constitutionally invalid (which put paid, as far as any Supreme Court decision can, to all the elaborate arguments for secession). \"The Constitution, in all its provisions,\" wrote Chase, \"looks to an indestructible Union, composed of indestructible States.\" But Chase also drew a line of demarcation between invalid rebel actions like the bond sale, and the everyday transactions needed \"to preserve the social community from anarchy and to maintain order.\" Whatever the Radicals' enthusiasm for a root-and-branch recreation of Southern society, \"acts necessary to peace and good order among citizens... which would be valid if emanating from a lawful government must be regarded in general as valid,\" and neither Congress nor the president had any authority to insert other substitute governments in their place.\n\n_Texas v. White_ was only the best known of the postwar claims cases that, in parallel with Andrew Johnson's amnesties, gradually gave back the legal ground that the Radical Republicans had claimed for reconstructing the Confederacy and presaged a more serious weakening to come\u2014on civil rights and the Reconstruction constitutional amendments. Salmon Chase suffered a stroke in August 1872, recovering sufficiently to rejoin the court but dying after a second stroke in May 1873. In his place, the newly reelected President Grant installed as chief justice another Ohio lawyer, Morrison Waite. But it would not be Waite who called the court's tune. \"Mr. Waite stands in the front rank of second-rate lawyers,\" snarled _The Nation_ , and he was easily overshadowed by Samuel Freeman Miller (\"the dominant personality now upon the bench\") and Joseph P. Bradley (who joined the court in 1869).\n\nThe direction of the Waite Court was set by Miller even before the new chief justice took the oath of office. In 1869, Louisiana's Republican governor, Henry Clay Warmoth, signed legislation restricting New Orleans slaughterhouses to the use of a single state-chartered \"grand slaughterhouse\" in order to control the dumping of offal into the Mississippi River. The Butchers Benevolent Association balked, claiming that the new law would deny non-cooperating butchers the \"privileges or immunities\" bestowed equally on all citizens by the Fourteenth Amendment. But when the Supreme Court heard the butchers' appeal in 1872, Miller issued a majority opinion that (in April 1873) confined the application of \"privileges or immunities\" strictly to federal matters. \"There is a citizenship of the United States,\" Miller wrote, which guarantees equal access to all the \"privileges and immunities\" the Constitution implies; but there is also \"a citizenship of a state,\" and the two \"are distinct from each other.\" The Fourteenth Amendment, Miller continued, protects only those \"privileges and immunities\" pertaining to federal citizenship, and only from interference by the federal government. The Louisiana statute interfered with neither; it governed only a function of state citizenship, which only the state could define.\n\nMiller's intention in the _Slaughterhouse Cases_ was an enlightened one\u2014to allow Louisiana to prevent the contamination of New Orleans' water supply\u2014and Miller explicitly criticized Southern whites' attempts to impose \"upon the colored race onerous disabilities and burdens.\" But it also reinforced the Chase Court's proclivity for clipping the wings of Congress's federalization of \"every little petty case of a civil character.\" And _Slaughterhouse_ would have long innings in a case that began the same week, in the same state. In March 1873, black Republicans in Grant Parish, a Unionist Louisiana parish, occupied the county courthouse in Colfax to ensure that the newly appointed county sheriff, recorder of deeds, and police jurors were sworn in without challenge. On April 13, a hastily assembled white militia, 140 strong, attacked Colfax, setting fire to the courthouse and gunning down more than 150 blacks, many of them after they had surrendered to the whites. The federal district attorney for Louisiana, James Roswell Beckwith, energetically used the Enforcement Acts to indict ninety-eight of the Colfax massacre's participants, beginning with the ringleader, Christopher Columbus Nash, and his lieutenants, James and John Hadnot, William Irwin, and William Cruikshank.\n\nHowever, the first trial of the Colfax defendants ended in deadlock before a mixed-race jury; a second trial in 1874 found only three of the defendants\u2014Cruikshank, James Hadnot, and Irwin\u2014guilty, and then only of conspiracy. When even that result was appealed to the Supreme Court as _US v. Cruikshank_ , Chief Justice Waite concluded that the original indictments had no ground whatsoever. The crimes in Colfax were committed by private individuals against other private individuals in malice (not statute), and involved local Louisiana issues; the Constitution, including the Bill of Rights and the Fourteenth Amendment, protected the \"privileges or immunities\" only of people acting as US citizens, and only against official state encroachments.\n\nThe _Slaughterhouse Cases_ , and then _Cruikshank_ (in consort with _Blyew v. US_ in 1872 and _US v. Reese_ in 1876), hobbled the power of both the Enforcement Acts and the Fourteenth Amendment to protect individuals from the actions of other individuals, and in many cases, to protect individuals from the actions of the states if no clear intention to violate federal law, or misconduct by federal actors, was involved. But it magnified the authority of the federal judiciary, even as it rang hollow for many of the people who had hoped that the federal courts would enlarge their rights, rather than merely the court's. John Mercer Langston, addressing a protest meeting that also featured Frederick Douglass and former Mississippi senator Blanche K. Bruce as speakers, was infuriated to find \"the Supreme Court... desirous of remanding us back to that old passed condition.... How long must we wait for State action to give us our rights.\" The editor of the African American newspaper the _New York Globe_ , T. Thomas Fortune, compared the court's decisions to being \"baptized in ice water.\" \"In the name of God, where now are the last two amendments to the federal constitution?\" wailed the _Natchez Brotherhood_. \"Will an American Congress permit a state to set up an oligarchy by reversing the intents and purposes of the organic law of the land?\" Waite's intentions were traditional and circumspect and concerned themselves more with the authority of the federal courts to prevent other areas of government from overextension. But the Waite Court nevertheless created the beginnings of an arc that successor courts in the 1880s and 1890s, under Chief Justice Melville Fuller, would use to abandon civil protections for African Americans entirely. The eclipse of Reconstruction was at hand.\n\n# Chapter 7\n\n#  Dissension: September 1872\u2013April 1877\n\nIt has become an unwritten political assumption that second presidential terms are marked by a steady decline of presidential authority and an increase in presidential scandal. That was not necessarily the expectation in 1872, as Ulysses Grant entered his second term, but the four years that followed did a great deal to provide the template for that pattern. And the price Grant would pay for a rise in scandal and a concomitant decline of credibility during his second term would, in turn, accrue to the lasting harm of Reconstruction.\n\nGrant survived a brush with scandal early in his first term, when two New York financiers, Jason \"Jay\" Gould and James Fisk, tried to use their connections to Grant's brother-in-law, Abel Corbin, to gain insider information on gold trading; he was embarrassed again by charges of corruption lodged against Ely Parker as commissioner of Indian Affairs in 1871. Grant had no direct involvement in either imbroglio, but the ooze of corruption fouled the political atmosphere all the same. Then, in September 1872, the _New York Sun_ published revelations about Cr\u00e9dit Mobilier of America, the financial corporation that handled the subcontracting for the construction of the transcontinental railroad. On paper, Cr\u00e9dit Mobilier channeled the purchasing and contracting for the Union Pacific Railroad corporation; in reality, it was a dummy edifice that billed the railroads for its services at hugely inflated prices, while passing the profits back to Union Pacific's officers\u2014and to twelve members of Congress, who accepted a bribe of shares in Cr\u00e9dit Mobilier and, in return, agreed to ignore Cr\u00e9dit Mobilier's shady dealings.\n\nThe Cr\u00e9dit Mobilier scandal did not touch Grant directly, but it set the wind blowing in a direction that did not bode well for him. Interior Secretary Columbus Delano had to be dismissed after charges of fraud were leveled at him; Grant's private secretary, Orville Babcock, was indicted for collusion with tax evaders; and Grant's secretary of war, William Worth Belknap, hurriedly resigned in advance of charges that he had taken bribes (only to have his replacement, George Robeson, charged with having profited from procurement deals for the navy). Before the Civil War, Mark Twain would write, \"there was nothing resembling a worship of money or its possessor\" among Americans. But the politicians and financiers \"reversed the commercial morals of the United States.... The people had desired money before his day,\" but now they had been \"taught... to fall down and worship it.\"\n\nTwain, who was not guiltless of his own money-lust, was exaggerating at both ends of his recollections. The American world of the 1870s was still one in which more than half the manufacturing firms of a city like Boston employed fewer than seven laborers and a third employed fewer than twenty, where shop owners in Cincinnati \"are generally men who are thoroughly acquainted with the practical features themselves. They are mechanics themselves.\" Lewis Latimer, who had enlisted in the Union navy at age sixteen, began work as an office boy, taught himself mechanical drafting, and assisted Alexander Graham Bell in patenting the telephone in 1876. Latimer's ascent could have been the textbook for the genius of free labor:\n\nHe believed then that whatever a man knew he had put in a book, so when he saw the [draughts]man making drawings he watched to find out what tools he used, then he went to a second hand book store and got a book on drawing and soon had a set of drawing instruments. He then looked over the draughtsman's shoulder to see how he used his instruments, and practiced with them at home until he felt thoroughly master of them, then one day he asked the draughtsman to let him do some drawing for him, the man laughed at him but finally consented to look at what he could do on another piece of paper and to his surprise found that Lewis was a real draughtsman, so he let him do some of his work from time to time and one day the boss saw him at work and was so pleased that he let him work everyday and gradually raised his wages so that from three dollars when he went to work he rose in eleven years to twenty dollars a week.\n\nWhat Twain had right, however, was that the world of American finance had changed. The Civil War had caused a stampede of withdrawals from the American economy by European investors, a vacuum filled by an array of self-invented brokers, bond-sellers, and bankers\u2014Jay Cooke, Jesse Seligman, George Higginson, Emanuel and Mayer Lehman, Junius S. Morgan, Jacob Schiff, Harvey Fisk, Alfred S. Hatch\u2014many of them newly arrived immigrants who aroused the hauteur of both northeastern Brahmins and Southern julep-sippers. When the new financiers were caught distributing bribes and kickbacks to Grant's officeholders and his party, their follies fed a nativist xenophobia and achieved a visibility never before seen in American economic life. The ensuing revulsion fueled the Liberal Republican Party's insurrection in 1872, and it paved the way for Democrats to regain control of the House of Representatives in the 1874 elections for the first time in twenty years, trading a Republican majority of 114 seats for a Democratic majority of 61. Connecticut elected a Democratic governor and a Democratic state legislature; New York elected a Democratic governor, secretary of state, attorney general, and majority in the state assembly; Indiana and Massachusetts elected a Democratic governor; and New Hampshire and Ohio gave both houses of their state legislatures to Democratic majorities.\n\nThe stench of political corruption easily drifted over into Reconstruction, since it became easy to portray Southern whites as yet more long-suffering victims of the new-model financial rapacity. In 1873, James Shepherd Pike, a veteran abolitionist and associate editor of the _New-York Tribune_ , began publishing \"The Prostrate State: South Carolina under Negro Government,\" based on his own observations of South Carolina politics, as a serial for the _Tribune_. Pike's \"The Prostrate State\" was a sort of reverse _Uncle Tom's Cabin_ , and it had approximately the same sensational effect. Standing in the state house in Columbia, Pike watched \"old aristocratic society\" replaced by a \"Black Parliament\" in which _arriviste_ carpetbaggers and treacherous scalawags, in league with their black dupes, created a carnival of corruption: legislative appropriations intended to purchase land for the freedpeople ended up in the pockets of the politicos; a special appropriation was made to reimburse the speaker of the state house, Franklin Moses, for his losses on a horse race; taxes had risen by 500 percent since 1860, even as the value of taxable property in the state had fallen by a third; and the governor, Robert Scott, \"spent $374,000 of the public money to get himself rechosen.\"\n\nPike's lurid account made no allusion to the prevalence of corruption in the South's prewar state governments, where \"expert transfers of balances to undiscovered bournes\" and \"august defalcations\" had been the order of the day. E. L. Godkin, the crusading editor of _The Nation_ , simply concluded that the Republican governments of South Carolina and Louisiana were \"a gang of robbers, making war on civilization and morality,\" and presidential intervention of the sort licensed by the Enforcement Acts was increasingly seen as just another species of corruption itself.\n\nNot that Grant or Congress could have easily paid the public costs for such interventions after 1873, even if they had wanted to. The transcontinental railroad spun off a network of smaller trunk lines into vast stretches of the West, on the assumption that railroads would generate demand for their services by their mere presence. But of the seventy-nine major railroad lines in the United States in 1872, twenty-eight had never managed to pay a dividend to investors. On September 18, 1873, the banking house of Jay Cooke & Co., which had invested heavily in the railroads, ran out of money: it could not meet the demands of its depositors, and closed the doors of its Philadelphia and Washington offices. New York stocks dropped 10 percent and the _New-York Tribune_ reported that \"general demoralization... characterized the transaction of business in Wall-st.\" By the end of the month, 101 banks across the nation had gone under, dragging down savings and erasing pools of capital for investment. As the economy struggled and slowed, so did government income. Federal budget receipts fell by 13.5 percent between 1873 and 1875, while spending followed it downward by 5.5 percent. Thanks to the Panic of 1873, there would be less and less room to finance federal oversight of Southern affairs.\n\nThe Panic fed Reconstruction fatigue. \"Let us have done with Reconstruction,\" the _New-York Tribune_ cried in 1870, \"The country is sick and tired of it.... We cannot forever keep the boy out of the water because he has not learned to swim.\" Much of the political energy behind Reconstruction had been generated by the public backlash against Andrew Johnson in 1866 (which led to Lyman Trumbull's 1866 Civil Rights Bill and the Reconstruction Acts), and through the election of Grant, who was determined to ensure voting rights through the Enforcement Acts. But Northern Democrats had never despaired of recovering their prewar hegemony in Washington. They bitterly opposed the Fifteenth Amendment, and even after the amendment's ratification, they employed Klan-like intimidation to suppress black votes at the polls in Northern cities. In Philadelphia, the black civic leader Octavius Catto was shot to death at Ninth and South Streets by a Democratic assassin on the day of the 1871 Pennsylvania state elections.\n\nNevertheless, many Republicans remained dedicated to the possibilities of a new racial egalitarianism. \"We believe the whole human race are one family,\" wrote Theodore Tilton in the _New York Independent_ , and \"we will have no permanent settlement of the negro question till our haughtier white blood, looking the negro in the face, shall forget that he is black, and remember only that he is a citizen.\" The Forty-First and Forty-Second Congress witnessed the arrival of six black congressmen (Benjamin Turner, Josiah Walls, Joseph Rainey, Robert Brown Elliott, Robert DeLarge, and Jefferson Long) and one black senator (Hiram Revels), elected by Republicans in Alabama, Mississippi, Louisiana, Florida, South Carolina, and Georgia. Overall, sixteen Southern blacks would serve in the Reconstruction Congresses; another six hundred would fill seats in state legislatures.\n\nBut for other Northern Republicans, black people remained something of an abstraction. \"The Northern people,\" smirked Confederate War Department clerk J. B. Jones at the end of the war, \"did not really like negro equality any better than we did,\" and John Emory Bryant was warned at the time he joined the Freedmen's Bureau that \"the North is no more concerned for the African than the South.\" Even the staunch Radical Republican Benjamin Wade never rid himself of the habit of referring to black people as \"niggers,\" and he complained that so much of the food in Washington was \"cooked by Niggers\" that \"I can smell and taste the Nigger.\" Northern blacks were disproportionately concentrated in urban areas (61 percent in towns of over 2,500, 55 percent in towns of over 10,000), which meant that for many Northern whites, it was entirely possible to live most of one's life without ever meeting a black person. One member of the Arkansas constitutional convention in 1868 remarked that \"I have seen, within fifty miles of St. Louis, the little children crowding the fence-rails, to look at a negro, and examining him with as much curiosity as if he had been a wild beast.\"\n\n9. The murder of Philadelphia black political organizer Octavius Catto on October 10, 1871, from _The Trial of Frank Kelly for the Assassination and Murder of Octavius V. Catto_ (1888). \"Mr. Catto, having closed down his school, was passing down South street, between 8th and 9th streets, when he was approached by a white man....Prof. Catto put his hand behind him as if to draw a pistol, when a car came up and he passed down the street a few paces....Catto again advanced and attempted to cross the street, when the assassin leveled his pistol again and shot at him three times....The first shot caused his death almost instantly.\"\n\nIt did not help, either, that the Northern free-labor apostles grew discouraged by the poor inroads they had made into Southern culture, and thus returned home disillusioned. They were, sighed a contributor to the _North American Review_ , only \"merchants, shopkeepers, mechanics, manufacturers, speculators, brokers, bankers\" and not \"barons after the fashion of the South.\" They were shunned and harassed, sometimes violently, \"a fashion set by the aristocracy,\" said John W. DeForest, a transplanted Union army veteran and Freedmens' Bureau commissioner. John Richard Dennett, a correspondent for _The Nation_ , discovered, \"When I am known to be a Northern man... I am made to feel that my company is not desired\"; a Southern physician sharpened the economic edge of Southern contempt by showing him a newspaper misprint\u2014\"Grand chance for Yankee enterprise and _thift_ \" (emphasis added)\u2014which the doctor thought would have better read \"grand chance for Yankee enterprise and theft.\" Harriet Beecher Stowe's colony in Florida, where she hoped to train \"immature minds such as those of our negroes\" to become a new version of \"the laboring class of England,\" limped along, battered by white suspicion and unexplained arson, but Stowe's presence gradually diminished, until by 1884 she and her ailing husband left for good. Albion Tourg\u00e9e lost his bid for a North Carolina congressional seat, and left his faltering law practice in Greensboro to write an embittered fictionalized memoir of his time in Reconstruction, appropriately titled _A Fool's Errand, By One of the Fools_. Having given up his hope of making headway for the \"dignity of labor, in liberty and equality before the sovereign law\" against the \"feudal civilization\" of the South, John Emory Bryant likewise departed Georgia for a new life in New York as a lawyer and realtor. The truth was, Benjamin Butler concluded, that \"capital is timid\":\n\nNo man will risk his capital where he does not believe he can get Justice before a Jury; where he does not believe that the community would look favorably upon his enterprise; and where he does believe every advantage will be taken from him and every wrong done him. And he now believes all that in the Southern States....New England is dotted all over with men who have gone down to Virginia, and bought farms... and who have been absolutely driven out by their neighbors.\n\nThe Reconstruction governments contributed mightily to their own demise with their incessant, self-weakening infighting. \"The Republicans are seriously weakened by faction,\" complained the _New-York Tribune._ \"Their leaders charge each other with all manner of treacheries and misdemeanors.\" In Mississippi, Republican governor James Alcorn tried to establish alliances with both the freedpeople and poor whites, only to be undermined by Klan violence on one side and Radical criticism of his perceived soft-handedness on the other. When Alcorn was elected to the US Senate, his most vocal critic, Adelbert Ames, snatched the gubernatorial nomination from Alcorn's would-be successor, thus forcing Alcorn to enter the 1873 governor's race himself\u2014which he then lost to Ames.\n\nBut Ames was himself embarrassed politically in December 1874, when Vicksburg's corrupt Republican mayor\u2014\"a white man at the time under indictment for twenty-three offenses\"\u2014was toppled by an alliance of reforming \"White Liners.\" Emboldened, the White Liners marched on Vicksburg to demand the resignation of the county's black sheriff, Peter Crosby. A pitched battle between Crosby's black supporters and the White Liners broke out, and Governor Ames was forced to call on President Grant for federal troops to restore order. The following November, Ames again asked Grant for federal troops to ensure order during the state elections. This time, Grant hesitated. His newest attorney general, Edwards Pierrepont, warned that any further military interventions in the South would only play into Northern Democrats' plans to use the military \"encroachments\" accusation in the 1876 presidential elections. \"The whole public was tired of these annual autumnal outbreaks in the South,\" Pierrepont advised Ames, and the Democrats were ready to use intervention \"to condemn any interference on the part of the Government.\" That did not prevent white Democrats from intimidating voters with their own display of paramilitary force: John Roy Lynch, an African American congressman who had his Mississippi seat snatched from under him by clever redistricting, testified before a Congressional inquiry that in Jefferson and Claiborne counties, Democrats had organized themselves as \"an armed military organization\" and \"would allow no republican meeting to be held and no republican speeches to be given by anybody.\" The result was a landslide for the Mississippi White Liners, and five months later, Ames resigned. The \"terrorism was so intense,\" testified Lynch, \"as in my judgment to make life, liberty, and happiness perfectly insecure except to democrats. I do not think there is any such thing as law in that society.\"\n\nThe same disastrous divisions sank the other Reconstruction Republican governments, one by one. Weakened by disputes between the Warmoth and Kellogg factions, and paralyzed by the Colfax massacre in 1873 and a fresh eruption of lethal race rioting in New Orleans in 1874, Louisiana wound up with two rival governors, one Republican and one Democrat, each demanding recognition from the federal government. Arkansas, similarly, ended 1874 with two rival Republican governors, Joseph Brooks and Elisha Baxter, and a Democratic legislature. Florida's Republican governor, Ossian Hart, skillfully managed to navigate between Florida's Republican factions and its unredeemed white Democrats; however, he died in March 1874, and in 1875 Democrats seized control of the state legislature. By the spring of 1876, only Louisiana and South Carolina still had Republican governments. \"We have endeavored to protect law and property... by the force of provisional state, by the force of provisional congressional law, and by the force of the armies of the United States,\" Vermont Republican George Edmunds admitted, \"and the result\" has been that every tool had broken in their hands: \"We have totally failed to secure the administration of justice.\"\n\n**End of Republican Reconstruction by State**\n\nStates reconstructed under the Reconstruction Acts | Effective ending of Reconstruction  \n---|---  \nVirginia  |  October 5, 1869   \nNorth Carolina  |  November 28, 1870   \nGeorgia  |  November 1, 1871   \nTexas  |  January 14, 1873   \nArkansas  |  November 10, 1874   \nAlabama  |  November 16, 1874   \nMississippi  |  January 4, 1876   \nLouisiana  |  January 2, 1877   \nFlorida  |  January 2, 1877   \nSouth Carolina  |  April 11, 1877\n\nBut it was not Republican factionalism alone that crippled the Reconstruction regimes. The freedpeople themselves were anything but a political monolith, and they wasted political energy on internecine quarrels as vigorously as white Republicans had. The first fault line was the reluctance of Southern blacks to accept white Republican leadership as unquestioningly as whites had expected. Northern white schoolteachers found that Southern blacks did not want whites to run either their politics or their schools. The teachers from the American Missionary Association who descended on the occupied Port Royal Sound to educate and uplift the freedpeople discovered, as Austa Malinda French noted, that \"nothing is more evident to those who actually know the Colored, than that while they respect, value, and revere, the good, they want little companionship with the whites.\"\n\nLawrence S. Berry, a freed slave turned journalist, urged his readers to \"forget our sable complexion\" and close ranks with progressive whites. But, as one Freedmen's Bureau agent discovered, \"their long experience of slavery has made them so distrustful of all whites, that on many plantations they persist still in giving credit only to the rumors set afloat by people of their own color, and believe that the officers who have addressed them are rebels in disguise.\" Moreover, the freedpeople resented the paternalistic tendencies of well-intentioned whites, especially when assistance calcified into orders and sympathy into control. When Georgia's short-lived Republican legislature tried to ban the admission of black state legislators, black Methodist leader Henry McNeal Turner frankly said, \"My colored friends, the white men are not to be trusted. They will betray you.\" And Hiram Revels, the first black US senator to be elected from Mississippi, denounced white Adelbert Ames and white Mississippi Republicans as \"notoriously corrupt and dishonest,\" and actually applauded the downfall of Adelbert Ames as a triumph over \"corruption, theft, and embezzlement.\"\n\nBut just as divisive were the fault lines that separated blacks from blacks. A racial hierarchy that bestowed privilege along a carefully graded spectrum of color had long existed in the black South. \"There is in the Southern States a great amount of prejudice in regards to color,\" the African American abolitionist and novelist William Wells Brown admitted in 1867, \"even among the negroes themselves. The nearer the negro or mulatto approaches to the white, the more he seems to feel his superiority over those of a darker hue.\" Louisiana's politics were more than sufficiently twisted by white factionalism, and made even worse by rivalries among factions led by Pinckney Benton Stewart Pinchback (one-quarter black, and married to a white woman), Oscar James Dunn (born a slave, to a slave mother and a free black carpenter), and Caesar Carpentier Antoine (a one-time business partner of Pinchback's, his father was a free _gens du couleur_ and his mother was West Indian).\n\nIn postwar Savannah, Aaron Bradley mounted a political smear campaign against his rival for a seat in Congress, Richard White, a mixed-race Union army veteran from Ohio. White, sneered Bradley, was a \"hybrid\" who did not deserve the support of true African Americans. \"What color will he represent himself?\" asked Bradley, who then answered his own question: \"The greasy color.\" Even Frederick Douglass and Martin Delany sparred, with Douglass (himself biracial) bitterly criticizing Delany's black racial purism for \"going about the same length in favor of blacks, as the whites have done in favor of the doctrine of white superiority.\" Delany was right to assert African Americans' \"need for dignity and self-respect,\" but not, Douglass warned, to the point where \"he stands up so straight that he leans back a little\" and ends up in a version of black racial triumphalism little different from white supremacy. Douglass also parted company with John Mercer Langston, despite Langston's agreement that \"this is no more a white man's country and government than it is the country and government of the black man.... It is the country and government of the American people.\" Nevertheless, Douglass accused Langston of \"mad political ambition.\"\n\nThese interracial feuds were a key factor in the most singular absence in black Reconstruction in the South: namely, the lack of a single commanding leader who could bind together the disparate threads of African American identity into a single movement. Slavery was certainly no useful training-ground for the game of politics, and the marginalized experience of free Northern blacks did not present much more in the ways of practical opportunities for honing political savvy. Given that only Louisiana and South Carolina had developed any substantial prewar populations of blacks who were property owners, business proprietors, and skilled craftsmen, the likeliest quarter from which such leadership could have developed was the Northern black community\u2014but even then, few Northern blacks made the attempt. And no wonder; it was doubtful if Southern blacks would feel obliged to follow Northern leadership, a reluctance the National Conference of Colored Men demonstrated when it debated a resolution \"That we pay no heed to such men as Fred. Douglass and his accomplices, for the simple reason that they are well-to-do Northern men who will not travel out of their way to benefit the suffering Southern Negro.\"\n\nAnd it was certain that Southern whites would make aggressive Northern blacks a target of choice. \"Write as you please, but _never go south_ , or killed you _most assuredly_ will be,\" warned Julia Griffiths Crofts, Douglass's British friend and supporter. \"You are, in many respects, _a marked man_.\" So, when Douglass was invited to set up a newspaper in Alexandria, Virginia, in 1866, he politely declined: \"It is not my duty to court violence or martyrdom or to act in any manner which can be construed into a spirit of bravado.... I think it wise to remain where I am, at least until the public mind of the South shall attain a more healthy tone than at present.\" Martin Delany and John Mercer Langston played highly visible roles as African American political leaders in the Reconstruction South, but those roles were plagued by division and distraction. Langston was only active in Reconstruction politics as an agent for the Freedmen's Bureau and the National Equal Rights League from 1865 till 1868, when he took up a faculty appointment at the new Howard University in Washington, DC. Delany grew so impatient with his white Republican allies in South Carolina that he accepted a judicial appointment from Democrats. \"What benefit,\" he asked in 1874, \"have the colored people in South Carolina derived from the propagation of Republican sentiments.\" None, in his estimate: \"Such a party is not worth the effort to keep it in existence.\"\n\nCharles Sumner might not have disagreed. Dumped by Grant and the Republican caucus from the chairmanship of the Senate Foreign Relations Committee, Sumner retaliated by leading the defection of the Liberal Republicans in 1872. He could sense a tidal wave of reaction heading toward Washington, and as his health deteriorated, he labored to persuade Congress to adopt a new civil rights bill. He died in March 1874, begging, \"Don't let my bill fail\"; the last session of the Forty-Third Congress passed it before the new Democratic majority in the House could assume power, but only by a tie-breaking vote in the Senate provided by Vice President Henry Wilson, and a disrespectful margin of 135\u2013114 in the House, with 38 abstentions. It was Reconstruction's last hurrah (and would be duly overturned by the Supreme Court in 1883).\n\nIn June 1876, the Republican National Convention met in Cincinnati, and this time a plethora of candidates battled one another over the course of six ballots, until on the seventh, the convention settled on an outsider, whom _The Nation_ deemed \"a man by no means conspicuous in public affairs\": Rutherford B. Hayes, a Union army veteran and second-term governor of Ohio. Hayes had loyally supported congressional Reconstruction and included black voting rights in his platform in his bid for the Ohio governorship. But by 1875, Hayes came to the conclusion that Reconstruction issues were costing him votes, and he backed away from endorsing Sumner's posthumous civil rights bill. The Democrats countered two weeks later with the nomination of Samuel Tilden, the reforming governor of New York, and almost at once the \"outbreaks\" Grant no longer felt eager to restrain with federal troops erupted in full force. The governors' races in Louisiana and South Carolina became violent referenda on Republican rule in those states. East Baton Rouge witnessed eighteen political murders in the month before the election. In Hamburg, South Carolina, an argument over right of way for a parade mushroomed into a skirmish between incumbent Republican governor Daniel Chamberlain's black militia and \"Red Shirts\" loyal to Democratic candidate (and ex-Confederate general) Wade Hampton; the fighting ended in the execution of four captured militiamen. Governor Chamberlain appealed to Grant for federal troops \"to repress the violence in this State during the present campaign.\"\n\nGrant, denouncing \"the late disgraceful and brutal slaughter of unoffending men in Hamburg, S.C.,\" issued a cease-and-desist proclamation and sent in three companies of infantry. \"The scene at Hamburg, as cruel, bloodthirsty, wanton, unprovoked, and as uncalled for as it was,\" Grant wrote to Chamberlain on May 26, \"would scarcely be accredited to savages, much less to civilized and Christian people.\" Nevertheless, Chamberlain informed the president that \"every republican meeting was interrupted by armed multitudes of democrats\" and \"the republican orators jeered at, interrupted, vilely insulted, and hissed down.\" More shoot-outs followed in Charleston on September 6, in Ellenton on September 15, and in Cainhoy on October 16. \"Who commit [ _sic_ ] these undeniable crimes against the very order and existence of society? Democrats,\" railed _Harper's Weekly._ \"Who condone [ _sic_ ] those crimes by silence, by incredulity, by calling them 'negro riots,' or by sneering at the bloody shirt? Democrats.\" In Edgefield County, Burton Stroud, a sharecropper, was told by his white employer that \"by this time another year there would not be a God damned radical found, and that we are determined to make away with every God damned one.\" When he persisted in attending \"the republican speech,\" white Democrats tracked him to his house, shouting:\n\n\"O, Burton, look out, God damn you, we are coming for you. We have been talking about you all year and telling you to come over on the Lord's side, and you wouldn't come; and now you shan't be a radical, and you shan't be a democrat, for, God damn you, we mean to kill you and cut your head off and put it up on a stick and make a mark to shoot at, and you will vote your next ticket in hell, God damn you.\" And about this time they were shooting about seventy-five yards from my house. I run, and there was a ditch by the side of which there was some weeds, and I went down the hill... and they went on past me and missed me.... My wife afterward came to me, after I got out in the body of woods, and I went over to a neighbor's of mine by the name of William Collins, a colored gentleman, and my wife brought my boots and hat to me; and I asked her where was the men, and she said around there yet; and I says, \"You come away from there, and come with me.\" Then I made for the village, and it was four weeks before I went back there any more.\n\n10. Massachusetts senator Charles Sumner died on March 11, 1874; in this memorial tableau, Columbia lays a wreath of laurel on his casket as freed slaves kneel in sorrow and an angel, leading an angelic chorus, carries a tablet with the motto \"Equal rights to all,\" a reference to the Civil Rights Bill that Sumner had been guiding toward passage at the time of his death.\n\nWhen voters went to the polls on November 7, in the largest voter turnout to that point in US history, Democrats appeared to have scored their greatest victory in two decades, starting at the top with Samuel Tilden. Even Grant admitted the next morning, \"Gentleman, it looks to me as if Mr. Tilden is elected.\"\n\nBut Grant was uncharacteristically premature in his judgment. Tilden carried New York by a whopping fifty thousand votes, but in other states, the vote count was much closer. Republican election judges in Louisiana, South Carolina, and Florida voided enough Democratic ballots to claim the electoral votes of those states for Hayes, and then to award him the election, 185 to 184. The howl of electoral robbery echoed loudly, so loudly that the editors of the _Atlantic Monthly_ feared that \"the national House of Representatives and Mr. Tilden have it in their power to cause an explosion in the South so terrific that the outbreak of 1860\u201361 will be almost forgotten.\" But the Democratic House and the still-Republican Senate could not agree on how to count the electoral votes, because rival totals were submitted in the three contested states, and in January 1877 Congress created a joint House-Senate Electoral Commission to determine the winner. After six weeks of bickering, Hayes was awarded all twenty contested electoral votes, just in time to be inaugurated. \"I have not yet seen a single Republican who had the face to stand up and defend the act,\" wrote Washington lawyer James William Denver in disgust. \"There will be no use in ever again holding an election.\" But as an olive branch to coax congressional Democrats into cooperation, Hayes signaled that he would recognize the election of Democratic governors in South Carolina and Louisiana\u2014namely, Hampton and Francis T. Nicholls\u2014and forbear the use of federal troops to intervene on behalf of Republican candidates and voters.\n\nThe Compromise of 1877 is often considered the end of Reconstruction\u2014and it was, in the sense that it finally closed the door on the possibility that the federal government would play a significant role in ensuring voting rights and fair elections in the South for another eight decades. But in truth, Republican Reconstruction was as good as dead even before Hayes's inauguration. Democratic control of the House ensured that, from 1875 onward, no funding for enforcement would be forthcoming, regardless of who occupied the presidency. Federal troops in the Department of the South comprised only thirty-seven companies from five regiments, numbering no more than 1,600 officers and men, and there were only 165 federal soldiers on duty in South Carolina, and 123 in Louisiana, in 1877. In fact, Hayes did not actually have to move any troops out of the South; he merely ordered the guard company assigned to the state capitol in Columbia and the St. Louis Hotel in New Orleans returned to their quarters.\n\nHayes soberly believed that his administration could \"be a government which guards the interests of both races carefully and equally.\" But it would have to operate \"by the united and harmonious efforts of both races, actuated by motives of mutual sympathy and regard\" which would \"forever wipe out in our political affairs the color line.\" Where Hayes expected that upwelling of mutual sympathy and regard to come from was unclear, but no Republicans in the South expected it to come from the Democrats, who had now completed their redemption of the old Confederacy. Daniel Chamberlain surrendered the governorship he claimed to have won in South Carolina to Wade Hampton on April 10, 1877, with what could be taken as the epitaph of Reconstruction: \"Today... by order of the President whom your votes alone rescued from overwhelming defeat, the Government of the United States abandons you.\"\n\n#  Epilogue\n\nOn June 15, 1869, Patrick Sarsfield Gilmore lifted his baton before an audience of forty thousand Bostonians and gave the downbeat for ten thousand singers and one thousand instrumentalists to open the National Peace Jubilee. The five-day festival had been Gilmore's project for two years, and it gave all those performers what Boston's premier music critic, John Sullivan Dwight, hailed as a \"new taste of the joy of unity of effort, a new love of co-operation, and a deeper sense of the divine significance and power of music than they ever had.\" Harvard's John Langdon Sibley thought it was \"exceedingly exciting... & nearly all have gone away jubilant over what they have seen and heard.\" It was also depressingly premature. On the day the festival opened, congressional Reconstruction had been in operation for only two years, and four of the one-time Confederate states were still under military rule. By the time Gilmore had organized an even bigger World Peace Jubilee in June 1872\u2014featuring a chorus of twenty thousand and orchestras and bands amounting to two thousand musicians\u2014all ten of the reconstructed states had been readmitted to the Union, but three of them had already fallen back into the hands of Democratic \"Bourbons\" and \"Redeemers,\" who were eagerly dismantling many of Reconstruction's accomplishments.\n\nGiven that dismantling, it is easy to miss what Reconstruction actually did manage to accomplish. Principally, it did, for better or worse, achieve the political reunion of the rebellious South with the rest of the nation, and did so with the basic shape of constitutional federalism reasonably intact and the chief object of the war\u2014the elimination of legalized slavery\u2014decided beyond challenge. The idea that states had the power to withdraw from, or break up, the American Union went down to a decisive defeat, but the importance of states within a federal republic did not. Charles Francis Adams Jr. admitted that \"what is known as the doctrine of States rights... fell into much unmerited odium through its abuse during the progress of the irrepressible conflict,\" but \"the American people still hold it in strong affection, and cling tenaciously to State lines and State authority.\" In time, \"when the wounds of Civil War are healed... and the republic restored to its normal condition,\" Wisconsin Republican senator James Doolittle predicted in 1879, \"the burdens of the federal government, now so crushing in their weight, will once more... rest as lightly upon our people, and its blessings will be as great to all sections as they were before that terrible convulsion\" and in \"our true normal condition, the burdens of the federal government will be as light as the air we breathe, though its blessings may be as vital to our well-being.\"\n\nEven the strongest measures taken by the US government, during both the war and Reconstruction, were deployed less with a view toward subjugating the states to a centralized authority, and more toward nudging them back into a federal alignment. Even Lyman Trumbull's Civil Rights Bill in 1866 and the Fourteenth Amendment were largely constructed so as to prod the states to carry out the work of equal rights themselves. Trumbull assured the Senate that his bill \"will have no operation in any State where the laws are equal, where all persons have the same civil rights without regard to color or race.\" All this was accomplished without resorting to the grimmer, retaliatory strategies of treason trials or military tribunals\u2014and even more to the point, without triggering a renewed civil war. \"Before I went South,\" wrote one British traveler in 1879, \"I expected to find that the Southern States had been for some time in a sort of Pandaemonium in which a white man could hardly live\"; instead, \"to my great surprise I found exactly the contrary.\"\n\nThe great losers in this process were Southern blacks who, in W. E. B. Du Bois's phrase, \"went free; stood a brief moment in the sun; then moved back into slavery.\" But even then, it was not entirely comparable to slavery; the Reconstruction amendments did recognize black citizenship and black voting rights, even if the Supreme Court acted swiftly to curtail their protection and enforcement. Black landownership not only became a reality, but increased from just 2.2 percent of Southern blacks in 1870 to 24 percent by 1910. Literacy rates rose over the same years, from 20 percent to 69 percent. These increases were not spread evenly: in the coastal plains and the mountains, 45 percent of Southern blacks owned their own land; in the Mississippi Valley and the vast \"Black Belt\" of northern Alabama, Georgia, and the Carolinas, only 8 percent of blacks owned their own land. In South Carolina, more than 75 percent of the lower Piedmont's twenty-four thousand tenant farmers were black, and fully half of them were sharecroppers by 1900. Even so, these circumstances, although far from ideal, were not slavery; indeed, the situation of blacks actually outstripped the condition of freed slaves and serfs in Brazil, Jamaica, and Russia.\n\nWhat Southern blacks lost in wholesale amounts was political agency. The tidal return of white Democratic rule to the South by 1877 allowed the disfranchisement of black Southerners through the imposition of literacy tests, poll taxes (starting with Georgia in 1871), physical segregation, property requirements, and sheer intimidation. Disfranchisement was a surprisingly slow, incremental process. There were 9 black legislators in South Carolina in 1882, and 11 in Mississippi; as late as 1890, there were 18 blacks in the Louisiana legislature. And Republicans in Virginia and North Carolina staged brief resurgences through alliances with poor white farmers in the 1880s and 1890s, campaigning, as Thomas Settle did in North Carolina, for \"no white man's party or colored man's party, but a party of principle; a party on whose banner is inscribed Liberty, union and Equality before the law; a party that proposes to elevate mankind of all races and colors.\" But after the turn of the century, only 1,300 blacks in Louisiana were managing to vote, and only 3,000 in Alabama and 5,000 in Texas. \"Here public opinion tolerates, when it does not justify, all crimes for the maintenance of the supremacy of the Democratic party,\" grumbled one federal supervisor of elections for North Carolina, continuing, \"Oligarchy sits enthroned by Fraud and Violence and dominates one-third of the Republic.\"\n\nAfter the Supreme Court overturned Sumner's Civil Rights Bill in the _Civil Rights Cases_ decision of 1883, Southern states began enacting statutes that banned interracial marriage, and segregated streetcars, trains, schools, dance halls, libraries, and hospitals\u2014all of it further sanctioned by the Supreme Court's _Plessy v. Ferguson_ decision in 1896. By 1920, black landownership had fallen to less than 20 percent; the 1930 census showed that of fifteen thousand locomotive engineers in the former Confederacy, only fifty were black, while 90 percent of the workers in furniture, textiles, and printing were white; by 1940, there were only eight black lawyers in Georgia and no black judges; not a single black policeman served in any of the deep South states. \"The colored man is the Jean Valjean of American society,\" Frederick Douglass protested on the twentieth anniversary of the outbreak of the Civil War. \"He has escaped from the galleys,\" but \"the workshop denies him work, and the inn denies him shelter; the ballot-box a fair vote, and the jury-box a fair trial. He may not now be bought and sold like a beast in the market, but he is the trammeled victim of a prejudice, well calculated to repress his manly ambition, paralyze his energies, and make him a dejected and spiritless man, if not a sullen enemy to society.\"\n\nThe Civil War redressed the unbalancing of federalism that had been threatened by nullification and secession; but federalism thus preserved rendered the federal government unable to ensure the political stability of Reconstruction and the incorporation of the South into a national free-labor system. Instead, federalism not only tied the hands, so to speak, of Radical Republicans for political purposes, but also guaranteed the persistence of Southern peonage. \"Although emancipated,\" John Mercer Langston warned in 1879, the freedman \"has not been given practical independence of the old slave-holding class, constituting the land-proprietors and employers in the section where he lives and labors for daily support.\" Southern elites saw little of benefit for themselves in the free-labor ideology that they wanted to embrace, nor were there many incentives for them to do so. \"Southerners used to look on the Northerners as coarse, money-getting people,\" complained the editors of the _Atlantic Monthly_ in 1877. \"Their contempt for the commercial character of the North originated... in the aristocratic training of the plantations, and their hatred of the liberty and equality doctrines... arose from the intolerance natural to all aristocracies.\" But it was not just the elites. Southern yeomen like Confederate apologist Leigh Robinson bundled together white supremacy and denunciations of the lure of the marketplace:\n\nWhat if the future shall say, that what the world called slavery, railed against as such, rolling up the whites of quite worldly eyes, in horror that such a thing should exist, stands forth as a patriarchal, beneficent relation, the kindest for the slave, as he came to us, not as France's \"rights of man\" fain would have him come; and what is now lauded to the skies, as \"freedom,\" be exhibited, as a cruel, grasping sauve qui peut, and Devil take the hindmost, the most sordid, the most heartless of all tyranny, the one which most degradingly, and least pitifully, shoves the weakest to the wall, and keeps him there.\n\nFree labor could even inspire a peculiarly gendered disdain, since (declared the Southern novelist Augusta Jane Evans in 1867) free labor made Northerners \"effeminate, selfish, most unscrupulously grasping\"; even their children were \"pitiable manikins already chanting praises to the Gold Calf.\"\n\nRedemption was an anti-free-labor strategy as much as it was a strategy of political exclusion. \"The nigger is going to be made a serf, sure as you live,\" prophesied one white Alabamian to John Townsend Trowbridge in 1865. \"It won't need any law for that.\" And not only blacks suffered from these exclusionary tactics: fully a quarter of all eligible whites in Virginia could not vote because they lacked the money to pay the state poll tax. When it was pointed out that South Carolina's \"eight box law\" (which required a voter to be able to read the names of candidates and the respective offices they were running for in order to place the correct ballot in one of eight ballot boxes) would disfranchise poor whites as much as blacks, the major general of the South Carolina militia merely replied, \"We care not if it does.\" The leader of the Republican minority protested that this law had no other purpose than \"keeping the middle classes and the poor whites, together with the negroes, from having anything to do with the elections,\" and he was not wrong. In North Carolina's Union county, not only agricultural acreage increased in the post-Reconstruction years; so too did the number of cotton plantations over one thousand acres (in Mississippi's Claiborne County, for example, the number of landholdings in the 50- to 199-acre bracket jumped from 73 to 170). Outside the principal cities, the use of cash as a medium of exchange entered only fitfully into Southern calculations. The New York Cash Store in Greenville, Alabama, advertised (despite its name) that \"we will take in exchange for goods, country produce, particularly Eggs, Chickens, Bees Wax, Dry Hides, Peas, Corn Meal, and anything else that we can dispose of.\"\n\nReconstruction aspired to be a bourgeois revolution, and it expected to triumph as effortlessly as the liberal notions of progress had promised. But time sometimes does indeed go backward. In the South, the bourgeois revolution was repelled. To have achieved a different outcome would have required two ingredients\u2014time and force\u2014that circumstances denied Reconstruction. Wendell Phillips certainly believed that Reconstruction could hardly last less than forty years. \"We have to... annihilate the old South, and put a new one there,\" and the best plan Phillips could imagine was long-term military occupation. \"When England conquered the Highlands, she held them,\u2014held them until she could educate them; and it took a generation. That is just what we have to do with the South.\" But Phillips was almost the only one willing to campaign for long-term military occupation. Instead, the same Romantic feudalism that had created the old Southern order reasserted its hegemony, and in another decade, the New South would link hands with Northern Progressivism in a comprehensive critique of the free-labor ideology. The bourgeoisie in this \"bourgeois revolution\" scenario did not flee to the aristocrats to create a \"counterrevolution of property\"; quite the opposite, the postwar Southern aristocrats ensured the destruction of the pure bourgeois revolution by appealing to a set of cultural and racial biases which safely defused the importance of property and sharply restricted access to it. Josephus Daniels, a North Carolina Progressive who would later serve in Woodrow Wilson's cabinet, hectored North Carolina's last black Congressman in the nineteenth century:\n\nIt is a sad commentary upon the political conditions that have obtained in this state... that North Carolina should have the only nigger Congressman....So far as this particular negro is personally concerned, he may be dismissed as beneath contempt....The negro in office regards himself as the enemy of the white man and is anxious to have his race share in that sentiment. Therefore he becomes a menace to the peace of the Commonwealth and a danger to the safety of both races....Venomous, forward, slanderous of the whites, appealing to the worst passions of his own race, he emphasizes anew the need of making an end of him and his kind.\n\nIn its passion for the organic notions of nationalism, Progressivism had scant patience for bourgeois rationalism. \"Very little reflection will serve to make it clear that a community which is striving toward individual liberty as an ideal is laboring to make it easier for the strong to exploit the weak,\" declared North Carolina Progressive and Duke University professor William Thomas Laprade in 1916, thus intersecting racial oppression with organic progressivism. In words that could have reawakened Jefferson Davis from his tomb, Laprade said, \"Nothing is more familiar to thoughtful persons than the practical antithesis between democracy and liberty,\" especially for a race characterized by \"docility\" and \"untrained minds.\" Slavery, Laprade had written in 1911, was a system of \"obligation or responsibility.\" While \"the freedom of a slave was... seriously limited, every such limitation had a definite basis in the practices of the community [and] protected him from the cruelty of his master.\" As a system of reciprocity, slavery had rendered the slave \"a free agent capable of acting for himself,\" even if the slave lacked bourgeois political rights.\n\nRather than a Union-approved narrative of triumphant _embougeoisment_ , the overthrow of Reconstruction allowed the South to develop a series of organic nationalist myths of its own, beginning with the cult of the \"Lost Cause,\" which argued that slavery had never been the real issue in the Civil War. \"The common belief that slavery was the cause of civil war is incorrect,\" announced former Confederate general Richard Taylor in his 1879 memoir, _Destruction and Reconstruction_ , \"and Abolitionists are not justified in claiming the glory and spoils of the conflict.\" Instead, Southerners had been rallying to defend a unique sectional culture, and had fought a noble, heroic, and constitutionally legal war, only to be overwhelmed by Northern might. The Lost Cause became a self-sealing narrative in which heroic Southrons, \"native to the soil,\" struggled to hold off a rapacious Yankee capitalism, and it was not unrelated to the emergence of another ideologically driven narrative of the nineteenth century, the Paris Commune. Both the legend of the Lost Cause and the legend of the Commune were expressions of a Eurocentric nostalgia for organized economies; both imagined themselves as the protectors of the oppressed, when in fact they were instruments of mayhem, leveling, and murder. Lost Causers ventured so far as to annex Abraham Lincoln, treating Lincoln's cautious political prudence as evidence that (as Kentucky journalist Henry Watterson claimed in 1902), \"if Lincoln had lived there would have been no era of reconstruction, with its repressive agencies and oppressive legislation.\"\n\nThe Lost Cause was matched by a rival but cooperating myth: that of the \"New South,\" which claimed, in Atlanta journalist Henry W. Grady's famous 1887 manifesto, that \"there is a New South, not through protest against the Old, but because of new conditions, new adjustments and, if you please, new ideas and aspirations.\" New Southerners feigned an admiration for markets, striving to persuade Northern investors that the South had learned its lesson, that \"we have found out that in the general summary the free Negro counts more than he did as a slave,\" and \"that one Northern immigrant is worth fifty foreigners, and have smoothed the path to south, wiped out the place where Mason and Dixon's line used to be, and hung our latch-string out to you and yours.\" Adolphe J. Lafargue, who represented Avoyelles Parish in the Louisiana House of Representatives, assured all hearers in 1887 that the South was \"new in brains, new in business enterprises, new in diversified industries and varied commercial pursuits, new in a reawakened agriculture, and an activity, push, thrift and a growth that is almost marvelous.\" This conviction allowed Woodrow Wilson to insist in 1896 that, \"There is nothing to apologize for in the past of the South... There is a great deal, however, cordially to accept in the present, and that is the consummation for which I pray and the consummation which has largely been brought about.\" But the assurances were hollow: the industrial renovation of the South was always an event of the future; per capita wealth in the South was less than half that of the North, and only 6 percent of the Southern labor force was involved in manufacturing by 1900. The New South was also the South of Jim Crow. And lynchings became the new way that blacks could be \"counted\" in the South: between 1885 and 1900, 210 lynchings took place in Alabama, 219 in Georgia, 253 in Mississippi, and 247 in Texas.\n\nThe end of Reconstruction is often spoken of in psychological terms, as a collapse of white Americans' nerve, or as a failure of Republican political will, when in cold truth Reconstruction did not fail so much as it was overthrown. Southern whites played the most obvious role in this overthrow, but they would never have succeeded without the consent of the Northern Democrats, who had never been in favor of an equitable Reconstruction, much less a bourgeois one. No such organized political opposition had arisen in Russia to obstruct Alexander II's emancipation of the serfs. Yet Democratic resistance played nearly as great\u2014if not greater\u2014a role in the demise of Reconstruction as Republican paralysis. From the time of Andrew Jackson, the \"Democracy\" had never been enthusiastic about market revolutions or bourgeois culture\u2014those had been the property of the Whigs and then the Republicans\u2014and they were no more enthusiastic after the war. Northern Democrats, wrote the veteran Baltimore politician John Pendleton Kennedy, \"flattered the lordly ambition of the aristocratic South, courted its favor, obeyed its behests, and found a satisfactory compensation in being permitted... to make it subservient to the selfish and sectional purpose of putting the whole Union at the foot of its slaveholding master.\" The Panic of 1873, the scandals of the Grant administration, and the reckless bubble of postwar finance all conspired to give new credence across the nation to the old Democratic complaints, and after the 1874 elections, Republicans would enter into more than twenty years of divided government, with neither party in control of the presidency and Congress at the same time and no hope of a rejuvenated effort to establish a national civil rights baseline along the lines of Sumner's Civil Rights Bill. The ability of the Democratic Party to rise, phoenix-like, from the ashes of near-self-destruction in 1860 to reconquer the House of Representatives in 1874, the Senate in 1878, and the presidency in 1884, is one of the least considered aspects of Reconstruction's demise, but it is also one of the most potent.\n\nIt may be too much to have expected Republicans, even Radical Republicans, to muster the political power to resist these forces. Much had been expected from Ulysses Grant as president. But Grant, while he was long on sympathy with the freedpeople, was temperamentally inclined to react to threats to civil rights rather than designing an overarching strategy for anticipating those threats; and in any case, his inauguration as president in 1868 came after three years of Andrew Johnson in which Reconstruction's momentum lost critical pressure. Much had also been expected of Southern Unionism, but Southern Unionists turned out to be too bruised, too racist, and too willing to strike compromises with the old plantation elite. Sumner and Stevens's plan to territorialize the Confederate states might have kept any reinvigorated Southern elite from storming back into Congress and joining forces with their one-time Northern Democratic allies. However, this could scarcely have been done without the continued maintenance of a sizeble military occupation force\u2014although given that such an occupation force would be responsible for 750,000 square miles and 9 million inhabitants, it would have required at least as many troops as were committed to pacifying the West.\n\nAs it was, the actual strength of the US Army had decreased dramatically since the last shots of the war were fired. The combined strength of the regular and volunteer forces had fallen between April 1865 and January 1866 from over 1 million men to just 90,000. Those who did remain were unenthusiastic about occupation duties. By July 1871, the army's strength stood at only 30,000, with most of those posted to the Western frontier; only 4,300 soldiers were on duty in posts in the South, and even then, mostly in major port or railroad cities. Grant used no more than 1,000 of them under the Enforcement Acts, but even that minuscule deployment did nothing to quell paranoia about \"military rule,\" and in 1878 House Democrats would ban the use of \"any part of the Army of the United States, as a posse comitatus\" in most domestic cases. (The supreme irony of this gesture was that, before the war, posse comitatus was exactly what slaveholders had been demanding as a mechanism for retrieving fugitive slaves).\n\nIt is dubious whether any Congress, Republican or Democrat, would have authorized the spending needed to support an effective army of occupation, capable of suppressing Southern white insurgencies, especially with the awesome debt of the war years looming. Nor was it likely, in simple ideological terms, that Republicans could have been persuaded by anything less than outright civil war or anyone less than Abraham Lincoln to have adopted transitional military dictatorships as an acceptable way of securing republican government for any significant length of time. Military occupations strained both the patience of taxpayers and the ingrained suspicion of military rule in American minds. After the Panic of 1873 and the Democratic takeover of the House the following year, the unlikelihood shrank to the vanishing point.\n\nIf it were possible to establish a scorecard for Reconstruction, the most promising tallies would be the successful restoration of the Union as a federal Union, the legal extirpation of secession as a political tool in settling national disputes, the raising of the freed slaves to citizenship through the Fourteenth and Fifteenth Amendments, and the avoidance of mass executions and imprisonments. That Reconstruction fell short of fully implementing most of these accomplishments is its tragedy, and that tragedy can be briefly and bluntly accounted for by six factors: the sheer unpreparedness of the victorious Union to undertake something as unprecedented as a political reconstruction of a third of its territory; the insurgent resistance of the defeated South; the unwillingness to prolong a military occupation to deal with that insurgency; the deaths and removal of the Radical Republican leadership (starting with Lincoln); the resurgence of the Northern Democrats; and, finally, the shortsighted decisions of the federal courts.\n\nIt is also possible to say that Reconstruction might have turned out a good deal worse than it did. Both the Civil War and Reconstruction were remarkable for their limited durations. By the standard of civil conflict, the American Civil War was comparatively short\u2014the English Civil Wars lasted seven years, and sporadic fighting continued in Scotland and Ireland for another five; the Taiping Rebellion lasted for fourteen; Sulla was the trigger for continual outbreaks of civil conflict for half a century, down to Octavian and Marc Antony. Indeed, some never seem to find any endpoint. Reflecting on the English Civil Wars, T. S. Eliot once remarked, \"The Civil War of the seventeenth century, in which Milton is a symbolic figure, has never been concluded.... I question whether any serious civil war ever does end. Throughout that period English society was so convulsed and divided that the effects are still felt.\"\n\nBut the American war did end, and in the span of only four years, and Reconstruction twelve years after that, and while it is always possible to wonder what might have happened if Reconstruction's issues had been pressed more firmly, or its overthrow contested more vigorously, it also has to be admitted that it might have spiraled onward in agony and insurgency for decades, and with every chance for a far more damaging, perhaps even genocidal, outcome. To assume that what was eventually achieved by federal authority in the \"Second Reconstruction\" in the 1960s would have taken no more real effort from federal authority in the 1870s pays insufficient attention to what had changed in the American experience as a whole over those nine decades. Any proper epitaph for Reconstruction must acknowledge that at least it managed to avoid the fate of other post\u2013civil war eras.\n\nMoreover, the good fight of Reconstruction was not overthrown entirely. Post-Reconstruction Republican presidents continued to appoint attorneys general who prosecuted voting-rights violations (although the number of convictions remained ludicrously small: 20 guilty verdicts out of 282 suits during the Hayes administration, 147 out of 692 cases under presidents James Garfield and Chester Arthur), and prosecutions of the Klan won a minor victory before the Supreme Court in ex parte _Yarbrough_ in 1884. For another generation after 1877, the struggles for black civil rights continued, albeit sporadically. (Union army veterans, for instance, organized the first racially integrated national society, the Grand Army of the Republic, and criticized the government's failure to give what was \"promised to our colored brethren, 'forty acres and a mule.' \")\n\nRepublican administrations also used the patronage they possessed for appointments of black Republicans to federal postmasterships and customs houses (especially along the long line of Southern coastal cities). As late as 1907, 244 black men and 36 black women held federal postmasterships from Republican administrations. Democrats repeatedly attempted to repeal the Enforcement Acts, only to be stymied by Republicans in the Senate and by President Hayes's vetoes, until the _Civil Rights Cases_ rendered them useless by ruling that the Fourteenth Amendment gave no power to guarantee \"social equality.\" Senate Republicans made valiant attempts to absorb equal funding for schools into the federal budget, only to have them beaten back by House Democrats. And in 1890, Massachusetts Republican Congressman Henry Cabot Lodge's elections bill (a revived Enforcement Act to provide federal oversight for mostly Southern elections) died a slow death, this time in the Senate.\n\nBut other Republicans wavered. The pressure to yield to reconciliation, to abandon black civil rights, to prevent defections by voters, and to concede talking points about race corroded the Reconstruction project. \"A new era of feeling and sympathy, the ties and associations of a common ancestry and a kindred destiny\" were what the _Atlantic Monthly_ hoped could now be \"fostered until the wounds of the past shall be cicatrized and forgotten\"\u2014by which was meant a common _white_ ancestry. Benjamin Harrison, another Midwestern Republican governor and Union army veteran who was elected president in 1888, assured the all-black Harrison League of Indianapolis that Republicans were dedicated to \"making all men free, and gave to you equal civil rights.\" But Harrison preferred, in the face of Democratic opposition (and with a view to swaying border state voters), to speak only of the \"common rights of American citizenship.\" When a Southern delegation confronted him in 1889, Harrison hastily promised to do nothing that would place \"them in positions where race instinct is sure to brew personal bitterness and social discord.\" He added, \"I would not like to see a Negro mayor or postmaster of Indianapolis,\" nor would he find it \"agreeable to me to practice my profession before a Colored federal judge.\"\n\nMerely to call Reconstruction a failure is too simplistic. Reconstruction was overthrown, subverted, and betrayed\u2014and then replicated, since many of the same hesitations over costs, internecine politics, and xenophobia led to dreary repetitions of these mistakes after the First World War and after the two Gulf Wars. In none of those cases would anything but lengthy and expensive occupations have sufficed to reinvent regimes that began conspiring to reverse battlefield defeat before the guns were barely cool; in none of those cases was enough of the country willing to use the force necessary to accompany liberation. \"Looking back over the whole policy of reconstruction, it seems to me that the wisest thing would have been to have continued for some time the military rule,\" said Ulysses S. Grant:\n\nThat was our right as a conqueror, and it was a mild penalty for the stupendous crime of treason. Military rule would have been just to all, to the negro who wanted freedom, the white man who wanted protection, the Northern man who wanted Union....The trouble about military rule in the South was that our people did not like it. It was not in accordance with our institutions. I am clear now that it would have been better for the North to have... held the South in a territorial condition.\n\nHe was surely right, but right will not stand as right if it is realized too late. \"I wonder if our white fellow-men realize the true sense or meaning of brotherhood?\" demanded Susie King Taylor, a slave-born teacher and military nurse. \"Was the war in vain? Has it brought freedom, in the full sense of the word, or has it not made our condition more hopeless.\" A century and a half later, her question still echoes horribly.\n\n#  Timeline\n\n1865  |   \n---|---  \nApril 9 | Surrender of Confederate General Robert E. Lee and his Army of Northern Virginia at Appomattox Court House  \nApril 14 | Assassination of Abraham Lincoln by John Wilkes Booth  \nApril 15 | Death of Abraham Lincoln\u2014Andrew Johnson sworn-in as 17th President  \nMay 10 | Capture of Jefferson Davis by federal cavalry near Irwinville, Georgia  \nMay 15 | Johnson appoints Oliver Otis Howard to head Freedmen's Bureau  \nMay 29 | Johnson's amnesty proclamation  \nJune 13-July 13 | New self-reconstructed state governments established  \nNovember 22 | Passage of Mississippi \"black code\"  \nDecember 4-July 28, 1866 | Meeting of 39th Congress (1st session)  \nDecember 13 | Establishment of Joint Congressional Committee on Reconstruction  \nDecember 18 | Secretary of State Seward announces ratification of 13th Amendment  \nDecember 19 | Passage of South Carolina \"black code\"  \n1866  \n---  \nJanuary 5 | Lyman Trumbull introduces bill to extend term of Freedmen's Bureau and a Civil Rights Bill  \nFebruary 19 | Johnson vetoes extension of Freedmen's Bureau Bill  \nMarch 27 | Johnson vetoes Civil Rights Bill  \nApril 2 | Johnson proclaims ending of military hostilities  \nApril 3 | Chief Justice Chase announces Supreme Court's decision in _ex parte Milligan_  \nApril 9 | Congress passes the Civil Rights Bill over Johnson's veto  \nMay 1-3 | Memphis, Tennessee, massacre  \nJune 13 | Passage of the 14th Amendment by House of Representatives  \nJuly 30 | New Orleans riot at the Mechanics Institute  \nAugust 14-16 | National Union Convention meets in Philadelphia  \nAugust 27-September 15 | Johnson's \"Swing 'Round the Circle\"  \nDecember 3-March 3, 1867 | Meeting of 39th Congress (2nd session)  \n1867   \n---  \nMarch 2 | Congress passes 1st Reconstruction Act over Johnson's veto  \nMarch 4-December 1  | Meeting of 40th Congress (1st session)  \nMarch 23 | Congress passes 2nd Reconstruction Act over Johnson's veto  \nApril 1-20 | Special session of 40th Congress  \nApril 12 | Chief Justice Chase delivers Supreme Court's opinion in _Texas v. White_  \nJuly 19  | Congress passes 3rd Reconstruction Act over Johnson's veto  \nDecember 2-November 10, 1868 | Meeting of 40th Congress (2nd session)  \n1868   \n---  \nMarch 2 | Articles of impeachment of Johnson approved and managers appointed by House of Representatives  \nMarch 11 | 4th Reconstruction Act becomes law after Johnson fails to sign or veto  \nMay 16 | First article of impeachment fails of guilty verdict in the Senate  \nJune 22-July 15 | Congress re-admits seven Southern states under the Reconstruction Acts  \nJuly 28 | Secretary of State Seward announces ratification of the 14th Amendment  \nAugust 11 | Death of Thaddeus Stevens  \nSeptember 19 | Camilla, Georgia, massacre  \nSeptember 28 | Opelousas, Louisiana, massacre  \nNovember 3 | Election of Ulysses S. Grant as 18th President  \nDecember 7-March 3, 1869 | Meeting of 40th Congress (3rd session)  \n1869   \n---  \nMarch 4 | Inauguration of Ulysses S. Grant as 18th President  \nMarch 4-April 10 | Meeting of 41st Congress (1st session)  \nMay 10 | Leland Stanford drives ceremonial \"golden spike\" to open transcontinental railroad at Promontory Point, Utah Territory  \nJune 15  | Opening of National Peace Jubilee in Boston  \nDecember 6-July 15, 1870 | Meeting of 41st Congress (2nd session)  \nDecember 6 | Grant's first annual message to Congress  \n1870   \n---  \nJanuary 24-July 15 | Re-admission of remaining Southern states  \nFebruary 25 | Hiram R. Revels, first African American to be sworn-in as a U.S. Senator, takes seat once occupied by Jefferson Davis  \nMay 31 | President Grant signs the 1st Force (or Enforcement) Act  \nJuly 19 | War declared between France and Prussia  \nOctober 12 | death of Robert E. Lee  \nDecember 5-March 3, 1871 | meeting of 41st Congress (3rd session)  \n1871   \n---  \nJanuary 8 | King Wilhelm I of Prussia proclaimed German Emperor at Versailles  \nFebruary 28 | Grant signs 2nd Force (or Enforcement) Act  \nMarch 4-April 20 | Meeting of 42nd Congress (1st session)  \nMarch 24 | Grant issues cease-and-desist proclamation to restrain political violence in South Carolina  \nApril 20 | Grant signs 3rd Force (or Enforcement) Act, also known as the Ku Klux Klan Act  \nJune 10 | U.S. Navy flotilla lands sailors and Marines in a punitive action against Korea  \nJuly 15 | Death of Thomas \"Tad\" Lincoln, fourth child of Abraham and Mary Lincoln  \nOctober 8-10 | Chicago's Great Fire  \nOctober 23 | Resignation of Rufus Bullock as governor of Georgia to avoid impeachment  \nOctober 27 | Arrest of William Marcy \"Boss\" Tweed in New York  \nNovember 10  | Henry Morton Stanley \"discovers\" Dr. David Livingstone near Ujiji, in east Africa  \nDecember 2 | Grant dismisses Amos Akerman as attorney-general  \nDecember 4-June 10, 1872 | Meeting of 42nd Congress (2nd session)  \n1872   \n---  \nFebruary 20 | Opening of New York City's Metropolitan Museumof Art  \nMay 1-3 | Liberal Republican national convention meets in Cincinnati and nominates Horace Greeley for president  \nSeptember 4 | Cr\u00e9dit Mobilier scandal published by the _New York Sun_  \nNovember 5 | Re-election of Ulysses Grant as President  \nNovember 29 | Death of Horace Greeley  \nDecember 2-March 3, 1873 | Meeting of 42nd Congress (3rd session)  \n1873   \n---  \nMarch 29 | James Pike begins serialization of _The Prostrate State_ in the _New-York Tribune_  \nApril 13 | Colfax, Louisiana, massacre  \nApril 14 | Justice Samuel Miller issues Supreme Court decision in _Slaughterhouse Cases_  \nMay 7 | Death of Chief Justice Salmon Chase  \nSeptember 18 | Jay Cooke & Co. closes doors of its Washington and Philadelphia offices\u2014beginning of Panic of 1873  \nOctober 10 | Assassination of Octavius V. Catto in Philadelphia  \nDecember 1-June 23, 1874 | Meeting of 43rd Congress (1st session)  \n1874   \n---  \nJanuary 4 | Inauguration of Adelbert Ames as Republican governor of Mississippi  \nMarch 11 | Death of Charles Sumner  \nApril 15-May 15 | 1st Impressionist exhibition opens in Paris  \nJune 1-September 7, 1875 | Congressional elections give Democrats 183-106 majority in the House of Representatives for 44th Congress  \nDecember 7-March 3, 1875 | Meeting of 43rd Congress (2nd session)  \n1875   \n---  \nMarch 1 | Grant signs Sumner's Civil Rights Bill  \nMarch 5-March 24 | Meeting of 44th Congress (special session)  \nMay 17 | Running of the 1st Kentucky Derby  \nJuly 31 | Death of Andrew Johnson  \nNovember 22 | Death of Vice-President Henry Wilson  \nDecember 6-August 15, 1876 | Meeting of 44th Congress (1st session)  \n1876   \n---  \nMarch 10 | Alexander Graham Bell conducts first successful experiment with Telephone  \nMarch 29 | Adelbert Ames agrees to resign as governor of Mississippi rather than face trumped-up charges by the Democratic state legislature  \nMarch 27 | Supreme Court, with Justice Waite writing for the majority, announces decision in _U.S. v. Cruikshank_  \nMay 10 | Opening of Centennial Exhibition in Philadelphia  \nJune 16 | Republican national convention nominates Rutherford B. Hayes as its candidate for president on the seventh ballot  \nJune 25  | Battle of the Little Big Horn  \nJune 28 | Democratic national convention nominates Samuel Tilden as its candidate for president  \nJuly 8 | Hamburg, South Carolina, massacre  \nNovember 7 | Election day fails to produce a clear presidential winner  \nDecember 4-March 3, 1877 | Meeting of 44th Congress (2nd session)  \n1877   \n---  \nJanuary 25-26 | Creation of Electoral Commission to determine winner of presidential election  \nMarch 2 | Designation of Rutherford B. Hayes as 19th President  \nApril 10 | Removal of federal protection for Chamberlain government in South Carolina and surrender of state capital to Wade Hampton the next day\n\n#  References\n\n#  Introduction\n\nAbraham Lincoln's comment on the idea of reconstruction are in his \"First Inaugural Address\" (March 4, 1861), \"Proclamation of Amnesty and Reconstruction\" (December 8, 1863), and \"Last Public Address\" (April 11, 1865), in _Collected Works of Abraham Lincoln_ , ed. R.P. Basler (New Brunswick, 1953), 4:252, 7:53\u20134, 8:403.\n\nCharles Sumner, \"Our Domestic Relations; or, How to Treat the Rebel States,\" _Atlantic Monthly_ 12 (October 1863), 521.\n\nThaddeus Stevens, _Reconstruction: Speech of the Hon. Thaddeus Stevens, Delivered in the City of Lancaster, September 7th, 1865_ (Lancaster, 1865), 5, and \"Reconstruction\" (December 18, 1865), _Congressional Globe_ , 39th Congress, 1st session, 74\n\nThe so-called Wade-Davis Bill is the \"Bill for Reconstruction,\" in _The Political History of the United States of America, During the Great Rebellion_ , ed. Edward McPherson (Washington, 1865), 317\u20138.\n\nCharles Sumner to John Bright (January 1, 1865), in _The Selected Letters of Charles Sumner_ , ed. Beverly Wilson Palmer (Boston, 1990), 2:262.\n\n#  Chapter One: Vengeance\n\nJacob Howard to Andrew Johnson (April 18, 1865), John W. Gorham to Johnson (June 3, 1865), and \"Interview with Charles Halpine\" (March 5, 1867), in _The Papers of Andrew Johnson: 1864\u20131865_ , ed. LeRoy P. Graf (Knoxville, 1986), 7:580, 8:173, 12:111.\n\n_Proceedings of the Union League of Philadelphia Regarding the Assassination of Abraham Lincoln, President of the United States_ (Philadelphia, 1865), 19.\n\nThe beginning of Andrew Johnson's slide toward reconciliation with the defeated Confederate states is in \"By the President of the United States of America\u2014A Proclamation\" (May 29, 1865), in _A Compilation of the Messages and Papers of the Presidents, 1789\u20131897_ , ed. James D. Richardson (Washington, 1897), 6:312\u2013314.\n\nThaddeus Stevens to Andrew Johnson (May 16, 1865), in _The Selected Papers of Thaddeus Stevens, Volume 2: April 1865-August 1868_ , eds. B.W. Palmer & H.B. Ochoa (Pittsburgh, 1998), 5.\n\nSumner, \"One Man Power vs. Congress\u2014The Present Situation\" (October 2, 1866), in _Charles Sumner: His Complete Works_ (Boston, 1900), 14:198, 202.\n\n\"Interview with George L. Stearns\" (October 3, 1865), in _The Political History of the United States of America During the Period of Reconstruction_ , ed. Edward McPherson (Washington, 1875), 49.\n\n\"Laws in Relation to Freedmen,\" 39th Congress, 2nd session, Senate Executive Doc. No. 6 (1867), 192\u2013199.\n\nHans L. Trefousse, _Thaddeus Stevens: Nineteenth-Century Egalitarian_ (1997; Mechanicsburg, PA, 2001), 175.\n\n#  Chapter Two: Alienation\n\nThe growing crescendo of opposition to Johnson can be seen in \"Reconstruction\" (December 4, 1865), \"Organization of the House\" (December 5, 1865), James Doolittle, \"Joint Committee on Reconstruction\" (December 12, 1865), Lyman Trumbull, \"Civil Rights\u2014Veto Message\" (April 4, 1866), and John B. Henderson, \"Reconstruction\" (June 8, 1866) in _Congressional Globe_ , 39th Congress, 1st session, 4\u201310, 26, 1755, 1760, 3034.\n\nJohnson's riposte to the Radicals begins with his \"First Annual Message\" (December 4, 1865), in _Messages and Papers of the Presidents_ , 6:353, 357, 360\u2013361, 365.\n\nLyman Trumbull's civil rights proposals appear in his \"An Act to protect all Persons in the United States in their Civil Rights, and furnish the Means of their Vindication\" (April 9, 1866), in _Statutes at Large_ , ed. George P. Sanger (Boston, 1868), 14:27\u201328.\n\nAndrew Johnson, \"Interview with a Colored Delegation respecting Suffrage\" (February 7, 1866), in McPherson, _Political History_ (1875), 53, 55.\n\n\"The Memphis Massacre,\" _Chicago Tribune_ (May 9, 1865).\n\n_James Speed; A Personality_ (Louisville, KY, 1914), 92\u201393. Underscores the disenchantment of members of Johnson's cabinet with the president.\n\n\"The President on the Stump,\" _North American Review_ 102 (April 1866), 532.\n\n\"Congress--Passage of the Civil Rights Bill in the House,\" _Chicago Tribune_ (April 10, 1866).\n\n\"The President's Message Vetoing the Freedmen's Bureau Bill--The Bill Immediately Passed Over the Veto by Both Houses,\" _New York Times_ (July 17, 1866).\n\n\"The Conspiracy at Washington,\" _Atlantic Monthly_ 20 (November 1867), 634, 636\u201337.\n\nThe Congressional Reconstruction legislation is contained in \"An Act to provide for the more efficient Government of the Rebel States\" (March 2, 1867), \"An Act Supplementary to an Act entitled 'An Act to provide for the more efficient Government of the Rebel States' \" (March 23, 1867), \"An Act Supplementary to an Act entitled 'An Act to provide for the more efficient Government of the Rebel States' (July 19, 1867) and \"An Act to amend the Act passed March twenty-third, eighteen hundred and sixty-seven\" (March 11, 1868), in _Statutes-at-Large, Treaties and Proclamations of the United States of America_ , ed. George P. Sanger (Boston, 1868), 14:428, 15:2, 14, 41.\n\n#  Chapter Three: Arrogance\n\nOn the costs of the war for the South, see Claudia D. Goldin and Frank D. Lewis, \"The Economic Cost of the American Civil War: Estimates and Implications,\" _Journal of Economic History_ 35 (June 1975), 317, and Jenny Bourne, \"Double Take: Abolition and the Size of Transferred Property Rights,\" in _Lincoln, Congress and Emancipation_ , eds. Paul Finkelman & Donald R. Kennon (Athens, OH, 2016), 228\u2013229.\n\nWhittier, \"Snow-Bound,\" in _American Poetry: The Nineteenth Century, Volume One: Freneau to Whitman_ (New York, 1993), 488\u2013489.\n\nSidney Andrews, \"Three Months Among the Reconstructionists,\" _Atlantic Monthly_ 17 (February 1866), 238, 245.\n\nFrederick Douglass, \"Reconstruction,\" _Atlantic Monthly_ 18 (December 1866), 762.\n\nOn the imposition of \"serfdom,\" see \"Report of an Assistant Superintendent, in Virginia\" (February 26, 1866), in Walter Lynwood Fleming, ed., _Documents Relating to Reconstruction_ (Morgantown, WV, 1904), 41.\n\nJohn Mercer Langston, \"Citizenship and the Ballot\" (October, 1865), in _Freedom and Citizenship: Selected Lectures and Addresses_ (Washington, 1883), 99\u2013100.\n\nExamination of Cullen A. Battle (October 18, 1871), in _Testimony Taken by the Joint Select Committee to Inquire Into the Condition of Affairs in the Late Insurrectionary States_ (Washington, 1872), 11:1061.\n\nFrances Butler Leigh, _Ten Years on a Georgia Plantation Since the War_ (London, 1883), 133.\n\n\"South Carolina Morals,\" _Atlantic Monthly_ 39 (April 1877), 474.\n\n\"The Political Condition of South Carolina,\" _Atlantic Monthly_ 39 (February 1877), 192\u20133.\n\nFor the Tenure of Office Act, see \"An Act regulating the Tenure of certain Civil Offices\" (March 2, 1867), and \"An Act Making Appropriations for the Support of the Army\" (March 2, 1867), in _Statutes at Large_ , 14:430\u201332, 485\u201387.\n\n\"Impeachment of the President (February 22\u201324, 1868), _Congressional Globe_ , 40th Congress, 2nd session, 1336\u20131369, 1382\u20131400.\n\n#  Chapter Four: Resistance\n\nOn the nomination of Grant, see _Official Proceedings of the National Republican Conventions of 1868, 1872, 1876 and 1880_ (Minneapolis, 1903), 28, 78.\n\n\"South Carolina\u2014The Reconstruction Convention,\" _New York Times_ (January 21, 1868).\n\nTourgee, \"Speech on Elective Franchise\" (February 22, 1868), in _Undaunted Radical: The Selected Writings and Speeches of Albion W. Tourg\u00e9e_ , eds. Mark Elliott & J.D. Smith (Baton Rouge, 2010), 39.\n\nAlbert T. Morgan, _Yazoo: Or, On the Picket Line of Freedom in the South_ (Washington, 1884), 25.\n\n\"The Mission of the War\u2014A Lecture by Frederick Douglass,\" _New-York Tribune_ (January 14, 1864).\n\n\"The Political Condition of South Carolina,\" _Atlantic Monthly_ 39 (February 1877), 186.\n\nExamination of Leander Bigger (July 15, 1871), in _Testimony Taken by the Joint Select Committee to Inquiry into the Condition of Affairs in the Late Insurrectionary States_ (Washington, 1872), 1:276.\n\nDepositions of Spencer Griffith, Charles E. Robert, S.C. Mercer and H.H. Aynett (August 27 and 29, 1868), in _Report of Evidence Taken Before the Military Committee: In Relation to Outrages Committed by the Ku Klux Klan in Middle and Western Tennessee_ (Nashville, 1868), 55\u201356, 59.\n\nT. Thomas Fortune, _After War Times: An African American Childhood in Reconstruction-Era Florida_ , ed. D.R. Weinfeld (Tuscaloosa, 2014), 17.\n\n#  Chapter Five: Distraction\n\nUlysses S. Grant to Joseph Hawley (May 29, 1868) and to Edward R.S. Canby (June 26, 1868), in _The Papers of Ulysses Grant_ , ed. John Y. Simon (Carbondale, IL, 1991), 18:264, 296; \"Inaugural Address,\" in _Messages and Papers of the Presidents_ , 7:6\u20138.\n\nOn Adolph \"Bovie,\" see William Livingstone, _Livingstone's History of the Republican Party_ (Detroit: Wm. Livingstone, 1900), 1:244.\n\n\"His mind seemed torpid\" appears in _The Education of Henry Adams: An Autobiography_ (Boston: Houghton Mifflin, 1918), 263\u2013264.\n\nGrant, \"First Annual Message\" (December 6, 1869), in _Messages and Papers of the Presidents_ , 7:28\u201329.\n\nOn the re-admission of Southern states, see \"Right of Representation\" (January 25, 1870), in _Congressional Globe_ , 41st Congress, 2nd session, 720, 759; \"An Act to admit the State of Virginia to Representation in the Congress of the United States\" (January 26, 1870), in _Statutes at Large_ , 16:62\u201363.\n\nOn landownership patterns, see Jonathan Weiner, \"Planter Persistence and Social Change: Alabama, 1850-1870,\" _Journal of Interdisciplinary History_ 7 (Autumn 1976), 237\u201338, 241, 257.\n\nThe Force Acts are itemized as \"An Act to enforce the Right of Citizens of the United States to Vote in the Several States of this Union, and for other Purposes\" (May 31, 1870), \"An Act to enforce the Rights of Citizens of the United States to Vote in the Several States of this Union\" (February 28, 1871), and \"An Act to enforce the Provisions of the Fourteenth Amendment to the Constitution of the United States\" (April 20, 1871) in _Statutes-at-Large_ , 16:140\u201346, 433\u201340 and 17:13\u201315.\n\nTestimony of Elias Hill (July 25, 1871), in _Testimony Taken by the Joint Select Committee to Inquire into The Condition of Affairs in the Late Insurrectionary States_ (Washington, 1872), 3:1411.\n\nEdward Townsend to Alfred Terry (September 11, 1871) and Grant to Akerman (December 12, 1871), in _Papers of Ulysses S. Grant_ , 22:180, 288.\n\nGeorge Ward Nichols, \"The Indian: What Should We Do With Him,\" _Harper's New Monthly Magazine_ 40 (April 1870), 733.\n\nConcerning Reconstruction in Mormon Utah, see \"The Mormons,\" _New York Herald_ (March 25, 1872), John Hay, \"The Mormon Prophet's Tragedy,\" _Atlantic Monthly_ 24 (December1869), 679, and \"Polygamy in Utah\" (March 23, 1870), _Congressional Globe_ , 41st Congress, 2nd session, 2181.\n\nOn the Liberal Republicans, see \"The Failure of the Confidence Game,\" _Harper's Weekly_ (November 9, 1872).\n\n#  Chapter Six: Law\n\nOn the reconstruction of the federal judiciary after the Civil War, see William M. Wiecek, \"The Reconstruction of Federal Judicial Power, 1863-1875,\" _American Journal of Legal History_ 13 (October 1969), 333.\n\nFor the role of Salmon Chase as chief justice, see Chase to J.M. Reid (January 29, 1865) to Lincoln (April 11, 1865), to Henry Hilliard (April 27, 1868) and to J.W. Schuckers (September 24, 1866), in Jacob W. Schuckers, _The Life and Public Services of Salmon Portland Chase_ (New York, 1874), 514\u20135, 520, 528, 541; see also Chase to William S. Rosecrans (August 16, 1869), in _The Salmon P. Chase Papers: Correspondence, 1865\u20131873_ , ed. John Niven (Kent, OH, 1998), 5:314.\n\n_Cummings v. The State of Missouri_ , in _Cases Argued and Adjudged in the Supreme Court of the United States, December Term, 1866_ , ed. J.W. Wall (Washington, 1870), 4:285.\n\nOn the 15th Amendment, see \"Suffrage\" (January 11 and 29, 1869), in _Congressional Globe_ , 40th Congress, 3rd session, 285, 723, and \"The Suffrage Amendments in Congress,\" _New York Times_ (February 15, 1869).\n\n_Texas v. White et al_ , in _Cases Argued and Adjudged in the Supreme Court of the United States, December Term 1868_ , 7:703.\n\n_Slaughter-House Cases_ (The Butchers' Benevolent Association of New Orleans v. The Crescent City Live Stock Landing and Slaughter-House Company), in _Reports of Cases Argued and Decided in the Supreme Court of the United States in the December Term, 1872, and October Term, 1873_, ed. Stephen K. Williams (Newark, NY, 1884), 21:408.\n\n_United States v. William J. Cruikshank et al_ (1876), in _Reports of Cases Argued and Decided in the Supreme Court of the United States in the October Terms, 1874, 1875, 1876_ , ed. Stephen K. Williams (Newark, NY, 1885), 23:588\u201397.\n\n#  Chapter Seven: Dissension\n\n\"The King of Frauds\u2014How the Credit Mobilier Bought Its Way Through Congress\" and \"The Credit Mobilier and the Bribery of Members of Congress,\" _New York Sun_ (September 4 and 10, 1872).\n\nJames S. Pike, _The Prostrate State: South Carolina Under Negro Government_ (New York, 1874), 12, 14.\n\n\"Wall-St. Panic-Stricken,\" _New-York Tribune_ (September 19, 1873).\n\nDaniel R. Biddle & Murray Dubin, _Tasting Freedom: Octavius Catto and the Battle for Equality in Civil War America_ (Philadelphia, 2010), 428\u2013429.\n\nTilton, \"One Blood of All Nations\" (February 27, 1864), in _Sanctum Sanctorum; or, Proof-Sheets from an Editor's Table_ (New York, 1870).\n\nDennett, \"The South As It Is,\" _The Nation_ (January 4, 1866). See also Current, _Those Terrible Carpetbaggers_ , 369\u201370 and Ruth Currie-McDaniel, _Carpetbagger of Conscience: A Biography of John Emory Bryant_ (Athens, GA, 1987),148, 176, 178.\n\n\"The Week,\" _The Nation_ (September 23, 1875).\n\n\"State of Georgia\" (March 15, 1870), in _Congressional Globe_ , 41st Congress, 2nd session, 1956.\n\nOn dissension within Reconstruction's ranks, see Mrs. A.M. French, _Slavery in South Carolina and the Ex-Slaves; or, The Port Royal Mission_ (New York, 1862), 136, 138, and Turner, in Currie-McDaniel, _Carpetbagger of Conscience_ , 95, and Jones, _A Dreadful Deceit_ , 177\u2013178.\n\nJulia Crofts to Douglass (April 13, 1863), in Douglas R. Egerton, _Thunder at the Gates: The Black Civil War Regiments that Redeemed America_ (New York, 2016), 77.\n\nSumner's Civil Rights Bill is contained in \"An Act to protect all citizens in their civil and legal rights\" (March 1, 1875), _Statutes-at-Large_ , 18:335\u2013337.\n\nJerry L. West, _The Bloody South Carolina Election of 1876: Wade Hampton III, the Red Shirt Campaign for Governor and the End of Reconstruction_ (Jefferson, NC, 2011), 69\u201372.\n\nChamberlain's farewell is part of \"To the Republicans of South Carolina\" (April 10, 1877), in Walter Allen, _Governor Chamberlain's Administration in South Carolina: A Chapter of Reconstruction in the Southern States_ (New York, 1888), 481.\n\n#  Epilogue\n\nC.F. Adams, \"The Government and the Railroad Corporations,\" _North American Review_ 112 (January 1871), 35, 51, and J.R. Doolittle, \"Law Address of Ex-Senator James R. Doolittle, Delivered before the Union College of Law at Chicago, June 6th, 1879,\" _Journal of the Illinois State Historical Society_ 19 (April-July 1926), 84.\n\nFrederick Douglass, \"National Colored Convention\" (April 15, 1872), in _The American Annual Cyclopedia and Register of Important Events of the Year 1872_ (New York, 1873), 775.\n\nJohn Mercer Langston, \"The Exodus\" (1879), in _Freedom and Citizenship: Selected Lectures and Addresses_ (Washington, 1883), 238.\n\nW.T. Laprade, \"A New Epoch,\" in _Proceedings of the Seventeenth Annual Session of the State Library and Historical Association of North Carolina, Raleigh, December 5\u20136, 1916_ (Raleigh, 1917), 51\u201353, 55, and \"Some Problems in Writing the History of American Slavery,\" _South Atlantic Quarterly_ 10 (April 1911), 138\u201339.\n\nOn the formulation of a \"Lost Cause\" and \"New South\" mythology, see Richard Taylor, _Destruction and Reconstruction: Personal Experiences of the Late War_ (London, 1879), 2, and Henry W. Grady, \"The New South\" (December 22, 1886), in Joel Chandler Harris, _Henry W. Grady: His Life, Writings, and Speeches_ (New York, 1890), 15\u201316.\n\n\"Over 3,000 Lynchings in Twenty Years,\" _Chicago Tribune_ (September 1, 1901).\n\nOn Grant's reflection on military occupation, see John Russell Young, _Around the World with General Grant_ (New York, 1879), 2:362\u201363\n\nSusie King Taylor, _Reminiscences of My Life in Camp with the 33 d United States Colored Troops_ (Boston, 1902), 61.\n\n#  Further reading\n\n#  Introduction\n\nWilliam Archibald Dunning gave birth to the school of Reconstruction interpretation which pictured white Southerners, not the freed slaves, as the victims of vengeful Northerners. Dunning's view, in _Reconstruction: Political & Economic, 1865\u20131877_ (1905), was challenged by W.E.B. DuBois in _Black Reconstruction in America_ , (1935), by Kenneth M. Stampp in _The Era of Reconstruction, 1865\u20131877_ (1967), and Eric Foner in _Reconstruction: America's Unfinished Revolution, 1863\u20131877_ (1988), all of which were shaped in varying degrees by Marxist influences. Barrington Moore, _Social Origins of Dictatorship and Democracy: Lord and Peasant in the Making of the Modern World_ (1967) proposed interpreting Reconstruction as a bourgeois revolution.\n\nLincoln's ideas on Reconstruction are treated in Louis P. Masur, _Lincoln's Last Speech: Wartime Reconstruction and the Crisis of Reunion_ (2015) and William C. Harris, _With Charity for All: Lincoln and the Restoration of the Union_ (1997). A more expansive view of Reconstruction is contained in Heather Cox Richardson's _The Death of Reconstruction: Race, Labor, and Politics in the Post-Civil War North, 1865\u20131901_ (2001) and _West from Appomattox: The Reconstruction of America after the Civil War_ (2008). Douglas Egerton's _The Wars of Reconstruction: The Brief, Violent History of America's Most Progressive Era_ (2014) and Mark Wahlgren Summers' _The Ordeal of the Reunion: A New History of Reconstruction_ (2014) illuminate the political upheavals of Reconstruction.\n\nFour important anthologies bring together an array of important primary sources on Reconstruction: Edward McPherson, _The Political History of the United States of America During the Period of Reconstruction_ (1875), Walter L. Fleming's _Documentary History of Reconstruction: Political, Military, Social, Religious, Educational, and Industrial_ (2 vols.,1906), John David Smith's _We Ask Only for Even-Handed Justice: Black Voices from Reconstruction, 1865\u20131877_ (2014), and Brooks Simpson's _Reconstruction: Voices from America's First Great Struggle for Racial Equality_ , in the Library of America series (2018).\n\n#  Chapter One: Vengeance\n\nThe most revealing memoirs of individuals at the center of the events in Washington which launched Reconstruction are _John Sherman's Recollections of Forty Years in the House, Senate and Cabinet_ (1895), George Boutwell's _Reminiscences of Sixty Years in Public Affairs_ (1902); James G. Blaine's _Twenty Years of Congress: from Lincoln to Garfield_ (1886), George W. Julian's _Political Recollections, 1840\u20131872_ (1884), the _Diary of Gideon Welles_ , ed. E.T. Welles (3 vols., Boston, 1911), and Chauncey M. Depew's _My Memories of Eighty Years_ (1922).\n\nAmong the first-hand views of the South immediately after the end of the war, Sidney Andrews' _The South Since the War: As Shown by Fourteen Weeks of Travel and Observation in Georgia and the Carolinas_ (Boston, 1866), John Townsend Trowbridge's _The South: A Tour of Its Battlefields and Ruined Cities_ (1867), and Whitelaw Reid's _After the War: A Southern Tour, May 1, 1865, to May 1, 1866_ (1866) are the most comprehensive.\n\nThe principal biographers of Andrew Johnson are Eric L. McKitrick, in _Andrew Johnson and Reconstruction_ (1960, 1988) and Hans L. Trefousse, in _Andrew Johnson: A Biography_ (1989). Gregory Downs, _After Appomattox: Military Occupation and the Ends of War_ (2015), offers the fullest survey of Union military occupation of the South.\n\n#  Chapter Two: Alienation\n\nThe Freedmen's Bureau still begs for a comprehensive history, but John and LaWanda Cox's \"General O.O. Howard and the 'Misrepresented Bureau',\" _Journal of Southern History_ 19 (November 1953) remains a fine introduction. Paul H. Bergeron, _Andrew Johnson and Reconstruction_ (2011), is an important guide to the steady deterioration of Johnson's authority in 1866\u201367. On the horrendous race riots in New Orleans and Memphis, which played a significant role in undercutting that authority, see Hannah Rosen, _Terror in the Heart of Freedom: Citizenship, Sexual Violence, and the Meaning of Race in the Postemancipation South_ (2009), James G. Hollandsworth, _An Absolute Massacre: The New Orleans Race Riot of July 30, 1866_ (2001), and Stephen V. Ash, _A Massacre in Memphis: The Race Riot That Shook the Nation One Year After the Civil War_ (2013). Joseph G. Dawson's _Army Generals and Reconstruction: Louisiana, 1862\u20131877_ (1982) is a model in describing the difficulties in implementing Congressional Reconstruction. Mark Wahlgren Summers's _A Dangerous Stir: Fear, Paranoia, and the Making of Reconstruction_ (2009) untangles the rumored plan of Andrew Johnson to stage a military coup of his own against Congress.\n\n#  Chapter Three: Arrogance\n\nFor the meaning of free labor, see Eric Foner, _Free Soil, Free Labor, Free Men: The Ideology of the Republican Party Before the Civil War_ (1970), Michael S. Green, _Freedom, Union, and Power: Lincoln and His Party during the Civil War_ (2004), and Heather Cox Richardson, _The Death of Reconstruction_. The struggles of the freed slaves to establish economic independence for themselves in Georgia and North Carolina are handled in Paul A. Cimbala, _Under the Guardianship of the Nation: The Freedmen's Bureau and the Reconstruction of Georgia, 1865\u20131870_ (1997) and Patricia C. Click, _Time Full of Trial: The Roanoke Island Freedmen's Colony, 1862\u20131867_ (2001). The standard works on the impeachment of Andrew Johnson are David O. Stewart, _Impeached: The Trial of President Johnson and the Fight for Lincoln's Legacy_ (2009) and Michael Les Benedict's classic, _The Impeachment and Trial of Andrew Johnson_ (1973). James Ashley's lead role in the impeachment itself can be followed in Robert F. Horowitz, _The Great Impeacher: A Political Biography of James M. Ashley_ (1979).\n\n#  Chapter Four: Resistance\n\nFew groups in American history have been more utterly misrepresented than the 'carpetbaggers,' whom the Dunning school routinely represented as Northern harpies who, in league with ignorant blacks and opportunistic Southern 'scalawags,' appeared at the end of the war to rip out profit from the body of a prostrate South. Revision of this farcical picture begins with Richard M. Current's _Those Terrible Carpetbaggers: A Reinterpretation_ (1988), David S. Cecelski, _The Fire of Freedom: Abraham Galloway and the Slaves' Civil War_ (2012), Canter Brown, _Florida's Black Public Officials, 1867\u20131924_ (1998), Mark Elliott, _Color-Blind Justice: Albion Tourg\u00e9e and the Quest for Racial Equality from the Civil War to Plessy v. Ferguson_ (2006), Nancy Koester, _Harriet Beecher Stowe: A Spiritual Life_ (2014), and Ruth Currie-McDaniel, _Carpetbagger of Conscience: A Biography of John Emory Bryant_ (1987). The 'scalawags' in particular have benefitted from sympathetic treatment through James Alex Baggett, _The Scalawags: Southern Dissenters in the Civil War and Reconstruction_ (2003), Sarah Woolfolk Wiggins, _The Scalawag In Alabama Politics, 1865\u20131881_ (1977), and Richard L. Hume and Jerry B. Gough, _Blacks, Carpetbaggers, and Scalawags: The Constitutional Conventions of Radical Reconstruction_ (2008).\n\nThe recreation of a resistant Southern sectional identity, continuous with the Confederacy, has been ably analyzed in Anne Sarah Rubin in _A Shattered Nation: The Rise and Fall of the Confederacy, 1861\u20131868_ (2005). The terrorist insurgency mounted by the Ku Klux Klan and other white supremacist militias can be followed in Wyn Craig Wade, _The Fiery Cross: The Ku Klux Klan in America_ (1998), Stephen Budiansky, _The Bloody Shirt: Terror After Appomattox_ (2006), Edward John Harcourt, \"Who Were the Pale Faces? New Perspectives on the Tennessee Ku Klux Klan,\" _Civil War History_ 51 (March 2005), Lee W. Formwalt, \"The Camilla Massacre of 1868: Racial Violence as Political Propaganda,\" _George Historical Quarterly_ , 71 (Fall 1987), James G. Dauphine, \"The Knights of the White Camellia and the Election of 1868: Louisiana's White Terrorists,\" _Louisiana History_ 30 (Spring 1989), and Carolyn E. DeLatte, \"The St. Landry Riot: A Forgotten Incident of Reconstruction Violence,\" _Louisiana History_ 17 (Winter 1976), 47\u201348.\n\n#  Chapter Five: Distraction\n\nOn the economic conditions which prevailed in the South during Reconstruction and after, the primary work is Edward L. Ayres, _The Promise of the New South: Life After Reconstruction_ (1992). But there are also a number of useful studies of specific components of that economy, starting with William J. Cooper, \"The Cotton Crisis in the Antebellum South: Another Look,\" _Agricultural History_ 49 (April 1975) and Robert Tracy McKenzie, \"Freedmen and the Soil in the Upper South: The Reorganization of Tennessee Agriculture, 1865-1880,\" _Journal of Southern History_ 59 (February 1993).\n\nThe reputation of Ulysses Grant as a president has suffered nearly as much obloquy as the 'carpetbaggers,' and with as little justification. Substantial work in retrieving Grant's reputation has been done by Brooks D. Simpson in _Let Us Have Peace: Ulysses S. Grant and the Politics of War and Reconstruction, 1861\u20131868_ (1991) and _The Reconstruction Presidents_ (1998), H.W. Brands, _The Man Who Saved the Union: Ulysses Grant in War and Peace_ (2012), Charles W. Calhoun, _The Presidency of Ulysses S. Grant_ (2017), and Ron Chernow, _Grant_ (2018). On Grant's suppression of the Ku Klux Klan in South Carolina, see Lou Falkner Williams, \"The South Carolina Ku Klux Klan Trials and Enforcement of Federal Rights, 1871-1872,\" _Civil War History_ 39 (March 1993), and Jerry Lee West, _The Reconstruction Ku Klux Klan in York County, South Carolina, 1865\u20131877_ (Jefferson, NC: McFarland, 2002).\n\nOn the unhappy and short-lived Liberal Republican movement, see Andrew L. Slap, _The Doom of Reconstruction: The Liberal Republicans in the Civil War Era_ (2006), and Hans L. Trefousse, _Carl Schurz: A Biography_ (1982).\n\n#  Chapter Six: Law\n\nThe federal courts were consigned to a curiously diminished role during the Civil War, as explained by Mark E. Neely, _Lincoln and the Triumph of the Nation: Constitutional Conflict in the American Civil War_ (2011) and Jonathan W. White, \"The Strangely Insignificant Role of the U.S. Supreme Court during the Civil War,\" _Journal of the Civil War Era_ 3 (June 2013). The determination of the jurists to re-capture their lost standing can be seen in Joseph G. Gambone, \"Ex Parte Milligan: The Restoration of Judicial Prestige,\" _Civil War History_ 16 (September 1970).\n\nThe dominant figures of the Reconstruction Supreme Court begin with Salmon Chase, as Chief Justice. Chase has benefited from several comprehensive biographies and studies, by Michael Les Benedict in \"Salmon P. Chase and Constitutional Politics,\" _Preserving the Constitution: Essays on Politics and the Constitution in the Reconstruction Era_ (2006), Jacob W. Schuckers, _The Life and Public Services of Salmon Portland Chase_ (1874), Frederick J. Blue, _Salmon P. Chase: A Life in Politics_ (1987) and John Niven, _Salmon P. Chase: A Biography_ (1995). Chase's papers (principally correspondence) have also been collected by Niven in five volumes, covering 1829 to 1873, and published by Kent State University Press. Michael A. Ross's _Justice of Shattered Dreams: Samuel Freeman Miller and the Supreme Court during the Civil War Era_ (2003) and Willard L. King's _Lincoln's Manager: David Davis_ (1960) capture the lives of two other influential justices of the Court, but Joseph P. Bradley, who played an outsize role in the decisions in _Slaughterhouse Cases_ and _U.S. v. Cruikshank_ still goes a-begging for a biographer. Those cases have been dramatically handled in Ronald M. Labb\u00e9 and Jonathan Lurie, _The Slaughterhouse Cases: Regulation, Reconstruction, and the Fourteenth Amendment_ (2003), Charles Lane, _The Day Freedom Died: The Colfax Massacre, the Supreme Court, and the Betrayal of Reconstruction_ (2008), and LeeAnna Keith, _The Colfax Massacre: The Untold Story of Black Power, White Terror, and the Death of Reconstruction_ (2008).\n\nThe 14th and 15th Amendments have found skillful examiners in John Mabry Mathews, _Legislative and Judicial History of the Fifteenth Amendment_ (1909), Pamela Brandwein, _Rethinking the Judicial Settlement of Reconstruction_ (2011), Kurt T. Lash, _The Fourteenth Amendment and the Privileges and Immunities of American Citizenship_ (2014), Gerard Magliocca, _American Founding Son: John Bingham and the Invention of the Fourteenth Amendment_ (2013) and Michael Les Benedict, \"At Every Fireside: Constitutional Politics in the Era of Reconstruction,\" in _Constitutionalism in the Approach and Aftermath of the Civil War_ , eds. Paul D. Moreno & Johnathan O'Neill ((2013).\n\n#  Chapter Seven: Dissension\n\nThe final stages of Reconstruction's overthrow, especially in Louisiana and South Carolina after the Hayes-Tilden presidential contest, have been told well and often in\n\nA.J. Langguth, _After Lincoln: How the North Won the Civil War and Lost the Peace_ (2014), Stetson Kennedy, _After Appomattox: How the South Won the War_ (1995), Nicholas Lemann, _Redemption: The Last Battle of the Civil War_ (2006), and Michael Perman, _The Road to Redemption: Southern Politics, 1869\u20131879_ (1984). Some of the wounds were self-inflicted, among both black and white Republicans, and these are described in Michael W. Fitzgerald, _Urban Emancipation: Popular Politics in Reconstruction Mobile, 1860\u20131890_ (2002), Philip B. Lyons, _Statesmanship and Reconstruction: Moderate versus Radical Republicans on Restoring the Union after the Civil War_ (2014) and Jacqueline Jones, _A Dreadful Deceit: The Myth of Race from the Colonial Era to Obama's America_ (2013). The crises in Louisiana and South Carolina are chronicled in Jerry L. West, _The Bloody South Carolina Election of 1876: Wade Hampton III, the Red Shirt Campaign for Governor and the End of Reconstruction_ (2011) and William Gillette, _Retreat from Reconstruction, 1869\u20131879_ (1979). Paul Leland Haworth laid out a detailed account of the contested 1876 presidential election in _The Hayes-Tilden Disputed Presidential Election of 1876_ (1906), to be followed by Roy Morris in _Fraud Of The Century: Rutherford B. Hayes, Samuel Tilden And The Stolen Election Of 1876_ (2004) and Michael F. Holt in _By One Vote: The Disputed Presidential Election of 1876_ (2008).\n\n#  Epilogue\n\nMight-have-beens cluster thickly around Reconstruction, especially concerning economic issues. Land re-distribution, as described by Claude F. Oubre, _Forty Acres and a Mule: The Freedmen's Bureau and Black Land Ownership_ (1978), might have been one solution; more aggressive federal defence of voting rights might have been another, as can be seen from Michael Perman, _Struggle for Mastery: Disfranchisement in the South, 1888\u20131908_ (2001).\n\nPeter Kolchin has provided the important reminder, though, that not everything in Reconstruction was a failure, especially when compared to other 19th-century emancipations, which he does in \"Comparative Perspectives on Emancipation in the U.S. South: Reconstruction, Radicalism and Russia,\" _Journal of the Civil War Era_ 2 (June 2012) and \"Thoughts on Emancipation in Comparative Perspective: Russia and the United States South,\" _Slavery and Abolition_ 11 (December 1990).\n\nOn the formulation of Southern regional myths after Reconstruction, the basic texts have long been Paul Gaston, _The New South Creed: A Study in Southern Mythmaking_ (1973) and C. Vann Woodward, _The Burden of Southern History_ (1968).\n\n#  Index\n\nA\n\nAdams, Charles Francis,\n\nAdams, Charles Francis, Jr.,\n\nAdams, Henry,\n\nAfrican Americans,  , , , , , , , , , , , , , , \u2013, , , , , , \u2013, , , \u2013, , , , , , , , \u2013, , \u2013, \u2013, \u2013, , \u2013\n\nAkerman, Amos Tappan (attorney general), \u2013\n\nAlabama,  , , , , , , , , , , , , , , , , , , , , , , ,\n\nAlcorn, James Lusk,  , , \u2013\n\nAllen, James S.,\n\nAllston, Elizabeth,\n\nAlston, James,\n\nAmerican Freedman's Union Commission,\n\nAmerican Missionary Association,  , ,\n\nAmes, Adelbert,  , , ,\n\nAntoine, Caesar Carpentier,\n\nAppomattox Court House, Virginia,\n\nArizona,\n\nArkansas,  , , , , , , , , , , , ,\n\nArthur, Chester Alan (twenty-\u200bfirst president),\n\nAshley, James Mitchell,  ,\n\n_Atlantic Monthly_ ,  , , , ,\n\nB\n\nBabcock, Orville,\n\n_Barron v. Baltimore_ (1833),\n\nBarton, Clara,\n\nBates, George,\n\nBattle, Cullen Andrews,  ,\n\nBaxter, Elisha,\n\nBeckwith, James Roswell,\n\nBelgium,\n\nBelknap, William Worth (secretary of war),\n\nBell, Alexander Graham,\n\nBelmont, August, 2\n\nBerry, Lawrence S.,\n\nBingham, John Armor,  .\n\n\"black codes,\"  , ,\n\nBlaine, James Gillespie,\n\nBooth, John Wilkes,  , , ,\n\nBoston, Massachusetts,  ,\n\n\"Bourbons,\"\n\nBorie, Adolph Edward (secretary of the Navy),\n\nBradley, Aaron,\n\nBradley, Joseph Philo,\n\nBramlette, E. L.,\n\nBrooks, Joseph,\n\nBrown, William Wells,\n\nBruce, Blanche Kelso,\n\nBryant, John Emory,  , ,\n\nBullock, Rufus,  , , ,\n\nButler, Benjamin Franklin,  ,\n\nC\n\nCaesar, Julius,\n\nCamilla, Georgia,\n\nCampbell, John Archibald,\n\nCampbell, Tunis,\n\nCardozo, Francis,\n\n\"carpetbaggers,\"  , , ,\n\nCatto, Octavius Valentine,  ,\n\nChamberlain, Daniel Henry,  ,\n\nChandler, Zachariah,\n\nCharles II (king of England),\n\nCharleston, South Carolina,  , , , , ,\n\nChase, Salmon Portland (chief justice),  , , , , ,\n\nCherokee (tribe),  ,\n\nChestnutt, Charles Waddell,\n\nChicago, Illinois,  , \u2013\n\n_Chicago Tribune_ ,  , , ,\n\nCincinnati, Ohio, \u2013 , ,\n\nCivil Rights Bill (1866), \u2013 , , , ,\n\nCivil Rights Bill (1875),  , , ,\n\n_Civil Rights Cases_ (1883),\n\nClayton, Powell,\n\nCleveland, Ohio,\n\nCoan, William L.,\n\nColfax, Schuyler (seventeenth vice president),\n\nColfax (Louisiana) massacre,  ,\n\nCollamer, Jacob,\n\nColumbia, South Carolina,  , ,\n\nConfederate States of America,  , , , ,\n\nconfiscation,  , , ,\n\nConnecticut,  ,\n\nConstitutional Guards,\n\nConstitution of the United States,  , \u2013, , , , , , ,\n\nand Fifteenth Amendment,\u2013, , , ,\n\nand Fourteenth Amendment,\u2013, \u2013, , ,\n\nand natural rights,\n\nand Thirteenth Amendment,\u2013,\n\n_Continental Monthly_ ,\n\nCooke, Jay,  ,\n\nCorbin, Abel,\n\nCorbin, David T., \u2013\n\n_Corfield v. Coryell_ (1823),\n\ncotton,  , , , , \u2013,\n\nCovode, John, \u2013\n\nCr\u00e9dit Mobilier, \u2013\n\nCrosby, Peter,\n\nCummings, John A., \u2013\n\nD\n\nDaniel, Peter Vivian,\n\nDaniels, Josephus,\n\nDavis, David,  , \u2013\n\nDavis, Henry Winter, \u2013 ,\n\nDavis, Jefferson,  , ,\n\nDeForest, John William,\n\nDelano, Columbus (secretary of the interior),\n\nDelany, Martin Robison,  , , ,\n\nDemocratic Party,  , , , , , , , , , , , , , , ,\n\nDennett, John Richard,\n\nDennison, William (postmaster general),\n\nDenver, James William,\n\nDoolittle, James,\n\nDorr Rebellion,\n\nDouglass, Frederick,  , , , , , ,\n\n_Dred Scott v. Sanford_ ,  , ,\n\nDu Bois, William Edward Burghardt, \u2013 , ,\n\nDunn, Oscar James,\n\nDunning, William Archibald, \u2013 ,\n\nDwight, John Sullivan,\n\nE\n\nEaton, John G.,\n\nEdisto Island,  ,\n\nEdmunds, George,\n\nEggleston, Beroth,\n\nEliot, Thomas Stearns,\n\nEmancipation Proclamation,  , ,\n\nEnforcement Acts,  , , , , , , ,\n\nEvans, Augusta Jane,\n\n_ex parte Merryman_ ,\n\n_ex parte Milligan_ , \u2013\n\n_ex parte Yarborough_ ,\n\nF\n\nFessenden, William Pitt,  ,\n\nField, Stephen Johnson,\n\nFish, Hamilton (secretary of state),\n\nFisk, James,\n\nFlorida,  , , , , , , , , , , , , ,\n\nForrest, Nathan Bedford,\n\nFortress Monroe, Virginia,  ,\n\nFortune, Timothy Thomas,  ,\n\nFreedmen's Bureau,  , , , , , , , , , , , , , , ,\n\nfree labor ideology, \u2013 , , , , , , , , , ,\n\nFrench, Austa Malinda,\n\nFrench, Benjamin Brown,\n\nFuller, Melville Weston (chief justice),\n\nG\n\nGarfield, James Abram (twentieth president),  ,\n\nGarrison, William Lloyd,\n\nGeorgia,  , , , , , , , , , , , , , , , , , , , , , , , , , , ,\n\nGeorgia Equal Rights Association,  ,\n\nGilmore, Patrick Sarsfield,\n\nGladstone, William Ewart,\n\nGodkin, Edwin Lawrence,  ,\n\nGordon, John Brown,\n\nGould, Jason \"Jay\",\n\nGrady, Henry Woodfin,\n\nGrand Army of the Republic,\n\nGrant, Ulysses Simpson (eighteenth president),  , , , , , \u2013, \u2013, , \u2013, , , \u2013, , \u2013, , , , , \u2013, \u2013\n\nGreeley, Horace, \u2013\n\nGrimes, James Wilson,\n\nGrosvenor, William,\n\nH\n\nhabeas corpus,  , , , ,\n\nHamburg, South Carolina,\n\nHamilton, Andrew Jackson,  ,\n\nHamlin, Hannibal,\n\nHampton, Wade,  , ,\n\nHancock, Winfield Scott,\n\nHarlan, James (secretary of the interior),\n\n_Harper's New Monthly Magazine_ ,\n\n_Harper's Weekly_ ,  , ,\n\nHarrison, Benjamin (twenty-second president),\n\nHart, Ossian,\n\nHay, John Milton, \u2013\n\nHayes, Rutherford Birchard (nineteenth president),  , , , ,\n\nHenri IV (king of France),\n\nHill, Elias,\n\nHoar, Ebenezer Rockwood (attorney general),  ,\n\nHoar, George Frisbie,\n\nHodgson, William,\n\nHolden, William Woods,  ,\n\nHomestead Act (1862),  ,\n\nHouse of Representatives,  , , , , , , \u2013, , , , , , , ,\n\nHovey, Alvan,\n\nHoward, Jacob,  ,\n\nHoward, Oliver Otis,  , , ,\n\nHoward University,\n\nHuggins, Allen P.,\n\nHumphreys, Benjamin,\n\nI\n\nIdaho,\n\nIllinois,  , , , ,\n\nIndiana,  , , ,\n\nJ\n\nJames, Horace,\n\nJay, John (chief justice),\n\nJim Crow,  ,\n\nJohnson, Andrew (seventeenth president),  , \u2013, \u2013, \u2013, , , \u2013, , , , \u2013, , , , , ,\n\nand amnesty proclamations,\u2013, , , , , ,\n\nand impeachment,\u2013,\n\nand provisional governors,\u2013, \u2013, , ,\n\nand rumors of coup,, \u2013\n\nand \"swing round the circle,\"\n\nJohnson, Herschel Vespasian,\n\nJohnson, James,\n\nJoint Committee on Reconstruction,  , , , ,\n\nJones, John Beauchamp,  , ,\n\nJulian, George Washington,\n\nK\n\nKelley, William Darrah,\n\nKellogg, William Pitt,  ,\n\nKennedy, John Pendleton,\n\nKnights of the White Camellia,\n\nKoerner, Gustave, \u2013\n\nKu Klux Klan,  , \u2013, , , \u2013, , ,\n\nL\n\nLafargue, Adolphe Joina,\n\nLangston, John Mercer,  , , , \u2013, ,\n\nLaprade, William Thomas,\n\nLatimer, Lewis, \u2013\n\nLee, Robert Edward,  ,\n\nLeigh, Frances Butler, \u2013\n\nLenin, Vladimir,\n\nLexington, Virginia,\n\nLincoln, Abraham (sixteenth president), \u2013 , , , , , , , , , , , , , , , \u2013, , , ,\n\n1863 plan of Amnesty and Reconstruction,\n\nLodge, Henry Cabot,\n\nLong, Jefferson,  ,\n\n\"Lost Cause\" ideology,  , ,\n\nLouisiana,  , , , , , , , \u2013, , , , , , , , , , , , , , , , , , , , ,\n\nLowry, Henry Berry,\n\nLudlow, Helen,\n\nLynch, John Roy,  ,\n\nlynching,\n\nM\n\nMacon, Georgia,\n\nMarshall, John (chief justice),  , ,\n\nMarvin, William,\n\nMaryland,  , , ,\n\nMatthews, Stanley,\n\nMcGowan, Samuel,\n\nMcLean, John,\n\nMcPherson, Edward,\n\nMeade, George Gordon,  ,\n\nMemphis, Tennessee, \u2013 ,\n\nMercer, S.C.,\n\nMethodist Freedman's Aid Society,\n\nMexico,\n\nMiller, Samuel Freeman,  , , ,\n\nMilligan, Lambdin Purdy, \u2013\n\nMinnesota,  ,\n\nMississippi,  , , , , , , , , , , , , , , , , , , , , , , , ,\n\nMissouri,  , ,\n\nMontana,\n\nMontgomery, Alabama,\n\nMorgan, Albert T.,\n\nMorton, Oliver Perry,\n\nMoses, Franklin J., Jr.,  , ,\n\nMurphy, Isaac,\n\nN\n\nNashville, Tennessee,\n\nNast, Thomas,\n\nNatchez, Mississippi,\n\n_Natchez Brotherhood_ ,\n\nNational Conference of Colored Men,\n\nNational Equal Rights League,\n\nNew Bern, North Carolina,\n\nNew Jersey,\n\nNew Mexico,\n\nNew Orleans, Louisiana,  , , , , , , , , , ,\n\n\"New South\" ideology, \u2013\n\nNew York,  , , , , , , , , , ,\n\nNew York Cash Store (Greenville, Alabama),\n\n_New York American,_\n\n_New York Globe_ ,\n\n_New York Herald_ ,  ,\n\n_New York Independent_ ,\n\n_New York Sun_ ,\n\n_New York Times,_  , ,\n\n_New-York Tribune_ ,  , , , , ,\n\nNicholls, Francis Redding Tillou,\n\nNorfolk, Virginia,\n\n_North American Review_ ,\n\nNorth Carolina,  , , , , , , , , , , , , , , , , , , ,\n\nO\n\nOberlin College,  ,\n\nOhio,  , , , , , , , , , , , , , , ,\n\nOpelousas, Louisiana,\n\nOrd, Edward Otho Cresap,  , ,\n\nOrr, James Lawrence,\n\nP\n\nPalmer, John McAuley,\n\nPanic of 1873,  , ,\n\nParker, Ely Samuel,  , ,\n\nParis Commune (1871),\n\nParsons, Lewis,\n\nPeck, Elisha,\n\nPennsylvania,  , , ,\n\nPerry, Benjamin Franklin,\n\nPettus, Edmund,\n\nPhelps, John Smith,\n\nPhiladelphia, Pennsylvania,  , , , ,\n\nPhillips, Wendell,\n\nPickett, George Edward,\n\nPierpont, Francis,  , ,\n\nPierrepont, Edwards (attorney general),\n\nPike, James Shepherd,\n\nPinchback, Pinckney Benton Stewart,  ,\n\nPlains Indian tribes,  , \u2013\n\nPleasants, Matthew,\n\n_Plessy v. Ferguson_ (1896),  ,\n\npolygamy, \u2013\n\nPompey, Gnaeus, Magnus,\n\nPope, John,  , , ,\n\n\"posse comitatus,\"\n\nProgressivism,  , ,\n\nR\n\nrailroads,  , , , , , ,\n\nUnion Pacific,\u2013\n\nWestern & Atlantic,\n\nRavenel, Henry William,\n\nRawlins, Hannah, \u2013\n\n\"Readjusters,\"\n\nReconstruction, accomplishments of, \u2013 , \u2013, \u2013\n\nas bourgeois revolution,\u2013, , \u2013\n\nand Compromise of 1877,\u2013\n\nhistory and interpretation of,, \u2013\n\noverthrow of,\n\nterminology,\n\nReconstruction Acts (1867\u201368), \u2013 , \u2013, , , , , ,\n\nReed, Harrison,  ,\n\nRed Cloud,\n\n\"Redeemers,\"  ,\n\nRed Shirts,  ,\n\nReid, Whitelaw,\n\nRepublican Party,  , , , , , , , , \u2013, \u2013, , , , , , , , , \u2013, , \u2013, , \u2013, , , \u2013, \u2013, \u2013, , , , , , , , \u2013, \u2013, , , , \u2013, \u2013\n\nand Liberal Republicans,\u2013, ,\n\nand Radical Republicans,, , , , \u2013, , \u2013, , \u2013, , , , \u2013, , , , , \u2013, , \u2013, , , , , , ,\n\nRevels, Hiram,  ,\n\nRhodes, James Ford,\n\nRichmond,  , ,\n\nRoanoke Island, \u2013\n\nRobert, Charles E.,\n\nRobeson, George (secretary of war),\n\nRobinson, Leigh,\n\nS\n\nSavannah, Georgia,\n\n\"scalawags,\" \u2013 ,\n\nSchofield, John,  ,\n\nSchurz, Carl,  , , , \u2013\n\nScott, Robert Kingston,  , , ,\n\nsecession,  , , , , , , , ,\n\nSenate of the United States,  , , , , , , , \u2013, , , \u2013, , , , , , , , , ,\n\nSettle, Thomas,\n\nSeward, William Henry (secretary of state),  ,\n\nSeymour, Horatio,\n\nSharkey, William Lewis,  ,\n\nShepley, George,\n\nSheridan, Philip Henry,  , \u2013\n\nSherman, John (U.S. senator),\n\nSherman, William Tecumseh,  , ,\n\nSibley, John Langdon,  , ,\n\n_Slaughterhouse Cases_ (1873), \u2013 ,\n\nSociety of Pale Faces,  ,\n\nSouth Carolina,  , , , , , , , , , , , , , , \u2013, \u2013, , , , , , , , \u2013, ,\n\nSpeed, James (attorney general),\n\nSpotted Tail,\n\nStanly, Edward, \u2013\n\nStanton, Edwin McMaster (secretary of war), \u2013\n\nSt. Catharine's Island,\n\nStephens, Alexander Hamilton,  , ,\n\nStevens, Thaddeus, \u2013 , , , \u2013, , ,\n\nStewart, Alexander, \u2013\n\nSt. Louis, Missouri,  , , ,\n\nStoneman, George,\n\nStowe, Harriet Beecher,  ,\n\nStroud, Burton, \u2013\n\nSulla, Lucius Cornelius,  ,\n\nSumner, Charles,  , , , , , , \u2013, , ,\n\nSupreme Court of the United States,  , \u2013, , \u2013,\n\nSwayne, Noah,\n\nSwisshelm, Jane Grey,\n\nT\n\nTaiping Rebellion,  ,\n\nTaney, Roger Brooke (chief justice),  , ,\n\nTaylor, Richard,\n\nTaylor, Susie King,\n\nTennessee,  , , , \u2013, , , , , , , , , , , , ,\n\nTenure of Office Act (1867), \u2013\n\nTexas,  , , , , , , , , , , ,\n\n_Texas v. White_ (1869), \u2013\n\nThomas, Lorenzo Dow,\n\nThomas, Samuel,\n\nTilden, Samuel,  ,\n\nTilton, Theodore,\n\nTourg\u00e9e, Albion Winegar,  ,\n\nTrowbridge, John Townsend,  ,\n\nTrumbull, Lyman,  , , , , , , , , ,\n\nTurner, Henry McNeal,\n\nTwain, Mark (Samuel Langhorne Clemens), \u2013\n\nTyler, John,  , , ,\n\nU\n\nUnion League,  ,\n\n_US v. Cruikshank_ (1875), \u2013\n\nUtah,  , \u2013\n\nV\n\nVallandigham, Clement Laird,\n\nand _ex parte Vallandigham_ ,\n\nVicksburg, Mississippi,  ,\n\nVirginia,  , , , , , , , , , , , , , , , , , , , , ,\n\nvoting rights,  , , , , \u2013, , , , , ,\n\nDistrict of Columbia bill (1866),,\n\nand Fifteenth Amendment,\u2013\n\nand registration,, \u2013, ,\n\nW\n\nWade, Benjamin Franklin, \u2013 , \u2013, ,\n\nand Wade-\u200bDavis Reconstruction plan,\u2013\n\nWaite, Morrison Remick (chief justice),  , , ,\n\nWalker, Gilbert,\n\nWarmoth, Henry Clay,  , , ,\n\nWashburne, Elihu,  ,\n\nWatterson, Henry,\n\nWelles, Gideon (secretary of the navy),  , , ,\n\nWells, James Madison,\n\nWest, George Benjamin,\n\nWhiskey Rebellion,\n\nWhite, Richard,\n\nWhite Leagues,  ,\n\nWhite Liners,\n\nWhitman, Walt,  ,\n\nWhittier, John Greenleaf,\n\nWilkinson, Morton,  ,\n\nWilliams, George Henry (attorney general),\n\nWilson, Henry (eighteenth vice president),  ,\n\nWilson, Reuban,\n\nWilson, Thomas Woodrow (28th president),  ,\n\nWinthrop, Robert Charles,\n\nWisconsin,  , ,\n\nWood, Fernando,\n\nWoodruff, Josephus,\n\nY\n\nYoung, Brigham, \u2013\n\n# Table of Contents\n\n  1. Cover page\n  2. Halftitle page\n  3. Series page\n  4. Title page\n  5. Copyright page\n  6. Contents\n  7. Acknowledgments\n  8. List of illustrations\n  9. Introduction\n  10. 1 Vengeance: April-December 1865\n  11. 2 Alienation: December 1865-March 1867\n  12. 3 Arrogance: March 1867- May 1868\n  13. 4 Resistance: May 1868- March 1869\n  14. 5 Distraction: March 1869- May 1872\n  15. 6 Law: 1866-1876\n  16. 7 Dissension: September 1872-April 1877\n  17. Epilogue\n  18. Timeline\n  19. References\n    1. Further reading\n  20. Index\n\n## Landmarks\n\n  1. Cover\n  2. Title page\n  3. Table of Contents\n\n## Pages\n\n  1. i\n  2. ii\n  3. iii\n  4. iv\n  5. v\n  6. vi\n  7. vii\n  8. viii\n  9. ix\n  10. x\n  11. xi\n  12. xii\n  13. xiii\n  14. xiv\n  15. xv\n  16. xvi\n  17. xvii\n  18. xviii\n  19. \n  20. \n  21. \n  22. \n  23. \n  24. \n  25. \n  26. \n  27. \n  28. \n  29. \n  30. \n  31. \n  32. \n  33. \n  34. \n  35. \n  36. \n  37. \n  38. \n  39. \n  40. \n  41. \n  42. \n  43. \n  44. \n  45. \n  46. \n  47. \n  48. \n  49. \n  50. \n  51. \n  52. \n  53. \n  54. \n  55. \n  56. \n  57. \n  58. \n  59. \n  60. \n  61. \n  62. \n  63. \n  64. \n  65. \n  66. \n  67. \n  68. \n  69. \n  70. \n  71. \n  72. \n  73. \n  74. \n  75. \n  76. \n  77. \n  78. \n  79. \n  80. \n  81. \n  82. \n  83. \n  84. \n  85. \n  86. \n  87. \n  88. \n  89. \n  90. \n  91. \n  92. \n  93. \n  94. \n  95. \n  96. \n  97. \n  98. \n  99. \n  100. \n  101. \n  102. \n  103. \n  104. \n  105. \n  106. \n  107. \n  108. \n  109. \n  110. \n  111. \n  112. \n  113. \n  114. \n  115. \n  116. \n  117. \n  118. \n  119. \n  120. \n  121. \n  122. \n  123. \n  124. \n  125. \n  126. \n  127. \n  128. \n  129. \n  130. \n  131. \n  132. \n  133. \n  134. \n  135. \n  136. \n  137. \n  138. \n  139. \n  140. \n  141. \n  142. \n  143. \n  144. \n  145. \n  146. \n  147. \n  148. \n  149. \n  150. \n  151. \n  152. \n  153. \n  154. \n  155. \n  156. \n  157. \n  158. \n  159. \n  160. \n  161. \n  162. \n  163. \n  164. \n  165. \n  166. \n  167. \n  168. \n  169. \n  170. \n  171. \n  172. \n  173. \n  174. \n  175. \n  176. \n  177. \n  178. \n  179. \n  180. \n  181. \n  182. \n  183. \n  184. \n  185. \n  186.\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":" \nDoes life go on when your heart is broken?\n\nSince her mother's sudden death, Emma has existed in a fog of grief, unable to let go, unable to move forward\u2014because her mother is, in a way, still there. She's being kept alive on machines for the sake of the baby growing inside her.\n\nEstranged from her stepfather and letting go of things that no longer seem important\u2014grades, crushes, college plans\u2014Emma has only her best friend to remind her to breathe. Until she meets a boy with a bad reputation who sparks something in her\u2014Caleb Harrison, whose anger and loss might just match Emma's own. Feeling her own heart beat again wakes Emma from the grief that has grayed her existence. Is there hope for life after death\u2014and maybe, for love?\n\nELIZABETH SCOTT\n\nTo Astrolabe, for over fifteen years  \nof being a bright and joyful light in my life. I miss you every day.\nContents\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\nChapter 29\n\nChapter 30\n\nChapter 31\n\nChapter 32\n\nChapter 33\n\nChapter 34\n\nChapter 35\n\nChapter 36\n\nChapter 37\n\nChapter 38\n\nChapter 39\n\nChapter 40\n\nChapter 41\n\nChapter 42\n\nChapter 43\n\nChapter 44\n\nChapter 45\n\nChapter 46\n\nChapter 47\n\nChapter 48\n\nChapter 49\n\nChapter 50\n\nAcknowledgments\n\nQuestions for Discussion\n\nQ&A with Elizabeth Scott\n\nTeaser\n\nExcerpt\n1\n\nI sit down with my mother. My smile is shaky as I tell her about my day.\n\n\"I think I did okay on my History test,\" I say. \"Oh, and Olivia wore her new pair of false eyelashes, the ones I told you about. She was batting them around so much that a teacher stopped and asked if she had something caught in her eyes.\"\n\nI laugh at the memory, and the sound is shaky too. \"Olivia wasn't super happy about that.\"\n\nThere's the slightest movement, but it's not on Mom's face. Her face never changes. But under the skin of Mom's stomach...I don't want to look but I can't help it, because there my mother's skin is moving.\n\nBecause the baby is moving.\n\nI close my eyes.\n\nWhen I open them, Mom's stomach is stretched out and still.\n\n\"Emma, are you ready to go?\" Dan says as he comes into the room, and I look up at him and nod.\n\n\"Did you two have a nice chat?\" he says, bending over to kiss Mom.\n\nI stare at him.\n\nHe must feel it because he straightens up, clearing his throat, and pats Mom's stomach. \"Look how big he's getting. Lisa, he's growing so much.\"\n\nMom doesn't say anything, not even to that.\n\nShe can't.\n\nShe's dead. Machines are keeping her alive. They breathe for her. They feed her. They regulate her whole body.\n\nMy mother is dead, but Dan is keeping her alive because of the baby.\n2\n\nDan and I don't talk on the ride home. As soon as I'm inside the house I head straight up to my room, and I lock the door.\n\nI never used to have a lock, but then, I used to have Mom. I used to think that Dan cared about what I thought. What I wanted. What Mom would have wanted. This way, all the talks he used to try to have, right after Mom first died, can't happen. Or at least, he can talk, but I don't have to see him and can put on music or headphones or even fingers in my ears to shut him out. Just like he shut me out.\n\nI don't have one of those wussy little turn-and-click locks. I have an actual lock, a bar with a padlock that I snap shut.\n\nClosing out the world.\n\nI put it in myself the day Dan told me what he was going to do to Mom. I walked out of the hospital, went to the hardware store and came home and put in the lock. My mother taught me how to do that. She believed women should know how to fix things. I'd seen her fix a broken toilet and watched her change the element in our hot water heater. She installed new locks on our doors when I was seven, after Olivia's family got robbed.\n\nI go over to my window and open it. On the roof, Olivia grins at me through her blond hair and then comes over and pushes herself inside.\n\n\"How did you know I was out there?\"\n\n\"I saw your hair when we came in. Also, your car down the road. Thanks for not parking...here.\"\n\n\"It makes things easier,\" she says. \"And clearly, I need a wig. Oooh, I could get a bunch. Red hair, blue hair\u2014\"\n\n\"That wouldn't stand out at all.\"\n\nShe sticks her tongue out at me. \"I'd get other ones too. Brown hair, black hair. I could be a spy, don't you think?\"\n\n\"Spies have to use computers, Olivia.\"\n\n\"No, they don't. They go on missions. They have tech people do the computer stuff for them.\"\n\n\"Someone's been watching Covert Ops.\"\n\n\"Like you don't watch it too. You know you love it. You and your mom both think Sebastian is...\" She trails off.\n\n\"Sebastian is cute,\" I say, and try not to think about how Mom and I used to watch the show together. \"But he's also fictional, plus even spies on TV have to use earpieces and stuff\u2014would you be willing to do that?\"\n\n\"For Sebastian I would,\" she says, grinning, and then flops on my bed. \"But I really wish I could be an old-fashioned spy. Like back when they had to write coded messages in invisible ink and speak a dozen languages.\"\n\n\"That sounds more like you,\" I say, and sit down next to her. \"I\u2014I saw the baby move today.\"\n\n\"Really?\"\n\n\"Yeah.\"\n\n\"Emma,\" she says, squeezing my hand, \"why do you even go to the hospital?\"\n\n\"Because I can see her. Because I want at least one person to be there for Mom and not for the baby.\"\n\n\"Dan\u2014\"\n\n\"Dan wants the baby. You know it, I know it. If Mom was alive...\" I stare at my dresser, at the photo of Mom and me. It was taken in Vermont when we went skiing. Mom is smiling and has one arm around me, holding me tight. It was the last vacation we took together, just her and me. She was thirty-five. I was ten.\n\nShe met Dan two weeks after we got back from Vermont. I was nice to him when I met him because he actually asked where I wanted to go to dinner when Mom suggested the three of us go out. I thought he was kind.\n\nI also thought he loved Mom.\n\n\"Hey,\" Olivia says, and I look at her.\n\n\"She'd love you for being there,\" she says. \"She does love you for being there. I know it.\"\n\nI hug her, and Olivia hugs me back.\n\nDan knocks on my door. \"Emma, you want some pizza? I made triple cheese.\"\n\nOf course he did. Dan doesn't order food. He makes it. \"The perfect man,\" Mom used to say. \"He can cook, he makes the bed and he remembers to put the toilet seat down.\" Then she'd laugh and kiss him.\n\nShe loved him so much.\n\n\"I'm not hungry,\" I say.\n\n\"I'll leave it by the door,\" he says with a sigh. \"Olivia, do you want me to leave you a slice too?\"\n\nOlivia looks at me. I shrug.\n\n\"Okay,\" Olivia says, and Dan says, \"Thanks for coming today, Emma.\" Like he does every day. Like I'm doing it for him. Like I'm somehow in this with him.\n\nI unbolt the door after five minutes. When I first started locking myself in, Dan would hang around and try to talk to me when I came out. I used to like how hard he tried, but I sure don't now. Not after what he's done to Mom. Now I wait until I'm sure he's gone.\n\nOlivia eats most of the pizza and then says she has to get home to make sure her parents eat.\n\n\"Wish me luck,\" she says. \"Prying their handheld whatevers away from them for longer than thirty seconds makes them both go into withdrawal. See you tomorrow?\"\n\n\"Yeah. You don't have to go out on the roof to leave, you know.\"\n\n\"I know,\" she says. \"But if I use the front door or try to go out any other way, I'll see Dan. And I know he'll ask me about you. He did the last time I left that way. I think he\u2014well, I think he's worried about you, you know?\"\n\n\"Why? Because my mother is dead and he's kept her body alive so he can try to save his precious son? Because I have to see her lying there\u2014\" I break off and open the window for Olivia.\n\nOlivia hugs me again and then leaves. After she does, I close my window and get into bed. It's early, but I don't care. In bed, I can look at my ceiling. It's yellow and the color is swirled around so there are a million patterns and shapes to get lost in. Mom painted it last year even though the doctor didn't want her \"exerting\" herself because she'd just had a blood clot taken out of her leg.\n\n\"Think about this instead of that boy,\" she'd said when I came in and lay down on the bed to look at it.\n\n\"I can't,\" I'd said. \"Anthony broke my heart.\"\n\n\"I know,\" she'd said, lying down next to me. \"But one day he won't matter.\"\n\n\"He said I was lovely.\" I'd looked up at the ceiling.\n\n\"They all say something like that,\" she'd told me. \"Trust the one who takes his time saying it.\"\n\n\"Dan said he was falling in love with you on your second date.\"\n\n\"Dan's different,\" she'd said. \"He's older, for one thing. And so am I. It's...you won't believe me, but one day Anthony will just be a memory and it won't hurt when you see him at all, I promise.\"\n\nShe was right. I wish I'd told her that. I could have. Anthony was nothing to me ages before she died.\n\nI wish I could tell Mom something, anything, and have her really hear it.\n\n\"I miss you,\" I whisper, and listen to Dan moving around downstairs. If I close my eyes, I can pretend I hear Mom, that this is just another night.\n\nThat she's still here.\n3\n\nDan drives me to school in the morning. He has done this since he and Mom got married, and I used to like it although I did start to ride with Olivia when she got her license.\n\nThat stopped when Mom died. I wanted Dan to remember I was around. I wanted him to remember Mom.\n\nLike, Mom worried about my grades. Not that they weren't good enough, but that I was working too hard. Dan told her that in order to grow up I had to be allowed to make my own choices.\n\nOh yes, Dan and his choices.\n\nWe drive to school in silence. At seventeen, I'm old enough to get my license, but the waiting list to get into any of the driver's ed classes within half an hour of the house stretches out for months. I'd planned to put my name on a list last year but never got around to it.\n\nLast year, before everything happened, Dan promised that over the summer he'd teach me how to drive and then I could just go get my license.\n\nI don't want him teaching me to drive now. What if something happens? What if I get hurt? If my body stops working, my brain stops functioning? Would he have machines keep me alive in case his son might one day need something? A lung, a kidney, bone marrow?\n\nBut I do ride in the car with him to school. I do it because it means he will have to pick me up afterward. That he will have to see me, that he will take me to see Mom. He works at home, so he can do that.\n\nOr at least, he used to work at home. I don't know if he still does, or if all the database consulting he did stopped when Mom did. Lately, he hasn't mentioned any two-hour phone calls to talk someone through using a new feature he's built.\n\nBut then, I haven't asked. I don't want to talk to him.\n\nHe was going to stay home with the baby, and Mom was going to go back to work. That was their plan. She was an assistant manager at BT&T bank. They sent flowers when she died. They didn't send anything for the baby. Maybe they didn't know what to do about it, but maybe they heard about what Dan's doing and think he's keeping a dead woman alive so he can get what he wants.\n\nIf they do, I love them for that. I mean, I know it's a baby and it's partly Mom, but I wish Dan had just once thought about what Mom would have wanted. It was so easy for him to choose to keep her here, dead, and it's so hard for me to think about, much less see.\n\n\"I got a call from your AP History teacher about how you're doing in class. Maybe we should talk about it,\" Dan says as we stop, one car in the many that are waiting to snake into the high school. Mostly freshman and sophomores get out here. Juniors get rides with their friends who have licenses or, better yet, get their own and a car to go with it.\n\nI could get a ride with Olivia, but I don't.\n\n\"See you later,\" I tell Dan and get out of the car. I won't talk to him about school just like I won't ride to school with Olivia anymore. If I did, then Dan would get to feel like things are normal and they're not. They are so not. Not while Mom is still...\n\nThe tears hit me hard, hot pressure behind my eyes, in my throat, in my chest. It's hard to breathe, to see, to think.\n\nI look down at the ground and walk, blinking hard once they've started to spill down my face.\n\nI cry without making a sound now. I have cried soundlessly, wordlessly, since I stood with Dan at the hospital and heard, \"I'm sorry, but...\"\n\nDan cried openmouthed then, sobbing, yelling his grief for everyone to see. I tried to hug him. I felt for him because I thought he loved her, because we were in the same place, because she was gone and he felt the gaping hole that had been born too, a Mom-shaped space in the universe.\n\nHe didn't hug me back. He didn't even seem to see me.\n\nAnd then the doctor told him about the baby.\n\n\"Hey,\" Olivia says, and I know it's her because I would know her voice anywhere. We've been friends since first grade, and we've been through period trauma, boy crap, bad hair, her parents and their ways. And now Dan and his baby.\n\n\"Hey,\" I say. I wipe my eyes and look at her. \"How's the car?\"\n\nOlivia makes a face at me but also wraps an arm around my shoulders, steering me toward our lockers. Her parents gave her a fully loaded convertible when she got her license, one with a built-in music player, phone, navigation system\u2014you name it, the car had it. Could do it, and all at the touch of a button.\n\nOlivia sold the car\u2014through the one newspaper left in the area, which is basically just ads\u2014and bought a used car. It's so old all it has is a CD player and a radio. We bought CDs at yard sales for a while, but all we could get was old music, which we both hate, and the radio is just people telling you that what they think is what you should think, so we mostly just drive around in silence.\n\nIt used to bother me sometimes but now I like it. The inside of my head is so full now that silence is...I don't know. There's just something about knowing Olivia is there, and that we don't have to talk. That she gets it. Gets me and what's going on.\n\nHer parents were unhappy about the car, though. Really unhappy, actually, but then there was a big crisis with one of their server farms at work and by the time they surfaced for air they hadn't slept in four days. And when they said, \"Olivia, that car was a gift,\" she said, \"Yes, it was. A gift, meaning something freely given, for the recipient to use as she wanted to, right?\"\n\nAs we hit her locker, we pass Anthony, and he says, \"Ladies,\" bowing in my direction. A real bow too, like it's the nineteenth century or something.\n\n\"Ass,\" Olivia says.\n\n\"A donkey is actually not as stupid as people believe. However, you are entitled to your own beliefs about asses. And me.\" He looks at me. \"Hello, Emma.\"\n\nI sigh. \"Hi, Anthony.\"\n\n\"If you ever want to talk about your grades, do know that I'm here.\"\n\nI can't believe I ever thought the way he talked was interesting. It's just stupid, like he's too good to speak like a normal person. \"I know, Anthony.\"\n\n\"I really would like to be of assistance to you. I believe in helping everyone. I'm talking to Zara Johns later. I think she feels threatened by the fact that I've been asked to help her organize the next school blood drive.\" Translation: he's butted in, and Zara's furious.\n\n\"Either that or she just doesn't like you. Emma, let's go,\" Olivia says, slamming her locker shut, and we head for mine.\n\n\"You okay?\" she says, and I nod. Anthony doesn't bother me at all anymore, just like Mom said would happen. I look at him and feel nothing. Well, some annoyance, but then, who wouldn't after listening to him talk?\n\nOf course, I didn't always think that he was annoying. I open my locker, deciding not to go down the Anthony road, and hear the guy next to me say, \"No way! I mean, everyone knows what'll happen to Caleb if he steals another car.\"\n\nOlivia and I glance at each other. If Anthony is the ass end of the smart part of the school, Caleb Harrison is the ass end of the stupid part. He's a total druggie and three years ago, when we were freshmen, he came to school so high he couldn't even talk. I heard that stopped last year, but then, as soon as school got out, his parents sent him off to some \"tough love camp,\" which is rich-people code for boot-camp rehab.\n\nHe came back seemingly off drugs but newly into stealing cars. He started by grabbing them at the mall and parking them in a different spot, but then he stole a teacher's car.\n\nAnd then he graduated to a school bus. It was empty at the time, but still, I heard that got him a couple of weeks in juvie, or would have except for his parents, who intervened. I guess now he's taken yet another step forward and by lunchtime, I know what Caleb stole.\n\nHis father's brand-new, limited-edition Porsche. And he didn't just steal it. He drove it into the lake over by the park, drove right off the highway and into the water. The police found him sitting on the lake's edge, watching the car sink. They were able to pull it out, but water apparently isn't good for the inside of a Porsche.\n\n\"You think he'll go to jail this time?\" Olivia asks as we sit picking at our lunches. I love that we have lunch together this semester, but it's the first lunch block, and it's hard to face food\u2014especially cafeteria food\u2014at 10:20 in the morning.\n\n\"I guess it depends on his parents,\" I say. \"Last time they talked to the judge or whatever. They'll probably just ship him off again. He must hate them, though.\"\n\n\"Yeah. To sit by the lake and watch the car sink like that\u2014\"\n\n\"Exactly.\"\n\n\"Even when my parents are sucking their lives away with all their computer crap, I'd never do anything like mess with their stuff,\" she says. \"How can you hate someone who raised you, who loves you so\u2014\" She breaks off.\n\n\"Dan didn't raise me,\" I say tightly. \"And he doesn't love me. Or Mom.\"\n\nOlivia nods and I think about hate. I understand what can make someone do what Caleb did, although I don't think a bored, rich druggie really gets hate. Not real hate.\n\nI do, though. If there were something I could do to Dan that would hurt him, I'd do it.\n4\n\nThe rest of school is like school always is. I sit, I pretend to listen, avoid my AP History teacher's attempt to try to talk to me after class and wait for the final bell to ring.\n\nI used to like school. I was the person\u2014along with Anthony\u2014who got A's on everything and so wrecked any possible grading curve. I did extra credit assignments for fun. I went out and did research about authors we were going to read. I learned about minor historical figures we'd discussed in passing.\n\nLast summer, I audited a biology class at the community college to make sure everything I'd learned in Advanced Bio stayed in my head. I was going to do the same thing with chemistry this summer, and maybe something in literature too.\n\nI was a great student. The kind of student everyone hates, actually. I didn't make friends in my classes, I had acquaintances that I blew away at everything, but I didn't care. I wanted great grades, the best grades, and I had Olivia, who was in regular classes and who knew there was a list of the top one hundred colleges out there but had no idea which was number one. Or eight. Or forty.\n\nI knew what the number one school was, and I knew I couldn't go there because one year of tuition cost an amount that was enough to support a family (or possibly two) for that year and they were stingy with scholarships, but I wanted a scholarship to one in the top ten. I wanted to be the best, not just for the scholarship I'd need to go to a great college, but because I could be.\n\nA lot of the time, I was. The best, I mean.\n\nAt school, anyway. Personally, my social life was...well, it was pretty poor. A few kisses at a few parties. Anthony.\n\nVery poor, really.\n\nI didn't mind. My dad\u2014my real dad\u2014was a history professor, and I wanted to be like him. Ever since I was little, that's what I wanted. To be what my dad was. To see my mother's face when I got my PhD in history.\n\nI don't care about school at all now. I sit in class and if I get called on, I say, \"I don't know.\" I don't do my homework and the leeway I got at first is gone. I'm getting F's on quizzes. On tests. I'm still ignored by my classmates, but now it's because I've fallen so far behind I'll never get back to where I was. I'm no threat anymore.\n\nI have a twenty-page paper on the New Deal that's beyond late. I haven't written a word of it.\n\nI'm not going to. I don't care about school right now, and if I ever do again, I'll never care about history. It's nothing but studying things that have happened. That are gone.\n\nHistory is full of death, and I've had enough of that.\n5\n\nIt took almost two years for my mother to get pregnant. Two years of planning, of Dan smiling and talking, hoping. Of Mom going to the fertility doctor's office over and over again.\n\nOf me hearing her crying sometimes.\n\n\"I'm sad,\" she'd say when I asked, and I would watch her, so drained-looking, and wonder why she was doing it.\n\nBut then Dan would show up, dry her tears, kiss her, and she'd smile and I'd know why. She wanted him to be happy. She loved him.\n\nSo she tried. And tried.\n\nAnd tried.\n\nShe was pregnant with me when she married my dad. She didn't know it, but she was. She used to say I was such a quiet baby that she didn't even know I was there until her clothes started getting tight.\n\n\"Of course,\" she'd say, \"you made up for it with the colic, but still, you were worth it.\" And then she'd kiss my forehead or my cheek. She used to talk about how easy it was, being pregnant with me, before she started trying for Dan.\n\nBefore trying to get pregnant took over her life.\n\nMy dad named me. His parents, who'd died when he was fourteen, left him in the care of his aunt Emma because one set of his grandparents was dead and the others were both well on their way to drinking themselves there. Emma loved history, just like my father, and took out a second mortgage on her house to send him to graduate school. She got sick with a cold the day before he got his doctorate and died of a massive internal infection a week after she saw him get it.\n\nDan's parents are dead, which is the only thing he and my dad have in common, besides the whole being married to Mom part. My mother's parents are both alive, but they live in Arizona and I've only met them twice. Both times were awful. They basically spent the entire visit telling my mother that she was such a disappointment and she needed to \"turn herself into someone better.\" The second time, Mom told them that maybe they needed to fix themselves and then we left. They didn't try to get in touch with her again and when she died, they called and left a message.\n\nI think that's why Mom was such a great mom. In spite of her parents, or maybe because of them, she taught herself how to love.\n\nAnd she did.\n\nShe loved so much, and she loved with everything, with her soul. I wanted to be like that.\n\nI don't anymore.\n\nMom did a lot of stuff for Dan, but what she did to get that baby...some of it sounded pretty gruesome. Painful, even. I once heard her tell Dan, \"I don't think I can do it. I just...my body is like this thing now.\"\n\n\"We'll talk to the doctor,\" Dan said. \"He did warn us that with that blood clot you had, things could be even riskier. And I know being on the drugs is hard. So if you're this unhappy...\"\n\n\"No, no,\" Mom said, but of course she'd say that. She loved Dan. She knew how much he wanted a baby. She knew that because she was over forty when they started trying, her best chances of having a baby\u2014\"The dream baby,\" she used to say with a smile\u2014lay with drugs and testing and all kinds of stuff. And risk. So much risk.\n\nDan began setting up a nursery in the guest bedroom about thirty seconds after the clinic called to say it looked like she was pregnant, and I can remember Mom saying, \"Dan, it hasn't even been a day yet. I don't want you to hope too much.\"\n\n\"Don't be afraid,\" Dan had said. \"I know it's true.\" He grinned at me as I stood in the corner of the already-changing guest room. \"You're going to have a brother or a sister, Emma!\"\n\n\"But if it hasn't even been a day\u2014\" I said, and then broke off as Mom looked at me, her face full of love and pleading.\n\n\"You need help with what you're doing?\" I said to him, and helped Dan box up the extra linens in the closet, sat with him while he drew up plans for what would go where and Mom sat, listening to him and smiling a little.\n\nShe was pregnant for real then, finally. But it was a hard pregnancy from the start. She was sick all the time, so much that she lost weight. Dan made her favorite meals to try to get her to eat but it didn't help much.\n\nAnd then, in the second month, she had some spotting and had to go the hospital. Dan rushed there from the house so fast he forgot to call the school and tell them to find me and tell me what was going on.\n\nI still remember coming home and finding Mom in bed.\n\n\"What happened?\" I said. \"Is something wrong with the baby?\"\n\n\"No, everything's fine,\" Mom said. \"I just\u2014I was bleeding some before and\u2014\" She broke off, her voice cracking, her eyes filling with tears.\n\n\"Lisa, honey, don't cry. You're okay. You're going to be fine,\" Dan said, and Mom nodded but she didn't look like she believed him. She looked scared.\n\nI waited until Dan left and sat down next to her. \"Mom, are you okay about the baby? Dan talks about it all the time, but you don't and I'm wondering if\u2014\"\n\nShe squeezed my hand and said, \"Emma, honey, I know what I want. I just...it was hard to get here. But now I am. I beat all the odds\u2014over forty, all the drugs, the warnings about the clot\u2014ugh, I already went over it. And over it.\"\n\nShe touched her stomach and I kissed her cheek and lay beside her.\n\n\"I could get used to this,\" I said a while later, stretching out with one foot to try to pull the TV remote up toward me.\n\n\"Not me,\" she said. \"I'd like to be able to get up and move around. I feel trapped just lying here. I mean, if I could paint your ceiling after the clot came out, why can't I walk downstairs?\"\n\n\"I heard that,\" Dan called from the hallway. \"No rest, no chocolate cake.\"\n\n\"Meanie,\" Mom said, grinning, but she was tapping her toes against the bed, like she heard a song and was following the beat. Like she wanted to move to it.\n\nShe was able to get out of bed after a week, and everything after that went okay. She still got sick, but not as much, and she started to finally gain some weight.\n\nAnd then, on a Wednesday morning, after I'd already left for school with Olivia, she went to grab a piece of toast in the kitchen and fell down.\n\nThat was it.\n\nThat's how she died.\n\nShe was getting breakfast, something she did every day, something normal, and her body just...stopped.\n\nDan ran right over to her and performed CPR until the ambulance came. She wouldn't\u2014couldn't\u2014open her eyes. Couldn't feel anything when she was touched. Couldn't talk.\n\nShe'd had a massive stroke caused by an embolism in her brain, the kind that\u2014\n\nThe kind that you don't come back from.\n\nMom was gone when she hit that floor. CPR kept her lungs going for a while, and then surgery and tubes and machines to try to figure out what was going on took over. And then the doctor came out and said, \"I'm sorry, but she's gone.\"\n\n\"Gone?\" Dan said. \"But she was breathing! I was with her. She was breathing!\"\n\nI tried to hug him, and then the doctor drew him aside. I found out later he told Dan that Mom was brain-dead, that without medical intervention she wouldn't be breathing, that her heart wouldn't be beating. That the baby was still alive and Dan could have everything turned off now\u2014and let Mom go\u2014or keep her hooked up to machines until the baby was old enough to maybe live on its own.\n\nMom never knew what happened to her. That's what I have to hold on to. That at least it was fast. That whatever pain there was didn't last long. That she reached for a piece of toast and left forever.\n\nExcept she's still here\u2014alive but not alive\u2014and I wonder if part of her is trapped in her broken body. A prisoner of the baby swimming around inside her.\n\nI think of how scared she was and wonder if this was what she saw coming. If she knew that no matter what happened to her, Dan would pick the baby\u2014that Dan would choose his baby over her. Over the family we'd had. Did she know that he would look me in the eye and say, \"Your mother would want this,\" even after I'd lain next to her in bed and heard how restless and scared not being able to move made her?\n\nHow having to lie still made her feel trapped.\n\nSometimes I hope she's gone, that she's in heaven looking down at all of this, but I've felt the weight of her hand in mine every day since she died. I've watched her fade, become smaller despite all the nutrients piped into her, the baby taking all it can.\n\nShe isn't gone. Not like she should be.\n\nMy mother's name was Lisa Davis Harold, and she was strong and beautiful. She was a person, she had her own thoughts, and I remember that. I remember how she was. Who she was.\n\nI remember her.\n\nI'm the only one who does.\n6\n\nAt the hospital, Dan always goes in and says hi to Mom first.\n\nActually, he wanted \"us\" to go in and \"say hello together,\" but the first and only time he asked me that, the night after I'd lain in bed, thinking of my mother lying in the hospital kept alive for the baby\u2014his baby\u2014I said, \"There isn't an us. There's you, and then there's me.\"\n\n\"But we're family.\"\n\n\"Were,\" I said. \"Go see what you're here for. And then I'm going to see Mom.\"\n\n\"I'm here every bit as much for your mother as I am for the baby.\"\n\n\"I know. After all, if her body can't be kept alive long enough, your baby won't survive, will it?\"\n\n\"Emma, that's not\u2014\"\n\n\"It's not? Then what is it?\"\n\n\"It's what your mother would want.\"\n\nI slapped him. Right there, in the hospital.\n\nSecurity was called, but Dan said nothing was wrong, that we were \"just struggling with our loss\" and that he'd sit with me outside for a while.\n\nHe did walk outside with me, and he actually put a hand on my arm and said, \"Emma, please. I don't think you're seeing\u2014\"\n\n\"Don't touch me,\" I said. \"Don't try to sell me your story. Mom loved you, I know that. You can kick me out of the house, send me to live with Mom's parents, maybe boarding school. Take your pick.\"\n\n\"I'd never do that. You're my family. Don't you know that? Don't you know I love you like you were my own\u2014\"\n\n\"Go see her,\" I said, cutting him off and making sure I was out of his reach.\n\n\"You should come too.\"\n\n\"I don't want to see her with you.\"\n\n\"Emma\u2014\" he said and then sighed.\n\nSo that's how I got to see Mom on my own. Dan goes in first while I sit in the waiting room outside the ICU, and then he goes and drinks some of the hospital's sludge coffee. I don't think he likes it, but then I don't care what Dan thinks or likes anymore.\n\nHe's in there now, doing his thing, and I'm staring at the ceiling. I did homework for the first few days, more out of numbness than anything else, and then I realized it was easier to just sit and look at the ceiling like I do at home. To think about how she'd painted it, to think about her, and not where I am.\n\nTo not think about Mom tethered to a bed by machines and IVs and the lump in her belly.\n\nOne of the volunteers comes in with the magazine cart. The thing is a joke because the hospital never has any new magazines. They just replace the old issues with slightly less old issues. But then I suppose most people in here aren't really that concerned with what's going on in the world.\n\nI know I'm not.\n\nThe magazine cart squeaks as it comes over to the last table in the room, the one that's at the far end of the bank of chairs where I like to sit. Not that there are a lot of people in the waiting room today. Or any day. The ICU is not a place where people come to stay for a long time. Not usually, anyway, but my mother is \"special.\"\n\nThe tears come again and I blink, watch the ceiling waterfall into little pieces as my throat gets tight.\n\nI don't want to see Mom like this, and I pinch the bridge of my nose hard. It makes my head hurt but stops the tears.\n\nMom used to do it whenever she thought she might cry. She hated to cry, and I can remember how, on the day she married Dan, she sat there getting her hair done and pinching her nose over and over so she wouldn't cry and mess up her makeup.\n\nI was part of the ceremony. Mom and I walked down the aisle together, and before Mom and Dan became husband and wife, Dan asked for my permission to be part of our family. He said, \"I'm so happy to have found you and your mother and I promise I'll always look out for you. I'll always want what you do, I'll always believe in you.\"\n\n\"Liar,\" I mutter, and wipe my eyes.\n\nI look away from the ceiling and see Caleb Harrison staring at me.\n7\n\nIt's definitely him. We aren't in any of the same classes but he's in the lunch block Olivia and I share and I've seen him getting food, shoving his perfectly wavy blond hair off his face as he waits to pay.\n\n\"What did you just say?\" he says, and if it wasn't for the snarl in his voice\u2014plus the fact that he steals cars (and now apparently drives them into lakes as well)\u2014he'd be cute.\n\nMore than cute, even.\n\nBut he does sound angry, and under his hair, blond curls falling all over his face, his eyes are narrowed and very pissed-off looking.\n\n\"Nothing,\" I mutter, and he grunts and turns away. I stare at his back and only then start to wonder why he's here.\n\nAnd why does what I said matter to him? Does he really think I would be sitting here, in the waiting room outside the ICU, and somehow be thinking about him? I mean, really? Yes, everyone knows who he is, but it's not like he's the kind of guy I'd go for. And besides, what do I have to be afraid of from him anyway? From anyone?\n\n\"Hey,\" I say. \"Can I have a Women's One?\"\n\nHe turns back around. \"What?\"\n\n\"A Women's One magazine,\" I say, and he really does think I was sitting here thinking about him, because he's glaring at me and plucking a magazine off the cart like it's diseased.\n\n\"What's your problem?\" he says, tossing it to me. It lands on the floor by my feet. \"If you have something you want to say to me\u2014\"\n\n\"Emma,\" Dan says, coming into the room, \"you can go in now.\"\n\n\"Great,\" I say and get up, step on the magazine and then push past Caleb Harrison like he isn't there.\n8\n\nMom is...she's the same as she's been since she was put in this room. She's still, so still, and I sit and look at her closed eyes, at her slightly downturned mouth. At the tube going into it.\n\nHer skin is strange-colored, almost waxy-looking, and her hand is warm but limp in mine.\n\n\"It wasn't much of a day,\" I tell her, and look around. The unit Mom's in has huge open windows by every door\u2014I don't know why\u2014but I can see people in other rooms. Most of them are sitting like I am, hunched by a bedside. A few are weeping. A few are just staring, lost-looking.\n\nI look away, look back at Mom. \"I turned in my paper on the New Deal,\" I say. \"And we've started a new book in English that I like a lot. Oh, and I got an A on my Algebra II quiz.\" I talk and talk, spinning a story of a day filled with academic success. Filled with lies.\n\nPart of it is because I am looking at her and I want her to think everything is okay even though I know she can't hear me.\n\nPart of it is because part of me thinks that maybe she can, that despite everything the doctor said she will somehow open her eyes and say, \"Emma, I know something's wrong. I can hear it in your voice. Start talking, okay?\"\n\nYes.\n\nYes, I want to talk about it so much; I love you and I miss you and I wish you were here but not like this, I don't want you here like this and I know I'm seventeen but I don't want you to be gone. I want you to open your eyes and tell me everything is going to be okay. I want you to squeeze my hand and tell me something, anything. I want to hear your voice, not the machines that beep all around us.\n\n\"Mom,\" I whisper, and kiss her hand, pressing my cheek to it, eyes closed as I imagine.\n\nI feel movement, a slight shift in her but I know it's not her.\n\nIt's the baby.\n\n\"Why did you do it, Mom?\" I ask. \"Why did you try so hard for this when it was so hard on you? When the risks were so many? When you cried so much? When you ended up\u2014when now you're here?\"\n\nI hear Dan's voice as he comes back into the ward. He always says hello to everyone, like he's so friendly. Like he's actually thinking about anything other than himself.\n\nI open my eyes and see a magazine cart in front of the door.\n\nAnd I see Caleb Harrison staring at me again.\n\n\"Hey there,\" Dan says to him as he comes to the room. \"I saw you earlier, right?\"\n\nCaleb nods, looking at him and then my mother. I see him stare at her stomach.\n\nDan walks into the room. \"Lisa, Emma and I are both here now, and I thought we'd all talk for a little while before we have to go.\" He pats Mom's stomach. \"I was thinking today we could talk about names.\"\n\n\"No,\" I say, and Dan looks at me.\n\nCaleb, still standing in the doorway, looks at me too, and Dan glances at me, then at him, and says, \"We don't need anything to read now, thanks.\"\n\nCaleb shrugs and moves off, the cart squeaking as he goes.\n\n\"You know him?\" Dan says.\n\n\"No, but it's not every day you see a girl sitting with her dead mother, is it? People would stare at that, don't you think?\"\n\n\"Emma, honey, your voice\u2014\"\n\n\"She can't hear me.\"\n\n\"The baby can, though, and I don't want\u2014\"\n\nI stand up so fast that I'm dizzy for a second. I don't want to hear more. I can't hear more.\n\n\"I don't feel good,\" I say. \"Can we go?\"\n\n\"I really was hoping we could talk about names. I'd like for you and I to...\" He sighs. \"Your little brother is in there, Emma. He's in there and he's fighting to stay alive.\"\n\nI walk out of the room then. I stop at the nurses' station and ask to use the phone. I hear Dan come out when he realizes I'm not coming back. I hang up the phone.\n\n\"Emma,\" he says, but I pretend I can't hear him and walk out. He follows me, of course.\n\n\"You're hurting your mother,\" he says when we're waiting for the elevator. \"She wanted this baby. She'd want you to be part of this. She'd be so sad to see how you're acting.\"\n\nI stay silent. I stay silent all the way to the car, all the way to the house. I don't think of it as home anymore.\n\n\"You say what she wants. What she thinks, what she feels,\" I say when we get there. \"She can't do anything now, and it's all because of you and what you want. So don't tell me how she feels, because she can't feel. She's dead. She died trying to have your baby, and if you want to think about feelings and Mom, how do you think she feels about that? How do you think being dead makes her feel?\"\n\n\"Emma,\" Dan says, and then \"Emma!\" but I'm out of the car and heading down the driveway, heading toward the car I know is waiting there.\n\nThe lights turn on as I reach it, and I open the passenger door and get in.\n\n\"Thank you,\" I say, and Olivia nods, squeezing my hand before we drive off.\n9\n\nThe phone at Olivia's house is blinking when we get in but she ignores it, sits me down in her parents' gleaming steel kitchen and puts a peanut butter sandwich in front of me.\n\n\"Just don't let me see you destroy it,\" she says, putting a bag of corn chips next to me, and then goes over to the phone.\n\nI hear her talking while I'm opening the sandwich and putting corn chips on top of the peanut butter.\n\n\"No, she called me from the hospital, and I said I'd come get her. I\u2014look, Dan, I think she just needs some decompression time. You know?\"\n\nI love Olivia. Not just for talking to Dan for me, but for a million little things. Like, she was okay that my mom loved peanut butter and corn chip sandwiches even before I was. I thought the idea was disgusting until I found myself wandering around the house three nights after she reached for toast and then broke. I was thinking about her, the things she did, like how she always had to put her wallet in her purse before she'd put anything else in or how much she hated peas.\n\nI was wandering, remembering, and I was alone. Dan was sleeping peacefully, no doubt dreaming of his baby.\n\nI thought about those sandwiches.\n\nI made one the way she always did, first pressing the slices of bread and peanut butter together, and then taking them apart to put the chips on before smooshing it back together, and it was good. As I ate it, for a moment I swear I could almost see her. Picture her smiling at me.\n\n\"Sure, she'll call later,\" Olivia says. \"Okay. Bye.\"\n\nShe comes back to the table, one arm extended. I hand her the chips and smile as she heads toward the pantry, eyes averted from my sandwich.\n\n\"It's not that bad. I've seen those gel things your parents eat.\"\n\n\"True,\" she says, coming back to the table and sitting down.\n\n\"You can see the sandwich now since you're sitting here, you know.\"\n\n\"Yeah, I know. What happened?\"\n\nI tell her.\n\n\"Oh,\" she says when I'm done. \"Names, huh? He must really think the baby's going to make it.\"\n\n\"I guess. All it has to do is lie there and suck everything out of Mom that's pumped in until it can survive long enough to live in an incubator.\"\n\n\"Emma,\" Olivia says, picking up my plate and walking over to the shiny steel sink. \"You know the baby's not a bug or anything. It's your brother.\"\n\n\"Half. And it's\u2014Mom is dead and it's not and I try not to see it but sometimes it moves and Mom's\u2014she's just lying there, you know? Her body is only there for the baby and Dan chose that. He said he loved her, that he'd do anything for her. What kind of love is that, Olivia? Would you want someone to keep your dead body breathing with tubes and machines because they wanted something from you?\"\n\nI'm yelling by the end and Olivia has come back to the table and puts her arms around me.\n\n\"I'm sorry,\" she says. \"I don't\u2014my parents\u2014our family's not like it was for you and your mom. And the baby, it'll never even know her. That's so strange and awful.\"\n\n\"When Dan finally gets around to thinking about that, he'll probably just say it's proof that science can work miracles and it's how Mom would have wanted it.\" By the time I'm done talking, I'm shaking so hard my teeth are chattering.\n\n\"I want to fix it for you, you know?\" Olivia says. \"You're so angry, Emma. And I don't know if it's with Dan or your mom.\"\n\n\"Dan. Definitely Dan.\"\n\n\"And the baby.\"\n\n\"I\u2014look, I do get that it didn't choose for Mom to die. But she did, you know? And the doctors say the embolism didn't happen because she was pregnant but it's just...\" I swallow. \"There was that clot and everything else\u2014she was so scared, you know, so scared, and now I see her every day and try not to wonder if she'll wake up even though I know she can't. That she won't.\"\n\n\"Maybe you should talk to someone.\"\n\n\"Dan said that too,\" I say. \"What's a shrink going to tell me that I don't already know? My mother's dead and I miss her. I'm angry at Dan for keeping her body alive so the baby he wants so badly can maybe survive. Mom would hate being trapped like she is and I can't\u2014won't\u2014forgive him for it. I can't forgive the baby either, and maybe that makes me awful, but I don't care.\"\n\n\"You really are angry. Like, I'm worried about you angry.\"\n\nI shrug and stare at the table again. Olivia knows me and she's right. I am angry. I am so angry I feel like it's all I am.\n\n\"At least I'm angry for a reason. At least I'm not running around stealing cars for fun like Caleb Harrison. I saw him at the hospital today. Twice, actually.\"\n\n\"Wow, so it is true,\" Olivia says.\n\n\"What?\"\n\n\"I heard his parents got him some emergency hearing and he got assigned community service for the thing with his dad's car,\" she says. \"You know, picking up trash and stuff. But I guess he's at the hospital instead. What did he say?\"\n\n\"Nothing,\" I say, thinking of his, What's your problem? and his stares. The second one was the worst. The way he was just looking at me and Mom, and how he must have seen me lying there, resting my head on her hand.\n\n\"Nothing? You sure?\"\n\n\"How do you know what happened to him, anyway? It's not like you'd have found out by going anywhere near a computer, so that means you talked to someone and that means...\"\n\n\"Yes, I saw Roger,\" she says, and blushes. \"But it's not what you think. I was getting gas and when I went to pay for it, he was inside getting a soda and we talked for a minute.\"\n\n\"Uh-huh. So you were getting gas.\"\n\n\"Yep.\"\n\n\"Even though you got it two days ago and you've only driven to school and back since.\"\n\n\"All right, fine,\" she says, mock-slapping my arm. \"I saw his car in the parking lot and I might have wanted to see him, and I did but it was no big deal. Okay?\"\n\n\"How long did you talk to him?\"\n\n\"Awhile.\"\n\nI grin at her. She stares at me for a moment and then grins back. \"I know! We talked! Do you think he likes me? I really want him to like me.\"\n\n\"What's not to like?\"\n\n\"The fact that most people think I'm a freak because I don't use computers or any of that stuff.\"\n\n\"Olivia, we go to school with people who steal buses. And their father's car. Oh, and that guy who always wears the same brown shirt. You're not a freak.\"\n\n\"Well, not compared to Caleb Harrison. Or Dennis and his shirt thing,\" she says. \"But neither of them have social lives and I'd like one.\"\n\n\"You have one. You talk to people in your classes. You dragged me to parties after the horrorfest that was Anthony. You went out with Pete last year. If you ever started using technology, you'd rule the school in a week.\"\n\n\"Nice try,\" she says, and grins at me. \"Roger said I have nice hair, but what does that mean? It just lies there.\"\n\n\"Olivia, you do have nice hair.\"\n\n\"It's flat.\"\n\n\"You'd like Caleb's hair,\" I say, and she blinks at me.\n\n\"What?\"\n\n\"I just\u2014it's wavy and stuff. Like how you're always saying you want yours to be.\"\n\n\"I thought you didn't talk to him.\"\n\n\"I didn't.\"\n\n\"But you noticed his hair.\"\n\n\"We were in the same room, Olivia. He was about two feet away from me. It was hard not to see him.\"\n\n\"He's cute,\" she says, and now I stare at her.\n\n\"No, not I think he's cute cute,\" she says. \"Scary druggies don't do it for me. But a lot of girls think he's hot.\"\n\n\"Not the ones in my classes!\"\n\n\"No, you all think guys like Anthony are hot. Caleb's got that whole quiet loner thing going, plus he has the cheekbone\/eye\/hair trifecta.\"\n\nWhen I stare at her she says, \"Awesome face, great eyes, amazing hair. A trifecta. What are you learning in your classes?\"\n\n\"Not that.\"\n\n\"Oh, right. How's the New Deal paper coming anyway?\"\n\n\"It's not.\"\n\nShe looks at me and then says, \"For real?\"\n\nI shrug.\n\n\"I know you haven't been buried in books like usual but I thought you wanted to go to one of those top ten schools. I thought you and Anthony were neck and neck to see who could have the best ranking and SAT score and all that stuff.\"\n\n\"Yeah, we were.\"\n\nOlivia frowns and starts to say something else, but her parents come in. They are both blond, like she is, but that's pretty much where any similarity stops. They work in IT support and their life\u2014their world, in fact\u2014is computers. I have never seen one of them without something that isn't electronic in one hand. It reminds me of how Olivia and I started the whole hanging out on my roof thing.\n\nA few years ago, they gave her some sort of \"does everything and can organize everything\" gadget for her birthday and she came over, climbed up the trellis on the side of our house onto our roof, knocked on my window (and scared the crap out of me), and when I came out onto the roof, she cried and we talked. And then I threw her gift off the roof.\n\nMom calmed Olivia's parents down, then calmed Olivia down, and then gave her a birthday gift from \"me and Emma\" and put foot rungs on the trellis so she could get up onto the roof easier. Since then, it was something we did once in a while for fun, but since Mom died, it's the only way she comes to see me.\n\nHer mother closes the door while typing out a message on some impossibly tiny thing, never looking up. Her father, who entered first, is using a strange-looking square, holding it in one hand and touching it with a plastic stylus, frowning as images flicker in and out.\n\n\"This cube isn't maximizing its storage capability or its potential speed,\" he says. \"It feels like more of a design idea than an actual product.\"\n\nOlivia rolls her eyes at me, gets up and gets two energy drinks out of the fridge. \"Hi, Mom. Hi, Dad.\"\n\n\"How are you?\" her dad says as her mother smoothes a hand over Olivia's shoulder. \"You have a good day?\"\n\n\"Yeah,\" Olivia says, and her Mom's device starts to beep.\n\n\"Have you eaten?\" her mother says, and Olivia rolls her eyes again.\n\n\"Yes. You?\"\n\nHer mother nods, and Olivia looks at her dad, who flushes. \"I'm going to,\" he says. \"But the cube came in and I wanted to see it. I'll eat later.\"\n\n\"Something without caffeine or the word Energy! in it?\"\n\nHer dad grins at her. \"Yes. And hey, we have a little more work to do, but then we're going to watch a movie.\"\n\n\"By yourselves?\" Olivia says. \"Without anything electronic in hand? Will you be all right?\"\n\n\"You,\" Olivia's mother says, and kisses her cheek. \"Want to join us?\" She looks at me. \"How about you, Emma? You in for a movie?\"\n\nI shake my head. Olivia's parents drive her crazy and they aren't around that much but they're here, truly here, even if it's not the way Olivia wishes they were, and I'm like a kid with her face pressed against the window, all the things I want and can't have right in front of me.\n\nI spy a family.\n\nI miss Mom so much.\n\n\"I'm going to hang out with Emma,\" Olivia says. \"She can spend the night, right?\"\n\n\"Sure,\" Olivia's dad says. \"Is it okay with Dan?\"\n\nI nod.\n\n\"Do you need anything?\" Olivia's mom says, and I shake my head because what I need isn't something anyone can give me. She looks at Olivia, kisses the top of her head, and then leaves, turning to the beeping gadget in her hand. Her dad grabs a package of crackers and wanders out, eating them one-handed as he starts to look at the cube again.\n\nSome people think Olivia's hatred of technology is an act, like she's pretending or whatever. But she really does hate it. It's not so much because of her parents\u2014although I think that's part of it\u2014as it's what she doesn't want her life to be. She thinks it's sad that people would rather talk without ever seeing each other.\n\n\"I just think life should be lived, you know?\" she's said to me more than once. \"And how can all the talking with a keyboard ever be like actually talking to someone? It can't. People need each other.\"\n\n\"I don't know,\" I always said. \"I think it just makes life bigger. People are closer, actually.\"\n\n\"I'd rather have an actual talk with my mom instead of having her send me messages,\" she'd say. \"Wouldn't you?\"\n\n\"Sometimes,\" I'd say, and she'd say, \"Okay, fine,\" but I get what she's saying now. Mom and I talked like everyone else does, in person and over the phone and in all the ways you can, but now that she's gone, I miss talking to her for real. Hearing her.\n\nI could call her cell phone and hear her voice mail, but it wouldn't be her. I could send her an email and get back the \"I'll be in soon!\" message she put up before she left work, but it wouldn't be her either. I would just hear and see electronic ghosts, and I already have a live one to face every day.\n\nI call Dan from Olivia's room.\n\n\"Hi,\" I say when he answers. \"I'm spending the night at Olivia's. I'll come home in the morning to get a ride to school.\"\n\n\"I don't know,\" Dan says. \"What about your homework? You left your bag in the car. Plus we still haven't talked about what happened\u2014\"\n\n\"There's nothing to say. You want a name. You pick it out.\"\n\n\"Emma, your mother would be so sad to hear you talk like this.\"\n\n\"She can't be sad though, can she?\" I say. \"She's dead. I'll see you in the morning.\"\n\nAnd then I hang up.\n\nHe doesn't call back. I know he won't. I know that despite everything he says, he knows what he's done. That he saw Mom die and made his choice.\n\nHe saw her die, and he still went ahead and decided that the baby was worth more than Mom and how scared she'd been about the pregnancy. About dying.\n\nAnd he didn't even ask me what I thought. Not about Mom. Not about the baby.\n\nNot once.\n\nHe just decided the baby was worth everything.\n10\n\nI wake up in the middle of the night and I can't fall back to sleep because I remember the day Mom came home with the official news that she was pregnant. Dan was with her and he was smiling so hard I thought his face must hurt. I'd never seen anyone so happy.\n\nMom didn't look like Dan did, and when he ran up to the nursery to get the sketches he'd been doing, she sat down at the kitchen table.\n\n\"Hey,\" I said. \"So what's it like to be knocked up?\"\n\n\"Scary,\" she said, and then bit her lip. \"I just...I'm not young like I was with you, Emma. It was easy then. I never thought about what could happen. How I might lose you.\"\n\n\"Where am I going?\"\n\n\"You know what I mean,\" she said. \"Being pregnant is risky. And it's really risky for me.\"\n\n\"Yeah, but you're disgustingly healthy. That clot didn't even slow you down even though you were supposed to rest. It's like when Dan and I got the flu. What did you get? Nothing. Not even a cough.\"\n\n\"You two were the worst,\" she said. \"Couldn't even have a fever at the same time, but what can I say? I love you.\"\n\n\"What's that?\" Dan said, coming into the room.\n\n\"The flu,\" Mom said. \"Remember?\"\n\n\"How could I forget?\" Dan said. \"Emma and I suffered, and you never even coughed.\"\n\n\"That's what I said!\" I said and Dan grinned at me. I looked over at Mom. She was staring at the kitchen table, but she wasn't looking at it. It was like she was looking at something far away.\n\n\"Mom?\"\n\n\"Hey,\" she said, blinking and looking at me. \"I spy a family.\"\n\n\"Yeah, you do. Three, soon to be four.\" I grinned at her.\n\nShe blinked again. \"I think I'd better go sit down. I don't want to take any chances.\"\n\n\"Oh honey,\" Dan said. \"You're already sitting down.\"\n\n\"I mean somewhere...I just...\" Mom trailed off.\n\n\"Everything's going to be fine,\" Dan said.\n\n\"Promise?\" Mom said, and her voice was shaking a little.\n\n\"Promise,\" Dan said and kissed her.\n\nI smiled and said, \"You guys,\" like I always had, like I thought I always would and the thing is, Mom was scared.\n\nShe was scared and I didn't see it. Not like I should have. I just thought it was the idea of the baby or the fact that she was over forty or maybe even giving birth itself, which sure didn't sound like fun to me.\n\nBut now I think Mom knew. I think that somehow, she knew that something was going to happen to us. That something was going to break our family.\n\nI grit my teeth and close my eyes. I don't want to think about this anymore.\n\nI stare at Olivia's dark ceiling and remember Caleb Harrison looking at me. Asking me what my problem was, and then staring at me and Mom and her stomach and then Dan and me.\n\nI think about what I saw in his eyes before I looked away.\n\nAnger.\n\nAnd, weirdly, understanding.\n11\n\nI get back to the house in the morning and find Dan sitting in the kitchen, hunched over a stack of papers. He's still there when I come back from showering and getting dressed.\n\n\"I need a ride,\" I say, looking at the muffins he's made, which are cooling on the counter. Chocolate chip, my favorite.\n\nI ignore my stomach's rumbling.\n\nDan looks at me.\n\n\"Do you want a muffin?\"\n\n\"No. I need a ride, like I said.\"\n\n\"I'd still pick you up after school even if you'd gone with Olivia,\" he says. \"I know how much you want to see\u2014\"\n\n\"I'll be in the car,\" I say, cutting him off. Having Dan take me to school sucks, but I want him to remember that I'm still here because after what happened to Mom, what's to stop him from deciding I'd be better off somewhere else? Maybe he really would ship me off to some boarding school or worse, Mom's parents. Not that I know they'd take me, which makes it even crappier.\n\nDan comes out in a few minutes, shuffle-walking like he's an old man.\n\n\"I need to tell you something,\" he says when we're on the road. \"It's about your mother's hospital bills.\"\n\nThat stops my worry fast, fast, fast. \"Let me guess. Someone else is paying them.\"\n\nHe blinks. \"How did you know?\"\n\n\"I saw the stack. How else could you?\"\n\n\"I\u2014well, I've been working, or trying to, but I'll never earn enough to pay for the house and everything plus your mother's care.\"\n\nI look away from him, stare out the window. \"Her care?\"\n\n\"Yes,\" he says, and I rest my head against the glass because he sounds like he means it, he really does. He really thinks that what he's done is caring. \"Luckily, some people have set up a fund. It's for the baby and your mother.\"\n\nSomething in his voice makes my stomach hurt, like it's being twisted around and then shoved up toward my throat. \"And what do they want in return?\"\n\n\"There's a court case in Florida. A woman just passed away. She was pregnant and her husband wants to try to save the baby, but her parents\u2014\"\n\n\"Let me guess, her husband wants you to talk to them,\" I say, cutting him off. \"Or are you going to talk in court about what you've done?\" I turn to stare at him, and Dan's cheeks blaze bright red.\n\n\"It's not that simple,\" he says slowly. \"He wants the baby, and her parents\u2014\"\n\n\"Fine. You should go down there and cry and say how sorry you are about Mom, how much you loved her, and how you're only trying to keep your little boy alive. Throw in something about how you know Mom would be so proud of you, covering your pain to focus on the baby.\"\n\n\"I am in pain,\" he says, his voice cracking. \"I loved her, and I'll love her forever. I understand that you don't want to hear this, but your mother wanted this baby, and I know she'd\u2014\"\n\n\"She's dead! You can't ask her what she thinks or how she feels and you never, ever did. You remember her being pregnant and happy. You don't remember how scared she was. You don't remember how things really were.\"\n\n\"I do, and\u2014\"\n\n\"She knew,\" I say. \"She knew something was going to happen. You don't remember how she looked when she had to go on bed rest. You don't remember how she'd just sit in her chair at night and hold her stomach like she knew it was going to break her. But you know what? I do. And I get to see what broke her every day. I get to see it and you want it and you'll get it and I hope...\"\n\nI trail off because Dan has pulled over, stopped on the side of the road, and is staring at me, white-faced.\n\n\"You hope what?\"\n\n\"I hope she forgives you,\" I say, but that's not what I was going to say and we both know it.\n\nDan blows out a breath and pulls back onto the road. His hands are shaking on the steering wheel. He doesn't say another word until we're at school.\n\n\"Your mother would be ashamed of you,\" he says quietly. \"Be angry at me, Emma, but don't ever be angry at your bro\u2014\"\n\nI get out of the car and slam the door shut on him. His words.\n\nHe's right, though. Mom would be shocked by what I almost said. By what I was thinking.\n\nMom was terrified of the pregnancy, but she loved Dan. She wanted to make him happy, and I know she would be sad to see how things are between us now. That she would tell me not to blame anyone, that things happen and choices are made.\n\nShe would tell me hate only destroys.\n\nI know this because she did.\n\n\"Hate almost killed me after your father died,\" she told me once, when I was nine and decided I wanted to know everything about him. \"I was so angry, Emma. Angry at your father for driving in the rain. Angry at him for not somehow knowing that there was going to be an accident. About a month afterward, I was sitting alone, just staring at nothing, and I was hit with this wave of...\" She trailed off.\n\n\"I went to his books\u2014I'd boxed them all up because I couldn't bear to see them,\" she said after a moment. \"I opened a box and got one out. I sat down with it and just started ripping the pages out. If he'd seen me, he'd have been so horrified. But he couldn't. And I thought 'Good, that's what you get for leaving me.' I missed him so much, I loved him so much, and yet I hated him for being gone.\"\n\n\"You hated him?\"\n\nShe nodded.\n\n\"Were you...were you sorry that you...?\"\n\n\"Oh, no,\" Mom said. \"Never. You were the best thing that ever happened to me. But even knowing that, I would look at you when you were little and know he'd never see all the things you were able to do. I knew that he'd want to be there. But he wasn't and I hated that and I hated him for it too, and the hate was so...it was like a pit, Emma. I couldn't ever see the bottom of it and I finally realized if I didn't stop, it would take over my whole life.\"\n\n\"You really hated him?\"\n\nMom looked at me. \"When someone you love...when they die, you want it undone. You'd do anything to have them back, and it's easy to believe that if only this had happened or that had happened, everything would be fine. And that's what makes you angry. What makes you hate. You don't want to believe that sometimes bad things happen just because they do.\"\n\n\"Mom, I'm sorry,\" I whisper now as I step into school, and I hope she hears me. That she forgives me. That she can help me find a way to untangle the knot of hate in my heart, because it's there.\n\nIt's there, and I feel it.\n\nIt's there, and I can't make it go away. I understand what she meant now about the edge and how hate can take over everything. I see it. I feel it.\n\nBut I don't know how to stop it.\n\nAnd the one person who could, the one person who'd be able to pull me back, is gone.\n12\n\nI'm in a pretty bad mood when Olivia finds me, and she takes one look at my face and wordlessly hands me a rubber band. Olivia's mom is a worrier who had a pretty messed-up childhood, and she always wears a rubber band around one wrist so that when she feels a burst of worry or a bad memory coming on, she can snap it against her skin and remind herself that she's here.\n\nI put the rubber band on and give it a good yank. It stings\u2014a lot\u2014but I don't feel better. I already know I'm here. I already know what's on my mind.\n\nAnger.\n\nI'm starting to get scared at how angry I am, though. At how, when I try to find a way out, even for a second, I can't.\n\nI snap the rubber band again as Olivia opens her locker. Still nothing. I do it again, and again, and then the band breaks, falls off my wrist and to the floor.\n\nI stare at it. Someone steps right on it, and then it's gone, trampled off down the hall.\n\nI look at my wrist. There's a red welt on it.\n\nMy mother has marks on her skin from the tubes and needles. She has to be turned and moved so her skin won't get sores.\n\n\"Okay, that clearly didn't work,\" Olivia says, grabbing my arm as she closes her locker door. \"Come on, we gotta get you to your locker before classes start.\"\n\n\"I left my books in Dan's car,\" I say, and look around as Olivia says something about finding a notebook for me to take to class.\n\nI see people walking by. Fast, slow, laughing, frowning. So normal. I hate that too.\n\nAnd then I see Caleb Harrison standing by a locker, staring at me. I see him look at my wrist, at my face, and I can't see anyone walking by anymore.\n\nHe saw me yesterday. He saw me with Mom yesterday.\n\nHe knows something's wrong with me.\n\n\"Here,\" Olivia says, sticking a notebook into my hands just as the bell rings. \"See you later.\"\n\nI nod. What happened to Mom isn't a secret, but the whole baby thing never really got much attention. I thought it would\u2014I thought it would pull in the nighttime reporters, the ones who are seen on TV everywhere\u2014but it didn't. There were a couple of things locally, sandwiched in between stories about allegations surrounding the governor, but that was it.\n\n\"Death...and life,\" they always said, like my mother and what happened to her could be boiled down into three words and a pause.\n\nSome people in my classes said they were sorry or asked how I felt, but that was right after it happened, and when I didn't break down and scream, when I kept coming to school, things went back to how they'd always been. Who was applying where, who needed what SAT score, who was going to hire someone to help them write their entrance essay and who was stressing out and how badly it would screw their grades.\n\nOn the news, there were murders and robberies and local sports heroes and possible rain showers.\n\nThat was it.\n\nMom got five days. Five days in the local news. Five days when people talked about her. When they noticed she was gone, when they asked questions, but those days passed. It was enough time for them but not for me, and it still isn't. The thirty-day mark since she died looms in front of me because if she makes it till then, if she stays dead but her body is still alive, chances are her body will hold out long enough for the baby to\u2014I can't think about this, not now. I don't want to think about it at all. I don't want to think about anything.\n\nExcept, as I head to class I wonder why Caleb Harrison was staring at me.\n\nWhat he sees that no one else seems to.\n13\n\n\"You haven't turned in your New Deal paper,\" Anthony says to me in second period, just before the teacher starts talking.\n\nI shrug. A month ago, Anthony would have been telling me his paper was longer or had more sources. Six months ago, he would have said the same thing and I would have said, \"And we'll see who gets the better grade, won't we?\" and he would have responded, \"Emma, your competitive streak worries me,\" and I'd have said, \"I know it does,\" and think we were flirting.\n\nThat's really what I thought. I thought Anthony\u2014so determined to get into the right school, to be king of everything\u2014was the guy. The boyfriend who could be mine if only I could get him to notice me.\n\nI was wrong, but I didn't know that until after I'd made the disastrous mistake of hooking up with him.\n\nIt was four months ago, I think. I used to know it to the day, even to the hour, but I don't now.\n\nWe'd been working on a project for Advanced Chemistry together, doing an experiment on chemical compounds and reactions at night, just me and him with our chemistry teacher down the hall in his office snoring so loudly we could hear him.\n\nSo there we were, writing about chemical reactions, talking about them, and Anthony said, \"I hope you realize I'm still going to get the best grade.\"\n\n\"I realize you think you will.\"\n\n\"I like your conviction,\" he said. \"That's not supposed to be a seven, by the way.\"\n\n\"It isn't a seven, it's a one. I guess I can't say I like your ability to read handwriting.\"\n\n\"I have other qualities,\" he said.\n\n\"Please,\" I said, and felt so....well, this is humiliating, but I felt so flirty. Sexy, even. Like everything around us was charged.\n\n\"You've got moxie,\" Anthony said, and only he could say something like that with a totally straight face.\n\n\"Well, jeepers,\" I said, and he grinned.\n\n\"You're lovely.\"\n\nLovely. No one had ever called me lovely. My mother said I was cute, which everyone knows is mother code for \"you look like a regular girl, sweetie,\" which is fine except regular is boring, regular gets you to seventeen without any boyfriends and a few nothing kisses at various parties.\n\n\"Lovely?\" I said.\n\n\"Everything about you is lovely,\" he said and I\u2014though I wish I hadn't now\u2014kissed him. Just like in the movies, I moved in and kissed him and he kissed me back and the experiment got ruined but I got felt up on a lab table and went home with the memory of Anthony saying, \"Well, I can't say I regret this outcome\" as he tucked in his shirt. I practically floated all the way there.\n\nThe next day, he acted like nothing had happened. When I first saw him, he was talking to a bunch of people about the need to promote student government awareness (the popular people won the elections, but Anthony was always treasurer and was always convinced everyone cared as much about how much money the yearbook got as he did) and he waved at me. Like he did to everyone else who walked by.\n\n\"He's distracted,\" Olivia said. She'd heard what had happened, of course, and after a millisecond of silence had said, \"Wow! That's...Anthony! And you! Anthony and you!\"\n\nExcept it wasn't. In English, when we got our latest papers back, he said, \"Look at this,\" and then motioned for me to show him mine.\n\nI wanted more than grade comparing. I wanted kissing. I wanted talking, and not just about who had the better grade. I wanted into his heart.\n\nStill, he acted like nothing had happened.\n\nI cried and I talked to Olivia. I told Mom, and she said, \"If he's acting like a jerk, he probably is one,\" and I told her I was telling things wrong, that it wasn't like that.\n\nShe said, \"Emma,\" like only she could, with so much love in those four letters, and that was it.\n\nAfter three days, I finally worked up the nerve to talk to him. I did it at the end of the day, went over to his locker after most everyone had left and he was talking to two freshmen, telling them that extracurriculars were important, but that they had to be the right ones.\n\n\"Anyone can sign up for drama,\" he said. \"Debate, now, that's different. That's an art form, and one colleges love. Do you like discussing issues?\"\n\n\"I don't know,\" one of the freshmen said and Anthony said, \"See, debate helps with that. You'll know things. You'll be able to talk about anything. Trust me.\"\n\n\"I mean, I don't know about public speaking,\" the freshman said and Anthony said, \"Well then, why would you want to sign up for drama? That's public speaking.\"\n\nThe freshman said, \"No, I mean...\" and then trailed off as Anthony said, \"Look, you clearly need to think about it. It's your future, though.\"\n\nHe turned to me after they left and said, \"Emma, what brings you by? Worried about the new assignment in history? I think you'll do fine.\"\n\n\"No,\" I said. \"I\u2014the other night.\"\n\n\"Oh,\" he said. \"I don't really need the extra credit now, but I can come in and help you if you can reschedule. I just can't make it on Thursday nights anymore because I'm starting to volunteer to deliver meals to old people. Did you know mileage for that is tax-deductible? Or would be, if I paid taxes.\"\n\n\"Right,\" I said, as my joy, which had already shrunk considerably, shriveled into nothing. \"You do remember that we made out, right? And you said I was...you said I was lovely.\"\n\n\"Of course,\" he said, smiling at me. \"It was a very memorable moment and you are lovely. You know I adore you, Emma. I mean, you must know that. You're intelligent enough, after all.\"\n\n\"You\u2014\" I said, but coming right on the heels of the declaring, he'd turned back to his locker and was grabbing his bag like everything was over, said and done.\n\n\"You adore me?\" I said, and he shut his locker, put an arm around me and said, \"Of course. What's not to like? You're sweet and you're quite smart. How could you not have a place in my heart? Plus you find me attractive, which is always a nice thing to know.\"\n\n\"I\u2014\"\n\n\"I can tell I've upset you. Will you walk with me?\"\n\nI did, and he said, \"I'm focused on college,\" as we stepped down the hall. Our footsteps were perfectly in sync. \"And being with someone\u2014well, at our age, can we really understand what a true relationship is? Do you feel you understand it? I know I don't.\"\n\nI'd heard Anthony say lots of things. Interesting things. Smart things. Some stupid things.\n\nBut never this kind of crap, and I stopped walking.\n\n\"I understand you said I was lovely so I'd let you feel me up,\" I said, and he stopped too and blinked at me.\n\n\"You're angry.\"\n\n\"Yes,\" I said, and then, oh, then my voice cracked and I said, \"I thought we\u2014I thought what happened meant something.\"\n\n\"It does,\" he said, taking my hand, walking again and I walked too, watching my hand in his as he talked. \"It did, and it does. But one moment doesn't mean everything. It can't. If it did, we'd be letting one act define who we are and you're bigger than that. I'm bigger than that. We both want the same things, Emma. We can talk about this more if you want, as I've always found the connection people make between physical action and emotion fascinating.\"\n\nWe were outside and I looked at him, at this guy I'd let shove his hands inside my shirt, and realized he believed everything he said and what had happened between us was...\n\nNothing.\n\nI pulled my hand free of his and said, \"See you later, Anthony.\"\n\nI was proud of that, of how I just pulled away and walked off.\n\nI didn't cry until I got to Olivia's house.\n\n\"One moment, my ass,\" Olivia said after I'd told her everything. \"Who else would ever even go near him? He should be kissing your feet and thanking you for touching his funky self.\"\n\n\"If that's true, then why did he blow me off?\"\n\n\"Because he's an ass. And he's a\u2014what do you call it? Obsessed with yourself.\"\n\n\"He's not a narcissist,\" I muttered, sniffling, and then dragged myself home, heart battered and self-confidence shot. I'd really thought he liked me, but it turned out he liked someone else more.\n\nHimself.\n\nOlivia was right. Anthony was\u2014and is\u2014a narcissist, and as time passed I saw it. All of what I thought was banter was really him worrying that I had better grades. All of him offering to help out and acting like someone out of a really old movie and speaking like it was 1850 was his way of reminding himself and everyone else that he was special.\n\nAnd the making out happened because Anthony was just a guy and I'd launched myself at him. Why would he turn that down? He was, after all, him, and to Anthony, he was pretty amazing.\n\n\"So, your paper?\" he says again to me now, and I look at him.\n\n\"You win,\" I say. \"You're the king of grades, the prince of GPAdom, the duke of whatever. Valedictoriandriandom, let's say.\"\n\n\"If you ever want to talk,\" he says, getting a handkerchief\u2014of course he would have one\u2014\"you know I'm here.\"\n\nI take the hankie. \"What should we talk about?\" I look at him and I know he sees my haunted eyes. My empty, furious smile.\n\nI know he sees that I don't give a crap about my grades, about school, about all the stuff that used to matter so much to me.\n\nHe blinks at me, opens his mouth, then closes it. I hand the hankie back.\n\nHe doesn't say another word to me.\n\nI look at him and try to remember that person I was not so long ago, but she seems so far away. She seems gone.\n14\n\nCaleb Harrison stares at me again at lunch.\n\nI don't see him at first. I'm sitting with Olivia, picking at the fried rice in front of me, which is basically rice and limp broccoli, and then I...I don't know.\n\nI feel it. Him.\n\nI look up and there, across the cafeteria, in a corner by himself, is Caleb Harrison. And he's looking at me.\n\nI elbow Olivia who says, \"Oof!\" and then follows my eyes.\n\n\"Wait, you said nothing happened yesterday.\"\n\n\"Nothing did.\"\n\n\"But he's\u2014oh, never mind. He's looking out the window now. Remember how popular he was in middle school? And then he was all freaky drug guy and then stealing cars guy and now he's really screwed-up loner guy who puts cars into lakes. Scary.\"\n\nNo, I think. Sad.\n\nCaleb Harrison is sad.\n\nI don't know why he is, or how I know it. I just do.\n\nI look at him.\n\nHe's looking at me again and I feel it, actually feel him looking at me, like from all the way across the cafeteria he's somehow able to see into me. That there's something in me he wants to see.\n\nI take a sip of my soda and he's watching. I am hyperaware of it, of him, of how I suck a drop that's clung to my lower lip off, pulling it into my mouth for a moment and how I open my lips a little to breathe, because it's strangely airless in here and I want to ask him why but when I picture that, my walking up to him and saying, \"Yeah, hi, I think you have a problem,\" he's standing up too and moving in, so close his hair brushes my face, so close his mouth is right next to mine and we're touching without saying a word, just standing there, close enough to kiss with his hands cupped around my waist, one sliding up, the other down and my hands are doing the same and he's breathing into me and I'm breathing into him and\u2014\n\n\"Hey,\" Olivia says, waving a hand in front of my face. \"Bell rang. What are you doing?\"\n\n\"Nothing.\" At least, that's what I wish I'd been doing. Thinking.\n\n\"Your face is red.\"\n\n\"It's hot in here.\" He's gone now. Thank goodness.\n\n\"Not really. You sure you're okay?\"\n\nI nod and get up. I tell her that I'm fine. I go to class. I meet Dan after school. I go to the hospital.\n\nI am not okay. I am not thinking about just Mom, like I have been for days, weeks.\n\nI am thinking about something else. Someone else. Someone I don't even know and I pictured\u2014\n\n\"Are you all right?\" Dan says as we're heading to where Mom is. \"Your face is a little red.\"\n\n\"It's from all the sex I'm having at school instead of going to class. It's tiring, but way more fun than trying to conjugate the past perfect tense of 'I see my dead mother every day.'\"\n\n\"Emma!\" Dan looks shocked and a little scared.\n\n\"Oh, relax. I won't do it in your bed when you're in Florida ruining someone else's life. When is that, by the way? I'll need to stock up on condoms.\"\n\n\"I've been worried and now I'm really concerned. I think you need help.\"\n\n\"Really? Well, talk it over with Mom and see how she feels about it. Oh, wait, she can't talk. You were concerned about her too, right? So maybe you can see why I'm not all that interested in your so-called concern.\"\n\n\"Emma,\" he says, sputters really, but I ignore him and head into the waiting room.\n\nWhere Caleb Harrison is standing with a magazine cart, having just heard everything I said.\n15\n\nCaleb sits down when I come in, sprawls on a chair with the cart in front of him, moving it back and forth with his feet. He doesn't look at me but then he doesn't need to, does he?\n\nWe both know what he just heard.\n\nI know my face is red\u2014it's so hot I can feel it\u2014and I brace for another stare or a snarl like yesterday or something even worse, something crude that will make me feel just as bad as Dan sounded. (And looked too, but I don't trust Dan anymore. I wish I'd never trusted him.)\n\n\"Your mom's dead, right?\" is what he says instead and it's so not what I expected and so blunt, so not covered in all the softness of I'm sorry and terrible to hear that now I just stare at him.\n\n\"I couldn't really tell yesterday,\" he says. \"But that's what I figured. I remember hearing something about it. About how there's a ba\u2014\"\n\n\"She's dead,\" I say before he can finish his sentence. \"But she's pregnant. So they're keeping her breathing and feeding her and everything else until she's twenty-five weeks along.\"\n\n\"They can do that? I mean, if she's dead\u2014\"\n\n\"Yes,\" I say, the word bitten off, sour and angry. \"My stepfather wanted...it's what he wants. So it's happened. You can be brain-dead and kept alive on machines. It just doesn't usually happen when there's...there's no hope for her.\"\n\nHe pushes the cart with his foot again. I find looking at it easier than looking at him, but when I dare a glance he's looking at the floor.\n\n\"How many weeks is she now?\" he says after a moment where the only noise is the cart squeaking.\n\nI swallow. \"Almost sixteen.\" A little past twelve that final morning, the morning she went to get toast and fell down and didn't get back up. A little past twelve weeks pregnant the morning she died and that was it, should have been it except it wasn't, isn't.\n\n\"So she'll just...\" He trails off. The cart squeaks again. He's frowning, but not like he's angry. Like he's thinking.\n\nIf he's thinking about my mother\u2014just my mother\u2014that would make, as far as I can tell, two of us who do that. Who've really done that.\n\n\"Yeah,\" I say, my voice tight. \"If she makes it to thirty days the way she is now she'll just lie there, dead, for another ten weeks. Then Dan gets what he wants.\"\n\n\"Dan?\"\n\n\"The guy in the hall. My stepfather.\"\n\n\"Oh,\" he says, and pushes the cart again. \"That's pretty screwed up.\"\n\nI sit down then, not facing him but not turned away either. High school teaches you lots of stuff, but one of the most important things is that you don't ever act like what someone's said has gotten to you, even if it has.\n\nThat's pretty screwed up.\n\nYeah, it is. It really is, and I know Olivia gets it, and I love her for it, but no one else has said it. No one. Sorry is all apologies but it isn't what this is about. No one's said the truth, the raw wound of what happened. What is happening.\n\nNot until now, when I'm sitting here with Caleb Harrison who takes drugs and steals cars and who gets that what's happened to the family I used to have has exploded into something huge and very, very screwed up.\n\nI miss Mom all over again then, wish she was here, that I could know I'd be able to walk into the house and it would be home. That I'd see her. The real her. I'd see her smile, push her hair back. Rub her forehead if she's had a bad day and asks for an aspirin, kicking her shoes off. She'd always ask what was for dinner, kissing my forehead before she'd turn to Dan and say, \"Well, Chef, what's on tap?\"\n\n\"You don't talk much,\" Caleb says.\n\nI look at him, and I see that understanding in his eyes again. I don't get it, and I don't know if I like it. It's scarier than pity or Dan's beseeching stares. Pity I get. Dan wanting me to want what he does\u2014it's Dan, so I get that too. But what do Caleb Harrison and I have in common? What could he understand about this? His parents are both alive and apparently dedicated to keeping him out of jail in spite of everything he does.\n\nIf there's one thing I learned from Anthony, it's the power of questions. I swallow, and hope my face isn't still as red as I'm afraid it is. \"Why were you staring at me today?\"\n\n\"Because I saw you with your mom yesterday.\" A magazine starts to fall off the cart and he reaches up, pushes it back in place. \"Do you miss her?\"\n\nI look at the ceiling. My eyes are burning. I wish I knew how to make things better. Or even bearable. I'd settle for that. For just being able to breathe without feeling like it hurts.\n\n\"I get that. Missing someone, I mean,\" Caleb says, his voice quiet.\n\nHe knows and now I don't get it, I don't get it at all, and when I look at him I can't see how he understands. I just see him, so blond and pretty, so safe from everything he does because he has his parents. He has them doing everything for him.\n\n\"Oh, so you know what it's like to have a dead mother being kept alive so her so-called husband can get what he wants more than anything else?\"\n\n\"No, but I know what it's like to live with a dead person.\"\n\n\"Your parents are both still...\" I say and trail off as I remember that there is a dead person in Caleb's life.\n\nHis sister.\n\nAnd then we just look at each other and I don't care that he's screwed up and gorgeous. I care that someone really does get what's going on. Sees it.\n\nAnd I think he feels the same. Although I don't fall into the gorgeous category.\n\n\"I\u2014\" I say, and then Dan comes in and looks at Caleb, then at me. \"Emma, you can go in now.\"\n\nI get up, glancing at Caleb.\n\nDan sees it. \"I'll just talk to your friend....\" he says and Caleb snorts, gets up and pulls the magazine cart past both of us, out into the hall.\n\n\"Who is he?\" Dan says. \"I'm not sure how to ask this, but\u2014?\"\n\n\"Then don't, because even if there was something to tell, I wouldn't want to say it to you.\"\n\nHe looks at me, hurt all over his face, and I wish I could believe in it. In him.\n\nBut I don't, and I go see Mom. I sit with her. I hold her hand.\n\nIt's not enough today. I don't just want to sit here with her hand lying still in mine. I wish I could curl up next to her. That I could lie with my head on her shoulder. But I can't because she's in a hospital bed and it's filled with taped tubes and wires and IVs.\n\nIt's filled with her stomach, with what's in it, and when I look at her face I can't see her, I just see the tubes going into her mouth and down her throat and up her nose and that's all she is now. That's what she is.\n\n\"Mom,\" I whisper, and pull away, wrap my arms around myself. I can't talk to her. I can't tell stories about school, about anything. I can't pretend that life is normal except for this. Not now. Not today. I can't find her under what's in front of me.\n\nI can't see anything of Mom. I just see a dead woman.\n\nI pinch the bridge of my nose, hard, over and over until I know I won't cry.\n\nWhen Dan comes to get me he tries to put a hand on my shoulder. I move away from it, from him.\n\n\"Emma, please,\" he says, his voice breaking, and I stare at him. He flinches and I know it's from the hate in my eyes.\n\nHe turns to Mom and I walk out, leave the hospital and go wait by the car. I see an ambulance come in, sirens wailing, before Dan comes out and gets in the car. I wonder what will happen to the person inside. I hope they are all right.\n\nI don't see Caleb. I'm not exactly looking for him, but I am thinking about him. About what he said, about living with a dead person.\n16\n\nDuring the ride home, I think about Caleb.\n\nI think about his sister, Minnie.\n\nShe died three years ago, when Caleb and I were both fourteen. She was seven, I think, or maybe only six. I know she was riding her bicycle and fell off.\n\nThat was it. That was how she died. That simply. There she was, on her bike, and then she was gone. Like getting up and going to get toast. Just a moment and then...gone.\n\nI guess, now that I think about it, that's when Caleb started doing drugs. He must have really loved his little sister. Maybe everything Caleb's done is how he gets through it.\n\nWhat would it be like, to live with that loss for three years? I can barely stand it, and it's been less than a month. How could anything be left inside you at all?\n\n\"Emma,\" Dan says, and I notice the car's stopped, that we're outside the house. I get out before he can say anything else, head up to my room and lock the door.\n\nHe knocks, but I pretend not to hear him. It gets easier every day. I go over to the window and open it.\n\nI know eventually Olivia will come in.\n17\n\nShe does, a while later, and frowns as soon as she sees me.\n\n\"What happened?\"\n\n\"The usual.\"\n\n\"Uh-huh. You saw Caleb, didn't you?\"\n\n\"Yeah.\"\n\n\"Did he say something?\"\n\nI shrug.\n\n\"Emma!\"\n\n\"Not really. Sort of.\"\n\n\"Sort of? Like he sort of spoke? Or he sort of said something that upset you?\"\n\n\"No, he didn't\u2014I just saw him again and he knew about Mom. He did see me with her yesterday.\"\n\n\"Oh.\" Even Olivia hasn't seen that. \"What did he say?\"\n\n\"That it was screwed up.\"\n\n\"Emma\u2014\"\n\nI shake my head. \"Do you remember his sister?\"\n\n\"His\u2014oh, yeah,\" Olivia says. \"Minnie. Fell off her bike and didn't have her helmet on, right? My parents were all, 'That's why we made you wear a helmet, Olivia.'\" She shakes her head. \"Remember how awful it was? I wanted a pink one, like yours, and had that weird-shaped silver one.\"\n\nI nod, and she shakes her head. \"Parents.\" She blushes, looks at me. \"Sorry.\"\n\n\"Don't. You can't be weird on me. For real, not now, okay?\"\n\n\"It was bad today, wasn't it?\"\n\n\"Yeah,\" I say and she hugs me and says, \"I'm sorry. I'm so sorry,\" and I cry like I always seem to be doing. I cry and when I'm done my eyes hurt and I have a headache and nothing's changed. You'd think I'd have learned that by now, that tears don't change anything.\n\nYou'd think I'd have run out of them.\n\nBut I haven't.\n18\n\nMom did it. Well, she didn't. Her body did it, because today is day thirty.\n\nThirty days of Mom being dead but kept alive and at the start of week sixteen it was okay, it was in the distance.\n\nThat distance is closed today.\n\nI have missed school or a weekend or maybe both. I don't know. It doesn't matter.\n\nOlivia came by a few times, but aside from, \"Emma, eat. EAT,\" I can't remember another thing she's said. I've just felt empty inside, silent, and the one time Dan knocked on my door and said, \"Emma?\" I said, \"Do you miss her at all?\" and listened to the drone of his voice, words buzzing over me.\n\nI know I can't stay in my room forever. The thing about Mom dying is that the world didn't stop. It didn't even slow down. It's flowers and cards and everyone understands but no one does because Mom wasn't Mom to them. Without her it's like I'm living inside a mirror. I see things, I do things, but they are just surfaces and nothing more.\n\nI'm numb, so numb, because thirty days is here but when I'm in the shower staring at the water, I wonder if Caleb feels like that about Minnie. Is loss this constant pain, not mental, but actual pain? It's like even my teeth hurt, but there's a fog over it, one that makes the pain hurt and yet leaves me carved out too.\n\nIs grief this forever wishing for what was even though I know I shouldn't?\n\nI shake my head, water splashing everywhere.\n\nMom loved to take baths. She actually took one the night before she died. I sat on the floor next to her tub, smelling the \"calming\" bath beads, oily bubbles, she'd put in. She'd put her hair up, stuck it in a half-knot that was already falling down and pushed the bubbles around as we sat there.\n\n\"You don't have to start on that paper now,\" she said. \"You can relax. Watch some TV. Hang out with your old pregnant mother.\"\n\n\"But\u2014\"\n\n\"You don't have to try so hard,\" she said, reaching one hand out and touching mine, her skin hot and slick. \"I love you no matter what.\"\n\n\"I know,\" I said and she squeezed my hand.\n\n\"So then let yourself breathe once in a while. Smell the roses. Go for it. Other inspiring things I can't remember now and so on.\" She grinned at me. \"You do know what I mean, right?\"\n\n\"Dan's going to come up and make you get out of the tub soon.\"\n\n\"You two.\" She sighed. \"Am I allowed to brush my hair?\"\n\n\"I don't know. Dan's got the pregnancy book. I'm going to read it after I get half of my New Deal paper done.\"\n\n\"Don't work on it tonight, okay? Watch Covert Ops with me.\"\n\nI didn't. I worked on my homework and started pulling together ideas for my paper, blew a kiss at Mom when she stuck her head in and said, \"Okay, I guess that's a no on the TV. Don't stay up too late.\"\n\n\"I won't,\" I said, half glancing at her as I read about FDR and his plans.\n\nHer hair was down and dry. The lamp in my room cast a shadow on her face.\n\nThat was the last time I ever saw her.\n\nThirty days ago today, and I turn off the shower and watch the water run down the drain, circling, circling until it's gone.\n\nI get dressed, putting on clothes from the pile scattered around my bed. I realize, for the first time, that the rest of my room is boxed off to me. I could pick up the books piled on my desk. The earrings on my dresser I'd bought shopping with Mom the week before she took that last bath.\n\nBut I can't.\n\nI unlock my door, the reminder of putting in the lock\u2014of Mom\u2014almost bringing me to my knees and then I go downstairs.\n\nIt's early\u2014I woke up before the sun rose and watched it come up, light tearing up the dark, and just thought thirty thirty thirty thirty. Thirty days. Mom's gone but still here and that's how it will be for another\u2014\n\nI can't. I just can't think it. I walk into the kitchen and Dan is there, looking like he hasn't slept. He's reading a book and I see it's the pregnancy book, the one he was reading the night before Mom died. The one I was going to read.\n\nHe sees me and puts the book down, pages up so I can't see the cover, the serene portrait of a woman holding her belly. A woman, alive.\n\n\"Do you want some breakfast?\" he says. \"I could make something.\"\n\n\"It's thirty days today,\" I say and he nods, his eyes filling with tears. I wish I could take all of my own back now. I don't want to be anything like Dan.\n\n\"I thought that might be you\u2014well, why you were in your room,\" he says. \"I asked one of the doctors at the hospital about it. She says she'd like to meet you. Talk to you. She thought it was interesting that you didn't come with me to see your mother during this period.\"\n\nInteresting? Really? All because I sat in my room and thought about her. The real her, not the one I have to see.\n\nNot that one who breaks me over and over again.\n\n\"I'd like a waffle,\" I say, and Dan looks at me, surprised, and then smiles, huge, bright, and gets up, moving around the kitchen. He gets out bowls and boxes, eggs and spoons, and plugs in the waffle iron I bought for him two Christmases ago.\n\n\"Here,\" he says a few minutes later and I look up from the table\u2014where did Mom touch it before she fell? I've never asked. Was it here? Over there? Did she not touch it at all, just walk straight to the toaster? Was Dan looking when she fell?\n\nWhen did he know what he would do? After the ambulance came? After the doctor said she was gone?\n\nOr did he know all along? Deep in his heart, had he made his choice the moment two years of drugs and testing and waiting brought Mom home to tell him what he so longed to hear?\n\nHe puts a waffle on the table. I hold on to the chair in front of me. I smell flour and eggs and milk and chocolate, which Dan always puts in waffles for me.\n\nMom would say, \"Dan, you shouldn't spoil her,\" and Dan would say, \"Chocolate isn't spoiling. Love isn't spoiling.\"\n\n\"You went shopping.\"\n\nHe nods. \"We didn't have much to eat.\"\n\nNo, we didn't. Food goes bad. It spoils.\n\nI swallow.\n\n\"I'm not hungry.\"\n\n\"But you said\u2014Emma, I can hear your stomach!\"\n\nDan is standing with his hands pressed together like he's praying and in the silence of the kitchen I hear a slight gurgle, a churning. It is my stomach, awake and moving.\n\nInside my dead mother there is a baby. It needs to eat. I suppose it has a stomach. I don't know. I don't want to think about it.\n\nThirty thirty thirty thirty.\n\n\"I'm going to the car,\" I say.\n\n\"You have to stop this,\" Dan says. \"She wouldn't want you to punish yourself like this. It's not your fault that she died.\"\n\n\"I know it's not my fault. Are you going to take me to school?\"\n\n\"Emma, I lost her too. We both lost her. I miss\u2014\"\n\nI leave then. I walk outside, past the car and down the driveway. I cannot be here. I cannot see him, not now. I can't say anything else and I'm afraid that if I do he will find a way to make it so I can't see her. That he will\u2014he could do anything to me because he's my guardian now. He could even have me committed.\n\nI feel sick.\n\nMom wanted him to be my guardian. I wanted him to be my guardian too.\n\nWe both thought he was good at taking care of things.\n\nAnd then she died.\n\nI hear the car, hear Dan. \"Emma, get in.\"\n\nI look at the car. \"Are you going to send me away?\"\n\n\"What?\" There's silence for a moment and then he sighs, a battered sound, and then I hear the horn beep over and over, the sound muffled, start-stop-start-stop.\n\nI look over and see he is hitting it. Dan is smacking his hands into the horn, face red, wet with tears.\n\nHe stops. \"I would never send you away,\" he says. \"You're my family.\"\n\nThirty thirty thirty thirty.\n\nI don't believe him, but I want to be here. I have to be here. Mom doesn't need me like I want her to, but I am all she has.\n\nI get in the car.\n\n\"Emma, please know I wanted your mother and you, I wanted our family. That hasn't changed a bit. I wish that we were all\u2014\"\n\n\"Yes,\" I say so he won't say anything more. He does, he's wishing and wishing, but I don't listen.\n\nI don't believe in wishes anymore.\n19\n\nCaleb isn't in school. I notice before first period and again at lunch. I don't say anything to Olivia, who saw Roger at the mall on Saturday and tells me all about it. I listen to her story of a fast-beating heart and flushed face and hesitant conversation like I am an old woman.\n\nHeartbeat. That's what's keeping Mom here. I try to not think about it, but I can't because she doesn't have one.\n\nHeartbeat. It's just a word. A word.\n\nIt's more than that.\n\n\"Hey,\" Olivia says, waving a hand in front of my face. \"You're just sitting there nodding,\" she says. \"This is really huge for me. He said it was great to see me. Not good, not fun. Great.\" She starts to say something else but then stops and says, \"What's going on?\"\n\n\"It's been thirty days since she died,\" I mutter.\n\n\"Thirty...oh,\" Olivia says. \"So then\u2014\"\n\n\"Yeah,\" I say. \"The doctors said if she made it through thirty days, the odds were stronger that she'd make it long enough for them to\u2014so that eventually they can go in and get the baby out.\" I wipe my stupid welling eyes. \"I'm sorry. I know everything with Roger is huge, I do. Tell me again.\"\n\nShe shakes her head. \"It sounded stupid when I was saying it. Who cares if he said it was great to see me?\"\n\n\"Great is better than good. Or fun.\"\n\n\"I could walk by his locker after lunch and see if he says anything.\" Olivia plays with a button on her shirt. \"Should I? No, wait, I can't do that. What if he doesn't say anything?\"\n\nI look at her, the one person who I know loves me, and think about Mom in the bath that last night.\n\n\"I think you should go for it.\"\n\n\"Really?\"\n\nI nod.\n\nWe go and he's there and he says, \"Hey, Olivia,\" and she says, \"Hey,\" back and squeezes my hand so hard it hurts after we've walked away, but it's a good hurt. Her smile makes me smile and it gets me through the rest of the day and to the hospital, up to the floor where Mom lies.\n\nTo Caleb and his cart passing us in the hall on the way to her, to me sitting in a chair in the waiting room.\n\nI realize I am waiting\u2014hoping\u2014to see him as I stand up and walk out into the hall. He's at the door as I get there, and we both stop and look at each other.\n\n\"I was waiting for you,\" I say, and immediately wish I could take it back. It's true, but I shouldn't have just said it like that. Or at all.\n\n\"Oh,\" he says, and looks at the cart. \"I\u2014uh\u2014\" His hair falls in his face and whatever it was before, those moments where I looked at him and thought he knows, they seem far away now. He's just a guy standing there, made awkward by a strange girl with a round face that doesn't have the curves and grace of his.\n\n\"Not like that,\" I say, because it isn't like that, or even if it was (just a little\u2014okay, a lot), I know it can't be. I know what goes together and it is not me and someone who steals cars and gets high. It's not me and someone who looks like him. It's not me and anyone like him.\n\n\"Not like what?\"\n\n\"Like whatever is normal for you.\"\n\n\"Nothing is ever normal in any hospital,\" he says.\n\n\"You've been to more than one?\"\n\n\"Five. Three for Minnie, two for me. Stomach pump and a dislocated shoulder, and every single hospital had the same color walls and the same awful lights. Not normal at all.\"\n\nThirty, I think, and swallow.\n\n\"You weren't in school,\" I say, to stop what I'm thinking. \"Are you okay?\"\n\nHe looks at me and then pushes his hair back with one hand. His eyes are wide and very green, even under the hospital lights. He looks so surprised.\n\n\"Emma?\" Dan says, and it's time for me to go, to see Mom. I leave without looking back because I'm embarrassed and just...I don't know. I shouldn't have said anything to Caleb.\n\n\"Hi, Mom,\" I say when I'm sitting next to her. I don't say thirty. I don't say anything about that at all.\n\n\"Today was okay,\" I say instead, and tell her about Olivia and Roger. I look at her still face. I know there is nothing to see, that who she was is gone but she's here. She's still here, right here.\n\nExcept she isn't.\n\nI should know this by now, but grief is slippery, a tangle of thorns that dig in so deep you don't know where they stop and you start. You don't know where you are.\n\nI think of Caleb and how he looked when I asked where he was. So surprised, like no one would ever wonder where he was. How he was.\n\nHe has thorns too, and I wonder if he knows where his end and he begins.\n20\n\nI see Caleb before first period the next morning. I'm alone because Olivia is at the orthodontist making sure her braces-free teeth are still straight. She got her braces off before high school started but I know she's afraid that she'll end up with them again. Her father had them twice, and so did her mother.\n\nI am at her locker, though, because she wanted me to grab one of her notebooks and give it to her later. I haven't picked up the notebook. I'm staring at a picture of me and Olivia from last summer, the two of us sitting in my backyard laughing at something I don't remember now.\n\nMom took the picture. I remember that.\n\n\"You missed the bell.\"\n\nI blink and look over at Caleb. He's looking at the picture, and then he looks at me.\n\n\"That was taken before, right?\" he says, and how can he tell?\n\n\"You're smiling,\" he says, like I've asked him.\n\nI stare at him.\n\nI stare at him because he's right. I can't remember the last time I smiled for real. I turn away, shut the locker on the picture, the me that was, and then glance at him. His head is bowed a little, all that blond hair falling over his face, part of the trifecta Olivia told me about.\n\nI see it, him, but I see something else too. His hair is something for him to hide behind. My own bangs are down past my nose now and I haven't gotten them trimmed. I don't push them back. I let them swing forward, shadow my sight.\n\nI'm hiding too.\n\n\"Aren't you going to first period?\" I say.\n\nHe shrugs. \"Study hall, so no. You?\"\n\n\"I\u2014\" I have Advanced French, or the class you take before French Literature, where you read French novels in French. I have taken French since the sixth grade and done well, beyond well, but now I just sit in class, the words washing over me, familiar but faraway sounding, like the echo of a song you can't quite remember. \"No.\"\n\nHe pushes his hair back with one hand then, those green eyes on me, and I think of the lake. Of how I'd heard he sat and watched the car sink. I am sure, suddenly, that he did. \"Yesterday I had...there was court stuff. Judge follow-up.\"\n\n\"Oh.\" I think of thorns, of grief growing deep. Becoming you. \"Do you miss her?\"\n\nHe looks away, and doesn't say a word.\n\nI know what that kind of silence means, and clear my throat. \"So what happened?\"\n\n\"To Min\u2014?\" he says, and then stops. \"Wait. You mean to me, right?\"\n\nI nod.\n\nHe heads toward the gym, the far end of the school, and stops after a second, looking back at me.\n\nI am walking toward him, and then I am walking with him.\n\n\"You've probably heard it already,\" he says. \"Drugs, theft, suck camp, my dad's car, and now community service at the hospital.\"\n\n\"For your dad's car?\"\n\nHe walks over to the vending machines. Our feet hit the floor at the same time with every step. I like the sound it makes. \"Yeah.\"\n\nI look at him. I want to ask about suck camp, which I'm guessing was the \"tough love\" place his parents sent him to, the one that brought him home all cleaned up (mostly) until now. Until his dad's car.\n\nBut it's his story to tell, and I know what it's like to have to live with one. How hard it can be to think about, much less talk about.\n\n\"You have a dollar?\" I say and he grins\u2014I see the flash of his teeth\u2014and digs into his pockets, pulling one out.\n\n\"You like cotton candy?\" I say and he nods, glancing at me. His hair has fallen into his face again.\n\nI put the dollar into the machine and press the button.\n\n\"Me too,\" I say as it falls. When I pull the bag out, it's silver and light, with a pink puff picture on it.\n\nHe pulls out another dollar and puts it in the machine.\n\n\"I didn't mean\u2014\"\n\n\"I know,\" he says, and buys another bag. He pulls it open right away, sweet smell all around us, sugar and fake color.\n\n\"I remember you,\" I say slowly, and I do, bits of things half-buried. Elementary school, and the beginning of middle school; a boy with blond curls and a big smile and huge green eyes that looked like they had a bit of the sun in them, strange but somehow so pretty that you noticed them. \"You talked a lot. You had\u2014you were\u2014\"\n\n\"I had friends,\" he says, smiling a little.\n\nI nod, opening my own bag. The cotton candy isn't sticky. It pulls apart in soft little clumps. \"And you used to chase Amy Gray all over the playground.\"\n\nHe laughs. Amy is still pretty chaseable, only she's been caught by a six-foot-tall volleyball player, Monica, and has been since we were freshmen.\n\n\"You used to try to trade your sandwiches,\" he says and I'd forgotten that, how when I was little Mom always bought the wrong kind of bread, the kind of stuff you see old people pick up at the supermarket and sigh when it thuds into their carts. Dan had slowly managed to get it out of the house after he and Mom started dating and I remember hugging him the first time I ate a piece of toast that didn't weigh half a pound.\n\n\"What?\" Caleb says, and I shake my head.\n\n\"I just\u2014my mom used to get that bread, you know? And I'd forgotten how much I hated it and how she would never let me try something else until\u2014\" That had been Dan too. He'd said that I should at least be allowed to try another kind of bread for a week. \"It can't hurt her, Lisa,\" he'd said. \"There's enough stuff in that bread she's been eating to keep her heart and the rest of her healthy until she's eighty.\"\n\nI swallow and hear the bag crumple in my hand.\n\n\"So what happened to you?\" I say again, and Caleb tosses his empty bag at the trash can. It misses and lands on the ground.\n\n\"I killed my sister,\" he says.\n21\n\n\"You\u2014she fell off her bike,\" I say, startled by the flatness of his voice. The sureness. How can he think Minnie falling off her bike was his fault?\n\nHow can he think her death was his fault?\n\n\"I was with her. Watching her. I had to do that a lot. Minnie was\u2014she didn't like any of the nannies we had.\"\n\n\"You had a nanny? But you were\u2014\" I break off, blushing.\n\n\"Yeah, I was too old,\" he says. \"I hated it. And Minnie knew it so she hated them too and she had 'impulse problems.' You know how there's a moment when you think about something before you do it? Minnie never had that. She'd just...anyway. She wanted to go bike riding and the last nanny had quit so we went to the park. My parents were always on me and her to make sure she wore a helmet but she hated it and I didn't want to deal with trying to get her to wear it so I didn't even bother trying. We got to the park and I was bored out of my mind sitting there, watching her ride around. Pissed off, you know? And then there was a car and I saw her...she flew off the bicycle and landed on the sidewalk. Her bike went in the lake and she just\u2014she just lay there. People were screaming and all I could do was look at her. If she'd had a helmet on, she'd have lived. Broken bones, sure. But she'd have lived.\"\n\nI don't know what to say. I look down at my crumpled bag of cotton candy. I pick up one of the bits I pulled free. The first bite is what I remember\u2014the pure, almost-crisp taste of sugar followed by a hint of bitter, like there's so much sweet my mouth can't process it all. It hurts, a little, to swallow. \"If I skip second period, will I get detention?\"\n\nHe nods.\n\n\"I can't get detention. The...you know. Seeing my mom. I guess you can't get it either. You have to be at the hospital or else you'll get in trouble, right? With the judge or whatever and your parents?\"\n\n\"How come you haven't said anything about me killing Millie?\" His voice is so quiet.\n\n\"Because you didn't. She was hit by a car. You couldn't have stopped that.\"\n\n\"If I'd made her wear her helmet...\"\n\n\"She knew she was supposed to wear one. You can't take the blame for that.\"\n\n\"Maybe.\" He swallows. \"My parents think I should.\"\n\n\"My mother thought having a baby would make my family better,\" I say, and I hear how my voice is shaking, dig my fingers into my palms. \"She was wrong. Parents can be wrong.\"\n\nThe bell rings then, and both of us jump a little. Both of us look at each other.\n\n\"See you, Emma,\" Caleb finally says, and then disappears down a hallway before I can say anything in reply.\n\nI don't see him at the hospital that afternoon, and Dan spends a long time talking to the doctor. I whisper to Mom about the bread and how I'd forgotten about it.\n\n\"'I met\u2014there's someone who understands,\" I say. I haven't talked to her about a guy since the Anthony nightmare.\n\n\"It's Caleb Harrison,\" I say, and everyone would have something to say about that. Even Dan tried to say something that first time.\n\nBut she doesn't reply. Her stomach doesn't even move.\n\nWhen it's time to go, Dan drives back to the house quickly, silently, and doesn't say anything to me when we get there, just gets out and goes inside.\n\nBad news about the baby, I bet.\n\nI thought I'd be happy about it, but I'm\u2014\n\nI'm not.\n\nI'm not happy. I'm not sad. I'm just...here. Numb.\n\nI go up to my room and open my window. There's a note taped to it, Olivia's scrawled Emma on the front.\n\nAnd on my roof is Caleb.\n22\n\n\"Caleb?\" I say, like he's not sitting right there, like I can't see him sitting with his legs crossed, his hair blowing a little in the wind.\n\n\"Hey,\" he says. \"I didn't see you at the hospital this afternoon. I got stuck working on another floor. And I...I heard you and Olivia talking about the whole roof thing a while ago. Not that I meant to or anything, but I\u2014\" He breaks off, and although the sun is setting, I swear Caleb Harrison, car thief, is blushing.\n\n\"Anyway,\" he says, \"I just came over. I don't know why.\"\n\nHe is blushing. Who would have thought Caleb Harrison could blush?\n\nI look at him, and then climb out onto the roof. I've been out here with Olivia, and it's not so different with Caleb. I sit down next to him and a strand of his hair blows across my face, making my skin prickle.\n\nIt might be a little different.\n\nI open the note from Olivia. She has pink stationery with little dots on it that she uses for important messages, and there are matching envelopes too. Those only get used for very special messages.\n\nThis message is in one of those envelopes, and apparently she's with Roger, who asked her out when she was leaving school. They're going to watch TV at his house.\n\nThis is something, right? she's written, and then Call me later!\n\nCall me later! is underlined three times and I know she must be happy and nervous because of how hard her pen has dug into the paper.\n\n\"So what's up with you getting letters on your window?\" Caleb says as I'm putting the note away. \"Nobody writes anymore.\"\n\n\"Olivia does. You know about her technology thing, right?\"\n\n\"Yeah, she was in my English class last year. Brought in her typewriter when we were supposed to be looking up stuff on library computers and asked about encyclopedias. I thought Mrs. Grimwood would have a fit. Is it true she doesn't even talk on the phone?\"\n\n\"No, she will, but it has to be a landline. You know, like a phone in a house or a store or something.\"\n\n\"And she writes letters.\"\n\n\"Yeah, but only for important stuff.\"\n\n\"Like what?\"\n\n\"Can't tell you.\"\n\nHe grins at me and it's a wide, beautiful thing. My heart does this weird stutter-skip in my chest and Olivia really wasn't kidding about what he looks like.\n\n\"So, where were you at the hospital?\" I say because it is the only thing I can think of, because the wind and his smile and today have left me feeling like everything inside me is visible and I don't know if I want to be seen.\n\n\"Fourth,\" he says. \"Lots of people waiting for surgery. Foot and gallbladders, mostly. I don't even know what a gallbladder is.\"\n\n\"It helps with digestion, but you don't have to have one, especially if it gets infected. Or if you get gallstones.\"\n\n\"Oh,\" Caleb says, leaning forward a little and looking at the ground. His hair falls over his face again. \"You really are smart. I mean, I knew you were because of all the classes you're in, but still.\"\n\n\"What good is knowing about a gallbladder? It's not\u2014I spent so much time worrying about stuff like that, stuff that doesn't...\" I look down at the ground now. In the dark, it's hard to see. \"Stuff that doesn't matter.\"\n\n\"Everything matters,\" Caleb says, and I look at him. He's looking up at the sky now, at the clouds, dark against the night sky, and the tiny bits of starlight that shine through them. \"Or at least that's what I was told at suck camp.\"\n\n\"That's the thing your parents sent you to?\"\n\n\"Yeah, after school ended last year they shipped me off to The Helping Center. I spent the summer walking up hills and camping\u2014which sucked\u2014and being told that everything matters.\"\n\nHe doesn't sound like he believes it. \"Does it?\"\n\nHe's silent for a moment and then leans back against my roof, still looking up at the faint light of the stars. \"No. If everything matters, then nothing can because it's all the same. And if it's all the same, then nothing is ever special. And shouldn't there be\u2014aren't there\u2014special things? People?\"\n\n\"Yes,\" I say, looking up at the sky too and thinking of Mom.\n\n\"Yeah, so that part of camp sucked too,\" he says with a short, sad little laugh. He sits up. \"Sorry. It's dumb.\"\n\n\"Why?\"\n\nHe glances at me and even though I can't see him that clearly I am sure I've somehow startled him again.\n\n\"Just because,\" he finally says. \"I already know what I did and what my parents think of me. But I guess they just wanted to be sure I knew. That I accepted it.\"\n\n\"But Minnie didn't die because you\u2014\"\n\n\"I gotta go,\" he says, standing up, and I look at his back.\n\n\"You don't accept it, do you?\" I say. \"You know it isn't true. That's why you stole your dad's car. Because you won't do what they want. Won't be what they want.\"\n\nHe freezes, and then looks over his shoulder at me.\n\n\"I was angry,\" he finally says. \"I've said I was sorry but that isn't enough for them. It'll never be enough.\"\n\nI stand up and walk over to him, sure-footed after years of me and Olivia hanging out up here. \"I told Dan not to do what he's done. Mom wouldn't want it. He said I was too upset to think straight and I didn't know what she wanted. He still says that. And I...\" I swallow. \"I loved him, you know? But he just wants the baby. And so now we're waiting, and I know he wants me to want what he does. But I can't. I won't.\"\n\nCaleb turns around. He's a little taller than I am, tall enough that I have to look up to see him.\n\n\"It's lonely,\" he says, not a question at all, and when I nod, he hugs me.\n\nAnd I stand there on my roof, in the dark, being held by Caleb Harrison. That's all he does. Olivia is around, always, and she's here for me, but she doesn't get it. Not like Caleb does.\n\nIn the dark, with his arms around me, I don't feel so empty.\n23\n\nI almost forget to call Olivia but remember as I'm brushing my teeth and getting ready for bed.\n\n\"Sorry,\" I say when she picks up the phone.\n\n\"For what?\" She sounds really happy.\n\n\"I should have called earlier.\" I should have done it right after Caleb left but I didn't. I just sat on the roof for a while, thinking.\n\nThinking about him.\n\n\"I wasn't home earlier,\" she says, and then laughs. \"Emma, he's so amazing! We ended up talking for hours. Did you know his middle name is Thomas? And he hates candy corn too! And he kisses\u2014\"\n\n\"Wait, wait, wait. Kisses?\"\n\n\"Yep,\" she says, and I can hear her smile. \"He kissed me! I actually wished I could have gotten in touch with you right then\u2014well, not right then, but right after\u2014because wow! Amazing! It was the best night of my life.\"\n\n\"So, are you two together?\"\n\n\"He wants to go out this weekend. I told him I'd think about it.\"\n\n\"Really?\"\n\n\"No.\" She laughs. \"I said 'Yes!' practically before he finished asking me. Was that dumb? It was dumb, wasn't it?\"\n\n\"Did he take it back?\"\n\n\"No.\"\n\n\"Then how dumb can it be?\"\n\n\"It's just\u2014I'm really happy and then I'm really nervous and then I'm happy and\u2014you get it,\" she says. \"I'm never going to be able to sleep tonight. Maybe I should do something about my hair.\"\n\n\"Olivia,\" I say. Olivia likes trying new stuff. Bright green eyeliner. Putting a tiny fake tattoo of a star near her mouth. Wearing fake eyelashes. But once in a while, she'll do something extreme, and it's always with her hair.\n\nAnd she always regrets it. Last year, she decided she wanted bangs and cut them herself. The result was beyond bad and took months to grow out. And the year before that, she decided she wanted to have black streaks in her hair but ended up with black spots.\n\n\"Nothing like before,\" she says. \"I was just thinking about curling it.\"\n\n\"With what?\"\n\n\"You can buy stuff to make it curly.\"\n\n\"Like a curling iron, or stuff in a box that comes with a list of instructions?\"\n\n\"Instructions.\"\n\n\"No way. Remember the black dye? You don't want something like that to happen again, do you?\"\n\n\"No, but I want to look...I want him to see me and think 'Wow.'\"\n\n\"Then don't put anything in your hair.\"\n\n\"It's straight.\"\n\n\"It's pretty and you're totally at the nervous end of things right now, right?\"\n\n\"Maybe. I just\u2014you know last year there was Pete, and it ended up being nothing and I like Roger so much more and\u2014\"\n\n\"Pete was a jerk. He was a total party boyfriend, super nice when he was drunk and wanted to get you alone but a troll the rest of the time.\"\n\nShe laughs. \"He was a total party boyfriend. And jerk. Okay, you're right, I'm just nervous. Roger isn't Pete. Right?\"\n\n\"Right.\"\n\n\"Okay, so I am going to be completely calm about this now. Or at least change the subject. How was the hospital?\"\n\n\"The usual.\"\n\n\"The usual?\"\n\n\"Well, Dan seemed a little upset on the way home, but\u2014well, you know. He does his thing. I do mine. And we both wait.\"\n\n\"You didn't see Caleb at the hospital?\"\n\n\"Nope.\"\n\n\"Something happened,\" she says. \"I can tell.\"\n\n\"I talked to Caleb at school. So, what happened at the orthodontist?\"\n\n\"My teeth are fine, but nice try. What did you talk about?\"\n\n\"Cotton candy.\"\n\n\"You and Caleb Harrison talked about cotton candy? Was it like something he ate when he got high? Is he doing stuff again? You know even the hard-core, have-to-get-high-in-the-bathroom-before-class fiends don't talk to him anymore, right?\"\n\n\"He wasn't high, Olivia.\"\n\n\"So you talked to him about cotton candy. Anything else?\"\n\nI think of him standing on my roof. Of the things he said about where he's been. What his parents thought he did.\n\nOf how he knows what being alone is like.\n\n\"Yeah.\"\n\n\"Emma, look, that 'yeah' sounded like it was a lot. I think you need to come out with Roger and me on Friday. He knows a lot of great guys.\"\n\n\"Olivia, you can't take me along on your date and I don't need one anyway.\"\n\n\"Look, I know things are awful. I think of your mom all the time too,\" she whispers. \"But I don't want you to forget about everything else, Emma. How about I talk to one of Roger's friends and see if he\u2014\"\n\n\"No. I can't think about guys like that now.\"\n\n\"But you're talking to Caleb Harrison.\"\n\n\"It's not...it's different.\"\n\n\"Okay,\" she says, and I really can't think about it, about dating and all the things Olivia is doing.\n\nBut I am thinking about Caleb. About how he is nothing like everyone says he is. About how I stood on the roof with him tonight and felt more at peace, more alive, than I have in ages.\n24\n\nDan is downstairs in the kitchen in the morning, like always, only today he isn't cooking or eating. He's sitting at the kitchen table, looking at something.\n\nLooking at a picture of Mom.\n\nI see it. I can't not see it, and I stop when I do, frozen by Mom's face. Her smile.\n\nIt's not a pregnant picture or anything. Just a picture of her.\n\n\"Do you remember this?\" Dan says.\n\n\"Yes,\" I say, because I do.\n\nThe photo is from her last birthday, when Dan and I surprised her with a dinner-and-gift-celebration. I made a banner that said Happy Birthday, and Remember to Thank Your Banner Maker, and when she saw it, she laughed so hard she cried. The picture is of her shaking her head, wiping her eyes with a huge smile on her face.\n\nI watch him smooth a finger over Mom's face, gently brushing over her smile.\n\nI want to touch the photo. I want to hold it, I want to look at Mom up close, see her as she was. But I don't want to ask Dan for it. I don't even want to think about him looking at it, or why he is.\n\nSo I just say, \"I'm ready to go now,\" and start to head for the door.\n\nDan clears his throat. \"She was happy, right?\"\n\nI stare at him. I want to hate him and part of me does. Part of me hates him so much I can hardly breathe because of it, but another part of me remembers the guy who came into her life, into our lives, and made things better.\n\n\"Yeah,\" I say. \"She was.\"\n\nHe must hear the break in my voice, the stress on was, the anger and grief I can't hide and won't because Dan did this, Dan is why we are here, like this, right now.\n\nHe must hear it because he looks at me.\n\n\"You and I. What's happened?\" he says. \"I look at this picture and all I can think about is how things were. How I've lost you too. Emma, I miss\u2014\"\n\n\"I really need to get to school,\" I say to stop him from talking more.\n\n\"Will you at least tell me how to make it right?\"\n\n\"You can't. It's too late.\"\n\n\"It's never too late.\"\n\n\"Yeah, it is,\" I say, thinking of Mom, of how there is no chance for her, how there's never been a chance for her, and still she lies in that hospital because of him.\n\nI walk outside and get into the car.\n\nAfter a moment, Dan comes out. And yet another day begins.\n\n\"Do you want the photo?\" he says when we get to school, and Dan knows me. He knows how much I love Mom and of course he saw what was written on my face when I saw it.\n\nOf course he knows how much I miss her.\n\n\"No,\" I say, and get out of the car.\n\nIn school, I don't look for Olivia. I can't handle seeing anyone, even her. I just walk around, step one two three four through all the halls, and end up by the vending machines. There are people waiting in line to buy caffeinated water and juice, our school's \"healthy\" answer to soda.\n\nThere are people buying candy bars too\u2014a couple of years ago, they put in fruit instead, but sales dropped so low the candy came back.\n\nI see a few people I know from class, and I see them glance at me and then turn away. I'm like Sara Walton was two years ago, the girl who would and could and did until she had a nervous breakdown.\n\nSara and I both went from being the person to beat to being the story of how you could\u2014can\u2014screw up, and I used to think Sara should have been stronger, that she should have focused on what was important: on getting the right grades, on doing everything possible to get into the absolute best school.\n\nNow I get how she felt. How you see everything you've done and think, Why? Why bother? Why\u2014 I feel the tears welling and dig around in my bag, find some change and hold it tightly in my fist, the coins digging into my palm.\n\nThere is no why. That's what Sara figured out, and that's what I know now.\n\nThings just happen and sometimes you wake up one morning and go to school and your mother dies.\n\nI buy a bag of cotton candy when I'm finally in front of the vending machine.\n\n\"Emma?\"\n\n\"Hey, Olivia,\" I say, and slide the candy out and into my bag, move aside so the next person can get what they're waiting for.\n\n\"Where have you been?\"\n\n\"Walking around.\"\n\n\"Around school? Why? Are you looking for Caleb?\"\n\n\"Olivia, no. I'm just\u2014this morning wasn't so great, okay? Dan had a picture of Mom and started talking about her and then said he felt like he'd lost me too, and I just couldn't handle it.\"\n\n\"Oh.\" The bell rings and she looks at me.\n\nShe looks at me and I see pity. I know she loves me, but she also feels sorry for me and I don't want that, not now.\n\n\"I gotta go to class,\" I say and smile at her before I walk off. I turn around at the end of the hall and see her watching me. I wave to show I'm okay, and she waves back and then disappears.\n\nThe halls empty fast and there I am. Here I am.\n\nI am Emma, seventeen. I live with my stepfather. My mother is dead. Inside her is a baby.\n\nI close my eyes. I do not want to think about this. Not about Mom or Dan's choice. Not any of it, but it is what I am\u2014all I am\u2014and how can I get away from myself?\n\nI go outside. Just like that, I leave school. It's easier than I thought it would be. The part of me that tried so hard, that wanted all those A's and worked so hard\u2014that part is truly gone.\n\nBut I've still never left school unless Mom\u2014or Dan, after he and Mom got married\u2014had to take me to the doctor or the dentist or I got sick. I've never just walked out. It seems so strange that you can do it.\n\nBut you can.\n\nOutside it is quiet, the school a low hum behind me, voices from classrooms carrying, bleeding out through the walls, out into the air around me, but softly, softly.\n\nIt's sunny. I don't remember when I stopped noticing stuff like that.\n\nI look up at the sky. There's the sun, bright and yellow. Clouds, white and fluffy.\n\nIt shouldn't be like this. The sun shouldn't be shining. There should be no weather, there should be nothing, but that's the worst thing about Mom dying. Nothing else stopped.\n\nOnly she did.\n\n\"What are you doing out here?\"\n\nCaleb. I look away from the sky and see him standing nearby, looking at me.\n\n\"I don't know. What are you doing out here?\"\n\nHe runs a hand through his hair, sends it sprawling over his face. \"I saw you come out and...I don't know. I guess I know the look on your face. I've just never seen it on someone else.\"\n\nI look at what I can see of his face. \"How does it look?\"\n\n\"Bad.\"\n\nI think of the people from my classes that I saw before, and how they looked at me and turned away. I think of Olivia and the pity in her eyes.\n\n\"It's all bullshit anyway,\" he says, gesturing back at the school, and then takes a step toward me, shoving his hands into his pockets. His hair parts enough that I can see his eyes.\n\nHe is looking at me, right at me. He sees me, and there is no pity in his eyes. He just sees me.\n\n\"You want to get out of here?\" he says.\n\n\"Yeah,\" I say. \"I do.\"\n25\n\nCaleb has a car, which surprises me. He sees my face and laughs, not at me but\u2014well, it sounds like he's laughing at himself. And it's not a happy sound.\n\n\"By special permission,\" he says with a flourish. \"My parents are too busy to drive me around, so I got to keep my license. It pays to be a lawyer who knows everyone around here.\"\n\nMaybe I should be nervous about getting into a car with him, but I'm not. As weird as it might sound, I trust Caleb Harrison. He's not\u2014who he was isn't who he is now.\n\nI get that.\n\nHe drives us to the rich part of town, the part where all the houses are practically piled on top of each other, separated by slivers of immaculate lawn to pretend there is space. They all look alike too, massive brick structures that stretch up and out and line streets with names like Park Avenue Lane.\n\nHe drives slowly, carefully, and I know which house is his the moment we turn onto Royal Crescent Court. It's huge\u2014the largest house on the street\u2014but it is also the loneliest-looking house I've seen besides the one Dan and I share. It's like the house itself knows something is missing. That someone is gone.\n\nHe pulls into the driveway, turns the car off, and then looks at me. \"Inside is\u2014there's stuff. My parents really loved Minnie.\"\n\nI think of Dan, holding the picture this morning.\n\n\"Lots of photos?\"\n\n\"Sort of,\" Caleb says, and his voice is strained.\n\nHe hesitates by the front door. I can't see his face through his hair and, on impulse, reach out to move it back a little. To see his face.\n\nIt's soft, not stiff with stuff like Anthony's or slick like Olivia's with the cream she rubs into hers every morning to make sure it doesn't frizz.\n\nIt's just hair, I know, but touching it sends a shock through me, a spark that makes my fingers tingle, sends my thoughts scattering, and when Caleb looks at me, I see he feels it too, whatever it is. I pull my hand away.\n\nMy fingers still tingle.\n\nThen Caleb opens the door.\n26\n\nI have seen cemeteries. I should have seen one many times by now, over the past thirty-one days. I thought I would see Mom there, that I'd sit by a stone with her name carved on it and wish she was here.\n\nInstead, everything went sideways, upside down, strange, and I have never ever seen anything like Caleb's house.\n\nAt first it looks ordinary. There are rooms. There is furniture. There are rugs. In the hall where we stand is a place to put umbrellas and a coatrack.\n\nThere is also a painting of Minnie. It hangs on a wall directly in front of us. It is as tall as I am, maybe taller. It is all that is on the wall.\n\nMinnie, grinning, one hand raised like she's saying hello. Asking you to come closer.\n\nThere's more. In every room Caleb and I walk through\u2014living room, dining room, kitchen, four other rooms that are filled with furniture but that I don't have a name for, there is Minnie. She is in little-girl things turned into art, a pair of sneakers resting on top of a table in a glass box, a set of barrettes mounted into a frame and put on an end table. She is in drawings, yellowed and curling at the edges all over the fridge, her name scrawled in the corner.\n\nShe is in paintings too. In every room, there is a painting. None as large as the first one, but they are the first thing your eye is drawn to. Among the furniture and the rugs and the things that make a room, a house, she is there, a girl in a picture, smiling.\n\nShe is blond and tiny, and behind her are grass and trees and bright, bright sky.\n\nShe is always smiling.\n\nWe stop in a room that I think is maybe another living room. A sitting room, maybe? A sofa and some chairs are placed by a fireplace with tiny pink ballet slippers on the mantel. A few photos sit on an end table. The windows look out onto what I guess is Caleb's backyard, a bit of grass and then a room of his neighbor's house. The curtains are green and match the rug. The sofa is soft-looking, the chairs wide.\n\nIt is like being inside a tomb. Everything about the room, the house, is about loss. It screams it without words. It scares me.\n\nI reach out blindly and touch Caleb's arm. I move my hand down, touch it to his.\n\nHis fingers are cold and I look at him. He is staring at the portrait of Minnie and I see what people at school see when they look at me. I see why they turn away.\n\nI see what grief does, how it strips you bare, shows you all the things you don't want to know. That loss doesn't end, that there isn't a moment where you are done, when you can neatly put it away and move on.\n\nIt never leaves you.\n\nI look at the portrait. I don't remember Minnie that well, but I don't remember her as blond and tiny.\n\n\"I thought Minnie's hair was darker.\"\n\n\"It was,\" Caleb says, and his hand, in mine, is shaking a little. \"And she was...my mother used to worry about her weight. She put Minnie on diets. The first one was when she was four. I remember that because Minnie cried. She didn't do that a lot. But she cried whenever Mom told her she was fat.\"\n\nHe looks at me. \"But they wanted to remember her like she was....\" His voice trails off, and I see he is looking at the end table. I walk over to it, his hand still in mine, and pick up one of the photos.\n\nThere are three people in it. The moment I see it, I can tell Caleb has his father's eyes and his mother's hair. They are standing together, smiling stiffly, and in front of them is a little girl\u2014blond and tiny, just like the paintings \u2014who is holding her arms out for someone. I see a hand reaching toward her, the edges of fingers, and one single blond curl.\n\n\"When she was little,\" he says, \"she was beautiful. Everyone said so. People used to stop my parents and tell them how pretty Minnie was. Then she got older and people didn't say it.\"\n\nHe takes the photo out of my hand, moving away from me, our hands no longer joined, and puts it back down. \"I wish...\"\n\nHe falls silent and stays that way, looking at the photo.\n\nI look at his back. At the way his shoulders are hunched.\n\nI look at him.\n\n\"You're in it. The picture.\"\n\nHe looks over his shoulder at me.\n\n\"I was.\"\n\n\"Your parents took you out of it?\"\n\n\"No, I did. My mother cried about not having any photos after the funeral, and I wanted her to have some. But we only ever took family photos on vacations. They didn't\u2014there weren't any pictures of just Minnie.\"\n\n\"So you made some.\"\n\n\"Yeah,\" he says and turns away, looks back at the photo again. \"I made some.\"\n\n\"Are there any of you?\"\n\n\"No. I'm here. They have to look at me every day. That's more than enough for them.\"\n\nI think of him taking his dad's car. Of how he watched it sink into the lake. The lake near where his sister died.\n\nI think about what he said to me last night.\n\nIt's lonely.\n\nHe didn't just lose his sister. He lost everything. He's alone.\n\nHe's been alone for years and he's still here. He carries ghosts and blame that shouldn't be. He got twisted all around but never broke.\n\nI walk over to him.\n\n\"Caleb,\" I say, and he looks at me.\n\nI take his hand and lead him out of the room. Away from the pictures, the portraits, the pieces of Minnie's life lying around.\n\nI don't know if I should take him out of the house or not. I can't believe he lives here, in this giant, furnished tomb. I can't believe I never wondered why he did the things he did, why I just accepted that he was a loser and never thought about it. About him.\n\n\"How do you do it?\"\n\nHe's silent for a moment, looking at the floor, and then he looks at me.\n\n\"I don't know.\"\n\n\"You don't know?\" My voice is shaky, cracking.\n\nI want an answer. I want something to help me understand why Mom died and why she's still here. I want to know how to fix things. Caleb has survived loss. He lives with grief all around him, in him. He must know how he did it. How he does it.\n\nHe lets go of my hand. \"I'm not\u2014Emma, I drove my dad's car into the lake. I don't want to be in this house, I don't want this to be how things are. But I want a lot of things that can't happen, that won't happen, and that's what I know.\"\n\nIt's not what I want him to say, but it's the truth. I hear it in his voice. I know it in my heart.\n\nUnder the idea that we can all make our fates, that we have choices, is the reminder that sometimes we don't. That sometimes life is bigger than our plans. Bigger than us.\n\n\"Okay,\" I say.\n\n\"Okay?\" he says and I smile because he isn't okay and I'm not okay, but what he's saying is okay. Is right. Sometimes you want things that can't happen. That won't happen. And it's just how things are.\n\n\"There need to be more words,\" I say. \"Like a way to get that okay isn't possible, but it is okay.\"\n\n\"I don't think there can be a word for that. Do you...do you maybe want something to eat?\"\n\nI look around the hall, back toward the rooms we've just been in. This quiet, strange, sad monument to a girl that exists only in her parents' mind. \"Here?\"\n\n\"No. Well, sort of.\"\n\n\"Sort of?\"\n\n\"Not in here,\" he says and looks back into the house too. \"I don't\u2014I've never brought anyone here before. You can see why, I guess.\"\n\n\"Yeah,\" I say, because I do, and we walk outside.\n27\n\nWe eat lunch in his room.\n\nHis room isn't in the house.\n\nIt's above the garage, a small white room with a window that has no curtains. There are no rugs on the floor. No soft chairs. There is a bed, a small TV, a bathroom and, tucked into the corner, a tiny kitchen.\n\n\"The nannies used to live here,\" he says, going over to a small fridge. \"And then my mom took a lot of pills after Minnie died. I started taking them too and then I found out if I took them and drank...\" He looks over at me. \"You know about this.\"\n\n\"I heard things.\"\n\n\"Right,\" he says, pulling out a pizza box and shutting the fridge. \"So there were pills and other stuff, and one night I lit my room on fire.\"\n\n\"On fire?\" I hadn't heard about that.\n\nHe nods. \"My mom was up\u2014she doesn't sleep much\u2014and she put it out. She, uh\u2014when I woke up she was just staring at me. I started to say I was sorry and stuff and she just kept staring at me.\"\n\n\"She didn't say anything?\"\n\nHe looks at the pizza box. \"Not until I stopped talking. Then she said, 'Haven't you done enough?' I moved over here after that.\"\n\n\"Is that why you stopped...you know.\"\n\n\"What, getting high? You can say it. You can't get high just from saying the word.\" He blows out a breath. \"Sorry. I...um...it's not why I stopped. The first car I stole, I\u2014I hit someone. A guy. He had this little backpack with him. He was going to see his kid, who was at her first sleepover and had forgotten some stuff. I got out of the car and he was just lying there and that backpack was lying there and I thought he was dead. That I'd done what...\"\n\nHe trails off and shifts so his hair falls forward, covering his face.\n\n\"The guy didn't die,\" he says after a moment. \"I didn't believe it, even though he sat up after a second and started screaming at me about his leg and what I'd done. I can still see him right before he sat up and started yelling, you know? Minnie...\" He shakes his head. \"That's when I stopped. Why I stopped.\"\n\n\"You hit someone with a car?\" I say, and I know he just said it, but I didn't\u2014I knew he'd done stuff, but I didn't know he'd done something like that.\n\nI knew he understood grief. I didn't know that it had spilled out of him and into the world.\n\nI didn't know he had hurt people other than himself.\n\nHe nods. \"And I\u2014well, you know I kept stealing cars after that. My parents kept as much of it as quiet as they could\u2014they paid off the guy I hit and managed to cut deals for the other cars up until the bus thing.\"\n\nMy throat feels tight, gummy. \"Why?\"\n\n\"Because my dad wants to be a judge and he had a pretty good chance of becoming one until the bus. I did that on a Tuesday, during school, and it caused a lot of problems.\"\n\n\"I remember,\" I say, thinking back. We'd all had to wait an extra hour to leave school, even the people with cars, because the bus had to be brought back, and even then the school board was afraid to use it until a group of mechanics looked it over. They ended up readjusting all the school bus routes so the kids on the bus Caleb stole could be put onto others and still get home. \"You got sent away after that, right?\"\n\n\"Yeah. I've never told anyone about the guy. Not even the therapists at suck camp. I...anyway, my parents have plenty of reasons to hate me that aren't about Minnie.\" He pushes his hair off his face and blinks hard, looks toward the door. \"You probably want to go back to school now.\"\n\nPart of me wants to. Caleb Harrison understands me, but he isn't\u2014he truly can't help me. I thought I got that before, but I didn't.\n\nI do now.\n\nCaleb understands me because he's still hurting too. I thought he was a victim, that he lashed out because his parents blamed him for his sister's death. And they did, and he did\u2014he took all their hate and tore himself apart over it\u2014but he isn't innocent. He did things, awful things.\n\nBut I look at him, standing in this tiny white room, and he is the loneliest person I have ever seen. He has nothing. No one.\n\nAnd I know how that feels.\n\nI get up and walk over to him. And then, like he did last night, I put my arms around him.\n\nHe stiffens for a second and then he hugs me back, his arms wrapping tight around me. I feel him shaking and rest my head on his shoulder.\n\n\"Emma,\" he says, just a whisper, and that's all he says. But I hear it, and I don't let go.\n28\n\nWe drive back to school the same way we left, in silence, but his voice catches me as I'm getting out of the car.\n\n\"Emma, I just want you to know that what I did, I wish it hadn't happened. That's the worst part. Because the liv\u2014\"\n\n\"The living with it is forever,\" I say, and when he looks at me, I see that he will always have ghosts.\n\nBut I also see that he has carried them by himself and that he knows what he's done, who he is, and he is trying to live with it. With himself, and with the world he has.\n\nI see that he is beautiful. Not just outside, but inside, under the mistakes and anger and grief, is a heart that beats pure and true. That loves his sister and his parents and wishes for things that can't ever be.\n\nHe wants things to be whole, but knows they can't. He sees that, and doesn't look away.\n\n\"I'm going to fail all my classes this term,\" I tell him. \"I won't do the things I thought I would, won't go to the college I used to want to. It all seems so stupid now that my mother's gone.\" I swallow. \"The night before she died, I blew her off to do homework. I thought I'd always...I thought there would be other times. But there aren't, and that's the worst. That's the part I hate most.\"\n\nI look at him, and I don't wonder if he'll understand. I know grades and the plans I'd made, the way I tried to build a future that would be glowing, perfect, is something he's never thought about. He never had the time. The chance.\n\nBut I know he understands about seeing all the things you lost, the moments you thought you'd have, the ones you were so sure of. I know he gets that there is a moment you never saw, a moment that just came, and that it's unmade and remade you. That you can't be who you used to be once it's happened.\n\nHe leans toward me and touches my hair, my nonbeautiful, noncurly hair, and looks at me. At my face, which has no trifecta, which is just a face and nothing more until he touches it, one finger sliding down my cheek to my jaw.\n\nAnd then, for a moment, I do feel different. Not beautiful, not something as simple as that.\n\nI feel special. I feel like there is the world, and then there is Caleb and me. Just us, a broken little party of two but it isn't so bad, being broken. Not with someone who understands. Not with him.\n\n\"I didn't expect you,\" he says, and then he presses his forehead against mine, his eyes closed. \"I don't know what to do.\"\n\nI look at him and I understand, but part of me wants there to be more. Wants to be special beyond the understanding we share.\n\nPart of me just wants him to want me.\n\n\"I should go,\" I say because I have felt want and it didn't do me any favors and Mom is dead and Caleb isn't going to save me and I don't know if I should trust myself because I used to believe that to have the perfect life, I just needed to study and work hard.\n\nAnd look where that got me.\n\nI get out of the car and head into school. I don't look back, but I spend what's left of the day hiding in one of the girls' bathrooms and thinking about him.\n\nIt's actually a relief when the final bell rings and I go out and find Dan. I don't understand this part of me, the thing inside me that has let me think about something other than Mom. I know that I am still living but it hasn't\u2014\n\nIt hasn't been like this.\n\nAt the hospital, I sit in the cafeteria, a fruit cup smelling of canned pineapple in front of me.\n\nI see Caleb's cart after Dan comes to get me, as I'm walking to Mom.\n\nI stop. I pull the package of cotton candy out of my bag, the one from this morning, put it on the cart and then I go see her.\n\nWhen I sit down, I tell her everything.\n\n\"Are you mad?\" I say, and I don't think she would be, but she would be worried. She wanted me to be safe, to be with a guy like Dan or my father.\n\nMy father, who died, and Dan.\n\nDan, who did this.\n\n\"Would you\u2014?\" My voice breaks, and I hold her hand tight. Will her to somehow hear me. To show me a sign. Something, anything.\n\nShe's silent, but her belly ripples. I swallow and close my eyes.\n29\n\nI see Caleb the next day, as I'm standing by Olivia's locker before first period.\n\n\"So you were really quiet last night,\" she's saying and I nod because I was. Olivia came over and talked about Roger and asked about Mom, and then she painted her fingernails with magic markers I bought to do a project I know I signed up for but remember nothing about.\n\nShe's still talking, and I see Caleb and I know he sees me and I like that I know that, that I see him seeing me, and then he is coming over.\n\nI am watching him walk toward me and we have talked, I have told him things I haven't even told Olivia, but we haven't really talked at school.\n\nWe haven't talked where everyone can see us.\n\n\"Hey,\" he says and his hair is in his face again. I tilt my own head back so I can see him a little better and catch his grin, slight and embarrassed, and he pushes his hair back, those cheekbones a little red. \"Thanks for...you know.\"\n\nI think he means yesterday but he could also mean the cotton candy or both, but I feel people looking at us and I know I have to say something. \"Sure.\"\n\nHis grin grows wider but his eyes get serious, and now I know what he means. He means yesterday. His house, his life. The truths he told.\n\nHow I stayed after I heard them.\n\nHe moves a little, his hair starting to fall forward, and I reach out and touch one side of it, the curls sliding over my fingers.\n\nWe both still and I feel the same jolt I did yesterday, only now it's stronger.\n\nIt's stronger and it's confusing but I don't want to turn away.\n\nHe stares at me and I am staring back and he leans in a little, leans in toward me, and I could touch his face, I could slide my hand so it would rest on the line of his jaw, like he touched me before. I could\u2014\n\n\"Emma?\" Olivia says, and she sounds beyond surprised.\n\nCaleb shakes his head, like he's surfacing from underwater, and his hair falls over his face as my hand slides away.\n\n\"See you later?\" he says.\n\nI nod.\n\n\"What was that?\" Olivia says after he goes.\n\n\"Talking.\"\n\n\"Uh-huh. You touched his hair! What's going on? Are you and Caleb hanging out now?\"\n\nI think of him asking me if I wanted to go yesterday. Of how he shook when I held him. Of how he'd whispered my name.\n\n\"Yeah, we are.\"\n\n\"You sure you want to?\" Olivia says, and when I don't say anything, she sighs. \"I know everything with your mom is terrible, but Caleb? He steals cars!\"\n\n\"There's more to him that that. He gets everything that's going on.\"\n\n\"And I don't.\"\n\n\"You do, but not like he does. He gets how it feels. Everything with the baby, it's\u2014\"\n\n\"He. You know that the baby is a boy, but you don't say that. Why?\"\n\nI don't say, because then it's not just about what Dan did. How he never even asked me, just made his choice. \"He, he, he. There. Better?\"\n\n\"No because it's like you don't think about anything but what Dan did and well\u2014it's weird how you don't talk about how you're going to have a brother, you know? And now Caleb Harrison?\"\n\n\"You don't even know how well I know all about it. About him. It's all about him, and you don't have to sit in the hospital and look at your dead mother every day.\"\n\n\"Okay, okay, sorry,\" Olivia says, her mouth trembling. \"I just...I don't know. I want to make things better for you.\"\n\nI put my arm through hers. \"I know you do.\"\n\nBut the thing is, she can't. Not like she wants to.\n\nNo one can make things better, but Caleb...Caleb gets it in a way no one else does.\n30\n\nThe afternoon drive to the hospital is the same as always, but the visit isn't. There is a doctor waiting for us when we get to the floor where Mom is, and I see Dan stiffen.\n\nThere is only one reason a doctor waits to see you.\n\n\"Mom?\" I say, my voice breaking. \"What's wrong with Mom?\"\n\nI know she's gone, I do, but I've gotten used to seeing her, to touching her even if she can't feel it, and I suddenly don't want what I've been so sure I do. I don't want to hear that she's gone forever and ever, that the machines stopped holding on to her, that she slipped away and all that's left is the ground and the goodbyes all over again but without her there, without me seeing her every day.\n\n\"No,\" Dan says, \"oh no, no, no,\" and I realize the doctor is talking, that he's saying, \"Lisa's holding steady,\" and then I know it's the baby.\n\nMom's belly, her swollen, fluttering belly, and what's inside, what Dan wants, what he's done, and the doctor is still talking but I'm not listening because Mom is still here, I can still see her. She's here, right here.\n\nI start to walk toward her. I want to see her now, I have to see her now, but a nurse touches my arm and steers me behind Dan, and the doctor is still talking, saying, \"fetal distress\" and \"fluid levels\" and Dan is crying.\n\n\"What's going to happen?\" he says, and all the talk afterward drifts over me because I forgot what I realized the day I was in Caleb's room, when I learned he wasn't going to save me, that he couldn't make all my pain go away.\n\nUnder the idea that we can all make our fates, that we have choices, is the reminder that sometimes we don't. That sometimes life is bigger than our plans. Bigger than us.\n\nI look at Dan, who is still crying, and I reach over and touch his hand. He grabs my fingers and holds on, tight.\n\nI don't try to pull away.\n\nWe wait, first with the doctor and then in the waiting room.\n\n\"Thank you, Emma,\" Dan says. \"Thank you for being here.\"\n\n\"I\u2014\" I start to say, and then the doctor comes back in. He is smiling now and then Dan is standing up, crying again but hugging the doctor and saying \"Thank you\" over and over again.\n\n\"We're still not out of the woods,\" the doctor says. \"But you can go see her now. Just one visitor, though. We still are monitoring the baby very closely.\"\n\n\"The baby,\" Dan says, grinning, and then he turns to me and says, \"I'm so glad you understand now. This is really what Lisa wanted, and now our son\u2014\"\n\n\"I want to see her,\" I say, and Dan, Dan who was just hugging me, who held my hand, who I once thought loved Mom more than anything, who I thought would put her and me first because he said he would says, \"Emma, I have to see her.\"\n\nI walk out of the waiting room then. I watch Dan come out. I watch him walk toward Mom's room. He looks at me before he does and I stare back at him until he turns away.\n\nHe doesn't ask me to come with him.\n\nHe is crying again, though, but I don't care, I don't care, I don't care.\n\nIt's not the first time he's done something without talking to me. It's not the first time he's acted like I don't matter to him.\n\nWhen he comes back out I walk toward the elevator because I know he will come, he will drive us both back to the house, and he got his way again. He gets his baby, his precious son, and sometimes life is bigger than our plans but Dan's are safe.\n\nCaleb is in the elevator, along with a woman in a wheelchair. He sees me and his eyes go wide. I see everything in them. Worry. Care.\n\nDan isn't even looking at me. He is staring at the floor, he is looking at the elevator ceiling, anywhere but at me, and I hate him. I hate that Mom ever met him and that I once loved him.\n\nI feel fingers touching mine and I know it's Caleb, that he's behind me and that he's reached out, stretched his fingers so they'll touch mine.\n\nI reach back and hold on.\n\nAnd when the elevator dings and the doors open, I turn around and hug him. I lean into him and say, \"Thank you,\" and then I pull away and walk past Dan, who is looking at me and I dare him to say something, anything, with my eyes.\n\nHe doesn't.\n31\n\nThat night I sleep in Mom's room. I lie in the bed she slept in, I put my head on her pillow. I knock Dan's pillow onto the floor. He doesn't sleep in here anymore anyway; he hasn't since the day Mom died. He sleeps downstairs or sometimes, I think, in the nursery.\n\nThe phone rings, and neither of us answers it. I think I hear a noise on the roof at one point but I don't get up. I just want to get through tonight, and there is no room in my heart or head for anything or anyone else but Mom.\n\nAnd my anger.\n\nI find Caleb as soon as I get to school in the morning, though, wade past the people in the halls and slip into a space next to his locker. He smiles when he sees me.\n\nI smile back, but I know it is shaking.\n\n\"Hey,\" he says. \"What happened yesterday?\"\n\n\"Problem with the baby.\"\n\n\"Oh. Is everything okay?\"\n\n\"For now.\"\n\n\"Your stepdad seemed pretty wrecked.\"\n\n\"He should,\" I say, and hear the bitterness in my voice.\n\n\"I came by last night,\" he says. \"But you didn't come out on the roof.\"\n\n\"No,\" I say, and Caleb touches my arm. I feel it under my clothes, under my skin, I feel his touch seep into my body, beat through me to the thrum of my blood pulsing.\n\n\"Angry or scared?\"\n\n\"Both.\"\n\n\"I'll be there today,\" he says. \"At the hospital, I mean.\" He clears his throat. \"You know that already. I\u2014uh, I really wanted to do more yesterday. I haven't felt like that in a long time.\"\n\nAnd then he says, \"Is it true that you and Anthony were together?\"\n\nI stare at him, shocked. \"Anthony?\" Why are we talking about Anthony?\n\n\"Yeah. The guy who knows everything and talks all the time. I just...I heard some stuff.\"\n\n\"Stuff?\" I knew Anthony would have talked\u2014that's what he does\u2014he talks and talks and I bet he said, \"I like Emma, but when she asked for a physical relationship, I felt that it was the wrong choice for both of us. She did understand that, eventually, although she was rather emotional about it.\"\n\nI thought I was okay with that. I really did. I certainly pictured him saying it. I just never pictured Caleb hearing it, or me caring about Caleb hearing it.\n\nBut I do. I really do.\n\n\"He didn't want me,\" I mutter. \"Okay? That's the whole story.\"\n\n\"Anthony's an idiot,\" he says.\n\n\"Oh.\" I know I am blushing, I can feel how hot my face is. I wish I hadn't told him Anthony didn't want me. Not that it matters, except it does.\n\n\"Emma?\" It's Olivia and I look around, see her staring at me and Caleb. At Caleb's hand, which is still on my arm, and my hand, which is somehow now resting on it.\n\n\"Hey,\" I say, and Olivia looks at Caleb, eyebrows raised, and Caleb sighs, not with anger, but with resignation, and whispers, \"I'll come find you later, okay?\"\n\n\"Hi, Olivia,\" he says as she comes up to us.\n\n\"Hi,\" she says, and she's not being mean but she definitely doesn't look or sound friendly and Caleb vanishes into the crowd of people moving through the hall.\n\n\"Okay, you and Caleb are more than just friends.\"\n\n\"We're not.\" Not really.\n\n\"Then what's with all the staring and the touching?\"\n\n\"He had his hand on my arm.\"\n\n\"I haven't seen him touch anyone in....well, forever,\" Olivia says. \"Look, are you sure he knows you're just friends? I mean, he's Caleb, you know? Druggie. Car thief.\"\n\n\"He's still a person.\"\n\n\"I know, but what if he's trying to\u2014?\"\n\n\"You aren't going to say 'take advantage,' are you?\"\n\n\"All right, that might be melodramatic, but\u2014\"\n\nI cut her off. \"Look, it's Caleb, the guy who is the trifecta, right? So Caleb and me? Anthony didn't even want me, Olivia. Come on.\"\n\n\"He doesn't look at you like Anthony did.\"\n\nAnthony's an idiot.\n\nI clear my throat, and know I'm blushing again because she's right.\n\nAnthony never, ever once looked at me like Caleb does.\n\nOlivia looks at me. \"I don't want you to get hurt.\"\n\n\"It's not like that,\" I say, and it isn't. Not like how Olivia means. Because of yesterday, and how I learned you can break all over again even when you think you can't. Because I'm not the girl I was, the one who looked at Anthony and saw what I wanted to see and nothing more. I had that luxury, that belief that the future could be whatever I wanted.\n\nI know better now. I saw who Anthony was\u2014too late, but I saw it\u2014and I see who Caleb is. He made me see all of him, everything, and I want to keep looking.\n32\n\nDan tries to say something to me on the way to the hospital after school.\n\n\"I was worried yesterday,\" he says. \"Upset. And if I scared you or made you angry, that's okay.\" He stares straight ahead at the traffic. \"I know you wanted to see your mother. Please don't think I don't know how much you love her.\"\n\n\"But you needed to see her,\" I say. \"Check on what's inside her.\"\n\n\"Your brother.\"\n\n\"You were there to see the baby as much as her. Maybe more,\" I say.\n\nDan shakes his head and says, \"That's not true. How can you even think that?\"\n\nI laugh and it's so harsh and bitter it makes my own heart contract.\n\nBut it does shut Dan up.\n\n\"Thirty-three days,\" I say when we get to the hospital. \"That's how I think it. Why I think it. Come find me when I'm allowed to see her. You'll let me do that today, won't you? Now that you've got the all-clear on what you want.\"\n\nI start to move away but Dan grabs my arm.\n\n\"Emma, instead of running away, why don't we talk?\"\n\nThe doctor comes out, catches Dan's eye, and I hear him suck in a breath. I pull away.\n\nI walk away.\n\nDan doesn't come after me.\n\nI don't sit in the waiting room. I can't. Not today, not with Dan and the empty words he wants me to hear.\n\nI'm seventeen. I'm supposed to have my future mapped out\u2014what college I'll go to, what major I'll have, what job I want. I'm supposed to even see past that, to the job I'll get, to the life I'll have on my own.\n\nAll that, and I can't see my own mother. Because I'm just a kid, and Dan is her husband. I'm not an adult, I don't have an equal say. I've never had one in what's happened. Not once, and why? Dan was her husband but I'm\u2014I was\u2014her daughter. It's not fair that all I've gotten to be is just here. I don't want that.\n\nI want someone to hear me.\n\nI want Dan to care what I think.\n\nNo, I wanted him to care.\n\nI can't take this.\n\nI head out of the hospital and walk into the parking lot. A parking lot should look the same everywhere but this one is somehow more depressing, as if all the cars are tainted by what's inside. Sickness, or the potential of it. The worry. The waiting. The fear. Even the light reflecting off the windshields isn't warm or comforting. It just hits my eyes and makes them sting.\n\nI rub them, and then pinch the bridge of my nose. I will see Mom today. I will. Somehow I will make it happen\u2014and then I see Caleb in the parking lot, standing by a blue car.\n\nHis car isn't blue.\n\nHe doesn't see me, and I look at him. He is staring at the car like it's more than a car, like he sees something in it or around it. That he somehow needs it.\n\nHe rubs a hand along one window, fingers tracing down to the door handle. He pulls on it like it will open but it doesn't.\n\n\"Hey,\" I say, and he doesn't turn away from the car. He is still touching the handle and his other hand is shaking. I can see that even from here.\n\n\"Caleb?\" I say, and walk over to him.\n\nHe doesn't seem to hear me.\n\n\"Caleb?\" I say again, and he turns, his eyes wild, wide and unfocused, like he's not really here. One hand tests the handle again and his other hand is still shaking. I watch his gaze skim over the ground, all flat and paved over except for a few places where the concrete edging has chipped away, fallen free.\n\nI swallow. \"Why this car?\"\n\n\"What?\" His voice is distant, like he isn't here.\n\n\"Why do you want to steal this car?\"\n\nHe looks at me then, and I look back at him steadily even though I'm shaking inside now, startled by this glimpse of a Caleb I knew about, but that I've never seen.\n\nHe still looks alone, but now there is anger in his eyes too.\n\n\"I...my mother,\" he says slowly, like the words are painful to say, like it hurts him to say them. \"She was here just now, visiting a client. I saw her in the hall. She said, 'Try not to screw this up,' and she didn't even look at me. She never looks at me.\" He laughs, and the sound is so fake and bitter that I get goose bumps.\n\n\"I'm sorry.\"\n\n\"Are you going to try to stop me?\"\n\n\"No. Someone I\u2014someone I like told me they couldn't tell me how to be okay. I'm pretty much in the same place.\"\n\nHe blinks, then looks back at the car. \"I'd probably get sent away again. I'd be out of the house. They wouldn't have to see me. I wouldn't have to see them.\"\n\nHe blows out a breath, and then looks at me again. \"I really want to do it, Emma.\"\n\n\"But suck camp sucked, remember?\"\n\n\"Yeah,\" he says. \"But I also wouldn't be able to talk to you anymore, and I'd miss that. I'd miss you.\"\n\nAnthony said I was lovely once, and I felt such warmth, such gratitude. I thought it was want.\n\nIt wasn't. Want is what hits me now, and it is not lovely or even pleasant. It's hard and fast and it hurts. My insides cramp with it, and a thousand images I didn't know I could even think of race through my mind.\n\n\"I'd miss you too,\" I say, because it is all I can say, because in my mind Caleb is not standing near me, he is next to me, twined around me, and there is no car, there is no pain, there is just us, and I\u2014\n\nI want a broken former car thief who still wants to steal cars but who just said he won't because if he did, he'd miss me.\n\n\"You didn't want to hear that,\" he says, and I shake my head because it's not that, it's so not that.\n\nBut yet it is because I've seen what wanting does.\n\nMy mother wanted Dan and when she died, he ignored how scared she was, how she used to rub her belly with her eyes closed like she was wishing for it all to work out for what he wants. For his son.\n\nI take a deep breath, and then I tell Caleb the truth. \"You confuse me.\"\n\nI look at him and my heart, which I was so sure was dead, burned out, beats hard and fast. Beats like I'm alive, like there is life in me, like I didn't die when Mom did.\n\n\"So I guess I'm not taking this car,\" Caleb says, and lets go of the handle. \"And just so you know, you confuse me too.\"\n\n\"I know,\" I say, and when he smiles at me I want things to be simple, I want him to be a guy, a regular guy, and I want to be a girl who knows she will go home to her mother and a stepfather she knows and loves but\u2014\n\nBut that's not how things are.\n\nIt's not how life is.\n\nI feel his smile then. I don't just see it anymore. It crawls inside me, curls up alongside all my grief. Confuses me, just like I said.\n\n\"We go inside now, right?\" Caleb says, and we do. Silent, together but not, walking side by side but not touching.\n\nHe takes the elevator up to the floor where Mom lies, and I look at him when I get off. He starts to wave, and then pushes his way off, stands next to me.\n\n\"Your mom,\" he says. \"How is she?\"\n\n\"I haven't seen her yet.\"\n\n\"Will you?\"\n\n\"I don't know.\"\n\n\"Do you want\u2014can I wait with you?\"\n\nI look at him and we smile at each other.\n\nAnd so we sit in the waiting room, his foot pushing the cart I saw so much significance in that first day, back and forth.\n\n\"Do you need to go to other floors?\"\n\n\"Yeah,\" he says, but he doesn't leave and I'm glad. It's nice not to be alone.\n\nDan comes in, looks at Caleb, and then says, \"You can see her, Emma, but just for a minute. She's tired.\"\n\nCaleb gets up when I do and Dan looks at both of us. Says, \"Emma, I haven't really met your friend.\"\n\n\"This is Caleb,\" I say, as Caleb nods at Dan and walks out. \"And Mom isn't tired. She's dead. You know it. I know it.\"\n\nI head toward Mom, and Dan follows me. I hear him breathing.\n\n\"Her body is tired, Emma,\" he says, and he sounds weary, on the edge of tears. \"The baby is still in trouble and things aren't as good as they could be. The risk of infection is even worse now and the doctors...\"\n\nHe keeps talking but I ignore him and sit next to Mom. She is even thinner now. I look at her belly. It's still swollen. Round.\n\n\"Hey, Mom,\" I say, and take her hand. It lies heavy and cool in mine. \"I missed you yesterday. I know you're gone, but when I see you I can...\" I trail off, look over at Dan.\n\nHe doesn't leave.\n\n\"Dan's here,\" I say. \"He's watching over what you worked so hard for. I still remember what you told me that time you had to go on bed rest. How you hated the thought of having to lie still. Of how trapped you felt. You wanted to go downstairs but Dan wouldn't let you. He was okay with you moving around after you got the clot taken out but once you were pregnant with his baby he...\"\n\nDan leaves then and I blow out a breath.\n\n\"Mommy,\" I whisper. \"I need you. I don't understand anything anymore. Yesterday, when the doctor said something to Dan I thought you were gone. I forgot that you are.\" I press my forehead to hers and pretend there are no tubes, no machines, that the skin under mine can feel my touch.\n\nIt's easy to. I want some part of her to be here. I hadn't realized how much.\n\nAnd even though she's gone, I don't want to let her go.\n\nI don't know if I can.\n33\n\n\"You know that I respect your privacy,\" Dan says on the way home. \"I let you lock yourself in your room all the time. But today, with your mother\u2014look, everything is very fragile right now.\"\n\n\"You mean with the baby,\" I say, and it costs me to get that last word out.\n\n\"Why do you do that?\" Dan says. \"Your brother is fighting for his life, and you talk about him like he's not real. I don't like that.\"\n\nWhen I don't say anything, he looks over at me, a quick, incredulous glance. \"You were excited about the baby, I know you were. Don't you see that he's family? That he might have your mother's eyes or her smile, that he'll be a way for us to always remember her?\"\n\n\"I know,\" I say, and I want to sound calm but I don't, I hear how high my voice is, I hear how angry it is. \"I know she'll live through him. I heard you back when you made your choice.\"\n\n\"It's not just my choice.\"\n\nIt was, and it is, and we both know it.\n\n\"Emma, it really wasn't and isn't just my choi\u2014\"\n\n\"Shut up,\" I say, and I'm screaming now, I feel the vibration of the words roaring up my throat and out of me. \"Shut up, shut up, shut up! I am so tired of you talking. I'm so tired of you.\"\n\n\"Tough, because I'm not going anywhere and you don't get to talk to me like that.\"\n\nDan pulls into the driveway.\n\n\"Yeah, I do,\" I say as I open my door and get out, not caring that the car isn't even stopped, that I fall as I do, that I tumble onto the ground. I just want to be away from him. He stops the car, gets out and calls after me but I'm already in the house, running up the stairs and into my room, bolting the door closed behind me.\n\nLet him call the police and have them come break down the door. Let him try it. I'm ready. I sit on my bed, fists clenched, waiting.\n\nDan knocks on the door.\n\n\"Emma, I know you're angry, but it can't be an excuse anymore.\" I hear the Dan I used to think I knew, the one who sat with me after Olivia and I went to the eighth grade dance and Olivia got asked to dance three times and I didn't get asked once. Dan didn't say things would be better in the morning or lies like that. He just hugged me and then served ice cream for breakfast, telling Mom, \"It's got calcium, and sometimes people deserve something fun because they're special.\"\n\nI'd told Mom about the dance too, and she'd rubbed my back and told me she loved me and it had been nice. But Dan really made me feel like everything was okay. Like I was okay. Like I was special.\n\nI miss him. I hate that.\n\nI grit my teeth and go out onto the roof just in time to see someone\u2014Caleb?\u2014walking toward the edge of it.\n\n\"Hey,\" I say.\n\n\"Hey.\"\n\n\"Olivia?\"\n\n\"Hi,\" she says, turning around and coming back. \"I was up here when you came in. Should I go?\"\n\n\"Why?\"\n\n\"You got out of a moving car, Emma. You looked pissed as hell and you ran into the house and I figured you and Dan would be talking.\"\n\n\"We're not.\"\n\n\"Oh,\" she says. \"So, do you want me to stay?\"\n\n\"Yeah. You know you don't even have to ask that.\"\n\nOlivia sits down. \"You sounded like maybe you wanted someone else to be out here.\"\n\nI clear my throat. \"Did I?\"\n\nShe looks at me. I rub one foot along the shingles, hear the scrape against the bottom of my shoe. \"All right. I saw Caleb at the hospital and I thought for a second that he was here. But I am glad to see you.\"\n\n\"Yeah?\"\n\n\"Yeah,\" I say, and she bumps her foot with mine. \"What happened today at the hospital?\"\n\n\"Mom's...well. You know. But there are some problems with the baby and Dan wasn't even going to let me see her by myself.\"\n\n\"Why?\"\n\n\"Stress. As if she can be stressed. It's like he forgets she's gone, and yesterday I realized I have too. I mean, I know she is, but I can see her, and...\" I wipe my eyes. \"I'm so tired of crying. Tell me something. Talk about you. Please.\"\n\n\"But\u2014\"\n\n\"Olivia, I need normal. You don't know how much I need it.\"\n\n\"Well my parents are both working, of course, and both coming home late\u2014server problems again. Or something. I try not to absorb the tech talk. Oh, and Roger and I made out after school.\" She says the last bit with so much glee she's practically vibrating with it.\n\n\"Where?\"\n\nShe blushes. \"My car.\"\n\n\"Your car? What happened?\"\n\n\"Well, we were talking after the last bell rang and he said he was going to get a ride home with his friend Ivan and I said, 'Oh, I can give you a ride home,' and he smiled at me like I was the last day of school and once we got in the car, stuff just happened.\"\n\n\"In the parking lot?\"\n\n\"A little bit,\" she says, blushing. \"And then I drove him to his house and we might have made out some more. Until his little brother came home and found us.\"\n\n\"He didn't!\"\n\n\"Yeah, he did. But Roger was so great about it. He was nice to his brother but also got him to, you know, go, and then he said he'd call me and that he wanted to see my phone. I've told him about it.\"\n\n\"So, he wants to see your room.\"\n\n\"He said phone,\" Olivia says, but she's grinning. \"It was just the most amazing afternoon.\" She looks at me. \"You can't really want to hear this. I sound so shallow.\"\n\n\"You sound happy. Normal, remember?\"\n\n\"But\u2014\"\n\n\"Nope. You sound happy and I'm glad you are and that you told me about it. And I want more updates, especially if room viewing is involved.\"\n\n\"You sure?\"\n\n\"I need this. I don't want to forget that people have actual lives.\" It's what I don't have, and hearing about Olivia washes over me like rain, covering the thing I limp through every day, the thing that passes for a life but isn't one.\n\n\"Well, then I promise you'll hear if there's any room viewing. Which there won't be because I want more than just a thing, you know? I'd like an actual boyfriend. Not that I'm ruling out room viewing, but it's way down the line and...okay, now I'm totally wondering what he'd think of my room. And what it would be like to have him in it.\"\n\nShe leans back, shaking her head. \"How am I supposed to be picking out which colleges I'm going to apply to when I just want to see if Roger is willing to go out with me this weekend. And next weekend. And also make out with me. Argh! I need to do something to get my mind off him.\"\n\n\"No haircuts. Or dying of hair.\"\n\n\"I wasn't going to do that. I could get bangs though, right? Bangs aren't a haircut.\"\n\n\"No bangs! You so need a different stress outlet, and no, I don't mean new eyeliner.\"\n\n\"I already bought some,\" she says and when I give her a look says, \"What? I was a little wound up after I left Roger's house. Do you think he'll call tonight?\"\n\n\"If he looked at you like you're the last day of school, then yeah, he'll call.\"\n\n\"I thought getting a guy to like you was the hard part,\" she mutters. \"But it's just as hard after that. How come no one ever tells you that?\"\n\n\"I think there's a lot of things that no one ever tells you,\" I say, and Olivia puts an arm around me, pulling me close.\n\nWe sit in silence until she says she has to go.\n\n\"Thank you,\" I say when she gets up, and she says, \"Oh yeah, you thank me. I haven't shut up, and all I've talked about is Roger.\"\n\n\"Thank you,\" I say again and she hugs me, hard, before she starts to climb off the roof. I watch until she's gone.\n\nUntil I can't see anything.\n34\n\nDan doesn't come back to my room and I go through the motions of getting ready for bed\u2014the face washing, the teeth brushing\u2014before I give up and crawl into bed, pissed-off. I know I won't sleep. I felt okay when Olivia was here, when I could hear about her life and not think about mine, but now I'm just hungry and still angry.\n\nAnd lonely.\n\nI always thought of grief as a blow that took everything out of you. And it is like that. But it stays, past that first hard hit. It stays and blows its breath into you.\n\nIt is always there, reminding you of what you've lost. What's gone.\n\nI sit up for a while, reading a book Mom got me last year at Christmas, a huge story about a girl and a dragon and a prince that she ends up having to save. \"You can't read school stuff all the time,\" she'd said, and I'd thanked her for the book and never opened it. I didn't want stories. I wanted grades, and good ones at that. I wanted a future.\n\nInside she'd written, A story can take you anywhere. XO Forever, Mom.\n\nI didn't see it until I opened the book just a few minutes ago and I wish my story wasn't\u2014\n\nI wish it wasn't this one, and that's grief blowing softly over me again. Reminding me all over again.\n\nI missed so many things with Mom.\n\nI drop the book on my bed. I won't do this, I won't cry, I will just be glad I saw what she wrote now.\n\nAnd I am. But I still wish I'd seen it before. That I'd known I needed to treasure every second I had, that I should focus on everything she said and did so I could always remember.\n\nMy stomach rumbles and I look at my door. Past it is the house. Past it is Dan.\n\nWhen my stomach rumbles again, I get up. It's after midnight now. Dan will be asleep, on the sofa or in his office or in the nursery. I don't know why he doesn't sleep in Mom's bedroom anymore. I'm just glad there's something of Mom he's left alone. That he hasn't used for what he wants.\n\nI open my door and go out into the hall. I go because I'm hungry and because I want to prove I can. I'm not afraid of anything Dan might say, because I've heard it all before and it's just words. Syllables and letters adding up to his choice and nothing more.\n\nAnd I don't miss him that much. Not really.\n\nThen I hear the humming.\n\nI freeze, sure I'm hearing things; a ghost, a dream, that I'm asleep and back in a different time, back in a different me.\n\nBut I'm not hearing things. I'm in the hallway, in the dark, and my mother is dead and Dan is in the nursery, humming.\n\nDan used to hum a lot. All the time, in fact. I noticed it when he and Mom first started going out, the weird thing he did, how he hummed sometimes, and then I realized when he did it.\n\nHe hummed when he was doing something that made him happy.\n\nHe hummed when he was happy, and so he was always humming around Mom. And even me. Mom made him happy. We made him happy, and I used to love to hear him hum, loved waking up and going downstairs to Mom blinking sleep from her eyes as she rushed off to work, Dan humming as he urged her to eat something and talked about what he'd be working on that day, in his office in the house. \"I like working at home,\" he'd say. \"It's quiet, I get a lot done, and whenever I want, I can just step out into the hall and see signs of my family.\"\n\nHe made Mom's habit of dropping things\u2014shoes, suit jackets, towels\u2014sound like it was beautiful. He picked up after her. He picked up after me. He never told me to clean my room like Mom did. He just...he hummed, and was happy, and I knew it meant he loved us.\n\nI lean against the wall now, my head spinning, my insides twisting, and yes, he really is humming.\n\nI creep down the hall, one step, two steps, silent as can be.\n\nHe's in the nursery.\n\nThere is one light on, the light Dan put in by the changing table. It casts shadows everywhere, up and down the walls, dark lines and shapes, but it shows Dan clearly.\n\nIt shows Dan, and he is putting a crib together.\n\nHe's putting it together and he looks happy and he's humming.\n\nHe is happy. I know what his humming means because I know him. I spent years loving him, trusting him, and now he is humming and then he whispers something.\n\nWhispers, \"Lisa.\"\n\nHe whispers Mom's name and he isn't humming anymore. His voice is soft, sad, sorry, and no, no, no, no. NO.\n\nHe said her name like he loves her, like he's sorry, and I am gone now, I am slipping away, quietly heading downstairs and out of the house, into the garage, my chest so tight I can't breathe.\n\nI am seeing spots, I am dying, this is what happened to Mom and now it's happening to me. You walk downstairs and you're fine, everything's fine except maybe you know something is wrong, maybe you've said something to your daughter but you've been smiling and trying and you reach for the toast and then you die.\n\nJust like that.\n\nExcept I don't.\n\nI stand in the garage, and the tightness in my chest passes.\n\nThe memory of Dan humming doesn't, and neither does the sadness with which he said Mom's name.\n\nI pace around the garage and find myself in front of the fridge where Mom kept all the things she wasn't allowed to have after she got pregnant. Dan, who could never resist a sale at the grocery store, used to have it stocked with things he'd gotten \"for practically free!\" Mom and I made him stop that after eating pea soup two weeks in a row, but the fridge stayed.\n\nIt was from Dan's old apartment and Mom would stash tiny chocolate bars in it, and Dan would store leftovers, let them sit in the freezer till he'd eat them for lunch.\n\nMom cleaned the kitchen out after the first official week of pregnancy, sighing as she put things into bags and had me take them out here.\n\nI never looked inside the bags. I didn't think about what pregnant women shouldn't eat. I didn't care. I just saw Mom, and she was there like she always was, doing what she always did, which was organizing things the way she wanted and I\u2014\n\nI never thought about what she gave up. Not until after she gave up her life.\n\nI stare at the fridge, almost shaking, which is weird because it's food, just food, how could she have thought that would hurt her?\n\nAnd then I open it.\n\nI want to know what she was afraid of besides the baby.\n\nAnd it's...it's cheese. A wheel of Brie, frozen and inedible. Some lunch meat. A piece of lamb all wrapped up and marked SMELLS HORRIBLE.\n\nI don't feel better seeing this. She put away cheese and lunch meat and lamb? I see coffee too, but I remember that, suffered through two weeks of Mom switching to decaf and growling in the mornings.\n\nI used to hide out in my room, wait until she was gone or just about to leave. I missed her on purpose.\n\nI stand there, staring at this food, and it is just food, there are no secrets here, there is nothing of Mom here, and Dan was putting the crib together, I saw him, I saw it.\n\nAnd then I see the bottle of wine.\n35\n\nMom wasn't much of a drinker. She liked sweet drinks, the ones that come in frosted glasses and are tinted blue or green. Once Dan was around, he found her true weakness, which was for wine that smelled like fruit punch. I still remember the faces he'd make when he'd bring a bottle home, how he'd mock grimace when he'd open it and pour her a glass, shaking his head when she'd offer him one.\n\n\"Too sweet,\" he'd say. \"Wine shouldn't taste like sugar.\"\n\nMom would shake her head and say, \"Why would I want to drink something that is supposed to taste like oak or peat? I like what I like.\"\n\n\"I know you do,\" Dan would say and Mom smelled sweet, like fruit, when she kissed me after drinking her wine, and her smile would be softer somehow, as if all the things she worried about had been lifted away.\n\nShe drank a wine that came in a bottle with a picture of strawberries on it, and I pick it up now and stare at the top, think of how Dan used to say wine should have corks and Mom would say, \"Okay for you, not okay for me.\" And they'd look at each other and there was so much love there.\n\nAt least, I thought there was.\n\nThe bottle isn't full. Mom must have had a glass, back before my bro\u2014the baby\u2014swam inside her. She came home from work and Dan opened the bottle and she drank a glass and maybe kissed me and smelled sweet and I don't remember it at all.\n\nI was probably doing homework and was annoyed by the distraction. Annoyed by my mother wanting to see me. To pay attention to me.\n\nI was so stupid.\n\nI pick up the bottle. I open it, unscrewing the top, and take a sip. It is sweet, just like fruit punch, just like Dan always said, and Mom drank from this bottle. I have another sip and another and another until I am loose-limbed, breathing easy and the air smells like fake strawberries and Dan's humming seems far away. It doesn't hurt so bad.\n\nI look at the bottle. It's almost empty, and I lift it up and swallow the last sip. The world is light now, I am light, and things are bad\u2014they are\u2014but I don't feel so bad. I feel like I could float up out of the house and into the sky. Up into Mom.\n\nI wish Caleb was here. He'd know how it feels to want to float up. And he's so pretty, those cheekbones, those eyes, the hair he hides behind that I touched but not like I wanted to.\n\nI can admit it now, it's easy. I want to have his hair in my hands while he holds me.\n\nI should go see him. It's not that far. I can walk there. I can leave through the garage side door and Dan doesn't keep me prisoner here. I spend more time worrying that he'll decide he doesn't want me than I want to think about and...\n\nNo.\n\nI don't want to think.\n\nI look at the strawberries on the wine bottle again. They are happy-looking. I didn't know strawberries could look happy.\n\nI want to be happy. That would be so nice, to be happy.\n\nI'd feel happier if I left the house, so I go outside, and there I am, just me and the night. Well, and the bottle, but it is so light in my hands and I am so light, I'm almost flying, sort of. I don't even have to walk down the street, my feet are just going, going, going and they know where to go.\n\nCaleb's house is even bigger in the night, all huge and dark and I was there, I went in there, and I held his hand and he is so alone and I am so alone. I look at the garage with my feet on the ground but not really on it because I feel so free, I am just me, I am not\u2014\n\nI am not sad.\n\nAnd I see things too. I see a door on the side of the garage, just like on mine, and I open it. It's dark inside and there are cars and I bet one of them replaced the one Caleb drove into the lake because his parents don't know it's not about the cars. They don't see, but then adults never see anything, it's like you get old and you don't see how things are, you just see how you want them to be. Dan does that, he did that, and I don't want to think about Dan.\n\nSo I won't. I'll just be here, in this garage, and I am walking up Caleb's stairs which are so tall and deep I don't quite know how to place my feet.\n\nI think I'm drunk, but the thought passes through and floats away like all the others. Like how I'm sad and I miss Mom and Dan. Just thoughts, and it's hard to get up the stairs and I wish there was more wine. I wish I could taste strawberries and feel like I do now forever, plus I am going to see Caleb sleeping. He will have his eyes closed and he'll be surprised except I'll be quiet, so quiet he'll never know I am here and\u2014\n\n\"Emma?\" Light, bright light, making me blink.\n\nIt's Caleb. And he's awake.\n36\n\n\"You're supposed to be asleep,\" I say, and Caleb stares at me.\n\n\"Are you\u2014Emma, are you drunk?\" he says and I shake my head because I don't drink, and how could a few sips of wine that tasted like fruit really make me drunk? To get drunk you have to drink beer or smelly liquor and lots of it. Standing in a garage with a bottle of wine can't get you drunk. Not that drunk, anyway.\n\n\"Yeah,\" Caleb says, \"It can,\" and I say, \"Wait, did I just say all that?\" and he nods. He isn't wearing a shirt and he has nice chest muscles and he's nice and I like him.\n\n\"Okay, Emma, okay, stop and drink this,\" Caleb says and I take the glass of water that is suddenly right in front of me. I'm thirsty, but not super thirsty, but somehow the water is all gone and then there is another glass and I drink that too.\n\n\"You put a shirt on,\" I say to Caleb, who runs a hand through his hair and looks at me like he hasn't ever done before, like I've made him happy and angry at the same time.\n\n\"You shouldn't have come here,\" he says and that is such a stupid thing to say that I tell him that and add, \"Where else would I go? Who else understands? And besides, I wanted to see you,\" and he blinks and his eyes are closed and he has a shirt on but under it is skin\u2014I know that\u2014and I can feel it\u2014yes, there it is, warm under my hand and he jerks back like I've hit him and says, \"Emma, please.\"\n\n\"What?\" I say, and I don't get it, doesn't he know how this feels? Doesn't he want to feel this way too?\n\nSomeone groans. It's not me. Is he sick? I look at him, but he looks okay. He looks great.\n\n\"I feel great,\" I say, and he shakes his head like he knows something I don't. I move toward him and he backs up but not fast, he backs up slow and I see his eyes and I'm not stupid.\n\n\"I never said you were stupid,\" he says and I guess I'm still talking but that's okay. I don't talk much, I keep so much inside, I am so full of things that hurt but they don't hurt now. I rest my head on his shoulder, brush my lips over his neck.\n\nHe tastes like salt, like tears. He tastes like pizza and grief and love and fear. He tastes like Caleb and I want more and there is a noise and it isn't me I am not saying anything now, I know I'm not, and I pull away and it's Caleb, he is making that noise, a broken, almost animal sound, his head thrown back and I can see a pulse beating in his neck. It's his heart and it is beating and I can make it beat fast and I like that and I move in again.\n\n\"Emma,\" he says, and he sounds so strange, so serious, and I poke his chest because he must get this, I know the things he used to do, he must have felt like this.\n\n\"Emma, you'll be sorry when you're sober,\" he says, and I look at him and say this\u2014I know I say this\u2014I say, \"Why would I ever be sorry for you?\" and his mouth brushes across my forehead, my cheek, my neck and I am turning toward him.\n\nHe is so close, he just needs to be a little closer, and my head hurts a little but not so bad. I still feel good and he feels good and I like that. I want it to keep going forever and ever.\n\n\"I can't,\" he says. \"I don't\u2014Emma, I don't want you to be sorry over me,\" and then we are lying down and he is closing his eyes and making soft, desperate noises as I press my face into his neck, his shoulder, and he is turning his mouth away from mine. When I say, \"Why?\" he says, \"Just sleep, okay? Just sleep.\"\n\n\"I wanted the baby,\" I tell him, and I never meant to say that. I like to pretend I never thought that but I did, and I have said it now and I don't like this, I don't like it at all, and he says, \"Hey, hey,\" his voice soft and I close my eyes and say, \"Do you think that's why she died? Because I wanted the baby too?\"\n\n\"No,\" he whispers. \"No, it wasn't because of you. Don't ever think that.\"\n\n\"You won't kiss me,\" I whisper because I can and my eyes are heavy and the room is spinning and I don't feel bad but it's not like before. I am not floating. I remember things. I remember Dan and the crib and why I had to go.\n\n\"You don't want to kiss me,\" I say, and right before darkness falls he says, \"I do. I really, really do,\" and I want to think about that but I can't, I can't focus like I want to.\n\nBut he said it.\n37\n\nI wake up during the night, twice, and throw up into Caleb's trash can.\n\nHe holds my hair back and I try not to feel like an idiot.\n\nI do anyway.\n\n\"You aren't an idiot,\" he says, his hand rubbing warm circles on my back and I should be embarrassed that I'm still talking.\n\nI'm not, though. I just feel...\n\nWell, awful.\n\nBut nice too.\n38\n\nI have gotten straight A's every grading period since they started giving A's until...well, until. I wish there could be a line for that day, that moment where Mom left, something solid people could see.\n\nSo they could see that there is before Mom died, and that there is after.\n\nThe two don't connect. They can't. The person who could have held that time together is the reason why it's separate.\n\nI always did the right thing. Before, before, before. That was my thing. Doing the right thing.\n\nI have never had a drink, because it wasn't the right thing, and I wake up with my head pounding, a vicious, throbbing pain over my right eye, and my mouth tastes like someone stuffed it with smelly cotton balls and I am not at home.\n\nI am lying on a narrow bed next to Caleb. He smells wonderful and looks wonderful and I know I don't. I also don't have that moment you're supposed to have after you've been drinking, the moment where you wonder what you did.\n\nI know exactly what I did, what I said, and I am so humiliated I can feel my face, nuclear hot and practically glowing, as I look at his closed eyes.\n\nWhich open and look right into mine.\n\n\"Hey,\" he says. \"Need something for your head?\" and it's like everything is fine, like I didn't come over and throw myself at him but I did, I know I did, and maybe I didn't rip off my clothes (thank goodness), but I wanted him and I know he knows it and now\u2014\n\n\"Emma,\" he says, and his hand is touching mine. He is holding my hand and I want to die and I also want him to keep holding my hand. \"Stop worrying.\"\n\n\"I'm not. It's just that I should go.\"\n\nI start to sit up but it hurts way too much to do that. Caleb keeps holding my hand, and he is looking at me. He remembers last night\u2014of course he does\u2014he wasn't the one drunk on a bottle of strawberry wine. He didn't act like an idiot.\n\nAnd then he touches me. One hand on my neck, a mirror of where my mouth touched his skin last night.\n\n\"Don't run,\" he says. \"I've done that, and you just end up back where you started. And I\u2014\" He breaks off, blushes and I see the color on his face, his chest, and my body stirs. It wasn't just last night and the wine that did it. It's him. He does this to me, melts me all over.\n\n\"I don't want you to go yet,\" he says.\n\nSo I stay.\n\nI stay, and we eat cold toaster pastries in the light of the rising sun and he gives me water and ibuprofen. We are silent but it's not a bad silent, it's a comfortable silent. I thought I'd have to run away, that being drunk and barging in to see him and all the other things meant I had to go.\n\nWhen I tell him that he grins and he is so beautiful that I swallow and look at my hands.\n\n\"Since when have you done what you're supposed to?\" he says.\n\n\"Since forever. At least until...\" I trail off and all the hurt comes back, everything from last night comes back, Dan and the nursery. The crib.\n\nI wish I hadn't seen it.\n\nI wish I'd never wanted the baby but I did, and look what it cost.\n\nCaleb hands me another cold toaster pastry.\n\n\"Here,\" he says, wrapping my fingers around it, but I can't eat anymore. I have to go home. I have to see Dan.\n\nI have to talk to him.\n\nThis time, when I get up, Caleb doesn't stop me and I see the bed for the first time\u2014really see it, I mean. It's tiny and we were there, we were both there, and I said\u2014\n\n\"Last night\u2014\" I start to say, and he stands up too and says, \"I know,\" and his voice is so serious and he is letting me know it's okay because this is what you do when you care about someone. You don't think me, me, me. You think you, you, you.\n\nI could think that about him. I do think that, and I could think past that.\n\nI could love him.\n\nI think I already might, but I don't want to think about that now.\n\n\"I meant it,\" I say in a rush because it's Caleb and it's true. \"I meant all of it.\" I know I am blushing, and I leave then because I can't say more, I don't want to say more, there is too much else going on and I shouldn't\u2014\n\nMy mother is dead and I shouldn't feel.\n\nBut I do, and I look back when I leave. I watch him smile and see him wave at me.\n\nAnd I smile and wave back.\n39\n\nI notice the sun again as I'm walking home. It surprises me like it did before, but this time I stop and look up at it. I know you're not supposed to do that, but all that radiant light. All that warmth. It's just there. Always there.\n\nMy vision is spotted when I look away, and it makes me a little dizzy. Well, that and the hangover, but it was worth it. All of it was worth it and today will be\u2014\n\nI haven't thought of a day that way, as what it will be, what it could be, since before Mom died. My days have just been what is.\n\nI'm not happy, but I'm something close to it.\n\nThen I walk into the house.\n\nDan is standing right by the door and says, \"Emma?\" like he doesn't know me. His eyes are bloodshot, almost worried-looking, and when I say, \"Yeah?\" he just looks at me.\n\n\"Dan,\" I say slowly, heading toward the kitchen after a moment has passed and he's still staring. \"Are you going to be okay to drive me to school?\"\n\n\"But you...\" Dan has followed me into the kitchen and he is speaking strangely, haltingly, like he is having a hard time finding words, and I know he knows that I left the house last night.\n\n\"I went for a walk,\" I say, because I have been walking and because the rest is mine. Dan has his nursery and his crib and his baby. I can have last night. The bad\u2014and the good.\n\n\"You were gone all night. I've been sitting down here for hours waiting and worrying. Where were you?\"\n\n\"Worrying?\"\n\n\"Yes, worrying. Where were you?\"\n\n\"I went to see a friend.\"\n\n\"Who? Not Olivia because when I called her she had no idea where you were.\" He leans in a little and then stills. \"You've been drinking! That's\u2014you can't do that!\"\n\nI shrug.\n\n\"I don't want this kind of thing to happen again. You need to talk to someone, and I think\u2014\"\n\n\"You've said that before. You say it a lot, actually. But you don't do anything, do you? It's all just pretend.\"\n\n\"Pretend? I've been up all night worried about you. Is that pretending?\"\n\n\"You were up putting the crib together for your son,\" I say, and he stiffens.\n\n\"Where else is he supposed to sleep? And why is it always 'your son'? Why isn't it ever 'my brother'? Because he is, and your mother would have wanted\u2014\"\n\n\"What? This? She would have wanted to be dead? Oh yeah, Dan, she'd be so happy to see her body now. She'd be thrilled to know she's a collection of wires and machines for what you want.\"\n\n\"Your mother wanted\u2014\"\n\n\"Don't. You can say she did all you want but that doesn't make it true. She was dead, Dan. Dead. You didn't get to ask her what she wanted. You didn't know, and you don't know. You're just pretending. Again.\"\n\n\"No, you don't know what your mother wanted,\" he says, and stares at me so firmly, with such belief in his eyes, that I see where he's been able to find the strength to do this, to keep Mom's body lying there, dead but run by machines for the baby.\n\nHe truly believes this is what Mom would have wanted.\n\nI shake my head at him. \"No, see, you know what you wanted. You were there, you saw how scared she was about being pregnant. And you didn't and don't care.\"\n\n\"She was scared,\" he says, and I stare at him, shocked, my head starting to pound again and my stomach churning because he knew, he knew, he knew.\n\nHe knew she was scared.\n\n\"Lisa was terrified she'd never get pregnant,\" he says. \"And then, when she did, she was terrified she'd lose the baby. She wanted him so much. Don't you remember how she'd sit with her hand over her stomach? Don't you know why she did that?\"\n\nMy stomach churns again. I don't like what he's saying; he's twisting it all around, he's not saying she was scared like she actually was, he's making it sound like she\u2014\n\nLike she knew what he'd do if she died, and that she'd want it.\n\nLike he knew her better than me.\n\n\"I saw her,\" I say, and I keep my words careful, because I did know her better. I was part of her. I was there every day all the years before Dan ever was.\n\nI knew she loved him before she told him. She told me first, because she wanted to know how I felt.\n\nRun, I should have said. Run.\n\nBut I didn't, and once upon a time Mom was with us and loved us and we loved her and I loved Dan and I believe he loved me and that should have been enough, that should have been everything, but it isn't and wasn't because life isn't like that.\n\n\"I saw her,\" I say again. \"I saw how she spent two years trying to get pregnant. I heard her crying when she wasn't, when you were out getting groceries or working in the yard or getting your hair cut. She never cried in front of you, you know. She always smiled and said she knew it would happen and you'd smile. You don't know what it cost her to say that to you. You got to hear about all the risks. She lived with them, even when the clot made things even riskier.\"\n\nHe pales, but I'm not done. \"And then she got pregnant. She wanted to make sure, but you started planning right away, you were celebrating when she was still waiting and she'd been waiting for years to be sure. Did you ever even think about what would have happened if she wasn't pregnant? Did you ever think about how she'd have felt, to see everything you wanted not come true?\"\n\n\"I didn't know she cried,\" Dan says and his voice is cracking but I don't care. I am beyond pity now. He should have heard all of this long ago. He should have seen it but he closed his eyes and just saw what he wanted. He didn't see what Mom wanted.\n\n\"She knew what was coming,\" I say. \"From the moment she got pregnant, she was worried, just like you say. But I saw everything, Dan. I saw how hard it was for her to smile. How she moved so slowly, like she was afraid to, and this was Mom, who could change a tire in under a minute and who refinished the floors all by herself who used to dance around the house. She never moved like she had to be afraid until she was pregnant, and you can pretend all you want, but she was scared. She sat with a hand on her belly because deep down, she knew. She knew she was going to die.\"\n\n\"She didn't ever think that,\" Dan says, standing up. \"You can't really believe that. You\u2014\" He pauses, stares at me. \"But you do. You think she believed our baby would kill her?\"\n\n\"She never even talked about having it. She let you handle the nursery, and when you and I tried to get her to talk about names, she wouldn't. Don't you see what that was? Don't you get it?\"\n\n\"She was scared,\" Dan roars, and then sinks to the floor, face in his hands. \"Emma, she was scared, but not like you think. She was scared she was going to lose the baby. She wanted to do all those things\u2014the nursery, talk about names\u2014but there had been so many lost babies by then. She used to lie next to me at night and say 'It was so easy with Emma. Maybe it was too easy. Maybe this is why it's so hard now.' She would say\u2014\"\n\nHe looks at me, tears in his eyes. \"She would say, 'I'm not supposed to be this lucky.' She didn't think she deserved you and her life and we did talk about the baby. She wanted him, and she never believed she was going to die.\"\n\n\"Right. So, she didn't believe it, and she also said, 'Oh, by the way, if I die, please make sure to keep my body going, please hook me up to machines and tubes so I can have your baby, Dan.' Is that what she told you after she said she thought she was too lucky?\"\n\n\"That's\u2014\"\n\n\"What? True? Not true? We both know she never got a say because she was scared and then she was dead. You chose for her and you want to pretend it's what she wanted but you don't know that. You can't ever know that.\"\n\n\"And you can't know she would have wanted your brother to die,\" he says. \"Because that's what would have happened, and did you ever hear her wish him away? Did she ever tell you he was a mistake? Did she ever whisper that she didn't want him? We both know she was scared, but I know why and I did what I thought she'd want, and that was to save our son.\"\n\n\"You never even asked me,\" I say and the words come out in a rush, ripped from me, and he blinks.\n\n\"What?\"\n\n\"You never asked me about Mom. You never\u2014it was like you didn't even notice I was there, in the hospital, when they told us she was dead and then you left because the doctor called you over. You just chose, you chose the baby, and I see my dead mother every day, Dan. I'm seventeen and I see my dead mother and I miss her and I want her back but she can't come back. She'll just lie there till the doctor cuts her open and then her body won't be needed anymore, you won't need her anymore and it'll be you and your son. The perfect little family.\"\n\n\"Stop this,\" Dan says, moving over to the stove, his whole body shaking. \"I have loved you since the moment you and your mother let me into your lives. I have always been proud to have you as my daughter but hearing you talk about your mother like this, like she's nothing to me, like it doesn't kill me to know that she's gone, like I don't miss her like you do or love her at all is bullshit. And you say 'me and my son,' but what about you? Is the idea of having a brother that bad? Are you that jealous?\"\n\n\"Jealous? You think I'm jealous? Did you hear anything I just said? You never asked me what I wanted for Mom. You never ever asked me anything about her. I've hated the baby because Mom died and he lived, but he'll never...\" I start to laugh then because he is so wrong, we are all so wrong and broken and I am not laughing so much as crying, I push away from the table and I am standing in the kitchen, I am looking at Dan and it is all pouring out, all of it, everything I've kept inside is pouring out.\n\n\"He'll never see her. Never, and when's the last time you hugged me? Do you know, Dan? Because I do. It was the morning Mom died. After she died, you were there and I was there but it was like we didn't know each other. You made your choices and we came back here and that was it. You go through the motions, you take me to see Mom and talk about how worried you are. But I saw you in that nursery. You were thinking about the baby. You whispered Mom's name. You were thinking about her. But me...you've said you won't get rid of me, but that's all. And I get it, I do. You've got a baby coming, and I was just part of the deal with Mom. You can forget me, and you did because when she died, you didn't talk to me about anything. You never even looked at me.\"\n\n\"Emma...\" Dan's face is pale. \"She'd just died and it wasn't supposed to happen. We had plans, so many plans, and then I saw her die, I saw my wife die and I didn't\u2014Emma, I'm sorry I didn't talk to you, but I do love you. Those aren't just words.\"\n\n\"But they are,\" I say. \"They're just words and that's all they are and I lost two people when Mom died. I lost her and I lost you. Your son will get you and I\u2014\" I break off and look at my hands, see how they are knotted together.\n\n\"I loved him,\" I say. \"I wanted a brother, I liked picking out nursery stuff, I even wanted Mom to be happier about it, I wanted her to be like us. But then she died and everything became about him. You get up every morning for him, not me, and I...the thing is, I did believe in those words once. In I love you. I believed that you did, and that you were\u2014you were my dad. But you stopped being that the day Mom died. You just stopped everything.\"\n\nI am so tired now. I have said everything inside me, and Dan is just standing there staring at me, and this is what broken is.\n\nI'm so tired. I just want to sleep. I just want to forget, even if it's only for a while, that I saw what I did last night.\n\nThat I have told Dan everything and my reply has been silence.\n\nI walk up to my room. I look back once and Dan is still standing in the kitchen. He is still silent.\n\nI don't bother to lock my door when I get to my room. There's no need. There never was. I just wanted to pretend that Dan wanted to come and get me. That I was the one keeping him out when the truth is he was gone from the moment I got to the hospital and said, \"What happened?\" and Dan said, \"Your mother,\" like he didn't know me.\n\nThe whole time, when we sat there and he talked to the doctor and made his choices, during all those terrible hours when she was first gone, he never once looked at me.\n40\n\nI wake up to the sun shining in my eyes and it's not like earlier, it's not pretty. I sit up and look at the clock.\n\nIt's almost time for Dan and me to be at the hospital. I take a shower, not bothering to wash my hair, and get dressed. As I walk out of my room, I take the padlock off and put it on my dresser. I don't need it.\n\nI just wanted to.\n\nI go downstairs knowing what I will see. I will see a house that is a house and not a home, and I will be alone because Dan and I have said everything and I'm sure he doesn't want to pretend anymore.\n\nThe time for pretending ended the second I called him on it.\n\nI take a deep breath.\n\nEverything is out now, everything about what we used to be and what we are now, and I finally get it. I see that this is what Caleb's life is like.\n\nCaleb is alone like I am and he can't fix it but he understands and together we are\u2014together, we are less alone. We are friends. We are\u2014\n\nI think of the noise he made last night when I kissed him, of the warmth of his skin.\n\nWe are something that could be.\n\nI will go to the hospital on my own. I'll see Caleb. I...will I even be allowed to see Mom? I don't think Dan would stop that. He's not cruel, he's just gone.\n\nExcept he isn't.\n\nHe is in the kitchen, still standing where I left him.\n\n\"Dan?\" I say, and he turns to me, looks at me like he used to, like he did back when I looked at him and saw part of my family.\n\n\"I'm sorry,\" he says. \"I'm so sorry,\" and then he is hugging me and I don't stiffen. I think I will but I don't because I know Dan's hugs and I have missed them. I have missed him.\n\nWords can lie but hugs can't. You know when they are real and this is real and Dan is here and that means he didn't leave me, that he's not going to send me away to Mom's parents or to some boarding school or just kick me out.\n\nIt means the Dan I knew is here. That I still matter to him.\n\n\"I got lost,\" he says. \"When Lisa died, what you said is true. I've been pretending with you and she was my heart, my world, and she would hate me for how I've acted.\" He pulls back and looks at me. \"I won't leave you again, I swear.\"\n\nI look at him. I want to believe him. I want to believe him more than anything because everything seems good now but I believed him before too.\n\n\"I\u2014all right,\" he says when I am silent. \"I'll prove it. I'll be here. And I'm going to keep being here. But there is one thing you have to know, that you have to hear. I'm not sorry for the choices I made about your mother and brother. I do know it's what she would have wanted. Do you hear me? I know.\"\n\n\"No. You believe.\"\n\nI wait for the withdrawal. The sigh, the words of concern that mean nothing, and the hurt look that will show just that. Just hurt. Nothing past it. Nothing for me.\n\n\"Okay, Emma, you have your beliefs and they...they hurt me,\" he says. \"You make it seem like I never loved your mother, and that I somehow chose the baby over her. Do you really believe I don't love your mother? That I wouldn't do anything to have her back?\"\n\nI look at him.\n\n\"No,\" I say, and my voice comes out rusty, the word squeezed from me. \"I know you loved her. I just\u2014why did she go like that? Why couldn't she have at least said goodbye?\"\n\nAnd then I cry. Dan hugs me again like he used to, like he's really here. Like I matter again. And it's not perfect; it'll never be like it was, there will always be the time I needed him and he wasn't there, and I still can't believe Mom would have ever chosen this silent, frozen life that isn't one. I will always know the fear Dan saw in her wasn't the fear that really was.\n\nI hug him back, though. I don't know if I will ever be able to live with how he shut me out when Mom died, but I do know that Mom loved him. That I loved him.\n\nI know Mom still loves him, because that's who she was.\n\nI don't love him like I did before, but I also know I've missed him even though I wouldn't admit it. I know I've missed how things were.\n\nAnd I believe he's missed that too. I believe he's missed me.\n\nI believe he wants me around and that makes me feel something I haven't felt in a while.\n\nIt makes me feel safe.\n41\n\nMom was not a good storyteller. When I was little, her fairy tales always got mixed up and forever ended with \"And then some things happened and everyone lived happily ever after. The end.\"\n\n\"What things?\" I'd say and she'd say, \"Fairy-tale things,\" and I started asking for stories from books, and I'm not sure who was more relieved by that, me or her.\n\nDan was always trying to get her to tell better stories. He would say, \"And then what happened?\" when they first started going out, and even after they got married, even after it was obvious Mom would say, \"Well, then we had a meeting and there's some stuff that has to be worked out,\" he kept trying.\n\nAnd then Mom got involved in this big deal at work and suddenly she was full of stories. Unfortunately, they were all about forms and meetings and phone calls, and for weeks Dan and I nodded as she said, \"And then I said, 'Tyler, you find that spreadsheet\u2014that's the C one, not the B one\u2014and get me those figures so I can fill out that 5673' and then I took a break and ate an apple and it was one of those awful ones that look great but are all mushy and\u2014\"\n\nShe kept talking and Dan and I looked at each other and sighed.\n\n\"Hey,\" Mom said. \"I saw that. Is my story that boring?\"\n\n\"Nope,\" Dan and I said at the same time and Mom said, \"You two,\" and shook her head. \"I feel like I'm on the outside here, the person who can't be in the 'tells good stories' group.\"\n\nDan and I both said, \"No, they're great!\"\u2014again at the same time\u2014and Mom laughed and said, \"Liars. But I love you both anyway. And this deal is important. The bit about the apple\u2014that, well, okay that not so much.\"\n\nDan laughed and I said, \"You can't be on the outside.\"\n\n\"You haven't seen you two making faces at my stories,\" she said.\n\n\"No,\" I said. \"I mean, you can't be on the outside because you're everything,\" and Mom smiled at me and said, \"Emma, a family is more than one person. That's why it's a family.\"\n\nAnd she was right.\n\nBut then again, she usually was.\n\nI can still remember her, pregnant, sitting and watching TV while rubbing her stomach. I actually thought that was normal. But then I realized it wasn't.\n\nI realized too late that she knew our family would change forever.\n\nBut now, things might be okay.\n\nShe'd like that.\n42\n\nWhen Dan and I get to the hospital, I head for the waiting room.\n\nDan says, \"Thank you,\" and smiles at me.\n\nI smile back, and who knew a day could be like this, so bad and so good? But then last night was the same too, so awful and then Caleb.\n\nCaleb, who I'm going to see.\n\nBut when I step into the waiting room, I see Olivia.\n\nI freeze, and Olivia sees it, I know she does, and it isn't that I don't want to see her or that I wish she was Caleb (although, a small, horrible part of me wishes she was), but that Olivia isn't\u2014\n\nShe isn't supposed to be here.\n\nOlivia is my link to the me from before. To how things used to be, and although everything has changed, the one thing that hasn't is us.\n\n\"You don't want to see me,\" she says and she's hurt, I know how her voice wobbles when she's upset, when something's happened that she doesn't want.\n\n\"No, it's not\u2014I just didn't think I'd see you,\" I say, but she gets up to leave, and I see she's done something to her hair after all, put in little braids along the front but they look nice, they have little bows at the bottom and I know Olivia probably wants them to be ironic but they look cute.\n\n\"Stop,\" I say, and she sniffs once and then does, turning her face to the side like she does when she's trying not to cry and this is the thing about her being here: she is so healthy. She is so normal. She doesn't belong here and this is what I didn't want. I didn't want what was left of who I was to disappear.\n\n\"When you're here I can't\u2014you're my link to before.\"\n\n\"Before? Oh. You mean before your mom died.\"\n\nI nod, and she says, \"That's why you're always asking me about me, right? Why you only tell me a little about what's going on with you?\"\n\n\"You know what's going on. My mother's dead and there's a baby.\"\n\n\"But you don't want me here.\"\n\n\"It's not that. It's just that\u2014\"\n\n\"I'm not Caleb Harrison?\"\n\n\"What?\"\n\n\"I'm not stupid,\" she says. \"I've seen you two and now everyone is talking about how you're hanging out with Caleb and you haven't said hardly anything to me about it even though I've asked and asked.\"\n\n\"Olivia\u2014\"\n\n\"I get it, you're talking to him and you're looking at him and he's looking at you and maybe he gets stuff I don't because of his sister, but do you know how it feels to think that your best friend would rather talk to some guy she barely knows than you? And that I have to hear about it from everyone else! I mean, Anthony tried to talk to me about it. Anthony!\"\n\n\"I'm sorry, so sorry, but it's not like that. You do know everything, I swear. I just wanted part of my life to not be about this.\" I gesture at the waiting room.\n\n\"So when you and Caleb talk, you just talk about your mom?\"\n\nShe's got me and she knows it, but then, Olivia knows me.\n\n\"More than just Mom.\"\n\nHer eyes fill with tears but she blinks twice, hard. \"You weren't in school today. What happened?\"\n\n\"I saw Dan in the nursery last night. He was putting together a crib and I just\u2014I had to get out of there.\"\n\n\"I know. He called.\" Olivia folds her arms across her chest, her little braids dancing as she does. \"You went out? Caleb?\"\n\n\"Okay, yes, I saw Caleb last night.\"\n\n\"Don't you want to know how I know? Don't you think he was talking about it in school today?\"\n\n\"No. I know he wouldn't do that. You know because you know me.\"\n\n\"And you know Caleb Harrison well enough to know what he would or wouldn't say? Are you forgetting this is the guy who drove his dad's car into the lake because he could?\"\n\n\"No, I know what he's done and yes, I know him,\" I say, and Olivia sighs.\n\n\"All right,\" she mutters. \"I guess you do, because he didn't say anything about you and him to anyone. You really weren't worried?\" I shake my head and she sits down. \"Why didn't you come see me last night?\"\n\nI sit next to her. \"Because you really are the only person in my life who's been there for everything. For before Mom died. For after. And you haven't changed. Everything else has, but you haven't, and I don't want...\" I swallow. \"I don't want you to change. I don't want us to change.\"\n\n\"But we have. Emma, we're still best friends, but when your mom died, everything did change. You've changed. For starters, you're for sure failing all your classes. And then there's Caleb. Don't you think I've seen all of that? What kind of friend would I be if I didn't?\"\n\n\"Olivia,\" I say helplessly, and she puts an arm around me.\n\n\"You're my best friend,\" she says. \"That hasn't changed. And I get that you need to not talk about your mom and your brother all the time and it's not awful for me to talk about Roger. But you don't have to go to Caleb Harrison to talk about stuff. You never would have done that\u2014\"\n\n\"Before. I know,\" I say. \"But that's just it, isn't it? That was before, not now. And he\u2014I like him.\"\n\n\"You do remember that he's really messed up, right?\"\n\n\"Who isn't?\" I say, and she looks at me for a moment.\n\n\"I want to say something really smart right now, but I got nothing. I hate that.\"\n\n\"You don't need to say something smart. You put up with me. You're here. You've always been here.\"\n\n\"Not here, though.\"\n\n\"No,\" I say. \"But now you are. What's it like?\"\n\n\"Scary. Sad. You do this every day?\"\n\n\"Yeah,\" I say, and rest my head on her shoulder. \"I like your braids.\"\n\n\"We shouldn't talk about me.\"\n\n\"Why not? I want it to be you and me, not just me. You remind me that life is still out there.\"\n\n\"And Caleb?\"\n\nI sit up and look at my hands. \"He reminds me of that too, but it's different with him.\"\n\n\"Because he wasn't there before. And because you think he's hot.\"\n\n\"I...yeah,\" I mutter, scooching down in my chair and she scooches down too, fiddling with the end of one of her braids.\n\n\"So you and Caleb?\"\n\n\"It's\u2014\"\n\n\"Complicated?\"\n\nI nod.\n\n\"It's always complicated,\" Olivia says. \"Roger's ex-girlfriend is calling him. It's not like what you're dealing with but\u2014\" She picks at one of her braids. \"And now I'm talking about myself again.\"\n\n\"And you came here. You're amazing, you know. Roger's ex doesn't stand a chance.\"\n\n\"Amazing?\" she says, and I grin at her.\n\n\"No, you can't do anything else to your hair.\"\n\n\"Should I leave?\"\n\n\"No,\" I say. \"Stay.\"\n\nSo she does, until Dan comes and it's time for me to go see Mom.\n43\n\nDan stops me before I go in to see Mom. He says, \"Emma?\" and I hear Something's wrong in his voice.\n\nMy stomach twists. I know there is more hurt to come, but I don't know what kind. I sag against the wall and look at him. I wish Caleb was here. I'm glad I talked to Olivia, I am, but Caleb knows what this is like.\n\n\"Is it Mom?\" I whisper, and Dan shakes his head but slowly, slowly.\n\n\"She's\u2014well, the machines are working fine,\" he says. \"Some toxins have built up in her bloodstream but the doctor says that's normal.\"\n\nHe looks at me. \"I hate how the doctor talks about her. It's like once I made the choice to\u2014once it was made, Lisa is...everyone here talks about her like she's here and I catch myself thinking that I need to tell her something or...\" He trails off.\n\nI can't do this. I can't say, No, Dan, you can't talk to her like I would have because I finally see he gets that she's gone too. I see her and I know she's dead but I still hope and now I know that he does too. I see that someone can be gone and you can still think they will come back.\n\nI see that even when there is no hope, you can still have it.\n\n\"Is it very bad?\" I say, and it hurts to say it. It really does. The words cut into my throat, slice over my tongue.\n\n\"The baby...\" Dan says and I tense, thinking of course, of course, and he sees that. He takes a deep breath and takes a step back. Says, \"You should go see her.\"\n\nI see how alone he is, and we found a way to reach each other but I can't get there now. I can't forget that Mom walked around holding her stomach and never talked about the baby. I can't forget what I know.\n\nAnd that's how we part. No words of consolation from either of us. To either of us, and I worry that what we've built will fall apart. I worry nothing will survive, that we all died the morning Mom did.\n\n\"Hey,\" I say when I walk into her room, when I see her. Her face is swollen, and there are more IVs in her arms.\n\nIt hurts to look at her.\n\n\"I remembered you telling stories,\" I say. \"I miss them. Even the ones about the business deal, the ones Dan and I...\" I trail off.\n\n\"I found all the food you put away,\" I tell her. \"I found it last night. I...Mom, I drank your bottle of wine.\" I look at the monitors beeping away. Beeping for her. \"I went to see Caleb after that. We slept together. Not like you think, okay? I swear. But I did stay all night and he was so\u2014Mom, I don't get it. I was sure he was glad I was there, even if I did throw up a lot, but I haven't seen him today and I'm scared. I was sure before too, remember? With Anthony. And I was wrong. I just...I just want to know what to do. You knew with Dad and with Dan. How did you know?\"\n\nThe monitors beep and that's it. That's my reply.\n\nI look at Mom, and she looks so wrong. So...no. I can't think it, even if it is. She's so bloated and her skin is weird-colored and my eyes are burning.\n\nI turn away, but not before I catch a glimpse of her stomach rippling, of the baby turning inside her.\n\nFor the first time, I wonder what it's like for him. I see her, but he lives inside her. He lived in her when she was alive. I wonder if he knows the difference.\n\nI wonder if he is scared.\n\n\"Hey,\" I say, and not to Mom. Her belly ripples again and I swallow. I should say something else but I can't. Everything is tangled up inside me, grief and love and hate and worry.\n\nI have never spoken to my brother before.\n\nI close my eyes and keep them closed until Dan comes.\n\nI watch as he kisses Mom goodbye, his lips to hers. I watch him put a hand on her stomach. \"Be back soon,\" he says, and there is no movement under her skin.\n\nNo reply.\n44\n\nDan and I walk in silence to the elevator and as we're waiting for it, I hear a familiar sound, a squeaky cart.\n\nCaleb.\n\nI look up, see him and watch him smile at me. Watch the curve of his mouth, the one I wanted to kiss last night.\n\nThe one I still want to kiss, and I fidget, start to look away, but then remember what he said. That he wanted to kiss me. That we were together, all night, and granted he spent a lot of it holding my hair while I puked, but he was there. He helped me and he held me and when I left I knew I'd see him later.\n\nAnd so I smile back and it's not like time stops\u2014I don't have that kind of life anymore because I used to believe that everything would be okay. That I could have a fairy-tale perfect life.\n\nI smile back and I'm simply happy to see him.\n\nI believe he's happy to see me, and that's enough.\n\n\"Emma?\" Dan says, and I look at him, see he's stepped into the elevator.\n\n\"Right, coming,\" I say and just as I get ready to step on, Caleb walks by, brushing a hand against mine and sliding a piece of paper into it.\n\n\"You didn't say hi to your friend,\" Dan says as we get off the elevator, when we're walking out to the car. \"Or is he not your friend? I don't know very much about what's going on with you now. I know you see Olivia.\"\n\nI nod, and I know the note isn't burning my skin. I know that. But it sure feels like it is because I want to read it so bad.\n\n\"So, I got a call from your AP History teacher and I also got a letter from the school. What's going on with you and your classes?\" Dan says, and I stop and look at him.\n\n\"Oh,\" he says. \"That bad?\"\n\n\"I'm done,\" I say.\n\n\"But you like school so much. Too much, Lisa always says. Said.\" His voice cracks on that word. \"It's different now, isn't it?\"\n\n\"Yes,\" I say. \"Everything's different now.\"\n\nDan's silent as we get into the car and then he says, \"So, that guy. Caleb, right?\"\n\nI nod.\n\n\"Friend? Not friend?\"\n\n\"Friend.\"\n\nDan taps one hand against the steering wheel. \"Is he who you were with last night?\"\n\n\"It's not like that.\"\n\n\"Like what?\"\n\n\"Like what you're thinking. I feel like I'm taped together most of the time, like I'm a shadow of the Emma that used to be.\"\n\n\"That's when things can happen,\" he says, and then blows out a breath. \"That's what I feel like I should say. But I know what you mean about being taped together and after this morning, I think that what happened last night was important because it made us talk for real. Although I do wish you hadn't gotten drunk.\"\n\n\"Not as much as I do.\"\n\n\"You got sick?\"\n\n\"More than once.\"\n\n\"What would she do?\" he says and I look at the road, my eyes filling with tears.\n\n\"I don't\u2014it would be different. None of it would have ever happened.\"\n\n\"Even last night with Caleb?\"\n\nI nod.\n\n\"Is he like Anthony?\"\n\n\"No.\"\n\n\"Good. Your mother would be happy about that,\" Dan says. \"I am too. So, he's nice?\"\n\nI think about what that question would have meant before, back when my life was grades and planning for the future. When I was so sure I knew what I wanted and I never looked at someone who wasn't in my classes, who didn't have the drive I did, the belief that grades and getting into the best possible college meant everything. When I was that girl, I would have heard the stories\u2014if I ever even noticed him\u2014and believed them, looked through Caleb if I happened to see him. He didn't get what was important, and so he wouldn't have mattered.\n\nI think about what it means now. How I missed so much trying to be the best student. How I could have spent time with Mom and didn't. How I never would have bothered to look past the surface because I was so busy chasing what I thought mattered more. How I thought I could create my future, how I believed I could shape all of it.\n\nI know better now, and Caleb matters to me. He matters to me in a way that's new. That no one else has and it's because he's seen everything, he has seen that I am made up of grief and fury and fear and held a hand out to me. Not to save me, but to just be there.\n\n\"He's nice,\" I say to Dan, who says, \"I'm glad, because Anthony\u2014having to listen to him talk when you were doing that debate thing last year\u2014Emma, that boy is an ass.\"\n\n\"He really is,\" I say, and then we are both laughing but it's a little too loud, a little too hard. A little too brittle.\n\nWe have forgotten how to do this normally. We are doing this without Mom and it's weird and we both know it.\n\nBut still, we try, and when we get back to the house Dan says, \"We're going to be okay.\"\n\nI look at him, and I can't nod. I can't say yes. But I can say, \"I hope so,\" because it's the truth. Because I can feel myself hoping, and it's scary but it's nice too.\n\nThe phone rings and Dan answers it. I head up to my room. I leave the door open.\n\nI sit on my bed and open my hand. I've kept it closed all the way home and my heart is pounding as I look at the piece of paper inside. It's just a note, but it matters.\n\nCaleb matters.\n\nI've never seen his handwriting before. It's cramped, the letters printed, no loops or curls of cursive.\n\nWant to see you, stuck down in recovery handing out magazines. Do you want to come see me tonight? I'm at home if you do.\n\nI fold the note back up and put it in my pocket, walk downstairs and realize I'm tracing over it with one hand, like I can feel Caleb's words through my jeans.\n\nDan's in the living room, still on the phone. \"Remember the Florida thing?\" he says to me and then \"Hang on,\" into the phone.\n\nI nod. I remember how he was going to go and talk about what's been done to Mom.\n\nThis morning and even the car ride back here suddenly seem very far away and I think again about how Dan never asked me about Mom. I was just her daughter. I just spent my entire life with her. I just love her.\n\n\"I've been thinking,\" Dan says, and I look at him. \"I'm not going to go down there.\"\n\n\"You're not?\" Does he get it? Does he really finally get it?\n\n\"I can't,\" he says. \"You've heard what's going on, and I can't leave the baby now. I can't leave your mother now. So I'm going to do some teleconference thing with the lawyers. It might be a while, so I was thinking that for dinner\u2014\"\n\nHe doesn't get it.\n\n\"I'm going out,\" I say.\n\n\"Emma, we've talked about this. You know how I feel, and\u2014\"\n\n\"Yeah, I know, and there's the money for doing it, right?\"\n\n\"This isn't about the money. At least...we do need it, but I truly believe that if there's a chance another baby can make it and her mother wanted it, then\u2014\"\n\n\"This mom said that before she died?\"\n\nDan looks away from me. \"Her husband knows what she wanted, and I think that a father's wishes count\u2014\"\n\n\"For everything.\"\n\n\"No, but he should have a say,\" he says, and I see the moment he gets it, that he remembers what I said to him this morning.\n\n\"Emma, I'm sorry. I was in so much pain, and I just wanted to do what your mother wanted, I just wanted to make sure that the baby\u2014\"\n\n\"I know,\" I say for what feels like the millionth time, and I am so tired of talking to Dan, of hearing his attempts to make what he's done okay. I'm tired of him pretending he never saw what Mom was really afraid of, that she clutched her belly and kept silent about the baby because she was scared.\n\nI'm tired of him pretending like I got any say in what happened. I'm tired of how he just left me and chose what he wanted. Once that happened, we weren't a family because families talk and he didn't ask me what I wanted. If he had, I\u2014I don't know. I just know I never got to say anything. My voice didn't matter.\n\nMom knew I was going to get chicken pox before I ever got my first spot. She knew Dan was special as soon as she met him. She knew she loved my dad on their first date. She knew I'd get over Anthony.\n\nI wish I'd spent that last night with her, that I'd put away my books before it was too late and sat with her. That I hadn't been so sure about making the future what I wanted that I forgot the present. That I had a memory of her right before she died besides being at the hospital and hearing she was dead. Before I stood there, alone, and realized I would never see her again.\n\nBefore I was told I could see her. That I ended up in this place, this here.\n\nDan says something as I leave but I don't stop to listen. I have heard it all before and I don't need to hear it again. I don't want this morning and the talk we had poisoned.\n\nI'm afraid it is, though.\n\nI walk and reach into my pocket. I feel the note, Caleb's words, and my heart flutters. Maybe it shouldn't do that, and maybe I shouldn't want it to.\n\nBut it does, and I do.\n45\n\nCaleb's house looks enormous in the dark, and I shiver a little, not from cold, but from memory of the place, as I head toward it.\n\nHe comes down to meet me.\n\n\"Hey,\" I say. \"How did you know it was me?\"\n\n\"I saw you coming,\" he says. \"Not that I was looking for\u2014never mind. I was, you know. Hoping.\"\n\nI walk over to him. \"Have you ever had a day that was good and then bad and then good and then bad and then good again?\"\n\n\"Your mom?\"\n\n\"I\u2014\" I say and then I'm spilling it all out, how I was happy this morning and then talked to Dan and then things seemed like they would be different but okay, and then we talked again and things were different but not okay. \"It felt just like it has since Mom died,\" I said. \"And I\u2014\"\n\n\"Hoped,\" Caleb says. \"You hoped. I've done that.\"\n\n\"Does it always suck?\"\n\n\"With my parents it does. But not with everything.\"\n\n\"You're right,\" I say softly and he is. Hope doesn't always suck because I felt it when I got his note. I felt it walking over here, at the idea of seeing him. I feel it now, when I am with him.\n\nHope is so simple and so hard to have but it's here, and we have it and it's about each other.\n\n\"You wanna come up?\" he says, and I nod because I do.\n\nAnd then, just in case he didn't see, I reach out and take his hand. I've felt it before but now it's not about fear of his house or trying to provide some sort of comfort. It's not a quick brush of fingers, the passing of a note.\n\nI take his hand just to hold it.\n\nHis fingers twine with mine and we head up to his room. It's as bare as I remember but smells, strangely enough, like sugar.\n\n\"Are you cooking or\u2014?\" I say and then break off as I see what's sitting on his counter.\n\nI see a tiny cotton candy machine, like people buy for kids, blue with little animals on it, and beside it are two cones of pink sugar, propped up in a coffee cup.\n\n\"Oh,\" I say because it is all I can say. He did this, he went out and bought a cotton candy machine and made cotton candy, real cotton candy, and it is the sweetest, most amazing thing.\n\n\"It's probably cold,\" he says. \"It's not as easy as you'd think to make it and I burned some sugar and it probably smells in here and I'm sorry about that and also, I made three cones but I ate one of them and\u2014\"\n\n\"Caleb,\" I say and he stops talking. He stops talking and he looks at me and then he is right there, he is right next to me, in front of me, all around me, and he smells like sugar and I thought I understood want when I was with Anthony that night in the lab but I didn't, it takes you over, your blood, your breath. It is you, it is the world, it is everything, and when his lips touch mine there is nothing but that. But us.\n\nHe tastes like sugar, he tastes like Caleb and I want more, I want him. I wrap my arms around him and that's where thought stops. I am all sensation; his breath, the taste of his mouth, his tongue against mine, his lips on my throat, his hair twining around my fingers, his hands on my waist, my hips, and he pulls back, breathing hard, and looks at me like I am\u2014\n\nHe looks at me like I am beautiful, and when he does, I am.\n\n\"Emma,\" he says, and he is shaking and I did that to him and I am shaking too and maybe I should be scared but I'm not, I feel, and for once it's not anger or sadness or worry it's just want and happiness and I didn't know it could be like this.\n\nEven before, I didn't know.\n\nI pick up one of the cones of cotton candy. He made this for me. I look at him and he is still looking at me as if I'm the only person in the world, as if I'm everything, and as I pick off a piece of cotton candy and eat it, I feel him watching me, I feel him watching my mouth and I hold out the cone, watch him blink at it, watch him sway a little and then grin, take it from my hand.\n\nI watch him eat a piece and I know I am looking at him but I don't care, I want to look at him, at this guy I never saw, who I would have written off and would have never known, and I can't bear the thought of that, of this not happening, and I say, \"Caleb,\" and I hear what is in my voice, all the wonder. All the joy.\n\nHe smiles and he is all I can see.\n\nSo I kiss him, and the world is just us again and I hope it stays that way forever and ever and ever.\n\nBut of course it doesn't.\n46\n\nHis parents come home. It's hard not to notice, even when you are in the middle of an incredible kiss, because his room shakes and I hear the car.\n\nSo does he and we pull apart and look at each other. His lips are redder than usual and his hair is tousled and I did that, I had my hands in his hair.\n\nI was touching him and it's enough to make me forget about the car.\n\nBut then another one comes in and the hot light in Caleb's eyes dims and he looks down at the floor as if he can see through it to the garage. To who is in the cars.\n\n\"Caleb,\" I hear, and it's a woman's voice, nice-sounding, sweet-sounding even, and it must be his mother. She doesn't sound like I thought she would, but then I see Caleb's face and remember the house and before I know it I have reached out and taken his hand again.\n\n\"I have to go see them,\" he says. \"You don't need to\u2014you shouldn't have to deal with them.\"\n\nI squeeze his hand gently and say, \"I want to.\"\n\n\"I can't,\" he says, and pulls his hand away. \"I can't.\"\n\n\"Oh.\"\n\n\"It's not you,\" he says. \"I know how that sounds, but it's not like that.\" He looks frantic, frightened.\n\n\"So you want me to go?\"\n\nHe looks at me but doesn't nod. Doesn't say yes. He just stands there.\n\n\"Caleb?\" I say and his mouth works but nothing comes out. It's like he can't get the words out, like they are too hard, and then he rubs a hand down his face and I see there are tears in his eyes.\n\n\"They'll make you hate me,\" he says. \"And if they make you hate me, I don't know what\u2014\"\n\n\"They can't make me hate you.\"\n\n\"I hate myself when I'm with them,\" he says, his voice breaking. \"So how will you be able to not hate me?\"\n\n\"Because I know you.\"\n\nHe shakes his head, like it can't be that simple, and says, \"No, I\u2014you don't get it.\"\n\nI step on his foot. Hard. Mom used to do it to Dan when she wanted his attention and he was busy cooking or daydreaming about some database thing. It seemed to work for her.\n\n\"Ouch,\" he says, and looks at me. Sees me looking at him.\n\nI guess it works for me too.\n\n\"I know you,\" I say again. \"And you know me. So trust me.\"\n\n\"Emma,\" Caleb starts, and I lift my foot up again, lift it up very high. He sees that, and I watch the shadows in his eyes fall back a bit. See the tiniest bit of a smile, fast and fleeting but there, on his face.\n\n\"Do you trust me?\" he says, low-voiced, and his hands are shaking.\n\nI wonder if he has ever asked anyone else this, and I think he probably has. I think he asked the people who are supposed to love him and I think they said No. I think they saw he was breaking and didn't care. I think they let him become broken.\n\nAnd still he pulled the pieces of who he is together. He is who he is because of who he is and nothing more and that makes him so special.\n\n\"I trust you,\" I say, and look at him. I let what is in my heart out, I let it into my eyes.\n\nI see him look at me. I see him hear what I've said. I see him look into my eyes.\n\n\"Emma,\" he says, and it's too soon to say it, way too soon, but what's in my heart is in his too and he touches my face, cupping the side of my jaw and my name is a prayer in his voice, four letters of joy.\n\nAnd then we head downstairs. We walk through the garage, which is dark again but warm and filled with the sounds of engines cooling, and then he opens a door to the house.\n\nWe walk through it.\n47\n\nThe house is just like I remember, perfect-looking.\n\nIncredibly sad, a monument to Minnie. It's as if even his parents don't even live here, like this really is a giant, gilded tomb.\n\nCaleb doesn't say anything as we walk through the house but he's holding my hand again and it's shaking a little, damp against my skin.\n\nIf you'd told me that I'd be walking hand in hand with Caleb Harrison just a few months ago, I'd have laughed at you. Said, \"Who?\" probably, and then, \"Oh, him. Yeah, sure. That'll happen. And also, everyone will get a magic pony.\"\n\nBut here I am, and there is nowhere else I want to be.\n\n\"Hi, Dad,\" Caleb says, stopping abruptly at the doorway of a room, and I get a glimpse of it over his shoulder. It's one of the ones he showed me, filled with untouched-looking furniture and a portrait of Minnie hanging on the wall with all the lights in the room shining on it.\n\nHis father says something. It might be \"Hello,\" but it might just be a grunt. He's looking through a folder of papers with one hand, a glass more than half-full of amber liquid in another. I don't need to wonder what it is. I can smell the alcohol from here.\n\n\"Caleb,\" someone else says. I see his mother stepping out of a dim corner of the room, stepping away from a chair covered with files. He does have her hair, but where his is free, wild and curly and falling all over the place, hers is pulled back into a tight knot with exactly two strands allowed loose, curling over her ears and down along her face.\n\n\"Hi, Mom,\" he says. \"How was work?\"\n\n\"Fine,\" she says, and the way she's looking at him is\u2014it's like he's some sort of animal that she's forced to see and doesn't want to.\n\nShe hates him for living. I see that, and the force of that hate makes me want to cry.\n\nI can't even imagine how it makes Caleb feel.\n\n\"Do you need something?\" she says, and she's so polite, but it's the kind of polite you are to someone you know you have to see but don't want to. The kind that is all surface.\n\nShe blinks, glancing at me, and for a moment, I see who she really is under the icy layers, the hate. I see pain so deep it's endless.\n\nI see that she is broken too, and that she blames Caleb for it.\n\n\"No,\" Caleb says, and his hand is really sweating in mine now, and he's holding on tight, holding on as if I am keeping him standing upright. \"I just heard you call when you came in. I'll go now.\"\n\nHis mother doesn't say a word, just shrugs, and I watch her eyes fall on Minnie's portrait as if they have to, as if it is the only thing she can see. I watch her face relax, soften.\n\nI realize Caleb is no longer around. Not in her mind. She saw him just now, but she didn't, not really. I bet she hasn't seen him since the night Minnie died.\n\nI knew grief could destroy you, but I didn't know it could turn you into the walking dead. A shiver crawls up my spine because there is something in her blankness and hatred I understand. Something I see in myself.\n\nIt would be so easy for me to make my entire life about Mom, about what happened to her. It would be so easy for my whole life to be hating Dan. Hating the baby.\n\nCaleb feels my shudder because he glances at me and then starts to turn away, starts to walk us away from his parents.\n\n\"It's polite to introduce your guests,\" his father says, and his voice is falsely chipper, laced with something bitter.\n\n\"I\u2014sorry,\" Caleb says. \"Mom, Dad, this is Emma.\"\n\n\"Emma,\" his father says, and drains the rest of his glass. \"Very nice to meet you.\"\n\nHe doesn't look at me when he says it.\n\n\"Emma?\" his mother says, turning away from the picture of Minnie, and her gaze is so needy and empty and angry and lonely it actually hurts to see it. \"Wait, I know that name. You're that girl, aren't you? The one whose mother died.\"\n\nCaleb's hand suddenly clamps down hard on mine and it's not just his fingers that are shaking now. He's shaking all over, trembling so hard I can feel it, and I get why just as she says, \"There was something else too...a baby?\"\n\n\"Yes,\" I manage to get out, and Caleb's father says, \"Are you the girl whose father is keeping the body alive for the baby?\"\n\n\"Stepfather, and yes,\" I say again, and he looks at me as if he didn't expect me to speak.\n\n\"I'm sorry,\" Caleb's mother says, and her voice is very stiff and very brittle, but I think she means it. \"I know how painful loss can be.\"\n\nShe looks at Caleb, and her gaze changes. So does her voice. \"When did you meet my son?\"\n\n\"We go to school together.\"\n\nHis father looks at me now. \"And you started talking after her mother died?\"\n\n\"I\u2014\" I say, and then realize he isn't talking to me. He's talking to Caleb.\n\n\"Yes,\" Caleb says, low-voiced. \"We started talking then.\"\n\n\"Of course you would. First the drugs, then the cars, but you will not hurt another girl. Do you understand me? You. Will. Not.\"\n\nHe looks at me then.\n\n\"You need to leave,\" Caleb's father says, and he sounds normal. But the light in his eyes isn't.\n\n\"Yes,\" his mother says. \"Go right now. You don't\u2014\" She makes a low, keening noise. \"You don't want to get hurt, do you?\"\n\nThey want me to go. No, more than that. They mean I should go and leave Caleb. They\u2014\n\nI think they think they are saving me in their own icy, twisted way.\n\nThey think he could hurt me.\n\nThat he would.\n\nCaleb, staring at the floor, starts to let go of my hand but I won't do it. I hold on.\n\nI hold on and look at him until he feels my gaze. I look at him until he looks at me and then I say, \"I'm happy to go, but I want Caleb to come with me. It's dark out, you know, and I'd feel better if he could walk me home. I know I'm safe with him.\"\n\nHis mother starts to say something but then stops and turns back to the picture of Minnie as if I'm no longer there at all.\n\nHis father doesn't look away, though. He closes his folder and says, \"Has he told you that he hit someone? That he stole a car and then ran someone down? That was at night. So I don't know how safe you'd be with him. His own sister certainly wasn't.\"\n\nI can't believe this. I can't believe them. How they talk to him. How they treat him.\n\nHow they don't see who he is at all. I take a deep breath.\n\n\"I know he hit someone because he told me. And Minnie? I think she was safe with him. She knew she was supposed to wear a helmet when she rode her bike but somehow you can't see that. Caleb did his best and all you do is blame him for it,\" I say. \"And me? I know I'm safe with Caleb.\"\n\nHis father shakes his head and gets up. He walks out of the room, careful not to touch either Caleb or me. His mother stays where she is, but she is looking at me as if I've spoken a foreign language.\n\nAnd for a second\u2014just a second\u2014I think she hears me.\n\nBut then she goes back to the dark corner. To her chair, where I realize she can look up and see only Minnie's portrait.\n\nCaleb doesn't move. He just stands there and I turn to him, worried about what I'll see.\n\nHe looks upset, but he also looks surprised, and when I start to walk away, ready to get out of this house, he follows, his hand still in mine.\n\n\"Emma,\" he says as we approach the door and then he puts his arms around me, right there in that frozen house. He's shaking, hard, and I realize that part of him still believed he killed Minnie, that even though he told himself it wasn't true, that even when he tried to live as if it wasn't, there was a part that screamed it was.\n\nThere are other things he's done, things he will have to live with, and there is no perfect ever-after coming for him, but I swear I can actually feel some of the anger and grief he carries flowing away, flowing out of him.\n\nIt lasts for a second, maybe two, but that's enough. I can tell because when he whispers, \"Let's get out of here,\" I know it's because he wants to and not because he has to.\n\nMaybe that one word\u2014wants\u2014doesn't seem like much of a difference.\n\nBut it is.\n\n\"You believe in me,\" he says when we're outside, out on the street and this dark is so much warmer than that house. \"You trust me.\" He touches my hair, my face, and then kisses me once, gently.\n\n\"Not enough,\" he murmurs, and kisses me again, kisses me harder, and I smile, open my arms to him, myself to him, and we stand there in the darkness, in its embrace, its warmth and fall into each other.\n\nIt's so much. It's everything.\n\nIt's late, after midnight, when my stomach rumbles and his does too and we go and get burgers, eat them in the parking lot after walking through the drive-through and laughing at the looks we get, and we are still kissing. In between bites of burger and sips of soda we are kissing like we can't stop.\n\nHe pulls away, breathing hard, and says, \"This is crazy.\"\n\n\"Yes.\"\n\n\"But right.\"\n\n\"Yes,\" I say again and we grin at each other. He looks at me, right at me, in that way he has, like he sees all the way inside me and what he sees is beautiful, and I close my eyes because I don't want this moment to end. I like being happy.\n\nIt's nice, and I don't think I ever enjoyed it like I should have.\n\nI didn't know happiness could be lost, but it can be.\n\n\"Emma, are you\u2014?\" He pauses. \"You should go home, right? Talk to Dan about everything again.\"\n\n\"It's not my home,\" I say, and it hasn't been, not since Mom died, but I want it to be. I want that house to be my home again, I want it to be more than a place where I sleep, where memories coil around me and dig their way under my skin, leaving space for grief's thorns to burrow deeper still.\n\nI want things to be right again and I thought they could be today and then I realized they couldn't and I still\u2014\n\nI still have to go back.\n\nI still have to see Dan. And I do have to talk to him again. Not for him. Not even for Mom. But for me.\n\n\"You're right. I do need to talk to Dan.\"\n\nIt feels strange to say that. I don't know when I last said that and meant it.\n\nIt feels strange but not false. It feels true, and that's because it is. Because life has come and changed things. Changed Dan, changed me, and maybe grief is all we will share. Maybe it's all we will ever share now and will break the family we were into nothing.\n\nBut I should know. I need to know.\n\nSo Caleb walks me back to the house. Home. That word still feels false when I think it. Feels false when I look at where I live. At where I once took everything inside for granted.\n\nI kiss Caleb before I walk up the driveway, but I don't say goodbye. I've had enough of that word forever. I don't want to have to say it again. Not for a very, very long time.\n\n\"Emma,\" he says, and I look back at him. Watch him walk toward me, and he is the trifecta, like Olivia said. He is beautiful, but the thing that truly makes him so is on the inside.\n\nIs his heart, and I touch his chest, feel it beat against my hand.\n\nIt beats fast but sure. Solid. I like that.\n\n\"See you tomorrow. Well, today,\" he says, and then he kisses me again. His heart beats faster when our lips meet.\n\n\"Hi,\" he says when we separate, and smiles at me. He understands what it's like to not want to say goodbye too.\n\nAnd this\u2014all of this, everything that's happened between us\u2014it feels like something. It is something.\n\nIt's a beginning.\n\n\"Hi,\" I whisper back and then turn away and walk to the house. I look back once, because I know it's important to do that now. To see the people that are in your heart.\n\nHe's there, and he waves. I wave back and then go forward.\n\nThe kitchen light is on and I see Dan standing by the stove like he's waiting for me.\n\nMaybe he needs to do this too, to figure out if what we were can be somehow saved or if we have to let it all go.\n\nI walk into the house. I walk into the kitchen.\n\n\"Dan,\" I say, and he looks at me.\n\nHe looks at me and I know something is wrong. Horribly wrong.\n\n\"There's been\u2014\" he says and his voice is thick with tears. \"There's a problem, and this time I want to talk to you about it. I want to know what you think we should do.\"\n48\n\nI almost say, Mom?\n\nIn spite of everything, I almost say it. I almost ask What's happened to her? even though nothing can happen to her now.\n\nOver a month of her being gone and I still see her.\n\nEvery day, I see her.\n\nI've hated Dan for making his choice without asking me, for holding Mom's body here for the baby.\n\nBut what I hate him for most of all is that he's made it so I can see her. So that I can sit with her, touch her, watch her chest rise and fall. Every day, I see her breathe and tell myself she's dead. I tell myself she's gone, that she left the moment a blood vessel in her brain burst without warning.\n\nI tell myself she's dead.\n\nI have told myself that all along, but part of me doesn't believe it.\n\nI didn't see her die. I didn't see her fall to the ground. I didn't see her leave.\n\nI never even saw her body. Not until after Dan had made his choice. Not until after machines made her heart beat again, made her lungs breathe again, made her body here again.\n\nNot until after I could sit with her. Hold her hand. Rest my head on her shoulder.\n\nBe with her.\n\nI know all that's left is her body, but there's so much power in that, in seeing the face and hands and hair of my mother. I know that there's a place machines can't reach, that there is no way to hold on to who a person is when they die, no way to capture what makes them who they are and keep it with you.\n\nYou can hold on to their body, but not them. Who they are slips away. Flies free.\n\nBut it's hard to remember that when you see someone you love. When you can touch them, talk to them, hold their hand.\n\nI have hated Dan for choosing the baby, for keeping Mom's body alive.\n\nI have hated him for making me hope when I know there isn't any.\n\nI hate him for what he's done to Mom, but I hate him just as much for what he's done to me.\n\nSo I almost say, Mom? and as the word bubbles scorching up my throat I feel my hands knot into fists because it isn't her, it can't be her, he can't be talking about her, and yet here I am almost asking. Hoping.\n\nThis is what it all comes back to. To what happened, and how I never got a chance to say goodbye. I never got a chance to say anything.\n\nDan made his choice. He picked the baby without any thought, without even asking me what I thought. It was like I didn't exist in that moment and now Mom's body beats and breathes for it. Her body exists and I can see it. I can see her. I have seen her.\n\nBut it isn't her at all.\n\n\"It's not your mother,\" Dan says, seeing my face, my clenched fists, and I think of how Caleb must have felt when he came home from his summer away and saw his parents looking at him just like they had when his sister died. How empty he must have felt.\n\nHow furious.\n\n\"I know it's not her. She's dead.\"\n\nDan flinches. \"I know she's dead too,\" and he saw what I didn't. He saw Mom\u2014my mother, God, my mother\u2014he saw her life end. He saw her stop breathing. He was there when she left. He got to say goodbye like I haven't.\n\nI was happy\u2014just now, I was happy. I know I was. I can remember it, being happy just now with Caleb.\n\nHow can it be gone so fast? How can life be so cold?\n\nHow could it take Mom away from me?\n\nI want my mother. I had her for seventeen years, but I thought I had forever. I thought I was ready for anything, I had my whole life planned, but seventeen years is nothing when it's gone in the time it takes someone to reach for a piece of toast.\n\nIt's nothing when the one person you were sure would be there, would always be there, is gone.\n\nI sink to the floor now, weightless, boneless. I wait to cry but the tears won't come. They have always come before. I have cried more than I ever thought I could, oceans of tears since the day she died and I first saw her again.\n\nBut I can't cry now. I want to, but grief has wrapped itself around me so tight that there's nothing left. Its thorns have closed around me, burrowed all the way inside. Grief has found the tiny thread of hope I held on to in spite of everything and snipped it, pinned it with a thorn to my still-beating heart.\n\nDan isn't going to talk to me about Mom. He is alive. I am alive.\n\nShe isn't.\n\n\"Emma?\" Dan says, and he is kneeling next to me now. I can feel him looking at me. I can hear him breathing, and here we are exactly where I thought we'd never be. I was going to finish school and leave.\n\nI wasn't going to be left.\n\n\"I hate you,\" I say, and the words come out as flat and empty as I am, and Dan sits down across from me.\n\n\"I know,\" he says, and his voice is as flat as mine too. As empty. \"But I meant what I said when you came in. I didn't ask you what to do about your mother before, but I'll ask you now. It's\u2014it's the baby, Emma.\"\n\nThe baby.\n\nI stare at him.\n\n\"It's his heartbeat,\" Dan says softly. \"There are problems. It's not strong enough, not all the time. The doctor says there's a chance he can make it to forty-three days, to the twenty-five week mark, but if his heartbeat doesn't stabilize he might die. And all the doctor says is that there's a chance he'll make it. That's all he'll say. That there's a chance and\u2014\" He breaks off and then grabs my hands. His fingers are cold and shaking so hard I feel their tremors rattling across my palms, up my arms.\n\n\"What do I do?\" he says. \"Do I let the doctors deliver him now, even though he'll almost never make it, or do I wait and hope that he'll make it to forty-three days? I already know he'll never see his mother. I don't\u2014the thought Emma, of him never seeing me or you...\" He squeezes my hands tight, and now my hands are cold too.\n\nSo cold, and he says, \"What should I do?\"\n\nI look at him and he is looking back at me and he doesn't know what to do, he is scared and lost and the baby could die, his heart could stop beating just like Mom's did.\n\nOne heartbeat, two heartbeats, three heartbeats, more, and you never know when you have used yours up.\n\nThat's the thing. You don't know.\n\nHow long will your heart beat for? How many heartbeats do you have?\n\nI look at Dan and see he is waiting for my answer. That he needs me. That he wants me to help him. To be with him.\n\nTo be his family.\n\nBut I don't know what to say.\n49\n\nI don't know what to say but I do. That's the thing.\n\nI do.\n\nIt has always been about Dan and what he wants. It has always been about the baby.\n\nHe's kept Mom's body alive for the baby, kept her lying in a hospital bed, here but not here.\n\nLet her go, I told him when I finally knew, when it was too late, and that's all I need to say now.\n\nLet her go.\n\nLet him go.\n\nLet it all go.\n\nI pull away from Dan and stand up.\n\nI just have to say it. I just have to say Let go. Let her go.\n\nLet them go.\n\n\"Emma?\" Dan says, \"Emma, please talk to me,\" but I'm not listening to him. I don't want to even hear him.\n\nI step into the living room. I used to sit with Mom on the sofa. Right here, right where I am sitting down now. I should have sat with her more. Been with her more.\n\nI wish I had, but I didn't. I only sat with her when I thought I could, when my \"schedule\" would allow it.\n\n\"Emma,\" Dan says again, pleading now, and I shake my head.\n\nI didn't spend enough time with Mom. I thought I would see her get old and even worried about it once in a while, about what would happen to her, vague thready thoughts about how I would manage my life and take care of her and Dan when they were old.\n\nI worried about the future and I waited for it too. I wanted Mom to see me do all the things I'd planned.\n\nI thought it would matter if I was first in my class, if I got into a school that made people jealous, if I had a GPA that was so high no one else could reach it.\n\nI was going to talk about dedication when I was valedictorian, planned on standing on stage and talking about how I wanted to follow in my father's footsteps, about how I knew I was making my mother proud.\n\nI'd planned all that and it's too late to go back, too late to take back all those nights I worked for extra credit I never needed, the nights I spent preparing for tests I was already prepared for.\n\nIt's too late to take them back and to just sit with her on this sofa. Be with her.\n\nShe's gone, and I sat beside her a few nights before she died, desperate to get to my homework, but she'd had a bad day at work, lots of phone calls that left her upset and tense, and she just wanted me to watch TV with her for a while, said, \"Come on, sit and rest for a minute. If I can do it, you can do it. Right?\"\n\nI remember it so clearly. I didn't want to, mumbled and muttered about homework but sat down, and she'd sighed and put her feet up on the coffee table.\n\nThey were swollen.\n\nI'd forgotten that until now.\n\nI remember that I could see the marks her shoes had left, and said, \"Mom, you need new shoes.\"\n\n\"No, it's just the baby,\" she'd said. \"It's what happens.\"\n\n\"Your feet must hurt,\" I'd said, and stopped thinking about homework and focused on her.\n\nI wish I'd done that more now.\n\n\"A little,\" she'd said, wiggling her toes, and flipped through the channels. She'd stopped on a sitcom we'd both seen a million times before, four friends living in \"the big city\" and trying to \"make their way in the world.\" They would, of course. That's the beauty of TV. Everything works out in the end.\n\nMom had flipped through the channels during a commercial and now I lean into the sofa, lean into where she was sitting, and remember her doing that.\n\nI remember it all because I was really and truly there with her then.\n\nI noticed.\n\nI saw her holding the remote. I saw images flickering by one after another. An infomercial, a bad movie, an okay movie, an ad, another comedy, more ads.\n\nShe stopped on an ad for a restaurant.\n\nI remember thinking that was weird because it was a restaurant ad and who hadn't seen one? We'd just seen ten alone, but the ad she watched\u2014\n\nThere was a family sitting in a booth, smiling as the waitress dropped off their enormous platters of food.\n\n\"Are you hungry?\" I'd said, thinking of how Mom had eaten all of her dinner and then gone back to the fridge to make a sandwich, Dan trying to help and her saying, \"I'm pregnant, Dan. I can handle bread. Look, here I am holding one slice! And now another! Next thing you know, I'll be able to pick up mustard all by myself! Amazing, isn't it?\"\n\n\"No,\" Mom had said to me as I asked her about the ad she'd stopped on, and her voice was quiet, so quiet. \"I'm not hungry.\"\n\n\"Are you sure?\"\n\n\"I'm sure.\"\n\nI looked at her because I wasn't sure\u2014she was eating a lot and still had a bit of mustard on her chin\u2014and she was staring at the TV.\n\nShe was staring at the ad, at the food on the table, at the waitress smiling.\n\nAt the family. The mother, the father, the daughter and the baby in the high chair reaching for the \"grown-up\" food and making everyone laugh.\n\nSpecial Deal! flashed across the screen and Mom sighed and put a hand on her belly. On the tiny rise that was there.\n\n\"We'll do that,\" she whispered. \"We'll do that.\"\n\n\"Not there,\" I said. \"Dan hates that place.\"\n\nShe smiled and said, \"He does, doesn't he? But we'll all go somewhere nice and it'll be just like that ad.\"\n\nI hurt. I actually hurt all over, and I see what I missed and I can't stop it even though I want to.\n\nI remember how she touched her stomach again, stroking over the tiny bump. She whispered, \"Do you hear that, baby? We'll do it. We will. You just hang in there for me. You stay with me.\"\n\n\"Where's he going to go?\" I said, and then \"Can we go back to the show?\" and she said, \"Sure,\" and we did, we went back to the sitcom and that was it.\n\nExcept it wasn't.\n\nI thought I knew what she wanted, but I didn't.\n\nShe wanted that picture the ad showed; she wanted that family\u2014the mother, the father, the daughter, the baby. She wanted all of us together, and when I saw her touching her stomach I thought she was hungry and then, later\u2014after\u2014scared.\n\nAnd she was scared, but not like I thought. Not for herself.\n\nShe didn't want to be kept still. She didn't want bed rest.\n\nBut she did it.\n\nShe did it because she wanted the baby to be here. She wanted the baby to be here and all those times I heard her crying when she wasn't pregnant or when she miscarried were because she was in pain.\n\nBut also because she wasn't pregnant.\n\nShe wanted what she saw in that ad that night I thought I was being with her. She wanted something so simple. She wanted all of us to be together. She wanted me and Dan and the baby and her together forever and ever.\n\nShe gave up so much to get pregnant and the thing is\u2014\n\nThe thing is she wanted it. She wanted him, the baby swimming inside her. I thought she was scared and she was, but not of him.\n\nShe was scared for him.\n\nShe agreed to bed rest, terrified of it, for him.\n\nShe wasn't afraid of dying.\n\nShe was afraid he would.\n\nI see her now, really see her, and all the moments I thought Mom was scared for herself, all those moments where she touched her stomach and I saw fear in her eyes I thought I knew what it was for. I thought she knew what was coming and didn't know how to escape it.\n\nI was wrong.\n\nShe touched her stomach because she wanted to know the baby was there. Because she wanted to feel him. Because she wanted him to stay with her. She didn't talk about names because she was afraid to pick one, afraid she'd lose him like all the other babies she'd almost had, and those hadn't swum inside her for nearly as long, those hadn't lived long enough for names to even be an issue. She'd wanted him to make it. To be with her and me and Dan.\n\nShe'd wanted him to live.\n\nMom, I think, Mom, but I am just a girl on a sofa, a girl whose mother woke up one morning and died because that's how life is.\n\nI hurt all over even more now, like someone has shattered my insides, like I've been torn apart and put back together but I'm missing something.\n\nHer.\n\nAnd him. My brother.\n\nI know that, and I know what I need to do. What Mom would want me to.\n\nA family is more than one person.\n\n\"Dan,\" I say, and wait till he turns toward me.\n50\n\nI'm at the hospital. I'm just...I'm just standing in a corner of a hall on one of the floors.\n\nIt's small. Quiet. It's where I want to be. Where I need to be.\n\nIt's been forty-three days since Mom died, and I'm finally going to do it.\n\nI'm finally going to say goodbye.\n\n\"I'm sorry,\" I say as I sit down and take her hand, which is cool and almost papery. I can't bring myself to look at the rest of her. I want to remember the way she was before all this happened. \"I'm so sorry I didn't see that you wanted the baby so badly. That you weren't scared about yourself at all. I wish I'd spent more time with you. I wish\u2014\" My voice cracks. I lace my fingers through hers, tears welling as I have to move her fingers to make it happen. \"I wish you were here but you aren't. I miss you, Mom. I miss you and I know I always will but...\" I bite my lip hard and make myself keep talking. \"But today everything changes. And even though you aren't here, I know you're happy. That you'll be happy with what's going to happen. With how things are going to be.\"\n\nI let my fingers fall away from hers, catch her hand and put it gently by her side. \"I love you. I miss you.\" I start to cry. I can't tell her goodbye. She's still with me. She'll be with me forever. I love her that much and she loved me that much too.\n\nI sit there with her for a little while longer, until I'm not crying. Until I can look at her stomach.\n\nShe's twenty-five weeks pregnant now.\n\nForty-three days, and the baby's heart is still beating.\n\nIn fact, he'll be born today. Dan and I picked out his name together.\n\nLiam.\n\nAnd now, the operation to bring him here has started. Dan wasn't sure how long it would last. The doctors\u2014there are more of them now, more of them here for this moment\u2014aren't even sure.\n\nLiam won't come into the world like most babies do. Mom's body will be there, but she won't, and when he comes out she'll be\u2014that will be it. He'll have to be taken to ICU right away.\n\nI blow out a breath as I think of what will happen.\n\nThe machines that have kept Mom's heart pumping, her lungs breathing, will be turned off. Her body will be empty. Silent.\n\nLiam will never hear her breathe on her own. Never touch her for real. Left unsaid is what might happen to him. Left unsaid is that no one has been able to say if he will ever draw his first breath.\n\nLeft unsaid is that he might slip away like Mom did. That he could be here. And then be gone.\n\n\"We don't know what will happen,\" the doctor who will head the operation told Dan and me last night. \"We think things look good, but\u2014\"\n\nI stopped listening then. I held Dan's hand.\n\nHe held mine back and when we got home we looked at the photos of Mom we've been collecting. They are for Liam. They are for us.\n\nI am still in the corner where I stood before I said goodbye to Mom when Caleb rubs my arm gently. He's been with me since I got to the hospital, walked with me to this corner and stood, waiting, for me to come back. He came to be with me.\n\nOlivia came over this morning. She brought doughnuts but no one ate them. We just sat there, around the table, talking about Mom. Dan told us how she used to call him at home from work and ask him what he was making for dinner, sighing and saying, \"If I'd known getting married would mean dinner every night, I would have done it years ago.\"\n\nOlivia told us about how she'd managed to get her parents to agree to stay home and meet Roger, who had not only told his ex to stop calling him but, when she came up to him at a party when he was with Olivia and acted like she wasn't there, said, \"This is my girlfriend, Olivia,\" which was pretty much perfect boy behavior and Olivia figured after that, her parents would be easy.\n\n\"I told them they needed to talk to him without any gadgets the whole time. I figure I've got two minutes, tops, but still, I want them to meet him.\" As soon as she said it, she added, \"But if you need me\u2014\"\n\n\"I do,\" I said. \"I'll need you when I get home and tomorrow and for the rest of my life. I also know I need you to be happy. And besides, it's not like I'm not going to tell you everything anyway.\"\n\n\"Promise?\"\n\nI gave her a look and she grinned, but still hugged me before we left and whispered, \"Emma, I'm here if you need anything. You know that, right? You can even call my mom or dad if it's an emergency.\"\n\nThat was big stuff and I hugged her because she was awesome and I knew she meant everything and because Olivia has been here through it all and she has always been my friend. Mom would call her \"a keeper.\"\n\nMom's right. But like I always knew\u2014she usually was.\n\nOlivia and I are fine, and she is going to have her parents meet Roger, just like she's finally starting to be okay about me and Caleb.\n\n\"I know you,\" she said to me last week, \"and you're happy when you're with him. Really happy and it's been a while. So I'm happy. I just...I mean, you, Miss I-Have-To-Be-Perfect and a car thief. I never would have guessed.\"\n\n\"Me either,\" I said, and it was true. The me I was before never would have guessed. Never would have seen.\n\nBut I\u2014the me I am now\u2014I do see. And I am happy with Caleb.\n\nCaleb.\n\nWhen I told him what I'd said to Dan, that I wanted to go ahead and do the very thing I'd been fighting against, that I wanted to keep Mom's body beating and breathing so that Liam might live, he didn't say anything. He just held me and I cried.\n\nI cried because I was happy and scared and still a little angry. I am all those things still and Caleb was there. Is here.\n\nWe are together. We walk with each other when we can between classes in school. We talk when we can. He even eats lunch with me and Olivia. People have said stuff, but he doesn't care and the girl I used to be, the girl who would have, is gone. I am not going to be valedictorian, I'm not going to even be in the top twenty. I'm probably going to summer school to make up a bunch of my classes. (Okay, I'm definitely going, and Dan has started homework checks, which are weird, but I get why he's doing it).\n\nIn the end, it's okay because now I walk through school and see that it isn't everything. That it's just a part of my life and that it shouldn't be everything.\n\nI know what matters.\n\nCaleb does too.\n\nWhen he kisses me, I am still here, I am still in this life, in my life, but I am glad to be there. Glad to be the me who saw him. Who gets to hold him.\n\nThings haven't changed for him. His parents are the same, and they won't change. That's another thing about life I've learned. The people who need to see things clearly the most tend to miss life, believe they are living it when they are only letting it pass by.\n\nCaleb still hurts in a way I understand, but not because of them. He hurts for his sister. That's how grief is.\n\nGrief holds you tight. It holds you forever.\n\nBut we can hold each other too, and we do. I know he is thinking of me and Minnie, just as I am thinking of him and Mom.\n\n\"Should we go?\" he says after a while, after it's been\u2014after we both know enough time has almost passed, and I nod.\n\nThis is the part of grief that hurts the most.\n\n\"She was the best mother,\" I tell him. \"Fun. Strong. She loved me. She loved Dan. She loved...she loved Liam. I miss her so much.\"\n\nBeside me, Caleb lets out a deep, shuddering breath. When he speaks, his voice is very low, just above a whisper.\n\n\"I miss Minnie,\" he says. \"I miss her waking up before everyone else and making so much noise we all got up too. I miss her coming into my room and going through my stuff. I miss her asking me to make her chocolate milk. I never realized how quiet\u2014it's so quiet in the house now, you know?\"\n\nWe should go, but neither of us do. We are both thinking of people we love. Of people we have lost. Of people who left us before we thought they would.\n\nBut we are here.\n\nWe are here, and we walk to the elevator together. Caleb kisses me while we wait for it.\n\n\"Hi,\" he says, and I smile at him, touch his hair, parting his curls so I can see his eyes.\n\n\"Hi,\" I say, and then whisper, \"I'm scared.\" I thought saying it out loud would make it\u2014I didn't think it would make it disappear, but I thought it would make it easier.\n\nIt doesn't.\n\n\"I know,\" Caleb says, and we get off the elevator, walk to the waiting room. We sit, two people among others whose faces are creased with worry too, and flip through old magazines.\n\nWe sit and wait and when I have looked through all the magazines he buys me chips and a soda, kisses me when I ask what time it is because I just asked before he went to the vending machines.\n\nHe kisses me and for that kiss, for that moment, I forget how worried I am. It comes back, of course, but with Caleb, I feel more whole\u2014I am more whole\u2014than I have been since Mom died.\n\nI love him.\n\nI love him because of who he is, who he really is past what everyone else sees\u2014the lost boy, the druggie, the car thief. I love him because he is strong and caring. I love him because he broke and put himself back together again. I love him because he is beautiful inside and out.\n\nI love him for being here with me. I love him for not telling me that everything will be all right. I love him because he knows what life is like, what it can do, and is always honest about it.\n\n\"Caleb,\" I say, and he looks at me. Smiles.\n\n\"I love you,\" he says, and that is how it is. How we are. He knows I love him without me saying it, just like I know he loves me.\n\nBut I do like it when he says it. I glow inside when he does.\n\n\"I love you too,\" I say, and I had no idea he'd come into my life. That being with him\u2014loving him\u2014could be so right. But it is. We are.\n\nI love you is just words but it is more than that when you mean them. When you feel them.\n\nSo yes, I love him, he loves me, and it matters. It matters a lot, but there is still no news and I knot my hands together, squeeze.\n\nNo news and no Dan, who I last saw getting ready to go back to be with Mom and Liam for the operation. Who said, \"You can still come.\"\n\nBut I couldn't. I had to talk to Mom for one last time, and I did.\n\nShe's gone, but I can be happy. I can be in love. I can be both those things and scared too, and I am. I am, and this is what life is.\n\nI sit next to Caleb, waiting and thinking about what life really is. About how it has its own will. How it shows you things that rip you open, tear your world apart. How it unfolds even when you think it can't. How it takes you places you never thought you'd be. Shows you things you never knew you wanted to see. Brings you pain\u2014and joy.\n\nWhere will it take me now?\n\nI don't know. And that's what life is. You can plan all you want, but you will never know what will be. Life just is, and I am here in it. I am waiting for what comes next.\n\nI hear a noise, footsteps, and Caleb says, \"Emma,\" wonder in his voice.\n\nI look and I see Dan.\n\nI see Dan, and he is smiling. He is gesturing, hands flying, and I know I have a brother. That Liam is here, he's made it, and when I can, when the doctors will let me, I will fly to the ICU to see him. To tell him all the things we will do together. To wait to be able to hold him for the first time.\n\nI will always carry Mom in my heart. I will always miss her. I will always wish she was here.\n\nI will always know what life can take, but I am ready to see what it can give.\n\nI'm ready to move forward.\n\n* * * * *\nAcknowledgments\n\nA very special thank you to Natashya Wilson and everyone at Harlequin Teen for their enthusiasm, encouragement and general all-around awesomeness.\n\nRobin, Beth, Diana and Jess\u2014thank you for all your support, kindness and for, well, being who you are.\n\nThanks also go out to the following fantastic members of my mailing list: Sebrina Parker Schultz, Petra (Safari Poet), Samantha Page Townsend, Nicole Hackett, Autumn Nelson, Renee Combs, Alexandria, Tess Puhak, Ashley Evans, Christi Aldellizzi, Nakoya Wilson, Hannah Joy Herring, Stephanie Fleischer, Meghan Dondero, Nancy Woodford, Lexi Welch, Genevieve Swords, Lauren Becker, and Lucile Ogie-Kristianson.\n\nA special shout-out to Christi Aldellizzi for not just being a mailing list star, but for being so generous in giving to help out victims of Hurricane Sandy. Christi is another reason I know that my readers are the BEST readers ever.\n\nLastly, thank you to all the librarians, booksellers and readers who talk up my books. Your support means so much to me!\nQuestions for Discussion\n\n  1. Heartbeat explores a relatively rare situation in which Emma's mother is dead but her body is being kept alive due to her pregnancy. What did you think about the situation? How did grief affect Emma's actions and feelings?\n  2. What do you think Emma would have done if Dan had made her part of the decision when her mother died?\n  3. Olivia is Emma's sole friend when the story begins. Why do you think Olivia stays by Emma's side throughout her grief and changing attitudes? What do you think makes a friendship strong?\n  4. Caleb and Emma are brought together initially through circumstance and then shared grief. Do you believe their relationship has a chance to last and grow? Why or why not?\n  5. Why did Emma fear that Dan might choose to send her away? What was at the root of her fears?\n  6. Emma's attitude toward school undergoes a 180-degree change before and after her mother's death. What do you think will happen with her attitude in the future and why?\n  7. Heartbeat explores Emma's journey from grief to hope through different kinds of love\u2014family love, friendship, and romantic love. How did each type affect Emma? What helped her, and did any hinder her? What do you think Emma will pass on to her baby brother as he grows up?\n\nQ&A with Elizabeth Scott\n\nQ: Tell us a little about yourself and how you came to be an author.\n\nA: I fell into writing fiction by total accident. When I was in school, I went out of my way to avoid any \"creative\" writing assignments because I didn't think I had any imagination. Also, all the writers I knew seemed very intense and talked a lot about \"the craft,\" and it seemed kind of scary. I just wanted to read books, not write them!\n\nI wrote what ended up being my first piece of fiction while I was at work, bored out of my mind in a meeting, and I wasn't intense and certainly didn't know anything about writing, but I did learn one thing\u2014writing stories was fun! I joined an online critique group and never looked back after that. I think the thing I love about writing the most, besides getting to tell stories, is that there's always more to learn. There are always new things to try and hopefully, ways to be better!\n\nQ: What inspired you to write Heartbeat?\n\nA: I'd read a newspaper article about the death of a pregnant woman and thought, \"What if she'd died but the baby hadn't?\" It turns out that once in a while, that does happen, and the moment I learned that, Emma sprang into my head.\n\nQ: Emma and Olivia have a strong friendship. What do you think is important in a friend?\n\nA: Someone who gets you. It might not sound like much, but everyone has a messy\/dark side, and my friends, the ones who have been in my life for a long time\u2014they get it. And they're okay with it.\n\nQ: What kind of research did you do while writing Heartbeat?\n\nA: Obviously, there's not a lot out there about Emma's situation, but I read about premature births and prenatal care. And I'd read several books about the \"camp\" Caleb mentions being sent to and knew I wanted him to have been to one.\n\nQ: As an author, you probably like many of your characters, but tell us, do you have a secret favorite?\n\nA: I don't, because I can't read my books! I mean, if I have to do a reading, I'll pick out a passage, but that's it. So many writers I know can read their own stuff once it's published, but I can't. I've tried a couple of times, but have yet to make it past the first few pages. I guess I'd rather just be writing something new!\n\nQ: What do you hope that readers will remember after finishing Heartbeat?\n\nA: Honestly, I just hope they like it! I'm not a big fan of \"message\" books. I just want to tell a story and if people like it, then YAY!\n\nQ: What advice do you have for aspiring authors?\n\nA: Read. I meet so many people who want to be writers, who have written stories\/novels, and when I ask them what they last read, they just look at me. I was a reader long before I became a writer and I think that if you don't love reading, then why are you writing? Also, don't just read what genre you write. Read everything you can get your hands on. Read stuff that you wish you could write. (In my case, it's poetry.)\n\nThank you, Elizabeth!\n\nTurn the page for an exclusive look at the next   \npowerful and romantic contemporary novel   \nfrom award-winning author Elizabeth Scott\n\nHOPE LIES\n1\n\nI was eating peas when the police came for my mother.\n\n\"Warren?\" my mother said when the doorbell rang. She didn't like interruptions during dinner, but they still happened sometimes. And they were always for my father, who could never quite leave work, even when he wasn't in the office.\n\nDad got up, giving my mother a rueful smile, then shooting a longing glance at his roast beef as he left the room.\n\n\"Victoria,\" he called out a moment later, his voice strangely tense, and my mother, frowning slightly, put her fork down and got up.\n\n\"I bet it's the club,\" I said. The Pleasantfield Country Club was always trying to get my parents to donate money, even though the only time they did was for charity fund-raisers.\n\n\"Eat your peas, Isabel,\" my mother said as she left the dining room. I guiltily rolled them out from under the pile of what I'd thought were artfully arranged meat scraps.\n\nAs soon as she left though, I picked up my plate and dumped as many as I could onto my father's. It wasn't a big deal. He liked peas.\n\nDad came back as I was putting my plate down. He looked strange, frightened and sick.\n\n\"Dad?\" I said, my stomach clenching.\n\n\"Your mother's had to leave with the police,\" he said. And that's how it all started. How I found out my mother wasn't who I thought she was. How I found out she was someone else. Someone who'd disappeared a long time ago.\n\nSomeone who was wanted for murder.\n\nWe hope you enjoyed this Harlequin ebook. Connect with us on Harlequin.com for info on our new releases, access to exclusive offers, free online reads and much more!\n\nOther ways to keep in touch:\n\nHarlequin.com\/newsletters  \nFacebook.com\/HarlequinBooks  \nTwitter.com\/HarlequinBooks  \nHarlequinBlog.com\nISBN-13: 9781460318133\n\nHEARTBEAT\n\nCopyright \u00a9 2014 by Elizabeth Spencer\n\nAll rights reserved. By payment of the required fees, you have been granted the non-exclusive, non-transferable right to access and read the text of this e-book on-screen. No part of this text may be reproduced, transmitted, down-loaded, decompiled, reverse engineered, or stored in or introduced into any information storage and retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereinafter invented, without the express written permission of publisher, Harlequin Enterprises Limited, 225 Duncan Mill Road, Don Mills, Ontario, Canada M3B 3K9.\n\nThis is a work of fiction. Names, characters, places and incidents are either the product of the author's imagination or are used fictitiously, and any resemblance to actual persons, living or dead, business establishments, events or locales is entirely coincidental. This edition published by arrangement with Harlequin Books S.A.\n\n\u00ae and \u2122 are trademarks of the publisher. Trademarks indicated with \u00ae are registered in the United States Patent and Trademark Office, the Canadian Trade Marks Office and in other countries.\n\nwww.Harlequin.com\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nTable of Contents\n\nCover\n\nTitle page\n\nCopyright page\n\nAcknowledgements\n\nPrologue: My Life as an Extremist\n\n1 Why Clothing is not Superficial\n\nTrinidad\n\nThe Sari\n\nLondon\n\nConclusion: A Good Fit?\n\n2 Theories of Things\n\nFrom Hedonism to Functionalism\n\nObjectification\n\nMateriality\n\n3 Houses: Accommodating Theory\n\nHousing and Power\n\nHome and Agency\n\nCaribbean Homes\n\n4 Media: Immaterial Culture and Applied Anthropology\n\nMaking Trinidadians on a Wing and a Prayer\n\nWhat is a Mobile Phone Relationship?\n\nMedia, Poverty and Welfare\n\n5 Matter of Life and Death\n\nBirth\n\nDeath\n\nBack to the Beginning\n\nIndex\n\nCopyright \u00a9 Daniel Miller, 2010\n\nThe right of Daniel Miller to be identified as Author of this Work has been asserted in accordance with the UK Copyright, Designs and Patents Act 1988.\n\nFirst published in 2010 by Polity Press\n\nPolity Press\n\n65 Bridge Street\n\nCambridge CB2 1UR, UK\n\nPolity Press\n\n350 Main Street\n\nMalden, MA 02148, USA\n\nAll rights reserved. Except for the quotation of short passages for the purpose of criticism and review, no 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 publisher.\n\nISBN-13: 978-0-7456-4423-3\n\nISBN-13: 978-0-7456-4424-0(pb)\n\nISBN-13: 978-0-7456-5497-3(Single-user ebook)\n\nISBN-13: 978-0-7456-5496-6(Multi-user ebook)\n\nA catalogue record for this book is available from the British Library.\n\nThe publisher has used its best endeavours to ensure that the URLs for external websites referred to in this book are correct and active at the time of going to press. However, the publisher has no responsibility for the websites and can make no guarantee that a site will remain live or that the content is or will remain appropriate.\n\nEvery effort has been made to trace all copyright holders, but if any have been inadvertently overlooked the publishers will be pleased to include any necessary credits in any subsequent reprint or edition.\n\nFor further information on Polity, visit our website: www.politybooks.com\nAcknowledgements\n\nI have always felt that a subject called Social Anthropology could manage to be a hell of a lot more social than is commonly the case. As a result, I changed my way of working many years ago. Increasingly I carry out almost all my research in collaboration. This means that, although this book is largely about my research, it presents results and ideas that have, in most cases, been developed only through my engagement with others. It is impossible, for example, to consider my discussion of the sari in chapter 1 without this being also credited simultaneously to Mukulika Banerjee, or that of the Jamaican mobile phone without this being equally seen as the work of Heather Horst. Other research collaborators have included Zuzana B\u00farikov\u00e1 for the work on au pairs, Alison Clarke for the work on shopping, Mirca Madianou for the work on long-distance relationships, Lucy Norris for the work on clothing and waste, Fiona Parrott for the work on loss, Don Slater for the work on the Internet in Trinidad, and Sophie Woodward for the work on denim.\n\nOne of the other delights of my work has been the collaboration with students and especially with PhD students. This creates a vicarious pride which is not entirely dissimilar from that of parents and children. As in the latter case, it is often the younger party that shows much more maturity than the older. Many years ago I established something we call a dinner\/drinking group, which meets monthly to drink, discuss pre-circulated papers by the students and then eat (and occasionally party). Sometimes in each other's homes, sometimes in my office, sometimes in restaurants.\n\nThanks to this format, friendships have often been formed, and also collaborative publications. The publications in turn reflect the cross-influence between what often started as quite different projects. This has made it easier to produce edited collections with relative coherence compared to most such volumes. So I owe a considerable amount to past and present PhD students: Ivana Bajic, Ira Berdichevsky, Julie Botticello, Achsah Carrier, Pat Clark, Alison Clarke, Magda Craciun, Dimitrios Dalakaglou, Inge Daniels, Adam Drazin, Justin Finden-Crofts, Pauline Garvey, Cleo Gougoulis, Martha Guarneros, Anat Hecht, Heather Horst, Gabrielle Hosein, Neil Jarman, Mark Johnson, Emma Lindblad, Tom McDonald, Desmond Mallikarachichi, Simone Mangal, Jean-S\u00e9bastien Marcoux, Marjorie Murray, Razvan Nicolescu, Kaori O'Connor, Bodil Olesen (honorary), Panarai Ostapirat, Andrea Pellegram, Anna Pertiera, Elia Petridou, Sigrid Rausing, Tom Rogers, Ed Ross, Florian Schlichting, Miran Shin, Andrew Skuse, Jo Tacchi, Chang-Kwo Tan, Sophie Woodward. In addition, over the years there have been constant visitors, students who came for just a few months, who joined this group, and then left again; I simply cannot mention all these and I hope they will forgive me. Others who have commented on this manuscript include Eleana Yalouri and Karen Ida Dannesboe as well as anonymous reviewers.\n\nI also owe a considerable debt to my colleagues over the years in material culture studies at University College London. That is, Barbara Bender, Victor Buchli, Paolo Favero, John Gledhill, Suzanne K\u00fcchler, Chris Pinney, Michael Rowlands, Chris Tilley and Graeme Ware. I have only ever worked at UCL and never once been tempted by any other potential job. There are also many colleagues around the world who have encouraged and inspired me from time to time, and helped especially when I was first developing these ideas, for example, Rob Foster, Nicky Gregson, Caroline Humphrey, Webb Keane, Marilyn Strathern, Nicholas Thomas and Rick Wilk. As well as others, sadly deceased, such as Judy Attfield, Alfie Gell and Marianne Gullestad.\n\nOn a more personal note, of particular importance has been the friendship and advice over the years of Mukulika Banerjee, Kathryn Earle and Stephen Frosh. Nothing I do has ever been possible without the support of Rickie, Rachel and David and I would like to thank Rickie, in particular, for her assistance in editing the text. Finally I am indebted to a vast number of people who have granted me such generous help during fieldwork in a dozen different countries over many many years. I can't name them but it is their ideas and actions that inform this book.\nPrologue: My Life as an Extremist\n\nDon't, just don't, ask for or expect a clear definition of 'stuff'. Sure, there are academic traditions that believe knowledge is best conveyed through such clear definitions. This is perhaps essential in the natural sciences. But personally I have always had a horror of what I think of as pedantic semantics. To try and determine the exact criteria by which some things would be excluded from stuff as perhaps less tangible, or too transient, would be a hopeless exercise. Does an email or a fashion count as stuff, a kiss or a leaf or polystyrene packing? This is a book about the variety of things that we might term stuff, but nowhere in this volume will you find any attempt at a definition of that term. Rather it is intended to introduce my own perspective on the study of material culture, a set of studies that are principally concerned with stuff. Material culture is no better defined than stuff is. In practice these studies have promiscuously picked up on interesting topics and perspectives that more self-absorbed, well-defined sub-disciplines felt were outside of their brief. Material culture thrives as a rather undisciplined substitute for a discipline: inclusive, embracing, original, sometimes quirky researches and observations.\n\nMaterial culture starts, after all, with mere happenstance: the history of the established academic disciplines. Some of these seem obvious; we have language and so we have linguistics. Others on reflection seem a trifle bizarre. The study of rocks is allocated to geography, and then somehow the study of traffic gets mixed into the pot. Yet no one thought to have an academic discipline whose specific area of study would be artefacts, the object world created by humanity. It could so easily have been otherwise. Consider the degree to which established academic disciplines, from archaeology to architecture, from sociology to design, require theories and perspectives on this material world. If material culture had existed for a century of established study in thousands of colleges, it would have been as taken for granted as linguistics is today. But this didn't happen and it is not the purpose of this book to create a discipline where there wasn't one. Instead it seeks to welcome this aberration; to embrace the openness and collegiality of not having to be overly disciplined. Stuff as a term serves just fine. Not establishing a discipline doesn't mean that a study of stuff lacks substance or consequence. Material culture studies are becoming recognized as a vital contribution to half a dozen established disciplines, from archaeology to design. We provide many theories, and analytical approaches, for example, in discussing the specific nature and consequences of materiality. Several of these are discussed in the course of this book.\n\nThis volume is not a textbook, nor a systematic review of the range of studies of material culture. Most of that work is barely mentioned here. It is instead, together with a future companion volume, intended as a retrospective examination of my own previous research. When I was first approached by a publisher to consider creating an edited collection based on my previously published work, I was ambivalent. I had always felt that, in some respects, such collections are lacking something. In general, an edited collection requires very little effort, partly because one is not required to re-think earlier material in the light of later work, while as soon as a book appears in press I tend to see only its deficiencies. Fortunately such introspections are limited by the fact that I write mainly as an anthropologist. This means that most of my writing is about something far more interesting than myself or my ideas \u2013 it is an account of other peoples, their practices and cosmologies and ideas. As accounts of others they hopefully have just as much to teach now as when I first wrote them, which helps give me confidence in the value of a return to these works. But I still preferred not to create a book of edited papers, but to wait until I could partially rewrite this past corpus. By reconsidering each publication in the light of others I could draw attention to some larger themes that connect them, and longer trajectories that they developed over time. With twenty-six previous volumes this task seemed better divided into two main themes.\n\nOther grounds for this rewriting of earlier work emerge in retrospect. Academia, as the anthropologist Bourdieu points out, in its structure of employment, creates an incentive for each generation to attempt the overthrow of its predecessors, demonstrating the intellectual agility of the young by repudiating established arguments. As a student in Cambridge, in a highly competitive environment, this desire to demonstrate cleverness, as opposed to understanding, was paramount. Amongst my first works, a jointly edited collection called Ideology, Power and Prehistory1 was a critique of Foucault. There was a need to demonstrate that I was comfortable with the work of figures such as Derrida and Lacan and clever enough to appropriate the opaque and often recondite terms and ideas that impressed, partly through obfuscating phraseology. I gained a great deal from that lively competition and ambition. But it has also taken me a considerable time to overcome the more negative aspects of that legacy: to appreciate that a discipline such as anthropology, devoted to the comprehension and welfare of people in society, was ultimately served much better by understanding than by cleverness, which merely betrayed the need to express the stories and suffering of others. The intention was that this book would attempt to retain some of these insights from earlier work, but to release them from the encrusting jargon in which they were first written and instead to try and convey them as clearly as possible. Another problem with an edited collection is its selectivity. By contrast, a re-composed retrospective examination allows one in hindsight to consider the corpus of work more holistically. So this volume seeks both to juxtapose and to relate theories of stuff, applied anthropology, ethnographic reportage and the analysis of particular genres of clothing, home and the media. Ultimately there are a good many reasons why you should give a stuff about stuff, and this book attempts to bring them together.\n\nFor all these reasons, I approached Polity Press with the idea of doing something rather different from a conventional edited volume: to write two new books that summarized and reviewed my own work in a clearer, less pretentious style that could examine my academic trajectory as a whole. This first volume speaks to around a dozen books and over a hundred papers about material culture. All this research was undertaken under the auspices of, and using the methodologies of, the discipline of anthropology. It is therefore a complement to other material culture work orientated more to disciplines such as design or philosophy.2 It is also very much beholden to the specific context of the department of Anthropology at University College London, from where all this research was conducted. My own research has occupied a specific niche within an ecological diversity of ideas that colonize other feeding grounds for material culture, grazed by my colleagues in the department. These include the anthropology of art, museum studies and culture heritage management, visual anthropology, and the anthropology of landscape and architecture. Our collectivity is represented in the Journal of Material Culture which we edit, and many other publications. I also run a blog at www.materialworldblog.com with colleagues at UCL and other universities. While this volume represents almost entirely my own work, rather than that of the larger group, I am happy to acknowledge the extent to which my research has constantly been informed by my colleagues. It is even more influenced by over forty PhD students, some of whose work is discussed here.\n\nThis book is intended as an interim report rather than a signing off. I have recently started several new projects on topics ranging from global denim to the use of media by migrants for the conduct of long-distance relationships. Hopefully consolidation can create a higher academic platform for the launch of future work. But the foundation is constructed from the building blocks of material culture itself. The central argument of this book is a paradox: that the best way to understand, convey and appreciate our humanity is through attention to our fundamental materiality. It might seem that this would only be true of our particular present \u2013 a consequence of the sheer scale of contemporary consumption and its implications for environmental and other ethical debates. I do not believe this to be the case. I will argue that non-industrial societies are just as much material cultures as we are. This volume treats consumer and non-consumer societies equally and in juxtaposition. A consideration of the specific issues of consumer society will be the subject of a second retrospective volume entitled Consumed by Doubt. If discussion of issues of political economy, indeed politics more generally, including a concern for the consequences of materialism for the future of the environment, are largely missing from this volume, it is because they are the subject of this second complementary work. Before that more political debate, however, I feel we need to consider the more foundational issues: the consequences of our materiality and of material culture for a more profound understanding of what we ourselves are.\n\nThe leitmotif of this book is a challenge to our common-sense opposition between the person and the thing, the animate and inanimate, the subject and the object. Certainly this is being achieved in some measure these days through science. New forms of biomedicine constantly challenge the way we think about such matters as the body and perception. Anthropologists including Donna Harraway, Emily Martin, Marilyn Strathern and Bruno Latour have considered these consequences of science, especially biomedical discoveries, as well as the languages and images we use to convey them. Here, by contrast, I am concerned with developments in social science, rather than natural science; with anthropology's qualitative encounter with the diversity of peoples, and the increasing diversity of things.\n\nStuff is ubiquitous, and problematic. But whatever our environmental fears or concerns over materialism, we will not be helped by either a theory of stuff, or an attitude to stuff, that simply tries to oppose ourselves to it; as though the more we think of things as alien, the more we keep ourselves sacrosanct and pure. The idea that stuff somehow drains away our humanity, as we dissolve into a sticky mess of plastic and other commodities, is really an attempt to retain a rather simplistic and false view of pure and prior unsullied humanity. There are good uses of anthropology and also terrible uses of anthropology. One of the latter is a primitivism which assumes that because tribal people didn't have much stuff they were necessarily less materialistic. On the contrary, some of the most sophisticated relationships to things may be found amongst peoples such as Australian Aboriginals or North West Coast American Indians, whose material possessions may seem paltry, compared to modern Londoners'.\n\nFurthermore, not having things is no evidence that you don't want them. An Amazonian Indian may be much more desirous of possessions than we are, but simply unable to obtain them. The anthropologist Stephen Hughes-Jones, who spent many years in Amazonian research, was shocked by the avaricious desire to obtain goods that he encountered amongst people who up to then had almost no possessions.3 It may not be what pop groups want us to believe of the peoples of Amazonia, but personally I turn to pop groups for music rather than for anthropology. Anyway, why should we deprive Amazonian Indians of the delights of pop music? This model of the noble, unmaterialistic savage is entirely unhelpful. All it achieves is an assumption of lost purity. It makes us feel alienated and polluted simply for being who we are. Instead, this book tries to face up to stuff: to acknowledge it, respect it and expose ourselves to our own materiality rather than to deny it. My starting point is that we too are stuff, and our use and identification with material culture provides a capacity for enhancing, just as much as for submerging, our humanity. My hope and intention is that this book will demonstrate how and why a more profound appreciation of things will lead to a more profound appreciation of persons.\n\nFor some time now I have had the ambition to become an extremist, something I consider to be a noble ambition, at least for an anthropologist. In fact being an extremist more or less sums up what I mean by being an anthropologist. While I am generally rather unsympathetic to disciplinary labels and work in collaboration with wonderful researchers from many other disciplines, I am committed to what I see as this extremist quality in anthropology. But, at this point, I suppose I had better explain why I would consider extremism, in this particular context, to be such a noble pursuit. I would think that one of the less controversial characterizations one could make about the modern world is that it shows an ever increasing tendency to greater particularity and simultaneously to greater universality. On the one hand there is a proliferation of diversity in occupations, in commodities, in experiences and in relationships. A Londoner might work in corporate PR or waste incineration, play with a Nintendo Wii or beach volleyball, and love karaoke or tequila. At the same time we devise institutions based on implied universals, such as children's rights, web technology, neo-liberalism, or a curriculum for chemistry, that have the ambition of reaching all the peoples of the world. At one end of the spectrum we become ever more particular as individuals in our work, tastes, practices and interests. At the other end we have recourse to ever more universal aspirations, some of which we subscribe to, but most of which do not depend on our active identification. Most, such as human rights or medical services, grow more in the form of institutions that strive for universality in practice. Of course, these two processes are connected. Institutions are more universal the more they encompass an increasing particularity.\n\nThis simultaneous growth in particularity and universality is one way of defining the modern world. It was central to that philosophical tradition which is usually called dialectics and is most closely associated with the work of Hegel, a philosophical foundation I will refer to several times in this volume, since it is the one which underpins much of my own work. How should academics respond to this condition? I find that, in general, most disciplines respond by searching out some kind of middle ground. Any social science or natural science of the person, such as psychology, that proceeds through hypothesis testing will tend to focus upon some small element of predictability. It will propose a theory or analysis that can be tested against a relatively limited and often controlled observation. In applying for a research grant most academics will present an argument based on a quite immediate relationship between a particular behaviour and a specific generalization or cause. Is class or ethnicity better correlated to educational performance in the UK? How do older people in Chicago respond when you look them in the eye directly or look at them indirectly? Do Serbians more often consider that they are infected with a cold when the weather is cold?\n\nThe anthropology I am committed to eschews such hypothesis testing. My problem in seeking research grants is that invariably my only real hypothesis is that I really have very little idea of what I am actually going to find when I go out to conduct fieldwork. This hypothesis has always proved correct. In going to live within another community I assume that the most important findings are going to be about things one didn't even suspect existed before going to live there. If you didn't know they existed, how could you have hypothesized about them? In my experience an inconsistent, opportunistic research student tends to be far more original and productive than consistent research students, who achieve merely what they set out to do.\n\nI don't then follow this search for the middle ground of academic research \u2013 the testing ground. This is because I see the primary purpose of anthropology as fulfilling an ideal I derive originally from Hegel. The central problem confronted by the modern world is that universalism and particularity can so easily lose touch with each other. Finance, theory or aspirations to be good can all take on a universalistic abstract form. But these then fall out of synch with the actual diversity, complexity and contradictions of business, analysis and the ethics of the everyday. So the purpose of anthropology is to bring these two back into conversation with and acknowledgement of each other. After all, there is probably no discipline as committed to particularism as anthropology. A principle of its work is relativism. We conduct fieldwork by spending a year participating in the lives of members of an Apostolic church in Zimbabwe, rice farmers of North West Thailand, separated Filipino families, or aspiring pop bands in Manchester. We are not supposed to claim authority to write about these communities until we have thoroughly immersed ourselves in the everyday life of these very specific groups. Our methodology, known as ethnography, typically consists of living with families for a year or more, speaking their language and participating in mundane activities such as cooking and cleaning.\n\nThis is a world apart from those who are satisfied with questionnaires and focus groups and seek to conduct experiments. Anthropologists can become so extreme that all generalization seems suspect. Yes, this Balinese family used a word we might translate as love, but the people in the next valley use it slightly differently, as do women as against men, or younger as against older women. Indeed on reflection they seem to be using it a bit differently this year from last year. So how could we ever say what that word love means to a people we drastically overgeneralize every time we use the term Balinese? Such extreme relativism seems doomed to extreme parochialism, a common affliction amongst anthropologists.\n\nYet, at the very same time, anthropology is a discipline dedicated to universals, and will continually throw up ideas such as L\u00e9vi-Strauss's structuralism, the principle of the gift, anthropological versions of Marxism or psychoanalysis, relationships as negotiated experience, the consequences of our materiality. Our theories seem to transcend all such particularities. Culture itself is viewed as binary, or founded in exchange or in kinship, or opposed to nature. We attempt to write a book about stuff. Some of our generalizations are so vast they are scary. The advantage of this simultaneous commitment to the extremes of particularity and of generality is that anthropology can make its major contribution to the understanding of humanity through constantly re-connecting these two without losing a commitment to both extremes. In that sense this discipline that many people regard as looking backwards to prior manifestations of humanity is actually uniquely geared to the central dilemmas of modern life.\n\nMost commonly, good anthropological work reveals the particular as a manifestation of the universal. So there is regional neo-liberalism. Coca-Cola is everywhere, but means slightly different things in each locality. The gift is based on an obligation to return the gift but this works differently within the Chinese Guanxi system than with the Maori Hau. My colleague Mukulika Banerjee is an expert on democracy. She has researched why people bother to vote in an Indian village. But this in turn means understanding how these villages respond to the universal pretensions of democracy as an ideal. Democracy as a practice and as a consequence varies from place to place, and yet cannot be understood without also acknowledging its authority as a global aspiration or global conceit. Perhaps there is no capitalism, only capitalisms. So the reason I desire to be an extremist is that this represents a commitment to keep in touch simultaneously with the extremes of universalism and particularism in modern life.\n\nWhy does this matter so much? It is because one of the major dangers that besets the world today lies in the increasing dissociation of the two extremes. The universals fly apart from the particular. Economists, psychologists and human rights lawyers come up with more and more general models that claim to represent humanity and assume increasing authority to be imposed upon humanity as a whole. All economies are increasingly expected to seek to accord with ever more abstract, often mathematical models that generate new financial instruments. All children are supposed now to have the same basic human rights. But these universals become detached from the very specifics of our humanity and cultural differences that remain meaningful. London may be gloriously cosmopolitan, but the devout Sikh remains very distinct from the retro-Goth living next door. So I subscribe to a discipline that is prepared to dedicate a year in order to know what it means to live as a devout Sikh, but equally what it means to live as a retro-Goth in London; to simultaneously consider the implications of neo-liberalism and universal human rights. But then there is always a third stage to this dialectic without which our task remains incomplete. This is to ask what the consequences of neo-liberalism are to the retro-Goth and the challenge a devout Sikh might represent to assumptions about universal human rights. Only at that point have we achieved the goal of being anthropologists.\n\nOnce upon a time anthropologists were associated only with the study of less developed or small-scale societies. I hope those days are gone forever. All people today are equal in their right to the burden of being studied by some or other anthropologist. My definition of the anthropologist is someone who seeks to demonstrate the consequences of the universal for the particular and of the particular for the universal by equal devotion to the empathetic understanding and encompassment of both. This is why I believe my extremism to be a noble cause. But it follows from this same premise that I have, within this universalistic ambition, my own particularism. My particular particularism is the field of material culture \u2013 the study of stuff.\n\nThis book charts a path towards material culture studies through several chapters, each of which takes responsibility for providing a different perspective upon this endeavour. The first chapter on clothing is a demolition of the most common academic and popular view of stuff \u2013 the idea that objects signify or represent us and that they are principally signs or symbols that stand for persons. Instead, I argue that in many respects stuff actually creates us in the first place. More specifically, I demonstrate why clothing is not superficial. The second chapter presents theories of material culture. Starting with a theory of objects per se, it moves steadily upwards to the more rarefied theory of objectification and thence to a plane of transcendence where we gain a perspective from which we can no longer distinguish subjects from objects. It then examines the consequences of our various beliefs about the properties of materiality itself. The third chapter takes these neat and clean abstract theories and drags them back down to the messy world of application in its consideration of our relationship with our homes and houses. It literally domesticates such theories through examining the process of 'accommodating', and reveals how such theories, when applied to specifics, have to incorporate wider factors such as the impact of governments, the history of styles, international migration, and the power and agency that lies in the houses themselves.\n\nThe fourth chapter examines the ambiguous materiality of media and communication. Stuff is not necessarily a thing we can hold or touch. This chapter also takes responsibility for issues not previously addressed, but which emerge in applied anthropology. What can and should we do with the knowledge and understanding gained by material culture studies, when it comes to the welfare of populations? How can we hope to improve their conditions and respect their aspirations while considering the negative impact of their desires? Finally, in the fifth chapter, we consider stuff as the matter of life and death; that which brings us into the world and that which helps us part from the world. In this final chapter I take what was introduced theoretically as the way objects construct subjects, and show how this is true for the everyday understanding of what it means to be human.\n\nNotes\n\n1 All references are to my own work, unless another author is specified. In this case, D. Miller and C. Tilley (eds), Ideology, Power and Prehistory (Cambridge: Cambridge University Press, 1984).\n\n2 For example, Judy Attfield's Wild Things (Oxford: Berg, 1992) is a textbook on material culture written from the perspective of design studies.\n\n3 S. Hugh-Jones, 'Yesterday's Luxuries, Tomorrow's Necessities: Business and Barter in Northwest Amazonia', in C. Humphrey and S. Hugh-Jones (eds), Barter, Exchange and Value (Cambridge: Cambridge University Press, 1992), pp. 42\u201374.\n1\n\nWhy Clothing is not Superficial\n\nWhen I began my career as an academic, committed to the study of material culture, the dominant theory and approach to the study of things was that of semiotics. We were taught that the best way to appreciate the role of objects was to consider them as signs and as symbols that represent us. The example that was most commonly employed to illustrate this perspective was that of clothing, since it seemed intuitively obvious that we choose clothing for precisely this reason. My clothing shows that I am sexy, or Slovenian, or smart, or all three. Through the study of the differentiation of clothing we could embark upon the study of the differentiation of us.1 Clothes might represent gender differences, but also class, levels of education, cultures of origin, confidence or diffidence, our occupational roles as against our evening leisure. Clothing was a kind of pseudo-language that could tell us about who we are. As such, material things were a neglected adjunct to the study of language: an apparently unspoken form of communication that could actually speak volumes once we had attuned ourselves to this capacity. Anthropological discussions by Mary Douglas and Marshall Sahlins, amongst others, that advocated this approach seemed to suggest a whole new significance to the study of stuff.\n\nThere is no doubt that material culture studies was significantly enhanced by the arrival of this semiotic perspective; but ultimately it became as much a limitation as an asset. This chapter aims to repudiate a semiotic approach to things in general and to clothing in particular. Consider one of the best-known clothing stories. 'The Emperor's New Clothes' is a morality tale about pretentiousness and vanity. The Emperor is persuaded by his tailors that the clothes they have stitched him are fine to the point of invisibility, leaving him to strut naked around his court. The problem with semiotics is that it makes the clothes into mere servants whose task it is to represent an Emperor \u2013 the human subject. Clothes do our bidding and represent us to the outside world. In themselves, clothes are pretty worthless creatures, superficial, of little consequence, mere inanimate stuff. It is the Emperor, the self, that gives them such dignity, glamour and refinement.\n\nBut what and where is this self that the clothes represent? In both philosophy and everyday life we imagine that there is a real or true self which lies deep within us. On the surface is found the clothing which may represent us and may reveal a truth about ourselves, but it may also lie. It is as though if we peeled off the outer layers we would finally get to the real self within. But what was revealed by the absence of clothes was not the Emperor's inner self but his outward conceit. Actually, as Ibsen's Peer Gynt observed, we are all onions. If you keep peeling off our layers you find \u2013 absolutely nothing left. There is no true inner self. We are not Emperors represented by clothes, because if we remove the clothes there isn't an inner core. The clothes were not superficial, they actually were what made us what we think we are. At first this sounds odd, unlikely, implausible or just plain wrong. To discover the truth of Peer Gynt, as applied to clothing, we need to travel to Trinidad, from there to India, and then to use these experiences to re-think our relationship to clothing back in London.\n\nTrinidad2\n\nThe problem with viewing clothing as the surface that represents, or fails to represent, the inner core of true being is that we are then inclined to consider people who take clothes seriously as themselves superficial. Prior to feminism, newspaper cartoons had few qualms in showing women as superficial merely by portraying their desire to shop for shoes or dresses. Young black males were superficial because they wanted expensive trainers that they were not supposed to be able to afford. By contrast, we student academics at places such as Cambridge were deep and profound because frankly we looked rubbish, and clearly didn't much care that we did. When I met my wife as fellow students, my trousers were held up at the top with string and their hem at the base with staples. She must have thought I was deep, because there certainly wasn't much to attract her on the surface. Such assumptions are fine within the confines of Cambridge but a problem for an anthropologist going out to Trinidad. Because the point of anthropology is to enquire empathetically into how other people see the world. Dismissing them as superficial would represent a rather disastrous start to such an exercise. For Trinidadians in general were devoted to clothes, and knew they were good at looking good. Colourful prints and butterfly belts were a priority.\n\nI worked much of my time in Trinidad with squatters who had neither a water supply nor electricity in the house. Yet women living in these squatters' camps might have a dozen or twenty pairs of shoes. A common leisure activity was to hold a fashion display, on a temporary catwalk, along one of the open spaces within the squatters' encampment. They would beg, borrow, make or steal clothes. It wasn't just the clothes, it was also the hair, the accessories and the way they strutted their stuff; knowing how to walk sexy and to look glamorous or beguiling. Movements were based on an exaggerated self-confidence and a strong eroticism, with striding, bouncy, or dance-like displays. In local parlance there should be something hot about the clothing and something hot about the performance. On evenings I could spend three hours with them, waiting as they got themselves ready to go out and party, trying on and discarding outfits until they got it right.\n\nThis association is hardly new for the region. Early accounts of slave society in the Caribbean include references to the particular devotion of slaves to clothing. A. C. Carmichael stated in 1833: 'Generally speaking, the coloured women have an insatiable passion for showy dresses and jewels... The highest class of females dress more showily and far more expensively than European ladies'.3 Freilich, carrying out ethnographic research in an impoverished village in 1957\u20138, reports, 'the wife of one of the peasants said \"every new function needs new clothes. I would not wear the same dress to two functions in the same district\" '.4 This desire was still more forcefully expressed during the 1970s oil boom in Trinidad when both seamstresses and their clients suggested that purchasing two new outfits a week was quite common for women in work. We do not necessarily condemn a population just because they show some devotion to stuff. Anthropologists celebrate, rather than demean, the devotion of Trobriand Islanders to canoe prows or of the Nuer to cattle. But curiously a devotion to clothing, as one can see from these descriptions by outsiders, was always viewed rather more harshly, especially for those without wealth.\n\nAs evident in the description of the local catwalk, what mostly concerned Trinidadians was not fashion \u2013 that is, the collective following of a trend, but style \u2013 that is, the individual construction of an aesthetic based not just on what you wear, but on how you wear it. There used to be a term saga boys for men who combined sartorial originality with ways of walking and talking that never let up from conspicuous display. Another local term gallerying gets it just right. Trinidad style, in turn, has two components, individualism and transience. The individual has to re-combine elements in their own way. The source of these elements is unimportant. They may be copied from the soap operas or the fashion shows which appear on television, sent from relatives abroad or purchased while abroad. They may simply re-combine local products. But the various elements should work together, be appropriate to the person who carries them off well, for ideally just one particular occasion. It didn't matter what clothes cost or even whether the clothes worn on the catwalk belonged to them or were borrowed for the occasion. This wasn't about accumulation, but about transience. The stylist may learn from fashion but only as the vanguard. Then they must move on. Trinidad's best known cultural export, Carnival, enshrines this transience. Individuals may spend weeks, if not months, creating elaborate and time-consuming costumes. But these must be discarded and re-made annually. What is celebrated is the event, the moment.\n\nThere are many possible reasons why Trinidadians might have developed this affinity with style as individual and transient expression. Some theories go back to slavery and beyond. Henry Lewis Gates in his book The Signifying Monkey argued for a West African aesthetic structure found especially amongst the Yoruba.5 He notes the way jazz musicians take up themes and develop them, but very often return to a pastiche of well-known rhythms and tunes from previous compositions. In ordinary speech both musicians and others will refer to Signifyin(g) upon someone or something. One can see a similar use of clothing in the development of vogueing made famous by Madonna and the film Paris is Burning, which treats clothing and high fashion in a similar referential manner. But Trinidadian clothing style doesn't return to classic themes in this way.\n\nAn alternative might be to look, not to some origin in Africa, but to the experience of slavery itself. The idea of keeping things on the surface as a defensive strategy against the condition of extreme degradation is brilliantly depicted by Toni Morrison in her novel Beloved: 'so you protected yourself and loved small. Picked the tiniest stars out of the sky to own... Anything bigger wouldn't do. A woman, a child, a brother \u2013 a big love that would split you wide open in Alfred Georgia.'6 The precarious existence given by the condition of slavery precluded any internalization of love, since there was no knowing when this love-object might be wrested away, resulting in a kind of adaptive tendency to keep things on the surface, to refuse any internalization and thus to minimize one's sense of loss. This is more plausible, and may remain relevant to many of those in the squatting area. But the majority of the population of Trinidad do not have origins in Africa and slavery. Many come from South Asia or have mixed backgrounds.\n\nSo, instead of trying to ask where such a relationship to style comes from, instead of seeing it as a problem that requires explanation, we can turn the lens back onto ourselves. Why do we think that a devotion to clothing is a problem anyway? Why do we see it as a sign of superficiality and what does the very term superficiality imply? The problem with a theory of semiotics and of treating clothing as superficial is that we presume a certain relationship between the interior and the exterior. We possess what could be called a depth ontology. The assumption is that being \u2013 what we truly are \u2013 is located deep inside ourselves and is in direct opposition to the surface. A clothes shopper is shallow because a philosopher or a saint is deep. The true core to the self is relatively constant and unchanging and also unresponsive to mere circumstance. We have to look deep inside ourselves to find ourselves. But these are all metaphors. Deep inside ourselves is blood and bile, not philosophical certainty. We won't find a soul by cutting deep into someone, though I suppose we might accidentally release it. My point is that there is simply no reason on earth why another population should see things this same way. No reason at all why they should consider our real being to be deep inside and falsity on the outside. The argument here is that Trinidadians by and large don't.\n\nIn stark contrast to this depth ontology Trinidadians seem to have almost a horror of things becoming interiorized, rather than kept on the surface. Perhaps the most popular leisure activity in Trinidad is the lime, in which a group of people either hang around a street corner or travel in a group, for example, into the countryside to make a cook. A feature of the lime is the genre of verbal insult which is known as picong or giving fatigue. The individual failings of a fellow limer would then be picked upon. An older male might be asked 'when you alive yet, you could cook?' or about whether any remaining hair is really his. An accident or mistake might be thrown back at the 'guilty' party many times, with appellations such as 'mother-cunt'. Such picong almost always remains good-humoured, because the recipient knows that they are being judged by their ability not to take this on. This often witty and always barbed invective between friends makes the lime a kind of training ground in which one is steeled against taking in the abuse which can be received in life. There is a version of madness called tabanca. This afflicts people not because they have lost a relationship, but because they then discover that they allowed that relationship to get inside them, and when it ended they became distracted and disorientated. One of the most common expressions heard in response to any misfortune, from a passing insult to the break-up of a relationship, is doh (don't) take it on. In other words implying don't take it in.\n\nMost Trinidadians would certainly assert humour and wit as central to their self-definition and would see it as contributing to their sense of cool and style. A person without a sense of humour, who can't take insults, is seen as ignorant and prone to violence, a label Trinidadians use of their Caribbean rivals, the Jamaicans. This keeping of things on the surface also means the freedom to construct oneself and not be categorized by circumstance. In London when two middle-class people meet they tend to ask each other 'and what do you do?' \u2013 meaning their employment. But most Trinidadians consider this highly inappropriate. One works simply because one needs to earn money, so this is entirely the wrong source of self-definition. Asking what work someone does tells you nothing significant about them. It is the things one chooses freely to do that should define you, not the things you have to do. Freedom in self-construction seems central.\n\nIt is again at Carnival that one comes to appreciate the further implications of not seeing the essential nature or truth of a person as a property located deep within. One of the main themes of Carnival is the revelation of truth. Carnival starts at night with a festival called Jouvert derived from the French jour d'ouvert or the opening of the day. People dress as creatures of the night, such as devils, or come out covered in mud (London's Notting Hill has an appealing variant: covering your body in chocolate). Sometimes they carry placards with scandals and accusations. Gradually they move towards the centre of town where they are revealed by the dawn. In 1988, one of the most striking costumes represented a current calypso and was called Bacchanal Woman. A huge figure wore a dress festooned with eyes. Bacchanal is the disorder that follows scandalous revelation. The classic example is where a strict schoolteacher has tried to portray herself as thoroughly respectable, until a pregnancy reveals something else going on. People try constantly not to reveal the truth about themselves but Carnival brings the things of the night into the light of revelation.\n\nThe point all this makes about lies is that people are constantly trying to hide them. And where is the obvious place to hide things? Well, deep inside where other people can't see them. So the truth that emerges at Carnival is premised on exactly the opposite set of metaphors as that of our own depth ontology. For Trinidadians it is entirely obvious that truth resides on the surface where other people can easily see it and attest to it, while lies are to be found in the hidden recesses deep within. A person's real being, then, is also on the surface, and evident. The deep person, who keeps things stored close to himself or herself and out of view, is viewed as just dishonest. The point, of course, is that truth is neither intrinsically deep nor on the surface. Neither set of metaphors can be judged as right or wrong. It is simply that there is no reason why any other population should have a concept of superficiality which sees the deep inside as true and significant and the surface as false and insignificant. In many ways the Trinidadians seem to have a rather more obvious logic of spatial metaphors of truth and being than we do.\n\nThese differences in metaphors, reflecting differences in the concept of being, may use idioms of time as well as of space. If the self is not deep inside it is also not viewed as constant. We see the self as growing, based on things that are accumulated. So occupation, social status and position create substance which is accumulated within. This comes from a historical preference for relatively fixed identities and hierarchies. In earlier times a person was defined by birth. We now prefer an apparently meritocratic ideal which defines them more by cumulative achievement. But Trinidadians may not be aiming for such a sense of an incremental self, which would be regarded as both false and imposed by position. A person today should not be judged by what they used to be, but what they are now. Instead we have to imagine a situation in which being is constantly re-created through a strategy of display and the response of that moment. In going to a party, or forming a relationship, the individual usually aims high. They attempt the best style, the wittiest verbal agility and, if possible, the most impressive partner. But one only finds out if this is actually who you are from the response of the day; how people react to you and appraise you. It is each particular and assumed transient activity that tells one who one is. It is the event itself that gives judgement. However, this is only a specific event or relationship, so that the position has to be recovered again on the next occasion.\n\nThe advantage of a transient self is that it is less subject to institutional construction and judgement. It is not given by for-mal recognition or occupation. This means that comparatively speaking it is a self that can feel free \u2013 which for Trinidadians is tantamount to saying it is more real or truthful. Men and women prefer to judge the state of their relationship by the way they treat each other at that time. They are suspicious of the way an institution, such as marriage, can lead to one side taking the other for granted, and no longer having to make constant the attention that signifies that the relationship remains true. For Trinidadians, marriage as an institution can easily make a relationship false, since one can mistake its formal nature for its reality, which lies in the actual way each treats the other in the present. This is one reason why people prefer to wait for marriage until they are very established in their relationship, often with several children.\n\nThere is a problem with the kind of historical determinism which always explains the present by a search for roots, by a narrative of how people got to be as we now find them. Social science tends to be more concerned with how things connect up with each other today. The past has certainly made its contribution. Perhaps the huge emphasis Trinidadians put on freedom, the freedom to construct themselves through style, rather than being defined by occupation, does indeed show an influence from slavery. The past experience of oppression might put a premium on freedom today. But anthropologists prefer to see things in comparative perspective, looking for analogies in other societies. Anthropologists writing about Papua New Guinea, for example, have argued that, there too, people prefer to judge by appearances.7 Members of a community parade and dance for considerable periods in front of others. By doing so, not only can the observers see how cohesive they are currently as a group, but also the individuals doing the parading see, in the eyes of those judging them, who they actually are. Have they been able to look good as individuals and cohesive as a group? My sense is that populations who are relatively egalitarian prefer metaphors which suggest that people are to be defined by their current abilities and achievements; that they should lose that achieved position when those abilities wane. It follows that they tend to see truth and being as on the surface. By contrast, strongly entrenched societies with long histories of institutionalization, whether of class or position, tend to see being and truth as deep within the self and relatively constant.\n\nIn such a context clothes do not just mean something quite different; they are something quite different. In these egalitarian societies, they are the very forms by which one can discover who one actually is. It then makes a great deal of sense to spend rather more time being concerned with how one looks, if how one looks is who one is. Not a reflection of who one is, but actually who one is. Our own notion of being as depth has some surprising consequences. For example, we have this very peculiar ideal about looking natural, which tends to imply that putting on make-up and clothes is false and superficial. But why should we assume this? Why should the fact that one person has freckles tell us who they are? Or that one person is born uglier than another person, and so can portray evil or a debased character on the stage? We see the natural just like the deep as being the truth of a person.The Trinidadian conception, by contrast, is that who we are is not at all given by the happenstance of physiognomy \u2013 our face when we wake up in the morning. Why on earth should the natural look of a person be a guide to who that person is? By contrast, a person who spends time, money, taste and attention in creating a look, where the final look is the direct result of all that activity and effort, can properly be discovered in their appearance. Because now one is judging what they have done, not what they happen to look like originally. We are judging them by their labour, not their birth. One aspires by the act of self-cultivation. Other people can then see if you are all that you claim thereby to be, or if actually you are clearly not up to that mark.\n\nTrinidad is a street corner society where such failures are going to be commented upon, often within a few hundred metres of setting out from your house. People love to pronounce in public upon how another person looks. But again, if we push ourselves beyond our usual metaphors, we can see that this can make perfectly good sense. Trinidadians do label people as plumbers or lawyers, rich or poor, or for that matter black or white, but regard this as superficial, compared to a judgement as to how they have constructed themselves, whether they can strut their stuff. My intention is not really to favour one view of being over the other, although it's hard not to see the Trinidadian view as rather clearer and more consistent than our own. The consequence, for present purposes, is that when Trinidadians spend far more attention on clothing, and make it more of a priority in their lives, they are not being deluded or more facile.\n\nThis first case study also provides a clear illustration as to what I mean when I claim an ambition to be an extremist. We go to an island in the Caribbean and find that there is a relationship to clothing unlike anything we had imagined. But we achieve this understanding through an extreme particularism. Anthropologists can always become still more parochial if we choose. It is something we do with relish because it proves our scholarship. Only after living there for a year could I start to understand the diversity of Trinidad itself. By then even the general term Trinidadian looked suspect. Because the people I am dealing with are Indo and Afro, big-shot and grass-roots, men and women, elderly and children. Actually they are Jonas and George, Simone and Rhoda. Ethnography is a devotion to the particular, what was special about these individual people at this time. Yet to understand one tiny microcosm of this population \u2013 why impoverished women had so many shoes \u2013 one needed to interrogate a basic philosophical assumption about what it is to be human. I needed to challenge our fundamental theory of ontology, that is, the philosophy of being, to expose the assumptions we make about where being is located and the multiplicity of metaphors and assumptions that flow in all directions from the presumption that being is deep.\n\nWe thereby gain an appreciation that what we had assumed to be a universal was itself a particular. That ontology is a cultural construction and not an inherent truth. But demolishing the foundations of Western philosophy in order to understand impoverished Trinidadian women's relationship to their shoes strikes me as entirely worthwhile. It justifies the struggle to bring the two extremes of universalism and particularism back into conversation with each other. After all, the aim of anthropology is understanding, in the sense of empathy. I suppose I was always trying to imagine what it would be like to be in their shoes.\n\nThere is another reason for starting with this particular example. The point made here relates to more than just clothing. The term superficiality and the assumptions we make about where being is located form part of a much larger denigration of material culture in our own society, where materialism itself is viewed as superficial. Becoming a consumer society is generally seen as symptomatic of a loss of depth in the world. This example of a universalizing attitude to modern mass consumption is clearly expressed in another branch of philosophy; the writings on postmodernism, which may be viewed as merely the most recent manifestation of aesthetic and philosophical ideologies which lie at the core of Western thought. Here though we are embarked on a journey that is intended to rescue, not just clothing, but the whole of material culture and the people who study material culture from this same accusation of superficiality. To show why people in places such as tribal Papua New Guinea may be resolutely more materialistic than we are, and may accept with little problem that people are constituted by things and appearances. So demolishing our presumptions about superficiality can only be a start. Next we need to look in more detail at how things, such as clothing, come not to represent people, but to actually constitute who they are. In pursuit of that goal we are now going to shift from Trinidad to India. Once that was hard to imagine. But thanks to Google Earth, just zoom out a little, move the globe a little, and now zoom right back in.\n\nThe Sari8\n\nA sari is a single piece of entirely unsewn cloth, usually around 6 metres, worn by being draped around the body. Usually these days it is worn in association with a petticoat, a blouse piece, and beneath these, pants and a bra. The aim of this section is not to tell you how Indian women wear a sari, or how the sari represents their identity. Quite the opposite. The intention is to explain how the sari wears the Indian woman, how it makes her what she is \u2013 both woman and Indian. If, in Trinidad, clothing challenges what we meant by the concept of being, in India we shall see how a phrase such as 'being a woman' is also subject to comparative analysis. As will become apparent, being a woman is quite different if it is accomplished through wearing a sari rather than through wearing a skirt or dress. Clothes are among our most personal possessions. They are the main medium between our sense of our bodies and our sense of the external world. First then let's consider how it feels to wear a sari.\n\nIn the now widespread Nivi style, the sari is draped from right to left, passing over the lower body twice \u2013 the second time in a cluster of fan-shaped pleats \u2013 and the upper body once. The pallu, the free and usually more decorated end of the sari, falls over the left shoulder down to the waist. Given the asymmetry of the sari, no sensation in one part of the body is repeated in any other. The right leg does not feel like the mirror of the left. The two shoulders and the two breasts are touched by the garment in quite different ways. The right shoulder can remain untouched by the sari, while the left bears the weight of the pallu. The right breast feels the pressure of the pleats of the pallu pulled across the bosom, whereas the left one feels exposed, covered from the front but visible from the side. The right side of the waist is hot from the pleats passing over it, but the left side is uncovered and cool.\n\nThe centre point of the sari \u2013 the navel \u2013 carries the sensation of being a focal pivot around which the security of the whole garment revolves. Here the pleats are tucked into the drawstring of the petticoat. About a metre of cloth is gathered, with a good 12 cm tucked inside the string, against the belly. This causes perspiration, and a zari or starched cotton border may scratch,9 but these sensations also give reassurance that the pleats are not spilling out. The thighs help define the graceful folds of the pleats that fan out from between the legs. Here again the sensations are asymmetrical. The pleats often lie from right to left in such a way that the first pleat rests on the right leg while the last pleat rests against the left leg. It is how one holds the right leg and knee that defines the shape of the pleats, but it is from the left leg that the sari curves from mid-thigh around towards the back of the body, resting in folds on the back of one's waist to then be brought around from the right side as the pallu. So the curve of a woman's hip and waist is accentuated on the left side. A slight bend at the knee can create a horizontal break in the vertical folds of the sari, giving a woman a more feminine and statuesque look. The ankles always feel slightly crowded as the gathers of the folds of the sari rest against them, their touch made heavier by the 'fall'. When walking, the right leg determines the length of the stride, which is kept in check by the warning tension at the ankle when the stride is too long for the sari. The left leg needs to move a little bit out and forward so as not to trap the pleats between the knees. The pallu may slide off with the movement, in which case the right arm comes up to restore it to the left shoulder, but carefully so as not to crush the cloth. After a few strides the sari may slip down from the left waist, and the left arm needs to pull it back up in order to retain the fan shape of the pleats.\n\nAs they sit down, women invariably check that the pallu has not slipped from covering the right breast, and that the waist is not too exposed by the folding of the torso. The folding of the body upon sitting down in a rickshaw or a car tends also to crush the sari in front and threatens the sequence of the pleats. A series of adjustments is required to even out the part of the pallu which is visible in the front; the pleats need to be rearranged to help them retain their order, and the pallu needs to be freed from being trapped beneath the bottom. Different sorts of weather create their own sensations and problems. The perspiration that accumulates where the sari is tucked in or densely bunched is mainly unseen. Indeed, the perspiration running down the legs meets the air circulating there, creating a pleasurable cooling sensation. When the waist becomes too damp and itchy, a woman can push the petticoat string to a fresh patch of skin on the waist. In wet weather the bottom of the sari tends to be soaked first, making it heavy and pulling the whole garment downwards. The sari loses its shape and is harder to control or to feel comfortable in. A wet sari also clings to the body at various points, accentuating curves and dips of its own accord, rather than as intended by the wearer.\n\nClearly then wearing a sari has a specific feel. But this is only a hint of much more profound differences. To appreciate these we need to zoom in upon just one part of the sari \u2013 the pallu, the often highly decorated end of the sari that falls over the shoulder. The pallu represents a prosthetic quality to the garment that is not shared by any Western clothing. This is most obvious in its functional usage. As a woman does her household chores, the pallu is in constant use as a kind of third hand, lifting hot vessels in the kitchen, wiping the seat she is about to sit on in a public place, cleaning her spectacles, gathering up rupee notes in a purse-like knot, or protecting her face from smoke and smog. The pallu's presence is so constant and available, so taken for granted, that it almost seems part of the body itself. Yet the same quality that extends the capacity of a person also gives the pallu the power to betray them. When something happens that represents the unwelcome intrusion of the external world upon the self, it may well have the pallu at the end of it. The pallu gets jammed in a car door, flies in your face so that you cannot see, or falls off your head when you are trying to be modest. The same pallu that is used to hold a hot karhai (cooking pot) of food may actually catch fire when cooking. Such accidents are all too common, and can result in horrible injury and death. They are not always accidents either. In many instances of dowry deaths, the groom's family claim that her pallu caught fire 'accidentally' while she was cooking. On the other hand, desperately unhappy brides typically end their misery by hanging themselves from the ceiling by their pallu.\n\nThe close identification between people and the pallu starts with the initial relationship between mother and infant. Most Indians have their first encounter with the sari before the time of memory. Mothers use it as a multi-purpose nursing tool. When breast-feeding they cradle the baby within it, veiling the operation from the outside world, and use the cloth to wipe the surplus milk from the baby's lips. The pallu retains this ability to appear as an extension of the mother, as one woman observed of her son:10\n\nWhen he falls asleep he puts my pallu twisted around his thumb into his mouth. If I disengage my sari he starts wailing. Sometimes people give children pieces of cloth like a hankie or a scarf to get them out of the habit of their mothers' pallu. But I never have. Several neighbours have suggested it but I think \u2013 what is the harm if he holds my sari, why give him a separate piece of cloth? He plays hide-and-seek with my sari. He keeps hiding behind it and showing his face. He keeps doing this until I smile back. He thinks I may be angry otherwise. When we sleep next to each other, I cover my face and pretend to be asleep. So he moves the pallu away to see if I am smiling or laughing. If I don't register any emotion he starts crying and keeps calling 'Ma! Ma!' then I have to give in and he is happy. And he starts doing the same by covering his face and playing the same game himself. He learnt to walk holding not my finger, but my pallu.\n\nFor the child, the pallu becomes a physical embodiment of their mother's love, a love they can literally take hold of. The pallu exemplifies what the psychoanalyst Winnicott has dubbed a 'transitional object'.11 A child takes time to understand that it is itself an individual thing, separate from the rest of the external world as first represented by the mother and her breast. The pallu helps bridge that awful separation and comforts the child during his growing awareness of it. A popular version of this idea is the 'comfort blanket' carried by the character Linus in the cartoon Peanuts. A middle-aged man recalls:12\n\nI had a very good childhood, I think of my mother and my grandmother, you know, showing their love, wiping the sweat off my forehead with the pallu, using it like a fan in the summer. My mother's mother \u2013 I would be lying in her lap with my head in her lap and she would just cover my head with the pallu. I would go to sleep in this way.\n\nFor adults, the ambiguity of the pallu being simultaneously part of someone, yet separate from them, continues when it comes to their own attempts to form relationships. Given the natural propensity of the pallu to slip down from the bosom, the action of constantly covering up one's chest can have the effect, if done well (and some do it very well), of constantly drawing attention to the area that is ostensibly being protected. So a man has no idea whether a woman is re-covering herself because of what she does not want him to see, or is pointing out what she does want him to see. Another provocative form of manipulation deftly exploits the property of a sari as a draped garment, and consists of swiftly tightening it to accentuate the tautness of the bottom or the smallness of the waist. There is the additional potential for beauty and eroticism in a fabric which may appear transparent when in a single layer but is opaque when worn in several layers. The same subtlety that applies to flirtation is even more commonly applied to the institution of modesty, in which the pallu plays a central role. Most Hindu women have to cover their heads in the presence of certain family members, while most Muslim women have to cover theirs before strangers. Women have considerable scope to manipulate the precise way the pallu is held between the teeth or placed over their head, leaving the observing male quite uncertain as to the attitude that lies behind the action: it might be demure or respectful, tantalizing or truculent. These nuances of modesty and eroticism are used to great effect by those making Bollywood films and television soap operas. Physical contact in public between men and women remains frowned upon. Even married couples promenading in Delhi's parks will not, for example, hold hands. In this context, the pallu becomes an important point of playful contact between grown-up siblings or similarly aged friends. You can tug at it in order to encourage a sister or friend to 'get a move on'. Touching the pallu allows for intimacy in the absence of touching the body itself. If someone gets too attached to another person, they are likely to be teased by having it said that they have 'attached themselves to the other's pallu' (as a television presenter said to a fan who kept phoning her show). On the other hand, and for much the same reasons, in a sexual context touching the pallu can be a very intimate and personal act indeed. In Bengal, for instance, it is traditional for a bride on her wedding night to tie the end of her pallu to her husband's dhoti. These associations also help explain the comment of a sex worker in Kolkata, who reported that she did not like her clients to touch the pallu because she felt it should be reserved for her relationship with her husband.\n\nThe full emotional repertoire of the pallu is most clearly seen in watching television melodrama. The scene with the woman applying for a job as a servant has her constantly fiddling with the end of her pallu, betraying her anxiety. The school teacher who thrusts the pallu into her midriff or holds it tight in her fist to ensure that her class knows that she means business. A girl laughing just a bit too hard covers her mouth with the pallu. A woman dabbing with it at her tears thereby also screens her eyes. The pallu is a haven for an embarrassed face and a cover for unseemly emotion. But, as with so many gestures made with the sari, these ambiguities can in turn give rise to an erotic element: often unintentionally, as when covering the mouth suddenly accentuates the eyes, or screening the eyes draws attention to the lips. Thus, while the series, plots and characters may change, the pallu's starring role remains constant.\n\nThis intimate relationship between person and cloth is not simply given, by virtue of being born in South Asia. All of this has to be learnt and mastered by each individual. A typical girl in Delhi is not brought up wearing a sari. She will first attempt this intimidating feat at the special 'school farewell' ceremony, which marks the end of the final school year, for girls aged around seventeen. The girls fumble around, scared stiff of the unfamiliar folds, dangerous-looking pins and sudden extrusions of loose cloth. They are continually worried about the risk of slippage, exposure and shame before highly competitive peers, their parents and their teachers. The girls' ability to tame and inhabit this fearsome flood of fabric will be taken as an indicator of their future ability to perform the social roles that will be expected of them. Not surprisingly at this point saris often stay in place thanks largely to discreetly placed safety pins. A girl may then wear the sari a few more times at the weddings of relatives, only to be faced with the occasion in her life when she is most subject to constant scrutiny, her own wedding, where her sari wearing becomes the centrepiece of the performance.\n\nOnce married, the neophyte sari wearer strives for social respectability. She must learn to move, drape, sit, fold, pleat and swirl the sari in an appropriate way. She may live in constant fear of embarrassment on moving to her husband's home. She can hardly sleep because she is so afraid that loss of consciousness will lead to her head or knees being uncovered, and as a result she feels stifled in the summer nights. For one woman her worry about the sari falling off leads her to tie her petticoat string so tightly that later on the doctor is convinced she will harm the baby now growing inside her. At the level of safety and practicality, she must try to avoid the frequent injuries that arise from getting the sari caught in doors, machines or, worst of all, the stove.\n\nAs an outsider entering a village where all the women wear a sari one just assumes that this has become natural as something obvious and straightforward. But this is not at all the case. One meets women who have worn a sari for forty years and still never feel that they are really in command of the garment. Think of this more on analogy with driving a car. Learning to drive and learning to wear a sari both mark a shift in one's own sense of age; there is a feeling of becoming an adult, with all the new freedoms, capacities, constraints and fears that growing up entails. Both are also public events, and one's competence \u2013 or lack of it \u2013 are open for others to scrutinize and criticize. Both share the involvement of family, the same pride and fears of the parents, the common assistance of the neighbour, friend or relative who kindly gives the girl extra practice and helps her to hone her skills. Some girls turn out to be 'naturals'. Some, having had the misfortune to fail their test at driving or by tripping over at their school farewell, never achieve real mastery and acquire a new and unwelcome identity as 'poor woman driver\/sari wearer' that will burden them for the rest of their lives. Most embark on the new task with difficulty, but through practice over the years gradually come to develop an automatic competence that allows them to proceed without much thought about the mechanics of what they are doing.\n\nJust as in driving a car, for most women, their actions in wearing the sari become increasingly natural and automatic and they lose awareness that they are inhabiting the sari and its requirements. If a woman is lucky and starts to dominate rather than be oppressed by her sari, then eventually such hard-won skills will give her a remarkable flexibility to hide or accentuate those features of her body which she wishes to expose, moderate or cover up completely. Whether she works in a field or an office, she may now have a little money, saved after household expenses have been met, with which to buy her own clothes. If she has the financial resources (many village women never earn enough money to participate fully), she is able to make choices, experiment with colours and materials and she learns to spot a bargain. In other words, she becomes an active consumer of saris. It is during this period that a woman really develops her individual style, making decisions about which colours suit her, which designs she prefers and what kinds of material she is able to best manage and enjoy. This is also the period of her life when she has to juggle the competing demands of children, husband, relatives, neighbours, friends and workplace. These may all constrain her choices, but where she is able, she dresses herself accordingly for these different audiences, occasions and moods.\n\nJust as the sari starts off as far more oppressive than most Western garments, it now has the capacity to be far more powerful. Men working in offices complained that they could not com-pete with some women, simply because men don't wear a sari. A woman at one with her sari knows exactly how to place her pallu. While everyone else looking at her thinks it is just about to fall from her shoulder, she knows it isn't. She has command of a tool that allows her to express a variety of subtle emotions and claims, manipulating the sari's particular capacity for ambiguity especially with respect to eroticism. The sari by this stage becomes an instrument of power. No one ever achieved such a political mastery of this garment as Indira Gandhi. Somehow her wardrobe represented every region, group and aspiration of hundreds of millions of ordinary Indian women.\n\nTo read this account as merely a report on a research project about how Indian women wear a sari would be to miss the key elements being conveyed here. The point of this study is that the sari plays a considerable and generally unacknowledged role in creating a specificity to being an Indian woman that is distinct. The fact that, every day, a person is expected to change her appearance in relation to ever shifting circumstance or encounter makes for a significant difference in how it feels to be a woman. There is a huge contrast with most stitched clothing, which, once put on in the morning, can be largely taken for granted for the rest of the day. Stitched clothing cannot be manipulated in this way and the wearer does not expect to do so. But the sari forces a continued engagement and conversation with its wearer, and a constant pressure to respond to changes in one's surrounding social environment. In the larger book, from which this account is taken, we show how the current struggle between the sari and its main alternative, the shalwar kamiz, has become a struggle over what it means to be a modern Indian.\n\nThe sari is like a fellow actor, constantly on stage, whose presence must always be remembered. The sari turns a woman into a person who interacts with others and with the self through this constantly shifting material. A sari can be extremely supportive when attended to, helping accomplish all manner of tasks, practical, social and emotional. But when neglected it can be quick to betray, causing others to judge you harshly for quirks of appearance that you did not intend. Such varied and ambivalent experiences with the sari have a far-reaching bearing upon a woman's sense of herself, as an individual. Simply saying that someone is an Indian woman is merely a labelling. By examining the minutiae of sari-wearing we can start to see that there are a multitude of different expectations and experiences that are a direct result of wearing a particular item of clothing. These all create the specific experience of being a sari-wearing Indian woman.\n\nLondon\n\nThe presumption of anthropology lies in such comparative analysis between Trinidad and India. Recognizing that we take for granted our own ways of doing things, and that it is only through coming to appreciate how other people have entirely different experiences and expectations that we can start to challenge our own. That is the intended logic of this chapter. It is only by paying attention to the relationship between people and clothing in Trinidad and in India that a Londoner, such as myself, can in turn find people's relationship to clothes in London extraordinary, exotic and in need of explanation. Why should we think clothing is superficial, why do we wear such static clothing when there are alternatives that are more dynamic? Finally, we become ready to acknowledge why, from other people's perspective, it is we who may be seriously weird.\n\nIn its initial conception anthropology tended to treat the places from which anthropologists then came, such as London, as given, relatively unmarked, contrasted with all other places that required explanation. Today we have an anthropology based on a global equality of amazement and exoticism. Just because I was born, brought up in and now live in London doesn't mean that I understand it. Quite the contrary, in some ways as a Londoner I am the least qualified person to engage with London, because it has, for me, this taken for granted quality. In other disciplines, the fashion is to suggest that if we seek to understand the experience of, for example, a transsexual Argentinian shop assistant, we need mainly to give voice to transsexual Argentinian shop assistants. But anthropology has always resisted this kind of politics. Insisting instead that we need an exchange of understanding, where it is the others who can see much more easily the conventions and suppositions upon which we found our daily lives. This implies that we need to treat cosmopolitan sites such as London with the same respect of intense observation and analysis as anywhere else. It is all too easy to sink back into glib generalization when we return to familiar surroundings. Even in Western Europe there are profound distinctions in regional relationships to clothing if we examine them systematically. An example emerged recently in the research of one of my PhD students, Marjorie Murray, who was born and grew up in Chile, but carried out her ethnographic fieldwork in Madrid.\n\nMurray found people's relationships to clothing in Madrid to be completely unlike either her own Chile, or indeed London, where she had come to study for her doctorate. She was struck by the extreme conformity of Madrilenians. When they go out, they wear what they consider to be classic, smart and generally quite expensive clothing, even if they are not wealthy. They spend a considerable amount of time preparing themselves and their appearance before setting forth. It is quite common to use designer labels as part of this ensemble. They also have a huge knowledge of what everyone else is wearing, both people they see on the street and celebrities. By contrast, when they return home, they immediately change into clothing which is generally extremely shabby, often old, even worn or torn. Clothing which makes them look really quite unattractive. In neither the public nor in this private dressing is there much sense of individual expression. Rather, people look to wear what most people will agree is proper and appropriate clothing. Of course this is a huge generalization; no one would have the slightest difficulty in finding endless diversity, subtlety and distinction. But to conduct anthropological enquiry we need to assert comparative generalizations. Murray brings considerable evidence to show that, at least as against London or Chile, Madrid can still be characterized as a place of astonishing convention and conformity. For all our sense of European cosmopolitanism, there are marked particularities to individual cities. Another of my PhD students found that her sister in Italy was so horrified by the kind of grungy appearance that London seemed to have inflicted upon her that she insisted she return to Italy for a kind of sartorial cure.\n\nOne reason the people of Madrid may wear still more classic and conservative clothing in public is very likely associated with their history as the capital of Spain, and previously the Spanish Empire. Rather than expressing themselves as individuals, they feel responsible for representing the grandeur of Spain itself, of which they are the centre. They make it very hard for any individual who tries not to conform to the public ideal of respectability. Indeed, wearing shabby clothes at home seems to confirm that the role of clothing is to demonstrate this public appearance and is not a medium for individual or private self-expression.\n\nIn London the situation is largely reversed. London is striving today to be a capital of cool partly in refutation of a previous time when it was a capital of Empire \u2013 now seen as extremely uncool. To merely conform to a generic public fashion, or to resort to designer labels, is likely to see the wearer branded as sartorially incompetent. Styles that deliberately repudiate respectability are often the vanguard of fashion. In London individuals are exhorted to express themselves, even find themselves, through their clothes. As most women in London will attest, this is a hell of a lot harder than it sounds. It does, however, lead to a very different understanding of what fashion itself is. Most theories of fashion, most studies of fashion, look to the fashion industry, its organization, pressures and patterns to account for fashion.13 But actually there is good evidence that most people choose their clothing as much despite as because of that fashion industry. If one glances across the range of clothing actually worn on London underground trains or in the street, things look very different from the fashion magazines that those same people are reading.\n\nCurrently, I am involved in a large-scale research project on denim,14 which is what, on any given day, around half the population actually wears. Certainly the fashion industry tries to incorporate denim by developing an expensive array of designer denim. This is the denim the magazines will display and write about. But the vast majority of denim that people actually wear is relatively nondescript. Over the last decade, the rise of cheap supermarket denim has been much more significant than the rise of designer denim. It is cheap denim that one mainly sees on a given street. Since the style barely changes from decade to decade, it is hard to argue that this shows us responding to the pressures of fashion in its restless search for new profitable niches.\n\nSo perhaps we need to look elsewhere for a theory of women's fashion in London. Just as in India and Trinidad, we need to take seriously the experience of the women who wear it. The critical moment in that experience may well be when a woman gets up in the morning, examines a full wardrobe and yet feels an overwhelming fear that she has 'absolutely nothing to wear'. Another PhD student, Sophie Woodward, wrote her thesis on this act of getting dressed in the morning.15 She reasoned that merely to observe what women wear on the street provides only a partial glimpse. Because so often, when they first get dressed, they try on a more adventurous set of clothing, maybe more than one, before losing confidence and returning to their default clothing. I decided never to enquire too deeply as to just how exactly she managed to conduct this fieldwork, but this is precisely what she found. Fashion in London is much more about anxiety than it is about industry. It is this anxiety that may explain the major changes in fashion over the last few decades.\n\nA paper called 'The little black dress is the solution. But what is the problem?' reflected on these changes.16 Why can every other colour only at best aspire to be the new black? Where and when did this happen? Black may have some association with the chromophobia that seems to have possessed modernism from time to time, but that didn't prevent a veritable explosion of colour in the 1960s, not far short of Austin Powers. Most of what I have written so far concerns women, but I too experienced these changes. I see them rather as those described by Salman Rushdie in his wonderful children's book Haroun and the Sea of Stories. The book's premise is that there is some evil mechanism that is taking away the vital stream of stories that course through the veins of our world. During my lifetime I have been witness to a similar dreadful loss. There has been a gradual leaching out of colour and print from the world of clothing in London. Just as in Rushdie's story, it is as though somewhere there is a vast hole through which colour and print is leaking out, leaving an increasingly grey, brown and black world of clothing that makes for a drab, colourless environment, only partially compensated for by a few exceptions such as sportswear and the little red dress.\n\nI feel personally affronted by this assault, since I too suffer from this same affliction. When I started lecturing I was still wearing a bright orange jersey and a necklace of shells retained from my fieldwork in the Solomon Islands. But I was already looking like an anachronistic hippie. Of course, being a hippie was itself merely conventional to that time, and I have shifted with all the subsequent movements towards the colourless. Today I seem to have ended up with conventional male brown, indigo and black, with classic Armani-emulating cuts for more formal wear, and jeans for the more informal. About the most exciting possibility left for me to discover is a new shade of grey.\n\nTo explain all this we need to both examine and then account for the underlying problem. My own investigation came from a year spent shopping with people from one street in North London.17 I found that women's anxieties about clothes come in all shapes and sizes. Sharon, for example, is reaching her fortieth birthday and sees this as a watershed in terms of consolidating her appearance and her wardrobe. For she is 'fed-up with getting it all wrong'. A full-time mother of three children living in state housing, she contrasts herself with a close friend who, she comments, 'could wear a piece of rag and look gorgeous'. Recent fashion disasters included being the only person to wear evening dress at a party for her mother-in-law, because she had believed her mother-in-law's claim that everyone else would dress up. Also driving 15 miles across London with her family to a christening, and then turning back since she simply had to replace the blue cotton summer dress she was wearing with another one that had less of a 'T-shirt' style neckline.\n\nSharon's anxieties could be labelled as mid-life. But the situation is not much different shopping with Charmaigne, a highly fashionable eighteen-year-old who is extremely self-conscious of her 'ideal' figure (she dances in videos for vanguard pop music), and excels in choosing items which she considers most appropriately draw attention to her body. But feeling she needs to look a little more grown up, she decides she needs a floral print dress. During the course of our expedition to Brent Cross shopping centre, we encounter hundreds of possible floral prints. The issue of cost is not raised, only of suitability. There is certainly no lack of choice. The problem for Charmaigne, again and again, was knowing whether she actually liked a print or not and therefore being able to determine what her taste was or could be. In the end she returns home empty-handed. Charmaigne certainly does not lack knowledge of clothes: her problem is knowing what her own taste in clothes might be with respect to an unfamiliar genre. One can have a fine sense of the nuances of language without knowing what one wants to say.\n\nThis is why in England there is such an appeal to those who can provide clear direction. To watch a programme called What not to Wear and pray there are some simple rules one can follow. Or, at a greater extreme, sign up to a company such as Colour Me Beautiful which claims a whole science of clothing that will tell you exactly what colour is right for you. So it is possible to document and characterize this anxiety at the core of fashion, but we also need to account for it. Why has it grown over the last few decades so that people feel unable to wear much of what is easily available in the shops? Here anthropology can step in with its comparative perspective. What makes the situation in London so different from the previous examples?\n\nIn India, women who wear a sari are subject to a very well established set of rules and social conventions. A woman who wears something inappropriate will soon be made well aware of her faux pas. By comparison, clothing in London is much less guided by order and social convention. It has become quite extraordinarily diverse and subject to rapid changes. So both the constraint, but also the support, of social convention have diminished. Advice from the fashion industry is much harder to use. Once there was some consistency and one was confident in how long skirts should be worn that year. Hemlines around the country went up and down in unison. Now advice is as voluminous as it is contradictory, and despite reading multiple magazines, most people are not too sure what fashion actually is at any point of time. This is despite some brilliant fashion 'agony aunts' (my own heroine being Hadley Freeman of The Guardian). But then people are much more suspicious of commercial statements about the current norms of fashion. Perhaps more importantly much of this disengagement from fashion comes from the overwhelming pressure to express the individual and not merely follow fashion whatever it may be. This leads to a suspicion in London of genres such as designer brands which for adults may be seen as vulgar or stupid. Entirely different from the situation in Madrid.\n\nIf in London people bear individual responsibility for developing their own sense of style, this makes London seem much more like Trinidad. But there are separate issues which make London very different from Trinidad. People in London are relatively circumspect in expressing their opinions about how others look. Strangers would never shout out comments on the street to a passing woman, in the way they seem to do in Trinidad every few minutes. Comments in London are rarely direct; they are more often based on banter, or irony or said to a third person, rather than directly to the individual in question. As a result, individuals in London find it much more difficult to gain a purchase on this external presentation of themselves. They simply feel unsure about what other people think about them, and then in turn they become increasingly insecure that they even know what they think about themselves. At least with overt criticism you know where you stand. In London this is replaced with paranoia about what people might be saying behind your back \u2013 from where they can get a clear view that one's bottom really is too big. It is perhaps no surprise that the key television programme was called not What to Wear but What not to Wear.\n\nWithout the social norms of India and the explicit critical comments of Trinidad, women feel a lack of support in developing their own personal preference. The expression 'I may not know about art but I know what I like' is replaced by the opposite feeling that they do know a great deal about clothing but are not at all sure what they like. In some ways the nail in the coffin of certitude and confidence has been feminism. While there have been some retreats and disavowals in recent years, there have clearly been greater advances in the effective impact of feminism in London as compared to India or Trinidad. Feminism is devoted to the empowerment of women to make choices for themselves and not be dictated to by external pressures. This is entirely positive in terms of equality. But it places still more of a burden on individuals to know for themselves what it is they want and who they want to be. And in the modern world that is becoming one of the hardest of tasks. All of this culminates in a situation where the heart of fashion becomes anxiety. So for London, paradoxically, freedom ends up as a conformity that can be quite drab, but a conformity that is entirely different in cause and effect from that of Madrid.\n\nAcademics take delight in repudiating what has been called the 'intuitively obvious'. We gain far more credit if our insights appear original and challenging. But that doesn't in itself make them wrong. The situation is full of ironic contradictions: freedoms that create anxiety, empowerment that feels oppressive, individualism that leads to conformity. Let's be clear as to the implications of this argument. On a political front the much larger and more significant problem with feminism is surely that it has not been successful enough, as yet. I am a student of the 1970s and 1980s, with the same commitments. I am extremely frustrated by the backtracking on what I took as irreversible progress towards equality. But that doesn't stop us appreciating feminism's contradictions \u2013 its contribution to this huge burden of freedom. I would hardly advocate a return to authoritarian sartorial codes and unwarranted respect for the voice of industry elites. An anxiety when shopping and dressing, or the feeling that a full wardrobe appears to have 'nothing in it', may, on reflection, be a price worth paying.\n\nConclusion: A Good Fit?\n\nThe intention here is merely to make sense of what, on reflection, is a relationship to clothes that now seems a good deal more peculiar in London than in India or Trinidad. London achieves a spectacular sense of drabness in the teeth of an abundant selection of colourful alternatives hanging in at least some shops, if only we would choose to buy them. What all this demonstrates is the vast range of possible relationships between the concept of the self, the person and clothing. There is nothing new in this. If we look at the longer term it becomes still clearer that it isn't just the clothing that is changing, it's the other side of the equation, the self, that is changing. Mostly this book summarizes my own work here, and that of my students, but no apology is required for a quick excursion to visit a brilliant account of this historical trajectory \u2013 The Fall of Public Man by Richard Sennett.18\n\nConcentrating on material from France, Sennett reflects on the period before the French Revolution; what is sometimes called the ancien r\u00e9gime. At this time people were largely expected to dress according to their station in life. People assumed that they would know from clothing who a person was: a butcher, servant, prostitute or gentlewoman. As in many parts of the world, at different times there might be sumptuary laws; making any attempt to dress above one's station in life was illegal. It was only right to make fun, in public, of a person dressing in a manner inappropriate to their occupation. So clothing at that time did not represent some authentic inner person. Rather, people were expected to live up to the conventional category that their clothing represented. Sennett traces various changes in this fundamental relationship. One is the rise of theatre and the idea of acting. One consequence of being conscious that people may be acting a part is the idea that a person who is not acting is being natural. A corollary is that people perhaps ought to be natural when not explicitly acting.\n\nThen the whole relationship between nature, the self and appearance is thrown into confusion by the French Revolution. This creates a slew of new ideals about truth and the person. At one time there was a cult of almost nakedness to express the ideal that the real self should not be hidden by artifice, which now includes clothing itself. From here we can see the start of the liberal trajectory that emphasizes the self itself as a goal in life. This leads eventually to the post-1960s Californian ideal of spending one's life looking for the true, natural, authentic self. So most people today, in places such as England or the US, come to believe that getting clothes right is part of an appropriate exploration of who one really is. Clothes don't so much change us as reveal us, even to ourselves: reveal the true inner and relatively constant self within. Typically we look for something in the shops that is original but not too showy, modest but not stuffy because \u2013 well, that's really what we are like, isn't it?\n\nThis brief discussion of Sennett's work has brought us full circle, back to the idea of authentic inner selves and the intrinsic superficiality of clothing; a philosophical supposition whose assumed universality was repudiated by the example of Trinidad. Hopefully there is a cumulative conclusion to these three case studies that supports the introduction to this chapter. In three very different instances it becomes clear that we cannot regard clothing as a form of representation, a semiotic sign or symbol of the person. It is this form of analysis, not the clothing, that may now appear to us as superficial. Instead we have discovered something really quite profound. That the concept of the person, the sense of the self, the experience of being an individual, are radically different at different times and in different places, partly in relation to differences in clothing. In each case we have found that clothing plays a considerable and active part in constituting the particular experience of the self, in determining what the self is.\n\nIn Trinidad people use clothes to find out who they are at that particular moment of time. In India the experience of being a woman is different, when one is expected to constantly shift appearance in relation to each shift in circumstance. In Madrid clothing helped to retain the cosmological ideal of Madrid as a centre of civilization. While in London clothing was found to be a source of anxiety, precisely because of the increasing pressure on individuals to express themselves, combined with the growing difficulty in determining one's own individual taste. So notwithstanding our increasingly common expectations of education and lifestyle, the supposed homogenizing effects of global capitalism, we have found striking regional differences in our relationship to clothing. This in turn has shown us that there remains also considerable diversity in who and what we think we are. That is not something that strikes me as superficial at all. So even if the same retail clothing chains are now to be found from S\u00e3o Paulo to Seoul this doesn't mean that the experience of wearing that clothing has been reduced to a single expressive form. One further conclusion is that the role of anthropology, committed to learning from comparative studies of humanity, is by no means diminished by global capitalism and modernity. In some ways the fun is only just beginning.\n\nThis chapter has also tried to exemplify the argument, introduced in the prologue, that anthropological studies of material culture should strive for a vision of extremism. But in this quest we still have a long way to go. The bulk of this chapter was devoted to an immersion in the rich lather of ethnography, luxuriating in the detail: the sensuality of touch, colour and flow. A study of clothing should not be cold; it has to invoke the tactile, emotional, intimate world of feelings. The first task of the anthropologist is to convey these feelings empathetically: what it feels like to wear a sari, where it presses on the body, and where you sweat. How you flirt and how to keep modest. How awful it is to be the only person dressing up for the party. How we can shop for hours and hours and not be able to buy any one of a thousand alternatives on offer. We have immersed ourselves in the minutiae of the intimate.\n\nThe next chapter moves us upwards to much more austere and colder climes. Up to the extremes of generalization, abstraction and philosophy, to ask if it is possible to have a theory of material culture and whether we can reach general conclusions about the nature of both humanity and our essential materiality. But even if we reach that peak, we don't want to be stuck there. The aim is not at all to become a philosopher. The aim of anthropology is to take any such pure, clean philosophy and drag it back down to the valley, to the muddy terrain of particularity and diversity. Much of this chapter has been concerned, not with clothing as material culture, but with clothing as clothing \u2013 with issues about wearing and texture that are highly specific to clothing and which will contrast with later chapters that concentrate on other genres such as housing or media. But clothing has achieved one task especially well. It has shown us how, in the hybrid world of everyday life, it is often these intimate and sensual realms that are most effective in determining the acceptability and plausibility of the regimes of thought that we call rationality or even ontology. Through the realm of clothing, we can see how, for most peoples, systems of thinking about the world also have to feel right.\n\nNotes\n\n1 Influential examples at that time included R. Barthes, Syst\u00e8me de la mode (Paris: Editions du Seuil, 1967); M. Douglas and B. Isherwood, The World of Goods (London: Allen Lane, 1978); and M. Sahlins, Culture and Practical Reason (Chicago: University of Chicago Press, 1976).\n\n2 The next section is based on 'Style and Ontology in Trinidad', in J. Friedman (ed.), Consumption and Identity (Amsterdam: Harwood, 1994), pp. 71\u201396; and Modernity \u2013 An Ethnographic Approach: Dualism and Mass Consumption in Trinidad (Oxford: Berg, 1994).\n\n3 A. C. Carmichael, Domestic Manners and Social Condition of the White, Colored, and Negro Population on the West Indies (New York: Negro University Press, 1833), p. 75.\n\n4 M. Freilich, 'Cultural Diversity among Trinidadian Peasants', PhD diss., Columbia University, p. 73.\n\n5 H. Gates, The Signifying Monkey (Oxford: Oxford University Press, 1988).\n\n6 T. Morrison, Beloved (New York: Knopf, 1987), p. 162.\n\n7 M. O'Hanlon, Reading the Skin: Adornment, Display and Society among the Wahgi (London: British Museum Publications, 1989); M. Strathern, 'The Self in Self-decoration', Oceania, 44 (1979), pp. 241\u201357.\n\n8 The next section is based on M. Banerjee and D. Miller, The Sari (Oxford: Berg, 2003), esp. ch. 2, pp. 23\u201344.\n\n9 Zari is embroidery, originally made using metal thread, usually gold-covered silver, but today mainly made with fake metal thread or plastic.\n\n10 Banerjee and Miller, The Sari, p. 32.\n\n11 D. Winnicott, Playing and Reality (London: Tavistock, 1971).\n\n12 Banerjee and Miller, The Sari, p. 34.\n\n13 Some of the following discussion is based on A. Clarke and D. Miller, 'Fashion and Anxiety', Fashion Theory, 6 (2002), pp. 191\u2013214.\n\n14 D. Miller and S. Woodward, 'A Manifesto for the Study of Denim', Social Anthropology, 15\/3 (2007), pp. 325\u201335; see also www.ucl.ac.uk\/global-denim-project.\n\n15 S. Woodward, Why Women Wear what they Wear (Oxford: Berg, 2007).\n\n16 'The Little Black Dress is the Solution. But What's the Problem?', in K. Ekstrom and H. Brembeck (eds), Elusive Consumption (Oxford: Berg, 2004), pp. 113\u201327.\n\n17 This material comes from Clarke and Miller, 'Fashion and Anxiety', and also A Theory of Shopping (Cambridge: Polity, 1998).\n\n18 R. Sennett, The Fall of Public Man (Cambridge: Cambridge University Press, 1976).\n2\n\nTheories of Things\n\nFrom Hedonism to Functionalism\n\nThe intention of this chapter is to argue for a theory that can explain the findings of the last. Already we are withdrawing from a comfortable idea that we start with people who make things which represent them or others. It is now clear that in material culture we are concerned at least as much with how things make people as the other way around. But that was the last chapter. By the end of this chapter the very conception of things as opposed to persons will come into question; rather, we will end up with processes that give us the illusion that these are discrete entities that make each other. This chapter is not intended to present the or the only such theory. Many of my colleagues in my own department and in other universities have complementary approaches to that taken here. Some give more emphasis, for example, to the properties of materials such as wood and stone, rather than of artefacts, or take a different route through phenomenology. But at least this chapter will hopefully demonstrate that it is possible to create plausible and helpful theories of stuff.\n\nThis demonstration will start in the form of a personal story, or search. Beginning with my earliest attempts at such a theory, it will then move upwards to the core theory of objectification as derived from the philosopher Hegel and as then transformed by later writers such as Marx and Simmel. Then we will detour to consider the nature of materiality itself. The chapter will conclude with a consideration of the wider consequence of these theories. Much of this is going to be hard work, even in the rather vulgarized versions of philosophy employed here. If you really can't bear such cold abstractions, you may even decide to skip forward a chapter. But don't despair. In the subsequent chapters we can hopefully once again get down and dirty.\n\nMy initial training was as a student in anthropology and archaeology but with an emphasis upon the latter, which to be honest seemed the more romantic of the two. I worked on several archaeological excavations; the very first of them was a Roman pottery factory in the surprisingly suburban environs of Muswell Hill, London. Later more exotic ventures included participation in an attempt to discover the lost empire of Srivijaya in the jungles of Sumatra. Seeing tigers in the wild, excavating in massive caves, watching the morning mists lift from valleys of rice terracing, being taught to swear in Indonesian by local maidens, holidaying in Bali. Life could have been worse. The fact that we conspicuously failed to find the lost empire of Srivijaya merely seemed a generous gift of future opportunities to others. I then attended lectures in London on ancient China, wrote papers in Cambridge about the Indus Civilization of South Asia and, with a fellow student, called ourselves an expedition and studied pottery-making in the Ambon region of Eastern Indonesia.\n\nAfter graduating, I was employed for two years as government archaeologist by the national museum of the Solomon Islands in the South Pacific through a voluntary scheme,1 one moment viewing tribal menstrual huts with the anthropologist Roger Keesing, who had written my student textbook,2 then collecting recipes for cooking children from people who had, in times gone by, eaten them (I will spare you the details, though it involved slaughtering young pigs of the opposite sex, and earth oven pits). Mostly trying to document and protect evidence for the past, partly through new legislation. Trekking barefoot with machetes through forests to discover abandoned hill forts. Then relaxing with fellow volunteers by snorkelling over coral, while lazily passing half empty flagons of Australian wine back and forth over the still surface of the reef-enclosed sea. Two more difficult years.\n\nSome stubborn streak of tedious academic concern somehow seemed to survive all this. The Ambon expedition had already focused upon pottery manufacture.3 It explored the way social and cultural differences, such as that between Muslim and Christian potters, were represented, not by different pots, which were identical, but through distinctions in the ways these same pots were manufactured. This interest in pottery derived from my point of transition between working as archaeologist and anthropologist. My start with that excavation in Muswell Hill made it clear why archaeologists have a special concern with pottery. Many ancient sites are strewn with pottery fragments, and sometimes not a whole lot else. So the route to the reconstruction of ancient times is usually rather too dependent upon the ways these pottery fragments were interpreted. The temptation was to surrender oneself to their extensive presence. So one would read, in those days, of prehistoric Europe being invaded by the Impressed Beaker People (on reflection they must have looked pretty odd). It seemed sensible to reconsider such simplistic identifications through analogy with contemporary populations and the significance (or insignificance) of pottery in their lives, such as could be observed by ethnographic study.\n\nLater, my return to Cambridge as a graduate student led to me re-locating to a village in north-west India where I was to live for a year in order to collect data for my PhD. The problem I set myself sounded quite simple. The potters in this village produced a range of over fifty pottery shapes, each with their own name and function. As such they were analogous with the assemblages of pottery that were the source of so much archaeological interpretation. Given the additional information available when working in a living society, I asked what precisely explained this particular range of pots. The book that resulted was called Artefacts as Categories,4 and was partly inspired by ideas at the time concerned with categorization processes. One of the book's arguments, however, bears repetition, since it dealt directly with perhaps the most protracted argument to be found in the study of material culture, which is the role of the term function. Function tends to remain our default gear in driving towards any explanation of why we have what we have. It is the way we label goods from frying pans to swimwear. But as well as dominating colloquial relationships to things it also provides a powerful trajectory of academic thinking. The adage 'form follows function' was prominent in modernism and remains a trope in design and architecture.\n\nJust when it seemed to have pretty much died out, the idea of function has recently made a rather spectacular comeback with a key role in the many explanatory models, based on evolution, that are becoming prevalent in both archaeology and anthropology. These models almost always depend upon function, as an aspect of humanity's adaptation to its environment. The analogy is with Darwin's Galapagos finches, each developing itself as a living tool to exploit some niche that ensured its survival. There is a huge current fashion for evolutionary style explanations of more or less any and every aspect of human behaviour. You will find such research constantly reported in respected journals such as The Economist and on television. It is well represented in my own department. Today this kind of evolutionary anthropology may well be more influential than social anthropology. To be honest I find it impossible to take much of it seriously. I become quite offended when some pundit for this approach appears on television under the title of anthropologist and explains in a deadpan voice why changes in the divorce rates over the last fifty years can be understood by factors connected with humanity's original descent from the trees.\n\nThe objection is really quite simple. If our social and cultural customs were indeed, in any way, connected to such functions, then that would have produced a relatively homogeneous humanity whose variation correlated largely with differences in its environment. But social anthropology exists because humanity developed quite otherwise. In one small area, such as Papua New Guinea, almost any generalization one might wish to make about human behaviour can be refuted. There are tribes who privilege homosexuality, eat their ancestors or their enemies, make men menstruate (by bleeding their penises), dance interminably, categorize their natural world systematically, fight constantly, decorate gaudily, develop brilliantly esoteric cosmological ideals about almost any philosophical supposition one could name. But each of these communities believes its customs to be entirely natural and proper, and assumes it is others who are bizarre and mistaken. The Solomon Islanders from whom I collected those recipes saw the killing and eating of a child as simply a necessary means of gaining a source of power (mana) for the tasks they undertook. In other words they saw this as a moral act rather than the reverse. Quite apart from the exotic distinctions represented by Melanesia, we are all more familiar with cultural differences between, say, the peoples of Japan and Amazonia, France and Sri Lanka, Kensington and King's Cross. I simply cannot see any correlation or even much effect on the environment. The climate remains fairly constant across London suburbs. Accounts that claim otherwise are hugely popular, but perhaps because their apparent explanations make life seem so much simpler.\n\nWhen a colleague informs me that, for a society in Africa with high infant mortality, grandparenting turns out to be critical to family reproduction I respect this as serious science. But only when it relates to a group that is struggling to reproduce, to behaviour that directly bears on that problem. To accept evolutionary effects on any other human behaviour would be to imply that we adjudicate one society as better adapted in evolutionary terms than another. But it is not the case that in New Guinea, or in the UK, one kinship system, ritual, housing type, propensity to warfare, or anything else is better able to secure social or cultural reproduction than any other, because of some environmental factor. Our behaviour is not a matter of adaptation. So the only thing evolutionary anthropology can tell us about, with any plausibility, is that which we all have and do in common as humanity. That is a perfectly respectable academic task. But this evolutionary explanation will never tell us much that is relevant to social anthropology, which focuses on the opposite problem of what we don't have in common. Evolution isn't going to help us much with understanding the diversity of stuff either. After all, in the twenty-first century the environmentalists are hardly celebrating humanity's increasingly adaptive technology, are they?\n\nLet's take a narrower focus, on the very word function. The problems with this concept should be evident from the last chapter. In Trinidad, India and London everyone uses clothes to keep warm and dry, but that tells you pretty much nothing of interest about our relationship to clothing. I suppose we could argue whether a sari was more or less functional than a skirt, but very quickly we would find that we were talking about function in a very loose fashion. If a sari is better in expressing modesty, is that a function? If a skirt signals gender distinction, is that a function? If most of the women in India who mend roads and work in the fields wear saris, does this imply they are more functional than skirts or trousers? As part of my research on Indian village pottery, one of the tasks I set myself was to investigate systematically the explanatory value of the word function in accounting for the shapes I encountered.5 I had seen such earthenware village pots exhibited in museums, accompanied by catalogues that waxed lyrical about how such traditional artefacts have evolved in perfect harmony with the needs of the villagers and their environment to produce the forms we now encounter. This is usually meant as a snide comment on our own industrial world, which is seen as shops full of superfluous nonsense that we should all regret.\n\nI therefore dutifully observed, for over a year, how water was taken from water pots, food was cooked in cooking pots and such like. Marking the precise position of fingers, forms of pouring and so forth. My conclusion was that while, not surprisingly, people did manage to store water and to cook, the precise shapes of the pots were quite remarkably ill-adapted to the purposes that they were expected to perform. People used large round water pots with narrow mouths, where the water is extracted using a vessel with a long handle. These are quite ineffective. It is impossible to get water out when they are still half-full. The oil lamps tended to go out, sometimes in a very short time, as the wicks sank into the oil. In any case, just as with the previous reflections on clothing, once we observe that they employed eleven different vessels all for exactly the same function of storing water, these differences obviously cannot be accounted for by function.\n\nActually there never was a functional society, and tribal or village peoples are probably rather less concerned with function per se than office workers or bankers.6 Yes, they want their boats to keep afloat, but so, when on holiday, do office workers. Trobriand Islanders do not grow massively long yams because they are easier to grow, or to eat. Instead we have to turn to quite other perspectives if we want to know why pots in an Indian village have their precise shape and range. First, the diversity of such pots is hardly random. Most of the fifty shapes observed were made to look different by exploiting just two or three dimensions of distinction. Just as my jackets focus on differences in, say, the number of buttons and shape of the lapel, so these pots vary according to the angle of the rim, how round they are or whether they have a flat base. In turn these differences can be systematically related to their symbolic significance. So, for example, if one thinks of foodstuffs in accordance with Hindu cosmology as a spectrum from the most pure to most impure, then the sequence of cooking pots makes sense. The most open-mouthed with the most angled rims are associated with the most polluting foodstuffs such as meat and beyond that pots for urination for the elderly, while the most closed and rounded forms are associated with milk, which in Hinduism has sacred connotations. A small distinction may mark the version of a pot intended for the festival of Divali, just as Christians might have a Christmas or Easter mug signified by special decoration. Pots that are given through formalized distributions, at certain times of the year, look systematically different from those that can be purchased for money at the market. In short, the reason why fifty different pots are produced reflects the complexity and elaboration of symbolic ritual and social distinctions. If you are still desperate to call these symbolic distinctions a function of the pots \u2013 well, no one can stop you.\n\nThe word culture tells us that societies elaborate what they are and what they do in many different ways. Through kinship, ritual and also through objects. But if we left our findings at this level we would merely return to our basic theory of representation, that of semiotics. The distinction in pots may not be reducible to function, but the last paragraph gives the impression that differences just represent distinctions in, for example, purity against pollution or festivals against secular times. In mainstream social anthropology the tendency was to reduce these symbolic representations, in turn, to some kind of fundamental social distinction. This follows the strictures of the foundational ancestors in the study of social science such as Durkheim. Following in this tradition came Mary Douglas, one of the most eloquent exponents of the study of categories and categorization. When studying, for example, food symbolism she always tried to relate these to prior social distinctions.7 This issue will be discussed in more detail shortly. A theory of representation, however, tells us little about the actual relationship between persons and things; it tends always to reduce the latter to the former. So this suggested the need for a further ambition: whether it was possible to develop a theory of things per se that did not reduce to social relations.\n\nThe first step was to produce a material culture theory that could apply to these pots. I employed two sources in embarking upon this quest. The first was the book Frame Analysis,8 in which the sociologist Goffman argued that much of our behaviour is cued by expectations, determined by the frames which constitute the context of action. We don't charge up on stage to rescue an actress playing Desdemona when she is in apparent distress. There are many elements of theatre which proclaim this as 'enacted' as against real violence. People occupy social roles, as fathers or as working in advertising. But then they also play around with such roles. We have to decide whether they are being ironic, or wanting to be taken 'at face value'. They distance themselves from their role in advertising in order to assert that really they are artists but just can't yet make a living that way. The important point to extract from this is that the cues that tell us how to interpret behaviour are usually unconscious. They may be in the place where the action is set, or the clothes being worn. When a lecturer suddenly starts a private conversation with a student in the middle of a lecture (I do this once a year to make this point), everyone suddenly becomes acutely aware of the underlying norms of the lecture as a genre. It is a framed activity that determines what is or is not appropriate behaviour. But mostly we remain unaware of this framing that constrains us.\n\nThe second source was The Sense of Order by the art historian Gombrich.9 Unlike all his other books, this focused, not upon the art work, but the frame in which the art work was set. Gombrich argued that when a frame is appropriate we simply don't see it, because it seamlessly conveys to us the mode by which we should encounter that which it frames. It is mainly when it is inappropriate (imagine a Titian framed in perspex, a Hirst in baroque gilt) that we are suddenly aware that there is indeed a frame. A more radical version of Gombrich's thesis could argue that art itself exists only in as much as frames, such as art galleries or the category of art itself, ensure that we pay particular respect, or pay particular money, for that which is contained within such frames. It is then perhaps the frame, rather than any quality independently manifested by the art work, that elicits the special response we give it as art. With respect to some contemporary art that conclusion seems to me pretty obvious.\n\nThese ideas of Goffman and Gombrich could be used to explain my experience of fieldwork. My problem was the degree to which the Indian villagers I lived with thought I was completely bonkers to be spending so much time looking at pottery. What they tried to tell me, again and again, was that \u2013 I guess 'lost the plot' would be a fair translation. Sure, the pots were all over the place, but they were a mere backdrop to the action. Sometimes this was literally the case. A village wedding has, as its most important ceremony, a moment where the bride and groom circumambulate a flame in a square space constituted by four towers of pots. These have become sufficiently iconic that a card just showing this tower of pottery works as an unambiguous wedding invitation. As such they exemplify this theory of frames. As the villagers were telling me, the pots are not the point, they are the frame. Material objects are a setting. They make us aware of what is appropriate and inappropriate. They tell us that this is a wedding, that an impure activity. But they work most effectively when we don't actually look at them, we just accept them. To start a discussion about the frame of a painting rather than the painting, the conventions of theatre rather than the play, the pots at a wedding rather than the marriage, the wallpaper of a room rather than the cosy chat we can have together in the room. It's all rather embarrassing; something that should be left unspoken has been inappropriately foregrounded. So between them, these ideas of Goffman and Gombrich constituted an argument for what I called the humility of things.10 The surprising conclusion is that objects are important, not because they are evident and physically constrain or enable, but quite the opposite. It is often precisely because we do not see them. The less we are aware of them, the more powerfully they can determine our expectations, by setting the scene and ensuring appropriate behaviour, without being open to challenge. They determine what takes place to the extent that we are unconscious of their capacity to do so.\n\nSo my first theory of things starts with exactly the opposite property of stuff than that we would expect. It is not that things are tangible stuff that we can stub our toe against. It is not that they are firm, clear foundations that are opposed to the fluffiness of the images of the mind or abstract ideas. They work by being invisible and unremarked upon, a state they usually achieve by being familiar and taken for granted. Such a perspective seems properly described as material culture since it implies that much of what makes us what we are exists, not through our consciousness or body, but as an exterior environment that habituates and prompts us.\n\nThis somewhat unexpected capacity of objects to fade out of focus and remain peripheral to our vision, and yet determinant of our behaviour and identity, had another important result. It helped explain why so many anthropologists looked down upon material culture studies as somehow either trivial or missing the point. The objects had managed to obscure their role and appear inconsequential. At a time when material culture studies had an extremely low status within the discipline, and we were seen by most anthropologists as rather pathetic, it seemed that objects had been very successful in achieving this humility, at least within anthropology. There is a wonderfully felicitous phrase, the 'blindingly obvious'. This implies that when something is sufficiently evident it can reach a point at which we are blinded to its presence, rather than reminded of its presence. One of the problems we have in persuading people that the study of blue denim is so significant is that its ubiquity seems to make people regard it as less of interest, rather than more of interest.\n\nIf my own contribution to the study of material culture rests on the precedent of one particular anthropologist, then it would certainly be that of Pierre Bourdieu. I believe that one of the premier publications within anthropology remains his book Outline of a Theory of Practice. To appreciate the contribution of Bourdieu we have first to track back a bit to the work of another consummate anthropologist, Claude L\u00e9vi-Strauss, who more or less invented the tradition that became known as structuralism. I am writing this text two weeks after celebrating his 100th birthday and acknowledging that no other anthropologist so entirely changed the way the world understands itself.\n\nThe central idea in structuralism was that we should not regard entities in isolation: a desk, a table, a dining table, a kitchen table. Rather we should start from the relationship between such things. What we are prepared to accept as a dining table depends in large part on the point at which, if it became any smaller, it would appear to be a kitchen table. Both the objects, and the words we use for them, achieve definition by contrast with what they are not, as much as from what they are. As such, structuralism focused on the relationship between things rather than the things themselves. We have just seen this with my Indian pots. A single pot is meaningless. As an example of culture they are an elaboration of shape along systematic dimensions of difference, so we can only understand each in relation to the whole system. Louis Dumont had argued much the same for Indian society itself.11 One couldn't comprehend people as simply belonging to a caste, that is, a group of people who are endogamous, marrying only amongst themselves, and usually associated with a particular position in Indian society and often also an occupation. That individual caste was part of a caste system, and behind that were the principles or relationships that generated that system. In the case of caste, the principle of hierarchy combined with a structure, such that each element makes sense only in relation to the whole. A caste was defined in part by the other castes that were higher or lower. For L\u00e9vi-Strauss, cosmologies and philosophies were representations of such systems of distinctions and dualisms that gave people the ability to see the world as meaningful.\n\nL\u00e9vi-Strauss used tribal myths as the paramount exemplification of his structuralism, but sometimes also objects. In a book called The Way of the Masks,12 he suggested that the masks of one tribe of the North West American coast could be a systematic inversion of another mask of the neighbouring tribe. Where the mask of the first tribe has a protuberance, the mask of its neighbours has a depression, and so forth. Up to then we thought we studied a society but, if there are patterns that play games passing from one society to another, suddenly even the term society was no longer an entity to be studied in isolation. Generally, though, for L\u00e9vi-Strauss this structuralism was an intellectual activity. What Bourdieu achieves is a demonstration of how these ideas can be used to create a much more satisfactory theory of how people come to be as they are and see the world in the particular way they do \u2013 that is, a theory of socialization. How does an individual grow up as a typical Parisian or Thai or Yanomami? In Bourdieu's account the key operator in making us characteristic of our own society is stuff. Most of his examples came from a North African Berber community called the Kabyle. He argues that what, in industrial societies, we now tend to inculcate through formal education happens to children born into Kabyle society through a process of habituation with the order of the things around them. Born in London, I learnt to eat with a knife and fork rather than chopsticks, to sit on chairs rather than squat, to pick my nose in private rather than in public, to fantasize about removing a woman's bra rather than her tunic, because these were the stuff that made up my environment.\n\nBourdieu, though, following L\u00e9vi-Strauss, argued that these things were not to be seen as unconnected entities. Rather, amongst the Kabyle, a child soon came to take for granted the systematic nature of the order of the home interior. One side was dark because another was light. It was high because another was low. This had to be placed on the right side, that on the left. Furthermore, a child would also soon appreciate that the same underlying order of things, discerned in the home interior, was also underlying the order of agricultural implements, or, for that matter, the order of kinship and marriage expectations. High against low, right against left, could apply as analogous in many different domains. A particular society elaborates its cultural practices through an underlying pattern which is manifested in a multitude of diverse forms. By learning to interact with a whole slew of different material cultures, an individual grows up assuming the norms that we call culture. The child doesn't learn these things as a passive set of categories, but through everyday routines that lead to consistent interaction with things, thereby providing for Bourdieu what he termed a theory of practice.\n\nThis seems to correspond very well to what I called the humility of things. Objects don't shout at you like teachers, or throw chalk at you as mine did, but they help you gently to learn how to act appropriately. This theory also gives shape and form to the idea that objects make people. Before we can make things, we are ourselves grown up and matured in the light of things that come down to us from the previous generations. We walk around the rice terraces or road systems, the housing and gardens that are effectively ancestral. These unconsciously direct our footsteps, and are the landscapes of our imagination, as well as the cultural environment to which we adapt. Bourdieu called the underlying unconscious order our habitus. There is nature, but culture gives us our second-nature, that which we habitually do without thought. Things, not, mind you, individual things, but the whole system of things, with their internal order, make us the people we are. And they are exemplary in their humility, never really drawing attention to what we owe them. They just get on with the job. But the lesson of material culture is that the more we fail to notice them, the more powerful and determinant of us they turn out to be. This provides a theory of material culture that gives stuff far far more significance than might have been expected. Culture comes above all from stuff. But, as often, when on a journey, to climb to the top of one hill then reveals an even higher peak ahead. So let's keep on climbing.\n\nObjectification\n\nEven if one has no intention of becoming a philosopher, many academics would still have an intellectual curiosity about trying to comprehend what all this might mean in more abstract terms. Philosophizing as a process may be seen as a series of steps upwards in abstraction. We start with the need for a theory of stuff as material culture. It seems then reasonable to see material culture as a subset of culture, as indicated in the last section. But then what is culture a subset of? What is the ultimate relationship between the orders of the external world and the constitution of persons? Can we reach a level of understanding which even dissolves away this distinction? At this level we want a philosophical theory that can account for every kind of stuff: bodies, streaming videos, a dream, a city, a sensation, a derivative, an ideology, a landscape, a decay, a philosophy. A theory of everything. If Bourdieu was my inspiration within anthropology, then in turning to philosophy I have an equally clear debt to Hegel. But if Bourdieu is a hard read, then Hegel is excruciating. So in treating with Hegel I am going to turn even more vulgar and academically sinful.\n\nThe system of thought Hegel developed is most fully elaborated in a book first published in 1807. This is sometimes translated as the Phenomenology of the Spirit, or the Phenomenology of Mind.13 The inconsistency of this translation is instructive. Because the thing the book is about changes during the course of the book, sometimes seeming more like mind and sometimes more like spirit. The Phenomenology is at one level a story, based on Bildung, or the stories about a hero's progress, popular in Germany at the time. Now the next few sentences are going to sound weird but, as they say, bear with me. The book starts, as good stories should, at the very beginning. Let's imagine something that can stand for a primitive condition without consciousness. An amoeba, perhaps. But this particular amoeba does have one thing going for it. A drive to have consciousness of something other than itself. So it posits the idea of something that could exist out there in the world it moves around in. By doing so it simultaneously achieves a kind of self-consciousness of itself. It becomes aware that it exists inside itself the more it is aware that there is an other that exists outside of itself. This makes our amoeba a rather unhappy little fellow, worried about this divided world it is starting to comprehend. Fortunately there comes to the rescue a further stage, in which it starts to appreciate that this very sense of externality only really came into being because of the development of its own consciousness. That doesn't seem so bad and it then becomes a bit more reconciled to this relationship. Once reconciled, it has achieved an increased level of sophistication (at least for an amoeba) which allows it to move upwards a little into a more developed form. It is then prepared to go to the next stage in positing a more complex sense of what exists out there, and so it goes through a similar sequence several times, increasing in sophistication at each stage.\n\nNow what the hell is this author on about \u2013 what gives with the amoeba? OK, before you press some button, and, as it were, turn off this book, let me provide a couple of examples that might start to show how and where this could make sense. Take, for example, the relationship between society and law. As a member of our society one might first experience The Law as something entirely external: an oppressive force, alien from oneself, that just makes life miserable and stops you having fun. Many teenagers feel like that about the law. But over time one might come to appreciate that law doesn't appear out of nothing. That it is your own society which has gradually accumulated a series of procedures that were intended to make social life more bearable. For example, you can have fun without someone breaking a bottle over your head with impunity. If the law is carried out in a way that accords with the original ideals and ideas behind it, then we might, after a while, identify ourselves as citizens who are prepared to be associated with this process and even help develop it further. Some might even contemplate becoming a magistrate. In abstract terms, here is something that we have created, but at first appears to us as an alien, external thing. Over time, once we realize that law exists because we created it, we become reconciled to it. We start to see ourselves in it. Our laws are better than their laws. We may even come to see law as an expression of some force of reason. As we achieve each enhanced level of consciousness about what law is, we mature through this increasingly sophisticated understanding of who and what we are. Ideally we then go on to create better laws.\n\nOn the other hand, it is clear that law works partly by having some autonomy from us, the citizens. While that should be positive and is necessary, it also creates a capacity for self-interest that no longer serves, but actually does oppress us. By virtue of this degree of necessary autonomy, law can become merely the means by which lawyers make money, or the threat of being sued so that schools cannot provide medicines to sick children, or the excuse for really boring television shows where a trial is really a trial. In his Phenomenology, and as further elaborated in The Philosophy of Right, Hegel argues that institutions, such as law, correspond to the manifestation and development of reason itself, as long as they remain true to their original purpose. So real education is that which fulfils the reason behind the idea and ideal of education, which is to enhance the capacities of those who are educated. This may not, of course, be the case with what happens to be taught at a particular school. At any given time, although we may make some minor contribution to the development of education or law, it is probably more accurate to say that education and law make us than that we make it. Being brought up within those institutions at a given period, schooled in a particular way, governed in a particular way, helps to make us typical of, say, medieval or nineteenth-century personages, or contemporary Latvians as opposed to Colombians. We are in great measure a product of history.\n\nFor Hegel, history consists of a sequence of such processes which we can call dialectical. A society gradually elaborates a custom or institution. Before we achieve law as we know it, we find a series of prior versions: an agreement between groups that, at the place of bartering goods, those taking place in the exchange should not be subject to violence. Then there emerges authority based on religion or kingship, before there develops an autonomous body of law. Similarly, for education, or aesthetics, or commerce or science or religion. But also for clothing, housing, cars and other stuff. OK, we don't have to go all the way back to the amoeba. We can afford to start with humanity's attempts to develop forms by which it comes to understand itself and to develop a world that accords with its sense of reason. However, each such development changes our consciousness and allows us to conceive of a still further development. So now it becomes clear that it isn't really very helpful just to say a person makes law, or that law makes a person. Or to suggest that people make a landscape, or growing up in that landscape makes people. What we end up with is the dynamic process itself, that simultaneously produces that which we colloquially come to talk about as objects and as subjects \u2013 medieval peasants and medieval landscapes, both of which are really the products of that same process that we might call medieval times.\n\nBut for Hegel there exists one more such process which occupies a rather higher level, and allows us to comprehend all these other processes. That is philosophy. For Hegel, philosophy is the means by which we gain our most advanced consciousness, the one that can comprehend the process I have just described. So the culmination of the history of philosophy is, of course, Hegel's own writings, which finally bring to consciousness this entire history of reason and its increasingly complex manifestations. Hegel saw himself as the final end point of philosophy. It's a rather extravagant claim. But I must admit, I happen to think he was largely right in this. The implication is that most effective philosophy, since his time, is in large measure a rewrite of his arguments. Again, from my limited knowledge of modern philosophy, I am tempted to agree. But I don't mind at all if I am wrong. Because the concern here is not ultimately with Hegel, but with stuff. The philosopher is here just as a means to an end. For this reason the point is not really to remain true to Hegel. We can happily appropriate those ideas that suit our purpose, while ditching others, for example his pseudo-evolutionary view of progress, without so much as an apology to the deceased.\n\nFew people today, even academics outside of philosophy, either read Hegel or are aware of his ideas. But forty years ago a great many people could have spelled out one particular version of these arguments that had come to be the creed followed by half the world. It was Karl Marx who claimed that Hegel had discovered something quite profound, but had made a fundamental mistake in thinking that the key to what humanity produced, or what produced humanity, was intellectual and would be resolved through philosophy. Marx suggested that Hegel had got the process right, but should have looked downwards, rather than upwards, and applied his ideas to our development of the material world. This is what made Marx a materialist. The way he developed from his more orthodox Hegelian roots can be traced in a body of writings variously called the 1844, the Paris, or the Economic and Philosophical manuscripts. These were written by Marx in his twenties (impressive that) but not published until 1932.14\n\nFor Marx, humanity starts with nature itself. This is the raw material from which we make our lives. Our social evolution consists not of advances in consciousness per se but in our increasing capacity to create an artifactual world from nature, first stone and pot, then agricultural systems, urban life and finally the industrial revolution, which represented a vast acceleration in our capacity to create stuff. But, just as with Hegel, each stage creates a new thing outside of ourselves, and we progress to the extent that we are able to see ourselves in this extension of ourselves, which is after all our own product. The reason we make things is because they potentially extend us as people. But we don't just posit them as external: through an act of consciousness we actually make them through labour. This new material world we have fashioned from nature allows us to travel, to improve our diet, to be entertained, to live longer. More than that, by seeing ourselves in this world we have created, we gain in complexity, sophistication and knowledge. The amoeba (or spore) can now play computer games where they can zap other creatures or turn them into civilizations which conquer other worlds. With these capacities some of us can then develop further our institutions such as education, law and philosophy. If that sounds like hard work, others can just stay there playing games that purport to do this on our behalf. But the initial process is always one of labour. It is human labour that transforms nature into objects, creating this mirror in which we can come to understand who we are. So labour produces culture in the form of stuff.\n\nMarx puts this as follows: 'The object of labour is therefore the objectification of the species-life of man: for man reproduces himself not only intellectually, in his consciousness, but actively and actually and he can therefore contemplate himself in a world he has created.'15 In reading Marx we have almost bypassed the need to create a theory of culture from Hegel and gone straight to a theory of material culture. There are, however, problems with the way it has been achieved. Hegel saw the process he described, where consciousness creates by positing something outside of itself, as a form of self-alienation. Now I would be the first to admit that the term self-alienation sounds more like the aftermath of a awful Saturday night out. But actually for Hegel this was an almost entirely positive and certainly a necessary process for our development.\n\nThis process involving self-alienation is what I refer to when I use the word objectification subsequently in this book. It is the way we enhance our capacity as human beings. By creating the film industry or cars we can grow ourselves. Every time we do such a thing, by the very same process we also create a contradiction, a possibility of oppressing ourselves if the thing we made then develops its own autonomous interests. By creating the car we also create pollution, road accidents and landscapes devastated by motorways. By creating the film industry we also create Rocky 18, Home Alone 7 and Emmanuelle Meets Godzilla (actually that last one might be worth watching). Nevertheless, we can see that films and cars can be and often are of tremendous benefit, although there is immediately a tension between something we may see as unequivocally good, say an ability to help disabled people get around, as against things we might regard as not so good, such as being encouraged to make polluting journeys to provide profits for the car industry. So within the process of objectification, self-alienating is essential, but it can become just plain alienating.\n\nWhat is also clear is that, as with clothing in the last chapter, cars, once they exist, become part of what we are. The humanity that existed before roads and traffic jams is not the same as that which exists afterwards. Cars have knock-on effects on other cultural forms. When I find that my family won't let me listen to Nirvana and Hendrix at what I see as the appropriate volume at home, I can blast them out at full volume when driving on a motorway. This is a little bit of what makes me who I am, as are less immediately personal but extremely powerful entailments of cars, such as the oil industry and sometimes war.\n\nThe intention here is to replace a theory of stuff as representation with stuff as one part of a process of objectification, or self-alienation. It is the theory that will give theoretical shape to the idea that objects make us, as part of the very same process by which we make them. That ultimately there is no separation of subjects and objects. Instead, in recognition of these thinkers, we can call this a dialectical theory of material culture. Making, growing up with and taking for granted a woven tunic, or fair-trade rice, makes us part of the society that understands itself as a tunic-wearing or fair-trade rice-using society. Making cars manufactures a new version of us. Humanity 4.3 (beta), as it were. But to clarify this further we need to do some backtracking from the direction in which these ideas were taken by Marx.\n\nMarx starts with a vision. Nature of itself belongs to no one. So the things we create from it should be to the benefit of those who are responsible for that creation. Yes, we start with objectification as a process by which we create ourselves. But as Hegel argued, once something is externalized, it can also become oppressive and we can lose consciousness that it ever was our creation. Marx argued that this happens in capitalism, which fools the workers into thinking that what makes this material world is not their labour, but the resources of capital. Even worse, their possession of nature itself is removed as private property. As a result, Marx stops writing much about the second part of the Hegelian process, where we regain this understanding. It gets shunted off into some utopian state called communism. Instead Marx and subsequent Marxists concentrate on this moment of rupture, when consciousness is taken away from us. Objects are reduced to the commodities provided by capitalism. Mostly Marxists used terms such as fetishism, reification and alienation. By now the word objectification has become more literal \u2013 subjects turned into objects. It implies that things are no longer fluid, but harden themselves against us to become the instruments of our oppression. By contrast, Hegel saw that we need to accept the integral contradictions to these processes. Private property, where people own and have rights to the place they live in, helps them to identify with that place. It's entirely different from private property where someone owns a property that another person lives in. Clearly stuff can be turned against us and become oppressive, but it is preferable to see this as a contradiction, rather than the only way to characterize our relationship to things. The theory of objectification used here is clearly Hegelian, rather than Marxist. So we still need an alternative route to stuff.\n\nFortunately Marx was not the only theorist to try and ground Hegel in our material world; he was simply the most influential. Another rather different trajectory is represented by Georg Simmel, one of the founding ancestors of modern sociology. This discussion again is restricted to a few elements of his vast output, The Philosophy of Money and some essays he wrote on the concept of culture.16 While Marx seemed to move us precipitously down from philosophy straight to material culture, Simmel allows us to stop and rest for a moment at the level of a theory of culture. This may be culture in its more particular sense such as art works, or its more general sense, which could then include education or law.\n\nFor Simmel, as opposed to Marx, culture itself is inherently tragic, that is, contradictory because, as Hegel argued, it always has this propensity to autonomy. Not just institutions such as education and law, but also stuff itself. Simmel was one of the very very few social theorists who thought deeply about the sheer quantitative increase in stuff that arose during the nineteenth century in which he lived. He could see that quantity, the increase in the amount of stuff we possess, itself posed such a contradiction. We have seen that objectification can be positive. Having more things might provide us with resources that enhance our capacity and experience and understanding. But this doesn't necessarily happen. Consider an individual living on the periphery of commodity culture, say an Australian Aboriginal, now settled into a suburb of Sydney. Their traditional culture had relatively few material things, but was hugely rich in the elaboration of cosmology and kinship. As a student in anthropology I quickly felt tied up in knots just trying to unravel the complexity of Australian Aboriginal kinship systems. We quite properly regard them as rich in culture. However, as individuals drift towards the fringes of urban life, they may lose much of that cultural wealth. But being on the margins, they may have little opportunity to possess, or even to understand, this vast material culture represented by the city. They may end up just reduced to alcoholism.\n\nThe way Simmel would have put this is that the subjective only gains when it can assimilate the expanding objective culture. What we cannot assimilate oppresses us. This is potentially just as true for someone born into city life. Simmel wrote several highly influential essays about the contradictions of the metropolis. A place with far more people which allows one to be far more lonely. A place where, if we try and relate to too many things, but have no substantial relationship to any one of them, we can become largely indifferent to the world and to ourselves. It is easy to become blas\u00e9 in the city. We are then being reduced, rather than expanded, by the sheer quantity of things.\n\nBut let's not pretend that the quantity of things is intrinsically oppressive. It is only that it has that capacity to become so. In general, peoples of many societies, who had their own integrity, see the attractions and potentials of this expanded cityscape, and many thrive in it and on it. They bitterly resent anyone telling them that they were better off without it, that they have some responsibility to remain peasants or tribal peoples so that city folk have someone to gawp at on their holidays and anthropologists can claim they are researching something exotic. They have another name for this prior state. They call it poverty. Loads of people around me bemoan some existential loss as a result of having too many things. But I can count the genuine ascetics I know on less than one hand. I simply don't know any at all. The point made by Simmel is not then to see the growth in stuff as intrinsically good or bad, but as intrinsically contradictory.\n\nFor Simmel the most significant example of this contradiction was to be found in money itself. Simmel was one of the most eloquent theorists of the positive qualities of money. Money is the basis of modern freedom. Without the capacity to be paid in wages we would have to be apprenticed to other people. Students might have to work for their lecturers directly rather than paying the university. Money is the basis of modern equality. Without it we couldn't have the redistribution of taxation, and the abstract sense of people as equal and quantitative, stripped of their specific qualities that make them unequal. It is that abstraction that permits us even to conceive of equality, whether in voting, in feminism as gender equality, or in using bureaucracy to treat everyone equally. How often do we acknowledge that all such positives depend upon money?\n\nAt the same time Simmel can sound rather more like Marx when it comes to the capacity of money to create also inequality as differential possession: to create an abstraction as capital that becomes increasingly divorced from the specifics of things to which we can relate. We are all familiar with a hundred other things we blame money for. This text is being written during a credit crunch crisis, based on the problems of a process called securitization, which is, in effect, a process that simply made capital too abstract for its own good and for our good. But, unlike Marx, Simmel does not separate this off into a period of capitalist money abstraction opposed to a kind of idealized post-commodity communism. Simmel is clear that such contradictions are intrinsic to culture. You simply can't have the benefits of one side of this coin without the risk of the other. Everything we produce has a tendency to autonomous interest and the potential to oppress rather than serve us. If culture, for Simmel, is this Hegelian process that potentially can either grow us or oppress us, then culture is a subset of dialectical philosophy, and material culture is a subset of culture. Stuff has adhering to it this tragic contradiction of culture itself. Commodities are not inherently good or bad, but you can't have their benefits without entailing the risk that they will oppress you. The good news is that awareness of this gives one an opportunity to address this contradiction: to eat, play sports, create transnational corporations, with some potential for moderation.\n\nOne of the (many) questions raised by these texts, but especially by this use of Simmel, is whether these ideas only apply to what might be called modern stuff, such as commodities: the deluge that followed from the industrial and consumer revolutions. This is what most people assume. Hegel certainly had a strong modernizing trajectory to his story. But, for an anthropologist, this would be a severe limitation to any theory of material culture, and this would be a very different book about stuff. Remember that Australian Aboriginal stranded on the outskirts of Sydney. It is only too easy to assume that previously he or she possessed, in prior Aboriginal culture, some kind of unsullied, pure, integrated cosmology that contained no contradictions at all. Indeed, this was traditionally the spirit in which much of anthropology was taught. As such, anthropology exhibited a kind of primitivism, but a subtle one. It obviously didn't denigrate other societies, but it did reduce them to a state of noble savagery whose main purpose had become a stick with which to beat ourselves. To make us think of ourselves as people who have lost all culture, and become merely snivelling victims of vast capitalist and modernist forces: a dualism that was to a degree reinforced by Marx's black and white portraits of capitalism and communism.\n\nBut in my understanding of Hegel, and then Simmel, stuff is intrinsically contradictory. Not just our stuff, all stuff. So, for a brief excursion, let us leave on one side cars and films, and see what this idea of objectification has to say about the Australian Aboriginals who had not been resettled in Sydney, with reference to the work of a Chicago-based anthropologist, Nancy Munn.17 Munn's first researches took her to the Walbiri, a group of Australian Aboriginals who possess relatively little by way of material culture, at least if we think of stuff as things such as they carry around for hunting or taking shelter. But, as many studies have shown, the few mobile objects they use for hunting and shelters are complemented by an intense relationship with the landscapes they inhabit: the rock formations, creeks and places where wallabies are found in abundance.\n\nIf most people know anything at all about such Australian tribes, then they have probably heard of the dreamtime; the time of the ancestors who lie at one remove from people, but whose actions and consequences remain an integral part of the lives of the living, ancestors remembered in myth. Stylistically, the myths of the Walbiri and the writings of Hegel do not appear to have a whole lot in common. But we can view these myths as at least analogous to Hegel's philosophy of objectification, just rather more appropriate to the cosmology of the Walbiri.\n\nMunn studied those myths which described the process by which ancestral beings created the landscape. They tell how this too was a process of self-alienation and externalization. There are three main genres of objectification: metamorphosis, by which an ancestor transformed his or her whole body into a feature, such as a hillock; imprinting, by which an ancestor might, for example, have dug a water hole with his penis; externalization, in which an ancestor might have taken out from within her body an object that is now a natural feature of the land. These myths serve for the Walbiri a similar role to that of history for Hegel. They recognize that there are prior forces which have already created the world in which we come to be socialized. But in turn we can come to act upon those forces. The Walbiri retain contact with ancestors through the dreamtime and re-enact this relationship in ceremony and ritual today. It is the same landscape and order which is used to legitimate the critical relationships of kinship and social order which give individuals a sense of who they are. So both ancestors and contemporary people externalize themselves as culture and recognize themselves in that which has been created. They objectify. And in their myths they possess also a 'theory' of culture that explains this process.\n\nMany people have heard of the Aboriginal dreamtime, but in the desperate desire to see them as the opposite of ourselves, we are much less aware that Aboriginal societies have an extremely elaborate conceptualization of law.18 For example, one group may have only the rights to represent a particular action of an ancestor who created some feature in the landscape, while another group has rights over the feature itself. It would be entirely possible to critique Aboriginal systems of law from the perspective of the common critiques of our own law. A feminist, or a communist Aboriginal, could describe these as self-mystifications that delude the tribe into failing to recognize the artificiality of cultural orders that legitimate inequalities of gender or power. The tribe will never achieve equality unless they repudiate them and take up universal human rights. This seems correct. There is surely just as much reification and fetishism in Aboriginal cosmology as in the idea that all profit is justified by risk taking. Such contradictions are fully evident today. These same tribal groups are commonly engaged with concepts of property and universal rights when defending claims made to the state with regard to their own territory or rights over their art works against counterfeits. Land disputes have become something of a major preoccupation. At the same time these peoples may be seeking to limit the penetration of such universalizing languages and laws that could be used to abolish all customary law. If the internal contradictions of their beliefs were not evident before, they certainly are now, in hundreds of contemporary Australian court cases.\n\nIf the Walbiri exemplify objectification within one group, then the next region studied by Munn illustrates the process by which society creates itself through objectification within a wider cultural sphere. This was the Kula ring, famously described by Malinowski in his 1922 book Argonauts of the Western Pacific,19 from the vantage point of one of its component island groups, the Trobriand Islands. What fascinated Malinowski, and continues to entrance us today, was finding that a whole series of island groups, off the coast of New Guinea, have formed themselves into a kind of circuit. This did not depend upon knowledge of the circuit as a whole but on the more immediate relationship between neighbouring groups that constituted the circuit. Around this ring circulated valuables, especially armshells in one direction and necklaces in the other direction. Munn undertook her research in an island called Gawa in which people contribute canoes that are used in this inter-island exchange. She traces how various kin relationships work through exchanges of labour and food to facilitate the construction of these canoes. Since the canoes are then launched into the Kula, one could see this as the alienation of the relationships and the labour that went into their production. But, in return for the canoes, these people receive valuables, which have been the product of similar processes elsewhere. These valuables are then traded downwards back through the same sets of relationships that produced the canoes.\n\nThe question behind all this is, frankly, why bother? Why not just sit at home and watch television, or at least sunsets? After all it clearly wasn't just to give research material to anthropologists. In her book The Fame of Gawa, Munn elaborates on this necessity. A couch potato (or couch yam) amongst the Gawa people would be someone who just wanted to grow yams and then eat them, without all this Kula ring palaver. The Gawa have a pretty effective response to this. They accuse their couch potatoes of witchcraft. If you just grow food and eat it, then no culture is created. If, by contrast, you grow food, exchange it in complex networks that create canoes, send these in turn to be exchanged for valuables from other islands, representing other networks, then something big happens. What comes back is a value, which Munn translates as Fame. An elaborate field is created through which people and islands grow not just food; they grow reputation, they can become Argonauts and have adventures, stories can be told, things heard about and seen. Life is richer. Witchcraft is that which shrinks and shrivels, not just bellies but also horizons, while exchange, such as Kula, allows people to grow fat with fame and fortune. Through exchange people learn how to 'get a life'. Seen from this vantage, objectification, the process of self-alienation, externalization and return, becomes a theory of what we can call culture, as much for Gawa as for Simmel's Berlin. But these are still pressures that can oppress people, especially couch potatoes. And if there weren't any such people, there probably wouldn't be all these sanctions of witchcraft. I have met loads of unhappy people in what are called tribal and village societies who could be said to find their own culture oppressive. Something we prefer to ignore in simple versions of us and them.\n\nA central role in this process called culture turns out to be that subset we can now properly call material culture. The Kula, at one level, is simply a circuit around which people circulate objects such as armshells and necklaces. It's not just people who become famous through their adventures in Kula. Individual shells gain their own reputation because of the travels they have passed through and then people strive to ensure that famous objects pass through their hands and not those of a rival. One of the most influential foundational theories for the discipline of anthropology is a simplified version of this theory. Mauss, partly influenced by Malinowski, wrote an essay on the gift.20 He suggested that the basis of most relationships is debt. That in giving something to another, whether a necklace, or one's sister as a wife, a bond is created between the society of givers and that of receivers. The bond consists of the obligation to return this gift. Not to remain in debt. The Maori have their own conceptualization of the force which lies in the object itself and creates the obligation to return the debt. This concept of hau is perhaps the most debated of all anthropological models. It seems analogous to a notion of the inalienable, that which is never really given away, because it demands some sort of return. If you give me something and I feel obliged one day to give something back, then we are in a relationship. It may be supportive or it may be competitive, but it is a relationship. Once again, in this theory of the gift, we do not start from what societies do with things, it is the circulation of things that creates society. Or better still, what we call society and stuff are actually artificial separations out of the same process. It may be a person that is given as gift; are they then an object? Or is it better sometimes to dissolve the distinction?\n\nSimmel marvelled at the incredible diversity of experience possible in a contemporary city, but marvelled even more at an individual's capacity to blithely ignore the same, or to be oppressed by it. The accusation of witchcraft suggests that, just because there is the Kula ring doesn't mean that an individual islander might not wish to escape from it. We cannot assume every wife 'swapped' in gift exchange in order to facilitate the reproduction of dominant kinship obligations feels comfortable, just as we cannot assume she doesn't. Objectification is a dialectical theory of culture, not just capitalist culture, because contradiction is not just a new feature of modern capitalism, or an aspect of living in cities. It is intrinsic to the very process we describe as culture.\n\nStill, creating culture is a pretty extraordinary process and it does indeed make anthropologists happy. The Walbiri provided an example of objectification within one group, the Kula the creation of a kind of super-group. Culture is contradictory rather than completely prescriptive. The Kula exists, but some Gawans become famous Argonauts, and others sit at home, put their feet up and risk being seen as a witch. To make these points we have climbed up from stuff, to culture to dialectical philosophy: to a peak in which the distinction of subjects and objects has disappeared. But the air is thin up there and it is cold and lonely. Most of the time we probably prefer talking what we generally term common sense, and being what philosophers (and others) might regard as extremely vulgar. In everyday life, sitting in a pub over a drink, we might prefer to be in conversations where there are clear and unproblematic entities to be called people and things, and we can tell the difference. That people act on and are represented or symbolized by things. We converse in all those ways which this chapter has suggested may be in some measure wrong. But this is a discrepancy to explore rather than ignore. For one thing, going up the philosophy mountain is great for a quick revelation as to how things 'really' are, but we wouldn't want to live up there. But, before we are ready to clamber back down we have one final peak to climb.\n\nMateriality\n\nSo far this chapter has shown that it is possible to have a theory of things, such as framing, and secondly how to progress towards a dialectical theory of objectification that transcends things and persons. But for a topic called stuff, we also need to tackle a third aspect, which is the meaning and implications of materiality itself.21 The problem with materiality is that for some reason we seem on the whole to be not at all keen on it. There is an underlying principle to be found in most of the religions that dominate recorded history. Wisdom has been accredited to those who claim that materiality represents the merely apparent, behind which lies that which is real. Perhaps the most systematic development of this belief arose over two millennia within South Asia. For religions such as Buddhism and Hinduism, theology has been centred upon the critique of materiality. At its simplest Hinduism, for example, rests upon the concept of maya, which proclaims the illusory nature of the material world. The aim of life is to transcend the apparently obvious; the stone we feel through the sole of our shoe, or the body as the core of our sensuous existence. Truth comes from our apprehension that this is mere illusion.\n\nNevertheless, paradoxically, material culture has been of considerable consequence as the means of expressing this conviction. In a Hindu temple the merely vestigial forms at the centre of a temple may be contrasted with the massive gates at the periphery. The sari of an elderly woman in its faded pastels presents a stark contrast to the bright and sensual sari of the bride. It thereby expresses, in material form, the goal of transcending our attachment to material life. So, to start on this seemingly vast question of what is materiality, we had better admit that, whatever materiality is, it is something we often profoundly don't want to be.\n\nYet the history of South Asia is not just the history of its religions. There is a parallel history, which tells of the endless struggle of cosmology with practice. This is the history of accumulation, taxation, wars and looting, empire and excess. It culminates in the integration of this region within a global political economy in which politics is increasingly subservient to an economics whose premise with respect to materiality could hardly be more different. In economic thought the accumulation of material commodities is itself the source of our extended capacity as humanity. Poverty is defined as the critical limit to our ability to realize ourselves as persons, consequent upon a lack of commodities. The focus upon materiality, though here in the form of accumulation, is therefore just as strong in economics as it is in Hinduism.\n\nFor a discipline such as anthropology, that is concerned with what it is to be human, we need therefore to start our discussion of this issue with an acknowledgement that the definition of humanity has often become almost synonymous with the position taken on the question of materiality. A good deal of what people take to be morality has followed in the wake of this. So either we desperately want to escape being material, or we spend our lives trying to accumulate more material or, most bizarrely, virtually all the people I live amongst in London seem to want to do both of these things simultaneously.\n\nIf world religions and global economics depend on their stance to materiality, then its investigation is clearly not just a minor indulgence of material culture studies as a subsection of anthropology. Hinduism and economics are not just beliefs about the world, but vast institutional forces that try to ensure that people live according to their tenets through priesthoods or through structural adjustment programmes. In this respect capitalism and several major religions are equal and analogous attempts to impose particular positions on materiality. On a somewhat smaller but still significant scale, materiality is obviously important to the study of media, the definition of art, the creation of new financial instruments, environmental politics and the respect we have for persons.\n\nWe will never get far in understanding materiality if we blunder into it head on. Let me remind you of my original stricture against trying to define stuff. Is a hard thing, such as a stone, more material than a soft thing such as a bubble? Is an idea that lasts more material than one that doesn't? It's that pedantic semantics again. It may be that in physics one can define materiality, but the definition would be of little value to social science. What we can do instead is to see what people in practice mean by this idea. What do religions which want to escape the material world do about it; how do people in practice employ ideas of more or less materiality? The following is a pr\u00e9cis of some papers in an edited book on the subject of materiality that each provides us with examples we can use to think more deeply about what is at stake here.\n\nWe can start by returning to the fundamental contradiction whereby religions find that the best way to express immateriality is through materiality. It's hard not to be entranced by the ancient Egyptians, largely because materiality is something at which they were spectacularly good. What we are entranced by are a trilogy of mummies, pyramids and monuments. Meskell tells us what they thought they were doing when they created these things in the first place.22 The ancient Egyptians believed, for example, that one helped give life to the gods by creating the precise statue appropriate to them. Similarly one gave extended life to the individual by getting the process of mummification right. Sheer materiality, expressed as one of the great pyramids, gave the very sense of 'being' a precise shape and form. So we are impressed by the legacy of their stuff. And yet, just as with Hinduism or Christianity, this cosmology rested upon a belief in the inherent superiority of the immaterial world. So the ancient Egyptians express our paradox with particular clarity. In essence it was their faith in the potential of monumentality to express immateriality that has created their legacy as a material presence in our own world. We continue to be enthralled by statues, mummies and pyramids because of the very exuberant faith that the Egyptians put into the process of materialization as a means for securing their own immortal transubstantiation.\n\nIf the ancient Egyptians express one logical implication of this paradox, then thanks to Engelke we can explore another,23 this time through following the logic of a contemporary religious group, the Apostolic Christians of Zimbabwe. In a way this has been a long time coming. The very founding of Protestantism saw thousands of people slain over esoteric arguments as to what exactly we mean by immateriality. Whether wine and bread were actually the blood and body of the Christian messiah or merely their representation. Whether churches should be full of icons as in Counter-Reformation churches, or entirely stripped of them. Ever since the Reformation there have been movements within Protestantism that have tended towards iconoclasm and asceticism as attempts to foreground the importance of immateriality to spirituality.\n\nThe Masowe Apostolics, studied by Engelke, take this to its logical conclusion in several respects. They feel that it has now become obvious that prior Christian forms failed in their quest for transcendence because they were betrayed by the very materiality of things such as bibles and churches. Protestantism had failed to follow through on its own logic of spirituality. So these Apostolics meet in fields rather than churches and memorize their teachings rather than holding forth with a bible. They try to repudiate systematically every material object that seems to mediate their relation to their Christ, so that they can achieve ultimately an unmediated relationship, soul to spirit. Yet even they cannot fully adhere to this ideal. The passion to repudiate things per se leads them to attach still more importance to the very few objects that remain to them. Engelke writes a whole essay on the ambiguity surrounding sacred honey and the constant temptation to use this lapse of immaterialism, the fact that they do see something special in honey, as a conduit that brings spiritual power back down to our instrumental earth.\n\nSo, the other side to the coin of materiality is immateriality \u2013 literally in the cases of the coinage analysed by Maurer.24 In religion the main purpose of the material is to express the immaterial. So, for a deeply religious Islamic society money is more important as a means to comprehend divinity than as a means to facilitate secular exchange. This is why in the early caliphate a consequence of the replacement of the Caliph's head by Qur'anic inscriptions on coins is the subordination of the issue of representation in coinage to that of the technologies for the imagination of the divine. The way a coin faces both sides, upwards to the transcendent and downwards to the functionalism of the market, is utilized to give words themselves (as in calligraphy) a role in objectification. The coin helps the believer to conceive of this Janus-faced relationship. Maurer argues that the effect of the new coinage is to bring the issue of how one understands the deity down to the somewhat safer question of how one understands the coinage. At the same time this secures the authority of coinage. In effect such a society has, by making coins a mode for the apprehension of the divine, thereby 'leveraged' their coinage as value, with the backing of divine authority. By removing the face of the coin it has actually 'countenanced' by the word. Once again, there are contemporary implications. Maurer's other example is the growth in Islamic banks that attempt to ensure that business operates in such a way that it declares its obeisance to religious strictures rather than stands in opposition to it.\n\nSo we can start with this paradox by which the immaterial can only be expressed through the material. We then find that most people prefer a gradation between these two, rather than a stark dualism. Some things and some people should be seen as more material than others. This is very well established in Hinduism as generally practised. In India we find a hierarchy starting from the mass of small and disparate images of regional spirits and divinities. These have been incorporated into the larger pantheon of Hindu deities. The minor deities are incorporated into mainstream religion by being viewed as avatars, expressive manifestations of the major deities such as Siva and Vishnu. The major deities in turn are seen by some as aspects of the one supreme deity. At higher philosophical levels the idea of a deity is seen as itself a vulgar rendition of a more transcendent sense of enlightenment, for those whose consciousness can achieve such heights. For them the very notion of God is a necessary simplification of something still less material.\n\nSo one can correctly label Hinduism as polytheistic, monotheistic and even atheistic, partly because each is seen as appropriate to the capacity of certain kinds of people to apprehend the 'reality' behind mere materiality. Ordinary people are ordinary partly because they require more materiality. Great sages are great partly because they can apprehend more immateriality. So if deities form a hierarchy, so do people. The lowly are those who need minor spirits, with more human-like foibles and myths, in order to help them comprehend religious ideals. By contrast, at the apex are the intellectual sages, who can contemplate transcendence in a more purely philosophical idiom.\n\nSo what helps us to understand materiality would not be a physicist's demonstration that monuments are more densely material than sponges in some absolute sense. What we employ is our sense of such a gradation between more or less material. Rowlands shows how some persons also can be seen as more material than others, by the people of the Cameroon grassfields.25 Appropriate metaphors abound \u2013 some persons and objects are seen as weighty, others are slight. Some people loom large, even when we might have preferred that they didn't. Others, however hard they try to gain our attention, we manage to leave at the periphery of our vision. For these people in Cameroon a chief has particular density. Materiality is gained by substances through the process of circulating through his body and presence. So, for example, his spit is itself efficacious in changing the order of things. By contrast, his subjects strive to have a presence as persons, but they simply do not possess the reality granted to the body of the chief. All other bodies are mere shadows of the one real body. Rowlands notes how these people had a parallel encounter with colonialism. They found new modes, such as bureaucratic form filling, which made some people more material than others, in relation to this alternative source of power. While Meskell indicates the extraordinary gulf between the God-like and ordinary in death and the afterlife, Rowlands draws our attention to the assertion of such distinctions in life. Sometimes people try to manipulate this aspect of themselves \u2013 consider the way a size zero model has more presence than a size eight.\n\nMany of us live in a world where more and more people are quite distant from any cosmology identified with established religion. But our secular world is just as haunted today by shadowy spirits that seem to perform strange magical feats that conjure vast powers out of the dross. People such as the Japanese traders in derivatives markets, studied by Mayazaki,26 are surely at least as incomprehensible to the uninitiated as any theological discussion of Talmud or the Sutras. Key processes in contemporary finance start not with doing something, but with an act of re-conceptualization. The most fundamental process is called securitization. This is the process that turns a potential future profit stream into something that can be traded. In turn, a derivative may be formed by trading the risk involved in speculating on what that profit stream will be. This sounds obscure enough, but what is deeply unsettling is to contemplate what is thereby produced.\n\nIn the trading known as arbitrage, experts will use models derived from theoretical physics to note tiny discrepancies between how a market should operate according to these models and how it is actually operating. But trading on the 'as though they existed' is sufficient to make these values exist. A million units can thereby be traded as a billion units. Well more, actually. The rabbits coming out of these hats reproduce at a rate that would even astonish rabbits. The notional amounts of derivatives contracts by the year 2000 were already more than a hundred trillion dollars, a figure most of us can no longer convert into a meaningful image. During a credit crunch it can even more magically be transformed back into a mere puff of smoke.\n\nNow for present purposes you don't have to understand modern finance (though perhaps we really should). The point is merely that the relationship between materiality and immateriality is no more straightforward in secular than in religious domains. Clearly this vast sum exists in some sense, but in what sense? Has derivative trading materialized something that didn't exist, or is it largely a juggling of immaterial concepts? One of the neat things about the Japanese arbitrage traders studied by Miyazaki is that they don't get paid by some share of the money they make in this trading. They accept a standard wage, because they see themselves as doing a good deed in the world. By spotting these discrepancies in the way markets operate, they are in a way punishing its imperfections, and in some small way helping to perfect the market itself. To make it as pure as its models. Is this then the secular opposite of religion? Hardly. Actually, religions were pretty good at making money on occasion also.\n\nThe volume on materiality, from which these case studies are drawn, contains many more examples of these paradoxical and ironic implications of the term. What they demonstrate is the varied and concrete consequences of this initial paradox. The more humanity reaches towards the conceptualization of the immaterial, the more important the specific form of its materialization. It's the same reason why modern art seems to sell for more money, the more it succeeds in its claim to esoteric conceptualization. If a work of art can fully encapsulate a politically radical repudiation of capitalism and materialism, it's probably worth a small fortune. We are not trying to assault common sense for the sake of it. All these examples show that contradiction and paradox and complexity are integral parts of the ordinary world we live in and essential to understanding it.\n\nThis chapter is by no means trying to exhaust the potential of theoretical approaches to materiality and to stuff. The theoretical perspectives that have been expounded here can hopefully complement, rather than be seen as in some kind of competition with many other significant developments in thinking about material culture. There are differences. Bruno Latour,27 for example, seems convinced that dialectical theory merely resurrects the opposition of subjects and objects, while I believe it succeeded in transcending them. Influential alternatives within anthropology include the work of Alfred Gell,28 who, as with Latour, found a route to transcending the opposition of things and persons through an emphasis on the way objects can be said to have agency, although Gell uses this to create an approach to art. Also influential are various versions of phenomenology, espoused, for example, by anthropologists such as Ingold and Tilley.29 Reading these additional approaches will provide a much broader, and more insightful, understanding of the possible ways in which we can go beyond our natural dualism of persons and things than would be achieved by reading this book alone. One of the trends in the work of some of these other anthropologists, which seems entirely appropriate, is to look more at substances such as stone or wood rather than, as in this volume, thinking of stuff mainly as artefacts or things.\n\nWhat all such approaches have in common is the desire to give greater attention and respect to materiality and material culture. Yet they all work in relation to a discipline that in Europe is conventionally described as social anthropology, not just as anthropology. This raises one final issue \u2013 the potential opposition, not between subjects and objects, but between social science and material culture. The possibility of a modern social science, such as anthropology, was at the least secured with the radical secularism that viewed religion as the emanation of the social collective. If there is a single founding ancestor to Western social science, it is probably Durkheim who most clearly enunciated this belief.30 The premise was a secularism that saw God and religious rules as fabrications. Societies had invented such beliefs and invented God because they needed to represent themselves to themselves. Religion was a collective social representation. But at the same moment that this theory desacralized religion by seeing it as fabrication, it actually also sacralized the social. The social sciences become devoted to the study of all phenomena that stand for what we now call society, social relations or indeed simply the subject.\n\nThe reason was simple enough. Durkheim felt that we could manage without God as long as we believed in society. Mere individuals bereft of this collective identity would lose a sense of purpose to life, as implied in his study of suicide. The problem, from the point of view of something that could have been called material culture studies, was not just that persons and societies were put on this semi-divine pedestal, but that this was achieved through distancing them from the vulgarity of the material. In this tradition, the study of persons' relationships to things should always finally be reduced back to social relations. This was not Durkheim's doing. In fact, as was previously noted, his compatriot Mauss was highly respectful of the potency of things in his analysis of the gift. This anti-materialism was something that long pre-dates Durkheim. It was rather the abiding legacy of the roots of this secular ideal of society in religion itself. As we have seen, religion constantly achieved its ideal of transcendence through its repudiation of the material. Enlightenment is tantamount to the separation from desire. When we look at ourselves today, we can see we have inherited much the same system of beliefs. The one thing we are never supposed to be is materialistic, because it is assumed that paying attention to the material is always at the expense of paying attention to the social, to the person. This will be a major theme in the companion to this volume, Consumed by Doubt.\n\nThis is also why approaches to material culture have tended to be held in some disdain by mainstream anthropology. The reason goes right back to the start of this chapter when my Indian informants watched me paying inordinate attention, as a PhD student, to pots and tried to persuade me that I had missed the point or indeed the plot. To pay too much attention to things is seen as a diminution in our capacity to appreciate persons as persons. Yet most secular people believe that the dethronement of God, the previous essential guarantor of morality, released, rather than suppressed, the development of a modern ethical sensibility.\n\nThe question is, after a couple of hundred years of secularism, whether we still need such a highfalutin ideal of society to help deal with the trauma of replacing the divinity. Indeed perhaps keeping that aura of the divine and attributing it to society has been more a problem than an asset? Personally I would rather have a divine God than pay obeisance to society as a kind of divine-lite. It could well be that the dethronement of society or humanity, or social relations or the subject, can be the premise for the further development of modern ethics, not its dissolution. This should not be confused with any opposition to society as an analytical term, as against, for example, the individual. There is very little in this book that would support a conceptualization of individuals outside of social relationships and indeed society. But we can speak of society, appreciate it, applaud it and work for it, without reifying it.\n\nSo if the first revolution consisted of Durkheim's enthroning society in the stead of religion, we now look to gain maturity by burying the corpse of our imperial majesty, society. Objects no longer have to be constantly reduced to subjects. But there is no reason at all for placing material culture on the pedestal that now looks empty. Academic work is best done from the more modest position of feet on the ground. But we can achieve this modesty partly through a greater acceptance of our own materiality as well as that of the world. Not materiality made exotic by the wondrous achievements of science, nor made criminal by the curses of environmentalists, though stuff has its achievements and its costs. But the emphasis in this chapter is on simply making stuff ordinary, as part and parcel of our existence in the world.\n\nNotes\n\n1 'Archaeology and Development', Current Anthropology, 21 (1980), pp. 709\u201326.\n\n2 R. Keesing, Cultural Anthropology: A Contemporary Perspective (New York: Holt, Rinehart and Winston, 1976).\n\n3 D. Miller and M. Spriggs, 'Ambon-Lease: A Study of Contemporary Pottery Making and its Archaeological Relevance', in M. Millett (ed.), Pottery and the Archaeologist, Institute of Archaeology Occasional Publications no. 4 (1979), pp. 25\u201334.\n\n4 Artefacts as Categories: A Study of Ceramic Variability in Central India (Cambridge: Cambridge University Press, 1985).\n\n5 Ibid., pp. 51\u201374.\n\n6 None of this is quite the same as the social science theory of functionalism, for which I provide a brief defence in the final chapter.\n\n7 e.g. M. Douglas, Implicit Meanings: Essays in Anthropology (London: Routledge, 1978); M. Douglas and B. Isherwood, The World of Goods (London: Allen Lane, 1979).\n\n8 E. Goffman, Frame Analysis (Harmondsworth: Penguin, 1975).\n\n9 E. Gombrich, The Sense of Order (London: Phaidon Press, 1979).\n\n10 Material Culture and Mass Consumption (Oxford: Blackwell, 1987), ch. 6, pp. 85\u2013108.\n\n11 L. Dumont, Homo Hierarchichus (London: Paladin, 1972).\n\n12 C. L\u00e9vi-Strauss, The Way of the Masks (Seattle: University of Washington Press, 1982).\n\n13 The two sources I have used for Hegel are G. Hegel, Phenomenology of Spirit (Oxford: Oxford University Press, 1977); and G. Hegel, The Philosophy of Right, trans. T. Knox (Oxford: Oxford University Press, 1967). My own discussion of Hegel is taken mainly from Material Culture and Mass Consumption, pp. 19\u201333, and to a lesser extent from The Dialectics of Shopping (Chicago: University of Chicago Press, 2001), pp. 176\u2013205.\n\n14 K. Marx, Early Writings (Harmondsworth: Penguin, 1975), pp. 379\u2013400. My discussion is found in Material Culture and Mass Consumption, pp. 34\u201349.\n\n15 K. Marx, Early Writings, p. 329.\n\n16 G. Simmel, The Philosophy of Money (London: Routledge, 1978), and G. Simmel, The Conflict in Modern Culture and Other Essays (New York: New York Teachers College Press, 1968). My discussion of Simmel is taken from Material Culture and Mass Consumption, pp. 68\u201382.\n\n17 N. Munn, The Fame of Gawa (Cambridge: Cambridge University Press, 1986); N. Munn, 'The Transformation of Subjects into Objects in Walbiri and Pitjantjatjara Myth', in R. Berndt (ed.), Australian Aboriginal Anthropology (Nedlands: University of Western Australia Press, 1986), pp. 141\u201363; N. Munn, Walbiri Iconography (Ithaca, NY: Cornell University Press, 1973); N. Munn, 'Spatiotemporal Transformations of Gawa Canoes', Journal de la Soci\u00e9t\u00e9 des Oc\u00e9anistes, 33 (1977), pp. 39\u201352. My discussion of Munn is found in Material Culture and Mass Consumption, pp. 50\u201367.\n\n18 The points made here derive from the work of F. Myers, in F. Myers, Pintupi Self: Sentiment, Place and Politics among Western Desert Aborigines (Washington: Smithsonian Institution Press, 1986); also F. Myers, 'Some Properties of Art and Culture: Ontologies of the Image and Economies of Exchange', in D. Miller (ed.), Materiality (Durham, NC: Duke University Press, 2005), pp. 88\u2013117.\n\n19 B. Malinowski, Argonauts of the Western Pacific (London: Routledge and Kegan Paul, 1922).\n\n20 M. Mauss, The Gift (London: Cohen and West, 1954).\n\n21 The section that follows is taken from my Editor's Introduction, in Materiality, pp. 1\u201350.\n\n22 L. Meskell, 'Objects in the Mirror Appear Closer than they Are', in Materiality, pp. 51\u201371.\n\n23 M. Engelke, 'Sticky Subjects and Sticky Objects: The Substance of African Christian Healing', in Materiality, pp. 118\u201339.\n\n24 B. Maurer, 'Does Money Matter? Abstraction and Substitution in Alternative Financial Forms', in Materiality, pp. 140\u201364.\n\n25 M. Rowlands, 'A Materialist Approach to Materiality', in Materiality, pp. 72\u201387.\n\n26 H. Miyazaki, 'The Materiality of Finance Theory', in Materiality, pp. 165\u201381.\n\n27 My favourite books by Latour would be B. Latour, We Have Never Been Modern (Hemel Hempstead: Harvester Wheatsheaf, 1993), and B. Latour, Pandora's Hope: An Essay on the Reality of Science Studies (Cambridge, Mass.: Harvard University Press, 1999).\n\n28 See esp. A. Gell, Art and Agency: An Anthropological Theory (Oxford: Oxford University Press, 1998).\n\n29 Examples include T. Ingold, The Perception of the Environment (London: Routledge, 2000), and C. Tilley, The Phenomenology of Landscape (Oxford: Berg, 1994).\n\n30 E. Durkheim, The Elementary Forms of Religious Life (Oxford: Oxford University Press, 2001).\n3\n\nHouses: Accommodating Theory\n\nHousing and Power\n\nOne of the delights of dialectical theory, alongside its cultivation of extremism, is that it includes within itself clear strictures about what one should do with theory. No one drives you further to the upper reaches of theoretical ambition than Hegel. There you will find ideas about Absolute Reason that can explain even more than The Hitchhiker's Guide to the Galaxy about life, the Universe and the rest of it (though I never found anything in Hegel about the number 42). But to be consistent, dialectical theory must imply that just theory alone, like everything else humanity produces, will always tend to become destructive and follow its own interests, oppressing rather than serving humanity. Unless there is a counter-movement that (to use its own terminology) negates this abstraction and brings it back into the service of our interests and welfare.\n\nThis is often the case. Theory, philosophy, modern art, economics and other movements that utilize obscure abstractions can easily degenerate into pretentious obfuscation and become oppressive playgrounds of academic divas and elites used to intimidate as much as to impress. Academics, tempted by the promise of an easy and assured claim to cleverness, create vast circulations of obscure and impressive citations. A scattering of names such as Lacan or Deleuze and Guattari is usually a good sign of such oppressive conceits. It is only through the subsequent processes of maturing and re-grounding theory in its application to everyday lives and languages that such cleverness becomes transformed into understanding and re-directed to a compassionate embrace, rather than an aloof distaste. To make matters worse, the most oppressive use of theory tends to come dressed in the guise of critical or radical political endeavour, a claimed concern with the actually oppressed conditions of our humanity.\n\nSo the task now is to take our artistic-looking idealized theory, once white marble, nowadays more angular perspex, and drag it back into the mud and murk of everyday life until it looks a lot less intimidating and more like something we feel at ease with bringing home to the folks. In this chapter the device for bringing theory home is through immersing it in the more general problem of what a home actually is. The aim is literally to domesticate theory, to help us feel at home with these abstractions by showing that the theoretical ideas discussed in the previous chapter can deliver genuine insights into how we ordinarily feel at home in this vast world. So, while less explicit in address to theory, this chapter also starts with issues of objectification and moves on to questions of agency and materiality.\n\nClothing raised the issue of the superficial, and our discussion proceeded through a substantive analysis of how we dress to demonstrate how clothing and persons mutually constitute each other. To a rather greater degree than clothing, housing implicates contours of power and scale that make such intimate issues as our personal relationships often contingent upon much grander forces. We are thereby confronted with a much broader and deeper context for our analysis. Consider the relationship between people and their homes in London. Just as in the case of clothing, people in London might want their homes to be part of a similar process of mutual constitution. But very often, if this is what they desire, it is a thwarted desire because housing brings with it powerful forces which are by no means under their control. I may have just made critical comments about the purity of certain academic radical politics, but I have no desire to eschew politics and my colours are firmly nailed to the mast of unashamedly left-orientated politics, whose values and core lie in a desire for equality of welfare.\n\nThe issues with clothing are simplified because, while there is a major fashion industry out there, we still retain a good deal of choice and autonomy with respect to both selecting and wearing clothes. Basically clothes are relatively cheap. But houses are altogether different. Houses are the elephants of stuff. Huge lumbering beasts that are excessively hard to control. You can buy a sweater at Primark for \u00a38 but to return with it to your own house in London will cost you more like \u00a3300,000. As a result, property attracts a great many more interested parties: the state, landholders, local councils, building societies and the like. Against these forces, any desire by us, the mere people who dwell in houses, to engage in a certain relationship to them can find us way down any pecking order of power. Sometimes these powers manifest themselves in quite extraordinary and unexpected ways, hidden within genres of material culture where we do not perceive them as power. This chapter therefore deals much more directly with issues of power than chapter 1. Beginning with the larger forces that operate on houses, such as global ideologies, it then moves downwards to smaller worlds of power and difference, expressed in gender and other relationships.\n\nTo start from the widest context, we have lived for quite some time now in a period of global housing, in that there are trends and forces that have radically determined housing stock in every part of the world. This includes the architecture of much of London's structures over the twentieth century.1 Earlier it was suggested that philosophy, often in collusion with art, has a tendency to clamber all over us in its zeal and thereby become oppressive. One instance of this came with the rise of modernism in the early part of the twentieth century. Stimulated by the advances of science, liberated by the repudiation of past convention by the arts, modernism forged its own proclamations of how the future should and would look. With respect to housing this was not some slight fashion that came and went. Marshalled by the zealotry of Le Corbusier and the Bauhaus, it became itself the orthodoxy of architecture and design. Fa\u00e7ade was a crime, form would follow function. Houses would be honest, objective and rational, and if need be even brutal. As an instrument of objectification, this new architecture would liberate its inhabitants, who would themselves become modern in the mirror of their new environment. Liberated from dusty crevices of history, exposed by shiny white surfaces, we would come to see so much more clearly who we could be.\n\nWe saw in the last chapter that, left to itself, stuff tends to modesty. There is a natural humility to things, in that they work best as the frame that guides our sense of what is appropriate, rather than as things we pay regard to in their own right. This tendency can make stuff quite powerful when put into the service of ideology. When someone tells us we should do this or be that, we bridle and feel put upon. When this message is carried, not by a hectoring voice, but well hidden within the mere substance of apparently silent stuff, we are less likely to sense our disempowerment. So it was in the early twentieth century. Officially politicians and the voice of ideology were engaged in continual battles of right as against left, socialism and the market. These we knew about and could contest. Yet, over exactly the same period, there grew up new modernist tower blocks to house the new modernist populations. Curiously, they did not listen to, or obey, these official distinctions between states and markets, between communism and capitalism. All the explicit contests of ideology were avoided. As a result, the spread of modernist architecture swept blissfully onwards regardless. It made not one iota of difference whether you were living in New York or Moscow. You were equally likely to be seeing a transformation of the urban landscape as it was taken over by modernism. These objects were vast, but we didn't see them nearly as clearly as expressions of official or political ideology.\n\nYet while forces did not marshal themselves on conventional lines of communism against capitalism, there was another remarkably quiet battle taking place around the field of architecture. Vast movements of resistance started to gather their forces to prevent modernism taking over the planet, yet very few people were even aware that such a war was being waged. Even in retrospect, we fail to pay regard to the evidence. We may even be mistaken as to who won. It is easy to think that modernity became ubiquitous as the form and style of modern housing. After all, it didn't just appear in the clusters of residential tower blocks. It was equally dominant in schools, offices and hospitals, replacing crenellations and spires with glass and concrete. But modernism isn't ubiquitous, and wandering around the streets of London one starts to realize that finally it was almost entirely confined to the housing that was directly built by the state and corporations.\n\nCome off the main road and wander around the interior streets of London and one finds that, apart from state housing built for workers, most of the privately built housing stock looks very different. It looks like suburbia. Who was the Le Corbusier of suburbia? Who rallied the population and told them to build barricades of bay windows, pebbledash and porches against the predation of this modernist beast? Well, no one. And yet if we interrogate these streets we can once again find they tell of an ideology just as consistent in its logic as that of modernism. After all, can it be coincidence that we are looking at the half-timbered, semi-detached, sub-urban dwellings of the middle classes? These terms are redolent of ambiguity and a refusal to choose between alternative categories. As Mary Douglas argued, we dislike such ambivalence; it seems unprincipled and it is no wonder that officially this style was condemned. Yet the more that authoritative discourses described suburbia as ugly and conservative and irrational, the more it flourished. Following the expansion of the London underground system, there were created a proliferation of suburban colonies that prided themselves on fa\u00e7ade and insisted on at least some attempt to add cottage front gardens to urban homes, that brought the country Cotswolds to Finchley. Just to reinforce the point, they were buttressed from the inside by cocktail cabinets and dressers. Steel and science and rationality had no chance against doorknobs and Dunroamin. Le Corbusier had no defence ready against nice and pretty.\n\nThe turning point for the UK came in 1968 with an accidental explosion in a residential tower block called Ronan Point, which revealed certain structural faults. And the struggle was largely over by the time that block was demolished in 1986. By this time, the same official voices that had so decried suburbia now directed their ire against these modernist residential blocks. Politicians of both right and left, who once rivalled each other in their desire to build such blocks, now united in their condemnation of high rise public housing, especially when it took the form of state-owned estates. Seen as a social failure, the people who lived there were increasingly regarded as urban detritus, responsible for everything that was bad about city life. Modern housing blocks bred crime and alcoholism. Yet who actually built these estates? Who had destroyed endless rows of small houses, condemned as slums, and replaced them with this vision of community that was supposed to rise inevitably from the careful construction by architects of an appropriate balance of communal and private space? It was never the people who lived within these dwellings; their opinions had never been solicited or voices heard. The people being condemned as urban failures had never been consulted in the decision to move them into such dwellings. They were being castigated by the very people who had made those decisions, the planners, council leaders and bureaucrats. People who actually ran the state and built modernist architecture, but who curiously when it came to their own private lives (apart from architects themselves) tended to prefer to live in semi-detached or even detached housing.\n\nTaken as a potted history of twentieth-century London housing, this appears a far cry from my mutually constituted collusions of persons and things happily making each other. It's almost the opposite. When first analysed this case study was intended as an exemplification of the Marxist concept of ideology. In that tradition, ideology stood for a paradox; that as a process of objectification people may well understand themselves in the mirror of the world they live within. But what happens when that world is created by others, rather than by themselves? If they see themselves in that world, then it would follow that they now misunderstand themselves. One powerful class built houses to objectify their own values, but also built the housing that came to represent the values against which they defined themselves. So, as in accordance with the Marxist concept of ideology the mass of the population were forced to see themselves in the mirror of the stuff produced by a dominant class, not their own stuff. Except that, contrary to Marxist assumptions, neither the proletariat nor the ruling class won the day. If anything, it was the middle class of suburbia, a class so universally derided by theorists that no one had bothered to theorize them properly.\n\nThis exemplifies the way housing can provide a critical caveat to the points made earlier about clothing. Yes, people are constructed by their material world, but often they are not themselves the agents behind that material world through which they must live. As we bring our theory of objectification back to the everyday world of housing, it's already looking pretty dirty; though perhaps muddled as much as muddy. Yet this is not a simple tale of recognizable villains and class interests. While right and left were equal in their zeal to build and their zeal to condemn, they need not have been conscious of the likely consequences of their actions. To find these values objectified in housing, you don't get very far by asking people, you have to examine the logic of the stuff itself, the form and underlying order of the built environment.\n\nWithin this example there is an architecture of modernism that spoke stridently of its stance to the world; and an architecture of suburbia that remained quiet and demure with respect to any underlying message. A material culture analysis has to listen to the explicit voices, but remain focused on messages that appear like invisible ink, emerging only under the wash of analysis. So far we have addressed only the exterior of houses, the way they face the public domain. If we want to consider issues of power on a more intimate scale we need to cross the thresholds and enter inside, to look beyond fa\u00e7ades to the process of interior decoration. Here we will find another field of struggle. But inside and outside the house are hardly entirely separate domains. The two remain deeply entwined, as the next case study will illustrate. Indeed for the time being we will stay with the urban modernist state housing of London, but look inside to see how that helps us re-think the meaning of this term objectification in relation to power.2 So let's pass through the living rooms and settle down to a cup of tea in the kitchen. After all, this is a working-class estate in North London, and tea in the kitchen is a proper place to have such a chat.\n\nAs you sit nursing your tea (coffee if you insist), reflect on the kitchens themselves. Since this was built as a single estate, under modernist guidance, the likelihood is that all the original tenants who moved in on completion of the building would have been allocated a flat with pretty much the identical kitchen to their neighbours'. So you could imagine this population as artists. Each is presented with a blank canvas \u2013 a white kitchen surface. After around thirteen years, at the time of my study, what had they painted onto this canvas and why? In particular I wanted to know whether some people had made radical changes and others not, and whether we could account for this difference in their ability to appropriate the things which they were allocated by the state.\n\nIn most cases there were considerable changes, but the explanation for these was not simple. The kitchen fitments provided by the state were actually quite good in quality. I consulted with an expert in ergonomics to establish this and also to confirm that when residents replaced them with expensive fitments, purchased from shops, they thereby gained no functional advantage. According to this expert, the newly purchased items were not ergonomically superior. Since these people were tenants, and not owners, they also gained no financial advantage by such a replacement \u2013 this was not an investment. So replacing one's kitchen with expensive new fittings does not seem to manifest some obvious form of rationality. But, at the other extreme, there were also kitchens that after thirteen years presented the very same decorative scheme that had originally been present when the flats were first occupied. But this did not mean the tenants had never undertaken decorative work. It often meant that when, every few years, they needed to re-paint their properties, they had chosen to retain this original style and appearance. Again there is no functional reason why they should choose to do so.\n\nTo understand what these artists had or had not done to their blank canvases we need to start with their very definite relationship to power represented by the state. This was a time when the UK government, under Margaret Thatcher, scorned to hide its disdain for state housing and tended to stigmatize those who occupied such flats. It was one of the reasons the people who lived there had come to believe that the kitchen fitments they had been supplied with couldn't possibly be any good and needed replacing. There is nothing intrinsically demeaning in living in state housing or renting rather than home ownership. In plenty of other countries rented accommodation may be preferred or even celebrated. But in the UK, at that time, these tenants were at the tail end of the story just told, where they were being blamed for the failures of modernist architects. Typically this estate had received awards when it was first built, but by now was seen as the pits. Under such conditions it is hardly surprising that for these families the flats themselves almost felt like the presence of an enemy. Simply by living there, they found they were branded with this now stigmatized identity of council tenants. This was not at all helped by the attitudes of local councils at that time. The bureaucracy remained extremely prohibitive about tenants doing anything that represented a major change to their flats. The local council had come to possess a big brother element that seemed to be in control of every last detail of one's life. In this modernist prison any change you made to the flat was seen as an attack on the architectural harmony of the whole. All of these factors meant that the relationship between people and the place they lived in reeked of alienation.\n\nIf this was the starting point that people were desperate to go beyond, the next question was which of them had been successful in their struggle and why? It appeared that the most important resource people had in changing their relationship to their own accommodation was not money. After all, it costs exactly the same amount to re-paint the original colours of the kitchen as to paint new colours on the walls. Very few people had actually gone out to purchase entirely new kitchens. Mostly it was a case of superficial decorations such as ornaments, posters and calendars on walls. These were used to extensively personalize some flats, but not all. What soon became clear was that the people who had been able to carry out this transformation were those who had good social relations and the support of others. This was the first of what became many studies of material culture, which concluded that it was the people blessed with strong relationships to other people that also had effective and fulfilling relationships to the material world, while those who felt they had failed in their relationships to either things or persons also tended to be unable to construct satisfactory relationships in the other domain. This was an important conclusion since it was the very opposite of that common assumption made in accusations of materialism, a word that implies that people who become focused on their relationships with things tend to do so at the expense of their relationships with other people.\n\nOf all the relationships that really helped people to overcome their sense of alienation to the flats they lived in, it was gender that turned out to be the most important. Single men, without female support, were typically the ones who had done practically nothing to change the flat over thirteen years. They had simply repainted back the original schema. But single women without male support were not in a much better state. Talking of support does not imply the kind of feminist-inspired support, where men start to help in traditionally female tasks. It reflected the maintenance of a very strict and traditional division of labour, where women are held responsible for knowing about, being interested in, and supplying an aesthetic sense of how the place should look, while in this conservative working-class arena men were expected to have no interest at all in any such issues of style. Instead men should cultivate an expertise with a DIY (do-it-yourself) tool set. A real man was a man with a power drill, the most conspicuously gendered object in the home. There is an obvious historical context to all this. Men who had traditionally spent rather more time at work in manual labour were facing a gradual reduction in working hours. This gave them more leisure time at home. But at home they were most likely to feel bored, or that they were getting under the feet of the females of the house. The rise of DIY in effect saved men's sense of their own masculinity and gave them an appropriate role within the house.\n\nWithout the aesthetic direction of women, men felt they could do nothing. Without the manual labour of men, women felt they could do nothing. But when they were prepared to exchange these skills, the result was a transformation of the kitchen that represents their combined efforts to overcome the initial state of alienation from the state. Sometimes the results were quite extreme, with Tudor-like half-timbering crossing over the ceiling; or absolutely everything from tiles to dishwashers in shades of blue. In such cases the couple had appropriated the original state fitments and personalized them so they could now identify positively with their own dwelling. Or at the extreme had thrown out the original fitments and purchased their own.\n\nThis conclusion suggested that two things that could easily have been studied in isolation really needed to be considered in relation to each other. On the one hand, there is the way the relationship to wider powers of the state and the market gives rise to a condition of alienation. On the other hand, to understand which households were able to overcome that condition of alienation, we have to examine more intimate relationships of gender. Stuff, in this case in the form of kitchens and their decoration, was integral to the way these two sets of relationship interconnect. Curiously, what emerged from this study of kitchens was a relationship of reciprocal exchange between males and females that looked for all the world like that which the anthropologist Marilyn Strathern had seen as the core to Melanesian societies in her book The Gender of the Gift.3 Strathern had used her Melanesian studies to argue that gender was not a given; gender studies should not be studies of what men do or what women do. Rather creating differences, that in turn create the conditions for exchange, ultimately creates what we recognize as gender. Making the transformation of the kitchen dependent upon an exchange of women's aesthetic concerns with male manual labour reinforced a clear definition of gender itself.\n\nThis account is highly generalized. Not every couple works things out in this way. Quite often couples originally form on the basis of the attraction between highly gendered individuals; a macho man being attracted to a conspicuously feminine woman and vice versa. When they become a couple living together, the opposition, which was once a source of mutual attraction, may or may not be reconciled in their development of their joint home. They may find a negotiated compromise. So, in one case, a woman gets the bedroom as her nest, evident in the luxurious bed with a teddy bear in the middle. Meanwhile the man can control the living room and populate it with computer games and a chair that takes a can of beer in the armrest while he is watching football on TV. Another couple may have a horror of what they regard as such gender stereotypes and strive for equality and sharing. This will produce a rather different, more compromised, aesthetic.\n\nBut often a woman's desire for equality is thwarted by larger power relations. As in the case of the kitchens, we may want to focus upon personal relationships, but find there constant echoes of much wider struggles of power. The problem for Georgina is that her husband is a professional artist with strong views on art and many other aesthetic matters. As a result, Georgina finds considerable difficulties in dealing with the aesthetics of the home, or trying to make it a medium for expressing their relationship. She is well aware that this activity of decoration could and should include far more compromise and shared projects than are likely to emerge within a sphere where his opinions are so clear and strong. She would have liked to include some romantic pictures, but he is tending towards minimalism. As a result, there are simply no decorations on the wall at all. Georgina knows full well that if the house has now emerged as a mirror of her social relationships, it throws into relief mainly their incompatibilities. Better to leave it be.\n\nA final example comes from a year's study carried out with Zuzana B\u00farikov\u00e1 on the experience of Slovak au pairs in London.4 When we started to look closely at the rooms which host families allocated and prepared for their au pairs to live in, it wasn't long before a clear pattern emerged. It seems that most families didn't think twice before deciding that IKEA represented the perfect source of au pair style. Not just IKEA in general, but specifically white melamine IKEA furniture, which was found in approximately half of the photo-documented au pairs' rooms we studied.\n\nThe reason lies surely in the very direct analogy these families see between their understanding of the au pair and their understanding of IKEA. Just like the au pair herself, white melamine from IKEA is generally seen as inexpensive, generically European in a young, modern poise, characterized by cleanliness, functionality and efficiency. Hopefully reasonably long-lasting, and quite easy to replace. When there is a change of au pair, the slate is wiped clean and one can begin again with the blank white surface, the impression of the previous occupant removed more or less instantly. In general the au pairs themselves are perfectly happy with the result. Such rooms are clean, easy to tidy, functional, modern, and bright. In Slovakia IKEA is considered somewhat upmarket. Somewhat bizarrely for a firm based in Sweden, but much of whose furniture is actually produced in Eastern Europe, many of these same items cost more in the Slovakian IKEA than in the IKEA in London. Only later on do many au pairs start to perceive another side to this functional modernism, its coldness and anonymity.\n\nIn all these cases the home has been used to bring into focus issues of power that were deliberately kept out of focus when discussing clothing. But there are reasons why each genre serves its respective illustrative purpose. Houses, the elephants of stuff, attract the attention of power, in a way that clothing, dismissed as mere trivia, mainly doesn't, but of course sometimes does. But we have also started to domesticate the concept of objectification itself. In the case of the kitchens it would be as true to say that the homes are tools for the construction of gender as to say it is men and women who combine to appropriate homes. Modernism creates design, but the middle class discover their own values through suburban design used as a bulwark against modernism. These stories are certainly about housing, but look beneath the surface and they remain also stories about objectification.\n\nHome and Agency5\n\nMany social science approaches seem to imply that, unless their analysis is all about these kinds of power relations, they have failed in their political duty. But the analysis of power relations, just as that of social relations, can easily decline into a kind of mechanical reductionism. The last chapter tried to show how a theory of objectification implies contradiction as intrinsic to culture. Such contradictions are not necessarily and not merely a reflection of these other relationships. So it is useful to examine also cases which are not necessarily about power and oppression. Instead perhaps we should turn to someone who is really rather fortunate. Who has had the ability to choose for himself pretty much exactly the house he would wish to occupy and who, with a decent salary and good fortune, has the means to cultivate a relationship to a house and home relatively free of the forces so far discussed. Take, for example, the author.\n\nI have been more than a little fortunate when it comes to house buying, starting with our very first house in Manchester, for which my wife and I paid out the princely sum of \u00a319,000. Mainly by dint of prices rising at the right time and place, we progressed from those beginnings in Manchester to my present London home which is well beyond any house I had ever expected to be able to occupy, especially in terms of style. I am not well versed in the arts and I don't have a lot of respect for my taste in such matters (nor does anyone else). However, I have always had a fondness for the arts and crafts period, ranging from jewellery by Archibald Knox to the architecture of Charles Rennie Mackintosh, which in the UK is a pretty commonplace affection. My house was built in 1906 and is quite unusual in that none of the rooms is a simple square. It also, in the immortal words of all estate agents, 'retains many original features' ranging from the oak staircase to various fireplaces and their surrounds. So, OK, the circular roses on the stained glass windows are not actually Mackintosh, but they are clearly sub-sub Mackintosh, and that's good enough for me.\n\nI am fortunate then in occupying that typically English idyll of suburban house and garden. Indeed I am writing this in a study at home looking out on squirrels, jays, foxes and blue tits. There is, however, one, I would admit relatively slight, problematic feature with this tiny tale of suburban bliss. The house is just a bit too good-looking, too close to the style I hugely admire, but could never hope to emulate. The problem is that, as any householders, we have to paint walls, choose carpets, add furnishings, put up curtains and so forth. And speaking for myself (not, I hasten to add, the rest of my family, whose tastes are impeccable) I constantly feel that anything I choose to do will fail to live up to the standards set by the house itself. Ideally I would buy and place things that complement and complete this aesthetic I admire. But inevitably, when I have chosen a cushion cover, when I have placed a new piece of furniture (some of these are from IKEA) I feel instead that I have betrayed and demeaned the house that really deserves someone with better taste than myself. I have not just let myself down; more importantly I have let my house down. I simply don't possess taste that equates with its original aesthetic.\n\nObviously I am not in any way trying to equate my good fortune, in living within an aesthetic that I perceive as superior to myself, with the plight of people forced to live within often soulless estates about which they had no choice whatsoever. I am contrasting, rather than equating, my position with this largely powerless population. The point is that, even when there are no such external or oppressive conditions, it does not therefore follow that we are returned to some idealized version of the theory presented in the last chapter. There are a wide range of reasons why people might not feel that the house is simply a vehicle by which they are positively constructed and developed. My alienation from the house derives solely from my sense of inferiority to the house itself.\n\nIn the last chapter it was suggested that we can find, within the myths, rituals and cosmologies of peoples in Melanesia or Australasia, evidence for the contradictions of objectification appropriate to their circumstance. Starting from my individual point of alienation, we can generalize this to English people as a whole by examining our own forms of myth. Take, for example, the genre of haunted houses. If we think for a moment of the classic haunted house, it is very unlikely that we have in mind just some very ordinary terraced property in a London street. If a property such as mine was really lucky, it might just manage to get itself a poltergeist, but it's most unlikely to attract a fully fledged ghost. For that honour we probably need to inhabit some ancestral pile that has been standing for generations and inherited its ghost from a tragic death, several centuries past; the little girl in blond ringlets who hid in a clothes chest and was never able to get out again, or a gallant knight who took his own life on hearing of his lover's demise.\n\nMy personal favourite within the genre is fictional, The Canterville Ghost by Oscar Wilde. The two key factors in this tale are first that it concerns an old house with its own name and history. The second is that the purchaser, Hiram B. Otis, was an American with little disposition to mind such things. Hiram B. Otis, who proclaims that he comes from a modern country, shows no respect for the well-established ghost. The first action by the new mistress of the house is to attempt to remove (with the assistance of Pinkerton's Champion Stain Remover) the bloodstains of Lady Eleanore de Canterville, murdered there by her husband in 1575. In subsequent pages the distinguished ghost becomes horrified by the materialism of Otis and his children's lack of respect for a pedigree such as he possesses and his many past achievements in the honourable act of haunting. To cut a short story very much shorter, it all ends well, of course. The ghost is finally laid to rest in the garden of death, as Otis's daughter, having fallen in love with the Duke of Cheshire, comes to respect and assist the historical tradition represented by the house.\n\nThe association of ghosts with stately homes is hardly an invention of Wilde. Basically the older the house the more chance that it is haunted. Very commonly the ghost gives evidence for aspects of the history of the house that were otherwise unknown, for example, a priest's hiding hole, or a lost hall or room. Ghosts clearly have a marked tendency to cling to their old haunts. Why, though, should the new owners of such houses find themselves encumbered with possessions beyond those they had knowingly purchased? The reason is that these ghosts are expressing precisely the theoretical point being made by our dialectical theory. They are even closer to what was introduced as alternative attempts to transcend the opposition of subjects and objects: theories associated with writers such as Latour and Gell, who talk in terms of the agency of things. That is, we think that we, human subjects, are free agents who can do this or that to the material culture we possess. But inevitably we can't. Objects can be obdurate little beasts, that fall from the mantelpiece and break, that refuse to grow in shady spots in the garden, that cause us to trip, and that crash their systems just as we were about to type something genuinely interesting. If, in all such cases, they are clearly not reflecting the agency that is represented by us, then it seems reasonable to start talking in terms of the agency represented by them \u2013 the gremlins. In short, one way of thinking about these theories of dialectics, or mutual constitution, is to imagine material culture as having agency all of its own. Things do things to us, and not just the things we want them to do.\n\nThis was the point I was making about my own house. Theoretically I own the house, and I should be able to do any damn thing I want to it, subject only to the feelings of my family and the laws of the state. But of course I can't. The wretched house is simply too good-looking and constantly humiliates me. But as long as I am prepared to be humble and respect its original features, I gain a great deal of pleasure from it. This is partly also a condition of my understanding that the house was built in 1906, had many occupants before me, and may have many after me. Compared to it I am quite transient. This is still more evident if one is living in some stately home built centuries ago. Then, the current householder may feel more like someone keeping it well for the future, rather than simply being the owner. We have a sense that the house has its own powers and properties that lie beyond us \u2013 something that may be easier to comprehend if we give that power and history an anthropomorphic form. Instead of saying that the house has agency, it is easier if we stick with the entities we normally think of as having agency, namely people, even if, in this case, they are dead people. A ghost may express this feeling that there is power in the mere longevity of the house, an agency which represents the limits of our own agency. A ghost can stand for that ancient history in a manner that we cannot. Or, to put it succinctly, perhaps we should realize that ghosts are the original estate agents (sorry!).\n\nIt is not surprising that this point about the sometimes obdurate nature of material things uses the example of the house rather than that of clothing. A house is much more likely to be experienced as an independent and autonomous agency than some lesser species of material culture. But the basic idea that things have agency holds for much more flimsy substances. Keane makes the point very effectively through an ethnographic study on the island of Sumba in Indonesia.6 One of his key examples is that of a piece of cloth that is torn when being transported by one group of people to another. No one intended the cloth to be torn, but inevitably people will interpret the fact that it was torn as significant. It may well be seen as a sign of undeclared agency of a person, such as a hidden reluctance to give the cloth, or it may be the agency of malevolent spirits as opposed to humans, intervening to ensure this less satisfactory outcome. What the haunted house story shows is that people in England, as well as in Sumba, conceive of spirits who pre-empt our modern theorization of agency as a possession of stuff.\n\nOf course most of us do not live in haunted houses and our strategies for dealing with the temporality of our homes are usually rather more mundane; however, they can still make manifest some stark contradictions. There are two main ways people generally work with the longevity of their homes when it comes to furnishing. One group tends to prefer genuine antiques, sometimes only those of the period of the home itself. Others respond by purchasing reproduction furniture. While some people combine these, quite often there is considerable antagonism and disdain between the two camps. For those who prefer reproduction items, true antiques may be seen as 'coffins' of furniture. The idea that people had died in that particular bed make it, for them, something to be avoided at all costs. For the buyer of antiques, the purchase of reproduction furniture is viewed as fundamentally dishonest. Those people are thought of as cheating, betraying the proper search for authenticity. By contrast, for people who buy reproduction furniture it is the purchasers of antiques who are dishonest since it is reproduction furniture that is being clear about these being contemporary additions to the site, since the antiques didn't actually come from or with that particular property. The debates go on beyond mere purchase. Does the inglenook fireplace have to be where it would originally have been or can it be made from a conversion of a space under the stairs? Or do you go so far as to renovate the house itself with antiquated methods such as oak beams with oak pegs?\n\nSuch contradictions are not a feature of the relationship of housing to the state and forms of power, but something intrinsic to certain properties of materiality, in this case longevity. We are forced into taking positions on wider cosmological issues of authenticity, truth and identity. This is all part of our need to come to terms with the agency of stuff itself. A useful way of thinking about this, in respect to housing, is to reflect upon the various meanings of the word accommodating. On the one hand it speaks to the need we all have, to find accommodation in the sense of a place to live, and accommodation can cover many such spaces, from the homes we might own, to temporary rented forms such as student accommodation. Secondly, this may also involve a process of accommodating in the sense of an appropriation of the home by its inhabitants. Less obvious is that this second meaning is reciprocal. It may imply our changing of a home to suit ourselves, but it can also imply the need to change ourselves in order to suit our accommodation. Thirdly, the term accommodating expresses a sense of willing, of benign agreement to compromise on behalf of the other, often the only spirit within which accommodation can be achieved. By considering our relationship to the home through the term accommodating we face the home not as a thing but as a process. Being accommodating and being accommodated is something with which we are constantly engaged. So how, then, does accommodation become accommodating? How do we, in practice, achieve this balance between our agency and that of the home? To explore this further, it helps if we split the problem up into a series of different registers or scales of accommodation. Fortunately this can be exemplified by a series of research projects that were carried out by some more of my then PhD students. While none of them was conceived with this sequence in mind, on juxtaposing them, they form a rather useful sequence. Each represents a different stage and scale in the study of the dynamics of accommodating.\n\nA project by Jean-S\u00e9bastien Marcoux examined the process of moving home in a site, Montreal, where this is done frequently and often on one particular day in the year.7 Marcoux notes that we start with many assumptions about moving house. It is supposed to be traumatic, destabilizing and even a cause of ill health. But his study tries to examine why people might often move for what they experience as quite positive reasons. Moving house allows for a kind of critical realignment of persons with their possessions. When moving house they discard some of their stuff but, in contrast to the house itself, many other possessions move with them. As such, moving house allows people to reconstruct their personal biography as represented in memories of associated objects and thereby the sense the family has of itself. Certain relationships with other people get discarded along with the objects that memorialized them, while others come to the fore and are used prominently in the decoration of the new home. So people have a chance to, as it were, work on and repair the way they represent themselves and their own histories to themselves, and to the world, in accordance with how they now want to see themselves. He concludes that moving home is a way of rewriting one's autobiography, inscribed in things.\n\nMoving down the scale, Alison Clarke (who also shared the shopping fieldwork with me) looked at changes that take place through the refurbishment of rooms within the home.8 This is an activity subject to its own simplistic and generalized clich\u00e9s, mainly about 'keeping up with the Joneses', which view interior decoration as a site of aspiration based on emulation of one's neighbours. In the context of a rise of privatized and more materialistic lifestyles, this is the image most journalists have of home decoration. Clarke replaces this clich\u00e9 with a far more informed and complex understanding of what the house performs in mediating the relationship to others. The context for her studies, that of working-class households within British state housing, presents an initial paradox. On the one hand, there is a clear concern with neighbours, 'the Joneses', and with what other people might think about one's home. But, in accordance with British working-class historical traditions, there is almost no actual visiting of homes by neighbours (visiting is almost exclusively by kin). Such families never actually see their neighbours' homes, so who exactly are they emulating?\n\nWhat Clarke discovered is that, in practice, the key relationship was to the home itself, rather than to neighbours. People feel more the discrepancy between the actual home they live in and the images they possess of aspirational or ideal homes, derived from various sources. Take, for example, an immigrant's aspirations towards assimilation or, the opposite, a chance to reconstruct one's ethnicity. Alternatively the aspirations of a single woman towards the place she will live in with her ideal partner, or her aspirations for the future of their children. It is the home itself that is used to objectify such goals: to create an imagined environment that achieves integration, or stands for some future state. The home as a mirror of aspirations also becomes a battlefield for the individual or couple as to whether they are living up to, or betraying, their own ideals. In short, we find they constantly judge themselves against the representation of who else they might be, as constructed by the refurbishment of their own homes.\n\nSometimes the ideal home is held as an external myth, and most ordinary actual homes betray a very different set of relationships. In a study called 'the untidy Japanese house' Inge Daniels shows, from her evidence living in a series of Japanese homes,9 that actually these are quite different from the external portrayal of such homes as a kind of natural and immaculate minimalism \u2013 the kind of Japanese homes portrayed in Sunday newspaper supplements in the UK. Actual Japanese homes bear little relationship to the Japanese home of our Orientalist projections. Furthermore, there is a deeper significance to the fact that most homes tend in practice to be quite untidy. This is equally true of the nature of the Japanese family and domestic relationships within the home, which also have tended to be portrayed in overly tidy and clear categories.\n\nFinally, this use of movement and change in the process of accommodation can be taken all the way down to the smallest scale. This became evident to Pauline Garvey while researching in a town in the south of Norway.10 Because she was present as an ethnographer, she saw not only the major changes that take place in the home, but the little movements, such as putting a chair or table in a slightly different position within a room. In itself trivial, but when seen over the longer term, such actions have two important consequences. Partly they bear on the horror people have of the idea that they are nowadays stuck in a rut, that nothing is ever going to change. Households may not want to have to contend with things that are significantly different. The chair might be put back the way it always was, after a while. But at least the inhabitants have expressed their agency over that of other persons and things. They can demonstrate to themselves that they still retain a capacity for change. Secondly, these are in effect small experiments, ways of re-thinking how things could be, different configurations, that mostly will not amount to anything, but occasionally become the catalyst for more significant changes in the environment within which people live.\n\nEach of these studies illustrates the process of accommodating as an example of objectification. In no case are people merely representing themselves in their homes as a static relationship. In each research project it is the dynamics of the home that is paramount, whether moving house, refurbishing a home, creating mess or merely moving stuff around. In each case the persons are once again creating themselves through the medium of stuff. The problem with such examples, as so often with material culture studies, is that they can feel relatively trivial, of limited consequence. We have travelled some distance from power and ideology in order not to be overwhelmed and reduced. But throughout this chapter there has been an insistence that the little details and the grand ideologies are usually linked. So the final example returns to global issues, in this case migration, but seen through these intricate details of home decoration.\n\nCaribbean Homes\n\nPeople don't just move homes, sometimes they move countries, and this brings into focus the potential gulf between home and homeland. The starting point for this example is the initial phase prior to such separations. Here we can reprise some of the earlier themes about how homes objectify values, before following the migration to London. The perspective then splits to examine, first, some of those who remain in London, and then those who become return migrants. The examples are taken from the two islands in which I have carried out fieldwork, Trinidad and Jamaica. In the case of Trinidad, the fieldwork has been intermittent over the course of twenty years.11 It started with a year living in the town of Chaguanas, near the centre of the island. But I was not the first to write about this town. It was the birthplace of the novelist V. S. Naipaul and the setting of several of his best-known novels, of which many would consider the finest of all to be the felicitously named A House for Mr Biswas. Chaguanas is not exactly a tourist destination and, as it happens, one of the very few houses that remain from an earlier period, and has any sort of character, is the very house which was the setting for that particular story.\n\nI have always been in awe of this superb novel that, with a poignancy I couldn't hope to even begin to convey, manages to capture one of the central arguments of this book, and I am sure was one of my inspirations. The theme of the novel is an increasingly desperate desire for objectification through one's own home. It concerns the poignancy of failure, of the pathetic nature of these attempts to objectify, when seen through the lens of a frustrated life, with limited resources and an unkind fate. The tragic failure that haunts Biswas comes from living in a succession of houses that he does not own and on which he leaves little mark. Without a house of one's own it was impossible to imagine a family and especially a descent line. The prologue ends with these sentiments:\n\nBut bigger than them all was this house, his house. How terrible it would have been, at this time, to be without it: to have died among the Tulsis,... to have left Shamas and the children among them in one room; to have lived without even attempting to lay claim to one's portion of the earth; to have lived and died as one had been born, unnecessary and unaccommodated.12\n\nThough fiction, Naipaul's book gives this abstract theme character and significance, and provides the perfect illustration of the points made in this chapter. Eventually Biswas does end up with a property, a ramshackle affair, jerry-built for a quick profit, full of faked and insubstantial elements, but a property nonetheless.\n\nBy the time I arrived in Chaguanas in 1987 Trinidad had experienced an oil boom, and the generation that followed Biswas had opportunities to fulfil these dreams and took them. I worked with four communities, based on a relatively wealthy residential area, a government housing sector, a village incorporated into the town and a squatters' camp on the periphery of the town. Amongst other studies I took around 800 photographs from 128 living rooms, for analysis. The remarkable quality of these living rooms was their homogeneity. The furnishings within them, often sofas and armchairs, tended to be upholstered either in maroon or in brown, and these colours were picked up by many of the other associated objects such as curtains, carpets and possessions. Most comprised a standard set of items such as the space saver \u2013 a set of open shelves, filled with ornaments, that does more or less the opposite of what its name implies. There would usually also be present a buffet (glass cabinet with kitchen and other wares), homilies often with religious ideals, artificial plastic flowers and stuffed animals or dolls, often still in their dusty plastic bags or boxes.\n\nThe subjects of decoration were most commonly fake tapestries of coniferous woodlands of the kind that would never grow in Trinidad. Religious themes such as the Last Supper were also common, as were items pertaining to mothers or Mother's Day, and weddings. I was never quite certain if the merchants holding or swigging tankards of beer were Portuguese or Dutch. The single most common motif was swans, perched in every kind of concrete, plastic, wood and stone, though I don't think I ever glimpsed the bird itself in Trinidad. Overall this characterization was regardless of income level or residential area. So, for example, both wealthy and impoverished Trinidadians collected quantities of cheap ceramic ornaments.\n\nThere are some clear themes in these interiors: the preference for artificial things, for filling up spaces, content that is orientated to family, religion and education. Above all, a layering and covering of things, from the crochet-style toilet roll covers to throws (cloth that covers the sofa seats so that you don't actually see them) but also the fact that the stuffed toys remain in their plastic bags, or that artificial lace covers a variety of different surfaces. It is not hard to see wider social values inscribed in these things. Complementing covering and layering are certain rituals for piercing these layers. For example, one of the most elaborated layers is the very professional cake decorations. The occasions when these are ceremonially cut with a knife are also about the only time when it is acceptable to kiss in public. This and several other examples start to suggest a more general opposition between sexuality, with its potential for uncovering and exposure, on the one hand, and this aesthetic of layering and covering. The filling of the living room with its layers and lace, the wedding photos and the educational certificates on the walls, all speak to the project of family longevity and respectability: exactly the values which are threatened by sexuality, associated with the outside and the street, the ever present threat of disruptive bacchanal introduced in chapter 1.\n\nThis spills over to genres other than the house itself. Inspired by Naipaul, the intention had been to work on the home, but once in Chaguanas it was impossible to avoid noticing how much this town was dominated by the car, the concern of 38 of the 176 commercial establishments in the area where I lived. There were garages and car part specialists, but they were dwarfed by three stores devoted to car upholstery, whose owners became wealthy enough to own much of the area's commercial property. Apart from domestic upholstery at Christmas, the trade was largely car upholstery, and the three major firms merely dominated a host of smaller car upholstery firms, making this probably the leading commercial concern of the town.\n\nDuring this fieldwork, which took place during the recession period of the 1980s, upholstery was dominated by repairs to vehicles such as taxis. But during the previous oil boom most new cars had their first outing to these upholsterers, who could transform everything from the car seat to the trunk (car boot) with designs such as fake snake skin or a black leatherette streaked bluish and silver, marketed under the title 'New York by Night'. Even the dashboard could be upholstered, cushions added, and the whole complemented by a variety of paraphernalia such as perfumes, religious icons and stickers. Cars could also be feminized, for example with heart-shaped satin cushions with projecting pink frills and central flower designs. The degree of such personalization was satirized by a journalist who recorded his removal of the accretions to his new second-hand car:\n\nThe tiger-skin covers came off on day one, as did the red plastic steering wheel cover; as did the little duckie. The white JPS emblems made it to day two, but no further; nor did the 'I love my Mazda' sticker. Presently slated for retirement are: the dashboard heart that lights up in red with the words love caressing; the pair of little green bordello cabin lights; the red hyphen lights above the front number plate; the fog lights inscribed Denji; and at least one of the three antennae.13\n\nThe commercial centrality of car upholstery made a bit more sense in the light of people's more general relationship to cars, which went well beyond anything one can experience in London. This was partly economic, what is locally called pulling bull, whereby non-commercially registered cars are actually used for taxi work. But the intimacy with cars went way beyond instrumentality. In Trinidad individuals may be located as often through the car parked in front of a house as by their house number. The local press constantly spread scandal and innuendo through reference to car ownership as in 'The leader has a nickname which resembles that of a popular large local fruit, and he drives a taxi which is neither too dark nor too light',14 or it might talk of an AIDS victim 'whose husband drives a Mazda'. Retailers, irrespective of what they were selling, routinely decided their expectations of particular customers entering their shop on the basis of their car, and so I would be told that a Laurel driver bought this, but a Cressida driver would not buy that. Occasionally people knew of another individual by their number plate but not their name. Trinidadians would queue in cars rather than walk the last few metres to deliver a child to school and would stay within their car while the child played in the park. A car owner would shudder in horror when a passenger slammed the car door too hard, and, when home, carefully cleaned the tread within the tyres.\n\nIf the upholsterers were devoted to car interiors, there were other shops in the same high streets who would foster equal care for the car exterior, tinting the window glass, adding stripes to the exterior paintwork or ensuring the car was up to date with current fashion by replacing metallic wheel hubs with white. On closer inspection it seemed that this division into two genres of car beauty products reflected two distinct groups of customers. As with so many dualisms in Trinidad, these are presented first in terms of ethnic stereotypes. A conversation described the Indian with gold on his fingers and hair greased back who wanted crushed velvet upholstery but could only afford short-pile acrylic so he spent ages brushing it the right way. By contrast, there was the Black dude, ideally with his deputy (mistress), though the tinted glass ensured that at least you think she might be there, projecting the person through the car thanks to the external paintwork and the very loud sound system.\n\nIn material culture analysis, one set of stylistic forms starts to make sense when it can be seen as consistent with others. The dominance of upholsterers suggested they had other interiors to transform as well as the car. An obvious extension is the home interior. Here too one saw material covered over by layers, but the values being expressed were also clearer. Less obvious was the link to the funeral parlours. But they also turned out to be a major outlet for upholstery since coffins, and the more expensive and luxurious caskets, are almost invariably lined in deep buttoned upholstery. Putting together the aesthetics of home, car and coffins started to account for the significance of upholsterers. The transformation of the car interior turned it into a vehicle not just in the sense of transporting people, but also a vehicle for transporting values. Specifically the values of the home interior: long-term family values. Values that could be continued aesthetically even after death and through death. In stark contrast the shops devoted solely to the car exterior suggested the car also served as a vehicle for the individualism and mobility associated with the world outside the home.\n\nFrom this vantage point we can see how a town dominated by the upholstery industry gives us insights into larger contradictions of Trinidadian society. On the one hand, it highlighted an orientation to family, to long-term reputation, to the past and tradition, to accumulation including education that gives solidity to the family. This cluster of values can be termed transcendence. They contrasted systematically to a constellation of values which can be called transience, the same values discussed in relation to style in chapter 1, expressing an always ephemeral and temporary achievement of freedom. An identity created in the moment of performance.\n\nThis in turn explained the way Trinidadian life is pivoted around two key festivals. Home decoration is central to the celebration of Christmas. Much of Christmas ritual consists of buying things for the living room, emptying it, tidying it, and ending Christmas Eve with the ceremonial re-hanging of the curtains that enclose it and constitute it as a given space. Even the street-wise young men, who live the life of transience, find that at Christmas time they have to locate themselves in the nexus of family, and help with the home decorations. One of my favourite fieldwork moments came when a man used me as an excuse to avoid helping his wife with Christmas domestic chores, and drove me to the house of his deputy (mistress) where he spent the evening \u2013 carrying out Christmas domestic chores for his deputy, of course.\n\nIf there is a systematic contrast between Christmas and Carnival, then there is also a remarkably clear point of transition. This takes place on 'Old Year's Night' (New Year's Eve), which starts with the most important Church service of the year and ends with the most important party of the year. So the study of stuff, and the patterns of what people do with stuff, is probably as much effective at revealing the underlying structural tension that objectifies Trinidadian culture order as anything Trinidadians might say about themselves. The reason an individual, such as Naipaul's Biswas, is so driven to achieve a life through taking possession of a home is because the home is foundational to objectifying that set of Trinidadian values with which he identifies. But if Naipaul brilliantly conveys the Caribbean aspiration to being accommodated, in contemporary Trinidad this tension is most eloquently expressed by that heavily upholstered living room on wheels that we call a car: that same population on the one hand wants to retain those family values, but on the other is also increasingly tempted by the allure of the car's mobility and its promise of freedom from the home and family.\n\nThis analysis of house and car illustrates the way material culture is studied as the objectification of values. But it remains an ethnographically based, static portrayal. From this starting point we can take off, as did so many Trinidadians, and travel in time and space to Trinidadians as migrants to London from the 1950s onwards. Not surprisingly, we find in London a dynamic synthesis of the two regions, to the extent that, within a generation, young people are quite uncertain as to what derives from where. For example, many younger Londoners of Caribbean origin think of the living room or front room as a kind of sacred space, a place they would be often banned from, except on Sundays. This owes little, though, to anything Caribbean, but owes much to working-class British traditions. In their search for respectability in London the migrants had copied the tradition of the parlour, a sphere of respectability, hardly used at all, outside of Christmas or for very special visitors.\n\nHome interiors pay little heed to whether their contents relate to any particular tradition. An exhibition recalling the home interiors of immigration juxtaposes reggae with Jim Reeves, souvenirs of St Lucia with souvenirs of the British seaside.15 Today there is a fashion for roots: the idea that if migrants remain true to their older traditions it may give them strength in resisting the racism and oppression they confront through the migration process. Actually this is highly unlikely, and one of the portraits in the book The Comfort of Things provides a clear example of why.16 The values that maintain respectability within the extended family of Caribbean traditions may just result in loneliness and the appearance of aloofness in London. What we call roots are likely to be part of the integrity of one kind of life. If people want to protect something of that life, it is likely that a creative synthesis sensitive to the change of context is going to be more effective than pure conservation or some model imposed by a state.\n\nThe world, however, is not so liberal and the end point of this story, so far, comes when some of these same Caribbean migrants, faced with problems such as racism in London, decide to return to the Caribbean. Or they simply follow their original intention, which is often to return to their place of origin on retirement. Here I am indebted to Heather Horst, whose PhD concerned the homes of those migrants who had returned from the UK to Jamaica.17 When her work is published I hope it will have as a frontispiece one of the graves she showed me in rural central Jamaica. The grave in question took the form of a rather dinky looking miniature concrete house, complete with doors, windows and gables. It looked more like something to put dolls inside than a corpse beneath. Some of these graves are surrounded by the style of ironwork identical to that which surrounds actual houses. She interprets these graves as marking the end of a long journey home, aspired to in life, but only achieved in death.\n\nFor most Jamaicans the project of building a house is not a one-off, single act. Most people could only afford to build their own house in stages. As money accumulated, one might lay a foundation for a new room, or complete the tiling of another. Building the home was a life's work and in turn became the primary mode by which life itself was marked as a progression. Despite all the pressure from the church, it was not having children that led people to marry; it was the ability to have some sort of house of one's own. This close association between life and building a home is obviously complicated when a Jamaican migrates to London. Most Jamaicans, given the nature of colonial education and governance, had come to regard Britain as a kind of second home, a source of their own cultural identity. In practice they experienced in Britain a sometimes violent repudiation of what they had seen as their own British identity.\n\nIn many cases this reinforced the initial intention of returning to Jamaica on retirement. Well over twenty thousand returnees have made this trip. But once they do return they face a second, even more unexpected, rejection. In general Horst found that the tendency was to migrate, not back to their original district of origin, but to an upland, relatively cool area of central Jamaica that seems somehow more English. Furthermore, over the decades of exile the migrants have developed an affection for certain elements of English life, such as an English-style garden. In turn it emerged that those who remained behind saw these returned migrants as indeed principally English rather than Jamaican. They worried that returnees with greater wealth would lay claim to land and authority at their expense. So ironically the people who had failed to become English through migration to England found that the moment they finally achieved this goal was when it was no longer desired. Many felt they were no more at home in Jamaica than they were in London.\n\nIf they could not create a home through a sense of homeland, it was once again the house they sought to transform into home. One of the principal incentives behind the initial migration was to earn the kind of salary that would build the house they had failed to construct on local incomes. These settlements in upland central Jamaica are full of such houses: houses whose scale and style are instantly recognized not as Jamaican, but as 'returnee' houses. Faced with this initial failure to become Jamaican again, Horst observed that many of these returned migrants increasingly spent their time going to, or being involved in, the organization of what become highly elaborate funerals. It was only in death, interred in the earth itself, beneath these miniature models of the perfect house, that the final return to Jamaica was successfully completed. A return blessed by deep religious faith in another final return to another heavenly place of origin.\n\nThese graves are the end point of a chapter that began with a discussion of modernism as ideology in the history of London housing. A chapter intended to exemplify the consequences of the theoretical ideas elaborated in the previous chapter. The result, I hope, is an absolute mess. We start trying to move into a nice, neat, newly built theory \u2013 minimalist and austere. The pure dialectic, in which we finally appreciate that subjects and objects exist only through the process of objectification in which each constitutes the other. An absolute theory that will give us mastery over explanation and understanding. The problem is that, as soon as we tried to furnish our theory with examples, they turned the whole thing upside down. The case studies now lie strewn higgledy piggledy all over the place.\n\nActually the theory is still there if you look hard enough. Whether it's Mr Biswas, Georgina, the Slovakian au pair in London, or the Caribbean migrants. Everyone is trying to engage in being accommodating as well as in being accommodated. They are all trying to find themselves, root themselves, create family descent or just a decent household through the stuff that is the home. The home is indeed a powerful instrument of objectification. But it is very rarely either a simple, let alone a heroic achievement. Hardly ever is the result a clear reflection of the intentions of the human actor. The process of accommodating is battered by floods of other factors: the agency of the house itself, haunted or otherwise, the dictates of the state, the profession of architecture, the limits of resources, the arguments between couples, the refusal of people to accept you back as Jamaican, the superiority of one's own house to one's own aesthetics, the contradictions of real antiques and reproduction furniture, the attempt to adhere to roots that betray you.\n\nSo the house as a form of objectification draws us into a maelstrom of power and instructional interests that often make us feel comparatively powerless. But to acknowledge these forces \u2013 governments, peer pressures, asymmetries of power by gender and class \u2013 doesn't mean that we now see people as their passive victims. We have witnessed all sorts of creative responses to these onslaughts. Often expressed in material objects, such as in the suburban rejection of modernism, cramming spaces with personal ornaments, or simply ignoring the local council and knocking down an interior wall.\n\nHouse interiors are often a mess because the compromises that we constantly have to make with authority add to the compromises that result from another of the critical components of our dialectical theory: the centrality of contradiction to culture. Trinidad is not to be understood as one overwhelming set of values, but rather as a constant tension between diametrically opposed values. London is neither modernist nor suburban, but a tension between these two. Just as the relationship between an individual's aspirations and the collectivity represented by talking in terms of the collective word Trinidadian is an irresolvable tension, so it is not surprising that in surveying actual homes and houses, we often find scenarios that are tragic, pathetic, or unresolved except through death. However, as made clear in the book The Comfort of Things they most often emerge as some kind of expressive logic that does indeed objectify their inhabitants, which in the circumstance is some kind of triumph.\n\nOne more conclusion follows.18 Except for the very first example of modernism and suburbia as fa\u00e7ades, all the evidence for this chapter depends on a willingness to step inside the private domain of other people. To study stuff, we need ourselves to be where stuff is. Right there, in the living room, the bathroom, the bedroom and the kitchen. This is where most of modern life is lived. Families are created in bedrooms and sometimes divorced there. Memories and aspirations are laid out in photographs and furniture. Yes, peering into the wardrobe you may be accused of voyeurism, of a lack of respect for privacy. But actually most people would face their anthropologist in the kitchen rather than standing on the doorstep with pen and clipboard. So if you spot stray, bedraggled anthropologists, out in the rain, looking for a dry living room to conduct research in \u2013 be kind \u2013 give them a cup of tea and they will listen to you all day and pay more respect to your choice of wallpaper than you could ever have guessed.\n\nNotes\n\n1 'Modernism and Suburbia as Contemporary Ideology', in D. Miller and C. Tilley (eds), Ideology, Power and Prehistory (Cambridge: Cambridge University Press, 1984).\n\n2 'Appropriating the State on the Council Estate', Man, 23 (1988), pp. 354\u201372.\n\n3 M. Strathern, The Gender of the Gift (Berkeley: University of California Press, 1988).\n\n4 Z. B\u00farikov\u00e1 and D. Miller, Au Pair (forthcoming); and Z. B\u00farikov\u00e1, 'The Embarrassment of Co-Presence: Au Pairs and their Rooms', Home Cultures, 3 (2005), pp. 99\u2013122.\n\n5 The next section combines 'Possessions', in D. Miller (ed.), Home Possessions (Oxford: Berg, 2001), pp. 107\u201321, with 'Accommodating', in C. Painter (ed.), Contemporary Art and the Home (Oxford: Berg, 2002), pp. 115\u201330.\n\n6 W. Keane, Signs of Recognition (Berkeley: University of California Press, 1997).\n\n7 J.-S. Marcoux, 'The Refurbishment of Memory', in Home Possessions, pp. 69\u201386.\n\n8 A. Clarke, 'The Aesthetics of Social Aspiration', in Home Possessions, pp. 23\u201345.\n\n9 I. Daniels, 'The \"Untidy\" House in Japan', in Home Possessions, pp. 201\u201329.\n\n10 P. Garvey, 'Organised Disorder: Moving Furniture in Norwegian Homes', in Home Possessions, pp. 47\u201368.\n\n11 Most of the following material is derived from Modernity: An Ethnographic Approach (Oxford: Berg, 1993); see also 'Migration, Material Culture and Tragedy: Four Moments in Caribbean Migration', in P. Basu and S. Coleman (eds), Mobilities, 3\/3 (2008), pp. 397\u2013413.\n\n12 V. S. Naipaul, A House for Mr Biswas (London: Andr\u00e9 Deutsch, 1961), p. 14.\n\n13 Trinidad Express, 20 Oct. 1988.\n\n14 The Bomb (a weekly Trinidadian newspaper), 21 Dec. 1990.\n\n15 M. McMillan, 'The West Indian \"Front Room\" in the African Diaspora', Fashion Theory, 7\/3 (2003), pp. 397\u2013414.\n\n16 The Comfort of Things (Cambridge: Polity, 2008).\n\n17 H. Horst, 'Back a Yaad: Constructions of Home among Jamaica's Return Migrant Community', PhD diss., University of London; H. Horst, 'A Pilgrimage Home', Journal of Material Culture, 9 (2004), pp. 11\u201326; H. Horst, 'Landscaping Englishness: The Postcolonial Predicament of Returnees in Mandeville, Jamaica', in R. Poter, D. Conway and J. Phillips (eds), The Experience of Return Migration: Caribbean Perspectives (Aldershot: Ashgate, 2005), pp. 207\u201323; and H. Horst, ' \"You Can't be Two Places at Once\": Rethinking Transnationalism through Jamaican Return Migration', Identities: Global Studies in Culture and Power, 14\/1 (2007).\n\n18 'Behind Closed Doors', in Home Possessions, pp. 1\u201322.\n4\n\nMedia: Immaterial Culture and Applied Anthropology\n\nThere is no point just adding media to clothes and houses. For starters, the media is rather less obviously a material thing, and is better addressed as a form of technology. There will also be some mention of commerce, which has been rather ignored so far, though this topic is mainly left to the companion volume Consumed by Doubt. But perhaps most importantly, I haven't so far asked why all this matters \u2013 beyond the intellectual curiosity of academics. Can such studies also lead to an attempt to intervene, advise and take responsibility for these new understandings? Is there a potential for research in material culture to contribute to the welfare of populations? Basically, can we start to make ourselves a bit useful? So this chapter ends on a more applied note concerned with the insights the study of stuff can contribute to the alleviation of poverty.\n\nBut we start with the technology. What kind of stuff is communication technology? For example, we may decide we want to initiate a discussion of the Internet, but what is the Internet?1 Using the word Internet we may forget the constantly changing configuration of things that people do online. The early genres of Internet use that were associated with activities such as flaming have pretty much disappeared. Today there are some regions where people know nothing about surfing for information and see the Internet as a place for chat, while in other regions they surf but don't chat. The Internet is not a thing, and has no clear material form except through the box and screen that is the computer. The Internet is not one of its particular usages. Rather it's a term we use to consolidate genres of usage that are linked through online access. Should the Internet be regarded then in terms of the capacities that seem inherent in it, in its actual usage, or perhaps the way it is understood \u2013 what we gloss as its meaning, when we use the term Internet?\n\nEven the physical link to the computer quickly breaks down. This association works in many countries, but when studying in Jamaica it soon became obvious that almost no low-income families could afford computers, so the very idea of chatting online, or surfing, was primarily seen as an attribute of the mobile phone. For people such as myself, used to keyboards, chatting by thumb looks remarkably clunky. But these Jamaicans simply did not see their Internet phones in these relative terms. They just saw the excitement of their first chat room on a mobile phone, their first music download, and all sorts of obscure information. The weather forecast in Jamaica, New York, has surely never before been quite so exciting.\n\nWe might feel more comfortable turning to the idea of inherent usage. Here the evidence from Jamaica points in the opposite direction, because I had really no appreciation of what the inherent usage of the mobile phone was until I went to work there.2 In London, while nearly everyone has a mobile phone, for most it is just an instrument for making phone calls. Others, usually far fewer than those desired by phone companies, take an interest in the phones' built-in capacity to surf the Internet, or to send videos. By contrast, in Jamaica, many people who have mobiles have ceased to wear a watch. Why bother when a phone can tell the time and sound an alarm? With a much greater facility for appreciating the advantages of free stuff, Jamaicans seize upon a multitude of secondary features. A phone is a handy calculator, quite often a diary and calendar and certainly the principal means of storing contacts and one's personal social network. It can record short memos to help in micro-management of work or personal affairs. Children play games as one would find in London. But it was in Jamaica that one could find people hand-crafting hardcore pornography, pixel by pixel, that could be sent by phone, though the results impressed more through their ingenuity than their realism.\n\nMany Jamaicans travel with their mobiles in the palm of their hand. Some exploit it as fashion, changing the fascia to match their clothes. Many women much prefer little pink clamshell phones. They might also do this in London, but there is no one in my neighbourhood similar to that shopkeeper in Kingston who routinely stores her phone within her ample cleavage. Men too, as one put it, may want 'the wickedest phone, the blingest phone'. Schoolchildren can match style so that they can 'look girlfriend and boyfriend through their phones'.\n\nThen there are ringtones which effectively dress the phone. Religious Christians, for example, often use common hymns and religious music to make their phones more appropriate. My favourite was an aspiring preacher who, without a trace of irony, included the tune 'Oh God you are the only one, that's why I am holding on so long'. The facility to assign ringtones to particular callers also appeals to those whose phone usage is dominated by keeping different relationships apart. Some even try to match the tone to what they see as the character of the caller, such as a 'sour' tune for a persistent ex-boyfriend. Jamaicans were equally adept at the performative potential for speaking on the phone. A young man on a crowded bus sitting with a female companion on either side of him receives a call from yet another woman. He talks as loudly as possible, drawing attention to his three women and celebrating the increasing annoyance and jealousy of one of the women sitting next to him who is poking him in the ribs and trying to get him to stop. When a man in a bar receives a call he may decide to stay in the bar because he wants people to hear the conversation, go outside because he wants them not to hear, or go outside because he wants them to think he has something that he does not want them to hear.\n\nSo the problem with the idea of inherent capacity is that usually we don't know what this is until it is manifested in usage and meaning. We know of many cases, such as texting, where something was built in as a capacity but the cultural genres it gave birth to were never anticipated by its designers. In the Philippines a relationship is not really a relationship without between 20 and 120 texts a day; this isn't an inherent capacity, but then it's not really a Filipino tradition either. It's something new. Texting has profoundly transformed what it means to be Filipino, because it has become pretty much the dominant practice of waking hours and redefined what relationships are.\n\nSo is there a more sophisticated way to see these ideas of usage and capacities and technologies?3 Anthropology can contribute its own understanding of what we mean by technology, if we are prepared to see mobile phones on a par with the way Trobriand Islanders use spells to grow crops in their gardens. Based on his reflections on a book by Malinowski called Coral Gardens and their Magic,4 Alfred Gell wrote an influential article entitled 'The Technology of Enchantment and the Enchantment of Technology' that was a precursor to his wonderful book Art and Agency.5 It's an approach that is particularly well suited to topics such as the Internet and phone. He noted that if you are interested in the technology of, say, an animal trap, this has to be designed around the shape, but also the behaviour, of the particular animal you have in mind to trap. For example, if you want to trap me, I recommend chocolate.\n\nThe Internet is best seen not as technology but as a platform which enables people to create technologies, and these in turn are designed for particular functions. So what people weave from the fibres of the Internet are the traps they use to catch particular kinds of passing surfers. They require a design that draws interest, attention and appreciation and thereby seduces its particular victims. When I first started looking at Internet websites I noticed that there was one genre that was particularly crude and ugly. These were the websites of Trinidadian wholesalers. Yet it is in this very ugliness that we can see the workings of the technology. When websites were first being developed in Trinidad, one problem was that something which goes on the web could potentially be seen by anyone. But if you are a wholesaler, getting the attention of individual consumers is going to be a huge waste of time. If possible it is much better to put them off. By contrast, the lack of styling sends a message they want to get across to retailers, who are their audiences, that money is not wasted on such fripperies. Rather this will be the cheapest source of the goods in question. The trap is designed to catch and hold fast those retailer animals, and not any other.\n\nAt the opposite end of the spectrum are a set of websites that are positively baroque in their ornamentation. As soon as you go on the site you are assailed with the sound of MP3s, as music is one of the most important aspects of their decoration. But then there is a blaze of colour, not still, but shifting as things pop up, open out, invite, surprise and joke around with the viewer. Somewhere beneath all this may lie a photograph or portrait that reveals an otherwise possibly quite shy schoolgirl or schoolboy, because these are the websites that circulated, prior to social networking sites, as the personal websites of adolescents. In this case they were in fierce competition, not to sell something, but to demonstrate, in this vicarious fashion, how attractive the individual was and how popular they could be amongst their peers. So the key function of the site was a section at the base that asked anyone visiting the site to sign on at the end and make a comment. This created a visual demonstration of just how many people had been attracted enough to post something on their site, as against horrible Charlene, who occupied the next desk and would never, and should never, be quite as popular, even on the Internet. Today this has been taken over by social networking. You just have to have more friends on Facebook than your mates. You also have to have a photo taken of you clutching your handbag, mascara running, asleep in the corridor of your student hostel, when you were too drunk to make it back to your room. Because otherwise you simply haven't been to Uni.\n\nSo we have quickly moved from thinking of technologies of communication as merely things, or capacities, and started to see them more as analogous with the arts of seduction: ways of making ourselves appear attractive to the person we are communicating with. Obviously seduction is only one of many things going on here. The larger point is that communication technologies are essentially cultural genres and we can best appreciate them in much the same way as other cultural genres. This brings us back to our consistent theme, that a material culture approach is defined by this dialectic of mutual creation. We cannot simply ask, as most people might enquire, 'How do Trinidadians use the Internet or how do Jamaicans use the phone?' because such phrasing implies that one end of this equation, the people, is a fixed entity. A Trinidadian who uses the Internet is at least, in some small measure, a different kind of entity than a Trinidadian who doesn't. This may even amount to a change in what we mean more generically when we use the general term Trinidadian. A cultural genre is something that we can appreciate makes Trinidadians what they are just as much as making the Trinidadian Internet what it is. In the next section we will explore another unexpected analogy in order to appreciate what we might mean by this phrase cultural genre. When the Internet first appeared, but had developed sufficiently for us to at least have a go at trying to work out what kind of animal we were dealing with, I worked with Don Slater on a wide range of genres of usage to see which would best help us comprehend it. Surprisingly we found that the most effective way of appreciating what the Internet had already become was through our study of Internet use in Trinidadian religion.6\n\nMaking Trinidadians on a Wing and a Prayer\n\nAs a means to summarize our findings on how the Internet had changed people's lives in Trinidad, we proposed four main modes of transformation. When a new medium arrives on the scene, the first and most important impact tends not to be anything which is itself radically new. Rather the tendency is to seize upon it in order to finally realize some desire that was already present, but so far had been frustrated, because people didn't have the means to fulfil it. For example, the Hindu community in Trinidad had seen itself as potentially a significant part of the global Hindu diaspora, but had felt itself to be very much on the periphery of this growing community. The Internet was seized upon as a means to assert a greater presence. Similarly many Christians wished to be part of the diocese of the Cathedral in Port of Spain, the capital of Trinidad, but often they lived too far away, or even abroad. Using the Internet they could become much more actively involved in this specific site. In each of these cases the people concerned had already formulated their desires, but prior to this moment they hadn't had the means to realize that sense of themselves. So finally, thanks to the Internet, I can achieve my previously thwarted goal of becoming \u2013 me. This is usually the first consequence of any new technology viewed as a cultural genre.\n\nIn the second instance we looked at the way new media creates both new freedoms but also the need for new controls. For example, the Catholic Church has a traditional hierarchy, so that the word of the Pope filtered down to the population through various intermediary positions such as bishops and priests. With the arrival of the Internet a lay individual gains direct and instant access to the centre without these intermediaries. Similarly young Muslim women could challenge the authority of a local imam over, for example, what they were and were not allowed to do as part of their wedding, by looking at discussions posted on the web. So traditional structures and hierarchies were under challenge. But at the same time few of those who formed such religious organizations wanted these to disintegrate. So a Pentecostal site first welcomed the ability of anyone to post ideas and commentary on their site, but later insisted these were filtered through the 'elders'.\n\nThe classic tale of this contradiction between freedom and control has become an essay called 'A Rape in Cyberspace' by Dibbell.7 He tells of how the Internet was first seen by Californian users as a welcome liberation from all authority. It represented a place where people could choose to become whoever they wished to be, often in the form of avatars, that is, online aspects of themselves, an idea that has since become more familiar with the rise of Second Life. But then the users studied by Dibbell found that a rogue member had worked out how to take control of other people's avatars and was making them do 'unspeakable' things to each other. Suddenly all these newly liberated Californians were casting about desperately for some version of the Internet police. This, then, is the second consequence of a new technology. We explore new things we can do, experienced often as new freedoms, but that also induces anxieties that we retain some control over how these freedoms and capacities are employed.\n\nIn our third model of Internet use, we examined the way the media is seen to mediate. For example, a group of Catholic Charismatics, situated on different islands, had developed a form of 'spiritual journeying' in which they helped each other deepen their exploration of their faith. As one woman noted, the Internet helped hugely. First, there was the advantage for women generally in that they felt liberated from having to concern themselves with their physical appearance. But more profound was the timescale of the Internet. Previously they had used letters but, by the time a reply had been received to some discussion, people had usually moved on from the position they were exploring when they first sent the letter. By contrast, in face to face dialogue people feel pressured to respond immediately, without having sufficient time to reflect upon what were experienced as very deep matters. The Internet seemed to allow, for the first time, an appropriate timescale. One could receive a message, ponder and reflect until a response had been properly formulated and then send this within a timeframe that still represented an active dialogue.\n\nReligion also allowed people to consider explicitly the materiality of the media as mediation. For example, these same Catholics created a lively debate as to whether the Internet could be used as a medium of confession. Some reasoned that, even within the church, there was a screen between confessor and penitent, so that this seemed to suggest that the Internet had a clear precedent and was appropriate. However, others reasoned that the priest still had to be present to give absolution, so that one could not employ the Internet for this process. So the very specific quality of this medium, its various forms of materiality, were of considerable significance to people who constantly debated these same issues as an integral part of their religious commitment. In this third instance we acknowledge the specific materiality of the medium and the impact that has on the communication it mediates.\n\nThe first model suggests that the initial use of the Internet is based on the fulfilment of prior desires. Clearly, however, the Internet can also create new, unprecedented imaginations, thoughts that had been barely conceivable prior to this technological possibility. This was clearest in the case of an Apostolic Church, partly because it fitted with the foundational theology of this church. When faced with the Internet, this church had one basic question. Why had God created the Internet at this time? Because for them, nothing that happened in the world was without divine significance. There was always God's purpose to be discerned. Mostly they reasoned that the church was designed to spread 'the word' and that all new developments were intended to facilitate that command. So previously, having seen how successful capitalism appeared to be in the world, they had tried to use that as their model for how their church should appear, as a kind of entrepreneurship. Now with the appearance of the Internet they believed that God had invented this thing at this time, basically for them. The only question was what exactly God was thinking of in choosing to do this right now. So, within a short time, they restyled their church The World Breakthrough Network. They largely abandoned traditional church services based on face to face community, and developed as an Internet-based church with Internet services. The most important aspect of the web was the way it facilitated international and global connectivity, and this was seen by the church as a sure sign that in a short while their message would become a global message. Thanks to the Internet they would thereby finally fulfil their purpose to establish God's plan for the whole planet.\n\nThese four processes illustrate what people can become through the Internet, or, in the terms of this volume, the nature of the Internet as a form of objectification. At first they strive largely to become the selves they hadn't previously been able to achieve. Later, they gain a new imagination of themselves as people whom previously they were not aware they even could become. Each of these processes comes with its own contradictions, most fully noted in those between freedom and constraint, but present in all cases. So the idea that objectification creates both its subject and its object here looks more concrete. It's not that Trinidadians use the Internet, or that the Internet creates Trinidadians. It's more that the Trinidadian Internet is something distinct from all other Internets and makes the Trinidadian who uses the Internet something beyond previous forms of being Trinidadian. There are new relationships, new aspirations and new idioms as well as new technologies.\n\nTo talk in terms of a new kind of Trinidadian cannot be reduced to the transformation of an individual Trinidadian. It implicates the very meaning of the word Trinidadian. Sometimes the capacity of the media actually to become the form by which people reflect back on who they are becomes quite explicit. To appreciate this, we can turn from this concern with new media to study a rather older medium, television.8 During the time of my earliest research in Trinidad, by far the most popular television programme was an imported US soap opera called The Young and the Restless. There are many soap operas shown in Trinidad, and when The Young and the Restless was introduced as a daily lunch-hour soap, it was not expected to have the same importance as established serials such as Dynasty and Dallas. But Trinidadians showed considerable ingenuity in arranging to see the show. There was a big trade in imported miniature televisions which can be set up in shops and offices. Those living in a squatting community without water or electricity would have a car battery recharged weekly to power the television in their home. I couldn't help studying the soap, since I found that for an hour a day no one would speak to me. I was reduced to studying them watching it. Friends who worked as seamstresses had to watch the show to find out what new clothing their clients were about to order.\n\nOne of the most intriguing findings was the way the soap opera was referred to and used in daily conversation. Anthropologists will listen out for wise old adages and sayings that seem to capture a certain sense of cultural expectation. But it was at those times when you would anticipate some such traditional saying that typically one would hear comments about the soap such as:\n\n'The same thing you see on the show will happen here, you see the wife blackmailing the husband or the other way around, I was telling my sister-in-law, Liana in the picture, just like some bacchanal woman.'\n\n'I believe marriage should be 50\u201350 not 30\u201370. The woman have to be strong, she have to believe in her vows no matter what... that make me remember The Young and the Restless, Nicky want her marriage to work but Victor is in love with somebody else, but she still holding on.'\n\n[as in a then current story] 'You always go back to the first person you loved, in my own family my elder sister went with a Muslim boy, and so was married off by parents to a Hindu man, but she left her husband, gone back to the first man and had a child by him.'\n\nIn Trinidad contemporary commentary is dominated by the songs that calypsonians bring out for Carnival, and, true to form in that year, a singer called Contender gave the reasons for the soap opera's success in the chorus of a calypso with the same title, whose content was largely a summary of the plot. The chorus alternated:\n\n'You talk of commess\n\ncheck the young and restless,'\n\nwith the words,\n\n'They like the bacchanal\n\nthey like the confusion.'\n\nIf one asks Trinidadians to summarize their country in one word, the likelihood is the response will be bacchanal. The term has a specific meaning in Trinidad, although, as elsewhere, it evokes the values celebrated in Carnival; that is, the general heightening of excitement, disorder and expressive sexuality. It is the same spirit that motivates the Jouvert ritual that was discussed in relation to Trinidad style in chapter 1. The idea is that people reveal themselves in the first light as the 'truth' by their satirical exposures of the pretensions of established order. The point previously established against our concept of superficiality was that truth should be on the surface where people can see it. The first synonym for bacchanal is clearly 'scandal', with lyrics such as 'Bacchanal Woman, sweet scandal where she walks', or 'We people like scandal. We people like bacchanal.' Scandal again implies the bringing into light of that which others want to remain hidden. Bacchanal also connotes confusion, disorder and wildness. The two ideas are linked by the term 'commess', which also means extreme confusion, but with connotations of it being caused by scandal.\n\nThe popularity of the soap opera makes sense in this context. Trinidad is an extraordinarily dynamic society. The oil boom gave a tremendous impetus to the growth of the middle class, to the extent that they emerged at its peak as dominant both numerically and culturally. With the recession, however, many of the more fragile pretensions of the nouveau element within this class were becoming exposed. There was a continual discourse about the financial plight that exists behind the closed doors of the home, only brought to light by events such as the cutting off of the phone through unpaid bills. Even in the suburban community, there were frequent rumours about how many properties were back in the hands of the banks or deserted by migrants to Canada. So this soap opera with its theme of scandalous revelations was not lending reassurance. The attraction of the programme was that it forced its point into the key fissure which manifested the basic contradictions of Trinidadian culture, at a time when this was especially sensitive.\n\nThis is precisely why Trinidadian television could not produce a programme of this kind. Constantly watched over, as a local product, it had to produce the respectable and serious discussions of things like problems of ministerial planning. It was left to the imported programmes to deal with these highly sensitive and difficult problems as faced in daily life. The soap opera may not have looked like Trinidad. It was made in the United States by people who had quite possibly never heard of Trinidad. But in its consumption it became more genuinely Trinidadian than anything which was locally made. Most writings on media studies tend to condemn such serials as a kind of cultural imperialism. Many Trinidadians expressed much the same view. But in this case, if one wanted to clearly establish what was generally seen by Trinidadians themselves as the essential character of being Trinidadian and addressed the most difficult local issues, the best source was this US soap opera. This goes beyond the point being made in terms of Trinidadian use of the Internet. There we saw how use of the Internet created new cultural genres that could be seen as characteristically Trinidad. But with the soap opera, we see television becoming the main medium by which the very definition of being a Trinidadian, the term bacchanal, becomes routinely illustrated. This may not be common, but the example shows it is certainly possible.\n\nWhat is a Mobile Phone Relationship?9\n\nRather more common than this act of collective self-definition is the gradual and rather more subtle development of new forms of relationship, especially in terms of media of communication, since after all they are largely an instrument of relationships. But at least some of the same issues arise. There will always be a dialectical process such that people are simultaneously creating a relationship with each other and with the media. This issue is clarified if we try and imagine what kind of relationship could be usefully termed a 'mobile phone relationship'. For example, a brother and sister living in London may develop a multifaceted relationship over fourteen years using many different media, and adopt mobile phones when they become cheap enough. This results in some minor changes, such as the micro-management of meetings in town, but these are generally insignificant when taken in the context of their relationship as a whole.\n\nBy contrast, let's hang out on a street in Manila. The Philippines established itself some years ago as the world's centre for prolific texting. Thanks to anthropological work by Raul Pertierra and others,10 I can imagine a young woman just getting into this medium. She likes the particular way it lends itself to short, ambiguous and suggestive, as well as creative, communication. At this stage she starts a texting relationship with a man she happened to meet when he overlooked her writing a text on a crowded bus. To be honest she had no real interest at all in the man at first. She simply wanted an excuse to practise and develop this medium of texting. But one of the treats of texting was flirting, and after a while she began to enjoy his facility with flirtatious texts, and realized this was a man she had quite a bit in common with. They did meet from time to time, there was even a brief sexual relationship between them, but that didn't work out so well, and soon they resumed what basically brought them and kept them together \u2013 the joys of text.\n\nA third example comes from our fieldwork on the mobile phone in Jamaica. A staunch Pentecostal woman loves bringing the word of Jesus to all and sundry. She also enjoys what Jamaicans call counselling, based on her conviction that she is inspired from above. A chance miscall introduces her to a teenager, whose hesitant phone manner betrayed a fragility she seizes upon to initiate a long conversation about this teenager's problems including her child's babyfather who had betrayed her. This depth of confession and counselling might have been difficult if they had met face to face. But the anonymity and chance of this initial encounter led to a sustained exchange, even though the cost of the calls had an impact on both their budgets.\n\nWhich of these three cases is a mobile phone relationship? Not the first, which just adds a rather inconsequential technology, of major interest to phone companies, but not much to the participants. The second case is unequivocally a Mobile Phone Relationship, as the primary concern is texting, and the other person is mostly there in order to facilitate the use of the phone. The core relationship is to texting itself. If the second case is a Mobile Phone Relationship, maybe the third is a lower-case mobile phone relationship, in that the phone doesn't constitute, but is instrumental to, the relationship. While it is tempting to concentrate on the extreme case, it is this third ambiguous example that is by far the most common.\n\nIn order to explore this further we also need to consider what we mean by a relationship. This is a topic explored in more detail in the final chapter and will re-emerge in the book Consumed by Doubt, since it is a theory first formulated with respect to shopping. But to summarize, I will argue that in many cases the term relationship refers to an inevitable tension between the general category that a person inhabits and the specifics of that person. In shopping we often buy things that make an actual father more like the category of father as we understand it, or an actual partner more like a proper partner as we had envisaged them.\n\nCurrently, together with Mirca Madianou of the University of Cambridge, I am carrying out a study of long-distance relationships, specifically those of separated families. We are examining the ability of particular media to sustain those relationships over time amongst Philippine and Caribbean families. Prior to this study we already possess an established literature. For example, in her book Children of Global Migration, Parre\u00f1as examines the consequences of contemporary Philippine migration.11 She notes there are approximately 9 million Filipino children under eighteen with at least one parent abroad as migrant labour. This represents 27 per cent of the youth population.\n\nThe situation seems to be one of extraordinary self-sacrifice. In her sample the average mother spent only 23.9 weeks out of the last eleven years with her children. This means that these women more or less missed out entirely on their children growing up. The mothers gave their reasons for taking up this work largely in terms of the children's welfare, such as to support their education, medical bills and augment income more generally. Parre\u00f1as's reasonable expectation was that the rise of regular and cheap communication would help to ameliorate the negative consequences of this separation, and enable the children to feel close again to their mothers and to better understand and appreciate their condition. She also anticipated that they would be influenced by the more egalitarian and modernist forms of gender relations by which the father would take on more domestic roles, in recognition of the mother taking on more of the traditionally male role of the breadwinner.\n\nHowever, most of the detailed description and analysis that is found in Parre\u00f1as's book suggests that these expectations are not fulfilled. The reason for this becomes much clearer in terms of the theory of relationships just described. What tends to happen is that the period of separation simply exacerbates the distinction between the idealized norms represented by mother and child, and their actual relationship, which in these cases has significantly diminished. These norms are held both individually and collectively. At a collective level, Parre\u00f1as shows there is a dismal view taken of the families where mothers have gone to work in other countries and left their children behind. These families tend to be stigmatized. There is a belief that such children, being brought up without their mothers, will behave badly. At an individual level, the greater proximity afforded by the development of mobile phones, if anything, reinforces the most conservative and traditional gender ideals about relationships and leads many of these children to concentrate less on the material benefits that accrue to them and more on their sense of abandonment by their mothers. To appreciate this failure of increased mobile phone use, we have to note that even mothers who return home more frequently are not necessarily regarded as better mothers.12 By contrast, it is perfectly possible for absent fathers to be seen as behaving adequately, in just keeping in touch by phone, unlike mothers, because for fathers this more occasional or distant relationship is closer to the normative expectations of fathers.\n\nWe have barely started our own research on this topic and our conclusions are likely to change. I include this here to make a further point. Now we have embarked on this study we spend much of our time talking both to the mothers who have spent many years in the UK, missing out on the development of their children, and we have now also travelled to Manila to talk to children who largely grew up without the presence of their mothers. On the one hand this is a neat academic exercise. If you want to know the consequences of new media for relationships, then such an extreme case is right on the button. But to merely exploit these mothers and children to improve academic knowledge would be pretty heartless. Nearly every week I come home with stories circulating around my head that as a parent I find appalling. These mothers and children don't just talk, they convey, they conjure, they force one to acknowledge, to try and imagine, to empathize, and finally to realize how distant one remains because privilege and good fortune have meant that one listens but does not directly share their experience. Yes, I sincerely believe this work will help us understand how relationships are affected by the media. But much more than this, it tells us why we really do need to understand such things.\n\nIn fact the reasoning that led to this study was not that this was a neat academic example. It was actually sitting down and looking at the literature on migration and Diaspora, and realizing that perhaps what academia had neglected was the one thing that mattered most to the migrants, the core relationships that were sundered by becoming a migrant. There is a vast literature on labour relationships, citizenship, identity and other such subjects. But we felt that if one actually asked the migrants themselves what mattered most to them in terms of the impact of becoming a migrant, they are more likely to respond with a very different topic: their core relationships of love and obligation, exemplified by mothers separated from children. People tend to care rather more about the people they love than the definition of who exactly they are. Yet relatively few studies focus explicitly upon this dimension of migration as against fashionable topics such as identity.\n\nMedia, Poverty and Welfare\n\nIn an edited collection of the work by my PhD students called Material Cultures,13 I argued that the study of material culture needed to move away from an overwhelming emphasis upon meaning, to a greater consideration of what matters. While some academics seem almost to prefer a clear separation between academic studies devoted to theoretical agendas, and applied academic work concerned with issues such as policy and development, I see this as exactly the kind of dualism that an extremist anthropology should seek to dissolve. I can think of no better testing ground for theory than the heat and heart of people's lives in the extremes of what we abstractly call development, but which signifies a maelstrom of transformation, struggle and aspiration. Nor is the justification for theory as a contribution to intellectual understanding in any way diminished by regarding it as a contribution to understanding and assisting in people's welfare.\n\nStuff matters. Its presence or absence is quite often the very definition of what people experience as poverty. It is the main reason these Filipina mothers give for working away from their children. So the final part of this chapter focuses on a study of the consequence of stuff that was motivated by these applied concerns with the welfare of low-income populations.14 The study of mobile phones in Jamaica, carried out with Heather Horst, was funded by DFID, the British Department for International Development. The concern was to establish the impact of new media on poverty and whether this should be a priority for aid policy in the future. Unusually, since most applied topics are more short-term, the grant was made to anthropologists who were committed to carrying out a traditional ethnography. Horst spent the entire year in Jamaica, having previously worked in Jamaica on several other projects over the course of ten years. She lived six months with a family in a rural site near the centre of the island and spent a further six months living with a family in Portmore, an outer district of the capital, Kingston. Both are low-income areas with high levels of unemployment. I lived with those same families for part of the year.\n\nThe question that DFID posed at the time remains hugely important today. The mobile phone has spread at such an astonishing pace. But 2004 was probably the first opportunity to make even an initial assessment of its impact on the lowest-income groups, because it was only by then that it had spread down to that lowest-income population, both in Jamaica, and many other parts of the world. It is entirely possible that the mobile phone is a new gimmick that represents a considerable cost and ultimately a drain on these very low incomes. Alternatively, it might represent a considerable boon and genuinely new way for people to improve upon those incomes. One problem, however, is already made evident by posing the question in these terms: is income itself the best measure of people's wellbeing? One of the arguments that anthropologists, such as Mary Douglas,15 had been waging upon those who develop measures of human welfare was with regard to the evaluation of communication in its own right. Under the influence of economists such as Amartya Sen the United Nations had been opening up its index of national wellbeing beyond the single criterion of income, to include factors such as education and health. Yet so far communication remained understood largely as a means to such ends rather than as an end in itself. Douglas, by contrast, argued that communication is such a basic human concern that it needed to be valued in its own terms. This may be particularly true for the Caribbean. Earlier anthropological studies of the region, with titles such as The Man of Words in the West Indies,16 had examined the place of performative conversation in society and showed in considerable detail the importance of the imperative to communicate. For example, in Trinidad, if you are a man standing on a street corner and a woman passes by, it is incumbent upon you to call out something to her \u2013 you just have to. So if media, as a genre of stuff, is to be made relevant to an assessment of people's welfare, it cannot be reduced to income. But to be taken seriously by economists, then its relationship to income must also be considered. So I will start with income and work outwards.\n\nAt the time of this study Jamaica was unusual in the Caribbean in terms of mobile phone impact. Unlike Trinidad, where our previous research had concentrated on the Internet, there was almost no Internet use amongst low-income populations in Jamaica. Yet by contrast there were more than three times the number of mobile phone subscribers than in Trinidad, an average of three phones per household. Many factors explain Jamaicans' preference for mobile phones over the Internet. The Jamaican government had been faster in responding to the World Bank call for liberalization of the telecoms sector. One result had been the arrival of a dynamic Irish media company, Digicel. Starting from scratch in 2001, by the end of 2004 Digicel had sold nearly 1.5 million mobile phones in Jamaica. So it might appear that the relative success of each new medium was mainly a result of actions by the state and commerce. However, our research was to reveal another story, which had to do more with people's usage of media. But first we had to tackle the agenda set by DFID.\n\nThe general assumption held by governments and economists at the time was that if the mobile phone was to be a major force for helping low-income peoples, then this would come through two main factors. The first would be in encouraging individuals to become entrepreneurs and the second in helping people to find employment. The desired outcome would be evident in a general increase in GDP. The Economist magazine, in particular, has tended to argue that mobile phones might deliver on these aims more effectively than the Internet, because they are relatively cheap and require fewer skills. With a mobile phone these nascent entrepreneurs would no longer need the capital required for having an office or fixed base, and could remain in contact with potential suppliers or customers at any time.\n\nThrough ethnographic observation we were able to reach conclusions which differed markedly from such assumptions and revealed a much more complex relationship between income and the spread of the phone. This is not to doubt that The Economist's logic works very well for some other countries, and our evidence is specific to Jamaica. But in Jamaica the use of mobile phones to either assist in becoming entrepreneurs or to obtain work was decidedly rare. There was no major new spirit of enterprise based on the phone, with two exceptions. One was the music business. Almost half the men we knew under the age of forty did see some new potential in playing music, hiring bands, making CDs and somehow making it 'big' in secular or religious music. Not that this often happened, but the aspiration was certainly there. Another exception was the taxi-driving business, which is so integrally linked to the phone as to form a vital part of what can be called their communicative ecology. That is to say, one type of communication, that of transport, is intimately linked with another. In our rural site you could request a taxi in which you were already a passenger to detour up the hills to what were often remote locations. But before the phone there was no way to summon a taxi in the first place. This matters a great deal; for example, in relation to the impact on health care. The phone was used less than we expected to summon doctors or nurses, but the link to the taxi meant that one could in emergency now summon transport to get you to a hospital.\n\nThese are exceptional cases. In general there was surprisingly little evidence for entrepreneurial use. Similarly in relation to obtaining employment. The general consensus was that the vast majority of jobs were accessed through social connections and patronage, and not through simply matching opportunity to qualifications. So if we had remained focused on income generation, then we would have concluded that the impact of mobile phones was disappointingly slight. But step outside that frame and the phone was found to have a major and positive impact on the struggle for survival by the very lowest income groups in Jamaica, fully justifying the way they prioritized the phone, even when living in conditions of destitution.\n\nAgain we can start with income, or the lack of it. We made very detailed investigations of the budgets of low-income families in both rural and urban settings. We went back day after day until we found we could actually match our data for household income with our data for household consumption, something certainly not true of official statistics about budgets. The result shocked us. So many people had effectively no income from either earnings or selling things. The only way they survived was from money that came from others; from those that have it. In the urban area (Kingston) 70 per cent of these low-income households received money from other family members, boyfriends and partners, babyfathers and friends in Jamaica and abroad. Of these households 36 per cent survived exclusively through social networks and the patronage of others.\n\nThe bottom line is that over a third of households have no income at all in the sense of anything they earn. They exist only through their ability to obtain money from others. The difference between being destitute and not being destitute is having friends or family that you can call upon in a time of need, even if it is US$1 for transport. The mobile phone is the ideal tool for a Jamaican trying to create and maintain these wider social networks which ultimately seem more reliable than a company, employer or even a parent or spouse alone. As many studies of poverty have shown, the problem is that people in such conditions are in an almost constant state of crisis posed by every new demand for funds, whether this is a school uniform for a child or filling a prescription given by a doctor. There is simply no surplus of funds available for additional or unanticipated expenditures. A great many of the short stories that punctuate our subsequent monograph on this research are demonstrations of the effectiveness of the phone in resolving these crises.\n\nSo, in response to the initial enquiry as conceived by economists, we reach a slightly perverse conclusion. Yes, the mobile phone is highly effective for ameliorating the worst forms of suffering associated with poverty. But most economists would associate the alleviation of poverty with a rise in GDP, promoted by increased entrepreneurialism. A critical point for the development of such entrepreneurialism is the accumulation of capital, at least sufficient to make a starting investment. The trouble is that the mobile phone seems to reinforce the tendency to distribute income within low-income areas from those who, at any given time, have it to those who have not. As a result, there is even less chance of an individual accumulating the kind of capital that allows them to start working as an entrepreneur. One effect of the mobile phone is to make entrepreneurship less feasible by dispersing capital downwards. A means for alleviating poverty amongst the lowest-income groups is not necessarily something that creates wealth for the country as a whole.\n\nAs anthropologists, we wanted to evaluate the impact of the phone on welfare that meant more than just economic development. Horst and myself found a local term link-up best described Jamaicans' most characteristic usage of the phone. Link-up comprises a very large number of very short conversations, in order to keep in occasional touch with the maximum number of people. The average length of a Jamaican mobile phone call was 19 seconds. Earlier anthropological research has demonstrated that Jamaicans knew of sometimes hundreds of people they would call relatives, though they might not make use of any particular connection until the need arose. I don't know anyone in the UK who could name hundreds of relatives. But the earlier anthropologists assumed this was just a feature of kinship. Halfway through our fieldwork we had one of those eureka moments. Instead of just asking people about their social networks, we could literally scroll through their phones and find everyone they communicated with (actually we discovered that wives were now routinely doing this to their husbands' phones). Through our analysis of the content of mobile phone address books, we found this emphasis on very extensive networks was true of all relationships, since only 15 per cent of the names in these address books were of kin, contrary to the expectations of anthropologists.\n\nIf you wish to maintain such a vast network of relationships, then short occasional calls just to keep in touch makes a good deal of sense. Mobile phone conversations in Jamaica have been pretty much stripped of the normal conventions and etiquette of conversation. There is no polite greeting, and there is often no ending at all; maybe just a word such as alright, or later. Both sides recognize the importance of keeping the call to the least cost, but see such calls as a kind of minimal unit of sociality. By minimum sociality I mean the least contact you need if you want to retain an option of making more substantial contact should the occasion arise. Think of the way people in the UK send Christmas cards to families they meet in other countries, sort of keeping things simmering on the lowest light. This may be their only communication for years. Unless they happen to visit the country, in which case these suddenly become dearest friends who spend time with each other. But something only possible if at least you send the card.\n\nAbout 98 per cent of calls in Jamaica are made on the basis of pre-paid cards. The transfer of money to the have nots can start even at this level. When someone needs money they will phone around to see who has some at that time. But even this costs more than they have. So they may start by begging a phonecard. A friend may not be able to afford to help you with your kids' education, but they may not begrudge the cost of a phonecard to help you find someone who can. The phone companies had soon come to see the importance of just keeping in touch. Digicel introduced a new facility called callme. This allowed a person with less than JA $3 (about UK 3p) on their phone to contact up to twenty-one other persons a week in the hope that the person they were calling would agree to accept the cost of the call on their credit. Within a few short months, callme accounted for over 80 per cent of all text messaging.\n\nIt isn't hard to explain why people ask for a phonecard or use callme. Typically anthropologists would predict some kind of generalized reciprocity lies behind this, but we found this played only a minor role. True, an individual who becomes known for always using callme, even when they could afford to pay for the call, is soon reviled for this abuse. But, in general, people who pass on credit and money to the lowest-income individuals have no real expectation that one day they will be repaid for this, or gain something in return. These are not just kin so it is not some very long-term sense of a wider collective interest either. Yet callme would probably not have lasted very long if it was rarely, if ever, successful. In fact it was in constant use.\n\nWe concluded that the reason for giving a small amount of money, or credit, is really the same reason for making a small and cheap phone call. It maintains and extends this very extensive network. Again this sets us against the economists, a perspective that tends to reduce phenomena such as extensive networking to some kind of economic need. Instead we prefer to return to our earlier point about the value of communication. Yes, this is used to help people in poverty, to get in touch with taxis, to put pressure on relatives abroad to send remittances \u2013 which is now one of the main sources of income. But these extensive networks existed prior to taxis and remittances, and seem to develop irrespectively of such functions. They flourish even when there is nothing at stake except another address to put in one's phone. While networking is vital in understanding coping strategies, the implications of cause and effect could also be reversed. People give and take not because they need to do so but also in order to facilitate connectedness. They give money in order to create the connections, as much as they develop connections in order to obtain the money.\n\nWe started by looking at the economic benefits of the mobile phone, but by the end we are coming to see that economic transactions may themselves be a means to facilitate communication, which now finally emerges as an end in itself. We may not go so far as Douglas and Ney who assert 'a social being has one prime need \u2013 to communicate'.17 But in the end we have a conclusion that justifies the inclusion of communication as a kind of stuff. The media is not merely a means to another end. People value communication in its own right as a thing they either can possess or feel bereft of. Many other sections in our book The Cell Phone are concerned with this wider context to communication.18 One of the chapters is devoted to the Jamaican concept of pressure, familiar from the lyrics of Bob Marley. Pressure also is often linked to a feeling of loneliness and in turn to the need for communication as an end in its own right.\n\nA problem with anthropology is that we have a tendency to end up with a story that is largely just a 'spoiler' to economics. Predictably the anthropologist informs you after a lengthy study that the whole situation is much more complex than you previously thought. Suddenly you have to deal with this thing called culture, which seems to defy economic models. Or they may tell you that if only you spend much more money on some infrastructure or concern of that population, the people would be better off. It is much easier to just be a critical anthropologist, showing all the ways development doesn't work, without taking responsibility for making feasible and costed proposals which can be subject to other people's criticisms.\n\nI was well aware of this tradition of critical anthropology when I started our project. My hope was that we could do something quite different. We tried to make proposals which would lead to the already heavily indebted Jamaican government spending less money, rather than more. For example, showing why a huge investment to develop community computers using a large loan of aid money was not the best use of that money, since it depended upon a concept of community that was not applicable to Jamaica. Similarly there were vast sums going into IT facilities for schools, most of which were failing; that is, both the IT and the schools. Our alternatives included putting IT money into the much more motivated sector of adult education, when people who had experience of unemployment became highly committed to training. Similarly we could see scope for using mobile phones to gain access to international standards of educational material that would be more effective than the large sums going into localizing content which were not well respected. Instead we needed to develop kitemarked educational materials under the auspices of a body such as UNESCO, since, without kitemarking, people used Google and ended up with a mess of supposedly educational material some of which was useful, much of which was crap. They desperately needed help in sorting the wheat from the chaff.\n\nSo we were quite prepared to come forth with proposals that were the opposite of those associated with mere critique. Proposals that cost less and looked feasible. I really do believe that material culture has a potential role in policy and in actually improving the welfare of populations. But that is not to say that I can claim that anything I have done or proposed actually has been useful to date. As far as I could tell no one either in government in Jamaica or DFID (who funded our work) took a blind bit of notice of any of our findings. My experience has been that people working in development listen to other people working in development, people working in bureaucracy listen to other people in bureaucracy, people in social anthropology ignore other people in material culture, and so on and so forth. No doubt this is a vice I also share.\n\nTo date I know of only one single instance in which my work has been directly applied to create something in the world. Many years ago I was giving a paper in Austria, and someone came to see me afterwards and said they were delighted to meet the person who was responsible for Big Foot. I hadn't the faintest idea what they were talking about. It turns out that someone in Austria had read Material Culture and Mass Consumption within which they noted a paper by Alison James that I had summarized, discussing children's sweets.19 On the basis of the logic of that argument they had devised a new ice cream in the shape of a large foot. Which at that time was selling quite well. So, much as I would like one day to concretely assist, for example, in the alleviation of poverty in the Caribbean, the only direct claim I can make so far to the practical advancement of the world is an Austrian ice cream. And even that was based on someone else's paper, that I had merely summarized in my work. Worst of all I forgot to go out and try a Big Foot, it might have been yummy.\n\nFortunately, though, academics don't depend entirely on direct impact. As one follows the careers of students occupied in a range of sectors from academia and commerce to planning and design, it is evident that material culture studies has become well established, and that its educational message has diffused quite widely. The results have included a more general interest in ethnography and qualitative studies in a range of fields concerned with the material world. But I would like to believe that the paramount influence has been our contribution to a significant change, evident in many professions over the last two or three decades: that groups such as designers, architects, media specialists and commercial bodies, once motivated more by prizes awarded for the quality of their products, have increasingly realized that they need to focus on something quite different. Ultimately what really matters are the consequences of those products for people.\n\nNotes\n\n1 The following section is taken from D. Miller and D. Slater, The Internet: An Ethnographic Approach (Oxford: Berg, 2000).\n\n2 The following section is taken from H. Horst and D. Miller, The Cell Phone: An Anthology of Communication (Oxford: Berg, 2006).\n\n3 The following is taken from 'The Fame of Trinis: Websites as Traps', Journal of Material Culture, 5 (2000), pp. 5\u201324.\n\n4 M. Malinowski, Coral Gardens and their Magic (Bloomington: Indiana University Press, 1935).\n\n5 A. Gell, 'The Technology of Enchantment and the Enchantment of Technology', in J. Coote and A. Sheldon (eds), Anthropology, Art, and Aesthetics (Oxford: Oxford University Press, 1992), pp. 40\u201366.\n\n6 The following section is taken from Miller and Slater, The Internet, ch. 7, pp. 173\u201393.\n\n7 J. Dibbell, 'A Rape in Cyberspace', in J. Dibbell, My Tiny Life (New York: Owl Books, 1998).\n\n8 The following is taken from 'The Young and the Restless in Trinidad: A Case of the Local and the Global in Mass Consumption', in R. Silverstone and E. Hirsch (eds), Consuming Technology (London: Routledge, 1992), pp. 163\u201382.\n\n9 The following section is taken from 'What is a Mobile Phone Relationship?', in E. Alampay (ed.), Living the Information Society in Asia (Singapore: Institute of South East Asian Studies, 2009), pp. 24\u201335.\n\n10 R. Pertierra, E. Ugarte, A. Pingol, J. Hernandez and N. Dacanay, TXT-ING Selves: Cellphones and Philippine Modernity (Manila: De La Salle University Press, 2002); and R. Pertierra, 'Mobile Phones, Identity and Discursive Intimacy', Human Technology, 1 (2005), pp. 23\u201344.\n\n11 R. Parre\u00f1as, Servants of Globalization (Stanford, Calif.: Stanford University Press, 2001); R. Parre\u00f1as, Children of Global Migration: Transnational Families and Gendered Woes (Stanford, Calif.: Stanford University Press, 2005); and R. Parre\u00f1as, 'Long Distance Intimacy: Class, Gender and Intergenerational Relations between Mothers and Children in Filipino Transnational Families', Global Networks, 5\/4 (2005), pp. 317\u201336.\n\n12 Parre\u00f1as, Children of Global Migration, p. 129.\n\n13 Material Cultures (Chicago: University of Chicago Press, 1998).\n\n14 The following section is taken from H. Horst and D. Miller, The Cell Phone: An Anthropology of Communication (Oxford: Berg, 2006).\n\n15 M. Douglas and S. Ney, 'Communication Needs of Social Beings', in M. Douglas, Missing Persons: A Critique of Personhood in Social Sciences (Berkeley: University of California Press, 1998), pp. 46\u201373.\n\n16 R. Abrahams, The Man-of-Words in the West Indies: Performance and the Emergence of Creole Culture (Baltimore: Johns Hopkins University Press, 1983).\n\n17 Douglas and Ney, 'Communication Needs', p. 46.\n\n18 H. A. Horst and D. Miller, The Cell Phone: An Anthropology of Communication (Oxford: Berg, 2006).\n\n19 A. James, 'Confections, Concoctions and Conceptions', in B. Waites, T. Bennett and G. Martin (eds), Popular Culture: Past and Present (London: Croom Helm, 1979), pp. 294\u2013307.\n5\n\nMatter of Life and Death\n\nThe premise for this book has been the observation that things make people just as much as people make things. But this can either sound like a glib sound bite, or remain an abstract theoretical proposition, as presented in chapter 2, unless it is shown more clearly how precisely things do in fact make people. The starting point for giving substance to this perspective is provided by Bourdieu's observations in Outline of a Theory of Practice.1 Individuals grow up to become, with varying degrees of typicality, members of a given society. This happens in most cases, not through formal education, but because they are inculcated into the general habits and dispositions of that society through the way they interact in their everyday practices with the order that is already prefigured in the objects they find around them. I live in a complex network of houses arrayed around streets, which facilitate cars. Cars are presumed not just by the roadways but by a rich paraphernalia of other devoted objects ranging from traffic lights to parking lots. I gain my understanding partly through sheer familiarity and partly through enforced consciousness when it comes to taking my driving test. Alternatively I might live in a society where I come to assume that the health of the crops I grow is dependent upon a host of spirits and ghosts from my ancestors, which I see in all the little shrines, offerings, patterns for planting crops and so forth, set up to ensure that ancestral ghosts are properly propitiated. I may be spared any formal examination but I will learn from priests and parents and from ridicule when I get things wrong. I will know in one case when to stop at a road junction and in the other case when to mutter a spell because I am passing an inauspicious tree.\n\nWe have explored the different ways this proposition may be understood using examples from clothing, housing and the media. We have abstracted this to the highest levels of theory. It seems appropriate in a final chapter to ask, though, just how stuff contributes to the very formation and loss of a person. To see how things are used to define matters of life and death. I guess this point only really became clear to me when I read the work of Linda Layne who had carried out an extensive study of late abortions and stillbirth.2 I had often taught the work of Mauss and the argument about how the exchange of things creates social relations. But I had never really felt the profound meaning of this, in my bones, until reading Layne. Layne showed how parents in the US dealing with late foetal loss and stillbirth insist that at Christmas time a gift is given to the person who would have been, or from the person who would have been. She tells of the trouble parents take to dispose of the layette, the things bought for the envisaged child, as part of mourning for the death of the child. The central fear of these parents is that other people will think that what they have lost was not a human being, a child, but a mere thing. The paradox is that it is primarily through material things that they find the most effective means for insisting upon the humanity of their child; that they were not just a thing. Only real people can give gifts or receive gifts. There is a poignancy to this example that helps make material culture studies more of a mission and less just a conceit. Yes, really, it is usually through the medium of things that we actually make people. In this chapter I want to confirm this point through two main case studies, that apply to people such as myself; people in London. The first will concentrate on birth and the second on death.\n\nBirth3\n\nAs a middle-class, Jewish academic living in North London it is not surprising to note that I have several friends who are involved in psychoanalysis, either professionally, or as clients. I confess that while I embrace the Jewish bit, I am not of the psychoanalytical faith. I find, in comparison to anthropology, that psychoanalysis is on the one hand too universalizing with a strong essentializing tendency in the more vulgar forms, and on the other hand too focused upon individuals as against social relations. It also has an infuriating habit of using the term object-relations to talk about persons entirely de-contextualized from things. The only objects this phrase recognizes seem to be other persons.\n\nStill, it seemed like an interesting experiment to juxtapose material culture perspectives on what produces people such as myself, with these psychoanalytical perspectives. Based on the little reading I have done in the area, mainly papers by Melanie Klein,4 I came up with a joke mainly aimed at my friends working in this field. I suggested that psychoanalysis is really a huge act of projection. The stages of development described by Klein are not really about infants at all. Instead they describe the various stages which a parent goes through in order to develop as a mature parent. This is the result of collective repression in which psychoanalysts all had such problems coming to terms with parenting that they shunted all the key problems onto the infant. I do not pretend to be knowledgeable about psychoanalysis. I have read little and understood less. But based on this joke I embarked on an examination of the use of consumption in bringing up children. I wanted to show that this does help us understand something of how these two categories of the person, infants and mothers, are given shape. A further inspiration was the work of Marilyn Strathern,5 who suggested that in Highland New Guinea it is understood that, in some sense, it is the child who grows the mother.\n\nThe research was a by-product of my study of shopping on a street in North London,6 for which I joined a group composed largely of middle-class mothers who belonged to the NCT. Although the letters NCT stand for National Childbirth Trust, it could more appropriately have been called the Natural Childbirth Trust since at that time its most active members and literature often expressed an almost obsessive concern with the concept of nature where applied to childbirth and childcare. At the most extreme its members were attracted to semi-cultic practices such as re-birthing, where the mother is expected to relive the experience of her own birth (although such practices are formally outside of the NCT). NCT members are encouraged to avoid any kind of assistance in the birth process such as pain relief or sometimes even doctors. These are described as 'interventions'. It was axiomatically preferable for both child and mother to follow a natural birth without these intrusions. The research took place in the early 1990s when these ideals were at their most zealous. Today things have relaxed \u2013 somewhat.\n\nIn avoiding pain relief such mothers become more directly exposed to the extreme pain of childbirth. This may have assisted in the construction of childbirth as a kind of rite, whose subject is not merely the birth of a new infant, but in equal measure the birth of a new form of adult \u2013 the mother. Childbirth is after all the most literal of the rites of passage, rituals which are commonly associated with pain.7 I observed that through natural childbirth these women are engaged in an act of recycling themselves to return to the world as natural.\n\nAccording to Klein,8 the starting point for infant development is the Paranoid-Schizoid position during which the infant is profoundly incapable of reconciling its experience of the good and bad breast. These represent two entirely opposed senses of the mother as the source of all positive and all negative feelings. I would argue that this is an entirely apt description of the perspectives, not of the infants, but of the newly born mothers. The goodness of the infant is intimately associated with its complete dependency upon the mother. It is regarded as helpless: the pure product of a natural birth. One of the key gestures of such births is that the newborn baby is placed immediately at the breast, preferably still covered in blood and other natural substances, that at all other times are considered dirty, but here emphasize continuity with her natural feeding of the foetus. The stress is on the infant as the biological extension of the mother. Furthermore, the NCT probably places almost as much emphasis on natural breast-feeding as on natural birth. All assistance is given to the mother who is prepared to breast-feed, and who avoids substitutes for as long as possible.\n\nAt this stage then, the mother has given birth to an infant who is perceived as largely an extension of her own biology, the main difference being its absolute purity and innocence. It represents a narcissistic version of the mother as refined and purified of all that she feels is bad about herself. The context is important. Most of these women were influenced by first-wave feminism in the 1970s. This gave them a strong sense of their personal potential and the importance of autonomous development as individuals. As students, their individuality was mainly expressed in the formation of left-of-centre political opinions, but as their incomes rose with employment, the emphasis turned to developing themselves as consumers and into people with taste. This feminism equally impacted upon their relationships, with partners seen as equal participants in growing a household income and dual career.\n\nIn Klein's work, some of the attributes most commonly ascribed to the new infant are rage, jealousy and above all a kind of primitive guilt. I believe these are equally applicable to these same mothers' attitudes to these infants.9 Although all the emphasis has been on the perfection of this ideal and purified version of themselves, at the same time the infant represents the total negation of everything that they have striven for: the quest for freedom from their own parents they developed as teenagers and sustained through feminism. These mothers saw themselves as the first generation able to compete with males. They felt a moral duty to strive to fulfil the new potential that history had given them, and their ambitions for their own careers were usually clear and explicit.\n\nWhat was therefore astonishing is the degree to which these same mothers allowed the infant to trample on all such aspirations. The ideology that made their infant into the good breast was based on its presentation as a pure and natural being. The infant's constant demands were accepted as essential priorities and it was felt that at no point should the mother's own desires prevent them being attended to. The infant was viewed as passing through natural stages with minimal parental 'intervention'. If they woke frequently at night they must grow out of this naturally and not be disciplined by some contrivance. There were many versions of this devotion to the natural. Some saw daily routine as unnatural, so feeding, sleeping and other activities were dictated by the apparent mood of the infant. Others saw routine as in some sense natural, and became obsessed that this should never be varied. It's not altogether clear why allowing a child to cry itself back to sleep is a contrivance to be avoided and bringing the baby back to your bed is, however, natural. But then I am perhaps reflecting my memory of resentment over many disrupted nights.\n\nThe idea of the pure and natural infant was fostered also by the most popular literature on childrearing at that time: books by Penelope Leach, Hugh Jolly and others with their vulgarized 'Winnicotian' emphasis upon the mother's recognition that there are always good reasons for all actions of infants (such as crying). These reinforced the dependency of the mother on the omnipotent infant who, by prolonging breast-feeding as long as possible, again extended its ability to determine her life. The loss of freedom was extreme. Women who before frequently went out at night for entertainment refused to go out even once for more than a year after the infant's birth. Even though they may have had male partners more willing than most British males to participate in childrearing, the emphasis upon the biological and natural relationship between mother and infant in some ways reconstructed them as gendered in just the sense that they had so far struggled to oppose.\n\nThese are just some instances of the way becoming a mother was based upon the systematic negation of the mother's previous self. It seems reasonable then to imagine that the infant is seen simultaneously as good breast and bad breast: that which simultaneously seems to extend and purify her, but also to destroy her autonomy. The mother may be said to have been passing through the Paranoid-Schizoid position in that at such a time it was extremely difficult for her to consider that the source of such benign and such negative emotions can really be the same object. First sacrifice and later guilt arise as mechanisms for preventing this contradiction in coming to the fore, since they allow her to project all the negativity represented by the child back onto herself. Alternatively she may deflect her anger by projecting it onto her partner or her own parents.\n\nAccording to Klein, the 'Paranoid-Schizoid position' develops into the 'Depressive position' (you have to admire her heartwarming terminology) when the infant learns to confront its realization that what previously had been clearly separated into the good and the bad are actually the attributes of the same object. Regarding this as a stage of mother development, this represents the start of a process of separation from an increasingly autonomous infant. At this point stuff starts to play a role, for example toys, or the transitional object (e.g. smelly blanket) that another psychoanalyst, Winnicott, saw as helping the infant separate from its identity with the external world.\n\nIf the infant's relationship to stuff emerges through play, the mother's prior skills with stuff will have developed partly through shopping. Many of these mothers noted that the pleasure they had developed in buying clothes and items for themselves was transferred directly onto the infant. While in other communities mothers are concerned to get back their figures and clothing style lost in pregnancy, these mothers tended to channel their knowledge and ability as consumers into the task of shopping for the baby. In the initial phase clothing the child became an act of pure projection. If the infant had had a say in how she was dressed she might not have picked those faux peasant lace smocks. So when she does develop some sense of agency, she immediately declares war.\n\nThe first battle relates to the substances which the infant is allowed to ingest. There is no problem at first since the infants are entirely breast-fed, and initial foods are usually home-made pulp from vegetables. Soon, however, a villain appears in the form of sugar against which the mother strives to protect her child. Sugar exemplifies a battle against all sorts of additives or substances that are seen as unnatural and therefore polluting to the pure nature of the infant. I saw mothers react to their toddlers reaching for a biscuit as one might respond if the infant were about to stick its fingers into the socket of an electric plug. Inevitably the battle ends in defeat, as sooner or later the infant acquires considerable access to a wide range of biscuits, sweets, chocolates and the dreaded fizzy drinks. Gradually the baby is seen to lose its organic status through the ingestion of artificial substances. The battle is repeated as home-made and healthy foods pushed by the mother are rejected in favour of a diet of fishfingers and baked beans or, if the infant is sufficiently victorious, burgers and pizzas. Parents do not give up without a struggle, within which their concept of biology plays a major role. It is very common for such parents to insist that their infants have an allergy to anything artificial. Infants are said to come out in spots as soon as they ingest any kind of additive or the wrong E-number. If the children do not oblige (with spots) then the parents may claim that these additives cause behavioural problems, such as hyper-activity, which is a harder claim to contest.\n\nThe babies of these mothers were largely de-gendered. Their parents resented, rather than welcomed, cards and gifts in pale pink and blue. If anything their children were green: the embodiments of natural childbirth and rearing. As the infant began to develop its own agency, the parent might expend considerable effort in trying to prevent any association with gendered toys. This included trying to prevent girls from becoming particularly interested in dolls, but the greater effort was usually expended on avoiding guns, swords and other weapons. In conversation parents might tell with pride of their young sons' enjoyment of dolls. But even in this world of imagination it turned out to be pretty hard to fight against guns. Mothers almost invariably told how finally there was no point in preventing their sons having access to toy guns, since he was found to be using every household object from pens to coathangers as gun substitutes, happily shooting adults and siblings.\n\nIn the 'Depressive position' mothers gradually come to terms with the simultaneously good and bad qualities of their infant. But while there is a growing acknowledgement of the separation of the infant, there is still the desire to remain the primary source of all potential gratification for the infant. At first the child remains a narcissistic projection of the better (or idealized) aspect of the mother. As one mother noted:\n\nThe difference is she has got a fantastic little figure and I have put on weight since having kids, and the honest thing is that I don't get so much pleasure, as I need to lose about two stone, that's the reason. Everything I look at I don't like myself in any more, but everything on her looks fantastic, so it is such a pleasure. I enjoy having a little girl for that reason.\n\nThe peasant smock gives way to a more youthful version of the mother \u2013 leggings, perhaps.\n\nIf there was one object that seemed to comprehensively spell defeat it had to be the Barbie doll. The particular significance of Barbie was evident given what has already been said about feminism and the passionate commitment to nature and the natural. The mothers do not object in principle to their infant dressing an anthropomorphic toy. This is seen as a learning and nurturing practice. But their preference would be for a figure that was both reasonably naturalistic and in many cases androgynous, such as to be suitable for both sons and daughters. Barbie, by contrast, is aggressively feminine and seems deliberately invented to anger such mothers. Not only does she represent the pre-feminist image of woman as sexualized bimbo addicted to fashion \u2013 she can't even stand up. This is a crushing defeat for parents who swore that their children would never succumb to such sexual stereotyping. Barbie is the unmitigated negation to the narcissistic projection of the infant as purified mother.\n\nThe capacity of children to establish radically opposed counter-cultures objectified in toys as vengeful insistence upon their autonomy is already well documented. Alison James's analysis of children's sweets (the true inspiration of the Big Foot ice cream) showed how these represent a systematic objectification of the adult's category of the inedible.10 Any visit to the local sweetshop will reveal that the very first objects children buy for themselves are systematic inversions of adults' ideas about acceptable food categories. Pocket-money sweets are most often in the form of snot, poo, corpses and similar transgressive forms. In the case of sweets, we can understand the victory of the child as reflecting the early use of pocket money as an independent resource. While such sweets are very cheap, Barbie dolls and their outfits are expensive and may become relevant at an earlier age than these sweets. It is the mother who often purchased the objects she detests. Why should she acquiesce so fully in this tragic defeat for her own desires?\n\nBarbie's victory is accomplished through her leadership of the barbarian horde of commodities. These pollute the mind with mindless consumerism. Many mothers claimed it was television advertising and similar external influences which turned their natural infant into a machine for absorbing endless quantities of garish goods. As a result, the infant becomes entranced by a world whose values are diametrically opposed to those intended by the parent. This is a world where foods are bright colours and full of artificial ingredients and additives, and where toys are equally garish, non-educational and non-functional. The toy shop as Early Learning Centre is soon replaced by Toys R Us, a site of utter revulsion to many of these mothers: a shop where there is no escape from the sense of toys as mass commodities reeking of materialism.\n\nThe poignancy of this decline derives from the fact that the corruption of the child by materialism directly invokes what the mothers see as their own major defeat in life. They recall their student life brimming with ideals and a certain purity forged out of their rejection of their own parents' values. This was followed by their decline into more materialistic concerns of home-making and self-styling prior to their re-birthing as mothers. Mothering had been intended to replace consumption as a superior form of self-construction through a new social relationship.\n\nIf it is the mother herself who buys the fifteen Barbies and the multitude of other toys that she had previously forsworn, this may be evidence that she has not changed her goal so much as developed a new strategy. Up to this time the mother had been seen as the major and the natural source of all that goes into the making of the infant. It may be that when faced with an opponent that threatens to overwhelm her, her response is to attempt to introject this enemy and make herself once again the primary source of pleasure for her child. Since she has had plenty of experience at constructing herself through commodities, it is not difficult for her to reinstate her role as the accomplished consumer, the means through which the desires of the child may be gratified, this time through the supply of commodities.\n\nThis development in mother\u2013child relations is likely to be redolent with contradictions. This was suggested when observing one mother shopping at the local Woolworths. She had just complained to me that her daughter wanted a Barbie for her birthday, but that her daughter already had about fifty Barbies (a large exaggeration). After looking around for some time she chose a Barbie using a dustbuster. Two minutes later she remarked that she doesn't have a dustbuster herself but wants one. Just after paying she noted that she regrets buying this since it would have been better if the child's grandmother (her own mother) had bought it.\n\nParenting becomes a form of tragic practice, experienced as a series of inevitable defeats. Parents obsessively attempt to build dams and repair breaches through which pour the growing agency and autonomy of their infants. This sequence often continues into a much more explicit series of conflicts as the child grows into teenage life. The battleground moves to areas such as computer games, sexuality, drugs, parties and other genres of teenage life. In many of these there will be a similar tension between direct opposition or the attempt to buy back children through becoming the primary source of commodity purchase. But just as the Kleinian tradition is prepared to regard these stages as necessary steps towards the development of the mature infant, they can also be viewed as the foundation for a mature parenthood that has learnt to deal with problems of separation that are intrinsic to relationships that start with intensive identity and are supposed to end in autonomy. I may be having fun with them, but my intention is not to oppose or denigrate these stages which probably result in mature relationships as long as the constant defeats are accepted. While I have largely referred to mothers, this was my story as a parent too.\n\nThis research may be used to complement the work of Bourdieu. It makes this stock phrase about how objects make people into something we can recognize in our own lives, as well as in anthropological accounts of other societies. One of the primary aims of anthropology is comparative. But one reason for this is that it is much harder for us to appreciate the underlying and unspoken structures and processes that make us typical of our own society, until we have seen how different they are from the foundations of other people's behaviour. If we can see how yams make people, then we can see how Barbie dolls make people. Throughout this volume it has been evident that this use of material culture in the construction of self and society is nothing specific to capitalist or commercial societies. It is found in all societies. But it is not surprising that what the Kabyle do with agricultural implements we do with shopping.\n\nDeath\n\nStuff is as much a matter of death as it is of life and this volume concludes by turning to another study of the role of objects, again derived from fieldwork in a London street. This time, though, the research is situated in South London rather than North London. If the study of shopping was concerned with how the accumulation of things creates social relations, then the other side of this coin seemed to require a study of how we divest ourselves from things and how this assists us in dealing with the loss of relationships.11\n\nAs it happens, there is a well-established anthropological theory about the role of stuff in relation to death. Put simply, it has been argued that one of the problems of death is that it is often unexpected and usually unplanned. A rather disruptive property, given the social significance that we reasonably attribute to it. If you have no chance to prepare for it, it's pretty tricky to do death right, from society's point of view. The solution to this problem is, relatively speaking, to ignore the actual moment of death as representing merely the ending of the biological entity that was a life. Instead that event is used as a signal for starting to get prepared for all the feasting and all the rituals that should properly accompany a death. Some societies bury the corpse at once, and dig 'em up again, when they are ready to feast the dead (usually this ends with a secondary re-burial). In such ways the unplanned biological death is replaced by the planned social death. Now in this regard stuff is pretty useful. Because, if you can't control the way you separate from the living body, you certainly can control the way you separate from, or divest yourself from, the objects that were once associated with that living body.\n\nThere are many ways in which societies separate from stuff. Some are quite extreme. For example, the French Gypsies studied by Williams,12 and the people of New Ireland studied by K\u00fcchler, more or less destroy everything that was once associated with the person. In the latter case they actually construct a substitute figure of the person and concentrate their ritual on the destruction and decay of this Malanggan as opposed to the body. 'The physical separation of body and soul is enacted analogically in performances that, step by step, undo the traces left by a person in the social matrix.'13 These measures are so effective that there is no use made of genealogy, or of individual memory; only the general idea of an ancestor remains.\n\nFor some reason, however, these observations about using stuff to deal with death were always made about other societies, and have not been seen as particularly relevant to our own. By contrast, there are many professional bodies that deal with death in a place such as London and many psychological theories that are used in the training of those professions. All of which have managed, for one reason or another, to blithely ignore even the possibility that stuff plays a significant role in the way we too deal with death. There is literature on stuff in relation to death in our society, but it is almost all concerned with memorialization of various kinds.14 What has been ignored is the very gradual process of divestment that may take place over many years. This allows us to return to the end of the last chapter, and the claim that the study of stuff could potentially contribute to the welfare of populations. Because when you turn from the professions and counsellors and theories that deal with death in our society and actually start to look at what most people do, in a place such as London, you suddenly find that stuff is actually very prominent indeed. Individuals, each in their own private domain, have found their own way to understand how they can use stuff in dealing with all kinds of loss.\n\nEarlier discussions of this process took place within a period of anthropological theory dominated by functionalism. At that time it was fine to suggest that the death was potentially destructive of society and society took these measures to prevent or to repair the potential breach in the fabric of society. Societies were considered as functioning wholes that took such collective actions. Today, to call an explanation functionalist in anthropology is a bit like accusing someone of original sin. Secretly, though, I have retained an affection for functionalism, and feel that in certain cases, and this may well be one of them, it is a perfectly reasonable way to think about how societies do actually \u2013 function. It's not that I want to fetishize a thing called society. But, as it happens, many of my informants used language that came close to those functionalist anthropological theories about divesting from objects as a kind of repair mechanism that made them feel whole again in dealing with rupture and trauma.\n\nThe third portrait in The Comfort of Things concerns Elia, who is of Greek descent.15 At first she can't bear to face the clothes left behind by her mother. She stores them at the base of her wardrobe. But over a period of fifteen years the way she comes to terms with these clothes and then the way she gives them to others, or sometimes wears them herself, plays a significant role in becoming reconciled to her loss. Another of our participants, Nadine, had to face the death of her young daughter in a fire in her flat, a fire which also destroyed all the photographs and household possessions associated with her daughter. Initially she couldn't bear to see any images or reminders of her daughter at all; the circumstances were too painful. She echoed the sentiment often evoked by others \u2013 what are mere things when a life has been taken away? It was in the second phase that she began to feel the loss of images and objects that could still give respect and form to the life that was lived. She started to collect, for example, books and other objects that were preserved because they had been at her daughter's school rather than at home. Without possessions she first had to accumulate before she could divest. This process of accumulation and subsequent divestment took many years, until finally the associated objects were reduced to those she could place within a briefcase, which she keeps in storage, and a single portrait on the wall. It is now important to her that these do not distract from her emphasis upon her other children.\n\nSo in London, just as much as in Melanesia, people use their divestment from things to maintain a control over the process of separation which is less violent and sudden than death itself. Since this is not socially recognized as a ritual of separation, they find their own routes to this process, and there are various ways in which it is accomplished. One of these could be called the natural economy of relationships. That is to say, to some degree, new relationships have a tendency to displace past ones. So the process of divesting from things after a death follows upon a pattern we have already established as a regular practice, when it comes to separating from past relationships. Recall the point made in chapter 3, based on the study of people moving homes, by Marcoux, about the positive side to house moving. Going to a new residence, leaving some things behind, provides an opportunity to, as it were, bring memory and memorialization up to date; based largely on deciding which past relationships still matter, and to what extent. 'Moving becomes a means to reshuffle relationships and memories by bringing them back into consciousness, by making them explicit and for deciding which ones to reinforce, which ones to abandon or put on hold.'16\n\nWe can only retain a certain number of objects as memories of ever more past relationships. The end point of this sequence was illustrated in the home of an elderly woman, Dora. We carried out an inventory of all the objects found in her living room. On reflection these turned out to be a r\u00e9sum\u00e9 of her full life. One of the most poignant is the bright red piggy bank that today she still fills with 20p coins. When full it contains \u00a350 which can be spent, a routine that reminds her of her origins in poverty. There is one photo of her as a little girl, which recalls a life made hard from birth when her father was gassed in the trenches in the First World War, and another photo of her as a girl guide. There is just one table inherited from her mother and, from the period when she first worked, there is her sewing machine. Then there is the decorated box and a valance from the Jewish family that ran the alteration shop where she was employed. There is a picture of the first wedding dress she made for herself, six decades ago, that sees its counterpoint in several examples of needlework from recent years.\n\nThough she has the two engagement rings from her two marriages, few possessions remain from the first marriage which was mired in poverty; only the government condolence letter for her husband's death. From the second marriage, which took her to Portugal and Spain, she has a table, a carpet and an ornament from Portugal. She displays a photo of herself with her husband at a dinner party, another of one of his ancestors and a decorated box from his family. Following his death she left most of their lavish belongings to his family, returning to England to unpack some of those things she had saved on her own account. Of these she treasured the stylish cutlery, egg cups and silver cups from a high-class London shop. With the thrift reflected in her piggy bank, she would buy cutlery one item at a time until she reached five, when the manager would give her the sixth for free. She has a certificate from the ambulance service she worked with during the war, and a photo of the luncheon room where she worked afterwards for twenty-five years, ending with a certificate of freedom from the City of London, where this was sited. There is a picture from the 1960s when she looked really good, a photo of a close friend, a letter from Mrs Thatcher and a picture from a holiday in France. There is no reason to imagine that Dora intended this r\u00e9sum\u00e9 effect. It is rather the result of this economy of relationships, such that each significant relationship, whether to persons or periods and events of her past, ultimately became reduced to just one or two objects, as other mementoes made way for other relationships. Clearly the more relationships one has lived through, the more any one relationship has to be pruned back to one or two totalizing mementoes in this thrift of memory.\n\nLikewise, the materiality of each of the genres of material objects is often employed to determine the temporality of divestment, which can apply to relationships that have ended for various reasons, not just death. The wearing out of clothes may present a complementary contrast to the fixity of jewellery. But there are many quite subtle strategies. Lucy, for example, knows that music plays a critical role in her memory of past relationships. If need be she will play a specific track over and over again until she is completely sick of it and finds it irritating, in order to pre-empt the way it would otherwise have worked as an involuntary reminder of a relationship from which she is now trying to distance herself. One man, who had been left unexpectedly by his five-year partner, could not bear either to look at or to delete retained emails. In fact, he exploited our project to give me the emails and digital photographs in question, as a way of finally separating from such materials. I became the excuse for separation he was looking for.\n\nA second aspect of this process of divestment returns us to my earlier theory of relationships, as including both an actual and an idealized component. In London, just as in many other areas, this process is also one in which an individual gradually transmutes into an ancestor by becoming largely remembered with respect to an idealized category. Most of the major divisions in social status and success that occur during life are entirely ignored in the relative homogeneity and equality expressed in most contemporary English grave practice. The size and quality of the gravestone tells us very little about the distinctions that pertained to the living. We replace actual hierarchy with imagined equality. Similarly Gran who is communicated to the grandchild is without faults and foibles, a highly idealized category, related to the stock figure of Granny often portrayed in popular media. Again material culture plays a central role in making these norms clear. The old clock or washing mangle eventually and effectively turn the deceased Gran into a kind of museum figure evocative as much of her period as of herself.\n\nThis works both individually and more generically. As we ourselves become older, we often relate more closely to the past as a category. We buy things from flea markets or antique shops that speak to this generic patina. That is, the objects were old and clearly previously used but there was no knowledge of any particular prior user.17 Most people possess personal and specific possessions that can help develop this transition from the specific to the generic. For example, one informant has a tin box on a shelf from the First World War. It was evident that this box stood at one level for the specific ancestry of his grandparent, but at other levels it stood, first, for British history and then for history itself.\n\nWe don't expect people to keep objects pertaining to their parent's long decline through Alzheimer's or incapacitation through illness. Instead they retain a few photos from the wedding, the holidays, the moments when the relationship came closest to its ideal.18 Deceased males are often memorialized by other males through items of technology: tools of their trade as a carpenter, their cricket bat or their best fly-fishing rod. These are objects which help fuse the memory of the specific ancestor with the idealized conceptualization of a man, especially the working man. A woman is more likely to be recalled through the tokens of her love and care that equally make her the generic woman. The same relationship to ideals in persons would be true of lost connections to places and things, as in the homeland represented by idealized souvenirs and postcard pictures rather than poverty. So people have both an economy of relationships that pares things down to a few key objects and also use this to transform the memory of that relationship from a more actual to a more idealized component.\n\nThe process also works in reverse. Just as those who have been lost become transformed into generic and idealized ancestors by those left behind, the older generation seeks to use family heirlooms as a kind of self-simplification to a few treasured states or idealized events. They hope that the objects they select will help their descendants become more as they would wish them to be. For example, religious parents may bequeath sacred objects to their children that, alongside their admonishments, attempt to secure a greater degree of religious observance amongst their descendants. This replaces a past English practice of having a stipulation in the will that allowed inheritance only if the descendant properly conformed to the wishes of the dead.\n\nSo, in this study of ordinary households in a street in South London, it appeared that, in almost every case, when people reflected on the loss of a relationship, they had somehow worked out for themselves that a process of divestment from the things associated with the lost person could play a major role in their strategies of dealing with this loss. These were largely of their own devising, making them typical of a more general argument about material culture. While this process of divestment seems obvious enough when described in this way, and I would expect almost all readers to think of examples from their own experience, it is found nowhere in the literature on how we should deal with loss or how we actually deal with loss. There is no public recognition; it does not appear in what is by now a quite voluminous literature on mourning and bereavement in our society.\n\nIn undertaking such a study of what are, in effect, random households, many of whom are immigrants to this country, there are other reasons for expecting diversity in observed practices. Nevertheless, after working with a hundred such individuals and households, it is possible to come up with generalizations about this process, such as implied by the economy of relationships. Similarly one can see that different material genres have different propensities to be used for longer or more transient processes of divestment. In writing these accounts of divestment the intention is not to impose such analytical generalizations as a normative model. I don't want to tell people that they should or should not use these processes of divestment in the way they respond to bereavement. We should be content that people find their own creative paths that may work best in the private anonymity of their individual and personal relationship to that loss. But just as in the last chapter there is an applied consequence to such research. The intention is not to teach ordinary people what they already appear to know, but to teach professionals that which they appear to have forgotten. Since there are many instances, for example, when people have to move to a care home or a hospice, where facilitating such a process, or making provision for such a process, would in fact help render the professions much more sensitive to that which people do for themselves. It is in this manner that such research can hope to become, not just educative, but also directly applied to the enhancement of the welfare of the population that has been researched.\n\nReturning to less applied and more academic concerns: in reflecting on these issues of life and death we see how the study of material culture is at least as effective a route to the anthropology of relationships and the constitution or dissolving of personhood as any attempt to confront the nature of relationships directly. I don't think we would gain more or truer insights if we asked these people directly about love, or parenting, or mourning. The study of material culture appears a rather circuitous route to understanding people and relationships, but we may arrive more swiftly at our destination, and reach much further, than many more tempting and more direct paths.\n\nBack to the Beginning\n\nIt is possible that some readers will view the contents of this book as not just an introduction to the analysis of stuff, but also a celebration. Although the companion volume will be directed to the negative implications of stuff as quantity, including issues of environmental degradation, there is nothing 'naff' about standing in awe and reverence in the face of stuff as spectacle. The extraordinary range of artefacts that we have created stands rival to nature. We feel no hesitation in celebrating the latter. There is a huge popular respect for the voice of David Attenborough directing our gaze at creatures and landscapes we could never have anticipated would one day appear in our living room. Many of us enjoy holidays snorkelling over coral reefs, luxuriating in the bright colours and myriad shapes. The boundary between this and artefacts has become quite fuzzy. Academics and office workers may enjoy long weekend walks through what we call nature, even when we know it is the product of centuries of cultivation and conflict. Culture itself has become far more democratized and I don't feel ashamed that I can feel lost in Radiohead through my iPod, indulge in a retrospective of Satyajit Ray or Akira Kurosawa at the cinema but also be dazzled by an MTV pop video of Beyonc\u00e9. I will also go and view commercial displays of stuff in the same manner that others go to art galleries: the design ideas expressed in the shop Liberty are a favourite; but also the creativity found in a market selling toys remade from cast-off drink cans in India or the Philippines. A domestic interior turns me into a sleuth examining every clue for understanding the people who have produced this gallery of images.\n\nAcademics by their profession, when they have finished gorging on colour and spectacle, expect to turn to knowledge and explanation. But this desire for learning and scholarship does not have to detract from being entranced by the aesthetic spectacle of difference. Amongst birds and insects gorgeous colours and extraordinary dances may have evolved as forms of mating. But in their diversity they transcend their cause in the same way as a 300-page descriptive novel, much of whose content, at one level, derives from a similar biological imperative. Studies termed social science have been stymied by the ambition to replicate, in the study of society, the reduction of the natural world to natural science. This book has tried to develop both interpretation and insight, but based on ethnography, which is a world apart from the replicable experiments of science. Modern attempts to create a reductionist science of stuff, whether in psychology or economics, seem inept and clumsy compared to the nuances of the actual stuff around us. What I would love to have been able to replicate in this book is something closer to what I hear in Attenborough's voice: a rapture that seems to balance perfectly his awe at what can be explained and in what transcends explanation.\n\nThis is a book then that seeks understanding and insight, but tries to avoid reductionism. It celebrates in clothing and cars, though it could have been custard, the quirky, the exuberant and the ridiculous in our passion for stuff. But it also shows how we can nevertheless seek to become philosophers of objectification who dissolve oppositions of persons and things, spies attempting to break the codes of difference and representation, and adjudicators of whether a new technology will or will not enhance the welfare of a population. The emphasis has been largely upon just one approach to material culture, although on several occasions the work of other anthropologists has been noted such as Gell, Keane and Latour, who tackle stuff in important and complementary ways.\n\nThe search for explanations of our contemporary material world has tended to revolve around the study of capitalism; systems of production and distribution that proliferate the stuff of everyday life. Yet there exist many discussions of stuff and the desire for goods set in entirely different worlds from that of modern capitalism. One of my favourite works as a student was the many-volumed Science and Civilisation in China by Joseph Needham. There are constant references to details of appearance in the 'world's first novel', The Tale of the Genji. The Roman poet Juvenal wrote verses with vivid curses of greed and luxury. The sequel to this book will be concerned with the causes and consequences of consumption, including capitalism. Consequences such as carbon footprints, but also labour footprints, waste footprints, transport footprints, social welfare footprints. It will acknowledge the sense we have sometimes of both ourselves and our planet drowning in ever rising levels of stuff and its progeny, waste; as well as the harrowing stories that emerge from research in the Caribbean, India or the Philippines, about the sacrifices, the wretched separations, that people make, more each year, as they seek to travel from the farm and huts they were born in, to try and become, or at least hope that their children might become, masters of the metropolis. In the light of which most academics willingly snap out of the reverie of spectacle, in order to appreciate that much of human history and contemporary motivation revolves around these desires for goods and their often baleful consequences.\n\nChapter 2 spoke of the humility of things. On the one hand, material culture appears as society made tangible, the hard, strong material presence that displays itself forthrightly, such that we can't escape its presence. Now, by the end of this volume, it has become apparent how often the very opposite is the case. Our common sense about our common senses betrays us and makes unwarranted assumptions. Stuff has a quite remarkable capacity for fading from view, and becoming naturalized, taken for granted, the background or frame to our behaviour. Indeed stuff achieves its mastery of us precisely because we constantly fail to notice what it does. Things act much more commonly as analogous to the frames around paintings than as paintings themselves. They guide us towards the appropriate way to behave and remain unchallenged since we have no idea that we are being so directed. They are particularly effective during the early phases of socialization, where they stand as our teachers, mentors and gurus, leading us to become examples of a specific society, class or gender more effectively than any explicit pedagogic exhortations.\n\nIt is this subtle, largely invisible, quality of humility that explains why we have linguistics, geography and sociology, but we have never developed an academic discipline devoted to the theory, analysis and acknowledgement of material culture. In some ways that has been rather a good thing. This volume is not intended to help found a discipline of stuff. The tendency of academic disciplines to become disciplinarian is too strong. I tend to avoid debates about 'how anthropological is it?' or 'is it sufficiently anthropological?', collaborating just as happily with geographers and sociologists as fellow anthropologists. Without a clear discipline academics concerned with material culture have been really rather free in our ability to roam the world and consider why some things matter, when and to whom; and why people claim that the things which we have argued are most important to them actually don't matter at all.\n\nAs an anthropologist my method could best be described as ethnographic involvement intended to lead to empathy; the desire to see things from other people's points of view. But I guess, at the end of my quest, I have become not just empathetic to people, but even empathetic to the things themselves. To feel rather sorry for the way ordinary stuff has been neglected and pushed to one side by more aggressive, forthright and self-regarding media, such as language and art, and the way things are constantly humiliated as the mere symbolic representation of persons and society. Because denigrating material things, and pushing them down, is one of the main ways we raise ourselves up onto apparent pedestals. From this height we make claims to a spirituality entirely divorced from our own materiality and the materiality of the world we live within. I am not sure about a spirituality that is obtained by ideals of purity and separation, or that enlightenment is reached through a denial of the material.\n\nI remain, as I began, an extremist. I may sit in front of a computer and conjure abstractions of theory and analysis. But then, after a while, I can't wait to take these theoretical abstractions and plunge them back into the fray of everyday life and the glorious mess of contradiction and ambivalence that is found there. My greatest admiration and respect is reserved for the ethics of pragmatism and compromise, forged amidst the ramshackle maze of stuff we live amongst. I do not seek for purity and principle. I certainly don't wish to place stuff itself on a pedestal, to ignore the harm it can cause us, or to demean our humanity. What this volume asks for is merely a consideration of things commensurate with the place they evidently have in our lives.\n\nNotes\n\n1 P. Bourdieu, Outline of a Theory of Practice (Cambridge: Cambridge University Press, 1977).\n\n2 L. Layne, 'He was a Real Baby with Baby Things: A Material Culture Analysis of Personhood, Parenthood and Pregnancy Loss', Journal of Material Culture, 5 (2000), pp. 321\u201345; and L. Layne, Motherhood Lost: The Cultural Construction of Miscarriage and Stillbirth in America (New York: Routledge, 2002).\n\n3 The following section is taken from 'How Infants Grow Mothers in North London', Theory, Culture and Society, 14 (1997), pp. 66\u201388.\n\n4 M. Klein, Envy and Gratitude and Other Works (London: Delacourte Press, 1975).\n\n5 Strathern, Gender of the Gift, pp. 250, 252.\n\n6 A Theory of Shopping (Cambridge: Polity, 1998).\n\n7 A. Van Gennep, The Rites of Passage (London: Routledge, 1977).\n\n8 Klein, Envy and Gratitude, pp. 1\u201324.\n\n9 See also R. Parker, Torn into Two: Mothering and Ambivalence (London: Virago, 1995).\n\n10 James, 'Confections, Concoctions and Conceptions'.\n\n11 The following is taken from D. Miller and F. Parrott, 'Loss and Material Culture in South London', Journal of the Royal Anthropological Institute, 15 (2009), 502\u201319.\n\n12 P. Williams, Gypsy World: The Silence of the Living and the Voices of the Dead (Chicago: University of Chicago Press, 2003).\n\n13 S. K\u00fcchler, Malanggan: Art, Memory and Material Culture (Oxford: Berg, 2002), p. 22; see also S. K\u00fcchler, 'Malanggan: Objects, Sacrifice and the Production of Memory', American Ethnologist, 15\/4 (1998), pp. 625\u201337.\n\n14 E. Hallam and J. Hockey, Death, Memory, and Material Culture (Oxford: Berg, 2001).\n\n15 The Comfort of Things, pp. 32\u201345.\n\n16 Marcoux, 'The Refurbishment of Memory', p. 83.\n\n17 N. Gregson and L. Crewe, Second Hand Cultures (Oxford: Berg, 2003).\n\n18 E. Goffman, Gender Advertisements (London: Macmillan, 1979).\nIndex\n\naccommodating\n\nadditives\n\nagency\n\nambiguity\n\nancien regime\n\nantiques\n\nApostolic Christians\n\napplied anthropology\n\narbitrage\n\narchaeology __\n\n _Argonauts of the Western Pacific_\n\nart\n\nAttenborough, D.\n\nau pair\n\nAustralian aboriginal culture\n\nBacchanal\n\nBacchanal woman\n\nBanerjee, M.\n\nBarbie doll __\n\n _Beloved_\n\nBig Foot (ice cream)\n\nBourdieu, P.\n\nBuddhism\n\nB\u00farikov\u00e1, Z. __\n\n _Call-me_\n\nCameroon\n\ncannibalism\n\nCanterville Ghost, the\n\ncapital\n\ncapitalism\n\nCarmichael A. C.\n\ncarnival\n\nCaribbean homes\n\ncars\n\ncar upholstery\n\ncaste\n\nCatholic Charismatics\n\nChaguanas\n\nchildbirth\n\nchildren's sweets\n\nChristianity\n\nChristmas\n\ncities\n\nClarke, A.\n\nclothing\n\ncolonialism __\n\n _Colour Me Beautiful_\n\ncommodities\n\ncommunication\n\ncommunicative ecology\n\nCommunism\n\nconsciousness\n\ncontradiction __\n\n _Coral Gardens and their Magic_\n\ncouncil housing\n\nculture\n\nDaniels, I.\n\ndeath\n\ndebt\n\nDelhi\n\ndenim\n\ndepressive position\n\ndepth ontology\n\nderivatives\n\ndesigner labels\n\ndialectics\n\nDibbell\n\ndivestment from things\n\nDo-it-yourself (DIY)\n\nDouglas, M.\n\ndowry deaths\n\ndreamtime\n\ndressing\n\ndriving\n\nDumont, L.\n\nDurkheim, E.\n\neconomics __\n\n _Economist, The_\n\neconomy of relationships\n\neducation\n\nEgyptians, ancient\n\nemployment\n\nEngelke, M.\n\nentrepreneurs\n\nequality\n\neroticism\n\nethics\n\nevolutionary anthropology\n\nexchange\n\nextremism\n\nFacebook __\n\n _Fall of Public Man_ , the\n\nfame __\n\n _Fame of Gawa_ , the\n\nfashion\n\nfashion and anxiety\n\nfashion displays\n\nfeminism\n\nFrench revolution\n\nfoetal loss\n\nfood __\n\n _Frame Analysis_\n\nframes\n\nfreedom\n\nfunction\n\nfunctionalism\n\nfuneral parlours\n\nGandhi, I.\n\nGarvey P.\n\nGates, H. L.\n\nGawa\n\nGell A.\n\ngender __\n\n _Gender of the Gift_ , the\n\nghosts\n\ngift, the\n\nGoffman, E.\n\nGombrich, E.\n\ngraves\n\nhabitus __\n\n _Haroun and the Sea of Stories_ __\n\n _Hau_\n\nhaunted house\n\nHegel, G. W. F.\n\nhippie\n\nHinduism\n\nhistory __\n\n _Hitchhiker's Guide to the Galaxy, The_\n\nhomes\/houses\n\nHorst, H. __\n\n _House for Mr Biswas_\n\nhouse moving\n\nHugh-Jones, S.\n\nhumility of things\n\nhumour\n\nhypothesis testing\n\nideal homes\n\nideology\n\nimmateriality\n\ninalienable\n\nIndia\n\ninfants\n\ninterior furnishing\n\nIslam\n\nIslamic banks\n\nInternet\n\nInternet through the phone\n\nJames, A.\n\nJamaica\n\nJapanese derivatives traders\n\nJapanese homes\n\njazz __\n\n _Journal of Material Culture_ , the\n\nJouvert\n\nJuvenal\n\nKabyle\n\nKeane W.\n\nkinship\n\nkitchens\n\nKlein, M.\n\nK\u00fcchler, S.\n\nKula ring\n\nlabour\n\nlandscape\n\nlanguage\n\nLatour, B.\n\nlaw\n\nlayering\n\nLayne, L.\n\nLe Corbusier\n\nLevi-Strauss, C. __\n\n _link-up_\n\nlime\n\nlittle black dress\n\nLondon\n\nloss\n\nMadianou, M.\n\nMadrid\n\nMalanggan\n\nMalinowski, B. __\n\n _Man of Words in the West Indies_ , the\n\nManila\n\nMarcoux, J-S.\n\nmarket, the\n\nmarriage\n\nmaterialism\n\nmateriality\n\nMaurer, B.\n\nMauss, M.\n\nMayazaki, H.\n\nmedia\n\nmemorialization\n\nMeskell L.\n\nmigration\n\nmobile phones\n\nmodernism\n\nmoney\n\nMontreal\n\nMorrison, T.\n\nmother\u2013infant relations\n\nmoving house\n\nMunn, N.\n\nMurray, M.\n\nMuswell Hill\n\nmyths\n\nNaipual, V. S.\n\nNational Childbirth Trust\n\nnature\n\nNeedham, J.\n\nnetworking\n\nNorway\n\nobjectification __\n\n _Outline of a Theory of Practice_ __\n\n _Pallu_\n\nPapua New Guinea\n\nparanoid-schizoid position __\n\n _Paris is Burning_\n\nParren\u02c7as, R.\n\nPeanuts (cartoon)\n\nPertierra, R.\n\nphenomenology __\n\n _Phenomenology of Spirit_ , the\n\nPhilippines\n\nphilosophy __\n\n _Philosophy of Money_ , the __\n\n _Philosophy of Right_ , the\n\npolicy\n\npost-modernism\n\npottery\n\npoverty\n\npower\n\nproperty\n\nProtestantism\n\npsychoanalysis\n\nreason\n\nrelationships\n\nreligion\n\nreproduction furniture\n\nRowlands, M.\n\nRushdie, S.\n\nsari\n\nscandal\n\nschool farewell\n\nscience __\n\n _Science and Civilisation in China_\n\nsecuritization\n\nself\n\nself-alienation\n\nsemiotics\n\nSennett, R. __\n\n _Sense of Order_ , the\n\nseparated families\n\nseparation\n\nsignifyin(g) __\n\n _Signifying Monkey_ , the\n\nSimmel, G.\n\nSlater, D.\n\nslavery\n\nsocialization\n\nsociety\n\nSolomon Islands\n\nsoap opera\n\nSpain\n\nSrivijaya\n\nstate, the\n\nstillbirth\n\nStrathern M.\n\nstructuralism\n\nstyle\n\nsumptuary laws\n\nsuburbia\n\nsuperficiality\n\nstuff, definition __\n\n _Tabanca_\n\ntechnology\n\nTelecoms sector\n\ntelevision melodrama\n\ntexting\n\ntheatre\n\ntoys\n\ntraps\n\nTrinidad\n\ntruth\n\nUniversity College London\n\nvogueing\n\nWalbiri __\n\n _Way of the Masks_ , the\n\nwebsites\n\nweddings\n\nwellbeing __\n\n _What not to Wear_\n\nWilde, O.\n\nWinnicott, D.\n\nWitchcraft\n\nWoodward, S. __\n\n _Young and the Restless, The_\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nHIGH PRAISE FOR HARLAN COBEN'S MYRON BOLITAR NOVELS\n\n**DEAL BREAKER**\n\n\"COBEN'S CHARACTERS COME TO LIFE THROUGH THE WRITTEN WORD, AND IT TAKES A REAL WRITING TALENT TO BE ABLE TO ACCOMPLISH THIS FEAT.\"\n\n\u2014 _The Springfield Leader_\n\n\"[ _DEAL BREAKER_ ] WAS A BLAST\u2014LOTS OF FUN TO READ AND THE ENDING WAS A REAL SURPRISE! I want more! Take my word for it, you don't have to know anything about football to enjoy this book!\"\n\n\u2014Maggie Griffin, co-owner,  \nPartners & Crime bookstore (New York City)\n\n\"WHAT SETS HARLAN COBEN ABOVE THE CROWD ARE WIT AND... AN ENTERTAINING PLOT.\"\n\n\u2014 _Los Angeles Times Book Review_\n\n**BACK SPIN**\n\n\"COBEN... SCORES A HOLE IN ONE! Golf takes center stage, but the sharp plotting and emotional density, as well as nonstop wisecracks, make this a book even for the golf-adverse. The characters are deftly etched and the details keenly observed.\"\n\n\u2014 _Publishers Weekly_\n\n\"ROLLICKING GOOD FUN... The fourth installment in Harlan Coben's stellar mystery series is another winner, one sure to widen the appeal of this consistently funny and perceptive writer. Pungent observations, indelibly drawn characters and a twisting, surprise-laden plot.\"\n\n\u2014 _The Atlanta Journal and Constitution_\n\n**FADE AWAY**\n\n\"FANS WHO ENJOY GRITTY, WITTY AND HARD-BOILED CHARACTERS... WILL ADORE EDGAR AWARD\u2013WINNING HARLAN COBEN'S LATEST NOVEL.\"\n\n\u2014 _The Midwest Book Review_\n\n\"FAST ACTION, SNAPPY DIALOGUE and plenty of insider hoops material make this a fast, enjoyable read.\"\n\n\u2014 _Toronto Star_\n\n**DROP SHOT**\n\n\"A FAST-PACED PLOT, WITTY DIALOGUE, more than a few surprises, and a you'll-never-guess-whodunit denouement. Harlan Coben brings a new and exciting voice to the mystery genre.\"\n\n\u2014 _The Armchair Detective_\n\n\"SUPERB WRITING... HARLAN COBEN SERVES UP AN ACE WITH _DROP SHOT._ \"\n\n\u2014John Lutz, author of _Death by Jury_\n**_Books by Harlan Coben_**\n\nDEAL BREAKER  \nDROP SHOT  \nFADE AWAY  \nBACK SPIN  \nONE FALSE MOVE  \nTHE FINAL DETAIL  \nDARKEST FEAR  \nTELL NO ONE  \nGONE FOR GOOD\n\n# Table of Contents\n\nCover\n\nOther Books By This Author\n\nTitle Page\n\nDedication\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\nChapter 29\n\nChapter 30\n\nChapter 31\n\nChapter 32\n\nChapter 33\n\nChapter 34\n\nChapter 35\n\nChapter 36\n\nChapter 37\n\nChapter 38\n\nChapter 39\n\nChapter 40\n\nChapter 41\n\nChapter 42\n\nChapter 43\n\nChapter 44\n\nChapter 45\n\nChapter 46\n\nChapter 47\n\nChapter 48\n\nChapter 49\n\nAbout the Author\n\nCopyright\n_This, like everything else, is for Anne._\n\nThe author wishes to thank Sunandan B. Singh, M.D.,  \nChief Medical Examiner of Bergen County, New Jersey;  \nBob Richter; Rich Henshaw; Richard Curtis; Jacob Hoye;  \nShawn Coyne; and, of course, Dave Bolt.\n\n# Chapter 1\n\nOtto Burke, the Wizard of Schmooze, raised his game another level.\n\n\"Come on, Myron,\" he urged with neoreligious fervor. \"I'm sure we can come to an understanding here. You give a little. We give a little. The Titans are a team. In some larger sense I would like all of us to be a team. You included. Let's be a real team, Myron. What do you say?\"\n\nMyron Bolitar steepled his fingers. He had read somewhere that steepled fingers made you look like a thoughtful person. He felt foolish.\n\n\"I'd like nothing more, Otto,\" he said, returning the pointless volley for the umpteenth time. \"Really I would. But we've given as much as we can. It's your turn now.\"\n\nOtto nodded vigorously, as if he had just heard some philosophical whimsy that put Socrates to shame. He tilted his head, angling the painted-on smile toward his team's general manager. \"Larry, what do you think?\"\n\nTaking his cue, Larry Hanson pounded the conference table with a fist hairy enough to be a gerbil. \"Bolitar can go to hell!\" he shouted, playing enraged to the hilt. \"You hear me, Bolitar? You understand what I'm telling you? Go to hell.\"\n\n\"Go to hell,\" Myron repeated with a nod. \"Got it.\"\n\n\"You being a wiseass with me? Huh? Answer me, dammit! You being a wiseass?\"\n\nMyron looked at him. \"You have a poppy seed stuck in your teeth.\"\n\n\"Goddamn wiseass.\"\n\n\"And you're beautiful when you're angry. Your whole face lights up.\"\n\nLarry Hanson's eyes widened. He swung his line of vision toward his boss, then back to Myron. \"You're out of your league here, Bolitar. And you fucking know it.\"\n\nMyron said nothing. The truth of the matter was, Larry Hanson was partially right. Myron was out of his league. He had been in sports representation for only two years now. Most of his clients were borderline cases\u2014guys who were lucky to make the cut and grabbed the league's minimum. And football was far from his specialty. He had only three NFL players, only one of whom was a starter. Now Myron sat across from thirty-one-year-old wunderkind Otto Burke, the youngest owner in the NFL, and Larry Hanson, former-football-legend-turned-exec, negotiating a contract that even in his inexperienced hands would be the biggest rookie contract in NFL history.\n\nYes, he\u2014Myron Bolitar\u2014had landed Christian \"Hot Prop\" Steele. Two-time Heisman trophy\u2014winning quarterback. Three straight AP and UPI number-one rankings. All-American four years in a row. If that wasn't enough, the kid was an endorsement wet dream. An A student, good-looking, articulate, polite, and white (hey, it mattered).\n\nBest of all, he was Myron's.\n\n\"The offer is on the table, gentlemen,\" Myron continued. \"We think it's more than fair.\"\n\nOtto Burke shook his head.\n\n\"It's a load of crap!\" Larry Hanson shouted. \"You're a goddamn idiot, Bolitar. You're going to flush this kid's career down the toilet.\"\n\nMyron spread his arms. \"How about a group hug?\"\n\nLarry was about to offer up another expletive, but Otto stopped him with a raised hand. In Larry's playing days Dick Butkus and Ray Nitschke couldn't stop him with body blows. Now this one-hundred-fifty-pound Harvard grad could silence him with but a wave.\n\nOtto Burke leaned forward. He hadn't stopped smiling, hadn't stopped the hand gestures, hadn't stopped the eye contact\u2014like an Anthony Robbins Personal Power infomercial come to life. Disconcerting as all hell. Otto was a small, fragile-looking man with the tiniest fingers Myron had ever seen. His hair was dark and heavy-metal long, flowing to his shoulders. He was baby-faced with a silly goatee that looked as if it'd been sketched on in pencil. He smoked a very long cigarette, or maybe it just appeared long against his tiny fingers.\n\n\"Now, Myron,\" Otto said, \"let's speak rationally here, okay?\"\n\n\"Rationally. Let's.\"\n\n\"Great, Myron, that'll be helpful. The truth is, Christian Steele is an unknown, untested quantity. He hasn't even put on a pro uniform yet. He may be the bust of the century.\"\n\nLarry snorted. \"You should know something about that, Bolitar\u2014about players who amount to nothing. Who crap out.\"\n\nMyron ignored him. He had heard the insult before. It no longer bothered him. Sticks and stones and all that. \"We are talking about perhaps the greatest quarterback prospect in history,\" he replied steadily. \"You made three trades and gave up six players to get his rights. You didn't do all that if you didn't believe he has what it takes.\"\n\n\"But this proposal\"\u2014Otto stopped, looked up as though searching the ceiling tiles for the right word\u2014\"it's not sound.\"\n\n\"Crap is more like it,\" Larry added.\n\n\"It's final,\" Myron said.\n\nOtto shook his head, the smile unfazed. \"Let's talk this through, okay? Let's look at it from every conceivable angle. \"You're new at this, Myron\u2014an ex-jock reaching for the executive brass ring. I respect that. You're a young guy trying to give it a go. Heck, I admire that. Really.\"\n\nMyron bit down. He could have pointed out that he and Otto were the same age, but he so loved being patronized. Didn't everybody?\n\n\"If you make a mistake on this,\" Otto continued, \"it could be the sort of thing that destroys your career. Do you know what I mean? Plenty of people already feel that you're not up to this\u2014to handling such a high-profile client. Not me, of course. I think you're a very bright guy. Shrewd. But the way you're acting...\" He shook his head like a teacher disenchanted with a favorite pupil.\n\nLarry stood, glowering down at Myron. \"Why don't you give the kid some good advice?\" he said. \"Tell him to get a real agent.\"\n\nMyron had expected this whole good-cop, bad-cop routine. He had, in fact, expected worse; Larry Hanson had not yet attacked the sexual appetites of anyone's mother. Still, Myron preferred the bad cop to the good cop. Larry Hanson was a frontal assault, easily spotted and handled. Otto Burke was the snake-infested high grass with buried land mines.\n\n\"Then I guess we have nothing more to discuss,\" Myron said.\n\n\"I believe a holdout would be unwise, Myron,\" Otto said. \"It might soil Christian's squeaky-clean image. Hurt his endorsements. Cost you both a great deal of money. You don't want to lose money, Myron.\"\n\nMyron looked at him. \"I don't?\"\n\n\"No, you don't.\"\n\n\"Can I jot that down?\" He picked up a pencil and began scribbling. \"Don't... want... to... lose... money.\" He grinned at both men. \"Am I picking up pointers today or what?\"\n\nLarry mumbled, \"Goddamn wiseass.\"\n\nOtto's smile remained locked on autopilot. \"If I may be so bold,\" he continued, \"I would think Christian would want to collect quickly.\"\n\n\"Oh?\"\n\n\"There are those who have serious reservations about Christian Steele's future. There are those\"\u2014Otto drew deeply on his cigarette\u2014\"who believe he may have had something to do with that girl's disappearance.\"\n\n\"Ah,\" Myron said, \"that's more like it.\"\n\n\"More like what?\"\n\n\"You're starting to fling mud. For a second there I thought I wasn't asking for enough.\"\n\nLarry Hanson stuck a thumb in Myron's direction. \"Do you believe this fucking sliver of pond film we're sitting with? You raise a legitimate issue about Christian's ex-bimbo, one that goes to the heart of his value as a public relations commodity\u2014\"\n\n\"Pitiful rumors,\" Myron interrupted. \"No one believed them. If anything, they made the public more sympathetic to Christian's tragedy. And don't call Kathy Culver a bimbo.\"\n\nLarry raised an eyebrow. \"Well, well, aren't we touchy,\" he said, \"for a low-life pissant.\"\n\nMyron's expression did not change. He had met Kathy Culver five years ago when she was a sophomore in high school, already a budding beauty. Like her sister Jessica. Eighteen months ago Kathy had mysteriously vanished from the campus of Reston University. To this day no one knew where she was or what had happened to her. The story had all the media's favorite tasty morsels\u2014a gorgeous co-ed, the fianc\u00e9e of football star Christian Steele, the sister of novelist Jessica Culver, a strong hint of sexual assault for extra seasoning. The press could not help themselves. They attacked like ravenous relatives around a buffet table.\n\nBut just recently a second tragedy had befallen the Culver family. Adam Culver, Kathy's father, had been murdered three nights earlier in what police were calling a \"botched robbery.\" Myron wanted very much to contact the family, to do more than merely offer simple condolences, but he had decided to stay away, not knowing if he was welcome, fairly certain he wasn't.\n\n\"Now if\u2014\"\n\nThere was a knock on the door. It opened a crack, and Esperanza stuck her head in. \"Call for you, Myron,\" she said.\n\n\"Take a message.\"\n\n\"I think you'll want to take it.\"\n\nEsperanza stayed in the doorway. Her dark eyes gave away nothing, but he understood.\n\n\"I'll be right there,\" he said.\n\nShe slipped back through the door.\n\nLarry Hanson gave an appreciative whistle. \"She's a babe, Bolitar.\"\n\n\"Gee, thanks, Larry. That means a lot coming from you.\" He rose. \"I'll be right back.\"\n\n\"We don't have all goddamn day to jerk off here.\"\n\n\"I'm sure you don't.\"\n\nHe left the conference room and met up with Esperanza at her desk.\n\n\"The Meal Ticket,\" she told him. \"He said it was urgent.\"\n\nChristian Steele.\n\nFrom her petite frame most would not guess that Esperanza used to be a professional wrestler. For three years she had been known on the circuit as Little Pocahontas. The fact that Esperanza Diaz was Latina, without a trace of American Indian blood, did not seem to bother the FLOW (Fabulous Ladies of Wrestling) organization. A minor detail, they said. Latino, Indian, what's the difference?\n\nAt the height of her pro wrestling career, the same script was played out every week in arenas all over the U.S. of A. Esperanza (\"Pocahontas\") would enter the ring wearing moccasins, a suede-fringed dress, and a headband that lassoed her long black hair away from her dark face. The suede dress came off before the fight, leaving a somewhat flimsier and less traditional Native American garb in its stead.\n\nProfessional wrestling has a pretty simple plot with painfully few variations. Some wrestlers are bad. Some are good. Pocahontas was good, a crowd favorite. She was cute and small and quick and had a tight little body. Everyone loved her. She would always be winning the fight on skill when her opponent would do something illegal\u2014throw sand in her eyes, use a dreaded foreign object that everyone in the free world except the referee could see\u2014to turn the tide. Then the bad wrestler would bring in a couple of extra cronies, ganging up three against one on poor Pocahontas, pounding mercilessly on the brave beauty to the unequivocal shock and chagrin of the announcers, who had seen the same thing happen last week and the week before.\n\nJust when it seemed there was no hope, Big Chief Mama, a mammoth creature, charged out of the locker room and threw the beasts off the defenseless Pocahontas. Then together Big Chief Mama and Little Pocahontas would defeat the forces of evil.\n\nMassively entertaining.\n\n\"I'll take it in my office,\" Myron said.\n\nAs he entered he saw the nameplate on his desk, a gift from his parents.\n\nMYRON BOLITAR  \nSPORTS AGENT\n\nHe shook his head Myron Bolitar. He still couldn't believe someone would name a kid Myron. When his family first moved to New Jersey, he had told everyone in his new high school that his name was Mike. Nope, no dice. Then he tried to nickname himself Mickey. Unhunh. Everyone reverted to Myron; the name was like a horror-movie monster that would not die.\n\nTo answer the obvious question: No, he never forgave his parents.\n\nHe picked up the phone. \"Christian?\"\n\n\"Mr. Bolitar? Is that you?\"\n\n\"Yes. And please call me... Myron.\" Acceptance of the inevitable, a sign of a wise man.\n\n\"I'm sorry to interrupt you. I know how busy you are.\"\n\n\"I'm busy negotiating your contract. I have Otto Burke and Larry Hanson in the next room.\"\n\n\"I appreciate that, Mr. Bolitar, but this is very important.\" His voice was trembling. \"I have to see you right away.\"\n\nHe switched hands. \"Something wrong, Christian?\" Mr. Perceptive.\n\n\"I\u2014I'd rather not discuss it over the phone. Would you be able to meet me at my room on campus?\"\n\n\"Sure, no problem. What time?\"\n\n\"Now, please. I\u2014I don't know what to make of this. I want you to see it.\"\n\nMyron took a deep breath. \"No problem. I'll throw Otto and Larry out. It'll be good for the negotiations. I'll be there in an hour.\"\n\nIt took a lot longer.\n\nMyron entered the Kinney garage on Forty-sixth Street, not too far from his Park Avenue office. He nodded to Mario, the garage attendant, walked past the pricing sheet, which had a small disclaimer on the bottom that read \"not including 97% tax,\" and headed to his car on the lower level. A Ford Taurus. Your basic Babe Magnet.\n\nHe was about to unlock the door when he heard a hissing sound. Like a snake. Or more likely, air escaping from a tire. The sound emanated from his back right tire. A quick examination told Myron that it had been slashed.\n\n\"Hi, Myron.\"\n\nHe spun around. Two men grinned at him. One was the size of a small Third World nation. Myron was big\u2014nearly six-four and two hundred twenty pounds\u2014but he guessed that this guy must have been six-six and closing in on three hundred. A heavy-duty weight lifter, his whole body was puffed up as if he were wearing inflatable life vests under his clothes. The second man was of average build. He wore a fedora.\n\nThe big man lumbered toward Myron's car. His arms swung stiffly at his sides. He kept tilting his head, cracking the part of the anatomy that on a normal human being might be called a neck.\n\n\"Having some car trouble?\" he asked with a chuckle.\n\n\"Flat tire,\" Myron said. \"There's a spare in the trunk. Change it.\"\n\n\"I don't think so, Bolitar. This was just a little warning.\"\n\n\"Oh?\"\n\nThe human edifice grabbed the lapels of Myron's jacket. \"Stay away from Chaz Landreaux. He's already signed.\"\n\n\"First change my tire.\"\n\nThe grin increased. It was a stupid, cruel grin. \"Next time I won't be so nice.\" He grabbed a little tighter, bunching up the suit and tie. \"Understand?\"\n\n\"You are aware, of course, that steroids make your balls shrink.\"\n\nThe man's face reddened. \"Oh, yeah? Maybe I oughta smash your face in, huh? Maybe I oughta pulverize you into oatmeal.\"\n\n\"Oatmeal?\"\n\n\"Yeah.\"\n\n\"Nice image, really.\"\n\n\"Fuck you.\"\n\nMyron sighed. Then his whole body seemed to snap into motion at the same time. He started with a head-butt that landed square on the big man's nose. There was a squelching noise like beetles being stepped on. Blood gushed from the nose.\n\n\"Son of a\u2014\"\n\nMyron cradled the back of the big man's head for leverage and smashed his elbow into the sweet of the Adam's apple, nearly caving the windpipe all the way in. There was a painful, gurgling choke. Then silence. Myron followed up with a knife-hand strike to the back of the neck below the skull.\n\nThe big man slid to the ground like wet sand.\n\n\"Okay, that's enough!\"\n\nThe man with the fedora stepped closer, a gun drawn and pointed at Myron's chest.\n\n\"Back away from him. Now!\"\n\nMyron squinted at him. \"Is that really a fedora?\"\n\n\"I said, back off!\"\n\n\"Okay, okay, I'm backing.\"\n\n\"You didn't have to do that,\" the smaller man said with almost childlike hurt. \"He was just doing his job.\"\n\n\"A misunderstood youth,\" Myron added. \"Now I feel terrible.\"\n\n\"Just stay away from Chaz Landreaux, okay?\"\n\n\"Not okay. Tell Roy O'Connor I said it's not okay.\"\n\n\"Hey, I ain't hired to get no answer. I'm just delivering.\"\n\nWithout another word the man with the fedora helped his fallen colleague to his feet. The big man stumbled to their car, one hand on his nose, the other massaging his windpipe. His nose was busted, but his throat would hurt even worse, especially when he swallowed.\n\nThey got in and quickly drove away. They did not stop to change Myron's tire.\n\n# Chapter 2\n\nMyron dialed Chaz Landreaux's number on his car phone.\n\nNot being what one would call mechanically inclined, it had taken Myron half an hour to change the tire. He rode slowly for the first few miles, fearing his handiwork would encourage the tire to slip off and flee. When he felt more confident, he accelerated and started back on the road to Christian's.\n\nWhen Chaz answered, Myron quickly explained what happened.\n\n\"They was already here,\" Chaz told him. Lots of noise in the background. An infant cried. Something fell and broke. Children laughed. Chaz shouted for quiet.\n\n\"When?\" Myron asked.\n\n\"Hour ago. Three men.\"\n\n\"Did they hurt you?\"\n\n\"Nah. Just held me down and made threats. Said they was going to break my legs if I didn't honor my contract.\"\n\nBreaking legs, Myron thought. How original.\n\nChaz Landreaux was a senior basketball player at Georgia State and a probable first-round NBA pick. He was a poor kid from the streets of Philadelphia. He had six brothers, two sisters, no father. The ten of them lived in an area that\u2014if daringly improved\u2014might one day be charitably dubbed \"poor ghetto.\"\n\nDuring his freshman year, an underling of a big-time agent named Roy O'Connor had approached Chaz\u2014four years before Chaz was eligible to talk to an agent. The man offered Chaz a five-thousand-dollar \"retainer\" up front, with monthly payments of $250, if he signed a contract making O'Connor his agent when he turned pro.\n\nChaz was confused. He knew that NCAA rules forbade him from signing a contract while he still had eligibility. The contract would be declared null and void. But Roy's man assured him this would be no problem. They would simply postdate the contract to make it appear Chaz had signed on after his final year of eligibility. They'd keep the contract in a safety deposit box until the proper time arrived. No one would be the wiser.\n\nChaz was not sure. He knew it was illegal, but he also knew what that kind of money would mean to his mom and eight siblings living in a two-room hellhole. Roy O'Connor then entered the picture and pitched the final inducement: If Chaz changed his mind at some future date, he could repay the money and tear up the contract.\n\nFour years later Chaz changed his mind. He promised to pay back every cent. No way, said Roy O'Connor. You have a contract with us. You'll stick with it.\n\nThis was not an uncommon setup. Dozens of agents did it. Norby Walters and Lloyd Bloom, two of the country's biggest agents, had been arrested for it. Threats too were not uncommon. But that was where it usually ended: with threats. No agent wanted to risk being exposed. If a kid stood firm, the agent backed off.\n\nBut not Roy O'Connor. Roy O'Connor was using muscle. Myron was surprised.\n\n\"I want you out of town for a little while,\" Myron continued. \"You got someplace to lay low?\"\n\n\"Yeah, I'll crash with a friend in Washington. But what we going to do?\"\n\n\"I'll take care of it. Just stay out of sight.\"\n\n\"Okay, yeah, I hear ya.\" Then: \"Oh, Myron, one other thing.\"\n\n\"What?\"\n\n\"One of the dudes who held me down said he knew you. A monster, man. I mean, huge. Slick-looking motherfucker.\"\n\n\"Did he say his name?\"\n\n\"Aaron. He said to tell you Aaron said hi.\"\n\nMyron's shoulders slumped. Aaron. A name from his past. Not a good name either. Roy O'Connor not only had muscle behind him\u2014he had serious muscle.\n\nThree hours after leaving his office, Myron shook off all thoughts about the garage incident and knocked on Christian's door. Despite the fact that he'd graduated two months earlier, Christian still lived in the same campus dorm he had occupied throughout his senior year, working as a counselor at Reston U's football summer camp. The Titans minicamp, however, started in two days, and Christian would be there. Myron had no intention of having Christian hold out.\n\nChristian opened the door immediately. Before Myron had a chance to explain his tardiness, Christian said, \"Thanks for getting here so fast.\"\n\n\"Uh, sure. No problem.\"\n\nChristian's face was completely devoid of its usual healthy color. Gone were the rosy cheeks that dimpled when Christian smiled. Gone was the wide-open, aw-shucks smile that made the co-eds swoon. Even the famed steady hands were noticeably quaking.\n\n\"Come on in,\" he said.\n\n\"Thanks.\"\n\nChristian's room looked more like a 1950s sitcom set than a modern-day campus dorm room. For one thing, the place was neat. The bed was made, the shoes in a row beneath it. There were no socks on the floor, no underwear, no jock straps. On the walls were pennants. Actual pennants. Myron couldn't believe it. No posters, no calendars with Claudia Schiffer or Cindy Crawford or the Barbi twins. Just old-fashioned pennants. Myron felt as if he'd just stepped into Wally Cleaver's dormitory.\n\nChristian didn't say anything at first. They both stood there uncomfortably, like two strangers stuck together at some cocktail party with no drinks in their hands. Christian kept his eyes lowered to the floor like a scolded child. He hadn't commented on the blood on Myron's suit. He probably hadn't noticed it.\n\nMyron decided to try one of his patented silver-tongued ice-breakers. \"What's up?\"\n\nChristian began to pace\u2014no easy accomplishment in a room slightly larger than the average armoire. Myron could see that Christian's eyes were red. He'd been crying, his cheeks still showing small traces of the tear tracks.\n\n\"Did Mr. Burke get mad about canceling the meeting?\" Christian asked.\n\nMyron shrugged. \"He had a major conniption, but he'll survive. Means nothing, don't worry about it.\"\n\n\"Minicamp starts Thursday?\"\n\nMyron nodded. \"Are you nervous?\"\n\n\"A little, maybe.\"\n\n\"Is that why you wanted to see me?\"\n\nChristian shook his head. He hesitated and then said, \"I\u2014I don't understand it, Mr. Bolitar.\"\n\nEvery time he called him mister, Myron looked for his father.\n\n\"Don't understand what, Christian? What's this all about?\"\n\nHe hesitated again. \"It's...\" He stopped, took a deep breath, started again. \"It's about Kathy.\"\n\nMyron thought he'd heard wrong. \"Kathy Culver?\"\n\n\"You knew her,\" Christian said. Myron couldn't tell if it was a statement or a question.\n\n\"A long time ago,\" Myron replied.\n\n\"When you were with Jessica.\"\n\n\"Yes.\"\n\n\"Then maybe you'll understand. I miss Kathy. More than anyone can ever know. She was very special.\"\n\nMyron nodded, encouraging. Very Phil Donahue.\n\nChristian took a step back, nearly banging his head into a bookshelf. \"Everybody sensationalized what happened to her,\" he began. \"They put it in tabloids, had stories about the disappearance on _A Current Affair_. It was like a game to everyone. A TV show. They kept calling us 'idyllic,' the 'idyllic couple.' \" He made quote marks in the air with his fingers. \"As if _idyllic_ meant unreal. Unfeeling. Everyone kept saying I was young, I'd get over it quickly. Kathy was just a pretty blonde, millions more like her for a guy like me. I was expected to get on with my life. She was gone. It was over and done with.\"\n\nChristian's boyish quality\u2014something that Myron thought would help make him the future endorsement king\u2014had suddenly taken on a new dimension. Instead of the shy, gee-whiz, modest little Kansas boy, Myron saw reality: a scared child huddled in a corner, a child whose parents were dead, who had no real family, probably no real friends, just hero-worshipers and those who wanted a piece of him (like Myron himself?).\n\nMyron shook his head. No way. Other agents, yes, but not him. Myron wasn't like that. But still something akin to guilt stayed there, poking a sharp finger into his ribs.\n\n\"I never really believed Kathy was dead,\" Christian continued. \"That was part of the problem, I guess. The not-knowing gets to you after a while. Part of me\u2014part of me almost hoped they'd find her body already, anything to end it. Is that an awful thing to say, Mr. Bolitar?\"\n\n\"I don't think so, no.\"\n\nChristian looked at him solemnly. \"I kept thinking about the panties. You know about that?\"\n\nMyron nodded. The lone clue in the mystery was Kathy's ripped panties, found on top of a campus Dumpster. Rumor had it that they were covered with semen and blood. To the world at large, the panties had confirmed what had long been suspected: Kathy Culver was dead. It was a sad though not uncommon story. She had been raped and murdered by a random psychopath. Her body would probably never be found\u2014or maybe some hunters would stumble across the skeletal remains in the woods one day, giving the press a great eleven o'clock commercial teaser, bringing the cameras back into the story with undying hopes of catching a grief-stricken relative on film.\n\n\"They made it seem like it was a dirty thing,\" Christian continued. \" 'Pink,' they said. 'Silk,' they said. They never called them underwear or undergarments or even just plain panties. It was always pink silk panties. Like that was important. One TV station even interviewed a Victoria's Secret model for her comment on them. Pink silk panties. Like that meant she was asking for it. Trashing Kathy like that...\"\n\nHis voice sort of faded away then. Myron said nothing. Christian was working up to something. Myron only hoped it wasn't a breakdown.\n\n\"I guess I should get to the point,\" Christian finally said.\n\n\"Take your time. I'm not going anywhere.\"\n\n\"I saw something today. I\u2014\" Christian stopped and swung his eyes toward Myron's. They looked at him, pleading. \"Kathy may still be alive.\"\n\nHis words hit Myron like a wet slap. Whatever Myron had been preparing himself for, whatever he imagined Christian was leading up to, hearing Kathy Culver might still be alive was not a part of the equation.\n\n\"What?\"\n\nChristian reached behind him and opened his desk drawer. The desk too was something out of _Leave It to Beaver_. Completely uncluttered. Two cans, one with Bic pens, the other with sharpened number-two pencils. Gooseneck lamp. Desk blotter with calendar. Dictionary, thesaurus, and _The Elements of Style_ all in a row between two globe bookends.\n\n\"This came in the mail today.\"\n\nHe handed Myron a magazine. On the cover was a naked woman. Calling her well-endowed would be tantamount to calling World War II a skirmish. Most men are somewhat mammary obsessed, and Myron was not above having similar sentiments, but this was positively freakish. The woman's face was far from pretty, kind of harsh looking. She was giving the camera a look that was supposed to be come-hither but looked more like constipation. Her tongue was licking her lips, her legs spread, her finger beckoning the reader to come closer.\n\nVery subtle effect, Myron thought.\n\nThe magazine was called _Nips_. The lead story, according to the words emblazoned across her right breast: \"How to Get Her to Shave Dat Thang.\"\n\nMyron looked up sharply. \"What's this all about?\"\n\n\"The paper clip.\"\n\n\"What?\"\n\nBut Christian seemed too weak to repeat it. He just pointed. On the top of the magazine Myron spotted a glint of silver. A paper clip was being used as a bookmark.\n\n\"It came with that on there,\" Christian said by way of explanation.\n\nMyron fingered through the pages, catching quick glimpses of flesh, until he arrived at the page marked off by the paper clip. His eyes squinted in confusion. It was an ad page, though it had as many erotic photos as any other. The top of the page read:\n\n**Live Fantasy Phone\u2014Pick Your Girl!**\n\nThere were three rows, four girls in each row, all the way down the page. Myron's eyes scanned down. He could not believe what he was seeing. \"Oriental Girls Are Waiting!\" \"Wet and Juicy Lesbos!\" \"Spank Me, Please!\" \"Bitches in Heat!\" \"Tiny Titties!\" (for those who didn't like the cover shot, no doubt) \"I Want You to Ride Me!\" \"Pick My Cherry!\" \"Make Me Beg for More!\" \"Wanted: Robocock.\" \"Mistress Savannah Demands You Call Now!\" \"Horny Housewife!\" \"Overweight Men Wanted.\" Each with matching photo\u2014provocative poses involving telephones.\n\nThere were some that were far more raunchy. Cross-dressers. Women with men's equipment. There were some Myron could not even understand. Like unfathomable science experiments. The telephone numbers were what you'd expect. 1-800-888-SLUT. 1-900-46-TRAMP. 1-800-REAM-MEE. 1-900-BAD-GIRL.\n\nMyron made a face. He wanted to wash his hands.\n\nThen he saw it.\n\nIt was in the bottom row, second from the right. It read, \"I'll Do Anything!\" The phone number was 1-900-344-LUST. $3.99 per minute. Discreetly billed to your telephone or charge card. Visa\/MC accepted.\n\nThe woman in the picture was Kathy Culver.\n\nMyron felt a coldness seep into him. He turned back to the cover and checked the date. It was the current issue.\n\n\"When did you get this?\"\n\n\"It came in today's mail,\" Christian said, picking up an envelope. \"In this.\"\n\nMyron's head began to swim. He tried to fight the dizziness and get some kind of footing, but the picture of Kathy kept tipping him back over. The envelope was plain manila. There was no return address\u2014that would have been too easy. It was not postmarked and had no stamps, merely reading:\n\nCHRISTIAN STEELE  \nBOX 488\n\nNo city, no state. That meant it'd been mailed on campus. The address had been handwritten.\n\n\"You get lots of fan mail, right?\" Myron asked.\n\nChristian nodded. \"But they go somewhere else. This was in my private box. The number is unlisted.\"\n\nMyron handled the envelope carefully, trying not to smudge any potential fingerprints. \"It could be trick photography,\" Myron added. \"Someone might have superimposed her head on\u2014\"\n\nChristian stopped him with a shake of his head. His eyes were back on the floor. \"It's not just her face, Mr. Bolitar,\" he said, embarrassed.\n\n\"Oh,\" Myron said, ever swift on the uptake. \"I see.\"\n\n\"Do you think we should give this to the police?\" Christian asked.\n\n\"Perhaps.\"\n\n\"I want to do the right thing,\" Christian said, his hands balling into fists. \"But I won't let them drag Kathy through the mud again. You saw what they did when she was the victim. What will they do when they see this?\"\n\n\"They'll go animal,\" he agreed.\n\nChristian nodded.\n\n\"But it's probably just a prank,\" Myron continued. \"I'll check it out before we do anything else.\"\n\n\"How?\"\n\n\"Let me worry about that.\"\n\n\"There's one other thing,\" Christian said. \"The handwriting on the envelope.\"\n\nMyron glanced at it again. \"What about it?\"\n\n\"I can't say for sure, but it looks a lot like Kathy's.\"\n\n# Chapter 3\n\nMyron stopped short when he saw her.\n\nHe had just stumbled into the bar in something of a daydream, his mind like a movie camera that couldn't stay in focus. He tried to sift through what he had just seen and learned from Christian, tried to compute the facts and form a solid, well-conceived conclusion.\n\nHe came up with nothing.\n\nThe magazine was jammed into the right pocket of his trench coat. Porn mag and trench coat, Myron thought. Jesus. The same questions echoed ad nauseam in his head: Could Kathy Culver still be alive? And if she was, what had happened to her? What could have led Kathy from the innocence of her dorm room to the back pages of _Nips_ magazine?\n\nThat was when he spotted the most beautiful woman he had ever seen.\n\nShe was sitting on a stool, her long legs crossed, sipping gently at her drink. She wore a white blouse opened at the throat, a short gray skirt, and black stockings. Everything clung just right. For a fleeting moment Myron thought she was just a by-product of his daydream, a dazzling vision to tantalize the senses. But the knot in his stomach made him quickly dismiss that notion. His throat went dry. Deep, dormant emotions crashed down upon him like a surprise wave at the beach.\n\nHe managed to swallow and commanded his legs to move forward. She was, quite simply, breathtaking. Everything else in the bar faded into the background, as though they were only stage props set for her.\n\nMyron approached. \"Come here often?\" he asked.\n\nShe looked at him like he was an old man jogging in a Speedo. \"Original line,\" she said. \"Very creative.\"\n\n\"Maybe not,\" he said. \"But what a delivery.\" He smiled. Winningly, he thought.\n\n\"Glad you think so.\" She turned back to her drink. \"Please leave.\"\n\n\"Playing hard to get?\"\n\n\"Get lost.\"\n\nMyron grinned. \"Stop it already. You're embarrassing yourself.\"\n\n\"Pardon me.\"\n\n\"It's obvious to everyone in this bar.\"\n\n\"Oh?\" she remarked. \"Do enlighten.\"\n\n\"You want me. Bad.\"\n\nShe almost smiled. \"That obvious, huh?\"\n\n\"Don't blame yourself. I'm irresistible.\"\n\n\"Uh-huh. Catch me if I swoon.\"\n\n\"I'm right here, sweetcakes.\"\n\nShe sighed deeply. She was as beautiful as ever, as beautiful as the day she had walked out on him. He hadn't seen her in four years, but it still hurt to think about her. It hurt even more to look at her. Their weekend at Win's house on Martha's Vineyard came to him. He could still remember the way the ocean breeze blew her hair, the way she tilted her head when he spoke, the way she looked and felt in his old sweatshirt. Simple fragile bliss. The knot in his stomach tightened.\n\n\"Hello, Myron,\" she said.\n\n\"Hello, Jessica. You're looking well.\"\n\n\"What are you doing here?\" she asked.\n\n\"My office is upstairs. I practically live here.\"\n\nShe smiled. \"Oh, that's right. You represent athletes now, don't you?\"\n\n\"Yes.\"\n\n\"Better than working all that undercover stuff?\"\n\nMyron did not bother answering. She glanced at him but did not hold the gaze.\n\n\"I'm waiting for someone,\" Jessica said suddenly.\n\n\"A male someone?\"\n\n\"Myron...\"\n\n\"Sorry. Old reflex.\" He looked at her left hand. His heart back-flipped when he saw no rings. \"You never married what's-his-name?\" he asked.\n\n\"Doug.\"\n\n\"That's right. Doug. Or was it Dougie?\"\n\n\"You're making fun of someone's name?\"\n\nMyron shrugged. She had a point. \"So what happened to him?\"\n\nHer eyes studied a beer ring on the bar. \"It wasn't about him,\" she said. \"You know that.\"\n\nHe opened his mouth and then closed it. Rehashing the bitter past was not going to do any good. \"So what brings you back to the city?\"\n\n\"I'm going to be teaching a semester at NYU.\"\n\nHis heart sped up again. \"You moved back to Manhattan?\"\n\n\"Last month.\"\n\n\"I'm really sorry about your father's\u2014\"\n\n\"We got your flowers,\" she interrupted.\n\n\"I wanted to do more.\"\n\n\"Better you didn't.\" She finished her drink. \"I have to go. It was nice seeing you.\"\n\n\"I thought you were meeting someone.\"\n\n\"My mistake, then.\"\n\n\"I still love you, you know.\"\n\nShe stood, nodded.\n\n\"Let's try again,\" he said.\n\n\"No.\"\n\nShe walked away.\n\n\"Jess?\"\n\n\"What?\"\n\nHe considered telling her about her sister's picture in the magazine. \"Can we have lunch sometime?\" he asked. \"Just talk, okay?\"\n\n\"No.\"\n\nJessica turned and left him. Again.\n\nWindsor Horne Lockwood III listened to Myron's story with his fingers steepled. Steepling looked good on Win, a lot better than on Myron. When Myron finished, Win said nothing for a few moments, doing more of that steepled-hands-concentration thing. Finally he rested his hands on the desk.\n\n\"My, my, haven't we had a special day?\"\n\nMyron rented his space from his old college roommate, Windsor Horne Lockwood III. People often said that Myron looked nothing like his name\u2014an observation Myron took as high praise; Windsor Horne Lockwood III, however, looked exactly like his name. Blond hair, perfect length, parted on the right side. His features were classical patrician, almost too handsome, like something crafted in porcelain.\n\nHis attire was always thoroughbred prep\u2014pink shirts, polo shirts, monogrammed shirts, khaki pants, golf (read, ugly) pants, white bucks (Memorial Day to Labor Day) or wing tips (Labor Day to Memorial Day) on his feet. Win even had that creepy accent, the one that did not originate from any particular geographical location as much as from certain prep schools like Andover and Exeter. (Win had gone to Exeter.) He played a mean game of golf. He had a three handicap and was the fifth-generation member of stuffy Merion Golf Club in Philadelphia and third-generation at equally stuffy Pine Valley in southern New Jersey. He had a perennial golf tan, one of those where the color could be found only on the arms (short-sleeve shirts) and a V-shape in the neck (open alligator shirt), though Win's lily-white skin never tanned. It burned.\n\nWin was full-fledged whitebread. He made star quarterback Christian Steele look like a Mediterranean houseboy.\n\nMyron had hated Windsor on sight. Most people did. Win was used to it. People liked to form and keep an immediate impression. In Win's case that impression was old money, elitist, arrogant\u2014in a phrase, a flaming asshole. There was nothing Win could do about it. People who relied solely on first impressions meant little to him.\n\nWin gestured to the magazine on his desk. \"You chose not to tell Jessica about this?\"\n\nMyron stood, paced, and then sat back down. \"What was I going to say? 'Hi, I love you, come back to me, here's a photo of your supposedly dead sister advertising a sex phone in a porno mag?' \"\n\nWin thought a moment. \"I'd refine the wording a bit,\" he said.\n\nHe flipped through the porno rag, his eyebrow arched as if to say Hmmm. Myron watched. He had decided not to tell Win about Chaz Landreaux or the incident in the garage. Not yet anyway. Win had a funny way of reacting when someone tried to hurt Myron. It wasn't always pretty. Better to save it for later, when Myron would know exactly how he wanted to handle Roy O'Connor. And Aaron.\n\nWin dropped the magazine on his desk. \"Shall we begin?\"\n\n\"Begin what?\"\n\n\"Investigating. That is what you planned for us, correct?\"\n\n\"You want to help?\"\n\nWin smiled. \"But of course.\" He turned his phone around so that it faced Myron. \"Dial.\"\n\n\"The number in the magazine?\"\n\n\"Well, golly, Myron, I thought we'd call the White House,\" Win said dryly. \"See if we can get Hillary to talk dirty.\"\n\nMyron took hold of the phone. \"You ever call one of these lines?\"\n\n\"I?\" Win feigned shock. \"The Debutantes' Darling? The Society Stud? Surely you jest.\"\n\n\"Neither have I.\"\n\n\"Perhaps you'd like to be alone, then,\" Win said. \"Loosen your belt, pull down your trousers, that kind of thing.\"\n\n\"Very funny.\"\n\nMyron dialed the 900 number under Kathy's photograph. He had made thousands of investigative calls, both during his years in the FBI and in his private work for team owners and commissioners. But this was the first time he'd felt self-conscious.\n\nAn awful beeping noise blasted his ear, followed by an operator: \"We're sorry. Your call is being blocked.\"\n\nMyron looked up. \"The call won't go through.\"\n\nWin nodded. \"I forgot. We have a block on all 900 calls. Employees were calling them all the time and ringing up quite a bill\u2014not just the sex phones but astrologers, sports lines, psychics, recipes, even dial-a-prayers.\" He reached behind him and pulled out another phone. \"Use this one. It's my private line. No blocks.\"\n\nMyron redialed. The phone rang twice before being picked up. A woman's voice on tape husked, \"Hello, there. You've reached the fantasy phone line. If you're under eighteen or do not wish to pay for this call, please hang up now.\" Less than a second passed before she continued. \"Welcome to the fantasy phone line, where you can talk to the sexiest, most willing, most beautiful, most desirous women in the world.\"\n\nMyron noticed that the taped voice was speaking far more slowly now, as if she were reading to a kindergarten class. Each word was its own sentence.\n\n_\"Welcome. To. The. Fantasy...\"_\n\n\"In a moment you will talk directly to one of our wondrous, gorgeous, voluptuous, hot girls who are here to heighten your pleasure to new boundaries of ecstasy. One-on-one private conversation. Discreetly billed to your phone. You will talk live with your personal fantasy girl.\" The voice droned on with its own form of iambic pentameter. Finally the tape gave instructions: \"If you have a touch-tone phone, press one if you'd like to talk about the secret confessions of a naughty schoolteacher. Press two if...\"\n\nMyron looked up at Win. \"How long have I been on?\"\n\n\"Six minutes.\"\n\n\"Twenty-four dollars already,\" Myron said. \"Does the term 'total scam' mean anything to you?\"\n\nWin nodded. \"Talk about jerking off.\"\n\nMyron pressed a button, anything to get off this revolving tape. The phone rang ten times\u2014Christ, they knew how to stretch the time\u2014before he heard another female voice say, \"Hi, there. How are you today?\"\n\nHer voice was exactly what Myron had expected. Low and husky.\n\n\"Uh, hi,\" Myron fumbled. \"Look, I'd like\u2014\"\n\n\"What's your name, honey?\" she asked.\n\n\"Myron.\" He slapped his forehead and held back a profanity. Had he really been stupid enough to use his real name?\n\n\"Mmmmm, Myron,\" she said as if testing it out. \"I like that name. It's so sexy.\"\n\n\"Yeah, well, thanks\u2014\"\n\n\"My name is Tawny.\"\n\nTawny. Sure.\n\n\"How did you get my number, Myron?\"\n\n\"I saw it in a magazine.\"\n\n\"What magazine, Myron?\"\n\nThe constant use of his name was beginning to unnerve him. _\"Nips.\"_\n\n\"Oooo. I like that magazine. It makes me so, you know.\"\n\nA way with words. \"Listen, uh, Tawny, I'd like to ask you about your ad.\"\n\n\"Myron?\"\n\n\"Yes.\"\n\n\"I love your voice. You sound really hot. Do you want to know what I look like?\"\n\n\"No, not real\u2014\"\n\n\"I have brown eyes. I have long brown hair, kinda wavy. I'm five-six. And I'm a 36-24-36. C cup. Sometimes a D.\"\n\n\"You must be very proud but\u2014\"\n\n\"What do you like to do, Myron?\"\n\n\"Do?\"\n\n\"For fun.\"\n\n\"Look, Tawny, you seem very nice, really, but can I talk to the girl in the ad?\"\n\n\"I am the girl in the ad,\" she said.\n\n\"No, I mean, the girl whose picture is in the magazine on top of this phone number.\"\n\n\"That's me, Myron. I'm that girl.\"\n\n\"The girl in the photo is a blonde with blue eyes,\" Myron said. \"You said you had brown eyes and brown hair.\"\n\nWin gave him a thumbs-up, scoring one for the detailed eye of Myron Bolitar, ace investigator.\n\n\"Did I say that?\" Tawny asked. \"I meant blonde with blue eyes.\"\n\n\"I need to talk to the girl in the ad. It's very important.\"\n\nHer voice went down another octave. \"I'm better, Myron. I'm the best.\"\n\n\"I don't doubt that, Tawny. You sound very professional. But right now I need to talk to the girl in the ad.\"\n\n\"She's not here, Myron.\"\n\n\"When will she be back?\"\n\n\"I'm not sure, Myron. But just sit back and relax. We're going to have fun\u2014\"\n\n\"I don't want to be rude, but I'm really not interested. Can I talk to your boss?\"\n\n\"My boss?\"\n\n\"Yes.\"\n\nHer tone was different now. More matter-of-fact. \"You're kidding, right?\"\n\n\"No. I'm serious. Please put your boss on.\"\n\n\"Okay, then,\" she said. \"Hold on a second.\"\n\nA minute passed. Then two. Win said, \"She's not coming back. She's just going to see how long the chump will stay on the line and pour dollars down her pants.\"\n\n\"I don't think so,\" Myron said. \"She liked my voice, said I sounded hot.\"\n\n\"Oh, I didn't realize. Probably the first time she's ever said that.\"\n\n\"My thinking exactly.\" A few minutes later Myron put the receiver back in its cradle. \"How long was I on for?\"\n\nWin looked at his watch. \"Twenty-three minutes.\" He grabbed a calculator. \"Twenty-three minutes times three ninety-nine per minute.\" He punched in the numbers. \"That call cost you ninety-one dollars and seventy-seven cents.\"\n\n\"A rare bargain,\" Myron said. \"You want to hear something weird? She never said anything dirty.\"\n\n\"What?\"\n\n\"The girl on the phone. She never said anything dirty.\"\n\n\"And you're disappointed.\"\n\n\"Don't you find that a bit strange?\"\n\nWin shrugged, skimming through the magazine. \"Have you looked through this at all?\"\n\n\"No.\"\n\n\"Half the pages are advertisements for sex phones. This is clearly big business.\"\n\n\"Safe sex,\" Myron said. \"The safest.\"\n\nThere was a knock on the door.\n\n\"Enter,\" Win called out.\n\nEsperanza opened the door. \"Call for you. Otto Burke.\"\n\n\"Tell him I'll be right there.\"\n\nShe nodded and left.\n\n\"I have some time on my hands,\" Win said. \"I'll try to find out who placed the ad. We'll also need a sample of Kathy Culver's handwriting for comparison.\"\n\n\"I'll see what I can come up with.\"\n\nWin resteepled his hands, bouncing the fingertips gently against one another. \"You do realize,\" he began, \"that this photograph probably means nothing. Chances are there is a very simple explanation for all this.\"\n\n\"Maybe,\" Myron agreed, rising from his chair. He had been telling himself the same thing for the past two hours. He no longer believed it.\n\n\"Myron?\"\n\n\"What?\"\n\n\"You don't think it was a coincidence\u2014Jessica being in the bar downstairs, I mean.\"\n\n\"No,\" Myron said. \"I guess I don't.\"\n\nWin nodded. \"Be careful,\" he said. \"A word to the wise.\"\n\n# Chapter 4\n\nDamn him.\n\nJessica Culver sat in her family's kitchen, in the very seat she had sat in innumerable times as a child.\n\nShe should have known better. She should have thought it through, should have come prepared for any occurrence. But what had she done instead? She had gotten nervous. She had hesitated. She had stopped for a drink in the bar below his office.\n\nStupid, stupid.\n\nBut that wasn't all. He had surprised her, and she had panicked.\n\nWhy?\n\nShe should have told Myron the truth. She should have told him in a plain unemotional voice the real reason she was there. But she hadn't. She had been drinking unaware, and suddenly he had appeared, looking so handsome and yet so hurt and\u2014\n\n_Oh shit, Jessie, you are one fucked-up chick_.\n\nShe nodded to herself. Yup. Fucked-up. Self-destructive. And a few other hyphenated words she couldn't come up with right now. Her publisher and agent did not see it that way, of course. They loved her \"foibles\" (their term\u2014Jessie preferred \"fuck-ups\"), even encouraged them. They were what made Jessica Culver such an exceptional writer. They were what gave Jessica Culver's writing that certain \"edge\" (again, their term).\n\nPerhaps that was so. Jessie really couldn't say. But one thing was certain: These foibling fuck-ups had turned her life to shit.\n\n_Oh, pity the suffering artist! Thy heart bleeds for such torment!_\n\nShe dismissed the mocking tone with a shake of her head. She was unusually introspective today, but that was understandable. She had seen Myron, and that had led to a lot of \"what if-ing\"\u2014a verifiable avalanche of useless \"what if-ing\" from every conceivable height and angle.\n\nWhat if. She pondered it yet again.\n\nIn her typically self-centered way, she had seen the \"what if-ing\" only in terms of herself, not Myron. Now she wondered about him, about what his life had really been like since the world crumbled down upon him\u2014not all at once\u2014but in small, decaying bits. Four years. She had not seen him in four years. She had shoved Myron into some back closet in her mind and locked the door. She'd thought (hoped?) that would be the end of it, that the door could stand up to a little pressure without opening. But seeing him today, seeing the kind, handsome face high above those broad shoulders, seeing the still why-me stare in his eyes\u2014the door had blown off its hinges like something in a gas explosion.\n\nJessica had been overwhelmed by her feelings. She wanted to be with him so badly that she knew she had to get out right away.\n\n_Makes sense_ , she thought, _if you're a total fuck-up_.\n\nJessica glanced out the window. She was waiting for Paul's arrival. Bergen County police Lieutenant Paul Duncan\u2014Uncle Paul to her, since infancy\u2014was two years away from retirement. He had been her father's closest friend, the executor of Adam Culver's will. They had both worked in law enforcement\u2014Paul as a cop, Adam as the county medical examiner\u2014for more than twenty-five years.\n\nPaul was coming to finalize the details for her father's memorial service. No funeral for Adam Culver. He wouldn't hear of it. But Jessica wanted to talk to Paul about another matter. Alone. She did not like what was going on.\n\n\"Hi, honey.\"\n\nShe turned to the voice. \"Hi, Mom.\"\n\nHer mother came up through the basement. She was wearing an apron, her fingers fiddling with the large wooden cross around her neck. \"I put his chair in storage,\" she explained in a forced matter-of-fact tone. \"Just cluttering space up here.\"\n\nFor the first time Jessica realized that her father's chair\u2014the one her mother must have been referring to\u2014was gone from the kitchen table. The simple unpadded four-legged chair her father had sat in for as long as Jessica could remember, the one closest to the refrigerator, so close that her father could turn around, open the door, and stretch for the milk on the top shelf without getting up, had been taken away, stored in some cob-webbed corner of the basement.\n\nBut not so Kathy's.\n\nJessie's gaze touched down on the chair to her immediate right. Kathy's chair. It was still here. Her mother had not touched it. Her father, well, he was dead. But Kathy\u2014who knew? Kathy could, in theory, walk through the back door right this very minute, banging it against the wall as she always did, smile brightly, and join them for dinner. The dead were dead. When you lived with a medical examiner, you understood just how useless the dead were. Dead and buried. The soul, well, that was another matter. Jessie's mom was a devout Catholic, attending mass every morning, and during crises like these her religious tenacity paid off\u2014like someone who spent time in a gym finally finding a use for their new muscles. She could believe without question in a divine and joyous afterlife. Such a comfort. Jessica wished she could do the same, but over the years her religious fervor had become a strict couch potato.\n\nExcept, of course, Kathy might not be dead. Ergo the chair\u2014Mom's lantern kept lit to guide her youngest back home.\n\nJessica awoke most mornings bolting upright in her bed, thinking about\u2014no, inventing new possibilities for\u2014her younger sister. Was Kathy lying dead in a pit somewhere? Buried under brush in the woods? A skeleton gnawed on by animals and inhabited by maggots? Was Kathy's corpse stuck in some cement foundation? Was it weighed down in the bottom of some river like the little undersea man in the living-room aquarium? Had she died painlessly? Had she been tortured? Had her body been chopped into small bits, burned, broken down with acid...\n\nOr was she still alive?\n\nThat eternal spring.\n\nHad Kathy possibly been kidnapped? Was she living in white slavery under the thumb of some Middle East sheikh? Or was she living chained to a radiator on a farm in Wisconsin like something on _Geraldo_? Could she have banged her head, forgotten who she was, and was now living as a street person with amnesia? Or had she simply run away to a different world?\n\nThe possibilities were endless. Even those lacking creativity can come up with a million different horrors when their loved one suddenly vanishes\u2014or more painfully, a million different hopes.\n\nJessica's thoughts were chased away by the tired chugging of a car engine. A familiar Chevy Caprice blanketed with tiny dents pulled up. It looked like a retrieval car at a driving range. She stood and hurried out the front door.\n\nPaul Duncan was a stocky man, compact, with salt-and-pepper hair now turning defiantly toward salt. He walked purposely, the way cops do. He greeted her on the front stoop with a big smile and kiss on the cheek. \"Hey, beautiful! How are you?\"\n\nShe hugged him. \"I'm okay, Uncle Paul,\" she said.\n\n\"You look great.\"\n\n\"Thanks.\"\n\nPaul shaded his eyes from the sun. \"Come on, let's go inside. It's hot as hell out here.\"\n\n\"In a minute,\" she said, putting a hand on his forearm. \"I want to talk to you first.\"\n\n\"What about?\"\n\n\"My father's case.\"\n\n\"I'm not handling that, honey. I don't do homicides anymore, you know that. Besides, it would be a conflict of interest\u2014me being Adam's friend and all.\"\n\n\"But you have to know what's going on.\"\n\nPaul Duncan nodded slowly. \"I do.\"\n\n\"Mom said the police think he was killed in a robbery attempt.\"\n\n\"That's right.\"\n\n\"You don't believe that, do you?\"\n\n\"Your father was robbed,\" he said. \"His wallet was gone. His watch. Even his rings. The guy stripped him clean.\"\n\n\"To make it look like a robbery.\"\n\nPaul smiled then, gently\u2014the way, she remembered, he had at her confirmation and Sweet Sixteen party and high school graduation. \"What are you getting at, Jess?\"\n\n\"You don't find this whole thing odd?\" she asked. \"You don't see a connection between this and Kathy?\"\n\nHe stumbled a step back, as if her words had given him a gentle push. \"What connection? Your sister vanished from her college campus. Your father was murdered by a robber a year and a half later. Where do you see a connection?\"\n\n\"Do you really believe that they have nothing to do with each other?\" she asked. \"Do you honestly believe that lightning struck twice in the same place?\"\n\nHe put his hands in his pockets. \"If you mean do I think your family has been the victim of two separate awful tragedies, the answer is yes. It happens all the time, Jess. Life is rarely fair. God doesn't go around divvying out the bad in equal doses. Some families go through life with nary a scratch. Some get too much. Like yours.\"\n\n\"So it's fate,\" she said. \"That's your answer. Fate.\"\n\nHe threw his hands up. \"Fate, lightning striking twice\u2014these are your phrases. You're the writer here, not me. I just call it a tragedy. I just call it a tragic, somewhat bizarre coincidence. I've seen a lot stranger. So had your dad.\"\n\nThe front door opened. Mom stood in the doorway. \"What's going on?\"\n\n\"It's nothing, Carol. We were just talking.\"\n\nCarol looked at her daughter. \"Jessica?\"\n\nHer eyes stayed on Paul's, probing. \"Just talking, Mom.\"\n\nJessica turned away and stepped back inside. Paul Duncan watched her, letting loose a silent breath. He had suspected she would be a problem\u2014Jessica never accepted easy solutions to anything in life, even when the answer was simple. Yep, he had hoped it wouldn't happen, but he had definitely foreseen this possibility.\n\nHe just wasn't sure what he should do about it.\n\nMidnight.\n\nAt ten P.M. Christian Steele had crawled under the blanket, read for ten minutes, and then switched off the light. Since then he had lain on his back in the dark, staring at the ceiling, not moving, not fooling himself into even hoping that sleep was imminent.\n\n\"Kathy,\" he said out loud.\n\nHis mind floated about aimlessly, settling like a butterfly for only brief moments before moving on. Darkness surrounded him, but not silence. There was no such thing as silence at football camp. Christian heard kegs being thrown, loud music, laughter, singing, swearing. He could distinctly hear Charles and Eddie, his offensive tackles, in the next room. They were permanently set on loud, like a radio turned up before the knob was ripped out. Christian was not above partying too, having fun by consuming alcohol until he hugged the porcelain god and puked up his offering. But not tonight.\n\nGod, not tonight.\n\n\"Kathy,\" he said again.\n\nWas it possible? After all this time...\n\nSo many things were happening at once. School was over. The Titans' minicamp began the day after tomorrow. The scrutiny of the press had grown more intense than ever. He liked the attention, liked being on the cover of _Sports Illustrated_ , liked the awe in people's faces when they spoke to him. Nice kid, they always said. Real nice. As though they expected him to be rude just because he could throw a pigskin with precision. As though he should somehow feel as though he belonged to a higher species, far above them, because he happened to be a good athlete.\n\nChristian was excited. He was scared. He knew he had to think about the future. Myron had told him of the dangers and of how short-lived fame could be. Myron was, after all, a classic example. He had told Christian about the importance of cashing in now, that his career would at best last ten years. So much was at stake. So much. He was famous now, but there was a big difference between college famous and pro famous. Soon he'd have it all. Competition. Fame. Real money\u2014not just the alumni secret handouts....\n\nBut so what?\n\n\"Kathy...\"\n\nHis phone rang.\n\nChristian shot up, his heart beating like a rabbit's. Fast reflexes. Sometimes they played against you. It was only the phone. Probably Charles or Eddie telling him, hey, it's party time! They'd both gotten drafted too. Charles had gone in the second round to Dallas. Eddie in the fifth to the Rams.\n\nHe picked up the phone. \"Hello?\"\n\nNo response.\n\n\"Hello?\" he said again.\n\nNothing. But the phone had not been hung up. Someone was there, silently holding the receiver to their ear.\n\n\"Who is this?\"\n\nNothing.\n\nChristian hung up. He began to lie back down when the phone rang again. He picked up the receiver.\n\n\"Hello?\"\n\nSilence again. Christian tried to listen more closely. Nothing. Or\u2014or was that breathing? Panic seized him. He couldn't say why. It was just a prankster calling on his unlisted phone. It might even be Charles or Eddie playing some kind of joke. Nothing to get upset about.\n\nExcept he was upset.\n\nHe cleared his throat. \"What do you want?\"\n\nStill nothing.\n\n\"If you call back again, I'll call the cops.\"\n\nHe slammed the phone down. His hand shook. He was just about to try to settle back down when he remembered something.\n\nStar. Six. Nine.\n\nThe phone company had sent something in the mail today. There had been advertisements on the TV\u2014a pregnant woman trying to get to the ringing phone, trudging across the room toward the phone, but when she arrived the caller had already hung up. Then what? She picked up the phone and the voice-over\u2014Cliff Robertson's or someone like that\u2014said something like \"You just missed the call. Was it important? Was it someone you wanted to talk to? There is only one way to find out. Press the star and then six and nine.\" They demonstrated it on the screen now, in case anyone wasn't sure how to use a phone. Then the voice-over continued. \"You'll be connected to your previous caller, even if the number is busy. We'll keep dialing for you, leaving your phone line free to make or receive other calls.\"\n\nThe pregnant woman listened to a phone ring and then spoke to her relieved husband, who was working on some drafting board at work.\n\nChristian picked up the phone. Then he hit the star, the six, and the nine.\n\nThe phone rang.\n\nHe rubbed his chin. A moment later a robotic operator came on. \"The number is currently busy. We will ring you back when the line is free. Thank you.\"\n\nChristian replaced the receiver. He sat up and waited. The partying was still going on. He could hear three or four distinct partying areas. Someone shouted, \"Yahooo!\" A window crashed. People cheered. His larger teammates were playing keg toss, a sort of discus throw involving beer kegs.\n\nThe phone rang.\n\nHe snatched the receiver as if it were a loose ball on the turf. The phone was ringing back the number\u2014just like the pregnant lady's on the television. After the fourth ring the phone was picked up.\n\nAn answering machine.\n\nA voice said, \"Hi. We're not in right now. Please leave a message at the beep, and we'll be sure to call you back. Thanks.\"\n\nThe phone slipped from Christian's grip. A chilly hand caressed the back of his neck. A sound\u2014some kind of choking noise\u2014escaped his lips. Christian tried to form words but he couldn't.\n\nThe answering machine. The voice.\n\nIt was Kathy.\n\n# Chapter 5\n\nMyron staggered into his office, punch-drunk from lack of sleep. He had not even bothered climbing into bed the night before. He tried to read, but the words swam in front of his eyes in meaningless waves. He put on the television. Nick at Nite, the cultural equivalent of aerosol cheese. Back-to-back episodes of _F Troop_ for three hours. Larry Storch's portrayal of Agarn was, in a phrase, pure thespian genius. Who knew that hitting someone repeatedly with a big hat could be so funny?\n\nBut not even such highbrow entertainment could stop his mind from going back to one thought: Jess was back. And like Win had said, it was no coincidence.\n\nAt midnight his mother had come down in her robe.\n\n\"Hon, you all right?\"\n\n\"I'm fine, Mom.\"\n\n\"You seemed distracted all night.\"\n\n\"It's nothing. Just have a lot of work.\"\n\nShe looked at him with her a-mother-is-psychic-and-knows look of disbelief. \"Whatever you say.\"\n\nAt the age of thirty-one Myron still lived at home. True, he had his own space, his own bedroom and bathroom in the basement. But there was no denying it. Myron still lived with Mommy and Daddy.\n\nFive minutes after his mother had gone back to bed, Christian Steele called Myron on his private line, the one that rang softly in the basement so as not to wake up his parents, both of whom slept so lightly, Myron was sure they'd been some kind of ghetto lookouts in a previous life. He filled Myron in on the weird phone calls.\n\nMyron was familiar with the star-six-nine, known as Return Call. The phone company charged on a \"pay-per-use\" basis\u2014around seventy-five cents per use. The problem was, Return Call did not trace the number. It automatically redialed the number of the last incoming call received, not letting you know the number. Star-five-seven\u2014Call Trace\u2014would have done the job, though the number is merely reported to the local phone company, which gives it only to the proper authorities.\n\nStill, Myron would call some of his old sources at the phone company, see what he could find out. He knew that star-six-nine worked only for certain local areas. That meant the call was not long distance. A start. Better than nothing. He would also put Caller ID or a trace on Christian's phone. Taps were no longer like you saw on television, the hero anxiously trying to get the caller to stay on the line until it was completed. They were automatic. Caller ID actually showed you the incoming number before you picked up the phone.\n\nBut of course, none of that answered the larger questions:\n\nWas it really Kathy's voice Christian had heard? And if so, what did that mean?\n\nLots of _preguntas._ Not too many answers.\n\nHe approached Esperanza's desk. \"How's it going?\"\n\nShe pierced him with a glare, shook her head in disgust, and looked back down at her desk.\n\n\"Back on decaf?\" he asked.\n\nAnother glare. Myron shrugged. \"Any messages?\"\n\nA head shake. Esperanza muttered something. Myron thought he picked up the Spanish equivalent of \"asswipe.\"\n\n\"You want to tell me why you're so upset?\"\n\n\"Right,\" she said bitingly. \"Like you don't know.\"\n\n\"I don't.\"\n\nThe glare was back. Women had a talent for glares. Esperanza had a divine gift.\n\n\"Forget it,\" he said. \"Just get me Otto Burke on the phone.\"\n\n\"Now?\" Esperanza said, her voice dripping with sarcasm. \"Won't you be busy?\"\n\n\"Just do it, please, okay? You're starting to piss me off.\"\n\n\"Oooo. I'm quaking.\"\n\nMyron shook his head. He had no time for her moods right now. He crossed the room and opened his office door. He stopped short.\n\n\"Hi.\"\n\nHe cleared his throat and closed the door behind him. \"Hello, Jessica.\"\n\nFor most athletes, Jessica thought, the spotlight fades slowly. But for a tragic few, it vanishes as though from a sudden power failure, bathing the athlete in dazzling darkness.\n\nSuch was the case with Myron.\n\nFor most athletes the expectation game helps dim the light gradually. A high school star becomes a college bench warmer. The light dims. A college starter realizes he will not be the team's high scorer. The light dims. The college superstar realizes he will never make it to the pros. The light dims. And then there are those very few, those who are one in a million, those with almost Wolfean \"right stuff,\" who become professional athletes.\n\nFor those, the light is blinding, forever damaging the vision of the ones who stare directly into it. That was what made the dimming so important. An athlete could get used to losing the light slowly. His career would peak before tapering off just slightly. He would brighten from the inexperienced rookie to the player in his prime, and then the light would begin to fade as he moved past seasoned vet.\n\nFor Myron that had not happened.\n\nHe had been one of those select few who basked in the most potent wattage imaginable, as if the spotlight shone on him and from inside of him. His basketball talent had first became apparent in the sixth grade. He had gone on to break every scoring and rebounding record in Essex County, New Jersey, a perennial basketball stronghold. Myron was short for a forward, a program six-six (really only six-four), but he was a physical brute, a bull, and a hell of a leaper for a white man. He was highly recruited, chose Duke, and won two NCAA titles in four years.\n\nThe Boston Celtics had drafted him in the first round, the eighth pick overall. Myron's spotlight grew impossibly bright.\n\nAnd then the fuse blew.\n\nA freak injury, they called it. It was a preseason game against the Washington Bullets. Two players weighing a combined six hundred pounds sandwiched the rookie Myron Bolitar. The doctors threw all kind of terms at the man-child who had never been injured before, not even a twisted ankle. Multiple fractures, they said. Shattered kneecap. Casts. Wheelchair. Crutches. Cane.\n\nYears.\n\nSixteen months later Myron could walk, though the limp lasted another two years. He never came back. His career was over. The only life he had ever known had been stripped from him. The press had done a story or two, but Myron was quickly forgotten.\n\nComplete blackout.\n\nJessica frowned. Spotlight. Bad metaphor. Too clich\u00e9 and inaccurate. She shook her head and looked up at him.\n\n\"That explains it,\" Myron said.\n\n\"Explains what?\"\n\n\"Esperanza's mood.\"\n\n\"Oh.\" She smiled at him. \"I told her we had an appointment. She didn't seem pleased to see me.\"\n\n\"No kidding.\"\n\n\"She'd still kill me for a nickel, huh?\"\n\n\"Or half that much,\" he replied. \"Want some coffee?\"\n\n\"Sure.\"\n\nHe picked up his phone. \"Can you get me a black coffee? Thanks.\" He put the receiver back in its cradle and looked up at her.\n\n\"How's Win?\" she asked.\n\n\"Good.\"\n\n\"His family owns the building?\"\n\n\"Yes.\"\n\n\"I understand Win's become quite a financial whiz\u2014despite himself.\"\n\nMyron nodded, waited.\n\n\"So you're still hanging around with Win,\" she continued. \"You still have Esperanza. Not a lot changes.\"\n\n\"Plenty changes,\" he said.\n\nEsperanza appeared at the door, the scowl still on her face. \"Otto Burke was in a meeting.\"\n\n\"Try Larry Hanson.\"\n\nShe handed the coffee to Jessica, smiled eerily, and left. Jessica studied the cup. \"Think she spat in it?\"\n\n\"Probably,\" Myron replied.\n\nShe put it down. \"I need to cut back anyway.\"\n\nMyron moved around his desk and sat down. The wall behind him was covered with theater posters. All musicals. His fingers drummed the desk.\n\n\"I'm sorry about yesterday,\" she said. \"I wanted to surprise you, catch you off guard. Not the other way around.\"\n\n\"Still seeking the upper hand?\"\n\n\"I guess so, yeah. Old habit.\"\n\nHe shrugged but said nothing.\n\n\"I need your help,\" she said.\n\nHe waited.\n\nShe took a breath and plunged. \"The police say my father was killed in a robbery attempt. I don't believe it.\"\n\n\"What do you believe?\" he asked.\n\n\"I think his murder has something to do with Kathy.\"\n\nMyron was not surprised. He leaned forward, his eyes never staying on hers for very long. \"What makes you say that?\"\n\n\"The police dismiss it as a coincidence,\" she said simply. \"I'm not big on coincidences.\"\n\n\"What about your dad's friend on the force, what's-his-name?\"\n\n\"Paul Duncan.\"\n\n\"Right, him. Have you spoken to him?\"\n\n\"Yes.\"\n\n\"And?\"\n\nShe began tapping her foot, an old, subconscious, annoying habit. She made herself stop. \"Paul says it was a robbery too. He spews out all the facts about the crime scene, the missing wallet, the missing jewelry, that kind of thing. He is perfectly logical and objective, which is not his way.\"\n\n\"What do you mean?\"\n\n\"Paul Duncan is a passionate man. A hothead. Here his best friend has been murdered, and he seems almost blas\u00e9 about it. It's not like him.\" She stopped, shifted in her chair. \"Something isn't right here, I don't know how else to explain it.\"\n\nMyron rubbed his chin but kept quiet.\n\n\"Look, you know I was never very close to my father,\" she continued. \"He wasn't an easy man to love. He was far better with his corpses than with breathing entities. He liked the idea of family, the concept\u2014it was the actual execution he found wearisome. But I still have to find out the truth. For Kathy.\"\n\n\"How did your father and Kathy get along?\" Myron asked.\n\nShe thought about it a moment. \"Better lately. When we were kids, they weren't very close. Kathy was a mama's girl, always hanging around my mom, wanting to be like her, the whole bit. But when she vanished, I'd venture to guess she was closer to my dad than my mom. He was crushed when she disappeared. He became obsessed. No, 'obsessed' isn't strong enough. All of us were obsessed, of course. But not like my father. It consumed him entirely. Everything about him changed. He had always been the quiet county medical examiner, the man who made no waves. Now he was using his position to keep the pressure on twenty-four hours a day. He became paranoid, convinced the police weren't doing all they could do to find her. He even started his own investigation.\"\n\n\"Did he find anything?\"\n\n\"No. Not that I know of.\"\n\nMyron looked away. At the far wall. A movie still of the Marx Brothers. _A Night at the Opera_. Groucho looked back but offered no answers.\n\n\"What is it?\" she asked.\n\n\"Nothing. Go on.\"\n\n\"There isn't much else. I can only tell you that my father was acting very strangely the past few weeks. He started calling me all the time when previously we'd only talked maybe three times a year, sounding a little teary. It was like he was play-acting the part of perfect Daddy with renewed vigor. I couldn't tell if it was a serious change or just a phase.\"\n\nMyron nodded, looking off again. He said nothing. Jessica almost thought he'd completely drifted off when he finally spoke, his voice almost inaudibly soft. \"What do you think happened to Kathy?\" he asked.\n\n\"I don't know.\"\n\n\"Do you think she's dead?\"\n\n\"I\u2014\" She stopped. \"I miss her. It's... I don't want to think she's dead.\"\n\nHe nodded again. \"So what do you want me to do?\"\n\n\"Look into it. Find out what's going on.\"\n\n\"Assuming something is going on.\"\n\n\"Right.\"\n\n\"Why me?\"\n\nShe thought about it a moment. \"I'm not sure,\" she said. \"I thought you'd believe me. I thought you'd help.\"\n\n\"I'll help,\" he said. \"But understand one thing: I have an important business interest in settling this whole thing.\"\n\n\"Christian?\"\n\n\"I'm his agent,\" he continued. \"I'm responsible for his well-being.\"\n\n\"He still misses my sister,\" she said.\n\n\"Yes.\"\n\n\"Is he okay?\"\n\nMyron's face remained set. \"He's fine.\"\n\n\"He's a good kid. I like him.\"\n\nMyron nodded.\n\nJessica rose and stepped toward the window. Myron averted his eyes. He did not like to look at her for too long at one time. She understood. It hurt her too. She looked down at Park Avenue, twelve stories below. A taxi driver with a turban was shaking his fist at an old woman with a cane. The old woman whacked him and ran. The driver fell. The turban did not even shift.\n\n\"Hiding your feelings from me has never been your forte,\" she said, still staring out the window. \"What don't you want to tell me?\"\n\nHe did not reply.\n\n\"Myron...\"\n\nEsperanza saved him, bursting through the door without knocking. \"Larry Hanson is out of the office,\" she said.\n\nWin came in behind her. \"I got something for us on that magazine....\" His voice died out when he saw Jessica.\n\n\"Hi, Win,\" she said.\n\n\"Hello, Jessica Culver.\" They embraced. \"My goodness, you look utterly fantastic. I read an article on you the other day, calling you the Literary Sex Symbol.\"\n\n\"You shouldn't read such trash.\"\n\n\"It was in my dentist's waiting room. Honest.\"\n\nAn uncomfortable pause followed. Esperanza broke it by pointing at Jessica, making a gagging motion by sticking her finger in her mouth, and then storming out.\n\n\"Ever the enchantress,\" Jessica muttered.\n\nMyron stood. \"Where are you staying?\"\n\n\"At my mom's.\"\n\n\"Same number?\"\n\n\"Yes.\"\n\n\"I'll call you later. Right now I've got to go with Win.\"\n\nJessica looked toward Win. He grinned at her. His face, as always, gave away nothing. \"I have a meeting with my editor this afternoon,\" she said. \"But I'll be home all night.\"\n\n\"Fine. I'll call you then.\"\n\nAn awkward impasse. No one knew exactly how to say good-bye. A wave? A handshake? A kiss?\n\n\"We've got to go,\" Myron said. He sprinted past her, never getting too close. Win shrugged at her in a what-can-you-do fashion and followed. She watched them disappear around the corner. Batman and Robin heading to the Bat-poles.\n\nShe left then. She had seen Myron twice now, and they had not yet touched\u2014not even brushed up against one another.\n\nIt was an odd thing to wonder about.\n\n# Chapter 6\n\n\"What did you find out?\" Myron asked.\n\nWin whipped the wheel to the right. The Jag XJR responded with nary a squeal. They had been driving without speaking for the past ten minutes, Win's CD player the only sound. Win favored show tunes. _Man of La Mancha_ was on now. Don Quixote serenaded his beloved Dulcinea.\n\n\" _Nips_ magazine is published by HDP,\" Win answered.\n\n\"HDP?\"\n\n\"Hot Desire Press.\" Another Bat-turn. The Jag accelerated past eighty.\n\n\"Speed limits,\" Myron said. \"Heard of them?\"\n\nWin ignored him. \"Their editorial office is located in Fort Lee, New Jersey.\"\n\n\"Editorial office?\"\n\n\"Whatever. We have an appointment with Mr. Fred Nickler, managing editor.\"\n\n\"His mother must be proud.\"\n\n\"Moralizing,\" Win mused. \"Nice.\"\n\n\"What did you tell Mr. Nickler?\" Myron asked.\n\n\"Nothing. I called and asked if we could see him. He said yes. Seemed like a very pleasant fellow.\"\n\n\"I'm sure he's a prince.\" Myron looked out the window. Buildings blurred. They fell back into silence. \"You're probably wondering what Jessica was doing in my office.\"\n\nWin gave a halfhearted shrug. It was not his way to pry.\n\n\"It's her father's murder. The police say it was a robbery. She thinks otherwise.\"\n\n\"How does she see it?\"\n\n\"She thinks there's a connection between his murder and Kathy.\"\n\n\"So the plot thickens. Are we going to help her?\"\n\n\"Yes.\"\n\n\"Goodie. So do we think there is a connection?\"\n\n\"Yes.\"\n\n\"Yes,\" Win agreed.\n\nThey pulled into the driveway of a building that could have been either a nice warehouse or low-rent office space. No elevator, but then again, only three levels. HDP, Inc., was on the second floor. When they entered the outer office, Myron was a bit surprised. He was not sure what he'd expected, but he had thought the dwellings of a sleaze merchant would not be so... nondescript. The walls were white with inexpensive but tastefully framed art posters\u2014McKnight, Fanch, Behrens. Mostly scenery shots of beaches and sunset. Nothing with naked breasts. Surprise number one. Surprise number two was the unremarkable receptionist. She was strictly standard issue, not an overaged, bleach-blond, flabby ex-bunny\/sexpot\/porno starlet with a breathy giggle and seductive wink.\n\nMyron was almost disappointed.\n\n\"May I help you?\" the receptionist asked.\n\nWin said, \"We're here to see Mr. Nickler.\"\n\n\"Your names, please?\"\n\n\"Windsor Lockwood and Myron Bolitar.\"\n\nShe picked up the phone, buzzed in, and a moment later said, \"Right through that door.\"\n\nNickler greeted them with a firm handshake. He was dressed in a blue suit, red tie, white shirt\u2014conservative as a Republican senatorial candidate. Surprise number three. Myron had expected gold chains or a Joey Buttafuoco earring or at the very least a pinkie ring. But Fred Nickler wore no jewelry, except for a plain wedding band. His hair was gray, his complexion a bit washed out.\n\nWin whispered, \"He looks like your uncle Sid.\"\n\nIt was true. The publisher of _Nips_ magazine looked like Sidney Griffin, popular suburban orthodontist.\n\n\"Please have a seat,\" Nickler said, moving back behind his desk. He smiled at Myron. \"I was at the Final Four when you guys beat Kansas. Twenty-seven points including the game winner. Hell of a performance Incredible.\"\n\n\"Thank you,\" Myron said.\n\n\"Never seen anything like it. The way that final shot kissed the backboard.\"\n\n\"Thank you.\"\n\n\"Just incredible.\" Nickler renewed his smile, shaking his head in awe at the memory. Then he sat back. \"So, what can I do for you gentlemen?\"\n\nMyron said, \"We have a couple of questions about an ad in one of your, uh, publications.\"\n\n\"Which one?\"\n\n\" _Nips._ \" Saying the word felt grungy. Myron tried not to make a face.\n\n\"Interesting,\" Nickler replied.\n\n\"What makes you say that?\"\n\n\" _Nips_ is a relatively new publication, and it's doing poorly\u2014far and away the worst of HDP's monthlies. I'm going to give it another month or two, and then it'll probably fold.\"\n\n\"How many magazines do you publish?\"\n\n\"Six.\"\n\n\"Are they all like _Nips_?\"\n\nNickler chuckled lightly. \"They are all pornographic magazines, yes. And they are all completely legal.\"\n\nMyron handed him the magazine Christian had given him. \"When was this printed?\"\n\nFred Nickler barely glanced at it. \"Four days ago.\"\n\n\"That's all?\"\n\n\"It's our most recent issue\u2014they've barely hit the stands. I'm surprised you found one.\"\n\nMyron opened to the proper page. \"We'd like to know who paid for this advertisement.\"\n\nNickler put on a pair of half-moon glasses. \"Which one?\"\n\n\"Bottom row. The Lust Line.\"\n\n\"Oh,\" he said. \"A sex phone.\"\n\n\"Is there a problem?\"\n\n\"No. But this ad wasn't paid for.\"\n\n\"What do you mean?\"\n\n\"It's the nature of the business,\" Nickler explained. \"Someone calls me up to place an ad for a dial-a-porn line. I tell him it costs X amount. He says, wow, I'm just starting out, I can't afford it. So if it looks like a good idea, I go in fifty-fifty with him. In other words, I take care of the marketing, if you will, while my partner takes care of the technical side\u2014phones, cables, girls to work the phones, whatever else. Then we split it down the middle. It limits both of our risks.\"\n\n\"Do you do this a lot?\"\n\nHe nodded. \"Ninety percent of my advertising comes for fantasy lines. I'd say I have a piece of the action in three-quarters of them.\"\n\n\"Can you give us the name of your partner on this particular venture?\"\n\nNickler studied the picture in the magazine. \"You're not with the police, are you?\"\n\n\"No.\"\n\n\"Private investigators?\"\n\n\"No.\"\n\nHe took off his glasses. \"I'm fairly small-time,\" he said. \"I have my own little niche. It's the way I like it. No one bothers me, and I don't bother anybody else. I have no interest in a lot of publicity.\"\n\nMyron shot a glance at Win. Nickler had a family, probably a nice house in Tenafly, told the neighbors he was in publishing. Pressure could be applied. \"I'll be frank with you,\" Myron said. \"If you don't help us out, it may blow up into something major. Newspapers, TV, the works.\"\n\n\"Is that a threat?\"\n\n\"Absolutely not.\" Myron reached into his wallet and took out a fifty-dollar bill. He placed it on the desk. \"We just want to know who put this ad in.\"\n\nNickler pushed the bill back toward Myron, his expression suddenly irritated. \"What is this, a movie? I don't need a payoff. If the guy has done something wrong, I want no part of him. This business has enough problems as it is. I run a straight operation. No underage girls, nothing illegal in any way, shape, or form.\"\n\nMyron looked at Win. \"Told you he was a prince.\"\n\n\"Think what you want,\" Nickler said in a voice that said he'd been down this road many times before. \"This is a business like any other. I'm just an honest guy trying to make an honest buck.\"\n\n\"Real American of you.\"\n\nHe shrugged. \"Look, I don't defend everything about this business. But there are plenty of worse. IBM, Exxon, Union Carbide\u2014these are the real monsters, the real exploiters. I don't steal. I don't lie. I satisfy a societal need.\"\n\nMyron had a quick comeback, but Win stopped him with a shake of his head. He was right. What was the point in antagonizing the guy?\n\n\"Could we have the name and address, please?\" Myron asked.\n\nNickler opened a drawer behind him and pulled out a file. \"Is he in some sort of trouble?\"\n\n\"We just need to talk to him.\"\n\n\"Can you tell me why?\"\n\nWin spoke to Nickler for the first time. \"You don't want to know.\"\n\nFred Nickler hesitated, saw Win's steady gaze, then nodded. \"The company is called ABC. They have a p.o. box in Hoboken, number 785. The guy's name is Jerry. I don't know anything else about him.\"\n\n\"Thanks,\" Myron said, standing. \"One more question if you don't mind: Have you ever seen the girl in the ad?\"\n\n\"No.\"\n\n\"You're sure?\"\n\n\"Positive.\"\n\n\"If you do or if you think of anything else, will you give me a call?\" Myron handed him a card.\n\nNickler looked as if he wanted to ask a question, his gaze continually drifting back to Kathy's photograph, but he settled for saying \"Sure.\"\n\nOnce outside, Win asked, \"What do you think?\"\n\n\"He's lying,\" Myron said.\n\nBack in the car Myron asked, \"Can I use the phone?\"\n\nWin nodded, his foot not slackening on the pedal. The speedometer was hovering at seventy-five. Myron watched it as if it were a taxi meter on a long ride, keeping his gaze averted from the blur of a street.\n\nMyron dialed the office. Esperanza answered the phone after one ring.\n\n\"MB SportReps.\"\n\nMB SportReps. The M stood for Myron, the B for Bolitar. Myron had thought of the name himself, though he rarely bragged about it. \"Did Otto Burke or Larry Hanson call?\"\n\n\"No. But you have lots of messages.\"\n\n\"Nothing from Burke or Hanson?\"\n\n\"You deaf?\"\n\n\"I'll be back in a little while.\"\n\nMyron hung up. Otto and Larry should have called by now. They were avoiding him. The question was, why?\n\n\"Trouble?\" Win asked.\n\n\"Maybe.\"\n\n\"I believe we need a rejuvenation.\"\n\nMyron looked up. He recognized the street immediately. \"Not now, Win.\"\n\n\"Now.\"\n\n\"I have to get back to the office.\"\n\n\"It'll keep. You need inner energy. You need focus. You need balance.\"\n\n\"I hate it when you talk like that.\"\n\nWin smiled, pulling into the parking lot. \"Come along now. I'd hate to kick your ass right here in the car.\"\n\nThe sign read MASTER KWAN'S TAE KWON DO SCHOOL. Kwan was nearing seventy now and rarely conducted classes any longer, choosing instead to hire well-tutored underlings to handle that work. Master Kwan stayed in his high-tech office, surrounded by four television screens so he could monitor the classes. Occasionally he leaned forward and barked something into a microphone, scaring some poor student into attention. Like something out of _The Wizard of Oz_.\n\nIf Master Kwan's English improved a bit, it might reach the level of pidgin. Win had brought him over from Korea fourteen years ago, when Win was only seventeen. It seemed to Myron that Master Kwan had spoken better English back then.\n\nWin and Myron changed into their white uniforms called _dobok_. Both men wrapped black belts around their stomachs. Win was a sixth-degree black belt, about as high a ranking as anybody in the United States. He had been studying tae kwon do since the age of seven. Myron had picked it up in college, giving him a dozen years of studying and a third-degree black belt.\n\nThey approached Master Kwan's door, paused in the doorway until he acknowledged their presence, then bowed at the waist. \"Good afternoon, Master Kwan,\" they said in unison.\n\nKwan smiled toothlessly. \"You here early.\"\n\n\"Yes, sir,\" Win replied.\n\n\"Need help?\"\n\n\"No, sir.\"\n\nKwan dismissed them by spinning back to his television screens. Myron and Win bowed once more and moved into the private _dojang_ for the upper-ranked black belts. They began with meditation, something Myron had never quite gotten the full grasp of. Win loved it. He did it for at least an hour a day. Win folded himself into a lotus position. Myron settled for sitting Indian style. Both men closed their eyes, placed their thumbs on the palm directly below the pinkie, and tilted their palms toward the ceiling. They rested their hands on their knees. Instructions echoed through Myron's mind like a mantra. Back straight. Bottom of tongue curled up against the back of the upper teeth. He breathed in through his nose for six seconds, concentrating on pushing the air down into the pit of his gut, making sure that his chest did not move, that only his abdomen expanded. Then he held the air down deep, counting to himself to prevent his mind from wandering. After seven seconds he slowly released the air through the mouth for a ten count, making sure to empty completely his contracting gut. Then he waited four seconds before breathing in again.\n\nWin did this painlessly. He did not count. His mind went blank. Myron always counted, needing it to keep his mind from wandering back to the problems of the day\u2014especially on a day like today. But in spite of himself he began to relax, to feel the tension leave his body with every long exhalation. It almost tingled.\n\nThey meditated for ten minutes before Win opened his eyes and said, _\"Barro.\"_ Korean for stop.\n\nThey performed deep stretching exercises for the next twenty minutes. Win had the flexibility of a ballet dancer, performing full splits effortlessly. Myron had gained a lot of flexibility since taking up tae kwon do. He believed it had helped him gain six inches on his vertical leap in college. He could almost do a full split, but he couldn't hold it long.\n\nIn short Myron was flexible; Win was Gumby.\n\nThey went through their forms or _poomse_ next, a complicated set of moves not unlike a violent dance step. What many exercised-crazed junkies never realized is that the martial arts are the ultimate aerobic workout. You are in constant motion\u2014jumping, turning, spinning\u2014propelling both arms and both legs nonstop for a half hour at a time. Low block and front kick, high block and punch, middle block and roundhouse kick. Inside blocks, outside blocks, knife-hand, fists, palm strikes, knees and elbows. It was an exhausting and exhilarating workout.\n\nWin moved through his routine flawlessly\u2014ever the contradiction and deception. See Win on the street, and people said arrogant Waspy wimp who couldn't bruise a peach with his best punch. See him in a _dojang_ , and he struck fear and awe. Tae kwon do is considered a martial art. Art. The word was not used by accident. Win was an artist, the best Myron had ever seen.\n\nMyron remembered the first time he'd seen Win demonstrate his talent. They were freshmen in college. A group of large football players decided to shave Win's blond locks because they didn't like the way he looked. Five of them sneaked into Win's room late at night\u2014four to hold down his arms and legs and one to carry the razor and shaving cream.\n\nSimply put, the football team had a poor season that year. Too many guys on the injured list.\n\nMyron and Win finished up with light free sparring. Then they dropped to the mat and performed one hundred push-ups on their fists\u2014Win counting out loud in Korean. That done, they sat again for meditation, this time lasting fifteen minutes.\n\n_\"Barro,\"_ Win said.\n\nBoth men opened their eyes.\n\n\"Feeling more focused?\" Win asked. \"Feeling the flow of energy? The balance?\"\n\n\"Yes, Grasshopper. You want me to snatch the pebble from your hand now?\"\n\nWin moved from his lotus position into a full stance in one graceful, effortless move. \"So,\" he said, \"have you reached any decisions?\"\n\n\"Yes.\" Myron struggled to stand in one motion, tipping from side to side as he ascended. \"I'm going to tell Jessica everything.\"\n\n# Chapter 7\n\nYellow stick-on phone messages swarmed Myron's phone like locusts on a carcass. Myron peeled them away and shuffled through them. Nothing from Otto Burke or Larry Hanson or anyone in the Titans organization.\n\nNot good.\n\nHe strapped on his headset telephone. He had resisted using one for a long time, figuring they were more suited for air traffic controllers than agents, but he quickly learned that an agent is but a fetus, his office a womb, his telephone an umbilical cord. It was easier with the headset. He could walk around; he could keep his hands free; he could forgo neck cramps from cradling a phone against his shoulder.\n\nHis first call was to the advertising director for BurgerCity, a new fast-food chain. They wanted to sign up Christian and were offering pretty good money, but Myron wasn't sure about it. BurgerCity was only regional. A national chain might come up with a better offer. Sometimes the hardest part of the job was saying no. He'd discuss the pros and cons with Christian, let him make the final decision. In the end it was his name. His money.\n\nMyron had already signed Christian to several very lucrative endorsement deals. Wheaties would have Christian's likeness on cereal boxes starting in October. Diet Pepsi was coming up with some promotion involving Christian throwing a two-liter bottle on a perfect spiral to nubile women. Nike was developing a sweatsuit line and cleats known as the Steele Trap.\n\nChristian stood to earn millions from endorsements, far more than he would make playing for the Titans, no matter how reasonable Otto Burke wanted to be. It was strange in a way. Fans grew agitated at the idea of a player trying to get the most out of his playing contract. They called him boorish, selfish, and egomaniacal when he demanded a great deal of money from a wealthy team owner\u2014but they had no problem when he grabbed vault-loads from Pepsi or Nike or Wheaties for promoting products he'd probably never used or even liked. It made no sense. Christian would make more money for spending three days shooting a thirty-second hypocritical spot than he would for spending the season getting blindsided by drooling men with overactive pituitary glands\u2014and that was how the fans wanted it.\n\nNo agent minded that setup. Most agents got between three and five percent of their players' total negotiated salary (Myron took four percent), compared with twenty or twenty-five percent for all endorsement money. (Myron took fifteen percent\u2014hey, he was new.) In other words, sign a million-dollar deal with a team, and the agent gets around forty grand. Sign him for a million-dollar commercial, and the agent can nab as much as a quarter mil.\n\nMyron's second call was to Ricky Lane, a running back for the New York Jets and a former college teammate of Christian's. Ricky was one of his most important clients, and Myron was fairly certain it had been Ricky who'd convinced Christian to hire him in the first place.\n\n\"I have a kids' camp appearance for you,\" Myron began. \"They're paying five grand.\"\n\n\"Sounds good,\" Ricky said. \"How long do I have to be there?\"\n\n\"Couple hours. Do a little talk, sign a few autographs, that kind of thing.\"\n\n\"When?\"\n\n\"A week from Saturday.\"\n\n\"What about that mall appearance?\"\n\n\"That's Sunday,\" Myron said. \"Livingston Mall. Morley's Sporting Goods.\" Ricky would get paid another five thousand dollars for sitting at a table for two hours and signing autographs.\n\n\"Cool.\"\n\n\"You want me to send a limo to pick you up?\"\n\n\"No, I'll drive. You hear anything about next year's contract yet?\"\n\n\"We're getting there, Ricky. Another week at the most. Listen, I want you to come in and see Win soon, okay?\"\n\n\"Yeah, sure.\"\n\n\"You in shape?\"\n\n\"The best of my life,\" Ricky said. \"I want that starting job.\"\n\n\"Keep working. And don't forget to make that appointment with Win.\"\n\n\"Will do. Later, Myron.\"\n\n\"Yeah, later.\"\n\nThe calls continued, one blurring into another. He returned calls from the press. They all wanted to know about a pending deal between the Titans and Christian. Myron politely no-commented. Occasionally it was good to use the media as leverage in negotiating, but not with Otto Burke. Matters were proceeding, he told them. An agreement could be expected at any time.\n\nHe then called Joe Norris, an old-time Yankee who appeared almost every weekend at a baseball card show. Joe made more in a month now than he had in an entire season in his heyday.\n\nNext up was Linda Regal, a tennis pro who had just cracked the top ten. Linda was worried about aging, offended because a broadcaster had referred to her as a \"familiar veteran.\" Linda was almost twenty.\n\nEric Kramer, a UCLA senior and probable second round NFL draft pick, was in town. Myron managed to arrange a dinner with him. That meant Myron was a finalist\u2014he and a zillion other agents. The competition was incredible. Example: There are twelve hundred NFL-authorized agents who court the two hundred college players who will be drafted in April. Something has to give. It's usually ethics.\n\nMyron called the New York Jets general manager, Sam Logan, to discuss Ricky Lane's contract.\n\n\"The kid is in the best shape of his career,\" Myron raved. He stood and paced. Myron had a large, fairly gorgeous office on Park Avenue between Forty-sixth and Forty-seventh streets. It impressed people, and appearance was important in a business dominated by sleazeballs \"I've never seen anything like it. I'm telling you, Sam, the kid is Gayle Sayers all over again. It's amazing, really.\"\n\n\"He's too small,\" Logan said.\n\n\"What are you talking about? Is Barry Sanders too small? Is Emmitt Smith too small? Ricky's bigger than both of them. And he's been lifting. I'm telling you, he's going to be a great one.\"\n\n\"Uh-huh. Look, Myron, he's a nice kid. He works hard. But I can't go any higher than...\"\n\nThe number was still too low. But it was better.\n\nThe calls continued without a break. Sometime during the day Esperanza brought him a sandwich, which he inhaled.\n\nAt eight o'clock Myron placed his final office call of the day.\n\nJessica answered. \"Hello?\"\n\n\"I'll be at your house in an hour,\" Myron said. \"We need to talk.\"\n\nMyron watched Jessica's face for a reaction. She kept looking at the magazine as if it were just another issue of _Newsweek_ , her expression frighteningly passive. Every once in a while she nodded, looked over the rest of the page, and glanced at the front and back covers of the magazine, always returning to the picture of Kathy. She was so nonchalant, Myron almost expected her to whistle.\n\nOnly her knuckles gave her away. They were bloodless white, the pages crinkling in her death grip.\n\n\"Are you okay?\" he asked.\n\n\"I'm fine,\" she said, her voice calm, almost soothing. \"You said Christian got this in the mail?\"\n\n\"Yes.\"\n\n\"And you and Win spoke to the man who publishes this\"\u2014she hesitated, her face finally showing some signs of disgust\u2014\"this thing?\"\n\n\"Yes.\"\n\nShe nodded. \"Did he give you the address of whoever put this ad in?\"\n\n\"Just a p.o. box. I'm going to scout it out tomorrow, see who picks up the mail.\"\n\nShe looked up for the first time. \"I'll go with you.\"\n\nHe almost protested but stopped himself. He didn't stand a chance. \"Okay.\"\n\n\"When did Christian give this to you?\"\n\n\"Yesterday.\"\n\nThat got her attention. \"You knew about this yesterday?\"\n\nHe nodded.\n\n\"And you didn't tell me?\" she snapped. \"I was pouring my heart out to you, feeling like some paranoid schizophrenic, and you knew about this the whole time?\"\n\n\"I wasn't sure how to tell you.\"\n\n\"Anything else you haven't told me?\"\n\n\"Christian got a phone call last night. He thinks it was from Kathy.\"\n\n\"What?\"\n\nHe quickly told her about it. When he reached the part about Christian hearing Kathy's voice, her face drained of all color.\n\n\"Has your friend at the phone company learned anything?\" she asked.\n\n\"No. But we know Return Call only works for specific towns within the 201 area codes.\"\n\n\"How many towns?\"\n\n\"About three-quarters of them.\"\n\n\"So you're talking about three-quarters of the northern part of New Jersey, the most densely populated state in the U.S.? That limits it down to what, two, three million people?\"\n\n\"It's not a big help,\" he admitted, \"but it's something.\"\n\nHer eyes settled back on the magazine. \"I didn't mean to jump all over you. It's just\u2014\"\n\n\"Forget it.\"\n\n\"You're the best person I've ever known,\" she said. \"I mean that.\"\n\n\"And you're the biggest pain in the ass.\"\n\n\"Tough to argue that one,\" she said, but there was a hint of a smile.\n\n\"Do you want to tell the police about this?\" he asked. \"Or Paul Duncan?\"\n\nShe thought a moment. \"I'm not sure.\"\n\n\"The press will eat it up,\" he said. \"They'll drag Kathy through the mud.\"\n\n\"I don't give a rat's ass what the press does.\"\n\n\"I'm just telling you,\" Myron said.\n\n\"They can call her a slut a million different ways. I don't care.\"\n\n\"What about your mom?\"\n\n\"I don't give a rat's ass what she wants either. I just want Kathy found.\"\n\n\"So you want to tell them,\" Myron said.\n\n\"No.\"\n\nHe looked at her, confused. \"Care to elaborate?\"\n\nHer words came slow, measured, the ideas coming to her even as she spoke. \"Kathy has been gone for more than a year now,\" she began. \"In all that time the cops and the press have come up with zip. Not one thing. She's just vanished without a trace.\"\n\n\"So?\"\n\n\"But now we get this magazine. Someone sent it to Christian, which means someone\u2014maybe Kathy, maybe not\u2014is trying to make contact. Think about it. For the first time in over a year there is some form of communication. I don't want that taken away. I don't want a lot of attention scaring away whoever is out there. Kathy might disappear again. This\"\u2014she held up the magazine\u2014\"this thing is disgusting, but it's also encouraging. It's something. Don't get me wrong. I'm shocked by this. But it's a solid thread\u2014a thread as confusing as all hell, but nonetheless a thread of hope. If the cops and the press are called in, whoever did this might get scared and vanish again. Permanently this time. I can't risk that. We have to keep this to ourselves.\"\n\nMyron nodded. \"Makes sense.\"\n\n\"So what's next?\" she asked.\n\n\"We go to the post office in Hoboken. I'll pick you up early. Say six.\"\n\n# Chapter 8\n\nJessica smelled great.\n\nThey were at Uptown Station in Hoboken. She stood very close to him. Her hair had that freshly washed smell he had tried for four years to forget. Inhaling made him feel light-headed.\n\n\"So this is playing detective,\" she said.\n\n\"Exciting, isn't it?\"\n\nThey had been trying to look inconspicuous\u2014no easy task when a man is six-four and a woman is a total knee-knocker\u2014for the better part of an hour, having arrived at the post office at six-thirty in the morning. No one had touched Box 785 yet.\n\nBoredom set in quickly. Jessica looked over the prices of different mailing containers. Not very interesting. She read the wanted ads, all of them, found them a bit more interesting. Wanted posters in a post office. Like they wanted you to write the guy a letter.\n\n\"You sure know how to show a girl a good time,\" she said.\n\n\"That's why they call me Captain Fun.\"\n\nShe laughed. The melodic sound twisted his stomach.\n\n\"Do you like being an agent, Captain Fun?\"\n\n\"Very much.\"\n\n\"I always thought of agents as a bunch of sleazeballs.\"\n\n\"Thank you.\"\n\n\"You know what I mean. Leeches. Vipers. Greedy, money-hungry, bloodsucking parasites, swindling na\u00efve jocks, doing lunch at Le Cirque, destroying everything that's good about sports\u2014\"\n\n\"The problems in the Middle East,\" he interrupted. \"That's our fault too. And the budget deficit.\"\n\n\"Right. But you're not any of those things.\"\n\n\"Not a leech, viper, or parasite. That's quite a rave.\"\n\n\"You know what I mean.\"\n\nHe shrugged. \"There are plenty of sleazy agents. There are also plenty of sleazy doctors, lawyers\u2014\" He stopped, the words sounding familiar. Hadn't Fred Nickler used the same argument in justifying his magazines? \"Agents are a necessary evil,\" he continued. \"Without them, athletes get taken advantage of.\"\n\n\"By whom?\"\n\n\"Owners, management. Agents have done some good for the athletes. They've helped raise their salaries, assure free agency, get them endorsement money.\"\n\n\"So what's the problem?\"\n\nMyron thought a moment. \"Two things,\" he said. \"First of all, some agents are crooks. Plain and simple. They see a young, rich kid, and they take advantage. But as the athletes get more sophisticated, as more stories like what happened to Kareem Abdul-Jabar become known, most of the crooks will be weeded out.\"\n\n\"And second?\"\n\n\"Agents have to wear too many hats,\" he said. \"We're negotiators, accountants, financial planners, hand-holders, travel agents, family counselors, marriage counselors, errand boys, lackeys\u2014whatever it takes to get the business.\"\n\n\"So how do you do it all?\"\n\n\"I give two of the biggest hats to Win\u2014accountant and financial planner. I'm the lawyer. He's the MBA. Plus we have Esperanza, who can do almost anything. It works well. We all check and balance one another.\"\n\n\"Just like the branches of the federal government.\"\n\nHe nodded. \"Jefferson and Madison would be proud.\"\n\nA hand reached out and opened Box 785.\n\n\"Show time,\" Myron said.\n\nJessica snapped her head around to look. The man was slim. Everything about him was too long, eerily elongated, as if he had spent time on a medieval rack. Even his face seemed stretched like a cartoon imprint on Silly Putty.\n\n\"Recognize him?\" Myron asked.\n\nShe hesitated. \"Something about him... but I don't think so.\"\n\n\"Come on, let's get out of here.\"\n\nThey hurried down the steps and got in the car. Myron had parked illegally in front of the building, putting a police emergency sign in his front windshield. A gift from a friend on the force. The emergency sign came in handy\u2014especially during sale days at the mall.\n\nThe slim man came out two minutes later. He got into a yellow Oldsmobile. New Jersey plates. Myron shifted into drive and followed. Slim took Route 3 to the Garden State Parkway north.\n\n\"We've been driving almost twenty minutes,\" Jessica said. \"Why would he go to a mailbox so far from his home?\"\n\n\"Could be that he's not going to his house. Maybe he's going to work.\"\n\n\"The dial-a-porn office?\"\n\n\"Maybe,\" Myron said. \"Or it could be that he travels a long way so no one will see him.\"\n\nHe got off at Exit 160, jumped on Route 208 heading north, and pulled off at Lincoln Avenue, Ridgewood.\n\nJessica sat up. \"This is my exit,\" she said.\n\n\"I know.\"\n\n\"What the hell is going on here?\"\n\nThe yellow Oldsmobile turned left at the end of the ramp. They were now within three miles of Jessica's house. If he took Lincoln Avenue all the way to Godwin Road, they'd be...\n\nNope.\n\nMr. Slim turned on Kenmore Road, a half-mile before the Ridgewood border. They were still in the heart of suburbia\u2014the suburb in question being Glen Rock, New Jersey. Glen Rock was so named because of a giant rock that sat on Rock Road. The key word here is _rock_.\n\nThe yellow Olds pulled into a driveway. 78 Kenmore Drive.\n\n\"Look casual,\" he said. \"Don't stare.\"\n\n\"What?\"\n\nHe didn't answer. He drove past the house without pausing, turned at the next street, and stopped the car behind some shrubs. He picked up the car phone and dialed the office. It was picked up midway through the first ring.\n\n\"MB SportReps,\" Esperanza said.\n\n\"Get me all you can on 78 Kenmore Street, Glen Rock, New Jersey. Owner's name, credit check, the works.\"\n\n\"Got it.\" Click.\n\nHe dialed another number. \"My friend at the phone company,\" he explained to Jessica. Then: \"Lisa? It's Myron. Look, I need a favor. Seventy-eight Kenmore Road, Glen Rock, New Jersey. I don't know how many lines the guy has, but I need you to check them all. I want to know every number he calls for the next two hours. Right. Hey, what did you find out about that 900 number? What? Oh, okay, I understand. Thanks.\"\n\nHe hung up.\n\n\"What did she say?\"\n\n\"The 900 number isn't operated by the phone company. Some small outfit out of South Carolina takes care of it. She can't get anything on it.\"\n\n\"So what do we do now?\" she asked. \"Just watch his house?\"\n\n\"No. I go inside. You wait here.\"\n\nShe arched an eyebrow. \"Excuse me?\"\n\n\"You were the one who didn't want to scare anyone away,\" he continued. \"If this guy has something to do with your sister, how do you think he'll react to seeing you?\"\n\nShe folded her arms across her chest and fumed. She knew he was right, but that didn't mean she had to be happy about it. \"Go,\" she said.\n\nHe got out of the car. It was one of those no-variety neighborhoods, each house cookie-cut from the same mold\u2014split-levels on three-quarters of an acre. Sometimes the house was backward, the kitchen on the right instead of the left. Most had aluminum siding. The street reeked of middle class.\n\nMyron knocked. The thin man opened the door.\n\n\"Jerry?\"\n\nSlim's face registered confusion. Up close he was better looking, his face more brooding than freakish. Give him a cigarette and a black turtleneck, and he could be reading poetry in a village caf\u00e9. \"May I help you?\"\n\n\"Jerry, I'm\u2014\"\n\n\"You must have the wrong house. My name isn't Jerry.\"\n\n\"You look like Jerry.\"\n\nSomething dark crossed his face. \"I'm sorry,\" he said, closing the door. \"I really don't have time right now.\"\n\n\"Sure about that, Jer?\"\n\n\"I already told you\u2014\"\n\n\"Do you know Kathy Culver?\"\n\nIt was a sneak attack. And it drew blood. \"Wha\u2014what's this all about?\" he snapped.\n\n\"I think you know.\"\n\n\"Who are you?\"\n\n\"My name is Myron Bolitar.\"\n\n\"Am I supposed to know you?\"\n\n\"Well, if you're a big basketball fan... actually, no. But I'd like to ask you a few questions.\"\n\n\"I have nothing to say.\"\n\nAce of spades time. Myron pulled out the magazine. \"Sure about that, Jerry?\"\n\nThe whites of Slim's eyes grew tenfold, looking like Wedgwood china on the elongated face. \"You have me mixed up with someone else. Good-bye.\"\n\nHe slammed the door.\n\nMyron shrugged, headed back to the car.\n\n\"Well?\" Jessica asked.\n\n\"We shook him,\" Myron said. \"Let's see what falls out.\"\n\nThe neighborhood newsstand.\n\nWin remembered a time when the phrase conjured up nostalgia and Rockwellian images of real America. No more. Any street, any corner, any hickville town was the same. Candy, newspapers, greeting cards\u2014and porno mags. Kids could pick up a Snickers bar and get an eyeful, all in one. Porno had become a staple of American life. Hardcore porn. The kind of porn that made _Penthouse_ look like _Highlights_ magazine.\n\nWin approached the man behind the lottery ticket dispenser. \"Pardon me,\" Win said.\n\n\"Yeah?\"\n\n\"Would you be able to tell me if you have the most recent issues of _Climaxx, Jiz, Orgasm Today, Licks, Quim, and Nips_?\"\n\nAn elderly woman gasped and gave him an icy stare. Win smiled at her. \"Let me guess,\" he said. \"Playmate of the Month, June 1926?\"\n\nShe made a harumph noise and turned away.\n\n\"Check over there,\" the man said. \"Between the comic books and Disney videos.\"\n\n\"Thank you.\"\n\nWin found three of them\u2014 _Climaxx, Orgasm Today_ , and _Quim_. He tried three other newsstands and was able to pick up _Lick_ , but there was no sign of _Jiz_ or _Nips._ He finally found copies of them at a hardcore shop on Forty-second Street called King David's Smut Palace. They had a big sign out front that said OPEN 24 HOURS. How very convenient. Win considered himself fairly worldly, but the items and photographs in the \"palace\" proved that both his life experiences and his imagination had at best been limited.\n\nIt was almost noon when he exited the palace. A productive and quasi-educational morning.\n\nWith a total of six magazines lodged under his arm, Win caught a taxi to midtown. He skimmed through a few in the backseat.\n\n\"So far so good,\" he said out loud.\n\nThe driver glanced at him in the rearview mirror, shrugged, looked back to the road.\n\nWhen Win arrived at his office, he spread the magazines across the vast breadth of his desk. He studied them closely, comparing them. Incredible. His suspicion had been sound. It was just as he thought.\n\nFive minutes later, Win put the magazines in his desk drawer. Then he buzzed Esperanza.\n\n\"Kindly send Myron to my office as soon as he comes in.\"\n\n# Chapter 9\n\n\"I have a confession,\" Jessica said.\n\nThey were coming out of the Kinney garage on Fifty-second Street, the smell of fumes and urine dissipating as they hit the relatively fresh air on the sidewalk. They turned down Fifth Avenue. The line for passports stretched past the statue of Atlas. A black man with long dreadlocks sneezed repeatedly, his hair flapping about like dozens of snakes. A woman behind him tsk-tsked a complaint. Many of the people waiting faced St. Patrick's across the street as though pleading for divine intervention, their faces lined with anguish. Japanese tourists took pictures of both the statue and the line.\n\n\"I'm listening,\" Myron replied.\n\nThey kept walking. Jessica did not face him, her gaze fixed on nothing straight ahead. \"We weren't close anymore. In fact, Kathy and I barely spoke.\"\n\nMyron was surprised. \"Since when?\"\n\n\"The last three years or so.\"\n\n\"What happened?\"\n\nShe shook her head, but she still did not look at him. \"I don't know exactly. She changed. Or maybe she just grew up and I couldn't handle it. We just drifted apart. When we saw each other, it was as if she couldn't stand to be in the same room with me.\"\n\n\"I'm sorry to hear that.\"\n\n\"Yeah, well, it's no big thing. Except Kathy called me the night she disappeared. First time in I don't know how long.\"\n\n\"What did she want?\"\n\n\"I don't know. I was on my way out the door. I rushed her off.\"\n\nThey fell into silence the rest of the way to Myron's office.\n\nWhen they got off the elevator, Esperanza handed him a sheet of paper and said, \"Win wants to see you right away.\" She glared at Jessica the way a linebacker might glare at a limping quarterback on a blindside blitz.\n\n\"Otto Burke or Larry Hanson call?\" Myron asked.\n\nShe swerved her glare toward Myron. \"No. Win wants to see you right away.\"\n\n\"I heard you the first time. Tell him I'll be up in five minutes.\"\n\nThey moved into Myron's office. He closed the door and skimmed over the sheet. Jessica sat in front of him. She crossed her legs the way few women could, turning an ordinary event into a moment of sexual intrigue. Myron tried not to stare. He also tried not to remember the luscious feel of those legs in bed. He was unsuccessful in both endeavors.\n\n\"What's it say?\" she asked.\n\nHe snapped to. \"Our slim friend on Kenmore Street in Glen Rock is named Gary Grady.\"\n\nJessica squinted. \"The name sounds familiar.\" She shook her head. \"But I can't place it.\"\n\n\"He's been married seven years, wife Allison. No kids. Has a $110,000 mortgage on that house, pays it on time. Nothing else yet. We should know more in a little while.\" He put the paper on his desk. \"I think we have to start attacking this on a few different fronts.\"\n\n\"How?\"\n\n\"We have to go back to the night your sister disappeared. Start with that, and move forward. The whole case needs to be reinvestigated. The same with your father's murder. I'm not saying the cops weren't thorough. They probably were. But we now know some things they don't.\"\n\n\"The magazine,\" she said.\n\n\"Exactly.\"\n\n\"How can I help?\" she asked.\n\n\"Start finding out all you can about what she was up to when she disappeared. Talk to her friends, roommates, sorority sisters, fellow cheerleaders\u2014anyone.\"\n\n\"Okay.\"\n\n\"Also get her school records. Let's see if there's anything there. I want to see what courses she was taking, what activities she was involved with, anything.\"\n\nEsperanza threw open the door. \"Meal Ticket. Line two.\"\n\nMyron checked his watch. Christian should be in the middle of practice by now. He picked up the phone. \"Christian?\"\n\n\"Mr. Bolitar, I don't understand what's going on.\"\n\nMyron could barely hear him. It sounded as if he were standing in a wind tunnel. \"Where are you?\"\n\n\"A pay phone outside Titans Stadium.\"\n\n\"What's the matter?\"\n\n\"They won't let me in.\"\n\nJessica stayed in the office to make a few calls. Myron rushed out. Fifty-seventh Street to the West Side Highway was unusually clear. He called Otto Burke and Larry Hanson from the car. Neither one was in. Myron was not astounded.\n\nThen he dialed an unlisted phone in Washington. Few people had this particular number.\n\n\"Hello?\" the voice answered politely.\n\n\"Hi, P.T.\"\n\n\"Ah shit, Myron, what the fuck do you want?\"\n\n\"I need a favor.\"\n\n\"Perfect. I was just telling someone, gee, I wish Bolitar would call so I could do him a favor. Few things bring me such joy.\"\n\nP.T. worked for the FBI. FBI chiefs come and go. P.T was a constant. The press didn't know about him, but every president since Nixon had had his number on their speed dial.\n\n\"The Kathy Culver case,\" Myron said. \"Who's the best guy to talk to about it?\"\n\n\"The local cop,\" P.T. answered without hesitation.\n\n\"He's an elected sheriff or something. Great guy, good friend of mine. I forget his name.\"\n\n\"Can you get me an appointment?\" Myron asked.\n\n\"Why not? Serving your needs gives my life a sense of purpose.\"\n\n\"I owe you.\"\n\n\"You already owe me. More than you can pay. I'll call you when I have something.\"\n\nMyron hung up. The traffic was still clear. Amazing. He crossed the Washington Bridge and arrived at the Meadowlands in record time.\n\nThe Meadowlands Sports Authority was built on useless swampland off the New Jersey Turnpike in a place called East Rutherford. From west to east stood the Meadowlands Race Track, Titans Stadium, and the Brendan Byrne Arena, named for the former governor who was about as well liked as a whitehead on prom night. Angry protests equal to the French Revolution had erupted over the name, but to no avail. Mere revolutions are hardly worthy adversaries for a politician's ego.\n\n\"Oh, Christ.\"\n\nChristian's car\u2014or he assumed it was Christian's\u2014was barely visible under the blanket of reporters. Myron had expected this. He had told Christian to lock himself in his car and not say a word. Driving away would have been useless. The press would have just followed, and Myron was not up for a car chase.\n\nHe parked nearby. The reporters turned toward him like lions smelling a wounded lamb.\n\n\"What's going on, Myron?\"\n\n\"Why isn't Christian at practice?\"\n\n\"You pulling a holdout or what?\"\n\n\"What's happening with his contract?\"\n\nMyron no-commented them, swimming through the sea of microphones, cameras, and flesh, squeezing his way into the car without allowing any of the slime to ooze in with him.\n\n\"Drive off,\" Myron said.\n\nChristian started the car and pulled out. The reporters parted grudgingly. \"I'm sorry, Mr. Bolitar.\"\n\n\"What happened?\"\n\n\"The guard wouldn't let me in. He said he had orders to keep me out.\"\n\n\"Son of a bitch,\" Myron muttered. Otto Burke and his damn tactics. Little weasel. Myron should have been looking for something like this. But a lockout? That seemed a tad extreme, even by Otto Burke's standards. Despite the posturing, they had been fairly close to signing. Burke had expressed strong interest in getting Christian to minicamp as soon as possible, to get him ready for the season.\n\nSo why would he lock Christian out?\n\nMyron didn't like it.\n\n\"Do you have a car phone?\" he asked.\n\n\"No, sir.\"\n\nIt didn't matter. \"Turn back around,\" Myron said. \"Park by Gate C.\"\n\n\"What are you going to do?\"\n\n\"Just come with me.\"\n\nThe guard tried to stop them, but Myron pushed Christian past him. \"Hey, you're not allowed in there!\" he called after them. \"Hey, stop!\"\n\n\"Shoot us,\" Myron said without stopping.\n\nThey strode onto the field. Players were hitting the tackle dummies hard. Very hard. No one was holding back. These were tryouts. Most of these guys were fighting for a spot on the team. Most had been high school and college superstars, accustomed to unadulterated greatness on the field. Most would get cut. Most would not allow the dream to end there, scrounging other teams' rosters for a possible opening, holding on, slipping endlessly, dying slowly all the while.\n\nA glamour profession.\n\nThe coaches blew whistles. The running backs practiced wind sprints. Kickers were knocking down field goals at the far goal post. Punters boomed slow lazy arcs high into the air. Several players turned and spotted Christian. A buzz developed. Myron ignored it. He had spotted his target, sitting in the first row on the fifty-yard line.\n\nOtto Burke sat like Caesar at the Colosseum, that damn smile still plastered to his face, his arms spread over the seats on either side of him. Behind him sat Larry Hanson and a few other executives. Caesar's senate. Occasionally Otto would lean back and award his entourage a comment that brought on aneurysm-like fits of laughter.\n\n\"Myron!\" Otto called out pleasantly, waving one of those tiny hands. \"Come on over. Have a seat.\"\n\n\"Wait here,\" Myron told Christian. He climbed the steps. The entourage, led by Larry Hanson, stood in unison and marched away.\n\nMyron snapped a salute at them. \"Hut two, three, four. Right face.\" No one laughed. Big surprise.\n\n\"Sit down, Myron,\" Otto said, beaming. \"Let's have a chat.\"\n\n\"You haven't been returning my calls,\" Myron said.\n\n\"Did you call?\" He shook his head. \"I'll have to get on my secretary about that.\"\n\nMyron let out a deep breath and sat. \"Why was Christian locked out?\"\n\n\"Well, Myron, it's pretty simple, actually. Christian hasn't signed his contract yet. The Titans don't have time to invest in someone who may not be part of our future.\" He nodded toward the field. \"Do you see who's here for a tryout? Neil Decker from Cincinnati. Fine quarterback.\"\n\n\"Yeah, he's great. He can almost throw a spiral.\"\n\nOtto chuckled. \"That's funny, Myron. You're a very amusing man.\"\n\n\"I'm so glad you think so. Mind telling me what's going on?\"\n\nOtto Burke nodded. \"That's fair, Myron. So let's talk frankly, shall we?\"\n\n\"Rationally, frankly, whatever you want.\"\n\n\"Great. We'd like to renegotiate your client's contract,\" he said. \"Downward.\"\n\n\"I see.\"\n\n\"We feel your client's value has depreciated.\"\n\n\"Uh-huh.\"\n\nBurke studied him. \"You don't seem surprised, Myron.\"\n\n\"So what is it this time?\" Myron asked.\n\n\"What is what this time?\"\n\n\"Well, let's start with Benny Keleher. You invited him to your house, plied him with booze, then had a cop arrest him on his ride home for drunk driving.\"\n\nOtto looked properly shocked. \"I had nothing to do with that.\"\n\n\"Amazing how he signed the next day. And then there's Eddie Smith. You had compromising photographs of him taken by a private eye and threatened to send them to his wife.\"\n\n\"Another lie.\"\n\n\"Fine, a lie. So let's cut to the chase, then. What has caused this sudden devaluation?\"\n\nOtto sat back. He withdrew a cigarette from a gold case with a Titans emblem on the cover. \"It's something I saw in a rather lewd magazine,\" he said. \"Something that truly disheartened me.\" He didn't look disheartened. He looked rather pleased.\n\n\"A new low,\" Myron said. \"You should be proud.\"\n\n\"Pardon me?\"\n\n\"You set it up. The magazine.\"\n\nOtto smiled. \"Ah, so you knew about it.\"\n\n\"How did you get that picture?\"\n\n\"What picture?\"\n\n\"The one in the ad.\"\n\n\"I had nothing to do with it.\"\n\n\"Sure,\" Myron said. \"I guess you're just a charter subscriber to _Nips._ \"\n\n\"I had nothing to do with that ad, Myron. Honestly.\"\n\n\"Then how did you get a hold of the magazine?\"\n\n\"Someone pointed it out to me.\"\n\n\"Who?\"\n\n\"I am not at liberty to discuss it.\"\n\n\"Very convenient.\"\n\n\"I'm not sure I like your tone, Myron. And let me tell you something else: You're the one who has done wrong in the case. If you knew about the magazine, you had an ethical responsibility to tell me.\"\n\nMyron looked up at the sky. \"You used the word _ethical_. Lightning did not strike. There is no God.\"\n\nThe smile flickered but stayed on. \"Much as we'd like to, Myron, we can't just wish this away. The magazine exists, and it must be dealt with. So let me tell you what I've come up with.\"\n\n\"I'm all ears.\"\n\n\"You're going to take our current offer and knock it down by a third. If not, the picture of Ms. Culver goes public. Think about it. You have three days to decide.\" Otto watched Neil Decker throw a pass. It looked like a duck with a broken wing, crashing well short of the receiver. He frowned, stroked his goatee. \"Make that two days.\"\n\n# Chapter 10\n\nDean of Students Harrison Gordon made sure the door to his office was locked. Double-locked, in fact. He was taking no chances. Not with this.\n\nHe sat back down and stared out his office window. Esteemed Reston University in all its glory. The view was a mesh of green grass and brick buildings. No ivy adorned these towers of learning, but it should have. The students were gone for summer break, but the commons still had a sprinkling of people on it\u2014campers from the football and tennis camps, local people who used the campus as a park, the old throwback hippies who pilgrimage to liberal arts institutions like Moslems to Mecca. Lots of red bandannas and ponchos and granola-types. A bearded man tossed a Frisbee. A small boy caught it.\n\nHarrison Gordon saw none of it. He had not spun his chair around to enjoy the view. He had done so to avert his gaze from the... thing on his desk. He wanted simply to destroy the damn thing and forget about it. But he couldn't. Something held him back. And something kept drawing him toward it, toward that page near the back....\n\n_Destroy it, you fool. If somebody finds it..._\n\nWhat?\n\nHe did not know. He spun his chair back around, keeping his eyes away from the magazine. The student file marked CULVER, KATHERINE lay to the right. He swallowed. With a shaking hand he sifted through the stacks of transcripts and recommendation letters. It was an impressive file, but Harrison had no time for that now.\n\nThe buzz of his intercom\u2014a horrid noise\u2014startled him upright.\n\n\"Dean Gordon?\"\n\n\"Yes,\" he said, nearly shouting. His heart was beating like a rabbit's.\n\n\"I have someone here to see you. She doesn't have an appointment, but I thought you might want to see her.\"\n\nEdith's voice was hushed, a church-whisper.\n\n\"Who is it?\" he asked.\n\n\"It's Jessica Culver. She's Kathy's sister.\"\n\nPanic punctured his heart like an icicle.\n\n\"Dean Gordon?\"\n\nHe clamped his hand over his mouth, afraid he might scream.\n\n\"Dean Gordon? Are you there?\"\n\nThere were no true options. He would have to see her and find out what she wanted. To act in any other manner would raise suspicion.\n\nHe opened his bottom drawer and scooped the contents of his desk into it. He shut it, took out his key ring, and locked his desk. Better safe than sorry. Last, he unbolted his door.\n\n\"Send Ms. Culver in,\" he said.\n\nJessica was at least as beautiful as her sister, which was saying something quite extraordinary. He debated on how to greet her and settled for funeral director mode\u2014detached sympathy, warm professionalism.\n\nHe shook her hand with gentle firmness. \"Miss Culver, I'm so sorry we have to meet under such circumstances. Our prayers are with your family during this difficult time.\"\n\n\"Thank you for seeing me without an appointment.\"\n\nHe waved his hand as if to scoff, It's nothing. \"Please have a seat. Can I get you something to drink? Coffee, soda?\"\n\n\"No, thank you.\"\n\nHe moved back to his chair. He sat and folded his hands on the desk. \"Is there something I can do for you?\"\n\n\"I need my sister's file,\" Jessica replied.\n\nHarrison felt his fingers bunch, but he kept his face steady. \"Your sister's school file?\"\n\n\"Yes.\"\n\n\"May I inquire as to why?\"\n\n\"It involves her disappearance.\"\n\n\"I see,\" he said slowly. His voice, he was surprised to hear, remained calm. \"I believe the police were very thorough with the file. They made copies of everything in it\u2014\"\n\n\"I understand that. I'd like to see the file for myself.\"\n\n\"I see,\" he said again.\n\nSeveral seconds passed. Jessica shifted in her chair. \"Is there a problem?\" she asked.\n\n\"No, no. Well, perhaps. I'm afraid it may not be possible to give you the file.\"\n\n\"What?\"\n\n\"What I mean to say is, I'm not sure you have any legal right to it. Parents certainly do. But I'm not sure about siblings. I'll need to check this with a university attorney.\"\n\n\"I'll wait,\" Jessica said.\n\n\"Uh, fine. Would you mind waiting in the other room, please?\"\n\nShe stood, turned, stopped. She looked back over her shoulder at him. \"You knew my sister, didn't you, Dean Gordon?\"\n\nHe managed a smile. \"Yes, I did. Wonderful young lady.\"\n\n\"Kathy worked for you.\"\n\n\"Filing, answering the phone, that sort of thing,\" he said quickly. \"She was a terrific worker. We all miss her very much.\"\n\n\"Did she seem okay to you?\"\n\n\"Okay?\"\n\n\"Before she disappeared,\" Jessica continued, her eyes boring into his. \"Was she acting strangely?\"\n\nBeads of sweat popped onto his forehead, but he dared not wipe them away. \"No, not that I could see. She seemed perfectly fine. Why do you ask?\"\n\n\"Just checking. I'll wait out front.\"\n\n\"Thank you.\"\n\nShe closed the door.\n\nHarrison let loose a long breath. What now? He would have to give her the file; to do otherwise would do far more than merely raise suspicion. But he could not, of course, just pull the file out of his bottom drawer and hand it to Jessica. No, he would wait a few minutes, walk down to the filing room to handle her case \"personally,\" then return with the file.\n\nWhy, he wondered, did Jessica Culver need the file? Was there something he had missed?\n\nNo. He was sure of that.\n\nHarrison had spent the last year hoping, praying, that it was over. But he should have known better. Matters like this never truly die. They hide, take root, grow stronger, prepare for a fresh onslaught.\n\nKathy Culver was not dead and buried. Like some gothic ghost, she had arisen, haunting him, crying out from some great beyond.\n\nFor vengeance.\n\nMyron returned to the office.\n\n\"Win buzzed down twice,\" Esperanza said. \"He wants to see you. Now.\"\n\n\"On my way.\"\n\n\"Myron?\"\n\n\"What?\"\n\nEsperanza's lovely dark eyes were solemn. \"Is she back? Jessica, I mean.\"\n\n\"No. She's just visiting.\"\n\nHer face registered doubt. Myron did not press it. He no longer knew what to think himself.\n\nHe ran up the stairs two at a time. Win was two floors above him, but he might as well have been in another dimension. As soon as he opened the big steel door, the tireless clamor swarmed in, attacking. The large open space was in perpetual motion. Two, maybe three hundred desks covered the huge floor like throw rugs. Every desk had at least two computer terminals on it. There were no partitions. Hundreds of men sat and stood at every angle, each wearing a white button-down shirt with tie and suspenders, suit jackets draped from the back of their chairs. There were painfully few women. The men were all on the phone, most covering the mouthpiece to scream at someone else. They all looked alike. They all sounded alike. They were all pretty much the same person.\n\nWelcome to Lock-Horne Investments & Securities.\n\nAll six floors were exactly the same. In fact, Myron often suspected that Lock-Horne had only one floor and that the elevator was set to stop on the same floor no matter which number you hit from floor fourteen to floor nineteen, giving the illusion of a bigger company.\n\nOffice after office made up the compound's perimeter. These were saved for the head honchos, the top dogs, the numero unos, or in securities talk, the Big Producers. The BPs all had windows and sunshine, unlike the peons on the inside, who sickened and paled from the unnatural light.\n\nWin had a corner office with a view of both Forty-seventh Street and Park Avenue\u2014a view that screamed major dinero. His office was decorated in Early American Wasp. Dark-paneled walls. Forest green carpet. Wing-back chairs. Paintings of a fox hunt on the wall. Like Win had ever seen a fox.\n\nWin looked up from his massive oak desk when Myron entered. The desk weighed slightly less than a cement mixer. He'd been studying a computer print-out, one of those never-ending reams with green and white stripes. The desk was blanketed with them. They sort of matched the carpet.\n\n\"How did your morning rendezvous go with our friend Jerry the Phone-icator?\" Win asked.\n\n\"Phone-icator?\"\n\nWin smiled. \"I spent the whole morning working that one.\"\n\n\"It was worth it,\" Myron said.\n\nHe filled Win in on his encounter with Gary \"Jerry\" Grady. Win sat back and steepled. Myron then filled him in on his encounter with Otto Burke. Win leaned forward and unsteepled.\n\n\"Otto Burke,\" Win said, his voice measured, \"is a scoundrel. Perhaps I should pay him a private visit.\" He looked up at Myron hopefully.\n\n\"No. Not yet Please.\"\n\n\"Are you quite positive?\"\n\n\"Yes. Promise me, Win. No visits.\"\n\nHe was clearly disappointed. \"Fine,\" Win said, grudgingly.\n\n\"So what did you want to see me about?\"\n\n\"Ah.\" Win's face lit up again. \"Take a look at this.\"\n\nHe lifted the reams of computer print-outs and unceremoniously dumped them on the floor. Underneath were a pile of magazines. The top one was called _Climaxx_. The subheadline read, \"Double Xs for Double the Pleasure.\" Nifty sales technique. Win fanned them out as if he were doing a card trick.\n\n\"Six magazines,\" he said.\n\nMyron read the titles. _Climaxx, Licks, Jiz, Quim, Orgasm Today_ , and of course, _Nips._ \"Nickler's publications?\"\n\n\"God, you are good,\" Win said.\n\n\"Years of training. So what about them?\"\n\n\"Take a look at the pages I have marked off.\"\n\nMyron started with _Climaxx_. The cover featured another freakishly endowed woman, this time licking her own nipple. Handy. Win had used leather bookmarks to mark the page. Leather bookmarks in porno magazines. Like cigarettes in an aerobics class.\n\nThe page marked off was already too familiar. Myron felt his stomach churn all over again.\n\n**Live Fantasy Phone\u2014Pick Your Girl**\n\nThere were still three rows, still four in each row. His eyes immediately moved down to the bottom row, second from the right. It still read, \"I'll Do Anything!\" The phone number was still 1-900-344-LUST. Still $3.99 per minute. Still discreetly billed to your telephone or charge card, Visa and MasterCard accepted.\n\nBut the woman in the picture was not Kathy Culver.\n\nHe quickly scanned the rest of the page. Nothing else was different. The same Oriental girl was still waiting. The same buttock still craved a spanking. \"Tiny Titties\" had not pubesced.\n\n\"This same advertising page is in all six magazines,\" Win explained. \"But only _Nips_ has Kathy Culver's picture.\"\n\n\"Interesting.\" Myron thought a moment. \"Nickler probably sells package deals to advertisers\u2014buy space in six for the price of three, that kind of thing.\"\n\n\"Precisely. I would venture to say that all six magazines have the exact same ads.\"\n\n\"But someone stuck Kathy's picture in _Nips._ \" Myron was getting used to saying the name of the magazine. It no longer felt grimy on his lips, which in itself made him feel grimier.\n\nWin said, \"Do you remember Nickler telling us that _Nips_ was doing poorly?\"\n\nMyron nodded.\n\n\"Well, I had a devil of a time locating it. Most of the other rags were fairly easy to find on corner newsstands. But I had to go to a hardcore porno palace on Forty-second Street to come up with _Nips._ \"\n\n\"Yet,\" Myron added, \"Otto Burke was able to get a copy.\"\n\n\"Precisely. I am sure you've considered the possibility that Mr. Burke is behind it.\"\n\n\"The idea has crossed my mind.\"\n\nThere was a knock on the door. Esperanza entered.\n\n\"Your handwriting expert is on the phone,\" she said. \"I put it on Win's line.\"\n\nWin picked up the receiver and handed it to Myron.\n\n\"Hello.\"\n\n\"Hey, Myron, it's Swindler. I just went over the two samples you gave me.\"\n\nMyron had given Swindler the envelope _Nips_ had come in as well as a letter in Kathy's handwriting.\n\n\"Well?\"\n\n\"They match. It's her or a very professional forgery.\n\nMyron felt his stomach dive. \"You're sure?\"\n\n\"Positive.\"\n\n\"Thanks for calling.\"\n\n\"Yeah, no problem.\"\n\nMyron handed the receiver back to Win.\n\n\"A match?\" Win asked.\n\n\"Yep.\"\n\nWin tilted back in his chair and smiled. \"Yowzer.\"\n\n# Chapter 11\n\nMyron ran into Ricky Lane in the corridor. He hadn't seen him in three months. Ricky looked a lot bigger. The Jets would be pleased.\n\n\"What are you doing here?\" Myron asked.\n\n\"I made an appointment with Win,\" Ricky said with a big grin. \"Just like my agent advised.\"\n\n\"Good to see you listen to your agent.\"\n\n\"Always. The man is brilliant.\"\n\n\"And he never argues with a client.\"\n\nRicky laughed. \"Say, I heard Christian got locked out of camp.\"\n\nNews traveled fast. \"Where did you hear that?\"\n\n\"The FAN.\"\n\nWFAN was New York's all-sports radio station. \"Have you spoken to him lately?\"\n\nRicky made a face. \"Christian?\"\n\n\"Yeah.\"\n\n\"Not since my last college football game, what, year and a half ago.\"\n\n\"I thought you were friends.\" Myron had, in fact, assumed that Ricky had recommended his services to Christian.\n\n\"We were teammates,\" Ricky replied steadily. \"We were never friends.\"\n\n\"You don't like him?\"\n\nRicky shrugged. \"Not really. None of us did.\"\n\n\"Who is 'us'?\"\n\n\"Guys on the team.\"\n\n\"What's wrong with him?\"\n\n\"Long story, man. Not worth telling.\"\n\n\"I'd be interested.\"\n\n\"Put it like this,\" Ricky said. \"Christian was a little too perfect for most of us, okay?\"\n\n\"An egomaniac?\"\n\nRicky paused, considering. \"Not really. I mean, to be straight, I guess a lot of it was jealousy. Christian wasn't just good. Shit, he wasn't even just great. He was incredible. Best I ever seen.\"\n\n\"So?\"\n\n\"So he expected the same from everyone else.\"\n\n\"He got on people's case when they made mistakes?\"\n\nRicky paused again, shook his head. \"No, that ain't it either.\"\n\n\"You're being a tad obtuse, Ricky.\"\n\nRicky Lane looked up, looked down, looked left, looked right, looked very uneasy. \"I can't explain it,\" he said. \"It's going to sound like a lot of griping, but guys weren't crazy about all the attention he was getting. I mean, we won two national championships, and the only guy they ever talked to was Christian.\"\n\n\"I heard those interviews. He always gave his teammates all the credit.\"\n\n\"Yeah, a real gentleman,\" Ricky replied with more than a hint of sarcasm. \"All that 'it's a team effort' bullshit just made the press love him even more. Guys on the team thought he was a promo-hog, you know? His own best PR firm. They blamed him for being too popular.\"\n\n\"Did you?\"\n\n\"I don't know. Maybe. Truth was, I just didn't really like him. We had nothing in common except football. He's a pure Midwest white-boy. I'm a city-slicking black man. It ain't a winning combination.\"\n\n\"That's all it was?\"\n\nHe gave a half-shrug. \"I guess so. But man, this is all ancient history. I don't know why I brought it up. It don't matter no more. Christian just didn't fit in, okay. He was a nice guy, I guess. He was always polite. But that don't play so well in a locker room, you know?\"\n\nMyron knew. Juvenile, sexist, homophobic bantering\u2014that was the stuff of locker-room popularity.\n\n\"I gotta go, man. Win will be wondering where I am.\"\n\n\"Okay. I'll see you around.\"\n\nRicky had almost turned away when Myron thought of something else. \"What can you tell me about Kathy Culver?\"\n\nRicky's face blanched. \"What about her?\"\n\n\"Did you know her?\"\n\n\"A little, I guess. I mean, she was a cheerleader and dated the quarterback. But we never hung out or anything.\" He looked very unhappy now. \"Why you asking?\"\n\n\"Was she popular? Or was she hated too?\"\n\nRicky's eyes darted about like birds trying to find a safe place to land. \"Look, Myron, you always been straight with me, I always been straight with you, right?\"\n\n\"Right.\"\n\n\"I don't want to say nothing else. She's dead. Might as well let her be.\"\n\n\"What does that mean?\"\n\n\"Nothing. I just don't like talking about her, okay. It's kinda creepy. I'll see you later.\"\n\nRicky hurried down the corridor as if Reggie White were chasing him. Myron watched him. He debated following him but decided against it. Ricky would say no more today.\n\n# Chapter 12\n\nEsperanza stuck her head in the door. \"Someone\u2014or something\u2014is here to see you.\"\n\nMyron held up a silencing hand. The headset had been on since his return to his office. \"Look, I have to go,\" he said. \"See if you can get him upgraded to first class. He's a big guy. Thanks.\" He took off the headset. \"Who is it?\"\n\nShe made a face. \"Aaron. He didn't give a last name.\"\n\nHe didn't have to. \"Send him in.\"\n\nSeeing Aaron was like falling into a time warp. He was as big as Myron remembered, as big as the lummox in the garage. He was dressed in a freshly pressed white suit, but he wore no shirt with it, displaying plenty of tan pectoral cleavage. He wasn't wearing socks either. Nifty haircut, the swept-back look \u00e0 la Pat Riley. A saunter for a walk. Designer sunglasses. Designer cologne that smelled suspiciously like insect repellent. Aaron was the pure definition of \"supersmooth\"\u2014just ask him, he'd tell you.\n\nHe smiled widely. \"Nice to see you, Myron.\"\n\nThey shook hands. Myron did not squeeze. He was far too mature. That, and Aaron could probably squeeze harder. \"Have a seat.\"\n\n\"Wonderful.\" Aaron made a production of it, spreading out his arms as if he were wearing a cloak. He removed his sunglasses with an audible snap. \"I like your office. It's really great.\"\n\n\"Thank you.\"\n\n\"Great address. Great view.\"\n\nThe password is _great._ \"You looking to rent space?\"\n\nAaron laughed as if that were the gem of gems. \"No,\" he said. \"I don't like being cooped in an office. It's not my style. I like my freedom. I like being out on my own, on the road. I wouldn't do well chained to a desk.\"\n\n\"Wow, that's fascinating, Aaron. Really.\"\n\nHe laughed again. \"Ah, Myron, you haven't changed a bit. I'm glad to see it.\"\n\nThey hadn't seen each other since high school. Myron had gone to Livingston High School in New Jersey. Aaron had gone to his archenemy, West Orange High. The teams played each other twice a year, and it was rarely a pleasant encounter.\n\nIn those days Myron's best friend was a huge ox named Todd Midron. Todd was a big, softhearted, simple kid with a lisp. He played Lenny to Myron's George. He was also the toughest kid Myron had ever met.\n\nTodd never lost a fight. Never. No one ever came close to him. He was just too powerful. During a game their senior year, Aaron undercut and nearly injured Myron. Todd took exception. He went after Aaron. Aaron destroyed him. Myron tried to help his friend, but Aaron shrugged Myron off like a dandruff flake. He continued to pulverize Todd, steadily, methodically, glaring at Myron the whole time, not even glancing at his limp victim. The beating was ferocious. By the time it ended, Todd's face was an unrecognizable pulpy mess. Todd spent four months in a hospital. His jaw was wired shut for nearly a year.\n\n\"Hey,\" Aaron said. He pointed to a movie still on the wall. \"That's Woody Allen and what's-her-name.\"\n\n\"Diane Keaton.\"\n\n\"Right, Diane Keaton.\"\n\n\"Is there something I can do for you?\" Myron asked.\n\nAaron turned his whole body toward Myron. The glare from his shaved chest was nearly blinding. \"I think there is, Myron. In fact, I think there's something we can do for each other.\"\n\n\"Oh?\"\n\n\"I represent a competitor of yours. A certain dispute has arisen between the two of you. My client wishes to settle it peacefully.\"\n\n\"Are you an attorney now, Aaron?\"\n\nHe smiled. \"Not likely.\"\n\n\"Oh.\"\n\n\"I am referring to a young man named Chaz Landreaux. He recently signed a contract with your company, MB SportReps.\"\n\n\"I thought of the name myself.\"\n\n\"Pardon me.\"\n\n\"MB SportReps. I came up with the name by myself.\"\n\nAaron renewed his smile. It was a good smile. Lots of teeth. \"There is a problem with the contract.\"\n\n\"Do tell.\"\n\n\"You see, Mr. Landreaux has also signed a contract with Roy O'Connor at TruPro Enterprises, Incorporated. The contract predates yours. So you see the problem: Your contract is invalid.\"\n\n\"Why don't we let a court of law decide that?\"\n\nHe sighed deeply. \"My client feels it is in everyone's best interest to avoid litigation.\"\n\n\"Gee, what a surprise. So what does your client suggest?\"\n\n\"Mr. O'Connor would be willing to pay you for your time.\"\n\n\"Very generous of him.\"\n\n\"Yes.\"\n\n\"And if I say no?\"\n\n\"We hope it won't come to that.\"\n\n\"But if it does?\"\n\nAaron sighed, stood, leaned on Myron's desk. \"I'll be forced to make you disappear.\"\n\n\"Like in a magic trick?\"\n\n\"Like in dead.\"\n\nMyron put his hand to his chest. \"Gasp. Oh. Gasp.\"\n\nAaron laughed again, this time without humor. \"I know all about your tae kwon do display in the garage. But that guy was a stupid musclehead. I am not. I boxed professionally. I'm a black belt in jujitsu and a grand master in aikido. I've killed people.\"\n\n\"I bet that looks good on a r\u00e9sum\u00e9,\" Myron said.\n\n\"Let me put this in very simple terms for you, Myron: You fuck with us, I'll kill you.\"\n\n\"Shiver. Tremble.\" Myron was not quite as confident as his sarcasm, but he knew better than to show fear. Guys like Aaron are like dogs. They smell fear, they pounce.\n\nAaron laughed again. He was laughing a lot today. He was either very amused or had been sniffing gas. He turned his back and walked to the door. \"This is your final warning,\" he said. \"Landreaux honors his contract with Mr. O'Connor, or both of you end up worm food.\"\n\nWorm food. First oatmeal. Now worm food.\n\n\"I like you, Myron. I'd really hate to see something bad happen. But you understand.\"\n\n\"Business is business.\"\n\n\"Exactly.\"\n\nEsperanza appeared at the door.\n\nAaron gave her a sharklike smile. \"Well, well,\" he said. Then he followed up with his best big-guy wink. Esperanza managed to keep her clothes on. Amazing restraint.\n\n\"Pick up line two,\" she said.\n\n\"Listen to this call closely, Myron,\" Aaron added with a final grin. \"Appreciate the gravity of the situation. And remember. Worm food.\"\n\n\"Worm food. I'll keep it in mind.\"\n\nAaron winked at Esperanza again, blew her a kiss, and left.\n\n\"Charming,\" she said.\n\n\"Who's on the phone?\"\n\n\"Chaz Landreaux.\"\n\nMyron picked up the headset. \"Hello.\"\n\n\"Motherfuckers were at my mom's!\" Chaz shouted. \"They told her they were going to cut off my nuts and send them to her in a box! My mother, man! They said this to my mother!\"\n\nMyron felt his fingers tighten into fists. \"I'll take care of it,\" he said slowly. \"They won't bother her again.\"\n\nEnough game playing. It was time to act.\n\nIt was time to tell Win about Roy O'Connor.\n\nWin smiled like a kid on a snow day listening to the radio for a school closing. \"Roy O'Connor,\" he said.\n\n\"I don't want him hurt. Promise me.\"\n\nWin's eyes drifted dreamily. He might have nodded a yes, but Myron couldn't say for sure.\n\n# Chapter 13\n\nBaumgart's on Palisades Avenue. Their old stomping grounds.\n\nPeter Chin greeted them at the door, his eyes widening in delight and surprise when he spotted Jessica. \"Miss Culver! How wonderful to see you again.\"\n\n\"Nice to see you, Peter.\"\n\n\"You look as lovely as ever. You beautify my restaurant.\"\n\nMyron said, \"Hi, Peter.\"\n\n\"Yeah, whatever.\" He dismissed Myron with a hand wave. His full attention was on Jessica; a crocodile gnawing on his foot wouldn't have changed that. \"You look a little too thin, Miss Culver.\"\n\n\"The food's not as good in Washington.\"\n\n\"Funny,\" Myron said. \"I was thinking she looked a little chunky.\"\n\nJessica eyed him. \"Dead man.\"\n\nBaumgart's was an institution in Englewood, New Jersey. For fifty years it was an old Jewish deli and soda fountain, noted for its superb ice cream and desserts. When Peter Chin bought it eight years ago, he kept all of the tradition but added the best nouvelle Chinese cuisine in the state. The combination was a smash. The normal order might consist of Peking duck, sesame noodles, chocolate milk shake, french fries and a death-by-chocolate sundae for dessert. When Myron and Jessica had lived together, they ate at Baumgart's at least once a week.\n\nMyron still came once a week. Usually with Win or Esperanza. Sometimes alone. He never brought a date here.\n\nPeter walked them past the soda fountain and put them in a booth under a huge painting. Modern art. It was a portrait of either Cher or Barbara Bush. Maybe both. Very esoteric.\n\nMyron and Jessica sat across the table from each other, silently. The moment seemed weighed down, overwhelming. Being here together again\u2014they had expected it to generate some light nostalgia. But the effect was more like a body blow.\n\n\"I've missed this place,\" she said.\n\n\"Yes.\"\n\nShe reached her hand across the table and took his. \"I've missed you.\"\n\nHer face was aglow, the way it used to be when she looked at him as though he were the only person in the entire world. Myron felt something squeeze his heart, making it nearly impossible to breathe. The rest of the world broke apart, diffused. There were only the two of them.\n\n\"I'm not sure what to say.\"\n\nShe smiled. \"What? Myron Bolitar at a loss for words?\"\n\n\"Ripley's, huh?\"\n\nPeter came by. Without preamble, he said, \"You'll start with the crispy duck appetizer and squab package with pine nuts. For your main course you'll have soft-shell crab in special sauce and the Baumgart lobster and shrimp.\"\n\n\"Can we choose dessert?\" Myron asked.\n\n\"No. Myron, you'll have the pecan pie \u00e0 la mode. And for Miss Culver.\" He stopped, building suspense like a game-show host.\n\nShe smiled expectantly. \"You don't mean...\"\n\nPeter nodded. \"Banana pudding cake with vanilla wafers. There's only one piece left, but I put it away for you.\"\n\n\"Bless you, Peter.\"\n\n\"Each man does what he can. You didn't bring wine?\" Baumgart's was BYO.\n\n\"We forgot,\" Jessica said. She was dazzling Peter with her smile. Not fair. Jessica's looks were like a _Star Trek_ laser set on stun. Her smile, kill.\n\n\"I'll send someone across the street to get a bottle. Kendall-Jackson Chardonnay?\"\n\n\"You have a good memory,\" she said.\n\n\"No. I just remember what is important.\" Myron rolled his eyes. Peter bowed slightly and left.\n\nShe turned the smile back to Myron. He felt frightened and helpless and deliriously happy.\n\n\"I'm sorry,\" she said.\n\nHe shook his head. He was afraid to open his mouth.\n\n\"I never meant\u2014\" She was unsure how to continue. \"I made a lot of mistakes in my life,\" she said. \"I am dumb. I am self-destructive.\"\n\n\"No,\" Myron said. \"You're perfect.\"\n\nHer voice grew dramatic, her hand against her chest. \" 'Take the blinders from your eyes and see me as I really am.' \"\n\nHe thought a moment. \"Dulcinea to Don Quixote in _Man of La Mancha._ And it's 'take the clouds,' not blinders.\"\n\n\"Very impressive.\"\n\n\"Win was playing it in the car.\" This was an old game of theirs. Guess the Quote.\n\nShe fiddled with her water glass, making little water circles and then inspecting them for clarity and definition. Eventually she created an aquatic Olympics logo. \"I'm not sure what I'm trying to say to you,\" she said at last. \"I'm not sure what I want to happen here.\" She looked up. \"One last confession, okay?\"\n\nHe nodded.\n\n\"I came to you because I thought you would help. That was true. But that wasn't the only reason.\"\n\n\"I know,\" he said. \"I try not to think about it too much. It terrifies me.\"\n\n\"So what do we do now?\"\n\nHis chance. He hoped there would be others. \"Did you get your sister's file?\"\n\n\"Yes.\"\n\n\"Have you gone through it yet?\"\n\n\"No. I just picked it up.\"\n\n\"Then why don't we open it now?\"\n\nShe nodded. The crispy duck and squab package with pine nuts appeared. Jessica took out a manila envelope and slit the seal. \"Why don't you look at it first?\"\n\n\"Okay,\" he said. \"But save me some food.\"\n\n\"Chance.\"\n\nHe started sifting through the papers. The top page was Kathy's high school transcript. After her junior year her ranking had been twelfth in a class of three hundred. Not bad. But by the end of senior year her ranking had slipped considerably\u2014to fifty-eighth.\n\n\"Her grades dropped senior year of high school,\" Myron said.\n\n\"Whose didn't drop senior year?\" Jess countered. \"She was probably just goofing off.\"\n\n\"Probably.\" But usually that meant A students got B's or C's. Kathy had gotten one A, three D's and an F in her final semester. Her clean record was also muddied with several detentions\u2014all in her senior year. Strange. But probably meaningless.\n\n\"Do you want to fill me in on what happened today?\" Jessica asked between bites.\n\nShe was even beautiful when pigging out. Amazing. He started by telling her about Win's discovery in the six magazines.\n\n\"So what does it mean,\" she asked, \"her picture being only in that one rag?\"\n\n\"I'm not sure.\"\n\n\"But you have an idea?\"\n\nHe did. But it was too early to say anything. \"Not yet.\"\n\n\"Did you hear from your friend at the phone company?\"\n\nHe nodded. \"Gary Grady placed two calls after we left. One was to Fred Nickler's office at Hot Desire Press. The other was someplace in the city. There was no answer when we called it. We got the information kind of late in the day.\"\n\n\"And the handwriting analyst?\"\n\nBest to dive right in. \"The handwriting matches. It's either Kathy's or a very good forger.\"\n\nThat slowed her chopsticks. \"My God.\"\n\n\"Yes.\"\n\n\"Then she's alive?\"\n\n\"It's still just a possibility. Nothing more. That envelope could have been written before she died. Or like I said, it could be a clever forgery.\"\n\n\"You're reaching.\"\n\n\"I'm not so sure,\" he said. \"If she's alive, where is she? Why is she doing all this?\"\n\n\"Maybe she's been kidnapped. Maybe she's being forced to.\"\n\n\"Forced to address envelopes? Now who's reaching?\"\n\n\"Do you have a better explanation?\" she asked.\n\n\"Not yet. But I'm working on it.\" He started looking through the file again. \"You ever hear of a guy named Otto Burke?\"\n\n\"The big record company magnate who owns the Titans?\"\n\n\"Right. He also knew about the magazine.\" Myron quickly summarized his visit to Titans Stadium.\n\n\"So you think Otto Burke might be behind it?\" she asked.\n\n\"Otto has a motive: knocking down Christian's asking price. He certainly has the resources: lots of money. And it would also explain why Christian got a copy in the mail.\"\n\n\"He was sending Christian a message,\" she added.\n\n\"Right.\"\n\n\"But how would Burke forge my sister's handwriting?\"\n\n\"He could have hired an expert.\"\n\n\"Where did he get a writing sample?\"\n\n\"Who knows? It can't be that difficult.\"\n\nHer eyes glazed over. \"So this was all a hoax? This was all some plot to gain leverage in a negotiation?\"\n\n\"It's possible. But I don't think so.\"\n\n\"Why not?\"\n\n\"Something just doesn't mesh. Why would Burke go through all that trouble? He could have blackmailed us with just the photo. He didn't have to put it in a magazine. The photo was enough.\"\n\n\"She grasped on to his hope as if it were a life preserver. \"Good point,\" she said.\n\n\"The question then becomes,\" he continued, \"how did Otto get a copy of the magazine?\"\n\n\"Maybe someone in his organization picked up a copy at a newsstand.\"\n\n\"Very unlikely. _Nips_ \"\u2014the word felt grungy again, good\u2014\"has a very low circulation rate. The chances that someone in the Titans organization bought that particular magazine, had time to read it carefully, somehow spotted Kathy's picture in the bottom row on a page of ads in the back\u2014it's fairly remote at best.\"\n\nJessica snapped her fingers. \"Someone mailed it to him too.\"\n\nHe nodded. \"Why should Christian have been the only one? For all we know, dozens of people were sent that magazine.\"\n\n\"How do we find out?\"\n\n\"I'm working on it.\"\n\nHe managed to salvage a sliver of crispy duck before it was sucked into the black hole. It was delicious. He turned his attention back to Kathy's files. Her bad grades continued during her first semester at Reston. By second semester, her grades had picked up considerably. He asked Jessica about this.\n\n\"She settled into college life, I guess,\" she said. \"She joined the drama group, became a cheerleader, started dating Christian. She went through culture shock in her first semester. It's not uncommon.\"\n\n\"No. I guess not.\"\n\n\"You don't sound convinced.\"\n\nHe shrugged. Myron Bolitar, Se\u00f1or Skepticalo.\n\nKathy's recommendation letters were next. Three of them. Her high school guidance counselor called her \"unusually gifted.\" Her tenth-grade history teacher said, \"Her enthusiasm for life is contagious.\" Her twelfth-grade English teacher said, \"Kathy Culver is bright, witty, and fun-spirited. She will be a welcome addition to any institution of learning.\" Nice comments. He scanned down to the bottom of the page.\n\n\"Uh-oh,\" he said.\n\n\"What is it?\"\n\nHe handed her the glowing recommendation letter from Kathy's twelfth-grade English teacher at Ridgewood High School. A Mr. Grady.\n\nA Mr. Gary, aka \"Jerry\" Grady.\n\n# Chapter 14\n\nMyron was startled awake by the telephone. He'd been dreaming about Jessica. He tried to remember specifics, but the details disintegrated into small pieces and blew away, leaving behind only a few frustrating snippets. The clock on his nightstand read seven o'clock. Someone was calling him at home at seven o'clock in the morning. Myron had a pretty good idea who it was.\n\n\"Hello?\"\n\n\"Good morning, Myron. I hope I didn't wake you.\"\n\nMyron recognized the voice. He smiled and asked, \"Who is this?\"\n\n\"It's Roy O'Connor.\"\n\n\" _The_ Roy O'Connor?\"\n\n\"Uh, yes, I guess so. Roy O'Connor, the agent.\"\n\n\"The superagent,\" Myron corrected. \"To what do I owe this honor, Roy?\"\n\n\"Would it be possible for us to meet this morning?\" The voice had a discernible quake to it.\n\n\"Sure thing, Roy. My office, okay?\"\n\n\"Uh, no.\"\n\n\"Your office, Roy?\"\n\n\"Uh, no.\"\n\nMyron sat up. \"Should I keep guessing places and you can say hotter or colder?\"\n\n\"You know Reilly's Pub on Fourteenth Street?\"\n\n\"Yes.\"\n\n\"I'll be in the booth in the back right-hand corner. One o'clock. We'll have lunch. If that's okay with you.\"\n\n\"Peachy, Roy. Want me to wear anything special?\"\n\n\"Uh, no.\"\n\nMyron hung up, smiled. A night visit from Win, usually while sleeping soundly in your bedroom, your innermost sanctuary. Worked every time.\n\nHe got out of bed. He heard his mother in the kitchen above him, his father in the den watching television. Early morning at the Bolitar house. The basement door opened.\n\n\"Are you awake, Myron?\" his mother shouted.\n\nMyron. What a goddamn awful name. He hated it with a passion. The way he looked at it, he'd been born with all his fingers and toes, he didn't have a harelip or a cauliflower ear or a limp of any kind\u2014so to compensate for his lack of ill fortune, his parents had christened him Myron.\n\n\"I'm awake,\" he answered.\n\n\"Daddy bought some fresh bagels. They're on the table.\"\n\n\"Thanks.\"\n\nHe got out of bed and climbed the steps. With one hand he felt the rough beard he'd have to shave; with the other he picked the yellow sleep-buggers out of the corner of his eyes. His father was sprawled on the den couch like a wet sock, wearing an Adidas sweatsuit and eating a bagel oozing with whitefish spread. As he did every morning, Myron's father was watching a videocassette of people exercising. Getting in shape through osmosis.\n\n\"Good morning, Myron. There're some bagels on the table.\"\n\n\"Uh, thanks.\" It was like one parent never heard the other.\n\nHe entered the kitchen. His mom was nearly sixty, but she looked much younger. Say, forty-five. She acted much younger too. Say, sixteen.\n\n\"You came in late last night,\" she said.\n\nMyron made a grunting noise.\n\n\"What time did you finally get home?\"\n\n\"Really late. It was almost ten.\" Myron Bolitar, the late-night scream machine.\n\n\"So,\" Mom began, struggling to look and sound casual, \"who were you out with?\" Mistress of the Subtle.\n\n\"Nobody,\" he said.\n\n\"Nobody? You were out all night with nobody?\"\n\nMyron looked left and right. \"When are you going to bring in the hot lights and jumper cables?\"\n\n\"Fine, Myron. If you don't want to tell me\u2014\"\n\n\"I don't want to tell you.\"\n\n\"Fine. Was it a girl?\"\n\n\"Mom...\"\n\n\"Okay, forget I asked.\"\n\nMyron reached for the phone and dialed Win's number. After the eighth ring he began to hang up when a weak, distant voice coughed. \"Hello?\"\n\n\"Win?\"\n\n\"Yeah.\"\n\n\"You okay?\"\n\n\"Hello?\"\n\n\"Win?\"\n\n\"Yeah.\"\n\n\"What took you so long to answer the phone?\"\n\n\"Hello?\"\n\n\"Win?\"\n\n\"Who is this?\"\n\n\"Myron.\"\n\n\"Myron Bolitar?\"\n\n\"How many Myrons do you know?\"\n\n\"Myron Bolitar?\"\n\n\"No, Myron Rockefeller.\"\n\n\"Something's wrong,\" Win said.\n\n\"What?\"\n\n\"Terribly wrong.\"\n\n\"What are you talking about?\"\n\n\"Some asshole is calling me at seven in the morning pretending to be my best friend.\"\n\n\"Sorry, I forgot the time.\" Win was not what one would call a morning person. During their years at Duke, Win was never out of bed before noon\u2014even on the days he had a morning class. He was, in fact, the heaviest sleeper Myron had ever known or imagined. Myron's parents, on the other hand, woke up when somebody in the Western hemisphere farted. Before Myron moved into the basement, the same scenario was played out nightly:\n\nAround three in the morning, Myron would get out of bed to go to the bathroom. As he tiptoed past his parents' bedroom, his father would stir ever so slowly, as though someone had dropped a Popsicle on his crotch.\n\n\"Who's that?\" his father would shout.\n\n\"Just me, Dad.\"\n\n\"Is that you, Myron?\"\n\n\"Yes, Dad.\"\n\n\"Are you okay, son?\"\n\n\"Fine, Dad.\"\n\n\"What are you doing up? You sick or something?\"\n\n\"I'm just going to the bathroom, Dad. I've been going to the bathroom by myself since I was fourteen.\"\n\nDuring their sophomore year at Duke, Myron and Win lived in the smallest double on campus with a bunk bed that Win said \"creaks slightly\" and Myron said \"sounds like a duck being run over by a back hoe.\" One morning, when the bed was quiet and he and Win were asleep, a baseball crashed through their window. The noise was so deafening that their entire dorm jumped out of bed and rushed to see if Myron and Win had survived the wrath of whatever gigantic meteorite had fallen through the roof. Myron rushed to the window to yell obscenities. Dorm members stamped across the underwear-carpeted floor to join in the tirade. The ensuing reverberations were loud enough to disturb a diner waitress on her coffee break.\n\nWin just lay asleep, a blanket of broken glass strewn over his blanket.\n\nThe next night, Myron called through the darkness of his bottom bunk. \"Win?\"\n\n\"Yes.\"\n\n\"How do you sleep so soundly?\" But Win didn't answer because he'd fallen asleep.\n\nOn the phone Win asked, \"What do you want?\"\n\n\"Did all go well last night?\"\n\n\"Mr. O'Connor hasn't called you yet?\"\n\n\"He has.\" End of subject. Myron didn't want details.\n\n\"I know,\" Win continued, \"that you did not awaken me to question my effectiveness.\"\n\n\"Kathy Culver got only one A in her senior year at Ridgewood High. Guess who her teacher was.\"\n\n\"Who?\"\n\n\"Gary Grady.\"\n\n\"Hmm. Dial-a-porn and high school English. Interesting vocational mix.\"\n\n\"I was thinking we could go see Mr. Grady this morning.\"\n\n\"At the school?\"\n\n\"Sure. The two of us can pretend we're concerned parents.\"\n\n\"For the same kid?\"\n\n\"Putting the rainbow curriculum to the test.\"\n\nWin laughed. \"This is going to be fun.\"\n\n# Chapter 15\n\n\"How do we find him?\" Win asked.\n\nThey arrived at Ridgewood High School at nine-thirty. It was a warm June day, the kind of day where you stared at the window and daydreamed about the end of school. Not much movement around the building\u2014as though the entire school, even the edifice, were coasting toward summer vacation.\n\nMyron remembered how miserable such days were. It gave him an idea.\n\n\"Let's pull the fire alarm,\" he said.\n\n\"I beg your pardon.\"\n\n\"We'll get everyone outside. It'll be easier to spot him.\"\n\n\"Idiotically ingenious,\" Win said.\n\n\"Besides, I always wanted to pull a fire alarm.\"\n\n\"Walk on the wild side.\"\n\nNo one noticed them when they entered the school. There were no guards, no locks on the door, no hall monitors of any kind. This was not an urban high school. Myron found a fire alarm not too far from the entrance.\n\n\"Kids, don't try this at home,\" Myron said. He pulled. Bells went off. Then cheers from the kids. Myron felt good about his deed. He thought about pulling alarms more often but decided some might construe the act as immature.\n\nWin held the door open and pretended to be a fire marshal. \"Single file,\" he told the students. \"And remember: Only you can prevent fires.\"\n\nMyron spotted Grady. \"Bingo.\"\n\n\"Where?\"\n\n\"Turning the corner. On the left. Mr. Fashion.\"\n\nGary Grady was wearing a yellow Century 21\u2013like blazer with Keith Partridge orange-striped pants. Win looked visibly pained at the sight. They made their approach.\n\n\"Hi, Jerry.\"\n\nGrady's head shot around. \"That's not my name.\"\n\n\"Yeah, you told me. It's your alias, right? When you do business with Fred Nickler. Your real name is Gary Grady.\"\n\nNearby students stopped walking.\n\n\"Keep moving!\" Gary snapped.\n\nThe students restarted their grudging trudge.\n\n\"Impatient teachers,\" Myron said.\n\n\"Sad,\" Win agreed.\n\nGary's thin face seemed to stretch even further. He stepped closer so that no one could overhear.\n\n\"Perhaps we can continue this conversation later,\" he whispered.\n\n\"I don't think so, Gary.\"\n\n\"I'm in the middle of a class.\"\n\n\"Tough tittie,\" Myron said.\n\nWin arched an eyebrow. \"Tough tittie?\"\n\n\"Something about being back in high school,\" Myron said. \"Besides, I thought it appropriate considering the situation.\"\n\nWin considered for a moment. \"Okay, I can accept that.\"\n\nMyron turned back to Gary. \"The fire drill will last a little while. Then it will take a little while for the kids to file back in. Then they'll want to goof around in the halls for a while. By then we'll be all done.\"\n\nGary crossed his arms over his chest. \"No.\"\n\n\"Option two, then.\" Myron took out a copy of _Nips._ \"We can play Show and Tell with the principal.\"\n\nGrady coughed into his fist. A loud fire whistle sounded. Sirens came closer. \"I don't know what you're talking about,\" he said, taking a few more steps away from the kids.\n\n\"I followed you.\"\n\n\"What?\"\n\nMyron sighed, gave him exasperation. \"You were in Hoboken yesterday morning. You picked up the mail at an address used for advertising sex lines in porno rags. Then you went home to Glen Rock, saw me, panicked, and called Fred Nickler, the managing editor of said rags.\"\n\n\"Amateur,\" Win added with disgust.\n\n\"Now, we can discuss this with you or with the school board. Up to you.\"\n\nGary glanced at his watch. \"You have two minutes.\"\n\n\"Fine.\" Myron gestured to the right. \"Why don't we step into the teachers' lavatory? I assume you have a key.\"\n\n\"Yes.\"\n\nHe opened the door. Myron had always wanted to see a teachers' bathroom, see how the other half lives. It was unremarkable in every way.\n\n\"Okay, you have me here,\" Gary said. \"What do you want?\"\n\n\"Tell me about this ad.\"\n\nGary swallowed. His enlarged Adam's apple bobbed up and down like a boxer's head avoiding jabs. \"I don't know anything about it.\"\n\nMyron and Win exchanged a glance.\n\n\"Can I stick his head in a toilet?\" Win asked.\n\nGary straightened his back. \"If you are trying to frighten me, it won't work.\"\n\nWin's voice was semipleading. \"One quick dunk?\"\n\n\"Not yet.\" Myron turned his attention back to Gary. \"I have no interest in busting you, Gary. You're a perv, that's your business. I want to know about your connection with Kathy Culver.\"\n\nSweat appeared above Gary's upper lip. \"She was a student of mine.\"\n\n\"I know. Why is her picture in _Nips_? In your ad?\"\n\n\"I have no idea. I saw it for the first time yesterday.\"\n\n\"But that's your ad, right?\"\n\nHe hesitated, giving silent half-shrugs to no one in particular. \"Okay,\" he said, \"I admit it. I advertise in Mr. Nickler's publications. No law against that. But I did not put that picture of Kathy in the ad.\"\n\n\"Who did?\"\n\n\"I don't know.\"\n\n\"But you admit operating sex lines?\"\n\n\"Yes. It's harmless. I do it to make extra money. Nobody gets hurt.\"\n\n\"Another prince,\" Myron said. \"How much extra money?\"\n\n\"In the business's heyday I was making twenty thousand dollars a month.\"\n\nMyron wasn't sure he heard right. \"Twenty thousand dollars a month from phone sex?\"\n\n\"In the mid-eighties, yes. Before the government got involved and began to crack down on 900 lines. Now I'm lucky to clear eight grand a month.\"\n\n\"Damn bureaucrats,\" Myron said. \"So how does Kathy Culver fit into all this?\"\n\n\"What do you mean?\"\n\n\"Gee, Gary, a naked picture of her is in your ad this month. Maybe that's what I mean.\"\n\n\"I already told you. I had nothing to do with that.\"\n\n\"Then I guess it's a coincidence, her being a student of yours and all.\"\n\n\"Yes.\"\n\n\"I won't hold him under long,\" Win promised. \"Please.\"\n\nMyron shook his head. \"You wrote her a glowing recommendation letter for college, correct?\"\n\n\"Kathy was a wonderful student,\" Gary replied.\n\n\"And what else?\"\n\n\"If you are suggesting that my relationship with Kathy was something other than student-teacher\u2014\"\n\n\"That's exactly what I'm suggesting.\"\n\nOnce again he crossed his arms over his chest. \"I will not dignify that with a response. And I am now terminating this conversation.\"\n\nGary was addressing them in that way teachers do. Sometimes teachers forget that life is not a classroom.\n\n\"Dunk him,\" Myron said.\n\n\"With pleasure.\"\n\nGary probably had two inches on Win. He leaned up on his toes and gave Win his most withering glare.\n\n\"I'm not afraid of you,\" Gary said.\n\n\"Mistake number one.\"\n\nWin moved with a speed that videocameras would not catch. He took hold of Gary's hand, twisted it, and pulled down. Hapkido move. Gary dropped to the tile floor. Win pressed his knee against the point of Gary's elbow. Gently. Not too much pain. Just enough to let him know who was in control.\n\n\"Damn,\" Win said.\n\n\"What?\"\n\n\"All the toilets are clean. I hate when that happens.\"\n\n\"Anything to add before the dunk?\" Myron asked.\n\nGary's face was white. \"Promise me you won't tell anyone,\" he managed.\n\n\"You'll tell us the truth?\"\n\n\"Yes. But you have to swear you won't tell anyone. Not the principal, no one.\"\n\n\"Okay.\" Myron nodded to Win. Win let go. Gary took back his hand and caressed it as though it were an abused puppy.\n\n\"Kathy and I had an affair,\" he said.\n\n\"When?\"\n\n\"Her senior year. It lasted a few months, that's all. I haven't seen her since, I swear.\"\n\n\"And that's everything?\"\n\nHe nodded. \"I don't know anything else. Somebody else put that picture in the ad.\"\n\n\"If you're lying, Gary\u2014\"\n\n\"I'm not. Hand to God.\"\n\n\"Okay,\" Myron said. \"You can go.\"\n\nGary rushed out. He had not even paused to check his hair in the mirror.\n\n\"Scum,\" Myron said. \"The man is pure unadulterated scum. Seduces his students, operates a dial-a-porn line.\"\n\n\"But a snappy dresser,\" Win said. \"So what next?\"\n\n\"We finish the investigation. Then we go to the school board. We tell them all about Mr. Grady's extracurricular activities.\"\n\n\"Didn't you just promise him you weren't going to tell?\"\n\nMyron shrugged. \"I lied.\"\n\n# Chapter 16\n\nIn something of a trance, Jessica thanked Myron and hung up the phone. She half-stumbled into the kitchen and sat down. Her mother and her younger brother Edward looked up.\n\n\"Honey,\" Carol Culver began, \"are you okay?\"\n\n\"Fine,\" she managed.\n\n\"Who was on the phone?\"\n\n\"Myron.\"\n\nSilence.\n\n\"We were talking about Kathy,\" she continued.\n\n\"What about her?\" Edward asked.\n\nHer brother had always been Edward, not Ed or Eddie or Ted. He was only a year out of college and already he owned a successful computer business, IMCS (Interactive Management Computer Systems), which developed software systems for several prestigious corporations. Edward wore only jeans, even in the office, and obnoxious T-shirts, the kind with chintzy iron-on decals that say stuff like \"Keep on Truckin'.\" He didn't own a tie. He had a wide, almost-feminine face with delicate porcelain features. Women would kill for his eyelashes. Only the buzz-cut hair\u2014and the pithy phrase on his T-shirt\u2014hinted at what Edward was proud to be: COMPUTER WEENIES HAVE THE BEST HARDWARE.\n\nJessica took a deep breath. She could not be concerned with delicacies or feelings anymore. She opened her purse and pulled out a copy of _Nips._ \"This magazine hit the stands a few days ago,\" she said.\n\nShe tossed it on the table, cover up. A cross between puzzlement and disgust blanketed her mother's face.\n\nEdward remained stoic. \"What the hell is this?\" he asked.\n\nJessica flipped to the page in the back. \"There,\" she said simply, pointing to the picture of Kathy in the bottom row.\n\nIt took a few moments for them to comprehend what they were seeing, as though the information had been waylaid somewhere between the eye and brain. Then Carol Culver let out a groan. Her hand flew to her mouth, smothering a scream. Edward's eyes narrowed into thin slits.\n\nJessica did not give her time to recover. \"There's more,\" she said.\n\nHer mother looked up at her with hollow, haunted eyes. There was no life behind them anymore, as though a final cold gust had put out a flickering flame.\n\n\"A handwriting expert checked the envelope it came in. The writing matches Kathy's.\"\n\nEdward inhaled sharply. Carol's legs finally gave out, folding at the knees. She landed hard in her chair and crossed herself. Tears came to her eyes.\n\n\"She's alive?\" Carol managed.\n\n\"I don't know.\"\n\n\"But there's a chance?\" Edward followed up.\n\nJessica nodded. \"There's always been a chance.\"\n\nStunned silence.\n\n\"But I need some information,\" Jessica continued. \"I need to know what happened to Kathy. What made her change.\"\n\nEdward's eyes narrowed again. \"What do you mean?\"\n\n\"Kathy had an affair with her high school English teacher. Senior year.\"\n\nMore silence. Jessica was not so sure it was stunned.\n\n\"The teacher, a maggot named Gary Grady, has admitted it.\"\n\n\"No,\" her mother said weakly. She lowered her head, her crucifix dangling like a pendulum. She began to weep. \"Sweet Jesus, not my baby...\"\n\nEdward stood. \"That's enough, Jess.\"\n\n\"It's not enough.\"\n\nEdward grabbed his jacket. \"I'm out of here.\"\n\n\"Wait. Where are you going?\"\n\n\"Good-bye.\"\n\n\"We need to talk this out.\"\n\n\"The hell we do.\"\n\n\"Edward\u2014\"\n\nHe ran out the back door, slamming it behind him.\n\nJessica turned back to her mother. Her sobs were gut-wrenching. Jessica watched for a minute or two. Then she turned and left the kitchen.\n\nRoy O'Connor was already in the back booth when Myron arrived. His glass was empty, and he was sucking on an ice cube. He sounded like an aardvark near an anthill.\n\n\"Hey, Roy.\"\n\nO'Connor nodded to the seat across the table, not bothering to stand. He wore gold rings that disappeared under the folds of flesh in his chubby unstained hands. His fingernails were manicured. He was somewhere between forty-five and fifty-five years old, but it was impossible to tell where. He was balding, wearing the ever-desirable swept-over look, parting his hair just below the armpit.\n\n\"Nice place, Roy,\" Myron said. \"A table in the back, low lights, soft romantic music. If I didn't know better\u2014\"\n\nO'Connor shook his head. \"Look, Bolitar, I know you think you're a regular Buddy Hackett, but give it a rest, okay?\"\n\n\"I guess flowers are out, then.\" Pause. Then: \"Buddy Hackett?\"\n\n\"We need to talk.\"\n\n\"I'm all ears.\"\n\nA waitress came over. \"Can I get you gentlemen something to drink?\"\n\n\"Another,\" Roy said, pointing to his glass.\n\n\"And for you?\"\n\n\"Do you have Yoo-Hoo?\" Myron asked.\n\n\"I think so.\"\n\n\"Great. I'll have one.\"\n\nShe left. Roy shook his head. \"A fucking Yoo-Hoo,\" he mumbled.\n\n\"Did you say something?\"\n\n\"Your goon visited me last night.\"\n\n\"Your goons visited me first,\" Myron said.\n\n\"I had nothing to do with that.\"\n\nMyron gave him his best \"come off it\" look of pure skepticism. The waitress put down the drinks Roy scooped up his martini as if it held a life-saving antidote. Myron, by contrast, sipped his Yoo-Hoo daintily. Ever the gentleman.\n\n\"Look, Myron,\" O'Connor continued, \"it's like this. I signed Landreaux. I gave him money up front. I gave him money every month. I kept my part of the bargain.\"\n\n\"You signed him illegally.\"\n\n\"I'm not the first guy to do it,\" he said.\n\n\"Nor the last. What's your point, Roy?\"\n\n\"Look, you know me. You know how I operate.\"\n\nMyron nodded. \"You're a chicken-shitted crook.\"\n\n\"I might have threatened the kid. Fine. I've done that before. But that's it. I'd never really hurt anybody.\"\n\n\"Uh-huh.\"\n\n\"Word would get out to the athletes. I'd be ruined.\"\n\n\"Damn shame that would be.\"\n\n\"Bolitar, you're not making this any easier.\"\n\n\"I'm not trying to.\"\n\nO'Connor grabbed the drink again. He finished it and signaled to the waitress for another. \"I've gotten involved with the wrong people,\" he said.\n\n\"What do you mean?\"\n\n\"I worked up some big-time gambling debts. Debts I couldn't pay off.\"\n\n\"So they took a piece of your business.\"\n\nRoy nodded. \"They control me now. Your\u2014your friend from last night.\" A Geiger counter could have registered the quake in his voice when he mentioned Win. \"I want to do just what he said, but I don't have the power anymore.\"\n\nMyron took another sip of his Yoo-Hoo, hoping he wasn't getting one of those chocolate mustaches. \"My friend won't be pleased to hear that.\"\n\n\"You have to tell him it's not me.\"\n\n\"Then who is it?\"\n\nRoy sat back, shaking his head. \"I can't say. But I can tell you they play for keeps. And they don't understand a thing about this business. They think they can just scare everyone into compliance. They want to make an example out of someone.\"\n\n\"And Landreaux is the example?\"\n\n\"Landreaux. And you. They want to hurt Landreaux. They want to kill you. They're putting out a contract on your head.\"\n\nAnother cool sip. Myron said nothing.\n\n\"You don't seem very worried,\" Roy said.\n\n\"I laugh in the face of death,\" Myron replied. \"Well, maybe not laugh. More like a snicker. A quiet snicker.\"\n\n\"Jesus, you're a lunatic.\"\n\n\"And I wouldn't do it directly in death's face. So it's more like a quiet snicker behind his back.\"\n\n\"Bolitar, this isn't funny.\"\n\n\"No,\" Myron agreed. \"It's not. I strongly suggest you call them off.\"\n\n\"Haven't you heard a word I've said? I got no control here.\"\n\n\"If something happens to me, my friend will be very upset. He'll take it out on you.\"\n\nRoy swallowed. \"But I'm powerless. You have to believe that.\"\n\n\"Then tell me who's calling the shots.\"\n\n\"I can't.\"\n\nMyron shrugged. \"Maybe we can be buried next to one another. One of those romantic tragedy things.\"\n\n\"They'll kill me if I say anything.\"\n\n\"What do you think my friend will do to you?\"\n\nRoy shuddered. He sucked on the ice again, trying to salvage the last remnants of the whiskey. \"Where is that damn bimbo with my drink?\"\n\n\"Who's calling the shots, Roy?\"\n\n\"You didn't hear it from me, right?\"\n\n\"Right.\"\n\n\"You won't tell them?\"\n\n\"Mum's the word.\"\n\nOne more ice suck. Then Roy said, \"Ache.\"\n\n\"Herman Ache?\" Myron asked, surprised. \"Herman Ache is behind this?\"\n\nRoy shook his head. \"His younger brother. Frank. He's out of control. I don't know what the psycho will do next.\"\n\nFrank Ache. It made sense. Herman Ache was one of New York's leading mobsters, responsible for countless misery. But next to his younger brother Frank, Herman was an Alan Alda clone. Aaron would enjoy working for someone like Frank.\n\nThis was not good news. Myron toyed with the idea of dropping the snicker altogether. \"Anything else you can tell me?\"\n\n\"No. I just don't want anyone hurt.\"\n\n\"You're some guy, Roy. So selfless.\"\n\nO'Connor stood. \"I got nothing more to say.\"\n\n\"I thought we were going to have lunch.\"\n\n\"Have it by yourself,\" O'Connor said. \"It's on my tab.\"\n\n\"Won't be the same without your company.\"\n\n\"Yet somehow you'll muddle through.\"\n\nMyron picked up the menu. \"I'll try.\"\n\n# Chapter 17\n\nWho else to call?\n\nThe answer, Jessica realized, was obvious.\n\nNancy Serat. Kathy's roommate and closest friend.\n\nJessica sat at her father's desk. The lights were turned off, the shades were pulled down, but the sunlight was still strong enough to sneak through and cast shadows.\n\nAdam Culver had done everything he could to make his home office radically different from the cement, institutional, macabre feel of the county morgue. The results were mixed. The converted bedroom had bright yellow walls, plenty of windows, silk flowers, white Formica desk. Teddy bears encircled the room. William Shakesbear. Rhett Beartler with Scarlett O'Beara. Bear Ruth. Bearlock Holmes. Humphrey Beargart with Lauren Bearcall. The whole atmosphere was cheerful, albeit a forced cheerful, like a clown you laugh at but find a little scary.\n\nShe took her phone book from her purse. Nancy had sent the family a card a few weeks ago. She had won some fellowship and was staying on campus to work in admissions. Jessica looked up her number and dialed.\n\nOn the third ring the answering machine picked up. Jessica left a message and hung up. She was about to start going through the drawers when a voice stopped her.\n\n\"Jessica.\"\n\nShe looked up. Her mother stood in the doorway. Her eyes were sunken, her face a skeletal death mask. Her body swayed as though she were about to topple over.\n\n\"What are you doing in here?\" Carol asked.\n\n\"Just looking around,\" she said.\n\nCarol nodded, her head bobbing on the string that was her neck. \"Find anything?\"\n\n\"Not yet.\"\n\nCarol sat down. She stared straight ahead, her eyes unfocused. \"She was always such a happy child,\" she said slowly. Her fingers fiddled with prayer beads, her gaze still far off. \"Kathy never stopped smiling. She had such a wonderful, happy smile. It lit up any room she entered. You and Edward, well, you were both more brooding. But Kathy\u2014she had a smile for everyone and everything. Do you remember?\"\n\n\"Yes,\" Jessica said. \"I remember.\"\n\n\"Your father used to joke that she had the personality of a born-again cheerleader,\" Carol added, chuckling at the memory. \"Nothing ever brought her down.\" She stopped, the chuckle fading away. \"Except, I guess, me.\"\n\n\"Kathy loved you, Mom.\"\n\nShe sighed deeply, her chest heaving as though even a sigh took great effort. \"I was a strict mother with you girls. Too strict, I guess. I was old-fashioned.\"\n\nJessica did not reply.\n\n\"I just didn't want you or your sister to...\" She lowered her head.\n\n\"To what?\"\n\nShe shook her head. Her fingers moved across the beads at a more fervid pace. For a long time neither of them spoke. Then Carol said, \"You were right before, Jessica. Kathy changed.\"\n\n\"When?\"\n\n\"Her senior year.\"\n\n\"What happened?\"\n\nTears sprang to Carol's eyes. Her mouth tried to form words, her hands moving in gestures of helplessness. \"The smile,\" she replied with something like a shrug. \"One day it was gone.\"\n\n\"Why?\"\n\nHer mother wiped her eyes. Her lower lip quivered. Jessica's heart reached out to her, but for some reason the rest of her couldn't. She sat and watched her suffering, strangely uninvolved, as if she were watching a late-night tearjerker on cable.\n\n\"I'm not trying to hurt you,\" Jessica said. \"I just want to find Kathy.\"\n\n\"I know, sweetheart.\"\n\n\"I think,\" Jessica continued, \"that whatever changed Kathy is connected to her disappearance.\"\n\nHer mother's shoulders sagged. \"Merciful God.\"\n\n\"I know it hurts,\" Jessica said. \"But if we can find Kathy, if we can find who killed Dad\u2014\"\n\nCarol's head shot up. \"Your father was killed in a robbery.\"\n\n\"I don't think so. I think it's all connected. Kathy's disappearance, Dad's murder, everything.\"\n\n\"But\u2014how?\"\n\n\"I don't know yet. Myron is helping me find out.\"\n\nThe doorbell rang.\n\n\"That'll be Uncle Paul,\" her mother said, heading for the door.\n\n\"Mom?\"\n\nCarol stopped but did not turn around.\n\n\"What's going on? What are you afraid to tell me?\"\n\nThe doorbell rang again.\n\n\"I better get that,\" Carol said. She hurried down the stairs.\n\n\"So,\" Win began, \"Frank Ache wants to kill you.\"\n\nMyron nodded. \"Seems so.\"\n\n\"A shame.\"\n\n\"If he'd only get to know me. The real me.\"\n\nThey sat in the front row at Titans Stadium. Out of the goodness of his heart, Otto had agreed to let Christian practice. That, and the fact that veteran quarterback Neil Decker was beyond horrendous.\n\nThe morning session had been a lot of wind sprints and walking through plays. The afternoon session, however, was a bit of a surprise. The players were in full gear, almost unheard of this early on in the year.\n\n\"Frank Ache is not a kind fellow,\" Win said.\n\n\"He likes torturing animals.\"\n\n\"Excuse me?\"\n\n\"A friend of mine knew him growing up,\" Myron explained. \"Frank Ache's favorite hobby was to chase down cats and dogs and bash in their heads with a baseball bat.\"\n\n\"I bet that impressed the girls,\" Win said.\n\nMyron nodded.\n\n\"I assume, then, that you will be in need of my unique services.\"\n\n\"For a few days, anyway,\" Myron replied.\n\n\"Goodie. May I also assume that you have a plan?\"\n\n\"I'm working on it. Feverishly.\"\n\nChristian jogged out on the field. He moved in that effortless way great athletes do. He got into the huddle, broke it, and approached the line of scrimmage.\n\n\"Full contact!\" a coach yelled out.\n\nMyron looked at Win. \"I don't like this.\"\n\n\"What?\"\n\n\"Full contact on the first day.\"\n\nChristian started calling out numbers. Then he gave a few hut-huts before the ball was snapped to him. He faded back to pass.\n\n\"Oh, shit,\" Myron said.\n\nTommy Lawrence, the Titans' All-Pro linebacker, charged forward unblocked. Christian saw him too late. Tommy placed his helmet into Christian's sternum and slammed him to the ground\u2014the kind of tackle that hurts like hell but doesn't do any permanent damage. Two other defenders piled on.\n\nChristian got up, wincing and holding his chest. Nobody helped him.\n\nMyron stood.\n\nWin stopped him with a shake of his head. \"Sit down, Myron.\"\n\nOtto Burke came down the stairs, entourage in tow.\n\nMyron glared at him. Otto smiled brightly. He made a tsk-tsk noise. \"I traded a lot of popular veterans to get him,\" he said. \"It looks like some of the guys aren't too thrilled.\"\n\n\"Sit down, Myron,\" Win repeated.\n\nMyron hesitated, then complied.\n\nChristian limped back to the huddle. He called the next play and again approached the line of scrimmage. He surveyed the defense, yelled out numbers and hut-huts, then took the snap from the center. He stepped back. Tommy Lawrence blitzed again over left guard, completely untouched. Christian froze. Tommy bore down on him. He leaped like a panther, his arms stretched out for a bone-crushing tackle. Christian moved at the very last moment. Not a big move. Just a slight shift, actually. Tommy flew by him and landed on the ground. Christian pumped and threw a bomb.\n\nComplete pass.\n\nMyron turned around, grinning. \"Hey, Otto?\"\n\n\"What?\"\n\n\"Kiss my grits.\"\n\nOtto's smile did not falter. Myron wondered how he did that, if his mouth was frozen that way, like the threat a little kid hears from his mom when he's making faces. Otto nodded and walked away. His entourage followed in a row, like a family of mallard ducks.\n\nWin looked at Myron \"Kiss my grits?\"\n\nShrug. \"Paying homage to Flo on _Alice_.\"\n\n\"You watch too much television.\"\n\n\"Listen, I've been thinking.\"\n\n\"Oh?\"\n\n\"About Gary Grady,\" Myron said.\n\n\"What about him?\"\n\n\"He has an affair with a student. She vanishes a year or so later. Time passes and her picture ends up in a porno ad he runs.\"\n\n\"Your point being?\"\n\n\"It's crazy.\"\n\n\"So is everything about this case.\"\n\nMyron shook his head. \"Think about it. Grady admits having an affair with Kathy, right? So what would be the last thing he'd want to do?\"\n\n\"Publicize it.\"\n\n\"Yet her picture ends up in his ad.\"\n\n\"Ah.\" Win nodded. \"You believe someone is setting him up.\"\n\n\"Exactly.\"\n\n\"Who?\"\n\n\"Fred Nickler would be my bet,\" Myron said.\n\n\"Hmm. He did hand over Grady's p.o. box without much debate.\"\n\n\"And he has the power to switch photos in his own magazine.\"\n\n\"So what do you suggest?\" Win asked.\n\n\"I'd like you to check out Mr. Fred Nickler very thoroughly Maybe talk to him again. _Talk_ ,\" Myron repeated. \"Not visit.\"\n\nOn the field Christian was fading back again. For the third straight time Tommy Lawrence blitzed over left guard untouched. In fact, the left guard stood with his hands on his hips and watched.\n\n\"Christian's own lineman is setting him up,\" Myron said.\n\nChristian side-stepped Tommy Lawrence, cocked his arms, and whipped the ball with unearthly velocity directly into his left guard's groin. There was a short _oomph_ sound. The left guard collapsed like a folding chair.\n\n\"Ouch,\" Win said.\n\nMyron almost clapped. \" _The Longest Yard_ revisited.\"\n\nThe left guard was, of course, wearing a cup. But a cup was far from full protection against a speeding missile. He rolled on the ground, back curved fetal-like, eyes wide. Every man in the general vicinity gave a collective, sympathetic \"Ooo.\"\n\nChristian walked over to his left guard\u2014a man weighing in excess of 275 pounds\u2014and offered him a hand. The left guard took it. He limped back to the huddle.\n\n\"Christian has balls,\" Myron said.\n\nWin nodded. \"But can the same be said of the left guard?\"\n\n# Chapter 18\n\nAs soon as Myron entered the Reston University campus, his car phone rang.\n\n\"Listen, putz, I got what you want,\" P.T. said. \"My friend's name is Jake Courter. He's the town sheriff.\"\n\n\"Sheriff Jake,\" Myron said. \"You're kidding, right?\"\n\n\"Hey, don't let the title fool you. Jake used to work homicide in Philly, Boston, and New York. Good man. He said he'd meet with you today at three.\"\n\nMyron checked his watch. It was one o'clock now. The station was five minutes away. \"Thanks, P.T.\"\n\n\"Can I ask you something, Myron?\"\n\n\"Shoot.\"\n\n\"Why you looking into this?\"\n\n\"It's a long story, P.T.\"\n\n\"This have to do with her sister? That great piece of tail you used to nail?\" He cackled.\n\n\"You're all class, P.T.\"\n\n\"Hey, Myron, I want to hear about it sometime. The whole story.\"\n\n\"It's a promise.\"\n\nMyron parked the car and headed into the old athletic center. The corridor was a bit more beaten up than Myron had expected. Three rows of framed photographs of past athletic teams\u2014some from as far back as a hundred years ago\u2014lined the walls. Myron approached a beaded-glass door that looked like something out of an old Sam Spade film. The word FOOTBALL was stenciled in black. He knocked.\n\nThe voice was like an old tire on an unpaved road. \"What?\"\n\nMyron stuck his head. \"Busy, Coach?\"\n\nReston University football coach Danny Clarke looked up from his computer. \"Who the hell are you?\" he rasped.\n\n\"Fine, thanks. But let's dispense with the pleasantries.\"\n\n\"That supposed to be funny?\"\n\nMyron tilted his head. \"You didn't think so?\"\n\n\"I'll ask one more time: Who the hell are you?\"\n\n\"Myron Bolitar.\"\n\nThe coach's scowl did not change. \"Am I supposed to know you?\"\n\nIt was a hot summer day, the campus was practically empty, and here sat the school's legendary football coach wearing a suit and tie, watching videotapes of high school prospects. A suit and tie and no air conditioning. If the heat bothered Danny Clarke, it didn't show. Everything about him was well groomed and tidy. He was shelling and eating peanuts, but no mess was visible. His jaw muscles bunched as he chewed, making little knobs appear and disappear near his ears. He had a prominent vein in his forehead.\n\n\"I'm a sports agent.\"\n\nHe flicked his eyes away like a ruler dismissing an underling. \"Get out of here. I'm busy.\"\n\n\"We need to talk.\"\n\n\"Out of here, asshole. Now.\"\n\n\"I just\u2014\"\n\n\"Listen up, shithead.\" He pointed a coach finger at Myron. \"I don't talk to bottom-feeders. Ever. I run a clean program with clean players. I don't take payoffs from so-called agents or any of that bullshit. So if you got an envelope stuffed with green, you can go shove it up your ass.\"\n\nMyron clapped. \"Beautiful. I laughed, I cried, it became a part of me.\"\n\nDanny Clarke looked up sharply. He wasn't used to having his orders questioned, but part of him seemed almost amused by it. \"Get the hell out of here,\" he growled, but more gently now. He turned back to the television. On the screen a young quarterback threw a long, tight spiral. Caught. Touchdown.\n\nMyron decided to disarm him with tact. \"The kid looks pretty good,\" he said.\n\n\"Yeah, well, it's a good thing you're a scum-sucking leech and not a scout. The kid can't play a lick. Now take a hike.\"\n\n\"I want to talk to you about Christian Steele.\"\n\nThat got his attention. \"What about him?\"\n\n\"I'm his agent.\"\n\n\"Oh,\" Danny Clarke said. \"Now I remember. You're the old basketball player. The one who hurt his knee.\"\n\n\"At your service,\" Myron said.\n\n\"Is Christian okay?\"\n\nMyron tried to look noncommittal. \"I understand he didn't get along with his teammates.\"\n\n\"So? You his social coordinator?\"\n\n\"What was the problem?\"\n\n\"I can't see how it matters now,\" he said.\n\n\"Then humor me.\"\n\nIt took the coach some time to relax his glare. \"It was a lot of things,\" he said. \"But I guess Horty was the main problem.\"\n\n\"Horty?\" Clever interrogation techniques. Pay attention.\n\n\"Junior Horton,\" he explained. \"A defensive lineman. Good speed, good size, good talent. The brains of a citrus beverage.\"\n\n\"So what does this Horty have to do with Christian?\"\n\n\"They didn't see eye to eye.\"\n\n\"How come?\"\n\nDanny Clarke thought a moment. \"I don't know. Something to do with that girl who disappeared.\"\n\n\"Kathy Culver?\"\n\n\"Right. Her.\"\n\n\"What about her?\"\n\nHe turned back to the VCR and changed tapes. Then he typed something on his computer. \"I think maybe she dated Horty before Christian. Something like that.\"\n\n\"So what happened?\"\n\n\"Horty was a bad apple from the get-go. In his senior year I found out he was pushing drugs to my players: cocaine, dope, Lord knows what else. So I bounced him. Later, I heard he'd been supplying the guys with steroids for three years.\"\n\nLater my ass, Myron thought. But for once he kept the thought to himself. \"So what does this have to do with Christian?\"\n\n\"Rumors started circulating that Christian had gotten Horty thrown off the team. Horty fueled them, you know, telling the guys that Christian was turning them all in for using steroids, stuff like that.\"\n\n\"Was that true?\"\n\n\"Nope. Two of my best players showed up game day so stoned, they could barely see. That's when I took action. Christian had nothing to do with it. But you know how it is. They all figured Christian was the star. If he wanted his ass wiped, the coaches asked Charmin or Downy.\"\n\n\"Did you tell your guys Christian had nothing to do with it?\"\n\nHe made a face. \"You think that would have helped? They would have thought I was covering for him, protecting him. They would have hated him even more. As long as it didn't affect their play\u2014and it didn't\u2014it was not my concern. I just let it be.\"\n\n\"You're a real character developer, Coach.\"\n\nHe gave Myron his best intimidate-the-freshman glare. The forehead vein started pulsing. \"You're out of line, Bolitar.\"\n\n\"Wouldn't be the first time.\"\n\n\"I care about my boys.\"\n\n\"Yeah, I can tell. You let Horty stay as long as he pumped your boys with dangerous albeit play-enhancing drugs. When he graduated to the big leagues\u2014to the stuff that had a negative on-the-field impact\u2014all of a sudden you became a righteous drug czar.\"\n\n\"I don't have to listen to this bullshit,\" Danny Clarke ranted. \"Especially from a no-good, bloodsucking vampire. Get the hell out of my office. Now.\"\n\nMyron said, \"You want to catch a movie together sometime? Maybe a Broadway show?\"\n\n\"Out!\"\n\nMyron left. Another day, another friend. Charm was the key.\n\nHe had plenty of time to kill before he visited Sheriff Jake, so he decided to take a stroll. The campus was like a ghost town, except no tumbleweeds were skittering along the ground. The students were gone for summer break. The buildings stood lifeless and sad. In the distance a stereo was playing Elvis Costello. Two girls appeared. Co-ed types wearing crotch-riding shorts and halter tops. They were walking a hairy, little dog\u2014a Shih Tzu. It looked like Cousin It after one too many spins in the dryer. Myron smiled and nodded as the girls passed him. Neither one fainted or disrobed. Astonishing. The little dog, however, snarled at him. Cujo.\n\nHe was nearly at his car when he spotted the sign:\n\nCAMPUS POST OFFICE\n\nHe stopped, looked around the grounds, saw nobody. Hmm. It was worth a try.\n\nThe inside of the post office was painted institutional green, the same color as the school bathroom. A long V-shaped corridor was wallpapered with p.o. boxes. He heard the distant sound of a radio. He couldn't make out the song, just a strong, monotonous bass beat.\n\nMyron approached the mail window. A kid sat with his feet up. The music was coming from the kid's ears. He was listening to one of those Walkman clones with the minispeakers that bypass the ears and plug directly into the cerebrum. His black high-tops rested on a desk, his baseball hat tipped down like a sombrero at siesta time. There was a book on his lap. Philip Roth's _Operation Shylock._\n\n\"Good book,\" Myron said.\n\nThe kid did not look up.\n\n\"Good book,\" Myron said again, this time yelling.\n\nThe kid pulled the speakers out of his ears with a sucking pop. He was pale and red-haired. When he took off his hat, his hair was Afro-wild. Bernie from _Room 222._\n\n\"What?\"\n\n\"I said, good book.\"\n\n\"You read it?\"\n\nMyron nodded. \"Without moving my lips.\"\n\nThe kid stood. He was tall and lanky.\n\n\"You play basketball?\" Myron asked.\n\n\"Yeah,\" the kid said. \"Just finished my freshman year. Didn't play much.\"\n\n\"I'm Myron Bolitar.\"\n\nThe kid looked at him blankly.\n\n\"I played ball for Duke.\"\n\nBlink, blink.\n\n\"No autographs, please.\"\n\n\"How long ago did you play?\" the kid asked.\n\n\"Graduated ten years ago.\"\n\n\"Oh,\" the kid replied, as though that explained everything. Myron did some quick math in his head. The kid had been seven or eight when Myron won the national title. He suddenly felt very old.\n\n\"We used peach baskets back then.\"\n\n\"What?\"\n\n\"Never mind. Can I ask you a few questions?\"\n\nThe kid shrugged. \"Go ahead.\"\n\n\"How often are you on duty in the post office?\"\n\n\"Five days a week in the summer, nine to five.\"\n\n\"Is it always this quiet?\"\n\n\"This time of year, yeah. No students, so there's almost no mail.\"\n\n\"Do you do the mail sorting?\"\n\n\"Sure.\"\n\n\"Do you do pick-ups?\"\n\n\"Pick-ups?\"\n\n\"Campus mail.\"\n\n\"Yeah, but there's only that slot by the front door.\"\n\n\"That's the only campus mailbox?\"\n\n\"Um-hmm.\"\n\n\"Been getting a lot of campus mail lately?\"\n\n\"Next to none. Three, four letters a day.\"\n\n\"Do you know Christian Steele?\"\n\n\"Heard of him,\" the kid said. \"Who hasn't?\"\n\n\"He got a big manila envelope in his box a few days ago. There was no postmark, so it had to be mailed from campus.\"\n\n\"Yeah, I remember. What about it?\"\n\n\"Did you see who mailed it?\" Myron asked.\n\n\"No,\" the kid said. \"But they were the only pieces of mail I got that whole day.\"\n\nMyron cocked his head. \"They?\"\n\n\"What?\"\n\n\"You said 'they.' They were the only pieces.' \"\n\n\"Right. Two big envelopes. Exact same except for the address.\"\n\n\"Do you remember who the other one was addressed to?\"\n\n\"Sure,\" the kid said. \"Harrison Gordon. He's the dean of students.\"\n\n# Chapter 19\n\nNancy Serat dropped her suitcase on the floor and rewound the answering machine. The tape raced back, shrieking all the way. She had spent the weekend in Canc\u00fan, a final vacation before starting her fellowship at Reston University, her alma mater.\n\nThe first message was from her mother.\n\n\"I don't want to disturb you on vacation, dear. But I thought you'd want to know that Kathy Culver's father died yesterday. He was stabbed by a mugger. Awful. Anyway, I thought you'd want to know. Give us a call when you get back in. Your father and I want to take you out to dinner for your birthday.\"\n\nNancy's legs felt weak. She collapsed into the chair, barely hearing the next two messages\u2014one from her dentist's office reminding her of a teeth cleaning on Friday, the other from a friend planning a party.\n\nAdam Culver was dead. She couldn't believe it. Her mother had said it was a mugger. Nancy wondered. Was it really random? Or did it have something to do with his visit on...?\n\nShe calculated the days.\n\nKathy's father had visited on the day he died.\n\nA voice on the machine jarred her back to the present.\n\n\"Hello, Nancy. This is Jessica Culver, Kathy's sister. When you get in, please give me a call. I need to talk to you as soon as possible. I'm staying with my mom. The number here is 555-1477. It's kind of important. Thank you.\"\n\nNancy suddenly felt very cold. She listened to the rest of the messages. Then she sat without moving for several minutes, debating her options. Kathy was dead\u2014or so everyone believed. And now her father, hours after talking to Nancy, was dead too.\n\nWhat did it mean?\n\nShe remained very still, the only sound her own breaths coming in short, hitching gasps. Then she picked up the phone and dialed Jessica's number.\n\nThe dean's office was closed, so Myron proceeded straight to his house. It was an old Victorian with cedar shingles on the west end of the campus. He rang the doorbell. A very attractive woman opened the door. She smiled solicitously.\n\n\"May I help you?\"\n\nShe wore a tailored cream suit. She was not young, but she had a grace and beauty and sex appeal that made Myron's mouth a little dry. In front of such a lady Myron wanted to remove his hat, except he wasn't wearing one.\n\n\"Good afternoon,\" he said. \"I'm looking for Dean Gordon. My name is Myron Bolitar, and\u2014\"\n\n\"The basketball player?\" she interrupted. \"Of course. I should have recognized you right away.\"\n\nTo grace, beauty, and sex appeal, add knowledge of basketball.\n\n\"I remember watching you in the NCAAs,\" she continued. \"I cheered you all the way.\"\n\n\"Thank you\u2014\"\n\n\"When you got hurt\u2014\" She stopped, shook the head attached to the Audrey Hepburn neck. \"I cried. I felt like a part of me was hurt too.\"\n\nGrace, beauty, sex appeal, basketball knowledge, and alas, sensitivity. She was also long-legged and curvy. All in all, a nice package.\n\n\"That's very kind of you, thank you.\"\n\n\"It's a pleasure to meet you, Myron.\"\n\nEven his first name sounded good coming from those lips. \"And you must be Dean Gordon's wife. The lovely dean-nessa.\"\n\nShe laughed at the Woody Allen rip-off. \"Yes, I'm Madelaine Gordon. And no, my husband is not home at the moment.\"\n\n\"Are you expecting him soon?\"\n\nShe smiled as though the question were a double entendre. Then she gave him a look that flushed his cheeks. \"No,\" she said slowly. \"He won't be home for hours.\"\n\nHeavy accent on the word _hours._\n\n\"Well then, I won't bother you anymore.\"\n\n\"It's no bother.\"\n\n\"I'll come by another time,\" he said.\n\nMadelaine (he liked that name) nodded demurely. \"I'll look forward to it.\"\n\n\"Nice meeting you.\" With Myron, every line was a lady-slayer.\n\n\"Nice meeting you too,\" she singsonged. \"Good-bye, Myron.\"\n\nThe door closed slowly, teasingly. He stood there for another moment, took a few deep breaths, and hurried back to his car. Whew.\n\nHe checked his watch. Time to meet Sheriff Jake.\n\nJake Courter was alone in the station, which looked like something out of Mayberry RFD. Except Jake was black. There were never any blacks in Mayberry. Or Green Acres. Or any of those places. No Jews, Latinos, Asians, ethnics of any kind. Would have been a nice touch. Maybe have a Greek diner or a guy named Abdul working for Sam Drucker at the grocery store.\n\nMyron estimated Jake to be in his mid-fifties. He was in plainclothes, his jacket off, his tie loosened. A big gut spilled forward like something that belonged to someone else. Manila files were scattered across Jake's desk, along with the remnants of what might be a sandwich and an apple core. Jake gave a tired shrug and wiped his nose with what looked like a dishrag.\n\n\"Got a call,\" he said by way of introduction. \"I'm supposed to help you out.\"\n\n\"I'd appreciate it,\" Myron said.\n\nJake leaned back and put his feet on the desk. \"You played ball against my son. Gerard. Michigan State.\"\n\n\"Sure,\" Myron said, \"I remember him. Tough kid. Monster on the boards. Defensive specialist.\"\n\nJake nodded proudly. \"That's him. Couldn't shoot worth a lick, but you always knew he was there.\"\n\n\"An enforcer,\" Myron added.\n\n\"Yep. He's a cop now. In New York. Made detective second grade already. Good cop.\"\n\n\"Like his old man.\"\n\nJake smiled. \"Yeah.\"\n\n\"Give him my regards,\" Myron said. \"Better yet, give him an elbow to the rib cage. I still owe him a few.\"\n\nJake threw back his head and laughed. \"That's Gerard. Finesse was never his forte.\" He blew his nose into the dishrag. \"But I'm sure you didn't come all this way to talk basketball.\"\n\n\"No, I guess not.\"\n\n\"So why don't you tell me what this is all about, Myron?\"\n\n\"The Kathy Culver case,\" he said. \"I'm looking into it. Very surreptitiously.\"\n\n\"Surreptitiously,\" Jake repeated with a raised eyebrow. \"Awfully big word, Myron.\"\n\n\"I've been listening to self-improvement tapes in the car.\"\n\n\"That right?\" Jake blew his nose again. Sounded like a ewe's mating call. \"So what's your interest in this\u2014aside from the fact that you represent Christian Steele and you used to have a thing for Kathy's sister.\"\n\nMyron said, \"You're thorough.\"\n\nHe took a bite out of the half-eaten sandwich on his desk, smiled. \"Man does love to be flattered.\"\n\n\"It's like you said. Christian Steele. He's a client. I'm trying to help him out.\"\n\nJake studied him, waiting again. It was an old trick. Stay silent long enough, and the witness would start talking again, elaborating. Myron did not bite.\n\nAfter a full minute had passed Jake said, \"So let me get this straight. Christian Steele signs on with you. One day you start chatting. He says, 'You know, Myron, the way you been licking my lily-white ass and all, I'd like you to go play Dick Fuckin' Tracy and find my old squeeze who's been missing for the last year and a half and the cops and feds can't find.' That how it went, Myron?\"\n\n\"Christian doesn't curse,\" Myron said.\n\n\"Okay, fine, you want to skip the dance? Let's skip it. You want me to give, you have to give back.\"\n\n\"That's fair enough,\" Myron said. \"But I can't. Not yet, anyhow.\"\n\n\"Why not?\"\n\n\"It could hurt a lot of people,\" Myron said. \"And it's probably nothing.\"\n\nHe made a face. \"What do you mean, hurt?\"\n\n\"I can't elaborate.\"\n\n\"Fuck you can't.\"\n\n\"I'm telling you, Jake. I can't say anything.\"\n\nJake studied him again. \"Let me tell you something, Bolitar. I'm no glory hound. I'm like my son was on the court. Not flashy but a workhorse. I don't look for clippings so I can climb up the ladder. I'm fifty-three years old. My ladder don't go no higher. Now this may seem a bit old-fashioned to you, but I believe in justice. I like to see truth prevail. I've lived with Kathy Culver's disappearance for eighteen months. I know her inside and out. And I have no idea what happened that night.\"\n\n\"What do you think happened?\" Myron asked.\n\nJake picked up a pencil and tapped it on the desk. \"Best guess based on the evidence?\"\n\nMyron nodded.\n\n\"She's a runaway.\"\n\nMyron was surprised. \"What makes you say that?\"\n\nA slow smile spread across Jake's face. \"That's for me to know and you to find out.\"\n\n\"P.T. said you would help.\"\n\nJake shrugged and took another bite from yet another sandwich scrap. \"What about Kathy's sister? I understand you two were pretty heavy.\"\n\n\"We're friends now.\"\n\nJake gave a low whistle. \"I've seen her on TV,\" he said. \"Hard to be friends with a woman who looks like that.\"\n\n\"You're a real nineties guy, Jake.\"\n\n\"Yeah, well, I forgot to renew my subscription to _Cosmo_.\"\n\nThey stared at each other for a while. Jake settled back in his chair and examined his fingernails. \"What do you want to know?\"\n\n\"Everything,\" Myron said. \"From the beginning.\"\n\nJake folded his arms across his chest. He took a deep breath and let it loose slowly. \"Campus security got a call from Kathy Culver's roommate, Nancy Serat. Kathy and Nancy lived in the Psi Omega sorority house. Nice house. All pretty white girls with blond hair and white teeth. Kind that all look alike and sound alike. You get the picture.\"\n\nMyron nodded. He noticed that Jake was not reading or even consulting a file. This was coming from memory.\n\n\"Nancy Serat told the rent-a-cop that Kathy Culver hadn't returned to her room for three days.\"\n\n\"Why did Nancy wait so long to call?\" Myron asked.\n\n\"Seems Kathy wasn't spending too many nights in the sorority house anyhow. She slept in your client's room most of the time. You know, the one who doesn't like to curse.\" Brief smile. \"Anyhow, your boy and Nancy got to talking one day, both figuring Kathy had been spending all her time with the other. That's when they realized she was missing and called campus security.\n\n\"Campus security told us about it, but no one got very excited at first. A co-ed missing for a few days is hardly an earth-shattering event. But then one of the rent-a-cops found the panties on top of a waste bin, and well, you know what happened then. The story spread like a grease stain on Elvis's pillow.\"\n\n\"I read there was blood on the panties,\" Myron said.\n\n\"A media exaggeration. There was a bloodstain, dry, probably from a menstrual cycle. We typed it. B negative. Same type as Kathy Culver's. But there was also semen. Enough antibodies for a DNA and blood test.\"\n\n\"Did you have any suspects?\"\n\n\"Only one,\" Jake said. \"Your boy, Christian Steele.\"\n\n\"Why him?\"\n\n\"Usual reasons. He was the boyfriend. She was on her way to see him when she vanished. Nothing very specific or damaging. But the DNA test on the semen cleared him.\" He opened a small refrigerator behind him. \"Want a Coke?\"\n\n\"No, thanks.\"\n\nJake grabbed a can and snapped it open. \"Here's what you probably read in the papers,\" he continued. \"Kathy is at a sorority cocktail party. She has a drink or two, nothing serious, leaves at ten P.M. to meet Christian, and disappears. End of story. But now let me fill it in a little.\"\n\nMyron leaned forward. Jake took a swig of Coke and wiped his mouth with a forearm the size of an oak trunk.\n\n\"According to several of her sorority sisters,\" he said, \"Kathy was distracted. Not herself. We also know she got a phone call a few minutes before she left the house. She told Nancy Serat the call was from Christian and she was going to meet him. Christian denies making the call. These were all intracampus calls, so there is no way for us to tell. But the roommate says Kathy sounded strained on the phone, not like she was talking to her true love, Mr. Clean-Mouth.\n\n\"Kathy hung up the phone and went back downstairs with Nancy. Then she posed for the now-famous last photograph before leaving for good.\"\n\nHe opened his desk drawer and handed Myron the photograph. Myron had, of course, seen it countless times before. Every media outlet in the country had run the photograph with morbid fascination. A picture of twelve sorority sisters. Kathy stood second from the left. She wore a blue sweater and skirt. Pearls adorned her neck. Very preppy. According to Kathy's sorority sisters, Kathy left the house alone immediately after the picture was taken. She never returned.\n\n\"Okay,\" Jake said, \"so she leaves the cocktail party. Only one person saw her for sure after that.\"\n\n\"Who?\" Myron asked.\n\n\"Team trainer. Guy named Tony Gardola. He saw her, strangely enough, entering the team's locker room around quarter after ten. The locker room was supposed to be empty at that hour. Only reason Tony was there was that he forgot something. He asked her what she was doing there, and she said she was meeting Christian. Tony figured what the hell, kids today. Might be having a kinky locker-room encounter. Tony decided it was in his best interest not to ask too many questions.\n\n\"That's our last _firm_ report on her whereabouts. We have a possible sighting of her on the western edge of campus at around eleven P.M. Someone saw a blond woman wearing a blue sweater and skirt. It was too dark to make a positive ID. The witness said he wouldn't have even noticed, except she seemed in a rush. Not running but doing one of those quick-walks.\"\n\n\"Where on the western edge of campus?\" Myron asked.\n\nJake opened a file and took out a map, still studying Myron's face as though it held a clue. He spread the map out and pointed. \"Here,\" he said. \"In front of Miliken Hall.\"\n\n\"What's Miliken Hall?\" Myron asked.\n\n\"Math building. Locked tight by nine o'clock. But the witness said she was moving west.\"\n\nMyron's eyes traced a path to the west. There were four other buildings labeled FACULTY HOUSING. Myron remembered the spot.\n\nIt was where Dean Gordon lived.\n\n\"What is it?\" Jake asked.\n\n\"Nothing.\"\n\n\"Bullshit, Bolitar. You see something.\"\n\n\"It's nothing.\"\n\nJake's eyebrows furrowed. \"Fine. You want to play it that way? Then get the fuck out. I still got my ace in the hole, and I ain't showing it.\"\n\nMyron had planned for this. Jake Courter would have to be given something. That was fine, as long as Myron could turn it to his advantage.\n\n\"It seems to me,\" Myron said slowly, \"that Kathy was walking in the general direction of the dean's house.\"\n\n\"So?\"\n\nMyron said nothing.\n\n\"She worked for him,\" Jake said.\n\nMyron nodded.\n\n\"What's the connection?\"\n\n\"Oh, I'm sure it's completely innocent,\" Myron said. \"But you might want to ask him about it. You being so thorough and all.\"\n\n\"Are you saying\u2014\"\n\n\"I'm not saying anything. I am merely making an observation.\"\n\nAgain Jake studied him. Myron looked back coolly. A visit from Jake Courter would probably not crack Dean Gordon, but it should soften him a bit. \"Now about that ace in the hole...?\"\n\nJake hesitated. \"Kathy Culver inherited money from her grandmother,\" he said.\n\n\"Twenty-five grand,\" Myron added. \"All three kids got the same. They're sitting in a trust account.\"\n\n\"Not exactly,\" Jake said. He stood, hitched his pants up. \"You want to know why I said the evidence pointed to Kathy being a runaway?\"\n\nMyron nodded.\n\n\"The day Kathy Culver vanished, she visited the bank,\" Jake continued. \"She cleared out her inheritance. Every penny.\"\n\n# Chapter 20\n\nMyron started back toward New York. He flipped on the radio. Wham's classic hit \"Careless Whisper\" was playing. George Michael was bemoaning the fact that he would never dance again because \"guilty feet have got no rhythm.\" Deep, Myron thought. Very deep.\n\nHe picked up the car phone and dialed Esperanza.\n\n\"What's up?\" he asked.\n\n\"You coming back to the office?\"\n\n\"I'm on my way there now.\"\n\n\"I wouldn't make any stops,\" she said.\n\n\"Why?\"\n\n\"You have a surprise client waiting for you.\"\n\n\"Who?\"\n\n\"Chaz Landreaux.\"\n\n\"He's supposed to be hiding in Washington.\"\n\n\"Well, he's here. And he looks like shit.\"\n\n\"Tell him to sit tight. I'm on my way.\"\n\n\"It's like this,\" Chaz began. \"I want to cancel our contract.\"\n\nHe paced the office like an expectant father, and he did indeed look like shit. The cocky grin was nowhere to be seen. The swagger was more like a hunch. He kept licking his lips, darting his eyes, bunching and unbunching his fingers.\n\n\"Why don't you start at the beginning?\" Myron tried.\n\n\"Ain't no beginning,\" Chaz snapped. \"I want out. You gonna fight me on it?\"\n\n\"What happened?\"\n\n\"Nothing happened. I changed my mind, is all. I want to go with Roy O'Connor at TruPro now. They're big-time. You're a nice guy, Myron, but you don't have their connections.\"\n\n\"Uh-huh.\"\n\nSilence. More pacing.\n\n\"Can I have the contract or what?\"\n\n\"How did they get to you, Chaz?\"\n\n\"I don't know what the fuck you're talking about. How many times do I have to say it? I don't want you, okay?\" Chaz was on the edge and teetering. \"I want TruPro.\"\n\n\"It's not that easy,\" Myron said.\n\n\"You gonna fight me on this?\" he asked again.\n\n\"They won't stop with this, Chaz. You're in over your head. You have to let me help you.\"\n\nHe stopped. \"Help me? You wanna help me? Then give me back my contract. And don't pretend you give a shit about me. You just want your piece.\"\n\n\"Do you really believe that?\" Myron asked.\n\nHe shook his head. \"You don't get it, man. I don't want you. I want to go with TruPro.\"\n\n\"I get it. And like I said before, it's not that easy. These guys got you by the balls. You think you can make them let go by doing what they say. But you can't. Not for good anyway. Whenever they want something, they'll just reach back into your pants and give another squeeze. They won't stop, Chaz. Not until they've squeezed you for everything they can.\"\n\n\"Man, you don't know shit. I don't have to explain nothing to you.\" He approached the desk, but his eyes looked away. \"I want that goddamn contract. I want it now.\"\n\nMyron picked up his phone. \"Esperanza, bring me Chaz's contract. The original.\" He hung up. \"It'll just be a moment.\"\n\nChaz said nothing.\n\n\"You don't know what you're mixed up in,\" Myron continued.\n\n\"Fuck off, man. I know exactly what I'm mixed up in.\"\n\n\"Let me help, Chaz.\"\n\nHe snorted. \"What can you do?\"\n\n\"I can stop them.\"\n\n\"Oh yeah, I can tell. You done a great job so far.\"\n\n\"What happened?\"\n\nBut he just shook his head.\n\nEsperanza came in and handed Myron the contract. Myron in turn handed it to Chaz. He grabbed it and hurried to the door.\n\n\"Sorry, Myron. But this is business.\"\n\n\"You can't beat them, Chaz. Not on your own. They'll suck you dry.\"\n\n\"Don't worry about me. I can take care of myself.\"\n\n\"I don't think you can.\"\n\n\"Just stay the fuck out. It ain't your business no more.\"\n\nHe took off without a backward glance. When he was gone, Win opened the door between the conference room and Myron's office. \"Interesting conversation,\" Win said.\n\nMyron nodded, thinking.\n\n\"We've lost a client,\" Win said. \"Too bad.\"\n\n\"It's not that simple, Win.\"\n\n\"That's where you're mistaken,\" Win replied steadily. \"It's just that simple. He dumped you for another agency. As he so eloquently put it, 'It ain't your business no more.' \"\n\n\"Chaz is being pressured.\"\n\n\"And you offered to help him. He refused.\"\n\n\"He's a scared kid.\"\n\n\"He's an adult who makes his own decisions. One of which was to tell you to fuck off.\"\n\nMyron looked up. \"You know what they'll do to him.\"\n\n\"It's a world of free will, Myron. Landreaux chose to take the money in college. And he chose to go back to them now.\"\n\n\"Will you follow him?\"\n\n\"Pardon?\"\n\n\"Follow Chaz. See where he takes those contracts.\"\n\n\"You complicate the simple, Myron. Just let it be.\"\n\n\"I can't. You know I can't.\"\n\nWin nodded. \"I guess I do.\" He thought a moment. \"I'll do it for the sake of our business,\" he said. \"For the added revenue. If we get Landreaux back in our stable, it will be very profitable. You may enjoy playing superhero, but as far as I'm concerned, this is no moral crusade. I am doing this for the money. That is the only reason. The money.\"\n\nMyron nodded. \"I wouldn't want it any other way.\"\n\n\"Fine. As long as we are clear on that point. And I want you to take this.\"\n\nWin handed him a Smith & Wesson .38 and a shoulder holster. Myron put it on. Carrying a gun was incredibly uncomfortable, yet the weight felt good, like a reminder of some kind of protective bubble. Sometimes the sensation made you feel heady, invincible even.\n\nThat was usually when you got popped.\n\n\"Be extra careful,\" Win said. \"The word has hit the streets.\"\n\n\"What word?\"\n\n\"A price has officially been put on your head,\" Win said, as if it were amusing cocktail conversation. \"Thirty thousand dollars to the man who takes you out.\"\n\nMyron made a face. \"Thirty thousand? Hell, I used to be a fed. I should be worth sixty, seventy grand minimum.\"\n\n\"Bad economy. Times are tough.\"\n\n\"I'm being discounted?\"\n\n\"Appears so, yes.\"\n\nMyron opened the revolver and checked the bullets. Just as he suspected. Win had loaded the gun with dumdums\u2014bullets with cross-hatched tips to expose the lead. Wasn't enough to be using hollow-point Winchester Silvertip bullets. Win had to doctor them for that extra little crunch. \"These are illegal.\"\n\nWin put his hand against his chest. \"My. Oh. My. How. Awful.\"\n\n\"And unnecessary.\"\n\n\"If you say so.\"\n\n\"I say so.\"\n\n\"They are effective.\"\n\n\"I don't want them,\" Myron said.\n\n\"Fine\" He handed Myron uncut bullets. \"Be a wimp.\"\n\n# Chapter 21\n\nJessica listened to the message on the answering machine.\n\n\"Hi, Jessica. It's Nancy Serat. I'm so sorry to hear about your father. He was such a nice man. I can't believe it. He was here the morning he died. So weird. He was so nostalgic that day. He told me all about that favorite yellow sweater he gave Kathy. Such a sweet story. I wish I could have been more helpful. I just can't believe\u2014well, I'm rambling, sorry. I do that when I'm nervous Anyway I'll be out until ten o'clock tonight. You can come by then or give me a call. Bye.\"\n\nJessica rewound the message and played it back. Then a third time. Nancy Serat had seen her father on the morning of his murder.\n\nAnother coincidence?\n\nShe thought not.\n\nMyron called his mother. \"I won't be home for a few days.\"\n\n\"What?\"\n\n\"I'm going to stay with Win.\"\n\n\"In the city?\"\n\n\"Yes.\"\n\n\"New York City?\"\n\n\"No, Mom. Kuwait City.\"\n\n\"Don't be such a wise guy with your mother, save it for your friends,\" she said. \"So why are you staying in the city?\"\n\nHmm. Should he tell her the truth? _Because, Mom, a mobster has a contract out on my head and I don't want to put you and Dad in danger_. Nah. Might make her worry. \"I'm going to be working late the next few nights.\"\n\n\"You sure about this?\"\n\n\"Yes.\"\n\n\"Be careful, Myron. Don't walk around alone at night.\"\n\nEsperanza opened the door. \"Urgent call on line three,\" she said, loud enough for Myron's mother to hear.\n\n\"Mom, I gotta go. Urgent call.\"\n\n\"Call us.\"\n\n\"I will.\" He hung up and looked up at Esperanza. \"Thanks.\"\n\n\"Don't mention it.\"\n\n\"Is there anyone on the phone?\"\n\nShe nodded. \"Timmy Simpson again. I tried to handle it, but he says his problem needs your particular expertise.\"\n\nTimmy Simpson was a rookie shortstop for the Red Sox. A major-league pain in the ass.\n\n\"Hi, Timmy.\"\n\n\"Hey, Myron, I've been waiting here two goddamn hours for your call.\"\n\n\"I was out. What's the problem?\"\n\n\"I'm here in Toronto, okay, at the Hilton. And this hotel's got no hot water.\"\n\nMyron waited. Then he said, \"Did I hear you correctly, Timmy? Did you say\u2014\"\n\n\"Unfuckinbelievable, ain't it?\" Timmy shouted. \"I go in the shower, right, wait five minutes, then ten minutes. The water's fucking freezing, Myron. Ice cold. So finally I call down to the front desk, right? Some pissant manager tells me they're having some kind of plumbing problem. Plumbing problem, Myron, like I'm staying in a fuckin' trailer park or something. So I say, when's it going to be fixed? He gives me this whole long spiel how he don't know. Can you believe this shit?\"\n\nNo, Myron thought. \"Timmy, why exactly are you calling me?\"\n\n\"Jesus Christ, Myron, I'm a pro, right? And I'm stuck in this hellhole with no hot water. I mean, isn't there something in my contract about that?\"\n\n\"A hot water clause, perhaps?\" Myron tried.\n\n\"Or something. I mean, come on. Where do they get off? I need a shower before a game. A _hot_ shower. Is that too much to expect? I mean, what am I going to do?\"\n\nStick your head in the toilet and flush, Myron thought, massaging his temples with his fingertips. \"I'll see what I can do, Timmy.\"\n\n\"Talk to the hotel manager, Myron. Make him understand the importance.\"\n\n\"As far as I'm concerned,\" Myron said, \"those orphans in Eastern Europe are a minor annoyance in comparison to this. But if the hot water doesn't come back on soon, check into another hotel. We'll send the bill to the Red Sox.\"\n\n\"Good idea. Thanks, Myron.\"\n\nClick.\n\nMyron stared at the phone. Unbelievable. He leaned back and wondered how to handle his three big problems: Chaz Landreaux's sudden departure, Kathy Culver's possible re-emergence, and the Toronto Hilton's plumbing. He decided to forgo the last. Only so much one man can do.\n\nProblem 1: Chaz Landreaux was climbing into bed with Frank Ache. There was only one way out of that. Big brother Herman.\n\nMyron picked up the phone and dialed. He still knew the number by heart. It was picked up on the first ring. \"Clancy's Tavern.\"\n\n\"It's Myron Bolitar. I'd like to see Herman.\"\n\n\"Hold on.\" Five minutes passed before the voice came back on. \"Tomorrow. Two o'clock.\"\n\nClick. No need to wait for an answer. Whatever time Herman Ache agreed to see you, you were free.\n\nProblem 2: Kathy Culver. _Nips_ magazine had been mailed from a campus box. It had been mailed not only to Christian Steele but also to Dean Harrison Gordon. Why? Myron knew that Kathy had worked for the dean. Was there more to her job than just filing? An affair, perhaps? And what about the dean's lovely wife? Did she wear underwear?\n\nBut Myron was digressing.\n\nThe catalyst of this whole thing was the ad in _Nips._ Gary Grady claimed he had nothing to do with it. Maybe. Maybe not. But either way the picture had to go through Fred Nickler. Good ol' Freddy was at the center of this.\n\nMyron looked up the number and dialed.\n\n\"HDP. May I help you?\"\n\n\"I'd like to speak to Fred Nickler.\"\n\n\"Whom shall I say is calling?\"\n\n\"Myron Bolitar.\"\n\n\"Please hold.\"\n\nA minute passed. Then Fred Nickler came on. \"Hello?\"\n\n\"Mr. Nickler, this is Myron Bolitar.\"\n\n\"Yes, Myron. What can I do for you?\"\n\n\"I'd like to come by and ask you a few more questions about the ad.\"\n\n\"I'm afraid I'm quite busy right now, Myron. Why don't you give me a call tomorrow? Maybe we can set something up.\"\n\nSilence.\n\n\"Myron? You there?\"\n\n\"Do you know who took that picture, Mr. Nickler?\"\n\n\"Of course not.\"\n\n\"Your friend Jerry denies any knowledge of it.\"\n\n\"Myron, please. You're a man of the world. What did you expect him to say?\"\n\n\"He says he had nothing to do with putting that picture in the ad.\"\n\n\"Well, that's quite impossible. He was the advertiser. He submitted the photograph.\"\n\n\"Then you have a copy of the photo?\"\n\nPause. \"It has to be in the file somewhere.\"\n\n\"Maybe you can pull it out, and I'll come pick it up.\"\n\n\"Listen, Myron, I hate to be rude, but I'm really busy right now. It will just be the same photograph you already saw.\"\n\n\"Kathy's picture was only in _Nips_ ,\" Myron said.\n\n\"Pardon me?\"\n\n\"Her picture. It wasn't in any of your other magazines. Only _Nips_.\"\n\nPause. \"So?\" But his voice was suddenly tottery.\n\n\"So the same ad was in all six magazines. The same exact page with the same exact pictures. Except for one small change in _Nips_. Someone had changed just one photograph in the bottom row. Someone had switched pictures for just that one magazine and not the others. Why?\"\n\nFred Nickler coughed. \"I really don't know, Myron. Tell you what: I'll check on it and let you know. Gotta zillion calls waiting. Gotta run. Bye.\"\n\nAnother click.\n\nMyron sat back. Fred Nickler was starting to panic.\n\nWith a shaking hand Fred Nickler dialed the number. After three rings the phone was picked up.\n\n\"County police.\"\n\nFred cleared his throat. \"Paul Duncan, please.\"\n\n# Chapter 22\n\nNine P. M.\n\nMyron called Jessica. He filled her in on his dean discovery.\n\n\"Do you really think Kathy was having an affair with the dean?\" Jessica asked.\n\n\"I don't know. But after seeing his wife, I'd tend to doubt it.\"\n\n\"Good-looking?\"\n\n\"Very,\" Myron said. \"And she knows her basketball. She even cried when I got hurt.\"\n\nJessica made a noise. \"The perfect woman.\"\n\n\"Do I detect a note of jealousy?\"\n\n\"Dream on,\" Jessica said. \"The fact that a man is married to a beautiful woman does not preclude him from having affairs with pretty co-eds.\"\n\n\"True enough. So the question is: How did Dean Gordon get his name on this infamous mailing list?\"\n\n\"I haven't got a clue,\" she said. \"But I too found out something interesting today. My father visited Nancy Serat, Kathy's roommate, the morning he died.\"\n\n\"Why?\"\n\n\"I don't know yet. Nancy just left a message on my machine. I'm meeting her in an hour.\"\n\n\"Good. Call me if you hear anything else.\"\n\n\"Where are you going to be?\" she asked.\n\n\"I work nights at Chippendale's,\" Myron said. \"Stage name Zorro.\"\n\n\"Should be Tiny.\"\n\n\"Ouch.\"\n\nAn uncomfortable silence engulfed them. Jessica finally broke it. \"Why don't you come by the house tonight?\" she asked, struggling to keep her tone level.\n\nMyron's heart pounded. \"It'll be late.\"\n\n\"That's okay. I'm not sleeping much. Just knock on my bedroom window. Zorro.\"\n\nShe hung up. For the next five minutes Myron sat perfectly still and thought about Jessica. They had first started dating a month before his career ended. She stayed with him. She nursed him. She loved him. He pushed her away under some macho disguise of protecting her. But she wouldn't leave. Not then, anyway.\n\nEsperanza opened the door without knocking. She looked at him and snapped, \"Stop it.\"\n\n\"What?\"\n\n\"You're making that face again.\"\n\n\"What face?\"\n\nShe imitated him. \"That repulsive lovesick-puppy face.\"\n\n\"I wasn't making any face.\"\n\n\"Right. You disgust me, Myron.\"\n\n\"Thank you.\"\n\n\"You know what I think? I think you're more interested in getting back in Jessica's pants than you are in finding her sister.\"\n\n\"Jesus, what the hell is with you?\"\n\n\"I was there, remember? When she left.\"\n\n\"Hey, I'm a big boy. I can take care of myself.\"\n\nEsperanza shook her head. \"D\u00e9j\u00e0 vu all over again.\"\n\n\"What?\"\n\n\"Take care of yourself. Bullshit. You sound just like Chaz Landreaux. Both of you have your head up your ass.\"\n\nEsperanza's dark face reminded him of Spanish nights, golden sand, full moons against starless skies. There had been moments of temptation between them, but one or the other had always realized what it would mean and stopped it. Such temptations no longer came their way anymore. Aside from Win, Esperanza was his closest friend. Her concern, Myron knew, was genuine.\n\nHe changed subjects. \"Was there a reason for your unannounced entrance?\"\n\n\"I found something.\"\n\n\"What?\"\n\nShe read from a steno pad. Why she had a steno pad he could not say. She could not take dictation or type a lick. \"I finally tracked down the other number Gary Grady called after your visit. It belongs to a photography studio called\u2014get this\u2014Global Globes Photos. Located off Tenth Avenue, near the tunnel.\"\n\n\"Sleazy area.\"\n\n\"The sleaziest,\" she said. \"I think the studio specializes in pornography.\"\n\n\"Nice to have a specialty.\" Myron checked his watch. \"Any word from Win?\"\n\n\"Not yet.\"\n\n\"Leave the photographer's address on his voice mail. Maybe he'll finish in time to meet me.\"\n\n\"You going tonight?\" she asked.\n\n\"Yes.\"\n\nEsperanza closed the pad with a snap. \"Mind if I tag along?\"\n\n\"To the photography studio?\"\n\n\"Yes.\"\n\n\"Don't you have class tonight?\" Esperanza was getting her law degree from NYU at night.\n\n\"No. And I've done all my homework, Daddy. Really I have.\"\n\n\"Shut up and come on.\"\n\n# Chapter 23\n\nHookerville.\n\nThere were all kinds. White, Black, Asian, Latino\u2014a verifiable United Nations of prostitutes. Most were young, very young, stumbling on too-high heels, like children playing dress-up, which in a real sense they were. Most were thin, dried-up, needle tracks covering their arms like dozens of tiny insects, their skin pulled tightly around cheekbones, giving their faces a haunted skull look. Their eyes were hollow and set deep, their hair lifeless and strawlike.\n\nMyron muttered, \"Don't they know they're making love to what's already dead?\"\n\nEsperanza paused, thinking. \"Don't know that one.\"\n\n\"Fontine in _Les Mis\u00e9rables_. The musical.\"\n\n\"I can't afford Broadway musicals. My boss is cheap.\"\n\n\"But cute.\"\n\nHe watched a blond girl in sixties hot-pants negotiate with a sleazeball in a Ford station wagon. He knew her story. He had seen girls (boys sometimes) just like her get off the bus at the Port Authority, a Greyhound bus that had originated in West Virginia or western Pennsylvania or that great, barren mono-expanse New Yorkers simply referred to as the Midwest. She had run away from home\u2014maybe to avoid abuse, but more likely because she was bored and \"belonged\" in a big city. She had high-stepped off the bus with a wide smile, mesmerized, without a penny. Pimps would eye her and wait with the patience of a vulture. When the time was right, they would sweep down and claim their carcass. They'd introduce her to the Big Apple, get her a place to stay, some food, a hot shower, maybe a room with a Jacuzzi and dazzling lights and a cool CD player and cable TV with a remote. They'd promise to set her up with a photographer, get her a few modeling gigs. Then they'd teach her how to party, _really_ party, not that candy-ass shit she'd done in Hicks Falls with some beer and a zit-infested senior pawing at her in the backseat of a pickup. They'd show her how to have a good time with the prime stuff, the numero-uno white powder.\n\nBut things would change. Someone would have to pay for all these good times. The modeling job would fall through, and she couldn't just be a freeloader. Besides, the partying was more a need now than a luxury. Like food or breathing. She could no longer exist without a snort or a pinch from her favorite needle.\n\nIt didn't take long to plummet and hit bottom. And once there she didn't have the strength\u2014not even the desire, really\u2014to get up.\n\nShe ended up here.\n\nMyron parked. He and Esperanza got out of the car silently. Myron felt his stomach churn. It was night, of course. Places like this existed only at night. They fled with the onslaught of sunlight.\n\nMyron had never been with a whore, but he knew Win had engaged their services on plenty of occasions. Win liked the convenience. His favorite spot was an Asian whorehouse on Eighth Street called Noble House. Back in the mid-eighties, Win and a few friends would have what they called \"Chinese night\" in Win's apartment\u2014Hunan Garden would deliver food, Noble House women. The truth was, Win had no feelings for women. He didn't trust them. Whores were what he wanted. It wasn't just the lack of attachment. Win never let women attach. But prostitutes were throwaways. Disposable.\n\nMyron didn't think Win still partook in such events\u2014not in this disease-ridden era\u2014but he didn't know for sure. They never talked about it.\n\n\"Pretty spot,\" Myron said. \"Scenic.\"\n\nEsperanza nodded.\n\nThey passed a nightclub of some sort. The music was loud enough to crack the sidewalk. A teen\u2014Myron couldn't say if it was male or female\u2014with green spiked hair bumped into him. Looked like the Statue of Liberty. There were lots of motorcycles, ear and nipple rings, tattoos, chain jewelry. A constant whore chorus of \"Hey, baby\" pelted him from every conceivable angle, their faces blurring into one mass of human debris. The place was like a carnival freak show.\n\nThe sign above the door read CLUB F. U. The logo was a raised middle finger. Subtle. A chalkboard read the following:\n\nHEAVY \"MEDICAL\" NIGHT!  \nLIVE BANDS!  \nFeaturing the only local appearances by:  \nPAP SMEAR  \nand RECTAL THERMOMETER\n\nMyron could see through the open door. People weren't dancing. They were jumping up and down, heads lolling lifelessly as if their necks were rubber bands, their arms tucked against their sides. Myron focused in on one kid, maybe fifteen years old, lost in the violet bliss, sweat matting his long hair to his face. He wondered if the group onstage was Pap Smear or Rectal Thermometer. Didn't matter. Sounded like someone had jammed a rutting pig into a Cuisinart.\n\nThe whole scene was like Dickens meets _Blade Runner_.\n\n\"The studio is next door,\" Esperanza said.\n\nThe building was either a disastrous brownstone or a small warehouse. Whores hung out the windows like shreds of leftover Christmas decorations.\n\n\"This is it?\" Myron asked.\n\n\"Third floor,\" Esperanza answered. She did not seem intimidated by the surroundings in the least, but she had come from streets not much better than this. Her face remained a placid pool. Esperanza never showed weakness. Her temper flared often, but for all their times together, Myron had never seen her cry. She could not say the same of him.\n\nMyron approached the stoop. An overweight whore stuffed into a bodysuit that doubled as sausage casing licked her lips and stepped in front of him.\n\n\"Hey, yo, want a blow job? Fifty bucks.\"\n\nMyron tried not to close his eyes. \"No,\" he said softly, lowering his head. He wanted to offer words of wisdom, words that could transform her, change her circumstances. But he just said, \"I'm sorry,\" and hurried past. The fat girl shrugged and moved on.\n\nIt was a walk-up. No surprise there. The stairwells were littered with people, most unconscious or maybe dead. Myron and Esperanza carefully climbed over them. A cacophony of music\u2014everything from Neil Diamond to what might have been Pap Smear bellowed through the corridor. There were other sounds too Broken bottles, shouts, curses, crashing, a baby crying. An orchestra from hell.\n\nWhen they reached the third floor, they saw a glassed-in office. No one was inside, but the pictures on the wall\u2014not to mention the bullwhip and handcuffs\u2014left little doubt that they had arrived at the right place Myron tried the knob. It turned.\n\n\"You stay out here,\" he said.\n\n\"Okay.\"\n\nHe moved in \"Hello?\"\n\nNo one answered him, but music was coming from the other room. Sounded like calypso music. He called out again and stepped into the studio.\n\nMyron was struck by how professional the setup was. It was clean, brightly lit, with one of those big white umbrella things you always see in photo studios. There were half a dozen cameras set up on tripods, and overhead was a variety of different-colored lights.\n\nOf course, the setting was not the first thing that struck him. Other things caught his eye first. The naked woman sitting on a motorbike, for example. To be accurate, she wasn't fully naked\u2014she had on a pair of black boots. Nothing else. Not a look every woman could pull off, but it seemed to work for her. She had not seen him yet, intensely studying the magazine in her hand. _The National Sun_. Headline: Boy 16 Becomes Grandmother. Hmm. He stepped closer. She was big-breasted, very Russ Meyer, but Myron could see scars under the large swellings. Implants, the fashion accessory of the eighties.\n\nShe looked up, startled.\n\nMyron smiled warmly. \"Hi.\"\n\nShe screamed Piercingly. \"Get the fuck out of here!\" she shrieked, covering her chest. Modesty. So rare nowadays. It was nice to see.\n\nMyron said, \"My name\u2014\"\n\nAnother piercing scream. Myron heard a noise behind him and spun. A skinny kid wearing no shirt stood smiling. He popped open a switchblade, a maniacal grin plastered across his face. His Bruce Lee\u2013like build shimmered in the light. He crouched low and beckoned Myron forward. Very _West Side Story_. If only the kid would snap his fingers.\n\nAnother door opened, and red light leaked out. A woman stepped into view. She had what looked like curly red hair, but Myron couldn't be sure if that was her color or if it just appeared red because of the light from the darkroom.\n\n\"You're trespassing,\" she said to Myron. \"Hector has the right to kill you where you stand.\"\n\n\"I don't know where you got your law degree,\" Myron said, \"but if Hector isn't careful, I'm going to take away his toy and shove it where the sun don't shine.\"\n\nHector giggled. He began to toss the knife back and forth between his hands.\n\n\"Wow,\" Myron said.\n\nThe topless model fled to the dressing room, which was cleverly marked UNDRESSING ROOM. The woman from the darkroom stepped fully into the studio and closed the darkroom door. Her hair was indeed red, more like burnt auburn actually. Her skin was what some might call peaches and cream. She was maybe thirty and looked, strange as it might sound, perky. The Katie Couric of the porno world.\n\n\"Are you the owner?\" Myron asked.\n\n\"Hector is very good with a blade,\" she replied coolly. \"He could slice out a man's heart and show it to him before he died.\"\n\n\"That must liven a party.\"\n\nHector stepped closer Myron did not move.\n\n\"I could demonstrate my skills in the martial arts,\" Myron began. He quickly withdrew his gun and aimed it at Hector's chest. \"But I just showered.\"\n\nHector's eyes widened in surprise.\n\n\"Let this be a lesson to you, Blade Boy,\" Myron continued. \"Half the people in this building probably carry guns. You go around waving that toy, and someone without my tender heart will ace you.\"\n\nThe redhead did not seem taken aback by the gun. \"Get out of here,\" she said to Myron \"Now.\"\n\n\"Are you the owner?\" Myron tried again.\n\n\"You got a warrant?\"\n\n\"I'm not a cop.\"\n\n\"Then get your ass out of here.\" She undulated a lot when she talked. Her hips and legs in constant motion. She signaled to Hector, who closed up the switchblade. \"You can go, Hector.\"\n\n\"Not so fast, Hector,\" Myron said. \"Get in the darkroom. I don't want you getting any ideas about coming back with a gun.\"\n\nHector looked toward the redhead. She nodded, and he went.\n\n\"Close the door,\" Myron said.\n\nHe closed it. Myron walked over and pulled the dead bolt.\n\nThe redhead put her hands on her hips. \"Happy now?\"\n\n\"Nearly ecstatic.\"\n\n\"Now get out.\"\n\n\"Listen,\" Myron said with his melt-'em, warm smile, \"I don't want any trouble. I'm just here to buy some photographs. My name is Bernie Worley. I work for a new porno magazine.\"\n\nShe made a face. \"Do I really look that stupid? Bernie Worley, here to buy some photographs. Give me a fuckin' break.\"\n\nThere was a sudden noise. People. Lots of them. A commotion, even by this place's standard. In the corridor. Right where he had left Esperanza. Alone.\n\nMyron turned and ran, feeling his heart leap to his throat. If something had happened to her\u2014\n\nHe threw open the door. Dozens of people surrounded Esperanza, most kneeling. She stood in the middle, smiling and\u2014he couldn't believe it\u2014signing autographs.\n\n\"It's Pocahontas!\" someone shouted.\n\n\"Make mine out 'With love to Manuel.' \"\n\n\"You're still my favorite!\"\n\n\"I remember when you beat Queen Carimba. What a fight!\"\n\n\"That Highway Hannah. Such a dirty fighter. When she threw salt in your eyes, I could have killed her.\"\n\nEsperanza caught Myron's eye, shrugged, went back to signing old matchbooks and scraps of paper. The redhead followed him out the door. When she saw Esperanza, her entire being lit up. \"Poca?\"\n\nEsperanza looked back up. \"Lucy?\"\n\nThey hugged. They stepped back into the studio, Myron following.\n\n\"Where you been, girl?\" Lucy asked.\n\n\"Here, there.\"\n\nThe two women kissed. On the lips. A little too long. Esperanza turned around. \"Myron?\"\n\n\"Huh?\"\n\n\"Your eyes are bulging.\"\n\n\"They are?\"\n\n\"I don't tell you everything.\"\n\n\"Apparently not,\" he said. \"But at least I know why my startling good looks didn't faze your friend.\"\n\nBoth women found that laughable. \"Lucy, this is Myron Bolitar.\"\n\nLucy looked him up and down. \"He your boyfriend?\"\n\n\"No. Just a good friend. And my boss.\"\n\n\"He looks like a guy I know, worked a kinky show at a club down the street. He had this act where he peed on different women.\"\n\n\"It wasn't me,\" Myron assured her. \"I have enough trouble peeing in a public urinal.\"\n\nLucy turned her attention to Esperanza. \"You look good, Poca.\"\n\n\"Thanks.\"\n\n\"Out of the wrestling game, huh?\"\n\n\"Completely.\"\n\n\"But you're still working out?\"\n\n\"As often as I can.\"\n\n\"Nautilus?\"\n\n\"Um-hmm.\"\n\n\"It shows,\" Lucy said with a wicked smile. \"You really look hot.\"\n\nMyron cleared his throat. \"Hey, how about those Knicks?\"\n\nThe women ignored him. \"You still taking pictures of the wrestlers?\" Esperanza asked.\n\n\"Not much anymore I'm mostly into this shit.\"\n\nEsperanza looked back at Myron. \"Lucy\u2014that isn't her real name, we just call her that because of her hair\u2014she used to do the promo photos of all the wrestlers.\"\n\n\"So I gathered,\" Myron said. \"Do you think she can help us out?\"\n\n\"What do you want to know?\" Lucy asked.\n\nMyron handed her the copy of _Nips_. He pointed to Kathy's picture. \"I want to know about this,\" he said.\n\nLucy studied the photograph for a second. \"He a cop?\" she asked Esperanza.\n\n\"A sports agent.\"\n\n\"Oh.\" She did not ask for further elaboration. \"Because this could get us in trouble.\"\n\n\"How so?\" Myron asked.\n\n\"The photograph. The girl is topless.\"\n\n\"So?\"\n\n\"So it's illegal. Topless girls aren't allowed in 900 ads. We're going to get screwed if the government sees this.\"\n\n\"We?\" Myron repeated. Again the clever interrogation techniques.\n\n\"I'm one of the owners of these dial-a-porn companies. A lot of the lines work out of this building.\"\n\n\"I'm not sure I understand,\" Myron said. \"What do you mean, topless girls are illegal? Almost every girl in that magazine is naked.\"\n\n\"Not in the ads for 900 lines,\" Lucy corrected. \"Couple years back a law was passed. Nine hundred lines had to go clean. Look here.\" She turned a page and pointed at another ad. \"The girl might look suggestive, but she can't be naked. And look at the name of the lines. Stuff like 'Secret Confessions' or 'Talk to Girls.' Now look at the ones for the 800 lines. Hard core. 'Cum Between My Tits,' stuff like that.\"\n\nMyron remembered his conversation with Tawny on the 900 line. He had been struck by the fact that she said nothing dirty. \"So you can only have phone sex on the other lines?\"\n\n\"Right. You see, you need real permission for those. That's how the government sees it. Any asshole can call a 900 line. The charges are automatic. They start almost immediately after your call is answered. But not with an 800 line or one of the other numbers. You have to use either your credit card or a callback. That's the way you get billed.\"\n\n\"So all that talk about 900 lines being dirty\u2014\"\n\n\"Is bullshit,\" Lucy finished. \"They're cons. We can't say one dirty thing on those lines. We use them as lures mostly, because they're so easy to use. A guy just has to dial. No credit card. No callbacks. Most of the time we talk about skinny-dipping or massages\u2014suggestive but not sexual. Get him excited, you know what I mean?\"\n\n\"I think so, yes.\"\n\n\"These guys call horny anyway. I mean, most are so hard up, they'll stick it in a knothole to get relief. What we try to do is get him to say the first dirty word, which usually isn't too difficult. Once he does, we say, 'Oh, baby, I can't talk dirty on this line, but you should call me back at X number with a credit card.' The guys call it and get charged all over again.\"\n\n\"Aren't they afraid of how it'll look on their credit card bill?\" Myron asked.\n\nLucy shook her head. She was still undulating. It was a combination of irritating and erotic. \"The company names are usually pretty discreet,\" she explained. \"We bill under names like Norwood Incorporate or Telemark\u2014not Hot Lesbos or Sucking Starlet. You want to see it?\"\n\n\"See what?\"\n\n\"The operation upstairs. Where we answer some of the calls. Lots of people work out of their homes, but I got a crew of six or seven working the lines now.\"\n\nMyron shrugged \"Yeah, sure.\"\n\nLucy took them up one level. Some sort of sickening stench engulfed the stairwell. When they reached the landing, Lucy opened a door. They stepped through and quickly closed it behind them.\n\n\"This is Fantasies Forever Lines,\" Lucy said. \"Not to mention Dick-a-Lick, Hootersline, Telefun, and a dozen others.\"\n\nMyron could not believe what he was seeing. His mouth dropped open. He had expected ugly women or fat women or old women. But he had not expected this.\n\nThey were men. All but one of the workers were male.\n\n\"Gay lines?\" Myron asked.\n\nLucy shook her head, smiling. \"Very few gay calls come in. Maybe one in a hundred.\"\n\n\"But... these are men.\"\n\nMyron Bolitar, the essence of keen observation.\n\nHe heard a man in a gruff, truck-driver voice say, \"Yeah, big man, slide it all the way in. That's it. Oh, yeah, that feels good.\"\n\nLucy smiled at the man. The man rolled his eyes and continued, \"Don't stop, Stallion. Ride me.\"\n\nEsperanza, Myron was glad to see, looked equally confused. \"What's going on, Lucy?\" she asked.\n\n\"It's the times,\" Lucy said. \"In this economy men are a cheaper source of labor. Most of the girls are on the streets. These are brothers, cousins, street kids.\"\n\n\"But their voices\u2014\"\n\n\"They use a voice changer. Sharper Image sells them, but I get them cheaper in the Village. You can make little girls sound like Barry White, or vice versa. These guys can become a husky woman, a teenage virgin, a little girl\u2014whatever the line calls for.\"\n\nMyron was stunned. \"Do the customers know this?\"\n\n\"Of course not.\" She turned to Esperanza. \"Dumb. But he is kinda cute.\"\n\nMyron Bolitar, Lesbian Fantasy Man.\n\nThe room looked like any telemarketing office. The phones were high-tech. Dozens of lines lit up, each marked for what role was to be played. Horny Housewife. Dominatrix. Cross-dressers. Busty Babes. Even Foot Fetish. Each employee also had another phone for Visa and MasterCard verification.\n\n\"The lines with a C next to them got to be kept clean,\" Lucy explained. \"We also have another hundred or so people working phones from their homes. Most of those are women.\"\n\n\"Horny housewives?\"\n\n\"Some of them. Most are just plain housewives. Anyway, that's why I found the ad strange. A 900 line shouldn't have a topless girl.\"\n\nThey left the room and walked back down to the studio. Myron almost tripped over a wino who chose the moment Myron was stepping over him to stand up.\n\n\"Is ABC one of the companies upstairs?\" Myron asked.\n\n\"Yeah.\"\n\n\"And we know Gary Grady called you yesterday. Can you tell us why?\"\n\n\"Who?\"\n\n\"Gary Grady.\"\n\nLucy shook her head. \"Don't know him.\"\n\n\"How about Jerry?\"\n\n\"Oh yeah, him.\" She gave a small laugh. \"I figured that wasn't his real name. He was always real secretive.\"\n\n\"So what did he want?\"\n\nShe nodded as though something had just occurred to her. \"I get it now.\"\n\n\"Get what?\"\n\n\"He was asking me about a photograph I'd taken a couple years back.\"\n\n\"This one?\" Myron asked, pointing to Kathy's picture again.\n\n\"Yeah. One of his girls.\"\n\nMyron and Esperanza exchanged a glance. \"You mean there were others?\"\n\n\"Few. Half dozen, maybe more.\"\n\nMyron felt the rage consume again. \"Underage girls?\"\n\n\"How the fuck am I supposed to know?\"\n\n\"You didn't ask?\" Myron asked.\n\n\"Do I look like a cop? Look, man, if you're here to hassle me\u2014\"\n\n\"He's not,\" Esperanza said. \"You can trust him.\"\n\n\"The fuck I can, Poca. He comes busting in here with a fucking gun, scares the piss out of my model.\"\n\n\"We need your help,\" Esperanza said. \" _I_ need your help.\"\n\n\"I don't want to hurt you, Lucy,\" Myron said. \"I'm just interested in the girl in the picture.\"\n\nLucy hesitated. \"All right,\" she said at last. \"But back off.\"\n\nMyron gave a quick nod of agreement. \"Jerry brought this girl to you?\"\n\n\"Yeah, when I had my other studio a couple blocks away. Like I said, he brought in a few girls over the years. He wanted their photos for all kinds of stuff. Porno mags, smut film stills, that kind of thing. Most were a cut or two above the average hosebag who comes through the door. But he usually keeps the photos under wraps until they're a little older. Legal age, I guess.\"\n\nThe rage again. Myron's hands tightened into fists. \"So Jerry asked you about this picture yesterday?\"\n\n\"Yeah.\"\n\n\"What did he want to know?\"\n\n\"If I sold any copies recently.\"\n\n\"Have you?\"\n\nPause. \"Yeah. Couple months ago.\"\n\n\"Who bought them?\"\n\n\"You think I keep records?\"\n\n\"A he or a she?\"\n\n\"A he.\"\n\n\"Do you remember what he looks like?\"\n\nShe took out a cigarette, lit it, took a deep puff. \"I'm not real good with faces.\"\n\n\"Anything, Lucy,\" Esperanza added. \"Young, old, anything you can remember.\"\n\nAnother puff. Then: \"Old. Not ancient, but not a young guy. Might have been my father's age. And he knew what he was doing.\" She looked at Myron. \"Not like you. Bernie Worley. Jesus.\"\n\nMyron pressed on. \"What do you mean, he knew what he was doing?\"\n\n\"The man paid me top dollar under one condition: I hand over every photo and negative in front of him right now. Smart. He wanted to make sure I didn't have time to make any extra copies or an extra set of negatives.\"\n\n\"How much did he pay you?\"\n\n\"Sixty-five hundred altogether. In cash. Five grand for the photos and negatives. Plus another grand for Jerry's phone number. Said he wanted to get in touch with the girl personally. Then he gave me another five hundred if I didn't say anything to Jerry.\"\n\nIn the background there was yet another bloodcurdling scream. It went ignored. \"Would you know the man if you saw him again?\" Myron asked.\n\n\"I don't know,\" she said. \"I can't picture him now, but if we met up face to face... who knows?\" There was a pounding noise from the darkroom. \"Mind if I let Hector out now?\"\n\n\"We were just leaving,\" Myron said. He handed her a card. \"If you remember anything else\u2014\"\n\n\"Yeah, I'll call.\" She looked over to Esperanza. \"Don't be a stranger, Poca.\"\n\nEsperanza nodded but said nothing. They were quiet the entire way down. When they stepped into the hot air, surrounded by the night street, she said, \"Didn't mean to shock you in there.\"\n\n\"Not my business,\" he said. \"I was a little surprised, that's all.\"\n\n\"Lucy is a lesbian. I experimented with it a little. Long time ago.\"\n\n\"You don't have to explain,\" he said. But he was glad she told him. Myron had no secrets from Esperanza. He didn't like thinking she had some from him.\n\nThey were about to head back to the car when Myron felt the muzzle of a gun against his ribs.\n\nA voice said, \"Stay cool, Myron.\"\n\nIt was the man with the fedora hat from the garage. He reached into Myron's jacket and took out the 38. A second man, this one with a Gene Shalit\u2013like mustache, grabbed Esperanza and pressed his gun against her temple.\n\n\"If Myron moves,\" Fedora said to the other man, \"blow the bitch's brains all over the sidewalk.\"\n\nThe man nodded, half-smiling.\n\n\"Come on,\" Fedora said, nudging Myron forward with the gun. \"Let's take a little walk.\"\n\n# Chapter 24\n\nJessica parked in front of the house Nancy Serat was renting for the semester. It was more a cottage really, located at the end of a dark street about a mile from the campus of Reston University. Even at night Jessica could see the house's salmon-pink hue, which seemed to clash with the planet earth. The landscape looked like the trees had vomited\u2014the front yard of _The Munsters_. A faded 118 ACRE STREET was stenciled on the weather-beaten sign. A blue Honda Accord with a Reston University bumper sticker sat in the driveway.\n\nJessica headed down the broken remnants of what must have once been a cement path. She rang the bell and immediately heard a scurrying sound. Several seconds passed. No one approached the door. She tried again. No scurrying sound this time. No sound at all.\n\n\"Nancy?\" she called out. \"It's Jessica Culver.\"\n\nShe hit the bell a few more times, though in a house this small there was not much chance she hadn't been heard. Unless Nancy was in the shower. A possibility. The lights, she could see through the window shades, were on. The car was in the driveway. Jessica had heard movement.\n\nNancy had to be home.\n\nJessica reached out for the knob. Under normal conditions some filter in her mind would probably have stopped her from simply trying to open the door of a virtual stranger (she had only met Nancy once). But these conditions were hardly normal. She took hold of the knob and turned.\n\nLocked.\n\nNow what?\n\nShe stood at the door five more minutes ringing the bell. Still nothing. Jessica circled the house, using a distant streetlight and the house's glow-in-the-dark properties to guide her. She stumbled over a tricycle that looked like something recovered from an archaeological dig. Her feet got tangled in the high grass, the prickly ends tickling her calves. As she circled, Jessica peeked through the small openings in the window shades. She could make out rooms and spotted an occasional piece of furniture or wall hanging, but no people.\n\nIn the backyard she saw the shades were not pulled down in the kitchen. The lights were off too. It was pitch black here, the pink not getting the illumination of the streetlight to cast its glow. She peered through the kitchen window, cupping her hands around her face to cut off the reflection. A sliver of light from the front room slashed across the room. On the table sat a purse. And a set of keys.\n\nSomeone was home.\n\nA sound behind her made her jump. Jessica spun, but it was too dark to make out what it was. Her heart beat wildly in her chest. Crickets singsonged unceasingly. She pounded on the door with both fists.\n\n\"Nancy! Nancy!\"\n\nShe heard the panic in her voice and scolded herself for it. _Get a grip. You're spooking yourself._\n\nShe stopped, took a few deep breaths, felt herself relax. She took another look through the window, pressing her face right up against the glass. She was watching the sliver of light when it happened.\n\nSomeone walked by.\n\nJessica jumped back. She hadn't seen the person, hadn't seen anything, except the sliver of light disappear for the briefest of seconds. She looked again. Nothing. But someone had gone past and blocked off the light. She put her hand on the kitchen doorknob.\n\nThis time the door was not locked. The knob turned easily.\n\n_Don't just go in, dodo! Call the cops!_\n\nAnd say what? I knocked on a door and no one answered? That I then started peeking through windows and saw someone moving around?\n\n_That doesn't sound so bad_.\n\nSounds bad enough to me. Besides, I'd have to find a phone. By the time I do that, whatever is going on may be over. I may have lost my one opportunity....\n\n_Opportunity for what?_\n\nShe pushed the voice away. Then she opened the door. She waited for the door to squeak madly, but it slid open with remarkable silence. She stepped into the kitchen and left the door open. Better for the quick getaway.\n\n\"Nancy?\n\n_\"Kathy?\"_\n\nShe clasped her hand over her mouth. She hadn't meant that. Kathy wasn't here. Jessica wished like hell she were, but that would be too easy. Kathy wasn't here. And if she were, she certainly would not be afraid to open the door for her sister. Her baby sister. The sister with the bright smile. The sister whom she loved....\n\n_The sister you let slip away. The sister you impatiently rushed off the line the night she vanished._\n\nFor several minutes Jessica just stayed in the kitchen. There were no sounds, except those maddening crickets. No running water. No shower. No scurrying. No footsteps. She opened the purse and extracted the wallet. Driver's license and assorted credit cards\u2014all in the name of Nancy Serat. She flipped to the back and stopped suddenly at a wallet-size photograph.\n\n_The_ picture. The sorority sisters picture. The last picture of Kathy.\n\nShe dropped the wallet as though it were something scaly and alive. Enough, Jessica said to herself. She moved toward the light. One foot slid out, the other followed. In a matter of seconds Jessica was at the door. It was open a crack, allowing the light to cast its sliver now unimpeded. She pushed through, crouching like a cop with a gun, preparing for the worst.\n\nAnd the worst was what she got.\n\nJessica stumbled back. \"Jesus Christ\u2014\"\n\nNancy lay flat on her back, her hands at her sides. Her eyes stuck out like two golf balls, staring at Jessica. Her face was a deep purplish-blue, like a giant bruise. Her mouth was wide and twisted in pure agony. The tongue lolled out like a dead fish. Nancy Serat's entire expression was still frozen in a look that begged and screamed with its every cell for oxygen. A thin line of still-wet saliva clung to her chin.\n\nA cord of some kind\u2014no, a wire\u2014was wrapped around Nancy's neck, barely visible. Most of it had sliced clean through the skin and was embedded deep in the flesh. A circling lash of blood marked the spot where the wire had entered.\n\nJessica stared, lost. The world vanished for several moments, leaving behind only the horror. She forgot about the scurrying when she first rang the bell. She forgot the shadow that had cut off the sliver of light.\n\nJessica did not hear the approaching footsteps. Still staring at Nancy's face, unable to tear her eyes away, she felt a sudden, sharp pain in her head. She saw white flashes. Her body folded at the waist and pitched forward. A tingling numbness followed.\n\nThen nothing.\n\n# Chapter 25\n\nFedora Hat knew what he was doing.\n\n\"Stay a few steps behind me,\" he barked at his new partner.\n\nIn the garage Fedora and Musclehead (who, Myron was happy to see, seemed to be out of commission) had underestimated Myron. Fedora would not make the same mistake twice. Not only had he always kept eyes and gun on Myron but he was making sure that his new partner (the Mustache) kept both himself and Esperanza a safe distance away.\n\nSmart.\n\nMyron had been tempted to make a move, but even his best move was useless in this circumstance. If he managed to get the gun away from Fedora, there was no way he'd be able to turn it on the Mustache before he'd shoot either him or Esperanza.\n\nHe would have to wait and watch. He knew what Fedora and Mustache intended to do. They hadn't been hired to buy him ice cream or teach line dancing or even beat him up. Not this time.\n\n\"Let her go,\" Myron said. \"She has nothing to do with this.\"\n\n\"Keep moving,\" Fedora replied.\n\n\"You don't need her.\"\n\n\"Move.\"\n\nMustache spoke for the first time. \"I might want a little company later,\" he sneered. Then he stopped and pressed the gun against Esperanza's right cheek while he licked\u2014actually licked\u2014her left cheek with a wet cow-like tongue. Esperanza stiffened. Mustache looked at Myron. \"You got a problem with that, pal?\"\n\nMyron knew words would be either superfluous or harmful at this stage. He kept his mouth shut.\n\nThey turned a corner. The stench of garbage was overwhelming. It was piled at least six feet high on both sides of the narrow alley. Fedora quickly scanned the area. It appeared to be abandoned.\n\n\"Go,\" he said, giving Myron another poke with the gun. \"End of the alley.\"\n\nMyron felt as if he were walking a plank. He tried to take it as slowly as possible.\n\n\"What are we going to do with the piece of ass?\" Mustache asked.\n\nFedora's eyes never left Myron. \"She's seen us,\" he said. \"She's a witness.\"\n\n\"But we weren't hired to ace her,\" Mustache whined.\n\n\"So?\"\n\n\"So let's not just waste a piece like this\"\u2014he smiled\u2014\"especially when we can fuck it first.\"\n\nMustache laughed at his suggestion. Fedora did not. He stepped back, aiming the gun at Myron's back. Myron turned to face him. They were separated by about six feet. Myron was against the back wall. There was no avenue of escape. The nearest window was at least twelve feet off the ground. No room to move at all.\n\nFedora raised the gun so that it stared Myron right in the face. Myron did not blink. He looked into Fedora's eyes.\n\nAnd then they were gone. Fedora's eyes were gone. Along with half his head.\n\nThe bullet had ripped off the skull at the midway point, splitting Fedora's head open like a coconut. He slid to the ground, the fedora floating down after him.\n\nA dum-dum bullet.\n\nMustache cried out and dropped the gun. He held his hands up. \"I surrender!\"\n\nMyron ran forward. \"Don't! He's surren\u2014\"\n\nBut the gun exploded again. Mustache's face disappeared in a spray of red mist. Myron stopped, closed his eyes. Mustache joined Fedora on the filthy cement. Esperanza came over and wrapped her arms around Myron. They both turned toward the alley's entrance.\n\nWin stepped into view, studying his handiwork as though it were a statue he wasn't sure he liked. He was dressed in a gray suit, his red tie still in a perfect Windsor knot. His blond hair was neat, conservative, parted as always on the left. The .44 was in his right hand. His cheeks were rosy, and there was just a hint of a smile on his face.\n\n\"Good evening,\" Win said.\n\n\"How long have you been here?\" Myron asked. He hadn't spotted. Win when they exited the photography studio. But he had known he was there. With Win you just knew. One of life's constants.\n\n\"I arrived as you entered the dwelling of ill repute,\" Win answered. He smiled. \"But I wanted my appearance to have that flair of drama.\"\n\nMyron let go of Esperanza.\n\n\"We better get moving,\" Win said. \"Before the authorities arrive.\"\n\nThey walked away from the corpses in silence. Esperanza was shaking. Myron did not feel so hot either. Only Win seemed completely unaffected by what had transpired. As they approached the car, the same fat young prostitute clad in sausage casing approached Win.\n\n\"Hey, yo, want a blow job? Fifty bucks.\"\n\nWin looked at her. \"I would rather have my semen sucked out with a catheter.\"\n\n\"Okay,\" the girl said. \"Forty bucks.\"\n\nWin laughed and walked away.\n\n# Chapter 26\n\n\"All units. One-eighteen Acre Street. All units One-eighteen Acre Street.\"\n\nPaul Duncan heard the call on his police scanner. He was only a few blocks from the scene, but this was not his district. Far from it. He could certainly not answer the call. That would only draw attention and questions. Questions like what was he doing here.\n\nPieces were starting to come together. Fred Nickler, the publisher of those sleazy rags, had called him earlier in the day. What he had told Paul explained a lot. Not everything. Not by a long shot. But he now understood Jessica's behavior the other night. She had learned about Kathy's picture. Myron Bolitar must have told her.\n\nBut how had Myron gotten a copy of it?\n\nNot important. Not really. What was important was that Myron Bolitar was involved. He could not be underestimated. Jessica was a big enough pain in the ass on her own. But now she had Myron on her side and probably that Win Lockwood, Myron's psychotic Tonto. Paul knew something about their past work for the feds. Not a lot. Myron and Win had answered only to top government officials. Their work was almost always classified. But Paul knew their reputations. That was enough.\n\nA police car sped past Paul, sirens screaming. They were probably on their way to 118 Acre Street. Paul turned up his scanner. He wanted to hear every word that was said.\n\nHe debated calling Carol, but what could he tell her? She hadn't been specific on the phone, just telling him about the phone message from Nancy to Jessica. So what did Jessica know? How had she found out?\n\nAnd what would Carol ultimately be pressured into saying?\n\nTwo ambulances flew by him. They too had their sirens on full blast. Paul swallowed. He wanted to pull over, but he wanted more to drive as far away as possible.\n\nOnce again Paul Duncan thought of his friend Adam Culver. Dead. Murdered. With everything that had happened, there had been no time for Paul to mourn.\n\nYes, mourn.\n\nThat might sound strange\u2014Paul Duncan mourning Adam Culver. Especially if anyone knew how Adam had spent the last precious hours of his life.\n\nWin and Myron dropped Esperanza off at the apartment she shared with her sister and cousin in the east part of Greenwich Village. Myron escorted her to the door.\n\n\"You okay?\"\n\nShe nodded. Her face was deathly pale. She had not spoken a word since the shooting. \"Win\u2014\" She stopped, shook her head. It took her a full minute to pull herself together. \"He saved us. I guess that's what counts.\"\n\n\"Yes.\"\n\n\"I'll see you in the morning.\"\n\nMyron returned to the car. He called Jessica. She wasn't home yet, but Myron did manage to wake her mother. They drove to a twenty-four-hour diner on Sixth Avenue\u2014one of those Greek diners with a menu the approximate length of a Tolstoy novel. Win was a vegetarian. He ordered a salad and french fries. Myron ordered a Diet Coke. He couldn't eat.\n\nAfter they were settled in, Myron asked, \"What happened with Chaz?\"\n\nWin was picking at a basket of stale bread. His face registered displeasure, but he settled on a small packet of Saltines. \"Mr Landreaux hurried straight from our esteemed offices to a building at 466 Fifth Avenue,\" he began. \"He took the elevator to the eighth floor, which is rented by Roy O'Connor and TruPro Enterprises. When Landreaux entered the elevator, he had your contract tightly clutched in his paw. When he exited, the contract was no longer visible. He had no pockets that could hold such a document. Conclusion: Mr Landreaux gave the contract to someone at TruPro Enterprises.\"\n\n\"Your powers of deduction,\" Myron said. \"In a word: uncanny.\"\n\nWin smiled. \"I assume you are feeling better.\"\n\nMyron shrugged.\n\n\"We are not the same, you and I,\" Win added. \"You call it execution, what I did to that vermin. I call it extermination.\"\n\n\"You didn't have to kill him.\"\n\n\"I _wanted_ to kill him,\" Win said with flat inflection. \"And I doubt any of us will mourn his death for very long.\"\n\nTrue enough, but the argument did not ease Myron's mind. He wanted to drop the subject. \"Where did Chaz go after he left TruPro?\"\n\nWin took a dainty bite out of the corner of the Saltine. \"Before I get into that, I should point out that Mr. Landreaux was escorted from the building by a large man who fit the description of your friend Aaron. Large. Confident. Athletic. Suit with no shirt. Sunglasses, though the sun had already set.\"\n\n\"Sounds like Aaron.\"\n\n\"They split up on the street. Aaron got into a stretch limousine. Chaz Landreaux walked to the Omni Hotel.\"\n\n\"Which Omni?\" Myron asked. Manhattan had several.\n\n\"The one near Carnegie Hall. Landreaux met up with his mother in the lobby. Their reunion was rather moving. Mother and son embraced. Both were crying.\"\n\n\"Hmm,\" Myron said.\n\nThe waitress arrived with the food and drinks. She put them down, scratched her butt with a pencil, and returned to the kitchen.\n\n\"So where did they go after that?\"\n\n\"Upstairs. They ordered room service.\"\n\nMyron thought a moment. \"What is Chaz's mother doing up from Philadelphia?\"\n\n\"I would assume,\" Win said, pulling a napkin out of the dispenser and spreading it on his lap, \"based on their mutual anguish, that Frank Ache reached Chaz Landreaux through a family member.\"\n\n\"A kidnapping?\"\n\nWin shrugged. \"A possibility. Frank just sent two men to try to kill you. I highly doubt he is going to become squeamish over a ghetto abduction.\"\n\nSilence.\n\n\"We're wading in some deep doo-doo,\" Myron said.\n\n\"Indeed. Too deep.\"\n\nChaz had a big family. If Frank really wanted to hit him where he lived, he'd take one of his siblings. \"We'll settle it tomorrow,\" Myron said. \"I scheduled a meeting with Herman Ache. Two o'clock. Usual place.\"\n\n\"Should I attend?\"\n\n\"Most definitely.\"\n\nWin ate his salad. \"You do know that this won't be easy.\"\n\nMyron nodded.\n\n\"Herman Ache does not like to intervene in his brother's business.\"\n\n\"I know.\"\n\nWin put down his fork. \"If I may be so bold as to offer a suggestion.\"\n\n\"I'm listening.\"\n\n\"Frank Ache sent two professionals after you. Their untimely deaths will not dissuade him from trying again.\"\n\n\"Uh-huh. So what's your suggestion?\"\n\n\"Cut your losses now. Make an exchange. You let them keep Landreaux. They call off the contract on your head.\"\n\n\"I can't do that.\"\n\n\"You can. You choose not to.\"\n\n\"Semantics.\"\n\n\"You don't have to help him.\"\n\n\"I _want_ to help him,\" Myron answered.\n\nWin sighed. \"A man must try to illuminate even those who prefer to sit in the darkness. Do you have a plan yet?\"\n\n\"I'm still working on it.\"\n\n\"Feverishly?\"\n\nMyron nodded.\n\n\"In the meantime,\" Win said, \"what did you learn from the photographer?\"\n\nMyron filled him in on the meeting with Lucy.\n\n\"So who bought the nude pictures?\" Win asked.\n\n\"A name springs to mind,\" Myron said.\n\n\"Who?\"\n\n\"Adam Culver.\"\n\n\"Kathy's father?\"\n\nMyron nodded. \"Think about it. The buyer was in his fifties. He wanted all copies and all negatives on the spot. He left nothing to chance.\"\n\n\"The father protecting the daughter?\"\n\n\"It makes sense,\" Myron said.\n\n\"But Kathy was missing for over a year. How did Adam Culver suddenly learn about the photographs?\"\n\n\"Maybe he knew about them all along.\"\n\n\"Then why did he wait so long to buy them?\"\n\nMyron shrugged. \"We'll know more tomorrow. I'm going to send Esperanza over to the studio with a picture of Adam, see if Lucy recognizes him.\"\n\nWin took another bite of his salad. \"It's a rather strange development.\"\n\n\"Yes.\"\n\n\"But\"\u2014Win stopped to finish chewing\u2014\"here is something else you may not have considered: If Adam Culver purchased all the pictures and negatives in order to protect his daughter, how did her photograph end up in the magazine?\"\n\nMyron had considered that. He just didn't have an answer.\n\nThe waitress put down the check. Myron picked up the tab for both of them. The total was $8.50. Mr. Magnanimous. They drove uptown. Win lived in the San Remo building overlooking Central Park West. Very fancy address. They were on Seventy-second Street when the car phone rang.\n\nMyron looked at his multicolored Swatch. A gift from Esperanza.\n\nPast midnight.\n\n\"Rather late for a call to your car,\" Win noted.\n\nMyron picked up the phone. \"Hello?\"\n\nThe voice came fast. \"Bolitar, it's Jake Courter. Get your ass down to St. Barnabas Hospital in Livingston right away.\"\n\n\"What happened?\"\n\n\"Just get down here. Now.\"\n\n# Chapter 27\n\n\"We got the call around eleven-thirty,\" Jake said, ushering Myron through the lobby of St. Barnabas. Jake's face was set, his eyes red and puffy. They hurried past the circular visitors' desk and waited for an elevator.\n\n\"Is Jessica okay?\" Myron asked.\n\n\"She is going to be fine,\" he said. Then he added, \"Wish I could say the same about Nancy Serat.\"\n\n\"What happened?\"\n\n\"She was garroted with a wire.\" The elevator arrived. Jake pressed the button for the fifth floor. \"When no one answered the door, Jessica let herself in through the back. The killer must have still been there. He knocked her over the head and ran. When she came to, she called us. I'd say she's pretty lucky the perp didn't waste her.\"\n\nThe elevator opened with a _ding_. \"What room is she in?\" Myron asked.\n\n\"Five fifteen.\"\n\nMyron sprinted down the corridor. He turned the corner. Jessica was in the bed, her face ashen. A doctor stood next to her, preparing a needle. Jake came up behind Myron but stayed in the doorway.\n\nHer voice was wobbly. \"Myron?\"\n\n\"I'm here,\" he said, taking her hand. She looked small and frail and alone. \"I won't leave.\"\n\nThe doctor pricked her with the needle. \"You need your rest,\" he said.\n\n\"I'm fine,\" Jessica insisted weakly. \"I want to get out of here.\"\n\n\"We think it's best if you stay overnight for observation.\"\n\n\"But\u2014\"\n\n\"Listen to him, Jess,\" Myron interrupted. \"There's nothing we can do tonight.\"\n\nThe drug began to take effect. Her eyes fluttered back. \"Nancy...\"\n\n\"It's okay,\" Myron soothed.\n\n\"Her face was blue...\"\n\n\"Shhh.\"\n\nJessica slipped into unconsciousness. Myron looked up at the doctor. \"Is she going to be okay?\"\n\n\"She'll be fine. I think the shock of what she saw was worse than the blow to her head.\"\n\nJake put his hand on Myron's shoulder. \"Come on, I'll buy you a cup of coffee.\"\n\n\"I want to stay.\"\n\n\"You can come back later. Right now we need to talk.\"\n\nMyron gazed down at Jessica. She was deep in sleep.\n\n\"She'll be out for a while,\" the doctor assured him.\n\nThey walked down the corridor silently and took the elevator back to the lobby. The place had that hospital smell\u2014that unique combo of something antiseptic and the hospital food. Win had parked the car and was now sitting in the waiting area. He stood when he saw them.\n\n\"That your friend Win?\" Jake asked, motioning with his chin. \"The one P.T. told me about?\"\n\n\"Yes.\"\n\n\"Tell him to stay here. I want to talk to you alone.\"\n\nMyron signaled to Win. Win nodded, sat back down, picked up a newspaper, crossed his legs. Jake looked him over for a minute. \"He as crazy as P.T. says?\"\n\n\"Pretty much.\"\n\n\"Come on.\"\n\nThey grabbed coffee and found a table in the corner. \"The crime scene unit is going over Nancy's house now. They'll beep me if they find anything.\"\n\n\"So what do you know so far?\" Myron asked.\n\n\"Not much. Nancy spent the last few days in Canc\u00fan\u2014a graduation present from her parents.\"\n\n\"Have they been told?\"\n\nHe shook his head. \"I'm going over there right after we talk.\"\n\nSilence. Jake broke it. \"So how did Jessica get involved in this?\"\n\n\"She wanted me to look into her father's murder. She didn't buy the fact that he was killed in a botched robbery.\"\n\nJake nodded. \"She thought her old man's murder had something to do with her sister.\"\n\n\"Yes.\"\n\n\"I figured as much. I got the file in the car.\"\n\nMyron sat up. \"Adam Culver's homicide file?\"\n\n\"Hey, I ain't an idiot, Bolitar. You start investigating after eighteen months. Why? Had to be the father's murder. You saw a connection. But I gotta be honest. I don't see it. No connections in that file at all. A few inconsistencies maybe. But no connection.\"\n\n\"What sort of inconsistencies?\" Myron asked.\n\n\"Adam Culver was supposed to be in Denver when he was killed. At a medical examiners' conference at the Hyatt Regency. But he never showed, missed his morning flight.\"\n\n\"Does the file say why?\"\n\n\"Adam didn't feel well. A reasonable explanation.\"\n\n\"Who told them that?\"\n\n\"His wife.\"\n\nPause. \"What else?\"\n\n\"Nothing else. The crime scene\u2014a quiet street\u2014was unremarkable. He was stabbed through the heart.\"\n\n\"What was he doing out?\"\n\n\"The wife said he went out to buy some groceries.\"\n\nMyron chewed that one over for a moment. \"Odd thing to do,\" he said, \"when you're not feeling well.\"\n\n\"Yeah, that's easy for us to say, sitting here like this. But the cops were concentrating on finding a mugger. No one really gave a shit about a missed flight or what it might mean.\"\n\n\"Any witnesses to the murder?\"\n\n\"None. The file is pretty bare-bone.\" Jake leaned forward and tried to stare Myron down. Myron did not look away. \"Now,\" Jake said slowly, \"you start talking to me. And don't give me no 'I don't want no one hurt' crap. Too late for that now. Why are you really involved in all this?\"\n\n\"I told you Jessica.\"\n\nJake leaned farther forward until their faces were only inches apart. \"Stop jerking me around,\" he spat out. \"I ain't blind. I can see Jessica Culver is great tail. But don't start giving me this bullshit that you just decided to drop everything and help on a whim. You ain't that hard up.\"\n\n\"There was also Christian to consider,\" Myron said.\n\n\"What about him?\"\n\n\"He's my top client. He was still upset about his fianc\u00e9e's disappearance.\"\n\nJake made a snorting nose. \"Yeah, I bet.\"\n\n\"What's that supposed to mean?\"\n\n\"It means,\" Jake said, \"that I'm not convinced Christian is completely innocent in all this.\"\n\n\"But you said the DNA test on the semen\u2014\"\n\n\"I'm not saying he raped her.\"\n\n\"Then what are you saying?\"\n\n\"That he might be involved,\" Jake replied. \"Your client had no solid alibi for the time of the disappearance. He claims he was in bed at eleven o'clock, but no one can confirm it.\"\n\n\"He has a single room,\" Myron said. \"Who's going to confirm he was in bed when he lived alone?\"\n\n\"It's suspicious,\" Jake replied.\n\n\"How? Kathy Culver was seen entering the team locker room after ten, right?\"\n\nJake nodded.\n\n\"And you know Christian was meeting with the offensive coordinator until ten-thirty,\" Myron continued. \"That's confirmed.\"\n\n\"But that's where his alibi ends.\"\n\n\"He went to bed after that. Kathy was seen wandering around on the other side of the campus at eleven o'clock. I don't see the connection.\"\n\n\"Maybe there is none,\" Jake said simply. \"But he's the boyfriend. The boyfriend is always a prime suspect. And there was something else.\"\n\n\"What?\"\n\n\"His teammates.\"\n\n\"What about them?\"\n\nJake finished his coffee. He tapped the cup to get the last few drops. \"They were cooperative, I guess, but some of them seemed awfully vague. Nothing I could pin down, but some of them looked more nervous than they should. Like they were covering something up. Like maybe, just maybe, they were protecting their star quarterback before the big game.\"\n\nExcept, Myron thought, nobody on the team liked Christian. His teammates would not have gone out of their way to protect him. Just the opposite, in fact.\n\nSo why were they nervous?\n\nJake settled back and smiled, marking a change in tactics. \"Now, Myron, I've been awfully sweet, haven't I? I've told you all I know, and you're still holding back on me. That ain't nice. Something else\u2014something you haven't shared with me yet\u2014put a real hairy bug in your ass. Now I visited our friend Dean Gordon a few hours ago, just like you suggested. The man was cordial, friendly, not at all a pompous ass. Which ain't like him. In fact, I think he was scared shitless. Now why's that?\"\n\n\"Did he tell you anything?\"\n\n\"Oh, he was real helpful. Kathy was a wonderful girl, an honor student, a hard worker, blah, blah, blah. Oh, yeah. He also told me your ex upstairs paid him a visit. Seems Jessica wanted her sister's file. Imagine that.\"\n\n\"We were trying to gather as much info as possible.\"\n\n\"Information on what?\"\n\nMyron eyed his coffee. It looked like sewer sludge. \"On the morning Adam Culver was murdered, he visited Nancy Serat.\"\n\nJake's eyes widened a bit. \"How do you know that?\"\n\n\"Nancy left a message on Jessica's phone to meet her at ten o'clock tonight. She also said that she'd seen Adam Culver on the morning of the murder.\"\n\n\"Jesus Christ.\" Jake crossed his arms, resting them on his belly. \"So Adam Culver visits Nancy Serat in the morning. He finds something out. Something big. Something so big he cancels his trip.\"\n\n\"Something so big,\" Myron added, \"it gets him killed.\"\n\nJake nodded, thinking. \"Then the killer has to get rid of the source.\"\n\n\"Nancy Serat.\"\n\n\"Right.\" Jake stopped. \"But I questioned that girl for hours. I asked her everything.\" His voice faded off, and a shadow crossed his face. Myron knew what he was wondering. Any cop worth a damn would be asking the same questions. Did I fuck up? Did I miss something? Is a young girl dead because of me?\n\n\"If Nancy knew something that important,\" Myron said, \"the killer wouldn't have waited eighteen months to silence her. I think it's a little more complicated than our scenario. I think Adam Culver had already put most of it together. Nancy had the final piece, a piece that by itself meant nothing to anyone\u2014except Adam Culver.\"\n\n\"You trying to make me feel better?\"\n\n\"No. It's how I see it. If I thought you fucked up, I'd say so.\"\n\n\"You didn't see her body,\" Jake said quietly. \"Strangulation ain't pretty. The damn wire nearly sliced her head off. Not a nice way to go, Myron.\" He stopped, shook his head. \"After seeing that, I know what Jessica is asking herself, because I keep asking myself the same thing.\"\n\n\"What's that?\"\n\n\"Did Kathy meet a similar fate?\"\n\nSilence. They drank some coffee. Myron's was already cold, but he didn't complain. Cold, sludgelike coffee seemed to fit the occasion.\n\n\"P.T. told me all about you,\" Jake said after a massive slurp. \"Said you were smart, that I could trust you. He don't say that about too many folks. Said you and that Win fella were as good as they come. A little too maverick, but right now I could use that. I'm a cop. I have to follow rules. You don't. More power to you. But this is my territory, and I ain't gonna sit around like some fucking movie extra.\" He put his hands on the table. They were big and callused and had no rings. \"So now I want you to tell me everything, Myron. Right now. Just you and me. It won't get out, you have my word. Don't hold anything back. You understand?\"\n\nMyron nodded.\n\n\"So start talking, boy I'm all ears.\"\n\nMyron took out the magazine and handed it to Jake. \"It all started with this.\"\n\n# Chapter 28\n\nThe morning papers had no mention of Nancy Serat's murder, but the radio was beginning to pick up early reports of a murdered woman. Just a question of time. Myron took Route 280 east to the New Jersey Turnpike north. Scenic road. Like driving through west Beirut on a good day. Problem was, people unfairly judged New Jersey by this road. It was like judging a woman's beauty by the size of her feet.\n\nBilly Joel was on the radio, singing, \"I love you just the way you are.\" Big talk, Myron mused, when you've been married to Christie Brinkley.\n\nExit 16W led him directly into the Meadowlands parking lot. Murder and intrigue were all well and good, but agenting paid the bills. He had a meeting with Otto Burke. Otto was expecting a response to his demand vis-\u00e0-vis Christian's contract. Myron had prepared one for him.\n\nHe had spent the night in Jessica's hospital room, trying to get comfortable in a chair that doubled as a medieval torture device. But he had not minded. He liked watching her sleep. It brought back memories. He'd always hoped they'd one day sleep together again, though last night was not precisely what he had had in mind.\n\nJess had woken up two hours ago. Belligerent. Testy. Demanding. In a word: herself. Before her brother Edward took her home, Myron had told her all he knew\u2014especially about his visit to Lucy's photo studio. She had given him a photograph of her father to show Lucy. Myron was surprised to see Jessica carried one in her wallet. But he was far more surprised to catch a fleeting glimpse of a picture from four summers ago\u2014a picture she tried to skip past without his seeing. But he had seen it, and he remembered the precise moment it had been taken. Their last weekend in Martha's Vineyard. Just the two of them. Tan, happy, relaxed. A barbecue at Win's summer house. The pinnacle before the inevitable slide.\n\nMyron had not had a chance to change clothes. He looked as if he'd spent the evening in the bottom of a laundry hamper.\n\nOtto was waiting for him in the owner's box on Titans Stadium mezzanine level. Larry Hanson was with him. Otto greeted Myron with a bony handshake and a wide smile. Mr. Sunshine Larry offered a quick wave. He did not meet Myron's eye. It was no wonder. Larry Hanson was a tough guy, a loud brute even, but he tried to play fair. He didn't like to cheat, and he did not like what Otto was doing now. He looked, in fact, as if he wanted to blend into the wall.\n\n\"Please, Myron,\" Otto said, spreading his arms like Carol Merrill on _Let's Make a Deal_ , \"sit wherever you like.\"\n\n\"Always the perfect host, Otto.\"\n\n\"I do try, Myron. Thank you for noticing.\"\n\n\"Sarcasm, Otto. It's called sarcasm.\"\n\nOtto kept the smile aglow. His goatee was exactly the same as always, never heavier or lighter. Must trim it every day, Myron thought. They sat in two seats facing the field. Fifty yard line. Fans would kill for these seats. Down below, players were scattered across the field. Myron spotted Christian walking toward the sideline. His helmet was off, his head held high. Christian didn't know about Nancy Serat's murder\u2014her name had not yet been released\u2014but the press would be all over him soon enough. Myron could protect him only so much, though he did entertain hopes that the news of Christian's signing would deflect some attention away from the murder.\n\n\"So,\" Otto said with a clap of his hands, \"are you ready to sign?\"\n\nDown on the field Christian was being introduced to a bunch of long-haired men. Myron recognized the men from a video on MTV. They were Otto Records' latest find. A group called StillLife. Good sound, but did they have the raw talent, of, say, Pap Smear?\n\n\"Sure,\" Myron said. \"We would like nothing more.\"\n\n\"Great I have a pen.\"\n\n\"How handy. I have a contract.\" He handed it to Otto. Otto read it quickly. His mouth was smiling, but his eyes frowned. He passed it to Larry Hanson.\n\n\"I'm confused, Myron. This looks like your last offer.\"\n\n\"Very perceptive, Otto.\"\n\n\"I thought we had an agreement,\" he said.\n\n\"We do. There it is.\"\n\n\"I think you're forgetting\"\u2014he paused, searching for the right word\u2014\"Christian's sudden devaluation.\"\n\n\"You make him sound like a foreign currency.\"\n\nOtto laughed. He looked over to Larry as if to say, laugh too. Larry could only muster a smile. \"Okay, Myron, I'll accept that. We are all, to some extent, commodities. Your client, however, is now trading at a lower rate against the U.S. dollar.\"\n\n\"Thanks for keeping within the metaphor, Otto, but I don't see it that way.\" Myron looked at Larry Hanson. \"How's his play been, Larry?\"\n\n\"Well, it's very early,\" Larry said, clearing his throat. \"You really can't tell too much after such a short time period.\"\n\n\"But if you had to grade him so far?\"\n\nAnother throat clear. \"Let's just say,\" he replied, \"that Christian's play has not been a disappointment.\"\n\n\"There you go,\" Myron said, matching Otto's smile. \"His value has, if anything, increased with his recent on-the-field display. You have now had a tasty morsel of his potential. I don't see how you could expect us to drop our asking price.\"\n\nOtto rose, nodding his head. He clasped his hands behind his back and walked to the bar. \"Care for a drink, Myron?\"\n\n\"Do you have any Yoo-Hoo?\"\n\n\"No, I don't.\"\n\n\"Nothing, then.\"\n\nOtto poured himself a 7-Up. He did not ask Larry Hanson if he wanted anything. \"I will admit,\" Otto said, \"that Christian's play so far has been impressive, though I must caution you, Myron\u2014and you too, Larry\u2014that there is a big difference between practice and games. Between how an athlete performs in a scrimmage and how he performs in a pressure situation.\"\n\nMyron and Larry exchanged a glance. The glance said, Pretentious asshole.\n\n\"But let me also add,\" Otto continued, \"that our product is dependent on more than just performance. If, for example, our team were to win the Super Bowl but were also involved in a major drug or sex scandal, the overall value of the product may decline.\"\n\n\"Can you demonstrate that with a graph?\" Myron asked. \"I'm not sure I understand.\"\n\n\"It means,\" Otto said, \"that the photograph in that sleazy publication makes Christian worth less money to us.\"\n\n\"But it's not a picture of him.\"\n\n\"It's a picture of his fianc\u00e9e.\"\n\n\"Ex-fianc\u00e9e.\"\n\n\"His fianc\u00e9e who vanished under mysterious circumstances.\"\n\n\"Christian and I are willing to take the chance,\" Myron said. \"It was in a small publication. It hasn't gotten out so far. We don't think it will.\"\n\nOtto sipped his 7-Up. He seemed to enjoy it, even adding an \"aaah\" like he was taping a commercial. \"But the press might find out.\"\n\n\"I don't think so,\" Myron said. \"I've discussed it with Christian. We both feel the same.\"\n\n\"Then you are both fools.\"\n\nThe facade had dropped open a crack.\n\n\"Now, Otto, that wasn't very nice.\"\n\nThe facade slid back up, smooth as an electric car window. \"Let me remind you of our previous discussion on this very subject, Myron. See if you can follow this. You were to take our agreement and knock it down by a third. If not, the picture of the au naturel Ms. Culver goes public, thereby ruining your player's endorsement career.\"\n\n\"But he didn't do anything, Otto. It's only a picture of Kathy Culver.\"\n\n\"It doesn't matter. Advertisers do not like the smallest whiff of controversy. Remember this, Myron: In business, appearance is far more important than reality.\"\n\n\"Appearance versus reality,\" Myron said. \"That I have to write down.\"\n\nOtto took out a contract of his own. \"Sign it,\" he said. \"Now.\"\n\nMyron just smiled at him.\n\n\"Sign it, Myron. Or I'll ruin you.\"\n\n\"I don't think so, Otto.\"\n\nMyron began to unbutton his shirt.\n\n\"What do you think you're doing?\"\n\n\"Don't get excited, Otto. I'm stopping after the third button. Just enough to show you this.\" He pointed to the small microphone on his chest.\n\n\"What the hell\u2014?\"\n\n\"It's a wire, Otto. It leads to a tape recorder stuck in my belt. You can make the picture public, that's up to you. It may damage Christian, it may not. I, in turn, will make this tape public. I will also sue your sorry ass for any damages Christian may have suffered because of your actions, and I will also see to it that you are arrested for extortion and blackmail.\" Myron smiled. \"I always wanted to own a record company. Chicks dig that, don't they, Otto?\"\n\nOtto looked at him coolly. \"Larry?\"\n\n\"Yes, Mr. Burke.\"\n\n\"Take the tape away from him. Forcibly, if necessary.\"\n\nMyron looked at Hanson \"You're a big guy, Larry,\" Myron said. \"And I know you were one of the toughest fullbacks ever to play this game. But if you get out of that chair, I'll put you in a body cast.\"\n\nLarry Hanson merely nodded. Not afraid, but not moving either.\n\n\"There are two of us,\" Otto urged. \"I can call in security guards to help.\"\n\n\"I don't think so, Mr. Burke.\" Larry was almost smiling. \"And I don't think a few security guards are going to scare him very much. Are they, Myron?\"\n\n\"Not likely.\"\n\n\"I think we should sign his contract, Mr. Burke. I think it's best for all.\"\n\n\"I've even drawn up a press release,\" Myron said. \"Says how happy Christian is to be playing for such an outstanding and reputable organization as the Titans.\"\n\nOtto thought a moment. \"If I sign,\" he said, \"you'll hand over the tape?\"\n\n\"Not likely.\"\n\n\"Why not?\"\n\n\"You keep the magazine and I keep the tape. Think of it as our own little balance of terror. A throwback to the cold war.\"\n\n\"But you have my word\u2014\"\n\n\"Please, Otto, it hurts when I laugh.\"\n\nOtto thought a moment. He was shaken but calm. A guy his age doesn't reach this level without learning to take a few knocks.\n\n\"Myron?\"\n\n\"Yes.\"\n\n\"I can't tell you how thrilled the Titans are to have Christian Steele, the quarterback of the future, with us.\"\n\n\"Just sign right here, Otto.\"\n\n\"My pleasure, Myron.\"\n\n\"No, Otto. Mine.\"\n\nOtto signed. Myron and Otto shook hands. The deal was done.\n\n\"Shall we meet the press jointly, Myron?\"\n\n\"Sounds wondrous, Otto.\"\n\n\"There's a shower downstairs. I'll make sure you're provided with shaving equipment, if you like.\"\n\n\"Very kind of you.\"\n\nOtto's smile was back. The man was never down long. He picked up the phone. \"Christian Steele has been signed,\" Otto said. Then, looking back and winking at Myron, he added, \"At the highest salary ever given to a rookie.\"\n\nMyron winked back and gave him the thumbs up. Lifelong chums. He checked his watch. There would be just enough time to shower and do the press conference before he would have to head back into the city for his meeting with Herman Ache.\n\nHe had no idea how he was going to handle the evil Ache brothers. But he was still working on it. Feverishly.\n\n# Chapter 29\n\nJessica arrived at the house in Ridgewood at ten o'clock. The doctor had wanted to run some more tests in the morning. Jessica refused. They finally reached a compromise whereby Jessica promised to visit him in his office sometime during the week. Edward had driven her home in silence.\n\nWhen they arrived, Jessica noted that her mother's car was not in the driveway. Good. Not in much of a mood to handle a hysterical mother on top of everything else, Jessica had insisted that no one tell her mother about last night's incident. Mom had enough on her mind. No reason to get her unnecessarily upset.\n\nJessica headed straight for the study. Her father had been up to something, that much was clear. There were too many weird happenings for it to have been any other way. He had visited Nancy Serat on the morning of his death. He had skipped out on a medical examiners' convention in Denver because he hadn't felt well\u2014something he would never do. He had possibly even purchased nude photographs of Kathy.\n\nYou didn't have to be Sherlock Holmes to realize something was amiss.\n\nShe flicked on the track lights, illuminating the room a bit too harshly for her taste. She used the dimmer. Downstairs, Edward was in the kitchen opening the refrigerator.\n\nShe began to rifle through her father's drawers. She had no idea what she was looking for. Perhaps a small box with the words BIG CLUE scrawled across the top. That would be nice. She tried not to think about Nancy Serat, about her blue face frozen in terror, but the thought stayed anchored front and center. She thought of more pleasant things, like waking to see Myron folded up in that hospital chair like a contortionist from Le Cirque du Soleil. The image made her smile.\n\nIn the file drawer she found a folder marked CMA. Her father's Merrill Lynch Cash Management Account. She pulled it out. The CMA statement is a financial instrument of great beauty. Everything in one statement\u2014your stocks, bonds, other holdings, checks, Visa card transactions. Jessica had one of her own.\n\nShe checked the charges and checks cleared on the most recent statement. Nothing unusual. Problem was, the statement ended three weeks ago. She needed something more recent.\n\nShe flipped to the last page. On the bottom in small print it read \"You have an alphabetic character in your Merrill Lynch account number. Please use nine-eight-two-three-three-four as your account access number for CMA-DATA.\"\n\nCMA-DATA. The 800 line. She had used it before with her own account, whenever she found a discrepancy. She dialed the number and immediately heard a taped voice say, \"Welcome to the Merrill Lynch Financial Service Center. Enter your Merrill Lynch account number or your account access number.\"\n\nJessica entered the number.\n\n\"Enter your selection. You may interrupt the dialogue at any time. For your current balance and purchasing power, enter one. For check clearing information, enter two. For most recent funds received, enter three. For most recent Visa transactions, enter six.\"\n\nShe decided to start with the charges and then look at the checks. She pressed six.\n\nThe voice said, \"Visa draft for $28.50 is on delay debit as of May twenty-eighth. Visa draft for $14.75 is on delay debit as of May twenty-eighth.\"\n\nThe machine was not telling her where the charges were coming from. The same would be true for the checks. Knowing just the amounts would do her no good.\n\n\"Visa draft for $3,478.44 is on delay debit as of May twenty-seventh.\"\n\nShe froze. Three thousand dollars? For what? She hung up, hit the redial button, and put in the account access number.\n\n\"Enter your selection.\"\n\nThis time she pressed zero for a customer service representative.\n\n\"Good morning,\" a pleasant-voiced woman singsonged. \"May I help you?\"\n\n\"Yes, there's a Visa charge on my account for over three thousand dollars. I'd like to know where the charge came from.\"\n\n\"Your account number, please?\"\n\n\"Nine-eight-two-three-three-four.\"\n\nThere was some keyboard clacking in the background. \"And you are?\" the rep asked.\n\nJessica checked the statement. A joint account, thank God. \"Carol Culver,\" she said.\n\n\"Hold one moment, Mrs. Culver.\"\n\nMore clacking. \"Yes, I have it here. $3,478.44. Eye-Spy Shop in Manhattan.\"\n\nEye-Spy? What the hell was that all about?\n\n\"Thank you,\" Jessica said.\n\n\"Anything else today, Mrs. Culver?\"\n\n\"Yes. My husband and I have all our records on a personal computer, and I'm afraid the computer has had a disk failure. Can I ask you to give me the most recent checks that have been written against the account?\"\n\n\"Certainly.\"\n\nMore clacking. \"Check one-nineteen for $295 to Volvo Finance, written on May twenty-fifth.\"\n\nCar payment.\n\n\"Check one-eighteen for $649 to Getaway Realty, also written on May twenty-fifth.\"\n\nHold the phone. \"Did you say Getaway Realty?\"\n\n\"Yes, that's correct.\"\n\n\"Does it say where they're located?\"\n\n\"I'm afraid I don't have that information.\"\n\nThey went through the rest of the month's checks. Nothing unusual. Jessica thanked the woman and hung up.\n\n$649 to Getaway Realty? $3,478.44 to Eye-Spy? More and more amiss.\n\nEdward knocked on the door. \"Hi,\" he said.\n\n\"Hi.\"\n\nHe stepped into their father's study, head lowered.\n\n\"I'm sorry about the other day,\" Edward said. He blinked several times, his to-die-for eyelashes waving up and down. \"About running out like that.\"\n\n\"It's okay.\"\n\n\"You hit a raw nerve,\" he said. \"Asking all those questions and everything.\"\n\n\"They need to be asked,\" she replied. \"I think everything is connected. What happened to Kathy. What happened to Dad. What made Kathy change.\"\n\nEdward flinched at the word _change._ Then he shook his head. His T-shirt of the day featured Beavis and Butthead. \"You're wrong,\" he said. \"It doesn't have anything to do with what happened to her.\"\n\n\"Maybe,\" she said. \"Only way to find out is if you tell me.\"\n\n\"I don't feel comfortable about it. It's painful.\"\n\n\"I'm your sister. You can trust me.\"\n\n\"We were never very close,\" he said bluntly. \"Not like you and Kathy.\"\n\n\"Or you and Kathy,\" Jessica said. \"But I still love you.\"\n\nShe waited.\n\n\"I don't know where to begin exactly,\" he said. \"It started her senior year of high school. You had just moved to Washington. I was at Columbia. I was living off campus with my friend Matt. Remember him?\"\n\n\"Of course. Kathy dated him for two years.\"\n\n\"Almost three,\" Edward corrected. \"Matt and Kathy were like something out of another century. They were together three years, and he never got, well, below the neck. I mean, never. And it wasn't just from a lack of trying. Matt was as straightlaced as any guy I knew, but that didn't mean he didn't push it now and again. But Kathy held him off.\"\n\nJessica nodded, remembering. Kathy had still been confiding in her at that stage.\n\n\"Mom loved Matt,\" Edward continued. \"She thought he was the greatest. She used to invite him over for tea like something out of _The Glass Menagerie._ A gentleman caller sitting on the porch with the youngest daughter. Dad liked him too. Everything seemed to be going well. They planned on getting engaged in another year, married after he graduated, the whole Chevy-and-apple-pie love story. Then one day Kathy called him on the phone and just dumped him. No explanation.\n\n\"Matt was shocked. He tried to talk to her, but Kathy wouldn't see him. I tried to talk to her too, but she just blew me off. Then I started hearing rumors.\"\n\nJessica shifted in her chair. \"What kind of rumors?\" she asked.\n\n\"The kind,\" Edward said slowly, \"a brother doesn't like to hear about his sister.\"\n\n\"Oh.\"\n\n\"Worse than oh. Guys were trashing her nonstop. Someone had finally found the key to Miss Prude's chastity belt, they said, and now they couldn't get it back closed. I even got into a fight. Got the shit beaten out of me protecting Kathy's honor.\" He spat out the word _honor_ as though it had an offensive taste.\n\n\"She changed at home too. She never went to mass anymore. I thought Mom would have a stroke\u2014you know how she gets about stuff like that.\"\n\nJessica nodded. She knew only too well.\n\n\"But she never said a word. Kathy started staying out late. She went to college parties. Some nights she wouldn't even come home.\"\n\n\"Didn't Mom stop her?\" Jessica asked.\n\n\"She couldn't, Jess. It was unbelievable. Kathy had spent her entire life in fear of the woman. Now it was like Kathy had found Kryptonite. Mom couldn't touch her.\"\n\n\"What about Dad?\"\n\n\"He was never as strict as Mom, you know that. He wanted to be everyone's buddy, not the bad guy. But strangely enough, Kathy grew closer to Dad during all this. He was thrilled by the sudden attention. I think he was afraid if he laid down the law, he'd push her away from him.\"\n\nSounded like her father. \"What did you do?\" she asked.\n\n\"I confronted her.\"\n\n\"What did she say?\"\n\n\"Nothing really. She wouldn't deny it or admit it. She would just stand there and smile eerily. She said I didn't understand, that I was 'na\u00efve.' Na\u00efve. Can you believe Kathy could call someone else na\u00efve?\"\n\nJessica thought a minute. \"But none of that explains what started it, what made her change in the first place.\"\n\nEdward opened his mouth, stopped. He spread his hands, then dropped them back to his sides as though they were too heavy to hold up. His voice was barely audible \"Something with Mom,\" he said.\n\n\"What with Mom?\"\n\n\"I don't know. I think maybe Mom does. Kathy became withdrawn from you and me. But she still loved us. It was Mom who got the brunt of it.\"\n\nJessica leaned back in her father's chair, considering his last comment. \"I knew Kathy had changed the last couple of years, but I had no idea...\" Her voice sort of faded away.\n\n\"But it ended, Jess. You have to remember that.\"\n\n\"What ended?\" she asked.\n\n\"This stage Kathy went through. That's why I don't think it's related to her disappearance. By the time she disappeared, it was all in the past.\"\n\n\"What do you mean, in the past?\"\n\n\"She changed back. Oh, I don't mean she started going to mass every Sunday or became buddies with Mom. But whatever had twisted her out of shape had finally let go. She was regaining her old self. I think Christian had a lot to do with that. I think he helped bring her back from the edge. The slutty behavior certainly stopped. So did the drugs, the drinking, the partying. Other things too. The smile even came back a little.\"\n\nJessica remembered Kathy's school transcript. The terrible grades in her senior year and the beginning of college. Then the sudden turnabout back toward excellence that had started her second semester freshman year\u2014when she met Christian. It added up with what Edward was saying.\n\nSo was the past irrelevant? Was this period of her life, as Edward had insisted, all behind her? Perhaps. But Jessica doubted it. If it were truly dead and buried, why was her picture now appearing in a pornographic magazine? And that of course led to the central question in all this:\n\nWhat had made Kathy change in the first place?\n\nJessica still did not know. But she now had a pretty good idea who might.\n\n# Chapter 30\n\nThere were several things Myron enjoyed more than visiting Herman Ache. Having his eyeball removed with a grapefruit spoon, for example.\n\n\"I heard your press conference on the radio,\" Win said. The top was down on Win's racing-green Jaguar XJR. Myron was not big on having the top down. It was just a question of time before a bug got stuck in his teeth. \"I trust that Christian was pleased with the deal.\"\n\n\"Very.\"\n\n\"The press still hasn't picked up on Nancy Serat.\"\n\n\"Jake hasn't released her name yet. Once they do\u2014\"\n\n\"Party time.\"\n\n\"Exactly.\"\n\n\"Does Christian know?\" Win asked.\n\n\"Not yet. He was so damn happy. I just wanted to let him enjoy it a little longer.\"\n\n\"You should warn him.\"\n\n\"I will. Jake promised to let me know the second it got out.\"\n\n\"You seem to like this Jake fellow,\" Win noted.\n\n\"He's a good man. We can trust him.\"\n\nWin wiggled his fingers, regripped the wheel, accelerated. \"I don't trust officers of the law,\" Win said. \"It's safer that way.\"\n\nThe car was going very fast. The West Side Highway was not built for such speed\u2014a four-lane highway with traffic lights every twenty yards. Plus the \"ongoing\" construction didn't help. The construction had been going on for as long as anyone could remember. History books stated that Peter Minuit, the Dutchman who purchased Manhattan from the Indians in 1626, often complained about the delays around Fifty-seventh Street.\n\nBut none of that deterred Win's hefty accelerator foot. The Javits Center was a blur. So was the Hudson River, for that matter.\n\nMyron said, \"Could you slow down a tad?\"\n\n\"No need to worry. The car has a driver-side air bag.\"\n\n\"Wonderful.\"\n\nThey were getting closer to Ache's office. Myron's stomach knotted\u2014not helped by the smog blasting into his face because the top was down. His nerves were as taut as a freshly strung tennis racket. Win, on the other hand, looked relaxed. Then again, Frank Ache didn't have a contract out on his head.\n\nWin's car phone rang. He picked it up. \"Hello?\" He handed the phone to Myron. \"It's P.T.\"\n\nMyron took the receiver. \"What's up?\"\n\n\"Hey, Myron, how you feeling today?\"\n\n\"Can't complain.\"\n\n\"Glad to hear it. Say, you'll never guess what happened last night.\"\n\n\"What?\"\n\n\"Two of New York's finest hit men were found dead in an alley. Sad, ain't it?\"\n\n\"Tragic,\" Myron agreed.\n\n\"They worked for Frank Ache.\"\n\n\"That a fact?\"\n\n\"Forty-four Magnum with dum-dum bullets were used. Blew their heads clean off.\"\n\n\"Such a loss.\"\n\n\"Yeah, I'm losing sleep over it too. Anyway, word out on the street is, this ain't over. Corpses don't exactly waylay the wants of a guy like Frank Ache. The contract is still out on whatever ugly slob pissed Frank off.\"\n\nMyron said, \"Ugly?\"\n\n\"Well, it's been nice talking to you, Myron. Take care.\"\n\n\"You too, P.T.\"\n\nMyron hung up.\n\n\"The contract is still in place?\" Win asked.\n\n\"Yep.\"\n\n\"They won't hit you in Herman's office,\" Win said. \"He would never allow it.\"\n\nMyron knew that was true. There was a certain code, even among men who have probably ordered the deaths of hundreds of people. Some idiots believed that these codes were based on some sort of ethics. Not even close. The codes were two things to mobsters: (1) a device to make them appear almost human, and (2) a way of protecting themselves and their position. Ethics are to a mobster what honesty is to a politician.\n\nA construction site slowed them near Twelfth Street, but they still made it with time to spare. The air smelled of pizza\u2014probably because they parked in front of a pizzeria called The First Original Ray's Pizza of New York, Really, We're Not Kidding, Honest, We're It. A tall woman in a blue business suit and fancy sunglasses strolled purposefully down the sidewalk. Myron smiled at her, and she returned it. He would have preferred a faint or even a small swoon, but you can't have everything.\n\nAt two in the afternoon Clancy's Tavern was already in full swing. Myron stopped right outside the door, fixed his hair, turned left, smiled, turned right, smiled, looked up, smiled.\n\nWin looked a question at him.\n\n\"The feds take pictures of everyone who comes in here,\" Myron said. \"I just wanted to look my best.\"\n\n\"Now you tell me. I look like hell.\"\n\nClancy's patrons were all men. Not exactly a swinging pick-up joint. A jukebox played Bob Seger. The decor was Early American Beer. Lots of those neon signs, the ones that spell out company names. Budweiser, Bud Light, Miller, Miller Lite, Schlitz. A clock courtesy of Michelob. A mirror from Coors Coasters from Pabst. The mugs had Rolling Rock logos emblazoned across them.\n\nMyron knew that there were probably a million FBI bugging devices in here. Herman Ache didn't care. Anybody who said something truly damaging in the tavern itself was beyond stupid and deserved to get nailed. The real talk went on in the back rooms. Ache made sure they were swept for bugs every day.\n\nWin drew a few curious glances when they entered. Prep was not exactly the \"in\" style of Clancy's clientele. But no one stared too long. This was a bar where no one stared at anyone too long.\n\n\"Is that your friend Aaron?\" Win asked.\n\nAaron was at the back of the bar wearing his customary white suit. This time he wore a shirt, albeit one of those pectoral-displaying sleeveless muscle T's. It was as if Aaron's wardrobe had entered some molecular transformer with issues of _GQ_ and _Pumping Iron_. Aaron waved them to come forward with a hand the size of a manhole cover.\n\n\"Hello, Myron,\" Aaron said. \"A genuine pleasure to see you again.\"\n\nMyron Bolitar, Mr. Popularity. \"Aaron, I'd like you to meet Win Lockwood.\"\n\nAaron angled the smile at Win. \"Pleasure, Win.\" They shook hands with death stares, each sizing the other man up. Neither flinched.\n\n\"They're waiting in the back,\" Aaron said. \"Come on.\"\n\nAaron led them to a locked door with a one-way mirror. The door opened immediately. They entered. Two hoods stood stonefaced. In front of them was a long corridor. There was\u2014and this was new\u2014a metal detector, like at the airport.\n\nAaron shrugged, as if to say, A sign of the times. \"Hand over your weapons, if you'd be so kind. Then step through.\"\n\nMyron took out his thirty-eight, Win a brand-new forty-four. Last night's forty-four had no doubt been destroyed. They stepped through. The metal detector did not ding, but the two hoods still searched with one of those gizmos that looked suspiciously like a vibrator. Then they searched again, this time by hand.\n\n\"Very thorough,\" Win said.\n\n\"Almost enjoyable,\" Myron added. \"I thought he was going to ask me to turn my head and cough.\"\n\n\"Hey, funny man,\" one of the hoods groused, \"this way.\"\n\nThe two hoods took over, escorting them down the corridor. Aaron stayed back and watched. Myron did not like that. The walls were white, the carpet office-orange. Lithographs of the French Riviera lined the walls. The front of Clancy's Tavern looked like a dive; the back like a dentist's office.\n\nTwo other men appeared at the other end of the corridor. They were both carrying guns.\n\nMyron leaned toward Win's ear. \"Uh-oh.\"\n\nWin nodded.\n\nThe two men pointed their guns at Myron and Win. One barked, \"Hey, you, Goldilocks. Get over here.\"\n\nWin looked at Myron. \"Goldilocks?\"\n\n\"I think he means you.\"\n\n\"Oh. The blond hair. I get it now.\"\n\n\"Yeah, Goldie, get your butt down here.\"\n\n\"Later,\" Win said. He moved down the corridor. The two hoods from the metal detector took out their guns. Four men, four guns. Lots of firepower. Not taking any chances after last night.\n\n\"Hands on your head. Let's go.\"\n\nWin and Myron, separated by approximately ten feet, did as they were told. One of the hoods from the metal detector approached Myron. Without warning, he punched the butt of his gun against Myron's kidney.\n\nMyron dropped to his knees. Nausea swam through him. The man followed up with a kick to the ribs. Then another. Myron slid to the ground. The other man joined in. He stomped on Myron's upper legs like they were small brushfires. One stomp landed on the already-sore kidney. Myron thought he was going to vomit.\n\nIn something of a haze Myron spotted Win. He had not moved, his face displaying something akin to noninterest. Win had sized up the situation and made a quick determination: There was nothing he could do to help. Worrying and fretting were worthless. Win was spending his time calmly studying the men. He didn't like to forget a face.\n\nThe kicks came in a nonstop flurry. Myron curled into a fetal position and tried to ride it out. The kicks hurt like hell, but they were too rushed to do serious damage. One landed near his eye. He'd have a shiner for sure.\n\nThen a voice shouted, \"What the hell\u2014 Stop this moment!\"\n\nThe kicks halted immediately.\n\n\"Get away from him!\"\n\nThe men backed off. \"Sorry, Mr. Ache.\"\n\nMyron rolled onto his back. With some effort he managed to sit up. Herman Ache stood by an open door. \"Are you okay, Myron?\"\n\nMyron winced. \"Never better, Herman.\"\n\n\"I can't tell you how sorry I am,\" Herman Ache said. Then glaring at his men. \"But some people will be even sorrier.\"\n\nThe men cowered away from the older man. Myron almost rolled his eyes. This was all an act. Herman Ache's men did not beat up men in Herman's corridor without permission. This had been a setup. Now Myron supposedly owed Herman, even before the negotiating started. Not to mention the fact that pain is a great fear-inducer, the perfect prenegotiation cocktail.\n\nAaron came down the hall. He helped Myron to his feet and sort of half-shrugged as if to say Cheap move, but what can you do?\n\n\"Come,\" Herman beckoned. \"Let's talk in my office.\"\n\nMyron moved tentatively into the office. He had not been here in several years, but not much had changed. Golf was still the theme. LeRoy Neiman painting of some golf course on the main wall. Lots of those stupid cartoon\/artworks of old-fashioned golfers. Aerial photographs of golf courses. In one corner of the office was a movie screen showing a shot of a fairway. In front of the screen was a golf tee. The player hits the ball against the screen. A computer then calculates where it would have landed and changes the image on the screen to match that. Then the player takes his second shot. Fun city.\n\n\"Nice office,\" Win said.\n\nFigures.\n\n\"Thank you, son.\" Herman Ache smiled. Capped teeth. He was in his early sixties, tan, fit, wearing white pants and a yellow golf shirt with a Nicklaus golden bear where an alligator normally went\u2014as if he were on his way to a gin tournament in Miami Beach. Herman Ache had gray hair. Not his own. A toupee or one of those Hair Club systems, a good one, one most people would probably not spot. He had liver spots on his hands. His face was wrinkle free, probably from collagen shots or a face-lift. The neck gave him away. The flesh was baggy and Reaganesque. Looked like a big scrotum.\n\n\"Please, gentlemen, have a seat.\"\n\nThey did so. The door was closed behind them. Aaron, two new hoods, and Herman Ache. Nausea's grip on Myron's stomach began to slacken.\n\nHerman picked up a golf club and sat on the edge of his desk. \"I understand,\" he said, \"that you and Frank are having a misunderstanding, Myron.\"\n\n\"That's what I wanted to talk to you about.\"\n\nHerman nodded. \"Frank?\"\n\nThe door opened. Frank entered. You could tell that they were brothers, both having almost identical facial features, but that was where the similarities ended. Frank had at least twenty pounds on his older brother. He was pear-shaped with small Paul Schaefer shoulders and a rubber tire that would be the envy of the Michelin Man. Frank was completely bald, forgoing the hair weave. His teeth were black with spaces between them. His face was permanently set on angry scowl.\n\nBoth brothers had grown up on the streets. Both had started out as small-time hoods and worked their way up. Both had seen their own children gunned down over the years. Both had gunned down plenty of other people's children. Herman liked to pretend that he dwelled on a loftier plane than his coarse younger brother\u2014a plane of fine books, the arts, golf. But the escape was not that easy. Two sides of the same coin. Frank gratingly reminded Herman of his origins and perhaps true nature. But Frank was comfortable and accepted in his world. Herman was not.\n\nFrank was dressed in a powder blue sweat suit with neon yellow trim. The jacket was unzippered and\u2014taking a fashion tip from Yves St. Aaron\u2014he wore no shirt. His chest hairs were matted with either some type of oil or sweat. Quite a turn-on. The form-fitting pants were a few sizes too small, outlining a bulge in his crotch. Myron started feeling nauseous again.\n\nFrank did not speak. He sat at his brother's desk and waited.\n\n\"Now, Myron,\" Herman continued, \"I understand this is all about some black boy who plays basketball.\"\n\n\"Chaz Landreaux,\" Myron said. \"And I'm not sure he'd be crazy about being called 'boy.' \"\n\n\"Pardon an old man who is not up on all the politically correct terms. I meant no disrespect.\"\n\nWin sat quietly, studying his surroundings.\n\n\"Let me tell you how I see it,\" Herman continued. \"And I'm trying to be objective here. Your Mr. Landreaux made a deal. He took the money. For four years he helped his family with that money. Then when it was time to pay up, he reneged.\"\n\n\"That's objective? Chaz Landreaux is just a kid\u2014\"\n\n\"Spare me the lecture,\" Herman interrupted gently. \"We're not social workers here. You know that. We are businessmen. We made an investment in this young man. We risked several thousand dollars on him. The investment was finally about to pay dividends when you interfered.\"\n\n\"I didn't interfere. He came to me. He's a scared kid. O'Connor got his hooks in him when he was eighteen. There are rules against approaching kids that young for a reason. Now the kid's trying to get out before he slides in too deep.\"\n\nHerman looked skeptical. \"Oh, come on now, Myron. Kids grow up fast nowadays. He knew exactly what he was doing. So it was against the rules\u2014big deal. The kid knew the rules. He wanted the money anyway.\"\n\n\"He'll pay it back.\"\n\nFrank Ache spoke for the first time. \"Fuck he will.\"\n\nMyron waved. \"Hi, Frank. Boss threads.\"\n\n\"And fuck you too, bug shit. Deal's a deal.\"\n\nMyron turned to Win. \"Bug shit?\"\n\nWin shrugged.\n\n\"The deal,\" Myron continued, \"was that Chaz could back out at any time and pay back the money. Roy O'Connor told him that.\"\n\n\"I don't give a fuck what O'Connor said.\"\n\nHerman said, \"Please, Frank, we don't need to get hostile.\"\n\n\"Ah, fuck him, Herman. This asshole wants to fuck me over. He wants to steal food off my fucking table. Not just this Landreaux nigger. That's just the start. We got dozens of prospects signed like this. We lose one, we lose them all. I say we let the other agents know we ain't to be messed with. I say we waste Bolitar right now.\"\n\nMyron said, \"I don't like that idea.\"\n\n\"Who the fuck asked you?\"\n\n\"Just giving my opinion.\"\n\n\"Please, Frank, this isn't helping. You promised to let me handle this.\"\n\n\"Handle what? Kill the son of a bitch. End of story.\"\n\n\"Wait in the other room. I'll take care of it, I promise.\"\n\nFrank glared at Myron. Myron did not bother glaring back. He knew this was part of the act. He knew that they were trying to intimidate him in much the same way Otto Burke and Larry Hanson had. But for some odd reason, the air of death gave the Mutt and Jeff routine a whole new dynamic.\n\nWin, however, remained pensive.\n\n\"Come on, Aaron,\" Frank growled. \"Let's get the fuck out of here.\" He stood. \"But the contract is still on.\"\n\n\"Fine,\" Herman said. \"If you want to kill him, I won't get in the way.\"\n\n\"He's as good as dead.\"\n\nFrank and Aaron left. Frank slammed the door. Overacting, Myron thought, but an effective cameo appearance.\n\nMyron said, \"He's fun.\"\n\nHerman moved to the corner of the room. He took a slow practice swing with the club. \"I wouldn't mess with him, Myron. Frank is really angry. Me, I've always liked you. From the early days. But I'm not sure I can help you on this one.\"\n\nThe \"early days\" had begun Myron's sophomore year at Duke. It was not something he liked to remember. His father had been gambling. And losing. On the day before a game against Georgia State, Myron returned to his dorm to find his father and two of Herman Ache's hoods. The two hoods told Myron that if Georgia State did not cover the twelve-point spread, his father would lose a finger. His father was crying, the first time Myron had ever seen his father cry. Myron made three turnovers in the last forty seconds to make sure Duke won by only ten.\n\nFather and son never talked about it.\n\n\"Why is this kid, this Chaz Landreaux, so important to you, Myron?\"\n\n\"I think he's worth saving.\"\n\n\"Saving from what?\"\n\n\"He's just a kid, Herman. Frank is putting the screws to him. I want it to stop.\"\n\nHerman smiled, changed clubs, took a few more swings. Then he picked up his putter. \"Still a crusader, eh, Myron?\"\n\n\"Hardly. I'm just trying to help the kid.\"\n\n\"And yourself.\"\n\n\"Fine. And myself.\"\n\nMyron realized that Herman Ache was wearing golf cleats. Jesus. To most people golf is an idiotic excuse for a sport. For others it's a life-consuming obsession. There is no in between.\n\n\"I don't think,\" Herman said, reading the break in his carpet, \"I can stop Frank. He's very determined.\"\n\n\"You run the show,\" Myron said. \"Everyone knows that.\"\n\n\"But Frank is my brother. I don't step on his toes unless it's absolutely necessary. I don't think that's the case here.\"\n\n\"What did Frank do to him?\"\n\n\"Pardon?\"\n\n\"How did he scare the kid?\"\n\n\"Oh,\" Herman said. Another club changed. This time he exchanged the putter for a wood. \"He kidnapped his sister. Twin sister, I think.\"\n\nMyron felt his stomach dive anew. They'd been right. Not much satisfaction in that. \"Is she okay?\"\n\n\"Oh, I wouldn't worry,\" Herman said, as if that were a truly foolish question. \"They won't hurt her. Long as Landreaux continues to cooperate.\"\n\n\"When are they going to let her go?\"\n\n\"Two more days. Something about making sure the contract is official and Landreaux doesn't have second thoughts.\"\n\n\"What do you want, Herman? What's it going to cost to get Frank off?\"\n\nHe put on a golf glove and took a very deliberate swing, watching his hands. \"I'm an old man, Myron. A _rich_ old man. What could you possibly give me?\"\n\nWin sat forward, moving for the first time. \"Your club is too far open on your swing, Mr. Ache. Try turning your wrists a little more. Shift your grip to the right a little.\"\n\nThe sudden change in subject caught everyone by surprise. Herman looked at Win. \"I'm sorry. I never caught the name.\"\n\n\"Windsor Home Lockwood III.\"\n\n\"Ah, so you are the immortal Win. Not exactly what I expected.\" He tested the new grip. \"Feels odd.\"\n\n\"Give it a few weeks,\" Win said. \"Do you play often?\"\n\n\"As often as I can. It's more than just a game to me. It's...\"\n\n\"Sacred,\" Win finished for him.\n\nHis eyes livened. \"Exactly. You play, Mr. Lockwood?\"\n\n\"Yes.\"\n\n\"Nothing like it, is there?\"\n\n\"Nothing,\" Win agreed. \"Where do you play?\"\n\n\"Not easy for my kind to find good courses. I joined a club in Westchester. St. Anthony's. You know it?\"\n\n\"No.\"\n\n\"It's not much of a course. Eighteen holes, of course. Very rocky. You have to be half mountain goat.\"\n\nGolf stories. Myron loved them. Didn't everyone?\n\n\"I don't understand something,\" Myron said, playing along. \"With all your, uh, influence, why don't you play anywhere you want?\"\n\nHerman and Win looked at him as though he were a naked infidel praying in the Vatican. \"Excuse him,\" Win said. \"Myron does not understand golf. He thinks a nine iron is a vitamin supplement.\"\n\nHerman laughed. The hoods joined suit. Myron didn't get it.\n\n\"I understand fine,\" Myron said. \"Golf is a bunch of silly-dressed men using massive tracts of real estate to play with a ball and stick.\"\n\nMyron laughed. No one joined suit. Golfers are not known for their sense of humor.\n\nHerman put the club back in the bag. \"A man does not force or buy or bully his way onto a golf course,\" he explained. \"I have too much respect for the game, for the traditions, to do anything so crass. It would be like putting a gun against a priest's head to get the front pew.\"\n\n\"Sacrilege,\" Win said.\n\n\"Exactly. No _real_ golfer would do it.\"\n\n\"He has to be invited,\" Win added.\n\n\"Right. And you don't merely play a great course. You pay homage to it. I'd love to be invited to one of the world's great courses. It would be my dream. But it is not meant to be.\"\n\n\"How about being invited to two of them?\" Win asked.\n\n\"Two\u2014\" Herman stopped. His eyes widened for a millisecond, then quickly dimmed as though afraid he was being teased. \"What do you mean?\"\n\nWin pointed to a picture on the left wall. \"Merion Golf Club,\" he said. Then he pointed to a picture on the far wall. \"And Pine Valley.\"\n\n\"What about them?\"\n\n\"I assume you've heard of them?\"\n\n\"Heard of them?\" Herman repeated. \"They're the top two courses on the East Coast, two of the best in the world. Name a hole. Go ahead, any hole, either course.\"\n\n\"Sixth hole at Merion.\"\n\nHerman's face glowed like a little kid's on Christmas morning. \"One of the most underrated holes anywhere. It sets up with a semiblind tee-shot to a fairway that favors a soft fade. Start your tee-shot at middle bunker, then cut back to the center, keeping clear of the boundary, which comes in on the right. Long-to-middle iron to the modestly elevated green, careful of the bunkers on the left and right.\"\n\nWin smiled. \"Very impressive.\"\n\nSnore.\n\n\"Don't tell me, Mr. Lockwood, that you've played Merion and Pine Valley.\" Something well past awe resonated in Herman's voice.\n\n\"I'm a member of both.\"\n\nHerman inhaled sharply. Myron half-expected him to cross himself. \"A member,\" he began incredulously, \"of both?\"\n\n\"I'm a three handicap at Merion,\" Win continued. \"A five handicap at Pine Valley. And I'd like you to be my guest at both for a weekend. We'll try to get in seventy-two holes a day, thirty-six at each course. We'll start at five A.M. Unless that's too early.\"\n\nHerman shook his head. Myron thought his eyes looked teary. \"Not too early,\" he managed.\n\n\"Next weekend okay for you?\" Win asked.\n\nHerman picked up the phone. \"Let the girl go,\" he said. \"And the contract is off. Anyone touches Myron Bolitar, they're dead.\"\n\n# Chapter 31\n\nWin and Myron went back to the office. Myron felt sore from the beating, but nothing was broken. He would preserve. He was that kind of guy. Terribly brave.\n\nEsperanza said, \"You look like shit.\"\n\n\"You're so hung up on appearances.\"\n\nHe tossed her the photograph of Adam Culver. \"See if your friend Lucy recognizes him.\"\n\nShe snapped a salute. _\"Jawohl, Kommandant.\"_ Of all the old shows, Esperanza's favorite was _Hogan's Heroes._ Myron was not a big fan, though he always wished he could have been there when some young TV hotshot said, \"Hey, I got an idea for a sitcom! Set it in a POW camp in Nazi Germany. Laughs galore.\"\n\n\"How many calls?\" he asked.\n\n\"About a million. Mostly the press wanting your comments on Christian's signing.\" She smiled. \"Nice job on that one.\"\n\n\"Thank you.\"\n\n\"That Otto Burke,\" she said, a pencil near her mouth. \"Is he single?\"\n\nMyron looked at her, horrified. \"Why would you want to know?\"\n\n\"He's kinda cute.\"\n\nThe nausea was back. \"You're hitting me up for a raise, aren't you? Please say yes.\"\n\nEsperanza smiled coyly but said nothing. He started for his office.\n\n\"Hold it,\" she said. \"A strange message just came in for you a few minutes ago.\"\n\n\"From?\"\n\n\"A woman named Madelaine. Wouldn't give her last name. Sounded sultry.\"\n\nThe dean-nessa. Hmm.\n\n\"She leave a number?\"\n\nEsperanza nodded, handed it to him. \"Remember: The condom is your friend.\"\n\n\"Thanks, Mom.\"\n\n\"Speaking of which, your mother called twice, your father once. I think they're worried about you.\"\n\nHe entered his office. His little private sanctuary. He liked it in here. Myron held most of his negotiations and important meetings in the traditionally decorated conference room, freeing him up to make his office whatever he wanted it to be. He had, of course, his view of the Manhattan skyline to his left. On the wall behind his desk he had framed posters from Broadway musicals: _Fiddler on the Roof, The Pajama Game, How to Succeed in Business Without Really Trying, Man of La Mancha, Les Mis\u00e9rables, La Cage aux Folles, A Chorus Line, West Side Story, Phantom._\n\nAnother wall had movie stills: Humphrey Bogart and Ingrid Bergman in _Casablanca_ , Woody Allen and Diane Keaton in _Annie Hall_. Katharine Hepburn and Spencer Tracy in _Adam's Rib_. Groucho, Chico, and Harpo in _A Night at the Opera._ Adam West and Burt Ward in _Batman_ , the TV show, the real Batman, the one where Burgess Meredith played the Penguin and Cesar Romero played the Joker. The Golden Age of Television.\n\nThe final wall had photographs of Myron's clients. In a few days Christian Steele cloaked in Titan blue would join the group.\n\nHe dialed Madelaine Gordon's number. The answering machine picked up. Her silky voice. Hearing it again made his throat dry. He hung up, not leaving a message. He checked the time on the far wall. The clock was shaped like a giant watch with a Boston Celtics insignia in the center.\n\nThree-thirty.\n\nStill time to get to the campus. Madelaine was not important, but Myron very much wanted to see the dean. And he wanted to show up unexpectedly.\n\nAt Esperanza's desk he said, \"I'm going out for a while. You can reach me in the car.\"\n\n\"Are you limping?\" she asked.\n\n\"A little. Ache's men roughed me up.\"\n\n\"Oh. See you later.\"\n\n\"Hurts like hell, but I can take it.\"\n\n\"Uh-huh.\"\n\n\"Don't make a scene.\"\n\n\"Inside,\" she said. \"I'm dying.\"\n\n\"Please see if you can reach Chaz Landreaux. Tell him we need to talk.\"\n\n\"Okay.\"\n\nHe left. He picked up his car in the garage. Win was into cars. He loved his racing-green Jag. Myron drove a blue Ford Taurus. He was not what one might call a car man. A car got him from point A to point B, that was all. It was not a status symbol. It was not a second home. It was not his baby.\n\nThe drive didn't take long. Myron took the Lincoln Tunnel. He passed the famed York Motel. Long sign:\n\n$11.99 PER HOUR  \n$95 PER WEEK  \nMIRRORED ROOMS  \nNOW FEATURING SHEETS!\n\nHe paid the toll on the Parkway. The woman in the booth was very friendly. She almost looked at him when she tossed him the change.\n\nHe called his mother on the car phone and reassured her he was okay. She told him to call his father, he was the worried one. Myron called his father and reassured him he was okay. He told him to call his mother, she was the worried one. Great communication. The secret to a happy marriage.\n\nHe thought about Kathy Culver. He thought about Adam Culver. He thought about Nancy Serat. He tried to draw little lines, connecting them. The lines were tenuous at best. He was sure Fred Nickler, Sir Sleaze Rag, was one line. That picture hadn't sneaked into _Nips_ by itself. Fred seemed to run a tight operation. He had to know more than he was saying. Win was digging into his background, seeing what he could unearth.\n\nHalf an hour later, Myron arrived at the campus. Extra-deserted today. No one on the commons. Very few cars. He parked near the dean's house and knocked on the door. Madelaine (he still liked the name) answered. She smiled when she saw him, clearly pleased, tilting her head a little. \"Well, hello, Myron.\"\n\n\"Hi.\" The Return of Mr. Smooth.\n\nMadelaine Gordon was dressed for tennis. Short white skirt. Great legs. White shirt. He noticed that the shirt was see-through. Keen observation, the sign of a master investigator. Madelaine noticed him noticing. She did not seem particularly offended.\n\n\"I'm sorry to intrude,\" Myron said.\n\n\"No intrusion,\" she said. \"I was just about to take a shower.\"\n\nHmm. \"Your husband's not in, is he?\"\n\nShe crossed her hands under her breasts. \"Not for hours yet,\" she said. \"You got my message?\"\n\nHe nodded.\n\n\"Would you care to come inside?\"\n\nMyron said, \" 'Mrs. Robinson, you're trying to seduce me, aren't you?' \"\n\n\"Pardon me?\"\n\n_\"The Graduate.\"_\n\n\"Oh.\" Madelaine wet her lips. She had a very sexy mouth. People overlook the mouth. They talk about the nose, the chin, the eyes, the cheekbones. Myron was a mouth man. \"I guess I should be offended,\" she continued. \"I mean, I'm not that much older than you, Myron.\"\n\n\"Good point. Quote withdrawn.\"\n\n\"So,\" she said. \"I'll ask again. Would you like to come inside?\"\n\nMyron said, \"Sure.\" Bowling her over with quick wit. What chance did she have against such sparkling repartee?\n\nShe disappeared back into the house, creating an air vacuum that sucked Myron\u2014against his will, of course\u2014in after her. The inside was nice, the kind of house that obviously saw plenty of company. Big open room on the left. Tiffany lamps. Persian rugs. Busts of French guys with long, curly hair. Grandfather clock. Painted portraits of stern-faced men.\n\n\"Care to sit down?\" she said.\n\n\"Thank you.\"\n\n_Sultry_. That had been the word Esperanza used. It fit. Not just Madelaine's voice but her mannerisms, her walk, her eyes, her persona.\n\n\"How about a drink?\" she asked.\n\nHe noticed she already had one made for herself. \"Sure, whatever you're having.\"\n\n\"A vodka tonic.\"\n\n\"Sounds good.\" Myron hated vodka.\n\nShe mixed the drink. He sipped it, trying not to make a face. He wasn't sure if he was successful. She sat down next to him. \"I've never been this forward before,\" she said.\n\n\"That a fact?\"\n\n\"But I'm very attracted to you. It's one of the reasons I loved watching you play. You're really very handsome. I'm sure you're sick of hearing that.\"\n\n\"Well, I don't know if _sick_ is the right word.\"\n\nMadelaine crossed her legs. It wasn't Jessica's leg cross, but it was still worth watching. \"When you came to the door yesterday, I didn't want to miss out on the opportunity. I decided to throw caution to the wind and just go for it.\"\n\nMyron could not stop grinning. \"I see.\"\n\nShe stood and reached out her hand to him. \"Now how about that shower?\"\n\n\"Uh, can we talk first?\"\n\nPuzzlement shadowed her face. \"Is there something wrong?\"\n\nMyron feigned embarrassment. \"Aren't you married?\"\n\n\"And that bothers you?\"\n\nNot really. \"Yes. I guess it does.\"\n\n\"Admirable,\" she said.\n\n\"Thank you.\"\n\n\"Stupid too.\"\n\n\"Thank you.\"\n\nShe laughed. \"Actually, it's sweet. But Dean Gordon and I have what we call a semi-open marriage.\"\n\nHmm. \"Could you elaborate a little?\"\n\n\"Elaborate?\"\n\n\"Just to make me feel more comfortable about all this.\"\n\nShe sat back down. The white skirt might as well not have been there. Her legs could best be described as scrumptious. \"I've never had to elaborate before,\" she said.\n\n\"I realize that. But I'm interested.\"\n\nArched eyebrow. \"In?\"\n\n\"Can we start with your definition of _semi-open_?\"\n\nShe sighed. \"My husband and I have been close friends since childhood. Our parents summered together in Hyannis Port. We were both from the 'right families.' \" She made little quote marks in the air when she said \"right families.\" \"We thought that would be enough. But it wasn't.\"\n\n\"So why not divorce?\"\n\nShe looked a question. \"Why am I telling you this?\"\n\n\"My honest blue eyes,\" he said. \"They're hypnotic.\"\n\n\"Maybe they are.\"\n\nNow Myron gave her aw-shucks modesty. Mr. Adaptable Face.\n\n\"My husband is politically connected. He was an ambassador. He's next in line to be university president. If we get divorced\u2014\"\n\n\"That ends,\" Myron finished.\n\n\"Yes. Even these days, the hint of scandal can destroy a career and a lifestyle. But more than that, Harrison and I are still dear friends. Best friends, really. It's just that we need limited outside stimulation.\"\n\n\"Limited?\"\n\n\"Once every two months,\" she said.\n\nYikes. \"How did you come up with that number?\" he asked. \"Some kind of new algorithm, perhaps?\"\n\nShe smiled. \"Lots of discussions. Negotiations, really. Once a month seemed like too much. Once a semester too little.\"\n\nMyron nodded at her. _Toto, we're not in Kansas anymore_.\n\n\"And we always use a condom,\" she added. \"That's part of the arrangement.\"\n\n\"I see.\"\n\n\"Do you have one?\" she asked. \"A condom.\"\n\n\"On?\"\n\nShe smiled. \"I have some upstairs.\"\n\n\"Can I ask one more thing?\"\n\n\"If you must.\"\n\n\"How do you and your husband know that the other has kept to their, er, limit?\"\n\n\"Easy,\" she said. \"We tell each other. Everything. Helps spice things up a little.\"\n\nMadelaine was seriously strange, which only made her more attractive to Myron.\n\n\"Your husband. Does he ever fool around with co-eds?\"\n\nShe leaned forward and put her hand on his thigh. Upper thigh. Upper, upper thigh. \"That kind of thing turn you on?\"\n\n\"Yeah.\" He tried a rakish smile. But rakish was not him. He could see in her eyes that she wasn't buying it.\n\nMadelaine took back her hand. \"What are you up to, Myron?\" she asked.\n\n\"Up to?\"\n\n\"I feel like I'm being used,\" she said. \"But not the way I had in mind.\"\n\nMan. \"Just getting in the mood.\"\n\n\"I don't think so, Myron.\" She studied him for moment. \"Be honest for a second. Are we going to go to bed?\"\n\n\"No,\" he said. \"We're not.\"\n\n\"I've never been turned down before.\"\n\n\"And I've never turned down a proposition like this before,\" Myron said. \"Come to think of it, I've never had a proposition like this before.\"\n\n\"Is it because I'm married?\"\n\n\"No.\"\n\n\"Are you involved with someone else?\" she asked.\n\n\"Worse. I'm on the cusp of something that means a great deal to me. I don't know which way I'm going to fall. I'm confused.\"\n\n\"That's sweet.\"\n\nAgain he gave her aw-shucks.\n\n\"If it doesn't work out...?\" she said.\n\n\"I'll be back.\"\n\nShe kissed him then. Hard. It was a damn good kiss. He felt it in his toes.\n\n\"Just the overture,\" she said.\n\nHe'd be dead before the second scene. \"I really do have to talk to your husband. Do you know when he'll be home?\"\n\n\"Not for a while. But he's at the office across campus. By himself. You'll have to knock loudly for him to hear you.\"\n\nHe rose. \"Thanks.\"\n\n\"Myron?\"\n\n\"Yes?\"\n\n\"We never use names when we discuss our affairs. I don't know if Harrison fools around with co-eds. I would doubt it highly.\"\n\n\"How about Kathy Culver?\"\n\nShe visibly jumped. Her face stiffened. \"I think you better leave now.\"\n\n\"The honest blue eyes,\" Myron said. \"Watch the honest blue eyes.\"\n\n\"Not this time. And when I watched you play, it wasn't your eyes I looked at.\"\n\n\"Oh?\"\n\n\"Your ass,\" she said. \"It looked nice in those little shorts.\"\n\nMyron felt cheap. Or ecstatic. Probably ecstatic. \"Were they having an affair?\" he asked.\n\nShe said nothing.\n\n\"I'll shake my ass if I have to.\"\n\n\"They weren't having an affair,\" she said firmly. \"That much I know.\"\n\n\"So why did you get all bent out of shape?\"\n\n\"You were asking if my husband had an illicit affair with a co-ed who was probably murdered. I was taken aback.\"\n\n\"Did you know Kathy Culver?\"\n\n\"No.\"\n\n\"Did your husband ever talk about her?\"\n\n\"Not really. I just know she worked in his office.\" She looked at the grandfather clock, stood, and led him to the door. \"Talk to my husband, Myron. He's a good man. He'll tell you everything you need to know.\"\n\n\"Like?\"\n\nShe shook her head. \"Thanks for visiting.\"\n\nMadelaine was in shutdown mode. Probably hurt by his interrogation technique. Using his brawny body to get his way. Myron had never done that before. He liked it. Better than pistol-whipping a suspect, anyway.\n\nHe turned and left. Madelaine was probably watching his ass. He put a little wiggle in his step and hurried across campus.\n\n# Chapter 32\n\nJessica found Getaway Realty in the Bergen County Yellow Pages. Their office was a converted cottage next to a McDonald's off Route 17 on the New Jersey side of the New York\u2013New Jersey border. The drive was only twenty minutes, but it felt as if she'd arrived in the rural past. She actually saw a feed store.\n\nOnly one person was in the office.\n\n\"Well, hello there,\" the man said with a too-wide smile. He was mid-fifties, bald, with a long, scraggly gray beard, like a college professor's. He wore a flannel shirt, black tie, Levi's jeans, and red Chuck Taylor Converse sneakers.\n\n\"I'm Tom Corbett, president of Getaway Realty.\" He handed her a card. \"What can I do for you today?\"\n\n\"I'm Dr. Adam Culver's daughter,\" she said. \"He wrote a check to your office on May twenty-fifth for $649.\"\n\n\"Yeah, so?\"\n\n\"He passed away recently. I'd like to know what it was for.\"\n\nCorbett took a step back. \"I'm awfully sorry to hear that,\" he said. \"Nice man, your father.\"\n\n\"Thank you. Can you tell me why he came to you?\"\n\nHe thought a moment, shrugged. \"Don't see why not. He rented a cabin.\"\n\n\"Near here?\"\n\n\"Five, six miles. In the woods.\"\n\n\"For how long?\"\n\n\"A month. Starting May twenty-fifth. Still has it for a few more weeks, if you'd like to use it.\"\n\n\"What kind of cabin?\" she asked.\n\n\"What kind? Well, it's pretty small. One bedroom, one bathroom with shower stall, living room, kitchenette.\"\n\nThis made no sense. \"Do you think you could give me the directions and a spare key?\"\n\nHe thought that one over too, chewing on the inside of his mouth. \"It's a bit remote,\" he said. \"Kinda hard to find, darling.\"\n\nAside from _babe_ and _honey-bun_ , there were few things Jessica enjoyed being called more than _darling._ But now was not the time to explain her sentiments. She bit her lip and held back.\n\n\"The cottage's away from it all,\" Tom continued. \"Way away, if you know what I mean. A little hunting, a little fishing, but mostly just peace and quiet.\" He picked up a key chain as heavy as a barbell. \"I'll drive you.\"\n\n\"Thank you.\"\n\nHe drove a Toyota LandCruiser and chatted the whole way, as though she were a client. \"Here's our local grocery store.\"\n\nIt was an enormous A&P Superstore.\n\nShe was surprised when he turned onto an unpaved road. They were heading straight into the woods.\n\n\"Nice, ain't it? Real pretty.\"\n\n\"Uh-huh.\"\n\nGreen foliage surrounded them. Jessica was not much of the outdoor sort. To her, the great outdoors meant bugs and humidity and dirt and no running water and no bathroom. Man had evolved for millions of years to escape the woods. Why rush back? But more important, her father had felt the same. He hated the woods.\n\nWhy would he rent a cabin out here?\n\nTom pointed to a gully up ahead. \"Two years ago, guy got killed by a hunter over there. Accident. The hunter thought he was a deer, shot him in the head.\"\n\n\"Uh-huh.\"\n\n\"Couple of dead bodies been found in the woods. Three in the past two years, I think. Found one girl just a couple months back. Runaway, they guessed. Hard to tell 'cause she was all decayed and stuff.\"\n\n\"You're a hell of a salesman, Tom.\"\n\nHe laughed. \"Yeah, well, I can tell when someone ain't a buyer.\"\n\nJessica, of course, knew all about the bodies. The police hadn't caught the killer, but the general consensus was that the psychopath had gotten hold of one more young girl, one that had not yet been found:\n\nKathy Culver.\n\nCould Kathy's fate have been that simple and that horrible? Had she been another victim of a random psychopath, just as everyone thought?\n\nNo, Jessica told herself. Too many holes.\n\n\"When I was a kid growing up around here,\" Tom said, \"these woods were filled with legends. Guy with a hook hand lived in here, the old-timers said, used to kidnap bad little boys and gut them with his hook.\"\n\n\"Charming.\"\n\n\"Sometimes I wonder if he moved on to young ladies.\"\n\nJessica said nothing.\n\n\"Used to call him Dr. Hook,\" he continued.\n\n\"What?\"\n\n\"Dr. Hook. That's what we all called him.\"\n\n\"Isn't that a singer?\" she asked.\n\n\"A what?\"\n\n\"Never mind.\"\n\nThey drove another mile away from civilization. \"That's the house,\" Tom said. \"Up there behind the trees.\"\n\nIt was a small wooden cabin with a big front porch.\n\n\"Rustic, ain't it?\"\n\n_Decrepit_ would have been a better adjective. Jessica checked the porch, but there were no toothless hillbillies playing dueling banjos.\n\n\"Did my father say why he wanted to rent this cabin?\"\n\n\"Just said he needed someplace to get away from it all in these woods.\"\n\nIt still made no sense. Dad was going to be gone at a medical examiners' conference for a week out of the month, anyway. And Adam Culver was not the get-away-from-it-all type. He dealt with the dead. On vacations he wanted to be in Vegas or Atlantic City or someplace with lots of people and action. Now he was renting the Waltons' cabin.\n\nTom used the key to unlock the door. He pushed it open and said, \"After you.\"\n\nJessica stepped into the living room. And stopped short.\n\nTom came in behind her. His voice was a whisper. \"What the hell is this?\" he asked.\n\n# Chapter 33\n\nDean Gordon's office was in Compton Hall. The building was only three stories high but wide. Greek columns out front screamed House of Learning. Brick exterior. White double doors. Directly inside was a bulletin board filled with old notices. Meetings of the usual campus groups: the African American Change Committee, the Gay-Lesbian Alliance, the Liberators of Palestine, the Coalition to Stop the Domination of Womyn (never spelled _women_ , for the sexism the name implies), the South African Freedom Fighters\u2014all taking the summer off. College fun days.\n\nThere was no one inside the huge lobby. The motif was marble. Marble floors, banisters, columns. The walls were covered with huge portraits of men in graduation robes, most of whom would flip if they could read the bulletin board. All the lights were on. Myron's footsteps clacked and reverberated in the still room. He wanted to shout \"Echo,\" but was far too adult.\n\nThe dean of students' office suite was at the end of the left corridor. The door was locked. Myron knocked hard. \"Dean Gordon?\"\n\nShuffling behind the dark-paneled doors. Several seconds later, the door opened. Dean Gordon was wearing tortoiseshell glasses. He had wispy hair, conservatively cut, a handsome face with clear brown eyes. His features were gentle, as though the facial bones had been rounded off to soften his appearance. He looked kind, trustworthy. Myron hated that.\n\n\"I'm sorry,\" the dean said. \"The office is closed until tomorrow morning.\"\n\n\"We need to talk.\"\n\nConfusion crossed his face. \"Do I know you?\"\n\n\"I don't think so.\"\n\n\"You're not a student here.\"\n\n\"Hardly.\"\n\n\"May I ask who you are?\"\n\nMyron looked at him steadily. \"You know who I am. And you know what I want to talk about.\"\n\n\"I don't have the slightest idea to what you are referring, but I am really quite busy\u2014\"\n\n\"Read any good magazines lately?\"\n\nDean Gordon's whole body twitched. \"What did you say?\"\n\n\"I guess I could come back when the office was crowded. Maybe bring some reading material for the school's trustees, though I understand they only read the articles.\"\n\nNo response.\n\nMyron smiled\u2014knowingly. At least, he hoped that was how it looked. Myron had no idea what part the dean played in this little mystery. He had to step tentatively here.\n\nThe dean coughed into his fist. Not a real cough or throat-clear. Just something to stall, give him a chance to think. Finally he said, \"Please come in.\"\n\nHe disappeared back into his office. No sucking vacuum this time, but Myron still followed. They passed a few chairs in the waiting room, a secretary's desk. The typewriter was hidden by a khaki-colored dust cover. Camouflaged in the event of war.\n\nDean Gordon's office was cookie-cut university executive. Lots of wood. Diplomas. Old sketches of the Reston University chapel. Lucite blocks with clippings or awards on the desk. Bookshelves with all nonfiction titles. The books hadn't been touched. They were props, creating the mood of tradition, professionalism, competence. The prerequisite picture of the family. Madelaine and a girl who looked about twelve or thirteen years old. Myron picked up the photograph.\n\n\"Nice family,\" he said. Nice wife.\n\n\"Thank you. Please have a seat.\"\n\nMyron sat. \"Say, where did Kathy work?\"\n\nThe dean stopped in midseat. \"Pardon me?\"\n\n\"Where was her desk?\"\n\n\"Whose?\"\n\n\"Kathy Culver's.\"\n\nDean Gordon lowered himself the rest of the way, slowly, as into a hot tub of water. \"She shared a desk with another student in the room next door.\"\n\nMyron said, \"Convenient.\"\n\nDean Gordon's eyebrows frowned. \"I'm sorry. I missed your name.\"\n\n\"Deluise. Dom Deluise.\"\n\nThe dean allowed himself a small brittle smile. He looked tight enough to pop a wine cork with his butt. No doubt being sent the magazine had put the screws in. No doubt Jake's visit yesterday had tightened them a little. \"What, Mr. Deluise, can I do for you?\"\n\n\"I think you know.\" Again the knowing smile. Combined with the honest blue eyes. If Dean Gordon were female, he'd be naked by now.\n\n\"I'm afraid I don't have the slightest idea,\" the dean said.\n\nMyron continued the knowing smile. He felt like an idiot or a morning network weatherman, if there was a difference. This was an old trick he was trying. Pretend you know more than you do. Get him talking. Play it by ear. Impromptu.\n\nThe dean folded his hands and put them on his desk. Trying to look as if he were in control. \"This whole conversation is very strange. Perhaps you could explain why you're here.\"\n\n\"I thought we should chat.\"\n\n\"About?\"\n\n\"Your English department, for starters. Do you still make students read _Beowulf_?\"\n\n\"Please, whatever your name is, I don't have time for games.\"\n\n\"Neither do I.\" Myron took out his copy of _Nips_ and tossed it on the desk. The magazine was starting to look creased and worn from all the handling, as if it belonged to a hormonal adolescent.\n\nThe dean barely glanced at it. \"What is this?\"\n\n\"Now who's playing games?\"\n\nDean Gordon leaned back, his fingers fiddling with his chin. \"Who are you?\" he asked. \"Really.\"\n\n\"It's not important. I am merely a messenger.\"\n\n\"Messenger for who?\"\n\n\"For _whom_ ,\" Myron corrected. \"Prepositional phrase. And you a college dean.\"\n\n\"I don't need any smart talk, young man.\"\n\nMyron looked at him. \"Get real.\"\n\nThe dean sucked in air as if he were about to plunge underwater. \"What do you want?\"\n\n\"Isn't the pleasure of your company enough?\"\n\n\"This is not a joking matter.\"\n\n\"No, it's not.\"\n\n\"So kindly stop playing games. What do you want with me?\"\n\nMyron tried the knowing smile again. Dean Gordon looked puzzled for a brief moment but then returned the smile. It too was knowing.\n\n\"Or should I say,\" the dean added, \"how much?\"\n\nHe seemed more in control now. He had dealt with the blow and was carrying on. A problem had arisen. But there was a solution. There always was in his world.\n\nMoney.\n\nHe took out a checkbook from his top drawer. \"Well?\"\n\n\"Not that simple,\" Myron said.\n\n\"What do you mean?\"\n\n\"Don't you think someone should pay?\"\n\nHe shrugged. \"Let's talk figures.\"\n\n\"Don't you think this is worth something more than just money?\"\n\nHe looked bewildered, as though Myron had just denied the existence of gravity. \"I don't understand what you mean.\"\n\n\"What about justice?\" Myron asked. \"Kathy is owed. Big-time.\"\n\n\"I agree. And I am willing to pay. But what good is revenge going to do her now? You are the messenger, are you not?\"\n\n\"I am.\"\n\n\"Then go back and tell Kathy to take the money.\"\n\nMyron's heart collapsed. This man, a man who was clearly involved in what had happened that night, believed Myron was a messenger for a living, breathing Kathy Culver. Tread gently, fair Myron. Ever gently.\n\nBut how to play this.\n\n\"Kathy is not happy with you,\" he tried.\n\n\"I meant her no harm.\"\n\nMyron put his hand on his chest and lifted his head dramatically. \"Be thy intents wicked or charitable, thou com'st in such a questionable shape.\"\n\n\"What's that supposed to mean?\"\n\nMyron shrugged. \"I like to work Shakespeare into conversations. Makes me sound smart, don't you think?\"\n\nThe dean made a face. \"Can we return to the matter at hand?\"\n\n\"Sure.\"\n\n\"You say Kathy does not want money.\"\n\n\"Yup.\"\n\n\"What then does she want?\"\n\nGood question. \"She wants the truth to come out.\" Noncommittal, vague, open-ended.\n\n\"What truth?\"\n\n\"Stop playing dumb,\" Myron snapped, feigning annoyance. \"You weren't about to write a check to her favorite charity, were you?\"\n\n\"But I didn't do anything,\" he half-whined. \"Kathy took off that night. I haven't seen her since. How was I supposed to know what to think or do?\"\n\nMyron gave him a skeptical look. He did that because he had no idea what else to do. He was now playing Jake's game, the keep-silent-and-hope-he-ties-his-own-noose game. This worked especially well with political types. They're born with a defective chromosome that will not allow for prolonged silence.\n\n\"She has to understand,\" he continued. \"I did my best. She disappeared. What was I supposed to do? Go to the police? Was that what she wanted? I didn't know anymore. I was thinking of her. She might have changed her mind. I didn't know. I was trying to consider her interests.\"\n\nThe skeptical look came easier after that last sentence. Myron only wished he knew what the hell the dean was talking about. They sat there staring at one another. Then something happened to Dean Gordon's face. Myron wasn't sure exactly what it was, but his whole demeanor seemed to slump. His eyes grew twisted, pained. He shook his head.\n\n\"Enough,\" he said in a quiet voice.\n\n\"What's enough?\"\n\nHe closed the checkbook. \"I won't pay,\" he said. \"Tell Kathy I'll do whatever she wants. I'll stand by her no matter what the cost. This has gone on long enough. I can't live like this. I am not an evil man. She's a sick girl. She needs help. I want to help.\"\n\nMyron had not expected this. \"Do you mean that?\"\n\n\"Yes. Very much.\"\n\n\"You want to help your former lover?\"\n\nHis head shot up. \"What did you say?\"\n\nMyron had been skating blindly on thin ice. His last comment, it seemed, had been something of a blowtorch.\n\n\"Did you say 'lover'?\"\n\nUh-oh.\n\n\"Kathy didn't send you,\" he continued. \"She has nothing to do with you, does she?\"\n\nMyron said nothing.\n\n\"Who are you? What is your real name?\"\n\n\"Myron Bolitar.\"\n\n\"Who?\"\n\n\"Myron Bolitar.\"\n\n\"Are you a police officer?\"\n\n\"No.\"\n\n\"Then what exactly are you?\"\n\n\"A sports agent.\"\n\n\"A what?\"\n\n\"I represent athletes.\"\n\n\"You\u2014 So what do you have to do with this?\"\n\n\"I'm a friend,\" Myron said. \"I'm trying to find Kathy.\"\n\n\"Is she alive?\"\n\n\"I don't know. But you seem to think so.\"\n\nDean Gordon opened his bottom drawer, took out a cigarette, lit it.\n\n\"Bad for you,\" Myron said.\n\n\"I quit smoking five years ago. Or so everyone thinks.\"\n\n\"Another little secret?\"\n\nHe smiled without humor. \"So you were the one who sent me the magazine.\"\n\nMyron shook his head. \"Nope.\"\n\n\"Then who?\"\n\n\"I don't know. I'm trying to figure that out. But I know about it. And now I also know you're hiding something about Kathy's disappearance.\"\n\nHe inhaled deeply and let loose a long stream of smoke. \"I could deny it. I could deny everything we said here today.\"\n\n\"You could,\" Myron countered. \"But of course I have the magazine. I have no reason to lie. And I also have a friend in Sheriff Jake Courter. But you're right. In the end it would be my word against yours.\"\n\nDean Gordon took off his glasses and rubbed his eyes. \"No,\" he said slowly, \"it won't come down to that. I meant what I said before. I want to help her. I _need_ to help her.\"\n\nMyron was not sure what to think. The man looked in genuine pain, but Myron had seen performances that would put Olivier to shame. Was his guilt real? Was his sudden catharsis the result of having a conscience, or was it self-preservation? Myron didn't know. He didn't much care either, as long as he got to the truth.\n\n\"When was the last time you saw Kathy?\" Myron asked.\n\n\"The night she vanished,\" he said.\n\n\"She came to your house?\"\n\nHe nodded. \"It was late. I guess around eleven, eleven-thirty. I was in my study. My wife was upstairs in bed. The doorbell rang. Not once. Repeatedly, urgently. Interspersed with heavy door-pounding. It was Kathy.\"\n\nHis voice was on autopilot, as if he were reading a fairy tale to a child. \"She was crying. Or rather she was sobbing uncontrollably. So much so that she couldn't speak. I brought her into my study. I poured her some brandy and wrapped an afghan around her shoulders. She looked\"\u2014he stopped, considered\u2014\"very small. Helpless. I sat down across from her and took her hand. She jerked it back. That was when the tears stopped. Not slowly, but all at once, as though a switch had been thrown. She became very still. Her face was completely blank, no emotion whatsoever. Then she started talking.\"\n\nHe reached into the drawer for another cigarette. He put it in his mouth. The match lit on the fourth try.\n\n\"She started from the beginning,\" he continued. \"Her voice was remarkably steady. It never cracked or wavered\u2014uncanny, when you consider the fact that she was hysterical just moments earlier. But her words belied her placid tone. She told me stories\u2014\" He stopped again, shook his head. \"They were surprising, to say the least. I had known Kathy for almost a year. I considered her a thoughtful, sweet, proper young woman. I am not making moral judgments here. But she had always been what I considered old-fashioned. And here she was telling me stories that would make a sailor blush.\n\n\"She started by telling me that she used to be everything I always thought she was. The girl next door. Everyone's favorite. But then she changed. She became, in her own words, 'a free-wheeling slut.' She started with some boys in her high school class. But she quickly moved onto bigger things. Adults, teachers, friends of her parents. Biracial, homosexual, two-on-ones, even orgies. She took pictures of her encounters. For posterity, she said with a sneer.\"\n\n\"Did she mention any names?\" Myron asked. \"Of the teachers or adults or anyone?\"\n\n\"No. No names.\"\n\nThey fell into silence. Dean Gordon looked exhausted.\n\n\"What happened next?\" Myron prompted.\n\nHe lifted his head slowly, as though it took great effort. \"Her story began to change direction,\" he said. \"For the better. She said she realized that what she was doing was wrong and stupid. She began, she said, to work through her problems. That was when she met Christian and fell in love. She wanted to put it all behind her, but it wasn't easy. The past wouldn't just go away. She tried and tried, and then...\" His voice trailed off.\n\n\"And then?\" Myron prompted.\n\n\"Then Kathy just looked at me\u2014I'll never forget this\u2014and she said, 'I was raped tonight.' Just like that. Out of nowhere. I was stunned, of course. There were six of them, she said. Or seven, she wasn't sure. A gang-rape in the locker room. I asked her when. She told me it had started less than an hour ago. She had gone to the locker room to meet someone. A blackmailer, she said. A former, uh, suitor, who had threatened to reveal her past. She was going to pay for his silence.\"\n\nThe big cash withdrawal from her trust account, Myron thought.\n\n\"But when she got to the locker room, the blackmailer wasn't alone. Several of his teammates were with him, including another past suitor. They didn't hit her, she said. They didn't beat her. And she didn't fight. There were too many of them, and they were too strong.\" He closed his eyes, his voice a whisper. \"They took turns with her.\"\n\nSilence.\n\n\"As I said before, Kathy told me all this in the most dispassionate tone I had ever heard her use. Her eyes were clear, determined. She told me there was only one way to bury her past. Once and for all. She would have to confront it head-on. She'd have to push it out into the bright sunshine where it would wither and die like a medieval vampire. She said she knew what she had to do.\"\n\nMore silence.\n\n\"What?\" Myron asked.\n\n\"Prosecute the boys who raped her. Face up to her past and then put it behind her. Otherwise it would follow her around for the rest of her life.\"\n\n\"What did you say?\"\n\nDean Gordon winced at the question. He stamped out the cigarette. He glanced down at the bottom drawer but didn't reach for another. \"I told her to calm down.\" He laughed at the memory. \"Calm down. By now, the girl was so unemotional, so detached, that she could have been reading a telephone directory. And I told her to calm down. Jesus.\"\n\n\"What else?\"\n\n\"I told her that I thought she was still in shock. I meant that too. I told her that she should consider everything, weigh all her options, not rush into a decision that would undoubtably affect the rest of her life. I told her to think about what it would mean to have her past dragged out\u2014to her family, to her friends, to her fianc\u00e9, to herself.\"\n\n\"In other words,\" Myron said, \"you tried to talk her out of pressing charges.\"\n\n\"Perhaps. But I never said what I was really thinking: A self-described free-wheeling slut who had gotten involved in pornography and wild sex was going to claim she was raped by a group of college boys, two of whom she admitted having past liaisons with. I wanted her to think about all that before she did something rash.\"\n\n\"Don't be so easy on yourself,\" Myron said. \"You didn't give a damn about her. She came to you for help, and you thought about everything but her. You thought about your precious institution. You thought about the scandal. You thought about the football team on the eve of a national championship. You thought about your own career, how it would come out that she worked for you, how she felt comfortable visiting your house late at night. You'd be tied in. People would investigate you closer, maybe unearth your unusual marital arrangement.\"\n\nThat prodded him upright. \"What about my marital arrangement?\"\n\n\"Does the phrase 'once every two months' mean anything to you?\"\n\nHis mouth dropped open. \"How...?\" He stopped, almost smiled. \"You are a very well-informed young man.\"\n\n\"All-knowing,\" Myron corrected. \"Godlike.\"\n\n\"I won't comment on my marriage, but I would be less than honest if I did not admit that those selfish considerations crossed my mind. But I was also concerned for Kathy. A mistake like this\u2014\"\n\n\"A rape, Dean. Not a mistake. Kathy was raped. She didn't make a 'mistake.' She wasn't the victim of an indiscretion. A bunch of football players pinned her down in a locker room and took turns with her against her will.\"\n\n\"You're simplifying the situation.\"\n\n\"You're the one who simplified the situation. You just put Kathy last.\"\n\n\"That's not true.\"\n\nMyron shook his head. No time for this now. \"So what happened after you bestowed your stellar counsel upon Kathy?\"\n\nHe tried to shrug but couldn't pull it off. \"She looked at me funny, as though I had betrayed her when all I was trying to do was help. Or maybe she saw in my words the same thing you did. I don't know. She stood up then and said that she would be back tomorrow morning to press charges. Then she left. I never heard from her again until that magazine came in the mail. And the phone call a few nights ago.\"\n\n\"What phone call?\"\n\n\"A few nights ago, very late, I got a phone call. A female voice\u2014maybe Kathy's, maybe not\u2014said, 'Enjoy the magazine. Come and get me. I survived.' \"\n\n\" 'Come and get me. I survived'?\"\n\n\"Something like that, yes.\"\n\n\"What did she mean?\"\n\n\"I haven't the slightest idea.\"\n\n\"What did you think when you first heard about Kathy's disappearance?\"\n\n\"That she ran away. Decided it was all too much. I thought she'd come back when she was ready. The police thought that too, until they found her undergarments. Then they suspected violence. But I knew the undergarments were probably from the rape, not the disappearance. So in my mind I still considered her a runaway.\"\n\n\"Didn't the possibility that the rapists wanted to silence her cross your mind?\"\n\n\"It crossed my mind, yes. But these boys weren't capable of\u2014\"\n\n\"Rapists,\" Myron corrected. \" 'Boys' who gang-raped a young girl who never did them any harm. You didn't think they had the capability to commit murder?\"\n\n\"If they wanted her dead, they would never have let her go,\" the dean countered steadily. \"That's what I thought.\"\n\n\"So you kept your mouth shut.\"\n\nHe nodded. \"That was a mistake. I know that now. I was hoping she had just run away for a few days to straighten herself out. When a week passed, I realized it was too late to say anything.\"\n\n\"You chose to live with the lie.\"\n\n\"Yes.\"\n\n\"She was just a student, after all. She came to you for help during the hardest time of her life. And you turned her away.\"\n\n\"Don't you think I know that?\" he shouted. \"Don't you think this has been tearing me apart for the past year and a half?\"\n\n\"Yeah, you're a real humanitarian.\"\n\n\"What the hell do you want from me, Bolitar?\"\n\nMyron stood. \"Resign. Immediately.\"\n\n\"And if I refuse?\"\n\n\"Then I'll drag you down, and it'll be uglier than you ever imagined. First thing tomorrow morning. Turn in your letter of resignation.\"\n\nHe looked up, his fingers supporting his chin. Time passed. His face began to soften as though from a masseur's touch. His eyes closed, and his shoulders slumped. Then he nodded slowly. \"All right,\" he said. \"Thank you.\"\n\n\"This isn't penitence. You don't get off that easy.\"\n\n\"I understand.\"\n\n\"One last thing: Did Kathy mention any names at all?\"\n\n\"Names?\"\n\n\"Of the rapists?\"\n\nHe hesitated. \"No.\"\n\n\"But you have a guess?\"\n\n\"It's not based on anything concrete.\"\n\n\"Go on.\"\n\n\"A few days after she disappeared, I noticed a certain student was tossing around a lot of money. A troublemaker. He bought a new BMW convertible that came to my attention because he drove it across the commons. Ripped up a lot of grass.\"\n\n\"Who?\"\n\n\"An ex-football player. He was kicked off the team for selling drugs. His name was Junior Horton. They call him\u2014\"\n\n\"Horty.\"\n\nMyron left without another word, hurrying to get out of the building. It was a beautiful day. Warm but not humid, the sun weakening in the late afternoon but not quite ready to set. The air smelled of freshly cut grass and blooming cherry blossom trees. Myron wanted to spread out a blanket. He wanted to lie down and think about Kathy Culver.\n\nNo time.\n\nThe phone in his Ford Taurus was ringing when he unlocked the door. It was Esperanza.\n\n\"Dead end with Lucy,\" she said. \"Adam Culver wasn't the guy who bought the pictures.\"\n\nAnother theory blown to hell. He was about to start his car when he heard Jake Courter's voice.\n\n\"Thought I might find you here.\"\n\nMyron looked out the open window. \"What's up, Jake?\"\n\n\"We're about to release Nancy Serat's name to the press.\"\n\nMyron nodded. \"Thanks for letting me know.\"\n\n\"That's not why I'm here.\"\n\nMyron did not like his tone.\n\n\"We also have a suspect,\" Jake continued. \"We've brought him in for questioning.\"\n\n\"Who?\"\n\n\"Your client,\" Jake said. \"Christian Steele.\"\n\n# Chapter 34\n\n\"What about Christian?\" Myron asked.\n\n\"Nancy Serat had just rented that house a week ago,\" Jake replied, \"a day or two before she left for Canc\u00fan. She hadn't even unpacked yet.\"\n\n\"So?\"\n\n\"So how come Christian Steele's fingerprints\u2014clean, fresh prints\u2014are all over the place? On the front doorknob. On a drinking glass. On the fireplace mantel.\"\n\nMyron tried not to looked stunned. \"Come on, Jake. You can't make an arrest on something like that. The press will eat him alive.\"\n\n\"Like I give a flying shit.\"\n\n\"You have nothing.\"\n\n\"We can place him at the scene.\"\n\n\"So what? You can place Jessica at the scene. Gonna arrest her too?\"\n\nJake unbuttoned his jacket, allowing his belly to expand. He was wearing a brown suit, circa 1972. In a word: lapels. No slave to fashion, that Jake. \"Okay, smart-boy,\" he said, \"you want to tell me what your client was doing at Nancy Serat's house?\"\n\n\"We'll ask him. He'll talk to you. Christian's a good kid, Jake. Don't ruin him on speculation.\"\n\n\"Yeah. I'd hate to ruin your commissions.\"\n\n\"Low blow, Jake.\"\n\n\"You're not objective, Bolitar. The kid's your most valuable client, your ticket to the bigs. You don't want him to be guilty.\"\n\nMyron looked at him but said nothing.\n\n\"Leave your car here,\" Jake said. \"I'll drive you to the station.\"\n\nIt was only a mile away. When they pulled into the lot, Jake said, \"The new DA is here. Young hotshot named Roland.\"\n\nUh-oh. \"Cary Roland?\" Myron asked. \"Curly hair?\"\n\n\"You know him?\"\n\n\"Yeah.\"\n\n\"He's a publicity hound,\" Jake said. \"Gets a hard-on watching himself on TV. He practically creamed when he heard Christian's name.\"\n\nMyron could imagine. Old buddies, he and Cary Roland. This was not a good development. \"Has he released Christian's name?\"\n\n\"Not yet,\" Jake said. \"Cary decided to put it off until eleven. Gets a live feed from all the networks that way.\"\n\n\"And plenty of time to tighten the perm.\"\n\n\"That too.\"\n\nChristian was sitting in a small room, no bigger than eight by eight. He sat in a chair behind a desk. No hot lights. No one else was in the room.\n\n\"Where's Roland?\" Myron asked.\n\n\"Behind the mirror.\"\n\nOne-way glass, even in a rinky-dink station like this. Myron stepped into the room, looked in the mirror, adjusted his tie, and refrained from giving Roland the finger. Mr. Mature strikes again.\n\n\"Mr. Bolitar?\"\n\nMyron turned. Christian waved to him as if he'd spotted a familiar face in the stands.\n\n\"You okay?\" Myron asked.\n\n\"I'm fine,\" Christian said. \"I just don't understand what I'm doing here.\"\n\nA uniformed officer came in with a tape recorder. Myron turned to Jake. \"Is he under arrest?\"\n\nJake grinned. \"I almost forgot, Bolitar. You're a lawyer too. Nice to be dealing with a professional.\"\n\n\"Is he under arrest?\" Myron repeated.\n\n\"Not yet. We'd just like to ask him a few questions.\"\n\nThe uniformed officer took care of the preliminaries. Then Jake started.\n\n\"My name is Sheriff Jake Courter, Mr. Steele. Do you remember me?\"\n\n\"Yes, sir. You're handling my fianc\u00e9e's disappearance.\"\n\n\"That's correct. Now, Mr. Steele, do you know a woman named Nancy Serat?\"\n\n\"She was Kathy's roommate at Reston.\"\n\n\"Are you aware that Nancy Serat was murdered last night?\"\n\nChristian's eyes widened. He turned to Myron. Myron nodded. \"My God... no.\"\n\n\"Were you friends with Nancy Serat?\"\n\nHis voice was hollow. \"Yes, sir.\"\n\n\"Mr. Steele, can you tell us where you were last night?\"\n\nMyron interrupted. \"What time last night?\"\n\n\"From the time he left practice till he went to sleep.\"\n\nMyron hesitated. This was a trap. He could try to defuse it, or he could let Christian handle it on his own. Under most circumstances Myron would have stepped in and sounded a subtle warning of what the wrong answer might mean. But this time he sat back and watched.\n\n\"If you want to know if I was with Nancy Serat last night,\" Christian said slowly, \"the answer is yes.\"\n\nMyron breathed again. He looked back at the one-way mirror and stuck out his tongue. The demise of Mr. Mature.\n\n\"What time was that?\" Jake asked.\n\n\"Around nine o'clock.\"\n\n\"Where did you see her?\"\n\n\"At her house.\"\n\n\"The one at 118 Acre Street?\"\n\n\"Yes, sir.\"\n\n\"What was the purpose of your visit?\"\n\n\"Nancy returned from a trip that morning. She called and said she needed to talk to me.\"\n\n\"Did she tell you why?\"\n\n\"She said it had something to do with Kathy. She wouldn't tell me anything else over the phone.\"\n\n\"What happened when you arrived at the house at 118 Acre Street?\"\n\n\"Nancy practically shoved me out the door. She said I had to leave right away.\"\n\n\"Did she say why?\"\n\n\"No, sir. I asked Nancy what was going on, but she was insistent. She promised to call me in a day or two and tell me everything, but for now I had to go.\"\n\n\"What did you do?\"\n\n\"I argued with her for a minute or two. She started getting upset and saying stuff that made no sense. I finally just gave up and left.\"\n\n\"What sort of 'stuff' was she saying?\"\n\n\"Something about sisters reuniting.\"\n\nMyron sat up.\n\nJake asked, \"What about sisters reuniting?\"\n\n\"I don't remember exactly. Something like 'Time for sisters to reunite.' She really wasn't making much sense, sir.\"\n\nJake looked at Myron. Myron looked back.\n\n\"Do you remember anything else she said?\"\n\n\"No, sir.\"\n\n\"Did you go straight home after that?\"\n\n\"Yes, sir.\"\n\n\"What time did you arrive home?\"\n\n\"Ten-fifteen, I guess. Maybe a little later.\"\n\n\"Is there anybody who can confirm the time?\"\n\n\"I don't think so. I just moved into a condominium in Englewood. Maybe a neighbor saw me, I don't know.\"\n\n\"Would you mind waiting here for a minute?\"\n\nJake signaled for Myron to follow him. Myron nodded, leaned over to Christian. \"Don't say another word until I get back.\"\n\nChristian nodded.\n\nThey stepped into the other room. The other side of the mirror, so to speak. County District Attorney Cary Roland had gone to Harvard Law School with Myron. A bright guy. Law review. Clerk for a Supreme Court justice. Cary Roland had first shown signs of political ambition while exiting his mother's womb.\n\nHe looked the same. Gray suit with vest (yes, he'd worn suits to class). Hook nose. Small, dark eyes. Loose curly hair, like a seventies Peter Frampton's, only shorter.\n\nRoland shook his head. Then he made a noise of disgusted belief. \"Creative client, Bolitar.\"\n\n\"Not as creative,\" Myron said, \"as your barber.\"\n\nJake held back a laugh.\n\n\"I say we book him,\" Roland continued. \"We'll announce it at the press conference.\"\n\n\"Now I see it,\" Myron said.\n\n\"See what?\"\n\n\"The hard-on. When you said 'press.' \"\n\nSnickers.\n\nRoland fumed. \"Still a comedian, eh, Bolitar? Well, your client is about to go down.\"\n\n\"I don't think so, Cary.\"\n\n\"I don't care what you think.\"\n\nMyron sighed. \"Christian gave you a reasonable explanation for being at Nancy Serat's house. You got nothing else, ergo you got nothing. Besides, imagine the headlines if Christian's innocent. Young DA Makes Major Blunder. Tarnishes Name of Local Hero for Own Gain. Hurts Titans' Chances for Superbowl. Becomes Most Hated Man in State.\"\n\nRoland swallowed. He hadn't considered that. Blinded by the lights. The TV lights. \"Sheriff Courter, what do you think?\"\n\nBackpedal time.\n\n\"We have no choice,\" Jake said. \"We have to let him go.\"\n\n\"Do you believe his story?\"\n\nJake shrugged. \"Who the hell knows? But we don't have enough to keep him.\"\n\n\"Okay,\" Roland said with a weighty nod. Important man. \"He's free to go. But he better not leave town.\"\n\nMyron looked at Jake. \"Not leave town?\" He laughed. Hard. \"Did he just say not to leave town?\"\n\nJake was trying to hold it in. But his lip was quivering pretty good.\n\nRoland's face turned red. \"Infantile,\" he spat out. \"Sheriff, I want daily updates on this case.\"\n\n\"Yes, sir.\"\n\nRoland gave everyone his most frightening glare. No one fell to their knees. He stormed out.\n\n\"Must be nonstop laughs,\" Myron said, \"working with him.\"\n\n\"Gobs of fun.\"\n\n\"Can Christian and I go now?\"\n\nJake shook his head. \"Not until I hear all about your visit with Dean Gordon.\"\n\n# Chapter 35\n\nMyron filled Jake in. Then he drove Christian home. On the way he filled Christian in too. On everything. Christian wanted to know. Myron wanted to spare him, but he knew he didn't have the right to keep things from him.\n\nChristian did not interrupt with questions. In fact, he said nothing. On the field he was famous for his composure under any situation. Right now, Christian had on his best game face.\n\nWhen Myron finished, neither spoke for several minutes. Then Myron said, \"Are you okay?\"\n\nChristian nodded. His face was pale. \"Thank you for being up-front with me,\" he said.\n\n\"Kathy loved you,\" Myron said. \"Very much. Don't forget that.\"\n\nHe nodded again. \"We have to find her.\"\n\n\"I'm trying.\"\n\nChristian shifted in the car seat so he could face Myron. \"When I was being wooed by all these big agencies, the whole process felt\u2014I don't know\u2014so impersonal. It was all about money. Still is, I know that. I'm not being na\u00efve here, but you were different. I instinctively knew I could trust you. I guess what I'm trying to say is, you've become more than just an agent to me. I'm glad I chose you.\"\n\n\"Me too,\" Myron said. \"This might not the best time to ask, but how did you hear about me in the first place?\"\n\n\"Someone gave you a glowing recommendation.\"\n\n\"Who?\"\n\nChristian smiled. \"You don't know?\"\n\n\"A client?\"\n\n\"No.\"\n\nMyron shook his head. \"I have no idea.\"\n\nChristian settled back in his seat. \"Jessica,\" he said. \"She told me your life history. About your playing days, your injury, what you went through, how you worked for the FBI, how you went back to school. She said you were the best person she knew.\"\n\n\"Jessica doesn't get out much.\"\n\nThey fell back into silence. The New Jersey Turnpike had a center-lane closure, slowing them down to a crawl. Should have taken the western spur. Myron was about to change lanes when Christian said something that almost made him slam on the brakes.\n\n\"My mother once posed in the nude.\"\n\nMyron thought he'd heard wrong. \"What?\"\n\n\"When I was a little kid. I don't know if they were ever printed in a magazine or anything. I doubt it. She wasn't very attractive by then. She was twenty-five but looked sixty. She worked as a prostitute in New York. On the streets. I don't know who my father was. She figured he was one of the guys at a bachelor party, but she had no idea which one.\"\n\nMyron sneaked a glance at him. Christian stared straight ahead. The game face was still on.\n\n\"I thought your mother and father raised you in Kansas,\" Myron said carefully.\n\nChristian shook his head. \"Those were my grandparents. My mom died when I was seven. They legally adopted me. We had the same last name, so I just pretended they were my real parents.\"\n\nMyron said, \"I didn't know. I'm sorry.\"\n\n\"Don't be. They were wonderful parents. I guess they made a lot of mistakes with my mom, the way she ended up and all. But they were kind and loving to me. I miss them a lot.\"\n\nThe silence was heavier now. They drove past the Meadowlands. Myron paid the toll at the end of the turnpike and followed the signs to the George Washington Bridge. Christian had bought a place two miles before the bridge, six miles from Titans Stadium. A set of three hundred prefab condos loftily labeled Cross Creek Pointe, one of those New Jersey housing developments that looked like something out of _Poltergeist._\n\nAs they cruised to a stop, the car phone rang. Myron picked it up.\n\n\"Hello?\"\n\n\"Where are you?\"\n\nIt was Jessica.\n\n\"In Englewood.\"\n\n\"Take Route four west to seventeen north,\" she said quickly. \"I'll meet you in the Pathmark parking lot in Ramsey.\"\n\n\"What's going on?\"\n\n\"Just meet me there. Now.\"\n\n# Chapter 36\n\nThe moment Myron saw Jessica standing in the dusky glow of the Pathmark fluorescent parking lights, looking achingly beautiful in a pair of hip-hugging blue jeans and a red blouse open at the throat, he knew there was trouble. Big trouble.\n\n\"Very bad?\" he asked her.\n\nShe opened the car door and slid in next to him. \"Worse.\"\n\nHe couldn't help it. He couldn't stop thinking of how beautiful she was. She looked a little pale, her eyes a bit too sunken. She did not have crow's-feet quite yet, but new lines had etched their way into her face. Had they been there yesterday or the day she visited his office? He wasn't sure. But he thought she had never looked so devastating. The imperfections, if you wanted to call them that, just made her more real and hence more desirable. Myron had thought Dean-nessa Madelaine was attractive, but she was nary a penlight next to Jessica's blinding beacon.\n\n\"Want to tell me about it?\"\n\nShe shook her head. \"I'd rather just show you.\" She started giving directions. When they reached a road appropriately called Red Dirt Path, she said, \"My father rented a cabin out here.\"\n\n\"In these woods?\"\n\n\"Yes.\"\n\n\"When?\"\n\n\"Two weeks ago. He had it for the month. According to the realtor, he wanted some peace and quiet. A place to get away from it all.\"\n\n\"Doesn't sound much like your father,\" Myron said.\n\n\"Not like him at all,\" she agreed.\n\nA few minutes later, they arrived at the cabin. Myron had a hard time believing that Adam Culver, a man he had gotten to know fairly well during his time with Jessica, would want to vacation out here. The man liked to gamble. He liked the ponies, the roulette wheel, the blackjack table. He liked action. His idea of a quiet time was a Tony Bennett concert at the Sands.\n\nJessica got out of the car. Myron followed. Her posture was arrow-perfect. So was the walk, something Myron had always loved to watch in the past. But there was an unmistakable teeter in her step, as though her legs were not sure they could sustain the lovely torso over the long haul.\n\nTheir footsteps creaked on the steps of the wooden porch. Myron spotted plenty of dry rot. Jessica unlocked the front door and pushed it open.\n\n\"Take a look,\" she said.\n\nHe did. He said nothing. He could feel her eyes on him.\n\n\"I checked his charge card,\" she said. \"He spent over three thousand dollars at a place in the city called Eye-Spy.\"\n\nMyron knew the store. This was definitely their handiwork. Three videocameras were sprawled across the couch. Panasonic. All with mounting material, so they could be hung up somewhere. There were also three small television monitors. Also Panasonic. The kind you might see at a high-rise's security station. Two VCRs. Toshiba. Lots of cables and wires and stuff like that.\n\nBut that stuff wasn't the most bothersome thing he saw. Alone, those electronic goods could have meant one of several things. But two other items\u2014items that drew Myron's eye and held it like a baby near a shiny coin\u2014changed everything. They were the added catalyst. They completed a mixture that was far too noxious to be ignored.\n\nPropped against the wall was a rifle. And on the floor next to it, a set of handcuffs.\n\nJessica said, \"What the hell was he doing?\"\n\nHe knew what she was thinking. The dead girls found near here. The television images of their battered, decayed bodies hovered above them like the most haunting of ghosts.\n\n\"When did he buy this stuff?\" Myron asked.\n\n\"Two weeks ago.\" Her eyes were clear, controlled. \"Listen, I've had time to think about this. Even if our worst fears are true, it doesn't explain anything. What about the picture in the magazine? Or Kathy's handwriting on that envelope? Or the phone calls? Or for that matter his murder?\"\n\nMyron looked at her. He knew she was seeking an explanation\u2014any explanation but the one that stared them straight in the face. \"Are you okay?\" he asked.\n\nShe crossed her arms under her breasts, a hand on each elbow, as if she were hugging herself. \"I feel,\" she said, \"unanchored.\"\n\n\"Can you take more?\"\n\nHer hands dropped to her sides. \"Why? What is it?\"\n\nHe hesitated.\n\nShe exploded. \"Goddamn it, don't coddle me!\"\n\n\"Jess\u2014\"\n\n\"You know I hate that protect-the-little-lady bullshit of yours! Tell me what the hell is going on!\"\n\n\"Kathy was gang-raped by some of Christian's teammates on the night she disappeared.\"\n\nJessica looked as if she'd just been slapped with an open hand. Myron reached out. \"I'm sorry,\" he said.\n\n\"Just tell me what happened. Everything.\"\n\nHe did. Her clear, controlled eyes went blank, lifeless. She remained uncharacteristically silent.\n\n\"Bastards,\" she managed. \"The goddamn bastards.\"\n\nHe nodded.\n\n\"One of them killed her,\" she said. \"Or all of them. To shut her up.\"\n\n\"It's possible.\"\n\nShe paused, thinking. Then the eyes came back to life. \"Suppose,\" she began slowly, \"that my father learned about the rape.\"\n\nMyron nodded.\n\n\"What would he do?\" she continued. \"How would you react\u2014if it was your daughter?\"\n\n\"I'd be enraged,\" Myron replied.\n\n\"Would you be able to control yourself?\"\n\n\"Kathy is not my daughter,\" he said. \"And I'm still not sure I can control myself.\"\n\nJessica nodded. \"So maybe, just maybe, that explains this whole setup. The electronics, the cuffs, the rifle. Maybe he was using this hideaway, deep in the woods, so he could grab a rapist and exact a little private justice.\"\n\n\"Kathy was gang-raped. There were six of them. This place looks built for one.\"\n\n\"But,\" she continued with the hint of an eerie smile, \"suppose my father was in the exact same position we are in now.\"\n\n\"I don't follow.\"\n\n\"Suppose he knew the name of only one rapist. Maybe this Horton guy. What might he do then? What might _you_ do then?\"\n\n\"I might,\" Myron said, \"kidnap him and make him tell.\"\n\n\"Exactly.\"\n\n\"But it's a hell of a reach. Why would I videotape it? Why would I need cameras and monitors?\"\n\n\"Tape the confession, make sure no one comes down the road, I don't know. You have a better scenario?\"\n\nHe did not. \"Have you gone through the rest of the house yet?\"\n\n\"I didn't have a chance. The realtor brought me here. He practically burst a blood vessel when he saw this stuff.\"\n\n\"What did you tell him?\"\n\n\"That I knew all this was here. That my father was a private investigator working undercover.\"\n\nMyron made a face.\n\n\"Hey, it was the best I could come up with.\"\n\n\"And he bought that?\"\n\n\"I think so.\"\n\nMyron shook his head. \"I thought you were a writer.\"\n\n\"I'm not good with spur-of-the-moment. I'm a lot better with the written than the oral.\"\n\n\"Based on past experience,\" he said, \"I'd have to disagree.\"\n\n\"Nice time,\" she said, \"for a come-on.\"\n\nHe shrugged. \"Just trying to keep things loose.\"\n\nShe almost smiled.\n\n\"Let's look around,\" he said.\n\nThere wasn't much to search. The living room had no drawers or closets. Everything was in plain view\u2014the electronic equipment, the handcuffs, the rifle. The kitchenette held no surprises. Same with the bathroom. That left the bedroom.\n\nIt was small. The size of a guest bedroom at a beach house. The double bed took up almost the entire room. There were reading lights on either side of the bed, attached to the wall because there was no room for night tables. No dressers either. The bed was made with flannel sheets. They checked the closet.\n\nBingo.\n\nBlack pants, black T-shirt, black sweatshirt. And worst of all, a black ski mask.\n\n\"Ski mask in June?\" Myron said.\n\n\"He might have needed it to kidnap Horton,\" she tried. But her tone would not make the leap.\n\nMyron dropped to the floor and looked underneath the bed. He saw a plastic bag. He stretched out his hand, grabbed it, and dragged it along the dust-blanketed floor toward him. The bag was red. The initials BCME were emblazoned across the front.\n\n\"Bergen County Medical Examiner,\" Jessica explained.\n\nIt looked like one of those old Lord and Taylor's bags, the kind that snapped closed on the top. Myron pulled it back. The bag opened with a pop. He pulled out a pair of gray no-frills sweat pants with a drawstring. Then he reached back in and withdrew a yellow pullover with the letter T in red. Both were covered with caked-on dirt.\n\n\"Recognize these?\" he asked.\n\n\"Just the yellow sweater,\" she said. \"It's my dad's old varsity sweater from Tarlow High School.\"\n\n\"Funny thing to hide under a bed up here.\"\n\nJessica's eyes lit up. \"Nancy's message! Jesus Christ, she said my dad told her all about Kathy's yellow sweater.\"\n\n\"Whoa, slow down a second. What did Nancy say exactly?\"\n\n\"She said\u2014and I quote verbatim\u2014'He told me all about that favorite yellow sweater he gave Kathy. Such a sweet story.' Those were her exact words. My father never wore it. Kathy did. Like a nightshirt or kick-around-the-house shirt.\"\n\n\"Did your dad give it to her?\"\n\n\"Yes.\"\n\n\"So how did he get it back?\"\n\n\"I don't know. I imagine it was in her personal belongings at school.\"\n\n\"Which doesn't explain why he asked Nancy Serat about it. Or why it's hidden under his bed.\"\n\nThey stood in silence.\n\n\"We're missing something here,\" she said.\n\n\"Maybe your father saw something in these clothes we can't see yet.\"\n\n\"What do you mean?\"\n\n\"I don't know,\" Myron admitted. \"But these clothes were clearly significant to him. Maybe he found them somewhere unusual. Or maybe the police found them.\"\n\n\"But Kathy was wearing blue the night she left. That's been established.\"\n\nMyron remembered the testimony of the sorority sisters and the photograph. But then again...\n\n\"One way to check on that.\"\n\n\"How?\"\n\nHe ran out to the car. Darkness had finally laid claim on the long summer day. He turned on the phone, hoping they weren't too far out of a calling area. Three of those little bars lit up. Enough for the phone to work. He tried Dean Gordon's office. It rang twenty times. No answer. He tried the dean's house. It was picked up on the third ring.\n\nDean Gordon said, \"Hello?\"\n\n\"What was Kathy wearing when she came to your house?\" No need for identification or pleasantries.\n\n\"Wearing? A blouse and skirt of some kind.\"\n\n\"What color?\"\n\n\"Blue. I think the blouse was ripped a bit.\"\n\nMyron hung up.\n\nJessica said, \"Back to square one.\"\n\nMaybe, Myron thought. But the flash of an image seared across his mind. He couldn't grasp it, couldn't even make out what it was exactly. But it had been there, and it would come back.\n\n\"Let's go,\" she said softly, taking his hand. The car light provided enough illumination to see the look in her eyes. They were beautiful eyes, so light colored they were almost yellow. \"I want to get away from here.\"\n\nHe closed the car door, feeling suddenly choked up. The car light went out, basking them in darkness. He couldn't see her face anymore. \"Where do you want to go?\"\n\nFrom the darkness he heard her voice. \"Someplace,\" she said, \"where we can be alone.\"\n\n# Chapter 37\n\nThey found a high-rise Hilton in Mahwah.\n\nMyron checked them in to the best available suite. Jessica stood next to him. The hotel concierge swung his line of vision from Myron to Jessica, eyeing her lustily and Myron jealously. A formal affair was in full swing in the lobby. Men in tuxes, women in long gowns. But every man stared agog at Jessica, who was dressed in jeans and a button-down red blouse.\n\nMyron was used to it. When they were first together, he had taken an almost perverse pleasure in seeing men stare, the familiar you-look-but-I-touch-ha-ha school of macho sneering. But then he started seeing things in the looks that weren't there, and the even more familiar male insecurity burrowed through his rationality.\n\nJessica was practiced at this. She knew how to ignore the looks without looking cold, bothered, or interested.\n\nTheir room was on the sixth floor. They had barely closed the door when they kissed. Jessica's tongue circled and gently darted, making his whole body spasm helplessly. He began to unbutton her blouse. His mouth went dry. He actually gasped when he saw her again. Breathlessness made him heady. He cupped a warm breast, feeling the delicious weight in his hand. She moaned into his mouth.\n\nThey moved to the bed.\n\nTheir lovemaking had always been intense, all-consuming, but this was somehow more animalistic, needier, and yet more tender.\n\nLater, much later, Jessica sat up, kissed him gently on the cheek. \"That,\" she said, \"was awesome.\"\n\nMyron shrugged. \"Not bad.\"\n\n\"Not bad?\"\n\n\"For me. For you it was awesome.\"\n\nShe swung her legs out of bed and slipped into a hotel robe. \"I did enjoy myself,\" she said.\n\n\"Sounded like it.\"\n\n\"I was a tad noisy, huh?\"\n\n\"The Who in concert is a tad noisy. You were loud.\"\n\nShe stood above the bed, smiling. The robe was tied loosely, showing plenty of cleavage and legs that were so long, they were almost intimidating. \"I didn't hear you complain.\"\n\n\"How could you,\" Myron said, \"over all your screaming?\"\n\n\"What time is it?\"\n\n\"Midnight.\" He reached for the phone. \"Hungry?\"\n\nShe gave him a look he felt in his toes. Well, not exactly his toes. \"Famished,\" she said.\n\n\"For food, Jess. Food.\"\n\n\"Oh.\"\n\n\"Ever learn about the male's 'time for recovery' in health class?\"\n\n\"Must have been absent that day.\"\n\n\"The three R's. Replenishment, restoration, recuperation.\" He looked at the menu. \"Damn.\"\n\n\"What?\"\n\n\"No oysters.\"\n\n\"Myron?\"\n\n\"Yes.\"\n\n\"There's a hot tub in the bathroom.\"\n\n\"Jess...\"\n\nShe looked at him with who-me innocence. \"We can soak until the food comes. Recuperate. One of the three R's.\"\n\n\"Just soak?\"\n\n\"Just soak.\"\n\nShe had said soak. He was sure of it. Soak. Not soap. But that was how it started. She soaped him back to life. Myron tried to fight it, almost afraid of how good it felt. But he couldn't. Jess toyed with him, pushed him to the edge, let him teeter, then pulled him back. Myron was helpless. Words like _heaven, ecstasy, paradise, ambrosia_ floated through his mind.\n\nTotal surrender.\n\nWith a whispered \"Now,\" she let him go. His nerve endings surged and sang. The white-hot explosion was so powerful, his ears popped. The bright light hurt his eyes.\n\n\"Awesome,\" he managed.\n\nShe lay back, smiling. \"Not bad.\"\n\nThere was a knock on the door. Probably room service. Neither one of them moved.\n\n\"Why don't you get it,\" she said.\n\n\"My legs,\" he said. \"They can't move. I may never walk again.\"\n\nAnother knock.\n\n\"I'm not dressed,\" she said.\n\n\"And what am I, ready for a press conference?\"\n\n\"Bet you'd get good coverage.\"\n\nMyron moaned at the joke.\n\nAnother knock.\n\n\"Come on, Myron. Just throw a towel around your shapely ass and get moving.\"\n\nThe second woman to mention his ass in the same day. Yowzer. He grabbed the bath towel and headed for the door. Another knock.\n\n\"One second.\"\n\nHe opened the door. It wasn't their food.\n\n\"Maid service,\" Win said. \"May I turn down your bed?\"\n\n\"Didn't you see the Do Not Disturb sign?\"\n\nWin glanced at the doorknob. \"Sorry. No speaka da English.\"\n\n\"How the hell did you find us?\"\n\n\"I traced down your charge card,\" he said, as though it were the most natural thing in the world. \"You checked in here at eight twenty-two P.M.\" Win leaned his head in the doorway. \"Hello, Jessica.\"\n\nFrom the bathroom. \"Hi, Win.\" Myron heard her stepping out of the Jacuzzi. The image of water cascading down her naked body came to him like a deep punch.\n\n\"Come on in,\" he grumbled.\n\n\"Thank you.\" Win handed him a manila folder. \"Thought you might want to take a look at this.\"\n\nJessica came in from the bathroom. The robe was tied tighter. She was drying her hair with a towel. \"What's up?\" she asked.\n\n\"The police rap sheet of one Fred Nickler, aka Nick Fredericks,\" Win said.\n\n\"Imaginative alias,\" Myron said.\n\n\"For an imaginative fellow.\"\n\nJessica sat on the bed. \"He's the porno publisher, right?\"\n\nMyron nodded. The rap sheet was not very long. He started with the most recent dates. Traffic violations, two DWIs, one arrest for mail fraud.\n\n\"Nineteen seventy-eight,\" Win said.\n\nMyron skipped down. June 30, 1978. Fred Nickler had been arrested for endangering the welfare of a child. Charges dropped.\n\n\"So?\"\n\n\"Mr. Nickler was involved in kiddie porn,\" Win explained. \"He was only a small-time photographer back then. But he was nabbed with his hand, so to speak, in the cookie jar. More precisely, taking photographs of an eight-year-old boy.\"\n\nJessica said, \"Jesus.\"\n\nMyron remembered their meeting. \" 'Just an honest guy trying to make an honest buck.' \"\n\n\"Indeed.\"\n\nJessica asked, \"Why were the charges dropped?\"\n\n\"Ah,\" Win said, pointing a finger in the air, \"that's where things get interesting. In many ways it's not an uncommon story. Fred Nickler was only the photographer. A little fish. The authorities wanted the bigger fish. The little fish ratted out the big fish in exchange for leniency.\"\n\n\"And they dropped the charges completely?\" Myron said \"Not even a misdemeanor?\"\n\n\"Not even. It seems that Mr. Nickler also agreed to help out the police from time to time.\"\n\n\"So what's the significance?\"\n\n\"This entire arrangement was negotiated between Nickler and the officer in charge of the investigation,\" Win said. He shot a quick glance at Jessica.\n\n\"The officer in charge of the investigation was your friend Paul Duncan.\"\n\n# Chapter 38\n\n\"That's our man,\" Win said. \"Mr. Junior Horton.\"\n\nHorty looked like an ex\u2013football player. Big and wide, all veins and bulges. His arms looked like corded wood. He was dressed for a rap video. His button-down St. Louis Cardinals baseball shirt was untucked. His baggy shorts reached down past his knees. No socks. Black Reebok high-tops. A Chicago White Sox baseball cap. Dark sunglasses and lots of jewelry.\n\nIt was nine in the morning. One Hundred Thirty-second Street in Manhattan. The street was quiet. Horty was making a drug deal. He had been in and out of jail plenty of times, his one long stint of freedom during his time at Reston U. Drugs, mostly. Armed robbery, once. Two sexual assault charges. Twenty-four years old and a complete punk. Like most inmates he had spent his prison time lifting weights. Pumping iron. Our penal institutions develop violent men's physical strength, so when they get out, they'll be able to intimidate and maim with far greater skill. Nice system.\n\nJessica was not with them. She was packing her father's office\u2014that is, the morgue\u2014and checking for any additional bombshells. Myron had managed to talk her out of confronting Paul Duncan until they knew a little more. She listened grudgingly, but that was how Jessica usually listened anyway.\n\nHorty finished the transaction with a kid who looked no older than twelve, slapped him five, headed west. He wasn't wearing a Walkman, but he walked as though he were. Very jittery. His eyes were red. Every few steps he would snort the air and wipe his nose with the back of his hand.\n\n\"Boys and girls, can you say 'Cokehead'?\"\n\n\"Probably has the flu,\" Win said.\n\n\"The Colombian strain.\"\n\nThey ducked out of sight as he approached. When Horty reached the lip of the alley, Myron stepped in front of him.\n\n\"Junior Horton?\"\n\nHorty gave him a scornful street glare. \"Who the fuck wants to know?\"\n\n\"Snappy comeback,\" Myron said.\n\n\"Get the fuck out of my way or I kick your ass.\" He spotted Win. \"Both your ass.\"\n\n\"Asses,\" Win corrected. \"One ass. Two asses. Plural.\"\n\n\"What the fuck\u2014\"\n\n\"We want to talk to you,\" Myron said.\n\n\"Hey, fuck you, man.\"\n\nMyron turned to Win. \"He's a real badass.\"\n\n\"Indeed,\" Win said. \"I may wet myself.\"\n\nHorty stepped toward Win. He had at least six inches and sixty pounds on him. Horty probably thought he was being clever, going after and intimidating the little guy. Myron tried not to smile when Horty spat, \"Gonna fuck you up big-time.\"\n\n\"If you curse again,\" Win said in the tone of a preschool teacher, \"I will be forced to silence you.\"\n\n\"You?\" Horty laughed heartily. He flexed for a moment and then lowered his nose until it almost touched Win's. Win did not move. \"Little piece of upper-crust whitebread gonna shut me up? Fuck\u2014\"\n\nWin barely moved. His arm shot up, delivered a palm strike to the solar plexus, and was back at his side in what seemed like a tenth of a second. Horty stumbled back, gasping, unable to get any oxygen into his lungs.\n\n\"I asked you not to curse,\" Win said.\n\nIt took Horty nearly half a minute to recover. When he did, the lips started flapping again. \"Fucking cheap-shot motherfucker,\" he said rising. \"I gonna tear you a brand-new asshole.\"\n\nHe charged Win, his arms outstretched as though tackling a fullback. Win sidestepped him and delivered a quick roundhouse kick, again hitting the solar plexus. Horty folded and went down. His face was a mixture of fury, pain, surprise, and of course, embarrassment. He looked around to make sure nobody was watching. He was, after all, getting his butt whipped by Mr. Wonderbread.\n\n\"There are two hundred and six bones in the body,\" Win said evenly. \"Next time I break one.\"\n\nBut Horty wasn't listening. His eyes bulged. Rage twisted his face\u2014not to mention his limited ability to reason. Horty stood, stumbling, pretending he was more hurt than he was. The element of surprise. When Horty was close enough, he made his move.\n\nHe must have been really coked up, Myron mused. Or really stupid. Probably both.\n\nWin leaned away and snapped a sidekick toward Horty's lower leg. There was a cracking sound, like stepping on a dry twig. Horty screamed and went down. Win raised his leg for an ax kick, but Myron stopped him with a shake of his head.\n\n\"Two hundred five,\" Win said, lowering his foot gently, \"and counting.\"\n\n\"You broke my f\u2014\" He stopped, holding his leg and rolling back and forth. \"You broke my leg!\"\n\n\"Your right tibia,\" Win corrected.\n\n\"Who the\u2014who are you?\"\n\nMyron said, \"We're going to ask you a few questions. You're going to answer them.\"\n\n\"My leg, man. I need a doctor.\"\n\n\"When we're finished.\"\n\n\"Look, I just work for Terrell. He gave me this territory. You gotta a problem with that, you speak to him, okay?\"\n\n\"We don't want to talk to you about that.\"\n\n\"Please, man, I'm begging you. My leg.\"\n\n\"You used to attend Reston University.\"\n\nA surprised look replaced the pained one. \"Yeah, so? You want my r\u00e9sum\u00e9?\"\n\n\"You knew Kathy Culver.\"\n\nPanic now. \"You guys cops?\"\n\n\"No.\"\n\nSilence.\n\n\"You knew Kathy Culver.\"\n\n\"Kathy who?\"\n\nWin said, \"Number two-oh-five. The left femur. The femur is the largest bone in the body\u2014\"\n\n\"Okay, I knew her. So what?\"\n\n\"How did you meet?\" Myron asked.\n\n\"At a party. Her first week of school.\"\n\n\"Did you ever date?\"\n\n\"Date?\" Horty laughed at that one. \"No. She wasn't the kind you date.\"\n\n\"What kind was she?\"\n\n\"The kind who sucked off my Johnson first night. Willie's too.\"\n\n\"Who is Willie?\"\n\n\"My roomie.\"\n\n\"He play football?\"\n\n\"Yeah.\" Then he added, \"But only special teams,\" as if that made him a lower species of being.\n\n\"Go on.\"\n\n\"Man, why you want to hear this?\"\n\n\"Go on.\"\n\nHorty shrugged. The leg was swelling badly, but the coke was numbing the pain enough to keep him going. \"You see, we had this party. At Moore House. Where all the brothers lived. Kathy, she was like the only white chick there. So she comes in dressed like a prime-time ho. I mean, she was all that, you know? We start rapping and shit, you know. Did a little nose-candy like a Hoover vac. She liked the stuff. Then we start slow-dancing.\" The grin returned with the memory. \"Grinding, you know. She put her hand on the Black Blade right there on the dance floor. Starts rubbing it and shit. So I take her upstairs, and she sucks me off. But that ain't all. She takes a camera\u2014a fucking camera!\u2014out of her bag and asks me to take pictures. No shit! Close-ups, she wants, of her and the Black Blade.\"\n\nMyron's stomach began to churn again. Win looked on with his usual noninterest.\n\nHorty continued. \"Next night, she come back. Takes on me and Willie at the same time. We take more pictures, have a good old time. 'Cept this time I had my camera too.\"\n\n\"So you took some pictures of your own.\"\n\n\"Shit, yeah.\"\n\n\"Did you and Kathy have any more, uh, encounters?\"\n\n\"Nope. She moved on to other dudes, though. Primelooking babe for such a ho. All blond and built and shit.\"\n\n\"You talk to her after that?\"\n\nHe shrugged. \"Little. Not much. But once she started up with Christian, man, it was a whole other story.\"\n\n\"What do you mean?\"\n\n\"She be all nose up in the air, like her shit don't stink no more. Two of them all lovey-dovey and shit, like they was going steady on a TV show. All of a sudden the slut thinks she's some fucking pure-ass cherry. I mean, the ho been riding the Blade like a fucking bronco, and now she don't even say how-do. That ain't right. That just ain't right.\"\n\nMr. Etiquette.\n\n\"So you decided to blackmail her,\" Myron said.\n\n\"No way. Unh-unh.\"\n\n\"We know about it, Horty. We know she paid you for the pictures.\"\n\nHorty made a snorting sound. \"Aw, shit, that ain't blackmail. That's a business transaction. I just called her one day and told her I might have to knock her down a few pegs. And then I said a picture was worth a thousand words. She kinda agreed with all that and said she'd be willing to pay for such wonderful pictures. I told her they was real valuable to me. Had a lot of sentimental value and shit. But we finally reached an agreement. A mutually beneficial agreement,\" he stressed, \"not blackmail.\" He took hold of his leg and winced. \"End of story, man.\"\n\n\"You left something out.\"\n\n\"What?\"\n\n\"The gang-rape in the locker room.\"\n\nHe did not seem surprised. He half-smiled and said, \"Rape? Man, you ain't listening. This woman had Horty's Three H's: Hot, Horny, Ho. Shit, she'd jump naked into a rock pile if she thought there'd be a snake in it. She loved it. We all had a good time.\"\n\nWin looked at Myron. The look said Keep your cool.\n\n\"How many of you?\" Myron asked.\n\n\"Six.\"\n\n\"Why,\" he said in a low voice, \"didn't you just take the money, Horty? Why did you have to rape her?\"\n\n\"I just told you, man\u2014\"\n\n\"She didn't come to that locker room for consensual sex with six people. You raped her.\"\n\n\"Can't be, man,\" he said with a shake of his head. \"She a ho through and through. And once a ho, always a ho. That just the way it is. Fucking cunt acting all high and prissy and shit. Quarterback's girl. Miss fucking all-American cheerleader. Who the fuck did she think she was? So yeah, I showed her. I reminded her where she come from, what she really is. Not some fucking prom queen. A slut. A dick-loving ho.\"\n\nWin now stepped in front of Myron. Preventive measure.\n\n\"'Sides,\" Horty continued, \"I owed her boyfriend. Big-time.\"\n\n\"Christian Steele?\"\n\n\"Yeah. He did me wrong. I did him wrong. Passed around his little ho. Just a little payback, my man. To the prick who got me thrown off the team.\"\n\n\"No,\" Myron said. \"It wasn't Christian.\"\n\n\"What you talking about?\"\n\n\"I spoke with Coach Clarke. Two guys showed up for a game high. That's why you were thrown off. Christian had nothing to do with it.\"\n\n\"Oh,\" Horty said with a shrug. \"Ain't that something.\"\n\n\"Your remorse,\" Myron said, \"is very touching.\"\n\n\"I gotta get to a doctor, man. My leg is killing me.\"\n\n\"Weren't you worried about getting caught?\"\n\n\"What?\"\n\n\"Weren't you afraid she'd report the rape?\"\n\nHorty made a face as if Myron had suddenly started speaking Japanese. \"You crazy, man? Who she gonna tell? She just gave me major cash to keep it all quiet. She say anything, it all gets out. The whole ugly truth. Everyone would know\u2014Christian, her mammy, her pappy, her teachers. Everyone would know what she just paid all that money to hide. And what if she was dumb enough to tell? There were pictures and witnesses of her doing Willie and me at the party. Who gonna believe she was raped after seeing that?\"\n\nDean Gordon had made the same argument, Myron remembered. Great minds thinking alike.\n\n\"Hey, look, man, my leg's killing me.\"\n\n\"Did you ever see Kathy again?\" Myron managed.\n\n\"Nope.\"\n\n\"Were you the one who threw away the panties?\"\n\n\"Nope. One of the other guys had them. Thought he'd keep them as a souvenir. When he heard she was missing, he got scared, threw them away.\"\n\n\"Who?\"\n\n\"I ain't giving names.\"\n\n\"Yes,\" Win said. \"You are.\" He rested his foot against the broken tibia. That was enough.\n\n\"Okay, okay. Like I said, they was six of us. Three brothers, two white dudes, one chink.\"\n\nEqual opportunity rapists.\n\n\"One was the place kicker. Guy named Tommy Wu. Then there was Ed Woods, Bobby Taylor, Willie and me.\"\n\n\"That's five.\"\n\nHorty hesitated. \"Give me a break, man. The other dude was the one who threw away the panties. But he's a friend, man. Still gives me money when I'm down, you know. I can't just give him up. He's big-time.\"\n\n\"What do you mean, big-time?\"\n\n\"Plays pro ball and shit. I can't give you his name.\"\n\nWin put the slightest pressure on the leg. Horty bucked.\n\n\"Ricky Lane.\"\n\nMyron froze. \"The running back for the Jets?\" Dumb question. How many Ricky Lanes who now play pro football went to Reston University?\n\n\"Yeah. Now look, man, that's all I know.\"\n\nWin said to Myron, \"Do you have any other questions for him?\"\n\nMyron shook his head.\n\n\"Then leave,\" Win said.\n\nMyron did not move.\n\n\"I said,\" Win continued, \"leave.\"\n\n\"No.\"\n\n\"You heard what he said. You'll never convict him. He pushes drugs to kids, rapes innocent women, blackmails, steals, whatever, and laughs about it.\"\n\nHorty sat up. \"What the fuck is this?\"\n\n\"Leave,\" Win repeated.\n\nMyron hesitated.\n\n\"Yo, man, I told you everything I know.\" There was a tremble in Horty's voice.\n\nMyron did not move.\n\nHorty shouted, \"Don't leave me alone with this crazy motherfucker!\"\n\n\"Leave,\" Win said.\n\nMyron shook his head. \"No. I'll stay.\"\n\nWin studied Myron. Then he nodded and approached Horty, who was trying to claw away but not getting far.\n\n\"Don't kill him,\" Myron said.\n\nWin nodded. He went to work with the careful precision of a surgeon. His face never changed expression. If he heard Horty's cries, he never showed it.\n\nAfter a short time Myron told him to stop. Reluctantly Win stepped away.\n\nThey left.\n\n# Chapter 39\n\nRicky Lane lived in a New Jersey condo development similar to Christian's. Win waited in the car. As Myron approached the door, he felt rather than heard the bass from Ricky's stereo. It took three rings of the bell and several knocks before Ricky appeared.\n\n\"Hey, Myron.\"\n\nHe was wearing a silk shirt that was either very fashionable or a pajama top. Hard to tell. The shirt was unbuttoned, revealing a well-defined physique. His pants were held up by a drawstring. He was also wearing slippers. Maybe they were pajamas. Or lounging clothes. Or he was trying out for a walk-on role on _I Dream of Jeannie._\n\n\"We need to talk,\" Myron said.\n\n\"Come on in.\"\n\nThe music was deafening and awful. Made Pap Smear sound like Brahms. The motif was sleek modern. Lots of fiberglass. Lots of black and white. Lots of rounded edges. The stereo took up a whole wall. The lights on the equalizer looked like something on _Star Trek_.\n\nRicky flipped the stereo off. The silence was abrupt. Myron felt his chest stop vibrating.\n\n\"So what's up?\" Ricky asked.\n\nMyron tossed him a glass jar. Ricky caught it, looked a question.\n\n\"Pee in it,\" Myron said.\n\n\"What?\"\n\n\"I want you to urinate into this jar.\"\n\nRicky looked at the jar. Then at Myron. \"I don't get it.\"\n\n\"Your new size,\" Myron said. \"You're taking steroids.\"\n\n\"No way, man. Not me.\"\n\n\"Then give me a urine sample. Right now. I'll have it tested at a lab.\"\n\nRicky stared at the jar. He said nothing.\n\n\"Go ahead, Ricky. I don't have all day.\"\n\n\"You're my agent, Myron. You ain't my mother.\"\n\n\"True enough. Are you taking steroids?\"\n\n\"It's none of your business.\"\n\n\"I'll take that as a yes.\"\n\n\"Take it any way you'd like.\"\n\n\"Did Horty sell them to you? Or have you gotten a new supplier since college?\"\n\nSilence.\n\nRicky said, \"You're fired, Myron.\"\n\n\"I'm devastated. Now tell me about raping Kathy Culver.\"\n\nMore silence. Ricky was struggling to look casual, but his body language was all wrong.\n\n\"I know all about it,\" Myron continued. \"Your buddy Horty told all. Nice guy, by the way. A real sweetheart.\"\n\nRicky stumbled back. He put the jar down on a shiny cube that Myron guessed was a table. He turned away. His voice was barely audible. \"I never touched her.\"\n\n\"Bullshit. You and five other guys jumped her in the locker room. You took turns raping her.\"\n\n\"No. That's not how it happened.\"\n\nMyron waited. Ricky buttoned his shirt, his back still facing Myron. He took a CD out of the stereo and tucked it back into its case.\n\n\"I was there,\" Ricky began, his voice low. \"In the locker room. I was stoned. We all were. Stoned out of our minds. Horty had just gotten in a new supply, and...\" He sort of shrugged away the rest of the sentence.\n\n\"It started as a dare, you know. We knew we'd never go through with it. We figured we'd walk right to the edge but never jump. We kept waiting for someone to call it off.\" He stopped again.\n\nMyron said, \"But no one called it off.\"\n\nHe nodded slowly. \"It stopped. But too late. It stopped when it was my turn, and I said no.\"\n\n\"After all the others had gone?\"\n\n\"Yes. I stood there and watched them. I even cheered.\"\n\nSilence.\n\n\"You kept her panties?\"\n\n\"Yes.\"\n\n\"When you heard the police were investigating, you tossed them in that garbage bin.\"\n\nHe faced Myron. \"No,\" he said with something close to a hint of a smile. \"I wouldn't have been stupid enough to leave them on top of a Dumpster. I'd have burned them.\"\n\nMyron considered that for a moment. It was, he thought, an excellent point. \"Then who threw them away?\"\n\nRicky shrugged. \"Kathy, I guess. I gave them to her.\"\n\n\"When?\"\n\n\"Later.\"\n\n\"What time later?\"\n\n\"Around midnight, I think. After it happened... after she left the locker room, it was like someone had given us the antidote. Or like someone turned on the lights, and we finally saw what we'd done. We all went silent and just drifted away. Except Horty. He was laughing like a goddamn hyena, getting more and more stoned. The rest of us went back to our rooms. None of us said one word. I got into bed, for a little while anyway. Then I got dressed and went back out. I didn't have a plan. Not really. I just wanted to find her. Say something to her. I just wanted to... shit, I don't know.\"\n\nHis fingers were playing with his hair, twisting it like a little kid. He looked smaller now. \"I finally found her.\"\n\n\"Where?\"\n\n\"Crossing the campus.\"\n\n\"Where specifically?\"\n\n\"The middle, I guess. On the commons.\"\n\n\"What direction was she walking in?\"\n\nHe thought a moment. \"South.\"\n\n\"Like maybe she was coming from the faculty housing?\"\n\n\"Yes.\"\n\nAfter she left Dean Gordon's, he thought.\n\n\"Go on.\"\n\n\"I approached her. Called out her name. I thought she'd just run away, you know. It was dark and all. But she didn't. She just turned and stared at me. She wasn't scared. She wasn't shaking. She just stood there and stared me down. I said I was sorry. She didn't say anything. I gave her the panties. I told her she could use them as evidence. I even told her I'd testify. I didn't plan on saying that. It just came out. Kathy took the panties and walked away. She never said anything.\"\n\n\"Was that the last time you saw her?\"\n\n\"Yes.\"\n\n\"What was she wearing?\"\n\n\"Wearing?\"\n\n\"When you last saw her?\"\n\nHe looked up, trying to recall. \"Something blue, I think.\"\n\n\"Not yellow?\"\n\n\"No. Definitely not yellow.\"\n\n\"She hadn't changed clothes since the rape?\"\n\n\"I don't think so. No, they were the same clothes.\"\n\nMyron headed for the door. \"You're going to need more than a new agent, Ricky. You're also going to need a good lawyer.\"\n\n# Chapter 40\n\nJake was sitting next to Esperanza in the waiting area. He stood when Myron and Win entered.\n\n\"Got a minute?\"\n\nMyron nodded. \"My office.\"\n\nJake said, \"Alone.\"\n\nWithout a word Win spun and left.\n\n\"Nothing personal,\" Jake said. \"But the guy gives me the creeps.\"\n\n\"Come on in.\" He stopped at Esperanza's desk.\n\n\"Did you reach Chaz?\"\n\n\"Not yet.\"\n\nHe handed her an envelope. \"There's a photograph inside. Bring it to Lucy. See if she recognizes him.\"\n\nEsperanza nodded.\n\nMyron followed Jake into his office. The air conditioning was on full blast. It felt good.\n\n\"So what brings you to the Big Apple, Jake?\"\n\n\"I was over at John Jay,\" he said, \"checking something out.\"\n\n\"The crime lab?\"\n\n\"Yup.\"\n\n\"Find something?\" Myron asked.\n\nJake did not reply. He examined the pictures on the client wall, leaning forward and squinting. \"Heard of some of these guys,\" he said. \"But no superstars up here.\"\n\n\"No, no superstars.\"\n\n\"Nothing like Christian Steele.\"\n\nMyron sat down. He threw his legs up on the desk. \"You still think he killed Nancy Serat?\"\n\nJake did something with his shoulders. Might have been a shrug. \"Let's just say Christian is no longer our main suspect.\"\n\n\"Who is?\"\n\nJake moved away from the client wall. He sat down and crossed his legs. \"I've been poking into Adam Culver's homicide. Found out something interesting. Seems the cops concentrated solely on the murder scene and surrounding neighborhood. No reason for them to check anything else. They were convinced he was a victim of random street violence. I took a different avenue. I canvassed Culver's neighborhood in Ridgewood. Nice town. Real white. No brothers at all. You been there, I assume?\"\n\nMyron nodded.\n\n\"Anyway, I talked to a guy who lives two houses down from the Culvers. He says he was walking his dog on the night in question. He wasn't sure of the time, but he guessed it was eight o'clock or so. Seems he heard a big fight going on at the Culvers' house. Major blowup. He said he'd never heard anything like that before. It was so bad he almost called the cops, but he didn't want to pry. They'd been neighbors for twenty years and all. So he just let it slide.\"\n\n\"Did he know what the fight was about?\"\n\nJake shook his head. \"Nope. Just loud voices. Adam's and Carol's.\"\n\nMyron sat quietly, still leaning back in his chair. Adam and Carol Culver had fought hours before Adam's murder. Myron tried to put it together with what he already knew. For the first time things were beginning to fit.\n\n\"What else do you got?\" Myron asked.\n\n\"On Adam Culver's murder? Nothing.\"\n\nSilence.\n\n\"There were,\" Jake continued, \"a few hairs found at Nancy Serat's murder scene. On the body itself. More specifically, clutched in Nancy's hand.\"\n\nMyron sat up. \"Like maybe she tore them off the killer?\"\n\n\"Maybe,\" Jake said. \"But we checked the hairs at our own facilities and got a confirmation this morning at John Jay. There's no question. The hairs belong to Kathy Culver.\"\n\nMyron felt his flesh turn to cold stone. He couldn't speak.\n\n\"We had some of her hairs on file,\" Jake continued. \"From before. In case we ever found a body or wanted to check a location. Got them from her hairbrush at school. Both labs have done every comparison test conceivable. Neither one has any doubt. They're Kathy's hairs.\"\n\nMyron shook his head. He felt dizzy. Inside his head the Robot from _Lost in Space_ was shouting \"That does not compute!\" over and over again.\n\n\"You have any thoughts on this, Myron?\"\n\n\"Just the same ones you're having.\"\n\nJake nodded. \"What Christian said.\"\n\n\" 'Time for sisters to reunite,' \" Myron quoted.\n\n\"Yup. Kinda takes on a whole new meaning now, don't it.\"\n\n\"But it still doesn't explain anything,\" Myron said. \"Let's assume Kathy Culver is alive. Let's assume that Nancy Serat knows this. Why would Kathy kill her?\"\n\nJake shrugged. \"Sounds to me like Kathy may have gone off the deep end. I mean, first she's got this whole weird past. Then she falls in love with a guy. Then she's blackmailed. Then she's gang-raped. Then the dean turns his back on her. She cracks. Has a breakdown. Runs away. Maybe she tells Nancy Serat, maybe she doesn't. But somehow Nancy finds out. Nancy arranges a reunion\u2014probably a surprise reunion\u2014between sisters. Kathy gets there early. She's not happy about Nancy's surprise.\"\n\n\"So she kills her?\"\n\n\"Could be,\" Jake said. \"Kathy's loony-tunes. She doesn't want to be found. Shit, she probably killed her old man for the same reason. She's nuts. Maybe she wants revenge for some reason. On her father, on her best friend\u2014even on Christian and Dean Gordon and whoever else she sent that nutty magazine to.\"\n\nDidn't feel right to Myron. \"Then what about the big fight between Adam and Carol Culver? How does that fit in?\"\n\n\"Hell if I know,\" Jake said. \"I'm making this shit up as I go along. Maybe the fight was just a coincidence. Maybe ol' Adam was on edge because he was about to meet with his daughter. Maybe the mother knows more than she's saying.\"\n\nMyron thought about it. It was confusing, but the last part made sense. Maybe Carol Culver did know more than she was saying. More than maybe. Myron even had some idea now of what she was hiding.\n\nIt was time to pay Carol Culver a visit.\n\n# Chapter 41\n\nMyron pulled up in front of the familiar Victorian house on Heights Road in Ridgewood. He hesitated. He should have told Jessica about this, but there are things a woman might be more willing to tell a casual acquaintance than a daughter. This might be one of them.\n\nCarol Culver answered the door. She was wearing an apron and those industrial rubber gloves. She smiled when she saw him, but the smile did not reach her eyes. \"Hello, Myron.\"\n\n\"Hello, Mrs. Culver.\"\n\n\"Jessica isn't home right now.\"\n\n\"I know. I wanted to talk to you, if you have a minute.\"\n\nThe smile stayed. But a shadow crossed over the face. \"Come on in,\" she said. \"Can I get you something to drink? Maybe a little tea?\"\n\n\"That would be nice.\"\n\nHe stepped inside. He and Jessica had not visited here often during their time together. A major holiday or two, that was it. Myron never liked the house. Something about it was stifling, as though the air were too heavy for normal breathing.\n\nHe sat down on a couch that was hard as a park bench. The decor was solemn. Lots of religious memorabilia. Lots of madonnas and crosses and gold-leaf paintings. Lots of halos and serene faces looking skyward.\n\nTwo minutes later Carol reappeared, minus the gloves and apron, plus some tea and shortbread cookies. She was an attractive woman. She didn't really look like her daughters, but Myron had seen pieces of her in both of them. Jessica's straight posture. Kathy's shy laugh.\n\n\"So how have you been?\" she asked.\n\n\"Fine, thank you.\"\n\n\"It's been a long time since we've seen you, Myron.\"\n\n\"Yes.\"\n\n\"Are you and Jessica...?\" She feigned embarrassment. She did that a lot. \"I'm sorry. That's none of my business.\"\n\nShe poured the tea. Myron sipped it and nibbled on a cookie. Carol Culver did likewise.\n\n\"Tomorrow's the memorial service,\" she said. \"Adam donated his corpse to a medical school, you know. The spirit was all that mattered to him. The body was worthless tissue. I guess that's part of being a pathologist.\"\n\nMyron nodded, took another sip.\n\n\"Well, I just can't believe this weather,\" she rambled, a distracted smile frozen to her face. \"It's so hot out. If we don't have rain soon, the whole front lawn will be brown. And we just paid to have it reseeded last season\u2014\"\n\n\"The police will be here soon,\" Myron interrupted. \"I thought we should talk first.\"\n\nShe put her hand to her chest. \"The police?\"\n\n\"They'll want to talk to you.\"\n\n\"Me? What about?\"\n\n\"They know about the fight,\" he said. \"A neighbor was walking a dog. He heard you and Dr. Culver.\"\n\nShe stiffened. Myron waited, but she said nothing.\n\n\"Dr. Culver wasn't feeling sick that night, was he?\"\n\nThe color ebbed from her face. She put down her cup of tea and dabbed the corners of her mouth with a cloth napkin.\n\n\"He never intended to go to that medical conference in Denver, isn't that right, Mrs. Culver?\"\n\nShe lowered her head.\n\n\"Mrs. Culver?\"\n\nNo movement.\n\n\"I know this isn't easy,\" Myron said gently. \"But I'm trying to find Kathy.\"\n\nHer eyes remained on the floor. \"Do you really think you can, Myron?\"\n\n\"It's possible. I don't want to give you false hope, but I think it's possible.\"\n\n\"Then you think she might be alive?\"\n\n\"There's a chance, yes.\"\n\nShe finally raised her head. The eyes were wet. \"You do what you have to do to find her, Myron.\" Her voice was surprisingly steady and strong. \"She's my daughter. My baby. She has to come first. No matter what.\"\n\nMyron waited for Carol Culver to continue, but she fell back into silence. After nearly a full minute, Myron said, \"Dr. Culver just pretended he was going to that medical conference.\"\n\nShe took a deep breath and nodded.\n\n\"You thought he'd left that morning.\"\n\nAnother androidlike nod.\n\n\"Then he surprised you here.\"\n\n\"Yes.\"\n\nMyron's soft voice seemed to boom in the room. An antique clock ticked maddeningly. \"Mrs. Culver, what did he see when he arrived?\"\n\nTears began to flow. She lowered her head again.\n\n\"Did he see you,\" Myron continued, \"with another man?\"\n\nNothing.\n\n\"Was the man Paul Duncan?\"\n\nShe lifted her head. Her eyes met his. \"Yes,\" she said. \"I was with Paul.\"\n\nMyron waited again.\n\n\"Adam set a trap,\" she continued, \"and we got caught.\" The words were once again steady and strong. \"He had become suspicious. I don't know how. So he did just what you said\u2014pretended to go to a conference in Denver. He even had me arrange his flights, so I would be sure he was gone.\"\n\n\"What happened when your husband saw you?\"\n\nShaking fingers rubbed her cheeks. She stood, turned away. \"Exactly what you'd expect to happen when a man finds his wife and best friend in bed. Adam went crazy. He'd been drinking pretty heavily, which didn't help matters. He shouted at me, called me horrible names. I deserved that. I deserved a lot worse. He threatened Paul. We tried to calm him down, but of course that was impossible.\"\n\nShe picked up the tea again. Each word was making her a little stronger, making it a little easier to breathe. \"Adam stormed out. I was scared. Paul went after him. But Adam drove off. Paul left after that.\"\n\n\"How long have you and Paul Duncan...?\" His voice just sort of mumbled away.\n\n\"Six years.\"\n\n\"Did anybody else know?\"\n\nHer composure gave way. Not slowly. But as if a small bomb had blown it off her face. She crumbled, weeping freely. A realization came to Myron. He felt his blood freeze.\n\n\"Kathy,\" he whispered. \"Kathy knew.\"\n\nThe sobbing grew more intense.\n\n\"She found out,\" he continued, \"during her senior year.\"\n\nCarol tried to stop her tears, but that took time. Myron remembered how Kathy had worshiped her mother, the perfect woman, the woman who balanced old-fashioned values with a sense of the modern. Carol Culver had been a homemaker and a shop owner. She had raised three beautiful children. She had instilled in her children more than just a sense of what is now popularly called \"family values.\" For her values had been a rigid doctrine that she insisted her children follow. Jessica had rebelled. So had Edward. Only Kathy had been successfully locked in, like a lion kept in too small a cage.\n\nAnd she had finally broken free.\n\n\"Kathy...\" Carol Culver stopped, shut her eyes tightly. \"She walked in on us.\"\n\n\"And that was when she changed,\" Myron finished.\n\nCarol Culver nodded, her eyes still squeezed closed. \"I did that to her. Everything that happened was because of me. God forgive me.\" Then she shook her head. \"No. I don't deserve forgiveness. I don't want it. I just want my baby back.\"\n\n\"What did Kathy do when she saw you two?\"\n\n\"Nothing. At first. She just turned and ran away. But the next day she broke up with her boyfriend Matt. And from there\u2014she made sure I paid for what I'd done. For all the years I'd been a hypocrite. For all the years I lied to her. She wanted to hurt me in the worst way possible.\"\n\n\"She began to sleep around,\" Myron said.\n\n\"Yes. And she made sure I knew all about it.\"\n\n\"By telling you?\"\n\nCarol Culver shook her head. \"Kathy wouldn't talk to me anymore.\"\n\n\"So how did you find out?\"\n\nShe hesitated. Her face was drawn, her skin pulled tight against her cheekbones. \"Photographs,\" she said simply.\n\nSomething else clicked into place. Horty and the camera. \"She gave you photos of herself with men.\"\n\n\"Yes.\"\n\n\"White men, black men, sometimes more than one.\"\n\nHer eyes closed again, but she managed to say, \"And not just men. It started slowly. A couple of nude pictures of her. Like the one in that magazine.\"\n\n\"You saw that same picture before?\"\n\n\"Yes. It even had the name of a photographer stamped on the back.\"\n\n\"Global Globes Photos?\"\n\n\"No. It was something like Forbidden Fruit.\"\n\n\"Do you still have the picture?\"\n\nShe shook her head.\n\n\"You threw them away?\"\n\nShe shook her head again. \"I wanted to destroy them. I wanted to burn them and pretend I'd never seen them. But I couldn't. Kathy was punishing me. Keeping them was a form of penitence. I never told anyone about them, but I couldn't just throw them away. You see that, Myron, don't you?\"\n\nHe nodded.\n\n\"So I hid them in the attic. In an old storage box. I thought they'd be safe there.\"\n\nMyron saw where this was going. \"Your husband found them.\"\n\n\"Yes.\"\n\n\"When?\"\n\n\"A few months ago. He never told me about it. But of course I knew by the way he was acting. I checked the attic. The pictures were gone. Adam assumed that Kathy had hidden them up there. He had no idea she'd sent them to me. Or maybe he did. Maybe that's how he became suspicious of Paul and me. I don't know.\"\n\n\"Do you know what your husband did with those pictures, Mrs. Culver?\"\n\n\"No. They were so awful. So painful to look at. I think Adam destroyed them.\"\n\nMyron doubted it. They both sat in silence for several minutes. Finally Myron said, \"Jessica is going to want to know.\"\n\nCarol Culver nodded. \"You tell her, Myron.\"\n\nShe showed him to the door. He stopped at his car and turned back around. He studied the gray Victorian house. Twenty-six years ago a young family had moved in. They'd put up swings in the backyard and a basketball hoop in the driveway. They'd owned a station wagon, carpooled to Little League and choir practice, attended PTA meetings, hosted birthday parties. Myron could almost see it all happening, like a life insurance commercial playing in his head.\n\nHe slid into his car and drove away.\n\n# Chapter 42\n\nMyron was thinking about threads again.\n\nThreads like Gary Grady. Dean Gordon. Nancy Serat. Carol Culver. Christian Steele. Fred Nickler. Paul Duncan. Ricky Lane. Horty and the thugs. But there was one thread he had overlooked.\n\nOtto Burke.\n\nSuppose Jake was right. Suppose the magazines had been sent out to wreak vengeance or maybe to satisfy some misguided or irrational anger. Either way, it meant that everyone who had received a copy of _Nips_ was in some way connected to Kathy Culver.\n\nExcept Otto Burke.\n\nHow did he fit in? Otto hadn't even known Kathy Culver.\n\nOr had he?\n\nMyron got off Route 4 at the Garden State Plaza Mall and took Route 17 south to Route 3. New Jersey, land of routes. He pulled into the Meadowlands and parked near the Titans' executive offices. He found the general manager's office and asked for Larry Hanson.\n\nHe was let in almost immediately. He quickly explained the reason for his visit.\n\nLarry Hanson watched him without expression. His huge hands were folded on his desk. His neck strained the top button. Larry was about fifty, but he hadn't gone to flab. He looked, Myron thought not for the first time, like Sergeant Rock in the old comic strips. Should have been chewing on a big cigar.\n\nThe office was adorned with trophies. Larry had been named league MVP twice. He'd been All-Pro twelve times. He had been elected into the Football Hall of Fame on the first ballot. There were plenty of his old football photos, from high school through college and into the pros. Black-and-whites and colors. Same crew cut. Same gritty smile. Different poses, including plenty of knee-up, straight-arm favorites from yesteryear.\n\nWhen Myron finished, Larry studied his big hands for a minute, as if they were something he'd never noticed before.\n\n\"Why ask me?\" he said. \"Why don't you ask Otto Burke about the magazine?\"\n\n\"Because he won't tell me.\"\n\n\"And what makes you think I will?\"\n\n\"Because you're not a complete asshole.\"\n\nLarry's mouth twitched toward a smile, but he caught himself. \"Coming from you,\" he said, \"that really means a lot.\"\n\nMyron said nothing.\n\n\"This is important, huh?\"\n\nMyron nodded.\n\nLarry sat back. \"Burke didn't get the magazine in the mail. He heard about it from a private detective.\"\n\nMyron shifted in his chair. \"Otto was having Christian investigated?\"\n\nLarry's tone was flat. \"A man of Otto Burke's unquestionable integrity would never stoop to such a level.\"\n\n\"Under the desk,\" Myron said, \"you're crossing your fingers.\"\n\nAgain the twitch\/smile. \"This doesn't leave this room, Bolitar. You understand?\"\n\n\"Cross my heart.\" Myron motioned such with his hand.\n\n\"Burke has a whole security division,\" Larry explained. \"They poke into everyone on the payroll. Including yours truly. They also have a source network all over the place. The credo is pretty simple: If you got dirt on a Titan, Burke will pay top dollar for it. So one of these sources came across the magazine.\"\n\n\"How?\"\n\n\"I don't know. Maybe he's a steady reader.\"\n\n\"Do you know his name?\"\n\n\"Brian Sanford. A true sleazeball. He works out of Atlantic City. The casino route. Spies on gamblers, that kind of thing. A Titan puts a quarter in a slot machine, he reports it, especially since that whole Michael Jordan thing started. Burke likes to be kept informed. Gives him the edge in negotiating.\"\n\nMyron stood. \"Thanks. I appreciate it.\"\n\n\"Hey, Bolitar. This don't make us buddies or nothing. We talk again, I still hate your guts. You got it?\"\n\nMyron said, \"We're having a warm moment now, aren't we, Larry?\"\n\nHanson leaned his elbows on the desk, pointing a finger at Myron. \"I still think you're a little pissant piece of dog shit. And next time I see you, I'll prove it.\"\n\nMyron spread his arms. \"Come on, Larry. How about that hug now?\"\n\n\"Wiseass.\"\n\n\"Does that mean no?\"\n\n\"Do me a favor, Bolitar.\"\n\n\"Name it, bright eyes.\"\n\n\"Get the fuck out of my office.\"\n\n# Chapter 43\n\nMyron called Brian Sanford. Answering machine. Myron said he had a real big case, one that paid ten grand, and he'd stop by his office tonight at seven o'clock. Brian Sanford would be there. For ten grand, a guy like Sanford would let his mother take a bullet in the gut.\n\nMyron dialed his office.\n\nEsperanza said, \"MB SportReps.\"\n\n\"Did you show Lucy the photo?\"\n\n\"Yep.\"\n\n\"And?\"\n\n\"You found your buyer.\"\n\nMyron said, \"Lucy was sure?\"\n\n\"Positive.\"\n\n\"Thanks.\"\n\nHe hung up. An hour to kill. Myron headed over to the county medical examiner's office\u2014Dr. Adam Culver's old office. Just a hunch, but worth checking out.\n\nThe building was a one-level brick building. Institutional, almost like a small elementary school. The furniture was metal chairs with thin padding, again like a schoolteacher's. The waiting room magazines were pre-Watergate. The tiled floor was worn and yellowed with age, like the \"before\" shot on a Mr. Clean commercial. There was nothing even remotely decorative.\n\n\"Is Dr. Li in?\" he asked the receptionist.\n\n\"I'll buzz her.\"\n\nSally Li was dressed in hospital scrubs, but there was no blood or anything on them. She was Chinese, approaching forty, but she could have passed for much younger. She wore bifocals. A pack of cigarettes was stashed in her front pocket. Cigarettes with a surgeon's gown. Like bowling shoes with a tuxedo.\n\nThey had met a couple of times in the past. Sally Li came to many Culver family functions. She had been Adam's right-hand woman for the past decade. Myron greeted her with a kiss on the cheek.\n\n\"Jessica told me you were looking into Adam's death,\" she said without preamble.\n\nHe nodded. \"Can we talk for a minute?\"\n\n\"Sure.\" She led him to her office. Again, institutional. No personal stuff. Lots of pathology textbooks. A metal desk. Metal chair. A small tape recorder she probably used during autopsies. Her degrees on the wall. She wasn't married, had no children, so there was no picture on the desk. Big ashtray, though. Overflowing.\n\nShe struck a match, lit up, and said, \"How's tricks?\"\n\n\"An MD smoking,\" Myron said. \"Tsk, tsk.\"\n\n\"My patients never complain.\"\n\n\"Good point.\"\n\nShe took a deep drag. \"So what do you want to know?\"\n\n\"Did you and Adam ever have an affair?\"\n\n\"Yes.\" No hesitation. She looked him right in the eye. \"About four years ago. Lasted a week.\"\n\n\"Did Adam have a lot of affairs?\"\n\n\"Got me. A few, I guess. Why do you ask?\"\n\n\"I'm just trying to put a few things together.\"\n\n\"Vis-\u00e0-vis his murder?\"\n\n\"Right.\"\n\nShe took off her glasses. \"What does Adam's love life have to do with it?\"\n\n\"Probably nothing,\" Myron admitted. \"How had Adam been acting the last couple months?\"\n\n\"A bit wacko,\" she said. Again no hesitation.\n\n\"In what way?\"\n\nShe gave that one some thought. \"Businesswise, he wasn't letting me help him on a lot of big cases. He was keeping them all to himself.\"\n\n\"And that was unusual?\"\n\n\"That was unheard of. We always worked on big cases together.\"\n\n\"These cases,\" Myron said. \"Were they the girls found in the woods upstate?\"\n\nShe looked at him. \"You want to tell me how you knew that?\"\n\n\"Just a guess.\"\n\n\"Hell of a guess, Myron.\"\n\n\"You said big cases. I read the papers. Those are the big cases they keep talking about.\"\n\nSally didn't believe him, but she didn't push it either.\n\nMyron said, \"So what else was there?\"\n\nShe took another deep drag. \"He was very distracted. You'd talk to him, he'd nod, but he wouldn't listen.\"\n\n\"Anything else?\"\n\nSally crushed out the cigarette, though it still had plenty to go. She lit another. \"A new way to quit smoking,\" she said. \"I smoke the same amount of cigarettes, but I take less puffs each day. Gradual slowdown until I quit entirely. At this rate it should take no more than twelve years.\"\n\n\"Good luck.\"\n\n\"Thanks.\"\n\n\"So what else was there?\"\n\nAnother puff. \"Adam was ordering a lot of weird tests on the last girl they found in the woods.\"\n\n\"What do you mean, weird tests?\"\n\n\"Superfluous tests. In my opinion, anyway.\"\n\nMyron said, \"You never got a positive ID on her, right?\"\n\n\"Right.\"\n\n\"So maybe he was running the tests to see if he could get a handle on her whereabouts.\"\n\n\"Maybe. But he sent them out one at a time. He'd wait for one test to come back before he'd ask for the next one. Anthropological measurements, shape and size of cranium, pelvic bones, ossification of the bones, fusing of sutures on the skull\u2014all one at a time.\"\n\n\"So what do you make of that?\"\n\nShe shrugged again. \"I don't make anything out of that. It's just an example of what I meant by acting strangely. Distracted. The case was a weird one to start off. The girl's skull had been crushed by the perp, but that wasn't what killed her. In other words, she had been buried alive in those woods. She died trying to claw her way out.\"\n\nSilence.\n\n\"This girl,\" Myron said, \"what was she wearing?\"\n\nSally stiffened a little. Then she leaned forward. \"Okay, Myron, what's going on?\"\n\n\"Nothing. Why?\"\n\n\"You know why.\"\n\nMyron stopped. \"The girl's clothes are missing.\"\n\n\"Yes.\"\n\nHe felt his heart crash into the pit of his stomach, like a skydiver with a ripped parachute. \"Oh, shit.\"\n\n\"What is it?\"\n\n\"Sally, I need you to run a test for me.\"\n\n# Chapter 44\n\nThe address of Brian Sanford, private investigator, was a go-go bar conveniently located one block from Merv Griffin's Resorts. Atlantic City was like that. The big hotels were like beautiful flowers untouched and unbothered by the unseemly weeds of poverty and sleaze that surrounded them. The big flowers had not beautified the neighborhood as promised by the casino owners. The contrast, if anything, had made the weeds more glaringly hideous.\n\nThe go-go bar was called Eager Beaver, and it was exactly what one would expect. Blinking sign with missing letters on the outside. Lots of lowlights around the bar, lots of bright spotlights on the stage. Bored women danced in shifts, most of them unattractive. Lots of flab. Lots of implants. Lots of herpes.\n\nMyron made the key mistake of entering what might loosely be designated a bathroom. The urinals were stuffed with ice cubes\u2014an adequate substitute, Myron supposed, for an actual flushing mechanism. No doors were on the stalls, which did not deter the defecators at all. One man smiled and waved to Myron from a squat.\n\nMyron decided he could wait.\n\nHe called over a bartender. \"Could you tell me how to get to Brian Sanford's office?\"\n\n\"Michelob, Bud, Bud Light, Coors.\"\n\n\"I just want to know\u2014\"\n\n\"Michelob, Bud, Bud Light, Coors.\"\n\nMyron took out five dollars. The bartender pocketed it.\n\n\"Door in the back. Take the stairs up a level.\"\n\nHe didn't wait for Myron to thank him. Capitalism.\n\nA dancer on break approached him. She smiled. Each tooth was angled in a different direction, as if her mouth were the masterwork of a mad orthodontist.\n\n\"Hi,\" she said.\n\n\"Hi.\"\n\n\"You're really cute.\"\n\n\"I don't have any money.\"\n\nShe spun and walked away. Ah, romance.\n\nThe stairs did not creak. They cracked. Myron kept waiting for them to collapse. On the landing there was only one door. It was open. Myron knocked on the wall and peeked in.\n\nMyron called out, \"Hello.\"\n\nA man he assumed was Brian Sanford came to the door. All smiles. Dressed in a beige suit that had last been pressed during the Bay of Pigs. \"You the guy who left the message?\"\n\n\"Yes.\"\n\nThe office was a minicasino. No desk but a roulette table. A one-armed bandit in the corner. Decks of cards everywhere. Souvenir dice, the kind that have a hole drilled in them, littered the floor. So did racing forms. Keno cards too.\n\nThe man put out his hand. \"Brian Sanford. But everyone calls me Blackjack. You know who gave me that nickname?\"\n\nMyron shook his head.\n\n\"Frankie. That's what I call Frank Sinatra. Frankie. Not Frank. Frankie, I call him.\" He paused, waited.\n\nMyron said, \"Good nickname.\"\n\n\"See, Frankie and me were playing at the Sands one night, right, and I was on one of my streaks, you know. And Frankie turns to me and says, 'Yo, check out Blackjack. He can't lose.' Just like that. Frankie says, 'Hey, Blackjack.' Out of nowhere. The name stuck. Now everyone calls me Blackjack. All 'cause of Frankie.\"\n\n\"Great story,\" Myron said.\n\n\"Yeah, well, you know how it is. So what can I do for you, Mr....?\"\n\n\"Olson. Merlin Olson.\"\n\nBlackjack smiled knowingly. \"Okay, I can play it that way. Have a seat, Mr. Olson.\"\n\nMyron sat.\n\n\"But before we start, Mr. Olson, I have to tell you one thing right up front.\"\n\nHe was holding dice in his hand, moving them around in his hands the way some people do with those Chinese balls that are supposed to help circulation.\n\n\"What's that?\"\n\n\"I'm a very busy man. Lots of big stuff going on right now. You know how I started in this business?\"\n\nMyron shook his head.\n\n\"I used to be chief of security for Caesars Palace in Vegas. Head chief. You know how it is. I was in Vegas, right? But Donny\u2014that's what I call Donald Trump, Donny\u2014Donny asked me to head up security for his first hotel on the strip. Then he started nagging me to set up the Taj Mahal's security. I told him, I said, 'Donny, I got too much on my plate, you know?' \"\n\nMyron looked up. A small crop plane flew overhead, leaving mucho cow manure in its wake.\n\n\"So my problem is this, you see. I got a meeting tomorrow morning with Stevie\u2014Steve Wynn. First thing, seven A.M. sharp. Great guy, Stevie. Morning guy. Up at five every day. You know he's practically blind? Got cataracts or something. He keeps it hidden. Only tells his closest friends. So anyway Stevie wants me to do something for him. Normally I'd tell him no, but it's a personal favor and Stevie's a good friend. Not like Donny. I'm not crazy about Donny. Thinks he's some hot stud now that he's got Marla.\"\n\n\"Mr. Blackjack\u2014\"\n\n\"Please,\" he said throwing up his hands, \"just call me Blackjack.\"\n\n\"I'd like to ask you a few questions, uh, Blackjack. I need your particular expertise on an important matter.\"\n\nHe nodded. Very understanding. He didn't hitch up his pants importantly, but he should have. \"What's this all about?\"\n\n\"You performed some work for a friend of mine recently,\" Myron said. \"Mr. Otto Burke.\"\n\nA big smile now. \"Sure. Otto. Swell kid. Smart as a whip. He calls me whenever he comes down.\"\n\nProbably calls him Ottie, Myron thought.\n\n\"You gave him a magazine a few days ago. An issue of _Nips_.\"\n\nBlackjack looked wary now. He rolled the dice on the table. A three. \"What about it?\"\n\n\"We need to know how you located it.\"\n\n\"Who is 'we'?\"\n\n\"I work with Mr. Burke.\" Even saying it made Myron feel nauseous.\n\n\"So why didn't Ken call? He's the usual contact.\"\n\nMyron leaned forward. Conspiratorial. \"This is bigger than Ken, Blackjack. We don't feel anyone can be trusted with this but you.\"\n\nHe nodded. Again very understanding.\n\n\"Frankly, Blackjack\u2014and this has to remain hush-hush.\"\n\n\"Of course.\"\n\n\"You're our first choice to replace Ken. But we know how busy you are.\"\n\nHis eyes gleamed a bit. \"I appreciate that, Mr. Olson, but for someone like Otto Burke, I could try to open\u2014\"\n\n\"Let's talk about this case first, okay? How did you come across the magazine?\"\n\nThe wary look again. \"Don't take this the wrong way,\" he said, \"but how do I know you work with Otto? How do I know you're not some schmo off the street?\"\n\nMyron smiled. \"I knew it.\"\n\n\"What?\"\n\n\"I told Otto you were the right guy for the job. You're not sloppy. You're careful. We like that. We need that.\"\n\nBlackjack shrugged. He picked up the dice, gave another roll. Snake eyes. \"I'm a professional,\" he said.\n\n\"Clearly,\" Myron agreed. \"So why don't you call Otto yourself on the private line? He'll confirm everything. I'm sure you know the number.\"\n\nThat slowed him down a bit. He swallowed, trying to disguise it, looked around like a cornered rabbit. Myron could see the wheels churning. \"Uh, no reason to bother Otto with this,\" Blackjack said. \"You know how he hates that. I can tell you're an honest Joe. Besides, how would you know about the magazine if Otto hadn't told you?\"\n\nMyron shook his head. \"You're an amazing man, Blackjack.\"\n\nHe waved a hand of modesty.\n\n\"How did you find the magazine?\" Myron asked.\n\n\"Shouldn't we talk about my fee? On the phone you said something about ten grand.\"\n\n\"Otto said you were a trustworthy guy. He said to bill him through Ken. Whatever you think is fair.\"\n\nAnother nod. He picked up the dice. Rolled again. Another three. Practice, practice. \"I didn't find the magazine,\" Blackjack said. \"It found me.\"\n\n\"What do you mean?\"\n\n\"I was hired to do a job. Part of it was to send out copies of that mag to some people.\"\n\n\"Was Christian Steele one of those people?\"\n\n\"Yup. That's how I got suspicious. I mean, the envelopes were given to me already addressed and sealed. I didn't recognize any of the names except Christian's. Otto had already put out word he wanted anything, _anything_ , on Steele. So I opened it up and took a peek. That's when I saw the picture.\"\n\n\"Who hired you to mail out the magazines?\"\n\nBlackjack placed one chip on red, one chip on odd. He spun the roulette wheel. \"You wanna put down a couple of chips?\"\n\n\"No. Who hired you?\"\n\n\"Well, that's the weird part. I don't know. I got this big package in the mail with very specific instructions. Plus cash. But no name.\"\n\n\"Any return address?\"\n\n\"Nope. Just a postmark.\"\n\n\"From where?\"\n\n\"Right here in Atlantic City. I got it about ten, twelve days ago.\"\n\nThe roulette wheel stopped. Twenty-two. Black.\n\nBlackjack said, \"Damn.\"\n\n\"Do you still have the instructions?\"\n\n\"Yeah, sure.\" He opened a drawer and handed him a piece of paper. \"Here.\"\n\nThe letter had been typed:\n\n_Dear Mr. Sanford_ ,\n\n_For the sum of $5,000 (plus expenses) I would like you to perform the following services:_\n\n_1. Enclosed find seven envelopes. Two of them should be mailed from the campus mail box at Reston University on Friday. The other three should be mailed from a post office box in their respective towns._\n\n_2. Please mail out the following New Jersey Bell literature to each person on the list at the same time._\n\n_3. Please arrange a phone number in the 201 area code, one that will work on Return Call. This number should be immediately disconnected should anyone call it back or answer it. I would like you to hook up an answering machine with the enclosed tape to that phone. I would then like you to make calls to each of the numbers listed below from that number. On the first two nights\u2014Saturday and Sunday\u2014you will simply call repeatedly, hold the line when they answer, and say nothing until they_ _hang up. On Monday, you will call and say the following: \"Enjoy the magazine. Come and get me. I survived.\" Please make your voice sound female and vague. (As you know, there are phones that can disguise voices and make them sound female.)_\n\n_4. Enclosed is a money order for $3,000. Upon completion of this exercise I will contact you personally on or around the ninth of the month and pay off the remaining $2,000 plus expenses._\n\n_My name must remain anonymous. Thank you for understanding._\n\nMyron looked up. \"I assume the New Jersey Bell literature explained Return Call.\"\n\nNod.\n\n\"Who were the seven people?\"\n\nBlackjack shrugged. The dice were rolled yet again. Another snake eyes. The guy had the touch. \"I don't remember. Christian was one. Some dean was another. I mailed another from a town called Glen Rock.\"\n\n\"To a Gary Grady.\"\n\n\"Yeah, that's the name. I also mailed three from New York.\"\n\n\"One of those to Junior Horton?\"\n\n\"Uh, yeah, I think so. Junior. That rings a bell.\"\n\n\"And the last one?\"\n\n\"Some other place in New Jersey. Near Glen Rock.\"\n\nMyron stopped. \"Ridgewood?\"\n\n\"Yeah. Something-wood anyway. A woman's name. I remember because all the rest were men.\"\n\nMyron said, \"Carol Culver?\"\n\nHe thought a moment. \"Yeah. That's it. A name with two C's.\"\n\nMyron's shoulders slumped.\n\n\"Hey, buddy, you all right?\"\n\n\"Fine,\" he said softly. \"What about the phone calls?\"\n\n\"The numbers were on another page. I threw them away when I finished. I called Steele and hung up a few times. By the time I called him back to give him the message calls, the line was disconnected. Guess he'd moved.\"\n\nMyron nodded. Christian had moved from the campus to the condo.\n\n\"The guy in New York\u2014Junior\u2014he was never home so I never reached him either. The others all got hang-ups and then the message calls.\"\n\n\"How many of them used Return Call?\"\n\n\"Just two. Christian and the guy from Glen Rock. It wouldn't have worked for the guys in New York anyhoo. Return Call only works for that area code.\"\n\n\"Have you heard from your client yet?\"\n\n\"Nope. And yesterday was the ninth. I tell you, he better not stiff Blackjack Sanford.\" Another mental pants-hitch. \"If he knows what's good for him.\"\n\n\"Uh-huh. Anything else you can tell me?\"\n\n\"About this case? Nope. Hey, you wanna go over to Merv's? They know me over there. I can get us on a good table. Play a little blackjack maybe. Watch the legend in action.\"\n\nTempting, Myron thought. Like having electrolysis performed on his testicles. \"Maybe some other time.\"\n\n\"Yeah, okay. Say, how much you think I should bill Otto for? Like you said, I want to be fair.\"\n\n\"Oh, I'd bill him for the full amount.\"\n\n\"The whole ten G's?\"\n\n\"Yes. You've been very helpful, Blackjack. Thank you.\"\n\n\"Yeah, take care. Come by anytime.\"\n\n\"Oh, one more thing.\"\n\n\"What's that?\"\n\nMyron said, \"Mind if I use your bathroom?\"\n\n# Chapter 45\n\nIt was ten-thirty when Myron arrived at Paul Duncan's house. Lights were still on. Myron had not called to make an appointment. He wanted the element of surprise.\n\nThe house was a simple Cape Cod. Nice. Needed a new coat of paint maybe. The front yard had lots of budding flower beds. Myron remembered that Paul liked gardening in his down time. Lot of cops did.\n\nPaul Duncan answered the door holding a newspaper. A pair of reading glasses were low on his nose. His gray hair was neatly combed. He wore navy-blue Hagar slacks and a twist-a-flex Speidel watch. The casual man from Sears. A television played in the background. An audience applauded wildly. Paul was alone, except for a sleeping golden retriever curled in front of the television as if it were a fire on a snowy night.\n\n\"We need to talk, Paul.\"\n\n\"Can't this wait until the morning?\" His voice was strained. \"After Adam's memorial service?\"\n\nMyron shook his head and stepped into the den. The television audience applauded again. Myron glanced at the screen. Ed McMahon's _Star Search._ The spokes-models weren't on, so Myron turned away.\n\nPaul closed the door. \"What's this all about, Myron?\"\n\nA coffee table had _National Geographic_ and _TV Guide_. Also two books\u2014the latest Robert Ludlum and the King James Bible. Everything was very neat. A portrait of the golden retriever in its younger days hung on the wall. Lots of little porcelain figurines adorned the room. A couple of Rockwell plates too. Hardly a swinging bachelor pad or den of lust.\n\n\"I know about your affair with Carol Culver,\" Myron said.\n\nPaul Duncan played stiff-lip. \"I don't know what you're talking about.\"\n\n\"Then let me try to clarify myself. The affair's been going on for six years. Kathy caught you and Mommy a couple years back. Adam also caught you two on the night he was murdered. Any of this ring a bell?\"\n\nHis face went ashen. \"How...?\"\n\n\"Carol told me.\" Myron sat. He picked up the Bible and flipped through it. \"Guess you skipped the part about not coveting your neighbor's wife, huh, Paul?\"\n\n\"It's not what you think.\"\n\n\"What's not what I think?\"\n\n\"I love Carol. She loves me.\"\n\n\"That sounds swell, Paul.\"\n\n\"Adam treated her awfully. He gambled. He whored. He was cold to his family.\"\n\n\"So why didn't Carol divorce him?\"\n\n\"She couldn't. We're both devout Catholics. The Church wouldn't allow it.\"\n\n\"The Church prefers marital infidelity?\"\n\n\"That's not funny.\"\n\n\"No, it's not.\"\n\n\"Who are you to judge us? You think any of this was easy?\"\n\nMyron shrugged. \"You didn't stop. Not even after Kathy saw you.\"\n\n\"I love Carol.\"\n\n\"So you say.\"\n\n\"Adam Culver was my closest friend. He meant a great deal to me. But when it came to his family, he was a bastard. He provided for them materially, but that's it. Ask Jessica, Myron. She'll tell you. I've always been there. From the time she was a little girl. Who took her to the hospital when she fell off her bike? Me. Who built her swingset? Me. Who drove her down to Duke her freshman year? Me.\"\n\n\"Did you also dress up as the Easter Bunny?\" Myron asked.\n\nHe shook his head. \"You don't understand.\"\n\n\"Correction: I don't give a shit. There's a difference. Now let's go back to the day Kathy caught you two. Tell me what happened.\"\n\nHis face became irritated. \"You know what happened. She walked in on us.\"\n\n\"Were you naked?\"\n\n\"What?\"\n\n\"Were you and Mrs. Culver in the throes of passion?\"\n\n\"I won't dignify that with an answer.\"\n\nTime to rattle his cage a bit. \"What position? Missionary, doggie, what? Were either of you wearing handcuffs or a pig's mask?\"\n\nHe moved so he was standing directly over Myron. Everyone thought this was tremendously intimidating, towering over a seated foe. Fact was, Myron could deliver a palm strike to the groin before an ordinary man could even cock his fist.\n\n\"Watch it, son,\" Paul said.\n\n\"How did Kathy react to seeing you two lovebirds?\"\n\n\"There was no reaction. She ran away.\"\n\n\"Did either of you follow her?\"\n\n\"No. Frankly, we were both too shocked.\"\n\n\"I bet. Did you ever discuss the matter with Kathy?\"\n\nPaul stepped away, circled, sat in the chair next to Myron. \"She only mentioned it to me once.\"\n\n\"When?\"\n\n\"A few months later.\"\n\n\"What happened?\"\n\nHe looked away, his eyes darting about, searching for a safe place to land. \"This isn't easy to say.\"\n\nMyron nodded, feigned sympathy. \"Go on.\"\n\n\"Kathy made a pass at me.\"\n\n\"Did you catch it?\"\n\n\"What?\"\n\n\"As in 'catch her pass.' \"\n\nHe flashed the irritated face again. \"Of course not.\"\n\n\"You turned her down?\"\n\n\"I pretended I didn't know what she was talking about.\"\n\n\"Did she persist?\"\n\n\"Yes. But I kept ignoring her.\"\n\n\"Bet you were real excited, though. Mother and daughter. Both good-lookers. Your fantasies must have been in overdrive.\"\n\nIrritation turned to rage. He finally took off his reading glasses. Very dramatically. \"Last warning, pal.\"\n\n\"Uh-huh. So now tell me about Fred Nickler.\"\n\nPiss him off. Quick subject change. Keep him off balance.\n\n\"Who?\"\n\n\"For a cop,\" Myron said, \"you're a lousy liar. Nineteen seventy-eight. You let Nickler plea-bargain a kiddie porn charge. I know all about your connection with him, Paul. What I don't know is how he fits into all this.\"\n\n\"He helped me out from time to time. With cases.\"\n\n\"Including the disappearance of Kathy Culver?\"\n\n\"In a manner of speaking, yes.\"\n\n\"How?\"\n\n\"I guess there's no reason not to tell you.\" He coughed into a shaking fist. The golden retriever opened an eye but didn't move. \"Adam found photographs of Kathy in his attic. He brought them to me in the strictest confidence. On the back of one was the name of a photography studio called Forbidden Fruit. I couldn't find them anywhere. So Adam and I visited Nickler. Nickler told us that Forbidden Fruit was now called Global Globes. He gave me the address.\"\n\n\"Then you went and bought all the pictures and negatives of Kathy?\" A throwaway question. Lucy had already identified Paul Duncan from a photograph.\n\n\"Yes. We wanted to protect Kathy's name. But we also wanted the name of the animal that'd brought Kathy to the studio.\"\n\n\"Gary Grady.\"\n\n\"You know about that?\"\n\n\"I am,\" Myron said, \"well informed.\"\n\n\"Well, I checked Grady out completely. He was shady, no question about it. A high school teacher with all those sex lines. He advertised in at least fifty pornographic magazines. I tailed him for a couple of weeks, did a lot of it on my own time. I also had his phone tapped for a while. But in the end we came up with nothing.\"\n\n\"How did Adam react to that?\"\n\n\"Not well. Adam was always coming to me with some new angle on Kathy's case, mostly out of pure desperation. I don't blame him. She was his youngest daughter. The one child he had a decent relationship with. Adam was willing to do anything to find her. He even wanted to kidnap Grady and torture him until he talked. I told him I'd do anything to help, but that we had to keep within the limits of the law. He didn't like hearing that.\"\n\n\"Tell me about the night Adam died.\"\n\nPaul took a deep breath. \"He set us up beautifully.\"\n\n\"I know all about that. What happened after he caught you and Carol in bed?\"\n\nPaul Duncan rubbed his eyes with his palms. \"He went berserk. He started calling Carol names. Awful names. We tried to talk to him, but what could we say? After a while he told her he wanted a divorce and ran out.\"\n\n\"What did you do then?\"\n\n\"I went home.\"\n\n\"Did you stop on the way?\"\n\n\"No.\"\n\n\"Anybody who can confirm you were home?\"\n\n\"I live alone.\"\n\n\"Anybody who can confirm you were home?\" Myron repeated.\n\n\"No, dammit. That's why Carol and I didn't tell anyone. We knew how it would look.\"\n\n\"Not good,\" Myron agreed.\n\n\"I didn't kill him. I wronged him. I was a terrible friend. But I didn't kill him.\"\n\nMyron gave a small shoulder shrug. \"You seem like a pretty good candidate, Paul. You lied about the night of his murder. You were having a long-term affair with his wife, a wife who could marry you only if her husband died. He confronted you two in his bed on the night of the murder. His missing daughter was the only person who knew about your secret liaison. Her photograph appears in a magazine published by your source. No, Paul, I'd say it looks pretty goddamn shitty.\"\n\n\"I had nothing to do with any of that.\"\n\n\"What did you do with Kathy's pictures?\"\n\n\"I gave them to Adam, of course.\"\n\n\"Did you keep any for yourself? Maybe as a little souvenir?\"\n\n\"Of course not!\"\n\n\"And you never saw any of the pictures again?\"\n\n\"Never.\"\n\n\"Yet somehow Kathy's picture ended up in a porno mag.\"\n\nPaul nodded slowly.\n\n\"A porno mag published by your buddy Fred Nickler.\"\n\nAnother nod.\n\n\"So now comes the big question, Paul: How did Kathy's picture end up in Nickler's magazine?\"\n\nUsing both arms for leverage Paul Duncan stood. He moved to the television and flicked it off. The junior dancers faded away. The dog did not move. Paul studied the blank screen for a while and then said, \"It's going to sound crazy.\"\n\n\"I'm listening.\"\n\n\"Adam arranged it. He put Kathy's picture in that magazine.\"\n\nMyron waited. His spine began to tingle.\n\n\"I don't understand it either,\" Paul continued. \"Nickler called me yesterday. He was all upset, said you were nosing around and realizing something was up. I had no idea what he was talking about. Then he explained it to me. Adam had told Nickler to put that picture in his magazine. You see, Adam had met Nickler when we were trying to find the photographer's studio. So Adam went back to him, pretended he was still working on a case with me. He told Nickler to put Kathy's picture in Gary Grady's ad. He also told him not to say anything if anybody asked about it\u2014except to give out Gary's alias and address.\"\n\n\"Enough clues,\" Myron said, \"so someone would find Grady.\"\n\n\"It seems so, yes.\"\n\n\"Did Nickler tell you why he placed the picture only in _Nips_?\"\n\n\"No. I can call and ask him, if you'd like.\"\n\nMyron shook his head. \"Not necessary.\"\n\n\"That's all I know. I can't for the life of me figure out what Adam was doing. Maybe he wanted to set up Grady. Or maybe he just snapped. But the truth is, I have no idea why Adam would put his own daughter's picture in that magazine.\"\n\nMyron rose. He had a very good idea why.\n\n# Chapter 46\n\nWin gazed into the mirror. Despite the fact that the hour was closing in on midnight, his evening was just beginning. He patted his hair, smiled at his reflection, and said, \"God, I am handsome.\"\n\nMyron grunted.\n\n\"Are you going to call Jessica?\" Win asked.\n\n\"I want to go over it again.\"\n\n\"Now?\"\n\n\"Now.\"\n\n\"And make my nubile lass wait?\"\n\n\"She'll survive.\"\n\n\"You don't understand. This girl is very special to me.\"\n\n\"What's her last name?\"\n\nWin thought a moment, shrugged. \"Okay, what do you wish to review?\"\n\n\"I've told you everything I know,\" Myron said. \"I want to know how you see it.\"\n\nWin turned away from the antique mirror. His Central Park West apartment had been a gift from his grandfather. It was huge, worth millions, and decorated like Versailles. Myron was afraid to touch anything. He was sitting in an antique chair with wooden arms digging into his ribs.\n\n\"Do you mind if I break the case down into three separate entities?\" Win said.\n\n\"Whatever you want.\"\n\n\"Fine. Then let us begin. Entity one: Kathy Culver's disappearance. During her senior year of high school, Kathy's personality changed for reasons her mother has now revealed to you. Kathy then sought to hurt said mother with promiscuity. Ergo the lewd photographs, which Kathy mailed to Carol. But Kathy Culver did not see the danger in her actions. She took for granted that she could just end it whenever she so desired. But that was not the case. When she wanted to stop\u2014when she met Christian, it seems\u2014she could not just backslide out.\"\n\nMyron nodded.\n\n\"Enter Mr. Junior Horton. He decided to cash in on the new, unsullied Kathy Culver through blackmail. Kathy agreed to pay him in exchange for silence and photographs. On the night in question Mr. Horton called Kathy at her sorority house. She agreed to meet him in the locker room. Once there, she was gang-raped by Junior Horton and several cohorts.\"\n\nWin stopped and moved toward a decanter. \"Care for a little cognac?\"\n\n\"No, thanks.\"\n\nHe poured some into a snifter. \"The rape bent her past the breaking point,\" he continued. \"She snapped. She suddenly craved redemption and justice above all else. So she headed immediately to Dean Gordon's office to report the attack. Dean Gordon had been her employer, and she probably considered him a friend. She told him what had happened to her in the locker room. His reaction was either superfluous or detrimental to her resolve. Take your pick.\"\n\n\"Probably detrimental,\" Myron added.\n\n\"Yes, probably detrimental. Either way, Kathy left Dean Gordon's house disheartened. She walked around the campus in a sort of catatonic daze, I imagine. Ricky Lane approached her. He apologized and gave her the panties\u2014that is, evidence of the crime against her. After that\u2014who knows? We slam into a big brick wall. The only thing we know for sure is that the panties were found on top of a waste bin several days later. Are there any questions so far?\"\n\nMyron shook his head.\n\n\"Then let's move on to Entity two: Adam Culver's involvement. Sometime after Kathy disappears, her father finds the lewd photographs of his little princess in the attic. We know that they were hidden there by Carol Culver. But Adam, I am sure, did not realize that. He would have naturally assumed that Kathy had hidden them there. He would also have naturally assumed that the pictures were connected to his daughter's disappearance.\"\n\n\"Logical,\" Myron agreed.\n\n\"Yes, quite.\" Win twirled his cognac, studying the color. \"Adam Culver then enlists the aid of Paul Duncan in his investigation. They track down the photographs' place of origin with the help of Fred Nickler. They also find out about Gary Grady. They continue their investigation, but nothing new develops. Paul wants to give up. Adam is desperate\u2014so desperate that he tries to draw out the assailant in a most unorthodox manner.\"\n\nWin paused, considering. \"Here,\" he said, \"is where it gets very interesting. We know Adam Culver had the photographs. We know he arranged to have them put in a pornographic magazine. I find it significant that the picture was placed only in _Nips_ magazine.\"\n\nMyron leaned forward. They were on the same wavelength. \"The magazine with the smallest\u2014almost nonexistent\u2014circulation.\"\n\n\"That fact disturbed you from the beginning,\" Win said.\n\nMyron nodded. \"Someone didn't want that magazine seen by a lot of people.\"\n\n\"Like her father.\"\n\n\"Right.\"\n\n\"And,\" Win continued, \"we know that Adam Culver liked to frequent the casinos of Atlantic City. He might have met your friend Blackjack during one of his visits or at least heard his name. He could have hired someone else to forge his daughter's handwriting. He probably had a tape with her voice from an old answering machine. Ergo, Adam Culver set the whole thing up. He sent out the magazine to everyone who might have been involved in Kathy's disappearance. Her fianc\u00e9, for one. People in the picture, like Junior Horton.\"\n\n\"Why did he send one to his wife?\" Myron asked.\n\n\"I don't know.\"\n\n\"And Dean Gordon?\"\n\n\"Perhaps the dean was in one of those attic photographs. Or perhaps Adam found out about Kathy's visit to the dean's house that night. Most likely Adam was merely considering every possibility. But it's not really that relevant to the case. What is relevant, however, is the question of why Adam did not once again enlist the help of Paul Duncan.\"\n\n\"Because,\" Myron said, \"Adam found out that Paul was sleeping with his wife.\"\n\nWin nodded. \"Paul was no longer a friend or trustworthy. Adam was now on his own. He sent the package to Sir Blackjack, making sure it would never be traced back to him. Then Adam set up his second little sting operation, the one on his wife and Paul. He walked in on them, ran out, and was killed.\"\n\n\"So who murdered him?\" Myron said.\n\nWin put down the snifter on a harpsichord from the seventeenth century. He steepled his fingers, bouncing them gently off one another. \"There are two strong possibilities,\" he said. \"First, Paul Duncan. We cannot just dismiss him. He had motive and opportunity. Second, Adam wanted to stir up the killer, that much is clear. But perhaps the magazine stirred up more trouble than he'd anticipated.\"\n\n\"Except for one thing,\" Myron interjected. \"The magazines hadn't been sent out yet. Adam was dead two days before Blackjack mailed them.\"\n\n\"So perhaps someone discovered what Adam was up to before they were mailed.\"\n\n\"Otto Burke?\"\n\nWin shrugged.\n\n\"But Otto has no connection to Kathy Culver,\" Myron said.\n\n\"None that we are aware of. Which leads us to Entity three: the unknowns. A major unknown, as I see it, is Nancy Serat. We can assume that she gave Adam Culver valuable information. But we do not know who killed her. Or what she meant when she told Christian it was time for sisters to reunite. And we especially do not know why Kathy Culver's hair was found on her dead body.\"\n\nWin rechecked his hair. Perfect. He smiled, winked, did everything but kiss his own reflection. \"We also have no explanation for Adam Culver's cabin in the woods. He could have become desperate enough to grab suspects and do his own interrogations. Or he could have been seeking retribution for all in the wicked photographs. On someone like Gary Grady. Or Junior Horton. But for some reason my mind cannot fully accept either of these rationales.\"\n\nMyron nodded. It didn't feel right to him either.\n\n\"And so now we've reached the final unknown. The most significant unknown of all: Miss Kathy Culver herself. Is she alive? Is she behind all this? Is she involved in any way at all?\"\n\nWin picked the snifter off the harpsichord. He took a sip of cognac, let it roll around his tongue, swallowed. \"The end.\"\n\nThey both sat in silence. Myron churned the facts through his head yet again. None of them changed. Win studied his face.\n\n\"This was all a mental exercise,\" Win said. \"A test drive, as it were.\"\n\nMyron said nothing.\n\n\"You know what happened. You knew before I said a word.\"\n\nMyron handed Win the telephone. \"Cancel your date. We have a lot of work to do.\"\n\n# Chapter 47\n\nThe memorial service.\n\nMyron slipped in late and ducked behind a pillar. He was in desperate need of a shower, a shave, a nap. And he looked it.\n\nHe spotted Jessica in the front pew. She sat on one side of her mother, Edward on the other. All three were crying.\n\nThe priest delivered the standard death spiel like an actor who knew his lines too well. Nothing new or original was said. There was no coffin, no well-dressed corpse in peaceful repose. The priest seemed bothered by this, by the absence of his customary prop. He kept motioning down on cue, only to draw back when he realized that there was nothing in front of him.\n\nMyron stayed out of sight. The church was crowded. Paul Duncan sat in the second row, directly behind Carol. Every once in a while Paul would put his hand on her shoulder, but he'd never leave it there long. Appearances. Christian was next to him, head lowered in prayer. Otto Burke and Larry Hanson were a few rows back. Good PR move. The press would undoubtedly be made aware of Otto Burke's heartfelt concern for his players' personal plights. Again, appearances.\n\nWin was near the back. To his right sat Sally Li. Her face looked drawn, as if she could use a cigarette. Myron had spoken to her late last night. She had done the test. It had come out as he'd suspected.\n\nDean Gordon and his wife Madelaine were off to the left. Dean Gordon looked grim. Madelaine Gordon looked good in black. Myron recognized a few other faces in the crowd, but he couldn't put a name or place to any of them. It didn't matter.\n\nThe priest made a few last comments about the hereafter, God's will, and reuniting with the beloved in Heaven. Jessica's sob racked her whole body. No one put an arm around her. No one comforted her. She looked small and frail. Myron felt a lump rise in this throat.\n\n_Here we go._\n\nWhen the ceremony ended, Myron did not hesitate. He walked purposively down the aisle. Jessica ran toward him without hesitation. They hugged, both closing their eyes. The mourners turned away and began to head for the exit. Win kept close to Otto Burke, Larry Hanson, and Dean Gordon.\n\nJessica finally released her grip. \"Where were you?\" she asked.\n\nMyron swallowed. He nodded to Paul Duncan, shook hands with Edward and Christian, lightly kissed Carol on the cheek.\n\n\"I don't know how to say this,\" Myron said.\n\n\"What's the matter?\"\n\nHe looked her straight in the eye. \"I found Kathy. She's alive.\"\n\nThe group went silent.\n\nJessica opened her mouth, closed it.\n\n\"I'm meeting her tonight,\" Myron said.\n\nJessica finally found her voice. \"I don't understand.\"\n\n\"It's a long story. But she's alive. I'll bring her home to you tonight.\"\n\nJessica looked at Carol. Carol looked back. Everyone looked at everyone else.\n\n\"I'll go with you,\" Jessica said.\n\n\"You can't.\"\n\n\"Like hell I can't.\"\n\n\"I promised her,\" Myron said. \"Just me. Alone. She's scared.\"\n\n\"Of what?\"\n\n\"Of the person who tried to kill her.\"\n\n\"Who?\"\n\nMyron shook his head. \"She wouldn't tell me. Not on the phone.\" He took hold of Jessica's hand. It was cold and stiff. Like marble. \"I'll bring her right to the house. I promise. We'll all talk then. But we can't risk scaring her off.\"\n\nJessica shook her head. She looked lost. \"Where are you meeting her?\"\n\n\"It's in the woods.\"\n\n\"What woods?\" Jessica pulled back a little. \"You're not making any sense.\"\n\n\"I can't tell you, Jess. I promised her. Kathy said it's the spot where she was left for dead. She wants to show me where it happened.\"\n\nMore silence.\n\nPaul Duncan said, \"Dear God.\"\n\nCarol practically fainted into his arms.\n\n\"Where has she been?\" Jessica asked.\n\n\"I only know bits and pieces from my investigation. She spent most of the time recovering from her injuries. She also spent some time in the Caribbean. An island called Cura\u00e7ao. I picked up her trail from an entry that night in St. Mary Hospital's registry. On the night she vanished, a patient was found unconscious in the middle of a road. She gave her name as Katherine Pierce.\"\n\nCarol gasped. \"Pierce? That's my maiden name.\"\n\nMyron nodded. \"I don't know all the details yet. She was hit over the head. The blow cracked her skull. The assailant thought she was dead. But she wasn't. He buried her in the woods. She woke up and managed to dig herself out. It's a miracle she survived.\"\n\nJessica's eyes filled with tears. \"She's alive?\"\n\n\"Yes.\"\n\n\"You're sure?\"\n\n\"Yes.\"\n\nJessica hugged her mother then. Edward joined in. Christian and Paul watched dumbfounded. Myron turned toward the door. Win was standing there. His nod was almost imperceptible.\n\n# Chapter 48\n\nMyron parked his car on the dirt road. He was alone. The car's clock read 8:30 P.M. He grabbed his flashlight and headed toward the meeting spot.\n\nThe brush was thick. Several branches whipped across his face. He listened for other sounds. Crickets hummed away. Nothing else. The flashlight sliced through the heavy darkness, carving a path for him to follow. Myron heard his feet crunch on twigs and leaves. His mouth felt bone-dry. It always felt that way at moments like this.\n\nHe was getting close now, no more than twenty or thirty yards away.\n\n\"Kathy?\" he called out.\n\nNo answer.\n\n\"It's Myron, Kathy. I'm alone.\"\n\nNo reply. But then Myron heard a shuffling from in front of him. Something came into view. A head. A head of long blond hair.\n\n\"It's okay,\" Myron said gently. \"I'm here alone.\"\n\nShe stepped toward him tentatively. Her right hand shaded her eyes from the flashlight's harsh glare. Myron pointed the beam away. \"It's all right,\" he said.\n\nShe continued to move toward him, a dim silhouette. Her steps were slow, plodding, like a B-movie monster come to life.\n\n\"It's okay,\" Myron said again. \"No one is going to hurt you.\"\n\n\"I wish that were true.\"\n\nThe voice had not come from her. It had come from behind him. Myron closed his eyes. His shoulders slumped. \"Hello, Christian.\"\n\n\"Don't move, Mr. Bolitar. Put your hands up.\"\n\n\"Why bother?\"\n\n\"What?\"\n\n\"You're going to kill us. Just like you tried to kill Kathy. Just like you killed her father and Nancy.\"\n\n\"I never meant to hurt anybody,\" he said.\n\n\"But you did.\"\n\nChristian cocked the gun. \"Hands up. Now.\"\n\nMyron raised his hands slowly. \"Kathy opened up to you that night. She told you everything\u2014every sordid detail of her past. She wanted to clean the slate.\"\n\n\"She lied to me!\" Christian shouted. \"All the time we were together\u2014it was all a lie.\"\n\n\"So you tried to kill her.\"\n\n\"Kathy wanted me to still love her, Mr. Bolitar. But don't you see? I never loved her. I loved a lie. She wanted me to stand beside that lie while she told her story to the world. She wanted me to sell out my teammates, toss away a chance at a national championship and Heisman trophy\u2014all for the sake of a lying whore.\"\n\n\"A lying whore,\" Myron said, \"like your mother.\"\n\nHe nodded. \"Mr. Bolitar, tell her. Tell her what that game meant. In terms of money, fame, pride. You understand, Mr. Bolitar. It helped get me that contract.\"\n\n\"So you hit her over the head.\"\n\n\"I didn't mean to. It just happened. I thought she was dead. I couldn't find a pulse.\"\n\n\"So you drove her out here and buried the body. You hoped she'd never be found, but if she were, it'd be blamed on a serial killer.\"\n\nChristian stepped closer. He raised the gun. \"Enough talk,\" he said. \"I'm not going to let you stall around until someone shows up.\"\n\n\"No need. Someone's been here all the time.\"\n\nWin came out from behind a tree, no more than a yard away from Christian. He pressed the .44 against Christian's ear and said, \"Drop it, or your brain becomes squirrel lunch.\"\n\nChristian dropped the gun.\n\n\"It's over,\" Myron shouted.\n\nFrom a farther distance two uniformed police arrived. They handcuffed Christian.\n\nJake Courter stumbled behind them, high-stepping through the long grass. \"Too old for this shit,\" he mumbled. When he reached the clearing he said, \"Nice setup, Bolitar.\"\n\n\"Lots of details. The secret to a good scam.\"\n\n\"Gonna tell me what's going on now?\"\n\n\"Sure. Jess?\"\n\nJessica took off the blond wig and stepped forward.\n\nChristian's mouth dropped open. \"What the\u2014\"\n\n\"You killed Kathy,\" Myron said, \"but not from the blow to the head. She suffocated trying to claw her way out of the dirt.\"\n\nJake looked confused. \"Where's the body?\"\n\n\"In the morgue. Where it's been since the police found it two months ago. Sally Li confirmed the identity last night.\"\n\n\"So why hadn't it been identified before?\"\n\n\"Because the county medical examiner was Kathy's father. He knew who it was right away, but he pretended otherwise.\"\n\n\"Why?\"\n\n\"Think about it a second, Jake. From Adam Culver's perspective. Your case had gone nowhere in eighteen months. Adam knew that. He also knew the body provided no new clues. So he figured that the only way to catch Kathy's killer was to draw him out. How? By making the killer think Kathy might still be alive. After all, she'd been alive when he dumped her in the woods. So Adam kept the corpse's identity a secret from everyone\u2014the police, his friends, even his own family. He also figured that the nude photographs were tied into all this. So he used them.\"\n\n\"You mean he put that ad in the magazine?\"\n\nMyron nodded. \"Adam Culver arranged everything. Even the mysterious phone calls saying 'Come and get me. I survived.' He did everything he could to make it look like Kathy was alive.\"\n\nJake nodded. \"So what you guys were just doing\u2014\"\n\n\"Was finishing up Adam Culver's plan. Our performance at the church this morning sowed the final seeds of doubt.\"\n\n\"You were forcing Christian to make a play for you.\"\n\n\"Exactly.\"\n\n\"Incredible. So everyone was in on on this?\"\n\n\"Jessica was,\" Myron said. \"So were her mother and brother. It would have been too cruel to lie to them. But Paul Duncan didn't know. Neither did anybody else, and Win made sure that all the suspects\u2014Otto, the dean, even Gary Grady\u2014knew about Kathy's 'survival.' \"\n\n\"Then you weren't sure it was Christian?\"\n\n\"No, I was sure.\"\n\n\"You were trying to play it fair.\"\n\nMyron nodded. \"That's why I didn't tell you anything. I wanted you to see what happened without any preconceived notions.\"\n\n\"Fair enough,\" Jake said. \"Go on.\"\n\n\"Adam Culver understood that only the killer would know this spot. If he made the killer think Kathy could still be alive, he or she would have to come back here\u2014just to make sure Kathy was dead. That was why Adam rented that cabin nearby. That was why he had all that electronic equipment. To tape him. To have proof.\"\n\n\"Catching the killer returning to the scene of the crime,\" Jake said.\n\n\"Right.\"\n\n\"But I don't get something. Adam was killed before the magazine was mailed out. How did Christian find out about it?\"\n\n\"He didn't. Remember, Adam was a pathologist. He wasn't an investigator. He overlooked a very important clue. At first anyway.\"\n\n\"What clue?\"\n\n\"Kathy's clothes.\"\n\n\"What about them?\"\n\n\"When Kathy's body was found, she was wearing a yellow sweater and a pair of gray sweat pants. Yet the sorority sisters said she was wearing blue when she left the house. The rapists said she was wearing blue. Dean Gordon said she was wearing blue. Ricky Lane said she was wearing blue. The sorority sisters were also positive that Kathy never returned to the house. So the question was: Where did the yellow sweater and gray sweat pants come from?\"\n\nJake shrugged.\n\n\"It took Adam a while to realize the significance of the clothes. But when he did, he went to the most obvious source. Kathy's roommate.\"\n\n\"Nancy Serat.\"\n\n\"Right. But he didn't want to let on that Kathy's body had been found. So he asked Nancy where he could find her favorite yellow sweater, pretending to be a typical dad on some kind of nostalgic tour. But think about it. If Kathy didn't go back to her sorority house, where did she change clothes?\"\n\nJake saw it now. \"At Christian's,\" he said with a snap of the fingers. \"Kathy slept there all the time. She must have kept clothes there.\"\n\n\"Right.\"\n\n\"And Nancy and Christian were friends,\" Jake said, picking up the thread. \"She'd see nothing wrong with telling Christian all about Adam's visit. Probably thought the whole thing was kinda cute.\"\n\nMyron turned toward Christian. \"You got scared when you heard Adam had been asking about the yellow sweater. You knew he was getting close. So you followed him that night. You heard him fight with his wife. You saw him storm out of the house, and you figured this was the ideal opportunity to kill him. Another perfect misdirection.\"\n\nChristian said nothing.\n\nJake said, \"What do you mean, 'another perfect misdirection'?\"\n\n\"When your investigation of Kathy's disappearance began,\" Myron said, \"who did you focus in on?\"\n\n\"Christian,\" Jake said. \"Like I said, we always check out the boyfriend.\"\n\n\"So what did Christian do? With campus security combing the campus for clues, he planted the panties on the top of a garbage bin.\"\n\n\"The panties,\" Jake added, \"with someone else's semen.\"\n\n\"Proof he didn't do it.\"\n\n\"Well, I'll be damned.\"\n\n\"He also misdirected us with Nancy Serat. He strangled Nancy. Then he planted one of Kathy's hairs at the scene.\"\n\n\"But where did he get the hair?\"\n\n\"Kathy slept in his room all the time, right? She would have kept other stuff there besides her clothes. Stuff like a hairbrush.\"\n\n\"Son of a bitch.\"\n\n\"It was almost perfect. Blame someone who was dead. And if Kathy wasn't dead\u2014if she had indeed survived\u2014he'd make her look like a lunatic. Who'd believe the ravings of a girl who'd killed her old roommate? But Christian didn't count on Jessica showing up at Nancy's. He panicked. He hit her over the head and ran. Problem was, he'd left his fingerprints behind. But Christian was quick. He even used that to his advantage. When you dragged him in the next morning, he immediately admitted to being at Nancy's house. And then he came up with that wonderful story about sisters reuniting.\"\n\n\"Another perfect misdirection,\" Jake said.\n\n\"Except he forgot about the glass.\"\n\n\"What glass?\"\n\n\"His fingerprints were found in several spots in the house, including a drinking glass. Yet Christian told us Nancy barely let him in the door, that she practically pushed him away mumbling about the reuniting sisters. Under those circumstances, isn't it odd she'd offer him a drink?\"\n\nMyron looked at Christian. He lowered his eyes.\n\n\"I\u2014I didn't mean to hurt any of them, Mr. Bolitar,\" he said.\n\n\"You were manipulative and calculating,\" Myron said. \"You covered all the bases, even when you hired me. I was small-time. I could be controlled. You knew about my background, that I was an experienced investigator. You knew if any trouble arose, I'd keep things quiet. That I'd keep you informed. That I'd try to protect you. You played me for a sucker.\"\n\nEveryone remained silent until Jake said, \"All right. Get him out of here.\"\n\nThe uniformed officers led Christian away.\n\nMyron looked back at Jessica. She still hadn't said a word. Tears slid down her cheeks. None of this morning's tears had been for her father. Maybe some of these were.\n\nWin shook his head. \" 'Squirrel lunch.' I can't believe I said 'squirrel lunch.' \"\n\nJessica stopped crying. She even smiled a little. Myron put his arm around her and pulled her in close. Together they made their way back to the car.\n\n# Chapter 49\n\nThree days later Myron drove Jessica to the airport.\n\n\"Just drop me off at the terminal,\" she said.\n\n\"I'll wait with you at the gate.\"\n\n\"You should head back.\"\n\n\"I have time.\"\n\n\"The traffic will be murder.\"\n\n\"I don't care.\"\n\n\"Myron?\"\n\n\"What?\"\n\n\"Just drop me off. Please. You know I hate scenes.\"\n\n\"I won't make a scene.\"\n\n\"You always make a scene.\"\n\nSilence.\n\n\"What's going to happen to Gary Grady?\" she asked.\n\n\"I've sent all the information to the school board and the local press. I don't know if he'll spend any time in jail, but he's finished.\n\n\"What about Dean Gordon?\"\n\n\"He resigned this morning. He's going to enter the private sector.\"\n\n\"And the rapists?\"\n\n\"Cary Roland is the DA. This case means big headlines. He'll do his best. Ricky Lane is going to turn state's evidence.\"\n\n\"You dumped Ricky as a client?\"\n\nMyron nodded.\n\n\"And you lost Christian.\"\n\nAnother nod.\n\n\"All in all,\" she said, \"this case hasn't had a real positive economic effect on you.\"\n\n\"I'm more worried about the personal effect.\"\n\n\"Meaning?\"\n\n\"Meaning you're back in my life.\"\n\n\"Isn't that a good thing?\"\n\n\"It is. Except you're leaving.\"\n\n\"Just for a month or two. It's a book tour.\"\n\nHe pulled up to the front of the terminal.\n\n\"I'll be back,\" she said.\n\nHe nodded.\n\nJessica kissed him. He held on. She finally pushed him back. He released her grudgingly.\n\n\"I love you,\" he said.\n\n\"I love you too.\" She stepped out of the car. \"And I'll be back.\"\n\nHe watched her walk toward the entrance. He watched her pass through the sliding glass doors, watched her walk to the ticket gate, watched her disappear down an escalator. When she was out of sight, he still watched until a security guard knocked on his window.\n\n\"Unloading zone, bub. Move it!\"\n\nMyron looked back one more time. Then he drove back to the office.\n\n# About the Author\n\n**Harlan Coben** is the winner of the Edgar, Shamus, and Anthony awards. His critically acclaimed novels have been published in thirty-three languages around the world and have been number one bestsellers in more than half a dozen countries. In addition to the Myron Bolitar series _(Deal Breaker, Drop Shot, Fade Away, Back Spin, One False Move, The Final Detail, Darkest Fear,_ and _Promise Me)_ , he is also the author of _Tell No One, Gone for Good, No Second Chance, Just One Look,_ and _The Innocent_. His website is www.harlancoben.com.\n**_Look for Harlan Coben's  \nnext novel of suspense_**\n\n**GONE FOR GOOD**\n\n**NOW AVAILABLE FROM DELL**\n\n# **1**\n\n**T hree days before her death, my mother told** me\u2014these weren't her last words, but they were pretty close\u2014that my brother was still alive.\n\nThat was all she said. She didn't elaborate. She said it only once. And she wasn't doing very well. Morphine had already applied its endgame heart squeeze. Her skin was in that cusp between jaundice and fading summer tan. Her eyes had sunken deep into her skull. She slept most of the time. She would, in fact, have only one more lucid moment\u2014if indeed this had been a lucid moment, which I very much doubted\u2014and that would be a chance for me to tell her that she had been a wonderful mother, that I loved her very much, and good-bye. We never said anything about my brother. That didn't mean we weren't thinking about him as though he were sitting bedside too.\n\n\"He's alive.\"\n\nThose were her exact words. And if they were true, I didn't know if it would be a good thing or bad.\n\nWe buried my mother four days later.\n\nWhen we returned to the house to sit shivah, my father stormed through the semi-shag in the living room. His face was red with rage. I was there, of course. My sister, Melissa, had flown in from Seattle with her husband, Ralph. Aunt Selma and Uncle Murray paced. Sheila, my soul mate, sat next to me and held my hand.\n\nThat was pretty much the sum total.\n\nThere was only one flower arrangement, a wonderful monster of a thing. Sheila smiled and squeezed my hand when she saw the card. No words, no message, just the drawing\n\nDad kept glancing out the bay windows\u2014the same windows that had been shot out with a BB gun twice in the past eleven years\u2014and muttered under his breath, \"Sons of bitches.\" He'd turn around and think of someone else who hadn't shown. \"For God's sake, you'd think the Bergmans would have at least made a goddamn appearance.\" Then he'd close his eyes and look away. The anger would consume him anew, blending with the grief into something I didn't have the strength to face.\n\nOne more betrayal in a decade filled with them.\n\nI needed air.\n\nI got to my feet. Sheila looked up at me with concern. \"I'm going to take a walk,\" I said softly.\n\n\"You want company?\"\n\n\"I don't think so.\"\n\nSheila nodded. We had been together nearly a year. I've never had a partner so in sync with my rather odd vibes. She gave my hand another I-love-you squeeze, and the warmth spread through me.\n\nOur front-door welcome mat was harsh faux grass, like something stolen from a driving range, with a plastic daisy in the upper left-hand cover. I stepped over it and strolled up Downing Place. The street was lined with numbingly ordinary aluminum-sided split-levels, circa 1962. I still wore my dark gray suit. It itched in the heat. The savage sun beat down like a drum, and a perverse part of me thought that it was a wonderful day to decay. An image of my mother's light-the-world smile\u2014the one before it all happened\u2014flashed in front of my eyes. I shoved it away.\n\nI knew where I was headed, though I doubt if I would have admitted it to myself. I was drawn there, pulled by some unseen force. Some would call it masochistic. Others would note that maybe it had something to do with closure. I thought it was probably neither.\n\nI just wanted to look at the spot where it all ended.\n\nThe sights and sounds of summer suburbia assaulted me. Kids squealed by on their bicycles. Mr. Cirino, who owned the Ford\/Mercury dealership on Route 10, mowed his lawn. The Steins\u2014they'd built up a chain of appliance stores that were swallowed up by a bigger chain\u2014were taking a stroll hand in hand. There was a touch football game going on at the Levines' house, though I didn't know any of the participants. Barbecue smoke took flight from the Kaufmans' backyard.\n\nI passed by the Glassmans' old place. Mark \"the Doof\" Glassman had jumped through the sliding glass doors when he was six. He was playing Superman. I remembered the scream and the blood. He needed over forty stitches. The Doof grew up and became some kind of IPO-start-up zillionaire, I don't think they call him the Doof anymore, but you never know.\n\nThe Marianos' house, still that horrid shade of phlegm yellow with a plastic deer guarding the front walk, was on the bend. Angela Mariano, our local bad girl, was two years older than us and like some superior, awe-inducing species. Watching Angela sunning in her backyard in a gravity-defying ribbed halter top, I had felt the first painful thrusts of deep hormonal longing. My mouth would actually water. Angela used to fight with her parents and sneak smokes in the toolshed behind her house. Her boyfriend drove a motorcycle. I ran into her last year on Madison Avenue in midtown. I expected her to look awful\u2014that was what you always hear happens to that first lust-crush\u2014but Angela looked great and seemed happy.\n\nA lawn sprinkler did the slow wave in front of Eric Frankel's house at 23 Downing Place. Eric had a space-travel-themed bar mitzvah at the Chanticleer in Short Hills when we were both in seventh grade. The ceiling was done up planetarium style\u2014a black sky with star constellations. My seating card told me that I was sitting at \"Table Apollo 14.\". The centerpiece was an ornate model rocket on a green fauna launching pad. The waiters, adorned in realistic space suits, were each supposed to be one of the Mercury 7. \"John Glenn\" served us. Cindi Shapiro and I had sneaked into the chapel room and made out for over an hour. It was my first time. I didn't know what I was doing. Cindi did. I remember it was glorious, the way her tongue caressed and jolted me in unexpected ways. But I also remember my initial wonderment evolving after twenty minutes or so into, well, boredom\u2014a confused \"what next?\" along with a naive \"is that all there is?\"\n\nWhen Cindi and I stealthily returned to Cape Kennedy's Table Apollo 14, ruffled and in fine post-smooch form (the Herbie Zane Band serenading the crowd with \"Fly Me to the Moon\"), my brother, Ken, pulled me to the side and demanded details. I, of course, too gladly gave them. He awarded me with that smile and slapped me five. That night, as we lay on the bunk beds, Ken on the top, me on the bottom, the stereo playing Blue Oyster Cult's \"Don't Fear the Reaper\" (Ken's favorite), my older brother explained to me the facts of life as seen by a ninth-grader. I'd later learn he was mostly wrong (a little too much emphasis on the breast), but when I think back to that night, I always smile.\n\n_\"He's alive...\"_\n\nI shook my head and turned right at Coddington Terrace by the Holders' old house. This was the same route Ken and I had taken to get to Burnet Hill Elementary School. There used to be a paved path between two houses to make the trip shorter. I wondered if it was still there. My mother\u2014everyone, even kids, had called her Sunny\u2014used to follow us to school quasi-surreptitiously. Ken and I would roll our eyes as she ducked behind trees. I smiled, thinking about her overprotectiveness now. It used to embarrass me, but Ken would simply shrug. My brother was securely cool enough to let it slide. I wasn't.\n\nI felt a pang and moved on.\n\nMaybe it was just my imagination, but people began to stare. The bicycles, the dribbling basketballs, the sprinklers and lawn mowers, the cries of touch footballers\u2014they all seemed to hush as I passed. Some stared out of curiosity because a strange man strolling in a dark gray suit on a summer evening was something of an oddity. But most, or again so it seemed, looked on in horror because they recognized me and couldn't believe that I would dare tread upon this sacred soil.\n\nI approached the house at 47 Coddington Terrace without hesitation. My tie was loosened. I jammed my hands in my pockets. I toed the spot where curb met pavement. Why was I here? I saw a curtain move in the den. Mrs. Miller's face appeared at the window, gaunt and ghostlike. She glared at me. I didn't move or look away. She glared some more\u2014and then to my surprise, her face softened. It was as though our mutual agony had made some sort of connection. Mrs. Miller nodded at me. I nodded back and felt the tears begin to well up.\n\nYou may have seen the story on _20\/20_ or _PrimeTime Live_ or some other television equivalent of fish wrap. For those who haven't, here's the official account: On October 17 eleven years ago, in the township of Livingston, New Jersey, my brother, Ken Klein, then twenty-four, brutally raped and strangled our neighbor Julie Miller.\n\nIn her basement. At 47 Coddington Terrace.\n\nThat was where her body was found. The evidence wasn't conclusive as to if she'd actually been murdered in that poorly finished subdwelling or if she'd been dumped postmortem behind the water-stained zebra-striped couch. Most assume the former. My brother escaped capture and ran off to parts unknown\u2014at least, again, according to the official account.\n\nOver the past eleven years, Ken has eluded an international dragnet. There have however been \"sightings.\"\n\nThe first came about a year after the murder from a small fishing village in northern Sweden. Interpol swooped in, but somehow my brother evaded their grasp. Supposedly he was tipped off. I can't imagine how or by whom.\n\nThe next sighting occurred four years later in Barcelona. Ken had rented\u2014to quote the newspaper accounts\u2014\"an oceanview hacienda\" (Barcelona is not on an ocean) with\u2014again I will quote\u2014\"a lithe, dark-haired woman, perhaps a flamenco dancer.\" A vacationing Livingston resident, no less, reported seeing Ken and his Castilian paramour dining beachside. My brother was reportedly tan and fit and wore a white shirt opened at the collar and loafers without socks. The Livingstonite, one Rick Horowitz, had been a classmate of mine in Mr. Hunt's fourth-grade class. During a three-month period, Rick entertained us by eating caterpillars during recess.\n\nBarcelona Ken yet again slipped through the law's fingers.\n\nThe last time my brother was purportedly spotted he was skiing down the expert hills in the French Alps (interestingly enough, Ken never skied before the murder). Nothing came of it, except a story on _48 Hours._ Over the years, my brother's fugitive status had become the criminal version of a VH1 _Where Are They Now_ , popping up whenever any sort of rumor skimmed the surface or, more likely, when one of the network's fish wraps was low on material.\n\nI naturally hated television's \"team coverage\" of \"suburbia gone wrong\" or whatever similar cute moniker they came up with. Their \"special reports\" (just once, I'd like to see them call it a \"normal report, everyone has done this story\") always featured the same photographs of Ken in his tennis whites\u2014he was a nationally ranked player at one time\u2014looking his most pompous. I can't imagine where they got them. In them Ken looked handsome in the way that people hate right away. Haughty, Kennedy hair, suntan bold against the whites, toothy grin, Photograph Ken looked like one of those people of privilege (he was not) who coasted through life on his charm (a little) and trust account (he had none).\n\nI had appeared on one of those magazine shows. A producer reached me\u2014this was pretty early on in the coverage\u2014and claimed that he wanted to present \"both sides fairly.\" They had plenty of people ready to lynch my brother, he noted. What they truly needed for the sake of \"balance\" was someone who could describe the \"real Ken\" to the folks back home.\n\nI fell for it.\n\nA frosted-blond anchorwoman with a sympathetic manner interviewed me for over an hour. I enjoyed the process actually. It was therapeutic. She thanked me and ushered me out and when the episode aired, they used only one snippet, removing her question (\"But surely, you're not going to tell us that your brother was perfect, are you? You're not trying to tell us he was a saint, right?\") and editing my line so that I appeared in nose-pore-enhancing extreme close-up with dramatic music as my cue, saying, \"Ken was no saint, Diane.\"\n\nAnyway, that was the official account of what happened.\n\nI've never believed it. I'm not saying it's not possible. But I believe a much more likely scenario is that my brother is dead\u2014that he has been dead for the past eleven years.\n\nMore to the point, my mother always believed that Ken was dead. She believed it firmly. Without reservation. Her son was not a murderer. Her son was a victim.\n\n_\"He's alive.... He didn't do it.\"_\n\nThe front door of the Miller house opened. Mr. Miller stepped through it. He pushed his glasses up his nose. His fists rested on his hips in a pitiful Superman stance.\n\n\"Get the hell out of here, Will,\" Mr. Miller said to me.\n\nAnd I did.\n\nThe next big shock occurred an hour later.\n\nSheila and I were up in my parents' bedroom. The same furniture, a sturdy, faded swirling gray with blue trim, had adorned this room for as long as I could remember. We sat on the king-size bed with the weak-springed mattress. My mother's most personal items\u2014the stuff she kept in her bloated nightstand drawers\u2014were scattered over the duvet. My father was still downstairs by the bay windows, staring out defiantly.\n\nI don't know why I wanted to sift through the things my mother found valuable enough to preserve and keep near her. It would hurt. I knew that. There is an interesting correlation between intentional pain infliction and comfort, a sort of playing-with-fire approach to grieving. I needed to do that, I guess.\n\nI looked at Sheila's lovely face\u2014tilted to the left, eyes down and focused\u2014and I felt my heart soar. This is going to sound a little weird, but I could stare at Sheila for hours. It was not just her beauty\u2014hers was not one would call classical anyway, her features a bit off center from either genetics or, more likely, her murky past\u2014but there was an animation there, an inquisitiveness, a delicacy too, as if one more blow would shatter her irreparably. Sheila made me want to\u2014bear with me here\u2014be brave for her.\n\nWithout looking up, Sheila gave a half-smile and said, \"Cut it out.\"\n\n\"I'm not doing anything.\"\n\nShe finally looked up and saw the expression on my face. \"What?\" she asked.\n\nI shrugged. \"You're my world,\" I said simply.\n\n\"You're pretty hot stuff yourself.\"\n\n\"Yeah,\" I said. \"Yeah, that's true.\"\n\nShe feigned a slap in my direction. \"I love you, you know.\"\n\n\"What's not to love?\"\n\nShe rolled her eyes. Then her gaze fell back onto the side of my mother's bed. Her face quieted.\n\n\"What are you thinking about?\" I asked.\n\n\"Your mother.\" Sheila smiled. \"I really liked her.\"\n\n\"I wish you had known her before.\"\n\n\"Me too.\"\n\nWe started going through the laminated yellow clippings. Birth announcements\u2014Melissa's, Ken's, mine. There were articles on Ken's tennis exploits. His trophies, all those bronze men in miniature in mid-serve, still swarmed his old bedroom. There were photographs, mostly old ones from before the murder. Sunny. It had been my mother's nickname since childhood. It suited her. I found a photo of her as PTA president. I don't know what she was doing, but she was onstage and wearing a goofy hat and all the other mothers were cracking up. There was another one of her running the school fair. She was dressed in a clown suit. Sunny was the favorite grown-up among my friends. They liked when she drove the carpool. They wanted the class picnic at our house. Sunny was parental cool without being cloying, just \"off\" enough, a little crazy perhaps, so that you never knew exactly what she would do next. There had always been an excitement\u2014a crackle if you will\u2014around my mother.\n\nWe kept it up for more than two hours. Sheila took her time, looking thoughtfully at every picture. When she stopped at one in particular, her eyes narrowed. \"Who's that?\"\n\nShe handed me the photograph. On the left was my mother in a semi-obscene yellow bikini, I'd say 1972ish, looking very curvy. She had her arm around a short man with a dark mustache and happy smile.\n\n\"King Hussein,\" I said.\n\n\"Pardon me?\"\n\nI nodded.\n\n\"As in the kingdom of Jordan?\"\n\n\"Yep. Mom and Dad saw him at the Fontainebleau in Miami.\"\n\n\"And?\"\n\n\"Mom asked him if he'd pose for a picture.\"\n\n\"You're kidding.\"\n\n\"There's the proof.\"\n\n\"Didn't he have guards or something?\"\n\n\"I guess she didn't look armed.\"\n\nSheila laughed. I remember Mom telling me about the incident. Her posing with King Hussein, Dad's camera not working, his muttering under his breath, his trying to fix it, her glaring at him to hurry, the king standing patiently, his chief of security checking the camera, finding the problem, fixing it, handing it back to Dad.\n\nMy mom Sunny.\n\n\"She was so lovely,\" Sheila said.\n\nIt's an awful clich\u00e9 to say that a part of her died when they found Julie Miller's body, but the thing about clich\u00e9s is that they're often dead-on. My mother's crackle quieted, smothered. After hearing about the murder, she never threw a tantrum or cried hysterically. I often wish she had. My volatile mother became frighteningly even. Her whole manner became flat, monotone\u2014 _passionless_ would be the best way to describe it\u2014which in someone like her was more agonizing to witness than the most bizarre histrionics.\n\nThe front doorbell rang. I looked out the bedroom window and saw the Eppes-Essen deli delivery van. Sloppy joes for the, uh, mourners. Dad had optimistically ordered too many platters. Delusional to the end. He stayed in this house like the captain of the _Titanic_. I remember the first time the windows had been shot out with the BB gun not long after the murder\u2014the way he shook his fist with defiance. Mom, I think, wanted to move. Dad would not. Moving would be a surrender in his eyes. Moving would be admitting their son's guilt. Moving would be a betrayal.\n\nDumb.\n\nSheila had her eyes on me. Her warmth was almost palpable, more sunbeam on my face, and for a moment I just let myself bathe in it. We'd met at work about a year before. I'm the senior director of Covenant House on 41st Street in New York City. We're a charitable foundation that helps young runaways survive the streets. Sheila had come in as a volunteer. She was from a small town in Idaho, though she seemed to have very little small-town-girl left in her. She told me that many years ago, she too had been a runaway. That was all she would tell me about her past.\n\n\"I love you,\" I said.\n\n\"What's not to love?\" she countered.\n\nI did not roll my eyes. Sheila had been good to my mother toward the end. She'd take the Community Bus Line from Port Authority to Northfield Avenue and walk over to the St. Barnabas Medical Center. Before her illness, the last time my mom had stayed at St. Barnabas was when she delivered me. There was probably something poignantly life-cycling about that, but I couldn't see it just then.\n\nI had however seen Sheila with my mother. And it made me wonder. I took a risk.\n\n\"You should call your parents,\" I said softly.\n\nSheila looked at me as though I'd just slapped her across the face. She slid off the bed.\n\n\"Sheila?\"\n\n\"This isn't the time, Will.\"\n\nI picked up a picture frame that held a photo of my tanned parents on vacation. \"Seems as good as any.\"\n\n\"You don't know anything about my parents.\"\n\n\"I'd like to,\" I said.\n\nShe turned her back to me. \"You've worked with runaways,\" she said.\n\n\"So?\"\n\n\"You know how bad it can be.\"\n\nI did. I thought again about her slightly off-center features\u2014the nose, for example, with the telltale bump\u2014and wondered. \"I also know it's worse if you don't talk about it.\"\n\n\"I've talked about it, Will.\"\n\n\"Not with me.\"\n\n\"You're not my therapist.\"\n\n\"I'm the man you love.\"\n\n\"Yes.\" She turned to me. \"But not now, okay? Please.\"\n\nI had no response to that one, but perhaps she was right. My fingers were absently toying with the picture frame. And that was when it happened.\n\nThe photograph in the frame slid a little.\n\nI looked down. Another photograph started peeking out from underneath. I moved the top one a little farther. A hand appeared in the bottom photograph. I tried pushing it some more, but it wouldn't go. My finger found the clips on back. I slid them to the side and let the back of the frame drop to the bed. Two photographs floated down behind it.\n\nOne\u2014the top one\u2014was of my parents on a cruise, looking happy and healthy and relaxed in a way I hardly ever remember them being. But it was the second photograph, the hidden one, that caught my eye.\n\nThe red-stamped date on the bottom was from less than two years ago. The picture was taken atop a field or hill or something. I saw no houses in the background, just snow-capped mountains like something from the opening scene of _The Sound of Music._ The man in the picture wore shorts and a backpack and sunglasses and scuffed hiking boots. His smile was familiar. So was his face, though it was more lined now. His hair was longer. His beard had gray in it. But there was no mistake.\n\nThe man in the picture was my brother, Ken.\nPublished by  \nDell Publishing  \na division of  \nRandom House, Inc.\n\nCopyright \u00a9 1995 by Harlan Coben\n\nAll rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without the written permission of the Publisher, except where permitted by law.\n\nThe trademark Dell\u00ae is registered in the U S Patent and Trademark Office.\n\neISBN: 978-0-307-48363-8\n\nv3.0_r1\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nMAD MOTHERS, BAD MOTHERS, & WHAT A \"GOOD\" MOTHER WOULD DO\nMAD MOTHERS, BAD MOTHERS, & WHAT A \"GOOD\" MOTHER WOULD DO\n\n_The Ethics of Ambivalence_\n\nSARAH LaCHANCE ADAMS\n\n_Columbia University Press_\n\n_New York_\n\nColumbia University Press\n\n_Publishers Since 1893_\n\nNew York Chichester, West Sussex\n\ncup.columbia.edu\n\nCopyright \u00a9 2014 Columbia University Press\n\nAll rights reserved\n\nLibrary of Congress Cataloging-in-Publication Data\n\nLaChance Adams, Sarah.\n\nMad mothers, bad mothers, and what a good mother would do: the ethics of ambivalence \/ Sarah LaChance Adams.\n\npages cm\n\nIncludes bibliographical references and index.\n\nISBN 978-0-231-16674-4 (cloth: alk. paper)\u2014ISBN 978-0-231-16675-1 (pbk.: alk. paper)\u2014ISBN 978-0-231-53722-3 (e-book)\n\n1. Mothers\u2014Conduct of life. 2. Ambivalence. I. Title.\n\nBJ1610.L1455 2014\n\n306.874'3\u2014dc23\n\n2013035268\n\nA Columbia University Press E-book.  \nCUP would be pleased to hear about your reading experience with this e-book at cup-ebook@columbia.edu.\n\nJacket Design: Jordan Wannemacher\n\nJacket Image: \u00a9Shutterstock\n\nReferences to Web sites (URLs) were accurate at the time of writing. Neither the author nor Columbia University Press is responsible for URLs that may have expired or changed since the manuscript was prepared.\nFor Geneva Louise LaChance Adams & Nancy Bird LaChance\nLonging for the further, the higher, the brighter.\n\n\"I want heirs,\" so saith everything that suffereth,\n\n\"I want children, I do not want myself.\"\n\n\u2014NIETZSCHE, _THUS SPAKE ZARATHUSTRA_\n_A Rain-Brain in a Broom-Room_\n\n_\"Write another one, Mum,\" she says_ ,\n\n_and wriggles her pants to her waist_ ,\n\n_and pulls me down to a Henry-hug_ ,\n\n_and squinches her soft-nosed face_.\n\n_Seeing me raise the pen again_ ,\n\n_she clasps tiny fingers in delight_\n\n_and turns up the ends of her rose-bud mouth_\n\n_and bounces her pudgy knees tight_.\n\n_Earlier with her mini'ture tool_ ,\n\n_She'd set out to sweep up her room_ ,\n\n_Needing the dust pan in the back hall asked_ ,\n\n_\"Why don't you use your broom?\"_\n\n_I'm hoping that someday she'll fully comprehend_ ,\n\n_that I'll no longer have to explain_ ,\n\n_when using my broom I don't get the smiles_\n\n_that I get when I'm using my brain!_\n\n\u2014NANCY BIRD LACHANCE, 1979\nCONTENTS\n\n_Acknowledgments_\n\n1. MAD MOTHERS, BAD MOTHERS, AND WHAT A \"GOOD\" MOTHER WOULD DO\n\n2. THE MOTHER AS ETHICAL EXEMPLAR IN CARE ETHICS\n\n3. MOTHERHOOD'S JANUS HEAD\n\nPsychology and Psychoanalysis\n\nThe Two Faces of Maternal Ambivalence\n\nMutuality\u2014My Child, Myself\n\nConflict\u2014Either the Kid Goes or I Go\n\nAmbiguous Intersubjectivity\u2014More and Less Than One\n\n4. MATERNITY AS VULNERABILITY IN THE PHILOSOPHY OF EMMANUEL LEVINAS\n\nA Sketch of Femininity and Maternity in Levinas\n\nAmbiguous Intersubjectivity in Levinas\n\nEthical Ambivalence\n\n5. MATERNITY AS DEHISCENCE IN THE FLESH IN THE PHILOSOPHY OF MAURICE MERLEAU-PONTY\n\nAmbiguous Intersubjectivity and Maternal Flesh\n\nFeminist Readings of Maurice Merleau-Ponty\n\nThe Logic of the Flesh\n\nMerleau-Ponty's Ethics (Implied or Inferred)\n\n6. MATERNITY AS NEGOTIATING MUTUAL TRANSCENDENCE IN THE PHILOSOPHY OF SIMONE DE BEAUVOIR\n\nIntersubjective Ambiguity and Existential Morality\n\nThe Existential and Ethical Significance of Motherhood\n\n\"Without Failure, No Ethics\"\n\nCONCLUSION: THE STRANGER OF MY FLESH\u2014AN EXISTENTIAL PHENOMENOLOGICAL ETHICS\n\n_Notes_\n\n_Bibliography_\n\n_Index_\nACKNOWLEDGMENTS\n\nFor Simone de Beauvoir, the realization of freedom is to exceed who one is and who one has been through engaging in projects that change the future in a direction she values. Yet the exercise of freedom is not just the glorious, uninhibited expression of free will. In taking action one is faced with disruptive questions about the boundaries between oneself and others and the implications of one's actions for them. Making use of one's freedom entangles one in the lives of others. Having a child and writing a book are two activities that make this starkly apparent. Doing both at the same time can instill in one enough gratitude to last a lifetime. One cannot do these things alone, but requires comrades, caregivers, collaborators, worthy rivals, people to stand guard over one's solitude and to knock on the door when it is time for a break. My chief hope is that they find the finished product to be worthy of their sacrifices.\n\nI'd like to thank those who funded my dissertation work, conference travel to discuss this work, and the book project: Mount Holyoke College Alumnae Association for the Mary E. Wooley Fellowship; the University of Wisconsin, Superior for a Faculty Development Grant; the Oregon Humanities Center; the Center for the Study of Women in Society; as well as the following fellowships administered by the University of Oregon Graduate School: the Charles A. Reed Fellowship, the John L. and Naomi M. Luvaas, and Betty Foster McCue fellowships.\n\nI wish to acknowledge my faculty mentors at the University of Oregon, especially Mark Johnson, John Lysaker, Bonnie Mann, and Beata Stawarska.\n\nFellow graduate students who were integral to my intellectual development and sanity include David Craig, Carolyn Culbertson, Caroline Lundquist, Jos\u00e9 Mendoza, Lucy Schultz, and Jessica Sims.\n\nFurther thanks are owed to my long-distance teachers and advisers: Ted George, Lisa Guenther, Dorothea Olkowski, Brian Schroeder, Eva Simms, Gail Weiss, Talia Welsh, William Wilkerson, and Jason Wirth.\n\nAll of my colleagues in the Department of Social Inquiry at the University of Wisconsin, Superior have been incredibly supportive, especially Deb Augsberger, Janet Blair, and Joel Sipress. My gratitude also goes out to Kaelene Arvidson-Hicks and Lisa Mattsson.\n\nStudents at University of Wisconsin who have been my philosophical community and genuine interlocutors include Eric Chan Burr, Yasmina Antcliff, Erin Blood, Taylor Gambos, and Benjamin Jesberg.\n\nI'd like to thank Hattie Peterson (and Amber) for loaning me their little cottage on the lake in which to complete the final revisions.\n\nThank you, Scott Poupore-Haats, for helping me negotiate life's many ambiguities.\n\nThe anonymous reviewers for Columbia University Press provided invaluable comments.\n\nA profound debt is owed to my family; including those nearby\u2014Robert Adams, Geneva LaChance Adams, Nancy Bird LaChance; those far away\u2014Joan Adams, Diane LaChance, Danielle Liberty, Elizabeth Parslow, and Kristan Stringer; and those who've passed on\u2014Charles Adams Jr., William Bird Sr., Marilyn Hawkins, Geneva LaChance, Raymond LaChance Sr., and Wayne LaChance. Special thanks to my Mamacita for letting me add her poem to the book.\n\nI feel deeply grateful to my dear compatriots in motherhood, Courtney Bates and Christa Badorek, who bravely share their own complex journeys with me.\n\nBeauvoir writes of the child drawing: \"By himself, he would not have dared to put confidence in those hesitant lines.\" These are the words that most aptly describe how I feel about your contributions to this work. Thank you.\n_One_\n\nMAD MOTHERS, BAD MOTHERS, & WHAT A \"GOOD\" MOTHER WOULD DO\n\nWoman seems to differ from man in her greater tenderness and less selfishness. Woman, owing to her maternal instincts, displays these qualities toward her infants in an eminent degree; therefore it is likely that she would often extend them toward her fellow creatures.\n\n\u2014CHARLES DARWIN, _THE DESCENT OF MAN, AND SELECTION IN RELATION TO SEX_\n\nWomen make us poets, children make us philosophers.\n\n\u2014MALCOLM DE CHAZAL\n\nOn April 14, 2011, just after 7:30 AM, LaShanda Armstrong deliberately drove her minivan into the Hudson River. Her four children, ten-year-old La'Shaun, five-year-old Landen, two-year-old Lance, and eleven-month-old Lainaina, were with her in the van. Only La'Shaun survived; he was able to roll down a window and climb out. He made his way to a nearby fire station shivering and incoherent. By the time rescuers were able to find and reach the vehicle, one hour later, the rest of the family had drowned. In a response to this incident, Cheryl Meyer, coauthor of _Mothers Who Kill Their Children_ recently made the \"conservative\" estimate that, in the United States, a mother kills a child once every three days. Clearly American mothers and children are in a state of emergency that calls for _immediate_ action.\n\nUpon hearing of such horrific incidents as that of Armstrong and her children, we must ask ourselves: how could a mother do such a thing? Maternal love and care are supposed to be a given. This is one of those rare, mutual assumptions shared by scientists and poets alike. We would prefer to think that such instances must be rare and pathological, the actions of a deeply disturbed or evil individual. We would like to believe that a mother like Armstrong has nothing in common with \"good mothers,\" that most mothers never share in the feelings and impulses that drove Armstrong to act as she did. However, as I will argue here, this is simply not the case.\n\nIn _Mothers Who Kill Their Children_ , Cheryl Meyer and Michelle Oberman devote a chapter to cases like Armstrong's, incidents of what they call purposeful filicide. This category is opposed to unintentional killing through abuse or neglect, murders that are assisted or coerced, and neonaticide (within twenty-four hours postpartum). Meyer and Oberman had initially intended to divide purposeful filicide into two categories\u2014cases with mental illness ( _mad_ mothers) and cases without mental illness ( _bad_ mothers). However, it soon became apparent that the cases were not so clear-cut. \"we realized that these women did not easily fit into a dichotomy (i.e., 'mad versus bad') but represented a diverse continuum, covering the entire spectrum of mental illness, ethnic and cultural group distinctions, and socioeconomic strata.\" Although in all cases the mother was experiencing some kind of emotional distress, the majority would not meet the legal requirement of insanity\u2014they were or would be considered competent to stand trial\u2014and the majority were deemed legally sane at the time of the murders. Only 8 percent suffered from postpartum disorders. Meyer and Oberman also found that these mothers could not simply be categorized as bad mothers; they were not neglectful, wicked, or uncaring. On the contrary, they were generally very dedicated to their children. Meyer and Oberman write:\n\nAt first glance, the mothers within this category seem like premeditated murderers who violently kill their children. However, upon deeper examination one of the most distinctive features of these women's stories was their devotion toward their children. While it may seem like an oxymoron to describe women who kill their children as loving mothers, by all accounts that is exactly what most of them were. The overwhelming majority of them had no history of abuse or neglect toward their children and most people who knew them spoke of their undying love for their children.\n\nThis seemed to be the case with Armstrong. In news reports neighbors describe her as a loving mother who was stretched beyond her limits. The twenty five year old was taking classes at a community college, holding down a job, and caring for her four children alone. A supervisor at the children's day care center described Armstrong as under enormous stress. The woman said that when Armstrong picked up her children the day before the murder \"the only thing she'd say was that she was so alone.\" A woman at Armstrong's neighborhood church stated: \"Sometimes she'd be holding the baby on her hip, and one child in each hand and trying to walk with her groceries at the same time, and she'd drop the diapers or something on the ground. She couldn't handle everything at once.\"\n\nInterestingly, out of mothers who committed purposeful filicide studied by Meyer and Oberman, more than half, 68 percent, were murder-suicides. Researchers believe that in many of these cases, women believe they are killing their children to spare them from a worse fate. Lita Linzer Schwartz, author of _Endangered Children: Neonaticide, Infanticide, Filicide_ , states that \"we see cases where the mother thinks the child would be better off in heaven than on this miserable earth. They think it's a good deed, a blessing.\" In some cases mothers fear leaving their children to an abusive father or to a broken foster care system. Oberman says that when some mothers feel driven to suicide \"she thinks, 'what would a good mother do?'\" and killing her children seems to be the lesser evil than leaving them without a mother. Jill Korbin, an anthropologist who has interviewed incarcerated mothers who killed their children, says that many of the mothers she spoke to believed that they were good mothers. Korbin states: \"And it's that very ideal of being a 'good mother' that is holding our society back from taking preventive action or intervening in a potentially abusive situation before it's too late.\"\n\nIndeed, the pregiven fact of maternal love is so widely assumed to be the case that scholars rarely question the existence, structure, and significance of maternal care. They attribute it to the essence of motherhood and\/or label it maternal instinct. In so doing, they ignore the complexity of mothers' responses to their children and foreclose inquiry into its significance. Maternal love, in reality, is part of a much more complex picture that includes contradictory impulses and emotions. Philosophers, too, are guilty of frequently trivializing maternity. Many who do consider the mother-child relationship treat it metaphorically, giving little consideration to its lived experience. These accounts idealize the mother's relationship to her child, emphasizing love, connection, and fecundity. In this case, philosophical metaphors of motherhood generally shed more light on long-standing stereotypes than the phenomenon itself. Those who resist this poetic impulse tend to consider pregnancy, childbirth, and mothering as obstacles to women's participation in public life and to their financial independence.\n\nThroughout this book I offer an alternative to the usual treatments of motherhood in philosophy and explore its significance in connection with the philosophies of care ethics, Emmanuel Levinas, Maurice Merleau-Ponty, and Simone de Beauvoir. I move the discussion from abstract notions of the feminine and maternal to consideration of the concrete experiences of mothers. My intention is to provide a more nuanced characterization of the mother-child relationship, one that highlights its conflicted nature. I argue that the ambiguity of human relationships results in an ambivalent ethical orientation, contingent as it is on negotiating the interrelated yet separable interests of the self and the other. Central to this discussion is a phenomenological description of maternal ambivalence (mothers' simultaneous desires to nurture and violently reject their children). My research shows that such feelings are fairly common and are the result of valid conflicts that exist between mothers' and children's interests. I argue that ethical ambivalence is morally productive insofar as it helps one to recognize the alterity of others, attend to the particularities of situation, and negotiate one's own needs and desires with those of other people. The maternal example brings human interdependence into relief while also affirming our independent (and often conflicting) interests. I maintain that it is _because of_ , not in spite of, the tensions inherent to mothering that it is an instructive case for ethics. This paradigm example demonstrates our need for an account of ethics situated in a world held in common where self and other share a relation of interdependent co-origination, yet conflict between self and other can still be made sense of. This understanding of _what is_ (phenomenologically speaking) allows us to more fairly assert _what ought to be_ and how we might get there.\n\nOne of my key concerns is how to keep ambivalence from finding its expression in filicide. To better appreciate this problem, I take two overlapping perspectives on the phenomenon\u2014the existential and the social. First, I assert that maternal ambivalence is the result of our existential condition, one that is perhaps impossible (and even undesirable) to change. Nevertheless, there are also certain oppressive social structures at play that heighten ambivalence to mortally dangerous levels. These existential and social factors operate in a dynamic interdependent tension. One cannot be understood without reference to the other; but there is a relationship of _motivation_ rather than determination between the two.\n\nIn the following chapter I examine the mother as ethical exemplar, as she is portrayed in care ethics. Care ethics invites theorists to expand their concerns to more intimate relationships. They insist we consider the material and social conditions of ethics and embolden us to think beyond atomistic individualism. In some ways my own project is an affirmation of care ethics, but my approach and many of my questions differ. I am concerned with a traditionally feminine enterprise, motherhood, and how this practice can expand our notions of the human being, the good, and the ethical. Since the counterpoint to traditional ethical theory has been so well achieved in feminist ethics already, I do not feel called to address certain questions that have become standard in care ethics. I am not concerned with whether or not mothers are sufficiently feminist or about feminine versus masculine virtues. Using the model of mothers does not imply that I wish to set up standards of care or criteria of ethical behavior. I will assume, however, that some behaviors are ethical and others are not (such as taking care of children versus killing them). I will not debate whether justice or care is more primary or how the two orientations relate to one another.\n\nAlthough care ethics sets important precedents for my research, it pays inadequate attention to the contradictions of mother-child relationships, especially what might drive a mother to murder a child for whom she deeply cares. In care ethicists' emphasis on interdependence, they sometimes ignore the need for individual flourishing and conflicts of interest between mother and child. We require an ethical theory that accounts for the needs to care, to be cared for, _and_ to maintain independence. Although care ethics does not, at this point, provide the full account, I believe that collaboration with feminist phenomenology can fill in some of the blanks. One of my hopes is to provide some new insight and directions for care ethics, and my own ideas about the future of ethical theory and action are compatible with some of theirs.\n\nThis book is my contribution to a phenomenology of ethics\u2014an exploration of ethical experience as it appears, paying attention to the meaning that emerges through description and inquiring into its philosophical significance. In chapter 3 I present an account of maternal experience, pointing out some themes and questions that arise from these descriptions. Using sources and examples from first-person narratives, psychiatry, psychology, sociology, anthropology, and history, I provide a description of maternal subjectivity and ambivalence, my claim being that clashes between mother and child frequently act as a rupture within the woman herself between her competing desires to nurture and to be independent. Maternal experience challenges the assumption that subjectivity is singular and reveals that the ethical draw of another can disrupt one's sense of self-coherence. Such conflicts are not unique to motherhood, but are especially intense because of the child's dependence and vulnerability, societal expectations of women (such as their being primarily responsible for children), the shared embodiment between mother and child, and our society's systematic neglect of caregivers and their dependents. It is my contention that ethical theory should be able to encompass the ambiguity of this relationship, including the fact that many mothers resist their impulses to abuse, neglect, and abandon their children in spite of both opportunity and motive to the contrary.\n\nMy approach is that of a feminist phenomenologist and has precedent in the works of Simone de Beauvoir, Iris Marion Young, Linda Fisher, and many others. Maurice Merleau-Ponty has also been influential in this branch of phenomenology. In his preface to _Phenomenology of Perception_ he writes: \"all [phenomenology's] efforts concentrate upon re-achieving a direct and primitive contact with the world, and endowing that contact with philosophical significance.\" Phenomenologists do not adhere to the point of view of the detached consciousness; instead it is through the first-person perspective that we gain the most basic insights into the world's meaning for human beings. This method is already consistent with some other feminists' manners of proceeding. For example, Sara Ruddick describes her method as follows: \"As an 'anthropologist,' I begin by remembering as honestly and deeply as I can my own experience as a mother and daughter and that of my closest friends. I then extend my memory as responsibly as I am able, by eavesdropping, by looking at films, and, most of all by mother-watching.\" Ruddick's \"anthropology\" is similar to that of phenomenology in that both are concerned with how individual experiences are shared by those in similar situations and give rise to meanings in common. Linda Fisher affirms this overlap in approaches in \"Phenomenology and Feminism: Perspectives on Their Relation\": \"Phenomenology and feminism share this commitment to descriptive and experiential analysis, where the systematic examination and articulation of the nature of lived experience, along with the attendant theoretical and practical implications, functions as the basis for reflective discourse.\" The best way to avoid the idealization of motherhood is by turning to its reality. With this in mind, I find it essential to consider the accounts of a number of mothers. They will be the measure of the theories' validity. As Edmund Husserl advocates, we must continually go \"back to the things themselves\": women's experiences as mothers.\n\nWhen the issue of maternal ambivalence arises, an immediate question that many people pose is \"What is the cause of maternal ambivalence?\" In both popular culture and scholarly work, maternal rage is frequently considered either a result of postpartum hormones, the product of a more pervasive pathology, or the consequence of living in a racist, heterosexist, and classist patriarchy that idealizes the nuclear family. These diagnoses shed necessary light on the hard facts that women confront, the situation they are responding to, and thus add to our understanding of their experiences. However, to begin the inquiry in this way invites us to _diagnose_ maternal ambivalence. This assumes that maternal ambivalence is first and foremost an atypical _problem_ to be overcome. Instead, I see it as primarily a phenomenon to be understood, one that can shed light on fundamental structures of our relationships with others. In considering this as a problem from the outset, too much has been decided beforehand. That is, we miss out on looking into its potentially positive, creative aspects. Moreover, I believe that pointing to specific causes of human behavior and feelings is suspect. I am skeptical that causes, whether linear or complex, can be adequately proven when it comes to our own \"nature.\" At best, I think that we can point to a propitious environment or a potential motivation. To diagnose a problem or to look for its cause from the outset is to move beyond the phenomenon at question. It is to look underneath it to find out \"what is really going on.\" In taking such an approach one forecloses the search to understand the inner coherence of the phenomenon itself, what it means to the person who is experiencing it and what is it that _inherently makes sense_ about this phenomenon in and of itself. Ultimately, if we jump to a search for causes, we forget to provide an account of the thing itself.\n\nThis is different from saying that the phenomenon can be separated from the situation in which it arises. We cannot take the threads from a fabric and still have a pattern within it. Situation includes all the material and social conditions in which a phenomenon arises. Without the situation, there is no phenomenon, but this is not to say that the situation _causes_ the phenomenon any more than threads can _cause_ the pattern in fabric to emerge. In both cases the resourcefulness, creativity, and agency of the human beings involved must be recognized. This is not to be ahistorical, to decontextualize the phenomenon. We must assume that context and history are vital and looking more deeply into a phenomenon ought to shed further light on the context itself.\n\nSome of the complexities of maternal life can be understood through the philosophy of Emmanuel Levinas, as I explain in chapter 4. Levinas argues that to be a subject is to be incessantly undermined by one's proximity to the other. Yet, in spite of this nearness, one cannot overcome the insurmountable alterity of the other. He asserts that we are drawn to respond ethically when we witness infinity and vulnerability in the face of another and illustrates his theory with images of femininity and maternity. While Levinas accurately depicts the ambiguous intersubjectivity described by mothers, he fails to acknowledge the practical significance and context of their conflicted ethical stance. He portrays the mother as an ethical prototype, a natural and willing \"hostage\" in his ideal of infinite responsibility; in his universe, maternal filicide is impossible. Levinas's philosophy advocates values that essentialize gender roles as they have been prescribed by patriarchy and elevates them to an ethical imperative. In addition, his emphasis on the powerful call of the face of the other, while important, denies the subtle interplay of active and passive ethical agency. Deliberative agency, I argue, is especially necessary to account for our ability to resist violence even when we feel a compulsion toward it. Finally, Levinas ignores the practical limitations to caring for others and fails to explore how one might negotiate conflicting interests. I conclude that his principle of infinite responsibility places an unfair burden on the individual as ethical agent, as it minimizes the importance of social and material factors.\n\nWhat remains extremely important in Levinas's account for our understanding of maternal experience is that _having dependents_ is a key factor in what makes us capable of ethics. Care ethics has thoroughly demonstrated that _to be dependent_ is central to being human and that the philosophical tradition has overlooked this. However, I contend, with Levinas's support (or more precisely with the support of Lisa Guenther's reading of Levinas), that _having dependents_ is perhaps even more threatening to our sense of enjoyment and independence. One's own dependence can be somewhat more easily concealed, and the feminist expos\u00e9 of the philosophical cannon, and of patriarchy more broadly, demonstrates this quite well. Historically, men have quite comfortably concealed their own dependency on the work of women and slaves, as wealthy countries continue to conceal their dependency on nature and exploited countries. However, I think that it is much more difficult to shake the dependence of others upon oneself. As Levinas aptly describes, the face of the other keeps popping up at the most inopportune moments. One is watching a late night movie and the ASPCA commercials with their neglected and abused animals comes on, or a Save the Children commercial with an impoverished child in Africa, India, or Latin America interrupts one's relaxation. Or one is out on a date and the homeless drunk asks for change. These interruptions are temporary, but, whether we decide to help or not help, their faces tend to linger. The reason they inspire ire is because they impose needs on us that are far greater than what we can fulfill, even if we gave them everything we have. They force us to confront our own perpetual ethical failure. As Levinas teaches, they impose on our self-centered enjoyments, sense of personal integrity and make us feel hostage to their needs. To have dependents is something that is simultaneously psychologically threatening and necessary for our full humanity. One need not have children to feel the dependence of others upon oneself, but they are what I call the primal parasite that makes this dynamic undeniably visible.\n\nTo deepen our understanding of intersubjective ambiguity as a promising way to connect it with the ethical ambivalence that mothers experience, I next turn to the work of Maurice Merleau-Ponty. Merleau-Ponty claims that each person experiences a prereflective coherence between herself and the world, between himself and others. In his later work he deems the things of the world, including people, to be expressions of one sensible flesh. While flesh establishes our continuity, it also involves an inherent dehiscence between the sensing and the sensed, and between the self and the other. Merleau-Ponty uses metaphors of maternity to illustrate these concepts, which do accurately capture the fundamental ambiguity of mother-child relations. Moreover, in his lectures on child psychology and pedagogy he makes empirical claims about maternal experience that are consistent with my findings. Upon comparing maternal experience to Merleau-Ponty's notion of the flesh, I propose that maternity can be aptly characterized as a case of dehiscence in the flesh.\n\nTo elaborate on his notion of ambiguous intersubjectivity and counter some of his critics, I explicate Merleau-Ponty's use of Hegel's \"immanent logic of human existence.\" I also refer to the Hegelian-style logic at work in Merleau-Ponty to demonstrate the manner in which alterity and interconnection relate in our ambiguous relations and how this is a more nuanced and useful view over those held by Levinas, Jean Paul Sartre, and Antonio Damasio. I claim that, although he neither wrote an ethics nor prescribed a vision of the good life, Merleau-Ponty does describe a life-world that comprehends intersubjective ambiguity, provokes the question \"What should I do?\" and indicates that the answer will often be conflicted and founded in ambivalence.\n\nWhile Merleau-Ponty looks at intersubjectivity in concert with our embodiment, Beauvoir adds to this hearty consideration of the social and material conditions of our ethical relations. This is the subject of chapter 6. Beauvoir argues that, when freely chosen, motherhood is a vital commitment to care for another and, moreover, that a person's ethical standing is indicated by how she negotiates the ambiguity between her independence and her responsibility to others. Motherhood heightens the possibilities for \"existential evil,\" since it provides the opportunities to dominate a vulnerable person and\/or to escape one's freedom in devotion to another. Whether or not the mother will be able to carry out her obligation depends greatly on how she responds to a typically highly constraining situation. Beauvoir posits that individuals and society as a whole should embrace an active responsibility for children; mothers must have opportunities to engage in other meaningful, enduring activities. I assert that having reliable support will not eliminate maternal ambivalence, but will enable mother and child to negotiate mutual existential transcendence.\n\nOf all the philosophers discussed herein, Beauvoir provides one of the most important contributions to understanding the ethical life of mothers; she explicitly acknowledges the likelihood of ethical failure. The romanticizaton of maternity has divided mothers into the categories of either naturally good or pathologically bad. This has meant that mothers feel an exaggerated guilt over the slightest feeling of aggression or impulse toward violence against their children. In contrast, Beauvoir realizes that our ethical obligations often outstrip our ability to meet them, that others tax our reserves, and that one may _justifiably_ need to assert one's own wellbeing over another. Recognition of this possibility for failure need not lead to despair. It is the beginning of considering how to reasonably meet competing demands when possible. It is to affirm that it is often unclear what choice should be made before examining the complex relationships involved.\n\nThroughout this book I consider feminists' engagements with each of the central figures. I aim to highlight the lively dialogue between feminists and existential phenomenology and to showcase feminist phenomenology. I often disagree with both critics and supporters on each of the figures. For example, in Beauvoir's case, I oppose critics who deem her account overly strife ridden and supporters who minimize the importance of conflict to Beauvoir's philosophy as a whole. In contrast to this, I maintain that Beauvoir's conflicted account of maternity (and of intersubjectivity more generally) is neither in deference to Sartre's philosophy nor is it a rhetorical strategy; it is an insightful recognition of maternity's joint potential to enrich and to devastate. In this case, the firsthand accounts of mothers provide support for Beauvoir's theory and breathe new life into the debates in secondary scholarship.\n\nFor Levinas, Merleau-Ponty, and Beauvoir, ambiguous intersubjectivity is the context of our ethical life, and it has a number of consequences we find to be true in the case of mothers. The needs of the other are integral to one's own, since my well-being is wrapped up in hers. I am compelled to respond to her cries and become almost helplessly drawn to meet her needs. Yet her desires will conflict with mine. I revolt against her demands and\/or struggle to balance them with my own. This results in an ambivalent ethical orientation; I want to both care for and abandon the other. In this instance passive responsiveness is insufficient. Conscious effort, skill, resources, and thought are all required to negotiate our competing, yet intertwined, needs. Levinas, Merleau-Ponty, and Beauvoir each provide an integral piece of this account.\n\nI view philosophy as a hermeneutic dialogue across generations (in the manner that Han-Georg Gadamer describes in _Truth and Method_ ). The works of a philosopher have a special relevance in the time and place they were conceived, but also have a unique potential to open up the meaning of contemporary concerns. In this investigation I seek to enable mutual critique and enlightenment between maternal experience and existential phenomenology. Bringing them into dialogue both tests and demonstrates the concrete relevance of philosophy to everyday experience. Attention to mothering reveals paradoxical truths that require a complex logic to comprehend them. In this regard, philosophy may be a critical player in challenging and expanding our conceptions of motherhood in society. Therefore, although I favor an interdisciplinary approach to understanding motherhood, I think that philosophers can have an especially important role in challenging long-standing misconceptions.\n\nChildren's vulnerability is profound and obvious; their dependence is undeniable and seems justified to even the staunchest individualists. Care ethics places such dependency relations at the foundation of ethical theory. They find that the mother-child relationship can help us recognize important factors we may overlook in other cases. We are all fragile bodies that require the goodwill and generosity of others. Human bonds are essential to human life, but those bonds also place us at risk. The boundaries between self and other are often unclear, but this can be more easily overlooked in the supposedly \"autonomous\" adult. Recognizing our interdependence helps us to understand the true context for ethical life. As Kelly Oliver states, \"When the relation between self and other becomes ambiguous, when identity is an exchange between self and other, only then can we begin to talk about ethics. Ethics requires a relationship between two that are neither identical nor autonomous.\" When considered in the context of the mother-child relation, because it can be among the most intimate and the most conflicted, our intersubjective ambiguity and ethical ambivalence come into most striking relief.\n_Two_\n\nTHE MOTHER AS ETHICAL EXEMPLAR IN CARE ETHICS\n\nEarly movements for women's emancipation sought specific rights for women, such as those for suffrage and the right to own property. Since the 1960s women have waged a more comprehensive battle for inclusion in public and political life. As a part of this struggle, feminists undermine the myth of binary genders that serves to separate us and our concerns from public notice. We fight against the fallacy that emotion, nature, and caretaking are the strict purview of women and that only men use reason in their affairs. We insist that the personal is political, that what happens in the political realm impacts the personal and vice versa. For example, feminists claim that domestic violence should be prevented by legislation and police enforcement and argue that when women are systematically denied access to education and finances they are not able to develop at full capacity. While women seek freedom from the imprisonment of feminine roles, we also point out that the domestic realm is ethically and politically significant; this work had been devalued in spite of the fact that all people rely on it.\n\nDrawing on the same claims as the wider women's movement, the feminist revolution also changed the field of ethics. Innovations first began as grassroots applications and then worked their way into academia. The term _feminist ethics_ came into use in the 1970s and 1980s. Today feminist ethics continues to challenge androcentrism in the ethical tradition through attention to a wide range of issues: the idea that women cannot be full moral agents, the fact that virtues traditionally attributed to women (for example, obedience and nurturance) give advantages to men, the devaluation of the domestic realm and its ostracism from the political, ignorance of the need to care for dependents, and traditional ethical theory's contradiction with \"a distinctively feminine moral experience or sensibility.\"\n\nIn 1982 a new branch of feminist ethics, care ethics, began with the publication of Carol Gilligan's influential and hotly debated book, _In a Different Voice_. In it Gilligan presents a challenge to Lawrence Kohlberg's moral stage theory by noting differences in the ethical orientations of males and females. According to Gilligan's studies, men and boys emphasize individual rights and consider _interference_ with another to be the primary violation. Meanwhile, women and girls tend to believe that ethical wrongdoing is the failure to take care of another in need; noninterference is a greater offense than interference. As such, the girls and women in Gilligan's studies repeatedly give voice to \"the ethics of care\": \"Her world is a world of relationships and psychological truths where an awareness of the connection between people gives rise to a recognition of responsibility for one another, a perception of the need for response.\" With this work Gilligan inaugurated a concern with uncovering and affirming a quintessentially feminine approach to ethics.\n\nWhile there is great diversity among care ethicists, they agree on several fundamental principles: caring is assuming an ongoing responsibility for and commitment to another. As Joan Tronto states succinctly, \"caring involves responding to the particular, concrete, physical, spiritual, intellectual, psychic, and emotional needs of others.\" These theorists affirm a relational ontology; relationships are deemed more primary than individuals. Individual rights, therefore, become secondary to responsibilities between people in relationship to one another. Care is based on interpersonal connection and can neither be performed from a detached, objective perspective nor deduced from abstract principles or hypothetical situations. They further posit that it involves taking on the point of view of another and cannot be generalized from one's own experience. Most care ethicists do not attribute this \"feminine\" approach strictly to women, but do agree that it emerges in the context of practices that are historically a woman's domain, especially caretaking of children, the infirm, and the elderly; women's traditional roles have made them more aware of the prevalence and importance of interpersonal dependence. Care ethicists encourage us to rethink personal and social accountability on the basis of an ethics of care and have brought this perspective to larger political and societal concerns. The addition of this perspective to international relations, for example, has enabled care ethics to propose innovative solutions to problems such as global poverty, environmental degradation, and widespread incarceration that a justice or rights-based perspective overlooks.\n\nThe fact that women have often been forced into the caretaking role has caused some feminists to worry that care ethics is a \"slave morality.\" They argue that differences between feminine and masculine ethics are induced by men's dominance of women and that to affirm these differences is to affirm the injustice that caused them. Nevertheless, every human being needs to be taken care of by someone else, at least in infancy. This is a basic fact of human existence, and, as I will continue to argue throughout this book, it is the _denial_ of the need for human care that allows care work to become as exploitative as it has been. For this reason, it is one of the major concerns of care ethics to make this hidden truth visible. This is the first step toward affirming the needs and rights of caretakers and fighting for the just treatment and support of both dependency workers and their charges. If this work is recognized as essential to the good of both individuals and society, shared more equitably and compensated fairly, then caregiving need _not_ be equated with slavery. In this case, the values of care ethics neither have to arise under nor contribute to oppressive circumstances; women's affirmation of care need not be a kind of Stockholm syndrome that is \"transforming vices of subservience into virtues of redemption.\" On the contrary, care ethics insists that ethical theorists consider the material and social conditions that both constrain and enable ethics and emboldens them to think beyond the staunch individualism of the past. This more honest conception of the human being enables ethical theory to be more realistic, practical, and expansive. As long as theorists acknowledge the historical and contemporary contexts in which \"caring\" has been a mechanism of oppression, these changes in ethical theory will contribute to the emancipation of women.\n\nThe conception of the human being as autonomous has provided for an impractical conception of ethics in which detachment and objectivity have been considered the only proper ethical orientations. While a human being might die alone, it is preposterous to say that anyone is born alone. We do not pop out of the ground like mushrooms as Thomas Hobbes would have us imagine, but out of the womb of a woman. Without a mother, or someone acting as a mother, no human infant would survive for a day. Thus, in care ethics, the example of mother and child appears time and again as the prototype for a caring relationship. What happens if we do not start with a mushroom mythos, but with the plain fact that we are all of woman born? What difference does this make for our ethical, social, and political thought if we recognize that our \"independence\" is made possible through reliance on others?\n\nTheorists have taken this tenet in several directions. Virginia Held employs the mother-child relationship in a thought experiment to challenge beliefs about the structure of relationships in our society. She finds that this relation, once it occurs, is not contractual or voluntary; it does not presuppose symmetrical reciprocity; the individuals involved are not strictly separate entities; harmony, love, and cooperation are valued; and a policy of noninterference is obviously morally insufficient. With different suppositions about relationships, the goals and motives of our society and its ethical theory can be rethought as well. Attention to mothering brings the value of care more clearly into view and advocates an ethical paradigm in which interdependence and engagement are the appropriate points of departure for theory and practice.\n\nEva Kittay proposes a principle around which such a paradigm might be constructed. She argues that since we are all \"some mother's child,\" each person is worthy of care and respect. This first principle recognizes the necessity of acknowledging the value of the caregiver's work. Those who are most vulnerable need respect and assistance, but so do their caregivers. Caregiving unfolds in a far-reaching social situation. The dependent individual is the catalyst for building an edifice of care in which caretakers serve as the pillars, but are themselves in need of the foundation of supportive friends, family, and society. The caregiver will not be able to help others if she does not receive sufficient nurturance. This apparently obvious point is frequently ignored to the detriment of everyone.\n\nOne of the most detailed and well-known considerations of maternal ethics is Sara Ruddick's _Maternal Thinking: Toward a Politics of Peace_. Mothers, she claims, are those women or men who meet a child's demands for preservation, growth, and social acceptability. These demands lead to the mother's activities of preservative love, nurturance, and training. She argues that reason arises in the contexts of particular practices and thus maternal practice encourages a unique brand of thinking. Since maternal practice relies on _both_ emotion and reason, maternal ethics mends the schism found between them in most traditional ethics. Although Ruddick asserts that mothers garner a distinctive intelligence, she is cautious not to romanticize them: \"Mothers are not any more or less wonderful than other people\u2014they are not especially sensible or foolish, noble or ignoble, courageous or cowardly.\"\n\nWhile there is considerable consensus among maternal ethicists, some important disagreements remain. Of particular concern is the question of whether or not mothers are naturally inclined to care for their children. Nel Noddings is especially insistent that natural inclination is the original impetus for all moral behavior and prescribes our understanding of the good. Again and again, Noddings affirms that natural caring is epitomized in the mother's response to her child: \"Responding to my own child crying in the night may require a physical effort, but it does not usually require what might be called an ethical effort. I naturally want to relieve my child's distress\" (17). \"Mothers quite naturally feel with their infants\" (31). The example of mother love, above all others, is used to reaffirm the distinction between her idea of natural caring and principle-oriented morality. Reason, the ethics of the father\u2014\"the detached one\"\u2014only becomes necessary when inclination fails (2). But, even then, our ideals are always grounded in a previous \"pre-conscious\" affirmation of caring (28). She claims that it is the father's objectivity that would have allowed Abraham to kill Isaac. \"But for the mother, for us, this is horrendous\" (43).\n\nAs I'll demonstrate more fully in the next chapter, Noddings's squishy conception of maternal care is a quintessential example of a destructive (nonfeminist) feminine ethics. Her supposedly pivotal, mind-exploding ethical question, \"what if this were my child?\" (36) is in foolish refusal of the obvious\u2014that real mothers, good mothers, do not always seamlessly care for their children. She underestimates the complexity of maternal emotion and the necessity of maternal thinking. As she depicts it, there is no actual ethical _conflict_ in need of resolution. She maintains the same old tiresome dichotomies of masculine versus feminine and reason versus emotion. Finally, her view is pitifully stereotypical. She depicts mothers as joyful and bottomless founts of loving compassion and thus places no limit on the care that one can or should provide for another.\n\nNoddings's formulation is not just simpleminded; it is dangerous. If maternal care simply comes naturally, then there is little reason to advocate for the support of mothers. Noddings's \"natural inclination\" assumption is oblivious to familial, cultural, and social factors that enable or disable women's caring. Secondly, she does not recognize that providing care may require the use of _all_ available human faculties, including reason and deliberative choice. She might as well side with Immanuel Kant in denying women access to education because the development of their ability to reason will only interfere with the services they provide to the family. Third, in naming the father \"the detached one\" she effectively denies men's nurturing, discourages them from developing these capacities, and ignores the fact that emotional distance and care are not always mutually exclusive. In fact, detachment may be the _best_ response of a loving parent. In moments of rage, as we will see in the next chapter, a mother may disconnect from her emotions as a coping mechanism to prevent her from harming herself or her child.\n\nFortunately, most care ethicists are not so naive. They are cautious not to reinforce the oppression of women by romanticizing women's traditional roles. For those who read her carefully, Gilligan deftly undercuts this potential criticism. Some of her subjects are women considering abortion, and Gilligan claims that for some of them a greater degree of ethical maturity coincides with the decision to have an abortion. While Gilligan does not necessarily advocate abortion as a caring practice, she also does not eliminate it. Abortion remains an option because Gilligan rightfully acknowledges that a mature and sustainable ethical position must include self-care. Even mothers must secure their own oxygen masks before helping another.\n\nRuddick provides further options for thinking about the structure of maternal care outside of natural inclination. She declares that \"mothers\" are not necessarily women; a mother is anyone who meets a child's needs for preservative love, nurturance, and training. Furthermore, she thinks there is no necessary reason why mothers care for their children upon birth. She opposes the idea of a state of fusion often associated with mother and child, insisting that women have a _need_ for individuation from their children. Instead of describing birth as a moment of sympathy between mother and infant, she emphasizes that it is a separation. Given this division of mother and infant, Ruddick believes that any mother's care depends on her \"adoption\" of the child\u2014an event after which she feels a lifelong commitment to care for it. This notion of adoption asserts that choice is involved in the care of a child; it is not automatic, immediate, or necessarily grounded in nature. It is a commitment that must recur repeatedly. As a mother who legally adopted her son writes: \"Thankfully, you can't get arrested for your private thoughts (I would have been behind bars long ago). I've since learned that loads of new parents\u2014both the adoptive and biological variety\u2014fear that they've ruined their lives at some point as their children evolve. Somehow, perhaps because all kids eventually will fall asleep and ultimately will grow up, you sign on for another day. And then another.\" Ruddick does not oversimplify or make light of this responsibility. As Kittay also affirms, it is a bond, a duty, and a commitment all in one. It is both voluntary and compelled, both reasoned and felt. The concept of adoption supports Ruddick's claim that reason and inclination do not need to be pitted against one another in order to affirm that the practice of mothering has something unique to offer ethics.\n\nUltimately, what does it mean to take the mother and child as presenting an exemplary ethical relationship? Kittay says that the mothering relation is meant to be an _analogy_ \"for social relations in which vulnerability is central.\" The analogy affirms a connection-based equality\u2014we are equal on the basis that we are all \"some mother's child\" and it is on this ground that we are entitled to care. For Held, the mother-child paradigm is a _thought experiment_ demonstrating that the competitive dominance-driven notion of the human does not cover all moral relations. Both these approaches are viable. Still, we must keep in mind that to make an analogy is not the same as declaring equivalence. Even when some common features can be identified, distinctions will remain between different types of relationships. While Held's thought experiment is also a useful approach in some respects, as I show in chapter 3, the mother-child relationship can also be competitive. That is, there can be very real conflicts between the needs and desires of mothers and those of her child.\n\nIn fact, it is not because the mother-child relationship is typically harmonious that it is an instructive case for ethics. Contrary to the idea that maternal care is primarily \"natural\" stands the well-known fact that mothers abuse and neglect their children. Such occurrences are commonly deemed aberrations. Noddings argues that in such cases the woman is either sick or in pathological circumstances, but this analysis lacks subtlety. First, it is true that conflicts are heightened because of conditions such as racism, poverty, ignorance, mental illness, and so on. However, even sane, loving, middle- to upper-class mothers with supportive families and communities may have shaken a baby or slapped a teenager. Second, maternal abuse, neglect, and discontent occur in varying degrees. Is a frustrated mother who occasionally yells at a child to be deemed mentally unsound? Clearly such occurrences need further investigation, but the negative aspects of mothering are not given enough consideration in maternal ethics.\n\nHeld says that she is not concerned about perpetuating an idealization of motherhood as long it is not \"more severely idealized than the relation between rational contractors that it is to replace; for the purposes of my exploration, it does not need to be less of an idealization.\" This should raise an alarm. The romanticization of mothering can be difficult to resist. Motherhood carries heavy symbolism and mothering\/being mothered is a significant formative experience for most people. Still, mothers are not straightforward moral figures. They are subject to the same failings as other people. And although they are charged with the social training of children, they are often still in the process of figuring out their own value system; raising children can cause them to question their beliefs even further.\n\nCare ethics theorists must remain vigilant against generalizing about the family and idealizing its relations. They must take into account the complexity and diversity of family relationships and social environment. If they cannot address injustices within and about the family, then they will reinforce them. It is also important to recognize that there are unique features to the mother-child relationship. The child's needs and demands are distinctive, the child is unequal to the parent in skills and knowledge, the parental role is a teaching role, the parent must tolerate and accept behavior that would be unacceptable in another adult, parents are responsible for children's well-being and not the reverse, parents are supposed to behave better than children, and they must sometimes make executive orders over their children. All of this means that any discoveries regarding the mother-child relationship cannot be blindly applied to all relationships. However, we also cannot assume that an asymmetrical ethics is always suitable even in this relationship.\n\nIn order to act as a useful example, this relationship must be understood as thoroughly as possible. This is why I strongly advocate an interdisciplinary approach to studying mothering. We ought to be making use of all available perspectives and avoid relying on popular assumptions about motherhood or on the experiences of a few if this investigation is to be adequate from a feminist point of view. None are harmed more by the maternal ideal than mothers and children.\n\nAs I will demonstrate more fully in the following chapters, attention to the phenomenon of maternal ambivalence brings to light some questions that maternal ethics does not address: Human beings _do_ rely on each other for survival, but what are the ramifications of this for the self that is constituted in these relations? If mothers recognize human interdependence, then why do they often crave freedom and independence? If mothers are more inclined, as some would believe, toward a care-oriented perspective, then why do they sometimes abuse, neglect, and kill their children? Given that mothers sometimes feel hatred and rage toward their children, why do they frequently act in contradiction to these emotions? If mothers need to _adopt_ their children, what motivates them to do so?\n\nWhen these questions are considered, new aspects of mothering will emerge. One discovery that will have ramifications for ethics as a whole is the fact that mothers need independence from their children\u2014including alone time, a sense of boundaries, and opportunities to flourish outside of mothering activities. In its emphasis on interdependence, care ethics often ignores the need for personal flourishing. However, maternal accounts reveal a fundamental ambiguity for all human beings; we have _simultaneous_ needs to nurture, to be nurtured, and to maintain independence. This tension between ourselves as independent and ourselves as entangled in indissoluble bonds with others becomes acutely apparent in maternal ambivalence. The conflicts that can arise between caring for another and preserving one's independence can be exceedingly distressing. Even women who had previously been very independent typically find their participation in public life and the pursuit of their own projects, at least temporarily, curtailed. This seems especially difficult for women who have fought hard for their emancipation. Even those who are freed of the rule of the father may find themselves chafing under the yoke of children.\n\nCare ethics validates the inquiry into familial relationships against traditional philosophical disinterest in these concerns. They require us to rethink the sources of moral values and interactions by expanding our notion of the origins of morality to include nurturance and care. Some theorists rest care ethics on the maternal relationship, making it the quintessential model for our ethical attitudes, motives, and acts. One problem with this approach is that philosophers lack an adequate understanding of mothering itself. On this point, care ethics would benefit from greater collaboration with other disciplines in the humanities and social sciences. Most vitally, we must attend to the voices of more mothers on this topic. If mothering is to be a useful example for ethics, we need to understand how the nurturance of children happens for diverse groups of women. As I will demonstrate in the next chapter, the experience of maternal ambivalence reveals a more nuanced account of our interpersonal relationships than that provided by care ethics. Its existence demonstrates that one can act ethically (at the minimum, not kill one's child and give it what it needs to survive another day) in spite of the overwhelming desire to do otherwise.\n\nCare ethics and traditional rights-based ethical theory seem to serve different masters. Care ethics provides for the person in need and for her caretakers. Traditional theory enables the individual to pursue her own ends without interference. Maternal experience points to the value of _both_ of these approaches. In accounts of maternal ambivalence we witness an ambiguous subject who finds herself caught in a bind; providing for the other can frustrate her own pursuits, yet satisfying the needs of the other can be as vital to her as fulfilling her own. Thus this enigmatic experience reveals implications for ethical theory more broadly\u2014the need to reconcile care with individual freedom. It shows that doing the right thing does not always come naturally; thought and sometimes the resistance of impulse are necessary. It affirms the needs to care, to be cared for, _and_ to maintain independence. We need an ethics that argues for the value of all three. No simple amalgamation of care ethics and traditional ethics will bring these two aspects of the human condition together. What is needed is a third approach that can conceive of the _ambiguity_ between these apparently contradictory motives. Such an ethics will be aware of the fundamental ambiguity at the heart of our existence; it will contend with the fact that distance and relation with other people is always being negotiated.\n_Three_\n\nMOTHERHOOD'S JANUS HEAD\n\nFor finding your mother\n\nThere's one certain test.\n\nYou must look for the creature\n\nWho loves you the best.\n\n\u2014DAVID KIRK, _LITTLE MISS SPIDER_\n\nI did realize before [I had children] that people come up to you and say silly things like how can anybody ever batter children? I would say that it must be the easiest thing in the world.\n\n\u2014GILLIAN HARTLEY, QUOTED IN OAKLEY, _BECOMING A MOTHER_\n\nAt the Uffizi Gallery in Florence there is a series of six panels by the Renaissance painter Antonio Pollaiolo. Each one depicts a different virtue as a woman\u2014prudence, faith, hope, justice, temperance, and charity. Charity is the only one of these figures who holds a child. Women have been idealized in many ways, but the mother is the ultimate model for generosity, compassion, and altruism. She is counted on to help the helpless, even when it is to her own detriment. On the other hand, it is common knowledge that mothers sometimes neglect, abuse, and even murder their own children. Such incidences generate feelings of disgust, horror, and severe judgment. Mothers like this are thought to be abominations, pathological, violations of the very definition of _mother_.\n\nIn this chapter I will describe the experiences of women who are neither self-immolating saints nor pathological murderesses. Their feelings about their children are mixed and varied\u2014not just from woman to woman but within the selfsame woman. A loving mother can be mean, a giving mother can be selfish, a content mother can be filled with rage. In accounts of maternal ambivalence, women describe a passionate love for their children paired with fantasies of suicide, infanticide, or accidental death as the only way out of motherhood. They wish they could, somehow, reverse the fact of their child's existence:\n\nI could remember little except anxiety, physical weariness, anger, self-blame, boredom, and division within myself: a division made more acute by the moments of passionate love, delight in my children's spirited bodies and minds, amazement at how they went on loving me in spite of my failures to love them wholly and selflessly. . . . My children cause me the most exquisite suffering of which I have any experience. It is the suffering of ambivalence: the murderous alternation between bitter resentment and raw-edged nerves, and blissful gratification and tenderness. Sometimes I seem to myself, in my feelings toward these tiny guiltless beings, a monster of selfishness and intolerance. Their voices wear away at my nerves, their constant needs, above all their need for simplicity and patience, fill me with despair at my own failures, despair too at my fate, which is to serve a function for which I was not fitted. And I am weak sometimes from held-in rage. There are times when I feel only death will free us from one another, when I envy the barren woman who has the luxury of her regrets but has a life of freedom and privacy.\n\nI sometimes think she's a cow. I _hated_ her at first, not so much in hospital, but the first week or two here: I just wished she'd never been born, couldn't _bear_ her. It was a mixture really, half and half, I'd be either madly in love with her, she's ever so good, dear little baby, terribly protective and all that, and then sometimes I'd think oh: I'd wake up and remember there was a child there . . . I just felt _so_ miserable . . . I just wanted to die. I would happily have committed suicide.\n\nRomantic ideals of motherhood tell us that our children bring out the best in us. True enough, I've done things for my kids that I never would have dreamed I was capable of doing; they've showed me my deep capacity to love, to value someone else's life more than I value my own. But . . . I wish there wasn't a but. Who can imagine feeling hatred for one's own child? Who can imagine anger\u2014not just fleeting anger but a smoldering rage\u2014toward an infant? I didn't know it was possible until I experienced it. . . . Sometimes mothers are mired in regret over the decision to have a child. Good mothers. Normal mothers. Mothers who don't hit or starve their children, who never lock them in the closet or leave them in the car while they go into the store. Some mothers who love their children also hate them.\n\nI was becoming convinced that I was the only mother in the world who had such hateful feelings for the child I also loved so intensely, who wished over and over that it had never happened, who, finally, could understand those women I had met when working for the Welfare Department who had burnt their babies' arms, beat their faces, killed them. But I would never breathe a word of such identification, I decided. I would hide my real feelings in order to avoid the terrible looks which say, I am not like you nor have I ever been.\n\nAccounts of maternal ambivalence, depression, and aggression can be found in a variety of sources\u2014first-person narratives, psychiatry, psychology, sociology, anthropology, and history. Each one attests to the hidden prevalence of these responses to motherhood. In _The Mother Knot_ , her memoir of the first year of motherhood, Jane Lazarre writes: \"The only thing which seems to me to be eternal and natural in motherhood is ambivalence and its manifestation in the ever ongoing cycles of separation and unification with our children.\" In _Maternal Desire_ , psychologist Daphne de Marneffe writes of ambivalence: \"What is remarkable about these moments is not that they are a rarity or crisis, but rather that they are so routine. If there is any feature we can reliably assign to the experience of mothering, it is that it will encompass extreme states of feeling.\" In _Mothers: Their Power and Influence_ , psychiatrist Ann Dally argues that maternal ambivalence is widespread, but such feelings are so unacceptable that many women cannot consciously admit them: \"Mothers are not infrequently distressed by sudden desires to injure or torture their babies, and they are not necessarily bad or unsuccessful mothers. Other mothers have similar thoughts but are unable to acknowledge them as their own. The hostile ideas seem to come out of the blue and from elsewhere. Such mothers usually feel a great need to reassure others that they love their babies, that such thoughts are 'utterly alien' to them and so on.\" In _Of Woman Born_ , poet and feminist theorist Adrienne Rich also notes the commonness of such feelings. She describes an evening meeting of women poets in 1975 who discussed Joanne Michulski\u2014a mother of eight who had murdered and decapitated her two youngest children on her suburban front lawn. Rich comments that \"every woman in that room who had children, every poet, could identify with her.\" In \"Normal Motherhood: An Exercise in Self-Control?\" Ann Oakley, a sociologist, explains that two-thirds of mothers in her study \"expressed negative feelings and ambivalence.\" In _Becoming a Mother_ she also saw a strong correlation between being a mother and feeling depressed: \"Surveying the mental health of women, researchers in London have found that one in three women have definite psychiatric symptoms of depression and that the likelihood of becoming depressed is crucially related to motherhood.\"\n\nA number of scholars have discovered maternal aggression and neglect in diverse historical and cultural contexts. In _Mother Love: Myth and Reality_ , historian Elisabeth Badinter disputes the existence of maternal instinct or any inevitable desire among women to care for and love children. She cites the many forms maternal indifference has taken throughout history. A sampling of these are insensitivity to the death of a small child, the refusal to nurse, and sending infants off to be raised by strangers (where their chances of survival were very poor). Badinter concludes that, before the end of the eighteenth century in Europe, parental care was erratic at best.\n\nIn _Mother Nature: Maternal Instincts and How They Shape the Human Species_ , Sarah Blaffer Hrdy, an anthropologist and primatologist, provides an analysis of maternal hate and infanticide from the perspective of evolutionary biology. Hrdy notes the widespread denial of maternal ambivalence: \"Many people snatch at implausible straws so as to cling to the conviction that the emotional ambivalence many mothers feel about investing in infants is 'unnatural,' and hence very rare, and completely separate from more common, or 'normal,' maternal emotions.\" Hrdy affirms that the murder and abandonment of children are not an historical or cultural anomaly:\n\nI still recall the crisp autumn day in the old cathedral city of Durham, England, when at a conference on abandoned children, the full extent of a phenomenon I had been aware of for years sank in. The talks were routine scientific fare. Overhead projectors flashed graphs and charts onto a screen. The black lines sprawling across the grid summarized data from European foundling homes, tracking changes in infant mortality rates over time. As the morning wore on, the phenomenon of child abandonment was described, country by country, epoch by epoch, for England, Sweden, Italy, even Portugal's colony in the Azores. Gradually it dawned on me that this phenomenon affected not tens of thousands or even hundreds of thousands of infants, as I had long assumed, but millions of babies.\n\nTo say that maternal aggression, depression, and ambivalence can be found in a variety of times and places is not to claim its universality; it is not to deny a relationship to cultural pressures and constraints. Perhaps there is one happy land where mothers are so well supported that they never experience a negative feeling toward their children. However, in this research, which spans cultures, classes, races, and historical time periods, maternal ambivalence is found to be pervasive. Even if it is impossible to assert universality, it nevertheless _does_ contradict the pervasive denial of maternal ambivalence, the tendency to label ambivalent mothers unnatural or pathological, and the taking of mothers' love and care for granted. Ultimately, I claim that maternal ambivalence results from mother's efforts to achieve both intimacy and separation in relation to their children and that this is likely to be an aspect of all mother-child relationships. Nevertheless, it is critically important to recognize the role of situation in either exacerbating conflicts between mother and child or supporting women in their attempts to negotiate them.\n\nPSYCHOLOGY AND PSYCHOANALYSIS\n\nDetailed theories about the structures of maternal ambivalence come largely from psychology and psychoanalysis. Rozsika Parker's _Torn in Two: The Experience of Maternal Ambivalence_ is the definitive contemporary resource on maternal ambivalence. Parker, a clinical psychologist, draws on her psychotherapy practice, interviews, personal experience, and feminist and psychoanalytic texts in her account of maternal ambivalence: \"I refer to the fleeting (or not so fleeting) feelings of hatred for a child that can grip a mother, the moment of recoil from a much-loved body, the desire to abandon, to smash the untouched plate of food in a toddler's face, to yank a child's arm while crossing the road, scrub too hard with a facecloth, change the lock against an adolescent, or the fantasy of hurling a howling baby out of the window.\" Parker states from the outset that while maternal ambivalence is experienced by all mothers at some point, their reactions to it vary broadly: \"There is, however, a spectrum of maternal feeling, from thoughtful containment to retaliatory abuse, a spectrum from pulling a zipper roughly to outright acts of violence. It is perhaps reassuring to think in neatly bifurcated terms of those mothers who contain and those who retaliate, but small moments of aggression will well up and pepper even the most caring of mothers . . . I am suggesting that most mothers do indulge in small acts of revenge.\"\n\nEarly psychoanalysts also recognize the pervasiveness of maternal animosity. In \"Hate in the Countertransference,\" D. W. Winnicott draws a comparison between the psychoanalysts' work with a psychotic and a mother's relationship to her baby, defining the infant as one who \"cannot understand what he owes to his mother.\" Winnicott declares, \"The mother . . . hates the infant from the word go\" and gives a list of eighteen reasons why she would, including: \"The baby is a danger to her body in pregnancy and at birth. . . . He is ruthless, treats her as scum, an unpaid servant, a slave. . . . He is suspicious, refuses her good food, and makes her doubt herself, but eats well with his aunt. . . . If she fails him at the start she knows he will pay her out forever.\" Winnicott shows respect for the difficulties of motherhood, but does not acknowledge that mothers sometimes fail to contain their frustration: \"A mother has to be able to tolerate hating her baby without doing anything about it. . . . The most remarkable thing about a mother is her ability to be hurt so much by her baby and to hate so much without paying the child out, and her ability to wait for rewards that may or may not come at a later date.\"\n\nIn _Totem and Taboo_ , Sigmund Freud identifies not only the existence of maternal enmity, but true ambivalence:\n\nEvery psychoanalyst knows how infallibly this anxious excess of tenderness can be resolved even under the most improbable circumstances, as for instance, when it appears between mother and child, or in the case of affectionate married people. Applied to the treatment of privileged persons this theory of an ambivalent feeling would reveal that their veneration, their very deification, is opposed in the unconscious by an intense hostile tendency, so that, as we had expected, the situation of an ambivalent feeling is here realized.\n\nFreud explains that mothers may display excessive affection toward their children as a reaction formation to conceal their hatred. This by no means _resolves_ a mother's ambivalence; her animosity remains concealed in the unconscious.\n\nMelanie Klein is well-known for her account of infantile ambivalence. She thinks that the infant regards his mother as two separate people\u2014the mother who gratifies his needs (the good breast) and the mother who frustrates them (the bad breast). Meanwhile, the _mother_ ' _s_ feelings toward her child are not a reflection of her current relationship with that child, they reflect her childhood feelings toward her own mother: \"We already know that this early relationship is characterized by the conflicts between love and hate. Unconscious death-wishes which the child bears towards her mother are carried over to her own child when she becomes a mother.\" While Klein recognizes the existence of maternal ambivalence, she minimizes its significance and force. She claims that a girl's desire for a baby, present from early childhood, diminishes the aggressive tendencies she might feel toward her actual baby. Her child's helplessness and vulnerability fulfill her lifelong desire to offer \"more love than can be given to any other person.\"\n\nThe most detailed psychoanalytic account of maternal ambivalence is provided by Helene Deutsch in the second volume of _Psychology of Women_. In chapter 9, which is devoted to the mother-child relation, Deutsch declares that \"woman's two greatest tasks as a mother are to shape her unity with the child in a harmonious manner and later to dissolve it harmoniously.\" Yet the chapter details the many pitfalls mothers experience in attempting to achieve this \"harmony,\" and much attention is given to mothers' various neuroses.\n\nIn spite of these few examples treating maternal ambivalence, classical psychoanalysts rarely consider the psychic life of the mother. The mother-child relationship is considered in retrospect, from the point of view of the child and for the sake of diagnosing adult pathology. Ambivalence is typically attributed to the infant, child, neurotic, and\/or psychotic.\n\nIn \"A Case of Obsessional Neurosis,\" Freud declares that \"in every neurosis we come upon the same suppressed instincts behind the symptoms . . . hatred kept suppressed in the unconscious by love.\" In \"Thoughts for the Times on War and Death,\" he makes a similar claim: \"It might be said that we owe the fairest flowering of our love to the reaction against the hostile impulse which we sense within us.\" Ambivalence is one of Freud's explanations for equinophobia in the case study of Little Hans. Freud thought that horses were a substitute for Hans's father, toward whom he was ambivalent. Little Hans's negative feelings toward horses were copresent with feelings of curiosity and esteem. Although classical psychoanalysts regard emotional ambivalence as a predictable element of psychic life, when repressed it is to blame for fixations, phobias, and obsessions.\n\nNevertheless, ambivalence also has a constructive potential. As Parker correctly asserts, maternal ambivalence is neither a moral nor a psychological failure; it can positively contribute to the relationship between mother and child and enhance the mother's own psychological development. However, before presenting these conclusions, it is necessary to provide a more thorough description of maternal ambivalence.\n\nTHE TWO FACES OF MATERNAL AMBIVALENCE\n\nMost simply, maternal ambivalence can be described as the simultaneous and contradictory emotional responses of mothers toward their children\u2014love and hate, anger and tenderness, pity and cruelty, satisfaction and rage. But it would be incomplete to think of maternal ambivalence as simply an emotional reaction. It can be better understood through Martin Heidegger's concept of _Befindlichkeit_. _Befindlichkeit_ , sometimes translated as mood, more literally means \"how you find yourself in the world.\" It is not just an internal state, but emphasizes that _how you find yourself_ is in your orientation to the world; it is your relationship to people, events, and facts. Consider the way one relates to the world when depressed. Everything is exhausting, hopeless, and ugly. It is not just the _person_ who is depressed, that person's _world_ is depressed. It is, _in fact_ , more arduous to take action in the world. Depression is not just a subjective perspective on the world; it changes _who one is_ in relation to the world and how _the world is_ in relation to oneself. Similarly, the ambivalent mother is not just an internal subject who is having a feeling about the objective world. When we consider maternal ambivalence as _Befindlichkeit_ , we ask how the ambivalent mother _finds herself_. _Where is she_ in relationship to her children?\n\nThe ambivalent mother finds herself in two opposite orientations toward her child(ren). They share both a relationship of conflict and a relationship of happy mutuality. On one side, both mother and child receive some reward from the relationship. Their interests and desires are held in common; what benefits one, benefits both. We can see this positive reciprocity when a child asks to be held and the mother wants to hold her, when both enjoy an outing together, or during a mutually enlivening conversation. On the other side, the interests of mother and child are at odds with one another, e.g., when a child needs help with homework, but the mother has her own work to do, when a mother needs to relax after a long day and a child wants to tell a rambling story about her day, or when a baby wakes her exhausted mother in the middle of the night. In some cases it is impossible for both mother and child to get what they want and\/or need. They interfere with each another, and their feelings are discordant.\n\nYet even this portrayal is too straightforward. Mothers often feel as though their own desires are directed against themselves when they are in opposition to their children's needs and wishes. When one's beloved child cries in despair at one's departure, one may _both_ want _and_ not want to leave. When the mother simultaneously desires intimacy and distance in relation to her child; when she feels the impulses to both harm and protect, to both abandon and nurture; this is when maternal ambivalence is at its perplexing height. At these moments a woman can feel most displaced by motherhood, as she simultaneously looses and finds herself in relation to her child. First, I will discuss each of these moments\u2014of mutuality and of conflict\u2014individually and then I will describe how they come together in an ambiguous intersubjectivity within which the child is experienced as both intimately tied to oneself and as a stranger.\n\nMUTUALITY\u2014MY CHILD, MYSELF\n\nTo suffer with and for and against a child\u2014maternally, egotistically, neurotically, sometimes with a sense of helplessness, sometimes with the illusion of learning wisdom\u2014but always, everywhere, in body and soul, with that child\u2014because that child is a piece of oneself.\n\n\u2014ADRIENNE RICH, _OF WOMAN BORN_\n\nThe maternal body seems perhaps more permeable than most. However, all bodily boundaries are permeable by necessity; the skin and orifices provide means of exit and entry for food, air, drink, lovers, and children. This penetrability means the body is open to both danger and pleasure\u2014viruses, wounds, sex, and so on. In spite of the body's accessibility, it typically maintains its integrity and provides a relatively stable location for the sense of self. In _The Feeling of What Happens_ , Antonio Damasio argues that \"things are [either] in or out of you\" and that for each body there is no more than one person. The idea that \"one body goes with one self\" is what he calls the first principle of the singularity of self: \"One key to understanding living organisms, from those that are made up of one cell to those that are made up of billions of cells, is the definition of their boundary, the separation of what is _in_ and what is _out_. The structure of the organism is inside the boundary and the life of the organism is defined by the maintenance of internal states within the boundary. Singular individuality depends on the boundary.\" Damasio's claim seems obvious enough, but maternal experience complicates this ostensibly straightforward \"first principle.\"\n\nFrom the perspective of a pregnant mother, another person, literally, inhabits her body with her. In \"Pregnant Embodiment,\" Iris Young considers the fact that in pregnancy another person exists within the boundaries of one's body. She contemplates how the presence of the fetus within the mother challenges the unity of the mother's body by conflating her sense of interiority and exteriority. Young states,\n\nThe integrity of my body is undermined in pregnancy not only by this externality of the inside, but also by the fact that the boundaries of my body are themselves in flux. In pregnancy I literally do not have a firm sense of where my body ends and the world begins.\n\nThe pregnant subject, I suggest, is decentered, split, doubled in several ways. She experiences her body as both herself and not herself. Its inner movements belong to another being, yet they are not other, because her body boundaries shift and because her bodily self-location is focused on her trunk in addition to her head.\n\nYoung points to the fact that, to some degree, to share one's body is to share one's self. To muddle the physical boundary of the self is to muddle the psychological boundary between self and other. Damasio is correct that \"singular individuality depends on the boundary,\" but in pregnancy it is not accurate to say that \"things are either in or out of you.\" In pregnancy, neither subjectivity nor the body is singular.\n\nThis intertwining between self and other is also experienced in breastfeeding. Through nursing, two bodies share one physiological process. Consider these descriptions of this alliance between mother and child:\n\nFaith [the author's daughter] is the consumer, and I am the consumed . . . I am milk. My milk is me. When Faith cries for me, she cries for milk, for the breast stuffed into the mouth. It is all the same thing. I am eaten. I have never felt more alive. I am eaten. I have never felt more abolished. The oneness of breastfeeding is totalizing. . . . We even sleep and wake in tandem, my eyes opening just before she begins to stir in the bed next to me. I pull her down to my breast, check for a wet diaper. I'm getting good at this. We drift back to sleep, conjoined.\n\nThe day after Lea was born I watched her sleep next to me on the big bed. Even before she stirred I could feel the milk come down into my breasts. Lea whimpered, opened her eyes, and I picked her up. Greedily she nuzzled around for the target, locked on, and began sucking.\n\nThe need of a mother for her baby to consume the milk she produces is a very strange phenomenon. My body was ruled by her need, her need called forth from it an immediate and uncontrollable response. . . . I made milk, smelled like milk, was sticky with this stuff that was me, but not me, that produced in me the need to give it away. . . . The miracle is that she and my body were one, that she, more than myself, controlled what my body made in milk. . . . I often looked at her and marveled that even though she had left my body, she still grew through it. Milk was the line that tied us together.\n\nAfter only several days of his life, we both felt that the breast was his. As he drew milk out of me, my inner self seems to shrink into a very small knot, gathering intensity under a protective shell, moving away, further and further away, from the changes being wrought by this child who was at once separate and a part of me. Frightened that he would claim my life completely I tried to cling to my boundaries. Yet I held him very close, stroked his skin, and imagined that we were still one person.\n\nMany mothers revel in the embodied connection of nursing. Even in times of extreme alienation and crisis in the relationship, it can provide a bond:\n\nBut what nobody understood was that the breast-feeding was my only real connection to the baby. If I were to eliminate that, I might have no hope of coming through this nightmare. I was hanging on to the breast-feeding as my lifeline. It was the only thing that made me unique in terms of caring for her, and it created an undeniable connection, even if only a physical one. Without it, she might be lost to me forever. If I continued, maybe I would eventually become less cut off from my own flesh and blood. I wanted to recede into the distance, but her mouth on my body kept forcing me to be present. I refused to quit.\n\nThe thoughtful responsiveness of these women to their infants mirrors the way nursing mothers' and babies' bodies reply to one another. Mother and child _together_ create the possibility for making milk. The breast responds to the baby's hunger by producing the amount that she needs. Operating by supply and demand, the breast is \"a self-replenishing vessel.\" Nursing also can help to suppress ovulation in the mother (preventing another pregnancy from occurring too soon) and to contract the uterus to its prepregnancy size. Meanwhile, breast-feeding does much more than simply feed the child. The breast-fed baby receives the mother's _live_ white blood cells capable of killing giardia, trichomonad protozoans, dysentery amoebas, and E. coli bacteria on contact (231). Many of the changes that breast-feeding brings impact both mother and child for the rest of their lives. Women who breast-feed dramatically reduce their chances of getting ovarian or breast cancer (228). Breast milk also contains chemical signals that direct the baby's development and impact every system in the child's body (234). This can best be observed in the health advantages enjoyed by breast-fed children into adulthood. They have fewer respiratory, gastrointestinal, urinary tract, and ear infections. Nursing also dramatically reduces their chances of getting bacterial meningitis, high blood pressure, allergies, asthma, juvenile diabetes, Crohn's disease, ulcerative colitis, juvenile rheumatoid arthritis, and Hodgkin's lymphoma (225\u2013228).\n\nThe times of pregnancy and nursing are brief and cannot simply stand in as symbols for the entirety of motherhood. But mothers attest that embodied mutuality exists beyond the physiological collaborations of pregnancy and breast-feeding:\n\nLovers make you a gift of your body; so do children. The body again becomes distinct, edged, a marvel you'd forgotten, retrieved by the unexpected: your belly slimy with gel. . . . Or your newly shaved leg, buffed by the smooth ecstatic face of a sybaritic three-year-old. . . . My daughter, now two, has started licking me. The start of a hug, arms around my neck; or I'm bending down, picking up little plastic kitty cats off the floor, and her tongue laps my cheekbone, flicks the crater of my eye. . . . Her licks have the power to completely alter my mood, capture my full attention, put me under her spell, which is no doubt part of the reason she loves to lick me. . . . As a young woman, I thought about my body all the time: how to disguise it, to shape it, to present it, to comfort it; what everyone else thought of it. Motherhood taught me to live in it.\n\nA child can get under one's skin (in a positive _and_ negative sense) regardless of their age: \"Eight years in, I can't always tell the difference between my children's needs and my own. . . . The needs of our children and our world and our selves merge and divide and merge again, until sometimes you can't tell one strand from another.\"\n\nThe \"undeniable connection\" described by mothers cannot be reduced to mere mother-child enmeshment. As a healthy child develops, every milestone achieved is a step toward a more independent existence. Mothers are certainly aware of this fact. Even as they cherish the intimacy with their children it is already slipping away. \"Now that she sits up, has cut two bottom teeth, and is beginning to enjoy solid food, I know my milk supply will wane as more and more of her calories come directly from the rest of the world, unfiltered through my body. I'm beginning to perceive motherhood as a long, slow letting go, of which birth is just the first step.\"\n\nTo some women, however, the first step in this differentiation is actually conception. Julia Kristeva, for example, considers pregnancy to be, first and foremost, a rupture within the mother's body:\n\nMy body is no longer mine, it doubles up, suffers, bleeds, catches cold, puts its teeth in, slobbers, coughs, is covered with pimples, and it laughs. . . . There is him, however, his own flesh, which was mine yesterday. . . . There is this other abyss that opens up between the body and what had been its inside: there is the abyss between the mother and child. What connection is there between myself, or even more unassumingly between my body and this internal graft and fold, which, once the umbilical cord has been severed, is an inaccessible other? My body . . . and him. No connection. Nothing to do with it. . . . I confront the abyss between what was once mine and is henceforth but irreparably alien. Trying to think through that abyss: staggering vertigo. No identity holds up. A mother's identity is maintained only through the well-known closure of consciousness within the indolence of habit, when a woman protects herself from the borderline that severs her body and expatriates it from her child.\n\nFor many women, the sense of differentiation from the fetus increases as their pregnancy progresses. They sense an individuated intentionality in the infant's movements. They feel a difference between a kick, hiccup, or a stretch, and believe that the fetus's touch communicates something to them\u2014annoyance, boredom, playfulness, and so on. They say that the baby touches them from within and they know whether it was a foot, a hand, or an elbow that touched them. Furthermore, the fetus seems to reply to the mother's voice, music, a poke to the belly, rubbing the belly, and so on. This responsiveness is interpreted by some mothers as the purposefulness of another human being:\n\nIt's exciting when the baby kicks. I just love it! Each kick has a different meaning to it. You can really feel it stretching out, or when you are in an uncomfortable position for it, it will give you a different type of kick from anything else.\n\nBefore the baby is born you know the personality. With Danielle I couldn't sleep at night because at night the baby woke up. Even from the beginning, I could feel her flutter at night.\n\nIt's really strange. When the baby started kicking, all of a sudden it seemed real and I could believe it. Like you said, when the baby's active, he's doing this and that. . . . It's one thing to be pregnant and it's another to start thinking about who's in there.\n\nNor, in pregnancy, did I experience the embryo as decisively internal in Freud's terms, but rather, as something inside and of me, yet becoming hourly and daily more separate, on its way to becoming separate from me and of-itself. In early pregnancy the stirring of the fetus felt like ghostly tremors of my own body, later like the movements of a being imprisoned in me . . .\n\nResearch has confirmed mothers' intuitions that a fetus can hear, move voluntarily, swallow, taste, and touch. Fetuses respond to pressure and pain and loud or soothing noises. They establish patterns of rest and movement in the womb. This means the responsiveness of a fetus to her mother is indeed both possible and likely.\n\nThere is another sense in which the pregnant mother engages in interaction with her fetus as a separate person. Many women respond differently to their own bodies when they realize there is a distinctly other presence within them. Kristan Stringer, a mother of four, describes a sudden change in her self-care as a result of being pregnant. \"When I first became pregnant with Elizabeth, I was involved with drug use at the time. Then, suddenly, I was pregnant. A baby breathes through you and digests your food. I felt more responsible to my body.\"\n\nWhen a woman takes care of her body differently in pregnancy, she is not just taking care of herself in a new way; she is taking care of her baby to be. In \"Women's Experiences of Pregnancy: A Tapestry of Joy and Suffering,\" Teresa Bondas and Katie Eriksson explain that Stringer's changed attitude toward her own health is a typical response to being pregnant. The fact that many women become extra vigilant toward their health during pregnancy and do not maintain this stance afterward suggests that their healthful behaviors are largely directed toward the developing fetus and less so toward themselves. They seek to give their child a chance at healthy development, not an abstract medicalized conception of a fetus, but the specific baby developing within her. Francine Wynn states that \"although the birth-giver can take care of the fetus by taking care of herself, the point of her caretaking, and some of her details, depend upon the specific needs of the potential infant to whom she aims to give birth.\"\n\nPerhaps mothers are accused of mindless enmeshment with their children because the pleasure of caring for a child can be a visceral and ecstatic experience. From playing with a young child to having an adult conversation, the joy of mothering is lived in the body. In _Maternal Desire_ , Daphne De Marnefffe describes the carnal character of this experience: \"I feel it in my throat, in my belly, in my involuntary smile, and in a startling, ridiculous welling up of tears. . . . I am not in command of my own joy. . . . It seems that on the simplest level, the desire to mother is rooted in this experience\u2014the expectation of it, the living of it, the longing for it. It is a rush of connection, a feeling that both deepens and exceeds us.\"\n\nConnection to one's child is not merely that of one monad to another, but can lead to a blissful (even if temporary) self-forgetting. Children, especially young children and infants, live a direct relation to the world, unmediated by reflection and judgment. Eva Simms, a developmental psychologist, describes it this way: \"The newborn lives the pre-reflective and unmediated subjectivity 'wild Being' of the flesh, which adults experience consciously only in moments of great ecstasy or breakdown of the cogito, but which nevertheless remains the continual substratum of adult experience.\" In moments of communion with their children, mothers can connect to this immediacy and spontaneity. It is not surprising that some women describe it as akin to a religious experience: \"When I saw him and heard him cry, I was overwhelmed with emotion, and when the nurse placed him in my arms, I felt that I had _knowledge_ of something very powerful that made life completely comprehensible. I remember feeling very light, as if every burden was lifted from me. It made me understand why some people search for ecstatic experiences of revelation in religion.\" While this woman describes holding her child after giving birth, others have the same response to first meeting their adopted child.\n\nAntonio Damasio argues that \"the tendency toward one single self and its advantage to the healthy mind are undeniable.\" Yet the experiences of mothers gesture toward a self that exceeds simple singularity. While the body is, indeed, the location of the self, this is a body that is pregnable in more ways than one. As Simms eloquently explains, human embodiment is fundamentally intertwined with others: \"It is through understanding our fundamental housedness in the flesh and the dance we are engaged in with the (m)other that we can see human consciousness and selfhood arising out of its bodily, co-existential substratum.\" Although intertwining begins in the womb, it is not merely infantile. For a physically typical human female, part of what it means to become an _adult_ is to be capable of interdependent embodiment, to develop breasts and to ovulate, to be equipped for pregnancy and breast-feeding. Yet coexistence is not only a matter of physiological symbiosis. Temporarily forgetting oneself, yielding to the needs of a child, being swept away by passionate love\u2014these are experienced in the body and can occur with children of various ages. This \"undeniable connection,\" \"totalizing oneness,\" \"staggering vertigo,\" and \"rush of connection\" are not blind enmeshment; they occur across the abyss between mother and child. This is why it is so surprising and wonderful when, within the flux of these ever-shifting borders, mothers encounter the ecstasy of mutuality.\n\nCONFLICT\u2014EITHER THE KID GOES OR I GO\n\nIt's a good thing we love Carter, or else we'd be banging him on the floor.\n\nWhile feminists have sought to increase women's opportunities for independence and self-determination, they have also argued that women exist in relation. In _The Reproduction of Mothering_ , Nancy Chodorow claims: \"that women experience a sense of self-in-relation that is in contrast to men's creation of a self that wishes to deny relation and connection. . . . Chief among the conscious outcomes of these processes are the ways in which many women feel intuitively connected to others, able to empathize, and embedded in or dependent upon relationships, on the one side, and on the other side, many men's counterphobically asserted independence and anxiety about intimacy if it signals dependence.\" Chodorow is undeniably correct in her assertion that relationships are fundamentally important to women. However, she seems to ignore that women also feel anxiety about their interdependence and that this anxiety is well founded. Dependence on others and their dependence upon oneself can seriously impede one's plans and ambitions.\n\nMothers do not always revel in the self-forgetting and sacrifices that frequently characterize motherhood. A mother may have a longing for independence that is at odds with caretaking, revolt against the relentless empathy with one's child, and mourn the loss of her maiden self. Depending on the age and needs of the child and other adults involved in the child's life, a mother's self-assertion and desire for distance can be a real detriment to the child. Yet even if a mother _wants_ to care for her children, when she loses freedom of movement, guilt-free work time, recreation, adult-interactions and alone-time, this is a serious deprivation that can drive her to despair.\n\nA mother's sense of conflict with her child can begin as early as pregnancy. Many women attest to experiencing the fetus as a monstrous _Other_ who threatens her body, eats away at her vitality, upsets her sense of self, invades her home, and ultimately seeks to commandeer her life. In these cases the child may appear in dreams, fantasies, or in her thoughts as a parasite or other monstrous figure. One pregnant woman had the recurring dream that her infant sat in the middle of a playpen doing nothing and that the locus of the child's subjectivity\u2014the face\u2014was missing. Some women describe pregnancy as \"a kind of hijacking\u2014an invasion from within . . . as if by some oversight one has put on the wrong skin.\"\n\nOther mothers feel a sense of harmony with their pregnancy, but upon the birth of the child feel as though it is a threatening stranger. One woman said of her newborn: \"I feel that there is this intruder in the house who is never going to go away.'\" Another admitted that her daughter seemed \"like a complete alien.\" As Parker notes, these feelings of alienation may come as a surprise to the new mother:\n\nImmediately after giving birth, most women expect to experience a joyful sense of love and oneness with their baby: some do but many don't and warnings that it may not happen tend to fall on deaf ears. For women carry babies for nine months within a culture which represents the postnatal mother-child social relationship as if it replicated the intrauterine state of antenatal union. The nineteenth-century tradition of presenting a new mother with a pincushion bearing pins arranged to spell out the words \"Welcome Little Stranger\" was in many ways a more appropriate representation of the state of affairs.\n\nThe pregnant or breast-feeding mother provides her child's sustenance from her own body and this can exhaust and deplete her. On average, a mother will spend six hours a day nursing a baby who does not eat solid foods yet. The lack of appropriate places to feed a baby means that breast-feeding mothers are often isolated in the private sphere. Breastfeeding my newborn daughter took about ten hours a day. It can seem that the baby is not just sucking nutrients from the mother, but her vitality as well: \"The maiden 'I' sometimes has to weep with the sure, coming death of the maiden-self, the self that could 'arise and go' at will; _the self that is not food for others, but eats and drinks the world_.\" The sublime violence of childbirth or a cesarean section can also cause a woman to feel that she has been more of a hostage than a host to the child and that she will be left a deflated husk of her former self: \"I pictured the scene in _Alien_ when the guy's chest opens up and the creature comes out. . . . Had I fulfilled my purpose here, and in life? Was I purely a vessel, a pod, and now I was no longer necessary?\" Some mothers feel that their children, like young cannibals, threaten to consume them:\n\nMy life right now is just all theirs. Sometimes it's a depressing thought because I think \"Where am I? I want my life back.\" . . . I mean, they are totally selfish. It's like an ice cream. They just gobble that down and say, \"Let me have the cinnamon roll now.\"\n\nThe children remind me of a litter of kittens. They would rather have a starving dead mother than allow her to go away and feed. It doesn't matter how long that cat has been lying there without eating, when she gets up and moves away, the kittens feel she is a bad mother. And she feels she's a bad mother. As soon as I feel OK about myself going and doing something for myself, they make me feel I've abandoned them. And the sense they give me of being a rotten selfish mum destroys whatever ability I have to do anything besides being a wife and mother.\n\nWhen Martin was a year old he would only go to sleep if he could hold my hand. I would sit there by his [crib] and lose all sense of myself. My physical boundaries would feel to be disintegrating. I would want to scream at him \"Give me my space.\" I felt desperate for physical space, let alone time for myself.\n\nAn image often stayed in the forefront of [one mother's] thoughts, of a teapot tilted over, with the proverbial last drop hanging from its lip: \"Tip me over and pour me out,\" the motto beneath it might read. Other times she imagined herself to be a little generator with another tiny appliance plugged into her, sucking energy. And yet her own power source had been disconnected.\n\nMothers responses to these feelings vary, but many describe an impulse to either fight or flight. They have a sudden urge to kill their child, to run away, to kill themselves, or otherwise reverse the fact of their child's existence.\n\nI can remember going in one night, going in where she was in the middle of colic [and saying] \"Take this baby before I throw it through the window.\"\n\nIf I hadn't had anyone to help me, I mean these girls that _do_ turn against their babies, I think it would be quite easy to do it. . . . I mean I think if I'd been left on my own I would have just _left_ him.\"\n\nSometimes . . . I feel my child has destroyed my marriage. Sometimes I want my child to disappear so I can have my husband back.\n\nI hit traffic on the drive home and started getting the terrible feeling that I was going to ram the car into the wall on the side of the freeway. The baby was in the back, and though I didn't think of hurting her, I wanted to smash violently through the window myself.\n\nUsually in the first few months you feel pretty down about things. It's just trying to cope with less sleep and feeling like a zombie, and not being able to speak to people properly because you're too tired to think. . . . You teeter on the edge and come back a bit, and then back to the edge and think, oh, will you jump or not?\n\nAt other times, during and after feedings, he would cry and scream. . . . Sometimes I hated him for rejecting me so completely; \"Shut up! I'll kill myself if you don't shut up!\" I'd yell. Then I would try to shove my nipple into his mouth and he would push it away, his face distorted with pain.\n\nI have felt many times over the years that I was capable of hurting him. I have not done this yet. Or at any rate, I have hurt him only a little\u2014I have spanked him a few times, yanked him and grabbed him too hard. I have managed to stay on this side of the line. . . . At one of my lowest points, a friend\u2014a teacher\u2014told me that she looks at her child and thinks: I gave you life. So if I kill you, it's a wash.\n\nMany of these testaments to maternal rage feature pregnant mothers or mothers of infants. This might lead one to think that they are simply examples of postpartum depression, that these responses to mothering are the result of pregnant and\/or postpartum hormones. Depression is known to be the foremost cause of disease-related disability among women. Pregnancy and new motherhood are well-known risk factors, significantly increasing the likelihood that a woman will be depressed; studies have estimated that postpartum depression is very common, affecting as many as 5 to 25 percent of new mothers. Nevertheless, there is no strong consensus on the cause of postpartum depression. No correlation to the hormonal changes of pregnancy has been substantiated. In addition, some new parents who were never pregnant display the same symptoms, including fathers and adoptive parents. Numerous Web sites provide resources and support for parents suffering from postadoption depression (though it is not an accepted diagnosis by the American Psychiatric Association). As one mother discusses, adoption brings its own added complications to maternal ambivalence:\n\nGrateful though we should be, adoptive parents are not always. I was anything but grateful one night when my son was two. It had been a three-tantrum day, and there he was in his high chair, throwing spaghetti directly at my face. I could feel the tomato sauce sinking into my hair and can remember thinking the most horrible thoughts: Who is this kid? Where did he come from? Are we dealing with a bad seed here? If he's this angry now, wait till he understands that he was adopted, that he doesn't look a thing like his parents. What will he throw at me then? Would my biological child have flung spaghetti at me? And worse: Can we return him?\n\nMothers of adolescents and young adults have also described such feelings. The mother of an adolescent boy says: \"I feel torn in two. I want to be a loving mother and to offer him a good home. Yet, at the same time, I just want him out of here.\" Suffice to say that negative responses to motherhood cannot be univocally reduced to a single diagnosis or biological cause.\n\nAnne Oakley argues that maternal depression is related to the expectation that motherhood requires abnegation of the self. She believes that, in part, \"postnatal depression is a form of bereavement reaction in which a woman mourns her former identity\" and that motherhood seems \"to lead to a sense of lowered ('depressed') self-worth: children take the centre of the stage; the mother is merely a supporting player. Her role is static, theirs dynamic: having no time to herself, her self is quiescent.\" Indeed, many mothers' lives are directed around the lives of their children:\n\nI would feel the futility of any attempt to salvage myself, and also the inequality between us: my needs always balanced against those of a child, and always loosing.\n\nMy outings are [limited]. I'm excited when I have to go grocery shopping. Everything I pick up is what they eat, everything they like, or what they should eat. Me, I'm just _there_. I'm there for them. I feel that I'm here for them.\n\nOne sometimes forgets that one is actually a person other than [the child's] mother.\n\nBeing a mother is a terrible thing. It ruins your relationship with your husband. It ruins your life. You can't leave them because you love them and when you're with them you hate them. I used to be a damn good nurse. Very competent. I've nursed people all over the world. I ran an entire ward in Boston. Now I'm a mother, and that means that I'm nothing.\n\nYet, even when mothers have other activities and interests, many women may still feel that motherhood undermines the identity that they have spent a lifetime creating. Quoting from her pregnancy journal, Rich says that she fears \"the crumbling to death of that scarcely-born physiognomy which my whole life has been a battle to give birth to\u2014a recognizable, autonomous self, a creation in poetry and in life.\"\n\nJane Lazarre's breathtaking memoir, _The Mother Knot_ , explores the misery of this loss:\n\nI turned to that self inside of me, that girlwoman who had once been all I needed to know of myself, whom I had fought to understand, to love, to free\u2014I turned to her now and I banished her. Into a protective shell tied in a knot, she retreated, four, five, six times a day, whenever Benjamin wanted to nurse. Soon, even when I sought her, she would not come, but began to stay out of reach longer and longer, sometimes not reappearing for whole days. For if she was present when the baby needed me, she was of necessity pushed aside, sent to go hungry. She who had been in my life, whom I know I had to nourish daily in order to be fed in return, hid for weeks, hoarding her gentleness and her strength, placing no gifts in my outstretched hands.\n\nNaomi Wolf also describes this loss of self as peculiar to maternity:\n\nI realized that my identity was about to be cloven in two, my independence cut by half. It was the first time I could see it spelled out for me in all its sweetness and regret, in all its ambiguity. . . . The tension was this: I was mourning, protesting a point of departure in the road that I could never retrace. An \"I\" would go forward, swept irrevocably on by the tide of the natural order. . . . And the \"I\" would reconfigure eventually around [the baby's] need, and take joy in it, and spin a new identity. But it would never again be the \"I\" it had been before.\n\nThere are times when a woman's self-determination is deferred in favor of the child's concrete and immediate needs. Wolf describes one of these moments:\n\nOn another day, I became engaged in writing again for the first time in months. It was one of those rare good days when a writer can lose herself in time. Caught up in my excitement, I stayed too long at the office. It meant that I was late for a feeding. By the time I snapped out of my state of concentration and raced out of the building to run the few blocks to cross the street to reach our house, I was very late indeed. Christine, our new caregiver, was standing on the other side of the busy boulevard, holding the baby, on her way over to come get me. The wind was whipping Christine's hair and skirt, and the baby was howling. I tried to lunge across, but with six lane of traffic roaring between us, I couldn't. I felt guilty almost to the point of crying at the baby's hunger, but in the midst of my guilt, intertwined with it, was a sheer vexation, almost a childlike anger, at having had to interrupt my work. It wasn't just the one interruption, but the realization that this was a condition: interruption was now my life. I was crying because I could not win. Because, as a worker, I was turning away from my work at exactly the most important moment; yet at the same time, as a mother, I had already stayed too long at the fair.\n\nWolf's desires and her child's needs are not in straightforward opposition. While she feels a yearning for uninterrupted concentration, she also describes \"a longing for caring,\" even a desire to run into traffic to reach her daughter. Whereas her desires used to be more strictly her own, her child's needs have an irresistible pull. There is no simple solution to this contradiction. More time away from one's child is not necessarily a feasible or attractive option. Wolf does not want to sacrifice either work or motherhood, as both are deeply meaningful to her. Thus the problem of her \"self-sacrifice\" is not just a matter of yielding to the child's demands over her own. Lazarre, too, describes this aspect of the conflict:\n\nFor two days every week I would be in New York attending classes. For two days and one night every week I would be away from Benjamin. I felt peaceful and expectant standing alone, frightened and exhilarated by this attempt to embrace once again my own life. But the deepest part of me leaned toward that moment on the following afternoon when I would return to my baby: I was changed forever . . . except for the two days I am in New York, there is a real danger of losing faith in my existence.\n\nFor Wolf and Lazarre, becoming a mother means that they yearn in two opposite directions. In wanting time to themselves, they are also directed against that part of themselves that wants to be with their children. The conflicted aspect of maternal ambivalence is not just a fissure between the mother and her child; it is a rupture within the woman herself, between her own competing desires, between equally valued parts of herself. Many women find this conflict to be a disturbing splitting of their self-coincidence. In her study of maternal identity across economic classes, sociologist Martha McMahon comes to similar conclusions: \"To these women, motherhood, I argue did not mean self-sacrifice. Rather it meant the 'death' of an earlier self and the emergence of a new self through a process of moral reform. In this context, the notion of self-sacrifice for one's child is rhetorical in that the self has been transformed in ways in which self and other are integrated. Thus when women spoke of self-sacrifice, it was the sacrifice of other selves rather than the self as mother to which they were referring.\"\n\nIn Wolf's narrative, Wolf-the-writer must contest with Wolf-the-mother. Whereas Wolf's identity used to be based primarily on her own achievements, actions, and qualities, this is no longer the case. For a loving mother, her sense of self, groundedness, and well-being are, at least in part, dependent on meeting the needs of her child. In such cases, one who obstructs her self-determination is also integral to her sense of self. Though one is never fully transparent to oneself, self-alienation is dramatic when one is in symbiotic relation to one's saboteur.\n\nWhere women expect to have an independent identity and yet, as mothers, are fundamentally intertwined with their children, there may be no way to avoid feeling torn. De Marneffe agrees with this analysis: \"From time to time, we try to fortify ourselves in our choices by disowning our competing motivations: by elaborately insisting on our contentment at home with children, or our belief in the importance of mother's work. But the reality for most of us is that we are torn, and we live with a sense of conflict, sometimes flickering, sometimes flaring, hour by hour, day by day.\"\n\nThere is both internal and external pressure for women to be creative and unique individuals (at least in Western culture). Yet opportunities for self-expression frequently mean hours away from children. Mothers are, of course, blamed for any neglect of children, whether actual or imagined. But women also suffer separation from their children: \"One professional woman I know spends her Monday mornings in her office, like clockwork, crying over leaving her children for another work week.\" The mother who works, whether out of necessity or desire, misses opportunities to develop the relationship with her children, to teach them about the world, and to contribute to who they are becoming. If being a mother means to engage in the practice of mothering, then being away from one's child for long hours can be a threat to that identity.\n\nMierle Laderman Ukele, a performance and environmental artist, deals with this conflict. In her self-titled \"Maintenance Art,\" Ukele showcases daily life-sustaining activities that generally go unnoticed. In a performance titled \"Handshake and Thanking Ritual,\" she shook the hands of all eighty-five hundred New York City sanitation workers and said \"Thank you for keeping New York City alive.\" Ukele's earliest work deals with maternal care. For example, in a photograph titled \"Maintenance Art: Rinsing a B.M. Diaper,\" she is shown rinsing a cloth diaper in the toilet.\n\nWhile Ukele's thesis is that our daily care activities can be art, she also deals with the conflicted aspect of being both an artist and a mother. In: \"Some Kinds of Maintenance Cancel Others Out, Keep Your Head Together\u20141,000 Times, or Babysitter Hangup\u2014Incantation Ritual,\" a performance at the Institute of Contemporary Art in Boston, she carried out the following \"procedure\":\n\n1) Begin: Call New York long distance & ask babysitter if babies (3) are o.k.? o.k.\n\n2) Hang string back and forth across full length of gallery (30 ft.).\n\n3) Read each set of statements out loud on each page.\n\n4) Stamp each page with Maintenance Art Stamp.\n\n5) Clip it to string.\n\n6) Move on to next identical page.\n\n7) Fill up whole space with words and sheets, move up the stairs and outside to boundary of I.C.A\u2014street.\n\n8) If I say it enough times, (maybe) it will come true.\n\n9) Call N.Y.C. and ask babysitter if children are o.k.? o.k.  \nEND\n\n(Repeat approx. 500 times)\n\nUkele calls the babysitter a total of one thousand times to ask if her children are OK. According to art theorist Andrea Liss, Ukele acts out her anxiety over leaving her children with a babysitter in New York. Ukele is not just asking if the children are OK, but if it is OK to leave them with a babysitter. As the \"mantra\" in number 8 reveals: \"If I say it enough times, (maybe) it will come true,\" it really does _not_ feel ok to leave her children so far away. Liss claims that Ukele's art, \"poetically and unabashedly addressed maternal issues and created a feminist statement in its acceptance that children are individual human beings both connected to and distinct from their mothers. To make such a statement is an admission of love, a succumbing to difference, and a difficult acceptance of the ambiguity of intimacy within the mother-child bond . . . the secret code of the mother's mourning.\"\n\nIn order to flourish, or even to survive, young children are dependent on someone's intense investment in them. So often, this responsibility rests on the shoulders of the mother. Conflict arises when caring for children competes with women's self-care, development, and expression. But, for the mother who derives at least part of her identity from her relationship with her child, this struggle is not just directed toward her child; it is part of her self-relation. This is further complicated by the fact that a child's successful development typically leads to increased independence. From the time one begins to care for a child, whether through pregnancy or adoption, he is taking steps away. Even as the mother longs for independence, her child is already leaving. The \"secret code of the mother's mourning\" is that the relationship with her child, even when it is experienced as an invasion, is already an uneasy letting go:\n\nI see the accelerating future approach. I've done this already, after all, with two boys\u2014grieved for the silly three-year-olds, the gap-toothed six-year-olds, the willowy nine-year-olds. They've died and will never return, and I grieve. . . . I remember the world fading into the universe made by Two, and I remember how it shatters when that ends\u2014and then, how it begins again, brand new. So much to do, so many mistakes to make for the first time. And what is there to regret? This is how the world goes, this is how it must be. I don't grieve for her\u2014I grieve for me, sitting by some poolside in a few years, pretending to read so my sunglasses hide my hungry, tearful eyes as she glides by, oblivious.\n\nMothers identify innumerable moments that signal the ongoing, increasing loss\u2014from childbirth, to starting school, to the rebellions of adolescence:\n\nZachary's body isn't mine anymore, not like it once was. We've been moving in this direction since the day he was born, when I held him swaddled in flannel and watched every possible emotion roll and brighten and twitch over his face, like a symphony warming up, like clouds passing in front of the moon.\n\nFrom this point on, I knew, our separation would continue to grow over the years, as she experienced things in her life that I wouldn't even know about. Imagining her in the world without needing the protection of a mother's body, I felt a small moment of panic, a sudden, clenching cramp, not unlike the muted pain experienced at her birth.\n\n\"Stay close by,\" I call to her as she runs. \"Stay by Mama.\" Each day, she runs a little farther away, turns her head and laughs.\"\n\nAnd there is also the possibility that one's child will die:\n\nA woman carries a baby in her body and when that child dies, no matter how old he or she is, the mother feels that something has been cut out from inside her\u2014the loss is as permanent and profound as becoming a parent. I asked Rose and Suzanne how childbirth and child death changed their lives. \"A woman named Elizabeth Stone wrote that having a child is 'to decide forever to have your heart go walking around outside your body,' Suzanne says. \"That's what becoming a parent is like. And now I've lost that piece of myself.\"\n\nEven an intimate moment can highlight the child's departure:\n\nCarefully, very consciously, gauging my reaction, he lowers his soapy head to my chest, leaning his chin on my breast and looking up into my face, then lifts his hand from the water and places it over my heart. A trigger of concern goes off in my brain\u2014physical boundaries: when to impose\u2014but when he sees my even expression (my poker face, hitched to genuine pleasure at the closeness of my child), I'm just his mommy gazing back at him, and all the tension goes out of his body. He knows he's my little boy, curled up with his mother in the bathtub for one of the last times; we both know it, and relax.\n\nEyes closed, her tiny fists clenched, she mewled and rooted against my breast, already instinctively searching for reconnection. Exhausted, I was overwhelmed with the force of her perfection and individuality: a tiny being separate from and yet still so intimately tied to me.\n\nConnection and loss are the yin and yang of motherhood and, nevertheless, make it difficult to achieve a sense of balance, coherence, and unity. If the woman, as a more or less singular individual, is sacrificed to motherhood, then what is it that takes her place? The mother who finds herself joined to her child's needs, joys, and pains (whether this is the source of pleasurable mutuality or painful conflict) has increased herself in number. She is more than one. But, to the extent that her child is at the center of herself, she has lost her self-coincidence, and she is less than one. How are we to conceive of this dichotomy?\n\nAMBIGUOUS INTERSUBJECTIVITY\u2014MORE AND LESS THAN ONE\n\nIn \"On the Maternal Order,\" an interview with biologist H\u00e9l\u00e8ne Rouch, Luce Irigaray considers the placenta as a model for the relationship between mother and child. Rouch and Irigaray note that, contrary to the psychoanalytic idea that mother and fetus are fused, the placenta is a vital intermediary that preserves differentiation within the relationship. Indeed, the placenta is an amazing example of symbiosis, central to the survival of both pregnant woman and fetus. Its growth is initiated by the embryo, but it is made up of tissues from both individuals. Rouch states:\n\nOn the one hand, it's the mediating space between mother and fetus, which means that there's never a fusion of maternal and embryonic tissues. On the other hand, it constitutes a system regulating exchanges between the two organisms, not merely quantitatively regulating the exchanges (nutritious substances from mother to fetus, waste matter in the other direction), but also modifying the maternal metabolism: transforming, storing, and redistributing maternal substances for both her own and the fetus' benefit. It thus establishes a relationship between mother and fetus, enabling the latter to grow without exhausting the mother in the process, and yet not simply being a means for obtaining nutritious substances.\n\nWhile a woman is pregnant, the placenta controls her hormone production. This ensures that the fetus is recognized as an other, but not as a hostile other to be rejected (as in the case of a transplanted organ). According to Rouch, \"there has to be a recognition of the other, of the non-self by the mother, and therefore an initial reaction from her, in order for placental factors to be produced. The difference between the 'self' and other is, so to speak, continuously negotiated. It's as if the mother always knew that the embryo (and thus the placenta) was other, and that she lets the placenta know this, which then produces the factors enabling the maternal organism to accept it as other.\" Sandra Steingraber compares the placenta to the symbiotic relationships between organisms and the synergetic ecology of a forest:\n\nThe placenta is made up of the cells of two individuals\u2014indeed, it is the only mammalian organ with this characteristic. In that sense, it is like the lichen. . . . Part fungus, part algae, lichens represent a symbiosis so complete that the two organisms are, for all intents and purposes, one creature. Likewise, a placenta is an intertwining of mother and child in the closest kind of embrace biologically possible. Yet, because the child's portion of the placenta is made up of cells genetically different from those of the mother, it should be identified as non-self by the mother's body and rejected, like any other implanted tissue.\n\nIn fact, the internal anatomy of the human placenta closely resembles a maple grove: the long column of cells sent out by the embryo into the uterine lining during the first few weeks of pregnancy quickly branch and branch again until, by the third month of pregnancy, the treetops of an entire forest press up against the deepest layers of the womb. Meanwhile, the open taps of the uterus's spiral arteries send jets of blood spurting between those arboreal structures. As the mother's blood trickles through the canopy of placental branches all kinds of important transactions take place. Most notable, carbon dioxide and metabolic wastes are swapped for oxygen, water, minerals, antibodies, and nutrients. This is the same process by which the blood inside our own capillaries is cleansed and refueled but with one major difference: rather than relying on simple diffusion, the placenta actively pumps much of what it needs out of the percolating raindrops of maternal blood . . . Many larger molecules transported into the placenta are picked apart before they are allowed to cross the border between mother and child. Some proteins, for example, are disassembled and carried over brick by amino-acid brick. Then they are rebuilt on the other side.\n\nSteingraber describes pregnancy as a \"subtle hijack[ing].\" \"In all cases, a pregnant woman aids and abets her own infiltration.\" Meanwhile Irigaray refers to this relation as a \"placental economy\" and deems it \"almost ethical.\"\n\nWhy is placental economy _almost_ ethical? For some biological mothers, being disinherited from their bodies in pregnancy initiates this process in their lives as a whole. However, biological processes cannot stand in for conscious and deliberative acts. The exact relationship between the reflective mind and prereflective body, especially within pregnancy, is insufficiently understood. It could be that the body's reaction to pregnancy is already affected by the mother's attitude toward it. For example, in \"Being Torn: Toward a Phenomenology of Unwanted Pregnancy,\" Caroline Lundquist notes that in denied pregnancies the \"normal symptoms\" of pregnancy, such as weight gain and morning sickness, are suppressed or reduced. This seems to imply that the mother's conscious acknowledgment of her pregnancy is necessary in order for it to take its \"normal\" course. In addition, denied fetuses are frequently underweight. This is an extraordinary finding since, as Lundquist also notes, the human female is unique in that, when they are limited, her body will direct resources toward the fetus first. This means that an unwanted fetus cannot be deliberately starved; the mother's body will be deprived first. Yet, in denied pregnancy, the fetus often _will_ be denied vital nutrients, even when there are enough for both. Clearly, ethical mothering is not merely a compliance with biological precedents, but it is interesting to recognize that the body has its own way of negotiating the ambiguity between mother and child. Pregnancy, as ambiguous intersubjectivity, mirrors the relationship between a child and the mother(s) who raise(s) her. In _The Mask of Motherhood_ , Ann Maushart comes to similar conclusions: \"Indeed, by its very nature pregnancy is ambivalence made flesh. It's the classic instance of 'to have and have not,' 'to be and not to be.' The blurring of the boundaries of selfhood, of where 'I' leaves off and 'You' begins, is a primary biological fact of pregnancy. Later it will become the primary emotional reality of parenthood. The delicate yet strenuous agenda of symbiosis\u2014to exist at once separately and in seamless, self-less merger with another\u2014is the central paradox of motherhood.\" The biological aspects of motherhood do not _cause or explain_ ambivalence, but they are one more manifestation of the ambiguity between mother and child.\n\nDue to the enigmatic nature of ambiguous intersubjectivity, many women attempt to understand it through creative nonfiction, poetry, and art. Rishma Dunlop does so in the essay \"Written on the Body\" and in the poem \"Child\": \"Child of mine. Separate and strange. Separated in the beginning by childbirth, yet inseparable from my flesh. The recognition of the abyss between mother and child.\"\n\nIn my baby's eyes,\n\nI am locked\n\nin self-extension.\n\nI do not know\n\nwhere I finish\n\nand she begins;\n\nshe is my pact\n\nwith life and death\n\nand I must dance for her,\n\nso she will know the\n\nuncommon\n\nsteps.\n\nCreative work, even when it is concerned with motherhood, often confronts the irony that reflecting on motherhood requires a step back from it. This is true at a practical level because children can be a tremendous impediment to thoughtful concentration. Ruth Nemzoff describes this in the following poem:\n\nMy one-year-old peed on the floor\n\nMy seven-year-old threw-up in the car\n\nMy nine-year-old has chicken pox\n\nAnd I wonder\n\n\"Why can't I write my thesis?\"\n\nHowever, a mother's creativity and her relationship to her children are not always in plain opposition when both are central to who she is. One mother courageously tries to explain this dynamic to her daughter:\n\nAdriane, age four, would sidle up to me: \"Do you still love me when you're doing that work?\" Or, more confrontationally, at age seven: \"Which do you love more\u2014me or your work?\" I've been told that I was seriously remiss in my answer to the latter. But I had made a pledge of honesty to my daughters from the very beginning. And so I would say, \"You can't make me choose. I have to do my work, not just for money but for me. But I have to be with you too. I love you and I love my work.\" My answer is deeply etched in memory by the surrounding cultural guilt that told me there was only one right answer: \"I love you most of all my darling.\" But I was obdurate in my pledge to be honest. Truly, I loved my work, even as my daughters were at the core of my being.\n\nIn her essay on maternal subjectivity, Mielle Chandler also describes the complexity of this dynamic. Chandler explains that the _act_ of writing about motherhood, not just _what_ is written, is an exploration of ambiguous intersubjectivity:\n\nTo write a paper is to leave mothering, or, rather, it is to leave the type of subjectivity I engage in while mothering. A clean break is neither possible nor desirable, mothering being my topic, and so integral to my identity. Indeed, to leave it would be to become someone completely different. As I write, a child is asleep in the next room, part of me is still on duty, ready to battle dream monsters or change wet sheets, making a mental note to tell the babysitter (that is, the stand-in mother whose labor affords me, temporarily, a paper-writing subjectivity) that the child can't go swimming tomorrow (I'm worried the cold might turn into an ear infection), adding granola bars to the shopping list. The process is one of traveling between an individuated and separated subjectivity which allows me to write, and an actively in-relation subjectivity (if it can be called a subjectivity) which is born of mothering. It is an existence fraught with tension, for while each site demands my attention, the former requiring quiet sustained concentration, the latter the alertness of a catcher behind home plate, neither allows me to inhabit the other adequately.\n\nPerhaps the most brilliant demonstration of a mother living out ambiguous intersubjectivity through writing is Julia Kristeva's \"Stabat Mater.\" This piece is actually two intertwined essays: one presents a reasoned argument against the conceptions of motherhood as they are posited by science and Christianity and the other enacts the experience of being a mother poetically. The first account is often interrupted mid-sentence by the poetic, not to be taken up again until a few pages later. In this way, Kristeva presents the interconnected, yet contradictory nature of what she calls the symbolic (formal aspects of signification, such as judgment, grammar, and structure) and semiotic (rhythms and tones that discharge bodily drives, that channel the unconscious and provide catharsis). For Kristeva, to be a mother is to be on intimate terms with the semiotic, to have one's symbolic thought constantly interrupted. Recall Wolf's writing being halted by the hungry wails of her infant. In \"Stabat Mater,\" Kristeva brilliantly demonstrates this disruption for the reader. What is more, Kristeva argues that the ongoing development of the symbolic requires intrusions of the semiotic. Thus, for Kristeva, maternity is not a mere disruption; it is the well of prereflective, unconscious, and embodied understanding from which the symbolic must drink.\n\nKristeva, a psychoanalyst and philosopher, highlights the great creative potential inherent in mothering, but classical psychoanalysis gives little consideration to its transformative powers. The mother is typically a static figure in tales of children's development who either enables or impedes separation. A good mother provides care and stability; she holds on, but with enough slack so that the child can achieve gradual, age-appropriate independence. Having a child has largely been considered the pinnacle of her life, rather than instigation to further development. However, accounts of maternal ambivalence repeatedly affirm that motherhood entails dramatic transformations of the self. In particular, conflict with one's child guards against a potentially consuming intimacy and brings the mother's need for differentiation into view. Rich affirms that a mother must \"struggle from that one-to-one intensity into new realization, or reaffirmation, of her being-unto-herself. . . . The mother has to wean herself from the infant and the infant from herself. In psychologies of childrearing the emphasis is placed on 'letting the child go' for the child's sake. But the mother needs to let it go as much or more for her own. . . . It is not enough to let our children go; we need selves of our own to return to.\" Maternal aggression is often a sign that the mother is in need of literal and psychological distance. Regarding the mother whose son will only sleep while holding her hand, Parker says: \"But what she does not consider is the possibility that her desire to pull back, to reclaim her sense of bodily boundaries, may also signify an appropriate and productive move towards separation on her part as a mother.\"\n\nNegotiating the double binds of mothering can lead to homicidal and\/or suicidal depression and acts of violent retribution or neglect. Yet Parker repeatedly asserts that maternal ambivalence, though painful, can be a psychological _achievement_.\n\nThe mother's achievement of ambivalence\u2014the awareness of her coexisting love and hate for the [child]\u2014can promote a sense of concern and responsibility towards, and differentiation of self from, the [child]. Maternal ambivalence signifies the mother's capacity to know herself and to tolerate traits in herself she may consider less than admirable\u2014and to hold a more complete image of her baby. Accordingly, idealization and\/or denigration of self and, by extension, her baby, diminish. But the sense of loss and sorrow that accompanies maternal ambivalence cannot be avoided. Acknowledging that she hates where she loves is acutely painful for a mother.\n\nInsofar as she imagines mothering to entail only a picturesque mutuality, as long as she cannot reconcile herself to the full 360 degrees of her responses, ambivalence increases a mother's denial and rage. In a society that alternately romanticizes and condemns mothers, it is a brave overcoming of cultural denial to recognize the duality of maternal experience. The ability to think through ambivalence and ambiguity seems to make all the difference in whether or not a mother's rage can be checked. When reflection is possible, a woman can begin to understand herself and her _actual_ child through their interactions, rather than trying to be the perfect mother. The child comes into relief as a particular, _concrete_ other as one whose needs, desires, and perspective change radically as she develops. The acknowledgment of ambivalence makes possible this genuine discovery of relationship as it unfolds and sensitivity to the developmental needs of both mother and child; but the mother who denies her own complexity cannot expect to recognize her child's.\n\nReflection's importance to mothering tells us that being a mother is far more than being a womb, a placenta, or a breast. While these organs share a kind of symbiosis with a biological child, this is insufficient for appropriate responsiveness to a child. It does not keep one from responding to a child out of one's own projected desires. Furthermore, a child, no matter how beloved, is also radically other with discrepant needs, experiences, and desires. A sense of _unanimity_ with one's child may motivate an ethical response, but _dispute_ forces recognition of interpersonal divergence. When taken together, these attitudes entreat a mother to expand and practice self-knowledge, sound judgment, and problem-solving abilities in order to respond appropriately to her child.\n\nSara Ruddick provides a detailed account of maternal deliberation in _Maternal Thinking_. Ruddick believes that thought develops in the context of a given practice. Since each practice is directed toward particular ends, thought is cultivated in deliberating about the means to those ends. Over time, a certain style of thought will prove to be more conducive to achieving the ends of a given practice. In this way, mothering, as a practice, will engender a unique manner of thought in its practitioners\u2014maternal thinking.\n\nMothering entails meeting three demands, according to Ruddick: to preserve the lives of children, to foster their physical, emotional, and intellectual growth, and to teach them to be socially acceptable. Thus maternal practice involves preservative love, nurturance, and training. To accomplish the tasks of mothering, mothers must strategize about how to meet the demands of caring for children. They make judgments, attempt to ask appropriate questions, assume certain standards for truth, consider their own methods, set priorities and aims, and consciously affirm certain values, thereby developing their intellectual capacities. Indeed, studies indicate that parenting stimulates brain activity and development in the prefrontal cortex. It has been noted that women typically have a more developed prefrontal cortex, the part of the brain associated with complex thinking. But research suggests that men who are actively involved in child rearing can improve their own capacity in this area.\n\nMaternal thinking involves the use of seemingly opposed faculties: reason, passion, and love, challenging the idea that reason is detached and impersonal. Reason is not there simply to dominate and control emotion. Emotion can be thought provoking, but it is also thoughtful and reflective in itself. The practical nature of motherhood's ends encourages concrete thought over abstraction. Dilemmas, conflicts, and struggles do not detract from this process, but rather they prompt the need for further thought.\n\nExamples of how the practice of mothering encourages maternal thinking can be seen when we consider the complexities of training a child to be socially acceptable. To avoid ostracism, a child must be able to understand and get along with others in their society. However, a mother's own values may often be at odds with the dominant culture. Thus the dilemma may arise as to how she can train her children to be socially acceptable, while also teaching them to be people whom she herself will like and respect. This is further complicated by the fact that a mother's own values are not necessarily stable and may be brought into question by the very practice of mothering. In addition, a mother often finds that, as a child matures, her values may come to conflict with hers. For instance, the daughter of a pacifist mother may join the military, or a mother who values taking care of one's belongings may have a son who finds this to be materialistic. In such cases a mother must consider how she can respect her child's values while feeling passionately opposed to them.\n\nQuoting the experience of Julie Olsen Edwards, Ruddick tells a striking story of maternal ambivalence to illustrate the importance of maternal thinking. Edwards looked forward to having her baby. Ruddick quotes her as saying: \"You were such a wanted baby. I would rub my belly in awe of your growing. Sit motionless waiting for your kicks and stretches. Think of you, wonder about you, wait for you\u2014totally caught in the miracle of your coming to life. First child, child of hope, child of commitment.\" When born, however, the baby was sick with croup and bronchitis and could not sleep. Since the father had to both work and study, Edwards was frequently left alone with the baby. One night when the father could not come home and the baby was up yet again, Edwards woke up and went into her room:\n\nMy knees buckle beneath me and I find myself huddled on the floor. \"Please do not cry. Oh child I love, please do not cry. Tonight you can breathe, so let me breathe.\" And I realize my chest is locked and I am gasping for breath. I picture myself walking towards you, lifting your tininess in both my hands and flinging you at the window. Mixed with my choking I can almost hear the glass as it would smash and I see your body, your perfect body, swirl through the air and land three stories below on the pavement.\n\nRuddick continues the story: \"Sickened by her vision, Julie vomited and then felt calmer. After changing the baby's diaper and propping a warm bottle so that she could drink, Julie shut the door to the baby's room, barricading it against herself with a large armchair. Later that night, she carefully wrapped her daughter in blankets, carried her downstairs and rode a bus from one end of the city to another, 'thinking you would be safe with me if we were not alone.'\" Ruddick presents this example to demonstrate that, in mothering, passion works in concert with careful thought, attention, judgment, and deliberation. In extreme cases mothers may need to strategize to protect the lives of their children, even from themselves.\n\nWe have seen examples of mothers _thinking through_ their conflicted feelings toward their children via art, nonfiction writing, and poetry. When I use this expression _thinking through_ , I mean to indicate that the emotions themselves contain an understanding, even if it is not yet fully formulated. This engagement between reason and passion is not a hierarchical relation. The emotions themselves must be contemplated and turned over in order to gain greater understanding of them. Yet it is not the place of reason, in these accounts, to merely squelch unruly emotions. As Kristeva has asserted, the unformulated semiotic can be a fount of insight, telling us things that our rational accounts have hitherto overlooked. Maternal thinking assists women not only to get past specific instances of conflict but also to understand the ambiguity of their needs for both distance _from_ and intimacy _with_ their children. This is true at all stages of parenting. Regarding successfully negotiating the efforts of adolescent girls toward differentiation from their mothers, de Marneffe says: \"On closer view, the necessary balance between separation and connectedness relates, again, to the capacity for reflectiveness so basic to satisfying relationships between children and caregivers. The core of reflectiveness is the capacity to perceive the difference between one's own mind and another's and ultimately to be able to think about, and respond to, one's own and other's needs and desires . . . interpretive space, or the room for surprise, or the discovery of the other\u2014is the heart of true relating.\"\n\nAs psychiatrists Michael Kerr and Murray Bowen observe in _Family Evaluation_ , it seems to be a common problem that when one person in a relationship withdraws the other person pursues. Likewise, if one person pursues, the other then withdraws. Kerr and Bowen's fundamental assumptions are that all people crave both intimacy and distance, but usually only recognize the need for one or the other at a time. The attainment of either more distance or more relation creates an appreciation for the other. The achievement of both intimacy and distance simultaneously is a paradox, but one that can be embraced nonetheless. A dyad has unlimited alternatives for dealing with this paradox, including achieving a satisfying balance within their relationship, vacillating between extremes, or finding their equilibrium in establishing a triangle with a third person. This tension, however, can also cause much pain and confusion, especially when it appears as a force beyond their control.\n\nIn \"Distance and Relation,\" Martin Buber takes this thought further. He believes that in order for a connection to occur between two individuals, they must set each other at a distance. \"One can enter into relation only with being which has been set at a distance.\" This means that, paradoxically, the more aware a mother is of her child's separateness, the more striking their intimacy will be. Likewise, if she loses sight of their separateness, her enjoyment of their intimacy will be jeopardized. \"One cannot stand in relation to something that is not perceived as contrasted and existing for itself.\" When a mother realizes her child's impending departure, it is conducive to their current closeness. The briefness of childhood can make a mother's awareness of the end even more pressing. Buber says:\n\nHere and now for the first time does the other become a self for me, and the making independent of his being which was carried out in the first movement of distancing is shown in a new highly pregnant sense as a presupposition\u2014a presupposition of this \"becoming a self for me,\" which is, however, to be understood not in a psychological but in a strictly ontological sense, and should therefore be called \"becoming a self with me.\" But it is ontologically complete only when the other knows that he is made present by me in his self and when this knowledge induces the process of his inmost self-becoming.\n\nThe child's independence from the mother, the many occasions when she misunderstands her, the shortness of their time together, the mother's powerlessness to protect the child, and the permanent separation they will someday suffer are all partially responsible for the child's \"becoming a self for\/with\" the mother. In recognizing the interpersonal abyss, a mother can induce the \"process of [her child's and her own] inmost self-becoming.\"\n\nMothering imparts harsh lessons about the boundaries of the self\u2014the constraints on what one can give to another, the limits of self-determination, and the way that others can become integral to the self. These borders may remain permeable and ambiguous, expanding and constricting throughout the life of the relationship. When a mother is able to face her conflicted feelings and recognize the distance between herself and her child, then the tumult indigenous to mothering invites a _thoughtful_ response to her child. Reflection helps to mediate between mothers' murderous fantasies toward their children and their commitment to protect them. Undermining the usual idealization of maternal self-sacrifice, it gives attention to the concrete and specific needs of a unique child and mother and thus supports both their efforts toward intertwined individuation.\n\nAttention to maternal ambivalence brings to light more than mothers' contradictory emotional responses (love and hate) and impulses (to nurture and do violence). These mixed feelings\u2014passionate rage, love, profound boredom, ecstasy, and so on\u2014are not mere reactions. They contain wisdom of their own that tells us what is going on in a given situation and they act as forces that can bind as well as liberate. The impulse to shove a child away indicates the need for space, just as the desire to hold him close points to the awareness of abyssal separation. Rather than suppressing these lessons, we need to turn them over in our minds and conversations. It is time we realized, on a broad cultural scale, that if we care for the well-being of children we _must_ care for the well-being of their caregivers.\n\nMotherhood is an especially striking example for ethical theory because of the child's intense dependence and vulnerability, society's high expectations of mothers, and the shared embodiment between mother and child. Yet, as I will explain more fully in the coming chapters, the insights of maternal ambivalence are relevant beyond the mother-child relationship. Such claustrophobia, ecstasy, and longing are present in other relationships as well; and this teaches us that _both_ intimacy and individuation are vital. Motherhood demonstrates, even in cases of cherished nearness, that one cannot overcome the insurmountable alterity of the other. Although there is a prereflective coherence between self and other, there is also an inherent dehiscence. Maternal experience also illustrates the possibilities for existential good or evil\u2014we can dominate vulnerable people and\/or escape our freedom in devotion to them. Nevertheless, whether or not one will be able to carry out her obligation depends on the larger political, cultural, and material context. Both individuals and society as a whole must embrace an active responsibility for the defenseless and provide caregivers with opportunities to engage in other meaningful activities. Our ambivalence toward others teaches us that the failure to help them is always a risk, but, recognizing the possibility of this failure, is itself a critical part of an ethical orientation.\n\nIn contrast to Pollaiolo's Charity, I propose the Janus head as a more accurate image of maternity. Janus, the Roman god of entrances and exits (especially those of the home), is a figure with two faces looking in opposite directions. The ambivalent mother holds two perspectives at once; this part of the comparison seems obvious. However, Janus is most like a mother in that he is always positioned at gates or doorways; he oversees new beginnings, paths and journeys; he has the gift of seeing both past and future. The maternal body is less a container for the self than a threshold for her own life and the life of another. As her \"heart goes walking around outside,\" her self is both internal and external to her body. As her body also envelopes another, it both encircles and exceeds her. As she anticipates the expiration date on childhood dependence, the mother inhabits both the _here_ of intimacy within and between bodies and the _there_ of physical distance and the future. It is rooted in the present, but always anticipating that which does not yet exist. It is a relation of becoming. As the mother attempts to live in these contradictions, she may feel that she is suspended across a chasm. But true relating requires this simultaneous sympathy and reflective distance. It compels the recognition that while the mother's body is the child's first home, it is also her passageway to the wider world.\n\nIn the next three chapters I engage Levinas, Merleau-Ponty, and Beauvoir because they include key elements missing in a care ethics approach to mothers' ethical orientations. Each one contributes to a robust account of ambiguous intersubjectivity and describes (or suggests, in Merleau-Ponty's case) ethical life as inherently conflicted in a manner consistent with maternal experience. In the works of these three philosophers the subject is always already in relation. This is, of course, a claim already made by care ethics, but it is not explored in the depth that we see in existential phenomenology. The other is a primordial disturbance in any selfsameness I might enjoy. Our relationships to others are so intimate that they pervade us to the core, influencing our very perceptions. Moreover, this closeness is not only cause for mutuality and care but also rebellion, frustration, and even rage. Perhaps this ambiguity is even a central factor in the high percentage of murder-suicides in cases of purposeful filicide. Yet there is also a chasm between self and other, a radical alterity that Levinas has placed at the origin of ethics, a divide that keeps us from ever being one and the same. As in the 32 percent of cases in which a mother kills her children and _not_ herself, the other might die while I live on.\n_Four_\n\nMATERNITY AS VULNERABILITY IN THE PHILOSOPHY OF EMMANUEL LEVINAS\n\nOver the thirty years span in which Emmanuel Levinas wrote _Time and the Other_ (1947), _Totality and Infinity_ (1961), and _Otherwise Than Being or Beyond Essence_ (1974), the feminine and maternity became increasingly central to his ethics. Feminist scholarship has demonstrated that there is tremendous ambiguity in his philosophy, including multiple solid interpretations of his work that are nevertheless at odds with one another. This makes it quite difficult to evaluate the feminist potential of his philosophy. Differences in interpretation frequently stem from whether or not Levinas is taken as describing real, empirical women or if he explains aspects of the feminine and maternal that could be present within everyone. When read as a descriptive phenomenology, his account is incomplete and may even be used to justify the submission of women and their isolation in the home. Were women to recognize themselves in his account of the feminine, they would find themselves in a merely supportive role. His depiction of maternity reveals a profound ignorance of its diverse forms and of the central importance of a mother's situation in her ability to care for her child. Nevertheless, I think that whether or not Levinas intends to describe real women may not be the most important question. Regardless of his intent, we can see a very clear appropriation of some patriarchal ideals in his philosophy. As he talks _about_ femininity (whether or not he means women), he frequently fails to _address women_ and their concerns.\n\nWhen read as a prescriptive (even if non-normative) ethics, the damage may be even worse. Since Levinas affirms the need to care for the other, even to the extent of self-abnegation, his ethic of infinite responsibility advocates a destructive asymmetry. Women traditionally hold the responsibility for care giving at the sacrifice of their own needs and are especially vulnerable to being oppressed under this ideal. Contrary to Levinas's expectation and intention, mothers find that when they live toward such an unbalanced servitude, it devastates their capacity to care for their children. His solution to conflicts of interest\u2014submission to the needs of the other\u2014denies his own great insight into intersubjective intertwining. And, although he sheds necessary light on the passive element in ethical agency, he does so at the cost of denying the importance of deliberative agency. Given this passivity\u2014that, according to Levinas, being compelled by the epiphany of the face is _the_ ethical motivator\u2014and his acknowledgment that one may also feel incited to violence by others' vulnerability, it is unclear how he expects the peaceful impulse will win.\n\nSo why include Levinas in this book at all? Levinas's philosophy contains some vital insights as well as some critical half-truths that will contribute to a robust view of ethics. Chief among these is the coercive power that others' needs and vulnerability exert over us. This helps to make sense of why we might rebel with hostility against the dependency of others. Levinas also describes the radical alterity of others, which points to the insufficiency of mere empathy in meeting other's needs. This aggression and radical divergence between self and other receive insufficient attention in care ethics, but are starkly prevalent in accounts of maternal ambivalence. Ruddick's notion of adoption could also be augmented through consideration of the anachronistic temporality of ethical obligation described by Levinas. Both birth and adoptive mothers describe a baffling temporality to their commitment; it hesitates, resists, jerks forward, repeats itself, and yet also finds itself to be always already in place. While care ethics focuses on connection between mothers and children, Levinas makes sense of the feelings that children disrupt our enjoyment and self-possession. Care ethics understands that it is human to be dependent, but they do not make it as explicit that we depend on our dependents to make us ethical creatures.\n\nA SKETCH OF FEMININITY AND MATERNITY IN LEVINAS\n\nLevinas's Early Work\n\nIn _Time and the Other_ , Levinas claims that solitude, which is \"an absence of time,\" is overcome in the relationship with the other. \"The other is the future. The very relationship with the other is the relationship with future\" (77). After a brief discussion of eros, Levinas concludes that one cannot overcome solitude via the erotic alone (84\u201390). The subject remains a subject; communication fails (88). Eros remains important, however, in that it leads to fecundity and \"time is essentially a new birth\" (81). That is, another's natality can give one time, the future. This birth is construed as the birth of a son to a father:\n\nHow can the ego become other to itself? This can happen only in one way: through paternity. Paternity is the relationship with a stranger who, entirely whole being Other, is myself, the relationship of the ego with a myself who is nonetheless a stranger to me. . . . I do not _have_ my child; I _am_ in some way my child. But the words \"I am\" here have significance different from an Eleatic or Platonic significance. . . . The son is an ego, a person. Lastly, the alterity of the son is not that of an alter ego. Paternity is not a sympathy through which I can put myself in the son's place. It is through my being, not through sympathy, that I am my son.\n\n(91)\n\nIt is because the son is so similar to the father, because he seems to be the father himself, that the father identifies with him. However, the father also realizes that the son is not himself, he is a distinctly _other_ person who will inhabit a future that the father will never know. This ambiguity in the relationship between father and son causes the father to recognize his own ambiguity in relationship to himself. Just as his son is both self and other to himself, so he too is self and other to himself. This is how the father discovers the truth of his subjectivity, that he is a stranger to himself.\n\nIn _Totality and Infinity_ , Levinas recognizes women more explicitly than in previous work, although little is added to the feminine role in ethics. In this later book he identifies the feminine with voluptuosity and hospitality. Feminine voluptuosity is important to ethics because its _telos_ is the birth of the son, and this birth initiates the father into ethical responsibility. Still, the feminine beloved appears without a voice and without a face, which means, in Levinas's scheme, that she has no ethical standing and no personhood.\n\nThe beloved is opposed to me not as a will struggling with my own or subject to my own, but on the contrary as an irresponsible animality which does not speak true words. The beloved, returned to the stage of infancy without responsibility\u2014this coquettish head, this youth, this pure life \"a bit silly\"\u2014has quit her status as a person. The face fades, and in its impersonal and inexpressive neutrality is prolonged, in ambiguity, into animality. The relations with the Other are enacted in play; one plays with the Other as with a young animal.\n\nIn this passage, the feminine other is a plaything, not taken seriously in her face or voice, having no will, an irresponsible nonperson.\n\nWhile feminine hospitality is vital to ethics, it does not admit women to participate in ethics. Hospitality is important, according to Levinas, because it prepares the place for the welcoming of the needy stranger. The woman provides that which the man has to offer\u2014good soup, a warm fire. Yet ultimately it is the man's place to invite the stranger into the home, while the woman remains invisible and inaudible: \"And the other whose presence is discretely an absence, with which is accomplished the primary hospitable welcome which describes the field of intimacy, is the Woman.\" Though the absent-presence of the woman is essential, she remains an ethical nonentity. She is neither receiver of the stranger, nor is she received. She sets the stage in her domestic service and then fades into the background.\n\nIn this exposition I have referred to the feminine as interchangeable with empirical women, however, feminist scholars actively debate whether or not the feminine represents _women_ for Levinas. Some scholars argue that he is not referring to concrete women but to a principle that can be found, to a greater or lesser extent, in every person. There is some support for this interpretation in _Ethics and Infinity_ (1982). Speaking of _Time and the Other_ , Levinas says: \"Perhaps, on the other hand, all these allusions to the ontological difference between the masculine and the feminine would appear less archaic if, instead of dividing humanity into two species (or into two genders), they would signify that the participation in the masculine and the feminine were the attribute of every human being.\" This statement seems to indicate Levinas's own uncertainty about the meaning of the feminine and the masculine. Even though it was spoken by Levinas eight year after the publication of _Otherwise Than Being_ , many scholars see it as an invitation to rethink the meaning of the feminine in his work. Diane Perpich argues: \"In fact, the feminine is nowhere identified by Levinas with concrete, really existing beings. It is not a set of characteristics or qualities attributed to a certain class of beings (namely, women); it is not a _type_ , of which individual women would be tokens.\"\n\nHowever, I argue that there are passages in which Levinas may be identifying the feminine with empirical women. Consider this translation of a quote from \"Judaism and the Feminine\" (1969) in which the feminine, woman, and wife are described interchangeably:\n\nTo light eyes that are blind, to restore to equilibrium . . . should be the ontological function of the feminine [ _du feminine_ ], the vocation of the one \"who does not conquer.\" Woman [ _La femme_ ] does not simply come to someone deprived of companionship to keep him company. She answers to a solitude . . . the strange flow of gentleness must enter into the geometry of infinite and cold space. Its name is woman [ _femme_ ]. . . . The wife [ _La femme_ ], the betrothed [ _fianc\u00e9e_ ], is not the coming together in a human being of all the perfections of tenderness and goodness which subsist in themselves. Everything indicates that the feminine [ _le feminin_ ] is the original manifestation of these perfections, of gentleness itself, the origin of all gentleness on earth.\n\n_La femme_ has been translated alternately as the wife and woman. It could also be that Levinas intended _la femme_ as an archetypal (rather than empirical) woman. However, the translation of _la femme_ as wife, when it is juxtaposed with _fianc\u00e9e_ , is appropriate; and this would seem to point toward actual women. Ultimately, the idea that the feminine is the origin of all gentleness on earth, whether it is intended as archetypal or empirical, is problematic is either case. I disagree with Perpich that Levinas is not describing a set of characteristics attributed to women, since his description is perfectly in line with a historical, patriarchal image of women. Real women are continually held up against these ideals. Thus, regardless of Levinas's _intention_ , he perpetuates a limited view of woman as nurturer, erotic beloved, idealized wife, and mother. Woman's kindness, her passivity, and her conjugal vocation to be the precondition for all virtue. On this view, women's engagements in activities outside the kitchen, bedroom, and nursery are a threat to the very existence of ethics; women's submission is an ethical imperative. This becomes even more disturbing when paired with Levinas's statements that exclude women from participation in the ethical relation. Recall that the feminine beloved's face \"fades,\" that she has \"quit her status as a person.\" As Irigaray says, \"For him, the feminine does not stand for an other to be respected in her human freedom and human identity. The feminine other is left without her own specific face. On this point, his philosophy falls radically short of ethics.\" If Levinas's philosophy is to be read in this light, then it would not just be inadequate as a theory of ethics, but would be, itself, unethical. It not only validates but also enforces the traditional feminine role that has been oppressive to so many women. Moreover, since women are without a \"face,\" this leaves them no recourse to seek justice in response to their exploitation.\n\nTina Chanter acknowledges a slippage between Levinas's references to the feminine and to women. \"So while Levinas cannot be accused of definitively or intentionally marking the inferiority of the feminine sex, there are ways in which this inferiority is marked in his texts, despite his best intentions.\" In spite of this equivocation, she believes that Levinas ultimately intends to upset the traditional values of egoism and male dominance. Donna Brody affirms that Levinas can be read along either interpretation (feminine = woman, or feminine = an aspect of everyone), but that each is problematic in its own way. If we read Levinas as referring to women when he speaks of the feminine, then we must condemn his patriarchal characterization. But if we interpret the masculine and feminine to be aspects in each of us, then he lacks any theory of sexual difference. Given Brody's estimation that he vacillates between these two positions, it would seem that Levinas remains _dually_ problematic for feminism.\n\nSome scholars applaud Levinas's ethics as providing a necessary _feminine_ counterpoint to the overly masculine history of ethics for its emphasis on care, charity, and the home; but ultimately the concern remains that it may not be sufficiently _feminist_. Levinas's account is problematic whether he intends to speak of actual women _or_ of a feminine principle. His characterization of femininity is anachronistic; it evokes and affirms conditions that feminists have rightly denounced as oppressive. Levinas maintains that he does not want to \"ignore the legitimate claims of feminism.\" Yet I see no evidence that Levinas is even familiar with their arguments. Feminists have many landmines to negotiate within his texts because he draws on sexist portrayals of women for his characterization of the feminine. In his early works, Levinas writes from a heterosexual, patriarchal point of view. The significant players are all masculine; the feminine and maternal are always depicted from a masculine point of view. In _Time and the Other_ , the father's relationship to the son is the central human relation; woman's most important function is to provide a son. Mothers and daughters do not participate in egoism or ambiguity, in time or the future. In this work Levinas disappoints women in the classic way, he fails to acknowledge their participation in humanity. _Totalityand Infinity_ adds little to improve this. Even though Levinas seems to try to honor women in his descriptions of hospitality, they are still regarded within their traditional restrictive role. Furthermore, his characterizations of voluptuousness are overtly sexist; insofar as the feminine beloved is voluptuous for Levinas, she has no ethical status.\n\nOtherwise Than Being, or Beyond Essence\n\nIn _Time and the Other_ and _Totality and Infinity_ , ethics and the emergence of self-understanding begins with the birth of a son; in _Otherwise Than Being_ , they begin before birth, in the maternal body. In contrast to Levinas's previous focus on the \" _concrete situation_ \" of paternity, he turns to maternity _as a metaphor_ : \"The evocation of maternity in this metaphor suggests to us the proper sense of oneself. The oneself cannot form itself; it is already formed with absolute passivity. In this sense it is the victim of a persecution that paralyzes any assumption that could awaken it, so that it would posit itself for itself. This passivity is that of an attachment that has already been made, as something irreversibly past, prior to all memory and recall.\" Levinas is not referring to the fetus or child when he says that the self is passively formed via maternity in a time irrevocably lost to memory. While the mother gives physical existence to the child, the child bestows the mother with the \"proper sense of oneself,\" \"complete being 'for the other'\" (108). It is the mother (with child) who is this absolute passivity, the \"victim\" who is the prototypical ethical subject, the sufferer of this inevitable amnesia; and _pregnancy_ \u2014\"the gestation of the other in the same\"\u2014is the archetype for responsibility (105). In this view, to be a mother means to be \"devoted to the others, without being able to resign,\" \"incarnated in order to offer itself, to suffer and to give\" (105). The maternal metaphor helps Levinas to make clear how subjectivity, properly understood, is \"an irreplaceable hostage\"; how as ethical beings we are _all_ victimized, held captive, and persecuted (124). He believes that serving the other in this way is not an alienation from oneself because \"I am summoned as someone irreplaceable\" (114). That is, ethical responsibility is the true foundation of the self; the self is epiphenomenal to its responsibility for the other. Thus, in this text, pregnancy and motherhood are the image of all ethical relations. It is significant, however, that maternity is invoked by Levinas specifically _as a metaphor_. Men are not excluded from these relations as women were from the \"concrete situation\" of paternity in the earlier works.\n\nWhy does Levinas shift his focus from paternity to maternity? In _The Gift of the Other: Levinas and the Politics of Reproduction_ , Lisa Guenther points out that by introducing maternity Levinas addresses his previous \"neglect of the flesh.\" \"The maternal body _gives herself_ , her skin stretching to make room for an Other; she not only provides but actually _becomes_ a dwelling-place for the stranger. For Levinas, the maternal body gives despite herself, without having chosen to give.\" It is true that the maternal body gives to the fetus without the mother needing to will it. The very calcium from her bones will be leached for the developing baby if necessary. Empirical facts such as these can support the belief that instances of involuntary generosity exist. The example of maternity also helps Levinas to break more completely from the notion of the independently existing ego than he could before. When Levinas works from the example of the father, the subject always exists (albeit imperfectly) before the child arrives, but the mother embodies permeability to the other. In _Time and the Other_ , the self begins in solitude until the other makes communication and, most importantly, _time_ possible; in _Totality and Infinity_ , the other gives one ethics and freedom. However, in _Otherwise Than Being_ , responsibility for the other is the source of one's own uniqueness; the child is conceived before the mother can be born.\n\nThere is no simple conclusion about the feminist (or sexist) potential of _Otherwise Than Being_. Initially it might appear to be a dramatic overturning of Levinas's earlier ideas. A relational ontology takes hold. Many feminists have also proclaimed the necessity of such an ontology. Care ethicists, for one, argue that individualist notions of the self stand in stark denial of the mother-child relationship, and of the fact that our existence is dependent on others before we can even entertain the possibility of solipsism. In _Otherwise Than Being_ , ethics and intersubjectivity seem to be a matriarchal affair. However, it is a problem that women gain their selfhood and ethical standing through bearing children. Is a woman's identity to be entirely derivative? What about women who are childless by choice, by chance, or because of infertility? What about adoptive mothers who did not gestate the other within? Are they not models of the \"proper\" sense of oneself? Are women not to be as valued for their other achievements and abilities? Is their independence only a _barrier_ to their ethical responsiveness?\n\nAnother problem with _Otherwise Than Being_ is that its characterizations of maternity seem to be at odds with women's reproductive freedom. For example, Sonia Sikka notes that abortion could never be ethical under Levinas's picture of maternity: \"Notice what happens, in this regard, if one introduces, into the portrait of ideal maternity painted by Levinas, the image of a woman who, in order to secure her own well-being, chooses to eject the other whom she harbors within her body. From the perspective of Levinas's ideal mother, must not this latter woman, whether or not her choice is guaranteed by the law of the land, be judged as monstrous? It must, I think.\" Obviously, the fact that abortion becomes a quintessentially unethical act in Levinas's philosophy would be problematic for many feminists. (I will have more to say about this later in this chapter.) Likewise, if a woman is able to use birth control, give her children up for adoption, or even ask for help with the children, then this gives the \"irreplaceable hostage\" a way out of her bondage. Even when women _do_ want children, they do not necessarily want to become their hostage. In _Maternal Ethics and Other Slave Moralities_ , Cynthia Willett expresses concern that the figure of the mother in Levinas is one of a self-sacrificing martyr that \"effaces the female self. . . . The verticality of the approach of the Other obliquely renders the mother supine before her infant's every demand.\" This problem is compounded by the fact that \"Levinas never theorizes the maternal subject as the recipient of care.\"\n\nLevinas's image of the suffering, captive mother looks even worse when we pair it with his language, which is evocative of sexual violence. As Chlo\u00e9 Taylor indicates:\n\nEvoking as Cynthia Willett has noted, both the \"violation\" of a feminized subject and \"pregnancy against one's will,\" Levinas writes that the subject has the other \"in his skin,\" is \"penetrated-by-the-other,\" and this involuntarily, \"despite itself,\" \"a sacrificed rather than sacrificing itself.\" One is \"torn up from oneself in the core of one's unity,\" in a \"nudity more naked than all destitution,\" and \"whitens under the harness\" in the \"form of a corporeal life devoted to expressing and giving. It is devoted, and does not devote itself.\" Drawing intentionally or otherwise on metaphors of concubinism, rape and involuntary impregnation, this subject is responsible for the other who is not in \"his\" skin, like a phallus, like a fetus, whether or not \"he\" chose to have it there.\n\nEven if we assume that Levinas does not intend to advocate violence against women, what is to prevent his philosophy from being used to rationalize misogyny? In light of these concerns, Levinas's nod to feminism in _Time and the Other_ , is not very reassuring\n\nIs it possible that, while he invokes women's servitude as a metaphor, Levinas does not intend to enslave women or romanticize their oppression? Certainly, his supporters could argue that he does not intend to single women out. What Levinas says of the mother, he intends for every person: not just mothers' subjectivity, but _everyone_ ' _s_ subjectivity is derivative. It is not that women, in particular, are passive helpers, rather ethical \"agency\" is constituted in radical passivity and responsiveness to others. Thus, if women appear submissive in Levinas's account, this actually demonstrates that they are ethically empowered. Levinas's allies could argue that although women find themselves in their traditional place in Levinas's writings, he elevates the status of these roles. In this regard, Levinas is in agreement with care ethics theorists who seek to affirm the value of feminine caretaking.\n\nHowever, it is important to recognize that the call to prostrate oneself has diverse implications for people who are differently situated. While it may be a radical move to obligate a privileged person to serve others, to make this decree to an exploited person or group merely reinforces an oppressive status quo. In this regard, Levinas's ethics can be charged with the same criticism that the valuing of \"feminine tenderness\" conceals a \"slave morality.\" Some feminists argue that to value feminine ethics is to affirm the oppressive conditions in which they arise. In this view, women's position of social inferiority has damaged them morally, and such women should serve primarily as cautionary tales. What of mothers who do not want to submit to the \"proper sense of oneself\" according to Levinas? What about women's desires for political freedom and support in caregiving? Is it never justified to desire some independence, to have a life that is not completely commandeered by the other? Levinas might say that these women suffer from the same delusion as Western philosophy.\n\nTo be without a choice can seem to be violence only to an abusive or hasty and imprudent reflection, for it precedes the freedom non-freedom couple, but thereby sets up a vocation that goes beyond the limited and egoist fate of him who is only for-himself, and washes his hands of the faults and misfortunes that do not begin in his own freedom or in his present. It is the setting up of a being that is not for itself, but is for all, is both being and disinterestedness. . . . Responsibility for the other, this way of answering without prior commitment, is human fraternity itself, and it is prior to freedom.\n\nFeminists have struggled for the rights to make choices, to enjoy freedoms, to be \"for-herself\" in spite of the compulsion to live for others. Historically, their existence as \"responsibility for others\" has not been the basis for participation in \"human fraternity,\" but rather exclusion from it. Levinas does not acknowledge that the desire for freedom from the demands of the other is contextualized by different material, historical, and cultural situations.\n\nThe best defense for Levinas comes from Lisa Guenther's _The Gift of the Other_ and \"'Like a Maternal Body': Emmanuel Levinas and the Motherhood of Moses.\" Guenther argues that Levinas is not trying to dictate women's roles, rights, or behavior; instead, he describes some parallels between the responsibility of mothers and the obligations we all bear. She supports this conclusion through analysis of some of the biblical passages cited by Levinas.\n\nIn _Totality and Infinity_ , Levinas repeatedly invokes Isaiah 49 where God compares himself to a nursing mother. In this verse, god is addressing Zion who has asked why God has abandoned her (Isaiah 49:14):\n\nCan a mother forget her nursing child,\n\nOr show no compassion for the child of\n\nher womb?\n\nEven these may forget,\n\nYet I will not forget you.\n\n(ISAIAH 49:15)\n\nIn reassuring Zion that he will take even better care of her than a nursing mother, god speaks to her as a daughter who still needs the care of a mother. God is a feminine other to Zion and the care that he gives to her can be passed onto the next generation. The implication is that a mother's care is received from her own mother and is to be paid forward to her children, grandchildren, great grandchildren, and so on; the mother-child relationship extends beyond the present dyad to an indefinite number of generations forward and backward.\n\nZion is much like the imperfect mother to whom God compares himself. She does not feel compassion for her children, she has forgotten them, and they are like strangers to her.\n\nThen you will say in your heart,\n\n\"Who has borne me these?\n\nI was bereaved and barren,\n\nExiled and put away\u2014\n\nSo who has reared these?\n\nI was left all alone\u2014\n\nWhere then have these come from?\"\n\n(ISAIAH 49:21).\n\nHowever, God's extended generosity applies not only to one's own descendents but to strangers as well. Thus God's generosity allows Zion to care for her own children, even if they seem like strangers. Similarly, Zion can care for the stranger as though it were her own child. \"In this sense God becomes for Zion like a 'feminine Other' in _Totality and Infinity_ ; She gives the capacity to welcome strangers whose arrival on the doorstep commands me to give beyond my present resources for hospitality.\"\n\nInterestingly, when Levinas refers to Isaiah, he is discussing _paternity_ , not maternity. To describe the father in _Totality and Infinity_ , he draws on the model of a mother\u2014Zion\u2014who relies on the strength and compassion of god the father. For Guenther, this is key to undoing Levinas's seeming rigidity regarding sex and gender roles: \"When Levinas cites _Isaiah_ 49 in his discussion of paternity, he makes room for a reader to suggest that the 'father' in paternity might also be a mother, that the 'son' might also be a daughter, and that the parent's own parent, God, might be like a mother who inscribes the memory of his child on her hands, and never fails to show 'compassion for the child of her womb.'\"\n\nGuenther's claim that Levinas was probably not thinking of maternity in strictly biological terms is further supported by Levinas's references to Moses in _Otherwise Than Being_. The context for this discussion comes from the book of Numbers:\n\nMoses heard the people weeping throughout their families, all at the entrances of their tents. The Lord became very angry and Moses was displeased. So Moses said to the Lord, \"Why have you treated your servant so badly? Why have I not found favor in your sight, that you lay the burden of all this people on me? Did I conceive all these people? Did I give birth to them, that you should say to me, 'Carry them in your bosom, as a nurse carries a suckling child, to the land that you promised on oath to their ancestors'?\"\n\n(NUMBERS 11:10\u201312)\n\nMoses clearly invokes motherhood as the model of responsibility. Like Zion, Moses is alienated from the people he is supposed to care for; though he did not conceive or give birth to these people, he must care for them like a mother for her baby. Guenther indicates that in these passages invoked by Levinas caring for others is not strictly mothers' work; it is only _similar_ to it: \"In _Otherwise Than Being_ , Levinas suggests that responsibility for the Other obligates me to bear her 'like a maternal body' (OB 67; AE109), even though she is a stranger whom I have 'neither conceived nor given birth to' (Num. 11:12, cited in OB 91; AE 145). . . . In this sense, maternity would not refer to a biological or social imperative for women to reproduce, but rather an ethical imperative for each of us to bear the stranger _as if_ she were already under my skin, gestating in my own flesh.\" Guenther argues that Levinas's use of Moses also points to the multiple quasi-mothers that made Moses' own maternity possible. First, there is god, who acts as a mother to Moses. Second, there is the pharaoh's daughter who finds Moses as a baby and raises him like her own son. Most interestingly, even Moses's biological mother, Jochebed, is only able raise him by acting _like_ a mother to him. According to the narrative in Exodus, at the time Moses was born, the Israelites were beginning to outnumber their Egyptian captors. So Pharaoh ordered that newborn Israelite boys be killed. Jochebed, unwilling to kill her son, placed him in a basket that gently carried him to the river. After the pharaoh's daughter finds and adopts Moses, she unknowingly hires the child's biological mother as his wet nurse. Just as Ruddick claims, even Moses's biological mother must adopt him. Among these biblical figures who hold the predictable maternal ideal, maternal responsibility is often unbidden, shifting, and unpredictable.\n\nGuenther argues that the appearance of such a wide variety of mothers in these important biblical passages suggests that Levinas's figure of maternity should not be thought of narrowly. She claims that Levinas denies \"any strict correlation between women and mothers, or even between motherhood and responsibility.\" Thus, she concludes,\n\nThe maternity of Moses and of god suggests that one is not born, but rather becomes _like_ a mother. The biological fact of incarnation in a female body need not condemn me to a destiny of childbirth, nor does incarnation in a male body free me from the responsibility of bearing the Other \"like a maternal body.\" In this sense, maternity becomes more than a social role or fixed biological destiny, either of which would bind the identity of women to childbirth and child rearing. By understanding maternity _ethically_ as the embodied response to an Other whom I may or may not have \"conceived and given birth to,\" we recognize maternity as a locus of responsibility, without expecting women to bear that responsibility alone.\n\nFor Guenther, Levinas neither sentences women to maternity and gender-based obligations nor frees men of ethical responsibility.\n\nAnother of Guenther's key claims is that since the ethical calling makes one _like_ a mother, then this means that the ideal of mothering need not map onto its reality. Indeed, she acknowledges that while Levinas does not characterize motherhood realistically, she considers this is an appropriate gap between _what is_ and _what ought to be_ :\n\nMotherhood is not always a patient, generous, compassionate practice of ethical substitution. Maternal generosity can spill over into resentment and anger, even to the point of becoming violent. Women and men who raise children, like anyone else, have moments of impatience, fatigue, distraction, selfishness; these moments do not indicate a failure of particular mothers to reach the ideal of maternal sainthood, but rather a confirmation of the difficulty of ethical life and the vulnerability of human creatures who are both limited and responsible for Others who push their limits. If the human were unfailingly saintly, there would be no need for hagiography nor even for an ethics of responsibility; knowing the good, doing the good, and being good would all neatly coincide. . . . To be _like_ a maternal body even when you are one is to admit a gap between mothering as an ethical and political practice, and the mother as an ontological, biological, or social identity. By recognizing that it is possible to be _like_ (or also _unlike_ ) a mother even while one \"is\" a mother, we recognize a difference between ontology and ethics, between being and the otherwise-than-being.\n\nFor Guenther, while we can aspire to ethical sainthood, lapses are not really a failure; they are confirmation of our all-too-human limitations. While Levinas helps us to _know_ the good, we remain limited in how good we can actually _be_. Mothers, like other mortals, are imperfect, but the figure of the maternal remains instructive for Guenther. The biological relation to another is not the _cause_ of one's responsibility toward that other, but the gift of care that is passed from one generation to the next is the condition for the possibility of ethics. In stating that ethics is similar to the maternal relationship, Guenther argues, Levinas affirms the possibility that even a stranger can be taken in and cared for like family. As in the poignant ending of the _Grapes of Wrath_ in which a young mother whose baby has died feeds a starving man from her breast, by making one into a mother, the child calls on a radical generosity that can be applied to the stranger.\n\nWhile Guenther embraces the fallibility of mothers, nowhere does Levinas acknowledge that a mother might lose patience, act selfishly, be violent toward her children, or be limited in any way in her capacity to give. He never equivocates; the mother's vulnerability, permeability, responsibility, and devotion are complete. In my view, this oversight is even more stark given his invocation of Zion and Moses. Both figures are fallible and unwilling mothers, but Levinas does not mention these limitations in his characterizations of maternity. On the contrary, the mother that Levinas presents is more like the god that supports Zion and Moses when they falter. She is the mother who is always compassionate and always remembers her child.\n\nIt might seem that in order to read Levinas as acceptable to feminists our best option is to claim that when he speaks of the feminine, women, and the maternal, he does not intend to invoke or implicate actual female-identified people. In fact, Levinas equivocates a great deal on this point, and arguments on both sides are probably valid. Yet, in some respects, this is beside the point. The deeper concern is that he advocates values that naturalize gendered roles as they have been prescribed by patriarchy and elevates them to an ethical imperative.\n\nAs I argued in chapter 3, mothers' feelings toward their children are much more complex than Levinas would have us believe. Although he attempts to illuminate the meaning of maternity in _Otherwise Than Being_ , his ignorance of the conflicts between mother and child is striking. Nevertheless, it could be argued that Levinas's use of maternity is only metaphorical: he does not overtly claim to give an accurate description of maternal experience; thus it is inappropriate to critique him on this basis. In contradicting this claim, Jacques Derrida's analysis of philosophical uses of metaphor in \"White Mythology: Metaphor in the Text of Philosophy\" is helpful. As Derrida asserts, the philosophical use of metaphor is never neutral; philosophers _actively establish_ the meaning of the metaphors they employ.\n\nDerrida argues that, in philosophy, a metaphor \"promises more than it gives.\" And this is clearly the case in _Otherwise Than Being_ , where the metaphorical characterization of motherhood obscures its actual complexity in favor of an obfuscating and worn-out straw figure. Derrida argues that such figures are treated as modes of expression of the idea, but in fact they have a history of their own. In fact, the biblical mothers are not, as Levinas treats them, evidence of their lasting truth. They are examples of a _historical_ understanding of motherhood, which, in this case, reveals that certain one-sided tropes of maternity are very long-standing indeed. They cannot be relied on to provide the truth of motherhood, but are instead evidence of past attitudes and impressions.\n\nDerrida claims that when philosophers use metaphors they are not used as mere ornaments; they are treated as an _essential_ , internal link to the phenomenon they are used to elaborate. This is definitely true in Levinas's case. He practices a circular hermeneutics between his notions of motherhood and ethical subjectivity in which elaboration on one phenomenon is also intended to elucidate the other. In the end, we \"know\" motherhood differently than we initially did, and, to a great extent, this is overdetermined by Levinas's theory of subjectivity. The _image_ he presents of motherhood is really a concealed metaphysics, an essentialist depiction of motherhood. Levinas has not so much unveiled a truth as he has created one.\n\nEspecially given his apparent ignorance of the complexities of motherhood, it is disturbing that Levinas appropriates the maternal perspective and attributes it to all ethical beings. Derrida poses this concern in speaking of Levinas's quotation of the Song of Songs, \"[the] phrase translated and quoted . . . it is torn from the mouth of a woman, so as to be given to the other. Why doesn't he clarify that in this work?\" This question could be directed to the references to Zion as well. Guenther also mentions this worry with regard to the use of Moses: \"We could argue that by referring to Moses as an example of maternity\u2014especially in the absence of any female examples of maternity\u2014Levinas appropriates the female capacity to give birth to the male body.\" This anxiety is put most vehemently by Donna Brody: \"It might also be objected that the female capacity for child-bearing is poached, colonized, and substituted from her to him where it figures as a kind of agamic or parthenogenic [sic] reproduction of other-in-the-same. In other words, she may be read as redetermined according to the most eminent meaning\u2014or she may be read as _altogether_ obliterated, exiled even from the significance of maternity.\"\n\nIt is a common and tiresome practice of male philosophers, poets, and other meaning makers to borrow the experiences of women via metaphor. In Beauvoir's \"Myth\" chapter in _The Second Sex_ , she gives innumerable examples of how men make woman the other, then identify with that otherness. At least since Plato's _Theaetetus_ , maternal experience has been exemplary for philosophy, and yet has been appropriated in a way that excludes women from that which they represent. (Women only give birth to _bodies_ , while _men_ give birth to _ideas_.) Male philosophers have displayed their womb envy in a manner that denigrates the unique capacities of female bodies. A slightly more authentic response than this reaction formation can be seen in the essay \"Yogurt Couvade\" by Rob Hardy as he describes the experience of observing his pregnant partner's body:\n\nShe crossed the room, loose-jointed and front-heavy, and lifted the yellow maternity blouse up over her head. She looked like a daffodil gone to seed, the withered corona twisting above a plump ovary bursting with seed. As she dropped her blouse into the laundry basket, the brown line under her belly button reminded me of the rust stain in the bathtub stretching from leaky faucet down to the drain. Her body itself was a living metaphor, a noun curled up inside another noun. It was only by metaphors that I could relate to the experience of pregnancy. As a man, I could never experience the thing itself, from the inside; I could only press my hand against the skin of metaphor.\n\nHardy claims that it is only through _metaphor_ that men may understand the experience of pregnancy, but this is not fully the case. They may also understand it by asking and listening to women themselves. When philosophers make metaphors about pregnancy, childbirth, and mothering, divorced from any serious dialogue with those who have lived through it, they don't come to understand women's bodily experience, they _appropriate_ it, and often very poorly.\n\nUltimately, one should approach Levinas's maternal metaphor with skepticism, but this is not to say that Levinas is of _no_ use to feminists and others interested in maternal ethics. As Guenther points out, to be _like_ a maternal body is to have dependents that are very hard to shake. The fetus is the vampiric primal parasite\u2014a profound dependency that can suck the nutrients from one's blood\u2014that suggests the grasping and bottomless needs of others. The biblical quotes referenced by Levinas demonstrate the frustration one might feel at the burden of responsibility for others even though Levinas's passages on maternity do not directly thematize this problem. Oddly enough, he is most helpful in understanding mothering when he is either not speaking directly of mothers, women, or the feminine or when they are only implied peripherally. One example is his theory of an ambiguous intersubjectivity that is a more accurate reflection of mothers' lives, as discussed previously in chapter 3.\n\nAMBIGUOUS INTERSUBJECTIVITY IN LEVINAS\n\nProximity and Subjectivity\n\nIn _Otherwise Than Being_ Levinas denies the idea that subjectivity is primarily self-coincidence or self-knowledge and posits that \"a subject is a hostage.\" For Levinas, subjectivity is incarnate consciousness; it is exposure and proximity to sensibility and to the other. The other is not just another point of view, but a primordial disturbance in my appropriation and enjoyment of the world. Levinas's subject is radically open, destabilized, and, above all, vulnerable. Even sensibility is a radical disruption of the self: \"Sensibility] is being torn up from oneself, being less than nothing, a rejection into the negative, behind nothingness; it is maternity, a gestation of the other in the same.\" For Levinas, to be a subject is to be incessantly undermined by one's proximity to the other. While some may find Levinas's language hyperbolic, it has much in common with firsthand accounts of motherhood. As explained in [chapter 3, mothers frequently find themselves hostages to their children. A mother sometimes experiences her child as a monstrous _Other_ who keeps her captive, threatens her body, eats away at her vitality, upsets her sense of self, invades her home, and commandeers her life. In many respects, the birth of the child is the death of the mother as she has known herself.\n\nAccording to Levinas, proximity to the other is substitution, giving one's material substance to the other, putting myself in her place not as an alter ego, but as the food in her stomach. And my obligation to feed the other is absolute. If we are to understand Levinas concretely, proximity begins in the intertwining of bodies. In pregnancy and nursing, the mother's bodily boundaries expand to include another. Her body, her blood, and much of her food become sustenance for the other. Yet mothering remains an embodied experience, even beyond these short stages. As argued in chapter 3, older children\u2014adult children as well\u2014get under their mother's skin in both positive and negative ways.\n\nProximity describes the way the other draws me toward her, to provide for her, to give over myself. Yet, for Levinas, it extends beyond mere physical contact or seeing the other, beyond \"handshakes, caresses, struggle, collaboration, commerce, conversation,\" beyond consciousness or knowledge. This undermining proximity can even persist in spite of a child's death. Recall the mother in chapter 3 who describes her experience of her adult child's death: \"'A woman named Elizabeth Stone wrote that having a child is 'to decide forever to have your heart go walking around outside your body,' Suzanne says. 'That's what becoming a parent is like. And now I've lost that piece of myself.'\" Like this woman, Levinas describes the other as the pulsing of breath and heartbeat: \"the body which makes giving possible makes one other without alienating. For this other is the heart, and the goodness of the same, the inspiration or the very psyche in the soul.\" This statement seems sentimental enough to be embroidered on a pillow. But in fact it is gruesome and cruel for another to be one's heart, goodness, breathe, psyche, and soul. All that might serve as the center of oneself belongs to another. It is to have one's vital organs outside one's body exposed to all the cruelty and indifference in the world, to be \"an open cauldron of blood.\" To become a mother is to be turned inside out and never to be quite right-side in again. According to Levinas, there is no \"right-side in;\" there is no subject that exists before vulnerability to others. The self only arises in relation to others; there is no preexistent, pure self to return to. Thus, to revert to oneself, to try to find oneself as autonomous, would be to find no self at all; \"to revert to oneself . . . is to be like a stranger.\"\n\nAlterity, Ethics, and Freedom\n\nWhile it is as near as any intimacy, proximity occurs across an abyss and cannot overcome the insurmountable alterity of the other. Describing the encounter with radical alterity is a central focus in _Totality and Infinity_. Levinas explains that the face of the other indicates the presence of an unknown infinity. Realized suddenly, it provides an abrupt leap of understanding. However, this understanding is not _knowledge_ of the person, as though one suddenly knows what that person is about. On the contrary, it is an awareness that one does _not_ know, that this person is wholly other to oneself, utterly unique. The other's infinity is always more than one comprehends it to be. When one sees the other as a body, a personality, a set of thoughts, symptoms, behaviors, attitudes, or feelings, these are signs of an unreachable world. According to Levinas, to have a proper respect for the other means to recognize that one can never fully know him. The mere presence of the other challenges the free reign of one's spontaneity, because when one moves about the world without regard for the other, one bumps into her. Thus the face tells us that to be impetuous is to be violent. The face _exposes_ the other to us. To expose another is to leave him in a dangerous situation; to expose a baby means to leave it outside to die. Thus the other's face reveals him to us, but he is revealed as unprotected. For this reason, Levinas says \"the epiphany of the face is ethical.\" The appearance of infinity in the face is accompanied by the awareness of vulnerability and thus the call to be ethical.\n\nMothers provide numerous accounts of the ethical epiphany of the face. In _Down Came the Rain: My Journey Through Postpartum Depression_ , Brooke Shields describes the moment when she felt suddenly compelled to protect and care for her daughter:\n\nAt the hospital exit, as expected, we were met by cameras and comments. The photographers asked the baby's name and tried to get a close-up of our five-day-old daughter. I was struck by how much of an intrusion this was. I was accustomed to it, but it felt different where she was concerned. It felt extremely personal and even more invasive. Suddenly I didn't want them to know her name, nor did I want the cameras in her little face. I regretted putting her through this but felt as if I didn't have a choice. A wave of fear washed over me, and I instantly thought something was going to happen to her out there. I sought the refuge of the locked car. Rowan didn't cry once. As I entered the waiting vehicle, a smile frozen on my face, Rowan looked straight up at me as if to say, \"Mom, these people are too close.\" I looked down at her and was taken aback by the directness of her gaze. Once again, she seemed wiser than I felt. Silently I made a promise to protect her.\n\nShields's daughter is exposed by the cameras. They reveal both the vulnerability and the wisdom that are displayed on her face, and, in Levinas's terms, Rowan's infinity calls Shields to protect her.\n\nEva Kittay describes a similar experience in \"'Not _My_ Way Sesha, _Your_ Way, Slowly': 'Maternal Thinking' in the Raising of a Child with Profound Intellectual Disabilities.\" Kittay describes an interaction between her disabled daughter Sesha and Sesha's caregiver Peggy. Peggy had not wanted to take the job, but agreed to a trial week. During this week, Peggy was working on some exhausting physical therapy with Sesha:\n\nI sat her down in her stroller and sat down on a park bench. I realized I was simply too exhausted from the effort. I thought, how am I going to do this? How can I possibly do this job? When I looked down at Sesha and saw her little head pushed back against her stroller moving first to one side then to another. I couldn't figure out what she was doing. Until I traced what her eyes were fixed on. She had spotted a leaf falling, and she was following its descent. I said 'Thank you for being my teacher, Sesha. I see now. Not _my_ way, _your_ way, slowly.' After that I fully gave myself over to Sesha. That forged the bond.\n\nIn both these accounts the women feel compelled to care for the child after witnessing the child's vulnerability and uniqueness. As Levinas predicts, the child herself induces the mothering person to care. Their vulnerability has a certain power: \"Infinity presents itself as a face in the ethical resistance that paralyzes my powers and from the depths of defenceless eyes rises firm and absolute in its nudity and destitution.\" George Kunz has named this phenomenon \"the paradox of power and weakness.\" The weak person is powerful because she calls the powerful to respond to her needs. \"The child and the ill person calling me to their aid, those suffering poverty inspiring my compassion, even the irresponsibility of an enemy urging me to help them become responsible points to this paradox.\" In this sense the ethical person is rendered passive by the other. This susceptibility compels protection. Sesha's observance of the leaf and the expression of wisdom in Rowan's eyes are indications of their unique perceptions of the world, suggestions of an inaccessible infinity. For Levinas, to have a proper respect for the other means to recognize that one can never fully know her. In both cases the mothering person witnesses something in the face of the child\u2014wisdom, intentionality, and also vulnerability. This epiphany compels her to give herself over, to promise to protect.\n\nAccording to Levinas, we are constantly constrained and motivated by others. Yet this does not mean that we are not free. No person is a sovereign subject, and the ability to assert one's will against others is domination, not freedom. Instead he argues that freedom is made possible by ethical responsibility. \"Conscience . . . is the revelation of a resistance to my powers that does not counter them as a greater force, but calls in question the na\u00efve right of my powers, my glorious spontaneity as a living being. Morality begins when freedom, instead of being justified by itself, feels itself to be arbitrary and violent.\" Ethics is freedom called into question, but freedom is also defined by the opportunity to be ethical. Freedom begins when one suspends her rights in favor of another, when he controls his caprices, when she restricts her reign. Freedom is not an exertion of power, but the exercise of restraint. And this possibility for ethical freedom is not recognized until one notices that one has already violated the other. Levinas writes: \"The first consciousness of my immorality is not my subordination to facts, but to the Other, to the Infinite.\" The other causes me to realize my \"moral unworthiness,\" that I have already harmed her; thus the possibility of restrained freedom becomes possible. It is the reconfiguring of \"innocent spontaneity\" as arbitrary, violent, and shameful. This is a paradoxical opportunity; one's response to the other is not _enforced_ , but it is _coerced_ by her vulnerability. In Shields's case, as for many others, she cares for her daughter even when the difficulty of doing so leads her to suicidal and murderous thoughts.\n\nPeggy's choice to become Sesha's caregiver was also freely obligatory. In caring for Sesha, she enters voluntary servitude. Kittay says that Peggy had not wanted to take the job, but that by the end of the one-week trial \"it was already nearly too late to quit. Sesha had worked her way into Peggy's heart.\" As Kittay describes it here, Peggy's commitment to Sesha was not a simple choice. She had not wanted to take the job, and it was not forced on her, yet she felt compelled to take it. Her adoption of Sesha, in Ruddick's terms, had an odd temporality to it. \"It was _already nearly_ too late too quit.\" In fact, the time to commit or quit would have been right then, at the end of the one-week trial. While she could point to a moment of commitment\u2014when she witnessed Sesha's infinity\u2014it was also as though the obligation began before any deliberate or reflective choice.\n\nThe awkward temporality of adoption, whether by a biological parent, a paid caregiver, or a legally adoptive parent, describes for Levinas the temporal structure of all ethical commitment. He does not think that responsibility arises from commitment, but that commitment already presupposes obligation. \"The relationship of proximity . . . [is] anachronously prior to any commitment.\" Ambiguous intersubjectivity, characterized by both proximity and alterity, is obligation. And obligation is perpetually anachronistic; its time is never the present. Indeed, this could also be said of Shields's commitment to Rowan. When she pledges to care for Rowan outside the hospital, she is already obligated to her. But when did this obligation begin? When she decided to conceive? When she began fertility treatments? When Rowan was conceived? When she was born? There is no clear answer, no clear origin for her obligation. \"This is an anarchic plot, for it is neither the underside of a freedom, a free commitment undertaken in a present or a past that could be remembered, nor slave's alienation, despite the gestation of the other in the same, which responsibility for the other signifies\" (105). Obligation is an-archic; it is without _arch\u00e9_. An _arch\u00e9_ establishes the beginning of linear time, but we are always already within time. We cannot witness the _arch\u00e9_.\n\nFor Levinas, proximity and alterity form an irresolvable ambivalence, \"not a dialectical unity of unity and difference\" (83). Proximity cannot overcome alterity; alterity cannot abolish proximity. Dialectical unity is a mere exercise of reason (84), but the relation to the other precedes and defies reason. This ambivalence is \"the thorn in the flesh of reason, which is the shudder of subjectivity\" (84). Subjectivity seeks unity with itself; it seeks understanding, but self-coherence would eliminate the necessary intrusion of the other. As there is no unity with oneself, there is also no unity to be found with the other; this would imply reciprocity between two equal terms\u2014self and other. But the relationship to the other is always asymmetrical, \"going to the other without concerning oneself with his movement toward me\" (84). The other and oneself are not two magnets that are mutually drawn to one another. The other is a magnet that draws oneself, an uncharged piece of metal, closer. Yet, despite this attraction, these two never overcome the chasm of alterity. According to Levinas, the allure of ethics is never satiated.\n\nETHICAL AMBIVALENCE\n\nLevinas thinks that the other appears to be in conflict with the self when one understands the self through enjoyment and self-creation (72\u201373). From this perspective, alterity is the interruption of self-fulfillment and nourishment. Levinas illustrates this repeatedly with the example of taking the bread from one's own mouth to feed another. To be sure, mothers frequently blame an inability to nurture themselves (both literally and metaphorically) for the loss of their independent selves. Recall Naomi Wolf's lament for \"the self that is not food for others, but eats and drinks the world\" and Jane Lazarre's \"mother knot\":\n\nI turned to that self inside of me, that girlwoman who had once been all I needed to know of myself, whom I had fought to understand, to love, to free\u2014I turned to her now and I banished her. Into a protective shell tied in a knot, she retreated, four, five, six times a day, whenever Benjamin wanted to nurse. Soon, even when I sought her, she would not come, but began to stay out of reach longer and longer, sometimes not reappearing for whole days. For if she was present when the baby needed me, she was of necessity pushed aside, sent to go hungry. She who had been in my life, whom I know I had to nourish daily in order to be fed in return, hid for weeks, hoarding her gentleness and her strength, placing no gifts in my outstretched hands.\n\nThe part of one's identity that finds itself in satisfaction of its own needs, in the \"singularization\" of the ego, encounters the needs of others to be an intrusion. As I demonstrate in chapter 3, mothers frequently set aside self-creation, fulfillment, and expression to respond to their children's needs, and, as Levinas predicts, it is a responsibility that cannot easily be shifted to another. Yet, for Levinas, the self that mourns her lost independence has a mistaken view of freedom; we are only free insofar as we are ethically obligated to others.\n\nAlthough we may remember a commitment made to the other, the obligation that is its origin is immemorial. This emphasizes the passivity of the ethical \"agent,\" but it also stresses that the obligation arising from the other's face is not _causal_. Causal relationships presuppose a linear temporality, but obligation does not submit to this temporality. Ethical obligation has no present, it has always already happened. So, although we are compelled by it, we are not _determined_ by the face of the other. The opportunity to be ethical establishes freedom, and with this freedom comes the possibility to abandon or even to murder.\n\nIn _Totality and Infinity_ , Levinas describes murder as a paradoxical power. Murder seeks to annihilate the other, but the other's face \"expresses my moral impossibility of annihilating.\" This \"impossibility\" is not complete; it contains within it the option to kill. Yet it remains unclear in Levinas's philosophy how murder can actually occur. In \"Peace and Proximity\" he writes: \"By starting with this extreme straightforwardness of the face of the other ( _autrui_ ), we have previously been able to write that the face of the other in its precariousness and defenselessness, is for me at once the temptation to kill and the call to peace, the 'You shall not kill.' The face which already accuses me makes me suspicious but already claims me and demands me.\" On the one hand, it is because violence would be so easy that the command \"thou shalt not kill\" makes sense. The other's vulnerability, therefore, is no straightforward call to be ethical. We may want to make way for the passing cyclist and also to push him down. We may want to soothe the crying baby and also toss her out the window. Mothers are often shocked by their desire to do violence to their children, especially when they are small and defenseless, but weakness can inspire malice as well as nurturance. On the other hand, according to Levinas, the impulse to harmonious coexistence is more fundamental: \"War presupposes peace, the antecedent and non-allergic presence of the Other; it does not represent the first event of the encounter.\" He thinks that one does not usually feel hostile toward the other; that existence does not chafe us. Pacifism, not murder, is precedent. To say that war presupposes peace begs the question; it does not explain how violent impulses can be overcome.\n\nIn these passages Levinas seems to deny the fact that ethical living is often a _struggle_. Certainly his language is the language of struggle, and it would not be accurate to say that he is dismissive of the devastating impact of the other's demands. Yet his simple assertion that war presupposes peace seems rather weak when juxtaposed with mothers' accounts of murderous rage. Although he understands that maternity involves sacrifice, that it undoes a person at both visceral and metaphysical levels, he is relatively unconcerned with the devastation caused by the demands of the other. In his account the mother is always compassionate, she always remembers and cares for her child. In this respect, Levinas thinks that ethics is not really a struggle at all, that there should be no maternal ambivalence. In his model, then, mothers ought to be rather poor ethical exemplars. At this point, we are compelled to return to the earlier question of whether or not Levinas is describing what _is_ or what _ought to be_. Is he describing the way mothers are, even if incompletely? Or is he describing the way that they should be, even if his demands are impossible?\n\nIt is a common and justifiable reading of Levinas to consider his work as descriptive phenomenology. In the _Stanford Encyclopedia of Philosophy_ , Bettina Bergo begins her discussion of Levinas as follows:\n\nLevinas's philosophy has been called ethics. If ethics means rationalist self-legislation and freedom (deontology), the calculation of happiness (utilitarianism), or the cultivation of virtues (virtue ethics), then Levinas's philosophy is not an ethics. Levinas claimed, in 1961, that he was developing a \"first philosophy.\" This first philosophy is neither traditional logic nor metaphysics, however. It is an interpretive, phenomenological description of the rise and repetition of the face-to-face encounter, or the intersubjective relation at its precognitive core; viz., being called by another and responding to that other. . . . He proposes phenomenological description and a hermeneutics of lived experience in the world. He lays bare levels of experience described neither by Husserl nor by Heidegger. These layers of experience concern the encounter with the world, with the human other, and a reconstruction of a layered interiority characterized by sensibility and affectivity.\n\nIf we are to take Levinas as an existential phenomenologist, as describing intersubjectivity and its implications for ethics, then he does capture the intersubjective ambiguity between mother and child, but he does not recognize that infinite demands actually _deplete_ us, making it difficult or impossible to serve the other. His attitude is reminiscent of Melanie Klein, who claims that a girl's desire for a baby, present from early childhood, diminishes the aggressive tendencies she might feel toward her actual baby. Her child's helplessness and vulnerability fulfill her lifelong desire to offer \"more love than can be given to any other person.\"\n\nPerhaps Levinas slips between describing the world as it appears and as it should be. As Lisa Guenther argues, there may be an appropriate gap between _what is_ and _what ought to be_. This is also a viable interpretation, as there is much in Levinas that is not verifiable through phenomenological description. Consider the fact that, for Levinas, our responsibility for the other is infinite. Most people hold that there are limits to one's responsibility for others. One can only be responsible for what one is aware of and when one has the power and resources to help; it is fair to weigh one's own well-being against what would be sacrificed for the other; we cannot be responsible for what others choose to do. In short, people generally believe that what we _should_ do is limited by what we _can_ do. The question of what we _can_ do may be addressed empirically. However, for Levinas, the _ought_ exceeds the _can_ and is therefore outside the purview of descriptive phenomenology.\n\nYet if Levinas describes an ideal to which mothers can aspire, then he lets us down in this regard as well. Idealizations of mothering only make its practice that much more difficult. When mothers think they should be able to give to their children infinitely and are unable to do so (as no human could), they feel guilty, angry, worthless, ashamed, depressed, and fearful of the judgment of others. These feelings paralyze mind and body, making it difficult to respond to one's child at all, let alone appropriately. As Alice Miller writes in _For Your Own Good: Hidden Cruelty in Childrearing and the Roots of Violence_ : \"I cannot listen to my child with empathy if I am inwardly preoccupied with being a good mother; I cannot be open to what she is telling me.\" Levinas's denial of maternal ambivalence, or any substantial ethical conflict, takes him too far from _what is_ to help us with _what ought to be_. His idealization of maternal sainthood is out of touch with the struggles of mothers, and this will only be counterproductive. What is more helpful to women is an accurate understanding of their own experiences; for this the maternal icon must be dethroned.\n\nIn _Torn in Two: The Experience of Maternal Ambivalence_ , Rozsika Parker describes ambivalence as an achievement that can ultimately enhance the mother's responsivity to her child. However, this requires that she let go of the maternal ideal and recognize her own limitations in the face of the tremendous weight of her responsibility. One mother, Selma, attests to this: \"Last week when he wouldn't stop crying, I shook him, which I know is a really bad thing to do. Afterwards I thought about it, and found myself admitting, really for the first time, the full weight of his dependence on me\u2014acknowledging that the baby is entirely dependent on me for twenty-four hours a day\u2014and what a huge drain that is. The funny thing is that since then his crying has not got to me in the same way.\"\n\nAs Selma lets go of the maternal ideal and its accompanying guilt, she recognizes her own limitations. This diminishes her hostility toward her son and enables her to respond more appropriately to his needs. Parker states: \"I think the conflict between love and hate actually spurs mothers on to struggle to understand and know their [child]. In other words, the suffering of ambivalence can promote thought\u2014and the capacity to think about the baby and child is arguably the single most important aspect of mothering.\" Thoughtful recognition of one's limitations as a mother is essential to a practical ethical response to one's child. Yet, as I will argue, deliberation is a critical dimension of ethical life that Levinas mistakenly denies in his emphasis on the passivity of ethical responsiveness.\n\nUnderstanding maternal ambivalence is not only necessary to the psychological well-being of individual women. There is a political dimension to the naturalizing of maternal care as well. By eliminating any sense of conflict that mothers might justifiably feel against their children, Levinas supports a status quo that demonizes women who fail at mothering. In \"Monster Stories: Women Charged with Perinatal Endangerment,\" Anna Lowenhaupt Tsing describes the persecution of women in the courts and the media who may have been responsible for the deaths of their perinatal infants. One example is \"Marlene Harris,\" a twenty-five-year-old maid. She had planned to give her baby up for adoption, but was having second thoughts at the time labor began. She was afraid that the baby would be taken away from her before she got to hold him. Instead of driving to the hospital, she went to a motel where she gave birth alone. Cold and disoriented after giving birth, she got into a warm bath with the baby and fell asleep. The baby drowned. Tsing also describes the case of \"Donna Sloan,\" a nineteen-year-old college student who gave birth to her baby in her dormitory bathroom. She had not realized she was pregnant and thought that her labor pains were gas. The baby was stillborn. At the heart of the legal cases against each of these women was the assumption that she should have \"direct intuitive knowledge gained from her bodily experiences. Within this convention, a woman unaware of her pregnancy or labor must be a physical or psychological anomaly.\" The courts and media believed that Donna Sloan could not have been unaware of her pregnancy and Marlene Harris had to have known she put her baby at risk by getting into the bath with him. Maternal instinct should have alerted them to the correct course of action. Thus, they wondered: \"What kind of a woman could endanger her own offspring at the moment of birth, the moment which should excite her most maternal sentiments? This question generated its own answer: Only a person completely lacking in parental\u2014and human\u2014sensibilities could commit such an act. To those who considered their cases, these women were, as one prosecutor put it, 'unnatural,' 'bizarre,' and 'without basic human emotion.'\" The accusation that women like Sloan and Harris are \"without basic human emotion\" is the flip side of the claim that mothers are, or should be, always able to respond appropriately to their children.\n\nLevinas denies that there are any practical limitations in women's abilities to care for their children. Yet there are innumerable factors that play into this capacity to care for another, including the material, physical, and psychological. Decent caregiving takes skill, thought, emotional intelligence, and resources. Recognition of these necessities is absolutely critical to providing proper support for mothers and children. In many cases the abuse, neglect, and abandonment of children is a failure of the social system that ought to support mothering work, not a failure of mothers themselves. For example, currently in Great Britain decreased social support for families with disabled children has put many families in the tragic position of having to place their disabled children in institutions. Riven Vincent, a mother of a child with severe quadriplegic cerebral palsy and epilepsy, was recently quoted in the _Guardian_ as saying: \"I have no wish to put my daughter into a home. We want to look after her, all I am asking for is a little more support. Without this we simply cannot cope and nor can families up and down the country just like ours.\" In order for children to receive proper care in our society, the limitations of mothers absolutely _must_ be recognized.\n\nWhen mothers lack the resources to care for a child, giving that child up may be an important act of ethical mothering. The options available to _some_ women (alternatives that Levinas denies) include giving a child up for adoption or having an abortion. In \"Abortion Is a Motherhood Issue,\" Judith Arcana argues that when a woman has an abortion she is not eschewing accountability; she is, in fact, taking responsibility for the life of her child:\n\n[Women] considered the conditions surrounding the mother, and the probability of her child's life being a strong one, including joy, good work, health, maturity, and usefulness. In their considerations then, as now, women sometimes judged that that probability was too slight, uncertain or simply absent. Their choices, like our own when we abort were never made in a vacuum; even in our time, in this woman-hating, mother-blaming society, there is always the decision made on balance, weighing both the potential years of the child's life and the mother's struggle to nurture it against great odds.\n\nSometimes it is simply not possible for a mother to provide for a child's well-being. Ending a new life, instead of leaving it to an existence of pain, may be the most ethical choice. Above all, the choice to have an abortion is not a means to escape responsibility; many women find it to be the best way to meet their obligations. In Levinas's universe, we are not allowed to think about our limitations in caring for others, and so these factors never come into play.\n\nLevinas's metaphorical use of the maternal body as a model for ethics is obsolete in the face of some women's abilities to have abortions, use birth control, or give their children up for adoption. He prefers the image of the maternal body since it gives without _willing_ to give. We need not dispense with the idea that the obligation to care begins before deliberation, but this is an insufficient account of ethical motivation. The ethical response does not always take the form of an epiphany. The daily drudgery and joys of mothering require all of one's creativity and deliberation. It requires a daily commitment that is usually more mundane than transcendent, but Levinas does not speak to the earthly tedium of breasts leaking milk at inopportune moments, the smells of dirty diapers and sweaty socks, and fights over appropriate clothing and curfews.\n\nIt is not getting pregnant that makes one a mother, in this way being _like a maternal body_ is not a comprehensive image for ethical responsibility. Being pregnant does not necessarily compel one to take care of a child. It is the daily commitment to take care of a child that makes a woman (broadly defined) into a mother. Importantly, this expands the idea of who can be considered a mother. A woman who gives up her child for adoption or has an abortion can still be regarded as a mother, if she does so with a sense that this is the best way to provide for her child. Adoptive mothers are just as authentic as those who give birth to their children and perhaps provide a better metaphor of ethical responsibility. People seem more willing to understand the adoptive child as initially a stranger toward whom one might have mixed feelings, but for whom one feels drawn to care due to her vulnerability. Adoption asserts that choice is involved in the care of a child; it is not automatic, immediate, or grounded in maternal instinct. As Kittay says, it is a bond, a duty, and a commitment all in one. It is both voluntary and compelled, both reasoned and felt. Of course, Ruddick's solution might not appeal to Levinas, since his vision of a subject as a hostage means that she is \"obsessed with responsibilities which did not arise in decisions taken by a subject 'contemplating freely.'\" The paradigm of adoption might make it clearer that an epiphany is an insufficient motivation for ongoing commitment to care for another. While Ruddick wants to ally ethical behavior with ethical _thought_ , Levinas seems largely indifferent to this. Yet maternal experience with the phenomenon of adoption confirms that a conscious commitment to the other works in concert with the repeated realization of the other's infinity and vulnerability as well as other factors. There is no reason why we must affirm epiphany at the cost of denying the centrality of deliberation and daily commitment to ethical life.\n\nLevinas's account of ambiguous intersubjectivity is, in some aspects, accurate to the experience of mothers. Ethical obligation indicates the limits of self-knowledge and disrupts the complacence within ourselves. It is an interruption of personal projects and aspirations. It shows how we are deeply indebted to others, not just as a baby is indebted to his mother, for providing for our needs, securing our survival, and flourishing. We are also indebted to those who need us, those who interrupt our flourishing, our enjoyment, and compel us to face our responsibility to others. Ultimately, ethics can lead us to a more robust identity, an identity that is not just seeing who we expect to see when we look in the mirror and finding a consistent unity across time. Others force us to recognize that we must continually take up the question Who am I? As Levinas says, \"The oneself is a creature, but an orphan by birth or an atheist no doubt ignorant of its Creator, for if it knew it it would again be taking up its commencement.\" We are all infants in the sense defined by Winnicott, one who \"cannot understand what he owes to his mother.\" True, we cannot fathom what we owe our mothers, but we also cannot comprehend what we owe the other who upsets our self-satisfaction.\n\nWhile some scholars applaud Levinas's ethics as replacing a tradition of individualism with a relational ontology, as providing a necessary _feminine_ counterpoint to the overly masculine history of ethics, the concern remains that it may not be sufficiently _feminist_. In Levinas's account women gain selfhood and ethical status in their roles as housekeepers, lovers to men, and mothers. He has been thoroughly and rightly criticized by feminists for these failures and therefore no simple appropriation of his ideas is possible. If Levinas's hagiography is meant to show us the way to be good mothers, then his efforts are also potentially destructive. It is more helpful to mothers to understand the reality of their situation, and, while Levinas's account of ambiguity does contribute to this aim, his ignorance of maternal ambivalence does not.\n\nUnfortunately, Levinas appropriates the maternal perspective without consideration of the experience for actual women. Careful attention to the phenomenon might have helped him to avoid his easy appropriation of patriarchal values. Instead, he ignores vital aspects of maternity that are part of all of ethical life\u2014the desire to flee from one's obligations, fantasies of murder and suicide, and feelings of rage. Mothers have the intense desire to protect, to even sacrifice themselves for their children, but these same children can become sacrificial victims both in fantasy and on suburban front lawns. Even when a mother loves her child as deeply as Levinas predicts, this does not eliminate the intensity of her conflicted feelings. Mothers do have an individuated sense of self that existed before the birth of the child and that does not exist only to serve, but also to flourish in and of itself. And the flourishing of a woman can sometimes be _at odds_ with the raising of children, even when it is also necessary to it. Levinas's philosophy could be amended by taking _adoption_ as the paradigm example. This removes the assumption that mothers will be the automatic and natural caregiver. In this view, all children are strangers\u2014orphans in need\u2014and their care is not to be taken for granted by anyone.\n\nLevinas believes that conflicts of interests can be resolved by yielding to the Other; the self has no robust ethical standing. To take this stance is to ignore one of his own best insights, our simultaneous proximity and distance. If we take seriously the intertwining of self and other, it is not possible to simply put the other before oneself. An unbalanced asymmetry renders ethics impossible, especially when there is no broader context of social support. He misses the important insight, articulated by Kittay and other care ethicists, that the caregiver must be provided for. In order to be edible, one must also eat. Levinas is missing several critical components that can be furnished by Merleau-Ponty and Beauvoir: a more careful logic of ambiguous intersubjectivity, an understanding of how asymmetrical relations can nevertheless be balanced, the factor of social context and support, and, finally, a better idea of how ethical failure relates to ethical success.\n_Five_\n\nMATERNITY AS DEHISCENCE IN THE FLESH IN THE PHILOSOPHY OF MAURICE MERLEAU-PONTY\n\nDehiscence occurs when a body opens and spills its internal contents. It happens naturally when a plant comes to reproductive maturity: an anther opens displaying its pollen to the wind; a fruit splits and sheds its seeds; the operculum of moss, fungi, or algae is discarded to release its spores; the seedpods of milkweed burst releasing their tiny downy parachutes. Dehiscence also means the opening of a wound along a surgical suture that causes fluid to leak. Much like being a mother, dehiscence indicates a breach in one's internal integrity that can be life giving and\/or life threatening.\n\nMaurice Merleau-Ponty appropriates the term _dehiscence_ to describe one aspect of our ontological and existential condition. He thinks that each person experiences a prereflective coherence between herself and the world, between himself and others. We do not feel ourselves to be the author of our experiences; they come to us as though written \"anonymously\" as _real_ beyond individual perception. This sense of the world's veracity is supported by our social life. Although we do not share the exact outlook of others, our differing perspectives converge in one and the same world. This means that together we inhabit a sensuous unity, but we also suffer a division. While the other is not hidden from me, locked within interiority, she also is not fully laid out for my examination. These structures, described phenomenologically in the _Phenomenology of Perception_ , become ontological in _The Visible and the Invisible_. In his later work, Merleau-Ponty deems the things of the world, including people, to be expressions of one _sensible_ flesh. While flesh establishes our continuity, it also involves an inherent dehiscence between the sensing and the sensed and between the self and the other. This is not dualism, but a description of the fundamental ambiguity of our relations.\n\nIn this chapter I propose that maternity can be aptly characterized as a case of dehiscence in the flesh. From pregnancy to caring for adopted and older children, the mother-child relation illustrates how human beings coconstitute the world with one another. The mother, in particular, is the child's first environment. Some scholars worry that in taking the maternal as flesh, the mother is equated with nature and her subjectivity is erased. However, this concern is eased if the concept of flesh is properly understood as involving dehiscence, an _\u00e9cart_ , a necessary separation. For Merleau-Ponty, reversibility is never a perfect or harmonious unity. Likewise, the mother's relation to her child, as well as her relation to nature, is necessarily ambiguous.\n\nTo elaborate on Merleau-Ponty's notion of ambiguous intersubjectivity, I explicate his use of Hegel's \"immanent logic of human existence.\" This discussion of Hegel also helps to explain how Merleau-Ponty differs from theorists discussed earlier and highlights the implications of his philosophy for our considerations of ethics. The ambiguity of our intersubjective relations, as described both by Merleau-Ponty and in maternal accounts, results in an ambivalent ethical orientation, contingent as it is on negotiating the interrelated yet discrete interests of the self and the other. Although Merleau-Ponty never wrote an ethics, he describes a life world that comprehends intersubjective ambiguity, provokes the question \"What should I do?\" and indicates that the answer will often be conflicted and founded in ambivalence.\n\nAMBIGUOUS INTERSUBJECTIVITY AND MATERNAL FLESH\n\nIn the _Phenomenology of Perception_ , Merleau-Ponty emphasizes our sense of coherence with the world. Prior to reflection, one does not stand out as situated over and against the world and others. We are immersed in a reality whose qualities and meanings appear as both shared and self-evident: \"My consciousness . . . is hardly distinguishable from what is offered to it.\" We inhale the scents of the world, drink its fluids, eat its fruit, we bump into its hard surfaces, absorb its heat, we ache at its beauty and gag at its ugliness, all without a thought of doing so. The sensory is open to me \"through a gift of nature, with no effort made on my part\" ( _Phenomenology_ 251). When I put on some music, _there is music_. When someone hits me, _there is pain_. When I eat, _hunger is sated_. There is no immediate distinction between myself and these perceptions. There is no need to imagine an _I_ or a _world_ over and against one another, in order to perceive the sensory phenomena. \"Nothing here is thematized. Neither subject nor object is _posited_ \" (281). Even when we do abstract ourselves from this world via reflection, our primary engagement with it is already presupposed and carries on in the background. While I entertain abstract thoughts like my own nonexistence or the illusory quality of reality, the ground retains its solidity, the air remains breathable, coffee still tastes bitter. I do not experience myself as the enactor of these facts, 'I' remain _anonymous_. I am the blind spot in my vision that allows me to believe in the world naively, to not see my own seeing.\n\nFor Merleau-Ponty, the sense of continuity with the world extends to other people as well. In the natural attitude others are not minds trapped in an inaccessible interiority. We are all _sentient_ creatures in a _sensible_ world, and we meet in this public world that we both hold in common. Imagine that my gaze shifts from a book on the table to my friend sitting in front of it. She is as apparent to me as the book. As with the rest of the world, I take for granted that she is as she appears to me. This is _not_ to say that to perceive another person is the same as perceiving an inanimate object. I recognize hers as another view upon the world. Though we perceive the world differently, it is, nevertheless, a common world we behold. At the Chicago Institute of Art my friend and I stand on opposite ends of a sculpture of Hyacinth. I am examining the girlish curls tied in a knot on the top of his head. He notes how the right toe delicately scratches the left calf. He sees what I do not, yet I never question if the sculpture is one and the same. Together we take for granted that _there is_ the sculpture of Hyacinth. We do not \"conceive our perspective views as independent from one another; we know that they slip into each other and are brought together finally in the thing\" ( _Phenomenology_ 411). I know my friend by virtue of our inhabiting this same self-evident world. \"We are brought together in the one single world in which we all participate as anonymous subjects of perception\" (411). When riding our bikes through town, without speaking, my friend and I both stop at an intersection. She observes, as well as I do, that it is a dangerous street to cross. We hear the roar of cars together, feel the breeze they blow in our faces, and smell the exhaust. The street, like the museum, and every other place a human being can go is a public space that we inhabit together. Typically, I experience her in this world, which is common to us both, without needing to conceptualize her as either separate from or similar to me.\n\nOf course, our sharing a world in common does not mean that it always strikes us in the same way. I listen to a new band by myself, but I am eager for my friend to hear it. I try a new dish, and I want her to try it. Why? It is not because her impressions will inevitably be the same as my own. Rather, her differing view adds another dimension to things that they could not have for me alone. Merleau-Ponty explains: \"No sooner has my gaze fallen upon a living body in the process of acting than the objects surrounding it immediately take on a fresh layer of significance: they are no longer simply what I myself could make of them, they are what this other pattern of behavior is about to make of them. . . . Already the other body has ceased to be a mere fragment of the world, and become the theater of a certain process of elaboration, and, as it were, a certain 'view' of the world\" ( _Phenomenology_ 411\u2013412).\n\nWithout the other, objects have only one \"side\" at a time, the side that I see of them, but through another's pattern of behavior\u2014the things that she laughs at, her expressions of exclamation, what she prefers and dislikes\u2014things gain an added contour for me. Her gestures speak to me directly, conveying meaning without need for reflection. As Merleau-Ponty notes, the gesture is understood immediately; there is no need for analogy or interpretation. \"The gesture does not make me think of anger, it is anger itself\" ( _Phenomenology_ 214). Consider, for example, your earliest childhood memory of a parent getting angry with you. The facial gestures of anger are as immediately understood as a slap in the face.\n\nMerleau-Ponty emphasizes that communication with others introduces us to the world's truth. Indeed, before I met my friend, the world did not have the same meaning. Moreover, he rightly claims that there is no time for me before which others were there. We do not begin as a singular point of view to which others are added. Every human infant is born into a human _community_ without which she would not survive. To ask when others' views came to influence my own is like asking when I realized that objects have a third dimension. The gestures of others are not only acts of _communication_ but also _establish_ a common meaning ( _Phenomenology_ 215\u2013216). The faces of one's primary caregivers are the original source of significance. We can see this in laboratory experiments when an infant or toddler is introduced to an unfamiliar, potentially scary toy (such as a noisy robot with flashing lights): securely attached children look at their caregivers to see how they ought to react. If a mother reacts positively to the toy, the child will typically respond in the same manner. If she responds with a fearful facial expression, the child will retreat. When left alone in an unfamiliar place, infants will not explore, but spend the majority of the time crying; those who are left in the room with their mothers take an interest in the toys and the room. Young children look to a trusted person to tell them the meaning of objects, people, and situations. Infants with no one to rely upon, such as those who are neglected, abused, or raised in institutions, do not explore their environment. For a human being, there was never a time before others. They contribute to the constitution of the world as I know it before I am aware of it.\n\nThe communication of meaning between mother and child is not unidirectional; the child instills the mother's world with renewed significance. One obvious example is the child's cry. It impacts one on a visceral level causing an instant feeling of urgency, compassion, and\/or irritation. Although one cannot always tell precisely what a child needs when he cries, nevertheless, one is immediately called to action. A child's smile or even a simple gaze can also suddenly change one's orientation. Daphne de Marnefffe describes this experience when looking at the face of a nursing baby: \"I am not in command of my own joy. The delight I feel at being at the receiving end of those eyes is almost overwhelming, like a gush of uncontrollable laughter.\" The meaning conveyed by a child's gesture can insight more than a momentary shift of mood. Recall the Kate Moses story quoted in chapter 3: \"My daughter, now two, has started licking me. The start of a hug, arms around my neck; or I'm bending down, picking up little plastic kitty cats off the floor, and her tongue laps my cheekbone, flicks the crater of my eye. . . . Her licks have the power to completely alter my mood, capture my full attention, put me under her spell, which is no doubt part of the reason she loves to lick me.\" The lick is a gesture in which the child achieves an over _lapping_ of bodies. She conquers the distance of adult preoccupations and brings her mother into the immediacy of the shared moment. Just like an adult, the child reveals her perspective through the gesture. To her daughter, Moses's body is lickable; it is likable and lovable, an opportunity for play. Moses's daughter endows her body with new meaning: \"As a young woman, I thought about my body all the time: how to disguise it, to shape it, to present it, to comfort it; what everyone else thought of it. Motherhood taught me to live in it.\"\n\nMerleau-Ponty argues that babies and young children are fully immersed in the public world, unimpeded by a belief in personal interiority; and that the earlier we look into the child's development the more pervasive is her or his sense of anonymity ( _Phenomenology_ 413). In _the Child_ ' _s Relations with Others_ , drawing on Jacques Lacan and Jean Piaget, he describes the child as egocentric, meaning that he is not reflectively aware of himself as a unique point of view, that she inhabits a continuous anonymity. The infant is \"unaware of himself and the other as different beings . . . egocentricism is not at all the attitude of a me that grasps itself (as the term 'egocentrism' might lead us to believe). Rather it is the attitude of a me which is unaware of itself and lives as easily in others as it does in itself\u2014but which, being unaware of others in their own separateness as well, in truth is no more conscious of them than of itself.\" The young child experiences a sense of unity with the world, just as when \"I contemplate the blue sky . . . I am the sky myself\" ( _Phenomenology_ 249). According to Merleau-Ponty, all of the child's relations have this character, but, with the mother, he experiences total identification: \"The maternal relation is (as the psychoanalysts say) a relation of identification, in which the subject projects on his mother what he himself experiences and assimilates the attitudes of the mother.\" Whether children are fundamentally egocentric and\/or completely identified with the mother is a topic for endless debate. However, it is clear that sometimes the horizons of mother and child can vividly overlap. Mothers can even be drawn into the immediacy of the child's perspective, a blissful, temporary self-forgetting free from adult concerns and judgments. If psychologists and philosophers can claim that the child experiences a sense of unity with the world, it is because adults can sometimes share the child's perspective.\n\nAs I show in chapter 3, this connection between mother and child is not to be taken for granted; indeed, cases of extreme neglect demonstrate its importance. The infant experiences the world through the expressions of others, and, without sufficient social interaction, her world appears small, dull, and flat. For lonely infants the world is not enticing. It is well documented that when a child is raised in relative solitude her sensory development is delayed or curtailed. For example, in _Blind Infants and Their Mothers: An Examination of the Sign System_ , Selma Fraiberg describes a (socially but not physically) neglected infant, Lenny. Lenny had given up any attempts to explore the world visually. Before Fraiburg intervened, caseworkers had assumed that Lenny was blind. Eva Simms provides another example. She describes a neglected infant who showed only very passive sensory engagement. As Simms says,\n\nMoreover, infants who are not cared for show signs of severe sensory and intellectual impairment after only a year. Research in the relatively new field of sensory integration has shown that the development of our sense unfolds in interaction with the world made inviting by caring adults. Children who are kept in non-stimulating, sterile, and isolated environments have difficulty seeing, hearing, touching, and tasting. The infants' sense, if not engaged by caring adults, will restrict their engagement with the world to familiar structures: the minute shadings of the white crib, a distance that spans a few feet, the touch of others that restricts itself to the feeling of a spoon on one's lips or maybe the grasp of a hand under one's arms when placed on a potty chair. . . . The contrast between the development of neglected children and those who receive adequate care shows that feeding a child does not necessarily mean that the child is nourished.\n\nIn _The Child_ ' _s Relations with Others_ , Merleau-Ponty also cites studies of the perception of eleven to sixteen year olds that describe how these are influenced by others: \"Recent studies have tended to show that even external perception of sense qualities and space\u2014at first glance the most disinterested, least affective of all the functions\u2014is profoundly modified by the personality and by the interpersonal relationships in which the child lives.\"\n\nMerleau-Ponty does believe that as we mature we come to realize our differentiation from others. Yet he maintains that anonymous intersubjectivity underlies all our interactions: \"the unsophisticated thinking of our earliest years remains as an indispensable acquisition underlying that of maturity, if there is to be for the adult one single perspective\" ( _Phenomenology_ 414). Effectively, this means that on the level of prereflective experience I do not extricate my view from others, but this does not mean I have an all-access pass to their thoughts, feelings, and perspectives. I still perceive from my point of view, and they do so from theirs. Merleau-Ponty states that \"although I am outrun on all sides by my own acts, submerged in generality, the fact remains that I am the one by whom they are experienced\" ( _Phenomenology_ 417). Each body has its own integrity, and, thanks to its boundaries, perception is possible. The softness of my pillow brushes against my cheek. Flavor travels across the surface of my tongue. While the body's surfaces delimit it, the boundaries of my body are also the point of contact between myself and the sensible world; without them, no world would exist for me. As our commonalities are discovered through the shared world, so one's separation from others becomes apparent. I like cilantro, but my friend thinks it tastes like soap. I like wheat beer, but my friend thinks it smells like medicine. These seemingly small differences are also indications of a fundamental separateness. Although _together_ we inhabit a sensuous unity, we also suffer a division. She tastes with her tongue and I with mine. Even the pregnant mother maintains her own bodily integrity in this manner. As the fetus breathes her amniotic fluid, the mother does not taste it. The same bodies that permit us sensory experience dictate that we shall each have this access only via our _own_ body. The same bodies that allow us contact keep us from merging. While the other is not hidden from me locked within interiority, she also is not obvious to me.\n\nMerleau-Ponty continues this line of thinking in _The Visible and the Invisible_. There he also argues that perception happens in a public place, a world that is accessible to one and all. In my relations to others, I do not have access to a \"hidden realm of interiority.\" Instead \"it is in the world that we rejoin one another.\" In the _Phenomenology of Perception_ , Merleau-Ponty describes anonymity as, first and foremost, a phenomenon of lived experience. However, in _The Visible and the Invisible_ this phenomenology becomes ontology. Not only do we _experience_ ourselves as \"connatural with the world\" (252) we _are_. He writes: \"It is to experience therefore that the ultimate ontological power belongs, and the essences, the necessities by essence, the internal or logical possibility, solid and incontestable as they may be under the gaze of the mind, have finally their force and their eloquence only because all my thoughts and the thoughts of others are caught up in the fabric of one sole Being\" ( _The Visible_ 110). This ontological continuity between self, other and world is named the _flesh_ (123).\n\nThe things of the world, including people, are expressions of this one _sensible_ flesh. Flesh is sensible in multiple respects: First, it can be perceived via the senses, smelled, touched, seen, and so on. Second, it is also capable of perception. My nose, ears, brain, etc. are all of the flesh; yet they are also articulations through which the world can bend back upon itself, sense itself, and discern itself. \"Color that sees itself, surface that touches itself\" ( _The Visible_ 135). To take a mundane example, after a strenuous workout I can smell my own sweat. The flesh which sweats is the same flesh that smells it. The skin and nose are apertures upon a world of which they are also a part. Thus perception is not evidence of our dominion over the sensory; it is a possibility inherent to the flesh itself realized in human beings as a part of the flesh of the world. Vision and the visible, touch and the tactile, hearing and the sonorous, are \"flesh offered to flesh\" ( _The Visible_ 131). \"He who sees cannot possess the visible unless he is possessed by it, unless he _is of it_ \" (134).\n\nThe concept of flesh describes more than just our immersion in the world. Also inherent to the flesh is a gap, an _\u00e9cart_ , a dehiscence between the sentient and the sensed. In Merleau-Ponty's classic example of reversibility, when the right hand touches the left hand the two do not merge. \"I palpate with my left hand only [my right hand's] outer covering. . . . I am always on the same side of my body; it presents itself to me in one invariable perspective\" ( _The Visible_ 148). This noncoincidence, the \"hiatus between my right hand touching and my right hand touched,\" is not a \"failure;\" it is the distinction without which no relation would be possible (148). When I look at a painting, I may temporarily lose myself, but I do not really _become_ the painting. Reversibility between oneself and the world is never wholly complete: \"reversibility [is] always imminent and never realized in fact. . . . I never reach coincidence; the coincidence eclipses at the moment of realization\" (147). Nevertheless, _\u00e9cart_ does not reestablish classic dualism either; it does not indicate things that are different _in essence_. The smell of my sweat is certainly _mine_ , but I can still encounter it as an object or other. I may even be surprised by it. This is neither a contradiction, nor a failure to understand my proper relation to my body and world. I can take my own body as both subject and object at once, without these two perspectives merging and without completely disconnecting them. Similarly, the ambiguity of togetherness and separation are fundamental to the flesh itself.\n\nIn _The Visible and the Invisible_ , Merleau-Ponty describes one's relationship with others as parallel to the reversible relationship with the sensible; \"we situate ourselves in ourselves _and_ in the things, in ourselves _and_ in the other, at the point where, by a sort of _chiasm_ , we become the others and we become world\" (160). This is an ambiguous intersubjectivity in which I overlap with others, and yet remain distinct. It is analogous to the chiasma of the optic nerves intersecting to create one single view out of binocular vision. Our two eyes maintain their separate viewpoints, but their perspectives converge to reveal the world's depth. Similarly, the other's view is set against one's own. As in the example of the statue of Hyacinth, if one person's position differs from another, this does not indicate the existence of two realities. Instead, the world gains added dimension and appears that much more real. While the mother shares her child's perspective, she does not completely lose sight of her own. When my child cries out in the night, I may jump up and go to her feeling full of sympathy, but I may simultaneously resent my own loss of sleep. It is important to recognize that when \"we become the others and we become the world\" an _\u00e9cart_ still remains: \"It is from the lawn before me that I think I catch sight of the impact of the green on the vision of another, it is through the music that I enter into his musical emotion, it is the thing itself that opens unto me the access to the private world of another. But the thing itself, we have seen, is always for me the thing that _I_ see\" ( _The Visible_ 11). Without this divergence between self and other, the world would lack the \"objectivity\" endowed by intersubjectivity ( _The Visible_ 142).\n\nMerleau-Ponty often uses metaphors of pregnancy and motherhood to illustrate the meaning of the flesh and to explicate his notion of intersubjectivity. For example, in his first mention of the flesh in _The Visible and the Invisible_ he states, \"Each landscape of my life, because it is not a wondering troop of sensations or a system of ephemeral judgments but a segment of the durable flesh of the world, is qua visible, pregnant with many other visions besides my own\" (123). To be pregnant is to nurture another's horizon within oneself. This is paradoxical and yet it is our constant condition. Our fleshly incarnation means that we can both perceive and be perceived, and that everything I see is suffused with the visions of others\u2014my partner's, my child's, a friend's, a stranger's, and so on. Pregnancy and motherhood in general, put this reality into relief. The other, whether one's child or not, is both familiar and strange. She is made from the same flesh as I am. Like me, she displays her feelings and some of her thoughts on her face with smiles, tears and a furrowed brow. We both encounter the world through smell, taste, vision and thoughts. She sleeps, she dreams, and wakes up, just as I do. Yet there is always something of her that I cannot reach.\n\nMerleau-Ponty's imagery of pregnancy and motherhood is precisely parallel to his characterization of ambiguous intersubjectivity: He declares that we are of the same flesh, suffused with one another, even while we remain distinct. This relation is embodied in multiple ways as I encounter the other through my body, through my embodied relation to the world, and as the other impacts the manner in which I experience my body. While it is possible to understand this other, _to some extent_ , through analogy with myself, she still remains entirely different from me, both surprising and mysterious.\n\nMerleau-Ponty also briefly considers the relationship between the biological mother and fetus as an excellent _literal_ example of the flesh. He describes the connection between the body and the world as two parts of a complete circuit, and claims that the human fetus in the womb has \"prepared but unemployed circuits . . . the seer is being premeditated in counter-point in the embryonic development\" ( _The Visible_ 147). According to Merleau-Ponty, the possibility of the child's uniquely human existence and her co-extension with the flesh of the world is already _pre_ figured in the fetal body. In this passage, he describes the fetus as _anticipating_ her emergence into the world, seeming to neglect that the fetus is already part of a living circuit with a world provided through and by the pregnant woman's body\u2014the sounds of the mother's heartbeat and voice, the muffled voices and white noise of the world outside her body, the taste of amniotic fluid that is breathed in and out, the swaddling sensation of being nestled in the womb, the rocking motions of the mother's walk, and so on. Research confirms that the maternal body is a sensory world that engages the fetus, demonstrating that she can hear, move voluntarily, swallow, taste, and touch\u2014that fetuses respond to pressure and pain and loud or soothing noises. When the baby is born, she is taken out of a complete environment to which he or she has clearly become accustomed. This fact is recognized by contemporary parenting gurus who advise caregivers to think of the newborn as inhabiting a fourth trimester\u2014still a fetus in terms of its needs and expectations. This fact is especially capitalized on by Harvey Karp's _The Happiest Baby on the Block_ books and DVDs. He teaches parents to use the \"5 Ss\"\u2014sshhing (saying sshh in the baby's ear), swaddling, swinging, side or stomach lying, and sucking\u2014to soothe a crying newborn. The first three of these techniques are intended to mimic the womb environment, and my own experience with these methods indicates that they work remarkably well. Indeed, both folk wisdom and scholarly research agree that neonates are most content when nested in and warmed by another's body, where it is not too light or loud, hearing and feeling the rhythms of the other's heartbeat and breath and smelling her familiar smells.\n\nThe mother herself is both sensing and sensed in a reversible relation with the fetus who alternately seems to be a part of her own body and to be the body of another. She is the first world with which the infant is in chiasmic relation. In this respect, this quote rings especially true: \"we situate ourselves in ourselves _and_ in the things, in ourselves _and_ in the other, at the point where, by a sort of _chiasm_ , we become the others and we become world\" ( _The Visible_ 160). However, in this case a greater passivity is at work than this statement implies. The fetus does not situate herself; she _is situated_ in the gestational mother who, at least initially, _is_ all things, all others, and the entire world to the fetus. Thinking the birth mother as flesh suggests that intersubjectivity extends beyond sharing perspectives on a common world, beyond our filtering, highlighting, and obscuring the world for one another. It suggests that, from the start, we _are_ this common world for one another. More specifically, the gestational mother (not just _any_ other) is the first landscape within which we find ourselves.\n\nUpon birth, the infant is exposed to a much wider world than before, but if cared for she continues to be nestled close to the bodies of her caregivers; they continue to constitute the world for her. For example, as I describe it in chapter 3, mother and child share one physiological process in breast-feeding. Eva Simms also describes this \"primal housedness and well-being\" drawing on Bachelard, Merleau-Ponty, and her own experience. Simms reflects on how she was her daughter's first environment: \"I was her house, but more than that I was the field that nourished her, the rain that quenched her thirst, the sun that warmed her skin . . . milk is the visible sign of the invisible, the in-between body, the _chiasm_ , mother-infant flesh.\" Breast-feeding is a symbiotic relationship as, for example, the rooting reflex already suggests the existence, location, and shape of the breast and as the amount of milk produced meets the baby's demand for it. \"Implicit in the first cry, the first turning of the head toward the (m)other's voice, the first search for the breast is intentionally a directedness that presupposes that there is something to turn to.\" Simms illustrates how the maternal can indeed be thought as first flesh: \"In the beginning, the maternal world and thing-world are the same.\"\n\nIn the case of mother and child, shared flesh is deeply ambiguous. Simms describes the aspect of breast-feeding in which the perspectives and bodies of mother and child are aligned. Of course, as I have explained in chapter 3, women also feel that breast-feeding and other aspects of childcare threaten to consume their vitality. They liken these same beloved children to parasites and cannibals. Clearly there is a divergence of perspectives as well\u2014the _\u00e9cart_ that Merleau-Ponty describes as integral to intersubjectivity on the levels of experience and ontology.\n\nMerleau-Ponty may have planned to consider the maternal body an instantiation of the flesh, as his working notes for _The Visible and the Invisible_ seem to imply: \"Do a psychoanalysis of Nature: it is the flesh, the mother\" (267). However, there is some concern among feminist scholars that drawing a parallel between the mother and flesh erases the subjectivity of the mother and conflates her with nature. In the next section of this chapter, I will review these concerns. Then in the following segment I will address this concern by further clarifying the logic of his theory of ambiguity intersubjectivity. There is textual evidence that Merleau-Ponty employs principles from Hegel's logic; investigating these principles will help us to better understand his notion of ambiguous intersubjectivity and address feminist concerns.\n\nFEMINIST READINGS OF MAURICE MERLEAU-PONTY\n\nIn \"Only Nature is Mother to the Child\" Dorothea Olkowski critiques Merleau-Ponty for denying the sociality of human origins. According to Olkowski, Merleau-Ponty denies maternal subjectivity, placing our origins in a wild nature. \"All signs of the female, the woman who gives birth, who cradles, caresses, nurses, and cares for the infant, are erased. As such, in place of the personal, intimate relation with this fully human woman, the infant comes to itself from out of the prepersonal and anonymous realm of nature.\" In Olkowski's view, Merleau-Ponty conflates the maternal with nature, reinstating women's traditional exclusion from culture.\n\nA similar objection is made by Luce Irigaray in \"The Invisible of the Flesh: A Reading of Merleau-Ponty, _The Visible and the Invisible_ , 'The Intertwining\u2014the Chiasm.'\" She criticizes Merleau-Ponty for failing to consider that only the maternal-feminine gestates and gives birth. As a result, she claims, he misunderstands perception, sexual difference, and the flesh. Irigaray argues for the radically passive origin of perception: the formation of the eyes, optic nerve, and brain in \"this mystery of the prenatal night where he was palpated without seeing\"; vision developing within the obscurity of intrauterine life, \"the infant's nocturnal abode\" (152) where nothing can be seen and the sense of touch (and perhaps hearing) is primary. One's origin necessarily remains in darkness, an obstacle between the passive and active dimensions of perception, an interruption in one's continuity with the world. In opposition to this, Irigaray says that Merleau-Ponty denies the obscurity of our beginnings, reducing the (m)other to the same, and therefore \"nothing new happens, only this permanent weaving between the world and the subject which presupposes that the subject sees the whole\" (170). Within his \"closed system\" (161), this \"solipsistic relation\" (180), it is as though Merleau-Ponty's seer has never been born, because he remains in a prenatal continuity with the world (173). \"It has no spacing or interval for the freedom of questioning between two\" (183). Irigaray claims that to overcome his \"labyrinthine solipsism\" a radical other is needed\u2014the other of sexual difference, the maternal-feminine, an other who cannot be seen (as the fetus cannot see the mother; 157).\n\nAccording to Irigaray, the maternal reveals a fundamental _ir_ reversibility at the source of the reversibility. The child is unequal to the mother, since the mother gives birth to the child and never the reverse. The child (especially the male child) can never appreciate her perspective; he can never witness the same landscape. But Merleau-Ponty, she claims, mistakenly assumes a persistent mutuality, eliminating alterity, and thus never escapes solipsism. For Irigaray, fecundation is not mutual; we do not diverge from a common flesh. The child comes from the mother's flesh, and this indicates the source of all difference\u2014sexual difference.\n\nScholars agree that Merleau-Ponty does not have any explicit theory of sexual difference, and he has been thoroughly critiqued for this neglect. For example, Judith Butler worries that while Merleau-Ponty wants to describe concrete, existential structures, at times \"he offers a description of bodily experience clearly abstracted from the concrete diversity that exists.\" In \"Through the Wild Region: An Essay in Phenomenological Feminism,\" Jeffner Allen also claims that Merleau-Ponty ignores the uniqueness of women's embodied experience. She believes that this omission does not simply create a lack in Merleau-Ponty's philosophy; it reaffirms the masculinist assumption that men's experiences can serve as the model for all: \"Women are the concealed term that is purportedly included in the term 'human beings.' In androcentric culture, where women's situation is that of the other, 'human being' is not a term that pertains directly to women, for we are unlike human beings in general, i.e. men. . . . Neuter neutral inquiry, including the claim that 'I am my body,' is not harmless. It acts, rather, to further conceal women's experiences of our embodiment.\" Shannon Sullivan also critiques Merleau-Ponty along these lines in _Living Across and Through Skins_ and \"Domination and Dialogue in Merleau-Ponty's _Phenomenology of Perception_.\" She agrees with Allen that Merleau-Ponty assumes a gender neutral body unmarked by race, class, or sexuality. She specifically targets the concept of the anonymous body arguing that it is, in fact, the masculine body posing as \"body for all.\" Sullivan says that Merleau-Ponty is guilty of \"ethical solipsism,\" because he merely generalizes his own experiences.\n\nA related but different criticism is that Merleau-Ponty may be dependent on an implicit theory of sexual difference for his notion of the flesh (as Irigaray implies). Elizabeth Grosz raises this concern in \"Merleau-Ponty and Irigaray in the Flesh\":\n\nMerleau-Ponty does not explicitly address the question of sexual difference in _The Visible and the Invisible_ , yet, if Irigaray's reading is appropriate, it is clear that his work derives much from a kind of implicit sexualization of ontology, the utilization of a whole series of metaphors embedded within and derived from relations between the sexes. The metaphors are the condition of the possibility of his understanding of the flesh, which is itself the condition of the possibility of the division and interaction of subjects and objects. In this sense, the feminine may be said to be the unspoken, disembodied underside of the flesh: the flesh as a point-for-point convergence with the attributes of femininity.\n\nThis denial of sexual difference connects to the concern that Merleau-Ponty is in denial of alterity more broadly speaking. In \"From the Body Proper to Flesh: Merleau-Ponty on Intersubjectivity,\" Beata Stawarska, claims that Merleau-Ponty fails to preserve the other's \"free subjectivity and autonomy\" and collapses bodily relations into a single universal. She considers this move as emblematic of the way that \"the masculine frame of mind continuously misrepresents the structures of intersubjectivity (and not only gender), in its tendency to reduce the other to the same.\" In particular, she objects to his classic example of reversibility\u2014one's hands touching each other. \"Merleau-Ponty takes a specific, intrasubjective experience of the body, as in my right hand touching my other hand, to be a paradigm for relations to other embodied subjects, as in my right hand touching the hand of the other when we greet each other by shaking hands.\" Stawarska argues that a relationship to one's own body should not be used as the exemplar for relations with others. In doing so, she thinks that Merleau-Ponty forgets the particularity of perspectives and that \"this basic distinction between self and other is disregarded or regarded as a secondary phenomenon.\"\n\nThe scholars discussed raise a number of important concerns, which I will address in the next few pages. First, Merleau-Ponty has no theory of sexual difference. This is undeniably true. Yet the fact that he does not explicitly deal with sexual difference does not mean that he considers all bodies to be the same. Butler, Sullivan, and Allen mistake anonymity for universality. Merleau-Ponty is defended against this view by Silvia Stoller in \"Reflections on Feminist Merleau-Ponty Skepticism\" and Gail Weiss in \"The Anonymous Intentions of Transactional Bodies.\" Stoller argues that anonymous corporeality is not neutral: \"In equating 'anonymous body' and 'neutral body' [Sullivan] confuses anonymity with neutrality; anonymous does not mean that subjects face each other 'neutrally,' nor does it mean that their bodily identity is completely 'unknown,' or even one of 'anyone and thus of everyone' (5). Given the fact that for Merleau-Ponty the human subject is always a situated subject\u2014and this applies also to its anonymous mode of being\u2014such a conclusion is unfounded.\" Similarly, Weiss asserts:\n\nTo say that an experience operates anonymously, then, is not equivalent to saying that it is universal or that it is trans-historical. It is worth noting that the examples of anonymity provided by Merleau-Ponty are always grounded in the experience of a particular body in its concrete engagement with the world. . . . To say that these transactions tend to be anonymous does not mean that we are not, at the same time, two particular bodies, marked by our race, class, gender, etc., engaged in a social relationship. The point is that this very social relationship is predicated on its anonymity.\n\nMy view of anonymity supports that of Weiss and Stoller. For Merleau-Ponty the body is always situated within a biological, social, and historical context, even in its anonymous mode. This means that sex and gender will _necessarily_ be a factor in embodied experience.\n\nAlthough Merleau-Ponty did not directly address sexual difference, this is not grounds for dismissing his feminist potential. His fascination with unique experiences of embodiment is demonstrated throughout _The Structure of Behavior_ and _The Phenomenology of Perception_. His insightful remarks on pregnancy in his lectures on child psychology and pedagogy (which I will present further on in this chapter) are good evidence for a specific interest in female embodiment. Furthermore, his metaphorical treatment of maternity is not the \"unspoken, disembodied underside of the flesh\"; to employ images of a phenomenon is not to ignore it. Rather, these depictions of motherhood demonstrate an unusually subtle understanding of maternity, especially when compared to Levinas's attempt. Although Merleau-Ponty did not complete the work for us, many feminists have found that he contributes invaluable resources for thinking about embodied sexual difference. Their excellent work demonstrates that, while he does not address sexual difference himself, Merleau-Ponty provides some helpful theoretical groundwork for feminists to think about women's embodied experience in its particularity.\n\nThe second major concern of Merleau-Ponty's feminist detractors is that he eliminates alterity. Unlike Irigaray, Merleau-Ponty does not found all difference in _sexual_ difference. For this reason, Merleau-Ponty's philosophy could never satisfy her. However, reading Merleau-Ponty on his own terms, I think it is clear that Irigaray and his other critics underestimate the importance of alterity to his philosophy. As I have explained, Merleau-Ponty's notion of intersubjectivity is ambiguous: I overlap with others and yet remain distinct. It is precisely because the two (or more) views that come together are _different_ that they add greater depth to the world; the solitary individual, were such a creature even possible, would perceive a flattened world in which objects have no backside. Although there is a prereflective sense of unity with others and the world, an _\u00e9cart_ between perceiver and perceived and between self and other remains a necessary aspect of these relations. The world's depth is a consequence of there always being an inaccessible aspect to people and things that can be perceived by others while remaining (at least temporarily) invisible to me.\n\nIrigaray claims that for Merleau-Ponty the subject sees the whole, but this is not true to his account. The prereflective experience of continuity does not deny the fact of alienation. He writes: \"Reversibility between oneself and the world is never wholly complete: \"reversibility [is] always imminent and never realized in fact. . . . I never reach coincidence; the coincidence eclipses at the moment of realization\" ( _The Visible_ 147). Even as we are in and of the world, the world still escapes us, and for this reason he would agree with her claim that there is _ir_ reversibility at the heart of reversibility. Reversibility never achieves complete or harmonious unity.\n\nIn \"Vision, Mirror, and Expression: The Genesis of the Ethical Body in Merleau-Ponty's Later Works,\" Alia Al-Saji makes a similar claim about reversibility. She explains that a difference remains between hands when one touches another. This is true whether or not it is my two hands touching each other or my hand touching another person's hand. \"In Merleau-Ponty's account, there is some slight difference ( _\u00e9cart_ ) between two bodies that resists the cycle of totalizing vision, a singularity that cannot be incorporated into it. For my hand and that of the other remain distinct, as do my two hands that touch. Al-Saji would agree with Stawarska that the relationship to one's own body is distinct from that of the relationship to another's body. However, Al-Saji thinks that Merleau-Ponty also agrees with them that an _\u00e9cart_ exists in both cases, but that one relationship is not reducible to the other. I would argue, and I think Al-Saji would agree, that the case of my hand touching my other hand is not meant to be _paradigmatic_ of all relations of reversibility. Rather, my self-touching is _analogous_ to my touching another. To say that they are analogous is to say that there is both similarity _and_ difference between these two cases.\n\nThere is more noteworthy support in secondary literature for my claim that alterity is a vital element of the Merleau-Pontian universe. In \" _\u00c9cart_ : The Space of Corporeal Difference,\" Gail Weiss describes _\u00e9cart_ as \"a space of noncoincidence that resists articulation\"\u2014at the heart of our relations. She agrees that Irigaray does not realize how foundational _\u00e9cart_ is for Merleau-Ponty. Weiss describes its role in his philosophy as follows: \" _\u00c9cart_ , as the moment of disincorporation that makes all forms of corporeal differentiation possible, is also precisely what allows us to establish boundaries between bodies, boundaries that must be respected in order to respect the agencies that flow from them. From a Merleau-Pontian perspective, these boundaries can best be respected by not artificially viewing bodies as isolated from one another but by acknowledging the reversible relationships that are exhibited within and across them.\" Interestingly, Weiss turns Irigaray's criticism back upon itself and says that one could accuse her of ignoring nonreversible difference between the mother and fetus: \"It is also possible to take Irigaray to task for offering an idealized account of intrauterine existence characterized by fluid interactions between mother and fetus, in which each resonates, aggressively as well as lovingly, to the movements and demands of the other (the same could be said of her depiction of female sexuality).\" According to Weiss, it is Irigaray who presents us with a closed system, not Merleau-Ponty, leaving no central role for others in the mother-child relationship. Weiss insists that it is always mediated by others. I will make this case more strongly in chapter 6, but for now I will say that social support, or its lack, can make a dramatic difference in the woman's attitude toward the fetus.\n\nIn \"Grounding Agency in Depth: The Implications of Merleau-Ponty's Thought for the Politics of Feminism,\" Helen Fielding also insists that _\u00e9cart_ is critical to Merleau-Ponty: \"In my contact with others there is a coming close, a contact with the more determinate qualities, but this is accompanied by a spreading and a fading away of our backgrounds, our pasts and our futures. Others always remain distant from me in their own depth.\" She reaffirms this view again in \"Envisioning the Other: Lacan and Merleau-Ponty on Subjectivity\" and adds that the _\u00e9cart_ between people parallels the _\u00e9cart_ between different senses.\n\nIn Merleau-Ponty's understanding of reversibility, there could never be a fusing together of sensibilities but nor is there an absolute isolation from others. Rather, our relations with others involves a reciprocity and mirroring that demand not a possession of the other but a dispossession (Merleau-Ponty 1968, 266). Although there is a \"surface of separation\" between self and the other, this is also the place where people come together\u2014it is the \"geometrical locus of the projections and introjections.\" The negativity to which they both refer is, then, for Merleau-Ponty, the hinge or the _\u00e9cart_ that separates the senses, that allows our lives to \"rock into one another\" while remaining distinct (Merleau-Ponty, 1968, 234).\n\nFielding argues that for Merleau-Ponty, in regards to both the senses and other human beings, there is neither fusion with nor isolation from the other. In both perception and human interactions, there is a \"surface of separation\" that makes both distance and proximity possible. Indeed, the depth of the other is witnessed precisely in her distance, just as the thickness of things only becomes apparent when held at a distance from the eyes.\n\nAlthough there are significant disparities of interpretation in the secondary literature on Merleau-Ponty, on the issue of alterity there is primarily a difference in _emphasis_. That is, while some scholars concentrate on Merleau-Ponty's overcoming of dichotomy and theory of connection between self and other, others focus on the _\u00e9cart_ between self and other. Merleau-Ponty's philosophy demonstrates that _both_ these aspects are vital; both alterity and interconnection are characteristic of self-other relations. Understanding this requires a speculative moment in the Hegelian sense. That is, both alterity and interconnection describe a way of relating one term to another, but we need to understand the relationship between these two relationships\u2014the relationship between alterity and interconnection. It is important to look at the logic at work in Merleau-Ponty's philosophy in order then to address how the mother can be understood as flesh in a nonreductive manner.\n\nTHE LOGIC OF THE FLESH\n\nMerleau-Ponty presents us with a view of human relations that may seem paradoxical. On one hand, the other appears to me as a pattern of behavior, publicly observable in a world that we share. Yet, she also remains inaccessible to me, perceiving the world in unique and surprising ways. This seeming contradiction begins with his early work. For example, in the _Phenomenology of Perception_ , he declares, \"But _this alien life_ , _like mine_ with which it is in communication, is an _open life_ \" (412). How can an _alien_ life be _like mine_? How can another be both _open to me_ and _alien_? How can one _communicate_ with an alien? To this, Merleau-Ponty answers, \"Solitude and communication cannot be the two horns of a dilemma, but two 'moments' of one phenomenon, since in fact other people do exist for me\" (418). In these passages Merleau-Ponty's thinking and language are distinctly Hegelian. He asks us not only to take opposing terms together: intimacy and alienation, similarity and difference, community and individuality, and so on, but he also asks us to think about the relationship between these \"moments.\" For Merleau-Ponty such oppositions reveal the immanent logic of human experience. In this section I will show the textual evidence that Merleau-Ponty employs Hegel's logic in his thinking of ambiguity. Clarifying how these principles work will help address feminist concerns and further illuminate the ambiguous intersubjectivity of mother and child.\n\nAccording to Merleau-Ponty's essay \"Hegel's Existentialism,\" in the _Phenomenology of Spirit_ , Hegel proposes \"to reveal the immanent logic of human experience in all its sectors. . . . Experience here no longer simply means our entirely contemplative contact with the world as it did in Kant; the word reassumes the tragic resonance it has in ordinary language when a man speaks of what he has lived through. It is no longer a laboratory test but is a trial of life.\" According to Merleau-Ponty, Hegelian logic is not arbitrary or abstract, but comprehends the challenges of human existence: \"We will not admit a preconstituted world, a logic, except for having seen them arise from our experience of brute being, which is as it were the umbilical cord of our knowledge and the source of meaning for us\" ( _The Visible_ 157).\n\nMerleau-Ponty judges as _bad dialectic_ a logic that imposes its own laws upon the ineluctable ( _The Visible_ 94). Here Merleau-Ponty may be referring to \"the late Hegel, who treated history as the visible development of a logical system. . . . But if the Hegel of 1827 may be criticized for his idealism, the same cannot be said of the Hegel of 1807.\" He might also be referring to that flimsy oversimplification of Hegel's logic which describes it as a process of thesis, antithesis, synthesis (terms that Hegel does not use), assumed to be applicable to any dichotomy. As I will indicate, he could be describing Sartre's appropriation of the master-slave dialectic. To better understand what Merleau-Ponty means by good and bad dialectic, we will look to the mature (though abbreviated) formulation of Hegel's logic, _The Encyclopedia Logic_.\n\nIn the _Logic_ , Hegel declares that thinking is not subject to a preconceived set of formal laws but its own self-given laws and determinations, the preconstituted logic that Merleau-Ponty also eschews. Hegel does not believe that one can make whatever connections they see fit between one thought and another and call it logic. For Hegel, such a thinker takes the reins from thought itself. He believes that the thinker must humbly follow where thought leads; conducting what is essentially a phenomenology of thinking. As we will see in more detail, these determinations and laws\u2014the _Logic_ \u2014arise as thought reflects upon itself; the _Logic_ is conceived by thinking as thinking conceives of itself. The three moments of the logical are the side of abstraction or of the understanding, the dialectical or negatively rational side, and the speculative or positively rational one. Note that these are the actual terms used by Hegel in the _Logic_ , not the shorthand terms of thesis, antithesis, and synthesis critiqued by Merleau-Ponty.\n\nUnderstanding helps one make the straightforward determination between what a thing _is_ and what it _is not_. This moment of thinking is necessary and helpful to the practical sphere and includes such particular realizations as \"This is a chair, not a table.\" It allows us to form universal categories, as in \"All chairs are objects that are made for sitting upon.\" The understanding also enables us to make distinctions and to identify, for example, elements of a theory\u2014matter, color, magnitude, etc. Hegel also affirms that ordinary understanding has many useful functions: it allows us to focus our interests, to choose a profession, to make judgments (as in jurisprudence), to distinguish between the duties of a government's branches, to make decisions, to stay on topic, and to think in a linear fashion. Without the understanding we could not distinguish one thing from another, determine a preference, devise a theory, or carry on practical business of all sorts. In short, the understanding is what enables us to be reasonable in the everyday sense of the word. \"It should not be absent, and . . . to the degree that it is absent, its absence must be considered a defect\" (127).\n\nNevertheless, Hegel insists that thinking must go further. This is not meant as a moral imperative for the thinker. Rather, this means that the understanding is thinking in its most stilted form and quickly reveals itself as such. Ordinary understanding is merely one-sided; it is not the whole truth, but only one moment of it. Since the understanding is only one-sided, the abstractions of the understanding will turn into their opposite. This is when we encounter the dialectical moment\u2014\"the self-sublation of these finite determinations on their own part, and their passing into their opposites\" (128).\n\nThe dialectical moment is when we find that the one-sided determination of the understanding is insufficient. In this moment it becomes apparent that while a thing _is what it is_ , it also _is what it is not_. Consider this classic example: the oracle at Delphi stated that Socrates was the wisest man in Athens. According to Plato, Socrates stated that he did not understand how this could be, since he knew nothing at all. After some dialogue and thought, Socrates believed that the oracle must have considered him wise because Socrates was aware of his own ignorance. Ordinary understanding would indicate that either Socrates is wise or he is not. However, this rigid way of thinking does little to reveal the true nature of Socrates' wisdom. In fact, and as the dialectical moment would have it, Socrates is both wise and ignorant at the same time. Furthermore, _Socrates could not be wise were he not also ignorant_. Here we see that wisdom also includes ignorance and vice versa; the two cannot be considered mutually exclusive.\n\nOne might say that, according to the dialectical moment, one thing can never be distinguished from another. Hegel anticipates this unfortunate interpretation: \"The dialectical moment, taken separately on its own by the understanding, constitutes _skepticism_ [simply that one doubts what others take to be true]. . . . Dialectic is usually considered as an external art, which arbitrarily produces a confusion and a mere _semblance_ of _contradictions_ in determinate concepts\" (128). In ordinary understanding one thing is often explained in opposition to what it is not\u2014its negative. But for Hegel, negation is the movement when one thing _becomes_ its opposite. Nevertheless, the two do not collapse into one another. In the dialectical moment, both difference and sameness must be admitted. In this sameness the two do not become equivalent, rather their intimate relationship is revealed. We can already see here much in common with Merleau-Ponty's characterization of intersubjectivity.\n\nThe dialectical is not superior, or opposed, to the understanding. Rather the two are meant to be taken together. \"The dialectic is the genuine nature that properly belongs to the determinations of the understanding, to things, and to the finite in general\" (128). Thus the dialectical moment does not raze the understanding. Rather there is an ambivalent toggling back and forth between one thought (Socrates is wise) and its negation (Socrates is ignorant) in order to have a better understanding of both (wisdom embraces ignorance, ignorance can indicate wisdom). Clearly this is an unsteady state, and the _Logic_ does not rest in the dialectical moment. However, in order to better understand the impetus to move beyond the dialectical we must consider one of the _Logic_ 's actual thought determinations.\n\nHegel's _Logic_ begins with _pure being_. \"Pure being makes the beginning, because it is pure thought as well as the undetermined\" (136). Without presupposing too much, the most that can be said about thought in the beginning is that it _is_. To comprehend what we mean when we say that _thought is_ , we must know what is meant by the _is_ itself. In order to understand thought, therefore, we must understand _being_. Since we are concerned with thought itself, pure being must not be considered a representation of anything else. In other words, pure being must not refer to _what_ is, but to the _is_ itself. This is why, at this moment, we speak of being as _pure_ being. _Pure_ being exists only in the moment of understanding. As Hegel states above, it is totally undetermined\u2014it has, as yet, no other, no relation to any dialectical opposite. For these same reasons, pure being is the most abstract thought in the _Logic_. It is being in its initial one-sided determination.\n\nIn trying to understand pure being, it quickly becomes apparent that pure being is so abstract that it is, in fact, nothing at all. Since pure being has no actual content, when one thinks of pure being one thinks of nothing. \"The true situation is that being as such is not firm and ultimate, but rather something that overturns dialectically into its opposite\u2014which, taken in the same immediate way, is _nothing_. . . . This pure being is the _pure abstraction_ , and hence it is the _absolutely negative_ , which when taken immediately is _nothing_ \" (139). Nevertheless, thinking of nothingis still _not_ not-thinking. Nothing _is_ that of which one thinks. The being of nothingis indicated in the thought of nothing. Thus, when we think of nothing, we have continued to think of pure being. At this point, we are left moving between the two thoughts \" _pure being is nothing_ ,\" and \" _nothing is pure being_.\"\n\nWe seem to be left in limbo\u2014unable to proceed and unable to go back. Still, one can strain too much in trying to distinguish pure being from nothing. Hegel predicts our inclination to move beyond these abstractions and explains that this is a force behind the progression of the logic: \"All that really matters here is consciousness about these beginnings: that they are nothing but these empty abstractions, and that each of them is as empty as the other; the drive to find in being and in both [being and nothing] a stable meaning is this very necessity, which leads being and nothing further along and endows them with a true, i.e., concrete meaning\" (140). The fact that being and nothing are difficult to distinguish is precisely that which we ought to observe. We are not meant to simply think \"being and nothing,\" but to reflect upon the process of thinking them. Each one vanishes into its opposite. As we think being, it becomes nothing. As we think nothing, it becomes being. It is the particular relation between being and nothing that is now worthy of note\u2014this is a relationship of _becoming_. \"As their unity, _becoming_ is the truest expression of the result of being and nothing; it is not just the unity of being and nothing, but its inward _unrest_ \u2014a unity which in its self-relation is not simply motionless, but which, in virtue of the diversity of being and nothing which it contains, is inwardly turned against itself\" (143). The moment in which becoming is discovered is the speculative moment. As Hegel explains, the nature of the speculative moment is a sustained unrest. It is neither anxious, like the ambivalence of the understanding and the dialectic, nor is it static, like remaining in either ordinary understanding or the skeptical. \"The _speculatively_ or _positively rational_ apprehends the unity of the determinations in their opposition, the affirmative that is contained in their dissolution and in their transition\" (131). In the speculative moment, the opposition between the understanding and the dialectical becomes a totality.\n\nWe can see why this moment is called speculative, because it is the moment when we take a step back from thinking and reflect upon it. We cannot just think it _through_ , but must also think it _over_. This is how movement is gained in the _Logic_. It is how we come to understand how thought proceeds and it is the most truly philosophical moment of the _Logic_ : \"But it is one thing to have feelings and representations that are _determined_ and _permeated_ by _thinking_ , and another to have _thoughts about them_. The thoughts _about_ these modes of consciousness\u2014generated by _thinking them over_ \u2014are what reflection, argumentation, and the like, as well as philosophy are comprehended under\" (25). In the _Logic_ , thought is its own nourishment. \"But what we have here is the free act of thinking putting itself at the standpoint where it is for its own self, _producing its own object for itself_ thereby, and _giving it to itself_. . . . Philosophy shows itself as a circle that goes back into itself\" (141). For Hegel, speculative philosophy does not simply mean to contemplate one's reflection, but to catch one's reflection contemplating itself. In philosophy, for Hegel, thought comes to self-knowledge simultaneously as it comes into being (becomes).\n\nBad dialectic, for both Merleau-Ponty and Hegel, rests in the negative moment; it \"ends up at cynicism at formalism, for having eluded its own double meaning\" ( _The Visible_ 94). The cynic or skeptic as described in the _Phenomenology of Spirit_ experiences a false sense of freedom which arises from a sense of division from the object: \"What skepticism causes to vanish is not only objective reality as such, but its own relationship to it.\" The skeptic actively negates the possibility of objective reality; she is in control of what qualifies as truth, and the only truth she recognizes is the power of her own negation. Thus the skeptical is merely the negative dialectical moment. The speculative moment in Hegel's logic goes beyond this to what Merleau-Ponty refers to as _good_ or _hyper_ dialectic. He writes: \"The bad dialectic begins almost with the dialectic, and there is no good dialectic but that which criticizes itself and surpasses itself as a separate statement; the only good dialectic is the hyperdialectic. . . . What we call hyperdialecitc is a thought that on the contrary is capable of reaching truth because it envisages without restriction the plurality of the relationships and what has been called ambiguity\" ( _The Visible_ 94). Ironically, bad dialectic is a moment within good dialectic. It is a one-sided negation, a short-sighted ambivalence. Hyperdialectic observes this ambivalence and sees the truth in the relation between terms\u2014neither being nor nothingness, but becoming. The result is an ambiguous and complex manifold; a relation that should not be reified.\n\nThe association of Merleau-Ponty with Hegel will not be sufficient for most readers to overcome the charge that Merleau-Ponty eliminates alterity; Hegel has been frequently and famously critiqued on this point. Gilles Deleuze, for instance, argues that Hegel subsumes difference to identity. Similarly, William Desmond claims that Hegel subsumes the other within the greater whole. However, other scholars defend Hegel on the grounds that speculative logic is about the relationship between opposites, which makes no sense if there is no real difference between them. For example, in \"Identity, Difference, and the Logic of Otherness,\" William Maker writes that the \"view of Hegel's system as an intolerant omnivore . . . suffers from one defect. It is wrong.\" As I have argued, my reading of Hegel is consistent with this more recent scholarship that \"not the elimination but the proliferation of difference is the heart of Hegel's system.\" In my view, Merleau-Ponty and Hegel privilege neither identity nor difference. Both _Aufhebung_ and Merleau-Ponty's philosophy of ambiguity highlight the relation between opposites, a relation that disappears if they are merely synthesized.\n\nLike Hegel, Merleau-Ponty fetishizes neither identity nor difference. For the purposes of this book, it is most relevant to see how speculative logic is at work in his account of intersubjectivity. This also provides the opportunity to review some of the characterizations of intersubjectivity presented in the previous chapters of this book and their relation to Merleau-Ponty's notion.\n\nThe first movement of the logic, the understanding, dictates that _the self_ = _the self_. Antonio Damasio's principle of the singularity of self (\"one body goes with one self\") exemplifies this type of thinking. Recall, from chapter 3, in _The Feeling of What Happens_ , Damasio argues that \"things are [either] in or out of you\" and that for each body there is no more than one person. Of course, Damasio is not entirely wrong. This understanding is true, necessary, and practical. It is important that one not continually confuse oneself with others, not to live in a diffuse fusion with the world as Merleau-Ponty claims the infant does. If one goes to the doctor, one needn't bring one's entire family. But this characterization of the self is incomplete. It raises problems of egoism, solipsism, and narcissism. It invites the question: why be ethical? It does not recognize the contributions of others to my own existence or how integral they are to who I am. As Irigaray and Kristeva have argued, it is also a forgetting of the maternal and of nature.\n\nThe second moment is the dialectical, when the first determination is revealed as insufficient. In chapter 3 maternal experience was shown to contradict Damasio's first principle. In pregnancy, breast-feeding, and beyond to relationships with older and adopted children, maternal subjectivity is not simply singular. Care ethicists also argue that individualist notions of the self stand in stark denial of the mother-child relationship, of the fact that our existence is dependent on others before we can even entertain the possibility of individuality. In this moment it becomes apparent that while a mother _is who she is_ , she also _is who she is not; the self_ = _the other_.\n\nThere is truth to this view, as Levinas's characterization of epiphenomenal subjectivity demonstrates. The maternal metaphor helps him clarify how subjectivity can be understood as \"an irreplaceable hostage\"; how as ethical beings we are captivated by the needs of others. In his respect, Levinas accurately (though inadvertently) characterizes the experience of maternal ambivalence: the way a mother often experiences her child as a monstrous _Other_ who keeps her captive, threatens her body, eats away at her vitality, upsets her sense of self, invades her home, and commandeers her life. Levinas explains how when subjectivity seeks internal self-coherence it seeks the understanding that _the self_ = _the self_ , but this attempted self-satisfaction will inevitably be foiled (except perhaps in the case of a sociopath) by the intrusion of the Other. As there is no unity with oneself, there is also no union to be found with the other. For Levinas, our relations are fundamentally asymmetrical, the Other is held high above the self. Thus, in his account, proximity and alterity form an irresolvable ambivalence, \"not a dialectical unity of unity and difference.\" Though Levinas seeks to provide the conditions for the possibility of ethics, he effectively demonstrates why one would vacillate between wanting to destroy or nurture the other.\n\nLevinas is an example of how one can become stuck in the dialectical by failing to take the speculative view. Unfortunately, he understands the third movement of Hegel's _Logic_ in its simplified form, as synthesis, and claims that \"dialectical unity\" is a mere exercise of reason. Unlike Merleau-Ponty, who sees the immanent logic of human experience at work in (early) Hegel, Levinas believes that the relation to the Other precedes and defies reason. In elevating the Other above the self, Levinas institutes a perverse and tormenting infinite responsibility that devastates women's actual abilities to care for their children, undermining the very ethics he would establish.\n\nThe dialectical movement is an overcorrection; \"the philosophy of the negative bypasses the goal\" ( _The Visible_ 88). No mere synthesis would correct this course. This is evidenced in Sartre's partial appropriation of Hegel in _Being and Nothingness_. When Sartre attempts to bring self and other together, intersubjectivity becomes an eternal conflict. In the encounter with the other, I am objectified. Without the other's existence I could never conceive myself from an outside perspective. Yet the other's look leaves me feeling judged, defenseless, enslaved, endangered, fixed, and irredeemable. \"Fear is . . . the discovery of my being-as-object on the occasion of the appearance of another in my perceptive field.\" Thus, with the other's look, a world that was once mine now escapes me and I must instead take responsibility for the stranger that I have become to myself. Sartre then adds that in order to cope with this situation and free myself from being objectified, I objectify the other in response. For, by objectifying or negating the other, her negation of me is no longer central to my experience. In _Being and Nothingness_ intersubjectivity is reciprocal negation. Adding the negation of the other to one's own negation becomes an eternal power struggle. Sartre provides no way of moving forward from this difficulty. Like Levinas, Sartre neglects the speculative moment in _Being and Nothingness_.\n\nMerleau-Ponty critiques this \"philosophy of negativity\" in the chapter \"Interrogation and Dialectic\" from _The Visible and the Invisible_. He addresses Sartre directly in this passage:\n\nWhence comes this uneasiness that a philosophy of the negative leaves: it described our factual situation with more penetration than had ever before been done\u2014and yet one retains the impression that this situation is one that is being surveyed from above, and indeed it is: the more one describes experience as a compound of being and nothingness, the more their absolute distinction is confirmed; the more the thought adheres to experience, the more it keeps it at a distance. Such is the sorcery of the thought of the negative.\n\n(187)\n\nPhilosophy of negativity is ambivalent, unable to integrate multiple meanings ( _The Visible_ 91), ignorant of \"where the two moments cross, there where 'there is' something\" (95). In contrast, Merleau-Ponty's (and Hegel's) logic demonstrates that two seemingly contradictory aspects of human relations can be taken together; both alterity and interconnection are characteristic of self-other relations. Solitude and communication are two moments of _one_ phenomenon\u2014ambiguous intersubjectivity. As in perception, a mother can temporarily lose herself in visceral and ecstatic relation to her child. Yet, even as \"we become the others,\" irreversibility remains ( _The Visible_ 160). Every intimacy is also a slipping away.\n\nWe can see this \"immanent logic of human existence\" in the lives of mothers. Mothers are strikingly aware of their distinction from their children; this is the point of view of the understanding. The mother has her own needs for sleep, food, work, and play, which are frequently interrupted by her child. In some cases women are able to find time for themselves and negotiate healthy boundaries between themselves and their children.\n\nNevertheless, women recognize that meeting their own needs is more complex than simply seeking space from their children. This is the dialectical moment. Just as Socrates is both ignorant and wise, the mother is both more and less than one. If she feels an investment in her child's well-being, then her own desires often feel directed against herself when they are in opposition to her child's needs and wishes. At these moments a woman can feel most displaced by motherhood, as she simultaneously loses and finds herself in relation to her child. This is the negative moment, the bad dialectic, when one can become trapped in ambivalence, torn between one's own needs and the needs of a beloved child. It can persist in the form of Sartrean-style struggle for dominance or Levinasian-style maternal self-sacrifice.\n\nBut this is not the end of it. Mothers too entertain a speculative moment, a stepping back to reflect on the contradictions inherent to the relationship. In chapter 3 we saw examples of the speculative moment in art, poetry, and philosophy. Mierle Laderman Ukele's performance piece \"Some Kinds of Maintenance Cancel Others Out, Keep Your Head Together\u20141,000 Times, or Babysitter Hangup\u2014Incantation Ritual,\" is an excellent example. As a part of this performance Ukele calls the babysitter a total of one thousand times to ask if her children are OK. Ukele's piece is not just a synthesis of her art and mothering, it is a simultaneous reflection on and enactment of the tensions in the ambiguity of her relationship with her children and her own ambivalent _Befindlichkeit_. Kristeva's \"Stabat Mater\" is another brilliant example, where the author's poetic description of motherhood continually interrupts her scholarly writing on the topic. Creative work by mothers on motherhood enacts the simultaneous immersion within the mothering relation and the reflective distance needed from it.\n\nThe speculative moment is at once a preservation of contradictions and a transformation of them into something more useful and true. Indeed, this reflective distance can help to realize the creative force of maternal ambivalence. The ability to think through ambivalence and ambiguity seem to make all the difference in whether or not a mother's rage can be checked. Indeed, Merleau-Ponty argues that we must have an understanding and acceptance of ambiguity in order to deal with emotional ambivalence. In \"The Child's Relations with Others,\" Merleau-Ponty writes:\n\nIn sum, the subjects who carry within themselves extremely strong conflicts are precisely those who reject, in their views of external things, the admission that there are particular situations that are ambiguous, full of conflicts, and mixed in value. This occurs in such a way that one can say that a very strong emotional ambivalence shows up, at the level of understanding or perception, as a very weak ambiguity in the things and in his view of things. Emotional ambivalence is what demands the denial of intellectual ambiguity. In subjects whose intellectual ambiguity is strong it often happens that the emotional foundation is much more stable than in other subjects.\n\nAn understanding of ambiguity is necessary to emotional maturity and entertaining the speculative moment can be one method of coming to this realization and dealing with maternal ambivalence. This reinforces my earlier claim that when reflection is possible a woman can better understand herself and her _actual_ child; a purely emotive response will be insufficient. Maternal thinking is the vehicle through which mothers can negotiate the contradictions of their relationship, recognition of both union and radical divergence. Hegel's logic helps to clarify how the ambivalence engendered by the dialectic between mother and child is not overcome through a synthesis that would be an ossification. Its tension cannot be resolved, but it can become productive insofar as it reveals our fundamental ambiguity.\n\nAccusations that Merleau-Ponty denies maternal subjectivity are based on one-sided readings that overemphasize continuity over alterity. The concept of the flesh neither eliminates individual subjectivity nor declares an easy unity with the world and with others. A key dialectic of the body is the dehiscence between subject and object, the outcome of which is not an either\/or but a both\/and. Yet the \"constitutive paradox\" remains ( _The Visible_ 136). \"The body is a being of two leaves\" ( _The Visible_ 137). This \"double reference\" informs us \"that each one calls for the other\" (137). As the subject and object call for one another, likewise the baby calls for the breast and the breast calls for the baby. These two aspects do not dissolve into one another; they work in concert. The body, both sensed and sentient, is \"two segments of one sole circular course which goes above from left to right and below from right to left, but which has one sole movement in two phases\" (138). Merleau-Ponty points to this circularity of dialectical relations between organism and environment as early as _The Structure of Behavior_.\n\nThis dialectic of dehiscence is especially evident in the embodied and intersubjective lives of mothers, where the child is both a subject and object to the mother. There is no reason to think that even in prenatal life there is pure unity. Every cell is created through its own division. Likewise, the human being comes to exist through splitting and dividing. Childbirth is itself a form of dehiscence. The body is \"born of segregation\" from the flesh of the world and from the flesh of a woman ( _The Visible_ 136). This ambiguity, complementarity, and divergence persists: \"Eight years in, I can't always tell the difference between my children's needs and my own. . . . The needs of our children and our world and our selves merge and divide and merge again, until sometimes you can't tell one strand from another.\" Even for the mother whose child has died, the dialectic is not resolved. She loses a piece of herself, one of her own vital organs. Contrary to critics such as Irigaray, the dehiscence of motherhood is consistent with the orientation of Merleau-Ponty's philosophy.\n\nMerleau-Ponty's logic also addresses the criticism that he equates the mother with nature. He argues that all human beings are natural beings. We are embodied creatures who cannot escape the needs for clean air, water, and food. We were gestated within another, feeding, breathing, and excreting through her body. These bodies will someday return to inert matter; they will become food for other creatures or be burned to ashes and smoke. Merleau-Ponty describes this aspect of our relation to nature as anonymous because it is not the result of my own volition or reflection. Yet this prereflective life describes only half of our relation to nature; we are also alienated from it. This alienation is evidenced in our ignorance of our intimacy with it. We do not always recognize that which is personal, unique, and deliberate, founded in a prereflective nonvolitional body. We pollute our soil, water, and air. We struggle through science and philosophy to understand nature as though it were a stranger. Indeed, we are never completely able to thematize nature or our relation to it; it is irreducible to our attempts to know it. Our relation to nature is ambiguous because it exceeds us, yet it _is_ us.\n\nThe mother's place in nature, like all human beings, is also ambiguous. Yet her ties to embodied existence are especially difficult to ignore. As one mother said regarding her pregnancy and delivery: \"It was the most intensely physical experience I had ever gone through. For a long time I was totally reduced to my 'body.'\" In the case of a biological mother, her immersion in bodily nature becomes acutely obvious because of a combination of bodily changes and social context. Consider breast-feeding. When a woman breast-feeds, she experiences an additional bodily process, one that is not shared by the majority, considered disgusting by many people, and inadequately accommodated by society. The production of breast milk depends on its expression, either by the baby, by hand, or by a breast pump. If the milk is not expressed at regular intervals, not only does its supply diminish, but it causes leaking, painful engorgement of the breasts, and mastitis (breast infection). The expression of milk, including direct breast-feeding, cannot be done just anywhere. It is unsanitary to express milk or breast-feed in a restroom (you would not consider eating your lunch there), and women who breast-feed in the open risk exclamations of outrage, looks of disgust, and charges of indecency. Unless a woman can afford the four-hundred-dollar-hand-free breast pump, expressing her milk necessarily entails an interruption of whatever else she is doing for at least twenty minutes, approximately every three hours (depending on the age and needs of the child). The breast-feeding mother may rightfully find her personal existence overwhelmed by the demands of nature and the limitations of society.\n\nIf you have never breast-fed it may be difficult to identify with this example. Compare it to an imaginary scenario in which you are one of only a few people who needs to urinate. It is a bodily process that you must attend to periodically or suffer embarrassing leaks, pain, and risks to your health. Since the majority of people do not urinate, most public places have no place for you to do it. Obviously, you cannot just urinate anywhere or you risk arrest and the revulsion of others. Now imagine that it takes at least twenty minutes for you to urinate, and it can take a total of approximately eight hours per day (as breast-feeding a newborn can). Anyone suffering these conditions would find himself newly \"submerged in nature\" and isolated in the home. Not only would one be spending a considerable amount of time attending to one's body, it would limit one's ability to engage in activities outside the home or a few urinating-friendly locations. Since it takes up so much time and can only be done in certain locations, it would interrupt one's usual activities, work, and social life. One could come to feel that one's life was devoted to merely caring for the body, in the private realm, not participating in public meaning-making activities.\n\nOf course, we need not refer to this imaginary scenario to understand the situation of the breast-feeding woman. Consider the person in a wheelchair trying to get into a building that has stairs at every access point. Think of the person with a mental disability whose only access to income requires filling out complicated forms. In each of these cases, society neglects to consider the brute facts of an individual's life. This lack of accommodation means that the individual's difference becomes a grave disability. A person in such a situation may feel herself trapped in an unforgiving body, punished by nature, but in fact human beings built the stairs and designed the bureaucratic forms. Societal practices impinge on the body and are at least partially responsible for disabling many people.\n\nIt is not only the breast-feeding or pregnant mother who finds herself subject to her body\/nature. The mother of older children may also be acutely aware of her embodiment. Her ability to meet her own needs for sleep, nourishment, and exercise can greatly impact her ability to interact with her children positively. This, of course, is not always a negative change. As Kate Moses describes it in chapter 3, having children can _retrieve_ one's living relation to the body, transforming it from one of domination to cooperation.\n\nEven to the extent that motherhood is \"natural,\" this does not exclude it from human significance, reflection, and resistance to \"biological destiny.\" As the above examples illustrate, mothers can justifiably find themselves at odds with their bodies, especially when societal practices exacerbate their physical needs and limitations. This is a key factor in maternal ambivalence. Merleau-Ponty himself acknowledges this difficulty in his lectures on child psychology and pedagogy. In the lecture titled \"The Adult's View of the Child,\" Merleau-Ponty describes ambivalence in pregnancy:\n\n[The pregnant woman] feels her own body to be alienated from her; it is no longer the simple extension of her own activity. Her body ceases to be entirely hers; it is systematically inhabited by another being. Her body will shortly bring another consciousness to the world. Her own pregnancy is not an act like others she accomplishes with her body. Pregnancy is more an anonymous process which happens through her and of which she is only the seat. . . . In addition, pregnancy is accompanied by all sorts of anxieties, worries, and ambivalent feelings. The woman's sentiments regarding her pregnancy are always mixed sentiments, since there is always a latent conflict between her personal life and the invasion of what could well be called the species-life.\n\nIn this passage Merleau-Ponty acknowledges the bodily burden of the pregnant woman, but does not equate her with nature. On the contrary, her sense of alienation is heightened by the \"invasion of . . . species-life.\" Even while she is the \"seat\" of a process that she cannot control (short of having an abortion), she remains an individual who can reflect on and object to her own role in nature or \"species-life.\" Yet Merleau-Ponty argues that human interrogations of nature are another instance of nature's dehiscence, that nature is necessarily divided within and against itself. Thus even this resistance to species life does not sever the mother from nature.\n\nTo say that the maternal is a place of fleshly dehiscence is not to turn the female reproductive role into an ontological principle. The maternal body is _one_ place where the physical and metaphysical, and the human and nature, immanence and transcendence intersect. It is true that the maternal body and mothering person is the first world of the fetus and the infant. And it may also be true that the fetus, infant, and young child live in a sense of union with the mother or other primary caregivers. But this does _not_ mean that the mother _is_ or _experiences herself_ as undifferentiated flesh. Merleau-Ponty directly denies any simple fusion between pregnant mother and embryo in _The Visible and the Invisible_ : \"The vortex of the embryogenesis suddenly centers itself upon the interior hollow it was preparing\u2014a certain fundamental divergence, a certain constitutive dissonance emerges\" (233\u2013234). If the maternal body is to be thought of as flesh, then the child comes to be through a process of self-othering, a radical divergence in which the terms _self_ and _other_ are not, themselves, mutually exclusive. This relation overcomes the bad dialectic of self versus other and of being and nothingness. As there is no simple unity here, there also is no harmony without \"constitutive dissonance.\"\n\nMERLEAU-PONTY'S ETHICS (IMPLIED OR INFERRED)\n\nSome Merleau-Ponty scholars have inferred an ethics from his philosophy and believe that, although he does not directly prescribe rules of behavior, he makes ethics philosophically possible and psychologically explicable. In secondary literature we can again see the difference in emphasis between those who see alterity as most critical and those who stress commonality.\n\nNumerous scholars agree that by upsetting the distinction between self and other Merleau-Ponty undercuts the egoism\/altruism dichotomy and opens new possibilities for thinking about ethics. In \"Vision, Violence, and the Other: A Merleau-Pontian Ethics,\" Jorella Andrews makes this case: \"First, then, a truly ethical situation for Merleau-Ponty can only emerge when participants understand that they are internally related to each other in ways that are as yet indeterminate, and that they are profoundly _together_ in their respective differences.\" Similarly, in \"The Foundations of Merleau-Ponty's Ethical Theory,\" Douglas Low declares that connection makes ethics possible, \"[Ethics] involves recognizing the other as a being like oneself, who can experience and suffer like oneself. There is here an empathy, and identification with the other, an overlapping of ego boundaries to include others.\" In \"Sociobiological and Social Constructionist Accounts of Altruism: A Phenomenological Critique,\" Edwin Gantt and Jeffrey Reber claim that in describing human beings as fundamentally interconnected, \"always already being-for-the-other,\" Merleau-Ponty undermines the assumption of humans beings as disparate egos that are at base selfishly motivated\u2014an assumption that renders ethics impossible.\n\nIn \"Responsivity of the Body: Traces of the Other in Merleau-Ponty's Theory of Body and Flesh,\" Bernhard Waldenfels makes a similar but more subtle argument that there is a proto-ethics in the human's responsiveness to and primordial unity with the world. He writes: \"Stretching throughout the work of Merleau-Ponty like a red thread is what I call the motif of responsivity (or answerability) . . . between the organism and its world.\" According to Waldenfels, since Merleau-Ponty's \"subject\" is constituted in terms of its responsiveness to its environment, the \"alien\" is thus fundamentally constitutive of the \"self.\" Waldenfels claims that this responsiveness goes beyond that notion of a subject seeking knowledge; it has a distinctly ethical flavor. Responsiveness is \"consenting to the offers ( _Angebote_ ) and demands ( _Anspr\u00fcche_ ) of the other. In this expanded sense, demands have to be understood in the double sense of an 'appeal to someone' ( _Anspruch an_ ) and a claim to something ( _Anspruch auf_ ).\" Waldenfels claims that our relationship to the natural world and our relationships with other human beings are parallel. As our perceptual responses make us who we are, so do our responses to other people. He thinks that responsiveness eliminates the need to posit egoism and altruism as the only alternatives, by upsetting the radical distinction between oneself and the alien. The other's demands are \"intertwined with my own being . . . otherness penetrates into the heart of self-presence.\" \"Bad ambiguity\" is when otherness is subsumed to a totalizing whole, but he thinks the notions of flesh and reversibility overcome this. For Waldenfels it might be said that perceptual responsiveness is necessary but not sufficient for ethical responsibility.\n\nThese accounts prioritize connection for a viable theory of ethics. As Waldenfels notes, even if sympathy with and responsiveness to others is necessary for ethics, it is not _sufficient_. The phenomenon of maternal ambivalence demonstrates that the desire to commit violence can occur even when there is a feeling of sympathy and connection with the other. In fact, intense intimacy can lead to feelings of claustrophobia and thus _instigate_ violence. Misappropriations of Merleau-Ponty that base ethics solely in a sympathetic response, do not explain why it is often a _struggle_ to respond to others, why our actions and desires can conflict, and why we can respond to others' needs even when we don't _want_ to do so.\n\nIn contrast, Alia Al-Saji claims that the possibility of a Merleau-Pontian ethics would be found in his theory of alterity:\n\nThere is no longer a mere difference of number between my body and that of the other, but a differentiation that spans the dimension of the flesh. This new dimensionality, this multiplicity of dimensions of the flesh, represents for us the concrete possibility of an ethics. It opens up between myself and the other, beyond the numerical distinction of our bodies or our two looks, all the possible differentiations that can be generated between us, and that express the singularity of each. Hence, voice, gesture and speech, as well as thoughts, are inserted between us.\n\nAccording to Al-Saji, our differences are discovered through our expressions. The immediacy of the other's anger, for example, does not simply unite us to one another; it demonstrates a division. It makes the other before me a _particular_ other. If we inhabited an undifferentiated group life, I would necessarily be angry as well, and the anger of one would be the same as the anger of another. But my responses to an expression of anger will vary; I can be sad, reasonable, indifferent or terrified, and this will depend on the particular person who is angry with me. This communication between us is also a divergence. This recognition of difference means that our vision is not hegemonic \"that vision can become part of ethics.\"\n\nSimilarly, in \"'All Things Considered': Sensibility and Ethics in the Later Merleau-Ponty and Derrida,\" Ann Murphy says that Merleau-Ponty's account of sensibility provides an _opening_ for ethics\u2014\"it is an elaboration of embodiment that provokes the question of response but no definitive or prescriptive answer. Hence the structure of sensibility begs the question of ethics and the problem of response but can provide nothing by way of a normative ethics.\" Murphy acknowledges that both violence and hospitality are possible under Merleau-Ponty's philosophy; the opening to responsibility provides the opportunity and the compulsion to _both_ of these options.\n\nFor each of these scholars Merleau-Ponty's characterizations of sensibility and intersubjectivity already imply a latent Merleau-Pontian ethics. Andrews, Low, Gantt, and Reber think that he explains why human beings are not just selfishly motivated, why we can be generous. They believe that to be ethical means to be connected and to have empathy. Unfortunately, these four do not rigorously observe the is\/ought distinction. They read Merleau-Ponty as describing humans in communion and then claim that we ought to be in harmony with one another. Waldenfels, AlSaji, and Murphy are more cautious in their conclusions. They do not think Merleau-Ponty ordains specific attitudes or behaviors, but rather that he opens up the possibility of ethical theory. There are some indications Merleau-Ponty intended to provide such a foundation. At the end of his posthumously published prospectus for _The Visible and the Invisible_ , he states that to establish \"good ambiguity . . . a spontaneity which gathers together the plurality of monads . . . would be metaphysics itself and would at the same time give us a principle of an ethics.\" What principle would this be? What ethics would it establish? He did not elaborate.\n\nThe overarching question of Merleau-Ponty's work is our place in the world and our responsiveness to it. Any responsibility we do take up necessitates the ability to respond, which in Merleau-Ponty is revealed through consideration of the sensible. But does being able to _respond_ necessarily mean that we are _responsible_? Merleau-Ponty's philosophy describes our social and physical embeddedness. With this intertwining comes a vulnerability to one another. We are dependent on others for our very perception of reality. The character of our fleshly ontological condition means that we are capable of responding to others both as we interconnect and as we differ. This means that the other can be disruptive. Consider the false friend. Given that I depend on others for my sense of \"objectivity\" and the solidity of the world, finding out that I have been lied to by a trusted friend undermines the very reality that we have coconstituted. \"The relations with the other are always complicated,\" and ethical life is thus necessarily conflicted. We cannot take others' or our own positive ethical response for granted. Even when our intentions are good, we may not know how to get the best result. In short, it is a fact of ethical life that we will sometimes fail to help the other and that she may fail us. Recognizing this inevitability (overlooked by Levinas) is paramount.\n\nIn his Sorbonne lectures on child development, Merleau-Ponty claims that ambivalence undergirds our conscious experience. He advocates replacing Freud's notion of the unconscious with ambivalence: \"We should prefer this notion of ambivalence, which paints perfectly all that is equivocal in certain behaviors, 'resistances' to treatment, in which the subject is partially complicit, attitudes of hate that are at the same time love, desires that express themselves as agony, and so forth\" (73). Emotional ambivalence is the prereflective underside to intellectual ambiguity. Thus to the extent that that intersubjectivity is togetherness, this mutuality will involve love as well as hate.\n\nMerleau-Ponty describes the mother-child relationship as a paramount instance of this ambivalence. He writes:\n\nEvidently major troubles will arise from the manner in which [the mother] envisages her connection with the child. This connection is essentially ambivalence: the infant belongs to her, but she also belongs to him. A variety of positive and negative feelings are activated by this connection. . . . Pregnancy returns the mother with another being to the current of common life in communion. At the same time, in a negative sense, it becomes necessary to renounce many personal projects, to endure exhaustion and fear of deformation. She feels that this mysterious operation which is occurring in her puts her in danger (i.e., the symmetry between birth and death). Moreover, in this difficult situation, a situation over which she has no control, she must passively await its development. She is accompanied by all sorts of fantasies. The child is sometimes imagined to be a hero, sometimes a monster. . . . Hence, Hegel [states], \"the birth of the child is the death of the parents.\" The birth of the child is at the same time the fulfillment of their union as well as its transformation.\n\n(79)\n\nMerleau-Ponty realizes that the mother's relationship to her child will be among the most conflicted. And it is clear that Merleau-Ponty sees a combination of natural-physical factors and cultural factors conspiring to amplify maternal ambivalence: the shared embodiment of mother and child, the physical dangers of pregnancy, the uncertainty of the child's future, and the interruption of the mother's personal aspirations all seem to be inevitable facts. Merleau-Ponty does not take for granted that a mother will bond to her infant once it is born. He writes: \"At the same time, we must give the mother flexible time, long enough to take possession of her child, to identify with the infant, and to love him as her own. The relation between mother and child is partly a weak instinctual relation and also a human relation\" (80). The woman's socially imposed role is another critical factor: \"The father's attitude toward his children is as ambivalent as the mother's. He identifies with the child and oscillates between domination and sacrifice. . . . However, the father's problems with the child are much less acute than those of the mother for several reasons . . . our customs give the man a greater serenity in his conflicts with the child from the fact that he is separated from the child most of the day\" (82).\n\nMerleau-Ponty recognizes that the maternal-child relation is even more ambiguous than most relations and that the mother will be highly ambivalent in her relationship to her child. Because of her physical and psychic ties to a dependent other, her embodied existence is more difficult to ignore than average and she is more likely to feel at odds with it. Even in pregnancy, when mother and child seem most closely joined, there is \"constitutive dissonance\" between them. Though the mother finds herself to be a place of fleshly dehiscence, and while this can serve to undermine her personal life, it can never erase her subjectivity or her humanity. It is the flesh that makes possible both individual and social life. And maternal experience demonstrates that the ambivalence incited by our ambiguous social life can be a productive tension.\n\nThus far in this book we have explored different moments that characterize the phenomenon of intersubjectivity. According to the more straightforward understanding of intersubjectivity, I am me and you are you; _self_ \u2260 _other_. It is important to be able to distinguish oneself from others in this way. Yet maternal experience demonstrates that this understanding is only a half-truth; maternal subjectivity is not merely singular. The other side of this coin is the dialectical move made by Levinas; _the self_ = _the other_. This is no simple equation, but it does mean that to the extent that there is subjectivity it is attributable to the other's ethical appeal. This position illustrates our fundamental interconnection with others and makes our ethical being more comprehensible. For Levinas, this view affirms an infinite responsibility of the self for the other, which I argue is ultimately detrimental. Such a self-sacrificing take on ethical obligation has been especially damaging to women. The negative dialectical approach to intersubjectivity sets up an irresolvable ambivalence between self and other for either dominance (as in Sartre) or submission (as in Levinas). The negative dialectical position merely reinstates a dualism between self and other. Through taking the speculative or hyperdialectical view, Merleau-Ponty helps us to think how these apparently opposing truths can be taken as two sides of the same phenomenon. Merleau-Ponty's logic \"envisages without restriction the plurality of the relationships and what has been called ambiguity\" ( _The Invisible_ 94).\n\nIt is Merleau-Ponty who best characterizes the ambiguity of our intersubjective relations. He enables us to see how we can communicate with one another across the abyss between us. He renders moot the question \"Why be ethical?' by demonstrating that we are already intertwined with others, hence already immersed in ethical relations of some kind. What remains is to better understand one's place in the world. This view of the human condition is precisely what Merleau-Ponty believes he has in common with Hegel. He writes, \"Hegel's thought is existentialist in that it views man not as being from the start a consciousness in full possession of its own clear thoughts but as a life which is its own responsibility and which tries to understand itself.\" The meaning of our ethical life must follow from our orientation to our existence and to others, and this is what Merleau-Ponty helps us to understand more clearly.\n\nOf course, being in ethical relations does not mean that we will necessarily care for one another. And it does not indicate the best course of action. Since ambivalence undergirds our relations with one another, these relations are always complicated and conflicted. Merleau-Ponty demonstrates the possibilities for both responsibility and irresponsibility. As Mauro Carbone says in \"FLESH: Towards the History of a Misunderstanding\": \"The stranger, as flesh of my flesh, is just because of that my brother. But my brother could well be Cain. I myself could be such. As a condition for _all_ these possibilities, as a condition of 'a reversibility always imminent and never realised in fact, flesh founds _every_ possible ethics and _every_ possible politics, that is, does not found _any_ particular ethics or politics.\" In \"The Ineradicable Danger of Ambiguity at Ch[i]asm's Edge,\" Phillip E. Young even demonstrates how Merleau-Ponty's philosophy could be co-opted by totalitarians. All this is to say that Merleau-Ponty's philosophy does not inevitably dictate a moral scheme that we would embrace. He invites us to ask the question \"What should I do?\" but does not answer it for us. To ask this question for oneself is to begin to take responsibility.\n\nMerleau-Ponty writes: \"The very productivity or freedom of human life, far from denying our situation, utilizes it and turns it into a means of expression.\" One's situation provides her with the means to expression, thus one's own guilt and responsibility cannot be merely individual. Still, one's freedom is an indication of personal accountability. \"There is equal weakness in blaming ourselves alone and in believing only in external causes.\" This ambiguity between our immanence and transcendence in our ethical life invites us to ask different kinds of questions with regards to our treatment of one another. For example, instead of just asking why an individual man raped an individual woman (What makes _him_ do something so horrible?), we need to look at the bigger picture, a society in which such acts are not only common, but both subtly and bluntly condoned. What was the situation that enabled or encouraged this act? Clearly, ethics are not _only_ a matter of individual virtue. Even Aristotle, the original virtue ethicist, insisted that a certain amount of _luck_ is required to enable one to live the good life. If we take luck to mean the situation that surrounds a person, her influences, opportunities, and resources, then it is not something to be left to the fates alone.\n\nMerleau-Ponty shows that ethical discourse is relevant to creatures such as us, beings who are knitted within their context, who are points of convergence and divergence in the flesh of the world, who are situated within relations that are individual, societal, and historical. One's situation will undoubtedly impact one's answers to the ethical questions posed by life. Merleau-Ponty says \"A moral imperative only emerges in contact with a situation.\" This principle certainly applies to mothers.\n\nIn this book I argue that maternal ambivalence is both predictable and understandable. However, I take it for granted that harming a child is not. Indeed, it seems that a great number of women experience maternal ambivalence without acting out their aggressions. How does a woman's situation influence this outcome? It is never easy to draw a cause-and-effect relationship between a certain situation and a particular action. Since the individual person is a variable factor in this equation, it would be imprecise to claim that _Situation A_ causes _Behavior B_. How then are we to consider the situation of mothers with respect to their (inter)subjectivity, ambivalence, and ethical orientations?\n\nThe way in which our ethical response is situated is considered more robustly by Simone de Beauvoir. In the next chapter we will consider how Beauvoir helps to further develop Merleau-Ponty's emphasis on situation and makes it more concrete. She articulates how immanence and transcendence converge in our ethical relations and specifically considers the conditions for maternal responsibility: \"The mother's relation with her children is defined within the overall context of her life; it depends on her relations with her husband, her past, her occupations, herself; it is a fatal and absurd error to claim to see a child as a panacea. . . . The young woman must be in a psychological, moral, and material situation that allows her to bear the responsibility; if not, the consequences will be disastrous.\" Given our interdependence, Beauvoir articulates how the burden of ethical life should not rest solely on the shoulders of the individual. Yet she also asserts that the individual is strictly responsible for the way in which she _takes up_ her situation.\n_Six_\n\nMATERNITY AS NEGOTIATING MUTUAL TRANSCENDENCE IN THE PHILOSOPHY OF SIMONE DE BEAUVOIR\n\nWithout failure, no ethics. For a being who, from the outset, is an exact coincidence with himself, a perfect plenitude, the notion \"ought to be\" [devoir \u00eatre] would not make sense. One does not propose ethics to a god. It is impossible to propose one to man if he is defined as nature, as given. . . . This means that there can be an \"ought to be\" [devoir \u00eatre] only for a being who, according to the existentialist definition, questions himself within his being.\n\n\u2014SIMONE DE BEAUVOIR, \"INTRODUCTION TO AN ETHICS OF AMBIGUITY\"\n\nBeauvoir finds that our ethical life is inherently tragic and full of insoluble contradictions. Neither the meaning of our existence nor our ethical standing is assured at any moment. The liberation of others from mechanisms of oppression is the precondition for their freedom, and the freedom of others is the precondition for ours. But these same freedoms are in conflict with one another. We depend on others, but we cannot count on their generosity or collaboration. Beauvoir finds that it is precisely _because_ of our inability to achieve self-coincidence that it makes sense to have an ethics. This lack is _vulnerability_ to others and to a future that always remains unknown. Each time we struggle to overcome who we have been and what we have accomplished, we create a new position to be surpassed. Though many of our strivings are ultimately futile, they are also precisely the manner in which we become present to our existence. Our efforts on behalf of ourselves and others are the thrust of ethical life; and the alternative to acknowledging their potential failure is inauthenticity\u2014the denial of my own and\/or the other's freedom and\/or limitations.\n\nThe failure to coincide with oneself is more than an inability to achieve one's self-serving goals\u2014to acquire wealth, academic degrees, fame, and so on. It also indicates the impossibility of caring for others and liberating them completely\u2014to answer every time they call, to be present to all of their hopes and pains, to rescue them from every crisis. More than any other person, this is what a mother is expected to do for her children; thus ethical failure is a constant threat to her. Beauvoir richly describes the ways in which motherhood amplifies the painful tensions of ethical life and leads to an attitude of ambivalence toward one's children. Indeed, we find that maternal ambivalence is an ethical \"failure\" in the positive sense that it has for Beauvoir. It is a failure to deny the openness of the future, to ignore the needs and values of others, to plow through to one's ends without regard for their full consequences, and to adhere to abstract principles that may be inappropriate to the circumstances at hand. Maternal ambivalence is a robust exemplar of the contradictions of our ethical life.\n\nMaternity is a unique ethical condition, but it brings into sharp relief the pattern of interdependence that is relevant to all human relationships. It highlights how tragic ethical failure can be. It is not always possible to work for the good of all; even mothers must sometimes sacrifice the freedom or care of one for another. The acknowledgment that someone will be (even temporarily) neglected or abandoned requires that we continue to question the validity of our projects and the costs of achieving them.\n\nINTERSUBJECTIVE AMBIGUITY AND EXISTENTIAL MORALITY\n\nOur freedoms support each other like the stones in an arch . . .\n\n\u2014BEAUVOIR, \"PYRRHUS AND CINEAS\"\n\nBeauvoir's philosophical ethical project, including \"Pyrrhus and Cineas\" (1944), \"Moral Idealism and Political Realism\" (1945),\" Introduction to an Ethics of Ambiguity\" (1946), _The Ethics of Ambiguity_ (1948) and _The Second Sex_ (1949), calls for us to \"assume our fundamental ambiguity. It is in the knowledge of our genuine conditions that we must draw our strength to live and our reason for acting.\" She indicates the many contradictions at the heart of human existence, acknowledges the conflicts both within ourselves and in our relationships, and demands that we act from out of the knowledge of this ambiguity rather than its denial. While it may be possible to willfully ignore the human condition, Beauvoir demonstrates that if one exercises her freedom then she will continually confront ambiguity. For Beauvoir the realization of freedom is transcendence\u2014exceeding who one is in the present and who one was in the past through engaging in projects that change the future in a direction she values. The desire for transcendence prevents our complacency, our contentment with past achievements. A project completed becomes a past to surpass. Yet the exercise of freedom is not just the glorious, uninhibited expression of free will. In taking action one is faced with disruptive questions about the boundaries between oneself and others and the implications of one's actions for them. Works of transcendence bring to light the highly constrained nature of human freedom; it arises within a context of interdependence. \"Man can find a justification of his own existence only in the existence of other men. . . . I concern others and they concern me. There we have an irreducible truth.\"\n\nThere are a number of ways that the exercise of one's freedom depends on others. For one, we require others to give our projects (and thus ourselves) validity and solidity. According to Beauvoir, we are like children who rush to their parents for approval of our every creation. It is not enough that our works please us individually; we want them to have an intersubjective importance. What is the point of a great painting or poem that no one sees? Even the works of Vincent van Gogh, who was relatively solitary in his lifetime, are incomplete until others engaged with them. Still, we need more than just appreciators of our works, we also need collaborators. Everything from creating a more efficient bureaucracy to liberating Paris from the Nazis involves cooperation with others. Where would van Gogh be without the financial and emotional support of his brother Theo? What is more, if the full realization of our projects is in their impact on the future, then we rely on others to continue them beyond our lifetimes. The continuation of Beauvoir scholarship and the offering of courses in feminist phenomenology mean that the realization of Beauvoir's transcendence continues to our present day. Of course, there is no guarantee, even if others take up our project, that it will continue in the spirit we intend: case in point, Parshley's infamous translation of _The Second Sex_. As Beauvoir writes: \"An action thrown into the world is not propagated infinitely like the wave in classical physics. . . . The action does not stop the instant we accomplish it. It escapes us toward a future, where it is immediately grasped again by foreign consciousness. It is never a blind constraint for the other but a given to be surpassed, and the other surpasses it, not I.\" Although we cannot control or predict the continued impact of our works, without the investments of others, our efforts would not even make a ripple. \"Thus, we see that no existence can be validly fulfilled if it is limited to itself. It appeals to the existence of others.\"\n\nThe investment of others in my projects requires that they voluntarily share the values that I wish to assert. Others can be enslaved to build a pyramid, but, once the whip stops cracking, what is to prevent them from robbing the tomb? The long-standing value of my projects requires that others not be compelled by force or necessity to join my cause. They must have adequate health care, food, shelter, and security so that their lives are not consumed with bare survival. This is one reason why we have an interest in the well-being of others. To oppress others would be to fundamentally misunderstand the nature of our own freedom. If others are not free, I have no colleagues, comrades, or allies and my projects will lead a short life. For that reason she states: \"To will oneself moral and to will oneself free are one and the same decision.\" Our own freedom depends on the welfare of others.\n\nIf this concern for the other seems egoistic, it's important to remember that for Beauvoir the self is not pregiven. Our transcendent projects are the essence of our existence. Since we depend on others for their realization, we have a most fundamental need for other people. \"It is because my subjectivity is not inertia, folding in on itself, separation, but, on the contrary, movement toward the other that the difference between me and the other is abolished, and I can call the other mine.\" It would be a solipsistic error to think that any project, from constructing a house to building a fortune, is merely the distinction of an individual. It is also an attachment with the others who are involved. Paradoxically, any individuality is founded through intertwining with others, and as the other becomes _mine_ , I have responsibility for her. \"Is this kind of ethics individualistic or not? Yes, if one means by that it accords to the individual an absolute value and that it recognizes in him alone the power of laying the foundations of his own existence. . . . But it is not solipsistic, since the individual is defined only by his relationship to the world and to other individuals; he exists by transcending himself, and his freedom can be achieved only through the freedom of others.\"\n\nFor Beauvoir, each person's entanglement in the web of human relations is an existential _fact_ , preexisting the willing of one's freedom. Others are necessary to one's ethical freedom, but they are also integral to one's natural freedom. Natural freedom\u2014the possibility for transcendence initiated by others' objectification of oneself and one's rebellion against this objectification\u2014is the kind of freedom we cannot escape. This is the idea of freedom shared by Sartre in _Being and Nothingness_ that \"every man is free, that there is no way of his not being free.\" For example, Beauvoir claims that the victim of torture is free in the natural sense; she may still assume an attitude of compliance, forgiveness, or rebellion. Clearly, this notion of freedom attends less to the constraining power of situation than ethical freedom does, even though it acknowledges the importance of the existence of others. For Beauvoir, without the other's objectifying viewpoint, one would be trapped in immanence: \"It is the existence of other men that wrests each man from his immanence and enables him to accomplish himself as transcendence, as flight toward the object, as a project.\" For Beauvoir, this objectification precedes our morally free acts.\n\nBeauvoir believes that natural freedom is consistent with ethical freedom, even though it sets up a conflict in which subjects try to assert themselves over and against each other: \"But this foreign freedom, which confirms my freedom, also enters into conflict with it: this is the tragedy of the unhappy consciousness; each consciousness seeks to posit itself alone as sovereign subject. Each one tries to accomplish itself by reducing the other to slavery.\" The existences of others, who attempt to overpower me, judge me, or treat me as a means to their ends, places my identity continually under threat. The struggle against this reification is an original impetus to realize my freedom. In the context of this ongoing challenge, I seek to reestablish myself by realizing my perspective and values through my actions. For that reason, Beauvoir believes that natural freedom does not suffice for an ethical life; it ought to drive one toward greater accomplishments.\n\nWhat meaning can there be _to will oneself free_ [ethical freedom], since at the beginning we _are_ free? . . . This objection would mean something only if freedom were a thing or a quality naturally attached to a thing. Then, in effect, one would either have it or not have it. But the fact is that it merges with the very movement of this ambiguous reality which is called existence and which _is_ only by making itself be; to such an extent that it is precisely only by having to be conquered that it gives itself. To will oneself free is to effect the transition from nature to morality by establishing a genuine freedom on the original upsurge of existence.\n\nThe torture victim's attitude of forgiveness or rebellion is an incomplete response. It is her concrete efforts to eradicate torture, to liberate other victims, or perhaps to take revenge on its perpetrators that fully realize her freedom. Of course, one can be too broken, lazy, cowardly, defeated, or passive to will her freedom and thus fail to recognize the importance of others. However, the existence of our natural freedom means that we will always be confronted by others and the denial of their significance will be a hard-won delusion.\n\nConflict with others is also inevitable at the practical level for the simple reason that peoples' desires and interests clash: \"I am not dealing with one freedom but with several freedoms. And precisely because they are free, they do not agree among themselves.\" Since transcendent projects extend beyond oneself, they can impinge on other people. A project that benefits some may devastate others. The dam built on a river provides electricity, but might also ruin fishing downstream. \"What is good for different men differs. Working for some often means working against the others. One cannot stop at this tranquil solution: wanting _the_ good of _all_ men. We must define _our_ own good. . . . The respect of the human person in general cannot suffice to guide us because we are dealing with separate and opposed individuals.\"\n\nIn spite of the fact that some toes will inevitably be stepped on, Beauvoir declares it a moral error to assert oneself without concern for others, bulldozing through to one's ends or soaking up the privileges of the oppressor class without regard for the impact on others. This is because one's actions create the circumstances within which others will be either impeded or enabled to act. \"What concerns me is the other's situation, as something founded by me. . . . I am the facticity of his situation.\" This worry is a central basis for Beauvoir's \"existential morality,\" which is given its most mature expression in _The Second Sex_.\n\nEvery subject posits itself as a transcendence concretely, through projects; it accomplishes its freedom only by perpetual surpassing toward other freedoms; there is no other justification for present existence than its expansion toward an indefinitely open future. Every time transcendence lapses into immanence, there is degradation of existence into \"in-itself,\" of freedom into facticity; this fall is a moral fault if the subject consents to it; if this fall is inflicted on the subject, it takes the form of frustration and oppression; in both cases it is an absolute evil.\n\nThe key existential good is an ever-reaching freedom upon an open future of possibilities. The existential ethic, the realization of one's freedom, consists in striving to exceed that which one currently is, to achieve transcendence through engaging in projects\u2014meaningful work that affirms one's own perspective, values, and creativity. The individual who surrenders to her objectification and immanence, who quits striving toward the creative possibilities available to her, has failed morally. In addition, a moral society, institution, or person creates greater opportunities for human beings' ingenuity, expansiveness, and potential. When restrictions are imposed on the individual that foreclose her opportunities, she is \"frustrated and oppressed.\" And Beauvoir maintains that it is unqualifiedly immoral to destroy another's ability to live toward her own ends: \"Whatever the problems raised for [man], the setbacks that he will have to assume, and the difficulties with which he will have to struggle, he must reject oppression at any cost.\"\n\nUltimately, this means that we owe others more than a policy of noninterference; we owe them our vigorous effort toward their liberation. Beauvoir maintains that it is insufficient to grant that the other has ontological freedom. The enslaved, impoverished or colonized person is severely limited by his situation; opportunities for the practical realization of freedom differ tremendously for the child born in an American suburb and one raised in a Brazilian _favela_. The concept of ontological freedom is largely blind to these differences. To remain ignorant of such inequalities is disingenuous, to deny the importance of situation and leave the marginalized to their own struggle is to remain complicit with their oppressors. Often, any tool with which an oppressed person might have attacked the civilization that subjugates him has been confiscated: \"having been kept in a state of servitude and ignorance, they have no means of breaking the ceiling which is stretched over their heads.\" To be politically detached is simply another way of siding with whomever is strongest at the time. Beauvoir insists that caring for the oppressed should not merely be a form of detached paternalistic charity. Since the other's freedom is integral to one's own, it is everyone's cause. \"It is this interdependence which explains why oppression is possible and why it is hateful.\"\n\nAlthough at first it may seem to contradict her emphasis on interpersonal strife and individual freedom, Beauvoir holds generosity as the highest human achievement: \"The conflict can be overcome by the free recognition of each individual in the other, each one positing both itself and the other as object and as subject in a reciprocal movement. But friendship and generosity, which accomplish this recognition of freedoms concretely, are not easy virtues; they are undoubtedly man's highest accomplishment; this is where he is in truth: but this truth is a struggle endlessly begun, endlessly abolished; it demands that man surpass himself at each instant.\" This notion of generosity involves an enlightened recognition of the others' and one's own simultaneous objectivity and subjectivity, which one must continually labor to maintain. This generosity is not at odds with personal liberty; it is precisely the recognition of the other's freedom. Thus the greatest generosity would be to advocate for the other's concrete freedom, to be a platform for her transcendence: \"A lucid generosity is what should guide our actions. We will assume our own choices and posit as our ends the situations that will be new points of departure for others. But we must not delude ourselves with the hope that we can do anything _for_ others.\" The best we can do for another is to put him in a position of being capable of striving in the direction he wills.\n\nRecent Beauvoir scholarship agrees that, for Beauvoir, human relations are marked by mutual affirmation and reciprocity; freedom is always relational. For example, Barbara Andrew claims that \"recognition of the others' freedom is the only acceptable moral attitude or attunement towards the world.\" She aligns this view with care ethics: \"recognition of the other's freedom is a response to the ethical needs of those others as well as one's own ethical needs, and thus is a form of care.\" Andrew minimizes the importance of interpersonal conflict to Beauvoir's philosophy: \"[Beauvoir] does not see a necessary struggle or a need to destroy the other to establish one's own subjectivity. . . . Transcendence occurs only through reaching out to others.\"\n\nNumerous scholars agree that Beauvoir's work has been interpreted falsely as an elaboration of Jean-Paul Sartre's _Being and Nothingness_. When her work is considered on its own merits, they think, it becomes clear that for Beauvoir human relationships are not primarily characterized by struggle, that negation of the other is not fundamental for her as it is for Sartre. Ursula Tidd argues this point: \"In terms of the Sartrean account of the structure of consciousness\u2014that it entails a double negation: the Self negates itself as not the Other, and negates the Other as object to be transcended\u2014Beauvoir accepts the first negation but not the second. In her account of consciousness, the Other cannot be a mere object of consciousness to be transcended; the Other is already incorporated within the Self's transcendental movement.\" Fedrika Scarth also attributes the conflictual model to Sartre. She states that \"[Beauvoir's ethics] is driven by her insistence, contra Sartre, that ethical relationships with others are possible and that relating to others ethically will require the acceptance of our ambiguity as situated and embodied subjects.\" Julie Ward draws the same conclusion from Beauvoir's autobiographical work. Ward thinks that harmonious relationships between female friends are Beauvoir's ideal model:\n\nAmong female friends, Beauvoir feels recognized and importantly, _preserved_ , whereas with her male love-objects, she experiences a loss of self. So, the early diary not only shows that Beauvoir possesses a positive notion of being seen by the other but also that the model of this positive relation to another is evident in some of Beauvoir's female friendships. For Beauvoir, it is precisely with Zaza and H\u00e9l\u00e8ne that being with the other is a comfort, not a conflict, and provides continuity, not disruption, to being a subject.\n\nIn \"Teaching Sartre About Freedom,\" Sonia Kruks argues that these ideas align Beauvoir more with Merleau-Ponty than Sartre: \"There is for Merleau-Ponty, unlike Sartre, an undifferentiated or 'general' being, a 'primordial layer at which both things and ideas come into being,' which is 'anterior to the ideas of subject and object' and in which each of us participates as embodied existence. It is through this common participation that we escape the solipsism implied in the Cartesian cogito\u2014a solipsism still lurking in _Being and Nothingness_.\"\n\nAs it should be obvious by now, contemporary Beauvoir scholarship is strongly motivated to distinguish Beauvoir as a philosopher in her own right, apart from Sartre. Kate and Edward Fullbrook's _Simone de Beauvoir and Jean-Paul Sartre: The Remaking of a Twentieth-Century Legend_ is often credited with spearheading this effort. The Fullbrooks argue that many of Sartre's insights from _Being and Nothingness_ originate in his reading of Beauvoir's _She Came to Stay_. Margaret Simons also makes this claim and supports it with research on Beauvoir's diaries that reveal the variety and chronology of her intellectual influences and origination of her ideas.\n\nIn many cases Beauvoir scholars are attempting to retrieve her from critics of her pessimism toward maternity. These detractors believe that Beauvoir describes maternity in negative terms because she appropriates a Sartrean repulsion toward the female body and a one-sidedly conflictridden (Sartrean and Hegelian) account of consciousness. For example, Genevieve Lloyd critiques Beauvoir's account of the female body as \"an intrinsic obstacle to transcendence. . . . Underlying de Beauvoir's descriptions of female biology is the original Hegelian opposition between the individuality of self-consciousness and the inchoate generality of life.\" Lloyd believes that, for Beauvoir, the female body and its accompanying role must be transcended to achieve authenticity and freedom. Similarly, regarding Beauvoir's decision to write books instead of have children, Alice Jardin claims: \"Beauvoir's decision not to have children in the world might be seen as but an acting out of her complete denial of the maternal, of her refusal of the maternal body's most intimate influences upon her own body and body of work.\"\n\nIn response to this critique of Beauvoir, some supporters \"creatively reinterpret\" her views deemphasizing the importance of conflict. According to Scarth, Beauvoir's highly conflictual picture of pregnancy is a \"rhetorical strategy\" designed to overturn the patriarchal myths of pregnant women as passive vessels. For Scarth, Beauvoir's talk of alien invaders and parasites illuminates the transgression of boundaries in pregnancy as \"an intense experience of otherness within.\" Scarth further states: \"Beauvoir's language _is_ excessive, in the sense that many women probably would not recognize themselves and their experience of maternity in it; it is unfamiliar, strange, and disturbing.\"\n\nIn _The Philosophy of Simone de Beauvoir: Gendered Phenomenologies, Erotic Generosities_ , Debra Bergoffen tries to rescue Beauvoir in a similar manner. She claims that Beauvoir and Irigaray have similar ends in mind, although they go about it differently: \"Where Beauvoir turns to the model of the generous man and the generous mother and to the phenomenon of the gift to explore the possibilities of immanence, subjective ambiguity\/undecidability, and the couple, Irigaray turns to the pregnant body.\" Along with Irigaray, Bergoffen thinks that the pregnant woman is engaged in an ethical relation originating in the body's response, that the pregnant body recognizes the fetus as a \"gift of generosity, abundance and plenitude to which nothing is owed,\" and that this bodily response is also the source of the intentional generosity of the mother. \"[Beauvoir] does not seem to realize that the maternal body has already expressed [the desire for the happiness of the other].\" According to Bergoffen, Beauvoir overlooks the inherent generosity of the maternal body and thereby fails to realize the potential in her own work: \"Whether Beauvoir has been taken in by a patriarchal imagery which divests pregnancy of its activity or is overreacting to patriarchy's distorting 'religion of maternity,' the effect is the same, she misses the radical implications of her affirmations of the erotic body.\"\n\nBergoffen implies that there is a unidirectional causality between the pregnant mind and body\u2014that the pregnant body's generosity inclines the mother herself toward generosity, but that the body's response is unaffected by the mother's existent feelings. However, as I discuss in chapter 4, on Levinas, I am skeptical of the idea that maternal munificence originates in the responses of the maternal body. Too little is known about the relationship between the pregnant body and mind. It could be that the body's reaction to pregnancy is already affected by the mother's attitude toward it. For example, in \"Being Torn: Toward a Phenomenology of Unwanted Pregnancy,\" Caroline Lundquist notes that in denied pregnancies the \"normal symptoms\" of pregnancy, such as weight gain and morning sickness, are suppressed or reduced. This seems to imply that the mother's conscious acknowledgment of her pregnancy is necessary in order for it to take its typical course. In addition, denied fetuses are frequently underweight. This is an extraordinary finding since, as Lundquist also notes, the human female is unique in that when resources are limited, her body will direct them toward the fetus first. This means an unwanted fetus cannot be deliberately starved; the mother's body will be deprived first. Yet in denied pregnancy the fetus often _will_ be deprived of vital nutrients, even when there are enough for both. While no simple conclusions can be drawn from these facts, I think it is safe to say that they complicate any easy assumptions about the generosity of the maternal body.\n\nEven if reciprocity and generosity are paramount to Beauvoir's ethics, this is insufficient cause to eliminate the strong emphasis she also places on conflict. In _The Second Sex_ , for example, Beauvoir often gives voice to the importance of interpersonal struggles: \"Once the subject attempts to assert himself, the other, who limits and denies him, is nonetheless necessary for him: he attains himself only through the reality that he is not. That is why man's life is never plenitude and rest, it is lack and movement, it is _combat_.\" While initially recognizing the ambiguity of our human relations, many scholars ultimately side with only one aspect; the calling to be ethical is affirmed, but the central importance of conflict is denied. However, for Beauvoir, the reason why generosity is so laudable is not because it is necessarily common or prudent, but because it also works against one's own interests: \"To will oneself free is also to will others free. This will is not an abstract formula. It points out to each person the concrete action to be achieved. But others are separate even opposed, and the man of good will sees concrete and difficult problems arising in relations with them.\" The contradictory nature of this dilemma should not to be diminished.\n\nIt is especially troubling that Beauvoir scholars take recourse to the mother-child relationship to deny the importance of interpersonal conflict. Beauvoir's descriptions of pregnancy, childbirth, and motherhood are not a mere rhetorical strategy. On the contrary, she considers the real threat to women's freedom that childbearing and rearing presents and considers motherhood as a concrete illustration of her theory. Scarth's assertion that Beauvoir's account of maternity is excessive, that most women would not recognize themselves in it, is simply untrue. While _some_ women may not feel this way, many _do_ feel as though they are \"at war\" with their children and have used similarly strong language to describe their experiences. In attempting to make Beauvoir's characterization of motherhood more palatable, Beauvoir's supporters reject her great insights into the intersubjective and ethical lives of mothers.\n\nAs I will demonstrate in the next section, this oversight is a serious error. Beauvoir's characterization of the mother-child relationship is a vivid, and at times raw, example of her philosophy of intersubjective ambiguity. Beauvoir realizes mothers often find themselves in violent opposition to their children, whose well-being is also essential to them. Contrary to scholars such as Scarth and Bergoffen, I argue that Beauvoir offers an accurate depiction of the conflicts central to the maternal condition. Mothers who strive simultaneously for their own freedom and the liberation of their children often face tragic choices. As such, it is clearly an ethical situation, possibly even paradigmatic within an ethics of ambiguity. To minimize the importance of conflict in her account of motherhood is not just to misunderstand Beauvoir on this one count but to iron out the paradoxical nature of her ethics of ambiguity.\n\nTHE EXISTENTIAL AND ETHICAL SIGNIFICANCE OF MOTHERHOOD\n\nBeauvoir's ethics is very useful to considering the complexities of motherhood. As described in chapter 3, mothers often experience dramatic losses of self, a newfound passivity and vulnerability, and clashes between their freedom and the needs of their children. Considering Beauvoir's account of maternity in light of her philosophy, we find that she provides a means of understanding these difficulties in a positive light. Beauvoir's supposed antinatalism highlights an existentially significant paradox in motherhood\u2014motherhood can simultaneously hold a woman back due to its demands and physical impairments as well as open her to her own authentic becoming. Ultimately, the mother's ethical \"failure\"\u2014her ongoing struggle to meet the conflicting, yet interdependent, needs for care and freedom of both self and other\u2014can contribute to both their flourishing. Nevertheless there is a tragic element to ethics, that sometimes the needs of everyone cannot be met. No amount of generosity or good intention can overcome this fact. And mothers, because they are typically responsible for the life and death of their children, _especially_ cannot escape this reality.\n\nThe ambiguity of our intersubjective existence comes into sharp relief in the experiences of pregnancy and mothering. As Iris Young famously notes in \"Pregnant Embodiment,\" to share one's body is to share one's self. Many mothers and pregnant women report that their sense of self is no longer simply singular. They oscillate between defining themselves as an individual person to experiencing a sense of coconstitutionality with their child or fetus. According to Beauvoir, this ambiguity and the ambivalent feelings it gives rise to are not even escaped by the woman who has an abortion: \"This intervention she demands is one she often rejects in her own heart. She is divided inside herself. It might appear that her spontaneous desire is to keep this child whose birth she is preventing; even if she does not positively want this motherhood, she feels ill at ease with the ambiguity of the act she is about to perform. . . . In addition to women who think they tried to kill a living thing, there are women many who feel they have mutilated a part of themselves.\" In fact, women's experiences of abortions vary, and this is no doubt deeply influenced by their situation. Nevertheless, the fact that many women experience their abortions with these mixed feelings indicates that it is not easy to escape a sense of coconstitutionality with one's fetus or child.\n\nIn _The Second Sex_ , speaking against a tradition that romanticizes the mother-child relationship, Beauvoir emphasizes the negative aspects of this ambiguity. She describes motherhood as immersing women into immanence and characterizes the fetus as a hostile invader who threatens the mother's life and freedom. She claims that the nausea, dizziness, diarrhea, constipation, and other common problems associated with pregnancy are largely psychosomatic\u2014expressing the conflict between woman's own interests and those of the fetus. She writes:\n\nBut pregnancy is above all a drama playing itself out in the woman between her and herself. She experiences it both as an enrichment and a mutilation; the fetus is part of her body, and it is a parasite exploiting her; she possesses it, and she is possessed by it; it encapsulates the whole future, and in carrying it, she feels as vast as the world; but this very richness annihilates her, she has the impression of not being anything else. A new existence is going to manifest itself and justify her own existence, she is proud of it; but she also feels like the plaything of obscure forces, she is tossed about, assaulted. What is unique about the pregnant woman is that at the very moment her body transcends itself, it is grasped as immanent: it withdraws into itself in nausea and discomfort; it no longer exists for itself alone and then becomes bigger than it has ever been. The transcendence of an artisan or a man of action is driven by subjectivity, but for the future mother the opposition between subject and object disappears; she and this child who swells in her form an ambivalent couple that life submerges; snared by nature, she is plant and animal, a collection of colloids, an incubator, an egg.\n\nAccording to Beauvoir, motherhood, and especially pregnancy, sap a woman's energy, often forcing her to give up her independent projects in order to serve another's needs. The mother feels the tremendous weight of responsibility for another, the force of another making itself through her in ways she cannot control, and alienation from her body and former life. In pregnancy the mother and child are immersed together in processes of nature, and the woman does not necessarily welcome her newfound vulnerability. Many of the ways in which the child is made, and the mother remade, cannot be willed. This is not to say that they are merely passive instruments of nature, but that the pains, illnesses, and dangers of pregnancy, plus the surprising changes in her body, make a new dimension of the passivity that we all experience especially evident. Although some find Beauvoir's views to be harsh, they do resonate with other women's accounts, as I explain in chapter 3. Beauvoir understands a pregnant woman or mother's negative feelings toward her fetus or child is not merely the emotional reaction of a woman under increased pressures and strains; it is an insightful comprehension of her existential condition.\n\nNevertheless, in an interview with Margaret Simons, when asked if she thought motherhood was a negative activity, Beauvoir stated:\n\nNo, I didn't say that exactly. I said that there could be a human relation, even a completely interesting and privileged relation between mother and child but that, in many cases, it was on the order of narcissism or tyranny or something like that. But I didn't say that motherhood in itself was always something to be condemned, no, I didn't say that. No, something that has dangers, but obviously, any human adventure has its dangers, such as love or anything. I didn't say that motherhood was something negative.\n\nIt is true that Beauvoir did not fully denigrate maternity. In this passage family life is described as a possible venue for transcendence: \"[Some people] have been able to realize their freedom; they have given it a content without disavowing it. They have engaged themselves, without losing themselves, in political action, in intellectual or artistic research, in family or social life.\" For Beauvoir, it is not the specific content of one's projects that mark them as works of freedom, but the manner of going about them. Even those who undertake great political or artistic projects may be surrendering to a false seriousness and immanence if they consider the values toward which they work to be absolute: \"By virtue of the fact that he refuses to recognize that he is freely establishing the value of the end he sets up, the serious man makes himself the slave of that end. He forgets that every goal is at the same time a point of departure and that human freedom is the ultimate, the unique end which man should destine himself.\" If one works toward the \"unique end\" of freedom, regardless of the setting, one affirms transcendence.\n\nIn fact, at times Beauvoir seems positively optimistic about the prospects of motherhood. In _The Second Sex_ she says: \"These examples all prove that there is no such thing as maternal 'instinct': the word does not in any case apply to the human species. The mother's attitude is defined by her total situation and by the way she accepts it. It is, as we have seen, extremely variable. But the fact is that if circumstances are not positively unfavorable, the mother will find herself enriched by a child.\" Yet it is important to note Beauvoir's insistence that a mother's attitude is a response to her _situation_. This can account, in part, for some of her negative evaluations of motherhood's impact on women. Throughout _The Second Sex_ , Beauvoir places great emphasis on the oppressive circumstances in which women find themselves. For example, Beauvoir discusses the hypocrisy around abortion in France in the 1940s. The subject was taboo, even though, she claims, \"in France every year there are as many abortions as births.\" It was made illegal because it was considered dangerous, though when conducted by a knowledgeable specialist it was much safer than back-alley abortions. In addition, Beauvoir notes that \"the same society so determined to defend the rights of the fetus shows no interest in children after they are born.\" Once born, children were considered the sole responsibility and property of their parents, regardless of the abuses and neglect suffered as a result.\n\nThe situation of mothers today remains very difficult to navigate. According to Sharon Lerner's _The War on Moms: On Life in a Family-Unfriendly nation_ , 16 percent of families with the lowest incomes get less than two weeks paid vacation and sick leave, and only 59 percent of those in the highest income bracket get that much time paid time off; 71 percent of the poorest working adults cannot take off sick leave for sick children; 34 percent of those in the highest income bracket have the same problem. Families below the poverty line spend a quarter of their income on childcare. And the birth of a baby is the primary cause of a poverty spell for American families.\n\nConditions such as these place women in the position of making impossible decisions. Consider one example described by Lerner. Gina St. Aubin is a married mother of three and formerly worked as a victim's advocate. After her oldest son was diagnosed with cerebral palsy, childcare became an issue. Her first sitter quit and the second was fired after admitting to frequently leaving the baby out of earshot when he cried. A qualified caregiver would cost as much as her own salary, so St Aubin quit her job. The increasing costs of medical care, gas, food, and clothing caused further financial strain. Lerner writes:\n\nGina started to panic. She had already cut back on almost everything she could. The family dressed in secondhand clothes. They hadn't renewed their zoo pass and didn't buy any birthday presents or cards. Rather than making a plate for herself at mealtimes and risking the possibility of wasting food, she usually ate the children's leftovers. [Her son] even stopped participating in the Special Olympics because St. Aubin felt they couldn't afford his entry fee. Finally she decided to cut back on her son's therapies, several of which were not covered by insurance. Although she fears that his prospects will dwindle as a result, it had come to those out-of-pocket costs or the mortgage. . . . Facing the bleakest economy in decades, many families are grappling with similarly unimaginable choices: childcare or food? Tuition or nursing home payment? Rent or medicine? The collapse of the housing market, combined with widespread job loss and the skyrocketing costs of food, gas, and other such essentials, has made the reality of many American families that much harsher.\n\nFacing choices as stark as these, a woman's concerns for her own flourishing are pushed beyond the back burner right off the stove entirely.\n\nBeauvoir asserts that the mother's own well-being is, in fact, important, but, insofar as her welfare is intertwined with her children's, discerning _the good_ requires a complex calculus. Moreover, sometimes the decision regards not only the good of the mother versus the good of the child but between multiple children. This was the case for one mother featured on MTV's recent documentary _No Easy Decision_. Markai, the teenage mother of an infant under one, finds herself pregnant after misunderstanding the proper use of her birth control. Believing that she could not follow through with giving her child up for adoption, she faces the decision of having an abortion or raising the new baby. Markai and the baby's father, James, have just reached a level of financial stability that allows them to have their own apartment. Having both suffered poverty while growing up, their key concern is that, if they keep the second pregnancy, the daughter they already have will suffer. They decide that having an abortion is the responsible solution. However, this decision is made with regret. The day after the abortion they struggle with what to call the terminated pregnancy. They find it upsetting to call it a baby, but the solution offered by the counselor at the abortion clinic\u2014calling it \"a little bunch of cells\"\u2014feels painfully inadequate. Pointing tearfully to their daughter, Markai says \"Nothing but a bunch of cells can turn out just like her.\" This example shows that the good of mother, child, and the rest of the family cannot be divorced from one another. What would be the ethical failure here? To keep the pregnancy or to abort it? In either case, the family suffers. Both Markai and Gina St. Aubin confront the tragic face of ethical failure. They recognize that, whatever their decision, someone _will_ suffer. They are left with figuring out who will be neglected and how much, facing their inability to be infinitely responsible for others.\n\nAccording to Levinas' philosophy, an abortion could never be ethically justified, because one's responsibility for the other is infinite. But infinite responsibility is an unsupportable fantasy. To deny that, in some cases, someone's well-being _will_ be sacrificed is just bad faith. It denies both the realities of human limitation _and_ freedom. As the existentialism truism affirms: choosing not to choose is still a choice. In Markai's case, choosing not to choose would mean opting for a financially intolerable situation. In Gina's situation it would mean losing her house. In fact, in order to care for their children, mothers _must_ recognize their limitations. This is why it is not paradoxical for a woman to claim (as many do) that having an abortion is a _mothering_ decision, not shirking their responsibility. In chapter 3 we saw that Judith Arcana argues for this view in her essay \"Abortion Is a Motherhood Issue.\" This perspective is also affirmed in _No Easy Decision_. Following the documentary portion of the show, there is an interview with Markai and two other young women (who do not have children) who have had abortions. They each claim that having an abortion is acknowledging, not evading, their responsibilities. One of the women states: \"The decision to terminate my pregnancy was a parenting decision. It was a parenting choice and I feel like that dialogue is not out there at all. People assume that if you have an abortion it's because you are denying the fact that you are a parent, but it's not. It's not at all.\" These women agreed that in their circumstances, having an abortion was the most responsible option. Beauvoir agrees that making an ethical choice sometimes, unfortunately, involves leaving one person's needs aside. To deny this fact, as Levinas does, is to evade the true gravity of our choices.\n\nIf Beauvoir seems ambivalent about motherhood's potential, I maintain that this is the appropriate attitude, given its potentials to initiate both growth and devastation. Beauvoir is apparently aware of the contradictions of her own position: \"As the meaning of pregnancy is thus ambiguous, it is natural for the woman's attitude to be ambivalent as well\"; \"pregnancy and motherhood are experienced in different ways depending on whether they take place in revolt, resignation, satisfaction or enthusiasm.\" It is not that some mothers are satisfied and some are resigned, that some revolt and some are enthusiastic. As we read on, we find that Beauvoir appreciates that an individual woman may experience motherhood in most or all of these manners. She may feel fulfilled at times and revolt at others:\n\nOne must keep in mind that the decisions and feelings the young mother expresses do not always correspond to her deep desires. An unwed mother can be overwhelmed in material terms by the burden suddenly imposed on her, be openly distressed by it, and yet find in the child the satisfaction of secretly harbored dreams; inversely, a young married woman who joyfully and proudly welcomes her pregnancy can fear it in silence, hate it with obsessions, fantasies, and infantile memories that she refuses to recognize.\n\nThe mother expresses her ambivalence in contradictory fantasies about the child's future. He may be a hero, a monster, or a little bit of each.\n\nFor mothers, as for anyone, Beauvoir arguees that the recognition of her ambiguous condition is the measure of her authenticity. In parenting, the alternatives to this honest assessment are inauthentic denial of one's own independent needs or an oppressive attitude toward one's child. _Devotion_ is one form of this self-delusion and seriousness that Beauvoir finds especially odious:\n\nLet's suppose the other needed me and that his existence had an absolute value. Then my being is justified since I am for a being whose existence is justified. I am released from both the risk and the anguish. By positing an absolute end before me, I have abdicated my freedom; questions are no longer posed; I no longer want to be anything but a response to that appeal which requires me. The master is hungry and thirsty; the devoted slave desires only to be the dish that he prepares and the glass of water that he brings to appease the hunger and thirst; he makes himself into a docile instrument. . . . Many men, and even more women, wish for such a rest: let us devote ourselves.\n\nThe utter dependency of children combined with the cultural belief that women are naturally responsible for them conspire to yoke mothers to their children. Indeed, maternal devotion is the most culturally glorified escape from freedom for women. It provides a false release from the risks of having one's own projects and facing their potential failure and futility. For a mother to surrender her transcendence to the care of a child is immoral, in Beauvoir's terms, even if the mother thinks she is doing it for the good of her child. Taking up an attitude of devotion exacerbates mothers' resentment toward their children, clouding their ability to parent effectively. Recall from chapter 3 that Freud proposed that an excess of devotion was actually repressed ambivalence:\n\nEvery psychoanalyst knows how infallibly this anxious excess of tenderness can be resolved even under the most improbable circumstances, as for instance, when it appears between mother and child, or in the case of affectionate married people. Applied to the treatment of privileged persons this theory of an ambivalent feeling would reveal that their veneration, their very deification, is opposed in the unconscious by an intense hostile tendency, so that, as we had expected, the situation of an ambivalent feeling is here realized.\n\nFreud believes that mothers who display a disproportionate amount of fondness for their children do so as a reaction formation to conceal their hatred. Repressed hostility will still find its expression, even if the person who experiences it does not recognize it as such. Beauvoir noticed the conjoined attitudes of self-abnegation and bitterness in her own mother, the only cure for which was being on her deathbed. Beauvoir writes: \"No question of renunciation or sacrifice any more: her first duty was to get better and so to look after herself; giving herself up to her own wishes and her own pleasures with no holding back, she was at last freed from resentment.\"\n\nOn this point as well, Beauvoir's attitude is rightly in opposition to Levinas, who thinks that bowing to the other is the solution to ethical conflicts. Levinas proposes that the other should always win out, no matter the cost to oneself, but, as Beauvoir points out, there is a moral cost to this solution. Even when a mother loves her child as deeply as Levinas predicts, this does not mediate the validity of conflicting interests. Mothers do have a more or less individuated sense of self that existed before the birth of the child and that exists not only to serve but also to flourish in and of itself. The person who is denied such opportunities becomes unable to help others thrive. Beauvoir safeguards the importance of caring for both the self and the other and shows how complex this effort really is. When the needs of self and other are intertwined as well as in conflict, to propose Levinas's solution\u2014infinite responsibility for the other\u2014is overly simplistic.\n\nBeauvoir categorically refuses the idealization of maternal self-immolation, revealing its sinister potential. This argument is illustrated in \"Throwing the Book,\" Sara Levine's essay about her mother-in-law's devotion to her son. As Beauvoir predicts, the devoted mother clings to her child's dependence as the source of her own meaning. As a result, she cannot see her son beyond his babyhood: \"The woman holds a storehouse of anecdotes about the man I love, anecdotes I longed to hear until I realized all the stories took place before he turned three. Not a single nickname has been allowed to grow stale, though on my first visit to the Cotwold cottage, I observed that 'our best boy,' our 'Small,' routinely bumped his head on the bedroom ceiling, having grown to six-foot-two.\" In spite of having moved several times, Levine's mother-in-law kept her son's baby dishes in the cabinet and his first Wellies on the doormat. Levine describes a turn in her husband's baby book in which her mother-in-law begins to write from the baby's point of view. \"Right after she acknowledges the challenges of raising a baby, almost as if it were too difficult to negotiate the tensions between _I_ and _he_ , she throws over the vexed I and begins to write the book as if she were the baby herself. . . . Reading this was like watching a woman get swallowed alive.\" Later in the essay Levine discusses the oppressive tone of the baby book received as a gift for her own daughter. It encouraged her to give easy meaning to the \"welter of feelings and sensations\" she experienced, from her \"big honking mother love\" to \"the very information (ambivalence, conflict, the seeds of a storyteller's art) that, my mother-in-law instinctively understood, a baby book is obliged to purge.\" Levine finds that even a seemingly harmless artifact, such as a baby book, encourages women to repress the realities of motherhood and deny their own perspective. \"Unfortunately, as I saw my handwriting loop across that paper, a part of me that I liked\u2014the independent thinker part\u2014seemed to sicken and die.\" Mothers are surrounded by propaganda\u2014from diaper commercials to childcare manuals\u2014that encourages them to focus on trivialities and vague sentimentalities, to cling to their children's dependence as their source of meaning, and ultimately to lose themselves in maternal devotion.\n\nBeauvoir claims that maternal devotion does not end well for mother _or_ child. The mother's surrender and despotism go hand in hand: \"many so called devotions . . . are in truth tyrannies.\" If the mother loses her sense of herself as an individual, then she will develop a need for the child to depend on her in order to maintain her identity. To Beauvoir, the self-sacrificing mother becomes like a nurse who resents the patient that gets well. She may work to keep the child dependent; denying him what is needed to realize his potential and to respond to a future that remains unknown. \"And in practice raising a child as one cultivates a plant which one does not consult about its needs is very different from considering it as a freedom to whom the future must be opened. Thus we can set up point number one: the good of an individual or group of individuals requires that it be taken as an absolute end of our action; but we are not authorized to decide upon this end _a priori_.\" A plant strives toward light and water, and the way a particular plant will strive can be known in advance, but a child's ends cannot be known in advance. \"The mother can have _her_ reasons for wanting _a_ child, but she cannot give to _this_ other\u2014who tomorrow is going to be\u2014his own _raisons d_ ' _\u00eatre_ ; she engenders him in the generality of his body, not in the specificity of his existence.\"\n\nOn this topic, her views seem to be in agreement with Sartre's _Notebooks for an Ethics_. He states that, in order to be an authentic parent, one must \"consider . . . everyday emancipation as the real end . . . as freedom, [the child] ought to be recognized for himself.\" The authentic parent gives the child a concrete sense of her freedom and eventually makes herself, as caretaker, obsolete. This does not mean that the authentic parent overlooks the child's need for protection and nurturance. In \"Sartre, Embodied Minds, Authenticity, and Childhood,\" Adrian Mirvish provides a helpful illustration of how parents negotiate between protecting the child in her fragility and encouraging the free pursuit of her own projects. When the parent teaches the child to ride a bicycle, at first he hangs onto it while she gets the feeling of it. Eventually, however, when the child seems ready, the parent must let go. As Beauvoir insists, a parent cannot control who a child becomes: \"To bring a child into the world is not to found him.\" \"I bring a child into the world; if he becomes a criminal, I am not an evildoer.\" Though one may be deeply invested in another's freedom and well-being, another person's life cannot justifiably be one's own project.\n\nIdeally, the mother gives the child a concrete sense of her freedom. As a healthy child develops, every milestone achieved is a step toward a more independent existence. According to Beauvoir, one can only be a point of departure for another, a platform for her transcendence. \"I never create anything for the other except points of departure. . . . I am not the one who founds the other; I am only the instrument upon which the other founds himself. He alone makes himself be by transcending my gifts.\" \"And that is exactly what makes for the touching character of maternal love, properly understood.\" This moving maternal love requires the maintenance of the mother's own individuality, provision for her needs, and meaningful work beyond the care of her children:\n\nIn a properly organized society where the child would in great part be taken care of by the group, where the mother would be cared for and helped, motherhood would absolutely not be incompatible with women's work. . . . It is the woman who has the richest personal life who will give the most to her child and who will ask for the least, she who acquires real human values through effort and struggle will be the most fit to bring up children. If too often today a woman has a hard time reconciling the interests of her children with a profession that demands long hours away from home and all her strength, it is because, on the one hand, woman's work is still too often a kind of slavery; on the other hand, no effort has been made to ensure children's health, care, and education outside the home. This is social neglect: but it is solipsism to justify it by pretending that a law was written in heaven or in the bowels of the earth that requires that the mother and child belong to each other exclusively; this mutual belonging in reality only constitutes a double and harmful oppression.\n\nBeauvoir argues that given proper support women can have both satisfying family life and careers, but the social neglect of mothers continues. For example, in the United States, women do twice as much housework as men, many women are forced by their employers to choose between working more than full time and quitting their jobs entirely, and single women with children earn fifty-six cents of every dollar that married men make. Only one in seven of the fifteen million children who qualified for federally funded subsidized childcare in 2000 received it. Women who are attempting to do it all, in spite of these conditions, are simply worn out. There is what Lerner calls \"an epidemic of exhaustion\" among mothers \"sweeping through cubicle, cluttered kitchens, and child-care centers around the country. Call these women the maxed-out generation, if you like; victims of a family-unfriendly nation; or simply 'hosed.'\" For most mothers, a \"rich personal life\" is but a flight of the imagination. Thus, if Beauvoir infrequently addresses the elevating potential of motherhood, this is in keeping with her emphasis on the importance of social, political, and material context. She rightly points out that, with conditions as they are, having children threatens a woman's well-being in all respects.\n\nBeauvoir's emphasis on a _situated_ ethics is a critical improvement, especially over Levinas. As mentioned in chapter 5, Merleau-Ponty briefly notes the importance of a mother's context in her responses to her child. For Levinas, however, one's responsibility for the other is infinite regardless of social location. In contrast, Beauvoir considers our ethical responses as they are positioned in larger circumstances, especially with regard to maternal responsibility: \"The mother's relation with her children is defined within the overall context of her life; it depends on her relations with her husband, her past, her occupations, herself; it is a fatal and absurd error to claim to see a child as a panacea. . . . The young woman must be in a psychological, moral, and material situation that allows her to bear the responsibility; if not, the consequences will be disastrous.\" Given our interdependence, Beauvoir articulates how the burden of ethical life should not rest solely on the shoulders of the individual. As she demonstrates, the inherently conflicted nature of the mother-child relationship is exacerbated by insufficient social support. Given the contemporary constraints faced by mothers, they need a way to understand the feelings and impulses they are most ashamed of beyond the discourse of individual moral failure or success.\n\nThe double binds of mothering can lead to homicidal and\/or suicidal depression and acts of violent retribution or neglect. Maternal aggression, a phenomenon that is much more common than recognized, is often a sign that the mother is in need of physical and psychological distance. The desire to pull back, to reclaim one's bodily boundaries, can be an appropriate and productive move toward separation on the mother's part. Her need for independence guards against a potentially consuming intimacy. As Adrienne Rich writes, mothers find the need to \"struggle from that one-to-one intensity into new realization, or reaffirmation, of her being-unto-herself. . . . The mother has to wean herself from the infant and the infant from herself. In psychologies of child-rearing the emphasis is placed on 'letting the child go' for the child's sake. But the mother needs to let it go as much or more for her own . . . it is not enough to let our children go; we need selves of our own to return to.\"\n\nBeauvoir rightly thinks that whether or not a mother is ethical in the existential sense depends both on her situation and how she takes it up. In a society that alternately romanticizes and condemns mothers, it is a brave overcoming of cultural denial to recognize the duality of maternal experience. For this reason, maternal ambivalence, though painful, is a psychological and ethical _achievement_. As Rozsika Parker writes: \"the mother's achievement of ambivalence\u2014the awareness of her co-existing love and hate for the [child]\u2014can promote a sense of concern and responsibility towards, and differentiation of self from, the [child]. Maternal ambivalence signifies the mother's capacity to know herself and to tolerate traits in herself she may consider less than admirable\u2014and to hold a more complete image of her baby. Accordingly, idealization and\/or denigration of self and, by extension, her baby, diminish.\"\n\nThe potential of motherhood to enrich comes from the fact that others can keep us from \"hardening in the absurdity of facticity.\" A mother's identity may have been based primarily on her own achievements, actions, and qualities before she had children, but, to the extent that she cares for her child's well-being, this is no longer true. For a loving mother, her sense of self, groundedness, and well-being are, at least in part, dependent on meeting the needs of her child. In such cases, one who obstructs her self-determination is also integral to her. Though one is never fully transparent to oneself, self-alienation is dramatic when one is in symbiotic relation to one's saboteur. Beauvoir believes in the transformative power of disturbances to one's self-concept. And, indeed, there is an abundance of recent motherhood memoirs describing how the experience of motherhood changed a woman for the better. In a recent review of a number of these autobiographies, Kristin Kovacic writes: \"In each book, it takes a lot of time, a lot of therapy, and a baby, to get to full maturity.\" Mothering inspires many women to take charge of their own lives for the first time. In two examples, _From Beer to Maternity_ by Maggie Lamond Simone and _Lit_ by May Karr, motherhood helps the mother, a chronic drunk, to get sober. But as Kovacic notes, it is not mothering alone that does the trick. Motherhood inspires these women to find meaningful projects of their own.\n\nThe other invites one to see oneself from an outside perspective, from the point of view of _their_ needs and desires. Mothers inhabit a powerful tension between their self-identities and the force of the child's and society's views of her. At many turns women are invited to instrumentalize themselves for the care of others. Should she resist that pull, she will acquire an identity that exists in authentic ambiguity. Where women expect to have an independent identity and yet, as mothers, are fundamentally intertwined with their children, there may be no way to avoid feeling torn.\n\n\"WITHOUT FAILURE, NO ETHICS\"\n\n\u2014SIMONE DE BEAUVOIR, _THE ETHICS OF AMBIGUITY_\n\nFor Beauvoir, a successful ethical life means that one strives toward one's own transcendence and contributes to the opportunities of others. She believes that we are meant to reach beyond our limitations and thus we will often fall short of what we want to be. We require the liberation of others to succeed, but this is far from guaranteed. Caring for other people will often drain us of the energy to pursue our own projects, and when we do manage to help them we may fail to fully liberate them. Those who have opportunities for free action will often not support my own projects, but rather assert their values and desires in opposition to mine. Likewise, since everyone's interests are not the same, those projects of mine that _do_ reach fruition are likely to impede some people, even if they aid others. To top it off, these conflicts with others are both fundamental to and frustrate my own freedom. As Beauvoir sees it, failure is entwined in the braid of ethical life.\n\nBeauvoir makes clear the importance of acknowledging our limitations in \"Moral Idealism and Political Realism.\" In this essay she describes the problems with political action, which she conceives as one kind of transcendent action. Political action is tricky because in changing the world to benefit some, others are likely to get hurt. The moral idealist and political realist wish to deny the significance of these facts; both make an absolute object of their ends, which allows them to deny the sacrifices of their causes. This recognition of the possibility for misjudgment in one's ethical and political life is paramount. We must acknowledge our actions entail risks that no absolute authority can safely justify: \"This means that the political man cannot avoid making decisions or choices; things will not give him ready-made answers, neither on the level of being nor on the level of values. In each new situation he must question himself anew about his ends, and he must choose and justify them without assistance. But it is precisely in this free engagement that morality resides.\" Beauvoir requires that we continue to question the validity of our projects and the costs of achieving them. The likelihood of ethical failure does not mean that our existence is absurd and meaningless, but it does indicate that life's significance is not pregiven. \"Ethics is not the ensemble of constituted values and principles; it is the constituting movement through which values and principles are posited; it is the movement that an authentically moral man must reproduce himself.\" The meaning of life must be striven toward and asserted, but also interrogated and justified. To this end, a certain amount of self-doubt is essential.\n\nThe fact that we do not achieve moral perfection means that we must continue to strive, and for an existential ethics this is certainly a good thing. Beauvoir rejoices at the tension between failure and success in \"Introduction to an Ethics of Ambiguity\" as bringing one in touch with a central truth of the human experience. She merrily declares, \"I delight in this very effort toward an impossible possession.\" In this case she focuses on the impossibility of complete self-determination as \"delicious torment, cruel happiness, blessed torture.\" Nevertheless, a bit of self-deception accompanies her happiness in this early work. As she later finds, the disappointment of failing oneself and others cannot be salved. She writes more intimately of the failure to be completely there for a beloved other in _A Very Easy Death_ \u2014the chronicle of her mother's last days. In this work the nettles sting cruelly, not sweetly. She writes: \"When someone you love dies you pay for the sin of outliving her with a thousand piercing regrets . . . you feel that she should have had more room in your life\u2014all the room, if need be. You snatch yourself away from this wildness: she was only one among many. But since you never do all you might for anyone\u2014not even within the arguable limits that you have set yourself\u2014you have plenty of room left for self-reproach.\"\n\nOur moral obligations to others are weighty, sometimes to the point of becoming oppressive. For numerous reasons this is even more strongly the case in a mother's relationship to her children\u2014the economic situation of mothers is often so bleak, their responsibility is nearly complete (as the obligation to care for children is insufficiently shared), and the consequences of their actions can make the difference between the life and death of another person. The interconnection between the interests of the self and other is typically a core feature of the mother-child relationship. Yet, because the neediness of children can be so overwhelming, conflicts of interest are even more prominent and inevitable than in any other kind of relationship. Thus the possibility of grave moral failure is a real and terrifying prospect to mothers. The unique features of this relationship mean opportunities for regret are perhaps more numerous than in any other kind. She finds herself continually under question. Perhaps this is why Adrienne Rich once said that \"guilt, guilt, guilt\" is the mother's real G-spot.\n\nThe denial of pre-established ideals requires that one attend to the particular situation. For example, some think that a mother should always attend her child's needs before her own, but life is rarely that simple. Many women who think that they would never have an abortion find themselves in circumstances in which they can not sustain a child's life or provide for her flourishing. Sometimes it is necessary to secure one's own oxygen mask before others. And sometimes \"saving\" one child might mean aborting the life of another. Assuming we can know the right thing to do in advance is a denial of the unpredictable nature of the future and the needs and desires of other individuals. That is why parents often find unsolicited childcare advice from friends, relatives, and strangers to be so obnoxious. What works for one baby and family does not always work for another. When left to cry himself to sleep, one baby learns to self-soothe while another cries until he vomits. For this reason, Beauvoir refuses to give her readers advice: \"It will be said that these considerations remain quite abstract. What must be done practically? Which action is good? Which is bad? To ask such a question is also to fall into na\u00efve abstraction. We don't ask the physicist, 'Which hypotheses are true?' Nor the artist, 'By what procedures does one produce a work whose beauty is guaranteed?' Ethics does not furnish recipes any more than do science and art. One can merely propose methods.\"\n\nBeauvoir helps to make clear that even the most generous mother will rightfully feel a conflict between her child's needs and desires and her own. Since there is no guarantee that children will recognize their mothers as independent of them and having needs of their own, it may be necessary and appropriate for the mother to assert her interests against those of the child. While a sense of _mutuality_ may motivate the desire to nurture, _dispute_ forces recognition of interpersonal divergence. One's child, no matter how beloved, is also radically Other, with discrepant goals, experiences, and sometimes even values. A mother's self-coincidence is unsettled by unclear boundaries between herself and her child. Yet the inability to gain full self-certainty is a _good_ failure insofar as it opens a gap for ethical self-questioning. Phenomenologically the conflict between mother and child is there. Recognizing this clash is the beginning of negotiating _mutual_ transcendence\u2014the only kind of transcendence that is really possible.\nCONCLUSION\n\n_The Stranger of My Flesh\u2014an Existential Phenomenological Ethics_\n\nAn existential-phenomenological ethics is written from the perspective of an incarnate consciousness that is born in proximity to others _and_ radical distinction from them. From this point of view, we find that intersubjective existence is a living contradiction. Our connections to others are profound and visceral; we share intimate space, intersect in embodiment, and coestablish the world's meaning, dimensions, and veracity. Our freedom and our life's unique meaning are dependent on our responsiveness to others. We need each other's generosity and collaboration; we are their facticity, and they are ours. Nevertheless, we suffer the abyss of our divergent bodies and perspectives. Although our bodies overlap and interpenetrate, we still remain within our own skin. Even though the other is integral to who I am, she also exceeds my comprehension. We can be drawn into the outlook of another, but we are never in her place. Her alterity is insurmountable. Even the child born of one's own body is estranged flesh.\n\nThe other threatens any enjoyment of self-sameness that I might experience; she undermines the complacent familiarity I may temporarily have with myself. And it is this interruption that is the opening for ethics. It is motivated by the needs of others, by their vulnerability, and thus it originates with them, not me. Since I am entangled with others before I have the opportunity to will or deny it, I am drawn to care for them before I can consider whether or not it is in my own interests. In this respect my ethical involvement is passive. Yet full responsivity also requires an active concern and effort. This becomes apparent because the neediness of others can also invite aggression or neglect. The phenomenon of maternal ambivalence vividly demonstrates that the desire to commit violence can occur even when there is a feeling of sympathy and connection with the other. In fact, intense intimacy can lead to feelings of claustrophobia and an oppressive level of responsibility and thus _instigate_ hostility. Acting on impulse alone can harm as well as hurt. Passive responsiveness is insufficient; intelligence, strategy, and resourcefulness are needed to navigate these difficulties.\n\nIn all cases there are practical limitations on what one can do for another. Caring for others threatens one's own well-being, even as it is absolutely integral to it, because their needs and desires sometimes clash with one's own. There is no easy way out of this problem; devotion, detachment, and self-centeredness are all flawed orientations. The ambiguous nature of our intersubjective life undermines the purity of either egoism or altruism. Such an either\/or attitude disregards the ways in which we are integral to one another\u2014that one's own well-being and freedom depends on that of others. Yet the preservation of independence is also integral to ethical responsiveness.\n\nConducting a phenomenology of ethics, I aim to describe ethical life as it is experienced. Ethics involves an orientation toward others and oneself, and, with its careful attention to lived experience, phenomenology can reveal the structure of these relations. Yet this orientation cannot simply be assumed; there are many obstacles to fulfilling one's duties to oneself and others. To be ethical is something toward which we must strive. And we may often find that ethical success and failure are intertwined.\n\nLike everyone else, mothers are capable of the basest crimes\u2014murder, rape, criminal neglect, and so on. Yet when a mother commits such crimes against her child our horror is exponentially greater. It is not just anyone who is expected to care for a child, to be the person who makes their basic trust in the world possible. Of course, other people can care for a child and help that child to thrive, but the fact of our current situation is that, if a mother is ignorant, insane, overwhelmed, indifferent, or malicious, that child's chances of flourishing (or even surviving) are dramatically reduced. Ultimately, it should come as no surprise that we romanticize this relationship; we do not want to dwell on the gruesome acts committed against the most vulnerable and innocent. Nevertheless, filicide may simply be one extreme result of a fairly normal state.\n\nConflicts of interest are an unavoidable fact of human existence exacerbated by living in a society that privileges some and exploits others. Thus an active ethical response requires the deliberation of someone who can, with limited resources, balance other-care with self-care. Not every mother is capable of this. And, even in the best of circumstances, and in spite of our best efforts, the needs of everyone cannot always be met; every life may not be sustainable. Love, even maternal love, does not conquer all. But this failure should invite both outrage and action. We must struggle against inequalities and fight for the liberation and care of every human being for our own sake as well as theirs. Of course, it is beyond the ability of a single individual to redress this alone; individual moral responsibility can only be _part_ of the solution. The fight for justice must be a collective human effort.\n\nConsider the case of Angela McAnulty. On February 24, 2011, McAnulty, a mother from Eugene, Oregon was sentenced to death by a jury for murdering her fifteen-year-old daughter, Jeanette Maples. McAnulty admitted to having tortured the girl for several years. Testimony revealed that the mother denied her daughter food (placing locks on the cupboards) and water (so that she had to drink water from the toilet in order to survive). The girl had been starved to such an extent that she weighed only fifty pounds when she collapsed from cardiac arrest. Her brain was bleeding from massive head injuries, she had flesh torn from her body with belts and sticks, and suffered hundreds of other injuries, some of them infected all the way to the bone. According to McAnulty's husband, the mother would turn up the television in the living room or the vacuum cleaner to cover the sounds of her daughter's screaming and then she would strip the girl naked and whip her. In his opening statement, Lane County Deputy District Attorney Erik Hasselman stated: \"The one woman who could have saved her was the very person who was responsible for her torture, her mother.\"\n\nClearly, in the eyes of the judge and jury, McAnulty did something unforgivable and irredeemable; she is the first woman to be sentenced to execution since the death penalty was reinstated in Oregon in 1984 and she will be only the second woman to be executed by the state in history. McAnulty's abuse of her daughter is undeniably horrific, and she should be held responsible for what she did. However, when we remove the assumption that mothers _alone_ are responsible for the well-being of their children, we must challenge the prosecutor's argument that only her mother could have saved her. Many other people could have saved Jeanette Maples: her father, her stepfather, her step-grandmother, state officials in Oregon and California, teachers and peers at Jeanette's school. Every one of these people had some indication that the girl could be in trouble. Anthony Maples, Jeanette's father, hadn't seen or talked to his daughter in nearly a decade. McAnulty and Maples's two sons had been removed from their care and placed in foster care by the state of California, yet Jeanette was left with McAnulty. Lynn McAnulty, the girl's step-grandmother, said that she anonymously called Oregon state child welfare officials several times because she suspected that the girl was being abused. The _Seattle Weekly_ reported that \"friends and administrators at Cascade Middle School in Eugene remember seeing Maples coming to school in torn clothes\u2014sometimes they'd catch glimpses of terrible bruises.\" Worst of all, Richard McAnulty, the condemned woman's husband, was fully aware that Jeanette was being severely beaten and starved. He also pleaded guilty to murder by abuse for his role in the girl's death, but was sentenced to twenty-five years' imprisonment (instead of receiving the death penalty), though the prosecutor stated that Mr. McAnulty was just as responsible for the girl's death as her mother. All these people could have done more to help Jeanette. Indeed, it takes a village to raise a child, but it also takes a village to allow one to be so severely abused. To merely blame the mother, to pass her off as pathological, is grossly insufficient. It is not enough to ask the question \"Why did she do it?\" We must also ask: how did _we_ let this happen in our community (this community of Eugene, Oregon where I myself was a graduate student at the time)? What might have been done to McAnulty herself that might have contributed to the rage she vented on her daughter? As Meyer and Oberman state: \"these cases often leave one with the sense that there is blood on more than one pair of hands.\"\n\nWe need to take collective responsibility for the lives of children, not rely on the resources of a single individual, often largely on her own, because our cultural _myths_ say that she is naturally suited to the job. It is urgent that we overcome our denial and realize that the impulses moving women to kill their children are surprisingly commonplace. Our society constantly proclaims the importance of caring for children, and yet we can find children without food, proper clothing, education, health care, and basic safety everywhere. We need to channel our outrage and blame into something more productive, into finding, protecting, and caring for those children who are, right now, suffering.\n\nPhilosophers can make a unique contribution to clarifying the complexities of the maternal situation, but in order to do so they must listen carefully to the experiences of mothers to understand their material and social conditions. Generally speaking, the philosophical canon is guilty of either ignoring or mistreating mothers. If philosophy is to have concrete relevance, then it must go beyond the metaphorical and romanticized perspective. The metaphors we use are not neutral; they advocate an interpretation of the phenomenon in question. Thus it is only right to check these metaphorical understandings against the lives they intend to invoke. When we take seriously the true complexity of motherhood, we find that the mother-child relationship is philosophically rich indeed.\n\nAt the beginning of this book I said that I would demonstrate that it is _because of_ , not in spite of, the tensions inherent to mothering that it is an instructive case for ethics. And, indeed, these discoveries extend beyond the mother-child relationship. We have found that caring for others, though it is something that one sometimes feels compelled to do, does not come naturally. Romanticizing any type of relationship or setting up impossible ideals is counterproductive. Mothers dealing with their ambivalence demonstrate how important it is to recognize one's limitations. The acknowledgment of hostility toward someone in need is central to being able to respond more appropriately to her call. Emotion can be thought provoking, but it is also thoughtful and reflective in itself; it reveals its own understanding of a situation before it is fully reflected upon.\n\nConflicted mothers teach us that deliberative agency is part of what enables one to resist violence and to nurture another. Empathy and connection are a minute aspect of what is necessary for ethical responsiveness. It is usually more tedious than transcendent. Thus there must be a continually renewed commitment or adoption; one must _repeatedly_ choose to care for another. Even in genuine mutuality the insurmountable alterity of the other cannot be overcome. While a sense of _unanimity_ with someone may motivate an ethical response, _dispute_ forces recognition of interpersonal divergence. When taken together, these attitudes entreat us to expand and practice self-knowledge, sound judgment, and problem-solving abilities in order to respond fittingly to one another.\n\nThe ethical successes and failures of mothers demonstrate that although our circumstances do not dictate our behaviors, they highly constrain what is possible. Ethical responsiveness requires skills, intelligence, resources, and emotional support. If an ethical response is asymmetrical (as it typically is in the mother-child relation), then this must be balanced by other factors in a person's life. This is why meeting our responsibilities depends on the familial, political, cultural, and material context. An individualistic model of ethical responsibility is both inaccurate and destructive. It has enabled and encouraged the denial of the need for care and thereby made care work needlessly exploitative.\n\nAcknowledging both our dependence and dependents is critical to becoming ethical and fully realizing our capacities. Children are the primal parasites that make the necessity of caregiving undeniably visible. Yet their dependency has something even greater to offer. They invite a fuller expression of our freedom. This view of freedom is not just the ability to do whatever we want. Rather, by making use of our freedom (in order to write a book, for instance) we become entangled in the lives and needs of others. This freedom is articulated not only in overt acts but also in restraint, in permitting the interruption of our enjoyment and self-possession. This generosity is not at odds with personal liberty; it is precisely the recognition of our mutual value and freedom.\n\nMaternal experience also demonstrates that the likelihood of failure must be a part of any ethical theory. Being generous is not always prudent, beautiful, or fun. The coercive vulnerability of others can incite hostility, resentment, and even violence. But acknowledging this can lead us to consider how to reasonably meet competing demands. Recognizing the prevalence of ethical ambivalence means we cannot so easily dismiss, pathologize, or demonize those who fail. The necessity of adoption means that no one is absolved of responsibility if someone is neglected, abused, or murdered. Nietzsche wrote: \"All evil do I accredit to thee: therefore do I desire of thee the good.\" Mothers have opportunity for great crimes and great heroism. Maternal ethics are illustrative precisely because both these options are possible; we are capable of being both better and worse than we typically imagine.\nNOTES\n\n1. MAD MOTHERS, BAD MOTHERS, AND WHAT A GOOD MOTHER WOULD DO\n\n. <http:\/\/news.yahoo.com\/s\/ap\/20110416\/ap_on_he_me\/us_when_mothers_kill>.\n\n. Meyer and Oberman, _Mothers Who Kill Their Children_ , 93.\n\n. Ibid., 89.\n\n. <http:\/\/www.latimes.com\/news\/nationworld\/nation\/la-na-mother-drowns-kids-20110414,0,221941.story>.\n\n. Ibid.\n\n. Ibid.\n\n. <http:\/\/news.yahoo.com\/s\/ap\/20110416\/ap_on_he_me\/us_when_mothers_kill>.\n\n. Ibid.\n\n. Ibid.\n\n. For examples see LaChance Adams and Lundquist \"Introduction: The Philosophical Significance of Pregnancy, Childbirth, and Mothering,\" 3\u20135.\n\n. For further examples of feminist phenomenologists see ibid., 13\u201315.\n\n. Merleau-Ponty, _Phenomenology_ , vii.\n\n. Ruddick, _Maternal Thinking_ , 62.\n\n. Fisher, \"Phenomenology and Feminism,\" 33.\n\n. For more details on this approach and its philosophical background see ibid., 6\u201313.\n\n. At this point we may begin to ask whether ethical ambivalence and intersubjective ambiguity exist beyond the Western context. I argue that these structures will exist wherever there is the attempt to negotiate intimacy and separation. Is there a culture on earth or in history in which this was not at issue? This is a question to answer in dialogue with anthropologists and historians. Yet even at this point there is some evidence that these findings will be relevant across cultures. According to some writings by post\u2013World War II Japanese feminists, concerns of maternal ambivalence and intersubjective ambiguity do extend to their context. Some scholars argue that pregnancy, childbirth, and child rearing have given Japanese women experiences of ambiguity and intense self-alienation such as those I have described here, which they describe as akin to the Zen realization of no-self. Sakiko, \"Living as a Woman and Thinking as a Mother in Japan.\" They have also claimed, as I have, that this alienation possesses a positive transformative power. Yusa, \"Women Rocking the Boat.\" Clearly more research would be required to fully consider the applicability of these terms to additional contexts.\n\n. Some will already note that I differ from scholars who claim that Levinas does not address _actual_ mothers. I will give my full reasons for this in the chapter on Levinas.\n\n. I'm thinking here of care ethics, but of others as well, such as Tuana's _Women and the History of Philosophy_ and Pateman's _The Sexual Contract_.\n\n. Although I don't think that she fully agrees with me on this point, this inspiration came during a presentation by Claire Katz at the _Psychology for the Other_ symposium at Seattle University in November 2012.\n\n. The idea of the \"primal parasite\" came to me during a reading of a paper by David Alexander Craig on primal parricide at the International Association for Environmental Philosophy, October 2012.\n\n. Oliver, _Womanizing Nietzsche_ , 189.\n\n2. THE MOTHER AS ETHICAL EXEMPLAR IN CARE ETHICS\n\n. Jagger, \"Feminist Ethics,\" 81.\n\n. Gilligan, _In a Different Voice_ , 30.\n\n. Gilligan found that both men and women apply both ethical models. Although differences between the groups were not absolute, there were significant recognizable patterns of disparity.\n\n. Tronto, \"Women and Caring,\" 103.\n\n. Noddings is one exception to this. She believes that feminine caretaking arises from women's \"deep feminine psychological structure\" and that women should \"retain their natural orientation.\" Meanwhile, men should not adopt women's relational patterns but instead \"bring their best human and masculine qualities to the experience of parenting.\" Noddings, _Caring_ , 129.\n\n. Bartky, \"Feeding Egos and Tending Wounds\"; Card, \"Gender and Moral Luck\"; MacKinnon, _Feminism Unmodified_ ; Puka, \"The Liberation of Caring.\"\n\n. See especially Kittay, _Love_ ' _s Labor_.\n\n. Puka, \"The Liberation of Caring,\" 65.\n\n. Ruddick, _Maternal Thinking_ , 25.\n\n. I find it curious that most care ethicists give very little consideration to this radical aspect of Gilligan's work or to abortion at all.\n\n. Ruddick, _Maternal Thinking_ , 210.\n\n. Ibid., 39.\n\n. Ibid., 218.\n\n. Wolff, \"Secret Thoughts of an Adoptive Mother.\"\n\n. I'll elaborate on the manner in which reason and passion work in concert further on in this chapter.\n\n. Kittay, _Love_ ' _s Labor_ , 69.\n\n. Held, _Feminist Morality_ , 196.\n\n. I am thinking especially of Held's essay \"Care and Justice in the Global Context\" where she brings the perspective of care to provide more balanced and sensible solutions to global issues such as those considered in international relations.\n\n. Ruddick is one exception to this. I will explain this aspect of her account in the next chapter.\n\n. Held, _Feminist Morality_ , 198.\n\n. As an example, consider Manninen's \"The Pro-Choice Pro-Lifer.\"\n\n. Grimshaw, _Philosophy and Feminist Thinking_ , 250\u2013251.\n\n. As Ruddick states: \"Any idealized figure of the Good Mother casts a long shadow on many actual mothers' lives. Our days include few if any perfect moments, perfect children perfectly cared for. . . . Many mothers who live in the Good Mother's shadow, knowing that they have been angry and resentful and remembering episodes of violence and neglect, come to feel that their lives are riddled with shameful secrets that even the closest friends can't share.\" Ruddick, _Maternal Thinking_ , 31.\n\n3. MOTHERHOOD'S JANUS HEAD\n\n. Rich, _Of Woman Born_ , 15 and 21.\n\n. Quoted in Oakley, _Becoming a Mother_ , 142.\n\n. Jones, \"Love with Teeth,\" 40\u201341.\n\n. Lazarre, _The Mother Knot_ , 46.\n\n. Ibid., xxii.\n\n. De Marneffe, _Maternal Desire_ , 120.\n\n. Dally, _Mothers_ , 186.\n\n. Rich, _Of Woman Born_ , 24.\n\n. Oakley, \"Normal Motherhood,\" 99.\n\n. Oakley, _Becoming a Mother_ , 12.\n\n. Badinter, _Mother Love_ , 4.\n\n. Hrdy, _Mother Nature_ , 290.\n\n. \"Consider early-twentieth century admission records for Broadmoor, Britain's state asylum for the criminally insane. Forty-eight percent of the women hospitalized between 1902 and 1927 were women who had committed infanticide.\" Ibid., 280. \"As in much of Catholic Europe, a _ruota_ , or rotating barrel [in which babies could be left], was installed in 1660 to replace the old marble basin at Florence's main foundling home, the Innocenti. By 1699, however, it was necessary to place a grill across the opening to prevent parents from shoving in older children as well.\" Ibid., 304.\n\n. Ibid., 303.\n\n. Parker, _Torn in Two_ , 5.\n\n. Ibid., 1.\n\n. Ibid., 103.\n\n. Winnicott, \"Transitional Objects,\" 202.\n\n. Winnicott, \"Hate in Countertransference,\" 201.\n\n. Ibid., 202.\n\n. Freud, _Totem and Taboo_ , 66.\n\n. Klein, \"Love, Guilt and Reparation,\" 322.\n\n. Ibid., 318.\n\n. Both Beauvoir and Merleau-Ponty rely a great deal on Deutsch's work in their characterizations of maternal ambivalence.\n\n. Deutsch, _The Psychology of Women_ , 294.\n\n. Freud, \"A Case of Obsessional Neurosis,\" 240.\n\n. Freud, \"Thoughts for the Times on War and Death,\" 299.\n\n. Freud writes: \"The emotional life of man is in general made up of pairs of contraries such as these. Indeed, if it were not so, repression and neurosis would perhaps never come about.\" Freud, \"Notes on a Case of Obsessional Neurosis,\" 10:113.\n\n. Gendlin, \"Befindlichkeit.\"\n\n. From this point forward I will usually speak of the mother-child relationship (as opposed to mother\/children). This is for the sake of simplicity, and not intended to imply that there is always necessarily only _one_ child involved. Having more than one child obviously adds more complexity to the family dynamics. For example, sometimes a mother must decide between meeting the needs of one child versus another, or sometimes a mother's ambivalence is split between children (one child is experienced as only loved, the other as only hated). In spite of my nominal reference to the relationship as a dyad, I will also explore the dynamics involving more than two people.\n\n. Damasio, _The Feeling of What Happens_ , 145.\n\n. Ibid., 142.\n\n. Ibid., 135\u2013136.\n\n. Young, _Throwing Like a Girl_ , 50.\n\n. Ibid., 160.\n\n. Steingraber, _Having Faith_ , 215.\n\n. Simms, \"Milk and Flesh,\" 25\u201326.\n\n. Lazarre, _The Mother Knot_ , 28.\n\n. Shields, _Down Came the Rain_ , 81.\n\n. Steingraber, _Having Faith_ , 225.\n\n. Moses, \"A Mother's Body,\" 177.\n\n. Weaver-Zercher, \"Afterbirth,\" 42\u201344.\n\n. Steingraber, _Having Faith_ , 248.\n\n. Kristeva, \"Stabat Mater,\" 315, 316, 324, and 325.\n\n. Quoted in Ballou, _The Psychology of Pregnancy_ , 90.\n\n. Stringer.\n\n. Quoted in Ballou, _The Psychology of Pregnancy_ , 97.\n\n. Rich, _Of Woman Born_ , 63.\n\n. Trevarthan, \"The Self Born\"; Piontelli, \"Infant Observation.\"\n\n. Stringer.\n\n. Bondas and Eriksson, \"Women's Lived Experiences of Pregnancy,\" 836.\n\n. Wynn, \"The Early Relationship,\" 6.\n\n. De Marneffe, _Maternal Desire_ , 90\u201391.\n\n. Simms, \"Milk and Flesh,\" 33\u201334.\n\n. Quoted in de Marneffe, _Maternal Desire_ , 93\u201394.\n\n. Ibid., 94.\n\n. Damasio, _The Feeling of What Happens_ , 226.\n\n. Simms, \"Milk and Flesh,\" 31.\n\n. When her infant son will not stop crying at dinner, an older child gives words to what the mother is feeling. Jones, \"Love with Teeth,\" 40.\n\n. Chodorow, _The Reproduction of Mothering_ , viii.\n\n. Ballou, _The Psychology of Pregnancy_ , 161.\n\n. Maushart, _The Mask of Motherhood_ , 49.\n\n. Wolf, _Misconceptions_ , 2\u20133.\n\n. Parker, _Torn in Two_ , 81.\n\n. Ibid., 44.\n\n. <http:\/\/www.whattoexpect.com\/first-year\/breastfeeding\/breastfeeding-guide\/tackling-the-technique.aspx> (accessed December 8, 2009).\n\n. Wolf, _Misconceptions_ , 200 (emphasis added).\n\n. Shields, _Down Came the Rain_ , 41 and 45.\n\n. Quoted in Hays, _The Cultural Contradictions of Motherhood_ , 137. Hays remarks: \"Mothers, she seems to be saying, are like confections that the kids just gobble down\u2014and then they ask for more.\"\n\n. Quoted in Parker, _Torn in Two_ , 131\u2013132.\n\n. Quoted ibid., 120.\n\n. Wolf, _Misconceptions_ , 247.\n\n. Quoted in Brown et al., _Missing Voices_ , 179.\n\n. Quoted in Oakley, _Becoming a Mother_ , 144.\n\n. Quoted in Wolf, _Misconceptions_ , 247.\n\n. Shields, _Down Came the Rain_ , 131.\n\n. Quoted in Brown et al., _Missing Voices_ , 182.\n\n. Lazarre, _Mother Knot_ , 29.\n\n. Lamott, \"Maternal Anger,\" 90.\n\n. Kessler, \"Epidemiology of Women and Depression,\" 5\u201313.\n\n. Robins and Regier, _Psychiatric Disorders in America_ ; Burke et al. \"Comparing Age at Onset of Major Depression\"; Gaynes, \"Perinatal Depression.\"\n\n. Harris, \"Biological and Hormonal\"; O'Hara, _Postpartum Depression_.\n\n. Goodman, \"Paternal Postpartum Depression.\" Also Wolff, \"Secret Thoughts of an Adoptive Mother.\"\n\n. See, for example, www.rainbowkids.com\/expertarticledetails.aspx?id=272; www.adoptionissues.org\/post-adoption-depression.html; <http:\/\/library.adoption.com\/articles\/post-adoption-depression-.html>.\n\n. Wolff, \"Secret Thoughts of an Adoptive Mother.\"\n\n. Parker, _Torn in Two_ , 5.\n\n. Oakley, \"Normal Motherhood,\" 89.\n\n. Ibid., 99\u2013100; Oakley, _Becoming a Mother_ , 13.\n\n. Rich, _Of Woman Born_ , 23.\n\n. Quoted in Hays, _The Cultural Contradictions of Motherhood_ , 136.\n\n. Quoted in Oakley, _Becoming a Mother_ , 252\n\n. Quoted in Lazarre, _Mother Knott_ , 68.\n\n. Rich, _Of Woman Born_ , 29.\n\n. Lazarre, _Mother Knot_ , 28.\n\n. Wolf, _Misconceptions_ , 60 and 106.\n\n. Ibid., 211.\n\n. Lazarre, _Mother Knot_ , 50 and 56.\n\n. McMahon, _Engendering Motherhood_ , 168.\n\n. De Marneffe, _Maternal Desire_ , 49.\n\n. Ibid., 52.\n\n. Liss, \"Maternal CARE,\" 43\u201367.\n\n. Ibid., 65.\n\n. Tisdale, \"Double Dare,\" 253\u2013254.\n\n. Moses, \"A Mother's Body,\" 179.\n\n. Keller, \"You'll Get Used to It,\" 119.\n\n. Moses, \"A Mother's Body,\" 180.\n\n. Peri, \"Dancing with Death,\" 271.\n\n. Moses \"A Mother's Body,\" 176.\n\n. Keller, \"You'll Get Used to It,\" 11.\n\n. Chandler writes, \"[Mothering] is a series of responses to the fundamental needs of another who is so interconnected with the self that there exists no definitive line of differentiation. When one mothers one is not one's own person. This is most acute when mothering an infant. Although one is not one's own person, one is not someone else's person either, for the infant is both neither 'other' than the mother nor the same thing. . . . In summation, 'mother' is an identity formed through a repetition of practices which constitute one as so profoundly interconnected that one is not one, _but is simultaneously more and less than one_.\" Chandler, \"Emancipated Subjectivities,\" 274.\n\n. Irigaray, \"On the Maternal Order,\" 39.\n\n. Rouch quoted ibid., 39.\n\n. Ibid., 41.\n\n. Steingraber, _Having Faith_ , 32.\n\n. Ibid., 31.\n\n. Ibid.\n\n. Ibid., 32.\n\n. Irigaray, \"On the Maternal Order,\" 41.\n\n. Maushart, _The Mask of Motherhood_ , 40\u201341.\n\n. Dunlop, \"Child,\" 105.\n\n. Dunlop, \"Written on the Body,\" 106.\n\n. Nemzoff, \"Untitled Poem,\" 20.\n\n. Frye. \"Making a Living,\" 21.\n\n. Chandler, \"Emancipated Subjectivities,\" 271.\n\n. Consider, for example, Margaret Mahler, Fred Pine, and Anni Bergman's _The Psychological Birth of the Human Infant_ , where they chronicle the \"individuation-separation\" process in infant development.\n\n. Rich, _Of Woman Born_ , 36\u201337.\n\n. Parker, _Torn in Two_ , 121.\n\n. Ibid., 22.\n\n. Smith, _The Daddy Shift_ , 129.\n\n. Quoted in Ruddick, _Maternal Thinking_ , 65.\n\n. Ibid., 67.\n\n. Ibid. Next to this account, Noddings's theory is a gross oversimplification. In particular, this statement: \"Responding to my own child crying in the night may require a physical effort, but it does not usually require what might be called an ethical effort. I naturally want to relieve my child's distress.\" Noddings, _Caring_ , 17.\n\n. De Marneffe, _Maternal Desire_ , 199 and 200.\n\n. Martin Buber, \"Distance and Relation,\" in _Martin Buber_ , 4.\n\n. Ibid., 6.\n\n. Ibid., 15.\n\n4. MATERNITY AS VULNERABILITY IN THE PHILOSOPHY OF EMMANUEL LEVINAS\n\nSome of these ideas appeared previously in LaChance Adams, \"The Pregnable Subject.\"\n\n. This interpretation is owed, at least in part, to Nancy Tuana's suggestions on how to \"read philosophy as a woman\" in _Women and the History of Philosophy_.\n\n. Levinas, _Time and the Other_ , 57.\n\n. Levinas, _Totality and Infinity_ , 263.\n\n. Ibid., 155.\n\n. See, for example, Ainley, \"Levinas and Kant\"; Chanter, \"Introduction\"; Brody, \"Levinas's Maternal Method\"; Perpich, \"From the Caress to the Word.\"\n\n. Levinas, _Time and the Other_ , 68.\n\n. Perpich, \"From the Caress,\" 47.\n\n. Levinas, \"Judaism and the Feminine,\" 33, _Difficile libert\u00e9_ , 53\u201354.\n\n. This critique is parallel to that of social contract origin myths in Carole Pateman's _The Sexual Contract_. Pateman argues that, according to social contract, it is only through subjugating a woman that a man can gain status as an individual.\n\n. Levinas, _Totality and Infinity_ , 263.\n\n. Irigaray, \"Questions to Emmanuel Levinas,\" 113.\n\n. Chanter, \"Introduction,\" 17.\n\n. Chanter, \"Feminism and the Other,\" 36.\n\n. Levinas, _Time and the Other_ , 86.\n\n. This problem has been thoroughly documented. See, for example: Beauvoir, _The Second Sex_ ; Bernasconi and Critchley, _Re-reading Levinas_ ; Chanter, _Feminist Interpretations of Emmanuel Levinas_ ; Irigaray, \"The Fecundity of the Caress\" and _An Ethics of Sexual Difference_.\n\n. Levinas, _Time and the Other_ , 78 (emphasis added).\n\n. Levinas, _Otherwise Than Being_ , 104.\n\n. Guenther, _The Gift of the Other_ , 95.\n\n. Ibid., 95\u201396.\n\n. It is worth noting that many women do not experience such happenings as an example of the generosity of the maternal body, but instead as a parasitic leaching by the fetus. Furthermore, as discussed in the previous chapter, in cases of denied pregnancy, a fetus may be deprived of nutrients even when they are available. This suggests that a woman's conscious acknowledgment of her pregnancy (even if not her deliberate _choice_ ) is central to the \"generosity\" of the pregnant body.\n\n. Sikka, \"The Delightful Other,\" 109.\n\n. Willett, _Maternal Ethics_ , 84\u201385.\n\n. Taylor, \"Levinasian Ethics,\" 222.\n\n. Ibid., 229. Taylor quotes Willett, _Maternal Ethics_ , 84; Levinas, _Otherwise Than Being_ , 49\u201351; and Levinas, _Entre Nous_ , 49.\n\n. Levinas, _Time and the Other_ , 86.\n\n. Some examples include Kittay, _Love_ ' _s Labor_ ; and Noddings, _Caring_. Still, it is important to note some important differences between these theorists, such as Noddings's affirmation of caregiving as women's role versus Kittay's extension of caregiving responsibilities to men. The actual content of what is meant by the value of feminine caretaking will differ among these theorists.\n\n. See, for example, Bartky, \"Feeding Egos and Tending Wounds\"; Card, \"Gender and Moral Luck\"; MacKinnon, _Feminism Unmodified_ ; Puka, \"The Liberation of Caring.\"\n\n. Levinas, _Otherwise Than Being_ , 116.\n\n. Levinas, _Totality and Infinity_ , 278, 267, and 277.\n\n. Guenther, _The Gift of the Other_ , 92.\n\n. Ibid., 94.\n\n. Ibid., 6\u20137.\n\n. Guenther, \"Like a Maternal Body,\" 119.\n\n. Ibid., 131.\n\n. Ibid., 128 and 132.\n\n. Thanks to Lisa Guenther for pointing out this possible critique of my work.\n\n. Derrida, \"White Mythology,\" 209.\n\n. Ibid., 223.\n\n. Ibid., 221\u2013222.\n\n. Derrida, \"At this Moment,\" 19.\n\n. Guenther, \"Like a Maternal Body,\" 124.\n\n. Brody, \"Levinas's Maternal Method,\" 74.\n\n. Hardy, \"Yogurt Couvade,\" 22.\n\n. Levinas, _Otherwise Than Being_ , 83, 99, 100, and 112.\n\n. \"Sensibility is exposedness to the other.\" Ibid., 75.\n\n. Ibid.\n\n. Ibid., 83.\n\n. Peri, \"Dancing with Death,\" 271.\n\n. Levinas, _Otherwise Than Being_ , 109.\n\n. Wolf's description of the appearance of her abdomen during her cesarean section in progress. Wolf, _Misconceptions_ , 141.\n\n. Levinas, _Otherwise Than Being_ , 92.\n\n. Levinas, _Totality and Infinity_ , 199.\n\n. Shields, _Down Came the Rain_ , 59\u201360.\n\n. Kittay, \"Not _My_ Way Sesha,\" 12.\n\n. Levinas, _Totality and Infinity_ , 199\u2013200.\n\n. Kunz, _The Paradox of Power and Weakness_ .\n\n. Ibid., 133.\n\n. Levinas, _Totality and Infinity_ , 84.\n\n. Ibid., 83.\n\n. Ibid.\n\n. Kittay, \"'Not _My_ Way Sesha,\" 12.\n\n. Levinas, _Otherwise Than Being_ , 100\u2013101.\n\n. Wolf, _Misconceptions_ , 200 (emphasis added).\n\n. Lazarre, _Mother Knot_ , 28.\n\n. Levinas, _Otherwise Than Being_ , 76.\n\n. Levinas, _Totality and Infinity_ , 232.\n\n. Levinas, \"Peace and Proximity,\" 167.\n\n. Levinas, _Totality and Infinity_ , 199.\n\n. Bergo, \"Emmanuel Levinas.\"\n\n. Klein, \"Love, Guilt and Reparation,\" 318.\n\n. Guenther, \"Like a Maternal Body,\" 128.\n\n. Miller, _For Your Own Good_ , 258.\n\n. Quoted in Parker, _Torn in Two_ , 23.\n\n. Ibid., 8\u20139.\n\n. Tsing, \"Monster Stories,\" 284.\n\n. Ibid., 283.\n\n. Gentleman, \"Cameron to Look at Case of Mother.\"\n\n. Arcana, \"Abortion Is a Motherhood Issue,\" 226.\n\n. Levinas, _Otherwise Than Being_ , 112.\n\n. Consider, for example, the Tennessee woman who sent her adoptive Russian son back.\n\n. Levinas, _Otherwise Than Being_ , 105.\n\n. Winnicott, \"Transitional Objects,\" 202.\n\n5. MATERNITY AS DEHISCENCE IN THE FLESH IN THE PHILOSOPHY OF MAURICE MERLEAU-PONTY\n\n. He writes, \"a sort of dehiscence opens my body in two.\" Merleau-Ponty, _The Visible and the Invisible_ , 123.\n\n. Merleau-Ponty, _Phenomenology of Perception_ , 278.\n\n. Karen, _Becoming Attached_ , 143\u2013161; and Ainsworth et al., _Patterns of Attachment_.\n\n. Ibid.\n\n. Karen, _Becoming Attached_ , 13\u201325; and James, _Handbook for the Treatment of Attachment-Trauma_.\n\n. De Marneffe, _Maternal Desire_ , 90.\n\n. Moses, \"A Mother's Body,\" 77.\n\n. Ibid.\n\n. Merleau-Ponty, \"The Child's Relations with Others,\" 119.\n\n. It is not entirely clear if Merleau-Ponty gets the idea that the child has an anonymous relation to the mother from psychoanalysis or if he has simply found a like-minded theory here. In \"Anonymity and Sociality,\" Stawarska argues that the developmental psychology of Merleau-Ponty's day influenced his theory of intersubjectivity. \"Specifically, the psychological hypothesis about the anonymous and fusional form initially taken by human sociality appears to play a determining role in his conception of interpersonal life formulated on the ontological plane\" (1).\n\n. Simms, \"Milk and Flesh,\" 29. Simms cites Ayers, \"Sensory Integration and the Child\"; and Dennis, _Children of the Creche_.\n\n. Merleau-Ponty, \"The Child's Relations with Others,\" 100.\n\n. However, as I argue in chapter 2, the \"integrity\" of the body can vary dramatically from one person to another, both in its distinctness from others and in its sense of internal unity.\n\n. Merleau-Ponty, _The Visible and the Invisible_ , 10.\n\n. Lisa Guenther provides a thorough account and analysis of these metaphors in her essay \"The Birth of Sexual Difference.\" She writes:\n\nOn a number of occasions throughout his work, Merleau-Ponty compares the relation between self and other to a pregnancy. In 'The Philosopher and his Shadow,' for example, he claims that 'each one of us is pregnant with the others and confirmed by them in his body.' In 'Dialogue and the Perception of the Other,' Merleau-Ponty describes the other as 'reproduced from me,' born through 'that strange filiation which makes the other forever my second, even when I prefer him to myself.'\" The Birth of Sexual Difference,\" 89. In \"Dialogue,\" he asks: \"To the infinity that was me something else still adds itself; a sprout shoots forth, I grow; I give birth, this other is made from my flesh and blood and yet is no longer me. How is that possible? How can the cogito emigrate beyond me, since it is me?\"\n\nIbid., 90.\n\nMerleau-Ponty's use of maternity as a metaphor is not so obnoxious as with the typical philosopher, probably because Merleau-Ponty did some research on the experiences of mothers. I will refer to some of this research further on in this chapter.\n\n. Trevarthan, \"The Self Born\"; and Piontelli, \"Infant Observation from Before Birth.\"\n\n. Sears, Sears, and Holt, _The Pregnancy Book_ ; and Sears et al., _The Baby Book_.\n\n. Simms, \"Milk and Flesh,\" 23.\n\n. Ibid., 26.\n\n. Ibid., 27.\n\n. Ibid., 35.\n\n. Olkowski, \"Only Nature Is Mother to the Child,\" 51.\n\n. Irigaray, \"The Invisible of the Flesh,\" 154.\n\n. See Guenther quoted in \"The Birth of Sexual Difference,\" 88\u201394.\n\n. Butler, \"Sexual Ideology and Phenomenological Description,\" 95.\n\n. Allen, \"Through the Wild Region,\" 245.\n\n. Sullivan, _Living Across and Through Skins_ , 66.\n\n. Grosz, \"Merleau-Ponty and Irigaray in the Flesh,\" 155.\n\n. Stawarska, \"From the Body Proper to Flesh,\" 104, 103.\n\n. Ibid., 92.\n\n. Ibid., 93.\n\n. Interestingly, Levinas has the same reading of reversibility in \"Intersubjectivity.\"\n\n. Stawarska, \"From the Body Proper to Flesh,\" 94.\n\n. Stoller, \"Reflections on Feminist Merleau-Ponty Skepticism,\" 176.\n\n. Weiss, \"The Anonymous Intentions of Transactional Bodies,\" 194.\n\n. Grosz, \"Merleau-Ponty and Irigaray in the Flesh,\" 155.\n\n. See, for example, Beauvoir, _The Second Sex_ ; Bigwood, \"Renaturalizing the Body\"; Grosz, _Volatile Bodies_ ; Guenther, _The Gift of the Other_ ; Hein\u00e4maa, _Toward a Phenomenology of Sexual Difference_ ; Simms, \"Milk and Flesh\"; Wynn, \"The Early Relationship of Mother and Pre-Infant\"; and Young, _Throwing Like a Girl_.\n\n. Al-Saji, \"Vision, Mirror, and Expression,\" 53.\n\n. Weiss, \" _Ecart_ ,\" 205.\n\n. Ibid., 214.\n\n. Ibid., 208.\n\n. Fielding, \"Grounding Agency in Depth,\" 180.\n\n. Fielding, \"Envisioning the Other,\" 194.\n\n. Some of these ideas appeared previously in LaChance Adams, \"The Need of Philosophy in Hegel.\"\n\n. Merleau-Ponty, \"Hegel's Existentialism,\" 656.\n\n. Ibid., 64.\n\n. \"The bad dialectic is that which thinks it recomposes being by a thetic thought, by an assemblage of statements, by thesis, antithesis, synthesis.\" Merleau-Ponty, _The Visible and the Invisible_ , 94.\n\n. See for example Hegel, _The Encyclopedia Logic_ , 127\u2013128.\n\n. Hegel, _Phenomenology of Spirit_ , 123.\n\n. Deleuze, _Difference and Repetition_.\n\n. Desmond, _Beyond Hegel and Dialectic_ , 2.\n\n. See part 1 of Grier, _Identity and Difference_.\n\n. Maker, \"Identity, Difference, and the Logic of Otherness,\" 15.\n\n. Ibid., 18.\n\n. Damasio, _The Feeling of What Happens_ , 142.\n\n. Ibid., 145.\n\n. Levinas, _Otherwise Than Being_ , 124.\n\n. Ibid., 83.\n\n. Ibid., 84.\n\n. Sartre, _Being and Nothingness_ , 288.\n\n. As I argue in \"Becoming with Child,\" Sartre provides us with a manner of understanding these conflicts in a positive light in _Notebooks for an Ethics_.\n\n. Merleau-Ponty, \"The Child's Relations with Others,\" 105.\n\n. Taminiaux, \"Merleau-Ponty.\"\n\n. Weaver-Zercher, \"Afterbirth,\" 42\u201344.\n\n. Peri, \"Dancing with Death,\" 271.\n\n. Quoted in Yusa, \"Women Rocking the Boat,\" 155.\n\n. Tuvel, \"Exposing the Breast.\"\n\n. Ibid.\n\n. Thanks to Bonnie Mann for this point.\n\n. Moses, \"A Mother's Body,\" 177.\n\n. Merleau-Ponty, _Child Psychology and Pedagogy_ , 78.\n\n. Note the difference between this statement and Carol Bigwood's appropriation of Merleau-Ponty in \"Renaturalizing the Body (with the Help of Merleau-Ponty).\" She claims that Merleau-Ponty's _chiasm_ or _intertwining_ provides a helpful alternative to the subject-object dichotomy in making sense of the mother's relation to her fetus. She states that \"it becomes especially clear in the case of pregnancy that, as Merleau-Ponty argues, the metaphysical dichotomous categories of subject and object, and self and other, fail to describe the incarnate situation, for the 'subject' is blurred and diffused in pregnancy. A woman is inhabited by a growing sentience that is not truly 'other' to herself.\" Ibid., 68.\n\n. Andrews, \"Vision, Violence, and the Other,\" 177.\n\n. Low, \"The Foundations of Merleau-Ponty's Ethical Theory,\" 181.\n\n. Gantt and Reber, \"Sociobiological and Social Constructionist Accounts,\" 26.\n\n. Waldenfels, \"Responsivity of the Body,\" 94.\n\n. Ibid., 99.\n\n. Ibid., 102.\n\n. Al-Saji, \"Vision, Mirror, and Expression,\" 54.\n\n. Ibid., 56.\n\n. Murphy, \"All Things Considered,\" 436.\n\n. Ibid., 447.\n\n. Merleau-Ponty, \"An Unpublished Text,\" 11.\n\n. Merleau-Ponty, _Child Psychology and Pedagogy_ , 83.\n\n. Merleau-Ponty, _Hegel_ ' _s Existentialism_ , 65.\n\n. Carbone, \"FLESH,\" 57.\n\n. Merleau-Ponty, \"An Unpublished Text,\" 6.\n\n. Merleau-Ponty, _Signs_ , 35.\n\n. Merleau-Ponty, _Child Psychology and Pedagogy_ , 68.\n\n. Beauvoir, _The Second Sex_ , 566.\n\n6. MATERNITY AS NEGOTIATING MUTUAL TRANSCENDENCE IN THE PHILOSOPHY OF SIMONE DE BEAUVOIR\n\n. Beauvoir, \"Introduction to an Ethics of Ambiguity,\" 292.\n\n. Beauvoir, _The Ethics of Ambiguity_ , 9.\n\n. As Beauvoir writes,\n\nOne cannot fulfill a man; he is not a vessel that docilely allows itself to be filled up. His condition is to surpass everything given. . . . Since man is project, his happiness, like his pleasures, can only be projects. The man who has made a fortune immediately dreams of making another. Pascal said it perfectly: it is not the hare that interests the hunter, it is the hunt. . . . A goal is always the meaning and the result of an effort. Separated from that effort, no reality is a goal but only a given made to be surpassed. . . . The notion of end is ambiguous since every end is a point of departure at the same time. But this does not prevent it from being seen as an end. Man's freedom resides in this power.\n\nBeauvoir, \"Pyrrhus and Cineas,\" 98\u201399.\n\n. \"And it is not true that the recognition of the freedom of others limits my own freedom: to be free is not to have the power to do anything you like; it is to be able to surpass the power given toward an open future; the existence of others as a freedom defines my situation and is even the condition of my own freedom.\" Beauvoir, _The Ethics of Ambiguity_ , 91.\n\n. Ibid., 72.\n\n. \"As soon as a child has finished a drawing or a page of writing, he runs to show them to his parents. He needs their approval as much as candy or toys; the drawing requires an eye that looks at it. These disorganized lines must become a boat or a horse for someone. . . . By himself, he would not have dared to put confidence in those hesitant lines.\" Beauvoir, \"Pyrrhus and Cineas,\" 116.\n\n. Ibid., 109.\n\n. \"Men are free, and I am thrown into the world among these foreign freedoms. I need them because once I have surpassed my own goals, my actions will fall back upon themselves, inert and useless, if they have not been carried off toward a new future by new projects.\" Beauvoir, \"Pyrrhus and Cineas,\" 135. \"What is mine is therefore first what I do. But as soon as I have done it, the object goes and separates itself from me; it escapes me. The thought that I expressed a moment ago, is it still my thought? In order for it to be mine, I must make it mine again each instant by taking it toward my future.\" Ibid., 93.\n\n. Beauvoir, _The Ethics of Ambiguity_ , 67.\n\n. \"The sick man who wears himself out by struggling against sickness or the slave against slavery care for neither poetry, nor astronomy, nor the improvement of aviation. They first need health, leisure, security, and the freedom to do with themselves what they want.\" Beauvoir, \"Pyrrhus and Cineas,\" 137).\n\n. \"A man alone in the world would be paralyzed by the manifest vision of the vanity of all his goals. He would undoubtedly not be able to stand living. But man is not alone in the world.\" Ibid., 115.\n\n. Beauvoir, _The Ethics of Ambiguity_ , 24.\n\n. Beauvoir, \"Pyrrrhus,\" 93.\n\n. Beauvoir, _The Ethics of Ambiguity_ , 156.\n\n. Ibid., 24.\n\n. Beauvoir, _The Second Sex_ , 159.\n\n. Ibid.\n\n. Beauvoir, _The Ethics of Ambiguity_ , 25.\n\n. Beauvoir, \"Pyrrhus and Cineas,\" 131.\n\n. Ibid., 127.\n\n. Beauvoir, _The Ethics of Ambiguity_ , 61, 65\u201366, 83, and 96.\n\n. Beauvoir, \"Pyrrhus and Cineas,\" 126.\n\n. I claim that _The Second Sex_ is the most mature of Beauvoir's ethical texts because she is the least moralistic in tone (a tone that she herself found irritating in her earlier work) and more understanding of the power of situation over an individual's ability to be ethical in her sense of the term.\n\n. Beauvoir, _The Second Sex_ , 16.\n\n. Beauvoir, _The Ethics of Ambiguity_ , 96.\n\n. Ibid., 37.\n\n. Ibid., 76.\n\n. Ibid., 82.\n\n. Beauvoir, _The Second Sex_ , 159\u2013160.\n\n. Beauvoir, \"Pyrrhus and Cineas,\" 124.\n\n. See Andrew, \"Care, Freedom, and Reciprocity\"; Gothlin, \"Beauvoir and Sartre on Appeal\"; Kruks, \"Teaching Sartre About Freedom\"; Scarth, _The Other Within_ ; Tidd, \"The Self-Other Relation\"; Ward, \"Reciprocity and Friendship in Beauvoir's Thought.\"\n\n. Andrew, \"Care, Freedom, and Reciprocity,\" 291.\n\n. Ibid., 290.\n\n. Ibid.\n\n. Tidd, \"The Self-Other Relation,\" 230.\n\n. Scarth, _The Other Within_ , 7.\n\n. Ward, \"Reciprocity and Friendship in Beauvoir's Thought,\" 41.\n\n. Kruks, \"Teaching Sartre About Freedom,\" 88.\n\n. See, for example, Simons, \"Confronting an Impasse\" and \"Beauvoir's Philosophical Independence in a Dialogue with Sartre.\"\n\n. See, for example, Collins and Pierce, \"Holes and Slime\"; Gatens, _Feminism and Philosophy_ and \"Toward a Feminist Philosophy of the Body\"; Greens, \"Sartre, Sexuality and _The Second Sex_ \"; Lloyd, _Man of Reason_ ; Spelman, \"Woman as Body\" and _Inessential Woman_ ; Suleiman, _The Female Body in Western Culture_ ; Strickling, \"Simone de Beauvoir and the Value of Immanence.\"\n\n. Lloyd, _Man of Reason_ , 99\u2013100.\n\n. Ibid., 101\n\n. Jardin, \"Death Sentences, 90.\n\n. Scarth, _The Other Within_ , 156. This is Scarth's own description of her project.\n\n. Ibid., 140 and 165.\n\n. Ibid., 156.\n\n. Ibid., 140.\n\n. Bergoffen, _The Philosophy of Simone de Beauvoir_ , 207.\n\n. Ibid., 208.\n\n. Ibid., 209.\n\n. Ibid.\n\n. Beauvoir, _The Second Sex_ , 159 (emphasis added).\n\n. Beauvoir, _The Ethics of Ambiguity_ , 73.\n\n. Iris Marion Young, \"Pregnant Embodiment: Subjectivity and Alienation,\" in _Throwing Like a Girl_ , 46.\n\n. Beauvoir, _The Second Sex_ , 531.\n\n. Ibid., 524.\n\n. Ibid., 539.\n\n. Simons, \"Two Interviews with Simone de Beauvoir,\" 18.\n\n. Beauvoir, _The Ethics of Ambiguity_ , 55.\n\n. Ibid., 48\u201349.\n\n. Beauvoir, _The Second Sex_ , 554.\n\n. Ibid., 524.\n\n. Ibid., 525.\n\n. The same critique is often made regarding current discourse on children and abortion.\n\n. Lerner, _The War on Moms_ , 63.\n\n. U.S. Dept of Health and Human Services, \"Indicators of Welfare Dependence.\"\n\n. Lerner, _The War on Moms_ , 70.\n\n. MTV, _No Easy Decision_.\n\n. Beauvoir, _The Second Sex_ , 540.\n\n. Ibid., 533.\n\n. Ibid.\n\n. Beauvoir, \"Pyrrhus and Cineas,\" 117.\n\n. Freud, _Totem and Taboo_ , 66.\n\n. Beauvoir, _A Very Easy Death_ , 71.\n\n. Levine, \"Throwing the Book,\" 20.\n\n. Ibid., 21.\n\n. Ibid., 21\u201322.\n\n. Ibid., 22.\n\n. Beauvoir, \"Pyrrhus and Cineas,\" 119.\n\n. Ibid., 118.\n\n. Beauvoir, _The Ethics of Ambiguity_ , 142.\n\n. Beauvoir, _The Second Sex_ , 539.\n\n. Sartre, _Notebooks for an Ethics_ , 194.\n\n. Mirvish, \"Sartre, Embodied Minds, Authenticity, and Childhood,\" 27.\n\n. Beauvoir, \"Pyrrhus and Cineas,\" 138.\n\n. Ibid., 117.\n\n. Ibid., 121.\n\n. Ibid., 123.\n\n. Beauvoir, _The Second Sex_ , 569.\n\n. Lerner, _The War on Moms_ , 2, 39, 63.\n\n. Mezey, Greenberg, and Schumacher, \"The Vast Majority.\"\n\n. Lerner, _The War on Moms_ , 20.\n\n. Beauvoir, _The Second Sex_ , 556.\n\n. Parker, _Torn in Two_ , 121.\n\n. Rich, _Of Woman Born_ , 36\u201337.\n\n. Parker, _Torn in Two_ , 22.\n\n. Beauvoir, _The Ethics of Ambiguity_ , 71.\n\n. Kovacic, \"Salvation Road,\" 61.\n\n. \"Without failure, no ethics. For a being who, from the outset, is an exact coincidence with himself, a perfect plenitude, the notion 'ought to be' [ _devoir \u00eatre_ ] would not make sense. One does not propose ethics to a god. It is impossible to propose one to man if he is defined as nature, as given. . . . This means that there can be an 'ought to be' [ _devoir \u00eatre_ ] only for a being who, according to the existentialist definition, questions himself within his being.\" Beauvoir, _The Ethics of Ambiguity_ , 291.\n\n. Beauvoir, \"Moral Idealism and Political Realism,\" 187.\n\n. Ibid., 188.\n\n. Beauvoir, \"Introduction to an Ethics of Ambiguity,\" 292.\n\n. Ibid.\n\n. Beauvoir, _A Very Easy Death_ , 108.\n\n. Beauvoir, _The Ethics of Ambiguity_ , 134.\n\nCONCLUSION\n\n. <http:\/\/blogs.seattleweekly.com\/dailyweekly\/2011\/02\/angela_mcanulty_eugene_mom_ple.php>.\n\n. <http:\/\/www.klcc.org\/Feature.asp?FeatureID=2246>.\n\n. <http:\/\/special.registerguard.com\/csp\/cms\/sites\/web\/news\/cityregion\/24249180\u201341\/maples-family-jeanette-mcanulty-death.csp>.\n\n. Ibid.\n\n. Ibid.\n\n. <http:\/\/blogs.seattleweekly.com\/dailyweekly\/2011\/02\/angela_mcanulty_eugene_mom_ple.php>\n\n. Meyer and Oberman, _Mothers Who Kill Their Children_ , 175.\n\n. Nietzsche, _Thus Spake Zarathustra_ , 112.\nBIBLIOGRAPHY\n\nAinley, Alison. \"Levinas and Kant: Maternal Morality and Illegitimate Offspring.\" In _Feminist Interpretations of Emmanuel Levinas_ , ed. Tina Chanter, 203\u2013223. University Park: Pennsylvania State University Press, 2001.\n\nAinsworth, Mary, M. C. Blehar, E. Waters, and S. Wall, _Patterns of Attachment: A Psychological Study of the Strange Situation_. 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Nagoya: Nanzan Institute for Religion and Culture, 2009.\nINDEX\n\nPage numbers refer to the print edition but are hyperlinked to the appropriate location in the e-book.\n\nAbandonment, 99\u2013100; _ruota_ , 200 _n_ 13\n\nAbortion, , , 105\u20136, 213 _n_ 64; Beauvoir on circumstances of, 173\u201374; \"Being Torn: Toward a Phenomenology of Unwanted Pregnancy,\" , 168\u201369; care ethicists' nonconsideration of, 199 _n_ 10; coconstitutionality with fetus, ; in _No Easy Decision_ , 175\u201376\n\n\"Abortion Is a Motherhood Issue\" (Arcana), ,\n\nAbraham and Isaac,\n\nAdoption, 105\u20136, 207 _n_ 80; ecstasy of, ; postpartum depression after, 49\u201350; as temporal structure of all ethical commitment, 97\u201398\n\n\"The Adult's View of the Child\" (Merleau-Ponty),\n\nAlien invaders, 167\u201368\n\nAllen, Jeffner,\n\nAllergies,\n\n\"All Things Considered': Sensibility and Ethics in the Later Merleau-Ponty and Derrida\" (Murphy), 149\u201350\n\nAlterity: as interruption of selffulfillment, 98\u201399; Levinas on, , 94\u201398; in Merleau-Ponty's philosophy, , 118\u201319, 128\u201330, ; mother's insurmountable, 189\u201395; Al-Saji on Merleau-Ponty's theory of,\n\nAmbiguity,\n\nAmbiguous intersubjectivity: Beauvoir on assuming our fundamental, ; Beauvoir on existential morality and, 159\u201370, 211 _n_ 4, 214 _n_ 99; being more, yet less, than one, 58\u201372, , 203 _n_ 110; breast-feeding and, 143\u201344; caregiving intertwined with self needs, 10\u201313; \"Emancipated Subjectivities and the Subjugation of Mothering Practices,\" 62\u201363; flesh continuity and, 10\u201311; \"The Ineradicable Danger of Ambiguity at Ch[i]asm's Edge,\" ; \"Introduction to an Ethics of\n\nAmbiguity,\" , , 185\u201386, 211 _n_ 4, 214 _n_ 99; in Kristeva's \"Stabat Mater,\" 63\u201364; in Levinas, 92\u201394, , 100\u2013101, 107\u20138; Lundquist on pregnancy, ; Merleau-Ponty's, , 111\u201323, 127\u201328, 208 _n_ 32; Nature exceeds us yet is us, 143\u201344; in Ukele's maintenance art, 54\u201355; beyond Western context, 197 _n_ 16\n\nAmbivalence, ethical: alterity as interruption of self-fulfillment, 98\u201399; Merleau-Ponty's, , 111\u201323; _The Mother Knot_ , , , ; beyond Western context, 197 _n_ 16\n\nAmbivalence, maternal, , , ; as achievement, 64\u201365, ; Beauvoir on, 177\u201378; biblical perspective of, 84\u201387, ; conflict versus mutuality, 35\u201336; dependent's faces inspiring, 9\u201310, ; insights from, 70\u201371; Levinas's denial of, 100\u20132; mutuality of self and child, 36\u201345; Parker on, , 102\u20133; pervasiveness of, 31\u201332; as phenomenon to be understood, ; political dimension of, 103\u20134; situated, 8\u20139; thinking through one's, 63\u201370; in _Torn in Two: The Experience of Maternal Ambivalence_ , , 102\u20133\n\nAndrew, Barbara,\n\nAndrews, Jorella,\n\nAndrocentric culture, ,\n\nAnger, maternal: bitter resentment and, , ; faces of dependents inspiring, 9\u201310, ; facial gestures of, ; at having to interrupt one's work, ; Al-Saji on immediacy of other's, ; smoldering rage, ; tenderness and,\n\nAnonymity, , 125\u201326\n\n\"The Anonymous Intentions of Transactional Bodies\" (Weiss),\n\nAnthropologic approach,\n\nArcana, Judith, ,\n\n_Arch\u00e9_ ,\n\nAristotle,\n\nArmstrong, LaShanda, 1\u20133\n\nAsthma,\n\n_Aufhebung_ (to eliminate or remove),\n\nBachelard, Gaston,\n\nBadinter, Elisabeth, 30\u201331\n\nBeauvoir, Simone de, , , ; on actions guided by generosity, 165\u201369; on circumstance and situation, , 161\u201362, 173\u201374, 212 _n_ 23; on flight toward one's own project, 160\u201363, , 211 _n_ 3; against idealization of motherhood, 171\u201379; on limitations and failure, 11\u201312, , , 185\u201388, 214 _n_ 99; on male philosophers borrowing from women's experiences, ; on maternal ambivalence, 177\u201378; on maternal devotion, 171\u201381; on maternal love and creation of freedom, 181\u201382; rejoicing in failure and success, , 186\u201387; Sartre's work in relation to, 165\u201367; _The Second Sex_ , , , , 163\u201365, 169\u201372, 212 _n_ 23; on self-doubt as essential, 186\u201388; on social support, 181\u201383\n\nBeauvoir scholarship, , 164\u201366; Jardin's critique, ; Lloyd's critique, ; responses to critique, 167\u201368\n\n_Becoming a Mother_ (Oakley),\n\n\"Becoming with Child\" (Sartre), 209 _n_ 61\n\n_Befindlichkeit_ (how you find yourself in the world),\n\n_Being and Nothingness_ (Sartre), 139\u201340; Beauvoir's work in relation to, 165\u201367\n\n\"Being Torn: Toward a Phenomenology of Unwanted Pregnancy\" (Lundquist), , 168\u201369\n\nBergman, Anni, 203 _n_ 125\n\nBergo, Bettina,\n\nBergoffen, Debra, ,\n\nBigwood, Carol, 210 _n_ 72\n\n\"The Birth of Sexual Difference\" (Guenther), 208 _n_ 15\n\n_Blind Infants and Their Mothers: An Examination of the Sign System_ (Fraiberg),\n\nBondas, Teresa,\n\nBowen, Murray,\n\nBreast cancer,\n\nBreast-feeding: as alignment of perspectives and bodies, ; ambiguous because it exceeds us yet is us, 143\u201344; both mother and child impacted by, 39\u201340; Damasio's singularity of self contradicted, , , 137\u201338; depletion from, 46\u201347; fight or flight response to, ; as intertwining of self and other, 38\u201339\n\nBrody, Donna, ,\n\nBuber, Martin, 69\u201370\n\nButler, Judith,\n\nCarbone, Mauro,\n\n\"Care and Justice in the Global Context\" (Held), 199 _n_ 18\n\nCare ethics theorists, ; on feminine caretaking, , 205 _n_ 26; nonconsideration of abortion, 199 _n_ 10\n\nCare giving, 4\u20136, ; care ethics theorists on feminine, , 205 _n_ 26; children's vulnerability, ; collaboration with other disciplines, ; Damasio's singularity of self contradicted, , , 137\u201338; differences in ethical orientations, ; fundamental principles of, 16\u201317; having dependents as key factor in, 9\u201310; international relations perspective, ; personal independence ignored, ; prefrontal cortex development from, ; rights-based ethical theory of, ; St. Aubin's, 174\u201375; Tronto on,\n\n\"A Case of Obsessional Neurosis\" (Freud),\n\nCesarean section, , 206 _n_ 50\n\nChandler, Mielle, 62\u201363; on mother as more and less than one, 58\u201372, , 203 _n_ 110\n\nChanter, Tina,\n\nChazal, Malcolm de,\n\nChiasm, 119\u201322; \"The Invisible of the Flesh: A Reading of Merleau-Ponty, _The Visible and the Invisible_ , 'The Intertwining\u2014the Chiasm,'\" 123\u201324; P. E. Young on,\n\n_The Child's Relations with Others_ (Merleau-Ponty), 113\u201314,\n\nChoosing not to choose,\n\nCoconstitutionality, with fetus,\n\nCommunity, ; human survival and, 113\u201314,\n\nCraig, David Alexander, 198 _n_ 20\n\nCrohn's disease,\n\n_The Cultural Contradictions of Motherhood_ (Hays), 202 _n_ 69\n\nDally, Ann,\n\nDamasio, Antonio, ; on singularity of self, , , 137\u201338\n\nDarwin, Charles,\n\nDehiscence, , 111\u201323\n\nDeleuze, Gilles,\n\nDe Marneffe, Daphne, 29\u201330, , 53\u201354\n\nDepletion, 46\u201347\n\nDepression: from abnegation of self, ; Beauvoir's emphasis on social support, 181\u201383; _Down Came the Rain: My Journey Through Postpartum Depression_ , ; Heidegger's concept of _Befindlichkeit_ , ; postpartum, , 49\u201350,\n\nDerrida, Jacques, 89\u201390; Murphy on, 149\u201350\n\nDesmond, William,\n\nDeutsch, Helene,\n\n_devoir \u00eatre_ (ought to be), ; Beauvoir on, , , 214 _n_ 99; Guenther on, ; Levinas on, 101\u20132,\n\nDevotion, maternal: Beauvoir on, 171\u201381; Freud on, 178\u201379; Levinas on, ; Levine's essay on, 179\u201380; _see also_ love, maternal\n\nDialectic: of dehiscence, 142\u201344; in Hegel's logic, 135\u201337; Merleau-Ponty on, 131\u201332, 136\u201337\n\nDisabilities, children with, 95\u201396, 104\u20135\n\n\"Distance and Relation\" (Buber), 69\u201370\n\nDomestic violence,\n\n\"Domination and Dialogue in Merleau-Ponty's _Phenomenology of Perception_ \" (Sullivan),\n\n_Down Came the Rain: My Journey Through Postpartum Depression_ (Shields),\n\nDunlop, Rishma,\n\nDysentery amoebas,\n\n_\u00e9cart_ (necessary separation), , 118\u201319, 128\u201330\n\n\" _\u00c9cart_ :The Space of Corporeal Difference\" (Weiss), 128\u201329\n\nE. coli bacteria,\n\nEdwards, Julie Olsen,\n\n\"Emancipated Subjectivities and the Subjugation of Mothering Practices\" (Chandler), 62\u201363, 203 _n_ 110\n\n_The Encyclopedia Logic_ (Hegel), 131\u201346\n\n_Endangered Children: Neonaticide, Infanticide, Filicide_ (Schwartz),\n\n\"Envisioning the Other: Lacan and Merleau-Ponty on Subjectivity\" (Fielding), 129\u201330\n\nEriksson, Katie,\n\nEthics, , , ; alterity as interruption of self-fulfillment, 98\u201399; Aristotle on luck and, ; Beauvoir on actions guided by generosity, 165\u201369; Beauvoir on importance of interpersonal struggles, 169\u201372; Beauvoir's, applied to motherhood, 170\u201385; Beauvoir's existential morality, 159\u201370, 211 _n_ 4, 214 _n_ 99; biblical perspective, 84\u201387, ; choosing not to choose, ; dependency relations at core of ethical theory, ; ethical life as ongoing struggle, 100\u20131; ethical solipsism, , ; ethical theory as mechanism of oppression, 17\u201318; human beings as always being-for-the-other, 147\u201348; \"The Ineradicable Danger of Ambiguity at Chi]asm's Edge,\" ; infinite responsibility, , [96\u201397, , ; insurmountable alterity as opening for, 189\u201395; Levinas on, 94\u201398; Levinas's undermining his own, 138\u201339; _Maternal Ethics and Other Slave Moralities_ , ; Merleau-Ponty's ethical ambivalence, , 111\u201323; in _The Mother Knot_ , , , ; negative aspects of mothering, , 199 _n_ 19; overly masculine history of, ; role of self-reproach in, 186\u201387; Ruddick on, ; in _The Visible and the Invisible_ , 147\u201352; beyond Western context, 197 _n_ 16\n\n_Ethics and Infinity_ (Levinas),\n\nExistential condition, ; Beauvoir on pregnancy, 172\u201373; Merleau-Ponty on, 109\u201310, 111\u201323\n\nExodus,\n\nFace of the other, 9\u201310, , ; ethical epiphany of, 94\u201395\n\n_Family Evaluation_ (Bowen),\n\nFather: in Abraham and Issac, ; absent, ; abusive, ; daughter's emancipation from, ; in Freudian interpretation, ; in Levinas's _Time and the Other_ , 79\u201381; in Levinas's _Totality and Infinity_ , ; postpartum depression of, ; son's birth as extension into future, 75\u201376; as \"the detached one,\" 19\u201320; _see also_ patriarchal values\n\n_The Feeling of What Happens_ (Damasio), , 137\u201338\n\nFemale friends,\n\nFeminist ethics, , 198 _n_ 3; care ethics as \"slave morality,\" , ; challenging androcentrism, , ; on choices, ; \"Grounding Agency in Depth: The Implications of Merleau-Ponty's Thought for the Politics of Feminism,\"\n\nFeminist scholarship: Beauvoir scholarship, , 164\u201368; Levinas as problematic for, , 77\u201380, ; Merleau-Ponty as resource for, 125\u201327, 207 _n_ 10; on parallel between mother and flesh, 122\u201323; \"Phenomenology and Feminism: Perspectives on Their Relation,\" ; \"Through the Wild Region: An Essay in Phenomenological Feminism,\"\n\nFetus: coconstitutionality with, ; as parasite, , , , 198 _n_ 20, 205 _n_ 20; responsiveness of, 41\u201342; as separate person, 42\u201343\n\nFielding, Helen, 129\u201330\n\nFight or flight response,\n\nFilicide: Meyer and Oberman on, ; Michulski case of, ; purposeful, 1\u20132; to spare from worse fate, 3\u20134\n\nFisher, Linda,\n\nFlesh, continuity of, ; ambiguous intersubjectivity and, 10\u201311; \"From the Body Proper to Flesh: Merleau-Ponty on Intersubjectivity,\" 125\u201326, 207 _n_ 10; \"The Invisible of the Flesh: A Reading of Merleau-Ponty, _The Visible and the Invisible_ , 'The Intertwining\u2014the Chiasm,'\" 123\u201324; maternity as dehiscence in, 109\u201310, 111\u201323; \"Merleau-Ponty and Irigaray in the Flesh,\" ; \"Responsivity of the Body: Traces of the Other in Merleau-Ponty's Theory of Body and Flesh,\" ; to share one's body is to share one's self,\n\n\"FLESH: Towards the History of a Misunderstanding\" (Carbone),\n\n_For Your Own Good: Hidden Cruelty in Childrearing and the Roots of Violence_ (Miller),\n\n\"The Foundations of Merleau-Ponty's Ethical Theory\" (Low),\n\nFraiberg, Selma,\n\nFreedom: Andrew on recognition of others', ; Beauvoir on maternal love and creation of, 181\u201382; Beauvoir on morality and, 160\u201365; in _The Encyclopedia Logic_ , 134\u201336; homicide and possibility of, 99\u2013100; letting go, 56\u201358, , 183\u201384; Levinas on alterity and, , 94\u201398; in _Phenomenology of Spirit_ , 136\u201337; in \"Pyrrhus and Cineas,\" 211 _n_ 8; restricted during pregnancy,\n\nFreud, Sigmund: on maternal devotion, 178\u201379; on pairs of contraries, 200 _n_ 28; \"Thoughts for the Times on War and Death,\" 34\u201335; _Totem and Taboo_ ,\n\n_From Beer to Maternity_ (Simone), 184\u201385\n\n\"From the Body Proper to Flesh: Merleau-Ponty on Intersubjectivity\" (Stawarska), 125\u201326, 207 _n_ 10\n\nFullbrook, Edward,\n\nFullbrook, Kate,\n\nGantt, Edwin, 147\u201348\n\nGenerosity: Beauvoir on actions guided by, 165\u201369; of pregnant body, , , 198 _n_ 20, 205 _n_ 20\n\nGiardia,\n\n_The Gift of the Other_ (Guenther),\n\nGilligan, Carol, , 20\u201321, 198 _n_ 3, 199 _n_ 10\n\nGood Mother idealization: Beauvoir's rejection of, 171\u201381; Levine on oppressive nature of, 179\u201380; Ruddick on, 199 _n_ 23\n\n_Grapes of Wrath_ (Steinbeck),\n\nGrosz, Elizabeth,\n\n\"Grounding Agency in Depth: The Implications of Merleau-Ponty's Thought for the Politics of Feminism\" (Fielding),\n\n_The Guardian_ ,\n\nGuenther, Lisa: \"The Birth of Sexual Difference,\" 208 _n_ 15; on _devoir \u00eatre_ , ; _Grapes of Wrath_ analogy by, ; on Levinas, 84\u201387, 90\u201392,\n\nGuilt, ; Beauvoir on, 171\u201381; Good Mother idealization causing, ; Levine on oppressive nature of Good Mother ideal, 179\u201380; role of self-reproach in ethics, 186\u201387; Ruddick on, 199 _n_ 23\n\n\"Handshake and Thanking Ritual\" (Ukele),\n\n_The Happiest Baby on the Block_ (Karp),\n\nHardy, Rob,\n\nHarris, Marlene, 103\u20134\n\nHartley, Gillian,\n\nHasselman, Erik,\n\n\"Hate in the Countertransference\" (Winnicott), 32\u201333\n\nHatred: in Hrdy's analysis, ; love and, , , 200 _n_ 30; those who contain versus those who retaliate, 32\u201333; _see also_ ambivalence, maternal\n\nHays, Sharon, 202 _n_ 69\n\nHegel, Georg Wilhelm Friedrich, ; critiques of, 137\u201338; _The Encyclopedia Logic_ , 131\u201346; immanent logic of human existence, ; Merleau-Ponty's Hegelian thinking and language, 130\u201346, ; _Phenomenology of Spirit_ , 131\u201337\n\n\"Hegel's Existentialism\" (Merleau-Ponty),\n\nHeidegger, Martin,\n\nHeld, Virginia, , ; \"Care and Justice in the Global Context,\" 199 _n_ 18\n\nHermeneutic dialogue,\n\nHigh blood pressure,\n\nHobbes, Thomas,\n\nHodgkin's lymphoma,\n\nHomicide: Beauvoir's emphasis on social support, 181\u201383; filicide, 1\u20134, ; freedom and possibility of, 99\u2013100; infanticide, 200 _n_ 13; intense intimacy and violence, 148\u201349; \"Monster Stories: Women Charged with Perinatal Endangerment,\" ; neonaticide, ; -suicide, 1\u20133,\n\nHrdy, Sarah Blaffer,\n\nHusserl, Edmund,\n\nHyperdialectic, 136\u201337,\n\n_In a Different Voice_ (Gilligan),\n\nIndividuation-separation process, in infant development, 203 _n_ 125\n\n\"The Ineradicable Danger of Ambiguity at Ch[i]asm's Edge\" (P. E. Young),\n\nInfanticide, 200 _n_ 13\n\nIntegrity, bodily, 207 _n_ 13\n\nInterpersonal dependence,\n\n\"Interrogation and Dialectic\" (Merleau-Ponty),\n\n\"Introduction to an Ethics of Ambiguity\" (Beauvoir), , , 185\u201386, 211 _n_ 4, 214 _n_ 99\n\n\"The Invisible of the Flesh: A Reading of Merleau-Ponty, _The Visible and the Invisible_ , 'The Intertwining\u2014the Chiasm'\" (Irigaray), 123\u201324\n\nIrigaray, Luce, , 123\u201324; Bergoffen on, ; on placental economy, ; Weiss on, 128\u201329\n\nIsaiah, 84\u201386,\n\nIsraelites, 86\u201387, ; Abraham and Isaac,\n\nJanus (Roman god of entrances and exits),\n\nJardin, Alice,\n\nKant, Immanuel, ,\n\nKarp, Harvey,\n\nKarr, May, 184\u201385\n\nKatz, Claire Elise, , 198 _n_ 19\n\nKerr, Michael,\n\nKirk, David,\n\nKittay, Eva, 18\u201319, , , 205 _n_ 26; on child with disabilities, 95\u201396; on mother-child relationship, 21\u201322,\n\nKlein, Melanie, 33\u201334, 101\u20132\n\nKohlberg, Lawrence,\n\nKorbin, Jill, 3\u20134\n\nKovacic, Kristin, 184\u201385\n\nKristeva, Julia, , 63\u201364\n\nKruks, Sonia,\n\nKunz, George,\n\nLacan, Jacques, ; \"Envisioning the Other: Lacan and Merleau-Ponty on Subjectivity,\" 129\u201330\n\nLazarre, Jane, , 51\u201353,\n\nLerner, Sharon, 174\u201375\n\nLetting go, 56\u201358, ; Rich on, 183\u201384\n\nLevinas, Emmanuel: on alterity, ethics, and freedom, , 94\u201398; ambiguous intersubjectivity in, 92\u201394, , 100\u20131, 107\u20138; Biblical references, 84\u201386, ; circular hermeneutics of, 89\u201390; denial of maternal ambivalence, 100\u2013102; denial of mothers' limitations, 104\u20135; on the feminine, 73\u201374, 76\u201379; feminist scholars on, , 77\u201380, ; Guenther on, 84\u201387, 90\u201392, ; on infinite responsibility, , 96\u201397, , ; on lost independence, 99\u2013100; maternity as metaphor, 80\u201381, 89\u201390; on paternity, 75\u201376, ; as stuck in dialectical, 138\u201339; on subjectivity and proximity, 92\u201393, 97\u201398, ; _Time and the Other_ , , , , , , ; _Totality and Infinity_ , , 79\u201380, , ,\n\n_Levinas, Judaism and the Feminine: The Silent Footsteps of Rebecca_ (Katz),\n\nLevine, Sara, 179\u201380\n\n\"Like a Maternal Body: Emmanuel Levinas and the Motherhood of Moses,\"\n\nLiss, Andrea,\n\n_Lit_ (Karr), 184\u201385\n\n_Living Across and Through Skins_ (Sullivan), 124\u201325\n\n\"Living as a Woman and Thinking as a Mother in Japan\" (Sakiko), 197 _n_ 16,\n\nLloyd, Genevieve,\n\nLove, maternal: Beauvoir on, 181\u201382; hatred and, , , 200 _n_ 30; _Love's Labor_ , 205 _n_ 26; _Mother Love: Myth and Reality_ , 30\u201331; _see also_ devotion, maternal\n\n_Love's Labor_ (Kittay), 205 _n_ 26\n\nLow, Douglas,\n\nLundquist, Caroline, , 168\u201369\n\nMahler, Margaret, 203 _n_ 125\n\n\"Maintenance Art: Rinsing a B.M. Diaper\" (Ukele),\n\nMaker, William,\n\nManninen, Bertha Alvarez, 199 _n_ 21\n\nMaples, Jeanette, 191\u201393\n\n_The Mask of Motherhood_ (Maushart), 60\u201361\n\nMaternal ambivalence, _see_ ambivalence, maternal\n\n_Maternal Desire_ (De Marneffe),\n\n_Maternal Ethics and Other Slave Moralities_ (Willett),\n\nMaternal instinct, 29\u201331, 103\u20134; Beauvoir on, 173\u201374\n\n_Maternal Instinct_ (de Marneffe), 29\u201330\n\n_Maternal Thinking: Toward a Politics of Peace_ (Ruddick), , 65\u201366\n\nMaushart, Ann, 60\u201361\n\nMcAnulty, Angela, 191\u201393\n\nMcMahon, Martha,\n\nMeningitis,\n\nMental competency, to stand trial,\n\nMerleau-Ponty, Maurice, , 10\u201311, ; on acceptance of contradictions, 141\u201342; \"The Adult's View of the Child,\" ; Bigwood on, 210 _n_ 72; critique of Sartre, ; on dialectic, 131\u201332, 136\u201337; on dialectic of dehiscence, 142\u201344; \"Envisioning the Other: Lacan and Merleau-Ponty on Subjectivity,\" 129\u201330; ethics of, 147\u201352; on existential condition, 109\u201310, 111\u201323; as feminist resource, 125\u201327, 207 _n_ 10; Hegelian logic and language in, 130\u201346, ; on incompleteness of reversibility, 127\u201330, 208 _n_ 32; Irigaray on, 123\u201324; _The Phenomenology of Perception_ , , 109\u201310, 111\u201323, ; \"The Philosopher and his Shadow,\" 208 _n_ 15; Stawarska on, 125\u201326, 207 _n_ 10; _The Visible and the Invisible_ , 109\u201310, 117\u201323, ; Waldenfels on,\n\n\"Merleau-Ponty and Irigaray in the Flesh\" (Grosz),\n\nMeyer, Cheryl, 1\u20132, ; on filicide, ; on murder-suicides, 3\u20134\n\nMichulski, Joanne,\n\nMiller, Alice,\n\nMirvish, Adrian,\n\n_Misconceptions_ (Wolf), 51\u201352, 206 _n_ 50\n\n\"Monster Stories: Women Charged with Perinatal Endangerment\" (Tsing), 103\u20134\n\n\"Moral Idealism and Political Realism\" (Beauvoir), , 185\u201386\n\nMoral stage theory,\n\nMoses, 86\u201387,\n\nMoses, Kate,\n\nMother: as child's first environment, , ; depletion of, 46\u201347; epidemic of exhaustion among, 182\u201383; insurmountable alterity as opening for ethics, 189\u201395; limitations of, 104\u20135; maternal body, 205 _n_ 20; resistance to biological destiny, 145\u201346; as simultaneously more and less than one, 58\u201372, , 203 _n_ 110; social neglect of, 182\u201383\n\nMother-child relationship, , 200 _n_ 30; Beauvoir on importance of interpersonal struggles, 169\u201372; common meanings constituting, 113\u201314; complexities of, 4\u20139; Kittay on, 21\u201322, ; letting go inherent in, 56\u201358, , 183\u201384; role of social support in, 128\u201329; Ruddick on, ; unique features of, ; _see also specific characteristic_\n\nMotherhood: Beauvoir's ethics applied to, 170\u201385; checking with true complexities of, ; daily commitment in, 105\u20136; dependency relations at core of ethical theory, ; ecstasy of mutuality, ; embodied sexual difference, 125\u201327, 207 _n_ 10; Held on, , 199 _n_ 18; idealization of, 7\u20138, 102\u20133, 171\u201379; interdisciplinary approach to, 23\u201324\n\n_The Mother Knot_ (Lazarre), , ,\n\n_Mother Love: Myth and Reality_ (Badinter), 30\u201331\n\n_Mother Nature: Maternal Instincts and How They Shape the Human Species_ (Hrdy),\n\n_Mothers: Their Power and Influence_ (Dally),\n\n_Mothers Who Kill Their Children_ (Meyer), 1\u20132\n\nMurder: Beauvoir's emphasis on social support, 181\u201383; freedom and possibility of, 99\u2013100; infanticide, 200 _n_ 13; intense intimacy and violence, 148\u201349; \"Monster Stories: Women Charged with Perinatal Endangerment,\" ; neonaticide, ; -suicide, 1\u20134\n\nMurphy, Ann, 149\u201350\n\nMutuality, of self and child, 36\u201343; dispute compared to, 187\u201388; ecstasy of, 44\u201345; insurmountable alterity compared to,\n\nNature: in \"The Adult's View of the Child,\" ; exceeds us yet is us, 143\u201344; maternal instinct, 29\u201331, 103\u20134, 173\u201374; natural inclination, 19\u201320, 198 _n_ 5; psychoanalysis of, 122\u201323\n\nNemzoff, Ruth, 61\u201362\n\nNeonaticide,\n\nNietzsche, Friedrich,\n\nNoddings, Nel: on abuse and neglect, ; _Caring_ , 205 _n_ 26; on natural inclination, 19\u201320, 198 _n_ 5, 204 _n_ 132\n\n_No Easy Decision_ , 175\u201376\n\n\"Normal Motherhood: An Exercise in Self-Control\" (Oakley),\n\n_Notebooks for an Ethics_ (Sartre), , 209 _n_ 61\n\n\"Not My Way, Sesha, Your Way, Slowly: Maternal Thinking in the Raising of a Child with Profound Intellectual Disabilities\" (Kittay), 95\u201396\n\nNumbers, Old Testament,\n\nNursing, _see_ breast-feeding\n\nOakley, Ann, ,\n\nOberman, Michelle, 1\u20132, ; on filicide, ; on murder-suicides, 3\u20134\n\n_Of Woman Born_ (Rich),\n\nOlder children, , , , 200 _n_ 13, 201 _n_ 59\n\nOliver, Kelly,\n\nOlkowski, Dorothea,\n\n\"Only Nature Is Mother to the Child\" (Olkowski),\n\n\"On the Maternal Order\" (Rouch), 58\u201359\n\nOppression: Beauvoir on, 160\u201365; ethical theory as mechanism of, 17\u201318; _see also_ freedom\n\n_Otherwise Than Being or Beyond Essence_ (Levinas), , , ; maternity metaphor in, 80\u201381, 89\u201390; proximity and subjectivity in, 92\u201394, 97\u201398,\n\nOxygen mask metaphor,\n\nPacificsm,\n\nPairs of contraries, 200 _n_ 28\n\nParasite, , , , , 198 _n_ 20, 205 _n_ 20; Beauvoir on existential condition of pregnancy, 172\u201373\n\nParker, Rozsika, ; on mother's achievement of ambivalence, 64\u201365, ; _Torn in Two: The Experience of Maternal Ambivalence_ , , 102\u20133; on \"Welcome Little Stranger,\"\n\nParshley, H. M.,\n\nPateman, Carole, 198 _n_ 18, 204 _n_ 9\n\nPatriarchal values: Beauvoir's overturning of, 167\u201368; dependence and independence, 9\u201310; idealization of, 8\u201310; Levinas's appropriation of, 75\u201376, 78\u201379, , , ; _see also_ father\n\n\"Peace and Proximity\" (Levinas),\n\nPerpich, Diane, ,\n\n\"Phenomenology and Feminism: Perspectives on Their Relation\" (Fisher),\n\n_The Phenomenology of Perception_ (Merleau-Ponty), , 109\u201310, 111\u201323,\n\n_Phenomenology of Spirit_ (Hegel), 131\u201337\n\n\"The Philosopher and His Shadow\" (Merleau-Ponty), 208 _n_ 15\n\n_The Philosophy of Simone de Beauvoir: Gendered Phenomenologies, Erotic Generosities_ (Bergoffen),\n\nPiaget, Jean,\n\nPine, Fred, 203 _n_ 125\n\nPlacenta, 58\u201360\n\nPlato,\n\nPollaiolo, Antonio, ,\n\nPostpartum depression, ; after adoption, 49\u201350; _Down Came the Rain:My Journey Through Postpartum Depression_,\n\nPoverty: Beauvoir on freedom and, ; maternity amid, , 174\u201375, , ; in United States, 182\u201383\n\nPrefrontal cortex,\n\nPregnancy: \"The Adult's View of the Child,\" ; Beauvoir on alien invaders, 167\u201368; coconstitutionality with fetus, ; Damasio's singularity of self contradicted, , , 137\u201338; as exception to singular individuality, 38\u201344; fetal responsiveness, 41\u201342; fetus as parasite, , , , 198 _n_ 20, 205 _n_ 20; fetus as separate person, 42\u201343; Kristeva on, ; Lundquist on, ; self-care during, ; to share one's body is to share one's self, ; subject-object dichotomy, 210 _n_ 72; _The Visible and the Invisible_ , 117\u201323\n\n\"Pregnant Embodiment\" (I. M. Young), , 37\u201338,\n\nPrescriptive ethics,\n\n\"The Pro-Choice Pro-Lifer\" (Manninen), 199 _n_ 21\n\nProximity and subjectivity, 92\u201393, 97\u201398; \"Peace and Proximity,\"\n\nPsychoanalysis: Freudian, 33\u201334; Freud on maternal devotion, 178\u201379; Freud on pairs of contraries, 200 _n_ 28; mother as static figure in, 63\u201364; of Nature, 122\u201323; \"Only Nature is Mother to the Child,\"\n\n_The Psychological Birth of the Human Infant_ (Mahler, Pine, and Bergman), 203 _n_ 125\n\n_Psychology of Women_ (Deutsch),\n\nPure being, 134\u201336\n\n\"Pyrrhus and Cineas\" (Beauvoir), 211 _n_ 3, 211 _n_ 6, 211 _n_ 8, 211 _nn_ 10\u201311; on assuming our fundamental ambiguity,\n\nReber, Jeffrey, 147\u201348\n\n\"Reflections on Feminist Merleau-Ponty Skepticism\" (Stoller),\n\nRelational ontology,\n\n\"Renaturalizing the Body (with the Help of Merleau-Ponty)\" (Bigwood), 210 _n_ 72\n\n_The Reproduction of Mothering_ (Chodorow),\n\nResponsiveness, 147\u201348\n\n\"Responsivity of the Body: Traces of the Other in Merleau-Ponty's Theory of Body and Flesh\" (Waldenfels),\n\nReversibility, 127\u201330, 208 _n_ 32\n\nRich, Adrienne, , , ; on guilt, ; on letting go, 183\u201384\n\nRouch, H\u00e9l\u00e8ne, 58\u201359\n\nRuddick, Sara, , , 65\u201366, , ; on Edwards's maternal thinking, ; on Good Mother idealized image, 199 _n_ 23; on mother-child relationship, ,\n\n_Ruota_ (rotating barrel in which babies could be left), 200 _n_ 13\n\nAl-Saji, Alia, ,\n\nSakiko, Kitagawa, 197 _n_ 16,\n\nSartre, Jean Paul, , , , 209 _n_ 61; Beauvoir's work in relation to, 165\u201367; _Being and Nothingness_ , 139\u201340, 165\u201367\n\n\"Sartre, Embodied Minds, Authenticity, and Childhood\" (Mirvish),\n\nScarth, Fedrika, , 167\u201368,\n\nSchwartz, Lita Linzer,\n\n_The Second Sex_ (Beauvoir), , , 212 _n_ 23; existential morality in, 163\u201365; importance of interpersonal struggles, 169\u201372; Parshley's translation of,\n\nSelf and other: breast-feeding as intertwining of, 38\u201339; female friends as ideal of, ; _The Gift of the Other_ , ; Levinas's _Time and the Other_ , , , , , , ; _Living Across and Through Skins_ , 124\u201325; maternal ambivalence as achievement of coexisting, 64\u201365, ; \"Responsivity of the Body: Traces of the Other in Merleau-Ponty's Theory of Body and Flesh,\" ; to share one's body is to share one's self, ; as subjectobject dichotomy, 210 _n_ 72\n\nSelf care: care-giving intertwined with, 10\u201313; other-care balanced with, ; during pregnancy,\n\nSelf determination, deferred,\n\nSelf-forgetting, 43\u201344\n\nSelfhood: abnegation of, 50\u201353; lament for self that is not food for others, , , 206 _n_ 50; maiden transformed, 46\u201347; singularization of ego,\n\nSelf-reproach, 186\u201387\n\nSelf-sacrifice, 52\u201353\n\n_The Sexual Contract_ (Pateman), 198 _n_ 18, 204 _n_ 9\n\n_She Came to Stay_ (Beauvoir),\n\nShields, Brooke, ,\n\nSikka, Sonia,\n\nSimms, Eva, 43\u201344; child isolation and sensory integration, 115\u201316; mother as child's first environment, ,\n\nSimone, Maggie Lamond, 184\u201385\n\nSimons, Margaret, ; Beauvoir's interview with,\n\nSingular individuality: Damasio on, ; pregnancy as exception to, 38\u201344\n\nSingularity of self, , , 137\u201338\n\nSingularization, of ego,\n\nSituation: of andocentric culture, ; Aristotle on, ; Beauvoir on social support, 181\u201383; Beauvoir's emphasis on power of, , 160\u201365, 173\u201374, 212 _n_ 23; maternal ambivalence and, 8\u20139; maternity amid poverty, , 174\u201375, , ; moral imperative emerging from, 154\u201355\n\nSlavery, ; Beauvoir on freedom, 160\u201365; devotion akin to, ; in \"Pyrrhus and Cineas,\" 211 _n_ 8; Willett's _Maternal Ethics and Other Slave Moralities_ ,\n\nSocial contract, 204 _n_ 9\n\nSocial support: Beauvoir's emphasis on, 181\u201383; lack of, 191\u201393; Weiss on, 128\u201329\n\n\"Sociobiological and Social Constructionist Accounts of Altruism: A Phenomenological Critique\" (Gantt and Reber), 147\u201348\n\nSolitude: Beauvoir on, 211 _nn_ 11; Merleau-Ponty on communication and,\n\n\"Some Kinds of Maintenance Cancel Others Out, Keep Your Head Together\u20141,000 Times, or Babysitter Hangup\u2014Incantation Ritual\" (Ukele), 54\u201355,\n\nSong of Songs,\n\nSpeculative moment, , 135\u201337; examples of, 54\u201355, 63\u201364, ; as transformation of contradictions, 141\u201342\n\n\"Stabat Mater\" (Kristeva), 63\u201364\n\n_Stanford Encyclopedia of Philosophy_ ,\n\nSt. Aubin, Gina, 174\u201375\n\nStawarska, Beata, 125\u201326, 207 _n_ 10\n\nSteinbeck, John,\n\nSteingraber, Sandra, 59\u201360\n\nStockholm syndrome,\n\nStoller, Silvia,\n\nStone, Elizabeth, ,\n\nStringer, Kristan,\n\n_The Structure of Behavior_ (Merleau-Ponty),\n\nSubjectivity and proximity, 92\u201393, 97\u201398,\n\nSubject-object dichotomy, 210 _n_ 72\n\nSuicide: Beauvoir's emphasis on social support, 181\u201383; Meyer on, 3\u20134\n\nSullivan, Shannon, 124\u201325\n\nTaylor, Chlo\u00e9, 82\u201383\n\n\"Teaching Sartre About Freedom\" (Kruks),\n\nTeapot metaphor,\n\n_Theaetetus_ (Plato),\n\n\"Thoughts for the Times on War and Death\" (Freud), 34\u201335\n\nThou shalt not kill,\n\n\"Through the Wild Region: An Essay in Phenomenological Feminism\" (Allen),\n\n\"Throwing the Book\" (Levine), 179\u201380\n\nTidd, Ursula,\n\n_Time and the Other_ (Levinas), , , , , ,\n\n_Torn in Two: The Experience of Maternal Ambivalence_ (Parker), , 102\u20133\n\n_Totality and Infinity_ (Levinas), , 79\u201380, , ; on alterity, , 94\u201398; paradoxical power of murder,\n\n_Totem and Taboo_ (Freud),\n\nTrichomonad protozoans,\n\nTronto, Joan,\n\nTsing, Anna Lowenhaupt, 103\u20134\n\nTuana, Nancy, 198 _n_ 18, 204 _n_ 1\n\nUkele, Mierle Laderman, 54\u201355,\n\nUlcerative colitis,\n\nUnited States: maternity amid poverty, , ; poverty in, 182\u201383; _The War on Moms: On Life in a Family-Unfriendly Nation_ ,\n\n_A Very Easy Death_ (Beauvoir), , 186\u201387\n\nVincent, Riven, 104\u20135\n\nViolence, , ; domestic, ; intense intimacy and, 148\u201349; \"Monster Stories: Women Charged with Perinatal Endangerment,\" ; weakness inspiring, ; _see also_ murder\n\n_The Visible and the Invisible_ (Merleau-Ponty), 109\u201310, 117\u201323; critique of Sartre in,\n\n\"Vision, Mirror, and Expression: The Genesis of the Ethical Body in Merleau-Ponty's Later Works\" (Al-Saji),\n\n\"Vision, Violence, and the Other: A Merleau-Pontian Ethics\" (Andrews),\n\nVulnerability: coercive power of, ; dependency relations at core of ethical theory,\n\nWaldenfels, Bernhard,\n\nWar: \"Thoughts for the Times on War and Death,\" 34\u201335; thou shalt not kill, ; _The War on Moms: On Life in a Family-Unfriendly Nation_ ,\n\nWard, Julie,\n\n_The War on Moms: On Life in a Family-Unfriendly Nation_ (Lerner),\n\nWeiss, Gail, , 128\u201329\n\n\"Welcome Little Stranger\" pincushion,\n\nWhite blood cells,\n\n\"White Mythology: Metaphor in the Text of Philosophy\" (Derrida), 89\u201390\n\nWillett, Cynthia,\n\nWinnicott, D. W., 32\u201333,\n\nWolf, Naomi, 51\u201352, , 206 _n_ 50\n\nWomb environment, mimicking of,\n\nWomb envy,\n\n_Women and the History of Philosophy_ (Tuana), 198 _n_ 18, 204 _n_ 1\n\n\"Women Rocking the Boat\" (Yusa), 197 _n_ 16,\n\nWomen's movement,\n\n\"Written on the Body\" (Dunlop),\n\nWynn, Francine,\n\n\"Yogurt Couvade\" (Hardy),\n\nYoung, Iris Marion, , 37\u201338,\n\nYoung, Phillip E.,\n\nYusa, Michiko, 197 _n_ 16,\n\nZen realization, 197 _n_ 16\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n\n\nProduced by Paul Murray, Chuck Greif and the Online\nDistributed Proofreading Team at http:\/\/www.pgdp.net\n\n\n\n\n\n\n\n\n\n\n[Illustration: image of book's spine]\n\n[Illustration: image of book's cover]\n\n[Illustration: DON JOHN OF AUSTRIA.\n\nFROM THE ORIGINAL IN THE ROYAL MUSEUM AT MADRID.\n\nLondon: George Routledge & Sons, Broadway, Ludgate Hill.]\n\n\n\n\nHISTORY OF THE REIGN\n\nOF\n\nPHILIP THE SECOND\n\n_KING OF SPAIN_\n\nVOLUME THE THIRD\n\nAND\n\nBIOGRAPHICAL & CRITICAL MISCELLANIES\n\nBY\n\nWILLIAM H. PRESCOTT\n\n\nLONDON\n\nGEORGE ROUTLEDGE AND SONS\n\nBROADWAY, LUDGATE HILL\n\nNEW YORK: 416, BROOME STREET\n\n   PRESCOTT'S WORKS.\n\n  _One-Volume Edition._\n\n  FERDINAND AND ISABELLA, 5s.\n  CONQUEST OF MEXICO. 5s.\n  CONQUEST OF PERU. 5s.\n  PHILIP THE SECOND. Vols. I. and II. in One Vol., 5s.\n  PHILIP THE SECOND. Vol. III., and ESSAYS, in One Vol., 5s.\n  CHARLES THE FIFTH. 5s.\n\n\n\n\nCONTENTS\n\nOF\n\nTHE THIRD VOLUME.\n\n\nBOOK V.\n\n\nCHAPTER I.\n\n                                                 PAGE\n\nTHE MOORS OF SPAIN                                  1\n\nConquest of Spain by the Arabs                      1\n\nHostility between the Two Races                     2\n\nThe Country recovered by the Spaniards              2\n\nEffect of the Struggle on the National Character    2\n\nReligious Intolerance of the Spaniards              3\n\nAttempts to convert the Moslems                     3\n\nPolicy of Ximenes                                   3\n\nSuppression of the Mahometan Worship                4\n\nOutward Conformity to Christianity                  4\n\nMoors abandon their National Habits                 4\n\nTheir Condition under Philip the Second             5\n\nTheir Industry and Commerce                         5\n\nTreatment by the Government                         6\n\nOrdinance of 1563                                   8\n\nStringent Measures called for by the Clergy         9\n\nPrepared by the Government                          9\n\nSeverity of the Enactments                         10\n\nApproval of them by Philip                         11\n\nProclamation at Granada                            12\n\nIndignation of the Moriscoes                       12\n\nRepresentations to Deza                            12\n\nAppeal to the Throne                               13\n\nRejection of their Prayers                         14\n\n\nCHAPTER II.\n\nREBELLION OF THE MORISCOES                         14\n\nThe Edict enforced                                 14\n\nPlans for Resistance by the Moriscoes              15\n\nTheir Descent on Granada                           16\n\nFailure of the Attempt                             16\n\nGeneral Insurrection                               17\n\nElection of a King                                 17\n\nCharacter of Aben-Humeya                           18\n\nHis Coronation                                     18\n\nHis Preparations for Defence                       19\n\nThe Christian Population                           19\n\nUnsuspicious of their Danger                       19\n\nAttacked by the Moors--Panic                       20\n\nGeneral Massacre                                   21\n\nHorrible Cruelties                                 21\n\nFate of the Women and Children                     22\n\nFierceness of Aben-Farax                           23\n\nDeposed from his Command                           23\n\n\nCHAPTER III.\n\nREBELLION OF THE MORISCOES                         24\n\nConsternation in the Capital                       24\n\nMutual Fears of the Two Races                      24\n\nGarrison of the Alhambra strengthened              25\n\nTroops mustered by Mondejar                        25\n\nCivic Militia--Feudal Levies                       25\n\nWarlike Ecclesiastics                              26\n\nMarch of the Army                                  26\n\nPass of Tablate                                    27\n\nBridge crossed by a Friar                          27\n\nThe Army follows                                   28\n\nThe Moriscoes withdraw                             28\n\nEntrance into the Alpujarras                       28\n\nNight Encampment at Lanjaron                       29\n\nRelief of Orgiba                                   29\n\nMondejar pursues his March                         30\n\nGloom of the Mountain Scenery                      30\n\nDefile of Alfajarali                               30\n\nSudden Attack                                      30\n\nBravery of the Andalusian Knights                  31\n\nPrecipitate Retreat of the Moriscoes               31\n\nCapture of Bubion                                  31\n\nHumanity of Mondejar                               31\n\nSufferings of the Army                             32\n\nCapture of Jubiles                                 33\n\nPrisoners protected by Mondejar                    33\n\nMassacred by the Soldiers                          33\n\nChristian Women sent to Granada                    34\n\nWelcomed by the Inhabitants                        34\n\n\nCHAPTER IV.\n\nREBELLION OF THE MORISCOES                         35\n\nMondejar's Policy                                  35\n\nAben-Humeya at Paterna                             35\n\nOffers to Surrender                                36\n\nFlight to the Sierra Nevada                        36\n\nDisposition of the Moorish Prisoners               37\n\nAttack on Las Guajaras                             38\n\nEvacuated by the Garrison                          38\n\nMassacre ordered by Mondejar                       38\n\nCruelty of the Count of Tendilla                   39\n\nAttempt to capture Aben-Humeya                     39\n\nHis Escape                                         40\n\nHeroism of Aben-Aboo                               40\n\nThe Marquis of Los Velez                           40\n\nHis Campaign in the Alpujarras                     41\n\nCruelties committed by the Troops                  41\n\nCelebration of a religious _Fete_                  42\n\nLicentiousness of the Soldiery                     42\n\nContrast between Mondejar and Los Velez            43\n\nAccusations against the former                     44\n\nDecision arrived at in Madrid                      44\n\nEffect on the Army                                 45\n\nMoorish Prisoners in Granada                       45\n\nRumours circulated in the Capital                  45\n\nNight Attack on the Prisoners                      46\n\nFearful Struggle and Massacre                      46\n\nApathy of the Government                           47\n\nRenewal of the Insurrection                        47\n\n\nCHAPTER V.\n\nREBELLION OF THE MORISCOES                         48\n\nDon John of Austria                                48\n\nBirth and Early History                            49\n\nPlaced under the Care of Quixada                   49\n\nSecresy in regard to his Origin                    50\n\nThe young Geronimo at Yuste                        50\n\nTestamentary Depositions of the Emperor            51\n\nThe Boy presented to the Regent                    51\n\nCurious Scene                                      52\n\nMeeting appointed with the King                    53\n\nPhilip acknowledges his Brother                    53\n\nAssigns him an Establishment                       54\n\nRoyal Triumvirate at Alcala                        54\n\nChivalrous Character of Don John                   55\n\nHis adventurous Disposition                        55\n\nHe is entrusted with the Command of a Fleet        56\n\nHis Cruise in the Mediterranean                    56\n\nHe is selected for the Command in Granada          57\n\nRestrictions on his Authority                      57\n\nHis Reception at Granada                           57\n\nAnswers to Petitioners                             58\n\nDiscussions in the Council of War                  59\n\nNew Levies summoned                                59\n\nIncreased Power of Aben-Humeya                     60\n\nForays into the Christian Territory                60\n\nMovements of Los Velez                             61\n\nExtension of the Rebellion                         61\n\nSuccessful Expedition of Requesens                 61\n\nMoriscoes lay Siege to Seron                       62\n\nSurrender and Massacre of the Garrison             62\n\nDecree for removing the Moriscoes from Granada     63\n\nTheir Consternation and Grief                      63\n\nExpulsion from the City                            64\n\nFarewell to their ancient Home                     64\n\nDistribution through the Country                   64\n\nRuinous Effects on Granada                         65\n\nCharacter of the Transaction                       66\n\n\nCHAPTER VI.\n\nREBELLION OF THE MORISCOES                         66\n\nState of the Troops under Los Velez                66\n\nEncounter with Aben-Humeya                         67\n\nFlight of the Morisco Prince                       67\n\nDesertions from the Spanish Camp                   68\n\nMondejar recalled to Court                         68\n\nHis Character                                      68\n\nExterminating Policy of the Government             69\n\nSensual Tyranny of Aben-Humeya                     69\n\nTreachery towards Diego Alguazil                   70\n\nPlan of Revenge formed by Alguazil                 71\n\nConspiracy against Aben-Humeya                     71\n\nHis Assassination                                  72\n\nHe is succeeded by Aben-Aboo                       72\n\nEnergy of the new Chief                            73\n\nRepulse at Orgiba                                  73\n\nThe Place evacuated by the Garrison                74\n\nContinual Forays                                   74\n\nConflicts in the _Vega_                            75\n\nDon John's desire for Action                       75\n\nPhilip yields to his Entreaties                    76\n\nPreparations for the Campaign                      76\n\nSurprise of Guejar                                 76\n\nMortification of Don John                          77\n\nMendoza the Historian                              77\n\n\nCHAPTER VII.\n\nREBELLION OF THE MORISCOES                         79\n\nPhilip's Instructions to his Brother               80\n\nDon John takes the Field                           80\n\nDiscontent of Los Velez                            80\n\nHis Meeting with Don John                          81\n\nHe retires from the War                            81\n\nInvestment of Galera                               82\n\nDescription of the Place                           82\n\nMunitions and Garrison                             83\n\nEstablishment of Batteries                         84\n\nThe Siege opened                                   84\n\nFirst Assault                                      84\n\nSpaniards repulsed                                 85\n\nMines opened in the Rock                           86\n\nSecond Assault                                     86\n\nExplosion of the Mine                              87\n\nTroops rash to the Attack                          87\n\nStruggle at the Ravelin                            87\n\nBravery of the Morisco Women                       87\n\nIll Success of Padilla                             87\n\nFailure of the Attack                              88\n\nInsubordination of the Troops                      88\n\nSevere Loss of the Spaniards                       88\n\nBloody Determination of Don John                   89\n\nPrudent Advice of Philip                           89\n\nCondition of the Besieged                          89\n\nPreparations for a last Attack                     90\n\nCannonade and Explosions                           91\n\nThird Assault                                      91\n\nIrresistible Fury of the Spaniards                 91\n\nStruggle in the Streets and Houses                 92\n\nDesperation of the Inhabitants                     92\n\nInhumanity of the Conqueror                        92\n\nWholesale Massacre                                 92\n\nThe Town demolished                                94\n\nTidings communicated to Philip                     94\n\nReputation gained by Don John                      94\n\n\nCHAPTER VIII.\n\nREBELLION OF THE MORISCOES                         95\n\nSeron reconnoitred                                 95\n\nSudden Attack by the Moriscoes                     95\n\nArmy thrown into Confusion                         96\n\nIndignation of Don John                            96\n\nDeath of Quixada                                   97\n\nHis Character                                      98\n\nDona Magdalena de Ulloa                            98\n\nRapid Successes of Don John                        98\n\nNegotiations opened with El Habaqui                99\n\nMerciless Pursuit of the Rebels                    99\n\nGuerilla Warfare                                   99\n\nConferences at Fondon                             100\n\nAben-Aboo consents to treat                       100\n\nArrangement concluded                             100\n\nSubmission tendered by El Habaqui                 101\n\nDissatisfaction with the Treaty                   102\n\nVacillation of Aben-Aboo                          102\n\nEl Habaqui engages to arrest him                  103\n\nFate of El Habaqui                                103\n\nMission of Palacios                               104\n\nHis Interview with Aben-Aboo                      104\n\nSpirited Declaration of that Chief                104\n\nStern Resolve of the Government                   104\n\nWar of Extermination                              105\n\nExpedition of the Duke of Arcos                   105\n\nMarch across the Plain of Calaluz                 106\n\nEngagement with the Moriscoes                     106\n\nThe Rebellion crushed                             106\n\nEdict of Expulsion                                106\n\nRemoval of the Moriscoes                          107\n\nDon John's Impatience to Resign                   108\n\nHis Final Dispositions                            108\n\nHiding-place of Aben-Aboo                         109\n\nPlot formed for his Capture                       109\n\nHis Interview with El Senix                       109\n\nHis Murder                                        110\n\nHis Body brought to Granada                       110\n\nHis Head placed in a Cage                         110\n\nRemarks on his Career                             111\n\nWasted Condition of the Country                   112\n\nThe scattered Moriscoes                           112\n\nCruelly treated by the Government                 112\n\nTheir Industry and Cheerfulness                   113\n\nIncrease of their Numbers                         113\n\nThey preserve their National Feeling              114\n\nMutual Hatred of the Two Races                    114\n\nExpulsion of the Moriscoes from Spain             114\n\nWorks of Marmol and Circourt                      114\n\n\nCHAPTER IX.\n\nWAR WITH THE TURKS                                116\n\nSultan Selim the Second                           116\n\nDetermines on the Conquest of Cyprus              116\n\nSpirit of Pius the Fifth                          117\n\nHis Appeal to Philip                              117\n\nKing's Entrance into Seville                      117\n\nDetermines to join the League                     118\n\nCapture of Nicosia                                118\n\nVacillating Conduct of Venice                     118\n\nMeeting of Deputies at Rome                       119\n\nTreaty of Confederation                           119\n\nRatified and proclaimed                           120\n\nTurkish Fleet in the Adriatic                     120\n\nPapal Legate at Madrid                            120\n\nConcessions to the Crown                          121\n\nFleets of Venice and Rome                         121\n\nPreparations in Spain                             121\n\nEnthusiasm of the Nation                          122\n\nDon John's Departure                              122\n\nHis Reception at Naples                           128\n\nHis noble Appearance                              123\n\nAccomplishments and Popularity                    123\n\nPresentation of the Consecrated Standard          124\n\nArrival at Messina                                124\n\nGrand Naval Spectacle                             124\n\nStrength and Condition of the Fleets              125\n\nDiscretion of the Generalissimo                   125\n\nCommunications from the Pope                      126\n\nDeparture from Messina                            126\n\n\nCHAPTER X.\n\nWAR WITH THE TURKS                                126\n\nArrival at Corfu                                  127\n\nCouncil of War                                    127\n\nResolution to give Battle                         127\n\nArbitrary Conduct of Veniero                      128\n\nPassage across the Sea of Iona                    128\n\nFall of Famagosta                                 128\n\nThe Enemy in Sight                                129\n\nPreparations for Combat                           129\n\nFinal Instructions of Don John                    129\n\nApproach of the Turkish Fleet                     130\n\nIts Form and Disposition                          130\n\nChange in the order of Battle                     131\n\nLast Preparation of the Christians                131\n\nBattle of Lepanto                                 132\n\nLeft Wing of the Allies turned                    132\n\nRight Wing, under Doria, broken                   132\n\nDon John and Ali Pasha engaged                    133\n\nSuperior Fire of the Spaniards                    133\n\nBird's-eye View of the Scene                      134\n\nVenetians victorious on the Left                  134\n\nContinued Struggle in the Centre                  135\n\nTurkish Admiral boarded                           135\n\nDeath of Ali Pasha                                135\n\nVictory of the Christians                         136\n\nFlight of Uluch Ali                               137\n\nChase and Escape                                  137\n\nAllies take Shelter in Petala                     137\n\n\nCHAPTER XI.\n\nWAR WITH THE TURKS                                137\n\nLosses of the Combatants                          137\n\nTurkish Armada annihilated                        138\n\nRoll of Slaughter and Fame                        138\n\nExploits of Farnese                               138\n\nNoble Spirit of Cervantes                         139\n\nSons of Ali Pasha Prisoners                       139\n\nGenerously treated by Don John                    139\n\nHis Conduct towards Veniero                       140\n\nOperations suspended                              141\n\nTriumphant Return to Messina                      141\n\nCelebrations in Honour of the Victory             141\n\nTidings despatched to Spain                       142\n\nPhilip's reception of them                        142\n\nAcknowledgments to his Brother                    143\n\nDon John's Conduct criticised                     144\n\nReal Fruits of the Victory                        145\n\nDelay in resuming Operations                      145\n\nDeath of Pius the Fifth                           145\n\nPhilip's Distrust                                 146\n\nPermits his Brother to Sail                       146\n\nTurks decline to accept Battle                    147\n\nAnniversary of Lepanto                            147\n\nAllies disband their Forces                       147\n\nPerfidy of Venice                                 147\n\nThe League dissolved                              148\n\nTunis taken by Don John                           148\n\nHe provides for its Security                      149\n\nReturns to Naples                                 149\n\nHis Mode of Life there                            150\n\nHis Schemes of Dominion                           150\n\nTunis retaken by the Moslems                      150\n\nDon John's Mission to Genoa                       151\n\nHe prepares a fresh Armament                      151\n\nHis Disappointment and Return to Madrid           151\n\n\nBOOK VI.\n\n\nCHAPTER I.\n\nDOMESTIC AFFAIRS OF SPAIN                         153\n\nInternal Administration                           153\n\nRevolutions under Isabella and Charles V.         153\n\nAbsolute Power of the Crown                       154\n\nContrast between Charles and Philip               154\n\nThe latter wholly a Spaniard                      154\n\nThe Royal Councils                                155\n\nPrincipal Advisers of the Crown                   155\n\nCharacter of Ruy Gomez de Silva                   155\n\nFigueroa, Count of Feria                          157\n\nCardinal Espinosa                                 157\n\nTwo Parties in the Council                        159\n\nBalance held by Philip                            159\n\nHis Manner of transacting Business                159\n\nHis Assiduity                                     160\n\nHis Mode of dividing the Day                      161\n\nHis Love of Solitude                              161\n\nExtent of his Information                         161\n\nPartial Confidence in his Ministers               162\n\nHis Frugality                                     162\n\nHis magnificent Establishment                     162\n\nHis fatal Habit of Procrastination                163\n\nRemonstrances of his Almoner                      164\n\nHabits of the great Nobles                        164\n\nManners of the Court                              165\n\nDegeneracy of the Nobles                          165\n\nSplendour of their Households                     165\n\nLoss of Political Power                           166\n\nDepressed Condition of the Commons                166\n\nPetitions of the Cortes                           166\n\nTheir Remonstrance against Arbitrary Government   167\n\nTheir Regard for the National Interests           167\n\nErroneous Notions respecting Commerce             168\n\nSumptuary Laws                                    168\n\nEncouragement of Bull-Fights                      169\n\nVarious Subjects of Legislation                   169\n\nSchools and Universities                          170\n\nRoyal Pragmatics                                  170\n\nPhilip's Replies to the Cortes                    170\n\nFreedom of Discussion                             171\n\nStanding Army                                     171\n\nGuards of Castile                                 171\n\n\nCHAPTER II.\n\nDOMESTIC AFFAIRS OF SPAIN                         172\n\nPhilip the Champion of the Faith                  172\n\nEndowments of the Church                          172\n\nAlienations in Mortmain                           172\n\nDisputed Prerogatives                             173\n\nAppointments to Benefices                         173\n\nThe Clergy dependent on the Crown                 174\n\nThe Escorial                                      174\n\nMotives for its Erection                          174\n\nSite selected                                     175\n\nConvent founded                                   175\n\nRoyal Humility                                    176\n\nBuilding commenced                                176\n\nPhilip's Interest in it                           177\n\nHis Architectural Taste                           177\n\nHis Oversight of the Work                         177\n\nHe governs the World from the Escorial            178\n\nThe Edifice endangered by Fire                    178\n\nMaterials used in its Construction                179\n\nArtists employed                                  179\n\nPhilip's Fondness for Art                         180\n\nCompletion of the Escorial                        180\n\nThe Architects                                    180\n\nCharacter of the Structure                        181\n\nIts Whimsical Design                              181\n\nIts Magnitude                                     181\n\nInterior Decorations                              182\n\nRavages it has undergone                          182\n\nIts present Condition                             182\n\nAnne of Austria                                   183\n\nHer Reception in Spain                            183\n\nHer Marriage with Philip                          184\n\nHer Residence at the Escorial                     185\n\nHer Character and Habits                          185\n\nHer Death                                         185\n\n\n\n\nHISTORY\n\nOF\n\nPHILIP THE SECOND.\n\n\n\n\nBOOK V\n\n\n\n\nCHAPTER I.\n\nTHE MOORS OF SPAIN.\n\nConquest of Spain by the Arabs.--Slow Recovery by the\nSpaniards.--Efforts to convert the Moslems.--Their Homes in the\nAlpujarras.--Their Treatment by the Government.--The Minister\nEspinosa.--Edict against the Moriscoes.--Their ineffectual Remonstrance.\n\n1566, 1567.\n\n\nIt was in the beginning of the eighth century, in the year 711, that the\nArabs, filled with the spirit of conquest which had been breathed into\nthem by their warlike apostle, after traversing the southern shores of\nthe Mediterranean, reached the borders of those straits that separate\nAfrica from Europe. Here they paused for a moment, before carrying their\nbanners into a strange and unknown quarter of the globe. It was but for\na moment, however, when, with accumulated strength, they descended on\nthe sunny fields of Andalusia, met the whole Gothic array on the banks\nof the Guadalete, and, after that fatal battle, in which King Roderick\nfell with the flower of his nobility, spread themselves, like an army of\nlocusts, over every part of the Peninsula. Three years sufficed for the\nconquest of the country,--except that small corner in the north, where a\nremnant of the Goths contrived to maintain a savage independence, and\nwhere the rudeness of the soil held out to the Saracens no temptation to\nfollow them.\n\nIt was much the same story that was repeated, more than three centuries\nlater, by the Norman conquerors in England. The battle of Hastings was\nto that kingdom what the battle of the Guadalete was to Spain; though\nthe Norman barons, as they rode over the prostrate land, dictated terms\nto the vanquished of a sterner character than those granted by the\nSaracens.\n\nBut whatever resemblance there may be in the general outlines of the two\nconquests, there is none in the results that followed. In England the\nNorman and the Saxon, sprung from a common stock, could not permanently\nbe kept asunder by the barrier which at first was naturally interposed\nbetween the conqueror and the conquered; and in less, probably, than\nthree centuries after the invasion, the two nations had imperceptibly\nmelted into one; so that the Englishman of that day might trace the\ncurrent that flowed through his veins to both a Norman and a Saxon\norigin.\n\nIt was far otherwise in Spain, where difference of race, of religion, of\nnational tradition, of moral and physical organization, placed a gulf\nbetween the victors and the vanquished too wide to be overleaped. It is\ntrue, indeed, that very many of the natives, accepting the liberal terms\noffered by the Saracens, preferred remaining in the genial clime of the\nsouth to sharing the rude independence of their brethren in the\nAsturias, and that, in the course of time, intermarriages, to some\nextent, took place between them and their Moslem conquerors. To what\nextent cannot now be known. The intercourse was certainly far greater\nthan that between our New-England ancestors and the Indian race which\nthey found in possession of the soil,--that ill-fated race, which seems\nto have shrunk from the touch of civilization, and to have passed away\nbefore it like the leaves of the forest before the breath of winter. The\nunion was probably not so intimate as that which existed between the old\nSpaniards and the semi-civilized tribes that occupied the plateau of\nMexico, whose descendants, at this day, are to be there seen filling the\nhighest places, both social and political, and whose especial boast it\nis to have sprung from the countrymen of Montezuma.\n\nThe very anxiety shown by the modern Spaniard to prove that only the\n_sangre azul_--\"blue blood\"--flows through his veins, uncontaminated by\nany Moorish or Jewish taint, may be thought to afford some evidence of\nthe intimacy which once existed between his forefathers and the tribes\nof Eastern origin. However this may be, it is certain that no length of\ntime ever served, in the eye of the Spaniard, to give the Moslem invader\na title to the soil; and after the lapse of nearly eight centuries,--as\nlong a period as that which has passed since the Norman conquest,--the\nArabs were still looked upon as intruders, whom it was the sacred duty\nof the Spaniards to exterminate or to expel from the land.\n\nThis, then, was their mission. And it is interesting to see how\nfaithfully they fulfilled it; and during the long period of the Middle\nAges, when other nations were occupied with base feudal quarrels or\nborder warfare, it is curious to observe the Spaniard intent on the one\ngreat object of reclaiming his country from the possession of the\ninfidel. It was a work of time; and his progress, at first almost\nimperceptible, was to be measured by centuries. By the end of the ninth\ncentury it had reached as far as the Ebro and the Douro. By the middle\nof the eleventh, the victorious banner of the Cid had penetrated to the\nTagus. The fortunes of Christian Spain trembled in the balance on the\ngreat day of Navas de Tolosa, which gave a permanent ascendancy to the\nCastilian arms; and by the middle of the thirteenth century the\ncampaigns of James the First of Aragon, and of St. Ferdinand of Castile,\nstripping the Moslems of the other southern provinces, had reduced them\nto the petty kingdom of Granada. Yet on this narrow spot they still\ncontinued to maintain a national existence, and to bid defiance for more\nthan two centuries longer to all the efforts of the Christians. The\nfinal triumph of the latter was reserved for the glorious reign of\nFerdinand and Isabella. It was on the second of January, 1492, that,\nafter a war which rivalled that of Troy in its duration, and surpassed\nit in the romantic character of its incidents, the august pair made\ntheir solemn entry into Granada; while the large silver cross which had\nserved as their banner through the war, sparkling in the sunbeams on the\nred towers of the Alhambra, announced to the Christian world that the\nlast rood of territory in the Peninsula had passed away for ever from\nthe Moslem.\n\n[Sidenote: EFFORTS TO CONVERT THEM.]\n\nThe peculiar nature of the war in which the Spaniard for eight centuries\nhad thus been engaged, exercised an important influence on the national\ncharacter. Generation after generation had passed their lives in one\nlong uninterrupted crusade. It had something of the same effect on the\ncharacter of the nation that the wars for the recovery of Palestine had\non the Crusaders of the Middle Ages. Every man learned to regard himself\nas in an especial manner the soldier of Heaven,--for ever fighting the\ngreat battle of the Faith. With a mind exalted by this sublime\nconviction, what wonder that he should have been ever ready to discern\nthe immediate interposition of Heaven in his behalf--that he should have\nseen again and again the patron saint of his country, charging on his\nmilk-white steed at the head of his celestial chivalry, and restoring\nthe wavering fortunes of the fight? In this exalted state of feeling,\ninstitutions that assumed elsewhere only a political or military aspect\nwore here the garb of religion. Thus the orders of chivalry, of which\nthere were several in the Peninsula, were founded on the same principles\nas those of Palestine, where the members were pledged to perpetual war\nagainst the infidel.\n\nAs a consequence of these wars with the Moslems, the patriotic principle\nbecame identified with the religious. In the enemies of his country the\nSpaniard beheld also the enemies of God; and feelings of national\nhostility were still further embittered by those of religious hatred. In\nthe palmy days of the Arabian empire, these feelings, it is true, were\ntempered by those of respect for an enemy who, in the various forms of\ncivilization, surpassed not merely the Spaniards, but every nation in\nChristendom. Nor was this respect wholly abated under the princes who\nafterwards ruled with imperial sway over Granada, and who displayed, in\ntheir little courts, such a union of the courtesies of Christian\nchivalry with the magnificence of the East, as shed a ray of glory on\nthe declining days of the Moslem empire in the Peninsula.\n\nBut as the Arabs, shorn of their ancient opulence and power, descended\nin the scale, the Spaniards became more arrogant. The feelings of\naversion with which they had hitherto regarded their enemies, were now\nmingled with those of contempt. The latent fire of intolerance was\nfanned into a blaze by the breath of the fanatical clergy, who naturally\npossessed unbounded influence in a country where religious\nconsiderations entered so largely into the motives of action as they did\nin Spain. To crown the whole, the date of the fall of Granada coincided\nwith that of the establishment of the Inquisition,--as if the hideous\nmonster had waited the time when an inexhaustible supply of victims\nmight be afforded for its insatiable maw.\n\nBy the terms of the treaty of capitulation, the people of Granada were\nallowed to remain in possession of their religion and to exercise its\nrights; and it was especially stipulated that no inducements or menaces\nshould be held out to effect their conversion to Christianity.[1] For a\nfew years the conquerors respected these provisions. Under the good\nTalavera, the first archbishop of Granada, no attempt was made to\nconvert the Moslems, except by the legitimate means of preaching to the\npeople and of expounding to them the truths of revelation. Under such a\ncourse of instruction the work of proselytism, though steadily, went on\ntoo slowly to satisfy the impatience of some of the clergy. Among\nothers, that extraordinary man, Cardinal Ximenes, archbishop of Toledo,\nwas eager to try his own hand in the labour of conversion. Having\nreceived the royal assent, he set about the affair with characteristic\nardour, and with as little scruple as to the means to be employed as the\nmost zealous propagandist could have desired. When reasoning and\nexpostulation failed, he did not hesitate to resort to bribes, and, if\nneed were, to force. Under these combined influences the work of\nproselytism went on apace. Thousands were added daily to the Christian\nfold; and the more orthodox Mussulmans trembled, at the prospect of a\ngeneral defection of their countrymen. Exasperated by the unscrupulous\nmeasures of the prelate, and the gross violation they involved of the\ntreaty, they broke out into an insurrection, which soon extended along\nthe mountain ranges in the neighbourhood of Granada.\n\nFerdinand and Isabella, alarmed at the consequences, were filled with\nindignation at the high-handed conduct of Ximenes. But he replied, that\nthe state of things was precisely that which was most to be desired. By\nplacing themselves in an attitude of rebellion, the Moors had renounced\nall the advantages secured by the treaty, and had, moreover, incurred\nthe penalties of death and confiscation of property! It would be an act\nof grace in the sovereigns to overlook their offence, and grant an\namnesty for the past, on condition that every Moor should at once\nreceive baptism or leave the country.[2] This precious piece of\ncasuistry, hardly surpassed by anything in ecclesiastical annals, found\nfavour in the eyes of the sovereigns, who, after the insurrection had\nbeen quelled, lost no time in proposing the terms suggested by their\nminister as the only terms of reconciliation open to the Moors. And, as\nbut few of that unhappy people were prepared to renounce their country\nand their worldly prospects for the sake of their faith, the result was,\nthat in a very short space of time, with but comparatively few\nexceptions, every Moslem in the dominions of Castile consented to abjure\nhis own faith and receive that of his enemies.[3]\n\nA similar course of proceeding was attended with similar results in\nValencia and other dominions of the crown of Aragon, in the earlier part\nof Charles the Fifth's reign; and before that young monarch had been ten\nyears upon the throne, the whole Moorish population--_Moriscoes_, as\nthey were henceforth to be called--were brought within the pale of\nChristianity,--or, to speak more correctly, within that of the\nInquisition.[4]\n\nSuch conversions, it may well be believed, had taken too little root in\nthe heart to bear fruit. It was not long before the agents of the Holy\nOffice detected, under the parade of outward conformity, as rank a\ngrowth of infidelity as had existed before the conquest. The blame might\nin part, indeed, be fairly imputed to the lukewarmness of the Christian\nlabourers employed in the work of conversion. To render this more\neffectual, the government had caused churches to be built in the\nprincipal towns and villages occupied by the Moriscoes, and sent\nmissionaries among them to wean them from their errors and unfold the\ngreat truths of revelation. But an act of divine grace could alone work\nan instantaneous change in the convictions of a nation. The difficulties\nof the preachers were increased by their imperfect acquaintance with the\nlanguage of their hearers; and they had still further to overcome the\nfeelings of jealousy and aversion with which the Spaniard was naturally\nregarded by the Mussulman. Discouraged by these obstacles, the\nmissionary became indifferent to the results. Instead of appealing to\nthe understanding, or touching the heart, of his hearer, he was willing\nto accept his conformity to outward ceremony as the evidence of his\nconversion. Even in his own performance of the sacred rites, the\necclesiastic showed a careless indifference, that proved his heart was\nlittle in the work; and he scattered the purifying waters of baptism in\nso heedless a way over the multitude, that it was not uncommon for a\nMorisco to assert that none of the consecrated drops had fallen upon\nhim.[5]\n\n[Sidenote: THEIR HOMES IN THE ALPUJARRAS.]\n\nThe representations of the clergy at length drew the attention of the\ngovernment. It was decided that the best mode of effecting the\nconversion of the Moslems was by breaking up those associations which\nconnected them with the past,--by compelling them, in short, to renounce\ntheir ancient usages, their national dress, and even their language. An\nextraordinary edict to that effect, designed for Granada, was\naccordingly published by Charles in the summer of 1526; and all who did\nnot conform to it were to be arraigned before the Inquisition. The law\nwas at once met, as might have been expected, by remonstrances from the\nmen of most consideration among the Moriscoes, who, to give efficacy to\ntheir petition, promised the round sum of eighty thousand gold ducats to\nthe emperor in case their prayers should be granted. Charles, who in his\nearly days did not always allow considerations of religion to supersede\nthose of a worldly policy, lent a favourable ear to the petitioners; and\nthe monstrous edict, notwithstanding some efforts to the contrary, was\nnever suffered to go into operation during his reign.[6]\n\nSuch was the state of things on the accession of Philip the Second.\nGranada, Malaga, and the other principal cities of the south, were\nfilled with a mingled population of Spaniards and Moriscoes, the latter\nof whom,--including many persons of wealth and consideration,--under the\ninfluence of a more intimate contact with the Christians, gave evidence,\nfrom time to time, of conversion to the faith of their conquerors. But\nby far the larger part of the Moorish population was scattered over the\nmountain-range of the Alpujarras, south-east of Granada, and among the\nbold sierras that stretch along the southern shores of Spain. Here,\namidst those frosty peaks, rising to the height of near twelve thousand\nfeet above the level of the sea, and readily descried, from their great\nelevation, by the distant voyager on the Mediterranean, was many a\ngreen, sequestered valley, on which the Moorish peasant had exhausted\nthat elaborate culture which, in the palmy days of his nation, was\nunrivalled in any part of Europe.[7] His patient toil had constructed\nterraces from the rocky soil, and, planting them with vines, had clothed\nthe bald sides of the sierra with a delicious verdure. With the like\nindustry he had contrived a network of canals along the valleys and\nlower levels, which, fed by the streams from the mountains, nourished\nthe land with perpetual moisture. The different elevations afforded so\nmany different latitudes for agricultural production; and the fig, the\npomegranate, and the orange grew almost side by side with the hemp of\nthe north and the grain of more temperate climates. The lower <DW72>s of\nthe sierra afforded extensive pastures for flocks of merino sheep;[8]\nand the mulberry-tree was raised in great abundance for the manufacture\nof silk, which formed an important article of export from the kingdom of\nGranada.\n\nThus, gathered in their little hamlets among the mountains, the people\nof the Alpujarras maintained the same sort of rugged independence which\nbelonged to the ancient Goth when he had taken shelter from the Saracen\ninvader in the fastnesses of the Asturias. Here the Moriscoes, formed\ninto communities which preserved their national associations, still\ncherished the traditions of their fathers, and perpetuated those usages\nand domestic institutions that kept alive the memory of ancient days. It\nwas from the Alpujarras that, in former times, the kings of Granada had\ndrawn the brave soldiery who enabled them for so many years to bid\ndefiance to their enemies. The trade of war was now at an end. But the\nhardy life of the mountaineer gave robustness to his frame, and saved\nhim from the effeminacy and sloth which corrupted the inhabitants of the\ncapital. Secluded among his native hills, he cherished those sentiments\nof independence which ill suited a conquered race; and, in default of a\ncountry which he could call his own, he had that strong attachment to\nthe soil which is akin to patriotism, and which is most powerful among\nthe inhabitants of a mountain region.\n\nThe products of the husbandman furnished the staples of a gainful\ncommerce with the nations on the Mediterranean, and especially with the\nkindred people on the Barbary shores. The treaty of Granada secured\ncertain commercial advantages to the Moors, beyond what were enjoyed by\nthe Spaniards.[9] This, it may be well believed, was looked upon with no\nfriendly eye by the latter, who had some ground, moreover, for\ndistrusting the policy of an intercourse between the Moslems of Spain\nand those of Africa, bound together as they were by so many ties--above\nall, by a common hatred of the Christians. With the feelings of\npolitical distrust were mingled those of cupidity and envy, as the\nSpaniard saw the fairest provinces of the south still in the hands of\nthe accursed race of Ishmael, while he was condemned to earn a scanty\nsubsistence from the comparatively ungenial soil of the north.\n\nIn this state of things, with the two races not merely dissimilar, but\nessentially hostile to one another, it will readily be understood how\ndifficult it must have been to devise any system of legislation by which\nthey could be brought to act in harmony as members of the same political\nbody. That the endeavours of the Spanish government were not crowned\nwith success would hardly surprise us, even had its measures been more\nuniformly wise and considerate.\n\n[Sidenote: THEIR TREATMENT BY THE GOVERNMENT.]\n\nThe government caused the Alpujarras to be divided into districts, and\nplaced under the control of magistrates, who, with their families,\nresided in the places assigned as the seats of their jurisdiction. There\nseem to have been few other Christians who dwelt among the Moorish\nsettlements in the sierra, except, indeed, the priests who had charge of\nthe spiritual concerns of the natives. As the conversion of these\nlatter was the leading object of the government, they caused churches to\nbe erected in all the towns and hamlets; and the curates were instructed\nto use every effort to enlighten the minds of their flocks, and to see\nthat they were punctual in attendance on the rites and ceremonies of the\nChurch. But it was soon too evident that attention to forms and\nceremonies was the only approach made to the conversion of the heathen;\nand that below this icy crust of conformity the waters of infidelity lay\nas dark and deep as ever. The result, no doubt, was to be partly charged\non the clergy themselves, many of whom grew languid in the execution of\na task which seemed to them to be hopeless.[10] And what task, in truth,\ncould be more hopeless than that of persuading a whole nation at once to\nrenounce their long-established convictions, to abjure the faith of\ntheir fathers, associated in their minds with many a glorious\nrecollection, and to embrace the faith of the very men whom they\nregarded with unmeasured hatred? It would be an act of humiliation not\nto be expected even in a conquered race.\n\nIn accomplishing a work so much to be desired, the Spaniards, if they\ncannot be acquitted of the charge of persecution, must be allowed not to\nhave urged persecution to anything like the extent which they had done\nin the case of the Protestant Reformers. Whether from policy or from\nsome natural regard to the helplessness of these benighted heathen, the\nbloodhounds of the Inquisition were not as yet allowed to run down their\ngame at will; and, if they did terrify the natives by displaying their\nformidable fangs, the time had not yet come when they were to slip the\nleash and spring upon their miserable victims. It is true there were\nsome exceptions to this more discreet policy. The Holy Office had its\nagents abroad, who kept watch upon the Moriscoes; and occasionally the\nmore flagrant offenders were delivered up to its tender mercies.[11] But\na more frequent source of annoyance arose from the teasing ordinances\nfrom time to time issued by the government, which could have answered no\nother purpose than to irritate the temper and sharpen the animosity of\nthe Moriscoes. If the government had failed in the important work of\nconversion, it was the more incumbent on it, by every show of confidence\nand kindness, to conciliate the good-will of the conquered people, and\nenable them to live in harmony with their conquerors, as members of the\nsame community. Such was not the policy of Philip, any more than it had\nbeen that of his predecessors.\n\nDuring the earlier years of his reign, the king's attention was too\nclosely occupied with foreign affairs to leave him much leisure for\nthose of the Moriscoes. It was certain, however, that they would not\nlong escape the notice of a prince who regarded uniformity of faith as\nthe corner-stone of his government. The first important act of\nlegislation bearing on these people was in 1560, when the Cortes of\nCastile presented a remonstrance to the throne against the use of <DW64>\nslaves by the Moriscoes, who were sure to instruct them in their\nMahometan tenets, and thus to multiply the number of infidels in the\nland.[12] A royal _pragmatic_ was accordingly passed, interdicting the\nuse of African slaves by the Moslems of Granada. The prohibition caused\nthe greatest annoyance; for the wealthier classes were in the habit of\nemploying these slaves for domestic purposes, while in the country they\nwere extensively used for agricultural labour.\n\nIn 1563 another ordinance was published, reviving a law which had fallen\ninto disuse, and which prohibited the Moriscoes from having any arms in\ntheir possession, but such as were duly licensed by the captain-general\nand were stamped with his escutcheon.[13] The office of captain-general\nof Granada was filled at this time by Don Inigo Lopez de Mendoza, count\nof Tendilla, who soon after, on his father's death, succeeded to the\ntitle of marquis of Mondejar. The important post which he held had been\nhereditary in his family ever since the conquest of Granada. The present\nnobleman was a worthy scion of the illustrious house from which he\nsprung.[14] His manners were blunt, and not such as win popularity; but\nhe was a man of integrity, with a nice sense of humour and a humane\nheart,--the last of not too common occurrence in the iron days of\nchivalry. Though bred a soldier, he was inclined to peace. His life had\nbeen passed much among the Moriscoes, so that he perfectly understood\ntheir humours; and, as he was a person of prudence and moderation, it is\nnot improbable, had affairs been left to his direction, that the country\nwould have escaped many of those troubles which afterwards befell it.\n\nIt was singular, considering the character of Mendoza, that he should\nhave recommended so ill-advised a measure as that relating to the arms\nof the Moriscoes. The ordinance excited a general indignation in\nGranada. The people were offended by the distrust which such a law\nimplied of their loyalty. They felt it an indignity to be obliged to sue\nfor permission to do what they considered it was theirs of right to do.\nThose of higher condition disdained to wear weapons displaying the\nheraldic bearings of the Mendozas instead of their own. But the great\nnumber, without regard to the edict provided themselves secretly with\narms, which, as it reached the ears of the authorities, led to frequent\nprosecutions. Thus a fruitful source of irritation was opened; and many,\nto escape punishment, fled to the mountains, and there too often joined\nthe brigands who haunted the passes of Alpujarras, and bade defiance to\nthe feeble police of the Spaniards.[15]\n\n[Sidenote: THE MINISTER ESPINOSA.]\n\nThese impolitic edicts, as they were irritating to the Moriscoes, were\nbut preludes to an ordinance of so astounding a character as to throw\nthe whole country into a state of revolution. The apostasy of the\nMoriscoes,--or, so to speak more correctly, the constancy with which\nthey adhered to the faith of their fathers,--gave great scandal to the\nold Christians, especially to the clergy, and above all to its head, Don\nPedro Guerrero, archbishop of Granada. This prelate seems to have been a\nman of an uneasy, meddlesome spirit, and possessed of a full share of\nthe bigotry of his time. While in Rome, shortly before this period, he\nhad made such a representation to Pope Pius the Fourth as drew from\nthat pontiff a remonstrance, addressed to the Spanish government, on the\nspiritual condition of the Moriscoes. Soon after, in the year 1567, a\nmemorial was presented to the government, by Guerrero and the clergy of\nhis diocese, in which, after insisting on the manifold back-slidings of\nthe \"New Christians,\" as the Moriscoes were termed, they loudly called\nfor some efficacious measures to arrest the evil. These people, they\nsaid, whatever show of conformity they might make to the requisitions of\nthe Church, were infidels at heart. When their children were baptized,\nthey were careful, on returning home, to wash away the traces of\nbaptism, and, after circumcising them, to give them Moorish names. In\nlike manner, when their marriages had been solemnized with Christian\nrites, they were sure to confirm them afterwards by their own\nceremonies, accompanied with the national songs and dances. They\ncontinued to observe Friday as a holy day; and what was of graver\nmoment, they were known to kidnap the children of the Christians, and\nsell them to their brethren on the coast of Barbary, where they were\ncircumcised, and nurtured in the Mahometan religion. This last\naccusation, however improbable, found credit with the Spaniards, and\nsharpened the feelings of jealousy and hatred with which they regarded\nthe unhappy race of Ishmael.[16]\n\nThe memorial of the clergy received prompt attention from the\ngovernment, at whose suggestion, very possibly, it had been prepared. A\ncommission was at once appointed to examine into the matter; and their\nreport was laid before a junta, consisting of both ecclesiastics and\nlaymen, and embracing names of the highest consideration for talent and\nlearning in the kingdom. Among its members we find the Duke of Alva, who\nhad not yet set out on his ominous mission to the Netherlands. At its\nhead was Diego de Espinosa, at that time the favourite minister of\nPhilip, or at least the one who had the largest share in the direction\nof affairs. He was a man after the king's own heart, and, from the\nhumble station of _colegial mayor_ of the college of Cuenca in\nSalamanca, had been advanced by successive steps to the high post of\npresident of the Council of Castile and of the Council of the Indies. He\nwas now also bishop of Siguenza, one of the richest sees in the kingdom.\nHe held an important office in the Inquisition, and was soon to succeed\nValdes in the unenviable post of grand inquisitor. To conclude the\ncatalogue of his honours, no long time was to elapse before, at his\nmaster's suggestion, he was to receive from Rome a cardinal's hat. The\ndeference shown by Philip to his minister, increased as it was by this\nnew accession of spiritual dignity, far exceeded what he had ever shown\nto any other of his subjects.\n\nEspinosa was at this time in the morning, or rather, the meridian of his\npower. His qualifications for business would have been extraordinary,\neven in a layman. He was patient of toil, cheerfully doing the work of\nothers as well as his own. This was so far fortunate that it helped to\ngive him that control in the direction of affairs which was coveted by\nhis aspiring nature. He had a dignified and commanding presence, with\nbut few traces of that humility which would have been graceful in one\nwho had risen so high by his master's favour as much as by his own\ndeserts. His haughty bearing gave offence to the old nobility of\nCastile, who scornfully looked from the minister's present elevation to\nthe humble level from which he had risen. It was regarded with less\ndispleasure, it is said, by the king, who was not unwilling to see the\npride of the ancient aristocracy rebuked by one whom he had himself\nraised from the dust.[17] Their mortification, however, was to be\nappeased ere long by the fall of the favourite--an event as signal and\nunexpected by the world, and as tragical to the subject of it, as the\nfall of Wolsey.\n\nThe man who was qualified for the place of grand inquisitor was not\nlikely to feel much sympathy for the race of unbelievers. It was\nunfortunate for the Moriscoes that their destinies should be placed in\nthe hands of such a minister as Espinosa. After due deliberation, the\njunta came to the decision that the only remedy for the present evil was\nto lay the axe to the root of it; to cut off all those associations\nwhich connected the Moriscoes with their earlier history, and which were\nso many obstacles in the way of their present conversion. It was\nrecommended that they should be interdicted from employing the Arabic\neither in speaking or writing, for which they were to use only the\nCastilian. They were not even to be allowed to retain their family\nnames; but were to exchange them for Spanish ones. All written\ninstruments and legal documents, of whatever kind, were declared to be\nvoid and of no effect unless in the Castilian. As time must be allowed\nfor a whole people to change its language, three years were assigned as\nthe period at the end of which this provision should take effect.\n\nThey were to be required to exchange their national dress for that of\nthe Spaniards; and, as the Oriental costume was highly ornamented, and\noften very expensive, they were to be allowed to wear their present\nclothes one year longer if of silk, and two years if of cotton, the\nlatter being the usual apparel of the poorer classes. The women,\nmoreover, both old and young, were to be required, from the passage of\nthe law, to go abroad with their faces uncovered,--a scandalous thing\namong Mahometans.\n\nTheir weddings were to be conducted in public, after the Christian\nforms; and the doors of their houses were to be left open during the day\nof the ceremony, that any one might enter and see that they did not have\nrecourse to unhallowed rites. They were further to be interdicted from\nthe national songs and dances with which they were wont to celebrate\ntheir domestic festivities. Finally, as rumours--most absurd ones--had\ngot abroad that the warm baths which the natives were in the habit of\nusing in their houses were perverted to licentious indulgences, they\nwere to be required to destroy the vessels in which they bathed, and to\nuse nothing of the kind thereafter.\n\nThese several provisions were to be enforced by penalties of the\nsternest kind. For the first offence the convicted party was to be\npunished with imprisonment for a month, with banishment from the country\nfor two years, and with a fine varying from six hundred to ten thousand\nmaravedis. For a second offence the penalties were to be doubled; and\nfor a third, the culprit, in addition to former penalties, was to be\nbanished for life. The ordinance was closely modelled on that of Charles\nthe Fifth, which, as we have seen, he was too politic to carry into\nexecution.[18]\n\n[Sidenote: EDICT AGAINST THE MORISCOES.]\n\nSuch were the principal provisions of a law which, for cruelty and\nabsurdity, has scarcely a parallel in history. For what could be more\nabsurd than the attempt by an act of legislation to work such a change\nin the long-established habits of a nation--to efface those\nrecollections of the past, to which men ever cling most closely under\nthe pressure of misfortune--to blot out by a single stroke of the pen,\nas it were, not only the creed, but the nationality of a people--to\nconvert the Moslem, at once, both into a Christian and into a Castilian?\nIt would be difficult to imagine any greater outrage offered to a people\nthan the provision compelling women to lay aside their veils--associated\nas these were in every Eastern mind with the obligations of modesty; or\nthat in regard to opening the doors of the houses, and exposing those\nwithin to the insolent gaze of every passer; or that in relation to the\nbaths--so indispensable to cleanliness and comfort, especially in the\nwarm climate of the South.\n\nBut the masterpiece of absurdity, undoubtedly, is the stipulation in\nregard to the Arabic language; as if by any human art a whole\npopulation, in the space of three years, could be made to substitute a\nforeign tongue for its own; and that, too, under circumstances of\npeculiar difficulty, partly arising from the total want of affinity\nbetween the Semitic and the European languages, and partly from the\ninsulated position of the Moriscoes, who, in the cities, had separate\nquarters assigned to them, in the same manner as the Jews, which cut\nthem off from intimate intercourse with the Christians. We may well\ndoubt, from the character of this provision, whether the Government had\nso much at heart the conversion of the Moslems as the desire to entangle\nthem in such violations of the law as should afford a plausible pretext\nfor driving them from the country altogether. One is strengthened in\nthis view of the subject by the significant reply of Otadin, professor\nof theology at Alcala, who, when consulted by Philip on the expediency\nof the ordinance, gave his hearty approbation of it, by quoting the\nappalling Spanish proverb, \"The fewer enemies, the better.\"[19] It was\nreserved for the imbecile Philip the Third to crown the disasters of his\nreign by the expulsion of the Moriscoes. Yet no one can doubt that it\nwas a consummation earnestly desired by the great body of the Spaniards,\nwho looked, as we have seen, with longing eyes to the fair territory\nwhich they possessed, and who regarded them with the feelings of\ndistrust and aversion with which men regard those on whom they have\ninflicted injuries too great to be forgiven.\n\nYet there were some in the junta with whom the proposed ordinance found\nno favour. Among these, one who calls to mind his conduct in the\nNetherlands may be surprised to find the duke of Alva. Here, as in that\ncountry, his course was doubtless dictated less by considerations of\nhumanity than of policy. Whatever may have been his reasons, they had\nlittle weight with Espinosa, who probably felt a secret satisfaction in\nthwarting the man whom he regarded with all the jealousy of a rival.[20]\n\nWhat was Philip's own opinion on the matter, we can but conjecture from\nour general knowledge of his character. He professed to be guided by the\ndecision of the \"wise and learned men\" to whom he had committed the\nsubject. That this decision did no great violence to his own feelings,\nwe may infer from the promptness with which he signed the ordinance.\nThis he did on the 17th of November, 1566, when the pragmatic became a\nlaw.\n\nIt was resolved, however, not to give publicity to it at once. It was\ncommitted to the particular charge of one of the members of the junta,\nDiego Deza, auditor of the Holy Office, and lately raised by Espinosa to\nthe important post of president of the chancery of Granada. This put him\nat once at the head of the civil administration of the province, as the\nMarquis of Mondejar was at the head of the military. The different views\nof policy entertained by the two men led to a conflict of authority\nwhich proved highly prejudicial to affairs. Deza, who afterwards rose to\nthe dignity of cardinal, was a man whose plausible manners covered an\ninflexible will. He showed, notwithstanding, an entire subserviency to\nthe wishes of his patron, Espinosa, who committed to him the execution\nof his plans.\n\nThe president resolved, with more policy than humanity, to defer the\npublication of the edict till the ensuing first of January, 1667, the\nday preceding that which the Spaniards commemorated as the anniversary\nof the surrender of the capital. This humiliating event, brought home at\nsuch a crisis to the Moriscoes, might help to break their spirits, and\ndispose them to receive the obnoxious edict with less resistance.\n\nOn the appointed day the magistrates of the principal tribunals, with\nthe corregidor of Granada at their head, went in solemn procession to\nthe Albaicin, the quarter occupied by the Moriscoes. They marched to the\nsound of kettle-drums, trumpets, and other instruments; and the\ninhabitants, attracted by the noise, and fond of novelty, came running\nfrom their houses to swell the ranks of the procession on its way to the\ngreat square of _Bab el Bonat_. This was an open space, of large extent,\nwhere the people of Granada, in ancient times, used to assemble to\ncelebrate the coronation of a new sovereign; and the towers were still\nstanding from which the Moslem banners waved, on those days, over the\nheads of the shouting multitude. As the people now gathered tumultuously\naround these ancient buildings, the public crier, from an elevated\nplace, read, in audible tones and in the Arabic language, the royal\nordinance. One may imagine the emotions of shame, sorrow, and\nindignation with which the vast assembly, consisting of both sexes,\nlistened to the words of an instrument, every sentence of which seemed\nto convey a personal indignity to the hearers--an outrage on all those\nideas of decorum and decency in which they had been nurtured from\ninfancy; which rudely rent asunder all the fond ties of country and\nkindred; which violated the privacy of domestic life, deprived them of\nthe use of their own speech, and reduced them to a state of utter\nhumiliation unknown to the meanest of their slaves. Some of the weaker\nsort gave way to piteous and passionate exclamations, wringing their\nhands in an agony of grief. Others, of sterner temper, broke forth into\nmenaces and fierce invective, accompanied with the most furious\ngesticulations. Others, again, listened with that dogged, determined air\nwhich showed that the mood was not the less dangerous that it was a\nsilent one. The whole multitude was in a state of such agitation that an\naccident might have readily produced an explosion which would have\nshaken Granada to its foundations. Fortunately there were a few discreet\npersons in the assembly, older and more temperate than the rest, who had\nsufficient authority over their countrymen to prevent a tumult. They\nreminded them that in their fathers' time the emperor Charles the Fifth\nhad consented to suspend the execution of a similar ordinance. At all\nevents, it was better to try first what could be done by argument and\npersuasion. When these failed, it would be time enough to think of\nvengeance.[21]\n\n[Sidenote: THEIR INEFFECTUAL REMONSTRANCE.]\n\nOne of the older Moriscoes, a man of much consideration among his\ncountrymen, was accordingly chosen to wait on the president and explain\ntheir views in regard to the edict. This he did at great length, and in\na manner which must have satisfied any fair mind of the groundlessness\nof the charges brought against the Moslems, and the cruelty and\nimpracticability of the measures proposed by the government. The\npresident, having granted to the envoy a patient and courteous hearing,\nmade a short and not very successful attempt to vindicate the course of\nthe administration. He finally disposed of the whole question by\ndeclaring that \"the law was too just and holy, and had been made with\ntoo much consideration, ever to be repealed; and that, in fine, regarded\nas a question of interest, his majesty estimated the salvation of a\nsingle soul as of greater price than all the revenues he drew from the\nMoriscoes.\"[22] An answer like this must have effectually dispelled all\nthoughts of a composition, such as had formerly been made with the\nemperor.\n\nDefeated in this quarter, the Moriscoes determined to lay their\nremonstrance before the throne. They were fortunate in obtaining, for\nthis purpose, the services of Don Juan Henriquez, a nobleman of the\nhighest rank and consideration, who had large estates at Beza, in the\nheart of Granada, and who felt a strong sympathy for the unfortunate\nnatives. Having consented, though with much reluctance, to undertake the\nmission, he repaired to Madrid, obtained an audience of the king, and\npresented to him a memorial on behalf of his unfortunate subjects.\nPhilip received him graciously, and promised to give all attention to\nthe paper. \"What I have done in this matter,\" said the king, \"has been\ndone by the advice of wise and conscientious men, who have given me to\nunderstand that it was my duty.\"[23]\n\nShortly afterwards, Henriquez received an intimation that he was to look\nfor his answer to the president of Castille. Espinosa, after listening\nto the memorial, expressed his surprise that a person of the high\ncondition of Don Juan Henriquez should have consented to take charge of\nsuch a mission. \"It was for that very reason I undertook it,\" replied\nthe nobleman, \"as affording me a better opportunity to be of service to\nthe king.\" \"It can be of no use,\" said the minister; \"religious men have\nrepresented to his majesty that at his door lies the salvation of these\nMoors; and the ordinance which has been decreed, he has determined shall\nbe carried into effect.\"[24]\n\nBaffled in this direction, the persevering envoy laid his memorial\nbefore the councillors of state, and endeavoured to interest them in\nbehalf of his clients. In this he met with more success; and several of\nthat body, among whom may be mentioned the duke of Alva and Luis de\nAvila, the grand commander of Alcantara, whom Charles the Fifth had\nhonoured with his friendship, entered heartily into his views. But it\navailed little with the minister, who would not even consent to delay\nthe execution of the ordinance until time should have been given for\nfurther inquiry, or to confine the operation of it, at the outset, to\none or two of the provisions, in order to ascertain what would probably\nbe the temper of the Moriscoes.[25] Nothing would suit the peremptory\nhumour of Espinosa but the instant execution of the law in all its\ndetails.\n\nNor would he abate anything of this haughty tone in favour of the\ncaptain-general, the marquis of Mondejar. That nobleman, with good\nreason, had felt himself aggrieved that, in discussions so materially\naffecting his own government, he should not have been invited to take a\npart. From motives of expediency, as much as of humanity, he was\ndecidedly opposed to the passage of the ordinance. It was perhaps a\nknowledge of this that had excluded him from a seat in the junta. His\nrepresentations made no impression on Espinosa; and when he urged that,\nif the law were to be carried into effect, he ought to be provided with\nsuch a force as would enable him to quell any attempt at resistance, the\nminister made light of the danger, assuring him that three hundred\nadditional troops were as many as the occasion demanded. Espinosa then\nperemptorily adjourned all further discussion, by telling the\ncaptain-general that it would be well for him to return at once to\nGranada, where his presence would be needed to enforce the execution of\nthe law.[26]\n\nIt was clear that no door was left open to further discussion, and that,\nunder the present government, no chance remained to the unfortunate\nMoriscoes of buying off the law by the payment of a round sum, as in the\ntime of Charles the Fifth. All negotiations were at an end. They had\nonly to choose between implicit obedience and open rebellion. It was not\nstrange that they chose the latter.\n\n\n\n\nCHAPTER II.\n\nREBELLION OF THE MORISCOES.\n\nResistance of the Moriscoes--Night Assault on Granada--Rising in the\nAlpujarras--Election of a King--Massacre of the Christians.\n\n1568.\n\n\nThe same day on which the ordinance was published in the capital, it was\nproclaimed in every part of the kingdom of Granada. Everywhere it was\nreceived with the same feelings of shame, sorrow, and indignation.\nBefore giving way to these feelings by any precipitate action, the\nMoriscoes of the Alpujarras were discreet enough to confer with their\ncountrymen in the Albaicin, who advised them to remain quiet until they\nshould learn the result of the conferences going on at Madrid.\n\nBefore these were concluded, the year expired after which it would be\npenal for a Morisco to wear garments of silk. By the president's orders\nit was proclaimed by the clergy, in the pulpits throughout the city,\nthat the law would be enforced to the letter. This was followed by more\nthan one edict relating to other matters, but yet tending to irritate\nstill further the minds of the Moriscoes.[27]\n\n[Sidenote: RESISTANCE TO THE EDICT.]\n\nAll hope of relieving themselves of the detested ordinance having thus\nvanished, the leaders of the Albaicin took counsel as to the best mode\nof resisting the government. The first step seemed to be to get\npossession of the capital. There was at this time in Granada a Morisco\nnamed Farax Aben-Farax, who followed the trade of a dyer. But though he\nwas engaged in this humble calling, the best blood of the Abencerrages\nflowed in his veins. He was a man of a fierce, indeed ferocious nature,\nhating the Christians with his whole heart, and longing for the hour\nwhen he could avenge on their heads the calamities of his countrymen. As\nhis occupation earned him frequently into the Alpujarras, he was\nextensively acquainted with the inhabitants. He undertook to raise a\nforce there of eight thousand men, and bring them down secretly by night\ninto the _vega_, where, with the aid of his countrymen in the Albaicin,\nhe might effect an entrance into the city, overpower the garrison in the\nAlhambra, put all who resisted to the sword, and make himself master of\nthe capital. The time fixed upon for the execution of the plan was Holy\nThursday, in the ensuing month of April, when the attention of the\nSpaniards would be occupied with their religious solemnities.\n\nA secret known to so many could not be so well kept, and for so long a\ntime, but that some information of it reached the ears of the\nChristians. It seems to have given little uneasiness to Deza, who had\nanticipated some such attempt from the turbulent spirit of the\nMoriscoes. The captain-general, however, thought it prudent to take\nadditional precautions against it; and he accordingly distributed arms\namong the citizens, strengthened the garrison of the Alhambra, and\nvisited several of the great towns on the frontiers, which he placed in\na better posture of defence. The Moriscoes, finding their purpose\nexposed to the authorities, resolved to defer the execution of it for\nthe present. They even postponed it to as late a date as the beginning\nof the following year, 1569. To this they were led, we are told, by a\nprediction found in their religious books, that the year of their\nliberation would be one that began on a Saturday. It is probable that\nthe wiser men of the Albaicin were less influenced by their own belief\nin the truth of the prophecy, than by the influence it would exert over\nthe superstitious minds of the mountaineers, among whom it was\ndiligently circulated.[28]\n\nHaving settled on the first of January for the rising, the Moslems of\nGranada strove, by every outward show of loyalty, to quiet the\nsuspicions of the government. But in this they were thwarted by the\ninformation which the latter obtained through more trustworthy channels.\nStill surer evidence of their intentions was found in a letter which\nfell by accident into the hands of the marquis of Mondejar. It was\naddressed by one of the leaders of the Albaicin to the Moslems of the\nBarbary coast, invoking their aid by the ties of consanguinity and of a\ncommon faith. \"We are sorely beset,\" says the writer, \"and our enemies\nencompass us all around like a consuming fire. Our troubles are too\ngrievous to be endured. Written,\" concludes the passionate author of the\nepistle, \"in nights of tears and anguish, with hope yet lingering,--such\nhope as still survives amidst all the bitterness of the soul.\"[29]\n\nBut the Barbary powers were too much occupied by their petty feuds to\ngive much more than fair words to their unfortunate brethren of Granada.\nPerhaps they distrusted the efficacy of any aid they could render in so\nunequal a contest as that against the Spanish monarchy. Yet they allowed\ntheir subjects to embark as volunteers in the war; and some good service\nwas rendered by the Barbary corsairs, who infested the coasts of the\nMediterranean, as well as by the _monfis_,--as the African adventurers\nwere called,--who took part with their brethren in the Alpujarras,\nwhere they made themselves conspicuous by their implacable ferocity\nagainst the Christians.\n\nMeanwhile the hot blood of the mountaineers was too much inflamed by the\nprospect of regaining their independence to allow them to wait patiently\nfor the day fixed upon for the outbreak. Before that time arrived,\nseveral acts of violence were perpetrated,--forerunners of the bloody\nwork that was at hand. In the month of December, 1568, a body of Spanish\nalguazils, with some other officers of justice, were cut off in the\nneighbourhood of Granada, on their way to that city. A party of fifty\nsoldiers, as they were bearing to the capital a considerable quantity of\nmuskets,--a tempting prize to the unarmed Moriscoes,--were all murdered,\nmost of them in their beds, in a little village among the mountains\nwhere they had halted for the night.[30] After this outrage Aben-Farax,\nthe bold dyer of Granada, aware of the excitement it must create in the\ncapital, became convinced it would not be safe for him to postpone his\nintended assault a day longer.\n\nAt the head of only a hundred and eighty followers, without waiting to\ncollect a larger force, he made his descent, on the night of the\ntwenty-sixth of December, a week before the appointed time, into the\n_vega_ of Granada. It was a dreadful night. A snow-storm was raging\nwildly among the mountains, and sweeping down in pitiless fury on the\nplains below.[31] Favoured by the commotion of the elements, Aben-Farax\nsucceeded, without attracting observation, in forcing an entrance\nthrough the dilapidated walls of the city, penetrated at once into the\nAlbaicin, and endeavoured to rouse the inhabitants from their slumbers.\nSome few came to their windows, it is said, but, on learning the nature\nof the summons, hastily closed the casements and withdrew, telling\nAben-Farax that \"it was madness to undertake the enterprise with so\nsmall a force, and that he had come before his time.\"[32] It was in vain\nthat the enraged chief poured forth imprecations on their perfidy and\ncowardice, in vain that he marched through the deserted streets,\ndemolishing crucifixes and other symbols of Christian worship which he\nfound in his way, or that he shouted out the watchword of the faithful,\n\"There is but one God, and Mahomet is the prophet of God!\" The uproar of\nthe tempest, fortunately for him, drowned every other noise; and no\nalarm was given till he stumbled on a guard of some five or six\nsoldiers, who were huddled round a fire in one of the public squares.\nOne of these Farax despatched; the others made their escape, raising the\ncry that the enemy was upon them. The great bell of St. Salvador rang\nviolently, calling the inhabitants to arms. Dawn was fast approaching;\nand the Moorish chief, who felt himself unequal to an encounter in which\nhe was not to be supported by his brethren in the Albaicin, thought it\nprudent to make his retreat. This he did with colours flying and music\nplaying, all in as cool and orderly a manner as if it had been only a\nholiday parade.\n\n[Sidenote: RISING IN THE ALPUJARRAS.]\n\nMeantime the citizens, thus suddenly startled from their beds, gathered\ntogether, with eager looks, and faces white with fear, to learn the\ncause of the tumult; and their alarm was not diminished by finding that\nthe enemy had been prowling round their dwellings, like a troop of\nmountain wolves, while they had been buried in slumber. The marquis of\nMondejar called his men to horse, and would have instantly given chase\nto the invaders, but waited until he had learned the actual condition of\nthe Albaicin, where a population of ten thousand Moriscoes, had they\nbeen mischievously inclined, might, notwithstanding the timely efforts\nof the government to disarm them, have proved too strong for the slender\nSpanish garrison in the Alhambra. All, however, was quiet in the Moorish\nquarter; and, assured of this, the captain-general sallied out, at the\nhead of his cavalry and a small corps of foot, in quest of the enemy.\nBut he had struck into the mountain-passes south of Granada; and\nMendoza, after keeping on his track, as well as the blinding tempest\nwould permit, through the greater part of the day, at nightfall gave up\nthe pursuit as hopeless, and brought back his wayworn cavalcade to the\ncity.[33]\n\nAben-Farax and his troop, meanwhile, traversing the snowy skirts of the\nSierra Nevada, came out on the broad and populous valley of Lecrin,\nspreading the tidings everywhere, as they went, that the insurrection\nwas begun, that the Albaicin was in movement, and calling on all true\nbelievers to take up arms in defence of their faith. The summons did not\nfall on deaf ears. A train had been fired which ran along the mountain\nregions to the south of Granada, stretching from Almeria and the Murcian\nborders on the east to the neighbourhood of Velez Malaga on the west. In\nthree days the whole country was in arms. Then burst forth the fierce\npassions of the Arab,--all that unquenchable hate which seventy years of\noppression had nourished in his bosom, and which now showed itself in\none universal cry for vengeance. The bloody drama opened with the\nmassacre of nearly every Christian man within the Moorish borders,--and\nthat too with circumstances of a refined and deliberate cruelty, of\nwhich, happily, few examples are to be found in history.\n\nThe first step, however, in the revolutionary movement had been a false\none, inasmuch as the insurgents had failed to secure possession of the\ncapital, which would have furnished so important a _point d'appui_ for\nfuture operations. Yet, if contemporary chroniclers are correct, this\nfailure should rather be imputed to miscalculation than to cowardice.\nAccording to them, the persons of most consideration in the Albaicin\nwere many of them wealthy citizens, accustomed to the easy, luxurious\nway of life so well suited to the Moorish taste. They had never intended\nto peril their fortunes by engaging personally in so formidable a\ncontest as that with the Castilian crown. They had only proposed to urge\ntheir simple countrymen in the Alpujarras to such a show of resistance\nas should intimidate the Spaniards, and lead them to mitigate, if not\nindeed to rescind, the hated ordinance.[34] If such was their\ncalculation, as the result showed, it miserably failed.\n\nAs the Moriscoes had now proclaimed their independence, it became\nnecessary to choose a sovereign in place of the one whose authority they\nhad cast aside. The leaders in the Albaicin selected for this dangerous\npre-eminence a young man who was known to the Spaniards by his Castilian\nname of Don Fernando de Valor. He was descended in a direct line from\nthe ancient house of the Omeyas,[35] who for nearly four centuries had\nsat with glory on the throne of Cordova. He was but twenty-two years of\nage at the time of his election, and, according to a contemporary who\nhad seen him, possessed a comely person and engaging manners. His\ncomplexion was of a deep olive; his beard was thin; his eyes were large\nand dark, with eyebrows well defined, and nearly approaching each other.\nHis deportment was truly royal; and his lofty sentiments were worthy of\nthe princely line from which he was descended.[36] Notwithstanding this\nflattering portrait from the pen of a Castilian, his best\nrecommendation, to judge from his subsequent career, seems to have been\nhis descent from a line of kings. He had been so prodigal in his way of\nlife that, though so young, he had squandered his patrimony, and was at\nthis very time under arrest for debt. He had the fiery temperament of\nhis nation, and had given evidence of it by murdering, with his own\nhand, a man who had borne testimony against his father in a criminal\nprosecution. Amidst his luxurious self-indulgence he must be allowed to\nhave shown some energy of character and an unquestionable courage. He\nwas attached to the institutions of his country; and his ferocious\nnature was veiled under a bland and plausible exterior, that won him\ngolden opinions from the multitude.[37]\n\nSoon after his election, and just before the irruption of Aben-Farax,\nthe Morisco prince succeeded in making his escape from Granada, and,\nflying to the mountains, took refuge among his own kindred, the powerful\nfamily of the Valoris, in the village of Beznar. Here his countrymen\ngathered round him, and confirmed by acclamation the choice of the\npeople of Granada. For this the young chieftain was greatly indebted to\nthe efforts of his uncle, Aben-Jahuar, commonly called El Zaguer, a man\nof much authority among his tribe, who, waiving his own claims to the\nsceptre, employed his influence in favour of his nephew.\n\nThe ceremony of the coronation was of a martial kind, well suited to the\nrough fortunes of the adventurer. Four standards, emblazoned with the\nMoslem crescent, were spread upon the ground, with their spear-heads\nseverally turned towards the four points of the compass. The Moorish\nprince, who had been previously arrayed in a purple robe, with a crimson\nscarf or shawl, the insignia of royalty, enveloping his shoulders, knelt\ndown on the banners, with his face turned towards Mecca, and, after a\nbrief prayer, solemnly swore to live and die in defence of his crown,\nhis faith, and his subjects. One of the principal attendants,\nprostrating himself on the ground, kissed the footprints of the\nnewly-elected monarch, in token of the allegiance of the people. He was\nthen raised on the shoulders of four of the assistants, and borne aloft\namidst the waving of banners and the loud shouts of the multitude,\n\"Allah exalt Muley-Mohammed-Aben-Humeya, lord of Andalucia and\nGranada!\"[38]\n\n[Sidenote: MASSACRE OF THE CHRISTIANS.]\n\nSuch were the simple forms practised in ancient times by the\nSpanish-Arabian princes, when their empire, instead of being contracted\nwithin the rocky girdle of the mountains, stretched over the fairest\nportions of the Peninsula.[39]\n\nThe first act of Aben-Humeya was to make his appointments to the chief\nmilitary offices. El Zaguer, his uncle, he made captain-general of his\nforces. Aben-Farax, who had himself aspired to the diadem, he removed to\na distance, by sending him on an expedition to collect such treasures as\ncould be gathered from the Christian churches in the Alpujarras. He\nappointed officers to take charge of the different _tahas_, or\ndistricts, into which the country was divided. Having completed these\narrangements, the new monarch--the _reyezuelo_, or \"little king,\" of the\nAlpujarras, as he was contemptuously styled by the\nSpaniards--transferred his residence to the central part of his\ndominions, where he repeated the ceremony of his coronation. He made a\nrapid visit to the most important places in the sierra, everywhere\ncalling on the inhabitants to return to their ancient faith, and to\nthrow off the hated yoke of the Spaniards. He then established himself\nin the wildest parts of the Alpujarras, where he endeavoured to draw his\nforces to a head, and formed the plan of his campaign. It was such as\nwas naturally suggested by the character of the country, which, broken\nand precipitous, intersected by many a deep ravine and dangerous pass,\nafforded excellent opportunities for harassing an invading foe, and for\nentangling him in those inextricable defiles, where a few mountaineers\nacquainted with the ground would he more than a match for an enemy far\nsuperior in discipline and numbers.\n\nWhile Aben-Humeya was thus occupied in preparing for the struggle, the\nwork of death had already begun among the Spanish population of the\nAlpujarras; and Spaniards were to be found, in greater or less numbers,\nin all the Moorish towns and hamlets that dotted the dark sides of the\nsierras, or nestled in the green valleys at their base. Here they dwelt\nside by side with the Moriscoes, employed probably less in the labours\nof the loom, for which the natives of this region had long been famous,\nthan in that careful husbandry which they might readily have learned\nfrom their Moorish neighbours, and which, under their hands, had clothed\nevery spot with verdure, making the wilderness to blossom like the\nrose.[40] Thus living in the midst of those who professed the same\nreligion with themselves, and in the occasional interchange, at least,\nof the kind offices of social intercourse, which sometimes led to nearer\ndomestic ties, the Christians of the Alpujarras dwelt in blind security,\nlittle dreaming of the mine beneath their feet.\n\nBut no sooner was the first note of insurrection sounded, than the scene\nchanged as if by magic. Every Morisco threw away his mask, and, turning\non the Christians, showed himself in his true aspect, as their avowed\nand mortal enemy.\n\nA simultaneous movement of this kind, through so wide an extent of\ncountry, intimates a well-concerted plan of operations; and we may share\nin the astonishment of the Castillan writers, that a secret of such a\nnature, and known to so many individuals, should have been so long and\nfaithfully kept,--in the midst, too, of those who had the greatest\ninterest in detecting it,[41]--some of them, it may be added, spies of\nthe Inquisition, endowed, as they seem to have been, with almost\nsupernatural powers for scenting out the taint of heresy.[42] It argues\nan intense feeling of hatred in the Morisco, that he could have been so\nlong proof against the garrulity that loosens the tongue, and against\nthe sympathy that so often, in similar situations, unlocks the heart, to\nsave some friend from the doom of his companions. But no such instance,\neither of levity or lenity, occurred among this extraordinary people.\nAnd when the hour arrived, and the Christians discerned their danger in\nthe menacing looks and gestures of their Moslem neighbours, they were as\nmuch astounded by it as the unsuspecting traveller on whom, as he\nheedlessly journeys through some pleasant country, the highwayman has\ndarted from his covert by the roadside.\n\nThe first impulse of the Christians seems to have been very generally to\ntake refuge in the churches; and every village, however small, had at\nleast one church, where the two races met together to join in the forms\nof Christian worship. The fugitives thought to find protection in their\nholy places and in the presence of their venerated pastors, whose\nspiritual authority had extended over all the inhabitants. But the wild\nanimal of the forest, now that he had regained his freedom, gave little\nheed to the call of his former keeper,--unless it were to turn and rend\nhim.\n\nHere crowded together, like a herd of panic-stricken deer with the\nhounds upon their track, the terrified people soon found the church was\nno place of security, and they took refuge in the adjoining tower, as a\nplace of greater strength, and affording a better means of defence\nagainst an enemy. The mob of their pursuers then broke into the church,\nwhich they speedily despoiled of its ornaments, trampling the crucifixes\nand other religious symbols under their feet, rolling the sacred images\nin the dust, and desecrating the altars by the sacrifice of swine, or by\nsome other act denoting their scorn and hatred of the Christian\nworship.[43]\n\nThey next assailed the towers, the entrances to which the Spaniards had\nbarricaded as strongly as they could; though, unprovided as they were\nwith means of defence, except such arms as they had snatched in the\nhurry of their flight, they could have little hope of standing a siege.\nUnfortunately, these towers were built more or less of wood, which the\nassailants readily set on fire, and thus compelled the miserable inmates\neither to surrender or to perish in the flames. In some instances they\nchose the latter; and the little garrison--men, women, and\nchildren--were consumed together on one common funeral pile. More\nfrequently they shrank from this fearful death, and surrendered at the\nmercy of their conquerors,--such mercy as made them soon regret that\nthey had not stayed by the blazing rafters.\n\n[Sidenote: MASSACRE OF THE CHRISTIANS]\n\nThe men were speedily separated from the women, and driven with blows\nand imprecations, like so many cattle, to a place of confinement. From\nthis loathsome prison they were dragged out, three or four at a time,\nday after day, the longer to protract their sufferings; then, with their\narms pinioned behind them, and stripped of their clothing, they were\nthrown into the midst of an infuriated mob, consisting of both sexes,\nwho, armed with swords, hatchets, and bludgeons, soon felled their\nvictims to the ground, and completed the bloody work.\n\nThe mode of death was often varied to suit the capricious cruelty of the\nexecutioners. At Guecija, where the olive grew abundantly, there was a\nconvent of Augustine monks, who were all murdered by being thrown into\ncaldrons of boiling oil.[44] Sometimes the death of the victim was\nattended with circumstances of diabolical cruelty, not surpassed by\nanything recorded of our North-American savages. At a place called\nPitres de Ferreyra, the priest of the village was raised by means of a\npulley to a beam that projected from the tower, and was then allowed to\ndrop from a great height upon the ground. The act was repeated more than\nonce in the presence of his aged mother, who, in an agony of grief,\nembracing her dying son, besought him \"to trust in God and the blessed\nVirgin, who through these torments would bring him into eternal life.\"\nThe mangled carcase of the poor victim, broken and dislocated in every\nlimb, was then turned over to the Moorish women, who, with their\nscissors, bodkins, and other feminine implements, speedily despatched\nhim.[45]\n\nThe women, indeed, throughout this persecution, seem to have had as\nrabid a thirst for vengeance as the men. Even the children were\nencouraged to play their part in the bloody drama; and many a miserable\ncaptive was set up as a target to be shot at with the arrows of the\nMoorish boys.\n\nThe rage of the barbarians was especially directed against the priests,\nwho had so often poured forth anathemas against the religion which the\nMoslems loved, and who, as their spiritual directors, had so often\ncalled them to account for offences against the religion which they\nabhorred. At Coadba the priest was stretched out before a brazier of\nlive coals until his feet, which had been smeared with pitch and oil,\nwere burned to a cinder. His two sisters were compelled to witness the\nagonies of their brother, which were still further heightened by the\nbrutal treatment which he saw them endure from their tormentors.[46]\n\nFire was employed as a common mode of torture, by way of retaliation, it\nmay be, for similar sufferings inflicted on the Infidel by the\nInquisition. Sometimes the punishments seemed to be contrived so as to\nform a fiendish parody on the exercises of the Roman Catholic religion.\nIn the town of Filix the pastor was made to take his seat before the\naltar, with his two sacristans, one on either side of him. The bell was\nrung, as if to call the people together to worship. The sacristans were\neach provided with a roll containing the names of the congregation,\nwhich they were required to call over, as usual, before the services, in\norder to see that no one was absent. As each Morisco answered to his\nname, he passed before the priest, and dealt him a blow with his fist,\nor the women plucked his beard and hair, accompanying the act with some\nbitter taunt expressive of their mortal hate. When every one had thus\nhad the opportunity of gratifying his personal grudge against his\nancient pastor, the executioner stepped forward, armed with a razor,\nwith which he scored the face of the ecclesiastic in the detested form\nof the cross, and then, beginning with the fingers, deliberately\nproceeded to sever each of the joints of his wretched victim![47]\n\nBut it is unnecessary to shock the reader with more of these loathsome\ndetails, enough of which have already been given, not merely to prove\nthe vindictive temper of the Morisco, but to suggest the inference that\nit could only have been a long course of cruelty and oppression that\nstimulated him to such an awful exhibition of it.[48] The whole number\nof Christians who, in the course of a week, thus perished in these\nmassacres--if we are to receive the accounts of Castilian writers--was\nnot less than three thousand![49] Considering the social relations which\nmust to some extent have been established between those who had lived so\nlong in the neighbourhood of one another, it might be thought that, on\nsome occasions, sympathy would have been shown for the sufferers, or\nthat some protecting arm would have been stretched out to save a friend\nor a companion from the general doom. But the nearest approach to such\nan act of humanity was given by a Morisco, who plunged his sword in the\nbody of a Spaniard in order to save him from the lingering death that\notherwise would await him.[50]\n\nOf the whole Christian population very few of the men who fell into the\nhands of the Moslems escaped with life. The women were not always\nspared. The Morisco women, especially, who had married Christian\nhusbands and embraced Christianity, which they refused to abjure, became\nthe objects of vengeance to their own sex. Sad to say, even the\ninnocence and helplessness of childhood proved no protection against the\nfury of persecution. The historians record the names of several boys,\nfrom ten to twelve or thirteen years of age, who were barbarously\nmurdered because they would not renounce the religion in which they had\nbeen nurtured for that of Mahomet. If they were too young to give a\nreason for their faith, they had at least learned the lesson that to\nrenounce it was a great sin; and, when led out like lambs to the\nslaughter, their mothers, we are told, stifling the suggestions of\nnatural affection in obedience to a higher law, urged their children not\nto shrink from the trial, nor to purchase a few years of life at the\nprice of their own souls.[51] It is a matter of no little gratulation to\na Catholic historian, that, amongst all those who perished in these\nfrightful massacres, there was not one of any age or either sex who\ncould be tempted to secure personal safety by the sacrifice of religious\nconvictions.[52] On the contrary, they employed the brief respite that\nwas left them in fortifying one another's courage, and in bearing\ntestimony to the truth in so earnest a manner that they might almost\nseem to have courted the crown of martyrdom. Yet among these martyrs\nthere were more than one, it is admitted, whose previous way of life\nshowed but a dim perception of the value of that religion for which,\nthey were thus prepared to lay down their lives.[53]\n\n[Sidenote: MASSACRE OF THE CHRISTIANS.]\n\nThe chief blame of these indiscriminate proscriptions has been laid on\nAben-Farax, the famous dyer of Granada, whose appetite for blood seems\nto have been as insatiable as that of any wild beast in the Alpujarras.\nIn executing the commission assigned to him by Aben-Humeya, he was\nobliged to visit all parts of the country. Wherever he came, impatient\nof the slower movements of his countrymen in the work of destruction, he\ncaused the prisons to be emptied, and the wretched inmates to be\nbutchered before his eyes. At Ugijar he thus directed the execution of\nno less than two hundred and forty Christians, laymen and\necclesiastics.[54] His progress through the land was literally over the\ndead bodies of his victims.\n\nFierce as he was, Aben-Humeya had some touches of humanity in his\nnature, which made him revolt at the wholesale murders perpetrated by\nhis lieutenant. He was the more indignant when, on hastening to Ugijar\nto save the lives of some of the captives, his friends, he found that he\nhad come too late, for the man of blood had been there before him. He\nsoon after summoned his officer into his presence, not with the\nimpolitic design of taxing him with his cruelties, but to call him to a\nreckoning for the treasure he had pillaged from the churches; and\ndissatisfied, or affecting to be so, with his report, he at once deposed\nAben-Farax from his command. The ferocious chief submitted without a\nmurmur. He descended into the common file, and no more appears on the\nscene. He was one of those miscreants who are thrown on the surface by\nthe turmoil of a revolution, and, after floating there for a while,\ndisappear from sight, and the wave of history closes over them for\never.\n\n\n\n\nCHAPTER III.\n\nREBELLION OF THE MORISCOES.\n\nPanic in Granada--Muster of Troops--Mondejar takes the Field--Bold\nPassage at Tablate--Retreat of the Moriscoes--Combat at\nAlfajarali--Perilous March--Massacre at Jubiles--The Liberated\nChristians.\n\n1568, 1569.\n\n\nAs day after day brought tidings to the people of Granada of the\nbarbarities perpetrated in the Alpujarras, the whole city was filled\nwith grief and consternation. The men might be seen gathered together in\nknots in the public squares; the women ran about from house to house,\ntelling the tale of horrors which could hardly be exaggerated in the\nrecital. They thronged to the churches, where the archbishop and the\nclergy were all day long offering up prayers to avert the wrath of\nheaven from Granada. The places of business were abandoned. The shops\nand booths were closed.[55] As men called to mind the late irruption of\nAben-Farax, they were filled with apprehensions that the same thing\nwould be attempted again; and rumours went abroad that the mountaineers\nwere plotting another descent on the city, and, with the aid of their\ncountrymen in the Albaicin, would soon deluge the streets with the blood\nof the Christians. Under the influence of these fears, some took refuge\nin the fortress of the Alhambra; others fled into the country. Many kept\nwatch during the long night, while those who withdrew to rest started\nfrom their slumbers at the least noise, supposing it to be the war-cry\nof the Moslem, and that the enemy was at the gates.\n\nNor was the alarm less that was felt by the Moriscoes in the city, as it\nwas certainly better founded,--for the Moriscoes were the weaker party\nof the two. They knew the apprehensions entertained of them by the\nChristians, and that, when men have the power to relieve themselves of\ntheir fears, they are not apt to be very scrupulous as to the means of\ndoing so. They were afraid to venture into the streets by day, and at\nnight they barricaded their houses as in a time of siege.[56] They well\nknew that a single act of imprudence on their part, or even the merest\naccident, might bring the Spaniards upon them, and lead to a general\nmassacre. They were like the traveller who sees the avalanche trembling\nabove him, which the least jar of elements, or his own unwary movements,\nmay dislodge from its slippery basis, and bring down in ruin on his\nhead. Thus the two races, inhabitants of the same city, were like two\nhostile camps, looking on each other with watchful and malignant eyes,\nand ready at any moment to come into deadly conflict.\n\nIn this stage of things the Moriscoes, anxious to allay the\napprehensions of the Spaniards, were profuse in their professions of\nloyalty, and in their assurances that there was neither concert nor\nsympathy between them and their countrymen in the Alpujarras. The\ngovernment, to give still greater confidence to the Christians, freely\ndistributed arms among them, thus enabling them, as far as possible, to\nprovide for their own security. The inhabitants enrolled themselves in\ncompanies. The citizen was speedily converted into the soldier, and\nevery man, of whatever trade or profession,--the mechanic, the\nmerchant, the lawyer,--took his turn of military service. Even the\nadvocates, when attending the courts of justice, appeared with their\nweapons by their side.[57]\n\n[Sidenote: MUSTER OF TROOPS.]\n\nBut what contributed above all to revive the public confidence was the\ncare of the government to strengthen the garrison in the Alhambra by the\naddition of five hundred regular troops. When, by these various means,\nthe marquis of Mondejar saw that tranquillity was restored to the\ncapital, he bestowed all his thoughts on an expedition into the\nAlpujarras, desirous to crush the insurrection in its bud, and to rescue\nthe unfortunate captives, whose fate there excited the most dismal\napprehensions amongst their friends and relatives in Granada. He sent\nforth his summons accordingly to the great lords and the cities of\nAndalusia, to furnish him at once with their contingents for carrying on\nthe war. The feudal principle still obtained in this quarter, requiring\nthe several towns to do military service for their possessions, by\nmaintaining, when called upon, a certain number of troops in the field,\nat their own expense for three months, and at the joint expense of\nthemselves and the government for six months longer.[58] The system\nworked well enough in those ancient times, when a season rarely passed\nwithout a foray against the Moslems. But since the fall of Granada, a\nlong period of inactivity had followed, and the citizen, rarely summoned\nto the field, had lost all the essential attributes of the soldier. The\nusual term of service was too short to supply the experience and the\ndiscipline which he needed; and far from entering on a campaign with the\npatriotic or the chivalrous feeling that gives dignity to the profession\nof arms, he brought with him the mercenary spirit of a trader, intent\nonly on his personal gains, and eager, as soon as he had enriched\nhimself by a lucky foray, or the sack of some ill-fated city, to return\nhome, and give place to others, as inexperienced and possessed of as\nlittle subordination as himself.[59]\n\nBut, however deficient this civic militia might be in tactics, the men\nwere well provided with arms and military accoutrements; and, as the\nmotley array of troops passed over the _vega_, they made a gallant show,\nwith their gay uniforms and bright weapons glancing in the sun, while\nthey proudly displayed the ancient banners of their cities, which had\nwaved over many a field of battle against the infidel.[60]\n\nBut no part of the warlike spectacle was so brilliant as that afforded\nby the chivalry of the country; the nobles and cavaliers who, with their\nretainers and household troops, had taken the field with as much\nalacrity on the present occasion as their fathers had ever shown when\nroused by the cry that the enemy was over the borders.[61] They were\nmuch inferior in numbers to the militia of the towns. But inferiority\nof numbers was more than compensated by excellence of discipline, by\ntheir perfect appointments, and by that chivalrous feeling which made\nthem discard every mercenary consideration in the pursuit of glory. Such\nwas the feeling of Luis Paer de Castillego, the ancient regidor of\nCordova. When offered an independent command, with the emoluments\nannexed to it, he proudly replied: \"I want neither rank nor pay. I, my\nsons, my kindred, my whole house, will always be found ready to serve\nour God and our king. It is the title by which we hold our inheritance\nand our patent of nobility.\"[62]\n\nWith such loyal and high-mettled cavaliers to support him, Mondejar\ncould not feel doubtful of the success of his arms. They had, however,\nalready met with one reverse; and he received tidings that his\nadvance-guard, sent to occupy a strong pass that led into the mountains,\nhad been driven from its position, and had sustained something like a\ndefeat. This would have been still more decisive, had it not been for\nthe courage of certain ecclesiastics, eight in number--four of them\nFranciscans, and four of the Society of Jesus--who, as the troops gave\nway, threw themselves into the thick of the fight, and by their example\nshamed the soldiers into making a more determined resistance. The\npresent war took the form of a religious war; and many a valiant\nchurchman, armed with sword and crucifix, bore his part in it as in a\ncrusade.\n\nHastening his preparations, the captain-general, without waiting for\nfurther reinforcements, marched out of Granada on the second of January,\n1569, at the head of a small body, which did not exceed in all two\nthousand foot and four hundred horse. He was speedily joined by levies\nfrom the neighbouring towns--from Jaen, Loja, Alhama, Antequera, and\nother places--which in a few days swelled his little army to double its\noriginal size. The capital he left in the hands of his son, the count of\nTendilla; a man of less discretion than his father, of a sterner and\nmore impatient temper, and one who had little sympathy for the Morisco.\nBy his directions, the peasantry of the _vega_ were required to supply\nthe army with twenty thousand pounds of bread daily.[63] The additional\ntroops stationed in the city, as well as those who met there, as in a\nplace of rendezvous, on their way to the sierra, were all quartered on\nthe inhabitants of the Albaicin, where they freely indulged in the usual\nhabits of military licence. The Moriscoes still retained much of that\njealous sensibility which leads the natives of the East to seclude their\nwives and daughters from the eye of the stranger. It was in vain,\nhowever, that they urged their complaints in the most respectful and\ndeprecatory terms before the governor. The haughty Spaniard only\nanswered them with a stern rebuke, which made the Moriscoes too late\nrepent that they had not profited by the opportunity offered them by\nAben-Farax of regaining their independence.[64]\n\nLeaving Granada, the captain-general took the most direct route, leading\nalong the western slant of the Sierra Nevada, that mountain-range which,\nwith its frosty peaks glistening in the sun like palisades of silver,\nfences round the city on the south, and screens it in the summer from\nthe scorching winds of Africa. Thence he rapidly descended into the\nbeautiful vale of Lecrin, which spreads out, like a gay carpet\nembroidered with many a wild flower, to the verge of the Alpujarras. It\nwas now, however, the dead of winter, when the bright colouring of the\nlandscape, even in this favoured region, watered as it was by numerous\nfountains and running streams, had faded into the sombre tints more in\nharmony with the rude scenes on which the Spaniards were about to enter.\n\n[Sidenote: BOLD PASSAGE AT TABLATE.]\n\nHalting a night at Padul to refresh his troops, Mondejar pressed forward\nto Durcal, which he reached barely in time to save his advance-guard\nfrom a more shameful discomfiture than it had before experienced; for\nthe enemy, pressing it on all sides, was in possession of the principal\navenues to the town. On the approach of the main body of the Spaniards,\nhowever, he made a hasty retreat, and established himself in a strong\nposition at the pass of Tablate. The place was defended by a _barranca_,\nor ravine, not formidable from its width, but its rocky side swept sheer\ndown to a depth that made the brain of the traveller giddy as he looked\ninto the frightful abyss. The chasm extended at least eight leagues in\nlength, thus serving, like a gigantic ditch scooped out by the hand of\nNature, to afford protection to the beautiful valley against the inroads\nof the fierce tribes of the mountains.\n\nAcross this gulf a frail wooden bridge had been constructed, forming the\nonly means of access from this quarter to the country of the Alpujarras.\nBut this structure was now nearly demolished by the Moriscoes, who had\ntaken up the floor, and removed most of the supports, till the passage\nof the tottering fabric could not safely be attempted by a single\nindividual, much less by an army.[65] That they did not destroy the\nbridge altogether, probably arose from their desire to re-establish as\nsoon as possible their communications with their countrymen in the\nvalley.\n\nMeanwhile the Moslems had taken up a position which commanded the\nfarther end of the bridge, where they calmly awaited the approach of the\nSpaniards. Their army, which greatly fluctuated in its numbers at\ndifferent periods of the campaign, was a miscellaneous body, ill\ndisciplined and worse armed. Some of the men carried fire-arms, some\ncrossbows; others had only slings or javelins, or even sharp-pointed\nstakes; any weapon, in short, however rude, which they had contrived to\nsecrete from the Spanish officials charged with enforcing the laws for\ndisarming the Moriscoes. But they were a bold and independent race,\ninured to a life of peril and privation; and, however inferior to the\nChristians in other respects, they had one obvious advantage, in their\nfamiliarity with the mountain wilds in which they had been nurtured from\ninfancy.\n\nAs the Spaniards approached the ravine, they were saluted by the enemy,\nfrom the other side, with a shower of balls, stones, and arrows, which,\nfalling at random, did little mischief. But as soon as the columns of\nthe Christians reached the brow of the _barranca_, and formed into line,\nthey opened a much more effective fire on their adversaries; and when\nthe heavy guns with which Mendoza was provided were got into position,\nthey did such execution on the enemy that he thought it prudent to\nabandon the bridge, and take post behind a rising ground, which screened\nhim from the fire.\n\nAll thoughts were now turned on the mode of crossing the ravine; and\nmany a look of blank dismay was turned on the dilapidated bridge, which,\nlike a spider's web, trembling in every breeze, was stretched across the\nformidable chasm. No one was bold enough to venture on this pass of\nperil. At length a Franciscan monk, named Christoval de Molina, offered\nhimself for the emprise. It was again an ecclesiastic who was to lead\nthe way in the path of danger. Slinging his shield across his back, with\nhis robe tucked closely around him, grasping a crucifix in his left\nhand, and with his right brandishing his sword, the valiant friar set\nhis foot upon the bridge.[66] All eyes were fastened upon him, as,\ninvoking the name of Jesus, he went courageously but cautiously forward,\npicking his way along the skeleton fabric, which trembled under his\nweight, as if about to fall in pieces and precipitate him into the gulf\nbelow. But he was not so to perish; and his safe arrival on the farther\nside was greeted with the shouts of the soldiery, who, ashamed of their\nhesitation, now pressed forward to follow in his footsteps.\n\nThe first who ventured had the same good fortune as his predecessor. The\nsecond, missing his step or becoming dizzy, lost his foothold, and,\ntumbling headlong, was dashed to pieces on the bottom of the ravine. One\nafter another, the soldiers followed, and with fewer casualties than\nmight have been expected from the perilous nature of the passage. During\nall this time they experienced no molestation from the enemy,\nintimidated, perhaps, by the unexpected audacity of the Spaniards, and\nnot caring to come within the range of the deadly fire of their\nartillery. No sooner had the arquebusiers crossed in sufficient\nstrength, than Mondejar, putting himself at their head, led them against\nthe Moslems. He was received with a spirited volley, which had well-nigh\nproved fatal to him; and had it not been for his good cuirass, that\nturned the ball of an arquebuse, his campaign would have been brought to\na close at its commencement. The skirmish lasted but a short time, as\nthe Moriscoes, already disheartened by the success of the assailants, or\nin obedience to the plan of operations marked out by their leader,\nabandoned their position, and drew off rapidly towards the mountains. It\nwas the intention of Aben-Humeya, as already noticed, to entangle his\nenemies in the defiles of the sierra, where, independently of the\nadvantage he possessed from a knowledge of the country, the rugged\ncharacter of the ground, he conceived, would make it impracticable for\nboth cavalry and artillery, with neither of which he was provided.[67]\n\nThe Spanish commander, resuming his former station, employed the night\nin restoring the bridge, on which his men laboured to such purpose, that\nby morning it was in a condition for both his horse and his heavy guns\nto cross in safety. Meanwhile he received tidings that a body of a\nhundred and eighty Spaniards, in the neighbouring town of Orgiba, who\nhad thrown themselves into the tower of the church on the breaking out\nof the insurrection, were still holding their position, and anxiously\nlooking for succour from their countrymen. Pushing forward, therefore,\nwithout loss of time, he resumed his march across the valley, which was\nhere defended on either side by rugged hills, that, growing bolder as he\nadvanced, announced his entrance into the gorges of the Alpujarras. The\nweather was tempestuous. The roads were rendered worse than usual by the\nheavy rains, and by the torrents that descended from the hills. The\nSpaniards, moreover, suffered much from straggling parties of the enemy,\nwho had possession of the heights, whence they rolled down huge rocks,\nand hurled missiles of every kind on the heads of the invaders. To rid\nhimself of this annoyance, Mondejar ordered detachments of horse--one of\nthem under the command of his son, Don Antonio de Mendoza--to scour the\ncrests of the hills and dislodge the skirmishers. Pioneers were sent in\nadvance, to level the ground and render it practicable for cavalry. The\nservice was admirably performed; and the mountaineers, little acquainted\nwith the horse, which they seemed to have held in as much terror as did\nthe ancient Mexicans, were so astounded by seeing the light-footed\nAndalusian steed scaling the rough sides of the sierra, along paths\nwhere the sportsman would hardly venture, that, without waiting for the\ncharge, they speedily quitted the ground and fell back on the main body\nof their army.\n\n[Sidenote: RETREAT OF THE MORISCOES.]\n\nThis was posted at Lanjaron, a place but a few miles off, where the\nMoriscoes had profited by a gentle eminence that commanded a narrow\ndefile, to throw up a breastwork of stone and earth, behind which they\nwere entrenched, prepared, as it would seem, to give battle to the\nSpaniards.\n\nThe daylight had begun to fade, as the latter drew near the enemy's\nencampment; and, as he was unacquainted with the ground, Mondejar\nresolved to postpone his attack till the following morning. The night\nset in dark and threatening. But a hundred watchfires blazing on the\nhill-tops illumined the sky, and sent a feeble radiance into the gloom\nof the valley. All night long the wild notes of the musical instruments\npeculiar to the Moors, mingling with their shrill war-cries, sounded in\nthe ears of the Christians, keeping them under arms, and apprehensive\nevery moment of an attack.[68] But a night attack was contrary to the\nusual tactics of the Moors. Nor, as it appeared, did they intend to join\nbattle with the Spaniards at all in this place. At least, if such had\nbeen their design, they changed it. For at break of day, to the surprise\nof the Spaniards, no vestige was to be seen of the Moriscoes, who,\nabandoning their position, had taken flight, like their own birds of\nprey, into the depths of the mountains.\n\nMondejar, not sorry to be spared the delay which an encounter must have\ncaused him at a time when every moment was so precious, now rapidly\npushed forward to Orgiba, where he happily arrived in season to relieve\nthe garrison, reduced almost to the last extremity, and to put to flight\nthe rabble who besieged it.\n\nIn the fulness of their hearts, and with the tears streaming from their\neyes, the poor prisoners came forth from their fortress to embrace the\ndeliverers who had rescued them from the most terrible of deaths. Their\napprehensions of such a fate had alone nerved their souls to so long and\nheroic a resistance. Yet they must have sunk ere this from famine, had\nit not been for their politic precaution of taking with them into the\ntower several of the Morisco children whose parents secretly supplied\nthem with food, which served as the means of subsistence--scanty though\nit was--for the garrison. But as the latter came forth into view, their\nwasted forms and famine-stricken visages told a tale of woe that would\nhave softened a heart of flint.[69]\n\nThe situation of Orgiba pointed it out as suitable for a fortified post,\nto cover the retreat of the army, if necessary, and to protect the\nconvoys of supplies to be regularly forwarded from Granada. Leaving a\nsmall garrison there, the captain-general, without longer delay, resumed\nhis pursuit of the enemy.\n\nAben-Humeya had retreated into Poqueira, a rugged district of the\nAlpujarras. Here he had posted himself, with an army amounting to more\nthan double its former numbers, at the extremity of a dangerous defile,\ncalled the Pass of Alfajarali. Behind lay the town of Bubion, the\ncapital of the district, in which, considering it as a place of safety,\nmany of the wealthier Moriscoes had deposited their women and their\ntreasures.\n\nMondejar's line of march now took him into the heart of the wildest\nregions of the Alpujarras, where the scenery assumed a character of\nsublimity very different from what he had met with in the lower levels\nof the country. Here mountain rose beyond mountain, till their hoary\nheads, soaring above the clouds, entered far into the region of eternal\nsnow. The scene was as gloomy as it was grand. Instead of the\nwide-spreading woods that usually hang round the skirts of lofty\nmountains, covering up their nakedness from the eye, nothing here was to\nbe seen but masses of shattered rock, black as if scathed by volcanic\nfires, and heaped one upon another in a sort of wild confusion, as if\nsome tremendous convulsion of nature had torn the hills from their\nfoundations, and thrown them into primitive chaos. Yet the industry of\nthe Moriscoes had contrived to relieve the savage features of the\nlandscape, by scooping out terraces wherever the rocky soil allowed it,\nand raising there the vine and other plants, in bright patches of\nvariegated culture, that hung like a garland round the gaunt and swarthy\nsierra.\n\nThe temperature was now greatly changed from what the army had\nexperienced in the valley. The wind, sweeping down the icy sides of the\nmountains, found its way through the harness of the cavaliers and the\nlight covering of the soldiers, benumbing their limbs, and piercing them\nto the very bone. Great difficulty was experienced in dragging the\ncannon up the steep heights, and along roads and passes, which, however\neasily traversed by the light-footed mountaineer, were but ill suited to\nthe movements of an army clad in the heavy panoply of war.\n\nThe march was conducted in perfect order, the arquebusiers occupying the\nvan, and the cavalry riding on either flank, while detachments of\ninfantry, the main body of which occupied the centre, were thrown out to\nthe right and left, on the higher grounds along the route of the army,\nto save it from annoyance from the mountaineers.\n\nOn the thirteenth of January, Mondejar entered the narrow defile of\nAlfajarali, at the farther end of which the motley multitude that had\ngathered round the standard of Aben-Humeya were already drawn up in\nbattle-array. His right wing rested on the bold side of the sierra; the\nleft was defended by a deep ravine, and his position was strengthened by\nmore than one ambuscade, for which the nature of the ground was\neminently favourable.[70] Indeed, ambushes and surprises formed part of\nthe regular strategy of the Moorish warrior, who lost heart if he failed\nin these,--like the lion, who, if balked in the first spring upon his\nprey, is said rarely to attempt another.\n\n[Sidenote: COMBAT AT ALFAJARALI.]\n\nPutting these wily tactics into practice, the Morisco chief, as soon as\nthe Spaniards were fairly entangled in the defile, without waiting for\nthem to come into order of battle, gave the signal; and his men,\nstarting up from glen, thicket, and ravine, or bursting down the\nhill-sides like their own winter-torrents, fell at once on the\nChristians,--front, flank, and rear,--assailing them on every\nquarter.[71] Astounded by the fiery suddenness of the assault, the\nrear-guard retreated on the centre, while the arquebusiers in the van\nwere thrown into still greater disorder. For a few moments it seemed as\nif the panic would become general. But the voice of the leader was heard\nabove the tumult, and by his prompt and sagacious measures he\nfortunately succeeded in restoring order, and reviving the confidence of\nhis men. He detached one body of cavalry, under his son-in-law, to the\nsupport of the rear, and another to the front under the command of his\nson, Antonio de Mendoza. Both executed their commissions with spirit;\nand Mendoza, outstripping his companions in the haste with which he\ngalloped to the front, threw himself into the thickest of the fight,\nwhere he was struck from his horse by a heavy stone, and was speedily\nsurrounded by the enemy, from whose grasp he was with difficulty, and\nnot till after much hard fighting, rescued by his companions. His\nfriend, Don Alonso Portocarrero, the scion of a noble house in\nAndalusia, whose sons had always claimed the front of battle against the\ninfidel, was twice wounded by poisoned arrows; for the Moors of the\nAlpujarras tipped their weapons with a deadly poison distilled from a\nweed that grew wild among the mountains.[72]\n\nA fierce struggle now ensued; for the Morisco was spurred on by hate and\nthe recollection of a thousand wrongs. Ill provided with weapons for\nattack, and destitute of defensive armour, he exposed himself to the\nhottest of his enemy's fire, and endeavoured to drag the horsemen from\ntheir saddles, while stones and arrows, with which some musket-balls\nwere intermingled, fell like rain on the well-tempered harness of the\nAndalusian knights. The latter, now fully roused, plunged boldly into\nthe thickest of the Moorish multitude, trampling them under foot, and\nhewing them down, right and left, with their sharp blades. The\narquebusiers, at the same time, delivered a well-directed fire on the\nflank of the Moriscoes, who, after a brave struggle of an hour's\nduration, in which they were baffled on every quarter, quitted the\nfield, covered with their slain, as precipitately as they had entered\nit, and, vanishing among the mountains, were soon far beyond\npursuit.[73]\n\nFrom the field of battle Mondejar marched at once upon Bubion, the\ncapital of the district, and now left wholly unprotected by the Moslems.\nYet many of their wives and daughters remained in it; and what rejoiced\nthe heart of Mondejar more than all, was the liberation of a hundred and\neighty Christian women, who came forth, frantic with joy and gratitude,\nto embrace the knees of their deliverers. They had many a tale of horror\nto tell their countrymen, who had now rescued them from a fate worse\nthan that of death itself; for arrangements had been made, it was said,\nto send away those whose persons offered the greatest attractions, to\nswell the harems of the fierce Barbary princes in alliance with the\nMoriscoes. The town afforded a rich booty to the victorious troops, in\ngold, silver, and jewels, together with the finest stuffs, especially of\nsilk, for the manufacture of which the people of the country were\ncelebrated. As the Spanish commander, unwilling to be encumbered with\nunnecessary baggage, had made no provision for transporting the more\nbulky articles, the greater part of them, in the usual exterminating\nspirit of war, was consigned to the flames.[74] The soldiers would\nwillingly have appropriated to themselves the Moorish women whom they\nfound in the place, regarding them us the spoils of victory; but the\nmarquis, greatly to the disgust of his followers, humanely interfered\nfor their protection.\n\nMondejar now learned that Aben-Humeya, gathering the wreck of his forces\nabout him, had taken the route to Jubiles,--a place situated in the\nwildest part of the country, where there was a fortress of much\nstrength, in which he proposed to make a final stand against his\nenemies. Desirous to follow up the blow before the enemy had time to\nrecover from its effects, Mondejar resumed his march. He had not\nadvanced many leagues before he reached Pitres, the principal town in\nthe district of Ferreiras. It was a place of some importance, and was\nrich in the commodities usually found in the great Moorish towns, where\nthe more wealthy of the inhabitants rivalled their brethren of Granada\nin their taste for sumptuous dress and in the costly decorations of\ntheir houses.\n\nThe conquerors had here the satisfaction of releasing a hundred and\nfifty of their poor countrywomen from the captivity in which they had\nbeen held, after witnessing the massacre of their friends and relatives.\nThe place was given up to pillage; but the marquis, true to his\nprinciples, notwithstanding the murmurs, and even menaces, of his\nsoldiers, would allow no injury to be done to the Moorish women who\nremained in it. In this he acted in obedience to the dictates of sound\npolicy, no less than of humanity, which indeed, happily for mankind, can\nnever be dissevered from each other. He had no desire to push the war to\nextremities, or to exterminate a race whose ingenuity and industry were\na fruitful source of revenue to the country. He wished, therefore, to\nleave the door of reconciliation still open; and while he carried fire\nand sword into the enemy's territory, he held out the prospect of grace\nto those who were willing to submit and return to their allegiance.\n\nThe route of the army lay through a wild and desolate region, which,\nfrom its great elevation, was cool even in midsummer, and which now, in\nthe month of January, wore the dreary aspect of a polar winter. The\nsnow, which never melted on the highest peaks of the mountains, lay\nheavily on their broad shoulders, and, sweeping far down their sides,\ncovered up the path of the Spaniards. It was with no little difficulty\nthat they could find a practicable passage, especially for the train of\nheavy guns, which were dragged along with incredible toil by the united\nefforts of men and horses. The soldiers, born and bred in the sunny\nplains of Andalusia, were but ill provided against an intensity of cold\nof which they had never formed a conception. The hands and feet of many\nwere frozen. Others, benumbed, and exhausted by excessive toil,\nstraggled in the rear, and sunk down in the snow-drifts, or disappeared\nin the treacherous ravines and crevices, which, under their glittering\nmantle, lay concealed from the eye. It fared still worse with the\nMoriscoes, especially with the women and children, who, after hanging on\nthe skirts of the retreating army, had, the better to elude pursuit,\nscaled the more inaccessible parts of the mountains, where, taking\nrefuge in caverns, they perished, in great numbers, of cold and\nhunger.[75]\n\nMeanwhile Aben-Humeya, disheartened by his late reverses, felt too\nlittle confidence in the strength of his present position to abide there\nthe assault of the Spaniards. Quitting the place, therefore, and taking\nwith him his women and effects, he directed his course by rapid marches\ntowards Paterna, his principal residence, which had the advantage, by\nits neighbourhood to the Sierra Nevada, of affording him, if necessary,\nthe means of escaping into its wild and mysterious recesses, where none\nbut a native would care to follow him. He left in the castle of Jubiles\na great number of Morisco women, who had accompanied the army in its\nretreat, and three hundred men, who, from age or infirmity, would be\nlikely to embarrass his movements.\n\n[Sidenote: MASSACRE AT JUBILES.]\n\nOn reaching Jubiles, therefore, the Spanish general met with no\nresistance from the helpless garrison who occupied the fortress, which,\nmoreover, contained a rich booty in gold, pearls, and precious stones,\nto gratify the cupidity of the soldiers.[76] Yet their discontent was\nexpressed in more audacious terms than usual at the protection afforded\nby their commander to the Morisco women, of whom there were more than\ntwo thousand in the place. Among the women found there was also a good\nnumber of Christian captives, who roused the fierce passions of their\ncountrymen by their piteous recital of the horrors they had witnessed,\nof the butchery of fathers, husbands, and brothers, and of the\npersecutions to which they had themselves been subjected in order to\nconvert them to Islamism. They besought the captain-general to take pity\non their sufferings, and to avenge their wrongs by putting every man and\nwoman found in the place to the sword.[77] It is evident that, however\nprepared they may have been to accept the crown of martyrdom rather than\nabjure their faith, they gave little heed to the noblest of its\nprecepts, which enjoined the forgiveness of their enemies. In this\nrespect Mondejar proved himself decidedly the better Christian; for\nwhile he listened with commiseration to their tale of woe, and did all\nhe could to comfort them in their affliction,[78] he would not abandon\nthe protection of his captives, male or female, nor resign them to the\nbrutality of his soldiers.\n\nHe provided for their safety during the night by allowing them to occupy\nthe church. But as this would not accommodate more than a thousand\npersons, the remainder, including all the men, were quartered in an open\nsquare in the neighbourhood of the building. The Spanish troops encamped\nat no great distance from the spot.\n\nIn the course of the night one of the soldiers found his way into the\nquarters of the captives, and attempted to take some freedoms with a\nMorisco maiden. It so happened that her lover, disguised in woman's\nattire, was at her side, having remained with her for her protection.\nHis Moorish blood fired at the insult, and he resented it by striking\nhis poniard into the body of the Spaniard. The cry of the latter soon\nroused his comrades. Rushing to the place, they fell on the young\nMorisco, who, now brandishing a sword which he had snatched from the\ndisabled man, laid about him so valiantly that several others were\nwounded. The cry rose that there were armed men, disguised as women,\namong the prisoners. More soldiers poured in to the support of their\ncomrades, and fell with fury on their helpless victims. The uproar was\nuniversal. On the one side might be heard moans and petitions for mercy;\non the other, brutal imprecations, followed by deadly blows, that showed\nhow little prayers for mercy had availed. The hearts of the soldiers\nwere harder than the steel with which they struck; for they called to\nmind the cruelties inflicted on their own countrymen by the Moriscoes.\nStriking to the right and left, they hewed down men and women\nindiscriminately,--both equally defenceless. In their blind fury they\neven wounded one another; for it was not easy to discern friend from foe\nin the obscurity, in which little light was to be had, says the\nchronicler, except such as came from the sparks of clashing steel or the\nflash of fire-arms.[79] It was in vain that the officers endeavoured to\ncall off the men from their work of butchery. The hot temper of the\nAndalusian was fully roused; and it would have been as easy to stop the\nexplosion of the mine when the train has been fired, as to stay his\nfury. It was not till the morning light showed the pavement swimming in\ngore, and the corpses of the helpless victims lying in heaps on one\nanother, that his appetite for blood was satisfied. Great numbers of the\nwomen, and nearly all the men, perished in this massacre.[80] Those in\nthe church succeeded in making fast the doors, and thus excluding their\nenemies, who made repeated efforts to enter the building. The marquis of\nMondejar, indignant at this inhuman outrage perpetrated by his\nfollowers, and at their flagrant disobedience of orders, caused an\ninquiry into the affair to be instantly made; and the execution of three\nof the most guilty proved a salutary warning to the Andalusian soldier\nthat there were limits beyond which it was not safe to try the patience\nof his commander.[81]\n\nBefore leaving Jubiles, Mondejar sent off to Granada, under a strong\nescort, the Christian captives who, since their liberation, had remained\nwith the army. There were eight hundred of them, women and children,--a\nhelpless multitude, whose wants were to be provided for, and whose\npresence could not fail greatly to embarrass his movements. They were\nobliged to perform that long and wearisome journey across the mountains\non foot, as there were no means of transportation. And piteous was the\nspectacle which they presented when they reached the capital. As the\nwayworn wanderers entered by the gate of Bib-arranbla, the citizens came\nforth in crowds to welcome them. A body of cavalry was in the van,--each\nof the troopers holding one or two children on the saddle before him,\nwith sometimes a third on the crupper clinging to his back. The infantry\nbrought up the rear; while the centre of the procession was occupied by\nthe women,--a forlorn and melancholy band, with their heads undefended\nby any covering from the weather; their hair, bleached by the winter's\ntempests, streaming wildly over their shoulders; their clothes scanty,\ntattered, and soiled with travel; without stockings, without shoes, to\nprotect their feet against the cold and flinty roads; while in the lines\ntraced upon their countenances the dullest eye might read the story of\ntheir unparalleled sufferings. Many of the company were persons who,\nunaccustomed to toil, and delicately nurtured, were but poorly prepared\nfor the trials and privations of every kind to which they had been\nsubjected.[82]\n\n[Sidenote: SITUATION OF ABEN-HUMEYA.]\n\nAs their friends and countrymen gathered round them, to testify their\nsympathy and listen to the story of their misfortunes, the voices of the\npoor wanderers were choked with sobs and lamentations. The grief was\ncontagious; and the sorrowing and sympathetic multitude accompanied the\nprocession like a train of mourners to the monastery of Our Lady of\nVictory, in the opposite quarter of the city, where services were\nperformed with much solemnity, and thanks were offered up for their\ndeliverance from captivity. From the church they proceeded to the\nAlhambra, where they were graciously received by the marchioness of\nMondejar, the wife of the captain-general, who did what she could to\nalleviate the miseries of their condition. Those who had friends and\nrelations in the city, found shelter in their houses; while the rest\nwere kindly welcomed by the archbishop of Granada, and by the\ncharitable people of the town, who provided them with raiment and\nwhatever was necessary for their comfort.[83] The stories which the\nfugitives had to tell of the horrid scenes they had witnessed in the\nAlpujarras, roused a deeper feeling of hatred in the Spaniards towards\nthe Moriscoes, that boded ill for the security of the inhabitants of the\nAlbaicin.\n\n\n\n\nCHAPTER IV.\n\nREBELLION OF THE MORISCOES.\n\nSituation of Haben-Humeya--Fate of the Moorish Prisoners--Storming of\nGuajaras--Escape of Haben-Humeya--Operations of Los Velez--Cabal against\nMondejar--Licence of the Soldiers--Massacre in Granada--The Insurrection\nrekindled.\n\n1569.\n\n\nBefore the marquis of Mondejar quitted Jubiles, he received a visit from\nseventeen of the principal Moriscoes in that part of the country, who\ncame to tender their submission, exculpating themselves, at the same\ntime, from any share in the insurrection, and humbly suing for the\ncaptain-general's protection. This, agreeably to his policy, he promptly\naccorded, granting them a safe-conduct, with instructions to tell their\ncountrymen what he had done, and persuade them, if possible, to return\nto their allegiance, as the only way of averting the ruin that else\nwould speedily overtake them. This act of clemency, so repugnant to the\nfeelings of the Spaniards, was a new cause of disgust to his soldiers,\nwho felt that the fair terms thus secured by the rebels were little less\nthan a victory over themselves.[84] Yet the good effects of this policy\nwere soon made visible, when the marquis resumed his march; for, as his\nfavourable dispositions became more generally known, numbers of the\nMoriscoes, and several places on the route, eagerly tendered their\nsubmission, imploring his mercy, and protection against his followers.\n\nAben-Humeya, meanwhile, who lay at Paterna, with his wives and his\nwarriors gathered around, saw with dismay that his mountain throne was\nfast sliding away from beneath him. The spirit of distrust and\ndisaffection had crept into his camp. It was divided into two parties;\none of these, despairing of further resistance, would have come\ninstantly to terms with the enemy; the other still adhered to a bolder\npolicy; but its leaders, if we may trust the Castilian writers, were\nless influenced by patriotic than by personal motives, being for the\nmost part men who had borne so conspicuous a part in the insurrection,\nthat they could scarcely hope to be included in any amnesty granted by\nthe Spaniards. Such, in particular, were the African adventurers, who\nhad distinguished themselves above all others by their ferocious\npersecution of the Christians. They directed, at this time, the counsels\nof the Moorish prince, filling his mind with suspicions of the loyalty\nof some of his followers, especially of the father of one of his\nwives,--a person of much authority among the Moriscoes. To suspect and\nto slay were words of much the same import with Aben-Humeya. He sent for\nhis relative, and, on his entering the apartment, caused him to be\ndespatched before his eyes.[85] He would have followed this up by the\nmurder of some others of the family, if they had not eluded his grasp;\nthus establishing his title to a descent from those despots of the East\nwith whom the lives of their kindred were of as little account as the\nvermin in their path.[86]\n\nHe was still at the head of a numerous army; its number, indeed,\namounting to six thousand men, constituted its greatest strength; for,\nwithout discipline, almost without arms, it was made up of such rude,\nincongruous materials, that, as he already had experience, it could\nnever abide the shock of battle from the militia of Castile. The Moorish\nprince had other causes for discouragement in the tidings he was hourly\nreceiving of the defection of his subjects. The clemency shown by the\nconqueror was doing more for him than his arms,--as the snow which the\nblasts of winter have only bound more closely to the hill-side loosens\nits hold and falls away under the soft touch of spring. Notwithstanding\nhis late display of audacity, the unhappy young man now lost all\nconfidence in his own fortunes and in his followers. Sorely perplexed,\nhe knew not where to turn. He had little of the constancy or courage of\nthe patriot who has perilled his life in a great cause; and he now had\nrecourse to the same expedient which he had so lately punished with\ndeath in his father-in-law.\n\nHe sent a message to the marquis of Mondejar, offering to surrender,\nand, if time were given, to persuade his people to follow his example.\nMeanwhile he requested the Spanish commander to stay his march, and thus\nprevent a collision with his troops. Mondejar, though he would not\nconsent to this, advanced more leisurely, while he opened a negotiation\nwith his enemy. He had already come in sight of the rebel forces, when\nhe consented, at the request of Aben-Humeya, to halt for a night in the\nneighbouring village of Iniza, in order to give time for a personal\ninterview. This required the troops, some of whom had now advanced\nwithin musket-range of the enemy, to fall back, and take up ground in\nthe rear of their present position. In executing this manoeuvre, they\ncame almost in contact with a detachment of the Moorish army, who, in\ntheir ignorance of its real object, regarding the movement as a hostile\ndemonstration, sent a shower of arrows and other missiles among the\nSpaniards, which they returned, with hearty goodwill, by a volley of\nmusketry. The engagement soon became general. Aben-Humeya at the time\nwas reading a letter, which he had just received from one of Mondejar's\nstaff, arranging the place for the interview, when he was startled by\nthe firing, and saw with consternation his own men warmly engaged with\nthe enemy. Supposing he had been deceived by the Spaniards, he flung the\nletter on the ground, and throwing himself into the saddle, without so\nmuch as attempting to rally his forces, which were now flying over the\nfield in all directions, he took the road to the Sierra Nevada, followed\nby only five or six of his attendants.[87] His horse was fleet, and he\nsoon gained the defiles of the mountains. But he was hotly pursued; and,\nthinking it safer to trust to himself than to his horse, he dismounted,\ncut the hamstrings of the animal, to prevent his being of service to his\npursuers, and disappeared in the obscure depths of the sierra, where it\nwould have been fruitless to follow him.\n\n[Sidenote: THE FALL OF JUBILES.]\n\nThe rout of his army was complete; and the victors might have inflicted\nan incalculable loss on the fugitives, had not the marquis of Mondejar\ncalled off his troops, and put a stop to the work of death. He wished to\nkeep open as widely as possible the door of reconciliation. His conduct,\nwhich was not understood, and could not have been appreciated by his\nmen, was stigmatized by them as treachery. They found some amends for\ntheir disappointment in the pillage of Paterna, the residence of\nAben-Humeya, which well provided with the costly finery so much loved by\nthe Moriscoes, furnished a welcome booty to the conquerors.[88]\n\nAmong the Moorish captives were Aben-Humeya's mother, two of his\nsisters, and one of his wives, to whom, as usual, Mondejar extended his\nprotection.\n\nYet the disposal of his prisoners was a subject of perplexity to the\nSpanish commander. His soldiers, as we have seen, would have settled it\nat once, had their captain consented, by appropriating them all as the\nspoils of victory. There were many persons, higher in authority than\nthese soldiers, who were of the same way of thinking on the subject with\nthem. The question was one of sufficient importance to come before the\ngovernment. Philip referred it to the council of state; and, regarding\nit as a case of conscience, in which the interests of religion were\nconcerned, he asked the opinion of the Royal Audience of Granada, over\nwhich Deza presided. The final decision was what might have been\nexpected from tribunals with inquisitors at their head. The Moriscoes,\nmen and women, were declared to have incurred by their rebellion the\ndoom of slavery. What is more remarkable is the precedent cited for this\njudgment, it being no other than a decision of the Council of Toledo, as\nfar back as the time of the Visigoths, when certain rebellious Jews were\nheld to have forfeited their liberty by an act of rebellion.[89] The\nMorisco, it was said, should fare no better than the Jew, since he was\nnot only, like him, a rebel and an infidel, but an apostate to boot. The\ndecision, it was understood, was very satisfactory to Philip, who,\nhowever, \"with the pious moderation that distinguished so just and\nconsiderate a prince,\"[90] so far mitigated the severity of the\nsentence, in the pragmatic which he published, as to exempt from its\noperation boys under ten years of age and girls under eleven. These were\nto be placed in the care of responsible persons, who would give them the\nbenefits of a Christian education. Unhappily, there is reason to think\nthat the good intentions of the government were not very conscientiously\ncarried out in respect to this provision by those intrusted with the\nexecution of it.[91]\n\nWhile the question was pending, Jubiles fell into the hands of the\nvictors; and Mondejar, not feeling himself at liberty to release his\nfemale captives, of whom more than a thousand, by this event, had come\ninto his possession, delivered them in charge to three of the principal\nMoriscoes, to whom, it may be remembered, he had given letters of\nsafe-conduct. They were allowed to restore the women to their families,\non condition that they should all be surrendered on the demand of the\ngovernment. Such an act, it must be admitted, implies great confidence\nin the good faith of the Moslems,--a confidence fully justified by the\nresult. When, in obedience to the pragmatic, they were claimed by the\ngovernment, they were delivered up by their families,--with the\nexception of some who had died in the meantime,--and the greater part of\nthem were sold by public auction in Granada.[92]\n\nThe only place of any importance which now held out against Mondejar was\nLas Guajaras, situated in the plains of Salobrena, in the direction of\nVelez Malaga. This was a rocky, precipitous hill, on the summit of\nwhich, nature, with little assistance from art, had constructed a sort\nof rude fortress. It was held by a fierce band of Moriscoes, who,\ndescending from the heights, swept over the plains, carrying on\ndevastating forays, that made them the terror of the surrounding\ncountry. Mondejar, moved by the complaints of the inhabitants, left\nUgijar on the fifth of February, at the head of his whole array, now\nmuch augmented by the arrival of recent levies, and marched rapidly on\nGuajaras. He met with a more formidable resistance than he had expected.\nHis first attempt to carry the place was repulsed with a heavy loss on\nthe part of the assailants. The Moorish garrison, from its elevated\nposition, poured a storm of missiles on their heads, and, what was\nworse, rolled down huge masses of rock, which, ploughing through the\nCastilian ranks, overthrew men and horses, and did as great execution as\nwould have been done by artillery. Eight hundred Spaniards were left\ndead on the field: and many a noble house in Andalusia had to go into\nmourning for that day's disaster.\n\nMondejar, stung by this repulse,--the first reverse his arms had\nexperienced,--determined to lead the attack in person on the following\nday. His approaches were made with greater caution than before; and,\nwithout much injury, he succeeded in bringing his arquebusiers on a\nhigher level, where their fire swept the enemy's intrenchments and\ninflicted on him a terrible loss. Still the sun went down, and the place\nhad not surrendered. But El Zamar, its brave defender, without\nammunition, almost without arms, felt that there was no longer hope for\nhis little garrison. Silently evacuating the place, therefore, at dead\nof night, the Moriscoes, among whom were both women and children,\nscrambled down the precipice with the fearlessness of the mountain goat,\nand made their escape without attracting the notice of the Spaniards.\nThey left behind only such as, from age or infirmity, were unable to\nfollow them in their perilous descent.\n\nOn the next day, when the Spanish general prepared to renew the assault,\ngreat was his astonishment to find that the enemy had vanished, except\nonly a few wretched beings incapable of making any resistance. All the\nevil passions of Mondejar's nature had been roused by the obstinate\ndefence of the place, and the lives it had cost him. In the heat of his\nwrath, he ordered the helpless garrison to be put to the sword. No\nprayer for mercy was heeded. No regard was had to age or to sex. All\nwere cut down in the presence of the general, who is even said to have\nstimulated the faltering soldiers to go through with their bloody\nwork.[93] An act so hard to be reconciled with his previous conduct has\nbeen referred by some to the annoyance which he felt at being so\nfrequently taxed with excessive lenity to the Moriscoes, an accusation\nwhich was carried, indeed, before the crown, and which the present\noccasion afforded him the means of effectually disproving. However this\nmay be, the historian must lament the tarnished honour of a brave and\ngenerous chief, whose character up to this time had been sullied by\nnone of those acts of cruelty which distinguished this sanguinary\nwar.[94]\n\n[Sidenote: CAPTURE AND DEATH OF EL ZAMAR.]\n\nBut even this cruelty was surpassed by that of his son, the count of\nTendilla. El Zamar, the gallant defender of the fortress, wandered about\namong the crags with his little daughter, whom he carried in his arms.\nFamished and fainting from fatigue, he was at length overtaken by his\nenemies, and sent off as a prisoner to Granada, where the fierce\nTendilla caused the flesh to be torn from his bones with red-hot\npincers, and his mangled carcase, yet palpitating with life, to be\nafterwards quartered. The crime of El Zamar was that he had fought too\nbravely for the independence of his nation.\n\nHaving razed the walls of Guajaras to the ground, Mondejar returned with\nhis blood-stained laurels to his head-quarters at Orgiba. Tower and town\nhad gone down before him. On every side his arms had proved victorious.\nBut one thing was wanting--the capture of Aben-Humeya, the \"little king\"\nof the Alpujarras. So long as he lived, the insurrection, now smothered,\nmight be rekindled at any time. He had taken refuge, it was known, in\nthe wilds of the Sierra Nevada, where, as the captain-general wrote, he\nwas wandering from rock to rock with only a handful of followers.[95]\nMondejar sent two detachments of soldiers into the sierra, to discover\nhis haunts, if possible, and seize upon his person.\n\nThe commander of one of these parties, named Maldonado, ascertained that\nAben-Humeya, secreting himself among the fastnesses of the mountains by\nday, would steal forth at night, and repair, with a few of his\nfollowers, to a place called Mecina, on the skirts of the sierra. Here\nhe found shelter in the house of his kinsman, Aben-Aboo, one of those\nMoriscoes who, after the affair of Jubiles, had obtained a safe-conduct\nfrom Mondejar. Having gained this intelligence, and learned the\nsituation of the house, the Spanish captain marched, with his little\nband of two hundred soldiers, in that direction. He made his approach\nwith the greatest secrecy. Travelling by night, he reached undiscovered\nthe neighbourhood of Aben-Aboo's residence. Advancing under cover of the\ndarkness, he had arrived within gunshot of the dwelling, when, at this\ncritical moment, all his precautions were defeated by the carelessness\nof one of his company, whose arquebuse was accidentally discharged. The\nreport, reverberating from the hills in the silence of night, roused the\ninmates of the house, who slept as the wearied mariner sleeps when his\nship is in danger of foundering. One of them, El Zaguer, the uncle of\nAben-Humeya, and the person who had been mainly instrumental in securing\nhim his crown--a crown of thorns--was the first roused, and, springing\nto the window, he threw himself down, though the height was\nconsiderable, and made his way to the mountains.\n\nHis nephew, who lay in another part of the building, was not so\nfortunate. When he reached the window, he saw with dismay the ground in\nfront occupied by a body of Castilian troops. Hastening to another\nwindow, he found it still the same; his enemies were everywhere around\nthe house. Bewildered and sorely distressed, he knew not where to turn.\nThus entrapped, and without the means of making any terms with his\nenemies, he knew he had as little to hope from their mercy as the wolf\nhas from the hunters who have caught him in his lair. The Spaniards,\nmeanwhile, were thundering at the door of the building for admittance.\nFortunately it was well secured. A sudden thought occurred to\nAben-Humeya, which he instantly put in execution. Hastening down stairs,\nhe took his station behind the door, and gently drew the bolts. The\nnoise was not heard amidst the din made by the assailants, who, finding\nthe door give way, supposed they had forced the fastenings, and pouring\nin, soon spread themselves in every direction over the house in search\nof the fugitive. Aben-Humeya, ensconced behind the door, escaped\nobservation; and, when his enemies had disappeared, stole out into the\ndarkness, and, under its friendly mantle, succeeded in finding his way\nto the mountains.\n\nIt was in vain that the Spaniards, enraged at the loss of the quarry,\nquestioned Aben-Aboo as to the haunts of his kinsman, and of El Zaguer,\nhis uncle, in the sierra. Nor could the most excruciating tortures shake\nhis constancy. \"I may die,\" said the brave Morisco, \"but my friends will\nlive.\" Leaving him for dead, the soldiers returned to the camp, taking\nwith them a number of prisoners, his companions. There was no one of\nthem, however, that was not provided with a safe-conduct from the\nmarquis, who accordingly set them at liberty; showing a respect for his\nengagements, in which unhappily, as we shall see hereafter, he was not\ntoo well imitated by his soldiers. The heroic Aben-Aboo, though left for\ndead, did not die, but lived to head another insurrection, and to take\nample vengeance on his enemies.[96]\n\nWhile the arms of the marquis of Mondejar were thus crowned with\nsuccess, the war raged yet more fiercely on the eastern <DW72>s of the\nAlpujarras, where a martial race of mountaineers threatened a descent on\nAlmeria and the neighbouring places, keeping the inhabitants in\nperpetual alarm. They accordingly implored the government at Granada to\ntake some effectual measures for their relief. The president, Deza, in\nconsequence, desired the marquis of Los Velez, who held the office of\n_adelantado_ of the adjoining province of Murcia, to muster a force and\nprovide for the defence of the frontier. This proceeding was regarded by\nMondejar's friends as an insult to that nobleman, whose military\nauthority extended over the country menaced by the Moriscoes. The act\nwas the more annoying, that the person invited to assume the command was\na rival, between whose house and that of the Mendozas there existed an\nancient feud. Yet the king sanctioned the proceeding, thinking perhaps\nthat Mondejar was not in sufficient force to protect the whole region of\nthe Alpujarras. However this may be, Philip, by this act, brought two\ncommanders of equal authority on the theatre of action; men who, in\ntheir characters and habitual policy, were so opposed to each other,\nthat little concert could 'be expected between them.\n\nDon Luis Fajardo, marquis of Los Velez, was a nobleman somewhat advanced\nin years, most of which had been passed in the active duties of military\nlife. He had studied the art of war under the great emperor, and had\nacquired the reputation of a prompt and resolute soldier, bold in\naction, haughty, indeed overbearing, in his deportment, and with an\ninflexible will, not to be shaken by friend or foe. The severity of his\nnature had not been softened under the stern training of the camp; and,\nas his conduct in the present expedition showed, he was troubled with\nnone of those scruples on the score of humanity which so often turned\nthe edge of Mondejar's sword from the defenceless and the weak. The\nMoriscoes, who understood his character well, held him in terror, as\nthey proved by the familiar _sobriquet_ which they gave him of the\n\"iron-headed devil.\"[97]\n\n[Sidenote: OPERATIONS OF LOS VELEZ.]\n\nThe marquis, on receiving the invitation of Deza, lost no time in\ngathering his kindred and numerous vassals around him; and they came\nwith an alacrity which showed how willingly they obeyed the summons to a\nforay over the border. His own family was a warlike race, reared from\nthe cradle amidst the din of arms. In the present expedition he was\nattended by three of his sons, the youngest of whom a boy of thirteen,\nhad the proud distinction of carrying his father's banner.[98] With the\nlevies promptly furnished from the neighbouring places, Los Velez soon\nfound himself supported by a force of greater strength than that which\nfollowed the standard of Mondejar. At the head of this valiant but\nill-disciplined array, he struck into the gloomy gorges of the\nmountains, resolved on bringing the enemy at once to battle.\n\nOur limits will not allow room for the details of a campaign which in\nits general features bears so close a resemblance to that already\ndescribed. Indeed the contest was too unequal to afford a subject of\nmuch interest to the general reader, while the details are of still less\nimportance in a military view, from the total ignorance shown by the\nMoriscoes of the art of war.\n\nThe fate of the campaign was decided by three battles, fought\nsuccessively at Huecija, Filix, and Ohanez, places all lying in the\neastern ranges of the Alpujarras. That of Filix was the most sanguinary.\nA great number of stragglers hung on the skirts of the Morisco army; and\nbesides six thousand--many of them women[99]--left dead upon the field,\nthere were two thousand children, we are told, butchered by the\nSpaniards.[100] Some fled for refuge to the caves and thickets; but they\nwere speedily dragged from their hiding-places, and massacred by the\nsoldiers in cold blood. Others, to escape death from the hands of their\nenemies, threw themselves headlong down the precipices,--some of them\nwith their infants in their arms,--and thus miserably perished. \"The\ncruelties committed by the troops,\" says one of the army, who chronicled\nits achievements, \"were such as the pen refuses to record.[101] I\nmyself,\" he adds, \"saw the corpse of a Morisco woman, covered with\nwounds, stretched upon the ground, with six of her children lying dead\naround her. She had succeeded in protecting a seventh, still an infant,\nwith her body, and though the lances which pierced her had passed\nthrough its clothes, it had marvellously escaped any injury. It was\nclinging,\" he continues, \"to its dead mother's bosom, from which it drew\nmilk that was mingled with blood. I carried it away and saved it.\"[102]\nFor the credit of human nature he records some other instances of the\nlike kind, showing that a spark of humanity might occasionally be struck\nout from the flinty breasts of these marauders.\n\nThe field of battle afforded a rich harvest for the victors, who\nstripped the dead, and rifled the bodies of the women of collars,\nbracelets, ornaments of gold and silver, and costly jewels, with which\nthe Moorish female loved to decorate her person. Sated with plunder, the\nsoldiers took the first occasion to leave their colours and return to\ntheir homes. Their places were soon supplied, as the display of their\nriches sharpened the appetites of their countrymen, who eagerly floaked\nto the banner of a chief that was sure to lead them on to victory and\nplunder. But that chief, with all his stern authority, was no match for\nthe spirit of insubordination that reigned among his troops; and, when\nhe attempted to punish one of their number for a gross act of\ndisobedience, he was made to understand that there were three thousand\nin the camp ready to stand by their comrade and protect him from\ninjury.[103]\n\nThe wild excesses of the soldiery were strangely mingled with a respect\nfor the forms of religion, that intimated the nature of the war in which\nthey were engaged. Before entering into action the whole army knelt down\nin prayer, solemnly invoking the protection of Heaven on its champions.\nAfter the battle of Ohanez, where the mountain streams were so polluted\nwith the gore that the Spaniards found it difficult to slake their\nthirst, they proceeded to celebrate the _fete_ of the Purification of\nthe Virgin.[104] A procession was formed to the church, which was headed\nby the marquis of Los Velez and his chivalry, clad in complete mail, and\nbearing white tapers in their hands. Then came the Christian women, who\nhad been rescued from captivity, dressed, by the general's command, in\nrobes of blue and white, as the appropriate colours of the Virgin.[105]\nThe rear was brought up by a body of friars and other ecclesiastics, who\nhad taken part in the crusade. The procession passed slowly between the\nfiles of the soldiery, who saluted it with volleys of musketry as it\nentered the church, where _Te Deum_ was chanted, and the whole company\nprostrated themselves in adoration of the Lord of Hosts, who had given\nhis enemies into their hands.\n\n[Sidenote: CABAL AGAINST MONDEJAR.]\n\nFrom this solemn act of devotion the troops proceeded to the work of\npillage, in which the commander, unlike his rival, the marquis of\nMondejar, joined as heartily as the meanest of his followers. The\nMoorish captives, to the number of sixteen hundred, among whom, we are\ntold, were many young and beautiful maidens, instead of meeting with the\nprotection they had received from the more generous Mondejar, were\ndelivered up to the licentious soldiery; and for a fortnight there\nreigned throughout the camp a carnival of the wildest riot and\ndebauchery.[106] In this strange confusion of the religious sentiment\nand of crimes most revolting to humanity, we see the characteristic\nfeatures of the crusade. Nowhere do we find such a free range given to\nthe worst passions of our nature as in the wars of religion,--where each\nparty considers itself as arrayed against the enemies of God, and where\nthe sanctity of the cause throws a veil over the foulest transgressions\nthat hides their enormity from the eye of the transgressor.\n\nWhile the Moriscoes were stunned by the fierce blows thus dealt in rapid\nsuccession by the iron-hearted marquis, the mild and liberal policy of\nhis rival was still more effectually reducing his enemies to obedience.\nDisheartened by their reverses, exhausted by fatigue and hunger, as they\nroved among the mountains, without raiment to clothe or a home to\nshelter them, the wretched wanderers came in one after another to sue\nfor pardon. Nearly all the towns and villages in the district assigned\nto Mondejar, oppressed with like feelings of despondency, sent\ndeputations to the Spanish quarters, to tender their submission and to\nsue for his protection. While these were graciously received, the\ngeneral provided for the future security of his conquests, by\nestablishing garrisons in the principal places, and by sending small\ndetachments to different parts, to act as a sort of armed police for the\nmaintenance of order. In this way, says a contemporary, the tranquillity\nof the country was so well established, that small parties of ten or a\ndozen soldiers wandered unmolested from one end of it to the other.[107]\n\nMondejar, at the same time, wrote to the king, to acquaint him with the\nactual state of things. He besought his master to deal mercifully with\nthe conquered people, and thus afford him the means of redeeming the\npledges he had given for the favourable dispositions of the\ngovernment.[108] He made another communication to the marquis of Los\nVelez, urging that nobleman to co-operate with him in the same humane\npolicy, as the one best suited to the interests of the country. But his\nrival took a very different view of the matter; and he plainly told the\nmarquis of Mondejar, that it would require more than one pitched battle\nyet to break the spirit of the Moriscoes; and that, since they thought\nso differently on the subject, the only way left was for each commander\nto take the course he judged best.[109]\n\nUnfortunately, there were others--men, too, of influence at the\ncourt--who were of the same stern way of thinking as the marquis of Los\nVelez; men acting under the impulse of religious bigotry, of implacable\nhatred of the Moslems, and of a keen remembrance of the outrages they\nhad committed. There were others who, more basely, thought only of\nthemselves and of the profit they should derive from the continuance of\nthe war.\n\nAmong those of the former class was the president Deza, with the members\nof the Audience and the civil authorities in Granada. Always viewing\nthe proceedings of the captain-general with an unfriendly eye, they\nloudly denounced his policy to the king, condemning his ill-timed lenity\nto a crafty race, who would profit by it to rally from their late\ndisasters and to form new plans of rebellion. It was not right, they\nsaid, that outrages like those perpetrated against both _divine and\nhuman majesty_ should go unpunished.[110] Mondejar's enemies did not\nstop here, but accused him of defrauding the exchequer of its dues, the\nfifth of the spoils of war gained in battle from the infidel. Finally,\nthey charged him with having shown want of respect for the civil\nauthorities of Granada, in omitting to communicate to them his plan of\noperations.\n\nThe marquis, advised by his friends at court of these malicious attempts\nto ruin his credit with the government, despatched a confidential envoy\nto Madrid, to present his case before his sovereign and to refute the\naccusations of his enemies. The charge of peculation seems to have made\nno impression on the mind of a prince who would not have been slow to\nsuspect, had there been any ground for suspicion. There may have been\nstronger grounds for the complaint of want of deference to the civil\nauthorities of Granada. The best vindication of his conduct in this\nparticular must be found in the character and conduct of his\nadversaries. From the first, Deza and the municipality had regarded him\nwith jealousy, and done all in their power to thwart his plans and\ncircumscribe his authority. It is only confidence that begets\nconfidence. Mondejar, early accustomed to command, was probably too\nimpatient of opposition.[111] He chafed under the obstacles and\nannoyances thrown in his way by his narrow-minded rivals. We have not\nthe means before us of coming to a conclusive judgment on the merits of\nthe controversy, but from what we know of the marquis's accusers, with\nthe wily inquisitor at their head, we shall hardly err by casting our\nsympathies into the scale of the frank and generous-hearted soldier,\nwho, while those that thus censured him were living at ease in the\ncapital, had been fighting and following up the enemy, amidst the\nwinter's tempests and across mountains covered with snow; and who, in\nlittle more than a month, without other aid than the disorderly levies\nof the cities, had quelled a dangerous revolt, and restored tranquillity\nto the land.\n\nPhilip was greatly perplexed by the different accounts sent to him of\nthe posture of affairs in Granada. Mondejar's agent suggested to the\ncouncil of state that it would be well if his majesty would do as his\nfather, Charles the Fifth, would have done in the like case--repair\nhimself to the scene of action, and observe the actual state of things\nwith his own eyes. But the suggestion found no favour with the minister,\nEspinosa, who affected to hold the Moriscoes in such contempt, that a\nmeasure of this kind, he declared, would be derogatory to the royal\ndignity. A better course would be for his majesty to send some one as\nhis representative, clothed with full powers to take charge of the war,\nand of a rank so manifestly pre-eminent, that neither of the two\ncommanders now in the field could take umbrage at his appointment over\ntheir heads.\n\nThis suggestion, as the politic minister doubtless had foreseen, was\nmuch more to Philip's taste than that of his going in person to the\nscene of strife; for, however little he might shrink from any amount of\nlabour in the closet, he had, as we have seen, a sluggish temperament,\nthat indisposed him to much bodily exertion. The plan of sending some\none to represent the monarch at the seat of war was accordingly\napproved; and the person selected for this responsible office was\nPhilip's bastard brother, Don John of Austria.[112]\n\n[Sidenote: LICENCE OF THE SOLDIERS.]\n\nRumours of what was going on in the cabinet at Madrid, reaching Granada\nfrom time to time, were followed by the most mischievous consequences.\nThe troops, in particular, had no sooner learned that the marquis of\nMondejar was about to be superseded in the command, than they threw off\nthe little restraint he had been hitherto able to impose on them, and\nabandoned themselves to the violence and rapine to which they were so\nwell disposed, and which seemed now to be countenanced by the president\nand the authorities in Granada. The very patrols whom Mondejar had\ncommissioned to keep the peace were the first to set the example of\nviolating it. They invaded the hamlets and houses they were sent to\nprotect, plundered them of their contents, and committed the foulest\noutrages on their inmates. The garrisons in the principal towns imitated\ntheir example, carrying on their depredations, indeed, on a still larger\nscale. Even the capital, under the very eyes of the count of Tendilla,\nsent out detachments of soldiers, who with ruthless violence trampled\ndown the green plantations in the valleys, sacked the villages, and\ndragged away the inhabitants from the midst of their blazing dwellings\ninto captivity.[113]\n\nIt was with the deepest indignation that the marquis of Mondejar saw the\nfine web of policy he had been so busily contriving thus wantonly rent\nasunder by the very hands that should have protected it. He now longed\nas ardently as any in the province for the coming of some one entrusted\nwith authority to enforce obedience from the turbulent soldiery; a task\nof still greater difficulty than the conquest of the enemy. While such\nwas the state of things, an event occurred in Granada which, in its\ngeneral character, may remind one of some of the most atrocious scenes\nof the French Revolution.\n\nIn the beginning of the troubles, the president had caused a number of\nMoriscoes, amounting to not less than a hundred and fifty, it is said,\nto be arrested and thrown into the prison of the Chancery. Certain\ntreasonable designs, of which they had been suspected for a long time,\nfurnished the feeble pretext for this violent proceeding. Some few,\nindeed, were imprisoned for debt. But the greater number were wealthy\nmen, who enjoyed the highest consideration among their countrymen. They\nhad been suffered to remain in confinement during the whole of the\ncampaign; thus serving, in some sort, as hostages for the good behaviour\nof the people of the Albaicin.\n\nEarly in March, a rumour was circulated that the mountaineers, headed by\nAben-Humeya, whose father and brother were among the prisoners, were\nprepared to make a descent on the city by night, and, with the\nassistance of the inhabitants of the Albaicin, to begin the work of\ndestruction by assaulting the prison of the Chancery and liberating\ntheir countrymen. This report, readily believed, caused the greatest\nalarm among the citizens, boding no good to the unhappy prisoners. On\nthe evening of the seventeenth, Deza received intelligence that lights\nhad been seen on some of the neighbouring mountains, which seemed to be\nof the nature of signals, as they were answered by corresponding lights\nin some of the houses in the Albaicin. The assault, it was said, would\ndoubtless be made that very night. The president appears to have taken\nno measures for the protection of the city, but, on receiving the\ninformation, he at once communicated it to the alcayde of the prison,\nand directed him to provide for the security of his prisoners. The\nalcayde lost no time in gathering his friends about him, and caused arms\nto be distributed among a body of Spaniards, of whom there appears to\nhave been a considerable number confined in the place at this time. Thus\nprepared, they all remained, as in silent expectation of some great\nevent.\n\nAt length, some time before midnight, the guard posted in the Campana,\none of the towers of the Alhambra, struck the bell with a succession of\nrapid strokes, such as were used to give an alarm. In a moment every\nSpaniard in the prison was on his feet; and, the alcayde throwing open\nthe doors and leading the way, they fell at once on their defenceless\nvictims, confined in another quarter of the building. As many of these\nwere old and infirm, and most of them inoffensive citizens, whose quiet\nway of life had little fitted them for brawl or battle, and who were now\ndestitute of arms of any kind, they seemed to be as easy victims as the\nsheep into whose fold the famishing wolves have broken in the absence of\nthe shepherd. Yet they did not give up their lives without an effort to\nsave them. Despair lent them strength, and snatching up chairs, benches,\nor any other article of furniture in their cells, they endeavoured to\nmake good their defence against the assailants. Some, exerting a vigour\nwhich despair only could have given, succeeded in wrenching stones from\nthe walls or iron bars from the windows, and thus supplied themselves\nwith the means, not merely of defence, but of doing some mischief to the\nassailants in their turn. They fought, in short, like men who are\nfighting for their lives. Some, however, losing all hope of escape,\npiled together a heap of mats, bedding, and other combustibles, and,\nkindling them with their torches, threw themselves into the flames,\nintending in this way to set fire to the building, and to perish in one\ngeneral conflagration with their murderers.[114] But the flames they had\nkindled were soon extinguished in their own blood, and their mangled\nremains were left to blacken among the cinders of their funeral pile.\n\nFor two hours the deadly conflict between parties so unequally matched\nhad continued; the one shouting its old war-cry of \"Saint Iago,\" as if\nfighting on an open field; the other, if we may take the Castilian\naccount, calling on their prophet to come to their assistance. But no\npower, divine or human, interposed in their behalf; and, notwithstanding\nthe wild uproar caused by men engaged in a mortal struggle, by the sound\nof heavy blows and falling missiles, by the yells of the victors and the\ndying moans and agonies of the vanquished, no noise to give token of\nwhat was going on--if we are to credit the chroniclers--found its way\nbeyond the walls of the prison. Even the guard stationed in the\ncourt-yard, we are assured, were not roused from their slumbers.[115]\n\nAt length some rumour of what was passing reached the city, where the\nstory ran that the Moriscoes were in arms against their keepers, and\nwould soon probably get possession of the gaol. This report was enough\nfor the people, who, roused by the alarm-bell, were now in a state of\nexcitement that disposed them to any deed of violence. Snatching up\ntheir weapons, they rushed, or rather flew, like vultures snuffing the\ncarrion from afar, to the scene of slaughter. Strengthened by this\nreinforcement, the assailants in the prison soon completed the work of\ndeath; and, when the morning light broke through the grated windows, it\ndisclosed the full extent of the tragedy. Of all the Moriscoes only two\nhad escaped,--the father and brother of Aben-Humeya, over whom a guard\nhad been especially set. Five Spaniards were slain, and seventeen\nwounded; showing the fierce resistance made by the Moslems, though\ndestitute of arms.[116]\n\n[Sidenote: THE INSURRECTION REKINDLED.]\n\nSuch was the massacre in the prison of the Chancery of Granada, which,\nas already intimated, nowhere finds a more fitting parallel than in the\nmurders perpetrated on a still larger scale during the French\nRevolution, in the famous massacres of September. But the miscreants who\nperpetrated these enormities were the tools of a sanguinary faction,\nthat was regarded with horror by every friend of humanity in the\ncountry. In Granada, on the other hand, it was the government itself, or\nat least those of highest authority in it, who were responsible for the\ndeed. For who can doubt that a proceeding, the success of which depended\non the concurrence of so many circumstances as to preclude the idea of\naccident, must have been countenanced, if not contrived, by those who\nhad the direction of affairs?\n\nAnother feature, not the least striking in the case, is the apathy shown\nby contemporary writers,--men who on more than one occasion have been\nwilling to testify their sympathy for the sufferings of the Moriscoes.\nOne of these chroniclers, after telling the piteous tale, coolly remarks\nthat it was a good thing for the alcayde of the prison, who pocketed a\nlarge sum of money which had been found on the persons of the wealthy\nMoors. Another, after noticing the imputation of an intended rising on\nthe part of the prisoners as in the highest degree absurd, dismisses the\nsubject by telling us that \"the Moriscoes were a weak, scatter-brained\nrace, with just wit enough to bring on themselves such a _mishap_,\"--as\nhe pleasantly terms the massacre.[117] The government of Madrid received\nthe largest share of the price of blood. For when the wives and families\nof the deceased claimed the inheritance of their estates, in some cases\nvery large, their claims were rejected--on what grounds we are not\ntold--by the alcaldes of the Court of Audience in Granada, and the\nestates were confiscated to the use of the crown. Such a decision,\nremarks a chronicler, may lead one to infer that the prisoners had been\nguilty of even more heinous offences than those commonly imputed to\nthem.[118] The impartial reader will probably come to a very different\nconclusion; and since it was the opulent burghers who were thus marked\nout for destruction, he may naturally infer that the baser passion of\navarice mingled with the feelings of fear and hatred in bringing about\nthe massacre.\n\nHowever this may be, so foul a deed placed an impassable gulf between\nthe Spaniards and the Moriscoes. It taught the latter that they could no\nlonger rely on their perfidious enemy, who, while he was holding out to\nthem one hand in token of reconciliation, was raising the other to smite\nthem to the ground. A cry of vengeance ran through all the borders of\nthe Alpujarras. Again the mountaineers rose in arms. They cut off\nstragglers, waylaid the patrols whom Mondejar had distributed throughout\nthe country, and even menaced the military posts of the Spaniards. On\nsome occasions, they encountered the latter with success in the open\nfield, and in one instance defeated and slew a large body of Christians,\nas they were returning from a foray laden with plunder. Finally they\ninvited Aben-Humeya to return and resume the command, promising to stand\nby him to the last. The chief obeyed the call and, leaving his retreat\nin the Sierra Nevada, again took possession of his domains, and,\nplanting his blood-red flag on his native hills,[119] soon gathered\naround him a more formidable host than before. He even affected a\ngreater pomp than he had before displayed. He surrounded himself with a\nbody-guard of four hundred arquebusiers.[120] He divided his army into\nbattalions and companies, and endeavoured to introduce into it something\nof the organization and tactics of the Spaniards.[121] He sent his\nbrother Abdallah to Constantinople, to represent his condition to the\nSultan, and to implore him to make common cause with his Moslem brethren\nin the Peninsula. In short, rebellion assumed a more audacious front\nthan at any time during the previous campaign; and the Christians of\nAndalusia and Granada looked with the greatest anxiety for the coming of\na commander possessed of sufficient authority to infuse harmony into the\ncounsels of the rival chiefs, to enforce obedience from the turbulent\nsoldiery, and to bring the war to a speedy conclusion.\n\n\n\n\nCHAPTER V.\n\nREBELLION OF THE MORISCOES.\n\nEarly life of Don John of Austria--Acknowledged by Philip--His Thirst\nfor Distinction--His Cruise in the Mediterranean--Made\nCommander-in-chief--The War renewed--Removal of the Moriscoes.\n\n1569.\n\n\nAs Don John of Austria is to occupy an important place, not only in the\nwar with the Moriscoes, but in some of the most memorable scenes in the\nremainder of this history, it will be proper to acquaint the reader with\nwhat is known of the earlier part of his career. Yet it is precisely\nover this part of it that a veil of mystery hangs, which no industry of\nthe historian has been able wholly to remove.\n\nIt seems probable that he was born in the year 1547.[122] The\ntwenty-fourth of February is assigned by common consent--I hardly know\non what ground--as the day of his birth. It was also, it may be\nremembered, the birthday of his father, Charles the Fifth. His mother,\nBarbara Blomberg, was an inhabitant of Ratisbon, in Germany. She is\ndescribed as a beautiful young girl, who attracted the emperor's notice\nseveral years after the death of the empress Isabella.[123] The Spanish\nchroniclers claim a noble descent for Barbara.[124] Indeed, it would go\nhard but a Spaniard could make out a pedigree for his hero. Yet there\nare several circumstances which suggest the idea that the mother of Don\nJohn must have occupied a very humble position.\n\n[Sidenote: DON JOHN OF AUSTRIA.]\n\nSubsequently to her connexion with Charles she married a German named\nKegell, on whom the emperor bestowed the office of commissary.[125] The\nonly other notice, so far as I am aware, which Charles took of his\nformer mistress was the settlement on her of a yearly pension of two\nhundred florins, which he made the day before his death.[126] It was\ncertainly not a princely legacy, and infers that the object of it must\nhave been in a humble condition in life to have rendered it important to\nher comfort. We are led to the same conclusion by the mystery thrown\naround the birth of the child, forming so strong a contrast to the\npublicity given to the birth of the emperor's natural daughter, Margaret\nof Parma, whose mother could boast that in her veins flowed some of the\nbest blood of the Netherlands.\n\nFor three years the boy, who received the name of Geronimo, remained\nunder his mother's roof, when, by Charles's order, he was placed in the\nhands of a Fleming, named Maffi, a musician in the imperial band. This\nman transferred his residence to Leganes, a village in Castile, not far\nfrom Madrid. The instrument still exists that contains the agreement by\nwhich Maffi, after acknowledging the receipt of a hundred florins,\nengages for fifty florins annually, to bring up the child with as much\ncare as if he were his own.[127] It was a moderate allowance, certainly,\nfor the nurture of one who was some day to come before the world as the\nson of an emperor. It showed that Charles was fond of a bargain, though\nat the expense of his own offspring.\n\nNo instruction was provided for the child except such as he could pick\nup from the parish priest, who, as he knew as little as Maffi did of the\nsecret of Geronimo's birth, probably bestowed no more attention on him\nthan on the other lads of the village. And we cannot doubt that a boy of\nhis lively temper must have preferred passing his days in the open\nfields, to confinement in the house and listening to the homilies of his\nteacher. As he grew in years, he distinguished himself above his young\ncompanions by his courage. He took the lead in all their rustic sports,\nand gave token of his belligerent propensities by making war on the\nbirds in the orchards, on whom he did great execution with his little\ncrossbow.[128]\n\nFour years were passed in this hardy way of life, which, if it did\nnothing else for the boy, had the advantage of strengthening his\nconstitution for the serious trials of manhood, when the emperor thought\nit was time to place him in a situation where he would receive a better\ntraining than could be found in the cottage of a peasant. He was\naccordingly transferred to the protection of Luis Quixada, Charles's\ntrusty major-domo, who received the child into his family at\nVillagarcia, in the neighbourhood of Valladolid. The emperor showed his\nusual discernment in the selection of a guardian for his son. Quixada,\nwith his zeal for the faith, his loyalty, his nice sentiment of honour,\nwas the very type of the Castilian hidalgo in his best form; while he\npossessed all those knightly qualities which made him the perfect mirror\nof the antique chivalry. His wife, Dona Magdalena de Ulloa, sister of\nthe marquis of Mota, was a lady yet more illustrious for her virtues\nthan for her rank. She had naturally the most to do with the training of\nthe boy's earlier years; and under her discipline it was scarcely\npossible that one of so generous a nature should fail to acquire the\ncourtly breeding and refinement of taste which shed a lustre over the\nstern character of the soldier.\n\nHowever much Quixada may have reposed on his wife's discretion, he did\nnot think proper to try it, in the present instance, by communicating to\nher the secret of Geronimo's birth. He spoke of him as the son of a\ngreat man, his dear friend, expressing his desire that his wife would\nreceive him as her own child. This was the less difficult, as Magdalena\nhad no children of her own. The solicitude shown by her lord may\npossibly have suggested to her the idea that the boy was more nearly\nrelated to him than he chose to acknowledge,--in short, that he was the\noffspring of some intrigue of Quixada previous to his marriage.[129] But\nan event which took place not long after the child's introduction into\nthe family, is said to have awakened in her suspicions of an origin more\nin accordance with the truth. The house at Villagarcia took fire; and,\nas it was in the night, the flames gained such head that they were not\ndiscovered till they burst through the windows. The noise in the street\nroused the sleeping inmates; and Quixada, thinking first of his charge,\nsprang from his bed, and, rushing into Geronimo's apartment, snatched up\nthe affrighted child, and bore him in his arms to a place of safety. He\nthen reentered the house, and, forcing his way through the smoke and\nflames, succeeded in extricating his wife from her perilous situation.\nThis sacrifice of love to loyalty is panegyrized by a Castilian\nchronicler as \"a rare achievement, far transcending any act of heroism\nof which antiquity could boast.\"[130] Whether Magdalena looked with the\nsame complacency on the proceeding we are not informed. Certain it is,\nhowever, that the interest shown by her husband in the child had no\npower to excite any feeling of jealousy in her bosom. On the contrary,\nit seemed rather to strengthen her own interest in the boy, whose\nuncommon beauty and affectionate disposition soon called forth all the\ntenderness of her nature. She took him to her heart, and treated him\nwith all the fondness of a mother,--a feeling warmly reciprocated by the\nobject of it, who, to the day of his death, regarded her with the truest\nfeelings of filial love and reverence.\n\nIn 1558, the year after his retirement to Yuste, Charles the Fifth,\nwhether from a wish to see his son, or, as is quite as probable, in the\nhope of making Quixada more contented with his situation, desired his\nmajor-domo to bring his family to the adjoining village of Cuacos. While\nthere, the young Geronimo must doubtless sometimes have accompanied his\nmother, as he called Dona Magdalena, in her visits to the monastery.\nIndeed, his biographer assures us that the sight of him operated like a\npanacea on the emperor's health.[131] We find no allusion to him,\nhowever, in any of the letters from Yuste; and, if he did go there, we\nmay be sure that Charles had sufficient control over himself not to\nbetray, by any indiscreet show of fondness, his relationship to the\nchild.[132] One tradition respecting him lingered to a late period\namong the people of Cuacos, where the peasants, it is said, pelted him\nwith stones as he was robbing their orchards. It was the first lesson in\nwar of the future hero of Lepanto.\n\n[Sidenote: DON JOHN OF AUSTRIA.]\n\nThere is no reason to doubt that the boy witnessed the obsequies of the\nemperor. One who was present tells us that he saw him there, dressed in\nfull mourning, and standing by the side of Quixada, for whose page he\npassed among the brethren of the convent.[133] We may well believe that\na spectacle so solemn and affecting as these funeral ceremonies must\nhave sunk deep into his young mind, and heightened the feelings of\nveneration with which he always regarded the memory of his father. It\nwas, perhaps, the appearance of Geronimo as one of the mourners that\nfirst suggested the idea of his relationship to the emperor. We find a\nletter from Quixada to Philip, dated soon after, in which he speaks of\nrumours on the subject as current in the neighbourhood.[134]\n\nAmong the testamentary papers of Charles was found one in an envelope\nsealed with his private seal, and addressed to his son Philip, or in\ncase of his death, to his grandson Carlos, or whoever might be in\npossession of the crown. It was dated in 1554, before his retirement to\nYuste. It acknowledged his connexion with a German maiden, and the birth\nof a son named Geronimo. The mother's name was not given. He pointed out\nthe quarter where information could be got respecting the child, who was\nthen living with the violin-player at Leganes. He expressed the wish\nthat he should be trained up for the ecclesiastical profession, and\nthat, when old enough, he should enter a convent of one of the reformed\norders. Charles would not, however, have any constraint put on the\ninclinations of the boy, and in case of his preferring a secular life,\nhe would have a suitable estate settled on him in the kingdom of Naples,\nwith an annual income of between thirty and forty thousand ducats.\nWhatever course Geronimo might take, the emperor requested that he\nshould receive all the honour and consideration due to him as his son.\nHis letter concluded by saying that, although for obvious reasons he had\nnot inserted these directions in his will, he wished them to be held of\nthe same validity as if he had.[135] Philip seems from the first to have\nso regarded them, though, as he was then in Flanders, he resolved to\npostpone the public acknowledgment of his brother till his return to\nSpain.\n\nMeanwhile, the rumours in regard to Geronimo's birth had reached the\nears of the regent, Joanna. With natural curiosity, she ordered her\nsecretary to write to Quixada and ascertain the truth of the report. The\ntrusty hidalgo endeavoured to evade the question, by saying that some\nyears since a friend of his had entrusted a boy to his care; but as no\nallusion whatever was made to the child in the emperor's will, the story\nof their relationship to each other should be treated as idle\ngossip.[136] The reply did not satisfy Joanna, who seems to have settled\nit in her own mind that the story was well founded. She took an\noccasion soon after to write to Dona Magdalena, during her husband's\nabsence from home, expressing her wish that the lady would bring the boy\nwhere she could see him. The place selected was at an _auto de fe_ about\nto be celebrated in Valladolid. Dona Magdalena, reluctant as she was,\nfelt herself compelled to receive the request from such a source as a\ncommand, which she had no right to disobey. One might have thought that\na ceremony so heartrending and appalling in its character as an _auto de\nfe_ would be the last to be selected for the indulgence of any feeling\nof a light and joyous nature. But the Spaniard of that and of a much\nlater age regarded this as the sweetest sacrifice that could be offered\nto the Almighty; and he went to it with the same indifference to the\nsufferings of the victim--probably with the same love of\nexcitement--which he would have felt in going to a bull-fight.\n\nOn the day which had been named, Magdalena and her charge took their\nseats on the carpeted platform reserved for persons of rank, in full\nview of the scaffold appropriated to the martyrs who were to suffer for\nconscience' sake. It was in the midst of the august company here\nassembled, that the son of Charles the Fifth was to receive his first\nlesson in the school of persecution; that he was to learn to steel his\nheart against sympathy with human suffering; to learn, above all, that\ncompassion for the heretic was a crime of the deepest dye. It was a\nterrible lesson for one so young--of an age when the mind is most open\nto impressions; and the bitter fruits of it were to be discerned ere\nlong in the war with the Moriscoes.\n\nAs the royal train approached the place occupied by Dona Magdalena, the\nregent paused and looked around for the boy. Magdalena had thrown her\nmantle about him, to conceal him as much as possible from the public\neye. She now drew it aside; and Joanna looked so long and earnestly on\nthe child, that he shrunk abashed from her gaze. It was not, however,\nbefore she had recognized in his bright blue eyes, his ample forehead,\nand the rich yellow locks that clustered round his head, some of the\npeculiarities of the Austrian line, though happily without the deformity\nof the protruding lip, which was no less its characteristic. Her heart\nyearned with the tenderness of a sister, as she felt convinced that the\nsame blood flowed in his veins as in her own; and, stooping down, she\nthrew her arms around his neck, and, kissing him, called him by the\nendearing name of brother.[137] She would have persuaded him to go with\nher and sit by her side, but the boy, clinging closely to his\nfoster-mother, refused to leave her for the stranger lady.\n\nThis curious scene attracted the attention of the surrounding\nspectators, which was hardly diverted from the child by the appearance\nof the prisoners on the scaffold to receive their sentences. When these\nhad been pronounced, and the wretched victims led away to execution, the\nmultitude pressed so eagerly round Magdalena and the boy, that it was\nwith difficulty the guards could keep them back, till the regent, seeing\nthe awkwardness of their situation, sent one of her train, the count of\nOsorno, to their relief; and that nobleman, forcing his way through the\ncrowd, carried off Geronimo in his arms to the royal carriage.[138]\n\n[Sidenote: DON JOHN ACKNOWLEDGED BY PHILIP.]\n\nIt was not long before all mystery was dispelled by the public\nacknowledgment of the child as the son of the emperor. One of the first\nacts of Philip, after his return to Spain in 1559, was to arrange an\ninterview with his brother. The place assigned for the meeting was an\nextensive park, not far from Valladolid, in the neighbourhood of the\nconvent of _La Espina_, a spot much resorted to by the Castilian princes\nof the older time for the pleasures of the chase.\n\nOn the appointed day, Quixada, richly dressed, and mounted on the best\nhorse in his stables, rode forth, at the head of his vassals, to meet\nthe king, with the little Geronimo, simply attired, and on a common\npalfrey, by his side. They had gone but a few miles when they heard,\nthrough the woods, the sound of horses' hoofs, announcing the approach\nof the royal cavalcade. Quixada halted, and alighting, drew near to\nGeronimo, with much deference in his manner, and, dropping on one knee,\nbegged permission to kiss his hand. At the same time he desired his ward\nto dismount, and take the charger which he had himself been riding.\nGeronimo was sorely bewildered by what he would have thought a merry\njest on the part of his guardian, had not his sedate and dignified\ncharacter forbidden the supposition. Recovering from his astonishment,\nhe complied with his guardian's directions; and the vision of future\ngreatness must have flashed on his mind, if, as we are told, when\npreparing to mount, he turned round to Quixada, and with an affected air\nof dignity, told him that, \"since things were so, he might hold the\nstirrup for him.\"[139]\n\nThey had not proceeded far when they came in sight of the royal party.\nQuixada pointed out the king to his ward, adding that his majesty had\nsomething of importance to communicate to him. They then dismounted; and\nthe boy, by his guardian's instructions, drawing near to Philip, knelt\ndown and begged leave to kiss his majesty's hand. The king, graciously\nextending it, looked intently on the youth; and at length broke silence\nby asking \"if he knew who was his father.\" Geronimo, disconcerted by the\nabruptness of the question, and, indeed, if the reports of his origin\nhad ever reached his ears, ignorant of their truth, cast his eyes on the\nground and made no answer. Philip, not displeased with his\nembarrassment, was well satisfied, doubtless, to read in his intelligent\ncountenance and noble mien an assurance that he would do no discredit to\nhis birth. Alighting from his horse, he embraced Geronimo, exclaiming,\n\"Take courage, my child, you are descended from a great man. The emperor\nCharles the Fifth, now in glory, is your father as well as mine.\"[140]\nThen, turning to the lords who stood around, he presented the boy to\nthem as the son of their late sovereign, and his own brother. The\ncourtiers, with the ready instinct of their tribe, ever prompt to\nworship the rising sun, pressed eagerly forward to pay their obeisance\nto Geronimo. The scene was concluded by the king's buckling a sword on\nhis brother's side, and throwing around his neck the sparkling collar of\nthe Golden Fleece.\n\nThe tidings of this strange event soon spread over the neighbourhood,\nfor there were many more witnesses of the ceremony than those who took\npart in it; and the king and his retinue found, on their return, a\nmultitude of people gathering along the route, eager to get a glimpse of\nthis newly discovered gem of royalty. The sight of the handsome youth\ncalled forth a burst of noisy enthusiasm from the populace, and the air\nrang with their tumultuous _vivas_ as the royal party rode through the\nstreets of the ancient city of Valladolid. Philip expressed his\nsatisfaction at the events of the day, by declaring that \"he had never\nmet better sport in his life, or brought back game so much to his\nmind.\"[141]\n\nHaving thus publicly acknowledged his brother, the king determined to\nprovide for him an establishment suited to his condition. He assigned\nhim for his residence one of the best mansions in Madrid. He was\nfurnished with a numerous band of retainers, and as great state was\nmaintained in his household as in that of a prince of the blood. The\ncount of Priego acted as his chief major-domo; Don Luis Carrillo, the\neldest son of that noble, was made captain of the guard; and Don Luis de\nCordova master of the horse. In short, nobles and cavaliers of the best\nblood in Castile did not disdain to hold offices in the service of the\npeasant boy. With one or two exceptions, of little importance, he\nenjoyed all the privileges that belonged to the royal _infantes_. He did\nnot, like them, have apartments in the palace; and he was to be\naddressed by the title of \"Excellency,\" instead of \"Highness,\" which was\ntheir peculiar prerogative. The distinction was not always scrupulously\nobserved.[142]\n\nA more important change took place in his name, which from _Geronimo_\nwas now converted into _John of Austria_,--a lofty name, which intimated\nhis descent from the imperial house of Hapsburg, and on which his deeds\nin after-life shed a lustre greater than the proudest title that\nsovereignty could confer.\n\nLuis Quixada kept the same place after his pupil's elevation as before.\nHe continued to be his _ayo_, or governor, and removed with Dona\nMagdalena to Madrid, where he took up his residence in the house of Don\nJohn. Thus living in the most intimate personal relations with him,\nQuixada maintained his influence unimpaired till the hour of his own\ndeath.\n\nPhilip fully appreciated the worth of the faithful hidalgo, who was\nfortunate in thus enjoying the favour of the son in as great a degree as\nhe had done that of the father,--and, as it would seem, with a larger\nrecompense for his services. He was master of the horse to Don Carlos,\nthe heir to the crown; he held the important post of president of the\nCouncil of the Indies; and he possessed several lucrative benefices in\nthe military order of Calatrava. In one of his letters to the king, we\nfind Quixada remarking that he had endeavoured to supply the\ndeficiencies of his pupil's early education by training him in a manner\nbetter suited to his destinies in after-life.[143] We cannot doubt that,\nin the good knight's estimate of what was essential to such a training,\nthe exercises of chivalry must have found more favour than the monastic\ndiscipline recommended by the emperor. However this may have been,\nPhilip resolved to give his brother the best advantages for a liberal\neducation by sending him to the University of Alcala, which, founded by\nthe great Ximenes, a little more than a century before, now shared with\nthe older school of Salamanca the glory of being the most famous seat of\nscience in the Peninsula. Don John had for his companions his two\nnephews, Don Carlos and Alexander Farnese, the son of Margaret of Parma.\nThey formed a triumvirate, each member of which was to fill a large\nspace in the pages of history; Don Carlos from his errors and\nmisfortunes, and the two others from their military achievements. They\nwere all of nearly the same age. Don John, according to a writer of the\ntime, stood foremost among the three for the comeliness, or rather\nbeauty of his person, no less than for the charm of his manners;[144]\nwhile the soul was filled with those nobler qualities which gave promise\nof the highest excellence.[145]\n\n[Sidenote: DON JOHN'S THIRST FOR DISTINCTION.]\n\nHis biographers tell us that Don John gave due attention to his studies,\nbut the studies which found most favour in his eyes were those connected\nwith the art of war. He was perfect in all chivalrous accomplishments;\nand he sighed for some field on which he could display them. The\nknowledge of his real parentage filled his soul with a generous\nambition, and he longed by some heroic achievement to vindicate his\nclaim to his illustrious descent.\n\nAt the end of three years, in 1564, he left the university. The\nfollowing year was that of the famous siege of Malta; and all\nChristendom hung in suspense on the issue of the desperate conflict,\nwhich a handful of warriors, on their lonely isle, were waging against\nthe whole strength of the Ottoman empire. The sympathies of Don John\nwere roused in behalf of the Christian knights; and he resolved to cast\nhis own fortunes into the scale with theirs, and win his maiden laurels\nunder the banner of the Cross. He did not ask the permission of his\nbrother. That he knew would be refused to him. He withdrew secretly from\nthe court, and with only a few attendants took his way to Barcelona,\nwhence an armament was speedily to sail, to carry succour to the\nbesieged. Everywhere on the route he was received with the respect due\nto his rank. At Saragossa he was lodged with the archbishop, under whose\nroof he was detained by illness. While there he received a letter from\nthe king, who had learned the cause of his departure, commanding him to\nreturn, as he was altogether too young to take part in this desperate\nstrife. Don John gave little heed to the royal orders. He pushed on to\nBarcelona, where he had the mortification to find that the fleet had\nsailed. He resolved to cross the mountains and take ship at Marseilles.\nThe viceroy of Catalonia could not dissuade the hot-headed youth from\nhis purpose, when another despatch came from court, in which Philip, in\na more peremptory tone than before, repeated his orders for his brother\nto return, under pain of his severe displeasure. A letter from Quixada\nhad warned him of the certain disgrace which awaited him, if he\ncontinued to trifle with the royal commands. Nothing remained but to\nobey; and Don John, disappointed in his scheme of ambition, returned to\nthe capital.[146]\n\nThis adventure caused a great sensation throughout the country. The\nyoung nobles and cavaliers about the court, fired by Don John's example,\nwhich seemed like a rebuke on their own sluggishness, had hastened to\nbuckle on their armour, and follow him to the war.[147] The common\npeople, peculiarly sensible in Spain to deeds of romantic daring, were\ndelighted with the adventurous spirit of the young prince, which gave\npromise that he was one day to take his place among the heroes of the\nnation. This was the beginning of the popularity of John of Austria with\nhis countrymen, who in time came to regard him with feelings little\nshort of idolatry. Even Philip, however necessary he may have thought it\nto rebuke the insubordination of his brother, must in his heart have\nbeen pleased with the generous spirit he had exhibited. At least, the\nfavour with which he continued to regard the offender showed that the\nroyal displeasure was of no long continuance.\n\nThe sudden change in the condition of Don John might remind one of some\nfairy tale, where the poor peasant boy finds himself all at once\nconverted by enchantment into a great prince. A wiser man than he might\nwell have had his head turned by such a rapid revolution of the wheel of\nfortune; and Philip may naturally have feared that the idle dalliance of\na court, to which his brother was now exposed, might corrupt his simple\nnature and seduce him from the honourable path of duty. Great,\ntherefore, must have been his satisfaction, when he saw that, far from\nthis, the elevation of the youth had only served to give a wider\nexpansion to his views, and to fill his bosom with still higher and\nnobler aspirations.\n\nThe discreet conduct of Don John in regard to his nephew, Don Carlos,\nwhen the latter would have engaged him in his wild and impracticable\nschemes, established him still more firmly in the royal favour.[148]\n\nIn the spring of the year 1568, an opportunity occurred for Philip to\ngratify his brother's ambition, by entrusting him with the command of a\nfleet then fitting out, in the port of Carthagena, against the Barbary\ncorsairs, who had been making alarming depredations of late on the\nSpanish commerce. But, while giving him this appointment, the king was\ncareful to supply the lack of experience in his brother by naming as\nsecond in command an officer in whose abilities he perfectly confided.\nThis was Antonio de Zuniga y Requesens, grand commander of St. James, an\neminent personage, who will come frequently before the reader in the\nprogress of the narrative. Requesens, who at this time filled the post\nof ambassador at Rome, was possessed of the versatility of talent so\nimportant in an age when the same individual was often required to\nexchange the duties of the cabinet for those of the camp. While Don John\nappeared before the public as the captain of the fleet, the actual\nresponsibility for the conduct of the expedition rested on his\nlieutenant.\n\nOn the third of June, Don John sailed out of port, at the head of as\nbrave an armament as ever floated on the waters of the Mediterranean.\nThe prince's own vessel was a stately galley, gorgeously fitted up, and\ndecorated with a profusion of paintings, the subjects of which, drawn\nchiefly from ancient history and mythology, were of didactic import,\nintended to convey some useful lesson to the young commander. The moral\nof each picture was expressed by some pithy maxim inscribed beneath it\nin Latin. Thus, to whatever quarter Don John turned his eyes, they were\nsure to fall on some homily for his instruction; so that his galley\nmight be compared to a volume richly filled with illustrations, that\nserve to impress the contents on the reader's memory.[149]\n\nThe cruise was perfectly successful; and Don John, on his return to\nport, some eight months later, might boast that, in more than one\nengagement, he had humbled the pride of the corsairs, and so far\ncrippled them that it would be long before they could resume their\ndepredations; that, in fine, he had vindicated the honour of his\ncountry's flag throughout the Mediterranean.\n\nHis return to Madrid was welcomed with the honours of a triumph.\nCourtier and commoner, men of all classes, in short, vied with each\nother in offering up the sweet incense of adulation, filling his young\nmind with lofty visions of the future, that beckoned him forward in the\npath of glory.\n\n[Sidenote: DON JOHN MADE COMMANDER-IN-CHIEF.]\n\nWhen the insurrection of the Moriscoes broke out in 1568, the eyes of\nmen naturally turned on Don John of Austria, as the person who would\nmost likely be sent to suppress it. But Philip thought it would be\nsafer to trust the command to those who, from their long residence in\nthe neighbourhood, were better acquainted with the character of the\ncountry and of its inhabitants. When, however, the dissensions of the\nrival chiefs made it necessary to send some one invested with such\npowers as might enable him to overawe this factious spirit and enforce\ngreater concert of action, the council of state recommended Don John to\nthe command. Their recommendation was approved by the king, if, indeed,\nit was not originally made at his suggestion.\n\nStill the \"prudent\" monarch was careful not to invest his brother with\nthat independent command which the public supposed him to possess. On\nthe contrary, his authority was restricted within limits almost as\nnarrow as those which had curbed it in the Mediterranean. A council of\nwar was appointed, by whose opinions Don John was to be guided in every\nquestion of moment. In case of a division of opinion, the question was\nto be referred to the decision of Philip.[150]\n\nThe chief members of this body, in whom the supreme power was virtually\nlodged, were the marquis of Mondejar, who from this time does not appear\nto have taken the field in person; the duke of Sessa, grandson of the\ngreat captain, Gonsalvo de Cordova, and endowed with no small portion of\nthe military talent of his ancestor; the archbishop of Granada, a\nprelate possessed of as large a measure of bigotry as ever fell to the\nlot of a Spanish ecclesiastic; Deza, president of the Audience, who\nhated the Moriscoes with the fierce hatred of an inquisitor; and,\nfinally, Don John's faithful _ayo_, Quixada, who had more influence over\nhim than was enjoyed by any other, and who had come to witness the first\nof his pupil's campaigns, destined, alas! to be the closing one of his\nown.[151]\n\nThere could hardly have been a more unfortunate device than the\ncontrivance of so cumbrous a machinery as this council, opposed as it\nwas, from its very nature, to the despatch so indispensable to the\nsuccess of military operations. The mischief was increased by the\nnecessity of referring every disputed point to the decision of the king.\nAs this was a contingency that often occurred, the young prince soon\nfound almost as many embarrassments thrown in his way by his friends as\nby his foes,--embarrassments which nothing but an uncommon spirit of\ndetermination on his own part could have overcome.\n\nOn the sixth of April, 1569, Don John took leave of the king, then at\nAranjuez, and hastened towards the south. His coming was eagerly\nexpected by the inhabitants of Granada; by the Christians, from their\nhopes that it would remedy the disorders in the army and bring the war\nto a speedy conclusion; by the Moriscoes, from the protection they\nanticipated he would afford them against the violence of the Spaniards.\nPreparations were made in the capital for giving him a splendid\nreception. The programme of the ceremonies was furnished by Philip\nhimself.[152] At some miles from the city, Don John was met by the count\nof Tendilla, at the head of a small detachment of infantry, wearing\nuniforms partly of the Castilian fashion, partly of the\nMorisco,--presenting altogether a strange and picturesque spectacle, in\nwhich silks, velvets, and rich embroidery floated gaily amidst the iron\nmail and burnished weapons of the warrior.[153] As the prince proceeded\nalong his route, he was met by a long train of ecclesiastical and civic\nfunctionaries, followed by the principal cavaliers and citizens of\nGranada. At their head were the archbishop and the president, the latter\nof whom was careful to assert his rank by walking on the right of the\nprelate. Don John showed them both the greatest deference; and as they\ndrew near, he dismounted from his horse, and, embracing the two\nchurchmen, stood with hat in hand, for some moments, while conversing\nwith them.[154] As their train came up, the president presented the most\neminent persons to the prince, who received them with that frank and\ngraceful courtesy which won the hearts of all who approached him. He\nthen resumed his route, escorted on either side by the president and the\narchbishop. The neighbouring fields were covered with spectators, and on\nthe plains of Beyro he found a large body of troops, not less than ten\nthousand, drawn up to receive him. As he approached, they greeted him\nwith salvoes of musketry, delivered with admirable precision. As Don\nJohn glanced over their beautiful array, and beheld their perfect\ndiscipline and appointments, his eyes brightened and his cheek flushed\nwith a soldier's pride.\n\nHardly had he entered the gates of Granada, when he was surrounded by a\nthrong of women, who gathered about him in an attitude of supplication.\nThey were the widows, the mothers, and the daughters of those who had so\nmiserably perished in the massacres of the Alpujarras. They were clad in\nmourning, some of them so scantily as too plainly to reveal their\npoverty. Falling on their knees, with tears streaming from their eyes,\nand their words rendered almost inarticulate by their sobs, they\ndemanded justice,--justice on the murderers of their kindred. They had\nseen their friends fall, they said, beneath the blows of their\nexecutioners; but the pain with which their hearts were then rent was\nnot so great as what they now felt on learning that the cruel acts of\nthese miscreants were to go unpunished.[155] Don John endeavoured to\ncalm their agitation by expressions of the deepest sympathy for their\nmisfortunes,--expressions of which none who saw his countenance could\ndoubt the truth; and he promised that he would do all in his power to\nsecure them justice.\n\nA livelier scene awaited him as the procession held its way along the\nstreets of the ancient capital. Everywhere the houses were gaily\ndecorated with tapestries of cloth of gold. The multitude who thronged\nthe avenues filled the air with their loyal acclamations. Bright eyes\nglanced from balconies and windows, where the noblest matrons and\nmaidens of Granada, in rich attire, were gathered to look upon the\nsplendid pageant, and the young hero who was the object of it.[156] In\nthis state he moved along until he reached the palace of the Royal\nAudience, where, by the king's command, apartments had been sumptuously\nfitted up for his accommodation.[157]\n\n[Sidenote: DISCUSSIONS OF THE COUNCIL.]\n\nThe following day, a deputation waited on Don John from the principal\nMoriscoes of the city, claiming his protection against the injuries and\ninsults to which they were exposed whenever they went abroad. They\ncomplained especially of the Spanish troops quartered on them, and of\nthe manner in which they violated the sanctity of their dwellings by the\nfoulest outrages. Don John replied in a tone that expressed little of\nthe commiseration which he had shown to the female petitioners on the\npreceding day. He told the Moriscoes that he had been sent to restore\norder to Granada, and that those who had proved loyal would find\nthemselves protected in all their rights. Those, on the contrary, who\nhad taken part in the late rebellion, would be chastised with unsparing\nrigour.[158] He directed them to state their grievances in a memorial,\nwith a caution to set down nothing which they could not prove, or it\nwould go hard with them. The unfortunate Moriscoes found that they were\nto expect such justice only as comes from the hand of an enemy.\n\nThe first session of the council showed how defective was the system for\nconducting the war. In the discussions that ensued, Mondejar remarked\nthat the contest, in his opinion, was virtually at an end; that the\nMoriscoes, for the most part, were in so favourable a mood, that he\nwould undertake, if the affair were placed in his hands, to bring them\nall to submission in a very short time. This proposal was treated with\ncontempt by the haughty president, who denounced them as a false-hearted\nrace, on whose promises no one could rely. The war, he said, would never\nbe ended so long as the Moriscoes of the capital were allowed to\ncommunicate with their countrymen in the mountains, and to furnish them\nwith secret intelligence respecting what was passing in the Christian\ncamp. The first step was to remove them all from Granada into the\ninterior; the second, to make such an example of the miscreants who had\nperpetrated the massacres in the Alpujarras as should strike terror into\nthe hearts of the infidels, and deter them from any further resistance\nto authority. In this division of opinion the members took different\nsides, according to the difference of their tempers. The\ncommander-in-chief and Quixada both leaned to Mondejar's opinion. After\na protracted discussion, it became necessary to refer the question to\nthe king, who was by no means distinguished for the promptness with\nwhich he came to his conclusions. All this required much time, during\nwhich active operations could not be resumed.[159]\n\nYet Don John did not pass it idly. He examined the state of the works in\nGranada and its neighbourhood; he endeavoured to improve the condition\nof the army, and to quell the spirit of insubordination which had risen\nin some portions of it; finally, he sent his commands for enforcing\nlevies, not merely in Andalusia and the adjoining provinces, but in\nCastile. The appeal was successful; and the great lords in the south,\nmore particularly, gathering their retainers, hastened to Granada, to\ndraw their swords under this popular chieftain.[160]\n\nMeanwhile the delay was attended with most mischievous consequences, as\nit gave the enemy time to recover from the disasters of the previous\ncampaign. Aben-Humeya had returned, as we have seen in the former\nchapter, to his mountain throne, where he soon found himself in greater\nstrength than before. Even the \"Moriscoes of the peace,\" as they were\ncalled, who had resumed their allegiance to the crown, exasperated by\nthe outrages of the Spanish soldiery, and the contempt which they showed\nfor the safe-conduct of the marquis of Mondejar, now came in great\nnumbers to Aben-Humeya's camp, offering their services, and promising to\nstand by him to the last. Other levies he drew from Africa. The Moslem\nprinces to whom he had applied for succour, though refusing to embark\nopenly in his cause, as he had desired, allowed such of their subjects\nas chose to join his standard. In consequence a considerable body of\nBarbary Moors crossed the sea, and entered into the service of the\nMorisco chief. They were a fierce, intrepid race, accustomed to a life\nof wild adventure, and possessing a better acquaintance with military\ntactics than belonged to the Spanish mountaineers.[161]\n\nWhile strengthened by these recruits, Aben-Humeya drew a much larger\nrevenue than formerly from his more extended domains.[162] Though showy\nand expensive in his tastes, he did not waste it all on the maintenance\nof the greater state which he now assumed in his way of living. He\nemployed it freely in the pay of foreign levies, and in procuring arms\nand munitions for his own troops; and he profited by his experience in\nthe last campaign, and by the example of his African mercenaries, to\nintroduce a better system of tactics among his Morisco warriors. The\npolicy he adopted, as before, was to avoid pitched battles, and to\nconfine himself chiefly to the _guerilla_ warfare, better suited to the\ngenius of the mountaineer. He fell on small detachments of Spaniards,\nwho were patrolling the country, cut off the convoys, and thus greatly\nstraitened the garrisons in their supplies. He made forays into the\nChristian territories, penetrating even into the _vega_, and boldly\ncarried the war up to the walls of Granada.\n\nHis ravages in this quarter, it is true, did not continue long after the\narrival of Don John, who took effectual measures for protecting the\ncapital from insult. But the prince was greatly chagrined by seeing the\nrapid extension of the Morisco domain. Yet he could take no decisive\nmeasures to check it until the council had determined on some plan of\noperations. He was moreover fettered by the king's orders not to take\nthe field in person, but to remain and represent him in Granada, where\nhe would find enough to do in regulating the affairs and providing for\nthe safety of the city.[163] Philip seems to have feared that Don John's\nadventurous spirit would lead him to some rash act that might\nunnecessarily expose him to danger. He appears, indeed, as we may gather\nfrom numerous passages in his letters, to have been more concerned for\nthe safety of his brother than for the success of the campaign.[164] He\nmay have thought, too, that it was better to trust the war to the hands\nof the veteran chief, the marquis of Los Velez, who could boast so much\nlarger experience than Don John, and who had possessed the king with a\nhigh idea of his military talents.\n\n[Sidenote: THE WAR RENEWED.]\n\nThis nobleman still held the command of the country east of the\nAlpujarras, in which lay his own large property. He had, as we have\nseen, a hard and arrogant nature, which could ill brook the paramount\nauthority of the young commander-in-chief, to whom he rarely\ncondescended to write, preferring to make his communications directly to\nthe king.[165] Philip, prompted by his appetite for power, winked at\nthis irregular proceeding, which enabled him to take a more direct part\nin the management of affairs than he could otherwise have done. It was a\nmost injudicious step, and was followed, as we shall see, by disastrous\nconsequences.\n\nThe marquis, without waiting for orders, resolved to open the campaign\nby penetrating into the Alpujarras with the small force he had under his\ncommand. But a body of some four hundred troops, which he had caused to\noccupy the pass of Ravaha, was cut off by the enemy, and the haughty\nchieftain reluctantly obeyed the orders of Don John to abandon his\ndesign. Aben-Humeya's success encouraged him to attack the marquis in\nhis new quarters at Verja. It was a well-concerted enterprise, but\nunfortunately, before the time arrived for its execution, it was\nbetrayed by a prisoner to the Spanish commander. It consequently failed.\nAben-Humeya penetrated into the heart of the town, where he found\nhimself in the midst of an ambuscade, and with difficulty, after a heavy\nloss, effected his retreat. But if the victory remained with the\nSpaniards, the fruits of it fell to the Moriscoes. The spirit shown by\nthe Moslem prince gave new life to his countrymen, and more than\ncounterbalanced the effects of his defeat. The rich and populous country\nof the Rio de Almanzora rose in arms. The marquis of Los Velez found it\nexpedient to abandon his present position, and to transfer his quarters\nto Adra, a seaport on the Mediterranean, which would afford him greater\nfacilities for receiving reinforcements and supplies.[166]\n\nThe spirit of insurrection now spread rapidly over other parts of the\nAlpujarras, and especially along the sierra of Bentomiz, which stretches\nfrom the neighbourhood of Alhama towards the south. Here the\nmountaineers, who had hitherto taken no part in the troubles of the\ncountry, ranging themselves under the crimson banner of Aben-Humeya,\nbroke forth into open rebellion. The inhabitants of Velez and of the\nmore important city of Malaga were filled with consternation, trembling\nlest the enemy should descend on them from the mountains and deluge\ntheir streets with blood. They hastily mustered the militia of the\ncountry, and made preparations for their defence.\n\nFortunately, at this conjuncture, they were gladdened by the sight of\nthe grand-commander, Requesens, who sailed into the harbour of\nVelez-Malaga with a squadron from Italy, having on board several\nbattalions of Spanish veterans, who had been ordered home by the\ngovernment to reinforce the army of the Alpujarras. There were no better\ntroops in the service, seasoned as they were by many a hard campaign,\nand all under the most perfect discipline. The first step of\nRequesens,--the same officer, it will be remembered, who had acted as\nthe lieutenant of Don John of Austria in his cruise in the\nMediterranean,--was to request of his young general the command of the\nexpedition against the rebels of Bentomiz. These were now gathered in\ngreat force on the lofty table-land of Fraxiliana, where they had\nstrengthened the natural defences of the ground by such works as\nrendered the approach to it nearly impracticable. The request was\nreadily granted; and the grand-commander of St. James, without loss of\ntime, led his battalions into the heart of the sierra.\n\nWe have not space for the details. It is enough to say that the\nexpedition was one of the best-conducted in the war. The enemy made a\ndesperate resistance; and, had it not been for the timely arrival of the\nbold burghers of Malaga, the grand-commander would have been driven from\nthe field. The Morisco women fought by the side of their husbands; and\nwhen all was lost, many threw themselves headlong from the precipices\nrather than fall into the hands of the Spaniards.[167] Two thousand of\nthe enemy were slain, and three thousand captives, with an immense booty\nof gold, silver, jewels, and precious stuffs, became the spoil of the\nvictors. The spirit of rebellion was effectually crushed in the sierra\nof Bentomiz.\n\nYet it was not a bloodless victory. Full six hundred of the Christians\nfell on the field of battle. The loss bore most heavily on the troops\nfrom Italy. Nearly every captain in this valiant corps was wounded.[168]\nThe bloody roll displayed, moreover, the name of more than one cavalier\nas distinguished for his birth as for his bravery. Two thousand\nMoriscoes succeeded in making their escape to the camp of Aben-Humeya.\nThey proved a seasonable reinforcement, for that chief was meditating an\nassault on Seron.[169]\n\nThis was a strongly-fortified place, perched like an eagle's eyry on the\nsummit of a bold cliff that looked down on the Rio de Almanzora, and\ncommanded its formidable passes. It was consequently a most important\npost, and at this time was held by a Spanish garrison under an officer\nnamed Mirones. Aben-Humeya sent a strong detachment against it,\nintending to carry it by storm. But the Moriscoes had no battering\ntrain, and, as it soon appeared, were little skilled in the art of\nconducting a siege. It was resolved, therefore, to abandon the present\nplan of operations, and to reduce the place by the slower but surer way\nof blockade. Five thousand men, accordingly, sat down before the town on\nthe 18th of June, and effectually cut off all communication from abroad.\n\nThe garrison succeeded in conveying intelligence of their condition to\nDon John, who lost no time in ordering Alonso de Carbajal to march with\na body of troops and a good supply of provisions to their relief. But,\njust after his departure, Don John received information that the king\nhad entrusted the marquis of Los Velez with the defence of Seron. He,\ntherefore, by Quixada's advice, countermanded his orders to Carbajal,\nand directed him to return. That officer, who had approached within a\nshort distance of the place, reluctantly obeyed, and left Seron to its\nfate. The marquis of Los Velez, notwithstanding the jealousy he\ndisplayed of the interference of Don John in the affair, showed so\nlittle alacrity in providing for the safety of the beleaguered fortress,\nthat the garrison, reduced to extremity, on the eleventh of July,\nsurrendered on honourable terms. But no sooner had they given up the\nplace, than the victors, regardless of the terms of capitulation,\nmurdered in cold blood every male over twelve years of age, and made\nslaves of the women and children. This foul act was said to have been\nperpetrated by the secret command of Aben-Humeya. The Morisca chief\nmight allege, in vindication of his perfidy, that he had but followed\nthe lesson set him by the Spaniards.[170]\n\n[Sidenote: REMOVAL OF THE MORISCOES.]\n\nThe loss of Seron caused deep regret to the army. Nor could this regret\nbe mitigated by the reflection, that its loss was to be attributed not\nso much to the valour of the Moslems as to the misconduct of their own\ncommanders, or rather to the miserable system adopted for carrying on\nthe war. The triumph of the Moriscoes, however, was greatly damped by\nthe intelligence which they had received, shortly before the surrender\nof Seron, of disasters that had befallen their countrymen in Granada.\n\nPhilip, after much hesitation, had given his sanction to Deza's project\nfor the removal of the Moriscoes from the capital into the interior of\nthe country. The day appointed for carrying the measure into effect was\nthe twenty-third of June. A large body of troops, with the principal\ncommanders, was secretly assembled in the capital to enforce the\nexecution of the plan. Meanwhile, rumours were current that the\nMoriscoes in the city were carrying on a secret communication with their\ncountrymen in the Alpujarras; that they supplied the mountaineers with\narms and money; that the young men were leaving Granada to join their\nranks; finally, that a conspiracy had been planned for an assault on the\ncity, and even that the names of the leaders were given. It is\nimpossible, at this time, to say what foundation there was for these\ncharges; but the reader may recollect that similar ones had been\ncirculated previous to the barbarous massacre in the prison of the\nChancery.\n\nOn the twenty-third of the month, on the eve of St John's, an edict was\npublished, commanding all the Morisco males in Granada between ten and\nsixty years of age, to repair to the parish churches to which they\nrespectively belonged, where they were to learn their fate. The women\nwere to remain some time longer in the city, to dispose of the most\nvaluable effects, such as could not easily be transported. This was not\ndifficult, at the low prices for which, in their extremity, they were\nobliged to part with their property. We are left in ignorance of the\nfate of the children, who, no doubt, remained in the hands of the\ngovernment, to be nurtured in the Roman Catholic faith.[171]\n\nNothing could exceed the consternation of the Moriscoes on the\npublication of this decree, for which, though so long suspended by a\nthread, as it were, over their heads, they were wholly unprepared. It is\nnot strange, as they recalled the atrocious murders perpetrated in the\nprison of the Chancery, that they should have been led to believe that\nnothing less than a massacre of the whole Moorish population was now\ndesigned. It was in vain that the marquis of Mondejar endeavoured to\nallay their fears. They were somewhat comforted by the assurance of the\nPresident Deza, given under his own hand, that their lives were in no\ndanger. But their apprehensions on this point were not wholly quieted\ntill Don John had pledged his royal word that no harm should come to\ntheir persons; that, in short, the great object of the government was to\nsecure their safety. They then submitted without any attempt at\nresistance. Resistance, indeed, would have been hardly possible,\ndestitute as they were of weapons or other means of defence, and\nsurrounded on all quarters by the well-armed soldiery of Castile. They\naccordingly entered the churches assigned to them, at the doors of which\nstrong guards were stationed during the night.\n\nOn the following morning the Moriscoes were marched out and formed into\na procession, which was to take its way to the great hospital in the\nsuburbs. This was a noble building, erected by the good Queen Isabella\nthe Catholic, not long after the Conquest. Here they were to stay till\nthe arrangements were completed for forming them into divisions\naccording to their several places of destination. It was a sad and\nsolemn spectacle, that of this company of exiles, as they moved with\nslow and uncertain step, bound together by cords,[172] and escorted, or\nrather driven along like a gang of convicts, by the fierce soldiery.\nThere they were, the old and the young, the rich and the poor, now,\nalas! brought to the same level, the forms of most of them bowed down,\nless by the weight of years than of sorrow, their hands meekly folded on\ntheir breasts, their cheeks wet with tears, as they gazed for the last\ntime on their beautiful city, the sweet home of their infancy, the proud\nseat of ancient empire, endeared to them by so many tender and glorious\nrecollections.[173]\n\nThe march was conducted in an orderly manner, with but a single\ninterruption, which, however, was near being attended by the most\ndisastrous consequences. A Spanish alguazil, offended at some words that\nfell from one of the prisoners--for so they might be called--requited\nhim with a blow from his staff. But the youth whom he struck had the\nfiery blood of the Arab in his veins. Snatching up a broken tile, he\ndealt such a blow on the offender's head as nearly severed his ear from\nit. The act cost him his life. He was speedily cut down by the\nSpaniards, who rushed to the assistance of their wounded comrade. A\nrumour now went round that the Moriscoes had attempted the life of Don\nJohn, whose dress resembled in its colour that of the alguazil. The\npassions of the soldiery were roused. They flocked to the scene of\nviolence, uttering the most dreadful imprecations. Their swords and\nlances glittered in the air, and in a few moments would have been\nsheathed in the bodies of their terrified victims.\n\nFortunately, the quick eye of Don John discerned the confusion.\nSurrounded by a body-guard of arquebusiers, he was there in person to\nsuperintend the removal of the Moriscoes. Spurring his horse forward\ninto the midst of the tumult, and showing himself to the troops, he\nexclaimed that no one had offered him any harm. He called on them to\nreturn to their duty, and not to dishonour him as well as themselves, by\noffering violence to innocent men, for whose protection he had so\nsolemnly pledged his word. The soldiers, abashed by the rebuke of their\nyoung chief, and satisfied with the vengeance they had taken on the\noffender, fell back into their ranks. The trembling Moriscoes gradually\nrecovered from their panic, the procession resumed its march, and\nwithout further interruption reached the hospital of Isabella.[174]\n\n[Sidenote: REMOVAL OF THE MORISCOES.]\n\nThere the royal _contadores_ were not long in ascertaining the number of\nthe exiles. It amounted to thirty-five hundred. That of the women, who\nwere soon to follow, was much greater.[175] The names, the ages, and the\noccupations of the men were all carefully registered. The following day\nthey were marched into the great square before the hospital, where they\nwere distributed into companies, each under a strong escort, to be\nconducted to their various places of destination. These, far from being\nconfined to Andalusia, reached into New Castile. In this arrangement we\nmay trust that so much respect was paid to the dictates of humanity, as\nnot to separate those of the same kindred from one another. But the\nchroniclers give no information on the subject; probably regarding\ndetails of this sort, in regard to the fallen race, as below the dignity\nof history.\n\nIt was on the twenty-fifth of June, 1569, that, bidding a sad farewell\nto the friends and companions of their youth, from whom they were now to\nbe for ever parted, they set forth on their doleful pilgrimage. The\nmorning light had broken on the red towers of the Alhambra, as the bands\nof exiles, issuing from the gates of their beloved capital, the spot\ndearest to them upon earth, turned their faces towards their new\nhomes,--homes which many of them were destined never to behold. The\ngovernment, with shameful indifference, had neglected to provide for the\npoor wanderers the most common necessaries of life. Some actually\nperished of hunger by the way. Others, especially those accustomed from\ninfancy to a delicate nurture, sank down and died of fatigue. Some were\nseized by the soldiers, whose cupidity was roused by the sight of their\nhelplessness, and were sold as slaves. Others were murdered by their\nguards in cold blood.[176] Thus reduced far below their original number,\nthey reached their appointed places, there to linger out the remainder\nof their days in the midst of a population who held them in that\nabhorrence with which a good Catholic of the sixteenth century regarded\n\"the enemies of God.\"[177]\n\nBut the evils which grew out of this stern policy of the government were\nnot wholly confined to the Moriscoes. This ingenious people were so far\nsuperior to the Spaniards in the knowledge of husbandry, and in the\nvarious mechanical arts, that they formed the most important part of the\npopulation of Granada. The only art in which their rivals excelled them\nwas that which thrives at the expense of every other--the art of war.\nAware of this, the government had excepted some of the best artisans in\nthe capital from the doom of exile which had fallen on their countrymen,\nand they had accordingly remained in the city. But their number was too\nsmall to produce the result desired; and it was not long before the\nquarter of the town which had been occupied by the Moriscoes exhibited a\nscene of woeful desolation. The light and airy edifices, which displayed\nin their forms the fantastic graces of Arabian architecture, fell\nspeedily into decay. The parterres and pleasure-grounds, filled with\nexotics, and glowing in all the exuberance of southern vegetation,\nbecame a wilderness of weeds; and the court-yards and public squares,\nwhere tanks and sparkling fountains, fed by the streams of the Sierra\nNevada, shed a refreshing coolness over the atmosphere in the sultriest\nmonths of summer, were soon converted into a melancholy heap of rubbish.\n\nThe mischiefs growing out of the removal of the Moriscoes fell sorely on\nthe army. The men had been quartered, as we have seen, in the houses of\nthe Moriscoes. From the present occupants, for the most part needy and\nthriftless speculators, they met with very different fare from what they\nhad enjoyed under the former wealthy and luxurious proprietors. The\ntroops supplied the deficiency, as far as they could, by plundering the\ncitizens. Hence incessant feuds arose between the people and the army,\nand a spirit of insubordination rapidly grew up in the latter, which\nmade it more formidable to its friends than to its foes.[178]\n\nAn eyewitness of these troubles closes his narrative of the removal of\nthe Moriscoes by remarking that it was a sad spectacle to one who\nreflected on the former policy and prosperity of this ill-starred race;\nwho had seen their sumptuous mansions in the day of their glory, their\ngardens and pleasure-grounds, the scene of many a gay revel and jocund\nholiday, and who now contrasted all this with the ruin into which\neverything had fallen.[179] \"It seems,\" he concludes, \"as if Providence\nhad intended to show, by the fate of this beautiful city, that the\nfairest things in this world are the most subject to decay.\"[180] To the\nphilosopher of the present age it may seem rather the natural result of\nthat system of religious intolerance which had converted enemies those\nwho, under a beneficent rule, would have been true and loyal subjects,\nand who by their industry and skill would have added incalculably to the\nresources of the country.\n\n\n\n\nCHAPTER VI.\n\nREBELLION OF THE MORISCOES.\n\nOperations of Los Velez--Conspiracy against Aben-Humeya--His\nAssassination--Election of Aben-Aboo--Vigorous Prosecution of the\nWar--Fierce Combats in the Vega--Impetuous Spirit of Don John--Surprise\nof Guejar.\n\n1569.\n\n\nWhile the events related in the preceding chapter were occurring, the\nmarquis of Los Velez lay, with a considerable force, at Adra, a port on\nthe Mediterranean, at the foot of the Alpujarras, which he had selected\nchiefly from the facilities it would afford him for getting supplies for\nhis army. In this he was disappointed. Before the month of June had\nexpired, his troops had begun to be straitened for provisions. The evil\nwent on increasing from day to day. His levies, composed chiefly of raw\nrecruits from Andalusia, were full of that independent, and indeed\nturbulent spirit, which belongs to an ill-disciplined militia. There was\nno lack of courage in the soldiery. But the same men who had fearlessly\nbraved the dangers of the campaign, now growing impatient under the\npinch of hunger, abandoned their colours in great numbers.\n\nThere were various causes for the deficiency of supplies. The principal\none of these may probably be found in the remissness of the council of\nwar, several of whose members regarded the marquis with an evil eye, and\nwere not sorry to see his embarrassments.\n\n[Sidenote: OPERATIONS OF LOS VELEZ.]\n\nSome vigorous measures were instantly to be taken, or the army, it was\nevident, would soon altogether melt away. By the king's command, orders\nwere despatched to Requesens, who lay with his squadron off the port of\nVelez-Malaga, to supply the camp with provisions, while it received\nreinforcements, as before, principally from the Andalusian militia. The\narmy received a still more important accession in the well-disciplined\nveterans who had followed the grand-commander from Italy. Thus\nstrengthened, and provisioned for a week or more, Los Velez, at the head\nof twelve thousand men, set forth on the twenty-sixth of July, and\nstruck at once into the Alpujarras. He had been directed by the council\nto establish himself at Ugibar, which, by its central position, would\nenable him to watch the movements of Aben-Humeya, and act on any point\nas occasion required.\n\nThe marquis, without difficulty, defeated a force of some five or six\nthousand men, who had been stationed to oppose his entrance into the\nmountain country. He then pressed forward, and on the high lands beyond\nUgibar--which place he had already occupied--he came in sight of\nAben-Humeya, with the flower of his troops drawn up to receive him.\n\nThe two chiefs, in their characters, their persons, and their\nequipments, might be considered as no bad types of the European and the\nArab chivalry. The marquis, sheathed in complete mail, of a sable\ncolour, and mounted on his heavy war-horse, also covered with armour,\nwas to be seen brandishing a lance which, short and thick, seemed rather\nlike a truncheon, as he led his men boldly on, prepared to plunge at\nonce into the thick of the fight.[181] He was the very emblem of brute\nforce. Aben-Humeya, on the other hand, gracefully managing his\nswift-footed, snow-white Andalusian, with his Morisco mantle of crimson\nfloating lightly from his shoulders, and his Turkish turban wreathed\naround his head,[182] instead of force, suggested the opposite ideas of\nagility and adroitness, so characteristic of the children of the East.\n\nRiding along his lines, the Morisco prince exhorted his followers not to\nfear the name of Los Velez: for, in the hour of danger, God would aid\nHis own; and better was it, at any rate, to die like brave men in the\nfield, than to live dishonoured.[183] Notwithstanding these magnanimous\nwords, it was far from Aben-Humeya's wish to meet his enemy in a fair\nfield of fight. It was contrary to the genius and the habit of his\nwarfare, which was of the guerilla kind, abounding in sallies and\nsurprises, in which, seeking some vulnerable point, he could deal his\nblow and retreat precipitately among the mountains.\n\nYet his followers, though greatly inferior in numbers to the enemy,\nbehaved with spirit; and the field was well contested, till a body of\nAndalusian horse, making a _detour_ under cover of some rising ground,\nfell unexpectedly on the rear of the Moriscoes, and threw them into\nconfusion. The marquis pressing them at the same time vigorously in\nfront, they broke, and soon gave way on all sides. Aben-Humeya,\nperceiving the day lost, gave the rein to his high-mettled genet, who\nswiftly bore him from the field; and, though hotly pursued, he soon left\nhis enemies behind. On reaching the foot of the Sierra Nevada, the chief\ndismounted, and hamstringing his noble animal, plunged into the depths\nof the mountains, which again opened their friendly arms to receive\nhim.[184] Yet he did not remain there long before he was joined by his\nfollowers; and no sooner was he in sufficient strength, than he showed\nhimself on the eastern skirts of the sierra, whence, like an eagle\nstooping on his prey, he rushed down upon the plains below, sweeping\nthrough the rich valley of the Rio de Almanzora, and carrying fire and\nsword to the very borders of Murcia. Here he revenged himself on Los\nVelez by falling on his town of Las Cuevas, firing his dwellings,\nravaging his estates, and rousing his Morisco vassals to rebellion.[185]\n\nMeanwhile the marquis, instead of following up his victory, remained\ntorpid within the walls of Calahorra. Here he had desired the council to\nprovide stores for the subsistence of his army. To his dismay, none had\nbeen provided; and as his own attempts to procure them were\nunsuccessful, he soon found himself in the same condition as at Adra.\nThe famine-stricken troops, with little pay and less plunder, first\nbecame discontented, then mutinous, and at length deserted in great\nnumbers. It was in vain that the irascible old chief poured out his\nwrath in menaces and imprecations. His arrogant temper had made him\nhated even more than he was feared by his soldiers. They now went off,\nnot stealthily and by night, but in the open day, whole companies at a\ntime, their arquebuses on their shoulders, and their matches\nlighted.[186] When Don Diego Fajardo, the marquis's son, endeavoured to\nstay them, one, more audacious than the rest, lodged a musket-ball in\nhis body. It was not long before the gallant array with which the\nmarquis had so proudly entered the Alpujarras, was reduced to less than\nthree thousand men. Among them were the Italian veterans, who refused to\ntarnish their well-earned laurels by thus basely abandoning their\ncommander.\n\nThe council of war complained loudly to the king of the fatal inactivity\nof the marquis, and of his neglect to follow up the advantages he had\ngained. Los Velez angrily retorted by throwing the blame on that body,\nfor neglecting to furnish him with the supplies which would have enabled\nhim to do so. Philip, alarmed, with reason, at the critical aspect of\naffairs, ordered the marquis of Mondejar to repair to court, that he\nmight confer with him on the state of the country. This was the avowed\nmotive for his recall. But, in truth, it seems probable that the king,\naware of that nobleman's leaning to a pacific policy, and of his\npersonal hostility to Los Velez, deemed it best to remove him altogether\nfrom any share in the conduct of the war. This he did most effectually,\nby sending him into honourable exile, first appointing him Viceroy of\nValentia, and afterwards raising him to the important post of Viceroy of\nNaples. From this period the name of Mondejar no more appears on the\ntheatre of the Morisco war.[187]\n\n[Sidenote: DECLINE OF ABEN-HUMEYA'S POPULARITY.]\n\nThe marquis did not win the favour to which he was entitled by his\ndeserts. He seems to have possessed some of the best qualities of a good\ncaptain. Bold in action, he was circumspect in council. Slow and\nsagacious in the formation of his plans, he carried them out with\nsingular perseverance. He knew the country well which was the seat of\nthe insurrection, and perfectly understood the character of its\ninhabitants. What was more rare, he made allowance for the excesses into\nwhich they had been drawn by a long course of insult and oppression. The\nhumanity of his disposition combined with his views of policy to make\nhim rely more on conciliatory measures than on fear, for the reduction\nof the enemy. How well this worked we have seen. Had he been properly\nsupported by those engaged with him in the direction of affairs, we can\nhardly doubt of his ultimate success. But, unhappily, the two most\nprominent of these, the President Deza and the Marquis of Los Velez,\nwere narrow-minded, implacable bigots, who, far from feeling compassion\nfor the Moriscoes, looked on the whole race as \"God's enemies.\"\nUnfortunately, these views found favour with the government; and\nPhilip, who rightly thought that the marquis of Mondejar would only\nprove a hindrance to carrying on hostilities with vigour, acted\nconsistently in sending him from the country. Yet, while he was thus\nremoved from the conduct of the war, it may be thought an unequivocal\nacknowledgment of Mondejar's deserts, that he was transferred to the\nmost considerable post in the gift of the crown.\n\nBefore the marquis's departure, Philip had transferred his court to\nCordova, in order to facilitate his communication with the seat of war.\nHe hoped, too, that the knowledge of his being so near would place some\ncheck on the disorderly temper of the soldiery, and animate them with\nmore loyal and patriotic feelings. In this way of proceeding he\nconsidered himself as imitating the example of his great ancestors,\nFerdinand and Isabella, who, during the war of Granada, usually\ntransferred their court to one of the capitals of the South. He did not,\nhowever, think it necessary, like them, to lead his armies in person,\nand share in the toils of the campaign.\n\nOn the nineteenth of October, Philip published an edict, which intimated\nhis design of following up the war with vigour. It commanded that such\nof the Moriscoes as had hitherto been allowed to remain in Granada\nshould now be removed from it, in order that no means of communication\nmight be left to them with their brethren in the mountains. It was\nfurther proclaimed, that the war henceforth was to be carried on with\n\"fire and blood;\"[188] in other words, that no mercy was to be shown the\ninsurgents. This was the first occasion on which this fierce\ndenunciation had been made by the government. To reconcile the militia\nof the towns to the service, their pay was to be raised to a level with\nthat of the Italian volunteers; and to relieve the towns, the greater\npart of the expense was to be borne by the crown. Before the publication\nof this ordinance the king had received intelligence of an event\nunexpected alike by Christian and by Moslem--the death of Aben-Humeya,\nand that by the hands of some of his own followers.\n\nThe Morisco prince, after carrying the war up to the borders of Murcia,\nlaid siege to two or three places of strength in that quarter. As might\nhave been expected, he failed in these attempts, from his want of\nbattering artillery. Thus foiled, he led back his forces into the\nAlpujarras, and established his quarters in the ancient Moorish palace\nof Lanjaron, on the <DW72>s of the mountains commanding the beautiful\nvalley of Lecrin. Here the torpid condition of the Spaniards under Los\nVelez allowed the young monarch to remain, and give himself up to those\nsensual indulgences with which the Moslem princes of the East were apt\nto solace their leisure in the intervals of war. His harem rivalled that\nof any Oriental satrap in the number of its inmates. This was strange to\nthe Moriscoes, who, since their nominal conversion to Christianity, had\nof course repudiated polygamy. In the eyes of the Moslems, it might pass\nfor good evidence of their prince's orthodoxy.\n\nEver since Aben-Humeya's ascent to the throne he had been declining in\npopularity. His handsome person, the courtesy of his manners, his\nchivalrous spirit, and his devotion to the cause, had easily won him the\naffections of his subjects. But a too sudden elevation had unfortunately\nthat effect on him which it is wont to have on weak minds, without any\nsettled principles or lofty aim to guide them. Possessed of power, he\nbecame tyrannical in the use of it.[189] His arbitrary acts created\nenemies, not the less dangerous that they were concealed. The\nconsciousness of the wrongs he had committed made him suspicious. He\nsurrounded himself with a body-guard of four hundred men. Sixteen\nhundred more were quartered in the place where he was residing; and the\nprincipal avenues to it, we are told, were defended by barricades.[190]\nThose whom he suspected he treated with particular kindness. He drew\nthem around his person, overwhelmed them with favours, and, when he had\nwon them by a show of confidence, he struck the fatal blow.[191] During\nthe short period of his reign, no less than three hundred and fifty\npersons, we are assured, fell victims to his jealousy or his\nrevenge.[192]\n\nAmong Aben-Humeya's officers was one named Diego Alguazil, who had a\nbeautiful kinswoman, with whom he lived, it is said, on terms of greater\nintimacy than was justified by the relationship of the parties. As he\nwas one day imprudently speaking of her to Aben-Humeya in the glowing\nlanguage of a lover, the curiosity of the king was so much inflamed by\nit that he desired to see her. In addition to her personal charms, the\nfair Zahara was mistress of many accomplishments which rendered her\nstill more attractive. She had a sweet voice, which she accompanied\nbewitchingly on the lute, and in her dancing displayed all the soft and\nvoluptuous movements of the dark-eyed beauties of Andalusia.[193] When\nbrought before the king, she did her best to please him; for though\nattached, as it seems, to her kinsman, the ambitious coquette had no\nobjection to having a royal suitor in her chains. In this she perfectly\nsucceeded; and the enamoured prince intimated his desire to Alguazil\nthat he would resign to him the possession of his mistress. But the\nMorisco loved her too well; and neither threats nor promises of the most\nextravagant kind were able to extort his consent. Thus baffled, the\nreckless Aben-Humeya, consulting only his passion, caused the perhaps\nnot reluctant Zahara to be taken by force and lodged in his harem. By\nthis act he made a mortal enemy of Alguazil.\n\nNor did he long enjoy the favour of his new mistress, who, come of an\nancient lineage in Granada,[194] had hoped to share the throne of the\nMorisco monarch. But Aben-Humeya's passion did not carry him to this\nextent of complaisance; and Zahara, indignant at finding herself\ndegraded to the rank and file of the seraglio, soon breathed only a\ndesire for vengeance. In this state of things she found the means of\ncommunicating with her kinsman, and arranged with him a plan for\ncarrying their murderous intent into execution.\n\n[Sidenote: CONSPIRACY AGAINST ABEN-HUMEYA.]\n\nThe most important corps in the Morisco army was that of the Turkish\nmercenaries. But they were so fierce and turbulent a race that\nAben-Humeya paid dear for their services. A strong body of these troops\nlay on the frontiers of Orgiba, under the command of Aben-Aboo--a near\nrelative of the Morisco prince, whose life, it may be remembered, he had\nonce saved by submitting to every extremity of torture rather than\nbetray his lurking-place. To this commander Aben-Humeya despatched a\nmessenger, directing him to engage the Turks in a certain expedition,\nwhich would serve both to give them employment, and to satisfy their\nappetite for plunder.\n\nThe time named for the messenger's departure was communicated by Zahara\nto her kinsman, who caused him to be waylaid and murdered, and his\ndespatches to be secured. He then had a letter written to Aben-Aboo,\nwhich bore apparently the royal signature. This was counterfeited by his\nnephew, a young man then holding the post of secretary to Aben-Humeya,\nwith whom he had lately conceived some cause of disgust. The letter\nstated that the insubordination of the Turks made them dangerous to the\nstate; and that in some way or other they must be removed, and that\nspeedily. With this view, Aben-Aboo was directed to march them to\nMecina, on the frontiers of the Sierra Nevada, where he would be joined\nby Diego Alguazil, with a party of soldiers, to assist him in carrying\nthe plan into execution. The best mode, it was suggested, of getting rid\nof the Turks, would be by poison.\n\nThis letter was despatched by a courier, who was speedily followed by\nAlguazil and a hundred soldiers, as the cunning conspirator desired to\npresent himself before Aben-Aboo without leaving him time for\nconsideration.\n\nHe found that commander in a state of the utmost perplexity and\nconsternation. Alguazil declared that he had come in consequence of\ncertain instructions he had received from the king, of too atrocious a\nnature for him to execute. Aben-Aboo had as little mind to perform the\nbloody work assigned to him. He had no distrust of the genuineness of\nthe letter. Hosceyn, the commander of the Turks, happening to pass the\nhouse at that time, was called in, and the despatches were shown to him.\nThe fiery chief insisted on communicating them to some of his comrades.\nThe greatest indignation prevailed among the Turkish leaders, outraged\nby this base treachery of the very man whom they had come to serve at\nthe peril of their lives. They one and all demanded, not his deposition,\nbut his death. Diego Alguazil saw that his scheme was working well. He\nartfully fanned the flame, and professed to share deeply in the\nindignation of the Moslems. It was at length agreed to put the tyrant to\ndeath, and to offer the crown to Aben-Aboo.\n\nThis chieftain enjoyed a high reputation for sagacity and prudence. His\npassions, unlike those of Aben-Humeya, seemed ever under the control of\nhis reason; and, far from indulging an ill-regulated ambition, he had\nbeen always faithful to his trust. But the present temptation was too\nstrong for his virtue. He may have thought that, since the throne was to\nbe vacant, the descendant of the Omeyas had a better claim to it than\nany other. Whatever may have been the sophistry to which he yielded, he\nknew that those who now promised him the crown had the power to make\ntheir promise good. He gave his assent on condition that, in the course\nof three months, his election should be confirmed by the dey of Algiers,\nas the representative of the Turkish sultan.\n\nHaving arranged their plans, the conspirators lost no time in putting\nthem in execution. They set out that very hour, on the evening of the\nthird of October, for Lanjaron, with a body of four hundred troops--one\nhalf being Turks, the other Moriscoes. By midnight they reached their\nplace of destination. Diego Alguazil and the Turkish captains were too\nwell known as enjoying the confidence of Aben-Humeya to meet with any\nopposition to their entrance into the town. Nor, though the Morisco king\nhad retired to rest, did the guard oppose any difficulty to their\npassing into his dwelling. Proceeding to his chamber, they found the\ndoors secured, but speedily forced an entrance. Neither arm nor voice\nwas raised in his defence.[195]\n\nAben-Humeya, roused from sleep by the tumult, would have sprung from his\ncouch; but the faithless Zahara held him fast in her embrace, until\nDiego Alguazil and some others of the conspirators, rushing in, bound\nhis arms together with a Moorish veil.[196] Indeed, he was so much\nbewildered as scarcely to attempt resistance.\n\nThe Turkish commander then showed him the letter. Aben-Humeya recognized\nthe writing of his secretary, but declared that he had never dictated\nsuch a letter, nor was the signature his. How far his assertion gained\ncredit we are not informed. But the conspirators had already gone too\nfar to be forgiven. To recede was death. Either Aben-Humeya or they must\nbe sacrificed. It was in vain that he protested his innocence, and that\nhe offered to leave the question to the sultan, or to the dey of\nAlgiers, or to any person competent to decide it. But little heed was\ngiven to his protestations, as the conspirators dragged him into an\nadjoining apartment. The unhappy young man perceived that his hour was\ncome--that there was no one of all his friends or menials to interpose\nbetween him and his fate. From that moment he changed his tone, and\nassumed a bearing more worthy of his station. \"They are mistaken,\" he\nsaid, \"who suppose me to be a follower of the Prophet. I die, as I have\nlived, in the Christian faith. I accepted the post of head of the\nrebellion that I might the better avenge the wrongs heaped on me and my\nfamily by the Spaniards. They have been avenged in full measure, and I\nam now ready to die. Neither,\" said he, turning to Aben-Aboo, his\ndestined successor, \"do I envy you. It will not be long before you will\nfollow me.\" He then, with his own hands, coolly arranged around his neck\nthe cord with which he was to be strangled, adjusted his robes, and,\ncovering his face with his mantle, submitted himself without a struggle\nto his executioners.[197]\n\nHis body was thrown into a neighbouring sewer, with as little concern as\nif it had been that of a dog. There it continued, till Don John of\nAustria, hearing that Aben-Humeya had died a Christian, caused his\nremains to be removed to Guadix, and laid in the ground with the\nsolemnities of Christian burial.[198]\n\nThat Aben-Humeya should have come to so miserable an end is not strange.\nThe recklessness with which he sacrificed all who came between him and\nthe gratification of his passions, surrounded him with enemies, the more\ndangerous in a climate where the blood is hot, and the feeling of\nrevenge is easily kindled in the bosom. At the beginning of his reign\nhis showy qualities won him a popularity which, however, took no root in\nthe affections of the people, and which faded away altogether when the\ndefects of his character were more fully brought to light by the\nexigencies of his situation; for he was then found to possess neither\nthe military skill necessary to insure success in the field, nor those\nhigher moral attributes which command respect and obedience at home.\n\n[Sidenote: CHARACTER OF ABEN-ABOO.]\n\nVery different was the character of his successor, Aben-Aboo. Instead\nof displaying the frivolous and licentious tastes of Aben-Humeya, his\nprivate life was without reproach. He was much older than his\npredecessor; and if he had not the same fiery enthusiasm and dashing\nspirit of adventure which belonged to Aben-Humeya, he discovered both\nforecast in the formation of his plans, and singular courage in carrying\nthem into execution. All confided in his integrity; while the decorum\nand gravity of his demeanour combined with the more substantial\nqualities of his character to inspire a general feeling of reverence in\nthe people.[199] It was not till the time of his proposed elevation to\nthe supreme power, that the lustre of these qualities was darkened by\nthe perpetration of one foul deed,--his connivance at the conspiracy\nagainst his sovereign. But if he were really the dupe, as we are told,\nof Alguazil's plot, he might plead, to some extent, the necessity of\nself-preservation; for he may well have believed that, if he refused to\naid Aben-Humeya in the execution of his bloody purpose in reference to\nthe Turks, the tyrant would not long suffer him to live in possession of\na secret so perilous to himself. At all events, the part he had taken in\nthe conspiracy seems to have given no disgust to the people, who, weary\nof the despotism under which they had been living, welcomed with\nenthusiasm the accession of the new sovereign. Many places which had\nhitherto taken no part in the struggle for independence, now sent in\ntheir adhesion to Aben-Aboo, who soon found himself the ruler over a\nwider extent of territory than, at any time, had acknowledged the sway\nof his predecessor.\n\nIt was not long before the confirmation of his election arrived from\nAlgiers; and Aben-Aboo, assuming the regal name of Muley Abdallah\nMohammed as a prefix to his own, went through the usual simple forms of\na coronation of a king of Granada. In his right hand on this occasion,\nhe bore a banner inscribed with the legend, \"More I could not\ndesire--less would not have contented me.\"[200] Such an inscription\nmaybe thought to intimate that a more aspiring temper lurked within his\nbosom than the world had given him credit for.\n\nThe new sovereign did not, like his predecessor, waste his time in\neffeminate sloth. He busied himself with various important reforms,\ngiving especially a new organization to the army, and importing a large\nquantity of arms and munitions from Barbary. He determined not to allow\nhis men time for discontent, but to engage them at once in active\nservice. The first object he proposed was the capture of Orgiba, a\nfortified place, which commanded the route to Granada, and which served\nas a point of communication between that capital and remoter parts of\nthe country.\n\nAben-Aboo got everything in readiness with such despatch, that on the\ntwenty-sixth of October, a few weeks only after the death of\nAben-Humeya, he set out on his expedition at the head of a\nwell-appointed army, consisting of more than ten thousand men, partly\nforeign mercenaries and partly natives. Hastening his march, he soon\npresented himself before Orgiba, and laid siege to the place. He pushed\nmatters forward so vigorously, that in a few days he was prepared to\nstorm the works. Four times he brought his men to the assault; but\nthough, on the fourth, he succeeded in throwing himself, with a small\nbody of troops, on the ramparts, he was met with such determined\nresistance by the garrison and their brave commander, Francisco de\nMolina, that he was obliged to fall back with loss into his trenches.\nThus repulsed, and wholly destitute of battering ordnance, the Morisco\nchief found it expedient to convert the siege into a blockade.\n\nThe time thus consumed gave opportunity to Don John of Austria to send a\nstrong force, under the duke of Sesa, to the relief of the garrison.\nAben-Aboo, desirous to intercept his enemy's march, and occupy one of\nthose defiles that would give him the advantage of position, silently\nbroke up his encampment, under cover of the night, and took the\ndirection of Lanjaron. Here he came so suddenly on the advanced guard of\nthe Christians, that, taken by surprise, it gave way, and falling back,\nafter considerable loss, on the main body of the army, threw the whole\ninto confusion. Happily the duke of Sesa, though labouring at the time\nunder a sharp attack of gout, by extraordinary exertions was enabled to\nrally his men, and inspire them with courage to repulse the enemy, thus\nretrieving his own honour and the fortunes of the day.\n\nMeanwhile, the brave Molina and his soldiers no sooner learned that the\nbesiegers had abandoned their works, than, eager to profit by their\ntemporary absence, the cause of which they suspected, they dismantled\nthe fortress, and, burying their guns in the ground, hastily evacuated\nthe place. The duke of Sesa, finding that the great object of his\nexpedition--the safety of the garrison--was now accomplished, and not\nfeeling himself in sufficient strength to cope with the Morisco chief,\ninstantly began his retreat on Granada. In this he was not molested by\nAben-Aboo, who was only too glad to be allowed without interruption to\nfollow up the siege of Orgiba. But, finding this place, to his surprise,\nabandoned by the enemy, he entered it without bloodshed, and with\ncolours flying, as a conqueror.[201]\n\nThese successes in the commencement of his reign furnished a brilliant\naugury for the future. The fame of Aben-Aboo spread far and wide through\nthe country; and the warlike peasantry thronged from all quarters to his\nstandard. Tidings now arrived that several of the principal places on\nthe eastern skirts of the Alpujarras had proclaimed their adherence to\nthe Morisco cause; and it was expected that the flame of insurrection\nwould soon spread to the adjoining provinces of Murcia and Valencia. So\nwidely, indeed, had it already spread, that, of all the Morisco\nterritory south of Granada, the country around Malaga and the sierra of\nRonda, on the extreme west, were the only portions that still\nacknowledged the authority of Castile.[202]\n\nThe war now took the same romantic aspect that it wore in the days of\nthe conquest of Granada. Beacon-fires were to be seen along the highest\npeaks of the sierra, throwing their ominous glare around for many a\nleague, and calling the bold mountaineers to the foray. Then came the\ngathering of the wild militia of the country, which, pouring down on the\nlower levels, now in the faded green of autumn, swept away herds and\nflocks, and bore them off in triumph to their fastnesses.\n\nSometimes marauders penetrated into the _vega_, the beautiful _vega_,\nevery inch of whose soil was fertilized with human blood, and which now,\nas in ancient times, became the battle-ground of Christian and Moslem\ncavaliers. Almost always it was the former who had the advantage, as was\nintimated by the gory trophies,--the heads and hands of the vanquished,\nwhich they bore on the points of their lances, when, amidst the shouts\nof the populace, they came thundering on through the gates of the\ncapital.[203]\n\n[Sidenote: IMPETUOUS SPIRIT OF DON JUAN.]\n\nYet sometimes fortune lay in the opposite scale. The bold infidels,\nafter scouring the _vega_, would burst into the suburbs, or even into\nthe city of Granada, filling the place with consternation. Then might be\nseen the terror-stricken citizens hurrying to and fro, while the great\nalarm-bell of the Alhambra sent forth its summons, and the chivalry,\nmounting in haste, shouted the old war-cry of _Saint Jago_, and threw\nthemselves on the invaders, who, after a short but bloody fray, were\nsure to be driven in confusion across the _vega_, and far over the\nborders.\n\nDon John, on these occasions, was always to be descried in the front of\nbattle, as if rejoicing in his element, and courting danger like some\npaladin of romance. Indeed, Philip was obliged, again and again, to\nrebuke his brother for thus wantonly exposing his life, in a manner, the\nking intimated, wholly unbecoming his rank.[204] But it would have been\nas easy to rein in the war-horse when the trumpet was sounding in his\nears, as to curb the spirits of the high-mettled young chieftain when\nhis followers were mustering to the charge. In truth, it was precisely\nthese occasions that filled him with the greatest glee; for they opened\nto him the only glimpses he was allowed of that career of glory for\nwhich his soul had so long panted. Every detachment that sallied forth\nfrom Granada on a warlike adventure was an object of his envy; and as he\ngazed on the blue mountains that rose as an impassable barrier around\nhim, he was like the bird vainly beating its plumage against the gilded\nwires of its prison-house, and longing to be free.\n\nHe wrote to the king in the most earnest terms, representing the forlorn\ncondition of affairs,--the Spaniards losing ground day after day, and\nthe army under the marquis of Los Velez wasting away its energies in\nsloth, or exerting them in unprofitable enterprises. He implored his\nbrother not to compel him to remain thus cooped up within the walls of\nGranada, but to allow him to have a real as well as nominal command, and\nto conduct the war in person.[205]\n\nThe views presented by Don John were warmly supported by Requesens, who\nwrote to Philip, denouncing, in unqualified terms, the incapacity of Los\nVelez.\n\nPhilip had no objection to receive complaints, even against those whom\nhe most favoured. He could not shut his eyes to the truth of the charges\nnow brought against the hot-headed old chief, who had so long enjoyed\nhis confidence, but whose campaigns of late had been a series of\nblunders. He saw the critical aspect of affairs, and the danger that the\nrebellion, which had struck so deep root in Granada, unless speedily\ncrushed, would spread over the adjoining provinces. Mondejar's removal\nfrom the scene of action had not brought the remedy that Philip had\nexpected.\n\nYet it was with reluctance that he yielded to his brother's wishes;\nwhether distrusting the capacity of one so young for an independent\ncommand, or, as might be inferred from his letters, apprehending the\ndangers in which Don John's impetuous spirit would probably involve him.\nHaving formed his plans, he lost no time in communicating them to his\nbrother. The young warrior was to succeed Los Velez in the command of\nthe eastern army, which was to be strengthened by reinforcements, while\nthe duke of Sesa, under the direction of Don John, was to establish\nhimself, with an efficient corps, in the Alpujarras, in such a position\nas to cover the approaches to Granada.\n\nA summons was then sent to the principal towns of Andalusia, requiring\nthem to raise fresh levies for the war, who were to be encouraged by\npromises of better pay than had before been given. But these promises\ndid not weigh so much with the soldiers as the knowledge that Don John\nof Austria was to take charge of the expedition; and nobles and\ncavaliers came thronging to the war, with their well-armed retainers, in\nsuch numbers that the king felt it necessary to publish another\nordinance, prohibiting any, without express permission, from joining the\nservice.[206]\n\nAll now was bustle and excitement in Granada, as the new levies came in,\nand the old ones were receiving a better organization. Indeed, Don John\nhad been closely occupied for some time with introducing reforms among\nthe troops quartered in the city, who, from causes already mentioned,\nhad fallen into a state of the most alarming insubordination. A similar\nspirit had infected the officers, and to such an extent, that it was\ndeemed necessary to suspend no less than thirty-seven out of forty-five\ncaptains from their commands.[207] Such were the difficulties under\nwhich the youthful hero was to enter on his first campaign.\n\nFortunately, in the retainers of the great lords and cavaliers, he had a\nbody of well-appointed and well-disciplined troops, who were actuated by\nhigher motives than the mere love of plunder.[208] His labours,\nmoreover, did much to restore the ancient discipline of the regiments\nquartered in Granada. But the zeal with which he had devoted himself to\nthe work of reform had impaired his health. This drew forth a kind\nremonstrance from Philip, who wrote to his brother not thus to overtask\nhis strength, but to remember that he had need of his services; telling\nhim to remind Quixada that he must watch over him more carefully. \"And\nGod grant,\" he concluded, \"that your health may be soon re-established.\"\nThe affectionate solicitude constantly shown for his brother's welfare\nin the king's letters, was hardly to have been expected in one of so\nphlegmatic a temperament, and who was usually so little demonstrative in\nthe expression of his feelings.\n\nBefore entering on his great expedition, Don John resolved to secure the\nsafety of Granada, in his absence, by the reduction of \"the robber's\nnest,\" as the Spaniards called it, of Guejar. This was a fortified\nplace, near the confines of the Alpujarras, held by a warlike garrison,\nthat frequently sallied out over the neighbouring country, sometimes\ncarrying their forays into the _vega_ of Granada, and causing a panic in\nthe capital. Don John formed his force into two divisions, one of which\nhe gave to the duke of Sesa, while the other he proposed to lead in\nperson. They were to proceed by different routes, and, meeting before\nthe place, to attack it simultaneously from opposite quarters.\n\n[Sidenote: CAPTURE OF GUEJAR.]\n\nThe duke, marching by the most direct road across the mountains,\nreached Guejar first, and was not a little surprised to find that the\ninhabitants, who had received notice of the preparations of the\nSpaniards, were already evacuating the town; while the garrison was\nformed in order of battle to cover their retreat. After a short skirmish\nwith the rear-guard, in which some lives were lost on both sides, the\nvictorious Spaniards, without following up their advantage, marched into\nthe town, and took possession of the works abandoned by the enemy.\n\nGreat was the surprise of Don John, on arriving some hours later before\nGuejar, to see the Castilian flag floating from its ramparts; and his\nindignation was roused as he found that the laurels he had designed for\nhis own brow had been thus unceremoniously snatched from him by another.\n\"With eyes,\" says the chronicler, \"glowing like coals of fire,\"[209] he\nturned on the duke of Sesa, and demanded an explanation of the affair.\nBut he soon found that the blame, if blame there were, was to be laid on\none whom he felt that he had not the power to rebuke. This was Luis\nQuixada, who, in his solicitude for the safety of his ward, had caused\nthe army to be conducted by a circuitous route, that brought it thus\nlate upon the field. But though Don John uttered no word of rebuke, he\nmaintained a moody silence, that plainly showed his vexation; and, as\nthe soldiers remarked, not a morsel of food passed his lips until he had\nreached Granada.[210]\n\nThe constant supervision maintained over him by Quixada, which, as we\nhave seen, was encouraged by the king, was a subject of frequent remark\namong the troops. It must have afforded no little embarrassment and\nmortification to Don John, alike ill-suited, as it was, to his age, his\naspiring temper, and his station. For his station as commander-in-chief\nof the army made him responsible, in the eyes of the world, for the\nmeasures of the campaign. Yet, in his dependent situation, he had the\npower neither to decide on the plan of operations, nor to carry it into\nexecution. Not many days were to elapse before the death of his\nkind-hearted monitor was to relieve him from the jealous oversight that\nso much chafed his spirit, and to open to him an independent career of\nglory, such as might satisfy the utmost cravings of his ambition.\n\n     One of the authorities of the greatest importance, and most\n     frequently cited in this book, as the reader may have noticed, is\n     Diego Hurtado de Mendoza. He belonged to one of the most\n     illustrious houses in Castile--a house not more prominent for its\n     rank than for the great abilities displayed by its members in the\n     various walks of civil and military life, as well as for their rare\n     intellectual culture. No one of the great families of Spain has\n     furnished so fruitful a theme for the pen of both the chronicler\n     and the bard.\n\n     He was the fifth son of the marquis of Mondejar, and was born in\n     the year 1503, at Granada, where his father filled the office held\n     by his ancestors, of captain-general of the province. At an early\n     age he was sent to Salamanca, and passed with credit through the\n     course of studies taught in its venerable university. While there\n     he wrote--for, though printed anonymously, there seems no good\n     reason to distrust the authorship--his famous \"Lazarillo de\n     Tormes,\" the origin of that class of _picaresco_ novels, as they\n     are styled, which constitutes an important branch of Castilian\n     literature, and the best specimen of which, strange to say, was\n     furnished by the hand of a foreigner,--the \"Gil Blas\" of Le Sage.\n\n     Mendoza had been destined to the Church, for which the extensive\n     patronage of his family offered obvious advantages. But the taste\n     of the young man, as might be inferred from his novel, took another\n     direction, and he persuaded his father to allow him to enter the\n     army, and take service under the banner of Charles the Fifth.\n     Mendoza's love of letters did not desert him in the camp; and he\n     availed himself of such intervals as occurred between the\n     campaigns to continue his studies, especially in the ancient\n     languages, in the principal universities of Italy.\n\n     It was impossible that a person of such remarkable endowments as\n     Mendoza, the more conspicuous from his social position, should\n     escape the penetrating eye of Charles the Fifth, who, independently\n     of his scholarship, recognized in the young noble a decided talent\n     for political affairs. In 1538 the emperor appointed him ambassador\n     to Venice, a capital for which the literary enterprises of the Aldi\n     were every day winning a higher reputation in the republic of\n     letters. Here Mendoza had the best opportunity of accomplishing a\n     work which he had much at heart,--the formation of a library. It\n     was a work of no small difficulty in that day, when books and\n     manuscripts were to be gathered from obscure, often remote sources,\n     and at the large cost paid for objects of _virtu_. A good office\n     which he had the means of rendering the sultan, by the redemption\n     from captivity of a Turkish prisoner of rank, was requited by a\n     magnificent present of Greek manuscripts, worth more than gold in\n     the eyes of Mendoza. It was from his collection that the first\n     edition of Josephus was given to the world. While freely indulging\n     his taste for literary occupations in his intervals of leisure, he\n     performed the duties of his mission with an ability that fully\n     vindicated his appointment as minister to the wily republic. On the\n     opening of the Council of Trent, he was one of the delegates sent\n     to represent the emperor in that body. He joined freely in the\n     discussions of the conclave, and enforced the views of his\n     sovereign with a strength of reasoning and a fervid eloquence that\n     produced a powerful impression on his audience. The independence he\n     displayed recommended him for the delicate task of presenting the\n     remonstrances of Charles the Fifth to the papal court against the\n     removal of the council to Bologna. This he did with a degree of\n     frankness to which the pontifical ear was but little accustomed,\n     and which, if it failed to bend the proud spirit of Paul the Third,\n     had its effect on his successor.\n\n     Mendoza, from whatever cause, does not seem to have stood so high\n     in the favour of Philip the Second as in that of his father.\n     Perhaps he had too lofty a nature to stoop to that implicit\n     deference which Philip exacted from the highest as well as the\n     humblest who approached him. At length, in 1568, Mendoza's own\n     misconduct brought him, with good reason, into disgrace with his\n     master. He engaged in a brawl with another courtier in the palace;\n     and the scandalous scene, of which the reader will find an account\n     in the preceding volume, took place when the prince of Asturias,\n     Don Carlos, was breathing his last. The offending parties were\n     punished first by imprisonment, and then by banishment from Madrid.\n     Mendoza, who was sixty-five years of age at this time, withdrew to\n     Granada, his native place. But he had passed too much of his life\n     in the atmosphere of a court to be content with a provincial\n     residence. He accordingly made repeated efforts to soften his\n     sovereign's displeasure, and to obtain some mitigation of his\n     sentence. These efforts, as may be believed, were unavailing; and\n     the illustrious exile took at length the wiser course of submitting\n     to his fate and seeking consolation in the companionship of his\n     books,--steady friends, whose worth he now fully proved in the hour\n     of adversity. He devoted himself to the study of Arabic, to which\n     he was naturally led by his residence in a capital filled with the\n     monuments of Arabic art. He also amused his leisure by writing\n     verses, and his labours combined with those of Boscan and\n     Garcilasso de la Vega to naturalize in Castile those more refined\n     forms of Italian versification that made an important epoch in the\n     national literature.\n\n     But the great work to which he devoted himself was the history of\n     the insurrection of the Moriscoes, which, occurring during his\n     residence in Granada, may be said to have passed before his eyes.\n     For this he had, moreover, obvious facilities, for he was the near\n     kinsman of the captain-general, and was personally acquainted with\n     those who had the direction of affairs. The result of his labours\n     was a work of inestimable value, though of no great bulk--being\n     less a history of events than a commentary on such a history. The\n     author explores the causes of these events. He introduces the\n     reader into the cabinet of Madrid, makes him acquainted with the\n     intrigues of the different factions, both in the court and in the\n     camp, unfolds the policy of the government and the plans of the\n     campaigns--in short, enables him to penetrate into the interior,\n     and see the secret working of the machinery, so carefully shrouded\n     from the vulgar eye.\n\n     The value which the work derived from the author's access to these\n     recondite sources of information is much enhanced by its\n     independent spirit. In a country where few dared even think for\n     themselves, Mendoza both thought with freedom and freely expressed\n     his thoughts. Proof of this is afforded by the caustic tone of his\n     criticism on the conduct of the government, and by the candour\n     which he sometimes ventures to display when noticing the wrongs of\n     the Moriscoes. This independence of the historian, we may well\n     believe, could have found little favour with the administration.\n     It may have been the cause that the book was not published till\n     after the reign of Philip the Second, and many years after its\n     author's death.\n\n     [Sidenote: MENDOZA.]\n\n     The literary execution of the work is not its least remarkable\n     feature. Instead of the desultory and gossiping style of the\n     Castilian chronicler, every page is instinct with the spirit of the\n     ancient classics. Indeed, Mendoza is commonly thought to have\n     deliberately formed his style on that of Sallust; but I agree with\n     my friend Mr. Ticknor, who, in a luminous criticism on Mendoza, in\n     his great work on Spanish Literature, expresses the opinion that\n     the Castilian historian formed his style quite as much on that of\n     Tacitus as of Sallust. Indeed, some of Mendoza's most celebrated\n     passages are obvious imitations of the former historian, of whom he\n     constantly reminds us by the singular compactness and energy of his\n     diction, by his power of delineating a portrait by a single stroke\n     of the pencil, and by his free criticism on the chief actors of the\n     drama, conveyed in language full of that practical wisdom which, in\n     Mendoza's case, was the result of a large acquaintance with public\n     affairs. We recognize also the defects incident to the style he has\n     chosen--rigidity and constraint, with a frequent use of ellipsis,\n     in a way that does violence to the national idiom, and, worst of\n     all, that obscurity which arises from the effort to be brief.\n     Mendoza hurts his book, moreover, by an unseasonable display of\n     learning, which, however it may be pardoned by the antiquary, comes\n     like an impertinent episode to break the thread of the narrative.\n     But, with all its defects, the work is a remarkable production for\n     the time, and, appearing in the midst of the _romantic_ literature\n     of Spain, we regard it with the same feeling of surprise which the\n     traveller might experience who should meet with a classic Doric\n     temple in the midst of the fantastic structures of China or\n     Hindostan.\n\n     Not long after Mendoza had completed his history, he obtained\n     permission to visit Madrid, not to reside there, but to attend to\n     some personal affairs. He had hardly reached the capital when he\n     was attacked by a mortal illness, which carried him off in April,\n     1575, in the seventy-third year of his age. Shortly before his\n     death he gave his rich collection of books and manuscripts to his\n     obdurate master, who placed them, agreeably to the donor's desire,\n     in the Escorial, where they still form an interesting portion of a\n     library of which so much has been said, and so little is really\n     known by the world.\n\n     The most copious notice with which I am acquainted, of the life of\n     Mendoza, is that attributed to the pen of Inigo Lopez de Avila, and\n     prefixed to the Valencian edition of the \"Guerra de Granada,\"\n     published in 1776. But his countrymen have been ever ready to do\n     honour to the memory of one who, by the brilliant success which he\n     achieved as a statesman, a diplomatist, a novelist, a poet, and an\n     historian, has established a reputation for versatility of genius\n     second to none in the literature of Spain.\n\n\n\n\nCHAPTER VII.\n\nREBELLION OF THE MORISCOES.\n\nDon John takes the Field--Investment of Galera--Fierce\nAssaults--Preparations for a last Attack--Explosion of the\nMines--Desperation of the Moriscoes--Cruel Massacre--Galera demolished.\n\n1570.\n\n\nDon John lost no time in completing the arrangements for his expedition.\nThe troops, as they reached Granada, were for the most part sent forward\nto join the army under Los Velez, on the east of the Alpujarras, where\nthat commander was occupied with the siege of Galera, though with but\nlittle prospect of reducing the place. He was soon, however, to be\nsuperseded by Don John.\n\nPhilip, unable to close his ears against the representations of his\nbrother, as well as those of more experienced captains in the service,\nhad at length reluctantly come to a conviction of the unfitness of Los\nVelez for the command. Yet he had a partiality for the veteran; and he\nwas willing to spare him, as far as possible, the mortification of\nseeing himself supplanted by his young rival. In his letters, the king\nrepeatedly enjoined it on his brother to treat the marquis with the\nutmost deference, and to countenance no reports circulated to his\nprejudice. In an epistle filled with instructions for the campaign,\ndated the twenty-sixth of November, the king told Don John to be\ndirected on all occasions by the counsels of Quixada and Requesens. He\nwas to show the greatest respect for the marquis, and to give him to\nunderstand that he should be governed by his opinions. \"But, in point of\nfact,\" said Philip, \"should his opinion clash at any time with that of\nthe two other counsellors, you are to be governed by theirs.\"[211]\n\nOn Quixada and Requesens he was indeed always to rely, never setting up\nhis own judgment in opposition to theirs. He was to move with caution,\nand, instead of the impatient spirit of a boy, to show the\ncircumspection of one possessed of military experience. \"In this way,\"\nconcluded his royal monitor, \"you will not only secure the favour of\nyour sovereign, but establish your reputation with the world.\"[212] It\nis evident that Philip had discerned traits in the character of Don John\nwhich led him to distrust somewhat his capacity for the high station in\nwhich he was placed. Perhaps it may be thought that the hesitating and\ntimid policy of Philip was less favourable to success in military\noperations than the bold spirit of enterprise which belonged to his\nbrother. However this may be, Don John, notwithstanding his repeated\nprotestations to the contrary, was of too ardent a temperament to be\nreadily affected by these admonitions of his prudent adviser.\n\nThe military command in Granada was lodged by the prince in the hands of\nthe duke of Sesa, who, as soon as he had gathered a sufficient force,\nwas to march into the western district of the Alpujarras, and there\ncreate a diversion in favour of Don John. A body of four thousand troops\nwas to remain in Granada; and the commander-in-chief, having thus\ncompleted his dispositions for the protection of the capital, set forth\non his expedition on the twenty-ninth of December, at the head of a\nforce amounting only to three thousand foot and four hundred horse. With\nthese troops went a numerous body of volunteers, the flower of the\nAndalusian chivalry, who had come to win renown under the banner of the\nyoung leader.\n\nHe took the route through Guadix, and on the third day reached the\nancient city of Baza, memorable for the siege it had sustained under his\nvictorious ancestors, Ferdinand and Isabella. Here he was met by\nRequesens, who, besides a reinforcement of troops, brought with him a\ntrain of heavy ordnance and a large supply of ammunition. The guns were\nsent forward, under a strong escort, to Galera; but, on leaving Baza,\nDon John received the astounding tidings that the marquis of Los Velez\nhad already abandoned the siege, and drawn off his whole force to the\nneighbouring town of Guescar.\n\n[Sidenote: LOS VELEZ RESIGNS HIS COMMAND.]\n\nIn fact, the rumour had no sooner reached the ears of the testy old\nchief, that Don John was speedily coming to take charge of the war, than\nhe swore in his wrath that if the report were true, he would abandon the\nsiege and throw up his command. Yet those who knew him best did not\nthink him capable of so mad an act. He kept his word, however; and when\nhe learned that Don John was on the way, he broke up his encampment and\nwithdrew, as above stated, to Guescar. By this course he left the\nadjacent country open to the incursions of the Moriscoes of Galera;\nwhile no care was taken to provide even for the safety of the convoys\nwhich, from time to time, came laden with supplies for the besieging\narmy.\n\nThis extraordinary conduct gave no dissatisfaction to his troops, who,\nlong since disgusted with the fiery yet imbecile character of their\ngeneral, looked with pleasure to the prospect of joining the standard of\nso popular a chieftain as John of Austria. Even the indignation felt by\nthe latter at the senseless proceeding of the marquis was forgotten in\nthe satisfaction he experienced, at being thus relieved from the\nembarrassments which his rival's overweening pretensions could not have\nfailed to cause him in the campaign. Don John might now, with a good\ngrace, and without any cost to himself, make all the concessions to the\nveteran so strenuously demanded by Philip. It was in this amiable mood\nthat the prince pushed forward his march, eager to prevent the\ndisastrous consequences which might arise from the marquis's abandonment\nof his post.\n\nAs he drew near to Guescar, he beheld the old nobleman riding towards\nhim at the head of his retainers, with a stiff and stately port, like\none who had no concessions or explanations to make for himself. Without\nalighting from his horse, as he drew near the prince, he tendered him\nobeisance by kissing the hand which the latter graciously extended\ntowards him. \"Noble marquis,\" said Don John, \"your great deeds have shed\na lustre over your name. I consider myself fortunate in having the\nopportunity of becoming personally acquainted with you. Fear not that\nyour authority will be in the least abridged by mine. The soldiers under\nmy command will obey you as implicitly as myself. I pray you to look on\nme as a son, filled with feelings of reverence for your valour and your\nexperience, and designing on all occasions to lean on your counsels for\nsupport.\"[213]\n\nThe courteous and respectful tone of the prince seems to have had its\neffect on the iron nature of the marquis, as he replied, \"There is no\nSpaniard living who has a stronger desire than I have to be personally\nacquainted with the distinguished brother of my sovereign, or who would\nprobably be a greater gainer by serving under his banner. But to speak\nwith my usual plainness, I wish to withdraw to my own house; for it\nwould never do for me, old as I am, to hold the post of a\nsubaltern.\"[214] He then accompanied Don John back to the town, giving\nhim, as they rode along, some account of the siege and of the strength\nof the place. On reaching the quarters reserved for the\ncommander-in-chief, Los Velez took leave of the prince; and, without\nfurther ceremony, gathering his knights and followers about him, and\nescorted by a company of horse, he rode off in the direction of his town\nof Velez Blanco, which was situated at no great distance, amidst the\nwild scenery stretching toward the frontiers of Murcia. Here among the\nmountains he lived in a retirement that would have been more honourable\nhad it not been purchased by so flagrant a breach of duty.[215]\n\nThe whole story is singularly characteristic, not merely of the man, but\nof the times in which he lived. Had so high-handed and audacious a\nproceeding occurred in our day, no rank, however exalted, could have\nscreened the offender from punishment. As it was, it does not appear\nthat any attempt was made at an inquiry into the marquis's conduct. This\nis the more remarkable, considering that it involved such disrespect to\na sovereign little disposed to treat with lenity any want of deference\nto himself. The explanation of the lenity shown by him on the present\noccasion may perhaps be found, not in any tenderness for the reputation\nof his favorite, but in Philip's perceiving that the further prosecution\nof the affair would only serve to give greater publicity to his own\negregious error in retaining Los Velez in the command, when his conduct\nand the warnings of others should long ago have been regarded as proof\nof his incapacity.\n\nOn the marquis's departure, Don John lost no time in resuming his march\nat the head of a force which now amounted to twelve thousand foot and\neight hundred horse, besides a brilliant array of chivalry, who, as we\nhave seen, had come to seek their fortunes in the war. A few hours\nbrought the troops before Galera; and Don John proceeded at once to\nreconnoitre the ground. In this survey he was attended by Quixada,\nRequesens, and the greater part of the cavalry. Having completed his\nobservations, he made his arrangements for investing the place.\n\nThe town of Galera occupied a site singularly picturesque. This,\nhowever, had been selected, certainly not from any regard to its\nromantic beauty, still less for purposes of convenience, but for those\nof defence against an enemy,--a circumstance of the first importance in\na mountain country so wild and warlike as that in which Galera stood.\nThe singular shape of the rocky eminence which it covered was supposed,\nwith its convex summit, to bear some resemblance to that of a galley\nwith its keel uppermost. From this resemblance the town had derived its\nname.[216]\n\nThe summit was crowned by a castle, which in the style of its\narchitecture bore evident marks of antiquity. It was defended by a wall,\nmuch of it in so ruinous a condition as to be little better than a mass\nof stones loosely put together. At a few paces from the fortress stood a\nravelin. But neither this outwork nor the castle itself could boast of\nany other piece of artillery than two falconets, captured from Los Velez\nduring his recent siege of the place, and now mounted on the principal\nedifice. Even these had been so injudiciously placed as to give little\nannoyance to an enemy.\n\nThe houses of the inhabitants stretched along the remainder of the\nsummit, and descended by a bold declivity the north-western side of the\nhill to a broad plain known as the _Eras_, or \"Gardens.\" Through this\nplain flowed a stream of considerable depth, which, as it washed the\nbase of the town on its northern side, formed a sort of moat for its\nprotection on that quarter. On the side towards the Gardens, the town\nwas defended by a ditch and a wall now somewhat dilapidated. The most\nremarkable feature of this quarter was a church with its belfry or\ntower, now converted into a fortress, which, in default of cannon, had\nbeen pierced with loopholes and filled with musketeers,--forming\naltogether an outwork of considerable strength, and commanding the\napproaches to the town.\n\n[Sidenote: INVESTMENT OF GALERA.]\n\nOn two of its sides, the rock on which Galera rested descended almost\nperpendicularly, forming the walls of a ravine fenced in on the opposite\nquarter by precipitous hills, and thus presenting a sort of natural\nditch on a gigantic scale for the protection of the place. The houses\nrose one above another, on a succession of terraces, so steep that in\nmany instances the roof of one building scarcely reached the foundation\nof the one above it. The houses which occupied the same terrace, and\nstood therefore on the same level, might be regarded as so many\nfortresses. Their walls, which, after the Moorish fashion, were\nill-provided with lattices, were pierced with loopholes, that gave the\nmarksmen within the command of the streets on which they fronted; and\nthese streets were still further protected by barricades thrown across\nthem at only fifty paces' distance from each other.[217] Thus the whole\nplace bristled over with fortifications, or rather seemed like one great\nfortification itself, which nature had combined with art to make\nimpregnable.\n\nIt was well victualled for a siege, at least with grain, of which there\nwas enough in the magazines for two years' consumption. Water was\nsupplied by the neighbouring river, to which access had been obtained by\na subterranean gallery, lately excavated in the rock. These necessaries\nof life the Moriscoes could command. But they were miserably deficient\nin what, in their condition, was scarcely less important,--fire-arms and\nammunition. They had no artillery except the two falconets before\nnoticed; and they were so poorly provided with muskets as to be mainly\ndependent on arrows, stones, and other missiles, such as had filled the\narmories of their ancestors. To these might be added swords, and some\nother weapons for hand-to-hand combat. Of defensive armour they were\nalmost wholly destitute. But they were animated by an heroic spirit, of\nmore worth than breastplate or helmet, and to a man they were prepared\nto die rather than surrender.\n\nThe fighting men of the place amounted to three thousand, not including\nfour hundred mercenaries, chiefly Turks and adventurers from the Barbary\nshore. The town was, moreover, encumbered with some four thousand women\nand children; though, as far as the women were concerned, they should\nnot be termed an incumbrance in a place where there was no scarcity of\nfood; for they showed all the constancy and contempt of danger possessed\nby the men, whom they aided not only by tending the sick and wounded,\nbut by the efficient services they rendered them in action. The story of\nthis siege records several examples of these Morisco heroines, whose\nferocious valour emulated the doughtiest achievements of the other sex.\nIt is not strange that a place so strong in itself, where the women were\nanimated by as brave a spirit as the men, should have bid defiance to\nall the efforts of an enemy like Los Velez, though backed by an army in\nthe outset at least as formidable in point of numbers as that which now\nsat down before it under the command of John of Austria.[218]\n\nHaving concluded his survey of the ground, the Spanish general gave\norders for the construction of three batteries, to operate at the same\ntime on different quarters of the town. The first and largest of these\nbatteries, mounting ten pieces of ordnance, was raised on an eminence on\nthe eastern side of the ravine. Though at a greater distance than was\ndesirable, the position was sufficiently elevated to enable the guns to\ncommand the castle and the highest parts of the town.\n\nThe second battery, consisting of six heavy cannon, was established\nlower down the ravine, towards the south, at the distance of hardly more\nthan seventy paces from the perpendicular face of the rock. The\nremaining battery, composed of only three guns of smaller calibre, was\nerected in the Gardens, and so placed as to operate against the tower\nwhich, as already noticed, was attached to the church.\n\nThe whole number of pieces of artillery belonging to the besiegers did\nnot exceed twenty. But they were hourly expecting a reinforcement of\nthirteen more from Cartagena. The great body of the forces was disposed\nbehind some high ground on the east, which effectually sheltered the men\nfrom the fire of the besieged. The corps of Italian veterans, the flower\nof the army, was stationed in the Gardens, under command of a gallant\nofficer named Pedro de Padilla. Thus the investment of Galera was\ncomplete.\n\nThe first object of attack was the tower in the Gardens, from which the\nMoorish garrison kept up a teasing fire on the Spaniards, as they were\nemployed in the construction of the battery, as well as in digging a\ntrench, in that quarter. No sooner were the guns in position than they\ndelivered their fire, with such effect that an opening was speedily made\nin the flimsy masonry of the fortress. Padilla, to whom the assault was\ncommitted, led forward his men gallantly to the breach, where he was met\nby the defenders with a spirit equal to his own. A fierce combat ensued.\nIt was not a long one; for the foremost assailants were soon reinforced\nby others, until they overpowered the little garrison by numbers, and\nsuch as escaped the sword took refuge in the defences of the town that\nadjoined the church.\n\nFlushed with his success in thus easily carrying the tower, which he\ngarrisoned with a strong body of arquebusiers, Don John now determined\nto make a regular assault on the town, and from this same quarter of the\nGardens, as affording the best point of attack. The execution of the\naffair he entrusted, as before, to Juan de Padilla and his Italian\nregiment. The guns were then turned against the rampart and the\nadjoining buildings. Don John pushed forward the siege with vigour,\nstimulating the men by his own example, carrying fagots on his shoulders\nfor constructing the trenches, and, in short, performing the labours of\na common soldier.[219]\n\nBy the twenty-fourth of January, practicable breaches had been effected\nin the ancient wall; and at the appointed signal, Padilla and his\nveterans moved swiftly forward to the attack. They met with little\ndifficulty from the ditch or from the wall, which, never formidable from\nits height, now presented more than one opening to the assailants. They\nexperienced as little resistance from the garrison. But they had not\npenetrated far into the town before the aspect of things changed. Their\nprogress was checked by one of those barricades already mentioned as\nstretched across the streets, behind which a body of musketeers poured\nwell-directed volleys into the ranks of the Christians. At the same\ntime, from the loopholes in the walls of the buildings, came incessant\nshowers of musket-balls, arrows, stones, and other missiles, which swept\nthe exposed files of the Spaniards, soon covering the streets with the\nbodies of the slain and the wounded. It was in vain that the assailants\nstormed the houses, and carried one entrenchment after another. Each\nhouse was a separate fortress; and each succeeding barricade, as the\nascent became steeper, gave additional advantage to its defenders, by\nplacing them on a greater elevation above their enemy.\n\n[Sidenote: FIERCE ASSAULTS.]\n\nThus beset in front, flank, and rear, the soldiers were completely\nblinded and bewildered by the pitiless storm which poured on them from\ntheir invisible foe. Huddled together, in their confusion they presented\nan easy mark to the enemy, who shot at random, knowing that every\nmissile would carry its errand of death. It seemed that the besieged had\npurposely drawn their foes into the snare, by allowing them to enter the\ntown without resistance, until, hemmed in on all sides, they were\nslaughtered like cattle in the shambles.\n\nThe fight had lasted an hour, when Padilla, seeing his best and bravest\nfalling around him, and being himself nearly disabled by a wound, gave\nthe order to retreat; an order obeyed with such alacrity, that the\nSpaniards left numbers of their wounded comrades lying in the street,\nvainly imploring not to be abandoned to the mercy of their enemies. A\ngreater number than usual of officers and men of rank perished in the\nassault, their rich arms making them a conspicuous mark amidst the\nthrong of assailants. Among others was a soldier of distinction named\nJuan de Pacheco. He was a knight of the order of St. James. He had\njoined the army only a few minutes before the attack, having just\ncrossed the seas from Africa. He at once requested Padilla, who was his\nkinsman, to allow him to share in the glory of the day. In the heat of\nthe struggle, Padilla lost sight of his gallant relative, whose\ninsignia, proclaiming him a soldier of the Cross, made him a peculiar\nobject of detestation to the Moslems; and he soon fell, under a\nmultitude of wounds.[220]\n\nThe disasters of the day, however mortifying, were not a bad lesson to\nthe young commander-in-chief, who saw the necessity of more careful\npreparation before renewing his attempt on the place. He acknowledged\nthe value of his brother's counsel, to make free use of artillery and\nmines before coming to close quarters with the enemy.[221] He determined\nto open a mine in the perpendicular side of the rock, towards the east,\nand to run it below the castle and the neighbouring houses on the\nsummit. For this he employed the services of Francesco de Molina, who\nhad so stoutly defended Orgiba, and who was aided in the present work by\na skilful Venetian engineer. The rock, consisting of a light and brittle\nsandstone, was worked with even less difficulty than had been expected.\nIn a short time the gallery was completed, and forty-five barrels of\npowder were lodged in it. Meanwhile the batteries continued to play with\ngreat vivacity on the different quarters of the town and castle. A small\nbreach was opened in the latter, and many buildings on the summit of the\nrock were overthrown. By the twenty-seventh of January all was ready for\nthe assault.\n\nIt was Don John's purpose to assail the place on opposite quarters.\nPadilla, who still smarted from his wound, was to attack the town, as\nbefore, on the side towards the Gardens. The chief object of this\nmanoeuvre was to create a diversion in favour of the principal\nassault, which was to be made on the other side of the rock, where the\nspringing of the mine, it was expected, would open a ready access to the\ncastle. The command on this quarter was given to a brave officer named\nAntonio Moreno. Don John, at the head of four thousand men, occupied a\nposition which enabled him to overlook the scene of action.\n\nOn the twenty-seventh, at eight in the morning, the signal was given by\nthe firing of a cannon; and Padilla, at the head of his veterans, moved\nforward to the attack. They effected their entrance into the town with\neven less opposition than before; for the cannonade from the Gardens had\nblown away most of the houses, garrisoned by the Moslems, near the wall.\nBut as the assailants pushed on, they soon became entangled, as before,\nin the long and narrow defiles. The enemy, entrenched behind their\nredoubts thrown across the streets, poured down their murderous volleys\ninto the close ranks of the Spaniards, who were overwhelmed, as on the\nformer occasion, with deadly missiles of all kinds from the occupants of\nthe houses. But experience had prepared them for this; and they had come\nprovided with mantelets, to shelter them from the tempest. Yet, when the\nannoyance became intolerable, they would storm the dwellings; and a\nbloody struggle usually ended in putting their inmates to the sword.\nEach barricade, too, as the Spaniards advanced, became the scene of a\ndesperate combat, where the musket was cast aside, and men fought hand\nto hand with sword and dagger. Now rose the fierce battle-cries of the\ncombatants, one party calling on St. Jago, the other on Mahomet, thus\nintimating that it was still the same war of the Cross and the Crescent\nwhich had been carried on for more than eight centuries in the\nPeninsula.[222] The shouts of the combatants, the clash of weapons, the\nreport of musketry from the adjoining houses, the sounds of falling\nmissiles, filled the air with an unearthly din, that was reverberated\nand prolonged in countless echoes through the narrow streets, converting\nthe once peaceful city into a Pandemonium. Still the Spaniards, though\nslowly winning their way through every obstacle, were far from the\ntable-land on the summit, where they hoped to join their countrymen from\nthe other quarter of the town. At this crisis a sound arose which\noverpowered every other sound in this wild uproar, and for a few moments\nsuspended the conflict.\n\nThis was the bursting of the mine, which Don John, seeing Padilla well\nadvanced in his assault, had now given the order to fire. In an instant\ncame the terrible explosion, shaking Galera to its centre, rending the\nportion of the rock above the gallery into fragments, toppling down the\nhouses on its summit, and burying more than six hundred Moriscoes in the\nruins. As the smoke and dust of the falling buildings cleared away, and\nthe Spaniards from below beheld the miserable survivors crawling forth,\nas well as their mangled limbs would allow, they set up a fierce yell of\ntriumph. The mine, however, had done but half the mischief intended; for\nby a miscalculation in the direction, it had passed somewhat to the\nright of the castle, which, as well as the ravelin, remained uninjured.\nYet a small breach had been opened by the artillery in the former; and\nwhat was more important, through the shattered sides of the rock itself\na passage had been made, which, though strewn with the fallen rubbish,\nmight afford a practicable entrance to the storming party.\n\n[Sidenote: FIERCE ASSAULTS.]\n\nThe soldiers, seeing the chasm, now loudly called to be led to the\nassault. Besides the thirst for vengeance on the rebels who had so long\nset them at defiance, they were stimulated by the desire of plunder; for\nGalera, from its great strength, had been selected as a place of deposit\nfor the jewels, rich stuffs, and other articles of value belonging to\nthe people in the neighbourhood. The officers, before making the attack,\nwere anxious to examine the breach and have the rubbish cleared away, so\nas to make the ascent easier for the troops. But the fierce and\nill-disciplined levies were too impatient for this. Without heeding the\ncommands or remonstrances of their leaders, one after another they broke\ntheir ranks, and, crying the old national war-cries, \"_San Jago!_\"\n\"_Cierra Espana!_\" \"St. James!\" and \"Close up Spain!\" they rushed madly\nforward, and, springing lightly over the ruins in their pathway, soon\nplanted themselves on the summit. The officers, thus deserted, were not\nlong in following, resolved to avail themselves of the enthusiasm of the\nmen.\n\nFortunately the Moriscoes, astounded by the explosion, had taken refuge\nin the town, and thus left undefended a position which might have given\ngreat annoyance to the Spaniards. Yet the cry no sooner rose, that the\nenemy had scaled the heights, than, recovering from their panic, they\nhurried back to man the defences. When the assailants, therefore, had\nbeen brought into order and formed into column for the attack, they were\nreceived with a well-directed fire from the falconets, and with volleys\nof musketry from the ravelin, that for a moment checked their advance.\nBut then rallying, they gallantly pushed forward through the fiery\nsleet, and soon found themselves in face of the breach which had been\nmade in the castle by their artillery. The opening, scarcely wide enough\nto allow two to pass abreast, was defended by men as strong and\nstout-hearted as their assailants. A desperate struggle ensued, in which\nthe besieged bravely held their ground, though a Castilian ensign, named\nZapata, succeeded in forcing his way into the place, and even in\nplanting his standard on the battlements. But it was speedily torn down\nby the enemy, while the brave cavalier, pierced with wounds, was thrown\nheadlong on the rocky ground below, still clutching the standard with\nhis dying grasp.\n\nMeanwhile the defenders of the ravelin kept up a plunging fire of\nmusketry on the assailants; while stones, arrows, javelins, fell thick\nas rain-drops on their heads, rattling on the harness of the cavaliers,\nand inflicting many a wound on the ill-protected bodies of the soldiery.\nThe Morisco women bore a brave part in the fight, showing the same\nindifference to danger as their husbands and brothers, and rolling down\nheavy weights on the ranks of the besiegers. These women had a sort of\nmilitary organization, being formed into companies. Sometimes they even\njoined in hand-to-hand combats with their enemies, wielding their swords\nand displaying a prowess worthy of the stronger sex. One of these\nAmazons, whose name became famous in the siege, was seen on this\noccasion to kill her antagonist, and bear away his armour as the spoils\nof victory. It was said that, before she received her mortal wound,\nseveral Spaniards fell by her hand.[223]\n\nThus, while the besieged, secure within their defences, suffered\ncomparatively little, the attacking column was thrown into disorder.\nMost of its leaders were killed or wounded. Its ranks were thinned by\nthe incessant fire from the ravelin and castle; and, though it still\nmaintained a brave spirit, its strength was fast ebbing away. Don John,\nwho from his commanding position had watched the field, saw the\nnecessity of sending to the support of his troops six companies of the\nreserve, which were soon followed by two others. Thus reinforced, they\nwere enabled to keep their ground.\n\nMeanwhile the Italian regiment under Padilla had penetrated far into the\ntown. But they had won their way inch by inch, and it had cost them\ndear. There was not an officer, it was said, that had not been wounded.\nFour captains had fallen. Padilla, who had not recovered from his former\nwound, had now received another, still more severe. His men, though\nshowing a bold front, had been so roughly handled, that it was clear\nthey could never fight through the obstacles in their way, and join\ntheir comrades on the heights. While little mindful of his own wounds,\nPadilla saw with anguish the blood of his brave followers thus poured\nout in vain; and, however reluctantly, he gave the order to retreat.\nThis command was the signal for a fresh storm of missiles from the\nenemy. But the veterans of Naples, closing up their ranks as a comrade\nfell, effected their retreat in the same cool and orderly manner in\nwhich they had advanced, and, though wofully crippled, regained their\nposition in the trenches.\n\nThus disengaged from the conflict on this quarter, the victorious\nMoslems hastened to the support of their countrymen in the castle, where\nthey served to counterbalance the reinforcement received by the\nassailants. They fell at once on the rear of the Christians, whose front\nranks were galled by the guns from the enemy's battery--though clumsily\nserved--while their flanks were sorely scathed by the storm of musketry\nthat swept down from the ravelin. Thus hemmed in on all sides, they were\nindeed in a perilous situation. Several of the captains were killed. All\nthe officers were either killed or wounded; and the narrow ground on\nwhich they struggled for mastery was heaped with the bodies of the\nslain. Yet their spirits were not broken; and the tide of battle, after\nthree hours' duration, still continued to rage with impotent fury around\nthe fortress. They still strove, with desperate energy, to scale the\nwalls of the ravelin, and to force a way through the narrow breach in\nthe castle. But the besieged succeeded in closing up the opening with\nheavy masses of stone and timber, which defied the failing strength of\nthe assailants.\n\nAnother hour had now elapsed, and Don John, as from his station he\nwatched the current of the fight, saw that to prolong the contest would\nonly be to bring wider ruin on his followers. He accordingly gave the\norder to retreat. But the men who had so impetuously rushed to the\nattack, in defiance of the commands of their officers, now showed the\nsame spirit of insubordination when commanded to leave it; like the\nmastiff who, maddened by the wounds he has received in the conflict,\nrefuses to loosen his hold on his antagonist, in spite of the chiding of\nhis master. Seeing his orders thus unheeded, Don John, accompanied by\nhis staff, resolved to go in person to the scene of action, and enforce\nobedience by his presence. But on reaching the spot, he was hit on his\ncuirass by a musket-ball, which, although it glanced from the\nwell-tempered metal, came with sufficient force to bring him to the\nground. The watchful Quixada, not far distant, sprang to his aid; but it\nappeared he had received no injury. His conduct, however, brought down\nan affectionate remonstrance from his guardian, who, reminding him of\nthe king's injunctions besought him to retire, and not thus expose a\nlife so precious as that of the commander-in-chief to the hazards of a\ncommon soldier.\n\nThe account of the accident soon spread, with the usual exaggerations,\namong the troops, who, after the prince's departure, yielded a slow and\nsullen obedience to his commands. Thus for a second time the field of\nbattle remained in possession of the Moslems; and the banner of the\ncrescent still waved triumphantly from the battlements of Galera.[224]\n\n[Sidenote: PREPARATIONS FOR A LAST ATTACK.]\n\nThe loss was a heavy one to the Spaniards, amounting, according to their\nown accounts--which will not be suspected of exaggeration--to not less\nthan four hundred killed and five hundred wounded. That of the enemy,\nscreened by his defences, must have been comparatively light. The loss\nfell most severely on the Spanish chivalry, whose showy dress naturally\ndrew the attention of the well-trained Morisco marksmen. The bloody roll\nis inscribed with the names of many a noble house in both Andalusia and\nCastile.\n\nThis second reverse of his arms stung Don John to the quick. The eyes of\nhis countrymen were upon him; and he well knew the sanguine\nanticipations they had formed of his campaign, and that they would hold\nhim responsible for its success. His heart was filled with mourning for\nthe loss of his brave companions in arms. Yet he did not give vent to\nunmanly lamentation; but he showed his feelings in another form, which\ndid little honour to his heart. Turning to his officers, he exclaimed:\n\"The infidels shall pay dear for the Christian blood they have spilt\nthis day. The next assault will place Galera in our power; and every\nsoul within its walls--man, woman, and child--shall be put to the sword.\nNot one shall be spared. The houses shall be razed to the ground, and\nthe ground they covered shall be sown with salt.\"[225] This inhuman\nspeech was received with general acclamations. As the event proved, it\nwas not an empty menace.\n\nThe result of his operations showed Don John the prudence of his\nbrother's recommendation,--to make good use of his batteries and his\nmines before coming to close quarters with the enemy. Philip, in a\nletter written some time after this defeat, alluding to the low state of\ndiscipline in the camp, urged his brother to give greater attention to\nthe morals of the soldiers,--to guard especially against profanity and\nother offences to religion, that by so doing he might secure the favour\nof the Almighty.[226] Don John had intimated to Philip, that, under some\ncircumstances, it might be necessary to encourage his men by leading\nthem in person to the attack. But the king rebuked the spirit of the\nknight-errant, as not suited to the commander, and admonished his\nbrother that the place for him was in the rear; that there he might be\nof service in stimulating the ardour of the remiss; adding, that those\nwho went forward promptly in the fight, had no need of his presence to\nencourage them.[227]\n\nDon John lost no time in making his preparations for a third and last\nassault. He caused two new mines to be opened in the rock on either side\nof the former one, and at some thirty paces' distance from it. While\nthis was going on, he directed that all the artillery should play\nwithout intermission on the town and castle. His battering-train,\nmeantime, was reinforced by the arrival of fourteen additional pieces of\nheavy ordnance from Cartagena.\n\nThe besieged were no less busy in preparing for their defence. The women\nand children toiled equally with the men in repairing the damages in the\nworks. The breaches were closed with heavy stones and timber. The old\nbarricades were strengthened, and new ones thrown across the streets.\nThe magazines were filled with fresh supplies of stones and arrows. Long\npractice had made the former missile a more formidable weapon than usual\nin the hands of the Moriscoes. They were amply provided with water, and,\nas we have seen, were well victualled for a siege longer than this was\nlikely to prove. But, in one respect, and that of the last importance,\nthey were miserably deficient. Their powder was nearly all expended.\nThey endeavoured to obtain supplies of ammunition, as well as\nreinforcements of men, from Aben-Aboo. But the Morisco prince was fully\noccupied at this time with maintaining his ground against the duke of\nSesa, in the west. His general, El Habaqui, who had charge of the\neastern army, encouraged the people of Galera to remain firm, assuring\nthem that before long he should be able to come to their assistance. But\ntime was precious to the besieged.[228]\n\nThe Turkish auxiliaries in the garrison greatly doubted the possibility\nof maintaining themselves, with no better ammunition than stones and\narrows, against the well-served artillery of the Spaniards. Their\nleaders accordingly, in a council of war, proposed that the troops\nshould sally forth and cut their way through the lines of the besiegers,\nwhile the women and children might pass out by the subterranean avenue\nwhich conducted to the river, the existence of which, we are told, was\nunknown to the Christians. The Turks, mere soldiers of fortune, had no\nlocal attachment or patriotic feeling to bind them to the soil. But when\ntheir proposal was laid before the inhabitants, they all, women as well\nas men, treated the proposition with disdain, showing their\ndetermination to defend the city to the last, and to perish amidst its\nruins rather than surrender.\n\nStill sustained by the hope of succour, the besieged did what they could\nto keep off the day of the assault. They did not, indeed, attempt to\ncounter-mine; for, if they had possessed the skill for this, they had\nneither tools nor powder. But they had made sorties on the miners, and,\nthough always repulsed with loss, they contrived to hold the camp of the\nbesiegers in a constant state of alarm.\n\nOn the sixth of February, the engineers who had charge of the mines gave\nnotice that their work was completed. The following morning was named\nfor the assault. The orders of the day prescribed that a general\ncannonade should open on the town at six in the morning. It was to\ncontinue an hour, when the mines were to be sprung. The artillery would\nthen play for another hour, after which the signal for the attack would\nbe given. The signal was to be the firing of one gun from each of the\nbatteries, to be followed by a simultaneous discharge of all. The orders\ndirected the troops to show no quarter to man, woman, or child.\n\n[Sidenote: EXPLOSION OF THE MINES.]\n\nOn the seventh of February, the last day of the Carnival, the besiegers\nwere under arms with the earliest dawn. Their young commander attracted\nevery eye by the splendour of his person and appointments. He was armed\n_cap-a-pie_, and wore a suit of burnished steel, richly inlaid with\ngold. His casque, overshadowed by brilliant plumes, was ornamented with\na medallion displaying the image of the Virgin.[229] In his hand he\ncarried the baton of command; and as he rode along the lines addressing\na few words of encouragement to the soldiers, his perfect horsemanship,\nhis princely bearing, and the courtesy of his manners reminded the\nveterans of the happier days of his father, the emperor. The cavaliers\nby whom he was surrounded emulated their chief in the richness of their\nappointments; and the Murcian chronicler, present on that day, dwells\nwith complacency on the beautiful array of southern chivalry gathered\ntogether for the final assault upon Galera.[230]\n\nFrom six o'clock till seven, a furious cannonade was kept up from the\nwhole circle of batteries on the devoted town. Then came the order to\nfire the mines. The deafening roar of ordnance was at once hushed into a\nsilence profound as that of death, while every soldier in the trenches\nwaited, with nervous suspense, for the explosion. At length it came,\noverturning houses, shaking down a fragment of the castle, rending wider\nthe breach in the perpendicular side of the rock, and throwing off the\nfragments with the force of a volcano. Only one mine, however, exploded.\nIt was soon followed by the other, which, though it did less damage,\nspread such consternation among the garrison, that, fearing there might\nstill be a third in reserve, the men abandoned their works, and took\nrefuge in the town.\n\nWhen the smoke and dust had cleared away, an officer with a few soldiers\nwas sent to reconnoitre the breach. They soon returned with the tidings\nthat the garrison had fled, and left the works wholly unprotected. On\nhearing this, the troops, with furious shouts, called out to be led at\nonce to the assault. It was in vain that the officers remonstrated,\nenforcing their remonstrances, in some instances, by blows with the flat\nof their sabres. The blood of the soldiery was up; and, like an\nill-disciplined rabble, they sprang from their trenches in wild\ndisorder, as before, and, hurrying their officers along with them, soon\nscaled the perilous ascent, and crowned the heights without opposition\nfrom the enemy. Hurrying over the _debris_ that strewed the ground, they\nspeedily made themselves masters of the deserted fortress and its\noutworks,--filling the air with shouts of victory.\n\nThe fugitives saw their mistake, as they beheld the enemy occupying the\nposition they had abandoned. There was no more apprehension of mines.\nEager to retrieve their error, they rushed back, as by a common impulse,\nto dispute the possession of the ground with the Spaniards. It was too\nlate. The guns were turned on them from their own battery. The\narquebusiers who lined the ravelin showered down on their heads missiles\nmore formidable than stones and arrows. But, though their powder was\nnearly gone, the Moriscoes could still make fight with sword and dagger,\nand they boldly closed, in a hand-to-hand contest with their enemy. It\nwas a deadly struggle, calling out--as close personal contest is sure to\ndo--the fiercest passions of the combatants. No quarter was given; none\nwas asked. The Spaniard was nerved by the confidence of victory, the\nMorisco by the energy of despair. Both fought like men who knew that on\nthe issue of this conflict depended the fate of Galera. Again the\nwar-cries of the two religions rose above the din of battle, as the one\nparty invoked their military apostle, and the other called on Mahomet.\nIt was the same war-cry which for more than eight centuries had sounded\nover hill and valley in unhappy Spain. These were its dying notes, soon\nto expire with the exile or extermination of the conquered race.\n\nThe conflict was at length terminated by the arrival of a fresh body of\ntroops on the field with Padilla. That chief had attacked the town by\nthe same avenue as before; everywhere he had met with the same spirit of\nresistance. But the means of successful resistance were gone. Many of\nthe houses on the streets had been laid in rains by the fire of the\nartillery. Such as still held out were defended by men armed with no\nbetter weapons than stones and arrows. One after another, most of them\nwere stormed and fired by the Spaniards; and those within were put to\nthe sword, or perished in the flames.\n\nIt fared no better with the defenders of the barricades. Galled by the\nvolleys of the Christians, against whom their own rude missiles did\ncomparatively little execution, they were driven from one position to\nanother; as each redoubt was successively carried, a shout of triumph\nwent up from the victors, which fell cheerily on the ears of their\ncountrymen on the heights; and when Padilla and his veterans burst on\nthe scene of action, it decided the fortunes of the day.\n\nThere was still a detachment of Turks, whose ammunition had not been\nexhausted, and who were maintaining a desperate struggle with a body of\nSpanish infantry, in which the latter had been driven back to the very\nverge of the precipice. But the appearance of their friends under\nPadilla gave the Spaniards new heart; and Turk and Morisco, overwhelmed\nalike by the superiority of the numbers and of the weapons of their\nantagonists, gave way in all directions. Some fled down the long avenues\nwhich led from the summit of the rock. They were hotly pursued by the\nSpaniards. Others threw themselves into the houses, and prepared to make\na last defence. The Spaniards scrambled along the terraces, letting\nthemselves down from one level to another by means of the Moorish\nladders used for that purpose. They hewed openings in the wooden roofs\nof the buildings, through which they fired on those within. The helpless\nMoriscoes, driven out by the pitiless volleys, sought refuge in the\nstreet. But the fierce hunters were there, waiting for their miserable\ngame, which they shot down without mercy,--men, women, and children;\nnone were spared. Yet they did not fall unavenged; and the corpse of\nmany a Spaniard might be seen stretched on the bloody pavement, lying\nside by side with that of his Moslem enemy.\n\nMore than one instance is recorded of the desperate courage to which the\nwomen as well as the men were roused in their extremity. A Morisco girl,\nwhose father had perished in the first assault in the Gardens, after\nfiring her dwelling, is said to have dragged her two little brothers\nalong with one hand, and, wielding a scimitar with the other, to have\nrushed against the foe, by whom they were all speedily cut to pieces.\nAnother instance is told, of a man who, after killing his wife and his\ntwo daughters, sallied forth, and calling out, \"There is nothing more to\nlose; let us die together!\" threw himself madly into the thick of the\nenemy.[231] Some fell by their own weapons, others by those of their\nfriends, preferring to receive death from any hands but those of the\nSpaniards.\n\nSome two thousand Moriscoes were huddled together in a square not far\nfrom the gate, where a strong body of Castilian infantry cut off the\nmeans of escape. Spent with toil and loss of blood, without ammunition,\nwithout arms, or with such only as were too much battered or broken for\nservice, the wretched fugitives would gladly have made some terms with\ntheir pursuers, who now closed darkly around them. But the stag at bay\nmight as easily have made terms with his hunters and the fierce hounds\nthat were already on his haunches. Their prayers were answered by volley\nafter volley, until not a man was left alive.\n\nMore than four hundred women and children were gathered together without\nthe walls, and the soldiers, mindful of the value of such a booty, were\nwilling to spare their lives. This was remarked by Don John, and no\nsooner did he observe the symptoms of lenity in the troops, than the\nflinty-hearted chief rebuked their remissness, and sternly reminded them\nof the orders of the day. He even sent the halberdiers of his guard and\nthe cavaliers about his person to assist the soldiers in their bloody\nwork; while he sat a calm spectator, on his horse, as immovable as a\nmarble statue, and as insensible to the agonizing screams of his victims\nand their heart-breaking prayers for mercy.[232]\n\n[Sidenote: CRUEL MASSACRE.]\n\nWhile this was going on without the town, the work of death was no less\nactive within. Every square and enclosure that had afforded a temporary\nrefuge to the fugitives was heaped with the bodies of the slain. Blood\nran down the kennels like water after a heavy shower. The dwellings were\nfired, some by the conquerors, others by the inmates, who threw\nthemselves madly into the flames rather than fall into the hands of\ntheir enemies. The gathering shadows of evening--for the fight had\nlasted nearly nine hours[233]--were dispelled by the light of the\nconflagration, which threw an ominous glare for many a league over the\ncountry, proclaiming far and wide the downfall of Galera.\n\nAt length Don John was so far moved from his original purpose as to\nconsent that the women, and the children under twelve years of age,\nshould be spared. This he did, not from any feeling of compunction, but\nfrom deference to the murmurs of his followers, whose discontent at\nseeing their customary booty snatched from them began to show itself in\na way not to be disregarded.[234] Some fifteen hundred women and\nchildren, in consequence of this, are said to have escaped the general\ndoom of their countrymen.[235] All the rest, soldiers and citizens,\nTurks, Africans, and Moriscoes, were mercilessly butchered. Not one man,\nif we may trust the Spaniards themselves, escaped alive! It would not be\neasy, even in that age of blood, to find a parallel to so wholesale and\nindiscriminate a massacre.\n\nYet, to borrow the words of the Castilian proverb, \"If Africa had cause\nto weep, Spain had little reason to rejoice.\"[236] No success during the\nwar was purchased at so high a price as the capture of Galera. The loss\nfell as heavily on the officers and men of rank as on the common file.\nWe have seen the eagerness with which they had flocked to the standard\nof John of Austria. They showed the same eagerness to distinguish\nthemselves under the eye of their leader. The Spanish chivalry were sure\nto be found in the post of danger. Dearly did they pay for that\npre-eminence; and many a noble house in Spain wept bitter tears when the\ntidings came of the conquest of Galera.[237]\n\nDon John himself was so much exasperated, says the chronicler, by the\nthought of the grievous loss which he had sustained through the\nobstinate resistance of the heretics,[238] that he resolved to carry at\nonce into effect his menace of demolishing the town, so that not one\nstone should be left on another. Every house was accordingly burnt or\nlevelled to the ground, which was then strewed with salt, as an accursed\nspot, on which no man was to build thereafter. A royal decree to that\neffect was soon afterwards published; and the village of straggling\nhouses, which, undefended by a wall, still clusters round the base of a\nhill, in the Gardens occupied by Padilla, is all that now serves to\nremind the traveller of the once flourishing and strongly fortified city\nof Galera.\n\nIn the work of demolition Don John was somewhat retarded by a furious\ntempest of sleet and rain, which set in the day after the place was\ntaken. It was no uncommon thing at that season of the year. Had it come\non a few days earlier, the mountain torrents would infallibly have\nbroken up the camp of the besiegers, and compelled them to suspend\noperations. That the storm was so long delayed, was regarded by the\nSpaniards as a special interposition of Heaven.\n\nThe booty was great which fell into the hands of the victors; for\nGalera, from its great strength, had been selected by the inhabitants of\nthe neighbouring country as a safe place of deposit for their\neffects,--especially their more valuable treasures of gold, pearls,\njewels, and precious stuffs. Besides these, there was a great quantity\nof wheat, barley, and other grain, stored in the magazines, which\nafforded a seasonable supply to the army.\n\nNo sooner was Don John master of Galera, than he sent tidings of his\nsuccess to his brother. The king was at that time paying his devotions\nat the shrine of Our Lady of Guadalupe. The tidings were received with\nexultation by the court,--by Philip with the stolid composure with which\nhe usually received accounts either of the success or the discomfiture\nof his arms. He would allow no public rejoicings of any kind. The only\nway in which he testified his satisfaction was by offering up thanks to\nGod and the Blessed Virgin, \"to whom,\" says the chronicler, \"he thought\nthe cause should be especially commended, as one in which more glory was\nto be derived from peace than from a bloody victory.\"[239] With such\nhumane and rational sentiments, it is marvellous that he did not\ncommunicate them to his brother, and thus spare the atrocious massacre\nof his Morisco vassals at Galera.\n\n[Sidenote: DISASTER AT SERON.]\n\nBut, however revolting this massacre may appear in our eyes, it seems to\nhave left no stain on the reputation of John of Austria in the eyes of\nhis contemporaries. In reviewing this campaign, we cannot too often call\nto mind that it was regarded not so much as a war with rebellious\nvassals, as a war with the enemies of the Faith. It was the last link in\nthat long chain of hostilities which the Spaniard for so many centuries\nhad been waging for the recovery of his soil from the infidel. The\nsympathies of Christendom were not the less on his side, that now, when\nthe trumpet of the crusader had ceased to send forth its notes in other\nlands, they should still be heard among the hills of Granada. The\nMoriscoes were everywhere regarded as infidels and apostates; and there\nwere few Christian nations whose codes would not at that day have\npunished infidelity and apostasy with death. It was no harder for them\nthat they should be exterminated by the sword than by the fagot. So far\nfrom the massacre of the Moriscoes tarnishing the reputation of their\nconqueror, it threw a gloomy _eclat_ over his achievement, which may\nhave rather served to add to its celebrity. His own countrymen, thinking\nonly of the extraordinary difficulties which he had overcome, with pride\nbeheld him entering on a splendid career, that would place his name\namong those of the great paladins of the nation. In Rome he was hailed\nas the champion of Christendom; and it was determined to offer him the\nbaton of generalissimo of the formidable league which the pope was at\nthis time organizing against the Ottoman empire.[240]\n\n\n\n\nCHAPTER VIII.\n\nREBELLION OF THE MORISCOES.\n\nDisaster at Seron--Death of Quixada--Rapid Successes of Don\nJohn--Submission of the Moriscoes--Fate of El Habaqui--Stern Temper of\nAben-Aboo--Renewal of the War--Expulsion of the Moors--Don Juan returns\nto Madrid--Murder of Aben-Aboo--Fortunes of the Moriscoes.\n\n1570, 1571.\n\n\nDon John was detained some days before Galera by the condition of the\nroads, which the storm had rendered impassable for heavy waggons and\nartillery. When the weather improved he began his march, moving south,\nin the direction of Baza. Passing through that ancient town, the scene\nof one of the most glorious triumphs of the good Queen Isabella the\nCatholic, he halted at Caniles. Here he left the main body of his army,\nand, putting himself at the head of a detachment of three thousand foot\nand two hundred horse, hastened forward to reconnoitre Seron, which he\npurposed next to attack.\n\nSeron was a town of some strength, situated on the <DW72> of the sierra,\nand defended by a castle held by a Morisco garrison. On his approach,\nmost of the inhabitants, and many of the soldiers, evacuated the place,\nand sought refuge among the mountains. Don John formed his force into\ntwo divisions, one of which he placed under Quixada, the other under\nRequesens. He took up a position himself, with a few cavaliers and a\nsmall body of arquebusiers, on a neighbouring eminence, which commanded\na view of the whole ground.\n\nThe two captains were directed to reconnoitre the environs, by making a\ncircuit from opposite sides of the town. Quixada, as he pressed forward\nwith his column, drove the Morisco fugitives before him, until they\nvanished in the recesses of the mountains. In the meantime, the\nbeacon-fires, which for some hours had been blazing from the topmost\npeaks of the sierra, had spread intelligence far and wide of the coming\nof the enemy. The whole country was in arms; and it was not long before\nthe native warriors, mustering to the number of six thousand, under the\nMorisco chief, El Habaqui, who held command in that quarter, came\npouring through the defiles of the mountains, and fell with fury on the\nfront and flank of the astonished Spaniards. The assailants were soon\njoined by the fugitives from Seron; and the Christians, unable to\nwithstand this accumulated force, gave way, though slowly, and in good\norder, before the enemy.\n\nMeanwhile, a detachment of Spanish infantry, under command of Lope de\nFigueroa, _maestro del campo_, had broken into the town, where they were\nbusily occupied in plundering the deserted houses. This was a part of\nthe military profession which the rude levies of Andalusia well\nunderstood. While they were thus occupied, the advancing Moriscoes,\nburning for revenge, burst into the streets of the town, and, shouting\ntheir horrid war-cries, set furiously on the marauders. The Spaniards,\ntaken by surprise, and encumbered with their booty, offered little\nresistance. They were seized with a panic, and fled in all directions.\nThey were soon mingled with their retreating comrades under Quixada,\neverywhere communicating their own terror, till the confusion became\ngeneral. It was in vain that Quixada and Figueroa, with the other\ncaptains, endeavoured to restore order. The panic-stricken soldiers\nheard nothing, saw nothing, but the enemy.\n\nAt this crisis, Don John, who from his elevated post had watched the\nimpending ruin, called his handful of brave followers around him, and at\nonce threw himself into the midst of the tumult. \"What means this,\nSpaniards?\" he exclaimed. \"From whom are you flying? Where is the honour\nof Spain? Have you not John of Austria, your commander, with you? At\nleast, if you retreat, do it like brave men, with your front to the\nenemy.\"[241] It was in vain. His entreaties, his menaces, even his\nblows, which he dealt with the flat of his sabre, were ineffectual to\nrouse anything like a feeling of shame in the cowardly troops. The\nefforts of his captains were equally fruitless, though in making them\nthey exposed their lives with a recklessness which cost some of them\ndear. Figueroa was disabled by a wound in the leg. Quixada was hit by a\nmusket-ball on the left shoulder, and struck from his saddle. Don John,\nwho was near, sprang to his assistance, and placed him in the hands of\nsome troopers, with directions to bear him at once to Caniles. In doing\nthis the young commander himself had a narrow escape; for he was struck\non his helmet by a ball, which, however, fortunately glanced off without\ndoing him injury.[242] He was now hurried along by the tide of\nfugitives, who made no attempt to rally for the distance of half a\nleague, when the enemy ceased his pursuit. Six hundred Spaniards were\nleft dead on the field. A great number threw themselves into the houses,\nprepared to make good their defence. But they were speedily enveloped by\nthe Moriscoes, the houses were stormed or set on fire, and the inmates\nperished to a man.[243]\n\nDon John, in a letter dated the nineteenth of February, two days after\nthis disgraceful affair, gave an account of it to the king, declaring\nthat the dastardly conduct of the troops exceeded anything he had ever\nwitnessed, or indeed could have believed, had he not seen it with his\nown eyes. \"They have so little heart in the service,\" he adds, \"that no\neffort that I can make, not even the fear of the galleys or the gibbet,\ncan prevent them from deserting. Would to Heaven I could think that they\nare moved to this by the desire to return to their families, and not by\nfear of the enemy.\"[244] He gave the particulars of Quixada's accident,\nstating that the surgeons had made six incisions before they could\nascertain where the ball, which had penetrated the shoulder, was lodged;\nand that, with all their efforts, they had as yet been unable to extract\nit. \"I now deeply feel,\" he says, \"how much I have been indebted to his\nmilitary experience, his diligence, and care and how important his\npreservation is to the service of your majesty. I trust in God he may be\npermitted to regain his health, which is now in a critical\ncondition.\"[245]\n\n[Sidenote: DEATH OF QUIXADA.]\n\nIn his reply to this letter, the king expressed his sense of the great\nloss which both he and his brother would sustain by the death of\nQuixada. \"You will keep me constantly advised of the state of his\nhealth,\" he says. \"I know well it is unnecessary for me to impress upon\nyou the necessity of watching carefully over him.\" Philip did not let\nthe occasion pass for administering a gentle rebuke to Don John for so\nlightly holding the promise he had made to him from Galera, not again to\nexpose himself heedlessly to danger. \"When I think of your narrow escape\nat Seron, I cannot express the pain I have felt at your rashly incurring\nsuch a risk. In war, every one should confine himself to the duties of\nhis own station; nor should the general affect to play the part of the\nsoldier, anymore than the soldier that of the general.\"[246]\n\nIt seems to have been a common opinion, that Don John was more fond of\ndisplaying his personal prowess than became one of his high rank; in\nshort, that he showed more the qualities of a knight-errant, than those\nof a great commander.[247]\n\nMeanwhile, Quixada's wound, which from the first had been attended with\nalarming symptoms, grew so much worse as to baffle all the skill of the\nsurgeons. His sufferings were great, and every hour he grew weaker.\nBefore a week had elapsed, it became evident that his days were\nnumbered.\n\nThe good knight received the intelligence with composure,--for he did\nnot fear death. He had not the happiness in this solemn hour to have her\nnear him on whose conjugal love and tenderness he had reposed for so\nmany years.[248] But the person whom he cherished next to his wife, Don\nJohn of Austria, was by his bedside, watching over him with the\naffectionate solicitude of a son, and ministering those kind offices\nwhich soften the bitterness of death. The dying man retained his\nfaculties to the last, and dictated, though he had not the strength to\nsign, a letter to the king, requesting some favour for his widow, in\nconsideration of his long services. He then gave himself up wholly to\nhis spiritual concerns; and on the twenty-fourth of February, 1570, he\ngently expired, in the arms of his foster-son.\n\nQuixada received a soldier's funeral. His obsequies were celebrated with\nthe military pomp suited to his station. His remains, accompanied by the\nwhole army, with arms reversed, and banners trailing in the dust, were\nborne in solemn procession to the church of the Jeronymites in Caniles;\nand \"we may piously trust,\" says the chronicler, \"that the soul of Don\nLuis rose up to heaven with the sweet incense which burned on the altars\nof St. Jerome; for he spent his life, and finally lost it, in fighting\nlike a valiant soldier the battles of the faith.\"[249]\n\nQuixada was austere in his manners, and a martinet in enforcing\ndiscipline. He was loyal in his nature, of spotless integrity, and\npossessed so many generous and knightly qualities, that he commanded the\nrespect of his comrades; and the regret for his loss was universal.\nPhilip, writing to Don John, a few days after the event, remarks: \"I did\nnot think that any letter from you could have given me so much pain as\nthat acquainting me with the death of Quixada. I fully comprehend the\nimportance of his loss, both to myself and to you, and cannot wonder you\nshould feel it so keenly. It is impossible to allude to it without\nsorrow. Yet we may be consoled by the reflection that, living and dying\nas he did, he cannot fail to have exchanged this world for a\nbetter.\"[250]\n\nQuixada's remains were removed, the year following, to his estate at\nVillagarcia, where his disconsolate widow continued to reside.\nImmediately after her lord's decease, Don John wrote to Dona Magdalena,\nfrom the camp, a letter of affectionate condolence, which came from the\nfulness of his heart: \"Luis died as became him, fighting for the glory\nand safety of his son, and covered with immortal honour. Whatever I am,\nwhatever I shall be, I owe to him, by whom I was formed, or rather\nbegotten in a nobler birth. Dear sorrowing widowed mother! I only am\nleft to you; and to you, indeed, do I of right belong, for whose sake\nLuis died, and you have been stricken with this woe. Moderate your grief\nwith your wonted wisdom. Would that I were near you now, to dry your\ntears, or mingle mine with them! Farewell, dearest and most honoured\nmother! and pray to God to send, back your son from these wars to your\nbosom.\"[251]\n\nDona Magdalena survived her husband many years, employing her time in\nacts of charity and devotion. From Don John she ever experienced the\nsame filial tenderness which he evinces in the letter above quoted.\nNever did he leave the country or return to it without first paying his\nrespects to his mother, as he always called her. She watched with\nmaternal pride his brilliant career; and when that was closed by an\nearly death, the last link which had bound her to this world was snapped\nfor ever. Yet she continued to live on till near the close of the\ncentury, dying in 1598, and leaving behind her a reputation for goodness\nand piety little less than that of a saint.\n\nDon John, having paid the last tribute of respect to the memory of his\nguardian, collected his whole strength, and marched at once against\nSeron. But the enemy, shrinking from an encounter with so formidable a\nforce, had abandoned the place before the approach of the Spaniards. The\nSpanish commander soon after encountered El Habaqui in the\nneighbourhood, and defeated him. He then marched on Tijola, a town\nperched on a bold cliff, which a resolute garrison might have easily\nheld against an enemy. But the Moriscoes, availing themselves of the\ndarkness of the night, stole out of the place, and succeeded, without\nmuch loss, in escaping through the lines of the besiegers.[252] The fall\nof Tijola was followed by that of Purchena. In a short time the whole\nRio de Almanzora was overrun, and the victorious general, crossing the\nsouth-eastern borders of the Alpujarras, established his quarters, on\nthe second of May, at Padules, about two leagues from Andarax.\n\n[Sidenote: NEGOTIATIONS WITH EL HABAQUI.]\n\nThese rapid successes are not to be explained simply by Don John's\nsuperiority over the enemy in strength or military science. Philip had\nturned a favourable ear to the pope's invitation to join the league\nagainst the Turk, in which he was complimented by having the post of\ncommander-in-chief offered to his brother, John of Austria. But before\nengaging in a new war, it was most desirable for him to be released from\nthat in which he was involved with the Moriscoes. He had already seen\nenough of the sturdy spirit of that race to be satisfied that to\naccomplish his object by force would be a work of greater time than he\ncould well afford. The only alternative, therefore, was to have recourse\nto the conciliatory policy which had been so much condemned in the\nmarquis of Mondejar. Instructions to that effect were accordingly sent\nto Don John, who, heartily weary of this domestic contest, and longing\nfor a wider theatre of action, entered warmly into his brother's views.\nSecret negotiations were soon opened with El Habaqui, the Morisco chief,\nwho received the offer of such terms for himself and his countrymen as\nleft him in no doubt, at least, as to the side on which his own interest\nlay. As a preliminary step, he was to withdraw his support from the\nplaces in the Rio de Almanzora; and thus the war, brought within the\nnarrower range of the Alpujarras, might be more easily disposed of. This\npart of his agreement had been faithfully executed; and the rebellious\ndistrict on the eastern borders of the Alpujarras had, as we have seen,\nbeen brought into subjection, with little cost of life to the Spaniards.\n\nDon John followed this up by a royal proclamation, promising an entire\namnesty for the past to all who within twenty days should tender their\nsubmission. They were to be allowed to state the grievances which had\nmoved them to take up arms, with an assurance that these should be\nredressed. All who refused to profit by this act of grace, with the\nexception of the women, and of children under fourteen years of age,\nwould be put to the sword without mercy.\n\nWhat was the effect of the proclamation we are not informed. It was\nprobably not such as had been anticipated. The Moriscoes, distressed as\nthey were, did not trust the promises of the Spaniards. At least we find\nDon John, who had now received a reinforcement of two thousand men,\ndistributing his army into detachments, with orders to scour the country\nand deal with the inhabitants in a way that should compel them to\nsubmit. Such of the wretched peasantry as had taken refuge in their\nfastnesses were assailed with shot and shell, and slaughtered by\nhundreds. Some, who had hidden with their families in the caves in which\nthe country abounded, were hunted out by their pursuers, or suffocated\nby the smoke of burning fagots at the entrance of their retreats.\nEverywhere the land was laid waste, so as to afford sustenance for no\nliving thing. Such were the conciliatory measures employed by the\ngovernment for the reduction of the rebels.[253]\n\nMeanwhile the duke of Sesa had taken the field on the northern border of\nthe Alpujarras, with an army of ten thousand foot and two thousand\nhorse. He was opposed by Aben-Aboo with a force which in point of\nnumbers was not inferior to his own. The two commanders adopted the same\npolicy; avoiding pitched battles, and confining themselves to the\ndesultory tactics of _guerilla_ warfare, to skirmishes and surprises;\nwhile each endeavoured to distress his adversary by cutting off his\nconvoys and by wasting the territory with fire and sword. The Morisco\nchief had an advantage in the familiarity of his men with this wild\nmountain fighting, and in their better knowledge of the intricacies of\nthe country. But this was far more than counterbalanced by the\nsuperiority of the Spaniards in military organization, and by their\npossession of cavalry, artillery, and muskets, in all of which the\nMoslems were lamentably deficient. Thus, although no great battle was\nwon by the Christians, although they were sorely annoyed, and their\nconvoys of provisions frequently cut off, by the skirmishing parties of\nthe enemy, they continued steadily to advance, driving the Moriscoes\nbefore them, and securing the permanency of their conquests by planting\na line of forts, well garrisoned, along the wasted territory in their\nrear. By the beginning of May, the duke of Sesa had reached the borders\nof the Mediterranean, and soon after united his forces, greatly\ndiminished by desertion, to those of Don John of Austria at\nPadules.[254]\n\nNegotiations, during this time, had been resumed with El Habaqui, who\nwith the knowledge, if not the avowed sanction, of Aben-Aboo, had come\nto a place called Fondon de Andarax, not far distant from the\nhead-quarters of the Spanish commander-in-chief. He was accompanied by\nseveral of the principal Moriscoes, who were to take part in the\ndiscussions. On the thirteenth of May they were met by the deputies from\nthe Castilian camp, and the conference was opened. It soon appeared that\nthe demands of the Moriscoes were wholly inadmissible. They insisted,\nnot only on a general amnesty, but that things should be restored to the\nsituation in which they were before the edicts of Philip the Second had\ngiven rise to the rebellion. The Moorish commissioners were made to\nunderstand that they were to negotiate only on the footing of a\nconquered race. They were advised to prepare a memorial preferring such\nrequests as might be reasonably granted; and they were offered the\nservices of Juan de Soto, Don John's secretary, to aid them in drafting\nthe document. They were counselled, moreover, to see their master,\nAben-Aboo, and obtain full powers from him to conclude a definitive\ntreaty.\n\nAben-Aboo, ever since his elevation to the stormy sovereignty of the\nAlpujarras, had maintained his part with a spirit worthy of his cause.\nBut as he beheld town after town fall away from his little empire, his\npeople butchered or swept into slavery, his lands burned and wasted,\nuntil the fairest portions were converted into a wilderness,--above all,\nwhen he saw that his cause excited no sympathy in the bosoms of the\nMoslem princes, on whose support he had mainly relied,--he felt more and\nmore satisfied of the hopelessness of a contest with the Spanish\nmonarchy. His officers, and indeed the people at large, had come to the\nsame conviction; and nothing but an intense hatred of the Spaniards, and\na distrust of their good faith, had prevented the Moriscoes from\nthrowing down their arms and accepting the promises of grace which had\nbeen held out to them. The disastrous result of the recent campaign\nagainst the duke of Sesa tended still further to the discouragement of\nthe Morisco chief; and El Habaqui and his associates returned with\nauthority from their master to arrange terms of accommodation with the\nSpaniards.\n\nOn the nineteenth of May, the commissioners from each side again met at\nFondon de Andarax. A memorial, drafted by Juan de Soto, was laid before\nDon John, whose quarters, as we have seen, were in the immediate\nneighbourhood. No copy of the instrument has been preserved, or at least\nnone has been published. From the gracious answer returned by the\nprince, we may infer that it contained nothing deemed objectionable by\nthe conquerors.\n\n[Sidenote: SUBMISSION OF THE MORISCOES.]\n\nThe deputies were not long in agreeing on terms of accommodation--or\nrather of submission. It was settled that the Morisco captain should\nproceed to the Christian camp, and there presenting himself before the\ncommander-in-chief, should humbly crave forgiveness, and tender\nsubmission on behalf of his nation; that, in return for this act of\nhumiliation, a general amnesty should be granted to his countrymen, who,\nthough they were no longer to be allowed to occupy the Alpujarras, would\nbe protected by the government wherever they might be removed. More\nimportant concessions were made to Aben-Aboo and El Habaqui. The\nlast-mentioned chief, as the chronicler tells us, obtained all that he\nasked for his master, as well as for himself and his friends.[255] Such\npolitic concessions by the Spaniards had doubtless their influence in\nopening the eyes of the Morisco leaders to the folly of protracting the\nwar in their present desperate circumstances.\n\nThe same evening on which the arrangement was concluded, El Habaqui\nproceeded to his interview with the Spanish commander. He was\naccompanied by one only of the Morisco deputies. The others declined to\nwitness the spectacle of their nation's humiliation. He was attended,\nhowever, by a body of three hundred arquebusiers. On entering the\nChristian lines, his little company was surrounded by four regiments of\nCastilian infantry, and escorted to the presence of John of Austria, who\nstood before his tent, attended by his officers, from whom his princely\nbearing made him easily distinguished.\n\nEl Habaqui, alighting from his horse, and prostrating himself before the\nprince, exclaimed, \"Mercy! We implore your highness, in the name of his\nmajesty, to show us mercy, and to pardon our transgressions, which we\nacknowledge have been great!\"[256] Then unsheathing his scimitar, he\npresented it to Don John, saying that he surrendered his arms to his\nmajesty in the name of Aben-Aboo and the rebel chiefs for whom he was\nempowered to act. At the same time the secretary, Juan de Soto, who had\nborne the Moorish banner, given him by El Habaqui, on the point of his\nlance, cast it on the ground before the feet of the prince. The whole\nscene made a striking picture, in which the proud conqueror, standing\nwith the trophies of victory around him, looked down on the\nrepresentative of the conquered race as he crouched in abject submission\nat his feet. Don John, the predominant figure in the _tableau_, by his\nstately demeanour tempered with a truly royal courtesy, reminded the old\nsoldiers of his father the emperor, and they exclaimed, \"This is the\ntrue son of Charles the Fifth!\"\n\nStooping forward, he graciously raised the Morisco chief from the\nground, and, returning him his sword, bade him employ it henceforth in\nthe service of the king. The ceremony was closed by flourishes of\ntrumpets and salvoes of musketry, as if in honour of some great victory.\n\nEl Habaqui remained some time after his followers had left the camp,\nwhere he met with every attention, was feasted and caressed by the\nprincipal officers, and was even entertained at a banquet by the bishop\nof Guadix. He received however, as we have seen, something more\nsubstantial than compliments. Under these circumstances, it was natural\nthat he should become an object of jealousy and suspicion to the\nMoriscoes. It was soon whispered that El Habaqui, in his negotiations\nwith the Christians, had been more mindful of his own interests than of\nthose of his countrymen.[257]\n\nIndeed, the Moriscoes had little reason to congratulate themselves on\nthe result of a treaty which left them in the same forlorn and degraded\ncondition as before the breaking out of the rebellion,--which in one\nimportant respect, indeed, left them in a worse condition, since they\nwere henceforth to become exiles from the homes of their fathers. Yet,\ncruel and pitiable in the extreme as was the situation of the Moriscoes,\nthe Spanish monks, as Don John complains to his brother, inveighed\nopenly in their pulpits against the benignity and mercy of the\nking;[258] and this too, he adds, when it should rather have been their\nduty to intercede for poor wretches who, for the most part, had sinned\nthrough ignorance.[259] The ecclesiastic on whom his censure most\nheavily falls, is the President Deza,--a man held in such abhorrence by\nthe Moriscoes as to have been one principal cause of their insurrection;\nand he beseeches the king to consult the interests of Granada by\nbestowing on him a bishopric, or some other dignity, which may remove\nhim from the present scene of his labours.[260]\n\nAmong those disappointed at the terms of the treaty, as it soon\nappeared, was Aben-Aboo himself. At first he affected to sanction it,\nand promised to all he could to enforce its execution. But he soon\ncooled, and, throwing the blame on El Habaqui, declared that this\nofficer had exceeded his powers, made a false report to him of his\nnegotiations, and sacrificed the interests of the nation to his own\nambition.[261] The attentions lavished on that chief by the Spaniards,\nhis early correspondence with them, and the liberal concessions secured\nto him by the treaty, furnished plausible grounds for such an\naccusation.\n\nAccording to the Spanish accounts, however, Aben-Aboo at this time\nreceived a reinforcement of two hundred soldiers from Barbary, with the\nassurance that he would soon have more effectual aid from Africa. This,\nwe are told, changed his views. Nor is it impossible that the Morisco\nchief, as the hour approached, found it a more difficult matter than he\nhad anticipated to resign his royal state and descend into the common\nrank-and-file of the vassals of Castile,--the degraded caste of Moorish\nvassals, whose condition was little above that of serfs.\n\nHowever this maybe, the Spanish camp was much disquieted by the rumours\nwhich came in of Aben-Aboo's vacillation. It was even reported that, far\nfrom endeavouring to enforce the execution of the treaty, he was\nsecretly encouraging his people to further resistance. No one felt more\nindignant at his conduct than El Habaqui, who had now become as loyal a\nsubject as any other in Philip's dominions. Not a little personal\nresentment was mingled with his feeling towards Aben-Aboo; and he\noffered, if Don John would place him at the head of a detachment, to go\nhimself, brave the Morisco prince in his own quarters, and bring him as\na prisoner to the camp. Don John, though putting entire confidence in El\nHabaqui's fidelity,[262] preferred, instead of men, to give him money;\nand he placed eight hundred gold ducats in his hands, to enable him to\nraise the necessary levies among his countrymen.\n\n[Sidenote: FATE OF EL HABAQUI.]\n\nThus fortified, El Habaqui set out for the head-quarters of Aben-Aboo,\nat his ancient residence in Mecina de Bombaron. On the second day the\nMorisco captain fell in with a party of his countrymen lingering idly by\nthe way, and he inquired, with an air of authority, why they did not go\nand tender their submission to the Spanish authorities, as others had\ndone. They replied, they were waiting for their master's orders. To this\nEl Habaqui rejoined, \"All are bound to submit: and if Aben-Aboo, on his\npart, shows unwillingness to do so, I will arrest him at once, and drag\nhim at my horse's tail to the Christian camp.\"[263] This foolish vaunt\ncost the braggart his life.\n\nOne of the party instantly repaired to Mecina and reported the words to\nAben-Aboo. The Morisco prince, overjoyed at the prospect of having his\nenemy in his power, immediately sent a detachment of a hundred and fifty\nTurks to seize the offender and bring him to Mecina. They found El\nHabaqui at Burchal, where his family were living. The night had set in,\nwhen the chieftain received tidings of the approach of the Turks; and\nunder cover of the darkness he succeeded in making his escape into the\nneighbouring mountains. The ensuing morning the soldiers followed\nclosely on his track; and it was not long before they descried a person\nskulking among the rocks, whose white mantle and crimson turban proved\nhim to be the object of their pursuit. He was immediately arrested and\ncarried to Mecina. His sentence was already passed. Aben-Aboo,\nupbraiding him with his treachery, ordered him to be removed to an\nadjoining room, where he was soon after strangled. His corpse, denied\nthe rights of burial, having been first rolled in a mat of reeds, was\nignominiously thrown into a sewer; and the fate of the unhappy man was\nkept a secret for more than a month.[264]\n\nHis absence, after some time, naturally excited suspicions in the\nSpanish camp. A cavalier, known to Aben-Aboo, wrote to him to obtain\ninformation respecting El Habaqui, and was told, in answer, by the wily\nprince, that he had been arrested and placed in custody for his\ntreacherous conduct, but that his family and friends need be under no\nalarm, as he was perfectly safe. Aben-Aboo hinted, moreover, that it\nwould be well to send to him some confidential person with whom he might\narrange the particulars of the treaty,--as if these had not been already\nsettled. After some further delay, Don John resolved to despatch an\nagent to ascertain the real dispositions of the Moriscoes towards the\nChristians, and to penetrate, if possible, the mystery that hung round\nthe fate of El Habaqui.\n\nThe envoy selected was Hernan Valle de Palacios, a cavalier possessed of\na courageous heart, yet tempered by a caution that well fitted him for\nthe delicate and perilous office. On the thirteenth of July he set out\non his mission. On the way he encountered a Morisco, a kinsman of the\nlate monarch, Aben-Humeya, and naturally no friend to Aben-Aboo. He was\nacquainted with the particulars of El Habaqui's murder, of which he gave\nfull details to Palacios. He added, that the Morisco prince, far from\nacquiescing in the recent treaty, was doing all in his power to prevent\nits execution. He could readily muster, at short notice, said the\ninformer, a force of five thousand men, well armed, and provisioned for\nthree months; and he was using all his efforts to obtain further\nreinforcements from Algiers.\n\nInstructed in these particulars, the envoy resumed his journey. He was\ncareful, however, first to obtain a safe-conduct from Aben-Aboo, which\nwas promptly sent to him. On reaching Mecina, he found the place\noccupied by a body of five hundred arquebusiers; but by the royal order\nhe was allowed to pass unmolested. Before entering the presence of \"the\nlittle king of the Alpujarras,\" as Aben-Aboo, like his predecessor, was\nfamiliarly styled by the Spaniards, Palacios was carefully searched, and\nsuch weapons as he carried about him were taken away.\n\nHe found Aben-Aboo stretched on a divan, and three or four Moorish girls\nentertaining him with their national songs and dances. He did not rise,\nor indeed change his position, at the approach of the envoy, but gave\nhim audience with the lofty bearing of an independent sovereign.\n\nPalacios did not think it prudent to touch on the fate of El Habaqui.\nAfter expatiating on the liberal promises which he was empowered by Don\nJohn of Austria to make, he expressed the hope that Aben-Aboo would\nexecute the treaty, and not rekindle a war which must lead to the total\ndestruction of his country. The chief listened in silence; and it was\nnot till he had called some of his principal captains around him, that\nhe condescended to reply. He then said, that God and the whole world\nknew it was not by his own desire, but by the will of the people, that\nhe had been placed on the throne. \"I shall not attempt,\" he said, \"to\nprevent any of my subjects from submitting that prefer to do so. But\ntell your master,\" he added, \"that, while I have a single shirt to my\nback, I shall not follow their example. Though no other man should hold\nout in the Alpujarras, I would rather live and die a Mussulman than\npossess all the favours which King Philip can heap on me. At no time,\nand in no manner, will I ever consent to place myself in his\npower.\"[265] He concluded this spirited declaration by adding, that, if\ndriven to it by necessity, he could bury himself in a cavern, which he\nhad stowed with supplies for six years to come, during which it would go\nhard but he would find some means of making his way to Barbary. The\ndesperate tone of these remarks effectually closed the audience.\nPalacios was permitted to return unmolested, and to report to his\ncommander the failure of his mission.\n\nThe war, which Don John had flattered himself he had so happily brought\nto a close, now, like a fire smothered, but not quenched, burst forth\nagain with redoubled fury. The note of defiance was heard loudest among\nthe hills of Ronda, a wild sierra on the western skirts of the\nAlpujarras, inhabited by a bold and untamed race, more formidable than\nthe mountaineers of any other district of Granada. Aben-Aboo did all he\ncould to fan the flame of insurrection in this quarter, and sent his own\nbrother, El Galipe, to take the command.\n\nThe Spanish government, now fully aroused, made more vigorous efforts to\ncrush the spirit of rebellion than at any time during the war. Don John\nwas ordered to occupy Guadix, and thence to scour the country in a\nnortherly direction. Another army, under the Grand-Commander Requesens,\nmarching from Granada, was to enter the Alpujarras from the north, and\ntaking a route different from that of the duke of Sesa, in the previous\ncampaign, was to carry a war of extermination into the heart of the\nmountains. Finally, the duke of Arcos, the worthy descendant of the\ngreat marquis of Cadiz, whose name was so famous in the first war of\nGranada, and whose large estates in this quarter he had inherited, was\nentrusted with the operations against the rebels of the Serrania de\nRonda.\n\n[Sidenote: RENEWAL OF THE WAR.]\n\nThe grand-commander executed his commission in the same remorseless\nspirit in which it had been dictated. Early in September, quitting\nGranada, he took the field at the head of five thousand men. He struck\nat once into the heart of the country. All the evils of war in its most\nhorrid form followed in his train. All along his track, it seemed as if\nthe land had been swept by a conflagration. The dwellings were sacked\nand burned to the ground. The mulberry and olive groves were cut down;\nthe vines were torn up by the roots; and the ripening harvests were\ntrampled in the dust. The country was converted into a wilderness.\nOccasionally small bodies of the Moriscoes made a desperate stand. But\nfor the most part, without homes to shelter or food to nourish them,\nthey were driven, like unresisting cattle, to seek a refuge in the\ndepths of the mountains, and in the caves in which this part of the\ncountry abounded. Their pursuers followed up the chase with the fierce\nglee with which the hunter tracks the wild animal of the forest to his\nlair. There they were huddled together, one or two hundred frequently in\nthe same cavern. It was not easy to detect the hiding-place amidst the\nrocks and thickets which covered up and concealed the entrance. But when\nit was detected, it was no difficult matter to destroy the inmates. The\ngreen bushes furnished the materials for a smouldering fire, and those\nwithin were soon suffocated by the smoke, or, rushing out, threw\nthemselves on the mercy of their pursuers. Some were butchered on the\nspot; others were sent to the gibbet or the galleys; while the greater\npart, with a fate scarcely less terrible, were given up as the booty of\nthe soldiers, and sold into slavery.[266]\n\nAben-Aboo had a narrow escape in one of these caverns, not far from\nBerchul, where he had secreted himself with a wife and two of his\ndaughters. The women were suffocated, with about seventy other persons.\nThe Morisco chief succeeded in making his escape through an aperture at\nthe farther end, which was unknown to his enemies.[267]\n\nSmall forts were erected at short intervals along the ruined country. No\nless than eighty-four of these towers were raised in different parts of\nthe land, twenty-nine of which were to be seen in the Alpujarras and the\nvale of Lecrin alone.[268] There they stood, crowning every peak and\neminence in the sierra, frowning over the horrid waste, the sad\nmemorials of the conquest. This was the stern policy of the victors.\nWithin this rocky girdle, long held as it was by the iron soldiery of\nCastile, it was impossible that rebellion should again gather to a head.\n\nThe months of September and October were consumed in these operations.\nMeanwhile the duke of Arcos had mustered his Andalusian levies, to the\nnumber of four thousand men, including a thousand of his own vassals. He\ntook with him his son, a boy of not more than thirteen years of\nage,--following in this, says the chronicler, the ancient usage of the\nvaliant house of Ponce de Leon.[269] About the middle of September he\nbegan his expedition into the Sierra Vermeja, or Red Sierra. It was a\nspot memorable in Spanish history for the defeat and death of Alonso de\nAguilar, in the time of Ferdinand and Isabella, and has furnished the\ntheme of many a plaintive _romance_ in the beautiful minstrelsy of the\nSouth. The wife of the duke of Arcos was descended from Alonso de\nAguilar, as he himself was the grandson of the good count of Urena, who,\nwith better fortune than his friend, survived the disasters of that day.\nThe route of the army led directly across the fatal field. As they\ntraversed the elevated plain of Calaluz, the soldiers saw everywhere\naround the traces of the fight. The ground was still covered with\nfragments of rusty armour, bits of broken sword-blades, and heads of\nspears. More touching evidence was afforded by the bones of men and\nhorses, which, in this solitary region, had been whitening in the blasts\nof seventy winters. The Spaniards knew well the localities, with which\nthey had become familiar from boyhood in the legends and traditions of\nthe country. Here was the spot where the vanguard, under its brave\ncommander, had made its halt in the obscurity of the night. There were\nthe faint remains of the enemy's entrenchments, which time had nearly\nlevelled with the dust; and there, too, the rocks still threw their dark\nshadows over the plain, as on the day when the valiant Alonso da Aguilar\nfell at their base in combat with the renowned Feri de Ben Estepar. The\nwhole scene was brought home to the hearts of the Spaniards. As they\ngazed on the unburied relics lying around them, the tears, says the\neloquent historian who records the incident, fell fast down their iron\ncheeks; and they breathed a soldier's prayer for the repose of the noble\ndead. But these holier feelings were soon succeeded by others of a\nfierce nature, and they loudly clamoured to be led against the\nenemy.[270]\n\nThe duke of Arcos, profiting by the errors of Alonso de Aguilar, had\nmade his arrangements with great circumspection. He soon came in sight\nof the Moriscoes, full three thousand strong. But, though well posted,\nthey made a defence little worthy of their ancient reputation, or of the\nnotes of defiance which they had so boldly sounded at the opening of the\ncampaign. They indeed showed mettle at first, and inflicted some loss on\nthe Christians. But the frequent reverses of their countrymen seemed to\nhave broken their spirits; and they were soon thrown into disorder, and\nfled in various directions into the more inaccessible tracts of the\nsierra. The Spaniards followed up the fugitives, who did not attempt to\nrally. Nor did they ever again assemble in any strength, so effectual\nwere the dispositions made by the victorious general. The insurrection\nof the Sierra Vermeja was at an end.[271]\n\nThe rebellion, indeed, might be said to be everywhere crushed within the\nborders of Granada. The more stout-hearted of the insurgents still held\nout among the caves and fastnesses of the Alpujarras, supporting a\nprecarious existence until they were hunted down by detachments of the\nSpaniards, who were urged to the pursuit by the promise from government\nof twenty ducats a head for every Morisco. But nearly all felt the\nimpracticability of further resistance. Some succeeded in making their\nescape to Barbary. The rest, broken in spirit, and driven to extremity\nby want of food in a country now turned into a desert, consented at\nlength to accept the amnesty offered them, and tendered their\nsubmission.\n\n[Sidenote: EXPULSION OF THE MOORS.]\n\nOn the twenty-eighth of October Don John received advices of a final\nedict of Philip, commanding that all the Moriscoes in the kingdom of\nGranada should be at once removed into the interior of the country. None\nwere to be excepted from this decree, not even the _Moriscos de la\nPaz_, as those were called who had loyally refused to take part in the\nrebellion.[272] The arrangements for this important and difficult step\nwere made with singular prudence, and, under the general direction of\nDon John of Austria, the Grand-Commander Requesens, and the dukes of\nSesa and Arcos, were carried into effect with promptness and energy.\n\nBy the terms of the edict, the lands and houses of the exiles were to be\nforfeited to the crown. But their personal effects--their flocks, their\nherds, and their grain--would be taken, if they desired it, at a fixed\nvaluation by the government. Every regard was to be paid to their\npersonal conveniences and security; and it was forbidden, in the\nremoval, to separate parents from children, husbands from wives; in\nshort, to divide the members of a family from one another;--\"an act of\nclemency,\" says a humane chronicler, \"which they little deserved; but\nhis majesty was willing in this to content them.\"[273]\n\nThe country was divided into districts, the inhabitants of which were to\nbe conducted, under the protection of a strong military escort, to their\nseveral places of destination. These seem to have been the territory of\nLa Mancha, the northern borders of Andalusia, the Castiles, Estremadura,\nand even the remote province of Galicia. Care was taken that no\nsettlement should be made near the borders of Murcia or Valencia, where\nlarge numbers of the Moriscoes were living in comparative quiet on the\nestates of the great nobles, who were exceedingly jealous of any\ninterference with their vassals.\n\nThe first of November, All-Saints' Day, was appointed for the removal of\nthe Moriscoes throughout Granada. On that day they were gathered in the\nprincipal churches of their districts, and after being formed into their\nrespective divisions, began their march. The grand-commander had\noccupied the passes of the Alpujarras with strong detachments of the\nmilitary. The different columns of emigrants were placed under the\ndirections of persons of authority and character. The whole movement was\nconducted with singular order,--resistance being attempted in one or two\nplaces only, where the blame, it may be added, as intimated by a\nCastilian chronicler, was to be charged on the brutality of the\nsoldiers.[274] Still, the removal of the Moriscoes on the present\noccasion was attended with fewer acts of violence and rapacity than the\nformer removal, from Granada. At least this would seem to be inferred by\nthe silence of the chroniclers; though it is true such silence is far\nfrom being conclusive, as the chroniclers, for the most part, felt too\nlittle interest in the sufferings of the Moriscoes to make a notice of\nthem indispensable. However this may be, it cannot be doubted that,\nwhatever precautions may have been taken to spare the exiles any\nunnecessary suffering, the simple fact of their being expelled from\ntheir native soil is one that suggests an amount of misery not to be\nestimated. For what could be more dreadful than to be thus torn from\ntheir pleasant homes, the scenes of their childhood, where every\nmountain, valley, and stream were as familiar friends,--a part of their\nown existence,--to be rudely thrust into a land of strangers, of a race\ndiffering from themselves in faith, language, and institutions, with no\nsentiment in common but that of a deadly hatred? That the removal of a\nwhole nation should have been so quietly accomplished, proves how\nentirely the strength and spirit of the Moriscoes must have been broken\nby their reverses.[275]\n\nThe war thus terminated, there seemed no reason for John of Austria to\nprolong his stay in the province. For some time he had been desirous to\nobtain the king's consent to his return. His ambitious spirit, impatient\nof playing a part on what now seemed to him an obscure field of action,\npent up within the mountain barrier of the Alpujarras, longed to display\nitself on a bolder theatre before the world. He aspired, too, to a more\nindependent command. He addressed repeated letters to the king's\nministers,--to the Cardinal Espinosa and Gomez de Silva in\nparticular,--to solicit their influence in his behalf. \"I should be\nglad,\" he wrote to the latter, \"to serve his majesty, if I might be\nallowed, on some business of importance. I wish he may understand that I\nam no longer a boy. Thank God, I can begin to fly without the aid of\nothers' wings, and it is full time, as I believe, that I was out of\nswaddling-clothes.\"[276] In another letter he expresses his desire to\nhave some place more fitting the brother of such a monarch as Philip,\nand the son of such a father as Charles the Fifth.[277] On more than one\noccasion he alludes to the command against the Turk as the great object\nof his ambition.\n\nHis importunity to be allowed to resign his present office had continued\nfrom the beginning of summer, some months before the proper close of the\ncampaign. It may be thought to argue an instability of character, of\nwhich a more memorable example was afforded by him at a later period of\nlife. At length he was rejoiced by obtaining the royal consent to resign\nhis command and return to court.\n\n[Sidenote: MURDER OF ABEN-ABOO.]\n\nOn the eleventh of November, Don John repaired to Granada. Till the\nclose of the month he was occupied with making the necessary\narrangements preparatory to his departure. The greater part of the army\nwas paid off and disbanded. A sufficient number was reserved to garrison\nthe fortresses and to furnish detachments which were to scour the\ncountry and hunt down such Moriscoes as still held out in the mountains.\nAs Requesens was to take part in the expedition against the Ottomans,\nthe office of captain-general was placed in the hands of the valiant\nduke of Arcos. On the twenty-ninth of November, Don John, having\ncompleted his preparations, quitted Granada and set forth on his\njourney to Madrid, where the popular chieftain was welcomed with\nenthusiasm by the citizens, as a conqueror returned from a victorious\ncampaign. By Philip and his newly-married bride, Anne of Austria, he was\nno less kindly greeted; and it was not long before the king gave a\nsubstantial proof of his contentment with his brother, by placing in his\nhands the baton, offered by the allies, of generalissimo in the war\nagainst the Turks.\n\nThere was still one Morisco insurgent who refused to submit, and who had\nhitherto eluded every attempt to capture him, but whose capture was of\nmore importance than that of any other of his nation. This was\nAben-Aboo, the \"little king\" of the Alpujarras. His force of five\nthousand men had dwindled to scarcely more than four hundred. But they\nwere men devoted to his person, and seemed prepared to endure every\nextremity rather than surrender. Like the rest of his nation, the\nMorisco chief took refuge in the mountain caves, in such remote and\ninaccessible districts as had hitherto baffled every attempt to detect\nhis retreat. In March, 1571, an opportunity presented itself for making\nthe discovery.\n\nGranada was at this time the scene of almost daily executions. As the\nmiserable insurgents were taken, they were brought before Deza's\ntribunal, where they were at once sentenced by the inexorable president\nto the galleys or the gibbet, or the more horrible doom of being torn in\npieces with red-hot pincers. Among the prisoners sentenced to death, was\none Zatahari, who was so fortunate as to obtain a respite of his\npunishment at the intercession of a goldsmith named Barredo, a person of\nmuch consideration in Granada. From gratitude for this service, or\nperhaps as the price of it, Zatahari made some important revelations to\nhis benefactor respecting Aben-Aboo. He disclosed the place of his\nretirement and the number of his followers, adding, that the two persons\non whom he most relied were his secretary, Abou-Amer, and a Moorish\ncaptain named El Senix. The former of these persons was known to\nBarredo, who, in the course of his business, had frequent occasion to\nmake journeys into the Alpujarras. He resolved to open a correspondence\nwith the secretary, and, if possible, win him over to the Spanish\ninterests. Zatahari consented to bear the letter, on condition of a\npardon. This was readily granted by the president, who approved the\nplan, and who authorized the most liberal promises to Abou-Amer in case\nof his co-operation with Barredo.\n\nUnfortunately--or, rather, fortunately for Zatahari, as it proved,--he\nwas intercepted by El Senix, who, getting possession of the letter,\ncarried it to Abou-Amer. The loyal secretary was outraged by this\nattempt to corrupt him. He would have put the messenger to death, had\nnot El Senix represented that the poor wretch had undertaken the mission\nonly to save his life.\n\nPrivately the Moorish captain assured the messenger that Barredo should\nhave sought a conference with him, as he was ready to enter into\nnegotiations with the Christians. In fact, El Senix had a grudge against\nhis master, and had already made an attempt to leave his service and\nescape to Barbary.\n\nA place of meeting was accordingly appointed in the Alpujarras, to which\nBarredo secretly repaired. El Senix was furnished with an assurance,\nunder the president's own hand, of a pardon for himself and his friends,\nand of an annual pension of a hundred thousand maravedis, in case he\nshould bring Aben-Aboo, dead or alive, to Granada.\n\nThe interview could not be conducted so secretly but that an intimation\nof it reached the ears of Aben-Aboo, who resolved to repair at once to\nthe quarters of El Senix, and ascertain the truth for himself. That\nchief had secreted himself in a cabin in the neighbourhood. Aben-Aboo\ntook with him his faithful secretary and a small body of soldiers. On\nreaching the cave, he left his followers without, and, placing two men\nat the entrance, he, with less prudence than was usual with him, passed\nalone into the interior.\n\nThere he found El Senix, surrounded by several of his friends and\nkinsmen. Aben-Aboo, in a peremptory tone, charged him with having held\na secret correspondence with the enemy, and demanded the object of his\nlate interview with Barredo. Senix did not attempt to deny the charge,\nbut explained his motives by saying that he had been prompted only by a\ndesire to serve his master. He had succeeded so well, he said, as to\nobtain from the president an assurance that, if the Morisco would lay\ndown his arms, he should receive an amnesty for the past, and a liberal\nprovision for the future.\n\nAben-Aboo listened scornfully to this explanation; then, muttering the\nword, \"Treachery!\" he turned on his heel, and moved towards the mouth of\nthe cave, where he had left his soldiers, intending probably to command\nthe arrest of his perfidious officer. But he had not given them, it\nappears, any intimation of the hostile object of his visit to El Senix;\nand the men, supposing it to be on some matter of ordinary business, had\nleft the spot to see some of their friends in the neighbourhood. El\nSenix saw that no time was to be lost. On a signal which he gave, his\nfollowers attacked the two guards at the door, one of whom was killed on\nthe spot, while the other made his escape. They then all fell upon the\nunfortunate Aben-Aboo. He made a desperate defence. But though the\nstruggle was fierce, the odds were too great for it to be long. It was\nsoon terminated by the dastard Senix coming behind his master, and with\nthe butt-end of his musket dealing him a blow on the back, of his head\nthat brought him to the ground, where he was quickly despatched by a\nmultitude of wounds.[278]\n\nThe corpse was thrown out of the cavern. His followers, soon learning\ntheir master's fate, dispersed in different directions. The faithful\nsecretary fell shortly after into the hands of the Spaniards, who, with\ntheir usual humanity in this war, caused him to be drawn and quartered.\n\nThe body of Aben-Aboo was transported to the neighbourhood of Granada,\nwhere preparations were made for giving the dead chief a public entrance\ninto the city, as if he had been still alive. The corpse was set astride\non a mule, and supported erect in the saddle by a wooden frame, which\nwas concealed beneath ample robes. On one side of the body rode Barredo;\non the other, El Senix, bearing the scimitar and arquebuse of his\nmurdered master. Then followed the kinsmen and friends of the Morisco\nprince, with their arms by their side. A regiment of Castilian infantry\nand a troop of horse brought up the rear. As the procession defiled\nalong the street of Zacatin, it was saluted by salvoes of musketry,\naccompanied by peals of artillery from the ancient towers of the\nAlhambra, while the population of Granada, with eager though silent\ncuriosity, hurried out to gaze on the strange and ghastly spectacle.\n\nIn this way the company reached the great square of Vivarambla, where\nwere assembled the president, the duke of Arcos, and the principal\ncavaliers and magistrates of the city. On coming into their presence, El\nSenix dismounted, and, kneeling before Deza, delivered to him the arms\nof Aben-Aboo. He was graciously received by the president, who confirmed\nthe assurance which had been given him of the royal favour. The\nmiserable ceremony of public execution was then gone through with. The\nhead of the dead man was struck off. His body was given to the boys of\nthe city, who, after dragging it through the streets with scoffs and\nimprecations, committed it to the flames. Such was one of the lessons by\nwhich the Spaniards early stamped on the minds of their children an\nindelible hatred of the Morisco.\n\n[Sidenote: CHARACTER OF ABEN-ABOO.]\n\nThe head of Aben-Aboo, enclosed in a cage, was set up over the gate\nwhich opened on the Alpujarras. There, with the face turned towards his\nnative hills, which he had loved so well, and which had witnessed his\nbrief and disastrous reign, it remained for many a year. None ventured,\nby removing it, to incur the doom which an inscription on the cage\ndenounced on the offender: \"This is the head of the traitor Aben-Aboo.\nLet no one take it down, under penalty of death.\"[279]\n\nSuch was the sad end of Aben-Aboo, the last of the royal line of the\nOmeyades who ever ruled in the Peninsula. Had he lived in the peaceful\nand prosperous times of the Arabian empire in Spain, he might have\nswayed the sceptre with as much renown as the best of his dynasty.\nThough the blood of the Moor flowed in his veins, he seems to have been\nremarkably free from some of the greatest defects in the Moorish\ncharacter. He was temperate in his appetites, presenting in this respect\na contrast to the gross sensuality of his predecessor. He had a lofty\nspirit, was cool and circumspect in his judgments, and, if he could not\nboast that fiery energy of character which belonged to some of his\nhouse, he had a firmness of purpose not to be intimidated by suffering\nor danger. Of this he gave signal proof when, as the reader may\nremember, the most inhuman tortures could not extort from him the\ndisclosure of the lurking-place of his friends.[280] His qualities, as I\nhave intimated, were such as peculiarly adapted him to a time of\nprosperity and peace. Unhappily, he had fallen upon evil times, when his\ncountry lay a wreck at his feet; when the people, depressed by long\nservitude, were broken down by the recent calamities of war; when, in\nshort, it would not have been possible for the wisest and most warlike\nof his predecessors to animate them to a successful resistance against\nodds so overwhelming as those presented by the Spanish monarchy in the\nzenith of its power.\n\nThe Castilian chroniclers have endeavoured to fix a deep stain on his\nmemory, by charging him with the murder of El Habaqui, and with the\nrefusal to execute the treaty to which he had given his sanction. But,\nin criticising the conduct of Aben-Aboo, we must not forget the race\nfrom which he sprung, or the nature of its institutions. He was a\ndespot, and a despot of the Oriental type. He was placed in a\nsituation--much against his will, it may be added--which gave him\nabsolute control over the lives and fortunes of his people. His word was\ntheir law. He passed the sentence, and enforced its execution. El\nHabaqui he adjudged to be a traitor; and, in sentencing him to the\nbowstring, he inflicted on him only a traitor's doom.\n\nWith regard to the treaty, he spoke of himself as betrayed, saying that\nits provisions were not such as he had intended. And when we consider\nthat the instrument was written in the Spanish tongue; that it was\ndrafted by a Spaniard; finally, that the principal Morisco agent who\nsubscribed the treaty was altogether in the Spanish interest, as the\nfavours heaped on him without measure too plainly proved, it can hardly\nbe doubted that there were good grounds for the assertion of Aben-Aboo.\nFrom the hour of his accession, he seems to have devoted himself to the\ngreat work of securing the independence of his people. He could scarcely\nhave agreed to a treaty which was to leave that people in even a worse\nstate than before the rebellion. From what we know of his character, we\nmay more reasonably conclude that he was sincere when he told the\nSpanish envoy, Palacios, who had come to press the execution of the\ntreaty, and to remind him of the royal promises of grace, that \"his\npeople might do as they listed, but, for himself, he would rather live\nand die a Mussulman than possess all the favours which the king of Spain\ncould heap on him.\" His deeds corresponded with his words; and,\ndesperate as was his condition, he still continued to bid defiance to\nthe Spanish government, until he was cut off by the hand of a traitor.\n\nThe death of Aben-Aboo severed the last bond which held the remnant of\nthe Moriscoes together. In a few years the sword, famine, and the\ngallows had exterminated the outcasts who still lurked in the fastnesses\nof the mountains. Their places were gradually occupied by Christians,\ndrawn thither by the favourable terms which the government offered to\nsettlers. But it was long before the wasted and famine-stricken\nterritory could make a suitable return to the labours of the colonists.\nThey were ignorant of the country, and were altogether deficient in the\nagricultural skill necessary for turning its unpromising places to the\nbest account. The Spaniard, adventurous as he was, and reckless of\ndanger and difficulty in the pursuit of gain, was impatient of the\nhumble drudgery required for the tillage of the soil; and many a valley\nand hill-side which, under the Moriscoes, had bloomed with all the rich\nembroidery of cultivation, now relapsed into its primitive barrenness.\n\nThe exiles carried their superior skill and industry into the various\nprovinces where they were sent. Scattered as they were, and wide apart,\nthe presence of the Moriscoes was sure to be revealed by the more minute\nand elaborate culture of the soil, as the secret course of the\nmountain-stream is betrayed by the brighter green of the meadow. With\ntheir skill in husbandry they combined a familiarity with various kinds\nof handicraft, especially those requiring dexterity and fineness of\nexecution, that was unknown to the Spaniards. As the natural result of\nthis superiority, the products of their labour were more abundant, and\ncould be afforded at a cheaper rate than those of their neighbours. Yet\nthis industry was exerted under every disadvantage which a most cruel\nlegislation could impose on it. It would be hard to find in the pages of\nhistory a more flagrant example of the oppression of a conquered race,\nthan that afforded by the laws of this period in reference to the\nMoriscoes. The odious law of 1566, which led to the insurrection, was\nput in full force. By this the national songs and dances, the peculiar\nbaths of the Moriscoes, the _fetes_ and ceremonies which had come down\nto them from their ancestors, were interdicted under heavy penalties. By\nanother ordinance, dated October 6, 1572, still more cruel and absurd,\nthey were forbidden to speak or to write the Arabic, under penalty of\nthirty days' imprisonment in irons for the first offence, double that\nterm for the second, and for the third a hundred lashes and four years'\nconfinement in the galleys. By another monstrous provision in the same\nedict, whoever read, or even had in his possession, a work written or\nprinted in the Arabic, was to be punished with a hundred stripes and\nfour years in the galleys. Any contract or public instrument made in\nthat tongue was to be void, and the parties to it were condemned to\nreceive two hundred lashes and to tug at the oar for six years.[281]\n\n[Sidenote: FORTUNES OF THE MORISCOES.]\n\nBut the most oppressive part of this terrible ordinance related to the\nresidence of the Moriscoes. No one was allowed to change his abode, or\nto leave the parish or district assigned to him, without permission from\nthe regular authorities. Whoever did so, and was apprehended beyond\nthese limits, was to be punished with a hundred lashes and four years'\nimprisonment in the galleys. Should he be found within ten leagues of\nGranada, he was condemned, if between ten and seventeen years of age, to\ntoil as a galley-slave the rest of his days; if above seventeen, he was\nsentenced to death![282] On the escape of a Morisco from his limits, the\nhue and cry was to be raised, as for the pursuit of a criminal. Even his\nown family were required to report his absence to the magistrate; and\nin case of their failure to do this, although it should be his wife or\nhis children, says the law, they incurred the penalty of a whipping and\na month's imprisonment in the common gaol.[283]\n\nYet, in the face of these atrocious enactments, we find the Moriscoes\noccasionally making their escape into the province of Valencia, where\nnumbers of their countrymen were living as serfs on the estates of the\ngreat nobles, under whose powerful protection they enjoyed a degree of\ncomfort, if not of independence, unknown to their race in other parts of\nthe country. Some few, also, finding their way to the coast, succeeded\nin crossing the sea to Barbary. The very severity of the law served in\nsome measure to defeat its execution. Indeed, Philip, in more than one\ninstance in which he deemed that the edicts pressed too heavily on his\nMoorish vassals, judged it expedient to mitigate the penalty, or even to\ndispense with it altogether,--an act of leniency which seems to have\nfound little favour with his Castilian subjects.[284]\n\nYet, strange to say, under this iron system, the spirit of the\nMoriscoes, which had been crushed by their long sufferings in the war of\nthe rebellion, gradually rose again as they found a shelter in their new\nhomes, and resumed their former habits of quiet industry. Though\ndeprived of their customary amusements, their _fetes_, their songs, and\ntheir dances,--though debarred from the use of the language which they\nhad lisped from the cradle, which embodied their national traditions,\nand was associated with their fondest recollections,--they were said to\nbe cheerful, and even gay. They lived to a good age, and examples of\nlongevity were found among them, to which it was not easy to find a\nparallel among the Spaniards. The Moorish stock, like the Jewish, seems\nto have thriven under persecution.[285]\n\nOne would be glad to find any authentic data for an account of the\nactual population at the time of their expulsion from Granada. But I\nhave met with none. They must have been sorely thinned by the war of the\ninsurrection and the countless woes it brought upon the country. One\nfact is mentioned by the chroniclers, which shows that the number of the\nexiles must have been very considerable. The small remnant still left in\nGranada, with its lovely _vega_ and the valley of Lecrin, alone\nfurnished, we are told, over six thousand.[286] In the places to which\nthey were transported they continued to multiply to such an extent that\nthe Cortes of Castile, in the latter part of the century, petitioned the\nking not to allow the census to be taken, lest it might disclose to the\nMoriscoes the alarming secret of their increase of numbers.[287] Such a\npetition shows, as strongly as language can show, the terror in which\nthe Spaniards still stood of this persecuted race.\n\nYet the Moriscoes were scattered over the country in small and isolated\nmasses, hemmed in all around by the Spaniards. They were transplanted to\nthe interior, where, at a distance from the coast, they had no means of\ncommunicating with their brethren of Africa. They were without weapons\nof any kind; and, confined to their several districts, they had not the\npower of acting in concert together. There would seem to have been\nlittle to fear from a people so situated. But the weakest individual,\nwho feels that his wrongs are too great to be forgiven, may well become\nan object of dread to the person who has wronged him.\n\nThe course of the government in reference to the Moriscoes was clearly a\nfailure. It was as impolitic as it was barbarous. Nothing but the\nblindest fanaticism could have prevented the Spaniards from perceiving\nthis. The object of the government had been to destroy every vestige of\nnationality in the conquered race. They were compelled to repudiate\ntheir ancient usages, their festivals, their religion, their\nlanguage,--all that gave them a separate existence as a nation. But this\nserved only to strengthen in secret the sentiment of nationality. They\nwere to be divorced for ever from the past. But it was the mistake of\nthe government that it opened to them no future. Having destroyed their\nindependence as a nation, it should have offered them the rights of\ncitizenship, and raised them to an equality with the rest of the\ncommunity. Such was the policy of ancient Rome towards the nations which\nshe conquered; and such has been that of our own country towards the\ncountless emigrants who have thronged to our shores from so many distant\nlands. The Moriscoes, on the contrary, under the policy of Spain, were\ncondemned to exist as foreigners in the country,--as enemies in the\nmidst of the community into which they were thrown. Experience had\ntaught them prudence and dissimulation; and in all outward observances\nthey conformed to the exactions of the law. But in secret they were as\nmuch attached to their national institutions as were their ancestors\nwhen the caliphs of Cordova ruled over half the Peninsula. The\nInquisition rarely gleaned an apostate from among them to swell the\nhorrors of an _auto da fe_; but whoever recalls the facility with which,\nin the late rebellion, the whole population had relapsed into their\nancient faith, will hardly doubt that they must have still continued to\nbe Mahometans at heart.\n\nThus the gulf which separated the two races grew wider and wider every\nday. The Moriscoes hated the Spaniards for the wrongs which they had\nreceived from them. The Spaniards hated the Moriscoes the more, that\nthey had themselves inflicted these wrongs. Their hatred was further\nembittered by the feeling of jealousy caused by the successful\ncompetition of their rivals in the various pursuits of gain,--a\ncircumstance which forms a fruitful theme of complaint in the petition\nof the Cortes above noticed.[288] The feeling of hate became in time\nmingled with that of fear, as the Moriscoes increased in opulence and\nnumbers; and men are not apt to be over scrupulous in their policy\ntowards those whom they both hate and fear.\n\nWith these evil passions rankling in their bosoms, the Spaniards were\ngradually prepared for the consummation of their long train of\npersecutions by that last act, reserved for the reign of the imbecile\nPhilip the Third,--the expulsion of the Moriscoes from the\nPeninsula,--an act which deprived Spain of the most industrious and\ningenious portion of her population, and which must be regarded as one\nof the principal causes of the subsequent decline of the monarchy.\n\n[Sidenote: MARMOL--CIRCOURT.]\n\n     An historian less renowned than Mendoza, but of more importance to\n     one who would acquaint himself with the story of the Morisco\n     rebellion, is Luis del Marmol Carbajal. Little is known of him but\n     what is to be gathered from brief notices of himself in his works.\n     He was a native of Granada, but we are not informed of the date of\n     his birth. He was of a good family, and followed the profession of\n     arms. When a mere youth, as he tells us, he was present at the\n     famous siege of Tunis, in 1535. He continued in the imperial\n     service two-and-twenty years. Seven years he was a captive, and\n     followed the victorious banner of Mohammed, Scherif of Morocco, in\n     his campaigns in the west of Africa. His various fortunes and his\n     long residence in different parts of the African continent,\n     especially in Barbary and Egypt, supplied him with abundant\n     information in respect to the subjects of his historical inquiries;\n     and, as he knew the Arabic, he made himself acquainted with such\n     facts as were to be gleaned from books in that language. The fruits\n     of his study and observation he gave to the world in his\n     \"_Descripcion General de Africa_,\" a work in three volumes folio,\n     the first part of which appeared at Granada in 1573. The remainder\n     was not published till the close of the century.\n\n     The book obtained a high reputation for its author, who was much\n     commended for the fidelity and diligence with which he had pushed\n     his researches in a field of letters into which the European\n     scholar had as yet rarely ventured to penetrate.\n\n     In the year 1600 appeared, at Malaga, his second work, the\n     \"_Historia del Rebelion y Castigo de los Moriscos del Reyno de\n     Granada_,\" in one volume, folio. For the composition of this\n     history the author was admirably qualified, not only by his\n     familiarity with all that related to the character and condition of\n     the Moriscoes, but by the part which he had personally taken in the\n     war of the insurrection. He held the office of commissary in the\n     royal army, and served in that capacity from the commencement of\n     the war to its close. In the warm colouring of the narrative, and\n     in the minuteness of its details, we feel that we are reading the\n     report of one who has himself beheld the scenes which he describes.\n     Indeed, the interest which, as an actor, he naturally takes in the\n     operations of the war, leads to an amount of detail which may well\n     be condemned as a blemish by those who do not feel a similar\n     interest in the particulars of the struggle. But if his style have\n     somewhat of the rambling, discursive manner of the old Castilian\n     chronicler, it has a certain elegance in the execution, which\n     brings it much nearer to the standard of a classic author. Far from\n     being chargeable with the obscurity of Mendoza, Marmol is\n     uncommonly perspicuous. With a general facility of expression, his\n     language takes the varied character suited to the theme, sometimes\n     kindled into eloquence and occasionally softened into pathos, for\n     which the melancholy character of his story afforded too many\n     occasions. Though loyal to his country and his faith, yet he shows\n     but few gleams of the fiery intolerance that belonged to his\n     nation, and especially to that portion of it which came into\n     collision with the Moslems. Indeed, in more than one passage of his\n     work we may discern gleams of that Christian charity which, in\n     Castile was the rarest, as it was, unhappily, the least precious of\n     virtues, in the age in which he lived.\n\n     In the extensive plan adopted by Marmol, his history of the\n     rebellion embraces a preliminary notice of the conquest of Granada,\n     and of that cruel policy of the conquerors which led to the\n     insurrection. The narrative, thus complete, supplied a most\n     important hiatus in the annals of the country. Yet notwithstanding\n     its importance in this view, and its acknowledged merit as a\n     literary composition, such was the indifference of the Spaniards to\n     their national history, that it was not till the close of the last\n     century, in 1797, that a second edition of Marmol's work was\n     permitted to appear. This was in two volumes, octavo, from the\n     press of Sancha, at Madrid,--the edition used in the preparation of\n     these pages.\n\n     The most comprehensive, and by far the most able history of the\n     Moors of Spain with which I am acquainted, is that of the Count\n     Albert de Circourt,--\"_Histoire des Arabes en Espagne_.\" Beginning\n     with the beginning, the author opens his narrative with the\n     conquest of the Peninsula by the Moslems. He paints in glowing\n     colours the magnificent empire of the Spanish caliphs. He dwells\n     with sufficient minuteness on those interminable feuds which,\n     growing out of a diversity of races and tribes, baffled every\n     attempt at a permanent consolidation under one government. Then\n     comes the famous war of Granada, with the conquest of the country\n     by the \"Catholic Kings;\" and the work closes with the sad tale of\n     the subsequent fortunes of the conquered races until their final\n     expulsion from the Peninsula. Thus the rapidly shifting scenes of\n     this most picturesque drama, sketched by a master's hand, are\n     brought in regular succession before the eye of the reader.\n\n     In conducting his long story, the author, far from confining\n     himself to a dry record of events, diligently explores the causes\n     of these events. He scrutinizes with care every inch of debateable\n     ground which lies in his path. He enriches his narrative with\n     copious disquisitions on the condition of the arts, and the\n     progress made by the Spanish Arabs in science and letters; thus\n     presenting a complete view of that peculiar civilization which so\n     curiously blended together the characteristic elements of European\n     and Oriental culture.\n\n     If, in pursuing his speculations, M. de Circourt may be sometimes\n     thought to refine too much, it cannot be denied that they are\n     distinguished by candour and by a philosophical spirit. Even when\n     we may differ from his conclusions, we must allow that they are the\n     result of careful study, and display an independent way of\n     thinking. I may regret that in one important instance--the policy\n     of the government of Ferdinand and Isabella--he should have been\n     led to dissent from the opinions which I had expressed in my\n     history of those sovereigns. It is possible that the predilection\n     which the writer, whether historian or novelist, naturally feels\n     for his hero when his conduct affords any ground for it, may have\n     sometimes seduced me from the strict line of impartiality in my\n     estimate of character and motives of action. I see, however, no\n     reason to change the conclusions at which I had arrived after a\n     careful study of the subject. Yet I cannot deny that the labours of\n     the French historian have shed a light upon more than one obscure\n     passage in the administration of Ferdinand and Isabella, for which\n     the student of Spanish history owes him a debt of gratitude.\n\n\n\n\nCHAPTER IX.\n\nWAR WITH THE TURKS.\n\nLeague against the Turks--Preparations for the War--Don John\nCommander-in-Chief--His Reception at Naples--His Departure from Messina.\n\n1570-1571.\n\n\nWhile Philip was occupied with the Morisco insurrection, his attention\nwas called to another quarter, where a storm was gathering that menaced\nSpain in common with the rest of Christendom. In 1566, Solyman the\nMagnificent closed his long and prosperous reign. His son and successor,\nSelim the Second, possessed few of the qualities of his great father.\nBred in the seraglio, he showed the fruits of his education in his\nindolent way of life, and in the free indulgence of the most licentious\nappetites. With these effeminate tastes, he inherited the passion for\nconquest which belonged not only to his father, but to the whole of his\nwarlike dynasty. Not that, like them, he headed his armies in the field.\nThese were led by valiant commanders, who had learned the art of war\nunder Solyman. Selim was, above all, fortunate in possessing for his\ngrand vizier a minister whose untiring industry and remarkable talents\nfor business enabled him to bear on his own shoulders the whole burden\nof government. It was fortunate for the state, as well as for the\nsultan, that Mahomet had the art to win the confidence of his master,\nand to maintain it unshaken through the whole of his reign.\n\nThe scheme which most occupied the thoughts of Selim was the conquest of\nCyprus. This island, to which nature had been so prodigal of her gifts,\nbelonged to Venice. Yet, placed at the extremity of the Mediterranean,\nit seemed in a manner to command the approaches to the Dardanelles,\nwhile its line of coast furnished convenient ports, from which swarms of\ncruisers might sally forth in time of war, and plunder the Turkish\ncommerce.\n\nSelim, resolved on the acquisition of Cyprus, was not slow in devising a\npretext for claiming it from Venice as a part of the Ottoman empire. The\nrepublic, though willing to make almost any concession rather than come\nto a rupture with the colossal power under whose shadow she lay, was not\nprepared to surrender without a struggle the richest gem in her colonial\ndiadem. War was accordingly declared against her by the Porte, and vast\npreparations were made for fitting out an armament against Cyprus.\nVenice, in her turn, showed her usual alacrity in providing for the\nencounter. She strained her resources to the utmost. In a very short\ntime she equipped a powerful fleet, and took measures to place the\nfortifications of Cyprus in a proper state of defence. But Venice no\nlonger boasted a navy such as in earlier days had enabled her to humble\nthe pride of Genoa, and to ride the unquestioned mistress of the\nMediterranean. The defences of her colonies, moreover, during her long\nrepose, had gradually fallen into decay. In her extremity, she turned to\nthe Christian powers of Europe, and besought them to make common cause\nwith her against the enemy of Christendom.\n\n[Sidenote: LEAGUE AGAINST THE TURKS.]\n\nFortunately the chair of St. Peter was occupied, at this crisis, by Pius\nthe Fifth, one of those pontiffs who seem to have been called forth by\nthe exigencies of the time, to uphold the pillars of Catholicism, as\nthey were yet trembling under the assaults of Luther. Though he was near\nseventy years of age, the fire of youth still glowed in his veins. He\npossessed all that impetuous eloquence which, had he lived in the days\nof Peter the Hermit, would have enabled him, like that enthusiast, to\nrouse the nations of Europe to a crusade against the infidel. But the\ndays of the crusades were past; and a summons from the Vatican had no\nlonger the power to stir the souls of men like a voice from heaven. The\ngreat potentates of Europe were too intent on their own selfish schemes\nto be turned from these by the apprehension of a danger so remote as\nthat which menaced them from the East. The forlorn condition of Venice\nhad still less power to move them; and that haughty republic was now\nmade to feel, in the hour of her distress, how completely her perfidious\nand unscrupulous policy had estranged from her the sympathies of her\nneighbours.\n\nThere was one monarch, however, who did not close his ears against the\nappeal of Venice,--and that monarch, one of more importance to her cause\nthan any other, perhaps all others united. In the spring of 1570, Luigi\nTorres, clerk of the apostolic chamber, was sent to Spain by Pius the\nFifth, to plead the cause of the republic. He found the king at Ecija,\non the route from Cordova, where he had been for some time presiding\nover a meeting of the Cortes. The legate was graciously received by\nPhilip, to whom he presented a letter from his holiness, urging the\nmonarch, in the most earnest and eloquent language, to give succour to\nVenice, and to unite with her in a league against the infidel. Philip\ndid not hesitate to promise his assistance in the present emergency; but\nhe had natural doubts as to the expediency of binding himself by a\nleague with a power on whose good faith he had little reliance. He\npostponed his decision until his arrival at Seville. Accompanied by the\nlegate, on the first of May, he made his solemn entry into the great\ncommercial capital of the South. It was his first visit there, and he\nwas received with tumultuous joy by the loyal inhabitants. Loyalty to\ntheir monarchs has ever been a predominant trait of the Spaniards; and\nto none of their princes did they ever show it in larger measure than to\nPhilip the Second. No one of them, certainly, was more thoroughly\nSpanish in his own nature, or more deeply attached to Spain.\n\nAfter swearing to respect the privileges of the city, the king received\nthe homage of the authorities. He then rode through the streets under a\ngorgeous canopy, upheld by the principal magistrates, and visited the\nchurches and monasteries, hearing _Te Deum_, and offering up his prayers\nin the cathedral. He was attended by a gay procession of nobles and\ncavaliers, while the streets of the populous city were thronged with\nmultitudes, filled with enthusiasm at the presence of their sovereign.\nBy this loyal escort Philip was accompanied to the place of his\nresidence, the royal alcazar of Seville. Here he prolonged his stay for\na fortnight, witnessing the shows and festivals which had been prepared\nfor his entertainment. At his departure he received a more substantial\nproof of the attachment of the citizens, in a donation of six hundred\nthousand ducats. The object of this magnificent present was to defray,\nin part, the expenses of the king's approaching marriage with his fourth\nwife, Anne of Austria, the daughter of his cousin, the emperor\nMaximilian. The fair young bride had left her father's court, and was\nalready on her way to Madrid, where her nuptials were to be celebrated,\nand where she was to take the place of the lovely Isabella, whose death,\nnot two years since, had plunged the nation in mourning.[289]\n\nWhile at Seville, Philip laid the subject of the league before his\nministers. Some of these, and among the number Espinosa, president of\nthe council of Castile, entertained great doubts as to the policy of\nbinding Spain by a formal treaty with the Venetian republic. But, with\nall his distrust of that power, Philip took a broader view of the matter\nthan his ministers. Independently of his willingness to present himself\nbefore the world as the great champion of the Faith, he felt that such\nan alliance offered the best opportunity for crippling the maritime\npower of Turkey, and thus providing for the safety of his own colonial\npossessions in the Mediterranean. After much deliberation, he dismissed\nthe legate with the assurance that, notwithstanding the troubles which\npressed on him both in the Low Countries and in Granada, he would\nfurnish immediate succours to Venice, and would send commissioners to\nRome, with full powers to unite with those of the pope and the republic\nin forming a treaty of alliance against the Ottoman Porte. The papal\nenvoy was charged with a letter to the same effect, addressed by Philip\nto his holiness.\n\nThe ensuing summer, the royal admiral, the famous John Andrew Doria, who\nwas lying with a strong squadron off Sicily, put to sea by the king's\norders. He was soon after reinforced by a few galleys which were\nfurnished by his holiness, and placed under the command of Mark Antonio\nColonna, the representative of one of the most ancient and illustrious\nhouses in Rome. On the last of August, 1570, the combined fleet effected\nits junction with the Venetians at Candia, and a plan of operations was\nimmediately arranged. It was not long before the startling intelligence\narrived that Nicosia, the capital of Cyprus, had been taken and sacked\nby the Turks, with all the circumstances of cruelty which distinguish\nwars in which the feeling of national hostility is embittered by\nreligious hatred. The plan was now to be changed. A dispute arose among\nthe commanders as to the course to be pursued. No one had authority\nenough to enforce compliance with his own opinion. The dispute ended in\na rupture. The expedition was abandoned; and the several commanders\nreturned home with their squadrons, without having struck a blow for the\ncause. It was a bad omen for the success of the league.[290]\n\nStill the stout-hearted pontiff was not discouraged. On the contrary, he\nendeavoured to infuse his own heroic spirit into the hearts of his\nallies, giving them the most cheering assurances for the future, if they\nwould but be true to themselves. Philip did not need this encouragement.\nOnce resolved, his was not a mind lightly to be turned from its purpose.\nVenice, on the other hand, soon showed that the Catholic king had good\nreason for distrusting her fidelity. Appalled by the loss of Nicosia,\nwith her usual inconstancy, she despatched a secret agent to\nConstantinople, to see if some terms might not yet be made with the\nSultan. The negotiation could not be managed so secretly, however, but\nthat notice of it reached the ears of Pius the Fifth. He forthwith\ndespatched an envoy to the republic to counteract this measure, and to\npersuade the Venetians to trust to their Christian allies rather than to\nthe Turks, the enemies of their country and their religion. The person\nselected for this mission was Colonna, who was quite as much\ndistinguished for his address as for his valour. He performed his task\nwell. He represented so forcibly to the government that the course he\nrecommended was the one dictated not less by interest than by honour,\nthat they finally acquiesced, and recalled their agent from\nConstantinople. It must be acknowledged that Colonna's arguments were\ngreatly strengthened by the cold reception given to the Venetian envoy\nat Constantinople, where it was soon seen that the conquest of the\ncapital had by no means tended to make the sultan relax his hold on\nCyprus.[291]\n\n[Sidenote: LEAGUE AGAINST THE TURKS.]\n\nTowards the close of 1570, the deputies from the three powers met in\nRome to arrange the terms of the league. Spain was represented by the\ncardinals Granvelle and Pacheco, together with the ambassador, Juan de\nZuniga, all three at that time being resident in Rome. It will readily\nbe believed that the interests of Spain would not suffer in the hands of\na commission with so skilful a tactician as Granvelle to direct it.\n\nYet though the parties seemed to be embarked in a common cause, there\nwas found much difficulty in reconciling their different pretensions.\nThe deputies from Venice, in the usual spirit of her diplomacy, regarded\nthe league as exclusively designed for her benefit; in other words, for\nthe protection of Cyprus against the Turks. The Spanish commissioners\ntook a wider view, and talked of the war as one waged by the Christian\nagainst the Infidel; against the Moors no less than the Turks. In this\npolitic view of the matter, the Catholic king was entitled to the same\nprotection for his colonies on the coast of Africa as Venice claimed for\nCyprus.\n\nAnother cause of disagreement was the claim of each of the parties to\nselect a commander-in-chief for the expedition from its own nation. This\npre-eminence was finally conceded to Spain, as the power that was to\nbear the largest share of the expenses.\n\nIt was agreed that the treaty should be permanent in its duration, and\nshould be directed against the Moors of Tunis, Tripoli, and Algiers, as\nwell as against the Turks; that the contracting parties should furnish\ntwo hundred galleys, one hundred transports and smaller vessels, fifty\nthousand foot, and four thousand five hundred horse, with the requisite\nartillery and munitions; that by April, at farthest, of every succeeding\nyear, a similar force should be held in readiness by the allies for\nexpeditions to the Levant; and that any year in which there was no\nexpedition in common, and either Spain or the republic should desire to\nengage in one on her own account against the Infidel, the other\nconfederates should furnish fifty galleys towards it; that if the enemy\nshould invade the dominions of any of the three powers, the others\nshould be bound to come to the aid of their ally; that three-sixths of\nthe expenses of the war should be borne by the Catholic king, two-sixths\nby the republic, the remaining sixth by the Holy See; that the Venetians\nshould lend his holiness twelve galleys, which he was to man and equip\nat his own charge, as his contribution towards the armament; that each\npower should appoint a captain-general; that the united voices of the\nthree commanders should regulate the plan of operations; that the\nexecution of this plan should be entrusted to the captain-general of the\nleague, and that this high office should be given to Don John of\nAustria; that, finally, no one of the parties should make peace, or\nenter into a truce with the enemy, without the knowledge and consent of\nthe others.[292]\n\nSuch were the principal provisions of the famous treaty of the Holy\nLeague. The very first article declares this treaty perpetual in its\nnature. Yet we should be slow to believe that the shrewd and politic\nstatesmen who directed the affairs of Spain and the republic could for a\nmoment believe in the perpetuity of a contract which imposed such\nburdensome obligations on the parties. In fact, the league did not hold\ntogether two years. But it held together long enough to accomplish a\ngreat result, and as such occupies an important place in the history of\nthe times.\n\nAlthough a draft of the treaty had been prepared in the latter part of\nthe preceding year, it was not ratified till 1571.[293] On the\ntwenty-fourth of May, the pope caused it to be read aloud in full\nconsistory. He then, laying his hand on his breast, solemnly swore to\nthe observance of it. The ambassadors of Spain and Venice made oath to\nthe same effect, on behalf of their governments, placing their hands on\na missal with a copy of the Gospels beneath it. On the day following,\nafter mass had been performed, the treaty was publicly proclaimed in the\nchurch of St. Peter.[294]\n\nThe tidings of the alliance of the three powers caused a great sensation\nthroughout Christendom. Far from dismaying the sultan, however, it only\nstimulated him to greater exertions. Availing himself of the resources\nof his vast empire, he soon got together a powerful fleet, partly drawn\nfrom his own dominions, and in part from those of the Moslem powers on\nthe Mediterranean, who acknowledged allegiance to the Porte. The armada\nwas placed under the command of Selim's brother-in-law, the Pacha Piali,\na man of an intrepid spirit, who had given many proofs of a humane and\ngenerous nature; qualities more rare among the Turks, perhaps among all\nnations, than mere physical courage.\n\nEarly in the spring of 1571, the Ottoman admiral sailed out of the\nGolden Horn, and directed his course towards Candia. Here he remained\nuntil joined by a strong Algerine force under the redoubtable corsair\nUluch Ali,--a Calabrian renegade, who had risen from the humblest\ncondition to the post of dey of Algiers. Early in the season the\ncombined fleets sailed for the Adriatic; and Piali, after landing and\nlaying waste the territory belonging to the republic, detached Uluch\nwith his squadron to penetrate higher up the gulf. The Algerine, in\nexecuting these orders, advanced so near to Venice as to throw the\ninhabitants of that capital into a consternation such as they had not\nfelt since the cannon of the Genoese, two centuries before, had\nresounded over their waters. But it was not the dey's purpose to engage\nin so formidable an enterprise as an assault upon Venice; and soon\ndrawing off, he joined the commander-in-chief at Corfu, where they\nwaited for tidings of the Christian fleet.[295]\n\nThe indefatigable Pius, even before the treaty was signed, had\ndespatched his nephew, Cardinal Alessandrino, to the different courts,\nto rouse the drooping spirits of the allies, and to persuade other\nprinces of Christendom to join the league. In the middle of May, the\nlegate, attended by a stately train of ecclesiastics, appeared at\nMadrid. Philip gave him a reception that fully testified his devotion to\nthe Holy See. The king's brother, Don John, and his favourite minister,\nRuy Gomez de Silva, with some of the principal nobles, waited at once on\nthe cardinal who had taken up his quarters in the suburbs, at the\nDominican monastery of Atocha, tenanted by brethren of his own order. On\nthe following morning the papal envoy made his entrance, in great state,\ninto the capital. He was mounted on a mule, gorgeously caparisoned, the\ngift of the city. John of Austria rode on his right; and he was escorted\nby a pompous array of prelates and grandees, who seemed to vie with one\nanother in the splendour of their costumes. On the way he was met by the\nroyal cavalcade. As the legate paid his obeisance to the monarch, he\nremained with his head uncovered; and Philip, with a similar act of\ncourtesy, while he addressed a few remarks to the churchman, held his\nhat in his hand.[296] He then joined the procession, riding between the\nlegate on the right and his brother on the left, who was observed, from\ntime to time, to take part in the conversation,--a circumstance\noccasioning some surprise, says an historian, as altogether contrary to\nthe established etiquette of the punctilious Castilian court.[297]\n\n[Sidenote: PREPARATIONS FOR THE WAR.]\n\nThe ceremonies were concluded by religious services in the church of\nSanta Maria, where the legate, after preaching a discourse, granted all\npresent a full remission of the pains of purgatory for two hundred\nyears.[298] A gift of more worth, in a temporal view, was the grant to\nthe king of the _cruzada_, the _excusada_, and other concessions of\necclesiastical revenue, which the Roman see knows so well how to bestow\non the champions of the Faith. These concessions came in good time to\nsupply the royal coffers, sorely drained by the costly preparations for\nthe war.\n\nMeanwhile, the Venetians were pushing forward their own preparations\nwith their wonted alacrity,--indeed, with more alacrity than\nthoroughness. They were prompt in furnishing their quota of vessels, but\ndiscreditably remiss in their manner of equipping them. The fleet was\nplaced under the charge of Sebastian Veniero, a noble who had grown grey\nin the service of his country. Zanne, who had had the command of the\nfleet in the preceding summer, was superseded on the charge of\nincapacity, shown especially in his neglect to bring the enemy to\naction. His process continued for two years, without any opportunity\nbeing allowed to the accused of appearing in his own vindication. It was\nfinally brought to a close by his death,--the consequence, as it is\nsaid, of a broken heart. If it were so, it would not be a solitary\ninstance of such a fate in the annals of the stern republic. Before\nmidsummer the new admiral sailed with his fleet, or as much of it as was\nthen ready, for the port of Messina, appointed as the place of\nrendezvous for the allies. Here he was soon joined by Colonna, the papal\ncommander, with the little squadron furnished by his holiness; and the\ntwo fleets lay at anchor, side by side, in the capacious harbour,\nwaiting the arrival of the rest of the confederates and of John of\nAustria.\n\nPreparations for the war were now going actively forward in Spain.\nPreparations on so large a scale had not been seen since the war with\nPaul the Fourth and Henry the Third, which ushered in Philip's\naccession. All the great ports in the Peninsula, as well as in the\nkingdom of Naples, in Sicily, in the Balearic Isles, in every part of\nthe empire in short, swarmed with artisans, busily engaged in fitting\nout the fleet which was to form Philip's contingent to the armament. By\nthe terms of the treaty, he was to bear one-half of the charges of the\nexpedition. In his naval preparations he spared neither cost nor care.\nNinety royal galleys, and more than seventy ships of small dimensions,\nwere got in readiness in the course of the summer. They were built and\nequipped in that thorough manner which vindicated the pre-eminence in\nnaval architecture claimed by Spain, and formed a strong contrast to the\nslovenly execution of the Venetians.[299]\n\nLevies of troops were at the same time diligently enforced in all parts\nof the monarchy. Even a corps of three thousand German mercenaries was\nsubsidized for the campaign. Troops were drawn from the veteran\ngarrisons in Lombardy and the kingdom of Naples. As the Morisco\ninsurrection was fortunately quelled, the forces engaged in it, among\nwhom were the brave Neapolitan battalion and its commander, Padilla,\ncould now be employed in the war against the Turk.\n\nBut it can hardly be said to have required extraordinary efforts to fill\nthe ranks on the present occasion; for seldom had a war been so popular\nwith the nation. Indeed, the Spaniards entered into it with an alacrity\nwhich might well have suggested the idea that their master had engaged\nin it on his own account, rather than as an ally. It was, in truth, a\nwar that appealed in a peculiar manner to the sensibilities of the\nCastilian, familiar from his cradle with the sound of the battle-cry\nagainst the Infidel. The whole number of infantry raised by the\nconfederates amounted to twenty-nine thousand. Of this number Spain\nalone sent over nineteen thousand well-appointed troops, comprehending\nnumerous volunteers, many of whom belonged to the noblest houses of the\nPeninsula.[300]\n\nOn the sixth of June, Don John, after receiving the last instructions of\nhis brother, set out from Madrid on his journey to the south. Besides\nhis own private establishment, making a numerous train, he was escorted\nby a splendid company of lords and cavaliers, eager to share with him in\nthe triumphs of the Cross. Anxious to reach the goal, he pushed forward\nat a more rapid rate than was altogether relished by the rest of the\ncavalcade. Yet, notwithstanding this speed on the road, there were\nmatters that claimed his attention in the towns through which he passed\nthat occasioned some delay. His journey had the appearance of a royal\nprogress. The castles of the great lords were thrown open with princely\nhospitality to receive him and his suite. In the chief cities, as\nSaragossa and Barcelona, he was entertained by the viceroys with all the\npomp and ceremony that could have been shown to the king himself. He\nremained some days in the busy capital of Catalonia, and found there\nmuch to engage his attention in the arsenals and dockyards, now alive\nwith the bustle of preparation. He then made a brief pilgrimage to the\nneighbouring hermitage of our Lady of Montserrat, where he paid his\ndevotions, and conversed with the holy fathers, whom he had always\ndeeply reverenced, and had before visited in their romantic solitudes.\n\n[Sidenote: DON JOHN'S RECEPTION AT NAPLES.]\n\nEmbarking at Barcelona, he set sail with a squadron of more than thirty\ngalleys,--a force strong enough to guard against the Moslem corsairs in\nthe Mediterranean, and landed, on the twenty-fifth, at Genoa. The doge\nand the senate came out to welcome him, and he was lodged during his\nstay in the palace of Andrew Doria. Here he received embassies and\ncongratulatory addresses from the different princes of Italy. He had\nalready been greeted with an autograph letter, couched in the most\nbenignant terms, from the sovereign pontiff. To all these communications\nDon John was careful to reply. He acquainted his holiness, in\nparticular, with the whole course of his proceedings. While on the way,\nhe had received a letter from his brother, giving him a full catalogue\nof the appropriate titles by which each one of his correspondents should\nbe addressed. Nor was this list confined to crowned heads, but\ncomprehended nobles and cavaliers, of every degree.[301] In no country\nhas the perilous code of etiquette been more diligently studied than in\nSpain, and no Spaniard was better versed in it than Philip.\n\nPursuing his route by water, Don John, in the month of August, dropped\nanchor in the beautiful bay of Naples. Arrangements had been made in\nthat city for his reception on a more magnificent scale than any he had\nwitnessed on his journey. Granvelle, who had lately been raised to the\npost of viceroy, came forth, at the head of a long and brilliant\nprocession, to welcome his royal guest. The houses that lined the\nstreets were hung with richly-tinted tapestries, and gaily festooned\nwith flowers. The windows and verandahs were graced with the beauty and\nfashion of that pleasure-loving capital; and many a dark eye sparkled as\nit gazed on the fine form and features of the youthful hero, who at the\nage of twenty-four had come to Italy to assume the baton of command, and\nlead the crusade against the Moslems. His splendid dress of white velvet\nand cloth of gold set off his graceful person to advantage. A crimson\nscarf floated loosely over his breast; and his snow-white plumes,\ndrooping from his cap, mingled with the yellow curls that fell in\nprofusion over his shoulders. It was a picture which the Italian maiden\nmight love to look on. It was certainly not the picture of the warrior\nsheathed in the iron panoply of war. But the young prince, in his\ngeneral aspect, might be relieved from the charge of effeminacy, by his\ntruly chivalrous bearing and the dauntless spirit which beamed from his\nclear blue eye. In his own lineaments he seemed to combine all that was\nmost comely in the lineaments of his race. Fortunately he had escaped\nthe deformity of the heavy Burgundian lip, which he might perhaps have\nexcused, as establishing his claims to a descent from the imperial house\nof Hapsburg.[302]\n\nDon John had found no place more busy with preparations for the campaign\nthan Naples. A fleet was riding at anchor in her bay, ready to sail\nunder the command of Don Alvaro Bazan, first marquis of Santa Cruz, a\nnobleman who had distinguished himself by more than one gallant\nachievement in the Mediterranean, and who was rapidly laying the\nfoundations of a fame that was one day to eclipse that of every other\nadmiral in Castile.\n\nTen days Don John remained at Naples, detained by contrary winds. Though\nimpatient to reach Messina, his time passed lightly amidst the _fetes_\nand brilliant spectacles which his friendly hosts had provided for his\nentertainment. He entered gaily into the revels; for he was well skilled\nin the courtly and chivalrous exercises of the day. Few danced better\nthan he, or rode, or fenced, or played at tennis with more spirit and\nskill, or carried off more frequently the prizes of the tourney. Indeed,\nhe showed as much ambition to excel in the mimic game of war as on the\nfield of battle. With his accomplishments and personal attractions, we\nmay well believe that Don John had little reason to complain of coldness\nin the fair dames of Italy. But he seems to have been no less a\nfavourite with the men. The young cavaliers, in particular, regarded him\nas the very mirror of chivalry, and studiously formed themselves on him\nas their model. His hair clustered thickly round his temples, and he was\nin the habit of throwing it back, so as to display his fine forehead to\nadvantage. This suited his physiognomy. It soon became the mode with\nthe gallants of the court; and even those whose physiognomies it did not\nsuit were no less careful to arrange their hair in the same manner.\n\nWhile at Naples he took part in a ceremony of an interesting and\nsignificant character. It was on the occasion of the presentation of a\nstandard sent by Pius the Fifth for the Holy War. The ceremony took\nplace in the church of the Franciscan convent of Santa Chiara. Granvelle\nofficiated on the occasion. Mass was performed by the cardinal-viceroy\nin his pontificals. _Te Deum_ was then chanted, after which Don John,\napproaching the altar with a slow and dignified step, gracefully knelt\nbefore the prelate, who, first delivering to him the baton of\ngeneralissimo, in the name of his holiness, next placed in his hands the\nconsecrated standard. It was of azure damask. A crucifix was embroidered\non the upper part of the banner, while below were the arms of the\nChurch, with those of Spain on the right, and of Venice on the left,\nunited by a chain, from which were suspended the arms of John of\nAustria. The prelate concluded the ceremony by invoking the blessing of\nHeaven on its champion, and beseeching that he might be permitted to\ncarry the banner of the Cross victorious over its enemies. The choir of\nthe convent then burst forth into a triumphant peal, and the people from\nevery quarter of the vast edifice shouted \"Amen!\"[303]\n\nIt was a striking scene, pregnant with matter for meditation to those\nwho gazed on it. For what could be more striking than the contrast\nafforded by these two individuals,--the one in the morning of life, his\neye kindling with hope and generous ambition, as he looked into the\nfuture and prepared to tread the path of glory under auspices as\nbrilliant as ever attended any mortal; the other drawing near to the\nevening of his day, looking to the past rather than the future, with\npale and thoughtful brow, as of one who, after many a toilsome day and\nsleepless night, had achieved the proud eminence for which his companion\nwas panting,--and had found it barren!\n\nThe wind having become more favourable, Don John took leave of the gay\ncapital of the South, and embarked for Messina, which he reached on the\ntwenty-fifth of August. If in other places he had seen preparations for\nwar, here he seemed to be brought on the very theatre of war. As he\nentered the noble port, he was saluted with the thunders of hundreds of\npieces of ordnance from the combined fleets of Rome and Venice, which\nlay side by side awaiting his arrival. He landed beneath a triumphal\narch of colossal dimensions, embossed with rich plates of silver, and\ncuriously sculptured with emblematical bas-reliefs, and with\ncomplimentary legends in Latin verse, furnished by the classical poets\nof Italy.[304] He passed under two other arches of similar rich and\nelaborate construction, as he rode into the town amidst the ringing of\nbells, the cheers of the multitude, the waving of scarfs and\nhandkerchiefs from the balconies, and other lively demonstrations of the\npublic joy, such as might have intoxicated the brain of a less ambitious\nsoldier than John of Austria. The festivities were closed in the evening\nby a general illumination of the city, and by a display of fireworks\nthat threw a light far and wide over the beautiful harbour and the\ncountless ships that floated on its waters.\n\n[Sidenote: THE ARMADA OF THE ALLIES.]\n\nNothing could be finer, indeed, whether by day or by night, than the\nspectacle presented by the port of Messina. Every day a fresh\nreinforcement of squadrons, or of single galleys or brigantines, under\nsome brave adventurer, entered the harbour to swell the numbers of the\ngreat armada. Many of these vessels, especially the galleys, were richly\ncarved and gilt, after the fashion of the time, and with their\nmany- streamers, and their flags displaying the arms of their\nseveral states, made a magnificent show as they glanced over the waters.\nNone, in the splendour of their decorations, exceeded the _Real_, as the\ngalley of the commander-in-chief was termed. It was of great size, and\nhad been built in Barcelona, famous for its naval architecture all the\nworld over. The stern of the vessel was profusely decorated with emblems\nand devices drawn from history. The interior was furnished in a style of\nluxury that seemed to be designed for pleasure, rather than for the\nrough duties of war. But the galley was remarkable for both strength and\nspeed,--the two most essential qualities in the construction of a ship.\nOf this she gave ample evidence in her contest with the Turk.[305]\n\nThe whole number of vessels in the armada, great and small, amounted to\nsomething more than three hundred. Of these full two-thirds were \"royal\ngalleys.\" Venice alone contributed one hundred and six, besides six\n_galeazzas_. These were ships of enormous bulk, and, as it would seem,\nof clumsy construction, carrying each more than forty pieces of\nartillery. The Spaniards counted a score of galleys less than their\nVenetian confederates. But they far exceeded them in the number of their\nfrigates, brigantines, and vessels of smaller size. They boasted a still\ngreater superiority in the equipment of their navy. Indeed, the Venetian\nsquadron was found so indifferently manned, that Don John ordered\nseveral thousand hands to be drafted from the ships of the other Italian\npowers, and from the Spanish, to make up the necessary complement. This\nproceeding conveyed so direct a censure on the remissness of his\ncountrymen, as to give great disgust to the admiral, Veniero. But in the\npresent emergency he had neither the power to resist nor to resent\nit.[306]\n\nThe number of persons on board of the fleet, soldiers and seamen, was\nestimated at eighty thousand. The galleys, impelled by oars more than by\nsails, required a large number of hands to navigate them. The soldiers,\nas we have seen, did not exceed twenty-nine thousand; of which number\nmore than nineteen thousand were furnished by Spain. They were\nwell-appointed troops, most of them familiar with war, and officered by\nmen, many of whom had already established a high reputation in the\nservice. On surveying the muster-roll of cavaliers who embarked in this\nexpedition, one may well believe that Spain had never before sent forth\na fleet in which were to be found the names of so many of her sons\nillustrious for rank and military achievement. If the same can be said\nof Venice, we must consider that the present war was one in which the\nprosperity, perhaps the very existence, of the republic was involved.\nThe Spaniard was animated by the true spirit of the Crusades, when,\ninstead of mercenary motives, the guerdon for which men fought was glory\nin this world and paradise in the next.\n\nSebastian Veniero, trembling for the possessions of the republic in the\nAdriatic, would have put to sea without further delay, and sought out\nthe enemy. But Don John, with a prudence hardly to have been expected,\ndeclined moving until he had been strengthened by all his\nreinforcements. He knew the resources of the Ottoman empire; he could\nnot doubt that in the present emergency they would be strained to the\nutmost to equip a formidable armament; and he resolved not to expose\nhimself unnecessarily to the chances of defeat, by neglecting any means\nin his power to prepare for the encounter. It was a discreet\ndetermination, which must have met the entire approbation of his\nbrother.\n\nWhile he was thus detained at Messina, a papal nuncio, Odescalco, bishop\nof Pena, arrived there. He was the bearer of sundry spiritual favours\nfrom the pontiff, whose real object, no doubt, was to quicken the\nmovements of John of Austria. The nuncio proclaimed a jubilee; and every\nman in the armada, from the captain-general downwards, having fasted\nthree days, confessed and partook of the communion. The prelate, in the\nname of his holiness, then proclaimed a full remission of their sins;\nand he conceded to them the same indulgences as had been granted to the\ndeliverers of the Holy Sepulchre. To Don John the pope communicated\ncertain revelations and two cheering prophecies from St. Isadore, which\nhis holiness declared had undoubted reference to the prince. It is\nfurther stated, that Pius appealed to more worldly feelings, by\nintimating to the young commander that success could not fail to open\nthe way to the acquisition of some independent sovereignty for\nhimself.[307] Whether this suggestion first awakened so pleasing an idea\nin Don John's mind, or whether the wary pontiff was aware that it\nalready existed there, it is certain that it became the spectre which\nfrom this time forward continued to haunt the imagination of the\naspiring chieftain, and to beckon him onward in the path of perilous\nambition to its melancholy close.\n\nAll being now in readiness, orders were given to weigh anchor; and on\nthe sixteenth of September the magnificent armament--unrivalled by any\nwhich had rode upon these waters since the days of imperial Rome--stood\nout to sea. The papal nuncio, dressed in his pontificals, took a\nprominent station on the mole; and as each vessel passed successively\nbefore him, he bestowed on it his apostolic benediction. Then, without\npostponing a moment longer his return, he left Messina and hastened back\nto Rome to announce the joyful tidings to his master.[308]\n\n\n\n\nCHAPTER X.\n\nWAR WITH THE TURKS.\n\nPlan of Operations--Tidings of the Enemy--Preparations for\nCombat--Battle of Lepanto--Rout of the Turkish Armada.\n\n1571.\n\n\n[Sidenote: PLAN OF OPERATIONS.]\n\nAs the allied fleet coasted along the Calabrian shore, it was so much\nbaffled by rough seas and contrary winds that its progress was slow. Not\nlong before his departure Don John had sent a small squadron under a\nSpanish captain, Gil de Andrada, to collect tidings of the enemy. On his\nreturn that commander met the Christian fleet, and reported that the\nTurks, with a powerful armament, were still in the Adriatic, where they\nhad committed fearful ravages on the Venetian territories. Don John now\nsteered his course for Corfu, which, however, he did not reach till the\ntwenty-sixth of September. He soon had ample opportunities of seeing for\nhimself the traces of the enemy, in the smoking hamlets and desolated\nfields along the coast. The allies were welcomed with joy by the\nislanders, who furnished them with whatever supplies they needed. Here\nDon John learned that the Ottoman fleet had been standing into the Gulf\nof Lepanto, where it lay as if waiting the coming of the Christians.\n\nThe young commander-in-chief had now no hesitation as to the course he\nought to pursue. But he chose to call a council of his principal\ncaptains before deciding. The treaty of alliance, indeed, required him\nto consult with the other commanders before taking any decisive step in\nmatters of importance; and this had been strenuously urged on him by the\nking, ever afraid of his brother's impetuosity.\n\nThe opinions of the council were divided. Some who had had personal\nexperience of the naval prowess of the Turks appeared to shrink from\nencountering so formidable an armament, and would have confined the\noperations of the fleet to the siege of some place belonging to the\nMoslems. Even Doria, whose life had been spent in fighting with the\ninfidel, thought it was not advisable to attack the enemy in his present\nposition, surrounded by friendly shores, whence he might easily obtain\nsuccour. It would be better, he urged, to attack some neighbouring\nplace, like Navarino, which might have the effect of drawing him from\nthe gulf, and thus compel him to give battle in some quarter more\nadvantageous to the allies.\n\nBut the majority of the council took a very different view of the\nmatter. To them it appeared that the great object of the expedition was\nto destroy the Ottoman fleet, and that a better opportunity could not be\noffered than the present one, while the enemy was shut up in the gulf,\nfrom which, if defeated, he would find no means of escape. Fortunately,\nthis was the opinion, not only of the majority, but of most of those\nwhose opinions were entitled to the greatest deference. Among these were\nthe gallant marquis of Santa Cruz, the Grand-Commander Requesens, who\nstill remained near the person of Don John, and had command of a galley\nin his rear, Cardona, general of the Sicilian squadron, Barbarigo, the\nVenetian _provveditore_, next in authority to the captain-general of his\nnation, the Roman Colonna, and Alexander Farnese, the young prince of\nParma, Don John's nephew, who had come, on this memorable occasion, to\ntake his first lesson in the art of war,--an art in which he was\ndestined to remain without a rival.\n\nThe commander-in-chief, with no little satisfaction, saw himself so well\nsupported in his own judgment; and he resolved, without any unnecessary\ndelay, to give the Turks battle in the position they had chosen. He was\ndesirous, however, to be joined by part of his fleet, which, baffled by\nthe winds, and without oars, still lagged far behind. For the galley,\nwith its numerous oars in addition to its sails, had somewhat of the\nproperties of a modern steamer, which so gallantly defies both wind and\nwave. As Don John wished also to review his fleet before coming into\naction, he determined to cross over to Comenizza, a capacious and\nwell-protected port on the opposite coast of Albania.\n\nThis he did on the thirtieth of September. Here the vessels were got in\nreadiness for immediate action. They passed in review before the\ncommander-in-chief, and went through their various evolutions, while the\nartillerymen and musketeers showed excellent practice. Don John looked\nwith increased confidence to the approaching combat. An event, however,\noccurred at this time, which might have been attended with the worst\nconsequences.\n\nA Roman officer, named Tortona, one of those who had been drafted to\nmake up the complement of the Venetian galleys, engaged in a brawl with\nsome of his crew. This reached the ears of Veniero, the Venetian\ncaptain-general. The old man, naturally of a choleric temper, and still\nsmarting from the insult which he fancied he had received by the\nintroduction of the allies on board of his vessels, instantly ordered\nthe arrest of the offender. Tortona for a long while resisted the\nexecution of these orders; and when finally seized, with some of his\ncompanions, they were all sentenced by the vindictive Veniero to be hung\nat the yardarm. Such a high-handed proceeding caused the deepest\nindignation in Don John, who regarded it, moreover, as an insult to\nhimself. In the first moments of his wrath he talked of retaliating on\nthe Venetian admiral by a similar punishment. But, happily, the\nremonstrances of Colonna--who, as the papal commander, had in truth the\nmost reason to complain--and the entreaties of other friends, prevailed\non the angry chief to abstain from any violent act. He insisted,\nhowever, that Veniero should never again take his place at the\ncouncil-board, but should be there represented by the _provveditore_\nBarbarigo, next in command,--a man, fortunately, possessed of a better\ncontrol over his temper than was shown by his superior. Thus the cloud\npassed away, which threatened for a moment to break up the harmony of\nthe allies, and to bring ruin on the enterprise.[309]\n\nOn the third of October, Don John, without waiting longer for the\nmissing vessels, again put to sea, and stood for the Gulf of Lepanto. As\nthe fleet swept down the Ionian Sea, it passed many a spot famous in\nancient story. None, we may imagine, would be so likely to excite an\ninterest at this time as Actium, on whose waters was fought the greatest\nnaval battle of antiquity. But the mariner probably gave little thought\nto the past, as he dwelt on the conflict that awaited him at Lepanto. On\nthe fifth, a thick fog enveloped the armada, and shut out every object\nfrom sight. Fortunately, the vessels met with no injury, and, passing by\nIthaca, the ancient home of Ulysses, they safely anchored off the\neastern coast of Cephalonia. For two days their progress was thwarted by\nheadwinds. But on the seventh, Don John, impatient of delay, again put\nto sea, though wind and weather were still unfavourable.\n\nWhile lying off Cephalonia he had received tidings that Famagosta, the\nsecond city of Cyprus, had fallen into the hands of the enemy, and this\nunder circumstances of unparalleled perfidy and cruelty. The place,\nafter a defence that had cost hecatombs of lives to the besiegers, was\nallowed to capitulate on honourable terms. Mustapha, the Moslem\ncommander, the same fierce chief who had conducted the siege of Malta,\nrequested an interview at his quarters with four of the principal\nVenetian captains. After a short and angry conference, he ordered them\nall to execution. Three were beheaded. The other, a noble named\nBragadino, who had held the supreme command, he caused to be flayed\nalive in the market-place of the city. The skin of the wretched victim\nwas then stuffed; and with this ghastly trophy dangling from the yardarm\nof his galley, the brutal monster sailed back to Constantinople, to\nreceive the reward of his services from Selim.[310] These services were\ngreat. The fall of Famagosta secured the fall of Cyprus, which thus\nbecame permanently incorporated in the Ottoman empire.[311]\n\n[Sidenote: PREPARATIONS FOR COMBAT.]\n\nThe tidings of these shocking events filled the breast of every Venetian\nwith an inextinguishable thirst for vengeance. The confederates entered\nheartily into these feelings; and all on board of the armada were\nimpatient for the hour that was to bring them hand to hand with the\nenemies of the Faith.\n\nIt was two hours before dawn, on Sunday, the memorable seventh of\nOctober, when the fleet weighed anchor. The wind had become lighter; but\nit was still contrary, and the galleys were indebted for their progress\nmuch more to their oars than their sails. By sunrise they were abreast\nof the Curzolari,--a cluster of huge rocks, or rocky islets, which on\nthe north defends the entrance of the Gulf of Lepanto. The fleet moved\nlaboriously along, while every eye was strained to catch the first\nglimpse of the hostile navy. At length the watch on the fore-top of the\n_Real_ called out \"A sail!\" and soon after declared that the whole\nOttoman fleet was in sight. Several others, climbing up the rigging,\nconfirmed his report; and in a few moments more, word was sent to the\nsame effect by Andrew Doria, who commanded on the right. There was no\nlonger any doubt; and Don John, ordering his pennon to be displayed at\nthe mizen-peak, unfurled the great standard of the League, given by the\npope, and directed a gun to be fired, the signal for battle. The report,\nas it ran along the rocky shores, fell cheerily on the ears of the\nconfederates, who, raising their eyes towards the consecrated banner,\nfilled the air with their shouts.[312]\n\nThe principal captains now came on board the _Real_, to receive the last\norders of the commander-in-chief. Even at this late hour, there were\nsome who ventured to intimate their doubts of the expediency of engaging\nthe enemy in a position where he had a decided advantage. But Don John\ncut short the discussion. \"Gentlemen,\" he said, \"this is the time for\ncombat, not for counsel.\" He then continued the dispositions he was\nmaking for the attack.\n\nHe had already given to each commander of a galley written instructions\nas to the manner in which the line of battle was to be formed in case of\nmeeting the enemy. The armada was now disposed in that order. It\nextended on a front of three miles. Far on the right, a squadron of\nsixty-four galleys was commanded by the Genoese admiral, Andrew\nDoria,--a name of terror to the Moslems. The centre, or _battle_, as it\nwas called, consisting of sixty-three galleys, was led by John of\nAustria, who was supported on the one side by Colonna, the\ncaptain-general of the pope, and on the other by the Venetian\ncaptain-general, Veniero. Immediately in the rear was the galley of the\nGrand-Commander Requesens, who still remained near the person of his\nformer pupil; though a difference which arose between them on the\nvoyage, fortunately now healed, showed that the young commander-in-chief\nwas wholly independent of his teacher in the art of war.\n\nThe left wing was commanded by the noble Venetian, Barbarigo, whose\nvessels stretched along the AEtolian shore, to which he approached as\nnear as, in his ignorance of the coast, he dared to venture, so as to\nprevent his being turned by the enemy. Finally, the reserve, consisting\nof thirty-five galleys, was given to the brave marquis of Santa Cruz,\nwith directions to act in any quarter where he thought his presence most\nneeded. The smaller craft, some of which had now arrived, seem to have\ntaken little part in the action, which was thus left to the galleys.\n\nEach commander was to occupy so much space with his galley as to allow\nroom for manoeuvring it to advantage, and yet not enough to allow the\nenemy to break the line. He was directed to single out his adversary, to\nclose with him at once, and board as soon as possible. The beaks of the\ngalleys were pronounced to be a hindrance rather than a help in action.\nThey were rarely strong enough to resist a shock from an antagonist, and\nthey much interfered with the working and firing of the guns. Don John\nhad the beak of his vessel cut away. The example was followed\nthroughout the fleet, and, as it is said, with eminently good effect. It\nmay seem strange that this discovery should have been reserved for the\ncrisis of a battle.[313]\n\nWhen the officers had received their last instructions, they returned to\ntheir respective vessels; and Don John, going on board of a light\nfrigate, passed rapidly through the part of the armada lying on his\nright, while he commanded Requesens to do the same with the vessels on\nhis left. His object was to feel the temper of his men, and to rouse\ntheir mettle by a few words of encouragement. The Venetians he reminded\nof their recent injuries. The hour for vengeance, he told them, had\narrived. To the Spaniards and other confederates he said--\"You have come\nto fight the battle of the Cross; to conquer or to die. But whether you\nare to die or conquer, do your duty this day, and you will secure a\nglorious immortality.\" His words were received with a burst of\nenthusiasm which went to the heart of the commander, and assured him\nthat he could rely on his men in the hour of trial. On returning to his\nvessel, he saw Veniero on his quarter-deck; and they exchanged\nsalutations in as friendly a manner as if no difference had existed\nbetween them. At this solemn hour both these brave men were willing to\nforget all personal animosity in a common feeling of devotion to the\ngreat cause in which they were engaged.[314]\n\nThe Ottoman fleet came on slowly and with difficulty. For, strange to\nsay, the wind, which had hitherto been adverse to the Christians, after\nlulling for a time, suddenly shifted to the opposite quarter, and blew\nin the face of the enemy.[315] As the day advanced, moreover, the sun,\nwhich had shone in the eyes of the confederates, gradually shot its rays\ninto those of the Moslems. Both circumstances were of good omen to the\nChristians, and the first was regarded as nothing short of a direct\ninterposition of Heaven. Thus ploughing its way along, the Turkish\narmament, as it came more into view, showed itself in greater strength\nthan had been anticipated by the allies. It consisted of nearly two\nhundred and fifty royal galleys, most of them of the largest class,\nbesides a number of smaller vessels in the rear, which, like those of\nthe allies, appear scarcely to have come into action. The men on board\nof every description were computed at not less than a hundred and twenty\nthousand.[316] The galleys spread out, as usual with the Turks, in the\nform of a regular halfmoon, covering a wider extent of surface than the\ncombined fleets, which they somewhat exceeded in number. They presented,\nindeed, as they drew nearer, a magnificent array, with their gilded and\ngaudily-painted prows, and their myriads of pennons and streamers,\nfluttering gaily in the breeze; while the rays of the morning sun\nglanced on the polished scimitars of Damascus and on the superb\naigrettes of jewels which sparkled in the turbans of the Ottoman chiefs.\n\n[Sidenote: PREPARATIONS FOR COMBAT.]\n\nIn the centre of the extended line, and directly opposite to the station\noccupied by the captain-general of the League, was the huge galley of\nAli Pasha. The right of the armada was commanded by Mahomet Sirocco,\nviceroy of Egypt, a circumspect as well as courageous leader; the left,\nby Uluch Ali, dey of Algiers, the redoubtable corsair of the\nMediterranean. Ali Pasha had experienced a difficulty like that of Don\nJohn, as several of his officers had strongly urged the inexpediency of\nengaging so formidable an armament as that of the allies. But Ali, like\nhis rival, was young and ambitious. He had been sent by his master to\nfight the enemy; and no remonstrances, not even those of Mahomet\nSirocco, for whom he had great respect, could turn him from his purpose.\n\nHe had, moreover, received intelligence that the allied fleet was much\ninferior in strength to what it proved. In this error he was fortified\nby the first appearance of the Christians; for the extremity of their\nleft wing, commanded by Barbarigo, stretching behind the AEtolian shore,\nwas hidden from his view. As he drew nearer, and saw the whole extent of\nthe Christian lines, it is said his countenance fell. If so, he still\ndid not abate one jot of his resolution. He spoke to those around him\nwith the same confidence as before, of the result of the battle. He\nurged his rowers to strain every nerve. Ali was a man of more humanity\nin his nature than often belonged to his nation. His galley-slaves were\nall, or nearly all, Christian captives; and he addressed them in this\nbrief and pithy manner: \"If your countrymen are to win this day, Allah\ngive you the benefit of it; yet if I win it, you shall certainly have\nyour freedom. If you feel that I do well by you, do then the like by\nme.\"[317]\n\nAs the Turkish admiral drew nearer, he made a change in his order of\nbattle, by separating his wings further from his centre; thus conforming\nto the dispositions of the allies. Before he had come within\ncannon-shot, he fired a gun by way of challenge to his enemy. It was\nanswered by another from the galley of John of Austria. A second gun\ndischarged by Ali was as promptly replied to by the Christian commander.\nThe distance between the two fleets was now rapidly diminishing. At this\nsolemn moment a deathlike silence reigned throughout the armament of the\nconfederates. Men seemed to hold their breath, as if absorbed in the\nexpectation of some great catastrophe. The day was magnificent. A light\nbreeze, still adverse to the Turks, played on the waters, somewhat\nfretted by the contrary winds. It was nearly noon; and as the sun,\nmounting through a cloudless sky, rose to the zenith, he seemed to\npause, as if to look down on the beautiful scene, where the multitude of\ngalleys, moving over the water, showed like a holiday spectacle rather\nthan a preparation for mortal combat.\n\nThe illusion was soon dispelled by the fierce yells which rose on the\nair from the Turkish armada. It was the customary war-cry with which the\nMoslems entered into battle. Very different was the scene on board of\nthe Christian galleys. Don John might be there seen, armed _cap-a-pie_,\nstanding on the prow of the _Real_, anxiously awaiting the conflict. In\nthis conspicuous position, kneeling down, he raised his eyes to heaven,\nand humbly prayed that the Almighty would be with His people on that\nday. His example was followed by the whole fleet. Officers and men, all\nprostrating themselves on their knees, and turning their eyes to the\nconsecrated banner which floated from the _Real_, put up a petition like\nthat of their commander. They then received absolution from the priests,\nof whom there were some in every vessel; and each man, as he rose to his\nfeet, gathered new strength, as he felt assured that the Lord of Hosts\nwould fight on his side.[318]\n\nWhen the foremost vessels of the Turks had come within cannon-shot, they\nopened their fire on the Christians. The firing soon ran along the whole\nof the Turkish line, and was kept up without interruption as it\nadvanced. Don John gave orders for trumpet and atabal to sound the\nsignal for action; which was followed by the simultaneous discharge of\nsuch of the guns in the combined fleet as could be brought to bear on\nthe enemy. The Spanish commander had caused the _galeazzas_, those\nmammoth war-ships of which some account has been already given, to be\ntowed half a mile ahead of the fleet, where they might intercept the\nadvance of the Turks. As the latter came abreast of them, the huge\ngalleys delivered their broadsides right and left; and their heavy\nordnance produced a startling effect. Ali Pasha gave orders for his\ngalleys to open their line and pass on either side, without engaging\nthese monsters of the deep, of which he had had no experience. Even so,\ntheir heavy guns did considerable damage to several of the nearest\nvessels, and created some confusion in the pacha's line of battle. They\nwere, however, but unwieldy craft, and, having accomplished their\nobject, seem to have taken no further part in the combat.\n\nThe action began on the left wing of the allies, which Mahomet Sirocco\nwas desirous of turning. This had been anticipated by Barbarigo, the\nVenetian admiral, who commanded in that quarter. To prevent it, as we\nhave seen, he lay with his vessels as near the coast as he dared.\nSirocco, better acquainted with the soundings, saw there was space\nenough for him to pass; and darting by with all the speed that oars\ncould give him, he succeeded in doubling on his enemy. Thus placed\nbetween two fires, the extreme of the Christian left fought at terrible\ndisadvantage. No less than eight galleys went to the bottom, and several\nothers were captured. The brave Barbarigo, throwing himself into the\nheat of the fight, without availing himself of his defensive armour, was\npierced in the eye by an arrow, and, reluctant to leave the glory of the\nfield to another, was borne to his cabin. The combat still continued\nwith unabated fury on the part of the Venetians. They fought like men\nwho felt that the war was theirs, and who were animated not only by the\nthirst for glory, but for revenge.[319]\n\nFar on the Christian right a manoeuvre similar to that so successfully\nexecuted by Sirocco was attempted by Uluch Ali, the dey of Algiers.\nProfiting by his superiority in numbers, he endeavoured to turn the\nright wing of the confederates. It was in this quarter that Andrew Doria\ncommanded. He had foreseen this movement of his enemy, and he succeeded\nin foiling it. It was a trial of skill between the two most accomplished\nseamen in the Mediterranean. Doria extended his line so far to the right\nindeed, to prevent being surrounded, that Don John was obliged to remind\nhim that he left the centre too much exposed. His dispositions were so\nfar unfortunate for himself, that his own line was thus weakened, and\nafforded some vulnerable points to his assailant. These were soon\ndetected by the eagle eye of Uluch Ali; and, like the king of birds\nswooping on his prey, he fell on some galleys separated by a\nconsiderable interval from their companions, and, sinking more than one,\ncarried off the great _Capitana_ of Malta in triumph as his prize.[320]\n\n[Sidenote: BATTLE OF LEPANTO.]\n\nWhile the combat opened thus disastrously to the allies both on the\nright and on the left, in the centre they may be said to have fought\nwith doubtful fortune. Don John had led his division gallantly forward.\nBut the object on which he was intent was an encounter with Ali Pasha,\nthe foe most worthy of his sword. The Turkish commander had the same\ncombat no less at heart. The galleys of both were easily recognized, not\nonly from their position, but from their superior size and richer\ndecoration. The one, moreover, displayed the holy banner of the League;\nthe other, the great Ottoman standard. This, like the ancient standard\nof the caliphs, was held sacred in its character. It was covered with\ntexts from the Koran, emblazoned in letters of gold, and had the name of\nAllah inscribed upon it no less than twenty-eight thousand nine hundred\ntimes. It was the banner of the sultan, having passed from father to son\nsince the foundation of the imperial dynasty, and was never seen in the\nfield unless the Grand Seigneur or his lieutenant was there in\nperson.[321]\n\nBoth the chiefs urged on their rowers to the top of their speed. Their\ngalleys soon shot ahead of the rest of the line, driven through the\nboiling surges as by the force of a tornado, and closed with a shock\nthat made every timber crack, and the two vessels quiver to their very\nkeels. So powerful, indeed, was the impetus they received, that the\npacha's galley, which was considerably the larger and loftier of the\ntwo, was thrown so far upon its opponent that the prow reached the\nfourth bench of rowers. As soon as the vessels were disengaged from each\nother, and those on board had recovered from the shock, the work of\ndeath began. Don John's chief strength consisted in some three hundred\nSpanish arquebusiers, culled from the flower of his infantry. Ali, on\nthe other hand, was provided with an equal number of janizaries. He was\nfollowed by a smaller vessel, in which two hundred more were stationed\nas a _corps de reserve_. He had, moreover, a hundred archers on board.\nThe bow was still as much in use with the Turks as with the other\nMoslems.\n\nThe pacha opened at once on his enemy a terrible fire of cannon and\nmusketry. It was returned with equal spirit and much more effect: for\nthe Turks were observed to shoot over the heads of their adversaries.\nThe Moslem galley was unprovided with the defences which protected the\nsides of the Spanish vessels; and the troops, crowded together on the\nlofty prow, presented an easy mark to their enemy's balls. But though\nnumbers of them fell at every discharge, their places were soon supplied\nby those in reserve. They were enabled, therefore, to keep up an\nincessant fire, which wasted the strength of the Spaniards; and as both\nChristian and Mussulman fought with indomitable spirit, it seemed\ndoubtful to which side victory would incline.\n\nThe affair was made more complicated by the entrance of other parties\ninto the conflict. Both Ali and Don John were supported by some of the\nmost valiant captains in their fleets. Next to the Spanish commander, as\nwe have seen, were Colonna and the veteran Veniero, who, at the age of\nseventy-six, performed feats of arms worthy of a paladin of romance. In\nthis way a little squadron of combatants gathered round the principal\nleaders, who sometimes found themselves assailed by several enemies at\nthe same time. Still the chiefs did not lose sight of one another; but,\nbeating off their inferior foes as well as they could, each, refusing to\nloosen his hold, clung with mortal grasp to his antagonist.[322]\n\nThus the fight raged along the whole extent of the entrance to the Gulf\nof Lepanto. The volumes of vapour rolling heavily over the waters\neffectually shut out from sight whatever was passing at any considerable\ndistance, unless when a fresher breeze dispelled the smoke for a moment,\nor the flashes of the heavy guns threw a transient gleam on the dark\ncanopy of battle. If the eye of the spectator could have penetrated the\ncloud of smoke that enveloped the combatants, and have embraced the\nwhole scene at a glance, he would have perceived them broken into small\ndetachments, separately engaged one with another, independently of the\nrest, and indeed ignorant of all that was doing in other quarters. The\ncontest exhibited few of those large combinations and skilful\nmanoeuvres to be expected in a great naval encounter. It was rather an\nassemblage of petty actions, resembling those on land. The galleys,\ngrappling together, presented a level arena, on which soldier and\ngalley-slave fought hand to hand; and the fate of the engagement was\ngenerally decided by boarding. As in most hand-to-hand contests, there\nwas an enormous waste of life. The decks were loaded with corpses,\nChristian and Moslem lying promiscuously together in the embrace of\ndeath. Instances are recorded where every man on board was slain or\nwounded.[323] It was a ghastly spectacle, where blood flowed in rivulets\ndown the sides of the vessels, staining the waters of the gulf for miles\naround.\n\nIt seemed as if a hurricane had swept over the sea, and covered it with\nthe wreck of the noble armaments which a moment before were so proudly\nriding on its bosom. Little had they now to remind one of their late\nmagnificent array, with their hulls battered, their masts and spars gone\nor splintered by the shot, their canvas cut into shreds and floating\nwildly on the breeze, while thousands of wounded and drowning men were\nclinging to the floating fragments, and calling piteously for help. Such\nwas the wild uproar which succeeded the Sabbath-like stillness that, two\nhours before, had reigned over these beautiful solitudes.\n\nThe left wing of the confederates, commanded by Barbarigo, had been\nsorely pressed by the Turks, as we have seen, at the beginning of the\nfight. Barbarigo himself had been mortally wounded. His line had been\nturned. Several of his galleys had been sunk. But the Venetians gathered\ncourage from despair. By incredible efforts, they succeeded in beating\noff their enemies. They became the assailants in their turn. Sword in\nhand, they carried one vessel after another. The Capuchin was seen in\nthe thickest of the fight, waving aloft his crucifix, and leading the\nboarders to the assault.[324] The Christian galley-slaves, in some\ninstances, broke their fetters, and joined their countrymen against\ntheir masters. Fortunately, the vessel of Mahomet Sirocco the Moslem\nadmiral, was sunk; and though extricated from the water himself, it was\nonly to perish by the sword of his conqueror, Giovanni Contarini. The\nVenetian could find in his heart no mercy for the Turk.\n\n[Sidenote: BATTLE OF LEPANTO.]\n\nThe fall of their commander gave the final blow to his followers.\nWithout further attempt to prolong the fight, they fled before the\navenging swords of the Venetians. Those nearest the land endeavoured to\nescape by running their vessels ashore, where they abandoned them as\nprizes to the Christians. Yet many of the fugitives, before gaining the\nland, perished miserably in the waves. Barbarigo, the Venetian admiral,\nwho was still lingering in agony, heard the tidings of the enemy's\ndefeat, and, uttering a few words expressive of his gratitude to Heaven,\nwhich had permitted him to see this hour, he breathed his last.[325]\n\nDuring this time the combat had been going forward in the centre between\nthe two commanders-in-chief, Don John and Ali Pasha, whose galleys\nblazed with an incessant fire of artillery and musketry, that enveloped\nthem like \"a martyr's robe of flames.\" The parties fought with equal\nspirit, though not with equal fortune. Twice the Spaniards had boarded\ntheir enemy, and both times they had been repulsed with loss. Still\ntheir superiority in the use of fire-arms would have given them a\ndecided advantage over their opponents, if the loss they had inflicted\nhad not been speedily repaired by fresh reinforcements. More than once\nthe contest between the two chieftains was interrupted by the arrival of\nothers to take part in the fray. They soon, however, returned to each\nother, as if unwilling to waste their strength on a meaner enemy.\nThrough the whole engagement both commanders exposed themselves to\ndanger as freely as any common soldier. In such a contest even Philip\nmust have admitted that it would be difficult for his brother to find,\nwith honour, a place of safety. Don John received a wound in the foot.\nIt was a slight one, however, and he would not allow it to be dressed\ntill the action was over.\n\nAgain his men were mustered, and a third time the trumpets sounded to\nthe attack. It was more successful than the preceding. The Spaniards\nthrew themselves boldly into the Turkish galley. They were met with the\nsame spirit as before by the janizaries. Ali Pasha led them on.\nUnfortunately, at this moment, he was struck in the head by a\nmusket-ball, and stretched senseless in the gangway. His men fought\nworthily of their ancient renown. But they missed the accustomed voice\nof their commander. After a short but ineffectual struggle against the\nfiery impetuosity of the Spaniards, they were overpowered, and threw\ndown their arms. The decks were loaded with the bodies of the dead and\nthe dying. Beneath these was discovered the Turkish commander-in-chief,\nseverely wounded, but perhaps not mortally. He was drawn forth by some\nCastilian soldiers, who, recognizing his person, would at once have\ndespatched him. But the disabled chief, having rallied from the first\neffects of his wound, had sufficient presence of mind to divert them\nfrom their purpose, by pointing out the place below where he had\ndeposited his money and jewels; and they hastened to profit by the\ndisclosure, before the treasure should fall into the hands of their\ncomrades.\n\nAli was not so successful with another soldier, who came up soon after,\nbrandishing his sword, and preparing to plunge it into the body of the\nprostrate commander. It was in vain that the latter endeavoured to turn\nthe ruffian from his purpose. He was a convict, one of those\ngalley-slaves whom Don John had caused to be unchained from the oar and\nfurnished with arms. He could not believe that any treasure would be\nworth so much as the head of the pacha. Without further hesitation, he\ndealt him a blow which severed it from his shoulders. Then, returning to\nhis galley, he laid the bloody trophy before Don John. But he had\nmiscalculated on his recompense. His commander gazed on it with a look\nof pity mingled with horror. He may have thought of the generous conduct\nof Ali to his Christian captives, and have felt that he deserved a\nbetter fate. He coldly inquired \"of what use such a present could be to\nhim;\" and then ordered it to be thrown into the sea. Far from the order\nbeing obeyed, it is said the head was stuck on a pike, and raised aloft\non board of the captured galley. At the same time the banner of the\nCrescent was pulled down; while that of the Cross, run up in its place,\nproclaimed the downfall of the pacha.[326]\n\nThe sight of the sacred ensign was welcomed by the Christians with a\nshout of \"Victory!\" which rose high above the din of battle.[327] The\ntidings of the death of Ali soon passed from mouth to mouth, giving\nfresh heart to the confederates, but falling like a knell on the ears of\nthe Moslems. Their confidence was gone. Their fire slackened. Their\nefforts grew weaker and weaker. They were too far from shore to seek an\nasylum there, like their comrades on the right. They had no resource but\nto prolong the combat or to surrender. Most preferred the latter. Many\nvessels were carried by boarding, others were sunk by the victorious\nChristians. Ere four hours had elapsed, the centre, like the right wing,\nof the Moslems might be said to be annihilated.\n\nStill the fight was lingering on the right of the confederates, where,\nit will be remembered, Uluch Ali, the Algerine chief, had profited by\nDoria's error in extending his line so far as greatly to weaken it.\nUluch Ali, attacking it on its most vulnerable quarter, had succeeded,\nas we have seen, in capturing and destroying several vessels; and would\nhave inflicted still heavier losses on his enemy had it not been for the\nseasonable succour received from the marquis of Santa Cruz. This brave\nofficer, who commanded the reserve, had already been of much service to\nDon John when the _Real_ was assailed by several Turkish galleys at once\nduring his combat with Ali Pasha; for at this juncture the marquis of\nSanta Cruz arriving, and beating off the assailants, one of whom he\nafterwards captured, enabled the commander-in-chief to resume his\nengagement with the pacha.\n\nNo sooner did Santa Cruz learn the critical situation of Doria, than,\nsupported by Cardona, \"general\" of the Sicilian squadron, he pushed\nforward to his relief. Dashing into the midst of the _melee_, the two\ncommanders fell like a thunderbolt on the Algerine galleys. Few\nattempted to withstand the shock. But in their haste to avoid it, they\nwere encountered by Doria and his Genoese galleys. Thus beset on all\nsides, Uluch Ali was compelled to abandon his prizes, and provide for\nhis own safety by flight. He cut adrift the Maltese _Capitana_, which he\nhad lashed to his stern, and on which three hundred corpses attested the\ndesperate character of her defence. As tidings reached him of the\ndiscomfiture of the centre, and of the death of Ali Pasha, he felt that\nnothing remained but to make the best of his way from the fatal scene of\naction, and save as many of his own ships as he could. And there were no\nships in the Turkish fleet superior to his, or manned by men under more\nperfect discipline. For they were the famous corsairs of the\nMediterranean, who had been rocked from infancy on its waters.\n\n[Sidenote: ROUT OF THE TURKISH ARMADA.]\n\nThrowing out his signals for retreat, the Algerine was soon to be seen,\nat the head of his squadron, standing towards the north, under as much\ncanvas as remained to him after the battle, and urged forward through\nthe deep by the whole strength of his oarsmen. Doria and Santa Cruz\nfollowed quickly in his wake. But he was borne on the wings of the wind,\nand soon distanced his pursuers. Don John, having disposed of his own\nassailants, was coming to the support of Doria, and now joined in the\npursuit of the viceroy. A rocky headland, stretching far into the sea,\nlay in the path of the fugitive; and his enemies hoped to intercept him\nthere. Some few of his vessels were stranded on the rocks. But the rest,\nnear forty in number, standing more boldly out to sea, safely doubled\nthe promontory. Then, quickening their flight, they gradually faded from\nthe horizon, their white sails, the last thing visible, showing in the\ndistance like a flock of Arctic sea-fowl on their way to their native\nhomes. The confederates explained the inferior sailing of their own\ngalleys on this occasion by the circumstance of their rowers, who had\nbeen allowed to bear arms in the fight, being crippled by their wounds.\n\nThe battle had lasted more than four hours. The sky, which had been\nalmost without a cloud through the day, began now to be overcast, and\nshowed signs of a coming storm. Before seeking a place of shelter for\nhimself and his prizes, Don John reconnoitred the scene of action. He\nmet with several vessels too much damaged for further service. These,\nmostly belonging to the enemy, after saving what was of any value on\nboard, he ordered to be burnt. He selected the neighbouring port of\nPetala, as affording the most secure and accessible harbour for the\nnight. Before he had arrived there, the tempest began to mutter, and\ndarkness was on the water. Yet the darkness rendered only more visible\nthe blazing wrecks, which, sending up streams of fire mingled with\nshowers of sparks, looked like volcanoes on the deep.\n\n\n\n\nCHAPTER XI.\n\nWAR WITH THE TURKS.\n\nLosses of the Combatants--Don John's Generosity--Triumphant\nReturn--Enthusiasm throughout Christendom--Results of the\nBattle--Operations in the Levant--Conquest of Tunis--Retaken by the\nTurks.\n\n1571--1574.\n\n\nLong and loud were the congratulations now paid to the young\ncommander-in-chief by his brave companions-in-arms, on the success of\nthe day. The hours passed blithely with officers and men, while they\nrecounted to one another their manifold achievements. But feelings of\ngloom mingled with their gaiety, as they gathered tidings of the loss of\nfriends who had bought this victory with their blood.\n\nIt was, indeed, a sanguinary battle, surpassing, in this particular, any\nsea-fight of modern times. The loss fell much the most heavily on the\nTurks. There is the usual discrepancy about numbers; but it may be safe\nto estimate their loss at nearly twenty-five thousand slain and five\nthousand prisoners. What brought most pleasure to the hearts of the\nconquerors was the liberation of twelve thousand Christian captives,\nwho had teen chained to the oar on board the Moslem galleys, and who now\ncame forth, with tears of joy streaming down their haggard cheeks, to\nbless their deliverers.[328]\n\nThe loss of the allies was comparatively small,--less than eight\nthousand.[329] That it was so much less than that of their enemies, may\nbe referred in part to their superiority in the use of fire-arms; in\npart also to their exclusive use of these, instead of employing bows and\narrows, weapons on which, though much less effective, the Turks, like\nthe other Moslem nations, seem to have greatly relied. Lastly, the Turks\nwere the vanquished party, and in their heavier loss suffered the almost\ninvariable lot of the vanquished.\n\nAs to their armada, it may almost be said to have been annihilated. Not\nmore than forty galleys escaped out of near two hundred and fifty which\nentered into the action. One hundred and thirty were taken and divided\namong the conquerors. The remainder, sunk or burned, were swallowed up\nby the waves. To counterbalance all this, the confederates are said to\nhave lost not more than fifteen galleys, though a much larger number,\ndoubtless, were rendered unfit for service. This disparity affords good\nevidence of the inferiority of the Turks in the construction of their\nvessels, as well as in the nautical skill required to manage them. A\ngreat amount of booty, in the form of gold, jewels, and brocade, was\nfound on board several of the prizes. The galley of the\ncommander-in-chief alone is stated to have contained one hundred and\nseventy thousand gold sequins,--a large sum, but not large enough, it\nseems, to buy off his life.[330]\n\nThe losses of the combatants cannot be fairly presented without taking\ninto the account the quality as well as the number of the slain. The\nnumber of persons of consideration, both Christians and Moslems, who\nembarked in the expedition, was very great. The roll of slaughter showed\nthat in the race of glory they gave little heed to their personal\nsafety. The officer second in command among the Venetians, the\ncommander-in-chief of the Turkish armament, and the commander of its\nright wing, all fell in the battle. Many a high-born cavalier closed at\nLepanto a long career of honourable service. More than one, on the other\nhand, dated the commencement of their career from this day. Such was\nAlexander Farnese, prince of Parma. Though he was but a few years\nyounger than his uncle, John of Austria, those few years had placed an\nimmense distance between their conditions, the one filling the post of\ncommander-in-chief, the other being only a private adventurer. Yet even\nso, he succeeded in winning great renown by his achievements. The galley\nin which he sailed was lying yardarm and yardarm alongside of a Turkish\ngalley, with which it was hotly engaged. In the midst of the action\nFarnese sprang on board of the enemy, and with his good broadsword hewed\ndown all who opposed him, opening a path into which his comrades poured\none after another, and, after a short but murderous contest, succeeded\nin carrying the vessel. As Farnese's galley lay just astern of Don\nJohn's, the latter could witness the achievement of his nephew, which\nfilled him with an admiration he did not affect to conceal. The\nintrepidity displayed by the young warrior on this occasion gave augury\nof his character in later life, when he succeeded his uncle in command,\nand surpassed him in military renown.[331]\n\n[Sidenote: DON JOHN'S GENEROSITY.]\n\nAnother youth was in that fight, who, then humble and unknown, was\ndestined one day to win laurels of a purer and more enviable kind than\nthose which grow on the battle-field. This was Cervantes, who, at the\nage of twenty-four, was serving on board the fleet as a common soldier.\nHe had been confined to his bed by a fever; but, notwithstanding the\nremonstrances of his captain, he insisted, on the morning of the action,\nnot only on bearing arms, but on being stationed in the post of danger.\nAnd well did he perform his duty there, as was shown by two wounds on\nthe breast, and by another in the hand, by which he lost the use of it.\nFortunately it was the left hand. The right yet remained to indite those\nimmortal productions which were to be known as household words, not only\nin his own land, but in every quarter of the civilized world.[332]\n\nA fierce storm of thunder and lightning raged for four-and-twenty hours\nafter the battle, during which time the fleet rode safely at anchor in\nthe harbour of Petala. It remained there three days longer. Don John\nprofited by the delay to visit the different galleys and ascertain their\ncondition. He informed himself of the conduct of the troops, and was\nliberal of his praises to those who deserved them. With the sick and the\nwounded he showed the greatest sympathy, endeavouring to alleviate their\nsufferings, and furnishing them with whatever his galley contained that\ncould contribute to their comfort. With so generous and sympathetic a\nnature, it is not wonderful that he should have established himself in\nthe hearts of his soldiers.[333]\n\nBut the proofs of this kindly temper were not confined to his own\nfollowers. Among the prisoners were two sons of Ali, the Turkish\ncommander-in-chief. One was seventeen, the other only thirteen years of\nage. Thus early had their father desired to initiate them in a\nprofession which, beyond all others, opened the way to eminence in\nTurkey. They were not on board of his galley; and when they were\ninformed of his death, they were inconsolable. To this affliction was\nnow to be added the doom of slavery.\n\nAs they were led into the presence of Don John, the youths prostrated\nthemselves on the deck of his vessel. But raising them up, he\naffectionately embraced them, and said all he could to console them\nunder their troubles. He caused them to be treated with the\nconsideration due to their rank. His secretary, Juan de Soto,\nsurrendered his quarters to them. They were provided with the richest\napparel that could be found among the spoil. Their table was served with\nthe same delicacies as that of the commander-in-chief; and his\nchamberlains showed the same deference to them as to himself. His\nkindness did not stop with these acts of chivalrous courtesy. He\nreceived a letter from their sister Fatima, containing a touching appeal\nto Don John's humanity, and soliciting the release of her orphan\nbrothers. He had sent a courier to give their friends in Constantinople\nthe assurance of their personal safety; \"which,\" adds the lady, \"is\nheld by all this court as an act of great courtesy,--_gran\ngentileza_;--and there is no one here who does not admire the goodness\nand magnanimity of your highness.\" She enforced her petition with a rich\npresent, for which she gracefully apologized, as intended to express her\nown feelings, though far below his deserts.[334]\n\nIn the division of the spoil, the young princes had been assigned to the\npope. But Don John succeeded in obtaining their liberation.\nUnfortunately, the elder died--of a broken heart, it is said--at Naples.\nThe younger was sent home, with three of his attendants, for whom he had\na particular regard. Don John declined keeping Fatima's present, which\nhe gave to her brother. In a letter to the Turkish princess, he remarked\nthat he had done this, not because he undervalued her beautiful gift,\nbut because it had ever been the habit of his royal ancestors freely to\ngrant their favours to those who stood in need of them, but not to\nreceive aught by way of recompense.[335]\n\nThe same noble nature he showed in his conduct towards Veniero. We have\nseen the friendly demonstration he made to the testy Venetian on\nentering into battle. He now desired his presence on board his galley.\nAs he drew near, Don John came forward frankly to greet him. He spoke of\nhis desire to bury the past in oblivion, and complimenting the veteran\non his prowess in the late engagement, saluted him with the endearing\nname of \"father.\" The old soldier, not prepared for so kind a welcome,\nburst into tears; and there was no one, says the chronicler who tells\nthe anecdote, that could witness the scene with a dry eye.[336]\n\nWhile at Petala, a council of war was called to decide on the next\noperations of the fleet. Some were for following up the blow by an\nimmediate attack on Constantinople. Others considered that, from the\nwant of provisions and the damaged state of the vessels, they were in no\ncondition for such an enterprise. They recommended that the armada\nshould be disbanded, that the several squadrons of which it was composed\nshould return to their respective winter quarters, and meet again in the\nspring to resume operations. Others, again, among whom was Don John,\nthought that before disbanding, they should undertake some enterprise\ncommensurate with their strength. It was accordingly determined to lay\nsiege to Santa Maura, in the island of Leucadia, a strongly-fortified\nplace, which commanded the northern entrance into the Gulf of Lepanto.\n\n[Sidenote: DON JUAN'S TRIUMPHANT RETURN]\n\nThe fleet, weighing anchor on the eleventh of October, arrived off Santa\nMaura on the following day. On a careful reconnaissance of the ground,\nit became evident that the siege would be a work of much greater\ndifficulty than had been anticipated. A council of war was again\nsummoned; and it was resolved, as the season was far advanced, to\nsuspend further operations for the present, to return to winter\nquarters, and in the ensuing spring to open the campaign under more\nfavourable auspices.\n\nThe next step was to make a division of the spoil taken from the enemy,\nwhich was done in a manner satisfactory to all parties. One half of the\ngalleys and inferior vessels, of the artillery and small arms, and also\nof the captives, was set apart for the Catholic king. The other half was\ndivided between the pope and the republic, in the proportion settled by\nthe treaty of confederation.[337] Next proceeding to Corfu, Don John\npassed three days at that island, making some necessary repairs of his\nvessels; then, bidding adieu to the confederates, he directed his course\nto Messina, which he reached, after a stormy passage, on the\nthirty-first of the month.\n\nWe may imagine the joy with which he was welcomed by the inhabitants of\nthat city, which he had left but little more than six weeks before, and\nto which he had now returned in triumph, after winning the most\nmemorable naval victory of modern times. The whole population, with the\nmagistrates at their head, hurried down to the shore to witness the\nmagnificent spectacle. As the gallant armament swept into port, it\nshowed the results of the late contest in many a scar. But the\nconsecrated standard was still proudly flying at the masthead of the\n_Real_; and in the rear came the long line of conquered galleys, in much\nworse plight than their conquerors, trailing their banners ignominiously\nbehind them in the water. On landing at the head of his troops, Don John\nwas greeted with flourishes of music, while salvoes of artillery\nthundered from the fortresses which commanded the city. He was received\nunder a gorgeous canopy, and escorted by a numerous concourse of\ncitizens and soldiers. The clergy, mingling in the procession, broke\nforth into the _Te Deum_; and thus entering the cathedral, they all\njoined in thanksgivings to the Almighty, for granting them so glorious a\nvictory.[338]\n\nDon John was sumptuously lodged in the castle. He was complimented with\na superb banquet,--a mode of expressing the public gratitude not\nconfined to our day,--and received a more substantial guerdon in a\npresent from the city of thirty thousand crowns. Finally, a colossal\nstatue in bronze was executed by a skilful artist, as a permanent\nmemorial of the conqueror of Lepanto. Don John accepted the money, but\nit was only to devote it to the relief of the sick and wounded soldiers.\nIn the same generous spirit, he had ordered that all his own share of\nthe booty taken in the Turkish vessels, including the large amount of\ngold and rich brocades found in the galley of Ali Pasha, should be\ndistributed among the captors.[339]\n\nThe news of the victory of Lepanto caused a profound sensation\nthroughout Christendom; for it had been a general opinion that the Turks\nwere invincible by sea. The confederates more particularly testified\ntheir joy by such extraordinary demonstrations as showed the extent of\ntheir previous fears. In Venice, which might be said to have gained a\nnew lease of existence from the result of the battle, the doge, the\nsenators, and the people met in the great square of St. Mark, and\ncongratulated one another on the triumph of their arms. By a public\ndecree, the seventh of October was set apart, to be observed for ever as\na national anniversary.\n\nThe joy was scarcely less in Naples, where the people had so often seen\ntheir coasts desolated by the Ottoman cruisers; and when their admiral,\nthe marquis of Santa Cruz, returned to port with his squadron, he was\nwelcomed with acclamations such as greet the conqueror returning from\nhis campaign.\n\nBut even these honours were inferior to those which in Rome were paid to\nColonna, the Captain-general of the papal fleet. As he was borne in\nstately procession, with the trophies won from the enemy carried before\nhim, and a throng of mourning captives in the rear, the spectacle\nrecalled the splendours of the ancient Roman triumph. Pius the Fifth\nhad, before this, announced that the victory of the Christians had been\nrevealed to him from Heaven. But when the tidings reached him of the\nactual result, it so far transcended his expectations, that, overcome by\nhis emotions, the old pontiff burst into a flood of tears, exclaiming in\nthe words of the Evangelist, \"There was a man sent from God; and his\nname was John.\"[340]\n\nWe may readily believe that the joy with which the glad tidings were\nwelcomed in Spain fell nothing short of that with which they were\nreceived in other parts of Christendom. While lying off Petala, Don John\nsent Lope de Figueroa with despatches for the king, together with the\ngreat Ottoman standard, as the most glorious trophy taken in the\nbattle.[341] He soon after sent a courier with further letters. It so\nhappened that neither the one nor the other arrived at the place of\ntheir destination till some weeks after the intelligence had reached\nPhilip by another channel. This was the Venetian Minister, who on the\nlast of October received despatches from his own government, containing\na full account of the fight. Hastening with them to the palace, he found\nthe king in his private chapel, attending vespers on the eve of\nAll-Saints. The news, it cannot be doubted, filled his soul with joy;\nthough _it is said_ that, far from exhibiting this in his demeanour, he\ncontinued to be occupied with his devotions, without the least change of\ncountenance, till the services were concluded. He then ordered _Te Deum_\nto be sung.[342] All present joined with overflowing hearts in pouring\nforth their gratitude to the Lord of Hosts for granting such a triumph\nto the Cross.[343]\n\n[Sidenote: ENTHUSIASM THROUGHOUT CHRISTENDOM.]\n\nThat night there was a grand illumination in Madrid. The following day\nmass was said by the papal legate in presence of the king, who\nafterwards took part in a solemn procession to the church of St. Mary,\nwhere the people united with the court in a general thanksgiving.\n\nIn a letter from Philip to his brother, dated from the Escorial, the\ntwenty-ninth of November, he writes to him out of the fulness of his\nheart, in the language of gratitude and brotherly love:--\"I cannot\nexpress to you the joy it has given me to learn the particulars of your\nconduct in the battle, of the great valour you showed in your own\nperson, and your watchfulness in giving proper directions to others--all\nwhich has doubtless been a principal cause of the victory. So to you,\nafter God, I am to make my acknowledgments for it, as I now do; and\nhappy am I that it has been reserved for one so near and so dear to me\nto perform this great work, which has gained such glory for you in the\neyes of God and of the whole world.\"[344]\n\nThe feelings of the king were fully shared by his subjects. The\nenthusiasm roused throughout the country by the great victory was\nwithout bounds. \"There is no man,\" writes one of the royal secretaries\nto Don John, \"who does not discern the hand of the Lord in it;--though\nit seems rather like a dream than a reality, so far does it transcend\nany naval encounter that the world ever heard of before.\"[345] The best\nsculptors and painters were employed to perpetuate the memory of the\nglorious event. Amongst the number was Titian, who in the time of\nCharles the Fifth had passed two years in Spain, and who now, when more\nthan ninety years of age, executed the great picture of \"The Victory of\nthe League,\" still hanging on the walls of the _Museo_ at Madrid.[346]\nThe lofty theme proved a fruitful source of inspiration to the Castilian\nmuse. Among hecatombs of epics and lyrics, the heroic poem of\nErcilla[347] and the sublime _cancion_ of Fernando de Herrera perpetuate\nthe memory of the victory of Lepanto in forms more durable than canvas\nor marble--as imperishable as the language itself.\n\nWhile all were thus ready to render homage to the talent and bravery\nwhich had won the greatest battle of the time, men, as they grew cooler,\nand could criticise events more carefully, were disposed to ask, where\nwere the fruits of this great victory. Had Don John's father, Charles\nthe Fifth, gained such a victory, it was said, he would not thus have\nquitted the field, but, before the enemy could recover from the blow,\nwould have followed it up by another. Many expressed the conviction,\nthat the young generalissimo should at once have led his navy against\nConstantinople.\n\nThere would indeed seem to be plausible ground for criticising his\ncourse after the action. But we must remember, in explanation of the\nconduct of Don John, that his situation was altogether different from\nthat of his imperial father. He possessed no such absolute authority as\nthe latter did over his army. The great leaders of the confederates were\nso nearly equal in rank, that they each claimed a right to be consulted\non all measures of importance. The greatest jealousy existed among the\nthree commanders, as there did also among the troops whom they\ncommanded. They were all united, it is true, in their hatred to the\nTurk. But they were all influenced, more or less, by the interest of\ntheir own states, in determining the quarter where he was to be\nassailed. Every rood of territory wrung from the enemy in the Levant\nwould only serve to enlarge the domain of Venice; while the conquests in\nthe western parts of the Mediterranean would strengthen the empire of\nCastile. This feeling of jealousy between the Spaniards and the\nVenetians was, as we have seen, so great in the early part of the\nexpedition, as nearly to bring ruin on it.\n\nThose who censured Don John for not directing his arms against\nConstantinople would seem to have had but a very inadequate notion of\nthe resources of the Porte--as shown in the course of that very year.\nThere is a remarkable letter from the duke of Alva, written the month\nafter the battle of Lepanto, in which he discusses the best course to be\ntaken in order to reap the full fruits of the victory. In it he\nexpresses the opinion, that an attempt against Constantinople, or indeed\nany part of the Turkish dominions, unless supported by a general\ncoalition of the great powers of Christendom, must end only in\ndisappointment--so vast were the resources of that great empire.[348] If\nthis were so,--and no better judge than Alva could be found in military\naffairs,--how incompetent were the means at Don John's disposal for\neffecting this object--confederates held together, as the event proved,\nby a rope of sand, and a fleet so much damaged in the recent combat that\nmany of the vessels were scarcely seaworthy!\n\nIn addition to this, it may be stated, that Don John knew it was his\nbrother's wish that the Spanish squadron should return to Sicily to pass\nthe winter.[349] If he persisted, therefore, in the campaign, he must do\nso on his own responsibility. He had now accomplished the great object\nfor which he had put to sea. He had won a victory more complete than the\nmost sanguine of his countrymen had a right to anticipate. To prolong\nthe contest under the present circumstances, would he in a manner to\nprovoke his fate, to jeopard the glory he had already gained, and incur\nthe risk of closing the campaign with melancholy cypress, instead of the\nlaurel-wreath of victory. Was it surprising that even an adventurous\nspirit like his should have shrunk from hazarding so vast a stake with\nthe odds against him?\n\n[Sidenote: RESULTS OF THE BATTLE.]\n\nIt is a great error to speak of the victory of Lepanto as a barren\nvictory, which yielded no fruits to those who gained it. True, it did\nnot strip the Turks of an inch of territory. Even the heavy loss of\nships and soldiers which it cost them was repaired in the following\nyear. But the loss of reputation--that tower of strength to the\nconqueror--was not to be estimated. The long and successful career of\nthe Ottoman princes, especially of the last one, Solyman the\nMagnificent, had made the Turks to be thought invincible. There was not\na nation in Christendom that did not tremble at the idea of a war with\nTurkey. The spell was now broken. Though her resources were still\nboundless, she lost confidence in herself. Venice gained confidence in\nproportion. When the hostile fleets met in the year following the battle\nof Lepanto, the Turks, though greatly the superior in numbers, declined\nthe combat. For the seventy years which elapsed after the close of the\npresent war, the Turks abandoned their efforts to make themselves\nmasters of any of the rich possessions of the republic, which lay so\ntemptingly around them. When the two nations came next into collision,\nVenice, instead of leaning on confederates, took the field\nsingle-handed, and disputed it with an intrepidity which placed her on a\nlevel with the gigantic power that assailed her. That power was already\non the wane; and those who have most carefully studied the history of\nthe Ottoman empire date the commencement of her decline from the battle\nof Lepanto.[350]\n\nThe allies should have been ready with their several contingents early\nin the spring of the following year, 1572. They were not ready till the\nsummer was well advanced. One cause of delay was the difficulty of\ndeciding on what quarter the Turkish empire was to be attacked. The\nVenetians, from an obvious regard to their own interests, were for\ncontinuing the war in the Levant. Philip, on the other hand, from\nsimilar motives, would have transferred it to the western part of the\nMediterranean, and have undertaken an expedition against the Barbary\npowers. Lastly, Pius the Fifth, urged by that fiery enthusiasm which\nmade him overlook or overleap every obstacle in his path, would have\nmarched on Constantinople, and then carried his conquering banners to\nthe Holy Land. These chimerical fancies of a crusader provoked a\nsmile--it may have been a sneer--from men better instructed in military\noperations than the pontiff.[351]\n\nPius again laboured to infuse his own spirit into the monarchs of\nChristendom. But it was in vain that he urged them to join the League.\nAll, for some reason or other, declined it. It is possible that they may\nhave had less fear of the Turk, than of augmenting the power of the king\nof Spain. But the great plans of Pius the Fifth were terminated by his\ndeath, which occurred on the first of May, 1572. He was the true author\nof the League. It occupied his thoughts to the latest hour of his\nexistence; and his last act was to appropriate to its uses a\nconsiderable sum of money lying in his coffers.[352] He may be truly\nsaid to have been the only one of the confederates who acted solely for\nwhat he conceived to be the interests of the Faith. This soon became\napparent.\n\n[Sidenote: WAR WITH THE TURKS.]\n\nThe affairs of Philip the Second were at this time in a critical\nsituation. He much feared that one of the French faction would be\nraised to the chair of St. Peter. He had great reason to distrust the\npolicy of France in respect to the Netherlands. Till he was more assured\non these points, he was not inclined to furnish the costly armament to\nwhich he was pledged as his contingent. It was in vain that the allies\ncalled on Don John to aid them with his Spanish fleet. He had orders\nfrom his brother not to quit Messina; and it was in vain that he chafed\nunder these orders, which threatened thus prematurely to close the\nglorious career on which he had entered, and which exposed him to the\nmost mortifying imputations. It was not till the sixth of July that the\nking allowed him to send a part of his contingent, amounting only to\ntwenty-two galleys and five thousand troops, to the aid of the\nconfederates.\n\nSome historians explain the conduct of Philip, not so much by the\nembarrassments of his situation, as by his reluctance to afford his\nbrother the opportunity of adding fresh laurels to his brow, and\npossibly of achieving for himself some independent sovereignty, like\nthat to which Pius the Fifth had encouraged him to aspire. It may be\nthought some confirmation of this opinion--at least, it infers some\njealousy of his brother's pretensions--that, in his despatches to his\nministers in Italy, the king instructed them that, while they showed all\nproper deference to Don John, they should be careful not to address him\nin speech or in writing by the title of _Highness_, but to use that of\n_Excellency_; adding, that they were not to speak of this suggestion as\ncoming from him.[353] He caused a similar notice to be given to the\nambassadors of France, Germany, and England. This was but a feeble\nthread by which to check the flight of the young eagle as he was soaring\nto the clouds. It served to show, however, that it was not the will of\nhis master that he should soar too high.\n\nHappily Philip was relieved from his fears in regard to the new pope, by\nthe election of Cardinal Buoncampagno to the vacant throne. This\necclesiastic, who took the name of Gregory the Thirteenth, was\npersonally known to the king, having in earlier life passed several\nyears at the court of Castile. He was well affected to that court, and\nhe possessed in full measure the zeal of his predecessor for carrying on\nthe war against the Moslems. He lost no time in sending his \"briefs of\nfire,\"[354] as Don John called them, to rouse him to new exertions in\nthe cause. In France, too, Philip learned with satisfaction that the\nGuises, the devoted partisans of Spain, had now the direction of public\naffairs. Thus relieved from apprehensions on these two quarters, Philip\nconsented to his brother's departure with the remainder of his squadron.\nIt amounted to fifty-five galleys and thirty smaller vessels. But when\nthe prince reached Corfu, on the ninth of August, he found that the\nconfederates, tired of waiting, had already put to sea, under the\ncommand of Colonna, in search of the Ottoman fleet.\n\nThe Porte had shown such extraordinary despatch, that in six months it\nhad built and equipped a hundred and twenty galleys, making, with those\nalready on hand, a formidable fleet.[355] It was a remarkable proof of\nits resources, but suggests the idea of the wide difference between a\nTurkish galley of the sixteenth century and a man-of-war in our day. The\ncommand of the armament was given to the Algerine chieftain, Uluch Ali,\nwho had so adroitly managed to bring off the few vessels which effected\ntheir escape at the battle of Lepanto. He stood deservedly high in the\nconfidence of the sultan, and had the supreme direction in maritime\naffairs.\n\n[Sidenote: OPERATIONS IN THE LEVANT.]\n\nThe two fleets came face to face with each other off the western coast\nof the Morea. But though the Algerine commander was much superior to the\nChristians in the number and strength of his vessels, he declined an\naction, showing the same adroitness in eluding a battle that he had\nbefore shown in escaping from one.\n\nAt the close of August the confederates returned to Corfu, where they\nwere reinforced by the rest of the Spanish squadron. The combined fleet,\nwith this addition, amounted to some two hundred and forty-seven\nvessels, of which nearly two-thirds were galleys. It was a force\nsomewhat superior to that of the enemy. Thus strengthened, Don John,\nunfurling the consecrated banner as generalissimo of the League, weighed\nanchor, and steered with his whole fleet in a southerly direction. It\nwas not long before he appeared off the harbours of Modon and Navarino,\nwhere the two divisions of the Turkish armada were lying at anchor. He\nwould have attacked them separately, but, notwithstanding his efforts,\nfailed to prevent their effecting a junction in the harbour of Modon. On\nthe seventh of October, Uluch Ali ventured out of port, and seemed\ndisposed to give battle. It was the anniversary of the fight of Lepanto;\nand Don John flattered himself that he should again see his arms crowned\nwith victory, as on that memorable day. But if the Turkish commander was\nunwilling to fight the confederates when he was superior to them in\nnumbers, it was not likely that he would fight them now that he was\ninferior. After some manoeuvres which led to no result, he took refuge\nunder the castle of Modon, and again retreated into port. There Don John\nwould have followed him, with the design of forcing him to a battle. But\nfrom this he was dissuaded by the other leaders of the confederates, who\nconsidered that the chances of success in a place so strongly defended\nby no means warranted the risk.\n\nIt was in vain that the allies prolonged their stay in the\nneighbourhood, with the hope of enticing the enemy to an engagement. The\nseason wore away with no prospect of a better result. Meantime\nprovisions were failing, the stormy weather of autumn was drawing nigh,\nand Don John, disgusted with what he regarded as the timid counsels of\nhis associates, and with the control which they were permitted to\nexercise over him, decided, as it was now too late for any new\nenterprise, to break up and postpone further action till the following\nspring, when he hoped to enter on the campaign at an earlier day than he\nhad done this year. The allies, accordingly, on reaching the island of\nPaxo, late in October, parted from each other, and withdrew to their\nrespective winter-quarters. Don John, with the Spanish armament,\nreturned to Sicily.[356]\n\nThe pope and the king of Spain, nowise discouraged by the results of the\ncampaign, resolved to resume operations early in the spring on a still\nmore formidable scale than before. But their intentions were defeated by\nthe startling intelligence, that Venice had entered into a separate\ntreaty with the Porte. The treaty, which was negotiated, it is said,\nthrough the intervention of the French ambassador, was executed on the\nseventh of March, 1573. The terms seemed somewhat extraordinary,\nconsidering the relative positions of the parties. By the two principal\narticles the republic agreed to pay the annual sum of one hundred\nthousand ducats for three years to the sultan, and to cede the island of\nCyprus, the original cause of the war. One might suppose it was the\nTurks, and not the Christians, who had won the battle of Lepanto.[357]\n\nVenice was a commercial state, and doubtless had more to gain from peace\nthan from any war, however well conducted. In this point of view, even\nsuch a treaty may have been politic with so formidable an enemy. But a\nnation's interests, in the long run, cannot, any more than those of an\nindividual, be divorced from its honour. And what could be more\ndishonourable than for a state secretly to make terms for herself with\nthe enemy, and desert the allies who had come into the war at her\nsolicitation and in her defence? Such conduct, indeed, was too much in\nharmony with the past history of Venice, and justified the reputation\nfor bad faith which had made the European nations so reluctant to enter\ninto the League.[358]\n\nThe tidings were received by Philip with his usual composure. \"If\nVenice,\" he said, \"thinks she consults her own interests by such a\nproceeding, I can truly say that in what I have done I have endeavoured\nto consult both her interests and those of Christendom.\" He, however,\nspoke his mind more plainly afterwards to the Venetian ambassador. The\npope gave free vent to his feelings in the consistory, where he\ndenounced the conduct of Venice in the most bitter and contemptuous\nterms. When the republic sent a special envoy to deprecate his anger,\nand to excuse herself by the embarrassments of her situation, the\npontiff refused to see him. Don John would not believe in the defection\nof Venice when the tidings were first announced to him. When he was\nadvised of it by a direct communication from her government, he replied\nby indignantly commanding the great standard of the League to be torn\ndown from his galley, and in its place to be unfurled the banner of\nCastile.[359]\n\nSuch was the end of the Holy League, on which Pius the Fifth had so\nfully relied for the conquest of Constantinople and the recovery of\nPalestine. Philip could now transfer the war to the quarter he had\npreferred. He resolved, accordingly, to send an expedition to the\nBarbary coast. Tunis was selected as the place of attack,--a thriving\ncity, and the home of many a corsair who preyed on the commerce of the\nMediterranean. It had been taken by Charles the Fifth, in the memorable\ncampaign of 1535, but had since been recovered by the Moslems. The\nSpaniards, however, still retained possession of the strong fortress of\nthe Goletta, which overlooked the approaches to Tunis.\n\nIn the latter part of September, 1574, Don John left the shores of\nSicily at the head of a fleet consisting of about a hundred galleys, and\nnearly as many smaller vessels. The number of his troops amounted to not\nless than twenty thousand.[360] The story of the campaign is a short\none. Most of the inhabitants of Tunis fled from the city. The few who\nremained did not care to bring the war on their heads by offering\nresistance to the Spaniards. Don John, without so much as firing a shot,\nmarched in at the head of his battalions, through gates flung open to\nreceive him. He found an ample booty awaiting him,--nearly fifty pieces\nof artillery, with ammunition and military stores, large quantities of\ngrain, cotton and woollen cloths, rich silks and brocades, with various\nother kinds of costly merchandise. The troops spent more than a week in\nsacking the place.[361] They gained, in short, everything--but glory;\nfor little glory was to be gained where there were no obstacles to be\novercome.\n\n[Sidenote: DON JOHN AT TUNIS.]\n\nDon John gave orders that no injury should be offered to the persons of\nthe inhabitants. He forbade that any should be made slaves. By a\nproclamation, he invited all to return to their dwellings, under the\nassurance of his protection. In one particular his conduct was\nremarkable. Philip, disgusted with the expenses to which the maintenance\nof the castle of the Goletta annually subjected him, had recommended, if\nnot positively directed, his brother to dismantle the place, and to\ndemolish in like manner the fortifications of Tunis.[362] Instead of\nheeding these instructions, Don John no sooner saw himself in possession\nof the capital, than he commanded the Goletta to be thoroughly repaired,\nand at the same time provided for the erection of a strong fortress in\nthe city. This work he committed to an Italian engineer, named\nCerbelloni, a knight of Malta, with whom he left eight thousand\nsoldiers, to be employed in the construction of the fort, and to furnish\nhim with a garrison to defend it.\n\nDon John, it is said, had been urged to take this course by his\nsecretary, Juan de Soto, a man of ability, but of an intriguing temper,\nwho fostered in his master those ambitious projects which had been\nencouraged, as we have seen, by Pius the Fifth. No more eligible spot\nseemed likely to present itself for the seat of his dominion than\nTunis,--a flourishing capital surrounded by a well-peopled and fruitful\nterritory. Philip had been warned of the unwholesome influence exerted\nby De Soto; and he now sought to remove him from the person of his\nbrother by giving him a distinct position in the army, and by sending\nanother to replace him in his post of secretary. The person thus sent\nwas Juan de Escovedo. But it was soon found that the influence which\nEscovedo acquired over the young prince was both greater and more\nmischievous than that of his predecessor; and the troubles that grew out\nof this new intimacy were destined, as we shall see hereafter, to form\nsome of the darkest pages in the history of the times.\n\nHaving provided for the security of his new acquisition, and received,\nmoreover, the voluntary submission of the neighbouring town of Biserta,\nthe Spanish commander returned with his fleet to Sicily. He landed at\nPalermo, amidst the roaring of cannon, the shouts of the populace, and\nthe usual rejoicings that announce the return of the victorious\ncommander. He did not, however, prolong his stay in Sicily. After\ndismissing his fleet, he proceeded to Naples, where he landed about the\nmiddle of November. He proposed to pass the winter in this capital,\nwhere the delicious climate and the beauty of the women, says a\ncontemporary chronicler, had the attractions for him that belonged\nnaturally to his age.[363] His partiality for Naples was amply requited\nby the inhabitants, especially that lovelier portion of them whose\nsmiles were the well-prized guerdon of the soldier. If his brilliant\nexterior and the charm of his society had excited their admiration when\nhe first appeared among them as an adventurer in the path of honour, how\nmuch was this admiration likely to be increased when he returned with\nthe halo of glory beaming around his brow, as the successful champion of\nChristendom?\n\nThe days of John of Austria glided merrily along in the gay capital of\nSouthern Italy. But we should wrong him did we suppose that all his\nhours were passed in idle dalliance. A portion of each day, on the\ncontrary, was set apart for study. Another part was given to the\ndespatch of business. When he went abroad, he affected the society of\nmen distinguished for their science, or still more for their knowledge\nof public affairs. In his intercourse with these persons he showed\ndignity of demeanour tempered by courtesy; while his conversation\nrevealed those lofty aspirations which proved that his thoughts were\nfixed on a higher eminence than any he had yet reached. It was clear to\nevery observer that ambition was the moving principle of his\nactions,--the passion to which every other passion, even the love of\npleasure, was wholly subordinate.\n\nIn the midst of the gaieties of Naples his thoughts were intent on the\nbest means of securing his African empire. He despatched his secretary,\nEscovedo, to the pope, to solicit his good offices with Philip. Gregory\nentertained the same friendly feelings for Don John which his\npredecessor had shown, and he good-naturedly acquiesced in his petition.\nHe directed his nuncio at the Castilian court to do all in his power to\npromote the suit of the young chief, and to assure the king that nothing\ncould be more gratifying to the head of the Church than to see so worthy\na recompense bestowed on one who had rendered such signal services to\nChristendom. Philip received the communication in the most gracious\nmanner. He was grateful, he said, for the interest which the pope\ncondescended to take in the fortunes of Don John; and nothing,\ncertainly, would be more agreeable to his own feelings than to have the\npower to reward his brother according to his deserts. But to take any\nsteps at present in the matter would be premature. He had received\ninformation that the sultan was making extensive preparations for the\nrecovery of Tunis. Before giving it away, therefore, it would be well to\nsee to whom it belonged.[364]\n\nPhilip's information was correct. No sooner had Selim learned the fate\nof the Barbary capital, than he made prodigious efforts for driving the\nSpaniards from their conquests. He assembled a powerful armament, which\nhe placed under the command of Uluch Ali. As lord of Algiers, that chief\nhad a particular interest in preventing any Christian power from\nplanting its foot in the neighbourhood of his own dominions. The command\nof the land forces was given to Sinan Pasha, Selim's son-in-law.\n\nEarly in July, the Ottoman fleet arrived off the Barbary coast. Tunis\noffered as little resistance to the arms of the Moslems as it had before\ndone to those of the Christians. That city had been so often transferred\nfrom one master to another, that it seemed almost a matter of\nindifference to the inhabitants to whom it belonged. But the Turks found\nit a more difficult matter to reduce the castle of the Goletta and the\nfort raised by the brave engineer Cerbelloni, now well advanced, though\nnot entirely completed. It was not till the middle of September, after\nan incredible waste of life on the part of the assailants, and the\nextermination of nearly the whole of the Spanish garrisons, that both\nthe fortresses surrendered.[365]\n\n[Sidenote: DON JOHN ON A MISSION TO GENOA.]\n\nNo sooner was he in possession of them, than the Turkish commander did\nthat which Philip had in vain wished his brother to do. He razed to the\nground the fortress of the Goletta. Thus ended the campaign, in which\nSpain, besides her recent conquests, saw herself stripped of the strong\ncastle which had defied every assault of the Moslems since the time of\nCharles the Fifth.\n\nOne may naturally ask, Where was John of Austria all this time? He had\nnot been idle, nor had he remained an indifferent spectator of the loss\nof the place he had so gallantly won for Spain. But when he first\nreceived tidings of the presence of a Turkish fleet before Tunis, he was\nabsent on a mission to Genoa, or rather to its neighbourhood. That\nrepublic was at this time torn by factions so fierce, that it was on the\nbrink of a civil war. The mischief threatened to extend even more\nwidely, as the neighbouring powers, especially France and Savoy,\nprepared to take part in the quarrel, in hopes of establishing their own\nauthority in the state. At length Philip, who had inherited from his\nfather the somewhat ill-defined title of \"Protector of Genoa,\" was\ncompelled to interpose in the dispute. It was on this mission that Don\nJohn was sent, to watch more nearly the rival factions. It was not till\nafter this domestic broil had lasted for several months, that the\nprudent policy of the Spanish monarch succeeded in reconciling the\nhostile parties, and thus securing the republic from the horrors of a\ncivil war. He reaped the good fruits of his temperate conduct in the\nmaintenance of his own authority in the counsels of the republic; thus\nbinding to himself an ally whose navy, in time of war, served greatly to\nstrengthen his maritime resources.[366]\n\nWhile detained on this delicate mission, Don John did what he could for\nTunis, by urging the viceroys of Sicily and Naples to send immediate aid\nto the beleaguered garrisons.[367] But these functionaries seem to have\nbeen more interested in the feuds of Genoa than in the fate of the\nAfrican colony. Granvelle, who presided over Naples, was even said to be\nso jealous of the rising fame of John of Austria, as not to be unwilling\nthat his lofty pretensions should be somewhat humbled.[368] The supplies\nsent were wholly unequal to the exigency.\n\nDon John, impatient of the delay, as soon as he could extricate himself\nfrom the troubles of Genoa, sailed for Naples, and thence speedily\ncrossed to Sicily. He there made every effort to assemble an armament,\nof which he prepared, in spite of the remonstrances of his friends, to\ntake the command in person. But nature, no less than man, was against\nhim. A tempest scattered his fleet: and when he had reassembled it, and\nfairly put to sea, he was baffled by contrary winds, and taking refuge\nin the neighbouring port of Trapani, was detained there until tidings\nreached him of the fall of Tunis. They fell heavily on his ear; for they\nannounced to him that all his bright visions of an African empire had\nvanished, like the airy fabric of an Eastern tale. All that remained was\nthe consciousness that he had displeased his brother by his scheme of\nindependent sovereignty, and by his omission to raze the fortress of the\nGoletta, the unavailing defence of which had cost the lives of so many\nof his brave countrymen.\n\nBut Don John, however chagrined by the tidings, was of too elastic a\ntemper to yield to despondency. He was a knight-errant in the true sense\nof the term. He still clung as fondly as ever to the hope of one day\ncarving out with his good sword an independent dominion for himself. His\nfirst step, he considered, was to make his peace with his brother.\nThough not summoned thither, he resolved to return at once to the\nCastilian court,--for in that direction, he felt, lay the true road to\npreferment.\n\n\n\n\nBOOK VI.\n\n\n\n\nCHAPTER I.\n\nDOMESTIC AFFAIRS OF SPAIN.\n\nInternal Administration of Spain--Absolute Power of the Crown--Royal\nCouncils--Alva and Ruy Gomez--Espinoza--Personal Habits of Philip--Court\nand Nobles--The Cortes--The Guards of Castile.\n\n\nSeventeen years had now elapsed since Philip the Second ascended the\nthrone of his ancestors,--a period long enough to disclose the policy of\nhis government; longer, indeed, than that of the entire reigns of some\nof his predecessors. In the previous portion of this work, the reader\nhas been chiefly occupied with the foreign relations of Spain, and with\nmilitary details. It is now time to pause, and, before plunging anew\ninto the stormy scenes of the Netherlands, to consider the internal\nadministration of the country and the character and policy of the\nmonarch who presided over it.\n\nThe most important epoch in Castilian history since the great Saracen\ninvasion in the eighth century, is the reign of Ferdinand and Isabella,\nwhen anarchy was succeeded by law, and from the elements of chaos arose\nthat beautiful fabric of order and constitutional liberty which promised\na new era for the nation. In the assertion of her rights, Isabella, to\nwhom this revolution is chiefly to be attributed, was obliged to rely on\nthe support of the people. It was natural that she should requite their\nservices by aiding them in the recovery of their own rights,--especially\nof those which had been usurped by the rapacious nobles. Indeed, it was\nthe obvious policy of the crown to humble the pride of the aristocracy\nand abate their arrogant pretensions. In this it was so well supported\nby the commons, that the scheme perfectly succeeded. By the depression\nof the privileged classes and the elevation of the people, the different\norders were brought more strictly within their constitutional limits;\nand the state made a nearer approach to a well-balanced limited monarchy\nthan at any previous period of its history.\n\nThis auspicious revolution was soon, alas! to be followed by another, of\na most disastrous kind. Charles the Fifth, who succeeded his grandfather\nFerdinand, was born a foreigner,--and a foreigner he remained through\nhis whole life. He was a stranger to the feelings and habits of the\nSpaniards, had little respect for their institutions, and as little love\nfor the nation. He continued to live mostly abroad; was occupied with\nforeign enterprises; and the only people whom he really loved were those\nof the Netherlands, his native land. The Spaniards requited these\nfeelings of indifference in full measure. They felt that the glory of\nthe imperial name shed no lustre upon them. Thus estranged at heart,\nthey were easily provoked to insurrection by his violation of their\nrights. The insurrection was a failure; and the blow which crushed the\ninsurgents on the plains of Villalar, deprived them for ever of the few\nliberties which they had been permitted to retain. They were excluded\nfrom all share in the government, and were henceforth summoned to the\nCortes only to swear allegiance to the heir apparent, or to furnish\nsubsidies for their master. They were indeed allowed to lay their\ngrievances before the throne. But they had no means of enforcing\nredress; for, with the cunning policy of a despot, Charles would not\nreceive their petitions until they had first voted the supplies.\n\nThe nobles, who had stood by their master in the struggle, fared no\nbetter. They found too late how short-sighted was the policy which had\nled them to put their faith in princes. Henceforth they could not be\nsaid to form a necessary part of the legislature. For as they insisted\non their right to be excused from bearing any share in the burdens of\nthe state, they could take no part in voting the supplies; and as this\nwas almost the only purpose for which the Cortes was convened, their\npresence was no longer required in it. Instead of the powers which were\nleft to them untouched by Ferdinand and Isabella, they were now amused\nwith high-sounding and empty titles, or with offices about the person of\nthe monarch. In this way they gradually sank into the unsubstantial\nthough glittering pageant of a court. Meanwhile the government of\nCastile, assuming the powers of both making the laws and enforcing their\nexecution, became in its essential attributes nearly as absolute as that\nof Turkey.\n\nSuch was the gigantic despotism which, on the death of Charles, passed\ninto the hands of Philip the Second. The son had many qualities in\ncommon with his father. But among these was not that restless ambition\nof foreign conquest which was ever goading the emperor. Nor was he, like\nhis father, urged by the love of glory to military achievement. He was\nof too sluggish a nature to embark readily in great enterprises. He was\ncapable of much labour; but it was of that sedentary kind which belongs\nto the cabinet rather than the camp. His tendencies were naturally\npacific: and up to the period at which we are now arrived, he had\nengaged in no wars but those into which he had been drawn by the revolt\nof his vassals, as in the Netherlands and Granada, or those forced on\nhim by circumstances beyond his control. Such was the war which he had\ncarried on with the pope and the French monarchy at the beginning of his\nreign.\n\nBut while less ambitious than Charles of foreign acquisitions, Philip\nwas full as tenacious of the possessions and power which had come to him\nby inheritance. Nor was it likely that the regal prerogative would\nsuffer any diminution in his reign, or that the nobles or commons would\nbe allowed to retrieve any of the immunities which they had lost under\nhis predecessors.\n\nPhilip understood the character of his countrymen better than his father\nhad done. A Spaniard by birth, he was, as I have more than once had\noccasion to remark, a Spaniard in his whole nature. His tastes, his\nhabits, his prejudices, were all Spanish. His policy was directed solely\nto the aggrandisement of Spain. The distant races whom he governed were\nall strangers to him. With a few exceptions, Spaniards were the only\npersons he placed in offices of trust. His Castilian countrymen saw with\npride and satisfaction that they had a native prince on the throne, who\nidentified his own interests with theirs. They contrasted this conduct\nwith that of his father, and requited it with a devotion such as they\nhad shown to few of his predecessors. They not only held him in\nreverence, says the Venetian minister Contarini, but respected his laws,\nas something sacred and inviolable.[369] It was the people of the\nNetherlands who rose up against him. For similar reasons it fared just\nthe opposite with Charles. His Flemish countrymen remained loyal to the\nlast: it was his Castilian subjects who were driven to rebellion.\n\n[Sidenote: ALVA AND RUY GOMEZ.]\n\nThough tenacious of power, Philip had not the secret consciousness of\nstrength which enabled his father, unaided as it were, to bear up so\nlong under the burden of empire. The habitual caution of the son made\nhim averse to taking any step of importance without first ascertaining\nthe opinions of others. Yet he was not willing, like his ancestor, the\ngood Queen Isabella, to invoke the co-operation of the Cortes, and thus\nawaken the consciousness of power in an arm of the government which had\nbeen so long smitten with paralysis. Such an expedient was fraught with\ntoo much danger. He found a substitute in the several councils, the\nmembers of which, appointed by the crown and removable at its pleasure,\nwere pledged to the support of the prerogative.\n\nUnder Ferdinand and Isabella there had been a complete reorganization of\nthese councils. Their number was increased under Charles the Fifth, to\nsuit the increased extent of the empire. It was still further enlarged\nby Philip.[370] Under him there were no less than eleven councils, among\nwhich may be particularly noticed those of war, of finance, of justice,\nand of state.[371] Of these various bodies the council of state, charged\nwith the most important concerns of the monarchy, was held in highest\nconsideration. The number of its members varied. At the time of which I\nam writing, it amounted to sixteen.[372] But the weight of the business\ndevolved on less than half that number. It was composed of both\necclesiastics and laymen. Among the latter were some eminent jurists. A\nsprinkling of men of the robe, indeed, was to be found in most of the\ncouncils. Philip imitated in this the policy of Ferdinand and Isabella,\nwho thus intended to humble the pride of the great lords, and to provide\nthemselves with a loyal militia, whose services would be of no little\nadvantage in maintaining the prerogative.\n\nAmong the members of the council of state, two may be particularly\nnoticed for their pre-eminence in that body. These were the duke of Alva\nand Ruy Gomez de Silva, prince of Eboli. With the former the reader is\nwell acquainted. His great talents, his ample experience both in civil\nand military life, his iron will, and the fearlessness with which he\nasserted it, even his stern and overbearing manner, which seemed to\nproclaim his own superiority, all marked him out as the leader of a\nparty.\n\nThe emperor appears to have feared the ascendancy which Alva might one\nday acquire over Philip. \"The duke,\" wrote Charles to his son in a\nletter before cited, \"is the ablest statesman and the best soldier I\nhave in my dominions. Consult him, above all, in military affairs. But\ndo not depend on him entirely in these or any other matters. Depend on\nno one but yourself.\" The advice was good; and Philip did not fail to\nprofit by it. Though always seeking the opinions of others, it was the\nbetter to form his own. He was too jealous of power to submit to the\ncontrol, even to the guidance, of another. With all his deference to\nAlva, on whose services he set the greatest value, the king seems to\nhave shown him but little of that personal attachment which he evinced\nfor his rival, Ruy Gomez.\n\nThis nobleman was descended from an ancient house in Portugal, a branch\nof which had been transplanted to Castile. He had been early received as\na page in the imperial household, where, though he was several years\nolder than Philip, his amiable temper, his engaging manners, and above\nall, that tact which made his fortune in later life, soon rendered him\nthe prince's favourite. An anecdote is reported of him at this time,\nwhich, however difficult to credit, rests on respectable authority.\nWhile engaged in their sports, the page accidentally struck the prince.\nThe emperor, greatly incensed, and conceiving that such an indignity to\nthe heir-apparent was to be effaced only by the blood of the offender,\ncondemned the unhappy youth to lose his life. The tears and entreaties\nof Philip at length so far softened the heart of his father, that he\nconsented to commute the punishment of death for exile. Indeed, it is\nhard to believe that Charles had ever really intended to carry his cruel\nsentence into execution. The exile was of no long duration. The society\nof Gomez had become indispensable to the prince, who, pining under the\nseparation, at length prevailed on his father to recall the young noble,\nand reinstate him in his former situation in the palace.[373]\n\nThe regard of Philip, who was not of a fickle disposition, seemed to\nincrease with years. We find Ruy Gomez one of the brilliant suite who\naccompanied him to London on his visit there to wed the English queen.\nAfter the emperor's abdication, Ruy Gomez continued to occupy a\ndistinguished place in Philip's household, as first gentleman of the\nbedchamber. By virtue of this office he was required to attend his\nmaster both at his rising and his going to rest. His situation gave him\nready access at all hours to the royal person. It was soon understood\nthat there was no one in the court who exercised a more important\ninfluence over the monarch; and he naturally became the channel through\nwhich applicants for favours sought to prefer their petitions.[374]\n\nMeanwhile the most substantial honours were liberally bestowed on him.\nHe was created duke of Pastrana, with an income of twenty-five thousand\ncrowns--a large revenue, considering the value of money in that day. The\ntitle of Pastrana was subsequently merged in that of Eboli, by which he\nhas continued to be known. It was derived from his marriage with the\nprincess of Eboli, Anna de Mendoza, a lady much younger than he, and,\nthough blind of one eye, celebrated for her beauty no less than her wit.\nShe was yet more celebrated for her gallantries, and for the tragic\nresults to which they led--a subject closely connected with the personal\nhistory of Philip, to which I shall return hereafter.\n\nAmong his other dignities Ruy Gomez was made a member of the council of\nstate, in which body he exercised an influence not inferior, to say the\nleast of it, to that of any of his associates. His head was not turned\nby his prosperity. He did not, like many a favourite before him, display\nhis full-blown fortunes in the eye of the world; nor, though he\nmaintained a state suited to his station, did he, like Wolsey, excite\nthe jealousy of his master by a magnificence in his way of living that\neclipsed the splendours of royalty. Far from showing arrogance to his\ninferiors, he was affable to all, did what he could to serve their\ninterests with the king, and magnanimously spoke of his rivals in terms\nof praise. By this way of proceeding he enjoyed the good fortune, rare\nfor a favourite, of being both caressed by his sovereign and beloved by\nthe people.[375]\n\n[Sidenote: FIGUEROA AND ESPINOSA.]\n\nThere is no evidence that Ruy Gomez had the moral courage to resist the\nevil tendency of Philip's policy, still less that he ventured to open\nthe monarch's eyes to his errors. He had too keen a regard to his own\ninterests to attempt this. He may have thought, probably with some\nreason, that such a course would avail little with the king, and would\nbring ruin on himself. His life was passed in the atmosphere of a court,\nand he had imbibed its selfish spirit. He had profoundly studied the\ncharacter of his master, and he accommodated himself to all his humours\nwith an obsequiousness which does little honour to his memory. The duke\nof Alva, who hated him with all the hatred of a rival, speaking of him\nafter his death, remarked: \"Ruy Gomez, though not the greatest statesman\nthat ever lived, was such a master in the knowledge of the humours and\ndispositions of kings, that we were all of us fools in comparison.\"[376]\n\nYet the influence of the favourite was, on the whole, good. He was\nhumane and liberal in his temper, and inclined to peace,--virtues which\nwere not too common in that iron age, and which in the council served\nmuch to counteract the stern policy of Alva. Persons of a generous\nnature ranged themselves under him as their leader. When John of Austria\ncame to court, his liberal spirit prompted him at once to lean on Ruy\nGomez as his friend and counsellor. The correspondence which passed\nbetween them when the young soldier was on his campaigns, in which he\naddressed the favourite by the epithet of \"father,\" confessing his\nerrors to him and soliciting his advice, is honourable to both.\n\nThe historian Cabrera, who had often seen him, sums up the character of\nRuy Gomez by saying: \"He was the first pilot who in these stormy seas\nboth lived and died secure, always contriving to gain a safe port.\"[377]\nHis death took place in July, 1573. \"Living,\" adds the writer, in his\npeculiar style, \"he preserved the favour of his sovereign;--dead, he was\nmourned by him,--and by the whole nation, which kept him in its\nrecollection as the pattern of loyal vassals and prudent\nfavourites.\"[378]\n\nBesides the two leaders in the council, there were two others who\ndeserve to be noticed. One of these was Figueroa, count, afterwards\ncreated by Philip duke, of Feria, a grandee of Spain. He was one of\nthose who accompanied the king on his first visit to England. He there\nmarried a lady of rank, and, as the reader may remember, afterwards\nrepresented his master at the court of Elizabeth. He was a man of\nexcellent parts, enriched by that kind of practical knowledge which he\nhad gained from foreign travel and a familiarity with courts. He lived\nmagnificently, somewhat encumbering his large estates indeed by his\nprofusion. His person was handsome; and his courteous and polished\nmanners made him one of the most brilliant ornaments of the royal\ncircle. He had a truly chivalrous sense of honour, and was greatly\nesteemed by the king, who placed him near his person as captain of his\nSpanish guard. Feria was a warm supporter of Ruy Gomez; and the long\nfriendship that subsisted between the two nobles seems never to have\nbeen clouded by those feelings of envy and jealousy which so often arise\nbetween rivals contending for the smiles of their sovereign.\n\nThe other member of the council of state was a person of still more\nimportance. This was the Cardinal Espinosa, who, though an ecclesiastic,\npossessed such an acquaintance with affairs as belonged to few laymen.\nPhilip's eye readily discovered his uncommon qualities, and he heaped\nupon him offices in rapid succession, any one of which might well have\nengrossed his time. But Espinosa was as fond of labour as most men are\nof ease; and in every situation he not only performed his own share of\nthe work, but very often that of his associates. He was made president\nof the council of Castile, as well as that of the Indies, and finally a\nmember of the council of state. He was inquisitor-general, sat in the\nroyal chancery of Seville, and held the bishopric of Siguenca, one of\nthe richest sees in the kingdom. To crown the whole, in 1568, Pius the\nFifth, on the application of Philip, gave him a cardinal's hat. The king\nseems to have taken the greater pleasure in this rapid elevation of\nEspinosa, that he sprang from a comparatively humble condition; and thus\nthe height to which he raised him served the more keenly to mortify the\nnobles.\n\nBut the cardinal, as is too often the case with those who have suddenly\nrisen to greatness, did not bear his honours meekly. His love of power\nwas insatiable; and when an office became vacant in any of his own\ndepartments, he was prompt to secure it for one of his dependents. An\nanecdote is told in relation to a place in the chancery of Granada,\nwhich had become open by the death of the incumbent. As soon as the news\nreached Madrid, Hernandez de Cordova, the royal equerry, made\napplication to the king for it. Philip answered that he was too late,\nthat the place had been already given away. \"How am I to understand your\nmajesty?\" said the petitioner; \"the tidings were brought to me by a\ncourier the moment at which the post became vacant, and no one could\nhave brought them sooner unless he had wings.\" \"That may be,\" said the\nmonarch; \"but I have just given the place to another, whom the cardinal\nrecommended to me as I was leaving the council.\"[379]\n\nEspinosa, says a contemporary, was a man of noble presence. He had the\nair of one born to command. His haughty bearing, however, did little for\nhim with the more humble suitors, and disgusted the great lords, who\nlooked down with contempt on his lowly origin. They complained to the\nking of his intolerable arrogance; and the king was not unwilling to\nreceive their charges against him. In fact, he had himself grown to be\ndispleased with his minister's presumption. He was weary of the\ndeference which, now that Espinosa had become a cardinal, he felt\nobliged to pay him; of coming forward to receive him when he entered the\nroom; of taking off his cap to the churchman, and giving him a seat as\nhigh as his own; finally, of allowing him to interfere in all\nappointments to office. It seemed incredible, says the historian, that a\nprince so jealous of his prerogatives should have submitted to all this\nso long.[380] Philip was now determined to submit to it no longer; but\nto tumble from its pride of place the idol which he had raised with his\nown hands.\n\nHe was slow in betraying his intention, by word or act, to the\ncourtiers, still more to the unfortunate minister, who continued to show\nthe same security and confidence as if he were treading the solid\nground, instead of the crust of a volcano.\n\n[Sidenote: THE COUNCIL OF STATE.]\n\nAt length an opportunity offered when Espinosa, in a discussion\nrespecting the affairs of Flanders, made a statement which the king\ndeemed not entirely conformable to truth. Philip at once broke in upon\nthe discourse with an appearance of great indignation, and charged the\nminister with falsehood. The blow was the more effectual, coming from\none who had been scarcely ever known to give way to passion.[381] The\ncardinal was stunned by it. He at once saw his ruin, and the vision of\nglory vanished for ever. He withdrew, more dead than alive, to his\nhouse. There he soon took to his bed; and in a short time, in September\n1572, he breathed his last. His fate was that of more than one minister\nwhose head had been made giddy by the height to which he had\nclimbed.[382]\n\nThe council of state under its two great leaders, Alva and Ruy Gomez,\nwas sure to be divided on every question of importance. This was a\nfruitful source of embarrassment, and to private suitors, especially,\noccasioned infinite delay. Such was the hostility of the parties to each\nother, that, if an applicant for favour secured the good-will of one of\nthe chiefs, he was very certain to encounter the ill-will of the\nother.[383] He was a skilful pilot who in such cross seas could keep his\ncourse.\n\nYet the existence of these divisions does not seem to have been\ndiscouraged by Philip, who saw in them only the natural consequence of\nrivalry for his favour. They gave him, moreover, the advantage of seeing\nevery question of moment well canvassed, and, by furnishing him with the\nopposite opinions of his councillors, enabled him the more accurately to\nform his own.\n\nIn the mean time, the value which he set on both the great chiefs made\nhim careful not to disgust either by any show of preference for his\nrival. He held the balance adroitly between them; and if on any occasion\nhe bestowed a mark of his favour on the one, it was usually followed by\nsome equivalent to the other.[384] Thus, for the first twelve years of\nhis reign, their influence may be said to have been pretty equally\nexerted. Then came the memorable discussion respecting the royal visit\nto the Netherlands, Alva, as the reader may remember, was of the opinion\nthat Philip should send an army to punish the refractory and bring the\ncountry to obedience, when the king might visit it with safety to his\nown person. Ruy Gomez, on the other hand, recommended that Philip should\ngo at once, without an army, and by mild and conciliatory measures win\nthe malcontents back to their allegiance. Each advised the course most\ncongenial to his own temper, and the one, moreover, which would have\nrequired the aid of his own services to carry into execution.\nUnfortunately, the violent measures of Alva were more congenial to the\nstern temper off the king, and the duke was sent at the head of his\nbattalions.\n\nBut if Alva thus gained the victory, it was Ruy Gomez who reaped the\nfruits of it. Left without a rival in the council, his influence became\npredominant over every other. It became still more firmly established,\nas the result showed that his rival's mission was a failure. So it\ncontinued, after Alva's return, till the favourite's death. Even then\nhis well-organized party was so deeply rooted, that for several years\nlonger it maintained an ascendancy in the cabinet, while the duke\nlanguished in disgrace.\n\nPhilip, unlike most of his predecessors, rarely took his seat in the\ncouncil of state. It was his maxim that his ministers would more freely\ndiscuss measures in the absence of their master than when he was there\nto overawe them. The course he adopted was for a _consulta_, or a\ncommittee of two or three members, to wait on him in his cabinet, and\nreport to him the proceedings of the council.[385] He more commonly,\nespecially in the later years of his reign, preferred to receive a full\nreport of the discussion, written so as to leave an ample margin for his\nown commentaries. These were eminently characteristic of the man, and\nwere so minute as usually to cover several sheets of paper. Philip had a\nreserved and unsocial temper. He preferred to work alone, in the\nseclusion of his closet, rather than in the presence of others. This may\nexplain the reason, in part, why he seemed so much to prefer writing to\ntalking. Even with his private secretaries, who were always near at\nhand, he chose to communicate by writing; and they had as large a mass\nof his autograph notes in their possession, as if the correspondence had\nbeen carried on from different parts of the kingdom.[386] His thoughts\ntoo--at any rate his words--came slowly; and by writing he gained time\nfor the utterance of them.\n\nPhilip has been accused of indolence. As far as the body was concerned,\nsuch an accusation was well founded. Even when young, he had no\nfondness, as we have seen, for the robust and chivalrous sports of the\nage. He never, like his father, conducted military expeditions in\nperson. He thought it wiser to follow the example of his\ngreat-grandfather, Ferdinand the Catholic, who stayed at home and sent\nhis generals to command his armies. As little did he like to\ntravel,--forming too in this respect a great contrast to the emperor. He\nhad been years on the throne before he made a visit to his great\nsouthern capital, Seville. It was a matter of complaint in Cortes that\nhe thus withdrew himself from the eyes of his subjects. The only sport\nhe cared for--not by any means to excess--was shooting with his gun or\nhis crossbow such game as he could find in his own grounds at the wood\nof Segovia, or Aranjuez, or some other of his pleasant country seats,\nnone of them at a great distance from Madrid.\n\nOn a visit to such places he would take with him as large a heap of\npapers as if he were a poor clerk, earning his bread; and after the\nfatigues of the chase, he would retire to his cabinet and refresh\nhimself with his despatches.[387] It would, indeed, be a great mistake\nto charge him with sluggishness of mind. He was content to toil for\nhours, and long into the night, at his solitary labours.[388] No\nexpression of weariness or of impatience was known to escape him. A\ncharacteristic anecdote is told of him in regard to this. Having written\na despatch, late at night, to be sent on the following morning, he\nhanded it to his secretary to throw some sand over it. This functionary,\nwho happened to be dozing, suddenly roused himself, and, snatching up\nthe ink-stand, emptied it on the paper. The king, coolly remarking that\n\"it would have been better to use the sand,\" set himself down, without\nany complaint, to rewrite the whole of the letter.[389] A prince so much\naddicted to the pen, we may well believe, must have left a large amount\nof autograph materials behind him. Few monarchs, in point of fact, have\ndone so much in this way to illustrate the history of their reigns.\nFortunate would it have been for the historian who was to profit by it,\nif the royal composition had been somewhat less diffuse and the\nhandwriting somewhat more legible.\n\n[Sidenote: PERSONAL HABITS OF PHILIP.]\n\nPhilip was an economist of time, and regulated the distribution of it\nwith great precision. In the morning, he gave audience to foreign\nambassadors. He afterwards heard mass. After mass came dinner, in his\nfather's fashion. But dinner was not an affair with Philip of so much\nmoment as it was with Charles. He was exceedingly temperate both in\neating and drinking, and not unfrequently had his physician at his side,\nto warn him against any provocative of the gout,--the hereditary disease\nwhich at a very early period had begun to affect his health. After a\nlight repast, he gave audience to such of his subjects as desired to\npresent their memorials. He received the petitioners graciously, and\nlistened to all they had to say with patience,--for that was his virtue.\nBut his countenance was exceedingly grave,--which, in truth, was its\nnatural expression; and there was a reserve in his deportment which made\nthe boldest feel ill at ease in his presence. On such occasions he would\nsay, \"Compose yourself,\"--a recommendation that had not always the\ntranquillizing effect intended.[390] Once when a papal nuncio forgot, in\nhis confusion, the address he had prepared, the king coolly remarked:\n\"If you will bring it in writing, I will read it myself, and expedite\nyour business.\"[391] It was natural that men of even the highest rank\nshould be overawed in the presence of a monarch who held the destinies\nof so many millions in his hands, and who surrounded himself with a veil\nof mystery which the most cunning politician could not penetrate.\n\nThe reserve so noticeable in his youth increased with age. He became\nmore difficult of access. His public audiences were much less frequent.\nIn the summer he would escape from them altogether, by taking refuge in\nsome one of his country places. His favourite retreat was his\npalace-monastery of the Escorial, then slowly rising under his\npatronage, and affording him an occupation congenial with his taste. He\nseems, however, to have sought the country not so much from the love of\nits beauties as for the retreat it afforded him from the town. When in\nthe latter, he rarely showed himself to the public eye, going abroad\nchiefly in a close carriage, and driving late, so as to return to the\ncity after dark.[392]\n\nThus he lived in solitude even in the heart of his capital, knowing much\nless of men from his own observation than from the reports that were\nmade to him. In availing himself of these sources of information he was\nindefatigable. He caused a statistical survey of Spain to be prepared\nfor his own use. It was a work of immense labour, embracing a vast\namount of curious details, such as were rarely brought together in those\ndays.[393] He kept his spies at the principal European courts, who\nfurnished him with intelligence; and he was as well acquainted with what\nwas passing in England and in France, as if he had resided on the spot.\nWe have seen how well he knew the smallest details of the proceedings in\nthe Netherlands, sometimes even better than Margaret herself. He\nemployed similar means to procure information that might be of service\nin making appointments to ecclesiastical and civil offices.\n\nIn his eagerness for information, his ear was ever open to accusations\nagainst his ministers, which, as they were sure to be locked up in his\nown bosom, were not slow in coming to him.[394] This filled his mind\nwith suspicions. He waited till time had proved their truth, treating\nthe object of them with particular favour till the hour of vengeance had\narrived. The reader will not have forgotten the terrible saying of\nPhilip's own historian, \"His dagger followed close upon his smile.\"[395]\n\nEven to the ministers in whom Philip appeared most to confide, he often\ngave but half his confidence. Instead of frankly furnishing them with a\nfull statement of facts, he sometimes made so imperfect a disclosure,\nthat, when his measures came to be taken, his counsellors were surprised\nto find of how much they had been kept in ignorance. When he\ncommunicated to them any foreign despatches, he would not scruple to\nalter the original, striking out some passages and inserting others, so\nas best to serve his purpose. The copy, in this garbled form, was given\nto the council. Such was the case with, a letter of Don John of Austria,\ncontaining an account of the troubles of Genoa; the original of which,\nwith its numerous alterations in the royal handwriting, still exists in\nthe archives of Simancas.[396]\n\nBut though Philip's suspicious nature prevented him from entirely\ntrusting his ministers,--though with chilling reserve he kept at a\ndistance even those who approached him nearest,--he was kind, even\nliberal, to his servants, was not capricious in his humours, and seldom,\nif ever, gave way to those sallies of passion so common in princes\nclothed with, absolute power. He was patient to the last degree, and\nrarely changed his ministers without good cause. Ruy Gomez was not the\nonly courtier who continued in the royal service to the end of his days.\n\nPhilip was of a careful, or, to say truth, of a frugal disposition,\nwhich he may well have inherited from his father; though this did not,\nas with his father in later life, degenerate into parsimony. The\nbeginning of his reign, indeed, was distinguished by some acts of\nuncommon liberality. One of these occurred at the close of Alva's\ncampaigns in Italy, when the king presented that commander with a\nhundred and fifty thousand ducats, greatly to the discontent of the\nemperor. This was contrary to his usual policy. As he grew older, and\nthe expenses of government pressed more heavily on him, he became more\neconomical. Yet those who served him had no reason, like the emperor's\nservants, to complain of their master's meanness. It was observed,\nhowever, that he was slow to recompense those who served him until they\nhad proved themselves worthy of it. Still it was a man's own fault, says\na contemporary, if he was not well paid for his services in the\nend.[397]\n\n[Sidenote: THE ROYAL ESTABLISHMENT.]\n\nIn one particular he indulged in a most lavish expenditure. This was his\nhousehold. It was formed on the Burgundian model,--the most stately and\nmagnificent in Europe. Its peculiarity consisted in the number and\nquality of the members who composed it. The principal officers were\nnobles of the highest rank, who frequently held posts of great\nconsideration in the state. Thus the duke of Alva was chief major-domo;\nthe prince of Eboli was first gentleman of the bedchamber; the duke of\nFeria was captain of the Spanish guard. There was the grand equerry, the\ngrand huntsman, the chief muleteer, and a host of officers, some of whom\nwere designated by menial titles, though nobles and cavaliers of\nfamily.[398] There were forty pages, sons of the most illustrious houses\nin Castile. The whole household amounted to no less than fifteen hundred\npersons.[399] The king's guard consisted of three hundred men, one-third\nof whom were Spaniards, one-third Flemings, and the remainder\nGermans.[400]\n\nThe queen had also her establishment on the same scale. She had\ntwenty-six ladies-in-waiting, and, among other functionaries, no less\nthan four physicians to watch over her health.[401]\n\nThe annual cost of the royal establishment amounted to full two hundred\nthousand florins.[402] The Cortes earnestly remonstrated against this\nuseless prodigality, beseeching the king to place his household on the\nmodest scale to which the monarchs of Castile had been accustomed.[403]\nAnd it seems singular that one usually so averse to extravagance and\npomp should have so recklessly indulged in them here. It was one of\nthose inconsistencies which we sometimes meet with in private life, when\na man, habitually careful of his expenses, indulges himself in some,\nwhich taste, or, as in this case, early habits, have made him regard as\nindispensable. The emperor had been careful to form the household of his\nson, when very young, on the Burgundian model; and Philip, thus early\ntrained, probably regarded it as essential to the royal dignity.\n\nThe king did not affect an ostentation in his dress corresponding with\nthat of his household. This seemed to be suited to the sober-\nlivery of his own feelings, and was almost always of black velvet or\nsatin, with shoes of the former material. He wore a cap garnished with\nplumes after the Spanish fashion. He used few ornaments, scarce any but\nthe rich jewel of the Golden Fleece, which hung from his neck. But in\nhis attire he was scrupulously neat, says the Venetian diplomatist who\ntells these particulars; and he changed his dress for a new one every\nmonth, giving away his cast-off suits to his attendants.[404]\n\nIt was a capital defect in Philip's administration, that his love of\npower and his distrust of others made him desire to do everything\nhimself; even those things which could be done much better by his\nministers. As he was slow in making up his own opinions, and seldom\nacted without first ascertaining those of his council, we may well\nunderstand the mischievous consequences of such delay. Loud were the\ncomplaints of private suitors, who saw month after month pass away\nwithout an answer to their petitions. The state suffered no less, as the\nwheels of government seemed actually to stand still under the\naccumulated pressure of the public business. Even when a decision did\ncome, it often came too late to be of service; for the circumstances\nwhich led to it had wholly changed. Of this the reader has seen more\nthan one example in the Netherlands. The favourite saying of Philip,\nthat \"time and he were a match for any other two,\" was a sad mistake.\nThe time he demanded was his ruin. It was in vain that Granvelle, who at\na later day came to Castile to assume the direction of affairs,\nendeavoured, in his courtly language, to convince the king of his error,\ntelling him that no man could bear up under such a load of business,\nwhich sooner or later must destroy his health, perhaps his life.[405]\n\nA letter addressed to the king by his grand almoner, Don Luis Manrique,\ntold the truth in plainer terms, such as had not often reached the royal\near. \"Your majesty's subjects everywhere complain,\" he says, \"of your\nmanner of doing business; sitting all day long over your papers, from\nyour desire, as they intimate, to seclude yourself from the world, and\nfrom a want of confidence in your ministers.[406] Hence such\ninterminable delays as fill the soul of every suitor with despair. Your\nsubjects are discontented that you refuse to take your seat in the\ncouncil of state. The Almighty,\" he adds, \"did not send kings into the\nworld to spend their days in reading or writing, or even in meditation\nand prayer,\"--in which Philip was understood to pass much of his\ntime,--\"but to serve as public oracles, to which all may resort for\nanswers. If any sovereign have received this grace, it is your majesty;\nand the greater the sin, therefore, if you do not give free access to\nall.\"[407] One may be surprised to find that language such as this was\naddressed to a prince like Philip the Second, and that he should have\nborne it so patiently. But in this the king resembled his father.\nChurchmen and jesters--of which latter he had usually one or two in\nattendance--were privileged persons at his court. In point of fact, the\nhomilies of the one had as little effect as the jests of the other.\n\nThe pomp of the royal establishment was imitated on a smaller scale by\nthe great nobles living on their vast estates scattered over the\ncountry. Their revenues were very large, though often heavily burdened.\nOut of twenty-three dukes, in 1581, only three had an income so low as\nforty thousand ducats a year.[408] That of most of the others ranged\nfrom fifty to a hundred thousand; and that of one, the duke of Medina\nSidonia, was computed at a hundred and thirty-five thousand. Revenues\nlike these would not easily have been matched in that day by the\naristocracy of any other nation in Christendom.[409]\n\n[Sidenote: POMP OF THE NOBLES.]\n\nThe Spanish grandees preferred to live on their estates in the country.\nBut in the winter they repaired to Madrid, and displayed their\nmagnificence at the court of their sovereign. Here they dazzled the eye\nby the splendour of their equipages, the beauty of their horses, their\nrich liveries, and the throng of their retainers. But with all this the\nCastilian court was far from appearing in the eyes of foreigners a gay\none; forming in this respect a contrast to the Flemish court of Margaret\nof Parma. It seemed to have imbibed much of the serious and indeed\nsombre character of the monarch who presided over it. All was stately\nand ceremonious, with old-fashioned manners and usages. \"There is\nnothing new to be seen there,\" write the Venetian envoys. \"There is no\npleasant gossip about the events of the day. If a man is acquainted with\nany news, he is too prudent to repeat it.[410] The courtiers talk\nlittle, and for the most part are ignorant; in fact, without the least\ntincture of learning. The arrogance of the great lords is beyond belief;\nand when they meet a foreign ambassador, or even the nuncio of his\nholiness, they rarely condescend to salute him by raising their\ncaps.[411] They all affect that imperturbable composure, or apathy,\nwhich they term _sosiego_.\"[412]\n\nThey gave no splendid banquets, like the Flemish nobles. Their chief\namusement was gaming,--the hereditary vice of the Spaniard. They played\ndeep, often to the great detriment of their fortunes. This did not\ndisplease the king. It may seem strange that a society so cold and\nformal should be much addicted to intrigue.[413] In this they followed\nthe example of their master.\n\nThus passing their days in frivolous amusements and idle dalliance, the\nSpanish nobles, with the lofty titles and pretensions of their\nancestors, were a degenerate race. With a few brilliant exceptions, they\nfilled no important posts in the state or in the army. The places of\nmost consideration to which they aspired were those connected with the\nroyal household; and their greatest honour was to possess the empty\nprivileges of the grandee, and to sit with their heads covered in the\npresence of the king.[414]\n\nFrom this life of splendid humiliation they were nothing loth to escape\ninto the country, where they passed their days in their ancestral\ncastles, surrounded by princely domains, which embraced towns and\nvillages within their circuit, and a population sometimes reaching to\nthirty thousand families. Here the proud lords lived in truly regal\npomp. Their households were formed on that of the sovereign. They had\ntheir major-domos, their gentlemen of the bedchamber, their grand\nequerries, and other officers of rank. Their halls were filled with\nhidalgos and cavaliers, and a throng of inferior retainers. They were\nattended by body-guards of one or two hundred soldiers. Their dwellings\nwere sumptuously furnished, and their sideboards loaded with plate from\nthe silver quarries of the New World. Their chapels were magnificent.\nTheir wives affected a royal state: they had their ladies of honour; and\nthe page who served as cupbearer knelt while his mistress drank. Even\nknights of ancient blood, whom she addressed from her seat, did not\nrefuse to bend the knee to her.[415]\n\nAmidst all this splendour, the Spanish grandees had no real power to\ncorrespond with it. They could no longer, as in the days of their\nfathers, engage in fends with one another; nor could they enjoy the\nprivilege, so highly prized, of renouncing their allegiance and\ndeclaring war upon their sovereign. Their numerous vassals, instead of\nbeing gathered as of yore into a formidable military array, had sunk\ninto the more humble rank of retainers, who served only to swell the\nidle pomp of their lord's establishment: they were no longer allowed to\nbear arms, except in the service of the crown; and after the Moriscoes\nhad been reduced, the crown had no occasion for their services, unless\nin foreign war.[416]\n\nThe measures by which Ferdinand and Isabella had broken the power of the\naristocracy had been enforced with still greater rigour by Charles the\nFifth, and were now carried out even more effectually by Philip the\nSecond; for Philip had the advantage of being always in Spain, while\nCharles passed most of his time in other parts of his dominions. Thus\never present, Philip was as prompt to enforce the law against the\nhighest noble as against the humblest of his subjects.\n\nMen of rank commanded the armies abroad, and were sent as viceroys to\nNaples, Sicily, Milan, and the provinces of the New World. But at home\nthey were rarely raised to civil or military office. They no longer\nformed a necessary part of the national legislature, and were seldom\nsummoned to the meetings of the Cortes; for the Castilian noble claimed\nexemption from the public burdens, and it was rarely that the Cortes\nwere assembled for any other purpose than to impose those burdens. Thus,\nwithout political power of any kind, they resided like so many private\ngentlemen on their estates in the country. Their princely style of\nliving gave no umbrage to the king, who was rather pleased to see them\ndissipate their vast revenues in a way that was attended with no worse\nevil than that of driving the proprietors to exactions which made them\nodious to their vassals.[417] Such, we are assured by a Venetian\nenvoy--who, with great powers of observation, was placed in the best\nsituation for exerting them--was the policy of Philip. \"Thus,\" he\nconcludes, \"did the king make himself feared by those who, if they had\nmanaged discreetly, might have made themselves feared by him.\"[418]\n\nWhile the aristocracy was thus depressed, the strong arm of Charles the\nFifth had stripped the Castilian commons of their most precious rights.\nPhilip, happily for himself, was spared the odium of having reduced them\nto this abject condition. But he was as careful as his father could have\nbeen, that they should not rise from it. The legislative power of the\ncommons--that most important of all their privileges--was nearly\nannihilated. The Castilian Cortes were, it is true, frequently convoked\nunder Philip--more frequently, on the whole, than in any preceding\nreign; for in them still resided the power of voting supplies for the\ncrown. To have summoned them so often, therefore, was rather a proof of\nthe necessities of the government than of respect for the rights of the\ncommons.\n\n[Sidenote: THE CORTES.]\n\nThe Cortes, it is true, still enjoyed the privilege of laying their\ngrievances before the king; but as they were compelled to vote the\nsupplies before they presented their grievances, they had lost the only\nlever by which they could effectually operate on the royal will. Yet\nwhen we review their petitions, and see the care with which they watched\nover the interests of the nation, and the courage with which they\nmaintained them, we cannot refuse our admiration. We must acknowledge\nthat, under every circumstance of discouragement and oppression, the\nold Castilian spirit still lingered in the hearts of the people. In\nproof of this, it will not be amiss to cite a few of these petitions,\nwhich, whether successful or not, may serve at least to show the state\nof public opinion on the topics to which they relate.\n\nOne, of repeated recurrence, is a remonstrance to the king on the\nenormous expense of his household--\"as great,\" say the Cortes, \"as would\nbe required for the conquest of a kingdom.\"[419] The Burgundian\nestablishment, independently of its costliness, found little favour with\nthe honest Castilian; and the Cortes prayed his majesty to abandon it,\nand to return to the more simple and natural usage of his ancestors.\nThey represented \"the pernicious effects which this manner of living\nnecessarily had on the great nobles and others of his subjects, prone to\nfollow the example of their master.\"[420] To one of these petitions\nPhilip replied, that \"he would cause the matter to be inquired into, and\nsuch measures to be taken as were most for his service.\" \"No alteration\ntook place during his reign; and the Burgundian establishment, which in\n1562 involved an annual charge of a hundred and fifty-six millions of\nmaravedis, was continued by his successor.\"[421]\n\nAnother remonstrance of constant recurrence--a proof of its\ninefficacy--was that against the alienation of the crown lands, and the\nsale of offices and the lesser titles of nobility. To this the king made\nanswer in much the same equivocal language as before. Another petition\nbesought him no longer to seek an increase of his revenue by imposing\ntaxes without the sanction of the Cortes, required by the ancient law\nand usage of the realm. Philip's reply on this occasion was plain\nenough. It was, in truth, one worthy of an eastern despot. \"The\nnecessities,\" he said, \"which have compelled me to resort to these\nmeasures, far from having ceased, have increased, and are still\nincreasing, allowing me no alternative but to pursue the course I have\nadopted.\"[422] Philip's embarrassments were indeed great,--far beyond\nthe reach of any financial skill of his ministers to remove. His various\nexpedients for relieving himself from the burden which, as he truly\nsaid, was becoming heavier every day, form a curious chapter in the\nhistory of finance. But we have not yet reached the period at which they\ncan be most effectively presented to the reader.\n\nThe commons strongly urged the king to complete the great work he had\nearly undertaken, of embodying in one code the municipal law of\nCastile.[423] They gave careful attention to the administration of\njustice, showed their desire for the reform of various abuses,\nespecially for quickening the despatch of business, proverbially slow in\nSpain, and, in short, for relieving suitors, as far as possible, from\nthe manifold vexations to which they were daily exposed in the\ntribunals. With a wise liberality they recommended that, in order to\nsecure the services of competent persons in judicial offices, their\nsalaries--in many cases wholly inadequate--should be greatly\nincreased.[424]\n\nThe Cortes watched with a truly parental care over the great interests\nof the state--its commerce, its husbandry, and its manufactures. They\nraised a loud, and as it would seem not an ineffectual, note of\nremonstrance against the tyrannical practice of the crown in seizing for\nits own use the bullion which, as elsewhere stated, had been imported\nfrom the New World on their own account by the merchants of Seville.\n\nSome of the petitions of the Cortes show what would be thought at the\npresent day a strange ignorance of the true principles of legislation in\nrespect to commerce. Thus, regarding gold and silver, independently of\ntheir value as a medium of exchange, as constituting in a peculiar\nmanner the wealth of a country, they considered that the true policy was\nto keep the precious metals at home, and prayed that their exportation\nmight be forbidden. Yet this was a common error in the sixteenth century\nwith other nations besides the Spaniards. It may seem singular, however,\nthat the experience of three-fourths of a century had not satisfied the\nCastilian of the futility of such attempts to obstruct the natural\ncurrent of commercial circulation.\n\nIn the same spirit, they besought the king to prohibit the use of gold\nand silver in plating copper and other substances, as well as for\nwearing-apparel and articles of household luxury. It was a waste of the\nprecious metals, which were needed for other purposes. This petition of\nthe commons may be referred in part, no doubt, to their fondness for\nsumptuary laws, which in Castile formed a more ample code than could be\neasily found in any other country.[425] The love of costly and\nostentatious dress was a passion which they may have caught from their\nneighbours, the Spanish Arabs, who delighted in this way of displaying\ntheir opulence. It furnished accordingly, from an early period, a\nfruitful theme of declamation to the clergy, in their invectives against\nthe pomp and vanities of the world.\n\nUnfortunately Philip, who was so frequently deaf to the wiser\nsuggestions of the Cortes, gave his sanction to this petition; and in a\n_pragmatic_ devoted to the object, he carried out the ideas of the\nlegislature as heartily as the most austere reformer could have desired.\nAs a state paper, it has certainly a novel aspect, going at great length\ninto such minute specifications of wearing-apparel, both male and\nfemale, that it would seem to have been devised by a committee of\ntailors and milliners, rather than of grave legislators.[426] The\ntailors, indeed, the authors of these seductive abominations, did not\nescape the direct animadversion of the Cortes. In another petition they\nwere denounced as unprofitable persons, occupied with needlework, like\nwomen, instead of tilling the ground or serving his majesty in the wars,\nlike men.[427]\n\nIn the same spirit of impertinent legislation, the Cortes would have\nregulated the expenses of the table, which, they said, of late years had\nbeen excessive. They recommended that no one should be allowed to have\nmore than four dishes of meat and four of fruit served at the same meal.\nThey were further scandalized by the increasing use of coaches, a mode\nof conveyance which had been introduced into Spain only a few years\nbefore. They regarded them as tempting men to an effeminate indulgence,\nwhich most of them could ill afford. They considered the practice,\nmoreover, as detrimental to the good horsemanship for which their\nancestors had been so renowned. They prayed, therefore, that,\nconsidering \"the nation had done well for so many years without the use\nof coaches, it might henceforth be prohibited.\"[428] Philip so far\ncomplied with their petition, as to forbid any one but the owner of four\nhorses to keep a coach. Thus he imagined that, while encouraging the\nraising of horses, he should effectually discourage any but the more\nwealthy from affecting this costly luxury.\n\n[Sidenote: THE CORTES.]\n\nThere was another petition, somewhat remarkable, and worth citing, as it\nshows the attachment of the Castilians to a national institution which\nhas often incurred the censure of foreigners. A petition of the Cortes\nof 1573 prayed that some direct encouragement might be given to\nbull-fights, which of late had shown symptoms of decline. They advised\nthat the principal towns should be required to erect additional\ncircuses, and to provide lances for the combatants, and music for the\nentertainments, at the charge of the municipalities. They insisted on\nthis as important for mending the breed of horses, as well as for\nfurnishing a chivalrous exercise for the nobles and cavaliers. This may\nexcite some surprise in a spectator of our day, accustomed to see only\nthe most wretched hacks led to the slaughter, and men of humble\ncondition skirmishing in the arena. It was otherwise in those palmy days\nof chivalry, when the horses employed were of a generous breed, and the\ncombatants were nobles, who entered the lists with as proud a feeling as\nthat with which they would have gone to a tourney. Even so late as the\nsixteenth century it was the boast of Charles the Fifth, that, when a\nyoung man, he had fought like a _matador_, and killed his bull. Philip\ngave his assent to this petition, with a promptness which showed that he\nunderstood the character of his countrymen.\n\nIt would be an error to regard the more exceptionable and frivolous\npetitions of the Cortes, some of which have been above enumerated, as\naffording a true type of the predominant character of Castilian\nlegislation. The laws, or, to speak correctly, the petitions of that\nbody, are strongly impressed with a wise and patriotic sentiment,\nshowing a keen perception of the wants of the community, and a tender\nanxiety to relieve them. Thus we find the Cortes recommending that\nguardians should be appointed to find employment for such young and\ndestitute persons as, without friends to aid them, had no means of\ngetting a livelihood for themselves.[429] They propose to have visitors\nchosen, whose duty it should be to inspect the prisons every week, and\nsee that fitting arrangements were made for securing the health and\ncleanliness of the inmates.[430] They desire that care should be taken\nto have suitable accommodations provided at the inns for\ntravellers.[431] With their usual fondness for domestic inquisition,\nthey take notice of the behaviour of servants to their masters, and,\nwith a simplicity that may well excite a smile, they animadvert on the\nconduct of maidens who, \"in the absence of their mothers, spend their\nidle hours in reading romances full of lies and vanities, which they\nreceive as truths for the government of their own conduct in their\nintercourse with the world.\"[432] The books thus stigmatized were\ndoubtless the romances of chivalry, which at this period were at the\nheight of their popularity in Castile. Cervantes had not yet aimed at\nthis pestilent literature those shafts of ridicule which did more than\nany legislation could have done towards driving it from the land.\n\nThe commons watched over the business of education as zealously as over\nany of the material interests of the state. They inspected the condition\nof the higher seminaries, and would have provision made for the\nfoundation of new chairs in the universities. In accordance with their\nviews, though not in conformity to any positive suggestion, Philip\npublished a pragmatic in respect to these institutions. He complained of\nthe practice, rapidly increasing among his subjects, of going abroad to\nget their education, when the most ample provision was made for it at\nhome. The effect was eminently disastrous; for while the Castilian\nuniversities languished for want of patronage, the student who went\nabroad was pretty sure to return with ideas not the best suited to his\nown country. The king, therefore, prohibited Spaniards from going to any\nuniversity out of his dominions, and required all now abroad to return.\nThis edict he accompanied with the severe penalty of forfeiture of their\nsecular possessions for ecclesiastics, and of banishment and\nconfiscation of property for laymen.[433]\n\nThis kind of pragmatic, though made doubtless in accordance with the\npopular feeling, inferred a stretch of arbitrary power that cannot be\ncharged on those which emanated directly from the suggestion of the\nlegislature. In this respect, however, it fell far short of those\nordinances which proceeded exclusively from the royal will, without\nreference to the wishes of the commons. Such ordinances--and they were\nprobably more numerous than any other class of laws during this\nreign--are doubtless among the most arbitrary acts of which a monarch\ncan be guilty; for they imply nothing less than an assumption of the\nlaw-making power into his own hands. Indeed, they met with a strong\nremonstrance in the year 1579, when Philip was besought by the commons\nnot to make any laws but such as had first received the sanction of the\nCortes.[434] Yet Philip might vindicate himself by the example of his\npredecessors--even of those who, like Ferdinand and Isabella, had most\nat heart the interests of the nation.[435]\n\nIt must be further admitted, that the more regular mode of proceeding,\nwith the co-operation of the Cortes, had in it much to warrant the idea,\nthat the real right of legislation was vested in the king. A petition,\nusually couched in the most humble terms, prayed his majesty to give his\nassent to the law proposed. This he did in a few words; or, what was\nmuch more common, he refused to give it, declaring that, in the existing\ncase, \"it was not expedient that any change should be made.\" It was\nobserved that the number of cases in which Philip rejected the petitions\nof the commons was much greater than had been usual with former\nsovereigns.\n\n[Sidenote: THE GUARDS OF CASTILE.]\n\nA more frequent practice with Philip was one that better suited his\nhesitating nature and habit of procrastination. He replied in ambiguous\nterms, that \"he would take the matter into consideration,\" or \"that he\nwould lay it before his council, and take such measures as would be best\nfor his service.\" Thus the Cortes adjourned in ignorance of the fate of\ntheir petitions. Even when he announced his assent, as it was left to\nhim to prescribe the terms of the law, it might be more or less\nconformable to those of the petition. The Cortes having been dismissed,\nthere was no redress to be obtained if the law did not express their\nviews, nor could any remonstrance be presented by that body until their\nnext session, usually three years later. The practice established by\nCharles the Fifth, of postponing the presenting of petitions till the\nsupplies had been voted, and the immediate adjournment of the\nlegislature afterwards, secured an absolute authority to the princes of\nthe house of Austria, that made a fearful change in the ancient\nconstitution of Castile.\n\nYet the meetings of the Cortes, shorn as that body was of its ancient\nprivileges, were not without important benefits to the nation. None\ncould be better acquainted than the deputies with the actual wants and\nwishes of their constituents. It was a manifest advantage for the king\nto receive this information. It enabled him to take the course best\nsuited to the interests of the people, to which he would naturally be\ninclined when he did not regard them as conflicting with his own. Even\nwhen he did, the strenuous support of their own views by the commons\nmight compel him to modify his measures. However absolute the monarch,\nhe would naturally shrink from pursuing a policy so odious to the people\nthat, if persevered in, it might convert remonstrance into downright\nresistance.\n\nThe freedom of discussion among the deputies is attested by the\nindependent tone with which in their petitions they denounce the\nmanifold abuses in the state. It is honourable to Philip, that he should\nnot have attempted to stifle this freedom of debate; though perhaps this\nmay be more correctly referred to his policy, which made him willing to\nleave this safety-valve open for the passions of the people. He may have\nbeen content to flatter them with the image of power, conscious that he\nalone retained the substance of it. However this may have been, the good\neffect of the exercise of these rights, imperfect as they were, by the\nthird estate, must be highly estimated. The fact of being called\ntogether to consult on public affairs gave the people a consideration in\ntheir own eyes which raised them far above the abject condition of the\nsubjects of an Eastern despotism. It cherished in them that love of\nindependence which was their birthright, inherited from their ancestors,\nand thus maintained in their bosoms those lofty sentiments which were\nthe characteristics of the humbler classes of the Spaniards beyond those\nof any other nation in Christendom.\n\nOne feature was wanting to complete the picture of absolute monarchy.\nThis was a standing army,--a thing hitherto unknown in Spain. There was,\nindeed, an immense force kept on foot in the time of Charles the Fifth,\nand many of the troops were Spaniards. But they were stationed abroad,\nand were intended solely for foreign enterprises. It is to Philip's time\nthat we are to refer the first germs of a permanent military\nestablishment, designed to maintain order and obedience at home.\n\nThe levies raised for this purpose amounted to twenty companies of\nmen-at-arms, which, with the complement of four or five followers to\neach lance, made a force of some strength. It was further swelled by\nfive thousand _ginetes_, or light cavalry.[436] These corps were a heavy\ncharge on the crown. They were called \"the Guards of Castile.\" The\nmen-at-arms, in particular, were an object of great care, and were under\nadmirable discipline. Even Philip, who had little relish for military\naffairs, was in the habit of occasionally reviewing them in person. In\naddition to these troops there was a body of thirty thousand militia,\nwhom the king could call into the field when necessary. A corps of some\nsixteen hundred horsemen patrolled the southern coast of Andalusia, to\nguard the country from invasion by the African Moslems; and garrisons\nestablished in fortresses along the frontiers of Spain, both, north and\nsouth, completed a permanent force for the defence of the kingdom\nagainst domestic insurrection, as well as foreign invasion.\n\n\n\n\nCHAPTER II.\n\nDOMESTIC AFFAIRS OF SPAIN.\n\nThe Clergy--Their Subordination to the Crown--The Escorial--Queen Anne.\n\n\nA review of the polity of Castile would be incomplete without a notice\nof the ecclesiastical order, which may well be supposed to have stood\npre-eminent in such a country, and under such a monarch as Philip the\nSecond. Indeed, not only did that prince present himself before the\nworld as the great champion of the Faith, but he seemed ever solicitous\nin private life to display his zeal for religion and its ministers. Many\nanecdotes are told of him in connection with this. On one occasion,\nseeing a young girl going within the railing of the altar, he rebuked\nher, saying, \"Where the priest enters is no place either for me or\nyou.\"[437] A cavalier who had given a blow to a canon of Toledo he\nsentenced to death.[438]\n\nUnder his protection and princely patronage, the Church reached its most\npalmy state. Colleges and convents--in short, religious institutions of\nevery kind--were scattered broadcast over the land. The good fathers\nloved pleasant and picturesque sites for their dwellings; and the\ntraveller, as he journeyed through the country, was surprised by the\nnumber of stately edifices which crowned the hill-tops, or rested on\ntheir <DW72>s, surrounded by territories that spread out for many a\nleague over meadows and cultivated fields and pasture-land.\n\nThe secular clergy, at least the higher dignitaries, were so well\nendowed as sometimes to eclipse the grandees in the pomp of their\nestablishments. In the time of Ferdinand and Isabella, the archbishop of\nToledo held jurisdiction over fifteen principal towns and a great number\nof villages. His income amounted to full eighty thousand ducats a\nyear.[439] In Philip's time the income of the archbishop of Seville\namounted to the same sum, while that of the see of Toledo had risen to\ntwo hundred thousand ducats, nearly twice as much as that of the richest\ngrandee in the kingdom.[440] In power and opulence, the primate of Spain\nranked next in Christendom to the pope.\n\nThe great source of all this wealth of the ecclesiastical order in\nCastile, as in most other countries, was the benefactions and bequests\nof the pious--of those, more especially, whose piety had been deferred\ntill the close of life, when, anxious to make amends for past\ndelinquencies, they bestowed the more freely that it was at the expense\nof their heirs. As what was thus bequeathed was locked up by entail, the\nconstantly accumulating property of the Church had amounted, in Philip's\ntime, if we may take the assertion of the Cortes, to more than one-half\nof the landed property in the kingdom.[441] Thus the burden of providing\nfor the expenses of the state fell with increased heaviness on the\ncommons. Alienations in mortmain formed the subject of one of their\nearliest remonstrances after Philip's accession, but without effect; and\nthough the same petition was urged in very plain language at almost\nevery succeeding session, the king still answered that it was not\nexpedient to make any change in the existing laws. Besides his goodwill\nto the ecclesiastical order, Philip was occupied with the costly\nconstruction of the Escorial; and he had probably no mind to see the\nstreams of public bounty, which had hitherto flowed so freely into the\nreservoirs of the Church, thus suddenly obstructed, when they were so\nmuch needed for his own infant institution.\n\n[Sidenote: THE CLERGY.]\n\nWhile Philip was thus willing to exalt the religious order, already far\ntoo powerful, he was careful that it should never gain such a height as\nwould enable it to overtop the royal authority. Both in the Church and\nin the council--for they were freely introduced into the\ncouncils--theologians were ever found the most devoted servants of the\ncrown. Indeed, it was on the crown that they were obliged to rest all\ntheir hopes of preferment.\n\nPhilip perfectly understood that the control of the clergy must be\nlodged with that power which had the right of nomination to benefices.\nThe Roman see, in its usual spirit of encroachment, had long claimed the\nexercise of this right in Castile, as it had done in other European\nstates. The great battle with the Church was fought in the time of\nIsabella the Catholic. Fortunately the sceptre was held by a sovereign\nwhose loyalty to the Faith was beyond suspicion. From this hard struggle\nshe came off victorious; and the government of Castile henceforth\nretained possession of the important prerogative of appointing to vacant\nbenefices.\n\nPhilip, with all his deference to Rome, was not a man to relinquish any\nof the prerogatives of the crown. A difficulty arose under Pius the\nFifth, who contended that he still had the right, possessed by former\npopes, of nominating to ecclesiastical offices in Milan, Naples, and\nSicily, the Italian possessions held by Spain. He complained bitterly of\nthe conduct of the councils in those states, which refused to allow the\npublication of his bulls without the royal _exequatur_. Philip, in mild\nterms, expressed his desire to maintain the most amicable relations with\nthe see of Rome, provided he was not required to compromise the\ninterests of his crown. At the same time he intimated his surprise that\nhis holiness should take exceptions at his exercise of the rights of his\npredecessors, to many of whom the Church was indebted for the most\nsignal services. The pope was well aware of the importance of\nmaintaining a good understanding with so devoted a son of the Church;\nand Philip was allowed to remain henceforth in undisturbed possession of\nthis inestimable prerogative.[442]\n\nThe powers thus vested in the king he exercised with great discretion.\nWith his usual facilities for information he made himself acquainted\nwith the characters of the clergy in the different parts of his\ndominions. He was so accurate in his knowledge, that he was frequently\nable to detect an error or omission in the information he received. To\none who had been giving him an account of a certain ecclesiastic, he\nremarked--\"You have told me nothing of his amours.\" Thus perfectly\napprised of the characters of the candidates, he was prepared, whenever\na vacancy occurred, to fill the place with a suitable incumbent.[443]\n\nIt was his habit, before preferring an individual to a high office, to\nhave proof of his powers by trying them first in some subordinate\nstation. In his selection he laid much stress on rank, for the influence\nit carried with it. Yet frequently, when well satisfied of the merits of\nthe parties, he promoted those whose humble condition had made them\nlittle prepared for such, an elevation.[444] There was no more effectual\nway to secure his favour than to show a steady resistance to the\nusurpations of Rome. It was owing, in part at least, to the refusal of\nQuiroga, the bishop of Cuenca, to publish a papal bull without the royal\nassent, that he was raised to the highest dignity in the kingdom, as\narchbishop of Toledo. Philip chose to have a suitable acknowledgment\nfrom the person on whom he conferred a favour; and once, when an\necclesiastic, whom he had made a bishop, went to take possession of his\nsee without first expressing his gratitude, the king sent for him back,\nto remind him of his duty.[445] Such an acknowledgment was in the nature\nof a homage rendered to his master on his preferment.\n\nThus gratitude for the past and hopes for the future were the strong\nties which bound every prelate to his sovereign. In a difference with\nthe Roman see, the Castilian churchman was sure to be found on the side\nof the sovereign, rather than, on that of the pontiff. In his own\ntroubles, in like manner, it was to the king, and not to the pope, that\nhe was to turn for relief. The king, on the other hand, when pressed by\nthose embarrassments with which he was too often surrounded, looked for\naid to the clergy, who for the most part rendered it cheerfully and in\nliberal measure. Nowhere were the clergy so heavily burdened as in\nSpain.[446] It was computed that at least one-third of their revenues\nwas given to the king. Thus completely were the different orders, both\nspiritual and temporal, throughout the monarchy, under the control of\nthe sovereign.\n\nA few pages back, while touching on alienations in mortmain, I had\noccasion to allude to the Escorial, that \"eighth wonder of the world,\"\nas it is proudly styled by the Spaniards. There can be no place more\nproper to give an account of this extraordinary edifice, than the part\nof the narrative in which I have been desirous to throw as much light as\npossible on the character and occupations of Philip. The Escorial\nengrossed the leisure of more than thirty years of his life; it reflects\nin a peculiar manner his tastes, and the austere character of his mind;\nand whatever criticism may be passed on it as a work of art, it cannot\nbe denied that, if every other vestige of his reign were to be swept\naway, that wonderful structure would of itself suffice to show the\ngrandeur of his plans and the extent of his resources.\n\nThe common tradition that Philip built the Escorial in pursuance of a\nvow which he made at the time of the great battle of St. Quentin, the\n10th of August, 1557, has been rejected by modern critics, on the ground\nthat contemporary writers, and amongst them the historians of the\nconvent, make no mention of the fact. But a recently-discovered document\nleaves little doubt that such a vow was actually made.[447] However this\nmay have been, it is certain that the king designed to commemorate the\nevent by this structure, as is intimated by its dedication to St.\nLawrence, the martyr on whose day the victory was gained. The name given\nto the place was _El Sitio de San Lorenzo el Real_. But the monastery\nwas better known from the hamlet near which it stood,--_El Escurial_, or\n_El Escorial_,--which latter soon became the orthography generally\nadopted by the Castilians.[448]\n\n[Sidenote: THE ESCORIAL.]\n\nThe motives which, after all, operated probably most powerfully on\nPhilip, had no connection with the battle of St. Quentin. His father,\nthe emperor, had directed by his will that his bones should remain at\nYuste, until a more suitable place should be provided for them by his\nson. The building now to be erected was designed expressly as a\nmausoleum for Philip's parents, as well as for their descendants of the\nroyal line of Austria. But the erection of a religious house on a\nmagnificent scale, that would proclaim to the world his devotion to the\nFaith, was the predominant idea in the mind of Philip. It was, moreover,\na part of his scheme to combine in the plan a palace for himself; for,\nwith a taste which he may be said to have inherited from his father, he\nloved to live in the sacred shadows of the cloister. These ideas,\nsomewhat incongruous as they may seem, were fully carried out by the\nerection of an edifice dedicated at once to the threefold purpose of a\npalace, a monastery, and a tomb.[449]\n\nSoon after the king's return to Spain, he set about carrying his plan\ninto execution. The site which, after careful examination, he selected\nfor the building, was among the mountains of the Guadarrama, on the\nborders of New Castile,[450] about eight leagues north-west of Madrid.\nThe healthiness of the place and its convenient distance from the\ncapital combined with the stern and solitary character of the region, so\ncongenial to his taste, to give it the preference over other spots,\nwhich might have found more favour with persons of a different nature.\nEncompassed by rude and rocky hills, which sometimes soar to the\ngigantic elevation of mountains, it seemed to be shut out completely\nfrom the world. The vegetation was of a thin and stunted growth, seldom\nspreading out into the luxuriant foliage of the lower regions; and the\nwinds swept down from the neighbouring sierra with the violence of a\nhurricane. Yet the air was salubrious, and the soil was nourished by\nsprings of the purest water. To add to its recommendations, a quarry,\nclose at hand, of excellent stone, somewhat resembling granite in\nappearance, readily supplied the materials for building,--a\ncircumstance, considering the vastness of the work, of no little\nimportance.\n\nThe architect who furnished the plans, and on whom the king relied for\nsuperintending their execution, was Juan-Bautista de Toledo. He was born\nin Spain, and, early discovering uncommon talents for his profession,\nwas sent to Italy. Here he studied the principles of his art, under the\ngreat masters who were then filling their native land with those\nmonuments of genius that furnished the best study to the artist. Toledo\nimbibed their spirit, and under their tuition acquired that simple,\nindeed severe taste, which formed a contrast to the prevalent tone of\nSpanish architecture, but which, happily, found favour with his royal\npatron.\n\nBefore a stone of the new edifice was laid, Philip had taken care to\nprovide himself with the tenants who were to occupy it. At a general\nchapter of the Jeronymite fraternity, a prior was chosen for the convent\nof the Escorial, which was to consist of fifty members, soon increased\nto double that number. Philip had been induced to give the preference to\nthe Jeronymite order, partly from their general reputation for ascetic\npiety, and in part from the regard shown for them by his father, who had\nchosen a convent of that order as the place of his last retreat. The\nmonks were speedily transferred to the village of the Escorial, where\nthey continued to dwell until accommodations were prepared for them in\nthe magnificent pile which they were thenceforth to occupy.\n\nTheir temporary habitation was of the meanest kind, like most of the\nbuildings in the hamlet. It was without window or chimney, and the rain\nfound its way through the dilapidated roof of the apartment which they\nused as a chapel; so that they were obliged to protect themselves by a\ncoverlet stretched above their heads. A rude altar was raised at one end\nof the chapel, over which was scrawled on the wall, with charcoal, the\nfigure of a crucifix.[451]\n\nThe king, on his visits to the place, was lodged in the house of the\ncurate, in not much better repair than the other dwellings in the\nhamlet. While there, he was punctual in his attendance at mass, when a\nrude seat was prepared for him near the choir, consisting of a\nthree-legged stool, defended from vulgar eyes by a screen of such old\nand tattered cloth that the inquisitive spectator might, without\ndifficulty, see him through the holes in it.[452] He was so near the\nchoir, that the monk who stood next to him could hardly avoid being\nbrought into contact with the royal person. The Jeronymite who tells the\nstory assures us that Brother Antonio used to weep as he declared that\nmore than once, when he cast a furtive glance at the monarch, he saw his\neyes filled with tears. \"Such,\" says the good father, \"were the devout\nand joyful feelings with which the king, as he gazed on the poverty\naround him, meditated his lofty plans for converting this poverty into a\nscene of grandeur more worthy of the worship to be performed\nthere.\"[453]\n\nThe brethren were much edified by the humility shown by Philip when\nattending the services in this wretched cabin. They often told the story\nof his one day coming late to matins, when, unwilling to interrupt the\nservices, he quietly took his seat by the entrance, on a rude bench, at\nthe upper end of which a peasant was sitting. He remained some time\nbefore his presence was observed, when the monks conducted him to his\ntribune.[454]\n\nOn the twenty-third of April, 1563, the first stone of the monastery was\nlaid. On the twentieth of August following, the corner-stone of the\nchurch was also laid, with still greater pomp and solemnity. The royal\nconfessor, the bishop of Cuenca, arrayed in his pontificals, presided\nover the ceremonies. The king was present, and laid the stone with his\nown hands. The principal nobles of the court were in attendance, and\nthere was a great concourse of spectators, both ecclesiastics and\nlaymen; the solemn services were concluded by the brotherhood, who\njoined in an anthem of thanksgiving and praise to the Almighty, to whom\nso glorious a monument was to be reared in this mountain\nwilderness.[455]\n\n[Sidenote: THE ESCORIAL.]\n\nThe rude sierra now swarmed with life. The ground was covered with tents\nand huts. The busy hum of labour mingled with the songs of the\nlabourers, which, from their various dialects, betrayed the different,\nand oftentimes distant, provinces from which they had come. In this\nmotley host the greatest order and decorum prevailed; nor were the\npeaceful occupations of the day interrupted by any indecent brawls.\n\nAs the work advanced, Philip's visits to the Escorial were longer and\nmore frequent. He had always shown his love for the retirement of the\ncloister, by passing some days of every year in it. Indeed, he was in\nthe habit of keeping Holy Week not far from the scene of his present\nlabours, at the convent of Guisando. In his present monastic retreat he\nhad the additional interest afforded by the contemplation of the great\nwork, which seemed to engage as much of his thoughts as any of the\nconcerns of government.\n\nPhilip had given a degree of attention to the study of the fine arts\nseldom found in persons of his condition. He was a connoisseur in\npainting, and, above all, in architecture, making a careful study of its\nprinciples, and occasionally furnishing designs with his own hand.[456]\nNo prince of his time left behind him so many proofs of his taste and\nmagnificence in building. The royal mint at Segovia, the hunting-seat of\nthe Pardo, the pleasant residence of Aranjuez, the alcazar of Madrid,\nthe \"Armeria Real,\" and other noble works which adorned his infant\ncapital, were either built or greatly embellished by him. The land was\ncovered with structures both civil and religious, which rose under the\nroyal patronage. Churches and convents--the latter in lamentable\nprofusion--constantly met the eye of the traveller. The general style of\ntheir execution was simple in the extreme. Some, like the great\ncathedral of Valladolid, of more pretension, but still showing the same\naustere character in their designs, furnished excellent models of\narchitecture to counteract the meretricious tendencies of the age.\nStructures of a different kind from these were planted by Philip along\nthe frontiers in the north and on the southern coasts of the kingdom;\nand the voyager in the Mediterranean beheld fortress after fortress\ncrowning the heights above the shore, for its defence against the\nBarbary corsair. Nor was the king's passion for building confined to\nSpain. Wherever his armies penetrated in the semi-civilized regions of\nthe New World, the march of the conqueror was sure to be traced by the\necclesiastical and military structures which rose in his rear.\n\nFortunately, similarity of taste led to the most perfect harmony between\nthe monarch and his architect, in their conferences on the great work\nwhich was to crown the architectural glories of Philip's reign. The king\ninspected the details, and watched over every step in the progress of\nthe building, with as much care as Toledo himself. In order to judge of\nthe effect from a distance, he was in the habit of climbing the\nmountains at a spot about half a league from the monastery, where a kind\nof natural chair was formed by the crags. Here, with his spyglass in his\nhand, he would sit for hours, and gaze on the complicated structure\ngrowing up below. The place is still known as the \"king's seat.\"[457]\n\nIt was certainly no slight proof of the deep interest which Philip took\nin the work, that he was content to exchange his palace at Madrid for a\nplace that afforded him no better accommodations than the\npoverty-stricken village of the Escorial. In 1571 he made an important\nchange in these accommodations, by erecting a chapel which might afford\nthe monks a more decent house of worship than their old weather-beaten\nhovel; and with this he combined a comfortable apartment for himself. In\nthese new quarters he passed still more of his time in cloistered\nseclusion than he had done before. Far from confining his attention to\na supervision of the Escorial, he brought his secretaries and his papers\nalong with him, read here his despatches from abroad, and kept up a busy\ncorrespondence with all parts of his dominions. He did four times the\namount of work here, says a Jeronymite, that he did in the same number\nof days in the capital.[458] He used to boast that, thus hidden from the\nworld, with a little bit of paper, he ruled over both hemispheres. That\nhe did not always wisely rule, is proved by more than one of his\ndespatches relating to the affairs of Flanders, which issued from this\nconsecrated place. Here he received accounts of the proceedings of his\nheretic subjects in the Netherlands, and of the Morisco insurgents in\nGranada. And as he pondered on their demolition of church and convent,\nand their desecration of the most holy symbols of the Catholic faith, he\ndoubtless felt a proud satisfaction in proving his own piety to the\nworld by the erection of the most sumptuous edifice ever dedicated to\nthe Cross.\n\nIn 1577, the Escorial was so far advanced towards its completion as to\nafford accommodations not merely for Philip and his personal attendants,\nbut for many of the court, who were in the habit of spending some time\nthere with the king during the summer. On one of these occasions, an\naccident occurred which had nearly been attended with most disastrous\nconsequences to the building.\n\nA violent thunderstorm was raging in the mountains, and the lightning\nstruck one of the great towers of the monastery. In a short time the\nupper portion of the building was in a blaze. So much of it,\nfortunately, was of solid materials, that the fire made slow progress.\nBut the difficulty of bringing water to bear on it was extreme. It was\neleven o'clock at night when the fire broke out, and in the orderly\nhousehold of Philip all had retired to rest. They were soon roused by\nthe noise. The king took his station on the opposite tower, and watched\nwith deep anxiety the progress of the flames. The duke of Alva was one\namong the guests. Though sorely afflicted with the gout at the time, he\nwrapped his dressing-gown about him, and climbed to a spot which\nafforded a still nearer view of the conflagration. Here the \"good duke\"\nat once assumed the command, and gave his orders with as much promptness\nand decision as on the field of battle.[459]\n\nAll the workmen, as well as the neighbouring peasantry, were assembled\nthere. The men showed the same spirit of subordination which they had\nshown throughout the erection of the building. The duke's orders were\nimplicitly obeyed; and more than one instance is recorded of daring\nself-devotion among the workmen, who toiled as if conscious they were\nunder the eye of their sovereign. The tower trembled under the fury of\nthe flames; and the upper portion of it threatened every moment to fall\nin ruins. Great fears were entertained that it would crush the hospital,\nsituated in that part of the monastery. Fortunately, it fell in an\nopposite direction, carrying with it a splendid chime of bells that was\nlodged in it, but doing no injury to the spectators. The loss which bore\nmost heavily on the royal heart was that of sundry inestimable relics\nwhich perished in the flames. But Philip's sorrow was mitigated when he\nlearned that a bit of the true cross, and the right arm of St. Lawrence,\nthe martyred patron of the Escorial, were rescued from the flames. At\nlength, by incredible efforts, the fire, which had lasted till six in\nthe morning, was happily extinguished, and Philip withdrew to his\nchamber, where his first act, we are told, was to return thanks to the\nAlmighty for the preservation of the building consecrated to his\nservice.[460]\n\n[Sidenote: THE ESCORIAL.]\n\nThe king was desirous that as many of the materials as possible for the\nstructure should be collected from his own dominions. These were so\nvast, and so various in their productions, that they furnished nearly\nevery article required for the construction of the edifice, as well as\nfor its interior decoration. The grey stone, of which its walls were\nformed, was drawn from a neighbouring quarry. It was called\n_berroquena_,--a stone bearing a resemblance to granite, though not so\nhard. The blocks hewn from the quarries, and dressed there, were of such\nmagnitude as sometimes to require forty or fifty yoke of oxen to drag\nthem. The jasper came from the neighbourhood of Burgo de Osma. The more\ndelicate marbles, of a great variety of colours, were furnished by the\nmountain-ranges in the south of the Peninsula. The costly and elegant\nfabrics were many of them supplied by native artisans. Such were the\ndamasks and velvets of Granada. Other cities, as Madrid, Toledo, and\nSaragossa, showed the proficiency of native art in curious manufactures\nof bronze and iron, and occasionally of the more precious metals.\n\nYet Philip was largely indebted to his foreign possessions, especially\nthose in Italy and the Low Countries, for the embellishment of the\ninterior of the edifice, which, in its sumptuous style of decoration,\npresented a contrast to the stern simplicity of its exterior. Milan, so\nrenowned at that period for its fine workmanship in steel, gold, and\nprecious stones, contributed many exquisite specimens of art. The walls\nwere clothed with gorgeous tapestries from the Flemish looms. Spanish\nconvents vied with each other in furnishing embroideries for the altars.\nEven the rude colonies in the New World had their part in the great\nwork, and the American forests supplied their cedar and ebony and\nrichly-tinted woods, which displayed all their magical brilliancy of\ncolour under the hands of the Castilian workman.[461]\n\nThough desirous, as far as possible, to employ the products of his own\ndominions, and to encourage native art, in one particular he resorted\nalmost exclusively to foreigners. The oil-paintings and frescoes which\nprofusely decorated the walls and ceilings of the Escorial were executed\nby artists drawn chiefly from Italy, whose schools of design were still\nin their glory. But of all living painters, Titian was the one whom\nPhilip, like his father, most delighted to honour. To the king's\ngenerous patronage the world is indebted for some of that great master's\nnoblest productions, which found a fitting place on the walls of the\nEscorial.\n\nThe prices which Philip paid enabled him to command the services of the\nmost eminent artists. Many anecdotes are told of his munificence. He\nwas, however, a severe critic. He did not prematurely disclose his\nopinion. But when the hour came, the painter had sometimes the\nmortification to find the work he had executed, it may be with greater\nconfidence than skill, peremptorily rejected, or at best condemned to\nsome obscure corner of the building. This was the fate of an Italian\nartist, of much more pretension than power, who, after repeated failures\naccording to the judgment of the king--which later critics have not\nreversed--was dismissed to his own country. But even here Philip dealt\nin a magnanimous way with the unlucky painter. \"It is not Zuccaro's\nfault,\" he said, \"but that of the persons who brought him here;\" and\nwhen he sent him back to Italy, he gave him a considerable sum of money\nin addition to his large salary.[462]\n\nBefore this magnificent pile, in a manner the creation of his own taste,\nPhilip's nature appeared to expand, and to discover some approach to\nthose generous sympathies for humanity which elsewhere seemed to have\nbeen denied him. He would linger for hours while he watched the labours\nof the artist, making occasional criticisms, and laying his hand\nfamiliarly on his shoulder.[463] He seemed to put off the coldness and\nreserve which formed so essential a part of his character. On one\noccasion, it is said, a stranger, having come into the Escorial when the\nking was there, mistook him for one of the officials, and asked him some\nquestions about the pictures. Philip, without undeceiving the man,\nhumoured his mistake, and good-naturedly undertook the part of\n_cicerone_, by answering his inquiries, and showing him some of the\nobjects most worth seeing.[464] Similar anecdotes have been told of\nothers. What is strange is, that Philip should have acted the part of\nthe good-natured man.\n\nIn 1584, the masonry of the Escorial was completed. Twenty-one years had\nelapsed since the first stone of the monastery was laid. This certainly\nmust be regarded as a short period for the erection of so stupendous a\npile. St. Peter's church, with which one naturally compares it as the\nbuilding nearest in size and magnificence, occupied more than a century\nin its erection, which spread over the reigns of at least eighteen\npopes. But the Escorial, with the exception of the subterraneous chapel\nconstructed by Philip the Fourth for the burial-place of the Spanish\nprinces, was executed in the reign of one monarch. That monarch held in\nhis hands the revenues of both the Old World and the New; and as he\ngave, in some sort, a personal supervision to the work, we may be sure\nthat no one was allowed to sleep on his post.\n\nYet the architect who designed the building was not permitted to\ncomplete it. Long before it was finished, the hand of Toledo had\nmouldered in the dust. By his death it seemed that Philip had met with\nan irreparable loss. He felt it to be so himself; and with great\ndistrust consigned the important task to Juan de Herrera, a young\nAsturian. But though young, Herrera had been formed on the best models;\nfor he was the favourite pupil of Toledo, and it soon appeared that he\nhad not only imbibed the severe and elevated tastes of his master, but\nthat his own genius fully enabled him to comprehend all Toledo's great\nconceptions, and to carry them out as perfectly as that artist could\nhave done himself. Philip saw with satisfaction that he had made no\nmistake in his selection. He soon conferred as freely with the new\narchitect as he had done with his predecessor. He even showed him\ngreater favour, settling on him a salary of a thousand ducats a year,\nand giving him an office in the royal household, and the cross of St.\nIago. Herrera had the happiness to complete the Escorial. Indeed, he\nlived some six years after its completion. He left several works, both\ncivil and ecclesiastical, which perpetuate his fame. But the Escorial is\nthe monument by which his name, and that of his master, Toledo, have\ncome down to posterity as those of the two greatest architects of whom\nSpain can boast.\n\nThis is not the place for criticism on the architectural merits of the\nEscorial. Such criticism more properly belongs to a treatise on art. It\nhas been my object simply to lay before the reader such an account of\nthe execution of this great work as would enable him to form some idea\nof the object to which Philip devoted so large a portion of his time,\nand which so eminently reflected his peculiar cast of mind.\n\n[Sidenote: THE ESCORIAL.]\n\nCritics have greatly differed from each other in their judgments of the\nEscorial. Few foreigners have been found to acquiesce in the undiluted\npanegyric of those Castilians who pronounce it the eighth wonder of the\nworld.[465] Yet it cannot be denied that few foreigners are qualified to\ndecide on the merits of a work, to judge of which correctly requires a\nperfect understanding of the character of the country in which it was\nbuilt, and of the monarch who built it. The traveller who gazes on its\nlong lines of cold grey stone, scarcely broken by an ornament, feels a\ndreary sensation creeping over him, while he contrasts it with the\nlighter and more graceful edifices to which his eye has been accustomed.\nBut he may read in this the true expression of the founder's character.\nPhilip did not aim at the beautiful, much less at the festive and\ncheerful. The feelings which he desired to raise in the spectator were\nof that solemn, indeed sombre complexion, which corresponded best with\nhis own religious faith.\n\nWhatever defects may be charged on the Escorial, it is impossible to\nview it from a distance, and see the mighty pile as it emerges from the\ngloomy depths of the mountains, without feeling how perfectly it\nconforms in its aspect to the wild and melancholy scenery of the sierra.\nNor can one enter the consecrated precincts without confessing the\ngenius of the place, and experiencing sensations of a mysterious awe as\nhe wanders through the desolate halls, which fancy peoples with the\nsolemn images of the past.\n\nThe architect of the building was embarrassed by more than one\ndifficulty of a very peculiar kind. It was not simply a monastery that\nhe was to build. The same edifice, as we have seen, was to comprehend at\nonce a convent, a palace, and a tomb. It was no easy problem to\nreconcile objects so discordant, and to infuse into them a common\nprinciple of unity. It is no reproach to the builder that he did not\nperfectly succeed in this, and that the palace should impair the\npredominant tone of feeling raised by the other parts of the structure,\nlooking in fact like an excrescence, rather than an integral portion of\nthe edifice.\n\nAnother difficulty, of a more whimsical nature, imposed on the\narchitect, was the necessity of accommodating the plan of the building\nto the form of a gridiron--as typical of the kind of martyrdom suffered\nby the patron saint of the Escorial. Thus the long lines of cloisters,\nwith their intervening courts, served for the bars of the instrument.\nThe four lofty spires at the corners of the monastery, represented its\nlegs inverted; and the palace, extending its slender length on the east,\nfurnished the awkward handle.\n\nIt is impossible for language to convey any adequate idea of a work of\nart. Yet architecture has this advantage over the sister arts of design,\nthat the mere statement of the dimensions helps us much in forming a\nconception of the work. A few of these dimensions will serve to give an\nidea of the magnitude of the edifice. They are reported to us by Los\nSantos, a Jeronymite monk, who has left one of the best accounts of the\nEscorial.\n\nThe main building, or monastery, he estimates at seven hundred and forty\nCastilian feet in length by five hundred and eighty in breadth. Its\ngreatest height, measured to the central cross above the dome of the\ngreat church, is three hundred and fifteen feet. The whole circumference\nof the Escorial, including the palace, he reckons at two thousand nine\nhundred and eighty feet, or near three-fifths of a mile. The patient\ninquirer tells us there were no less than twelve thousand doors and\nwindows in the building; that the weight of the keys alone amounted to\nfifty _arrobas_, or twelve hundred and fifty pounds, and, finally, that\nthere were sixty-eight fountains playing in the halls and courts of this\nenormous pile.[466]\n\nThe cost of its construction and interior decoration, we are informed by\nFather Siguenca, amounted to very near six millions of ducats.[467]\nSiguenca was prior of the monastery, and had access, of course, to the\nbest sources of information. That he did not exaggerate, may be inferred\nfrom the fact that he was desirous to relieve the building from the\nimputation of any excessive expenditure incurred in its erection--a\ncommon theme of complaint, it seems, and one that was urged with strong\nmarks of discontent by contemporary writers. Probably no single edifice\never contained such an amount and variety of inestimable treasures as\nthe Escorial,--so many paintings and sculptures by the greatest\nmasters,--so many articles of exquisite workmanship, composed of the\nmost precious materials. It would be a mistake to suppose that, when the\nbuilding was finished, the labours of Philip were at an end. One might\nalmost say they were but begun. The casket was completed; but the\nremainder of his days was to be passed in filling it with the rarest and\nrichest gems. This was a labour never to be completed. It was to be\nbequeathed to his successors, who with more or less taste, but with the\nrevenues of the Indies at their disposal, continued to lavish them on\nthe embellishment of the Escorial.[468]\n\nPhilip the Second set the example. He omitted nothing which could give a\nvalue, real or imaginary, to his museum. He gathered at an immense cost\nseveral hundred cases of the bones of saints and martyrs, depositing\nthem in rich silver shrines, of elaborate workmanship. He collected four\nthousand volumes, in various languages, especially the Oriental, as the\nbasis of the fine library of the Escorial.\n\nThe care of successive princes, who continued to spend there a part of\nevery year, preserved the palace-monastery and its contents from the\nrude touch of Time. But what the hand of Time had spared, the hand of\nviolence destroyed. The French, who in the early part of the present\ncentury swept like a horde of Vandals over the Peninsula, did not\noverlook the Escorial. For in it they saw the monument designed to\ncommemorate their own humiliating defeat. A body of dragoons under La\nHoussaye burst into the monastery in the winter of 1808; and the ravages\nof a few days demolished what it had cost years and the highest efforts\nof art to construct. The apprehension of similar violence from the\nCarlists, in 1837, led to the removal of the finest paintings to Madrid.\nThe Escorial ceased to be a royal residence: tenantless and unprotected,\nit was left to the fury of the blasts which swept down the hills of the\nGuadarrama.\n\nThe traveller who now visits the place will find its condition very\ndifferent from what it was in the beginning of the century. The bare and\nmildewed walls no longer glow with the magical tints of Raphael and\nTitian, and the sober pomp of the Castilian school. The exquisite\nspecimens of art with which the walls were filled have been wantonly\ndemolished, or more frequently pilfered for the sake of the rich\nmaterials. The monks, so long the guardians of the place, have shared\nthe fate of their brethren elsewhere, since the suppression of religious\nhouses, and their venerable forms have disappeared.\n\n[Sidenote: QUEEN ANNE.]\n\nSilence and solitude reign throughout the courts, undisturbed by any\nsound save that of the ceaseless winds, which seem to be ever chanting\ntheir melancholy dirge over the faded glories of the Escorial. There is\nlittle now to remind one of the palace or of the monastery. Of the three\ngreat objects to which the edifice was devoted, one alone\nsurvives,--that of a mausoleum for the royal line of Castile. The spirit\nof the dead broods over the place,--of the sceptred dead, who lie in the\nsame dark chamber where they have lain for centuries, unconscious of the\nchanges that have been going on all around them.\n\nDuring the latter half of Philip's reign, he was in the habit of\nrepairing with his court to the Escorial, and passing here a part of the\nsummer. Hither he brought his young queen, Anne of Austria,--when the\ngloomy pile assumed an unwonted appearance of animation. In a previous\nchapter, the reader has seen some notice of his preparations for his\nmarriage with that princess, in less than two years after he had\nconsigned the lovely Isabella to the tomb. Anne had been already\nplighted to the unfortunate Don Carlos. Philip's marriage with her\nafforded him the melancholy triumph of a second time supplanting his\nson. She was his niece; for the empress Mary, her mother, was the\ndaughter of Charles the Fifth. There was, moreover, a great disparity in\ntheir years; for the Austrian princess, having been born in Castile\nduring the regency of her parents, in 1549, was at this time but\ntwenty-one years of age, less than half the age of Philip. It does not\nappear that her father, the emperor Maximilian, made any objection to\nthe match. If he felt any, he was too politic to prevent a marriage\nwhich would place his daughter on the throne of the most potent monarchy\nin Europe.\n\nIt was arranged that the princess should proceed to Spain by the way of\nthe Netherlands. In September, 1570, Anne bade a last adieu to her\nfather's court, and with a stately retinue set out on her long journey.\nOn entering Flanders, she was received with great pomp by the duke of\nAlva, at the head of the Flemish nobles. Soon after her arrival, Queen\nElizabeth despatched a squadron of eight vessels, with offers to\ntransport her to Spain, and an invitation for her to visit England on\nher way. These offers were courteously declined; and the German\nprincess, escorted by Count Bossu, captain-general of the Flemish navy,\nwith a gallant squadron, was fortunate in reaching the place of her\ndestination after a voyage of less than a week. On the third of October\nshe landed at Santander, on the northern coast of Spain, where she found\nthe archbishop of Seville and the duke of Bejar, with a brilliant train\nof followers, waiting to receive her.\n\nUnder this escort, Anne was conducted by the way of Burgos and\nValladolid to the ancient city of Segovia. In the great towns through\nwhich she passed she was entertained in a style suited to her rank; and\neverywhere along her route she was greeted with the hearty acclamations\nof the people: for the match was popular with the nation; and the Cortes\nhad urged the king to expedite it as much as possible.[469] The\nSpaniards longed for a male heir to the crown; and since the death of\nCarlos, Philip had only daughters remaining to him.\n\nIn Segovia, where the marriage ceremony was to be performed, magnificent\npreparations had been made for the reception of the princess. As she\napproached that city, she was met by a large body of the local militia,\ndressed in gay uniforms, and by the municipality of the place, arrayed\nin their robes of office and mounted on horseback. With this brave\nescort she entered the gates. The streets were ornamented with beautiful\nfountains, and spanned by triumphal arches, under which the princess\nproceeded, amidst the shouts of the populace, to the great\ncathedral.[470]\n\nAnne, then in the bloom of youth, is described as having a rich and\ndelicate complexion. Her figure was good, her deportment gracious, and\nshe rode her richly-caparisoned palfrey with natural ease and dignity.\nHer not very impartial chronicler tells us that the spectators\nparticularly admired the novelty of her Bohemian costume, her riding-hat\ngaily ornamented with feathers, and her short mantle of crimson velvet\nrichly fringed with gold.[471]\n\nAfter _Te Deum_ had been chanted, the splendid procession took its way\nto the far-famed _alcazar_, that palace-fortress, originally built by\nthe Moors, which now served both as a royal residence and as a place of\nconfinement for prisoners of state. Here it was that the unfortunate\nMontigny passed many a weary month of captivity; and less than three\nmonths had elapsed since he had been removed from the place which was so\nsoon to become the scene of royal festivity, and consigned to the fatal\nfortress of Simancas, to perish by the hand of the midnight executioner.\nAnne, it may be remembered, was said, on her journey through the Low\nCountries, to have promised Montigny's family to intercede with her lord\nin his behalf. But the king, perhaps willing to be spared the\nawkwardness of refusing the first boon asked by his young bride,\ndisposed of his victim soon after her landing, while she was yet in the\nnorth.\n\nAnne entered the _alcazar_ amidst salvoes of artillery. She found there\nthe good Princess Joanna, Philip's sister, who received her with the\nsame womanly kindness which she had shown twelve years before to\nElizabeth of France, when, on a similar occasion, she made her first\nentrance into Castile. The marriage was appointed to take place on the\nfollowing day, the fourteenth of November. Philip, it is said, obtained\nhis first view of his betrothed when, mingling in disguise among the\ncavalcade of courtiers, he accompanied her entrance into the\ncapital.[472] When he had led his late queen, Isabella, to the altar,\nsome white hairs on his temples attracted her attention.[473] During the\nten years which had since elapsed, the cares of office had wrought the\nsame effect on him as on his father, and turned his head prematurely\ngrey. The marriage was solemnized with great pomp in the cathedral of\nSegovia. The service was performed by the archbishop of Seville. The\nspacious building was crowded to overflowing with spectators, among whom\nwere the highest dignitaries of the Church and the most illustrious of\nthe nobility of Spain.[474]\n\nDuring the few days which followed, while the royal pair remained in\nSegovia, the city was abandoned to jubilee. The auspicious event was\ncelebrated by public illuminations and by magnificent _fetes_, at which\nthe king and queen danced in the presence of the whole court, who stood\naround in respectful silence.[475] On the eighteenth, the new-married\ncouple proceeded to Madrid, where such splendid preparations had been\nmade for their reception as evinced the loyalty of the capital.\n\nAs soon as the building of the Escorial was sufficiently advanced to\nfurnish suitable accommodations for his young queen, Philip passed a\npart of every summer in its cloistered solitudes, which had more\nattraction for him than any other of his residences. The presence of\nAnne and her courtly train diffused something like an air of gaiety over\nthe grand but gloomy pile, to which it had been little accustomed. Among\nother diversions for her entertainment, we find mention made of _autos\nsacramentales_, those religious dramas that remind one of the ancient\nMysteries and Moralities which entertained our English ancestors. These\n_autos_ were so much in favour with the Spaniards as to keep possession\nof the stage longer than in most other countries; nor did they receive\ntheir full development until they had awakened the genius of Calderon.\n\n[Sidenote: QUEEN ANNE.]\n\nIt was a pen, however, bearing little resemblance to that of Calderon\nwhich furnished these edifying dramas. They proceeded, probably, from\nsome Jeronymite gifted with a more poetic vein than his brethren. The\nactors were taken from among the pupils in the seminary established in\nthe Escorial. Anne, who appears to have been simple in her tastes, is\nsaid to have found much pleasure in these exhibitions, and in such\nrecreation as could be afforded her by excursions into the wild,\nromantic country that surrounded the monastery. Historians have left us\nbut few particulars of her life and character,--much fewer than of her\nlovely predecessor. Such accounts as we have, represent her as of an\namiable disposition, and addicted to pious works. She was rarely idle,\nand employed much of her time in needlework, leaving many specimens of\nher skill in this way in the decorations of the convents and churches. A\nrich piece of embroidery, wrought by her hands and those of her maidens,\nwas long preserved in the royal chapel, under the name of \"Queen Anne's\ntapestry.\"\n\nHer wedded life was destined not to be a long one,--only two years\nlonger than that of Isabella. She was blessed, however, with a more\nnumerous progeny than either of her predecessors. She had four sons and\na daughter. But all died in infancy or early childhood, except the third\nson, who, as Philip the Third, lived to take his place in the royal\ndynasty of Castile.\n\nThe queen died on the twenty-sixth of October, 1580, in the thirty-first\nyear of her age, and the eleventh of her reign. A singular anecdote is\ntold in connection with her death. This occurred at Badajoz, where the\ncourt was then established, as a convenient place for overlooking the\nwar in which the country was at that time engaged with Portugal. While\nthere the king fell ill. The symptoms were of the most alarming\ncharacter. The queen, in her distress, implored the Almighty to spare a\nlife so important to the welfare of the kingdom and of the Church, and\ninstead of it to accept the sacrifice of her own. Heaven, says the\nchronicler, as the result showed, listened to her prayer.[476] The king\nrecovered; and the queen fell ill of a disorder which in a few days\nterminated fatally. Her remains, after lying in state for some time,\nwere transported with solemn pomp to the Escorial, where they enjoyed\nthe melancholy pre-eminence of being laid in the quarter of the\nmausoleum reserved exclusively for kings and the mothers of kings. Such\nwas the end of Anne of Austria, the fourth and last wife of Philip the\nSecond.\n\n\nFOOTNOTES:\n\n[1] \"Que ningun Moro ni Mora seran apremiados a ser Christianos contra\nsu voluntad; y que si alguna doncella, o casada, o viuda, por razon de\nalgunas se quisiere tornar Christiana, tampoco sera recebida, hasta ser\ninterrogada.\" See the original treaty as given _in extenso_ by Marmol,\nRebelion de los Moriscos (Madrid, 1797), tom. i. pp. 88-98.\n\n[2] \"Y que pues habian sido rebeldes, y por ello merecian pena de muerte\ny perdimento de bienes, el perdon que les concediese fuese condicional,\ncon que se tornasen Christianos, o dexasen la tierra.\"--Ibid. p. 122.\n\n[3] The reader curious in the matter will find a full account of it in\nthe History of Ferdinand and Isabella, part II. chapters 6, 7.\n\n[4] Advertimientos de Don Geronimo Corella sobre la Conversion de los\nMoriscos del Reyno de Valencia, MS.\n\n[5] \"Sin tratar de instruir a cada uno en particular ni de examinar los\nni saber su voluntad los baptizaron a manadas y de modo que algunos de\nellos, segun es fama, pusieron pleito que no les avia tocado el agua que\nen comun les hechavan.\"--Ibid.\n\n[6] Marmol, Rebelion de los Moriscos, tom. i. pp. 133-155.--Bleda,\nCoronica de los Moros de Espana (Valencia, 1618), p.\n656.--Advertimientos de Corella, MS.--Ferreras, Hist. Generale\nd'Espagne, tom. ix. pp. 65, 68.--Vanderhammen, Don Juan de Austria, fol.\n55.\n\nThe last writer says that, besides the largess to the emperor, the\nMoriscoes were canny enough to secure the good-will of his ministers by\na liberal supply of doubloons to them also.--\"Sirvieron al Emperador con\nochenta mil ducados. Aprovecholes esto, y buena suma de doblones que\ndieron a los privados para que Carlos suspendiesse la execucion deste\nacuerdo.\"\n\n[7] Calderon, in his \"Amar despues de la Muerte,\" has shed the\nsplendours of his muse over the green and sunny spots that glitter like\nemeralds amidst the craggy wilds of the Alpujarras,\n\n    \"Porque entre puntas y puntas\n    Hay valles que la hermosean,\n    Campos que la fertilizan,\n    Jardines que la deleitan.\n    Toda ella esta poblada\n    De villages y de aldeas;\n    Tal, que, cuando el sol se pono\n    A las vislumbres que deja,\n    Parecen riscos nacidos\n    Concavos entre las penas,\n    Que rodaron de la cumbre\n    Aunque a la falda no llegan.\"\n\n[8] Senor de Gayangos, correcting a blunder of Casiri on the subject,\ntells us that the Arabic name of the Alpujarras was _Al-busherat_,\nsignifying \"mountains abounding in pastures.\"--See that treasure of\nOriental learning, the History of the Mohammedan Dynasties in Spain\n(London, 1843), vol. ii. p. 515.\n\n[9] Such was the exemption from certain duties paid by the Christians in\ntheir trade with the Barbary coast--a singular and not very politic\nprovision.--\"Que si los Moros que entraren debaxo de estas\ncapitulaciones y conciertos, quisieren ir con sus mercaderias a tratar y\ncontratar en Berberia, se les dara licencia para poderlo hacer\nlibremente, y lo mesmo en todos los lugares de Castilla y de la\nAndalucia, sin pagar portazgos, ni los otros derechos que los\nChristianos acostumbran pagar.\"--Marmol, Rebelion de los Moriscos, tom.\ni. p. 93.\n\n[10] Such is the opinion expressed by the author of the\n\"_Advertimientos_,\" whose remarks--having particular reference to\nValencia--are conceived in a spirit of candour, and of charity towards\nthe Moslems, rarely found in a Spaniard of the sixteenth century.--\"De\ndonde,\" he says, \"colije claramente que el no sanar estos enfermos hasta\nagora no se puede imputar a ser incurable la enfermedad, si no a averse\nerrado la cura, y tambien se vee que hasta oy no estan bastamente\ndescargados delante de Dios nuestro Senor aquellos a quien toca este\nnegocio, pues no han puesto los medios que Christo nuestro Senor tiene\nordenados para la cura de este mal.\"--MS.\n\n[11] \"Forzandoles con injurias y penas pecuniarias y justiciando a\nalgunos de ellos.\"--Ibid.\n\nMendoza, speaking of a somewhat later period, just before the outbreak,\nbriefly alludes to the fact that the Inquisition was then beginning to\nworry the Moriscoes more than usual:--\"Porque la Inquisicion los comenzo\na apretar mas de lo ordinario.\"--Guerra de Granada (Valencia, 1776), p.\n20.\n\n[12] Marmol, Rebelion de los Moriscos, tom. i. p. 135.\n\n[13] Ibid. tom. ii. p. 338.--Ordenanzas de Granada, fol. 375, ap.\nCircourt, Hist. des Arabes d'Espagne (Paris, 1846), tom. ii. p. 267.\n\nThe penalty for violating the above ordinance was six years' hard labour\nin the galleys. That for counterfeiting the stamp of the Mendoza arms\nwas death. _Vae victis!_\n\n[14] The name of Mendoza, which occupied for so many generations a\nprominent place in arms, in politics, and in letters, makes its first\nappearance in Spanish history as far back as the beginning of the\nthirteenth century.--Mariana, Historia de Espana, tom. i. p. 676.\n\n[15] M. de Circourt in his interesting volumes, has given a minute\naccount--much too minute for these pages--of the first developments of\nthe insurrectionary spirit of the Moriscoes, in which he shows a very\ncareful study of the subject.--Hist. des Arabes d'Espagne, tom. ii. pp.\n268 et seq.\n\n[16] Ferreras, Hist. d'Espagne, tom. ix. p. 524.--Marmol, Rebelion de\nlos Moriscos, tom. i. p. 142.--Vanderhammen, Don Juan de Austria, fol.\n55.\n\n[17] Such was the judgment of the acute Venetian who, as one of the\ntrain of the minister Tiepolo, obtained a near view of what was passing\nin the court of Philip the Second.--\"Levato di bassissimo stato dal re,\ne posto in tanta grandezza in pochi anni, per esser huomo da bene,\nlibero et schietto, et perche S. M. vuol tener bassi li grandi di\nSpagna, conoscendo l' altierissima natura loro.\"--Gachard, Relations des\nAmbassadeurs Venitiens sur Charles-Quint et Philippe II. (Bruxelles,\n1855), p. 175.\n\n[18] This remarkable ordinance may be found in the Nueva Recopilacion\n(ed. 1640), lib. viii. tit. 2, leyes 13-18.\n\nThe most severe penalties were those directed against the heinous\noffence of indulging in warm baths. For a second repetition of this, the\nculprit was sentenced to six years' labour in the galleys and the\nconfiscation of half his estates.\n\n[19] \"De los enemigos los menos.\"--Circourt gives a version of the whole\nof the professor's letter, with his precious commentary on this text.\n(Hist. des Arabes d'Espagne, tom. ii. p. 278.) According to Ferreras,\nPhilip highly relished the maxim of his ghostly counsellor.--Hist.\nd'Espagne, tom. ix. p. 525.\n\n[20] Cabrera, throwing the responsibility of the subsequent troubles on\nEspinosa and Deza, sarcastically remarks that \"two cowls had the\nordering of an affair which had been better left to men with helmets on\ntheir heads.\"--Cabrera, Filipe Segundo, lib. vii, cap. 21.\n\n[21] Marmol, Rebelion de los Moriscos, tom. i. pp. 147-151,--Circourt,\nHist. des Arabes d'Espagne, tom. ii. p. 283.--Ferreras, Hist. d'Espagne,\ntom. ix. p. 535.\n\nDr. Salazar de Mendoza considers that nothing but a real love of\nrebellion could have induced the Moriscoes to find a pretext for it in a\nmeasure so just and praiseworthy, and every way so conducive to their\nown salvation as this ordinance.--\"Tomaron par achaque esta accion tan\njustificada, y meritoria del Rey, y para sus almas tan provechosa y\nsaludable.\"--Monarquia de Espana, tom. ii. p. 137.\n\n[22] \"Y al fin concluyo con decirle resolutamente, que su Majestad\nqueria mas fe que farda, y que preciaba mas salvar una alma, que todo\nquanto le podian dar le renta los Moriscos nuevamente\nconvertidos.\"--Marmol, Rebelion de los Moriscos, tom. i. p. 163.\n\n[23] \"Que el habia consultado aquel negocio con hombres de ciencia y\nconciencia, y le decian que estaba obligado a hacer lo que\nhacia.\"--Ibid. p. 175.\n\n[24] \"Que el negocio de la prematica estaba determinado, y su Magestad\nresoluta en que se cumpliese.\"--Ibid, ubi supra.\n\n[25] Ibid. p. 176.--Cabrera. Filipe Segundo, lib. vii. cap.\n\n[26] \"A estas y otras muchas razones que el marques de Mondejar daba,\nDon Diego de Espinosa le respondio, que la voluntad de su Magestad era\naquella, y que se fuese al reyno de Granada, donde serio de mucha\nimportancia su persona, atropellando como siempre todas las dificultades\nque le ponian por delante.\"--Marmol, Rebelion de los Moriscos, tom. i.\np. 168.\n\n[27] An ordinance was passed at this time that the Moriscoes who had\ncome from the country to reside with their families in Granada should\nleave the city and return whence they came, under pain of\ndeath.--(Marmol, Rebelion de los Moriscos, tom. i. p. 169.) By another\nordinance, the Moriscoes were required to give up their children between\nthe ages of three and fifteen, to be placed in schools and educated in\nthe Christian doctrine and the Castillan tongue. (Ibid. p. 170.) The\n_Nueva Recopilacion_ contains two laws passed about this time, making it\na capital offence to hold any intercourse with Turks or Moors who might\nvisit Granada, even though they came not as corsairs, but for purposes\nof traffic. (Lib. viii. tit. 26, leyes 16, 18.) Such a law proves the\nconstant apprehensions in which the Spaniards lived of a treasonable\ncorrespondence between their Morisco subjects and the foreign Moslems.\n\n[28] Marmol Rebelion de los Moriscos, tom. i. pp. 223-233.--Mendoza,\nGuerra de Granada (Valencia, 1776), p. 43.--Hita, Guerras de Granada,\ntom. ii. p. 724.\n\n[29] \"Escrita en noches de augustia y de lagrimas corrientes,\nsustentadas con esperanza, y la esperanza deriva de la\namargura.\"--Marmol, Rebelion de los Moriscos, tom. i. p. 219.\n\n[30] Marmol, Rebelion de los Moriscos, tom. i. p. 235.\n\n[31]\n\n    \"La furia horrible de los torbellinos\n    Cada momento mas se vee yr creciendo;\n    Cubre la blanca nieve les caminos,\n    Tambien los hombres luego va cubriendo.\"\n\nSo sings, or rather says, the poet-chronicler Rufo, whose epic of four\nand twenty cantos shows him to have been much more of a chronicler than\na poet. Indeed, in his preface, he avows that strict conformity to truth\nwhich is the cardinal virtue of the chronicler.--See the Austriada\n(Madrid, 1584).\n\n[32] \"Pocos sois, i venis presto.\"--Mendoza, Guerra de Granada, p. 47.\n\nHita gives a _cancion_ in his work, the burden of which is a complaint\nthat the mountaineers had made their attack too late instead of too\nearly:--\n\n    \"Pocos sois, y venis tarde.\"\n\n(Guerras de Granada, tom. ii. p. 32.) The difference is explained by the\ncircumstance that the author of the verses--probably Hita\nhimself--considers that Christmas Eve, not New Year's Eve, was the time\nfixed for the assault.\n\n[33] Marmol, Rebelion de los Moriscos, tom. i. p. 238.--Mendoza, Guerra\nde Granada, pp. 45-52.--Miniana, Hist. de Espana, p. 367.--Herrera,\nHistoria General, tom. i. p. 726.--Ferreras, Hist. d'Espagne, tom. ix\npp. 573-575.\n\n[34] \"Creyendo que lo uno y lo otro seria parte para que por bien de paz\nse diese nueva orden en lo de la prematica, sin aventurar ellos sus\npersonas y haciendas.\"--Marmol, Rebelion de los Moriscos, tom. i. p.\n239.\n\n[35] Beni Umeyyah, in the Arabic, according to an indisputable\nauthority, my learned friend Don Pascual de Gayangos. See his Mohammedan\nDynasties in Spain, _passim_.\n\n[36] \"Era mancebo de veinte y dos anos, de poca barba, color moreno,\nverdinegro, cejijunto, ojos <DW64>s y grandes, gentil hombre de cuerpo:\nmostraba en su talle y garbo ser de sangre real, como en verdad lo era,\nteniendo los pensamientos correspondientes.\"--Hita, Guerras de Granada,\ntom. ii. p. 13.\n\nFew will be disposed to acquiesce in the savage tone of criticism with\nwhich the learned Nic. Antonio denounces Hita's charming volumes as\n\"Milesian tales, fit only to amuse the lazy and the listless.\"\n(Bibliotheca Nova, tom. i. p. 536.) Hita was, undoubtedly, the prince of\nromancers; but fiction is not falsehood; and when the novelist, who\nserved in the wars of the Alpujarras, tells us of things which he\nprofesses to have seen with his own eyes, we may surely cite him as an\nhistorical authority.\n\n[37] \"Usava de blandura general; queria ser tenido por Cabeza, i no por\nRei: la crueldad, la codicia cubierta engano a muchos en los\nprincipios.\"--Mendoza, Guerra de Granada, p. 129.\n\n[38] Ibid. p. 40.\n\nThe ceremonies of the coronation make, of course, a brave show in Rufo's\nepic. One stanza will suffice:--\n\n   \"Entonces con aplauso le pusieron\n    Al nuevo Rey de purpura un vestido,\n    Y a manera de beca le cineron\n    Al cuello y ombros un cendal brunido,\n    Quatro vanderas a sus pies tendieron,\n    Una hazia el Levante esclarecido,\n    Otra a do el sol se cubre en <DW64> velo,\n    Y otras dos a los polos dos del cielo.\"\n\n    La Austriada, fol. 24.\n\n[39] \"Tal era la antigua ceremonia con que eligian los reyes de la\nAndalucia, i despues los de Granada.\"--Mendoza, Guerra de Granada, p.\n40.\n\n[40]\n\n   \"Que en la agricultura tienen\n    Tal estudio, tal destreza,\n    Que a preneces de su hazada\n    Hacen fecundas las piedras.\"\n\n    Calderon, Amar despues de la Muerte, Jornada ii.\n\n[41]\n\n   \"Tres anos tuvo en silencio\n    Esta traicion encubierta\n    Tanto numero de gentes,\n    Cosa, que admira y eleva.\"--Ibid, ubi supra.\n\n[42] \"Una cosa mui de notar califica los principios desta rebelion, que\ngente de mediana condicion mostrada a guardar poco secreto i hablar\njuntos, callasen tanto tiempo, i tantos hombres, en tierra donde hai\nAlcaldes de corte i Inquisidores, cuya profesion es descubrir\ndelitos.\"--Mendoza, Guerra de Granada, p. 36.\n\n[43] Bleda, Cronica de Espana, p. 680--\"Robaron la iglesia, hicieron\npedazos los retablos y imagines, destruyeron todas las cosas sagradas, y\nno dexaron maldad ni sacrilegio que no cometieron.\"--Marmol, Rebelion de\nGranada, tom. i. p. 275.\n\n[44] \"Quemaron por voto un convento de Frailes Augustinos, que se\nrecogieron a la Torre echandoles por un horado de lo alto azeite\nhirviendo: sirviendose de la abundancia que Dios les dio en aquella\ntierra, para ahogar sus Frailes.\"--Mendoza, Guerra de Granada, p. 60.\n\n[45] Marmol, Rebelion de Granada, tom. i. p. 271.--Ferreras, Hist.\nd'Espagne, tom. ix. P. 582.\n\n[46] \"Y para darle mayor tormento traxeron alli dos hermanas doncellas\nque tenia, para que le viesen morir, y en su presencia las vituperaron y\nmaltrataron.\"--Marmol, Rebelion de Granada, tom. i. p. 316.\n\n[47] \"Llego un herege a el con una navaja, y le persino con ella,\nhendiendole el rostro de alto abaxo, y por traves; y luego le despedazo\ncoyuntura por coyuntura, y miembro a miembro.\"--Ibid. p. 348.\n\nAmong other kinds of torture which they invented, says Mendoza, they\nfilled the curate of Manena with gunpowder, and then blew him\nup.--Guerra de Granada, p. 60.\n\n[48] Of all the Spanish historians no one discovers so insatiable an\nappetite for these horrors as Ferreras, who has devoted nearly fifty\nquarto pages to an account of the diabolical cruelties practised by the\nMoriscoes in this persecution--making, altogether, a momentous\ncontribution to the annals of Christian martyrologv. One may doubt,\nhowever, whether the Spaniards are entirely justified in claiming the\ncrown of martyrdom for all who perished in this persecution. Those,\nundoubtedly, have a right to it who might have saved their lives by\nrenouncing their faith; but there is no evidence that this grace was\nextended to all; and we may well believe that the Moriscoes were\nstimulated by other motives besides those of a religious nature,--such\nmotives as would naturally operate on a conquered race, burning with\nhatred of their conquerors and with the thirst of vengeance for the\nmanifold wrongs which they had endured.\n\n[49] \"Murieron en pocos mas de quatro dias, con muertes exquesitas y no\nimaginados tormentos, mas de tres mil martires.\"--Vanderhammen, Don Juan\nde Austria, fol. 70.\n\n[50] \"Se adelanto un Moro, que solia ser grande amigo suyo, y haciendose\nencontradizo con el en el umbral de la puerta, le atraveso una espada\npor el cuerpo, diciendole: Toma, amigo, que mas vale que te mate yo que\notro.\"--Marmol, Rebelion de Granada, tom. i. p. 277.\n\n[51] Ferreras, Hist. d'Espagne, tom. ix. p. 617.\n\n[52] \"Fue gran testimonio de nuestra fe i de compararse con la del\ntiempo de los Apostoles; que en tanto numero de gente como murio a manos\nde infieles ninguno huvo que quisiese renegar.\"--Mendoza, Guerra de\nGranada, p. 61.\n\n[53] \"Todos estuvieron tan constantes en la fe, que si bien fueron\ncombidados con grandes riquezas y bienes a que la dejasen, con ninguno\nse pudo acabar; aunque entre los martyrizados huvo muchas mugeres,\nninos, y hombres que havian vivido descompuestamente.\"--Salazar de\nMendoza, Monarquia de Espana, tom. ii. p. 139.\n\n[54] \"Murieron este dia en Uxixar docientos y quarenta Christianos\nclerigos y legos, y entre ellos seis canonigos de aquella iglesia, que\nes colegial.\"--Marmol, Rebelion de Granada, tom. i. p. 297.\n\n[55] \"Estavan las casas yermas i tiendas cerradas, suspenso el trato,\nmudadas las horas de oficios divinos i humanos; atentos los Religiosos i\nocupados en oraciones i plegarias, como se suele en tiempo i punto de\ngrandes peligros.\"--Mendoza, Guerra de Granada, p. 54.\n\nMendoza paints the panic of Granada with the pencil of Tacitus.\n\n[56] Circourt, Hist. des Arabes d'Espagne, tom. ii. p. 322.\n\n[57] \"En un punto se mudaron todos los oficios y tratos en soldadesca,\ntanto que los relatores, secretarios, letrados, procuradores de la\nAudiencia entraban con espadas en los estrados, y no dexaban de pareseer\nmuy bien en aquella coyuntura.\"--Marmol, Rebelion de Granada, tom. i. p.\n358.\n\n[58] \"Servian tres meses pagados por sus pueblos enteramente, i seis\nmeses adelante pagavan los pueblos la mitad, i otra mitad el\nRei.\"--Mendoza, Guerra de Granada, p. 53.\n\n[59] Mendoza, with a few vigorous touches, has sketched, or rather\nsculptured in bold relief, the rude and rapacious character of the\nAndalusian soldiery.--\"Mal pagada i por esto no bien disciplinada;\nmantenida del robo, i a trueco de alcanzar o conservar este mucha\nlibertad, poca verguenza, i menos honra.\"--Mendoza, Guerra de Granada,\np. 103.\n\n[60] \"Toda gente lucida y bien arreada a punto de guerra, que cierto\nrepresentaban la pompa y nobleza de sus ciudades.\"--Marmol, Rebelion de\nGranada, tom. i. p. 396.\n\n[61]\n\n   \"Muchos capitanos fuertes,\n    muchos lucidos soldados,\n    ricos banderas tendidas,\n    y su estandarte dorado.\"\n\n    Hita, Guerras de Granada, tom. ii. p. 63.\n\n[62] Circourt, Hist. des Arabes d'Espagne, tom. ii. p. 326.\n\nSeville alone furnished two thousand troops, with one of the most\nillustrious cavaliers of the city at their head. They did not arrive,\nhowever, till a later period of the war.--See Zuniga, Annales de Sevilla\n(Madrid, 1677, fol.), p. 533.\n\n[63] \"Repartio los lugares de la vega en siete partidos, y mandoles, que\ncada uno tuviese cuidado de llevar diez mil panes amasados de a dos\nlibras al campo el dia que le tocase de la semana.\"--Marmol, Rebelion de\nGranada, tom. i. p. 404.\n\n[64] \"Paso este negocio tan adelante, que muchos Moriscos afrentados y\ngastados se arrepintieron por no haber tomado las armas cuando Abenfarax\nlos llamaba.\"--Ibid. p. 407.\n\n[65] \"Apenas podia ir por ella un hombre suelto; y aun este poco paso,\nle tenian descavado y solapado por los cimientos, de manera que si\ncargase mas de una persona, fuese abaxo.\"--Marmol, Rebelion de Granada,\np. 409.\n\n[66] \"Mas un bendito frayle de la orden del serafico padre San\nFrancisco, llamado fray Christoval de Molina, con un crucifixo en la\nmano izquierda, y la espada desnuda en la derecha, los habitos cogidos\nen la cinta, y una rodela echada a las espaldas, invocando el poderoso\nnombre de Jesus, llego al peligroso paso, y se metio determinadamente\npor el.\"--Marmol, Rebelion de Granada, tom. i. p. 410.\n\n[67] Ibid. p. 410, et seq.--Mendoza, Guerra de Granada, pp. 67,\n68.--Herrera, Historia General, tom. i. p. 736.\n\nHita has commemorated the bold passage of the bridge at Tablate in one\nof the _romances_, or ballads, with which he has plentifully besprinkled\nthe second volume of his work, and which present a sorry contrast to the\nballads in the preceding volume. These, which form part of the popular\nminstrelsy of an earlier age, have all the raciness and flavour that\nbelong to the native wild-flower of the soil. The ballads in the second\nvolume are, probably, the work of Hita himself,--poor imitations of the\nantique, and proving that, if his rich and redundant prose is akin to\npoetry, his poetry is still nearer allied to prose.\n\n[68] \"Estuvo alli aquella noche a vista de los enemigos, que teniendo\nocupado el paso con grandes fuegos por aquellos cerros, no hacian sino\ntocar sus atabalejos, dulzaynas, y xabecas, haciendo algazaras para\natemorizar nuestros Cristianos, que con grandisimo recato estuvieron\ntodos con las armas en las manos.\"--Marmol, Rebelion de Granada, tom. i.\np. 413.\n\n[69] Ibid. p. 414.--Herrera, Historia General, tom. i. p. 737.--Bleda,\nCronica de Espana, p. 684.--Mendoza, Guerra de Granada, pp. 69,\n70.--Ferreras, Hist. d'Espagne, tom. x. p. 17.\n\n[70] \"A la mano derecha cubiertos con un sierro, havia emboscados\nquinientos arcabuceros i vallesteros, demas desto otra emboscada en lo\nhondo del barranco de mucho mayor numero de gente.\"--Mendoza, Guerra de\nGranada, tom. i. p. 71.\n\n[71] \"Ellos quando pensaron que nuestra gente iva cansada acometieron\npor la frente, por el costado, i por la retaguardia, todo a un tiempo;\nde manera que quasi una hora se peleo con ellos a todas partes i a las\nespaldas, no sin igualdad i peligro.\"--Ibid. ubi supra.\n\n[72] This poison was extracted from the aconite, or wolf's-bane, that\ngrew rife among the Alpujarras. It was of so malignant a nature that the\nhistorian assures us that, if a drop mingled with the blood flowing from\na wound, the virus would ascend the stream and diffuse itself over the\nwhole system! Quince-juice was said to furnish the best\nantidote.--Mendoza, Guerra de Granada, tom. i. pp. 73, 74.\n\n[73] Ibid. pp. 71-74.--Cabrera, Filipe Segundo, p. 554.--Marmol,\nRebelion de Granada, tom. i. pp. 416-418.--Herrera, Historia General,\ntom. i. p. 737.--Bleda, Cronica de Espana, p. 684.\n\n[74] \"Mas la priesa de caminar en siguimiento de los enemigos, i la\nfalta de bagages en que la cargar i gente con que aseguralla, fue causa\nde quemar la mayor parte, porque ellos no se aprovechasen.\"--Mendoza,\nGuerra de Granada, p. 75.\n\n[75] \"Los Moros tomaron lo alto de la sierra, y no pararon hasta meterse\nen la nieve, donde perecieron cantidad de mugeres y de criatura de\nfrio.\"--Marmol, Rebelion de Granada, tom. i. p. 437.\n\n[76] \"El Marques les dio a saco todo el mueble, en que habia ricas cosas\nde seda, oro, plata, y aljofar, de que cupo la mejor y mayor parte a los\nque habian ido delante.\"--Marmol, Rebelion de Granada, tom. i. p. 444.\n\n[77] \"No tomen, senores, a vida hombre ni muger de aquestos hereges, que\ntan malos han sido, y tanto mal nos han hecho.\"--Ibid. p. 440.\n\n[78] \"El Marques se enternecio de ver aquellas pobres mugeres tan\nlastimadas, y consolandolas lo mejor que pudo,\" &c.--Ibid, ubi supra.\n\n[79] \"Hubo muchos soldados heridos, los mas que se herian unos a otros,\nentendiendo los que venian de fuera, que los que martillaban con las\nespadas eran Moros, porque solamente les alumbraba el centellear del\nacero, y el relampaguear de la polvora de los arcabuces en la tenebrosa\nescuridad de la noche.\"--Ibid. p. 445.\n\n[80] \"De los Moriscos quasi ninguno quedo vivo, de las Moriscas huvo\nmuchas muertas, de los nuestros algunos heridos, que con la escuridad de\nla noche se hacian dano unos a otros.\"--Mendoza, Guerra de Granada, p.\n77.\n\n[81] Ibid. ubi supra.--Bleda, Cronica de Espana, p. 685.--Herrera,\nHistoria General, tom. i. p. 787.--Marmol, Rebelion de Granada, tom. i.\np. 441 et seq.--Cabrera, Filipe Segundo, p. 558.\n\n[82] \"Habia entre ellas muchas duenas nobles, apuestas y hermosas\ndoncellas, criadas con mucho regalo, que iban desnudas y descalzas, y\ntan maltratadas del trabajo del captiverio y del camino, que no solo\nquebraban los corazones a los que las conocian, mas aun a quien no las\nhabia visto.\"--Marmol, Rebelion de Granada, tom. i. p. 448.\n\n[83] \"Y volviendo a las casas del Arzobispo, las que tenian parientes\nlas llevaron a sus posadas, y las otras fueron hospedadas con caridad\nentre la buena gente, y de limosna se les compro de vestir y de\ncalzar.\"--Marmol, Rebelion de Granada, tom. i. ubi supra.\n\n[84] \"Los soldados no podian llevar a paciencia ver que se tratase de\nmedios con los rebeldes; y quando otro dia se supo que los admitia, fue\ntan grande la tristeza en el campo, como si hubieran perdido la\njornada.\"--Ibid. p. 443.\n\n[85] Marmol, Rebelion de Granada, tom. i. p. 455.\n\n[86] Abderrahman--or, as spelt by Gayangos, Abdu-r-rhaman--the First,\nthe founder of the dynasty from which Aben-Humeya claimed his descent,\ntook refuge in Spain from a bloody persecution, in which every member of\nhis numerous family is said to have perished by the scimitar or the\nbowstring.\n\n[87] \"Y como vio que los Christanos iban la sierra arriba, y que los\nsuyos huian desvergonzadamente, entendiendo que todo lo que Don Alonso\nVenegas trataba era engano, echo las cartas en el suelo, y subiendo a\ngran priesa en un caballo, dexo su familia atras, y huyo tambien la\nvuelta de la sierra.\"--Marmol, Rebelion de Granada, tom. i. p. 460.\n\n[88] Marmol, Rebelion de Granada, tom. i. p. 458 et seq.--Ferreras,\nHist. d'Espagne, tom. x. PP. 28-31.--Mendoza, Guerra de Granada, pp. 80,\n81.--Cabrera, Filipe Segundo, pp. 560, 561.--Herrera, Historia General,\ntom. i. p. 737.\n\n[89] The decision referred to was, probably, one in the last Council of\nToledo, A.D. 690.--See Mariana, Hist. de Espana, tom. i. p. 452.\n\n[90] I quote the words of Marmol:--\"Con una moderacion piadosa, de que\nquiso usar como principe considerado y justo.\"--Rebelion de Granada,\ntom. i. p. 495.\n\n[91] Ibid. ubi supra.\n\n[92] Marmol, Rebelion de Granada, tom. i. pp. 465, 498.\n\nMendoza says they were all returned:--\"a thing never before seen,\nwhether it arose from fear or obedience, or that there was such an\nabundance of women that they were regarded as little better than\nhousehold furniture.\"--Guerra de Granada, p. 96.\n\n[93] \"Fue tanta la indignacion del Margues de Mondejar, que, sin\nperdonar a ninguna edad ni sexo, mando pasar a cuchillo hombres y\nmugeres, quantos habia en el fuerte; y en su presencia los hacia matar a\nlos alabarderos de su guardia, que no bastaban los ruegos de los\ncaballeros y capitanes, ni las piadosas lagrimas de las que pedian la\nmiserable vida.\"--Marmol, Rebelion de Granada, tom. i. p. 493.\n\n[94] Marmol, Rebelion de Granada, tom. i. p. 482 et seq.--Mendoza,\nGuerra de Granada, pp. 85-95.--Ferreras, Hist. d'Espagne, tom. x. pp.\n32-36.--Bleda, Cronica de Espana, p. 688 et seq.--Herrera, Historia\nGeneral, tom. i. p. 738.--Cabrera, Filipe Segundo, p. 589.\n\nThe storming of Guajaras is a favorite theme with both chroniclers and\nbards. Among the latter Hita has not failed to hang his garland of verse\non the tombs of more than one illustrious cavalier who perished in that\nbloody strife, and for whose loss \"all the noble dames of Seville,\" as\nhe tells us, \"went into mourning.\"--Guerras de Granada, tom. ii. pp.\n112-118.\n\n[95] \"Que no habia osado parar en la Alpuxarra, y con solos cincuenta o\nsesenta hombres, que le seguian, andaba huyendo de pena en\npena.\"--Marmol, Rebelion de Granada, tom. i. p. 464.\n\n[96] The Castillian chronicler Marmol refuse his admission--somewhat\nroughly expressed--to this brave Morisco,-\"este barbaro,\" as he calls\nhim, \"hijo de aspereza y frialdad indomable, y menospreciador de la\nmuerte.\"--(Marmol, Rebelion de Granada, tom. i. p. 503.) The story of\nthe escape of Aben-Humeya is also told, and with little discrepancy, by\nCabrera (Filipe Segundo, p. 573), and Ferreras (Hist. d'Espagne, tom. x.\npp. 39, 40).\n\n[97] \"Quando entendieron que peleaban contra el campo del Marques de los\nVelez, a quien los Moros de aquella tierra solian llamar Ibiliz Arraez\nel Hadid, que quiere decir, _diabolo cabeza de hierro_, perdieron\nesperanza de vitoria.\"--Marmol, Rebelion de Granada, tom. i. p. 451.\n\nHita, who was a native of Murcia, and followed Los Velez to the war,\ngives an elaborate portrait of this powerful chief, whom he extols as\none of the most valiant captains in the world, rivalling in his\nachievements the Cid, Bernardo del Carpio, or any other hero of greatest\nrenown in Spain.--Guerras de Granada, tom. ii. p. 68 et seq.\n\n[98] Circourt, Hist. des Arabes d'Espagne, tom. ii. p. 346.\n\n[99] \"Mas mugeres que hombres,\" says Mendoza, Guerra de Granada, p. 83.\n\n[100] \"En menos de dos horas fueron muertas mas de seis mil personas\nentre hombres y mugeres; y de ninos, desde uno hasta diez anos, habia\nmas de dos mil degollados.\"--Hita, Guerras de Granada, tom. ii. p. 126.\n\nWe may hope this is an exaggeration of the romancer. Mendoza says\nnothing of the children, and reduces the slain to seven hundred. But\nHita was in the action.\n\n[101] \"La soldadesca que andaba suelta por el lugar cometio crueldades\ninauditas, y que la pluma se resiste a transcribir.\"--Ibid. p. 125.\n\n[102] \"El nino arrastrando como pudo se llego a ella, y movido del deseo\nde mamar, se asio de los pechos de la madre, sacando leche mezclada con\nla sangre de las heridas.\"--Hita, Guerras de Granada, p. 126.\n\n[103] \"Advirtiendo al mismo tiempo que hay tres mil hombres paisanos\nsuyos puestos sobre las armas, y decididos a perder la vida por\nsalvarle.\"--Ibid. p. 132.\n\n[104] Hita has devoted one of the most spirited of his _romances_ to the\nrout of Ohanez. The opening stanza may show the tone of it:--\n\n    \"Las tremolantes banderas\n    del grande Fajardo parten\n    para las Nevadas Sierras,\n    y van camino de Ohanez.\n    Ay de Ohanez!\"\n\n\n[105] \"Todos los caballeros y capitanes en la procesion armados de todas\nsus armas, con velas de cera blanca en las manos, que se las habian\nenviado para aquel dia desde su casa, y todas las Christianas en medio\nvestidas de azul y blanco, que por ser colores aplicados a nuestra\nSenora, mando el Marques que las vistiesen de aquella manera a su\ncosta.\"--Marmol, Rebelion de Granada, tom. i. p. 469.\n\n[106] \"Trayendose muchas Moras hermosas, pues pasaron de trescientas las\nque se tomaron alli; y habiendolas tenido los soldados a su voluntad mas\nde quince dias, al cabo de ellos mando el marques que llevasen a la\niglesia.\"--Hita, Guerras de Granada, tom. ii. p. 155.\n\n[107] \"Por manera que estaba la Alpuxarra tan llana, que diez y doce\nsoldados iban de unos lugares en otros, sin hallar quien los\nenojase.\"--Marmol, Rebelion de Granada, tom. i. p. 498.\n\nMendoza fully confirms Marmol's account of the quiet state of the\ncountry.--Guerra de Granada, pp. 96, 97.\n\n[108] \"Le suplicase de su parte los admitiese, habiendose\nmisericordiosamente con los que no fuesen muy culpados, para que el\npudiese cumplir la palabra que tenia ya dada a los reducidos,\nentendiendo ser aquel camino el mas breve para acabar con ellos por la\nvia de equidad.\"--Marmol, Rebelion de Granada, tom. i. p. 483.\n\n[109] \"Que hiciese por su parte lo que pudiese, porque ansi haria el de\nla suya.\"--Ibid. p. 470.\n\n[110] \"Dexar sin castigo exemplar a quien tantos crimenes habian\ncometido contra la Magestad _divina y humana_.\"--Marmol, Rebelion de\nGranada, p. 499.\n\n[111] \"El Marques,\" says Mendoza, \"hombre de estrecha i rigurosa\ndisciplina, criado al favor de su abuelo i padre en gran oficio, sin\nigual ni contradictor, impaciente de tomar compania, communicava sus\nconsejos consigo mismo.\"--Guerra de Granada, p. 103.\n\n[112] Mendoza, Guerra de Granada, p. 115 et seq.--Marmol, Rebelion de\nGranada, tom. i. pp. 511-513.--Miniana, Historia de Espana, p.\n376.--Cabrera, Filipe Segundo, pp. 573, 574.\n\n[113] Marmol, Rebelion de Granada, tom. ii. p. 8 et seq.--Mendoza,\nGuerra de Granada, pp. 97, 128.--Miniana, Historia de Espana, p.\n376.--Cabrera, Filipe Segundo, pp. 575, 576.\n\n[114] \"Otros, como desesperados, juntando esteras, tascos, y otras cosas\nsecas, que pudiesen arder, so metian entre sus mesmas llamas, y las\navivaban, para que, ardiendo la carcel y la Audiencia, pereciesen todos\nlos que estaban dentro.\"--Marmol, Rebelion de Granada, tom. i. p. 517.\n\n[115] Ibid. ubi supra.\n\n[116] \"Los mataron a todos, sin dexar hombre a vida, sino fueron los dos\nque defendio la guardia que tenian.\"--Ibid. ubi supra. See also Mendoza,\nGuerra de Granada, p. 122; Herrera, Historia General, tom. i. p. 744.\n\n[117] \"Havia en ellos culpados en platicas i demonstraciones, i todos en\ndeseo; gente flaca, liviana, inhabil para todo, sino para dar ocasion a\nsu desventura.\"--Mendoza, Guerra de Granada, p. 122.\n\n[118] \"Las culpas de los quales debieron ser mayores de lo que aqui se\nescribe, porque despues pidiendo las mugeres y hijos de los muertos sus\ndotes y haciendas ante los alcaldes del crimen de aquella Audiencia, y\nsaliendo el fiscal a la causa, se formo proceso en forma; y por\nsentencias y revista fueron condenados, y aplicados todos sus bienes al\nreal fisco.\"--Marmol, Rebelion de Granada, tom. i. p. 517.\n\n[119] \"Levanto un estandarte bermejo, que mostrava el lugar de la\npersona del Rei a manera de Guion.\"--Mendoza, Guerra de Granada, p. 118.\n\n[120] \"Para seguridad de su persona pago arcabuceria de guardia, que fue\ncreciendo hasta quatrocientos hombres.\"--Mendoza, Guerra de Granada, ubi\nsupra.\n\n[121] \"Siguio nuestra orden de guerra, repartio la gente por escuadras,\njuntola en companias, nombro capitanes.\"--Ibid. ubi supra.\n\n[122] This, which is two years later than the date commonly assigned by\nhistorians, seems to be settled by the researches of Lafuente. (See\nHistoria General de Espana (Madrid, 1854), tom. xiii. p. 437, note.)\nAmong other evidence adduced by the historian is that of a medal struck\nin honour of Don John's victory at Lepanto, in the year 1571, the\ninscription on which expressly states that he was twenty-four years of\nage.\n\n[123] Vanderhammen, Don Juan de Austria, fol 3.--Villafane, Vida y\nVirtudes de Dona Magdalena de Ulloa (Salamanca, 1722), p. 36.--See also\nLafuente, Historia de Espana, tom. xiii. p. 432.\n\nThis last historian has made the parentage of John of Austria the\nsubject of a particular discussion in the Revista de Ambos Mundos, No.\n3.\n\n[124] Vanderhammen, alluding to the doubts thrown on the rank of his\nhero's mother, consoles himself with the reflection that, if there was\nany deficiency in this particular, no one can deny that it was more than\ncompensated by the proud origin of her imperial lover.--Don Juan de\nAustria, fol. 3.\n\n[125] Lafuente, Hist. de Espana, tom. xiii. p. 432, note.\n\n[126] Gachard, Retraite et Mort de Charles-Quint, tom. ii. p. 506.\n\nIn a private interview with Luis Quixada, the evening before his death,\nthe emperor gave him six hundred gold crowns to purchase the\nabove-mentioned pension.\n\n[127] This interesting document was found among the testamentary papers\nof Charles the Fifth. A copy of it has been preserved among the\nmanuscripts of Cardinal Granvelle.--Papiers d'Etat, tom. iv. pp. 499,\n500.\n\n[128] \"Gastava buena parte del dia en tirar con una ballestilla a los\npaxaros.\"--Vanderhammen, Don Juan de Austria, fol. 10.\n\n[129] \"Y puede ser llegase a sospechar, si acaso tendria por padre a su\nesposo.\"--Villefane, Vida de Magdalena de Ulloa, p. 38.\n\n[130] \"Accion singular y rara, y que dexa atras quantas la antiguedad\ncelebra por peregrinas.\"--Vanderhammen, Don Juan de Austria, fol. 31.\n\nAccording to another biographer, two fires occurred to Quixada, one in\nVillagarcia and one in Valladolid. On each of these occasions the house\nwas destroyed, but his ward was saved, borne off by the good knight in\nhis arms. (Villafane, Vida de Magdalena de Ulloa, pp. 44, 53.) The\ncoincidences are too much opposed to the doctrine of chances to commend\nthemselves readily to our faith. Vanderhammen's reflection was drawn\nforth by the second fire, the only one he notices. It applies, however,\nequally well to both.\n\n[131] Vanderhammen, Don Juan de Austria, fol. 16.\n\n[132] Indeed, Siguenza, who may have had it from the monks of Yuste,\ntells us that the boy sometimes was casually seen by the emperor, who\nwas careful to maintain his usual reserve and dignified demeanour; so\nthat no one could suspect his secret. \"Once or twice,\" adds the\nJeronymite father, \"the lad entered the apartment of his father, who\ndoubtless spoke to him as he would have spoken to any other\nboy.\"--Historia de la Orden de San Geronimo, tom. iii. p. 205.\n\n[133] Relation d'un Religieux de Yuste, ap. Gachard, Retraite et Mort de\nCharles-Quint, tom. ii. p. 55.\n\n[134] \"Hallo tan publico aqui lo que toca aquella persona que V. Mtad\nsabe que esta a mi cargo que me ha espantado, y espantame mucho mas las\nparticularidades que sobrello oyo.\"--Ibid. tom. i. p. 449.\n\n[135] A copy of this interesting document was found in the collection of\nGranvelle at Besancon, and has been lately published in the beautiful\nedition of the cardinal's papers.--Papiers d'Etat, tom. iv. p. 495 et\nseq.\n\n[136] \"Que pues su Mtad, en su testamento ni codecilo, no hazia memoria\ndel, que era razon tenello por burla, y que no sabia que poder responder\notra cosa, en publico ni en secreto.\"--Gachard, Retraite et Mort de\nCharles-Quint, tom. i. p. 446.\n\n[137] \"La Princesa al punto arrebatada del amor, lo abraco, y beso, sin\nreparar en el lugar que estava, y el acto que exercia. Llamole hermano y\ntratole de alteza.\"--Vanderhammen, Don Juan de Austria, fol. 23.\n\n[138] \"Llego el caso a estado, que le huvo de tomar en bracos el Conde\nOsorno hasta la carroca de la Princesa, porque le gozassen\ntodos.\"--Vanderhammen, Don Juan de Austria, fol. 25.\n\nThe story must be admitted to be a strange one, considering the\npunctilious character of the Castilian court, and the reserved and\ndecorous habits of Joanna. But the author, born and bred in the palace,\nhad access, as he tells us, to the very highest sources of information,\noral and written.\n\n[139] \"Vuelto ya en si de la suspension primera, alargo la mano, y monto\nen el caballo; y aun se dice que con airosa grandeza, anadio; Pues si\neso es asi tened el estribo.\"--Villafane, Vida de Dona Magdalena de\nUlloa, p. 51.\n\n[140] \"Macte, inquit, animo puer, praenobilis vire filius es tu; Carolus\nQuintus Imperator, qui coelo degit, utriusque nostrum pater\nest.\"--Strada, De Bello Belgico, tom. i. p. 608.\n\n[141] \"Jamas habia tenido dia de caza mas gustoso, ni logrado presa que\nle hubiese dado tanto contento.\"--Villafane, Vida de Dona Magdalena de\nUlloa, p. 52.\n\nThis curious account of Philip's recognition of his brother is told,\nwith less discrepancy than usual, by various writers of that day.\n\n[142] Vanderhammen, Don Juan de Austria, fol. 27.--\"Mandole llamar\nEcelencia; pero sus reales costunbres le dieron adelante titulo de\nAlteza i de senor entre los grandes i menores.\"--Cabrera, Filipe\nSegundo, lib. v. cap. 3.\n\n[143] \"Tengo mucho cuidado que aprenda y se le ensenen las cosas\nnecesarias, conforme a su edad y a la calidad de su persona, que, segun\nla estrecheza en que se crio y ha estado hasta que vino a mi poder, es\nbien menester con todo cuidado tener cuenta con el.\"--Gachard, Retraite\net Mort de Charles-Quint, tom. i. p. 450.\n\n[144] \"Longe tamen anteibat Austriacus et corporis habitudine, et morum\nsuavitate. Facies illi non modo pulchra, sed etiam venusta.\"--Strada, De\nBello Belgico, tom. i. p. 609.\n\n[145] \"Eminebat in adolescente comitas, industria, probitas, et, ut in\nnovae potentiae hospite, verecundia.\"--Ibid. loc. cit.\n\n[146] Strada, Be Bello Belgico, tom. ii. pp. 609, 610.--Vanderhammen,\nDon Juan de Austria, fol. 34-36.--Cabrera, Filipe Segundo, lib. vi. cap.\n24.\n\n[147] \"La fama de la partida de Don Juan saco del ocio a muchos\ncavalleros de la corte i reynos, que avergoncados de quedarse en el, le\nsiguieron.\"--Cabrera, Filipe Segundo, loc. cit.\n\n[148] Ante, vol. ii. book iv. ch. 6.\n\n[149] Vanderhammen has given a minute description of this royal galley,\nwith its pictorial illustrations. Among the legends emblazoned below\nthem, that of \"_Dolum reprimere dolo_\" savours strongly of the politic\nmonarch.--Don Juan de Austria, fol. 44-48.\n\n[150] \"Su comision fue sin limitacion ninguna; mas su libertad tan\natada, que de cosa grande ni pequena podia disponer sin comunicacion i\nparecer de los consegeros, i mandado del Rei.\"--Mendoza, Guerra de\nGranada, p. 139.\n\n[151] Ibid. p. 130 et seq.--Vanderhammen, Don Juan de Austria, fol.\n81.--Marmol, tom. i. pp. 511-513.--Villafane, Vida de Dona Magdalena de\nUlloa, p. 73.--Cabrera, Filipe Segundo, lib. ix. cap. 1.\n\n[152] \"Ya el Presidente tenia orden de su Magestad de la que se habia de\ntener en el recibimiento de su hermano.\"--Marmol, Rebelion de Granada,\ntom. ii. p. 17.\n\n[153] \"De manera que entre gala y guerra hacian hermosa y agradable\nvista.\"--Marmol, Rebelion de Granada, ubi supra.\n\n[154] \"El qual lo recibio muy bien, y con el sombrero en el mano, y le\ntuvo un rato abrazado. Y apartandose a un lado, llego el Arzobispo, y\nhizo lo mismo con el.\"--Ibid. tom. ii. p. 18.\n\n[155] \"Que no sintieron tanto dolor con oir los crueles golpes de las\narmas con que los hereges los mataban a ellos y a sus hijos, hermanos y\nparientes, como el que sienten en ver que han de ser perdonados.\"--Ibid.\np. 19.\n\nFrom this, it would seem that the love of revenge was a stronger feeling\nwith these Christian women than the love of friends.\n\n[156] \"Y mas galas y regocijos, porque estaban las ventanas de las\ncalles, por donde habia de pasar, entoldadas de panos de oro y seda, y\nmucho numero de damas y doncellas nobles en ellas, ricamente ataviadas,\nque habian acudido de toda la ciudad por verle.\"--Ibid. ubi supra.\n\n[157] Ibid. pp. 17-19.--Vanderhammen, Don Juan de Austria, fol.\n83.--Mendoza, Guerra de Granada, p. 133.\n\n[158] \"Juntamente con usar de equidad y clemencia con los que lo\nmerecieren, los que no hubieren sido tales seran castigados con\ngrandisimo rigor.\"--Marmol, Rebelion de Granada, tom. ii. p. 21.\n\n[159] Ibid. pp. 23, 24.--Vanderhammen, Don Juan de Austria, fol.\n85.--Cabrera, Filipe Segundo, lib. ix. cap. 1.--Herrera, Historia\nGeneral, tom. i. pp. 744, 745.\n\n[160] Mendoza, Guerra de Granada, p. 141.--Vanderhammen, Don Juan de\nAustria, fol. 85.--Marmol, Rebelion de Granada, tom. ii. p.\n27.--Cabrera, Filipe Segundo, lib. ix. cap. 1.\n\n[161] The historian of the Morisco rebellion tells us that these\nAfricans wore garlands round their heads, intimating their purpose to\nconquer or to die like martyrs in defence of their faith.--Marmol,\nRebelion de Granada, tom. ii. p. 73.\n\n[162] Besides a tenth of the produce of the soil, one source of his\nrevenue, we are told, was the confiscated property of such Moriscoes as\nrefused to yield him obedience. Another was a fifth of the spoil taken\nfrom the enemy.--Ibid. p. 35.--Also Mendoza, Guerra de Granada, p. 120.\n\n[163] \"Y la vuestra, ya yo os dixe que la queria para cosas mayores, y\nque asi agora yo no os embiaba a las de la guerra sino a esa ciudad a\ndar desde ella la orden en todo que combiniese: Pues y por otras\nocupaciones y cartas no lo podia hazer.\"--Carta del Rey a Don Juan de\nAustria, 10 de Mayo, 1569, MS.\n\n[164] Don John seems to have chafed under the restrictions imposed on\nhim by the king. At least we may infer so from a rebuke of Philip, who\ntells his brother that, \"though for the great love he bears him he will\noverlook such language this time, it will not be well for him to repeat\nit.\"--Ibid. 20 de Mayo, 1569, MS.\n\n[165] Vanderhammen, Don Juan de Austria, fol. 94.\n\nMarmol, with one or two vigorous _coups de pinceau_, gives the portrait\nof the marquis. \"No se podia determinar qual era en el mayor extremo, su\nesfuerzo, valentia y discrecion, o la arrogancia y ambicion de honra,\nacompanada de aspereza de condicion.\"--Rebelion de Granada, tom. ii. p.\n99.\n\n[166] Ibid. p. 73 et seq.--Vanderhammen, Don Juan de Austria, fol.\n94.--Mendoza, Guerra de Granada, p. 175 et seq.--Miniana, Historia de\nEspana, p. 377.\n\n[167] \"Quando vieron el fuerte perdido, se despenaron por las penas mas\nagrias, quiriendo mas morir hechas pedazos, que venir en poder de\nChristianos.\"--Marmol, Rebelion de Granada, tom. ii. p. 89.\n\n[168] \"Casi todos los capitanes.\"--Ibid. loc. cit.\n\n[169] The fierce encounter at Fraxiliana is given in great detail by\nMendoza (Guerra de Granada, pp. 165-169), and Marmol (Rebelion de\nGranada, tom. ii. pp. 86-90). No field of fight was better contested\nduring the war; and both historians bear testimony to the extraordinary\nvalour of the Moriscoes, worthy of the best days of the Arabian empire.\nPhilip, while he commends the generous ardour shown by the\ngrand-commander in the expedition, condemns him for having quitted his\nfleet to engage in it. \"El comendador mayor tubo buen suceso como\ndeseais, y como entiendo yo que lo merece su zelo y su intencion, mas\nsalir su persona en tierra, teniendo en vuestra ausencia el cargo de la\nmas fue cosa digna de mucha reprehension.\"--Carta del Rey a Don Juan, 25\nde Junio 1569, MS.\n\n[170] Marmol, Rebelion de Granada, tom. ii. pp. 108-111.--Ferreras,\nHist. d'Espagne, tom. x. pp. 83, 84--Cabrera, Filipe Segundo, lib. ix.\ncap. 6.\n\n[171] Mendoza, Guerra de Granada, p. 146--Marmol, Rebelion de Granada,\ntom. ii. p. 100.--Bleda (Cronica de Espana, p. 705), in the part of his\nwork, has done nothing more than transcribe the pages of Mendoza, and\nthat in so blundering a style as to mistake the date of this event by a\nmonth.\n\n[172] \"Puestos en la cuerda, con guarda de infanteria i cavalleria por\nuna i otra parte.\"--Mendoza, Guerra de Granada, p. 147.\n\n[173] \"Fue un miserable espectaculo,\" says an eyewitness; \"ver tantos\nhombres de todas edades, las cabezas baxas, las manos cruzadas y los\nrostros banados de lagrimas, con semblante doloroso y triste, viendo que\ndexaban sus regaladas casas, sus familias, su patria, y tanto bien como\ntenian, y aun no sabian cierto lo que se haria de sus cabezas.\"--Marmol,\nRebelion de Granada, tom. ii. p. 102.\n\n[174] Ibid. p. 103.--Mendoza, Guerra de Granada, p. 147. Both historians\nwere present on this occasion.\n\n[175] \"Los que salieron por todos tres mil i quinientos, el numero de\nmugeres mucho mayor.\"--Mendoza, Guerra de Granada, p. 147.\n\n[176] \"Muchos murieron por los caminos de trabajo, de cansancio, de\npesar, de hambre; a hierro, por mano de los mismos que los havian de\nguardar, robados, vendidos por cautivos.\"--Mendoza, Guerra de Granada,\np. 148.\n\n[177] \"Los enemigos de Dios,\"--the charitable phrase by which the\nMoriscoes, as well as Moors, came now to be denominated by the\nChristians.\n\n[178] Mendoza, Guerra de Granada, pp. 148-150.\n\n[179] \"Quedo grandisima lastima a los que habiendo visto la prosperidad,\nla policia, y el regalo de las casas, carmenes y guertas, donde los\nMoriscos tenian todas sus recreaciones y pasatiempos, y desde a pocos\ndias lo vieron todo asolado y destruido.\"--Marmol, Rebelion de Granada,\ntom. ii. p. 104.\n\n[180] \"Parecia bien estar sujeta aquella felicisima ciudad a tal\ndestruccion, para que se entienda que las cosas mas esplendidas y\nfloridas entre la gente estan mas aparejadas a los golpes de\nfortuna.\"--Marmol, ubi supra.\n\n[181] \"Armado de unas armas negras de la color del acero, y una celada\nen la cabeza llena de plumages, y una gruesa lanza en la mano mas recia\nque larga.\"--Marmol, Rebelion de Granada, tom. ii. p. 133.\n\n[182] \"Andaba Aben Umeya vistoso delante de todos en un caballo blanco\ncon una aljuba de grana vestida, y un turbante Turquesco en la\ncabeza.\"--Ibid. p. 134.\n\n[183] \"No temiesen el vano nombre del Marques de los Velez, porque en\nlos mayores trabajos acudia Dios a los suyos; y quando les faltase, no\nles podria faltar una honrosa muerte con las armas en las manos, que les\nestaba mejor que vivir deshonrados.\"--Ibid. p. 134.\n\n[184] \"Y apeandose del caballo, le hizo desjarretar, y se embreno en las\nsierras.\"--Ibid. loc. cit.\n\nHita commemorates the flight of the \"little king\" of the Alpujarras in\none of his ballads.--Guerras de Granada, tom. ii. p. 310.\n\n[185] Mendoza, Guerra de Granada, p. 209.--Marmol, Rebelion de Granada,\ntom. ii. p. 150.--Hita, Guerras de Granada, tom. ii. p. 233.\n\n[186] \"I tan adelante paso la desorden, que so juntaron quatrocientos\narcabuceros, i con las mechas en las serpentinas salieron a vista del\ncampo.\"--Mendoza, Guerra de Granada, p. 195.\n\n[187] Ibid. p. 198 et seq.--Marmol, Rebelion de Granada, tom. ii. p.\n146.\n\n[188] \"Que se publicase la guerra a fuego y a sangre.\"--Marmol, Rebelion\nde Granada, tom. ii. p. 160.\n\n[189] \"Vivia ya con estado de Rei, pero con arbitrio de\ntirano.\"--Mendoza, Guerra de Granada, p. 209.\n\n[190] \"Teniendo barreadas las calles del lugar de manera, que nadie\npudiese entrar en el sin ser visto o sentido.\"--Marmol, Rebelion de\nGranada, tom. ii. p. 163.\n\n[191] Mendoza, Guerra de Granada, p. 210.\n\nSuch is the Tiberius-like portrait given of him by an enemy--by one\nhowever, it may be added, who for liberal views and for discrimination\nof character was not surpassed by any chronicler of his time.\n\n[192] \"Los cuales pasaron de trescientos cincuenta, segun yo he sido\ninformado de varios Moriscos que seguian sus banderas; y de tal manera\nprocedia el reyecillo, que vino a ser odiosisimo a los suyos por sus\ncrueldades.\"--Hita, Guerras de Granada, tom. ii. p. 303.\n\n[193]\n\n    \"Que no la hay mas hermosa\n    en toda la Andalucia:\n    blanca es y colorada,\n    como la rosa mas fina;\n    Tane, danza, canta a estremo,\n    que es un encanto el oirla;\n    es moza, bella y graciosa\n    nadie vio tal en su vida.\"--Ibid. tom. ii. p. 324.\n\nThe severer pencil of Mendoza does not disdain the same warm colouring\nfor the portrait of the Morisco beauty.--Guerra de Granada, p. 213.\n\n[194] \"Muger igualmente hermosa i de linage.\"--Ibid.\n\n[195] \"Ninguno huvo que tomase las armas, ni bolviese de palabra por\nel.\"--Mendoza, Guerra de Granada, p. 217.\n\n[196] \"Ataronle las manos con un almaizar.\"--Ibid. p. 218.\n\n[197] \"El mismo se dio la buelta como le hiciesen menos mal; concerto la\nropa, cubriose el rostro.\"--Ibid. p. 219.\n\n[198] There is less discrepancy than usual in the accounts both of\nAben-Humeya's assassination and of the circumstances which led to it.\nThese circumstances have a certain Oriental colouring, which makes them\nnot the less probable, considering the age and country in which they\noccurred.--Among the different authorities in prose and verse, see\nMarmol, Rebelion de Granada, tom. ii. pp. 162-169; Mendoza, Guerra de\nGranada, pp. 212-220; Rufo, La Austriada, cantos 13, 14; Hita, Guerras\nde Granada, tom. ii. p. 337 et seq. Vanderhammen, Don Juan de Austria,\nfol. 103-105.\n\n[199] \"Con la reputacion de valiente i hombre del campo, con la\nafabilidad, gravedad, autoridad de la presencia, fue bien quisto,\nrespetado, obedecido, tenido como Rei generalmente de todos.\"--Mendoza,\nGuerra de Granada, p. 224.\n\nThis was painting him _en beau_. For a painting of an opposite\ncomplexion see Miniana, who represents him as \"audaz, perfido, suspicaz,\ny de pesimas costumbres.\" (Historia de Espana, p. 378.) Fortunately for\nAben-Aboo, the first-mentioned writer, a contemporary, must be admitted\nto be the better authority of the two.\n\n[200] \"No pude desear mas, ni contentarme con menos.\"--Marmol, Rebelion\nde Granada, tom. ii. p. 168.\n\nSee also, for the account of this martial ceremony, Mendoza, Guerra de\nGranada, p. 222.\n\n[201] Ferreras, Hist. d'Espagne, tom. x. pp. 111-118.--Marmol, Rebelion\nde Granada, tom. ii. pp. 169-189.--Mendoza, Guerra de Granada, p. 225 et\nseq.--Miniana, Hist. d'Espana, p. 378.\n\n[202] \"Desta manera quedaron levantados todos los Moriscos del Reino,\nsino los de la Hoya de Malaga i Serrania de Ronda.\"--Mendoza, Guerra de\nGranada, p. 241.\n\n[203] \"Llevando los escuderos las cabezas y las manos de los Moros en\nlos hierros de las lanzas.\"--Marmol, Rebelion de Granada, tom. ii. p.\n159.\n\nThe head of an enemy was an old perquisite of the victor--whether\nChristian or Moslem--in the wars with the Spanish Arabs. It is\nfrequently commemorated in the Moorish _romances_ as among the most\nhonourable trophies of the field, down to as late a period as the war of\nGranada. See, among others, the ballad beginning\n\n    \"A vista de los dos Reyes.\"\n\n\n[204] \"Y que salir a tales rebatos es desautoridad vuestra, siendo quien\nsois y teniendo el cargo que tenis.\"--Carta de Felipe Segundo a Don Juan\nde Austria, 30 de Setiembre, 1569, MS.\n\n[205] \"Le suplico mire que ni a quien soy, ni a la edad que tengo ni a\notra cosa alguna conviene encerrarme, cuando mas razon es que me\nmuestre.\"--Carta de Don Juan de Austria al Rey, 23 de Setiembre, 1569,\nMS.\n\n[206] \"Entendiose por Espana la fama de su ida sobre Galera, i moviose\nla nobleza della con tanto calor, que fue necesario dar al Rei a\nentender que no era con sua voluntad ir Cavalleros sin licencia a servir\nen aquella empresa.\"--Mendoza, Guerra de Granada, p. 256.\n\n[207] \"Havian las desordenes pasad tan adelante, que fue necesario para\nremediallas hacer demostracion no vista ni leida en los tiempos pasados,\nen la guerra: suspandar treinta i dos capitanes de quarenta i uno que\nhavia, con nombre de reformacion.\"--Ibid. p. 237.\n\n[208] \"Tambien la gente embiada por los senores, escogida, igual,\ndisciplinada, movidos por obligacion de virtud i deseo de acreditar sus\npersonas.\"--Ibid. p. 234.\n\n[209] \"Pusieronsele los ojos encendidos como brasa de puro\ncorage.\"--Marmol, Rebelion de Granada, tom. ii. p. 224.\n\n[210] \"Sin comer bocado en todo aquel dia se volvio a la ciudad de\nGranada.\"--Ibid. p. 225.\n\n[211] \"Y porque podria ser que ordenase al marques de los Velez que\nquedase con vos y os aconsejase, convendra en este caso que vos le\nmostreis muy buena cara y le trateis muy bien y le deis a entender que\ntomais su parecer, mas que en efecto tomeis el de los que he dicho\ncuando fuesen diferentes del suyo.\"--Carta del Rey a D. Juan de Austria,\n26 de Noviembre, 1569, MS.\n\n[212] \"Y que os goberneis como si hubiesedes visto mucha guerra y\nhalladoos en ella, que os digo que comigo y con todos ganeis harta mas\nreputacion en gobernaros desta manera, que no haciendo alguna mocedad\nque a todos nos costare caro.\"--Ibid. MS.\n\n[213] \"I que seais obedecido de toda mi gente, haciendolo yo asimismo\ncomo hijo vuestro, acatando vuestro valor i canas, i amparandome en\ntodas ocasiones de vuestros consejos.\"--Mendoza, Guerra de Granada, p.\n260.\n\n[214] \"Pues no conviene a mi edad anciana haver de ser cabo de\nesquadra.\"--Ibid. loc. cit.\n\n[215] The marquis of Los Velez was afterwards summoned to Madrid, where\nhe long continued to occupy an important place in the council of state,\napparently without any diminution of the royal favour.\n\nFor the preceding pages consult Marmol, Rebelion de Granada, tom. ii.\npp. 229-232; Mendoza, Guerra de Granada, pp. 257-260; Herrera, Hist.\nGeneral, tom. i, pp. 777, 778; Bleda, Cronica, pp. 733, 734.\n\n[216] The punning attractions of the name were too strong to be resisted\nby the ballad-makers of the day. See in particular the _romance_ (one of\nthe best, it may be added--and no great praise--in Hita's second volume)\nbeginning--\n\n    \"Mastredages marineros\n    de Huescar y otro lugar\n    han armado una Galera\n    que no la hay tal en la mar.\n    No tiene velas, ni remos,\n    y navegar, y hace mal,\"--\n\nand so on, for more stanzas than the reader will care to see.--Guerras\nde Granada, tom. ii. p. 469.\n\n[217] \"Las tenian los Moros barreadas de cincuenta en cincuenta pasos, y\nhechos muchos traveses de una parte y de otro en las puertas y paredes\nde las casas, para herir a su salvo a los que fuesen pasando.\"--Marmol,\nRebelion de Granada, tom. ii. p. 234.\n\nThe best and by far the most minute account of the topography of Galera\nis given by this author.\n\n[218] Ibid. p. 233 et seq.--Vanderhammen, Don Juan de Austria, fol. 112,\n113.--Hita, Guerras de Granada, tom. ii. p. 377 et seq.\n\nHita tells us he was not present at the siege of Galera; but he had in\nhis possession the diary of a Murcian officer named Tomas Perez de\nHevia, who served through the siege, and of whom Hita speaks as a person\nwell known for his military science. He says he has conformed implicitly\nto Hevia's journal which he commends for its scrupulous veracity.\nAccording to the judgment of some critics, the Murcian officer, if he\nmerits this encomium, may be thought to have the advantage of Hita\nhimself.\n\n[219] \"Para que los soldados se animasen al trabajo, iba delante de\ntodos a pie, y traia su haz acuestas como cada uno, hasta ponerlo en la\ntrinchea.\"--Marmol, Rebelion de Granada, tom. ii. p. 237.\n\n[220] Marmol, Rebelion de Granada, pp. 236-238.--Hevia, ap. Hita,\nGuerras de Granada, tom. ii. pp. 386, 387.--Vanderhammen, Don Juan de\nAustria, fol. 113.--Ferreras, Hist. d'Espagne, tom. x. p. 140.\n\n[221] \"Convendra por no aventurar mas gente buena que se haga todo lo\nque sea posible con las minas y artilleria, antes de venir a las\nmanos.\"--Carta del Rey a D. Juan de Austria, 6 de Febrero, 1570, MS.\n\n[222]\n\n   \"Unos llaman a Mahoma\n    otros dicen _Santiago_,\n    Otros gritan _cierra Espana,_\n    _muera el bando renegado_.\"\n\n    Romance, ap. Hita, Guerras de Granada.\n\n[223] No less than eighteen, according to Hevia. But this number,\nnotwithstanding Hita's warrant for the writer's scrupulous accuracy, is\nsomewhat too heavy a tax on the credulity of the reader.--\"Esta brava\nmora se llamaba a Zarzamodonia, era corpulenta, recia de miembros, y\nalcanzaba grandisima fuerza; se averiguo que en este dia mato ella sola\npor su mano a diez y ocho soldados, na de los peores del campo.\"--Hita,\nGuerras de Granada, tom. ii. p. 393.\n\n[224] For an account of the second assault see Mendoza, Guerra de\nGranada, pp. 264, 265; Marmol, Rebelion de Granada, tom. ii. pp.\n240-243; Vanderhammen, Don Juan de Austria, fol. 113, 114; Hevia, ap.\nHita, Guerras de Granada, tom. ii. p. 389 et seq.; Cabrera, Filipe\nSegundo, pp. 629, 630.\n\n[225] \"Yo hundire a Galera, y la asolare, y sembrare toda de sal; y por\nel riguroso filo de la espada pasaran chicos y grandes, quantos estan\ndentro, por castigo de su pertinacia, y en venganza de la sangre que han\nderramado.\"--Marmol, Rebelion de Granada, tom. ii. p. 244.\n\n[226] \"No puedo yo dejar de encargaros que le engais muy grande de que\nel no sea deservido en ese campo, ni haya las maldades y desordenes que\ndecis, que siendo tales no pueden hacer cosa buena, y asi lo procurad, y\nque no haya juramentos ni otras ofensas de Dios, que con esto el nos\nayudara y todo se hara bien.\"--Carta del Rey a D Juan de Austria, 6 de\nFebrero, 1570, MS.\n\n[227] \"Y con esa gente, segun lo que decis, mas importara estar detras\ndellos deteniendolos y castigandolos que no delante, pues para los que\nlo estan y hacen lo que deben no es menester.\"--Ibid.\n\n[228] It is singular that no one of the chroniclers gives us the name of\nthe Moorish chief who commanded in Galera. A romance of the time calls\nhim Abenhozmin.\n\n    \"Marinero que la rige\n    Sarracino es natural,\n    criado aca en nuestra Espana\n    por su mal y nuestro mal:\n    Abenhozmin ha por nombre,\n    y es hombre de gran caudal.\"\n\n    Hita, Guerras de Granada, tom. ii. p. 470.\n\n\n[229] \"Relumbrante y fortisimo morrion adornado de un penacho bello y\nelegante, sentado sobre una rica medalla de la imagen de nuestra Senora\nde la Concepcion.\"--Hevia, ap. Hita, Guerras de Granada, tom. ii. p.\n429.\n\n[230] \"Igualmente se arreo lo mejor que pado toda la caballeria, y era\ncosa digna de ver la elegancia y hermosura de un ejercito tan lucido y\ngallardo.\"--Hevia, ap. Hita, Guerras de Granada, loc. cit.\n\n[231] These anecdotes are given by Hevia, ap. Hita, Guerras de Granada,\ntom. ii. pp. 449-451.\n\n[232] \"Los quales mataron mas de quatrocientas mugeres y ninos... y ansi\nhizo matar muchos en su presencia a los alabarderos de su\nguardia.\"--Marmol, Rebelion de Granada, tom. ii. p. 248.\n\n[233] \"Duro el combate, despues de entrado el lugar, desde las ocho de\nla manana hasta las cinco de la tarde.\"--Hevia, ap. Hita, Guerras de\nGranada, tom. ii. p. 448.\n\n[234] \"Y no pararan hasta acabarlas a todas, si las quejas de los\nsoldados, a quien se quitaba el premio de la vitoria, no le movieran;\nmas esto fue quando se entendio que la villa estaba ya por nosotros, y\nno quiso que se perdonase a varon que pasase de doce anos.\"--Marmol,\nRebelion de Granada, tom. ii. p. 248.\n\n[235] \"Se cautivaron hasta otras mil y quinientas personas de mugeres y\nninos, porque a hombre ninguno se tomo con vida, habiendo muerto todos\nsin quedar uno en este dia, y en los asaltos pasados.\"--Hevia, ap. Hita,\nGuerras de Granada, tom. ii. p. 448.\n\nMarmol, while he admits that not a man was spared, estimates the number\nof women and children saved at three times that given in the text.\n\n[236] \"Si Africa llora, Espana no rie.\"\n\n[237] For the account of the final assault, as told by the various\nwriters, with sufficient inconsistency in the details, compare Marmol,\nRebelion de Granada, tom. ii. pp. 244-249; Mendoza, Guerra de Granada,\npp. 266-268; Vanderhammen, Don Juan de Austria, fol. 114, 115; Hevia,\nap. Hita, Guerras de Granada, tom. ii. p. 429 et seq.; Cabrera, Filipe\nSegundo, pp. 630, 631; Bleda, Cronica, p. 734; Ferreras, Hist.\nd'Espagne, tom. x. pp. 143, 144.\n\n[238] \"Tanto le crecia la ira, pensando en el dano que aquellos hereges\nhabian hecho.\"--Marmol, Rebelion de Granada, tom. ii. p. 248.\n\n[239] \"Solo dar gracias a Dios y a la gloriosa virgen Maria,\nencomendandoles el Catholico Rey aquel negocio, por ser de calidad, que\ndeseaba mas gloria de la concordia y paz, que de la vitoria\nsangrienta.\"--Marmol, Rebelion de Granada, tom. ii. p. 249.\n\n[240] \"Cela faict, par sa renommee qui voloit par le monde, tant des\nchrestiens que des infidelles, il fut faict general de la saincte\nligue.\"--Brantome, OEuvres, tom. i. p. 326.\n\n[241] \"Que es esto, Espanoles? de que huis? donde esta la honra de\nEspana? No teneis delante a Don Juan de Austria, vuestro capitan? de que\ntemeis? Retiraos con orden como hombres de guerra con el rostro al\nenemigo.\"--Marmol, Rebelion de Granada, tom. ii. p. 257.\n\n[242] \"Acudiendo a todas las necesidades con peligro de su persona,\nporque le dieron un escopetazo en la cabeza sobre una celada fuerte que\nllevaba, que a no ser tan buena, le mataran.\"--Ibid. p. 258.\n\n[243] Carta de D. Juan de Austria al Rey, 19 de Febrero, 1570,\nMS.--Marmol, Rebelion de Granada, tom. ii. p. 253 et seq.--Mendoza,\nGuerra de Granada, p. 273.--Villafane, Vida de Magdalena de\nUlloa.--Vanderhammen, Don Juan de Austria, fol. 116, 117.\n\n[244] \"Conforme a esto entendera V. M. la poca costancia y aficion que\ntienen a la guerra, estos que la dejan al mejor tiempo sin poderles\nreprimir galeras, ni horca ni cuantas diligencias se hacen. Y plega a\nDios que el amor de los hijos y parientes sea la causa y no miedo de los\nenemigos.\"--Carta de D. Juan de Austria al Rey, 19 de Febrero, 1570, MS.\n\n[245] Ibid.\n\n[246] \"Que cada uno ha de hacer su oficio y no el general de soldado, ni\nel soldado el de general.\"--Carta del Rey a D. Juan de Austria, 24 de\nFebrero, 1570, MS.\n\n[247] One evidence of this is afforded by the frankness of his friend,\nRuy Gomez de Silva. \"La primera,\" he writes to Don John, \"que por cuanto\nV. Ex.\u00aa esta reputado de atrevido y de hombre que quiere mas ganar\ncredito de soldado que de general, que mude este estilo y se deje\ngobernar.\"--(Carta de 4 de Marzo, 1570, MS.) It is to Don John's credit\nthat, in his reply, he thanks Ruy Gomez warmly for his admonition, and\nbegs his monitor to reprove him without hesitation, whenever he deems it\nnecessary, since, now that his guardian is gone, there is no other who\ncan take this liberty.--Carta de D. Juan de Austria a Ruy Gomez de\nSilva, MS.\n\n[248] According to Villafane, Dona Magdalena left Madrid on learning her\nhusband's illness, and travelled with such despatch that she arrived in\ntime to receive his last sighs. Hita also speaks of her presence at his\nbedside. But as seven days only elapsed between the date of the knight's\nwound and that of his death, one finds it difficult to believe that this\ncould have allowed time for the courier who brought the tidings, and for\nthe lady afterwards, whether in the saddle or litter, to have travelled\na distance of over four hundred and fifty miles, along execrable roads,\nwith much of the way lying through the wild passes of the Alpujarras.\n\n[249] \"Creemos piadosamente que el alma de D. Luis subiria al ciclo con\nel oloroso incienso que se quemo en los altares de S. Geronimo, porque\nsiempre habia empleado la vida en pelear contra enemigos de nuestra\nsanta fe, y por ultimo murio batallando con ellos como soldado\nvaleroso.\"--Hita, Guerras de Granada, tom. ii. p. 487.\n\n[250] Carta del Rey a D. Juan de Austria, 3 de Marzo, 1570, MS.\n\n[251] The letter is translated by Stirling from a manuscript, entitled\n\"Joannis Austriaci Vita, auctore Antonio Ossorio,\" in the National\nLibrary at Madrid.--See Cloister Life of Charles the Fifth (Am. ed.), p.\n286.\n\n[252] Tijola is the scene of the story, familiar to every lover of\nCastilian romance, and better suited to romance than history, of the\nMoor Tuzani and his unfortunate mistress, the beautiful Maleha. It forms\nthe most pleasing episode in Hita's second volume (pp. 523-540), and is\ntranslated with pathos and delicacy by Circourt, Hist. des Arabes\nd'Espagne, tom. iii. p. 345 et seq.\n\n[253] Marmol, Rebelion de Granada, tom. ii. pp. 290-320,\n340-346.--Vanderhammen, Don Juan de Austria, fol. 119 et seq.--Ferreras\nHist. d'Espagne, tom. x. p. 170 et seq.\n\n[254] Mendoza, Guerra de Granada, p. 271 et seq.--Marmol, Rebelion de\nGranada, tom. ii. pp. 283-289, 303-315, 321 et seq.\n\nIn a letter without date, of the duke of Sesa, forming part of a mass of\ncorrespondence which I was so fortunate as to obtain from the collection\nat Holland House, he insists on starvation as a much more effectual\nmeans of reducing the enemy than the sword. \"Esta guerra parece que no\npuede acabarse por medio mas cierto que el de la hambre que necesitara a\nlos enemigos a rendirse o perecer, y esta los acabara primero que el\nespada.\"--MS.\n\n[255] \"Con estas cosas y otras particulares que El Habaqui pidio para\nAben Aboo, y para los amigos, y para si mismo, que todas se le\nconcedieron.\"--Marmol, Rebelion de Granada, tom. ii. p. 360.\n\n[256] \"Misericordia, Senor, misericordia nos conceda vuestra Alteza en\nnombre de su Magestad, y perdon de nuestras culpas, que conocemos haber\nsido graves.\"--Ibid. p. 361.\n\n[257] The fullest account of these proceedings is to be found in Marmol,\nRebelion de Granada, tom. ii. pp. 355-362.\n\n[258] \"Predicando en los pulpitos publicamente contra la benignidad y\nclemencia que V. M. ha mandado usar con esta gente.\"--Carta de D. Juan\nde Austria al Rey, 7 de Junio, 1570, MS.\n\n[259] \"Que los religiosos que habrian de interceder con V. M. por estos\nmiserables, que cierto la mayor parte ha pecado con ignorancia, hagan su\nesfuerzo en reprender la clemencia.\"--Ibid.\n\n[260] \"The wise king,\" as Bleda tells us, \"did not forget Deza's eminent\nservices. He became one of the richest cardinals, passing the remainder\nof his days in Rome, where he built a sumptuous palace for his\nresidence.\"--(Cronica de Espana, p. 753.) Unfortunately this happy\npreferment did not take place till some time later--too late for the\npoor Moriscoes to profit by it.\n\n[261] \"Que El Habaqui habia mirado mal por el bien comun, contendandose\ncon lo que solamente Don Juan de Austria le habia querido conceder, y\nprocurando el bien y provecho para si y para sus deudos.\"--Marmol,\nRebelion de Granada, tom. ii. p. 390.\n\n[262] \"En lo que a esto toca, no tengo mas prendas que la palabra del\nHabaqui, el cual me podria enganar; pero certifico a V. M. que en su\nmanera de proceder ma paresce hombre que tracta verdad, y tal fama\ntiene.\"--Carta de D. Juan de Austria al Rey, 21 de Mayo, 1570, MS.\n\n[263] \"Que quando Aben Aboo de su voluntad no lo hiciese, le llevaria el\natado a la cola de su caballo.\"--Marmol, Rebelion de Granada, tom. ii.\np. 392.\n\n[264] \"Lo hizo ahogar secretamente, y mando echar el cuerpo en un\nmuladar envuelto en un zarzo de canas, donde estuvo mas de treinta dias\nsin saberse de su muerte.\"--Ibid. p. 393.\n\n[265] \"Que quando no quedase otro sino el en la Alpuxarra con sola la\ncamisa que tenia vestida, estimaba mas vivir y morir Moro, que todas\nquantas mercedes el Rey Filipe le podia hacer; y que fuese cierto, que\nen ningun tiempo, ni por ninguna manera, se pondria en su\npoder.\"--Marmol, Rebelion de Granada, tom. ii. p. 410.\n\n[266] It is the language of Marmol, who will not be suspected of\nexaggerating the cruelties of his countrymen. He does not seem, indeed,\nto regard them as cruelties. \"Unos enviaba el Comendador mayor a las\ngaleras, otros hacia justicia de ellos, y los mas consentia que los\nvendiesen los soldados para que fuesen aprovechados.\"--Rebelion de\nGranada, tom. ii. p. 436.\n\n[267] Ibid. p. 433.\n\n[268] Circourt gives a precise enumeration of the fortresses in\ndifferent districts of the country.--Hist. des Arabes d'Espagne, tom.\niii. pp. 135, 136.\n\n[269] \"Llevando cerca de si a su hijo, mozo quasi de trece anos Don Luis\nPonce de Leon, cosa usada en otra edad en aquella Casa de los Ponces de\nLeon, criarse los muchachos peleando con los Moros, i tener a sus padres\npor maestros.\"--Mendoza, Guerra de Granada, p. 318.\n\n[270] For the celebrated description of this event by Mendoza, see\nGuerra de Granada, pp. 301, 302. The Castilian historian, who probably\nborrowed the hint of it from Tacitus (Annales, lib. i. sec. 31), has\npainted the scene with a consummate art that raises him from the rank of\nan imitator to that of a rival. The reader may find a circumstantial\naccount of Alonso de Aguilar's disastrous expedition, in 1501, in the\nHistory of Ferdinand and Isabella, part ii. chap. 7.\n\n[271] Mendoza, Guerra de Granada, pp. 298-314.--Marmol, Rebelion de\nGranada, tom. ii. pp. 425-431.\n\n[272] Circourt quotes a remarkable passage from the _Ordenanzas de\nGranada_, which well illustrates the _conscientious manner_ in which the\ngovernment dealt with the Moriscoes. It forms the preamble of the law of\nFebruary 24, 1571. \"The Moriscoes who took no part in the insurrection\nought not to be punished. We should not desire to injure them; but they\ncannot hereafter cultivate their lands; and then it would be an endless\ntask to attempt to separate the innocent from the guilty. We shall\nindemnify them certainly. Meanwhile their estates must be confiscated,\nlike those of the rebel Moriscoes.\"--Hist. des Arabes d'Espagne, tom.\niii. p. 148.\n\n[273] \"Que las casas fuesen y estuviesen juntas; porque aunque lo\nmerecian poco, quiso su Magestad que se les diese esto\ncontento.\"--Marmol, Rebelion de Granada, tom. ii. p. 439.\n\n[274] \"Saquearon los soldados las casas del lugar, y tomaron todas las\nmugeres por esclavas; cosa que dio harta sospecha de que la desorden\nhabia nacido de su cudicia.\"--Ibid. p. 444.\n\nThe better feelings of the old soldier occasionally--and it is no small\npraise, considering the times--triumph over his national antipathies.\n\n[275] For the removal and dispersion of the Moriscoes, see Marmol,\nRebelion de Granada, tom. ii. pp. 437-444; Ferraras, Hist. d'Espagne,\ntom. x. pp. 227, 228; Vanderhammen, Don Juan de Austria, fol. 126.\n\nIt may well seem strange that an event of such moment as the removal of\nthe Moriscoes should have been barely noticed, when indeed noticed at\nall, by the general historian. It is still more strange that it should\nhave been passed over in silence by a writer like Mendoza, to whose\nnarrative it essentially belonged, and who could bestow thirty pages of\nmore on the expedition into the Serrania de Ronda. But this was a tale\nof Spanish glory. The haughty Castilian chronicler held the race of\nunbelievers in too great contempt to waste a thought on their\ncalamities, except so far as they enabled him to exhibit the prowess of\nhis countrymen.\n\n[276] \"Querria tambien que alla se entendiese que ya no soy mochacho, y\nque puedo, a Dios gracias, comenzar en alguna manera a volar sin alas\najenas, y sospecho ques ya tiempo de salir de panales.\"--Carta de D.\nJuan de Austria a Ruy Gomez de Silva, 16 de Mayo, 1570, MS.\n\n[277] \"No teniendo el lugar y auctoridad que ha de tener hijo de tal\npadre, y hermano de tal hermano.\"--Ibid., 4 de Junio, 1570, MS.\n\n[278] Marmol, Rebelion de Granada, tom. ii. pp. 449-454.--Mendoza,\nGuerra de Granada, pp. 324-327.--Bleda, Cronica de Espana, p.\n752.--Herrera, Historia General, tom. i. p. 781.--Vanderhammen, Don Juan\nde Austria, fol. 123.\n\n[279] \"Esta es la cabeza del traidor de Abenabo. Nadie la quite so pena\nde muerte.\"--Mendoza, Guerra de Granada, p. 329.--Marmol, Rebelion de\nGranada, tom. ii. pp. 455, 456--Bleda, Cronica de Espana, p.\n752.--Miniara, Hist. de Espana, p. 383.\n\n[280] Ante, p. 40.\n\n[281] Nueva Recopilacion, lib. viii. tit. ii. ley 19.\n\n[282] \"Si estos tales que se huyieren huydo, y ausentado fueren hallados\nen el dicho Reyno de Granada, o dentro de diez leguas cercanas a el,\ncaygan e incurran en pena de muerte que sea en sus personas\nexecutada.\"--Ibid. ubi supra.\n\n[283] Nueva Recopilacion, lib. viii. tit. ii. ley 19.\n\n[284] Examples of this are cited by Circourt, Hist. des Arabes\nd'Espagne, tom. iii. pp. 150, 151.\n\n[285] Ibid. p. 163.\n\nM. de Circourt has collected, from some authentic and not very\naccessible sources, much curious information relative to this part of\nhis subject.\n\n[286] Ferreras, Hist. d'Espagne, tom. x. p. 227.\n\n[287] \"Ils representerent que ce recensement allait leur reveler la\nsecret de leur nombre effrayant; qu'ils fourmillaient.\"--Circourt, Hist.\ndes Arabes d'Espagne, tom. iii. p. 164.\n\n[288] \"Qu'ils accapareaint tous les metiers, teut le commerce.\"--Ibid.\nloc. cit.\n\n[289] Ferreras, Hist. d'Espagne, tom. x. pp. 239, 240.--Cabrera, Filipe\nSegundo, p. 641.--Zuniga, Anales de Sevilla, pp. 536-538.\n\nThe chroniclers paint in glowing colours the splendours of the royal\nreception at Seville, which, enriched by the Indian trade, took its\nplace among the great commercial capitals of Christendom in the\nsixteenth century. It was a common saying,\n\n    \"Quien no ha visto a Sevilla\n     No ha visto a maravilla.\"\n\n\n[290] Herrera, Historia General, tom. i. p. 798 et seq.--Cabrera, Filipe\nSegundo, lib. vi. cap. 17.--Sagredo, Monarcas Othomanos, p. 277.\n\n[291] Cabrera, Filipe Segundo, p. 667.--Sagredo, Monarcas Othomanos, p.\n277.\n\n[292] A copy of the treaty in Latin, dated May 25, 1571, exists in the\nlibrary of the Academy of History, at Madrid. Senor Rosell has\ntransferred it to the appendix of his work, Historia del Combate Naval\nde Lepanto (Madrid, 1853), pp. 180-189.\n\n[293] A copy from the first draft of the treaty, as prepared in 1570, is\nincorporated in the Documentos Ineditos (tom. iii. pp. 337 et seq.). The\noriginal is in the library of the duke of Ossuna.\n\n[294] Rosell, Combate Naval de Lepanto, p. 56.\n\n[295] Paruta, Guerra di Cipro, p. 120 et seq.--Herrera, Hist. General,\ntom. ii. pp. 14, 15.\n\n[296] Cabrera, Filipe Segundo, lib. ix. cap. 22.--Ferreras, Hist.\nd'Espagne, tom. x. pp. 247, 248.--Vanderhammen, Don Juan de Austria,\nfol. 152.\n\n[297] \"No poco se maravillaron los curiosos, viendole, o por casualidad\no bien de intento, terciar llanamente en la conversacion, contra las\netiquetas hasta entonces observadas.\"--Rosell, Combate Naval de Lepanto,\np. 59.\n\n[298] \"Y concede dozientos anos de perdon a los\npresentes.\"--Vanderhammen, Don Juan de Austria, fol. 152.\n\n[299] \"_De las mejores que jamas se han visto_,\"--\"among the best\ngalleys that were ever seen,\"--says Don John in a letter, from Messina,\nto Don Garcia de Toledo.--Documentos Ineditos, tom. iii. p. 15.\n\nThe earlier part of the third volume of the Documentos Ineditos is taken\nup with the correspondence between John of Austria and Garcia de Toledo,\nin which the former asks information and advice in respect to the best\nmode of conducting the war. Don Garcia de Toledo, fourth marquis of\nVillafranca, was a man of high family, and of great sagacity and\nexperience. He had filled some of the highest posts in the government,\nand, as the reader may remember, was viceroy of Sicily at the time when\nMalta was besieged by the Turks. The coldness which on that occasion he\nappeared to show to the besieged, excited general indignation; and I\nventured to state, on an authority which I did not profess to esteem the\nbest, that in consequence of this he fell into disgrace, and was\nsuffered to pass the remainder of his years in obscurity. (Ante, vol.\nii. circ. fin.) An investigation of documents which I had not then seen\nshows this to have been an error. The ample correspondence which both\nPhilip the Second and Don John carried on with him, gives undeniable\nproofs of the confidence which he continued to enjoy at court, and the\nhigh deference which was paid to his opinion.\n\n[300] Authorities differ as usual as to the precise number both of\nvessels and troops. I have accepted the estimate of Rosell, who\ndiscreetly avoids the extremes on either side.\n\n[301] Vanderhammen has been careful to transcribe this precious\ncatalogue.--Don Juan de Austria, fol. 156 et seq.\n\n[302] Ibid. fol. 159 et seq.--Ferreras, Hist. d'Espagne, tom. x. p.\n251.--Herrera, Hist. General, tom. ii. p. 15 et seq.\n\n[303] \"Luego su Alteza, el Coro, y Pueblo dixeron con musica, vozes, y\nalegria; Amen.\"--Vanderhammen, Juan de Austria, fol. 159.\n\n[304] For a minute account of these arches and their manifold\ninscriptions, see Vanderhammen, Don Juan de Austria, fol. 160-162.\n\n[305] Rosell, Combate Naval de Lepanto, p. 84.\n\n[306] Don John, in his correspondence with his friend Don Garcia de\nToledo, speaks with high disgust of the negligence shown in equipping\nthe Venetian galleys. In a letter dated Messina, August 30, he says:\n\"Poneme cierta congoja ver que el mundo me obliga a hacer alguna cosa de\nmomento, contando las galeras pro numero y no por cualidad.\"--Documentos\nIneditos, tom. iii. p. 18.\n\n[307] Rosell, Combate Naval de Lepanto, p. 82.\n\nThe clearest and by far the most elaborate account of the battle of\nLepanto is to be found in the memoir of Don Cayetan Rosell, which\nreceived the prize of the Royal Academy of History of Madrid, in 1853.\nIt is a narrative which may be read with pride by Spaniards, for the\nminute details it gives of the prowess shown by their heroic ancestors\non that memorable day. The author enters with spirit into the stormy\nscene he describes. If his language may be thought sometimes to betray\nthe warmth of national partiality, it cannot be denied that he has\nexplored the best sources of information, and endeavoured to place the\nresult fairly before the reader.\n\n[308] Torres y Aguilera, Chronica de Guerra que ha acontescido en Italia\ny partes de Levante y Berberia desde 1570 en 1574 (Caragoca, 1579), fol.\n54.--Vanderhammen, Don Juan de Austria, fol. 165 et seq.--Cabrera,\nFilipe Segundo, lib. lx. cap. 23.\n\n[309] Torres y Aguilera, Chronica, fol. 64.--Vanderhammen, Don Juan de\nAustria, fol. 173.--Paruta, Guerra di Cipro, p. 149.--Relacion de la\nBatalla Naval que entre Christianos y Turcos hubo el ano 1571, MS.--Otra\nRelacion, Documentos Ineditos, tom. iii. p. 365.\n\n[310] Paruta, Guerra di Cipro, pp. 143, 144.--\"Despues hizo que lo\ndegollassen vivo, y lleno el pellejo de paja lo hizo colgar de la entena\nde una galeota, y desta manera lo llevo pol toda la ribera de la\nSuria.\"--Torres y Aguilera, Chronica, fol. 45.\n\n[311] Ibid. fol. 44, 45.--Paruta, Guerra di Cipro, pp.\n130-144.--Sagredo, Monarcas Othomanos, pp. 283-289.\n\n[312] Torres y Aguilera, Chronica, fol. 65.--Documentos Ineditos, tom.\niii. p. 241.--Rosell, Historia del Combate Naval, pp. 93, 94.\n\n[313] Torres y Aguilera, Chronica, fol. 53.--Herrera, Hist. General,\ntom. ii. p. 30.--Relacion de la Batalla Naval, MS.--Rosell, Historia del\nCombate Naval, pp. 95, 99, 100.\n\n[314] Torres y Aguilera, Chronica, fol. 67 et seq.--Relacion de la\nBatalla Naval, MS.--Otras Relaciones, Documentos Ineditos, tom. iii. pp.\n242, 262.\n\n[315] Most of the authorities notice this auspicious change of the wind.\nAmong others, see Relacion de la Batalla Naval, MS.; Relacion escrita\npor Miguel Servia, confesor de Don Juan, Documentos Ineditos, tom. xi.\np. 368: Torres y Aguilera, Chronica, fol. 75. The testimony is that of\npersons present in the action.\n\n[316] Amidst the contradictory estimates of the number of the vessels\nand the forces to the Turkish armada to be found in the different\nwriters, and even in official relations, I have conformed to the\nstatement given in Senor Rosell's _Memoria_, prepared after a careful\ncomparison of the various authorities.--Historia del Combate Naval, p.\n94.\n\n[317] \"Si hoy es vuestro dia, Dios os lo de; pero estad ciertos que si\ngano la jornada, os dare libertad: por lo tanto haced lo que debeis a\nlas obras que de mi habeis recebido.\"--Rosell, Historia del Combate\nNaval, p. 101.\n\nFor the last pages see Paruta, Guerra di Cipro, pp. 150, 151; Sagrado,\nMonarcas Othomanos, p. 292; Torres y Aguilera, Chronica, fol. 65, 66;\nRelacion de la Batalla Naval, MS.\n\n[318] This fact is told by most of the historians of the battle. The\nauthor of the manuscript so often cited by me further says, that it was\nwhile the fleet was thus engaged in prayer for aid from the Almighty\nthat the change of wind took place. \"Y en este medio, que en la oracion\nse pedia a Dios la victoria, estaba el mar alterado de que nuestra\narmada recibia gran dano y antes que se acabase la dicha oracion el mar\nestuvo tan quieto y sosegado que jamas se a visto, y fue fuerca a la\narmada enemiga amainar y venir al remo.\"\n\n[319] Torres y Aguilera, Chronica, fol. 71.--Paruta, Guerra di Cipro, p.\n156.--Cabrera, Filipe Segundo, p. 688.--Relacion de la Batalla Naval,\nMS.--Otra Relacion, Documentos Ineditos, tom. xi. p. 368.\n\nThe inestimable collection of the Documentos Ineditos contains several\nnarratives of the battle of Lepanto by contemporary pens. One of these\nis from the manuscript of Fray Miguel Servia, the confessor of John of\nAustria, and present with him in the engagement. The different\nnarratives have much less discrepancy with one another than is usual on\nsuch occasions.\n\n[320] Torres y Aguilera, Chronica, fol. 72.--Relacion de la Batalla\nNaval, MS.\n\nThe last-mentioned manuscript is one of many left us by parties engaged\nin the fight. The author of this relation seems to have written it on\nboard one of the galleys, while lying at Petala, during the week after\nthe engagement. The events are told in a plain, unaffected manner, that\ninvites the confidence of the reader. The original manuscript, from\nwhich my copy was taken, is to be found in the library of the University\nof Leyden.\n\n[321] A minute description of the Ottoman standard, taken from a\nmanuscript of Luis del Marmol, is given in the Colleccion de Documentos\nIneditos, tom. iii. pp. 270 et seq.\n\n[322] Documentos Ineditos, tom. iii. p. 265; tom. xi. p. 368.--Torres y\nAguilera, Chronica, fol. 70.--Paruta, Guerra di Cipro, pp. 156,\n157.--Relacion de la Batalla Naval, MS.\n\n[323] Herrera notices one galley, \"_La Piamontesa de Saboya_ degollada\nen ella toda la gente de cabo y remo y despedazado con once heridas D.\nFrancisco de Saboya.\" Another, \"_La Florencia_,\" says Rosell, \"perdio\ntodos los soldados, chusma, galeotes y caballeros de San Esteban que en\nella habia, excepto su capitan Tomas de Medicis y diez y seis hombres\nmas, aunque todos heridos y estropeados.\"--Historia del Combate Naval,\np. 113.\n\n[324] \"Tomo una Alabarda o Pertesana, y ligando en ella el Sancto\nCrucifixo, verdadera pendon, se puso delante de todos assi desarinado\ncomo estava, y fue el primero que entro en la Galera Turquesca, haziendo\ncon su Alabarda cosas que ponian admiracion.\"--Torres y Aguilera,\nChronicas, fol. 75.\n\n[325] \"Vivio hasta que sabiendo que la vitoria era ganada dijo: que daba\ngracias a Dios que lo hubiese guardado tanto que viese vencida la\nbatalla y roto aquel comun enemigo que tanto deseo ver\ndestruido.\"--Herrera, Relacion de la Guerra de Cipro, Documentos\nIneditos, tom. xxi. p. 360.\n\n[326] Relacion de la Batalla Naval, MS.--Herrera, Hist. General, tom.\nii. p. 33.--Paruta, Guerra di Cipro, pp. 157, 158.--Documentos Ineditos,\ntom. iii. p. 244.\n\nTorres y Aguilera tells a rather extraordinary anecdote respecting the\ngreat standard of the League in the _Real_. The figure of Christ\nemblazoned on it was not hit by ball or arrow during the action,\nnotwithstanding every other banner was pierced in a multitude of places.\nTwo arrows, however, lodged on either side of the crucifix, when a\nmonkey belonging to the galley ran up the mast, and, drawing out the\nweapons with his teeth, threw them overboard! (Chronica, fol. 75)\nConsidering the number of ecclesiastics on board the fleet, it is\nremarkable that no more miracles occurred on this occasion.\n\n[327] Torres y Aguilera, Chronica, fol. 72 et seq.--Relacion de la\nBatalla Naval, MS.--Vanderhammen, Don Juan de Austria, fol.\n182.--Documentos Ineditos, tom. iii. p. 247 et seq.--Paruta, Guerra di\nCipro, p. 160.--Cabrera, Filipe Segundo, lib. ix. cap. 25, 26.\n\n    \"Do el estandarte barbaro abatido\n    la Cruz del Redentor fue enarbolada\n    con un triunfo solene y grande gloria,\n    cantando abiertamente la vitoria.\"\n\n    Ercilla, La Araucana, par. ii. canto 24.\n\n\n[328] The loss of the Moslems is little better than matter of\nconjecture, so contradictory are the authorities. The author of the\nLeyden MS. dismisses the subject with the remark, \"La gente muerta de\nTurcos no se ha podido saber por que la que se hecho en la mar fuera de\nlos degollados fueron infinitos.\" I have conformed, as in my other\nestimates, to those of Senor Rosell, Historia del Combate Naval, p. 118.\n\n[329] Rosell computes the total loss of the allies at not less than\nseven thousand six hundred; of whom one thousand were Romans, two\nthousand Spaniards, and the remainder Venetians.--Ibid. p. 113.\n\n[330] Ibid. ubi supra.--Torres y Aguilera, Chronica, fol. 74 et\nseq.--Documentos Ineditos, tom. iii. pp. 246-249; tom. xi. p.\n370.--Sagredo, Monarcas Othomanos, pp. 295, 296.--Relacion de la Batalla\nNaval, MS.\n\n[331] Relacion de la Batalla Naval, MS.\n\nDon John notices this achievement of his gallant kinsman in the first\nletter which he wrote to Philip after the action. The letter, dated at\nPetala, October 10, is published by Aparici, Documentos Ineditos\nrelativos a la Batalla de Lepanto, p. 26.\n\n[332] Navarrete, Vida de Cervantes (Madrid, 1819), p. 19.\n\nCervantes, in the prologue to the second part of \"Don Quixote,\" alluding\nto Lepanto, enthusiastically exclaims, that, for all his wounds, he\nwould not have missed the glory of being present on that day. \"Quisiera\nantes haberme hallado en aquella faccion prodigiosa, que sano ahora de\nmis heridas, sin haberme hallado en ella.\"\n\n[333] This humane conduct of Don John is mentioned, among other writers,\nby the author of the Relacion de la Batalla Naval, whose language shows\nthat his manuscript was written on the spot: \"El queda visitando los\nheridos y procurando su remedio haziendoles merced y dandoles todo lo\nque aviase menester.\"--MS.\n\n[334] \"Lo qual toda esta corte tuvo a gran gentileza, y no hazen sino\nalabar la virtud y grandeza de vuestra Alteza.\"\n\nThe letter of Fatima is to be found in Torres y Aguilera, Chronica (fol.\n92). The chronicler adds a list of the articles sent by the Turkish\nprincess to Don John, enumerating, among other things, robes of sable,\nbrocade, and various rich stuffs, fine porcelain, carpets, and tapestry,\nweapons curiously inlaid with gold and silver, and Damascus blades\nornamented with rubies and turquoises.\n\n[335] \"El presente que me embio dexe de rescibir, y le huvo el mismo\nMahamet Bey, no por no preciarle como cosa venida de su mano, sino por\nque la grandeza de mis antecessores no acostumbra rescibir dones de los\nnecessitados de favor, sino darios y hazeries gracias.\"--Ibid. fol. 94.\n\n[336] According to some, Don John was induced, by the persuasion of his\nfriends, to make these advances to the Venetian admiral. (See Torres y\nAguilera, Chronica, fol. 75; Vanderhammen, Don Juan de Austria, fol.\n123.) It is certain he could not erase the memory of the past from his\nbosom, as appears from more than one of his letters, in which he speaks\nof the difficulty he should find, in another campaign, in acting in\nconcert with a man of so choleric a temper. In consequence the Venetian\ngovernment was induced, though very reluctantly, to employ Veniero on\nanother service. In truth, the conduct which had so much disgusted Don\nJohn and the allies seems to have found favour with Veniero's\ncountrymen, who regarded it as evidence of his sensitive concern for the\nhonour of his nation. A few years later they made ample amends to the\nveteran for the slight put on him, by raising him to the highest dignity\nin the republic. He was the third of his family who held the office of\ndoge, to which he was chosen in 1576, and in which he continued till his\ndeath.\n\n[337] The spoil found on board the Turkish ships was abandoned to the\ncaptors. There was enough of it to make many a needy adventurer rich.\n\"Assi por la victoria havida como porque muchos venian tan ricos y\nprosperados que no havia hombre que se preciasse de gastar moneda de\nplata sino Zequies, ni curasse de regatear en nada que\ncomprasse.\"--Torres y Aguilera, Chronica, fol. 79.\n\n[338] For the preceding pages see Vanderhammen, Don Juan de Austria,\nfol. 186; Torres y Aguilera, Chronica, fol. 79; Cabrera, Filipe Segundo,\np. 696; Herrera, Historia General, tom. ii. p. 37; Ferreras, Hist.\nd'Espagne, tom. x. p. 261.\n\n[339] An old _romance_ thus commemorates this liberal conduct of Don\nJohn:--\n\n    \"Y ansi seda como de oro\n    Ninguna cosa ha querido\n    Don Juan, como liberal,\n    Por mostrar do ha descendido,\n    Sino que entre los soldados\n    Fuese todo repartido\n    En premio de sus trabajos\n    Pues lo habian merecido.\"\n\n    Duran, Romancero General (Madrid, 1851), tom. ii. p. 185.\n\n\n[340] Lorea, Vida de Pio Quinto, cap. xxiv. Sec. ii.--Torres y Aguilera,\nChronica, fol. 80.--Rosell, Historia del Combate Naval, pp. 124, 125.\n\n[341] Philip, in a letter to his brother, dated from the Escorial in the\nfollowing November, speaks of his delight at receiving this trophy from\nthe hands of Figueroa. (See the letter, ap. Rosell, Hist. del Combate\nNaval, Apend. No. 15.) The standard was deposited in the Escorial, where\nit was destroyed by fire in the year 1671.--Documentos Ineditos tom.\niii. p. 256.\n\n[342] \"Y S. M. no se altero, ni demudo, ni hizo sentimiento alguno, y se\nestuvo con el semblante y serenidad que antes estaba, con el qual\nsemblante estuvo hasta que se acabaron de cantar las\nvisperas.\"--Memorias de Fray Juan de San Geronimo, Documentos Ineditos,\ntom. iii. p. 258.\n\n[343] The third volume of the Documentos Ineditos contains a copious\nextract from a manuscript in the Escorial written by a Jeronymite monk.\nIn this the writer states that Philip received intelligence of the\nvictory from a courier despatched by Don John, while engaged at vespers\nin the palace monastery of the Escorial. This account is the one\nfollowed by Cabrera (Filipe Segundo, p. 696) and by the principal\nCastilian writers. Its inaccuracy, however, is sufficiently attested by\ntwo letters written at the time to Don John of Austria, one by the royal\nsecretary Alzamora, the other by Philip himself. According to their\naccount, the person who first conveyed the tidings was the Venetian\nminister; and the place where they were received by the king was the\nprivate chapel of the palace of Madrid, while engaged at vespers on\nAll-Saints eve. It is worthy of notice, that the secretary's letter\ncontains no hint of the _nonchalance_ with which Philip is said to have\nheard the tidings. The originals of these interesting despatches still\nexist in the National Library at Madrid. They have been copied by Senor\nRosell for his memoir (Apend. Nos. 13, 15). One makes little progress in\nhistory before finding that it is much easier to repeat an error than to\ncorrect it.\n\n[344] \"Y ansi a vos (despues de Dios) se ha de dar el parabien y las\ngracias della, como yo os las doy, y a mi de que por mano de persona que\ntanto me toca como la vuestra, y a quien yo tanto quiero, se haya hecho\nun tan gran negocio, y ganado vos tanta honra y gloria con Dios y con\ntodo el mundo.\"--Rosell, Historia del Combate Naval, Apend. No. 15.\n\n[345] Carta del secretario Alzamora a Don Juan de Austria, Madrid, Nov.\n11, 1571, ap. Rosell, Historia del Combate Naval, Apend. No. 13.\n\n[346] See Ford, Handbook for Spain, vol. ii. p. 697.\n\n[347] Ercilla has devoted the twenty-fourth canto of the Araucana to the\nsplendid episode of the battle of Lepanto. If Ercilla was not, like\nCervantes, present in the fight, his acquaintance with the principal\nactors in it makes his epic, in addition to its poetical merits, of\nconsiderable value as historical testimony.\n\n[348] The letter, which is dated Brussels, Nov. 17, 1571, is addressed\nto Juan de Zuniga, the Castilian ambassador at the court of Rome. A copy\nfrom a manuscript of the sixteenth century, in the library of the duke\nof Ossuna, is inserted in the Documentos Ineditos, tom. iii. pp.\n292-303.\n\n[349] \"Ya havreis entendido la orden que se os ha dado de que inverneis\nen Mecina, y las causas dello.\"--Carta del Rey a su hermano, ap. Rosell,\nHistoria del Combate Naval, Apend. No. 15.\n\n[350] See Rosell, Historia del Combate Naval, p. 157; Lafuente, Historia\nde Espana (Madrid, 1850), tom. xiii. p. 538. Ranke, who has made the\nhistory of the Ottoman empire his particular study, remarks: \"The Turks\nlost all their old confidence after the battle of Lepanto. They had no\nequal to oppose to John of Austria. The day of Lepanto broke down the\nOttoman supremacy.\"--Ottoman and Spanish Empires (Eng. tr.), p. 23.\n\n[351] \"Su Santidad ha de querer que de gane Constantinopla y la Casa\nSanta, y que tendra muchos que le querran adular con facilitarselo, y\nque no faltaran entre estos algunos quo hacen profesion de soldados y\nque como su Beatitud no pueden entender estas cosas.\"--Carta del Duque\nde Alba, ap. Documentos Inedites, tom. iii. p. 300.\n\n[352] Ranke, History of the Popes (Eng. tr.), vol i. p. 384.\n\n[353] Lafuente, Historia de Espana, tom. xiii. p. 530.\n\n[354] \"Breves de fuego.\"--Ibid, p. 529.\n\n[355] \"E si e veduto, che quando gli fu data la gran rotta, in sei mesi\nrifabbrico canto venti galere, oltre quelle che si trovavano in essere,\ncosa che essendo preveduta e scritta da me, fu giudicata piuttosto\nimpossibile che creduta.\"--Relazione di Marcantino Barbaro 1573, Alberi,\nRelazioni Venete, tom. iii, p. 306.\n\n[356] For the preceding pages see Torres y Aguilera, Chronica, fol.\n87-89; Cabrera, Filipe Segundo, lib. x. cap. 5; Vanderhammen, Don Juan\nde Austria, fol. 159 et seq.; Paruta, Guerra di Cipro, p. 206 et seq.;\nSagredo, Monarcas Othomanos, pp. 301, 302.\n\n[357] It is Voltaire's reflection: \"Il semblait que les Turques eussent\ngagne la bataille de Lepante.\"--Essais sur les Moeurs, chap. 160.\n\n[358] The treaty is to be found in Dumont, Corps Diplomatique, tom. v.\npar. 1 pp. 218, 219.\n\n[359] Rosell, Historia del Combate Naval, p. 149.--Cabrera, Filipe\nSegundo, p. 747.--Torres y Aguilera, Chronica, fol. 95.\n\n[360] Vanderhammen, Don Juan de Austria, fol. 172.\n\n[361] Cabrera, Filipe Segundo, p. 765.--Vanderhammen, Don Juan de\nAustria, fol. 174, 175--Torres y Aguilera, Chronica, fol. 103 et\nseq.--The author last cited who was present at the capture of Tunis,\ngives a fearful picture of the rapacity of the soldiers.\n\n[362] The Castilian writers generally speak of it as the _peremptory\ncommand_ of Philip. Cabrera, one of the best authorities, tells us:\n\"Mandio el Rey Catolico a Don Juan de Austria enplear su armada en la\nconquista de Tunez, i que le desmantelase, i la Goleta.\" But soon after\nhe remarks: \"Olvidando el _buen acuerdo_ del Rey, por consejo de\nlisongeros determino de conservar la ciudad.\" (Filipe Segundo, pp. 763,\n764.) From this qualified language we may infer that the king meant to\ngive his brother his decided opinion, not amounting, however, to such an\nabsolute command as would leave him no power to exercise his discretion\nin the matter. This last view is made the more probable by the fact that\nin the following spring a correspondence took place between the king and\nhis brother, in which the former, after stating the arguments both for\npreserving and for dismantling the fortress of Tunis, concludes by\nreferring the decision of the question to Don John himself.\n\"Representadas todas estas dificultades, manda remitir S. M. al Senor\nDon Juan que el tome la resolucion que mas convenga.\"--Documentos\nIneditos, tom. iii. p. 139.\n\n[363] \"Porque la gentileza de la tierra i de las damas en su\nconservacion agradaba a su gallarda edad.\"--Cabrera, Filipe Segundo, p.\n755.--Also Vanderhammen, Don Juan de Austria, fol. 176.\n\n[364] Ferreras, Hist. d'Espagne, tom. x. p. 286.--Vanderhammen, Don Juan\nde Austria, fol. 178.\n\n[365] Torres y Aguilera, Chronica, fol. 116 et seq.--Relacion particular\nde Don Juan Sanogera, MS.\n\nVanderhammen states the loss of the Moslems at thirty-three thousand\nslain. (Don Juan de Austria, fol. 189.) But the arithmetic of the\nCastilian is little to be trusted as regards the infidel.\n\n[366] For a brief but very perspicuous view of the troubles of Genoa,\nsee San Migual, Hist. de Flipe Segundo (tom. ii. cap. 36). The care of\nthis judicious writer to acquaint the reader with contemporary events in\nother countries, as they bore more or less directly on Spain, is a\ncharacteristic merit of his history.\n\n[367] Torres y Aguilera, Chronica, fol. 113.\n\n[368] The principal cause of Granvelle's coldness to Don John, as we are\ntold by Cabrera (Filipe Segundo, p. 794), echoed, as usual, by\nVanderhammen (Don Juan de Austria, fol. 184), was envy of the fame which\nthe hero of Lepanto had gained by his conquests both in love and in war.\n\"La causa principal era el poco gasto que tenia de acudir a Don Juan,\ninvidioso de sus favores de Marte i Venus.\" Considering the cardinal's\nprofession, he would seem to have had no right to envy any one's success\nin either of these fields.\n\n[369] \"Questa oppinione, che di lui si ha, rende le sue leggi piu\nsacrosancte et inviolabili.\"--Relazione di Contarini, MS.\n\n[370] A manuscript, entitled \"_Origen de los Consejos_,\" without date or\nthe name of the author, in the library of Sir Thomas Phillips, gives a\nminute account of the various councils under Philip the Second.\n\n[371] \"Sono XI.; il consiglio dell' Indie, Castiglia, d'Aragona,\nd'inquisitione, di camera, dell' ordini, di guerra, di hazzienda, dl\ngiustizia, d'Italia, et di stato.\"--Sommario del' ordine che si tiene\nalla corte di Spagna circa il governo delli stati del Re Catholico, MS.\n\n[372] Ibid. The date of this manuscript is 1570.\n\n[373] Relazione di Badoer, MS.\n\n[374] Instead of \"Ruy Gomez,\" Badoer tells us they punningly gave him\nthe title of \"Rey Gomez,\" to denote his influence over the king. \"Il\ntitolo principal che gli vien dato e di Rey Gomez e non Ruy Gomez,\nperche pare che non sia stato mai alcun privato con principe del mondo\ndi tanta autorita e cosi stimato dal signor suo come egli e da questa\nMaesta.\"--Relazione, MS.\n\n[375] Cabrera, Filipe Segundo, pp. 712, 713.\n\nCabrera has given us, in the first chapter of the tenth book of his\nhistory, a finished portrait of Ruy Gomez, which for the niceness of its\ndiscrimination and the felicity of its language may compare with this\nbest compositions of the Castilian chroniclers.\n\n[376] \"El senor Ruy Gomez no fue de los mayores consejeros que ha\nhabido, pero del humor y natural de los reyes le roconozco por tan gran\nmaestro, que todos los que por aqui dentro andamos tenemos la cabeza\ndonde pensamos que traemos los pies.\"--Bermudez de Castro, Antonio Perez\n(Madrid, 1841), p. 28.\n\n[377] \"Fue Rui Gomez el primero piloto que en trabajos tan grandes vivio\ny murio seguro, tomando sienpre el mejor puerto.\"--Cabrera, p. 713.\n\n[378] \"Vivo conservo la gracia de su Rey, muerto le dolio su falta, i la\nlloro su Reyno, que en su memoria le a conservado paro exemplo de fieles\nvasallos i prudentes privados de los mayores Principes.\"--Ibid. ubi\nsupra.\n\n[379] \"Puede ser, pero el Cardenal Espinosa me consulto en saliendo del\nconsejo, i provei la placa.\"--Cabrera, p. 700.\n\n[380] \"Que en principe tan zeloso de su immunidad i oficio parecio\nincreible su tolerancia hasta alli.\"--Ibid. ubi supra.\n\n[381] The anonymous author of a contemporary relation speaks of the king\nas a person little subject to passions of any kind. The language is\nstriking: \"E questo Re poco soggetto alle pasioni, venga cio, o per\ninclinazione naturale, o per costume; e quasi non appariscono in lui i\nprimi movimenti ne dell' allegrezza, ne del dolore, ne dell' ira\nancora.\"--MS.\n\n[382] \"El Rey le hablo tan asperamente sobre el afinar una verdad, que\nle mato brevemente,\" says Cabrera emphatically.--Filipe Segundo, p. 699.\n\n[383] \"Perche chi vuole il favore del duoa d'Alva perde quello di Ruy\nGomez, e chi cerca il favore di Ruy Gomez, non ha quello del duca\nd'Alva.\"--Relazione di Soriano, MS.\n\n[384] Ranke has given some pertinent examples of this in an interesting\nsketch which he has presented of the relative positions of these two\nstatesmen in the cabinet of Philip.--Ottoman and Spanish Empires (Eng.\ntrans.), p. 38.\n\n[385] \"Non si trova mai S.M. presente alle deliberationi ne i consigli,\nma deliberato chiama una delle tre consulte.... alla qual sempre si\nritrova, onde sono lette le risolutioni del consiglio.\"--Relazione di\nTiepolo, MS.\n\n[386] Ranke, Ottoman and Spanish Empires, p. 32.\n\n[387] \"El dia que iva a caca bolvia con ansias de bolver al trabajo,\ncomo un oficial pobre que huviera de ganar la comida con ello.\"--Los\nDichos y Hechos, del Rey Phelipe II. (Brusselas, 1666), p. 214.--See\nalso Relazione di Pigafetta, MS.\n\n[388] Relazione di Vandramino, MS.--Relazione di Contarini, MS.\n\n\"Distribuia las horas del dia, se puede decir, todas en los negocios,\nquando yo lo conoci; porque aunque las tenia de ocio u ocupaciones\nforcosas de su persona, las gastava con tales criados elegidos tan a\nproposito que quanto hablava venia a ser informarse mucho, descanso en\nlo que a otro costara nota y fatiga.\"--MS. Anon. in the Library of the\ndukes of Burgundy.\n\n[389] Dichos y Hechos de Phelipe II., pp. 339, 340.\n\n[390] \"A estos estando turbados, y desalentados, los animava\ndiziendoles, Sossegaos.\"--Dichos y Hechos de Phelipe II., p. 40.\n\n[391] \"Diziendole si lo traeis escrito, lo vere, y os hare\ndespachar.\"--Ibid. p. 41.\n\n[392] \"Quando esce di Palazzo, suole montare in un cocchio coperto di\ntela incerata, et serrata a modo che non si vede..... Suole quando va in\nvilla ritornare la sera per le porte del Parco, senza esser veduto da\nalcuno.\"--Relazione di Pigafetta, MS.\n\n[393] Ranke, Ottoman and Spanish Empires, p. 32.\n\nInglis speaks of seeing this work in the library when he visited the\nEscorial.--Spain in 1830, vol. i. p. 348.\n\n[394] Ranke, Ottoman and Spanish Empires, p. 33.\n\n[395] See ante, vol. ii. circ. fin.\n\n[396] Lafuente, Historia de Espana, tom. xiv. p. 44.\n\nThe historian tells us he has seen the original letter with the changes\nmade in it by Philip.\n\n[397] \"Chi comincia a servirlo puo tener per certa la remunerazione, se\nil difetto non vien da rei.\"--Relazione Anon. MS.\n\n[398] Relazione della Corte di Spagna, MS.--Relazione di Badoer,\nMS.--Etiquetas de Palacio, MS.\n\n[399] Relazione di Badoer, MS.\n\n[400] \"Ha tre guardie die 100 persone l'una; la piu honorata e di\nBorgognoni e Fiamminghi, che hanno ad esser ben nati e servono a\ncavallo, e si dicono Arcieri accompagnando bene il Re per la citta a\npiede non in fila, ma alla rinfusa intorno alla persona reale; l'altri\nsono d'Albardieri 100 di nazion tedesca, et altri e tanti\nSpagnuoli.\"--Relazione della Corte di Spagna, MS.\n\n[401] Raumer, Sixteenth and Seventeenth Centuries, vol. i. p. 106.\n\n[402] Ibid. p. 105.\n\n[403] Cortes of 1558, peticion 4.\n\n[404] \"Questi habiti sempre sono nuovi et puliti, perche ogni mese se\ngli muta, et poi gli dona quando ad uno, e quando ad un\naltro.\"--Relazione di Pigafetta, MS.\n\n[405] Gachard cites a passage from one of Granvelle's unpublished\nletters, in which he says, \"Suplico a V. M., con la humildad qua devo,\nque considerando quanto su vida importa al principe nuestro senor, a\ntodos sus reynos y Estados, y vasallos suyos, y aun a toda la\nchristiandad, mirando en que miserando estado quedaria sin V. M., sea\nservido mirar adelante mas por su salud, descargandose de tan grande y\ncontinuo trabajo, que tanto dano le haze.\"--Rapport prefixed to the\nCorrespondance de Philippe II. (tom. i. p. li.), in which the Belgian\nscholar, with his usual conscientiousness and care, enters into an\nexamination of the character and personal habits of Philip.\n\n[406] \"Habiendo en otra ocasion avisado a vuestra magestad de la publica\nquerella y desconsuelo que habia del estilo que vuestra magestad habia\ntomado de negociar, estando perpetuamente asido a los papeles, por tener\nmejor titulo para huir de la gente, ademas de no quererse fiar de\nnadie.\"--Carta que escrivio al Senor Rey Felipe Segundo Don Luis\nManrique, su limosnero mayor, MS.\n\n[407] \"No embio Dios a vuestra magestad y a todos los otros Reyes, que\ntienen sus veces en la tierra, para que se extravien leyendo ni\nescribiendo ni aun contemplando ni rezando, si no para que fuesen y sean\npublicos y patentes oraculos a donde todos sus subditos vengan por sus\nrespuestas.... Y si a algun Rey en el mundo dio Dios esta gracia, es a\nvuestra magestad y por eso es mayor la culpa de no manifestarse a\ntodos.\"--Ibid.\n\nA copy of this letter is preserved among the Egerton MSS. in the British\nMuseum.\n\n[408] Nota di tutti li Titolati di Spagna con li loro casate et rendite,\n&c. fatta nel 1581, MS.\n\n[409] Ibid.\n\nThe Spanish aristocracy, in 1581, reckoned twenty-three dukes, forty-two\nmarquises, and fifty-six counts. All the dukes and thirteen of the\ninferior nobles were grandees.\n\n[410] \"La corte e muta; in publico non si ragiona di nuove, et chi pure\nle sa, se le trace.\"--Relazione di Pigafetta, MS.\n\n[411] \"Sono d'animo tanto elevato... che e cosa molto difficile da\ncredere.... e quando avviene che incontrino o nunzi del pontefice o\nambasciadori di qualehe testa cororata o d'altro stato, pochissimi son\nquelli che si levin la berreta.\"--Relazione di Badoero, MS.\n\n[412] \"Non si attende a lettere, ma la Nobilita e a maraviglia ignorante\ne ritirata, mantenenda una certa sua alterigia, ehe loro clriamano\n_sussiego_, che vuol dire tranquillita et sicurezza, et quasi\nserenita.\"--Relazione di Pigafette, MS.\n\n[413] \"Non si convita, non si cavalca, si giuoca, et si fa all'\namore.\"--Ibid.\n\nSee also the Relazioni of Badoero and Contarini.\n\n[414] Dr. Salazar y Mendoza takes a very exalted view of the importance\nof this right to wear the hat in the presence of the king,--\"a\nprerogative,\" he remarks, \"so illustrious in itself and so admirable in\nits effects, that it alone suffices to stamp its peculiar character on\nthe dignity of the grandee.\"--Dignidades de Castilla, p. 34.\n\n[415] Ranke, Ottoman and Spanish Empires, p. 57.\n\n[416] Relazione di Tiepolo, MS.--Relazione Anon. MS.--Relazione di\nContarini, MS.\n\n[417] \"Che per contrario affligiono i loro proprii sudditi ende\nincorrono nel loro odio.\"--Relazione di Contarini, MS.\n\n[418] \"Temono Sua Maesta, dove, quando si governassero prudentemente,\nsarieno da essa per le loro forze temuti.\"--Ibid.\n\n[419] \"Que bastaran para conquistar y ganar un reyno.\"--Cortes of\nValladolid of 1558, pet. 4.\n\n[420] Cortes of Toledo of 1559, pet. 3.\n\n[421] Lafuente, Historia de Espana, tom. xiii. p. 118.\n\n[422] Ibid. tom. xiv. p. 397.\n\n[423] Cortes of Valladolid of 1558, pet. 12.\n\n[424] Lafuente, Historia de Espana, tom. xiii. p. 125.\n\n[425] The history of luxury in Castile, and of the various enactments\nfor the restraint of it, forms the subject of a work by Sempere y\nGuarinos, containing many curious particulars, especially in regard to\nthe life of the Castilians at an earlier period of their\nhistory.--Historia del Luxo (Madrid, 1788, 2 tom. 12mo.).\n\n[426] \"Anssi mismo mandamos que ninguna persona de ninguna condicion ni\ncalidad que sea, no pueda traer ni traya en ropa ni en vestido, ni en\ncalzas, ni jubon, ni en gualdrapa, ni guarnicion de mula ni de cavallo,\nningun genero de bordado ni recamado, ni gandujado, ni entorchado, ni\nchaperia de oro ni de plata, ni de oro de canutillo, ni de martillo, ni\nningun genero de trenza ni cordon ni cordoncillo, ni franja, ni\npasamano, ni pespunte, ni perfil de oro ni plata ni seda, ni otra cosa,\naunque el dicho oro y plata sean falsos,\" &c.--Pracmatica expedida a\npeticion de la Cortes de Madrid de 1563.\n\n[427] \"Ocupados en este oficio y genero de vivienda de coser, que habia\nde se para las mugeres, muchos hombres que podrian servir a S. M. en la\nguerra dejaban de ir a ella, y dejaban tambien de labrar los\ncampos.\"--Cortes of 1573, pet. 75, ap. Lafuente, Hist. de Espana, tom.\nxiv. p. 407.\n\n[428] Cortes of 1573, pet. 75, ap. Lafuente, Hist. de Espana, tom. xiv.\np. 408.\n\n[429] Ranke, Ottoman and Spanish Empires, p. 59.\n\n[430] \"Que cada semana o cada mes se nombren en los ayuntamientos de\ncada ciudad o villa destos Reynos, dos Regidores, los quales se hallen a\nla vision y visitas de la carcel.\"--Cortes of Toledo of 1559, 1560, pet.\n102.\n\n[431] Provision real para que los mesones del reyno esten bien proveidos\nde los mantenimientos necesarios para los caminantes, Toledo, 20 de\nOctubre de 1560.\n\n[432] \"Como los mancebos y las donzellas por su ociosidad se\nprincipalmente ocupan en aquello [leer libros de mentiras y vanidades],\ndesvanecense y aficionanse en cierta manera a los casos que leen en\naquellos libros haver acontescido, ansi da amores como de armas y otras\nvanidades: y afficionados, quando se offrece algun caso semejante, danse\na el mas a rienda suelta que si no lo huviessen leydo.\"--Cortes of 1558,\npet. 107, cited by Ranke, Ottoman and Spanish Empires, p. 60.\n\n[433] Pracmatica para que ningun natural de estos reynos vaya a estudiar\nfuera de ellos, Aranjuez, 22 de Noviembre de 1559.\n\n[434] Marina, Teoria de las Cortes, tom. ii. p. 219.\n\n[435] See the \"Pragmaticas del Reyno,\" first printed at Alcala de\nHenares, at the close of Isabella's reign, in 1503. This famous\ncollection was almost wholly made up of the ordinances of Ferdinand and\nIsabella. After passing through several editions, it was finally\nabsorbed in the \"Nueva Recopilacion\" of Philip the Second.\n\n[436] Relazione di Contarini, MS.\n\n[437] \"Vos ni yo no avenios de subir donde los Sacerdotes.\"--Dichos y\nHechos de Phelipe II., p. 96.\n\n[438] Catrera, Filipe Segundo, p. 894.\n\n[439] L. Marineo Siculo, Cosas Memorabiles, fol. 23.\n\n[440] Nota di tutti li Titolati di Spagna, MS.\n\n[441] Lafuente, Historia de Espana, tom. xiv. p. 416.\n\n[442] Lafuente, Historia de Espana, tom. xiii. p. 261.--Cabrera, Filipe\nSegundo, pp. 432, 433.\n\n[443] Cabrera, Filipe Segundo, lib. xi. cap. 11; lib. xii. cap.\n21.--Relazione Anon. 1588, MS.\n\n[444] \"Otras vezes presentaba para Obispos Canonigos tan particulares i\npresbiteros tan apartados no solo de tal esperanca, mas pensamiento en\nsi mismos, i en la comun opinion, que la cedula de su presentacion no\nadmitia su rezelo de ser enganados o burlados. Eligia a quien no pedia,\ni merecia.\"--Cabrera, Filipe Segundo, p. 891.\n\n[445] Cabrera, Filipe Segundo, lib. xi. cap. 11.\n\n[446] Relazione di Contarini, MS.--Ranke, Ottoman and Spanish Empires,\np. 61.\n\n[447] The document alluded to is a letter, without date or signature,\nbut in the handwriting of the sixteenth century, and purporting to be\nwritten by a person entrusted with the task of drafting the necessary\nlegal instruments or the foundation of the convent. He inquires whether\nin the preamble he shall make mention of his majesty's vow. \"_El voto\nque S. M. hijo_, si S. M. no lo quiere poner ni declarar, bien puede,\nporque no hay para que; pero si S. M. quisiere que se declare en las\nescrituras, avisemelo v. m.\"--Documentos Ineditos, tom. xxviii. p. 567.\n\n[448] Examples equally ancient, of both forms of spelling the name, may\nbe found; though _Escorial_, now universal in the Castilian, seems to\nhave been also the more common from the first. The word is derived from\n_scoriae_, the dross of iron-mines, found near the spot.--See Ford,\nHandbook for Spain (3rd edition), p. 751.\n\n[449] A letter of the royal founder, published by Siguenca, enumerates\nthe objects to which the new building was to be specially\ndevoted.--Historia de la Orden de San Geronimo, tom. iii. p. 534.\n\n[450] \"The Escorial is placed by some geographers in Old Castile; but\nthe division of the provinces is carried on the crest of the _Sierra_\nwhich rises behind it.\"--Ford, Handbook for Spain, p. 750.\n\n[451] Siguenca, Hist. de la Orden de San Geronimo, tom. iii. p.\n549.--Memorias de Fray Juan de San Geronimo, Documentos Ineditos, tom.\nvii. p. 22.\n\n[452] \"Tenia de ordinario una banquetilla de tres pies, batisima y\ngrosera, por silla, y cuando iba a misa porque estuviese con algun\ndecencia se le ponia un pano viejo frances de Almaguer el contador, que\nya de gastado y deshilado hacia harto lugar por sus agujeros a los que\nquerian ver a la Persona Real.\"--Memorias de Fray Juan de San Geronimo,\nDocumentos Ineditos, tom. vii. p. 22.\n\n[453] \"Jurabame muchas veces llorando el dicho fray Antonio que muchas\nveces alzando cautamente los ojos vio correr por los de S. M. lagrimas;\ntanta era su devocion mezclada con el alegria de verse en aquella\npobreza y ver tras esto aquella alta idea que en su mente traia de la\ngrandeza a que pensaba levantar aquella pequenez del divino\nculto.\"--Ibid., ubi supra.\n\n[454] \"Para levantar tanta fabrica menester eran actos de humildad tan\nprofunda!\"--Ibid., p. 23.\n\n[455] Ibid., p. 25 et seq.--Siguenca, Hist. de la Orden de San Geronimo,\ntom. iii. p. 546.\n\n[456] \"Tenia tanta destreca en disponer las tracas de Palacios,\nCastillos, Jardines, y otras cosas, que quando Francisco de Mora mi Tio\nTracador mayor suyo, y Juan de Herrara su Antecessor le traian la\nprimera planta, assi mandava quitar, o poner, o mudar, como si fuera on\nVitrubio.\"--Dichos y Hechos de Phelipe II., p. 181.\n\n[457] Lafuente, Historia de Espana, tom. xiii. p. 253.\n\n[458] \"Sabese de cierto que se negociava aqui mas en un dia que en\nMadrid en quatro.\"--Siguenca, Hist. de la Orden de San Geronimo, tom.\niii. p. 575.\n\n[459] \"El buen Duque de Alba, aunque su vejez y gota no le daban lugar,\nse subio a lo alto de la torre a dar animo y esfuerzo a los oficiales y\ngente;.... y esto lo hacia S.E. como diestro capitan y como quien se\nhabia visto en otros mayores peligros en la guerra.\"--Memorias de Fray\nJuan de San Geronimo, Documentos Ineditos, tom. vii. p. 197.\n\n[460] Memorias de Fray Juan de San Geronimo, Documentos Ineditos, tom.\nvii. p. 201.\n\n[461] Siguenca, Hist. de la Orden de San Geronimo, tom. iii. p.\n596.--Dichos y Hechos de Phelipe II., p. 289.--Lafuente, Hist. de\nEspana, tom. xiv. p. 427.\n\n[462] Stirling, Annals of the Artists of Spain, tom. i. p. 211.\n\n[463] Stirling, Annals of the Artists of Spain, tom. i. p. 203.\n\n[464] Dichos y Hechos de Phelipe II., p. 81.\n\n[465] One of its historians, Father Francisco de los Santos, styles it\non his title-page, \"_Unica Maravilla del Mundo_.\"--Descripcion del Real\nMonasterio de San Lorenzo de el Escorial (Madrid, 1698).\n\n[466] Los Santos, Descripcion del Escorial, fol. 116.\n\n[467] Siguenca, Hist. de la Orden de San Geronimo, tom. iii. p. 862.\n\n[468] The enthusiasm of Fray Alonso de San Geronimo carries him so far,\nthat he does not hesitate to declare that the Almighty owes a debt of\ngratitude to Philip the Second for the dedication of so glorious a\nstructure to the Christian worship! \"Este Templo, Senor, deve a Filipo\nSegundo vuestra Grandeza; con que gratitud le estara mirando, en el\nImpireo, vuestra Divinidad!\"\n\nThis language, so near akin to blasphemy, as it would be thought in our\nday, occurs in a panegyric delivered at the Escorial on the occasion of\na solemn festival in honour of the hundredth anniversary of its\nfoundation. A volume compiled by Fray Luis de Santa Maria is filled with\na particular account of the ceremonies, under the title of \"Octava\nsagradamente culta, celebrada en la Octava Maravilla,\" &c. (Madrid,\n1664, folio).\n\n[469] Florez, Reynas Catholicas, tom. ii. p. 905.\n\n[470] Ibid. p. 908.\n\n[471] \"Realzada con gracia por el mismo trage del camino, sombrero alto\nmatizado con plumas, capotillo de terciopelo carmesi, bordado de oro a\nla moda Bohema.\"--Florez, Reynas Catholicas, tom. ii. p. 907.\n\n[472] Ibid., ubi supra.\n\n[473] Ante, vol. i. circ. fin.\n\n[474] Florez, Reynas Catholicas, tom. ii. p. 908.--Cabrera, Filipe\nSegundo, p. 661.\n\n[475] \"En el sarao bailaron Rey y Reyna, estando de pie toda la\nCorte.\"--Florez, Reynas Catholicas, tom. ii. p. 908.\n\n[476] \"El efecto dijo, que oyo Dios su oracion: pues mejorando el Rey,\ncayo mala la Reyna.\"--Ibid., p. 913.\n\n[Illustration: image of book's back cover]\n\n\n\n\n\n\nEnd of the Project Gutenberg EBook of History of The Reign of Philip The\nSecond King of Spain, by William H. Prescott\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"For those who liked Feeling Ghostly, I bring you an eclectic playlist of 50 songs for your upcoming Halloween festivities or just to enjoy at work or the privacy of your own home. Where else will you find Patsy Cline, Oingo Boingo, DJ Jazzy Jeff, and Jenny Lewis in the same mix. Enjoy!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"New video shared by the Guardian overnight has revealed the extent of the damage caused by Monday's fire at Notre Dame Cathedral.\nIt comes with the bold catchline that it will 'rewrite the rules of photography', but is Huawei's new smartphone as good as they say it is? We take a closer look.\nFrom snaps of surfing mates to award winning surrealism and dips with great white sharks, ocean photographer Warren Keelan is on the rise.\nTravel photographer Andrew Peacock shares the story behind his striking and vibrant images from Bhutan.\nImages need a pick me up? The Image Doctor will see you now. Get feedback on your work in our regular image doctor column.\nIn a world where innumerable images jostle for our attention, there's still something to be said for the old-school way of promoting and exhibiting on the gallery wall. Here's how to get your exhibition off the ground.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Micro Scan are proud to present you their second album 'Yes We Scan'. This album is a mix between full on, new school goa, and psychedelic trance vibes and characterised by hard yet pleasing leads, flying atmospheres, uplifting melodies and thundering yet precise rhythms. This new album both keeps in the style of Micro Scan that so many people have fallen in love with and also pushes new territory that not only defines new boundaries, it is truly a style unto itself! Previous to this release, their debut album 'Angels and Demons' has reached thousands of fans worldwide and is still considered a strong part of many DJs sets to this day. Microscan was one of the best kept secrets in France but now the mystery has been revealed! All the reviewers, critics and fans alike have been Raving about this project. Instant and massive dance floor appeal has driven their music to the leading DJs prime picks!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Moon-silvered clouds drift through the immensity of inky night skies. Far below, moonlighting activities are unfolding with astounding realism. The paintings in Peter Smets' latest body of work further explore how light variations transform the appearance of landscape. A grey, overcast day offers no definition and for Smets, light and the shadows cast in its absence are of prime importance. \"Light is like blood running through the veins, it makes a work live. Without light, a painting is dead,\" he says.\nNight time offers a different dimension of visual experience. It is a quieter, altogether less familiar world. In almost every one of the new works a full moon glimmers. The strange quality of its reflected light has prompted arcane associations throughout the ages. Both wisdom and irrationality have been attributed to the powerful force thought to emanate from a full moon. It is a dichotomy mirrored in the exhibition.\nThe contrast between works depicting a gentle fading of what was and those that chronicle an energetic surge towards the future is overt. Human figures, sinisterly grey cement trucks, conga lines of cable reels, expansive tarmacs, roads and leaf-littered pathways are ever heading towards the twinkling horizon of that future. Smets leaves it up to the viewer to interpret the implication of the gigantic red warning in the painting Right Way.\nMore than the pale glow of moonlight is required for the chiaroscuro Smets employs to create the illusion of depth and volume on a two dimensional surface. The glare of industrial spotlights and street lamps is needed to dispel the darkness of night and illuminate his narratives. Sense of place is always ambiguous and invariably the figures are devoid of emotional engagement with their surrounds. The paintings depicting an old man either sitting, head bowed and reading a newspaper, or hobbling away - walking stick in hand - are commentaries on the ageing population and the fact that progress is always conducted by new generations eager for change.\nCable reels are a recurring motif. These oversized drum-shaped objects carry the electrical wires necessary for any construction enterprise. Smets' enormous reels symbolically dominate the foreground but he tells that he has also chosen them for formal, pictorial reasons. They are spatial inventions that present vibrant colour opportunities against the predominantly cool, blue tones of the backgrounds; their great circular shapes complimenting vertical structures and the horizontality of distant skylines.\nA meticulous technician, Smets' realism is beyond any mere imitation of appearance. Although cultural and environmental interests are implicit, his paintings essentially manifest a quest for consummate composition and geometric purity. Every element has been measured, weighed and balanced to produce specific aesthetic responses. However dramatic and poignant the subject matter may be, it remains the vehicle for the expression of abstract principles.\nBorn in the Netherlands, Smets completed various Fine Art courses at the Maastricht Art School, Holland. Since migrating to Australia in 1986 he has been a finalist in numerous prestigious prizes including: Wynne Landscape Prize, AGNSW 2014, 2009, 2003; Sulman Art Prize, AGNSW 2012, 2009; Tattersall's Club Landscape Art Prize 2013, 2011, 2010, 2008, 2007, 2005, 2004; Prometheus Visual Art Award 2009, 2007; Doug Moran National Portrait Prize 2006; Stan and Maureen Duke Prize, GCCG 2013, 2012 (Winner Peoples Choice Award), 2008; Conrad Jupiter Art Award, GCCG 2004, 2003; Salon Des Refuses, 2013, 2011, 2003. His works have been collected by the Tweed River Regional Gallery, NSW; Gold Coast City Gallery, Queensland; Zurich Insurance, Sydney; Ministerie van LNV, Maastricht, the Netherlands; Gemeente Heerlen, Heerlen, the Netherlands; Nanao-shi Ishikawa, Osaka, Japan and various private collection in Germany, Switzerland, UK, USA and Mexico.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Are we closer to Heaven Or To Hell?\nAre the people on Earth closer to Heaven or to Hell?\nAt first I thought this question would be easily answered. My son loved a book as a toddler about a little girl who wants her Daddy to get the moon for her. Her Dad builds an enormous ladder and climbs up it to get the moon. It is a magical story by Eric Carle.\nThat threw me. I hadn't had time to buy a bottle of wine on the way home and such philosophical ruminations required a slightly numb sensation at the very least. But my son was persistent; strong coffee would have to do.\nAt first glance, it would seem that our actions as the human race on the whole seem more inclined to the hellish than the heavenly. Acts involving lying, cheating, stealing, violence, fill our news bulletins every day. The grim reality of existence can turn a blue sky grey, can pluck hope from a charitable heart. Yet as my son pointed out, for every bad act there must be a good one, otherwise the world would crumble. It's the passive principle of the Universe, the Yin and the Yang; the way of nature, the Tao.\nSo we noted, resolutely, the acts of humanity that bring us both closer to heaven and closer to hell.\n(And my personal hell \u2013 Paris Hilton releasing another album.\nSo are we closer to heaven or hell? I'd like to think that our heavenly acts come out ahead of our hellish ones. I'd like to think that the parable of the stranded starfish on the beach is how most people would choose to act.\nI can't think of a better way to look at life.\ninteresting post! but i would like to comment direct on the title \u2014 i would say, are closer to heaven because almost everything that is happening is like hell. It may mean that heaven is more closer to come. And i believe that we are so real close to the time when heaven comes down to this earth\u2026.\nNo news is good news; good news is no news; bad news SELLS!\nWow. Very thought provoking post. \u2026. I don't even have a comment\u2026 I have too much to think about now!\nThank you so much for your comments. I am not being 'born-again' or anything like that but have recently started researching the role of heaven and hell in classic literature for a guest lecture I'm doing at my old University. My son saw me reading Dante's 'La Divina Commedia' and wanted to know what it was about. I think that was what prompted his question. It's all a bit too much of a mindtrip for me, really. I need to watch something on TV to take my mind off it, like 'The Bold And The Beautiful.' Are Brooke and Ridge still together? I've wondered for ages.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I interviewed Henry Reichman, chair of the AAUP's Committee A on Academic Freedom and Tenure, about his new ebook, The Future of Academic Freedom (Johns Hopkins University Press). Reichman will talk about his ebook on a panel at the AAUP's Annual Convention on June 13, 2019 in Arlington, Virginia and different future occasions. You can too learn interviews with him on Democracy Now! and Inside Larger Ed.\nJohn Wilson: I'm just a little disturbed if you reward (albeit with very robust reservations) Stanley Fish and say \"he has a point when he argues that 'free speech is not an academic value.'\"(175) Why do we've to decide on between free speech and educational freedom when they are such shut cousins? Why can't we describe universities as having free speech plus educational freedom, the place in certain areas (like hiring, tenure, educating, and research) educational freedom is the prevailing worth, whereas in others (outdoors speakers, media, scholar rights, public expression), free speech is the dominant concept?\nHenry Reichman: I hardly assume that a truthful studying of my in depth critiques of Fish (in chapters 2 and 8) would characterize my remedy of him as reward with reservations. Extra accurate, perhaps, can be to say that I'm quite essential, while recognizing some constructive factors. Specifically, I feel his assertion that \"free speech is not an academic value\" is actually true, within the sense that it doesn't derive from the tutorial mission. As I put it in citing this statement, that is \"a valid point: academic freedom and freedom of speech are founded on differing justifications and should not be confused.\" (175) Free speech is, nevertheless, as I argue extensively, a democratic worth that can also be highly important to the university. Hence I don't actually disagree with your description of the connection between educational freedom and free speech. I merely phrase it somewhat in a different way. For example, in chapter 1 I wrote, \"The atmosphere for academic freedom on a given campus is inextricably linked to the broader atmosphere for freedom of expression.\" (23) Or, as I wrote in chapter 2, \"insofar as university administrations have abandoned the defense of academic freedom\u2014or simply pay lip service to the principle\u2014faculty members who wish to defend their own freedom as scholars will need to join with students in opposing restrictions on their freedom to speak.\" (49) And, of course, chapters 7 and eight, on scholar educational freedom and the rights of invited audio system, are virtually totally devoted to elaborating on the interdependence of school educational freedom and broader campus free expression rights.\nJW: The AAUP was founded in 1915, when the legal protections without spending a dime speech have been virtually nonexistent, the thought of unionizing school school was typically seen as abhorrent, educational freedom did not shield political speech in the classroom or criticism of the federal government throughout wartime, tenure was extra of a hope than a reality, and lots of other limitations on educational freedom existed. Why ought to we be tethered to archaic notions of educational freedom relatively than looking for to go away behind these previous limitations, because the AAUP did with unionization?\nHR: I don't perceive what archaic notions you're referring to. Tenure in the present day is as soon as once more, definitely for the majority of those that train in larger schooling, extra a hope than a reality, but simply as the AAUP fought for the principle anyway\u2014and gained actual positive factors, many of that are sadly being eroded\u2014I argue we will and must achieve this again. I no extra see tenure as \"archaic\" than the AAUP's founders noticed it as an unattainable pipe dream, which is what many at the time thought. As for unionization, I dedicate a whole chapter (based mostly on a Journal of Academic Freedom article) to how the AAUP over many years moved steadily to embrace the concept. Nevertheless it did so not in lieu of its conventional rules however as a mechanism for better implementing those. Moreover, because the guide suggests in several locations, the AAUP's interpretation of its 1915 rules has additionally advanced by means of numerous interpretive documents and investigation studies. Maybe the perfect example is the AAUP's expanded understanding of the school's right to extramural speech, which I chronicle in chapter 3 of the ebook. Finally, with respect to legal protections at no cost speech these are undoubtedly stronger at present than in 1915, however one shouldn't ignore the troubling tendency of conservative justices, who now maintain a majority on the Supreme Courtroom, to interpret the First Amendment in ways a lot nearer to the Lochner propertarian view of the early 20th century than to the custom of Justices Douglas, Brennan, and Marshall. Definitely, the judicial understanding of educational freedom, which privileges the institutional autonomy of the college over the professional rights of individual school members, is hardly a reliable defense. As I level out in chapter 2, a research by one scholar of some 339 selections involving school claims of educational freedom introduced on First Amendment grounds, revealed that 73% have been unsuccessful (45). Therefore, I am convinced, despite as we speak's extra strong understanding of free speech, the AAUP's longstanding strategy of pushing for the institution of professional norms by way of college policies and, where potential, collective bargaining agreements remains efficient\u2014offering school manage and converse out in help.\nJW: One difficulty you increase within the guide is the thought of utilizing union contracts to enforce educational freedom. Although it's a good suggestion, shouldn't that all the time be thought-about much less useful than changing campus policies? Coverage change works at all schools (not simply the uncommon ones with AAUP unions), it applies to the whole campus relatively than the unionized group (which typically doesn't embrace all instructors), and union contracts are compromises. If we will enact insurance policies that shield educational freedom, then the union doesn't want to give up one thing else or accept compromises in contract provisions. Shouldn't coverage change moderately than contracts be the main target of the AAUP on educational freedom?\nHR: I don't actually see the contradiction here. Contracts are themselves insurance policies, which have the benefit of being enforceable by regulation. The AAUP has lengthy advocated and nonetheless does\u2014as do I on this ebook\u2014the adoption of policies designed to protect educational freedom and shared governance. Our Beneficial Institutional Laws on Academic Freedom and Tenure present one model. We also urge, the place potential, that these insurance policies be included in union contracts to allow them to be enforced more successfully. To make certain, contracts are the outcome of compromise, however so too are coverage modifications. A school that is unable to win robust protections for tutorial freedom in its union contract is unlikely to have the ability to win administrative approval for institutional policies that achieve this. Moreover, it is inaccurate to say that each one campus policies apply to all teams. Take the University of California, for instance. Many of its insurance policies are fairly wonderful, but apply only to what they name \"Senate faculty,\" by which is supposed only the tenured and tenure-track school, who comprise less than half of all those that train. Ideally all schools and universities would undertake insurance policies with contractual standing that incorporate AAUP standards for all who train and conduct analysis. That is the aim. Attaining that objective, nevertheless, is sort of another matter and for that unionization is a strong weapon.\nJW: One critique I might make of your e-book is that you simply fall into what I think about the \"small number\" lure: You downplay assaults on campus free speech as \"much less severe than [FIRE] claims\" by mentioning that there are only a comparatively small quantity of disinvitations and audio system being shouted down. However can't that argument be turned towards you? In any case, there are only a small number of instances of unethical donor management or AAUP Committee A investigations annually, but certainly that doesn't mean corporatization isn't a problem in academia or that educational freedom is safe. Shouldn't we be dealing with all the issues of academia, even if we reject the exaggerated and one-sided ideological assessments of them, and acknowledge that even relatively uncommon abuses can have horrible effects and must be condemned?\nHR: Properly, I do assume that assaults on outdoors speakers are far much less in depth than FIRE claims and, extra necessary, much less of a menace to educational freedom and freedom of expression than other threats that each FIRE and, to a terrific extent, the media underplay. I'm arguing towards the place, articulated for example by Stephen Carter, that these are \"everyday events.\" They don't seem to be. But that hardly means I feel they don't seem to be a menace at all. That's one purpose why I dedicate 5 pages (175-179) to chronicling (and approving) the AAUP's longstanding defense of the rights of controversial audio system. In fact we should always handle \"all the problems of academia,\" though all those issues don't all the time even contain educational freedom or free expression. My ebook, for instance, says little about scholar debt or sexual harassment, however clearly these are major issues, extra essential for a lot of than challenges to professors' classroom speech or to outdoors audio system. But that's not what the guide is about.\nJW: Once we take a look at the longer term of educational freedom, for the first time within the historical past of American larger schooling, there's a massive push among conservatives to have robust protections at no cost speech on campus. Why shouldn't we be saying \"yes, thank you\" to these conservative campaigns, regardless of how hypocritical they're, and demand that universities undertake good policies that shield educational freedom and free expression? Why is the left now so suspicious of free speech just because their opponents are arguing for it? Why isn't the AAUP leaping at the alternative to hitch with the conservatives and convince schools to adopt robust protections for freedom on campus?\nHR: It's not simply that this push is hypocritical. In lots of instances it's itself an attack on free speech. I am totally unconvinced that the majority proposed free speech legislation aims at defending everybody's free speech. I feel it's primarily aimed toward policing speech, particularly the speech of students, but in addition of professors within the classroom. I supply considerable evidence for this in my guide. And I feel experience bears me out. To make certain, there are numerous principled libertarian conservatives who really defend both free speech and educational freedom, and I positively cite several of them in my ebook. Most of them, by the best way, are themselves teachers. And the AAUP has and will again in future, I'm positive, defend the rights of conservative professors ought to their opinions and expression endanger their employment status.\nHowever the conservative political movement for campus \"free speech\" is aimed mainly on the freedom only of sure audio system. When have we seen these conservatives converse out for the speech rights of their opponents? Hardly. Conservatives defend the rights of Koch-funded \"free enterprise\" centers, but when they do not be a part of in the assault they continue to be silent when \"liberal\" centers, as at the University of North Carolina, are shut down for political reasons. Too typically many of these politicians and campus\/media \"watchdogs\" who trumpet the rights of the Milos and the Coulters usually are not solely hypocritically silent when leftists are assaulted; they participate in the assault.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"These simple Energy Bites come together in no time and are an easy, filling and tasty snack!\nA couple of weeks ago, my sister and I had a girls night together when my husband went snowboarding. We had a blast. We went out to dinner, put Adalyn to bed, made these awesome energy bites and then had some wine and chatted. We made these energy bites using a bunch of tasty ingredients that I had on hand. They are high in protein, whole-grains and fiber.\nThey are good for anyone but are especially great for nursing moms because they contain flax and oats which can help increase milk production. These refrigerate well and made excellent snacks all week.\n1. Melt peanut butter in microwave.\n2. Combine all ingredients and mix very well.\n3. Refrigerate mixture for 1 hour or until dough is cold and firm.\n4. Roll into balls and store bites in fridge for up to 10 days.\nEnjoy! If you try these, let me know if you are addicted. I could not stop eating them! They are being added to my regular list of recipes to make during food prep.\nI was recently sent this Purely Inspired 100% plant based protein nutritional shake which worked well in this recipe to give a boost of protein and nutrients. It is dairy, gluten and soy free. One serving has 16 grams protein and 7 grams of fiber. It contains a concentrated greens blend with broccoli and spinach. It is a blend of proteins including pea protein, brown rice protein and alfalfa juice protein concentrate.\nThis protein could also be used in many recipes such as oatmeal, smoothies, baked goods, shakes, pancakes and more. I do not use protein powders all the time but think that they can be a good way to get some additional protein in your diet if you are lacking.\nThese bites are great on the go and provide a good amount of nutrients. I can eat these while driving even when I am in a hurry. Check out Purely Inspired on Facebook.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sacred Heart scored an impressive six goals during the first of the game, a fantastic response to going 1-0 down. Goalkeepers Harvey and Sam stopped some great shots while Dorian and Sam provided some vital assists. Dean was voted Player's player of the match after finishing well. It was a great performance by the team of Year 5 and 6 boys. Cameron worked really hard in midfield while captain Seb put in a disciplined performance.\nActive Fun at Lunchtime !!\nThe health and well-being of pupils is vitally important. Obesity is a major problem in ill health in children, mainly due to children nowadays going home and playing on computer games and leading sedentary lifestyles with a lack of knowledge on both physical exercise and maintaining a healthy and balanced diet.\nObesity in children is not something we can shy away from nowadays and it is set to become a worsening disaster for adults and children.\nTo stay healthy or to improve health, Children should be doing at least 60 minutes of physical activity every day.\nEncouraging children to eat a nutritious, balanced diet early on in life is so important so they get the correct amount of vitamins and minerals in their diet to help them grow and develop optimally. They are more likely to be energised and motivated, supporting their ability to learn.\nFitness sessions with KS2 have been implemented every Wednesday.\n\u2013 I explain to children why a healthy, balanced diet is important.\n\u2013 I give them more options to have a varied diet and a healthy relationship with food.\n\u2013 Fun, fitness sessions for children to participate in.\n\u2013 Giving children 15 minute plans to do at home and to try and encourage their parents to do it with them.\nI measure progress each term by getting children to complete different functional fitness tests. At the beginning of term we will complete the bleep test and re do the test at the end of term to see if any improvement has been made. These tests change every term to keep it fun and varied for the children.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"I've taken the big leap into the blog world. As soon as I hit \"create blog\" it hit me... I'm not a good writer, I have poor grammar, I cannot edit to save my life, I do not know (yet) how to take artsy photos, and I do not have people knocking on my door dying to know what I'm doing day to day. Why did I create my own blog? Maybe I should create a screening board to review entries before I post them.\nStay tuned. Non-interesting posts are sure to come.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In July we decided to clean out our studio storage unit. It involved a lot of work, but more significantly we felt emotional strings being tugged every time a box was opened. Since it was the studio's storage unit, it was filled with the evidence of our history since the very first time we put needle and thread to fabric. It's been awhile since we've looked back.\n\u2026it was filled with the evidence of our history since the very first time we put needle and thread to fabric.\nDuring those initial years we identified our media, learned how to collaborate, and invented a work of functional art. We were working full time \"day jobs\" and at night we'd return to the studio to create art and then sell it at festivals during the weekends. We continued like this for 3 years working 16-hour days, driven by our fear that we wouldn't be able to support ourselves solely with the earnings from our art.\nBy creating little iconic designs out of fabric and thread, framing them in plastic and adding a clock mechanism, we made unique timepieces that turned out to be immensely popular. Before we knew it, we were making hundreds at a time, selling them at art festivals, boutique stores and mail order gift catalogs. Their success allowed us to quit our day jobs though we felt like we became a two person factory. Having to spend all our time assembling the clocks while having very little time to pursue our dream of being fine artists left us feeling exhausted and empty.\nWe eventually reached a breaking point and decided to take some time off.\nWe eventually reached a breaking point and decided to take some time off. We drove to Cape Cod and sat and watched the ocean. We noticed how the autumn light cast over the cottages of the small New England town. We felt struck by the history of the little homes, representing hundreds of years of life lived in them. We felt everything slow down, and we were desperate to find a way to hold on to that feeling. But upon returning to Minneapolis, it wasn't long before we were back to our old habits. The stress of the production and lack of creativity was driving away our vision of being artists. We felt we couldn't be further away from what we initially aspired to.\nRealizing we needed to change, we decided to put our art before our fears.\nRealizing we needed to change, we decided to put art before our fears. We focused on creating only original fine art, and stopped allowing the worry of earning our livings drive us. With renewed enthusiasm we allowed ourselves to spend as much time on a single piece as it needed. We instantly felt excitement as we realized the immense potential of allowing the imagery to develop organically, reacting to the painter and printmaker in us that we had forgotten about. That first piece, without concerning ourselves with a subject or a composition, grew to resemble a neighborhood in Cape Cod. Technically it was a disaster, but we recognized its larger significance: to remember to take notice of our surroundings; slow down; trust and relax.\nMiraculously, by following our own suggestions, we've succeeded in working as full time artists long enough to fill a storage unit. It's been fun to rediscover the treasures of our past: hundreds of clock hands; starts of fabric designs that were obviously going nowhere; the shelving we constructed to hold the clocks while in stages of production; our traveling booth and show kit that included crossword books and walkie-talkies to use as we criss-crossed the country. Now that the storage unit is empty, it'll be interesting to see what we collect over the next 16 years.\nIt'll be interesting to see what we collect over the next 16 years.\nWhat have we been working on lately? See our available fiber art and paintings.\nYour note truly touched us Carolyn. There are so many things we feel grateful for, most are unexpected, but having ongoing conversations about art as well as the lives we all live is one of our greatest joys. Thank you for sharing the conversation with us. And we're glad you at least get to see Red Chair through your computer screen. Maybe one day it'll materialize.\nIf you ever decide to depart from the visual arts, you can always take up writing. Nicely done!\nThank you for reading our journal, and for the flattering suggestion Virlene. We find a certain comfort in writing down our thoughts \u2013 perhaps it's a byproduct of working so long with our hands. It'd be interesting to attempt writing as a practice, though it's easy to imagine it would have equal parts doodles and sketches to words. Until we find out, we'll keep journaling. We hope you'll continue to join us.\nYour description of our art and the pursuit of that which 'feeds' us is beautiful Krayna. Thank you for sharing your eloquent words with us. It's so clear to us that we are our best selves and, more importantly, best for the world when we do what we love. We hope everyone has a chance to find that in themselves and allow it to fuel and inform how they spend their precious time. Thank you for being that person Krayna.\nAbsolutely fascinating. And hooray! for your decision to slow down and do the wonderful art you are now producing.\nThank you for that encouragement Sarah. As we mentioned, it has been awhile since we looked back and remembered all the incredible opportunities that presented themselves once we allowed more time and space into our lives. As you know us now, we would not be where we are today if hadn't taken down our 'safety net'. Since then, we include meeting people like you and Dan as one of the rewards for pursuing our dream. Thanks for showing up!\nHi Lisa and Lori !\nThis is a wonderful message to those of us who love your art! I sent you a brief e-mail listing my favorite art pieces you have in your gallery.\nThanks for writing us Donna. We did get your email from weeks ago. We're sorry we haven't replied yet, but we were so glad to hear from you! Thank you for staying in touch with us, we'll 'talk' soon.\nWho is the writer of your articles? Excellent writing. We all love these updates. I know you'll be working, but I wanted you to know that you are warmly invited to the First Speak performance, 10-10-15 at noon to 2:00ish. I'll email the venue address when I get it. I will be performing an original oral story along with six other story tellers. I have written short stories most of my life, but recently realized I have dozens of stories that never made it to paper, because they are better told than written. I joined a story telling group (First Speak) and I am very happy to be back in the creative process. Hope to see you soon.\nWe're very glad you enjoy the articles Chris. We're finding sitting down to write our thoughts is one of our favorite things to do \u2013 it gives us a chance to deliberately take the time to reflect on our activities as artists. Thank you for tuning in to read about them. Thanks also for inviting us to your First Speak performance. We look forward to hearing more! Most importantly, congratulations! We think you're going to be wonderful!\nI love reading your emails and the gift you give of getting to know you better. This account of learning to let go and letting life happen as you focus on \"real art\" is precious.\nAnd it is a teaching for the rest of us since it conveys that art is not about doing but really about being (which then gets expressed as art through you) when you live fully in the moment. Thank you both for your sharing.\nWe're glad you're enjoying our stories, and we appreciate the time you spend with us as you read them. Living as artists, by its very definition, has taught us to open up to others. Sometimes we find it scary, but as you articulated so well, it gives us another opportunity to learn to live beyond our fears and be in the moment. Thank you for sharing your thoughts with us Rich.\nI am a scientist studying how people respond to nature, particularly improved health. What a coincidence \u2013 today I talked with a colleague about the evolution of my research over the past 2 decades. Remarkable similarities to the twin artists' story of creativity \u2013 discovering passion and satisfaction, dedicated intensive work for years, mastery of the media, realization of accomplishment. Different strokes, similar reflections. Love ya sisters!\nWhat you describe is a remarkable thing Kathy, in that we're finding ways to connect people with nature, each with our own expressions in art and science. We've always thought there's a thin line between the two, since they rest at opposite ends of the creative spectrum. We're continually intrigued by what you present to your audience, as you paint your picture with a composition that can only come from all your years of research, experimentation, diligence and talent. We look forward to investigating this topic more with you Kathy!\nIt was bitter sweet reading this wonderful journey that the two of you have been on over these past 16 years. Bitter sweet because both Ed and I miss seeing you and personally viewing your latest creations on the \"Friday Night Art Walks\", but are glad that we can stay in touch via e-mail. The choices that you have made, by leaving your fears behind and reaching for the unknown, has only enhanced the beauty and quality of your art work. Change is always good. It gives all of us a new perspective on life and moves us forward. You two are the VERY BEST. We miss you.\nWe're so glad you stay in touch with us Carol. Hopefully the e-newsletters and journal entries allow you to see the art and continue the conversation almost as if you're still here (though it's not the same as in person on First Friday). We appreciate having the opportunity to share with others the individual experiences we all have. We feel inspired when we hear how others embrace the inevitable surprise of change. Thanks to you and Ed for sharing your perspectives over the years!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Analyzes client requirements and organizational working processes to generate ERP configuration specifications.\nPrepare necessary project documentations associated with implementation like user manuals, customer query data, issue logs and their resolution.\nEstablish positive client relationships during engagement for high client satisfaction, quick deployment and adequate training.\nAssist clients with implementation and training.\nImplement and configure ERP application to customers.\nProvide pre and post-sales support as needed.\nTrain users in specific modules of ERP application.\nKeep information confidential and other duties as Assigned.\nBachelor's Degree in Information Technology, Management Information System (MIS), Computer Science, Engineering, Accounting, Finance, Business Administration or other related fields.\nAt least 2 year working experience as ERP consultants, having knowledge in Supply chain management (SCM), c or BI is will be given special considerations.\nGood leadership, strong analytical, problem solving and good interpersonal skill.\nHave positive attitude and be a good team player.\nExperienced in IFS System will be given special consideration.\nBenefits: Health Insurance, Transportation, OT, Bonus, Company trip, 1 hour for fitness gym in working hour, etc.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"New Jack City: Johnny Gill: Too hot to handle?\nJohnny Gill: Too hot to handle?\nLast New Jack City, things got a little hot and heavy on the dance floor. There must have been something in the air that night because the slow jamz were bringing folks together in ways that would make me blush if I were to attempt to describe them here. Let's just say 9 months from now don't be surprised if a few Ronnies, Bobbys, Rickys and Mikes are born in Portland.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There are a number of firms which concentrate in Custom composition writing. The synopsis does not require to be in virtually any conventional composition format. The perfect portion about creating an enlightening essay may be the huge variety of matters you can select from. Essay composing does not want to be tough.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The Presscheck Designated Marksman course is designed to equip patrol and tactical officers with the skills to apply accurate rifle fire in critical situations at varying distances and from various positions.\nThis 20 hour course stresses the importance of applying offset and holds at various distances and angles to ensure proper point of impact. Various techniques of supporting the rifle to mitigate recoil and stabilize the reticle for proper shot placement.\nDefinition reviews of Internal, External, and Terminal ballistics, including 100, 200, and 300 yard zero.\nZero\/confirm zero on all sights.\nEngagement of low percentage targets (small strike area often obscured with a hostage or other no-shoot area) from various distances (out to 500 meters) while applying proper hold offs.\nDrills used to demonstrate POI deviation when cant is applied to the rifle.\nThe student will understand the importance of critical hits and the consequences of a miss, and will be able to diagnose if they are capable of achieving hits at any given moment based upon pre shot diagnostics of their sights in real time, the hold off of their reticle based upon target distance AND body position and cant of the weapon.\nThis course will enable the student to continue to develop and maintain these skills through the use of clear, repeatable standards.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The 1981 Formula One season was the 32nd season of FIA Formula One motor racing. It featured the 1981 FIA Formula One World Championship which was contested over a fifteen race series that commenced on 15 March and ended on 17 October. Formula One cars also contested the 1981 South African Grand Prix, although this was technically a Formula Libre race and was not part of the Formula One World Championship.\nThe 1981 championship was the inaugural FIA Formula One World Championship, replacing the former World Championship for Drivers. Nelson Piquet won the Drivers' Championship, claiming the first of his three Formula One titles and Williams won the Constructors' Championship. (Wikipedia).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Get in, graduate, go far.\" To students seeking guidance during the hectic process of applying to college, the slogan of Bottom Line ensures them that it is possible.\nWhen Phyliss St-Hubert first heard about Bottom Line in the winter of last year, she applied within the following days. The oldest of three siblings, St-Hubert is the first member of her family to pursue a bachelor's degree.\nBottom Line has its students apply to their target schools, which are mostly colleges and universities in and around the Boston area. These include all UMass schools, Worcester State University, Boston College, Boston University, Northeastern University, and Suffolk University.\nSt-Hubert decided to apply to Suffolk University, and is now a freshman, class of 2018. Although Suffolk might not have always been her first choice, she couldn't be happier with her decision. During a phone interview last Thursday, St-Hubert explained her experience.\nBack row (L-R): Mike Wasserman, Bottom Line Mass. Executive Director and Landon Dickey, City of Boston Education Advisor to Mayor Walsh.\nFront row (L-R): Bottom Line students Phylliss St-Hubert, Jerika Adams-Harrison, and Sonia Rugwiza, all freshmen at Suffolk University.\nBottom Line is a privately funded, nonprofit organization that thrives in aiding low-income, first generation, at-risk urban youth in the Boston area to attend and graduate college. Its counselors are dedicated to aiding students throughout both the application process and the transition to college.\nWhen a high school junior or senior is interested in receiving help from Bottom Line, there is an online application on the organization's web site. Once the student is accepted, the first step of the organization's process, called College Access, begins. A counselor helps the student one-on-one with the application process, writing essays, and searching for scholarships. The experience can be extremely daunting for a student who might be the first in their family to attend college. Bottom Line takes the fear out and replaces it with inspiration, according to officials.\nBottom Line also helps students with financial concerns, registration questions, and even staying focused during school.\nSt-Hubert is living on campus this year in the Modern Theatre dormitories.\n\"It's good, I don't have my mom nagging me,\" she said with a laugh. \"There is always something to do.\" When thinking of some clubs she might join, she is considering the Residence Hall Association, Student Government Association, Black Student Union, and the Program Council, while she mentions she likes event planning.\nHer favorite class is a public speaking class, which assigned a presentation on the first day.\nAccording to a statement from Elevate Communications, Bottom Line's annual send-off of over 200 students was held on Aug. 13 at Hill Holiday, a marketing agency in downtown Boston. Hill Holiday has long been a partner with Bottom Line, and provides students with the opportunity for paid summer internships, and possibly full-time positions.\nBottom Line stresses that secondary education is essential for the growth and prosperity of individuals. According to their website, while only nine percent of the lowest income students earn a college degree by age 24, students who are enrolled in Bottom Line are twice as likely to graduate compared to their low-income peers.\nBottom Line currently has locations helping students in Boston, Chicago, Worcester, and New York.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"CFAllocator Documentation for this section has not yet been entered.\nCFAllocatorFlags Documentation for this section has not yet been entered.\nCFErrorDomain Documentation for this section has not yet been entered.\nCFException Documentation for this section has not yet been entered.\nCFExceptionDataKey Documentation for this section has not yet been entered.\nCFIndex Documentation for this section has not yet been entered.\nCFReadStream Documentation for this section has not yet been entered.\nCFRunLoopSource Documentation for this section has not yet been entered.\nCFRunLoopSourceCustom Documentation for this section has not yet been entered.\nCFSocket Documentation for this section has not yet been entered.\nCFSocket+CFSocketAcceptEventArgs Documentation for this section has not yet been entered.\nCFSocket+CFSocketConnectEventArgs Documentation for this section has not yet been entered.\nCFSocketCallBackType Documentation for this section has not yet been entered.\nCFSocketError Documentation for this section has not yet been entered.\nCFSocketException Documentation for this section has not yet been entered.\nCFSocketFlags Documentation for this section has not yet been entered.\nCFSocketNativeHandle Documentation for this section has not yet been entered.\nCFStream Documentation for this section has not yet been entered.\nCFStream+CFStreamCallback Documentation for this section has not yet been entered.\nCFStream+StreamEventArgs Documentation for this section has not yet been entered.\nCFStreamStatus Documentation for this section has not yet been entered.\nCFType Documentation for this section has not yet been entered.\nCFWriteStream Documentation for this section has not yet been entered.\nDispatchQueuePriority Documentation for this section has not yet been entered.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"These ordinances are here for your convenience. It is possible they are not the most recent versions. Please contact the Township secretary if you have any questions. The following ordinances are in PDF format.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzvbrq b/data_all_eng_slimpj/shuffled/split2/finalzzvbrq
new file mode 100644
index 0000000000000000000000000000000000000000..1f5e4fd5275f305ae6a87f2b7c6ba6dd0535abd7
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzvbrq
@@ -0,0 +1,5 @@
+{"text":"I just heard the news\u2026Kenya is ranked up there for providing bla bla opportunities for women\u2026thanks to the Constitutional a third quota requirement\u2026.\nBefore la femme movement comes after me na majora, let me confess that I don't mind being under a woman, I advocate their rise up the corporate ladder, heck, I would vote for a female president!I am an admirer of Wangari Maathai (a must-mention),Angela Merkel, Mother Teresa, Sirleaf\u2026.etc.\nArticle 81(b) Chapter 7 of the Constitution of Kenya states that \"no more than two thirds of public posts can be from one gender\".\nThis has been blown out of proportion by women who have hitherto been society underdogs,yapping around and about that this field has been levelled,the constitution now favours them,bla bla bla\u2026.\nFlashback to my 844 days when girls used to sing, \"What a man can do a woman can do!\"\u2013Those were good days,morale was high,competition stiff\u2026..now,that would have built. Great women in the society!\nNobody moves up without pressure,the constitution didn't have this clause when Wangari Maathai took to the world with her degrees!Otherwise,she would have rested at Undergraduate,knowing she would hold an uncontested Public Office! If we give the modern Kenya girl the drive-by opportunity,she will give books a wide berth\u2026why would you work if you had a plate guaranteed?\nAnd to the yapping women and female activists\u2026the boy child is yours too!Okay,I am not interpreting the Constitution as people are, I am not a lawyer,but it's clear on 'gender' and not 'female\/women'! But then,would I think the playground is tilted,and isn't motivating in any way!\nLets fight for positions,Lets look at the papers,the experience,the qualifications\u2026\u2026.\nAnd not the features below the belly button!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Consumers are exposed to blue light every day, whether from the sun or electronic devices. DSM offers day-long-total protection without affecting consumers activities, for both, in-and outdoors! Let's beat the blues!\nWe're committed to broadening horizons in the sun care industry, and with two brand new UV filters PARSOL\u00ae Max and PARSOL\u00ae Shield and our ambitious investment in PARSOL\u00ae products, we aim to revolutionize sun care.\nLog in here to start formulating your next sunscreen.\nCheck out our video tutorial and get acquainted with the tool.\nUnder ideal conditions, today's UV filters have the capacity to protect even the most sensitive skin against high levels of UV radiation. Yet the incidence of skin cancer continues to rise, with between 2 and 3 million case of non-melanoma and 132,000 cases of melanoma cancers globally each year. At DSM, we aim to reduce these numbers by maximizing the performance and sensorial profile of sun protection products. Formulating sunscreen formulations at a reduced cost and time is one step further to having more efficient sunscreens available to the consumers worldwide.\nDSM's SUNSCREEN OPTIMIZER\u2122 is a free tool allowing every formulator to develop the most efficient sunscreen formulation, optimize the existing one before going into a large number of in vivo SPF measurements.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Achiote Paste is a blend of Annatto seeds, garlic and spices. It is generally used in Central American and Mexican cuisines and is one of the main uses of annatto seeds - which provide the paste with its brick red colour. Used to create rubs, marinades, and as a general seasoning, the combination of ingredients creates a wonderful flavour for beef, lamb, poultry, pork, and seafood. Achiote Paste provides a bittersweet and tangy flavour that complements this large variety of dishes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Izy Kali is an ever curious designer that loves learning new skills to better answer the questions that design poses, working across Design, Illustration and Motion Graphics.\nWith several years of experience in agencies and as freelance, she is well-versed in print, digital, social media design, branding, animation, art direction as well as client pitches .\nAn international person, she has lived in 7 countries and speaks three languages, contributing to her unique design thinking and the quirky edge to her practice.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"SANDY POINT, St. Kitts and Nevis (April 26, 2016) \u2014 Chavez 'Beto' Belboda amassed 27 points, 6 rebounds, six steals as the Sandy Point PRIDE defeated the Achievers Wolf Pack, 79-70, Tuesday night at Basketball City.\nImpressive work: PRIDE (2-0 to begin the season) not only registered it's second victory, but they did so against a good team that has been clicking. Following a win over the 2015 'A' division runners up they recorded a season high 26 steals against a Wolf Pack Team that was only recently relegated to the 'A' division after playing in the premier division in 2015.\nAs it turned out, the long break between the team's first two games paid enormous dividends as there was more than enough juice available in the legs of the young lions to outlast the Wolves when coach Rambo ordered a fourth quarter defensive lock down to which the players responded with flashes of hard-nosed resistance.\nWild game: The PRIDE were terrific for most of the first 10 minutes of the first quarter in building a 20-15 cushion. Wolf Pack rallied to reduce the deficit to just two points, a 57-55 PRIDE lead at the end of the third quarter.\nPRIDE could have folded, but instead, the team's co-captains Taslin 'Mini' Belboda and Sheeno 'Jeezy' Liddie led by example and rallied the troupes en route to tallying 19 points 13 assists 8 steals and 16 points 15 rebounds 6 steals respectively.\n\"We had to dig deep tonight! Mel, Secada, Miguel, Dilyn and Bassue also gave us good contributions as they all made key hustle plays to help push us over the hump!\" Continued Coach Rambo.\nJoin the hunt and follow the PRIDE by searching 'Sandy Point Pride Basketball ' on Facebook, Instagram or Twitter, for the latest club news, scores and updates.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzvvnz b/data_all_eng_slimpj/shuffled/split2/finalzzvvnz
new file mode 100644
index 0000000000000000000000000000000000000000..0c554ea49c11acfce3dd7be091ee9f3ed7e59c79
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzvvnz
@@ -0,0 +1,5 @@
+{"text":"It's good to know what kind of breakfast is most effective in helping you to stick to a healthy diet and avoiding excess snacking during the rest of the day. Recent studies show that enjoying a fiber-rich breakfast is the answer. The next question is, of course, which kind of fiber-rich breakfast food is then best. \u2026 Continue readingResearch Results: Enjoy THIS BREAKFAST to Avoid Over-Eating Later in the Day!\nIf You're Hooked on \"Healthy\" BREAKFAST CEREALS from the Supermarket, Then These are the BEST OPTIONS!\nThe manufacturers of popular breakfast cereals have big marketing budgets to convince you that their products are nutritious and good for you. The truth is that most breakfast cereals are far from being truly nutritious, especially the brands designed to appeal to kids. Some cereals sold by health food stores may be good for you if they avoid the two ingredients typically found in the commercially-prepared cereals from the supermarket. \u2026 Continue readingIf You're Hooked on \"Healthy\" BREAKFAST CEREALS from the Supermarket, Then These are the BEST OPTIONS!\nWhich STORE-BOUGHT BREAKFAST CEREALS are Actually HEALTHY?\nSome cereals sold by health food stores may be good for you if they avoid two ingredients typically found in the\\'a0commercially-prepared cereals from the supermarket. Most breakfast cereals are far from being truly nutritious, especially the brands designed to appeal to kids. \u2026 Continue readingWhich STORE-BOUGHT BREAKFAST CEREALS are Actually HEALTHY?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Reading books with your children, especially before bedtime, is an excellent way to establish a solid parent-child bond whether they're preschool age or older. It's also a way to open the doors to infinite knowledge and natural curiosity. Reading stories to your children is a practical way to get them to relax, slow down, and to learn to pay attention.\nStudies reveal that reading books out loud to your children helps them develop both their language and evolving literacy skills that will in turn prepare them for going to school. Reading aloud to children is routine among virtually every child care or daycare facility today.\nAs a parent or day care provider, you're connecting and bonding with your child with every passing paragraph you read through colorful characters, interesting twists, and exciting new worlds. Once a child advances from picture books and stories to more complex concepts, the literature themes start to become even more essential to their overall personal development and growth.\nHere are some of the best books to read to your kids, no matter what their age.\n'The Snowy Day' is the perfect book to read to your child if their own school day is canceled due to inclement weather. This is a gentle, sweet story that conjures up a magical winter wonderland scene in the mind of a child. In reality, snow is often a pain to shovel. However, in this story, freshly fallen snow is the ideal canvas to make snow angels, footprints, and snowballs.\n'James and the Giant Peach' is a classic tale about a young boy who is forced to go and live with his two nasty, wicked aunts. However, he acquires magical crystals and accidentally drops them on his aunts' old peach tree. Suddenly, one of the peaches starts to grow exponentially until it finally rolls away with James on top. He sets off on a wonderful adventure while making unusual friends along the way.\nPerhaps more for an older child, this wonderful classic tale about children who are evacuated from their war-torn home and are forced to stay in an old rural house, has been loved by readers both young and old for generations. The story offers something for everyone, including magical creatures, brave rescues, betrayals, and an evil witch.\nAn unusually strong young girl, Pippi Longstocking makes her way to a Swedish town accompanied by a bag of gold coins and her horse and monkey. She reunites with her seafaring dad after making friends with two other children, Tommy and Annika, and must defend herself from thugs while astounding the grownups of the town with her highly odd and curious personality.\nCharlotte's Web tells the tale of a loveable friendship between Charlotte the grey spider and Wilbur the pig. Wilbur is very upset once he discovers that he's being fattened up in order to become someone's dinner in the fall. As a result, Charlotte is determined to save him. Charlotte and Wilbur become close friends and the book itself explores different themes, including the concept of friendship and loyalty.\nEven just reading aloud to your child for 15-20 minutes a day is one of the single most important ways to adequately prepare them for a lifetime of future learning. Not only does reading to your child enhance their capacity to read independently, but encourages their brain development overall.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Six years ago I wrote a book called \"Uncle Sam's Plantation.\" I wrote the book to tell my own story of what I saw living inside the welfare state and my own transformation out of it.\n\"This plan is more than a prescription for short-term spending-it's a strategy for America's long-term growth and opportunity in areas such as renewable energy, health care, and education.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You may have had a speech problem since childhood. Perhaps you had no access to therapy or the problem has grown worse over time. Regardless, as an adult your speech or language problem may be a barrier to achieving your personal or professional goals.\nAs a licensed and certified speech-language pathologist, Deborah Boswell works with adult clients who stutter or have voice disorders. Since 1991 she has provided caring and friendly speech therapy services to clients in Birmingham, Ala., Atlanta, GA. and throughout the Southeast.\nStuttering affects the fluency of speech. It begins during childhood and, in some cases, lasts throughout life. The disorder is characterized by disruptions in the production of speech sounds, also called \"disfluencies.\" Most people produce brief disfluencies from time to time. For instance, some words are repeated and others are preceded by \"um\" or \"uh.\" Disfluencies are not necessarily a problem; however, when a person who stutters avoids speaking situations because of their disfluent speech then speech therapy may be a solution.\nIn most cases, stuttering has an impact on at least some daily activities. The specific activities that a person finds challenging to perform vary by individual. For some people, communication difficulties only happen during specific activities, for example, talking on the telephone or talking before large groups. For most others, however, communication difficulties occur across a number of activities at home, school, or work.\nSome people may limit their participation in certain activities. Such participation restrictions often occur because the person is concerned about how others might react to disfluent speech. Other people may find that they are excluded from participating in certain activities because of stuttering. Clearly, the impact of stuttering on daily life can be directly related to how the person and others react to the disorder.\nDeborah Boswell has a passion for working with people who stutter, through individual sessions and through her monthly stuttering support group, which is sanctioned by the National Stuttering Association. This is an encouraging, supportive group of people. We welcome anyone who is interested to attend. Contact us for more information.\nIn addition to speech therapy services, Deborah Boswell offers SpeechEasy\u2122, the smallest, most advanced and effective treatment for the problem of stuttering. We are the only authorized provider of this product in North and Central Alabama. SpeechEasy\u2122 is small enough to fit completely in your ear canal, yet it uses an advanced microprocessor. It is digitally programmable with special software for a wide range of settings of Delayed Auditory Feedback (DAF) and Frequency Altered Feedback (FAF). We will work with your schedule to establish a three-hour SpeechEasy\u2122 consultation to see if this device will be of benefit to you.\nContact us today for a free phone or email consultation about Deborah's services.\nMany adults experience difficulty with spoken and written grammar as well as weak vocabulary skills. Deborah Boswell can work with you to improve your verbal expression so that you can speak with confidence in any situation.\nContact her for a free phone or email consultation with no obligation.\nMany people feel disadvantaged in the American workplace because their speaking voice differs from what is known as standard American English. It's proven\u2014the way you talk affects what others think of you, but the good news is you CAN change the way you speak.\nWhy would you want to change your accent?\nDialect reduction and accent reduction training is one of Deborah Boswell's areas of expertise, and she's eager to help you. Whether you speak with a heavy Southern accent or a foreign dialect, Deborah's proven approach to dialect reduction or accent modification coaching doesn't remove your accent completely, but it will give you tools to smooth out your speaking voice in a way that is authentic to you. This kind of personal improvement could result in greater confidence and respect in business or social situations. It could also have a big impact on possible career advancement.\nDeborah has extensive experience working with clients with various international dialects, including those from India, Spain, Turkey, Middle Eastern countries and others.\nWhat issues do you normally see when internationals come to you for accent modification services?\nDeborah and her expert trainers use one of the leading curricula for dialect reduction and accent reduction training, Compton's Pronouncing English as a Second Language. Seven-week and 13-week training programs are available, depending on the need. Session times are determined around your schedule. In special cases, Deborah Boswell can work on time-sensitive projects and for \"single sound\" speech difficulties.\nDeborah describes the process of helping a client reduce a foreign or regional accent.\nBecause of her experience taming (but not eliminating) her own \"wild Southern accent,\" she is passionate about providing this service either as coaching\/consulting or as a corporate seminar.\nIf you think you need speech or language therapy, there are three basic stages: assessment, treatment plan and treatment. The assessment involves sitting down with you and understanding your needs and your goals, and conducting diagnostic testing when applicable. We then develop a treatment plan that allows both therapist and client to agree on areas of focus, the frequency of sessions, and an estimated length of time you will be in treatment.\nThere are no cookie-cutter approaches to Deborah's services; every client is different. Your treatment will take place mostly in our office, but there will be homework consisting of practice and exercises you will do at home. Speech therapy can be hard work, but ultimately meeting your goals will have life-long effect: better communication with others and increased self-confidence.\nContact us for a free phone or email consultation with no obligation. If we determine, together, that an initial meeting is in order, we will evaluate your speech or language skills and consult with you on appropriate treatment. If treatment is necessary, we will set up a therapy program customized for your needs and schedule.\nWe host a monthly stuttering support group, which is sanctioned by the National Stuttering Association. We welcome anyone who is interested to attend. Contact us for more information.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Financial assistance is available for memberships and programs in instances where costs of participating would prevent participation. This is an application based program, with verification of income required. Financial assistance may not be available for certain programs \u2013 please contact membership services to determine if the desired program is eligible.\nApplications can be picked up and turned in at the Membership Services Desk.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzwcyn b/data_all_eng_slimpj/shuffled/split2/finalzzwcyn
new file mode 100644
index 0000000000000000000000000000000000000000..153c5d7a945e03c9d40e8b2361207bf30735454f
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzwcyn
@@ -0,0 +1,5 @@
+{"text":"Remove the tough stems from one bunch of kale. Tear leaves into bite-sized pieces and place in a large bowl. Drizzle your favorite healthy oil (olive oil or avocado oil are my favorites) over the leaves. Add an acidic medium, such as lemon juice or vinegar. The acid in citrus juice or vinegar helps to boost the absorption of iron and calcium from the kale and also breaks down the kale leaves during the massage.\nSprinkle with coarse salt. Roll up your sleeves and use your hands to massage the kale greens until they start to break down and become tender and a darker green color \u2013 the leaves should look wilted, almost like lightly saut\u00e9ed kale greens would look.\nAt this point, you can add any additional toppings or vinaigrette or just eat it plain as a kale salad. Enjoy!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Silk garment lovers can check out a wide array of traditional Indian weaves at Silk India 2016 Exhibition. The exhibition brings you specialties and Different varieties of silk that are reared based on climatic and geographic conditions of which only four namely Tassar, Eri, Mulberry and Muga are the prominent ones. The expo will showcase live demonstration of weaving.\nThe exhibition has all varieties of saris right from Hand Block Print Saris, Suit & Silk Bed Covers, Designer dress materials & Border, Lazes, Kurtis, Hand Woven Matka & Assam Muga Fabrics, Apoorva Silk Saris, Baluchari, Dhaka Masli, Gicha Saris, Boutique Saris, Kantha Saris, Zordoshi,.\nSo come along and get ready to be amazed by the beauty of these artistic pieces of saris!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"GeoCue Corporation is North American's largest provider of products, training and consulting services for airborne and mobile LIDAR applications. We are the North American authorized distributor for Terrasolid OY, the purveyor of the world's most widely used LIDAR production tools. We are also the parent company of QCoherent Software LLC, the world's only plug-in LIDAR exploitation solutions for the ESRI ArcGIS desktop products.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Yemeni armed forces spokesman said on Monday evening that the heroes of the army and popular committees carried out a very large operation in Haradh, where they controlled more than 6 square kilometers.\nHe confirmed that dozens of American-Saudi aggression's mercenaries were killed and wounded during the operation, pointing out a state of horror and confusing filled their ranks though they were support by the aggression's air force .\nHe explained that by the assistance of Allah , the army and popular committees could control the strategic mountain of al-Hesnain mountain , Hawthalah , Shaab al-Naba and the adjacent positions.\nHe mentioned that the operation was so wide and continued for 7 hours , led to make a number of what so-called the first brigade flee.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"With The Exquisite Hotel Collection easily recreate that deliciously crisp feeling of climbing into a hotel bed for the first time...for much less than the cost of a one night stay. Our MicronOne premium pillow provides a superior level of protection against dust, mites, even bedbugs, in an attractive, streamlined design. Featuring a double needle stitch with a light grey satin cord, this pillow is a superb combination of protection and style. Filled with 100% Polyester Gel, these superior pillows provide the softness of down with the perfect level of support for everyone, no matter your sleep style. Stomach sleeper? Select a soft pillow, which guarantees proper neck alignment, resulting in a restful night's sleep and pain-free morning or a firm pillow that will lovingly nestle your neck and head, allowing perfect alignment of your spine. Wake up feeling rejuvenated!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzwxwa b/data_all_eng_slimpj/shuffled/split2/finalzzwxwa
new file mode 100644
index 0000000000000000000000000000000000000000..e011810a8fbef8092ed18a41b0239af662eb36fc
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzwxwa
@@ -0,0 +1,5 @@
+{"text":"A local privilege escalation flaw in Windows is worrying and it is bad news if exploit code is readily available. What's even worse is when no software patch exists and there are no mitigations or workarounds to deploy while waiting for one.\nThat was the situation for many Windows administrators this week when a security researcher with the handle @SandboxEscaper publicized a local privilege execution vulnerability affecting the Advanced Local Procedure Call (ALPC) interface in Windows Task Scheduler (VU#906424) on Twitter and released a proof of concept on GitHub. The flaw lets a User account gain the privileges for a high-level System account.\nThe researcher did not notify Microsoft before publishing the proof-of-concept, which means Microsoft does not have an update to address the flaw, let alone an advisory discussing the vulnerability. Microsoft appears to be aware of the security flaw, as the company has released the following statement: \"Windows has a customer commitment to investigate reported security issues, and proactively update impacted devices as soon as possible.\" The next official patch release is set for Sept. 11, but it is not clear at this time whether an update for this vulnerability would be available on that date.\nThe fact that this vulnerability can be exploited reliably and that attack code already exists puts Windows administrators in a bind because there is no official guidance on what to do. To address this gap, security consultancy Acros Security released code that can block attempts to exploit this flaw on Windows 10 and Windows Server 2016 systems on its 0patch platform.\nAcros calls the code a micropatch, and it is designed to act as a hotfix that can be applied and removed quickly without impacting users. After the micropatch is downloaded from the server and stored on the machine, the code resides in memory and waits for a running process to load the exploit it was designed to block. The micropatch kicks in only when a process runs that exact exploit.\nA \"0patch loader\" component is \"in charge of intercepting every module load event and applying all micropatches for that module in memory before any code from that module gets executed,\" Acros CEO Mitja Kolsek said.\n\"Windows update simply replaces the vulnerable executable that we have a micropatch for with a corrected executable that has a different cryptographic hash.\"\nA Windows-internal mechanism, the ALPC interface allows client processes running within the operating system to ask a server process to provide some information or perform an action. The issue exists in an improperly secured API in the TaskScheduler service and allows an attacker with access to overwrite most files on the system, and when that file is loaded into a privileged System-level process, the contents of the file executes with the higher privileges. The proof-of-concept overwrote a file used by the printing subsystem on Windows so that if a user tried to print something, the operating system executed the attack code in the file with higher privileges.\nSince all code changes by the micropatch are done in memory, none of the files on the system get changed. This means all signatures on files remain valid so any integrity checks will remain unaffected, Kolsek said. When the Windows administrator eventually installs the official Windows update, the patch doesn't conflict with the micropatch.\nThese small updates\u2014this one is a mere 13 bytes\u2014can complement official vendor updates or address issues that won't receive official patches. Administrators interested in this micropatch would first need to install the 0patch Agent client but existing users should be able to see the micropatch automatically. And administrators don't have to worry about rolling back any code post-patch.\nThere is a lag time between when security updates become available and when enterprise IT teams finish testing and rolling them out into their environments. These micropatches are supposed to bridge that time interval so that enterprises can be protected while official patches are being tested.\nThere have been no attacks exploiting this vulnerability observed in the wild. It also helps that the severity of the zero-day vulnerability was limited by the fact that the targeted computer has to already be compromised and the attacker has to have access to a local account before exploiting the flaw. However, a user with local account privileges can exploit the flaw to then gain high-level privileges. An attacker could chain this privilege escalation flaw with a different bug, or trick a user into clicking on a phishing email or downloading malware.\nThis kind of privilege-escalation flaw can be used to break out of sandboxes and gain control of the system.\nAcros has also released the code it used so that administrators reluctant to get the micropatch can write their own code fix while waiting for Microsoft. This is one of those times when administrators don't have to wait for an official patch to defend themselves from potential attacks.\nMicrosoft has taken over 99 domains used by the Phosphorus attack group, which has ties to the Iranian government.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Homegrown Miami chef Jose Mendin has created an original blend of Old World elegance and New World vitality. Chef Mendin is fusing masterful Michelin-star techniques with local, farm-fresh ingredients to craft a menu that respects the elements and honors simplicity. Daily catch pulled fresh from the sea, vegetables plucked from the garden, and grass-fed quality meats are transformed into crudo specialties and dishes fire-grilled to perfection. At Habitat, the sophisticated service is as warm as the atmosphere and five-star cuisine is delivered family-style because great meals are meant to be shared. It's just more fun that way.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Through your generosity, Federation sends hundreds of young adults annually to experience our Jewish world firsthand. Here is Giana's story from her recent trip with March of the Living.\nAuschwitz. I know this word from pictures, videos, and the literary documentation of the events that took place at this concentration camp active during 1940 and 1945. Yet, this word was just simply a concentration camp, of course there are undeniable emotional responses evoked by the word alone, but there I was unaware that I had a major gap in understanding of the tragedy that occurred.\nAbout thirty minutes ago I was standing inside gas chamber one, facing the walls that were coated in scratches from top to bottom. Small etched Stars of David were weaved in with the marks that fingernails left on the cement. No picture can evoke the same experience as touching your own hand with the marks of desperation that were left by Jews experiencing their last moments gasping for air.\nLooking at an image through a screen can say a thousand words but standing on the same soil that contributed absorbed so much blood of my heritage. Nothing but this experience can close the gap in my understanding in how 11 million people who had futures and families were gone from the earth 6 years later.\nThe horrors that we saw today are unimaginable. No matter how educated you are on the holocaust I believe that no experience can replace the effects of standing where our ancestors once stood.\nEvery student on this trip can offer a particular moment that they can declare their gap in understanding closed. A moment that is unique to them because only they experienced it in that exact way. A moment that changed them forever. I experienced this moment today. Walking into the area that specifically was built for the execution through means of shooting and hanging I began to process. As I approached the wall of metal made to line the sentenced in front of I found myself fallen to my knees with tears dropping into the mud.\nMy surrendering was unexplainable. This sudden catharsis of emotion was full of fear sadness relief. All emotions surely felt by these people who were faced their brutal death in this same spot.\nI got to walk out of Auschwitz today. 11 million people didn't have that chance. It was stolen from them. Let it be our mission to honor their memories and celebrate those who did survive and walked out of those gates.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Object-Based Urban Detailed Land Cover Classification with High Spatia\" by Ruiliang Pu, Shawn Landry et al.\nImprovement in remote sensing techniques in spatial\/spectral resolution strengthens their applicability for urban environmental study. Unfortunately, high spatial resolution imagery also increases internal variability in land cover units and can cause a 'salt-and-pepper' effect, resulting in decreased accuracy using pixel-based classification results. Region-based classification techniques, using an image object (IO) rather than a pixel as a classification unit, appear to hold promise as a method for overcoming this problem. Using IKONOS high spatial resolution imagery, we examined whether the IO technique could significantly improve classification accuracy compared to the pixel-based method when applied to urban land cover mapping in Tampa Bay, FL, USA. We further compared the performance of an artificial neural network (ANN) and a minimum distance classifier (MDC) in urban detailed land cover classification and evaluated whether the classification accuracy was affected by the number of extracted IO features. Our analysis methods included IKONOS image data calibration, data fusion with the pansharpening (PS) process, Hue\u2013Intensity\u2013Saturation (HIS) transferred indices and textural feature extraction, and feature selection using a stepwise discriminant analysis (SDA). The classification results were evaluated with visually interpreted data from high-resolution (0.3 m) digital aerial photographs. Our results indicate a statistically significant difference in classification accuracy between pixel- and object-based techniques; ANN outperforms MDC as an object-based classifier; and the use of more features (27 vs. 9 features) increases the IO classification accuracy, although the increase is statistically significant for the MDC but not for the ANN.\nPu, Ruiliang; Landry, Shawn; and Yu, Qian, \"Object-Based Urban Detailed Land Cover Classification with High Spatial Resolution IKONOS Imagery\" (2011). School of Geosciences Faculty and Staff Publications. 361.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"No Wall Moisture. No Callbacks. No Problem.\nCommercial and residential architects know all about unintended consequences. In the drive to build ever-tighter, energy-conserving buildings, the result often includes an unwelcome outcome: wall failure from moisture. Learn more about our engineered rain screen drainage mat.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Protected area of hills and woodland.\nThe national park is a nice area with wome good walking paths. It takes about an hour to go from the entrance to the cliffs at the end of the park with some interesting WW2 ruins.\nThis is well worth an early morning or late trip. The scenery is stunning. It's quite and just lovely. You need your walking shoes to do the park justice and don't forget to take a picnic and plenty of water.\nLovely (but long!) walk around the national park with great views of the coast. Took the longer, flatter root and saw deer and donkeys. Take lots to drink as its further than it seems!\nWe have just spent a week in Alghero and visited the Parco di Porto Conte most days for a walk. Really pretty area and was pretty quiet generally.\nThis natural park is absolutely gorgeous this is another face (and our favorite) of Alghero. The nature is superb and there is a nice small beache. Quiet and in connection with the natural preservation you can observe birds and animals.\nGet quick answers from Parco di Porto Conte staff and past visitors.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"With its sweet fruity flavours Candy King is one of the most popular juices on the American market. We have all flavours and nicotine strengths in stock ready for quick dispatch for you to enjoy!\nAll e-liquid will arrive in a 100ml Shortfill bottle, simply then add your nicotine to boost it to 120ml (3mg\/1.8MG).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On the 24th April, Google launched an (algorithm) update to their live search engine. This changed the way that search results are filtered for ranking positions. Their intention for this update was to eliminate the presence of webspam in the top pages of their search results pages. Webspam in this case refers to websites that use questionable tactics to game search engine results and shortcut their way to successful rankings. In Google's words, such sites \"violate Google's webmaster guidelines\".\nWhat does this mean for ordinary businesses, entrepreneurs and webmasters?\nThis was a particularly significant and potent update, and reportedly impacted about 3% of search results.\nIn many cases, not only distinctly webspam sites were affected negatively, but also legitimate, good quality websites that do not engage in irresponsible promotional tactics, are neutral, or work hard for their positions as well.\nThis is because Google is still only a machine and is not perfect at deciding when someone is performing questionable tactics and when the aspects of the website or website promotion (such as links on other websites, to a website) occur naturally or simply by accident.\nSo the criteria that are considered for identifying \"webspam\" are still within a gray area, and as a net result, many small to large businesses have suffered losses in search traffic, without really having warranted it at all.\nWhile this seems unfair, it will remain a reality until Google improve the accuracy of their search ranking filters, over time.\nWhile some of the better sites may have been knocked down, it can be assumed that the actual spammers were likely hit even harder, and some may have slipped through the cracks. Time and the everyday users and quality raters of Google Search will tell.\nSo, how does a business move forward with search marketing?\nIt appears that while Google have made it more difficult for smaller, newer websites to compete for rankings on expensive, high-competition, high-traffic volume keywords, they have also rewarded sites that produce good quality, useful content, and gradually build up the authority of their website, via responsible promotional methods such as relevant link building and social media engagement.\nSo the way to move forward is to ensure that the marketing of a website is performed responsibly, and to expect that good results and success always require long term commitment.\nCan spammers get to my site?\nSince offsite factors may be looked at as signals for spam by Google, it is possible that an aggressive campaign could be employed by any given unscrupulous competitor to derail the performance of website in the search results.\nThis form released by the Google webspam team can be used to give feedback if you believe that your drop in performance is completely unwarranted, or due to an attack by black-hat search marketing saboteurs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Building or Strata Manager for a Strata.life property?\nClaim a building and offer your residents an always-on virtual concierge to help residents access useful information, services and people.\nOur automated concierge can reduce inbound resident service requests by 30% through resident self-service and better, more efficient building communications. Reduce emails, phone calls, FAQ and form requests as well as automate move-ins and problem reporting and support, particularly for common building problem requests.\nResidents can also subscribe to a building communication list to receive building notifications and updates.\nFor Strata Managers we can also receive and process property Strata Reports for your properties.\nWant Strata.life to support resident self-service for more than one building or an entire strata or facilities management portfolio? Contact us for more details.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The world premiere of Go Seek, the new down under snowboard flick from Transfer and Reason Films, went down at Revolver Bar in Queenstown on Friday night. There was a great turnout, as the kiwi shred community came together to enjoy the home grown movie and celebrate the start of the 2011 season.\nThe crowds began to drift in from 7.30pm, eagerly anticipating down under's latest snowboard release. According to the videographer, Ben Ryan, \"there were loads of people at the bar getting rowdy, I was stoked on the turnout!\" At about a quarter past ten, he fired up the projector to play the feature.\nThe Jindabyne premiere is this Wednesday night, 15 June, at 8pm at the Jindabyne cinema, followed by the after party at Clancy's Bar in the Banjo Patterson Hotel from 9.30pm. It promises to be an equally awesome night, as the Aussie crew gets their turn to catch up at the start of winter, and enjoy the latest home grown snowboard movie. On the night, just $10 will get you the new Transfer mag, Issue #7, plus entry to the premiere, plus entry to the after party.\nThe Queenstown snowboard community packed into Revolver Bar for the much awaited preview.\nAnticipation was at an all time high when the DVD title screen came up.\nThe crowd glued to the screen for the world premiere of Go Seek.\nGo Seek star, Nick Hyne, and friends enjoying the premiere.\nThe man behind it all, Ben 'Bugs' Ryan. Ben captured most of the Go Seek footage and was responsible for the mammoth task of editing the film. Once you see Go Seek, you'll realise he is one talented character!\nSpace cadet and a couple of girls enjoying the party.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzxkns b/data_all_eng_slimpj/shuffled/split2/finalzzxkns
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+{"text":"JCJNutrition is Calgary's newest comprehensive health and nutrition management company. Our goal is to be your first choice for nutrition planning and information for your family, residents, or your customers.\nJCJ can provide programs, recipes, and product development based on quality foods and a nutritional balance.\nVisit our online store to find the right plan for your family or company!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"While it is not at all clear just what improvements in cycling infrastructure we will really get with the rebuilding of Christchurch many of us are interested in the outcome, and have some ideas to share.\nAs with most things there is not always agreement. Thanks to a lot of work, some of it done here in Christchurch, we do have a pretty good idea of what people who would like to cycle want if they are to take it up. The key ingredient is safety.\nBy safety they mean that they want to be able to ride their bicycles to where they want to go without putting their lives at risk, hardly a surprising or unreasonable request. No doubt non cyclists have a one sided view of cycling's hazards. Looking out from a car window at cyclists kitted out in helmet and flouro and often being abused by drivers all encourages the image of cycling as dangerous. Regular cyclists and statistics show that cycling is not nearly as dangerous as it may appear. That it needs to be safer, even pleasant is obvious.\nAcademics and policy makers have identified three groups of cyclists. The \"Strong and Fearless\" cyclists are accustomed to putting up with the inadequate cycling infrastructure now provided. They make the best of poor road conditions and continue to ride regardless. The \"Enthused and Confident\" cyclists are up to most rides but would welcome well marked, even separated on road cycle lanes and intersection improvements.\nThe last group is the largest and often focused on. The \"Interested but Concerned\" are not likely to commit to serious utility cycling until a network of separated paths is provided for much of their cycling routes. This group is the largest at 30-40% of the population and research has found that they would cycle more if they felt safe. This pent up demand for cycling can help all road users, even motorists.\nProviding a complete network of separated paths is not likely to happen. Government has shown no appetite to commit adequate funding and there are practical problems with trying to install an alternative transport network in an established urban setting. Besides, streets do access most everywhere already and reallocating carriageway space and sharing what's there makes sense on many levels.\nWe can develop quite a few separated cycle networks, along rivers, railway lines and coastal frontage. These can be connected together by traffic calmed local streets with reduced speed limits. The current plans to have cycle networks share major arterials with cars, trucks and buses may be tolerated by the strong and fearless, but is not likely to encourage anyone.\nWhat is clear is that all of us will need to insist on support and funding for better cycling infrastructure. Personal preferences aside, if the infrastructure is not inherently doomed to disappoint, it is likely worth supporting.\nForward looking governments and academic researchers have been busy looking at what is being done to meet all road users' needs. They have also been looking at how we can get people free of their cars.\nThe \"Complete Streets Policy Workbook\" addresses the need for local governments to adopt policy which will achieve real transport choice for all. They found that complete networks for all modes are key. Building a few model streets may be good for publicity, but if it fails to meet actual needs it will only disappoint and discourage.\nWith all the talk of streets for people in the Central City Plan the Australian Cycling Resource Centre's Shared Zones report is of special interest. They found that such zones are of limited application, where pedestrian numbers are very high and speed limits are no more than 10km\/h. Road markings, kerbs and gutters are eliminated and the chaos necessitates care and cooperation. This can be a good approach for a high street with footpath cafes, a street scene and alternate routes provided to allow vehicles and bicycles to avoid it.\nBicycle Boulevards are a great way to deliver people friendly streets which also offer pedestrians and cyclists safe pleasant routes.\nBicycle boulevards meet peoples' need for safe spaces in which to live, work, commute and shop. Traffic is calmed by 30km\/h speed, intersection treatments and signage. Cyclists can safely use existing roads, thereby saving on costs and perhaps generate political support for this win-win solution. Other benefits include increased patronage of local merchants and enhanced real estate values. People are willing to pay more to live in a people friendly neighbourhood. This approach is also known as Local Street Bikeways, Bike\/Walk Streets, and Bicycle Priority Streets.\nThe actual approaches used, cycle lanes on the inside of parked cars, on the outside of parked cars, no parked cars, off road, on road, intersection treatments, etc are touched on in this report commissioned by the NZTA and written by University of Canterbury researchers.\nTheir findings from the literature, from surveys and from focus groups can help us understand each other and perhaps even learn to work together in lobbying for better infrastructure.\nWould be cyclists want a comprehensive easy to use network of cycle-only paths, with special provisions made for cyclists at intersections. Good end of trip facilities such as covered secure parking, lockers, showers, repair options were also cited. People also understand that efforts to educate all road users in how to safely use new infrastructure, how to behave, will be required.\nSome readers may scoff that government will never fund infrastructure at this level. Politicians will simply reject all cycle funding as it is simply too expensive to create a safe cycling environment. We saw this with the Christchurch City Council ban on new cycling infrastructure during the mid 2000's.\nBut this is what would be cyclists want, and there are a lot of them. Results from the study found that over 30% of non cyclists are open to cycling. It is not unreasonable to think that this number would grow if cycling were actually supported. Reminding would be cyclists of this desire come election time will be important.\nThe bottom line is safety. Safe Cycling for All ages and Abilities is what is required. Gil Penelosa, the former Director of Parks and Recreation in Bogota, Columbia describes '8 \u2013 80 cities', the idea being that if we build communities that suit 8 and 80 year olds then they will suit everyone. We need Rides of Neighbourhood Significance, not Roads of National Significance.\nThis report is a good source for links to the health, local economic, pollution reduction and community benefits of cycling. It provides citations which offer insights and details on many of the infrastructure approaches being used worldwide.\nAchieving the goal of high quality comprehensive cycling infrastructure is going to take all of us. While there are many possible approaches and solutions it will be important to strongly support all that are not clearly doomed to disappoint. No doubt we will need to build on, or even around, some solutions. Having some in place can help more people take up cycling. With their new found freedom and skills they will likely develop experience and confidence to demand more and to know what they need.\nShould I get the comments to indicate an interest I will explore cycling infrastructure in more detail in a future piece.\nI for one would appreciate more on infrastructure options. If we can get a consensus it strengthens the case for action. The feedback for the Christchurch Transport Plan shows that there is a wide range of ideas.out there.\nThanks for sharing this Jim. Refreshing ideas and good to see there is overwhelming support for cycling amongst submitters.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"After low-lying clouds scrubbed the first attempt, STS-130 took off at 4:14 am in what is scheduled to be the last night launch of a space shuttle. The Space Task Force was on hand for the historic launch of Endeavour\u2013as we will be for each of these 'Final Five' launches in 2010.\nSTS-130 is carrying the Tranquility module (formerly known as \"Node 3\") along with the special multi-window \"cupola\" viewport. The crew of Endeavour will spend their first two days catching up to the space station before docking to start their mission.\nThe orbiter is scheduled to return to Kennedy Space Center Stay late on Feb 20 after 13 days in space. Stay tuned to the Space Task Force as we continue our extended mission coverage of STS-130 from rollout to the pad to rollout on the runway.\nYou can also download the video here: STS-130 Launch!\nThis is what it looks like at 4:30am in the KSC Press Center when a launch is scrubbed.\nAll eyes are on the Flight Director, waiting for his call. The launch polling just ended with two \"NoGo\" calls due to weather. Things have been tentative and getting worse all day.\nFinally, he makes the tough call and scrubs the launch attempt. Everyone is a bit disappointed, but we know it is the safest thing to do. Now the press, much like the astronauts, must clean up enough to move out for a few hours and come back tomorrow.\nUnlike the astronauts, we can get in our cars and go back to hotels (or for some of us, home to our families). The crew of STS-130 has hours of work to do shutting-down the orbiter and getting out of their launch suits before they can retire\u2013in quarantine\u2013to the confines of the astronaut quarters.\nAnd yet, there is a bit of excitement in the air knowing that we get to do it all again. We'd all prefer to have a good launch the first time, but there is also a knowledge that we won't have many more shuttle launches. Very few of the Go\/NoGo pollings are left to call. We'll cherish every one and, if needed, come back and do it all again.\nGood night, everyone. See you tomorrow!\nWe were on hand for the incredible night launch of STS-128. The Space Shuttle Discovery took off just before midnight on August 28, 2009. It's the 128th shuttle flight.\nWe did something a little different for this video, we were at a viewing point three miles away. The camera is on a very smooth surface. The result is a pretty awesome representation of what the shockwaves feel like.\nTake the time to see one of the remaining shuttle launches, it's an experience you'll never forget. More importantly, if you have children, give them the experience of a lifetime, by taking them to a launch. There's plenty of information on the best viewing points for a shuttle launch. Check out SpaceFlightNow.com or NASA.gov for the schedule of the remaining flights.\nWait a minute, what's that in the water?\nA big preview of ISDC in Orlando, we'll take you on a visit to KSC to find two shuttles on the launch pads \u2013 and what's that in the water? Plus Tim takes us behind the scenes for a very special day at KSC. Get ready to spend some time in the \"no gravity zone\" as the Space Task Force launches right now!\nGeorge and Tim take you to the front lines of a shuttle launch at the Kennedy Space Center in their first ever video podcast.\nLaunch of STS-120 to the International Space Station, October 3, 2007.\nA very busy week in space. We cover developments on board the International Space Station and the mission of Space Shuttle Atlantis. Tim talks about watching a top-secret Atlas 5 launch from his driveway and the launch of The Wonder of it All, a new movie dedicated to the Apollo astronauts. For me \u2013 it's all about decorum in Zero-G (Bach versus Korn) and oceans on Mars (wonder what a Mars shark would have looked like?).\nBut wait \u2013 there's more \u2014 don't miss our review of \u2013 Mr. Bailey goes to Washington.\nAll ahead on Episode 8 of the Space Task Force.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"John Zimmer is from Canada and lives in Geneva. With degrees in International Relations and Law, he has 30 years of public speaking experience. John has worked for a major Canadian law firm, the United Nations and the World Health Organization (WHO). He is now a full-time speaker, giving keynotes, emceeing events, and helping companies and organizations give better presentations. His clients include the International Volleyball Federation, Roche, the WHO, the World Intellectual Property Organization, Zalando and Executive MBA programs at the University of Geneva, University of Lausanne and IESE. John is a 7-time European champion of Toastmasters public speaking contests and the co-creator of Rhetoric \u2013 The Public Speaking Game\u2122. He writes a public speaking blog, Manner of Speaking, that Prezi lists as one of the \"Top 100 Online Resources Every Presenter Should See\".","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bungalows do exist in Hawaii; it sometimes just takes a bit of effort to find them. You may also have to stretch the definition of \"bungalow\" in some cases, particularly for those attached to major hotels. Those of the modest variety are outnumbered by high-end bungalows where all you want and need is at your fingertips. Enjoy the hunt, and when you arrive, immerse yourself in the Hawaii vacation bungalow experience.\nFinding a bungalow on Oahu means moving out of high-rise-minded Waikiki. Turtle Bay Resort (turtlebayresort.com) on the North Shore offers a hotel setting and decadently designed beach cottages within steps of the water. The wall facing the beach completely opens up, providing an ocean view from inside the cottage or the outdoor lanai. Guests have full use of the resort amenities as well as a cottage concierge. Private, elegant and wired for sound, each cottage is surrounded by lush greenery, flowers and the sound of ocean waves.\nAlternatively, rent a bungalow at Black Point, an exclusive gated community in the upscale Kahala neighborhood. As an example, Hana Beach Rentals (obrhi.com) offers a three-bedroom, three-bathroom villa with ocean views, use of the private pool and access to the beach. Another option, the Waimanalo Beach Cottages (waimanalobeachcottages.com), offers studio, one-, two- and three-bedroom cottages. Though not as lavish as the Black Point property, they are more budget friendly. Waimanalo, on the eastern end of Oahu near Kailua, has some of the most stunning coastal views on the island.\nThe Fairmont Kea Lani (fairmont.com) and its distinctive sail-like roof line are fixtures of the Wailea area. The main hotel offers one-bedroom suites with kitchenettes and large balconies. The property's oceanfront two- and three-bedroom villas offer more space, privacy, easy beach access and even a private plunge pool. Top-of-the-line furnishings, marble bathrooms and high-end entertainment centers are part of the package.\nGoing to a private rental firm is another alternative on Maui. West Maui Lodging (westmauilodging.com) offers an assortment of cottages scattered along the western end of the island. Examples are the one-bedroom, one-bath Ohana cottage with ocean views or a two-bedroom, two-bath Ocean Cottage big enough for six people and within walking distance of the sands.\nThe Mauna Lani Bay Hotel and Bungalows (manalani.com) on the Big Island's west coast offers the ultimate bungalow experience. Set alongside the main arrow-shaped hotel, each 4,000-square-foot bungalow has two bedrooms, two bathrooms, a private pool, hot tub and sundeck. Depending on location, your bungalow is either oceanfront or ocean view. The hotel, which stocks the pantry before your arrival, includes round-trip transportation from the Kona International Airport, welcome massages and buffet breakfast in the rental rate. The entrance to the bungalows is separate from the hotel. Relax in your own personal space or take advantage of the hotel amenities; it's your choice. This is definitely a splurge property.\nThe Hapuna Beach Prince Hotel (princeresortshawaii.com), also on the Kohala Coast, has its own beachfront hideaway. The Hapuna Suite is an 8,000-square-foot villa on the hotel grounds. This four-bedroom property comes with its own entrance, floor-to-ceiling glass windows overlooking the Pacific and butler service. If you don't want to walk to the beach, take a dip in your own pool or work on that tan on your private deck.\nThe western end of Kauai was once home to sugar cane plantations and not much more. In the early 1900s it was common practice for plantation owners to build small cottages for the workers and their families. The Waimea Plantation Cottages (waimea-plantation.com) turned these low-rise, modest homes into rental cottages. They sit on a palm-studded grassy meadow fronting a sandy beach that's great for strolling but not so much for swimming. Each cottage is individually decorated with period furnishings and is charmingly simple. You might find a claw-foot tub, an old-fashioned kitchen or a lanai that looks bigger than the cottage. Some have oceanfront views, but most are hidden among the palms, offering views of the pool area or the grounds. This laid-back resort offers yoga, water aerobics and art classes. Dine at the on-site Grove Cafe or have the chef prepare a barbeque care package to cook on the grill outside your cottage. If you've seen the TV mini-series \"The Thorn Birds,\" the lobby might look familiar; it was used as the plantation home in the film.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzxkqe b/data_all_eng_slimpj/shuffled/split2/finalzzxkqe
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index 0000000000000000000000000000000000000000..4321c201f42e8df9344780162ccbf741a0e42f25
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@@ -0,0 +1,5 @@
+{"text":"CineSound AV allows your video and audio files to be experienced by the largest possible number of users by checking which multimedia player plugins are available on their browsers and supplying the appropriate embedded code to play the file. Upload separate files for each plugin and set common playback behaviours such as autostart, loop, show controls and hide on a per-clip basis. Plugin-specific behaviours can aso be set.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"House security checks \u2013 an officer inspects the home for security and makes recommendations to improve security.\nVacation checks \u2013 a resident's home is checked twice daily while on vacation.\nLockout service \u2013 if a person locks their keys in his or her vehicle, the police department will send an officer to attempt to open the vehicle.\nFinger printing of children \u2013 available for all grade school children.\nFinger printing for citizenship \u2013 residents may request that prints be taken for purposes of naturalization.\nCommunity Awareness Meetings \u2013 residents and police work together to solve problems and find solutions.\nCall (708) 202-4374 for information.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"All copyrights regarding the text, photographs, illustrations, videos, software, and other contents provided on this website (hereinafter known as the Contents) belong to Makino Milling Machine Co., Ltd. and its affiliated companies (hereinafter collectively known as Makino) and companies having business ties with Makino. The Contents may not be copied or reproduced without any authorization of the copyright owner. If the conditions of use of the Contents are individually indicated by the copyright owner, the conditions are also to be followed. Measures (warning, prosecution, claim for damages, injunction, and request for honor vindication measures) will be taken for persons who have violated the regulations with regard to the copyrights according to the copyright law.\nThe rights regarding to the trademarks, logos, and trade names provided on this website belong to Makino or the owner of each right. These rights may not be used without permission.\nWhen exporting (or providing to nonresidents) technologies and programs obtained from this website or products purchased from this website, comply with the Foreign Exchange and Foreign Trade Act and any other applicable laws and regulations.\nThis website records the information of visitors in the form of an access log. The access log includes the visitor's domain name, type of browser used, access date and others. However, it does not include personally identifiable information. The access log is used for maintenance of this website and statistical analysis regarding its utilization status, and it will not be used for any other purposes.\nAdministrators wishing to link to this website must first contact Makino providing an administrative overview of the site from which the link is to be established together with details on the reason for the link, the URL of said site, and all other pertinent information. Makino will not reply to the transmitted contents in principle. However, note that there are cases in which requests for links may be rejected. Please make the link to the top page of this website in principle unless there is a special reason. Makino shall be referred to as \"Makino Milling Machine Co., Ltd.\" in any links to this website.\nThis website may be linked to websites of third parties other than Makino. Please note that Makino provides these links only for convenience and does not bear any responsibility for the Contents of the linked websites. When using the linked websites, please observe the relevant regulations.\nMakino does not grant the copyright, patent right, trademark right, or any other rights of Makino or third parties with regard to the Contents of this website. Also, Makino does not guarantee the accuracy, usefulness, reliability or security of the Contents appearing on this website. Makino disclaims all responsibility for any loss or damage that may be incurred for any reason whatsoever as a result of accessing or being unable to access the Contents of this website. Makino may alter or delete the Contents or URLs of this website without prior notice. Please note that the information provided on this website may not always be the most up-to-date available. In addition, Makino may interrupt or terminate operation of the website or access to the Contents of this website without prior notice. Makino disclaims all responsibility for any loss or damage that may result for whatever reason from these events.\nMakino may change the Site Policy without prior notice. In this case, the updated Site Policy will apply. Your understanding is appreciated.\nCopyright \u00a9Makino All rights reserved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"With each new IBM-distributed software license you acquire you get affordable, renewable IBM Software Subscription and Support.\nBenefits : Streamlines budgeting for software upgrade and migration costs.\nDeep expertise in bite-size chunks for busy people. Register for our regular program of Webinars.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The bills in both states promote advanced recycling facilities that convert plastic scrap into raw material via chemical recycling.\nIowa and Tennessee have just passed bills supporting advanced recycling facilities that convert plastic waste into raw material via chemical recycling, which includes pyrolysis and depolymerization.\nRecycling Today reports that in Iowa, converting 25 percent of the state's post-consumer plastics into feedstocks and transportation fuels could support five advanced recycling and recovery facilities and generate $309 million in economic output annually. And in Tennessee, converting 25 percent of the state's post-consumer plastics into manufacturing feedstock and transportation fuels could support eight advanced recycling and recovery facilities and generate $264 million in economic output per year.\nA report released by the American Chemistry Council (ACC) in March revealed the potential economic impact of expanding advanced plastic recycling and recovery technologies in the United States to be nearly $10 billion. ACC examined a burgeoning class of technologies that convert used plastics into a range of products and raw materials such as chemicals and chemical feedstocks for new plastics, lower carbon transportation fuels and other petroleum-based commodities.\nThe Iowa House and Senate passed Senate File 534 and the Tennessee House and Senate passed Senate Bill 0923 in support of advanced recycling facilities that convert plastic scrap into raw material using pyrolysis or chemical recycling.\nThe legislation defines pyrolysis\u2014a process that converts post-consumer polymers into fuels, chemical feedstock, waxes or resin pellets\u2014and pyrolysis facilities as being separate from solid waste disposal facilities.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzyumr b/data_all_eng_slimpj/shuffled/split2/finalzzyumr
new file mode 100644
index 0000000000000000000000000000000000000000..a3ee88b49fca9de27ef7c7d22423ae01558109a6
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzyumr
@@ -0,0 +1,5 @@
+{"text":"Such an enjoyable legal caper featuring attorney Jane Larson!\nThere is some monkey-business going on in this story and Jane is determined to get to the bottom of it! Her star is rising in her New York Law Firm , but her heavy work load has taken a toll on her personal life. Now she has a new client whose husband may be involved in drugs and the mob. Looks like someone will be murdered to pay a drug debt and Jane discovers some disturbing clues as she digs to uncover the truth!\nI enjoyed the character of Jane and the NYC setting. The pace was good and it was a suspenseful murder mystery!\nThanks to the authors\/publisher for providing me with a copy for my honest review!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Strategic Management Services, LLC (Strategic Management) is a pioneer in helping the health care industry with timely and innovative regulatory compliance solutions. As one of the few consulting companies focusing exclusively on the health care compliance industry, Strategic Management differentiates itself by its unique expertise, qualifications and people.\nStrategic Management was founded over 25 years ago by Richard Kusserow, the former Inspector General for the Department of Health and Human Services (HHS). We were the first health care consulting firm to focus on corporate compliance and ethics initiatives for the health care industry \u2013 before the government had even issued any formal compliance program guidance documents for the industry.\nOver the last 25 years, Strategic Management has assisted thousands of health care organizations with regulatory compliance services, such as the development of compliance program infrastructure, evaluation of compliance programs, standard of conduct development and reviews, compliance training programs, hotline setup and operations and issue resolution assistance. While we continue to assist health care organizations with regulatory compliance services, our services have also expanded to include risk assessment and management, claims data analysis, assistance with the requirements of Corporate Integrity Agreements (CIAs), Independent Review Organization (IRO) duties, and litigation support, just to name a few.\nStrategic Management's leaders and key personnel have worked both in the highest levels of private sector health care organizations and with the executive enforcement and regulatory agencies of the federal government. Our staff includes senior level consultants from both industry and government, who are able to bring a broad scope and profound depth of knowledge and experience in regulatory enforcement and health care industry business and operations. This includes experience in federal government agencies such as U.S. Department of Health and Human Services, Office of Inspector General (HHS OIG), Centers for Medicare and Medicaid Services (CMS), US Department of Treasury, Department of Justice, Department of Labor, Federal Bureau of Investigation, as well as with private sector hospital systems, academic medical centers, long term care organizations, pharmaceutical companies, managed care organizations, and others.\nOur blend of knowledge and experience of how health care works from both inside the government and from private sector health care companies puts Strategic Management in the unique position to anticipate the needs of the health care marketplace, garner the best resources, provide innovative solutions, and build critical relationships.\nPlease get in touch with us if you have questions or would like to discuss the ways our company can help you. You can call us at (703) 683-9600 or fill out our online contact form.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The 2007 Bulldogs won a state title behind coach Paula Grande and pitcher Katleyn Stanley.\nNorth Mason had been close to a state softball title before, including a frustrating, nine-inning, 1-0 loss to Tumwater in the 2006 state 3A championship game. In 2007, the Bulldogs would not be denied.\nBehind the strong arm of sophomore Katelyn Stanley, the 2007 Bulldogs won the 2A Olympic League championship with a perfect 14-0 record (they finished 22-4 overall); won the district title; survived two exhausting, extra-inning games to open the state tournament; rallied against league foe Sequim in the semifinals; and knocked off a Centralia squad that was loaded with power hitters to finally get the monkey off their back and claim the state championship.\nDuring a ceremony at 6 p.m. Saturday in the school's commons area, that North Mason title squad and coach Paul Grande will be inducted into the school's Wall of Honor. All the team's players except Stanley are expected to be there.\nStanley lives in Odessa, Texas, where she played softball for Odessa College. She isn't able to attend, but will be represented by her parents: Deb, who was the team's scorekeeper, and Darrell Stanley.\nAlso to be inducted are Zac Joaquin, a 2011 state wrestling champion; Kimberly Wascher, a 2007 state champion swimmer; and the 1988 400-meter state champion relay team (Jeff Hinds, Jerry Joyce, Mike Calveard and Rob Atkinson).\nThe spark that propelled the 2007 softball team to its state title was lit in the moments after that dispiriting loss to Tumwater. As Tumwater players celebrated on the field, Grande gathered her emotionally depleted players in the dugout and gave the talk that started it all.\nGrande held three-hour practices and focused on the little things \u2013 situational hitting, proper form in fielding a ground ball and footwork. Outfielders were instructed on how to pivot to make a throw after a catch.\nGrande also practiced for extra-inning situations, which came in handy when the Bulldogs went nine innings in their first two state games and her players remained calm and poised. The Bulldogs beat R.A. Long of Longview in the first game, 1-0, on Dana Kuhn's RBI single; and beat Othello in the second game on Melissa Nys' RBI single.\nThe Bulldogs fell behind Olympic League rival Sequim 3-0 after one inning of their state semifinal game. But that poise and confidence kicked in and they rallied behind Stanley's pitching (a four-hitter) and bat (three hits) to win 5-3.\nGrande had the good fortune to have a pitcher in Stanley who was not overpowering, but was extremely good at locating pitches. That ability came in handy in the championship game.\nThe Tigers won their first three state games by the scores of 5-1, 12-2, and 8-1.\nWhen Grande and assistant coach Heather King scouted the Tigers, they realized Centralia's big hitters were living on the outside pitch.\nStanley followed instructions and the Bulldogs got a two-run home run from Molly Coppinger (now Johnson) in the third inning en route to a 3-0 victory.\nThere was a strange calmness over the Bulldogs as they began that 2007 season. They were on a mission and went about their business with an intensity to match their coach. Stanley would meet catcher Anna Heitstuman at 6 a.m. to throw, and team practices were intense and competitive and fun.\nThe 2006 championship game loss hurt, Stanley said.\nThe state championship inspired athletic director Mark Swofford to instantly add to his t-shirt he was wearing that listed all of the team's softball accomplishments through the years.\n\"They had a printer there (at Walla Walla Point Park, site of the championship in Wenatchee) and I had them add, '2007 State Champion' in big block letters,\" Swofford said.\nAnd now 2007 State Champion will be added to the school's Wall of Honor.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"CPU-Z 1.82 is a freeware that accumulates data on a portion of the primary gadgets of your framework. Processor name and number, codename, process, bundle, reserve levels. Mainboard and chipset. Memory sort, size, timings, and module particulars (SPD). Constant estimation of each center's interior recurrence, memory recurrence.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Jennifer Cluff: Galway on Moyse, Nagahara flutes etc.\nGalway on Moyse, Nagahara flutes etc.\nSee and hear Sir James Galway speak from his Weggis Masterclass 2009.\nGalway speaks about Marcel Moyse's playing and teaching, about Nagahara flutes, Brogger mechanism, and also touches upon several topics of interest (single tonguing, practising Moyse's technique books, buying a new flute etc).\nNote 2017: Link no longer works. Try looking on youtube for Weggis which is the name of the masterclasses Galway gives in the summer.\nThis is a great video! And very surprising as well. I play a Brannen with a Brogger mechanic, and it is the fastest mechanic I've ever tried. But I've never tried the Nagahara flutes. I'm interested now.\nI've going down to Boston after the convention to try the new La Fin headjoints at Brannen. I've played them once before and was very impressed.\nI have a Nagahara headjoint, and an Altus body.\nI've never tried a Nagahara body, however.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaaeyl b/data_all_eng_slimpj/shuffled/split2/finalzzzaaeyl
new file mode 100644
index 0000000000000000000000000000000000000000..9b2dd7d7fd8d0496c47291d2127343211c4cedc5
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaaeyl
@@ -0,0 +1,5 @@
+{"text":"Welcome to Innovation in the 'Burg, a podcast all about science and innovation in St. Petersburg. This week, host Alison Barlow, executive director of the St. Pete Innovation District, brings two professors from St. Petersburg College and University of South Florida, St. Petersburg together to talk innovations in STEM education. Professors Chad Mairn and Dr. David Rosengrant wax poetic on how technologies like 3D printing, virtual reality and augmented reality are helping students learn math, physics, and biology in new ways. They share thoughts on the future of STEM (hint: It's a lot like Magic School Bus!) and the most influential minds in science and technology.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Set includes an EVduty-40 charging station (model EVC30, NEMA 14-50P), a NEMA 14-50R outdoor outlet and an EV plug holster.\nKit includes an EVduty-40 charging station (model EVC30, NEMA 14-50P), a NEMA 14-50R outdoor outlet and an EV plug holster.\nYour EVduty charging station must be installed on a 40 amp circuit by a master electrician. Download the user manualand hand it to your electrician.\nConnected via outlet: portable installation with connection to a NEMA 14-50 wall outlet (not included), depending on selected variant.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Follow directions above, then drop for a teaspoon on to an ungreased cookie sheet. Bake at 375F for 12 minutes! Easy and Yummmmmy!\nRecipe from here, with slight changes.\nBerries picked freah out of the backyard!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A girl by the name of Ellen Harmon had a deep life-changing spiritual experience that had a dramatic influence on the rest of her life, and upon the lives of many who knew her, and the millions who have read her writings.\nShe was a lady of prayer, and knew what it was to agonise with God in prayer over matters that deeply disturbed her.\nAt an early age she exhibited some dramatic manifestations that were characteristic of people in the past who have been influenced in a supernatural way to convey messages from God to people. The visions she received often came to her while in prayer.\nAlthough she had minimal primary education, she wrote out the many messages she received, and they are available today for our benefit.\nShe married James White, and together spent their time in Christian ministry, initiating many of the ministries in the church they helped to pioneer.\nEllen White had incredible insights into things spiritual, including prayer. On these pages is a collection of extracts from her pen and publications that relate particularly to prayer. To be fully understood, each extract should be interpreted in the context in which it was given. They are included on this website for your encouragement.\nAll extracts on this site are used with the permission of the Ellen G White Estate and Research Centre.\nEnjoy Ellen White on prayer.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I had an opportunity to attend a meeting Thursday with Riley County officials and legislators who serve that area.\nI heard first-hand from four of the legislators who voted in favor of the bill to make it possible for TransCanada Pipeline Company to receive millions of dollars in property tax exemptions for the next 10 years.\nAll four indicated that they believed at the time they approved the exemption the company had met the terms of the exemption. All four also indicated that they would follow Rep. Carl Holmes and do whatever he told them to do.\nThe most troubling to me was that none of the four said they would correct the situation since TransCanada was not going to provide access to Kansas refineries, a requirement for the exemption.\nI don't fault legislators for voting for the exemption at the time because they were duped, maybe with Rep. Holmes, regarding the oil company's intentions. They were led to believe that the pipeline would go around Kansas if an exemption wasn't offered.\nWhat continues to baffle most of us who have been following the mayhem the past five years is why legislators would be willing to give away the farm to a foreign-based oil company that didn't make any bricks and mortar investment or expand the job market for Kansans.\nOn the one hand, the state continues to struggle to balance its budget \u2014 taking funds away from counties, cities, and school districts to do so. On the other hand, they give away millions of dollars during the next 10 years to a foreign company. It doesn't make sense.\nRiley County officials have the right idea with requesting a moratorium of sorts on tax exemptions. They are asking the legislature to not grant any more exemptions or rebates because it's costing local governments too much in the end. And they're right. The state isn't going to reduce their revenue. When they grant rebates or reduce taxes, it means less revenue being given to local entities. When local entities can't make ends meet, local taxes and services are increased to meet those responsibilities. So, we end up paying for the tax rebate or reduction.\nIt doesn't seem we're going to get to the bottom of this situation. Legislators are continuing to dig in, refusing to go against Rep. Holmes even if it has been proven they were duped.\nSigh. Indeed a head scratcher.\nWell, the worst is yet to come, folks. The pipeline company is planning another pipeline to run along the side of the current pipeline. And guess what they will be offered? Another exemption. The bill also allows exemptions for natural gas lines.\nWhen will it end? I think the nightmare will continue until someone with a conscious steps up and corrects this situation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzabupv b/data_all_eng_slimpj/shuffled/split2/finalzzzabupv
new file mode 100644
index 0000000000000000000000000000000000000000..3b11e15fb0b2e8af7f6bba9a6c61508dfe1e194b
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzabupv
@@ -0,0 +1,5 @@
+{"text":"Neeke Manasichchanu, a Telugu movie, is not currently screening in Hyderabad.\nWhat we own ends up owning us. The greatest benefactor in this regard is money. Money has become the energy that drives the universe. People would do anything for money - eat on time, respect elders, don't spit on the roads, and other politically correct stuff. But sometimes we need to be harshly reminded that money isn't everything in life. There are other cherished niceties of human emotions. Like the joy when one gets by tasting true love. Okay, so maybe pizza can be bought, but there are many things that money cannot buy. There are at least some things, though none come to mind.\nfullhyd.com has 700,000+ monthly visits. Tell Hyderabad what you feel about Neeke Manasichchanu, and then also see all user reviews for Neeke Manasichchanu!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Stream Doctor P feat. Eva Simons - Bulletproof (Vicetone Remix) [FREE DOWNLOAD] by Vicetone from desktop or your mobile device. Listen and download Doctor P feat. Eva Simons - Bulletproof (Vicetone Remix) [ FREE DOWNLOAD] by Vicetone for free on ToneDen. Eva Simons - Bulletproof (Original Mix) [ HD]. 9, views. Doctor P feat. Eva Simons - Bulletproof (Vicetone Remix). , views. Top Songs By Doctor .\nFree download Doctor P feat. Eva Simons - Bulletproof (Vicetone Remix) # mp3 or listen online music. Read about Bulletproof (Vicetone Remix) [FREE DOWNLOAD] by Doctor P feat. Eva Simons and see the artwork, lyrics and similar artists.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When we consider our health care, we tend to categorize dentistry separately, even though our dentist is also a doctor who makes an impact on our overall health and well being. As we learn more about the connections between oral health and overall health, it is becoming more evident that medicine and dentistry should be a partnership that works together for the benefit of patients.\nDetecting these concerns early on can help your physician recommend more effective treatment procedures and deliver better results.\nThere is also a definite connection between your periodontal health and other concerns in your body such as diabetes, cardiovascular disease, and stroke. We now understand that the inflammation that causes gum disease also contributes to these other health issues, and vice versa. Your best defense against gum disease is regular checkups with your dentist every six months as they can detect it very early on.\nYour mouth can be described as a window to your overall health. Hopefully, as we learn about more connections between general medicine and dentistry, soon they will work together as one.\nFor now, you can protect your oral health with a consistent and efficient home care routine, and regular visits for cleanings and checkups with your dentist. Dr. Reynolds Fischbach is the dentist in St. Paul, MN who can help you reach your goals. Give us a call today to schedule your next preventative appointment.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Eagle's Locksmith is a certified Locksmith in Cincinnati, Ohio with high technology services that provides door lock repair, keyless entry, access control, master key system, high security commercial locks and emergency assistance. Eagle's Locksmith is a full service locksmith company in the greater Cincinnati, locally owned and operated. Our locksmith technicians are fully insured and licensed to provide locksmith services by law in the state of Ohio. Our goal is to keep you safe and secure and solve any lock issue you may have. From the moment you pick up the phone to call us for assistance, you are becoming a valued customer and we will provide you the best service and make sure you are 100% satisfied. Our future with you is important to us, that's why we offer special discounts and rates for your next experience with our company.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"beautytek therapy is a powerful tissue remodelling treatment to effect total skin rejuvenation.\nHealthy skin is firm, plump, well-hydrated and smooth with a bright and translucent tone and with beautytek's facial rejuvenation programme this can be achieved by both men and women of all ages.\nClick here to view further face lifting results in our gallery.\nbeautytek uses the natural healing capacity of the body to stimulate the reproduction of healthy cells, repair the connective tissue within facial tissue and drain excess fluid and toxins from the body. Using a method of transdermal delivery known as iontophoresis the beautytek introduces active ingredients into the body which are readily prepared for absorption. Collagen and elastin levels are increased as well as the rate of perfusion (blood flow).\nFacial rejuvenation therapy can be tailored to match your precise requirements, so areas of concern to you can be treated more deeply than others. beautytek treatments are tailored to treat sagging skin, lines and wrinkles and poor skin tone at the same time, so the only choice you need to make is which areas to focus on!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzafmwa b/data_all_eng_slimpj/shuffled/split2/finalzzzafmwa
new file mode 100644
index 0000000000000000000000000000000000000000..2a79ef30681bebb046cb9598669d2c23ebe4f1b7
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzafmwa
@@ -0,0 +1,5 @@
+{"text":"15% off with this kit! Create your own Wonderoos v3 Starter Bundle! Our new Pick n Mix kit allows you to choose exactly what colours and prints go into your bundle, so you can customise it just as you like it.\nChoose any 5 Wonderoos v3 pocket nappies from the drop down menus, and we'll bundle them up in a free wet nappy bag for you with instructions on how to use everything.\nunique frilled leg design that prevents leaks \"like magic\"\nWonderoos have won multiple awards including most recently Gold awards from Prima Baby Magazine (2016) and Gentle Parenting (2016).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Grammy-award-winning country singer Charlie Daniels is perhaps best known for his 1979 hit \"The Devil Went Down to Georgia\". He has been active as a singer for more than 50 years, and in 2008 was inducted into the Grand Ole Opry, a lifelong dream. With dozens of successful albums and singles; his newest CD is \"Land That I Love\". We discuss his career, the stroke that he suffered early in 2010, and his surprisingly popular GEICO commercial.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Create timely, personalized messages in a scalable way to improve student engagement and retention.\nQuickly identify which students need additional support with flagged replies in the Signal Vine platform.\nUse information from your CRM to create personalized text messages to large groups of students.\nNot all students will reach out for help. Instead, proactively check-in with your students throughout the school year to surface any concerns and improve student engagement.\nHelp your students prepare for registration. First-year students may need more guidance as many have academic holds on their accounts or need guidance on courses.\nImprove your retention rates by engaging your students. Remind students about upcoming activity fairs, advising appointments, and other events that improve engagement.\nHi Sarah. We're halfway through the semester so I wanted to check in to see how you're feeling. Anything I can help you with?\nHi. Actually I'm having a tough time finding the right activities and meeting new people.\nThat's understandable. I can help with that. Would you like to schedule an appointment so we can review some ideas?\nThank you. I will set up an appointment.\nHi Tom, this is Lisa, your advisor. Course registration opens on Monday at 7am. Do you have an academic hold on your account?\nThat's okay. I can help with that. Stop by my office today or tomorrow, and we can take care of it.\nThanks. I will stop by today.\nHi Rachel. Are you ready for midterms next week? We have lots of study tips and guides for de-stressing here: austudytips.com. Do you want to connect with a tutor?\nYes, I'd like to find a tutor.\nHappy to help. You can choose a tutor from the list on our portal here: aututors.com.\nLearn how to incorporate personalized and relevant text messaging into your student retention strategy to retain more at-risk students.\nRead about a current college student's experience and how she thinks these academic advising techniques could've helped her.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome to Signmeister Inc, a family owned and operated Commercial Sign & Design Company, located in Winnipeg, Manitoba, Canada.\nOur experience and size affords us the ability to buy goods economically and fabricate efficiently, saving time and money.\nWe are very proud of our dedicated craftsmen who offer many years of experience to our projects.\n\u00a9 Signmeister 2014 All right reserved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Owners determined to sell! Attractive 4 bedroom family home overlooking Collins Park is on the market for the first time as the owners are planning to move to the country and want a definite sale. Designed by well-known JCY architects, the house is light, bright and airy with multiple access to sunny north facing terraces and garden. Features 4 brms; 2 \u00bd bathrooms; two spacious open plan living rooms; high vaulted ceilings; and fenced near level lawn. The house has 3 large double bedrooms upstairs and very large ground floor fourth bedroom with outdoor access suitable for teenager or extended family member. There is also a large internal double garage and room for children's cars or a boat on the 667 sq m section. The property is next to the walkway to the park, and is a short stroll to the Village shops and the pub.\nOffers a terrific family lifestyle in the heart of Greenhithe at the affordable end of the market - you won't get better than this!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzafwds b/data_all_eng_slimpj/shuffled/split2/finalzzzafwds
new file mode 100644
index 0000000000000000000000000000000000000000..472a6bcd1d0999dbe45d18721854c6a8251e2b5c
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzafwds
@@ -0,0 +1,5 @@
+{"text":"What a crazy week ahead of me! I expected Ktams to be out of town all week so I made all of these plans to keep busy, and then his trip was rescheduled. Now I'm backpedaling so I can spend more time around home with him. I can tell you I've got a fun happy hour at Andina coming up, and then the Portland Bloggers Event this Saturday!!\nI've got to keep my readers' bellies full though! Today I bring you what is becoming a go-to appetizer for grill season: Cheese and Bacon Stuffed Jalape\u00f1os. Y-U-M. What could be better than cream cheese, Colby Jack, bacon, and peppers!?\nWash and core your peppers. Many stands come with a corer, but it is only helpful if you've picked large enough peppers! This can prove challenging depending on the season. If you can't use the corer, I suggest a paring knife and a careful set of hands to cut out the insides (it's really easier than it sounds). Rinse out the inside of the pepper and (using a knife) spread in a healthy dollop of cream cheese. Follow that up with as much bacon as you deem necessary (I like 1\/2-3\/4 a piece per pepper), then finish it off with your sliced cheese. Repeat until all of your peppers are brimming with cheese and bacon. If you do end up with peppers that aren't quite the right size for your stand there's an easy little cheat that Ktams came up with\u2013simply stick a toothpick through about a half inch from the tops! See the picture below for a visual. Grill the peppers for 20-25 minutes or until they darken slightly and the cheese is fully melted. This recipe serves 2-4 people depending on their affinity toward grilled jalape\u00f1os, cheese, and bacon.\nSurprisingly enough this recipe could range in heat from a 2 up to a 3 based on the freshness of the peppers. Approach with some caution if you're looking for a very mild heat!\nWhat would you stuff a jalape\u00f1o with?\nHi! My name is Bee. I am a health conscious, adventurous food writer from Chicago. I reside in Portland, OR.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"An Eaton County Circuit Court judge has reversed a Charlotte City Council decision, paving the way for speakers at the Dairy Queen.\nCHARLOTTE - A judge says the owner of a Charlotte Dairy Queen where drive-thru speakers were previously prohibited should be allowed to install them.\nEaton County Circuit Court Judge Janice Cunningham has reversed a Charlotte City Council decision that upheld the prohibition of speakers at the South Cochran Avenue business.\nCunningham's Jan. 23 ruling comes a year after the City Council, as the acting Zoning Board of Appeals for the city, upheld a speaker restriction at the newly-constructed Dairy Queen.\nThe new building replaced the original 540-square-foot restaurant, which predated the city's zoning ordinance and sat among single family homes. Owner Brett Roberts' plans for the speakers drew opposition from neighboring residents.\nRoberts filed a lawsuit in Eaton County against the city through his company Roberts Real Estate of Charlotte LLC last March after it upheld a restriction put in place by the city's Planning Commission prohibiting drive-thru speakers.\nIn a Jan. 24 memo to Charlotte City Council, City Attorney Thomas Hitch said Cunningham, who issued a verbal opinion rather than a written one at the hearing last month, said she found its decision \"arbitrary and capricious.\"\n\"She went through in great detail what she considered to be the flaws in the findings of the Zoning Board of Appeals and concluded, under the case law, that the Court was of a firm belief that a mistake had been made,\" Hitch wrote.\nThe ruling clears the way for installation of speakers at the business drive-thru.\nRoberts didn't return a message from the State Journal Wednesday.\nCharlotte City Manager Gregg Guetschow is expected to issue a recommendation regarding whether the city should appeal Cunningham's reversal at its Feb. 11 City Council meeting.\nLast week Guetschow said he isn't sure the Michigan Court of Appeals would hear the matter, and that an appeal may not be \"the best use of city resources.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Great Location! Best School District # 207 & 79. Newer Heat & Ca, Roof & Windows. Nice Back Yard With Sun Deck. 1.5 Car Garage. Close To Blue Line Train And Kennedy X-way. Harlem\/Irving, Harlem\/Foster & Norridge Commans Shopping's All Closed By. New Mariano's On Lawrence And Many Banks And Restaurants Are Within Walking Distance. Great Place To Live And Raised Family.super For Kids: Main South H.s.& Pennoyer.\nFor more information about 7519 W Argyle, Harwood Heights, IL 60706 contact Urban Real Estate.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Decorative Resin can be a prime element in any design. AMS's approach to projects is to work with architects, designers, general contractors and homeowners to acheive the look that will give their project a uniqueness of its own. Such elements include: resin banding, cameos, filigrees, and brackets. In the past, we have been challenged with many different shapes, conditions, and custom mould creation. We have, and continue to have pride in ourselves on exceeding those demands. We would like to thank our customers for their continued support and for sharing their project photographs with us, so we may feature this cast stone gallery for your enjoyment.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ireland's Greenhouse Gas emissions fell by 210,000 tonnes (0.3%) in 2008 to 67.43 million tonnes.\nThe effects of the economic downturn are mainly evident in the 523,600 tonne reduction (4.4 %) in emissions from the Industry and Commercial sector with smaller decreases (less than 1%) from the Agriculture and Transport sectors.\nHowever an increase in emissions of 603,700 tonnes from the Residential sector (reflecting colder winter months) cancels the benefit of these reductions to a large extent. Energy sector emissions are largely unchanged.\nImproved methodologies in the Energy and Waste sectors have resulted in emissions decreasing by 0.56 million tonnes in 1990 and by 1.56 million tonnes in 2008.\nDistance above our Kyoto Limit is now 4.4 million tonnes (7 %).\nProvisional Greenhouse Gas emissions figures released today by the Environmental Protection Agency (EPA), for use in the annual Carbon Budget, include trends since 1990 and show Ireland's status in meeting our obligations set under the Kyoto Protocol. Ireland's Greenhouse Gas emissions fell by 210,000 tonnes (0.3 %) in 2008.\nAgriculture remains the single largest contributor to the overall emissions, at 27.5% of the total, followed by Energy (power generation & oil refining) at 21.6% and Transport at 21.3%. The remainder is made up by the Residential sector at 10.3%, Industry and Commercial at 17.6%, and Waste at 1.8%.\nThe figures show that, while Ireland's Kyoto limit in the period 2008-2012 is 62.84 million tonnes per annum, Ireland's emissions in 2008 were 4.4 million tonnes (7%) above this limit when account is taken of the EU Emissions Trading Scheme and of approved Forest Sinks.\nEmissions decreased by 523,600 tonnes (4.4 %) from 11.92 million tonnes in 2007 to 11.39 million tonnes in 2008 reflecting decreases in CO2 from combustion sources and cement production.\nThe emissions from agriculture decreased by 172,400 tonnes (0.9 %) in 2008, continuing the downward trend from the 1998 peak. The decline in emissions reflects lower sheep and cattle numbers and reduced use of fertiliser.\nTransport emissions were 121,100 tonnes lower in 2008 than in 2007. This represents a decrease of 0.8 %, following sustained increases in this sector since 1990. The decrease may reflect the initial impact of the economic downturn plus the changes in vehicle registration tax and road tax introduced in mid 2008. It may also reflect the effect of the on-going rollout of Transport 21. Emissions in 2008 were 176 % higher than the 1990 transport emissions.\nEmissions in 2008 were similar to 2007 with a small increase of 86,000 tonnes (0.6%).\nEmissions in 2008 increased by 603,700 tonnes (8.7 %) from the 2007 level. This was the largest sectoral change in 2008. It would appear to reflect increased use of domestic heating as a result of the winter months of 2008 being significantly colder than for the same period in 2007.\nEmissions for this sector show a decrease of 83,300 tonnes (7 %) below the 2007 level. Emissions in 2008 are 15 % lower than in 1990. In addition, the EPA has used a revised methodology based on the latest IPCC Guidelines to estimate methane generation in landfills together with the results of a major EPA funded research study on methane flaring to develop substantially revised estimates of emissions for all years.\nApproximately one third of Ireland's GHG emissions are covered by the EU Emissions Trading Scheme. Allocations to these installations impact on the calculation of the distance to target during the 2008 \u2013 2012 period. For detailed calculation see Table 1 in attached link.\nIncreases in approved Forest Sinks count towards achievement of our Kyoto limit and hence need to be included when estimating distance to target. In 2008 these amounted to 2.09 million tonnes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Prathinidhi (English: Representative) is a 2014 Telugu political thriller film produced by J Samba Siva Rao under Sudha cinemas and directed by debutant Prashanth Mandava. The film stars Nara Rohit and Shubra Aiyappa in the lead roles. Music is being composed by Sai Karthik. The film received positive reviews from critics and audience. This movie was remade in Tamil as Ko 2.\nThe filming was launched on 26 June 2013 at Hyderabad.It's also dub in hindi as a Pratinidhi - Ek Leader.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Tennessee Sen. Bob Corker, chairman of the Senate Foreign Relations Committee, said Wednesday he was taking himself out of the running to be Donald Trump's vice presidential pick.\nCorker had emerged in recent days as a contender but told The Washington Post he has withdrawn his name from consideration, saying he was better suited elsewhere.\n\"It's a highly political job, and that's not who I am. We had a very open conversation about that, and actually, we have been very candid about it from the very beginning of our meetings,\" Corker told The Post.\nThe Post reported that Corker had been under serious consideration by Trump and had submitted personal documents to the attorney managing the vetting process.\nTrump is expected to announce his choice ahead of the Republican Convention in Cleveland later this month.\nFormer House Speaker Newt Gingrich, New Jersey Gov. Chris Christie, Alabama Sen. Jeff Sessions and Indiana Gov. Mike Pence are among the short list of candidates, according to campaign sources.\nTrump spent part of his July 4th with Iowa Sen. Joni Ernst \u2013 fueling speculation that the freshman senator could be on the billionaire's short list.\nFox News' Kara Rowland and Christopher Snyder contributed to this report.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"National Bullying Prevention Center provides resources for students, parents, and educators.\nStop Bullying has information on prevention and response to bullying and cyber-bullying.\nThe Department of Education's Office of Special Education and Rehabilitative Services (OSEP) issued a \"Dear Collegue\" letter and companion piece on bullying entitled \"Effective Evidence-based Practices for Preventing and Addressing Bullying.\"\n\"Prohibited Disability Harassment\" is the topic of this letter from the U.S. Department of Education's Office for Civil Rights (OCR).\n\"Bullying: Facts for Schools and Parents\" by Andrea Cohn and Andrea Canter, Ph.D., NCSP, is posted by the National Association of School Psychologists.\n\"Addressing the Problem of Juvenile Bullying\" by Nels Ericson, is from the Office of Juvenile Justice and Delinquency Prevention at the U.S. Department of Justice.\n\"Bullying and Your Child\" resources are included on the Kids Health web site.\nOperation Respect is a non-profit organization working to transform schools, camps, and youth organizations into more respectful environments.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Kaczynski's lawyers attributed some of his emotional instability and dislike of mind control military police army records to his participation in this study. Click here to download the death certificate request form. California-based beatles tribute band, the fab four was cast to do the motion capture performance for the animated http:\/\/www.herbsforlove.com\/modules\/moultrie\/kent-county-michigan-police-records.html.\nMercer owned the property, the girl all but yelped. Indianola record herald police reports the month of july 2014, the harford county sheriff's office recovered three vehicles reported stolen from other states. Forgotpassword_sendbutton. Depending upon the type of warrant, that action can be visit web page arrest of a named individual or the search of a residence. Protect your rights and have peace of mind with matthew lopez law.\nNordisk conversationslexikon : indeholdende forklaring over vigtige navne, gjenstande og begreber, som forekomme i livet under laesning og i samtale. We are a team of volunteers and starting a new project in a community in the same niche. Of docs. Published by duncan (darby) macdonald, brockville, 1992 (expanded reprint2nd edition 1993). Beauvois (page images at hathitrust; Les deux quetzalcoatl espagnols: j.\nProximity to high performing public schools. Monday through friday, excluding legal holidays. Ben niva, goldendale, wash. Best acupuncture chinese herbs and pregnancy lambert klein is the owner of think healthy supplements acupuncture supply and products. Ca, our association is one of the longest running crafts and fine arts show in the usa. The united states has over 13,000 lodges, each one will extend the warm hand of brotherhood to the community and humankind. Et) near reno, nevada, after state highway patrol officers stopped his car and persuaded him to surrender without incident.\nDocumentos relativos a las obras del puerto de buenos aires \u00e9 informe del injeniero j. The massive so-called alexander sarcophagus found in sidon in modern lebanon, was probably made there at the start of the period by expatriate greek artists for a hellenized persian governor. The vaccine may indianola record herald police reports offer protection. For example, best electric acupuncture while pregnant many relatives acupuncture for headaches who admit anxiety declare this roast tricky. Business vehicle for personal use. Lisboa, typographia da revista popular, 1852), by francisco palha (page images at hathitrust). Of moreau, the existing feeder canal trail.\nI'd really appreciate it. Where can i http:\/\/www.herbsforlove.com\/modules\/moultrie\/local-police-records-check-toronto.html to view my credit report for free additionally, her radical faith and determination is providing a gateway so she credit bureau can birth out her gifts to be productive in the kingdom through workshops and seminars.\nTammany and washington parishes, including delinquency, indianola record herald police reports in need of care, family in need of services, and juvenile traffic offenses. Chez jean jaques dentand, 1690), by pierre richelet (page images at hathitrust). And head for the summit of binevenagh for one of ulster's finest, widest. You can find what you want. According to one survey, at indianola record herald police reports 79 inmates fled cca facilities nationwide between 1995 and 1998- compared to nine escapes from california prisons, which have more than twice as many inmates.\nHello web admin, i noticed that your on-page seo is is missing a few factors, for one you do not use all three h tags in indianola record herald police reports post, also i notice that you are not using bold or italics properly in your seo optimization. Likewise, after 1908, individuals who passed away in conneaut or ashtabula, the death records are located at their respective municipal health departments; And individuals passing away in any other area of the county, those death records are located at the ashtabula county health department.\nHolland (page images at hathitrust). Christmas in the city 2010. Go back into history, change one thing. Department of health and human services; Faina shtern, susan wood, carol krause. I'd genuinely appreciate it. Moda economico new balance formatori uomo arancione bianco 574. 23 months, mortgage insurance calculator the points do not gain you any advantage. For more information about the white river watershed, or to get involved in watershed activities, call josh kahan, basin steward, at 206-477-4721. Tully kingdon, d. Most women who deliver vaginally are cleared to go home after one or two nights, while women who deliver through cesarean section usually stay in the hospital three to four nights.\nI plead the blood of jesus for healing. Snell, with illustrations by george f. Garcinia cambogia ultra dosage hubert zandstra pure garcinia cambogia director general and dr.\nSeveral sites offer these graphics free, allowing times herald record police blotter 2015 to enhance everything from family tree charts to presentations and slideshows for your next family reunion.\nI've lost my bank card levofloxacin 750 mg cost another criminal records search clark county nevada massachusetts law man, state police col. But she doubted that either girl would have http:\/\/www.herbsforlove.com\/modules\/moultrie\/mesa-police-public-records.html her life to avoid that.\nBoyfriend Billy M Desiderio , place of birth Gainesville, DOB: 1 June 1949, emploument Frame Wirers, Central Office.\nCompany who recognized the impact a well-rounded 4-h public speaking. Personal injury attorney marshall county ky criminal records kansas city maybe negligence it's a combination of factors. Scott oliver's ordeal by battle, (london, t.\nWomen sculptors took longer to appear than women painters, and were clay county iowa jail records prominent until the 20th century.\nSpouse Marcellus Graeme Janish , place of birth San Francisco, DOB: 17 September 1950, job Telecommunications Facility Examiners.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"(d) all the returns or information required to be submitted or furnished to the Commissioner in terms of any Act administered by the Commissioner by the end of the relevant period within which the steps contemplated in this subsection must be taken, have been submitted or furnished or arrangements have been made with the Commissioner for the submission of any outstanding returns or furnishing of information.\nIn terms of the statutory merger provisions contained in the Companies Act, specifically section 116(5)(b), the Companies and Intellectual Property Commission (the CIPC) must, after receiving a notice of amalgamation or merger (form CoR89), deregister any of the amalgamating or merging companies that did not survive the amalgamation or merger. In effect, this notification serves the same purpose as a notice or request filed with CIPC in terms of section 80(2) or 82(3)(b)(ii) of the Companies Act (all three resulting in the company being wound up or deregistered).\nIt therefore appears that section 41(4) of the Income Tax Act effectively contains an unintended cross-referencing error or oversight, as the provision refers only to sections 80 and 82 of the Companies Act in the context of steps deemed to have been taken as part of a companys liquidation, winding up or deregistration. As stated above, a notice or request filed with the CIPC in terms of section 80(2) or 82(3)(b)(ii) of the Companies Act is not required for a company to be deregistered as a result of a statutory merger. Arguably, therefore, any transaction that gives rise to the automatic extinction of the transferor entity, may be unable to qualify for income tax relief under the reorganisation provisions.\nAlthough a number of Advance Tax Rulings have been published by the South African Revenue Service (SARS) regarding the application of sections 41 to 47 of the Income Tax Act to mergers, none provides sufficient detail to confirm that (1) the statutory merger provisions of the Companies Act have been utilised to implement those mergers, and (2) if so, SARS were satisfied that section 41(4) was compiled by way of the automatic termination of the company rather than by way of the filing of the notices or requests explicitly referenced in section 41(4).\nIn some corporate reorganisation rules, to qualify for the tax-neutral transfer of assets, one or more of the companies involved should cease to exist after the transaction. The legislation lists steps that show a taxpayer meeting this requirement. However, the steps do not take into account deregistration by operation of law. It is proposed that the rules be amended to include this option.\nIn addition to the above, a particularly interesting question that arises in this context is the impact of statutory mergers on certain carve outs to the de-grouping provisions, where the de-grouping in question occurs by way of a group companys liquidation, winding up or deregistration. Arguably, these terms do not include a companys extinction by operation of law.\nA pertinent example of such a carve out is contained in section 45(4)(c) of the Income Tax Act, which provides relief to companies from the relatively harsh six-year de-grouping rule contained in section 45(4)(b) in circumstances where the transferor and transferee company cease to form part of the same group of companies in relation to one another as a result of the transferor or transferee companys liquidation, winding up or deregistration. Should the recognition of a companys liquidation, winding up or deregistration be restricted to circumstances in which a notice or request is filed with CIPC in terms of section 80(2) or 82(3)(b)(ii) of the Companies Act, the application of the carve out to the de-grouping rules where such a liquidation, winding up or deregistration occurs by operation of law, leads to uncertainty for taxpayers.\nThe manner in which it is intended for this oversight to be addressed will be seen from the draft tax Bills which are due to be released during the course of this year.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzagtqb b/data_all_eng_slimpj/shuffled/split2/finalzzzagtqb
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+{"text":"The Indianapolis Home Show - Sincerely, Sara D.\nOh, guys! I had a crazy, exciting, stressful, fun, and exhausting couple of days! I am naturally more introverted (blogging is great because I can sit behind the screen), and I do not love to be the center of attention in any way. However, over the past few days I conquered my greatest fears.\nIf you have been following me, you've probably hear me mention that I was speaking at the Indianapolis Home Show. I was already a bundle of nerves on Friday (I was scheduled to speak Monday and Tuesday), when I received an email about appearing on the early morning news Monday.\nMy first thought was to run or pretend like I hadn't seen the email. However, I came to my senses and realized this is something you don't say no to. It was another way to stretch myself and a great way to face my fears head on. So, I did it! I woke up at 2 am and headed to the Indianapolis Fairgrounds to the site of the home show.\nI'm so thankful for friends like Jenny. This girl got up crazy early to support me, calm my nerves and pray with me.\nI'm so glad I did it. It ended up being fun, and I had the best news anchor to lead me!\nTo see a clip of from the news, click HERE.\nFrom there, I went back home and was back in time to put my kids on the bus. Then, after lunch, my mom and I picked up my friend Angela from Unexpected Elegance and headed back to the Home Sh0w.\nThe great thing about doing live TV before my presentations is that suddenly a planned speech was not nearly as nerve-wracking as it seemed before.\nI survived the first two presentations despite some technical glitches with my headset and powerpoint.\nOn day two I had the privilege of sharing the stage with the Diva of DIY, Leanne. This girl is not only crazy talented but she is SO nice!\nDay two went much better without the technical issues (and notice I ditched the notes).\nTuesday was also filled with amazing friends and families who came to listen and support me!\nI am so proud of myself for doing it \u2013 I spoke to a crowd AND survived!\nCongrats on conquering your fears! I am so bummed that I missed the show. It sounds like your session would have been right up my alley! Thanks for sharing your experience. Looks like it was a success!\nI have dabbled with chalk paint once before, and can't wait to try out those other paint types in your photo. I'd love to check out milk paint and white washing at some point!\nThanks, Jessica! I'm in the middle of a series on using 5 different types of painting furniture. I'll add the link to each of paint tutorials once they're published. You can see more here \u2013> https:\/\/www.sincerelysarad.com\/painting-furniture-tutorial-series\/ Thanks for your comment and good luck on your projects!\nCongratulations, Sara, on your presentations at the Indianapolis Home Show. It Would be inimidating to do what you did, but I bet you were terrific. You have faced the fear of speaking and have improved confidence for the next time you are presenting. Well done!\nThanks for the kind words and taking the time to comment. I appreciate you, Sandy!\nCongratulations, Sara! What an awesome feeling it must be to overcome that fear. I wish I could have been there to support you.\nMy husband and I were at your show at the Indianapolis Home Show. You did an outstanding job! I got a lot of great information from you. I was particularly interested in finding out more about painting fabrics. We stayed after the show awhile and talked to you. You were so gracious and helpful! I haven't had a chance to try it out yet, but every time I pass the 2 pieces of furniture that I want to redo, I think of your suggestions and I get excited to try them.\nI just ordered your eBook \"Painting Furniture\".\nLove your website and the free printables.\nThanks so much for taking the time to comment and for the kind words! Don't hesitate to email me with any questions, and thanks so much for ordering the book. I hope it's helpful and gives you the confidence you need to get started on some projects! Take care and enjoy the rest of your week. Thanks again!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Tumblr, Ugly, and Blog: when u short but u handsome as hell hereforpornandmemes: sappyassmemes: i mean i'm short but that's it I'm ugly and tall does that count too?\nhereforpornandmemes: sappyassmemes: i mean i'm short but that's it I'm ugly and tall does that count too?\nGraffiti, Discover, and Image: Graffiti removal guy comes back to discover image of himself. Ren utan Who is that handsome guy ?\nWho is that handsome guy ?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At IFA 2018, ASUS has announced a myriad of new consumer laptops, aimed at giving its customers the performance they need to do their tasks, combined with a slim profile and elegant looks. And now, these stylish and powerful machines have made their way onto local shores.\nThe new ZenBook 13, 14, and 15 all feature a slim profile, along with ASUS' signature NanoEdge displays that results to a 95% screen-to-body ratio for a near-borderless viewing experience.\nTheir ultra-compact design takes advantage of the ErgoLift hinge mechanism, which slightly raises the keyboard, giving you a more natural and comfortable typing experience.\nThe ZenBook 13 and 14 also feature the new NumberPad, which basically turns the touch pad into a numeric keypad. Under the hood are 8th Generation Intel Core i7 Processors, and up to an GTX 1050 with Max-Q design.\nThe ASUS ZenBook 13, 14 and 15 will be available in stores nationwide this October.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What is it? Turmeric is a bright yellow root that is used as a spice or seasoning. It's typically available dried and powdered, though the fresh root is becoming more widely available as well (we get it at our regular grocery store these days!). It has a subtle flavor\u2014warm, peppery, and a little astringent.\nWhy is it healthy? Turmeric is considered a superfood because it contains curcumin, a compound especially good at targeting inflammation, which is tied to a wide range of ailments, from cardiovascular disease, to Alzheimer's, to PTSD (Time magazine). It's also cited as a potential cancer-fighter, a remedy for colds and flu, a way to soothe stomachaches and indigestion, and more. Suraj remembers his grandmother applying it to his skin as a child to treat cuts and bruises (smart lady, as turmeric is indeed an antiseptic).\nWe use turmeric on near-daily basis in our kitchen, as it's a staple of Indian cooking. Most curries and vegetable preparations get at least a pinch, and I also like to boil my vegetables in turmeric water (potatoes especially, as it gives them an especially vibrant color). I add a teaspoon or so to most soups and broths as well, as it has a very subtle\/neutral flavor.\nWhen we're feeling under the weather, I make us this turmeric milk (also known as golden milk). Warmly spiced, creamy, and subtly sweet, it's easy to love. We like to have a cupful after dinner, to settle the stomach\/mind. It reminds me of one of those comforting foods you'd read about in a children's book, like the three bears' porridge, or bread & jam for Frances (anyone?). Golden milk for everyone!\nIf you don't drink dairy, you can substitute coconut milk. It's great that way too. I use fresh turmeric root (as in the above photo), but powder is fine too\u2014just stir well after adding so it doesn't clump. Jaggery is unrefined cane sugar; it's in the top left portion of the bowl in the photo above. All of these items can easily be found at any Indian grocery\u2026 for much cheaper than the regular grocery store. Leave out any spices you don't have\u2014this recipe is certainly open to interpretation!\nPlace all ingredients except the jaggery\/honey in a small saucepan. Bring to a low simmer (be careful not to let it boil, as it can quickly bubble up and scorch) and allow to simmer for just a minute or two. Remove from heat, cover, and allow to sit for at least 10 to 20 minutes to infuse flavors. Strain, discarding spices.\nAdd grated jaggery or honey, stirring to dissolve. Serve warm. Keeps in the fridge for about a week. Note that the color may become even more yellow over time\u2014that's the magic of turmeric.\nFresh turmeric root at a spice plantation we visited in Goa, India.\nTurmeric powder on sale at a market in Bangalore, India. It's used as both a food and a dye.\nI like to add a little turmeric to my pumpkin muffins since the flavor is so mild and I feel like it's a little bonus healthy pinch!\nOh that's a great idea Amy! Going to try that for sure.\nOh, I'm totally making your recipe now. Why have I never thought to add fennel?! You're brilliant. And may I say again just how happy I am that you're blogging!!!!\nYes, Lindsey, we love fennel! It's such a great spice for digestion, and flavor too.\nI use a combo of cinnamon, paprika and turmeric while saut\u00e9ing chopped onions before adding sliced or diced pare cooked potatoes. The best home fries ever.\nOh I love that combination! Trying it asap.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Description: This four-bedroom, three-full-bath home offers one-level living with a finished space upstairs. Features include an open floor plan, a family room with a high ceiling and a fireplace and an eat-in kitchen with an island.\nDescription: This four-bedroom 3\u00bd-bath home has a rocking-chair front porch. Features include upstairs and downstairs master suites, three bedrooms on the main level and plantation shutters. The house has a large flex space.\nDescription: This five-bedroom, 3\u00bd-bath home is close to schools. Features include a main-level master suite, four bedrooms upstairs and a kitchen with extensive cabinetry, a cooktop and an island. The house has a central vacuum.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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new file mode 100644
index 0000000000000000000000000000000000000000..6e80cbec54b5a67768a32c89d059075bdae5a1d6
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@@ -0,0 +1,5 @@
+{"text":"Mental health is something I am wary of every day. I live inside my head, my ability to express emotions through talking limited by my general dislike of feeling emotionally vulnerable. And that's okay.\nI am very thankful that I have writing as a way to get my thoughts out. Maybe they're not always the most cohesive thoughts, but that's okay.\nBecause of my training there are some days I feel physically low, tired, worn out, generally just a bit broken. I eat well to make sure I get my nutrients in, I sleep as much as possible, and I take a rest day to recover. And that's okay.\nIf it's a scheduled rest day I feel great, I lap it up, I treat it as a real part of my training plan. I embrace the rest and come back the next day ready to go.\nThe loop plays for hours over the day until I either, stupidly, crumble and do one or part of the workout, or I ride it out before crashing into the best sleep I've had in days.\nThere's guilt, there's annoyance, there's a feeling of weakness, wondering how much the competition are doing when I'm resting.\nToday was one of those days. And that's okay. Tomorrow I'll be ready to go.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"1.\u2003These regulations may be cited as the Bread (Control of Manufacture and Sale) Regulations 1988.\n\"MS 37\" means the Mauritian Standard specification for white and wholemeal wheat bread declared in General Notice No. 151 of 1985.\n3.\u2003(1)\u2003No person shall operate a bakery unless he holds a licence issued by the Minister.\n(b)\u2003be subject to such conditions as the Minister may determine.\n(b)\u2003cause to be manipulated in any day or night by every team he employs less than the minimum amount of bags of flour specified in paragraph (3) of the Second Schedule to the Baking Industry (Remuneration Order) Regulations 2003.\n(b)\u2003any bread which contains more moisture by weight than that specified in the fourth column of the Schedule.\n(2)\u2003(a)\u2003Any person who wishes to manufacture bread other than of the type specified in the first column of the Schedule shall apply in writing to the Minister.\n(b)\u2003The Minister may grant the application subject to such conditions as he may determine.\n5.\u2003The inspection, sampling and testing of bread shall be performed in the manner specified in MS 37.\n6.\u2003Every person who exposes, sells or offers for sale bread on his trade premises shall display in a conspicuous place a notice indicating the type, weight and price of the bread exposed, sold or offered for sale.\n7.\u2003No baker shall keep stale or rejected bread in that part of the bakery where bread is exposed, sold or offered for sale to the public.\n(c)\u2003the name and address of every bread seller who purchases bread from the baker and the quantity of bread supplied to the bread seller.\n(b)\u2003be produced on demand to any authorised officer.\n9.\u2003Every baker shall provide in his bakery a manual or mechanical sieve of a capacity of not less than 50 kilogrammes.\n10.\u2003Every baker shall install in that part of his bakery where bread is exposed, sold or offered for sale to the public a scale duly stamped under the Legal Metrology Act for the purpose of weighing bread when so requested by a purchaser.\n11.\u2003(1)\u2003An authorised officer may at any time enter any premises in which bread is manufactured, exposed, sold or offered for sale for the purpose of ensuring that these regulations are complied with.\n(2)\u2003Any person who prevents or obstructs an authorised officer in the performance of his functions under paragraph (1) shall commit an offence.\n12.\u2003The maximum retail price at which bread shall be sold in Mauritius and Rodrigues shall be the price specified in the fifth column of the Schedule.\n1.\u2003These regulations may be cited as the Commodities (Registration of Traders) Regulations 1978.\n(b)\u2003includes any person who, as agent, employee or otherwise, assists any such person specified in paragraph (a).\n3.\u2003(1)\u2003Subject to paragraphs (2) and (3), no trader shall deal in a commodity unless he holds a certificate of registration.\n(3)\u2003Where a trader is employed by a corporate body, soci\u00e9t\u00e9 civile or partnership, which is required to have a certificate of registration under these regulations, he shall not require a certificate of registration whilst being so employed.\n(c)\u2003such additional information as the Minister may require.\n(2)\u2003On receipt of an application under paragraph (1), the Minister may grant a certificate of registration subject to such conditions he may determine.\n(3)\u2003The Minister may revoke any certificate of registration for breach of any condition attached to it in addition to any other penalty that may be provided in any enactment.\n(b)\u2003be renewable for further periods of one year.\n(b)\u2003be made and treated as a fresh application under paragraph (1).\n(b)\u2003be produced on demand to an authorised officer.\n1.\u2003These regulations may be cited as the Consumer Protection (Consumer Goods) (Maximum Mark-up) Regulations 1998.\n\"wholesaler\", in relation to any consumer goods, means a person who purchases the consumer goods from an importer or manufacturer for purposes of resale to a retailer.\n(d)\u2003such other particulars or documents as the Minister may determine.\n(2)\u2003No person shall, in respect of any consignment of any consumer goods, sell or supply the consumer goods unless he submits the return referred to in paragraph (1).\n(b)\u2003such other information as the Minister may determine.\n5.\u2003(1)\u2003The Minister shall determine the maximum mark-up of, and the special allowance applicable to, any consumer goods.\n(2)\u2003The maximum mark-up determined by the Minister under paragraph (1) shall be the percentage of the cost price added to the cost price of a consumer goods and specified in the second column of the First Schedule.\n(c)\u2003be that specified in the third column of the First Schedule.\n6.\u2003(1)\u2003Where, on the date of submission of the return referred to in regulation 3 or 4, any importer or manufacturer has any existing stock of the same type of consumer goods, the Minister may direct that the weighted average cost price of the new consignment and of the existing stock shall, for the purposes of these regulations, be the cost price for the new consignment and the existing stock.\n(2)\u2003Paragraph (1) shall not apply to pharmaceutical products.\n(c)\u2003issue the original of the bill of sale to the retailer.\n(2)\u2003No retailer shall take delivery of any consumer goods which he has purchased from any importer or wholesaler of the consumer goods unless the importer or the wholesaler, as the case may be, has issued to him a bill of sale referred to in paragraph (1).\n8.\u2003For the purpose of determining the cost price of any consumer goods, the importer shall use the currency conversion rate specified in the bill of entry referred to in regulation 3.\n9.\u2003(1)\u2003Every importer shall affix a label to every pack, packet or container of the medicine, indicating legibly his name and the maximum retail price at which the medicine is to be dispensed, exposed, offered for sale or sold to consumers.\n(2)\u2003No retailer or wholesaler shall take delivery of a medicine which he has purchased unless the price label referred to in paragraph (1) is affixed to the medicine.\n(3)\u2003No person shall sell any free sample of a pharmaceutical product.\n(4)\u2003No person shall, without the approval of the Minister, alter, erase, obliterate, remove, or in any way, tamper with the price label to be affixed under paragraph (1).\n(b)\u2003the maximum retail price per half kilo at which the fresh fruit shall be sold.\nshall commit an offence and shall, on conviction, be liable to a fine not exceeding 100,000 rupees and to imprisonment for a term not exceeding 3 years.\nNote \u2013 *Items 5, 6, 7 and 8 shall apply only to importers of fresh fruits.\n(d)\u2003the Ministry will be notified in case such application is made and refund of duty is obtained.\n1.\u2003These regulations may be cited as the Consumer Protection (Consumer Goods) (Maximum Price) Regulations 1998.\n3.\u2003(1)\u2003Every person who imports or manufactures any consumer goods shall make an application to the Minister to fix the maximum price of the consumer goods.\n(2)\u2003The Minister may, on receipt of an application made under paragraph (1), request the applicant to furnish such additional information as he may determine.\n(b)\u2003where no time limit is specified in the request, within 15 days from the date of receipt of the request.\n(c)\u2003such other particulars or documents as the Minister may determine.\n(c)\u2003duly certified or authenticated.\n6.\u2003(1)\u2003On receipt of an application made under regulation 3, the Minister may, where he is satisfied that the maximum wholesale and maximum retail prices of the consumer goods have to be fixed, grant the application on such conditions as he may determine, and communicate his decision to the importer or manufacturer by way of a permit.\n(2)\u2003No importer shall, in respect of the consignment of any consumer goods, sell or supply the consumer goods unless he holds the permit specified in paragraph (1) in respect of that consignment.\n1.\u2003These regulations may be cited as the Consumer Protection (Control of Imports) Regulations 1999.\n\"value\" has the same meaning as in the Customs Act.\n3.\u2003No person shall import any prohibited goods.\n4.\u2003Subject to regulation 7, no person shall import any controlled goods unless he holds an import permit.\n5.\u2003(1)\u2003Any person who imports any controlled goods shall apply for an import permit from the Permanent Secretary.\n(b)\u2003be made in the form set out in the Fifth Schedule.\n(b)\u2003in relation to goods in the freeport zone, before the removal of any controlled goods from the freeport zone.\n(b)\u2003request the applicant to furnish such additional information or document as he may reasonably require.\n(5)\u2003The Permanent Secretary shall, in relation to the goods set out in the second column of the Sixth Schedule, grant an import permit, subject to the restrictions set out in the third column of the Sixth Schedule.\n(6)\u2003The Permanent Secretary shall, in relation to the goods set out in the second column of the Third Schedule, grant an import permit subject to the conditions set out in the third column of the Third Schedule.\n(6A)\u2003(a)\u2003The security required to be given by an authorised dealer pursuant to item 4A(iii) of the Third Schedule shall not, at any time, cover more than 20 import permits.\n(ii)\u2003already imported and which have not been certified to be free from removable radioactive surface contamination by the Radiation Protection Authority, exceed 20.\n(7)\u2003The Permanent Secretary may refuse to issue an import permit where the importer fails to comply with these regulations.\n(d)\u2003be subject to any condition that the Permanent Secretary may impose.\n(2)\u2003Any controlled goods shall be shipped on or after the date of the import permit but not after the date of validity of the permit.\n(g)\u2003the name of the bank.\n(4)\u2003An import permit issued under these regulations shall be without prejudice to the obligation of the importer to comply with any other enactment in force in Mauritius relating to the goods covered by the import permit.\n(5)\u2003The Permanent Secretary may cancel an import permit if he is satisfied that the holder of the permit no longer requires the permit.\n(6)\u2003Subject to regulation 8(2), no person other than the holder of the import permit shall retain the permit in his custody.\n(d)\u2003goods donated to the Government of Mauritius by a foreign Government.\n8.\u2003(1)\u2003No bank shall, in respect of the importation of any controlled goods, open a letter of credit or effect any payment in foreign currency in favour of any person except on production of the original import permit.\n(2)\u2003The bank shall retain the original import permit produced under paragraph (1).\n(b)\u20036,000 rupees in the case of any other controlled goods.\n(2)\u2003(a)\u2003The fee of 60,000 rupees payable under paragraph (1)(a) shall not apply to the categories of importers specified in Part I of the Seventh Schedule.\n(b)\u2003The fee of 6,000 rupees payable under paragraph (1)(b) shall not apply to the categories of importers specified in Part II of the Seventh Schedule.\n(b)\u2003he obtains the written approval of the Permanent Secretary.\n(2)\u2003The Permanent Secretary may, for the purpose of granting an approval under paragraph (1), request the importer to furnish the original and one copy of the relevant invoice of the goods together with the original and one copy of the related Bill of Lading or Airway Bill or other document of title acceptable to the Permanent Secretary, or postal receipt or the notice of arrival of goods issued by an organisation providing a courier service, as the case may be, and such additional information or document as he may reasonably require and consult such person as he considers necessary.\n(3)\u2003(a)\u2003Subject to subparagraphs (b), (c) and (f), the Permanent Secretary shall grant an approval subject to such conditions as he may determine and, in relation to goods specified in the second column of the Third Schedule, subject to the additional conditions specified in the third column of that Schedule.\n(iii)\u2003the submission of a certificate of inspection carried out under subparagraph (ii) which shall be in the form set out in the Eighth Schedule.\n(B)\u2003the imported vehicle is certified to be free from removable radioactive surface contamination by the Radiation Protection Authority.\n(B)\u2003a payment of 120,000 rupees is made to the Accountant-General.\n(iii)\u2003Where an imported second-hand, used\/reconditioned, motor vehicle is certified not to be free from removable radioactive surface contamination by the Radiation Protection Authority, the Permanent Secretary shall not give an approval.\n(ii)\u20036,000 rupees in the case of any controlled goods other than second-hand motor vehicles.\n(e)\u2003The fees of 60,000 rupees and 6,000 rupees payable under subparagraph (d) shall not apply to importers referred to in Parts I and II of the Seventh Schedule, respectively.\n(ii)\u2003the importer submits a certificate issued by an inspecting body, referred to in subparagraph (i), which indicates that the bunker fuels comply with the specifications set out in Parts II to IV of the First Schedule, as the case may be.\n(c)\u2003after consulting any person, the Permanent Secretary is satisfied that the removal of goods cannot be approved.\n(b)\u2003the original Customs provisional bill of entry and the final Customs bill of entry.\n11A.\u2003(1)\u2003Notwithstanding these regulations, the Permanent Secretary may authorise the performance of any act or thing required to be done under these regulations to be made or done electronically through such computer system via the TradeNet, as he may approve.\n\"TradeNet\" has the same meaning as in the Customs (Use of Computer) Regulations 1997.\n11B.\u2003(1)\u2003Where any imported second-hand, used\/reconditioned, motor vehicle or any second-hand motor vehicle parts and accessories originating from Japan are certified not to be free from removable radioactive surface contamination by the Radiation Protection Authority, the importer shall, at its own expense, return the contaminated vehicle or contaminated parts and accessories to its supplier not later than 2 months from the date the vehicle or parts and accessories are found to be contaminated.\n(2)\u2003Any person who, without reasonable excuse, fails to return any contaminated vehicle or contaminated parts and accessories within the period specified under paragraph (1) shall commit an offence and shall, on conviction, be liable to a fine not exceeding 120,000 rupees and to imprisonment for a term not exceeding 3 years.\n11C.\u2003No security by means of a bank guarantee, issued by the Bank in Mauritius for the purposes of these regulations shall be released unless the Permanent Secretary approves in writing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"[SANTA CRUZ, CA] \u2013 Members of SEIU Local 521 joined with other working people across the country today to declare their resolve to stick together in their union. Even though the U.S. Supreme Court delivered a long-expected decision in the Janus case, one that was backed by wealthy special interests seeking to rig the economy in favor of the greedy few, SEIU members remain dedicated to sticking together. Santa Cruz County workers, whether white, Black or brown, are standing together in unions and holding politicians accountable for creating more good, union jobs.\nSEIU members and community allies know that when working men and women join together in union, they gain the power in numbers to raise wages, win affordable healthcare and make a dignified retirement possible.\nComments about Union members in Santa Cruz County vow to stick together: \"No Court case will stand in our way\" are welcome. Off-topic comments and other violations of our community guidelines may be withheld or removed. Comments do not appear immediately after posting.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Previous Post How 50 Shades of Grey and Hollywood inspired me to create my SASSY line.\nNext Post Casual-Cool, L.A-Girl Style.\nLooks very nice! Great styles and inspirationa nd lovely blog you have!\nI love it. Thanks for sharing and check out the latest interview.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is not the Rocket Science, it very simple if you already have the knowledge of OOPS in any programming language.\nTo create a new instance of an object obj we use the statement new obj, assigning the result (which is of type obj) to a variable to access it later.\nIn the example below we define a class named Person and we create two instances (person1 and person2).\nProperties are variables contained in the class; every instance of the object has those properties. Properties should be set in the prototype property of the class (function) so that inheritance works correctly.\nIn the example below we define the gender property for the Person class and we define it at instantiation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzahqhu b/data_all_eng_slimpj/shuffled/split2/finalzzzahqhu
new file mode 100644
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@@ -0,0 +1,5 @@
+{"text":"As you know I like to mix and match my Bullet Journaling between paper and digital forms. It works really well for me as it means I can easily juggle my journal contents depending upon circumstances, location and what I have to hand.\nI've been playing with some formatting ideas for the Tailfish by Pyry Etsy store and over the weekend I designed a quick PDF for me to use in my digital OneNote journal. The great thing is I can easily duplicate the page, so even if I take a lot of notes that day, they can all be unicorn themed!\nIf you'd like to have your own Unicorn themed notes then you can download a copy by clicking the image below.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The goal of this lesson is to educate general industry employees who enter, supervise entry or attend the entry of a confined space about the practices and procedures necessary to protect employees from the hazards of permit-required confined spaces.\nYour company's Permit-required Confined Space Entry Program is its overall policy and plan for protecting you and other employees and contractors against confined space hazards, and for regulating entry into its permit-required spaces. Before you may enter a permit space, your supervisor and\/or the attendant must take any measures necessary to isolate the permit space and eliminate or control hazards. These measures include lockout or tagging of equipment to ensure that it does not present a hazard.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":">> p4080\/p5020 attaches the most of error interrupts to interrupt zero.\n>> And report error interrupts to mpic via interrupt 0.\n>> This patch can handle both of them.\n> Does this also work on T4 \/ PCIe controller rev3.0?\nNo, it does not work on T4.\n>> + * Copyright (c) 2013 Freescale Semiconductor, Inc.\n>> + * This function is for error interrupt ORed mechanism.\n>> + * This mechanism attaches most functions' error interrupts to interrupt 0.\n>> + * And report error interrupt to mpic via interrupt 0.\n>> + * EIMR0 - Error Interrupt Mask Register 0.\n>> + * bit in EIMR0.\n>> +\t * Unmask pcie error interrupt bit in EIMR0.\n>> +\t * Extend interrupt specifier has 4 cells.\n>> +\t *\t1 -- error interrupt.\n> Why is this all needed, we have handling of error interrupts in the kernel already?\nYes, you are right. This is not needed, and I will remove it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As Jio is setting its stage ready for the highly\u2013anticipated 4G service with a bouquet of offers and services, the biggest highlight, which grabbed everyone's eyeballs during the launch, was Reliance Jio's 'free voice calls', even on roaming. Yes, you read that right. Mukesh Ambani revealed that there will be no charges while making calls anywhere in India, and the users will only pay for mobile data. Reliance Jio 4G services are open for a while now for anyone to avail starting from September 5. However, here is still a lot of confusion in the market on whether the SIM is going to work with all smartphones.\nA simple answer is no. That's because the Reliance Jio 4G service is truely a 4G service, as it claims to be. There is no 2G or 3G signals so phones that are not 4G compatible will not work on this network. Although the sim is readily available to anyone now, it is important to know that the unlimited HD voice service is going to work only on handsets that support VoLTE (Voice over Long Term Evolution). This technology enables Reliance to transmit voice calls over data network which explains how they are able to deliver free-of-cost HD voice services and that too without any roaming charges. 99% of phones that are powered by a Qualcomm Snapdragon or mediatek chipset have support for VoLTE today. Only VoLTE supported mobiles will get access to true HD Voice call from Reliance Jio 4G network.\nReliance holds the license for 2300 MHz spectrum all over the country plus, 800 MHz spectrum in 10 circles and 1800 MHz spectrum in 6 circles for all kinds of services. A non-VoLTE phone can access 4G data seamlessly however may not be able to make voice calls. Reliance has come out with a workaround to fix this issue with an app called Jio Join (android only). By downloading this app on a 4G phone that does not support VoLTE you can manage to make voice calls over 4G LTE network but don't expect to get a HD clarity. It is to be noted that mobile data needs to be on continuously to make voice calls, which is not the case with VoLTE supported phones. So which are the phones that work on the service? Jio has already announced partnerships with Samsung, HTC, Micromax, Sony, Vivo, Intex, Lava, Gionee, Panasonic, Asus, Xolo, LG, Karbonn, Videocon, YU, Sansui, Alcatel and TCL. Here is the full list of compatible devices from Reliance Jio and its partners. Mentioned below are 177 TRUE 4G \/ VoLTE smartphones.\nNote: Will keep adding as and when new devices are discovered to support this service.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Tag Archives for \" improve your italian \"\nWondering how to say TO LIKE in Italian? This lesson is the part of a series of eight, where you will learn all there is to know about this verb!\nWondering how to say TO LIKE in Italian? This lesson is the part of a series, where you will learn all there is to know about this verb!\nWondering how to say TO LIKE in Italian?This lesson is the second for a series, where you will learn all there is to know about this verb!\nWondering how to say TO LIKE in Italian?This lesson is the first for a series where you will learn all there is to know about this verb!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzajapi b/data_all_eng_slimpj/shuffled/split2/finalzzzajapi
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index 0000000000000000000000000000000000000000..afb0daa19bb89dd57a15e817453649b5210547ca
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzajapi
@@ -0,0 +1,5 @@
+{"text":"Issue \"Derri\u00e8re le Miroir\". Poems by Apollinaire and by Blaise Cendrars, texts by Jean Wahl and by Lionello Venturi, illustrated with 2 lithographs in colors (1 in double page). Maeght Publisher, Paris. Second edition.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Unfortunately every time I try to do something on the website I am taken back to the login screen, and after logging in it takes me back to the login screen (even with the 'remember me' button checked.\nSorry to hear that the site is not behaving for you. You may benefit from clearing your browser's cache of cookies. Give that a go and let us know how you get on.\nGreat to hear. Sometimes the cookies can get confused!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Several research projects at our Division are focused on molecular and cellular mechanisms that contribute to the development of severe infections caused preferentially by bacteria, but also by viruses and fungi. The starting point for this work has often been the discovery of bacterial surface proteins and enzymes that interact with human defenses (innate and acquired immunity and blood coagulation). We also study the nature and significance of inflammatory responses in infectious diseases and other pathologic conditions. In this work we use a combination of different methods and techniques: protein chemistry, microbiology, molecular biology, cell biology, imaging techniques, animal experiments, and analysis of patient samples. Our aim is that the obtained knowledge will contribute to novel preventive, diagnostic, and therapeutic opportunities in clinical medicine.\nOur overall objective is to pursue experimental and fundamental medical research of high quality in order to tackle significant clinical problems. We are convinced that research performed at a high level of abstraction is required for breakthroughs generating novel biomedical and clinical implications. For more information, please see our sub-units and our projects.\nTilde Andersson, Gizem Ert\u00fcrk Bergdahl, Karim Saleh, Magn\u00fasd\u00f3ttir, H., St\u00f8dkilde, K., Andersen, C. B. F., Lundqvist, K., Jensen, A., Br\u00fcggemann, H. & Rolf Lood, 2019 Mar 5, In : Scientific Reports. 9, 3596.\nGizem Ert\u00fcrk Bergdahl, Tilde Andersson, Maria Allhorn, Sofie Yngman, Rainer Timm & Rolf Lood, 2019 Feb 27, In : ACS Sensors.\nVan Vlasselaer, A., Magnus Rasmussen, Johan Nilsson, Olaison, L. & Sigurdur Ragnarsson, 2019 Feb 27, In : Open Heart. 6, 1, e000926.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A thicker-gauge cuff in a solid team color a dual-colored pom and a warm fleece lining make this beanie a must-have for any fan. A tonal embroidered New Era flag adds subtle character. Fabric: 100% acrylic shell 100% polyester fleece lining. New Era products may not be resold without embellishment.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Becc Flags specialises in high quality vinyl stickers, like these Great Britain Wales national flags.\nHave your boat show an affiliation with these brightly coloured white, red and green vinyl stickers.\nThese ones suggest Welsh boats and ships.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzajjmb b/data_all_eng_slimpj/shuffled/split2/finalzzzajjmb
new file mode 100644
index 0000000000000000000000000000000000000000..a319686693d8a3b917f5359492122ba7e3210c8f
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzajjmb
@@ -0,0 +1,5 @@
+{"text":"How Singular is the Singularity?\nIf recent headlines are anything to go by, opinion on the likelihood \u2013 and impact \u2013 of the 'Technological Singularity' is diverging rapidly. Is this largely because we don't even agree on what it is?\n'Artificial Intelligence' (AI) is certainly in the news at lot at the moment. But so are robots; and Kurzweil's Singularity; and machine evolution; and transhumanism. Are these the same thing? Are they even related? If so, how? What exactly should we be arguing about? Are we worried precisely because we don't know this stuff?\nWell, perhaps to make a start, we should point out that intelligence isn't the same thing as evolution (in any sense). That's obvious and accepted for 'conventional' life-on-earth but we seem to be getting a bit confused between the two when it comes to machines. Developments in both may proceed in parallel and one may eventually lead to the other (although which way round is debatable) but they're not the same thing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"MEET OUR TEAM | Dr. Taylor Cox is the owner and dentist at Campbellsville Dental Care. Get to know him and his team better here.\nDr. Taylor Cox is a Campbellsville native and graduate of Taylor County High School. He attended the University of Kentucky, where he earned his Bachelor's degree, followed by a Doctorate in Dental Medicine from the UK's College of Dentistry. After completing residency training with the University of Florida, he and his wife, Morgan - also born and raised in Campbellsville - came home to Kentucky to open a practice and start a family.\nTaylor and Morgan have one son, Oliver, born January 2018. When not in the office, you might find them spending time on one of their family's farms, or enjoying the outdoors with their Scottish Terrier, Maxwell.\nSara Inman is a native of Washington County and currently resides in Campbellsville. She has worked as a dental assistant for the last ten years. During her free time, Sara enjoys spending time with her husband, Mark, and son, Logan.\nMisty Long is a native of Casey County and currently resides in the Elkhorn community outside Campbellsville. She is a formally trained dental assistant, and is excited to return to a role in dentistry. During her free time, Misty enjoys time outside with her husband, three sons and nine pets.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Viljami Kutvonen had to defend his TQ position hard in the first Modified A-main in Odense. The finn got off to a great start and instantly pulled out quite a lead, as Ronald V\u00f6lker struggled for grip on the opening laps.\nGradually, V\u00f6lker picked up the pace, and the german started gaining a lot on the leading Mugen Seiki driver \u2013 so much in fact that Ronald V\u00f6lker managed to challenge Viljami Kutvonen for the lead in the closing seconds of the heat.\nEventually, it came to a crash between the two as Ronald V\u00f6lker tried to pass Viljami Kutvonen. As a result, Ronald V\u00f6lker had to get back on the wheels while Viljami could race to the A1 victory.\nDionys Stadler finished third, while Super Stock leader Tim Benson fought his way from P9 on the grid to a P4 finish. David Ronnefalk finished fifth, followed by Patrick Danielsen and Patrick Hornum. Jakub Rozycki didn't have the best run in the A1, dropping from P3 to a P8 finish.\nViljami Kutvonen leads Ronald V\u00f6lker in the Modified A1.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Flower Form2\" double flowered with 3 - 7 racemes.\nFoliage TypeSerrate oval 3\" medium green leaves.\nPruning TimeAfter flowering, blooms on previous years wood.\nSoil RequirementWell drained, humus enriched soil is best.\nWatering RequirementMoist soil. Mulch area around shrub to help the soil retain moisture.\nPruningPrune annually after the blooming period, to help maintain its shape, in early summer, since the shrub blooms on the previous year's growth.\nUnique CharactersticsWonderfully fragrant double blooms on 6-8' tall plants. Ideal as a specimen planting or used to create a hedge.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Three of Frank Corcoran's 'tape' music masterworks are presented here for the first time in surround sound and as a complete body of work in a single concert. Each of these works reveals Corcoran's far-reaching vision of a distant, multi-layered, mythic past which is rendered so deftly here in late 20th century technology.\nWritten in the years spanning the peak of Corcoran's creative output, these pieces are in a consciously radiophonic style and, when heard together in a seamless concert performance, they accumulate a coherence and forcefulness of argument and bring to life Corcoran's mythological landscape, littered with visual metaphors, in a true 'cinema for the ear'.\nA public pre-concert talk will take place with Frank Corcoran at the Contemporary Music Centre Library, 19 Fishamble Street (just 100m from Smock Alley Theatre) at 6pm. This public event is free to attend and all are welcome. We encourage people to indicate their attendance by emailing rsvp@cmc.ie.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzajucu b/data_all_eng_slimpj/shuffled/split2/finalzzzajucu
new file mode 100644
index 0000000000000000000000000000000000000000..9c685ca1c7e59d2b23d86f9315b43f7844f99697
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@@ -0,0 +1,5 @@
+{"text":"Find companies that sell childrens furniture and decor in Gauteng. Whether it's a bunk bed, desk, bookshelf or cupboard or anything else you need for your childs room, you'll find it here.\nQuality, stylish and versatile baby and child-friendly furniture & Linen at affordable prices. Quality service guaranteed.Est 2000.\nStylish and well constructed life-size kitchen units for children from 3-10 years old. These innovative designs keep your budding little chefs engaged for hours, ideal as gifts for kids..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We were touched and honored to read this letter from the mother of a family that travels regularly with Artisans of Leisure. We hope you enjoy reading about how traveling together has become such an important part of their lives.\nAs my eldest daughter enters her senior year of high school, with her two sisters not far behind, I look back at how fast the time together as a family goes by. I am so grateful for the many, many travel experiences we were able to share; a time for just our family in unfamiliar and exciting places. Away from the blur of everyday life, and armed with a solid pact to \"get along for this special time\", we turned ourselves over to the competent guidance of Artisans of Leisure.\nOur first trip with Artisans of Leisure was to France and I thought that it would just be a special \"once in a lifetime\" trip. But the value they brought with their knowledgeable guides and drivers allowed us to see more places in a shorter amount of time. It was such an amazing trip that we scheduled more. The experience has been consistent throughout the many countries (11) we have traveled to with them.\nOur Artisans of Leisure travel consultant, who has come to know our family well, suggested we visit places we never dreamed of going, like Egypt and Russia. These adventures opened our eyes to incredible histories and made us more aware of key happenings in the world. Our children have learned an appreciation of other countries and cultures, as well being enlightened to the blessings of being born in America.\nIt is always a treat for me, the mom, not to worry about where we are going or whether we will be safe. I get to enjoy the trip along with everyone else in my family. The itineraries are always specifically tailored to our interests and energy levels. When my daughters were younger, we did history tours of amazing countries. Now, that they are teenagers, we have added more experiences into our explorations. This year, Artisans of Leisure recommended that we go to Africa. We are all looking forward to it.\nI often hear parents say that they want to wait until their kids are old enough to appreciate or remember overseas travel, but truth be told, the older they get, the harder it is to find the time and encourage the interest. They are invested in sports, school, friends. etc. None of us remember all the information presented or the significance of every landmark, but we do remember sharing the time together and the special bond created by the experience. My husband and I remember, and that is invaluable. We started traveling overseas when my daughters were 8, 9 and 11. Traveling with them when they are younger sets the stage for continuing as they get older. I would say the sweet spot for historical travel is between the ages of 8 and 12.\nThe family has since taken their tour of Kenya and Tanzania with Artisans of Leisure. See a slideshow of their amazing photographs in our gallery.\nContact Artisans of Leisure to learn more about customizing a tour for your family.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This extension enables you to syntax highlight code samples written in almost any language. It uses GeSHi - the Generic Syntax Highlighter.\nNB! GeSHi itself isn't included in the download - read the installation instructions in the extension documentation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"resin-based filler for square-edged floorboards.\nCellulose filler is mixed with the final fine sanding dust from the floor for an invisible repair on hardwood floors.\nResin filler is used for square edged floorboards because of its superior side bonding properties.\nWe usually recommend brushes over rollers for one off jobs as they give much better results in most domestic situations: you can control the amount of finish more accurately, spot the odd bit of debris, get into corners etc. with a brush.\nRollers can cause foaming and require some familiarisation to get good results.\nThe important thing to remember is to use a good floor brush, 5 or 6 inches wide. Regular paint brushes are not suitable as they don't 'load up' with finish. Floor brushes are designed for the thicker high-quality floor varnishes which we sell.\nIf you would like to order floor sanding materials or have questions about our floor accessories please don't hesitate to email or call.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"November 2018 - BROCK Information Systems and Information Technology Students teamed up to work on a community partnership project with Welland Toyota again this year. The project involved Goodman School of Business students who came into the dealership to research our service department in order to understand scheduling, data management and software capacity. They then spent the term developing an information system app based on the needs of this sector.\nWe are asking for your contributions for our Mo-team, Patrick Clarke, David Clarke and Niels Pedersen, who are growing their mustaches as they raise funds for men's health this month.\nNiels currently loves driving a 2018 RAV4 Platinum.\nAnna won a new Toyota Camry through Tim Horton's Roll Up the Rim Contest! Here's Anna getting the keys from Welland Toyota Sales Manager Rick Fauteux.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzakdvf b/data_all_eng_slimpj/shuffled/split2/finalzzzakdvf
new file mode 100644
index 0000000000000000000000000000000000000000..f013d5f20e2e91631d74a008e214b901b1c19418
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"(C)2018 Anuman Interactive SA. All rights reserved. Developed by Eden Games. Micro\u00efds is a registered trademark of Anuman Interactive SA. All rights reserved. The names, designs and logos of all products are the property of their respective owners and used by permission. Published and distributed by 3goo K.K.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Online \/ Digital BBC Poetry Viral \"Blake Signs Contract\"\nI am currently \"Juan Sheet\" on the Plenty advertising campaign. \"Route Canal\" which starred, written and directed by me is currently doing the film festival circuit. Winning prizes and being officially selected .\nProducer on Route Canal, Platos Breaking Point (feature film) and numerous theatre and short film projects.\nHorseriding Skills,Snowboarding, PADI Diving Skills, improve, Teaching & Coaching in Meisner and Method techniques.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Myeloproliferative neoplasms (MPNs) can increase the risks in pregnancy but planning makes a difference.\nMany people may be diagnosed with an MPN in their twenties and thirties, the time in life when many of us are establishing careers and starting families. It can come as an unwelcome surprise to learn not only that you have an MPN, but that this disorder may affect your ability to start a family. You might have learned you had an MPN when you had your first blood test after becoming pregnant. Understandably, this news may have come as a shock.\nMPNs can increase the risk of miscarriage both early and late in pregnancy. MPNs can also increase the risk of the mother developing pre-eclampsia, a dangerous condition in which a woman develops high blood pressure during pregnancy. Pregnant women who have MPNs need to be carefully monitored by their health care team including their haematologist, obstetrician and midwife to watch for and manage any problems that may develop.\nThe good news is that having an MPN doesn't have to prevent you from having children. The most important factor when you are thinking about starting a family is planning and working together with your health care team. It's key to discuss your plans together with your haematologist and your GP before you become pregnant (if possible) and while you are pregnant.\nNew mothers with MPNs can face medical and emotional challenges. The weeks after delivery (or after a miscarriage) can be a challenging time for women with MPNs. Your blood volume will decrease rapidly immediately after delivery and your blood counts may increase very fast to levels that may be high, even dangerous. Your haematologist will track you very carefully during this time to ensure that you stay healthy and safe.\nNew mothers with an MPN are at risk of clots after delivery. This is because blood volume decreases rapidly after delivery. New mothers with MPN may need to take heparin to prevent clots. This drug which is given by injection for several weeks after delivery is easy to self-administer or you can ask a family member for help. In addition to heparin, some new mothers need to continue with interferon therapy or may need to return to hydroxycarbamide or another cytoreductive drug. In some cases taking medication may mean that you will not be able to breastfeed.\nSome new mothers may not be able to breastfeed their baby if they need to take medication after delivery. Heparin is safe but hydroxycarbamide, anagrelide and other cytoreductive drugs can harm a baby if passed through the breast milk. The risks with interferon are less clear and you may be able to continue to breastfeed while on this drug but it would be important to discuss this with your own team. It can be upsetting to learn you cannot breastfeed if you hoped to breastfeed your child. You may also feel pressure from family members or other well-meaning people who would prefer that you breastfeed your baby. In all cases the most important consideration is the long-term health of both mother and baby. Talk with your midwife and doctor about your options. You may wish to talk with a buddy if this is upsetting or a difficult decision for you.\nIf you have had a caesarian section you might continue to bleed from your incision for some time, particularly if you are taking heparin. If bleeding is profuse, do not hesitate to ask your haematologist and GP for help. Keep your incision covered with sterile gauze bandages. Special tape is available that does not hurt your skin. Some mothers with MPNs have found that sterile, medical seaweed-based \"packs\" can assist in closing an incision. Ask your GP for advice.\nHaving a baby is always tiring, but fatigue can be a particular problem if you lost blood during delivery, if you continue to bleed after delivery, and if your baby was born early or small and requires around-the-clock care. It is very important to take good care of yourself during this time and to actively plan to have others help you. Consider hiring help if you can afford it, for instance a doula or a child-minder to help with older children. Ask for help from family members and friends. You may need extra help during the night so that you can get a few more hours of sleep. Ask your partner or another family member to give your baby a bottle feed with formula during the night so that you can recover and heal. Talk with your haematologist and your GP if you need help in coping with fatigue.\nVisit the section coping with fatigue for more practical tips about fatigue.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Naya Rivera, 29, has filed for divorce from husband Ryan Dorsey. They have been married for just over two years.\nThe former Glee star filed documents last week to end the marriage, according to TMZ.\nRivera is also reportedly seeking for primary physical custody of their 1-year-old son Josey, with Dorsey having visitation rights.\nRivera wed Dorsey in July 2014 just three months after her engagement to rapper Big Sean was called off.\nRivera and Dorsey, 33, dated in 2010, but she ended the relationship to commit herself more fully to her career.\nPosted on November 21, 2016, in ENTERTAINMENT NEWS. Bookmark the permalink.\tLeave a comment.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This stunning amber necklace, exclusive to Metalsmiths Sterling features Certified Baltic Amber from the Black Sea - a freeform pendant set in sterling silver.\nThis pendant features a modern asymmetrical bail and wrapped freeform sterling silver setting with a beautifully coloured elongated teardrop stone in shades of green and cognac. The pendant measures about 54mm in total length (including the bail), 18mm across the widest point and 9mm in overall thickness. The back is flat.\nYou will receive this specific piece of amber as pictured as this gemstone varies widely: for additional stones, consult select locations.\nThe necklace includes a 1.9mm sterling silver rolo chain in an 20\" length that fastens via a secure lobster clasp, complete with the Metalsmiths Sterling logo tag of quality.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzakhbf b/data_all_eng_slimpj/shuffled/split2/finalzzzakhbf
new file mode 100644
index 0000000000000000000000000000000000000000..f44311572cd5a8cbef0c8b8fd5397f3ef29fe311
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Know Newton Conover High Class of 2011 graduates that are NOT on this List? Help us Update the 2011 Class List by adding missing names.\nMore 2011 alumni from Newton Conover HS have posted profiles on Classmates.com\u00ae. Click here to register for free at Classmates.com\u00ae and view other 2011 alumni.\nAlumni from the Newton Conover High School class of 2011 that have been added to this alumni directory are shown on this page. All of the people on this page graduated in '11 from Newton Conover High. You can register for free to add your name to the NCHS alumni directory.\nDon't forget to upload your Newton Conover High School pictures and check on the latest info about any Newton Conover High School reunion.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The sound of flowing water is so pleasing; it's just one of the many delights of a tranquil water feature in your garden. But instead of having a simple pump in a pond, go one further and create a truly outstanding feature, hightlighted by an attractive mosaic tile wall. This project makes use of an existing freestanding wall and garden bed. If you have a wall but no bed, build one using brick pavers or blocks secured with mortar mix. Be sure to let it dry thoroughly before starting the project.\n1. You can liven your landscape with the dramatic foliage of elephant ears (Colocasia sp.).\n2. Make a statement in any garden with Agave attenuata's rosette of thick, fleshy leaves.\n3. Add year-round interest with Dianella 'Silver Streak'. Plant in clumps for showy impact.\n4. Forming a dense carpet of pretty flowers, Pratia pedunculata is a terrific filler and groundcover.\n5. This incredibly tough cycad (Zamia furfuracea) is low maintenance, too.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Personal style is a concept I embraced at an early age. From sporting my grandmother's vintage pieces to creating my own apparel, I always utilized style as a form of expression. Nothing made me smile more than seeing heads turn at my crafty, eclectic ensembles. I have proudly delved into the athletic, luxurious, tomboy, and even gothic style trends, each allowing me to experience the importance of fashion in both expression and perception. Today, my style is a sophisticated mix of bohemian and grunge, one I anticipate will change with my ever changing mind. I find that personal style is best represented when tailored not only to an individual's silhouette, but also to their true likes and preferences. With so much emphasis on celebrity trends and designer collections in the media, many are influenced to mock the looks they see on television and in print. Having immersed myself in a variety of styles, I am in full support of personalized style expression, with or without a budget.\nAlthough my clothing choice changes with the occasion and weather, it is my use of signature details that personalize my day-to-day wear. Weekdays at the office call for business casual attire, a look that is easily personalized with accessories. The pencil skirt is a staple piece of my wardrobe which can be dressed up or down with the appropriate blouse and accessories. On warmer days, I like to pair a dark or neutral-colored pencil skirt with a sleeveless blouse, and finish with a pair of suede heels or wedges. This look creates a professional, yet stylish and approachable appeal. My personal style is introduced with details such a fringe tote or strappy, studded wedges. I often dress for cooler days with a sleek pair of jeans or leggings and knee-high boots, completed with a scarf or blazer to keep the look business and weather appropriate.\nMy personal style is best observed when I dress for more casual occasions, such as shopping or meeting a friend for lunch. I prefer earth tones\u2014such as olive green, spice brown, and taupe\u2014for they provide a wardrobe that is easily mixed and matched. I avoid busy prints and graphics, but have an eye for fringe and purposely \"distressed\" pieces. I own many pairs of boots, and have a pair appropriate for almost every look\u2014from the casual and timeless trooper styled boot to the sleek and modern ankle boot. Style is not only determined by pieces of clothing, but also by the way one wears their hair and makeup. I am known for my signature brown smoky eye and long, dark (often beach-waved) hair. My style is not influenced by a particular celebrity or figure, but rather by my free-spirited nature and eye for the edgy and eccentric. It is the expressive aspect of fashion that has made me an enthusiast, and I hope to share this passion while encouraging others to stay true to themselves when creating their personal styles. If the mind bears it, allow the body to wear it!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Transact on Ardor, a scalable blockchain powered by the first Proof-of-Stake consensus network.\nArdor is a blockchain platform, that can be powered by the sun. It is a platform where businesses can setup their own chains without any need to code or secure it themselves. Ardor has a built in exchange between other chains on the platform, to easily exchange your token with others including fiat pegged ones like AEUR. Ardor offers Lightweight Smart Contracts that don't require the whole network for each dApp. Smart Transactions make features like a Deadman switch easy to implement. Fully developed features like built-in Voting, Messaging, Marketplace, Basic Cloud Storage and more.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Crowdfunding has the promise of becoming the most reliable real estate financing tool. It is therefore essential that anyone with vested interest in real estate learn more about this financing model. Crowdfunding exists in two major types as discussed below.\nThis refers to a crowding funding investment whereby an investor becomes a shareholder in the real estate project. Once one has invested in the project, it will be that they have bought an equity stake in the project. This has both its downside and its upside as explained by experts from Sun West Mortgage. The upside is that the investor benefits from rental income and any price appreciation at the time of sale. On the flipside, however, if the project does not break the bank, one is second in line for payout after other investors.\nIn this type of crowdfunding investment, realtors at Sun West Mortgage pose that an investor effectively loans out money to a project. The investor will then receive fixed returns that are based on the amount invested and the interest on the owner's mortgage loan. This is a less risky method of investing in a real estate project. The investor is given priority in payouts should the project fail to pay off.\nDepending on one's risk preference, crowdfunding offers a channel to invest in real estate. It is important to assess one's risk appetite before investing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzalvsq b/data_all_eng_slimpj/shuffled/split2/finalzzzalvsq
new file mode 100644
index 0000000000000000000000000000000000000000..6ceca5462050ab83a7098aee4cc9af25cf83da80
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"I have questions about the FDA RACC.\nSo for other candies, the RACC is 30g.\nWe currently are producing chocolate pieces in 50g each.\nWould it be 1 2\/3 servings per container, serving size 30g ( 1.06 oz)? I cannot find such examples anywhere on the FDA website.\n(6) A product that is packaged and sold individually that contains less than 200 percent of the applicable reference amount must be considered to be a single-serving container, and the entire content of the product must be labeled as one serving. In addition to providing a column within the Nutrition Facts label that lists the quantitative amounts and percent Daily Values per serving, for a product that is packaged and sold individually that contains more than 150 percent and less than 200 percent of the applicable reference amount, the Nutrition Facts label may voluntarily provide, to the left of the column that provides nutrition information per container (i.e., per serving), an additional column that lists the quantitative amounts and percent Daily Values per common household measure that most closely approximates the reference amount.\nBasically, if your product is more than 1.5 servings (RACC) but LESS than 3 servings (RACC) then you label it as 1 serving. In your case that means labeling it per 50g. You may choose if you wish (I wouldn't) to go dual column and list both per 28g and 50g on the nutritional panel.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Raspberry Pi 3 Spotted on Black Mirror Episode! @Raspberry_Pi #PiDay #RaspberryPi \u00ab Adafruit Industries \u2013 Makers, hackers, artists, designers and engineers!\nA Raspberry Pi 3 makes a guest appearance on the Season 4 Episode 5 of Black Mirror, which is titled Metalhead.\nHaven't seen this episode yet, waiting for my girl friend to finish the previous seasons!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Key highlights from budget 2014. For sure, the government is pro-entrepreneur friendly and has announced several measures to promote entrepreneurship in the country.\nOfcourse, for the first time we have seen words like startups, accelerators and incubators being mentioned in the budget speech. India definitely has come a long way!!\nStartup programme for rural youth in India with Rs 100-crore allocation.\nNationwide incubation and acceleration infrastructure for startups.\n10,000 crore fund to provide equity and soft loans for startups [though we think there are better ways to fuel entreprenuership in the country].\n100 crore allocated for rural entrepreneurship.\n200cr to encourage #startups by youth of SC\/ST.\nSpecial focus to ramp up software startups with allocation of Rs 500 cr.\n100 cr set aside for development of Technology Development Fund.\n5 more IITs and 5 more IIMs to be set up. New IITs proposed in Jammu, Chhatisgarh, Goa, AP & Kerala .\nPropose to set up 4 more AIIMS in AP, West Bengal, Vidharba and Purvanchal. Rs 500 crore to be allocated. The goal is to create an AIIMS in every state.\nNational Rural Internet and Technology Mission; Rs. 500 crores allocated.\nAll govt departments and ministries to be integrated through e-platform by 31st December 2014.\nOnline and mobile advertising to attract service tax. Print advertising is exempted.\nNational Industrial Corridor \u2013 HQ in Pune. Rs 100 crore has been allocated.\n100 'Smart Cities' \u2013 definitely a boost for cement, steel sectors!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"YRCI offers professional services in three specific areas: Human Capital and Human Resources, Acquisitions, and Financial Management. These are the only areas on which YRCI focuses. Many of our competitors are multi-service administrative support companies that provide everything from financial management to general facilities maintenance and housekeeping. We have chosen to focus on the professional services level in these three distinct areas and have become a leader in the provision of these services to Federal Government. This is our profession, and it is our mission to improve Government processes and programs while running a profitable business and being responsible citizens and taxpayers. This means we seek to provide sustained value.\nWe grew up in the staff augmentation business and have refined our talent sourcing model. We retain a strong recruiting staff backed up by experienced subject matter experts to assure that the talent we put in service to Government is the best qualified available within the parameters provided by the Government. This approach makes us unique in that we provide professionally screened, capable professionals that we have actually met, interviewed, and qualified to perform. All of our candidates are real, already working for us or validated, and ready to go to work.\nOur combined professional areas assure our customers that we provide extraordinary professionals and that we deliver an expertly administered contract, executed within budget and on time. We invite you to explore our services pages and learn more about YRCI. We look forward to contributing to your success by providing exceptional people that exceed your expectations in performance.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Surrounded by chic and modern decor, the atmosphere in Sai Woo is simply divine. Have a seat and enjoy one of their delicious cocktails, and prepare for an Asian fusion dining adventure. Here is where east meets west.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzalvwj b/data_all_eng_slimpj/shuffled/split2/finalzzzalvwj
new file mode 100644
index 0000000000000000000000000000000000000000..30bf3d4a450496110679bc0f2de07cd0d96bb4f1
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"original_post:2\/17\/2009____________________________ So Lydia is in a minimum security\u2026 jail? Prison. I'll admit, I was a little sketchy on the details because it just wasn't something I knew about. And I didn't want to spend a lot of time on her arc, while she was in prison. The main thing is that Rae visits her and boom, she is given the store. This surprises Rae, and suddenly her desire to make changes just got a lot more change ready.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pete and Phil fall into the category of early bookers. Contacting me around 18 months in advance is a pretty mid term outlook on things. But then if you knew Pete\u2026it wouldn't really surprise you. Meticulous planning and a keen understanding of the finer details were always going to make this a well run affair. Of course having two people who truly love each other coupled with some of the warmest friends and family I've ever seen at a wedding meant it could have been a campfire and some marshmallows, the result still would have been the same. I had two previous experiences with civil partnerships (or as Pete and Phil like to call it, Big Gay Weddings). Both times of the female persuasion. This would\/was\/is my first of the male. To be honest, if the two at the centre of it all love each other, themselves and their guests it really doesn't make a whole heck of a difference photographically. I can say that I'm a better photographer for the experience. I walked away with an enormous smile and a warm feeling inside. I hope all 'weddings' are just like this one. If you are interested in either Civil Partnership photographer or other weddings at the wonderful Cain Manor in Surrey please get in touch.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This morning in the nation's capitol, mourners said farewell to Dr. Dorothy Height, a life-long Civil Rights activist to whom this nation owes a debt of thanks. Rather than end my blogging stint on Harriet describing some of the revolutionary things that National Poetry Month has allowed this nation and its poets to accomplish, as I had originally planned, I have decided to dedicate this space to the memory of Dr. Dorothy Height. This is fundamentally about poetry, too, because I am curious about the ways we have and can and will memorialize the women and men who make this world the sort of place in which I want to live.\n- Camille Dungy on Harriet, the blog of the Poetry Foundation - read the full piece here, including a nice shout out to Split This Rock.\nWhat happened earlier this month happens almost every day somewhere in the world: Miners are killed at work. And why do they die--or for whom? Miners from Utah to sub-Saharan Africa to China's Shanxi province die, in part, for us. As consumers who walk the aisles at WalMarts, dollar stores, and suburban shopping malls, we fuel the extraction of coal and other minerals every time we purchase items that are intimately connected to miners around the world.\n- from \"Warning: Shopping May Prove Deadly to Miners,\" by Split This Rock 2010 featured poet Mark Nowak, published yesterday in Common Dreams. Read the full piece here: http:\/\/www.commondreams.org\/view\/2010\/04\/26-7 and add a comment or link to your FB profile to get the word out. Thanks!\nI urge the Council to resist the cuts to the Commission's budget proposed by the Mayor and to recognize the essential role that arts and culture play in the life of the District of Columbia and its residents, whose poetry explores the city's past, negotiates its present, and imagines its future. Without poetry, we are a city without a story.\nFrom Split This Rock Co-Director Sarah Browning's testimony in support of arts funding at the DC City Council hearings April 27, 2010. Read her full testimony on the web site of DC Advocates for the Arts here.\nPoet Colorado T. Sky reads at the 2008 Split This Rock Poetry Festival's Friday Open Mic.\nMelisa \"Misha\" Cahnmann-Taylor is the co-author of Teachers Act Up! Creating Multicultural Learning Communities Through Theatre (2010, Teachers College Press) and Arts-Based Research in Education: Foundations for Practice (2008, Routledge) and has published poetry and reviews in many journals including APR, Alaska Quarterly, Georgia Review, Barrow Street and Quarterly West. She won the 2004, 2005, and 2008 Dorothy Sargent Prizes. Cahnmann-Taylor is Associate Professor of Language and Literacy Education at the University of Georgia.\nCahnmann-Taylor appeared on the panel Birth and the Politics of Motherhood in Poetry during Split This Rock Poetry Festival: Poems of Provocation & Witness 2010.\nreform of health a victory?\nAt our table at AWP, we asked visitors to write Haiku postcards to President Obama. It was (and still is) an opportunity to let him hear the voices of poets and writers and all those who have the ability to imagine a world that is better.\nDespite my best intentions, I was too exhausted at the end of each day to post during the event. Here are some of my favorite Haiku from AWP. I hope to post more soon. If you're interested in writing a Haiku postcard to the president, please email me.\nAfter many years of fund raising, the memorial is only $14 million away from its $120 million goal. Check out the site to see how you can help build the dream.\nFrom Up Jump the Boogie (Cypher Books, 2010). Used by permission.\nJohn Murillo is the current Jay C. and Ruth Halls Poetry Fellow at the Wisconsin Institute for Creative Writing. A graduate of New York University's MFA program in creative writing, he has also received fellowships from Cave Canem, the New York Times, and the Fine Arts Work Center in Provincetown, Massachusetts. He is a two-time Larry Neal Writers Award winner, a former instructor with DCWritersCorps, and the author of the poetry collection, Up Jump the Boogie.\nMurillo appeared on the panel Aqui Estamos: A Sampling of Poetry from the Inaugural Acentos Poetry Festival during Split This Rock Poetry Festival: Poems of Provocation & Witness 2010.\nabout how 'qualified' you are.\"\nDan Vera is the author of a book of poems, The Space Between Our Danger and Delight (Beothuk Books, 2008). Poems of his have appeared in Delaware Poetry Review, The Amistad, Konch, and the anthologies DC Poets Against the War and Shaping Sanctuary. He is co-founder and co-publisher of Vrzhu Press, managing editor of the gay culture journal White Crane, founder of the Brookland Area Writers & Artists (BAWA), and co-hosts the monthly BAWA reading series.\nVera appeared on the panel Gay and Lesbian Poetry in the 40th Year Since Stonewall: History, Craft, Equality during Split This Rock Poetry Festival: Poems of Provocation & Witness 2010.\nSplit This Rock will be hangin' at AWP, the annual mash-up of Associated Writers and Writing Programs, taking over the Colorado Convention Center in Denver this week.\nCome say Hi and write a haiku post card to President Obama at Split This Rock's table at the bookfair. We'll be at Table #I21. (That's the letter \"I,\" followed by the number 21.) You know the good man needs to hear fro...m you!\nThen check out the Split This Rock panel, \"Don't You Hear This Hammer Ring? Socially Engaged Poetry in the Age of Obama,\" Saturday, April 10, 9-10:15 am, Colorado Convention Center, Street Level, 103-105. Discussing the current state of poetry of provocation and witness will be 2008 featured poet Naomi Ayala, John Murillo, and Split This Rock Co-Directors Sarah Browning and Melissa Tuckey.\nSarah will also be on an encore of a panel that was part of Split This Rock 2010, \"The War Is Not Over: Writing About Iraq and the Case of the Mutanabbi Street Coalition,\" organized by Persis Karim. This panel takes place on Thursday, April 8, at the more civilized hour of 4:30-5:45 pm, Mineral Hall, Hyatt Regency Denver, 3rd Floor. Other panelists are Beau Beausoleil, 2010 featured poet Sinan Antoon, and Evelyn So.\nWe'll be blogging from the conference, including posting some of our favorite haikus written to President Obama each day, so keep checking back.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Top 7 Apps to Root Android Phone Without PC 2018: As we know that play store is full of apps to root android 6.0.1 with PC , and most of the websites are providing the guide on how to root Android 4.4.2 without computer .... Top 14 Best Root Apps 2018 For Rooted Android Phones Once the Android is rooted it becomes very useful, and it is able to do many tasks. We have written many articles related to the same topic so you can Root your phone for any cause.\nSo, are you ready to explore the list of best apps for rooted android smartphones? Here we go. Top 40 Best Root Apps in 2018 (NEW) I have included a bunch of useful apps to be used on rooted phones.... VRoot dedicatedly root and un-roots almost every Android device running on the 2.2 (Froyo) to newer operating systems, which makes it among the best rooting apps for Android. The VRoot application was recently rebranded and named iRoot.\nBest tutorial to Root and Install Custom ROM in Honor View 10 2018. Hack Android Games Without Rooting . Usually, most of the hacking techniques will seek for the rooting process to complete.\nIf so, you are at the right place because, in this post, we are going to introduce you with the top 10 Best Game Hacker Apps for Android, using which you can get rid of all the limitations and enjoy your favorite game without paying a single penny from your pocket.\nRooting Android comes with hundreds of benefits. However, one of the most popular benefits is being able to access hundreds of new apps. On a normal Android device, apps have to go through a rigorous certification process before they get listed on Google Play and other Android marketplaces.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Give your dog the ultimate reward with Stella & Chewy's Carnivore Crunch Grass-Fed Beef Recipe Freeze-Dried Dog Treats. Based on the philosophy that pets thrive on the same diet they would eat in the wild, these treats give your dog what he naturally craves. Made from sustainably-sourced ingredients and raw protein, this recipe starts with 98% meat, organs, and bone\u2014offering the benefits of raw food nutrition with the convenience of freeze-dried preparation. And to ensure the highest quality, each batch is made with care in the USA in Stella & Chewy's own kitchen.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Ultraportable pocket cameras have largely been usurped by smartphones, but there are a select number of scenarios where an actual camera shines.\nThis, at least, is Sony's perception of the camera market.\nThe Japanese camera maker has announced the availability of its Cyber-shot RX100 VI premium compact camera, which \"combines portability, flexibility, fast responsiveness and professional image quality, making it an ideal choice for capturing daily life, cityscapes, portraiture, sports, and wildlife,\" Sony notes.\nFrom the outside, it might seem pointless to continue making these cameras \u2014 Sony currently offers five snappers of its RX compact range in South Africa \u2014 but the specs argue otherwise.\nThe RX100 VI sports a 20.1MP Exmor RS sensor and BIONZ X image processor, mated to a Fast Hybrid AF system with 315 phase detection autofocus points.\n\"The RX100 VI is the first from the series to include Sony's advanced High-density Tracking AF technology, which concentrates AF points around a subject to improve tracking and focus accuracy,\" the company continues.\nSony claims that the camera can focus on its subject within 0.03 seconds.\nFor video enthusiasts, the RX100 VI also records in 4K, and supports slow motion video at 960fps. And continuous snapping options at 24fps with a buffer limit of 233 photographs is a welcome addition for lovers of moving objects.\nA flippy screen is also included, while the camera itself features WiFi, NFC and Bluetooth support.\nThe notable difference between the RX100 VI and the slightly cheaper and older V is the zoom lens.\nWhile the RX100 V's lens is limited to 70mm, the VI extends to 200mm. The 24-200mm Zeiss lens features a f\/2.8-f\/4.5 aperture too.\nSo, how much does the camera cost?\nIt's not cheap. Sony's debuting the Cyber-shot RX100 VI debuts at R23 999, and is around R4000 more expensive than the older model.\nNevertheless, judging by its size and spec sheet, it should be an interesting option for serious hobbyists or travel photographers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Description: Cool wallpaper of art, wallpaper of girl, bird is Wallpapers for pc desktop,laptop or gadget. Cool wallpaper of art, wallpaper of girl, bird is part of the Arts & Culture collection Hight Quality HD wallpapers. Cool wallpaper of art, wallpaper of girl, bird wallpapers was with Tags: 3000x2460 px, Art, Bird, feathers, Girl, sitting, swimsuit, vulture .Image size this wallpaper is 3000 x 2460.This image posted by admin On October 9, 2014.You can download wallpaper Cool wallpaper of art, wallpaper of girl, bird for free here.Finally dont forget to share your opinion using the comment form below.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Despite increased polarization among the American people and the US Congress during the Obama administration, the Legislatures in Midwestern swing states like Pennsylvania, Ohio, and Michigan featured relatively stable levels of polarization within the chambers of both the House and the Senate. These measures focus on the average polarization between legislatures, and offer perspectives on how the country has changed during the first 6 years of the Obama Administration.\nThis contrasted slightly with polarization within the Senate, with more changes among each of these chambers. Specifically, the Michigan Senate has become less polarized since 2008, despite consistent control by Michigan Republicans, even so, it remains most polarized among the three state's chambers.\nMoving forward, there is potential to observe some interesting trends in these selected Midwestern legislatures. Questions to be aware of include whether the Trumpian wing of the Republican party can take a foothold within state legislatures and potentially further contribute to polarization, whether there will be increased or decreased turnover in deal-making within the more moderate state legislators. There is also, of course, significant potential for strong democratic mid-terms to alter the completion of both chambers and polarization. Alongside an increasingly polarized voting population and national congress, it will be interesting what effect these factors have moving forward in Midwestern legislature. Additionally, these three states will hold all gubernatorial elections in 2018, and could potentially offer perspective on the direction of the country and the foot-hold of the Trump administration within Rust Belt states.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Astonishing Small Bedroom Closet Size \u00bb Bathroom Size Vs Bedroom Closet Size Standard Walk In Closet Size Pictures - The image above with the title Astonishing Small Bedroom Closet Size \u00bb Bathroom Size Vs Bedroom Closet Size Standard Walk In Closet Size Pictures, is part of Standard Walk In Closet Size picture gallery. Size for this image is 1006 \u00d7 871, a part of Closets category and tagged with published February 27th, 2018 02:49:38 AM by admhalter. Find or search for images related to \"Astonishing Small Bedroom Closet Size \u00bb Bathroom Size Vs Bedroom Closet Size Standard Walk In Closet Size Pictures\" in another posts. Back to: Standard Walk In Closet Size.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Low-key comfort, meet urban access. We knew you two would get along.\nMalaysian caf\u00e9 has flavorful bites onsite for when you need more than caffeine to get you going.\nLooking for extras like a pool & gym? Just ask at the desk about guest access to partner facilities.\n2 restaurants onsite. The Fond(ue) Cafe and the Mustard Seed Malaysian Cafe & Restaurant serves regional cuisine.\nComfortable albeit basic rooms, nice location very walkable to much of what Auckland CBD has to offer. Staff was very friendly. The fondue cafe in the lobby also had a friendly staff and nice coffee and basic breakfast options. Overall a good value for the laid back traveller.\nsimple and well run, only down side is there don't have free unlimited wifi, only 500mb a day.\nBeds are tiny on roller wheels. So you can easily fall off and the bed slides. Do not get two beds -get a room with one-should be better.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Constructors are also referred to as First Parties: Microsoft, Sony, Nintendo, Apple\u2026 They all have their word to say regarding games made available on their platforms. Ever heard about certification process? The constructors need to make sure the games meet their quality standards and that's the job of their testers.\nDon't expect to be giving any feedback on the gameplay though. You're more likely to work with a checklist of very specific points to verify.\nFew video game tester jobs are available at those companies compared to the rest of the industry. The competition is quite high.\nConstructors can also own game development studio and act as publishers. Insuch scenario the game testing positions are pretty much the same as those described in the two following sections, except a the prestige of working for a big name.\nBeing employed by a game development studio generally means that you get to work on the same location as the developers. You will get to interact directly with them which make the job easier and more enjoyable.\nAs mentioned before, the main role of a game tester is to find defects rather than giving feedback on gameplay but a game development studio is the type of company that is more likely to encourage such feedback.\nNot all studio developments employ game testers. In smaller companies the developers may test themselves. Others will rely on the QA team of the publisher. Overall this is not the easiest place for job hunting, but the probably the best place to work at as a game tester.\nExample of famous game developers: Bioware, Rockstar, Ubisoft.\nThose companies publish video games created by others. They're responsible for manufacturing and marketing the games.\nWhile being employed by a publisher you might work on games from different studios which is nice. However it can be harder to feel really involved in the projects as the only interaction you will have with the developers are the defects you submit and the comments they add to them. Being farther from the development front lines makes the job less stimulating.\nOn a positive note, publishers typically have important QA teams and it's way easier to find a job there than at a development studio.\nTest labs are companies specialized in game testing. They provide outsourced testing services to game developers or publishers. It means than working there you would be as far as possible from where the game actually gets created. Clearly not the most exciting environment. On the other hand you get to test more varied games because those labs tend to offer testing services on a lot of platforms for a lot of clients.\nThe job security tends to be lower than in other types of companies because the work load depends heavily on the contracts that the lab can get from developers\/publishers. Just like testers are at the end of the food chain within a company, test labs are at the end of the food chain in the industry. So when hard times come and it's time to cut costs, game developers and publishers will first look at those contracts they have with test labs even before laying off their own testers. And test labs usually employ their own testers on a temporal basis to make sure they can let them go when the cash is not getting in.\nLast bad point about test labs: career progression is even more limited than elsewhere. When it comes to hiring the test labs usually consider quantity over quality. They will hire the cheapest and less experienced workers which often translates into a work place where people are bored and feel like they could be replaced at anytime.\nOne good thing about test labs though: they're the easiest point of entry in the video games industry.\nAs you can see, the best positions are typically with the game development studios and the less interesting ones with the testing labs. The publishers are somewhere in between. But wait it's not that simple.\nFirst: Some companies fall in multiple categories. Look at Bethesda: they are actually two companies. The original one, Bethesda Softworks started in the 1980's and used to be a game developer as well as its own publisher. But in 2001 they founded Bethesda Game Studio to be in charge game development and let Bethesda Softworks focus on publishing. Bethesda Softworks is now even publishing games from other studios.\nSecond: Size matters. If you were to work in a team of two testers for a small independent game studio you would have a completely different experience then if you were working in a QA team of 40 for a big name studio. It's easier to feel involved and committed in a small company. And sometimes it's also easier to negotiate a salary because the employer cannot just say \"no you will be paid like the 39 other testers\".\nStill, some people will feel more comfortable working for a bigger company. It's really a matter of taste.\nTo sum up: Be aware that the size and type of company can greatly impact your experience as a game tester. Consider those factors when job hunting but never assume anything. The job interview is the time to ask your prospective employer details about the role and to get a better idea of how things really work in a company.\nThanks for the article. I have only a couple questions. I have a phone interview coming up for a game QA tester position, and if I pass that, then an on-site interview in the company. I know it sounds like I'm trying to cheat, but I'm really not \u2014 I am really starting to worry about this, losing sleep and gaining stress. I just want to hear your experiences\/opinions on what the phone interview will consist of, and what the onsite interview will test me on. From what I've gathered, phone interview is a means of gathering more information about the resume and the cover letter. Job history, hypothetical questions, questions regarding your personality. Whereas the on-site interview's about the technical knowledge that is required to be a tester.\n3- Relax \u2013 Remember that at a job interview, both parties are there to find out whether they could work together.\nIt's hard to give more specific advice.\nYou might get presented with a test about finding bugs.\nAt some point I would myself interview candidates and have them perform such test. I then reviewed not only which bugs they had found, but how clearly they reported them.\nIf you don't know how to report bugs, I suggest you research it a bit.\n3- Expected behavior: That's how it should behave normally. You don't always have this information, especially in video game testing. Either you have a document stating clearly how it should behave, or you don't and then you can include a description of ho you think it should behave.\nLike I said, a bug report should be 100% clear. As a recruiter, I was fine with candidates missing non obvious bugs as long as the one they had found were well reported.\nBy going through this interview, you will improve at interviewing regardless of the outcome.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Have you ever wondered how players in the Ontario Player Development League are scouted and identified?\nSince the league was implemented in 2014, the OPDL has considerably improved the scouting and tracking of players within the league.\nClick here to read about the methods being used to assist in assessment.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"My Lara Croft killed about 657 people in a span of 30 hours, which works out to a brisk 22 kills per hour. As far as murder mileage goes, it's \u2026 is \"impressive\" the right word? I don't know, but it's hard to reconcile with Rise of the Tomb Raider's suggestion that Lara is a human being herself, never mind someone who also exudes shades of regret and fragility. But this is, again, my Lara Croft: the one who has to exist simultaneously as the central figure in a story, and as the actuator for my part in a commercial video game.\nAssuming we disregard Lara's sheer quantity of killing as 'gameplay only', as something deleted from the canon as we construct it in our heads, I wonder why it's important at all. Having the story play out just so, without dissonance, suggests that movies and books are telling the stories the familiar way - the 'right' way. Is it so bad to simply say \u2026 so what if this doesn't make sense? Is it really a debilitating problem for a video game character to be archaeologist, human, and skilled man-smasher? The game part is messing up a whole bunch of things to begin with.\nSomething as unremarkable as the difficulty level you choose in Rise of the Tomb Raider can warp its story in an unforeseen way. On more challenging modes, in which enemies pursue Lara ruthlessly and murder her with ease, you're more likely to believe the conceit that she is killing to survive. When the game pushes back against failure especially hard, Lara seems less like a vicious predator and more like a vulnerable person again. If you mess up, you get plenty of game-over evidence for the case against being a nice, fair, morally conflicted person.\nFurthermore, it emphasizes a genuinely interesting - and playable - statement buried inside Rise of the Tomb Raider. Violence, for someone as scarred and driven as Lara Croft, is the easy option. There are plenty of situations in Rise of the Tomb Raider which can be defused with patient observation, painstaking sneaking and out-of-the-box thinking. Except, right there at the campfire, all the goons are gathered near a flammable cannister and oooh, you're so tempted to just fire off the arrow and disintegrate them.\nIn recreating Lara Croft as a believable virtual character, both in terms of extravagant visual fidelity and her guttural reactions to this sort of game's obligatory perils, Crystal Dynamics has shifted our perceptions ever so slightly. It's a new form of the uncanny valley: the more realistic Lara becomes, and the more rooted she is in a harsh environment, the more unbelievable her game-mandated acts become. She is inherently more believable and more human than any of her peers, but seems to fall short moments before her game's grim realities take a toll on her mindset. She just seems too violent, and as a person we expect her to stop and wring her hands before she wrings some more necks. The problem is her game doesn't serve this desire for a thoughtful pause at all - often because it is, well, a game.\nIt turns out one death is not equal to another. Editing has a huge say in whether someone's demise is quick, gory, unimportant, funny, disturbing or drawn out (and quartered, if we're talking Eli Roth). Meanwhile, over in video games, the player is the one in charge of the camera, usually as they swivel it 'round and 'round a realistic and fluid world. Whereas a film would cut away and soften the tone of Lara's brutal survival tactics, we see everything in games for the sake of continuity and simulation. We see what's under the table, what's at the top of the mountain, and how a man squirms as he's choked into oblivion by a bow string.\nThe weird part is that you might enjoy the violence not for the gore, but as a product of successfully plundering the game's possibilities. Your victory is unavoidably represented in the form of a violent animation, which is probably why it seems as if Lara Croft enjoys a good neck-stab, even when the fiction protests. We might think of the game's open scenarios and stealth-action blend as a 'playground', a word that could be furthest from Lara's mind. But her mind is not the only one there.\nSince you're the one giving impetus to many of Lara's actions, it might be more important for you to consider a protagonist's moral implications, rather than having them do it on screen. The full story is not just represented in the game, but in the space between it and the player. Shrugging it off is less satisfying, though, when you're legitmately yearning to see Lara's decisions debated in the game - not through cutscenes, but action. Why isn't there some other way to interact that acknowledges both Lara's internal state and the reason we are likely to recoil from her? Obviously, Rise of the Tomb Raider isn't really built as a game that would do proper justice to a serious topic like post-traumatic stress disorder or obsession, but even a minor, playable diversion would be preferable to plastering it over with some cursory dialogue, which is what happens.\nStill, our own interference in the process just makes a mess of things, no matter how you look at it. The outlines of what can be cut without harming the game or the story become harder and harder to make out. Perhaps it's a writing problem, a missed opportunity to make the protagonist a more interesting person. Maybe it's a game problem, in that the protagonist we need couldn't exist as person inside the constraints of thrilling spectacle (which also has to sell well).\nI'm certainly not going to suggest a solution to The Problem - but I'd like to think that the quest to find for one is the very thing that makes video games so great. It's a game problem, not just a Lara problem. The story is a bug and a feature.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The new family has been living in my house now for five months. I like them better than the old family. They don't snap at each other or argue about money like the old family did. They're nicer, happier, warmer people. There are three of them: John, Lisa, and Maggie. They have a dog, too\u2014a Border Collie named Tuxedo. I gather from their dinner conversations that John works at a bank. He is gone from six in the morning until six at night almost every day, and when he comes home, he always looks dead tired. He is never unpleasant,though. He always kisses Lisa and asks her how her day was. He picks up Maggie, twirls her around, and tickles her. He's a good dad.\nLisa stays home with Maggie. She spends most days on the couch in the den, drinking coffee out of a glass mug, watching television, needlepointing pictures of foxes and rabbits in Victorian clothing. She isn't a hovering mother, but she isn't inattentive either. A few times a day, she puts down her needlepoint and does a puzzle with Maggie. Or the two of them sit at the dining room table and draw with crayons.\nMaggie loves to draw, and she's quite good for a four year old. She doesn't just draw stick figures. Her crayon people have discernible features\u2014eyes and noses, mouths and teeth, fingers and toes. They have bodies with proportional legs and arms. Her favorite subjects to draw are her mom and dad. She always draws her mom in a pink dress and her dad in his work clothes\u2014a gray suit and tie. She draws me, too. She draws me in a white dress. With long brown hair and blue eyes. I guess it looks like me. I don't exactly remember what I look like to be honest. I haven't seen myself for a long time. I can't see myself in mirrors anymore. I wish I could.\nWhen John and Lisa ask Maggie who the lady in the white dress is, she says it's her friend Mary. My name is actually Marianne, but Mary is close enough. I've stopped trying to correct her. John and Lisa think I'm Maggie's imaginary friend. Lisa thinks Maggie invented me because she doesn't have any friends her own age yet, and she's lonely. She's been pushing John particularly hard the past few weeks to enroll Maggie in preschool. She thinks that's the solution to the problem. John isn't as concerned about Maggie being lonely. He says Maggie is just creative. He says she would probably invent imaginary friends even if she had a dozen real friends.\nNeither John nor Lisa has ever considered the possibility that I might not be imaginary. At least they've never admitted to considering it. I can't blame them. They can't see me or hear me or sense me in any way. Only Maggie can. Maggie and Tuxedo. When they first moved in, Tuxedo would bark his head off whenever I came around, but he's gotten used to me. He doesn't even flinch when he sees me now. He knows I'm not a threat. He knows I mean well.\nMaggie has fantastic taste in movies. She likes Meet Me in St. Louis and Easter Parade and The Wizard of Oz just like I did when I was a little girl. She knows all of the words to \"Over the Rainbow,\" and she can sing it from beginning to end without hitting one bum note. She has beautiful little voice\u2014clear and high and lilting, like a flute. If she doesn't become an artist when she grows up, I could see her becoming a singer. Maybe she'll be the next Judy Garland. I think she has the talent. I wanted to be Judy Garland when I was a little girl, but I could never carry a tune. You can't really teach yourself to be a good singer. You either are or you aren't.\nThe way Anne talks, you'd think she was going to die tomorrow. I think she's probably got at least ten more years in her. She's pretty spry for a seventy-year-old. She turned seventy just last week. John and Lisa threw a party for her at the house. About twenty people came over\u2014an assortment of aunts, uncles, and cousins. All of them sidled up to Anne at some point in the evening and quietly assured her that she didn't look anywhere near seventy, which is true. She doesn't. She still has thick dark hair and a nice slim figure, and her face isn't nearly as lined or eroded as the faces of some seventy-year-olds you see. I would be eighty this year. I shudder to think about what I would look like. My grandmother looked about a hundred when she was seventy. There are some benefits to dying young. I wish I could have lived a little longer, though. Nineteen is too young.\nSometimes I get mad when I think about all that I missed. I wish I could have gotten married. I wish I could have been a mother. I wish I could have spent more time with my parents. I miss my parents very much. They were both marvelous people. My father was a smiling, affable Irishman who made friends easily and told the most delightful stories about his childhood in Donegal. My mother was a quiet, dark-haired woman from a respectable middle-class Italian family. She loved to read poetry and play the piano. Many a Sunday afternoon I spent lounging on the living room sofa while she played Liszt and Debussy, Chopin and Mozart. She had perfect time and exquisite control over the instrument. She tried to teach me to play, but I didn't have the patience for it. Like my father, if I wasn't good at something right away, I gave up. My mother was all patience\u2014patience and kindness.\nI don't know why my parents aren't here now. I don't know why it's just me in this house. I don't have any unfinished business. I don't have any interest in haunting anybody or chasing anybody out. If I could get out of here, I would. But every time I move toward the door, it pulls away from me. I can never reach it. I've stopped trying. Maybe I'll be here forever. It's a sad thought. At least I have Maggie.\nI hope Maggie doesn't stop seeing me, but she probably will. It's just a matter of time. No teenagers or adults have ever been able to see me. The Thompson's little boy, Jeremy, stopped seeing me when he was about six. I didn't mind, though. I didn't much enjoy talking to him. He was a cruel child\u2014always throwing the cat down the stairs and breaking his mother's dishes just to rile her up. He's probably in prison now\u2014if he's even still alive. I will definitely mind when Maggie stops seeing me. She's such a sweet girl. She has such a good heart. I'd like to think that if I'd been lucky enough to have a girl of my own, she would have been like Maggie.\nNights are the hardest\u2014when it's quiet and dark and there's no conversation or activity to keep my mind off myself, off my situation. I stay in Maggie's room. She knows I'm there, and she always says, \"Goodnight, Mary,\" after Lisa turns out the light and leaves. I don't sleep. I don't think I can sleep. But I wouldn't even if I could. I like to keep watch over Maggie. I feel like I'm protecting her. I'm not, of course. If there were any threat\u2014an intruder, a fire\u2014there is nothing I could do to save her. Still, I feel like she's safer when I'm there, as if my good will, my affection, my love were a sort of shield. I wish I could always be with her, but I know that can't be. She'll grow up. She'll leave home. She'll meet a man and have kids. She'll do great things in the world. And that is as it should be. I hope she has a happy, productive life. And I hope she lives a long, long time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Now based on the third-generation TT, the RS Coupe wrangles much more zip from Audi's 2.5-litre inline five-cylinder engine \u2014 400 horsepower and 354 pound-feet of torque at 1,700 rpm. The turbo five is mated with a seven-speed dual-clutch transmission, which then sends all that power through Audi's Quattro all-wheel-drive system. Audi's RS sport suspension with adaptive dampers uses magnetic ride technology to keep the 2+2 coupe flat in the corners when it's let loose. Audi's drive select also allows the driver to tweak the car's characteristics for comfort, auto, dynamic and individual modes. Each alters the steering, shift pattern, throttle response and exhaust tone. Inside, the virtual cockpit and its 12.3-inch screen allows the driver to pick one of three views including RS; this gives the tachometer prominence while displaying information on tire pressure, torque output and the lateral g-forces.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Looking for an old country farm house with beautiful mountain views? Look no further because you just found it! Located just minutes from Pikeville and Dunlap TN. we have approx. 1764 sq ft house with 3 bed, 2 full bath, Livingroom, Greatroom, and some of the original hardwood floors! It has central H&A ''gas heat'' and a huge 2 car garage with workshop all on a nice level lot with gorgeous mountain views! Property backs up to a farm. Handicap accessible ramp. Nice patio to enjoy back yard and take in those beautiful mountain views. I just don't think this property will last long on the market. Buyer to verify all information & measurements in order to make an informed offer.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"OnBase is all about making our customers more efficient. Learn some of the new process automation capabilities within OnBase 13. Forms can have documents attached to them and allow free-form drawing, to capture more information when a form is submitted. Documents in an OnBase process can now be signed in the cloud with our integrations with Docusign\u00ae eSignature\u00ae and CIC: SignatureOne\u00ae and Ceremony Server\u00ae. OnBase capture continues to automate data capture by introducing formless intelligent indexing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Online motion science-primarily based yoga lessons. Stream bulldog's online yoga courses from any system\u2026Any Time. Home Physique is superb for inexperienced persons and features a quick begin coaching on the way to get started with a yoga follow right in your home. My Health Yoga offers Yoga Courses on the Gold Coast and in Brisbane and a singular number of workshops, retreats, therapeutic massage, natural therapies and specialty Healers Courses.\nBy means of Akhanda Yoga On-line you may apply the complete vary of his distinctive class types alongside the Ganges River in Rishikesh, India and in different inspiring settings. Begin close creating a slant of key phrases coordinated to your small business. Every month we add new SATTVA Yoga sequences taking the practitioner through a systematic follow that targets the complete bodily body, bringing energy, flexibility, balance and an total sense of properly-being to your life.\nMy Health Yoga has been coaching yoga teachers since 2003 making us one of many original teacher training schools. You just graduated YTT or graduated a while again but aren't positive what to do, how to arrange what you are promoting, get your foot within the door at your local studio or how you can stand out within the sea of teachers.\nI am talking, interpersonal abilities, instructing methodologies, and qualities that will assist set you apart from different yoga teachers and create overflowing classes with a large, loyal pupil following. An active vinyasa flow apply that is a good way to begin your day; awakening your body and allowing your power to stream through you in a harmonious means.\nAs a yoga teacher, you wish to make your classes an exquisite expertise for every student that walks by your door. You are additionally in a position to download movies and add them to your calendar to create an everyday class schedule. If he want to buy a website with the news WEB OPTIMIZATION is I had to conceive a domain-polinfor positioning -> + or firm popularity phrase.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Overlooking the China-Russia border, Holiday Inn Suifenhe offers 4-star spacious accommodation just 5 minutes' from Suifenhe City Centre. Aside from an indoor pool, it also has a tennis court, spa and free wired internet.\nFeaturing seaside views, this Vladivostok hotel offers room service, free Wi-Fi and a local beach just 5 minutes away by foot. Vladivostok city centre is a 15-minute drive away.\nLocated in a scenic park, this hotel is a 20-minute drive from central Vladivostok and 300 metres from Sanatornaya Beach. It offers an international restaurant and rooms with satellite TV.\nThis small hotel in Vladivostok city centre is a 15-minute walk from the Amur Bay Coast. It offers brightly decorated rooms with free internet access and flat-screen TVs.\nLocated in the suburbs of Vladivostok, the Equator Hotel is a 5 minute walk from the Amur Bay and offers a tour desk, beauty salon and a cafe on site.\nThe Azimut Amur Bay Hotel, which allows pets, enjoys a privileged location on the banks of the Amur Bay. All rooms have private balconies overlooking the sea. Car Parking Free.\nJust 10 minute walk from the train station, the Azimut Hotel offers modern accommodation in the center of Vladivostok and offers free wireless internet access in public areas, airport shuttle and 24 hour reception 24.\nMudanjiang Airport hotels: great rates, no booking fees. Best price guaranteed!\nMaplandia.com in partnership with Booking.com offers highly competitive rates for all types of hotels near Mudanjiang Airport, China, from affordable family hotels to the most luxurious ones. Booking.com, being established in 1996, is longtime Europe's leader in online hotel reservations.\nWe have put together also a carefully selected list of recommended hotels near Mudanjiang Airport, only hotels with the highest level of guest satisfaction are included. Many photos and unbiased hotel reviews written by real guests are provided to help you make your booking decision. Luxury hotels (including 5 star hotels and 4 star hotels) and cheap Mudanjiang Airport, China hotels (with best discount rates and up-to-date airport hotel deals) are both available in separate lists. Always bear in mind that with Maplandia.com and Booking.com the best price is guaranteed!\nAccommodation in popular destinations near Mudanjiang Airport.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hadith : Main Muhammad hu, Main Ahmad hu, main Mahi hu, Main Hashir hu, Main Aaqib hu.\nJubair b. Mut'im may ALLAH be pleased with him reported that Allah's Messenger (may peace be upon him) as saying: I am Muhammad and I am Ahmad, and I am al-Mahi (the obliterator) by whom unbelief would be obliterated, and I am Hashir (the gatherer) at whose feet mankind will be gathered, and I am 'Aqib (the last to come) after whom there will be no Prophet.\nPrevious Post Hadees Roman Urdu: Jobhi Kharch karo apni Beewee par Allah ki raza ke lie, uska Badla (Neki) hay.\nNext Post Hadith: Al-Jahannamiyin ki Shafaat (the Hell Fire people).","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Grief brings you to your knees; shakes you to your core and rattles your senses.\nIn a blink of an eye, or over time, the end of a life leaves painful scars on the living. What ifs, whys and shoulds dapple the streaming light of life. Denial, anger, bargaining, depression, and acceptance are grief's unique gifts to us. These five stages of grief, however, are not meant as checkboxes to get through.\nGrief is an individual process with no closure.\nWe are meant to hold grief, to sit with our love, loss and pain, to process our feelings in our own way and in our own time. Writing helps me process my emotions and transition through tough times, but it doesn't get me over my grief. Writing helps me work through the grief of losing my uncle Ray, my dog Beck, and friends Karen, Tom and \u2013 most recently \u2013 Erin.\nGetting through grief is individual, meandering and messy.\nErin was near and dear, plucked too soon, without warning. I met Erin through a mutual friend soon after she and her growing family moved into the neighborhood. Our first-born boys had January birthdays, nine days apart. Blake and James went to school, played and watched sports (mostly hockey) together. Erin and I attended Little League games, school events, classroom parties and field trips. We connected with our shared PR experiences, client drama but, most importantly, the joy of raising our sons.\nOur boys shared a passion for different sports; hockey and soccer. Despite their differences, Blake and James had a mutual respect for each other's choices. It's a respect these boys were raised with by mothers who celebrate the similarities, differences and significant time commitment of youth sports, travel teams and putting up with other cranky parents.\nAs work and travel hockey swallowed more of Erin's time, I saw her less. A random text, a street conversation, Facebook posts, an email, a drive-by wave kept us connected. Erin was always willing to lend advice or help make a connection with a new neighbor or a potential client. Whenever I reached out to her, whether it was for advice or to interview for my back to school story, she always made time for me, supported me wholeheartedly and made me feel special.\nErin was great at relating to people \u2013 a gift that her son Brady also has.\nShe asked for my dog \"whisperer\" and I immediately sent her the information for my dog trainer, Melissa Coccola from Positive K-9 who had helped us train all four of our boxers.\n\"Thank you!!! We are all a bit desperate to get something started,\" replied Erin.\nOn June 3, Rob and I stood next to Erin, Connor and Brady at the middle school, chatting while waiting for Blake and James to return from their 8th grade trip to Washington D.C. Charlie had attended obedience training, but was still digging holes in the yard. Erin had just returned from a business trip to Germany. I was 60 days into my new job.\nIt was the last conversation I had with Erin.\nFriends, family, neighbors, acquaintances \u2013 have all been brought together by a profound and public loss. As we cope with our own messy lives, we give our best to love and support Erin's family by bringing a meal, stopping by to visit, offering a ride, sending a heart-felt note, or just giving a hug.\nI feel Erin in the things she left behind: her family, the flowers blooming on her front porch, a watering can, a beautiful heart-shaped stone tribute with Erin's initials \"E. H.\" made by Brady and her best friend Wendy.\nGrief floats freely like loose petals blowing in the wind.\nA memory, an image, or a single word can trigger my tears at the most unexpected of moments. My heart aches with the weight of my losses. I pause to breathe, close my eyes and soften into the pain so that I can live with my grief.\nThis is my best work today. Thanks for reading. If you like this post, please feel free to share it with your friends or send me a comment. You can also post a comment on my blog or Facebook, or tweet me @kristinebruneau.\nI too was so shocked and saddened by Erin's loss, especially at such a young age. Please know that you are not alone in your devastating thoughts and feelings as it relates to grief. When my dad died, the book \"Don't Take My Grief Away\" was so helpful to my mom and sisters and I. Thank you for this essay.\nThanks, Angie. I will look for the book.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The aloha inspired Kailani Leggings feature a tonal, flat lock overlap stitching, low to mid-rise elastic waistband, and tapered legs featuring beautiful Mesh accents on the back of the lower legs and front thighs, combining cutting edge functionality with modern aesthetics.\nThe Blue and White marble design is inspired by the majestic seas that surround the Hawaiian islands. \"Kai\" is the Hawaiian word for ocean. The back waistband features a reflective, Gray circular Koalani logo.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Brenda Mills, Family Service & Guidance Center CEO, said that any number of staff members could be tapped for the honor because of their professionalism and dedication to children and teens.\nAwards are presented in six staff categories each year. Honorees were nominated by their co-workers and selected by a committee of FSGC staff members representing all areas of the agency. Recipients of the award are selected for effectiveness in their work with children, families and co-workers; zeal in promoting the advancement of mental health; and professional interest and involvement.\nEach year, one in five children experiences a mental health issue that disrupts their life. Of those children, only one in three gets treatment.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Shwe Chan Thar Engineering & Construction CO.,Ltd (SCT) is one of the leading engineering services providers in Myanmar especially for Education industry.\nHaving the strengths of promising engineering services and training skills. SCT is standing as one of the leading companies in the educational and teaching aids supplies in Myanmar.\nWe concern for every engineering field especially in Aeronautical Engineering. Electrical Engineering, Electronic Engineering, Mechanical Engineering and Civil Engineering.\nWe can give the satisfactory engineering services in installation, overhauling, maintaining, replacing, manufacturing, instrumentations, repairing and trading of the engineering equipments and machines.\nSCT assigns professionally trained engineers and staffs either by recruiting internally or externally adaptively upon the requirements of the projects to have the quality of services for our customers.\nWe arrange and organize the engineering team for each and every specific project.\nWe can be able to fabricate some import substitute quality machinery parts locally. The customer can accept high quality, reliable and flexible advantages at reasonable cost with our services for long term.\nOur customers include Government Sectors, Government agencies, Industrial, Commercial entities, Private organization, Universities and Private Education Schools.\nIf the customer may face with difficulty in using their existing system, we can advice to upgrade with new technological services and devices.\nSCT can help to reduce the customers' risks by our well trained expertise and technical know-how.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Brightline\" redirects here. For other uses, see Bright line (disambiguation).\nVirgin Trains USA, formerly Brightline, is an express inter-city rail system in Florida, United States. The company is a subsidiary of Florida East Coast Industries (FECI), where it initially operated under the name, \"All Aboard Florida.\" It is currently owned and operated by Fortress Investment Group, with Virgin Group providing branding rights and acting as a minority investor. Virgin Trains USA is currently the United States' only privately owned and operated intercity passenger railroad.\nVirgin Trains USA's service runs between Miami and West Palm Beach with a single intermediate stop at Fort Lauderdale. The Fort Lauderdale\u2013West Palm Beach segment opened on January 13, 2018, followed by Fort Lauderdale\u2013Miami on May 19. An extension from West Palm Beach to Orlando via Cocoa is planned to open in late 2021 or early 2022, and more extensions are planned.\nVirgin Trains USA's diesel\u2013electric locomotive-hauled trains run alongside freight trains in a shared-use corridor that was upgraded from a pre-existing Florida East Coast Railway (FEC) freight train corridor. The future West Palm Beach\u2013Cocoa segment will be constructed in a similar fashion, while 40 mi (64 km) of new track will be constructed in the State Road 528 corridor for the remainder of the extension, between Cocoa and Orlando International Airport.\nOn September 18, 2018, Brightline announced the acquisition of XpressWest, a private company with the goal of connecting Las Vegas with Southern California via Victorville, California. Brightline announced the intent of purchasing 38 acres of land near the Las Vegas Strip for a station and following the Interstate 15 corridor from Las Vegas to Southern California with the initial journey projected for 2022.\nIn November 2018, it was announced that Brightline would have its name changed to Virgin Trains USA, coinciding with the Virgin Rail Group taking a minority shareholding in the concession, with Fortress continuing to hold controlling interest. It was formally rebranded on April 4, 2019.\nIn March 2012, Florida East Coast Industries (FECI) announced plans to operate a rail service between Miami and Orlando. The cost of all construction is projected at $1.5 billion. In March 2013, All Aboard Florida applied for a $1.6 billion Railroad Rehabilitation & Improvement Financing (RRIF) loan, which is administered by the Federal Railroad Administration. In late 2014, the company announced it had applied for a $1.75 billion private activity bond allocation, with proceeds from the bond sale substantially reducing or replacing entirely the amount of the RRIF loan request.\nThe company received a Finding of No Significant Impact from the Federal Railroad Administration in January 2013, effectively clearing way for work to begin between Miami and West Palm Beach. The Draft Environmental Impact Statement for the full build-out service was released in September 2014, and a series of public comment meetings followed. The Final Environmental Impact Statement was released on August 4, 2015. As of the start of 2015, the company has started site work at the Miami, Fort Lauderdale and West Palm Beach stations, plus right of way improvements along stretches of the corridor.\nService between Miami and West Palm Beach began on May 19, 2018, with service to Orlando projected to start in 2021. On November 10, 2015, All Aboard Florida announced that the service would operate under the name Brightline.\nSuccess in densely-populated southeast Florida does not mean that the planned service extension further to the north has met with universal approval. Two key Counties on the coastal route north of the West Palm Beach station have, for various reasons, been fighting the extension of the rail line through Martin and Indian River Counties in court. One of their objections is that Brightline is a private corporation, so they should not be allowed to issue tax-exempt bonds as if they were a municipality. As of Fall 2018 this legal fight has been going on for four years, during which time no progress on extending the line north of Palm Beach has been made. However in December, Federal District Judge Christopher Cooper on Christmas Eve threw out a suit by Indian River County that claimed the U.S. Department of Transportation improperly approved the bond allocation for Brightline, formerly All Aboard Florida. Dismissal of the lawsuit apparently clears the way for construction of the new rail corridor through the Treasure Coast and Space Coast. In April 2019 the company secured $1.75 billion in funding for the Orlando extension.\nConstruction began on the Miami to West Palm Beach section with the laying of new tracks and closure of the temporary surface lots in Government Center, Downtown Miami, in mid 2014. Preliminary work on the Miami station, such as site preparation and demolition, began later in the year. Suffolk Construction is the general contractor for the Miami station. Piles were being set on the four lots of MiamiCentral in early 2015.\nOn October 29, 2014, work on the Fort Lauderdale station began with the demolition of existing buildings on the site. A groundbreaking ceremony for the West Palm Beach station was held in November 2014. Moss & Associates, of Fort Lauderdale, has been named general contractor for the West Palm Beach and Fort Lauderdale stations.\nIn January 2015, crews started replacing track throughout the corridor. The system between Miami and West Palm Beach was projected to be operational by the first quarter of 2018, and to Orlando by late 2020. All Aboard Florida secured leasing of easement rights alongside the Beachline from the Central Florida Expressway Authority for $1.4 million in December 2015. As part of the contract, \"it is understood and agreed\" All Aboard Florida will be considering an extension to Jacksonville, Florida.\nPhase 2 of the construction between West Palm Beach and Orlando is behind the original schedule. Company officials now say construction may start in early 2019, with service to Orlando commencing in late 2021 or early 2022. The deadline for the associated bond sale has been extended to the end of 2018. The company is also seeking a federal loan to help finance phase 2 construction. On August 29, 2018, Brightline got the approval for $1.15 billion in tax-free bonds for expansion to Orlando. On December 13, 2018, Company President Patrick Goddard said during an update with the Central Florida Expressway Authority governing board that Brightline will give its contractor a 30-day notice to proceed on the $2.1 billion route on February 1. The company will then proceed with construction after those 30 days, with a goal of completing the route in 30 to 36 months. Completion of the project would be set for either fourth-quarter 2020 or first-quarter 2021. When asked about who the contractor would be, Brightline spokeswoman Ali Soule told Orlando Business Journal that more details would be revealed about the project closer to construction. Construction was scheduled to start in March 2019. and be finished by late 2021 or early 2022. Completion is now set for 2022.\nBrightline launched introductory services prior to January 7, 2018, though these runs were not open to the public. Public operations between West Palm Beach and Fort Lauderdale began Saturday, January 13.\nService between West Palm Beach and Miami began on May 19, 2018.\nDuring the first six days of operation, two people were killed in separate incidents at railroad crossings along the line, after they attempted to cross the tracks while the gates were down.\nDuring the first three months of operation, four people were killed by a Brightline train when they crossed the tracks illegally, with an additional three persons injured.\nThe 11th fatality associated with the Brightline train occurred on January 1, 2019, after a pedestrian was struck and thrown into a canal.\nUpon full buildout of the Miami\u2013Orlando route, trains will operate at up to 79 mph (127 km\/h) between Miami and West Palm Beach, up to 110 mph (177 km\/h) between West Palm Beach and Cocoa, and up to 125 mph (201 km\/h) between Cocoa and the Orlando International Airport. A future extension to Tampa from Orlando would, if constructed, also allow trains to operate at up to 125 mph (201 km\/h).\nTo cover the distance between Orlando and Miami in the desired time of about three hours, Brightline trains will have to operate with an overall average speed of 80 miles per hour (129 km\/h), which is similar to the overall average speed of the Acela Express operating on the Northeast Corridor between New York City and Washington, D.C. By comparison, the approximate driving time for this distance is about four hours, with an average speed of 60 mph (97 km\/h).\nThe project calls for more than $1.5 billion in upgrades to the rail corridor between Miami and Cocoa. The company is double tracking the corridor, improving signaling systems, and upgrading every grade crossing to meet the highest applicable safety standards set by the Florida Department of Transportation and Federal Railroad Administration. In January 2013, the Federal Railroad Administration issued a Finding of No Significant Impact (FONSI) for the Miami\u2013Cocoa phase of the project, effectively clearing the way for work to begin. Part of the corridor safety upgrades includes installing positive train control (PTC), which will enhance Brightline's ability to monitor and control train movements safely.\nResponding to citizen concerns about increased noise from additional horns, All Aboard Florida has stated that it will work with local communities to implement quiet zones where possible. Federal law requires quiet zone requests to originate from the local authority that has jurisdiction over the roadway, not the railroad company.\nIn August 2014, the company announced a partnership with the Broward and Palm Beach Metropolitan Planning Organizations to implement quiet zones between the city of Hallandale Beach and 15th Street in West Palm Beach. In December 2014, the Miami-Dade Metropolitan Planning Organization approved funding to construct quiet zones between PortMiami and the northern Miami-Dade County line. The quiet zones were originally planned to be in place when Brightline becomes operational between Miami and West Palm Beach by the end of 2017. Brightline started service on January 11, 2018, but various delays in constructing the quiet zones have stretched their in-service date to sometime in March.\nOn May 14, 2018, quiet zones went into effect in West Palm Beach, in Lake Worth on May 21, and in Boca Raton on May 30. The \"no train horn\" areas apply to freight and Brightline trains, though horns may still be used in emergency situations.\nThe FEC rail corridor includes a number of fixed-span bridges that will be replaced as part of the project. Most do not require United States Coast Guard (USCG) permitting as they do not span significant navigable waterways and clearances will not change. Twelve other bridges\u2014St. Johns River, Eau Gallie River, St. Sebastian River, Crane Creek, Turkey Creek, West Palm Beach Canal, Boynton Canal, Middle River (both the North and South Fork), Oleta River, Arch Creek and Hillsboro Canal\u2014will require permitting by the USCG. In addition, the project calls for significant investment and upgrades to three moveable bridges: St. Lucie, Loxahatchee, and New River. These improvements will ensure that bridge mechanical systems for raising and lowering the bridge spans are either fully upgraded or replaced. All Aboard Florida has stated that, prior to it becoming operational, it will start to regularly notify mariners of scheduled bridge closings via the internet, smart phone application and countdown signage on the bridges to enable mariners to have real-time information to decrease wait times at each bridge. Also, the company will station a bridge tender at the New River bridge.\nThe proposed line between Cocoa and Orlando is the only segment that does not have existing track or right-of-way owned by FEC. Originally, the Central Florida Expressway Authority (CFX) believed it could accommodate building new tracks for the project within the BeachLine Expressway's 300-foot (91 m) wide right-of-way. However, it determined that the right of way was too narrow to accommodate the tracks, additional utility lines, and any future roadway expansions. CFX began negotiations with Deseret Ranch, which owns the land just south of the BeachLine, to purchase additional land in order to widen the right-of-way. According to a pact made on July 16, 2013, CFX tentatively agreed to pay $12 million for an extra 200 ft (61 m) along the 22-mile (35 km) BeachLine corridor between Cocoa and Orlando International Airport. In early October 2013, CFX and All Aboard Florida reached a formal purchase agreement for the land required for the right-of-way. Although construction was slated to originally begin in early 2015, construction of the segment is now proposed to start in 2019.\nAlso in October 2013, the Greater Orlando Aviation Authority (GOAA) board approved development of a station and maintenance facility on Orlando International Airport property, as well as an easement to build track between the station and the mainline to the coast.\nThis segment of the proposed line will operate at speeds of up to 125 mph (201 km\/h) and will meet the United States Code's definition of High-speed rail, which includes rail services that are \"reasonably expected to reach sustained speeds of more than 125 miles per hour\". The Congressional Research Service uses the term \"higher\" speed rail for top speeds up to 150 mph (241 km\/h).\nSchematic of rapid transit and passenger rail service in the Miami metropolitan area in 2017. Tri-Rail Downtown Miami Link is scheduled to be operational in 2019.\nBetween Miami and West Palm Beach, there are 17 round trips Monday through Friday, and ten and nine on Saturdays and Sundays, respectively.\nDuring the first two and a half months of introductory service between West Palm Beach and Fort Lauderdale, ridership totaled 74,780, increasing from 17,800 in January to 32,900 in March 2018. The company itself announced that the ridership has been triple to what had been expected. The forecast provided to bond investors calls for 240,000 passengers per month by 2020 which includes service to Miami. Analyst Fitch Ratings has said that the company can break even at 56% of their ridership forecast.\nThe three South Florida stations were designed by Skidmore, Owings & Merrill in association with Zyscovich Architects. Rockwell Group designed the interiors. All three have adjacent parking garages which offer parking at the rate of $6 per day, eff. July 1, 2018.\nThe downtown Miami station, known as Miami (not to be confused with Miami Central Station, now known as Miami Intermodal Center, near Miami International Airport), spans nine acres located just east of Miami-Dade County Hall and includes 3 million square feet of mixed-use development with residential, office and commercial, and a retail concourse. The station connects Brightline with the Metrorail, Metromover, County bus and City of Miami trolley systems. The new Brightline station officially connects the following transit modes: 2 Metrorail stations, 2 Metromover stations, Metrobus, and the future Tri-Rail station. This increases connections to activities and tourist destinations, including the Performing Arts Center, Bayside Market and Bayfront Park. Service to Miami began on May 19, 2018.\nThe Fort Lauderdale station is located at NW 2nd Avenue between Broward Boulevard and NW 4th Street. The four-acre station site has a 60,000 sq ft (5,574 m2) station and platform. The Brightline train service in Ft. Lauderdale connects to the Sun Trolley and Broward County Transit system. Brightline also owns about three acres of land to the east of the Florida East Coast Railway corridor, where there are plans to build a transit-oriented development.\nThe West Palm Beach station is located between Datura and Evernia Streets and to the west of Quadrille. The two-acre station site has a 60,000 sq ft (5,574 m2) station and platform that connect with the neighborhood's existing vehicular, trolley and pedestrian networks and establish links to the Tri-Rail, Palm Tran Downtown Trolley and Amtrak West Palm Beach station.\nThe Orlando station will be included as part of the new Orlando International Airport Intermodal Terminal at Orlando International Airport. A proposed future extension of the Orlando area's SunRail commuter rail to the airport would connect at the terminal. Before the trains enter the Orlando Airport, Virgin Trains has also established a station at Walt Disney World where construction will take 30-36 months to build.\nIn September 2014, All Aboard Florida announced an order of five Siemens trainsets. Each Brightline train set initially consists of four passenger coaches, with a SCB-40 diesel-electric locomotive on each end. The coaches, with interiors designed by the LAB at Rockwell Group, feature ergonomic seating, Wi-Fi, level boarding, and meet ADA compliance standards. Each trainset holds 240 passengers. Working with All Aboard Florida, the LAB also conceived the Brightline name, brand platform, and visual identity. The entire trainset, including passenger cars, were manufactured by Siemens in its solar-powered plant in Florin, California. Once the route to Orlando is in operation, the trainsets will be expanded to seven coaches, and five more complete trainsets will be purchased. The first of five trainsets departed the Siemens factory on December 8, 2016, and arrived in West Palm Beach on December 14. The last trainset arrived in South Florida in October 2017.\nBrightline offers two classes of service: \"Select\" and \"Smart,\" with one and three coaches available on each trainset, respectively. \"Select\" offers 2x1 and four-to-a-table seating with 50 21-inch (530 mm)-wide seats per car and complimentary snacks and beverages, while the slightly less expensive \"Smart\" fare coaches seat 66 with narrower 19-inch (480 mm)-wide seats, with snacks and beverages available for purchase. Each trainset is able to hold 240 passengers.\nBrightline has expressed interest in adding a station on Florida's Treasure Coast between West Palm Beach and Orlando. In August 2018, the company asked cities in the area to submit proposals for station locations. Fort Pierce, which last had passenger train service in the 1960s, has expressed interest. The city of Stuart has also indicated that it will be negotiating for a potential station.\nBrevard County (comprising the Palm Bay\u2013Melbourne\u2013Titusville, FL Metropolitan Statistical Area) has been lobbying for a Brightline stop as part of the expansion to Orlando, as the line will pass through the county. In 2016, the Space Coast Transportation Planning Organization selected a site on Clearlake Road in Cocoa as its proposal to Brightline. As of February 2018, Brightline had not communicated a decision on the proposal.\nBrightline has announced intentions to extend the line to other areas when the initial phase is complete. Jacksonville is a likely expansion destination, as the Florida East Coast Railway already owns tracks running there.\nIn June 2018, Brightline began participation in a Request for Proposal process with the Florida Department of Transportation for the leasing of right-of-way along I-4 between Orlando and the Tampa Bay Area. Documents released preliminary to an IPO contained further details about the proposed line, including a possible stop at Walt Disney World, a fare of $35, and travel time of one hour. In November 2018, Florida DOT granted Brightline permission to negotiate with the Central Florida Expressway Authority for the leasing of the right-of-way. Brightline has also proposed connecting with the Sunrail Meadow Woods station as part of this line.\nOn September 18, 2018, Brightline announced the acquisition of XpressWest, a private company developing a high-speed rail project connecting Las Vegas with Southern California via Victorville, California. Brightline announced its intent of purchasing 38 acres of land near the Las Vegas Strip for a station and building the line following the Interstate 15 corridor from Las Vegas to Southern California, with the entry into service projected for 2022.\n^ \"Cummins QSK95 achieves Tier 4 certification\". Railway Gazette. April 6, 2016.\n^ a b c \"Brightline unveils Siemens locomotives and coaches\". June 8, 2016.\n^ a b \"All Aboard Florida \u2013 Miami to Orlando Passenger Rail Service\". Federal Railroad Authority. Retrieved February 17, 2015.\n^ Spear, Kevin. \"Brightline rebranded as Virgin Trains USA in new partnership\". OrlandoSentinel.com. Retrieved 5 April 2019.\n^ \"Palm Beach Post\". epaper.palmbeachpost.com.\n^ a b c Powers, Scott (December 13, 2018). \"Brightline expects trains to roll into Orlando as soon as 2021\". Florida Politics. Retrieved December 18, 2018.\n^ Branson, Richard (16 November 2018). \"Introducing Virgin Trains USA\". Virgin. Retrieved 16 November 2018.\n^ Lynch, Ryan (April 4, 2019). \"Sir Richard Branson-backed Virgin Trains USA Locks Up Funding for Orlando Leg\". Orlando Business Journal. Retrieved April 4, 2019.\n^ \"Quiet zones plans for FEC tracks get boost, as All Aboard Florida picks up costs for some safety upgrades\". Palm Beach Post. September 19, 2013.\n^ \"All Aboard Florida seeks federal loan\". Orlando Sentinel. March 18, 2013. Retrieved March 23, 2013.\n^ \"All Aboard Florida shocker: Rail seeks private financing\". Palm Beach Post. October 8, 2014.\n^ \"High-speed rail service heads south\". Florida Today. Associated Press. July 9, 2014.\n^ a b \"All Aboard Florida \u2013 Miami to Orlando Passenger Rail Service Draft Environmental Impact Statement\". Federal Railroad Administration. September 19, 2014. Retrieved February 11, 2015.\n^ a b c d e f Roustan, Wayne K. (May 19, 2018). \"All aboard as Brightline launches fast train service to Miami\". Sun-Sentinel. Retrieved May 20, 2018.\n^ a b c d Jacoby, Kenny (May 19, 2018). \"Brightline's inaugural West Palm-to-Miami trip draws crowd, celebration\". Retrieved May 20, 2018.\n^ a b \"F A Q\". gobrightline.com. Retrieved June 10, 2018.\n^ Hudak, Stephen (June 19, 2018). \"Brightline high-speed passenger train gets Orange County OK to lay tracks in wetlands\". OrlandoSentinel.com. Retrieved June 26, 2018.\n^ \"All Aboard Florida unveils branding of its rail service\". The Real Deal Miami. November 9, 2015. Retrieved November 10, 2015.\n^ \"Brightline spending reaches $7.2 million in Martin, Indian River counties, but for what?\". TCPalm (Treasure Coast Newspapers). Retrieved November 12, 2018.\n^ \"Judge tosses out Indian River County's lawsuit against All Aboard Florida\". TCPalm.\n^ \"Florida's Brightline clears legal hurdle, gets financing extension\". Bond Buyer.\n^ Chardy, Alfonso (August 25, 2014). \"Work begins \u2014 finally \u2014 on Miami-to-Orlando fast train\". Miami Herald. Retrieved August 26, 2014.\n^ Robbins, John Charles (September 27, 2016). \"Brightline passenger rail service 65% built\". Miami Today. Retrieved March 9, 2017.\n^ Judy, Scott (July 30, 2014). \"All Aboard Florida Taps Suffolk to Build Miami Station\". ENR Southeast. Retrieved February 11, 2015.\n^ Chardy, Alfonso (March 9, 2015). \"On track: Details for downtown Miami railway hub are revealed\". The Miami Herald. Retrieved March 9, 2015.\n^ Turnbell, Michael (October 29, 2014). \"Demolition begins for All Aboard Florida's new station\". SunSentinel. Retrieved April 8, 2018.\n^ Sorentrue, Jennifer (November 12, 2014). \"All Aboard Florida to begin construction today on West Palm Beach station\". Palm Beach Post. Retrieved February 11, 2015.\n^ Turnbell, Michael (January 20, 2015). \"Rail picks contractor for Fort Lauderdale, WPB stations\". Sun Sentinel. Retrieved February 11, 2015.\n^ Sorentrue, Jennifer (November 28, 2017). \"Brightline to start limited service in December\". Palm Beach Post. Retrieved December 28, 2017.\n^ Sorentrue, Jennifer (July 24, 2017). \"Brightline's Orlando leg delayed to 2020\". myPalmBeachPost. Cox Media Group. Retrieved July 25, 2017.\n^ Talley, Karen. \"All Aboard Florida gains easement, hints at Jacksonville in contract\". Orlando Sentinel. Retrieved April 8, 2018.\n^ \"Brightline wins extension to issue bonds for Orlando leg\". mypalmbeachpost. May 31, 2018. Retrieved June 26, 2018.\n^ Broadt, Lisa (August 29, 2018). \"Brightline gets approval for $1.15 billion in tax-free bonds for expansion to Orlando\". TC Palm. Retrieved September 28, 2018.\n^ Spear, Kevin. \"Brightline to start building rail from South Florida to Orlando in March while it rebrands as Virgin Trains\". OrlandoSentinel.com.\n^ \"Brightline seeks additional $950 million in funding for Orlando\". theledger.com.\n^ \"Brightline seeks additional $950 million in funding\". gainsvillesun.com.\n^ Sorentrue, Jennifer (January 7, 2018). \"Brightline allows passengers onboard for staff training exercises\". Palm Beach Post. Retrieved January 9, 2018.\n^ \"Happy rails to you: Brightline high-speed trains roll into action\". my Palm Beach Post. January 13, 2018. Retrieved May 20, 2018.\n^ \"Brightline slates fares and service date\". Railway Age. January 11, 2018. Retrieved May 20, 2018.\n^ Broadt, Lisa (January 17, 2018). \"Second person in less than a week killed by high-speed Brightline train in South Florida\". USA Today. Retrieved January 18, 2018.\n^ Darrah, Nicole (April 8, 2018). \"New high-speed Florida train kills 4th person since launching service\". Fox News. Retrieved April 8, 2018.\n^ \"Man's body recovered from canal after collision with Brightline train\", South Florida Sun-Sentinel, January 1, 2019.\n^ \"South Florida gets first peek at Brightline express train\". South Florida Sun-Sentinel. January 11, 2017. Retrieved February 16, 2019.\n^ \"Brightline-Virgin proposal for Tampa-to-Orlando rail service clears key state hurdle\". Tampa Bay Times. November 28, 2018. Retrieved February 15, 2019.\n^ \"FECI Bringing Private Passenger Rail To Florida By 2014\". Metro Jacksonville. Retrieved July 16, 2012. FECI envisions a three-hour trip between the cities at an average speed similar to the Acela between New York and Washington, DC... The initial 240-mile project between Orlando and downtown Miami is expected to cost $1 billion.\n^ \"ACELA High-Speed Rail Network System\". Railway Technology. Retrieved May 19, 2012. The 225 mile (362km) New York Penn Station to Washington, DC takes 2 hours 48 minutes, an average of 80mph (129km\/h).\n^ Gale, Kevin (March 22, 2012). \"Florida East Coast Industries plans Miami-to-Orlando passenger service\". South Florida Business Journal. Retrieved May 18, 2012.\n^ Parker, Terri (July 23, 2014). \"State requires All Aboard Florida to install millions in safety upgrades\". WPBF. Retrieved February 10, 2015.\n^ \"Private passenger train vision calls for double-tracking Florida East Coast\". Trains Magazine. March 23, 2012. Retrieved March 29, 2012. (Subscription required (help)).\n^ \"All Aboard Florida Passenger Rail Project FONSI\". Federal Railroad Administration. Retrieved February 11, 2015.\n^ \"Southeast Florida Passenger Rail Update: Sept. 20, 2013 Council Meeting\" (PDF). Trasurer Coast Regional Planning Council. Retrieved February 11, 2015.\n^ Stone, Rick (August 14, 2014). \"\"Quiet Zones\" Along The FEC Tracks May Silence Critics As Well As Train Whistles\". WLRN. Retrieved February 11, 2015.\n^ \"Federal Railroad Administration's Train Horn & Quiet Zone Rule\". Union Pacific. Retrieved February 11, 2015.\n^ Reiser, Emon (August 13, 2014). \"All Aboard Florida secures funds for Broward-Palm Beach quiet zone\". South Florida Business Journal. Retrieved February 11, 2015.\n^ Chardy, Alfonso (January 16, 2015). \"With \"quiet zones' funded for Miami-Dade, All Aboard Florida moves ahead\". Miami Herald. Retrieved February 11, 2015.\n^ Sorentrue, Jennifer (January 11, 2018). \"UPDATE: Quiet zone work to be completed in March, Brightline says\". Palm Beach Post. Retrieved February 9, 2018.\n^ Shatzman, Marci (May 30, 2018). \"Trains to silence horns at 10 Boca Raton crossings\". Sun Sentinel. Retrieved June 1, 2018.\n^ Tracy, Dan (July 16, 2013). \"Deal could lead to Miami-Orlando train, extended Osceola toll road\". Orlando Sentinel. Retrieved July 30, 2013.\n^ \"All Aboard Florida secures right-of-way, station agreements\". Railway Track and Structures. October 4, 2013. Retrieved October 9, 2013.\n^ \"49 U.S.C. 26105 \u2013 Definitions\". U.S. Government Publishing Office. Retrieved April 8, 2018.\n^ \"The Development of High Speed Rail in the United States: Issues and Recent Events\" (PDF). Congressional Research Service. Retrieved October 10, 2012.\n^ Seemuth, Mike (January 18, 2016). \"Concrete pours for Brightline facility in West Palm Beach\". The Real Deal Miami. Retrieved May 10, 2016.\n^ \"Schedule\". Brightline. 2018. Retrieved January 1, 2019.\n^ Broadt, Lisa (2018-06-29). \"Brightline financial documents reveal first quarter ridership, revenue\". TCPalm. Retrieved 2018-07-06.\n^ a b \"Brightline Ridership Triples Expectations; Miami To Start By End Of April\". The Next Miami. April 13, 2018.\n^ \"Here's How Many Passengers Have Been Riding Brightline Trains\". The Next Miami. April 30, 2018.\n^ \"SOM completes three new Brightline Florida rail stations\". Archinect. Retrieved June 10, 2018.\n^ \"Brightline \u2013 Rockwell Group\". www.rockwellgroup.com. Retrieved June 10, 2018.\n^ Kornfield, Meryl (2018-07-02). \"Brightline's garages start charging customers to park\". Sun-Sentinel.com. Retrieved 2018-07-06.\n^ \"Here Are Leasing Plans And New Renderings For All Aboard Florida's MiamiCentral Station Project\". The Next Miami. September 22, 2014. Retrieved February 11, 2015.\n^ \"Fort Lauderdale\". gobrightline.com. Retrieved June 10, 2018.\n^ Wallman, Brittany (May 4, 2014). \"Rail firm appears on track to transform Fort Lauderdale\". Sun Sentinel. Retrieved February 11, 2015.\n^ Sorentrue, Jennifer (November 11, 2014). \"All Aboard Florida updates WPB station design\". Palm Beach Post. Retrieved February 11, 2015.\n^ Fluker, Anjali (August 11, 2014). \"Exclusive: What All Aboard Florida rail station may include at Orlando airport\". Orlando Business Journal. Retrieved February 11, 2015.\n^ Trejos, Nancy (November 9, 2015). \"New Florida train service to whisk passengers between Miami and Orlando\". USA Today.\n^ \"Florida's Brightline to 'take the grey out of travel'\". Railway Gazette. November 10, 2015.\n^ \"All Aboard Florida Selects Siemens as Train Manufacturer\" (Press release). All Aboard Florida. September 11, 2014. Retrieved April 8, 2018.\n^ Brinkmann, Paul (September 11, 2014). \"Siemens to build All Aboard Florida trains\". Orlando Sentinel. Retrieved February 11, 2015.\n^ Hendrickson, Eric (December 13, 2016). \"First Brightline train on FEC rails\". Trains Magazine. Retrieved December 16, 2016.\n^ Viglucci, Andres (December 14, 2016). \"Take a gander at the very first finished Brightline train, now steaming to South Florida\". Miami Herald. Retrieved December 16, 2016.\n^ \"Fifth Trainset Delivered As Brightline Named Official Train Service Of Miami HEAT\" (Press release). Brightline. October 5, 2017. Retrieved May 20, 2018 \u2013 via PR Newswire.\n^ \"Brightline train service begins in South Florida\". WFTV 9. January 11, 2018. Retrieved February 9, 2018.\n^ \"Trains\". Brightline. Retrieved February 9, 2018.\n^ \"Brightline going biodiesel\". June 5, 2017. Retrieved May 20, 2018.\n^ \"Charger Diesel-Electric Locomotive: All Aboard Florida\" (PDF). Siemens Mobility. 2014. Archived from the original (PDF) on June 23, 2016. Retrieved January 3, 2016.\n^ Broadt, Lisa (2018-10-23). \"Stuart will pursue Brightline station; city is considering three locations\". TCPalm. Retrieved 2018-11-02.\n^ Berman, Dave (December 4, 2015). \"Brightline rail project moves ahead; Brevard seeks stop\". Florida Today. Retrieved May 9, 2016.\n^ Neale, Rick (March 10, 2016). \"Cocoa eyed for possible All Aboard Florida train station\". Florida Today. Retrieved May 9, 2016.\n^ Stephens, Henry (2018-02-09). \"Brevard backs Brightline, but stresses safety | Melbourne Beachsider | Vero News\". Vero News. Retrieved 2018-11-02.\n^ \"All Aboard Florida ticketing 15 months off\". Miami Today. October 13, 2015. Retrieved May 9, 2016.\n^ Davis, Ennis (December 23, 2015). \"High speed rail project has eye on Jacksonville\". Metro Jacksonville. Retrieved May 9, 2016.\n^ Kornfield, Meryl (2018-06-25). \"Brightline eyes Orlando to Tampa route\". Sun-Sentinel.com. Retrieved 2018-07-06.\n^ \"Orlando-to-Tampa Brightline service could feature a Disney World stop\". WESH. 2018-11-29. Retrieved 2018-11-30.\n^ \"Brightline can negotiate for land on Orlando-to-Tampa train corridor, state says\". miamiherald. 2019-11-29. Retrieved 2018-11-30.\n^ \"Brightline to Build Express Intercity Passenger Rail Connecting Southern California and Las Vegas\". Brightline. September 18, 2018.\nWikimedia Commons has media related to Virgin Trains USA.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"Recently got her. She's pretty close to the prototype and a very nice figure. However, her hair are slightly darker than official pictures.\nsailormatlac (Vor 1 Jahr) #29476073Thank you! She turned out better than I thought. I'm surprised how she looks good from various angles. Looks like I'll have to hunt her down on the second hand market next year.\nThank you! She turned out better than I thought. I'm surprised how she looks good from various angles. Looks like I'll have to hunt her down on the second hand market next year.\nsailormatlac (Vor 1 Jahr) #29435481Yep, and also on Mandarake too. Just curious to see the actual product.\nYep, and also on Mandarake too. Just curious to see the actual product.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I think we've spent enough time discussing the idea, and now it's time to get down to business.\nGreat job, everyone! If we continue at this rate, we'll be right on track to finish before the deadline.\nA: You haven't practiced for the presentation at all?!\nB: Nah, but don't worry! I'll just wing it.\nA: Jeremy is going to go ask his supervisor if he can take a month and a half off to go travel.\nB: That's not gonna fly!\nWe may not always see eye to eye in certain situations, but I still have faith in her to get the job done.\nHey Maxime, I'm going to be away for a couple days next week, but when I'm back we can touch base and figure out a way to get this project done.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hiz Kidz Spring Break Outing to Pump it Up!\nSpring Break Open Jump-Hiz Kidz ministry will be having a Spring Break outing on April 7th to Pump it Up! in Roanoke from 12 pm to 2 pm and then to McDonalds afterwards. The cost for Pump It Up is $7.50 per child (ages 10 and under) and parents can jump for free. Children under age 6 attending will need to have a guardian with them for the outing. This will be a fun way to work off the spring break energy! You can sign up using your connection card. Please see Ashley Rutter for more details.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Q: What do I need to view this site?\nFlash 9.0.115 or higher, 450kbps high speed connection or higher.\nWindows Media Player 9 or higher, 450kbps high speed connection or higher.\nWindows Media Player 9 for OSX or higher, 450kbps high speed connection or higher.\nQ: Where can I get free upgrades to meet this site's system requirements?\nGet free upgrades for Internet Explorer.\nGet free upgrades for Firefox.\nGet free upgrades for Safari.\nGet free upgrades for Flash.\nGet current system updates as recommended by Windows.\nGet Windows Media Player for Mac.\nGet WMV Player for Mac.\nQ: Are cookies required to view this site?\nA: Yes, you must have cookies enabled in order to access our site. For information on how to do this, visit How do I Enable Cookies? below.\nA: Following are step-by-step instructions to enable the cookies in your browser.\nNote that some firewalls and security tools also include cookie-blocking settings. We recommend you refer to your product manuals for instructions on how to disable this setting as it often resolves streaming issues.\nClose the Security window to save and apply the changes.\nClick the Privacy tab to display the Privacy panel.\nUse the vertical slide bar to adjust your privacy settings to no higher than Medium High.\nVerify that the Override Automatic Cookie Handling option is not checked.\nNote that if you prefer to use the Override Automatic Cookie Handling option, you must accept first-party cookies to enter the site. Though not required for entry, we strongly recommend also accepting third-party cookies for the best playback experience.\nClose all open Internet Explorer browsers.\nReopen the browser and return to the site.\nChoose Tools >Clear Private Data.\nCheck the box marked 'Cookies'.\nClick 'Clear Private Data Now'.\nClose all open Firefox browsers.\nQ: I receive an error message when trying to view this site.\nA: If you receive an error message from our site, visit our system requirements section for free upgrades and downloads, or for instructions on how to enable cookies.\nQ: I can enter the site, but the videos do not play.\nDisable ad-blocking settings in your personal firewall, browser, or ad-blocking software to determine if this allows the video to play.\nQ: I can see the video, but I cannot hear audio.\nA: Verify that your system volume and\/or speakers are not muted or turned to a very low setting.\nVerify that there are no other streaming applications running on your computer.\nVerify that your system has the updated AC-3 (dolby digital) codec installed for broadcast-quality downloads.\nQ: I can hear audio, but I cannot see the video.\nA: Verify that there are no other streaming applications running on your computer.\nQ: Video and\/or audio quality is poor or out of sync.\nA: For best performance, we recommend that you upgrade to the latest version of Macromedia Flash Player.\nQ: I can see the advertisements, but not the videos.\nA: Recommendations for fixing this issue are addressed in I can enter the site, but the videos do not play.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Use our Direct Internet Service (TIDIS) to connect your business to the internet through a dedicated access line. It is a cost-effective, stable and reliable way to move large amounts of data at high speeds without lag or down time.\nWhat is Direct Internet Service?\nAs the name suggests, TIDIS works through a direct line. You therefore have access to guaranteed uncontended bandwidth, without the interference of other users. The links are symmetrical and therefore upload and download speeds are the same. The service offers unlimited data usage and the line is dedicated to your business and your business premises. Access can only be provided through a Metro Ethernet fibre cable.\nHow Will Direct Internet Service Benefit My Business?\nTIDIS is designed to offer you constant bit-rate access without the hassle of congestion or usage limits. This allows you to set up carrier grade video conferencing and to share big data files quickly and easily. You are able to select the speed of connectivity based on the type of applications and number of end-users. We also offer redundancy on internet breakout and options on load balancing. Usage can be split up into national and international bandwidth, allowing you to gain access to international content to sync up with overseas clients, and national bandwidth for access to local content.n load balancing.Usage can be split up into national and international bandwidth, allowing you to gain access to international content to sync up with overseas clients, and national bandwidth for access to local content.\nWe have over 167 000 km of fibre installed throughout Africa.\nOur Service Level Agreement (SLA) includes SLA management and reporting on router or access issues. You will also be able to access the self-service portal to manage your system and keep on top of new developments through real time reports.\nThe system is built over an extremely stable, triple redundant core network.\nThe package is all-inclusive and is customised to suit your needs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzasrqs b/data_all_eng_slimpj/shuffled/split2/finalzzzasrqs
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index 0000000000000000000000000000000000000000..4ddc9ae1c98c878241800a7c16f89416ed711f4e
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+{"text":"Your Adventure Concierge. Whether it is a naturalist led hike, comfort camping, the Trans Catalina Trail, gear haul or an adventure expedition, Catalina Backcountry is proud to offer a personalized concierge experience to our guests. Our customized services are unique to each guest and are designed according to your own desires. With dedicated one-on-one service, we focus on all the minute details of your backcountry expedition, from transportation to the island and backcountry adventures to all the planning and coordination, while you enjoy the perfect blend of adventure and relaxation in the interior of our majestic Santa Catalina Island. Naturalist led hikes are one of the best ways to explore the backcountry of Catalina on limited time. Suitable for all ages and abilities, our guides are experienced locals who are excited to share their knowledge and love of the island with you. Comfort camping is designed to satisfy your craving for the outdoors and your desire for a good meal, nice glass of wine, and a comfortable bed like a hotel, only your room opens to a stunning Catalina Island landscape instead of a lobby! Our campsite concierge includes such things as: Camp Host | 5-Course Gourmet Meals |Hot Water On-Demand Showers | Private Tent Massages | Kayaking | Snorkeling | Mountain Biking | Yoga | Serenades By Local Musicians | and more! Gear Haul is ideal for hikers, bikers, and campers who dread carrying heavy gear such as tents, water, food, wood, etc. while exploring Catalina Island. Do all of your exploring with a simple day pack while we arrange transfers from campsite to campsite along the trail. Contact Catalina Backcountry for all your hiking, comfort camping, gear haul, customized adventure expeditions, and Trans-Catalina Trail needs. We look forward to providing you more information regarding your dream adventure on beautiful Catalina Island. Wedding Planning Imagine your dream wedding in the stunning backcounty wilderness of the island overlooking the beautiful turquoise ocean. As your own personal concierge we will plan and facilitate every detail of your special day..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Jason Fried discusses how the very places that were meant for productivity and work (offices) are in fact very detrimental to our ability to perform.\nJason Fried discusses how the very places that were meant for productivity and work (offices) are in fact very detrimental to our ability to perform. A lot of very good points about how to tweak your workspace (whether you work for a corporation or for yourself) to improve your productivity and ability to work.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Whenever you first start out on your own, lots of your family and friends offers you a hard time. Understanding there are others going by means of the very same course of as you'll be able to enable you to remain in a constructive way of thinking, so it's a good idea to build friendships and relationships with other online entrepreneurs. So join online boards, go to offline networking occasions, something to be sure to are mixing with the proper type of folks. It's essential to surround your self with people who find themselves going to repeatedly encourage you in your ventures.\nThere are lots of public who run a home-based mostly affair. However, usually household based mostly affair house owners and small affair homeowners are overlooking lone important thought with the intention of is estimate them thousands of dollars. The Fleet Security Innovation award, which recognises \"pioneering approaches\" to highway security was given to Royal Mail.\nWithin the age of globalization it is crucial for your business to remain on the cutting edge. Minor errors that set you again just a few days can value you big bucks. That is there are companies that guarantee enterprise retains flowing and that nothing ever gets back up.\nThink about installing a separate phone line. This separate line could also be essential so that your cellphone calls can be answered as professionally as attainable, with out things like youngsters, pets, other folks, or noisy distractions in the background. Perhaps take into account a better-quality voice-mail system as an alternative of the regular answering machine.\nCustoms is an authority which is in charge of the collection and safeguarding of customized duties. It's also responsible for controlling the movement of merchandise in and out of a specific country. Customs duty is a tax or tariff that's levied on the import and export of merchandise. The rules and rules fluctuate from nation to nation. The main purpose of this responsibility is to stop and eradicate unlawful import and export of goods.\nYou see your next money making alternative could possibly be right beneath your nose. You are just could also be too near your own life to acknowledge it. The fact is anything that you have quite a lot of knowledge and keenness about can be a solution to make cash; you simply must research it to be sure you wish to do it. And if you don't consider me just do a google search on any topic. I am going to bet you may find anyone that has had the keen thinking to make it a business.\nThis reduces the likelihood of downstream inconsistencies. Getting jobs overseas may be highly exhilarating and an attention-grabbing possibility for many individuals and is now being increasingly wanted by quite a few professionals and young graduates. You are ambitious. You might be pushed. You know the way to work hard. However, typically occasions our best strengths can work against us and turn out to be our greatest weak point. Beware nevertheless; all of us hit the point of diminishing returns.\n42 year-old Transport Organization Manager Spivery from Brentwood Bay, usually spends time with pursuits like astrology, Business and pc activities. Gains immense inspiration from life by going to destinations including Laurisilva of Madeira.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Charm, sophistication & curb appeal galore! This remodeled and ready to love Cape Cod in Seminary Valley has been transformed into a timeless treasure by the current owners. A stone walkway with path lights lead you to the front door to this home which sits perfectly elevated from the street. When you enter the first thing you will notice is the impressive and meticulously maintained hardwood floors which flow through the entire main and upper levels. The living room is lit by the huge front box windows and features a wood burning fireplace and hardwood mantle. Beautiful plantation shutters in the living room and dining room add architectural detail as well as privacy and energy conservation. The expanded area way into the kitchen allows for great entertaining opportunities and serving from the built in custom wine bar\/buffet. Store your crystal, china, wine or other spirits in a beautiful and useful space! The custom cabinets and updated appliances will lure you to cook and create in this gourmet worthy kitchen. Gas cooking, double ovens, under cabinet lighting and a breakfast bar makes everyday living easy. The dining room, too has huge windows overlooking the large backyard and the access to the rear deck is conveniently steps from the kitchen and dining area. The huge, secluded deck is constructed out of maintenance free material, is great for bbq's and even has a built in natural gas hook up. This is an amazing space to have small or large gatherings or for private relaxation. Sit and read or just chill to the sights and sounds of the natural setting. The two full bathrooms have both been completely renovated with tile and designer fixtures. The upstairs bath includes a jetted tub and heated floor. The master bedroom has an expanded closet with built in organization. The three secondary bedrooms are all large, bright and have tons of closet space. In the basement you will find the family room with space for tv viewing, a game room, exercise room, play room...you name it, there's space for it! There is also a bonus room great for an office, den or craft room. Lots of practical space that allows flexible use customizable to your lifestyle. The utility and storage area is perfect for projects with access to the backyard, has space for the Costco overflow AND enough room for a workshop! Seminary Valley is an ideal location for anyone who commutes, like to to dine out, shop, walk, bike, visit the city, enjoy the DC sports teams, loves history or is just looking for a great, mature, sought after neighborhood! The cul de sac street is quietand lined with sidewalks and street lights. Convenient to DC, Old Town Alexandria, minutes to Tysons Corner, 395, 495 & the BW Parkway.\nI was searching for a Property and found this listing (MLS\u00ae #VAAX233820). Please send me more information regarding 4651 Strathblane Pl, ALEXANDRIA, VA, 22304. Thank you!\nI'd like to request a showing of 4651 Strathblane Pl, ALEXANDRIA, VA, 22304 (MLS\u00ae #VAAX233820). Thank you!\nListing information last updated on April 22nd, 2019 at 12:46pm EDT.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Little Stranger (12A) Dr Faraday, the son of a housemaid, has built a life of quiet respectability as a country doctor. During the long hot summer of 1948, he's called to a patient at Hundreds Hall, where his mother once worked. The Hall has been home to the Ayres family for more than two centuries, but it's now in decline and its inhabitants \u2013 a mother and her two children \u2013 are haunted by a dying way of life and by something more ominous. When he takes on his new patient, Faraday has no idea how closely, the family's story is about to become entwined with his own. Today at 6pm.\nThe Seagull (12A) Today at 8.30pm.\nCrazy Rich Asians (12A) When Rachel Chu, a New York-born Chinese professor, jets off with boyfriend Nick Young to his home in Singapore for his best friend's wedding, she soon finds that she has a lot to take on board. Unbeknown to her, Nick is one of the wealthiest bachelors in the country and Rachel finds herself thrust into the world of the 'crazy rich' of the Singapore elite. A film that takes your average chick-lit turned rom-com for a spin. Friday at 5.30pm, Saturday and Wednesday at 8.30pm and Thursday at 6pm.\nCliff Richard Live Friday at 8pm and Sunday (encore) at 3pm.\nThe House with A Clock in Its Walls (12A) Jack Black and Cate Blanchett star in a tale of fantastical events in the most unexpected places. This magical adventure tells the spine-tingling story of ten-year-old Lewis, who goes to live with his uncle in a creaky old house with a tick-tocking heart. The sleepy facade of Lewis's new environment crumbles to reveal a secret world of warlocks and witches when he accidentally awakens the dead. Saturday, Tuesday, Wednesday and Thursday at 3.45pm, Sunday at 12.45pm and Monday at 4.45pm.\nMamma Mia! Here We Go Again (PG) Saturday, Sunday and Wednesday at 6pm, Tuesday and Thursday at 8.30pm.\nROH Live: Mayerling Monday at 7.15pm.\nThey Shall Not Grow Old (Cert TBC) Tuesday at 6pm.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Instreet Investments has launched a new website to improve social media functionality for advisers.\nThe new adviser portal has been developed based on adviser feedback and includes a more user friendly layout and improved access to information and social media.\n\"The financial planning industry is embracing the use of technology and accessing a lot more content digitally than ever before,\" Instreet managing director George Lucas said.\nInstreet said the new website will improve efficiency by allowing planners to access products and statement information for their clients without having to call Instreet directly.\nGoing forward, Mr Lucas said Instreet will continue to develop the educational focus of the site.\n\"Well-structured product solutions provide known outcomes, so investors can successfully manage their upside and downside in the pursuit of wealth maximisation,\" Mr Lucas said.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Carson Palmer rookie cards were in a rut for the past couple of years but have made a strong come back this year. Carson was hurt and missed most of the season lasy year but has ignited the Bengals to one of there best starts of all-time. After Carson Palmer was drafted from USC his rookie cards became very popular with collectors. His current football cards are very affordable and are very good buys right now. The long term outcome really depends how the Bengals do this year and how Carson finished out the season. He has amazing ability and is surrounded by outstanding talent. Some of his better rookie cards are his 2003 Topps Chrome and Bowman Chrome Rookies.\nInvestment Potential- Carson Palmer rookie cards are great buys in the current market. They are very undervalued and no wis a great oppurtunity for those who are looking long term. Carson Palmer rookies get a 3.5 out of 5 rating for your return on investment.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Do not push away friends and family, as you should use these people with regard to aid. Contact Rotunda Clinic for more detailed information on affordable IVF Infertility Surgery,IVF Treatment, Infertility Clinic India,IVF Clinic Mumbai,India, Rotunda Fertility Clinic,in vitro fertilisation. One intriquing, notable and useful aspect utilizing religious books may be the use of numbers which are scattered during the entire works. The waves of this frequency are even much slower than with the first two frequencies. You are no different from convicted murderers who slaughter persons without reservation. There may also be a lot of give up smoking products about the market. It is actually hard to remove an addiction from your brain chemical driven behaviours and sensory preferences and also DNA Two men were arrested Tuesday on arson and reckless endangerment charges after allegedly being involved in a 'tobacco treating' program that led to an explosion at the home in Hicksville, Nassau County police said. These unhealthy eating practices of fried food, oily meals overcooked vegetables and no physical workout is carried to adolescent and adulthood, where it rears its ugly head available as health problems, which later translate to impaired fertility. If you or someone you're keen on wants to quit smoking, let them in with this great idea Consider each of the tough things you have accomplished in other aspects of one's life, and make use of those memories to fuel your full dedication to stopping smoking. This may prevent any tobacco from spilling out when closed. Gingivitis leads to peridontitis, which will be the loss of teeth. However, anyone using Chantix that has experienced one of the adverse side effects may be shocked when this occurs to them.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Very well cared for 3+ bedroom raised bungalow on the edge of Wroxeter. 2 baths, finished rec room, 2 car garage, move in condition. 2 propane stoves are main source of heat.\nPlease send me more information on 1046 Harriston Road, Ontario. Thank you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We love books, and we love talking about books. Unfortunately, at this time we do not accept requests for review. We understand why you'd like to pitch us your latest project. However, if you do, it will just languish in the boundless void of our inbox. No one wants that.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"This is a reminder yet again Ebola is not away, people are still dying in their thousands in West Africa. I am hearing there is a 'rumoured' case in Belgium also.\nA SUSPECTED case of Ebola is being treated in a Scots hospital. The patient, believed to be a woman, recently returned from west Africa and began suffering from a fever. They were taken to the infectious diseases unit of the Western General Hospital inEdinburgh.\nIt's understood the affected person is from Auchtermuchty in Fife and can reveal a SORT ambulance \u2013 Special Operations Recovery Team \u2013 was despatched to an address in the town on Thursday afternoon. Specialists in full protection suits and equipment collected the patient who was described as a \"high possibility Ebola Case.\" This means the patient had a fever and must have returned from an 'outbreak' country such as Sierra Leone or Liberia in the last 21 days.\nThe patient was then transferred to the RIDU \u2013 Regional Infectious Diseases Unit \u2013 at the Western General Hospital, in Edinburgh where a sample of blood was taken that was sent to a specialist laboratory at the Royal Infirmary for testing. It is understood that whenever a SORT ambulance is requested the Scottish Government and Health Protection Scotland have to be notified. Unlike Nurse Pauline Cafferkey, from Blantyre, who was diagnosed at the end of last year, the Ebola case is not yet confirmed.\nMelanie Johnson, Director of Unscheduled Care, NHS Lothian, said: \"A patient who recently returned to Scotland from West Africa has been admitted to our Regional Infectious Diseases Unit (RIDU) at the Western General Hospital after they reported a raised temperature. \"As a precautionary measure, and in line with agreed procedures, the patient will be screened for possible infections and will be kept in isolation.\nEveryone on these planes and who they have met after now need watched and maybe tested.\nMiss Cafferkey, 39, was the UK's first confirmed case, having contracted the virus while volunteering at a medical centre in Sierra Leone. Diagnosed after returning to Britain, she was transported from Glasgow to an isolation unit at the Royal Free Hospital in London. Pauline, who is originally from Crossgates in Fife, is showing signs of improvement having been in a critical condition. Yesterday, a woman suspected of having Ebola tested negative for the disease at Northampton General hospital, and on Tuesday another suspected ebola case at North Manchester Hospital turned out to be a false alarm. And Stephen Powell, of Pontypridd South Wales, who was showing symptoms of the deadly virus was also given the all clear last week.\nAlthough, the number of suspected Ebola cases is slowly growing in the UK, health chiefs have moved to avoid panic. Britain's 'Ebola scare' is 'low risk', according to health chiefs as it can only be spread by direct contact with the bodily fluids of the infected person. It remains very unlikely that you will catch it.\nEbola starts with a flu-like illness which later develops into a high fever, severe headaches, muscle fatigue and chest pain. In about half of cases people can bleed from their mucous membranes, although its rarely heavy bleeding. You don't bleed to death from Ebola, instead the disease shuts down your organs through fluid redistribution, low blood pressure and widespread clotting. The virus is transmitted by direct contact with the bodily fluids of an infected person, such as blood, vomit or faeces. It has killed more than 7800 people, almost all in West Africa since the epidemic broke out a year ago.\nPauline Cafferkey contracted the deadly virus after going on a charity mission to Sierra Leone. She was diagnosed after returning to Glasgow having been allowed to travel back to her Cambuslang home via taxi before realising her symptoms. Pauline was initially admitted to Glasgow's Gartnavel Hospital on December 29, then transferred to the Royal Free Hospital in London the following day. On January 12, the hospital released a statement saying that Pauline was showing signs of improvement and was no longer critically ill.\nTHE first reported case in the Ebola outbreak ravaging west Africa dates back to December 2013, in Gu\u00e9ck\u00e9dou, a forested area of Guinea near the border with Liberia and Sierra Leone. Travellers took it across the border: by late March, Liberia had reported eight suspected cases and Sierra Leone six. By the end of June 759 people had been infected and 467 people had died from the disease, making this the worst ever Ebola outbreak. The numbers keep climbing. As of November 30th, 17,145 cases and 6,070 deaths had been reported worldwide, the vast majority of them in these same three countries. Many suspect these estimates are badly undercooked.\nDanger is still there\u2026or is it?\nThe rate at which cases give rise to subsequent cases, which epidemiologists call R0, is the key variable in the spread of Ebola. For easily transmitted diseases R0 can be high; for measles it is 18. Ebola is much harder to catch: estimates of R0 in different parts of the outbreak range from 1.5 to 2.2. Although there are some signs that the virus is gradually being brought under control in Guinea and Liberia, any R0 above 1 is bad news. The very high mortality rate of the disease, estimated at 60-70% in this outbreak, means that Ebola can quickly claim more lives than other, more established killers.\nThe inadequacies of the health-care systems in the three most-affected countries help to explain how the Ebola outbreak got this far. Spain spends over $3,000 per person at purchasing-power parity on health care; for Sierra Leone, the figure is just under $300. The United States has 245 doctors per 100,000 people; Guinea has ten. The particular vulnerability of health-care workers to Ebola is therefore doubly tragic: as of November 30th there had been 622 cases among medical staff in the three west African countries, and 346 deaths.\nWhy Are So Many Drs and Nurses Getting Ebola?\nWhy are we not closing borders or flights World wide?\nThe people treating the effected in Africa and America must use 100% perfect protocol when dealing with Ebola. We look and see pictures of sick\/vomit from the man in Dallas who died, Eric Duncan being cleaned by men in regular clothes with a water spray!! That vomit should have been cleaned by the CDC not rented clearers of a housing property, this is what happened in the case of Mr Duncan, his landlord was left to clean up the vomit and hired some company, People in the photos aren't in Level 4 Hazmat suits IN THE USA!\nThis lad look to be prepared to deal with the house of a dead Ebola patient? Because he did!!\nEveryone in Hazmat suits but one with a suit on. Makes no sense right?\nSo we must be real careful, we were told by the Major of New York, Mayor Bill de Blasio that is was really hard to catch Ebola and he is right, but when we see the infected and how their vomit is being cleaned that MUST send Alarm bells ringing with both the CDC and W.H.O, also in the elevator\/lift in the Hospital it was cleaned by cleaners not in proper suits, and if it doesn't send that message, then we are all doomed, VERY DOOMED, as one reader kindly told me earlier this week. 'Doom' now Miss kind reader? Yes one person thought they would come into my blog and poke fun and say \"Doom\" in a joking way. I wonder how that person feels today as they live in New York. I wonder how that person feels today. Ebola being in that person's area, I wonder if that person finds it oh so funny now? We all, every person, every city, every country treat Ebola like an enemy. Sadly and I know this as a fact, the reason Ebola is in the USA is because of money, see the USA don't spend high amount's of money unless it's on War. So Ebola, sadly, could be thriving all over the North East of the USA due to people acting like they are above the threat of the biggest threat to humanity when it comes to bacteria is SHOCKING!.. We can't see Ebola, we only see what it does. Sadly people are too relaxed over Ebola.\nNot what we want to hear, 5 days we heard no new cases outside West Africa, now we see one in New York and one in Mali North of Guinea, just as it was slowed down, lets hope this isn't in New York, I mean spreading. Pray!\nA New York hospital is testing a doctor who has returned from West Africa with \"Ebola-like\" symptoms, the city's health department said. The patient at Bellevue Hospital in Manhattan is a healthcare worker who came back to the US within the past 21 days, it said. Test results for the doctor, who has a 103-degree (39.4C) fever and gastrointestinal symptoms, are expected within hours. The medic has been named as 33-year-old Craig Spencer, originally from Detroit, Michigan. He was working with the aid group Doctors Without Borders, reportedly in Guinea, one of the three West African countries hardest hit by the deadly disease.\nA health ministry official said the girl's mother died in Guinea a few weeks ago and the child was brought to the Malian capital Bamako by relatives before travelling to Kayes. People who have come contact into the girl while in Kayes are being monitored and officials have appealed for those who may have had contact with her elsewhere to get in touch with them. In New York, hazmat-suited specialists have sealed off his fifth-floor apartment in Harlem. New York City Mayor Bill de Blasio told a news conference \"very few\" people had had direct contact with the doctor. The medic took an Uber taxi to a bowling alley in Williamsburg, Brooklyn, on Tuesday night, CNN reports.\nA woman who appeared agitated identified herself at Bellevue's front desk as the patient's fiancee, according to Reuters news agency. Bellevue said in a statement: \"The Health Department's team of disease detectives immediately began to actively trace all of the patient's contacts to identify anyone who may be at potential risk.\" Bellevue is one of three Manhattan hospitals and eight in the state of New York equipped to handle patients diagnosed with Ebola. From Wednesday, travellers to the US from Liberia, Sierra Leone or Guinea, the countries worst affected by the epidemic, were being funnelled through one of five major American airports conducting heightened screening for the virus. As the measures came into force, Connecticut placed six West Africans who arrived in the US on Saturday under quarantine for possible Ebola exposure.\nEight people are known to have been treated for Ebola in the US, six of whom contracted the deadly virus in West Africa. There have been only three diagnoses so far in the US \u2013 a now-deceased Liberian man, Thomas Duncan, in Dallas, Texas, and two nurses who treated him. The family of one nurse, Amber Vinson, said in a statement on Wednesday that health officials were \"no longer able to detect virus in her body\". The other infected nurse, Nina Pham, is said to be in good condition at a federal hospital in Bethesda, Maryland. Also on Wednesday an NBC freelance cameraman, Ashoka Mukpo, 33, was discharged from a Nebraska hospital after being cured of Ebola.\nThe 33-year-old, who caught the virus while working in Liberia, flew home to Rhode Island. The disease has killed nearly 4,900 people mainly in Liberia, Sierra Leone and Guinea.\nWell it looks like Ebola is calming down very fast in the USA, THANKFULLY! No reports now in 5 days since President Obama asked the Media to calm down with every story of a person sick on a plane and other stories, frightening people and giving people like me Ebola news to write about. https:\/\/acenewsdesk.wordpress.com\/2014\/10\/19\/obama-warns-americans-against-ebola-hysteria\/ Pretty much since that speech, there have been not one new report or scare. This is good, 2 Countries in Africa, Senegal and Nigeria are now Ebola free Countries, so I think we, as a World have beaten Ebola. I hope I am not speaking too soon. In West Africa a few Countries are still struggling badly, so we remain vigilant, but this here is fantastic news anyway you tell it, lets hope and pray this trend continues over the coming weeks, we must however still listen to the CDC and W.H.O The World Health Organisation said just 10 days ago it will spread worse.\nA photojournalist has been cured of Ebola as the United States tightens restrictions on travellers from West African countries. Ashoka Mukpo, who was working as a freelance cameraman for NBC News in Monrovia, Liberia, when he fell ill, will go home on Wednesday. In a statement, the 33-year-old American said he was humbled by his recovery.\nEight people including Mr Mukpo have or are being treated for ebola in the US; one of whom, a man from Liberia, has died.\nOne of the two nurses infected with the virus while treating a Liberian patient in Dallas is now faring better, the National Institutes of Health said on Tuesday.\nNina Pham's condition has been upgraded from fair to good. Emory Hospital in Atlanta, Georgia, which is treating the second nurse, Amber Vinson, has so far has not given an update on her condition. Her mother said she remained weak, without offering further details. The stories of recovery came amid news that passengers travelling to the US from three West African countries must enter the country through one of five airports. The designated airports are New York's Kennedy, Newark Liberty, Washington's Dulles, Chicago's O'Hare and Hartsfield-Jackson Atlanta. Each of these hospitals implemented enhanced ebola screenings earlier this month. Government officials said about 94% of the estimated 150 daily passengers from the stricken region pass through those five airports. Homeland Security chief Jeh Johnson said the decision was based on the department's \"ongoing response to prevent the spread of Ebola to the United States\".\nA Spanish nurse who was the first person to catch Ebola outside Africa has been cured of the deadly virus according to doctors treating her. \"She is now cured,\" Dr Jose Ramon Arribas, head of Madrid's Carlos III Hospital's infectious diseases unit, told reporters. The diagnosis of Teresa Romero, 44, on October 6 sparked fears that the disease would spread through Europe. But Dr Arribas gave the nurse the all clear and declared \"the criteria set by the World Health Organisation for curing the Ebola virus have been fulfilled\". She has had two negative tests in the past 48 hours indicating she was now clear of Ebola, after being in isolation in hospital since contracting the disease.\nMs Romero was one of the nursing staff at the Carlos III hospital who treated two elderly Spanish missionaries who caught the disease in West Africa and died in Madrid in August and September. It is believed she was given human serum containing antibodies from Ebola survivors and other drugs. Despite being cleared she will remain in hospital until she has fully recovered. Ms Romero's husband Javier Limon and another 14 people who had contact with her before she was diagnosed are under observation at the hospital but none has yet shown symptoms. Dr Arribas said a specialist laboratory confirmed that a fourth and final round of tests had shown her to be clear of the virus.\nThe news sparked relief after two tense weeks which raised public sympathy for the nurse and questions about safety procedures and public spending cuts in Spain's health sector. \"Someone surviving the Ebola virus is always a cause for celebration,\" said Marta Arsuaga, one of the doctors who have been working round the clock treating Ms Romero. Dr Arribas said that according to WHO guidelines, officials will have to wait until 42 days after the curing of the last infected patient to declare the country free of Ebola. Ebola, which begins with fever and can then escalate to diarrhoea, vomiting, internal bleeding and organ failure, has killed more than 4,500 people.\nMeanwhile, a US photojournalist infected with Ebola in Liberia is now free of the virus and will go home, the Nebraska hospital treating him said. \"Recovering from Ebola is a truly humbling feeling,\" said Ashoka Mukpo, who was working as a freelance cameraman for NBC News in Monrovia when he fell ill. Travellers entering the United States whose trips originated in Ebola-stricken Liberia, Sierra Leone or Guinea must now fly into one of five airports that have enhanced screening in place. The restrictions will subject affected travellers to temperature tests among other protocols at New York's JFK, New Jersey's Newark, Washington Dulles, Atlanta and Chicago's O'Hare international airports. Those airports account for about 94 per cent of the passengers flying to the United States from the Ebola-hit nations. The restrictions, announced by the US department of homeland security, will take effect from Wednesday (US time) and apply to all travellers, including US citizens and those who would have arrived by land or sea.\nUS health authorities said the condition of a nurse infected with Ebola has been upgraded to good. Nina Pham was one of two nurses in Dallas who became infected with Ebola while treating Liberian patient Thomas Duncan, who died of the disease on October 8. \"Ms Pham's clinical status has been upgraded from fair to good,\" the National Institutes of Health said in a statement. No additional details were available at this time, it added. Emory Hospital in Atlanta, Georgia, which is treating the second nurse, Amber Vinson, so far has not given an update on her condition. Her mother said on Monday she remained weak, without offering further details.\nEbola could spread globally if the world does not respond to the epidemic in Africa, Barack Obama has warned.\nThe President also said US monitoring of ebola must be \"much more aggressive\".\nHe insisted the second case of an infected nurse in Dallas highlights the need to ramp up efforts to confront the disease that has struck West Africa and has reached US shores.\nThe President spoke after meeting top Cabinet officials involved in the ebola response both in the US and in the West African region where the disease has been spreading at alarming rates.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"US automotive sales rose by nearly 8 percent or 9.86 million cars sold in March from 9.1 million in February, a far outcry from earlier prediction by Bloomberg analysts who said that US companies will only have a total sale of 8.8 million cars. Meanwhile, Japanese automakers post slumping revenue and weak demand.\nUS automotive sales rose by nearly 8 percent or 9.86 million cars sold in March from 9.1 million in February, a far outcry from earlier prediction by Bloomberg analysts who said that US companies will only have a total sale of 8.8 million cars.\nMeanwhile, Japanese automakers reported slumping sales and weak demand in March\u2014but at least slightly higher than what analysts earlier predicted.\nNissan Motor Co., Toyota Motor Corp., and Honda Motor's sales have significantly decreased last month as labor market continues to weaken and income becomes more dispensable in this rich Asian country.\nOn April 2, Honda's shares rocketed in the final closing minutes by as much as 14 percent; while Toyota's has an additional 8 percent increase. Meanwhile, Nissan's surged by as much as 21 percent.\nToyota has reported a 39 percent sales increase which can be attributed to the car maker's per vehicle incentives that helped its sales to jump 88 percent from last year's record.\nJapanese carmaker's executives said they did not expect the sales will get back higher than expected, saying that most of the people are now just \"sitting on the sidelines\" waiting for the economy to bounce back from deep recession.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Aiming to become an IAS officer? You need to prepare for IAS the right way. This IAS Preparation Guide contains all the resources and information required for IAS preparation. This guide is very detailed and contains a table of content for easy navigation. It is recommended that you bookmark this page (press Ctrl+D) to come back often for the latest IAS preparation tips.\nTo get started with IAS preparation you should first have some idea about the service itself. This will ensure your IAS preparation efforts are channelized in the right direction. The Indian Administrative Service (IAS) is a career nonpareil \u2013 one that cannot be compared to other jobs or services. First let us get acquainted with the basics of the IAS exam.\nNow that you have a good idea about the IAS exam, it's time to move to the next step in our IAS preparation journey which is laying a strong foundation with NCERT books. But why should you read NCERT book for IAS exam? Because NCERTs contain the fundamental knowledge to lay a strong foundation over which you can then refer the standard books for IAS. But you don't need to read each and every NCERT book. Rather you should only read only the essential NCERT books for IAS preparation which have been listed in the post below. The NCERT books can also be downloaded.\nOn the above page you will find standard reference books for Prelims as well as Mains. But since you are just starting with IAS preparation, it is recommended that you get the reference books for Prelims for the time being since this is what is required right now at this stage of your IAS preparation.\nYou are advised to prepare your own time table as per a schedule that suits you.\nOnce you have a time table ready and you have started preparing for IAS as per this routine you should short term and long term targets.\nLong term IAS preparation goals should ideally be a combination of Prelims and Mains preparation targets. Your daily time table should reflect this accordingly. Suppose you are preparing your optional subject side-by-side then in 6 months time you will want to prepare at least 50% syllabus of Paper 1 of the optional subject along with 4 topics of General Studies (for Prelims Paper 1 and Mains).\nPreparing current affairs for civil services exam is crucial since 8-15 questions are asked from the current affairs topic every year in the Prelims. Current affairs is also very important from Mains perspective for preparing GS Paper II and III (International relations, Economy, Technology, Biodiversity, Security and Disaster Management).\nYou should develop a habit of reading the newspaper for 60-90 minutes daily. Newspaper reading is indispensable for current affairs as well as boosting your general awareness knowledge-base. The editorials are one of the most important pages of a newspaper. Read the article below to know how to read the newspaper from the IAS exam perspective.\nOnce you know the right sources to prepare current affairs for IAS it is time to move to the next step of your IAS preparation ladder.\nNote making is an art which you should master if you are preparing for the IAS exam. Why is it so? Because you cannot read entire books and manuals 15-20 days before the exam, Prelims or Mains. So you need to prepare short, concise but effective notes that will help you revise quickly before the Prelims and Mains exams. Go through the following post to learn the exact steps to prepare effective notes for IAS exam.\nThese notes will prove to be a real boon when you revise for the Prelims and Mains.\nMost of us are used to working hard. Our education system is such that unless we cram tens of books each year, whether or not we understand what's inside the book is immaterial, we cannot progress to the higher level. So most of us are accustomed to working hard which in the context of IAS preparation refers to studying 10 hours or more, every day. So much for hard work.\nNow comes the smart part. Smartness could mean different things to different readers. It could mean reading selectively but reading well, reading many books selectively, mixing books and notes, making micro notes, proper time management, taking mock tests and so on.\nSmart work is all this and more. In fact smartness is the approach you adopt in a particular situation. To prepare for IAS smartly requires you to be flexible as opposed to rigid, to experiment as opposed to sticking with the familiar, to plan ahead as opposed to the short term only.\nThis brings us to the topic of starting IAS preparation. Is there an ideal time to start IAS preparation. Should you start IAS preparation after school or college or after graduation? You might have these questions in your might as you prepare for the IAS exam. All these questions are answered in the article below.\nThe next step when starting your IAS preparation is to get familiar with the Prelims syllabus. Doing this you will know exactly what you need to cover within this time frame to crack the prelims. The good part about the new IAS syllabus is that optional subjects have been done away with. So you can just concentrate on General Studies. And if you're like me then studying GS is like listening to music; there's no hard work involved. Just pure fun. Preparing for IAS need not necessarily mean monotony. The more you enjoy preparing GS the easier the Prelims goal will become. That's smart.\nOnce you know the syllabus get started with the actual preparation. Obviously everyone can't prepare for 10 hours, particularly working people. But even working people can crack the IAS. The exact time is not important here. Some can achieve in 6 hours what others can in 10 hours. It depends on you. If you're just starting out I'd suggest you start with 4-5 hours and scale up gradually.\nAs you know that Paper 2 (CSAT) is now only qualifying in scope and you require just 33% or 66 out of 200 marks to qualify this paper. The marks obtained in this paper are not counted for preparing the merit of candidates who will qualify the Prelims and write the Mains exam.\nSo your workload is considerably reduced, especially if you're from a non-Science and non-Engineering background. You can now mostly focus on preparing the Paper 1 topics.\nAnd you just need to focus on Comprehension, Data Interpretation, Interpersonal Skills, Logical Reasoning and Decision Making in Paper 2 to obtain 66 marks. These five topics are considerably easier to prepare and attempt as compared to mental ability and basic numeracy topics.\nAlso remember that there is no negative penalty for questions on Decision Making. At least 5 questions on Decision Making will be asked in Paper 2 so you can and should attempt these questions without any fear.\nAlso if you analyze the previous 3-4 year papers, you will know that at least 5 Comprehension passages are asked in every Prelims and these are easiest to score off in Paper 2. So you can easily score 50 marks from just Comprehension passages! Rest of the marks can be obtained from Data Interpretation, Decision Making and Interpersonal Skills questions.\nYes I know you need to prepare history, geography, current affairs, mental ability etc. Only thing is don't prepare all at once. Chunk it down. Pick one subject, let's say Polity, and combine it with current affairs which you should cover for some time everyday. Current affairs preparation consists of reading the newspaper, a good current events magazine and a year book.\nYou can read a good newspaper like The Hindu or the Indian Express, magazine like Civil Services Chronicle or Pratyogita Darpan and the Manorama Year Book. Schedule a particular time for newspaper reading, but in any case, don't read the newspaper for more than 90 minutes.\nNext, you can continue with the year book or start off with Indian polity. Whichever topic you choose get the right books only. Don't refer more books than are absolutely essential. You don't want a PhD in General Studies, you just need to clear it.\nBut before you start with the topic get the past 5 year's solved question papers and combine it with the syllabus as the question papers and syllabus are your best guide for IAS preparation.\nOnce you have started with polity or any other topic see it through completion. Don't try to multi-task. Mixing polity with history and geography will lead to loss of concentration and lower your output. Besides it will make your progress lower. On other hand once you've covered a topic in full you will gain confidence as you've pocketed x number of marks beforehand.\nThe JiT (Just in Time) Approach says we should seek information only when required at that moment. In this hyper-connected world there is an overdose of information and if you seek to acquire all information and knowledge before starting a task, you can never get started as you will lose yourself in the information maze.\nApplied to IAS preparation what this implies is don't try to cover all the books referred by your friends on a given topic, say GK. Since GK is so vast and constantly expanding you can never hope to \"master\" it. Rather, refer a good book like Manorama Year Book and a magazine along with the daily newspaper to build up your knowledge base.\nIf, while referring the past Prelims papers, you come across a new topic you can quickly refer the reference books at hand or head over to the library or internet and find out more. This way you retain the information for a much longer time rather than by trying to read everything at one go.\nTaking the above JiT approach further let's apply this to evaluating yourself. Instead of waiting till the last few days to take mock tests you should evaluate yourself right after completing a particular section in a topic. For instance refer the previous year's question papers before starting Quit India Movement in Modern Indian History and after completing it. I've already written about the benefits of this approach so I need not repeat it here.\nAdopting this JiT approach you know exactly where you stand, which topics have been covered well and which require more consideration. But more importantly it removes the uncertainty and anxiety to a certain level, if not completely. Before entering the examination hall you know what kind of questions to expect and this will boost your confidence and calm your nerves so you perform better than expected.\nBut going through the previous years papers is just the first step, I recommend you enroll for a good test series to know the latest trend of questions and also to prepare yourself for the forthcoming Prelims in a better way by attempting mock tests that cover the entire Prelims syllabus.\nMost IAS aspirants prefer to wait till the end moment to take mock tests or prepare for 2-3 years before making their first attempt waiting for the perfect preparation level. Unfortunately your preparation can never be perfect no matter how hard you try. The latest syllabus is such that questions will always be unpredictable. Gone are the days when you could rely on certain number of questions from a particular topic.\nSo instead of preparing for 2-3 years during which time the pattern (not the syllabus) could change so many times prepare for a year and jump into the fray. You can improvise along the way.\nThis also holds true for taking practice tests. I suggested constantly evaluating yourself after going through every topic rather than waiting till the end hoping to finish the entire syllabus before going through the question papers.\nGuess what, we are never able to finish the syllabus completely. There's always something left in the end, some topics that we wish we should have prepared differently. So shoot first then aim.\nYour friend tells you to refer MHE GS manual because that's the best one around, another friend refers Arihant, while a third one suggests XYZ classes notes. And you unwittingly jump from book to book hoping to cover every source that's considered important for that subject. Stop taking this information overload that you will never be able to process within the limited time at your disposal.\nRefer the standard books and supplement the missing information from other book for notes. Instead of reading three books for polity stick to one for detailed explanation and one for the bare acts. Similarly, I suggest just NCERTs for Ancient and Medival Indian History. No need for epic titles.\nWhether it's books for general studies, or public administration, sociology, political science or geography books stick to the above strategy.\nYou'll discover you can extract so much more by re-reading limited number of books than running after the next shiny book just launched.\nNo guide for IAS preparation can be complete unless you know important sources for finding information online.\n@lazylord thanks a lot bro\u2026i've jus planning to start my prep fr 2012 prelims\u2026.dis information wud b very helpful\u2026..\nSorry for writing such a long story.From the past 1 year I am following you.And I am sure your guidance is very effective.I am working in a software firm and get less time after coming from the office.In Mumbai I find it very difficult to get all the resources required for the Civil services like old NCERT books for History,magazines like \"frontline\" and \"The Hindu\" newspaper.As per your advice from the past 1 year I am reading \"The Indian Express\" and I find it very useful.For geography I regularly saw the map from \"Oxford School Atlas\" as per your advice.Yesterday while solving 2006 IAS-GS paper I am unable to answer many questions and hardly any question from geography.I don't thing Oxford school atlas is sufficient and i don't find many places,national parks and rivers like \"ken river which is tributary of yamuna\" \"zaskar river\" \"spiti river\",\"huangpu river\", \"karera\" \"jaisamand\" \"nahargarh\" national parks,places like baikonour,borobudur and so many things in the entire atlas.I lost all my confidence and now very tensed.I don't thing i scored more than 25-30 in the whole paper.One of the reason may be the current affairs questions are related to 2005-2006.In prelims 2011,in my first attempt i got 38 and 122 respectively in gs and apti.Sir please help me.\nI am of the opinion that for the optional paper of Geography ,the Oxford School Atlas is definetly not sufficient.It is of common sense that O.S.A. is for the pupil upto the secondary schooling.If you are based at Delhi or any other Metro then you are advised by me to go for the University level Oxford Atlas.It will be costly as it is being imported.Hence,yourfrustration is not worthier as far as O.S.A. is concerned for the preparation of map-work for the G.S. point of view.\nThanks kislay.Geography is not my optional.So I don't think it's good to lose another 20-30 days to start from a scratch to study another atlas.Correct me if I am wrong.It will help me.\nHi Vineet and apologies for the late reply.\nFirst of all never give up as you have a job to fall back so you can keep fighting.\nThe atlas is pretty comprehensive and contains all rivers including shyok, spiti ect. Only thing, it lacks in national parks and I've listed the reasons for it in the post.\nThere are different surveys appearing in the newspapers and news sources all the time. Just keep a tab on the important ones.\nThis approach gels with my style of working. An iterative method to prepare for the exam should definitively prove more effective than the brute-force approach of trying to read everything.\nYour emphasis on the syllabus has me intrigued. I never gave it anything more than a quick skim. I will remember to focus on it periodically to make sure that I am able to cover a major chunk of it.\nBased on your posts, I recently bought the topic wise analysis of the past 10 years mains paper. It is fantastic. Rather than working on the basis of intuition, one can see the general trend the mains paper has been taking over the years. I was wondering if you could suggest something similar for preparation of GS (prelims) or CSAT-1. I tried searching in flipkart, but did not find anything related.\nOnce again, thanks for the great post.\nGlad you found this post and other resources useful. yes the IAS exam requires sticking to the basics rather than trying everything. I will search something useful and let you know.\nApart from the current affairs from the newspapers and magazines,is the study material put up on your website enough for cracking the prelims and mains or are there some additional books required?\nI am a 10th class student and want to prepare for IAS wholeheartedly \u2026\u2026and i want some tips for that \u2026..could u plz\u2026.help me out\u2026.\nI am the student of class 10th and want some tips from u for how to start preparing for IAS as i want to it any how____________ i have read so many posts of yours i like them pretty much\u2026..\nfirst download the eBook. It lists the steps to start with IAS preparation!\nI am a medical graduate student and want to prepare for for IAS,could you please give me some tips that what subjects should i take to crack IAS in ist jerk as i have hold on Zoology.Sir you also tell me that does a complete hold on a particular subject can give us a pretty satisfaction in Prelims..\nHey\u2026\u2026thanks for ur valuable information .I am very new to this and planning to start my prelim for 2012..hoping ur support wil be there..\nYour information was of a great help.. but sir could you give some personal tips for me.. i have read all your posts.. but i wanted just a little help for starting my preperations.. sir, if you say then only i would post my problems.. thank u sir..\nplzz specify which coaching insstt. i hav to join for pub.administration?\nI have come to know about this site and found it very usseful. Now i will start the preparation. As i am working so dont have full time. But i will try my best.\nPlease tell me how i start the preparation. I am not getting right point.\nThanks for all that information, Lazylord. I was wondering if you have come across architects who have attempted the IAS exam. Because there's no one that I know of and that's the field I'm in and I do want to give this an honest shot but a lot of people tell me it's not advisable for some reason. Your thoughts? Just want an opinion. Thanks again!\nThanx a lot brother !!\nYou do a great job!!!!!!!!!!\nI am not exaggerating my expressions while SALUTING you.\nYou simply remove tons of burden from my head.\nthanks for ur information.Currently i am studying 6th sem of B.E. and will finish the course at may\/june 2013.So i am thinking of preparing for one whole year n then writing the exam at 2014.Is it advisable to take up exam at 2013 or 2014 batch, i have already selected my opts as pub ad and geography\u2026..pls advise me, so that i can start my preparation as soon as possible.\nsir i have seen your preparation tips for ias but sir by 2013 i will complete my phd course then only i will start my preparation so i want to know from you that is it too late for me for the preparation.pliz suggest me.\nI am planning to appear for the exam \u2013 2013 and have started preparing for the prelims. Could you just give an insight about how to go about the prepareation for Mains? I am a graduate in physics and post graduate in management. Planning to take physics and Administration as optionals-is it a good decision?\nI am looking for books for sociology and public administration in hindi.\ncan you please help me in finding the books which covers the whole mains syllabus.Please let me know the names of the books.\nUR GUIDANCE CAN MAKE ANOTHER IAS\u2026.I WILL WAITING FOR UR REPLY \u2026 SOON!!!\ni want to prepare ias from law because i just completed my ll.m. so please guide me, and share some of your words regarding IAS from law, my gk is not so good but i want to crack this paper. And i will crack , if i will get some proper guidance.\ni am in second yr of btech,really need help for ias preparation\u2026i don't know any thing regarding its prepration..i know there is no need of hurry but i want to know all..and i want to prepare in small parts from now\u2026i am a average student so i need to fix my basics..please help me.\nDidn't you get some idea from this post?\nwow!! what a blog!\u2026very very informative! God bless! Keep up the amazing work!\ni am in 2nd year b.tech . how should i start preparing from now ???\nThanks for the info.It was of great Help.\nVery useful tips for preparing and examining our-self in each and every step towards the goal..!! Thank you.!\nGreat article giving a lot of perspective on IAS preparation. I am a final year chemical engineering student from Mumbai planning to give IAS-13. When should i ideally start preparing for Mains? And can you please suggest some optionals?\nsir i want your guidance personally also. pls can u help me sir?\nsir i am not getting e-book. there is an error msg. pls look into and kindly fwd it to my email id. My id: sharmi.psa@gmail.com. thank you sir.\ni am mujibur\u2026.am also preparing for the ias\u2026.i want to take bengali as one of my main optional\u2026.can anyone suggest me where can i obtain previous question papers of bengali\u2026.\nhey lazylord,.,.well i am just 13 right now though i have made my firm decision and i want to be an ias , and the thing is that i have no idea about how do i start my preparations from the very starting,.which i think is now only , so i want your help, please do direct me how do i patch up with all the study material and what should be my next step in the preparation,,, and please i request you not to tell me that you are too young so enjoy ! cause i am already through all these ,.,.i just wanna start my preparations ,.,please guide me,.,and i am thinking that i'll choose humanities as my stream whatsoever, i am really very much interested in english so i am thinking of doing eng (hons.),.,,.i know u'll come up with great solutions to all my queries \u2026\u2026 THANKS ALOTT !!!\nhi..i am about to start my preparation from the new year . I am of 20 and am willing to appear on 2015 . please give a guidance of how to continue with d studies .\ni am doing bcom accts hons from South City College of kolkata .\nmera naam vicky hai aur main delhi institute of tool engineering se tool & die making main diploma kar raha hu aur sath hi main du se open se B.A kar raha hu.\nmeri graduation 2015 main puri hogi aur diploma bhi. sir mai ias ka exam dena chahta hu o bhi hindi medium main to sir kya aap mujhe kuch tips de sakte hai , ki main pre ki taiyari kaise karu aur mains ki ( public administration & geography as optional lena chahta hu ) sir mujhe iske bare main puri jankari chaiyeh aur hindi medium (optional ) ke writter ki details bhi sir please reply jarur karna.\ni was told we need to choose our optional subjects. but what i didn't understand was its relevance interms of wheter we need to study all subjects for the general paper and the subjective-objective gamit! kindly explain!\niam doing btech final year eee , my passion is to crack civils .iam in dialemma , to select the path , is it better to directly join coaching institutes after completion of my btech or while doing mtech is it better to prepare for i.a.s ??\nhow i have to select optionals sir ?? and from right now \" where i have to start\" ?? \" what i have to read \"??\nsir , please do reply me \u2026\u2026\u2026\u2026\u2026..\nThanks a ton for the in-depth article,it really helps in finding the way by which we want to start our journey for IAS preparation. For me i am an Engineer with certain no of years of experience but i was never an engineer with a choice,i always loved the history throughout my life and Geography was the subject which used to calm me,So do GS.I have even taught Geography in am school in the same class when i was studying.\nNow i am trying to recollect every piece of information and willing to correct the mistake which i made by opting a career which i never loved, though i am good in it too but just not my thing..\ni would like to know is it important to go through all the NCERT books from 6th to 10th or just TMH GS manual is sufficient.\nhello sir.. i am in b.com 1st yr and want to crack an ias after my graduation plz suggest me what to prepare from now m totally confused about it.\nthanks sir,it is most useful for who r upsc asiprent.\ni want prepare for IAS 2014 and i want to know the whole procedure what is prelims what is mains?\nwhat are the paper for prelims and what are the paper for mains?\nwhat is the strategy to start for the exam?\nwhat book should i follow from scratch ?\nsir,m in fybsc\u2026wana appear for 2015 examinations\u2026when should i start preparations and how? kindly reply asap :)..thanks..\nI am in MBA 2nd year and still m working with MNC company so i have come to know about this site and found it very usseful. Now i will start the preparation. As i am working so dont have full time. But i will try my best.\nPlease tell me how i start the preparation daily basis at home. I am not getting right point.\npleas help me what should i do.?\nI am an ECE student. I am in confusion to select the optional paper. Can you please help me sir?\nI am the student of class 12th and want some tips from u for how to start preparing for IAS .\nAnd hav'nt started the preperations yet.\nI am b.tech final year student (8 sem). Sir i am decided to prepare myself for IAS . I know it's not an esay task. Sir i need your help please tell me how i get started preparation for IAS . What the first step i should take to start preparation.\nPls refer the FREE eBook for getting started with IAS prep.\nSir, Would you suggest some online free material for prelims, also i am confused as to until when the current affairs will be from, i.e december 2012 or even later?\nWhat is your opinion on the current changes in the mains pattern?\nIsn't Pearson CSAT Manual 2013 suitable for prelims instead of different books you suggested?\nWhich sub is better for ias preparation arts or maths?\nbut I m very confusing from where 2 start.\nExactly what I was looking for\u2026\u2026 Thanks a lot!!\nI have done my 12th and am preparing for my admission in some good graduation college like DU or BHU\u2026.\nShouldn't I start preparing form now on?\nYes you can by reading some basic NCERT's, newspapers and magazines.\nI m an engineering graduate(ECE). I am ok with both engineering subjects as well as non technical subjects..I am confused between IAS and IES ..please help out..\ncnan i write my ias prelims after my 2nd puc!!!???????????and what are all the exams i need to face to become an ias officer!!???\nIts such a great write up for fresh ias aspirant like me. Infact the best among many I have read.Your great tips and the time you spend to help people is very valuable and much appreciated.God bless you ..\nhi:)m preparin for IAS but not gettin right path.kindly let me know which books to be referred for pub admin.n which other sub shud i choose for mains?\nhow i preapare for exam suggest some idea so that i cn fullfuil my duties n preapares too.\nwho i cover wide GS section.\nHave you downloaded the free eBook that was offered here till a few days back on IAS prep?\nHow should I prepare for preliminary examination? I'm really worried about the CSAT. I see the questions, I think I can solve them, but when I actually try to reach to the answers, I could not. What should I do?\nHow to maintain the balance between the study for GS and CSAT? How can I maximize the score in GS(Pre)?\nI'm in job and planning for civils. I've started with ncert 6th standard books. I'm studying maths, history a s science. Plz suggest me that what are the other subjects I should go for?\nIs it important to read all ncert books with minor details. Please suggest me for it.\nwell my question to you is that i'm still doing my engineering and i'm gonna start my 3rd year in a week and i want to be an IAS officer so what according to you i should do from now onwards so that i can build my foundation in IAS preparations what should i read what should i follow ??\nthere are so many questions that i have in my mind because i'm planning to do my preparations myself what should i do?\nI am a Engineer and working in GOVT Organisation. I want to Prepare for IAS. I am thinking to choose Electrical Engineering as my optional subject. I went through the last year paper and I am able to get good marks in this paper.\nI want to give my exam in HINDI medium as it will be more easy for to prepare for GS in hindi than English as I am more comfortable with Hindi and i can achieve more in Hindi rather than in English. but in Engineering Paper I am more Comfortable with English.\nAs per the rule I can give the answer of my question either in Hindi or English only in all paper.\nso now I am confused whether i should opt Hindi or English as my medium of writing.\nI feel a better bet in your case will be to write in Hindi since GS has 4 papers and Engineering has only 2. Also, in engineering papers mathematical problems can be written in roman numerals reducing the need for complete Hindi. So if you feel comfortable in Hindi, go with it.\nThank You for your reply. I think I should should go for Hindi. I was downloading NCERT books of History, Geography, Political Science and Economics of class 11th and 12th. Is it OK to start all these subject with NCERT or shall I refer other Books right now.\nI will post soon on which NCERTS to refer for which subjects.\nHello Sir, Im referring Arihant Publications Book for General Studies Paper 1. Is it sufficient to crack paper 1 .\nI will need to go through it before I can make a comment on this book.\nI will post something on Essay prep soon. It's currently not present on the site.\nsir please tell me, is there any need of coaching for IAS preparation. If yes then which coaching centre is best in India?\nIt depends on you. DO you feel need for coaching? For which subject do you require coaching?\nThis post should guide you on starting your prep.\nthanks sr ur tips are beyond expectation.\nSir currently I'm doing my engineering in mechanical stream(2nd year). How to start my preparations for IAS ? Please help !\nsir plz sugst me coching center for IAS \u2026.\nI am an mechanical engineer and now I m in job.I am planning for write IAS 2015.Due to my job I can not attained a coaching class but I have time to study at my place for self study.There are lots of study material available in market which make me confuse,and no one is there(friend or senior) who can guide me regarding my preparation.Now I have Taken first step for my preparation with news paper (THE HINDU), NCERT BOOK and Pratiyogita Darpan for IAS Pre Paper-I.In Paper-II(CSAT) I am feeling comfort.I would like to select Political Science as my optional.\nSir,you are requested to guide me from beginning to last stage of this examination basically for complete and best study material for both Pre and mains.\nI will wait your valuable suggestion.\nsir am going to stdy civil engineering from dis year\u2026.. but am planning to give my time for the preparation of IAS exam from d beginning\u2026.. ! but the proble is I hav no idea what to stdy for IAS and what subjects are required for an btech (civil) student to stdy for IAS exam\u2026.! so could u plz send me the subjects that are required to stdy for IAS exam for an btech student\u2026..!\nplz msg me in my email\u2026.. thnx!\nI completed my B.Tech in 2013,at present I am working with an MNC company.My age is 22 years.I am intrested in civil services.Please suggest me how to start it because I thought of appering to UPSC after 3 years.I'm thinking from now onwards I have to keep my basement for it,so can you please guide me how to plan with in these 3 years? as I(am also an employee) wanted to krack it in the first attempt i have taken the time span of 3 years.\ni'm in b.tech 2nd year.i want to prepare for ias please help me.\nPlease tell me that from how many years ago I should start preparation of current affairs to crack IAS?\nsir,my friend want to prepare for an ias ..he is now doing btech in electronics n communication 7 sem n next is trainee sem , so he wants to prpare at that tym ,, so plz guide us how to prepare for it he didnt get guideanc from anyone yet,, he is fav sub is physics n he wants opt for that ,, is that easy for to crack compartivly other sub\u2026 n how to prepare for premil?\nI have gone through your blog and i feel i can do it in parellal with job in PSU.Currently I am Master student of Indian Institute Of Science and i want to start my preparation from next year when my master will complete. But the biggest problem that i have is about choosing the language.Because i study till 10th in medium Hindi because of that every small term apart from Science and Computer Science i knows there hindi version but at the same time as a Engg student i also focused on english so common knowledge of daily life i know in English. And one biggest tragedy in IAS exam for hindi medium student is in language that is currently use in paper and also less no of selection in IAS from hindi medium. As far as my writing skill is concern i am good in hindi compare to english. In Englsih i do lots of grametical as well as spelling mistak. So from this point of view i should prefer hindi. Another importent thing is , i have studied Geography,Civics, History,Economics etc in hindi so every basics term i knows in hindi if i will prefer English then it will take lots of time to remember all terms in English so again i can not take risk.\nSo according to my situation what would you recommend to me???????\nIf you're comfortable in Hindi, go with it. English students don't have any particular advantage. It depends on individual's ability and expression as well. If people can qualify in regional languages, why can't you with Hindi?\nJust i need to know..\nSir, I am a B.Tech student of 3rd year. How should I have to start preparing for IAS. But have only 4-5 hours hardly for preparing it, because I also have to study my technical course. Please rply sir.\nSir I'm an engineering student in civil branch, i need information about preparation of IAS exam, so please help me\u2026..\nSo plz give me some questions about this. .\nDear sir,i need your valuable suggestions for reading a particular book.Is it required to cover every line of a book for a particular subject? If your answer is no,then how would i know the important part of that book.\nHi iam praveen plz i want plane to ias i dont know how to prepar \u2026.plez plez\u2026..give me help \u2026\u2026..\nI m a hindi medium aspirant\u2026 how can I crack this exam. What should be resources for exam? And from where I get.\nGood Read and very well explained.\nCan you also help us in choosing the subjects and then how to proceed with the preparation of the subjects that we chose. Really appreciate the help.\nFirstly, I feel very bad that I had reply after so many people views, but it is not matter that Sun is bright for everyone not only for one, as you are. After studying your posted information I wish to appear in I.A.S. examination in the year of 2016. I know the syllabus for I.A.S.(Pre. or Main) and also some referred books, but I am confused to take some decision that what should I start. My topic is History and I know the syllabus for History. I request to you that kindly tell me that what should I do?\n1. May I read all the books of N.C.E.R.T. (All Subjects like Polity, History, Geography, Economics, etc.,). or only One (History)for Preliminary examination or Main Examination (both examinations)?\n2. After completion of reading of N.C.E.R.T. books, may I read suggested books for History only or all topics like mentioned in Mains subjects G.S.-I, G.S.-II, G.S.-III and G.S.-IV?\n3. I know the topics of My optional subject History but I do not know the topics of above mentioned syllabus like G.S.-I, G.S.-II, G.S.-III and G.S.-IV.\nAs I am beginner for this examination and working in Part Time Job and as per advertisement I had seen all the eligibility criteria and selection procedure mannaul. But I am little bit confused regarding suggested books for these topics.\nI would like to request you that kindly give me the list of books for these all topics I can managed by myself but I want to list of suggested books I have tried many sites but I am unable to find such books, I found some books for few topics.\nAt last, I would again request to you kindly send\/post the suggested books for basic knowledge of all these topics, I am highly obliged to you. Please Sir\/Madam, please give me the response with suggested books of all topics like English, CSAT, GS-I, II, III, and IV and Optional Subject of above mentioned subjects. (JUST LIST OF SUGGESTED BOOKS) Please.\nHow should I start prepration for IAS, and which will be best books for prepration, and also tell me which sub wiil be suitable for me.\nI have read so many blogs and write-ups on effective IAS exam preparation, but the tips given in your blog are the best ones. I will always remember the 3 golden tips: Smartness, Persistency and Patience. I am taking coaching from one of the best institutes Vajiram and Ravi to get knowledge on different subjects. And now by combining the knowledge gained in the class with the tips given in this blog, I am sure I will be able to perform better in IAS exam. Every point given in the write up is helpful for every student aspiring to take IAS exam. Thanks again for such a motivational, helpful and effective blog.\nVery well said, and the right source to get this guidance is a good coaching institute. With abundance of online sources finding out the most helpful and updated source gets really difficult. I am lucky that I got support of helpful, experienced and supportive teachers at Vajiram and Ravi. The teachers not just provide me solution to my queries, but also equip me with the additional resources to explore more for effective preparation of IAS exam.\nSir I am a student of class 12 (PCM) and I have a dreamed to become an IAS officer when I was in class 7. But sir today I am little bit confused about which course I should choose after 12 so as preparation for IAS Continues alongside with graduation.\nsir, i m in final year B.tech cse student.\nplease tell me best coaching centre in Delhi or North India to prepare for ias.\nI want to prepare of IAS from beggin.what should i do first.\nWho is Gram Sevak? What are his functions?\nHello sir, i am going to start preparation for upsc. i am little confused.should i go for the prelims first or mains?\u2026or i can start both at the same time because after prelims i think time will be limited for the mains.\nI am an aspirant who has jus completed my 12th standard PCMB, i want to join BSc nd prepare for IAS. I don't even know single piece of information abt it except dat i hav a dream to crack it.\nSo can i get any help dat how to prepare?? Pls reply.\nSir i have g one through the preparation tips for a working personnel and i am well satisfied with your tips.\nsir i would like know some points in details like how to corelate newspaper and book(particular subject) and syllabus.\nsir i am fresher in this field.and i want to prepare ias.so what shall be my first steps.\nHi i am giving exam in 2018 and want an advice whether i should take maths as my optional subject or not as i have a lot of interest in maths and i like studying it\u2026..i have done bba as my college degree but i had maths in 11th and 12th standard an what books should i refer for it and can you give a brief strategy plan for it?\nshould i go for IAS preparation in hindi medium or english medium? because i belong to hindi medium UP board. Which one is beneficial?\nHow can you help to make strategy and prepare the notes for study, Could you tell us?\nThere is a link provided on this article on how to prepare notes for IAS exam. Please refer to it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In Tanjung Pinang city - Bintan Island, Laut Jaya Hotel, which is established since June 1997, has become one of the most popular Jasmine Hotel which provides the most comfortable services. There are 4 Laut Suite, 18 Sea View Deluxe, 24 Deluxe, 10 Standard rooms which offer basic facilities such as breakfast, full air conditioner, cable tv channels, hot & cold shower with bathtub, telephone, traditional massage for guests who wants to enjoy their weekends \/ holidays.\nGuest can enjoy the seasight view just by opening the window of room, where marine folks live on the boathouse & also the scenery of water village. Without going outside, guests can still enjoy Tanjung Pinang's specialties such as Fried Rice, Soto Ayam, Satay and more. As for the guests who need to go out, Laut Jaya Hotel provides them 24 hours Transportation service. Our fully trained receptionist will stand by to assist guests who experience any administration problems.\nLaut Jaya Hotel is located in the city's business district area not far from Sri Bintan Pura International Ferry Terminal (5 minutes from Laut Jaya Hotel). Ferry services that are offered here serve routes from & to Tanah Merah Ferry Terminal (Singapore) daily. It also serves routes from & to Stulang Laut Ferry Terminal (Johor Bahru, Malaysia).\nWith its strategic location, hotel guests can reach many attraction points without need to take a long journey. Places like Tepi Laut, Traditional market, Bestari mall, Bintan mall, Melayu Square, Penyengat Island, Raja Haji Fisabillah Monument, Grand Mosque of Sultan Riau, Senggarang Island, Banyan Tree Temple can be reach not more than 15 minutes from hotel. There's a pier located just in front of Laut Jaya Hotel in which the guests can charter for a boat taxi known as \"sampan\" to visit Senggarang Island.\nClick here to WhatsApp Chat with our staff directly for more information about our hotel rooms & tour packages.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Fluorescent analogs of biomolecules have been known as useful probes to study the structure, conformations and dynamics of cellular processes. These probes are more ideal than fluorescent labeled probes, as fluorescent analog probes retain the shape, size, conformation, and recognition element of the natural substrate, while giving useful intracellular information about detection and dynamics of biomolecules. The monoamine neurotransmitters control the central and periphery nervous systems. Serotonin (5-HT), in particular, is a versatile chemical messenger responsible for a multitude of biological processes, such as regulation of emotion, vasoconstriction, and bone metabolism. The study of serotonergic complex pathways is vital and essential in drug discovery for the diseases that result from the depletion and deregulation of serotonin in synapse. The extracellular concentration of serotonin is controlled by several transporters, most preferably the serotonin transporter (SERT). Selective serotonin reuptake inhibitors (SSRIs), along with dual- and triple-acting inhibitors, affect SERT and hence 5-HT in depression and related diseases. In this present investigation, firstly, a set of fluorescent analogs of neurotransmitter probes based on ethylamino-functionalized substrates were successfully designed and these fluorescent probes were synthesized by convenient synthetic methods. Secondly, optical properties of these fluorescent probes were investigated in organic medium, in order to test their suitability for screening and imaging the biological cells. Finally, their uptake was examined in the murine osteocytic cell line, MLO-Y4, platelets of blood sample and HEK-293 cells expressing the dopamine transporter (DAT), norepinephrine transporter (NET) or SERT. The fluorescent probes targeting bone-derived cell line expressing 5-HTT provide useful information in understanding the dynamics of 5-HT regulation with respect to SSRI treatment. A novel fluorescent analog of 5-HT probe was developed that may be utilized to study 5-HTT function in the context of 5-HT uptake or regulation in cell culture, tissue explants, or even in vivo.\nFluorescent; Cellular; Neurotransmitters; Inhibition; Monoamine; Uptake.\nBagale, Sharanappa Maduraya, \"Synthesis of Fluorescent Analogs of Neurotransmitters\" (2011). Open Access Theses. 241.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzauhnx b/data_all_eng_slimpj/shuffled/split2/finalzzzauhnx
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+{"text":"Trey Songz took to the stage at the Indigo2 on Sunday 29th March for a one-off 'Trigga Reloaded' show and our music editor Tasha Demi, went to check it out. Tickets to the show sold out within the hour of release and after witnessing the show, he did not disappoint.\nCapital Xtra's 'Name you can trust' DJ, Manny Norte, took to the decks to warm up the crowd before Ester Dean opened up the show as Trey's support act. Now, you may be unfamiliar with her name, but you definitely know her work. Ester Dean has penned some of the biggest tracks out there that we all know and love. Think of some of Rihanna's biggest hits: 'Rude Boy', 'What's My Name', 'Complicated', 'S & M'; and then Nicki's 'Pills N Potions' and 'Super Bass', and not forgetting Beyonce's 'Countdown', Katy Perry's 'Firework', Usher's 'Hot Tottie' and 'Lil Freak', Ciara's 'Never Ever' \u2013 and the list goes on. Ester was joined on stage by 2 male dancers, but with a back catalogue as strong as hers, she's all the hype man she needs!\nAfter Ester Dean's high energy performance, the excitement in the crowd was almost too much to bare for the hundreds of screaming girls cramped together trying to edge closer and closer to the stage to be just one centimetre closer to Trey when he finally arrived on stage.\nJust before Trey hit the stage, there was a sudden fan frenzy when everyone downstairs started screaming and taking photo's of the Kings Row balcony as they spotted Trey's brother Forrest, and Mum April, take their seats amongst the audience. Tremaine was about half an hour late for his stage time (that is kind of expected though, right?!), but the screams in the crowd when he appeared from the back of the stage in a blacked out get up, proved that nobody minded at all.\nNow, apart from stating the obvious (his charmingly good looks, and impressive physique), Trey pulled out all the stops last night to prove that he is a true talent and a really well respected artist. They're aren't many R&B singers out there that boast such a healthy collection of tracks \u2013 both old and new \u2013 and can say they have stayed relevant for all of those years with a super dedicated fanbase.\nTrey pulled out hit after hit, as he was backed by his band, and four impressive female dancers \u2013 who have without a doubt inspired us to lay off the McDonalds, and hit the gym hard!\nAs if the love in the audience wasn't strong enough, Trey treated us to a well thought out show, which saw him putting a twist on many of his well known, and well loved tracks. Taking to the stage for 'Already Taken', Trey broke down the end of the track and gave us a beautiful, acoustic rendition of the chorus full of quirky ad-libs which showed off his incredible voice.\nHe then moved onto perform, 'Invented Sex', 'All We Do', a beautiful rendition of 'Can't Be Friends', and a massive audience favourite, ' Can't Help But Wait'. Trey filled his time on stage by giving us some of his classic tracks like, 'Simply Amazing', '2 Reasons', 'Say Aah', 'Bottoms Up', as well as some of his new one's, including 'Na Na'.\nAs the show went on, Trey changed from being blacked out, to rocking an all-white-everything outfit, and teased his lady fans by progressively showing off more and more skin. When he finally took off his white tee during his 'Touchin, Lovin' performance, he threw it into the audience and the crowd went crazy. For the duration of tha song, the closing of the show, and even as the crowd made their way out, a group of girls were still left behind playing tug of war with what used to be Trey's t-shirt, but now looked like shreds of cotton\u2026 Dedication eh?!\nWe for one, are already looking forward to the next time Trey comes back to London. If you get a chance to see him perform live, take our word for it \u2013 go! He will not disappoint!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"More than half of Hispanic students leave school each year without the skills they need to succeed in the next phase of their lives. That's less than half who are on track to make their best contribution to the workforce, the nation, and the future we will all share. No one feels this systemic loss of potential more deeply than principals, who look in the faces of these students each day. The new leadership standards released last year (Professional Standards for Educational Leaders 2015) renew principals' focus on a culture of equity that eliminates student marginalization. By adopting those standards for principal development and evaluation, states can align resources and expectations with the renewed focus on building a culture of inclusion.\nAlso, as the data indicate and as NASSP has long advocated, there is nothing \"extra\" about extracurricular activities. Participation in high school activities such as student council and the National Honor Societies engages students more actively in the life of the school, reinforces behavioral sills, and provides students a platform to amplify their voice in constructive ways. The more marginalized a group of students is, the further school leaders must extend their reach to invite their participation. You simply can't improve student performance without student engagement.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We are the most full international list of the Vinyl Pressing Plants, Brokers, Lathe-cutting, and Lacquer Mastering Services.\nIf you want to find any information about Vinyl Pressing or Cutting services \u2013 you are in the right place.\nIf you are an owner of a vinyl record pressing plant \u2013 doesn't matter, a big or a small one, contact us.\nIf you find any incorrect information in our catalog - please, let us know.\nCatalog of companies that do pressing, lathe-cutting, embossing, mastering, and packaging of vinyl records.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"People come to Brighton Center from all walks of life, each with a unique set of circumstances, but what they all have in common is a desire for a better future. We Are: A Community of Support for thousands of individuals and families on the journey to achieving self-sufficiency and ultimately their dreams for a better life.\nTo donate gently used items, please contact Madison Smith or click here to visit the Clothing Closet.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Other name: SHOW BY ROCK!! \u3057\u3087~\u3068!!\nRyouma Sakamoto wants everyone to know about his passion for rock 'n' roll, so he roams around town with his electric guitar willing to show anyone he encounters that he's just as skilled as the famous Shinsengumi stars they admire. Unfortunately, Japan doesn't allow anything other than that group's Heaven's Songs, for writing or performing different types of music is forbidden and can lead to harsh consequences. Agitated by these strict rules and brainwashing, Ryouma does everything he can to show people that the music he loves will bring them the freedom they deserve. Along with his bandmates Shinsaku Takasugi and Kogoru Katsura, Ryouma works hard to find places for his rock 'n' roll group to perform. Refusing to back down until their music is accepted in Japan, the trio begin to realize that there's more to their passion than they had thought.\nOn the first day of junior high school, Mato Kuroi happens to run into Yomi Takanashi, a shy, withdrawn girl whom she immediately takes an interest in. Mato tries her best to make conversation with Yomi, wanting to befriend her. At first, she is avoided, but the ice breaks when Yomi happens to notice a decorative blue bird attached to Mato's phone, which is from the book \"Li'l Birds At Play.\" Discovering they have a common interest, the two form a strong friendship. In an alternate universe, the young girls exist as parallel beings, Mato as Black\u2605Rock Shooter, and Yomi as Dead Master. Somehow, what happens in one world seems to have an effect on the other, and unaware of this fact, the girls unknowingly become entangled by the threads of fate.\nOn her first day of junior high school, Mato Kuroi meets Yomi Takanashi. Though Yomi is initially taken aback by Mako's straightforward personality, the pair quickly becomes friends and begin to spend time together daily. As a sign of their friendship, Mato gives Yomi a cell phone charm\u2014a blue star, identical to her own. However, when the two enter their second year, their relationship starts to change. Placed in a different class, Mato begins to spend more time with Yuu Koutari instead, a girl she met through the basketball team. In fact, the former best friends drift apart so much so that Mato cannot find Yomi anywhere, as if she had disappeared entirely. Elsewhere, Black\u2605Rock Shooter is on a quest to vanquish the Dead Master. These two, while opposed, bear a connection not unlike Mato and Yomi. As their stories begin to cross, it seems Yomi's disappearance may have to do with the blue star-shaped charm and the legendary gunslinger herself.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Two young Goulburn women have made it into ArtExpress 2019.\nSophie Linton and Maddy Smith have been selected for this year's exhibition that showcases outstanding works of art created by NSW students for the Higher School Certificate (HSC) Visual Arts examination from 2018.\nThis annual exhibition, held at the NSW Art Gallery, provides an insight to the concerns and passions of young Australians, while introducing the artists and creative thinkers of tomorrow.\nSophie Linton, a Mulwaree High School graduate, is one of those creatives.\nShe attended the opening of the exhibition on February 6 in Sydney.\nLinton said for the HSC, her work Musician Man was listed under \"time-based forms\".\n\"It is a film that runs for about 3.5 minutes and plays on two different screens simultaneously,\" she said.\nIt was filmed on two iPhones and edited with Final Cut Pro.\n\"I have not had any experience with film before so I asked people about it and researched all I could about doing it,\" she said.\nLinton said her art teacher, Toni Campbell, was very proud that both she and Maddy Smith made it into ArtExpress.\nBeing selected has made her consider a possible career in the arts, she said.\n\"I was really surprised and excited that I got in, so it has made me consider art and film making as an option for a career,\" she said.\n\"I have just finished Year 12 and I am having a gap year and moving to Sydney. I might be able to access more courses in relation to film making up there.\n\"Film making is a big interest for young people and the quality of films being produced even on iPhones is becoming quite sophisticated.\nLinton has also been accepted into the University of Wollongong (UOW) to study communications and media, but says she has deferred that for a year.\nHer other interests include speech and drama.\nShe is also a former vice captain at Mulwaree High.\nYoung Goulburn woman Maddy Smith has also been selected for the ArtExpress 2019 exhibition.\nHer selected artwork was a piece called The Circle of Life, featuring coloured pencil on hardwood.\nSmith said the theme also determined the media she used.\n\"That is why I drew them on the circular wood panels,\" she said.\nThey will be mounted on a wall during the exhibition at the Armory Gallery at Sydney Olympic Park.\nThe artwork will also travel to Wagga Wagga as part of ArtExpress 2019.\nIronically, the image in the middle circle is of her friend and fellow art student Sophie Linton.\nSmith said she originally photographed the models and then drew a grid onto the wood to draw them.\nShe said she had always been drawing and loved art.\nShe has some big plans for her artistic future. \"I am enrolled at the National Art School in Sydney, but I also have an interview for the University of the Arts in London, which happens through the ArtExpress, so Sophie Linton has an interview with them, too,\" she said.\nArtExpress is an insight to the concerns and passions of young Australians, while introducing the artists and creative thinkers of tomorrow.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"To ensure you get the cheapest price possible for a flight to Miami, you should look to book at least 52 days in advance of your intended travel date. The price of your flight may increase if you delay and leave booking until a week or so before departure.\nThe cheapest day to fly to Miami is usually Sunday. At the moment, Saturday is the most expensive.\nTo get the best value, try booking a flight at Noon when visiting Miami. Generally the prices will increase for flights in the Evening as these tend to have higher demand.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Wavin's story begins in 1950s in Zwolle, the Netherlands. The local water utility found itself battling against serious (iron) pipe corrosion and a significant loss of water. Company founder Johan Keller decided that something needed to be done urgently; he succeeded in producing the world's first plastic pressure pipes for potable water. The Wavin company was founded to focus solely on plastic pipe production.\nWavin is now a global leader in the supply of plastic pipe systems and solutions. Since the 1950s, we have built an unrivalled reputation for continuous innovation, intelligent problem-solving, dedicated technical support and the highest standards.\nThe world does not stand still. Wavin has always been looking ahead \u2013 and innovating. These are our breakthrough innovations that changed the way installers and contractors work. These are Wavin's first-in-the-world innovations.\nPeople want the plumbing in their buildings to be silent. When low noise is a building requirement, the Wavin AS system gives the best performance of any plastic pipe.\nThe Wavin infiltration units were the answer to a growing demand to handle peaks in rainfall and to better control rainwater discharge. Wavin infiltration units attenuate rainwater and can have it slowly discharge into the surrounding soil.\nWavin Tigris K1 plastic press-fittings are made from high-performance PPSU (polyphenylsulfone) performing well in even the most challenging water conditions. These systems banish corrosion concern, delivering long-life performance at competitive cost.\nWavin's Tegra manholes and inspection chambers are easy to install, easy to access, versatile and durable, making these the ideal choice for all public sewer systems.\nWith Wavin Q-Bic plastic units you can build underground tanks quickly and easily. These units are suitable for both attenuation and infiltration of rainwater. Wavin Q-Bic is designed for trouble-free inspection and cleaning \u2013 ensuring maximum long-term performance. Each unit has generous 500 mm access channels for video inspection and cleaning.\nWavin TS DOQ is a co-extruded three-layer PE 100-RC pressure pipe. The outer layer protects the pipe against surface damages. Use Wavin's TS DOQ range when you seek maximum safety. It is the best-selling PE 100-RC system in the world.\nWavin's Hep2O pushfit range connects as easily to flexible polybutylene pipe as it does to copper, due to its intelligent pushfit technology. Its exceptional reliability is the result of four decades' experience.\nThe new Ekoplastik PPr Fiber Basalt pipe uses innovative materials to real advantage. The new generation P-RCT material gives the pipe an increased resistance to higher temperatures and high pressure. The basalt fibers add to the benefits by reducing thermal expansion.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Officially licensed, this Game of Thrones Hall Of Faces men's t-shirt is a striking addition to any fan's wardrobe. The design across the front of the t-shirt sees the poster artwork for the sixth season of the epic show. Featuring characters both currently alive as well as characters long deceased, this print depicts some of the series' key players as some of the masks available in the Hall of Faces in a colourful, distressed style. This Game of Thrones t-shirt comes in classic black and offers a crew neck and short sleeve combination; made for a standard fit this men's t-shirt has a high quality feel and finish. Please follow care label instructions carefully, note that different brands offer different sizing options. For additional sizing information, please contact a member of the team.\nOfficially licensed, this Game of Thrones Hall Of Faces men's t-shirt is a striking addition to any fan's wardrobe.\nThe design across the front of the t-shirt sees the poster artwork for the sixth season of the epic show.\nFeaturing characters both currently alive as well as characters long deceased, this print depicts some of the series' key players as some of the masks available in the Hall of Faces in a colourful, distressed style.\nThis Game of Thrones t-shirt comes in classic black and offers a crew neck and short sleeve combination; made for a standard fit this men's t-shirt has a high quality feel and finish.\nPlease follow care label instructions carefully, note that different brands offer different sizing options.\nFor additional sizing information, please contact a member of the team.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What Size Antigravity Battery Do I need?\nThe Small Case line of Antigravity batteries are our battery of choice for custom projects. They offer a significant weight advantage along with compact size and ability to be oriented in any direction allowing for very discreet placement on a bike. The LiFEPO4 cells provide a considerable amount of amperage for their comparative size and when sitting maintain their charge longer than traditional lead acid batteries.\nIf you're not already familiar with LiFEPO4 batteries when you buy an Antigravity Battery the first thing you should do upon receiving it is READ THE MANUAL. The very next thing you should to do is REGISTER YOUR WARRANTY. There is a window of time in for you to do this, if you procrastinate it may be a costly mistake.\nWhat size battery do I need? Or in other words, what is the smallest battery I can get away with?\nI've condensed the manufacturers recommendations into a single reference chart.\nWe've pushed these boundaries on our own custom builds and found that with the right circumstances you may be able get away with a battery smaller than what is recommended below. If you're running LED lights and the bike is getting plenty of ride time to charge up the battery you might be fine with the next size down but you need to consider a number of factors like your displacement, power requirements and ride style then use your own discretion.\nWe recommend that you install a quick connect harness and keep your bike on a float charger when not in use for a length of time. If you have an m.unit installed they have a very low draw in the milliamp range that over time will pull your battery down, additionally this will protect you battery in the event of a parasitic drain.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..9d7daf5083624e992e738e2761549d7f912bde93
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+{"text":"came up inside your mind?\nor have you lost your sight?\nAnd don't be feel ashamed, I simply enjoy it.\nBut I am free to do all the same old mistakes again and again and again.\nBut don't be afraid, I'm not a threat to you.\nWhich you consider less beautiful.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Y'sdom was introduced to Sri Lanka (Formerly Ceylon) by Mr. R. O. Buell, then General Secretary of the Colombo YMCA, on his return to Sri Lanka after attending an International Convention in Minneapolis, USA. Thus the first club (Colombo) began functioning in 1929 and was chartered on 3rd December 1930 with the late Sir Oliver Goonetilleka, former Governor General of Ceylon, as its charter president.\nThe last occasion when Colombo Club flag was displayed in India.\nThrough the Colombo club, Y'sdom was introduced to India where it spread rapidly and as a result, the India-Sri Lanka Region came into being. Today, the Y's Men's Club of Colombo has the distinction of being the oldest functioning Club in Asia. It may be also mentioned that in 1973, Colombo was the first Club in Asia to admit women as fully fledged members and Ms. Sylvia Perera was the first Lady Y's Man to be inducted.\nWith the charter of 5 more clubs -Kandy (28th July 1962), Moratuwa (30th March 1964), Amparai (November 1964), Jaffna (30th October 1971) and Badulla (1974), representations were made to International Headquarters for Sri Lanka to become a separate Region. Accordingly, at the International Convention in Atami, Japan in 1975, Sri Lanka became a Region of its own with Kenneth Somanader as the first Regional Director (RD). During his office, Kenneth chartered two Clubs, Chavakachcheri (22nd November 1975) and Hatton (March 1976). The first Regional Convention of the Sri Lanka Region was held at the National Seminary, Ampitiya on 10th, 11th, and 12th of September 1976 with a large gathering of over 200 present.\nStanley Arnold succeeded Kenneth as RD and during his term, he chartered two Clubs \u2013 Welimada (12th February 1977) and Kurunegala (28th May 1977). The 2nd Regional Convention was held in Colombo in October 1977 and Stanly Arnold was re-elected as Regional Director.He was subsequently appointed as an International Council Member (ICM) for the period 1978 to 1981. He was also appointed International Vice President of Asia Area for 1979\/80. Stanley was re-elected for a second term in office and continued as RD up to June 1978. During this period, a system of Regional Awards was introduced to spur the Clubs to greater efforts. These awards were presented for the first time at the 3rd Regional Convention held in June 1978 and Kurunegala, the youngest Club at that time was the first recipient of the 'Best Club Award'.\nWimsy Sinnathamby, took over the mantle of office from Stanley and also served as RD for 2 years from 1978 to 1980. He, together with IPRD Stanley and Lady Y's Man Sylvia Perera represented Sri Lanka Region for the first time at the 53rd International Convention in Melbourne, Australia in July 1978. During Wimsy's term of office, two Clubs were chartered \u2013 Wellawatte (10th November 1978) and Batticaloa (30th May 1980). The first International Brother Club triangle was also established during this period with Aars (Denmark), Arendal (Norway) and Moratuwa (Sri Lanka) combining to form the triangular relationship. The 4th Regional Convention in 1979 was held in Welimada and Moratuwa was adjudged the Best Club for 1979\/80.\nGladstone Thevathasan took over as RD at the 5th Regional Convention in 1980. During his term , two Clubs were chartered \u2013 Manipay (23rd August 1980) and Dehiwala (13th June 1981). Welimada was disaffiliated due to dormancy. Wellawatte won the 'Best Club' award at the Regional Convention. Also, though the Regional was barely 5 years old, it was honoured by being requested to host an Area Convention in Sri Lanka. Accordingly, a host Committee was appointed to plan and prepare for the event, with Gladstone as Chairman.\nInternational Council meeting \u2013 27th to 30th July 1981.\n9th Asia Area Convention \u2013 31st July 1981.\nAsia Area Council meeting \u2013 1st August 1981.\n150 Y's Men and Menettes from various parts of the world attended the meetings. PRD Stanley Arnold served as an Area Consultant and RD Darrell Perera served as a Councilor at the International Council Meeting. Among the distinguished visitors were Joseph Ohm of Korea who was installed as International President, IPE Max Larson, PIP's Bruce Price, Christian Bach Iverson and Tod Gunkelman, ISG Bjorn Pederson and International Treasurer Gerald Lee. Fifty years of Y'sdom in Sri Lanka was celebrated during the Asia Area Convention with the Colombo Club hosting a banquet.\nOne of the key issues arising at the meetings was the request by India to secede from the Asia Area and function as a separate Area. This resulted in the \"Parting of ways\" between colleagues, India and Sri Lanka, with Sri Lanka opting to continue as part of Asia Area.\nRD Darrell Perera conducted the 7th Regional Convention at the Theological College, Pilimathalawa from 18th to 20th June 1982, where Lanka Nesiah took over the mantle of leadership. His theme for the year was \"Reach out to new frontiers\". During his term of office, the 'Gladstone Thevathasan Fund' was established with money left over from the 9th Asian Y's Men's Convention. The interest earned from this fund was to be used to assist Sri Lankan Y's Men attend Conventions abroad. The Principles of Partnership between Y's Men and the YMCA was signed on 18th June 1983 by RD Lanka Nesiah on behalf of the Sri Lanka Region and Eraj Wijesinghe, President of the National Council of YMCAs on behalf of the YMCA. The Kopay Club was chartered in 1983 and the Moratuwa Club was adjudged the 'Best Club' for the year.\n8th Convention at Pannikuda Bay, Batticaloa on 14th & 15th May 1983 \u2013 Best Club - Wellawatte Elected RD Devane Perera.. Hatton Club was disaffiliated during the year. New Clubs chartered were Mt. Lavinia on 24th April 1984 and Kelaniya on 10th June 1984. Unfortunately, ethnic violence broke out in July 1983, and this disrupted the working of most Clubs.\n9th Convention at the Sri Lanka Technical Institute, Diyagala Boys Town, Ragama on 12th & 13th May 1984. Best Club - Kurunegala Elected RD Ercon Godlieb..\n10th Convention at Sun Island Tourist Village, Kahawa from 7th to 9th June 1985. Best Club \u2013 Moratuwa Elected RD George Chinnathamby.. Y's Ling Shyamila Perera of Wellawatte became the first Sri Lankan student to participate in the YEEP programme in Denmark.\n11th Convention at Hotel Thilanka, Kandy from 20th to 22nd June 1986. Best Club - Wellawatte Elected RD Joseph Antonirajah.. Young Moratuwa Club was chartered on 12th May 1987 and Kopay was disaffiliated.\n12th Convention at Hotel Mermaid, Kalutara from 12th to 14th June 1987. Best Club - Moratuwa & Wellawatte Elected RD Jackie Ratnam.. The Colombo South Club was chartered on 27th February 1988. Devane Perera served as ASD & ISD and was elected as an International Council Member for 3 years from July 1988.\n13th Convention at Hotel Mereviere, Kalutara from 17th to 19th June 1988. Best Club \u2013 Moratuwa Elected RD Clifford Ratnavibhusana.. The first entry from Sri Lanka in the Golden Book was made during the year in the name of Kenneth Somanader.\n14th Convention at Palm Village Hotel, Uswetakeiyawa from 16th to 18th June 1989. Best Club - Moratuwa Elected RD Jean Godlieb.. RD Jean thus became the first woman in Asia to hold the office of Regional Director.\n15th Convention at Palm Village Hotel, Uswetakeiyawa from 15th to 17th June 1990. Best Club - Colombo Elected RD Noreen Chinnathamby..\n16th Convention at Coral Gardens Hotel, Hikkaduwa from 14th to 16th June 1991. Best Club \u2013 Colombo Elected RD Hilary de Alwis..\n17th Convention at Barberyn Reef Hotel, Beruwala from 19th to 21st June 1992. Best Club - Kurunegala Elected RD Elizabeth Dwight..\n18th Convention at Barberyn Reef Hotel, Beruwala from 25th to 27th June 1993. Best Club \u2013 Wellawatte Elected RD Justin Perera..\n19th Convention at Grand Oriental Hotel, Colombo on 18th June 1994. Best Club - Kurunegala Elected RD Rajinda Wijemanne.. Sri Lanka received an invitation to host the Asia Area Convention and allied meetings in 1995. This was accepted and RD Elizabeth Dwight was appointed Chairperson of the Host Committee.\n20th Convention at the Town Hall, Kurunegala on 17th June 1995. Best Club - Young Moratuwa Elected RD Ercon Godlieb.\nChairman of Central Environmental Committee.\nThe Asia Area Convention took place from 15th to 17th September 1995 at the BMICH with 250 Y's Men & Menettes in attendance. Dr. Gamini Corea (former Secretary General of UNCTAD) was the . Keynote Speaker.\n21st Convention at Club Palm Gardens, Beruwala on 22nd & 23rd June 1996. Best Club - Kurunegala Chief Guest Elected RD Darrell Perera..\n22nd Convention \u2013 at Subhodhi Piliyandala 21st June 1997. Best Club \u2013Elected RD - V. Manoharan.\nY'sling Radhika Hettiarachchi left for Canada on YEEP.in July..\nLaid foundation to have an annual Fund raiser, the \"Sing-along\". Had the first \"Sing-along \u2013 Musical Memories-1\" at Galle Face Hotel in September.\n24th\u2026Convention \u2013 Club Palm Bay Marawila. 1999 Best Club \u2013 Elected RD - Lakshman Perera.\n25th\u2026convention \u2013 Hotel Camelot Negombo. 17th & 18th June 2000. Keynote address by Dr. Amal Kumarage. Best Club \u2013 Colombo Elected RD - Norman De Zilwa.\nChatered the Y'sMen's Club of Nuwara Eliya on 24th February 2001.\n26th\u2026Convention - Pegasus Reef Hotel. 2001 Keynote address by pastor Romesh Moraes Best Club \u2013 Dehiwela Elected RD - Nilu Fernandopulle.\nPlaced ground work for the formation of Kuliyapitiya Club.\n27th Convention \u2013 Club Hotel Dolphin. 15th & 16th June 2002. Best Club Colombo\u2013 Elected RD - Jayantha Fernando.\nChartered the Y's men's club of Kuliyapitiya on 23rd of November 2002.\nPlaced ground work for the formation of Mahanuwara and Negombo clubs.\nArea & International began to respond to an appeal for donations towards the Flood Relief.\n28th Convention \u2013 Kani Resorts Kalutara. 13th \u2013 15th June 2003. Best Club \u2013 Dehiwela Elected RD - Rita Hettiarachchi.\nKelaniya Club was disaffiliated due to dormancy.Chief Guest \u2013 Area President Janet Siew Guest of Honour IPE John Choa.\n22 houses for the Flood Victims were built in Hiniduma and Mandadeniya\/Ratnapura areas funded by International & area.\nThe membership training was taken to the door-step of the Clubs for the 1st time.\nMiss Marilivia came in as a YEEP Student from Brazil.\nChartering of Mahanuwara Club on 30th of May 2004 with 20 members took place.\nStarted ground work on chartering the Cinnomon Gardens Club.\nTraining Material on Y'sdom was prepared and transferred to Slides.\n29th Regional Convention - Pegasus Reef Hotel. 18th - 20th of June 2004. Best Club - Dehiwela. Elected RD Veraan De Mel.\"\nDonations received for Tsunami victims from International, Area and Donors.\n30th Regional Convention \u2013 Browns Beach Hotel. 25th \u2013 26th June 2005. Best Club \u2013 Dehiwela. Elected RD Felician Perera.\nChartering of Y Service Club of Cinnamon Gardens with 17 members on 28th May 2005.\nColombo the oldest club in the region celebrates its 75th Anniversary on 03rd December 2005.\nSri Lanka wins bid to host Asia Area Convention 2009. RD Felician appointed as Host Convention Committee Chairman. Chartering Seeduwa Club on 03rd March 2006 with 16 members.\n31st Regional Convention \u2013 Lihiniya Surf Hotel 16th \u2013 18th June 2006. Best Club \u2013 Dehiwela. Elected RD Anton Kandiah.\n15 houses built with assistance from International, Area and donors handed over to Tsunami victims at Weligama.\nChartering of the Menettes Club of Athugalpura on 18th November 2006 with 22 Members.\n32nd Regional Convention- Hotel Topaz \/Tourmaline Kandy. 22nd \u2013 24th June 2007. Keynote Speaker \u2013 Mr. Kushil Gunasekera. Best Club \u2013 Dehiwela. Elected RD Enrico Brown PRD Clifford Rathanavibhushana has been appointed as the Chairman of Asia Area Convention for 2009. Disaffiliated Seeduwa Club.\nGround work was started for chartering Bambalapitiya Club and Batticola City club by DG Ariyaratnam.\nChartering of Bambalapitiya Club on 17th August 2010 with 15 members.\nChartering of Singing Fish Batticoloa Club on 27th Novemeber 2010 with 26 members.\n38th regional convention was held at Global Towers hotel Colombo 6, 22nd June 2013. Best club \u2013 Dehiwela.\nElected RD \u2013 PRD Anton Kandiah. Chief Guest \u2013 AP Oliver wu.\n41st Regional Convention was held at Hotel Sapphire, Colombo 6, 18th June 2016. Best club \u2013 Young Moratuwa Elected RD \u2013PDG Prasantha De Siva. Chief Guest \u2013 APE Tung Ming Hsiao. Key Note Speaker - Mr.\n42nd Regional convention was held at GOH, Fort, Colombo 1 on 27th June 2017. Best Club \u2013 Young Moratuwa. Continued to be RD in the coming year. Chief Guest- AP Tung Ming Hsiao. Key Note Speaker - Mr. Geoffrey Alagaratnam .\n43rd Regional Convention was held at GOH, Fort, Colombo 1 on 23rd June 2018. Best Club- Young Moratuwa. Elected RD Ranarajah Serasinghe. Chief Guest \u2013 AP Tung Ming Hsiao. Chartering of Bandarawela Y's Menettes ' Club and Y's Youth Club took place during this Y's year.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The US infrastructure market is at an inflection point and it is poised for growth. It is expected to attract significant private capital in the coming years after decades of underinvestment.\nAnalysis of the 50 US state budgets shows that public spending on transport infrastructure is set to grow 11.6% over the next two years, up from 4.2% over the last two years. Infrastructure reinvestment is also expected to grow for the first time since the financial crisis. More states than ever now have public-private partnership (or P3) legislation and are opening dedicated P3 offices. States are increasingly publishing their infrastructure pipelines. They are moving forward with infrastructure and taking cues from Canada, Australia, the UK and Europe. Efforts are underway to streamline federal regulatory approvals.\nThe market in the United States is poised to attract new capital from institutional investors in the US and from abroad. There are proposals to broaden P3 regulations to allow more private capital into the market and investors are increasing their allocations to infrastructure. The sectors most likely to receive incremental private investment first are those that have clear user fees, are critical to the economy and are operating at a low standard. These include highways, airports and ports. Air traffic control is also high on the list because efforts are already underway for it to be privatized.\nThe electricity grid is facing many challenges from new sources and uses of power and has been identified by Congress and the administration as a priority. We expect continued near-record capital expenditures by utilities and additional government incentives to increase investment in the smart grid. Renewables have a well-functioning incentive plan in place by the states and the federal government and we do not anticipate any material changes. The freight rail industry is investing near record levels to prepare for future transportation growth.\nInfrastructure is an essential part of the economy. People use transportation infrastructure every day to commute to work, travel freely, and visit their friends and family. Businesses depend on a well-functioning infrastructure system to obtain supplies, manage inventories and deliver goods and services to market. This is true for US companies whose businesses rely directly on the infrastructure system, such as UPS and CSX. It is also true for businesses that indirectly rely on the infrastructure system, such as farmers who use publicly funded infrastructure to ship crops to buyers and dot.com companies that send goods purchased online to customers throughout the world.\nA modern transportation infrastructure network is necessary for an economy to function, and there is substantial empirical research that links the quality and quantity of infrastructure to competitiveness, productivity, economic growth and quality of life. In 1955 during his State of the Union address, President Eisenhower expressed his vision that \"A modern, efficient highway system is essential to meet the needs of our growing population, our expanding economy and our national security.\" This vision is even more relevant now than it was then. Today it would include making US highways and the nation's entire transportation, telecommunication and power infrastructure more efficient and effective.\nThis three-part series will focus on the state of the US infrastructure market. We will look at the amount of infrastructure spending taking place now and the amount that is needed. We will assess new initiatives being discussed to finance infrastructure with public and private capital and the sectors most likely to attract new investment.\nFor much of the last century America's infrastructure has supported and enabled innovation and has been a driver of US growth. During the Great Depression, federal funding supported construction of thousands of roads, bridges, airports and dams. During the 1950s, President Eisenhower spearheaded the National System of Interstate and Defense Highways. Under this 35-year program, 41,000 miles of critical roads were built by the states and funded mostly from fuel tax.\nThese massive federal programs are a thing of the past. Public officials lack the wherewithal to modernize US infrastructure. State and local governments still face funding gaps and tax revenues have not reached pre-crisis levels, although they are rising. The same is true of the federal government. What is needed now is a new way to plan for, pay for and deliver infrastructure. It is an opportune time for the US to pursue infrastructure because interest rates are low and funding is inexpensive. Economic growth is relatively strong and the availability of private capital looking for long-term steady returns is high. The challenge is enormous but the US has the resources to do it.\nAs shown in Exhibit 1, US public construction spending, a proxy for infrastructure reinvestment rates, reached a 20-year low of 21% of total tax receipts in 2016.(1) This is expected to change in the next two years. Goldman Sachs' most recent analysis of the Department of Transportation (DOT) budgets for the 50 US states points to higher public construction spending for the first time since the financial crisis (see Exhibit 2). It is being driven by 24 states that passed legislation in 2016 and 2017, adding new funding sources for transport infrastructure investment. Goldman Sachs expects growth of 5.5% and 6.1% in 2017 and 2018 (11.6% collectively), up from 0.1% and 4.1% for 2015 and 2016 (4.2% collectively). This represents a 175% increase for 2017 and 2018 vs. 2015 and 2016.\nCalifornia is leading the charge in new construction spending with a recent increase of 54% in its 2018 transport budget, which passed in April 2017 with 100% of the democratic majority voting to pass it. The bill calls to raise $52.4 billion over the next decade to pay for a massive program of rebuilding California's highways and bridges and improving public transit. It will be paid for by a combination of revenue measures, including an increase in the state's excise tax on gasoline from 28 cents to 40 cents per gallon, increased vehicle registration fees and a new $100 annual charge on emission-free vehicles.\nThe US is ripe for new private investment. Budget constraints, past underinvestment, and more states than ever having P3-enabling legislation will drive state and local governments to provide incentives to investors. More state and local governments are publishing their pipelines and opening government infrastructure offices. Investors are increasing their allocations to infrastructure. All of these factors suggest the US is at an inflection point.\nThe primary way for the government to tap private capital, other than through the municipal bond market, is through P3s. Private companies have long been involved in the construction of infrastructure, but their role has usually been that of a contractor with the ultimate responsibility for financing and delivering projects resting with a state or local agency. We believe this is on the cusp of changing.\nThe P3 contract was first introduced in the US in 1995 when State Route 91 was constructed in Southern California. Since then, 56 P3 projects have been funded with loans under the Transportation Infrastructure Finance and Innovation Act (TIFIA) and the number of projects has accelerated over the last 10 years (Exhibit 3).\nAlthough low vs. other countries, the trend toward greater use of the P3 market continued in 2016, mostly along avenues the Obama Administration paved through TIFIA, the Fixing America's Surface Transportation Act of 2015 (FAST), and the Water Infrastructure Finance and Innovation Act of 2014 (WIFIA). There were also amendments to WIFIA and the Water Infrastructure Improvements for the Nation (WIIN) Act. Federal agencies made progress establishing and expanding programs to administer federal funding for critical road and water infrastructure projects, although the federal government's commitment to P3 has remained limited to \"horizontal\" initiatives involving water systems and highways.\nThirty-five states, Washington, DC, and Puerto Rico now have P3 legislation, with Kentucky, New Hampshire and Tennessee being the most recent. States also have been more receptive to innovative P3 applications, with about two-thirds of the 35 states considering extending P3s from highway projects to transportation facilities such as airports and to \"vertical\" projects such as hospitals and schools.\nPublic-private partnerships are referred to as P3s in the US, public-finance initiatives (PFIs) in the UK and PPPs elsewhere.\nAvailability-payment P3s. Once construction is completed, the private developer is entitled to payments from the government, as long as contract conditions are fulfilled. Availability payments are sized to cover operating and maintenance costs, debt service costs and equity returns as a private entity operates the project. Availability payments are not subject to swings in demand such as traffic levels and are typically adjusted downward only for lack of performance or lack of availability of the asset to the public. The availability-payment P3 is prevalent in the UK, where they are also called private finance initiatives (or PFIs), and in Canada, Australia, France, the Netherlands and Portugal.\nDemand risk P3s. Demand risk P3s, or concessions, have a long history of public-private financing and have been used in many European countries and in Latin America, particularly for toll roads. Under a concession or demand risk P3, the project is largely financed by user fees and the government takes on no or only limited demand risk. This model is often used for toll roads, airports, public transport, and water, gas and electricity P3s.\nHybrid forms. P3 arrangements can have characteristics of both an availability-payment and a demand risk P3, exposing the government to a variety of potential liabilities: (1) explicit obligations such as availability payments; (2) contingent obligations, such as financial guarantees, termination payments and subsidies, if demand falls under certain thresholds; and (3) more remote contingent obligations, such as the risk of contract renegotiations or takeover of the project in the case of default of the special purpose vehicle.\nWe expect 2018 to usher in the launch of several significant P3 projects. It will also see the early impacts of the current administration, which has called for increased private investment in public projects and has hinted at changes to the legislative and regulatory framework that would lower the barriers of private capital looking to finance infrastructure.\nDespite the growing adoption, only $80 billion of TIFIA projects reached financial close in the last 12 years, with $8.9 billion in P3s reaching financial close in 2016. This represents only 0.05% of US GDP and is 74% lower than the median of 0.19% for other countries that use P3s. If the US used the same spending rate as the median, it would add $411 billion to incremental spending over 10 years.\nThere are barriers preventing more investment of private capital into the P3 market. One is the lack of an infrastructure pipeline. Many federal governments, such as in the EU, UK and Canada, and the new Asian Infrastructure Bank, publish their pipelines but the US does not.\nAnother barrier is that some US policies limit the scope with which public agencies can work with private investors. For example, 11 of the 35 states that have P3 legislation limit the types of projects that qualify for P3s. Much of this is due to a perception of political risk to incumbent politicians if a project does poorly.\nThese barriers are beginning to subside. Some states are now publishing their infrastructure pipelines (e.g., Virginia and California) and the administration is suggesting a US federal pipeline. A recent survey by JD Power showed US consumer attitudes toward P3s have become more favorable, and this will in turn decrease the perception of political risk. States also are increasingly turning to P3s because of concern over their bond ratings should they borrow in the public markets.\nThere are proposals for more US states to open offices of infrastructure investment that can serve as single points of contact where interested investors could begin a dialogue with a state or locality. These offices could also provide information to the public about P3 projects, thereby increasing transparency, countering misperceptions that might exist and increasing local political support. Currently seven states and Washington, DC, have P3 offices and the number is expected to grow. There is also a multi-state office on the West Coast, the West Coast Infrastructure Exchange (WCX), which serves California, Oregon, Washington and British Columbia.\nEconomic infrastructure consists of fixed, tangible assets that make up physical capital, one of the three factors of production. These assets are required for economic growth and are important in the capital formation and final consumption stages of GDP. Broad sectors include transportation, utilities, power and telecommunications. Characteristic of economic infrastructure projects is that they earn their revenues from demand, usage, volume and sometimes a partial availability payment from the government. Example of projects are toll roads, airports, ports, utilities and power generation plants.\nSocial infrastructure includes the system of networks and facilities that support people and the community. These assets are often operated by the private sector and are used to support and provide public services, such as public schools, parks, housing and large dams.\nCommercial infrastructure is a recent offshoot of the asset class and comprises assets for which the benefits of sharing infrastructure outweigh the competitive advantage of owning and operating one's own infrastructure. Often characterized by a high degree of competition, this category includes assets such as satellites, cable networks and mobile phone towers. Often commercial infrastructure is included in economic infrastructure.\nCommon characteristics of infrastructure are high barriers-to-entry, high capital costs, essential services, inelastic demand, diversified end-user base and long-term customer contracts.\nThere are a small but growing number of federal programs in the US that help certain infrastructure projects address credit needs.\nThe Transportation Infrastructure Finance and Innovation Act (TIFIA) offers letter of credit to surface infrastructure projects. TIFIA has an enviable track record. It is estimated that the federal government will get back 99.9% of its money from the $21.8 billion of current letters of credit it has issued, and P3s have been the main recipient of TIFIA financing. There are suggestions to expand TIFIA given its strong performance and to consolidate it into similar federal programs for the water and railroad sectors (WIFIA and RRIF, respectively). Transportation P3s would benefit from TIFIA expansion.\nPrivate Activity Bonds (PABs) are a federally supported program that can be a catalyst for P3s. PABs are issued by a public entity (e.g., a state or local government) on behalf of a private partner and they are aimed at projects that serve some public purpose, such as airport improvements, water facility upgrades and toll roads.\nInterest is tax-exempt for many PABs thus lowering the cost of debt. As a result, PABs are quite popular and many P3s take advantage of PABs as part of their overall finance package. There is a cap, however, on the amount of PABs that can be issued in each state. As of January 2016, there had been 21 P3s that used the PAB model. The Obama administration proposed a modified PAB in 2016 called a Qualified Public Infrastructure Bond (QPIB), which would not be subject to a cap and be available for a broader range of project types. Congress has recently made proposals for other types of innovative, federally sponsored financing programs such as Move America Bonds (MABs), an expanded version of PABs.\nInfrastructure is owned by the private sector or by the public sector (i.e., federal, state or local governments). The private sector tends to own demand-based infrastructure in which it can earn a required rate of return. This often includes economic infrastructure or projects that earn revenues from usage and volume. The public sector owns infrastructure that does not necessarily earn an explicit return but gives society a positive externality, e.g., a school affording someone an education. This often includes social infrastructure. Highways that do not have tolling are normally owned by the state unless they make an availability payment to the sponsor, in which case they are privately operated through a long-term concession agreement. However, the state retains ultimate ownership. The term \"sponsor\" means an infrastructure equity investor.\nBuild America Bonds (BABs) is another successful program for federal financing of infrastructure. Authorized only for the period 2009-2010, BABs utilized the municipal bond market but offered taxable bonds, in which either the issuer received a federal subsidy of 35% of the interest paid to bondholders (Direct Payment BABs) or the bondholder received a 35% federal tax credit (Tax Credit BABs). These subsidies brought the effective cost of debt to the issuer close to that of tax-exempt bonds.\nDirect Payment BABs were very successful and $181 billion were issued in 20 months, which financed one-third of all new state and local government long-term borrowings during this brief period. Unlike tax-exempt municipals, BABs were popular with investors who did not have a federal tax liability, such as foreign investors and US pension funds. There are recommendations to the administration that the BABs program be re-instituted given its past success.\nAsset recycling, a form of privatization or divestiture, is the leasing of existing infrastructure assets to private investors to raise funds for new investment. These assets have user-fee revenue streams, e.g., from toll roads and airports, and appeal to institutional investors that seek infrastructure investments that are both low risk and likely to generate stable returns. The proceeds are then reinvested into projects that do not have user fees and need direct government funding. Asset recycling has been popular in Australia, where the phrase was coined. Between 2013 and 2016, New South Wales leased A$15 billion of infrastructure assets to investors and allocated A$6 billion to new projects.\nThe administration has voiced its desire to emulate the Australian model as part of its $1 trillion infrastructure stimulus plan released on May 23, 2017. The plan included leasing specific assets such as the Power Marketing Administrations' transmission assets to the private sector. The plan also included the creation of an Air Traffic Control Corporation to manage the air traffic control system, and the expansion of the US Department of Veterans Affairs Administration's ability to lease out vacant assets. Other single-name assets that are prime candidates for privatization include TVA and Amtrak.\nPrivatization could play an important role in the US push to invest more in infrastructure. As of 2015, there were $52.3 trillion in fixed assets in the US, as shown in Exhibit 4. Although 78% of these are owned by the private sector, there are still $11.6 trillion owned by the government. This represents a large pool of assets, some of which could be leased or sold to help fund new infrastructure.\nThe idea behind privatization is that the shift from public to private management will produce improvements because the profit-seeking behavior of new, private sector managers will lead to cost cutting and greater attention to user satisfaction. It works best when firms can run assets or services more efficiently than the government or when competition between firms can bring down costs over time. Sometimes it is also easier for private companies to set prices properly. For example, America's airports charge planes to land in proportion to their weight, but if airports were privately owned it is likely that these tolls would be based on runway congestion, which small planes are prone to cause. Transportation economists have argued for years for a more efficient price mechanism to allocate existing runway capacity.\nWhat could the US government earn from asset recycling and privatization?\nGoldman estimates that state and local governments could unlock up to $120 billion of potential infrastructure investment if state-owned toll roads were divested and the proceeds were used for new projects (the Australian recycling model). This valuation excludes capital that could be unlocked by liberalizing tolling restrictions more generally, such as on Interstate highways that do not have tolls.\nOne impediment to this is that the 1956 law that created the Interstate highway system bans tolls on most Interstate highways. These roads are at the end of their useful lives and there is little appetite among politicians to raise gasoline taxes, so lifting the ban now seems appealing. A second impediment is that current laws require that all bonds on an asset to be recycled be refinanced from tax-exempt municipal bonds to taxable bonds, substantially raising the cost of capital for the private entity involved. Asset recycling would be more viable in the US if the government relaxed these regulations and allowed outstanding bonds on recycled assets to remain tax-exempt.\nAnother set of assets at the disposal of US local and state governments is the airport system. Over the past two decades, countries such as Italy, France and Spain have privatized their airports, while the US has almost exclusively maintained publicly owned and operated airports. In 2012, a pilot program was reauthorized for airport privatization but limits the number of private commercial airports to 10 vs. over 500 commercial airports in the US. As of April 2017, there were four airports in the program although several applications are pending. Like toll roads, airports are attractive to infrastructure investors for their consistent revenue generation and strong margins (40% EBITDA margins on average in Europe). Privatized airports in Europe rank significantly higher in customer satisfaction than do US public airports.\nSimilar to airports, ports in the US tend to be owned by the public sector. They represent another opportunity for asset divestitures to the private sector to generate additional capital for infrastructure investment. In addition to unlocking capital for reinvestment, privatization of ports could lead to increased operating efficiency. This is particularly important for US ports which rank in the lower quartile on both portfolio flow and port cost. Based on 2014 capacity, US ports generated an estimated $3.8 billion in sales in 2014, implying $23 billion in valuation using a similar multiple as used in airports.\nHighways, airports and ports alone could bring in an estimated $190 billion for US infrastructure. Other prime candidates for privatization are the air traffic control system, TVA and the four Power Marketing Administrations.\nSee Appendix 1 for more in-depth toll road, airport and port valuations.\nWe believe there is movement in the US infrastructure market. The US P3 market will get a boost over the next few years as barriers to infrastructure investment are lowered, existing financing tools are expanded, new financing tools are developed and current federal financing programs are built upon. There is a push for privatization and asset recycling of the air traffic control system, toll roads and airports and, if it is successful, other sectors would likely follow over the coming years.\nThe White House released details in May 2017 on its plan to incentivize $1 trillion in new investment. It included $200 billion of proposed federal financing with incentives for states and private investment for the remaining $800 billion. The proposal would expand existing credit guarantee programs, such as TIFIA and WIFA, and would lift the cap on PABs and restrictions on P3s. It would encourage privatization and liberalizing tolling restrictions, an important key to unlocking a large amount of capital. The is largely what was announced in the State of the Union address and what has recently been reported in the press.\nThere are two components to modernizing US infrastructure: money and permits. There needs to be a dramatic reduction in red tape so that infrastructure can be approved in two years or less, not 10 years or more as it often takes now. No one deliberately designed America's infrastructure approval system. It is an accident of legal accretion occurring over 50 years. This may be about to change. In an August 2017 press conference, the administration announced its intent to roll back the regulatory and permitting process. It included a two-year limit for federal environmental reviews in order to allow major projects to move forward. The intention is to follow the Canadian model, a move to reduce political risk that would be welcomed by investors. The US has a higher rate of project cancellations than other countries, which puts off investors, so this reform is an important component of attracting capital. It can be accomplished by consolidating decisions within a simplified framework with set deadlines and clear lines of accountability. Germany's approval process to air issues and get to a final decision is simple and effective. Canada recently overhauled its rules, by clearing a path to a two-year process for certain federal projects and by delegating other projects to the provinces. Australia is also working on an overhaul.\nProject finance is the method of financing infrastructure, industrial or public assets using limited recourse long-term debt, typically raised by a special purpose vehicle (SPV) created solely to complete the infrastructure project. Principal and interest payments are funded entirely from cash flows generated by the SPV, while the scope and duration of the project is defined in the SPV's contractual arrangements. Project finance debt can be an efficient way to fund long-term capital-intensive projects in which the underlying revenues are relatively stable and predictable. Project finance has also come to mean a sector in which issuers use this type of finance. In this regard, infrastructure is a subset of the larger project finance sector.\nThere is a great need for investment and a very large amount of capital within and outside of the US looking for suitable projects. Although the federal government plays a very important role, this opportunity is not completely dependent on the politics of Washington. State and local governments account for 75% of infrastructure spending and own 87% of public infrastructure (Exhibit 4). They own 98% of highways, including the entire US Interstate system. They own bridges, seaports, water and sewer systems, transit systems, schools, and police and fire stations. The federal government dominates in ownership of intellectual property, R&D assets and conservation, such as parks and recreation. It also owns the air traffic control system, most of Amtrak, postal buildings, large dams and other single name assets.\nThe federal government influences infrastructure at the state, local and private level \u2013 where most investment occurs \u2013 through incentives, taxation and regulations, not only by direct federal spending. Infrastructure in the US is much more decentralized than in many countries because the US is a large federation, similar to Canada and Australia, both of which have strong models of decentralized infrastructure coordination. The current politics of Washington will not derail the long-term trend of increased state, local and private capital investment in US infrastructure.\n$200 million in federal spending mostly through grants, and plans to incentivize state, local and private spending of $1.3 billion for a total of $1.5 billion over 10 years.\nEncourages PPP. Covers most main economic and social infrastructure sectors.\nMakes securing some funding means tested through a cost-benefit analysis to avoid boondoggle projects. This has been widely suggested.\nStrong emphasis on rural investment. Up to 25% of total federal spending expected.\nAdds to existing federal financing programs (TIFIA, RRIF, WIFIA, etc.).\nCreation of a new fund for public lands infrastructure. This will be funded by revenues due to the US government from mineral and energy development on federal lands and waters.\nExecutive order allowing for disposal of federal assets to improve overall allocation of economic resources in infrastructure investment. This could be a precursor to asset recycling and\/or privatization through asset disposals or the leasing of publicly-owned assets to the private sector. This could release a significant amount of capital.\nElimination of some of the important caps on PABs (e.g., volume by state and maximum amount allowed in transportation). Extends PABs eligibility to other sectors, e.g., ports and airports. This mirrors the Obama administration's 2016 proposal for a modified PAB called a Qualified Public Infrastructure Bond (QPIB).\nLiberalize tolling restrictions on Interstate highways and allow for commercialization of roadside rest areas. This could also release a significant amount of capital.\nApply fast act streamlining on all rail projects and subsidize short-line and passenger rail. There has been little emphasis on additional intercity passenger rail capacity in the US for years. Amtrak is used almost exclusively for this service.\nExpand federal credit assistance to state infrastructure banks.\nEliminate constraints on public transit PPP. Allow non-federal construction and operation of inland waterways. Currently dominated by the US Army Corps of Engineers.\nThe administration's plans have not come to fruition so far in 2018 but it is possible a plan will come together after the mid-term elections. Meanwhile, the private sector continues to invest, with non-bank investors increasingly taking up the slack.\nAn innovative proposal by Senator Delaney (D-Maryland) is to start an American Infrastructure Fund (AIF) that issues 50-year 1% coupon bonds through a reverse Dutch auction. Companies would be incentivised to repatriate by competing for a low tax rate. This would be achieved through a competitive bidding process where the 'price' would be a ratio of bond investment-to-repatriated profits. As companies bid the ratio decreases, i.e., the effective tax rate increases, until the target size is raised. The funds are then used as infrastructure investment. The early price talk, should AIF become a reality, is an offer of around 1-to-4 but some economists expect companies will bid in the 1-to-7 area.\nAnother plausible way repatriated profits could be put to work in infrastructure is through the US tax-equity market, which provides capital for a broad array of renewable projects in the US. This amounted to $13 billion in wind alone in 2016.\nCombining infrastructure with the Tax Cuts and Jobs Act of 2017 was expected by some analysts because historically Democrats have supported infrastructure spending and Republicans have supported tax cuts. Since it was omitted from tax reform, it seems likely to slip to 2019. At that point, it could be considered as a standalone issue, perhaps with the goal of attracting bipartisan support. It is possible that some infrastructure-focused initiatives could be addressed in the next 6-12 months, while leaving a broader infrastructure package until after the mid-term elections in November 2018.\nSee Appendix 2 on infrastructure elasticity for an explanation of how infrastructure quality can rank higher than its condition would suggest.\nToll road valuation $120 billion: Highway toll revenue generated $14.4 billion in 2014 from publicly owned toll roads according to the Federal Highway Administration. EBITDA margins are estimated by Goldman Sachs to be 65% based on a sample of the 12 largest toll roads. Precedent transactions in Europe and Australia have been at a 24 EV\/EBITDA multiple. This values existing US toll roads at $120 billion after estimated debt of $110 billion (5.7x sales). Seventy-five percent of toll revenues come from only six states. In order, these are New York, New Jersey, Texas, Florida, California and Illinois. They would stand to benefit the most from an asset recycling program.\nAirport valuation $50 billion: According to the FAA, the Airports and Airways Trust Fund (AATF) earned $14.4 billion in revenues in 2016. EBITDA margins at the 25 largest airports are 40% and are comparable to privatized airports outside of the US (e.g., best-in-class privatized airports in Europe have EBITDA margins of 55%). Implied EV\/EBITDA multiples are 20x based on historical IRRs. This gives a valuation of $50 billion after estimated debt of $65 billion.\nPort valuation $23 billion: Based on 2014 capacity, it is estimated that ports generated $3.8 billion in sales in 2014. Using a slightly higher multiple than airports (i.e., 21x), gives a valuation of $23 billion after estimated debt of $13 billion. Ports get a small premium on the multiple because their EBITDA margins are estimated at 45% vs. airports at 40%.\nAppendix 2. Good quality and average satisfaction, so why so much required investment?\nThere is an inconsistency when analyzing the quality of US infrastructure compared to the need for so much investment. The World Economic Forum (WEF) ranks US infrastructure eleventh globally out of 138 countries, and Americans have not expressed a high level of dissatisfaction with their infrastructure: 69% are satisfied with US highways and roads and 61% are satisfied with public transport. This is about the median of other countries. There are two explanations for this inconsistency: greater O&M spending vs. capital spending and high infrastructure elasticity.\nWe show the amount of required US investment in our second piece \"Outlook for United States infrastructure: US infrastructure market analysis,\" Autumn 2018.\nIt is widely reported in the press that US infrastructure spending is insufficient. Headline public spending has dropped 9% in real terms since 2003. When dissecting that number, however, it becomes clear that US capital spending has dropped 23% but operation and maintenance (O&M) spending has actually increased 6% (Exhibit 5). In fact, O&M spending has grown steadily at about 5% per annum since 1956.\nO&M spending is key to overall longevity and efficiency and has helped to maintain existing infrastructure performance. This explains why existing infrastructure gets good quality scores by Americans surveyed by the WEF despite large infrastructure needs estimated by the American Society of Civil Engineers (ASCE). We benchmarked 2014 numbers vs. 2003 in Exhibit 5 because real infrastructure spending dropped post 2003 due to a rapid increase in infrastructure-related input costs.\nInfrastructure is considered to be highly elastic between the extremes of condition (Exhibit 6). It is similar in concept to the price elasticity of demand (Ped), where Ped is > 1.0, except here it is the performance elasticity of condition (Pec), where Pec is > 1.0 and changes in condition (the independent variable) lead to changes in performance (the dependant variable). This high elasticity helps to explain the relatively good US quality scores. In this context condition can be thought of as age and performance as perceived quality.\nBy most accounts, existing infrastructure condition is below average but stable O&M spending has maintained it. This would put its condition in the middle-right part of the horizontal x-axis in Exhibit 6. This high elasticity also helps explain why its age has been able to increase over the years without a corresponding drop in quality. It also shows that infrastructure condition can only worsen to a certain degree until quality drops off rapidly.\nEven though infrastructure performance is considered elastic, the price elasticity of demand for the use of infrastructure (an entirely different matter) is typically inelastic. This is because often there are not substitutes, many infrastructure services are considered essential, and the user fees are often a small portion of disposable income and not discretionary (e.g., tolls on roads and bridges, user electricity bills and user fees on mass transit).\nRegular O&M spending can maintain infrastructure similar to other assets. On aircraft, for example, proper maintenance can significantly extend an airplane's service life but it gets more expensive with age, flying time and the number of take-offs and landings (or cycles). The serviceable lifespan of the 747 was thought to be 40 years when PanAm took to the skies with its first Jumbo Jet flight in 1970 and many older 747s are still in service today, especially in the military and firefighting units. Industry experts now estimate the 747 airframe can fly over 60 years because of today's improved inspection and maintenance programs.\nIn many cases the decision to retire an airframe is based on economics rather than age. America's infrastructure is similar: At some point it cannot be repaired anymore and just needs to be replaced.\n(1) Public construction spending as a percentage of total tax receipts. Tax receipts equal to total property, sales, income and corporate taxes from the US Census Bureau's quarterly summary.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Select Erase this disc and when prompted, click Next to erase the files. However, if an Erase this disc is not available, the disc cannot be re-written. But newer drives tend to be able to read the various formats. A common example of such a material is ordinary window glass, an amorphous form of silica.\nThe material used is chosen because the two solid states reflect light differently. CD-R compact discs are discs that can be recorded on only once.\nDestructive interference of the laser beam caused by the difference in path length between the bottom of the pit and the surrounding \"land\" causes the pits to appear darker than the background. If you only have one drive: As I said this whole question is out of curiosity, if I wanted to destroy a CD I'd hold it over a candle and admire the pretty colors it melts into.\nClears the disc registry and allows new files to overwrite the existing files. Switching the laser to a higher power, one sufficient to melt the material, enables the creation of a new mark. Type Control Panel in the search box and click Control Panel.\nInsert the disc in the optical disc drive. NightwishFan February 15th, Disc with this format may not be usable in a device with a different type of drive.\nDo you even know what Acronym means? Look for disks labeled CD-RW. Not Usable - Random dots add into the background and the lyric letters, worse on tracks 7 and higher. Go to your burning software and click clear disc Share to: A format that allows data to be written to the disc only once.\nOnce an area of the disc has been written to, it cannot be erased. Unrelated to the drive, the CDR media you use also determines if the copied karaoke discs will play. This is accomplished by heating the material to a lower temperature, which transforms it back to its crystalline state.\nHere the disk should be a rewritable CD or DVD disc which allows you to format and rewrite data multiple times. If there is data on the CD or DVD, you should first erase the data already on the disc and then format it for reuse.\nExisting data can be overwritten by turning the laser on continuously to the erase power, which will erase any existing marks. The data on a re-writable disc can be erased using either the standard Windows Explorer or most data burning software applications.\nDownload Now Windows 10 is supported Step 1. Thus, \"R\" format discs are suited to applications such as non-volatile data storage, audio, or video. Currently available in standard 4. In addition to a layer of metal, the media includes a thin layer of dye.\nDownload Now Windows 10 is supported Step 1. To make a recordable disc, you start with a plastic substrate that has blank grooves rather than a predefined pattern of pits. Click Start, type explorer in the search field, and select Windows Explorer from the list.\nCommercial burners such as Toast See links below have a simpler means of performing multiple burns. Here we will show you how to do it step by step.\nTo avoid this you would have to burn the negative of whatever the disk currently contains. A mini window pops out, on which you can edit the partition label and choose a file system. Phase change technology enables this rewriteability. Standard - The copy appears like the original with random dots in the background.\nThese marks, too, appear dark compared with the background. Use these steps to erase a rewritable disc: Repeat this procedure to add more stuff to the disc.Nov 23, \u00b7 \"I'm getting around the problem at the moment by erasing the CD each time I wish to place the updated file on disc, but that's becoming a bore.\" That is the only way you can rewrite to a CD-RW.\nJan 13, \u00b7 You format the disk for packet data so you can drag and drop data on and off when you need. It sounds like you are trying to do the latter, i.e. use your CD-RW as a disk to copy files from your computer by dragging and dropping them onto the calgaryrefugeehealth.com: Resolved.\nEven though general purpose CD-R and CD-RW discs and their consumer audio versions appear for all practical purposes identical, only blank media bearing the \"Compact Disc Digital Audio Recordable\" (CD-DA Recordable) and \"Compact Disc Digital Audio Rewritable\" (CD-DA Rewritable) logos can be written in consumer audio recorders.\nPlug and play The Dell External USB Slim DVD +\/\u2013 RW Optical Drive is a plug and play disc burning and disc playing solution that you can use with a USB port.\nPractice Questions Optical Media. STUDY. PLAY. You have a CD-RW drive that advertises speeds of 32x\/12x\/48x. What is the read speed of the drive? 48x the first number is the record speed, the second number is the rewrite speed, and the last number is the read speed.\nIf the drive does not perform rewrite functions, the middle number. No it is not possible to rewrite a DVD-r or DVD+r disc. The type of DVD that can do that is a DVD-rw or a DVD+rw.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Friends of the National Arboretum Canberra Inc. invite you to make a donation to our Fund to support the development of the National Arboretum Canberra. Donations over $2 are tax deductible.\nFriends of the National Arboretum Canberra Inc (ABN 33 630 033 978)is a separate incorporated association and is distinct from the Arboretum's Trust Account which largely focuses on Foundation Members. \"The Trustee for the Friends of the National Arboretum Trust Fund\" (the TFATF) is endorsed as a Deductible Gift Recipient (DGR) from 22 March 2011 and is a public ancillary fund covered by Item 2 of the table in section 30-15 of the Income Tax Assessment Act 1997. The Trust Fund is registered as a charity with the Australian Charities and Not-for-Profit Commission.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Faculty of Law students are invited to participate in the 2018 Dentons Canada LLP Negotiation Competition. In teams of two, students will compete to obtain the best possible results for their client through negotiation.\nTwo-person teams compete against each other in an attempt to obtain, through negotiation with the other team, the best possible result for their \"client\", as judged by competition judges. The judging criteria include not only the outcome achieved, but also a team's planning, flexibility, teamwork, ability to deal with ethical issues, and a self-analysis of the team's performance conducted at the end of the negotiation, outside of the presence of the opposing team.\nAs the competitions are unscripted, it is fast-paced and exciting to watch. This is largely a function of the fact that the participants themselves determine the format. There is no set period for either side or team member to talk (such as in a traditional moot competition); rather, the parties merely sit down (if they wish) and try to hash out a deal in the manner they see best. The students may attempt to be conciliatory, attempt a \"win-win\" outcome, take a \"hard-nosed\" position, refuse to discuss matters except on their terms, or storm out of the room. Further, the students may caucus, take notes on an easel, or do anything else to attempt to achieve or thwart a settlement. Just as in real negotiations that any lawyer or business person might be involved in, all of these tactics may, and are, used with varying degrees of success.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Renewed and unforeseen ergonomy, a pure and elegant design combined with a distinctive back that not only gives it an unique character and typology, but also offers additional comfort and a generous sense of shelter. Made in Europe.\nBlack or white steel base.\nPowder-coated black steel base with pads suitable for any type of flooring. Made in Europe.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome to the Scottish VMUG Community.\nWe are a group covering all of Scotland and are passionate about virtualisation. We run multiple events a year, usually made up of two large multi-tracked conferences and two social vBeer events. You can find specific details on upcoming events below.\nWe would particularly encourage you to join us in our active Slack team. It is a great way to both learn and share information and also to join in with our Scottish VMUG community.\nRegistration for the 2019 Glasgow and Edinburgh VMUG events are open, please book your place!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Democratic People's Republic of Korea, an actual hell-hole (that one day we'll have to explain to our children in the manner that people alive in 1940s had to explain why they didn't bomb Nazi concentration camps), where anyone who isn't a member of the ruling party is either starving to death or being tortured in a concentration camp, appeared to be taking tentative steps towards making life there slightly less awful.\nNorth Korea, where situations are so grim that even the CCP doesn't repatriate asylum seekers there, was expected to announce that farmers would be allowed to keep surplus food to sell or barter, rather than turn it over to the state.\nThe proposed directive appears aimed at boosting productivity at collective farms that have struggled for decades to provide for the country's 24 million people. By giving farmers such an incentive to grow more food, North Korea could be starting down the same path as China when it first began experimenting with a market-based economy.\nHowever, the parliamentary session, a rarity in itself, where it was assumed this radical policy would be debuted, has closed with no major reforms announced.\nThe North had fanned expectations of a big announcement by holding a meeting of its rubber-stamp parliament, the Supreme People's Assembly, which rarely convenes. But a state news agency report released after the meeting Tuesday said only that the assembly had authorized a few low-level personnel changes and extended the country's compulsory education period from 11 years to 12.\nSo for the time being, it looks like desperate North Koreans will have to go back to smuggling crystal meth into China while Kim Jong-Un and his hot wife lie in their luxurious bed, oblivious to the misery and suffering caused by their continued existence.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There were queues out the gates for the launch night of a new Cambridge biergarten yesterday (April 28).\nCrowds of eager residents flocked to the opening of Thirsty Riverside in the grounds of the city's Museum of Technology.\nRun by the team behind Chesterton Road's popular Thirsty bar, the new venue will provide open air food, drink and entertainment throughout the summer.\nSam Owens from Thirsty told the News: \"It was bonkers. By 6pm we had a queue of people waiting on the street. It was utterly mental. We never expected to get anywhere near what we had. But people were saying don't worry it was such a nice atmosphere we don't even mind queuing.\"\nOn Twitter the company thanked people for their patience on the launch night and insisted it will \"sort out\" the lengthy queues.\nThirsty's partnership with the Cambridge Museum of Technology will see a broad programme of events throughout the summer season.\nThis includes nightly outdoor food trucks, coffee and ice cream for sale, visiting DJs and plans for farmers' markets and private dining events.\nSam added: \"We want to make it a family friendly thing. You go to a proper biergarten in Germany and it's a place for families to come in the day time.\"\nThirsty Riverside has space to seat 200 customers outside the museum. Festoon lighting decorates the trees and railings, with umbrellas, tarpaulin and heaters keeping guests protected.\nFood will be provide by burger specialists Steak + Honour today (April 29), with Jack's Gelato, Hugh's Little Dutch Pancakes and Beanissimo coffee also on site.\nOn Sunday Provenance Kitchen and Proper Pizza Company from Norwich will be serving up food, while on Bank Holiday Monday, Honest Burgers and the Rural Coffee Project will be on the riverside.\nThirsty Riverside is open 11am-11pm on Saturdays and 11am-10pm on Sundays and the bank holiday.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Compliments and evaluations from our program are still coming in and every attendee has something good to say about you. We are hearing compliments such as, \"Very motivational and set a good tone for the conference.\"- \"Absolutely loved Dave Davlin!\", and when asked to rate your program 1 out of 5 (5 being the best) we are seeing 10's!!!\nOn behalf the employees of Crete-Monee School District 201U, Thank You! Your presentation was highlight of our Opening Day Event for the 2015-2016 school year. This year's session was voted the best since its inception. A large part of the success of Opening Day can be attributed to your message.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Rent BMW cars in Miami, Florida, U.S.\nCars BMW - it's quality and reliability, proven by many years of existence of the brand.\nIn the entire history of BMW quality of the cars do not affect non-financial and economic factors, the company strictly adheres to its image and products of consistently high quality.\nCompany Miami Car Rental offers rental in Miami, the German BMW cars in Florida, USA at the best prices.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"1438 heauto\u00fa (reflexive pronoun of the 3rd person) \u2013 1438 \/heauto\u00fa (\"himself, herself, itself,\" etc.) is the 3rd person reflexive (singular, plural) form which also functions as the reflexive for 1st and 2nd person (A-S).\naside* (1), conscious* (1), each other (2), herself (4), himself (91), itself (12), mind (1), none* (1), one another (13), oneself (1), ourselves (19), own (48), own estimation (2), own initiative (3), own persons (1), own...himself (1), senses (1), themselves (44), venture* (1), within* (1), yourselves (32).\n1. of the 3rd person singular and plural, to denote that the agent and the person acted on are the same; as, \u03c3\u1ff4\u03b6\u03b5\u03b9\u03bd \u1f11\u03b1\u03c5\u03c4\u03cc\u03bd, Matthew 27:42; Mark 15:31; Luke 23:35; \u1f51\u03c8\u03bf\u1fe6\u03bd \u1f11\u03b1\u03c5\u03c4\u03cc\u03bd, Matthew 23:12, etc. \u1f11\u03b1\u03c5\u03c4\u1ff7, \u1f11\u03b1\u03c5\u03c4\u03cc\u03bd are also often added to middle verbs: \u03b4\u03b9\u03b5\u03bc\u03b5\u03c1\u03af\u03c3\u03b1\u03bd\u03c4\u03bf \u1f11\u03b1\u03c5\u03c4\u03bf\u1fd6\u03c2, John 19:24 (Xenophon, mem. 1, 6, 13 \u03c0\u03bf\u03b9\u03b5\u1fd6\u03c3\u03b8\u03b1\u03b9 \u1f11\u03b1\u03c5\u03c4\u1ff7 \u03c6\u03af\u03bb\u03bf\u03bd); cf. Winers Grammar, \u00a7 38, 6; (Buttmann, \u00a7 135., 6). Of the phrases into which this pronoun enters we notice the following: \u1f00\u03c6' \u1f11\u03b1\u03c5\u03c4\u03bf\u1fe6, see \u1f00\u03c0\u03cc, II. 2 d. aa.; \u03b4\u1f30 \u1f11\u03b1\u03c5\u03c4\u03bf\u1fe6, of itself, i. e. in its own nature, Romans 14:14 (Tr L text read \u03b1\u1f50\u03c4\u03bf\u1fe6); \u1f10\u03bd \u1f11\u03b1\u03c5\u03c4\u1ff7, see in \u03b4\u03b9\u03b1\u03bb\u03bf\u03b3\u03af\u03b6\u03b5\u03c3\u03b8\u03b1\u03b9, \u03bb\u03ad\u03b3\u03b5\u03b9\u03bd, \u03b5\u1f30\u03c0\u03b5\u1fd6\u03bd. \u0391\u1f30\u03c2 \u1f11\u03b1\u03c5\u03c4\u03cc\u03bd \u1f14\u03c1\u03c7\u03b5\u03c3\u03b8\u03b1\u03b9, to come to oneself, to a better mind, Luke 15:17 (Diodorus 13, 95). \u039a\u03b1\u03b8' \u1f11\u03b1\u03c5\u03c4\u03cc\u03bd, by oneself, alone: Acts 28:16; James 2:17. \u03a0\u03b1\u1fe5 \u1f11\u03b1\u03c5\u03c4\u1ff7, by him, i. e. at his home, 1 Corinthians 16:2 (Xenophon, mem. 3, 13, 3). \u03a0\u03c1\u03cc\u03c2 \u1f11\u03b1\u03c5\u03c4\u03cc\u03bd, to himself i. e. to his home, Luke 24:12 (R G; T omits, WH (but with \u03b1\u1f51\u03c4\u03cc\u03bd) reject, L Tr (but the latter with \u03b1\u1f50\u03c4\u03cc\u03bd) brackets, the verse); John 20:10 (T Tr \u03b1\u1f50\u03c4\u03cc\u03bd (see \u03b1\u1f51\u03c4\u03bf\u1fe6)); with (cf. our to) himself, i. e. in his own mind, \u03c0\u03c1\u03bf\u03c3\u03b5\u03cd\u03c7\u03b5\u03c3\u03b8\u03b1\u03b9, Luke 18:11 (Tdf. omits) (2 Macc. 11:13); in the genitive, joined with a noun, it has the force of a possessive pronoun, as \u03c4\u03bf\u03cd\u03c2 \u1f11\u03b1\u03c5\u03c4\u1ff6\u03bd \u03bd\u03b5\u03ba\u03c1\u03bf\u03cd\u03c2: Matthew 8:22; Luke 9:60.\n2. It serves as reflexive also to the 1st and 2nd person, as often in classic Greek, when no ambiguity is thereby occasioned; thus, \u1f10\u03bd \u1f11\u03b1\u03c5\u03c4\u03bf\u1fd6\u03c2 equivalent to \u1f10\u03bd \u1f21\u03bc\u1fd6\u03bd \u03b1\u1f50\u03c4\u03bf\u1fd6\u03c2, Romans 8:23; \u1f11\u03b1\u03c5\u03c4\u03bf\u03cd\u03c2 equivalent to \u1f21\u03bc\u1fb6\u03c2 \u03b1\u1f50\u03c4\u03bf\u03cd\u03c2, 1 Corinthians 11:31; \u1f00\u03c6' \u1f11\u03b1\u03c5\u03c4\u03bf\u1fe6 equivalent to \u1f00\u03c0\u03cc \u03c3\u03b5\u03b1\u03c5\u03c4\u03bf\u1fe6 (read by L Tr WH), John 18:34; \u1f11\u03b1\u03c5\u03c4\u03cc\u03bd equivalent to \u03c3\u03b5\u03b1\u03c5\u03c4\u03cc\u03bd (read by L T Tr WH), Romans 13:9; \u1f11\u03b1\u03c5\u03c4\u03bf\u1fd6\u03c2 for \u1f51\u03bc\u1fd6\u03bd \u03b1\u1f50\u03c4\u03bf\u1fd6\u03c2, Matthew 23:31, etc.; cf. Matthiae, \u00a7 489 II.; Winers Grammar, \u00a7 22, 5; (Buttmann, \u00a7 127, 15).\n3. It is used frequently in the plural for the reciprocal pronoun \u1f00\u03bb\u03bb\u03ae\u03bb\u03c9\u03bd, \u1f00\u03bb\u03bb\u03ae\u03bb\u03bf\u03b9\u03c2, \u1f00\u03bb\u03bb\u03ae\u03bb\u03bf\u03c5\u03c2, reciprocally, mutually, one another: Matthew 16:7; Matthew 21:38; Mark 10:26 (Tr marginal reading WH \u03b1\u1f50\u03c4\u03cc\u03bd); ; Luke 20:5; Ephesians 4:32; Colossians 3:13, 16; 1 Peter 4:8, 10; see Matthine \u00a7 489 III.; K\u00fchner, ii., p. 497f; Bernhardy (1829), p. 273; (Lightfoot on Colossians 3:13).\nFrom a reflexive pronoun otherwise obsolete and the genitive case (dative case or accusative case) of autos; him- (her-, it-, them-, also (in conjunction with the personal pronoun of the other persons) my-, thy-, our-, your-) self (selves), etc. -- alone, her (own, -self), (he) himself, his (own), itself, one (to) another, our (thine) own(-selves), + that she had, their (own, own selves), (of) them(-selves), they, thyself, you, your (own, own conceits, own selves, -selves).\nNAS: the dead to bury their own dead.\nKJV: the dead bury their dead.\nNAS: divided against itself will not stand.\nNAS: and buy food for themselves.\nKJV: and buy themselves victuals.\nNAS: humbles himself shall be exalted.\nKJV: that shall humble himself shall be exalted.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How Much Does it Cost to Maintain a Lawn?\nThe size and condition of your yard and how often you want service calls will also influence the cost. Usually there is a minimum charge regardless of square footage.\nHire a landscaper to give you that great-looking lawn you dream about. Lawn care service is a convenience or even a necessity if you are too busy, don't want to invest in the equipment, or are away from the property for long periods. Schedule regular professional lawn care to increase curb appeal when you are in the process of selling your home, as well.\nEquipment -- Equipment needed will include a commercial lawn mower and fuel, lawn trimmer and\/or edger, aerator, garden tools such as rakes and shovels, and safety equipment including goggles and earplugs. This is normally owned by the lawn care service.\nSupplies -- Fertilizers, insecticides, weed killers, fungicides, and bags for grass clippings and leaves are supplies that your lawn care service may use. You may request organic, non-toxic varieties, but these are likely to come at a premium cost.\nLabor -- Lawn care is a labor-intensive project. The landscaper will cut your grass and neaten the edges, apply treatments, and aerate or dethatch as needed.\nCleanup -- Cleanup can include removing weeds, as well as grass cuttings or dead leaves. These items may be trashed or turned into mulch or compost.\nIt is essential to have the landscaper visually inspect the lawn before quoting an annual fee.\nPay an annual lawn care fee in advance for convenience and maybe a discount too. Talk to your lawn care pro about negotiating a lower charge if you refer new customers in your neighborhood.\nSave time, money, and water by having your landscaper install Xeriscaping, replacing part or all of your lawn with attractive low-care, low-water plants.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Before I was an author of a fiction book, I was a self-published author of a devotional book entitled Doses of Devotion. Now, my book has been nominated for a DIVA Award for best inspirational book by Book Divas. This book was some devotionals that I wrote for my blog Kisha's Daily Devotional that manifested itself into a book and it is available on Amazon for $0.99.\nPlease take a minute and vote for Doses of Devotion and then share it and ask everyone you know to vote as well.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Switzerland has been warned that its stock exchanges will be cut off from EU traders unless Bern approves a new framework for a future relationship. After 10 years of negotiating, patience appears to be wearing thin in Brussels.\nOn December 7, the seven-member Federal Council \u2013 which acts as the Swiss Government \u2013 must provide a clear answer on the proposal to avoid equivalence being withdrawn. If the framework is approved, the European Commission will offer a two-year extension allowing Swiss stock exchanges to operate within the bloc.\n\"It's deal time,\" an EU official told Reuters. \"Playing for time wouldn't change anything. The draft will not be changed anymore.\" Having skirted the issue last week, the Swiss Government enacted contingency measures to safeguard Swiss stock exchanges. According to the Financial Times, the SIX Swiss Exchange has stressed achieving equivalence is its \"highest priority\", as it seeks to secure a certain and transparent future for investors.\nThe Alpine nation, which signed a free-trade agreement with the then European Economic Community in 1972, rejected the chance to join the European Economic Area in 1992. Since then, ties between Switzerland and the EU have been held together by 120 bilateral agreements that allow the Swiss partial access to the European Single Market.\nThe ongoing spat has united Switzerland's traditionally pro-Europe left and anti-EU far-right, who agree that the deal infringes on Swiss sovereignty, leaving the governing four-party coalition short of a majority. The deal has proved a contentious issue; it would allow the European Court of Justice to play a role in conflict resolution and would align Swiss rules with EU law.\nEarlier in the year, Swiss voters rejected a proposal tabled by the right-wing populist Swiss People's Party to give domestic laws precedence over international legislation.\nThe prolonged Brexit negotiations have also complicated talks: the European Commission is determined to secure a deal that would send a message that the 'Swiss model' is not an option for third-country relationships.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Gastown has a new Pop-Up Park on the Carrall Street side of my store this weekend. It opened at noon Friday and continues Saturday, with Carrall Street closed between Water and Cordova from noon to 9 p.m. each Friday & Saturday until the end of August.\nThe Irish Heather gastro pub and Blarney Stone pub\/restaurant have added extra outdoor seating on the street, as have the Cork & Fin and Notturno. And there are picnic tables, benches and sun umbrellas for people to enjoy, as well as games of chess and checkers. You can also bring your own board games and have a pop-up picnic in this urban park, located in Vancouver's oldest neighbourhood. The street will also be closed Aug. 30 & 31 for the Gastown BBQ and Chili Festival, which has 12 local restaurants vying to win the title of best in four categories: chicken wings, pork ribs, beef brisket and slow-roasted pig.\nAngel Vancouver is continuing our summer clearance sale, with 30% off kids clothing from Desigual, the fun Barcelona-based fashion line, and up to 35% off select Desigual items. We also have a cool yellow Desigual coat designed by Christian Lacroix on sale for 20% off, and are offering 30% off Desigual T-shirts, shorts and pants for men.\nAngel Vancouver has the largest selection of Desigual for men, women and kids. We ship worldwide.\nI paint the shirts right in the store \u2014 you won't find a more unique, locally made item than that.\nLast night, I went to see the fabulous new film Boyhood by Rick Linklater, whose previous films include Slacker (his first film to catch my attention), Dazed and Confused, Waking Life and the trilogy Before Sunrise, Before Sunset and Before Midnight, which were made over a 17-year period. His new film was made over a 12-year period with the same cast, showing the evolution of a boy from age six to 18, when he goes off to college.\nThis entry was posted in Christian Lacroix for Desigual, Desigual, Desigual for kids and tagged Angel Handpainted clothing, Angel Vancouver, Christian Lacroix for Desigual, Desigual, Desigual for kids, Desigual for kids in Vancouver Canada, Desigual for men, Desigual Vancouver Canada, Gastown, La Vida es Chula (Life is Cool!). Bookmark the permalink.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Every night, at a 100-foot-wide crescent crack in the earth in south-central Texas, a phenomenon occurs that's as dependable as the eruptions of Old Faithful. Tens of millions of Mexican free-tailed bats, seasonal residents of Bracken Cave, stream out in a nightly ballet, swirling far and wide over Texas hill country. Over four hours, these hungry hunters emerge in search of insects to feed themselves and the millions of baby bats they birth in the warmth and safety of the cave.\nTo be sure, the spectacle of the bats leaving the cave is astonishing all on its own. But the bats' presence in Bracken also means that the cave itself is peculiar, a scientific, historic and ecological treasure whose mysteries are still being unraveled.\nRead on to find out how there's more to Bracken than just the bats that roost there.\n1. It's the largest known bat maternity colony in the world.\nBracken Cave hosts the largest known bat maternity colony in the world, playing nursery to millions of Mexican free-tailed bat babies each year. At a toasty 102 to 104 degrees Fahrenheit, the cave is a perfect incubator.\nIn March and April, pregnant females return to Bracken after overwintering in Mexico. In June, females pack the cave walls with naked, newborn pups\u2014as many as 500 per square foot.\nAll those hungry bat moms put a big dent in the bug population, too, collectively chowing down on 140 tons of flying beetles, winged ants and moths every night. One recent study also found that the Bracken bats save the region's cotton growers $741,000 every year in pesticide and crop damage costs.\n2. The bats are food, too.\nThe bats aren't the only animals feasting in the vicinity of the cave every night. From March to October, as the bats prepare to exit Bracken's mouth, they nervously circle around inside before emerging.\nAnd for good reason: outside, a host of predators including owls, Cooper's and red-tailed hawks, raccoons and ring-tailed cats await the chance to nab the bats as they make their evening dash. Western and Eastern coachwhip snakes gather in the cave's sinkhole, ready to pounce on any unlucky crash-landers.\n3. Lots of bats = lots of poop.\nAs the story goes: everybody poops. And in Bracken Cave, with more than 15 million Mexican free-tailed bats living there, you know that's got to be a lot!\nEach year, bats deposit an estimated 50 tons of guano on the floor of the cave. Historic accounts from guano miners suggested they'd dug down 30 to 60 feet in places without hitting bottom. But its full extent is still unknown.\nExtracting a \"core\" of guano with a hollow drill could answer this question. Scientists could use the core for radiocarbon dating to establish the age of the oldest layers, which could reach back centuries or millennia. It could also reveal what bats have been eating over time, and whether other species of bats have ever lived in Bracken.\nBut getting to the bottom of the deepest parts has proven to be tricky. A method to estimate guano thickness with electrical pulses can see that some areas of the cave are at least 59 feet deep, but it may extend another 33 feet. With such a depth, finding a drill long enough that can be brought into the cave is a problem that BCI is still trying to solve.\n4. The cave floor is a dangerous living organism.\nIf you were to enter Bracken Cave (wearing protective and breathing gear!), the floor would appear to be alive and moving.\n\"Guano is a food source for a lot of cave critters, including crickets, fleas, mites, and many other invertebrates,\" says Fran Hutchins, BCI's Director of Bracken Cave. In Bracken alone, there are six different species of flesh-eating, or dermastid, beetles; all of these organisms are important for keeping the cave from filling completely with guano.\nThe guano beetle, also called the lesser mealworm, is the major processor of the guano. Its larvae look similar to mealworms for sale in the pet store. In the summer, when guano increases dramatically, so does the beetle population, causing the floor of the cave to seethe with movement.\nThe flesh-eating beetles, as their name suggests, do not eat the guano, but debris and unlucky young bats. With millions of bats living in the cave, some are bound to fall to the floor and die; they are consumed in minutes.\nSo why do researchers need special protection? The beetles create toxic waste products including ammonia, and carbon dioxide from bat breath is often trapped near the floor\u2014two gases incompatible with human lungs. And you'll want to be covered head to toe to avoid attacks by those flesh-eating beetles.\n5. Guano also meant gunpowder.\nThose who have seen the bats' nightly emergence have described the sound as similar to steadily falling rain. They might not be rain, but they can sure look like it on Doppler weather radar scans\u2014check out the animation below.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We have had a young visitor at Stone Ledge for the past few weeks. A young fox has enjoyed sharing breakfast with Samson and Delilah. Not that the cats are all that keen about sharing, but feral cats, fox, and raccoons generally maintain a degree of mutual respect for one another. The cats will back off two or three feet to let the fox eat. Although, today, when I clapped my hands to encourage the fox to leave the feeding station, Delilah took it upon herself to chase the fox to the edge of the woodlands. Delilah is a very bossy cat!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You start blogging and work tirelessly on your content, but nobody reads it.\nIt's like a presentation in an empty room. No matter how much information you have to share, nobody will notice.\nYou are sure that if you can get more traffic on your blog, the content will leak.\nBut even if you go away from your last position, you'll have to restart the process next week.\nBut what would happen if there was a strategy that could bring traffic not only to your last message but also to the future?\nWhat if the same strategy were as effective as a guest blog to attract new visitors, but faster, easier and without you having to share your ideas with others?\nWhat if, thanks to this strategy, you could work with some of the biggest names on the network?\nDon't you want to know what the strategy is?\nNot only is Quora an incredibly fun and valuable product, but also it can help you get more traffic to your website. Quora is a question and answer site. The content is often of very high quality and there are many active viewers. There are also people who are very successful in certain industries and who are very active on the site.\nHere are my best tips on How to increase website traffic using Quora to your blog, product, website, etc.\nCheck the level of activity of your target audience to get an idea of the effectiveness of content creation in Quora. As a starting point, see the answers to the question \"What are the most followed Quora topics in 2013? You can see that technology, science, business, movies, travel, health and wellness, and encounters are among the most active.\nJust because your niche isn't among the most active doesn't mean you won't get value from Quora. Also ,do qualitative research. Review the topics and questions in your space to see how active they are and how many followers they have. Question statistics usually appear just below the fold on the right side.\nBefore diving to answer questions, ask questions, blogging, etc., follow some questions and navigate through the Quora content to learn about the \"tag\" and how people use it.\nWhen a person asks or follows a question, they have expressed a desire to receive information about it. By answering this question in a thoughtful manner, you will help them and establish yourself as an expert.\nMost people who follow the questions will receive an email with their answer in the body. Many people are very involved with your inbox, so you have a good chance of getting their attention. At least everyone who follows the question will be notified when they are in Quora.\nCheck the number of followers on specific questions (as noted above) to determine what type of content would be useful to the general public. These questions will probably give you the best return on your investment.\nQuora has her own blog product. You can use it to produce and distribute blog articles, as well as a blog on another platform. Blog posts, as well as questions and answers, appear in the message feed on the user's home page.\nYou can send messages from your main blog to your Quora blog, but keep in mind that Google may downgrade your search for duplicate content.\nChoose the appropriate categories for your blog so that it can be seen by people who follow these topics.\nPromote your content with credits.\nCredits are similar to the currency of the Quora. You will be rewarded for valuing the Quora community highly. You can use Quora credits to promote your content or answer questions. The cost you get by spending credits is the same as the cost you get by tweeting on your Twitter. You get more views, but without editing the current currency.\nQuora rewards behavior that improves the quality of Quora. Examples of behaviour that improves Quora include a highly rated answer or blog post, someone who followed your question, someone you \"asked for an answer\", a specific highly rated question, and some others.\nSo you can earn more trust by answering questions, asking questions and asking others questions. I received 1,720 credits, asking James Altusher to answer a question about desktop publishing. At the bottom of the answer there is also a link to my profile.\nGet more attention through comments, uncertainty, harassment, etc. Link to your website in your profile and your organic line. If you are more active in Quora, you will have more space. You increase the chances that someone will see your profile. When someone sees a link to your profile, they can click on it. When someone clicks on your profile, they can click on your page.\nVisit http:\/\/www.quora.com\/stats for the content and commitments. I have a decent number of visits to Quora's content and the percentage of these visits is clickable on my website.\nAs time goes by, I get more and more benefit from Quora. It takes time to understand what works and what does not. The more quality content you produce, the more people you will have. The more followers you have, the better you will answer.\nIn addition, Quora acts as a search engine for many people, including myself. When you have a question or want to learn something, Quora's question research is often an excellent starting point. Therefore, even if someone isn't following a given question, they can find its content by doing a search or browsing.\nAnswer questions that are relevant to and followed by your target audience. More specifically, answer the questions that someone who wants your product or service would ask you.\nFor example, I'm taking a video course on how to increase website traffic, so I answered What are some of the most ingenious ways to increase traffic to a website? The answer to this question gives readers the same value as my product. Someone who is interested in the answer to this question may also be interested in my product because it offers the same value.\nYou can also refer to questions such as \"What are the best resources to learn [your area of expertise],\" \"What are the best tools\/applications for [your customer or niche],\" \"What is the best [your product category],\" \"How can I [evaluate your offer],\" etc.?\nDon't answer questions by promoting your product without embarrassment. Offer value to your readers. Read Quora's spam policies for more information. Also, if you need to reveal your affiliation to any product you are promoting. Learn more about Quora's disclosure policy.\nIf someone finds you useful, they may want to know more about you or receive more of your content. If they do, they can browse your profile or click on the links in your response.\nTheir name and a brief biography will appear at the top of their response. You can modify it depending on the topic or question.\nLink to your site in your profile. If applicable, insert a link to your site at the end of your answer, or somewhere in the answer if it is contextual, and include a revelation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbbwdv b/data_all_eng_slimpj/shuffled/split2/finalzzzbbwdv
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+{"text":"As with any designed item, book design is a combination of form and function. The form relates to the creativity, to the way it makes you feel, to the strength of the design and how pleased the reader feels looking at it and reading it. The form is the beauty and art and creativity rolled into one that altogether produces a pleasant, excited or compelling reaction from the reader.\nThe function relates to the ease of the reader's experience. For example, the type of a printed book is typically eleven or twelve points with a line spacing or leading of fourteen or fifteen points. That is what readers expect. If you go too far outside of that comfort zone, readers will find another book.\nWithout question the cover of your book is its most critical and important design element, for it is there that the readers first meet you and your story. If they like what they see, they will continue the journey. When they turn to the back cover and then open the book, the design must continue to make them feel comfortable, in the presence of a \"real\" book in every way. The cover design may be loud, but typically the interior design is quieter or refined so as not to intrude upon the reader, but simply to make it right and easy for them to simply read the book. Good form and function working together.\nWhether a strong cover design or pagination or drop caps, the overall design of your book weighs heavily in its ultimate success. As is the case with editing, websites and marketing, engaging professionals to handle your design is critical. Do not make the mistake of many self-publishing authors who hire a professional editor but then have their book designed by a family member with little or no book experience.\nWe can help you get published.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We must know our surrounding to survive. Knowing your environment will allow you to use it as an advantage. When an attack occurs, it is too late to become aware of your surrounding and plan for survival. You must make a mental note of the Essential Elements of Information (EEI) that are crucial to your survival before an attack occurs. Prepare mentally prior to the emergency.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Socrates love of wisdom did not stop even with the threat of punishment or the promise of reward from Society.\nCarrots and sticks can corrupt some conclusions, secular and religious. Faith conclusions that are attached to reward and punishment are conclusions that can lead one away from genuine faith and genuine doubt. When you attach the carrot and the stick to a claim, evidence and reason can take back seats to primal desire, fear and superstition.\nIf a person came to me and said they believe in the Multiverse theory and that there are many Universes based on a combination of their study of science, philosophy, and a final leap of faith I can at least discuss this or even entertain this without any other motivations based in reward or punishment.\nThat would corrupt the pursuit of truth regarding the Multiverse theory because now I may have primal fear and desire directly impacting my decision making and thinking. Like Pascal's wager you may think it is better to believe in the Multiverse because of the carrots and sticks that have been attached to the claim. Threats of violence or promises of paradise are more likely to corrupt the pursuit of truth or abort it than it is to help encourage it in an honest and thoughtful way.\nSome religious claims are supported with the carrots of Love, Forgiveness, and Heaven. The opposite of that is to support the faith with the sticks of Judgment, Violence, and Hell.\nIn the Judeo Monotheistic world the fear of God is the way to wisdom. In the Greco-Roman Polytheistic world it goes straight to the love of wisdom.\nI side with the world of Classical Antiquity and Hellenism that the best way to pursue truth is not through the fear of punishment or the desire for reward but through the love of wisdom. Philo - Sophia.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"After being nearly wiped out from overhunting and disease, Bison have a great comeback story, now roaming in giant herds in the west. The wood bison is the largest land animal in the Americas. This design sprouted from one of those ads for Montana where I drew the animals in multiple angles in my sketchbook. One ended up with 2 heads. While paging through my sketchbook a friend thought it was interesting.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"He's a high school student from the town of Greensburg Kansas, and his enthusiasm for the green movement and the amazing things it's doing for his town after the devastation caused by the tornado that leveled it are an inspiring look at the future of green in America.\nAfter spending close to an hour speaking with him by phone I realized precisely why the rebuilding of Greensburg can become an uplifting model for us all. I trust you'll enjoy his insight and enthusiasm as much as I did.\nTreeHugger: What do you believe made the people of Greensburg decide to rebuild the town in a green fashion?\nTaylor Schmidt: Well after at least 96% of the town was destroyed there has obviously been a massive need for rebuilding, and the town has come together as a big family, really, and it's been one joint effort to rebuild the town better than it was and more sustainable and green than it was. So we've really been learning a lot about what we need to do to keep our town from dying again. And we've been learning about building and going green and implementing a lot of green into our rebuilding efforts.\nTH: And what have people in Greensburg learned that's struck you particularly?\nTS: Well we live in a very red, conservative state. It's the buckle of the Bible Belt. But we have become informed about green and see it as a universal concept. It's really a bipartisan issue, so I believe that parties shouldn't influence it. And green just makes sense to us, it's really simple switches. Simple ways you can build where it will last longer, save more energy, and use fewer resources. And there's been an incredible amount of folks helping us implement these ideas.\nTH: How well do the other kids in Greensburg understand all of this?\nTS: Kids have been the driving force for rebuilding. It's practically unprecedented. They've actually encouraged us to come into the process of rebuilding and haven't been shunning us like most people would. They've really embraced us; almost all of the youth have become involved in the rebuilding of Greensburg. We've been on committees with FEMA and there are around 20 students out of 100 involved on various committees and things like that.\nAnd because of that involvement a green club has formed at Greensburg High School. Basically it's a group of kids that want to learn more about green, what it is, how simple it is, how we can implement it in our lives. And what affect it has on our lives, finances, and the city.\nIt's just so exciting; I don't think you can find a person in the whole high school who doesn't know about going green.\nTH: How have the youth of Greensburg helped others in town understand the concept of going green?\nTS: We've been reading about it so we can help those who don't understand it as much in older generations; and as we learn more about things we can do ranging from emailing assignments or encouraging some people that are rebuilding to use CFL's instead of incandescent light bulbs we're really making a difference. The school community has really been supportive as well. The school is even going to be rebuilt to LEED platinum standards. It's a testimony to how much impact green has on our town right now. We've even gone with several of our teachers up to Chicago to the national green building convention and learned about how we can rebuild school and town with green roofs for water, and other practices ranging from all sorts of simple things like using efficient lighting and efficient water usage to installing wind turbines and solar and geothermal heating. Our school is actually going to be powered by its own large wind turbine.\nIt's going to be a sustainable, lasting, innovative, comfortable, and inspiring atmosphere that encourages education and works with environment instead of against it. That's what the entire town is trying to do; work with nature instead of against it.\nI think of myself as a 17-year old watching our town learn about how we can thrive again and even grow back better than we were before. Some people think how terrible it must be, but I think it's a blessing to live in such exciting times.\nTH: What have you come to believe makes the town of Greensburg so unique?\nTS: We've realized that the spirit of working together is what makes Greensburg so unique. Before this happened we all (the youth) wanted to leave, but now we want to stay. It's given us a reason to understand we have a long term affect on our community and the world.\nAnd we think that by going green we can show the rest of the world what we're made of and that we can come back again from tragedy.\nTH: And what impact do folks in town believe you'll all be having on society by choosing to be a part of the upcoming show?\nTS: We really want to show them what's next, and what's going to happen; the responsibility each of us have for the earth. We're an eco-town and a shining example of what we're all going to need to do if we want to continue living in a world that gives each of us so much.\nTH: If you had one question you could ask President Bush when he comes to speak at the high school graduation May 4th, what would it be?\nTS: How can the U.S. change its ways? How are we all going to become informed; as informed as we are here in Greensburg? Many people here thought that green was just this way out there radical idea that's never going to happen\u2026 Just this crazy little thing and that we're fine and nothing bad is ever going to happen.\nBut we've become informed that yes; there is something to be concerned about. Green isn't radical, it's simple. There are radical elements to it, but it's really simple and about changing bad habits to things that are more responsible with the result of making huge improvements on this world.\nEven these old Republicans here are realizing it's good on their finances, more comfortable than traditional methods, and it's really not California radical at all.\nYou don't have to give up everything; you can actually gain things by going green. Look at us; we're gaining a whole town by going green. It's extremely exciting!\nTH: Taylor, is there anything else you'd like everyone out there in America to know about going green?\nTS: Just that green is so simple; it's not a radical concept. It's just a simple switch to make a huge difference.\nThe rebuilding of Greensburg, Kansas is set to be the focus of an original television series airing on Discovery's Planet Green this summer.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The International Commission: The continued failure of the international community to activate international accounting tools contributes to encouraging the occupation forces to commit more crimes against the Palestinians in Gaza Strip.\nThe International Commission to Support Palestinians' Rights (ICSPR) condemns the violent Israeli military attacks that resumed on Monday, 12 November 2018, on a large scale on Gaza Strip, where the Israeli artillery and warplanes escalated their military targeting of agricultural lands and security locations all over Gaza strip, which until 20:30 led to killing 3 civilians: Muhammed Zakaria Altetri (27 yr), Mohammed Zuhdi Hassan Odeh (22 yr), and Hamad Mohammed Mousa al-Nahhal (23 yr).\nThe current Israeli escalation follows a bloody night in Gaza Strip as a result of Israeli special forces and warplanes targeted the eastern areas of Khan Yunis governorate, killing 7 Palestinians: Nour al-Din Muhammad Salameh Baraka (37 yr), Mohammed Majed Mousa al-Qara (23 yr), Khaled Mohammed Ali Quweider (29 yr), Mustafa Hassan Abu 'Odeh (21 yr), Mahmoud Atallah Musabeh (25 yr), Alaa Eddin Fawzi Mohammed Feseifis (19 yr) and Omar Naji Muslim Abu Khater (21 yr).\nICSPR confirms that the Israeli occupation forces continued their crimes against the citizens of Gaza Strip and their properties, which would not have happened if the international community broke its silence, which grants the leaders of the occupation state immunity in the face of accountability and international justice.\nICSPR urges the international community to immediately and urgently intervene to force the Israeli occupation to cease its ongoing hostilities on Gaza Strip and to prevent the escalation of the Israeli violence.\nICSPR urges the Palestinian leadership and diplomacy to exert more efforts with the International Criminal Court to urge it to move forward towards the opening of an international investigation into international crimes committed by the occupation forces to ensure that Israeli criminals do not escape punishment.\nICSPR urges the International Solidarity Movement with the Palestinian people and all Arab, regional and international organizations to act on all legal, political, diplomatic, human rights and popular levels to ensure the protection of Palestinian civilians in Gaza Strip.\nA bloody night in Gaza Strip, the martyrdom of 7 Palestinian civilians.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"JOHANNESBURG \u2013 A new elite Track and Field series will be held this year, Athletics South Africa has confirmed.\nWith our athletes preparing to qualify and compete at a host of international events, the four-leg Speed Series will assist them in launching their campaigns on the domestic circuit.\nThe new Track and Field series will be held in Durban (February 28), Bloemfontein (March 8), Potchefstroom (March 15) and Germiston (March 22).\nIn addition, athletes will have a chance to compete for national titles across all disciplines throughout the year.\n\"This is a big season for our athletes and we want to give them every chance to deliver their best both at home and abroad,\" said ASA president, Aleck Skhosana.\n\"We want to send the best possible teams across all age groups, in both individual and relay events, to carry the South African flag overseas, and we expect tough competition on the domestic circuit to propel our athletes forward as they prepare to take on the world.\nLater this year, the country's elite stars will turn out in various international competitions, including the World Relays in the Bahamas (April 22-23), the CAA African Junior Championships in Tlemcen (May 12-15) and the CAA African Youth Championships in Bata (June 29-July 2).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A. L. Breguet (1747 \u2013 1823) was born in Neuch\u00e2tel, France but spent most of his productive life in Paris. Breguet, both the man and the firm that bears his name, have had a profound effect on European culture. He is known as the architect of the greatest revolution in the science and art of time-keeping.\nHis career began with breakthroughs that included the development of self-winding perpetual watches, the introduction of gongs for repeating watches and the first shock-protection for balance pivots.\nLouis XVI and Marie-Antoinette were early enthusiasts of Breguet's watchmaking. The watches coming from his workshops had the latest improvements in original movements and most were fitted with the lever or ruby-cylinder escapements that he perfected.\nA.-L. Breguet took refuge in Switzerland during the French Revolution and returned to Paris with new ideas that resulted in the Breguet balance-spring, his first carriage clock (sold to Napol\u00e9on), the sympathique clock and its dependent watch, the tact watch, and finally the tourbillon, patented in 1801.\nBreguet became the indispensable watchmaker to scientific, military, financial and diplomatic elites of the age. His timepieces ruled the courts of Europe. For his most celebrated clients, Breguet designed exceptional timepieces. In 1810, for Caroline Murat, queen of Naples, he conceived the world's first wristwatch.\nHonors were bestowed on Breguet for his many important contributions to watchmaking a\/k\/a horology. Appointed to the Board of Longitude and as chronometer-maker to the navy, he entered the Academy of Sciences and received the Legion of Honour from Louis XVIII.\nToday, the Breguet name carries the same cachet as its founder: a capacity to innovate that remains vital and creative. Now overseen by Nicolas G. Hayek, Breguet's contemporary selection of watches and jewelry continue to contain bold innovations resulting in the Breguet firm filing a number of relatively new patent applications.\nBreguet holds a special place in our cultural heritage because A.-L. Breguet set the standard by which all fine watchmaking is now judged. Currently, the heirs at Breguet still make each timepiece as a model of supreme watchmaking art.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dr. Aniruddha Majumder is an eminent ENT specialist and ENT surgeon in Kolkata with 10 yrs of experience. He is trained in the UK for laser surgery and endoscopic sinus surgery. Currently, he is an assistant professor in medical education service.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Change can happen one degree at a time.\nIt's a week later, and we're back in General Hospital's emergency room for the Night Shift \u2014 Midnight to 6:00 am. This week the doctors and nurses are still reeling from the loss of their favorite habitual patient Mrs. Storch and shocked to discover it was caused by an injection to her IV. The paper is calling it a mercy killing and coupled with the brewing strike talks amongst the nurses and staff, as well as the continuing mix-ups with patient's charts, Dr. Ford and Miss Sneed are both worried a take over from Medcam is eminent.\nSpinelli showed up to fill in for Jason, only telling Toussaint the other man was working his day job. It quickly becomes apparent he is there to further his attempts to get closer to nurse Jolene, who is distraught and weepy for most of the episode. Many assume it's over the loss of the patient, but late in the episode she confides to Spin it is the seventh anniversary of her father's death. He died during surgery, she says, and while her mother blamed the health care system it drew her to it \u2013 she wanted to effect change. A noble cause and honorable tribute to his memory comforted Spinelli.\nAs during the day, Spinelli brings lighthearted fun to even the most dramatic of situations with his quirky and awkward movements as well as his way of paralleling every situation to a video game, but even more so on Night Shift we are seeing a depth and growth to the character, which is just as entertaining to watch. Hopefully we'll see this friendship between him and Jolene escalate and carry over to the daytime hours.\nStacy Sloan was back at the hospital, actually passing out in front amidst the striking employees. Thankfully Patrick noticed her and got her inside where Robin pressed to keep him involved in her case. With neither pregnancy nor HIV being his area of expertise, Patrick found himself feeling trapped and accused Robin of forcing the idea of having a baby on him.\nTwo patients were refused treatment in the emergency room this week. One was a nameless young woman, obviously very ill, but without health insurance. The other was Lainy Winter's father, who was transferred from Rose Lawn where he's being treated for his advanced Alzheimer's on the fear he has suffered a stroke. The Alzheimer's storyline was visited briefly awhile back on the parent show, but was dropped suddenly and I was happy to see it revisited on Night Shift.\nPatrick is able to diagnose a TIA or precursor stroke and advises that Lainey allow him to perform surgery soon to prevent a more serious one. Enter Dr. Ford who ruled out surgery as Lainey's father's health insurance would not cover it due to his pre-existing condition (the Alzheimer's). Lainey insisted she would pay out of pocket and Patrick offered to donate his services but Ford held firm, stating it was a waste of hospital resources to treat someone who in essence was untreatable. The scenes were quite disturbing yet unfortunately very realistic, reinforcing the need for a complete overhaul of the medical system as we know it.\nAs Lainey tried to help her father dress for his return to Rose Lawn, he became confused and combative. Cody, the Iraq war veteran she's been treating, came to the rescue, handling her father calmly and respectfully yet authoritatively. In the hall, Lainey thanked him, but added she hoped it was genuine kindness and not a grandiose attempt to get more pills out of her. Cody made an observation of his own: that she doesn't like to accept help from anyone. Is this a change in Cody? Is he finding his heart, hardened by his time in Iraq? Is Lainey getting through to him? I'm anxious to see how this develops both here and on General Hospital where Cody is still seeking his vengeance against Logan.\nAs Patrick ran off in a huff, frustrated by Dr. Ford's persist attention to the bottom line instead of heath care, he found himself trapped in the temperamental elevators with Toussaint. When the janitor began to sing \"When The Saints Go Marching In\" to pass the time, Patrick found the mysterious voice that had been haunting him these last four weeks he has worked the Night Shift. Further conversation revealed Toussaint was once a member of the group The Saints, a name Patrick recognized from discussions of music with his father. Later, Patrick confides Toussaint's previous life to Epiphany who is quite unnerved by it, and later quite tongue-tied when she runs him to him.\nBetween the medical cases Regina confronted a very offended Layla, telling her she had seen her enter Mrs Storch's room just minutes before she had passed. She can act offended all she wants, right now Layla is my top suspect too.\nKelly also found the time to roll another man, this time Pablo in the showers. When Robin tired to discuss her distant behavior with her, Kelly blew her off. Even later, last week's victim confronted her, trying to break through this wall she's building around her and the self destructive behavior she been demonstrating on GH too. I'm quite interested to see what's got Dr. Lee acting so out of character in the weeks to come.\nWhen Epiphany confronted Stan about the labor strike he had organized and vocalized how disappointed she was in the man he'd becomes. He voiced just as much disapproval in her showing her the woman who had early been removed from the ER for not having insurance. He suspected she had pneumonia and her apartment building had been 'red tagged'. She had turned to the hospital for compassion and was given none. Epiphany insisted she had rules to follow, but Stanford countered rules could be bent and broken, but medical staff should care.\nThis world is ever turning, nothing remains static and change is inevitable. How people and relationships deal with it becomes the question, and I will continue to turn in to the Night Shift hoping Dr. Drake and Dr. Scorpio come through just fine.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"We help school students, educators and support staff have all the help they need to plan their future, access the best chemical science resources and find their place in the chemical sciences. Below you can find the most relevant products and services to those working or studying at a school or college. Click here to view our full list of member benefits.\nWe help school students access study resources, information about careers and make confident decisions about their future in the chemical sciences.\nOur education resources page contains links to coursework support, experiments, fun games and an interactive periodic table.\nIf you are thinking about a career in chemistry, you can browse our website A Future in Chemistry. It highlights job profiles that help you narrow down what you want to do and explore your study options.\nIf you want to study chemistry at university, explore which courses have achieved accreditation. Accreditation of degree programmes by professional and statutory bodies is a mark of assurance that let you know that a course has met key standards.\nAs a school student, we do not have a membership category for you quite yet, but you can join us as a student member to receive information on careers in the chemical sciences and receive Chemistry World online and in print.\nEducators at schools and colleges encourage and enthuse the next generation of the chemical sciences. We support our teacher members with well-rounded professional development, ways to inspire their students with current developments in the chemical sciences and build a helpful network of like-minded peers.\nEngage with our education networks, get involved in outreach activities, apply for funding and connect with other education professionals. Our various events and conferences can also help you and your students meet other chemical science professionals.\nFurther your career with our Continuing Professional Development tools and services, and gain recognition for your expertise by applying for Chartered Science Teacher (CSciTeach). We also have a dedicated Career Management service with careers advisors, events and resources to support your career.\nSchool technicians are critical support staff that help to inspire the next generation of chemical scientists. Our support ensures they have access to a network of members in their local area, qualifications that acknowledge the application of their technical skills, and have opportunities to advance their careers.\nWe award Registered Science Technician (RSciTech) to those working in technical roles and Registered Scientist (RSci) to those in scientific and higher technical roles to recognise the skills and knowledge gained through ongoing professional development.\nTrack your progress with our Continued Professional Development tool to record your development and access online materials and regular webinars to support your career planning.\nYou can connect with your Local Section to meet and network with a diverse range of members in your area or contact your local Educational Coordinator to get involved in outreach and inspire activities in your school.\nOur members working in academia cover a variety of research interests and career stages. Below you can find key services and products tailored to career stage for those studying or working at a University. Click here to view our full list of member benefits.\nOur student member support starts before you even begin your degree and continues far beyond graduation.\nAs you settle in to university life we help you access the right materials with discounts on your core textbooks, track the skills you are developing with our Undergraduate Skills record and top up your knowledge with our specialist school-to-university transition resources.\nOnce you have found your feet we want to provide you with plenty of opportunities to strengthen your knowledge through our members' magazine Chemistry World and our ChemSpider database, connect with experts in your area through our local sections and get practical experience through an internship or placement found on our chemical science jobs board. Completion of a placement year can even help you get Registered Status, displaying the professional experience you have gained.\nTowards the end of your degree, you need to put all of your skills on show. We are here to help launch you into the next stage of your career with our special interest groups often closely linked to final year research projects, revisions resources and guides for thesis writing and viva advice, and our ChemCareers series of webinars preparing you for what comes next.\nOur postgraduate member support helps you gain recognition for your existing skills, make the right connections to support your research and help you plan for the future.\nBegin your professional development journey.\nAs an Associate Member, you can clearly demonstrate the wide range of skills and knowledge you have developed in your undergraduate degree and display your clear commitment to the chemical sciences. Combine this with an award such as Registered Scientist or Registered Science Technician for formal recognition of professional skills and you will have made your first step along the path to Chartered Chemist status.\nBy joining our special interest groups and divisions you can connect with people who want to share their experience and knowledge. Our community is all about sharing knowledge and ideas and our grants and funding include travel grants that exist to help you get where you need to be to develop yours.\nOur specialist careers management team can provide you with tailored support with one of our Catalyst consultations, CV Checks, online guidance and CPD resources to help you manage your professional development.\nWe support our members working in research and academia through opportunities to widen their exposure, contribute to the advancement of the chemical sciences and engage with like-minded communities.\nPromote your work and hear about the latest research at conferences and events. As a member you can receive reduced conference registration fees and will be eligible to apply for a wide variety of funding opportunities, including travel grants and research funding. You can also get 15% off the article processing charge when publishing in one of our open access journals.\nJoin our Divisions and Interest Groups to pursue and broaden your interests and build a valuable network of contacts. They run events, activities and even help advise on science policy.\nAchieving Chartered Chemist status clearly displays not only your expertise but also your ongoing commitment to developing your skills. Scientists with less than 6 years post-graduate experience can access our professional development programme, which provides a framework to meet 12 distinct attributes that make competent chemical scientist. If you hold more than 6 years post-graduate experience you can access our direct route to the qualification, in recognition of suitable knowledge and skills.\nAccess a variety of e-books, journals and databases from various publishers as well as content from our journals. You will also receive Chemistry World magazine every month which covers the latest developments in research, business and the job market.\nTechnicians in academia are critical to ensuring that the research undertaken in their institution happens in a safe and controlled manner. Our support ensures they have access to high-quality training courses, qualifications that acknowledge the application of their technical skills, and have opportunities to advance their careers.\nIn your role, you will need to maintain a strong awareness of the techniques and practices relevant to your institution. Our approved training courses database can help you make confident choices when identifying high quality training options.\nWe have a large number of communities, linked to specific areas of the chemical sciences and our analytical division interest groups in particular can help you exchange information and ideas at events and activities.\nMany of our members take a period of leave from work for to start a family or other commitments. For many this can be a time to take a break and consider what options are open to them. See our career break tab to see how we support our members during this time.\nOur members working in industry cover a wide range of sectors and career stages. Below you can find key services and products tailored to those working or studying in an industrial environment. Click here to view our full list of member benefits.\nAs an apprentice, you will be building your knowledge and applying it in an industrial setting. Using our Professional Development recording tool to track your progress, alongside online resources and webinars from our careers management team will help you launch your career.\nIf you have been tracking your development throughout your apprenticeship, you might have gained relevant professional experience that will allow you to apply for registered status. Registered Science Technician (RSciTech) and Registered Scientist (RSci) are awards that formally recognize these skills and how you have developed them in the working environment.\nThroughout your apprenticeship, our local section events, special interest groups and online forums can ensure that you have every opportunity to pursue your passion and connect with people that want to help you get ahead.\nTo support industry technicians we provide access to peer-reviewed training courses, qualifications that acknowledge the application of skills, and career advancing opportunities.\nIn your role, you will need to maintain a good awareness of the techniques and practices relevant to your organisation. Our approved training courses database help you make confident choices when identifying high quality training options.\nWe have a large number of communities, linked to specific areas of the chemical sciences and our analytical division interest groups in particular can help you exchange information and ideas at events and activities, with many specifically targeted toward younger members.\nAs your professional skillset develops, you can make sure that you receive the appropriate recognition by applying for registered status. The Registered Science Technician and Registered Science awards will begin your professional development journey and stand you in good stead for future career progression.\nThe support we provide to recent graduates and early career scientists helps them transfer the knowledge and skills they have gained into evidenced experience and career progress.\nHaving a one-to-one Catalyst consultation with one of our careers management team can launch you in the right career direction. The specially designed sessions provide you with a framework and career management tools to take your next career step with confidence.\nOur Early Career Network gives you opportunities to network with chemical scientists in a friendly and accessible environment. Offering peer-to-peer support, conferences, help and advice it furthers your personal and professional development. You could also engage the help of an experienced mentor to help you navigate the challenges that may arise through this early stage in your career.\nAs a practising scientist, succeeding in your role will mean enhancing your knowledge and skills whilst building your experience. Recording the evidence of the activities that you are involved in will help you gain Registered Scientist (RSci) status. If you already hold 3 years of professional experience, you may be ready to embark on our Professional Development Programme. This two-year programme gives you a clear set of attributes with which to develop and successful completion results in the Chartered Chemist Award.\nWe enable established scientists to maintain and enhance their skillsets and build the connections that help collaboration, exchange of information and ideas and allows them to stay informed of major developments in their field.\nAs a competent chemical scientist, achieving Chartered Chemist status clearly displays your expertise and your ongoing commitment to developing your skills. Scientists with professional experience of at least 6 years can access a direct route to the qualification, in recognition of suitable qualification and experience.\nWe have a large number of communities, linked to specific areas of the chemical sciences and our Industry and Technology division and its associated interest groups have a focus on supporting innovation, helping you collaborate, exchange information and share ideas at a range of events and activities.\nOur specialist career management team are here to help you with the career challenges you are currently facing and are available face-to-face, over the phone or by skype. If you have overcome various career challenges, you could train with us to become a Royal Society of Chemistry mentor. Our mentors share their experiences with younger members helping them set and achieve career goals whilst building greater self-confidence.\nWe recognise the additional skills needed and responsibilities held by managers in the chemical sciences. Our support enables them to develop leadership skills, share their experiences and gain recognition for the responsibility they hold.\nOur Management Group is a special interest group that runs events to develop your leadership skills and share ideas with other members who hold similar responsibilities to you.\nThe Royal Society of Chemistry can help you ensure that your staff are working towards becoming experts in the profession through the accreditation of company training. Our schemes provides you with the confidence that your staff are developing all of the attributes that make a competent chemical scientist and enables them to achieve various levels of professional qualification.\nConsultants offer organisations in the chemical science sector with expertise that has been developed over a career of experience. We help these members have these capabilities recognised and enable them to highlight their skills to the right audience.\nYou can access a Chartered Award that is appropriate to the field in which you offer your consultancy services. Holding one of these awards clearly displays to future clients the expertise that you hold and your commitment to enhancing your knowledge.\nOur Directory of Consultants helps organisations find the expertise they need. Making yourself and your knowledge discoverable on this directory will widen your exposure with the Royal Society of Chemistry community and beyond.\nThrough our networks and events you have access to endless opportunities including our flagship industry event, Chemistry Means Business, to meet potential organisations. You can also share best practices with our Consultancy group to help advance your business.\nThere are many reasons for an individual to take a break from their career. Family commitments, redundancy or as an opportunity to evaluate where their career is taking them. Whatever the reason, we provide our members with ways to keep in touch with the community, explore the options available to them and maintain their knowledge and skills.\nWe have support specific to those recent graduates who are looking for their first chemical science job role. Our support helps ease the pressure and find the right opportunities.\nOur Chemists' Community Fund can provide you with a breathing space grant designed to support recent graduates when looking for their first job. If you are struggling with student debts and living independently then you may be eligible for a one off payment of \u00a3300 to spend however you may need.\nThe Chemistry World jobs board has internships, placements and graduate roles from a wide range of well-known chemical science organisations, and you can tailor suggestions to your location, career stage or area of interest. We even hold Chemistry World Jobs Live, our annual recruitment fair, which gives you an opportunity to meet the hiring managers face to face.\nOur career management team can provide you with tools, advice and guidance specifically tailored to finding a role in the graduate jobs marketing. Their specialism in the chemical sciences means they can give you tips to tailor your CV, prepare for interviews and even deal with recruitment agencies.\nMany of our members take time away from work to start a family or look after loved ones. Our support ensures that during this time, you are able to take a step back whilst still keeping in touch with the chemical sciences and fully prepared to return to the workplace when you are ready.\nYou can access a reduced subscription rate for a period of extended leave and through our Chemists' Community Fund, access financial aid to help with unexpected costs or respite breaks.\nYou may be very happy to forget about the chemical sciences for a few months, but if you want to keep up to date with all the current chemical science information, you will receive your regular copies of Chemistry World in print or we can organise a talking version to arrive at your door. We can even arrange for one of our volunteer visitors to arrive with it. They are always keen to chat about similar interests in the chemical sciences.\nIf you are considering returning to the workplace, we can help you prepare with guidance from our careers management team or by connecting you with an experienced mentor. They can help you identify and achieve your chosen goals.\nMany of our members find themselves unemployed whether through choice or redundancy and look to us for support to help them back into the workplace. We can provide financial support, specialist careers advice and plenty of opportunities to build your skills, all with the aim to help you find your perfect role.\nYou can access a reduced subscription rate for a period of extended leave and through our Chemists' Community Fund, access financial aid to help with unexpected costs. We also provide grants to help you retrain for your ideal role and with our database of approved courses, you can ensure that you access the highest-quality training.\nPractical advice and specialist guidance.\nOur career management team can provide you with practical advice and guidance to help you find your idea role. They provide support at each stage of the process, from identifying professional development targets, building a network, using job boards and recruitment agencies, updating your CV, and preparing for interviews.\nParticipating in activities held by your local section can help you make contacts in your area. If you are passionate about particular areas of chemistry, our special interest groups will help you meet the influencers in those areas.\nOur local sections run a range of events and activities in your area ranging from retired member lunches through to general interest lectures. Getting to know your local section will help you meet new friends and continue your interest and passion in the chemical sciences.\nIf you are interested in volunteering, there are a wide range of options available ranging from one off events to providing long-term support to a project or to our community. It is a great way to keep in touch with the community and enjoy sharing your chemical science knowledge.\nWhen Chemistry World lands on your doorstep each month, we know our retired members are some of our most avid reader but you can also access daily updates online or even receive a talking version each month.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There are fewer things more beautiful than looking across a meadow of wildflowers but did you know you can create a mini meadow in your very own garden?\nThis guide will show you how\u2026.\nThere is much more to wildflowers than natural beauty. As well as adding spectacular colour to your garden they do a very important job. Wildflowers create a diverse environment that will attract and help support birds and insects including bees which are vital to our ecosystem.\nWhere to Sow Wildflower Seeds?\nWildflowers have a reputation for being difficult to grow. The reality is that they grow really well, as long as you have the right environment for them to grow.\nAreas of your garden that are naturally low in nutrients are ideal. Yes, that means they like poor soil! If the soil is too rich you will end up with mostly leaves and perhaps a few flowers if you're lucky. Avoid using fertilisers and weed killers near the area you wish to sow wildflowers. Make sure all clippings are removed from the area you wish to sow on as these will also pass nutrients back into the ground.\nThe nutrient levels will eventually come down creating the ideal environment in which to grow wildflowers.\nIf the chosen area for your wildflower meadow is heavily cultivated then it may need to be starved of nutrients for up to two years before it's ready for wildflower seeds.\nOnce you have the right soil conditions it time to prepare the ground. This is a key step and will give the wildflower seeds the best chance of germinating. First things first, make sure the area you plan to sow on is free of weeds. This will involve a bit of manual work but in the long term you'll be glad you did it. Weeds in the ground will compete for light, space and food, so help wildflowers to thrive by removing them.\nOnce all weeds have been removed, dig over the soil, firm it down then rake it.\nOnce the soil has been prepared as detailed above you are ready to sow. Regarding our own Wildflower Seed mixtures we recommend a sow rate of 5g\/m2.\nMost of our wildflower seeds can be sown in either spring or autumn but it's always important to check the usage guides as following these will help you get the best results.\nScatter your wildflower seeds by hand or you can use a seed spreader. Once down, rake into the soil lightly and give them a good soak. If you're concerned about birds or cats you can cover the area with netting to protect the seeds while they germinate.\nIt's important that the soil is warm and doesn't dry out. If there's a very hot spell of weather you'll need to water the seeds regularly.\nOne of the great things about a wildflower meadow is its low maintenance (not no maintenance). After the seeds germinate and start to grow you might see weeds coming up with wildflowers, we would recommend keeping your wildflower meadow weed free for the first year.\nAt the end of summer it's time to mow your wildflower meadow. The tell-tale sign for when to do this will be when your flowers have ripened and dropped their seeds. Set your mower to a high setting and carefully run it over the top of the meadow. Doing this will make sure your wildflower meadow is ready for the following year.\nThe final step is to rake up any clippings, again to prevent the nutrients being returned to the soil.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"FAQs How can I see a list of all events happening this month?\nHow can I see a list of all events happening this month?\nYou can see a monthly calendar view of all of the events that are happening at the library this month by visiting our Evanced calendar.\nTo get back to the library's website, you will need to click the Home button.\nNote: This option is not mobile-friendly. For optimum display, please view this page on a desktop computer.\nWhich events am I registered for?\nHow do I cancel my event registration?\nHow do I cancel my room reservation?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you're a college grad who's having trouble paying back your student loans, here's some good news: You may be able to qualify for student loan discharge or forgiveness if you meet certain eligibility requirements. But what is student loan forgiveness or discharge, and how are they different?\nDischarge vs. Forgiveness: What's the Difference?\nUnder normal circumstances, you must pay back your student loans even if you are unhappy with the education you received, don't finish college, or can't find a job in your field of study. Even if you are going through a bankruptcy, student loans are notoriously difficult to get rid of (though not impossible). But in some cases, you may qualify for loan discharge or forgiveness.\nTo qualify for forgiveness, you typically must work for a certain amount of time in a particular field, especially in high-need areas. Forgiveness can be granted regardless of the student's ability to repay. Discharge, on the other hand, is granted wen serious qualifications are met; these can include improprieties done on the part of your university, or other things that make it unlikely that you will ever be able to repay your debt.\nClosed School Discharge: If your school closes while you are enrolled and you can't complete your program because of the closure you may be able to discharge your student loans. This may also qualify if the school closes within 120 days after you withdraw. You would not qualify for discharge if you complete a similar program at another school by transferring credits or if you have completed all of the coursework for the program (even if you haven't received your degree!). This would apply to Direct Loans, FFEL Program Loans, and Perkins Loans.\nTotal and Permanent Disability Discharge: If you become totally and permanently disabled, you may receive what is called a \"TPD Discharge.\" To qualify, you must prove that you are unable to find employment due to some kind of disability. You may qualify if you were injured in the line of duty, if you are receiving Social Security Disability Insurance, or if you can prove through a physician that you are totally and permanently disabled in a way that might result in death, has lasted for at least 60 months, and can be expected to continue for at least another 60 months. This would apply to Direct Loans, FFEL Program Loans, and Perkins Loans.\nDeath Discharge: If the student borrower dies (which we hope isn't the case!) then the loans will be discharged. Similarly, if a parent borrower of a Parent PLUS loan (or the student that the loan was taken out for) dies, then the loan may be discharged. This would apply to Direct Loans, FFEL Program Loans, and Perkins Loans.\nFalse Certification of Student Eligibility or Unauthorized Payment Discharge: This type of discharge is related to misdeeds done by your university. These include things like your school falsely claiming how you may benefit from its training; the school signing your name on the application or promissory note without your authorization; or a few other reasons. This would apply to Direct Loans and FFEL Program loans, but not to Perkins Loans.\nUnpaid Refund Discharge: If you withdrew from your university within a certain time frame and your university did not refund the loan to the U.S. Department of Education (or other lender) then you may be eligible for discharge. This would apply to Direct Loans and FFEL Program loans, but not to Perkins Loans.\nBorrower Defense Discharge: Similar to the False Certification of Student Eligibility Discharge above, Borrower Defense Discharge may be granted when your university misleads you or engages in misconduct in violation of certain state laws. This can apply to Direct Loans and sometimes FFEL Program Loans, but not to Perkins Loans.\nTeacher Loan Forgiveness: This type of forgiveness impacts teachers who have been teaching full-time in a low-income elementary or secondary school or other educational service agency for five years, back-to-back. Up to $17,500 of your subsidized and unsubsidized loans may be forgiven in this way, but PLUS Loans cannot be included. Perkins Loans also cannot be forgiven in this way, but may qualify under Perkins Loan Cancellation and Discharge (see below).\nPublic Service Loan Forgiveness: Public Service Loan Forgiveness is available for those employed by governmental organizations and not-for-profit organizations. The program forgives whatever balance you still owe on your Direct Loans after you have made 120 \"qualifying monthly payments\" while working full-time for a qualified employer. FFEL Program Loans and Perkins Loans cannot qualify for this type of forgiveness.\nPerkins Loan Cancellation and Discharge: A portion of your Perkins Loans may qualify for forgiveness if you work in certain high-need fields (yes, it's called \"discharge,\" but it's really \"forgiveness\"). The total amount possible for forgiveness varies by occupation and length of time. This includes volunteers in the Peace Corps, teachers, members of the armed forces serving in areas of hostility, nurses or medical technicians, law enforcement or correction officers, \"Head Start\" workers, child or family services workers, and other professional providers of early intervention services. For a complete breakdown of forgiveness amounts on Perkins Loans, you should check out this chart.\nThe rules for various kinds of student loan discharge and forgiveness can be dramatically different, so before you assume that you qualify (or get your hopes up) it would be wise to contact your university or student loan servicer to ensure that you do. The information compiled here is a great jumping-off point, but you'll need to do your own research to make sure you qualify.\nIt's also important to note that if you have federal student loans that are forgiven or discharged, the government will consider this taxable income for the year in which the discharge or forgiveness took place. So if, for example, you received a discharge of $20,000 in student loans this year, then come tax season you would owe taxes on that $20,000. That can certainly make for a hefty tax bill, especially as the amount forgiven or discharged grows, so if you are building discharge or forgiveness into your student loan repayment plan then it's important to take it under consideration and plan for it.\nYou might have been told that refinancing your student loans could help you save money or lower your interest rates, which, if you're like any other human being, probably piqued your interest. While it's true that refinancing can make repaying your loans easier, you should know that if you refinance your federal student loans, they become private student loans\u2014at that means you are giving up certain safety nets, like the forgiveness options outlined above.\nIs that a reason not to refinance? I don't know, it might be. It completely depends on your own personal financial situation, needs, and goals. If you went to college to become a teacher, for example, knowing that you could discharge your student loans after five years of working in a low-income area, then you probably don't want to refinance in your fourth year, even if doing so would save you some money thanks to a lower interest rate. If, on the other hand, you didn't plan on using any of these forgiveness options, or if you don't qualify for any of them anyway, then you might want to consider refinancing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"with a short tar strip to smoke the tyres.\nI won this race in 2010 as well, wrote a cute story but never got to the race report.\nJames Cunnama (2nd) is back after ITB surgery in November and getting ready for a big 70.3 season in the overseas.\nWild trophies! (Pun intended) Caveman 1st, James Cunnama 2nd, and Stuart Marais \u2013 controversial 3rd. Stuart was leading by about 40\u2033 on the run, but took a wrong turn and was demoted to 3rd\u2026 Much more to come from this talented Stellenbosch athlete.\nCheck out my Movecount.com profile of my heart rate (and speed on the run) recorded on my Suunto Quest.\nI alway jump at the opportunity to spend time in Kraalbaai, Langebaan. I love that scenic, mystical part of the world so much, I even got engaged there! So when the WWWW came around I jumped at the opportunity to not only visit that part of paradise, but also do a fun triathlon with friends and like minded outdoorsy triathlony people.\nThe race itself was quite loaded, I went there with a fun weekend in mind, but then Raynard Tissink, James Cunnema and Stuart Marais rocked up. I think none of us were really prepared for a big show down, as its early season and James and myself were coming out of injuries. The swim is in the Kraalbaai cove- the water is quite warm for the Cape and you swim amongst anchored yachts and house boats. I lead out the swim with Stuart Marais and we set out on the scenic bike leg through the nature reserve and around the lagoon. I like rolling hills, but this being a road tri- (and me being used to mountain bikes) I wasnt quite used to the TT bike and I didnt quite have the ride I wanted. I got off the bike about 20 seconds behind James Cunnema and we tackled the run up and over a seemingly endless sand dune. Stuart came by us both just before the top of the dune and made a small gap. This run is almost completely off road, with large parts being on the beach and over some scenic (but TOUGH) dunes. I really enjoy the vastness of running on a foggy beach on the Cape West coast- especially if the turn around is right next to a huge, rusty shipwreck. Stuart missed the turn in the fog and I was somewhat undeservedly back in the lead. The last part of this run follows the lagoon shore again- past the famous \"Pulpit rock\" wit the finish right on the beach were I got engaged almost exactly a year ago. A fitting finish to such an amazing event.\nThis entry was posted\ton Monday, February 20th, 2012 at 9:00 am\tand is filed under west coast warm water weekend. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"You can combine this product with the Traxxas 5347 arm set. Using a 19 cm M4 threaded rod, you can easily connect the Traxxas couplings on both sides of the carbon tube. Generally you would need 6 tubes for a Delta printer.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A federal grand jury in Alexandria, Virginia, indicted former CIA agent John Kiriakou for releasing classified information to journalists that included the identities of a \"covert CIA officer\" and information on the role of \"another CIA employee in classified activities.\" The Justice Department charged him with one count of violating the Intelligence Identities Protection Act and three counts of violating the Espionage Act, along with a count for \"allegedly lying to the Publications Review Board of the CIA\" so he could include classified information in his book.\nThis is the official explanation for why Kiriakou is being charged, however, many recall he went public in 2007 and spoke out about the CIA's waterboarding of Abu Zubaydah in an ABC News broadcast. He did not oppose the \"enhanced interrogation technique,\" which was used on Zubaydah 83 times. He asserted the technique had worked and he was glad it had been used. But he also called waterboarding \"torture\" and wondered why this is what America had to do to be \"safe,\" leading many to consider him a whistleblower.\nThe charges result from an investigation that was triggered by a classified defense filing in January 2009, which contained classified information the defense had not been given through official government channels, and, in part, by the discovery in the spring of 2009 of photographs of certain government employees and contractors in the materials of high-value detainees at Guantanamo Bay, Cuba. The investigation revealed that, on multiple occasions, one of the journalists to whom Kiriakou is alleged to have illegally disclosed classified information, in turn, disclosed that information to a defense team investigator, and that this information was reflected in the classified defense filing and enabled the defense team to take or obtain surveillance photographs of government personnel. There are no allegations of criminal activity by any members of the defense team for the Guantanamo Bay detainees.\nAdd to that the tacked on charge of attempting but failing to \"trick\" the Publications Review Board into letting him publish classified information, the case seems to be wholly motivated by politics and retribution.\nThis prosecution further chills the climate making it even more difficult for journalists to get sources to talk to them for national security stories. It also sends a message to former government employees to not talk to or share information with Guantanamo defense attorneys. His book also showed CIA torture to be regularly used so it sends a message to former employees involved in the \"war on terror\" to be careful what they publish on counterterror operations for public consumption.\nKiriakou is the sixth person to be indicted under the Espionage Act by the Obama Administration. The previous five, all targeted in the span of a year, were: former NSA employee Thomas Drake, who shared details on the agency's warrantless wiretapping program; former CIA officer Jeffrey Sterling, who was charged with providing information with New York Times reporter James Risen on an operation that might have led to Iran getting information on how to build a nuclear bomb; former FBI linguist Shamai Leibowitz, who received twenty months in prison for who knows what exactly; former State Department arms expert Stephen Kim, who under assignment shared information with Fox News reporter James Rosen on North Korea's possible response to criticism of the country's nuclear program; and Pfc. Bradley Manning, who allegedly leaked classified information to WikiLeaks that included the \"Collateral Murder\" video, Afghan and Iraq War Logs, the US State Embassy cables, etc.\nPeter Van Buren, another victim in the war on whistleblowing whom the State Department recently moved to fire for linking to a WikiLeaks cable in a blog post and publishing a tell-all book on the State Department's Iraq reconstruction program, called the crackdown on whistleblowers \"free speech hypocrisy.\" The US government condemns other countries like China, Iran or Syria for cracking down on journalists and others who speak out, but Americans who speak out here at home are harshly prosecuted.\nThe use of the Espionage Act to go after individuals who have shared information with journalists or media organizations is a stunning attack on speech and press freedom. It is a calculated effort to make an example out of these individuals so that current and former employees do not get in the way of the work of agencies or departments in the US government. But, more importantly, this is where the Obama Justice Department is investing resources.\nHas the Justice Department prosecuted bank executives that engaged in massive fraud that helped create the economic collapse in 2008? Are they investing resources and taking risks to get these individuals? What about the people who committed war crimes, like torture? Is the Department going after these individuals or are they just interested in the people who blew the whistle on war crimes, including torture? Are Justice Department employees focused on ensuring there is no challenge by former government employees to the national security industrial complex of America?\nThe Obama Administration will spend capital to push the limits for prosecutions under the Espionage Act. It won't spend capital to push the limits for prosecutions of people who violated the Convention Against Torture or other treaties and legislation, which prohibits the cruel and inhuman treatment of individuals detained and interrogated in prison.\nTo read the full indictment against former CIA agent John Kiriakou, go here.\nThis is the official explanation for why Kiriakou is being charged; however, many recall he went public in 2007 and spoke out about the CIA's waterboarding of Abu Zubaydah in an ABC News broadcast. He did not oppose the \"enhanced interrogation technique,\" which was used on Zubaydah 183 times. He asserted the technique had worked and he was glad it had been used. But he also called waterboarding \"torture\" and wondered why this is what America had to do to be \"safe,\" leading many to consider him a whistleblower.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Following the Cambridge Analytica (CA) privacy train wreck that has been the past two weeks, Facebook says it's going to reach into the 20 or so dusty corners where it's tucked away privacy and security settings and pull them into a centralized spot for users to more easily find and edit whatever data it's got on them.\nThe changes are due to arrive over the coming weeks.\nIt gave details in a blog post on Wednesday.\nLast week, CEO Mark Zuckerberg announced a crackdown on abuse of Facebook's platform, strengthened policies, and pledged an easier way for people to revoke apps' ability to use their data.\nThe core of the data analytics personal data-gobbling scandal is, of course, how very, very easy it's been for apps to get at that data. \u2026 And how precious little Facebook has done to police those apps. \u2026 And the near-nil steps Facebook took to verify that the data of 50 million Facebook users inappropriately shared with data analytics firm CA had in fact been deleted (it hadn't).\nWe've heard loud and clear that privacy settings and other important tools are too hard to find, and that we must do more to keep people informed.\nThe changes, not surprisingly, put the onus on users to delve into what data Facebook has on them. The changes don't speak to the lack of vetting Facebook has put app developers through.\nPulling the Facebook plug will put you into good company: the #DeleteFacebook movement includes such luminaries as Elon Musk and comedian Will Ferrell, for example.\nPost-CA damage control at Facebook also includes diminishing its cozy relationship with data analytics firms such as CA and AggregateIQ (AIQ), an analytics firm tied to CA that recently, allegedly left its code lying around, open for all to access.\nAs CNN reports, Facebook announced, also on Wednesday, that it's cutting third-party data providers out of ad targeting by shutting down a tool that enables advertisers to target users with information gathered by external data brokers.\nWhile this is common industry practice, we believe this step, winding down over the next six months, will help improve people's privacy on Facebook.\nThe \"Move Fast and Break Things\" chickens have come home to roost. I kinda feel sorry for them.\nI can't see how Facebook could verify that the data had been properly deleted. Even if a facebook official scoured their computers and servers for any trace of their data, they could have a copy stashed on a DVD somewhere.\nBesides, I would be very hesitant to give a facebook employee or even a trusted third party investigator full access to my servers. Even most of my own employees shouldn't have that level of access.\nI was thinking along those lines, too \u2013 but Facebook could at least have demanded an explanation of the procedure used to delete the data, an affirmation that the procedure was followed, and some evidence to support the claim.\nOkay, stupid question, Lisa. Why does Sophos advise downloading my personal data from Facebook before deleting \u2013 not suspending \u2013 my Facebook account? Doesn't that data get deleted along with my account? Or does downloading it in some way take it away from Facebook (not likely)? I've long had my Facebook settings on maximum security \u2013 and I've never downloaded the Facebook or Messenger apps (I user the regular browser site even on my phone) but just being a part of that platform makes me uneasy these days.\nIt's just a precaution \u2013 we're saying take a backup when you can because you won't have the option later.\nAre we sure that fully deleting an account will actually Delete the data on facebook's servers?\nAs I understand, it never did previously as facebook said it became their property once it was posted\/uploaded. The data would just be flagged as not to be published. ???\nIIRC, Facebook offers \"deactivate your account\" (which means you can return later and your data will still be there if you change your mind) and \"delete your account\" (which really does zap your data).\nIt's Cambridge Analytica that you should be worried about having your data. They have it, they will not give it up, and they are using it to manipulate each person they can for the political agendas of their criminal customers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Home > Products > Get Well Soon!\nWhen your favorite college kid is under the weather, send them the Get Well Soon care package! These gifts of good health from Gumi- Aid Band Aids to Campbell's Chicken Noodle Soup can help deliver a little tender loving care all the way from home.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Linda Thompson has won awards for her poetry, short fiction and screenwriting. Like many writers, Linda spent years in full-time employment, writing in her spare time. All that changed when, in 2001, she won the BBC Talent Drama Writer competition. Since then, she has been able to earn her living as a writer and has over forty hours of television to her credit, becoming a member of the core writing teams for BBC1's Casualty and Doctors. Her poetry and short story successes include the Lake Aske Memorial Award and the Bridport Prize.\nLinda is an experienced creative writing lecturer, having devised and run a popular, well-attended course in scriptwriting for the University of Liverpool Continuing Education programme. She has also taught on short story modules at Liverpool John Moores University. She has an MA in Writing with Distinction from LJMU. lives with her husband, in an apartment overlooking the Mersey waterfront.\nLinda will be talking about her writing journey and what it is like to be a scriptwriter for one of daytime's most popular and well-loved shows.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"It was a time before Tesco and Next. A bygone era of boot-makers, tailors and ironmongers.\nLike other towns 100 years ago, Altrincham, in Cheshire, once thrived with a host of independent traders - rather than the generic chain stores we now see.\nSadly, today, it has become a 'ghost town' - littered with empty shops as the recession takes hold.\nAnd now, an amazing Edwardian map has been found which reveals just how much its high street has changed.\nDated 1910, the map of the historic market town is surrounded by adverts for clock-makers, horse and carriage hire, tobacconists, butchers and a theatre among others.\nLocal fish, game and poultry salesman WD Kenworthy promises: 'All kinds of poultry and game larded and boned' while the former Central Theatre vows: 'Always a first class entertainment free from vulgarity'.\nThe map had originally been published to coincide with the opening of the Altrincham and Sale Chamber of Commerce.\nNow over a century on, only one of the 17 family-run firms - the former Bowdon Hotel (then the Bowdon Hydro) - is still in existence but even that is now part of the Mercure chain.\nRecent figures showed more than 80 stores - 30 per cent of the town's retail buildings - stood empty and boarded up - topping a league table across the UK.\nThe map had been gathering dust in a garage before being discovered among a pile of old football programmes by retired businessman Noel White, 82, a life vice-president of the FA who has lived in the area all of his life.\nMr White, who was chairman and director of Altrincham FC for 25 years and director of Liverpool FC for 20 years, said: 'I've lived here all my life and it's still a lovely place to live. But when I was younger there were hardly any empty shops. Now there are many.\n'There are signs around Altrincham which say \"historic market town\" and you look at certain areas and there are big blocks of shops that are unoccupied and shuttered up.\n'I think one of the worse sites is there have been some disputes about planning permission for things to be built in place of the shops that are empty.\n'There used to be more local shops you would see going into the centre of Altrincham. There were little pockets of 4-5 shops.\nHigh inflation, low wage rises and less job security for shoppers has led to more than one in ten vacant shops on British high streets.\nThe figures from the British Retail Consortium last August revealed that 11.2 per cent of high street and shopping outlets were vacant at the end of May 2011.\nAlso in the figures, the number of people visiting shops, known as 'footfall', fell by 1 per cent in the three months to July.\nIt paints a grim picture for Britain's shopkeepers. High price inflation is reducing shoppers' spending power at a time when wages are rising by a lower amount, if at all.\nWhile overall levels of employment have held up better than expected during the recession, the threat of job losses has led to households cutting their spending.\nThe BRC said that retailers had suffered more in areas where there is high proportion of public sector workers, with Northern Ireland and Wales seeing the worst falls and also registering the highest level of shop vacancies.\nDiane Wehrle, research director at Springboard, the report's co-author, added that retail parks and malls are now being affected as much as town centres.\nHe added: 'I used to just go and get the rations. It was just local, just corner shops. These corner shops have gone, you could buy everything then with your ration allocation.\n'Some of these trades are still going, but in different forms and shapes, but obviously in different names.\n'There's none of them operating under the same name. Even the pub you would have thought that would have carried on - but that went into a shopping centre development at that end of the town about 35 years ago.\n'You get very few ironmongers. They're just absorbed into B&Qs and alike. It's a sad thing, but it's 2012 we're living in now so things do change.\n'That's today's world. Even the general post office has lost its individual main thoroughfare location.\nMr White himself opened a music shop, White and Swales, at 2 Cross Street, Altrincham with former Manchester City FC chairman Peter Swales and ended up with a total of 15 branches.\nHe added: 'The local chamber of trade does their best to encourage people to come, with special events in certain areas of Altrincham every few weeks, farmers markets and that kind of thing to bring people into the town centre.\n'They've revamped car parking charges to a lower rate which helps bring people in.\n'It's the era we live in, isn't it? You got footballers playing for \u00a38 a week 60 years ago; now they're playing for \u00a380,000 per week. So it's only part of life's cycle what's taken place. There's obviously more money circling now.\n'It's sad walking down virtually something like 200 yards of pavement with no shops there.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I got bit by a little black chihuahua!\nI know you are looking at those pictures of those chihuahuas and thinking \"how cute.\" Do not be fooled! One bit me the other day! Yep, you read that correctly. A cute little black chihuahua dog bit me on the back of my leg! He was very crafty and cunning. More cunning than the brown chihuahua that came out with him. Now you may wonder who gets bit by cute little chihuahua dogs. Me! That would be me! This time my enemy was not prowling around looking for someone to devour. This time my enemy attacked me from behind. He came out of nowhere, and he was quick! I heard his little pitty pat paw sounds running up behind me, but at that time I was not paying attention to them. My mind was pondering a certain worry that had been plaguing me for many months. I was completely caught off guard when I felt the pain of razor sharp teeth tearing through my calf muscle. Looking back over the event now, I can see I was out of position. Usually when I walk, I listen to praise and worship music and sing and praise the Lord. That makes my exercise time seem shorter. This day I just didn't want to do that. I wanted to worry about my circumstances. Now I am not saying had I been listening to worship, I would not have been bit; but yeah, if I had been spending my time worshiping, I might not have been bit. The enemy snuck up on me and mutilated my flesh.\n\"Watch out for those dogs, those evildoers, those mutilators of the flesh.\" Philippians 3:2 NLT.\nI wasn't on the look out for little black chihuahua dogs with razor sharp teeth that rush out at you from behind. God's word tells me to rejoice and have an attitude of gratitude in all circumstances.\nSo, I am going to rejoice in my suffering that day. God taught me very valuable spiritual lessons from that encounter with the ferocious chihuahua.\nDon't be caught off guard. Pay attention to warning signs. I heard the conniving little rascal coming but I just blew the sounds off. I never in a million years would have thought the cute little doggie would have bit me.\nDon't turn my back to the enemy. My son asked me why I didn't go up to the owner's home and tell them their little chihuahua just took a hunk out of my flesh. I couldn't do that because the only way I could keep the sweet little darling from attacking again was to turn around and face my accuser. (I mean my attacker!) Once he bit me, I whirled around and faced that chihuahua down. I wonder what would have happened had I said, \"No! In Jesus Name I say no!\" I wish I would have handled it that way, but I am not that quick-witted. Instead I faced my attacker and looked him squarely in the eye and commanded, \"No! No! Go home you filthy animal!\" I then backed away from him. I didn't dare turn and retreat. I had to face my enemy that was the only way he didn't attack again.\nThis was my spiritual lesson taught to me all because I got bit by a little black chihuahua. Every day I need to wear my full armor of God, and wear it properly. This armor was not used to protect my backside. The armor of a Roman soldier protected best when the soldier was facing and fighting the enemy. However, that day I could have used the thigh high metal sandals Roman soldiers used. These verses in Ephesians teach us to face our enemy, stand our ground, stand on every promise of God, and when we have done all that stand! Just stand there! Stand the ground that God has given you and don't let the enemy take it from you! Don't retreat and don't turn in defeat! We have a champion who is going to fight all our battles for us.\nThe New International Version. (2011). (Re 19:11\u201316). Grand Rapids, MI: Zondervan.\nThe New International Version. (2011). (Re 19:19\u201321). Grand Rapids, MI: Zondervan.\nI bow before the King of Kings and the Lord of Lords because He is my champion and he defeats all the beasts in my life!!!\nNext time you go on a walk, invite \"Charlie\" to come down and go with you! The little black dog would be a small treat for him.\nBut I do grasp your point, when you have challenges in life turn and face them because with God as your partner you have nothing to fear.\nAnd when the fight becomes to much stand still and watch God do His thing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"2 nights' half-board at the Ch\u00e2teau de Labessi\u00e8re, including drinks.\nYou'll be reimbursed for the cost of admission to Fort de La Falouse if you show your ticket at the ch\u00e2teau.\nBuilt in 1720, this ch\u00e2teau in the heart of a village on the banks of the River Meuse still has a soul. Owners Ren\u00e9 and Marie-Jos\u00e9 Eichenauer offer accommodation and half-board in 2 guest rooms with plenty of character, one of them with a fabulous balcony, and a family suite with two bedrooms. Sweet dreams ensured! There are no TVs here, or keys - a principle based on the simplicity and availability of the owners and their desire to create a sense of conviviality. Added to which there is a Louis XV lounge in which to enjoy a drink before or after dinner and a Louis XVI dining room in which to savour Marie-Jos\u00e9's mouthwatering, colourful cuisine. As to Ren\u00e9, he has plenty of tips for local sightseeing. There is a fabulous estate dotted with trees and boasting a heated swimming pool with retractable roof, recliners and an arbour, great amenities for anybody looking to enjoy a spot of relaxation. Last but not least, the owners speak German and English, and the ch\u00e2teau has secure parking but, if you don't have your own transport, you can also be picked up at the high-speed train station (Gare Meuse TGV).\nThe meals are well-known for their flavours and presentation. They are served in the Louis XVI dining room.\nExamples: Seafood Puff, Souffl\u00e9s, Truffle Ravioli, Duck Breast, Guinea Fowl Supreme, Fillet of Beef, Bergamot Cream Dessert, Profiteroles, Roast Peaches etc.\nAll the meals are plated up in the kitchen and served after guests have enjoyed a pre-dinner drink in the Louis XV lounge. They are followed by coffee and a liqueur.\nSecure parking in the inner courtyard.\nSmall park with private swimming pool in summer.\nExtensive documentation on regional sightseeing. Maps available.\nChambre parc avec terrasse priv\u00e9e 170 \u20ac par personne.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Do you recall that event Prince F\u00e9lix and Princess Claire hosted for Le Sourire de Lucie earlier this month? (If not, here's a refresher.) This week's issue of Point de Vue also has a report about the evening, which also includes new pictures of Princess Amalia. The daughter of F\u00e9lix and Claire just turned two on June 15 - and look how she has grown!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A healthier version of one of your favorite breakfast items.\n1. Whisk together milk, egg and butter. Separately mix the dry ingredients together.\n2. Add to the milk and egg mixture until just combined. Fold in some of your favorite fruit at this point.\n3. Cook in a teflon pan lightly rubbed with butter. Serve with fresh fruit and or syrup.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbirrv b/data_all_eng_slimpj/shuffled/split2/finalzzzbirrv
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+{"text":"Ground Floor Commerical Unit Of A Residential Hi-Rise Condo Building. Prime Bayview Frontage. Excellent Signage Exposure. Ample Parking In Rear Of Building With Easy Access. Public Transit At Your Doorstep. Suited For A Range Of Retail Uses.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our primary concern is the safety and well being of all of our students. The Verona Public Schools has taken a proactive approach regarding testing for the safety of lead levels in our school drinking water. The district collected samples from drinking sources in each of our six schools on April 8th in coordination with the Verona Township. Lead water testing was completed by Aqua Pro Tech Laboratories in Fairfield, NJ. A random sample of thirty (30) fixtures across the entire district were collected that included water fountains, kitchen sinks (where food may be prepared), and classroom sinks.\nThe district received the results of this testing on Friday, April 22nd. Of the thirty (30) fixtures tested, twenty-nine (29) fixtures were identified to be safe for drinking as established by the United States Environmental Protection Agency (US EPA) and the Safe Drinking Water Act. The results identified one (1) tested fixture above permissible levels for safe drinking water. This fixture was a classroom sink, not a water fountain, located at Laning Avenue School. As a result, the district immediately shut off the access to water to this fixture on Friday afternoon so the fixture cannot be used by any person. Furthermore, we collected an additional water sample at this fixture on April 23rd to certify the accuracy of the original test. We expect the results of this additional test by the laboratory in the next several days.\nWe have identified all of the water fountains in each school that have recently been tested and returned results of safe drinking water. These water fountains will be the only fixtures the district will permit students and staff to use. Any water fountain that has not been tested at this time will be shut off effective immediately as a precaution until results certifying the safety of the drinking water have been received.\nOn April 25th, the district will be continuing testing every water fountain and sink throughout the entire district. Children will only be permitted to drink from water fountains deemed safe by laboratory results.\nAlthough there is no legal requirement for school districts to test for lead in drinking water, we want to ensure the safety of each and every water fountain children may use. Please know we are taking every proactive measure possible as a safeguard for your children.\nThe district will remedy any issues identified as a result of additional testing. Please feel free to send your child to school with a bottle of water if you so choose. Each of our schools will have operational water fountains that have been tested and identified as safe drinking water sources. We will also be providing water bottles to our schools, where necessary. Thank you for your continued support and cooperation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As summer temperatures heat up, we want to inflate awareness about the importance of maintaining proper tire pressure.\nAnnually in the U.S., there are about 11,000 tire-related crashes that result in about 200 deaths, according to the National Highway Traffic Safety Administration under-inflation is a leading cause.\n\u2022 Inflate your tires according to the pressure specified in your vehicle's manual or tire information label on the driver's side door edge.\n\u2022 Properly inflating tires improves fuel efficiency, providing as much as 11 cents more value per gallon.\n\u2022 Follow your car's maintenance guidelines and rotate tires as specified. Also, be sure they are balanced and aligned correctly for optimal performance.\n\u2022 Check the tread on your tires. Tires should be replaced when tread is worn down to 2\/32 of an inch. Running out the clock on your tires can result in a blowout.\n\u2022 Many newer vehicles have automated tire pressure monitoring systems. Typically, these alerts are sent when a tire is significantly deflated, making regular checks is a good idea. Keep a tire pressure gauge in your glove box.\nNot only will properly maintained tires keep you safer, it will keep shredded tire carcasses \u2013 sometimes called \"road gators\" \u2013 from creating safety hazards on our highways, roads and streets.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our nutritional solutions are built on four solid pillars of R&D and innovative products, models and services, feed-to-food quality and safety, and sustainability. The four pillars help to develop our capacity as a one-stop solution provider. Further, our models and services have been developed in line with the needs of our customers to serve the objective of improving their productivity and profitability.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I got a new assignment a few weeks ago.\nI was feeling great. I had a comfortable office, a short commute, and to top it off, I felt like an expert.\nI felt like I had \"arrived\". I could spend my last decade of employment in that sweet spot of complacency.\nThen my boss comes in at the end of the day.\nNothing good ever happens when your boss comes to see you at the end of the day.\nNow I'm going to be honest and say that I really felt like it was a slap in the face.\nI was in a funk for a good two weeks.\nI got physically ill. I couldn't figure it out. Why me?\nI forgot Who was in control.\nThrough this experience, I have learned a lot about myself.\nI thought I had arrived, even though my journey is nowhere near complete.\nI thought I was humble, but my ego \u2013 that slippery little personality trait-it was still alive and in full force.\nAnd that's the part that hurt. Realizing that the issue was my ego. I tripped on it. I wasn't as humble as I thought.\nTo be clear, I am enjoying my new assignment. I am seeing folks that I used to work with and re-establishing old connections. My commute is longer, but less hectic. I don't have an office, but I get to engage with more people. I'm learning new things and my mind is being challenged again.\nIf I had continued to dwell on my perceived losses, my ego would have won, and my journey may have been short lived. By my choice or not.\nSometimes, especially as we get older, we believe that we shouldn't have to change. That our lives should remain comfortable and our minds at ease.\nGrowth is necessary. What does not grow, dies.\nSo, I am adjusting to my new assignment. And my ego is fine. It's not trippin' me anymore.\nOuch. Sometimes the Word stings me, too.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Discover the current value of the always high and attractive progressive Hit Super Jackpot!\nI consent to receiving news about special offers, discounts and loyalty program events using the e-mail address above.\nYou can withdraw the given consent anytime by clicking on the link at the bottom of any e-mail that you receive on the basis of this subscription.\nI confirm that I agree to the General Terms and that I am familiar with the HIT d.d. Nova Gorica's Data Protection Policy and my rights concerning data protection.\nAre you already a member of HIT Privilege Club? Log in here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Premiership will take a break this weekend as we look forward to some exciting fifth round FA Cup ties. The FA Cup is one of the most respected and well known domestic cups in Europe and is the ultimate arena for clubs from all leagues to do battle.\nHere we will take a brief look at the games due to play on Saturday.\nChelsea v Cardiff: Chelsea dropped some points through the week in the Premiership and will be looking to find that fine form which has seen them climb to the top of the league when they play Cardiff on Saturday. The Blues will be without their skipper John Terry who is away on compassionate leave, and Ashley Cole will also miss the match after fracturing his ankle during the 2-1 loss at Everton.\nCardiff's form has been up and down in the Championship. The Welsh team are in fourth spot but were hammered 5-1 by Newcastle, a result they got out of their system through the week when they beat Peterborough 2-0. Injury stricken Cardiff will be up against it on Saturday though.\nWhether Chelsea will play a full strength squad remains to be seen but Cardiff will be scrambling to find any squad so the ball is certainly in the Chelsea court.\nManchester City v Stoke City: Man City are busy in the league with their efforts to slip into fourth spot so they might rest a few players for their home tie with Stoke, but when you have a reserve team worth more than most clubs; this isn't an issue. Stoke are looking strong and may fancy their chances against the big spenders here, so expect a tough game.\nDerby v Birmingham: After thrashing Championship leaders Newcastle 3-0 during the week Derby's confidence will be sky-high when they meet Birmingham. Birmingham on the other hand are coming off a 2-0 defeat to West Ham but they have played beyond all expectations this season and may want a good cup run.\nThis should be a tight game, Derby will find it hard to break down Birmingham's tough defense and, buoyed on by the fans and looking for the win, may leave themselves open to some counterattacking.\nReading v West Brom: West Brom knocked out league rivals Newcastle in the last round and will be looking to take another Championship scalp against Reading who find themselves struggling in the relegation zone this season. Despite their league form Reading have been capturing all the headlines in this cup, knocking out Premiership teams Liverpool and Burnley to get this far, but can the underdogs keep up their winning streak against the leaders of their own league?\nSouthampton v Portsmouth: This is perhaps the biggest game of the round and has already attracted a lot of media attention. Struggling Portsmouth are hoping for a win against their rivals who are two leagues below them. The tie is promising to be aggressive and action packed, with a lot at stake for both teams; including south coast bragging rights.\nOn Sunday Bolton will be playing a lackluster Tottenham team, Crystal Palace will be hoping for a good performance to get them past Aston Villa whilst Notts County, who are the lowest ranked side left in the Cup, will need to get past Fulham at Craven Cottage if they are to continue on this amazing Cup run of theirs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We've used our brand development process to help all kinds companies reach their goals. But now we've turned our focus exclusively to nonprofits and public benefit corporations.\n\u00a9 2016 The Dave and Eddy Show. All rights reserved.\nHey\u2014where are all the pretty pictures?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"One of the first steps to manage your mortgage through Carrington Mortgage Services, LLC (CMS) is creating an online account.\nVisit our User Registration page.\nWhat is the process for making a payment after my loan transfers?\nWhen your loan has transferred to us, your Welcome Letter should show an effective date of the transfer. After this date, you can start sending your payments to CMS. For more information on payment methods, please visit this article.\nHow can I avoid late fees while my loan is being transferred? I have not received information on where to send my payment, and my payment due date is approaching.\nFor the first 60 days after the transfer effective date, we will not charge you a late fee, report to credit bureaus or treat the payment as late. We are here to help you through the service transfer and understand the confusion it may cause.\nWho will provide my Year End Tax Statement, my old or new loan servicer?\nPlease note that if your loan was transferred to CMS during a calendar year, you will also receive a statement from your previous servicer. If you have any questions about your prior servicer's statement, please contact them directly.\nA monthly statement will be generated no later than the 17th of the month (or the next business day). You can also view your statements online by logging into your account and selecting \"Billing and Tax Documents\".","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In a recent website redesign project for small business that does mainly in person CPR training we needed to replace paypal buttons with a more professional onsite checkout system. Since his classes were calendar based (past events automatically fall off the event page) we decided on Woocommerce for his store (75 products) and The Events Calendar with Events Ticket Plus to sell \"tickets\" to his classes which are limited to 9 students. This involved some customization. Although I knew upfront that The Events Calendar did not use shortcodes and generated the event page as a virtual page it did what I need most of all and that was integrate with woocommerce. The rest I would figure out.\nI thought I would share how I did one of the more vexing issues which was to put the class event list on a page underneath the existing content. My client had a SEO rich landing page which generated a lot of his conversions. We need to keep this page and have the events show below the content.\nMy plan was to utilize the list widget that already existed but I wanted the sidebar lists to just be the class name and the date and time information. I knew I could add the event content so it needed to be conditional to just the page I wanted the information to appear on.\nCopy the files to be edited to my theme folder.\nEdit the list-widget.php file to show content conditionally.\nCreate a custom widget area to appear below the content on the \"Open Class\" landing page.\nHook that custom widget area to that page.\nAdd the list widget to the widget area.\nApply any necessary style changes via CSS.\n1.The first step was to copy the needed files into my theme folder. I created a folder named \"tribe-events\" in my theme folder. Inside that I created another folder named \"widgets\" . I copied the file list-widget.php from the plugin folder to the widgets folder I just created via FTP. Whatever changes you make in this file will override the plugin file. See The Events Calendar Themer's Guide.\n2. I opened the list-widget.php file from the tribe-events\/widgets folder in my theme folder. Down around line 55 the code ends for the event duration (date and time). I added the code for adding the content along with the code to make it conditional \u2013 only show on one page. I still wanted the list widgets in my sidebar to the title and duration only. I showed the code before and after so you can see the placement of the code in context. My code starts on line 4 and ends on line 11.\n3. In my themes function.php file I created a widget area.\n4. Added a conditional hook to add the widget area to the specific page after the content. This is also in the functions.php file.(97 refers to the page ID). If you are not sure how to do this I suggest you google it or look here.\n5. Added the The Events Calendar List Widget to the new sidebar under Appearance > Widgets.\n6. Styled the button with css.\nIn this case I just needed to add some code to style the link to buy tickets.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Fighting the battle of the bulge on your body can be one of the hardest things you ever do. You can try every diet that comes along and do all kinds of extreme fitness regimens. If you don't find the right combination, you may fail. Otherwise, you may finally bring your weight down to your goal only to have stubborn fat that keeps hanging on. Weight loss surgery has finally helped you to reach your goal weight. It's time to take your results to the next level with post-bariatric surgery, which can contour your body and remove any additional fat and loose skin.\nIf you are at the point where you have undergone weight loss surgery but are still dealing with loose skin and some excess fat, post-bariatric surgery may be perfect for you. It's all about looking and feeling your best after your weight loss surgery. After all of your efforts, you deserve to have the body of your dreams. Post-bariatric surgery can optimize the results of your weight loss surgery, turning your body into something you can truly be proud of. Just because most of the excess weight is gone does not mean that you are not going to still have some excess skin, loose muscles and extra pockets of fat. During post-bariatric surgery, Dr. Stephens can address all of these issues.\nPost-bariatric surgery is an umbrella term for a combination of cosmetic procedures designed to contour the body. These procedures may include a tummy tuck, an arm lift or a breast lift, just to new a few. Common treatment areas include the back, hips, buttocks and abdomen. However, it is important to note that the procedure is completely customized for the individual.\nIf you have decided that it's time to take your weight loss surgery results to the next level, make an appointment at the office of J. Anthony Stephens, MD to learn more about how post-bariatric surgery can benefit you. You'll come in for a consultation to have your body and health history evaluated. From that point, our doctor can discuss your options. The main goal is to find the safest and most effective alternatives for you. You can finally get the body that you have always wanted. Let Dr. Stephens help you reach the finish line.\nThe office of J. Anthony Stephens, MD is located in Baton Rouge. Contact us today to schedule a consultation for post-bariatric surgery.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is my one hundred and seventy-ninth monthly teaching letter and continues my fifteenth year of publication. Since WTL #137, I have been continuing a series entitled The Greatest Love Story Ever Told, and have been expanding on its seven stages ever since: (1) the courtship, (2) the marriage, (3) the honeymoon, (4) the estrangement, (5) the divorce, (6) the reconciliation, and (7) the remarriage.\n\"1. [At Rutherford's writing], There is no expenditure on armaments in Iceland the money being spent on culture instead. Consequently, there is neither army nor navy and there are no fortifications.\n\"2. In proportion to the population, there are more books and periodicals published in Iceland than in any other country.\n\"3. The poor people speak the language of the country in the same manner as the university professors, hence there are no dialects or slang, and there is less difference between the commonly spoken speech and the written language than exists in other lands.\n\"5. Illiteracy is unknown in Iceland.\n\"6. Not only do the Icelanders accept Christianity, but the Established Church of Iceland is of the very best type, viz., evangelical and tolerant. [We in Christian Israel Identity, of course, have different standards for measuring the efficacy of churches. ed. W. Finck.] Iceland is the one country wherein the whole nation voluntarily support the Church and dwell together in Christian unity.\nSection A: \"BENJAMIN: 1. The Tribe of Benjamin was by far the smallest of all the Twelve Tribes of Israel.\n\"5. The Benjamites (Galilaeans) were the first to translate and preach the Scriptures in other languages (Acts 2:6-11).\nNote #1: \"Benjamin was born near Bethlehem, and of Jacob's twelve sons he was the only one born in the Holy Land. The name \"Benjamin' means 'son of my right hand.' Joseph and Benjamin were children of Rachel, who had no other sons. Although Reuben was Jacob's first-born he forfeited the birthright which was transferred to Joseph, the second youngest son (1 Chronicles 5:1, 2). The first-born was entitled to a double portion of the inheritance, hence Joseph's posterity formed not one tribe, but two, viz., Ephraim and Manasseh (these being the names of Joseph's two sons) and on the division of the land of Canaan these each received their own portion of territory in like manner to the other tribes. Hence, Ephraim, Manasseh and Benjamin were by blood more closely related to each other than to the other tribes of Israel.\nSection B: \"ICELAND: Iceland is the smallest civilised nation, not only in Europe, but in the whole World.\n\"Iceland became inhabited last of the European countries, and is therefore the youngest of them all.\nFrom The Youth Message, London, England. Reproduced from Destiny Magazine, January, 1948.\n\"In the following year he gave a series of lectures which proved to be most successful. Owing to their popularity, he published his lectures in book form in 1840 under the title Our Israelitish Origin, in which he traced the migrations of the peoples of Israel as they made their way across the continent of Europe to these Isles. He brings evidence to bear from Diodorus and from Ptolemy, supporting the earlier history of the Israelites. He studied the works of Rawlinson, Herodotus and Josephus and quotes extensively from Sharon Turner.\n\"The Icelanders are also very liberty-loving and tolerant in religion. The Established Church of Iceland is the Evangelical Lutheran Church. 'The Icelandic Church has always been liberal in her views, even the so-called orthodox branch of it.'4 As a result of the great tolerance exercised by this Church, sectarian bodies are extremely few and inconsiderable. The fact that the census of 1930 showed that there were only 1,503 people in the whole of Iceland who do not associate themselves with the Established Church, speaks very creditably for the nation.\n\"The oldest known translation of the Scriptures (O.T.) in any living language is that in Icelandic (12th century).\n\"Iceland is pre-eminently the enlightened and cultured nation of Christendom.\nNote #3: \"Iceland, Handbook by Thorsteinn Thorsteinsson, pp. 120-122.\nNote #4: \"Ibid., p. 122.\n\"From the foregoing it will be seen that we could appropriately speak of Iceland as the Benjamin of Christendom. As little Benjamin ultimately became the great light-bearer, can it be that little Iceland is destined to become a great light to the nations? The Biblical prophecies indicate that it will be so.\n\"Both Christ and Daniel inform us that the present age is to culminate in a climax of unprecedented trouble upon the nations (Matt. 24:21. Daniel 12:1). That terrible trouble which will smash to pieces all tyrannical and unjust institutions that oppress humanity (Zephaniah 3:8; James 5:1-4), is to be but the painful transition which will usher in a new and better age, the Golden Age of prophecy. This is repeatedly declared in the Bible. For example, that world-wide trouble and what follows is described in Zephaniah's prophecy just referred to: verse 8 depicts the trouble, but verse 9 following declares, 'For then (after the trouble) will I turn to the people a pure language, that they may all call upon the name of the Lord, to serve Him with one consent.' Again, Isaiah, chapter 34, gives a vivid picture of the world tribulation, whilst the whole of the next chapter, 35, tells of the glorious time to follow, and is one of the loveliest chapters in the whole Bible.\n\"1. The place is insular, for the songs of praise are stated to rise 'from the sea', 'from the isles of the sea'. The Old Testament of course was written in Hebrew, but the Hebrew word for 'sea' and for 'west' is the same, namely 'yom'. As there are no islands of any consequence that are not in the sea, the proper translation evidently is, 'isles of the west'. The only important islands in the west are the British Isles and Iceland, but the following considerations show that one of these islands in particular is referred to in the first place, and that it will be the people of that island who will ultimately incite the inhabitants of the other isles to sing also.\n\"2. These insular people are described in the prophecy as living amidst fires. In our world the only fires of nature that we know of are volcanoes, and Iceland is the only one of these islands that has active volcanoes. Not only so, but in proportion to its size, Iceland contains more volcanoes than any other country in the world. In modern times, the tremendous outpourings of white-hot lava in Iceland are unexampled anywhere else in the World. As comparatively recently as 1783 the great Laki Craters threw out a lava stream about 45 miles in length and 15 miles in breadth. Stef\u00e1n Stef\u00e1nsson in his book on Iceland says that this is 'by far the greatest on record and 'unparalleled on the Earth in historic times'.' So far as is known, this is the biggest fire the World has ever seen since man was upon the Earth. The whole country of Iceland was formed by fire; it owes its existence entirely to volcanic action, and the many geysers and hot springs to be found all over the island are connected with the volcanic fires below. Iceland is known to Europeans as 'The Land of Fire.'5 Truly, the Icelandic nation has lived amidst fires!\n\"The Hebrew word 'urim' translated 'fires\" in the above quotation from the English Bible also means 'lights'. The only great natural lights in our world are the Polar Lights (The Northern Lights or Aurora Borealis and the Southern Lights or Aurora Australis) and the only island nation living sufficiently near either Pole to see these lights vividly from all parts of their country is Iceland.\n\"In considering the Wolf as the emblem of Benjamin, we should remember that, after the division of the Twelve Tribes into two nations, the Tribe of Benjamin was so closely tied to the much larger Tribe of Judah, and for so long a time, that its own identity was partially obscured. Further, during the nearly four hundred years that these two tribes, Judah and Benjamin, formed the Kingdom of Judah, the national emblem of the kingdom was the Lion of Judah. Consequently, the Wolf was seldom used and in time became more of a tribal memory than a used emblem.\n\"However, there is some evidence which suggests that among the Northmen or Norsemen, the people who formed the northern wing of the Saxon migration across Europe, there were some who used the Wolf as an emblem. Many of these settled in Scandinavia, giving their name to Norway and later to Normandy in France.\n\"'It is a matter of the greatest interest that Hugh Lupus, nephew of William the Conqueror, used the Wolf as his personal device at the time that Heraldry was first introduced.\n\"As we continue our search among the Celto-Saxon nations for heraldic evidence of our Israelitish identity, we now turn to Iceland. While the number of emblems used by this section of the Celto-Saxon people is not large, those which they do have are very interesting.\n\"It will help our understanding of the significance of these emblems if we realize that, because of its geographical position in the midst of the stormy North Atlantic, there has been little immigration into Iceland since the country was first settled in the 9th and 10th centuries by Norsemen from Scandinavia and Denmark. Consequently, almost all of the people of Iceland have a Norse ancestry and, as such, are descendants of the northern wing of the great Saxon migration which swept north-westward across Europe from the western shore of the Caspian Sea, in the centuries immediately before the beginning of the Christian Era.\n\"Having had little contact with other peoples for a long period of time, the legends, folklore and emblems of the people of Iceland must have been derived from their Norse ancestors and before that from the Saxons. Evidence of this is to be seen in the fact that at least two of their national emblems are also the emblems of other sections of the Celto-Saxon people. Consequently, as the Saxons first appear in history in Iberia and Albania on the western shore of the Caspian Sea, it follows that the legends, folklore and emblems of the Icelandic people must go back to the approximate time and place where the Israel tribes disappeared and the history of the Saxons began.\n\"The importance of this becomes apparent when we consider the fact that when the Norsemen first arrived in Iceland they settled in, or were divided into, twelve communities or districts. While this may have been the result of local conditions, nevertheless, it is certainly reminiscent of the Twelve Tribes of Israel. Interesting, too, is the fact that even as the Twelve Tribes of Israel were divided into four Brigades of three tribes each, so also the twelve original settlements in Iceland were divided into four districts or provinces.\n\"Viewing this similarity from a heraldic standpoint, perhaps we should also note that the emblem on the shield in the original Icelandic Arms consisted of twelve bars. These could symbolize either the original twelve communities of Iceland, or the descent of the people of Iceland from Twelve-tribed Israel or both.\n\"Having noted the heraldic significance of the Man and the Bull in the Arms of Iceland, let us now consider another creature which appears above the Bull. It is called a Vulture. Yet, as represented in these Arms, it bears very little resemblance to a Vulture. Rather it has the appearance of an Eagle.\n\"The possibility that, in the ancient folklore of the Norse people, this third Land Guardian may have been an Eagle is supported by another of Iceland's traditional emblems. This one, called a Falcon, is not now used in the national Arms but it remains one of Iceland's traditional emblems. The use of the name Falcon, however, tends to obscure the fact that the falcon is the nearest relative of the eagle, and that the Eagle was one of the ancient emblems of the Norsemen from whom the Icelandic people are descended.\n\"Further, the Eagle was the chief emblem of Norway and Denmark during the years that people were leaving there to settle in Iceland.\n\"So, in view of the fact that two of Iceland's four emblems, the Man and the Bull, are identical with two of Israel's four Brigade emblems, that Iceland's Falcon emblem seems to have been derived from an older Norse emblem, the Eagle, and that one of Israel's other Brigade emblems was an Eagle, we accept it as a reasonable deduction that in early Icelandic or Norse times, the emblem now called a Falcon was an Eagle and that this is also the true origin of the Vulture emblem in the Arms. Leif Ericson's Raven emblem may also be only a variation of the ancient Eagle emblem of the Norsemen.\n\"What then of the Dragon, the fourth supporter in these Arms? Having seen that the Bull and the Man are two of Israel's four Brigade emblems, and if we accept the assumption that the Vulture emblem was originally an Eagle, then the fourth emblem should be a Lion for, as we have seen, the emblems of Israel's four Brigades were a Lion, a Man, an Ox and an Eagle. How then does a Dragon appear in the Arms of Iceland in the place of the Lion?\n\"We are aware, of course, of the legend of the Land Guardians, but this is hardly the answer. Rather this is probably a distortion of a Norse folk tale which was told before Iceland was settled. Further, it implies that the Land Guardian emblems were derived from the legend, whereas the truth is more likely to be that the legend was derived from the emblems.\n\"However, in returning to our question, we should note that at one time Iceland did have a Lion among its heraldic emblems. This was not the tawny [brownish-yellow] Lion of the Brigade of Judah but the Rampant Red Lion of the Zara Branch of the Tribe of Judah. Having used this emblem in the past, why was it not used in the Arms? Certainly there is no Dragon in the heraldry of ancient Israel and, as Iceland's other emblems proclaim its Israelitish identity, why was the Dragon chosen?\n\"To this question we have, so far, been unable to find a satisfactory answer. An interesting point about it, however, and one which may eventually provide a clue, is that this same substitution of a Dragon for a Rampant Red Lion also occurred in Wales. Originally the Rampant Red Lion was the emblem of all Britain, but in Wales, at some time in the past, this Lion was replaced by a Dragon.\n\"Hidden away in the early history of Iceland there is an interesting item which, though not a part of Icelandic heraldry, is, nevertheless, related to an emblem which appears in the heraldry of some of the other sections of the Celto-Saxon people.\n\"Though previously an independent republic, Iceland, in the year 1262, came under the rule of the King of Norway who appointed a viceroy (Earl) to govern the country for him. As evidence of his appointment and authority, the King gave this viceroy an emblem (a grant of Arms) and a Horn. Likewise, in the early history of York Minster, we have a record more than a thousand years old of a transaction by which a Saxon or Danish nobleman of Yorkshire gave a piece of land to York Minster by a Deed which consisted of a Horn filled with wine. They still have the Horn.\n\"While a further study of this use of a Horn as a Deed of land and as a symbol of authority might yield interesting results, we do know that the Horn is an heraldic device in the Arms of several places in Britain, the Netherlands and Denmark, and also in the Arms of several of the Scottish Clans. Further, as we saw in our study of Israel's tribal emblems, a Horn was one of the emblems of the Tribe of Ephraim.\n\"As an exclusive Israel emblem, a Horn would have no meaning for any other people. Consequently, its use by the King of Norway as a symbol of delegated authority and its recognition as such by the people of Iceland, is one more bit of evidence pointing to the Israelitish origin of the Norse people both in Iceland and Norway.\nBe that as it may, Rutherford has provided us with substantial evidence that indeed Benjamin served their allotted \"seven times\" punishment (actually their divorce period).\nWhat really happened in 1917-1918 was that the British drove the Turks out of Palestine, and it came under the rule of king George VI, a bona fide descendant of David. Therefore, neither Balfour, President Truman, nor the United Nations had any Biblically lawful authority to grant Palestine to the Edomite-jews! Today, the TRUE house of Israel and the TRUE house of Judah are in their \"appointed place\" to \"move no more\" (2 Sam. 7:10)!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Applied Biological Materials (ABM) Inc. is a Canadian company that is constantly on the lookout for the latest innovations for life science research and drug development. ABM develops novel products and services to researchers who want the highest quality at the most competitive pricing. Product lines comprehensively cover many cutting-edge technologies, such as CRISPR\/Cas9, cell immortalization, adenoviral and lentiviral expression systems.\nTranscriptional Activation by CRISPR\/Cas9 SAM New!\nFor transcriptional activation by the CRISPR\/CAS9 system (CRISPRa) a Synergistic Activation Mediator (SAM) is fused to the catalytically inactive dCas9 protein. dCas9-SAM can be targeted to the promoters of specific genes via sgRNA and strongly increase transcriptional activity.\nWith years of experience in cell immortalization, our partner ABM has developed the most comprehensive cell immortalization product line.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"YWCA Bucks County will host its 28th Annual Salute to Women Who Make a Difference Awards on Thursday, May 16, 2019, at 6:00 p.m., at Spring Mill Manor, Ivyland.\nThe Salute to Women Who Make a Difference is a hallmark event of the YWCA that proudly honors female leaders who live and\/or work in Bucks County. Each year, the YWCA Salute to Women recognizes women in Bucks County making exceptional contributions to their businesses, organizations, and their communities. Over the last 27 years, the YWCA has honored hundreds of women while raising much-needed funds to support our mission of eliminating racism, empowering women, and promoting peace, justice, freedom and dignity for all.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Traditions at Winter Haven is a new active adult community. This is the ideal location to sit by the lake or explore the many nature trails within the community. If you are feeling more active you can go for a dip in the pool or play tennis while you spend your days enjoying the Florida sunshine. Keep active with the on-site fitness center and aerobics studio, or simply socialize at the clubhouse. Whatever your interests, this stylish new active adult community will ensure a relaxing retirement.\nLocated in Central Florida, Traditions is connected to all of the things that Orlando has to offer. From world-class restaurants and shopping centers to theme parks and recreational spots, there are plenty of options to help you lead the life you want to live.\nTraditions offers residents all the luxury available with resort-style amenities, as well as a wonderful community to enjoy their retirement. The spacious homes at Traditions, with their beautiful interiors and fully maintained lawns are great for both cozy family dinners and entertaining friends. This community also offers a clubhouse with a fitness center and aerobics studio, a sparkling swimming pool to relax and unwind with after a long day, as well as a basketball and tennis court for those who enjoy a competitive match. There are also trails through the lush foliage and lakes that surround the community, where residents can enjoy nature and the natural beauty of Florida.\nThis active adult community features amenities designed specifically to keep residents on their feet. The state-of-the-art clubhouse includes a swimming pool and a fitness center with an aerobics studio. There are walking trails throughout the community as well as multiple sports courts for basketball and tennis. Traditions Estates also has an HOA.\nAerobics Studio: Located in the fitness center, the aerobics room hosts group fitness classes throughout the week.\nBasketball Court: One of the many sports courts located throughout the community.\nClubhouse: The 19,000 square foot clubhouse includes a fitness center, aerobics studio, library and computer room, ballroom, billiard tables, and a card and crafts.\nThe clubhouse also hosts planned social events such as movie nights, dinner dances, barbecues, yoga, and zumba.\nFitness Center: The fitness center includes an aerobics room with group exercise classes including yoga and zumba.\nLake\/Pond: The on-site lake includes a dock and boat slips. Perfect place for residents to try their hand at fishing.\nSport Court: Multiple sports courts include basketball, bocce, tennis, and shuffle board.\nSwimming Pool: The resort-style swimming pool is heated and includes a spa and sundeck.\nTennis Court: One of the many sports courts throughout the community.\nTrail: Walking trails throughout the community.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"This week The Admirable Admiral closes up early at the last video store near you to have another insightful Drink and Think. This week's co-host is none other than Mr. Editz, who enlightens us on his journey in life.\nThis week The Admirable Admiral covers a solo shift at the last video near you, to look back and review the first season of the Netflix series Voltron.\nThis week in the last video store near you The Admirable Admiral and Eric Praline get together to talk about and review the new Halloween, Luke Cage cancelation, and much more.\nThis week The Admirable Admiral and Eric Praline get together in the last video store near you to discuss and review the original 1978 Halloween, talk fun facts, and Iron fist being canceled and much more.\nThis week The Admiral and Eric Praline get together in the last video store near you to discuss the new Pet Sematary trailer, Venom news, and and tv news.\nThe week The Admirable Admiral and Eric Praline get together in the last video store near you to talk trailers, Mr Rogers Bio pic, and comic book movies.\nThis week The Admirable Admiral and Eric Praline close out their shifts at the last video store near you discussing 1991 classic The Adams Family, the Joker's new makeup, Marvel News, Space Jam, and much more.\nThis week The Admirable Admiral and Eric Praline celebrate 3 years of Cinema Gems with a review of \"The Predator\" the latest installment in the franchise. Join the crew in the last video near you to celebrate.\nThis week The Admirable Admiral and Eric Praline are back in a video store near you to talk about the original Predator from 1987 and they also get into some talk about James Bond, Robocop, and much more.\nThis week as the Admirable Admiral has a solo shift at the last video store near you he is joined remotely by Mr. Jim Strange Pork and they discuss and review Happy Time Murders and much more.\nThis week in the last video store near you The Admiral is joined this week by Commander Chaos to review the Robert Rodriguez family film, \"Spy Kids\"\nThis week the Admiral is joined by his special guest Dr Almond Kringle for a good ole fashioned drink and think!!!\nThis week The Admiral and Eric Praline (Yes Eric Praline is Back!!!) get together in the last video store near you to review Men In Black, James Bond debate, Dora The Explorer movie, and much more.\nThis week The Admirable Admiral is rocking another solo shift in a video store near you, talking about and reviewing Christopher Robin, The Facts Of Life reboot, and much more with The Honorable Tesla.\nOn a solo shift this week, with only the Honorable Tesla to keep him company, The Admirable Admiral talks shop without Mr. Praline this week, about the new Mission Impossible movie.\nThis week in the last video store near you, The Admiral and Eric gather together to discuss this past week's Comic Con trailers, James Bond Villains, & and much more!!\nThis week The Admiral and Eric Praline get together in a video store near you to talk about Baby Driver, Bill and Ted 3 news, Walking Dead News, and much more.\nThis week The Admirable Admiral and Eric Praline get together in a video store near you to discuss and review And-Man And The Wasp, Spiderman 2 news, and much more.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Ecstatic Birth Body Series was absolutely stunning. Sheri Winston blew the socks off the participants with her tour of a Woman's anatomy and all the connections she drew between the sexual and birthing processes. I received so many acknowledgments from participants like this FB post: \"First class of the Body Series and I'm already a whole new woman. The bliss door is open!!\" I am so proud of this baby being born, of myself for the part I play in its creation and nurture and growth. And yet, when the first session ended, I was trembling head to toe. Trembling with an overwhelming feeling of SHAME.\nShame. Can you believe it? There I was on the cusp of an incredible personal and professional moment and I had shame running through my veins. What was that? And I remembered my second birth, my daughter's ecstatic delivery that taught me that all of this is even possible. I had joy and sunshine beaming from my cells for weeks after that birth! But in the hours directly after her birth, I was trembling with an intense anger. Anger. What was that about?\nOh my, ladies. Birth, it cracks you wide open. It has the potential to be the deepest most profound healing experience of your life. It is a rite of passage, no doubt, to support your growth into becoming the mother you need to be for your child. I now know that the anger, after my daughter's birth, was the release of all the trauma I held in my tissues from my first birth. In opening wide, in being fully present in each moment, in allowing another baby to come through a place that had formerly been cut with a knife\u2026 yow\u2026all of that suppressed rage had been released, unbottled, and ultimately cleared. It's a wonderful thing that I don't walk with that anger within me anymore.\nAnd the shame? We live in a world that doesn't understand sexuality as sacred. Instead, it is seen as dirty, often perverted and distorted in its expression. Think about the biggest insults in the English language, and you can see we've got big issues valuing our sexuality. So, for me to have created a course that asserts that sexuality and birthing are part of the same beautiful process?!? To discuss how bringing sexual energy into the birthing room is not only ok, but to have Sheri Winston, an award winning sexual educator and former midwife, come in and outline in detail all the ways that it is actually part of our physiological design? Woah. That shame was all of our cultural conditioning jumping out of my skin\u2026.\nAnd right now, I have a choice, I can put my attention on the shame, and tell myself that I am not good enough, that I was practically hyperventilating during my talk, that I have embarrassed myself, and that I should slither back into my hole and allow centuries of cultural conditioning to keep me in check\u2026.\nOr I can put my attention on the deep pleasure of knowing that I did an incredible job, that I am sharing something so fully aligned with my soul, that women all over the world are receiving this critical information, and are deeply grateful for it.\nRemember ladies, the way we birth is the way we live. And Ecstatic Birth is teaching you a whole new and ecstatic way of doing both. Come connect with women from all over the globe. We're flipping the culture's conception of birth!\nPREVIOUS POST \u2014 What's an Ecstatic Birth anyway?\nNEXT POST \u2014 10 things you can do TODAY to invite ecstasy into your life and birth!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Are you seeking a fast paced position that offers overtime? Do you have a steel processing background?\nThe Advance Group has an excellent opportunity for Production Workers with Steel Processing experience in Taylor! This is a steel processing company that provides pickling, slitting, and blanking service to the flat rolled steel market.\nThis client is looking for high-energy candidates with steel processing experience to work on the production line. They run 3 shift, all employees would train on 1st shift, then be placed on 2nd or 3rd shift. This is a temp to hire opportunity, pay rate will start at $13\/hr.\nSteel Processing Candidates Apply today!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Let it Die by developers Grasshopper Manufacture and publisher Gung Ho Entertainment is the duo's first foray into the free-to-play market on the Playstation 4. Inspired by Dark Souls' combat and 80's arcade culture, the game has you climbing a gigantic tower on a quest to reach the top.\nYou start off choosing a male or female character aboard a train to the main hub station. Naked and bare-handed you'll work your way up the tower using whatever armor and weapons you find on your way. Eventually you'll gain the ability to craft your own weapons and armor with blueprints you get from defeating enemies or finding in loot cases during your escapades.\nThere is quite a bit of customization available in Let it Die in the way of choosing how your fighter spends their level upgrade points.\nNPC's in Let it Die are quite refreshing and bring a certain personality and charm you won't find in a lot of other games. You have a skateboarding skeleton as your guide through the tower. Did I mention he calls you Senpai in a Spanish accent? While you're back in the arcade you'll also meet Detox, a young girl who'll remind you of those 80's bubblegum popping cheerleaders you never got to date. Acting as your living tutorial you'll meet Meijin, a pro-gamer who decides to mentor you in the art of playing Let it Die.\nWhile I haven't finished climbing the tower and beating the game I can safely say that Let it Die is worthy of a download and is a fantastic free-to-play title that doesn't tread on the \"pay-to-win\" aspect that most gamers are familiar with. While you can optionally pay to get coins or death tokens (continues) they don't shove it down your throat and in the 15 hours I've poured into the game I haven't had a need to pay a dime to continue playing. Let it Die feels like a full-priced title and actually has more content than some AAA games. Overall if you own a Playstation 4 and haven't downloaded Let it Die you're honestly doing yourself a disservice. Head to the Playstation Store right now and download it Senpai!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Album description: Test FancyBox skin with mp4-movies, Vimeo and YouTube videos, a Webloc html page, EXIF data reports, Info window, pdf- and inc-files.\nThe menu is made with the Bootstrap library.\nThe html code of the menu at the top is in page-header_1.inc, it uses as menu a Bootstrap navbar.\nFile page-header_00.inc contains the link to the Bootstrap css file, this css code is loaded before the common.css code.\nFile page-footer_00.inc inserts links to js files just before the <\/body> tag.\nThis example shows also the position of the other 3 possible include files for the index page.\nIf you use a combination of photos and movies, select video support via Tools \/ Preferences \/ Album. On the Videos tab is resolution 'Original video' selected. The embedded Vimeo and YouTube videos are made with weblocations.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Online dating has grown by leaps and bounds as the method of choice for many singles looking for suitable dates. The usual ways of meeting potential mates in clubs, social settings, by chance or even through mutual acquaintances can be a waste of both time and energy. Today's singles are busier than ever -- they don't have the inclination or patience to strike up relationships with a number of different people in hopes of finding their \"one and only\".\nFinding someone to date through the internet might seem drastic to some; however, it has proven to be an efficient and safe way to make choices about relationships. We are fast becoming a society dependent on technology rather than personal face-to-face contact in many facets of our everyday lives. What better way to choose someone you would like to get to know than by reading a profile, seeing a photo or accessing audio and video clips from the comfort of your living room? If you find something encouraging, then you can move forward and take the next steps.\nOnline dating does not confine your prospects to a particular region either. You can meet new and exciting people from all over the world - if you so desire. Or, of course, you can meet someone near your own hometown if you use one of the larger dating services. Online matchmaking services have millions of singles in their databases, and thousands of new people sign up daily. One thing for sure, most of them are looking for compatible relationships just as you are - whether it be friendship, companionship or marriage.\nIf you are seriously considering getting involved in the online dating scene, visit some of the dating sites on the internet just to look around and become acquainted. Look at several of the sites just to see what they are about and what they have to offer. Many services give you a grace period just for this purpose -- giving you the chance to take a cursory look to decide if this is something you would be interested in before you join and have to make a financial commitment.\nAlso, it might be a good idea to visit discussion forums and read ebooks on the subject. There are many of both around.\nI am sure you realize there are people you come in contact with anywhere and everywhere who exaggerate and lie for all kinds of reasons. People you chat with online are no different.\nAs with all things, be cautious and use common sense if you do decide to meet and begin an on-going relationship with someone you've met online.\nS.J. Peel, webmaster and author. Visit http:\/\/www.nationalgay.com where you will find a complete information site about online dating.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"OCUFA has forwarded the Status of Women Committee Award of Distinction details as attached. The deadline for nominations is May 25, 2012. Nominators must be members of an OCUFA-affiliated faculty association. For nomination information, please see www.ocufa.on.ca, \"Awards\" section.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The cooperative arose thanks to the initiative of a group of farmers of the locality of Cieza (Murcia) that were convinced that through the cooperativism they would be able to achieve better its objectives combining its forces and leaving behind the competitiveness that years ago existed and that in greater or smaller measure did that some objectives not to be reached.\nSo, in 1973, was stablished on said locality the first Cooperative of Fruits of the Region of Murcia. In the begining, the number of associates they were only 19, but currently there are a total of 150. This increased due to the fact that the cooperative members realised that through cooperativism and not competitiveness they achieved better their objectives. This encouraged others to take part in the cooperative.\nAlthough it was created on the year 1973 as S.A.T. THADER N\u00ba 4697, in the year 1992 we transformed into THADER CIEZA S.C.L.\nAssuring the programming of the production and its adaptation to the demand,\tespecially as for the quantity and the quality.\nPromoting the concentration of the offer and putting the production of our members in the market.\nReducing costs of production and normalizing prices of the production.\nPromoting practices of cultivation and techniques of production and respectful\tmanagement with the environment, especially to protect the quality of the water, of\tthe floor and of the landscape and to preserve and promote the biodiversity.\nUse of our members of environment-respectful techniques, as for practices of cultivation and as to the management of the materials used.\nIncluding in the financial forecasts the technical media and necessary humans to guarantee the control of the compliance of the norms and phytosanitary dispositions and the authorized maximum contents of residues.\nIncrement of the commercial appraisal of the products of the associates.\nImproving the quality of the products of our associates.\nConsolidate and enlarge the personnel of the industry.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Breast augmentation surgery is one of the most popular and most desired plastic surgery procedures performed in the United States, as thousands upon thousands of women undergo breast surgery in hopes of attaining the perfect chest size and shape that they've always dreamed of. If you are disenchanted with the size of your breasts, the most effective solution is breast augmentation in Beverly Hills and Los Angeles with Dr. Ghavami.\nDr. Ghavami's specific approach to breast augmentation and the remarkable rapid recovery patients enjoy has been featured on \"The Doctors,\" \"The Ricki Lake Show,\" \"Extra,\" NBC, and Telemundo, just to name a few!!\nDr. Ghavami's extensive experience performing all facets of breast surgery has demonstrated his dedication as a breast augmentation surgeon. An aspect of Dr. Ghavami's professionalism that he takes great pride in is an unyielding commitment to performing cosmetic plastic surgery procedures that look natural and organic, and comprehensively complement a patient's innate features. With breast augmentation, this entails NOT selecting an implant that is too big for the patient's breast dimensions. Choosing an implant that is too big can have short-term and long-term consequences that are detrimental to the quality of the skin, as well as the rib and skeletal structures. In addition, re-operation and revisions may become more necessary and frequent.\nDr. Ghavami has special training in a technique that minimizes bleeding and trauma to the body and decreases recovery time (to less than 24 hours in most cases); he calls it \"Flash Recovery Breast Augmentation\u00ae.\" Similar approaches are known as \"rapid recovery\" or \"quick recovery breast augmentation.\" However, more important than the name is that Dr. Ghavami is not simply using this as a marketing or advertising gimmick; he truly delivers excellent results to individuals who are interested in bigger breasts and a shorter recovery time.\nAccurate breast dimensions, skin stretch, and thickness are measured and taken into account in choosing the implant size (volume or number of ccs).\nThe implant is placed under the pectoralis muscle (sub-pectoral or sub-muscular pocket).\nThe surgery is relatively bloodless, or minimal bleeding occurs. Therefore, there is less trauma to the breast tissues and ribcage when placing the implant under the muscle. No blunt dissection is used to make the pocket, ONLY heat cautery, so that vessels do not have a chance to bleed.\nEarly post-operative motion and return to activity is encouraged.\nMost patients return to most daily activities in 24 hours.\n**To see some of Dr. Ghavami's Flash Recovery Breast Augmentation\u00ae patients and their surgeries, please see the informative videos below. More videos can be seen in our video gallery.\nAt Ghavami Plastic Surgery we want to help each Beverly Hills patient see how breast augmentation might work for them. Digital simulation with Crisalix 3D technology allows Dr. Ghavami to show each patient what the results of breast augmentation might look like before committing to the surgery. This technology allows the patient to feel more comfortable and reassured that their desired outcome will be achieved.\nVideo: Dr. Ghavami showcases his Flash Recovery Breast Augmentation\u00ae procedure on The Ricki Lake show.\nMany breast implants patients who undergo breast augmentation in Beverly Hills with Dr. Ghavami are able to resume their routine activities only one day (24 hours) after their operation. Many of our patients describe going out to dinner the same night they have breast augmentation, and many choose to return to work within a few days.\nIf you are interested in achieving larger breasts through breast augmentation with Dr. Ghavami, please call and schedule a consultation today.\nFor more information on Beverly Hills breast implants and breast augmentation using implants, please contact Ghavami Plastic Surgery today. Dr. Ghavami is a Breast Augmentation specialist on RealSelf.com.\nTo view more Quick Recovery Breast Augmentation Before & After images, please visit our Breast Augmentation Photo Gallery. To Watch Videos from TV and News Programs on Dr. Ghavami's Breast Augmentation Patients, Visit our Video Gallery.\nDr. Ashkan Ghavami's breast augmentation method \u2014 known as Flash Recovery Breast Augmentation\u00ae \u2014 aims to offer patients a speedier operation, a shorter recovery period and substantially less post-operative pain or discomfort. Traditional breast augmentation procedures can require recovery periods that may not fit into a patient's busy lifestyle; and the pain, while manageable, can keep a patient in discomfort for longer than acceptable. Approaches to breast augmentation can involve a significant amount of jerking and prying (\"blunt dissection\") with both instruments and fingers, as the surgeon forms enough space for the implant to fit under the muscle or breast tissue. That kind of blunt method is effective, but it can also cause a considerable amount of trauma to the surrounding muscle and tissue.\nDr. Ghavami's Flash Recovery Breast Augmentation\u00ae, or quick recovery breast augmentation method, employs ONLY a cautery (heat) dissection that offers the surgeon the ability to control bleeding before it occurs (known as \"prospective hemostasis\"), allowing Dr. Ghavami to cut through the sub-muscular tissue and form a pocket without causing as much damage to the ribcage and other surrounding tissue. This type of approach to breast augmentation can takes less than 45 minutes to perform, and the patient is typically able to resume most daily activities within 24 hours.\nAdvancements in technology and extensive research into new methodologies have brought the field of plastic surgery very far in a very short period of time. Dr. Ghavami studied under Dr. John Tebbetts, whose own research paved the way for the improved Beverly Hills breast augmentation that Dr. Ghavami currently offers his patients. To learn more about rapid recovery breast augmentation, contact the office of Dr. Ghavami with any questions or concerns that you might have. Schedule a consultation today.\nAs a leading breast surgeon, Dr. Ghavami also offers other breast procedures, including breast reduction and breast lift in Beverly Hills.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\u25ba\u25bc pedestrian[Edit] : A person who is traveling on foot.\nCastilian Persona que viaja a pie.\nDutch Persoon, die te voet onderweg is.\nEnglish A person who is traveling on foot.\nGerman Person, die zu Fu\u00df unterwegs ist.\nItalian Persona che cammina a piedi.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbmrce b/data_all_eng_slimpj/shuffled/split2/finalzzzbmrce
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+{"text":"[S1145] E-mail from Diane Anderson to Stuart & Teresa Pask dated 25 April 2005.\n[S384] Website FamilySearch Internet (http:\/\/www.familysearch.org) : Parents: George W. PASK and Hegea BOYERS.\n[S384] Website FamilySearch Internet (http:\/\/www.familysearch.org) : Parents: George W. PASK and Keziah ROYER.\nOn August 13 1862, when 24, Joseph enlisted in the 3rd Wisconsin Cavalry, Company D. The Third Cavalry spent most of its time travelling in Missouri, Kansas, Arkansas and some of the Indian territories, trying to defeat any Confederate forces in those areas. They scouted and served on guard duty, and fought in what was described as \"guerilla warfare.\"\nJoseph G. Pask died at his home near Osseo, Mich., June 18, two days after his 92nd birthday. He was born in Lincolnshire, England and came with his parents to America when he was a lad of six years. The family settled in Richland county, afterwards coming to Defiance county, where he lived south of the Chickasaw for many years and where his four children, three daughters and one son were born.\nSeveral years ago, he moved to his Michigan home where he died.\nMr. Pask served through the Civil war, enlisting in 1862 at Madison, Wis., and serving until the end of the war.\nMrs. Field feels very grateful to the friends and neighbors at his Michigan home and also of his former home around Farmer for their many kindnesses during his death and burial.\n[S2] GRO Indexes to Births. \"1838\/Q2 PASK, Joseph George Sleaford 14 500.\"\nPhotograph & Information from Cindy Moore, dated 22 March 2017.\n[S5716] Website Facebook Group: Pask Family History: Information from Jules Wallace Pask via Messenger on 20 August 2017.\nInformation kindly provided by Dianne.\n[S5002] Website Ancestry: Year: 1870; Census Place: Washington, Defiance, Ohio; Roll: M593_1195; Page: 194B; Image: 316908; Family History Library Film: 552694.\n[S1259] Website FamilySearch Internet (http:\/\/www.familysearch.org) : Washington, Defiance, Ohio Family History Library Film 1255011 NA Film Number T9-1011 Page Number 242A.\n[S5002] Website Ancestry: Year: 1880; Census Place: Washington, Defiance, Ohio; Roll: 1011; Family History Film: 1255011; Page: 242A; Enumeration District: 241; Image: 0708.\n[S5002] Website Ancestry: Year: 1900; Census Place: Washington, Defiance, Ohio; Roll: 1263; Page: 14B; Enumeration District: 0019; FHL microfilm: 1241263.\nPhotograph & Information from Cindy Moore, dated 15 March 2017.\nPhotograph & Information from Cindy Moore, dated 14 March 2017.\nPhotograph & Information from Cindy Moore, dated 17 February 2017.\n[S3336] Website Find A Grave: Find A Grave Memorial# 144078420.\n[S5001] Website Ancestry: Ancestry.com. Michigan, Death Records, 1867-1950 [database on-line]. Provo, UT, USA: Ancestry.com Operations, Inc., 2015.\n[S3736] Website FamilySearch Internet (http:\/\/fsbeta.familysearch.org: ) Name Rosa Ann Mason Death date 23 Jan 1941 Death place Marshall, West Virginia Marital status Married Spouse name Frank P. Mason Father name Joseph George Pask Mother name Sarah Elizabeth Reed Film number 1983558 Digital GS number 4259837 Image number 782 Reference number 762 Source Statewide.\n[S384] Website FamilySearch Internet (http:\/\/www.familysearch.org) : Batch M514071.\n[S3336] Website Find A Grave: Find A Grave Memorial# 144078441 - shows her date of birth as 16 June 1848.\n[S3336] Website Find A Grave: Find A Grave Memorial# 144078441.\n[S384] Website FamilySearch Internet (http:\/\/www.familysearch.org) : Batch M514072.\n[S3336] Website Find A Grave: Find A Grave Memorial# 155817493.\n[S5002] Website Ancestry: Year: 1900; Census Place: Farmer, Defiance, Ohio; Roll: 1263; Page: 1A; Enumeration District: 0010; FHL microfilm: 1241263.\n[S3336] Website Find A Grave: Find A Grave Memorial# 155817484.\nPhotograph & Information from Mark Strand, dated 4 April 2017.\n[S5002] Website Ancestry: Year: 1930; Census Place: Bryan, Williams, Ohio; Roll: 1891; Page: 7A; Enumeration District: 0016; Image: 283.0; FHL microfilm: 2341625.\n[S5002] Website Ancestry: Year: 1940; Census Place: Glen Dale, Marshall, West Virginia; Roll: T627_4426; Page: 14A; Enumeration District: 26-34.\n[S3336] Website Find A Grave: Find A Grave Memorial# 155817505.\n[S3336] Website Find A Grave: Find A Grave Memorial# 144180359.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Twenty-five students, faculty members and recent alumni from East Stroudsburg University's department of psychology presented research at this year's Eastern Psychological Association annual meeting in New York City, held Feb. 28 to March 2.\nPeer reviewed presentations included: Laura Mariotti, a senior majoring in psychology from Old Forge, and Irina Khusid, Ph.D., associate professor of psychology, Digital World Impacts on Real World Friendships; Zoe Maa and Jyh-Hann Chang, Ph.D., professor of psychology, Shortened Form of The Compassion of Others Lives Scale; and Julia Sule, a senior majoring in communication sciences and disorders from Bethlehem, James Roe, a junior majoring in psychology from Milford, Dominic Brown-Andriulli, a junior majoring in psychology from Effort, and Chang, Compassion Levels of Speech-Language Pathology Students.\nAlso, Imani Williams, a senior majoring in psychology from Darby, Deena Dailey, Ph.D., adjunct professor of psychology, and Sussie Eshun, Ph.D., professor of psychology, Ambulatory Mobile Device Behaviors, Health Behaviors, Cognitive Dissonance and Optimism Bias; Raquel Sosa, a senior majoring in psychology from Wilkes-Barre, Sandi-Lynn Eshun, a graduate student at Yeshiva University studying psychology, Eshun and Dailey, Gender, Ethnicity, Contact, and Empathy as Predictors of Attitudes about Mental Illness; and Sosa and Dr. Eshun, The Impact of Language on Perceptions about Mental Illness.\nAlso, Carissa Ceballo, a junior majoring in psychology from Staten Island, New York, Sean McCann, a senior majoring in psychology from Stroudsburg, and Khusid; The Relationship Between Religiosity and Coping Strategies; and Keith Young-Smith, a graduate student studying public health from Philadelphia, Taylor Bess, a senior majoring in psychology from Phillipsburg, New Jersey, and Bonnie A. Green, Ph.D., professor of psychology, How a Class Assignment Uncovered Assumptions Related to Success and Privilege.\nAlso, Ariel A. Tucci, a graduate student studying public health from Douglassville, Green and John Darsinos, research assistant for Clear Path, Differences in Academic Self-Efficacy in Community College Students; Christopher Galanti, a senior majoring in psychology from Lafayette Hill, Stephanie Hawk, a junior majoring in psychology from Allentown, and Green, Oppositional Mindset: A New Variable of Success; Anthony Drago, Ed.D., professor and chair of psychology, Everything You Wanted to Know about Being a Department Chair but Were Afraid to Ask; and Green, Darsinos, and Dr. John Protzko, a post-doctoral fellow at University of California Santa Barbara, Translating Psychology Research for DR K-12 and HBCU NSF Grants.\nT. Michelle Jones-Wilson, Ph.D., professor of chemistry and Olivia Carducci, Ph.D., professor of mathematics were invited to participate in a symposium on how psychology faculty can work with other faculty to secure extramural funding for their research. Their talk is entitled: Functional Diversity Assembling a Disparate and Successful Team to Support Research and Obtain Funding for STEM Transfer Students.\nChang was selected as a fellow for Eastern Psychological Association.\nSarah Noonan of Long Pond was recently initiated into The Honor Society of Phi Kappa Phi at The University of Tampa, Baton Rouge, Louisiana.\nNoonan is among about 30,000 students, faculty, professional staff and alumni to be initiated into Phi Kappa Phi each year. Membership is by invitation only and requires nomination and approval by a chapter. Only the top 10 percent of seniors and 7.5 percent of juniors are eligible for membership. Graduate students in the top 10 percent of the number of candidates for graduate degrees may also qualify, as do faculty, professional staff and alumni who have achieved scholarly distinction.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As a part of my post-SoTL Institute reflection, I've been giving some thought to SoTL per se. In doing so, I thought of a brief mental exercise. I shared this with Spencer and he thought it would be worthwhile sharing this with the group.\n\"You are chosen to design Annual Reviews Vol. 1 for SoTL. What will be the title of the series, and what will be your choices for the topic areas of the first ten chapters? (Better yet, who will write them?)\"\nI'll be curious to hear your answers!\nSeries title: \"Beyond good teaching: the systematic exploration of our craft\"\n2. Disciplinary approaches to SoTL - how and why do the disciplines differ?\n4. What is this thing called active learning (and when are lectures still a valid approach)?\nPart II Cases and Studies - how do we do SoTL?\nHi - here's my $0.02. I think this would be cool on the Wiki, except that I wouldn't have seen it. But you could tell us in an email to go check it out?\nIn the introduction I would also include a section on why SoTL is \"valid\" research. Many of my tenure-track colleagues respond with, \"Oh, but you don't do any real research\" when I describe the research I do in my classroom. (Granted, until now, it has been a floundering attempt at research for my own benefit \u2014 I look forward to formalizing it in the year to come!) Of our 44 faculty in our biology department, I know of only 4 who conduct structured educational research. You all are familiar with the attitude. I would love a chapter I could photocopy and put in 44 mailboxes. If for no other reason, I would like our faculty to know there are tried and tested teaching methods published and available for use in our classrooms.\nInteresting question, Jim. And good start, Teri and Mary Beth. Do we have to limit it to 10 chapters? Seems we could also add chapters on other tools we learned about (rubrics, concept maps, etc.) and the publishing and funding of all this research.\nWhich brings me to an off-topic question I wish I had asked in DC: What sources of funding are available for SoTL research? What do you all plan on using? For this year, I plan on using myself and my personal funds \u2014 but in the years to come, what's available?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bend, OR \u2013 We are in the planning stages of a Suzuki only 4\u00d74 overland adventure and would like to invite all Suzuki enthusiasts to join in. We are currently targeting either dates in late May \/ June. Due to the nature of this area a late summer activity would not be advisable. UPDATE: We have selected the weekend of June 25th continuing onto June 28th. Background: Over the years, we have went on adventures where the itinerary and the goal was not all too structured and not the point. We've been fortunate and have found some amazing things out there. Awhile back we stumbled on the idea to hit the border convergence of three states. The first one we did was the California, Nevada and Oregon Borders.. That trip was such a fun success we carried on with creating another event following the same Latitude and culminating at the Idaho, Nevada, Oregon border. This trip\u2026 is the next one over\u2026.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This AGREEMENT is signed between CC FINANCE Ltd, a British Virgin Islands Company, with the registered office located at 3076 Sir Francis Drake's Highway, P.O. Box 3463, Road Town, Tortola, British Virgin Islands, hereinafter named \"CHAINCREATOR EXCHANGE\"), and the \"Sponsor\") as the FINANCIAL SPONSOR of the above mentioned crypto COIN\/TOKEN. The FINANCIAL SPONSOR of the COIN and CHAINCREATOR EXCHANGE, as well as the above mentioned COIN\/TOKEN, together and hereinafter named as the \"PARTIES \".\na) CHAINCREATOR EXCHANGE has approved the listing of the above mentioned TOKEN\/COIN on CHAINCREATOR EXCHANGE and possible white label versions of the main exchange.\nb) CHAINCREATOR EXCHANGE will allow its member community to trade the TOKEN\/COIN against other cryptocurrencies (to \"buy and sell\").\nc) CHAINCREATOR will aim to make the TOKEN\/COIN tradable within the shortest time frame after the listing gets the final acceptance from CHAINCREATOR Exchange. The listing approval will contain the listing time prediction, which varies based on the underlying protocol.\nd)The TOKEN SPONSOR will pay THE AGREED % of the OF THE TOTAL SUPPLY, expressed in the total number of the tokens.\nThe tokens which will be paid with immediate effect after the TOKEN\/COIN gets final acceptance for listing by CHAINCREATOR EXCHANGE. This will be paid to a dedicated address indicated in the final listing approval certificate from CHAINCREATOR Exchange.\ne) CHAINCREATOR EXCHANGE guarantees a TOKEN\/COIN listing of the minimum of 36 (thirty six) months, however CHAINCREATOR EXCHANGE reserves the right to delist the coin, in case it believes at its own discretion the coin is harmful or is in any way negative for the exchange.\nf) CHAINCREATOR works on a \"best efforts basis\" only. It is here with explicitly agreed and confirmed by the sponsor that CHAINCREATOR will have no legal and financial liability towards the TOKEN SPONSOR in case the token would not become a success or if the tokens were to become stolen or hacked. The financial risk is taken by the TOKEN SPONSOR, not by CHAINCREATOR who acts in a supportive service providing capacity only.\ng) The sponsor guarantees that they WILL NOT HOLD CHAINCREATOR, ITS PLATFORMS, AFFILIATES, ITS FIDUCIARIES, OR THEIR LICENSORS , ADVISORS SERVICE PROVIDERS, EMPLOYEES, AGENTS, OFFICERS OR DIRECTORS LIABLE FOR ANY DAMAGES OR CLAIMS OF ANY KIND.\nThis AGREEMENT is governed by Liechtenstein Law. In case of a dispute, only the Courts of the Vaduz will be the authorised Courts.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbnmey b/data_all_eng_slimpj/shuffled/split2/finalzzzbnmey
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+{"text":"recovery of investments can be quick with long-term benefits.\nLighting is an important aspect of a professional environment. It not only has a cost implication, but also has considerable impact on employee productivity and comfort. Lighting solutions have come a long way, with latest technologies taking centrestage. In this article we discuss how these technologies can benefit small- and medium-scale offices, and what will be the ROI for them.\nUntil recently, small- and medium-sized workplaces opted for generic lighting setups, due to the lower capex requirement. They wanted to avoid risk of low ROI. Lack of advanced lighting solutions with a strong business value and lack of knowledge on lighting for a small office space also factored in. In turn, they ended up spending much more on electricity with no way to customise their lighting to enhance employee comfort.\nToday, businesses are recognising the importance of employee experience in improving productivity. Personalisation of the workplace can be a major motivator, and smart lighting solutions can be a capable agent.\nThis benefit is derived from the capability of smart lighting solutions to mimic natural daylight and promote a healthy indoor ambience\u2014a concept called human-centric lighting.\nSmart lights have been modernised with wireless lighting solutions. Wiring and cables not only bring heavy installation hassles and costs, but also make it difficult for organisations to move their lighting infrastructure when needed. Wireless solutions, on the other hand, reduce installation and maintenance costs.\nPhilips Lighting has developed the InterAct series of smart lighting, focussed on small and medium offices. These are available in various lighting form factors, like 2\u00d72 luminaries, downlighters, spot lighting and TLEDs. With the help of a wireless network gateway (InterAct gateway), luminaires can be configured and controlled using an app (InterAct Pro) or over the Web.\nInterAct Pro dashboard delivers insights on real-time energy usage and various consumption patterns. Lighting sensors automatically dim or power off the lights based on occupancy. Moreover, occupants can remotely control the lights to set a personalised ambience in offices.\nWith real-time energy consumption measurement and centralised dashboard-based insight generation, smart lighting solutions allow workplaces to optimise their lighting usage. This, in turn, reduces electricity expenses. Moreover, scheduled lighting provides an additional sense of convenience, as office owners can power on or off their lights following specific timings.\nInitial investments in smart lighting may be slightly higher as compared to normal solutions, depending on scale. However, in a well-planned setup, recovery of investments can be quick.\nSumit Joshi, vice chairman and managing director, Philips Lighting India, says, \"Cost uplift for smart lighting can be anywhere between 1.5 and 1.7 times higher compared to conventional lights, depending on the number of light points. But the comparison should be done based on ROI. ROI should be calculated after adding the benefits of various features like occupancy-sensing, daylight harvesting, scheduling, zoning, dashboards and personalisation to assess the energy savings from LEDs.\n\"Keeping this in mind, the usual ROI of two years in a non-intelligent lighting system becomes around three years for the intelligent wireless lighting solution, with better returns going forward,\" adds Joshi.\nWith a clear picture of all technologies and expected benefits, offices can start investing in innovative lighting solutions with long-term profits in mind.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I've previously posted about people sharing personal information on Facebook without any regard for their own privacy. On Friday the BBC ran a story about how young people could be damaging their future careers with details they put on social network sites.\nSome 71% of 2,000 14 to 21-year-olds said they would not want colleges or employers to do a web search on them before they had removed some material.\nIf you ask me they deserve to have problems getting a job. There is a level of personal responsibility that everyone should be able to shoulder. There are very few jobs that don't involve keeping information secure, and if you fail to protect your own privacy how can you reasonably expect to be trusted to do the same for others.\nYou could argue that your Facebook profile is secure if you've used the privacy controls to prevent public access to your information. But is everyone aware that by default your profile is accessible by everyone in the networks and groups you join?\nMaybe the sites themselves do need to accept a certain amount of responsibility for not making these 'features' clear, but if you're sharing information you wouldn't want a potential employer to know with any third party you deserve whatever negative effect it has.\nYour information is your identity - protect it! Whenever you submit information to a website think about what it is, how much you trust the website and whether it's something you'd want the rest of the world to know.\nWhat's baffling for me is that people are generally very protective of their privacy when a stranger calls them on the telephone, and yet they're quite willing to reveal all to a website run by people who they know no more about than the person on the other end of the 'phone.\nMaybe it's the illusion that you're sharing stuff with your friends. Maybe people don't think about the middle-man because all they see is a request from someone they do know, rather than the website through which that request is being made.\nIt's like sending someone an embarassing story by postcard. Do you trust all of the people who'll see that postcard on its journey from you to your friend? How would you feel if you knew that the postal service was going to make several copies of your postcard just in case the original gets damaged? Would you still send your story by postcard?\nStretching the analogy a bit further, how about several months later you're looking for a job and you're worried that a potential employer may know someone in the postal service who's read your postcard. You ask the postal service to destroy all copies of said postcard. They tell you they have, but there may be some residual memories in their staff that will take a while to be purged - nothing they can do about it, that's just the way the system works.\nOk, that pushed the analogy a bit too far, but the principles are in there. Any information you put on the internet, whether it's in an email, on a website or via an instant messaging services is no longer under your control. Accept that fact and remember it whenever you put something out there.\nThe internet is a stranger on the other end of the telephone. It's the stranger sitting at the next table in a restaurant while you're telling your friends about your deviant exploits. It's the guy on the bus listening to your half of the conversation you're having on your mobile, and he's gonna compare notes later with the girl listening to the other half.\nI love a good analogy on a Sunday evening. Failing that I'll settle for a bunch of bad analogies.\nI think part of the problem is that the internet is still relatively young. Due to the pace of change I think it's all too easy to forget that. When it comes to the telephone, postcards and other well-established forms of communication the parents among us know all about the dangers and most have passed that on to their kids. The internet is a mystery to a lot of them, and it's the kids who are using this new-fangled technology in place of the traditional methods.\nIt may take a few generations for the internet-based privacy message to get through, and in the meantime it's gonna be a bumpy ride. Can anyone come up with a way to speed up the process?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I hardly ever have enough bananas to make regular banana bread so when I came across this recipe a few years ago, I tried it out. The original recipe needed some tweaking and the final result is an easy, moist bread that everybody loves.\nPreheat oven to 350 degrees. Lightly grease 9x5 bread loaf pan.\nIn a large bowl, combine flour, baking soda, salt, cinnamon and nutmeg. In a separate bowl, cream together butter and brown sugar, add eggs. Stir in bananas and berries until well blended. Combine flour mixture and banana mixture until just blended. Pour into the loaf pan.\nBake for 60-65 minutes or until a toothpick comes out clean. Do not overbake. Delicious!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Announces the creation of a signature look from the John Wolf collection of fabrics in New York.\nReports that Richloom Fabrics Group grabbed the top spot in the United States textile industry, following its acquisition of John Wolf Decorative Fabrics from Cone Mills Corp. Value of acquisition; Terms of agreement; Reciprocal advantages of the companies from their deal.\nReports the estimate made by Cone Mills that John Wolf Decorative Fabrics' restructuring could amount to $5 million. Performance of the stocks of the company; Comment of an analyst about the stock market performance of the company.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Salt Lake City \u2013 A new database launched by the non-partisan, good government group ABU Education Fund will shed light on campaign finance in Utah by creating a fully searchable collection of campaign disclosure statements.\nThe open-access Utah Disclosures Search consists of raw data gathered from campaign finance reports, available on the Utah Lt. Governor's website, that have been collated into an easily searchable database.\nMaryann Martindale, executive director of ABU Education and Alliance for a Better UTAH, has described the database as making financial disclosure documents more easily accessible.\nHowever, the data is still only as good as the campaign finance laws currently in place in Utah.\n\"If anything, this speaks to the need for more transparent campaign laws and stricter campaign finance laws if we truly want open and ethical elections in Utah,\" said Martindale.\nAttorney General John Swallow is currently being investigated by the Lt. Governor's Office over allegations that he broke election laws by failing to disclose his business interests before running for attorney general. Though this database could not have prevented his actions, it may have been able to shed light on the curious and concerning donations that Swallow and his predecessor, Mark Shurtleff, were soliciting.\nABU Education Fund is a 501(c)(3) non-profit organization dedicated to progressive ideas and causes. As a statewide grassroots organization, we work to provide a strong, educational voice by creating resources that advance civic engagement. ABU Education Fund is an affiliate of Alliance for a Better UTAH.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..96f003e69af4fc4161bf7d330c0d12fe200616b4
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbocns
@@ -0,0 +1,5 @@
+{"text":"LOS ANGELES (CNS) - In an effort to reduce jail overcrowding, the Board of Supervisors today approved a five-year, $75 million contract to house 512 county inmates at a jail run by the city of Taft.\nCommunity activists had spoken out against the move for months, arguing that it is a mistake to relocate inmates 118 miles away from downtown Los Angeles, separated from family and friends who provide a link to the world outside jail.\nSupervisors Mark Ridley-Thomas and Zev Yaroslavsky abstained from the 3-0 vote. Yaroslavsky expressed concern about the source of funding, while Ridley-Thomas tried to persuade his colleagues to postpone the decision to work on a more comprehensive solution, including alternatives to incarceration.\n\"To move this issue forward at this point is putting the cart before the horse,'' Ridley-Thomas said.\nBut the county is trying to ensure that convicts who pose a threat to the community serve the majority of their sentences, and Supervisor Gloria Molina argued that the time to take action was now.\nThe lack of space in county jails means that the lowest-level male offenders serve 20 percent, on average, of their court-mandated sentence, while female inmates serve 10 percent.\nThose convicted of lower-level offenses but originally charged with violent or serious crimes, such as manslaughter, kidnapping, mayhem, robbery and child molestation, serve an average of 40 percent of their sentences before being released, according to Assistant Sheriff Terri McDonald.\nChief Executive Officer William Fujioka said that if the county didn't commit in the next 30 to 60 days, state officials would likely contract for the space at Taft to alleviate their own problems with overcrowded correctional facilities.\nOpponents argued that low-level offenders, particularly those suffering from mental illness, should be moved out of jails and into community-based facilities.\nPeter Eliasberg, legal director of the American Civil Liberties Union of Southern California, said a pretrial release program could free up 700 beds.\nThe Taft Community Correctional Facility will be paid about $60 per inmate per day, versus the $118 per inmate per day average cost to keep someone behind bars in a county facility.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Adecco is currently assisting a beauty brand in their search for a Graphic Designer! This designer will have the opportunity to utilize their creative ability and help with in store wall displays. This is a 6 month contract in NYC. If you meet the following qualifications Please Apply Now!\n- Work with a small team and be collaborative as well as comfortable working independently.\nClick on Apply Now to be considered for this position or any other Graphic Designer related job opportunities with Adecco.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"2016 was a big year for collaborations in style, and 2017 is already looking to be another big one. Well, today we get a look behind the scenes of the upcoming collaboration collection between KAWS and the Jordan Brand. The behind the scenes look does not give us too much regarding the products but instead shows the creative process behind the kicks.\nThe images find an array of different retros including an eye-catching rendition of the Air Jordan 1. The word on the e-streets is that the KAWS x Air Jordan 4 will be releasing on March 1st. Today however the focus is behind the scenes. Check out the images below and speak your mind on it after the jump. Stay tuned for much more from the Jordan Brand and KAWS coming very soon!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We, \"Chiptronics Solutions\", are engaged in Manufacturing, Supplying and Trading a wide array of Security Solution Products like Time Attendance Machines, Access Control System and readers, Smart Cards, Access Control Accessories etc.\nThe company has been established by Mr. Urmish Thacker Our entire range is appreciated for its excellent performance, high efficiency and optimum functionality. These products are manufactured utilizing optimum grade raw components and sophisticated technology. Additionally, we specialize in modifying our range so as to accommodate different application requirements of our clients.\nFurther, within the short period of time, we have earned the trust of our clients based across the globe. To maintain this, we have set up a state-of-the-art manufacturing facility, which is equipped with latest technology and high-end machines that are used for effective production of our entire range. Moreover, we have formulated simple payment terms like D\/A & D\/P and payment modes like cash, cheque, credit card & DD for the convenience of our clients. We also offer after sale service for operational & installation training for our clients. Owing to our organized management, client-centric approach and qualitative product line.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"So if someone tests 60% of the cancer cells are PD-L1 does that mean 40% are not and can't be killed with Keytruda? So you can theoretically kill 60% of the tumor cells(if the med was 100% effective) but the other 40% can keep growing?\nAnyone ever ask their doctor that? I plan too my next visit.\nHi there I don't think it works that way found this on cancer website: Although helpful in determining which patients may respond to certain drugs, the test is not infallible. Some tumors that test for high levels of PD-L1 may not respond to checkpoint inhibitors, and those that test for low levels may have a strong response. Cancer cells are complex, and a variety of factors can influence how susceptible they are to such drugs.\nI have an 11 mm spiculated nodule iny right upper lobe. It had a suv of 6.8 on PET scan. Anyone else had this happen? Was it cancer?\nI responded on your other thread in the newcomer's forum. My nodule was very similar and was cancer, but that's a very small one, and likely to be treatable by surgery alone. I wrote more details on the other thread.\nAll NSCLC cells have the chance of being killed by the immune system. What anti PD-L1 immunotherapy does is remove the PD-L1 cancer cells' ability to hide from the immune system. All of the other cancer cells may not be \"hiding\" at all or \"hiding\" very little and can be seen by the immune system much easier. So when the immune system is unleashed, most, if not all, cancer cells are visible for the immune system to attack.\nI agree with Steff on the relationship between PD-L1 markers and lung cancer cells and how immunotherapy works to eliminate lung cancer cells.\nCool. I hope so. Saw dad last weekend and he was looking IMO better. He's still just lounging around but his last radiation treatment to the spine was 2 weeks ago so hopefully that will finish leaving his body here soon. We have our second Keytruda infusion in one week.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"Feeling mind-boggled by type 1 diabetes research? Our new toolkit demystifies the science involved. Download your copy below.\nYour support means we can fund vital type 1 research. But how does this research work? Have you ever read a dramatic headline about a research breakthrough and wondered about the science behind it? Have you ever wondered why the research process takes a long time? Do you want to learn how to find, read and assess the latest research reports and findings?\nWe want this research to be accessible so we created the Understanding Science Toolkit. This handy guide will help you get to grips with the research you work so hard to fund.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When an amount of data is stored in an organized way, that is called a database.\nIn computers, a database is managed by a software called Database Management System.\nA table is a set of data values. These values are organized using vertical columns and horizontal rows. Columns are identified by their names.\nIn MySQL, CREATE DATABASE statement creates a database with the given name. To use this statement, you must have the CREATE privilege for the database. You will get an error if the database exists and you did not specify IF NOT EXISTS clause.\nLimits on Number of Databases: MySQL has no limit on the number of databases. The underlying file system may have a limit on the number of directories.\nCREATE SCHEMA is a synonym for CREATE DATABASE.\ndatabase_name is the name of the new database.\nRules for valid database names are given in MySQL language structure \"Schema Object Names\" section.\ncreate_specification options specify database characteristics.\nThe CHARACTER SET clause specifies the default database character set.\nThe following statement creates 'bookinfo' database.\nThe database names are case sensitive under Unix but this restriction does not apply in Windows. This is also true for table names. The best practice is to use same letter case for creating a database as well as a table.\nNote: A database which has just been created is not current database. The user must have to instruct to make it the current database. A database needs to be created only once but a user must have to select it each time he intends to work with that database.\nMySQL use statement is used to change the database from default to the given database.\nSHOW statement displays a list of currently existing databases on the server.\nThe following MySQL statement will show the current database.\nThe list of databases shown bellow by the statement may be different to the other user's machine. SHOW DATABASES does not show databases for those you don't have SHOW DATABASES privilege.\nMySQL select database statement is used to see which database is currently selected.\nMySQL 'show tables' statement displays a list of the tables in the database in use. If there is no table in the database, it returns empty rows.\nThe following statement displays the list of tables in the database 'bookinfo'.\nShows the CREATE DATABASE statement that creates the given database. If the SHOW statement includes an IF NOT EXISTS clause, the output to includes such a clause. SHOW CREATE SCHEMA is a synonym for SHOW CREATE DATABASE.\nThe following statement shows the create database statement of 'world' database.\nFollowing query show you the database size in MySQL.\nSUM(data_length + index_length)\/1024\/1024 \"Size in MB\"\nThe following statement shows all the tables in 'hr' database with columns 'name' or 'department_id'. At first, see the tables in 'hr' database.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ecuador has amazing natural beauty!\nThe sightings will be extraordinary!\nPure and Sweet Natural Light! Respect Nature!\nSpecial thanks to all of my guests for joining these workshops!\nThis eagle has a large piece of meat in its talons, while the warthogs are looking for an easy meal! Amazing how the eagle stood his ground without flying off to safety. We enjoyed this scene for at least 10 minutes!\nReal nature at its finest in Tanzania!\nProtecting Natures habitat is the same as Protecting our Children's Future.\nThe forests in Ecuador are superb, especially in the rain!\nNo flash used with any bird or wildlife photography.\nThanks to India for protecting these Tigers!\nRoyal Bengal Tiger in some long grass.\nPlease respect and protect Nature.\nHelp the world protect the Tigers Please!\nI am back to my home in India!\nOntario's Nature at its Best!\nRespect for Nature always comes First.\nUntouched wild birds seem to make the most natural images.\nHow do we take sharp images?\nJoin me on a workshop!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Assortment of humbucker pickup wiring diagram. A wiring diagram is a simplified conventional pictorial representation of an electric circuit. It reveals the elements of the circuit as streamlined shapes, and the power as well as signal links between the devices.\nA wiring diagram usually gives details about the relative setting as well as setup of devices as well as terminals on the gadgets, to assist in building or servicing the gadget. This differs from a schematic diagram, where the arrangement of the components' interconnections on the representation typically does not match to the elements' physical locations in the completed tool. A pictorial representation would certainly show extra detail of the physical appearance, whereas a wiring diagram utilizes a more symbolic symbols to stress interconnections over physical look.\nA wiring diagram is frequently used to fix problems and making sure that all the links have been made which everything exists.\nAssortment of humbucker pickup wiring diagram. Click on the image to enlarge, and then save it to your computer by right clicking on the image.\nA wiring diagram is a kind of schematic which uses abstract pictorial symbols to reveal all the affiliations of parts in a system. Electrical wiring diagrams are made up of 2 points: icons that represent the parts in the circuit, and also lines that stand for the connections in between them.\nIt's simple to obtain puzzled concerning wiring diagrams and also schematics. Electrical wiring representations mainly shows the physical placement of elements as well as connections in the constructed circuit, but not necessarily in logic order. It highlights on the format of the cables. Schematics stress on exactly how circuits function rationally. It lowers integrated circuits into sub-components to earn the system's useful reasonings simpler to understand. It's most helpful for learning the general procedure of a system.\nTo check out a wiring diagram, initially you need to recognize what essential elements are included in a wiring diagram, and also which pictorial symbols are used to represent them. The typical elements in a wiring diagram are ground, power supply, cord and also connection, output tools, switches, resistors, logic gateway, lights, etc. A list of electrical signs as well as summaries can be discovered on the \"electric icon\" web page.\nA line stands for a cord. Cords are made use of to attach the elements together. There need to be a tale on the wiring diagram to inform you what each shade indicates.\nTypically circuits with greater than two parts have two basic sorts of links: series and also parallel. A series circuit is a circuit in which elements are connected along a single course, so the present flows via one component to obtain to the next one. In a series circuit, voltages add up for all elements connected in the circuit, and also currents are the very same via all elements. In a parallel circuit, each tool is straight linked to the source of power, so each tool receives the same voltage. The present in a parallel circuit streams along each parallel branch and also re-combines when the branches satisfy again.\nAn excellent wiring diagram requires to be technically proper and also clear to read. Care for every detail. The representation must reveal the appropriate instructions of the positive as well as negative terminals of each element.\nUtilize the ideal icons. Find out the meanings of the standard circuit icons and select the appropriate ones to use.\nAttract connecting cables as straight lines. Utilize a dot to suggest line junction, or usage line jumps to show cross lines that are not linked.\nTag parts such as resistors as well as capacitors with their values. Make certain the message placement looks tidy.\nIn basic it is good to place the favorable (+) supply at the top, as well as the adverse (-) supply at the bottom, and also the rational circulation from left to.\nAim to arrange the positioning minimizing cable crossings.\nRelated Posts of \"Humbucker Pickup Wiring Diagram\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This week, we continued our journey through the Bible using The Gospel Project\u00ae for Kids and studied Moses' reminder to God's people to worship only the one true God. God alone is worthy of worship and He is jealous of this.\nHelp your kids identify possible \"idols\" in their lives that draw their worship away from God and pray about ways to reject worshiping anything except God.\nGod's people were almost to the land God promised to give them.\nMoses reminded the Israelites of God's commands and to worship only God.\nWhat are some ways God has helped us in the past?\nHow does remembering how God has helped us before help us now?\nWhy do you think God gave His people rules?\ntalk about a fun memory you shared together. Why do you remember that special moment?\nTake it further: Write the key passage on your bathroom mirror with a dry erase marker to help you memorize the key passage as a family this week.","meta":{"redpajama_set_name":"RedPajamaC4"}}